{"commit":"0ca031c6d255e0d104161cf37703e54e4c2aca16","subject":"sum-properties: \u00b0","message":"sum-properties: \u00b0\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-properties.agda","new_file":"sum-properties.agda","new_contents":"module sum-properties where\n\nopen import Type\nimport Level as L\nopen import Data.Bool.NP\nopen Data.Bool.NP.Indexed\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Function.NP\nopen import sum\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nmodule _ {A : \u2605} (\u03bcA : SumProp A) (f g : A \u2192 \u2115) where\n    open \u2261.\u2261-Reasoning\n\n    sum-\u2293-\u2238 : sum \u03bcA f \u2261 sum \u03bcA (f \u2293\u00b0 g) + sum \u03bcA (f \u2238\u00b0 g)\n    sum-\u2293-\u2238 = sum \u03bcA f                          \u2261\u27e8 sum-ext \u03bcA (f \u27e8 a\u2261a\u2293b+a\u2238b \u27e9\u00b0 g) \u27e9\n              sum \u03bcA ((f \u2293\u00b0 g) +\u00b0 (f \u2238\u00b0 g))     \u2261\u27e8 sum-hom \u03bcA (f \u2293\u00b0 g) (f \u2238\u00b0 g) \u27e9\n              sum \u03bcA (f \u2293\u00b0 g) + sum \u03bcA (f \u2238\u00b0 g) \u220e\n\n    sum-\u2294-\u2293 : sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (f \u2294\u00b0 g) + sum \u03bcA (f \u2293\u00b0 g)\n    sum-\u2294-\u2293 = sum \u03bcA f + sum \u03bcA g               \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n              sum \u03bcA (f +\u00b0 g)                   \u2261\u27e8 sum-ext \u03bcA (f \u27e8 a+b\u2261a\u2294b+a\u2293b \u27e9\u00b0 g) \u27e9\n              sum \u03bcA (f \u2294\u00b0 g +\u00b0 f \u2293\u00b0 g)         \u2261\u27e8 sum-hom \u03bcA (f \u2294\u00b0 g) (f \u2293\u00b0 g) \u27e9\n              sum \u03bcA (f \u2294\u00b0 g) + sum \u03bcA (f \u2293\u00b0 g) \u220e\n\n    sum-\u2294 : sum \u03bcA (f \u2294\u00b0 g) \u2264 sum \u03bcA f + sum \u03bcA g\n    sum-\u2294 = \u2115\u2264.trans (sum-mono \u03bcA (f \u27e8 \u2294\u2264+ \u27e9\u00b0 g)) (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n\nmodule _M2 {A : \u2605} (\u03bcA : SumProp A) (f g : A \u2192 Bool) where\n    count-\u2227-not : count \u03bcA f \u2261 count \u03bcA (f \u2227\u00b0 g) + count \u03bcA (f \u2227\u00b0 not\u00b0 g)\n    count-\u2227-not rewrite sum-\u2293-\u2238 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                      | sum-ext \u03bcA (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                      | sum-ext \u03bcA (f \u27e8 to\u2115-\u2238 \u27e9\u00b0 g)\n                      = \u2261.refl\n\n    count-\u2228-\u2227 : count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (f \u2228\u00b0 g) + count \u03bcA (f \u2227\u00b0 g)\n    count-\u2228-\u2227 rewrite sum-\u2294-\u2293 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                    | sum-ext \u03bcA (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g)\n                    | sum-ext \u03bcA (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                    = \u2261.refl\n\n    count-\u2228\u2264+ : count \u03bcA (f \u2228\u00b0 g) \u2264 count \u03bcA f + count \u03bcA g\n    count-\u2228\u2264+ = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u2261.sym \u2218 (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g))))\n                         (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 p \u2192 Injective p \u2192 SumStableUnder \u03bc p\n\nCountStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nCountStableUnderInjection \u03bc = \u2200 p \u2192 Injective p \u2192 CountStableUnder \u03bc p\n\nmodule SumFinProperties where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb _ _ \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} p p-inj f rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc \u2192 CountStableUnderInjection \u03bc\n#-StableUnderInjection sui p p-inj f = sui p p-inj (to\u2115 \u2218 f)\n","old_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Function.NP\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 p \u2192 Injective p \u2192 SumStableUnder \u03bc p\n\nCountStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nCountStableUnderInjection \u03bc = \u2200 p \u2192 Injective p \u2192 CountStableUnder \u03bc p\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} p p-inj f rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc \u2192 CountStableUnderInjection \u03bc\n#-StableUnderInjection sui p p-inj f = sui p p-inj (to\u2115 \u2218 f)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6bd42bf5ef43bf32db7fb57583c7d7b7b052fce1","subject":"Fix wrong module name: A -> Extension-00","message":"Fix wrong module name: A -> Extension-00\n\nOld-commit-hash: f363f922f62f640837bc441403a2bae4c5aa71af\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/Extension-00.agda","new_file":"nats\/Extension-00.agda","new_contents":"{-\n\nChecklist of stuff to add when adding syntactic constructs\n\n- derive (symbolic derivation; most important!)\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- validity-of-derive\n- correctness-of-derive\n\n-}\n\nmodule Extension-00 where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- ad-hoc extensions\n  foldNat : \u2200 {\u0393 \u03c4} \u2192 (n : Term \u0393 nats) \u2192\n                      (f : Term \u0393 (\u03c4 \u21d2 \u03c4)) \u2192 (z : Term \u0393 \u03c4)\n                    \u2192 Term \u0393 \u03c4\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nweaken subctx (foldNat n f z) = {!!}\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\n\u27e6 foldNat n f z \u27e7Term \u03c1 = {!!}\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (foldNat n f z) \u03c1 = {!!}\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive (foldNat f n z) =\nderive (foldNat f n z) =  {!!}\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} (foldNat n f z) = {!!}\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive {\u0393} {\u03c4}\n  \u03c1 {consistency} (foldNat n f z) = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","old_contents":"{-\n\nChecklist of stuff to add when adding syntactic constructs\n\n- derive (symbolic derivation; most important!)\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- validity-of-derive\n- correctness-of-derive\n\n-}\n\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- ad-hoc extensions\n  foldNat : \u2200 {\u0393 \u03c4} \u2192 (n : Term \u0393 nats) \u2192\n                      (f : Term \u0393 (\u03c4 \u21d2 \u03c4)) \u2192 (z : Term \u0393 \u03c4)\n                    \u2192 Term \u0393 \u03c4\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nweaken subctx (foldNat n f z) = {!!}\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\n\u27e6 foldNat n f z \u27e7Term \u03c1 = {!!}\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (foldNat n f z) \u03c1 = {!!}\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive (foldNat f n z) =\nderive (foldNat f n z) =  {!!}\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} (foldNat n f z) = {!!}\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive {\u0393} {\u03c4}\n  \u03c1 {consistency} (foldNat n f z) = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"749ba243d78e595650cc5d4473375de38b55da9a","subject":"fromBinTree{,-ind}","message":"fromBinTree{,-ind}\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/BinTree.agda","new_file":"lib\/Explore\/BinTree.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.BinTree where\n\nopen import Data.Tree.Binary\n\nopen import Explore.Core\nopen import Explore.Properties\n\nfromBinTree : \u2200 {m} {A} \u2192 BinTree A \u2192 Explore m A\n-- fromBinTree empty      = empty-explore\nfromBinTree (leaf x)   = point-explore x\nfromBinTree (fork \u2113 r) = merge-explore (fromBinTree \u2113) (fromBinTree r)\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\n-- fromBinTree-ind empty      = empty-explore-ind\nfromBinTree-ind (leaf x)   = point-explore-ind x\nfromBinTree-ind (fork \u2113 r) = merge-explore-ind (fromBinTree-ind \u2113)\n                                               (fromBinTree-ind r)\n{-\nfromBinTree : \u2200 {m A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree (leaf x)   _ _   f = f x\nfromBinTree (fork \u2113 r) \u03b5 _\u2219_ f = fromBinTree \u2113 \u03b5 _\u2219_ f \u2219 fromBinTree r \u03b5 _\u2219_ f\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind (leaf x)   P _  P\u2219 Pf = Pf x\nfromBinTree-ind (fork \u2113 r) P P\u03b5 P\u2219 Pf = P\u2219 (fromBinTree-ind \u2113 P P\u03b5 P\u2219 Pf)\n                                           (fromBinTree-ind r P P\u03b5 P\u2219 Pf)\n-}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.BinTree where\n\nopen import Data.Tree.Binary\n\nopen import Explore.Core\nopen import Explore.Properties\n\nfromBinTree : \u2200 {m A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree (leaf x)   _ _   f = f x\nfromBinTree (fork \u2113 r) \u03b5 _\u2219_ f = fromBinTree \u2113 \u03b5 _\u2219_ f \u2219 fromBinTree r \u03b5 _\u2219_ f\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind (leaf x)   P _  P\u2219 Pf = Pf x\nfromBinTree-ind (fork \u2113 r) P P\u03b5 P\u2219 Pf = P\u2219 (fromBinTree-ind \u2113 P P\u03b5 P\u2219 Pf)\n                                           (fromBinTree-ind r P P\u03b5 P\u2219 Pf)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"86af75a4ffb1a4d9451d5976769c197b168dd27e","subject":"More sensible fix","message":"More sensible fix\n\nParametric\/Change\/Specification.agda still makes Agda crash.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d_\u208e {{change-algebra-base}} \u03b9\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n    change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n    change-algebra-family = family change-algebra\n\n  -- function changes\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d_\u208e {{environment-changes}} \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d_\u208e {{change-algebra-base}} \u03b9\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n    change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n    change-algebra-family = family change-algebra\n\n  -- function changes\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d_\u208e {{environment-changes}} \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"07045b7aa32a14de23ec233029b6fc6134b24685","subject":"Upgrade to Data.Two","message":"Upgrade to Data.Two\n","repos":"crypto-agda\/agda-nplib","old_file":"experiment\/liftVec.agda","new_file":"experiment\/liftVec.agda","new_contents":"open import Function\nopen import Data.Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin using (Fin)\nopen import Data.Vec\nopen import Data.Vec.N-ary.NP\nopen import Relation.Binary.PropositionalEquality\n\nmodule liftVec where\nmodule single (A : Set)(_+_ : A \u2192 A \u2192 A)where\n\n  data Tm : Set where\n    var : Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  eval : Tm \u2192 A \u2192 A\n  eval var x = x\n  eval (cst c) x = c\n  eval (fun tm tm') x = eval tm x + eval tm' x\n\n  veval : {m : \u2115} \u2192 Tm \u2192 Vec A m \u2192 Vec A m\n  veval var xs = xs\n  veval (cst x) xs = replicate x\n  veval (fun tm tm') xs = zipWith _+_ (veval tm xs) (veval tm' xs)\n\n  record Fm : Set where\n    constructor _=='_\n    field\n      lhs : Tm\n      rhs : Tm\n\n  s[_] : Fm \u2192 Set\n  s[ lhs ==' rhs ] = (x : A) \u2192 eval lhs x \u2261 eval rhs x\n\n  v[_] : Fm \u2192 Set\n  v[ lhs ==' rhs ] = {m : \u2115}(xs : Vec A m) \u2192 veval lhs xs \u2261 veval rhs xs\n\n  lemma1 : (t : Tm) \u2192 veval t [] \u2261 []\n  lemma1 var = refl\n  lemma1 (cst x) = refl\n  lemma1 (fun t t') rewrite lemma1 t | lemma1 t' = refl\n\n  lemma2 : {m : \u2115}(x : A)(xs : Vec A m)(t : Tm) \u2192 veval t (x \u2237 xs) \u2261 eval t x \u2237 veval t xs\n  lemma2 _ _ var = refl\n  lemma2 _ _ (cst c) = refl\n  lemma2 x xs (fun t t') rewrite lemma2 x xs t | lemma2 x xs t' = refl\n\n  prf : (t : Fm) \u2192 s[ t ] \u2192 v[ t ]\n  prf (lhs ==' rhs) pr [] rewrite lemma1 lhs | lemma1 rhs = refl\n  prf (l ==' r) pr (x \u2237 xs) rewrite lemma2 x xs l | lemma2 x xs r = cong\u2082 (_\u2237_) (pr x) (prf (l ==' r) pr xs)\n\nmodule MultiDataType {A : Set} {nrVar : \u2115} where\n  data Tm : Set where\n    var : Fin nrVar \u2192 Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  infixl 6 _+'_\n  _+'_ : Tm \u2192 Tm \u2192 Tm\n  _+'_ = fun\n\n  infix 4 _\u2261'_\n  record TmEq : Set where\n    constructor _\u2261'_\n    field\n      lhs rhs : Tm\n\nmodule multi (A : Set)(_+_ : A \u2192 A \u2192 A)(nrVar : \u2115) where\n  open MultiDataType {A} {nrVar} public\n\n  sEnv : Set\n  sEnv = Vec A nrVar\n\n  vEnv : \u2115 \u2192 Set\n  vEnv m = Vec (Vec A m) nrVar\n\n  Var : Set\n  Var = Fin nrVar\n\n  _!_ : {X : Set} \u2192 Vec X nrVar \u2192 Var \u2192 X\n  E ! v = lookup v E\n\n  eval : Tm \u2192 sEnv \u2192 A\n  eval (var x) G = G ! x\n  eval (cst c) G = c\n  eval (fun t t') G = eval t G + eval t' G\n\n  veval : {m : \u2115} \u2192 Tm \u2192 vEnv m \u2192 Vec A m\n  veval (var x) G = G ! x\n  veval (cst c) G = replicate c\n  veval (fun t t') G = zipWith _+_ (veval t G) (veval t' G)\n\n  lemma1 : {xs : Vec (Vec A 0) nrVar} (t : Tm) \u2192 veval t xs \u2261 []\n  lemma1 {xs} (var x) with xs ! x\n  ... | [] = refl\n  lemma1 (cst x) = refl\n  lemma1 {xs} (fun t t') rewrite lemma1 {xs} t | lemma1 {xs} t' = refl\n\n  lemVar : {m n : \u2115}(G : Vec (Vec A (suc m)) n)(i : Fin n) \u2192 lookup i G \u2261 lookup i (map head G) \u2237 lookup i (map tail G)\n  lemVar [] ()\n  lemVar ((x \u2237 G) \u2237 G\u2081) Data.Fin.zero = refl\n  lemVar (G \u2237 G') (Data.Fin.suc i) = lemVar G' i\n\n  lemma2 : {n : \u2115}(xs : vEnv (suc n))(t : Tm) \u2192 veval t xs \u2261 eval t (map head xs) \u2237 veval t (map tail xs)\n  lemma2 G (var x) = lemVar G x\n  lemma2 _ (cst x) = refl\n  lemma2 G (fun t t') rewrite lemma2 G t | lemma2 G t' = refl\n\n  s[_==_] : Tm \u2192 Tm \u2192 Set\n  s[ l == r ] = (G : Vec A nrVar) \u2192 eval l G \u2261 eval r G\n\n  v[_==_] : Tm \u2192 Tm \u2192 Set\n  v[ l == r ] = {m : \u2115}(G : Vec (Vec A m) nrVar) \u2192 veval l G \u2261 veval r G\n\n  prf : (l r : Tm) \u2192 s[ l == r ] \u2192 v[ l == r ]\n  prf l r pr {zero} G rewrite lemma1 {G} l | lemma1 {G} r = refl\n  prf l r pr {suc m} G rewrite lemma2 G l | lemma2 G r\n     = cong\u2082 _\u2237_ (pr (replicate head \u229b G)) (prf l r pr (map tail G))\n\nmodule Full (A : Set)(_+_ : A \u2192 A \u2192 A) (m : \u2115) n where\n  open multi A _+_ n public\n\n  solve : \u2200 (l r : Tm) \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  solve l r x = curry\u207f {n} {A = Vec A m} {B = curry\u207f\u2032 f}\n        (\u03bb xs \u2192 subst id (sym (curry-$\u207f\u2032 f xs))\n                   (prf l r (\u03bb G \u2192 subst id (curry-$\u207f\u2032 g G) (x $\u207f G)) xs))\n    where f = \u03bb G \u2192 veval {m} l G \u2261 veval r G\n          g = \u03bb G \u2192 eval l G \u2261 eval r G\n\n  mkTm : N-ary n Tm TmEq \u2192 TmEq\n  mkTm = go (tabulate id) where\n    go : {m : \u2115} -> Vec (Fin n) m \u2192 N-ary m Tm TmEq \u2192 TmEq\n    go [] f         = f\n    go (x \u2237 args) f = go args (f (var x))\n\n  prove :  \u2200 (t : N-ary n Tm TmEq) \u2192\n           let\n             l = TmEq.lhs (mkTm t)\n             r = TmEq.rhs (mkTm t)\n           in \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  prove t x = solve (TmEq.lhs (mkTm t)) (TmEq.rhs (mkTm t)) x\n\nopen import Data.Two\n\nmodule example where\n\n  open import Data.Fin\n  open import Data.Product using (\u03a3 ; proj\u2081)\n\n  module VecBoolXorProps {m} = Full \ud835\udfda _xor_ m\n  open MultiDataType\n\n  coolTheorem : {m : \u2115} \u2192 (xs ys : Vec \ud835\udfda m) \u2192 zipWith _xor_ xs ys \u2261 zipWith _xor_ ys xs\n  coolTheorem = VecBoolXorProps.prove 2 (\u03bb x y \u2192 x +' y \u2261' y +' x) Xor\u00b0.+-comm\n\n  coolTheorem2 : {m : \u2115} \u2192 (xs : Vec \ud835\udfda m) \u2192 _\n  coolTheorem2 = VecBoolXorProps.prove 1 (\u03bb x \u2192 (x +' x) \u2261' (cst 0\u2082)) (proj\u2081 Xor\u00b0.-\u203finverse)\n\ntest = example.coolTheorem (1\u2082 \u2237 0\u2082 \u2237 []) (0\u2082 \u2237 0\u2082 \u2237 [])\n","old_contents":"open import Function\nopen import Data.Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin using (Fin)\nopen import Data.Vec\nopen import Data.Vec.N-ary.NP\nopen import Relation.Binary.PropositionalEquality\n\nmodule liftVec where\nmodule single (A : Set)(_+_ : A \u2192 A \u2192 A)where\n\n  data Tm : Set where\n    var : Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  eval : Tm \u2192 A \u2192 A\n  eval var x = x\n  eval (cst c) x = c\n  eval (fun tm tm') x = eval tm x + eval tm' x\n\n  veval : {m : \u2115} \u2192 Tm \u2192 Vec A m \u2192 Vec A m\n  veval var xs = xs\n  veval (cst x) xs = replicate x\n  veval (fun tm tm') xs = zipWith _+_ (veval tm xs) (veval tm' xs)\n\n  record Fm : Set where\n    constructor _=='_\n    field\n      lhs : Tm\n      rhs : Tm\n\n  s[_] : Fm \u2192 Set\n  s[ lhs ==' rhs ] = (x : A) \u2192 eval lhs x \u2261 eval rhs x\n\n  v[_] : Fm \u2192 Set\n  v[ lhs ==' rhs ] = {m : \u2115}(xs : Vec A m) \u2192 veval lhs xs \u2261 veval rhs xs\n\n  lemma1 : (t : Tm) \u2192 veval t [] \u2261 []\n  lemma1 var = refl\n  lemma1 (cst x) = refl\n  lemma1 (fun t t') rewrite lemma1 t | lemma1 t' = refl\n\n  lemma2 : {m : \u2115}(x : A)(xs : Vec A m)(t : Tm) \u2192 veval t (x \u2237 xs) \u2261 eval t x \u2237 veval t xs\n  lemma2 _ _ var = refl\n  lemma2 _ _ (cst c) = refl\n  lemma2 x xs (fun t t') rewrite lemma2 x xs t | lemma2 x xs t' = refl\n\n  prf : (t : Fm) \u2192 s[ t ] \u2192 v[ t ]\n  prf (lhs ==' rhs) pr [] rewrite lemma1 lhs | lemma1 rhs = refl\n  prf (l ==' r) pr (x \u2237 xs) rewrite lemma2 x xs l | lemma2 x xs r = cong\u2082 (_\u2237_) (pr x) (prf (l ==' r) pr xs)\n\nmodule MultiDataType {A : Set} {nrVar : \u2115} where\n  data Tm : Set where\n    var : Fin nrVar \u2192 Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  infixl 6 _+'_\n  _+'_ : Tm \u2192 Tm \u2192 Tm\n  _+'_ = fun  \n\n  infix 4 _\u2261'_\n  record TmEq : Set where\n    constructor _\u2261'_ \n    field \n      lhs rhs : Tm\n\nmodule multi (A : Set)(_+_ : A \u2192 A \u2192 A)(nrVar : \u2115) where\n  open MultiDataType {A} {nrVar} public\n\n  sEnv : Set\n  sEnv = Vec A nrVar\n\n  vEnv : \u2115 \u2192 Set\n  vEnv m = Vec (Vec A m) nrVar\n\n  Var : Set\n  Var = Fin nrVar\n\n  _!_ : {X : Set} \u2192 Vec X nrVar \u2192 Var \u2192 X\n  E ! v = lookup v E\n\n  eval : Tm \u2192 sEnv \u2192 A\n  eval (var x) G = G ! x\n  eval (cst c) G = c\n  eval (fun t t') G = eval t G + eval t' G\n\n  veval : {m : \u2115} \u2192 Tm \u2192 vEnv m \u2192 Vec A m\n  veval (var x) G = G ! x\n  veval (cst c) G = replicate c\n  veval (fun t t') G = zipWith _+_ (veval t G) (veval t' G)\n\n  lemma1 : {xs : Vec (Vec A 0) nrVar} (t : Tm) \u2192 veval t xs \u2261 []\n  lemma1 {xs} (var x) with xs ! x\n  ... | [] = refl\n  lemma1 (cst x) = refl\n  lemma1 {xs} (fun t t') rewrite lemma1 {xs} t | lemma1 {xs} t' = refl\n\n  lemVar : {m n : \u2115}(G : Vec (Vec A (suc m)) n)(i : Fin n) \u2192 lookup i G \u2261 lookup i (map head G) \u2237 lookup i (map tail G)\n  lemVar [] ()\n  lemVar ((x \u2237 G) \u2237 G\u2081) Data.Fin.zero = refl\n  lemVar (G \u2237 G') (Data.Fin.suc i) = lemVar G' i\n\n  lemma2 : {n : \u2115}(xs : vEnv (suc n))(t : Tm) \u2192 veval t xs \u2261 eval t (map head xs) \u2237 veval t (map tail xs)\n  lemma2 G (var x) = lemVar G x\n  lemma2 _ (cst x) = refl\n  lemma2 G (fun t t') rewrite lemma2 G t | lemma2 G t' = refl\n\n  s[_==_] : Tm \u2192 Tm \u2192 Set\n  s[ l == r ] = (G : Vec A nrVar) \u2192 eval l G \u2261 eval r G\n\n  v[_==_] : Tm \u2192 Tm \u2192 Set\n  v[ l == r ] = {m : \u2115}(G : Vec (Vec A m) nrVar) \u2192 veval l G \u2261 veval r G\n\n  prf : (l r : Tm) \u2192 s[ l == r ] \u2192 v[ l == r ]\n  prf l r pr {zero} G rewrite lemma1 {G} l | lemma1 {G} r = refl\n  prf l r pr {suc m} G rewrite lemma2 G l | lemma2 G r\n     = cong\u2082 _\u2237_ (pr (replicate head \u229b G)) (prf l r pr (map tail G))\n\nmodule Full (A : Set)(_+_ : A \u2192 A \u2192 A) (m : \u2115) n where\n  open multi A _+_ n public\n\n  solve : \u2200 (l r : Tm) \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  solve l r x = curry\u207f {n} {A = Vec A m} {B = curry\u207f\u2032 f}\n        (\u03bb xs \u2192 subst id (sym (curry-$\u207f\u2032 f xs))\n                   (prf l r (\u03bb G \u2192 subst id (curry-$\u207f\u2032 g G) (x $\u207f G)) xs))\n    where f = \u03bb G \u2192 veval {m} l G \u2261 veval r G\n          g = \u03bb G \u2192 eval l G \u2261 eval r G\n\n  mkTm : N-ary n Tm TmEq \u2192 TmEq\n  mkTm = go (tabulate id) where\n    go : {m : \u2115} -> Vec (Fin n) m \u2192 N-ary m Tm TmEq \u2192 TmEq\n    go [] f         = f\n    go (x \u2237 args) f = go args (f (var x))\n\n  prove :  \u2200 (t : N-ary n Tm TmEq) \u2192 \n           let \n             l = TmEq.lhs (mkTm t)\n             r = TmEq.rhs (mkTm t)\n           in \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  prove t x = solve (TmEq.lhs (mkTm t)) (TmEq.rhs (mkTm t)) x\n\nopen import Data.Bool\n\nmodule example where\n\n  open import Data.Fin\n  open import Data.Bool.NP\n  open import Data.Product using (\u03a3 ; proj\u2081)\n\n  module VecBoolXorProps {m} = Full Bool _xor_ m\n  open MultiDataType\n\n  coolTheorem : {m : \u2115} \u2192 (xs ys : Vec Bool m) \u2192 zipWith _xor_ xs ys \u2261 zipWith _xor_ ys xs\n  coolTheorem = VecBoolXorProps.prove 2 (\u03bb x y \u2192 x +' y \u2261' y +' x) Xor\u00b0.+-comm \n\n  coolTheorem2 : {m : \u2115} \u2192 (xs : Vec Bool m) \u2192 _\n  coolTheorem2 = VecBoolXorProps.prove 1 (\u03bb x \u2192 (x +' x) \u2261' (cst false)) (proj\u2081 Xor\u00b0.-\u203finverse) \n \ntest = example.coolTheorem (true \u2237 false \u2237 []) (false \u2237 false \u2237 [])","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"56546fb3e9f31a8dfc5dd4dfbf825e8509a62b9c","subject":"Added TODO: Direct proof using of pred-N using N-least-pre-fixed.","message":"Added TODO: Direct proof using of pred-N using N-least-pre-fixed.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- From N as the least fixed-point to N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- We want to represent the total natural numbers data type\n--\n-- data N : D \u2192 Set where\n--   nzero : N zero\n--   nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n--\n-- using the representation of N as the least fixed-point.\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n  -- The higher-order version.\n  N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n  -- N is the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-least-pre-fixed :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n\n  -- Higher-order version (incomplete?)\n  N-least-pre-fixed-ho :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (NatF A n \u2192 A n) \u2192\n    N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- From\/to N-in\/N-in-ho.\n\nN-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\nN-in\u2081 = N-in-ho\n\nN-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\nN-in-ho\u2081 = N-in\u2081\n\n------------------------------------------------------------------------------\n-- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\nN-least-pre-fixed' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n  N n \u2192 A n\nN-least-pre-fixed' = N-least-pre-fixed-ho\n\nN-least-pre-fixed-ho' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (NatF A n \u2192 A n) \u2192\n  N n \u2192 A n\nN-least-pre-fixed-ho' = N-least-pre-fixed\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nnzero : N zero\nnzero = N-in (inj\u2081 refl)\n\nnsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nnsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-in and\n-- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN-post-fixed = N-least-pre-fixed A h\n  where\n  A : D \u2192 Set\n  A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n  h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  h (inj\u2081 prf)              = inj\u2081 prf\n  h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n    where\n    helper : A m' \u2192 N m'\n    helper (inj\u2081 prf')                = subst N (sym prf') nzero\n    helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 is {n} Nn = N-least-pre-fixed A h Nn\n  where\n  h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n  h (inj\u2081 prf)              = subst A (sym prf) A0\n  h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is helper An')\n    where\n    helper : N n'\n    helper with N-post-fixed Nn\n    ... | inj\u2081 n\u22610 = \u22a5-elim (0\u2262S (trans (sym n\u22610) prf))\n    ... | inj\u2082 (m' , prf' , Nm') = subst N (succInjective (trans (sym prf') prf)) Nm'\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 is {n} = N-least-pre-fixed A h\n  where\n  h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n  h (inj\u2081 prf)              = subst A (sym prf) A0\n  h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is An')\n\n------------------------------------------------------------------------------\n-- Example: We will use N-least-pre-fixed as the induction principle on N.\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192 zero + d      \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-least-pre-fixed A h Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  h : m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  h (inj\u2081 prf) = subst N (cong (flip _+_ n) (sym prf)) A0\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n  h (inj\u2082 (m' , m\u2261Sm' , Am')) = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n    where\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n------------------------------------------------------------------------------\n-- Example: Indirect proof using N-least-pre-fixed.\n\n-- TODO (18 December 2013): Direct proof using of pred-N using\n-- N-least-pre-fixed.\n\npred-N\u2081 : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N\u2081 = N-ind\u2081 A A0 is\n  where\n  A : D \u2192 Set\n  A i = N (pred\u2081 i)\n\n  A0 : A zero\n  A0 = subst N (sym pred-0) nzero\n\n  is : \u2200 {i} \u2192 N i \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ni ih = subst N (sym (pred-S i)) Ni\n\n------------------------------------------------------------------------------\n-- From\/to N as a least fixed-point to\/from N as data type.\n\nopen import FOTC.Data.Nat.Type renaming\n  ( N to N'\n  ; nsucc to nsucc'\n  ; nzero to nzero'\n  )\n\nN'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\nN'\u2192N nzero'      = nzero\nN'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n-- Using N-ind\u2081.\nN\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\nN\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n-- Using N-ind\u2082.\nN\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\nN\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n","old_contents":"------------------------------------------------------------------------------\n-- From N as the least fixed-point to N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- We want to represent the total natural numbers data type\n--\n-- data N : D \u2192 Set where\n--   nzero : N zero\n--   nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n--\n-- using the representation of N as the least fixed-point.\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n  -- The higher-order version.\n  N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n  -- N is the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-least-pre-fixed :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n\n  -- Higher-order version (incomplete?)\n  N-least-pre-fixed-ho :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (NatF A n \u2192 A n) \u2192\n    N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- From\/to N-in\/N-in-ho.\n\nN-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\nN-in\u2081 = N-in-ho\n\nN-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\nN-in-ho\u2081 = N-in\u2081\n\n------------------------------------------------------------------------------\n-- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\nN-least-pre-fixed' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n  N n \u2192 A n\nN-least-pre-fixed' = N-least-pre-fixed-ho\n\nN-least-pre-fixed-ho' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (NatF A n \u2192 A n) \u2192\n  N n \u2192 A n\nN-least-pre-fixed-ho' = N-least-pre-fixed\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nnzero : N zero\nnzero = N-in (inj\u2081 refl)\n\nnsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nnsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-in and\n-- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN-post-fixed = N-least-pre-fixed A h\n  where\n  A : D \u2192 Set\n  A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n  h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  h (inj\u2081 prf)              = inj\u2081 prf\n  h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n    where\n    helper : A m' \u2192 N m'\n    helper (inj\u2081 prf')                = subst N (sym prf') nzero\n    helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 is {n} Nn = N-least-pre-fixed A h Nn\n  where\n  h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n  h (inj\u2081 prf)              = subst A (sym prf) A0\n  h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is helper An')\n    where\n    helper : N n'\n    helper with N-post-fixed Nn\n    ... | inj\u2081 n\u22610 = \u22a5-elim (0\u2262S (trans (sym n\u22610) prf))\n    ... | inj\u2082 (m' , prf' , Nm') = subst N (succInjective (trans (sym prf') prf)) Nm'\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 is {n} = N-least-pre-fixed A h\n  where\n  h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n  h (inj\u2081 prf)              = subst A (sym prf) A0\n  h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is An')\n\n------------------------------------------------------------------------------\n-- Example: We will use N-least-pre-fixed as the induction principle on N.\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192 zero + d      \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-least-pre-fixed A h Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  h : m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  h (inj\u2081 prf) = subst N (cong (flip _+_ n) (sym prf)) A0\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n  h (inj\u2082 (m' , m\u2261Sm' , Am')) = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n    where\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n------------------------------------------------------------------------------\n-- Example: Indirect proof using N-least-pre-fixed.\n\npred-N\u2081 : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N\u2081 = N-ind\u2081 A A0 is\n  where\n  A : D \u2192 Set\n  A i = N (pred\u2081 i)\n\n  A0 : A zero\n  A0 = subst N (sym pred-0) nzero\n\n  is : \u2200 {i} \u2192 N i \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ni ih = subst N (sym (pred-S i)) Ni\n\n------------------------------------------------------------------------------\n-- From\/to N as a least fixed-point to\/from N as data type.\n\nopen import FOTC.Data.Nat.Type renaming\n  ( N to N'\n  ; nsucc to nsucc'\n  ; nzero to nzero'\n  )\n\nN'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\nN'\u2192N nzero'      = nzero\nN'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n-- Using N-ind\u2081.\nN\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\nN\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n-- Using N-ind\u2082.\nN\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\nN\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"16c3de301de3b361124574abbf79bebae8568e23","subject":"Cosmetic changes","message":"Cosmetic changes\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/FundLemma.agda","new_file":"Thesis\/SIRelBigStep\/FundLemma.agda","new_contents":"module Thesis.SIRelBigStep.FundLemma where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 k (lt1 k<n) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"module Thesis.SIRelBigStep.FundLemma where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 k (lt1 k<n) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ce0778e9e40ea86ade9d7a74235d225b5c0e7a05","subject":"Before changing SetLLRem","message":"Before changing SetLLRem\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/WellFormedLF.agda","new_file":"agda\/WellFormedLF.agda","new_contents":"module WellFormedLF where\n\nopen import Common\nopen import LinLogic\nopen import SetLL\nopen import SetLLRem hiding (drsize)\nopen import LinFun\nopen import Data.Product\n\n--data IndexLF : \u2200{u} \u2192 {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set where\n--  \u2193    : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF elf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  _\u2282\u2192_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF lf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  tr   : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u ll orll rll} \u2192 {{ltr : LLTr orll ll}} \u2192 {lf : LFun {u} {i} {j} {rll} {orll}}\n--         \u2192 IndexLF lf \u2192 IndexLF (tr {{ltr = ltr}} lf) \n--\n\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n\n--                                       \u2193 probably the subtrees that contain all the inputs. \n-- We need to keep truck of all the latest subtrees that are outputs of coms. We can then check whether a transformation permutates them. If so , the tr is acceptable.\n-- transformations inside a subtree are acceptable.\n-- tranformations between two subtrees are only acceptable if the next com depends on both of them.\n-- If more than one subtree depends on a specific subtree, but the two do not depend on each other, we first separate the elements for the first subtree and then for the other.\n-- Since these coms can be independently executed, there can be tranformations of one subtree that can not be done while the others are ready. --> We remove the separation transformation and consider that\n-- the unexecuted transformations are done on those separated elements.\n\n-- > The input tree contains subtrees that represent exactly the input of the coms it can execute.\n\n-- 2 TWO\n-- We need three structures.\n-- THe first is used to identify inputs from the same com. When a tranformation splits these coms, we also split the set into two sets. Next we track all the inputs that are part of the tranformation, be it sets of inputs of a specific com or individual inputs. From all these sets, at least one item from each set needs to be the input of the next com.\n-- The set of coms that are used to allow for commutation of inputs.\n\nmodule _ where\n\n  open import Data.Vec\n\n  mutual \n    data Descendant {u} : Set (lsuc u) where\n      orig  : Descendant\n      dec  : \u2115 \u2192 \u2200{i ll} \u2192 SetLLD {i} {u} ll \u2192 Descendant\n  \n  \n    \n    data SetLLD {i : Size} {u} : LinLogic i {u} \u2192 Set (lsuc u) where\n      \u2193     : \u2200{ll} \u2192  Descendant {u}              \u2192 SetLLD ll\n      _\u2190\u2227   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2227 rs)\n      \u2227\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2227 rs)\n      _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2227 rs)\n      _\u2190\u2228   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2228 rs)\n      \u2228\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2228 rs)\n      _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2228 rs)\n      _\u2190\u2202   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2202 rs)\n      \u2202\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2202 rs)\n      _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2202 rs)\n    \n  \n  \n  data MSetLLD {i : Size} {u} : LinLogic i {u} \u2192 Set (lsuc u) where\n    \u2205   : \u2200{ll}             \u2192 MSetLLD ll\n    \u00ac\u2205  : \u2200{ll} \u2192 SetLLD ll \u2192 MSetLLD ll\n  \n  -- TODO We shouldn't need this. When issue agda #2409 is resolved, remove this.\n  drsize : \u2200{i u ll} \u2192 {j : Size< \u2191 i} \u2192 SetLLD {i} {u} ll \u2192 SetLLD {j} ll\n  drsize (\u2193 mm)          = (\u2193 mm)\n  drsize (x \u2190\u2227)     = (drsize x) \u2190\u2227\n  drsize (\u2227\u2192 x)     = \u2227\u2192 (drsize x)\n  drsize (x \u2190\u2227\u2192 x\u2081) = (drsize x \u2190\u2227\u2192 drsize x\u2081)\n  drsize (x \u2190\u2228)     = (drsize x) \u2190\u2228\n  drsize (\u2228\u2192 x)     = \u2228\u2192 (drsize x)\n  drsize (x \u2190\u2228\u2192 x\u2081) = (drsize x \u2190\u2228\u2192 drsize x\u2081)\n  drsize (x \u2190\u2202)     = (drsize x) \u2190\u2202\n  drsize (\u2202\u2192 x)     = \u2202\u2192 (drsize x)\n  drsize (x \u2190\u2202\u2192 x\u2081) = (drsize x \u2190\u2202\u2192 drsize x\u2081)\n  \n\n  fillAllLowerD : \u2200{i u} \u2192 \u2200 ll \u2192 SetLLD {i} {u} ll\n  fillAllLowerD \u2205 = \u2193 orig\n  fillAllLowerD (\u03c4 x) = \u2193 orig\n  fillAllLowerD (ll \u2227 ll\u2081) = (fillAllLowerD ll) \u2190\u2227\u2192 fillAllLowerD ll\u2081\n  fillAllLowerD (ll \u2228 ll\u2081) = (fillAllLowerD ll) \u2190\u2228\u2192 fillAllLowerD ll\u2081\n  fillAllLowerD (ll \u2202 ll\u2081) = (fillAllLowerD ll) \u2190\u2202\u2192 fillAllLowerD ll\u2081\n  fillAllLowerD (call x) = \u2193 orig\n\n\n  compose : \u2200{u i} \u2192 {j : Size< \u2191 i} \u2192 \u2200 {oll ll} \u2192 SetLLD {i} {u} oll \u2192 SetLLRem {_} {j} oll ll \u2192 SetLLD ll \u2192 SetLLD oll\n  compose sdo sr (\u2193 x) = {!!}\n  compose sdo sr (sd \u2190\u2227) = {!!}\n  compose sdo sr (\u2227\u2192 sd) = {!!}\n  compose sdo sr (sd \u2190\u2227\u2192 sd\u2081) = {!!}\n  compose sdo sr (sd \u2190\u2228) = {!!}\n  compose sdo sr (\u2228\u2192 sd) = {!!}\n  compose sdo sr (sd \u2190\u2228\u2192 sd\u2081) = {!!}\n  compose sdo sr (sd \u2190\u2202) = {!!}\n  compose sdo sr (\u2202\u2192 sd) = {!!}\n  compose sdo sr (sd \u2190\u2202\u2192 sd\u2081) = {!!} \n\n  findNextCom : \u2200{u i} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 LFun {rll = rll} {ll = ll} \u2192 SetLLD ll\n  findNextCom {u} {i = pi} {ll = oll} lf = {!!} where\n    findNextCom` : \u2200{pi} \u2192 {oll : LinLogic pi {u}} \u2192 {i : Size< \u2191 pi} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 LFun {rll = rll} {ll = ll} \u2192 SetLLRem oll ll \u2192 SetLLRem oll rll \u00d7 SetLLD oll \n    findNextCom` {oll = oll} I sr = (sr , fillAllLowerD oll)\n    findNextCom` (_\u2282_ {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf\u2081 lf\u2082) sr with (findNextCom` lf\u2081 (fillAllLowerRem pll))\n    ... | (r\u2081 , r\u2082) with (findNextCom` lf\u2082 (fillAllLowerRem (replLL ll ind ell)))\n    ... | (g\u2081 , g\u2082) = {!!}\n    findNextCom` (tr lf\u2081) sr = {!!}\n    findNextCom` (obs lf\u2081) sr = {!!}\n    findNextCom` (com df lf\u2081) sr = {!!}\n    findNextCom` (call x lf\u2081) sr = {!!}\n\n\n\n","old_contents":"module WellFormedLF where\n\nopen import Common\nopen import LinLogic\nopen import SetLL\n\n--data IndexLF : \u2200{u} \u2192 {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set where\n--  \u2193    : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF elf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  _\u2282\u2192_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF lf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  tr   : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u ll orll rll} \u2192 {{ltr : LLTr orll ll}} \u2192 {lf : LFun {u} {i} {j} {rll} {orll}}\n--         \u2192 IndexLF lf \u2192 IndexLF (tr {{ltr = ltr}} lf) \n--\n\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n\n--                                       \u2193 probably the subtrees that contain all the inputs. \n-- We need to keep truck of all the latest subtrees that are outputs of coms. We can then check whether a transformation permutates them. If so , the tr is acceptable.\n-- transformations inside a subtree are acceptable.\n-- tranformations between two subtrees are only acceptable if the next com depends on both of them.\n-- If more than one subtree depends on a specific subtree, but the two do not depend on each other, we first separate the elements for the first subtree and then for the other.\n-- Since these coms can be independently executed, there can be tranformations of one subtree that can not be done while the others are ready. --> We remove the separation transformation and consider that\n-- the unexecuted transformations are done on those separated elements.\n\n-- > The input tree contains subtrees that represent exactly the input of the coms it can execute.\n\n-- 2 TWO\n-- We need three structures.\n-- THe first is used to identify inputs from the same com. When a tranformation splits these coms, we also split the set into two sets. Next we track all the inputs that are part of the tranformation, be it sets of inputs of a specific com or individual inputs. From all these sets, at least one item from each set needs to be the input of the next com.\n-- The set of coms that are used to allow for commutation of inputs.\n\nmodule _ where\n\n  open import Data.Vec\n  \n  data Ancestor : Set where\n    orig  : Ancestor\n    anc   : \u2115 \u2192 Ancestor \u2192 Ancestor\n    manc  : \u2115 \u2192 \u2200{n} \u2192 Vec Ancestor n \u2192 Ancestor \u2192 Ancestor\n\n\n  \n  data SetLLD {i : Size} {u} : LinLogic i {u} \u2192 Set (lsuc u) where\n    \u2193     : \u2200{ll} \u2192  Ancestor               \u2192 SetLLD ll\n    _\u2190\u2227   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2227 rs)\n    \u2227\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2227 rs)\n    _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2227 rs)\n    _\u2190\u2228   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2228 rs)\n    \u2228\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2228 rs)\n    _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2228 rs)\n    _\u2190\u2202   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2202 rs)\n    \u2202\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2202 rs)\n    _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2202 rs)\n    \n  \n  \n  data MSetLLD {i : Size} {u} : LinLogic i {u} \u2192 Set (lsuc u) where\n    \u2205   : \u2200{ll}             \u2192 MSetLLD ll\n    \u00ac\u2205  : \u2200{ll} \u2192 SetLLD ll \u2192 MSetLLD ll\n  \n  -- TODO We shouldn't need this. When issue agda #2409 is resolved, remove this.\n  drsize : \u2200{i u ll} \u2192 {j : Size< \u2191 i} \u2192 SetLLD {i} {u} ll \u2192 SetLLD {j} ll\n  drsize (\u2193 mm)          = (\u2193 mm)\n  drsize (x \u2190\u2227)     = (drsize x) \u2190\u2227\n  drsize (\u2227\u2192 x)     = \u2227\u2192 (drsize x)\n  drsize (x \u2190\u2227\u2192 x\u2081) = (drsize x \u2190\u2227\u2192 drsize x\u2081)\n  drsize (x \u2190\u2228)     = (drsize x) \u2190\u2228\n  drsize (\u2228\u2192 x)     = \u2228\u2192 (drsize x)\n  drsize (x \u2190\u2228\u2192 x\u2081) = (drsize x \u2190\u2228\u2192 drsize x\u2081)\n  drsize (x \u2190\u2202)     = (drsize x) \u2190\u2202\n  drsize (\u2202\u2192 x)     = \u2202\u2192 (drsize x)\n  drsize (x \u2190\u2202\u2192 x\u2081) = (drsize x \u2190\u2202\u2192 drsize x\u2081)\n  \n  fillAllLowerD : \u2200{i u} \u2192 \u2200 ll \u2192 SetLLD {i} {u} ll\n  fillAllLowerD ll = {!!}\n  \n  \n  \n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"120a6dd883e6567cf125c6d37bb76c3063ce2241","subject":"Fixed an theorem.","message":"Fixed an theorem.\n\nIgnore-this: a7c05b0bdd220240bb7c76cb5a83c29e\n\ndarcs-hash:20120303194550-3bd4e-12021126bfb9a318cdc8105eeab0d45f82ee7a8f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/TheoremsATP.agda","new_file":"src\/FOL\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- FOL theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module contains some examples showing the use of the ATPs for\n-- proving FOL theorems.\n\nmodule FOL.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A     : Set\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n      \u03c6(x)\n  -----------  \u2200-intro\n    \u2200x.\u03c6(x)\n\n    \u2200x.\u03c6(x)\n  -----------  \u2200-elim\n     \u03c6(t)\n\n      \u03c6(t)\n  -----------  \u2203-intro\n    \u2203x.\u03c6(x)\n\n   \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n ----------------------  \u2203-elim\n           \u03c8\n-}\n\npostulate\n  -- TODO: 2012-03-02. Fix? the universal introduction\n  \u2200-intro : (\u2200 x \u2192 A\u00b9 x) \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200-elim  : (\u2200 x \u2192 A\u00b9 x) \u2192 (t : D) \u2192 A\u00b9 t\n  -- It is necessary to assume a non-empty domain. See\n  -- FOL.NonEmptyDomain.TheoremsI\/ATP.\u2203I.\n  --\n  -- TODO: 2012-02-28. Fix the existential introduction rule.\n  -- \u2203-intro : ((t : D) \u2192 A\u00b9 t) \u2192 \u2203 A\u00b9\n  \u2203-elim'  : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n-- {-# ATP prove \u2200-intro #-}\n{-# ATP prove \u2200-elim #-}\n-- {-# ATP prove \u2203-intro #-}\n{-# ATP prove \u2203-elim' #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200 x \u2192 A\u00b9 x) \u2194 (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 A\u00b9)       \u2194 (\u2200 x \u2192 \u00ac (A\u00b9 x))\n  gDM\u2083 : (\u2200 x \u2192 A\u00b9 x)   \u2194 \u00ac (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2084 : \u2203 A\u00b9           \u2194 \u00ac (\u2200 x \u2192 \u00ac (A\u00b9 x))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : (\u2200 x y \u2192 A\u00b2 x y) \u2194 (\u2200 y x \u2192 A\u00b2 x y)\n  \u2203-ord : (\u2203[ x ] \u2203[ y ] A\u00b2 x y) \u2194 (\u2203[ y ] \u2203[ x ] A\u00b2 x y)\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate\n  -- TODO: 2012-02-03. The ATPs are not proving the theorem.\n  -- \u2200-erase-add : ((x : D) \u2192 A) \u2194 A\n  \u2203-erase-add : (\u2203[ x ] A \u2227 A\u00b9 x) \u2194 A \u2227 (\u2203[ x ] A\u00b9 x)\n-- {-# ATP prove \u2200-erase-add #-}\n{-# ATP prove \u2203-erase-add #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : (\u2200 x \u2192 A\u00b9 x \u2227 B\u00b9 x) \u2194 ((\u2200 x \u2192 A\u00b9 x) \u2227 (\u2200 x \u2192 B\u00b9 x))\n  \u2203-dist : (\u2203[ x ] A\u00b9 x \u2228 B\u00b9 x) \u2194 (\u2203 A\u00b9 \u2228 \u2203 B\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n\n-- Interchange of quantifiers.\n-- The related theorem \u2200x\u2203y.Axy \u2192 \u2203y\u2200x.Axy is not (classically) valid.\npostulate \u2203\u2200 : \u2203[ x ] (\u2200 y \u2192 A\u00b2 x y) \u2192 \u2200 y \u2192 \u2203[ x ] A\u00b2 x y\n{-# ATP prove \u2203\u2200 #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate\n  \u2203\u2192\u00ac\u2200\u00ac : \u2203[ x ] A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 \u00ac A\u00b9 x)\n  \u2203\u00ac\u2192\u00ac\u2200 : \u2203[ x ] \u00ac A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 A\u00b9 x)\n{-# ATP prove \u2203\u2192\u00ac\u2200\u00ac #-}\n{-# ATP prove \u2203\u00ac\u2192\u00ac\u2200 #-}\n\n-- \u2200 in terms of \u2203 and \u00ac.\npostulate\n  \u2200\u2192\u00ac\u2203\u00ac : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u00ac (\u2203[ x ] \u00ac A\u00b9 x)\n  \u2200\u00ac\u2192\u00ac\u2203 : (\u2200 {x} \u2192 \u00ac A\u00b9 x) \u2192 \u00ac (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200\u2192\u00ac\u2203\u00ac #-}\n{-# ATP prove \u2200\u00ac\u2192\u00ac\u2203 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- FOL theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module contains some examples showing the use of the ATPs for\n-- proving FOL theorems.\n\nmodule FOL.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A     : Set\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n      \u03c6(x)\n  -----------  \u2200-intro\n    \u2200x.\u03c6(x)\n\n    \u2200x.\u03c6(x)\n  -----------  \u2200-elim\n     \u03c6(t)\n\n      \u03c6(t)\n  -----------  \u2203-intro\n    \u2203x.\u03c6(x)\n\n   \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n ----------------------  \u2203-elim\n           \u03c8\n-}\n\npostulate\n  -- TODO: 2012-03-02. Fix? the universal introduction\n  \u2200-intro : (\u2200 x \u2192 A\u00b9 x) \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200-elim  : (\u2200 x \u2192 A\u00b9 x) \u2192 (t : D) \u2192 A\u00b9 t\n  -- It is necessary to assume a non-empty domain. See\n  -- FOL.NonEmptyDomain.TheoremsI\/ATP.\u2203I.\n  --\n  -- TODO: 2012-02-28. Fix the existential introduction rule.\n  -- \u2203-intro : ((t : D) \u2192 A\u00b9 t) \u2192 \u2203 A\u00b9\n  \u2203-elim'  : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n-- {-# ATP prove \u2200-intro #-}\n{-# ATP prove \u2200-elim #-}\n-- {-# ATP prove \u2203-intro #-}\n{-# ATP prove \u2203-elim' #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200 x \u2192 A\u00b9 x) \u2194 (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 A\u00b9)       \u2194 (\u2200 x \u2192 \u00ac (A\u00b9 x))\n  gDM\u2083 : (\u2200 x \u2192 A\u00b9 x)   \u2194 \u00ac (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2084 : \u2203 A\u00b9           \u2194 \u00ac (\u2200 x \u2192 \u00ac (A\u00b9 x))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : (\u2200 x y \u2192 A\u00b2 x y) \u2194 (\u2200 y x \u2192 A\u00b2 x y)\n  \u2203-ord : (\u2203[ x ] \u2203[ y ] A\u00b2 x y) \u2194 (\u2203[ y ] \u2203[ x ] A\u00b2 x y)\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate\n  -- TODO: 2012-02-03. The ATPs are not proving the theorem.\n  -- \u2200-erase-add : ((x : D) \u2192 A) \u2194 A\n  \u2203-erase-add : (\u2203[ x ] A \u2227 A\u00b9 x) \u2194 A \u2227 (\u2203[ x ] A\u00b9 x)\n-- {-# ATP prove \u2200-erase-add #-}\n{-# ATP prove \u2203-erase-add #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : \u2200 x \u2192 A\u00b9 x \u2227 B\u00b9 x \u2194 ((\u2200 x \u2192 A\u00b9 x) \u2227 (\u2200 x \u2192 B\u00b9 x))\n  \u2203-dist : (\u2203[ x ] A\u00b9 x \u2228 B\u00b9 x) \u2194 (\u2203 A\u00b9 \u2228 \u2203 B\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n\n-- Interchange of quantifiers.\n-- The related theorem \u2200x\u2203y.Axy \u2192 \u2203y\u2200x.Axy is not (classically) valid.\npostulate \u2203\u2200 : \u2203[ x ] (\u2200 y \u2192 A\u00b2 x y) \u2192 \u2200 y \u2192 \u2203[ x ] A\u00b2 x y\n{-# ATP prove \u2203\u2200 #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate\n  \u2203\u2192\u00ac\u2200\u00ac : \u2203[ x ] A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 \u00ac A\u00b9 x)\n  \u2203\u00ac\u2192\u00ac\u2200 : \u2203[ x ] \u00ac A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 A\u00b9 x)\n{-# ATP prove \u2203\u2192\u00ac\u2200\u00ac #-}\n{-# ATP prove \u2203\u00ac\u2192\u00ac\u2200 #-}\n\n-- \u2200 in terms of \u2203 and \u00ac.\npostulate\n  \u2200\u2192\u00ac\u2203\u00ac : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u00ac (\u2203[ x ] \u00ac A\u00b9 x)\n  \u2200\u00ac\u2192\u00ac\u2203 : (\u2200 {x} \u2192 \u00ac A\u00b9 x) \u2192 \u00ac (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200\u2192\u00ac\u2203\u00ac #-}\n{-# ATP prove \u2200\u00ac\u2192\u00ac\u2203 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6b698f52d57559a1c01ed1ef0ab1afb783796b9a","subject":"Strong-Fiat-Shamir: some progress and re-org","message":"Strong-Fiat-Shamir: some progress and re-org\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Strong-Fiat-Shamir.agda","new_file":"ZK\/Strong-Fiat-Shamir.agda","new_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; done; ask)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {RP R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  (v \u21c4 p) r Y c = \u03a3-Protocol.\u03a3-game (v , p) r Y c\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct (v , p) = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (mk (get-A r Y) c (get-f r Y c)) \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 Y : \u2605 where\n    constructor mk\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n  record Special-Soundness : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 Challenge \u00d7 \u03a3-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.One\nopen import Data.Two\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Strategy\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}{Prf : \u039b \u2192 \u2605}\n  {RS : \u2605}{Q : \u2605}{Resp :    \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n  -- (Q? : Decidable (_\u2261_ {A = Q}))\n  where\n\nRandom-Oracle : \u2605\nRandom-Oracle = Q \u2192 Resp\n\nState : (S A : \u2605) \u2192 \u2605\nState S A = S \u2192 A \u00d7 S\n\ndata Adversary-Query : \u2605 where\n  query-H : (q : Q) \u2192 Adversary-Query\n  query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\nChallenger-Resp : Adversary-Query \u2192 \u2605\nChallenger-Resp (query-H s) = Resp\nChallenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\nAdversary : \u2605 \u2192 \u2605\nAdversary A = Strategy Adversary-Query Challenger-Resp A\n\nTranscript = List (\u03a3 Adversary-Query Challenger-Resp)\n\nPrfs : \u2605\nPrfs = List (\u03a3 \u039b Prf)\n\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\nComplete-Proof-System : {RP : \u2605}{Prf : \u039b \u2192 \u2605} \u2192 Proof-System RP Prf \u2192 \u2605\nComplete-Proof-System PS = \u2200 rp w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n  where open Proof-System PS\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n\nrecord Simulator {RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192\n                      let \u03c0 = Simulate rs t Y\n                      in Verify Y \u03c0 \u2261 1\u2082\n\nmodule Is-Zero-Knowledge\n  {RP}\n  (PF : Proof-System RP Prf)\n  (sim : Simulator PF)\n  where\n\n  open Simulator sim\n\n  module _ (ro : Random-Oracle) where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {A} \u2192 \ud835\udfda \u2192 RS \u2192 Adversary A \u2192 A \u00d7 Transcript\n    Experiment \u03b2 rs adv = runT (challenger \u03b2 rs) adv []\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {RP R\u03a3P : \u2605}\n  (PF : Proof-System RP Prf)\n  where\n  open Proof-System PF\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  (v \u21c4 p) r Y c = \u03a3-Protocol.\u03a3-game (v , p) r Y c\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct (v , p) = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (mk (get-A r Y) c (get-f r Y c)) \u2261 1\u2082\n\n  module Special-Honest-Verifier-Zero-Knowledge where\n\n    record SHVZK (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n      open \u03a3-Protocol \u03a3-proto\n      field\n        Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n        Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n        -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n        -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  module Special-Soundness where\n\n    record SpSo : \u2605 where\n      field\n        Extract : (Y : \u039b)(t\u2080 t\u2081 : \u03a3-Transcript Y) \u2192 W\n        Extract-ok : \u2200 Y A c\u2080 c\u2081 f\u2080 f\u2081 \u2192 (c\u2080 \u2262 c\u2081) \u2192 L (Extract Y (mk A c\u2080 f\u2080) (mk A c\u2081 f\u2081)) Y\n\n  module Simulation-Sound-Extractability where\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-H _ , _) = 0\u2082\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 (query-create-proof _ _ , nothing) = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n    open StatefulRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (sim : Simulator PF)\n        (open Is-Zero-Knowledge PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Random-Oracle)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 rs (Adv w)\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge.SHVZK \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge.SHVZK shvzk\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 (Challenge \u00d7 \u03a3-Prf Y)\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 Challenge \u00d7 \u03a3-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge.SHVZK \u03a3-proto)\n              (ssound : Special-Soundness.SpSo)\n              where\n      open Simulation-Sound-Extractability\n      {-\n      S : Simulator PF\n      S = record { H-patch = {!!}\n                 ; Simulate = {!!}\n                 ; verify-sim-spec = {!!} }\n      -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"81d49a84159ca3aee47134efe64ff3235d29314e","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: b0f0381caf8a0f958cf0aadd900b632\n\ndarcs-hash:20120105172321-3bd4e-4cf58940f8066449afbaaf5df946f7aa92362520.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/PropertiesByInductionATP.agda","new_file":"src\/FOTC\/Data\/Nat\/PropertiesByInductionATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the FOTC natural numbers type)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the FOTC natural numbers. The following\n-- examples show some proofs using it.\n\nmodule FOTC.Data.Nat.PropertiesByInductionATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n-- The predicate is not inside the where clause because the\n-- translation of projection-like functions is not implemented.\n+-rightIdentity-P : D \u2192 Set\n+-rightIdentity-P i = i + zero \u2261 i\n{-# ATP definition +-rightIdentity-P #-}\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = indN +-rightIdentity-P P0 is Nn\n  where\n  postulate P0 : +-rightIdentity-P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 +-rightIdentity-P i \u2192 +-rightIdentity-P (succ\u2081 i)\n  {-# ATP prove is #-}\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = N (i + n)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  -- Metis 2.3 (release 20110926): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove is #-}\n\n+-assoc : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o} Nm Nn No = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + n + o \u2261 i + (n + o)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  -- Metis 2.3 (release 20110926): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  -- Metis 2.3 (release 20110926): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove is #-}\n\n-- A proof without use ATPs definitions.\n+-assoc' : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc' {n = n} {o} Nm Nn No = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + n + o \u2261 i + (n + o)\n\n  postulate P0 : zero + n + o \u2261 zero + (n + o)\n  {-# ATP prove P0 #-}\n\n  postulate\n    is : \u2200 {i} \u2192\n         i + n + o \u2261 i + (n + o) \u2192  -- IH.\n         succ\u2081 i + n + o \u2261 succ\u2081 i + (n + o)\n  {-# ATP prove is #-}\n\nx+Sy\u2261S[x+y] : \u2200 {m} n \u2192 N m \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] n Nm = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  -- Metis 2.3 (release 20110926): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove is #-}\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} Nm Nn = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + n \u2261 n + i\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  -- Metis 2.3 (release 20110926): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove P0 +-rightIdentity #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  -- Metis 2.3 (release 20110926): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove is x+Sy\u2261S[x+y] #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the FOTC natural numbers type)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the FOTC natural numbers. The following\n-- examples show some proofs using it.\n\n-- TODO: We have not tested which ATPs fail on the ATP conjectures.\n\nmodule FOTC.Data.Nat.PropertiesByInductionATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n-- The predicate is not inside the where clause because the\n-- translation of projection-like functions is not implemented.\n+-rightIdentity-P : D \u2192 Set\n+-rightIdentity-P i = i + zero \u2261 i\n{-# ATP definition +-rightIdentity-P #-}\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = indN +-rightIdentity-P P0 is Nn\n  where\n  postulate P0 : +-rightIdentity-P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 +-rightIdentity-P i \u2192 +-rightIdentity-P (succ\u2081 i)\n  {-# ATP prove is #-}\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = N (i + n)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  {-# ATP prove is #-}\n\n+-assoc : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o} Nm Nn No = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + n + o \u2261 i + (n + o)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  {-# ATP prove is #-}\n\n-- A proof without use ATPs definitions.\n+-assoc' : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc' {n = n} {o} Nm Nn No = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + n + o \u2261 i + (n + o)\n\n  postulate P0 : zero + n + o \u2261 zero + (n + o)\n  {-# ATP prove P0 #-}\n\n  postulate\n    is : \u2200 {i} \u2192\n         i + n + o \u2261 i + (n + o) \u2192  -- IH.\n         succ\u2081 i + n + o \u2261 succ\u2081 i + (n + o)\n  {-# ATP prove is #-}\n\nx+Sy\u2261S[x+y] : \u2200 {m} n \u2192 N m \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] n Nm = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  {-# ATP prove P0 #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  {-# ATP prove is #-}\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} Nm Nn = indN P P0 is Nm\n  where\n  P : D \u2192 Set\n  P i = i + n \u2261 n + i\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  {-# ATP prove P0 +-rightIdentity #-}\n\n  postulate is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n  {-# ATP prove is x+Sy\u2261S[x+y] #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"655a42bd4923f174009da823127bf76a5c29ee83","subject":"flipbased-tree: probability results for return, toss, map, zip, weaken","message":"flipbased-tree: probability results for return, toss, map, zip, weaken\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-tree.agda","new_file":"flipbased-tree.agda","new_contents":"module flipbased-tree where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import bintree\nopen import Data.Product\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\nreturn\u21ba : \u2200 {c a} {A : Set a} \u2192 A \u2192 \u21ba c A\nreturn\u21ba = leaf\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (leaf x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (leaf 0b) (leaf 1b)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (leaf x) = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 c {m a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ c = weaken\u2264 (m\u2264n+m _ c)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (leaf x) = leaf (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (leaf x)      = weaken+ c x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 leaf map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _*\/_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n  -- \u00b7\/1+_ : \u2115 \u2192 Set\n  -- \/+\/ : \u2115 \u2192 [0,1] \u2192 [0,1] \u2192 [0,1]\n\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n  _\/2+_\/2-comm : \u2200 x y \u2192 x \/2+ y \/2 \u2261 y \/2+ x \/2\n  1*q\u2261q : \u2200 q \u2192 1\/1 *\/ q \u2261 q\n  0*q\u2261q : \u2200 q \u2192 0\/1 *\/ q \u2261 0\/1\n  distr-*-\/2+\/2 : \u2200 x y z \u2192 (x *\/ z) \/2+ (y *\/ z) \/2 \u2261 (x \/2+ y \/2) *\/ z\n  distr-\/2-* : \u2200 p q \u2192 (p \/2) *\/ (q \/2) \u2261 ((p *\/ q) \/2) \/2\n  0\/2\u22610 : 0\/1 \/2 \u2261 0\/1\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\ndata Pr[return\u21ba_\u2261_]\u2261_ {a} {A : Set a} (x y : A) : [0,1] \u2192 Set a where\n  Pr-\u2261 : x \u2261 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 1\/1\n  Pr-\u2262 : x \u2262 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 0\/1\n\ninfix 2 Pr[_\u2261_]\u2261_\ndata Pr[_\u2261_]\u2261_ {a} {A : Set a} : \u2200 {c} \u2192 \u21ba c A \u2192 A \u2192 [0,1] \u2192 Set a where\n  Pr-return : \u2200 {c x y pr} (pf : Pr[return\u21ba x \u2261 y ]\u2261 pr) \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 pr\n\n  Pr-fork : \u2200 {c} {left right : \u21ba c A} {x p q r}\n            (eq : p \/2+ q \/2 \u2261 r)\n            (pf\u2080 : Pr[ left \u2261 x ]\u2261 p)\n            (pf\u2081 : Pr[ right \u2261 x ]\u2261 q)\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 r\n\nPr-fork\u2032 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x p q}\n            \u2192 Pr[ left \u2261 x ]\u2261 p\n            \u2192 Pr[ right \u2261 x ]\u2261 q\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 (p \/2+ q \/2)\nPr-fork\u2032 = Pr-fork refl\n\nPr-return-\u2261 : \u2200 {c a} {A : Set a} {x : A} \u2192 Pr[ return\u21ba {c = c} x \u2261 x ]\u2261 1\/1\nPr-return-\u2261 = Pr-return (Pr-\u2261 refl)\n\nPr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 0\/1\nPr-return-\u2262 = Pr-return \u2218 Pr-\u2262\n\nimport Function.Equality as F\u2261\nimport Function.Equivalence as F\u2248\n\n_\u2248\u21d2_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248\u21d2 p\u2082 = \u2200 {x pr} \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr \u2192 Pr[ p\u2082 \u2261 x ]\u2261 pr\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 {x pr} \u2192 (Pr[ p\u2081 \u2261 x ]\u2261 pr) F\u2248.\u21d4 (Pr[ p\u2082 \u2261 x ]\u2261 pr)\n\n\u2248-refl : \u2200 {c a} {A : Set a} \u2192 Reflexive {A = \u21ba c A} _\u2248_\n\u2248-refl = F\u2248.id\n\n\u2248-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 p\u2081 \u2248 p\u2082 \u2192 p\u2082 \u2248 p\u2081\n\u2248-sym \u03b7 = F\u2248.sym \u03b7\n\n\u2248-trans : \u2200 {c a} {A : Set a} \u2192 Transitive {A = \u21ba c A} _\u2248_\n\u2248-trans f g = g F\u2248.\u2218 f\n\nfork-sym\u21d2 : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248\u21d2 fork p\u2082 p\u2081\nfork-sym\u21d2 (Pr-fork {p = p} {q} refl pf\u2081 pf\u2080) rewrite p \/2+ q \/2-comm = Pr-fork\u2032 pf\u2080 pf\u2081\n\nfork-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\nfork-sym = F\u2248.equivalence fork-sym\u21d2 fork-sym\u21d2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ]\u2261 0\/1\n            \u2192 Pr[ right \u2261 x ]\u2261 p\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork\u2032 eq\u2081 eq\u2082\n\nex\u2081 : \u2200 x \u2192 Pr[ toss \u2261 x ]\u2261 1\/2\nex\u2081 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261\nex\u2081 0b = F\u2248.Equivalence.to fork-sym F\u2261.\u27e8$\u27e9 (Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261)\n\nPr-map : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {x pr} {f : A \u2192 B} \u2192\n  (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n  Pr[ Alg \u2261 x ]\u2261 pr \u2192\n  Pr[ \u27ea f \u00b7 Alg \u27eb \u2261 f x ]\u2261 pr\nPr-map f-inj (Pr-return (Pr-\u2261 refl)) = Pr-return (Pr-\u2261 refl)\nPr-map f-inj (Pr-return (Pr-\u2262 x\u2262y)) = Pr-return (Pr-\u2262 (x\u2262y \u2218 f-inj))\nPr-map f-inj (Pr-fork eq pf\u2080 pf\u2081) = Pr-fork eq (Pr-map f-inj pf\u2080) (Pr-map f-inj pf\u2081)\n\nPr-same : \u2200 {c a} {A : Set a} {Alg : \u21ba c A} {x pr\u2080 pr\u2081} \u2192\n    pr\u2080 \u2261 pr\u2081 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2080 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2081\nPr-same refl = id\n\nPr-weaken\u2264 : \u2200 {c\u2080 c\u2081 a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    (c\u2080\u2264c\u2081 : c\u2080 \u2264 c\u2081) \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken\u2264 c\u2080\u2264c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken\u2264 p (Pr-return pf) = Pr-return pf\nPr-weaken\u2264 (s\u2264s c\u2080\u2264c\u2081) (Pr-fork eq pf\u2080 pf\u2081)\n  = Pr-fork eq (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2080) (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2081)\n\nPr-weaken+ : \u2200 {c\u2080} c\u2081 {a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken+ c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken+ c\u2081 = Pr-weaken\u2264 (m\u2264n+m _ c\u2081)\n\nPr-map-0 : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {f : A \u2192 B} {x} \u2192 (\u2200 y \u2192 f y \u2262 x)\n           \u2192 Pr[ map\u21ba f Alg \u2261 x ]\u2261 0\/1\nPr-map-0 {Alg = leaf x} f-prop = Pr-return (Pr-\u2262 (f-prop x))\nPr-map-0 {Alg = fork Alg Alg\u2081} f-prop = Pr-fork (trans (0\/2+p\/2\u2261p\/2 0\/1) 0\/2\u22610)\n                                                (Pr-map-0 f-prop) (Pr-map-0 f-prop)\n\nPr-zip : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} {Alg\u2080 : \u21ba c\u2080 A} {Alg\u2081 : \u21ba c\u2081 B} {x y pr\u2081 pr\u2082} \u2192\n  Pr[ Alg\u2080 \u2261 x ]\u2261 pr\u2081 \u2192\n  Pr[ Alg\u2081 \u2261 y ]\u2261 pr\u2082 \u2192\n  Pr[ zip\u21ba Alg\u2080 Alg\u2081 \u2261 (x , y) ]\u2261 (pr\u2081 *\/ pr\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2261 refl)) pf\u2082\n  rewrite 1*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map {f = _,_ x} (cong proj\u2082) pf\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2262 pf)) pf\u2082\n  rewrite 0*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map-0 (\u03bb y x\u2081 \u2192 pf (cong proj\u2081 x\u2081)))\nPr-zip (Pr-fork refl pf\u2081 pf\u2082) pf\u2083 = Pr-fork (distr-*-\/2+\/2 _ _ _) (Pr-zip pf\u2081 pf\u2083) (Pr-zip pf\u2082 pf\u2083)\n\nex\u2082 : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ]\u2261 1\/4\nex\u2082 x y = Pr-same (trans (distr-\/2-* _ _) (cong (_\/2 \u2218 _\/2) (1*q\u2261q _)))\n                  (Pr-zip {Alg\u2080 = toss} {Alg\u2081 = toss} (ex\u2081 x) (ex\u2081 y))\n\npostulate\n  ex\u2083 : \u2200 {n} (x : Bits n) \u2192 Pr[ random \u2261 x ]\u2261 1\/2^ n\n","old_contents":"module flipbased-tree where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Vec\n\nimport flipbased\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a\n\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\ndata Tree {a} (A : Set a) where\n  return\u21ba : \u2200 {c} \u2192 A \u2192 \u21ba c A\n  fork    : \u2200 {c} \u2192 (left right : \u21ba c A) \u2192 \u21ba (suc c) A\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (return\u21ba x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (return\u21ba 0b) (return\u21ba 1b)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (return\u21ba x) = return\u21ba (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (return\u21ba x) = return\u21ba x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nweaken+ : \u2200 {m c a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ {m} {c} = weaken\u2264 (\u2115\u2264.trans (m\u2264m+n m c) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m c)))\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (return\u21ba x)      = weaken+ {_} {c} x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\npostulate\n  Pr[_\u2261_] : \u2200 {c a} {A : Set a} \u2192 \u21ba c A \u2192 A \u2192 [0,1]\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 x \u2192 Pr[ p\u2081 \u2261 x ] \u2261 Pr[ p\u2082 \u2261 x ]\n\npostulate\n  fork-sym : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\n\n  Pr-return-\u2261 : \u2200 {c a} {A : Set a} (x : A) \u2192 Pr[ return\u21ba {c = c} x \u2261 x ] \u2261 1\/1\n\n  Pr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ] \u2261 0\/1\n\n  Pr-fork : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p q}\n            \u2192 Pr[ left \u2261 x ] \u2261 p\n            \u2192 Pr[ right \u2261 x ] \u2261 q\n            \u2192 Pr[ fork left right \u2261 x ] \u2261 p \/2+ q \/2\n\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ] \u2261 0\/1\n            \u2192 Pr[ right \u2261 x ] \u2261 p\n            \u2192 Pr[ fork left right \u2261 x ] \u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork eq\u2081 eq\u2082\n\nex\u2081 : \u2200 x \u2192 Pr[ toss \u2261 x ] \u2261 1\/2\nex\u2081 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) (Pr-return-\u2261 1b)\nex\u2081 0b rewrite fork-sym {0} (return\u21ba 0b) (return\u21ba 1b) 0b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) (Pr-return-\u2261 0b)\n\npostulate\n  ex\u2082 : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ] \u2261 1\/2^ 2\n\n  ex\u2083 : \u2200 {n} (x : Bits n) \u2192 Pr[ random \u2261 x ] \u2261 1\/2^ n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a1713e8fd0cf292f0524a71143e066f20b571f3a","subject":"Nat: avoid a name conflict (temporary)","message":"Nat: avoid a name conflict (temporary)\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a5427687a4fde38a5d0017db3fe19a46e49f6ec2","subject":"[ test-suite ] Fixed.","message":"[ test-suite ] Fixed.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOL\/PropositionalLogic\/TheoremsATP.agda","new_file":"src\/fot\/FOL\/PropositionalLogic\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Propositional logic theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- This module contains some examples showing the use of the ATPs to\n-- prove theorems from propositional logic.\n\nmodule FOL.PropositionalLogic.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some propositional formulae (which are translated as\n-- 0-ary predicates).\npostulate P Q R : Set\n\n------------------------------------------------------------------------------\n-- The introduction and elimination rules for the intuitionist\n-- propositional connectives are theorems.\n\npostulate\n  \u2192I  : (P \u2192 Q) \u2192 P \u2192 Q\n  \u2192E  : (P \u2192 Q) \u2192 P \u2192 Q\n  \u2192I' : (P \u2192 Q) \u2192 P \u2192 Q\n  \u2192E' : (P \u21d2 Q) \u21d2 P \u21d2 Q\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2192I' #-}\n{-# ATP prove \u2192E' #-}\n\npostulate\n  \u2227I   : P \u2192 Q \u2192 P \u2227 Q\n  \u2227E\u2081  : P \u2227 Q \u2192 P\n  \u2227E\u2082  : P \u2227 Q \u2192 Q\n  \u2227I'  : P \u21d2 Q \u21d2 P \u2227 Q\n  \u2227E\u2082' : P \u2227 Q \u21d2 Q\n  \u2227E\u2081' : P \u2227 Q \u21d2 P\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2227I' #-}\n{-# ATP prove \u2227E\u2081' #-}\n{-# ATP prove \u2227E\u2082' #-}\n\npostulate\n  \u2228I\u2081  : P \u2192 P \u2228 Q\n  \u2228I\u2082  : Q \u2192 P \u2228 Q\n  \u2228E   : (P \u2192 R) \u2192 (Q \u2192 R) \u2192 P \u2228 Q \u2192 R\n  \u2228I\u2081' : P \u21d2 P \u2228 Q\n  \u2228I\u2082' : Q \u21d2 P \u2228 Q\n  \u2228E'  : (P \u21d2 R) \u21d2 (Q \u21d2 R) \u21d2 P \u2228 Q \u21d2 R\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u2228I\u2081' #-}\n{-# ATP prove \u2228I\u2082' #-}\n{-# ATP prove \u2228E' #-}\n\npostulate\n  \u22a5E  : \u22a5 \u2192 P\n  \u22a5E' : \u22a5 \u21d2 P\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u22a5E' #-}\n\n------------------------------------------------------------------------------\n-- Boolean laws (there are some non-intuitionistic laws)\n\npostulate\n  \u2227-ident : P \u2227 \u22a4       \u2194 P\n  \u2228-ident : P \u2228 \u22a5       \u2194 P\n  \u2227-dom   : P \u2227 \u22a5       \u2194 \u22a5\n  \u2228-dom   : P \u2228 \u22a4       \u2194 \u22a4\n  \u2227-idemp : P \u2227 P       \u2194 P\n  \u2228-idemp : P \u2228 P       \u2194 P\n  dn      : \u00ac \u00ac P       \u2194 P\n  \u2227-comm  : P \u2227 Q       \u2192 Q \u2227 P\n  \u2228-comm  : P \u2228 Q       \u2192 Q \u2228 P\n  \u2227-assoc : (P \u2227 Q) \u2227 R \u2194 P \u2227 Q \u2227 R\n  \u2228-assoc : (P \u2228 Q) \u2228 R \u2194 P \u2228 Q \u2228 R\n  \u2227\u2228-dist : P \u2227 (Q \u2228 R) \u2194 P \u2227 Q \u2228 P \u2227 R\n  \u2228\u2227-dist : P \u2228 Q \u2227 R   \u2194 (P \u2228 Q) \u2227 (P \u2228 R)\n  DM\u2081     : \u00ac (P \u2228 Q)   \u2194 \u00ac P \u2227 \u00ac Q\n  DM\u2082     : \u00ac (P \u2227 Q)   \u2194 \u00ac P \u2228 \u00ac Q\n  abs\u2081    : P \u2228 P \u2227 Q   \u2194 P\n  abs\u2082    : P \u2227 (P \u2228 Q) \u2194 P\n  \u2227-neg   : P \u2227 \u00ac P     \u2194 \u22a5\n  \u2228-neg   : P \u2228 \u00ac P     \u2194 \u22a4\n{-# ATP prove \u2227-ident #-}\n{-# ATP prove \u2228-ident #-}\n{-# ATP prove \u2227-dom #-}\n{-# ATP prove \u2228-dom #-}\n{-# ATP prove \u2227-idemp #-}\n{-# ATP prove \u2228-idemp #-}\n{-# ATP prove dn #-}\n{-# ATP prove \u2227-comm #-}\n{-# ATP prove \u2228-comm #-}\n{-# ATP prove \u2227-assoc #-}\n{-# ATP prove \u2228-assoc #-}\n{-# ATP prove \u2227\u2228-dist #-}\n{-# ATP prove \u2228\u2227-dist #-}\n{-# ATP prove DM\u2081 #-}\n{-# ATP prove DM\u2082 #-}\n{-# ATP prove abs\u2081 #-}\n{-# ATP prove abs\u2082 #-}\n{-# ATP prove \u2227-neg #-}\n{-# ATP prove \u2228-neg #-}\n\n------------------------------------------------------------------------------\n-- Theorems related with the definition of logical connectives in\n-- terms of others\n\npostulate\n  \u2194-def  : (P \u2194 Q) \u2194 (P \u2192 Q) \u2227 (Q \u2192 P)\n  \u2192-def  : P \u2192 Q   \u2194 \u00ac P \u2228 Q\n  \u2228\u2081-def : P \u2228 Q   \u2194 \u00ac P \u2192 Q\n  \u2228\u2082-def : P \u2228 Q   \u2194 \u00ac (\u00ac P \u2227 \u00ac Q)\n  \u2227-def  : P \u2227 Q   \u2194 \u00ac (\u00ac P \u2228 \u00ac Q)\n  \u00ac-def  : \u00ac P     \u2194 P \u2192 \u22a5\n  \u22a5-def  : \u22a5       \u2194 P \u2227 \u00ac P\n  \u22a4-def  : \u22a4       \u2194 \u00ac \u22a5\n{-# ATP prove \u2194-def #-}\n{-# ATP prove \u2192-def #-}\n{-# ATP prove \u2228\u2081-def #-}\n{-# ATP prove \u2228\u2082-def #-}\n{-# ATP prove \u2227-def #-}\n{-# ATP prove \u00ac-def #-}\n{-# ATP prove \u22a5-def #-}\n{-# ATP prove \u22a4-def #-}\n\n------------------------------------------------------------------------------\n-- Some intuitionistic theorems\n\npostulate \u2192-transposition : (P \u2192 Q) \u2192 \u00ac Q \u2192 \u00ac P\n{-# ATP prove \u2192-transposition #-}\n\npostulate\n  i\u2081 : P \u2192 Q \u2192 P\n  i\u2082 : (P \u2192 Q \u2192 R) \u2192 (P \u2192 Q) \u2192 P \u2192 R\n  i\u2083 : P \u2192 \u00ac \u00ac P\n  i\u2084 : \u00ac \u00ac \u00ac P \u2192 \u00ac P\n  i\u2085 : ((P \u2227 Q) \u2192 R) \u2194 (P \u2192 (Q \u2192 R))\n  i\u2086 : \u00ac \u00ac (P \u2228 \u00ac P)\n  i\u2087 : (P \u2228 \u00ac P) \u2192 \u00ac \u00ac P \u2192 P\n{-# ATP prove i\u2081 #-}\n{-# ATP prove i\u2082 #-}\n{-# ATP prove i\u2083 #-}\n{-# ATP prove i\u2084 #-}\n{-# ATP prove i\u2085 #-}\n{-# ATP prove i\u2086 #-}\n{-# ATP prove i\u2087 #-}\n\n------------------------------------------------------------------------------\n-- Some non-intuitionistic theorems\n\npostulate \u2190-transposition : (\u00ac Q \u2192 \u00ac P) \u2192 P \u2192 Q\n{-# ATP prove \u2190-transposition #-}\n\npostulate\n  ni\u2081 : ((P \u2192 Q) \u2192 P) \u2192 P\n  ni\u2082 : P \u2228 \u00ac P\n  ni\u2083 : \u00ac \u00ac P \u2192 P\n  ni\u2084 : (P \u2192 Q) \u2192 (\u00ac P \u2192 Q) \u2192 Q\n  ni\u2085 : (P \u2228 Q \u2192 P) \u2228 (P \u2228 Q \u2192 Q)\n  ni\u2086 : (\u00ac \u00ac P \u2192 P) \u2192 P \u2228 \u00ac P\n{-# ATP prove ni\u2081 #-}\n{-# ATP prove ni\u2082 #-}\n{-# ATP prove ni\u2083 #-}\n{-# ATP prove ni\u2084 #-}\n{-# ATP prove ni\u2085 #-}\n{-# ATP prove ni\u2086 #-}\n\n------------------------------------------------------------------------------\n-- The principle of the excluded middle implies the double negation\n-- elimination\npostulate\n  pem\u2192\u00ac\u00ac-elim : (P \u2228 \u00ac P) \u2192 \u00ac \u00ac P \u2192 P\n{-# ATP prove pem\u2192\u00ac\u00ac-elim #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Propositional logic theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- This module contains some examples showing the use of the ATPs to\n-- prove theorems from propositional logic.\n\nmodule FOL.PropositionalLogic.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some propositional formulae (which are translated as\n-- 0-ary predicates).\npostulate P Q R : Set\n\n------------------------------------------------------------------------------\n-- The introduction and elimination rules for the intuitionist\n-- propositional connectives are theorems.\n\npostulate\n  \u2192I  : (P \u2192 Q) \u2192 P \u2192 Q\n  \u2192E  : (P \u2192 Q) \u2192 P \u2192 Q\n  \u2192I' : (P \u2192 Q) \u2192 P \u2192 Q\n  \u2192E' : (P \u21d2 Q) \u21d2 P \u21d2 Q\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2192I' #-}\n{-# ATP prove \u2192E' #-}\n\npostulate\n  \u2227I   : P \u2192 Q \u2192 P \u2227 Q\n  \u2227E\u2081  : P \u2227 Q \u2192 P\n  \u2227E\u2082  : P \u2227 Q \u2192 Q\n  \u2227I'  : P \u21d2 Q \u21d2 P \u2227 Q\n  \u2227E\u2082' : P \u2227 Q \u21d2 Q\n  \u2227E\u2081' : P \u2227 Q \u21d2 P\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2227I' #-}\n{-# ATP prove \u2227E\u2081' #-}\n{-# ATP prove \u2227E\u2082' #-}\n\npostulate\n  \u2228I\u2081  : P \u2192 P \u2228 Q\n  \u2228I\u2082  : Q \u2192 P \u2228 Q\n  \u2228E   : (P \u2192 R) \u2192 (Q \u2192 R) \u2192 P \u2228 Q \u2192 R\n  \u2228I\u2081' : P \u21d2 P \u2228 Q\n  \u2228I\u2082' : Q \u21d2 P \u2228 Q\n  \u2228E'  : (P \u21d2 R) \u21d2 (Q \u21d2 R) \u21d2 P \u2228 Q \u21d2 R\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u2228I\u2081' #-}\n{-# ATP prove \u2228I\u2082' #-}\n{-# ATP prove \u2228E' #-}\n\npostulate\n  \u22a5E  : \u22a5 \u2192 P\n  \u22a5E' : \u22a5 \u21d2 P\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u22a5E' #-}\n\n------------------------------------------------------------------------------\n-- Boolean laws (there are some non-intuitionistic laws)\n\npostulate\n  \u2227-ident : P \u2227 \u22a4       \u2194 P\n  \u2228-ident : P \u2228 \u22a5       \u2194 P\n  \u2227-dom   : P \u2227 \u22a5       \u2194 \u22a5\n  \u2228-dom   : P \u2228 \u22a4       \u2194 \u22a4\n  \u2227-idemp : P \u2227 P       \u2194 P\n  \u2228-idemp : P \u2228 P       \u2194 P\n  dn      : \u00ac \u00ac P       \u2194 P\n  \u2227-comm  : P \u2227 Q       \u2192 Q \u2227 P\n  \u2228-comm  : P \u2228 Q       \u2192 Q \u2228 P\n  \u2227-assoc : (P \u2227 Q) \u2227 R \u2194 P \u2227 Q \u2227 R\n  \u2228-assoc : (P \u2228 Q) \u2228 R \u2194 P \u2228 Q \u2228 R\n  \u2227\u2228-dist : P \u2227 (Q \u2228 R) \u2194 P \u2227 Q \u2228 P \u2227 R\n  \u2228\u2227-dist : P \u2228 Q \u2227 R   \u2194 (P \u2228 Q) \u2227 (P \u2228 R)\n  DM\u2081     : \u00ac (P \u2228 Q)   \u2194 \u00ac P \u2227 \u00ac Q\n  DM\u2082     : \u00ac (P \u2227 Q)   \u2194 \u00ac P \u2228 \u00ac Q\n  abs\u2081    : P \u2228 P \u2227 Q   \u2194 P\n  abs\u2082    : P \u2227 (P \u2228 Q) \u2194 P\n  \u2227-neg   : P \u2227 \u00ac P     \u2194 \u22a5\n  \u2228-neg   : P \u2228 \u00ac P     \u2194 \u22a4\n{-# ATP prove \u2227-ident #-}\n{-# ATP prove \u2228-ident #-}\n{-# ATP prove \u2227-dom #-}\n{-# ATP prove \u2228-dom #-}\n{-# ATP prove \u2227-idemp #-}\n{-# ATP prove \u2228-idemp #-}\n{-# ATP prove dn #-}\n{-# ATP prove \u2227-comm #-}\n{-# ATP prove \u2228-comm #-}\n{-# ATP prove \u2227-assoc #-}\n{-# ATP prove \u2228-assoc #-}\n{-# ATP prove \u2227\u2228-dist #-}\n{-# ATP prove \u2228\u2227-dist #-}\n{-# ATP prove DM\u2081 #-}\n{-# ATP prove DM\u2082 #-}\n{-# ATP prove abs\u2081 #-}\n{-# ATP prove abs\u2082 #-}\n{-# ATP prove \u2227-neg #-}\n{-# ATP prove \u2228-neg #-}\n\n------------------------------------------------------------------------------\n-- Theorems related with the definition of logical connectives in\n-- terms of others\n\npostulate\n  \u2194-def  : (P \u2194 Q) \u2194 (P \u2192 Q) \u2227 (Q \u2192 P)\n  \u2192-def  : P \u2192 Q   \u2194 \u00ac P \u2228 Q\n  \u2228\u2081-def : P \u2228 Q   \u2194 \u00ac P \u2192 Q\n  \u2228\u2082-def : P \u2228 Q   \u2194 \u00ac (\u00ac P \u2227 \u00ac Q)\n  \u2227-def  : P \u2227 Q   \u2194 \u00ac (\u00ac P \u2228 \u00ac Q)\n  \u00ac-def  : \u00ac P     \u2194 P \u2192 \u22a5\n  \u22a5-def  : \u22a5       \u2194 P \u2227 \u00ac P\n  \u22a4-def  : \u22a4       \u2194 \u00ac \u22a5\n{-# ATP prove \u2194-def #-}\n{-# ATP prove \u2192-def #-}\n{-# ATP prove \u2228\u2081-def #-}\n{-# ATP prove \u2228\u2082-def #-}\n{-# ATP prove \u2227-def #-}\n{-# ATP prove \u00ac-def #-}\n{-# ATP prove \u22a5-def #-}\n{-# ATP prove \u22a4-def #-}\n\n------------------------------------------------------------------------------\n-- Some intuitionistic theorems\n\npostulate \u2192-transposition : (P \u2192 Q) \u2192 \u00ac Q \u2192 \u00ac P\n{-# ATP prove \u2192-transposition #-}\n\npostulate\n  i\u2081 : P \u2192 Q \u2192 P\n  i\u2082 : (P \u2192 Q \u2192 R) \u2192 (P \u2192 Q) \u2192 P \u2192 R\n  i\u2083 : P \u2192 \u00ac \u00ac P\n  i\u2084 : \u00ac \u00ac \u00ac P \u2192 \u00ac P\n  i\u2085 : ((P \u2227 Q) \u2192 R) \u2194 (P \u2192 (Q \u2192 R))\n  i\u2086 : \u00ac \u00ac (P \u2228 \u00ac P)\n  i\u2087 : (P \u2228 \u00ac P) \u2192 \u00ac \u00ac P \u2192 P\n{-# ATP prove i\u2081 #-}\n{-# ATP prove i\u2082 #-}\n{-# ATP prove i\u2083 #-}\n{-# ATP prove i\u2084 #-}\n{-# ATP prove i\u2085 #-}\n{-# ATP prove i\u2086 #-}\n{-# ATP prove i\u2087 #-}\n\n------------------------------------------------------------------------------\n-- Some non-intuitionistic theorems\n\npostulate \u2190-transposition : (\u00ac Q \u2192 \u00ac P) \u2192 P \u2192 Q\n{-# ATP prove \u2190-transposition #-}\n\npostulate\n  ni\u2081 : ((P \u2192 Q) \u2192 P) \u2192 P\n  ni\u2082 : P \u2228 \u00ac P\n  ni\u2083 : \u00ac \u00ac P \u2192 P\n  ni\u2084 : (P \u2192 Q) \u2192 (\u00ac P \u2192 Q) \u2192 Q\n  ni\u2085 : (P \u2228 Q \u2192 P) \u2228 (P \u2228 Q \u2192 Q)\n  ni\u2086 : (\u00ac \u00ac P \u2192 P) \u2192 P \u2228 \u00ac P\n{-# ATP prove ni\u2081 #-}\n{-# ATP prove ni\u2082 #-}\n{-# ATP prove ni\u2083 #-}\n{-# ATP prove ni\u2084 #-}\n{-# ATP prove ni\u2085 #-}\n{-# ATP prove ni\u2086 #-}\n\n------------------------------------------------------------------------------\n-- The principle of the excluded middle implies the double negation\n-- elimination\npostulate\n  pem\u2192\u00ac\u00ac-elim : \u2200 {P} \u2192 (P \u2228 \u00ac P) \u2192 \u00ac \u00ac P \u2192 P\n{-# ATP prove pem\u2192\u00ac\u00ac-elim #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1876902b81c7e19fc063a50114b2c1354fa59a0c","subject":"agda : wouter \/ PT : cleanup","message":"agda : wouter \/ PT : cleanup\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenWP : \u2115 -> Set\n    isEvenWP = wp0 (_+ 2) isEven\n\n    _        : isEvenWP \u2261 isEven \u2218 (_+ 2)\n    _        = refl\n\n    _        : Set\n    _        = isEvenWP 5\n\n    _        : isEvenWP 5 \u2261 isEven 7\n    _        = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_; s\u2264s; z\u2264n)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    _ : Expr\n    _ = Val 3\n    _ : Expr\n    _ = Div (Val 3) (Val 0)\n\n    _ : Val 0 \u21d3 0\n    _ = \u21d3Base\n\n    _ : Val 3 \u21d3 3\n    _ = \u21d3Base\n\n    _ : Div (Val 3) (Val 3) \u21d3 1\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Val 3) \u21d3 3\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    _ : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    _     : evv \u2261 Pure 3\n    _     = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    _     : evd \u2261 Pure 1\n    _     = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    _     : evd0 \u2261 Step Abort (\u03bb ())\n    _     = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    _ : Expr -> Nat -> Set\n    _ = _\u21d3_\n\n    _ : Set\n    _ = Val 1 \u21d3 1\n\n    _ : Expr -> Partial Nat -> Set\n    _ = mustPT _\u21d3_\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    _ : mustPT _\u21d3_ (Val 1) (Pure 1)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- a value of this type cannot be constructed\n    _ : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    _ = {!!}\n    -}\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    _ : Expr -> Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    _ = refl\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    _ = \u21d3Base\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    _ = {!!}\n    -}\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    _ : SafeDiv (Val 3) \u2261 \u22a4\n    _ = refl\n    _ : SafeDiv (Val 0) \u2261 \u22a4\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Val 0)\n\n    _ : SafeDiv (Div (Val 3) (Val 3))\n                 \u2261 Pair ((x : Val 3 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 3))\n\n    _ : SafeDiv (Div (Val 3) (Val 0))\n                 \u2261 Pair ((x : Val 0 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    _ = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- domain is well-defined Exprs (i.e., no div-by-0)\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module DomTest where\n    _ : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    _ = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    _ : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    _ = {!!}\n    -}\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n            -> (wpPartial f \u2291 wpPartial g) <-> (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement\n  - to relate Kleisli morphisms\n  - to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre and post condition\n  - to its implementation\n\n  add top two elements; fails if stack has too few elements\n\n  show how to prove definition meets its specification\n  -}\n\n  -- define specification in terms of a pre\/post condition\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre  :      a         -> Set l -- precondition on 'a'\n      post : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                     -- and the corresponding outputs\n\n  -- [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  -- for non-dependent examples (e.g., type b does not depend on x : a)\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function (to discard unused arg of type a)\n\n  -- describes desired postcondition for addition function\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  -- spec for addition function\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n  --            pre                     post\n  {-\n  need to relate spec to an implementation\n  'wpPartial' assigns predicate transformer semantics to functions\n  'wpSpec'    assigns predicate transformer semantics to specifications\n  -}\n  -- given a spec, Spec a b\n  -- computes weakest precondition that satisfies an arbitrary postcondition P\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\a -> (pre a)        -- i.e., spec\u2019s precondition should hold and\n                                  \u2227 (post a \u2286 P a) --       spec's postcondition must imply P\n\n  -- using 'wpSpec' can now find a program 'add' that \"refines\" 'addSpec'\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop      Nil  = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs0 =\n    pop xs0 >>= \\{(x1 , xs1) ->\n    pop xs1 >>= \\{(x2 , xs2) ->\n    return ((x1 + x2) :: xs2)}}\n\n  -- verify correct\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd _        (_ :: Nil) (s\u2264s        () , _)\n  correctnessAdd P a@(x :: y :: xs ) (s\u2264s (s\u2264s z\u2264n) , post_addSpec_a_\u2286P_a)\n                                                        -- wpPartial add P (x :: y :: xs)\n                                                         = post_addSpec_a_\u2286P_a (x + y :: xs) AddStep\n  -- paper version has \"extra\" 'Nil\" case\n--correctnessAdd P              Nil  (           () , _)\n--correctnessAdd P        (_ :: Nil) (s\u2264s        () , _)\n--correctnessAdd P   (x :: y :: xs ) (            _ , H) = H                   (x + y :: xs) AddStep\n\n  {-\n  repeat: this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition\n  - to its implementation\n\n  compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n   where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s          -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (    a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (    a \u00d7 s          -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n   where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven'  : isEvenPT \u2261 isEven \u2218 (_+ 2)\n    isEven'  = refl\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    xxx  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    xxx  = \u21d3Base\n\n    xxx' : Set\n    xxx' = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    yyy  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    yyy  = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    zzz  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    zzz  = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module DomTest where\n    domIs   : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    domIs   = refl\n    domIs'  : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    domIs'  = refl\n    domIs'' : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    domIs'' = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"c5a95702845021cedc3dfa83878c60e6f9654389","subject":"Rename","message":"Rename\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7TermCache \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a79e9939c6512db4827fc831f2689871d99cfc26","subject":"Move function-specific comment right before function","message":"Move function-specific comment right before function\n\nCurrently, this seems a comment about the whole module.\n\nOld-commit-hash: c25e918e712496981adfe4c74f94a8d80e8e5052\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Context.agda","new_file":"Base\/Change\/Context.agda","new_contents":"module Base.Change.Context\n    {Type : Set}\n    (\u0394Type : Type \u2192 Type) where\n\nopen import Base.Syntax.Context Type\n\n-- Transform a context of values into a context of values and\n-- changes.\n\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- like \u0394Context, but \u0394Type \u03c4 and \u03c4 are swapped\n\u0394Context\u2032 : Context \u2192 Context\n\u0394Context\u2032 \u2205 = \u2205\n\u0394Context\u2032 (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394Type \u03c4 \u2022 \u0394Context\u2032 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n","old_contents":"module Base.Change.Context\n    {Type : Set}\n    (\u0394Type : Type \u2192 Type) where\n\n-- Transform a context of values into a context of values and\n-- changes.\n\nopen import Base.Syntax.Context Type\n\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- like \u0394Context, but \u0394Type \u03c4 and \u03c4 are swapped\n\u0394Context\u2032 : Context \u2192 Context\n\u0394Context\u2032 \u2205 = \u2205\n\u0394Context\u2032 (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394Type \u03c4 \u2022 \u0394Context\u2032 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"06d988ce4bc3692e4e60e39bdb8f2e1daddb660e","subject":"[WIP] add type annotations to Denotation.Change.Popl14","message":"[WIP] add type annotations to Denotation.Change.Popl14\n\nOld-commit-hash: 90b06722aecf9d0b5c6c69547217cb2c522912be\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Change\/Popl14.agda","new_file":"Denotation\/Change\/Popl14.agda","new_contents":"module Denotation.Change.Popl14 where\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Popl14.Denotation.Value\nopen import Theorem.Groups-Popl14\nopen import Postulate.Extensionality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\n\n---------------\n-- Interface --\n---------------\n\n\u0394Val : Type \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\ninfixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n-- apply \u0394v v ::= v \u229e \u0394v\n_\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n--   diff u v ::= u \u229f v\n_\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n\n-- Lemma diff-is-valid\nR[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f v)\n\n-- Lemma apply-diff\nv+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let\n    _+_ = _\u229e_ {\u03c4}\n    _-_ = _\u229f_ {\u03c4}\n  in\n  v + (u - v) \u2261 u\n\n--------------------\n-- Implementation --\n--------------------\n\n-- \u0394Val \u03c4 is intended to be the semantic domain for changes of values\n-- of type \u03c4.\n--\n-- \u0394Val \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n-- Detailed motivation:\n--\n-- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n--\n-- \u0394Val : Type \u2192 Set\n\u0394Val (base base-int) = \u2124\n\u0394Val (base base-bag) = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229e_ {base base-int} n \u0394n = n + \u0394n\n_\u229e_ {base base-bag} b \u0394b = b ++ \u0394b\n_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192 f v \u229e \u0394f v (v \u229f v) (R[v,u-v] {\u03c3})\n\n-- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229f_ {base base-int} m n = m - n\n_\u229f_ {base base-bag} d b = d \\\\ b\n_\u229f_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e \u0394v) \u229f f v\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {base base-int} n \u0394n = \u22a4\nvalid {base base-bag} b \u0394b = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f \u0394f =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : \u0394Val \u03c3) (R[v,\u0394v] : valid v \u0394v)\n  \u2192 valid (f v) (\u0394f v \u0394v R[v,\u0394v])\n  -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n  \u00d7 (_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n\n-- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\nv+[u-v]=u {base base-int} {u} {v} = n+[m-n]=m {v} {u}\nv+[u-v]=u {base base-bag} {u} {v} = a++[b\\\\a]=b {v} {u}\nv+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n  let\n    _+_ = _\u229e_ {\u03c3 \u21d2 \u03c4}\n    _-_ = _\u229f_ {\u03c3 \u21d2 \u03c4}\n    _+\u2080_ = _\u229e_ {\u03c3}\n    _-\u2080_ = _\u229f_ {\u03c3}\n    _+\u2081_ = _\u229e_ {\u03c4}\n    _-\u2081_ = _\u229f_ {\u03c4}\n  in\n    ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n      begin\n        (v + (u - v)) w\n      \u2261\u27e8 refl \u27e9\n        v w +\u2081 (u (w +\u2080 (w -\u2080 w)) -\u2081 v w)\n      \u2261\u27e8 cong (\u03bb hole \u2192 v w +\u2081 (u hole -\u2081 v w)) (v+[u-v]=u {\u03c3}) \u27e9\n        v w +\u2081 (u w -\u2081 v w)\n      \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n        u w\n      \u220e) where\n      open \u2261-Reasoning\n\nR[v,u-v] {base base-int} {u} {v} = tt\nR[v,u-v] {base base-bag} {u} {v} = tt\nR[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n  let\n    w\u2032 = w \u229e \u0394w\n    _+_ = _\u229e_ {\u03c3 \u21d2 \u03c4}\n    _-_ = _\u229f_ {\u03c3 \u21d2 \u03c4}\n    _+\u2081_ = _\u229e_ {\u03c4}\n    _-\u2081_ = _\u229f_ {\u03c4}\n  in\n    R[v,u-v] {\u03c4}\n    ,\n    (begin\n      (v + (u - v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n      v w +\u2081 (u - v) w \u0394w R[w,\u0394w]\n    \u220e) where open \u2261-Reasoning\n\n-- `diff` and `apply`, without validity proofs\ninfixl 6 _\u27e6\u2295\u27e7_ _\u27e6\u229d\u27e7_\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n_\u27e6\u2295\u27e7_ {base base-int}  n \u0394n = n +  \u0394n\n_\u27e6\u2295\u27e7_ {base base-bag}  b \u0394b = b ++ \u0394b\n_\u27e6\u2295\u27e7_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192\n  let\n    _-\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n    _+\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  in\n    f v +\u2081 \u0394f v (v -\u2080 v)\n\n_\u27e6\u229d\u27e7_ {base base-int}  m n = m -  n\n_\u27e6\u229d\u27e7_ {base base-bag}  a b = a \\\\ b\n_\u27e6\u229d\u27e7_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v \u2192 _\u27e6\u229d\u27e7_ {\u03c4} (g (_\u27e6\u2295\u27e7_ {\u03c3} v \u0394v)) (f v)\n","old_contents":"module Denotation.Change.Popl14 where\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Popl14.Denotation.Value\nopen import Theorem.Groups-Popl14\nopen import Postulate.Extensionality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\n\n---------------\n-- Interface --\n---------------\n\n\u0394Val : Type \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\ninfixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n-- apply \u0394v v ::= v \u229e \u0394v\n_\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n--   diff u v ::= u \u229f v\n_\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n\n-- Lemma diff-is-valid\nR[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f v)\n\n-- Lemma apply-diff\nv+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n\n--------------------\n-- Implementation --\n--------------------\n\n-- \u0394Val \u03c4 is intended to be the semantic domain for changes of values\n-- of type \u03c4.\n--\n-- \u0394Val \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n-- Detailed motivation:\n--\n-- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n--\n-- \u0394Val : Type \u2192 Set\n\u0394Val int = \u2124\n\u0394Val bag = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229e_ {int} n \u0394n = n + \u0394n\n_\u229e_ {bag} b \u0394b = b ++ \u0394b\n_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192 f v \u229e \u0394f v (v \u229f v) (R[v,u-v] {\u03c3})\n\n-- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229f_ {int} m n = m - n\n_\u229f_ {bag} d b = d \\\\ b\n_\u229f_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e \u0394v) \u229f f v\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {int} n \u0394n = \u22a4\nvalid {bag} b \u0394b = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f \u0394f =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : \u0394Val \u03c3) (R[v,\u0394v] : valid v \u0394v)\n  \u2192 valid (f v) (\u0394f v \u0394v R[v,\u0394v])\n  \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n\n-- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\nv+[u-v]=u {int} {u} {v} = n+[m-n]=m {v} {u}\nv+[u-v]=u {bag} {u} {v} = a++[b\\\\a]=b {v} {u}\nv+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} = ext (\u03bb w \u2192\n  begin\n    (v \u229e (u \u229f v)) w\n  \u2261\u27e8 refl \u27e9\n    v w \u229e (u (w \u229e (w \u229f w)) \u229f v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e (u hole \u229f v w)) v+[u-v]=u \u27e9\n    v w \u229e (u w \u229f v w)\n  \u2261\u27e8 v+[u-v]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u-v] {int} {u} {v} = tt\nR[v,u-v] {bag} {u} {v} = tt\nR[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n  let\n    w\u2032 = w \u229e \u0394w\n  in\n    R[v,u-v] {\u03c4}\n    ,\n    (begin\n      (v \u229e (u \u229f v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v+[u-v]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u229e (u \u229f v) w \u0394w R[w,\u0394w]\n    \u220e) where open \u2261-Reasoning\n\n-- `diff` and `apply`, without validity proofs\ninfixl 6 _\u27e6\u2295\u27e7_ _\u27e6\u229d\u27e7_\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n_\u27e6\u2295\u27e7_ {int}  n \u0394n = n +  \u0394n\n_\u27e6\u2295\u27e7_ {bag}  b \u0394b = b ++ \u0394b\n_\u27e6\u2295\u27e7_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192 f v \u27e6\u2295\u27e7 \u0394f v (v \u27e6\u229d\u27e7 v)\n\n_\u27e6\u229d\u27e7_ {int}  m n = m -  n\n_\u27e6\u229d\u27e7_ {bag}  a b = a \\\\ b\n_\u27e6\u229d\u27e7_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v \u2192 g (v \u27e6\u2295\u27e7 \u0394v) \u27e6\u229d\u27e7 f v\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2c95a0ed6648adae17543219c076caabca2bf732","subject":"State proof obligations for constants","message":"State proof obligations for constants\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\nchangeSemVar : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSemVar t \u03c1 d\u03c1 = \u27e6 deriveVar t \u27e7Var (alternate \u03c1 d\u03c1)\n\nchangeSem : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSem t \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1))\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (v : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 v \u27e7Var (changeSemVar v)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1))\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term (changeSem t)\n\nsemConst-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 changeSem (const c) \u03c1 d\u03c1 \u2261 \u27e6 deriveConst c \u27e7Term \u2205\nsemConst-rewrite c \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 \u27e6 deriveConst c \u27e7Term \u2205\ncorrectDeriveConst ()\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6 deriveConst c \u27e7Term \u2205)\nvalidDeriveConst ()\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite semConst-rewrite c \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 changeSem t \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (changeSem t \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 changeSem t \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295\n       \u27e6 derive t \u27e7Term (alternate \u03c11 d\u03c11)\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term (a \u2022 \u03c1) \u2295 \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite semConst-rewrite c \u03c1 d\u03c1 = validDeriveConst c\n","old_contents":"module New.Correctness where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\nchangeSemVar : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSemVar t \u03c1 d\u03c1 = \u27e6 deriveVar t \u27e7Var (alternate \u03c1 d\u03c1)\n\nchangeSem : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSem t \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1))\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (v : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 v \u27e7Var (changeSemVar v)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1))\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term (changeSem t)\n\ncorrectDerive (const ()) \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 changeSem t \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (changeSem t \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 changeSem t \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295\n       \u27e6 derive t \u27e7Term (alternate \u03c11 d\u03c11)\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term (a \u2022 \u03c1) \u2295 \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const ()) \u03c1 d\u03c1 \u03c1d\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4a40c50c8d82aaf4b7461c877efc293d49fbd8c7","subject":"flipbased-counting: two new relations \u2248\u1d2c and \u2248\u1d2c\u2032","message":"flipbased-counting: two new relations \u2248\u1d2c and \u2248\u1d2c\u2032\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-counting.agda","new_file":"flipbased-counting.agda","new_contents":"open import Function\nopen import Data.Nat.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_; _\u2261_)\n\nopen import Data.Bits\nimport flipbased\n\nmodule flipbased-counting\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n   (count\u21ba : \u2200 {n} \u2192 \u21ba n Bit \u2192 \u2115)\n where\n\nopen flipbased \u21ba toss return\u21ba map\u21ba join\u21ba\n\ninfix 4 _\u2248\u21ba_ _\u224b\u21ba_ _\u2248\u2141?_ _\u2248\u1d2c_ _\u2248\u1d2c\u2032_\n\n-- f \u2248\u21ba g when f and g return the same number of 1 (and 0).\n_\u2248\u21ba_ : \u2200 {n} (f g : EXP n) \u2192 Set\n_\u2248\u21ba_ = _\u2261_ on count\u21ba\n\n_\u2248\u2141?_ : \u2200 {c} (g\u2080 g\u2081 : \u2141? c) \u2192 Set\ng\u2080 \u2248\u2141? g\u2081 = \u2200 b \u2192 g\u2080 b \u2248\u21ba g\u2081 b\n\n_\u2248\u1d2c_ : \u2200 {n a} {A : Set a} (f g : \u21ba n A) \u2192 Set _\n_\u2248\u1d2c_ {n} {A = A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n_\u2248\u1d2c\u2032_ : \u2200 {n a} {A : Set a} (f g : \u21ba n A) \u2192 Set _\n_\u2248\u1d2c\u2032_ {n} {A = A} f g = \u2200 {ca} (Adv : A \u2192 \u21ba ca Bit) \u2192 f >>= Adv \u2248\u21ba g >>= Adv\n\n{- Restore if we find more use of it\nRelatedEXP : (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\nRelatedEXP _\u223c_ {m} {n} f g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\nmodule ... z where\n  x \u223c y = dist x y > ... z ...\n-}\n\n_\u224b\u21ba_ : \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\n-- _\u224b\u21ba_ = RelatedEXP _\u2261_\n_\u224b\u21ba_ {m} {n} f g = \u27e82^ n * count\u21ba f \u27e9 \u2261 \u27e82^ m * count\u21ba g \u27e9\n\nSafe\u2141? : \u2200 {c} (f : \u2141? c) \u2192 Set\nSafe\u2141? f = f 0b \u2248\u21ba f 1b\n\nReversible\u2141? : \u2200 {c} (g : \u2141? c) \u2192 Set\nReversible\u2141? g = g \u2248\u2141? g \u2218 not\n\n\u2248\u21d2\u224b\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u2248\u21ba g \u2192 f \u224b\u21ba g\n\u2248\u21d2\u224b\u21ba eq rewrite eq = \u2261.refl\n\n\u224b\u21d2\u2248\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u224b\u21ba g \u2192 f \u2248\u21ba g\n\u224b\u21d2\u2248\u21ba {n} = 2^-inj n\n\nmodule \u2248\u2141? {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u2141? n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u2141?_\n\n        refl : Reflexive \u211b\n        refl b = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p b = \u2261.sym (p b)\n\n        trans : Transitive \u211b\n        trans p q b = \u2261.trans (p b) (q b)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u224b\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u224b\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u1d2c {n} {a} {A : Set a} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u21ba n A\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u1d2c_\n\n        refl : Reflexive \u211b\n        refl A = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p A = \u2261.sym (p A)\n\n        trans : Transitive \u211b\n        trans p q A = \u2261.trans (p A) (q A)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid\n  open Setoid setoid public\n\nmodule \u2248\u1d2c\u2032 {n} {a} {A : Set a} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u21ba n A\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u1d2c\u2032_\n\n        refl : Reflexive \u211b\n        refl A = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p A = \u2261.sym (p A)\n\n        trans : Transitive \u211b\n        trans p q A = \u2261.trans (p A) (q A)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid\n  open Setoid setoid public\n\nmodule \u2141? {n} where\n  join : EXP n \u2192 EXP n \u2192 \u2141? n\n  join f g b = if b then f else g\n\n  safe-sym : \u2200 {g : \u2141? n} \u2192 Safe\u2141? g \u2192 Safe\u2141? (g \u2218 not)\n  safe-sym {g} g-safe = \u2248\u21ba.sym {n} {g 0b} {g 1b} g-safe\n\n  Reversible\u21d2Safe : \u2200 {c} (g : \u2141? c) \u2192 Reversible\u2141? g \u2192 Safe\u2141? g\n  Reversible\u21d2Safe g g\u2248g\u2218not = g\u2248g\u2218not 0b\n\ndata Rat : Set where _\/_ : (num denom : \u2115) \u2192 Rat\n\nPr[_\u22611] : \u2200 {n} (f : EXP n) \u2192 Rat\nPr[_\u22611] {n} f = count\u21ba f \/ 2^ n\n","old_contents":"open import Function\nopen import Data.Nat.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_; _\u2261_)\n\nopen import Data.Bits\nimport flipbased\n\nmodule flipbased-counting\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n   (count\u21ba : \u2200 {n} \u2192 \u21ba n Bit \u2192 \u2115)\n where\n\nopen flipbased \u21ba toss return\u21ba map\u21ba join\u21ba\n\ninfix 4 _\u2248\u21ba_ _\u224b\u21ba_ _\u2248\u2141?_\n\n-- f \u2248\u21ba g when f and g return the same number of 1 (and 0).\n_\u2248\u21ba_ : \u2200 {n} (f g : EXP n) \u2192 Set\n_\u2248\u21ba_ = _\u2261_ on count\u21ba\n\n_\u2248\u2141?_ : \u2200 {c} (g\u2080 g\u2081 : \u2141? c) \u2192 Set\ng\u2080 \u2248\u2141? g\u2081 = \u2200 b \u2192 g\u2080 b \u2248\u21ba g\u2081 b\n\n_\u223c[_]EXP_ : \u2200 {m n} \u2192 EXP m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 EXP n \u2192 Set\n_\u223c[_]EXP_ {m} {n} f _\u223c_ g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\n_\u224b\u21ba_ : \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\nf \u224b\u21ba g = f \u223c[ _\u2261_ ]EXP g\n\nSafe\u2141? : \u2200 {c} (f : \u2141? c) \u2192 Set\nSafe\u2141? f = f 0b \u2248\u21ba f 1b\n\nReversible\u2141? : \u2200 {c} (g : \u2141? c) \u2192 Set\nReversible\u2141? g = g \u2248\u2141? g \u2218 not\n\n\u2248\u21d2\u224b\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u2248\u21ba g \u2192 f \u224b\u21ba g\n\u2248\u21d2\u224b\u21ba eq rewrite eq = \u2261.refl\n\n\u224b\u21d2\u2248\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u224b\u21ba g \u2192 f \u2248\u21ba g\n\u224b\u21d2\u2248\u21ba {n} = 2^-inj n\n\nmodule \u2248\u2141? {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u2141? n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u2141?_\n\n        refl : Reflexive \u211b\n        refl b = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p b = \u2261.sym (p b)\n\n        trans : Transitive \u211b\n        trans p q b = \u2261.trans (p b) (q b)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u224b\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u224b\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2141? {n} where\n  join : EXP n \u2192 EXP n \u2192 \u2141? n\n  join f g b = if b then f else g\n\n  safe-sym : \u2200 {g : \u2141? n} \u2192 Safe\u2141? g \u2192 Safe\u2141? (g \u2218 not)\n  safe-sym {g} g-safe = \u2248\u21ba.sym {n} {g 0b} {g 1b} g-safe\n\n  Reversible\u21d2Safe : \u2200 {c} (g : \u2141? c) \u2192 Reversible\u2141? g \u2192 Safe\u2141? g\n  Reversible\u21d2Safe g g\u2248g\u2218not = g\u2248g\u2218not 0b\n\ndata Rat : Set where _\/_ : (num denom : \u2115) \u2192 Rat\n\nPr[_\u22611] : \u2200 {n} (f : EXP n) \u2192 Rat\nPr[_\u22611] {n} f = count\u21ba f \/ 2^ n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a2a3722c55d4534ac22294785bb1d8d1e115042d","subject":"Add incrementalization lemma for binary functions","message":"Add incrementalization lemma for binary functions\n\nSurprisingly it seems this was never stated, even though it would have\nbeen useful in some proofs (and even though it's a natural lemma for\nbinary functions).\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set c)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c}\n    (Carrier : Set c) : Set (suc c) where\n  field\n    Change : Carrier \u2192 Set c\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} {A : Set a} (P : A \u2192 Set p): Set (suc p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 b)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (p \u2294 q)\nRawChange\u208d_,_\u208e x y f = \u2200 px (dpx : \u0394\u208d_\u208e x px) \u2192 \u0394\u208d_\u208e y (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} (A : Set a) (B : Set b) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g (a \u229e da \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n    funNil = \u03bb f \u2192 funDiff f f\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebraFun : ChangeAlgebra (A \u2192 B)\n\n      changeAlgebraFun = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff  (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n-- Reexport a few members with A and B marked as implicit parameters. This\n-- matters especially for changeAlgebra, since otherwise it can't be used for\n-- instance search.\nmodule _\n    {a} {b} {A : Set a} {B : Set b} {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    open FunctionChanges A B {{CA}} {{CB}} public\n      using\n        ( changeAlgebraFun\n        ; apply\n        ; correct\n        ; incrementalization\n        ; DerivativeAsChange\n        ; FunctionChange\n        )\n\nmodule BinaryFunctionChanges\n    {a} {b} {c} (A : Set a) (B : Set b) (C : Set c) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} {{CC : ChangeAlgebra C}} where\n    incrementalization-binary : \u2200 (f : A \u2192 B \u2192 C) df a da b db \u2192\n      (f \u229e df) (a \u229e da) (b \u229e db) \u2261 f a b \u229e apply (apply df a da) b db\n    incrementalization-binary f df x dx y dy\n      rewrite cong (\u03bb \u25a1 \u2192 \u25a1 (y \u229e dy)) (incrementalization f df x dx)\n      = incrementalization (f x) (apply df x dx) y dy\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 \u0394 pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 \u0394 pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebraListChanges : ChangeAlgebraFamily (All P)\n\n    changeAlgebraListChanges = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set c)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c}\n    (Carrier : Set c) : Set (suc c) where\n  field\n    Change : Carrier \u2192 Set c\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} {A : Set a} (P : A \u2192 Set p): Set (suc p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 b)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (p \u2294 q)\nRawChange\u208d_,_\u208e x y f = \u2200 px (dpx : \u0394\u208d_\u208e x px) \u2192 \u0394\u208d_\u208e y (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} (A : Set a) (B : Set b) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g (a \u229e da \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n    funNil = \u03bb f \u2192 funDiff f f\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebraFun : ChangeAlgebra (A \u2192 B)\n\n      changeAlgebraFun = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff  (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n-- Reexport a few members with A and B marked as implicit parameters. This\n-- matters especially for changeAlgebra, since otherwise it can't be used for\n-- instance search.\nmodule _\n    {a} {b} {A : Set a} {B : Set b} {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    open FunctionChanges A B {{CA}} {{CB}} public\n      using\n        ( changeAlgebraFun\n        ; apply\n        ; correct\n        ; incrementalization\n        ; DerivativeAsChange\n        ; FunctionChange\n        )\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 \u0394 pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 \u0394 pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebraListChanges : ChangeAlgebraFamily (All P)\n\n    changeAlgebraListChanges = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d9a3d68c58bd5c8ae49738e7a8f68fd5e89b4bce","subject":"Improve reuse (see #28).","message":"Improve reuse (see #28).\n\nOld-commit-hash: 7da11aabf971e192ab71542164b98c80cbe26325\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  cond : \u2200 {\u0393 \u03c4} \u2192 (e\u2081 : Term \u0393 bool) (e\u2082 e\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 cond t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  cond-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  cond-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (cond-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (cond-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} true = true\nweaken {\u0393\u2081} {\u0393\u2082} false = false\nweaken {\u0393\u2081} {\u0393\u2082} (cond e\u2081 e\u2082 e\u2083) = cond (weaken {\u0393\u2081} {\u0393\u2082} e\u2081) (weaken {\u0393\u2081} {\u0393\u2082} e\u2082) (weaken {\u0393\u2081} {\u0393\u2082} e\u2083)\n","old_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Denotational.Notation\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import Syntactic.Contexts Type public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  cond : \u2200 {\u0393 \u03c4} \u2192 (e\u2081 : Term \u0393 bool) (e\u2082 e\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 cond t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  cond-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  cond-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (cond-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (cond-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} true = true\nweaken {\u0393\u2081} {\u0393\u2082} false = false\nweaken {\u0393\u2081} {\u0393\u2082} (cond e\u2081 e\u2082 e\u2083) = cond (weaken {\u0393\u2081} {\u0393\u2082} e\u2081) (weaken {\u0393\u2081} {\u0393\u2082} e\u2082) (weaken {\u0393\u2081} {\u0393\u2082} e\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a249c6b82c824f612dd5b7102544cf27e5484428","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 9ee4877688487f11241cec13ff852b2f\n\ndarcs-hash:20120120153646-3bd4e-cec9166ec957acc3b1450bf6c3b823058fc5a184.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/GCD\/Total\/TotalityATP.agda","new_file":"src\/FOTC\/Program\/GCD\/Total\/TotalityATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality properties of the gcd\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.GCD.Total.TotalityATP where\n\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Base.Properties\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicATP\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Program.GCD.Total.Definitions\nopen import FOTC.Program.GCD.Total.GCD\n\n------------------------------------------------------------------------------\n-- gcd 0 0 is total.\npostulate gcd-00-N : N (gcd zero zero)\n{-# ATP prove gcd-00-N #-}\n\n------------------------------------------------------------------------------\n-- gcd 0 (succ n) is total.\npostulate gcd-0S-N : \u2200 {n} \u2192 N n \u2192 N (gcd zero (succ\u2081 n))\n{-# ATP prove gcd-0S-N #-}\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 n) 0 is total.\npostulate gcd-S0-N : \u2200 {n} \u2192 N n \u2192 N (gcd (succ\u2081 n) zero)\n{-# ATP prove gcd-S0-N #-}\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 m) (succ\u2081 n) when succ\u2081 m > succ\u2081 n is total.\npostulate\n  gcd-S>S-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n              N (gcd (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n)) \u2192\n              GT (succ\u2081 m) (succ\u2081 n) \u2192\n              N (gcd (succ\u2081 m) (succ\u2081 n))\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove gcd-S>S-N #-}\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 m) (succ\u2081 n) when succ\u2081 m \u226f succ\u2081 n is total.\npostulate\n  gcd-S\u226fS-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n              N (gcd (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m)) \u2192\n              NGT (succ\u2081 m) (succ\u2081 n) \u2192\n              N (gcd (succ\u2081 m) (succ\u2081 n))\n{-# ATP prove gcd-S\u226fS-N #-}\n\n------------------------------------------------------------------------------\n-- gcd m n when m > n is total.\ngcd-x>y-N :\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192\n  (\u2200 {o p} \u2192 N o \u2192 N p \u2192 LT\u2082 o p m n \u2192 N (gcd o p)) \u2192\n  GT m n \u2192\n  N (gcd m n)\ngcd-x>y-N zN          Nn          _    0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\ngcd-x>y-N (sN Nm)     zN          _    _     = gcd-S0-N Nm\ngcd-x>y-N (sN {m} Nm) (sN {n} Nn) accH Sm>Sn =\n  gcd-S>S-N Nm Nn ih Sm>Sn\n  where\n  -- Inductive hypothesis.\n  ih : N (gcd (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n))\n  ih = accH {succ\u2081 m \u2238 succ\u2081 n}\n            {succ\u2081 n}\n            (\u2238-N (sN Nm) (sN Nn))\n            (sN Nn)\n            ([Sx\u2238Sy,Sy]<[Sx,Sy] Nm Nn)\n\n------------------------------------------------------------------------------\n-- gcd m n when m \u226f n is total.\ngcd-x\u226fy-N :\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192\n  (\u2200 {o p} \u2192 N o \u2192 N p \u2192 LT\u2082 o p m n \u2192 N (gcd o p)) \u2192\n  NGT m n \u2192\n  N (gcd m n)\ngcd-x\u226fy-N zN         zN           _    _     = gcd-00-N\ngcd-x\u226fy-N zN         (sN Nn)      _    _     = gcd-0S-N Nn\ngcd-x\u226fy-N (sN {m} Nm) zN          _    Sm\u226f0  = \u22a5-elim $ S\u226f0\u2192\u22a5 Sm\u226f0\ngcd-x\u226fy-N (sN {m} Nm) (sN {n} Nn) accH Sm\u226fSn = gcd-S\u226fS-N Nm Nn ih Sm\u226fSn\n  where\n  -- Inductive hypothesis.\n  ih : N (gcd (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m))\n  ih = accH {succ\u2081 m}\n            {succ\u2081 n \u2238 succ\u2081 m}\n            (sN Nm)\n            (\u2238-N (sN Nn) (sN Nm))\n            ([Sx,Sy\u2238Sx]<[Sx,Sy] Nm Nn)\n\n------------------------------------------------------------------------------\n-- gcd m n is total.\ngcd-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (gcd m n)\ngcd-N = wfInd-LT\u2082 P istep\n  where\n  P : D \u2192 D \u2192 Set\n  P i j = N (gcd i j)\n\n  istep : \u2200 {i j} \u2192 N i \u2192 N j \u2192 (\u2200 {k l} \u2192 N k \u2192 N l \u2192 LT\u2082 k l i j \u2192 P k l) \u2192\n          P i j\n  istep Ni Nj accH =\n    [ gcd-x>y-N Ni Nj accH\n    , gcd-x\u226fy-N Ni Nj accH\n    ] (x>y\u2228x\u226fy Ni Nj)\n","old_contents":"------------------------------------------------------------------------------\n-- Totality properties of the gcd\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.GCD.Total.TotalityATP where\n\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Base.Properties\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicATP\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Program.GCD.Total.Definitions\nopen import FOTC.Program.GCD.Total.GCD\n\n------------------------------------------------------------------------------\n-- gcd 0 0 is total.\npostulate gcd-00-N : N (gcd zero zero)\n{-# ATP prove gcd-00-N #-}\n\n------------------------------------------------------------------------------\n-- gcd 0 (succ n) is total.\npostulate gcd-0S-N : \u2200 {n} \u2192 N n \u2192 N (gcd zero (succ\u2081 n))\n{-# ATP prove gcd-0S-N #-}\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 n) 0 is total.\npostulate gcd-S0-N : \u2200 {n} \u2192 N n \u2192 N (gcd (succ\u2081 n) zero)\n{-# ATP prove gcd-S0-N #-}\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 m) (succ\u2081 n) when succ\u2081 m > succ\u2081 n is total.\npostulate\n  gcd-S>S-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n              N (gcd (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n)) \u2192\n              GT (succ\u2081 m) (succ\u2081 n) \u2192\n              N (gcd (succ\u2081 m) (succ\u2081 n))\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove gcd-S>S-N #-}\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 m) (succ\u2081 n) when succ\u2081 m \u226f succ\u2081 n is total.\npostulate\n  gcd-S\u226fS-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n              N (gcd (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m)) \u2192\n              NGT (succ\u2081 m) (succ\u2081 n) \u2192\n              N (gcd (succ\u2081 m) (succ\u2081 n))\n{-# ATP prove gcd-S\u226fS-N #-}\n\n------------------------------------------------------------------------------\n-- gcd m n when m > n is total.\ngcd-x>y-N :\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192\n  (\u2200 {o p} \u2192 N o \u2192 N p \u2192 LT\u2082 o p m n \u2192 N (gcd o p)) \u2192\n  GT m n \u2192\n  N (gcd m n)\ngcd-x>y-N zN          Nn          _    0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\ngcd-x>y-N (sN Nm)     zN          _    _     = gcd-S0-N Nm\ngcd-x>y-N (sN {m} Nm) (sN {n} Nn) accH Sm>Sn =\n  gcd-S>S-N Nm Nn ih Sm>Sn\n  where\n  -- Inductive hypothesis.\n  ih : N (gcd (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n))\n  ih = accH {succ\u2081 m \u2238 succ\u2081 n}\n            {succ\u2081 n}\n            (\u2238-N (sN Nm) (sN Nn))\n            (sN Nn)\n            ([Sx\u2238Sy,Sy]<[Sx,Sy] Nm Nn)\n\n------------------------------------------------------------------------------\n-- gcd m n when m \u226f n is total.\ngcd-x\u226fy-N :\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192\n  (\u2200 {o p} \u2192 N o \u2192 N p \u2192 LT\u2082 o p m n \u2192 N (gcd o p)) \u2192\n  NGT m n \u2192\n  N (gcd m n)\ngcd-x\u226fy-N zN         zN           _   _      = gcd-00-N\ngcd-x\u226fy-N zN         (sN Nn)      _    _     = gcd-0S-N Nn\ngcd-x\u226fy-N (sN {m} Nm) zN          _    Sm\u226f0  = \u22a5-elim $ S\u226f0\u2192\u22a5 Sm\u226f0\ngcd-x\u226fy-N (sN {m} Nm) (sN {n} Nn) accH Sm\u226fSn =\n  gcd-S\u226fS-N Nm Nn ih Sm\u226fSn\n  where\n  -- Inductive hypothesis.\n  ih : N (gcd (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m))\n  ih = accH {succ\u2081 m}\n            {succ\u2081 n \u2238 succ\u2081 m}\n            (sN Nm)\n            (\u2238-N (sN Nn) (sN Nm))\n            ([Sx,Sy\u2238Sx]<[Sx,Sy] Nm Nn)\n\n------------------------------------------------------------------------------\n-- gcd m n is total.\ngcd-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (gcd m n)\ngcd-N = wfInd-LT\u2082 P istep\n  where\n  P : D \u2192 D \u2192 Set\n  P i j = N (gcd i j)\n\n  istep : \u2200 {i j} \u2192 N i \u2192 N j \u2192 (\u2200 {k l} \u2192 N k \u2192 N l \u2192 LT\u2082 k l i j \u2192 P k l) \u2192\n          P i j\n  istep Ni Nj accH =\n    [ gcd-x>y-N Ni Nj accH\n    , gcd-x\u226fy-N Ni Nj accH\n    ] (x>y\u2228x\u226fy Ni Nj)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"84495ac29d0b2817ceb8faeb8f9c51c0dc4c1d44","subject":"IP: update","message":"IP: update\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/IP.agda","new_file":"Control\/Protocol\/IP.agda","new_contents":"open import Type\nopen import Function\nopen import Data.One\nopen import Data.Two\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Maybe\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Protocol\n\nmodule Control.Protocol.IP (Query : \u2605) (Resp : Query \u2192 \u2605) where\n\nAccept? = \ud835\udfda\naccept! = 1\u2082\nreject! = 0\u2082\nAccept! = \u2713\n\n-- One round from the point of view of the verifier\nVerifierRound : Proto\nVerifierRound = send \u03bb (q : Query)  \u2192\n                recv \u03bb (r : Resp q) \u2192\n                end\n\nProverRound : Proto\nProverRound = dual VerifierRound\n\nmodule Rounds (#rounds : \u2115) where\n  Verifier-rounds\u1d3e = replicate\u1d3e #rounds VerifierRound\n\n  Verifier-end\u1d3e = send \u03bb (_ : Accept?) \u2192 end\n\n  Verifier\u1d3e = Verifier-rounds\u1d3e >> Verifier-end\u1d3e\n\n  Prover-end\u1d3e = dual Verifier-end\u1d3e\n\n  Prover\u1d3e = dual Verifier\u1d3e\n\n  Transcript\u1d3e = source-of Verifier\u1d3e\n\n  Prover   : \u2605\n  Prover   = \u27e6 Prover\u1d3e \u27e7\n\n  Verifier : \u2605\n  Verifier = \u27e6 Verifier\u1d3e \u27e7\n\n  Transcript : \u2605\n  Transcript = \u27e6 Transcript\u1d3e \u27e7\n\n  >>-Transcript : \u2605\n  >>-Transcript = Log Verifier-rounds\u1d3e \u00d7 \u27e6 Verifier-end\u1d3e \u27e7 \u00d7 \u27e6 Verifier-end\u1d3e \u22a5\u27e7\n\n  accepting-transcript? : >>-Transcript \u2192 Accept?\n  accepting-transcript? (_ , (a , _) , _) = a\n\n  _\u21c6_ : Prover \u2192 Verifier \u2192 Accept?\n  p \u21c6 v = accepting-transcript? (>>-telecom Verifier-rounds\u1d3e v p)\n\nrecord IP (#rounds : \u2115) {A : \u2605} (\u2112 : A \u2192 \u2605) : \u2605 where\n  open Rounds #rounds public\n\n  field\n    verifier : Verifier\n\n  Convincing : Prover \u2192 \u2605\n  Convincing = \u03bb p \u2192 Accept! (p \u21c6 verifier)\n\n  Complete : \u2605\n  Complete = \u2200 x \u2192 \u2112 x \u2192 \u03a3 Prover Convincing\n\n  Sound : \u2605\n  Sound = \u2200 x \u2192 \u00ac(\u2112 x) \u2192 (p : Prover) \u2192 \u00ac(Convincing p)\n\nNP-Verifier : \u2605\nNP-Verifier = \u03a3 Query \u03bb q \u2192 Resp q \u2192 Accept? \u00d7 End\n\nNP-Verifier\u2261Verifier1 : NP-Verifier \u2261 Rounds.Verifier 1\nNP-Verifier\u2261Verifier1 = refl\n\nNP : {A : \u2605} (\u2112 : A \u2192 \u2605) \u2192 \u2605\nNP = IP 1\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function\nopen import Control.Protocol.Choreography\nopen import Data.One\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Maybe\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nmodule Control.Protocol.IP (Query : \u2605) (Resp : Query \u2192 \u2605) where\n\n-- One round from the point of view of the verifier\nVerifierRound : Proto\nVerifierRound = \u03a3\u1d3e Query    \u03bb q \u2192\n                \u03a0\u1d3e (Resp q) \u03bb r \u2192\n                end\n\nProverRound : Proto\nProverRound = dual VerifierRound\n\nmodule Rounds (#rounds : \u2115) where\n  Verifier\u1d3e : Proto\n  Verifier\u1d3e = replicate\u1d3e #rounds VerifierRound >> (\u03a3\u1d3e Accept? \u03bb _ \u2192 end)\n\n  Prover\u1d3e   : Proto\n  Prover\u1d3e   = dual Verifier\u1d3e\n\n  Transcript\u1d3e : Proto\n  Transcript\u1d3e = source-of Verifier\u1d3e\n\n  Prover   : \u2605\n  Prover   = \u27e6 Prover\u1d3e \u27e7\n\n  Verifier : \u2605\n  Verifier = \u27e6 Verifier\u1d3e \u27e7\n\n  Transcript : \u2605\n  Transcript = \u27e6 Transcript\u1d3e \u27e7\n\n  _\u21c6_ : Prover \u2192 Verifier \u2192 Accept?\n  _\u21c6_ = \u03bb p v \u2192 case >>-com (replicate\u1d3e #rounds VerifierRound) v p of \u03bb { (_ , (a , _) , _) \u2192 a }\n\nrecord IP (#rounds : \u2115) {A : \u2605} (\u2112 : A \u2192 \u2605) : \u2605 where\n  open Rounds #rounds public\n\n  field\n    verifier : Verifier\n\n  Convincing : Prover \u2192 \u2605\n  Convincing = \u03bb p \u2192 Is-accept (p \u21c6 verifier)\n\n  Complete : \u2605\n  Complete = \u2200 x \u2192 \u2112 x \u2192 \u03a3 Prover Convincing\n\n  Sound : \u2605\n  Sound = \u2200 x \u2192 \u00ac(\u2112 x) \u2192 (p : Prover) \u2192 \u00ac(Convincing p)\n\nNP-Verifier = \u03a3 Query \u03bb q \u2192 Resp q \u2192 Accept? \u00d7 \ud835\udfd9\n\nNP-Verifier\u2261Verifier1 : NP-Verifier \u2261 Rounds.Verifier 1\nNP-Verifier\u2261Verifier1 = refl\n\nNP : {A : \u2605} (\u2112 : A \u2192 \u2605) \u2192 \u2605\nNP = IP 1\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fc40c04b0d9ef151e73902e8feab75d63d0b32b4","subject":"Modified the postulate about the post-fixed point of BISI.","message":"Modified the postulate about the post-fixed point of BISI.\n\nIgnore-this: b89225245b6b620ae7b127c781a39f03\n\ndarcs-hash:20101101113300-3bd4e-62d3bad9121044a9f69c855c0d7438b4261ab3aa.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bisimilarity\/BisimilarityPostFixedPoint.agda","new_file":"Draft\/Bisimilarity\/BisimilarityPostFixedPoint.agda","new_contents":"module BisimilarityPostFixedPoint where\n\n-- open import LTC.Minimal\n-- open import LTC.MinimalER\n\ninfixr 5 _\u2237_\ninfix 4 _\u2248_\ninfixr 4 _,_\ninfix  3 _\u2261_\ninfixr 2 _\u2227_\n\n------------------------------------------------------------------------------\n-- LTC stuff\n\n-- The universal domain.\npostulate D : Set\n\n  -- LTC lists.\npostulate\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- The existential quantifier type on D.\ndata \u2203D (P : D \u2192 Set) : Set where\n  _,_ : (d : D) (Pd : P d) \u2192 \u2203D P\n\n\u2203D-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203D P \u2192 D\n\u2203D-proj\u2081 (x , _ ) = x\n\n\u2203D-proj\u2082 : {P : D \u2192 Set} \u2192 (x-px : \u2203D P) \u2192 P (\u2203D-proj\u2081 x-px)\n\u2203D-proj\u2082 (_ , px) = px\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\nsubst : (P : D \u2192 Set){x y : D} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n\n-- The conjunction data type.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n\n------------------------------------------------------------------------------\n\nBISI : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\nBISI R xs ys =\n  \u2203D (\u03bb x' \u2192\n  \u2203D (\u03bb y' \u2192\n  \u2203D (\u03bb xs' \u2192\n  \u2203D (\u03bb ys' \u2192\n     x' \u2261 y' \u2227 R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 y' \u2237 ys'))))\n\npostulate\n  -- The bisimilarity relation.\n  _\u2248_ : D \u2192 D \u2192 Set\n\n  -- The bisimilarity relation is a post-fixed point of BISI.\n  -\u2248-gfp\u2081 : {xs ys : D} \u2192 xs \u2248 ys \u2192\n            \u2203D (\u03bb x' \u2192\n            \u2203D (\u03bb xs' \u2192\n            \u2203D (\u03bb ys' \u2192 xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))\n\n-\u2248\u2192BISI\u2248 : (xs ys : D) \u2192 xs \u2248 ys \u2192 BISI _\u2248_ xs ys\n-\u2248\u2192BISI\u2248 xs ys xs\u2248ys =\n  x' , x' , xs' , ys' , refl , xs'\u2248ys' , xs\u2261x'\u2237xs' , ys\u2261x'\u2237ys'\n  where\n    x' : D\n    x' = \u2203D-proj\u2081 (-\u2248-gfp\u2081 xs\u2248ys)\n\n    xs' : D\n    xs' = \u2203D-proj\u2081 (\u2203D-proj\u2082 (-\u2248-gfp\u2081 xs\u2248ys))\n\n    ys' : D\n    ys' = \u2203D-proj\u2081 (\u2203D-proj\u2082 (\u2203D-proj\u2082 (-\u2248-gfp\u2081 xs\u2248ys)))\n\n    aux : xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n    aux = \u2203D-proj\u2082 (\u2203D-proj\u2082 (\u2203D-proj\u2082 (-\u2248-gfp\u2081 xs\u2248ys)))\n\n    xs'\u2248ys' : xs' \u2248 ys'\n    xs'\u2248ys' = \u2227-proj\u2081 aux\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' = \u2227-proj\u2081 (\u2227-proj\u2082 aux)\n\n    ys\u2261x'\u2237ys' : ys \u2261 x' \u2237 ys'\n    ys\u2261x'\u2237ys' = \u2227-proj\u2082 (\u2227-proj\u2082 aux)\n","old_contents":"module BisimilarityPostFixedPoint where\n\n-- open import LTC.Minimal\n-- open import LTC.MinimalER\n\ninfixr 5 _\u2237_\ninfix 4 _\u2248_\ninfixr 4 _,_\ninfix  3 _\u2261_\ninfixr 2 _\u2227_\n\n------------------------------------------------------------------------------\n-- LTC stuff\n\n-- The universal domain.\npostulate D : Set\n\n  -- LTC lists.\npostulate\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- The existential quantifier type on D.\ndata \u2203D (P : D \u2192 Set) : Set where\n  _,_ : (d : D) (Pd : P d) \u2192 \u2203D P\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\nsubst : (P : D \u2192 Set){x y : D} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n\n-- The conjunction data type.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n------------------------------------------------------------------------------\nBISI : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\nBISI R xs ys =\n  \u2203D (\u03bb x' \u2192\n  \u2203D (\u03bb y' \u2192\n  \u2203D (\u03bb xs' \u2192\n  \u2203D (\u03bb ys' \u2192\n     x' \u2261 y' \u2227 R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 y' \u2237 ys'))))\n\n-- From Peter email:\n\n-- For the case of bisimilarity \u2248 we have\n\n-- (i) a post-fixed point of BISI, is the following first order axiom\n\n-- forall xs,ys,x,y. x = y & xs \u2248 ys -> x :: xs \u2248 y :: ys\n\npostulate\n  -- The bisimilarity relation.\n  _\u2248_ : D \u2192 D \u2192 Set\n\n  -- The bisimilarity relation is a post-fixed point of BISI.\n  -\u2248-gfp\u2081 : {x y xs ys : D} \u2192 x \u2261 y \u2227 xs \u2248 ys \u2192 x \u2237 xs \u2248 y \u2237 ys\n\nfoo : (xs ys : D) \u2192 xs \u2248 ys \u2192 BISI _\u2248_ xs ys\nfoo xs ys xs\u2248ys = {!!}\n\nbar : (xs ys : D) \u2192 BISI _\u2248_ xs ys \u2192 xs \u2248 ys\nbar xs ys (x' , y' , xs' , ys' , x'\u2261y' , xs'\u2248ys' , xs\u2261x'\u2237xs' , ys\u2261y'\u2237ys)\n  = subst (\u03bb t\u2081 \u2192 t\u2081 \u2248 ys)\n          (sym xs\u2261x'\u2237xs')\n          (subst (\u03bb t\u2082 \u2192 x' \u2237 xs' \u2248 t\u2082)\n                 (sym ys\u2261y'\u2237ys)\n                 (-\u2248-gfp\u2081 (x'\u2261y' , xs'\u2248ys')))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"feafec5e00a3ade91e928f7090883031876b8747","subject":"FunUniverse.Agda: Two can be left abstract now","message":"FunUniverse.Agda: Two can be left abstract now\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Agda.agda","new_file":"FunUniverse\/Agda.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.Agda where\n\nopen import Type\nopen import Data.Two using (proj\u2032)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using ([]; _\u2237_)\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two using (\ud835\udfda)\nopen import Data.Vec using (Vec)\nopen import Data.Bit using (0b; 1b)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Core\n\n-\u2192- : \u2605 \u2192 \u2605 \u2192 \u2605\n-\u2192- A B = A \u2192 B\n\nfunCat : Category -\u2192-\nfunCat = F.id , _\u2218\u2032_\n\nmodule Abstract\ud835\udfda (some\ud835\udfda : \u2605) where\n\n    private\n        \u2605-U' : Universe \u2605\u2080\n        \u2605-U' = mk \ud835\udfd9 some\ud835\udfda _\u00d7_ Vec\n\n    agdaFunU : FunUniverse \u2605\n    agdaFunU = \u2605-U' , -\u2192-\n\n    module AgdaFunUniverse = FunUniverse agdaFunU\n \n    funLin : LinRewiring agdaFunU\n    funLin = mk funCat\n                (\u03bb f \u2192 \u00d7.map f F.id)\n                \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n                (\u03bb f g \u2192 \u00d7.map f g) (\u03bb f \u2192 \u00d7.map F.id f)\n                (F.const []) _ (uncurry _\u2237_) V.uncons\n\n    funRewiring : Rewiring agdaFunU\n    funRewiring = mk funLin _ (\u03bb x \u2192 x , x) (F.const []) \u00d7.<_,_> proj\u2081 proj\u2082\n                     V.rewire V.rewireTbl\n\nopen Abstract\ud835\udfda \ud835\udfda public\n\nfunFork : HasFork agdaFunU\nfunFork = (\u03bb { (b , xy)    \u2192 proj\u2032 xy b })\n        , (\u03bb { f g (b , x) \u2192 proj\u2032 (f , g) b x })\n\nagdaFunOps : FunOps agdaFunU\nagdaFunOps = mk funRewiring funFork (F.const 0b) (F.const 1b)\n\nmodule AgdaFunOps = FunOps agdaFunOps\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.Agda where\n\nopen import Type\nopen import Data.Two using (proj\u2032)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using ([]; _\u2237_)\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bit using (0b; 1b)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Core\n\n-\u2192- : \u2605 \u2192 \u2605 \u2192 \u2605\n-\u2192- A B = A \u2192 B\n\nagdaFunU : FunUniverse \u2605\nagdaFunU = \u2605-U , -\u2192-\n\nmodule AgdaFunUniverse = FunUniverse agdaFunU\n\nfunCat : Category -\u2192-\nfunCat = F.id , _\u2218\u2032_\n\nfunLin : LinRewiring agdaFunU\nfunLin = mk funCat\n            (\u03bb f \u2192 \u00d7.map f F.id)\n            \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n            (\u03bb f g \u2192 \u00d7.map f g) (\u03bb f \u2192 \u00d7.map F.id f)\n            (F.const []) _ (uncurry _\u2237_) V.uncons\n\nfunRewiring : Rewiring agdaFunU\nfunRewiring = mk funLin _ (\u03bb x \u2192 x , x) (F.const []) \u00d7.<_,_> proj\u2081 proj\u2082\n                 V.rewire V.rewireTbl\n\nfunFork : HasFork agdaFunU\nfunFork = (\u03bb { (b , xy)    \u2192 proj\u2032 xy b })\n        , (\u03bb { f g (b , x) \u2192 proj\u2032 (f , g) b x })\n\nagdaFunOps : FunOps agdaFunU\nagdaFunOps = mk funRewiring funFork (F.const 0b) (F.const 1b)\n\nmodule AgdaFunOps = FunOps agdaFunOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"19e8dae6dd8e0a7eff78868e12bf17f07250eaa9","subject":"Complete correctness proof for calculus with bags (Fixes #51)","message":"Complete correctness proof for calculus with bags (Fixes #51)\n\nOld-commit-hash: 93097a6e0bdf65daa5b90ce77bb36b077316fff5\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/NatBag.agda","new_file":"experimental\/NatBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   X. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   X. Introduce addition\n   X. Add MapBags and map\n2. Test it out with `inc` as primitive\n3. Finish ExplicitNils\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n4. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = diff d b\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemmas\n\n-- diff-apply holds with propositional equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 sym (\u2205++b=b {{\u27e6 derive b \u27e7 \u03c1}}) \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 extract-\u0394equiv\n       (correctness-of-derive \u03c1 {consistency} b)\n       emptyBag tt tt \u27e9\n    emptyBag ++ (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 \u2205++b=b {{\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)}} \u27e9\n    \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Semantic brackets preserve \u2295 and \u229d\n\n\u2295-preservation : \u2200 {\u03c4 \u0393} \u2192\n  {t : Term \u0393 \u03c4} {dt : Term \u0393 (\u0394-Type \u03c4)} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u2295 dt \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1\n\n\u229d-preservation : \u2200 {\u03c4 \u0393} \u2192\n  {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u229d t \u27e7 \u03c1 \u2261 \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u27e7 \u27e6 t \u27e7 \u03c1\n\n\u2295-preservation {nats} {\u0393} {t} {dt} {\u03c1} = refl\n\u2295-preservation {bags} {\u0393} {t} {dt} {\u03c1} = refl\n\u2295-preservation {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {t} {dt} {\u03c1} = extensionality (\u03bb x \u2192\n  begin\n    \u27e6 app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) t) (var this) \u2295\n      app (app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) dt) (var this))\n      (var this \u229d var this) \u27e7 (x \u2022 \u03c1)\n  \u2261\u27e8 \u2295-preservation \u27e9\n    \u27e6 app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) t) (var this) \u27e7 (x \u2022 \u03c1)\n    \u27e6\u2295\u27e7\n    \u27e6 app (app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) dt) (var this))\n      (var this \u229d var this) \u27e7 (x \u2022 \u03c1)\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n       (trans (weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} t (x \u2022 \u03c1))\n              (cong \u27e6 t \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl)\n       (trans (weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} dt (x \u2022 \u03c1))\n              (cong \u27e6 dt \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl \u27e8$\u27e9 refl) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (\u27e6 (var this \u229d var this) \u27e7 (x \u2022 \u03c1))\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x hole)\n          (\u229d-preservation {\u03c4\u2081} {\u03c4\u2081 \u2022 \u0393} {var this} {var this}) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (x \u27e6\u229d\u27e7 x)\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x hole) (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f x) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (\u27e6derive\u27e7 x)\n  \u220e) where open \u2261-Reasoning\n\n\u229d-preservation {nats} {\u0393} {s} {t} {\u03c1} = extensionality (\u03bb f \u2192\n  cong\u2082 f\n    (trans (weaken-sound t (f \u2022 \u03c1)) (cong \u27e6 t \u27e7 identity-weakening))\n    (trans (weaken-sound s (f \u2022 \u03c1)) (cong \u27e6 s \u27e7 identity-weakening)))\n\u229d-preservation {bags} {\u0393} {s} {t} {\u03c1} = refl\n\u229d-preservation {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {s} {t} {\u03c1} =\n  extensionality (\u03bb x \u2192\n  extensionality (\u03bb dx \u2192\n  begin\n    \u27e6 app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n      (var (that this) \u2295 var this)\n      \u229d\n      app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)) \u27e7\n    (dx \u2022 x \u2022 \u03c1)\n  \u2261\u27e8 \u229d-preservation\n     {s = app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n              (var (that this) \u2295 var this)}\n     {t = app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t)\n               (var (that this))} \u27e9\n    \u27e6 weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s \u27e7 (dx \u2022 x \u2022 \u03c1)\n      (\u27e6 var (that this) \u2295 var this \u27e7 (dx \u2022 x \u2022 \u03c1))\n    \u27e6\u229d\u27e7\n    \u27e6 weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t \u27e7 (dx \u2022 x \u2022 \u03c1)\n      (\u27e6 (var (that this)) \u27e7 (dx \u2022 x \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (trans (weaken-sound s (dx \u2022 x \u2022 \u03c1))\n              (cong \u27e6 s \u27e7 identity-weakening)\n        \u27e8$\u27e9 \u2295-preservation {t = var (that this)} {dt = var this})\n       (trans (weaken-sound t (dx \u2022 x \u2022 \u03c1))\n              (cong \u27e6 t \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl) \u27e9\n    \u27e6 s \u27e7 \u03c1 (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 \u27e6 t \u27e7 \u03c1 x\n  \u220e)) where open \u2261-Reasoning\n\n-- Derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = validity-of-derive \u03c1 {consistency} t\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _++_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (sym ([a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} \n             {\u27e6 d \u27e7 (update \u03c1 {consistency})}\n             {\u27e6 b \u27e7 (ignore \u03c1)} {\u27e6 d \u27e7 (ignore \u03c1)})) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency})\n        ++ \u27e6 d \u27e7 (update \u03c1 {consistency}))\n        \\\\(\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (diff b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ \u27e6 derive b \u27e7 \u03c1 \\\\ \u27e6 derive d \u27e7 \u03c1\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _\\\\_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++  (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       \\\\ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       ([a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} {\u27e6 b \u27e7 (ignore \u03c1)}\n             {\u27e6 d \u27e7 (update \u03c1 {consistency})} {\u27e6 d \u27e7 (ignore \u03c1)}) \u27e9\n    c ++  (\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1))\n  \u220e) where open \u2261-Reasoning\n\n\ncorrectness-of-derive \u03c1 {consistency} (map f b) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u2295 derive f \u27e7 \u03c1)\n               (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1 ++ \u27e6 derive b \u27e7 \u03c1))\n       \\\\\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _\\\\_\n          (cong\u2082 mapBag\n            (\u2295-preservation { t = weaken \u0393\u227c\u0394\u0393 f}\n                            {dt = derive f} {\u03c1 = \u03c1})\n            (extract-\u0394equiv\n              (correctness-of-derive\u2032 \u03c1 {consistency} b)\n              (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1) tt tt))\n          (cong\u2082 mapBag\n            (weaken-sound f \u03c1) (weaken-sound b \u03c1))) \u27e9\n    c ++\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u03c1) -- \u2295-preservation\n               (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1 ++                 -- correctness\u2032\n                \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1))\n       \\\\\n       (mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1)))   -- weaken-sound\n  \u2261\u27e8 cong\u2082 (\u03bb hole-f hole-b \u2192 c ++ mapBag hole-f hole-b\n            \\\\ mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1)))\n       (trans\n         (extract-\u0394equiv\n           (correctness-of-derive\u2032 \u03c1 {consistency} f)\n           (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1)\n           (validity-of-derive\u2032 \u03c1 {consistency} f)\n           (R[f,g\u229df] (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 f \u27e7 (update \u03c1))))\n         (f\u2295[g\u229df]=g (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 f \u27e7 (update \u03c1))))\n       (b++[d\\\\b]=d {\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1} {\u27e6 b \u27e7 (update \u03c1)})\n        \u27e9\n    c ++  mapBag (\u27e6 f \u27e7 (update \u03c1)) (\u27e6 b \u27e7 (update \u03c1))\n       \\\\ mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1))\n  \u220e) where open \u2261-Reasoning\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","old_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemma: diff-apply holds with propositional\n-- equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 sym (\u2205++b=b {{\u27e6 derive b \u27e7 \u03c1}}) \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 extract-\u0394equiv\n       (correctness-of-derive \u03c1 {consistency} b)\n       emptyBag tt tt \u27e9\n    emptyBag ++ (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 \u2205++b=b {{\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)}} \u27e9\n    \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    identity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\n    identity-weakening {\u2205} {\u2205} = refl\n    identity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n      cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _++_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (sym ([a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} \n             {\u27e6 d \u27e7 (update \u03c1 {consistency})}\n             {\u27e6 b \u27e7 (ignore \u03c1)} {\u27e6 d \u27e7 (ignore \u03c1)})) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency})\n        ++ \u27e6 d \u27e7 (update \u03c1 {consistency}))\n        \\\\(\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fbc8ffee14d64846f584fcb0c90726b733b42e48","subject":"Removed unnecessary imports.","message":"Removed unnecessary imports.\n\nIgnore-this: 223d6f86e9cb2503a0a973330e080f71\n\ndarcs-hash:20110306132538-3bd4e-6d77c41b16ebbd00eb0ed7cc16b12f780f291484.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/List\/Induction\/Acc\/WellFounded.agda","new_file":"src\/FOTC\/Data\/List\/Induction\/Acc\/WellFounded.agda","new_contents":"------------------------------------------------------------------------------\n-- Generic well-founded induction on lists\n------------------------------------------------------------------------------\n\n-- Adapted from FOTC.Data.Nat.Induction.Acc.WellFounded.\n\nmodule FOTC.Data.List.Induction.Acc.WellFounded where\n\nopen import FOTC.Base\n\nopen import Common.Relation.Unary\n\nopen import FOTC.Data.List.Type\n\n------------------------------------------------------------------------------\n-- The accessibility predicate: x is accessible if everything which is\n-- smaller than x is also accessible (inductively).\ndata Acc (_<_ : D \u2192 D \u2192 Set) : D \u2192 Set where\n acc : \u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 ys < xs \u2192 Acc _<_ ys) \u2192\n       Acc _<_ xs\n\naccFold : {P : D \u2192 Set} (_<_ : D \u2192 D \u2192 Set) \u2192\n          (\u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 ys < xs \u2192 P ys) \u2192 P xs) \u2192\n          \u2200 {xs} \u2192 List xs \u2192 Acc _<_ xs \u2192 P xs\naccFold _<_ f Lxs (acc _ h) =\n  f Lxs (\u03bb Lys ys<xs \u2192 accFold _<_ f Lys (h Lys ys<xs))\n\n-- The accessibility predicate encodes what it means to be\n-- well-founded; if all elements are accessible, then _<_ is\n-- well-founded.\nWellFounded : (D \u2192 D \u2192 Set) \u2192 Set\nWellFounded _<_ = \u2200 {xs} \u2192 List xs \u2192 Acc _<_ xs\n\nWellFoundedInduction :\n  {P : D \u2192 Set} {_<_ : D \u2192 D \u2192 Set} \u2192\n  WellFounded _<_ \u2192\n  (\u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 ys < xs \u2192 P ys) \u2192 P xs) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 P xs\nWellFoundedInduction {_<_ = _<_} wf f Lxs = accFold _<_ f Lxs (wf Lxs)\n\n-- Adapted from the standard library.\nmodule Subrelation\n       {_<\u2081_ _<\u2082_ : D \u2192 D \u2192 Set}\n       (<\u2081\u21d2<\u2082 : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 xs <\u2081 ys \u2192 xs <\u2082 ys)\n       where\n\n  accessible : Acc _<\u2082_ \u2286 Acc _<\u2081_\n  accessible (acc Lxs rs) =\n    acc Lxs (\u03bb Lys ys<xs \u2192 accessible (rs Lys (<\u2081\u21d2<\u2082 Lys Lxs ys<xs)))\n\n  well-founded : WellFounded _<\u2082_ \u2192 WellFounded _<\u2081_\n  well-founded wf = \u03bb Lxs \u2192 accessible (wf Lxs)\n","old_contents":"------------------------------------------------------------------------------\n-- Generic well-founded induction on lists\n------------------------------------------------------------------------------\n\n-- Adapted from FOTC.Data.Nat.Induction.Acc.WellFounded.\n\nmodule FOTC.Data.List.Induction.Acc.WellFounded where\n\nopen import FOTC.Base\n\nopen import Common.Relation.Unary\n\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded as Nat using ()\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.List.Type\n\n------------------------------------------------------------------------------\n-- The accessibility predicate: x is accessible if everything which is\n-- smaller than x is also accessible (inductively).\ndata Acc (_<_ : D \u2192 D \u2192 Set) : D \u2192 Set where\n acc : \u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 ys < xs \u2192 Acc _<_ ys) \u2192\n       Acc _<_ xs\n\naccFold : {P : D \u2192 Set} (_<_ : D \u2192 D \u2192 Set) \u2192\n          (\u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 ys < xs \u2192 P ys) \u2192 P xs) \u2192\n          \u2200 {xs} \u2192 List xs \u2192 Acc _<_ xs \u2192 P xs\naccFold _<_ f Lxs (acc _ h) =\n  f Lxs (\u03bb Lys ys<xs \u2192 accFold _<_ f Lys (h Lys ys<xs))\n\n-- The accessibility predicate encodes what it means to be\n-- well-founded; if all elements are accessible, then _<_ is\n-- well-founded.\nWellFounded : (D \u2192 D \u2192 Set) \u2192 Set\nWellFounded _<_ = \u2200 {xs} \u2192 List xs \u2192 Acc _<_ xs\n\nWellFoundedInduction :\n  {P : D \u2192 Set} {_<_ : D \u2192 D \u2192 Set} \u2192\n  WellFounded _<_ \u2192\n  (\u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 ys < xs \u2192 P ys) \u2192 P xs) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 P xs\nWellFoundedInduction {_<_ = _<_} wf f Lxs = accFold _<_ f Lxs (wf Lxs)\n\n-- Adapted from the standard library.\nmodule Subrelation\n       {_<\u2081_ _<\u2082_ : D \u2192 D \u2192 Set}\n       (<\u2081\u21d2<\u2082 : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 xs <\u2081 ys \u2192 xs <\u2082 ys)\n       where\n\n  accessible : Acc _<\u2082_ \u2286 Acc _<\u2081_\n  accessible (acc Lxs rs) =\n    acc Lxs (\u03bb Lys ys<xs \u2192 accessible (rs Lys (<\u2081\u21d2<\u2082 Lys Lxs ys<xs)))\n\n  well-founded : WellFounded _<\u2082_ \u2192 WellFounded _<\u2081_\n  well-founded wf = \u03bb Lxs \u2192 accessible (wf Lxs)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1187c8464bbff433a750d054679ce26dee5061b3","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: acf1daaa3d8fe89d0f934b52de4b1aeb\n\ndarcs-hash:20100810032601-3bd4e-eb1f6b6da20a70834705252fd7b59db680523247.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/NonConjectures\/InternalTypes.agda","new_file":"Test\/Succeed\/NonConjectures\/InternalTypes.agda","new_contents":"module Test.Succeed.NonConjectures.InternalTypes where\n\n-- Agda internal types (from Agda.Syntax.Internal)\n\n-- -- | Raw values.\n-- --\n-- --   @Def@ is used for both defined and undefined constants.\n-- --   Assume there is a type declaration and a definition for\n-- --     every constant, even if the definition is an empty\n-- --     list of clauses.\n-- --\n-- data Term = Var Nat Args\n-- \t  | Lam Hiding (Abs Term)   -- ^ terms are beta normal\n-- \t  | Lit Literal\n-- \t  | Def QName Args\n-- \t  | Con QName Args\n-- \t  | Pi (Arg Type) (Abs Type)\n-- \t  | Fun (Arg Type) Type\n-- \t  | Sort Sort\n-- \t  | MetaV MetaId Args\n--   deriving (Typeable, Data, Eq, Ord, Show)\n\n-- data Type = El Sort Term\n--   deriving (Typeable, Data, Eq, Ord, Show)\n\n-- data Sort = Type Term   -- A term of type Nat\n-- \t  | Prop\n-- \t  | Lub Sort Sort\n-- \t  | Suc Sort\n-- \t  | MetaS MetaId Args\n--           | Inf\n--           | DLub Sort (Abs Sort)\n--             -- ^ if the free variable occurs in the second sort\n--             --   the whole thing should reduce to Inf, otherwise\n--             --   it's the normal Lub\n--   deriving (Typeable, Data, Eq, Ord, Show)\n\npostulate\n  D : Set\n\n------------------------------------------------------------------------------\n-- Testing data Term = Def ...\n\nmodule Def where\n\n  postulate\n    P\u2081    : D \u2192 Set\n    P\u2083    : D \u2192 D \u2192 D \u2192 Set\n    a b c : D\n\n  -- A definition with one argument\n  postulate\n    termDef\u2081 : P\u2081 a\n  {-# ATP axiom termDef\u2081 #-}\n\n  -- A definition with three or more arguments\n  postulate\n    termDef\u2083 : P\u2083 a b c\n  {-# ATP axiom termDef\u2083 #-}\n\n------------------------------------------------------------------------------\n-- Testing data Term = Fun ...\n\nmodule Fun where\n\n  postulate\n    P       : D \u2192 Set\n    a       : D\n    termFun : P a  \u2192 P a\n  {-# ATP axiom termFun #-}\n\n------------------------------------------------------------------------------\n-- Testing data Term = Pi ...\n\nmodule Pi where\n  postulate\n    P      : D \u2192 Set\n    termPi : (d : D) \u2192 P d\n  {-# ATP axiom termPi #-}\n","old_contents":"module Test.Succeed.NonConjectures.InternalTypes where\n\n-- Agda internal types (from Agda.Syntax.Internal)\n\n-- -- | Raw values.\n-- --\n-- --   @Def@ is used for both defined and undefined constants.\n-- --   Assume there is a type declaration and a definition for\n-- --     every constant, even if the definition is an empty\n-- --     list of clauses.\n-- --\n-- data Term = Var Nat Args\n-- \t  | Lam Hiding (Abs Term)   -- ^ terms are beta normal\n-- \t  | Lit Literal\n-- \t  | Def QName Args\n-- \t  | Con QName Args\n-- \t  | Pi (Arg Type) (Abs Type)\n-- \t  | Fun (Arg Type) Type\n-- \t  | Sort Sort\n-- \t  | MetaV MetaId Args\n--   deriving (Typeable, Data, Eq, Ord, Show)\n\n-- data Type = El Sort Term\n--   deriving (Typeable, Data, Eq, Ord, Show)\n\n-- data Sort = Type Term   -- A term of type Nat\n-- \t  | Prop\n-- \t  | Lub Sort Sort\n-- \t  | Suc Sort\n-- \t  | MetaS MetaId Args\n--           | Inf\n--           | DLub Sort (Abs Sort)\n--             -- ^ if the free variable occurs in the second sort\n--             --   the whole thing should reduce to Inf, otherwise\n--             --   it's the normal Lub\n--   deriving (Typeable, Data, Eq, Ord, Show)\n\npostulate\n  D     : Set\n\n------------------------------------------------------------------------------\n-- Testing data Term = Def ...\n\nmodule Def where\n\n  postulate\n    P\u2081    : D \u2192 Set\n    P\u2083    : D \u2192 D \u2192 D \u2192 Set\n    a b c : D\n\n    -- A definition with one argument\n  postulate\n    termDef\u2081 : P\u2081 a\n  {-# ATP axiom termDef\u2081 #-}\n\n  -- A definition with three or more arguments\n  postulate\n    termDef\u2083 : P\u2083 a b c\n  {-# ATP axiom termDef\u2083 #-}\n\n------------------------------------------------------------------------------\n-- Testing data Term = Fun ...\n\nmodule Fun where\n\n  postulate\n    P : D \u2192 Set\n    a : D\n\n  postulate\n    termFun : P a  \u2192 P a\n  {-# ATP axiom termFun #-}\n\n------------------------------------------------------------------------------\n-- Testing data Term = Pi ...\n\nmodule Pi where\n  postulate\n    P : D \u2192 Set\n\n  postulate\n    termPi : (d : D) \u2192 P d\n  {-# ATP axiom termPi #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a883bca1e4dcf18cbd172d5af888116a00518e15","subject":"Rename lift-diff to diff-term.","message":"Rename lift-diff to diff-term.\n\nOld-commit-hash: 64cb73fbed07060d460e4d9ee85fcec48281d13a\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x)\n      ,\n      abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  diff-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  diff-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 lift-apply\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x)\n      ,\n      abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 lift-diff\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 lift-apply\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b01d8abafe916c9d2d8af12b12d2adb5f2ae8340","subject":"Move parameters to module level.","message":"Move parameters to module level.\n\nAll definitions in this module depend on the parameters {B} and\n{C} which never change inside this module. So it is easier to\nparameterize the whole module instead of each separate\ndefinition.\n\nOld-commit-hash: 1303320f48d486d11152713c604275c2c7bcdc18\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"module Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen import Syntax.Type.Plotkin\nopen import Syntax.Context\n\ndata Term\n  {type-of : C \u2192 Type B}\n  (\u0393 : Context {Type B}) :\n  (\u03c4 : Type B) \u2192 Set\n  where\n  const : (c : C) \u2192 Term \u0393 (type-of c)\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term {type-of} \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term {type-of} \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term {type-of} (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u22a2 \u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term {\u22a2} \u0393\u2081 \u03c4 \u2192\n  Term {\u22a2} \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u22a2 \u0393 \u03c3 \u03c4} \u2192\n  Term {\u22a2} \u0393 \u03c4 \u2192 Term {\u22a2} (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term {\u22a2} \u0393 \u03c4 \u2192 Term {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term {\u22a2} \u0393 \u03c4 \u2192 Term {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","old_contents":"module Syntax.Term.Plotkin where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen import Syntax.Type.Plotkin\nopen import Syntax.Context\n\ndata Term\n  {B : Set {- of base types -}}\n  {C : Set {- of constants -}}\n  {type-of : C \u2192 Type B}\n  (\u0393 : Context {Type B}) :\n  (\u03c4 : Type B) \u2192 Set\n  where\n  const : (c : C) \u2192 Term \u0393 (type-of c)\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term {B} {C} {type-of} \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term {B} {C} {type-of} \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term {B} {C} {type-of} (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {B C \u22a2 \u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term {B} {C} {\u22a2} \u0393\u2081 \u03c4 \u2192\n  Term {B} {C} {\u22a2} \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {B C \u22a2 \u0393 \u03c3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8ce53ee068c02c69a5155613225457171211fc6c","subject":"Typo.","message":"Typo.\n\nIgnore-this: 6cfad8d5ef2b2bfcf3dcb3e0006081f1\n\ndarcs-hash:20110219041942-3bd4e-84f500519b0855700f521277dca81627dbc8ee10.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/McCarthy91\/MCR\/LT2MCR-ATP.agda","new_file":"src\/LTC\/Program\/McCarthy91\/MCR\/LT2MCR-ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Property LT2MCR which proves that the recursive calls of the\n-- McCarthy 91 function are on smaller arguments.\n------------------------------------------------------------------------------\n\nmodule LTC.Program.McCarthy91.MCR.LT2MCR-ATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\nopen import LTC.Program.McCarthy91.MCR\n\n------------------------------------------------------------------------------\n\nLT2MCR-helper : \u2200 {n m k} \u2192 N n \u2192 N m \u2192 N k \u2192\n                LT m n \u2192 LT (succ n) k \u2192 LT (succ m) k \u2192\n                LT (k \u2238 n) (k \u2238 m) \u2192 LT (k \u2238 succ n) (k \u2238 succ m)\nLT2MCR-helper zN Nm Nk p qn qm h = \u22a5-elim (x<0\u2192\u22a5 Nm p)\nLT2MCR-helper (sN Nn) Nm zN p qn qm h = \u22a5-elim (x<0\u2192\u22a5 (sN Nm) qm)\nLT2MCR-helper (sN {n} Nn) zN (sN {k} Nk) p qn qm h = prfS0S\n  where\n    postulate k\u2265Sn   : GE k (succ n)\n              k\u2238Sn<k : LT (k \u2238 (succ n)) k\n              prfS0S : LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ zero)\n    {-# ATP prove k\u2265Sn x<y\u2192x\u2264y #-}\n    {-# ATP prove k\u2238Sn<k k\u2265Sn x\u2265y\u2192y>0\u2192x\u2238y<x #-}\n    {-# ATP prove prfS0S k\u2238Sn<k #-}\n\nLT2MCR-helper (sN {n} Nn) (sN {m} Nm) (sN {k} Nk) p qn qm h =\n  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm (LT2MCR-helper Nn Nm Nk m<n Sn<k Sm<k k\u2238n<k\u2238m)\n  where\n    postulate\n      k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm : LT (k \u2238 succ n) (k \u2238 succ m) \u2192\n                                LT (succ k \u2238 succ (succ n))\n                                   (succ k \u2238 succ (succ m))\n    {-# ATP prove  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm #-}\n\n    postulate\n      m<n     : LT m n\n      Sn<k    : LT (succ n) k\n      Sm<k    : LT (succ m) k\n      k\u2238n<k\u2238m : LT (k \u2238 n) (k \u2238 m)\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn<k #-}\n    {-# ATP prove Sm<k #-}\n    {-# ATP prove k\u2238n<k\u2238m #-}\n\nLT2MCR : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE m one-hundred \u2192 LT m n \u2192 MCR n m\nLT2MCR zN Nm p h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nLT2MCR (sN {n} Nn) zN p h = prfS0\n  where\n    postulate prfS0 : MCR (succ n) zero\n    {-# ATP prove prfS0 x\u2238y<Sx #-}\n\nLT2MCR (sN {n} Nn) (sN {m} Nm) p h with x<y\u2228x\u2265y Nn N100\n... | inj\u2081 n<100 = LT2MCR-helper Nn Nm N101 m<n Sn\u2264101 Sm\u2264101\n                                 (LT2MCR Nn Nm m\u2264100 m<n)\n  where\n    postulate m\u2264100  : LE m one-hundred\n              m<n    : LT m n\n              Sn\u2264101 : LT (succ n) hundred-one\n              Sm\u2264101 : LT (succ m) hundred-one\n    {-# ATP prove m\u2264100 x<y\u2192x\u2264y #-}\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn\u2264101 #-}\n    {-# ATP prove Sm\u2264101 #-}\n... | inj\u2082 n\u2265199 = prf-n\u2265100\n  where\n    postulate\n      101\u2238Sn\u22610  : hundred-one \u2238 succ n \u2261 zero\n      0<101\u2238Sm  : LT zero (hundred-one \u2238 succ m)\n      prf-n\u2265100 : MCR (succ n) (succ m)\n    {-# ATP prove 101\u2238Sn\u22610 x\u2264y\u2192x-y\u22610 #-}\n    {-# ATP prove 0<101\u2238Sm x<y\u21920<x\u2238y #-}\n    {-# ATP prove prf-n\u2265100 101\u2238Sn\u22610 0<101\u2238Sm #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Property LT2MCR which proves that the recursive calls of the\n-- McCarthy 91 functions are on smaller arguments.\n------------------------------------------------------------------------------\n\nmodule LTC.Program.McCarthy91.MCR.LT2MCR-ATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\nopen import LTC.Program.McCarthy91.MCR\n\n------------------------------------------------------------------------------\n\nLT2MCR-helper : \u2200 {n m k} \u2192 N n \u2192 N m \u2192 N k \u2192\n                LT m n \u2192 LT (succ n) k \u2192 LT (succ m) k \u2192\n                LT (k \u2238 n) (k \u2238 m) \u2192 LT (k \u2238 succ n) (k \u2238 succ m)\nLT2MCR-helper zN Nm Nk p qn qm h = \u22a5-elim (x<0\u2192\u22a5 Nm p)\nLT2MCR-helper (sN Nn) Nm zN p qn qm h = \u22a5-elim (x<0\u2192\u22a5 (sN Nm) qm)\nLT2MCR-helper (sN {n} Nn) zN (sN {k} Nk) p qn qm h = prfS0S\n  where\n    postulate k\u2265Sn   : GE k (succ n)\n              k\u2238Sn<k : LT (k \u2238 (succ n)) k\n              prfS0S : LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ zero)\n    {-# ATP prove k\u2265Sn x<y\u2192x\u2264y #-}\n    {-# ATP prove k\u2238Sn<k k\u2265Sn x\u2265y\u2192y>0\u2192x\u2238y<x #-}\n    {-# ATP prove prfS0S k\u2238Sn<k #-}\n\nLT2MCR-helper (sN {n} Nn) (sN {m} Nm) (sN {k} Nk) p qn qm h =\n  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm (LT2MCR-helper Nn Nm Nk m<n Sn<k Sm<k k\u2238n<k\u2238m)\n  where\n    postulate\n      k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm : LT (k \u2238 succ n) (k \u2238 succ m) \u2192\n                                LT (succ k \u2238 succ (succ n))\n                                   (succ k \u2238 succ (succ m))\n    {-# ATP prove  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm #-}\n\n    postulate\n      m<n     : LT m n\n      Sn<k    : LT (succ n) k\n      Sm<k    : LT (succ m) k\n      k\u2238n<k\u2238m : LT (k \u2238 n) (k \u2238 m)\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn<k #-}\n    {-# ATP prove Sm<k #-}\n    {-# ATP prove k\u2238n<k\u2238m #-}\n\nLT2MCR : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE m one-hundred \u2192 LT m n \u2192 MCR n m\nLT2MCR zN Nm p h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nLT2MCR (sN {n} Nn) zN p h = prfS0\n  where\n    postulate prfS0 : MCR (succ n) zero\n    {-# ATP prove prfS0 x\u2238y<Sx #-}\n\nLT2MCR (sN {n} Nn) (sN {m} Nm) p h with x<y\u2228x\u2265y Nn N100\n... | inj\u2081 n<100 = LT2MCR-helper Nn Nm N101 m<n Sn\u2264101 Sm\u2264101\n                                 (LT2MCR Nn Nm m\u2264100 m<n)\n  where\n    postulate m\u2264100  : LE m one-hundred\n              m<n    : LT m n\n              Sn\u2264101 : LT (succ n) hundred-one\n              Sm\u2264101 : LT (succ m) hundred-one\n    {-# ATP prove m\u2264100 x<y\u2192x\u2264y #-}\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn\u2264101 #-}\n    {-# ATP prove Sm\u2264101 #-}\n... | inj\u2082 n\u2265199 = prf-n\u2265100\n  where\n    postulate\n      101\u2238Sn\u22610  : hundred-one \u2238 succ n \u2261 zero\n      0<101\u2238Sm  : LT zero (hundred-one \u2238 succ m)\n      prf-n\u2265100 : MCR (succ n) (succ m)\n    {-# ATP prove 101\u2238Sn\u22610 x\u2264y\u2192x-y\u22610 #-}\n    {-# ATP prove 0<101\u2238Sm x<y\u21920<x\u2238y #-}\n    {-# ATP prove prf-n\u2265100 101\u2238Sn\u22610 0<101\u2238Sm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"67345797da0ec7be69ae812423b36527aa5c04bf","subject":"Nat: more properties","message":"Nat: more properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0cd9ddebb8cf75d4df2bc04ee0c195c0587029af","subject":"Missing cosmetic change.","message":"Missing cosmetic change.\n\nIgnore-this: 6a19bce792411fcc36f0fc4e790efa77\n\ndarcs-hash:20110728171007-3bd4e-be138098f352695feefb52c5b0e2ae5403c7deea.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/List\/Type.agda","new_file":"src\/FOTC\/Data\/Nat\/List\/Type.agda","new_contents":"----------------------------------------------------------------------------\n-- The FOTC list of natural numbers type\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.List.Type where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type.\ndata ListN : D \u2192 Set where\n  nilLN  :                             ListN []\n  consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n{-# ATP axiom nilLN #-}\n{-# ATP axiom consLN #-}\n","old_contents":"----------------------------------------------------------------------------\n-- The FOTC list of natural numbers type\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.List.Type where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type.\ndata ListN : D \u2192 Set where\n  nilLN  :                                      ListN []\n  consLN : \u2200 {n ns} \u2192 N n \u2192 (LNns : ListN ns) \u2192 ListN (n \u2237 ns)\n{-# ATP axiom nilLN #-}\n{-# ATP axiom consLN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3376865b1a5c84c532cfba6c4ae34e4e90a0965f","subject":"Disable option '--universal-quantified-functions'.","message":"Disable option '--universal-quantified-functions'.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/LambdaLifting.agda","new_file":"notes\/LambdaLifting.agda","new_contents":"","old_contents":"------------------------------------------------------------------------------\n-- Example of lambda-lifting\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --universal-quantified-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule LambdaLifting where\n\ninfixl 9 _\u00b7_\ninfix  8 if_then_else_\ninfix  7 _\u2261_\n\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n\npostulate\n  D                : Set\n  zero true false  : D\n  succ iszero pred : D \u2192 D\n  _\u00b7_              : D \u2192 D \u2192 D\n  if_then_else_    : D \u2192 D \u2192 D \u2192 D\n  lam fix          : (D \u2192 D) \u2192 D\n\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Conversion rules\n\n-- Conversion rules for Booleans.\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true then d\u2081 else d\u2082  \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred zero     \u2261 zero\n  pred-S : \u2200 d \u2192 pred (succ d) \u2261 d\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\npostulate\n  iszero-0 :       iszero zero     \u2261 true\n  iszero-S : \u2200 d \u2192 iszero (succ d) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rule for the \u03bb-abstraction and the application.\npostulate beta : \u2200 f a \u2192 lam f \u00b7 a \u2261 f a\n{-# ATP axiom beta #-}\n\n-- Conversion rule for the fixed-pointed operator.\npostulate fix-eq : \u2200 f \u2192 fix f \u2261 f (fix  f)\n{-# ATP axiom fix-eq #-}\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192 zero + d     \u2261 d\n  +-Sx : \u2200 d e \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x +-Sx #-}\n\npostulate\n  _*_  : D \u2192 D \u2192 D\n  *-0x : \u2200 d \u2192 zero * d \u2261 zero\n  *-Sx : \u2200 d e \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x *-Sx #-}\n\n------------------------------------------------------------------------------\n-- The original fach\n-- fach : D \u2192 D\n-- fach f = lam (\u03bb n \u2192 if (iszero n) then (succ zero) else n * (f \u00b7 (pred n)))\n\n-- Lambda-lifting via super-combinators (Hughes. Super-combinators. 1982).\n\n\u03b1 : D \u2192 D \u2192 D\n\u03b1 f n = if (iszero n) then (succ zero) else n * (f \u00b7 (pred n))\n{-# ATP definition \u03b1 #-}\n\nfach : D \u2192 D\nfach f = lam (\u03b1 f)\n{-# ATP definition fach #-}\n\nfac : D \u2192 D\nfac n = fix fach \u00b7 n\n{-# ATP definition fac #-}\n\npostulate fac0 : fac zero \u2261 succ zero\n{-# ATP prove fac0 #-}\n\npostulate fac1 : fac (succ zero) \u2261 succ zero\n{-# ATP prove fac1 #-}\n\npostulate fac2 : fac (succ (succ zero)) \u2261 succ (succ zero)\n{-# ATP prove fac2 #-}\n\n------------------------------------------------------------------------------\n-- Ouput:\n--\n-- $ agda2atp -inotes --non-fol-function notes\/LambdaLifting.agda\n-- Proving the conjecture in \/tmp\/LambdaLifting\/95-fac1.tptp ...\n-- E 1.6 Tiger Hill proved the conjecture in \/tmp\/LambdaLifting\/95-fac1.tptp\n-- Proving the conjecture in \/tmp\/LambdaLifting\/99-fac2.tptp ...\n-- E 1.6 Tiger Hill proved the conjecture in \/tmp\/LambdaLifting\/99-fac2.tptp\n-- Proving the conjecture in \/tmp\/LambdaLifting\/91-fac0.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/LambdaLifting\/91-fac0.tptp\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"eeb64c8072203504365523df010e26b7cd75ace3","subject":"Revert \"+-interchange is now with implicit arguments...\"","message":"Revert \"+-interchange is now with implicit arguments...\"\n\nThis reverts commit dd40e9080437a9a681f55282bbd3b79a51c7b3d5.\n","repos":"crypto-agda\/crypto-agda","old_file":"Neglible.agda","new_file":"Neglible.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\nopen import Function.Extensionality\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-refl : \u2200 {f} \u2192 f \u2264\u2192 f\n_\u2264\u2192_.\u2264\u2192 \u2264\u2192-refl k = OR.refl\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (fN k) (gD k) (f'D k) (g'D k) \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange (gN k) (fD k) (g'D k) (f'D k))\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (f'N k) (fD k) (g'D k) (gD k))\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange (g'N k) (gD k) (f'D k) (fD k)\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n\n  fromNeg : {f : \u2115\u2192\u211a} \u2192 Is-Neg f \u2192 NegBounded f\n  \u03b5 (fromNeg f-neg) = _\n  \u03b5-neg (fromNeg f-neg) = f-neg\n  bounded (fromNeg f-neg) = \u2264\u2192-refl\n\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\n  ~-Inv : {{_ : FunExt}}{{_ : UA}}(\u03c0 : \u2200 n \u2192 R n \u2243 R n)(f g : \u2200 x \u2192 R x \u2192 \ud835\udfda)\n          (eq : \u2200 x (r : R x) \u2192 f x r \u2261 g x (proj\u2081 (\u03c0 x) r)) \u2192 f ~ g\n  _~_.~ (~-Inv \u03c0 f g eq) = \u2264-NB lemma (fromNeg 0\u2115\u211a-neg)\n    where\n      open \u2264-Reasoning\n      lemma : ~dist f g \u2264\u2192 0\u2115\u211a\n      _\u2264\u2192_.\u2264\u2192 lemma k = dist (# (f k)) (# (g k)) * 1\n                      \u2261\u27e8 proj\u2082 prop.*-identity _ \u27e9\n                        dist (# (f k)) (# (g k))\n                      \u2261\u27e8 ap (flip dist (# (g k))) (count-ext (R\u1d41 k) (eq k)) \u27e9\n                        dist (# (g k \u2218 proj\u2081 (\u03c0 k))) (# (g k))\n                      \u2261\u27e8 ap (flip dist (# (g k))) (sumStableUnder (R\u1d41 k) (\u03c0 k) (\ud835\udfda\u25b9\u2115 \u2218 g k)) \u27e9\n                        dist (# (g k)) (# (g k))\n                      \u2261\u27e8 dist-refl (# (g k)) \u27e9\n                        0\n                      \u220e\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\nopen import Function.Extensionality\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-refl : \u2200 {f} \u2192 f \u2264\u2192 f\n_\u2264\u2192_.\u2264\u2192 \u2264\u2192-refl k = OR.refl\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {fN k} {gD k} {f'D k} {g'D k} \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange {gN k} {fD k} {g'D k} {f'D k})\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {f'N k} {fD k} {g'D k} {gD k})\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange {g'N k} {gD k} {f'D k} {fD k}\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n\n  fromNeg : {f : \u2115\u2192\u211a} \u2192 Is-Neg f \u2192 NegBounded f\n  \u03b5 (fromNeg f-neg) = _\n  \u03b5-neg (fromNeg f-neg) = f-neg\n  bounded (fromNeg f-neg) = \u2264\u2192-refl\n\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\n  ~-Inv : {{_ : FunExt}}{{_ : UA}}(\u03c0 : \u2200 n \u2192 R n \u2243 R n)(f g : \u2200 x \u2192 R x \u2192 \ud835\udfda)\n          (eq : \u2200 x (r : R x) \u2192 f x r \u2261 g x (proj\u2081 (\u03c0 x) r)) \u2192 f ~ g\n  _~_.~ (~-Inv \u03c0 f g eq) = \u2264-NB lemma (fromNeg 0\u2115\u211a-neg)\n    where\n      open \u2264-Reasoning\n      lemma : ~dist f g \u2264\u2192 0\u2115\u211a\n      _\u2264\u2192_.\u2264\u2192 lemma k = dist (# (f k)) (# (g k)) * 1\n                      \u2261\u27e8 proj\u2082 prop.*-identity _ \u27e9\n                        dist (# (f k)) (# (g k))\n                      \u2261\u27e8 ap (flip dist (# (g k))) (count-ext (R\u1d41 k) (eq k)) \u27e9\n                        dist (# (g k \u2218 proj\u2081 (\u03c0 k))) (# (g k))\n                      \u2261\u27e8 ap (flip dist (# (g k))) (sumStableUnder (R\u1d41 k) (\u03c0 k) (\ud835\udfda\u25b9\u2115 \u2218 g k)) \u27e9\n                        dist (# (g k)) (# (g k))\n                      \u2261\u27e8 dist-refl (# (g k)) \u27e9\n                        0\n                      \u220e\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"57708b770c00d1ba6276cda0f02752e147f2f533","subject":"Added a non-required totality hypothesis.","message":"Added a non-required totality hypothesis.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- f\u2089\u2081) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < f\u2089\u2081 n + 11.\n-- 3. For all n > 100, then f\u2089\u2081 n = n - 10.\n-- 4. For all n <= 100, then f\u2089\u2081 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationPropertiesATP\n  using ( x>y\u2192x\u2264y\u2192\u22a5 )\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\n  using ( x>y\u2228x\u226fy\n        ; x\u226fy\u2192x\u2264y\n        ; x\u226fSy\u2192x\u226fy\u2228x\u2261Sy\n        ; x+k<y+k\u2192x<y\n        ; <-trans\n        )\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( \u2238-N\n        ; +-N\n        )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n  using ( x<x+11 )\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n  using ( 10-N\n        ; 11-N\n        ; 89-N\n        ; 90-N\n        ; 91-N\n        ; 92-N\n        ; 93-N\n        ; 94-N\n        ; 95-N\n        ; 96-N\n        ; 97-N\n        ; 98-N\n        ; 99-N\n        ; 100-N\n        )\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n-- For all n > 100, then f\u2089\u2081 n = n - 10.\n--\n-- N.B. (21 November 2013). The hypothesis N n is not necessary.\npostulate f\u2089\u2081-x>100 : \u2200 n \u2192 N n \u2192 n > 100' \u2192 f\u2089\u2081 n \u2261 n \u2238 10'\n{-# ATP prove f\u2089\u2081-x>100 #-}\n\n-- For all n <= 100, then f\u2089\u2081 n = 91.\nf\u2089\u2081-x\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f 100' \u2192 f\u2089\u2081 n \u2261 91'\nf\u2089\u2081-x\u226f100 = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d \u226f 100' \u2192 f\u2089\u2081 d \u2261 91'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = \u03bb m\u226f100 \u2192\n                     \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nm 100-N m>100 (x\u226fy\u2192x\u2264y Nm 100-N m\u226f100))\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-100 m\u2261100\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-99 m\u226199\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-98 m\u226198\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-97 m\u226197\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-96 m\u226196\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-95 m\u226195\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-94 m\u226194\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-93 m\u226193\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-92 m\u226192\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-91 m\u226191\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-90 m\u226190\n  ... | inj\u2081 m\u226f89 = \u03bb _ \u2192 f\u2089\u2081-m\u226f89\n    where\n    m\u226489 : m \u2264 89'\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    f\u2089\u2081-m+11 : m + 11' \u226f 100' \u2192 f\u2089\u2081 (m + 11') \u2261 91'\n    f\u2089\u2081-m+11 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n    f\u2089\u2081-m\u226f89 : f\u2089\u2081 m \u2261 91'\n    f\u2089\u2081-m\u226f89 = f\u2089\u2081-x\u226f100-helper m 91' m\u226f100\n                                (f\u2089\u2081-m+11 (x\u226489\u2192x+11\u226f100 Nm m\u226489))\n                                f\u2089\u2081-91\n-- The function always terminates.\nf\u2089\u2081-N : \u2200 {n} \u2192 N n \u2192 N (f\u2089\u2081 n)\nf\u2089\u2081-N {n} Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (f\u2089\u2081-x>100 n Nn n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226f100 = subst N (sym (f\u2089\u2081-x\u226f100 Nn n\u226f100)) 91-N\n\n-- For all n, n < f\u2089\u2081 n + 11.\nf\u2089\u2081-ineq : \u2200 {n} \u2192 N n \u2192 n < f\u2089\u2081 n + 11'\nf\u2089\u2081-ineq = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < f\u2089\u2081 d + 11'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x>100\u2192x<f\u2089\u2081-x+11 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let f\u2089\u2081-m+11-N : N (f\u2089\u2081 (m + 11'))\n        f\u2089\u2081-m+11-N = f\u2089\u2081-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + 11')\n        h\u2081 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<f\u2089\u2081m+11 : m < f\u2089\u2081 (m + 11')\n        m<f\u2089\u2081m+11 = x+k<y+k\u2192x<y Nm f\u2089\u2081-m+11-N 11-N h\u2081\n\n        h\u2082 : A (f\u2089\u2081 (m + 11'))\n        h\u2082 = f f\u2089\u2081-m+11-N (<\u2192\u25c1 f\u2089\u2081-m+11-N Nm m\u226f100 m<f\u2089\u2081m+11)\n\n    in (<-trans Nm f\u2089\u2081-m+11-N (x+11-N (f\u2089\u2081-N Nm))\n                m<f\u2089\u2081m+11\n                (f\u2089\u2081x+11<f\u2089\u2081x+11 m m\u226f100 h\u2082))\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- f\u2089\u2081) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < f\u2089\u2081 n + 11.\n-- 3. For all n > 100, then f\u2089\u2081 n = n - 10.\n-- 4. For all n <= 100, then f\u2089\u2081 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationPropertiesATP\n  using ( x>y\u2192x\u2264y\u2192\u22a5 )\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\n  using ( x>y\u2228x\u226fy\n        ; x\u226fy\u2192x\u2264y\n        ; x\u226fSy\u2192x\u226fy\u2228x\u2261Sy\n        ; x+k<y+k\u2192x<y\n        ; <-trans\n        )\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( \u2238-N\n        ; +-N\n        )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n  using ( x<x+11 )\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n  using ( 10-N\n        ; 11-N\n        ; 89-N\n        ; 90-N\n        ; 91-N\n        ; 92-N\n        ; 93-N\n        ; 94-N\n        ; 95-N\n        ; 96-N\n        ; 97-N\n        ; 98-N\n        ; 99-N\n        ; 100-N\n        )\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n-- For all n > 100, then f\u2089\u2081 n = n - 10.\npostulate f\u2089\u2081-x>100 : \u2200 n \u2192 n > 100' \u2192 f\u2089\u2081 n \u2261 n \u2238 10'\n{-# ATP prove f\u2089\u2081-x>100 #-}\n\n-- For all n <= 100, then f\u2089\u2081 n = 91.\nf\u2089\u2081-x\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f 100' \u2192 f\u2089\u2081 n \u2261 91'\nf\u2089\u2081-x\u226f100 = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d \u226f 100' \u2192 f\u2089\u2081 d \u2261 91'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = \u03bb m\u226f100 \u2192\n                     \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nm 100-N m>100 (x\u226fy\u2192x\u2264y Nm 100-N m\u226f100))\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-100 m\u2261100\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-99 m\u226199\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-98 m\u226198\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-97 m\u226197\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-96 m\u226196\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-95 m\u226195\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-94 m\u226194\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-93 m\u226193\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-92 m\u226192\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-91 m\u226191\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-90 m\u226190\n  ... | inj\u2081 m\u226f89 = \u03bb _ \u2192 f\u2089\u2081-m\u226f89\n    where\n    m\u226489 : m \u2264 89'\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    f\u2089\u2081-m+11 : m + 11' \u226f 100' \u2192 f\u2089\u2081 (m + 11') \u2261 91'\n    f\u2089\u2081-m+11 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n    f\u2089\u2081-m\u226f89 : f\u2089\u2081 m \u2261 91'\n    f\u2089\u2081-m\u226f89 = f\u2089\u2081-x\u226f100-helper m 91' m\u226f100\n                                (f\u2089\u2081-m+11 (x\u226489\u2192x+11\u226f100 Nm m\u226489))\n                                f\u2089\u2081-91\n-- The function always terminates.\nf\u2089\u2081-N : \u2200 {n} \u2192 N n \u2192 N (f\u2089\u2081 n)\nf\u2089\u2081-N {n} Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (f\u2089\u2081-x>100 n n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226f100 = subst N (sym (f\u2089\u2081-x\u226f100 Nn n\u226f100)) 91-N\n\n-- For all n, n < f\u2089\u2081 n + 11.\nf\u2089\u2081-ineq : \u2200 {n} \u2192 N n \u2192 n < f\u2089\u2081 n + 11'\nf\u2089\u2081-ineq = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < f\u2089\u2081 d + 11'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x>100\u2192x<f\u2089\u2081-x+11 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let f\u2089\u2081-m+11-N : N (f\u2089\u2081 (m + 11'))\n        f\u2089\u2081-m+11-N = f\u2089\u2081-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + 11')\n        h\u2081 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<f\u2089\u2081m+11 : m < f\u2089\u2081 (m + 11')\n        m<f\u2089\u2081m+11 = x+k<y+k\u2192x<y Nm f\u2089\u2081-m+11-N 11-N h\u2081\n\n        h\u2082 : A (f\u2089\u2081 (m + 11'))\n        h\u2082 = f f\u2089\u2081-m+11-N (<\u2192\u25c1 f\u2089\u2081-m+11-N Nm m\u226f100 m<f\u2089\u2081m+11)\n\n    in (<-trans Nm f\u2089\u2081-m+11-N (x+11-N (f\u2089\u2081-N Nm))\n                m<f\u2089\u2081m+11\n                (f\u2089\u2081x+11<f\u2089\u2081x+11 m m\u226f100 h\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"222bfb692d1e49956b8a6dc85fda49682f8aa1af","subject":"add \u00d7\u2243 to Identities","message":"add \u00d7\u2243 to Identities\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8909f0ee0dff75ff8a6461651e3b8a882d4a4964","subject":"Updated draft on setoids.","message":"Updated draft on setoids.\n\nIgnore-this: 5f576ada50b5215ca1047d8405d00d77\n\ndarcs-hash:20120323173333-3bd4e-10f837bce125614a6f86ac0911b2c66f81ff3c1b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n-- Tested with the development version of Agda on 23 March 2012.\n\n{-\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor = : The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n-}\n\nmodule FOTC where\n\n------------------------------------------------------------------------------\n\nmodule PeterEquality where\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  infix 7 _\u2261_\n\n  data _\u2261_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2261 x\n    sym   : \u2200 {x y} \u2192 x \u2261 y \u2192                           y \u2261 x\n    trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192                 x \u2261 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2261 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2261 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- It seems we cannot define the identity elimination using the setoid\n  -- equality.\n  --\n  -- subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\nmodule PeterD where\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n  data D : Set where\n    zero succ true false : D\n    _\u00b7_             : D \u2192 D \u2192 D\n    loop            : D\n\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A refl Ax = Ax\n\n  data N : D \u2192 Set where\n    zN :               N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  -- 2012-03-23: Why the inductive structure makes 0 + 0 different\n  -- from 0? How to define _+_ ?\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n  --   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n  --   2003\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2261_\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} x\u2261y A Ay = x\u2261y (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z x\u2261y y\u2261z A Ax = y\u2261z A (x\u2261y A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- but it seems we cannot prove the congruency\n\n  -- cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  -- cong x\u2081\u2261x\u2082 y\u2081\u2261y\u2082 A Ax\u2081y\u2081 = {!!}\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids to formalize the FOTC\n------------------------------------------------------------------------------\n\n-- Tested with the development version of Agda on 07 February 2012.\n\n{-\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor = : The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n-}\n\nmodule FOTC where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  K S : D\n  _\u00b7_ : D \u2192 D \u2192 D\n\nmodule PeterEquality where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  infix 7 _\u2250_\n\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- It seems we cannot define the identity elimination using the setoid\n  -- equality\n  -- subst : \u2200 {x y} (P : D \u2192 Set) \u2192 x \u2250 y \u2192 P x \u2192 P y\n  -- subst P x\u2250y Px = {!!}\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  -- Barthe et al. [*, p. 262] use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- [*] Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n  -- type theory. Journal of Functional Programming, 13(2):261\u2013293, 2003\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2250_\n\n  _\u2250_ : D \u2192 D \u2192 Set\u2081\n  x \u2250 y = (P : D \u2192 Set) \u2192 P x \u2192 P y\n\n  -- We can proof the setoids properties\n\n  \u2250-refl : \u2200 x \u2192 x \u2250 x\n  \u2250-refl x P Px = Px\n\n  \u2250-sym : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n  \u2250-sym {x} x\u2250y P Py = x\u2250y (\u03bb z \u2192 P z \u2192 P x) (\u03bb Px \u2192 Px) Py\n\n  \u2250-trans : \u2200 x y z \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n  \u2250-trans x y z x\u2250y y\u2250z P Px = y\u2250z P (x\u2250y P Px)\n\n  -- and the identity elimination\n\n  \u2250-subst : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2250 y \u2192 P x \u2192 P y\n  \u2250-subst P x\u2250y = x\u2250y P\n\n  -- but it seems we cannot prove the congruency\n\n  -- \u2250-cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n  -- \u2250-cong x\u2081\u2250x\u2082 y\u2081\u2250y\u2082 P Px\u2081y\u2081 = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"29e8dc288e99777b534e223466a1c17a2f3bf1f7","subject":"Missing file added","message":"Missing file added\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\n\n-- Lifting a SNR to a to a Square matrix of some shape\n-- TODO: need to look at shape to make this one work?\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr L = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s L\n\n  _+S_ : S \u2192 S \u2192 S\n  One x +S One x\u2081 = One (x +\u209b x\u2081)\n\n  _\u2219S_ : S \u2192 S \u2192 S\n  One x \u2219S One x\u2081 = One (x \u2219\u209b x\u2081)\n\n  0S = One 0\u209b\n\n  _\u2243S_ : S \u2192 S \u2192 Set\n  One x \u2243S One x\u2081 = x \u2243\u209b x\u2081\n\n  SNR : SemiNearRing\n  SNR = record\n          { s = S\n          ; _\u2243\u209b_ = _\u2243S_\n          ; 0\u209b = 0S\n          ; _+\u209b_ = _+S_\n          ; _\u2219\u209b_ = _\u2219S_\n          ; isCommMon = {!!}\n          ; zero\u02e1 = {!!}\n          ; zero\u02b3 = {!!}\n          ; _<\u2219>_ = {!!}\n          }\nSquare snr (B shape shape\u2081) = {!!}\n  -- where\n  -- open SemiNearRing snr\n\n  -- S = Sq s shape\n\n  -- open Operations s _\u2219\u209b_ _+\u209b_\n  --   renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  -- 0S = One 0\u209b\n\n  -- -- _\u2243S_ : S \u2192 S \u2192 Set\n  -- -- _\u2243S_ m n with shape\n  -- -- ... | sh = ?\n\n\n  -- SNR : SemiNearRing\n  -- SNR =\n  --   record\n  --     { s = S\n  --     ; _\u2243\u209b_ = {!!}\n  --     ; 0\u209b = {!!}\n  --     ; _+\u209b_ = {!!}\n  --     ; _\u2219\u209b_ = {!!}\n  --     ; isCommMon = {!!}\n  --     ; zero\u02e1 = {!!}\n  --     ; zero\u02b3 = {!!}\n  --     ; _<\u2219>_ = {!!}\n  --     }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\n-- TODO: imported module missing?\n\n-- Lifting a SNR to a to a Square matrix of some shape\n-- TODO: need to look at shape to make this one work?\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr L = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s L\n\n  _+S_ : S \u2192 S \u2192 S\n  One x +S One x\u2081 = One (x +\u209b x\u2081)\n\n  _\u2219S_ : S \u2192 S \u2192 S\n  One x \u2219S One x\u2081 = One (x \u2219\u209b x\u2081)\n\n  0S = One 0\u209b\n\n  _\u2243S_ : S \u2192 S \u2192 Set\n  One x \u2243S One x\u2081 = x \u2243\u209b x\u2081\n\n  SNR : SemiNearRing\n  SNR = record\n          { s = S\n          ; _\u2243\u209b_ = _\u2243S_\n          ; 0\u209b = 0S\n          ; _+\u209b_ = _+S_\n          ; _\u2219\u209b_ = _\u2219S_\n          ; isCommMon = {!!}\n          ; zero\u02e1 = {!!}\n          ; zero\u02b3 = {!!}\n          ; _<\u2219>_ = {!!}\n          }\nSquare snr (B shape shape\u2081) = {!!}\n  -- where\n  -- open SemiNearRing snr\n\n  -- S = Sq s shape\n\n  -- open Operations s _\u2219\u209b_ _+\u209b_\n  --   renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  -- 0S = One 0\u209b\n\n  -- -- _\u2243S_ : S \u2192 S \u2192 Set\n  -- -- _\u2243S_ m n with shape\n  -- -- ... | sh = ?\n\n\n  -- SNR : SemiNearRing\n  -- SNR =\n  --   record\n  --     { s = S\n  --     ; _\u2243\u209b_ = {!!}\n  --     ; 0\u209b = {!!}\n  --     ; _+\u209b_ = {!!}\n  --     ; _\u2219\u209b_ = {!!}\n  --     ; isCommMon = {!!}\n  --     ; zero\u02e1 = {!!}\n  --     ; zero\u02b3 = {!!}\n  --     ; _<\u2219>_ = {!!}\n  --     }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"145e1c87374595fe5964750f55adf91496b78092","subject":"lift-+ forgot to add this one","message":"lift-+ forgot to add this one\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\n\nimport Function as F\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\n\nInv-isEquivalence : IsEquivalence (Inv.Inverse {f\u2081 = L.zero} {f\u2082 = L.zero})\nInv-isEquivalence = record\n   { refl = Inv.id\n   ; sym = Inv.sym\n   ; trans = F.flip Inv._\u2218_ }\n\nSEToid : Set _\nSEToid = Setoid L.zero L.zero\n\nSEToid\u2081 : Setoid _ _\nSEToid\u2081 = record\n  { Carrier = Setoid L.zero L.zero\n  ; _\u2248_ = Inv.Inverse\n  ; isEquivalence = Inv-isEquivalence }\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         Carrier = SEToid ;\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv-isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B : SEToid} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         Carrier = SEToid ;\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv-isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B : SEToid} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { Carrier = SEToid\n  ; _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = from\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      from : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      from = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = (\u2081\u223c\u2081 B-rel , A-rel)\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection L.zero)\nmodule \u228e-CMon = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection L.zero)\nmodule \u00d7\u228e\u00b0    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection L.zero)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {A B} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = \u00d7-CMon.comm _ _\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = record { to = \u2192-to-\u27f6 lower\n                 ; from = \u2192-to-\u27f6 lift\n                 ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                       ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {A B : \u2605} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e = sym Maybe\u2194\u22a4\u228e \u2218 \u228e-CMon.assoc \u22a4 _ _ \u2218 (Maybe\u2194\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9 \n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n","old_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\n\nimport Function as F\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\n\nInv-isEquivalence : IsEquivalence (Inv.Inverse {f\u2081 = L.zero} {f\u2082 = L.zero})\nInv-isEquivalence = record\n   { refl = Inv.id\n   ; sym = Inv.sym\n   ; trans = F.flip Inv._\u2218_ }\n\nSEToid : Set _\nSEToid = Setoid L.zero L.zero\n\nSEToid\u2081 : Setoid _ _\nSEToid\u2081 = record\n  { Carrier = Setoid L.zero L.zero\n  ; _\u2248_ = Inv.Inverse\n  ; isEquivalence = Inv-isEquivalence }\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         Carrier = SEToid ;\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv-isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B : SEToid} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         Carrier = SEToid ;\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv-isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B : SEToid} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { Carrier = SEToid\n  ; _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = from\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      from : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      from = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = (\u2081\u223c\u2081 B-rel , A-rel)\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection L.zero)\nmodule \u228e-CMon = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection L.zero)\nmodule \u00d7\u228e\u00b0    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection L.zero)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nswap-iso : \u2200 {A B} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = \u00d7-CMon.comm _ _\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = record { to = \u2192-to-\u27f6 lower\n                 ; from = \u2192-to-\u27f6 lift\n                 ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                       ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {A B : \u2605} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e = sym Maybe\u2194\u22a4\u228e \u2218 \u228e-CMon.assoc \u22a4 _ _ \u2218 (Maybe\u2194\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b41896e427242ac85988c84587d5f0432db092c3","subject":"most of the type assignment rules","message":"most of the type assignment rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 \u00eb\n    \u2987_\u2988[_]   : \u00eb \u2192 (Nat \u00d7 subst) \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< e2' > t2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u27e6 u ::[ \u0393 ] \u2987\u2988 \u27e7\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< e' > t) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x t1 t2 e e' t2' \u0394 } \u2192\n              (\u0393 ,, (x , t1)) \u22a2 e \u21d0 t2 ~> e' :: t2' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' :: (t1 ==> t2') \u22a3 \u0394\n      EASubsume : \u2200{e u m \u0393 t' e' \u0394 t} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 m \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                  t ~ t' \u2192\n                  \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u t  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 t ~> \u2987\u2988[ u , id \u0393 ] :: t \u22a3 \u27e6 u ::[ \u0393 ] t \u27e7\n      EANEHole : \u2200{ \u0393 e u t e' t' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 t ~> \u2987 e' \u2988[ u , id \u0393 ] :: t \u22a3 (\u0394 ,, u ::[ \u0393 ] t)\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n    TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n    TAVar : \u2200{\u0394 \u0393 x t} \u2192 \u0394 , (\u0393 ,, (x , t)) \u22a2 X x :: t\n    TALam : \u2200{ \u0394 \u0393 x t1 e t2} \u2192\n            \u0394 , (\u0393 ,, (x , t1)) \u22a2 e :: t2 \u2192\n            \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e :: (t1 ==> t2)\n    TAAp : \u2200{ \u0394 \u0393 e1 e2 t1 t2 t} \u2192\n           \u0394 , \u0393 \u22a2 e1 :: t1 \u2192\n           t1 \u25b8arr (t2 ==> t) \u2192\n           \u0394 , \u0393 \u22a2 e2 :: t2 \u2192\n           \u0394 , \u0393 \u22a2 e1 \u2218 e2 :: t\n    -- TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' } \u2192 -- todo: overloaded form in the paper here\n    -- TANEHole : \u2200 { \u0394 \u0393 e t' \u0393' u \u03c3 t }\n    TACast : \u2200{ \u0394 \u0393 e t t'} \u2192\n           \u0394 , \u0393 \u22a2 e :: t' \u2192\n           t ~ t' \u2192\n           \u0394 , \u0393 \u22a2 < e > t :: t\n\n  -- todo: substition goes here\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_&_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_&_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< e2' > t2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u & id \u0393 ] \u22a3  \u27e6 u ::[ \u0393 ] \u2987\u2988 \u27e7\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u & id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< e' > t) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x t1 t2 e e' t2' \u0394 } \u2192\n              (\u0393 ,, (x , t1)) \u22a2 e \u21d0 t2 ~> e' :: t2' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' :: (t1 ==> t2') \u22a3 \u0394\n      EASubsume : \u2200{e u m \u0393 t' e' \u0394 t} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 m \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                  t ~ t' \u2192\n                  \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u t  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 t ~> \u2987\u2988[ u & id \u0393 ] :: t \u22a3 \u27e6 u ::[ \u0393 ] t \u27e7\n      EANEHole : \u2200{ \u0393 e u t e' t' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 t ~> \u2987 e' \u2988[ u & id \u0393 ] :: t \u22a3 (\u0394 ,, u ::[ \u0393 ] t)\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where -- todo not a context\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u & \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u & \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9bfafe6e37ae77bcf7e5a7466815272929d3c877","subject":"More Squared proofs","message":"More Squared proofs\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  setoidS : (r c : Shape) \u2192 Setoid _ _\n  setoidS r c =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = \u2243S r c\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S' 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning (setoidS a L)\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 L)\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS a c)\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a c\u2081)\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c)\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c\u2081)\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S' 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) , (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02b3 L b L x\n      ih\u2081 = zeroS\u02b3 L b\u2081 L x\u2081\n    in begin\n      Row x x\u2081 \u2219S Col (0S b L) (0S b\u2081 L)\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (x \u2219S 0S b L) +S (x\u2081 \u2219S 0S b\u2081 L)\n    \u2248\u27e8 <+S> L L {x \u2219S 0S b L} {0S L L} {x\u2081 \u2219S 0S b\u2081 L} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02b3 L b c x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n      0S L c +S 0S L c\n    \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n      0S L c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02b3 L b c\u2081 x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 (0S L c\u2081) \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) = zeroS\u02b3 a L L x , zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b L x\n      ih\u2081 = zeroS\u02b3 a b\u2081 L x\u2081\n    in begin\n      x \u2219S 0S b L +S x\u2081 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L (0S _ _) \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b L x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 L x\u2083\n    in begin\n      x\u2082 \u2219S 0S b L +S x\u2083 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 L (0S _ _) \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c (0S _ _) \u27e9\n      0S _ _\n    \u220e) ,\n    (let\n      -- open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c\u2081 x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c\u2081 x\u2081\n    in transS a c\u2081 (<+S> a c\u2081 ih ih\u2081) (identS\u02e1 a c\u2081 (0S _ _))\n    -- begin\n    --   x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    -- \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n    --   0S _ _ +S 0S _ _\n    -- \u2248\u27e8 identS\u02e1 a c\u2081 (0S _ _) \u27e9\n    --   0S a c\u2081\n    -- \u220e\n    ) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c +S x\u2083 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c (0S _ _) \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c\u2081 x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c\u2081 +S x\u2083 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 (0S _ _) \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  <\u2219S> : (a b c : Shape) {x y : M s a b} {u v : M s b c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x \u2219S u) \u2243S' (y \u2219S v)\n  <\u2219S> L L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <\u2219> q\n  <\u2219S> L L (B c c\u2081) {One x} {One x\u2081} {Row u u\u2081} {Row v v\u2081} p (q , q\u2081) =\n    (<\u2219S> L L c {One x} {One x\u2081} {u} {v} p q) ,\n    <\u2219S> L L c\u2081 {One x} {One x\u2081} {u\u2081} {v\u2081} p q\u2081\n  <\u2219S> L (B b b\u2081) L {Row x x\u2081} {Row y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    -- Row x x\u2081 \u2219S Col u u\u2081 \u2243S' Row y y\u2081 \u2219S Col v v\u2081\n    let\n      open EqReasoning (setoidS _ _)\n      ih = <\u2219S> _ _ _ {x} {y} {u} {v} p q\n      ih\u2081 = <\u2219S> _ _ _ {x\u2081} {y\u2081} {u\u2081} {v\u2081} p\u2081 q\u2081\n    in begin\n      Row x x\u2081 \u2219S Col u u\u2081\n    \u2261\u27e8 refl-\u2261 \u27e9\n      x \u2219S u +S x\u2081 \u2219S u\u2081\n    \u2248\u27e8 <+S> L L {x \u2219S u} {y \u2219S v} {x\u2081 \u2219S u\u2081} {y\u2081 \u2219S v\u2081} ih ih\u2081 \u27e9\n      y \u2219S v +S y\u2081 \u2219S v\u2081\n    \u220e\n  <\u2219S> L (B b b\u2081) (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081) (q , q\u2081 , q\u2082 , q\u2083) =\n    (let\n      ih = <\u2219S> L b c p q\n      ih\u2081 = <\u2219S> L b\u2081 c p\u2081 q\u2082\n    in <+S> L c ih ih\u2081) ,\n    <+S> L c\u2081 (<\u2219S> L b c\u2081 p q\u2081) (<\u2219S> L b\u2081 c\u2081 p\u2081 q\u2083)\n  <\u2219S> (B a a\u2081) L L {Col x x\u2081} {Col y y\u2081} {One x\u2082} {One x\u2083} (p , p\u2081) q =\n    <\u2219S> a L L p q ,\n    <\u2219S> a\u2081 L L p\u2081 q\n  <\u2219S> (B a a\u2081) L (B c c\u2081) {Col x x\u2081} {Col y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <\u2219S> a L c p q ,\n    <\u2219S> a L c\u2081 p q\u2081  ,\n    <\u2219S> a\u2081 L c p\u2081 q ,\n    <\u2219S> a\u2081 L c\u2081 p\u2081 q\u2081\n  <\u2219S> (B a a\u2081) (B b b\u2081) L {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Col u u\u2081} {Col v v\u2081} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081) =\n    <+S> a L (<\u2219S> a b L p q) (<\u2219S> a b\u2081 L p\u2081 q\u2081) ,\n    <+S> a\u2081 L (<\u2219S> a\u2081 b L p\u2082 q) (<\u2219S> a\u2081 b\u2081 L p\u2083 q\u2081 )\n  <\u2219S> (B a a\u2081) (B b b\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> a c (<\u2219S> a b c p q) (<\u2219S> a b\u2081 c p\u2081 q\u2082) ,\n    <+S> a c\u2081 (<\u2219S> a b c\u2081 p q\u2081) (<\u2219S> a b\u2081 c\u2081 p\u2081 q\u2083) ,\n    <+S> a\u2081 c (<\u2219S> a\u2081 b c p\u2082 q) (<\u2219S> a\u2081 b\u2081 c p\u2083 q\u2082) ,\n    <+S> a\u2081 c\u2081 (<\u2219S> a\u2081 b c\u2081 p\u2082 q\u2081) (<\u2219S> a\u2081 b\u2081 c\u2081 p\u2083 q\u2083)\n\n  idemS : (r c : Shape) (x : M s r c) \u2192 x +S x \u2243S' x\n  idemS L L (One x) = idem x\n  idemS L (B c c\u2081) (Row x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) L (Col x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (idemS _ _ x) , (idemS _ _ x\u2081 , (idemS _ _ x\u2082  , idemS _ _ x\u2083))\n\n  distlS : (a b c : Shape) (x : M s a b) (y z : M s b c) \u2192\n    (x \u2219S (y +S z)) \u2243S' ((x \u2219S y) +S (x \u2219S z))\n  distlS L L L (One x) (One y) (One z) = distl x y z\n  distlS L L (B c c\u2081) (One x) (Row y y\u2081) (Row z z\u2081) =\n    distlS L L c (One x) y z ,\n    distlS L L c\u2081 (One x) y\u2081 z\u2081\n  distlS L (B b b\u2081) L (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    {!!}\n  distlS L (B b b\u2081) (B c c\u2081) (Row x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    {!!} ,\n    {!!}\n  distlS (B a a\u2081) L L (Col x x\u2081) (One x\u2082) (One x\u2083) = {!!}\n  distlS (B a a\u2081) L (B c c\u2081) (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) = {!!}\n  distlS (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) (Col y y\u2081) (Col z z\u2081) = {!!}\n  distlS (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) = {!!}\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = <\u2219S> shape shape shape\n      ; idem = idemS shape shape\n      ; distl = {!!}\n      ; distr = {!!}\n      }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  setoidS : (r c : Shape) \u2192 Setoid _ _\n  setoidS r c =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = \u2243S r c\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S' 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning (setoidS a L)\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 L)\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS a c)\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a c\u2081)\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c)\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c\u2081)\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S' 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) , (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02b3 L b L x\n      ih\u2081 = zeroS\u02b3 L b\u2081 L x\u2081\n    in begin\n      Row x x\u2081 \u2219S Col (0S b L) (0S b\u2081 L)\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (x \u2219S 0S b L) +S (x\u2081 \u2219S 0S b\u2081 L)\n    \u2248\u27e8 <+S> L L {x \u2219S 0S b L} {0S L L} {x\u2081 \u2219S 0S b\u2081 L} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02b3 L b c x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n      0S L c +S 0S L c\n    \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n      0S L c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02b3 L b c\u2081 x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 (0S L c\u2081) \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) = zeroS\u02b3 a L L x , zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b L x\n      ih\u2081 = zeroS\u02b3 a b\u2081 L x\u2081\n    in begin\n      x \u2219S 0S b L +S x\u2081 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L (0S _ _) \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b L x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 L x\u2083\n    in begin\n      x\u2082 \u2219S 0S b L +S x\u2083 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 L (0S _ _) \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c (0S _ _) \u27e9\n      0S _ _\n    \u220e) ,\n    (let\n      -- open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c\u2081 x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c\u2081 x\u2081\n    in transS a c\u2081 (<+S> a c\u2081 ih ih\u2081) (identS\u02e1 a c\u2081 (0S _ _))\n    -- begin\n    --   x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    -- \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n    --   0S _ _ +S 0S _ _\n    -- \u2248\u27e8 identS\u02e1 a c\u2081 (0S _ _) \u27e9\n    --   0S a c\u2081\n    -- \u220e\n    ) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c +S x\u2083 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c (0S _ _) \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c\u2081 x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c\u2081 +S x\u2083 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 (0S _ _) \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  <\u2219S> : (a b c : Shape) {x y : M s a b} {u v : M s b c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x \u2219S u) \u2243S' (y \u2219S v)\n  <\u2219S> L L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <\u2219> q\n  <\u2219S> L L (B c c\u2081) {One x} {One x\u2081} {Row u u\u2081} {Row v v\u2081} p (q , q\u2081) =\n    (<\u2219S> L L c {One x} {One x\u2081} {u} {v} p q) ,\n    <\u2219S> L L c\u2081 {One x} {One x\u2081} {u\u2081} {v\u2081} p q\u2081\n  <\u2219S> L (B b b\u2081) L {Row x x\u2081} {Row y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    -- Row x x\u2081 \u2219S Col u u\u2081 \u2243S' Row y y\u2081 \u2219S Col v v\u2081\n    let\n      open EqReasoning (setoidS _ _)\n      ih = <\u2219S> _ _ _ {x} {y} {u} {v} p q\n      ih\u2081 = <\u2219S> _ _ _ {x\u2081} {y\u2081} {u\u2081} {v\u2081} p\u2081 q\u2081\n    in begin\n      Row x x\u2081 \u2219S Col u u\u2081\n    \u2261\u27e8 refl-\u2261 \u27e9\n      x \u2219S u +S x\u2081 \u2219S u\u2081\n    \u2248\u27e8 <+S> L L {x \u2219S u} {y \u2219S v} {x\u2081 \u2219S u\u2081} {y\u2081 \u2219S v\u2081} ih ih\u2081 \u27e9\n      y \u2219S v +S y\u2081 \u2219S v\u2081\n    \u220e\n  <\u2219S> L (B b b\u2081) (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081) (q , q\u2081 , q\u2082 , q\u2083) =\n    {!!} ,\n    {!!}\n  <\u2219S> (B a a\u2081) L L {Col x x\u2081} {Col y y\u2081} {One x\u2082} {One x\u2083} (p , p\u2081) q =\n    {!!} ,\n    {!!}\n  <\u2219S> (B a a\u2081) L (B c c\u2081) {Col x x\u2081} {Col y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    {!!} ,\n    {!!} ,\n    {!!} ,\n    {!!}\n  <\u2219S> (B a a\u2081) (B b b\u2081) L {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Col u u\u2081} {Col v v\u2081} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081) =\n    {!!} ,\n    {!!}\n  <\u2219S> (B a a\u2081) (B b b\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    {!!}\n\n  idemS : (r c : Shape) (x : M s r c) \u2192 x +S x \u2243S' x\n  idemS L L (One x) = idem x\n  idemS L (B c c\u2081) (Row x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) L (Col x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (idemS _ _ x) , (idemS _ _ x\u2081 , (idemS _ _ x\u2082  , idemS _ _ x\u2083))\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = <\u2219S> shape shape shape\n      ; idem = idemS shape shape\n      ; distl = {!!}\n      ; distr = {!!}\n      }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8c441c2498ec2d52054d910e48b8ba0d0e32b1e8","subject":"Fixed iterate-Stream.","message":"Fixed iterate-Stream.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/MapIterate\/MapIterateI.agda","new_file":"src\/fot\/FOTC\/Program\/MapIterate\/MapIterateI.agda","new_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using co-induction\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The map-iterate property (Gibbons and Hutton, 2005):\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\nmodule FOTC.Program.MapIterate.MapIterateI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\nopen import FOTC.Relation.Binary.Bisimilarity.Type\n\n------------------------------------------------------------------------------\n\nunfoldMapIterate : \u2200 f x \u2192\n                   map f (iterate f x) \u2261 f \u00b7 x \u2237 map f (iterate f (f \u00b7 x))\nunfoldMapIterate f x =\n  map f (iterate f x)               \u2261\u27e8 mapCong\u2082 (iterate-eq f x) \u27e9\n  map f (x \u2237 iterate f (f \u00b7 x))     \u2261\u27e8 map-\u2237 f x (iterate f (f \u00b7 x)) \u27e9\n  f \u00b7 x \u2237 map f (iterate f (f \u00b7 x)) \u220e\n\nmap-iterate-Stream : \u2200 f x \u2192 Stream (map f (iterate f x))\nmap-iterate-Stream f x = Stream-coind A h (x , refl)\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ y ] xs \u2261 map f (iterate f y)\n\n  h : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h (y , prf) = f \u00b7 y\n                , map f (iterate f (f \u00b7 y))\n                , trans prf (unfoldMapIterate f y)\n                , f \u00b7 y , refl\n\niterate-Stream : \u2200 f x \u2192 Stream (iterate f (f \u00b7 x))\niterate-Stream f x = Stream-coind A h (x , refl)\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ y ] xs \u2261 iterate f (f \u00b7 y)\n\n  h : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h (y , prf) = f \u00b7 y\n                , iterate f (f \u00b7 (f \u00b7 y))\n                , trans prf (iterate-eq f (f \u00b7 y))\n                , f \u00b7 y , refl\n\n-- The map-iterate property.\n\u2248-map-iterate : \u2200 f x \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-coind B h (x , refl , refl)\n  where\n  -- Based on the relation used by (Gim\u00e9nez and Cast\u00e9ran, 2007).\n  B : D \u2192 D \u2192 Set\n  B xs ys = \u2203[ y ] xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y)\n\n  h : \u2200 {xs} {ys} \u2192 B xs ys \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 B xs' ys'\n  h (y , prf\u2081 , prf\u2082) =\n      f \u00b7 y\n    , map f (iterate f (f \u00b7 y))\n    , iterate f (f \u00b7 (f \u00b7 y))\n    , trans prf\u2081 (unfoldMapIterate f y)\n    , trans prf\u2082 (iterate-eq f (f \u00b7 y))\n    , ((f \u00b7 y) , refl , refl)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Gim\u00e9nez, Eduardo and Caster\u00e1n, Pierre (2007). A Tutorial on\n-- [Co-]Inductive Types in Coq.\n--\n-- Gibbons, Jeremy and Hutton, Graham (2005). Proof Methods for\n-- Corecursive Programs. In: Fundamenta Informaticae XX, pp. 1\u201314.\n","old_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using co-induction\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The map-iterate property (Gibbons and Hutton, 2005):\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\nmodule FOTC.Program.MapIterate.MapIterateI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\nopen import FOTC.Relation.Binary.Bisimilarity.Type\n\n------------------------------------------------------------------------------\n\nunfoldMapIterate : \u2200 f x \u2192\n                   map f (iterate f x) \u2261 f \u00b7 x \u2237 map f (iterate f (f \u00b7 x))\nunfoldMapIterate f x =\n  map f (iterate f x)               \u2261\u27e8 mapCong\u2082 (iterate-eq f x) \u27e9\n  map f (x \u2237 iterate f (f \u00b7 x))     \u2261\u27e8 map-\u2237 f x (iterate f (f \u00b7 x)) \u27e9\n  f \u00b7 x \u2237 map f (iterate f (f \u00b7 x)) \u220e\n\nmap-iterate-Stream : \u2200 f x \u2192 Stream (map f (iterate f x))\nmap-iterate-Stream f x = Stream-coind A h (x , refl)\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ y ] xs \u2261 map f (iterate f y)\n\n  h : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h (y , prf) = f \u00b7 y\n                , map f (iterate f (f \u00b7 y))\n                , (trans prf ((unfoldMapIterate f y)))\n                , f \u00b7 y , refl\n\n-- TODO (23 December 2013).\n-- map-iterate-Stream\u2082 : \u2200 f x \u2192 Stream (iterate f (f \u00b7 x))\n\n-- The map-iterate property.\n\u2248-map-iterate : \u2200 f x \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-coind B h (x , refl , refl)\n  where\n  -- Based on the relation used by (Gim\u00e9nez and Cast\u00e9ran, 2007).\n  B : D \u2192 D \u2192 Set\n  B xs ys = \u2203[ y ] xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y)\n\n  h : \u2200 {xs} {ys} \u2192 B xs ys \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 B xs' ys'\n  h (y , prf\u2081 , prf\u2082) =\n      f \u00b7 y\n    , map f (iterate f (f \u00b7 y))\n    , iterate f (f \u00b7 (f \u00b7 y))\n    , trans prf\u2081 (unfoldMapIterate f y)\n    , trans prf\u2082 (iterate-eq f (f \u00b7 y))\n    , ((f \u00b7 y) , refl , refl)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Gim\u00e9nez, Eduardo and Caster\u00e1n, Pierre (2007). A Tutorial on\n-- [Co-]Inductive Types in Coq.\n--\n-- Gibbons, Jeremy and Hutton, Graham (2005). Proof Methods for\n-- Corecursive Programs. In: Fundamenta Informaticae XX, pp. 1\u201314.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"347a67ed51a8af9f918731c0f060d6e580fe7cf3","subject":"Success on Nehemiah.Syntax.Term","message":"Success on Nehemiah.Syntax.Term\n","repos":"inc-lc\/ilc-agda","old_file":"Nehemiah\/Syntax\/Term.agda","new_file":"Nehemiah\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Syntax.Term where\n\nopen import Nehemiah.Syntax.Type\n\nopen import Data.Integer\n\nimport Parametric.Syntax.Term Base as Term\n\ndata Const : Term.Structure where\n  intlit-const : (n : \u2124) \u2192 Const \u2205 int\n  add-const : Const (int \u2022 int \u2022 \u2205) int\n  minus-const : Const (int \u2022 \u2205) (int)\n\n  empty-const : Const \u2205 (bag)\n  insert-const : Const (int \u2022 bag \u2022 \u2205) (bag)\n  union-const : Const (bag \u2022 bag \u2022 \u2205) (bag)\n  negate-const : Const (bag \u2022 \u2205) (bag)\n\n  flatmap-const : Const ((int \u21d2 bag) \u2022 bag \u2022 \u2205) (bag)\n  sum-const : Const (bag \u2022 \u2205) (int)\n\n\nopen Term.Structure Const public\n\n-- Import Base.Data.DependentList again here, as part of the workaround for\n-- agda\/agda#1985; see Parametric.Syntax.Term for info.\n--\n-- XXX This import is needed to define the patterns in the right scope; if we\n-- don't, \u2022 gets bound to a different occurrence, and we notice that during\n-- typechecking, which happens at use site.\nopen import Base.Data.DependentList public\n\n-- Shorthands of constants\n\npattern intlit n = const (intlit-const n) \u2205\npattern add s t = const add-const (s \u2022 t \u2022 \u2205)\npattern minus t = const minus-const (t \u2022 \u2205)\npattern empty = const empty-const \u2205\npattern insert s t = const insert-const (s \u2022 t \u2022 \u2205)\npattern union s t = const union-const (s \u2022 t \u2022 \u2205)\npattern negate t = const negate-const (t \u2022 \u2205)\npattern flatmap s t = const flatmap-const (s \u2022 t \u2022 \u2205)\npattern sum t = const sum-const (t \u2022 \u2205)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Syntax.Term where\n\nopen import Nehemiah.Syntax.Type\n\nopen import Data.Integer\n\nimport Parametric.Syntax.Term Base as Term\n\ndata Const : Term.Structure where\n  intlit-const : (n : \u2124) \u2192 Const \u2205 int\n  add-const : Const (int \u2022 int \u2022 \u2205) int\n  minus-const : Const (int \u2022 \u2205) (int)\n\n  empty-const : Const \u2205 (bag)\n  insert-const : Const (int \u2022 bag \u2022 \u2205) (bag)\n  union-const : Const (bag \u2022 bag \u2022 \u2205) (bag)\n  negate-const : Const (bag \u2022 \u2205) (bag)\n\n  flatmap-const : Const ((int \u21d2 bag) \u2022 bag \u2022 \u2205) (bag)\n  sum-const : Const (bag \u2022 \u2205) (int)\n\n\n--open Term.Structure --works\nopen Term.Structure Const --fails, even with the rest disabled.\n{-\nAn internal error has occurred. Please report this as a bug.\nLocation of the error: src\/full\/Agda\/TypeChecking\/Substitute.hs:183\n-}\n\n{-\n-- Shorthands of constants\n\npattern intlit n = const (intlit-const n) \u2205\npattern add s t = const add-const (s \u2022 t \u2022 \u2205)\npattern minus t = const minus-const (t \u2022 \u2205)\npattern empty = const empty-const \u2205\npattern insert s t = const insert-const (s \u2022 t \u2022 \u2205)\npattern union s t = const union-const (s \u2022 t \u2022 \u2205)\npattern negate t = const negate-const (t \u2022 \u2205)\npattern flatmap s t = const flatmap-const (s \u2022 t \u2022 \u2205)\npattern sum t = const sum-const (t \u2022 \u2205)\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c94ea256c17bed12b70ebbdf415c500dd2705e28","subject":"Minor changes.","message":"Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-unf-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e.\n  --\n  -- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n  Conat-coind :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial predicate A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification\/proof for\n  -- this principle.\n  Conat-coind-stronger :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-unf and Conat-unf-ho are equivalents\n\nConat-unf' : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf' = Conat-unf-ho\n\nConat-unf-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-unf-ho' = Conat-unf\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind' = Conat-coind-ho\n\nConat-coind-ho' :\n  (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-coind-stronger to Conat-coind-stronger\/Conat-coind\n\nConat-coind'' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-coind-stronger A h An\n\n-- 22 December 2013: We cannot prove Conat-coind-stronger using\n-- Conat-coind.\nConat-coind-stronger'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-coind-stronger'' A h An = Conat-coind A {!!} An\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e.\n--\n-- NatF Conat \u2264 Conat.\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in h = Conat-coind A h' h\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  h' : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h' (inj\u2081 n\u22610)              = inj\u2081 n\u22610\n  h' (inj\u2082 (n' , prf , Cn')) = inj\u2082 (n' , prf , Conat-unf Cn')\n\nConat-in-ho : \u2200 {n} \u2192 NatF Conat n \u2192 Conat n\nConat-in-ho = Conat-in\n\n-- A different definition.\nConat-in' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n            \u2200 {n} \u2192 Conat n\nConat-in' h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\nConat-in-ho' : (\u2200 {n} \u2192 NatF Conat n) \u2192 \u2200 {n} \u2192 Conat n\nConat-in-ho' = Conat-in'\n","old_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-unf-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e.\n  --\n  -- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n  Conat-coind :\n    \u2200 (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    \u2200 (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification for this\n  -- principle.\n  Conat-coind-stronger :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-unf and Conat-unf-ho are equivalents\n\nConat-unf' : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf' = Conat-unf-ho\n\nConat-unf-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-unf-ho' = Conat-unf\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind' :\n  \u2200 (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind' = Conat-coind-ho\n\nConat-coind-ho' :\n  \u2200 (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-coind-stronger to Conat-coind-stronger\/Conat-coind\n\nConat-coind'' :\n  \u2200 (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-coind-stronger A h An\n\n-- 22 December 2013: We cannot prove Conat-coind-stronger using\n-- Conat-coind.\nConat-coind-stronger'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-coind-stronger'' A h An = Conat-coind A {!!} An\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e.\n--\n-- NatF Conat \u2264 Conat.\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in h = Conat-coind A h' h\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  h' : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h' (inj\u2081 n\u22610)              = inj\u2081 n\u22610\n  h' (inj\u2082 (n' , prf , Cn')) = inj\u2082 (n' , prf , Conat-unf Cn')\n\nConat-in-ho : \u2200 {n} \u2192 NatF Conat n \u2192 Conat n\nConat-in-ho = Conat-in\n\n-- A different definition.\nConat-in' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n            \u2200 {n} \u2192 Conat n\nConat-in' h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\nConat-in-ho' : (\u2200 {n} \u2192 NatF Conat n) \u2192 \u2200 {n} \u2192 Conat n\nConat-in-ho' = Conat-in'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1c48ce021e56df25a7ff43d8a72ede5b311b39f6","subject":"Minor changes.","message":"Minor changes.\n\nIgnore-this: ef30ed12e72587ba03034ce04f2cfc3c\n\ndarcs-hash:20110805170456-3bd4e-69989ba5a71269fe59cf5f6b2bc15a85ca16fadc.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/List\/PostulatesVersusDataTypes.agda","new_file":"Draft\/FOTC\/Data\/List\/PostulatesVersusDataTypes.agda","new_contents":"module Draft.FOTC.Data.List.PostulatesVersusDataTypes where\n\n-- See Agda mailing list.\n-- Subject: Agda's unification: postulates versus data types\n\nmodule M\u2081 where\n  data D : Set where\n    _\u2237_ : D \u2192 D \u2192 D\n\n  data List : D \u2192 Set where\n    cons : \u2200 x xs \u2192 List xs \u2192 List (x \u2237 xs)\n\n  tail : \u2200 {x xs} \u2192 List (x \u2237 xs) \u2192 List xs\n  tail {x} {xs} (cons .x .xs xsL) = xsL\n\nmodule M\u2082 where\n  postulate\n    D   : Set\n    _\u2237_ : D \u2192 D \u2192 D\n\n  data List : D \u2192 Set where\n    cons : \u2200 x xs \u2192 List xs \u2192 List (x \u2237 xs)\n\n  tail : \u2200 {x xs} \u2192 List (x \u2237 xs) \u2192 List xs\n  tail l = {!!}  -- C-c C-c fails\n\nmodule M\u2083 where\n  postulate D : Set\n\n  _\u2237_ : D \u2192 D \u2192 D\n  x \u2237 xs = {!!}\n\n  data List : D \u2192 Set where\n    cons : \u2200 x xs \u2192 List xs \u2192 List (x \u2237 xs)\n\n  tail : \u2200 {x xs} \u2192 List (x \u2237 xs) \u2192 List xs\n  tail l = {!!}  -- C-c C-c fails\n","old_contents":"module Draft.FOTC.Data.List.PostulatesVersusDataTypes where\n\ndata U : Set where\n  _::_ : U \u2192 U \u2192 U\n\ndata ListU : U \u2192 Set where\n  cons : \u2200 x xs \u2192 ListU xs \u2192 ListU (x :: xs)\n\ntailU : \u2200 {x xs} \u2192 ListU (x :: xs) \u2192 ListU xs\ntailU {x} {xs} (cons .x .xs xsL) = xsL\n\npostulate\n  D   : Set\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata List : D \u2192 Set where\n  cons : \u2200 x xs \u2192 List xs \u2192 List (x \u2237 xs)\n\ntail : \u2200 {x xs} \u2192 List (x \u2237 xs) \u2192 List xs\ntail l = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9f817edc266f0802e386fb96bb430f34dc7fe8e4","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 5f699f5008e64b7007bf8c7d75ecaf1b\n\ndarcs-hash:20120519150052-3bd4e-b7efba30df6ebeca5137647cff8f0bcd11e4c169.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Function.agda","new_file":"src\/Common\/Function.agda","new_contents":"------------------------------------------------------------------------------\n-- Operations on and with functions\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Common.Function where\n\n-- The same fixity used in the standard library.\ninfixr 0 _$_\n\n------------------------------------------------------------------------------\n-- The right associative application operator.\n--\n-- N.B. The operator is not first-order, so it cannot be used with\n-- types\/terms which will be translated to FOL.\n_$_ : {A : Set}{B : A \u2192 Set} \u2192 ((x : A) \u2192 B x) \u2192 (x : A) \u2192 B x\nf $ x = f x\n\n-- N.B. The functions is not first-order, so it cannot be used with\n-- types\/terms which will be translated to FOL.\nflip : {A : Set} \u2192 (A \u2192 A \u2192 A) \u2192 A \u2192 A \u2192 A\nflip f y x = f x y\n","old_contents":"------------------------------------------------------------------------------\n-- Operations on and with functions\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Common.Function where\n\n-- The same fixity used in the standard library.\ninfixr 0 _$_\n\n------------------------------------------------------------------------------\n-- The right associative application operator.\n-- N.B. It cannot be used with types\/terms which will be translated to FOL.\n_$_ : {A : Set}{B : A \u2192 Set} \u2192 ((x : A) \u2192 B x) \u2192 (x : A) \u2192 B x\nf $ x = f x\n\n-- N.B. It cannot be used with types\/terms which will be translated to FOL.\nflip : {A : Set} \u2192 (A \u2192 A \u2192 A) \u2192 A \u2192 A \u2192 A\nflip f y x = f x y\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9755ab9f86f1e00b28c36e99421e4e9be389a95f","subject":"Comment out holes to ensure Everything compiles again","message":"Comment out holes to ensure Everything compiles again\n\nOld-commit-hash: dfc28a00f9301015066e19d4be2c4179f8e0849b\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Products.agda","new_file":"Base\/Change\/Products.agda","new_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  -- The following is probably bullshit:\n  -- Does not handle products of functions - more accurately, writing the\n  -- derivative of fst and snd for products of functions is hard: fst' p dp must return the change of fst p\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  -- We should do the same for uncurry instead.\n\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032 a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of its own validity, and because after application it does not contain a proof\n\n  -- As a consequence, proving that's a derivative seems too insanely hard. We\n  -- might want to provide a proof schema for curried functions at once,\n  -- starting from the right correctness equation.\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n\n  _,_\u2032-real : \u0394 _,_\n  _,_\u2032-real = nil _,_\n  _,_\u2032-real-Derivative : Derivative {{CA}} {{B\u2192A\u00d7B}} _,_ (\u0394A\u2192B\u2192A\u00d7B.apply _,_\u2032-real)\n  _,_\u2032-real-Derivative =\n    FunctionChanges.nil-is-derivative A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}} _,_\n\n  _,_\u2032\u2032 :  (a : A) \u2192 \u0394 a \u2192\n      \u0394 {{B\u2192A\u00d7B}} (\u03bb b \u2192 (a , b))\n  _,_\u2032\u2032 a da = record\n    { apply = _,_\u2032 a da\n    ; correct = \u03bb b db \u2192\n      begin\n        (a , b \u229e db) \u229e (_,_\u2032 a da) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n        a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9\n        (a , b) \u229e (_,_\u2032 a da) b db\n      \u220e\n    }\n{-\n  _,_\u2032\u2032\u2032 : \u0394 {{A\u2192B\u2192A\u00d7B}} _,_\n  _,_\u2032\u2032\u2032 = record\n    { apply = _,_\u2032\u2032\n    ; correct = \u03bb a da \u2192\n      begin\n        update\n        (_,_ (a \u229e da))\n        (_,_\u2032\u2032 (a \u229e da) (nil (a \u229e da)))\n      \u2261\u27e8 {!!} \u27e9\n        update (_,_ a) (_,_\u2032\u2032 a da)\n      \u220e\n    }\n    where\n      -- This is needed to use update above.\n      -- Passing the change structure seems hard with the given operators; maybe I'm just using them wrongly.\n      open ChangeAlgebra B\u2192A\u00d7B hiding (nil)\n{-\n  {!\n      begin\n        (_,_ (a \u229e da)) \u229e _,_\u2032\u2032 (a \u229e da) (nil (a \u229e da))\n      {- \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db)) -}\n      \u2261\u27e8 ? \u27e9\n        {-a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9-}\n       (_,_ a) \u229e (_,_\u2032\u2032 a da)\n      \u220e!}\n    }\n-}\n  open import Postulate.Extensionality\n\n\n  _,_\u2032Derivative :\n    Derivative {{CA}} {{B\u2192A\u00d7B}} _,_  _,_\u2032\u2032\n  _,_\u2032Derivative a da =\n    begin\n      _\u229e_ {{B\u2192A\u00d7B}} (_,_ a) (_,_\u2032\u2032 a da)\n    \u2261\u27e8\u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    --ext (\u03bb b \u2192 cong (\u03bb \u25a1 \u2192  (a , b) \u229e \u25a1) (update-nil {{?}} b))\n    \u2261\u27e8 {!!} \u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    \u2261\u27e8 sym {!\u0394A\u2192B\u2192A\u00d7B.incrementalization _,_ _,_\u2032\u2032\u2032 a da!} \u27e9\n  --FunctionChanges.incrementalization A (B \u2192 A \u00d7 B) {{CA}} {{{!B\u2192A\u00d7B!}}} _,_ {!!} {!!} {!!}\n       _,_ (a \u229e da)\n    \u220e\n-}\n","old_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  -- The following is probably bullshit:\n  -- Does not handle products of functions - more accurately, writing the\n  -- derivative of fst and snd for products of functions is hard: fst' p dp must return the change of fst p\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  -- We should do the same for uncurry instead.\n\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032 a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of its own validity, and because after application it does not contain a proof\n\n  -- As a consequence, proving that's a derivative seems too insanely hard. We\n  -- might want to provide a proof schema for curried functions at once,\n  -- starting from the right correctness equation.\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n\n  _,_\u2032-real : \u0394 _,_\n  _,_\u2032-real = nil _,_\n  _,_\u2032-real-Derivative : Derivative {{CA}} {{B\u2192A\u00d7B}} _,_ (\u0394A\u2192B\u2192A\u00d7B.apply _,_\u2032-real)\n  _,_\u2032-real-Derivative =\n    FunctionChanges.nil-is-derivative A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}} _,_\n\n  _,_\u2032\u2032 :  (a : A) \u2192 \u0394 a \u2192\n      \u0394 {{B\u2192A\u00d7B}} (\u03bb b \u2192 (a , b))\n  _,_\u2032\u2032 a da = record\n    { apply = _,_\u2032 a da\n    ; correct = \u03bb b db \u2192\n      begin\n        (a , b \u229e db) \u229e (_,_\u2032 a da) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n        a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9\n        (a , b) \u229e (_,_\u2032 a da) b db\n      \u220e\n    }\n\n  _,_\u2032\u2032\u2032 : \u0394 {{A\u2192B\u2192A\u00d7B}} _,_\n  _,_\u2032\u2032\u2032 = record\n    { apply = _,_\u2032\u2032\n    ; correct = \u03bb a da \u2192\n      begin\n        update\n        (_,_ (a \u229e da))\n        (_,_\u2032\u2032 (a \u229e da) (nil (a \u229e da)))\n      \u2261\u27e8 {!!} \u27e9\n        update (_,_ a) (_,_\u2032\u2032 a da)\n      \u220e\n    }\n    where\n      -- This is needed to use update above.\n      -- Passing the change structure seems hard with the given operators; maybe I'm just using them wrongly.\n      open ChangeAlgebra B\u2192A\u00d7B hiding (nil)\n{-\n  {!\n      begin\n        (_,_ (a \u229e da)) \u229e _,_\u2032\u2032 (a \u229e da) (nil (a \u229e da))\n      {- \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db)) -}\n      \u2261\u27e8 ? \u27e9\n        {-a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9-}\n       (_,_ a) \u229e (_,_\u2032\u2032 a da)\n      \u220e!}\n    }\n-}\n  open import Postulate.Extensionality\n\n  _,_\u2032Derivative :\n    Derivative {{CA}} {{B\u2192A\u00d7B}} _,_  _,_\u2032\u2032\n  _,_\u2032Derivative a da =\n    begin\n      _\u229e_ {{B\u2192A\u00d7B}} (_,_ a) (_,_\u2032\u2032 a da)\n    \u2261\u27e8\u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    --ext (\u03bb b \u2192 cong (\u03bb \u25a1 \u2192  (a , b) \u229e \u25a1) (update-nil {{?}} b))\n    \u2261\u27e8 {!!} \u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    \u2261\u27e8 sym {!\u0394A\u2192B\u2192A\u00d7B.incrementalization _,_ _,_\u2032\u2032\u2032 a da!} \u27e9\n  --FunctionChanges.incrementalization A (B \u2192 A \u00d7 B) {{CA}} {{{!B\u2192A\u00d7B!}}} _,_ {!!} {!!} {!!}\n       _,_ (a \u229e da)\n    \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ee41ad5dcc5d4a31d83ce3eb986b4f2f5a9ee268","subject":"lift the left zero of matrix multiplication","message":"lift the left zero of matrix multiplication\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  setoidS : (r c : Shape) \u2192 Setoid _ _\n  setoidS r c =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = \u2243S r c\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S' 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9 -- TODO: make nicer?\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning (setoidS a L)\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 L)\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS a c)\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a c\u2081)\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c)\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c\u2081)\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = {!!}\n      ; _<\u2219>_ = {!!}\n      }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  lift<+> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  lift<+> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  lift<+> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    lift<+> L c p q , lift<+> L c\u2081 p\u2081 q\u2081\n  lift<+> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    lift<+> r L p q , lift<+> r\u2081 L p\u2081 q\u2081\n  lift<+> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    lift<+> r c p q , lift<+> r c\u2081 p\u2081 q\u2081 ,\n    lift<+> r\u2081 c p\u2082 q\u2082 , lift<+> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = lift<+> shape shape }\n\n\n  liftIdent\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  liftIdent\u02e1 L L (One x) = identity\u02e1\u209b x\n  liftIdent\u02e1 L (B c c\u2081) (Row x x\u2081) = liftIdent\u02e1 L c x , liftIdent\u02e1 L c\u2081 x\u2081\n  liftIdent\u02e1 (B r r\u2081) L (Col x x\u2081) = liftIdent\u02e1 r L x , liftIdent\u02e1 r\u2081 L x\u2081\n  liftIdent\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    liftIdent\u02e1 r c x , liftIdent\u02e1 r c\u2081 x\u2081 ,\n    liftIdent\u02e1 r\u2081 c x\u2082 , liftIdent\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = liftIdent\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  -- TODO: can I use \u2243S to 'rewrite' types?\n  liftZero\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    let 0\u02e1 = 0S a b\n        0\u02b3 = 0S a c\n    in (0\u02e1 \u2219S x) \u2243S' 0\u02b3\n  liftZero\u02e1 L L L (One x) = zero\u02e1 x\n  liftZero\u02e1 L L (B c c\u2081) (Row x x\u2081) = (liftZero\u02e1 L L c x) , (liftZero\u02e1 L L c\u2081 x\u2081)\n  liftZero\u02e1 L (B b b\u2081) L (Col x x\u2081) = {!!}\n  liftZero\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) = {!!} , {!!}\n  liftZero\u02e1 (B a a\u2081) L L (One x) = liftZero\u02e1 a L L (One x) , liftZero\u02e1 a\u2081 L L (One x)\n  liftZero\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    liftZero\u02e1 a L c x , liftZero\u02e1 a L c\u2081 x\u2081 ,\n    liftZero\u02e1 a\u2081 L c x , liftZero\u02e1 a\u2081 L c\u2081 x\u2081\n  liftZero\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) = {!!} , {!!}\n  liftZero\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    {!liftZero\u02e1 a b c x!} , ({!!} , ({!!} , {!!}))\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = liftZero\u02e1 shape shape shape\n      ; zero\u02b3 = {!!}\n      ; _<\u2219>_ = {!!}\n      }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"07b3b64ff9795f6941b64231e7b2a0dd630e0de4","subject":"Update the Examples","message":"Update the Examples\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Examples.agda","new_file":"lib\/Explore\/Examples.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.Examples where\n\nopen import Type\nopen import Level.NP\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Data.Sum.NP\nopen import HoTT using (UA)\nopen import Function.NP\nopen import Function.Extensionality using (FunExt)\nopen import Relation.Binary.PropositionalEquality hiding ([_])\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\nopen import Explore.Monad {\u2080} \u2080 public renaming (map to map-explore)\nopen import Explore.Two\nopen import Explore.Product\nopen Explore.Product.Operators\n\nmodule E where\n  open FromExplore public\n\nmodule M {Msg Digest : \u2605}\n         (_==_ : Digest \u2192 Digest \u2192 \ud835\udfda)\n         (H : Msg \u2192 Digest)\n         (exploreMsg : \u2200 {\u2113} \u2192 Explore \u2113 Msg)\n         (d : Digest) where\n\n  module V1 where\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = exploreMsg [] _++_ (\u03bb m \u2192 [0: [] 1: [ m ] ] (H m == d))\n\n  module ExploreMsg = FromExplore {A = Msg} exploreMsg\n\n  module V2 where\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = ExploreMsg.findKey (\u03bb m \u2192 H m == d)\n\n  module V3 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 \u03b5 _\u2295_ f = exploreMsg \u03b5 _\u2295_ (\u03bb m \u2192 [0: \u03b5 1: f m ] (H m == d))\n\n  module V4 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 = exploreMsg >>= \u03bb m \u2192 [0: empty-explore 1: point-explore m ] (H m == d)\n\n  module V5 where\n\n    explore-H\u207b\u00b9 : \u2200 {\u2113} \u2192 Explore \u2113 Msg\n    explore-H\u207b\u00b9 = filter-explore (\u03bb m \u2192 H m == d) exploreMsg\n\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = E.list explore-H\u207b\u00b9\n\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = E.first explore-H\u207b\u00b9\n\n  module V6 where\n    explore-H\u207b\u00b9 : \u2200 {\u2113} \u2192 Explore \u2113 Msg\n    explore-H\u207b\u00b9 = explore-endo (filter-explore (\u03bb m \u2192 H m == d) exploreMsg)\n\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = E.list explore-H\u207b\u00b9\n\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = E.first explore-H\u207b\u00b9\n\n    last-H\u207b\u00b9 : Maybe Msg\n    last-H\u207b\u00b9 = E.last explore-H\u207b\u00b9\n\nMsg = \ud835\udfda \u00d7 \ud835\udfda\nDigest = \ud835\udfda\n-- _==_ : Digest \u2192 Digest \u2192 \ud835\udfda\nH : Msg \u2192 Digest\nH (x , y) = x xor y\nMsg\u1d49 : \u2200 {\u2113} \u2192 Explore \u2113 Msg\nMsg\u1d49 = \ud835\udfda\u1d49 \u00d7\u1d49 \ud835\udfda\u1d49\nmodule N5 = M.V5 _==_ H Msg\u1d49\nmodule N6 = M.V6 _==_ H Msg\u1d49\ntest5 = N5.list-H\u207b\u00b9\ntest6-list : N6.list-H\u207b\u00b9 0\u2082 \u2261 (0\u2082 , 0\u2082) \u2237 (1\u2082 , 1\u2082) \u2237 []\ntest6-list = refl\ntest6-rev-list : E.list (E.backward (N6.explore-H\u207b\u00b9 0\u2082)) \u2261 (1\u2082 , 1\u2082) \u2237 (0\u2082 , 0\u2082) \u2237 []\ntest6-rev-list = refl\ntest6-first : N6.first-H\u207b\u00b9 0\u2082 \u2261 just (0\u2082 , 0\u2082)\ntest6-first = refl\ntest6-last : N6.last-H\u207b\u00b9 0\u2082 \u2261 just (1\u2082 , 1\u2082)\ntest6-last = refl\n-- -}\n\n\ud835\udfdb\u1d41 : U\n\ud835\udfdb\u1d41 = \ud835\udfd9\u1d41 \u228e\u1d41 \ud835\udfda\u1d41\n\nprop-\u2227-comm : \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\nprop-\u2227-comm (x , y) = x \u2227 y == y \u2227 x\n\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n  check-\u2227-comm : \u2200 x y \u2192 \u2713 (x \u2227 y == y \u2227 x)\n  check-\u2227-comm x y = check! (\ud835\udfda\u1d41 \u00d7\u1d41 \ud835\udfda\u1d41) prop-\u2227-comm (x , y)\n\nprop-\u2227-\u2228-distr : \ud835\udfda \u00d7 \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\nprop-\u2227-\u2228-distr (x , y , z) = x \u2227 (y \u2228 z) == x \u2227 y \u2228 x \u2227 z\n\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n  check-\u2227-\u2228-distr : \u2200 x y z \u2192 \u2713 (x \u2227 (y \u2228 z) == x \u2227 y \u2228 x \u2227 z)\n  check-\u2227-\u2228-distr x y z =\n    check! (\ud835\udfda\u1d41 \u00d7\u1d41 \ud835\udfda\u1d41 \u00d7\u1d41 \ud835\udfda\u1d41) prop-\u2227-\u2228-distr (x , y , z)\n\nlist22 = list (\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41)\nlist33 = list (\ud835\udfdb\u1d41 \u2192\u1d41 \ud835\udfdb\u1d41)\n\n{-\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n  module _ (f\u1d41 : El (\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41)) x where\n    f = \u2192\u1d41\u2192\u2192 \ud835\udfda\u1d41 \ud835\udfda\u1d41 f\u1d41\n    check22 : \u2713 (f x == f (f (f x)))\n    check22 = check! ((\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41) \u00d7\u1d41 \ud835\udfda\u1d41) (\u03bb { (f , x) \u2192 let f' = \u2192\u1d41\u2192\u2192 \ud835\udfda\u1d41 \ud835\udfda\u1d41 f in f' x == f' (f' (f' x)) }) {{!!}} ((f 0\u2082 , f 1\u2082) , x)\n  {-\n  check22 : \u2200 (f : \ud835\udfda \u2192 \ud835\udfda) x \u2192 \u2713 (f x == f (f (f x)))\n  check22 f x = let k = check! ((\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41) \u00d7\u1d41 \ud835\udfda\u1d41) (\u03bb { (f , x) \u2192 let f' = \u2192\u1d41\u2192\u2192 \ud835\udfda\u1d41 \ud835\udfda\u1d41 f in f' x == f' (f' (f' x)) }) {{!!}} ((f 0\u2082 , f 1\u2082) , x) in {!k!}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.Examples where\n\nopen import Type\nopen import Level.NP\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Two\nopen import Data.Product\nopen import Data.Sum.NP\nopen import Data.One\nopen import HoTT using (UA)\nopen import Function.Extensionality using (FunExt)\nopen import Relation.Binary.PropositionalEquality hiding ([_])\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Universe.Base\nopen import Explore.Monad {\u2080} \u2080 public renaming (map to map-explore)\nopen import Explore.Two\nopen import Explore.Product\nopen Explore.Product.Operators\n\nmodule E where\n  open FromExplore public\n\nmodule M {Msg Digest : \u2605}\n         (_==_ : Digest \u2192 Digest \u2192 \ud835\udfda)\n         (H : Msg \u2192 Digest)\n         (exploreMsg : \u2200 {\u2113} \u2192 Explore \u2113 Msg)\n         (d : Digest) where\n\n  module V1 where\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = exploreMsg [] _++_ (\u03bb m \u2192 [0: [] 1: [ m ] ] (H m == d))\n\n  module ExploreMsg = FromExplore {A = Msg} exploreMsg\n\n  module V2 where\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = ExploreMsg.findKey (\u03bb m \u2192 H m == d)\n\n  module V3 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 \u03b5 _\u2295_ f = exploreMsg \u03b5 _\u2295_ (\u03bb m \u2192 [0: \u03b5 1: f m ] (H m == d))\n\n  module V4 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 = exploreMsg >>= \u03bb m \u2192 [0: empty-explore 1: point-explore m ] (H m == d)\n\n  module V5 where\n\n    explore-H\u207b\u00b9 : \u2200 {\u2113} \u2192 Explore \u2113 Msg\n    explore-H\u207b\u00b9 = filter-explore (\u03bb m \u2192 H m == d) exploreMsg\n\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = E.list explore-H\u207b\u00b9\n\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = E.first explore-H\u207b\u00b9\n\n  module V6 where\n    explore-H\u207b\u00b9 : \u2200 {\u2113} \u2192 Explore \u2113 Msg\n    explore-H\u207b\u00b9 = explore-endo (filter-explore (\u03bb m \u2192 H m == d) exploreMsg)\n\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = E.list explore-H\u207b\u00b9\n\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = E.first explore-H\u207b\u00b9\n\n    last-H\u207b\u00b9 : Maybe Msg\n    last-H\u207b\u00b9 = E.last explore-H\u207b\u00b9\n\nMsg = \ud835\udfda \u00d7 \ud835\udfda\nDigest = \ud835\udfda\n-- _==_ : Digest \u2192 Digest \u2192 \ud835\udfda\nH : Msg \u2192 Digest\nH (x , y) = x xor y\nMsg\u1d49 : \u2200 {\u2113} \u2192 Explore \u2113 Msg\nMsg\u1d49 = \ud835\udfda\u1d49 \u00d7\u1d49 \ud835\udfda\u1d49\nmodule N5 = M.V5 _==_ H Msg\u1d49\nmodule N6 = M.V6 _==_ H Msg\u1d49\ntest5 = N5.list-H\u207b\u00b9\ntest6-list : N6.list-H\u207b\u00b9 0\u2082 \u2261 (0\u2082 , 0\u2082) \u2237 (1\u2082 , 1\u2082) \u2237 []\ntest6-list = refl\ntest6-rev-list : E.list (E.backward (N6.explore-H\u207b\u00b9 0\u2082)) \u2261 (1\u2082 , 1\u2082) \u2237 (0\u2082 , 0\u2082) \u2237 []\ntest6-rev-list = refl\ntest6-first : N6.first-H\u207b\u00b9 0\u2082 \u2261 just (0\u2082 , 0\u2082)\ntest6-first = refl\ntest6-last : N6.last-H\u207b\u00b9 0\u2082 \u2261 just (1\u2082 , 1\u2082)\ntest6-last = refl\n-- -}\n\n{-\n\ud835\udfdb\u1d41 : U\n\ud835\udfdb\u1d41 = \ud835\udfd9\u1d41 \u228e\u1d41 \ud835\udfda\u1d41\n\nlist22 = list (\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41)\nlist33 = list (\ud835\udfdb\u1d41 \u2192\u1d41 \ud835\udfdb\u1d41)\n-}\n\n{-\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n  check22 : \u2200 (f : \ud835\udfda \u2192 \ud835\udfda) x \u2192 \u2713 (f x == f (f (f x)))\n  check22 f x = {!check! ((\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41) \u00d7\u1d41 \ud835\udfda\u1d41) (\u03bb { (f , x) \u2192 let f' = \u2192\u1d41\u2192\u2192 \ud835\udfda\u1d41 \ud835\udfda\u1d41 f in f' x == f' (f' (f' x)) }) {{!!}} ((f 0\u2082 , f 1\u2082) , x)!}\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a2b2f80eb2dc1347647f543f266d07cf0d394055","subject":"Added subst\u2082 and subst\u2084.","message":"Added subst\u2082 and subst\u2084.\n\nIgnore-this: 1eeae450127ce8aab0562fe232b5adb6\n\ndarcs-hash:20101230033155-3bd4e-4bbb40106385aee572b763114a84201e41cb3489.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Relation\/Binary\/PropositionalEquality.agda","new_file":"src\/Common\/Relation\/Binary\/PropositionalEquality.agda","new_contents":"------------------------------------------------------------------------------\n-- Propositional equality\n------------------------------------------------------------------------------\n\nmodule Common.Relation.Binary.PropositionalEquality where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\nsubst : (P : D \u2192 Set){x y : D} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n\nsubst\u2082 : (P : D \u2192 D \u2192 Set){x\u2081 x\u2082 y\u2081 y\u2082 : D} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 P x\u2081 x\u2082 \u2192\n         P y\u2081 y\u2082\nsubst\u2082 P refl refl Px\u2081x\u2082 = Px\u2081x\u2082\n\nsubst\u2084 : (P : D \u2192 D \u2192 D \u2192 D \u2192 Set){x\u2081 x\u2082 x\u2083 x\u2084 y\u2081 y\u2082 y\u2083 y\u2084 : D} \u2192\n         x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2083 \u2261 y\u2083 \u2192 x\u2084 \u2261 y\u2084 \u2192 P x\u2081 x\u2082 x\u2083 x\u2084 \u2192\n         P y\u2081 y\u2082 y\u2083 y\u2084\nsubst\u2084 P refl refl refl refl Px\u2081x\u2082x\u2083x\u2084 = Px\u2081x\u2082x\u2083x\u2084\n","old_contents":"------------------------------------------------------------------------------\n-- Propositional equality\n------------------------------------------------------------------------------\n\nmodule Common.Relation.Binary.PropositionalEquality where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\nsubst : (P : D \u2192 Set){x y : D} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d049e0b6b3b5223ab566d6bf96810ae399bc3889","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: b8c55355628668b9f638d321205a4fb2\n\ndarcs-hash:20120217235342-3bd4e-3344406004ebcaedec8b6c2e5d45eb44b360df48.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Relation\/Binary\/PropositionalEquality.agda","new_file":"src\/Common\/Relation\/Binary\/PropositionalEquality.agda","new_contents":"------------------------------------------------------------------------------\n-- Propositional equality\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module is re-exported by (some) \"base\" modules.\n\nmodule Common.Relation.Binary.PropositionalEquality where\n\nopen import Common.Universe using ( D )\n\n------------------------------------------------------------------------------\n-- The propositional equality via pattern matching.\n\nmodule Inductive where\n  -- We add 3 to the fixities of the standard library.\n  infix 7 _\u2261_\n\n  -- The identity type on the universe of discourse.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  -- Identity properties\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym refl = refl\n\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans refl h = h\n\n  trans\u2082 : \u2200 {w x y z} \u2192 w \u2261 x \u2192 x \u2261 y \u2192 y \u2261 z \u2192 w \u2261 z\n  trans\u2082 refl refl h = h\n\n  subst : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst P refl Px = Px\n\n  subst\u2082 : (P : D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n           P x\u2081 x\u2082 \u2192\n           P y\u2081 y\u2082\n  subst\u2082 P refl refl Pxs = Pxs\n\n  subst\u2083 : (P : D \u2192 D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 x\u2083 y\u2081 y\u2082 y\u2083} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2083 \u2261 y\u2083 \u2192\n           P x\u2081 x\u2082 x\u2083 \u2192\n           P y\u2081 y\u2082 y\u2083\n  subst\u2083 P refl refl refl Pxs = Pxs\n\n  subst\u2084 : (P : D \u2192 D \u2192 D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 x\u2083 x\u2084 y\u2081 y\u2082 y\u2083 y\u2084} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2083 \u2261 y\u2083 \u2192 x\u2084 \u2261 y\u2084 \u2192\n           P x\u2081 x\u2082 x\u2083 x\u2084 \u2192\n           P y\u2081 y\u2082 y\u2083 y\u2084\n  subst\u2084 P refl refl refl refl Pxs = Pxs\n\n  cong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\n  cong f refl = refl\n\n  cong\u2082 : (f : D \u2192 D \u2192 D) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n          f x\u2081 x\u2082 \u2261 f y\u2081 y\u2082\n  cong\u2082 f refl refl = refl\n\n------------------------------------------------------------------------------\n-- The propositional equality via its induction principle.\n\nmodule NonInductive where\n\n  -- We add 3 to the fixities of the standard library.\n  infix 7 _\u2261_\n\n  -- The identity type on the universe of discourse.\n  postulate\n    _\u2261_  : D \u2192 D \u2192 Set\n    refl : \u2200 {x} \u2192 x \u2261 x\n    J    : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n\n  -- Identity properties\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h = J (\u03bb y' \u2192 y' \u2261 x) h refl\n\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans {x} h\u2081 h\u2082 = J (\u03bb y' \u2192 x \u2261 y') h\u2082 h\u2081\n\n  trans\u2082 : \u2200 {w x y z} \u2192 w \u2261 x \u2192 x \u2261 y \u2192 y \u2261 z \u2192 w \u2261 z\n  trans\u2082 h\u2081 h\u2082 h\u2083 = trans (trans h\u2081 h\u2082) h\u2083\n\n  subst : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst = J\n\n  subst\u2082 : (P : D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n           P x\u2081 x\u2082 \u2192\n           P y\u2081 y\u2082\n  subst\u2082 P {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 h\u2083 =\n    subst (\u03bb y\u2081' \u2192 P y\u2081' y\u2082) h\u2081 (subst (\u03bb y\u2082' \u2192 P x\u2081 y\u2082') h\u2082 h\u2083)\n\n  cong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\n  cong f {x} h = subst (\u03bb x' \u2192 f x \u2261 f x') h refl\n\n  cong\u2082 : (f : D \u2192 D \u2192 D) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n          f x\u2081 x\u2082 \u2261 f y\u2081 y\u2082\n  cong\u2082 f {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 =\n    subst (\u03bb x\u2081' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081' y\u2082)\n          h\u2081\n          (subst (\u03bb x\u2082' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081 x\u2082') h\u2082 refl)\n","old_contents":"------------------------------------------------------------------------------\n-- Propositional equality\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module is re-exported by (some) \"base\" modules.\n\nmodule Common.Relation.Binary.PropositionalEquality where\n\nopen import Common.Universe using ( D )\n\n------------------------------------------------------------------------------\n-- The propositional equality via pattern matching.\n\nmodule Inductive where\n  -- We add 3 to the fixities of the standard library.\n  infix 7 _\u2261_\n\n  -- The identity type on the universe of discourse.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  -- Identity properties\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym refl = refl\n\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans refl h = h\n\n  trans\u2082 : \u2200 {w x y z} \u2192 w \u2261 x \u2192 x \u2261 y \u2192 y \u2261 z \u2192 w \u2261 z\n  trans\u2082 refl refl h = h\n\n  subst : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst P refl Px = Px\n\n  subst\u2082 : (P : D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n           P x\u2081 x\u2082 \u2192\n           P y\u2081 y\u2082\n  subst\u2082 P refl refl Pxs = Pxs\n\n  subst\u2083 : (P : D \u2192 D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 x\u2083 y\u2081 y\u2082 y\u2083} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2083 \u2261 y\u2083 \u2192\n           P x\u2081 x\u2082 x\u2083 \u2192\n           P y\u2081 y\u2082 y\u2083\n  subst\u2083 P refl refl refl Pxs = Pxs\n\n  subst\u2084 : (P : D \u2192 D \u2192 D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 x\u2083 x\u2084 y\u2081 y\u2082 y\u2083 y\u2084} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2083 \u2261 y\u2083 \u2192 x\u2084 \u2261 y\u2084 \u2192\n           P x\u2081 x\u2082 x\u2083 x\u2084 \u2192\n           P y\u2081 y\u2082 y\u2083 y\u2084\n  subst\u2084 P refl refl refl refl Pxs = Pxs\n\n  cong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\n  cong f refl = refl\n\n  cong\u2082 : (f : D \u2192 D \u2192 D) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n          f x\u2081 x\u2082 \u2261 f y\u2081 y\u2082\n  cong\u2082 f refl refl = refl\n\n------------------------------------------------------------------------------\n-- The propositional equality via its induction principle.\n\nmodule NonInductive where\n\n  -- We add 3 to the fixities of the standard library.\n  infix 7 _\u2261_\n\n  -- The identity type on the universe of discourse.\n  postulate\n    _\u2261_  : D \u2192 D \u2192 Set\n    refl : \u2200 {x} \u2192 x \u2261 x\n    J    : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n\n  -- Identity properties\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h = J (\u03bb y' \u2192 y' \u2261 x) h refl\n\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans {x} h\u2081 h\u2082 = J (\u03bb y' \u2192 x \u2261 y') h\u2082 h\u2081\n\n  trans\u2082 : \u2200 {w x y z} \u2192 w \u2261 x \u2192 x \u2261 y \u2192 y \u2261 z \u2192 w \u2261 z\n  trans\u2082 h\u2081 h\u2082 h\u2083 = trans (trans h\u2081 h\u2082) h\u2083\n\n  subst : (P : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst = J\n\n  subst\u2082 : (P : D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192\n           x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n           P x\u2081 x\u2082 \u2192\n           P y\u2081 y\u2082\n  subst\u2082 P {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u2261y\u2081 x\u2082\u2261y\u2082 Px\u2081x\u2082 =\n    subst (\u03bb y\u2081' \u2192 P y\u2081' y\u2082)\n          x\u2081\u2261y\u2081\n          (subst (\u03bb y\u2082' \u2192 P x\u2081 y\u2082') x\u2082\u2261y\u2082 Px\u2081x\u2082)\n\n  cong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\n  cong f {x} h = subst (\u03bb x' \u2192 f x \u2261 f x') h refl\n\n  cong\u2082 : (f : D \u2192 D \u2192 D) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n          f x\u2081 x\u2082 \u2261 f y\u2081 y\u2082\n  cong\u2082 f {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 =\n    subst (\u03bb x\u2081' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081' y\u2082)\n          h\u2081\n          (subst (\u03bb x\u2082' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081 x\u2082') h\u2082 refl)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d48e16c2367a2ad929f94f7e393beb62c52409cb","subject":"layout","message":"layout\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits\/Search.agda","new_file":"lib\/Data\/Bits\/Search.agda","new_contents":"open import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bits\nopen import Data.Bool.Properties using (not-involutive)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Algebra.FunctionProperties.NP\n\nmodule Data.Bits.Search where\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n              | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open import Data.Bits.OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nopen SimpleSearch\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n","old_contents":"open import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bits\nopen import Data.Bool.Properties using (not-involutive)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Algebra.FunctionProperties.NP\n\nmodule Data.Bits.Search where\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n            | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n              | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open import Data.Bits.OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nopen SimpleSearch\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b4c39231ae725a5b759926a08d76642ed2df680d","subject":"Base.Change.Products: Simplify code","message":"Base.Change.Products: Simplify code\n\nThese simplifications are inspired by Base.Change.Sums. But they're\nstill modest.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Products.agda","new_file":"Base\/Change\/Products.agda","new_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  -- An interesting alternative definition allows omitting the nil change of a\n  -- component when that nil change can be computed from the type. For instance, the nil change for integers is always the same.\n\n  -- However, the nil change for function isn't always the same (unless we\n  -- defunctionalize them first), so nil changes for functions can't be omitted.\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-nil : (v : A \u00d7 B) \u2192 PChange v\n  p-nil (a , b) = (nil a , nil b)\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  p-update-nil : (v : A \u00d7 B) \u2192 v \u2295 (p-nil v) \u2261 v\n  p-update-nil (a , b) =\n    let v = (a , b)\n    in\n      begin\n        v \u2295 (p-nil v)\n      \u2261\u27e8\u27e9\n        (a \u229e nil a , b \u229e nil b)\n      \u2261\u27e8 cong\u2082 _,_ (update-nil a) (update-nil b)\u27e9\n         (a , b)\n      \u2261\u27e8\u27e9\n        v\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = p-nil\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      ; update-nil = p-update-nil\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n  -- Morally, the following is a change:\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032-realizer : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032-realizer a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of\n  -- its own validity, and because after application it does not contain a\n  -- proof.\n  --\n  -- However, the above is (morally) a \"realizer\" of the actual change, since it\n  -- only represents its computational behavior, not its proof manipulation.\n\n  -- Hence, we need to do some additional work.\n\n  _,_\u2032-realizer-correct _,_\u2032-realizer-correct-detailed :\n    (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192\n      (a , b \u229e db) \u229e (_,_\u2032-realizer a da (b \u229e db) (nil (b \u229e db))) \u2261 (a , b) \u229e (_,_\u2032-realizer a da b db)\n  _,_\u2032-realizer-correct a da b db rewrite update-nil (b \u229e db) = refl\n\n  _,_\u2032-realizer-correct-detailed a da b db =\n    begin\n      (a , b \u229e db) \u229e (_,_\u2032-realizer a da) (b \u229e db) (nil (b \u229e db))\n    \u2261\u27e8\u27e9\n      a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n      a \u229e da , b \u229e db\n    \u2261\u27e8\u27e9\n      (a , b) \u229e (_,_\u2032-realizer a da) b db\n    \u220e\n\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 \u0394 (_,_ a)\n  _,_\u2032 a da = record { apply = _,_\u2032-realizer a da ; correct = \u03bb b db \u2192 _,_\u2032-realizer-correct a da b db }\n\n  _,_\u2032-Derivative : Derivative _,_ _,_\u2032\n  _,_\u2032-Derivative a da = ext (\u03bb b \u2192 cong (_,_ (a \u229e da)) (update-nil b))\n    where\n      open import Postulate.Extensionality\n\n  -- Define specialized variant of uncurry, and derive it.\n  uncurry\u2080 : \u2200 {C : Set \u2113} \u2192 (A \u2192 B \u2192 C) \u2192 A \u00d7 B \u2192 C\n  uncurry\u2080 f (a , b) = f a b\n\n  module _ {C : Set \u2113} {{CC : ChangeAlgebra \u2113 C}} where\n    B\u2192C : ChangeAlgebra \u2113 (B \u2192 C)\n    B\u2192C = FunctionChanges.changeAlgebra B C\n    A\u2192B\u2192C : ChangeAlgebra \u2113 (A \u2192 B \u2192 C)\n    A\u2192B\u2192C = FunctionChanges.changeAlgebra A (B \u2192 C)\n    A\u00d7B\u2192C : ChangeAlgebra \u2113 (A \u00d7 B \u2192 C)\n    A\u00d7B\u2192C = FunctionChanges.changeAlgebra (A \u00d7 B) C\n    open FunctionChanges using (apply; correct)\n\n    uncurry\u2080\u2032-realizer : (f : A \u2192 B \u2192 C) \u2192 \u0394 f \u2192 (p : A \u00d7 B) \u2192 \u0394 p \u2192 \u0394 (uncurry\u2080 f p)\n    uncurry\u2080\u2032-realizer f df (a , b) (da , db) = apply (apply df a da) b db\n\n    uncurry\u2080\u2032-realizer-correct uncurry\u2080\u2032-realizer-correct-detailed :\n      \u2200 (f : A \u2192 B \u2192 C) (df : \u0394 f) (p : A \u00d7 B) (dp : \u0394 p) \u2192\n        uncurry\u2080 f (p \u2295 dp) \u229e uncurry\u2080\u2032-realizer f df (p \u2295 dp) (nil (p \u229e dp)) \u2261 uncurry\u2080 f p \u229e uncurry\u2080\u2032-realizer f df p dp\n\n    -- Hard to read\n    uncurry\u2080\u2032-realizer-correct f df (a , b) (da , db)\n      rewrite sym (correct (apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db)))\n      | update-nil (b \u229e db)\n      | {- cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) -} (sym (correct df (a \u229e da) (nil (a \u229e da))))\n      | update-nil (a \u229e da)\n      | cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (correct df a da)\n      | correct (apply df a da) b db\n      = refl\n\n    -- Verbose, but it shows all the intermediate steps.\n    uncurry\u2080\u2032-realizer-correct-detailed f df (a , b) (da , db) =\n      begin\n        uncurry\u2080 f (a \u229e da , b \u229e db) \u229e uncurry\u2080\u2032-realizer f df (a \u229e da , b \u229e db) (nil (a \u229e da , b \u229e db))\n      \u2261\u27e8\u27e9\n        f (a \u229e da) (b \u229e db) \u229e apply (apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8 sym (correct (apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db))) \u27e9\n        (f (a \u229e da) \u229e apply df (a \u229e da) (nil (a \u229e da))) ((b \u229e db) \u229e (nil (b \u229e db)))\n      \u2261\u27e8 cong-lem\u2080 \u27e9\n        (f (a \u229e da) \u229e apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n      \u2261\u27e8 sym cong-lem\u2082 \u27e9\n        ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n      \u2261\u27e8 cong-lem\u2081 \u27e9\n        (f \u229e df) (a \u229e da) (b \u229e db)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (correct df a da) \u27e9\n        (f a \u229e apply df a da) (b \u229e db)\n      \u2261\u27e8 correct (apply df a da) b db \u27e9\n        f a b \u229e apply (apply df a da) b db\n      \u2261\u27e8\u27e9\n        uncurry\u2080 f (a , b) \u229e uncurry\u2080\u2032-realizer f df (a , b) (da ,  db)\n      \u220e\n      where\n        cong-lem\u2080 :\n            (f (a \u229e da) \u229e apply df (a \u229e da) (nil (a \u229e da))) ((b \u229e db) \u229e (nil (b \u229e db)))\n            \u2261\n            (f (a \u229e da) \u229e apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n        cong-lem\u2080 rewrite update-nil (b \u229e db) = refl\n\n        cong-lem\u2081 :\n                  ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n                  \u2261\n                  (f \u229e df) (a \u229e da) (b \u229e db)\n        cong-lem\u2081 rewrite update-nil (a \u229e da) = refl\n\n        cong-lem\u2082 :\n                  ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n                  \u2261\n                  (f (a \u229e da) \u229e apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n        cong-lem\u2082 = cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (correct df (a \u229e da) (nil (a \u229e da)))\n\n    uncurry\u2080\u2032 : (f : A \u2192 B \u2192 C) \u2192 \u0394 f \u2192 \u0394 (uncurry f)\n    uncurry\u2080\u2032 f df = record\n      { apply = uncurry\u2080\u2032-realizer f df\n      ; correct = uncurry\u2080\u2032-realizer-correct f df }\n\n    -- Now proving that uncurry\u2080\u2032 is a derivative is trivial!\n    uncurry\u2080\u2032Derivative\u2080 : Derivative {{CB = A\u00d7B\u2192C}} uncurry\u2080 uncurry\u2080\u2032\n    uncurry\u2080\u2032Derivative\u2080 f df = refl\n\n    -- If you wonder what's going on, here's the step-by-step proof, going purely by definitional equality.\n    uncurry\u2080\u2032Derivative : Derivative {{CB = A\u00d7B\u2192C}} uncurry\u2080 uncurry\u2080\u2032\n    uncurry\u2080\u2032Derivative f df =\n      begin\n        uncurry\u2080 f \u229e uncurry\u2080\u2032 f df\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 uncurry\u2080 f (a , b) \u229e apply (uncurry\u2080\u2032 f df) (a , b) (nil (a , b))})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 f a b \u229e apply (apply df a (nil a)) b (nil b)})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 (f a \u229e apply df a (nil a)) b})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 (f \u229e df) a b})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 uncurry\u2080 (f \u229e df) (a , b)})\n      \u2261\u27e8\u27e9\n        uncurry\u2080 (f \u229e df)\n      \u220e\n","old_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  -- An interesting alternative definition allows omitting the nil change of a\n  -- component when that nil change can be computed from the type. For instance, the nil change for integers is always the same.\n\n  -- However, the nil change for function isn't always the same (unless we\n  -- defunctionalize them first), so nil changes for functions can't be omitted.\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-nil : (v : A \u00d7 B) \u2192 PChange v\n  p-nil (a , b) = (nil a , nil b)\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  p-update-nil : (v : A \u00d7 B) \u2192 v \u2295 (p-nil v) \u2261 v\n  p-update-nil (a , b) =\n    let v = (a , b)\n    in\n      begin\n        v \u2295 (p-nil v)\n      \u2261\u27e8\u27e9\n        (a \u229e nil a , b \u229e nil b)\n      \u2261\u27e8 cong\u2082 _,_ (update-nil a) (update-nil b)\u27e9\n         (a , b)\n      \u2261\u27e8\u27e9\n        v\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = p-nil\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      ; update-nil = p-update-nil\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n  -- Morally, the following is a change:\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032-realizer : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032-realizer a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of\n  -- its own validity, and because after application it does not contain a\n  -- proof.\n  --\n  -- However, the above is (morally) a \"realizer\" of the actual change, since it\n  -- only represents its computational behavior, not its proof manipulation.\n\n  -- Hence, we need to do some additional work.\n\n  _,_\u2032-realizer-correct _,_\u2032-realizer-correct-detailed :\n    (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192\n      (a , b \u229e db) \u229e (_,_\u2032-realizer a da (b \u229e db) (nil (b \u229e db))) \u2261 (a , b) \u229e (_,_\u2032-realizer a da b db)\n  _,_\u2032-realizer-correct a da b db rewrite update-nil (b \u229e db) = refl\n\n  _,_\u2032-realizer-correct-detailed a da b db =\n    begin\n      (a , b \u229e db) \u229e (_,_\u2032-realizer a da) (b \u229e db) (nil (b \u229e db))\n    \u2261\u27e8\u27e9\n      a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n      a \u229e da , b \u229e db\n    \u2261\u27e8\u27e9\n      (a , b) \u229e (_,_\u2032-realizer a da) b db\n    \u220e\n\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 \u0394 (_,_ a)\n  _,_\u2032 a da = record { apply = _,_\u2032-realizer a da ; correct = \u03bb b db \u2192 _,_\u2032-realizer-correct a da b db }\n\n  _,_\u2032-Derivative : Derivative _,_ _,_\u2032\n  _,_\u2032-Derivative a da = ext (\u03bb b \u2192 cong (_,_ (a \u229e da)) (update-nil b))\n    where\n      open import Postulate.Extensionality\n\n  -- Define specialized variant of uncurry, and derive it.\n  uncurry\u2080 : \u2200 {C : Set \u2113} \u2192 (A \u2192 B \u2192 C) \u2192 A \u00d7 B \u2192 C\n  uncurry\u2080 f (a , b) = f a b\n\n  module _ {C : Set \u2113} {{CC : ChangeAlgebra \u2113 C}} where\n    B\u2192C : ChangeAlgebra \u2113 (B \u2192 C)\n    B\u2192C = FunctionChanges.changeAlgebra B C\n    A\u2192B\u2192C : ChangeAlgebra \u2113 (A \u2192 B \u2192 C)\n    A\u2192B\u2192C = FunctionChanges.changeAlgebra A (B \u2192 C)\n    A\u00d7B\u2192C : ChangeAlgebra \u2113 (A \u00d7 B \u2192 C)\n    A\u00d7B\u2192C = FunctionChanges.changeAlgebra (A \u00d7 B) C\n    module \u0394B\u2192C = FunctionChanges B C {{CB}} {{CC}}\n    module \u0394A\u2192B\u2192C = FunctionChanges A (B \u2192 C) {{CA}} {{B\u2192C}}\n    module \u0394A\u00d7B\u2192C = FunctionChanges (A \u00d7 B) C {{changeAlgebra}} {{CC}}\n\n    uncurry\u2080\u2032-realizer : (f : A \u2192 B \u2192 C) \u2192 \u0394 f \u2192 (p : A \u00d7 B) \u2192 \u0394 p \u2192 \u0394 (uncurry\u2080 f p)\n    uncurry\u2080\u2032-realizer f df (a , b) (da , db) = \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df a da) b db\n\n    uncurry\u2080\u2032-realizer-correct uncurry\u2080\u2032-realizer-correct-detailed :\n      \u2200 (f : A \u2192 B \u2192 C) (df : \u0394 f) (p : A \u00d7 B) (dp : \u0394 p) \u2192\n        uncurry\u2080 f (p \u2295 dp) \u229e uncurry\u2080\u2032-realizer f df (p \u2295 dp) (nil (p \u229e dp)) \u2261 uncurry\u2080 f p \u229e uncurry\u2080\u2032-realizer f df p dp\n\n    -- Hard to read\n    uncurry\u2080\u2032-realizer-correct f df (a , b) (da , db)\n      rewrite sym (\u0394B\u2192C.incrementalization (f (a \u229e da)) (\u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db)))\n      | update-nil (b \u229e db)\n      | {- cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) -} (sym (\u0394A\u2192B\u2192C.incrementalization f df (a \u229e da) (nil (a \u229e da))))\n      | update-nil (a \u229e da)\n      | cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (\u0394A\u2192B\u2192C.incrementalization f df a da)\n      | \u0394B\u2192C.incrementalization (f a) (\u0394A\u2192B\u2192C.apply df a da) b db\n      = refl\n\n    -- Verbose, but it shows all the intermediate steps.\n    uncurry\u2080\u2032-realizer-correct-detailed f df (a , b) (da , db) =\n      begin\n        uncurry\u2080 f (a \u229e da , b \u229e db) \u229e uncurry\u2080\u2032-realizer f df (a \u229e da , b \u229e db) (nil (a \u229e da , b \u229e db))\n      \u2261\u27e8\u27e9\n        f (a \u229e da) (b \u229e db) \u229e \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8 sym (\u0394B\u2192C.incrementalization (f (a \u229e da)) (\u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db))) \u27e9\n        (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) ((b \u229e db) \u229e (nil (b \u229e db)))\n      \u2261\u27e8 cong-lem\u2080 \u27e9\n        (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n      \u2261\u27e8 sym cong-lem\u2082 \u27e9\n        ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n      \u2261\u27e8 cong-lem\u2081 \u27e9\n        (f \u229e df) (a \u229e da) (b \u229e db)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (\u0394A\u2192B\u2192C.incrementalization f df a da) \u27e9\n        (f a \u229e \u0394A\u2192B\u2192C.apply df a da) (b \u229e db)\n      \u2261\u27e8 \u0394B\u2192C.incrementalization (f a) (\u0394A\u2192B\u2192C.apply df a da) b db \u27e9\n        f a b \u229e \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df a da) b db\n      \u2261\u27e8\u27e9\n        uncurry\u2080 f (a , b) \u229e uncurry\u2080\u2032-realizer f df (a , b) (da ,  db)\n      \u220e\n      where\n        cong-lem\u2080 :\n            (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) ((b \u229e db) \u229e (nil (b \u229e db)))\n            \u2261\n            (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n        cong-lem\u2080 rewrite update-nil (b \u229e db) = refl\n\n        cong-lem\u2081 :\n                  ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n                  \u2261\n                  (f \u229e df) (a \u229e da) (b \u229e db)\n        cong-lem\u2081 rewrite update-nil (a \u229e da) = refl\n\n        cong-lem\u2082 :\n                  ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n                  \u2261\n                  (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n        cong-lem\u2082 = cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (\u0394A\u2192B\u2192C.incrementalization f df (a \u229e da) (nil (a \u229e da)))\n\n    uncurry\u2080\u2032 : (f : A \u2192 B \u2192 C) \u2192 \u0394 f \u2192 \u0394 (uncurry f)\n    uncurry\u2080\u2032 f df = record\n      { apply = uncurry\u2080\u2032-realizer f df\n      ; correct = uncurry\u2080\u2032-realizer-correct f df }\n\n    -- Now proving that uncurry\u2080\u2032 is a derivative is trivial!\n    uncurry\u2080\u2032Derivative\u2080 : Derivative {{CB = A\u00d7B\u2192C}} uncurry\u2080 uncurry\u2080\u2032\n    uncurry\u2080\u2032Derivative\u2080 f df = refl\n\n    -- If you wonder what's going on, here's the step-by-step proof, going purely by definitional equality.\n    uncurry\u2080\u2032Derivative : Derivative {{CB = A\u00d7B\u2192C}} uncurry\u2080 uncurry\u2080\u2032\n    uncurry\u2080\u2032Derivative f df =\n      begin\n        uncurry\u2080 f \u229e uncurry\u2080\u2032 f df\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 uncurry\u2080 f (a , b) \u229e \u0394A\u00d7B\u2192C.apply (uncurry\u2080\u2032 f df) (a , b) (nil (a , b))})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 f a b \u229e \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df a (nil a)) b (nil b)})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 (f a \u229e \u0394A\u2192B\u2192C.apply df a (nil a)) b})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 (f \u229e df) a b})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 uncurry\u2080 (f \u229e df) (a , b)})\n      \u2261\u27e8\u27e9\n        uncurry\u2080 (f \u229e df)\n      \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9808544b980c3cc5c39b301c6bee2f86b426bae6","subject":"Compile fix","message":"Compile fix\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set c)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c}\n    (Carrier : Set c) : Set (suc c) where\n  field\n    Change : Carrier \u2192 Set c\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} {A : Set a} (P : A \u2192 Set p): Set (suc p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 b)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (p \u2294 q)\nRawChange\u208d x , y \u208e f = \u2200 px (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\n-- For non-abelian groups\nmodule PlainGroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isGroup : IsGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsGroup isGroup\n      hiding\n        ( refl\n        ; sym\n        )\n      renaming\n        ( \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    _\u229d_ : A \u2192 A \u2192 A\n    v \u229d u = (u \u207b\u00b9) \u2295 v\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8\u27e9\n               v \u2295 ((v \u207b\u00b9) \u2295 u)\n            \u2261\u27e8  sym (assoc _ _ _) \u27e9\n               (v \u2295 (v \u207b\u00b9)) \u2295 u\n            \u2261\u27e8 proj\u2082 inverse v \u27e8\u2295\u27e9 refl \u27e9\n               \u03b5 \u2295 u\n            \u2261\u27e8 proj\u2081 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} (A : Set a) (B : Set b) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g (a \u229e da \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n\n    funNil : (f : A \u2192 B) \u2192 FunctionChange f\n    funNil = \u03bb f \u2192 funDiff f f\n    private\n      -- Realizer for funNil\n      funNil-realizer : (f : A \u2192 B) \u2192 RawChange f\n      funNil-realizer f = \u03bb a da \u2192 f (a \u229e da) \u229f f a\n      -- Note that it cannot use nil (f a), that would not be a correct function change!\n      funNil-correct : \u2200 f a da \u2192 funNil-realizer f a da \u2261 apply (funNil f) a da\n      funNil-correct f a da = refl\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebraFun : ChangeAlgebra (A \u2192 B)\n\n      changeAlgebraFun = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n-- Reexport a few members with A and B marked as implicit parameters. This\n-- matters especially for changeAlgebra, since otherwise it can't be used for\n-- instance search.\nmodule _\n    {a} {b} {A : Set a} {B : Set b} {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    open FunctionChanges A B {{CA}} {{CB}} public\n      using\n        ( changeAlgebraFun\n        ; apply\n        ; correct\n        ; incrementalization\n        ; DerivativeAsChange\n        ; FunctionChange\n        ; nil-is-derivative\n        )\n\nmodule BinaryFunctionChanges\n    {a} {b} {c} (A : Set a) (B : Set b) (C : Set c) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} {{CC : ChangeAlgebra C}} where\n    incrementalization-binary : \u2200 (f : A \u2192 B \u2192 C) df a da b db \u2192\n      (f \u229e df) (a \u229e da) (b \u229e db) \u2261 f a b \u229e apply (apply df a da) b db\n    incrementalization-binary f df x dx y dy\n      rewrite cong (\u03bb \u25a1 \u2192 \u25a1 (y \u229e dy)) (incrementalization f df x dx)\n      = incrementalization (f x) (apply df x dx) y dy\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 \u0394 pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 \u0394 pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebraListChanges : ChangeAlgebraFamily (All P)\n\n    changeAlgebraListChanges = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set c)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c}\n    (Carrier : Set c) : Set (suc c) where\n  field\n    Change : Carrier \u2192 Set c\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} {A : Set a} (P : A \u2192 Set p): Set (suc p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 b)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (p \u2294 q)\nRawChange\u208d x , y \u208e f = \u2200 px (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isGroup : IsGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsGroup isGroup\n      hiding\n        ( refl\n        ; sym\n        )\n      renaming\n        ( \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    _\u229d_ : A \u2192 A \u2192 A\n    v \u229d u = (u \u207b\u00b9) \u2295 v\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8\u27e9\n               v \u2295 ((v \u207b\u00b9) \u2295 u)\n            \u2261\u27e8  sym (assoc _ _ _) \u27e9\n               (v \u2295 (v \u207b\u00b9)) \u2295 u\n            \u2261\u27e8 proj\u2082 inverse v \u27e8\u2295\u27e9 refl \u27e9\n               \u03b5 \u2295 u\n            \u2261\u27e8 proj\u2081 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\nmodule AbelianGroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} (A : Set a) (B : Set b) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g (a \u229e da \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n\n    funNil : (f : A \u2192 B) \u2192 FunctionChange f\n    funNil = \u03bb f \u2192 funDiff f f\n    private\n      -- Realizer for funNil\n      funNil-realizer : (f : A \u2192 B) \u2192 RawChange f\n      funNil-realizer f = \u03bb a da \u2192 f (a \u229e da) \u229f f a\n      -- Note that it cannot use nil (f a), that would not be a correct function change!\n      funNil-correct : \u2200 f a da \u2192 funNil-realizer f a da \u2261 apply (funNil f) a da\n      funNil-correct f a da = refl\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebraFun : ChangeAlgebra (A \u2192 B)\n\n      changeAlgebraFun = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n-- Reexport a few members with A and B marked as implicit parameters. This\n-- matters especially for changeAlgebra, since otherwise it can't be used for\n-- instance search.\nmodule _\n    {a} {b} {A : Set a} {B : Set b} {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    open FunctionChanges A B {{CA}} {{CB}} public\n      using\n        ( changeAlgebraFun\n        ; apply\n        ; correct\n        ; incrementalization\n        ; DerivativeAsChange\n        ; FunctionChange\n        ; nil-is-derivative\n        )\n\nmodule BinaryFunctionChanges\n    {a} {b} {c} (A : Set a) (B : Set b) (C : Set c) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} {{CC : ChangeAlgebra C}} where\n    incrementalization-binary : \u2200 (f : A \u2192 B \u2192 C) df a da b db \u2192\n      (f \u229e df) (a \u229e da) (b \u229e db) \u2261 f a b \u229e apply (apply df a da) b db\n    incrementalization-binary f df x dx y dy\n      rewrite cong (\u03bb \u25a1 \u2192 \u25a1 (y \u229e dy)) (incrementalization f df x dx)\n      = incrementalization (f x) (apply df x dx) y dy\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 \u0394 pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 \u0394 pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebraListChanges : ChangeAlgebraFamily (All P)\n\n    changeAlgebraListChanges = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ddad4849616d2fea806aa829236f7932b15bd3a3","subject":"Tests","message":"Tests\n","repos":"np\/agda-parametricity","old_file":"lib\/Reflection\/Param\/Tests.agda","new_file":"lib\/Reflection\/Param\/Tests.agda","new_contents":"{-# OPTIONS -vtc.unquote.decl:20 -vtc.unquote.def:20 #-}\n{-# OPTIONS --without-K #-}\nopen import Level hiding (zero; suc)\nopen import Data.Unit renaming (\u22a4 to \ud835\udfd9; tt to 0\u2081)\nopen import Data.Bool\n  using    (not)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082)\nopen import Data.String.Core using (String)\nopen import Data.Float       using (Float)\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; refl) renaming (_\u2257_ to _~_)\n\nopen import Function.Param.Unary\nopen import Function.Param.Binary\nopen import Type.Param.Unary\nopen import Type.Param.Binary\nopen import Data.Two.Param.Binary\nopen import Data.Nat.Param.Binary\nopen import Reflection.NP\nopen import Reflection.Param\nopen import Reflection.Param.Env\n\nmodule Reflection.Param.Tests where\n\nimport Reflection.Printer as Pr\nopen Pr using (var;con;def;lam;pi;sort;unknown;showTerm;showType;showDef;showFunDef;showClauses)\n\n-- Local \"imports\" to avoid depending on nplib\nprivate\n  postulate\n    opaque : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2192 B \u2192 B\n    -- opaque-rule : \u2200 {x} y \u2192 opaque x y \u2261 y\n\n  \u2605\u2080 = Set\u2080\n  \u2605\u2081 = Set\u2081\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\n{-\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n-}\n\n_\u2261-no-hints_ : Term \u2192 Term \u2192 Set\nt \u2261-no-hints u = noHintsTerm t \u2261 noHintsTerm u\n\n_\u2261-def-no-hints_ : Definition \u2192 Definition \u2192 Set\nt \u2261-def-no-hints u = noHintsDefinition t \u2261 noHintsDefinition u\n\np[Set\u2080]-type = param-type-by-name (\u03b5 1) (quote [Set\u2080])\np[Set\u2080] = param-clauses-by-name (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl p[Set\u2080]-type)\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261-no-hints Get-term.from-clauses p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nu-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\nunquoteDef u-p[Set\u2080] = p[Set\u2080]\n\nFalse : Set\u2081\nFalse = (A : Set) \u2192 A\n\nparam1-False-type = param-type-by-name (\u03b5 1) (quote False)\nparam1-False-term = param-term-by-name (\u03b5 1) (quote False)\n\nparam1-False-type-check : [Set\u2081] False \u2261 unquote (unEl param1-False-type)\nparam1-False-type-check = refl\n\n[False] : unquote (unEl param1-False-type)\n[False] = unquote param1-False-term\n\n[Level] : [Set\u2080] Level\n[Level] _ = \ud835\udfd9\n\n[String] : [Set\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [Set\u2080] Float\n[Float] _ = \ud835\udfd9\n\n-- Levels are parametric, hence the relation is total\n\u27e6Level\u27e7 : \u27e6Set\u2080\u27e7 Level Level\n\u27e6Level\u27e7 _ _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6Set\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6Set\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda)      = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115)      = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float)  = quote [Float]\ndefDefEnv1 (quote \u2605\u2080)     = quote [Set\u2080]\ndefDefEnv1 (quote \u2605\u2081)     = quote [Set\u2081]\ndefDefEnv1 (quote False)  = quote [False]\ndefDefEnv1 (quote Level)  = quote [Level]\ndefDefEnv1 n              = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082)         = quote [0\u2082]\ndefConEnv1 (quote 1\u2082)         = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero)     = quote [zero]\ndefConEnv1 (quote \u2115.suc)      = quote [suc]\ndefConEnv1 (quote Level.zero) = quote 0\u2081\ndefConEnv1 (quote Level.suc)  = quote 0\u2081\ndefConEnv1 n                  = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda)      = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115)      = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote \u2605\u2080)     = quote \u27e6Set\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081)     = quote \u27e6Set\u2081\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float)  = quote \u27e6Float\u27e7\ndefDefEnv2 (quote Level)  = quote \u27e6Level\u27e7\ndefDefEnv2 n              = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082)         = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082)         = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero)     = quote \u27e6\u2115\u27e7.\u27e6zero\u27e7\ndefConEnv2 (quote \u2115.suc)      = quote \u27e6\u2115\u27e7.\u27e6suc\u27e7\ndefConEnv2 (quote Level.zero) = quote 0\u2081\ndefConEnv2 (quote Level.suc)  = quote 0\u2081\ndefConEnv2 n                  = opaque \"defConEnv2\" n\n\ndefEnv0 : Env' 0\ndefEnv0 = record (\u03b5 0)\n                 { pConT = con\n                 ; pConP = con\n                 ; pDef  = id }\n\ndefEnv1 : Env' 1\ndefEnv1 = record (\u03b5 1)\n  { pConP = con \u2218\u2032 defConEnv1\n  ; pConT = con \u2218\u2032 defConEnv1\n  ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record (\u03b5 2)\n  { pConP = con \u2218\u2032 defConEnv2\n  ; pConT = con \u2218\u2032 defConEnv2\n  ; pDef = defDefEnv2 }\n\nparam1-[False]-type = param-type-by-name defEnv1 (quote [False])\nparam1-[False]-term = param-term-by-name defEnv1 (quote [False])\n\n{-\nmodule Const where\n  postulate\n    A  : Set\u2080\n    A\u1d63 : A \u2192 A \u2192 Set\u2080\n  data Wrapper : Set where\n    wrap : A \u2192 Wrapper\n\n  idWrapper : Wrapper \u2192 Wrapper\n  idWrapper (wrap x) = wrap x\n\n  data \u27e6Wrapper\u27e7 : Wrapper \u2192 Wrapper \u2192 Set\u2080 where\n    \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7) wrap wrap\n\n  wrapperEnv = record (\u03b5 2)\n   { pDef = [ quote Wrapper       \u2254 quote \u27e6Wrapper\u27e7  ] id\n   ; pConP = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   ; pConT = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   }\n\n  unquoteDecl \u27e6idWrapper\u27e7 = param-rec-def-by-name wrapperEnv (quote idWrapper) \u27e6idWrapper\u27e7\n-}\n\ndata Wrapper (A : Set\u2080) : Set\u2080 where\n  wrap : A \u2192 Wrapper A\n\nidWrapper : \u2200 {A} \u2192 Wrapper A \u2192 Wrapper A\nidWrapper (wrap x) = wrap x\n\ndata [Wrapper] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Wrapper A \u2192 Set\u2080 where\n  [wrap] : (A\u209a [\u2192] [Wrapper] A\u209a) wrap\n\n[Wrapper]-env = record (\u03b5 1)\n  { pDef = [ quote Wrapper \u2254 quote [Wrapper] ] id\n  ; pConP = [ quote wrap \u2254 con (quote [wrap])  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 2 (quote [wrap]) ] con\n  }\n\nunquoteDecl [idWrapper] =\n  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]\n\n  {-\n[idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n-- [idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n-}\n\n{-\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6Wrapper\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) Wrapper Wrapper\n\nprivate\n  \u27e6Wrapper\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6Wrapper\u27e7 where\n  \u27e6wrap\u27e7 : unquote (\u27e6Wrapper\u27e7-ctor (quote Wrapper.wrap))\n-}\ndata \u27e6Wrapper\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Wrapper A\u2080 \u2192 Wrapper A\u2081 \u2192 Set\u2080 where\n  \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A\u1d63) wrap wrap\n\n\u27e6Wrapper\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] id\n  ; pConP = [ quote wrap \u2254 con (quote \u27e6wrap\u27e7)  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 3 (quote \u27e6wrap\u27e7) ] con\n  }\n\n\u27e6idWrapper\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\n\u27e6idWrapper\u27e71 {x0} {x1} (x2) {._} {._} (\u27e6wrap\u27e7 {x3} {x4} x5)\n  = \u27e6wrap\u27e7 {_} {_} {_} {x3} {x4} x5\n\n\u27e6idWrapper\u27e7-clauses =\n  clause\n  (arg (arg-info hidden  relevant) (var \"A0\") \u2237\n   arg (arg-info hidden  relevant) (var \"A1\") \u2237\n   arg (arg-info visible relevant) (var \"Ar\") \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info visible relevant)\n   (con (quote \u27e6wrap\u27e7)\n    (arg (arg-info hidden  relevant) (var \"x0\") \u2237\n     arg (arg-info hidden  relevant) (var \"x1\") \u2237\n     arg (arg-info visible relevant) (var \"xr\") \u2237 []))\n   \u2237 [])\n  (con (quote \u27e6wrap\u27e7)\n   (arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant)  (var 2 []) \u2237\n    arg (arg-info hidden relevant)  (var 1 []) \u2237\n    arg (arg-info visible relevant) (var 0 []) \u2237 []))\n  \u2237 []\n\n\u27e6idWrapper\u27e72 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\nunquoteDef \u27e6idWrapper\u27e72 = \u27e6idWrapper\u27e7-clauses\n\nunquoteDecl \u27e6idWrapper\u27e7 =\n  param-rec-def-by-name \u27e6Wrapper\u27e7-env (quote idWrapper) \u27e6idWrapper\u27e7\n\ndata Bot (A : Set\u2080) : Set\u2080 where\n  bot : Bot A \u2192 Bot A\n\ngobot : \u2200 {A} \u2192 Bot A \u2192 A\ngobot (bot x) = gobot x\n\ndata [Bot] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Bot A \u2192 Set\u2080 where\n  [bot] : ([Bot] A\u209a [\u2192] [Bot] A\u209a) bot\n\n[Bot]-env = record (\u03b5 1)\n  { pDef = [ quote Bot \u2254 quote [Bot] ] id\n  ; pConP = [ quote bot \u2254 con (quote [bot])  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 2 (quote [bot]) ] con\n  }\n\n[gobot]' : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n[gobot]' {x0} (x1) {._} ([bot] {x2} x3)\n  = [gobot]' {x0} x1 {x2} x3\n\n-- [gobot]' = {!showClauses \"[gobot]'\" (param-rec-clauses-by-name [Bot]-env (quote gobot) (quote [gobot]'))!}\n\n[gobot]2 : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n\n[gobot]2-clauses =\n  clause\n    (arg (arg-info hidden  relevant) (var \"A\u1d620\") \u2237\n     arg (arg-info visible relevant) (var \"A\u1d63\") \u2237\n     arg (arg-info hidden  relevant) dot \u2237\n     arg (arg-info visible relevant)\n     (con (quote [bot])\n      (arg (arg-info hidden  relevant) (var \"x\u1d620\") \u2237\n       arg (arg-info visible relevant) (var \"x\u1d63\") \u2237 []))\n     \u2237 [])\n    (def (quote [gobot]2)\n     (arg (arg-info hidden  relevant) (var 4 []) \u2237\n      arg (arg-info visible relevant) (var 3 []) \u2237\n      arg (arg-info hidden  relevant) (var 1 []) \u2237\n      arg (arg-info visible relevant) (var 0 []) \u2237 []))\n    \u2237 []\n\nunquoteDef [gobot]2 = [gobot]2-clauses\n\nunquoteDecl [gobot] =\n  param-rec-def-by-name [Bot]-env (quote gobot) [gobot]\n\ndata \u27e6Bot\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Bot A\u2080 \u2192 Bot A\u2081 \u2192 Set\u2080 where\n  \u27e6bot\u27e7 : (\u27e6Bot\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Bot\u27e7 A\u1d63) bot bot\n\n\u27e6Bot\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Bot \u2254 quote \u27e6Bot\u27e7 ] id\n  ; pConP = [ quote bot \u2254 con (quote \u27e6bot\u27e7)  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 3 (quote \u27e6bot\u27e7) ] con\n  }\n\n\u27e6gobot\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Bot\u27e7 A \u27e6\u2192\u27e7 A) gobot gobot\n\u27e6gobot\u27e71 {x0} {x1} x2 {._} {._} (\u27e6bot\u27e7 {x3} {x4} x5)\n  = \u27e6gobot\u27e71 {x0} {x1} x2 {x3} {x4} x5\n\nunquoteDecl \u27e6gobot\u27e7 =\n  param-rec-def-by-name \u27e6Bot\u27e7-env (quote gobot) \u27e6gobot\u27e7\n\nid\u2080 : {A : Set\u2080} \u2192 A \u2192 A\nid\u2080 x = x\n\n\u27e6id\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A \u27e6\u2192\u27e7 A) id\u2080 id\u2080\n\u27e6id\u2080\u27e7 = \u03bb {x\u2081} {x\u2082} x\u1d63 {x\u2083} {x\u2084} x\u1d63\u2081 \u2192 x\u1d63\u2081\n\ndata List\u2080 (A : Set) : Set where\n  []  : List\u2080 A\n  _\u2237_ : A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} (f : A \u2192 B) (xs : List\u2080 A) \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\n-- idList\u2080 : List\u2080 \u2115 \u2192 List\u2080 \u2115\nidList\u2080 []       = []\nidList\u2080 {A} (x \u2237 xs) = idList\u2080 {A} xs\n\ndata \u27e6List\u2080\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080) : List\u2080 A\u2080 \u2192 List\u2080 A\u2081 \u2192 Set\u2080 where\n  \u27e6[]\u27e7  : \u27e6List\u2080\u27e7 A\u1d63 [] []\n  _\u27e6\u2237\u27e7_ : (A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63) _\u2237_ _\u2237_\n\ncon\u27e6List\u2080\u27e7 = conSkip' 3\n\u27e6List\u2080\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] id\n  ; pConP = [ quote List\u2080.[]  \u2254 con (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  ; pConT = [ quote List\u2080.[]  \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  }\n\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\n-- \u27e6idList\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A) idList\u2080 idList\u2080\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n\n{-\n\u27e6map\u2080\u27e7 : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : List\u2080 (x0)} \u2192 {x10 : List\u2080 (x1)} \u2192 (x11 : \u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 \u27e6List\u2080\u27e7 {x3} {x4} (x5) (map\u2080 {x0} {x3} (x6) (x9)) (map\u2080 {x1} {x4} (x7) (x10))\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (\u27e6[]\u27e7 )  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (_\u27e6\u2237\u27e7_ {x11} {x12} (x13) {x14} {x15} (x16) )  = _\u27e6\u2237\u27e7_ {x6 (x11)} {x7 (x12)} (x8 {x11} {x12} (x13)) {map\u2080 {x0} {x3} (x6) (x14)} {map\u2080 {x1} {x4} (x7) (x15)} (\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {x14} {x15} (x16))\n-}\n\nunquoteDecl \u27e6map\u2080\u27e7\n = param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) \u27e6map\u2080\u27e7\n\n{-\nmap-nat : \u2200 (f : \u2200 {X} \u2192 List\u2080 X \u2192 List\u2080 X)\n            {A B : Set} (g : A \u2192 B)\n          \u2192 f \u2218 map\u2080 g ~ map\u2080 g \u2218 f\nmap-nat f g x = {!\u27e6map\u2080\u27e7 _\u2261_ _\u2261_ {g}!}\n\n  -- The generated type is bigger since it is a familly for no reason.\n  data \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\n  private\n    \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\n  data \u27e6List\u2080\u27e7 where\n    \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n    _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n  \u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                        [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n             (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n-}\n\ndata Maybe' (A : Set) : Set\u2081 where\n  nothing : Maybe' A\n  just    : A \u2192 Maybe' A\n\n{-\n-- Set\u2081 is here because \u27e6List\u2080\u27e7 is not using parameters, hence gets bigger.\n-- This only happens without-K given the new rules for data types.\ndata List\u2080 : (A : Set) \u2192 Set\u2081 where\n  []  : \u2200 {A} \u2192 List\u2080 A\n  _\u2237_ : \u2200 {A} \u2192 A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} \u2192 (A \u2192 B) \u2192 List\u2080 A \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\nidList\u2080 []       = []\nidList\u2080 (x \u2237 xs) = x \u2237 idList\u2080 xs\n\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\nprivate\n  \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6List\u2080\u27e7 where\n  \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n  _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n\u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                      [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n           (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n\n-- test = \u27e6[]\u27e7 {{!showType (type (quote List\u2080.[]))!}} {{!!}} {!!}\n-}\n\n{-\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n-}\n\n{-\n\u27e6map\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote map\u2080)))\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {[]} {[]} \u27e6[]\u27e7\n  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {._ \u2237 ._}\n  {._ \u2237 ._}\n  (_\u27e6\u2237\u27e7_ {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085 {x\u2081\u2086} {x\u2081\u2087} x\u2081\u2088)\n  = _\u27e6\u2237\u27e7_ {x\u2081\u2080 x\u2081\u2083} {x\u2081\u2081 x\u2081\u2084} (x\u2081\u2082 {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085)\n    {map\u2080 {x\u2084} {x\u2087} x\u2081\u2080 x\u2081\u2086} {map\u2080 {x\u2085} {x\u2088} x\u2081\u2081 x\u2081\u2087}\n    (\u27e6map\u2080\u27e7 {x\u2084} {x\u2085} {x\u2086} {x\u2087} {x\u2088} {x\u2089} {x\u2081\u2080} {x\u2081\u2081} x\u2081\u2082 {x\u2081\u2086} {x\u2081\u2087}\n     x\u2081\u2088)\n-}\n\n{-\nunquoteDef \u27e6map\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote \u27e6map\u2080\u27e7)\n-}\n\n\n{-\nfoo : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : Reflection.Param.List\u2080 (x0)} \u2192 {x10 : Reflection.Param.List\u2080 (x1)} \u2192 (x11 : Reflection.Param.\u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 Reflection.Param.\u27e6List\u2080\u27e7 {x3} {x4} (x5) (Reflection.Param.map\u2080 {x0} {x3} (x6) (x9)) (Reflection.Param.map\u2080 {x1} {x4} (x7) (x10))\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7 )  = Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x} {x} (x) {xs} {xs} (xs) )  = Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x0 (x0)} {x0 (x0)} (x0 {x0} {x0} (x0)) {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} (Reflection.Param.test' {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0))\n-}\n\n-- test' = {! showFunDef \"foo\" (param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote test'))!}\n\nopen import Function.Param.Unary\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\n\nrevealed-[\u2192]' : \u2200 (A : Set\u2080) (A\u209a : A \u2192 Set\u2080)\n                  (B : Set\u2080) (B\u209a : B \u2192 Set\u2080)\n                  (f : A \u2192 B) \u2192 Set\u2080\nunquoteDef revealed-[\u2192]' = Get-clauses.from-def revealed-[\u2192]\n\nrevelator-[\u2192] : ({A : Set} (A\u209a : A \u2192 Set) {B : Set} (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set)\n              \u2192  (A : Set) (A\u209a : A \u2192 Set) (B : Set) (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set\nunquoteDef revelator-[\u2192] = Revelator.clauses (type (quote _[\u2080\u2192\u2080]_))\n\np-[\u2192]-type = param-type-by-name    (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]      = param-clauses-by-name (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\n\np-[\u2192]' = \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n           {A\u209a : A \u2192 Set\u2080}  (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 A\u209a x \u2192 Set\u2080)\n           {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n           {B\u209a : B \u2192 Set\u2080}  (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 B\u209a x \u2192 Set\u2080)\n           {f : A \u2192 B}      (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n         \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f\n         \u2192 Set\n\np-[\u2192]'-test : p-[\u2192]' \u2261 unquote (unEl p-[\u2192]-type)\np-[\u2192]'-test = refl\n\n[[\u2192]] : unquote (unEl p-[\u2192]-type)\nunquoteDef [[\u2192]] = p-[\u2192]\n\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : [[\u2192]] [\u2115] [[\u2115]] [\u2115] [[\u2115]] [suc] [suc]\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7          \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\npred' : \u2115 \u2192 \u2115\npred' = \u03bb { zero    \u2192 zero\n          ; (suc m) \u2192 m }\n\n\u27e6pred'\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred' pred'\nunquoteDef \u27e6pred'\u27e7 = param-clauses-by-name defEnv2 (quote pred')\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7-type = unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7)))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : \u27e6\u27e6Set\u2080\u27e7\u27e7-type \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = param-clauses-by-name defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261-no-hints Get-term.from-clauses p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7'' : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\nunquoteDef \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\n\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' : _\u2261_ {A = \u27e6\u27e6Set\u2080\u27e7\u27e7-type} \u27e6\u27e6Set\u2080\u27e7\u27e7'' \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' = refl\n\ntest-p0-\u27e6Set\u2080\u27e7 : pTerm defEnv0 (quoteTerm \u27e6Set\u2080\u27e7) \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-p0-\u27e6Set\u2080\u27e7 = refl\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\np1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n[\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\ntest-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\ntest-p1\u2115\u2192\u2115 = refl\n\np2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\ntest-p2\u2115\u2192\u2115 = refl\n\np\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261-no-hints quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 = refl\nZERO : Set\u2081\nZERO = (A : Set\u2080) \u2192 A\n\u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n\u27e6ZERO\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\npZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\nq\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\ntest-pZERO : pZERO \u2261-no-hints q\u27e6ZERO\u27e7\ntest-pZERO = refl\nID : Set\u2081\nID = (A : Set\u2080) \u2192 A \u2192 A\n\u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n\u27e6ID\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n  \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\npID = pTerm (\u03b5 2) (quoteTerm ID)\nq\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\ntest-ID : q\u27e6ID\u27e7 \u2261-no-hints pID\ntest-ID = refl\n\n\u27e6not\u27e7' : (\u27e6\ud835\udfda\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) not not\nunquoteDef \u27e6not\u27e7' = param-clauses-by-name defEnv2 (quote not)\ntest-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 \u27e6not\u27e7' x\u1d63\ntest-not \u27e60\u2082\u27e7 = refl\ntest-not \u27e61\u2082\u27e7 = refl\n\n[pred]' : ([\u2115] [\u2192] [\u2115]) pred\nunquoteDef [pred]' = param-clauses-by-name defEnv1 (quote pred)\n\ntest-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\ntest-p1-pred [zero]     = refl\ntest-p1-pred ([suc] n\u209a) = refl\n\n\u27e6pred\u27e7' : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred pred\nunquoteDef \u27e6pred\u27e7' = param-clauses-by-name defEnv2 (quote pred)\n\ntest-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\ntest-p2-pred \u27e6zero\u27e7     = refl\ntest-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\np\/2 = param-rec-def-by-name defEnv2 (quote _\/2)\nq\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\nunquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\ntest-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261-def-no-hints q\u27e6\/2\u27e7\ntest-\/2 = refl\ntest-\/2' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\ntest-\/2' \u27e6zero\u27e7 = refl\ntest-\/2' (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\ntest-\/2' (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) rewrite test-\/2' n\u1d63 = refl\n\np+ = param-rec-def-by-name defEnv2 (quote _+\u2115_)\nq\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\nunquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\ntest-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261-def-no-hints q\u27e6+\u27e7\ntest-+ = refl\ntest-+' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\ntest-+' \u27e6zero\u27e7    n'\u1d63 = refl\ntest-+' (\u27e6suc\u27e7 n\u1d63) n'\u1d63 rewrite test-+' n\u1d63 n'\u1d63 = refl\n\n{-\nis-good : String \u2192 \ud835\udfda\nis-good \"good\" = 1\u2082\nis-good _      = 0\u2082\n\n\u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n\u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n\u27e6is-good\u27e7 {_}      refl = {!!}\n\nmy-good = unquote (lit (string \"good\"))\nmy-good-test : my-good \u2261 \"good\"\nmy-good-test = refl\n-}\n\n{-\n\u27e6is-good\u27e7' : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\nunquoteDef \u27e6is-good\u27e7' = param-clauses-by-name defEnv2 (quote is-good)\ntest-is-good = {!!}\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS -vtc.unquote.decl:20 -vtc.unquote.def:20 #-}\n{-# OPTIONS --without-K #-}\nopen import Level hiding (zero; suc)\nopen import Data.Unit renaming (\u22a4 to \ud835\udfd9; tt to 0\u2081)\nopen import Data.Bool\n  using    (not)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082)\nopen import Data.String.Core using (String)\nopen import Data.Float       using (Float)\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; refl) renaming (_\u2257_ to _~_)\n\nopen import Function.Param.Unary\nopen import Function.Param.Binary\nopen import Type.Param.Unary\nopen import Type.Param.Binary\nopen import Data.Two.Param.Binary\nopen import Data.Nat.Param.Binary\nopen import Reflection.NP\nopen import Reflection.Param\nopen import Reflection.Param.Env\n\nmodule Reflection.Param.Tests where\n\nimport Reflection.Printer as Pr\nopen Pr using (var;con;def;lam;pi;sort;unknown;showTerm;showType;showDef;showFunDef;showClauses)\n\n-- Local \"imports\" to avoid depending on nplib\nprivate\n  postulate\n    opaque : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2192 B \u2192 B\n    -- opaque-rule : \u2200 {x} y \u2192 opaque x y \u2261 y\n\n  \u2605\u2080 = Set\u2080\n  \u2605\u2081 = Set\u2081\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\n{-\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n-}\n\n_\u2261-no-hints_ : Term \u2192 Term \u2192 Set\nt \u2261-no-hints u = noHintsTerm t \u2261 noHintsTerm u\n\n_\u2261-def-no-hints_ : Definition \u2192 Definition \u2192 Set\nt \u2261-def-no-hints u = noHintsDefinition t \u2261 noHintsDefinition u\n\np[Set\u2080]-type = param-type-by-name (\u03b5 1) (quote [Set\u2080])\np[Set\u2080] = param-clauses-by-name (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl p[Set\u2080]-type)\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261-no-hints Get-term.from-clauses p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nu-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\nunquoteDef u-p[Set\u2080] = p[Set\u2080]\n\nFalse : Set\u2081\nFalse = (A : Set) \u2192 A\n\nparam1-False-type = param-type-by-name (\u03b5 1) (quote False)\nparam1-False-term = param-term-by-name (\u03b5 1) (quote False)\n\nparam1-False-type-check : [Set\u2081] False \u2261 unquote (unEl param1-False-type)\nparam1-False-type-check = refl\n\n[False] : unquote (unEl param1-False-type)\n[False] = unquote param1-False-term\n\n[Level] : [Set\u2080] Level\n[Level] _ = \ud835\udfd9\n\n[String] : [Set\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [Set\u2080] Float\n[Float] _ = \ud835\udfd9\n\n-- Levels are parametric, hence the relation is total\n\u27e6Level\u27e7 : \u27e6Set\u2080\u27e7 Level Level\n\u27e6Level\u27e7 _ _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6Set\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6Set\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda)      = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115)      = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float)  = quote [Float]\ndefDefEnv1 (quote \u2605\u2080)     = quote [Set\u2080]\ndefDefEnv1 (quote \u2605\u2081)     = quote [Set\u2081]\ndefDefEnv1 (quote False)  = quote [False]\ndefDefEnv1 (quote Level)  = quote [Level]\ndefDefEnv1 n              = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082)         = quote [0\u2082]\ndefConEnv1 (quote 1\u2082)         = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero)     = quote [zero]\ndefConEnv1 (quote \u2115.suc)      = quote [suc]\ndefConEnv1 (quote Level.zero) = quote 0\u2081\ndefConEnv1 (quote Level.suc)  = quote 0\u2081\ndefConEnv1 n                  = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda)      = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115)      = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote \u2605\u2080)     = quote \u27e6Set\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081)     = quote \u27e6Set\u2081\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float)  = quote \u27e6Float\u27e7\ndefDefEnv2 (quote Level)  = quote \u27e6Level\u27e7\ndefDefEnv2 n              = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082)         = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082)         = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero)     = quote \u27e6\u2115\u27e7.\u27e6zero\u27e7\ndefConEnv2 (quote \u2115.suc)      = quote \u27e6\u2115\u27e7.\u27e6suc\u27e7\ndefConEnv2 (quote Level.zero) = quote 0\u2081\ndefConEnv2 (quote Level.suc)  = quote 0\u2081\ndefConEnv2 n                  = opaque \"defConEnv2\" n\n\ndefEnv0 : Env' 0\ndefEnv0 = record (\u03b5 0)\n                 { pConT = con\n                 ; pConP = con\n                 ; pDef  = id }\n\ndefEnv1 : Env' 1\ndefEnv1 = record (\u03b5 1)\n  { pConP = con \u2218\u2032 defConEnv1\n  ; pConT = con \u2218\u2032 defConEnv1\n  ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record (\u03b5 2)\n  { pConP = con \u2218\u2032 defConEnv2\n  ; pConT = con \u2218\u2032 defConEnv2\n  ; pDef = defDefEnv2 }\n\nparam1-[False]-type = param-type-by-name defEnv1 (quote [False])\nparam1-[False]-term = param-term-by-name defEnv1 (quote [False])\n\n{-\nmodule Const where\n  postulate\n    A  : Set\u2080\n    A\u1d63 : A \u2192 A \u2192 Set\u2080\n  data Wrapper : Set where\n    wrap : A \u2192 Wrapper\n\n  idWrapper : Wrapper \u2192 Wrapper\n  idWrapper (wrap x) = wrap x\n\n  data \u27e6Wrapper\u27e7 : Wrapper \u2192 Wrapper \u2192 Set\u2080 where\n    \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7) wrap wrap\n\n  wrapperEnv = record (\u03b5 2)\n   { pDef = [ quote Wrapper       \u2254 quote \u27e6Wrapper\u27e7  ] id\n   ; pConP = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   ; pConT = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   }\n\n  unquoteDecl \u27e6idWrapper\u27e7 = param-rec-def-by-name wrapperEnv (quote idWrapper) \u27e6idWrapper\u27e7\n-}\n\ndata Wrapper (A : Set\u2080) : Set\u2080 where\n  wrap : A \u2192 Wrapper A\n\nidWrapper : \u2200 {A} \u2192 Wrapper A \u2192 Wrapper A\nidWrapper (wrap x) = wrap x\n\ndata [Wrapper] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Wrapper A \u2192 Set\u2080 where\n  [wrap] : (A\u209a [\u2192] [Wrapper] A\u209a) wrap\n\n[Wrapper]-env = record (\u03b5 1)\n  { pDef = [ quote Wrapper \u2254 quote [Wrapper] ] id\n  ; pConP = [ quote wrap \u2254 con (quote [wrap])  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 2 (quote [wrap]) ] con\n  }\n\nunquoteDecl [idWrapper] =\n  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]\n\n  {-\n[idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n-- [idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n-}\n\n{-\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6Wrapper\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) Wrapper Wrapper\n\nprivate\n  \u27e6Wrapper\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6Wrapper\u27e7 where\n  \u27e6wrap\u27e7 : unquote (\u27e6Wrapper\u27e7-ctor (quote Wrapper.wrap))\n-}\ndata \u27e6Wrapper\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Wrapper A\u2080 \u2192 Wrapper A\u2081 \u2192 Set\u2080 where\n  \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A\u1d63) wrap wrap\n\n\u27e6Wrapper\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] id\n  ; pConP = [ quote wrap \u2254 con (quote \u27e6wrap\u27e7)  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 3 (quote \u27e6wrap\u27e7) ] con\n  }\n\n\u27e6idWrapper\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\n\u27e6idWrapper\u27e71 {x0} {x1} (x2) {._} {._} (\u27e6wrap\u27e7 {x3} {x4} x5)\n  = \u27e6wrap\u27e7 {_} {_} {_} {x3} {x4} x5\n\n\u27e6idWrapper\u27e7-clauses =\n  clause\n  (arg (arg-info hidden  relevant) (var \"A0\") \u2237\n   arg (arg-info hidden  relevant) (var \"A1\") \u2237\n   arg (arg-info visible relevant) (var \"Ar\") \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info visible relevant)\n   (con (quote \u27e6wrap\u27e7)\n    (arg (arg-info hidden  relevant) (var \"x0\") \u2237\n     arg (arg-info hidden  relevant) (var \"x1\") \u2237\n     arg (arg-info visible relevant) (var \"xr\") \u2237 []))\n   \u2237 [])\n  (con (quote \u27e6wrap\u27e7)\n   (arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant)  (var 2 []) \u2237\n    arg (arg-info hidden relevant)  (var 1 []) \u2237\n    arg (arg-info visible relevant) (var 0 []) \u2237 []))\n  \u2237 []\n\n\u27e6idWrapper\u27e72 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\nunquoteDef \u27e6idWrapper\u27e72 = \u27e6idWrapper\u27e7-clauses\n\nunquoteDecl \u27e6idWrapper\u27e7 =\n  param-rec-def-by-name \u27e6Wrapper\u27e7-env (quote idWrapper) \u27e6idWrapper\u27e7\n\ndata Bot (A : Set\u2080) : Set\u2080 where\n  bot : Bot A \u2192 Bot A\n\ngobot : \u2200 {A} \u2192 Bot A \u2192 A\ngobot (bot x) = gobot x\n\ndata [Bot] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Bot A \u2192 Set\u2080 where\n  [bot] : ([Bot] A\u209a [\u2192] [Bot] A\u209a) bot\n\n[Bot]-env = record (\u03b5 1)\n  { pDef = [ quote Bot \u2254 quote [Bot] ] id\n  ; pConP = [ quote bot \u2254 con (quote [bot])  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 2 (quote [bot]) ] con\n  }\n\n[gobot]' : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n[gobot]' {x0} (x1) {._} ([bot] {x2} x3)\n  = [gobot]' {x0} x1 {x2} x3\n\n-- [gobot]' = {!showClauses \"[gobot]'\" (param-rec-clauses-by-name [Bot]-env (quote gobot) (quote [gobot]'))!}\n\n[gobot]2 : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n\n[gobot]2-clauses =\n  clause\n    (arg (arg-info hidden  relevant) (var \"A\u1d620\") \u2237\n     arg (arg-info visible relevant) (var \"A\u1d63\") \u2237\n     arg (arg-info hidden  relevant) dot \u2237\n     arg (arg-info visible relevant)\n     (con (quote [bot])\n      (arg (arg-info hidden  relevant) (var \"x\u1d620\") \u2237\n       arg (arg-info visible relevant) (var \"x\u1d63\") \u2237 []))\n     \u2237 [])\n    (def (quote [gobot]2)\n     (arg (arg-info hidden  relevant) (var 4 []) \u2237\n      arg (arg-info visible relevant) (var 3 []) \u2237\n      arg (arg-info hidden  relevant) (var 1 []) \u2237\n      arg (arg-info visible relevant) (var 0 []) \u2237 []))\n    \u2237 []\n\nunquoteDef [gobot]2 = [gobot]2-clauses\n\nunquoteDecl [gobot] =\n  param-rec-def-by-name [Bot]-env (quote gobot) [gobot]\n\ndata \u27e6Bot\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Bot A\u2080 \u2192 Bot A\u2081 \u2192 Set\u2080 where\n  \u27e6bot\u27e7 : (\u27e6Bot\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Bot\u27e7 A\u1d63) bot bot\n\n\u27e6Bot\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Bot \u2254 quote \u27e6Bot\u27e7 ] id\n  ; pConP = [ quote bot \u2254 con (quote \u27e6bot\u27e7)  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 3 (quote \u27e6bot\u27e7) ] con\n  }\n\n\u27e6gobot\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Bot\u27e7 A \u27e6\u2192\u27e7 A) gobot gobot\n\u27e6gobot\u27e71 {x0} {x1} x2 {._} {._} (\u27e6bot\u27e7 {x3} {x4} x5)\n  = \u27e6gobot\u27e71 {x0} {x1} x2 {x3} {x4} x5\n\nunquoteDecl \u27e6gobot\u27e7 =\n  param-rec-def-by-name \u27e6Bot\u27e7-env (quote gobot) \u27e6gobot\u27e7\n\nid\u2080 : {A : Set\u2080} \u2192 A \u2192 A\nid\u2080 x = x\n\n\u27e6id\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A \u27e6\u2192\u27e7 A) id\u2080 id\u2080\n\u27e6id\u2080\u27e7 = \u03bb {x\u2081} {x\u2082} x\u1d63 {x\u2083} {x\u2084} x\u1d63\u2081 \u2192 x\u1d63\u2081\n\ndata List\u2080 (A : Set) : Set where\n  []  : List\u2080 A\n  _\u2237_ : A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} (f : A \u2192 B) (xs : List\u2080 A) \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\n-- idList\u2080 : List\u2080 \u2115 \u2192 List\u2080 \u2115\nidList\u2080 []       = []\nidList\u2080 {A} (x \u2237 xs) = idList\u2080 {A} xs\n\ndata \u27e6List\u2080\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080) : List\u2080 A\u2080 \u2192 List\u2080 A\u2081 \u2192 Set\u2080 where\n  \u27e6[]\u27e7  : \u27e6List\u2080\u27e7 A\u1d63 [] []\n  _\u27e6\u2237\u27e7_ : (A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63) _\u2237_ _\u2237_\n\ncon\u27e6List\u2080\u27e7 = conSkip' 3\n\u27e6List\u2080\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ]\n          ([ quote \u2115     \u2254 quote \u27e6\u2115\u27e7 ]\n          ([ quote id\u2080   \u2254 quote \u27e6id\u2080\u27e7 ] id))\n  ; pConP = [ quote List\u2080.[]  \u2254 con (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  ; pConT = [ quote List\u2080.[]  \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  }\n\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\n-- \u27e6idList\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A) idList\u2080 idList\u2080\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n\n{-\n\u27e6map\u2080\u27e7 : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : List\u2080 (x0)} \u2192 {x10 : List\u2080 (x1)} \u2192 (x11 : \u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 \u27e6List\u2080\u27e7 {x3} {x4} (x5) (map\u2080 {x0} {x3} (x6) (x9)) (map\u2080 {x1} {x4} (x7) (x10))\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (\u27e6[]\u27e7 )  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (_\u27e6\u2237\u27e7_ {x11} {x12} (x13) {x14} {x15} (x16) )  = _\u27e6\u2237\u27e7_ {x6 (x11)} {x7 (x12)} (x8 {x11} {x12} (x13)) {map\u2080 {x0} {x3} (x6) (x14)} {map\u2080 {x1} {x4} (x7) (x15)} (\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {x14} {x15} (x16))\n-}\n\nunquoteDecl \u27e6map\u2080\u27e7 = param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) \u27e6map\u2080\u27e7\n\n{-\n  -- The generated type is bigger since it is a familly for no reason.\n  data \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\n  private\n    \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\n  data \u27e6List\u2080\u27e7 where\n    \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n    _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n  \u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                        [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n             (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n-}\n\ndata Maybe' (A : Set) : Set\u2081 where\n  nothing : Maybe' A\n  just    : A \u2192 Maybe' A\n\n{-\n-- Set\u2081 is here because \u27e6List\u2080\u27e7 is not using parameters, hence gets bigger.\n-- This only happens without-K given the new rules for data types.\ndata List\u2080 : (A : Set) \u2192 Set\u2081 where\n  []  : \u2200 {A} \u2192 List\u2080 A\n  _\u2237_ : \u2200 {A} \u2192 A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} \u2192 (A \u2192 B) \u2192 List\u2080 A \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\nidList\u2080 []       = []\nidList\u2080 (x \u2237 xs) = x \u2237 idList\u2080 xs\n\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\nprivate\n  \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6List\u2080\u27e7 where\n  \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n  _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n\u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                      [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n           (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n\n-- test = \u27e6[]\u27e7 {{!showType (type (quote List\u2080.[]))!}} {{!!}} {!!}\n-}\n\n{-\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n-}\n\n{-\n\u27e6map\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote map\u2080)))\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {[]} {[]} \u27e6[]\u27e7\n  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {._ \u2237 ._}\n  {._ \u2237 ._}\n  (_\u27e6\u2237\u27e7_ {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085 {x\u2081\u2086} {x\u2081\u2087} x\u2081\u2088)\n  = _\u27e6\u2237\u27e7_ {x\u2081\u2080 x\u2081\u2083} {x\u2081\u2081 x\u2081\u2084} (x\u2081\u2082 {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085)\n    {map\u2080 {x\u2084} {x\u2087} x\u2081\u2080 x\u2081\u2086} {map\u2080 {x\u2085} {x\u2088} x\u2081\u2081 x\u2081\u2087}\n    (\u27e6map\u2080\u27e7 {x\u2084} {x\u2085} {x\u2086} {x\u2087} {x\u2088} {x\u2089} {x\u2081\u2080} {x\u2081\u2081} x\u2081\u2082 {x\u2081\u2086} {x\u2081\u2087}\n     x\u2081\u2088)\n-}\n\n{-\nunquoteDef \u27e6map\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote \u27e6map\u2080\u27e7)\n-}\n\n\n{-\nfoo : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : Reflection.Param.List\u2080 (x0)} \u2192 {x10 : Reflection.Param.List\u2080 (x1)} \u2192 (x11 : Reflection.Param.\u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 Reflection.Param.\u27e6List\u2080\u27e7 {x3} {x4} (x5) (Reflection.Param.map\u2080 {x0} {x3} (x6) (x9)) (Reflection.Param.map\u2080 {x1} {x4} (x7) (x10))\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7 )  = Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x} {x} (x) {xs} {xs} (xs) )  = Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x0 (x0)} {x0 (x0)} (x0 {x0} {x0} (x0)) {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} (Reflection.Param.test' {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0))\n-}\n\n-- test' = {! showFunDef \"foo\" (param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote test'))!}\n\nopen import Function.Param.Unary\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\n\nrevealed-[\u2192]' : \u2200 (A : Set\u2080) (A\u209a : A \u2192 Set\u2080)\n                  (B : Set\u2080) (B\u209a : B \u2192 Set\u2080)\n                  (f : A \u2192 B) \u2192 Set\u2080\nunquoteDef revealed-[\u2192]' = Get-clauses.from-def revealed-[\u2192]\n\nrevelator-[\u2192] : ({A : Set} (A\u209a : A \u2192 Set) {B : Set} (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set)\n              \u2192  (A : Set) (A\u209a : A \u2192 Set) (B : Set) (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set\nunquoteDef revelator-[\u2192] = Revelator.clauses (type (quote _[\u2080\u2192\u2080]_))\n\np-[\u2192]-type = param-type-by-name    (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]      = param-clauses-by-name (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\n\np-[\u2192]' = \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n           {A\u209a : A \u2192 Set\u2080}  (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 A\u209a x \u2192 Set\u2080)\n           {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n           {B\u209a : B \u2192 Set\u2080}  (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 B\u209a x \u2192 Set\u2080)\n           {f : A \u2192 B}      (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n         \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f\n         \u2192 Set\n\np-[\u2192]'-test : p-[\u2192]' \u2261 unquote (unEl p-[\u2192]-type)\np-[\u2192]'-test = refl\n\n[[\u2192]] : unquote (unEl p-[\u2192]-type)\nunquoteDef [[\u2192]] = p-[\u2192]\n\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : [[\u2192]] [\u2115] [[\u2115]] [\u2115] [[\u2115]] [suc] [suc]\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7          \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\npred' : \u2115 \u2192 \u2115\npred' = \u03bb { zero    \u2192 zero\n          ; (suc m) \u2192 m }\n\n\u27e6pred'\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred' pred'\nunquoteDef \u27e6pred'\u27e7 = param-clauses-by-name defEnv2 (quote pred')\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7-type = unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7)))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : \u27e6\u27e6Set\u2080\u27e7\u27e7-type \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = param-clauses-by-name defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261-no-hints Get-term.from-clauses p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7'' : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\nunquoteDef \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\n\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' : _\u2261_ {A = \u27e6\u27e6Set\u2080\u27e7\u27e7-type} \u27e6\u27e6Set\u2080\u27e7\u27e7'' \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' = refl\n\ntest-p0-\u27e6Set\u2080\u27e7 : pTerm defEnv0 (quoteTerm \u27e6Set\u2080\u27e7) \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-p0-\u27e6Set\u2080\u27e7 = refl\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\np1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n[\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\ntest-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\ntest-p1\u2115\u2192\u2115 = refl\n\np2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\ntest-p2\u2115\u2192\u2115 = refl\n\np\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261-no-hints quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 = refl\nZERO : Set\u2081\nZERO = (A : Set\u2080) \u2192 A\n\u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n\u27e6ZERO\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\npZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\nq\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\ntest-pZERO : pZERO \u2261-no-hints q\u27e6ZERO\u27e7\ntest-pZERO = refl\nID : Set\u2081\nID = (A : Set\u2080) \u2192 A \u2192 A\n\u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n\u27e6ID\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n  \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\npID = pTerm (\u03b5 2) (quoteTerm ID)\nq\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\ntest-ID : q\u27e6ID\u27e7 \u2261-no-hints pID\ntest-ID = refl\n\n\u27e6not\u27e7' : (\u27e6\ud835\udfda\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) not not\nunquoteDef \u27e6not\u27e7' = param-clauses-by-name defEnv2 (quote not)\ntest-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 \u27e6not\u27e7' x\u1d63\ntest-not \u27e60\u2082\u27e7 = refl\ntest-not \u27e61\u2082\u27e7 = refl\n\n[pred]' : ([\u2115] [\u2192] [\u2115]) pred\nunquoteDef [pred]' = param-clauses-by-name defEnv1 (quote pred)\n\ntest-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\ntest-p1-pred [zero]     = refl\ntest-p1-pred ([suc] n\u209a) = refl\n\n\u27e6pred\u27e7' : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred pred\nunquoteDef \u27e6pred\u27e7' = param-clauses-by-name defEnv2 (quote pred)\n\ntest-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\ntest-p2-pred \u27e6zero\u27e7     = refl\ntest-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\np\/2 = param-rec-def-by-name defEnv2 (quote _\/2)\nq\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\nunquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\ntest-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261-def-no-hints q\u27e6\/2\u27e7\ntest-\/2 = refl\ntest-\/2' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\ntest-\/2' \u27e6zero\u27e7 = refl\ntest-\/2' (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\ntest-\/2' (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) rewrite test-\/2' n\u1d63 = refl\n\np+ = param-rec-def-by-name defEnv2 (quote _+\u2115_)\nq\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\nunquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\ntest-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261-def-no-hints q\u27e6+\u27e7\ntest-+ = refl\ntest-+' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\ntest-+' \u27e6zero\u27e7    n'\u1d63 = refl\ntest-+' (\u27e6suc\u27e7 n\u1d63) n'\u1d63 rewrite test-+' n\u1d63 n'\u1d63 = refl\n\n{-\nis-good : String \u2192 \ud835\udfda\nis-good \"good\" = 1\u2082\nis-good _      = 0\u2082\n\n\u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n\u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n\u27e6is-good\u27e7 {_}      refl = {!!}\n\nmy-good = unquote (lit (string \"good\"))\nmy-good-test : my-good \u2261 \"good\"\nmy-good-test = refl\n-}\n\n{-\n\u27e6is-good\u27e7' : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\nunquoteDef \u27e6is-good\u27e7' = param-clauses-by-name defEnv2 (quote is-good)\ntest-is-good = {!!}\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e4e6824d73e23b701e1ba92ba7fbe57713592123","subject":"Added examples for an issue.","message":"Added examples for an issue.\n\nIgnore-this: d41f2b18e5032bdc71396e32315466b7\n\ndarcs-hash:20120619011758-3bd4e-b614e3f5267122084c9b6e13b2bb79830a0f802a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/BadErase.agda","new_file":"Issues\/BadErase.agda","new_contents":"------------------------------------------------------------------------------\n-- The translation is badly erasing the universal quantification\n------------------------------------------------------------------------------\n\n-- Found on 14 March 2012.\n\nmodule Issues.BadErase where\n\npostulate\n  _\u2194_ : Set \u2192 Set \u2192 Set\n  D   : Set\n  A   : Set\n\npostulate bad\u2081 : ((x : D) \u2192 A) \u2192 A\n{-# ATP prove bad\u2081 #-}\n\npostulate bad\u2082 : A \u2192 ((x : D) \u2192 A)\n{-# ATP prove bad\u2082 #-}\n\npostulate bad\u2083 : ((x : D) \u2192 A) \u2194 A\n{-# ATP prove bad\u2083 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The translation is badly erasing the universal quantification\n------------------------------------------------------------------------------\n\n-- Found on 14 March 2012.\n\nmodule Issues.BadErase where\n\npostulate\n  D : Set\n  A : Set\n\npostulate bad : ((x : D) \u2192 A) \u2192 A\n{-# ATP prove bad #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c312b0e9806924742644194bb8a96393e20e9309","subject":"composable: More abstraction madness","message":"composable: More abstraction madness\n","repos":"crypto-agda\/crypto-agda","old_file":"composable.agda","new_file":"composable.agda","new_contents":"module composable where\n\nopen import Level\nopen import Function\nopen import Data.Unit using (\u22a4)\nopen import Relation.Binary\n\nArrow : \u2200 {i} \u2192 Set i \u2192 \u2200 j \u2192 Set (suc j \u2294 i)\nArrow = Rel\n\nComposition : \u2200 {a \u2113} {A : Set a} \u2192 Arrow A \u2113 \u2192 Set _\nComposition = Transitive\n\nIdentity : \u2200 {a \u2113} {A : Set a} \u2192 Arrow A \u2113 \u2192 Set _\nIdentity = Reflexive\n\nIArrow : \u2200 {i j t} {I : Set i} (_\u219d\u1d62_ : Arrow I j) (T : I \u2192 Set t) a \u2192 Set _\nIArrow _\u219d\u1d62_ T a = \u2200 {i\u2080 i\u2081} \u2192 i\u2080 \u219d\u1d62 i\u2081 \u2192 T i\u2080 \u2192 T i\u2081 \u2192 Set a\n\nIReflexivity : \u2200 {a i j t} {I : Set i} {R : Rel I j} {T : I \u2192 Set t} \u2192 Reflexive R \u2192 IArrow R T a \u2192 Set _\nIReflexivity R-refl Arr = \u2200 {i A} \u2192 Arr (R-refl {i}) A A\n\nIIdentity : \u2200 {a i j t} {I : Set i} {_\u219d\u1d62_ : Arrow I j} {T : I \u2192 Set t} \u2192 Identity _\u219d\u1d62_ \u2192 IArrow _\u219d\u1d62_ T a \u2192 Set _\nIIdentity = IReflexivity\n\nITrans : \u2200 {i j t a} {I : Set i} {R\u2080 R\u2081 R\u2082 : Rel I j} {T : I \u2192 Set t}\n           (R-trans : Trans R\u2080 R\u2081 R\u2082)\n           (Arr\u2080 : IArrow R\u2080 T a)\n           (Arr\u2080 : IArrow R\u2081 T a)\n           (Arr\u2080 : IArrow R\u2082 T a)\n         \u2192 Set _\nITrans R-trans Arr\u2080 Arr\u2081 Arr\u2082\n  = \u2200 {i\u2080 i\u2081 i\u2082 j\u2080 j\u2081} \u2192 Trans (Arr\u2080 j\u2080) (Arr\u2081 j\u2081) (Arr\u2082 (R-trans {i\u2080} {i\u2081} {i\u2082} j\u2080 j\u2081))\n\nITransitive : \u2200 {i j t a} {I : Set i} {R : Rel I j} {T : I \u2192 Set t}\n                \u2192 Transitive R \u2192 IArrow R T a \u2192 Set _\nITransitive {R = R} R-trans Arr = ITrans {R\u2080 = R} {R} {R} R-trans Arr Arr Arr\n\nIComposition : \u2200 {i j t a} {I : Set i} {_\u219d\u1d62_ : Arrow I j} {T : I \u2192 Set t}\n                   (_\u00b7_ : Composition _\u219d\u1d62_)\n                   (\u27e8_\u27e9_\u219d_ : IArrow _\u219d\u1d62_ T a) \u2192 Set _\nIComposition = ITransitive\n\nrecord IComposable {i j t a} {I : Set i} {_\u219d\u1d62_ : Arrow I j} {T : I \u2192 Set t}\n                   (_\u00b7_ : Composition _\u219d\u1d62_)\n                   (\u27e8_\u27e9_\u219d_ : IArrow _\u219d\u1d62_ T a)\n                 : Set (a \u2294 t \u2294 i \u2294 j) where\n  constructor mk\n  infixr 1 _>>>_\n  field\n--    _>>>_ : \u2200 {i\u2080 i\u2081 i\u2082} {ix\u2080 : i\u2080 \u219d\u1d62 i\u2081} {ix\u2081 : i\u2081 \u219d\u1d62 i\u2082} {A B C}\n--            \u2192 (\u27e8 ix\u2080 \u27e9 A \u219d B) \u2192 (\u27e8 ix\u2081 \u27e9 B \u219d C) \u2192 (\u27e8 ix\u2080 \u00b7 ix\u2081 \u27e9 A \u219d C)\n    _>>>_ : IComposition (\u03bb {\u03b7} \u2192 _\u00b7_ {\u03b7}) \u27e8_\u27e9_\u219d_\n\nopen import Relation.Binary.PropositionalEquality\nRefl-Unit : \u2200 {\u2113 a} {A : Set a} {R : Rel A \u2113} \u2192 Reflexive R \u2192 Transitive R \u2192 Set _\nRefl-Unit {R = R} R-refl R-trans = \u2200 {x y} (p : R x y) \u2192 R-trans R-refl p \u2261 p\n\n{-\nrecord ICat        {i j t a} {I : Set i} {_\u219d\u1d62_ : Arrow I j} {T : I \u2192 Set t}\n                   {_\u00b7_ : Composition _\u219d\u1d62_}\n                   {\u27e8_\u27e9_\u219d_ : IArrow a _\u219d\u1d62_ T}\n                   (comp : IComposable _\u00b7_ \u27e8_\u27e9_\u219d_)\n                   {id\u1d62 : Identity _\u219d\u1d62_}\n                   (id : IIdentity (\u03bb {\u03b7} \u2192 id\u1d62 {\u03b7}) \u27e8_\u27e9_\u219d_)\n                   (_\u2248\u1d62_ : \u2200 {a b} (i j : a \u219d\u1d62 b) \u2192 Set)\n                   (_\u2248_ : \u2200 {i j A B} \u2192 i \u2248\u1d62 j \u2192 \u27e8 i \u27e9 A \u219d B \u2192 \u27e8 j \u27e9 A \u219d B \u2192 Set) : Set\n          where\n  constructor mk\n  open IComposable comp\n  field\n    id-unit->>> :\n      \u2200 f \u2192 \u27e8 id >>> f \u2248 f\n-}\nopen import Data.Unit using (\u22a4)\n\nConstArr : \u2200 {a} (A : Set a) \u2192 \u22a4 \u2192 \u22a4 \u2192 Set a\nConstArr A _ _ = A\n\nComposable : \u2200 {t a} {T : Set t} (_\u219d_ : T \u2192 T \u2192 Set a) \u2192 Set (t \u2294 a)\nComposable _\u219d_ = IComposable {i = zero} {_\u219d\u1d62_ = ConstArr \u22a4} _ (const _\u219d_)\n\n{- Composable, unfolded:\nrecord Composable {t a} {T : Set t} (_\u219d_ : T \u2192 T \u2192 Set a) : Set (t \u2294 a) where\n  constructor mk\n  infixr 1 _>>>_\n  field\n    _>>>_ : \u2200 {A B C} \u2192 (A \u219d B) \u2192 (B \u219d C) \u2192 (A \u219d C)\n-}\n\nconstComp' : \u2200 {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) \u2192 Composition (ConstArr A)\nconstComp' _\u00b7_ = _\u00b7_\n\nconstComp : \u2200 {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) \u2192 Composable (ConstArr A)\nconstComp _\u00b7_ = mk _\u00b7_\n\nmodule Composable = IComposable\n\nixFunComp : \u2200 {ix t} {Ix : Set ix} (F : Ix \u2192 Set t) \u2192 Composable (\u03bb i o \u2192 F i \u2192 F o)\nixFunComp _ = mk (\u03bb f g x \u2192 g (f x))\n\nfunComp : \u2200 {t} \u2192 Composable (\u03bb (A B : Set t) \u2192 A \u2192 B)\nfunComp = ixFunComp id\n\nopComp : \u2200 {t a} {T : Set t} {_\u219d_ : T \u2192 T \u2192 Set a} \u2192 Composable _\u219d_ \u2192 Composable (flip _\u219d_)\nopComp (mk _>>>_) = mk (flip _>>>_)\n\nopen import Data.Vec\nvecFunComp : \u2200 {a} (A : Set a) \u2192 Composable (\u03bb i o \u2192 Vec A i \u2192 Vec A o)\nvecFunComp A = ixFunComp (Vec A)\n\nopen import Data.Bits\nbitsFunComp : Composable (\u03bb i o \u2192 Bits i \u2192 Bits o)\nbitsFunComp = ixFunComp Bits\n\n-- open import Data.Fin\n-- funRewireComp : Composable (\u03bb i o \u2192 Fin o \u2192 Fin i)\n-- FunRewireComp = opComp (ixFunComp Fin)\n\n{-\nopen import bintree\n\nopen import Data.Nat\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRewireTbl : CircuitType\nRewireTbl i o = Vec (Fin i) o\n\nrewireTblComp : Composable RewireTbl\nrewireTblComp = {!!}\n-}\n","old_contents":"module composable where\n\nopen import Level\nopen import Function\nopen import Data.Unit using (\u22a4)\n\n-- Monoid M \u2245 Cat (ConstArr M)\n\nComposition : \u2200 {i a} {I : Set i} (_\u219d_ : I \u2192 I \u2192 Set a) \u2192 Set (i \u2294 a)\nComposition _\u219d_ = \u2200 {A B C} \u2192 (A \u219d B) \u2192 (B \u219d C) \u2192 (A \u219d C)\n\nrecord IComposable {i j s t} {I : Set i} {_\u219d\u1d62_ : I \u2192 I \u2192 Set j} {S : I \u2192 Set s}\n                   (_\u00b7_ : Composition _\u219d\u1d62_)\n                   (\u27e8_\u27e9_\u219d_ : \u2200 {i\u2080 i\u2081} \u2192 i\u2080 \u219d\u1d62 i\u2081 \u2192 S i\u2080 \u2192 S i\u2081 \u2192 Set t) : Set (t \u2294 s \u2294 i \u2294 j) where\n  constructor mk\n  infixr 1 _>>>_\n  field\n    _>>>_ : \u2200 {i\u2080 i\u2081 i\u2082} {ix\u2080 : i\u2080 \u219d\u1d62 i\u2081} {ix\u2081 : i\u2081 \u219d\u1d62 i\u2082} {A B C}\n            \u2192 (\u27e8 ix\u2080 \u27e9 A \u219d B) \u2192 (\u27e8 ix\u2081 \u27e9 B \u219d C) \u2192 (\u27e8 ix\u2080 \u00b7 ix\u2081 \u27e9 A \u219d C)\n\nopen import Data.Unit using (\u22a4)\n\nConstArr : \u2200 {a} (A : Set a) \u2192 \u22a4 \u2192 \u22a4 \u2192 Set a\nConstArr A _ _ = A\n\nComposable : \u2200 {s t} {S : Set s} (_\u219d_ : S \u2192 S \u2192 Set t) \u2192 Set (s \u2294 t)\nComposable _\u219d_ = IComposable {i = zero} {_\u219d\u1d62_ = ConstArr \u22a4} _ (const _\u219d_)\n\n{- Composable, unfolded:\nrecord Composable {s t} {S : Set s} (_\u219d_ : S \u2192 S \u2192 Set t) : Set (s \u2294 t) where\n  constructor mk\n  infixr 1 _>>>_\n  field\n    _>>>_ : \u2200 {A B C : S} \u2192 (A \u219d B) \u2192 (B \u219d C) \u2192 (A \u219d C)\n-}\n\nconstComp' : \u2200 {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) \u2192 Composition (ConstArr A)\nconstComp' _\u00b7_ = _\u00b7_\n\nconstComp : \u2200 {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) \u2192 Composable (ConstArr A)\nconstComp _\u00b7_ = mk _\u00b7_\n\nmodule Composable = IComposable\n\nixFunComp : \u2200 {ix s} {Ix : Set ix} (F : Ix \u2192 Set s) \u2192 Composable (\u03bb i o \u2192 F i \u2192 F o)\nixFunComp _ = mk (\u03bb f g x \u2192 g (f x))\n\nfunComp : \u2200 {s} \u2192 Composable (\u03bb (A B : Set s) \u2192 A \u2192 B)\nfunComp = ixFunComp id\n\nopComp : \u2200 {s t} {S : Set s} {_\u219d_ : S \u2192 S \u2192 Set t} \u2192 Composable _\u219d_ \u2192 Composable (flip _\u219d_)\nopComp (mk _>>>_) = mk (flip _>>>_)\n\nopen import Data.Vec\nvecFunComp : \u2200 {a} (A : Set a) \u2192 Composable (\u03bb i o \u2192 Vec A i \u2192 Vec A o)\nvecFunComp A = ixFunComp (Vec A)\n\nopen import Data.Bits\nbitsFunComp : Composable (\u03bb i o \u2192 Bits i \u2192 Bits o)\nbitsFunComp = ixFunComp Bits\n\n-- open import Data.Fin\n-- funRewireComp : Composable (\u03bb i o \u2192 Fin o \u2192 Fin i)\n-- FunRewireComp = opComp (ixFunComp Fin)\n\n{-\nopen import bintree\n\nopen import Data.Nat\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRewireTbl : CircuitType\nRewireTbl i o = Vec (Fin i) o\n\nrewireTblComp : Composable RewireTbl\nrewireTblComp = {!!}\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ea35ca82c47da6c485dee266192a700be4ab6a64","subject":"Inline \u0394Type.","message":"Inline \u0394Type.\n\nOld-commit-hash: 01465077ba29a751a7f67c946f2516def3753a94\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff apply {base \u03b9} = diff , apply\n  lift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n  lift-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff apply {base \u03b9} = diff , apply\n  lift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n  lift-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5957a0a5f21ebc9ad88ea001c4afa6b381372c04","subject":"Document P.C.Type.","message":"Document P.C.Type.\n\nOld-commit-hash: 11e38d018843f21411d014020ce42181e1f77d32\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Type.agda","new_file":"Parametric\/Change\/Type.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Simply-typed changes (Fig. 3 and Fig. 4d)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\n\nmodule Parametric.Change.Type\n  (Base : Type.Structure)\n  where\n\nopen Type.Structure Base\n\n-- Extension point: Simply-typed changes of base types.\nStructure : Set\nStructure = Base \u2192 Base\n\nmodule Structure (\u0394Base : Structure) where\n  -- We provide: Simply-typed changes on simple types.\n  \u0394Type : Type \u2192 Type\n  \u0394Type (base \u03b9) = base (\u0394Base \u03b9)\n  \u0394Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394Type \u03c3 \u21d2 \u0394Type \u03c4\n\n  -- And we also provide context merging.\n  open import Base.Change.Context \u0394Type public\n","old_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Change.Type\n  (Base : Type.Structure)\n  where\n\nopen Type.Structure Base\n\nStructure : Set\nStructure = Base \u2192 Base\n\nmodule Structure (\u0394Base : Structure) where\n  \u0394Type : Type \u2192 Type\n  \u0394Type (base \u03b9) = base (\u0394Base \u03b9)\n  \u0394Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394Type \u03c3 \u21d2 \u0394Type \u03c4\n\n  open import Base.Change.Context \u0394Type public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"22c2c8987fb43682069ff83b47bf139b730261ee","subject":"Data.Maybe.NP: export monad plus in M?","message":"Data.Maybe.NP: export monad plus in M?\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc)\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc)\n\nmodule M? \u2113 where\n  open Cat.RawMonad (monad {\u2113}) public\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"27037592477c7d9112554d56337e06c5a2c6f0b9","subject":"Remove remaining duplication between abs\u2081 ... abs\u2086","message":"Remove remaining duplication between abs\u2081 ... abs\u2086\n\nThis relies on Agda 2.4.0's varying arity in a more essential way.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_])\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    -- A better type for absN but a mess to use due to the proofs (which aren't synthesized, even though maybe they should be?)\n    -- absN : (n : \u2115) \u2192 {_ : n > 0} \u2192 absNType n\n    -- XXX See \"how to keep your neighbours in order\" for tricks.\n\n    -- Please the termination checker by keeping this case separate.\n    absNBase : \u2200 {\u03c4\u2081} \u2192 absNType [ \u03c4\u2081 ]\n    absNBase {\u03c4\u2081} f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    -- Otherwise, the recursive step of absN would invoke absN twice, and the\n    -- termination checker does not figure out that the calls are in fact\n    -- terminating.\n\n    -- What I'd like to write, avoiding the need for absNBase, but can't because of the termination checker.\n    {-\n    absN {zero}  (\u03c4\u2081 \u2237 []) f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n      absN (\u03c4\u2081 \u2237 []) (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n    -}\n    --What I have to write instead:\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ (suc n)) \u2192 absNType \u03c4s\n    absN {zero}  (\u03c4\u2081 \u2237 []) = absNBase\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n      absNBase (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n    -- Using a similar trick, we can declare absV which takes the N implicit\n    -- type arguments individually, collects them and passes them on to absN.\n    -- This is inspired by what's shown in the Agda 2.4.0 release notes, and\n    -- relies critically on support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType 0       \u03c4s = absNType \u03c4s\n    absVType (suc n) \u03c4s = {\u03c4\u1d62 : Type} \u2192 absVType n (\u03c4\u1d62 \u2237 \u03c4s)\n\n    -- XXX\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType (suc n) \u03c4s\n    absVAux \u03c4s zero    {\u03c4\u1d62} = absN (\u03c4\u1d62 \u2237 \u03c4s)\n    absVAux \u03c4s (suc n) {\u03c4\u1d62} = absVAux (\u03c4\u1d62 \u2237 \u03c4s) n\n\n    absV = absVAux []\n\n  open AbsNHelpers using (absV) public\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n\n  abs\u2081 = absV 0\n  abs\u2082 = absV 1\n  abs\u2083 = absV 2\n  abs\u2084 = absV 3\n  abs\u2085 = absV 4\n  abs\u2086 = absV 5\n\n{-\nabs\u2081\nHave: {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      Term \u0393 (\u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\nabs\u2082\nHave: {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n       Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      Term \u0393 (\u03c4\u2082 Type.Structure.\u21d2 \u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n-}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr)\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    -- A better type for absN but a mess to use due to the proofs (which aren't synthesized, even though maybe they should be?)\n    -- absN : (n : \u2115) \u2192 {_ : n > 0} \u2192 absNType n\n    -- XXX See \"how to keep your neighbours in order\" for tricks.\n\n    -- Please the termination checker by keeping this case separate.\n    absNBase : \u2200 {\u03c4\u2081} \u2192 absNType (\u03c4\u2081 \u2237 [])\n    absNBase {\u03c4\u2081} f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    -- Otherwise, the recursive step of absN would invoke absN twice, and the\n    -- termination checker does not figure out that the calls are in fact\n    -- terminating.\n\n  open AbsNHelpers using (absNType; absNBase)\n\n  -- XXX: could we be using the same trick as above to take the N implicit\n  -- arguments individually, rather than in a vector? I can't figure out how,\n  -- at least not without trying it. But it seems that's what's shown in Agda 2.4.0 release notes!\n  absN : {n : \u2115} \u2192 (\u03c4s : Vec _ (suc n)) \u2192 absNType \u03c4s\n  absN {zero}  (\u03c4\u2081 \u2237 []) = absNBase\n  absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n    absNBase (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  -- What I'd like to write, avoiding the need for absNBase, but can't because of the termination checker.\n  {-\n  absN {zero}  (\u03c4\u2081 \u2237 []) f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n  absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n    absN (\u03c4\u2081 \u2237 []) (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n  -}\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n  module _ {\u03c4\u2081 : Type} where\n    \u03c4s\u2081 =  \u03c4\u2081 \u2237 []\n    abs\u2081 : absNType \u03c4s\u2081\n    abs\u2081 = absN \u03c4s\u2081\n    module _ {\u03c4\u2082 : Type} where\n      \u03c4s\u2082 = \u03c4\u2082 \u2237 \u03c4s\u2081\n      abs\u2082 : absNType \u03c4s\u2082\n      abs\u2082 = absN \u03c4s\u2082\n      module _ {\u03c4\u2083 : Type} where\n        \u03c4s\u2083 = \u03c4\u2083 \u2237 \u03c4s\u2082\n        abs\u2083 : absNType \u03c4s\u2083\n        abs\u2083 = absN \u03c4s\u2083\n        module _ {\u03c4\u2084 : Type} where\n          \u03c4s\u2084 = \u03c4\u2084 \u2237 \u03c4s\u2083\n          abs\u2084 : absNType \u03c4s\u2084\n          abs\u2084 = absN \u03c4s\u2084\n          module _ {\u03c4\u2085 : Type} where\n            \u03c4s\u2085 = \u03c4\u2085 \u2237 \u03c4s\u2084\n            abs\u2085 : absNType \u03c4s\u2085\n            abs\u2085 = absN \u03c4s\u2085\n            module _ {\u03c4\u2086 : Type} where\n              \u03c4s\u2086 = \u03c4\u2086 \u2237 \u03c4s\u2085\n              abs\u2086 : absNType \u03c4s\u2086\n              abs\u2086 = absN \u03c4s\u2086\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3b48cfb7880c48f67527b01c85ad20363d89e8fa","subject":"Function.NP: Two","message":"Function.NP: Two\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/NP.agda","new_file":"lib\/Function\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Algebra\nopen import Algebra.Structures\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      renaming (Bool to \ud835\udfda)\nopen import Data.Product\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen Relation.Unary.Logical public using (_[\u2192]_; [\u03a0]; [\u03a0]e; [\u2200])\nopen Relation.Binary.Logical public using (_\u27e6\u2192\u27e7_; \u27e6\u03a0\u27e7; \u27e6\u03a0\u27e7e; \u27e6\u2200\u27e7)\n\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\n\u03a0\u03a0 : \u2200 {a b c} (A : \u2605 a) (B : A \u2192 \u2605 b) (C : \u03a3 A B \u2192 \u2605 c) \u2192 \u2605 _\n\u03a0\u03a0 A B C = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n\n\u03a0\u03a0\u03a0 : \u2200 {a b c d} (A : \u2605 a) (B : A \u2192 \u2605 b)\n                  (C : \u03a3 A B \u2192 \u2605 c) (D : \u03a3 (\u03a3 A B) C \u2192 \u2605 d) \u2192 \u2605 _\n\u03a0\u03a0\u03a0 A B C D = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : \u2605 a) (B : \u2605 b) \u2192 \u2605 (a L.\u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : \u2605 a) (n : \u2115) (B : \u2605 b) \u2192 \u2605 (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nEndo A = A \u2192 A\n\n[Endo] : \u2200 {a} \u2192 ([\u2605] {a} a [\u2192] [\u2605] _) Endo\n[Endo] A\u209a = A\u209a [\u2192] A\u209a\n\n\u27e6Endo\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Endo Endo\n\u27e6Endo\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\nCmp : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nCmp A = A \u2192 A \u2192 \ud835\udfda\n\n{- needs [\ud835\udfda] and \u27e6\ud835\udfda\u27e7 potentially move these to Data.Two\n\n[Cmp] : \u2200 {a} \u2192 ([\u2605] {a} a [\u2192] [\u2605] _ [\u2192] [\u2605] _) Cmp\n[Cmp] A\u209a = A\u209a [\u2192] A\u209a [\u2192] [\ud835\udfda]\n\n\u27e6Cmp\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Endo Endo\n\u27e6Cmp\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7\n-}\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : \u2605 a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = \u2261.refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = \u2261.refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = \u2261.refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = \u2261.refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = \u2261.refl\n  nest-+ (suc m) n = \u2261.cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = \u2261.cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = \u2261.refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 \u2261.refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 \u2261.cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 \u2261.refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 \u2261.refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261.\u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : \u2605 a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\n\u2026 : \u2200 {a} {A : \u2605 a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : \u2605 i} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : (x : A) \u2192 B x \u2192 \u2605 c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : \u2605 a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 \u2605 \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : \u2605 a) = EndoMonoid-\u2248 {A = A} \u2261.isEquivalence (\u2261.cong\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : \u2605 a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2261.\u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 \u2261.trans (p (h x)) (\u2261.cong g (q x)))\n\n\u27e6id\u27e7 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) id id\n\u27e6id\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6\u2218\u2032\u27e7 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 C\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 C\u1d63) \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 C\u1d63)) _\u2218\u2032_ _\u2218\u2032_\n\u27e6\u2218\u2032\u27e7 _ _ _ f\u1d63 g\u1d63 x\u1d63 = f\u1d63 (g\u1d63 x\u1d63)\n","old_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Algebra\nopen import Algebra.Structures\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Product\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen Relation.Unary.Logical public using (_[\u2192]_; [\u03a0]; [\u03a0]e; [\u2200])\nopen Relation.Binary.Logical public using (_\u27e6\u2192\u27e7_; \u27e6\u03a0\u27e7; \u27e6\u03a0\u27e7e; \u27e6\u2200\u27e7)\n\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\n\u03a0\u03a0 : \u2200 {a b c} (A : \u2605 a) (B : A \u2192 \u2605 b) (C : \u03a3 A B \u2192 \u2605 c) \u2192 \u2605 _\n\u03a0\u03a0 A B C = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n\n\u03a0\u03a0\u03a0 : \u2200 {a b c d} (A : \u2605 a) (B : A \u2192 \u2605 b)\n                  (C : \u03a3 A B \u2192 \u2605 c) (D : \u03a3 (\u03a3 A B) C \u2192 \u2605 d) \u2192 \u2605 _\n\u03a0\u03a0\u03a0 A B C D = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : \u2605 a) (B : \u2605 b) \u2192 \u2605 (a L.\u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : \u2605 a) (n : \u2115) (B : \u2605 b) \u2192 \u2605 (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nEndo A = A \u2192 A\n\n[Endo] : \u2200 {a} \u2192 ([\u2605] {a} a [\u2192] [\u2605] _) Endo\n[Endo] A\u209a = A\u209a [\u2192] A\u209a\n\n\u27e6Endo\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Endo Endo\n\u27e6Endo\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\nCmp : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nCmp A = A \u2192 A \u2192 Bool\n\n{- needs [Bool] and \u27e6Bool\u27e7 potentially move these to Data.Bool.NP\n\n[Cmp] : \u2200 {a} \u2192 ([\u2605] {a} a [\u2192] [\u2605] _ [\u2192] [\u2605] _) Cmp\n[Cmp] A\u209a = A\u209a [\u2192] A\u209a [\u2192] [Bool]\n\n\u27e6Cmp\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Endo Endo\n\u27e6Cmp\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Bool\u27e7\n-}\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : \u2605 a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = \u2261.refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = \u2261.refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = \u2261.refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = \u2261.refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = \u2261.refl\n  nest-+ (suc m) n = \u2261.cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = \u2261.cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = \u2261.refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 \u2261.refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 \u2261.cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 \u2261.refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 \u2261.refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261.\u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : \u2605 a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\n\u2026 : \u2200 {a} {A : \u2605 a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : \u2605 i} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : (x : A) \u2192 B x \u2192 \u2605 c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : \u2605 a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 \u2605 \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : \u2605 a) = EndoMonoid-\u2248 {A = A} \u2261.isEquivalence (\u2261.cong\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : \u2605 a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2261.\u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 \u2261.trans (p (h x)) (\u2261.cong g (q x)))\n\n\u27e6id\u27e7 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) id id\n\u27e6id\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6\u2218\u2032\u27e7 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 C\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 C\u1d63) \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 C\u1d63)) _\u2218\u2032_ _\u2218\u2032_\n\u27e6\u2218\u2032\u27e7 _ _ _ f\u1d63 g\u1d63 x\u1d63 = f\u1d63 (g\u1d63 x\u1d63)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ffe54a9fc4523ccfe8c3343695d569a1de89cfa7","subject":"Constants are values, so evaluating them takes 0 steps!","message":"Constants are values, so evaluating them takes 0 steps!\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV nat v1 v2 n = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat v1 v2 vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) zero n1 ()\neval-const-dec (lit v) (suc n0) .n0 refl = \u2264-step \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) zero n1 ()\neval-const-mono (lit v) (suc n0) .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) zero n1 ()\neval-const-strengthen (lit v) (suc n0) .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV nat v1 v2 n = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat v1 v2 vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (const (lit v)) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , suc zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"55b2f0787bf88248a412e30594fbe37030561ef9","subject":"Typo.","message":"Typo.\n\nIgnore-this: f327954fb8615da978c4ee2b943b6e71\n\ndarcs-hash:20120228172002-3bd4e-bc34798e02a6e5bb6901a2a5416c36d60541a168.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Existential.agda","new_file":"notes\/Existential.agda","new_contents":"-- Tested with the development version of Agda on 27 February 2012.\n\n-- Thm: (\u2203x)A(x), (\u2200x)(A(x) \u21d2 B(x)) \u22a2 (\u2203x)B(x)\n\n-- From: Elliott Mendelson. Introduction to mathematical logic. Chapman &\n-- Hall, 4th edition, 1997, p. 155.\n\n-- Rule A4: (\u2200x)A(x) \u21d2 A(t) (with some conditions)\n-- Rule E4: A(t) \u21d2 (\u2203x)A(x) (with some conditions)\n-- Rule C: From (\u2203x)A(x) to A(t) (with some conditions)\n\n-- 1. (\u2203x)A(x)          Hyp\n-- 2. (\u2200x)(A(x) \u21d2 B(x)) Hyp\n-- 3. A(b)              1, rule C\n-- 4. A(b) \u21d2 B(b)       2, rule A4\n-- 5. B(b)              4,3 MP\n-- 6. (\u2203x)B(x)          5, rule E4\n\nmodule Existential where\n\npostulate\n  D   : Set\n  A B : D \u2192 Set\n\nmodule Witness where\n\n  infixr 7 _,_\n\n  data \u2203 (A : D \u2192 Set) : Set where\n    _,_ : \u2200 x \u2192 A x \u2192 \u2203 A\n\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 x \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (x , Ax) h = h x Ax\n\n  -- A proof using the existential elimination.\n  prf\u2081 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2081 h\u2081 h\u2082 = \u2203-elim h\u2081 (\u03bb x Ax \u2192 x , (h\u2082 Ax))\n\n  -- A proof using pattern matching.\n  prf\u2082 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2082 (x , Ax) h = x , h Ax\n\nmodule NonWitness\u2081 where\n\n  data \u2203 (A : D \u2192 Set) : Set where\n    \u2203-intro : \u2200 {x} \u2192 A x \u2192 \u2203 A\n\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (\u2203-intro Ax) h = h Ax\n\n  -- A proof using the existential elimination.\n  prf\u2081 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2081 h\u2081 h\u2082 = \u2203-elim h\u2081 (\u03bb Ax \u2192 \u2203-intro (h\u2082 Ax))\n\n  -- A proof using pattern matching.\n  prf\u2082 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2082 (\u2203-intro Ax) h = \u2203-intro (h Ax)\n\nmodule NonWitness\u2082 where\n\n  -- We add 3 to the fixities of the standard library.\n  infix 6 \u00ac_\n\n  -- The empty type.\n  data \u22a5 : Set where\n\n  \u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n  \u22a5-elim ()\n\n  \u00ac_ : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  \u2203 : (D \u2192 Set) \u2192 Set\n  \u2203 A = \u00ac (\u2200 x \u2192 \u00ac (A x))\n\n  prf : \u2203 A \u2192 (\u2200 x \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf h\u2081 h\u2082 h\u2083 = h\u2081 (\u03bb x Ax \u2192 h\u2083 x (h\u2082 x Ax))\n","old_contents":"-- Tested with the development version of Agda on 27 February 2012.\n\n-- Thm: (\u2203x)A(x), (\u2200x)(A(x) \u21d2 B(x)) \u22a2 (\u2203x)B(x)\n\n-- From: Elliott Mendelson. Introduction to mathematical logic. Chapman &\n-- Hall, 4th edition, 1997, p. 155.\n\n-- Rule A4: (\u2200x)A(x) \u21d2 B(t) (with some conditions)\n-- Rule E4: A(t) \u21d2 (\u2203x)A(x) (with some conditions)\n-- Rule C: From (\u2203x)A(x) to A(t) (with some conditions)\n\n-- 1. (\u2203x)A(x)          Hyp\n-- 2. (\u2200x)(A(x) \u21d2 B(x)) Hyp\n-- 3. A(b)              1, rule C\n-- 4. A(b) \u21d2 B(b)       2, rule A4\n-- 5. B(b)              4,3 MP\n-- 6. (\u2203x)B(x)          5, rule E4\n\nmodule Existential where\n\npostulate\n  D   : Set\n  A B : D \u2192 Set\n\nmodule Witness where\n\n  infixr 7 _,_\n\n  data \u2203 (A : D \u2192 Set) : Set where\n    _,_ : \u2200 x \u2192 A x \u2192 \u2203 A\n\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 x \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (x , Ax) h = h x Ax\n\n  -- A proof using the existential elimination.\n  prf\u2081 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2081 h\u2081 h\u2082 = \u2203-elim h\u2081 (\u03bb x Ax \u2192 x , (h\u2082 Ax))\n\n  -- A proof using pattern matching.\n  prf\u2082 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2082 (x , Ax) h = x , h Ax\n\nmodule NonWitness\u2081 where\n\n  data \u2203 (A : D \u2192 Set) : Set where\n    \u2203-intro : \u2200 {x} \u2192 A x \u2192 \u2203 A\n\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (\u2203-intro Ax) h = h Ax\n\n  -- A proof using the existential elimination.\n  prf\u2081 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2081 h\u2081 h\u2082 = \u2203-elim h\u2081 (\u03bb Ax \u2192 \u2203-intro (h\u2082 Ax))\n\n  -- A proof using pattern matching.\n  prf\u2082 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2082 (\u2203-intro Ax) h = \u2203-intro (h Ax)\n\nmodule NonWitness\u2082 where\n\n  -- We add 3 to the fixities of the standard library.\n  infix 6 \u00ac_\n\n  -- The empty type.\n  data \u22a5 : Set where\n\n  \u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n  \u22a5-elim ()\n\n  \u00ac_ : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  \u2203 : (D \u2192 Set) \u2192 Set\n  \u2203 A = \u00ac (\u2200 x \u2192 \u00ac (A x))\n\n  prf : \u2203 A \u2192 (\u2200 x \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf h\u2081 h\u2082 h\u2083 = h\u2081 (\u03bb x Ax \u2192 h\u2083 x (h\u2082 x Ax))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"136efbaf5221fe6e7c1dae7424af260ceef56102","subject":"Fixed bug. An error message was added to the agda2atp tool.","message":"Fixed bug. An error message was added to the agda2atp tool.\n\nIgnore-this: e3fe97343f77a9366ac14b0a92728536\n  ++ (see agda2atp\/Test\/Fail\/NotErasedProofTerm.agda).\n\ndarcs-hash:20110322144816-3bd4e-62f67beb1ec58cc25136c00d89668a68571b8e8e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bugs\/BadErrorMessage1.agda","new_file":"Draft\/Bugs\/BadErrorMessage1.agda","new_contents":"","old_contents":"-- Reported by Ana\n\nmodule Draft.Bugs.BadErrorMessage1 where\n\nopen import LTC.Base\nopen import LTC.Data.Nat\n\nS\u2238N2N : \u2200 {n} \u2192 N n \u2192 (\u2200 m \u2192 N m \u2192 N (n \u2238 m)) \u2192 \u2200 m \u2192 N m \u2192 N (succ n \u2238 m)\nS\u2238N2N {n} Nn h m = indN P' p0 pS\n  where P' : D \u2192 Set\n        P' x = N (succ n \u2238 x)\n        postulate p0 : P' zero\n                  pS : \u2200 {x} \u2192 N x \u2192 P' x \u2192 P' (succ x)\n        {-# ATP prove p0 #-}\n\n-- agda2atp reports:\n\n-- An internal error has occurred. Please report this as a bug.\n-- Location of the error: src\/AgdaLib\/Syntax\/DeBruijn.hs:286\n\n-- The internal type of p0 is\n\n--  \u2200 (n : D) (Nn : N n) (h : (\u2200 m \u2192 N m \u2192 N (n \u2238 m))) \u2192 ...\n\n-- The agda2atp can erase the proof term Nn, but it cannot erase the\n-- proof term h.\n\n-- TODO: To change the error message.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"abc963f0df7baaa7fe401fc451e1ad71041763d1","subject":"Search\/Searchable\/Fin.agda","message":"Search\/Searchable\/Fin.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Fin.agda","new_file":"Search\/Searchable\/Fin.agda","new_contents":"module Search.Searchable.Fin where\n\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat\n\nopen import Search.Type\nopen import Search.Searchable\n\nmodule _ {A : Set}(\u03bcA : Searchable A) where\n\n  sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f \u00acFin0\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  ind zero    P P\u2219 Pf = Pf _\n  ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  sumFun : \u2200 n \u2192 Sum (Fin n \u2192 A)\n  sumFun n = sFun n _+_\n\n  ade : \u2200 n \u2192 AdequateSum (sumFun n)\n  ade n = {!!}\n\n  \u03bcFun : \u2200 {n} \u2192 Searchable (Fin n \u2192 A)\n  \u03bcFun = mk _ (ind _) (ade _)\n","old_contents":"module Search.Searchable.Fin where\n\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat\n\nopen import Search.Type\nopen import Search.Searchable\n\nmodule _ {A : Set}(\u03bcA : Searchable A) where\n\n  sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f \u00acFin0\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  ind zero    P P\u2219 Pf = Pf _\n  ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 Searchable (Fin n \u2192 A)\n  \u03bcFun = mk _ (ind _) {!!}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2593d4971f53b881129ee934af1df7f0b139f956","subject":"Add curried version of Mu constructor to PropDesc model.","message":"Add curried version of Mu constructor to PropDesc model.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : (i : I) (xs : El I D (\u03bc I D) i) \u2192 All I D (\u03bc I D) P i xs \u2192 P i (con xs))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El I D\u2081 (\u03bc I D\u2081) i) \u2192 All I D\u2081 (\u03bc I D\u2081) P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nUncurried : (I : Set) (D : Desc I) (X : I \u2192 Set) \u2192 Set\nUncurried I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurried : (I : Set) (D : Desc I) (X : I \u2192 Set) \u2192 Set\nCurried I (`End i) X = X i\nCurried I (`Rec j D) X = (x : X j) \u2192 Curried I D X\nCurried I (`Arg A B) X = (a : A) \u2192 Curried I (B a) X\nCurried I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 Curried I D X\n\ncurry : (I : Set) (D : Desc I) (X : I \u2192 Set)\n  (cn : Uncurried I D X) \u2192 Curried I D X\ncurry I (`End i) X cn = cn refl\ncurry I (`Rec i D) X cn = \u03bb x \u2192 curry I D X (\u03bb xs \u2192 cn (x , xs))\ncurry I (`Arg A B) X cn = \u03bb a \u2192 curry I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurry I (`RecFun A B D) X cn = \u03bb f \u2192 curry I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurry : (I : Set) (D : Desc I) (X : I \u2192 Set)\n  (cn : Curried I D X) \u2192 Uncurried I D X\nuncurry I (`End i) X cn refl = cn\nuncurry I (`Rec i D) X cn (x , xs) = uncurry I D X (cn x) xs\nuncurry I (`Arg A B) X cn (a , xs) = uncurry I (B a) X (cn a) xs\nuncurry I (`RecFun A B D) X cn (f , xs) = uncurry I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 Curried I D (\u03bc I D)\ncon2 I D = curry I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : (i : I) (xs : El I D (\u03bc I D) i) \u2192 All I D (\u03bc I D) P i xs \u2192 P i (con xs))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El I D\u2081 (\u03bc I D\u2081) i) \u2192 All I D\u2081 (\u03bc I D\u2081) P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"239bfdccd7560ddd4c55a28bd209ee38842311cd","subject":"figured out the last cecastfail case","message":"figured out the last cecastfail case\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values and errors are disjoint\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  mutual\n    -- indeterminates and expressions that step are disjoint\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = final-sub-not-trans x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    -- final expressions don't step\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors and expressions that step are disjoint\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail cases\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x () x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITCastID x\u2084) (FHCast FHOuter))\n  err-not-step (CECastFail x () x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITCastSucceed x\u2084 x\u2085) (FHCast FHOuter))\n  err-not-step (CECastFail x GHole x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITGround x\u2084 x\u2085) (FHCast FHOuter)) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast (FHCast x\u2084)) x\u2085 (FHCast (FHCast x\u2086))) = final-not-step x (_ , Step x\u2084 x\u2085 x\u2086)\n\n    -- congruence cases\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- cyrus this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values and errors are disjoint\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  mutual\n    -- indeterminates and expressions that step are disjoint\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = final-sub-not-trans x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    -- final expressions don't step\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors and expressions that step are disjoint\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail caserr-not-step\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!}\n\n    -- congruence caserr-not-step\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"05536450804504640dcab4d2a372088bbc7ef7b3","subject":"Revert \"Added a non-required totality hypothesis.\"","message":"Revert \"Added a non-required totality hypothesis.\"\n\nThis reverts commit 57708b770c00d1ba6276cda0f02752e147f2f533.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- f\u2089\u2081) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < f\u2089\u2081 n + 11.\n-- 3. For all n > 100, then f\u2089\u2081 n = n - 10.\n-- 4. For all n <= 100, then f\u2089\u2081 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationPropertiesATP\n  using ( x>y\u2192x\u2264y\u2192\u22a5 )\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\n  using ( x>y\u2228x\u226fy\n        ; x\u226fy\u2192x\u2264y\n        ; x\u226fSy\u2192x\u226fy\u2228x\u2261Sy\n        ; x+k<y+k\u2192x<y\n        ; <-trans\n        )\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( \u2238-N\n        ; +-N\n        )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n  using ( x<x+11 )\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n  using ( 10-N\n        ; 11-N\n        ; 89-N\n        ; 90-N\n        ; 91-N\n        ; 92-N\n        ; 93-N\n        ; 94-N\n        ; 95-N\n        ; 96-N\n        ; 97-N\n        ; 98-N\n        ; 99-N\n        ; 100-N\n        )\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n-- For all n > 100, then f\u2089\u2081 n = n - 10.\npostulate f\u2089\u2081-x>100 : \u2200 n \u2192 n > 100' \u2192 f\u2089\u2081 n \u2261 n \u2238 10'\n{-# ATP prove f\u2089\u2081-x>100 #-}\n\n-- For all n <= 100, then f\u2089\u2081 n = 91.\nf\u2089\u2081-x\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f 100' \u2192 f\u2089\u2081 n \u2261 91'\nf\u2089\u2081-x\u226f100 = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d \u226f 100' \u2192 f\u2089\u2081 d \u2261 91'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = \u03bb m\u226f100 \u2192\n                     \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nm 100-N m>100 (x\u226fy\u2192x\u2264y Nm 100-N m\u226f100))\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-100 m\u2261100\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-99 m\u226199\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-98 m\u226198\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-97 m\u226197\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-96 m\u226196\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-95 m\u226195\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-94 m\u226194\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-93 m\u226193\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-92 m\u226192\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-91 m\u226191\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-90 m\u226190\n  ... | inj\u2081 m\u226f89 = \u03bb _ \u2192 f\u2089\u2081-m\u226f89\n    where\n    m\u226489 : m \u2264 89'\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    f\u2089\u2081-m+11 : m + 11' \u226f 100' \u2192 f\u2089\u2081 (m + 11') \u2261 91'\n    f\u2089\u2081-m+11 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n    f\u2089\u2081-m\u226f89 : f\u2089\u2081 m \u2261 91'\n    f\u2089\u2081-m\u226f89 = f\u2089\u2081-x\u226f100-helper m 91' m\u226f100\n                                (f\u2089\u2081-m+11 (x\u226489\u2192x+11\u226f100 Nm m\u226489))\n                                f\u2089\u2081-91\n-- The function always terminates.\nf\u2089\u2081-N : \u2200 {n} \u2192 N n \u2192 N (f\u2089\u2081 n)\nf\u2089\u2081-N {n} Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (f\u2089\u2081-x>100 n n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226f100 = subst N (sym (f\u2089\u2081-x\u226f100 Nn n\u226f100)) 91-N\n\n-- For all n, n < f\u2089\u2081 n + 11.\nf\u2089\u2081-ineq : \u2200 {n} \u2192 N n \u2192 n < f\u2089\u2081 n + 11'\nf\u2089\u2081-ineq = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < f\u2089\u2081 d + 11'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x>100\u2192x<f\u2089\u2081-x+11 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let f\u2089\u2081-m+11-N : N (f\u2089\u2081 (m + 11'))\n        f\u2089\u2081-m+11-N = f\u2089\u2081-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + 11')\n        h\u2081 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<f\u2089\u2081m+11 : m < f\u2089\u2081 (m + 11')\n        m<f\u2089\u2081m+11 = x+k<y+k\u2192x<y Nm f\u2089\u2081-m+11-N 11-N h\u2081\n\n        h\u2082 : A (f\u2089\u2081 (m + 11'))\n        h\u2082 = f f\u2089\u2081-m+11-N (<\u2192\u25c1 f\u2089\u2081-m+11-N Nm m\u226f100 m<f\u2089\u2081m+11)\n\n    in (<-trans Nm f\u2089\u2081-m+11-N (x+11-N (f\u2089\u2081-N Nm))\n                m<f\u2089\u2081m+11\n                (f\u2089\u2081x+11<f\u2089\u2081x+11 m m\u226f100 h\u2082))\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- f\u2089\u2081) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < f\u2089\u2081 n + 11.\n-- 3. For all n > 100, then f\u2089\u2081 n = n - 10.\n-- 4. For all n <= 100, then f\u2089\u2081 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationPropertiesATP\n  using ( x>y\u2192x\u2264y\u2192\u22a5 )\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\n  using ( x>y\u2228x\u226fy\n        ; x\u226fy\u2192x\u2264y\n        ; x\u226fSy\u2192x\u226fy\u2228x\u2261Sy\n        ; x+k<y+k\u2192x<y\n        ; <-trans\n        )\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( \u2238-N\n        ; +-N\n        )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n  using ( x<x+11 )\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n  using ( 10-N\n        ; 11-N\n        ; 89-N\n        ; 90-N\n        ; 91-N\n        ; 92-N\n        ; 93-N\n        ; 94-N\n        ; 95-N\n        ; 96-N\n        ; 97-N\n        ; 98-N\n        ; 99-N\n        ; 100-N\n        )\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n-- For all n > 100, then f\u2089\u2081 n = n - 10.\n--\n-- N.B. (21 November 2013). The hypothesis N n is not necessary.\npostulate f\u2089\u2081-x>100 : \u2200 n \u2192 N n \u2192 n > 100' \u2192 f\u2089\u2081 n \u2261 n \u2238 10'\n{-# ATP prove f\u2089\u2081-x>100 #-}\n\n-- For all n <= 100, then f\u2089\u2081 n = 91.\nf\u2089\u2081-x\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f 100' \u2192 f\u2089\u2081 n \u2261 91'\nf\u2089\u2081-x\u226f100 = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d \u226f 100' \u2192 f\u2089\u2081 d \u2261 91'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = \u03bb m\u226f100 \u2192\n                     \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nm 100-N m>100 (x\u226fy\u2192x\u2264y Nm 100-N m\u226f100))\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-100 m\u2261100\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-99 m\u226199\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-98 m\u226198\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-97 m\u226197\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-96 m\u226196\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-95 m\u226195\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-94 m\u226194\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-93 m\u226193\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-92 m\u226192\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-91 m\u226191\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = \u03bb _ \u2192 f\u2089\u2081-x\u2261y f\u2089\u2081-90 m\u226190\n  ... | inj\u2081 m\u226f89 = \u03bb _ \u2192 f\u2089\u2081-m\u226f89\n    where\n    m\u226489 : m \u2264 89'\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    f\u2089\u2081-m+11 : m + 11' \u226f 100' \u2192 f\u2089\u2081 (m + 11') \u2261 91'\n    f\u2089\u2081-m+11 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n    f\u2089\u2081-m\u226f89 : f\u2089\u2081 m \u2261 91'\n    f\u2089\u2081-m\u226f89 = f\u2089\u2081-x\u226f100-helper m 91' m\u226f100\n                                (f\u2089\u2081-m+11 (x\u226489\u2192x+11\u226f100 Nm m\u226489))\n                                f\u2089\u2081-91\n-- The function always terminates.\nf\u2089\u2081-N : \u2200 {n} \u2192 N n \u2192 N (f\u2089\u2081 n)\nf\u2089\u2081-N {n} Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (f\u2089\u2081-x>100 n Nn n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226f100 = subst N (sym (f\u2089\u2081-x\u226f100 Nn n\u226f100)) 91-N\n\n-- For all n, n < f\u2089\u2081 n + 11.\nf\u2089\u2081-ineq : \u2200 {n} \u2192 N n \u2192 n < f\u2089\u2081 n + 11'\nf\u2089\u2081-ineq = \u25c1-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < f\u2089\u2081 d + 11'\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u25c1 m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x>100\u2192x<f\u2089\u2081-x+11 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let f\u2089\u2081-m+11-N : N (f\u2089\u2081 (m + 11'))\n        f\u2089\u2081-m+11-N = f\u2089\u2081-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + 11')\n        h\u2081 = f (x+11-N Nm) (<\u2192\u25c1 (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<f\u2089\u2081m+11 : m < f\u2089\u2081 (m + 11')\n        m<f\u2089\u2081m+11 = x+k<y+k\u2192x<y Nm f\u2089\u2081-m+11-N 11-N h\u2081\n\n        h\u2082 : A (f\u2089\u2081 (m + 11'))\n        h\u2082 = f f\u2089\u2081-m+11-N (<\u2192\u25c1 f\u2089\u2081-m+11-N Nm m\u226f100 m<f\u2089\u2081m+11)\n\n    in (<-trans Nm f\u2089\u2081-m+11-N (x+11-N (f\u2089\u2081-N Nm))\n                m<f\u2089\u2081m+11\n                (f\u2089\u2081x+11<f\u2089\u2081x+11 m m\u226f100 h\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f04e665aed5f41deea97dd4b7160ad5ee957e41a","subject":"Update Nehemiah.Change.Validity","message":"Update Nehemiah.Change.Validity\n\n* Adapt to weaker instance resolution.\n* Omit module application that runs into agda\/agda#1985.\n","repos":"inc-lc\/ilc-agda","old_file":"Nehemiah\/Change\/Validity.agda","new_file":"Nehemiah\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Change.Validity where\n\nopen import Nehemiah.Syntax.Type\nopen import Nehemiah.Denotation.Value\n\nimport Parametric.Change.Validity \u27e6_\u27e7Base as Validity\n\nopen import Nehemiah.Change.Type \nopen import Nehemiah.Change.Value\n\nopen import Data.Integer\nopen import Structure.Bag.Nehemiah\nopen import Base.Change.Algebra\n\nopen import Level\n\nchange-algebra-base : \u2200 \u03b9 \u2192 ChangeAlgebra zero \u27e6 \u03b9 \u27e7Base\nchange-algebra-base base-int = GroupChanges.changeAlgebra \u2124 {{abelian-int}}\nchange-algebra-base base-bag = GroupChanges.changeAlgebra Bag {{abelian-bag}}\n\ninstance\n  change-algebra-base-family : ChangeAlgebraFamily zero \u27e6_\u27e7Base\n  change-algebra-base-family = family change-algebra-base\n\n--open Validity.Structure change-algebra-base-family public --XXX agda\/agda#1985.\nopen Validity.Structure public\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Change.Validity where\n\nopen import Nehemiah.Syntax.Type\nopen import Nehemiah.Denotation.Value\n\nimport Parametric.Change.Validity \u27e6_\u27e7Base as Validity\n\nopen import Nehemiah.Change.Type \nopen import Nehemiah.Change.Value\n\nopen import Data.Integer\nopen import Structure.Bag.Nehemiah\nopen import Base.Change.Algebra\n\nopen import Level\n\nchange-algebra-base : \u2200 \u03b9 \u2192 ChangeAlgebra zero \u27e6 \u03b9 \u27e7Base\nchange-algebra-base base-int = GroupChanges.changeAlgebra \u2124\nchange-algebra-base base-bag = GroupChanges.changeAlgebra Bag\n\nchange-algebra-base-family : ChangeAlgebraFamily zero \u27e6_\u27e7Base\nchange-algebra-base-family = family change-algebra-base\n\nopen Validity.Structure change-algebra-base-family public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0abef3e4e473b86ab625d27e20ce029bf65d4411","subject":"Desc model: induction implicit.","message":"Desc model: induction implicit.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d7f9323c845f517eaad0556192d53f826bb6063c","subject":"Desc model: cases implicit.","message":"Desc model: cases implicit.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"94c667f3d542af3d0d5f3f603fe4d1aeccaf6e95","subject":"Add change structure for changes","message":"Add change structure for changes\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Sum\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n      ; \u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 Ch A \u00d7 Ch B\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = Ch A \u00d7 Ch B\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp : \u2200 (s\u2081 s\u2082 : A \u228e B) \u2192 SValid s\u2081 (convert\u2081 (rp s\u2082))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp s1 s2\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp s1 s2\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    } }\n\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b} } }\n","old_contents":"module New.Changes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Unit\nopen import Data.Product\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n      ; \u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 Ch A \u00d7 Ch B\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = Ch A \u00d7 Ch B\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    } }\n\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b} } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1be6c300fb4dd0a219451a50d15f5bc25c730bc7","subject":"Model examples change","message":"Model examples change\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_contents":"{-# OPTIONS --type-in-type --no-pattern-matching #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\nNatN : String\nNatN = \"Nat\"\n\nNatP : Tel\nNatP = Emp\n\nNatI : Scope NatP \u2192 Tel\nNatI _ = Emp\n\nNatE : Enum\nNatE = \"zero\" \u2237 \"suc\" \u2237 []\n\nNatB : (p : Scope NatP) \u2192 BranchesD NatE (NatI p)\nNatB _ = End tt , Rec tt (End tt) , tt\n\nNat : Set\nNat = Form NatN NatP NatI NatE NatB\n\nzero : Nat\nzero = inj NatN NatP NatI NatE NatB here\n\nsuc : Nat \u2192 Nat\nsuc = inj NatN NatP NatI NatE NatB (there here)\n\nelimNat : (P : Nat \u2192 Set)\n  (pz : P zero)\n  (ps : (n : Nat) \u2192 P n \u2192 P (suc n))\n  (n : Nat) \u2192 P n\nelimNat = elim NatN NatP NatI NatE NatB\n\n----------------------------------------------------------------------\n\nVecN : String\nVecN = \"Vec\"\n\nVecP : Tel\nVecP = Ext Set \u03bb _ \u2192 Emp\n\nVecI : Scope VecP \u2192 Tel\nVecI _ = Ext Nat \u03bb _ \u2192 Emp\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\nVecB : (p : Scope VecP) \u2192 BranchesD VecE (VecI p)\nVecB = uncurryScope VecP (\u03bb p \u2192 BranchesD VecE (VecI p)) \u03bb A\n  \u2192 End (zero , tt)\n  , Arg Nat (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n  , tt\n\nVec : (A : Set) \u2192 Nat \u2192 Set\nVec = Form VecN VecP VecI VecE VecB\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = inj VecN VecP VecI VecE VecB A here\n\ncons : (A : Set) (n : Nat) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A = inj VecN VecP VecI VecE VecB A (there here)\n\nelimVec : (A : Set) (P : (n : Nat) \u2192 Vec A n \u2192 Set)\n  (pn : P zero (nil A))\n  (pc : (n : Nat) (x : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n x xs))\n  (n : Nat) (xs : Vec A n) \u2192 P n xs\nelimVec = elim VecN VecP VecI VecE VecB\n\n----------------------------------------------------------------------\n\nadd : Nat \u2192 Nat \u2192 Nat\nadd = elimNat\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : Nat \u2192 Nat \u2192 Nat\nmult = elimNat\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : (A : Set) (m n : Nat) (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend A m n = elimVec A\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb m' x xs ih ys \u2192 cons A (add m' n) x (ih ys))\n  m\n\nconcat : (A : Set) (m n : Nat) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m n = elimVec (Vec A m)\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  (nil A)\n  (\u03bb m' xs xss ih \u2192 append A m (mult m' m) xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : Nat\none = suc zero\n\ntwo : Nat\ntwo = suc one\n\nthree : Nat\nthree = suc two\n\n[1,2] : Vec Nat two\n[1,2] = cons Nat one one (cons Nat zero two (nil Nat))\n\n[3] : Vec Nat one\n[3] = cons Nat zero three (nil Nat)\n\n[1,2,3] : Vec Nat three\n[1,2,3] = cons Nat two one (cons Nat one two (cons Nat zero three (nil Nat)))\n\ntest-append : [1,2,3] \u2261 append Nat two one [1,2] [3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type --no-pattern-matching #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\nNatN : String\nNatN = \"Nat\"\n\nNatP : Tel\nNatP = Emp\n\nNatI : Scope NatP \u2192 Tel\nNatI _ = Emp\n\nNatE : Enum\nNatE = \"zero\" \u2237 \"suc\" \u2237 []\n\nNatB : (p : Scope NatP) \u2192 BranchesD NatE (NatI p)\nNatB _ = End tt , Rec tt (End tt) , tt\n\nNat : Set\nNat = Form NatN NatP NatI NatE NatB\n\ninjNat : CurriedScope NatP \u03bb p\n  \u2192 CurriedFunc (Scope (NatI p))\n      (sumD NatE (NatI p) (NatB p))\n      (FormUncurried NatN NatP NatI NatE NatB p)\ninjNat = inj NatN NatP NatI NatE NatB\n\nelimNat : CurriedScope NatP \u03bb p\n  \u2192 (M : CurriedScope (NatI p) (\u03bb i \u2192 FormUncurried NatN NatP NatI NatE NatB p i \u2192 Set))\n  \u2192 let unM = uncurryScope (NatI p) (\u03bb i \u2192 FormUncurried NatN NatP NatI NatE NatB p i \u2192 Set) M\n  in CurriedBranches NatE\n     (SumCurriedHyps NatN NatP NatI NatE NatB p M)\n     (CurriedScope (NatI p) (\u03bb i \u2192 (x : FormUncurried NatN NatP NatI NatE NatB p i) \u2192 unM i x))\nelimNat = elim NatN NatP NatI NatE NatB\n\nzero : Nat\nzero = injNat here\n\nsuc : Nat \u2192 Nat\nsuc = injNat (there here)\n\n----------------------------------------------------------------------\n\nVecN : String\nVecN = \"Vec\"\n\nVecP : Tel\nVecP = Ext Set \u03bb _ \u2192 Emp\n\nVecI : Scope VecP \u2192 Tel\nVecI _ = Ext Nat \u03bb _ \u2192 Emp\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\nVecB : (p : Scope VecP) \u2192 BranchesD VecE (VecI p)\nVecB = uncurryScope VecP (\u03bb p \u2192 BranchesD VecE (VecI p)) \u03bb A\n  \u2192 End (zero , tt)\n  , Arg Nat (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n  , tt\n\nVec : (A : Set) \u2192 Nat \u2192 Set\nVec = Form VecN VecP VecI VecE VecB\n\ninjVec : CurriedScope VecP \u03bb p\n  \u2192 CurriedFunc (Scope (VecI p))\n      (sumD VecE (VecI p) (VecB p))\n      (FormUncurried VecN VecP VecI VecE VecB p)\ninjVec = inj VecN VecP VecI VecE VecB\n\nelimVec : CurriedScope VecP \u03bb p\n  \u2192 (M : CurriedScope (VecI p) (\u03bb i \u2192 FormUncurried VecN VecP VecI VecE VecB p i \u2192 Set))\n  \u2192 let unM = uncurryScope (VecI p) (\u03bb i \u2192 FormUncurried VecN VecP VecI VecE VecB p i \u2192 Set) M\n  in CurriedBranches VecE\n     (SumCurriedHyps VecN VecP VecI VecE VecB p M)\n     (CurriedScope (VecI p) (\u03bb i \u2192 (x : FormUncurried VecN VecP VecI VecE VecB p i) \u2192 unM i x))\nelimVec = elim VecN VecP VecI VecE VecB\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = injVec A here\n\ncons : (A : Set) (n : Nat) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A = injVec A (there here)\n\n----------------------------------------------------------------------\n\nadd : Nat \u2192 Nat \u2192 Nat\nadd = elimNat\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : Nat \u2192 Nat \u2192 Nat\nmult = elimNat\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : (A : Set) (m n : Nat) (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend A m n = elimVec A\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb m' x xs ih ys \u2192 cons A (add m' n) x (ih ys))\n  m\n\nconcat : (A : Set) (m n : Nat) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m n = elimVec (Vec A m)\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  (nil A)\n  (\u03bb m' xs xss ih \u2192 append A m (mult m' m) xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : Nat\none = suc zero\n\ntwo : Nat\ntwo = suc one\n\nthree : Nat\nthree = suc two\n\n[1,2] : Vec Nat two\n[1,2] = cons Nat one one (cons Nat zero two (nil Nat))\n\n[3] : Vec Nat one\n[3] = cons Nat zero three (nil Nat)\n\n[1,2,3] : Vec Nat three\n[1,2,3] = cons Nat two one (cons Nat one two (cons Nat zero three (nil Nat)))\n\ntest-append : [1,2,3] \u2261 append Nat two one [1,2] [3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9af23cf22e1ab362777619cd58cbabe4e108b6db","subject":"Forgotten an ATP pragma.","message":"Forgotten an ATP pragma.\n\nIgnore-this: 258014be53497c276baad0e0bb8d150\n\ndarcs-hash:20100525132333-3bd4e-e9de1d5b04d7e26b8ba973256477a726f831a691.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Equalities\/Properties.agda","new_file":"LTC\/Relation\/Equalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Functions on equalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Equalities.Properties where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\npostulate\n  sym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\n{-# ATP prove sym #-}\n\npostulate\n  trans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n{-# ATP prove trans #-}\n\npostulate\n  \u00acS\u22610 : {d : D} \u2192 \u00ac (succ d \u2261 zero)\n{-# ATP prove \u00acS\u22610 #-}\n\nx\u2261y\u2192Sx\u2261Sy : {m n : D} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nx\u2261y\u2192Sx\u2261Sy refl = refl\n","old_contents":"------------------------------------------------------------------------------\n-- Functions on equalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Equalities.Properties where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\npostulate\n  sym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\n{-# ATP prove sym #-}\n\npostulate\n  trans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n\npostulate\n  \u00acS\u22610 : {d : D} \u2192 \u00ac (succ d \u2261 zero)\n{-# ATP prove \u00acS\u22610 #-}\n\nx\u2261y\u2192Sx\u2261Sy : {m n : D} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nx\u2261y\u2192Sx\u2261Sy refl = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ae4b3986e9fb874d12f823c58e64f325d8167830","subject":"Hutton's razor: nop, don't try to compute context, life's too short for that.","message":"Hutton's razor: nop, don't try to compute context, life's too short for that.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/RevisedHutton.agda","new_file":"models\/RevisedHutton.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\n-- Open term: holes are either values or variables in a context\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = \n    subst (cong (IMu closeTerm) (snd variable)) \n          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata List (A : Set) : Set where\n  [] : List A\n  _::_ : A -> List A -> List A\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\ndata Comp : Set where\n  LT : Comp\n  EQ : Comp\n  GT : Comp\n\ncomp : {n : Nat} -> Fin n -> Fin n -> Comp\ncomp fze fze = EQ\ncomp fze _ = LT\ncomp _ fze = GT\ncomp (fsu n) (fsu n') = comp n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : Nat -> Type -> Set\nVar n ty = Fin n\n\n-- Open term: holes are either values or variables in a context\nopenTerm : Nat -> Type -> IDesc Type\nopenTerm n = toIDesc Type (exprFree ** (\\ty -> Val ty + Var n ty))\n\ndata Maybe (A : Set) : Set where\n  Nothing : Maybe A\n  Just : A -> Maybe A\n\nmergeVars : {n : Nat} ->\n            Maybe (List (Fin n * Type)) ->\n            Maybe (List (Fin n * Type)) ->\n            Maybe (List (Fin n * Type))\nmergeVars Nothing _ = Nothing\nmergeVars _ Nothing = Nothing\nmergeVars (Just []) (Just y) = Just y\nmergeVars (Just x) (Just []) = Just x\nmergeVars (Just (x :: xs)) (Just (y :: ys)) with comp (fst x) (fst y) \n                                              | mergeVars (Just xs) (Just (y :: ys)) \n                                              | mergeVars (Just (x :: xs)) (Just ys)\n... | LT | _        | Nothing   = Nothing\n... | LT | _        | Just xys  = Just (y :: xys)\n... | EQ | _        | _         = Nothing\n... | GT | Nothing  | _         = Nothing\n... | GT | Just xys | _         = Just (x :: xys)\n\ncollectVars : {ty : Type}{n : Nat} -> IMu (openTerm n) ty -> Maybe (List (Fin n * Type))\ncollectVars {ty} {n} t = cata Type (openTerm n) (\\_ -> Maybe (List (Fin n * Type))) collectVarsHelp ty t\n  where collectVarsHelp : (i : Type) -> [| openTerm n i |] (\\ _ -> Maybe (List (Fin n * Type))) -> Maybe (List (Fin n * Type))\n        collectVarsHelp ty (EZe , r n) = Just ((n , ty) :: [])\n        collectVarsHelp ty (EZe , l _) = Just []\n        collectVarsHelp ty (ESu EZe , (t1 , ( t2 , t3))) = mergeVars (mergeVars t1 t2) t3\n        collectVarsHelp nat (ESu (ESu EZe) , (x , y)) = mergeVars x y\n        collectVarsHelp nat (ESu (ESu (ESu ())) , _) \n        collectVarsHelp bool (ESu (ESu EZe) , (x , y) ) = mergeVars x y\n        collectVarsHelp bool (ESu (ESu (ESu ())) , _) \n        collectVarsHelp (pair x y) (ESu (ESu ()) , _)\n        \n\nValidContext : {ty : Type}{n : Nat}(c : Context n)(tm : IMu (openTerm n) ty) -> Set\nValidContext {ty} {n} c tm  with collectVars tm\n... | Nothing = Zero\n... | (Just y) = checkContext y\n  where checkContext : List (Fin n * Type) -> Set\n        checkContext [] = Unit\n        checkContext ((x , ty) :: xs) = ((typeAt c x) == ty) * checkContext xs \n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var n ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = {!lookup c variable!}\n--          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var n ty) ->\n                     IMu closeTerm ty) ->\n            (tm : IMu (openTerm n) ty) ->\n            ValidContext context tm ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term valid = \n    substI (\\ty -> Val ty + Var n ty) Val exprFree sig ty term\n\n\n\n{-\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n-}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"052a571080bb4647e0d34a8cc8fb5454c47c7ec4","subject":"agda: prove completely that diff-is-valid (#33)","message":"agda: prove completely that diff-is-valid (#33)\n\nThe proof is much more different from the one of derive-is-valid than\nI'd have expected, but it was relatively easy to complete directly in\nAgda, without doing it on paper beforehand.\n\nderive-is-valid is proved as a corollary in one line.\n\nOld-commit-hash: fd47a2b1346fb45e5e94db8d814e83e9fdf1442d\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds valid-\u0394-s-ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I want to start developing a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (derive-is-valid (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid = {!!}\n\n-- This proof could be finished using diff-is-valid:\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid {bool} v = tt\nderive-is-valid {\u03c4\u2081 \u21d2 \u03c4\u2082} v =\n  \u03bb s ds valid-\u0394-s-ds \u2192 diff-is-valid (v (apply ds s)) (v s) , (\n    begin\n      (apply (derive v) v) (apply ds s)\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 x (apply ds s)) (apply-derive v) \u27e9\n      v (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v (apply ds s))) \u27e9\n      apply (derive v s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\n-- This is a postulate elsewhere, but here I want to start developing a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (derive-is-valid (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"34efeb45d59b4e286806101d366893665f25f04c","subject":"agda\/...: lift-term\u2032 \u0394 (see #22)","message":"agda\/...: lift-term\u2032 \u0394 (see #22)\n\nOld-commit-hash: 751e41dfecd3ae6383acc772da530f50c6b1ce83\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : --\u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = {!!}\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f1c2df98ed89a3de0f4b42407cb950eded1c5ed4","subject":"Increase overloading and confusion.","message":"Increase overloading and confusion.\n\nOld-commit-hash: df682306832e5b833f837c92dec32b3c428a7ab9\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 {{\u0393\u2032}} (abs t)\n  \u2248\u27e8  \u0394-abs {{\u0393\u2032}} t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  \u2261\u27e8\u27e9\n     derive-term {{\u0393\u2032}} (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app {{\u0393\u2032}} t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7 \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083) =\n  begin\n    \u0394 (if t\u2081 t\u2082 t\u2083)\n  \u2248\u27e8 \u0394-if {{\u0393\u2032}} t\u2081 t\u2082 t\u2083 \u27e9\n    if (\u0394 t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (\u0394 {{\u0393\u2032}} t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (\u0394 {{\u0393\u2032}} t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (\u0394 {{\u0393\u2032}} t\u2082)\n           (\u0394 {{\u0393\u2032}} t\u2083))\n  \u2248\u27e8 \u2248-if (derive-term-correct {{\u0393\u2032}} t\u2081)\n          (\u2248-if (\u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct {{\u0393\u2032}} t\u2083) \u2248-refl) \u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct {{\u0393\u2032}} t\u2082) \u2248-refl) \u2248-refl))\n          (\u2248-if (\u2248-refl)\n                (derive-term-correct {{\u0393\u2032}} t\u2082)\n                (derive-term-correct {{\u0393\u2032}} t\u2083)) \u27e9\n    if (derive-term {{\u0393\u2032}} t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (derive-term {{\u0393\u2032}} t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (derive-term {{\u0393\u2032}} t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (derive-term {{\u0393\u2032}} t\u2082)\n           (derive-term {{\u0393\u2032}} t\u2083))\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083)\n  \u220e where open \u2248-Reasoning\n\nderive-term-correct {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u2248-\u0394 {{\u0393\u2032}} (derive-term-correct {{\u0393\u2033}} t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 {{\u0393\u2032}} (abs t)\n  \u2248\u27e8  \u0394-abs {{\u0393\u2032}} t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  \u2261\u27e8\u27e9\n     derive-term {{\u0393\u2032}} (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app {{\u0393\u2032}} t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083) =\n  begin\n    \u0394 (if t\u2081 t\u2082 t\u2083)\n  \u2248\u27e8 \u0394-if {{\u0393\u2032}} t\u2081 t\u2082 t\u2083 \u27e9\n    if (\u0394 t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (\u0394 {{\u0393\u2032}} t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (\u0394 {{\u0393\u2032}} t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (\u0394 {{\u0393\u2032}} t\u2082)\n           (\u0394 {{\u0393\u2032}} t\u2083))\n  \u2248\u27e8 \u2248-if (derive-term-correct {{\u0393\u2032}} t\u2081)\n          (\u2248-if (\u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct {{\u0393\u2032}} t\u2083) \u2248-refl) \u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct {{\u0393\u2032}} t\u2082) \u2248-refl) \u2248-refl))\n          (\u2248-if (\u2248-refl)\n                (derive-term-correct {{\u0393\u2032}} t\u2082)\n                (derive-term-correct {{\u0393\u2032}} t\u2083)) \u27e9\n    if (derive-term {{\u0393\u2032}} t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (derive-term {{\u0393\u2032}} t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (derive-term {{\u0393\u2032}} t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (derive-term {{\u0393\u2032}} t\u2082)\n           (derive-term {{\u0393\u2032}} t\u2083))\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083)\n  \u220e where open \u2248-Reasoning\n\nderive-term-correct {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u2248-\u0394 {{\u0393\u2032}} (derive-term-correct {{\u0393\u2033}} t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f7dfdf9faeb8aa7389195922929e02b45dd68aed","subject":"ANormalUntyped: WIP progress","message":"ANormalUntyped: WIP progress\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalUntyped.agda","new_file":"Thesis\/ANormalUntyped.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set\n\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  term : \u2200 {v} n (t : Term \u0393) \u2192\n    \u03c1 \u22a2 t \u2193[ n ] v \u2192\n    \u03c1 F\u22a2 term t \u2193[ n ] v\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {dt : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs dt \u2193 dclosure dt \u03c1 d\u03c1\n  dterm : \u2200 {dv} (dt : DTerm \u0393) \u2192\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 F\u22a2 dterm dt \u2193 dv\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k <\u2032 n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 = <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n = proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n = proj\u2082 (rrelTV3 n)\n\n  s-rrelV3 :\n    \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n    \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  s-rrelV3 k rec-rrelTV3 rrelV3-k 0 _ = rrelV3-k\n  s-rrelV3 k rec-rrelTV3 rrelV3-k (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    -- rrelV3-Type n\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n      (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    \u2200 j (j<k : j <\u2032 k) n2 \u2192\n    (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 k rec-rrelTV3 rrelV3-k j j<k v1 dv v2\n    -- where\n    --   s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n    --   -- If j = 0, we can't do a recursive call on (k - j) because that's not\n    --   -- well-founded. Luckily, we don't need to, just use relV as defined at\n    --   -- the same level.\n    --   -- Yet, this will be a pain in proofs.\n    --   s-rrelV3 0 _ = rrelV3-k\n    --   s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelV3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 _ _ _ = \u22a5\n\nopen import Postulate.Extensionality\nmutual\n  rrelT3-equiv : \u2200 k {\u0393} {t1 : Fun \u0393} {dt t2 \u03c11 d\u03c1 \u03c12} \u2192\n    rrelT3 k t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2261\n    \u2200 j (j<k : j <\u2032 k) n2 \u2192\n    \u2200 (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 j v1 dv v2\n  rrelT3-equiv n = cong (\u03bb \u25a1 \u2192 \u2200 j \u2192 \u25a1 j) (ext (\u03bb j \u2192 {!!}))\n\n  rrelV3-equiv : \u2200 n {\u03931 \u03932 \u0394\u0393 t1 dt t2 \u03c11 \u03c1' d\u03c1 \u03c12} \u2192\n    rrelV3 n (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) \u2261 \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        rrelV3 k v1 dv v2 \u2192\n        rrelT3 k\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-equiv n = cong (\u03bb \u25a1 \u2192 \u03a3 (_ \u2261 _ \u00d7 _ \u2261 _) \u25a1) (ext (\u03bb { (eq1 , eq2) \u2192\n    cong\n      (\u03bb \u25a1 \u2192\n         (\u03bb { (refl , refl)\n                \u2192 \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192 \u25a1 k k<n v1 dv v2\n            })\n         eq1 eq2)\n      (ext (\u03bb x \u2192 ext {!!}))}))\n  --\n  -- cong (\u03bb \u25a1 \u2192 \u2200 k \u2192 (k<n : k <\u2032 n) \u2192 \u25a1 k k<n)\n  -- rrelV3-equiv k\n\nrrel\u03c13 : \u2115 \u2192 \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13 n \u2205 \u2205 \u2205 \u2205 = \u22a4\nrrel\u03c13 n (Uni \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) = rrelV3 n v1 dv v2 \u00d7 rrel\u03c13 n \u0393 \u03c11 d\u03c1 \u03c12\n\n--\nrfundamental3 : \u2200 {\u0393} (t : Fun \u0393) \u2192 \u2200 k \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 k \u0393 \u03c11 d\u03c1 \u03c12) \u2192 rrelT3 k t (derive-dfun t) t \u03c11 d\u03c1 \u03c12\nrfundamental3 (term x) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 j j<k n2 v1 v2 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 = {!!}\nrfundamental3 {\u0393} (abs t) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .0 j<k .0 (closure .t .\u03c11) (closure .t .\u03c12) abs abs = dclosure (derive-dfun t) \u03c11 d\u03c1 , 0 , dabs , (refl , refl) , {!body !}\n  where\n    body1 : \u2200 k\u2081 (k\u2081<k : k\u2081 <\u2032 k) \u2192 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 rrelV3 k\u2081 v1 dv v2 \u2192 rrelT3 k\u2081 t (derive-dfun t) t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12)\n    body1 = \u03bb k\u2081 k\u2081<k v1 dv v2 x v3 v4 j n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n    rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n    !}\n  -- \u03bb\n  -- { k\u2081 k<n v1 dv v2 vv v3 v4 0 n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- !}\n  -- -- (Some.wfRec-builder rrelTV3-Type rrelTV3-step k\u2081\n  --                     -- (<-well-founded\u2032 k k\u2081 k<n))\n\n  -- -- All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k k\u2081 k<n : rrelTV3-Type k\u2081\n  -- --\n  -- ; k\u2081 k<n v1 dv v2 vv v3 v4 (suc j) n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- !}\n  -- -- rfundamental3 t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 {!suc j !} n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- }\n\n  --   -- \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n  --   -- \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  --   foo : s-rrelV3 k\n  --     (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k)\n  --     (rrelV3-step k (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k))\n  --     j j<k v1 (dclosure (derive-dfun t) \u03c11 d\u03c1) v2\n  --   foo with j\n  --   ... | s = {!!}\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set\n\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  term : \u2200 {v} n (t : Term \u0393) \u2192\n    \u03c1 \u22a2 t \u2193[ n ] v \u2192\n    \u03c1 F\u22a2 term t \u2193[ n ] v\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {dt : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs dt \u2193 dclosure dt \u03c1 d\u03c1\n  dterm : \u2200 {dv} (dt : DTerm \u0393) \u2192\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 F\u22a2 dterm dt \u2193 dv\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k <\u2032 n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 = <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n = proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n = proj\u2082 (rrelTV3 n)\n\n  s-rrelV3 :\n    \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n    \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  s-rrelV3 k rec-rrelTV3 rrelV3-k 0 _ = rrelV3-k\n  s-rrelV3 k rec-rrelTV3 rrelV3-k (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    (v1 v2 : Val Uni) \u2192\n    \u2200 j n2 (j<k : j <\u2032 k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 k rec-rrelTV3 rrelV3-k j j<k v1 dv v2\n    -- where\n    --   s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n    --   -- If j = 0, we can't do a recursive call on (k - j) because that's not\n    --   -- well-founded. Luckily, we don't need to, just use relV as defined at\n    --   -- the same level.\n    --   -- Yet, this will be a pain in proofs.\n    --   s-rrelV3 0 _ = rrelV3-k\n    --   s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelT3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 _ _ _ = \u22a5\n\nrrel\u03c13 : \u2115 \u2192 \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13 n \u2205 \u2205 \u2205 \u2205 = \u22a4\nrrel\u03c13 n (Uni \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) = rrelV3 n v1 dv v2 \u00d7 rrel\u03c13 n \u0393 \u03c11 d\u03c1 \u03c12\n\n--\nrfundamental3 : \u2200 {\u0393} (t : Fun \u0393) \u2192 \u2200 k \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 k \u0393 \u03c11 d\u03c1 \u03c12) \u2192 rrelT3 k t (derive-dfun t) t \u03c11 d\u03c1 \u03c12\nrfundamental3 (term x) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j n2 j<k \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 = {!!}\nrfundamental3 {\u0393} (abs t) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) (closure .t .\u03c12) .0 .0 j<k abs abs = dclosure (derive-dfun t) \u03c11 d\u03c1 , 0 , dabs , (refl , refl) ,\n  \u03bb\n  { k\u2081 k<n v1 dv v2 vv v3 v4 0 n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {! rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2 !}\n  ; k\u2081 k<n v1 dv v2 vv v3 v4 (suc j) n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {! rfundamental3 t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 {!suc j !} n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2 !}\n  }\n--\n  -- where\n\n  --   -- \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n  --   -- \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  --   foo : s-rrelV3 k\n  --     (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k)\n  --     (rrelV3-step k (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k))\n  --     j j<k v1 (dclosure (derive-dfun t) \u03c11 d\u03c1) v2\n  --   foo with j\n  --   ... | s = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a9a8c5e1e837317e84169faf09a50616e194fe71","subject":"Actually prove congruence lemmas for d.o.e.","message":"Actually prove congruence lemmas for d.o.e.\n\nThe corollaries are indeed trivial, but it's still good to write them\ndown.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df : \u0394 f} \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects {df = df} dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (Derivative-f-df : Derivative f df) \u2192 DerivativeAsChange df Derivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df Derivative-f-df = derivative-is-nil (DerivativeAsChange df Derivative-f-df) Derivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    (f : A \u2192 B) \u2192 (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192 Derivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (Derivative-f-df : Derivative f df) \u2192 DerivativeAsChange df Derivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df Derivative-f-df = derivative-is-nil (DerivativeAsChange df Derivative-f-df) Derivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    (f : A \u2192 B) \u2192 (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192 Derivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"757995f811b789374410d560566e8717a68f1e24","subject":"Revert \"Show how badly reasoning for \u2259 and \u2261 combine\"","message":"Revert \"Show how badly reasoning for \u2259 and \u2261 combine\"\n\nThis reverts commit 24faa2c5edd91d1db64977d5124ef40e01de433f. The commit\nshowed how badly something worked, so let's leave in the old version for\nnow.\n\nOld-commit-hash: 9f5f49cf0319ba809356655a204ec83725dfceb4\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    trans\n      (proof (let open \u2259-Reasoning in\n        begin\n          dx\n        \u2259\u27e8 dx\u2259nil-x \u27e9\n          nil x\n        \u220e))\n      (let open \u2261-Reasoning in\n        begin\n          x \u229e (nil x)\n        \u2261\u27e8 update-nil x \u27e9\n          x\n        \u220e)\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"df0a589fd6ba99849aba42de50fadcedc5ab274c","subject":"agda: prove update-diff","message":"agda: prove update-diff\n\nAlso, add infix declaration needed to state the result conveniently.\n\nOld-commit-hash: 1a2ad0add537966641273bb1ce153a976e3e6703\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  -- Relation.Binary.EqReasoning is more convenient to use with _\u2261_ if\n  -- the combinators take the type argument (a) as a hidden argument,\n  -- instead of being locked to a fixed type at module instantiation\n  -- time.\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n{-\n      lemma : nil f \u2259 dg\n      -- Goes through\n      --lemma = sym (derivative-is-nil dg fdg)\n      -- Generates tons of crazy yellow ambiguities of equality type. Apparently, unifying against \u2259 does not work so well.\n      lemma = \u2259-symm {{changeAlgebra}} {f} {dg} {nil f} (derivative-is-nil dg fdg)\n-}\n\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  -- Relation.Binary.EqReasoning is more convenient to use with _\u2261_ if\n  -- the combinators take the type argument (a) as a hidden argument,\n  -- instead of being locked to a fixed type at module instantiation\n  -- time.\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n{-\n      lemma : nil f \u2259 dg\n      -- Goes through\n      --lemma = sym (derivative-is-nil dg fdg)\n      -- Generates tons of crazy yellow ambiguities of equality type. Apparently, unifying against \u2259 does not work so well.\n      lemma = \u2259-symm {{changeAlgebra}} {f} {dg} {nil f} (derivative-is-nil dg fdg)\n-}\n\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bd5737fe4a6205941af51d14021d29b56781ae08","subject":"Minor change.","message":"Minor change.\n\nIgnore-this: a8fb51b7be37e1ae1a4d30cb3ddc2f0c\n\ndarcs-hash:20120201152333-3bd4e-84e00d6e13b1bdffb892ef4a6040cb657dd398bb.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/Base.agda","new_file":"src\/GroupTheory\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Group theory base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfix  11 _\u207b\u00b9\ninfixl 10 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n-- The group universe\nopen import Common.Universe public renaming ( D to G )\n\n-- The equality\n-- The equality is the propositional identity on the group universe.\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.Inductive public\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Group theory axioms\npostulate\n  \u03b5   : G          -- The identity element.\n  _\u00b7_ : G \u2192 G \u2192 G  -- The binary operation.\n  _\u207b\u00b9 : G \u2192 G      -- The inverse function.\n\n  assoc         : \u2200 x y z \u2192 x \u00b7 y \u00b7 z    \u2261 x \u00b7 (y \u00b7 z)\n  leftIdentity  : \u2200 x     \u2192     \u03b5 \u00b7 x    \u2261 x\n  rightIdentity : \u2200 x     \u2192     x \u00b7 \u03b5    \u2261 x\n  leftInverse   : \u2200 x     \u2192  x \u207b\u00b9 \u00b7 x    \u2261 \u03b5\n  rightInverse  : \u2200 x     \u2192  x    \u00b7 x \u207b\u00b9 \u2261 \u03b5\n{-# ATP axiom assoc leftIdentity rightIdentity leftInverse rightInverse #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfix  11 _\u207b\u00b9\ninfixl 10 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n-- The group universe\nopen import Common.Universe public renaming ( D to G )\n\n-- The equality\n-- The equality is the propositional identity on the group universe.\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.Inductive public\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Group theory axioms\npostulate\n  \u03b5   : G          -- The identity element.\n  _\u00b7_ : G \u2192 G \u2192 G  -- The group binary operation.\n  _\u207b\u00b9 : G \u2192 G      -- The inverse function.\n\n  assoc         : \u2200 x y z \u2192 x \u00b7 y \u00b7 z    \u2261 x \u00b7 (y \u00b7 z)\n  leftIdentity  : \u2200 x     \u2192     \u03b5 \u00b7 x    \u2261 x\n  rightIdentity : \u2200 x     \u2192     x \u00b7 \u03b5    \u2261 x\n  leftInverse   : \u2200 x     \u2192  x \u207b\u00b9 \u00b7 x    \u2261 \u03b5\n  rightInverse  : \u2200 x     \u2192  x    \u00b7 x \u207b\u00b9 \u2261 \u03b5\n{-# ATP axiom assoc leftIdentity rightIdentity leftInverse rightInverse #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"df40931154a8e60ffb751bf5fb6436018582169b","subject":"Some unfinished experiment","message":"Some unfinished experiment\n","repos":"crypto-agda\/crypto-agda","old_file":"Solver\/Linear\/Examples.agda","new_file":"Solver\/Linear\/Examples.agda","new_contents":"module Solver.Linear.Examples where\n\nopen import Solver.Linear.Syntax\nopen import Solver.Linear\nimport Data.String as String\nopen import Data.Zero\nopen import Relation.Binary.NP\nopen import Relation.Nullary.Decidable\nopen import Data.Vec\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Unit\nopen import Function\nopen import FunUniverse.Agda\nopen import Relation.Binary.PropositionalEquality\n\nmodule #Vars {a} {A : Set a}\n             (_\u225f\u1d2c_ : Decidable (_\u2261_ {A = A})) where\n  #vars : Syn A \u2192 \u2115\n  #vars tt      = 0\n  #vars (var x) = 1\n  #vars (t , u) = #vars t + #vars u\n\n  open import Data.List\n  vars : (t : Syn A) \u2192 List A \u2192 List A\n  vars tt      = id\n  vars (var x) = _\u2237_ x\n  vars (t , u) = vars t \u2218 vars u\n\n  lookupVar : \u2200 {b} {B : Set b} (t : Syn A) \u2192 Vec B (#vars t)\n                                 \u2192 A \u2192 B \u2192 B\n  lookupVar (var x) bs a\u2081 = {!!}\n  lookupVar tt bs a\u2081 = {!!}\n  lookupVar (t , t\u2081) bs a\u2081 = {!lookupVar t bs!}\n\nmodule Syntax\u02e2' {a} {A : Set a} {funU} linRewiring where\n  open Syntax (\u03bb x y \u2192 \u230a String\u2264._<?_ x y \u230b) String._\u225f_ {a} {A} {funU} linRewiring public\n  open import Solver.Linear.Parser\n\n  open #Vars String._\u225f_\n\n  module _ s {s-ok} where\n    e = parseEq\u02e2 s {s-ok}\n    t = LHS _ e\n    v = vars t\n    module _ {\u0393 : Vec A (#vars t)} where\n      \u2113 = lookupVar v \u0393\n      module _ {e-ok : EqOk? \u2113 e} where\n        rewire\u02e2 : EvalEq \u2113 e\n        rewire\u02e2 = rewire \u2113 e {e-ok}\n\n{-\n(Vec A n \u2192 B) \u2192 N-ary n A B\n\nSyn String \u2192 String \u2192 A\n\n--((String \u2192 A) \u2192 B)\n\nmodule example1 where\n  open Syntax\u02e2 funLin\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire\u02e2 \u0393 \"(A,B),C\u21a6(B,A),C\"\n   where\n    \u0393 : String \u2192 _\n    \u0393 \"A\" = A\n    \u0393 \"B\" = B\n    \u0393 \"C\" = C\n    \u0393  _  = \ud835\udfd8\n\nmodule example2 where\n  open Syntax\u1da0 funLin\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire\u1da0 (A \u2237 B \u2237 C \u2237 []) (\u03bb a b c \u2192 (a , b) , c \u21a6 (b , a) , c)\n\n  {-\nmodule example3 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest {n} M = Syntax _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' {n} M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\nmodule example\u2083 where\n\n  open import Data.Unit\n  open import Data.Product\n  open import Data.Vec\n\n  open import Function using (flip ; const)\n  \n  open import Function.Inverse\n  open import Function.Related.TypeIsomorphisms.NP\n\n  open \u00d7-CMon using () renaming (\u2219-cong to \u00d7-cong ; assoc to \u00d7-assoc)\n\n  module STest {n} M = Syntax _\u00d7_ \u22a4 _\u2194_ id (flip _\u2218_) A\u00d7\u22a4\u2194A (sym A\u00d7\u22a4\u2194A) (A\u00d7\u22a4\u2194A \u2218 swap-iso) (swap-iso \u2218 sym A\u00d7\u22a4\u2194A)\n                            \u00d7-cong first-iso (\u03bb x \u2192 second-iso (const x))\n                            (sym (\u00d7-assoc _ _ _)) (\u00d7-assoc _ _ _) swap-iso {n} M\n\n  test : \u2200 A B C \u2192 ((A \u00d7 B) \u00d7 C) \u2194 (C \u00d7 (B \u00d7 A))\n  test A B C = rewire ((# 0 , # 1) , # 2) (# 2 , (# 1 , # 0)) _ where\n    open STest (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Solver.Linear.Examples where\n\nopen import Solver.Linear\nopen import Data.String\nopen import Data.Zero\nopen import Relation.Binary.NP\nopen import Relation.Nullary.Decidable\nopen import Data.Vec\nopen import Data.Product\nopen import Data.Unit\nopen import Function\nopen import FunUniverse.Agda\n\nmodule example1 where\n\n  -- need to etaexpand this because otherwise we get an error\n  open Syntax\u02e2 funLin\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire\u02e2 \u0393 \"(A,B),C\u21a6(B,A),C\"\n   where\n    \u0393 : String \u2192 _\n    \u0393 \"A\" = A\n    \u0393 \"B\" = B\n    \u0393 \"C\" = C\n    \u0393  _  = \ud835\udfd8\n\nmodule example2 where\n  open Syntax\u1da0 funLin\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire\u1da0 (A \u2237 B \u2237 C \u2237 []) (\u03bb a b c \u2192 (a , b) , c \u21a6 (b , a) , c)\n\n  {-\nmodule example3 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest {n} M = Syntax _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' {n} M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\nmodule example\u2083 where\n\n  open import Data.Unit\n  open import Data.Product\n  open import Data.Vec\n\n  open import Function using (flip ; const)\n  \n  open import Function.Inverse\n  open import Function.Related.TypeIsomorphisms.NP\n\n  open \u00d7-CMon using () renaming (\u2219-cong to \u00d7-cong ; assoc to \u00d7-assoc)\n\n  module STest {n} M = Syntax _\u00d7_ \u22a4 _\u2194_ id (flip _\u2218_) A\u00d7\u22a4\u2194A (sym A\u00d7\u22a4\u2194A) (A\u00d7\u22a4\u2194A \u2218 swap-iso) (swap-iso \u2218 sym A\u00d7\u22a4\u2194A)\n                            \u00d7-cong first-iso (\u03bb x \u2192 second-iso (const x))\n                            (sym (\u00d7-assoc _ _ _)) (\u00d7-assoc _ _ _) swap-iso {n} M\n\n  test : \u2200 A B C \u2192 ((A \u00d7 B) \u00d7 C) \u2194 (C \u00d7 (B \u00d7 A))\n  test A B C = rewire ((# 0 , # 1) , # 2) (# 2 , (# 1 , # 0)) _ where\n    open STest (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c1d53d1c793d91e002e6c8b4ffbbcc8dca4c6eda","subject":"README.agda: Update recommended version of Agda","message":"README.agda: Update recommended version of Agda\n\nRationale: 2.4.0 doesn't install any more.\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda (2.4.0.*), the Agda standard library (version 0.8), generate\n-- Everything.agda with the attached Haskell helper, and finally run Agda on\n-- this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n--\n-- The formalization has four parts:\n--\n--   1. A formalization of change structures. This lives in\n--   Base.Change.Algebra. (What we call \"change structure\" in the\n--   paper, we call \"change algebra\" in the Agda code. We changed\n--   the name when writing the paper, and never got around to\n--   updating the name in the Agda code).\n--\n--   2. Incrementalization framework for first-class functions,\n--   with extension points for plugging in data types and their\n--   incrementalization. This lives in the Parametric.*\n--   hierarchy.\n--\n--   3. An example plugin that provides integers and bags\n--   with negative multiplicity. This lives in the Nehemiah.*\n--   hierarchy. (For some reason, we choose to call this\n--   particular incarnation of the plugin Nehemiah).\n--\n--   4. Other material that is unrelated to the framework\/plugin\n--   distinction. This is all other files.\n\n\n\n-- FORMALIZATION OF CHANGE STRUCTURES\n-- ==================================\n--\n-- Section 2 of the paper, and Base.Change.Algebra in Agda.\n\nimport Base.Change.Algebra\n\n\n\n-- INCREMENTALIZATION FRAMEWORK\n-- ============================\n--\n-- Section 3 of the paper, and Parametric.* hierarchy in Agda.\n--\n-- The extension points are implemented as module parameters. See\n-- detailed explanation in Parametric.Syntax.Type and\n-- Parametric.Syntax.Term. Some extension points are for types,\n-- some for values, and some for proof fragments. In Agda, these\n-- three kinds of entities are unified anyway, so we can encode\n-- all of them as module parameters.\n--\n-- Every module in the Parametric.* hierarchy adds at least one\n-- extension point, so the module hierarchy of a plugin will\n-- typically mirror the Parametric.* hierarchy, defining the\n-- values for these extension points.\n--\n-- Firstly, we have the syntax of types and terms, the erased\n-- change structure for function types and incrementalization as\n-- a term-to-term transformation. The contents of these modules\n-- formalizes Sec. 3.1 and 3.2 of the paper, except for the\n-- second and third line of Figure 3, which is formalized further\n-- below.\n\nimport Parametric.Syntax.Type   -- syntax of types\nimport Parametric.Syntax.Term   -- syntax of terms\nimport Parametric.Change.Type   -- simply-typed changes\nimport Parametric.Change.Derive -- incrementalization\n\n-- Secondly, we define the usual denotational semantics of the\n-- simply-typed lambda calculus in terms of total functions, and\n-- the change semantics in terms of a change structure on total\n-- functions. The contents of these modules formalize Sec. 3.3,\n-- 3.4 and 3.5 of the paper.\n\nimport Parametric.Denotation.Value      -- standard values\nimport Parametric.Denotation.Evaluation -- standard evaluation\nimport Parametric.Change.Validity       -- dependently-typed changes\nimport Parametric.Change.Specification  -- change evaluation\n\n-- Thirdly, we define terms that operate on simply-typed changes,\n-- and connect them to their values. The concents of these\n-- modules formalize the second and third line of Figure 3, as\n-- well as the semantics of these lines.\n\nimport Parametric.Change.Term        -- terms that operate on simply-typed changes\nimport Parametric.Change.Value       -- the values of these terms\nimport Parametric.Change.Evaluation  -- connecting the terms and their values\n\n-- Finally, we prove correctness by connecting the (syntactic)\n-- incrementalization to the (semantic) change evaluation by a\n-- logical relation, and a proof that the values of terms\n-- produced by the incrementalization transformation are related\n-- to the change values of the original terms. The contents of\n-- these modules formalize Sec. 3.6.\n\nimport Parametric.Change.Implementation -- logical relation\nimport Parametric.Change.Correctness    -- main correctness proof\n\n\n\n-- EXAMPLE PLUGIN\n-- ==============\n--\n-- Sec. 3.7 in the paper, and the Nehemiah.* hierarchy in Agda.\n--\n-- The module structure of the plugin follows the structure of\n-- the Parametric.* hierarchy.  For example, the extension point\n-- defined in Parametric.Syntax.Term is instantiated in\n-- Nehemiah.Syntax.Term.\n--\n-- As discussed in Sec. 3.7 of the paper, the point of this\n-- plugin is not to speed up any real programs, but to show \"that\n-- the interface for proof plugins can be implemented\". As a\n-- first step towards proving correctness of the more complicated\n-- plugin with integers, bags and finite maps we implement in\n-- Scala, we choose to define plugin with integers and bags in\n-- Agda. Instead of implementing bags (with negative\n-- multiplicities, like in the paper) in Agda, though, we\n-- postulate that a group of such bags exist. Note that integer\n-- bags with integer multiplicities are actually the free group\n-- given a singleton operation `Integer -> Bag`, so this should\n-- be easy to formalize in principle.\n\n-- Before we start with the plugin, we postulate an abstract data\n-- type for integer bags.\nimport Postulate.Bag-Nehemiah\n\n-- Firstly, we extend the syntax of types and terms, the erased\n-- change structure for function types, and incrementalization as\n-- a term-to-term transformation to account for the data types of\n-- the Nehemiah language. The contents of these modules\n-- instantiate the extension points in Sec. 3.1 and 3.2 of the\n-- paper, except for the second and third line of Figure 3, which\n-- is instantiated further below.\n\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\nimport Nehemiah.Change.Type\nimport Nehemiah.Change.Derive\n\n-- Secondly, we extend the usual denotational semantics and the\n-- change semantics to account for the data types of the Nehemiah\n-- language. The contents of these modules instantiate the\n-- extension points in Sec. 3.3, 3.4 and 3.5 of the paper.\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\n\n-- Thirdly, we extend the terms that operate on simply-typed\n-- changes, and the connection to their values to account for the\n-- data types of the Nehemiah language. The concents of these\n-- modules instantiate the extension points in the second and\n-- third line of Figure 3.\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- Finally, we extend the logical relation and the main\n-- correctness proof to account for the data types in the\n-- Nehemiah language. The contents of these modules instantiate\n-- the extension points defined in Sec. 3.6.\n\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n\n\n-- OTHER MATERIAL\n-- ==============\n--\n-- We postulate extensionality of Agda functions. This postulate\n-- is well known to be compatible with Agda's type theory.\n\nimport Postulate.Extensionality\n\n-- For historical reasons, we reexport Data.List.All from the\n-- standard library under the name DependentList.\n\nimport Base.Data.DependentList\n\n-- This module supports overloading the \u27e6_\u27e7 notation based on\n-- Agda's instance arguments.\n\nimport Base.Denotation.Notation\n\n-- These modules implement contexts including typed de Bruijn\n-- indices to represent bound variables, sets of bound variables,\n-- and environments. These modules are parametric in the set of\n-- types (that are stored in contexts) and the set of values\n-- (that are stored in environments). So these modules are even\n-- more parametric than the Parametric.* hierarchy.\n\nimport Base.Syntax.Context\nimport Base.Syntax.Vars\nimport Base.Denotation.Environment\n\n-- This module contains some helper definitions to merge a\n-- context of values and a context of changes.\nimport Base.Change.Context\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda (2.4.0), the Agda standard library (version 0.8), generate\n-- Everything.agda with the attached Haskell helper, and finally run Agda on\n-- this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n--\n-- The formalization has four parts:\n--\n--   1. A formalization of change structures. This lives in\n--   Base.Change.Algebra. (What we call \"change structure\" in the\n--   paper, we call \"change algebra\" in the Agda code. We changed\n--   the name when writing the paper, and never got around to\n--   updating the name in the Agda code).\n--\n--   2. Incrementalization framework for first-class functions,\n--   with extension points for plugging in data types and their\n--   incrementalization. This lives in the Parametric.*\n--   hierarchy.\n--\n--   3. An example plugin that provides integers and bags\n--   with negative multiplicity. This lives in the Nehemiah.*\n--   hierarchy. (For some reason, we choose to call this\n--   particular incarnation of the plugin Nehemiah).\n--\n--   4. Other material that is unrelated to the framework\/plugin\n--   distinction. This is all other files.\n\n\n\n-- FORMALIZATION OF CHANGE STRUCTURES\n-- ==================================\n--\n-- Section 2 of the paper, and Base.Change.Algebra in Agda.\n\nimport Base.Change.Algebra\n\n\n\n-- INCREMENTALIZATION FRAMEWORK\n-- ============================\n--\n-- Section 3 of the paper, and Parametric.* hierarchy in Agda.\n--\n-- The extension points are implemented as module parameters. See\n-- detailed explanation in Parametric.Syntax.Type and\n-- Parametric.Syntax.Term. Some extension points are for types,\n-- some for values, and some for proof fragments. In Agda, these\n-- three kinds of entities are unified anyway, so we can encode\n-- all of them as module parameters.\n--\n-- Every module in the Parametric.* hierarchy adds at least one\n-- extension point, so the module hierarchy of a plugin will\n-- typically mirror the Parametric.* hierarchy, defining the\n-- values for these extension points.\n--\n-- Firstly, we have the syntax of types and terms, the erased\n-- change structure for function types and incrementalization as\n-- a term-to-term transformation. The contents of these modules\n-- formalizes Sec. 3.1 and 3.2 of the paper, except for the\n-- second and third line of Figure 3, which is formalized further\n-- below.\n\nimport Parametric.Syntax.Type   -- syntax of types\nimport Parametric.Syntax.Term   -- syntax of terms\nimport Parametric.Change.Type   -- simply-typed changes\nimport Parametric.Change.Derive -- incrementalization\n\n-- Secondly, we define the usual denotational semantics of the\n-- simply-typed lambda calculus in terms of total functions, and\n-- the change semantics in terms of a change structure on total\n-- functions. The contents of these modules formalize Sec. 3.3,\n-- 3.4 and 3.5 of the paper.\n\nimport Parametric.Denotation.Value      -- standard values\nimport Parametric.Denotation.Evaluation -- standard evaluation\nimport Parametric.Change.Validity       -- dependently-typed changes\nimport Parametric.Change.Specification  -- change evaluation\n\n-- Thirdly, we define terms that operate on simply-typed changes,\n-- and connect them to their values. The concents of these\n-- modules formalize the second and third line of Figure 3, as\n-- well as the semantics of these lines.\n\nimport Parametric.Change.Term        -- terms that operate on simply-typed changes\nimport Parametric.Change.Value       -- the values of these terms\nimport Parametric.Change.Evaluation  -- connecting the terms and their values\n\n-- Finally, we prove correctness by connecting the (syntactic)\n-- incrementalization to the (semantic) change evaluation by a\n-- logical relation, and a proof that the values of terms\n-- produced by the incrementalization transformation are related\n-- to the change values of the original terms. The contents of\n-- these modules formalize Sec. 3.6.\n\nimport Parametric.Change.Implementation -- logical relation\nimport Parametric.Change.Correctness    -- main correctness proof\n\n\n\n-- EXAMPLE PLUGIN\n-- ==============\n--\n-- Sec. 3.7 in the paper, and the Nehemiah.* hierarchy in Agda.\n--\n-- The module structure of the plugin follows the structure of\n-- the Parametric.* hierarchy.  For example, the extension point\n-- defined in Parametric.Syntax.Term is instantiated in\n-- Nehemiah.Syntax.Term.\n--\n-- As discussed in Sec. 3.7 of the paper, the point of this\n-- plugin is not to speed up any real programs, but to show \"that\n-- the interface for proof plugins can be implemented\". As a\n-- first step towards proving correctness of the more complicated\n-- plugin with integers, bags and finite maps we implement in\n-- Scala, we choose to define plugin with integers and bags in\n-- Agda. Instead of implementing bags (with negative\n-- multiplicities, like in the paper) in Agda, though, we\n-- postulate that a group of such bags exist. Note that integer\n-- bags with integer multiplicities are actually the free group\n-- given a singleton operation `Integer -> Bag`, so this should\n-- be easy to formalize in principle.\n\n-- Before we start with the plugin, we postulate an abstract data\n-- type for integer bags.\nimport Postulate.Bag-Nehemiah\n\n-- Firstly, we extend the syntax of types and terms, the erased\n-- change structure for function types, and incrementalization as\n-- a term-to-term transformation to account for the data types of\n-- the Nehemiah language. The contents of these modules\n-- instantiate the extension points in Sec. 3.1 and 3.2 of the\n-- paper, except for the second and third line of Figure 3, which\n-- is instantiated further below.\n\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\nimport Nehemiah.Change.Type\nimport Nehemiah.Change.Derive\n\n-- Secondly, we extend the usual denotational semantics and the\n-- change semantics to account for the data types of the Nehemiah\n-- language. The contents of these modules instantiate the\n-- extension points in Sec. 3.3, 3.4 and 3.5 of the paper.\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\n\n-- Thirdly, we extend the terms that operate on simply-typed\n-- changes, and the connection to their values to account for the\n-- data types of the Nehemiah language. The concents of these\n-- modules instantiate the extension points in the second and\n-- third line of Figure 3.\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- Finally, we extend the logical relation and the main\n-- correctness proof to account for the data types in the\n-- Nehemiah language. The contents of these modules instantiate\n-- the extension points defined in Sec. 3.6.\n\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n\n\n-- OTHER MATERIAL\n-- ==============\n--\n-- We postulate extensionality of Agda functions. This postulate\n-- is well known to be compatible with Agda's type theory.\n\nimport Postulate.Extensionality\n\n-- For historical reasons, we reexport Data.List.All from the\n-- standard library under the name DependentList.\n\nimport Base.Data.DependentList\n\n-- This module supports overloading the \u27e6_\u27e7 notation based on\n-- Agda's instance arguments.\n\nimport Base.Denotation.Notation\n\n-- These modules implement contexts including typed de Bruijn\n-- indices to represent bound variables, sets of bound variables,\n-- and environments. These modules are parametric in the set of\n-- types (that are stored in contexts) and the set of values\n-- (that are stored in environments). So these modules are even\n-- more parametric than the Parametric.* hierarchy.\n\nimport Base.Syntax.Context\nimport Base.Syntax.Vars\nimport Base.Denotation.Environment\n\n-- This module contains some helper definitions to merge a\n-- context of values and a context of changes.\nimport Base.Change.Context\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93a456c3a471b05be1db7d20a63cfa4aea8a6a1a","subject":"agda Wouter Predicate Transformer","message":"agda Wouter Predicate Transformer\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/x.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/x.agda","new_contents":"{-# OPTIONS --type-in-type   #-} -- NOT SOUND!\n\nopen import Data.Empty      using (\u22a5)\nopen import Data.List       hiding (map; [_])\nopen import Data.Nat        renaming (\u2115 to Nat)\nopen import Data.Nat.DivMod\nopen import Data.Product    hiding (map)\nopen import Data.Unit       using (\u22a4; tt)\n------------------------------------------------------------------------------\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl)\n\nmodule x where\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\nabstract\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to\n    - use this refinement relation to show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\n-- C          : type of commands\n-- Free C R   : returns an 'a' or issues command c : C\n-- For each c : C, there is a set of responses R c\n-- 2nd arg of Step is continuation : how to proceed after receiving response R c\ndata Free (C : Set) (R : C \u2192 Set) (a : Set) : Set where\n  Pure :                           a  \u2192 Free C R a\n  Step : (c : C) \u2192 (R c \u2192 Free C R a) \u2192 Free C R a\n\n-- show that 'Free' is a monad:\n\nmap : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 (a \u2192 b) \u2192 Free C R a \u2192 Free C R b\nmap f (Pure x)   = Pure (f x)\nmap f (Step c k) = Step c (\u03bb r \u2192 map f (k r))\n\nreturn : \u2200 {a C : Set} {R : C \u2192 Set} \u2192 a \u2192 Free C R a\nreturn = Pure\n\n_>>=_ : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 Free C R a \u2192 (a \u2192 Free C R b) \u2192 Free C R b\nPure   x >>= f = f x\nStep c x >>= f = Step c (\u03bb r \u2192 x r >>= f)\n\n{-\ndifferent effects choose C and R differently, depending on their ops\n\n--------------------------------------------------\nWeakest precondition semantics\n\nidea of associating weakest precondition semantics with imperative programs\ndates to Dijkstra\u2019s Guarded Command Language [1975]\n\nways to specify behaviour of function f : a \u2192 b\n- reference implementation\n- define a relation R : a \u2192 b \u2192 Set\n- write contracts and test cases\n- PT semantics\n\ncall values of type a \u2192 Set : predicate on type a\n\nPTs are functions between predicates\ne.g., weakest precondition:\n-}\n\n-- \"maps\"\n-- function f : a \u2192 b and\n-- desired postcondition on the function\u2019s output, b \u2192 Set\n-- to weakest precondition a \u2192 Set on function\u2019s input that ensures postcondition satisfied\n--\n-- non-dependent version\n-- note: definition is just reverse function composition\nwp0 : \u2200 {a : Set} {b :     Set} (f :      a  \u2192 b)   \u2192           (b   \u2192 Set) \u2192 (a \u2192 Set)\nwp0 f P = \u03bb x \u2192 P   (f x)\n{-\nabove wp semantics is sometimes too restrictive\n- no way to specify that output is related to input\n- fix via making f dependent:\n-}\n-- dependent version\nwp  : \u2200 {a : Set} {b : a \u2192 Set} (f : (x : a) \u2192 b x) \u2192 ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)\nwp  f P = \u03bb x \u2192 P x (f x)\n\n-- shorthand for working with predicates and predicates transformers\n_\u2286_ : \u2200 {a : Set} \u2192 (a \u2192 Set) \u2192 (a \u2192 Set) \u2192 Set\nP \u2286 Q = \u2200 x \u2192 P x \u2192 Q x\n\n-- refinement relation defined between PTs\n_\u2291_ : \u2200 {a : Set} {b : a \u2192 Set} \u2192 (pt1 pt2 : ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)) \u2192 Set\u2081\npt1 \u2291 pt2 = \u2200 P \u2192 pt1 P \u2286 pt2 P\n\n{-\nuse refinement relation\n- to relate PT semantics between programs and specifications\n- to show a program satisfies its specification; or\n- to show that one program is somehow \u2018better\u2019 than another,\n  where \u2018better\u2019 is defined by choice of PT semantics\n\nin pure setting, this refinement relation is not interesting:\nthe refinement relation corresponds to extensional equality between functions:\n\nlemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\nrefinement : \u2200 (f g : a \u2192 b) \u2192 (wp f \u2291 wp g) \u2194 (\u2200 x \u2192 f x \u2261 g x)\n\nthis paper defines PT semantics for Kleisli arrows of form\n\n    a \u2192 Free C R b\n\ncould use 'wp' to assign semantics to these computations directly,\nbut typically not interested in syntactic equality between free monads\n\nrather want to study semantics of effectful programs they represent\n\nto define a PT semantics for effects\ndefine a function with form:\n\n  -- how to lift a predicate on 'a' over effectful computation returning 'a'\n  pt : (a \u2192 Set) \u2192 Free C R a \u2192 Set\n\n'pt' def depends on semantics desired for a particulr free monad\n\nCrucially, choice of\n- pt and\n- weakest precondition semantics :  wp\ntogether give a way to assign weakest precondition semantics\nto Kleisli arrows representing effectful computations\n\n------------------------------------------------------------------------------\n3 PARTIALITY\n\nPartial computations : i.e., 'Maybe'\n\nmake choices for commands C and responses R\n-}\n\ndata C : Set where\n  Abort : C -- no continuation\n\nR : C \u2192 Set\nR Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\nPartial : Set \u2192 Set\nPartial = Free C R\n\n-- smart constructor for failure:\nabort : \u2200 {a : Set} \u2192 Partial a\nabort = Step Abort (\u03bb ())\n\n{-\ncomputation of type Partial a will either\n- return a value of type a or\n- fail, issuing abort command\n\nNote that responses to Abort command are empty;\nabort smart constructor abort uses this to discharge the continuation\nin the second argument of the Step constructor\n\n--------------------------------------------------\nExample: division\n\nexpression language, closed under division and natural numbers:\n-}\n\ndata Expr : Set where\n  Val : Nat  \u2192 Expr\n  Div : Expr \u2192 Expr \u2192 Expr\n\nexv : Expr\nexv = Val 3\nexd : Expr\nexd = Div (Val 3) (Val 3)\n\n-- semantics specified using inductively defined RELATION:\n-- def rules out erroneous results by requiring the divisor evaluates to non-zero\ndata _\u21d3_ : Expr \u2192 Nat \u2192 Set where\n  \u21d3Base : \u2200 {x : Nat}\n        \u2192 Val x \u21d3 x\n  \u21d3Step : \u2200 {l r : Expr} {v1 v2 : Nat}\n        \u2192     l   \u21d3  v1\n        \u2192       r \u21d3         (suc v2)\n        \u2192 Div l r \u21d3 (v1 div (suc v2))\n\nexb : Val 3 \u21d3 3\nexb = \u21d3Base\n\nexs : Div (Val 3) (Val 3) \u21d3 1\nexs = \u21d3Step \u21d3Base \u21d3Base\n\n-- alternatively, semantics specified by an INTERPRETER\n-- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n-- define operation used by \u27e6_\u27e7 interpreter\n_\u00f7_ : Nat \u2192 Nat \u2192 Partial Nat\nn \u00f7 zero    = abort\nn \u00f7 (suc k) = return (n div (suc k))\n\n\u27e6_\u27e7 : Expr \u2192 Partial Nat\n\u27e6 Val x \u27e7     = return x\n\u27e6 Div e1 e2 \u27e7 = \u27e6 e1 \u27e7 >>= \u03bb v1 \u2192 \u27e6 e2 \u27e7 >>= \u03bb v2 \u2192 v1 \u00f7 v2\n\nevv   : Free C R Nat\nevv   = \u27e6 Val 3 \u27e7\nevv'  : evv \u2261 Pure 3\nevv'  = refl\n\nevd   : Free C R Nat\nevd   = \u27e6 Div (Val 3) (Val 3) \u27e7\nevd'  : evd \u2261 Pure 1\nevd'  = refl\n\nevd0  : Free C R Nat\nevd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\nevd0' : evd0 \u2261 Step Abort (\u03bb ())\nevd0' = refl\n\n{-\nHow to relate two definitions:\n- std lib 'div' requires implicit proof that divisor is non-zero\n  - \u21d3 relation generates via pattern matching\n  - _\u00f7_ does explicit check\n- interpreter uses _\u00f7_\n  - fails explicitly with abort when divisor is zero\n\nAssign a weakest precondition semantics to Kleisli arrows of the form\n\n    a \u2192 Partial b\n-}\n\nmustPT    : \u2200 {a : Set} {b : a \u2192 Set}\n          \u2192 (P : (x : a) \u2192          b x \u2192 Set)\n          \u2192      (x : a)\n          \u2192                Partial (b x)\n          \u2192 Set\nmustPT P _ (Pure y)       = P _ y\nmustPT P _ (Step Abort _) = \u22a5\n\nwpPartial : \u2200 {a : Set} {b : a \u2192 Set}\n          \u2192 (f : (x : a) \u2192 Partial (b x))\n          \u2192 (P : (x : a) \u2192          b x \u2192 Set)\n          \u2192 (a \u2192 Set)\nwpPartial f P = wp f (mustPT P)\n{-\nTo call 'wp', must show how to transform\n-      predicate                  P :         b \u2192 Set\n- to a predicate on partial results : Partial b \u2192 Set\nDone via proposition 'mustPT P c'\n- holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\nparticular PT semantics of partial computations determined by def mustPT\nhere: rule out failure entirely\n- so case Abort returns empty type\n\nGiven this PT semantics for Kleisli arrows in general,\ncan now study semantics of above monadic interpreter\nvia passing\n- interpreter: \u27e6_\u27e7\n- desired postcondition : _\u21d3_\nas arguments to wpPartial:\n-}\n\nexwpp  : Expr \u2192 Set\nexwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\nexwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb z \u2192 mustPT _\u21d3_ z \u27e6 z \u27e7\nexwpp' = refl\n\nwppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\nwppv1  = \u21d3Base\nwppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\nwppd   = \u21d3Step \u21d3Base \u21d3Base\n\n\n{-\nresulting in a predicate on expressions\n\nfor all expressions satisfying this predicate,\nthe monadic interpreter and the relational specification, _\u21d3_,\nmust agree on the result of evaluation\n\nWhat does this say about correctness of interpreter?\nTo understand the predicate better, consider defining this predicate on expressions:\n-}\n\nSafeDiv : Expr \u2192 Set\nSafeDiv (Val x)     = \u22a4\nSafeDiv (Div e1 e2) = (e2 \u21d3 zero \u2192 \u22a5) {-\u2227-} \u00d7 SafeDiv e1 {-\u2227-} \u00d7 SafeDiv e2\n\nexsdv : SafeDiv (Val 3) \u2261 \u22a4\nexsdv = refl\nexsdd : SafeDiv (Div (Val 3) (Val 3)) \u2261 \u03a3 ((x : Val 3 \u21d3 zero) \u2192 \u22a5) (\u03bb _ \u2192 \u03a3 \u22a4 (\u03bb _ \u2192 \u22a4))\nexsdd = refl\n\n{-\nExpect : any expr e for which SafeDiv e holds\ncan be evaluated without division-by-zero\n\ncan prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n-- lemma relates the two semantics\n-- expressed as a relation and an evaluator\n-- for those expressions that satisfy the SafeDiv property\n-}\n\n-- correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n{-\ncorrect (Val n) tt = \u21d3Base\ncorrect (Div exprL exprR) (exprR\u21d3zero\u2192\u22a5 , saveDivExprL , safeDivExprR)\n  with (correct exprL) saveDivExprL\n... | mustPT_\u21d3_exL\u27e6exL\u27e7\n  with (correct exprR) safeDivExprR\n... | mustPT_\u21d3_exR\u27e6exR\u27e7\n  = {!!}\n-}\n{-\n----- another try\ncorrect (Val x)                                   tt                                        =\n  \u21d3Base\ncorrect (Div (Val x1)          (Val x2))          (r2\u21d3zero\u2192\u22a5 , tt , tt)                     =\n  {!!}\ncorrect (Div (Val x)           (Div expr2 expr3)) (r2\u21d3zero\u2192\u22a5 , tt , fst , snd)              =\n  {!!}\ncorrect (Div (Div expr1 expr3) (Val x))           (r2\u21d3zero\u2192\u22a5 , (fst , snd) , tt)            =\n  {!!}\ncorrect (Div (Div expr1 expr3) (Div expr2 expr4)) (r2\u21d3zero\u2192\u22a5 , safeDivExpr1 , safeDivExpr2) =\n  {!!}\n-}\n\n{-\nInstead of manually defining SafeDiv,\ndefine more general predicate characterising the domain of a partial function:\n-}\n\ndom : \u2200 {a : Set} {b : a \u2192 Set}\n    \u2192 ((x : a)\n    \u2192 Partial (b x))\n    \u2192 (a \u2192 Set)\ndom f = wpPartial f (\u03bb _ _ \u2192 \u22a4)\n\n{-\ncan show that the two semantics agree precisely on the domain of the interpreter:\n\nsound    : dom       \u27e6_\u27e7 \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\ncomplete : wpPartial \u27e6_\u27e7 _\u21d3_ \u2286 dom \u27e6_\u27e7\n\nboth proofs proceed by induction on the argument expression\n\n--------------------------------------------------\nRefinement\n\nweakest precondition semantics on partial computations give rise\nto a refinement relation on Kleisli arrows of the form a \u2192 Partial b\n\ncan characterise this relation by proving:\n\nrefinement : (f g : a \u2192 Maybe b)\n           \u2192 (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x \u2192 (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\nuse refinement to relate Kleisli morphisms,\nand to relate a program to a specification given by a pre- and postcondition\n\n--------------------------------------------------\nExample: Add (interpreter for stack machine)\n\nadd top two elements; can fail fail if stack has too few elements\n\nbelow shows how to prove the definition meets its specification\n\nDefine specification in terms of a pre\/post condition.\n-}\n-- specification of a function of type (x : a) \u2192 b x consists of:\nrecord Spec (a : Set) (b : a \u2192 Set) : Set where\n  constructor [_,_]\n  field\n    pre  : a \u2192 Set             -- precondition on 'a'\n    post : (x : a) \u2192 b x \u2192 Set -- postcondition relating inputs that satisfy this precondition\n                               -- and the corresponding outputs\n\n{-\n[ P , Q ] : specification consisting of precondition P and postcondition Q\n\nfor non-dependent examples (e.g., type b does not depend on x : a)\n-}\n\nK : {A B : Set} \u2192 A \u2192 B \u2192 A\nK x y = x\n\nSpecK : Set \u2192 Set \u2192 Set\nSpecK a b = Spec a (K b)\n\n-- specification for addition function : describes the desired postcondition\ndata Add : List Nat \u2192 List Nat \u2192 Set where\n  AddStep : {x1 x2 : Nat} {xs : List Nat}\n          \u2192 Add (x1 \u2237 x2 \u2237 xs) ((x1 + x2 ) \u2237 xs)\n\naddSpec : SpecK (List Nat) (List Nat)\naddSpec = [ (\u03bb xs \u2192 length xs > 1) , Add ]\n\n{-\nHow to relate this specification to an implementation?\nNote: 'wpPartial' assigns predicate transformer semantics to functions\n- but do not yet have a corresponding predicate transform semantics for specifications.\nwpSpec function does that:\n-}\n\n-- Given a specification, Spec a b\n-- computes the weakest precondition that satisfies an arbitrary postcondition P\n-- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\nwpSpec : {a : Set} {b : a \u2192 Set}\n       \u2192 Spec a b \u2192 (P : (x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)\nwpSpec [ pre , post ] P = \u03bb x \u2192 (pre x) \u00d7 {-\u2227-} (post x \u2286 P x)\n\n-- Can now formulate program\nadd : List Nat \u2192 Partial (List Nat)\nadd xs =\n  pop xs >>= \u03bb { (x1 , xs) \u2192\n  pop xs >>= \u03bb { (x2 , xs) \u2192\n  return ((x1 + x2 ) \u2237 xs) } }\n where\n  pop : {a : Set} \u2192 List a \u2192 Partial (a \u00d7 List a)\n  pop Data.List.[] = abort\n  pop (x \u2237 xs)     = return (x , xs)\n\n-- that refines the specification given by addSpec:\n{-\ncorrectness : wpSpec addSpec \u2291 wpPartial add\ncorrectness addstep xs (pre-addSpec-xs , post-addSpec-xs-\u2286-addstep-xs) =\n  {!!}\n-}\n{-\nThis example illustrates how to use the refinement relation\n- to relate a specification\n  - given in terms of a pre- and postcondition,\n- to its implementation.\n\nCompared to the refinement calculus\n- have not yet described how to mix code and specifications (see Section 7)\n\n--------------------------------------------------\nAlternative semantics\n-}\n","old_contents":"{-# OPTIONS --type-in-type   #-} -- NOT SOUND!\n\nopen import Data.Empty      using (\u22a5)\nopen import Data.Nat        renaming (\u2115 to Nat)\nopen import Data.Nat.DivMod\nopen import Data.Product    hiding (map)\nopen import Data.Unit       using (\u22a4)\n------------------------------------------------------------------------------\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl)\n\nmodule x where\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\nabstract\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nINTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to\n    - use this refinement relation to show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n2 BACKGROUND\n\nFree monads\n-}\n\n-- C          : type of commands\n-- Free C R   : returns an 'a' or issues command c : C\n-- For each c : C, there is a set of responses R c\n-- 2nd arg of Step is continuation : how to proceed after receiving response R c\ndata Free (C : Set) (R : C \u2192 Set) (a : Set) : Set where\n  Pure :                           a  \u2192 Free C R a\n  Step : (c : C) \u2192 (R c \u2192 Free C R a) \u2192 Free C R a\n\n-- show that 'Free' is a monad:\n\nmap : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 (a \u2192 b) \u2192 Free C R a \u2192 Free C R b\nmap f (Pure x)   = Pure (f x)\nmap f (Step c k) = Step c (\u03bb r \u2192 map f (k r))\n\nreturn : \u2200 {a C : Set} {R : C \u2192 Set} \u2192 a \u2192 Free C R a\nreturn = Pure\n\n_>>=_ : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 Free C R a \u2192 (a \u2192 Free C R b) \u2192 Free C R b\nPure   x >>= f = f x\nStep c x >>= f = Step c (\u03bb r \u2192 x r >>= f)\n\n{-\ndifferent effects choose C and R differently, depending on their ops\n\nWeakest precondition semantics\n\nidea of associating weakest precondition semantics with imperative programs\ndates to Dijkstra\u2019s Guarded Command Language [1975]\n\nways to specify behaviour of function f : a \u2192 b\n- reference implementation\n- define a relation R : a \u2192 b \u2192 Set\n- write contracts and test cases\n- PT semantics\n\ncall values of type a \u2192 Set : predicate on type a\n\nPTs are functions between predicates\ne.g., weakest precondition:\n-}\n\n-- \"maps\"\n-- function f : a \u2192 b and\n-- desired postcondition on the function\u2019s output, b \u2192 Set\n-- to weakest precondition a \u2192 Set on function\u2019s input that ensures postcondition satisfied\n--\n-- note: definition is just reverse function composition\n-- wp0 : \u2200 {a b : Set} \u2192 (f : a \u2192 b) \u2192 (b \u2192 Set) \u2192 (a \u2192 Set)\nwp0 : \u2200 {a : Set} {b :     Set} (f :      a  \u2192 b)   \u2192           (b   \u2192 Set) \u2192 (a \u2192 Set)\nwp0 f P = \u03bb x \u2192 P   (f x)\n{-\nabove wp semantics is sometimes too restrictive\n- no way to specify that output is related to input\n- fix via making f dependent:\n-}\nwp  : \u2200 {a : Set} {b : a \u2192 Set} (f : (x : a) \u2192 b x) \u2192 ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)\nwp  f P = \u03bb x \u2192 P x (f x)\n\n-- shorthand for working with predicates and predicates transformers\n_\u2286_ : \u2200 {a : Set} \u2192 (a \u2192 Set) \u2192 (a \u2192 Set) \u2192 Set\nP \u2286 Q = \u2200 x \u2192 P x \u2192 Q x\n\n-- refinement relation defined between PTs\n_\u2291_ : \u2200 {a : Set} {b : a \u2192 Set} \u2192 (pt1 pt2 : ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)) \u2192 Set\u2081\npt1 \u2291 pt2 = \u2200 P \u2192 pt1 P \u2286 pt2 P\n\n{-\nuse refinement relation\n- to relate PT semantics between programs and specifications\n- to show a program satisfies its specification; or\n- to show that one program is somehow \u2018better\u2019 than another,\n  where \u2018better\u2019 is defined by choice of PT semantics\n\nin pure setting, this refinement relation is not interesting:\nthe refinement relation corresponds to extensional equality between functions:\n\nlemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\nrefinement : \u2200 (f g : a \u2192 b) \u2192 (wp f \u2291 wp g) \u2194 (\u2200 x \u2192 f x \u2261 g x)\n\nthis paper defines PT semantics for Kleisli arrows of form\n\n    a \u2192 Free C R b\n\ncould use 'wp' to assign semantics to these computations directly,\nbut typically not interested in syntactic equality between free monads\n\nrather want to study semantics of effectful programs they represent\n\nto define a PT semantics for effects\ndefine a function with form:\n\n  -- how to lift a predicate on 'a' over effectful computation returning 'a'\n  pt : (a \u2192 Set) \u2192 Free C R a \u2192 Set\n\n'pt' def depends on semantics desired for a particulr free monad\n\nCrucially,\nchoice of pt and weakest precondition semantics, wp, together\ngive a way to assign weakest precondition semantics to Kleisli arrows\nrepresenting effectful computations\n\n3 PARTIALITY\n\nPartial computations : i.e., 'Maybe'\n\nmake choices for commands C and responses R\n-}\n\ndata C : Set where\n  Abort : C -- no continuation\n\nR : C \u2192 Set\nR Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\nPartial : Set \u2192 Set\nPartial = Free C R\n\n-- smart constructor for failure:\nabort : \u2200 {a : Set} \u2192 Partial a\nabort = Step Abort (\u03bb ())\n\n{-\ncomputation of type Partial a will either\n- return a value of type a or\n- fail, issuing abort command\n\nNote that responses to Abort command are empty;\nabort smart constructor abort uses this to discharge the continuation\nin the second argument of the Step constructor\n\nExample: division\n\nexpression language, closed under division and natural numbers:\n-}\n\ndata Expr : Set where\n  Val : Nat  \u2192 Expr\n  Div : Expr \u2192 Expr \u2192 Expr\n\nexv : Expr\nexv = Val 3\nexd : Expr\nexd = Div (Val 3) (Val 3)\n\n-- semantics specified using inductively defined RELATION:\n-- def rules out erroneous results by requiring the divisor evaluates to non-zero\ndata _\u21d3_ : Expr \u2192 Nat \u2192 Set where\n  Base : \u2200 {x : Nat}\n       \u2192 Val x \u21d3 x\n  Step : \u2200 {l r : Expr} {v1 v2 : Nat}\n       \u2192     l   \u21d3  v1\n       \u2192       r \u21d3         (suc v2)\n       \u2192 Div l r \u21d3 (v1 div (suc v2))\n\nexb : Val 3 \u21d3 3\nexb = Base\n\nexs : Div (Val 3) (Val 3) \u21d3 1\nexs = Step Base Base\n\n-- Alternatively\n-- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n-- op used by \u27e6_\u27e7 interpreter\n_\u00f7_ : Nat \u2192 Nat \u2192 Partial Nat\nn \u00f7 zero    = abort\nn \u00f7 (suc k) = return (n div (suc k))\n\n\u27e6_\u27e7 : Expr \u2192 Partial Nat\n\u27e6 Val x \u27e7     = return x\n\u27e6 Div e1 e2 \u27e7 = \u27e6 e1 \u27e7 >>= \u03bb v1 \u2192 \u27e6 e2 \u27e7 >>= \u03bb v2 \u2192 v1 \u00f7 v2\n\nevv   : Free C R Nat\nevv   = \u27e6 Val 3 \u27e7\nevv'  : evv \u2261 Pure 3\nevv'  = refl\n\nevd   : Free C R Nat\nevd   = \u27e6 Div (Val 3) (Val 3) \u27e7\nevd'  : evd \u2261 Pure 1\nevd'  = refl\n\nevd0  : Free C R Nat\nevd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\nevd0' : evd0 \u2261 Step Abort (\u03bb ())\nevd0' = refl\n\n{-\nHow to relate two definitions:\n- std lib 'div' requires implicit proof that divisor is non-zero\n  - \u21d3 relation generates via pattern matching\n  - _\u00f7_ does explicit check\n- interpreter uses _\u00f7_\n  - fails explicitly with abort when divisor is zero\n\nAssign a weakest precondition semantics to Kleisli arrows of the form\n\n    a \u2192 Partial b\n-}\n\nwpPartial : \u2200 {a : Set} {b : a \u2192 Set}\n          \u2192 (f : (x : a) \u2192 Partial (b x))\n          \u2192 (P : (x : a) \u2192          b x \u2192 Set)\n          \u2192 (a \u2192 Set)\nwpPartial f P = wp f (mustPT P)\n where\n  mustPT : \u2200 {a : Set} {b : a \u2192 Set}\n         \u2192 (P : (x : a) \u2192 b x \u2192 Set)\n         \u2192      (x : a)\n         \u2192 Partial       (b x) \u2192 Set\n  mustPT P _ (Pure y)       = P _ y\n  mustPT P _ (Step Abort _) = \u22a5\n{-\nTo call 'wp', must show how to transform\n-      predicate                  P :         b \u2192 Set\n- to a predicate on partial results : Partial b \u2192 Set\nDone via proposition 'mustPT P c'\n- holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\nparticular PT semantics of partial computations determined by def mustPT\nhere: rule out failure entirely\n- so case Abort returns empty type\n\nGiven this PT semantics for Kleisli arrows in general,\ncan now study semantics of above monadic interpreter\nvia passing\n- interpreter: \u27e6_\u27e7\n- desired postcondition : _\u21d3_\nas arguments to wpPartial:\n\n  wpPartial \u27e6_\u27e7 _\u21d3_ : Expr \u2192 Set\n\nresulting in a predicate on expressions\n\nfor all expressions satisfying this predicate,\nthe monadic interpreter and the relational specification, _\u21d3_,\nmust agree on the result of evaluation\n\nWhat does this say about correctness of interpreter?\nTo understand the predicate better, consider defining this predicate on expressions:\n-}\n\nSafeDiv : Expr \u2192 Set\nSafeDiv (Val x)     = \u22a4\nSafeDiv (Div e1 e2) = (e2 \u21d3 zero \u2192 \u22a5) {-\u2227-} \u00d7 SafeDiv e1 {-\u2227-} \u00d7 SafeDiv e2\n\n{-\nExpect : any expr e for which SafeDiv e holds\ncan be evaluated without division-by-zero\n\ncan prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n-- lemma relates the two semantics\n-- expressed as a relation and an evaluator\n-- for those expressions that satisfy the SafeDiv property\ncorrect : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n\nInstead of manually defining SafeDiv, define more general predicate\ncharacterising the domain of a partial function:\n-}\n\ndom : \u2200 {a : Set} {b : a \u2192 Set}\n    \u2192 ((x : a)\n    \u2192 Partial (b x))\n    \u2192 (a \u2192 Set)\ndom f = wpPartial f (\u03bb _ _ \u2192 \u22a4)\n\n{-\ncan show that the two semantics agree precisely on the domain of the interpreter:\n\nsound    : dom       \u27e6_\u27e7 \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\ncomplete : wpPartial \u27e6_\u27e7 _\u21d3_ \u2286 dom \u27e6_\u27e7\n\nboth proofs proceed by induction on the argument expression\n\nRefinement\n\nweakest precondition semantics on partial computations give rise\nto a refinement relation on Kleisli arrows of the form a \u2192 Partial b\n\ncan characterise this relation by proving:\n\nrefinement : (f g : a \u2192 Maybe b)\n           \u2192 (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x \u2192 (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\nuse refinement to relate Kleisli morphisms,\nand to relate a program to a specification given by a pre- and postcondition\n\n\nExample: Add (interpreter for stack machine)\n\nadd top two elements; can fail fail if stack has too few elements\n\nbelow shows how to prove the definition meets its specification\n\nDefine specification in terms of a pre\/post condition.\nThe specification of a function of type (x : a) \u2192 b x consists of\n-}\nrecord Spec (a : Set) (b : a \u2192 Set) : Set where\n  constructor [_,_]\n  field\n    pre  : a \u2192 Set             -- a precondition on a, and\n    post : (x : a) \u2192 b x \u2192 Set -- a postcondition relating inputs that satisfy this precondition\n                               -- and the corresponding outputs\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"71d08e755840c84e86e2846d5b94d249142a7219","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    num   : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2295_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2297_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- expressions\n  data e\u0307 : Set where\n    _\u00b7:_   : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X      : Nat \u2192 e\u0307\n    \u00b7\u03bb     : Nat \u2192 e\u0307 \u2192 e\u0307\n    N      : Nat \u2192 e\u0307\n    _\u00b7+_   : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]  : Nat \u2192 e\u0307\n    \u2987_\u2988[_] : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_    : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    inl    : e\u0307 \u2192 e\u0307\n    inr    : e\u0307 \u2192 e\u0307\n    case   : e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 e\u0307\n    [_,_]  : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    fst    : e\u0307 \u2192 e\u0307\n    snd    : e\u0307 \u2192 e\u0307\n\n  data subst : Set where -- todo; reuse contexts? do we need access to an enumeration of the domain?\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb       : Nat \u2192 \u00eb \u2192 \u00eb\n    N        : Nat \u2192 \u00eb\n    _\u00b7+_     : \u00eb \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_,_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_,_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    inl      : \u00eb \u2192 \u00eb\n    inr      : \u00eb \u2192 \u00eb\n    case     : \u00eb \u2192 Nat \u2192 \u00eb \u2192 Nat \u2192 \u00eb \u2192 \u00eb\n    [_,_]    : \u00eb \u2192 \u00eb \u2192 \u00eb\n    fst      : \u00eb \u2192 \u00eb\n    snd      : \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 ==> t2) ~ (t1' ==> t2')\n    TCSum : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2295 t2) ~ (t1' \u2295 t2')\n    TCPro : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2297 t2) ~ (t1' \u2297 t2')\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICNumArr1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 ==> t2)\n    ICNumArr2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 num\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n\n    ICNumSum1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2295 t2)\n    ICNumSum2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 num\n    ICSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSumArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 ==> t4)\n    ICSumArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2295 t4)\n\n    ICNumPro1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2297 t2)\n    ICNumPro2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 num\n    ICPro1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICPro2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICProArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 ==> t4)\n    ICProArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2297 t4)\n\n    ICProSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 \u2295 t4)\n    ICProSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 \u2297 t4)\n\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr (\u2987\u2988 ==> \u2987\u2988)\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) \u25b8arr (t1 ==> t2)\n\n  data _\u25b8sum_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MSHole  : \u2987\u2988 \u25b8sum (\u2987\u2988 \u2295 \u2987\u2988)\n    MSPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) \u25b8sum (t1 \u2295 t2)\n\n  data _\u25b8pro_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MPHole  : \u2987\u2988 \u25b8pro (\u2987\u2988 \u2297 \u2987\u2988)\n    MPPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) \u25b8pro (t1 \u2297 t2)\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr (t2 ==> t') \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SNum    :  {\u0393 : tctx} {n : Nat} \u2192\n                 \u0393 \u22a2 N n => num\n      SPlus   : {\u0393 : tctx} {e1 e2 : e\u0307}  \u2192\n                 \u0393 \u22a2 e1 <= num \u2192\n                 \u0393 \u22a2 e2 <= num \u2192\n                 \u0393 \u22a2 (e1 \u00b7+ e2) => num\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n\n    --todo: add rules for products in both jugements\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr (t1 ==> t2) \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n      AInl : {\u0393 : tctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t1 \u2192\n                 \u0393 \u22a2 inl e <= t+\n      AInr : {\u0393 : tctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 inr e <= t+\n      ACase : {\u0393 : tctx} {e e1 e2 : e\u0307} {t t+ t1 t2 : \u03c4\u0307} {x y : Nat} \u2192\n                 x # \u0393 \u2192\n                 y # \u0393 \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e => t+ \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e1 <= t \u2192\n                 (\u0393 ,, (y , t2)) \u22a2 e2 <= t \u2192\n                 \u0393 \u22a2 case e x e1 y e2 <= t\n\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete num         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2295 t2)   = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2297 t2)   = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete (N x)      = \u22a4\n  ecomplete (e1 \u00b7+ e2) = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (inl e)    = ecomplete e\n  ecomplete (inr e)    = ecomplete e\n  ecomplete (case e x e1 y e2)  = ecomplete e \u00d7 ecomplete e1 \u00d7 ecomplete e2\n  ecomplete [ e1 , e2 ] = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (fst e) = ecomplete e\n  ecomplete (snd e) = ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n\n  -- indeterminate\n  data _indet : \u00eb \u2192 Set where\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where -- todo not a context\n\n  -- final\n  data _final : \u00eb \u2192 Set where\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    num   : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2295_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2297_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- expressions\n  data e\u0307 : Set where\n    _\u00b7:_   : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X      : Nat \u2192 e\u0307\n    \u00b7\u03bb     : Nat \u2192 e\u0307 \u2192 e\u0307\n    N      : Nat \u2192 e\u0307\n    _\u00b7+_   : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]  : Nat \u2192 e\u0307\n    \u2987_\u2988[_] : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_    : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    inl    : e\u0307 \u2192 e\u0307\n    inr    : e\u0307 \u2192 e\u0307\n    case   : e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 e\u0307\n    [_,_]  : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    fst    : e\u0307 \u2192 e\u0307\n    snd    : e\u0307 \u2192 e\u0307\n\n  data subst : Set where -- todo\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb       : Nat \u2192 \u00eb \u2192 \u00eb\n    N        : Nat \u2192 \u00eb\n    _\u00b7+_     : \u00eb \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_,_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_,_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    inl      : \u00eb \u2192 \u00eb\n    inr      : \u00eb \u2192 \u00eb\n    case     : \u00eb \u2192 Nat \u2192 \u00eb \u2192 Nat \u2192 \u00eb \u2192 \u00eb\n    [_,_]    : \u00eb \u2192 \u00eb \u2192 \u00eb\n    fst      : \u00eb \u2192 \u00eb\n    snd      : \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 ==> t2) ~ (t1' ==> t2')\n    TCSum : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2295 t2) ~ (t1' \u2295 t2')\n    TCPro : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2297 t2) ~ (t1' \u2297 t2')\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICNumArr1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 ==> t2)\n    ICNumArr2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 num\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n\n    ICNumSum1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2295 t2)\n    ICNumSum2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 num\n    ICSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSumArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 ==> t4)\n    ICSumArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2295 t4)\n\n    ICNumPro1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2297 t2)\n    ICNumPro2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 num\n    ICPro1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICPro2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICProArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 ==> t4)\n    ICProArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2297 t4)\n\n    ICProSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 \u2295 t4)\n    ICProSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 \u2297 t4)\n\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr (\u2987\u2988 ==> \u2987\u2988)\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) \u25b8arr (t1 ==> t2)\n\n  data _\u25b8sum_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MSHole  : \u2987\u2988 \u25b8sum (\u2987\u2988 \u2295 \u2987\u2988)\n    MSPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) \u25b8sum (t1 \u2295 t2)\n\n  data _\u25b8pro_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MPHole  : \u2987\u2988 \u25b8pro (\u2987\u2988 \u2297 \u2987\u2988)\n    MPPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) \u25b8pro (t1 \u2297 t2)\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr (t2 ==> t') \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SNum    :  {\u0393 : tctx} {n : Nat} \u2192\n                 \u0393 \u22a2 N n => num\n      SPlus   : {\u0393 : tctx} {e1 e2 : e\u0307}  \u2192\n                 \u0393 \u22a2 e1 <= num \u2192\n                 \u0393 \u22a2 e2 <= num \u2192\n                 \u0393 \u22a2 (e1 \u00b7+ e2) => num\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n\n    --todo: add rules for products in both jugements\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr (t1 ==> t2) \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n      AInl : {\u0393 : tctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t1 \u2192\n                 \u0393 \u22a2 inl e <= t+\n      AInr : {\u0393 : tctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 inr e <= t+\n      ACase : {\u0393 : tctx} {e e1 e2 : e\u0307} {t t+ t1 t2 : \u03c4\u0307} {x y : Nat} \u2192\n                 x # \u0393 \u2192\n                 y # \u0393 \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e => t+ \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e1 <= t \u2192\n                 (\u0393 ,, (y , t2)) \u22a2 e2 <= t \u2192\n                 \u0393 \u22a2 case e x e1 y e2 <= t\n\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete num         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2295 t2)   = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2297 t2)   = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete (N x)      = \u22a4\n  ecomplete (e1 \u00b7+ e2) = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (inl e)    = ecomplete e\n  ecomplete (inr e)    = ecomplete e\n  ecomplete (case e x e1 y e2)  = ecomplete e \u00d7 ecomplete e1 \u00d7 ecomplete e2\n  ecomplete [ e1 , e2 ] = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (fst e) = ecomplete e\n  ecomplete (snd e) = ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n\n  -- indeterminate\n  data _indet : \u00eb \u2192 Set where\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where -- todo not a context\n\n  -- final\n  data _final : \u00eb \u2192 Set where\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"464a985aa70ecceb7a7917d62fc7cb3e05dd59c2","subject":"Update Desc model. Follow ICFP paper. With type-in-type, but termination-checker-friendly.","message":"Update Desc model. Follow ICFP paper. With type-in-type, but termination-checker-friendly.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms )) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\nrecord One : Set where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\n-- Inductive types are implemented as a Universe. Hence, in this\n-- section, we implement their code.\n\n-- We can read this code as follow (see Conor's \"Ornamental\n-- algebras\"): a description in |Desc| is a program to read one node\n-- of the described tree.\n\ndata Desc : Set where\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  -- Read a field in |X|; continue, given its value\n\n  -- Often, |X| is an |EnumT|, hence allowing to choose a constructor\n  -- among a finite set of constructors\n\n  Ind : (H : Set) -> Desc -> Desc\n  -- Read a field in H; read a recursive subnode given the field,\n  -- continue regarless of the subnode\n\n  -- Often, |H| is |1|, hence |Ind| simplifies to |Inf : Desc -> Desc|,\n  -- meaning: read a recursive subnode and continue regardless\n\n  Done : Desc\n  -- Stop reading\n\n\n--********************************************\n-- Desc decoder\n--********************************************\n\n-- Provided the type of the recursive subnodes |R|, we decode a\n-- description as a record describing the node.\n\n[|_|]_ : Desc -> Set -> Set\n[| Arg A D |] R = Sigma A (\\ a -> [| D a |] R)\n[| Ind H D |] R = (H -> R) * [| D |] R\n[| Done |] R = One\n\n\n--********************************************\n-- Functions on codes\n--********************************************\n\n-- Saying that a \"predicate\" |p| holds everywhere in |v| amounts to\n-- write something of the following type:\n\nEverywhere : (d : Desc) (D : Set) (bp : D -> Set) (V : [| d |] D) -> Set\nEverywhere (Arg A f) d p v =  Everywhere (f (fst v)) d p (snd v)\n-- It must hold for this constructor\n\nEverywhere (Ind H x) d p v =  ((y : H) -> p (fst v y)) * Everywhere x d p (snd v)\n-- It must hold for the subtrees \n\nEverywhere Done _ _ _ =  _\n-- It trivially holds at endpoints\n\n-- Then, we can build terms that inhabits this type. That is, a\n-- function that takes a \"predicate\" |bp| and makes it hold everywhere\n-- in the data-structure. It is the equivalent of a \"map\", but in a\n-- dependently-typed setting.\n\neverywhere : (d : Desc) (D : Set) (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : [| d |] D) ->\n           Everywhere d D bp v\neverywhere (Arg a f) d bp p v =  everywhere (f (fst v)) d bp p (snd v)\n-- It holds everywhere on this constructor\n\neverywhere (Ind H x) d bp p v = (\\y -> p (fst v y)) ,  everywhere x d bp p (snd v)\n-- It holds here, and down in the recursive subtrees\n\neverywhere Done _ _ _ _ =  One\n-- Nothing needs to be done on endpoints\n\n\n-- Looking at the decoder, a natural thing to do is to define its\n-- fixpoint |Mu D|, hence instantiating |R| with |Mu D| itself.\n\ndata Mu (D : Desc) : Set where\n  Con : [| D |] (Mu D) -> Mu D\n\n-- Using the \"map\" defined by |everywhere|, we can implement a \"fold\"\n-- over the |Mu| fixpoint:\n\nfoldDesc : (D : Desc) (bp : Mu D -> Set) ->\n         ((x : [| D |] (Mu D)) -> Everywhere D (Mu D) bp x -> bp (Con x)) ->\n         (v : Mu D) ->\n         bp v\nfoldDesc D bp p (Con v) =  p v (everywhere D (Mu D) bp (\\x -> foldDesc D bp p x) v) \n\n\n--********************************************\n-- Nat\n--********************************************\n\ndata NatConst : Set where\n  ZE : NatConst\n  SU : NatConst\n\nnatc : NatConst -> Desc\nnatc ZE = Done\nnatc SU = Ind One Done\n\nnatd : Desc\nnatd = Arg NatConst natc\n\nnat : Set\nnat = Mu natd\n\nzero : nat\nzero = Con ( ZE ,  _ )\n\nsuc : nat -> nat\nsuc n = Con ( SU , ( (\\_ -> n) ,  _ ) )\n\ntwo : nat\ntwo = suc (suc zero)\n\nfour : nat\nfour = suc (suc (suc (suc zero)))\n\nsum : nat -> \n      ((x : Sigma NatConst (\\ a -> [| natc a |] Mu (Arg NatConst natc))) ->\n      Everywhere (natc (fst x)) (Mu (Arg NatConst natc)) (\\ _ -> Mu (Arg NatConst natc)) (snd x) ->\n      Mu (Arg NatConst natc))\nsum n2 (ZE , _) p =  n2\nsum n2 (SU , f) p =  suc ( fst p _) \n\n\nplus : nat -> nat -> nat\nplus n1 n2 = foldDesc natd (\\_ -> nat) (sum n2) n1 \n\nx : nat\nx = plus two two\n\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"08b4ad7acb56f1425bfb337f9989dcc1ae4d2a5b","subject":"agda: start boilerplate for derive-term-correct (if ...) (#25)","message":"agda: start boilerplate for derive-term-correct (if ...) (#25)\n\nThe normalized goal is rather huge, suggesting this proof will be\nrather big, like the derivation of if. Separate lemmas will probably\nbe needed.\n\nOld-commit-hash: 487d21a8c626e591fe5804201a1109b104e97426\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"67653281f7cca9492f6f7c442a4d08a8909fd342","subject":"progress progress #3","message":"progress progress #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | Inl (_ , _ , _ , refl , f)                             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _))       = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inr (_ , \u03c4 , refl , _ , _)))\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (_ , ._ , refl , gnd , _))) | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (_ , _ , refl , gnd , _)))  | Inr neq = S (_ , Step FHOuter (ITCastFail gnd GBase neq) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u2987\u2988})) | I x = S (_ , Step FHOuter ITCastID FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inl (_ , _ , _ , refl , _)                             = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _))       = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (_ , ._ , refl , GBase , _))) = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (_ , ._ , refl , GHole , _))) = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c4 = \u03c4})) | BV x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole1) | BV x\u2081 | Inl g = BV (BVHoleCast g x\u2081)\n  progress (TACast wt (TCHole1 {b})) | BV x\u2081 | Inr x = abort (x GBase)\n  progress (TACast wt (TCHole1 {\u2987\u2988})) | BV x\u2081 | Inr x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2081 | Inr x\n    with (htype-dec  (\u03c41 ==> \u03c42) (\u2987\u2988 ==> \u2987\u2988))\n  progress (TACast wt (TCHole1 {.\u2987\u2988 ==> .\u2987\u2988})) | BV x\u2082 | Inr x\u2081 | Inl refl = BV (BVHoleCast GHole x\u2082)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2082 | Inr x\u2081 | Inr x = S (_ , Step FHOuter (ITGround (MGArr x)) FHOuter)\n\n\n  -- this is the case i was working on on friday; this part seems ok if maybe redundant\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with ground-decidable \u03c4 | canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inl x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter )\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x    = S (_ , Step FHOuter (ITCastFail gnd x\u2082 x) FHOuter )\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inr x  | d' , \u03c4' , refl , gnd , wt' -- this case goes off the rails here\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = abort (x\u2082 gnd)\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x = {!!} -- S (_ , Step {!!} (ITCastFail gnd {!!} x) {!!})\n\n  -- this is the beginning of the next case\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | BV x\n    with htype-dec (\u03c41 ==> \u03c42) (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inr x = BV (BVArrCast x x\u2081)\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S(_ , Step (FHFailedCast (FHCast x)) a (FHFailedCast (FHCast q)))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxed x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | Inl (_ , _ , _ , refl , f)                             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _))       = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inr (_ , \u03c4 , refl , _ , _)))\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (_ , ._ , refl , gnd , _))) | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (_ , _ , refl , gnd , _)))  | Inr neq = S (_ , Step FHOuter (ITCastFail gnd GBase neq) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u2987\u2988})) | I x = S (_ , Step FHOuter ITCastID FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inl (_ , _ , _ , refl , _)                             = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _))       = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (_ , ._ , refl , GBase , _))) = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (_ , ._ , refl , GHole , _))) = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c4 = \u03c4})) | BV x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole1) | BV x\u2081 | Inl g = BV (BVHoleCast g x\u2081)\n  progress (TACast wt (TCHole1 {b})) | BV x\u2081 | Inr x = abort (x GBase)\n  progress (TACast wt (TCHole1 {\u2987\u2988})) | BV x\u2081 | Inr x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2081 | Inr x\n    with (htype-dec  (\u03c41 ==> \u03c42) (\u2987\u2988 ==> \u2987\u2988))\n  progress (TACast wt (TCHole1 {.\u2987\u2988 ==> .\u2987\u2988})) | BV x\u2082 | Inr x\u2081 | Inl refl = BV (BVHoleCast GHole x\u2082)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2082 | Inr x\u2081 | Inr x = S (_ , Step FHOuter (ITGround (MGArr x)) FHOuter)\n\n\n  -- this is the case i was working on on friday; this part seems ok if maybe redundant\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with ground-decidable \u03c4 | canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inl x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter )\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x    = S (_ , Step FHOuter (ITCastFail gnd x\u2082 x) FHOuter )\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inr x  | d' , \u03c4' , refl , gnd , wt' -- this case goes off the rails here\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = abort (x\u2082 gnd)\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x = {!ITCastFail ? ? x!}\n\n  -- this is the beginning of the next case -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | BV x = {!!}\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S(_ , Step (FHFailedCast (FHCast x)) a (FHFailedCast (FHCast q)))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxed x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"543a4a08de01259162b6b65a657416b56f011a40","subject":"flipbased: \u25b9\u21ba as flip map\u21ba","message":"flipbased: \u25b9\u21ba as flip map\u21ba\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"open import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule flipbased\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (weaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n where\n\nCoins = \u2115\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = return\u21ba\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ x f = join\u21ba (map\u21ba f x)\n\n_=<<_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       (A \u2192 \u21ba n\u2081 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_=<<_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2081 n\u2082 = flip _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\n_>=>_ : \u2200 {n\u2081 n\u2082 a b c} {A : Set a} {B : Set b} {C : Set c}\n        \u2192 (A \u2192 \u21ba n\u2081 B) \u2192 (B \u2192 \u21ba n\u2082 C) \u2192 A \u2192 \u21ba (n\u2081 + n\u2082) C\n(f >=> g) x = f x >>= g\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken {m} x = return\u21ba {m} 0 >> x\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 x = x >>= return\u21ba\n\npure\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure\u21ba = return\u21ba\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = toss >>= return\u21ba\n\n_\u25b9\u21ba_ : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 \u21ba n A \u2192 (A \u2192 B) \u2192 \u21ba n B\nx \u25b9\u21ba f = map\u21ba f x\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\u21ba\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map\u21ba f x\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 \u27ea f \u00b7 mx \u27eb \n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map\u21ba f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\nzip\u21ba : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} \u2192 \u21ba c\u2080 A \u2192 \u21ba c\u2081 B \u2192 \u21ba (c\u2080 + c\u2081) (A \u00d7 B)\nzip\u21ba x y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\n_\u27e8,\u27e9_ = zip\u21ba\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\nT\u21ba : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\nT\u21ba p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicate\u21ba x \u27eb\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFun : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\ncostRndFun-lem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\ncostRndFun-lem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\ncostRndFun-lem (suc m) n rewrite costRndFun-lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 \u21ba (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 return\u1d30 =<< x \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map\u21ba swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"open import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule flipbased\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (weaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n where\n\nCoins = \u2115\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = return\u21ba\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ x f = join\u21ba (map\u21ba f x)\n\n_=<<_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       (A \u2192 \u21ba n\u2081 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_=<<_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2081 n\u2082 = flip _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\n_>=>_ : \u2200 {n\u2081 n\u2082 a b c} {A : Set a} {B : Set b} {C : Set c}\n        \u2192 (A \u2192 \u21ba n\u2081 B) \u2192 (B \u2192 \u21ba n\u2082 C) \u2192 A \u2192 \u21ba (n\u2081 + n\u2082) C\n(f >=> g) x = f x >>= g\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken {m} x = return\u21ba {m} 0 >> x\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 x = x >>= return\u21ba\n\npure\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure\u21ba = return\u21ba\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = toss >>= return\u21ba\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\u21ba\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map\u21ba f x\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 \u27ea f \u00b7 mx \u27eb \n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map\u21ba f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\nzip\u21ba : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} \u2192 \u21ba c\u2080 A \u2192 \u21ba c\u2081 B \u2192 \u21ba (c\u2080 + c\u2081) (A \u00d7 B)\nzip\u21ba x y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\n_\u27e8,\u27e9_ = zip\u21ba\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\nT\u21ba : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\nT\u21ba p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicate\u21ba x \u27eb\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFun : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\ncostRndFun-lem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\ncostRndFun-lem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\ncostRndFun-lem (suc m) n rewrite costRndFun-lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 \u21ba (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 return\u1d30 =<< x \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map\u21ba swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7f1a4e72fc0081b46fe06904e4627bb80b3240ac","subject":"Add Coq's \"simp\" tactic.","message":"Add Coq's \"simp\" tactic.\n","repos":"spire\/spire","old_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin : Tactic\nrm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin a (fin i) = proj\u2082\n  (rm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","old_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin : Tactic\nrm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin a (fin i) = proj\u2082\n  (rm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"09aec6bcdb623c07e28d0f2d1ebe65cda4a07269","subject":"Changes due to Agda issue 953.","message":"Changes due to Agda issue 953.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/Common\/FOL\/Relation\/Binary\/PropositionalEquality\/NoPatternMatchingOnRefl.agda","new_file":"notes\/FOT\/Common\/FOL\/Relation\/Binary\/PropositionalEquality\/NoPatternMatchingOnRefl.agda","new_contents":"------------------------------------------------------------------------------\n-- Propositional equality without using pattern matching on refl\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.Common.FOL.Relation.Binary.PropositionalEquality.NoPatternMatchingOnRefl where\n\nopen import Common.FOL.FOL using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2261_\n\n------------------------------------------------------------------------------\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Elimination rule.\nsubst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\nsubst A refl Ax = Ax\n\n-- Identity properties\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym {x} h = subst (\u03bb y' \u2192 y' \u2261 x) h refl\n\ntrans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans {x} h\u2081 h\u2082 = subst (_\u2261_ x) h\u2082 h\u2081\n\ntrans\u2082 : \u2200 {w x y z} \u2192 w \u2261 x \u2192 x \u2261 y \u2192 y \u2261 z \u2192 w \u2261 z\ntrans\u2082 h\u2081 h\u2082 h\u2083 = trans (trans h\u2081 h\u2082) h\u2083\n\nsubst\u2082 : (A : D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192\n         x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n         A x\u2081 x\u2082 \u2192\n         A y\u2081 y\u2082\nsubst\u2082 A {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 h\u2083 =\n  subst (\u03bb y\u2081' \u2192 A y\u2081' y\u2082) h\u2081 (subst (A x\u2081) h\u2082 h\u2083)\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f {x} h = subst (\u03bb x' \u2192 f x \u2261 f x') h refl\n\ncong\u2082 : (f : D \u2192 D \u2192 D) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n        f x\u2081 x\u2082 \u2261 f y\u2081 y\u2082\ncong\u2082 f {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 =\n  subst (\u03bb x\u2081' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081' y\u2082)\n        h\u2081\n        (subst (\u03bb x\u2082' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081 x\u2082') h\u2082 refl)\n","old_contents":"------------------------------------------------------------------------------\n-- Propositional equality without using pattern matching on refl\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule\n  FOT.Common.FOL.Relation.Binary.PropositionalEquality.NoPatternMatchingOnRefl\nwhere\n\nopen import Common.FOL.FOL using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2261_\n\n------------------------------------------------------------------------------\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Elimination rule.\nsubst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\nsubst A refl Ax = Ax\n\n-- Identity properties\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym {x} h = subst (\u03bb y' \u2192 y' \u2261 x) h refl\n\ntrans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans {x} h\u2081 h\u2082 = subst (_\u2261_ x) h\u2082 h\u2081\n\ntrans\u2082 : \u2200 {w x y z} \u2192 w \u2261 x \u2192 x \u2261 y \u2192 y \u2261 z \u2192 w \u2261 z\ntrans\u2082 h\u2081 h\u2082 h\u2083 = trans (trans h\u2081 h\u2082) h\u2083\n\nsubst\u2082 : (A : D \u2192 D \u2192 Set) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192\n         x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n         A x\u2081 x\u2082 \u2192\n         A y\u2081 y\u2082\nsubst\u2082 A {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 h\u2083 =\n  subst (\u03bb y\u2081' \u2192 A y\u2081' y\u2082) h\u2081 (subst (A x\u2081) h\u2082 h\u2083)\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f {x} h = subst (\u03bb x' \u2192 f x \u2261 f x') h refl\n\ncong\u2082 : (f : D \u2192 D \u2192 D) \u2192 \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192\n        f x\u2081 x\u2082 \u2261 f y\u2081 y\u2082\ncong\u2082 f {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 =\n  subst (\u03bb x\u2081' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081' y\u2082)\n        h\u2081\n        (subst (\u03bb x\u2082' \u2192 f x\u2081 x\u2082 \u2261 f x\u2081 x\u2082') h\u2082 refl)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0f86619e6453a8b94f6c1ee2eca3f697e673f834","subject":"Proved that parametricity implies causality for signals.","message":"Proved that parametricity implies causality for signals.\n","repos":"agda\/agda-frp-js,agda\/agda-frp-js","old_file":"src\/agda\/FRP\/JS\/Model.agda","new_file":"src\/agda\/FRP\/JS\/Model.agda","new_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.True using ( True )\nopen import FRP.JS.Nat using ( \u2115 ; zero ; suc )\n\nmodule FRP.JS.Model where\n\n-- Preliminaries\n\ninfixr 4 _+_\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n\n-- Relations on Set\n\n_\u220b_\u2194_ : \u2200 \u03b1 \u2192 Set \u03b1 \u2192 Set \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n\n-- RSets and relations on RSet\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\nRSet\u2080 = RSet o\nRSet\u2081 = RSet (\u2191 o)\n\n_\u220b_\u21d4_ : \u2200 \u03b1 \u2192 RSet \u03b1 \u2192 RSet \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u21d4 B = \u2200 t \u2192 (\u03b1 \u220b A t \u2194 B t)\n\n--- Level sequences\n\ndata Levels : Set where\n  \u03b5 : Levels\n  _,_ : \u2200 (\u03b1s : Levels) (\u03b1 : Level) \u2192 Levels\n\nmax : Levels \u2192 Level\nmax \u03b5 = o\nmax (\u03b1s , \u03b1) = max \u03b1s \u2294 \u03b1\n\n-- Sequences of Sets\n\nSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nSets \u03b5        = \u22a4\nSets (\u03b1s , \u03b1) = Sets \u03b1s \u00d7 Set \u03b1\n\n\u27e8_\u220b_\u27e9 : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Set (max \u03b1s)\n\u27e8 \u03b5 \u220b tt \u27e9 = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (As , A) \u27e9 = \u27e8 \u03b1s \u220b As \u27e9 \u00d7 A\n\n_\u220b_\u2194*_ : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Sets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u2194* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u2194* (Bs , B) = (\u03b1s \u220b As \u2194* Bs) \u00d7 (\u03b1 \u220b A \u2194 B)\n\n\u27e8_\u220b_\u27e9\u00b2 : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u2194* Bs) \u2192 (max \u03b1s \u220b \u27e8 \u03b1s \u220b As \u27e9 \u2194 \u27e8 \u03b1s \u220b Bs \u27e9)\n\u27e8 \u03b5        \u220b tt       \u27e9\u00b2 tt       tt       = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (\u211cs , \u211c) \u27e9\u00b2 (as , a) (bs , b) = (\u27e8 \u03b1s \u220b \u211cs \u27e9\u00b2 as bs) \u00d7 (\u211c a b)\n\n-- Sequences of RSets\n\nRSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nRSets \u03b5        = \u22a4\nRSets (\u03b1s , \u03b1) = RSets \u03b1s \u00d7 RSet \u03b1\n\n_\u220b_\u21d4*_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 RSets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u21d4* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u21d4* (Bs , B) = (\u03b1s \u220b As \u21d4* Bs) \u00d7 (\u03b1 \u220b A \u21d4 B)\n\n-- Concatenation of sequences\n\n_+_ : Levels \u2192 Levels \u2192 Levels\n\u03b1s + \u03b5        = \u03b1s\n\u03b1s + (\u03b2s , \u03b2) = (\u03b1s + \u03b2s) , \u03b2\n\n_\u220b_++_\u220b_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 \u2200 \u03b2s \u2192 RSets \u03b2s \u2192 RSets (\u03b1s + \u03b2s)\n\u03b1s \u220b As ++ \u03b5        \u220b tt       = As\n\u03b1s \u220b As ++ (\u03b2s , \u03b2) \u220b (Bs , B) = ((\u03b1s \u220b As ++ \u03b2s \u220b Bs) , B)\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u21d4* Bs) \u2192 \u2200 \u03b2s {Cs Ds} \u2192 (\u03b2s \u220b Cs \u21d4* Ds) \u2192 \n  ((\u03b1s + \u03b2s) \u220b (\u03b1s \u220b As ++ \u03b2s \u220b Cs) \u21d4* (\u03b1s \u220b Bs ++ \u03b2s \u220b Ds))\n\u03b1s \u220b \u211cs ++\u00b2 \u03b5        \u220b tt       = \u211cs\n\u03b1s \u220b \u211cs ++\u00b2 (\u03b2s , \u03b2) \u220b (\u2111s , \u2111) = ((\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) , \u2111)\n\n-- Intervals\n\n_[_,_] : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 Time \u2192 Time \u2192 Set \u03b1\nA [ s , u ] = \u2200 t \u2192 True (s \u2264 t) \u2192 True (t \u2264 u) \u2192 A t\n\n_[_,_]\u00b2 : \u2200 {\u03b1 A B} \u2192 (\u03b1 \u220b A \u21d4 B) \u2192 \u2200 s u \u2192 (\u03b1 \u220b A [ s , u ] \u2194 B [ s , u ])\n(\u211c [ s , u ]\u00b2) \u03c3 \u03c4 = \u2200 t s\u2264t t\u2264u \u2192 \u211c t (\u03c3 t s\u2264t t\u2264u) (\u03c4 t s\u2264t t\u2264u)\n\n-- Type variables\n\ndata TVar : Levels \u2192 Set\u2081 where\n  zero : \u2200 {\u03b1s \u03b1} \u2192 TVar (\u03b1s , \u03b1)\n  suc : \u2200 {\u03b1s \u03b1} \u2192 (\u03c4 : TVar \u03b1s) \u2192 TVar (\u03b1s , \u03b1)\n\n\u03c4level : \u2200 {\u03b1s} \u2192 TVar \u03b1s \u2192 Level\n\u03c4level (zero {\u03b1 = \u03b1}) = \u03b1\n\u03c4level (suc \u03c4) = \u03c4level \u03c4\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03b1s} (\u03c4 : TVar \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (\u03c4level \u03c4)\n\u03c4\u27e6 zero \u27e7  (As , A) = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (As , A) = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u03c4 : TVar \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (\u03c4level \u03c4 \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u21d4 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211cs , \u211c) t a b = \u211c t a b\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211cs , \u211c) t a b = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t a b\n\n-- Types\n\ndata Typ (\u03b1s : Levels) : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 Typ \u03b1s\n  _\u2227_ _\u21d2_ _\u22b5_ : (T : Typ \u03b1s) \u2192 (U : Typ \u03b1s) \u2192 Typ \u03b1s\n  tvar : (\u03c4 : TVar \u03b1s) \u2192 Typ \u03b1s\n  univ : \u2200 \u03b1 \u2192 (T : Typ (\u03b1s , \u03b1)) \u2192 Typ \u03b1s\n\ntlevel : \u2200 {\u03b1s} \u2192 Typ \u03b1s \u2192 Level\ntlevel \u27e8 A \u27e9      = o\ntlevel (T \u2227 U)    = tlevel T \u2294 tlevel U\ntlevel (T \u21d2 U)    = tlevel T \u2294 tlevel U\ntlevel (T \u22b5 U)    = tlevel T \u2294 tlevel U\ntlevel (tvar \u03c4)   = \u03c4level \u03c4\ntlevel (univ \u03b1 T) = \u2191 \u03b1 \u2294 tlevel T\n\nT\u27e6_\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (tlevel T)\nT\u27e6 \u27e8 A \u27e9 \u27e7    As t = A\nT\u27e6 T \u2227 U \u27e7    As t = T\u27e6 T \u27e7 As t \u00d7 T\u27e6 U \u27e7 As t\nT\u27e6 T \u21d2 U \u27e7    As t = T\u27e6 T \u27e7 As t \u2192 T\u27e6 U \u27e7 As t\nT\u27e6 T \u22b5 U \u27e7    As t = \u2200 u \u2192 True (t \u2264 u) \u2192 T\u27e6 T \u27e7 As [ t , u ] \u2192 T\u27e6 U \u27e7 As u\nT\u27e6 tvar \u03c4 \u27e7   As t = \u03c4\u27e6 \u03c4 \u27e7 As t\nT\u27e6 univ \u03b1 T \u27e7 As t = \u2200 (A : RSet \u03b1) \u2192 T\u27e6 T \u27e7 (As , A) t\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (T : Typ \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (tlevel T \u220b T\u27e6 T \u27e7 As \u21d4 T\u27e6 T \u27e7 Bs)\nT\u27e6 \u27e8 A \u27e9 \u27e7\u00b2    \u211cs t a       b       = a \u2261 b\nT\u27e6 T \u2227 U \u27e7\u00b2    \u211cs t (a , b) (c , d) = T\u27e6 T \u27e7\u00b2 \u211cs t a c \u00d7 T\u27e6 U \u27e7\u00b2 \u211cs t b d\nT\u27e6 T \u21d2 U \u27e7\u00b2    \u211cs t f       g       = \u2200 {a b} \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t a b \u2192 T\u27e6 U \u27e7\u00b2 \u211cs t (f a) (g b)\nT\u27e6 T \u22b5 U \u27e7\u00b2    \u211cs t f       g       = \u2200 u t\u2264u {\u03c3 \u03c4} \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs [ t , u ]\u00b2) \u03c3 \u03c4 \u2192 T\u27e6 U \u27e7\u00b2 \u211cs u (f u t\u2264u \u03c3) (g u t\u2264u \u03c4)\nT\u27e6 tvar \u03c4 \u27e7\u00b2   \u211cs t v       w       = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t v w\nT\u27e6 univ \u03b1 T \u27e7\u00b2 \u211cs t f       g       = \u2200 {A B} (\u211c : \u03b1 \u220b A \u21d4 B) \u2192 T\u27e6 T \u27e7\u00b2 (\u211cs , \u211c) t (f A) (g B)\n\n-- Contexts\n\ndata Ctxt (\u03b1s : Levels) : Set\u2081 where\n  \u03b5 : Ctxt \u03b1s\n  _,_at_ : (\u0393 : Ctxt \u03b1s) (T : Typ \u03b1s) (t : Time) \u2192 Ctxt \u03b1s\n\nclevels : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Levels\nclevels \u03b5            = \u03b5\nclevels (\u0393 , T at t) = (clevels \u0393 , tlevel T)\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03b1s} (\u0393 : Ctxt \u03b1s) \u2192 RSets \u03b1s \u2192 Sets (clevels \u0393)\n\u0393\u27e6 \u03b5 \u27e7          As = tt\n\u0393\u27e6 \u0393 , T at t \u27e7 As = (\u0393\u27e6 \u0393 \u27e7 As , T\u27e6 T \u27e7 As t)\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u0393 : Ctxt \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u2194* \u0393\u27e6 \u0393 \u27e7 Bs)\n\u0393\u27e6 \u03b5 \u27e7\u00b2          \u211cs = tt\n\u0393\u27e6 \u0393 , T at t \u27e7\u00b2 \u211cs = (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs , T\u27e6 T \u27e7\u00b2 \u211cs t)\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 TVar (\u03b1s + \u03b2s) \u2192 TVar ((\u03b1s , \u03b1) + \u03b2s)\n\u03c4weaken \u03b1 \u03b5        \u03c4       = suc \u03c4\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) zero    = zero\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) (suc \u03c4) = suc (\u03c4weaken \u03b1 \u03b2s \u03c4)\n\n\u27e6\u03c4weaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n-- Weakening of types\n\ntweaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 Typ (\u03b1s + \u03b2s) \u2192 Typ ((\u03b1s , \u03b1) + \u03b2s)\ntweaken \u03b1 \u03b2s \u27e8 A \u27e9      = \u27e8 A \u27e9\ntweaken \u03b1 \u03b2s (T \u2227 U)    = tweaken \u03b1 \u03b2s T \u2227 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (T \u21d2 U)    = tweaken \u03b1 \u03b2s T \u21d2 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (T \u22b5 U)    = tweaken \u03b1 \u03b2s T \u22b5 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (tvar \u03c4)   = tvar (\u03c4weaken \u03b1 \u03b2s \u03c4)\ntweaken \u03b1 \u03b2s (univ \u03b2 T) = univ \u03b2 (tweaken \u03b1 (\u03b2s , \u03b2) T)\n\nmutual\n\n  \u27e6tweaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u22b5 U)    As A Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (tvar \u03c4)   As A Bs t a       = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\n  \u27e6tweaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192\n    T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u22b5 U)    As A Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (tvar \u03c4)   As A Bs t a       = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u22b5 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (tvar \u03c4)   \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u22b5 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (tvar \u03c4)   \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Weakening of contexts\n\ncweaken : \u2200 {\u03b1s} \u03b1 \u2192 Ctxt \u03b1s \u2192 Ctxt (\u03b1s , \u03b1)\ncweaken \u03b1 \u03b5            = \u03b5\ncweaken \u03b1 (\u0393 , T at t) = (cweaken \u03b1 \u0393 , tweaken \u03b1 \u03b5 T at t)\n  \n\u27e6cweaken\u27e7 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) As A \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7 (As , A) \u27e9\n\u27e6cweaken\u27e7 \u03b1 \u03b5            As A tt       = tt\n\u27e6cweaken\u27e7 \u03b1 (\u0393 , T at t) As A (as , a) = (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as , \u27e6tweaken\u27e7 \u03b1 \u03b5 T As A tt t a)\n\n\u27e6cweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) {As Bs A B} \u211cs \u211c {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192\n  \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7\u00b2 (\u211cs , \u211c) \u27e9\u00b2 (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as) (\u27e6cweaken\u27e7 \u03b1 \u0393 Bs B bs)\n\u27e6cweaken\u27e7\u00b2 \u03b1 \u03b5            \u211cs \u211c tt            = tt\n\u27e6cweaken\u27e7\u00b2 \u03b1 (\u0393 , T at t) \u211cs \u211c (as\u211cbs , a\u211cb) = (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b5 T \u211cs \u211c tt t a\u211cb)\n\n-- Substitution into type variables\n\n\u03c4subst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 TVar ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\n\u03c4subst T \u03b5        zero    = T\n\u03c4subst T \u03b5        (suc \u03c4) = tvar \u03c4\n\u03c4subst T (\u03b2s , \u03b2) zero    = tvar zero\n\u03c4subst T (\u03b2s , \u03b2) (suc \u03c4) = tweaken \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4)\n\n\u27e6\u03c4subst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6tweaken\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t \n    (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u207b\u00b9\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t \n    (\u27e6tweaken\u207b\u00b9\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t a)\n\n\u27e6\u03c4subst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6tweaken\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t \n    (\u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t \n    (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t a\u211cb)\n\n-- Substitution into types\n\ntsubst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 Typ ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\ntsubst T \u03b2s \u27e8 A \u27e9      = \u27e8 A \u27e9\ntsubst T \u03b2s (U \u2227 V)    = tsubst T \u03b2s U \u2227 tsubst T \u03b2s V\ntsubst T \u03b2s (U \u21d2 V)    = tsubst T \u03b2s U \u21d2 tsubst T \u03b2s V\ntsubst T \u03b2s (U \u22b5 V)    = tsubst T \u03b2s U \u22b5 tsubst T \u03b2s V\ntsubst T \u03b2s (tvar \u03c4)   = \u03c4subst T \u03b2s \u03c4\ntsubst T \u03b2s (univ \u03b2 U) = univ \u03b2 (tsubst T (\u03b2s , \u03b2) U)\n\nmutual\n\n  \u27e6tsubst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u27e7 T \u03b2s (U \u22b5 V)    As Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tsubst\u27e7 T \u03b2s V As Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u27e7 T \u03b2s (tvar \u03c4)   As Bs t a       = \u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\n  \u27e6tsubst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u22b5 V)    As Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u27e7 T \u03b2s U As Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (tvar \u03c4)   As Bs t a       = \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tsubst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u22b5 V)    \u211cs \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (tvar \u03c4)   \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u22b5 V)    \u211cs \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (tvar \u03c4)   \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Variables\n\ndata Var {\u03b1s} (T : Typ \u03b1s) (t : Time) : (\u0393 : Ctxt \u03b1s) \u2192 Set\u2081 where\n  zero : \u2200 {\u0393 : Ctxt \u03b1s} \u2192 Var T t (\u0393 , T at t)\n  suc : \u2200 {\u0393 : Ctxt \u03b1s} {U : Typ \u03b1s} {u} \u2192 Var T t \u0393 \u2192 Var T t (\u0393 , U at u)\n\nv\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Var T t \u0393 \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\nv\u27e6 zero \u27e7  As (as , a) = a\nv\u27e6 suc v \u27e7 As (as , a) = v\u27e6 v \u27e7 As as\n\nv\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (\u03c4 : Var T t \u0393) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (v\u27e6 \u03c4 \u27e7 As as) (v\u27e6 \u03c4 \u27e7 Bs bs)\nv\u27e6 zero \u27e7\u00b2  \u211cs (as\u211cbs , a\u211cb) = a\u211cb\nv\u27e6 suc v \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb) = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Expressions\n\ndata Exp : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Typ \u03b1s \u2192 RSet\u2081 where\n  const : \u2200 {\u03b1s \u0393 A t} \u2192 (a : A) \u2192 Exp {\u03b1s} \u0393 \u27e8 A \u27e9 t\n  pair : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 T t) \u2192 (f : Exp \u0393 U t) \u2192 Exp {\u03b1s} \u0393 (T \u2227 U) t\n  fst : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 T t\n  snd : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 U t\n  abs : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp (\u0393 , T at t) U t) \u2192 Exp {\u03b1s} \u0393 (T \u21d2 U) t\n  app : \u2200 {\u03b1s \u0393 T U t} \u2192 (f : Exp \u0393 (T \u21d2 U) t) \u2192 (e : Exp \u0393 T t) \u2192 Exp {\u03b1s} \u0393 U t\n  var : \u2200 {\u03b1s \u0393 T t} \u2192 (v : Var T t \u0393) \u2192 Exp {\u03b1s} \u0393 T t\n  tabs : \u2200 {\u03b1s \u0393} \u03b1 {T t} \u2192 (e : Exp (cweaken \u03b1 \u0393) T t) \u2192 Exp {\u03b1s} \u0393 (univ \u03b1 T) t\n  tapp : \u2200 {\u03b1s \u0393 t} (T : Typ \u03b1s) {U} \u2192 (e : Exp \u0393 (univ (tlevel T) U) t) \u2192 Exp {\u03b1s} \u0393 (tsubst T \u03b5 U) t\n\ne\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Exp \u0393 T t \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\ne\u27e6 const a \u27e7                  As as = a\ne\u27e6 pair e f \u27e7                 As as = (e\u27e6 e \u27e7 As as , e\u27e6 f \u27e7 As as)\ne\u27e6 fst e \u27e7                    As as = proj\u2081 (e\u27e6 e \u27e7 As as)\ne\u27e6 snd e \u27e7                    As as = proj\u2082 (e\u27e6 e \u27e7 As as)\ne\u27e6 abs e \u27e7                    As as = \u03bb a \u2192 e\u27e6 e \u27e7 As (as , a)\ne\u27e6 app f e \u27e7                  As as = e\u27e6 f \u27e7 As as (e\u27e6 e \u27e7 As as)\ne\u27e6 var v \u27e7                    As as = v\u27e6 v \u27e7 As as\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7         As as = \u03bb A \u2192 e\u27e6 e \u27e7 (As , A) (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7 As as = \u27e6tsubst\u27e7 T \u03b5 U As tt t (e\u27e6 e \u27e7 As as (T\u27e6 T \u27e7 As))\n\ne\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (e : Exp \u0393 T t) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (e\u27e6 e \u27e7 As as) (e\u27e6 e \u27e7 Bs bs)\ne\u27e6 const a \u27e7\u00b2                  \u211cs as\u211cbs = refl\ne\u27e6 pair e f \u27e7\u00b2                 \u211cs as\u211cbs = (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs , e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 fst e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2081 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 snd e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2082 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 abs e \u27e7\u00b2                    \u211cs as\u211cbs = \u03bb a\u211cb \u2192 e\u27e6 e \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb)\ne\u27e6 app f e \u27e7\u00b2                  \u211cs as\u211cbs = e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 var v \u27e7\u00b2                    \u211cs as\u211cbs = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7\u00b2         \u211cs as\u211cbs = \u03bb \u211c \u2192 e\u27e6 e \u27e7\u00b2 (\u211cs , \u211c) (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7\u00b2 \u211cs as\u211cbs = \u27e6tsubst\u27e7\u00b2 T \u03b5 U \u211cs tt t (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 T \u27e7\u00b2 \u211cs))\n\n-- Surface level types\n\ndata STyp : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 STyp\n  _\u2227_ _\u21d2_ : (T : STyp) \u2192 (U : STyp) \u2192 STyp\n  \u25a1 : (T : STyp) \u2192 STyp\n\n-- Translation of surface level types into types\n\n\u27ea_\u27eb : STyp \u2192 Typ (\u03b5 , o)\n\u27ea \u27e8 A \u27e9 \u27eb = \u27e8 A \u27e9\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2227 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u21d2 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar zero \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : STyp \u2192 RSet\u2080\nT\u27ea \u27e8 A \u27e9 \u27eb t = A\nT\u27ea T \u2227 U \u27eb t = T\u27ea T \u27eb t \u00d7 T\u27ea U \u27eb t\nT\u27ea T \u21d2 U \u27eb t = T\u27ea T \u27eb t \u2192 T\u27ea U \u27eb t\nT\u27ea \u25a1 T \u27eb   t = \u2200 u \u2192 True (t \u2264 u) \u2192 T\u27ea T \u27eb u\n\nWorld : RSet\u2080\nWorld t = \u22a4\n\nmutual\n\n  trans : \u2200 T {t} \u2192 T\u27ea T \u27eb t \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) t\n  trans \u27e8 A \u27e9   a       = a\n  trans (T \u2227 U) (a , b) = (trans T a , trans U b)\n  trans (T \u21d2 U) f       = \u03bb a \u2192 trans U (f (trans\u207b\u00b9 T a))\n  trans (\u25a1 T)   \u03c3       = \u03bb u t\u2264u \u03c4 \u2192 trans T (\u03c3 u t\u2264u)\n\n  trans\u207b\u00b9 : \u2200 T {t} \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) t \u2192 T\u27ea T \u27eb t\n  trans\u207b\u00b9 \u27e8 A \u27e9   a       = a\n  trans\u207b\u00b9 (T \u2227 U) (a , b) = (trans\u207b\u00b9 T a , trans\u207b\u00b9 U b)\n  trans\u207b\u00b9 (T \u21d2 U) f       = \u03bb a \u2192 trans\u207b\u00b9 U (f (trans T a))\n  trans\u207b\u00b9 (\u25a1 T)   \u03c3       = \u03bb u t\u2264u \u2192 trans\u207b\u00b9 T (\u03c3 u t\u2264u _)\n\n-- Causality\n\n_at_\u220b_\u2248[_\u2235_]_ : \u2200 T t \u2192 T\u27ea T \u27eb t \u2192 \u2200 u \u2192 True (t \u2264 u) \u2192 T\u27ea T \u27eb t \u2192 Set\n\u27e8 A \u27e9   at t \u220b a       \u2248[ u \u2235 t\u2264u ] b       = a \u2261 b\n(T \u2227 U) at t \u220b (a , b) \u2248[ u \u2235 t\u2264u ] (c , d) = (T at t \u220b a \u2248[ u \u2235 t\u2264u ] c) \u00d7 (U at t \u220b b \u2248[ u \u2235 t\u2264u ] d)\n(T \u21d2 U) at t \u220b f       \u2248[ u \u2235 t\u2264u ] g       = \u2200 {a b} \u2192 (T at t \u220b a \u2248[ u \u2235 t\u2264u ] b) \u2192 (U at t \u220b f a \u2248[ u \u2235 t\u2264u ] g b)\n(\u25a1 T)   at s \u220b \u03c3       \u2248[ u \u2235 s\u2264u ] \u03c4       = \u2200 t s\u2264t t\u2264u \u2192 (T at t \u220b \u03c3 t s\u2264t \u2248[ u \u2235 t\u2264u ] \u03c4 t s\u2264t)\n\nCausal : \u2200 T U t \u2192 T\u27ea T \u21d2 U \u27eb t \u2192 Set\nCausal T U t f = \u2200 u t\u2264u {a b} \u2192 (T at t \u220b a \u2248[ u \u2235 t\u2264u ] b) \u2192 (U at t \u220b f a \u2248[ u \u2235 t\u2264u ] f b)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (o \u220b World \u21d4 World)\n\u211c[ u ] t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248 : \u2200 T t u t\u2264u {a b} \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) t a b \u2192\n    (T at t \u220b trans\u207b\u00b9 T a \u2248[ u \u2235 t\u2264u ] trans\u207b\u00b9 T b)\n  \u211c-impl-\u2248 \u27e8 A \u27e9   t u t\u2264u a\u211cb         = a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) t u t\u2264u (a\u211cc , b\u211cd) = (\u211c-impl-\u2248 T t u t\u2264u a\u211cc , \u211c-impl-\u2248 U t u t\u2264u b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) t u t\u2264u f\u211cg         = \u03bb a\u2248b \u2192 \u211c-impl-\u2248 U t u t\u2264u (f\u211cg (\u2248-impl-\u211c T t u t\u2264u a\u2248b))\n  \u211c-impl-\u2248 (\u25a1 T)   s v s\u2264v \u03c3\u211c\u03c4         = \u03bb u s\u2264u u\u2264v \u2192 \u211c-impl-\u2248 T u v u\u2264v (\u03c3\u211c\u03c4 u s\u2264u (\u03bb t s\u2264t t\u2264u \u2192 \u2264-trans t u v t\u2264u u\u2264v))\n\n  \u2248-impl-\u211c : \u2200 T t u t\u2264u {a b} \u2192\n    (T at t \u220b a \u2248[ u \u2235 t\u2264u ] b) \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) t (trans T a) (trans T b)\n  \u2248-impl-\u211c \u27e8 A \u27e9   t u t\u2264u a\u2248b         = a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) t u t\u2264u (a\u2248c , b\u2248d) = (\u2248-impl-\u211c T t u t\u2264u a\u2248c , \u2248-impl-\u211c U t u t\u2264u b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) t u t\u2264u f\u2248g         = \u03bb a\u211cb \u2192 \u2248-impl-\u211c U t u t\u2264u (f\u2248g (\u211c-impl-\u2248 T t u t\u2264u a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   s v s\u2264v \u03c3\u2248\u03c4         = \u03bb u s\u2264u \u03c1 \u2192 \u2248-impl-\u211c T u v (\u03c1 u s\u2264u (\u2264-refl u)) (\u03c3\u2248\u03c4 u s\u2264u (\u03c1 u s\u2264u (\u2264-refl u)))\n\n-- Every expression is causal\n\ne\u27ea_at_\u220b_\u27eb : \u2200 T t \u2192 Exp \u03b5 \u27ea T \u27eb t \u2192 T\u27ea T \u27eb t\ne\u27ea T at t \u220b e \u27eb = trans\u207b\u00b9 T (e\u27e6 e \u27e7 (tt , World) tt)\n\ncausality : \u2200 T U t f \u2192 Causal T U t e\u27ea (T \u21d2 U) at t \u220b f \u27eb \ncausality T U t f u t\u2264u = \u211c-impl-\u2248 (T \u21d2 U) t u t\u2264u (e\u27e6 f \u27e7\u00b2 (tt , \u211c[ u ]) tt)\n","old_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.True using ( True )\nopen import FRP.JS.Nat using ( \u2115 ; zero ; suc )\n\nmodule FRP.JS.Model where\n\n-- Preliminaries\n\ninfixr 4 _+_\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\n-- Relations on Set\n\n_\u220b_\u2194_ : \u2200 \u03b1 \u2192 Set \u03b1 \u2192 Set \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n\n-- RSets and relations on RSet\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\nRSet\u2080 = RSet o\nRSet\u2081 = RSet (\u2191 o)\n\n_\u220b_\u21d4_ : \u2200 \u03b1 \u2192 RSet \u03b1 \u2192 RSet \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u21d4 B = \u2200 t \u2192 (\u03b1 \u220b A t \u2194 B t)\n\n--- Level sequences\n\ndata Levels : Set where\n  \u03b5 : Levels\n  _,_ : \u2200 (\u03b1s : Levels) (\u03b1 : Level) \u2192 Levels\n\nmax : Levels \u2192 Level\nmax \u03b5 = o\nmax (\u03b1s , \u03b1) = max \u03b1s \u2294 \u03b1\n\n-- Sequences of Sets\n\nSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nSets \u03b5        = \u22a4\nSets (\u03b1s , \u03b1) = Sets \u03b1s \u00d7 Set \u03b1\n\n\u27e8_\u220b_\u27e9 : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Set (max \u03b1s)\n\u27e8 \u03b5 \u220b tt \u27e9 = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (As , A) \u27e9 = \u27e8 \u03b1s \u220b As \u27e9 \u00d7 A\n\n_\u220b_\u2194*_ : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Sets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u2194* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u2194* (Bs , B) = (\u03b1s \u220b As \u2194* Bs) \u00d7 (\u03b1 \u220b A \u2194 B)\n\n\u27e8_\u220b_\u27e9\u00b2 : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u2194* Bs) \u2192 (max \u03b1s \u220b \u27e8 \u03b1s \u220b As \u27e9 \u2194 \u27e8 \u03b1s \u220b Bs \u27e9)\n\u27e8 \u03b5        \u220b tt       \u27e9\u00b2 tt       tt       = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (\u211cs , \u211c) \u27e9\u00b2 (as , a) (bs , b) = (\u27e8 \u03b1s \u220b \u211cs \u27e9\u00b2 as bs) \u00d7 (\u211c a b)\n\n-- Sequences of RSets\n\nRSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nRSets \u03b5        = \u22a4\nRSets (\u03b1s , \u03b1) = RSets \u03b1s \u00d7 RSet \u03b1\n\n_\u220b_\u21d4*_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 RSets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u21d4* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u21d4* (Bs , B) = (\u03b1s \u220b As \u21d4* Bs) \u00d7 (\u03b1 \u220b A \u21d4 B)\n\n-- Concatenation of sequences\n\n_+_ : Levels \u2192 Levels \u2192 Levels\n\u03b1s + \u03b5        = \u03b1s\n\u03b1s + (\u03b2s , \u03b2) = (\u03b1s + \u03b2s) , \u03b2\n\n_\u220b_++_\u220b_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 \u2200 \u03b2s \u2192 RSets \u03b2s \u2192 RSets (\u03b1s + \u03b2s)\n\u03b1s \u220b As ++ \u03b5        \u220b tt       = As\n\u03b1s \u220b As ++ (\u03b2s , \u03b2) \u220b (Bs , B) = ((\u03b1s \u220b As ++ \u03b2s \u220b Bs) , B)\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u21d4* Bs) \u2192 \u2200 \u03b2s {Cs Ds} \u2192 (\u03b2s \u220b Cs \u21d4* Ds) \u2192 \n  ((\u03b1s + \u03b2s) \u220b (\u03b1s \u220b As ++ \u03b2s \u220b Cs) \u21d4* (\u03b1s \u220b Bs ++ \u03b2s \u220b Ds))\n\u03b1s \u220b \u211cs ++\u00b2 \u03b5        \u220b tt       = \u211cs\n\u03b1s \u220b \u211cs ++\u00b2 (\u03b2s , \u03b2) \u220b (\u2111s , \u2111) = ((\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) , \u2111)\n\n-- Type variables\n\ndata TVar : Levels \u2192 Set\u2081 where\n  zero : \u2200 {\u03b1s \u03b1} \u2192 TVar (\u03b1s , \u03b1)\n  suc : \u2200 {\u03b1s \u03b1} \u2192 (\u03c4 : TVar \u03b1s) \u2192 TVar (\u03b1s , \u03b1)\n\n\u03c4level : \u2200 {\u03b1s} \u2192 TVar \u03b1s \u2192 Level\n\u03c4level (zero {\u03b1 = \u03b1}) = \u03b1\n\u03c4level (suc \u03c4) = \u03c4level \u03c4\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03b1s} (\u03c4 : TVar \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (\u03c4level \u03c4)\n\u03c4\u27e6 zero \u27e7  (As , A) = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (As , A) = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u03c4 : TVar \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (\u03c4level \u03c4 \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u21d4 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211cs , \u211c) t a b = \u211c t a b\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211cs , \u211c) t a b = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t a b\n\n-- Types\n\ndata Typ (\u03b1s : Levels) : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 Typ \u03b1s\n  _\u2227_ _\u21d2_ : (T : Typ \u03b1s) \u2192 (U : Typ \u03b1s) \u2192 Typ \u03b1s\n  var : (\u03c4 : TVar \u03b1s) \u2192 Typ \u03b1s\n  univ : \u2200 \u03b1 \u2192 (T : Typ (\u03b1s , \u03b1)) \u2192 Typ \u03b1s\n\ntlevel : \u2200 {\u03b1s} \u2192 Typ \u03b1s \u2192 Level\ntlevel \u27e8 A \u27e9 = o\ntlevel (T \u2227 U) = tlevel T \u2294 tlevel U\ntlevel (T \u21d2 U) = tlevel T \u2294 tlevel U\ntlevel (var \u03c4) = \u03c4level \u03c4\ntlevel (univ \u03b1 T) = \u2191 \u03b1 \u2294 tlevel T\n\nT\u27e6_\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (tlevel T)\nT\u27e6 \u27e8 A \u27e9 \u27e7    As t = A\nT\u27e6 T \u2227 U \u27e7    As t = T\u27e6 T \u27e7 As t \u00d7 T\u27e6 U \u27e7 As t\nT\u27e6 T \u21d2 U \u27e7    As t = T\u27e6 T \u27e7 As t \u2192 T\u27e6 U \u27e7 As t\nT\u27e6 var \u03c4 \u27e7    As t = \u03c4\u27e6 \u03c4 \u27e7 As t\nT\u27e6 univ \u03b1 T \u27e7 As t = \u2200 (A : RSet \u03b1) \u2192 T\u27e6 T \u27e7 (As , A) t\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (T : Typ \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (tlevel T \u220b T\u27e6 T \u27e7 As \u21d4 T\u27e6 T \u27e7 Bs)\nT\u27e6 \u27e8 A \u27e9 \u27e7\u00b2    \u211cs t a       b       = a \u2261 b\nT\u27e6 T \u2227 U \u27e7\u00b2    \u211cs t (a , b) (c , d) = T\u27e6 T \u27e7\u00b2 \u211cs t a c \u00d7 T\u27e6 U \u27e7\u00b2 \u211cs t b d\nT\u27e6 T \u21d2 U \u27e7\u00b2    \u211cs t f       g       = \u2200 {a b} \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t a b \u2192 T\u27e6 U \u27e7\u00b2 \u211cs t (f a) (g b)\nT\u27e6 var \u03c4 \u27e7\u00b2    \u211cs t v       w       = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t v w\nT\u27e6 univ \u03b1 T \u27e7\u00b2 \u211cs t f       g       = \u2200 {A B} (\u211c : \u03b1 \u220b A \u21d4 B) \u2192 T\u27e6 T \u27e7\u00b2 (\u211cs , \u211c) t (f A) (g B)\n\n-- Contexts\n\ndata Ctxt (\u03b1s : Levels) : Set\u2081 where\n  \u03b5 : Ctxt \u03b1s\n  _,_at_ : (\u0393 : Ctxt \u03b1s) (T : Typ \u03b1s) (t : Time) \u2192 Ctxt \u03b1s\n\nclevels : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Levels\nclevels \u03b5            = \u03b5\nclevels (\u0393 , T at t) = (clevels \u0393 , tlevel T)\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03b1s} (\u0393 : Ctxt \u03b1s) \u2192 RSets \u03b1s \u2192 Sets (clevels \u0393)\n\u0393\u27e6 \u03b5 \u27e7          As = tt\n\u0393\u27e6 \u0393 , T at t \u27e7 As = (\u0393\u27e6 \u0393 \u27e7 As , T\u27e6 T \u27e7 As t)\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u0393 : Ctxt \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u2194* \u0393\u27e6 \u0393 \u27e7 Bs)\n\u0393\u27e6 \u03b5 \u27e7\u00b2          \u211cs = tt\n\u0393\u27e6 \u0393 , T at t \u27e7\u00b2 \u211cs = (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs , T\u27e6 T \u27e7\u00b2 \u211cs t)\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 TVar (\u03b1s + \u03b2s) \u2192 TVar ((\u03b1s , \u03b1) + \u03b2s)\n\u03c4weaken \u03b1 \u03b5        \u03c4       = suc \u03c4\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) zero    = zero\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) (suc \u03c4) = suc (\u03c4weaken \u03b1 \u03b2s \u03c4)\n\n\u27e6\u03c4weaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n-- Weakening of types\n\ntweaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 Typ (\u03b1s + \u03b2s) \u2192 Typ ((\u03b1s , \u03b1) + \u03b2s)\ntweaken \u03b1 \u03b2s \u27e8 A \u27e9      = \u27e8 A \u27e9\ntweaken \u03b1 \u03b2s (T \u2227 U)    = tweaken \u03b1 \u03b2s T \u2227 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (T \u21d2 U)    = tweaken \u03b1 \u03b2s T \u21d2 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (var \u03c4)    = var (\u03c4weaken \u03b1 \u03b2s \u03c4)\ntweaken \u03b1 \u03b2s (univ \u03b2 T) = univ \u03b2 (tweaken \u03b1 (\u03b2s , \u03b2) T)\n\nmutual\n\n  \u27e6tweaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (var \u03c4)    As A Bs t a       = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\n  \u27e6tweaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192 T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (var \u03c4)    As A Bs t a       = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (var \u03c4)    \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (var \u03c4)    \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Weakening of contexts\n\ncweaken : \u2200 {\u03b1s} \u03b1 \u2192 Ctxt \u03b1s \u2192 Ctxt (\u03b1s , \u03b1)\ncweaken \u03b1 \u03b5            = \u03b5\ncweaken \u03b1 (\u0393 , T at t) = (cweaken \u03b1 \u0393 , tweaken \u03b1 \u03b5 T at t)\n  \n\u27e6cweaken\u27e7 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) As A \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7 (As , A) \u27e9\n\u27e6cweaken\u27e7 \u03b1 \u03b5            As A tt       = tt\n\u27e6cweaken\u27e7 \u03b1 (\u0393 , T at t) As A (as , a) = (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as , \u27e6tweaken\u27e7 \u03b1 \u03b5 T As A tt t a)\n\n\u27e6cweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) {As Bs A B} \u211cs \u211c {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192\n  \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7\u00b2 (\u211cs , \u211c) \u27e9\u00b2 (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as) (\u27e6cweaken\u27e7 \u03b1 \u0393 Bs B bs)\n\u27e6cweaken\u27e7\u00b2 \u03b1 \u03b5            \u211cs \u211c tt            = tt\n\u27e6cweaken\u27e7\u00b2 \u03b1 (\u0393 , T at t) \u211cs \u211c (as\u211cbs , a\u211cb) = (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b5 T \u211cs \u211c tt t a\u211cb)\n\n-- Substitution into type variables\n\n\u03c4subst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 TVar ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\n\u03c4subst T \u03b5        zero    = T\n\u03c4subst T \u03b5        (suc \u03c4) = var \u03c4\n\u03c4subst T (\u03b2s , \u03b2) zero    = var zero\n\u03c4subst T (\u03b2s , \u03b2) (suc \u03c4) = tweaken \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4)\n\n\u27e6\u03c4subst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6tweaken\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t \n    (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u207b\u00b9\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t \n    (\u27e6tweaken\u207b\u00b9\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t a)\n\n\u27e6\u03c4subst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6tweaken\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t \n    (\u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t \n    (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t a\u211cb)\n\n-- Substitution into types\n\ntsubst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 Typ ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\ntsubst T \u03b2s \u27e8 A \u27e9 = \u27e8 A \u27e9\ntsubst T \u03b2s (U \u2227 V) = tsubst T \u03b2s U \u2227 tsubst T \u03b2s V\ntsubst T \u03b2s (U \u21d2 V) = tsubst T \u03b2s U \u21d2 tsubst T \u03b2s V\ntsubst T \u03b2s (var \u03c4) = \u03c4subst T \u03b2s \u03c4\ntsubst T \u03b2s (univ \u03b2 U) = univ \u03b2 (tsubst T (\u03b2s , \u03b2) U)\n\nmutual\n\n  \u27e6tsubst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u27e7 T \u03b2s (var \u03c4)    As Bs t a       = \u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\n  \u27e6tsubst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (var \u03c4)    As Bs t a       = \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tsubst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (var \u03c4)    \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (var \u03c4)    \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Variables\n\ndata Var {\u03b1s} (T : Typ \u03b1s) (t : Time) : (\u0393 : Ctxt \u03b1s) \u2192 Set\u2081 where\n  zero : \u2200 {\u0393 : Ctxt \u03b1s} \u2192 Var T t (\u0393 , T at t)\n  suc : \u2200 {\u0393 : Ctxt \u03b1s} {U : Typ \u03b1s} {u} \u2192 Var T t \u0393 \u2192 Var T t (\u0393 , U at u)\n\nv\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Var T t \u0393 \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\nv\u27e6 zero \u27e7  As (as , a) = a\nv\u27e6 suc v \u27e7 As (as , a) = v\u27e6 v \u27e7 As as\n\nv\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (\u03c4 : Var T t \u0393) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (v\u27e6 \u03c4 \u27e7 As as) (v\u27e6 \u03c4 \u27e7 Bs bs)\nv\u27e6 zero \u27e7\u00b2  \u211cs (as\u211cbs , a\u211cb) = a\u211cb\nv\u27e6 suc v \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb) = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Expressions\n\ndata Exp : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Typ \u03b1s \u2192 RSet\u2081 where\n  const : \u2200 {\u03b1s \u0393 A t} \u2192 (a : A) \u2192 Exp {\u03b1s} \u0393 \u27e8 A \u27e9 t\n  pair : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 T t) \u2192 (f : Exp \u0393 U t) \u2192 Exp {\u03b1s} \u0393 (T \u2227 U) t\n  fst : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 T t\n  snd : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 U t\n  abs : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp (\u0393 , T at t) U t) \u2192 Exp {\u03b1s} \u0393 (T \u21d2 U) t\n  app : \u2200 {\u03b1s \u0393 T U t} \u2192 (f : Exp \u0393 (T \u21d2 U) t) \u2192 (e : Exp \u0393 T t) \u2192 Exp {\u03b1s} \u0393 U t\n  var : \u2200 {\u03b1s \u0393 T t} \u2192 (v : Var T t \u0393) \u2192 Exp {\u03b1s} \u0393 T t\n  tabs : \u2200 {\u03b1s \u0393} \u03b1 {T t} \u2192 (e : Exp (cweaken \u03b1 \u0393) T t) \u2192 Exp {\u03b1s} \u0393 (univ \u03b1 T) t\n  tapp : \u2200 {\u03b1s \u0393 t} (T : Typ \u03b1s) {U} \u2192 (e : Exp \u0393 (univ (tlevel T) U) t) \u2192 Exp {\u03b1s} \u0393 (tsubst T \u03b5 U) t\n\ne\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Exp \u0393 T t \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\ne\u27e6 const a \u27e7                  As as = a\ne\u27e6 pair e f \u27e7                 As as = (e\u27e6 e \u27e7 As as , e\u27e6 f \u27e7 As as)\ne\u27e6 fst e \u27e7                    As as = proj\u2081 (e\u27e6 e \u27e7 As as)\ne\u27e6 snd e \u27e7                    As as = proj\u2082 (e\u27e6 e \u27e7 As as)\ne\u27e6 abs e \u27e7                    As as = \u03bb a \u2192 e\u27e6 e \u27e7 As (as , a)\ne\u27e6 app f e \u27e7                  As as = e\u27e6 f \u27e7 As as (e\u27e6 e \u27e7 As as)\ne\u27e6 var v \u27e7                    As as = v\u27e6 v \u27e7 As as\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7         As as = \u03bb A \u2192 e\u27e6 e \u27e7 (As , A) (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7 As as = \u27e6tsubst\u27e7 T \u03b5 U As tt t (e\u27e6 e \u27e7 As as (T\u27e6 T \u27e7 As))\n\ne\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (e : Exp \u0393 T t) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (e\u27e6 e \u27e7 As as) (e\u27e6 e \u27e7 Bs bs)\ne\u27e6 const a \u27e7\u00b2                  \u211cs as\u211cbs = refl\ne\u27e6 pair e f \u27e7\u00b2                 \u211cs as\u211cbs = (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs , e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 fst e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2081 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 snd e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2082 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 abs e \u27e7\u00b2                    \u211cs as\u211cbs = \u03bb a\u211cb \u2192 e\u27e6 e \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb)\ne\u27e6 app f e \u27e7\u00b2                  \u211cs as\u211cbs = e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 var v \u27e7\u00b2                    \u211cs as\u211cbs = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7\u00b2         \u211cs as\u211cbs = \u03bb \u211c \u2192 e\u27e6 e \u27e7\u00b2 (\u211cs , \u211c) (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7\u00b2 \u211cs as\u211cbs = \u27e6tsubst\u27e7\u00b2 T \u03b5 U \u211cs tt t (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 T \u27e7\u00b2 \u211cs))\n\n-- Surface level types\n\ndata STyp : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 STyp\n  _\u2227_ _\u21d2_ : (T : STyp) \u2192 (U : STyp) \u2192 STyp\n\n-- Translation of surface level types into types\n\n\u27ea_\u27eb : STyp \u2192 Typ (\u03b5 , o)\n\u27ea \u27e8 A \u27e9 \u27eb = \u27e8 A \u27e9\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2227 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u21d2 \u27ea U \u27eb\n\nT\u27ea_\u27eb : STyp \u2192 RSet\u2080\nT\u27ea \u27e8 A \u27e9 \u27eb t = A\nT\u27ea T \u2227 U \u27eb t = T\u27ea T \u27eb t \u00d7 T\u27ea U \u27eb t\nT\u27ea T \u21d2 U \u27eb t = T\u27ea T \u27eb t \u2192 T\u27ea U \u27eb t\n\nWorld : RSet\u2080\nWorld t = \u22a4\n\nmutual\n\n  trans : \u2200 T {s} \u2192 T\u27ea T \u27eb s \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) s\n  trans \u27e8 A \u27e9   a       = a\n  trans (T \u2227 U) (a , b) = (trans T a , trans U b)\n  trans (T \u21d2 U) f       = \u03bb a \u2192 trans U (f (trans\u207b\u00b9 T a))\n\n  trans\u207b\u00b9 : \u2200 T {s} \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) s \u2192 T\u27ea T \u27eb s\n  trans\u207b\u00b9 \u27e8 A \u27e9   a       = a\n  trans\u207b\u00b9 (T \u2227 U) (a , b) = (trans\u207b\u00b9 T a , trans\u207b\u00b9 U b)\n  trans\u207b\u00b9 (T \u21d2 U) f       = \u03bb a \u2192 trans\u207b\u00b9 U (f (trans T a))\n\n-- Causality\n\n_at_\u220b_\u2248[_\u2235_]_ : \u2200 T s \u2192 T\u27ea T \u27eb s \u2192 \u2200 u \u2192 True (s \u2264 u) \u2192 T\u27ea T \u27eb s \u2192 Set\n\u27e8 A \u27e9   at s \u220b a       \u2248[ u \u2235 s\u2264u ] b       = a \u2261 b\n(T \u2227 U) at s \u220b (a , b) \u2248[ u \u2235 s\u2264u ] (c , d) = (T at s \u220b a \u2248[ u \u2235 s\u2264u ] c) \u00d7 (U at s \u220b b \u2248[ u \u2235 s\u2264u ] d)\n(T \u21d2 U) at s \u220b f       \u2248[ u \u2235 s\u2264u ] g       = \u2200 {a b} \u2192 (T at s \u220b a \u2248[ u \u2235 s\u2264u ] b) \u2192 (U at s \u220b f a \u2248[ u \u2235 s\u2264u ] g b)\n\nCausal : \u2200 T U s \u2192 T\u27ea T \u21d2 U \u27eb s \u2192 Set\nCausal T U s f = \u2200 u s\u2264u {a b} \u2192 (T at s \u220b a \u2248[ u \u2235 s\u2264u ] b) \u2192 (U at s \u220b f a \u2248[ u \u2235 s\u2264u ] f b)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (o \u220b World \u21d4 World)\n\u211c[ u ] t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248 : \u2200 T s u s\u2264u {a b} \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) s a b \u2192\n    (T at s \u220b trans\u207b\u00b9 T a \u2248[ u \u2235 s\u2264u ] trans\u207b\u00b9 T b)\n  \u211c-impl-\u2248 \u27e8 A \u27e9   s u s\u2264u a\u211cb         = a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) s u s\u2264u (a\u211cc , b\u211cd) = (\u211c-impl-\u2248 T s u s\u2264u a\u211cc , \u211c-impl-\u2248 U s u s\u2264u b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) s u s\u2264u f\u211cg         = \u03bb a\u2248b \u2192 \u211c-impl-\u2248 U s u s\u2264u (f\u211cg (\u2248-impl-\u211c T s u s\u2264u a\u2248b))\n\n  \u2248-impl-\u211c : \u2200 T s u s\u2264u {a b} \u2192\n    (T at s \u220b a \u2248[ u \u2235 s\u2264u ] b) \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) s (trans T a) (trans T b)\n  \u2248-impl-\u211c \u27e8 A \u27e9   s u s\u2264u a\u2248b         = a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) s u s\u2264u (a\u2248c , b\u2248d) = (\u2248-impl-\u211c T s u s\u2264u a\u2248c , \u2248-impl-\u211c U s u s\u2264u b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) s u s\u2264u f\u2248g         = \u03bb a\u211cb \u2192 \u2248-impl-\u211c U s u s\u2264u (f\u2248g (\u211c-impl-\u2248 T s u s\u2264u a\u211cb))\n\n-- Every expression is causal\n\ne\u27ea_at_\u220b_\u27eb : \u2200 T t \u2192 Exp \u03b5 \u27ea T \u27eb t \u2192 T\u27ea T \u27eb t\ne\u27ea T at t \u220b e \u27eb = trans\u207b\u00b9 T (e\u27e6 e \u27e7 (tt , World) tt)\n\ncausality : \u2200 T U s f \u2192 Causal T U s e\u27ea (T \u21d2 U) at s \u220b f \u27eb \ncausality T U s f u s\u2264u = \u211c-impl-\u2248 (T \u21d2 U) s u s\u2264u (e\u27e6 f \u27e7\u00b2 (tt , \u211c[ u ]) tt)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"71b3a14edcb512975548e8df67858a30b1f62d68","subject":"Rename TermConstructor to CurriedTermConstructor.","message":"Rename TermConstructor to CurriedTermConstructor.\n\nThis rename prepares for introducing UncurriedTermConstructor which\nwill allow to clarify the role of lift-\u03b7-const-rec.\n\nOld-commit-hash: b9de3c413f5a2a2015d3114bfb8483ab3078cc06\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 TermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"58c240b5bb748fbe58ece4cf0646e3c2245c5642","subject":"bijection-syntax: fix","message":"bijection-syntax: fix\n","repos":"crypto-agda\/crypto-agda","old_file":"bijection-syntax\/Bijection-Fin.agda","new_file":"bijection-syntax\/Bijection-Fin.agda","new_contents":"-- NOTE with-K\nmodule bijection-syntax.Bijection-Fin where\n\n  open import Type\n\n  open import bijection-syntax.Bijection\n  open import Function.NP hiding (Cmp)\n  open import Relation.Binary.PropositionalEquality.NP\n\n  open import Data.Empty\n  open import Data.Nat.NP hiding (suc-injective)\n  open import Data.Two\n  open import Data.Fin.NP\n    using ( Fin ; zero ; suc ; \u2115\u25b9Fin ; inject\u2081; suc-injective; _\u2264F_; z\u2264i; s\u2264s\n          ; module From-mono-inj )\n  open import Data.Nat.BoundedMonoInj-is-Id hiding (module From-mono-inj)\n  open import Data.Vec hiding ([_])\n  open import Data.Sum\n  import Algebra.FunctionProperties.Eq\n  open Algebra.FunctionProperties.Eq.Implicits\n\n  data `Syn : \u2115 \u2192 Type where\n    `id   : \u2200 {n} \u2192 `Syn n\n    `swap : \u2200 {n} \u2192 `Syn (2 + n)\n    `tail : \u2200 {n} \u2192 `Syn n \u2192 `Syn (1 + n)\n    _`\u2218_  : \u2200 {n} \u2192 `Syn n \u2192 `Syn n \u2192 `Syn n\n\n  `Rep = Fin\n\n  `Ix = \u2115\n\n  `Tree : Type \u2192 `Ix \u2192 Type\n  `Tree X = Vec X\n\n  `fromFun : \u2200 {i X} \u2192 (`Rep i \u2192 X) \u2192 `Tree X i\n  `fromFun = tabulate\n\n  `toFun : \u2200 {i X} \u2192 `Tree X i \u2192 (`Rep i \u2192 X)\n  `toFun T zero    = head T\n  `toFun T (suc i) = `toFun (tail T) i\n\n  `toFun\u2218fromFun : \u2200 {i X}(f : `Rep i \u2192 X) \u2192 f \u2257 `toFun (`fromFun f)\n  `toFun\u2218fromFun f zero = refl\n  `toFun\u2218fromFun f (suc i) = `toFun\u2218fromFun (f \u2218 suc) i\n\n  fin-swap : \u2200 {n} \u2192 Endo (Fin (2 + n))\n  fin-swap zero = suc zero\n  fin-swap (suc zero) = zero\n  fin-swap (suc (suc i)) = suc (suc i)\n\n  fin-tail : \u2200 {n} \u2192 Endo (Fin n) \u2192 Endo (Fin (1 + n))\n  fin-tail f zero = zero\n  fin-tail f (suc i) = suc (f i)\n\n  `evalArg : \u2200 {i} \u2192 `Syn i \u2192 Endo (`Rep i)\n  `evalArg `id       = id\n  `evalArg `swap     = fin-swap\n  `evalArg (`tail f) = fin-tail (`evalArg f)\n  `evalArg (S `\u2218 S\u2081) = `evalArg S \u2218 `evalArg S\u2081\n\n  vec-swap : \u2200 {n}{X : Type} \u2192 Endo (Vec X (2 + n))\n  vec-swap xs = head (tail xs) \u2237 head xs \u2237 tail (tail xs)\n\n  vec-tail : \u2200 {n}{X : Type} \u2192 Endo (Vec X n) \u2192 Endo (Vec X (1 + n))\n  vec-tail f xs = head xs \u2237 f (tail xs)\n\n  `evalTree : \u2200 {i X} \u2192 `Syn i \u2192 Endo (`Tree X i)\n  `evalTree `id       = id\n  `evalTree `swap     = vec-swap\n  `evalTree (`tail f) = vec-tail (`evalTree f)\n  `evalTree (S `\u2218 S\u2081) = `evalTree S \u2218 `evalTree S\u2081\n\n  `eval-proof : \u2200 {i X} S (T : `Tree X i) \u2192 `toFun T \u2257 `toFun (`evalTree S T) \u2218 `evalArg S\n  `eval-proof `id       T i = refl\n  `eval-proof `swap T zero = refl\n  `eval-proof `swap T (suc zero) = refl\n  `eval-proof `swap T (suc (suc i)) = refl\n  `eval-proof (`tail S) T zero = refl\n  `eval-proof (`tail S) T (suc i) = `eval-proof S (tail T) i\n  `eval-proof (S `\u2218 S\u2081) T i rewrite\n    `eval-proof S\u2081 T i |\n    `eval-proof S (`evalTree S\u2081 T) (`evalArg S\u2081 i) = refl\n\n  `inv : \u2200 {i} \u2192 Endo (`Syn i)\n  `inv `id       = `id\n  `inv `swap     = `swap\n  `inv (`tail S) = `tail (`inv S)\n  `inv (S `\u2218 S\u2081) = `inv S\u2081 `\u2218 `inv S\n\n  `inv-proof : \u2200 {i} \u2192 (S : `Syn i) \u2192 `evalArg S \u2218 `evalArg (`inv S) \u2257 id\n  `inv-proof `id x       = refl\n  `inv-proof `swap zero          = refl\n  `inv-proof `swap (suc zero)    = refl\n  `inv-proof `swap (suc (suc x)) = refl\n  `inv-proof (`tail S) zero = refl\n  `inv-proof (`tail S) (suc x) rewrite `inv-proof S x = refl\n  `inv-proof (S `\u2218 S\u2081) x rewrite \n    `inv-proof S\u2081 (`evalArg (`inv S) x) |\n    `inv-proof S x = refl\n\n  `RC : \u2200 {i} \u2192 Cmp (`Rep i)\n  `RC zero zero = eq\n  `RC zero (suc j) = lt\n  `RC (suc i) zero = gt\n  `RC (suc i) (suc j) = `RC i j\n\n  insert : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 Vec X (1 + n)\n  insert X-cmp x [] = x \u2237 []\n  insert X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert X-cmp x (x\u2081 \u2237 xs) | lt = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | eq = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | gt = x\u2081 \u2237 insert X-cmp x xs\n\n  `sort : \u2200 {i X} \u2192 Cmp X \u2192 Endo (`Tree X i)\n  `sort X-cmp [] = []\n  `sort X-cmp (x \u2237 xs) = insert X-cmp x (`sort X-cmp xs)\n\n  insert-syn : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 `Syn (1 + n)\n  insert-syn X-cmp x [] = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | lt = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | eq = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | gt = `tail (insert-syn X-cmp x xs) `\u2218 `swap\n\n  `sort-syn : \u2200 {i X} \u2192 Cmp X \u2192 `Tree X i \u2192 `Syn i\n  `sort-syn X-cmp []       = `id\n  `sort-syn X-cmp (x \u2237 xs) = insert-syn X-cmp x (`sort X-cmp xs) `\u2218 `tail (`sort-syn X-cmp xs)\n\n  insert-proof : \u2200 {n X}(X-cmp : Cmp X) x (T : Vec X n) \u2192 insert X-cmp x T \u2261 `evalTree (insert-syn X-cmp x T) (x \u2237 T)\n  insert-proof X-cmp x [] = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) with X-cmp x x\u2081\n  insert-proof X-cmp x (x\u2081 \u2237 T) | lt = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | eq = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | gt rewrite insert-proof X-cmp x T = refl\n\n  `sort-proof : \u2200 {i X}(X-cmp : Cmp X)(T : `Tree X i) \u2192 `sort X-cmp T \u2261 `evalTree (`sort-syn X-cmp T) T\n  `sort-proof X-cmp [] = refl\n  `sort-proof X-cmp (x \u2237 T) rewrite \n    ! `sort-proof X-cmp T = insert-proof X-cmp x (`sort X-cmp T)\n\n  module Alt-Syn where\n\n    data ``Syn : \u2115 \u2192 Type where\n      `id : \u2200 {n} \u2192 ``Syn n\n      _`\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n      `swap : \u2200 {n} m \u2192 ``Syn (m + 2 + n)\n\n    swap-fin : \u2200 {n} m \u2192 Endo (Fin (m + 2 + n))\n    swap-fin zero zero = suc zero\n    swap-fin zero (suc zero) = zero\n    swap-fin zero (suc (suc i)) = suc (suc i)\n    swap-fin (suc m) zero = zero\n    swap-fin (suc m) (suc i) = suc (swap-fin m i)\n\n    ``evalArg : \u2200 {n} \u2192 ``Syn n \u2192 Endo (`Rep n)\n    ``evalArg `id       = id\n    ``evalArg (S `\u2218 S\u2081) = ``evalArg S \u2218 ``evalArg S\u2081\n    ``evalArg (`swap m) = swap-fin m\n\n    _``\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n    `id ``\u2218 y = y\n    (x `\u2218 x\u2081) ``\u2218 `id = x `\u2218 x\u2081\n    (x `\u2218 x\u2081) ``\u2218 (y `\u2218 y\u2081) = x `\u2218 (x\u2081 `\u2218 (y `\u2218 y\u2081))\n    (x `\u2218 x\u2081) ``\u2218 `swap m = x `\u2218 (x\u2081 ``\u2218 `swap m)\n    `swap m ``\u2218 y = `swap m `\u2218 y\n\n    ``tail : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn (suc n)\n    ``tail `id = `id\n    ``tail (S `\u2218 S\u2081) = ``tail S ``\u2218 ``tail S\u2081\n    ``tail (`swap m) = `swap (suc m)\n\n    translate : \u2200 {n} \u2192 `Syn n \u2192 ``Syn n\n    translate `id = `id\n    translate `swap = `swap 0\n    translate (`tail S) = ``tail (translate S)\n    translate (S `\u2218 S\u2081) = translate S ``\u2218 translate S\u2081\n\n    ``\u2218-p : \u2200 {n}(A B : ``Syn n) \u2192 ``evalArg (A ``\u2218 B) \u2257 ``evalArg (A `\u2218 B)\n    ``\u2218-p `id B x = refl\n    ``\u2218-p (A `\u2218 A\u2081) `id x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (B `\u2218 B\u2081) x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (`swap m) x rewrite ``\u2218-p A\u2081 (`swap m) x = refl\n    ``\u2218-p (`swap m) B x = refl\n\n    ``tail-p : \u2200 {n} (S : ``Syn n) \u2192 fin-tail (``evalArg S) \u2257 ``evalArg (``tail S)\n    ``tail-p `id zero = refl\n    ``tail-p `id (suc x) = refl\n    ``tail-p (S `\u2218 S\u2081) zero rewrite ``\u2218-p (``tail S) (``tail S\u2081) zero\n                                  | ! ``tail-p S\u2081 zero = ``tail-p S zero\n    ``tail-p (S `\u2218 S\u2081) (suc x) rewrite ``\u2218-p (``tail S) (``tail S\u2081) (suc x)\n                                     | ! ``tail-p S\u2081 (suc x) = ``tail-p S (suc (``evalArg S\u2081 x))\n    ``tail-p (`swap m) zero = refl\n    ``tail-p (`swap m) (suc x) = refl\n\n    `eval`` : \u2200 {n} (S : `Syn n) \u2192 `evalArg S \u2257 ``evalArg (translate S)\n    `eval`` `id       x = refl\n    `eval`` `swap zero = refl\n    `eval`` `swap (suc zero) = refl\n    `eval`` `swap (suc (suc x)) = refl\n    `eval`` (`tail S) zero = ``tail-p (translate S) zero\n    `eval`` (`tail S) (suc x) rewrite `eval`` S x = ``tail-p (translate S) (suc x)\n    `eval`` (S `\u2218 S\u2081) x rewrite ``\u2218-p (translate S) (translate S\u2081) x | ! `eval`` S\u2081 x | `eval`` S (`evalArg S\u2081 x) = refl\n\n\n  data Fin-View : \u2200 {n} \u2192 Fin n \u2192 Type where\n    max : \u2200 {n} \u2192 Fin-View (\u2115\u25b9Fin n)\n    inject : \u2200 {n} \u2192 (i : Fin n) \u2192 Fin-View (inject\u2081 i)\n\n  data Sorted {X}(XC : Cmp X) : \u2200 {l} \u2192 Vec X l  \u2192 Type where\n    []  : Sorted XC []\n    sing : \u2200 x \u2192 Sorted XC (x \u2237 [])\n    dbl-lt  : \u2200 {l} x y {xs : Vec X l} \u2192 lt \u2261 XC x y \u2192 Sorted XC (y \u2237 xs) \u2192 Sorted XC (x \u2237 y \u2237 xs)\n    dbl-eq  : \u2200 {l} x {xs : Vec X l} \u2192 Sorted XC (x \u2237 xs) \u2192 Sorted XC (x \u2237 x \u2237 xs)\n\n  opposite : Ord \u2192 Ord\n  opposite lt = gt\n  opposite eq = eq\n  opposite gt = lt\n\n  flip-RC : \u2200 {n}(x y : Fin n) \u2192 opposite (`RC x y) \u2261 `RC y x\n  flip-RC zero zero = refl\n  flip-RC zero (suc y) = refl\n  flip-RC (suc x) zero = refl\n  flip-RC (suc x) (suc y) = flip-RC x y\n\n  eq=>\u2261 : \u2200 {i} (x y : Fin i) \u2192 eq \u2261 `RC x y \u2192 x \u2261 y\n  eq=>\u2261 zero zero p = refl\n  eq=>\u2261 zero (suc y) ()\n  eq=>\u2261 (suc x) zero ()\n  eq=>\u2261 (suc x) (suc y) p rewrite eq=>\u2261 x y p = refl\n\n  insert-Sorted : \u2200 {n l}{V : Vec (Fin n) l}(x : Fin n) \u2192 Sorted {Fin n} `RC V \u2192 Sorted {Fin n} `RC (insert `RC x V)\n  insert-Sorted x [] = sing x\n  insert-Sorted x (sing x\u2081) with `RC x x\u2081 | dbl-lt {XC = `RC} x x\u2081 {[]} | eq=>\u2261 x x\u2081 | flip-RC x x\u2081\n  insert-Sorted x (sing x\u2081) | lt | b | _ | _ = b refl (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | eq | _ | p | _ rewrite p refl = dbl-eq x\u2081 (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | gt | b | _ | l = dbl-lt x\u2081 x l (sing x)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) with `RC x y | dbl-lt {XC = `RC} x y {y' \u2237 xs} | eq=>\u2261 x y | flip-RC x y\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | lt | b | p | l\u2081 = b refl (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | eq | b | p | l\u2081 rewrite p refl = dbl-eq y (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) | gt | b | p | l\u2081 with `RC x y' | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | lt | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | eq | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | gt | xs' = dbl-lt y y' prf xs'\n  insert-Sorted x (dbl-eq y {xs} xs\u2081) with `RC x y | inspect (`RC x) y | dbl-lt {XC = `RC} x y {y \u2237 xs} | eq=>\u2261 x y | flip-RC x y | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-eq y xs\u2081) | lt | _ | b | p | l | _ = b refl (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | eq | _ | b | p | l | _ rewrite p refl = dbl-eq y (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | gt | [ prf ] | b | p | l | ss  rewrite prf = dbl-eq y ss \n\n  sort-Sorted : \u2200 {n l}(V : Vec (Fin n) l) \u2192 Sorted `RC (`sort `RC V)\n  sort-Sorted [] = []\n  sort-Sorted (x \u2237 V) = insert-Sorted x (sort-Sorted V)\n\n  RC-refl : \u2200 {i}(x : Fin i) \u2192 `RC x x \u2261 eq\n  RC-refl zero = refl\n  RC-refl (suc x) = RC-refl x\n\n  STail : \u2200 {X l}{XC : Cmp X}{xs : Vec X (suc l)} \u2192 Sorted XC xs \u2192 Sorted XC (tail xs)\n  STail (sing x) = []\n  STail (dbl-lt x y x\u2081 T) = T\n  STail (dbl-eq x T) = T\n\n  module sproof {X}(XC : Cmp X)(XC-refl : \u2200 x \u2192 XC x x \u2261 eq)\n     (eq\u2261 : \u2200 x y \u2192 XC x y \u2261 eq \u2192 x \u2261 y)\n     (lt-trans : \u2200 x y z \u2192 XC x y \u2261 lt \u2192 XC y z \u2261 lt \u2192 XC x z \u2261 lt)\n     (XC-flip : \u2200 x y \u2192 opposite (XC x y) \u2261 XC y x)\n     where\n\n    open import Data.Sum\n\n    _\u2264X_ : X \u2192 X \u2192 Type\n    x \u2264X y = XC x y \u2261 lt \u228e XC x y \u2261 eq\n\n    \u2264X-trans : \u2200 {x y z} \u2192 x \u2264X y \u2192 y \u2264X z \u2192 x \u2264X z\n    \u2264X-trans (inj\u2081 x\u2081) (inj\u2081 x\u2082) = inj\u2081 (lt-trans _ _ _ x\u2081 x\u2082)\n    \u2264X-trans {_}{y}{z}(inj\u2081 x\u2081) (inj\u2082 y\u2081) rewrite eq\u2261 y z y\u2081 = inj\u2081 x\u2081\n    \u2264X-trans {x}{y} (inj\u2082 y\u2081) y\u2264z rewrite eq\u2261 x y y\u2081 = y\u2264z\n\n    h\u2264t : \u2200 {n}{T : `Tree X (2 + n)} \u2192 Sorted XC T \u2192 head T \u2264X head (tail T)\n    h\u2264t (dbl-lt x y x\u2081 ST) = inj\u2081 (! x\u2081)\n    h\u2264t (dbl-eq x ST) rewrite XC-refl x = inj\u2082 refl\n\n    head-p : \u2200 {n}{T : `Tree X (suc n)} i \u2192 Sorted XC T \u2192 head T \u2264X `toFun T i\n    head-p {T = T} zero ST rewrite XC-refl (head T) = inj\u2082 refl\n    head-p {zero} (suc ()) ST\n    head-p {suc n} (suc i) ST = \u2264X-trans (h\u2264t ST) (head-p i (STail ST))\n\n    toFun-p : \u2200 {n}{T : `Tree X n}{i j : Fin n} \u2192 i \u2264F j \u2192 Sorted XC T \u2192 `toFun T i \u2264X `toFun T j\n    toFun-p {j = j} z\u2264i ST = head-p j ST\n    toFun-p (s\u2264s i\u2264Fj) ST = toFun-p i\u2264Fj (STail ST)\n\n    sort-proof : \u2200 {i}{T : `Tree X i} \u2192 Sorted XC T \u2192 Is-Mono `RC XC (`toFun T)\n    sort-proof {T = T} T\u2081 zero zero rewrite XC-refl (head T) = _\n    sort-proof T\u2081 zero (suc y) with toFun-p (z\u2264i {i = suc y}) T\u2081\n    sort-proof T zero (suc y) | inj\u2081 x rewrite x = _\n    sort-proof T zero (suc y) | inj\u2082 y\u2081 rewrite y\u2081 = _\n    sort-proof {T = T} T\u2081 (suc x) zero with toFun-p (z\u2264i {i = suc x}) T\u2081 | XC-flip (head T) (`toFun (tail T) x)\n    sort-proof T (suc x) zero | inj\u2081 x\u2081 | l rewrite x\u2081 | ! l = _\n    sort-proof T (suc x) zero | inj\u2082 y | l rewrite y | ! l = _\n    sort-proof T\u2081 (suc x) (suc y) = sort-proof (STail T\u2081) x y\n\n  lt-trans-RC : \u2200 {i} (x y z : Fin i) \u2192 `RC x y \u2261 lt \u2192 `RC y z \u2261 lt \u2192 `RC x z \u2261 lt\n  lt-trans-RC zero zero zero x<y y<z = y<z\n  lt-trans-RC zero zero (suc z) x<y y<z = refl\n  lt-trans-RC zero (suc y) zero x<y ()\n  lt-trans-RC zero (suc y) (suc z) x<y y<z = refl\n  lt-trans-RC (suc x) zero zero x<y y<z = x<y\n  lt-trans-RC (suc x) zero (suc z) () y<z\n  lt-trans-RC (suc x) (suc y) zero x<y y<z = y<z\n  lt-trans-RC (suc x) (suc y) (suc z) x<y y<z = lt-trans-RC x y z x<y y<z\n\n  `sort-mono : \u2200 {i}(T : `Tree (`Rep i) i) \u2192 Is-Mono `RC `RC (`toFun (`sort `RC T))\n  `sort-mono T x y = sproof.sort-proof `RC RC-refl (\u03bb x\u2081 y\u2081 x\u2082 \u2192 eq=>\u2261 x\u2081 y\u2081 (! x\u2082)) lt-trans-RC flip-RC (sort-Sorted T) x y\n\n  move-to-RC : \u2200 {n}{x y : Fin n} \u2192 x \u2264F y \u2192 `RC x y \u2261 lt \u228e `RC x y \u2261 eq\n  move-to-RC {y = zero} z\u2264i = inj\u2082 refl\n  move-to-RC {y = suc y} z\u2264i = inj\u2081 refl\n  move-to-RC (s\u2264s x\u2264Fy) = move-to-RC x\u2264Fy\n\n  move-from-RC : \u2200 {n}(x y : Fin n) \u2192 lt \u2261 `RC x y \u228e eq \u2261 `RC x y \u2192 x \u2264F y\n  move-from-RC zero zero prf = z\u2264i\n  move-from-RC zero (suc y) prf = z\u2264i\n  move-from-RC (suc x) zero (inj\u2081 ())\n  move-from-RC (suc x) zero (inj\u2082 ())\n  move-from-RC (suc x) (suc y) prf = s\u2264s (move-from-RC x y prf)\n\n  mono-RC : \u2200 {n}(f : Endo (Fin n))(f-mono : Is-Mono `RC `RC f)\n              {x y} \u2192 x \u2264F y \u2192 f x \u2264F f y\n  mono-RC {n} f f-mono {x} {y} x\u2264Fy with `RC x y | `RC (f x) (f y) | move-to-RC x\u2264Fy | f-mono x y | move-from-RC (f x) (f y)\n  mono-RC f f-mono x\u2264Fy | .lt | lt | inj\u2081 refl | r4 | r5 = r5 (inj\u2081 refl)\n  mono-RC f f-mono x\u2264Fy | .lt | eq | inj\u2081 refl | r4 | r5 = r5 (inj\u2082 refl)\n  mono-RC f f-mono x\u2264Fy | .lt | gt | inj\u2081 refl | () | r5\n  mono-RC f f-mono x\u2264Fy | .eq | lt | inj\u2082 refl | () | r5\n  mono-RC f f-mono x\u2264Fy | .eq | eq | inj\u2082 refl | r4 | r5 = r5 (inj\u2082 refl)\n  mono-RC f f-mono x\u2264Fy | .eq | gt | inj\u2082 refl | () | r5\n\n\n  fin-view : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 Fin-View i\n  fin-view {zero} zero = max\n  fin-view {zero} (suc ())\n  fin-view {suc n} zero = inject _\n  fin-view {suc n} (suc i) with fin-view i\n  fin-view {suc n} (suc .(\u2115\u25b9Fin n)) | max = max\n  fin-view {suc n} (suc .(inject\u2081 i)) | inject i = inject _\n\n  absurd : {X : Type} \u2192 .\u22a5 \u2192 X\n  absurd ()\n\n  drop\u2081 : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 .(i \u2262 \u2115\u25b9Fin n) \u2192 Fin n\n  drop\u2081 i neq with fin-view i\n  drop\u2081 {n} .(\u2115\u25b9Fin n) neq | max = absurd (neq refl)\n  drop\u2081 .(inject\u2081 i) neq | inject i = i\n\n  drop\u2081\u2192inject\u2081 : \u2200 {n}(i : Fin (suc n))(j : Fin n).(p : i \u2262 \u2115\u25b9Fin n) \u2192 drop\u2081 i p \u2261 j \u2192 i \u2261 inject\u2081 j\n  drop\u2081\u2192inject\u2081 i j p q with fin-view i\n  drop\u2081\u2192inject\u2081 {n} .(\u2115\u25b9Fin n) j p q | max = absurd (p refl)\n  drop\u2081\u2192inject\u2081 .(inject\u2081 i) j p q | inject i = ap inject\u2081 q\n\n\n  `mono-inj\u2192id : \u2200{i}(f : Endo (`Rep i)) \u2192 Injective f \u2192 Is-Mono `RC `RC f \u2192 f \u2257 id\n  `mono-inj\u2192id f inj mono = From-mono-inj.f\u2257id f inj (mono-RC f mono)\n\n\n  interface : Interface\n  interface = record \n    { Ix            = `Ix\n    ; Rep           = `Rep\n    ; Syn           = `Syn\n    ; Tree          = `Tree\n    ; fromFun       = `fromFun\n    ; toFun         = `toFun\n    ; toFun\u2218fromFun = `toFun\u2218fromFun\n    ; evalArg       = `evalArg\n    ; evalTree      = `evalTree\n    ; eval-proof    = `eval-proof\n    ; inv           = `inv\n    ; inv-proof     = `inv-proof\n    ; RC            = `RC\n    ; sort          = `sort\n    ; sort-syn      = `sort-syn\n    ; sort-proof    = `sort-proof\n    ; sort-mono     = `sort-mono\n    ; mono-inj\u2192id   = `mono-inj\u2192id\n    }\n\n  count : \u2200 {n} \u2192 (Fin n \u2192 \u2115) \u2192 \u2115\n  count {n} f = sum (tabulate f)\n\n  count-ext : \u2200 {n} \u2192 (f g : Fin n \u2192 \u2115) \u2192 f \u2257 g \u2192 count f \u2261 count g\n  count-ext {zero} f g f\u2257g = refl\n  count-ext {suc n} f g f\u2257g rewrite f\u2257g zero | count-ext (f \u2218 suc) (g \u2218 suc) (f\u2257g \u2218 suc) = refl\n\n  #\u27e8_\u27e9 : \u2200 {n} \u2192 (Fin n \u2192 \ud835\udfda) \u2192 \u2115\n  #\u27e8 f \u27e9 = count (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  #-ext : \u2200 {n} \u2192 (f g : Fin n \u2192 \ud835\udfda) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n  #-ext f g f\u2257g = count-ext (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g) (ap \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  com-assoc : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n  com-assoc x y z rewrite \n    ! \u2115\u00b0.+-assoc x y z |\n    \u2115\u00b0.+-comm x y      |\n    \u2115\u00b0.+-assoc y x z   = refl\n    \n  syn-pres : \u2200 {n}(f : Fin n \u2192 \u2115)(S : `Syn n)\n           \u2192 count f \u2261 count (f \u2218 `evalArg S)\n  syn-pres f `id = refl\n  syn-pres f `swap = com-assoc (f zero) (f (suc zero)) (count (\u03bb i \u2192 f (suc (suc i))))\n  syn-pres f (`tail S) rewrite syn-pres (f \u2218 suc) S = refl\n  syn-pres f (S `\u2218 S\u2081) rewrite syn-pres f S = syn-pres (f \u2218 `evalArg S) S\u2081\n\n  count-perm : \u2200 {n}(f : Fin n \u2192 \u2115)(p : Endo (Fin n)) \u2192 Injective p\n         \u2192 count f \u2261 count (f \u2218 p)\n  count-perm f p p-inj = trans (syn-pres f (sort-bij p)) (count-ext _ _ f\u2218eval\u2257f\u2218p)\n   where\n     open abs interface\n     f\u2218eval\u2257f\u2218p : f \u2218 `evalArg (sort-bij p) \u2257 f \u2218 p\n     f\u2218eval\u2257f\u2218p x rewrite thm p p-inj x = refl\n\n\n  #-perm : \u2200 {n}(f : Fin n \u2192 \ud835\udfda)(p : Endo (Fin n)) \u2192 Injective p\n         \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 p \u27e9\n  #-perm f p p-inj = count-perm (\ud835\udfda\u25b9\u2115 \u2218 f) p p-inj\n\n  test : `Syn 8\n  test = abs.sort-bij interface (\u03bb x \u2192 `evalArg (`tail `swap) x)\n-- -}\n","old_contents":"-- NOTE with-K\nmodule bijection-syntax.Bijection-Fin where\n\n  open import Type\n\n  open import bijection-syntax.Bijection\n  open import Function.NP hiding (Cmp)\n  open import Relation.Binary.PropositionalEquality.NP\n\n  open import Data.Empty\n  open import Data.Nat.NP hiding (suc-injective)\n  open import Data.Two\n  open import Data.Fin.NP\n    using ( Fin ; zero ; suc ; \u2115\u25b9Fin ; inject\u2081; suc-injective; _\u2264F_; z\u2264i; s\u2264s\n          ; module From-mono-inj )\n  open import Data.Vec hiding ([_])\n  open import Data.Sum\n\n  data `Syn : \u2115 \u2192 Type where\n    `id   : \u2200 {n} \u2192 `Syn n\n    `swap : \u2200 {n} \u2192 `Syn (2 + n)\n    `tail : \u2200 {n} \u2192 `Syn n \u2192 `Syn (1 + n)\n    _`\u2218_  : \u2200 {n} \u2192 `Syn n \u2192 `Syn n \u2192 `Syn n\n\n  `Rep = Fin\n\n  `Ix = \u2115\n\n  `Tree : Type \u2192 `Ix \u2192 Type\n  `Tree X = Vec X\n\n  `fromFun : \u2200 {i X} \u2192 (`Rep i \u2192 X) \u2192 `Tree X i\n  `fromFun = tabulate\n\n  `toFun : \u2200 {i X} \u2192 `Tree X i \u2192 (`Rep i \u2192 X)\n  `toFun T zero    = head T\n  `toFun T (suc i) = `toFun (tail T) i\n\n  `toFun\u2218fromFun : \u2200 {i X}(f : `Rep i \u2192 X) \u2192 f \u2257 `toFun (`fromFun f)\n  `toFun\u2218fromFun f zero = refl\n  `toFun\u2218fromFun f (suc i) = `toFun\u2218fromFun (f \u2218 suc) i\n\n  fin-swap : \u2200 {n} \u2192 Endo (Fin (2 + n))\n  fin-swap zero = suc zero\n  fin-swap (suc zero) = zero\n  fin-swap (suc (suc i)) = suc (suc i)\n\n  fin-tail : \u2200 {n} \u2192 Endo (Fin n) \u2192 Endo (Fin (1 + n))\n  fin-tail f zero = zero\n  fin-tail f (suc i) = suc (f i)\n\n  `evalArg : \u2200 {i} \u2192 `Syn i \u2192 Endo (`Rep i)\n  `evalArg `id       = id\n  `evalArg `swap     = fin-swap\n  `evalArg (`tail f) = fin-tail (`evalArg f)\n  `evalArg (S `\u2218 S\u2081) = `evalArg S \u2218 `evalArg S\u2081\n\n  vec-swap : \u2200 {n}{X : Type} \u2192 Endo (Vec X (2 + n))\n  vec-swap xs = head (tail xs) \u2237 head xs \u2237 tail (tail xs)\n\n  vec-tail : \u2200 {n}{X : Type} \u2192 Endo (Vec X n) \u2192 Endo (Vec X (1 + n))\n  vec-tail f xs = head xs \u2237 f (tail xs)\n\n  `evalTree : \u2200 {i X} \u2192 `Syn i \u2192 Endo (`Tree X i)\n  `evalTree `id       = id\n  `evalTree `swap     = vec-swap\n  `evalTree (`tail f) = vec-tail (`evalTree f)\n  `evalTree (S `\u2218 S\u2081) = `evalTree S \u2218 `evalTree S\u2081\n\n  `eval-proof : \u2200 {i X} S (T : `Tree X i) \u2192 `toFun T \u2257 `toFun (`evalTree S T) \u2218 `evalArg S\n  `eval-proof `id       T i = refl\n  `eval-proof `swap T zero = refl\n  `eval-proof `swap T (suc zero) = refl\n  `eval-proof `swap T (suc (suc i)) = refl\n  `eval-proof (`tail S) T zero = refl\n  `eval-proof (`tail S) T (suc i) = `eval-proof S (tail T) i\n  `eval-proof (S `\u2218 S\u2081) T i rewrite\n    `eval-proof S\u2081 T i |\n    `eval-proof S (`evalTree S\u2081 T) (`evalArg S\u2081 i) = refl\n\n  `inv : \u2200 {i} \u2192 Endo (`Syn i)\n  `inv `id       = `id\n  `inv `swap     = `swap\n  `inv (`tail S) = `tail (`inv S)\n  `inv (S `\u2218 S\u2081) = `inv S\u2081 `\u2218 `inv S\n\n  `inv-proof : \u2200 {i} \u2192 (S : `Syn i) \u2192 `evalArg S \u2218 `evalArg (`inv S) \u2257 id\n  `inv-proof `id x       = refl\n  `inv-proof `swap zero          = refl\n  `inv-proof `swap (suc zero)    = refl\n  `inv-proof `swap (suc (suc x)) = refl\n  `inv-proof (`tail S) zero = refl\n  `inv-proof (`tail S) (suc x) rewrite `inv-proof S x = refl\n  `inv-proof (S `\u2218 S\u2081) x rewrite \n    `inv-proof S\u2081 (`evalArg (`inv S) x) |\n    `inv-proof S x = refl\n\n  `RC : \u2200 {i} \u2192 Cmp (`Rep i)\n  `RC zero zero = eq\n  `RC zero (suc j) = lt\n  `RC (suc i) zero = gt\n  `RC (suc i) (suc j) = `RC i j\n\n  insert : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 Vec X (1 + n)\n  insert X-cmp x [] = x \u2237 []\n  insert X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert X-cmp x (x\u2081 \u2237 xs) | lt = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | eq = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | gt = x\u2081 \u2237 insert X-cmp x xs\n\n  `sort : \u2200 {i X} \u2192 Cmp X \u2192 Endo (`Tree X i)\n  `sort X-cmp [] = []\n  `sort X-cmp (x \u2237 xs) = insert X-cmp x (`sort X-cmp xs)\n\n  insert-syn : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 `Syn (1 + n)\n  insert-syn X-cmp x [] = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | lt = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | eq = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | gt = `tail (insert-syn X-cmp x xs) `\u2218 `swap\n\n  `sort-syn : \u2200 {i X} \u2192 Cmp X \u2192 `Tree X i \u2192 `Syn i\n  `sort-syn X-cmp []       = `id\n  `sort-syn X-cmp (x \u2237 xs) = insert-syn X-cmp x (`sort X-cmp xs) `\u2218 `tail (`sort-syn X-cmp xs)\n\n  insert-proof : \u2200 {n X}(X-cmp : Cmp X) x (T : Vec X n) \u2192 insert X-cmp x T \u2261 `evalTree (insert-syn X-cmp x T) (x \u2237 T)\n  insert-proof X-cmp x [] = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) with X-cmp x x\u2081\n  insert-proof X-cmp x (x\u2081 \u2237 T) | lt = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | eq = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | gt rewrite insert-proof X-cmp x T = refl\n\n  `sort-proof : \u2200 {i X}(X-cmp : Cmp X)(T : `Tree X i) \u2192 `sort X-cmp T \u2261 `evalTree (`sort-syn X-cmp T) T\n  `sort-proof X-cmp [] = refl\n  `sort-proof X-cmp (x \u2237 T) rewrite \n    ! `sort-proof X-cmp T = insert-proof X-cmp x (`sort X-cmp T)\n\n  module Alt-Syn where\n\n    data ``Syn : \u2115 \u2192 Type where\n      `id : \u2200 {n} \u2192 ``Syn n\n      _`\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n      `swap : \u2200 {n} m \u2192 ``Syn (m + 2 + n)\n\n    swap-fin : \u2200 {n} m \u2192 Endo (Fin (m + 2 + n))\n    swap-fin zero zero = suc zero\n    swap-fin zero (suc zero) = zero\n    swap-fin zero (suc (suc i)) = suc (suc i)\n    swap-fin (suc m) zero = zero\n    swap-fin (suc m) (suc i) = suc (swap-fin m i)\n\n    ``evalArg : \u2200 {n} \u2192 ``Syn n \u2192 Endo (`Rep n)\n    ``evalArg `id       = id\n    ``evalArg (S `\u2218 S\u2081) = ``evalArg S \u2218 ``evalArg S\u2081\n    ``evalArg (`swap m) = swap-fin m\n\n    _``\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n    `id ``\u2218 y = y\n    (x `\u2218 x\u2081) ``\u2218 `id = x `\u2218 x\u2081\n    (x `\u2218 x\u2081) ``\u2218 (y `\u2218 y\u2081) = x `\u2218 (x\u2081 `\u2218 (y `\u2218 y\u2081))\n    (x `\u2218 x\u2081) ``\u2218 `swap m = x `\u2218 (x\u2081 ``\u2218 `swap m)\n    `swap m ``\u2218 y = `swap m `\u2218 y\n\n    ``tail : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn (suc n)\n    ``tail `id = `id\n    ``tail (S `\u2218 S\u2081) = ``tail S ``\u2218 ``tail S\u2081\n    ``tail (`swap m) = `swap (suc m)\n\n    translate : \u2200 {n} \u2192 `Syn n \u2192 ``Syn n\n    translate `id = `id\n    translate `swap = `swap 0\n    translate (`tail S) = ``tail (translate S)\n    translate (S `\u2218 S\u2081) = translate S ``\u2218 translate S\u2081\n\n    ``\u2218-p : \u2200 {n}(A B : ``Syn n) \u2192 ``evalArg (A ``\u2218 B) \u2257 ``evalArg (A `\u2218 B)\n    ``\u2218-p `id B x = refl\n    ``\u2218-p (A `\u2218 A\u2081) `id x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (B `\u2218 B\u2081) x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (`swap m) x rewrite ``\u2218-p A\u2081 (`swap m) x = refl\n    ``\u2218-p (`swap m) B x = refl\n\n    ``tail-p : \u2200 {n} (S : ``Syn n) \u2192 fin-tail (``evalArg S) \u2257 ``evalArg (``tail S)\n    ``tail-p `id zero = refl\n    ``tail-p `id (suc x) = refl\n    ``tail-p (S `\u2218 S\u2081) zero rewrite ``\u2218-p (``tail S) (``tail S\u2081) zero\n                                  | ! ``tail-p S\u2081 zero = ``tail-p S zero\n    ``tail-p (S `\u2218 S\u2081) (suc x) rewrite ``\u2218-p (``tail S) (``tail S\u2081) (suc x)\n                                     | ! ``tail-p S\u2081 (suc x) = ``tail-p S (suc (``evalArg S\u2081 x))\n    ``tail-p (`swap m) zero = refl\n    ``tail-p (`swap m) (suc x) = refl\n\n    `eval`` : \u2200 {n} (S : `Syn n) \u2192 `evalArg S \u2257 ``evalArg (translate S)\n    `eval`` `id       x = refl\n    `eval`` `swap zero = refl\n    `eval`` `swap (suc zero) = refl\n    `eval`` `swap (suc (suc x)) = refl\n    `eval`` (`tail S) zero = ``tail-p (translate S) zero\n    `eval`` (`tail S) (suc x) rewrite `eval`` S x = ``tail-p (translate S) (suc x)\n    `eval`` (S `\u2218 S\u2081) x rewrite ``\u2218-p (translate S) (translate S\u2081) x | ! `eval`` S\u2081 x | `eval`` S (`evalArg S\u2081 x) = refl\n\n\n  data Fin-View : \u2200 {n} \u2192 Fin n \u2192 Type where\n    max : \u2200 {n} \u2192 Fin-View (\u2115\u25b9Fin n)\n    inject : \u2200 {n} \u2192 (i : Fin n) \u2192 Fin-View (inject\u2081 i)\n\n  data Sorted {X}(XC : Cmp X) : \u2200 {l} \u2192 Vec X l  \u2192 Type where\n    []  : Sorted XC []\n    sing : \u2200 x \u2192 Sorted XC (x \u2237 [])\n    dbl-lt  : \u2200 {l} x y {xs : Vec X l} \u2192 lt \u2261 XC x y \u2192 Sorted XC (y \u2237 xs) \u2192 Sorted XC (x \u2237 y \u2237 xs)\n    dbl-eq  : \u2200 {l} x {xs : Vec X l} \u2192 Sorted XC (x \u2237 xs) \u2192 Sorted XC (x \u2237 x \u2237 xs)\n\n  opposite : Ord \u2192 Ord\n  opposite lt = gt\n  opposite eq = eq\n  opposite gt = lt\n\n  flip-RC : \u2200 {n}(x y : Fin n) \u2192 opposite (`RC x y) \u2261 `RC y x\n  flip-RC zero zero = refl\n  flip-RC zero (suc y) = refl\n  flip-RC (suc x) zero = refl\n  flip-RC (suc x) (suc y) = flip-RC x y\n\n  eq=>\u2261 : \u2200 {i} (x y : Fin i) \u2192 eq \u2261 `RC x y \u2192 x \u2261 y\n  eq=>\u2261 zero zero p = refl\n  eq=>\u2261 zero (suc y) ()\n  eq=>\u2261 (suc x) zero ()\n  eq=>\u2261 (suc x) (suc y) p rewrite eq=>\u2261 x y p = refl\n\n  insert-Sorted : \u2200 {n l}{V : Vec (Fin n) l}(x : Fin n) \u2192 Sorted {Fin n} `RC V \u2192 Sorted {Fin n} `RC (insert `RC x V)\n  insert-Sorted x [] = sing x\n  insert-Sorted x (sing x\u2081) with `RC x x\u2081 | dbl-lt {XC = `RC} x x\u2081 {[]} | eq=>\u2261 x x\u2081 | flip-RC x x\u2081\n  insert-Sorted x (sing x\u2081) | lt | b | _ | _ = b refl (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | eq | _ | p | _ rewrite p refl = dbl-eq x\u2081 (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | gt | b | _ | l = dbl-lt x\u2081 x l (sing x)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) with `RC x y | dbl-lt {XC = `RC} x y {y' \u2237 xs} | eq=>\u2261 x y | flip-RC x y\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | lt | b | p | l\u2081 = b refl (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | eq | b | p | l\u2081 rewrite p refl = dbl-eq y (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) | gt | b | p | l\u2081 with `RC x y' | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | lt | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | eq | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | gt | xs' = dbl-lt y y' prf xs'\n  insert-Sorted x (dbl-eq y {xs} xs\u2081) with `RC x y | inspect (`RC x) y | dbl-lt {XC = `RC} x y {y \u2237 xs} | eq=>\u2261 x y | flip-RC x y | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-eq y xs\u2081) | lt | _ | b | p | l | _ = b refl (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | eq | _ | b | p | l | _ rewrite p refl = dbl-eq y (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | gt | [ prf ] | b | p | l | ss  rewrite prf = dbl-eq y ss \n\n  sort-Sorted : \u2200 {n l}(V : Vec (Fin n) l) \u2192 Sorted `RC (`sort `RC V)\n  sort-Sorted [] = []\n  sort-Sorted (x \u2237 V) = insert-Sorted x (sort-Sorted V)\n\n  RC-refl : \u2200 {i}(x : Fin i) \u2192 `RC x x \u2261 eq\n  RC-refl zero = refl\n  RC-refl (suc x) = RC-refl x\n\n  STail : \u2200 {X l}{XC : Cmp X}{xs : Vec X (suc l)} \u2192 Sorted XC xs \u2192 Sorted XC (tail xs)\n  STail (sing x) = []\n  STail (dbl-lt x y x\u2081 T) = T\n  STail (dbl-eq x T) = T\n\n  module sproof {X}(XC : Cmp X)(XC-refl : \u2200 x \u2192 XC x x \u2261 eq)\n     (eq\u2261 : \u2200 x y \u2192 XC x y \u2261 eq \u2192 x \u2261 y)\n     (lt-trans : \u2200 x y z \u2192 XC x y \u2261 lt \u2192 XC y z \u2261 lt \u2192 XC x z \u2261 lt)\n     (XC-flip : \u2200 x y \u2192 opposite (XC x y) \u2261 XC y x)\n     where\n\n    open import Data.Sum\n\n    _\u2264X_ : X \u2192 X \u2192 Type\n    x \u2264X y = XC x y \u2261 lt \u228e XC x y \u2261 eq\n\n    \u2264X-trans : \u2200 {x y z} \u2192 x \u2264X y \u2192 y \u2264X z \u2192 x \u2264X z\n    \u2264X-trans (inj\u2081 x\u2081) (inj\u2081 x\u2082) = inj\u2081 (lt-trans _ _ _ x\u2081 x\u2082)\n    \u2264X-trans {_}{y}{z}(inj\u2081 x\u2081) (inj\u2082 y\u2081) rewrite eq\u2261 y z y\u2081 = inj\u2081 x\u2081\n    \u2264X-trans {x}{y} (inj\u2082 y\u2081) y\u2264z rewrite eq\u2261 x y y\u2081 = y\u2264z\n\n    h\u2264t : \u2200 {n}{T : `Tree X (2 + n)} \u2192 Sorted XC T \u2192 head T \u2264X head (tail T)\n    h\u2264t (dbl-lt x y x\u2081 ST) = inj\u2081 (! x\u2081)\n    h\u2264t (dbl-eq x ST) rewrite XC-refl x = inj\u2082 refl\n\n    head-p : \u2200 {n}{T : `Tree X (suc n)} i \u2192 Sorted XC T \u2192 head T \u2264X `toFun T i\n    head-p {T = T} zero ST rewrite XC-refl (head T) = inj\u2082 refl\n    head-p {zero} (suc ()) ST\n    head-p {suc n} (suc i) ST = \u2264X-trans (h\u2264t ST) (head-p i (STail ST))\n\n    toFun-p : \u2200 {n}{T : `Tree X n}{i j : Fin n} \u2192 i \u2264F j \u2192 Sorted XC T \u2192 `toFun T i \u2264X `toFun T j\n    toFun-p {j = j} z\u2264i ST = head-p j ST\n    toFun-p (s\u2264s i\u2264Fj) ST = toFun-p i\u2264Fj (STail ST)\n\n    sort-proof : \u2200 {i}{T : `Tree X i} \u2192 Sorted XC T \u2192 Is-Mono `RC XC (`toFun T)\n    sort-proof {T = T} T\u2081 zero zero rewrite XC-refl (head T) = _\n    sort-proof T\u2081 zero (suc y) with toFun-p (z\u2264i {i = suc y}) T\u2081\n    sort-proof T zero (suc y) | inj\u2081 x rewrite x = _\n    sort-proof T zero (suc y) | inj\u2082 y\u2081 rewrite y\u2081 = _\n    sort-proof {T = T} T\u2081 (suc x) zero with toFun-p (z\u2264i {i = suc x}) T\u2081 | XC-flip (head T) (`toFun (tail T) x)\n    sort-proof T (suc x) zero | inj\u2081 x\u2081 | l rewrite x\u2081 | ! l = _\n    sort-proof T (suc x) zero | inj\u2082 y | l rewrite y | ! l = _\n    sort-proof T\u2081 (suc x) (suc y) = sort-proof (STail T\u2081) x y\n\n  lt-trans-RC : \u2200 {i} (x y z : Fin i) \u2192 `RC x y \u2261 lt \u2192 `RC y z \u2261 lt \u2192 `RC x z \u2261 lt\n  lt-trans-RC zero zero zero x<y y<z = y<z\n  lt-trans-RC zero zero (suc z) x<y y<z = refl\n  lt-trans-RC zero (suc y) zero x<y ()\n  lt-trans-RC zero (suc y) (suc z) x<y y<z = refl\n  lt-trans-RC (suc x) zero zero x<y y<z = x<y\n  lt-trans-RC (suc x) zero (suc z) () y<z\n  lt-trans-RC (suc x) (suc y) zero x<y y<z = y<z\n  lt-trans-RC (suc x) (suc y) (suc z) x<y y<z = lt-trans-RC x y z x<y y<z\n\n  `sort-mono : \u2200 {i}(T : `Tree (`Rep i) i) \u2192 Is-Mono `RC `RC (`toFun (`sort `RC T))\n  `sort-mono T x y = sproof.sort-proof `RC RC-refl (\u03bb x\u2081 y\u2081 x\u2082 \u2192 eq=>\u2261 x\u2081 y\u2081 (! x\u2082)) lt-trans-RC flip-RC (sort-Sorted T) x y\n\n  move-to-RC : \u2200 {n}{x y : Fin n} \u2192 x \u2264F y \u2192 `RC x y \u2261 lt \u228e `RC x y \u2261 eq\n  move-to-RC {y = zero} z\u2264i = inj\u2082 refl\n  move-to-RC {y = suc y} z\u2264i = inj\u2081 refl\n  move-to-RC (s\u2264s x\u2264Fy) = move-to-RC x\u2264Fy\n\n  move-from-RC : \u2200 {n}(x y : Fin n) \u2192 lt \u2261 `RC x y \u228e eq \u2261 `RC x y \u2192 x \u2264F y\n  move-from-RC zero zero prf = z\u2264i\n  move-from-RC zero (suc y) prf = z\u2264i\n  move-from-RC (suc x) zero (inj\u2081 ())\n  move-from-RC (suc x) zero (inj\u2082 ())\n  move-from-RC (suc x) (suc y) prf = s\u2264s (move-from-RC x y prf)\n\n  module toNatRC n (f : Endo (Fin (suc n)))(f-inj : Is-Inj f)(f-mono : Is-Mono `RC `RC f) where\n    proper-mono : \u2200 {x y} \u2192 x \u2264F y \u2192 f x \u2264F f y\n    proper-mono {x} {y} x\u2264Fy with `RC x y | `RC (f x) (f y) | move-to-RC x\u2264Fy | f-mono x y | move-from-RC (f x) (f y)\n    proper-mono x\u2264Fy | .lt | lt | inj\u2081 refl | r4 | r5 = r5 (inj\u2081 refl)\n    proper-mono x\u2264Fy | .lt | eq | inj\u2081 refl | r4 | r5 = r5 (inj\u2082 refl)\n    proper-mono x\u2264Fy | .lt | gt | inj\u2081 refl | () | r5\n    proper-mono x\u2264Fy | .eq | lt | inj\u2082 refl | () | r5\n    proper-mono x\u2264Fy | .eq | eq | inj\u2082 refl | r4 | r5 = r5 (inj\u2082 refl)\n    proper-mono x\u2264Fy | .eq | gt | inj\u2082 refl | () | r5\n    open From-mono-inj f f-inj proper-mono public\n\n  fin-view : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 Fin-View i\n  fin-view {zero} zero = max\n  fin-view {zero} (suc ())\n  fin-view {suc n} zero = inject _\n  fin-view {suc n} (suc i) with fin-view i\n  fin-view {suc n} (suc .(\u2115\u25b9Fin n)) | max = max\n  fin-view {suc n} (suc .(inject\u2081 i)) | inject i = inject _\n\n  absurd : {X : Type} \u2192 .\u22a5 \u2192 X\n  absurd ()\n\n  drop\u2081 : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 .(i \u2262 \u2115\u25b9Fin n) \u2192 Fin n\n  drop\u2081 i neq with fin-view i\n  drop\u2081 {n} .(\u2115\u25b9Fin n) neq | max = absurd (neq refl)\n  drop\u2081 .(inject\u2081 i) neq | inject i = i\n\n  drop\u2081\u2192inject\u2081 : \u2200 {n}(i : Fin (suc n))(j : Fin n).(p : i \u2262 \u2115\u25b9Fin n) \u2192 drop\u2081 i p \u2261 j \u2192 i \u2261 inject\u2081 j\n  drop\u2081\u2192inject\u2081 i j p q with fin-view i\n  drop\u2081\u2192inject\u2081 {n} .(\u2115\u25b9Fin n) j p q | max = absurd (p refl)\n  drop\u2081\u2192inject\u2081 .(inject\u2081 i) j p q | inject i = ap inject\u2081 q\n\n\n  `mono-inj\u2192id : \u2200{i}(f : Endo (`Rep i)) \u2192 Is-Inj f \u2192 Is-Mono `RC `RC f \u2192 f \u2257 id\n  `mono-inj\u2192id {zero}  = \u03bb f x x\u2081 ()\n  `mono-inj\u2192id {suc i} = toNatRC.f\u2257id i\n\n\n  interface : Interface\n  interface = record \n    { Ix            = `Ix\n    ; Rep           = `Rep\n    ; Syn           = `Syn\n    ; Tree          = `Tree\n    ; fromFun       = `fromFun\n    ; toFun         = `toFun\n    ; toFun\u2218fromFun = `toFun\u2218fromFun\n    ; evalArg       = `evalArg\n    ; evalTree      = `evalTree\n    ; eval-proof    = `eval-proof\n    ; inv           = `inv\n    ; inv-proof     = `inv-proof\n    ; RC            = `RC\n    ; sort          = `sort\n    ; sort-syn      = `sort-syn\n    ; sort-proof    = `sort-proof\n    ; sort-mono     = `sort-mono\n    ; mono-inj\u2192id   = `mono-inj\u2192id\n    }\n\n  count : \u2200 {n} \u2192 (Fin n \u2192 \u2115) \u2192 \u2115\n  count {n} f = sum (tabulate f)\n\n  count-ext : \u2200 {n} \u2192 (f g : Fin n \u2192 \u2115) \u2192 f \u2257 g \u2192 count f \u2261 count g\n  count-ext {zero} f g f\u2257g = refl\n  count-ext {suc n} f g f\u2257g rewrite f\u2257g zero | count-ext (f \u2218 suc) (g \u2218 suc) (f\u2257g \u2218 suc) = refl\n\n  #\u27e8_\u27e9 : \u2200 {n} \u2192 (Fin n \u2192 \ud835\udfda) \u2192 \u2115\n  #\u27e8 f \u27e9 = count (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  #-ext : \u2200 {n} \u2192 (f g : Fin n \u2192 \ud835\udfda) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n  #-ext f g f\u2257g = count-ext (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g) (ap \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  com-assoc : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n  com-assoc x y z rewrite \n    ! \u2115\u00b0.+-assoc x y z |\n    \u2115\u00b0.+-comm x y      |\n    \u2115\u00b0.+-assoc y x z   = refl\n    \n  syn-pres : \u2200 {n}(f : Fin n \u2192 \u2115)(S : `Syn n)\n           \u2192 count f \u2261 count (f \u2218 `evalArg S)\n  syn-pres f `id = refl\n  syn-pres f `swap = com-assoc (f zero) (f (suc zero)) (count (\u03bb i \u2192 f (suc (suc i))))\n  syn-pres f (`tail S) rewrite syn-pres (f \u2218 suc) S = refl\n  syn-pres f (S `\u2218 S\u2081) rewrite syn-pres f S = syn-pres (f \u2218 `evalArg S) S\u2081\n\n  count-perm : \u2200 {n}(f : Fin n \u2192 \u2115)(p : Endo (Fin n)) \u2192 Is-Inj p\n         \u2192 count f \u2261 count (f \u2218 p)\n  count-perm f p p-inj = trans (syn-pres f (sort-bij p)) (count-ext _ _ f\u2218eval\u2257f\u2218p)\n   where\n     open abs interface\n     f\u2218eval\u2257f\u2218p : f \u2218 `evalArg (sort-bij p) \u2257 f \u2218 p\n     f\u2218eval\u2257f\u2218p x rewrite thm p p-inj x = refl\n\n\n  #-perm : \u2200 {n}(f : Fin n \u2192 \ud835\udfda)(p : Endo (Fin n)) \u2192 Is-Inj p\n         \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 p \u27e9\n  #-perm f p p-inj = count-perm (\ud835\udfda\u25b9\u2115 \u2218 f) p p-inj\n\n  test : `Syn 8\n  test = abs.sort-bij interface (\u03bb x \u2192 `evalArg (`tail `swap) x)\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6538c1e3a868c0dc095139fc2243a8683b77c995","subject":"Remove dead code of Data.NatBag.Properties Checkpoint: Recast property proofs using set theoretic methods (#55)","message":"Remove dead code of Data.NatBag.Properties\nCheckpoint: Recast property proofs using set theoretic methods (#55)\n\nOld-commit-hash: 04b32e5d92152cdab089a25c0d8f9f4c39279fb0\n","repos":"inc-lc\/ilc-agda","old_file":"Data\/NatBag\/Properties.agda","new_file":"Data\/NatBag\/Properties.agda","new_contents":"module Data.NatBag.Properties where\n\nimport Data.Nat as \u2115\nopen import Relation.Binary.PropositionalEquality\nopen import Data.NatBag\nopen import Data.Integer\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\n\n-- This import is too slow.\n-- It causes Agda 2.3.2 to use so much memory that cai's\n-- computer with 4GB RAM begins to thresh.\n--\n-- open import Data.Integer.Properties using (n\u2296n\u22610)\nn\u2296n\u22610 : \u2200 n \u2192 n \u2296 n \u2261 + 0\nn\u2296n\u22610 \u2115.zero    = refl\nn\u2296n\u22610 (\u2115.suc n) = n\u2296n\u22610 n\n\n\n----------------\n-- Statements --\n----------------\n\n\nb\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 empty\n\n\u2205++b=b : \u2200 {b : Bag} \u2192 empty ++ b \u2261 b\n\nb\\\\\u2205=b : \u2200 {b : Bag} \u2192 b \\\\ empty \u2261 b\n\n\u2205\\\\b=-b : \u2200 {b : Bag} \u2192 empty \\\\ b \u2261 map\u2082 -_ b\n\nb++[d\\\\b]=d : \u2200 {b d : Bag} \u2192 b ++ (d \\\\ b) \u2261 d\n\n\n------------\n-- Proofs --\n------------\n\ni-i=0 : \u2200 {i : \u2124} \u2192 (i - i) \u2261 (+ 0)\ni-i=0 {+ \u2115.zero} = refl\ni-i=0 {+ \u2115.suc n} = n\u2296n\u22610 n\ni-i=0 { -[1+ n ]} = n\u2296n\u22610 n\n\n-- Debug tool\n-- Lets you try out inhabitance of any type anywhere\nabsurd! : \u2200 {B C : Set} \u2192 0 \u2261 1 \u2192 B \u2192 {x : B} \u2192 C\nabsurd! ()\n\n-- Specialized absurdity needed to type check.\n-- \u03bb () hasn't enough information sometimes.\nabsurd : Nonzero (+ 0) \u2192 \u2200 {A : Set} \u2192 A\nabsurd ()\n\n-- Here to please the termination checker.\nneb\\\\neb=\u2205 : \u2200 {neb : NonemptyBag} \u2192 zipNonempty _-_ neb neb \u2261 empty\nneb\\\\neb=\u2205 {singleton i i\u22600} with nonzero? (i - i)\n... | inj\u2081 _ = refl\n... | inj\u2082 0\u22600 rewrite i-i=0 {i} = absurd 0\u22600\nneb\\\\neb=\u2205 {i \u2237 neb} with nonzero? (i - i)\n... | inj\u2081 _ rewrite neb\\\\neb=\u2205 {neb} = refl\n... | inj\u2082 0\u22600 rewrite neb\\\\neb=\u2205 {neb} | i-i=0 {i} = absurd 0\u22600\n\nb\\\\b=\u2205 {inj\u2081 \u2205} = refl\nb\\\\b=\u2205 {inj\u2082 neb} = neb\\\\neb=\u2205 {neb}\n\n\u2205++b=b {b} = {!!}\n\nb\\\\\u2205=b {b} = {!!}\n\n\u2205\\\\b=-b {b} = {!!}\n\nnegate : \u2200 {i} \u2192 Nonzero i \u2192 Nonzero (- i)\nnegate (negative n) = positive n\nnegate (positive n) = negative n\n\nnegate\u2032 : \u2200 {i} \u2192 (i\u22600 : Nonzero i) \u2192 Nonzero (+ 0 - i)\nnegate\u2032 { -[1+ n ]} (negative .n) = positive n\nnegate\u2032 {+ .(\u2115.suc n)} (positive n) = negative n\n\n0-i=-i : \u2200 {i} \u2192 + 0 - i \u2261 - i\n0-i=-i { -[1+ n ]} = refl -- cases are split, for arguments to\n0-i=-i {+ \u2115.zero}  = refl -- refl are different.\n0-i=-i {+ \u2115.suc n} = refl\n\nrewrite-singleton :\n  \u2200 (i : \u2124) (0-i\u22600 : Nonzero (+ 0 - i)) ( -i\u22600 : Nonzero (- i)) \u2192\n    singleton (+ 0 - i) 0-i\u22600 \u2261 singleton (- i) -i\u22600\nrewrite-singleton (+ \u2115.zero) () ()\nrewrite-singleton (+ \u2115.suc n) (negative .n) (negative .n) = refl\nrewrite-singleton ( -[1+ n ]) (positive .n) (positive .n) = refl\n\nnegateSingleton : \u2200 {i i\u22600} \u2192\n  mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\n  inj\u2082 (singleton (- i) (negate i\u22600))\n\n-- Fun fact:\n-- Pattern-match on implicit parameters in the first two\n-- cases results in rejection by Agda.\nnegateSingleton {i} {i\u22600} with nonzero? i | nonzero? (+ 0 - i)\nnegateSingleton | inj\u2082 (negative n) | inj\u2081 ()\nnegateSingleton | inj\u2082 (positive n) | inj\u2081 ()\nnegateSingleton {_} {i\u22600} | inj\u2081 i=0 | _ rewrite i=0 = absurd i\u22600\nnegateSingleton {i} {i\u22600} | inj\u2082 _ | inj\u2082 0-i\u22600 =\n  begin -- Reasoning done in 1 step. Included for clarity only.\n    inj\u2082 (singleton (+ 0 + - i) 0-i\u22600)\n  \u2261\u27e8 cong inj\u2082 (rewrite-singleton i 0-i\u22600 (negate i\u22600)) \u27e9\n    inj\u2082 (singleton (- i) (negate i\u22600))\n  \u220e  where open \u2261-Reasoning\n\nabsurd[i-i\u22600] : \u2200 {i} \u2192 Nonzero (i - i) \u2192 \u2200 {A : Set} \u2192 A\nabsurd[i-i\u22600] {+ \u2115.zero} = absurd\nabsurd[i-i\u22600] {+ \u2115.suc n} = absurd[i-i\u22600] { -[1+ n ]}\nabsurd[i-i\u22600] { -[1+ \u2115.zero ]} = absurd\nabsurd[i-i\u22600] { -[1+ \u2115.suc n ]} = absurd[i-i\u22600] { -[1+ n ]}\n\nannihilate : \u2200 {i i\u22600} \u2192\n  inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600)) \u2261 inj\u2081 \u2205\nannihilate {i} with nonzero? (i - i)\n... | inj\u2081 i-i=0 = \u03bb {i\u22600} \u2192 refl\n... | inj\u2082 i-i\u22600 = absurd[i-i\u22600] {i} i-i\u22600\n\nb++[\u2205\\\\b]=\u2205 : \u2200 {b} \u2192 b ++ (empty \\\\ b) \u2261 empty\nb++[\u2205\\\\b]=\u2205 {inj\u2081 \u2205} = refl\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (singleton i i\u22600)} =\n  begin\n    inj\u2082 (singleton i i\u22600) ++\n      mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\u27e8 cong\u2082 _++_ {x = inj\u2082 (singleton i i\u22600)} refl\n                (negateSingleton {i} {i\u22600}) \u27e9\n    inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600))\n  \u2261\u27e8 annihilate {i} {i\u22600} \u27e9\n    inj\u2081 \u2205\n  \u220e where open \u2261-Reasoning\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (i \u2237 y)} = {!!}\n\nb++[d\\\\b]=d {inj\u2081 \u2205} {d} rewrite b\\\\\u2205=b {d} | \u2205++b=b {d} = refl\nb++[d\\\\b]=d {b} {inj\u2081 \u2205} = b++[\u2205\\\\b]=\u2205 {b}\nb++[d\\\\b]=d {inj\u2082 b} {inj\u2082 d} = {!!}\n\n","old_contents":"module Data.NatBag.Properties where\n\nimport Data.Nat as \u2115\nopen import Relation.Binary.PropositionalEquality\nopen import Data.NatBag\nopen import Data.Integer\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\n\n-- This import is too slow.\n-- It causes Agda 2.3.2 to use so much memory that cai's\n-- computer with 4GB RAM begins to thresh.\n--\n-- open import Data.Integer.Properties using (n\u2296n\u22610)\nn\u2296n\u22610 : \u2200 n \u2192 n \u2296 n \u2261 + 0\nn\u2296n\u22610 \u2115.zero    = refl\nn\u2296n\u22610 (\u2115.suc n) = n\u2296n\u22610 n\n\n\n----------------\n-- Statements --\n----------------\n\n\nb\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 empty\n\n\u2205++b=b : \u2200 {b : Bag} \u2192 empty ++ b \u2261 b\n\nb\\\\\u2205=b : \u2200 {b : Bag} \u2192 b \\\\ empty \u2261 b\n\n\u2205\\\\b=-b : \u2200 {b : Bag} \u2192 empty \\\\ b \u2261 map\u2082 -_ b\n\n-- ++d=\\\\-d : \u2200 {b d : Bag} \u2192 b ++ d \u2261 b \\\\ map\u2082 -_ d\n\nb++[d\\\\b]=d : \u2200 {b d : Bag} \u2192 b ++ (d \\\\ b) \u2261 d\n\n\n------------\n-- Proofs --\n------------\n\ni-i=0 : \u2200 {i : \u2124} \u2192 (i - i) \u2261 (+ 0)\ni-i=0 {+ \u2115.zero} = refl\ni-i=0 {+ \u2115.suc n} = n\u2296n\u22610 n\ni-i=0 { -[1+ n ]} = n\u2296n\u22610 n\n\n-- Debug tool\n-- Lets you try out inhabitance of any type anywhere\nabsurd! : \u2200 {B C : Set} \u2192 0 \u2261 1 \u2192 B \u2192 {x : B} \u2192 C\nabsurd! ()\n\n-- Specialized absurdity needed to type check.\n-- \u03bb () hasn't enough information sometimes.\nabsurd : Nonzero (+ 0) \u2192 \u2200 {A : Set} \u2192 A\nabsurd ()\n\n-- Here to please the termination checker.\nneb\\\\neb=\u2205 : \u2200 {neb : NonemptyBag} \u2192 zipNonempty _-_ neb neb \u2261 empty\nneb\\\\neb=\u2205 {singleton i i\u22600} with nonzero? (i - i)\n... | inj\u2081 _ = refl\n... | inj\u2082 0\u22600 rewrite i-i=0 {i} = absurd 0\u22600\nneb\\\\neb=\u2205 {i \u2237 neb} with nonzero? (i - i)\n... | inj\u2081 _ rewrite neb\\\\neb=\u2205 {neb} = refl\n... | inj\u2082 0\u22600 rewrite neb\\\\neb=\u2205 {neb} | i-i=0 {i} = absurd 0\u22600\n\n{-\n++d=\\\\-d {inj\u2081 \u2205} {inj\u2081 \u2205} = refl\n++d=\\\\-d {inj\u2081 \u2205} {inj\u2082 (i \u2237 y)} = {!!}\n++d=\\\\-d {inj\u2082 y} {d} = {!!}\n++d=\\\\-d {inj\u2081 \u2205} {inj\u2082 (singleton i i\u22600)}\n  rewrite \u2205++b=b {inj\u2082 (singleton i i\u22600)}\n  with nonzero? i | nonzero? (+ 0 - i)\n... | inj\u2082 _ | inj\u2082 0-i\u22600 =\n  begin\n    {!!}\n  \u2261\u27e8 {!!} \u27e9\n    {!inj\u2082 (singleton (- i) ?)!}\n  \u220e where open \u2261-Reasoning\n... | inj\u2081 i=0 | _ rewrite i=0 = absurd i\u22600\n++d=\\\\-d {inj\u2081 \u2205} {inj\u2082 (singleton (+ 0) i\u22600)} | _ | _ = absurd i\u22600\n++d=\\\\-d {inj\u2081 \u2205} {inj\u2082 (singleton (+ (\u2115.suc n)) i\u22600)}\n  | inj\u2082 (positive .n) | inj\u2081 ()\n++d=\\\\-d {inj\u2081 \u2205} {inj\u2082 (singleton -[1+ n ] i\u22600)}\n  | inj\u2082 (negative .n) | inj\u2081 ()\n-}\n\nb\\\\b=\u2205 {inj\u2081 \u2205} = refl\nb\\\\b=\u2205 {inj\u2082 neb} = neb\\\\neb=\u2205 {neb}\n\n\u2205++b=b {b} = {!!}\n\nb\\\\\u2205=b {b} = {!!}\n\n\u2205\\\\b=-b {b} = {!!}\n\nnegate : \u2200 {i} \u2192 Nonzero i \u2192 Nonzero (- i)\nnegate (negative n) = positive n\nnegate (positive n) = negative n\n\nnegate\u2032 : \u2200 {i} \u2192 (i\u22600 : Nonzero i) \u2192 Nonzero (+ 0 - i)\nnegate\u2032 { -[1+ n ]} (negative .n) = positive n\nnegate\u2032 {+ .(\u2115.suc n)} (positive n) = negative n\n\n0-i=-i : \u2200 {i} \u2192 + 0 - i \u2261 - i\n0-i=-i { -[1+ n ]} = refl -- cases are split, for arguments to\n0-i=-i {+ \u2115.zero}  = refl -- refl are different.\n0-i=-i {+ \u2115.suc n} = refl\n\nrewrite-singleton :\n  \u2200 (i : \u2124) (0-i\u22600 : Nonzero (+ 0 - i)) ( -i\u22600 : Nonzero (- i)) \u2192\n    singleton (+ 0 - i) 0-i\u22600 \u2261 singleton (- i) -i\u22600\nrewrite-singleton (+ \u2115.zero) () ()\nrewrite-singleton (+ \u2115.suc n) (negative .n) (negative .n) = refl\nrewrite-singleton ( -[1+ n ]) (positive .n) (positive .n) = refl\n\nnegateSingleton : \u2200 {i i\u22600} \u2192\n  mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\n  inj\u2082 (singleton (- i) (negate i\u22600))\n\n-- Fun fact:\n-- Pattern-match on implicit parameters in the first two\n-- cases results in rejection by Agda.\nnegateSingleton {i} {i\u22600} with nonzero? i | nonzero? (+ 0 - i)\nnegateSingleton | inj\u2082 (negative n) | inj\u2081 ()\nnegateSingleton | inj\u2082 (positive n) | inj\u2081 ()\nnegateSingleton {_} {i\u22600} | inj\u2081 i=0 | _ rewrite i=0 = absurd i\u22600\nnegateSingleton {i} {i\u22600} | inj\u2082 _ | inj\u2082 0-i\u22600 =\n  begin -- Reasoning done in 1 step. Included for clarity only.\n    inj\u2082 (singleton (+ 0 + - i) 0-i\u22600)\n  \u2261\u27e8 cong inj\u2082 (rewrite-singleton i 0-i\u22600 (negate i\u22600)) \u27e9\n    inj\u2082 (singleton (- i) (negate i\u22600))\n  \u220e  where open \u2261-Reasoning\n\nabsurd[i-i\u22600] : \u2200 {i} \u2192 Nonzero (i - i) \u2192 \u2200 {A : Set} \u2192 A\nabsurd[i-i\u22600] {+ \u2115.zero} = absurd\nabsurd[i-i\u22600] {+ \u2115.suc n} = absurd[i-i\u22600] { -[1+ n ]}\nabsurd[i-i\u22600] { -[1+ \u2115.zero ]} = absurd\nabsurd[i-i\u22600] { -[1+ \u2115.suc n ]} = absurd[i-i\u22600] { -[1+ n ]}\n\nannihilate : \u2200 {i i\u22600} \u2192\n  inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600)) \u2261 inj\u2081 \u2205\nannihilate {i} with nonzero? (i - i)\n... | inj\u2081 i-i=0 = \u03bb {i\u22600} \u2192 refl\n... | inj\u2082 i-i\u22600 = absurd[i-i\u22600] {i} i-i\u22600\n\n{-\nleft-is-not-right : \u2200 {A B : Set} {a : A} {b : B} \u2192\n                    inj\u2081 a \u2261 inj\u2082 b \u2192 \u2200 {X : Set} \u2192 X\nleft-is-not-right = \u03bb {A} {B} {a} {b} \u2192 \u03bb ()\n\nnever-both : \u2200 {A B : Set} {sum : A \u228e B} {a b} \u2192\n               sum \u2261 inj\u2081 a \u2192 sum \u2261 inj\u2082 b \u2192 \u2200 {X : Set} \u2192 X\nnever-both s=a s=b = left-is-not-right (trans (sym s=a) s=b)\n\nempty-bag? : \u2200 (b : Bag) \u2192 (b \u2261 inj\u2081 \u2205) \u228e \u03a3 NonemptyBag (\u03bb neb \u2192 b \u2261 inj\u2082 neb)\nempty-bag? (inj\u2081 \u2205) = inj\u2081 refl\nempty-bag? (inj\u2082 neb) = inj\u2082 (neb , refl)\n-}\n\nb++[\u2205\\\\b]=\u2205 : \u2200 {b} \u2192 b ++ (empty \\\\ b) \u2261 empty\nb++[\u2205\\\\b]=\u2205 {inj\u2081 \u2205} = refl\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (singleton i i\u22600)} =\n  begin\n    inj\u2082 (singleton i i\u22600) ++\n      mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\u27e8 cong\u2082 _++_ {x = inj\u2082 (singleton i i\u22600)} refl\n                (negateSingleton {i} {i\u22600}) \u27e9\n    inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600))\n  \u2261\u27e8 annihilate {i} {i\u22600} \u27e9\n    inj\u2081 \u2205\n  \u220e where open \u2261-Reasoning\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (i \u2237 y)} = {!!}\n\nb++[d\\\\b]=d {inj\u2081 \u2205} {d} rewrite b\\\\\u2205=b {d} | \u2205++b=b {d} = refl\nb++[d\\\\b]=d {b} {inj\u2081 \u2205} = b++[\u2205\\\\b]=\u2205 {b}\nb++[d\\\\b]=d {inj\u2082 b} {inj\u2082 d} = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39ff610ece1a0f53d91047ee69dba0f42e646718","subject":"simplify","message":"simplify\n","repos":"crypto-agda\/crypto-agda","old_file":"dist-props.agda","new_file":"dist-props.agda","new_contents":"module dist-props where\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product\n\nopen import Function\n\nopen import Search.Type\nopen import Search.Sum\n\nopen import Search.Searchable.Sum\nopen import Search.Searchable.Product\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule GameFlipping (R : Set)(sum : Sum R)(sum-ind : SumInd sum)(X Y : R \u2192 Bool) where\n\n  R' = Bool \u00d7 R\n  sum' : Sum R'\n  sum' = searchBit _+_ \u00d7-sum sum\n\n  open FromSum sum' renaming (count to count')\n  open FromSumInd sum-ind\n\n  _==\u1d2e_ : Bool \u2192 Bool \u2192 Bool\n  true  ==\u1d2e y = y\n  false ==\u1d2e y = not y\n\n  G : R' \u2192 Bool\n  G (b , r) = b == (if b then X r else Y r)\n\n  1\/2 : R' \u2192 Bool\n  1\/2 = proj\u2081\n\n  -- TODO use the library\n  lemma : \u2200 X \u2192 sum (const 1) \u2261 count (not \u2218 X) + count X\n  lemma X = sum-ind P (\u03bb {a}{b} \u2192 part1 {a}{b}) part2\n    where\n      # = FromSum.count\n\n      P = \u03bb s \u2192 s (const 1) \u2261 # s (not \u2218 X) + # s X\n\n      part1 : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f)\n      part1 {s\u2080} {s\u2081} Ps\u2080 Ps\u2081 rewrite Ps\u2080 | Ps\u2081 = +-interchange (# s\u2080 (not \u2218 X)) (# s\u2080 X) (# s\u2081 (not \u2218 X)) (# s\u2081 X)\n\n      part2 : \u2200 x \u2192 P (\u03bb f \u2192 f x)\n      part2 x with X x\n      part2 x | true  = refl\n      part2 x | false = refl\n\n\n  thm : dist (count' G) (count' 1\/2) \u2261 dist (count X) (count Y)\n  thm = dist (count' G) (count' 1\/2)\n      \u2261\u27e8 cong (dist (count' G)) helper \u27e9\n        dist (count' G) (count (not \u2218 Y) + count Y)\n      \u2261\u27e8 refl \u27e9 -- count' definition\n        dist (count (_==_ false \u2218 Y) + count (_==_ true \u2218 X)) (count (not \u2218 Y) + count Y)\n      \u2261\u27e8 {!refl!} \u27e9 -- count' definition\n        dist (count (not \u2218 Y) + count X) (count (not \u2218 Y) + count Y)\n      \u2261\u27e8 dist-x+ (count (not \u2218 Y)) (count X) (count Y) \u27e9\n        dist (count X) (count Y)\n      \u220e\n    where\n      open \u2261-Reasoning\n\n      helper = count' 1\/2\n             \u2261\u27e8 refl \u27e9\n               sum (const 0) + sum (const 1)\n             \u2261\u27e8 cong (\u03bb p \u2192 p + sum (const 1)) sum-zero \u27e9\n               sum (const 1)\n             \u2261\u27e8 lemma Y \u27e9\n               count (not \u2218 Y) + count Y\n             \u220e\n","old_contents":"module dist-props where\n\nopen import Data.Bool\nopen import Data.Nat.NP\nopen import Data.Product\n\nopen import Function\n\nopen import Search.Type\nopen import Search.Sum\n\nopen import Search.Searchable.Sum\nopen import Search.Searchable.Product\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule GameFlipping (R : Set)(sum : Sum R)(sum-ind : SumInd sum)(X Y : R \u2192 Bool) where\n\n  R' = Bool \u00d7 R\n  sum' : Sum R'\n  sum' = searchBit _+_ \u00d7-sum sum\n\n  open FromSum sum' renaming (count to count')\n  open FromSumInd sum-ind\n\n  _==\u1d2e_ : Bool \u2192 Bool \u2192 Bool\n  true  ==\u1d2e y = y\n  false ==\u1d2e y = not y\n\n  G : R' \u2192 Bool\n  G (b , r) = b ==\u1d2e (if b then X r else Y r)\n\n  1\/2 : R' \u2192 Bool\n  1\/2 = proj\u2081\n\n  lemma : \u2200 X \u2192 sum (const 1) \u2261 count (not \u2218 X) + count X\n  lemma X = sum-ind P (\u03bb {a}{b} \u2192 part1 {a}{b}) part2\n    where\n      # = FromSum.count\n      \n      P = \u03bb s \u2192 s (const 1) \u2261 # s (not \u2218 X) + # s X\n      \n      part1 : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f)\n      part1 {s\u2080} {s\u2081} Ps\u2080 Ps\u2081 rewrite Ps\u2080 | Ps\u2081 = +-interchange (# s\u2080 (not \u2218 X)) (# s\u2080 X) (# s\u2081 (not \u2218 X)) (# s\u2081 X)  \n\n      part2 : \u2200 x \u2192 P (\u03bb f \u2192 f x)\n      part2 x with X x\n      part2 x | true  = refl\n      part2 x | false = refl\n      \n\n  thm : dist (count' G) (count' 1\/2) \u2261 dist (count X) (count Y)\n  thm = dist (count' G) (count' 1\/2)\n      \u2261\u27e8 refl \u27e9\n        dist (count (not \u2218 Y) + count X) (sum (const 0) + sum (const 1))\n      \u2261\u27e8 cong (\u03bb p \u2192 dist (count (not \u2218 Y) + count X) (p + sum (const 1))) sum-zero \u27e9\n        dist (count (not \u2218 Y) + count X) (sum (const 1))\n      \u2261\u27e8 cong (dist (count (not \u2218 Y) + count X)) (lemma Y) \u27e9\n        dist (count (not \u2218 Y) + count X) (count (not \u2218 Y) + count Y)\n      \u2261\u27e8 dist-x+ (count (not \u2218 Y)) (count X) (count Y) \u27e9\n        dist (count X) (count Y)\n      \u220e\n    where\n      open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ece000bf8e7a13f345d059fdfc157ac01316ae5b","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 87092ca63c87c904873052fce7315bf4\n\ndarcs-hash:20110301055010-3bd4e-486c9856c3c8f961ed8eb13de29c7833577f9bc4.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Induction\/Acc\/WellFoundedInduction.agda","new_file":"src\/FOTC\/Data\/Nat\/Induction\/Acc\/WellFoundedInduction.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Induction.Acc.WellFoundedInduction where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The relation LT is well-founded.\nmodule WF-LT\u2081\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; Sx<Sy\u2192x<y    : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n  ; <-trans      : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n  ; x\u2261y\u2192y<z\u2192x<z  : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n  ; x<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\n  )\n  where\n\n  wf-LT : WellFounded LT\n  wf-LT Nn = acc Nn (helper Nn)\n    where\n      -- N.B. The helper function is the same that the function used by\n      -- FOTC.Data.Nat.Induction.NonAcc.WellFoundedInductionATP.\n      helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LT m n \u2192 Acc LT m\n      helper zN     Nm m<0  = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\n      helper (sN _) zN 0<Sn = acc zN (\u03bb Nm' m'<0 \u2192 \u22a5-elim $ x<0\u2192\u22a5 Nm' m'<0)\n\n      helper (sN {n} Nn) (sN {m} Nm) Sm<Sn = acc (sN Nm)\n        (\u03bb {m'} Nm' m'<Sm \u2192\n          let m<n : LT m n\n              m<n = Sx<Sy\u2192x<y Sm<Sn\n\n              m'<n : LT m' n\n              m'<n = [ (\u03bb m'<m \u2192 <-trans Nm' Nm Nn m'<m m<n)\n                     , (\u03bb m'\u2261m \u2192 x\u2261y\u2192y<z\u2192x<z m'\u2261m m<n)\n                     ] (x<Sy\u2192x<y\u2228x\u2261y Nm' Nm m'<Sm)\n\n          in  helper Nn Nm' m'<n\n        )\n\n------------------------------------------------------------------------------\n-- A different proof that the relation LT is well-founded.\nmodule WF-LT\u2082\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; x\u2264y\u2192x<y\u2228x\u2261y  : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\n  ; x<Sy\u2192x\u2264y     : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\n  )\n  where\n\n  wf-LT : WellFounded LT\n  wf-LT zN      = acc zN (\u03bb Nm m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n  wf-LT (sN Nn) = acc (sN Nn)\n                      (\u03bb Nm m<Sn \u2192 helper Nm Nn (wf-LT Nn)\n                                          (x<Sy\u2192x\u2264y Nm Nn m<Sn))\n    where\n      helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 Acc LT m \u2192 LE n m \u2192 Acc LT n\n      helper {n} {m} Nn Nm (acc _ h) n\u2264m =\n        [ (\u03bb n<m \u2192 h Nn n<m)\n        , (\u03bb n\u2261m \u2192 helper\u2081 (sym n\u2261m) (acc Nm h))\n        ] (x\u2264y\u2192x<y\u2228x\u2261y Nn Nm n\u2264m)\n        where\n          helper\u2081 : \u2200 {a b} \u2192 a \u2261 b \u2192 Acc LT a \u2192 Acc LT b\n          helper\u2081 refl acc-a = acc-a\n\n------------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers.\nmodule WFInd-LT\n  ( wf-LT : WellFounded LT )\n  where\n\n  wfInd-LT : (P : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 LT m n \u2192 P m) \u2192 P n) \u2192\n             \u2200 {n} \u2192 N n \u2192 P n\n  wfInd-LT P = WellFoundedInduction wf-LT\n","old_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Induction.Acc.WellFoundedInduction where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The relation LT is well-founded.\nmodule WF-LT\u2081\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; Sx<Sy\u2192x<y    : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n  ; <-trans      : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n  ; x\u2261y\u2192y<z\u2192x<z  : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n  ; x<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\n  )\n  where\n\n  wf-LT : WellFounded LT\n  wf-LT Nn = acc Nn (helper Nn)\n    where\n      -- N.B. The helper function is the same that the function used by\n      -- FOTC.Data.Nat.Induction.NonAcc.WellFoundedInductionATP.\n      helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LT m n \u2192 Acc LT m\n      helper zN     Nm m<0  = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\n      helper (sN _) zN 0<Sn = acc zN (\u03bb Nm' m'<0 \u2192 \u22a5-elim $ x<0\u2192\u22a5 Nm' m'<0)\n\n      helper (sN {n} Nn) (sN {m} Nm) Sm<Sn = acc (sN Nm)\n        (\u03bb {m'} Nm' m'<Sm \u2192\n          let m<n : LT m n\n              m<n = Sx<Sy\u2192x<y Sm<Sn\n\n              m'<n : LT m' n\n              m'<n = [ (\u03bb m'<m \u2192 <-trans Nm' Nm Nn m'<m m<n)\n                     , (\u03bb m'\u2261m \u2192 x\u2261y\u2192y<z\u2192x<z m'\u2261m m<n)\n                     ] (x<Sy\u2192x<y\u2228x\u2261y Nm' Nm m'<Sm)\n\n          in  helper Nn Nm' m'<n\n        )\n\n------------------------------------------------------------------------------\n-- A different proof that the relation LT is well-founded.\nmodule WF-LT\u2082\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; x\u2264y\u2192x<y\u2228x\u2261y  : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\n  ; x<Sy\u2192x\u2264y     : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\n  )\n  where\n\n  wf-LT : WellFounded LT\n  wf-LT zN      = acc zN (\u03bb Nm m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n  wf-LT (sN Nn) = acc (sN Nn)\n                      (\u03bb Nm m<Sn \u2192 helper Nm Nn (wf-LT Nn)\n                                          (x<Sy\u2192x\u2264y Nm Nn m<Sn))\n    where\n      helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 Acc LT m \u2192 LE n m \u2192 Acc LT n\n      helper {n} {m} Nn Nm (acc _ h) n\u2264m =\n        [ (\u03bb n<m \u2192 h Nn n<m)\n        , (\u03bb n\u2261m \u2192 helper\u2081 (sym n\u2261m) (acc Nm h))\n        ] (x\u2264y\u2192x<y\u2228x\u2261y Nn Nm n\u2264m)\n        where\n          helper\u2081 : \u2200 {a b} \u2192 a \u2261 b \u2192 Acc LT a \u2192 Acc LT b\n          helper\u2081 refl acc-a = acc-a\n\n------------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers.\nmodule WFInd-LT\n  ( wf-LT : WellFounded LT )\n  where\n\n  wfInd-LT : (P : D \u2192 Set) \u2192\n             (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 LT n m \u2192 P n) \u2192 P m) \u2192\n             \u2200 {n} \u2192 N n \u2192 P n\n  wfInd-LT P = WellFoundedInduction wf-LT\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d24f2d1901ad576257dcea4bef46b8f0a57cfa06","subject":"Cleaning.","message":"Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda","new_contents":"","old_contents":"------------------------------------------------------------------------------\n-- Testing normalisation on arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Inequalities.ArithmeticPropertiesSL where\n\nmodule NonPrimitives where\n\n  open import Data.Nat\n  open import Relation.Binary.PropositionalEquality\n\n  101\u2261100+11\u223810 : 101 \u2261 100 + 11 \u2238 10\n  101\u2261100+11\u223810 = refl\n\n  91\u2261100+11\u223810\u223810 : 91 \u2261 100 + 11 \u2238 10 \u2238 10\n  91\u2261100+11\u223810\u223810 = refl\n\n  -- Agsy failed using a timeout of 5 minutes and it succeed with a\n  -- timeout of 10 minutes.\n  100+11>100 : 100 + 11 > 100\n  100+11>100 = s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      (s\u2264s\n                                                       (s\u2264s\n                                                        (s\u2264s\n                                                         (s\u2264s\n                                                          (s\u2264s\n                                                           (s\u2264s\n                                                            (s\u2264s\n                                                             (s\u2264s\n                                                              (s\u2264s\n                                                               (s\u2264s\n                                                                (s\u2264s\n                                                                 (s\u2264s\n                                                                  (s\u2264s\n                                                                   (s\u2264s\n                                                                    (s\u2264s\n                                                                     (s\u2264s\n                                                                      (s\u2264s\n                                                                       (s\u2264s\n                                                                        (s\u2264s\n                                                                         (s\u2264s\n                                                                          (s\u2264s\n                                                                           (s\u2264s\n                                                                            (s\u2264s\n                                                                             (s\u2264s\n                                                                              (s\u2264s\n                                                                               (s\u2264s\n                                                                                (s\u2264s\n                                                                                 (s\u2264s\n                                                                                  (s\u2264s\n                                                                                   (s\u2264s\n                                                                                    (s\u2264s\n                                                                                     (s\u2264s\n                                                                                      (s\u2264s\n                                                                                       (s\u2264s\n                                                                                        (s\u2264s\n                                                                                         (s\u2264s\n                                                                                          (s\u2264s\n                                                                                           (s\u2264s\n                                                                                            (s\u2264s\n                                                                                             (s\u2264s\n                                                                                              (s\u2264s\n                                                                                               (s\u2264s\n                                                                                                (s\u2264s\n                                                                                                 (s\u2264s\n                                                                                                  (s\u2264s\n                                                                                                   (s\u2264s\n                                                                                                    (s\u2264s\n                                                                                                     (s\u2264s\n                                                                                                      (s\u2264s\n                                                                                                       (s\u2264s\n                                                                                                        (s\u2264s\n                                                                                                         (s\u2264s\n                                                                                                          (s\u2264s\n                                                                                                           (s\u2264s\n                                                                                                            (s\u2264s\n                                                                                                             (s\u2264s\n                                                                                                              (s\u2264s\n                                                                                                               (s\u2264s\n                                                                                                                (s\u2264s\n                                                                                                                 (s\u2264s\n                                                                                                                  (s\u2264s\n                                                                                                                   z\u2264n))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n\nmodule Primitives where\n\n  open import Data.Bool\n  open import Data.Nat hiding ( _<_ )\n  open import Relation.Binary.PropositionalEquality\n\n  primitive primNatLess : \u2115 \u2192 \u2115 \u2192 Bool\n\n  _<_ : \u2115 \u2192 \u2115 \u2192 Bool\n  _<_ = primNatLess\n\n  100<100+11 : 100 < (100 + 11) \u2261 true\n  100<100+11 = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d9f39e3e94f6a92b25e057a7a737c8c19e8e86d7","subject":"IDesc model: implement lists","message":"IDesc model: implement lists","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- List\n--********************************************\n\ndata ListDConst (l : Level) : Set l where\n  cnil : ListDConst l\n  ccons : ListDConst l\n\nlistDChoice : (l : Level) -> Set l -> ListDConst l -> IDesc l Unit\nlistDChoice x X cnil = const Unit\nlistDChoice x X ccons = sigma X (\\_ -> var Void)\n\nlistD : (l : Level) -> Set l -> IDesc l Unit\nlistD x X = sigma (ListDConst x) (listDChoice x X)\n\nlist : (l : Level) -> Set l -> Set l\nlist x X = IMu x Unit (\\_ -> listD x X) Void\n\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2f05ffbdd1a1f1b61f4e4d2af1e9628cbe81971f","subject":"Initial commit: Agda bags (work in progress)","message":"Initial commit: Agda bags (work in progress)\n\nOld-commit-hash: 59c52d1fe7d503bcff1b4a9b7849ed02034c61d0\n","repos":"inc-lc\/ilc-agda","old_file":"Data\/NatBag.agda","new_file":"Data\/NatBag.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"044fe9f6ed008b2e5b99318e97a0383f1d6d3352","subject":"flat-funs: Reworked default definitions and shown the space complexity of foldl","message":"flat-funs: Reworked default definitions and shown the space complexity of foldl\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_; module \u2115\u00b0)\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Level as L\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\nopen \u2261 using (_\u2261_)\n\nopen import Data.Bits using (Bit; Bits; _\u2192\u1d47_; 0b; 1b)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nmodule Defaults {t} {T : Set t} (\u266dFuns : FlatFuns T) where\n  open FlatFuns \u266dFuns\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first-default : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first-default f = < f \u00d7 id >\n\n    second-default : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second-default f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_>-default : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g >-default = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_>-default\n\n    dup-default : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup-default = < id , id >\n\n    swap-default : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap-default = < snd , fst >\n\n    assoc-default : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc-default = < fst \u2218 fst , first-default snd >\n\n    <tt,>-default : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,>-default = < tt , id >\n\n    <tt,>\u2032-default : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    <tt,>\u2032-default = snd\n\n  module DefaultsGroup1\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (<tt,>\u2032 : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    open DefaultsFirstSecond id <_\u00d7_>\n    open CompositionNotations _\u2218_\n\n    <_,_>-default : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g >-default = dup \u204f < f \u00d7 g >\n\n    snd-default : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd-default = first-default tt \u204f <tt,>\u2032\n\n    fst-default : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst-default = swap \u204f snd-default\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    where\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_>-default : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g >-default = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 \"space\".\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032-default : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032-default = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  -- In case you wonder... one can also derive cond with fork\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond-default : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond-default = fork fst snd\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n\n  infixr 9 _\u2218_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Products (group 1)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Products (group 2)\n    dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,>\u2032 : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults \u266dFuns\n  open CompositionNotations _\u2218_ public\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\u22a4\n  <,tt> = <tt,> \u204f swap\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = < f \u00d7 id >\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = < id \u00d7 f >\n\n  -- This uses <_,_>, hence has space cost 2, while a cost of 1 is expected.\n  fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork f g = second < f , g > \u204f cond\n\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public renaming (assoc\u2032-default to assoc\u2032)\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > \u204f cons\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = < f \u2237 \u229b fs >\n\n  \u229b\u2032 : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 A `\u2192 `Vec B n\n  \u229b\u2032 []       = nil\n  \u229b\u2032 (f \u2237 fs) = < f \u2237\u2032 \u229b\u2032 fs >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 _A `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f xs = \u229b\u2032 (V.map f xs)\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = \u229b (V.replicate f)\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <,nil> : \u2200 {A B} \u2192 A `\u2192 A `\u00d7 `Vec B 0\n  <,nil> = <,tt> \u204f second nil\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = <,nil> \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id F._\u2218\u2032_\n             (F.const 0b) (F.const 1b) (\u03bb { (b , x , y) \u2192 if b then x else y }) _\n             \u00d7.<_,_> proj\u2081 proj\u2082 (\u03bb x \u2192 x , x) (\u03bb f g \u2192 \u00d7.map f g)\n             \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n             (F.const []) (uncurry _\u2237_) V.uncons\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _\u2218_\n                 (const [ 0b ]) (const [ 1b ]) cond\u1d47 (const [])\n                 <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 {x}) dup-default <_\u00d7_>-default\n                 (\u03bb {x} \u2192 swap-default {x}) (\u03bb {x} \u2192 assoc-default {x}) id id (const []) id id\n  where\n  open BitsFunTypes\n  open FunOps\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = V.take A\n  snd\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 {A} = V.drop A\n  <_,_>\u1d47 : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <_,_>\u1d47 f g x = f x ++ g x\n  cond\u1d47 : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond\u1d47 {A} (b \u2237 xs) = if b then take A xs else drop A xs\n  open Defaults bitsFun\u266dFuns\n  open DefaultsGroup2 id _\u2218_ (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 {x})\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond) (S.tt , T.tt)\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.dup , T.dup) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.swap , T.swap) (S.assoc , T.assoc)\n       (S.<tt,> , T.<tt,>) (S.<tt,>\u2032 , T.<tt,>\u2032)\n       (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FunOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond) (S.tt , T.tt)\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.dup , T.dup) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.swap , T.swap) (S.assoc , T.assoc)\n       (S.<tt,> , T.<tt,>) (S.<tt,>\u2032 , T.<tt,>\u2032)\n       (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FunOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1; tt = 0;\n            <_,_> = _+_; fst = 0; snd = 0;\n            dup = 0; <_\u00d7_> = _\u2294_; swap = 0; assoc = 0;\n            <tt,> = 0; <tt,>\u2032 = 0;\n            nil = 0; cons = 0; uncons = 0 }\n\nmodule TimeOps = FlatFunsOps timeOps\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 0; <1b> = 0; cond = 1; tt = 0;\n             <_,_> = \u03bb x y \u2192 1 + (x + y); fst = 0; snd = 0;\n             dup = 1; <_\u00d7_> = _+_; swap = 0; assoc = 0;\n             <tt,> = 0; <tt,>\u2032 = 0;\n             nil = 0; cons = 0; uncons = 0 }\n\nmodule SpaceOps where\n  Space = \u2115\n  open FlatFunsOps spaceOps public\n\n  singleton\u22610 : singleton \u2261 0\n  singleton\u22610 = \u2261.refl\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_)\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Level as L\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\nopen \u2261 using (_\u2261_)\n\nopen import Data.Bits using (Bit; Bits; _\u2192\u1d47_; 0b; 1b)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _\u2218_\n  infixr 1 _\u204f_\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- Products\n    dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n    <tt,> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  _\u204f_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f \u204f g = g \u2218 f\n\n  _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f >>> g = f \u204f g\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = dup \u204f < f \u00d7 g >\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  <_\u00d7_>-on-top-of-<,> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g >-on-top-of-<,> = < fst \u204f f , snd \u204f g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  swap-on-top-<,> : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap-on-top-<,> = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = < f \u00d7 id >\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = < id \u00d7 f >\n\n  fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork f g = second < f , g > \u204f cond\n\n  -- In case you wonder... one can also derive cond with fork\n  cond-on-top-of-fork : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond-on-top-of-fork = fork fst snd\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\u22a4\n  <,tt> = <tt,> \u204f swap\n\n  assoc-default : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc-default = < fst \u204f fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd \u204f snd >\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > \u204f cons\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = < f \u2237 \u229b fs >\n\n  \u229b\u2032 : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 A `\u2192 `Vec B n\n  \u229b\u2032 []       = nil\n  \u229b\u2032 (f \u2237 fs) = < f \u2237\u2032 \u229b\u2032 fs >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 _A `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f xs = \u229b\u2032 (V.map f xs)\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = \u229b (V.replicate f)\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <,nil> : \u2200 {A B} \u2192 A `\u2192 A `\u00d7 `Vec B 0\n  <,nil> = <,tt> \u204f second nil\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = <,nil> \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id F._\u2218\u2032_\n             (F.const 0b) (F.const 1b) (\u03bb { (b , x , y) \u2192 if b then x else y })\n             (\u03bb x \u2192 x , x) proj\u2081 proj\u2082 (\u03bb f g \u2192 \u00d7.map f g)\n             \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) _ (\u03bb x \u2192 _ , x) (F.const []) (uncurry _\u2237_) V.uncons\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _\u2218_\n                 (const [ 0b ]) (const [ 1b ]) cond\u1d47\n                 dup\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 {x}) <_\u00d7_>\u1d47\n                 (\u03bb {x} \u2192 swap\u1d47 {x}) (\u03bb {x} \u2192 assoc\u1d47 {x}) (const []) id (const []) id id\n  where\n  open BitsFunTypes\n  open FunOps\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = V.take A\n  snd\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 {A} = V.drop A\n  dup\u1d47 : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n  dup\u1d47 xs = xs ++ xs\n  <_\u00d7_>\u1d47 : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  <_\u00d7_>\u1d47 {A} f g x = f (take A x) ++ g (drop A x)\n  cond\u1d47 : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond\u1d47 {A} (b \u2237 xs) = if b then take A xs else drop A xs\n  swap\u1d47 : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap\u1d47 {A} xs = drop A xs ++ take A xs\n  assoc\u1d47 : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc\u1d47 {A} {B} xs = take A (take (A + B) xs) ++ (drop A (take (A + B) xs) ++ drop (A + B) xs) -- < fst\u1d47 \u204f fst\u1d47 , first snd\u1d47 >\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond)\n       (S.dup , T.dup) (S.fst , T.fst) (S.snd , T.snd) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.swap , T.swap) (S.assoc , T.assoc)\n       (S.tt , T.tt) (S.<tt,> , T.<tt,>)\n       (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FunOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond)\n       (S.dup , T.dup) (S.fst , T.fst) (S.snd , T.snd) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.swap , T.swap) (S.assoc , T.assoc)\n       (S.tt , T.tt) (S.<tt,> , T.<tt,>)\n       (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FunOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1;\n            dup = 0; fst = 0; snd = 0; <_\u00d7_> = _\u2294_; swap = 0; assoc = 0;\n            tt = 0; <tt,> = 0;\n            nil = 0; cons = 0; uncons = 0 }\n\nmodule TimeOps = FlatFunsOps timeOps\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 0; <1b> = 0; cond = 1;\n             dup = 1; fst = 0; snd = 0; <_\u00d7_> = _+_; swap = 0; assoc = 0;\n             tt = 0; <tt,> = 0;\n             nil = 0; cons = 0; uncons = 0 }\n\nmodule SpaceOps where\n  Space = \u2115\n  open FlatFunsOps spaceOps public\n\n  singleton\u22610 : singleton \u2261 0\n  singleton\u22610 = \u2261.refl\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f3ccc448958e9a29f4b3791191c8dcf51d8f46e2","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Properties\/UnprovedATP.agda","new_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Properties\/UnprovedATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Unproven PA properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Mendelson.Properties.UnprovedATP where\n\nopen import PA.Axiomatic.Mendelson.Base\nopen import PA.Axiomatic.Mendelson.PropertiesATP\n\n------------------------------------------------------------------------------\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2248 m + (n + o)\n+-asocc m n o = S\u2089 A A0 is m\n  where\n  A : \u2115 \u2192 Set\n  A i = i + n + o \u2248 i + (n + o)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-leftCong #-}\n\n  -- 25 November 2013: Vampire 0.6 (revision 903) proves the theorem\n  -- using the --mode\u00a0casc option and a time out of 300 sec.\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is +-leftCong #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Unproven PA properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Mendelson.Properties.UnprovedATP where\n\nopen import PA.Axiomatic.Mendelson.Base\nopen import PA.Axiomatic.Mendelson.PropertiesATP\n\n------------------------------------------------------------------------------\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2248 m + (n + o)\n+-asocc m n o = S\u2089 A A0 is m\n  where\n  A : \u2115 \u2192 Set\n  A i = i + n + o \u2248 i + (n + o)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-leftCong #-}\n\n  -- 25 November 2013: Vampire 0.6 proves the theorem using a time out\n  -- of 300 sec.\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is +-leftCong #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2971c097d16d13188929ef857808539310db0b8e","subject":"Improve printing of resolved overloading.","message":"Improve printing of resolved overloading.\n\nAfter this change, the semantic brackets will contain the syntactic\nthing even if Agda displays explicitly resolved overloaded notation.\n\nOld-commit-hash: c8cbc43ed8e715342e0cc3bccc98e0be2dd23560\n","repos":"inc-lc\/ilc-agda","old_file":"meaning.agda","new_file":"meaning.agda","new_contents":"module meaning where\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\nopen Meaning public\n  using (\u27e8_\u27e9\u27e6_\u27e7)\n","old_contents":"module meaning where\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b5190556d5bc58a768c39b9fa871dea8b2d6f0f1","subject":"Fix bug in diff-term.","message":"Fix bug in diff-term.\n\nOld-commit-hash: 01ada999b676d2a8a2ab289b3a0c79f1524620e5\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"011116b87218eb84d5010b97f93c495b4f090fae","subject":"Implement nil changes.","message":"Implement nil changes.\n\nOld-commit-hash: da546d60ec48fe5f6f8b261aac1fdbb71b18fa69\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","old_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f2fde70dc52c3050e537b787efd3c7efa9778118","subject":"Introduce HOAS-like smart constructors for lambdas","message":"Introduce HOAS-like smart constructors for lambdas\n\nOld-commit-hash: 6bf4939f8ec84c611d98f6f2395332943573dfe5\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nuncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\nuncurriedConst constant = const constant\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\ncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n-- HOAS-like smart constructors for lambdas, for different arities.\n\nabs\u2081 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 (x : Term \u0393\u2032 \u03c4\u2081) \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\nabs\u2081 {\u0393} {\u03c4\u2081} =  \u03bb f \u2192 abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n\nabs\u2082 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4))\nabs\u2082 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2081 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2083 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4))\nabs\u2083 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2082 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2084 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4))\nabs\u2084 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2083 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2085 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4))\nabs\u2085 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2084 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2086 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4\u2086 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4\u2086 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4\u2086 \u21d2 \u03c4))\nabs\u2086 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2085 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nuncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\nuncurriedConst constant = const constant\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\ncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurriedConst constant = curryTermConstructor (uncurriedConst constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25e6b7ac020ffceed37754195a08542060325d04","subject":"Missing a cosmetic change.","message":"Missing a cosmetic change.\n\nIgnore-this: 1ff79f43e9cf87ce4ef78941000ea0ce\n\ndarcs-hash:20110313151430-3bd4e-363c8c0166d6c156ba38551e0eca323df0b50f18.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/List\/Type.agda","new_file":"src\/FOTC\/Data\/List\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC list type\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.List.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- The FOTC list type.\ndata List : D \u2192 Set where\n  nilL  :                              List []\n  consL : \u2200 x {xs} \u2192 (Lxs : List xs) \u2192 List (x \u2237 xs)\n\n-- Induction principle for List.\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          (\u2200 x {xs} \u2192 List xs \u2192 P xs \u2192 P (x \u2237 xs)) \u2192\n          \u2200 {xs} \u2192 List xs \u2192 P xs\nindList P P[] is nilL          = P[]\nindList P P[] is (consL x Lxs) = is x Lxs (indList P P[] is Lxs)\n","old_contents":"------------------------------------------------------------------------------\n-- The FOTC list type\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.List.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- The FOTC list type.\ndata List : D \u2192 Set where\n  nilL  : List []\n  consL : \u2200 x {xs} \u2192 (Lxs : List xs) \u2192 List (x \u2237 xs)\n\n-- Induction principle for List.\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          (\u2200 x {xs} \u2192 List xs \u2192 P xs \u2192 P (x \u2237 xs)) \u2192\n          \u2200 {xs} \u2192 List xs \u2192 P xs\nindList P P[] is nilL          = P[]\nindList P P[] is (consL x Lxs) = is x Lxs (indList P P[] is Lxs)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"63e76c39ea74fe1aff6cfb352031867948f8b3ea","subject":"Only renaming.","message":"Only renaming.\n\nIgnore-this: 735181c9057b86e703241df8c7d1f7b2\n\ndarcs-hash:20110218162103-3bd4e-34dedb41e328516d8c83a9ef2291477f8f441e55.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111'  : eleven + one-hundred \u2261 hundred-eleven\n  n111   : one-hundred + eleven \u2261 hundred-eleven\n  n101'  : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101   : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'   : hundred-one \u2238 ten \u2261 ninety-one\n  n91    : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102'  : eleven + ninety-one \u2261 hundred-two\n  n102   : ninety-one + eleven \u2261 hundred-two\n  n100'  : eleven + eighty-nine \u2261 one-hundred\n  n100   : eighty-nine + eleven \u2261 one-hundred\n  n110   : ninety-nine + eleven \u2261 hundred-ten\n  n109   : ninety-eight + eleven \u2261 hundred-nine\n  n108   : ninety-seven + eleven \u2261 hundred-eight\n  n107   : ninety-six + eleven \u2261 hundred-seven\n  n106   : ninety-five + eleven \u2261 hundred-six\n  n105   : ninety-four + eleven \u2261 hundred-five\n  n104   : ninety-three + eleven \u2261 hundred-four\n  n103   : ninety-two + eleven \u2261 hundred-three\n  n101'' : ninety + eleven \u2261 hundred-one\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n{-# ATP prove n110 #-}\n{-# ATP prove n109 #-}\n{-# ATP prove n108 #-}\n{-# ATP prove n107 #-}\n{-# ATP prove n106 #-}\n{-# ATP prove n105 #-}\n{-# ATP prove n104 #-}\n{-# ATP prove n103 #-}\n{-# ATP prove n101'' #-}\n\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 n100 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8514aefef6973b813691fb922e6d44e072cfc75d","subject":"update generic ZK","message":"update generic ZK\n","repos":"crypto-agda\/crypto-agda","old_file":"generic-zero-knowledge-interactive.agda","new_file":"generic-zero-knowledge-interactive.agda","new_contents":"open import Type\nopen import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (\u03bc\u03c0          : SumProp Permutation)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (\u03bcR\u209a-xtra    : SumProp R\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    \u03bcR\u209a : SumProp R\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    \u03bcR : SumProp R\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl    : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-cong-\u2219  : \u2200 {\u03b1 \u03b2 b} \u03b3 \u2192 \u03b1 == \u03b2 \u2261 b \u2192 (\u03b3 \u2219 \u03b1) == (\u03b3 \u2219 \u03b2) \u2261 b)\n            (==-true    : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (\u03bc\u2124q        : SumProp \u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (\u03bcR\u209a-xtra   : SumProp R\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s = ==-cong-\u2219 (g^ \u03c0) check-p-s\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 \u03bc\u2124q R\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (sum\u03c0        : Sum Permutation)\n         (\u03bc\u03c0          : SumProp sum\u03c0)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (\u03bcR\u209a-xtra    : SumProp sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sum\u03c0 \u00d7Sum sumR\u209a-xtra\n\n    \u03bcR\u209a : SumProp sumR\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    sumR : Sum R\n    sumR = sumBit \u00d7Sum sumR\u209a\n\n    \u03bcR : SumProp sumR\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl    : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-cong-\u2219  : \u2200 {\u03b1 \u03b2 b} \u03b3 \u2192 \u03b1 == \u03b2 \u2261 b \u2192 (\u03b3 \u2219 \u03b1) == (\u03b3 \u2219 \u03b2) \u2261 b)\n            (==-true    : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (sum\u2124q      : Sum \u2124q)\n            (\u03bc\u2124q        : SumProp sum\u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (sumR\u209a-xtra : Sum R\u209a-xtra)\n            (\u03bcR\u209a-xtra   : SumProp sumR\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s = ==-cong-\u2219 (g^ \u03c0) check-p-s\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 sum\u2124q \u03bc\u2124q R\u209a-xtra sumR\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8569ad557abf314286b4a14381efb48631f72316","subject":"IDesc model: proof that embedded IDesc and defined IDesc are isomorph (with extensionality axiom).","message":"IDesc model: proof that embedded IDesc and defined IDesc are isomorph (with extensionality axiom).","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : (A : Set)(B : Set)(f : A -> B)(x y : A) -> x == y -> f x == f y\ncong A B f x .x refl = refl\n\ncong2 : (A B C : Set)(f : A -> B -> C)(x y : A)(z t : B) -> \n        x == y -> z == t -> f x z == f y t\ncong2 A B C f x .x z .z refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : (A : Set)(B : Set)(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc Unit (descD I) (IMu Unit (\u03bb _ -> descD I)))\n        (hs : desc (Sigma Unit (IMu Unit (\u03bb _ -> descD I)))\n              (box Unit (descD I) (IMu Unit (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 (IDesc I) \n                                                (IDesc I)\n                                                (IDesc I) \n                                                prod \n                                                (iso1 I (iso2 I D))\n                                                D \n                                                (iso1 I (iso2 I D'))\n                                                D' p q\nproof-iso1-iso2 I (pi S T) = cong (S \u2192 IDesc I)\n                                  (IDesc I)\n                                  (pi S) \n                                  (\\x -> iso1 I (iso2 I (T x)))\n                                  T \n                                  (reflFun S (IDesc I)\n                                             (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                             T\n                                             (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (S \u2192 IDesc I)\n                                     (IDesc I)\n                                     (sigma S) \n                                     (\\x -> iso1 I (iso2 I (T x)))\n                                     T \n                                     (reflFun S \n                                              (IDesc I)\n                                              (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- induction : (I : Set)\n--             (R : I -> IDesc I)\n--             (P : Sigma I (IMu I R) -> Set)\n--             (m : (i : I)\n--                  (xs : desc I (R i) (IMu I R))\n--                  (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n--                  P ( i , con xs)) ->\n--             (i : I)(x : IMu I R i) -> P ( i , x )\n\n\nP : (I : Set) -> Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb x \u2192 sigma DescDConst (descDChoice I))\n                                 (P I)\n                                 ( proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) hs with proof-iso2-iso1 I D | proof-iso2-iso1 I D' \n                      ... | p | q = cong2 (IDescl I) \n                                          (IDescl I) \n                                          (IDescl I)\n                                          (prodl I) \n                                          (iso2 I (iso1 I D))\n                                          D \n                                          (iso2 I (iso1 I D'))\n                                          D' p q\n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                          (IDescl I)\n                                                                          (pil I S) \n                                                                          (\\x -> iso2 I (iso1 I (T x)))\n                                                                          T \n                                                                          (reflFun S \n                                                                                   (IDescl I)\n                                                                                   (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   (\\s -> proof-iso2-iso1 I (T s)))\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                             (IDescl I)\n                                                                             (sigmal I S) \n                                                                             (\\x -> iso2 I (iso1 I (T x)))\n                                                                             T \n                                                                             (reflFun S \n                                                                                      (IDescl I)\n                                                                                      (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      (\\s -> proof-iso2-iso1 I (T s)))\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fc4cee02dceacd367ce3a7cbb29f46b050702018","subject":"finished lem2-5-2","message":"finished lem2-5-2\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/Reduction.agda","new_file":"Agda\/Reduction.agda","new_contents":"module Reduction where\n\nopen import Data.Empty\nopen import Data.Nat\nopen import Data.List\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import Core\nopen import Core-Lemmas\n\n_\u219d_ : Set -> Set -> Set\u2081\nA \u219d B = A -> B -> Set\n\n\ndata _->\u03b2_ : PTerm \u219d PTerm where\n  redL : \u2200 {n m m'} -> Term n \u2192 m ->\u03b2 m' -> app m n ->\u03b2 app m' n\n  redR : \u2200 {m n n'} -> Term m \u2192 n ->\u03b2 n' -> app m n ->\u03b2 app m n'\n  abs : \u2200 L {m m'} -> ( \u2200 {x} -> x \u2209 L -> (m ^' x) ->\u03b2 (m' ^' x) ) ->\n    lam m ->\u03b2 lam m'\n  beta : \u2200 {m n} -> Term (lam m) -> Term n -> app (lam m) n ->\u03b2 (m ^ n)\n  Y : \u2200 {m \u03c3} -> Term m -> app (Y \u03c3) m ->\u03b2 app m (app (Y \u03c3) m)\n\n-- test : \u2200 {x} -> lam (app (lam (bv 0)) (fv x)) ->\u03b2 lam (fv x)\n-- test = abs [] (\u03bb _ -> beta (lam [] (\u03bb x\u2209L -> var)) var)\n\ndata _->||_ : PTerm \u219d PTerm where\n  refl : \u2200 {x} -> fv x ->|| fv x\n  reflY : \u2200 {\u03c3} -> Y \u03c3 ->|| Y \u03c3\n  app : \u2200 {m m' n n'} -> m ->|| m' -> n ->|| n' -> app m n ->|| app m' n'\n  abs : \u2200 L {m m'} -> ( \u2200 {x} -> x \u2209 L -> (m ^' x) ->|| (m' ^' x) ) ->\n    lam m ->|| lam m'\n  beta : \u2200 L {m m' n n'} -> (cf : \u2200 {x} -> x \u2209 L -> (m ^' x) ->|| (m' ^' x) ) -> n ->|| n' ->\n    (app (lam m) n) ->|| (m' ^ n')\n  Y : \u2200 {m m' \u03c3} -> m ->|| m' -> app (Y \u03c3) m ->|| app m' (app (Y \u03c3) m')\n\n\ndata NotAbsY : PTerm -> Set where\n  fv : \u2200 {x} -> NotAbsY (fv x)\n  bv : \u2200 {n} -> NotAbsY (bv n)\n  app : \u2200 {t1 t2} -> NotAbsY (app t1 t2)\n\n\ndata _>>>_ : PTerm \u219d PTerm where\n  refl : \u2200 {x} -> fv x >>> fv x\n  reflY : \u2200 {\u03c3} -> Y \u03c3 >>> Y \u03c3\n  app : \u2200 {m m' n n'} -> NotAbsY m -> m >>> m' -> n >>> n' -> app m n >>> app m' n'\n  abs : \u2200 L {m m'} -> ( \u2200 {x} -> x \u2209 L -> (m ^' x) >>> (m' ^' x) ) ->\n    lam m >>> lam m'\n  beta : \u2200 L {m m' n n'} -> (cf : \u2200 {x} -> x \u2209 L -> (m ^' x) >>> (m' ^' x) ) -> n >>> n' ->\n    app (lam m) n >>> (m' ^ n')\n  Y : \u2200 {m m' \u03c3} -> m >>> m' -> app (Y \u03c3) m >>> app m' (app (Y \u03c3) m')\n\n\nsubst-Term : \u2200 {x e u} -> Term e -> Term u -> Term (e [ x ::= u ])\nsubst-Term {x} {_} {u} (var {y}) Term-u with x \u225f y\nsubst-Term var Term-u | yes refl = Term-u\nsubst-Term var Term-u | no p = var\nsubst-Term {x} {_} {u} (lam L {e} cf) Term-u = lam (x \u2237 L) body\n  where\n  body : {y : \u2115} -> y \u2209 x \u2237 L \u2192 Term ((e [ x ::= u ]) ^' y)\n  body {y} y\u2209x\u2237L rewrite subst-open-var y x u e (fv-x\u2260y y x y\u2209x\u2237L) Term-u =\n    subst-Term (cf (\u03bb z \u2192 y\u2209x\u2237L (there z))) Term-u\n\nsubst-Term (app Term-e Term-e\u2081) Term-u =\n  app (subst-Term Term-e Term-u) (subst-Term Term-e\u2081 Term-u)\nsubst-Term Y Term-u = Y\n\n\n^-Term : \u2200 {m n} -> Term (lam m) -> Term n -> Term (m ^ n)\n^-Term {m} {n} (lam L cf) Term-n = body\n  where\n  y = \u2203fresh (L ++ FV m)\n\n  y\u2209 : y \u2209 (L ++ FV m)\n  y\u2209 = \u2203fresh-spec (L ++ FV m)\n\n  body : Term (m ^ n)\n  body rewrite subst-intro y n m (\u2209-cons-r L (FV m) y\u2209) Term-n =\n    subst-Term {y} {m ^' y} {n} (cf (\u2209-cons-l L (FV m) y\u2209)) Term-n\n\n\n->||-Term-l : \u2200 {m m'} -> m ->|| m' -> Term m\n->||-Term-l refl = var\n->||-Term-l reflY = Y\n->||-Term-l (app m->||m' m->||m'') = app (->||-Term-l m->||m') (->||-Term-l m->||m'')\n->||-Term-l (abs L x) = lam L (\u03bb x\u2209L \u2192 ->||-Term-l (x x\u2209L))\n->||-Term-l (beta L x m->||m') =\n  app (lam L (\u03bb {x\u2081} x\u2209L \u2192 ->||-Term-l (x x\u2209L)))\n      (->||-Term-l m->||m')\n->||-Term-l (Y {m} {m'} m->||m') = app Y (->||-Term-l m->||m')\n\n\n->||-Term-r : \u2200 {m m'} -> m ->|| m' -> Term m'\n->||-Term-r refl = var\n->||-Term-r reflY = Y\n->||-Term-r (app m->||m' m->||m'') = app (->||-Term-r m->||m') (->||-Term-r m->||m'')\n->||-Term-r (abs L x) = lam L (\u03bb x\u2209L \u2192 ->||-Term-r (x x\u2209L))\n->||-Term-r (beta L {m} {m'} {n} {n'} cf m->||m') =\n  ^-Term {m'} {n'} (lam L (\u03bb {x\u2081} x\u2209L \u2192 ->||-Term-r (cf x\u2209L))) (->||-Term-r m->||m')\n->||-Term-r (Y m->||m') = app (->||-Term-r m->||m') (app Y (->||-Term-r m->||m'))\n\n\n>>>-Term-l : \u2200 {m m'} -> m >>> m' -> Term m\n>>>-Term-l refl = var\n>>>-Term-l reflY = Y\n>>>-Term-l (app nAbsY m>>>m' m>>>m'') = app (>>>-Term-l m>>>m') (>>>-Term-l m>>>m'')\n>>>-Term-l (abs L x) = lam L (\u03bb x\u2209L \u2192 >>>-Term-l (x x\u2209L))\n>>>-Term-l (beta L x m>>>m') =\n  app (lam L (\u03bb {x\u2081} x\u2209L \u2192 >>>-Term-l (x x\u2209L)))\n      (>>>-Term-l m>>>m')\n>>>-Term-l (Y {m} {m'} m>>>m') = app Y (>>>-Term-l m>>>m')\n\n\n>>>-Term-r : \u2200 {m m'} -> m >>> m' -> Term m'\n>>>-Term-r refl = var\n>>>-Term-r reflY = Y\n>>>-Term-r (app nAbsY m>>>m' m>>>m'') = app (>>>-Term-r m>>>m') (>>>-Term-r m>>>m'')\n>>>-Term-r (abs L x) = lam L (\u03bb x\u2209L \u2192 >>>-Term-r (x x\u2209L))\n>>>-Term-r (beta L {m} {m'} {n} {n'} cf m>>>m') =\n  ^-Term {m'} {n'} (lam L (\u03bb {x\u2081} x\u2209L \u2192 >>>-Term-r (cf x\u2209L))) (>>>-Term-r m>>>m')\n>>>-Term-r (Y m>>>m') = app (>>>-Term-r m>>>m') (app Y (>>>-Term-r m>>>m'))\n\n\nlem2-5-1 : \u2200 s s' t t' (x : \u2115) -> s ->|| s' -> t ->|| t' ->\n  (s [ x ::= t ]) ->|| (s' [ x ::= t' ])\nlem2-5-1 _ _ t t' y (refl {x}) t->||t' with x \u225f y\nlem2-5-1 .(fv x) .(fv x) t t' x refl t->||t' | yes refl rewrite\n  fv-subst-eq x x t refl | fv-subst-eq x x t' refl = t->||t'\nlem2-5-1 _ _ t t' y (refl {x}) t->||t'             | no x\u2260y rewrite\n  fv-subst-neq x y t x\u2260y | fv-subst-neq x y t' x\u2260y = refl\nlem2-5-1 _ _ t t' x reflY t->||t' = reflY\nlem2-5-1 _ _ t t' x (app {m} {m'} {n} {n'} ss' ss'') t->||t' = app (lem2-5-1 m m' t t' x ss' t->||t') (lem2-5-1 n n' t t' x ss'' t->||t')\nlem2-5-1 _ _ t t' x (abs L {m} {m'} cf) t->||t' = abs (x \u2237 L) body\n  where\n  x\u2209FVy : (y : \u2115) -> (y \u2209 x \u2237 L) -> x \u2209 FV (fv y)\n  x\u2209FVy y y\u2209x\u2237L x\u2208FVy with fv-x\u2261y x y x\u2208FVy\n  x\u2209FVy .x y\u2209x\u2237L x\u2208FVy | refl = y\u2209x\u2237L (here refl)\n\n  body : {y : \u2115} -> (y \u2209 x \u2237 L) -> ((m [ x ::= t ]) ^' y) ->|| ((m' [ x ::= t' ]) ^' y)\n  body {y} y\u2209x\u2237L rewrite\n    subst-fresh2 x (fv y) t (x\u2209FVy y y\u2209x\u2237L) |\n    subst-open2 x t 0 (fv y) m (->||-Term-l t->||t') |\n    subst-fresh x (fv y) t (x\u2209FVy y y\u2209x\u2237L) |\n    subst-fresh2 x (fv y) t' (x\u2209FVy y y\u2209x\u2237L) |\n    subst-open2 x t' 0 (fv y) m' (->||-Term-r t->||t') |\n    subst-fresh x (fv y) t' (x\u2209FVy y y\u2209x\u2237L) =\n      lem2-5-1 (m ^' y) (m' ^' y) t t' x (cf (\u03bb z \u2192 y\u2209x\u2237L (there z))) t->||t'\n\nlem2-5-1 _ _ t t' x (beta L {m} {m'} {n} {n'} cf n->||n') t->||t' rewrite\n  subst-open x t' 0 n' m' (->||-Term-r t->||t') =\n    beta (x \u2237 L) {_} {m' [ x ::= t' ]}  body body\u2082\n  where\n  body : {y : \u2115} -> y \u2209 x \u2237 L -> ((m [ x ::= t ]) ^' y) ->|| ((m' [ x ::= t' ]) ^' y)\n  body {y} y\u2209x\u2237L rewrite\n    subst-open-var y x t m (fv-x\u2260y y x y\u2209x\u2237L) (->||-Term-l t->||t') |\n    subst-open-var y x t' m' (fv-x\u2260y y x y\u2209x\u2237L) (->||-Term-r t->||t') =\n      lem2-5-1 (m ^' y) (m' ^' y) t t' x (cf (\u03bb z \u2192 y\u2209x\u2237L (there z))) t->||t'\n\n  body\u2082 : (n [ x ::= t ]) ->|| (n' [ x ::= t' ])\n  body\u2082 = lem2-5-1 n n' t t' x n->||n' t->||t'\n\nlem2-5-1 _ _ t t' x (Y {m} {m'} ss') t->||t' = Y (lem2-5-1 m m' t t' x ss' t->||t')\n\n\nlem2-5-1-^ : \u2200 s s' t t' L -> (\u2200 {x} -> x \u2209 L -> (s ^' x) ->|| (s' ^' x)) -> t ->|| t' ->\n  (s ^ t) ->|| (s' ^ t')\nlem2-5-1-^ s s' t t' L cf t->||t' = body\n  where\n  x = \u2203fresh (L ++ FV s ++ FV s')\n\n  x\u2209 : x \u2209 (L ++ FV s ++ FV s')\n  x\u2209 = \u2203fresh-spec (L ++ FV s ++ FV s')\n\n  body : (s ^ t) ->|| (s' ^ t')\n  body rewrite\n    subst-intro x t s (\u2209-cons-l (FV s) (FV s') (\u2209-cons-r L (FV s ++ FV s') x\u2209)) (->||-Term-l t->||t') |\n    subst-intro x t' s' (\u2209-cons-r (FV s) (FV s') (\u2209-cons-r L (FV s ++ FV s') x\u2209)) (->||-Term-r t->||t') =\n      lem2-5-1 (s ^' x) (s' ^' x) t t' x (cf (\u2209-cons-l L (FV s ++ FV s') x\u2209)) t->||t'\n\n\nNotAbsY-subst : \u2200 {x y m} -> NotAbsY m -> NotAbsY (m [ x ::= fv y ])\nNotAbsY-subst {x} (fv {y}) with x \u225f y\nNotAbsY-subst {x} {y} fv | yes refl rewrite\n  fv-subst-eq x x (fv y) refl = fv\nNotAbsY-subst fv | no _ = fv\nNotAbsY-subst bv = bv\nNotAbsY-subst app = app\n\n\nlem2-5-1>>> : \u2200 s s' (x y : \u2115) -> s >>> s' -> (s [ x ::= fv y ]) >>> (s' [ x ::= fv y ])\nlem2-5-1>>> _ _ x y (refl {z}) with x \u225f z\nlem2-5-1>>> .(fv x) .(fv x) x y refl | yes refl rewrite\n  fv-subst-eq x x (fv y) refl = refl\nlem2-5-1>>> _ _ x y (refl {z}) | no z\u2260x = refl\nlem2-5-1>>> _ _ x y reflY = reflY\nlem2-5-1>>> _ _ x y (app {m} {m'} {n} {n'} \u00acabsY ss' ss'') =\n  app (NotAbsY-subst \u00acabsY) (lem2-5-1>>> m m' x y ss') (lem2-5-1>>> n n' x y ss'')\nlem2-5-1>>> _ _ x y (abs L {m} {m'} cf) = abs (x \u2237 L) body\n  where\n  x\u2209FVz : (z : \u2115) -> (z \u2209 x \u2237 L) -> x \u2209 FV (fv z)\n  x\u2209FVz z z\u2209x\u2237L x\u2208FVz with fv-x\u2261y x z x\u2208FVz\n  x\u2209FVz .x z\u2209x\u2237L x\u2208FVz | refl = z\u2209x\u2237L (here refl)\n\n  body : {z : \u2115} -> (z \u2209 x \u2237 L) -> ((m [ x ::= fv y ]) ^' z) >>> ((m' [ x ::= fv y ]) ^' z)\n  body {z} z\u2209x\u2237L rewrite\n    subst-fresh2 x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) |\n    subst-open2 x (fv y) 0 (fv z) m var |\n    subst-fresh x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) |\n    subst-fresh2 x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) |\n    subst-open2 x (fv y) 0 (fv z) m' var |\n    subst-fresh x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) =\n      lem2-5-1>>> (m ^' z) (m' ^' z) x y (cf (\u03bb z\u2081 \u2192 z\u2209x\u2237L (there z\u2081)))\n\nlem2-5-1>>> _ _ x y (beta L {m} {m'} {n} {n'} cf n>>>n') rewrite\n  subst-open x (fv y) 0 n' m' var =\n    beta (x \u2237 L) {_} {m' [ x ::= fv y ]} body body\u2082\n  where\n  body : {z : \u2115} -> z \u2209 x \u2237 L -> ((m [ x ::= fv y ]) ^' z) >>> ((m' [ x ::= fv y ]) ^' z)\n  body {z} z\u2209x\u2237L rewrite\n    subst-open-var z x (fv y) m (fv-x\u2260y z x z\u2209x\u2237L) var |\n    subst-open-var z x (fv y) m' (fv-x\u2260y z x z\u2209x\u2237L) var =\n      lem2-5-1>>> (m ^' z) (m' ^' z) x y (cf (\u03bb z\u2081 \u2192 z\u2209x\u2237L (there z\u2081)))\n\n  body\u2082 : (n [ x ::= fv y ]) >>> (n' [ x ::= fv y ])\n  body\u2082 = lem2-5-1>>> n n' x y n>>>n'\n\nlem2-5-1>>> _ _ x y (Y {m} {m'} ss') = Y (lem2-5-1>>> m m' x y ss')\n\n\n*^-^->>> : \u2200 {x y t d k} -> t >>> d -> y \u2209 x \u2237 (FV t ++ FV d) -> ([ k >> fv y ] ([ k << x ] t)) >>> ([ k >> fv y ] ([ k << x ] d))\n*^-^->>> {x} {y} {t} {d} {k} t>>>d y\u2209 rewrite\n  *^-^\u2261subst t x y {k} (>>>-Term-l t>>>d) | *^-^\u2261subst d x y {k} (>>>-Term-r t>>>d) = lem2-5-1>>> t d x y t>>>d\n\n\n\u2203>>> : \u2200 {a} -> Term a -> \u2203(\u03bb d -> a >>> d)\n\u2203>>> (var {x}) = fv x , refl\n\u2203>>> (lam L {t} cf) = body\n  where\n  x = \u2203fresh (L ++ FV t)\n\n  x\u2209 : x \u2209 (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n\n  d-spec : \u2203(\u03bb d -> (t ^' x) >>> d)\n  d-spec = \u2203>>> (cf (\u2209-cons-l L (FV t) x\u2209))\n\n  d : PTerm\n  d = proj\u2081 d-spec\n\n  subst1 : \u2200 x y t k -> x \u2209 FV t -> (t ^' y) \u2261 (([ k << x ] ([ k >> fv x ] t)) ^' y)\n  subst1 x y t k x\u2209FVt rewrite fv-^-*^-refl x t {k} x\u2209FVt = refl\n\n  cf' : \u2200 {y} -> y \u2209 x \u2237 (FV d ++ FV t) -> (t ^' y) >>> ((* x ^ d) ^' y)\n  cf' {y} y\u2209 rewrite subst1 x y t 0 (\u2209-cons-r L (FV t) x\u2209) =\n    *^-^->>> {x} {y} {t ^' x} {d} {0} (proj\u2082 d-spec)\n      (\u2209-\u2237 x (FV (t ^' x) ++ FV d) (fv-x\u2260y y x y\u2209)\n        (\u2209-cons-intro (FV (t ^' x)) (FV d)\n          (fv-^ {0} {y} {x} t (\u2209-cons-r (FV d) _ (\u2209-\u2237-elim _ y\u2209)) (fv-x\u2260y _ _ y\u2209))\n          (\u2209-cons-l _ (FV t) (\u2209-\u2237-elim _ y\u2209))))\n\n  body : \u2203(\u03bb d -> (lam t) >>> d)\n  body = lam (* x ^ d) , (abs (x \u2237 (FV d ++ FV t)) cf')\n\n\u2203>>> (app {t1} {t2} trm-t1 trm-t2) with trm-t1 | \u2203>>> trm-t1 | \u2203>>> trm-t2\n\u2203>>> (app trm-t1 trm-t2) | (var {x}) | d1 , t1>>>d1 | d2 , t2>>>d2 = app (fv x) d2 , app fv refl t2>>>d2\n\u2203>>> (app {_} {t2} trm-t1 trm-t2) | lam L cf | _ , abs L\u2081 {t1} {d1} cf\u2081 | d2 , t2>>>d2 =\n  d1 ^ d2 , beta L\u2081 {t1} {d1} {t2} {d2} cf\u2081 t2>>>d2\n\u2203>>> (app {_} {t2} trm-t1 trm-t2) | app {p1} {p2} trm-p1 trm-p2 | d1 , t1>>>d1 | d2 , t2>>>d2 =\n  app d1 d2 , app app t1>>>d1 t2>>>d2\n\u2203>>> (app trm-t1 trm-t2) | (Y {t}) | d1 , t1>>>d1 | d2 , t2>>>d2 = app d2 (app (Y t) d2) , Y t2>>>d2\n\u2203>>> (Y {t}) = Y t , reflY\n\n\nY->||Y-\u2261 : \u2200 {t t' : Type} -> (Y t) ->|| (Y t') -> (PTerm.Y t) \u2261 (Y t')\nY->||Y-\u2261 reflY = refl\n\n>>>closes->|| : \u2200 {a b d} -> a >>> d -> a ->|| b -> b ->|| d\n>>>closes->|| refl refl = refl\n>>>closes->|| reflY reflY = reflY\n>>>closes->|| (app \u00acabsY al>>>ald ar>>>ard) (app al->||alb ar->||arb) =\n  app (>>>closes->|| al>>>ald al->||alb) (>>>closes->|| ar>>>ard ar->||arb)\n>>>closes->|| (app () al>>>ald ar>>>ard) (beta _ _ _)\n>>>closes->|| (app () al>>>ald ar>>>ard) (Y _)\n>>>closes->|| (abs L {a} {d} cf) (abs L\u2081 {.a} {b} cf\u2081) = abs (L ++ L\u2081) ih\n  where\n  ih : \u2200 {x : \u2115} \u2192 x \u2209 L ++ L\u2081 \u2192 (b ^' x) ->|| (d ^' x)\n  ih {x} x\u2209L++L\u2081 = >>>closes->|| (cf (\u2209-cons-l _ _ x\u2209L++L\u2081)) (cf\u2081 (\u2209-cons-r L _ x\u2209L++L\u2081))\n\n>>>closes->|| (beta _ _ _) (app {m' = bv _} () _)\n>>>closes->|| (beta _ _ _) (app {m' = fv _} () _)\n>>>closes->|| (beta L {_} {ald} {ar} {ard} cf ar>>>ard) (app {m' = lam alb} {n' = arb} (abs L\u2081 cf\u2081) ar->||arb) =\n  beta (L ++ L\u2081) {m' = ald} ih arb->||ard\n  where\n  arb->||ard : arb ->|| ard\n  arb->||ard = >>>closes->|| ar>>>ard ar->||arb\n\n  ih : \u2200 {x} -> x \u2209 L ++ L\u2081 -> (alb ^' x) ->|| (ald ^' x)\n  ih {x} x\u2209L++L\u2081 = >>>closes->|| (cf (\u2209-cons-l _ _ x\u2209L++L\u2081)) (cf\u2081 (\u2209-cons-r L _ x\u2209L++L\u2081))\n\n>>>closes->|| (beta _ _ _) (app {m' = app _ _} () _)\n>>>closes->|| (beta _ _ _) (app {m' = Y _} () _)\n>>>closes->|| (beta L {al} {ald} {ar} {ard} cf ar>>>ard) (beta L\u2081 {m' = alb} {n' = arb} cf\u2081 ar->||arb) =\n  lem2-5-1-^ alb ald arb ard (L ++ L\u2081) ih arb->||ard\n  where\n  arb->||ard : arb ->|| ard\n  arb->||ard = >>>closes->|| ar>>>ard ar->||arb\n\n  ih : \u2200 {x} -> x \u2209 L ++ L\u2081 -> (alb ^' x) ->|| (ald ^' x)\n  ih {x} x\u2209L++L\u2081 = >>>closes->|| (cf (\u2209-cons-l _ _ x\u2209L++L\u2081)) (cf\u2081 (\u2209-cons-r L _ x\u2209L++L\u2081))\n\n>>>closes->|| (Y _) (app {m' = bv _} () _)\n>>>closes->|| (Y _) (app {m' = fv _} () _)\n>>>closes->|| (Y _) (app {m' = lam _} () _)\n>>>closes->|| (Y _) (app {m' = app _ _} () _)\n>>>closes->|| (Y {m} {m'} {t} m>>>m') (app {m' = Y t'} {n' = n''} Yt->||Yt' m->||n'') rewrite\n  Y->||Y-\u2261 Yt->||Yt' = Y (>>>closes->|| m>>>m' m->||n'')\n>>>closes->|| (Y {m} {m'} {t} m>>>m') (Y {m' = m''} m->||m'') =\n  app (>>>closes->|| m>>>m' m->||m'') (app reflY (>>>closes->|| m>>>m' m->||m''))\n\n\nlem2-5-2 : \u2200 {a b c} -> a ->|| b -> a ->|| c -> \u2203(\u03bb d -> b ->|| d \u00d7 c ->|| d)\nlem2-5-2 {a} a->||b a->||c = d , ((>>>closes->|| a>>>d a->||b) , (>>>closes->|| a>>>d a->||c))\n  where\n  a>>>d-spec : \u2203(\u03bb d -> a >>> d)\n  a>>>d-spec = \u2203>>> (->||-Term-l a->||c)\n\n  d : PTerm\n  d = proj\u2081 a>>>d-spec\n\n  a>>>d : a >>> d\n  a>>>d = proj\u2082 a>>>d-spec\n","old_contents":"module Reduction where\n\nopen import Data.Empty\nopen import Data.Nat\nopen import Data.List\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import Core\nopen import Core-Lemmas\n\n_\u219d_ : Set -> Set -> Set\u2081\nA \u219d B = A -> B -> Set\n\n\ndata _->\u03b2_ : PTerm \u219d PTerm where\n  redL : \u2200 {n m m'} -> Term n \u2192 m ->\u03b2 m' -> app m n ->\u03b2 app m' n\n  redR : \u2200 {m n n'} -> Term m \u2192 n ->\u03b2 n' -> app m n ->\u03b2 app m n'\n  abs : \u2200 L {m m'} -> ( \u2200 {x} -> x \u2209 L -> (m ^' x) ->\u03b2 (m' ^' x) ) ->\n    lam m ->\u03b2 lam m'\n  beta : \u2200 {m n} -> Term (lam m) -> Term n -> app (lam m) n ->\u03b2 (m ^ n)\n  Y : \u2200 {m \u03c3} -> Term m -> app (Y \u03c3) m ->\u03b2 app m (app (Y \u03c3) m)\n\n-- test : \u2200 {x} -> lam (app (lam (bv 0)) (fv x)) ->\u03b2 lam (fv x)\n-- test = abs [] (\u03bb _ -> beta (lam [] (\u03bb x\u2209L -> var)) var)\n\ndata _->||_ : PTerm \u219d PTerm where\n  refl : \u2200 {x} -> fv x ->|| fv x\n  reflY : \u2200 {\u03c3} -> Y \u03c3 ->|| Y \u03c3\n  app : \u2200 {m m' n n'} -> m ->|| m' -> n ->|| n' -> app m n ->|| app m' n'\n  abs : \u2200 L {m m'} -> ( \u2200 {x} -> x \u2209 L -> (m ^' x) ->|| (m' ^' x) ) ->\n    lam m ->|| lam m'\n  beta : \u2200 L {m m' n n'} -> (cf : \u2200 {x} -> x \u2209 L -> (m ^' x) ->|| (m' ^' x) ) -> n ->|| n' ->\n    (app (lam m) n) ->|| (m' ^ n')\n  Y : \u2200 {m m' \u03c3} -> m ->|| m' -> app (Y \u03c3) m ->|| app m' (app (Y \u03c3) m')\n\n\ndata NotAbsY : PTerm -> Set where\n  fv : \u2200 {x} -> NotAbsY (fv x)\n  bv : \u2200 {n} -> NotAbsY (bv n)\n  app : \u2200 {t1 t2} -> NotAbsY (app t1 t2)\n\n\ndata _>>>_ : PTerm \u219d PTerm where\n  refl : \u2200 {x} -> fv x >>> fv x\n  reflY : \u2200 {\u03c3} -> Y \u03c3 >>> Y \u03c3\n  app : \u2200 {m m' n n'} -> NotAbsY m -> m >>> m' -> n >>> n' -> app m n >>> app m' n'\n  abs : \u2200 L {m m'} -> ( \u2200 {x} -> x \u2209 L -> (m ^' x) >>> (m' ^' x) ) ->\n    lam m >>> lam m'\n  beta : \u2200 L {m m' n n'} -> (cf : \u2200 {x} -> x \u2209 L -> (m ^' x) >>> (m' ^' x) ) -> n >>> n' ->\n    app (lam m) n >>> (m' ^ n')\n  Y : \u2200 {m m' \u03c3} -> m >>> m' -> app (Y \u03c3) m >>> app m' (app (Y \u03c3) m')\n\n\nsubst-Term : \u2200 {x e u} -> Term e -> Term u -> Term (e [ x ::= u ])\nsubst-Term {x} {_} {u} (var {y}) Term-u with x \u225f y\nsubst-Term var Term-u | yes refl = Term-u\nsubst-Term var Term-u | no p = var\nsubst-Term {x} {_} {u} (lam L {e} cf) Term-u = lam (x \u2237 L) body\n  where\n  body : {y : \u2115} -> y \u2209 x \u2237 L \u2192 Term ((e [ x ::= u ]) ^' y)\n  body {y} y\u2209x\u2237L rewrite subst-open-var y x u e (fv-x\u2260y y x y\u2209x\u2237L) Term-u =\n    subst-Term (cf (\u03bb z \u2192 y\u2209x\u2237L (there z))) Term-u\n\nsubst-Term (app Term-e Term-e\u2081) Term-u =\n  app (subst-Term Term-e Term-u) (subst-Term Term-e\u2081 Term-u)\nsubst-Term Y Term-u = Y\n\n\n^-Term : \u2200 {m n} -> Term (lam m) -> Term n -> Term (m ^ n)\n^-Term {m} {n} (lam L cf) Term-n = body\n  where\n  y = \u2203fresh (L ++ FV m)\n\n  y\u2209 : y \u2209 (L ++ FV m)\n  y\u2209 = \u2203fresh-spec (L ++ FV m)\n\n  body : Term (m ^ n)\n  body rewrite subst-intro y n m (\u2209-cons-r L (FV m) y\u2209) Term-n =\n    subst-Term {y} {m ^' y} {n} (cf (\u2209-cons-l L (FV m) y\u2209)) Term-n\n\n\n->||-Term-l : \u2200 {m m'} -> m ->|| m' -> Term m\n->||-Term-l refl = var\n->||-Term-l reflY = Y\n->||-Term-l (app m->||m' m->||m'') = app (->||-Term-l m->||m') (->||-Term-l m->||m'')\n->||-Term-l (abs L x) = lam L (\u03bb x\u2209L \u2192 ->||-Term-l (x x\u2209L))\n->||-Term-l (beta L x m->||m') =\n  app (lam L (\u03bb {x\u2081} x\u2209L \u2192 ->||-Term-l (x x\u2209L)))\n      (->||-Term-l m->||m')\n->||-Term-l (Y {m} {m'} m->||m') = app Y (->||-Term-l m->||m')\n\n\n->||-Term-r : \u2200 {m m'} -> m ->|| m' -> Term m'\n->||-Term-r refl = var\n->||-Term-r reflY = Y\n->||-Term-r (app m->||m' m->||m'') = app (->||-Term-r m->||m') (->||-Term-r m->||m'')\n->||-Term-r (abs L x) = lam L (\u03bb x\u2209L \u2192 ->||-Term-r (x x\u2209L))\n->||-Term-r (beta L {m} {m'} {n} {n'} cf m->||m') =\n  ^-Term {m'} {n'} (lam L (\u03bb {x\u2081} x\u2209L \u2192 ->||-Term-r (cf x\u2209L))) (->||-Term-r m->||m')\n->||-Term-r (Y m->||m') = app (->||-Term-r m->||m') (app Y (->||-Term-r m->||m'))\n\n\n>>>-Term-l : \u2200 {m m'} -> m >>> m' -> Term m\n>>>-Term-l refl = var\n>>>-Term-l reflY = Y\n>>>-Term-l (app nAbsY m>>>m' m>>>m'') = app (>>>-Term-l m>>>m') (>>>-Term-l m>>>m'')\n>>>-Term-l (abs L x) = lam L (\u03bb x\u2209L \u2192 >>>-Term-l (x x\u2209L))\n>>>-Term-l (beta L x m>>>m') =\n  app (lam L (\u03bb {x\u2081} x\u2209L \u2192 >>>-Term-l (x x\u2209L)))\n      (>>>-Term-l m>>>m')\n>>>-Term-l (Y {m} {m'} m>>>m') = app Y (>>>-Term-l m>>>m')\n\n\n>>>-Term-r : \u2200 {m m'} -> m >>> m' -> Term m'\n>>>-Term-r refl = var\n>>>-Term-r reflY = Y\n>>>-Term-r (app nAbsY m>>>m' m>>>m'') = app (>>>-Term-r m>>>m') (>>>-Term-r m>>>m'')\n>>>-Term-r (abs L x) = lam L (\u03bb x\u2209L \u2192 >>>-Term-r (x x\u2209L))\n>>>-Term-r (beta L {m} {m'} {n} {n'} cf m>>>m') =\n  ^-Term {m'} {n'} (lam L (\u03bb {x\u2081} x\u2209L \u2192 >>>-Term-r (cf x\u2209L))) (>>>-Term-r m>>>m')\n>>>-Term-r (Y m>>>m') = app (>>>-Term-r m>>>m') (app Y (>>>-Term-r m>>>m'))\n\n\nlem2-5-1 : \u2200 s s' t t' (x : \u2115) -> s ->|| s' -> t ->|| t' ->\n  (s [ x ::= t ]) ->|| (s' [ x ::= t' ])\nlem2-5-1 _ _ t t' y (refl {x}) t->||t' with x \u225f y\nlem2-5-1 .(fv x) .(fv x) t t' x refl t->||t' | yes refl rewrite\n  fv-subst-eq x x t refl | fv-subst-eq x x t' refl = t->||t'\nlem2-5-1 _ _ t t' y (refl {x}) t->||t'             | no x\u2260y rewrite\n  fv-subst-neq x y t x\u2260y | fv-subst-neq x y t' x\u2260y = refl\nlem2-5-1 _ _ t t' x reflY t->||t' = reflY\nlem2-5-1 _ _ t t' x (app {m} {m'} {n} {n'} ss' ss'') t->||t' = app (lem2-5-1 m m' t t' x ss' t->||t') (lem2-5-1 n n' t t' x ss'' t->||t')\nlem2-5-1 _ _ t t' x (abs L {m} {m'} cf) t->||t' = abs (x \u2237 L) body\n  where\n  x\u2209FVy : (y : \u2115) -> (y \u2209 x \u2237 L) -> x \u2209 FV (fv y)\n  x\u2209FVy y y\u2209x\u2237L x\u2208FVy with fv-x\u2261y x y x\u2208FVy\n  x\u2209FVy .x y\u2209x\u2237L x\u2208FVy | refl = y\u2209x\u2237L (here refl)\n\n  body : {y : \u2115} -> (y \u2209 x \u2237 L) -> ((m [ x ::= t ]) ^' y) ->|| ((m' [ x ::= t' ]) ^' y)\n  body {y} y\u2209x\u2237L rewrite\n    subst-fresh2 x (fv y) t (x\u2209FVy y y\u2209x\u2237L) |\n    subst-open2 x t 0 (fv y) m (->||-Term-l t->||t') |\n    subst-fresh x (fv y) t (x\u2209FVy y y\u2209x\u2237L) |\n    subst-fresh2 x (fv y) t' (x\u2209FVy y y\u2209x\u2237L) |\n    subst-open2 x t' 0 (fv y) m' (->||-Term-r t->||t') |\n    subst-fresh x (fv y) t' (x\u2209FVy y y\u2209x\u2237L) =\n      lem2-5-1 (m ^' y) (m' ^' y) t t' x (cf (\u03bb z \u2192 y\u2209x\u2237L (there z))) t->||t'\n\nlem2-5-1 _ _ t t' x (beta L {m} {m'} {n} {n'} cf n->||n') t->||t' rewrite\n  subst-open x t' 0 n' m' (->||-Term-r t->||t') =\n    beta (x \u2237 L) {_} {m' [ x ::= t' ]}  body body\u2082\n  where\n  body : {y : \u2115} -> y \u2209 x \u2237 L -> ((m [ x ::= t ]) ^' y) ->|| ((m' [ x ::= t' ]) ^' y)\n  body {y} y\u2209x\u2237L rewrite\n    subst-open-var y x t m (fv-x\u2260y y x y\u2209x\u2237L) (->||-Term-l t->||t') |\n    subst-open-var y x t' m' (fv-x\u2260y y x y\u2209x\u2237L) (->||-Term-r t->||t') =\n      lem2-5-1 (m ^' y) (m' ^' y) t t' x (cf (\u03bb z \u2192 y\u2209x\u2237L (there z))) t->||t'\n\n  body\u2082 : (n [ x ::= t ]) ->|| (n' [ x ::= t' ])\n  body\u2082 = lem2-5-1 n n' t t' x n->||n' t->||t'\n\nlem2-5-1 _ _ t t' x (Y {m} {m'} ss') t->||t' = Y (lem2-5-1 m m' t t' x ss' t->||t')\n\n\nlem2-5-1-^ : \u2200 s s' t t' L -> (\u2200 {x} -> x \u2209 L -> (s ^' x) ->|| (s' ^' x)) -> t ->|| t' ->\n  (s ^ t) ->|| (s' ^ t')\nlem2-5-1-^ s s' t t' L cf t->||t' = body\n  where\n  x = \u2203fresh (L ++ FV s ++ FV s')\n\n  x\u2209 : x \u2209 (L ++ FV s ++ FV s')\n  x\u2209 = \u2203fresh-spec (L ++ FV s ++ FV s')\n\n  body : (s ^ t) ->|| (s' ^ t')\n  body rewrite\n    subst-intro x t s (\u2209-cons-l (FV s) (FV s') (\u2209-cons-r L (FV s ++ FV s') x\u2209)) (->||-Term-l t->||t') |\n    subst-intro x t' s' (\u2209-cons-r (FV s) (FV s') (\u2209-cons-r L (FV s ++ FV s') x\u2209)) (->||-Term-r t->||t') =\n      lem2-5-1 (s ^' x) (s' ^' x) t t' x (cf (\u2209-cons-l L (FV s ++ FV s') x\u2209)) t->||t'\n\n\nNotAbsY-subst : \u2200 {x y m} -> NotAbsY m -> NotAbsY (m [ x ::= fv y ])\nNotAbsY-subst {x} (fv {y}) with x \u225f y\nNotAbsY-subst {x} {y} fv | yes refl rewrite\n  fv-subst-eq x x (fv y) refl = fv\nNotAbsY-subst fv | no _ = fv\nNotAbsY-subst bv = bv\nNotAbsY-subst app = app\n\n\nlem2-5-1>>> : \u2200 s s' (x y : \u2115) -> s >>> s' -> (s [ x ::= fv y ]) >>> (s' [ x ::= fv y ])\nlem2-5-1>>> _ _ x y (refl {z}) with x \u225f z\nlem2-5-1>>> .(fv x) .(fv x) x y refl | yes refl rewrite\n  fv-subst-eq x x (fv y) refl = refl\nlem2-5-1>>> _ _ x y (refl {z}) | no z\u2260x = refl\nlem2-5-1>>> _ _ x y reflY = reflY\nlem2-5-1>>> _ _ x y (app {m} {m'} {n} {n'} \u00acabsY ss' ss'') =\n  app (NotAbsY-subst \u00acabsY) (lem2-5-1>>> m m' x y ss') (lem2-5-1>>> n n' x y ss'')\nlem2-5-1>>> _ _ x y (abs L {m} {m'} cf) = abs (x \u2237 L) body\n  where\n  x\u2209FVz : (z : \u2115) -> (z \u2209 x \u2237 L) -> x \u2209 FV (fv z)\n  x\u2209FVz z z\u2209x\u2237L x\u2208FVz with fv-x\u2261y x z x\u2208FVz\n  x\u2209FVz .x z\u2209x\u2237L x\u2208FVz | refl = z\u2209x\u2237L (here refl)\n\n  body : {z : \u2115} -> (z \u2209 x \u2237 L) -> ((m [ x ::= fv y ]) ^' z) >>> ((m' [ x ::= fv y ]) ^' z)\n  body {z} z\u2209x\u2237L rewrite\n    subst-fresh2 x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) |\n    subst-open2 x (fv y) 0 (fv z) m var |\n    subst-fresh x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) |\n    subst-fresh2 x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) |\n    subst-open2 x (fv y) 0 (fv z) m' var |\n    subst-fresh x (fv z) (fv y) (x\u2209FVz z z\u2209x\u2237L) =\n      lem2-5-1>>> (m ^' z) (m' ^' z) x y (cf (\u03bb z\u2081 \u2192 z\u2209x\u2237L (there z\u2081)))\n\nlem2-5-1>>> _ _ x y (beta L {m} {m'} {n} {n'} cf n>>>n') rewrite\n  subst-open x (fv y) 0 n' m' var =\n    beta (x \u2237 L) {_} {m' [ x ::= fv y ]} body body\u2082\n  where\n  body : {z : \u2115} -> z \u2209 x \u2237 L -> ((m [ x ::= fv y ]) ^' z) >>> ((m' [ x ::= fv y ]) ^' z)\n  body {z} z\u2209x\u2237L rewrite\n    subst-open-var z x (fv y) m (fv-x\u2260y z x z\u2209x\u2237L) var |\n    subst-open-var z x (fv y) m' (fv-x\u2260y z x z\u2209x\u2237L) var =\n      lem2-5-1>>> (m ^' z) (m' ^' z) x y (cf (\u03bb z\u2081 \u2192 z\u2209x\u2237L (there z\u2081)))\n\n  body\u2082 : (n [ x ::= fv y ]) >>> (n' [ x ::= fv y ])\n  body\u2082 = lem2-5-1>>> n n' x y n>>>n'\n\nlem2-5-1>>> _ _ x y (Y {m} {m'} ss') = Y (lem2-5-1>>> m m' x y ss')\n\n\n*^-^->>> : \u2200 {x y t d k} -> t >>> d -> y \u2209 x \u2237 (FV t ++ FV d) -> ([ k >> fv y ] ([ k << x ] t)) >>> ([ k >> fv y ] ([ k << x ] d))\n*^-^->>> {x} {y} {t} {d} {k} t>>>d y\u2209 rewrite\n  *^-^\u2261subst t x y {k} (>>>-Term-l t>>>d) | *^-^\u2261subst d x y {k} (>>>-Term-r t>>>d) = lem2-5-1>>> t d x y t>>>d\n\n\n\u2203>>> : \u2200 {a} -> Term a -> \u2203(\u03bb d -> a >>> d)\n\u2203>>> (var {x}) = fv x , refl\n\u2203>>> (lam L {t} cf) = body\n  where\n  x = \u2203fresh (L ++ FV t)\n\n  x\u2209 : x \u2209 (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n\n  d-spec : \u2203(\u03bb d -> (t ^' x) >>> d)\n  d-spec = \u2203>>> (cf (\u2209-cons-l L (FV t) x\u2209))\n\n  d : PTerm\n  d = proj\u2081 d-spec\n\n  subst1 : \u2200 x y t k -> x \u2209 FV t -> (t ^' y) \u2261 (([ k << x ] ([ k >> fv x ] t)) ^' y)\n  subst1 x y t k x\u2209FVt rewrite fv-^-*^-refl x t {k} x\u2209FVt = refl\n\n  cf' : \u2200 {y} -> y \u2209 x \u2237 (FV d ++ FV t) -> (t ^' y) >>> ((* x ^ d) ^' y)\n  cf' {y} y\u2209 rewrite subst1 x y t 0 (\u2209-cons-r L (FV t) x\u2209) =\n    *^-^->>> {x} {y} {t ^' x} {d} {0} (proj\u2082 d-spec)\n      (\u2209-\u2237 x (FV (t ^' x) ++ FV d) (fv-x\u2260y y x y\u2209)\n        (\u2209-cons-intro (FV (t ^' x)) (FV d)\n          (fv-^ {0} {y} {x} t (\u2209-cons-r (FV d) _ (\u2209-\u2237-elim _ y\u2209)) (fv-x\u2260y _ _ y\u2209))\n          (\u2209-cons-l _ (FV t) (\u2209-\u2237-elim _ y\u2209))))\n\n  body : \u2203(\u03bb d -> (lam t) >>> d)\n  body = lam (* x ^ d) , (abs (x \u2237 (FV d ++ FV t)) cf')\n\n\u2203>>> (app {t1} {t2} trm-t1 trm-t2) with trm-t1 | \u2203>>> trm-t1 | \u2203>>> trm-t2\n\u2203>>> (app trm-t1 trm-t2) | (var {x}) | d1 , t1>>>d1 | d2 , t2>>>d2 = app (fv x) d2 , app fv refl t2>>>d2\n\u2203>>> (app {_} {t2} trm-t1 trm-t2) | lam L cf | _ , abs L\u2081 {t1} {d1} cf\u2081 | d2 , t2>>>d2 =\n  d1 ^ d2 , beta L\u2081 {t1} {d1} {t2} {d2} cf\u2081 t2>>>d2\n\u2203>>> (app {_} {t2} trm-t1 trm-t2) | app {p1} {p2} trm-p1 trm-p2 | d1 , t1>>>d1 | d2 , t2>>>d2 =\n  app d1 d2 , app app t1>>>d1 t2>>>d2\n\u2203>>> (app trm-t1 trm-t2) | (Y {t}) | d1 , t1>>>d1 | d2 , t2>>>d2 = app d2 (app (Y t) d2) , Y t2>>>d2\n\u2203>>> (Y {t}) = Y t , reflY\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"047282f00de5f7bd07a7ab82432a54bc466d77a0","subject":"Prove that \u0394-Context strictly increases the context (see #29).","message":"Prove that \u0394-Context strictly increases the context (see #29).\n\nOld-commit-hash: 528a74e6ca967a4ad12946c97f4f77491928beb1\n","repos":"inc-lc\/ilc-agda","old_file":"ChangeContexts.agda","new_file":"ChangeContexts.agda","new_contents":"module ChangeContexts where\n\n-- CHANGE CONTEXTS\n--\n-- This module defines change contexts, that is, contexts where\n-- for every value assertion x : \u03c4, we also have a change\n-- assertion ds : \u0394-Type \u03c4.\n\n-- CHANGE ENVIRONMENTS\n--\n-- This module describes operations on change environments, that\n-- is, environments where for every value binding x = \u27e6 \u03c4 \u27e7, we\n-- also have a change assertion dx = \u27e6 \u0394-Type \u03c4 \u27e7.\n\nopen import Syntactic.Types\nopen import Changes\nopen import Denotational.Values\nopen import Denotational.Notation\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import Syntactic.Contexts Type\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n-- Properties\n\nmodule _ where\n  open Subcontexts renaming (drop_\u2022_ to drop\u2032_\u2022_)\n\n  \u227c-\u0394-Context : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n  \u227c-\u0394-Context {\u2205} = \u2205\n  \u227c-\u0394-Context {\u03c4 \u2022 \u0393} = drop\u2032 (\u0394-Type \u03c4) \u2022 keep \u03c4 \u2022 \u227c-\u0394-Context\n\n-- OPERATIONS on CHANGE ENVIRONMENTS\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n-- \u0394-Context\u2032: behaves like \u0394-Context, but has an extra argument \u0393\u2032, a\n-- prefix of its first argument which should not be touched.\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\nopen import Relation.Binary.PropositionalEquality as P\n\n\u0394-Context-\u22ce : \u2200 \u0393\u2081 \u0393\u2082 \u2192\n  \u0394-Context (\u0393\u2081 \u22ce \u0393\u2082) P.\u2261 \u0394-Context \u0393\u2081 \u22ce \u0394-Context \u0393\u2082\n\u0394-Context-\u22ce \u2205 \u0393\u2082 = refl\n\u0394-Context-\u22ce (\u03c4 \u2022 \u0393\u2081) \u0393\u2082 rewrite \u0394-Context-\u22ce \u0393\u2081 \u0393\u2082 = refl\n\n\u0394-Context-\u22ce-expanded : \u2200 \u0393\u2081 \u03c4\u2082 \u0393\u2082 \u2192\n  \u0394-Context \u0393\u2081 \u22ce (\u0394-Type \u03c4\u2082 \u2022 \u03c4\u2082 \u2022 \u0394-Context \u0393\u2082) P.\u2261 \u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2082))\n\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2082 rewrite \u0394-Context-\u22ce \u0393\u2081 (\u03c4\u2082 \u2022 \u0393\u2082) = refl\n\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 : \u2200 \u0393 \u0393\u2032 \u2192\n  \u0394-Context\u2032 \u0393 \u0393\u2032 P.\u2261 take \u0393 \u0393\u2032 \u22ce \u0394-Context (drop \u0393 \u0393\u2032)\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u2205 \u2205 = refl\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u03c4 \u2022 \u0393) \u2205 = refl\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032)\n  rewrite take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032 = refl\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken = {!!}\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {.(\u0394-Type \u03c4) \u2022 .\u03c4 \u2022 _} {\u0393\u2082} (\u0394 {\u03c4 \u2022 \u0393\u2083} t) = ?\n--weaken {.(\u0394-Type \u03c4) \u2022 .\u03c4 \u2022 _} {\u0393\u2082} (\u0394 {\u03c4 \u2022 \u0393\u2083} t) = ?\n--weaken {(\u0394-Type \u03c4) \u2022 \u03c4 \u2022 _} {\u0393\u2082} (\u0394 {\u03c4 \u2022 \u0393\u2083} t) = ?\n--weaken {(\u0394-Context .\u0393\u2081)} {\u0393\u2082} (\u0394 {\u0393\u2081 \u22ce \u0393\u2083} t) = ?\n-}\n","old_contents":"module ChangeContexts where\n\n-- CHANGE CONTEXTS\n--\n-- This module defines change contexts, that is, contexts where\n-- for every value assertion x : \u03c4, we also have a change\n-- assertion ds : \u0394-Type \u03c4.\n\n-- CHANGE ENVIRONMENTS\n--\n-- This module describes operations on change environments, that\n-- is, environments where for every value binding x = \u27e6 \u03c4 \u27e7, we\n-- also have a change assertion dx = \u27e6 \u0394-Type \u03c4 \u27e7.\n\nopen import Syntactic.Types\nopen import Changes\nopen import Denotational.Values\nopen import Denotational.Notation\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import Syntactic.Contexts Type\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n-- \u0394-Context\u2032: behaves like \u0394-Context, but has an extra argument \u0393\u2032, a\n-- prefix of its first argument which should not be touched.\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\nopen import Relation.Binary.PropositionalEquality as P\n\n\u0394-Context-\u22ce : \u2200 \u0393\u2081 \u0393\u2082 \u2192\n  \u0394-Context (\u0393\u2081 \u22ce \u0393\u2082) P.\u2261 \u0394-Context \u0393\u2081 \u22ce \u0394-Context \u0393\u2082\n\u0394-Context-\u22ce \u2205 \u0393\u2082 = refl\n\u0394-Context-\u22ce (\u03c4 \u2022 \u0393\u2081) \u0393\u2082 rewrite \u0394-Context-\u22ce \u0393\u2081 \u0393\u2082 = refl\n\n\u0394-Context-\u22ce-expanded : \u2200 \u0393\u2081 \u03c4\u2082 \u0393\u2082 \u2192\n  \u0394-Context \u0393\u2081 \u22ce (\u0394-Type \u03c4\u2082 \u2022 \u03c4\u2082 \u2022 \u0394-Context \u0393\u2082) P.\u2261 \u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2082))\n\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2082 rewrite \u0394-Context-\u22ce \u0393\u2081 (\u03c4\u2082 \u2022 \u0393\u2082) = refl\n\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 : \u2200 \u0393 \u0393\u2032 \u2192\n  \u0394-Context\u2032 \u0393 \u0393\u2032 P.\u2261 take \u0393 \u0393\u2032 \u22ce \u0394-Context (drop \u0393 \u0393\u2032)\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u2205 \u2205 = refl\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u03c4 \u2022 \u0393) \u2205 = refl\ntake-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032)\n  rewrite take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032 = refl\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken = {!!}\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {.(\u0394-Type \u03c4) \u2022 .\u03c4 \u2022 _} {\u0393\u2082} (\u0394 {\u03c4 \u2022 \u0393\u2083} t) = ?\n--weaken {.(\u0394-Type \u03c4) \u2022 .\u03c4 \u2022 _} {\u0393\u2082} (\u0394 {\u03c4 \u2022 \u0393\u2083} t) = ?\n--weaken {(\u0394-Type \u03c4) \u2022 \u03c4 \u2022 _} {\u0393\u2082} (\u0394 {\u03c4 \u2022 \u0393\u2083} t) = ?\n--weaken {(\u0394-Context .\u0393\u2081)} {\u0393\u2082} (\u0394 {\u0393\u2081 \u22ce \u0393\u2083} t) = ?\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"241582d7212cda6ed5b4d48058af4392fc61bce7","subject":"Desc model: followed by the generic proof that map (f . g) = map f . map g","message":"Desc model: followed by the generic proof that map (f . g) = map f . map g\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A : Set}{B : A -> Set}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\nproof-map-compos : (D : Desc)(X Y Z : Set)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A : Set}{B : A -> Set}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c29bc25eaf29f5b062c9326b9ad3562ef87c382","subject":"Bits: more search properties","message":"Bits: more search properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u00b7 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule SimpleSearch {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u00b7_ public\n\n    search-\u00b7-\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-\u00b7-\u03b5\u2261\u03b5 \u03b5 \u03b5\u00b7\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module Interchange (\u00b7-interchange : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (x \u00b7 z) \u00b7 (y \u00b7 t)) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u00b7 f\u2081 x) \u2261 search f\u2080 \u00b7 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u00b7-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\nopen SimpleSearch public\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-\u2257 : \u2200 {n} (f g : Bits n \u2192 \u2115) \u2192 f \u2257 g \u2192 sum f \u2261 sum g\nsum-\u2257 = search-\u2257 _+_\n\nsum-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 \u2115) \u2192 sum f \u2261 sum (f \u2218 _\u2295_ pad)\nsum-comm = search\u2032-comm _+_ \u2115\u00b0.+-comm\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n-- #-ext\n#-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n#-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n#-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n#-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 = #-comm\n\n#-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n#-const n b = sum-const n (Bool.to\u2115 b)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n#-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n#-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n           \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n             2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n           \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n             2^ c \u2238 #\u27e8 f \u27e9\n           \u220e\n  where open \u2261-Reasoning\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero} _+_ _*_ f p hom = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {A = A} _\u00b7_ f = search\u2032 {A = const A} _\u00b7_ f\n\nsearch-\u2257 : \u2200 {n a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) (f g : Bits n \u2192 A) \u2192 f \u2257 g \u2192 search _\u00b7_ f \u2261 search _\u00b7_ g\nsearch-\u2257 _\u00b7_ f g f\u2257g = search\u2032-\u2257 _\u00b7_ f g f\u2257g\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-\u2257 : \u2200 {n} (f g : Bits n \u2192 \u2115) \u2192 f \u2257 g \u2192 sum f \u2261 sum g\nsum-\u2257 = search-\u2257 _+_\n\nsum-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 \u2115) \u2192 sum f \u2261 sum (f \u2218 _\u2295_ pad)\nsum-comm = search\u2032-comm _+_ \u2115\u00b0.+-comm\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n-- #-ext\n#-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n#-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n#-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n#-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 = #-comm\n\n#-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n#-const n b = sum-const n (Bool.to\u2115 b)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\nsearch-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\nsearch-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n  where\n    go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n    go zero = refl\n    go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n#-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n#-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n           \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n             2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n           \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n             2^ c \u2238 #\u27e8 f \u27e9\n           \u220e\n  where open \u2261-Reasoning\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op (|de-morgan| f g)\n\nsearch-comm :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-comm {zero} _+_ _*_ f p hom = refl\nsearch-comm {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-comm {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-comm _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"569e98b5b6d8a84329f54249ad3f83477357f717","subject":"Switch count\u21ba to be based on search","message":"Switch count\u21ba to be based on search\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-implem.agda","new_file":"flipbased-implem.agda","new_contents":"module flipbased-implem where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Vec\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Data.Fin as Fin\nopen \u2261 using (_\u2257_; _\u2261_)\nopen Fin using (Fin; suc)\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run\u21ba : Bits n \u2192 A\n\nopen \u21ba public\n\nprivate\n  -- If you are not allowed to toss any coin, then you are deterministic.\n  Det : \u2200 {a} \u2192 Set a \u2192 Set a\n  Det = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet f = run\u21ba f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn\u21ba = mk \u2218 const\n\nmap\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap\u21ba f x = mk (f \u2218 run\u21ba x)\n-- map\u21ba f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin\u21ba {n\u2081} x = mk (\u03bb bs \u2192 run\u21ba (run\u21ba x (take _ bs)) (drop n\u2081 bs))\n-- join\u21ba x = x >>= id\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\n_\u2257\u21ba_ : \u2200 {c a} {A : Set a} (f g : \u21ba c A) \u2192 Set a\nf \u2257\u21ba g = run\u21ba f \u2257 run\u21ba g\n\n\u2141 : \u2115 \u2192 Set\n\u2141 n = \u21ba n Bit\n\n_\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c) \u2192 Set\n\u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n\u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n\u2257\u2141-trans p q b R = \u2261.trans (p b R) (q b R)\n\ncount\u21ba\u1da0 : \u2200 {c} \u2192 \u2141 c \u2192 Fin (suc (2^ c))\ncount\u21ba\u1da0 f = #\u27e8 run\u21ba f \u27e9\u1da0\n\ncount\u21ba : \u2200 {c} \u2192 \u2141 c \u2192 \u2115\ncount\u21ba f = #\u27e8 run\u21ba f \u27e9\n\n\u2257\u21ba-cong-# : \u2200 {n} {f g : \u2141 n} \u2192 f \u2257\u21ba g \u2192 count\u21ba f \u2261 count\u21ba g\n\u2257\u21ba-cong-# {f = f} {g} = \u2257-cong-# (run\u21ba f) (run\u21ba g)\n\n_\u223c[_]\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141_ {m} {n} f _\u223c_ g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\n_\u223c[_]\u2141\u2032_ : \u2200 {n} \u2192 \u2141 n \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141\u2032_ {n} f _\u223c_ g = count\u21ba f \u223c count\u21ba g\n\n_\u2248\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\nf \u2248\u2141 g = f \u223c[ _\u2261_ ]\u2141 g\n\n_\u2248\u2141\u2032_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\nf \u2248\u2141\u2032 g = f \u223c[ _\u2261_ ]\u2141\u2032 g\n\n\u2248\u2141-refl : \u2200 {n} {f : \u2141 n} \u2192 f \u2248\u2141 f\n\u2248\u2141-refl = \u2261.refl\n\n\u2248\u2141-sym : \u2200 {n} \u2192 Symmetric {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-sym = \u2261.sym\n\n\u2248\u2141-trans : \u2200 {n} \u2192 Transitive {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-trans = \u2261.trans\n\n\u2257\u21d2\u2248\u2141 : \u2200 {c} {f g : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f \u2248\u2141 g\n\u2257\u21d2\u2248\u2141 pf rewrite \u2257\u21ba-cong-# pf = \u2261.refl\n\n\u2248\u2141\u2032\u21d2\u2248\u2141 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141\u2032 g \u2192 f \u2248\u2141 g\n\u2248\u2141\u2032\u21d2\u2248\u2141 eq rewrite eq = \u2261.refl\n\n\u2248\u2141\u21d2\u2248\u2141\u2032 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141 g \u2192 f \u2248\u2141\u2032 g\n\u2248\u2141\u21d2\u2248\u2141\u2032 {n} = 2^-inj n\n\n\u2248\u2141-cong : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141 f' \u2192 g \u2248\u2141 g'\n\u2248\u2141-cong f\u2257g f'\u2257g' f\u2248f' rewrite \u2257\u21ba-cong-# f\u2257g | \u2257\u21ba-cong-# f'\u2257g' = f\u2248f'\n\n\u2248\u2141\u2032-cong : \u2200 {c} {f g f' g' : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141\u2032 f' \u2192 g \u2248\u2141\u2032 g'\n\u2248\u2141\u2032-cong f\u2257g f'\u2257g' f\u2248f' rewrite \u2257\u21ba-cong-# f\u2257g | \u2257\u21ba-cong-# f'\u2257g' = f\u2248f'\n\ndata Rat : Set where _\/_ : (num denom : \u2115) \u2192 Rat\n\nPr[_\u22611] : \u2200 {n} (f : \u2141 n) \u2192 Rat\nPr[_\u22611] {n} f = count\u21ba f \/ 2^ n\n","old_contents":"module flipbased-implem where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Vec\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Data.Fin as Fin\nopen \u2261 using (_\u2257_; _\u2261_)\nopen Fin using (Fin; suc)\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run\u21ba : Bits n \u2192 A\n\nopen \u21ba public\n\nprivate\n  -- If you are not allowed to toss any coin, then you are deterministic.\n  Det : \u2200 {a} \u2192 Set a \u2192 Set a\n  Det = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet f = run\u21ba f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn\u21ba = mk \u2218 const\n\nmap\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap\u21ba f x = mk (f \u2218 run\u21ba x)\n-- map\u21ba f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin\u21ba {n\u2081} x = mk (\u03bb bs \u2192 run\u21ba (run\u21ba x (take _ bs)) (drop n\u2081 bs))\n-- join\u21ba x = x >>= id\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\n_\u2257\u21ba_ : \u2200 {c a} {A : Set a} (f g : \u21ba c A) \u2192 Set a\nf \u2257\u21ba g = run\u21ba f \u2257 run\u21ba g\n\n\u2141 : \u2115 \u2192 Set\n\u2141 n = \u21ba n Bit\n\n_\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c) \u2192 Set\n\u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n\u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n\u2257\u2141-trans p q b R = \u2261.trans (p b R) (q b R)\n\ncount\u21ba\u1da0 : \u2200 {c} \u2192 \u2141 c \u2192 Fin (suc (2^ c))\ncount\u21ba\u1da0 f = #\u27e8 run\u21ba f \u27e9\u1da0\n\ncount\u21ba : \u2200 {c} \u2192 \u2141 c \u2192 \u2115\ncount\u21ba = Fin.to\u2115 \u2218 count\u21ba\u1da0\n\n_\u223c[_]\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141_ {m} {n} f _\u223c_ g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\n_\u223c[_]\u2141\u2032_ : \u2200 {n} \u2192 \u2141 n \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141\u2032_ {n} f _\u223c_ g = count\u21ba f \u223c count\u21ba g\n\n_\u2248\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\nf \u2248\u2141 g = f \u223c[ _\u2261_ ]\u2141 g\n\n_\u2248\u2141\u2032_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\nf \u2248\u2141\u2032 g = f \u223c[ _\u2261_ ]\u2141\u2032 g\n\n\u2248\u2141-refl : \u2200 {n} {f : \u2141 n} \u2192 f \u2248\u2141 f\n\u2248\u2141-refl = \u2261.refl\n\n\u2248\u2141-sym : \u2200 {n} \u2192 Symmetric {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-sym = \u2261.sym\n\n\u2248\u2141-trans : \u2200 {n} \u2192 Transitive {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-trans = \u2261.trans\n\n\u2257\u21d2\u2248\u2141 : \u2200 {c} {f g : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f \u2248\u2141 g\n\u2257\u21d2\u2248\u2141 pf rewrite ext-# pf = \u2261.refl\n\n\u2248\u2141\u2032\u21d2\u2248\u2141 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141\u2032 g \u2192 f \u2248\u2141 g\n\u2248\u2141\u2032\u21d2\u2248\u2141 eq rewrite eq = \u2261.refl\n\n\u2248\u2141\u21d2\u2248\u2141\u2032 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141 g \u2192 f \u2248\u2141\u2032 g\n\u2248\u2141\u21d2\u2248\u2141\u2032 {n} = 2^-inj n\n\n\u2248\u2141-cong : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141 f' \u2192 g \u2248\u2141 g'\n\u2248\u2141-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n\n\u2248\u2141\u2032-cong : \u2200 {c} {f g f' g' : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141\u2032 f' \u2192 g \u2248\u2141\u2032 g'\n\u2248\u2141\u2032-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b590f8a3ed001a65516e5f2211013e1a46683692","subject":"Update binds","message":"Update binds\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/PartialHeliosVerifier.agda","new_file":"ZK\/PartialHeliosVerifier.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.PartialHeliosVerifier where\n\nopen import Type\nopen import Function hiding (case_of_)\nopen import Data.Bool.Base\nopen import Data.Product\nopen import Data.List.Base using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base  using (String)\n\nopen import FFI.JS as JS hiding (_+_; _\/_; _*_; join)\nopen import FFI.JS.BigI as BigI\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\nopen import FFI.JS.SHA1\nopen import FFI.JS.Proc\nopen import Control.Process.Type\n\njoin : String \u2192 List String \u2192 String\njoin sep []       = \"\"\njoin sep (x \u2237 xs) = x ++ foldr (\u03bb y z \u2192 sep ++ y ++ z) \"\" xs\n\n{-\ninstance\n  showBigI = mk \u03bb _ \u2192 showString \u2218 BigI.toString\n-}\n\nG  = BigI\n\u2124q = BigI\n\nBigI\u25b9G : BigI \u2192 G\nBigI\u25b9G = id\n\nBigI\u25b9\u2124q : BigI \u2192 \u2124q\nBigI\u25b9\u2124q = id\n\nnon0I : BigI \u2192 BigI\nnon0I x with equals x 0I\n... | true  = throw \"Should be non zero!\" 0I\n... | false = x\n\nmodule BigICG (p q : BigI) where\n  _^_  : G  \u2192 \u2124q \u2192 G\n  _\u00b7_  : G  \u2192 G  \u2192 G\n  _\/_  : G  \u2192 G  \u2192 G\n  _+_  : \u2124q \u2192 \u2124q \u2192 \u2124q\n  _*_  : \u2124q \u2192 \u2124q \u2192 \u2124q\n  _==_ : (x y : G) \u2192 Bool\n\n  _^_ = \u03bb x y \u2192 modPow x y p\n  _\u00b7_ = \u03bb x y \u2192 mod (multiply x y) p\n  _\/_ = \u03bb x y \u2192 mod (multiply x (modInv (non0I y) p)) p\n  _+_ = \u03bb x y \u2192 mod (add x y) q\n  _*_ = \u03bb x y \u2192 mod (multiply x y) q\n  _==_ = equals\n\n  sumI : List BigI \u2192 BigI\n  sumI = foldr _+_ 0I\n\nbignum : Number \u2192 BigI\nbignum n = bigI (Number\u25b9String n) \"10\"\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n{-\nprint : {A : Set}{{_ : Show A}} \u2192 A \u2192 Callback0\nprint = Console.log \u2218 show\n-}\n\nPubKey         = G\nEncRnd         = \u2124q {- randomness used for encryption of ct -}\nMessage        = G {- plain text message -}\nChallenge      = \u2124q\nResponse       = \u2124q\nCommitmentPart = G\nCipherTextPart = G\n\nmodule HeliosVerifyV3 (g : G) p q (y : PubKey) where\n  open BigICG p q\n\n  verify-chaum-pedersen : (\u03b1 \u03b2 : CipherTextPart)(M : Message)(A B : G)(c : Challenge)(s : Response) \u2192 Bool\n  verify-chaum-pedersen \u03b1 \u03b2 M A B c s\n      = trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n        (g ^ s) == (A \u00b7 (\u03b1 ^ c))\n      \u2227 (y ^ s) == (B \u00b7 ((\u03b2 \/ M) ^ c))\n\n  verify-individual-proof : (\u03b1 \u03b2 : CipherTextPart)(ix : Number)(\u03c0 : JSValue) \u2192 Bool\n  verify-individual-proof \u03b1 \u03b2 ix \u03c0\n      = trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"commitmentA=\" A \u03bb _ \u2192\n        trace \"commitmentB=\" B \u03bb _ \u2192\n        trace \"challenge=\" c \u03bb _ \u2192\n        trace \"response=\" s \u03bb _ \u2192\n        res\n      where\n        m  = bignum ix\n        M  = g ^ m\n        A  = bigdec (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"A\" \u00bb)\n        B  = bigdec (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"B\" \u00bb)\n        c  = bigdec (\u03c0 \u00b7\u00ab \"challenge\" \u00bb)\n        s  = bigdec (\u03c0 \u00b7\u00ab \"response\" \u00bb)\n        res = verify-chaum-pedersen \u03b1 \u03b2 M A B c s\n\n  -- Conform to HeliosV3 but it is too weak.\n  -- One should follow a \"Strong Fiat Shamir\" transformation\n  -- from interactive ZK proof to non-interactive ZK proofs.\n  -- Namely one should hash the statement as well.\n  --\n  -- SHA1(A0 + \",\" + B0 + \",\" + A1 + \",\" + B1 + ... + \"Amax\" + \",\" + Bmax)\n  hash-commitments : JSArray JSValue \u2192 BigI\n  hash-commitments \u03c0s =\n    fromHex $\n    trace-call \"SHA1(commitments)=\" SHA1       $\n    trace-call \"commitments=\"       (join \",\") $\n    decodeJSArray \u03c0s \u03bb _ \u03c0 \u2192\n      (castString (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"A\" \u00bb))\n      ++ \",\" ++\n      (castString (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"B\" \u00bb))\n\n  sum-challenges : JSArray JSValue \u2192 BigI\n  sum-challenges \u03c0s =\n    sumI $ decodeJSArray \u03c0s \u03bb _ \u03c0 \u2192 bigdec (\u03c0 \u00b7\u00ab \"challenge\" \u00bb)\n\n  verify-challenges : JSArray JSValue \u2192 Bool\n  verify-challenges \u03c0s =\n      trace \"hash(commitments)=\" h \u03bb _ \u2192\n      trace \"sum(challenge)=\"    c \u03bb _ \u2192\n      h == c\n    where\n      h = hash-commitments \u03c0s\n      c = sum-challenges   \u03c0s\n\n  verify-choice : (v : JSValue)(\u03c0s : JSArray JSValue) \u2192 Bool\n  verify-choice v \u03c0s =\n      trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n      trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n      verify-challenges \u03c0s\n      \u2227\n      and (decodeJSArray \u03c0s $ verify-individual-proof \u03b1 \u03b2)\n    where\n      \u03b1 = bigdec (v \u00b7\u00ab \"alpha\" \u00bb)\n      \u03b2 = bigdec (v \u00b7\u00ab \"beta\"  \u00bb)\n\n  verify-choices : (choices \u03c0s : JSArray JSValue) \u2192 Bool\n  verify-choices choices \u03c0s =\n    trace \"TODO: check the array size of choices \u03c0s together election data\" \"\" \u03bb _ \u2192\n    and $ decodeJSArray choices \u03bb ix choice \u2192\n      trace \"verify choice \" ix \u03bb _ \u2192\n      verify-choice choice (castJSArray (\u03c0s Array[ ix ]))\n\n  verify-answer : (answer : JSValue) \u2192 Bool\n  verify-answer answer =\n      trace \"TODO: CHECK the overall_proof\" \"\" \u03bb _ \u2192\n      verify-choices choices individual-proofs\n    where\n      choices           = answer \u00b7\u00ab \"choices\" \u00bbA\n      individual-proofs = answer \u00b7\u00ab \"individual_proofs\" \u00bbA\n      overall-proof     = answer \u00b7\u00ab \"overall_proof\" \u00bbA\n\n  verify-answers : (answers : JSArray JSValue) \u2192 Bool\n  verify-answers answers =\n    and $ decodeJSArray answers \u03bb ix a \u2192\n      trace \"verify answer \" ix \u03bb _ \u2192\n      verify-answer a\n\n  {-\n  verify-election-hash =\n  \"\"\n  -- notice the whitespaces and the alphabetical order of keys\n  -- {\"email\": [\"ben@adida.net\", \"ben@mit.edu\"], \"first_name\": \"Ben\", \"last_name\": \"Adida\"}\n    computed_hash = base64.b64encode(hash.new(election.toJSON()).digest())[:-1]\n    computed_hash == vote.election_hash:\n  \"\"\n  -}\n\n  -- module _ {-(election : JSValue)-} where\n  verify-ballot : (ballot : JSValue) \u2192 Bool\n  verify-ballot ballot =\n        trace \"TODO: CHECK the election_hash: \" election_hash \u03bb _ \u2192\n        trace \"TODO: CHECK the election_uuid: \" election_uuid \u03bb _ \u2192\n        verify-answers answers\n      where\n        answers       = ballot \u00b7\u00ab \"answers\" \u00bbA\n        election_hash = ballot \u00b7\u00ab \"election_hash\" \u00bb\n        election_uuid = ballot \u00b7\u00ab \"election_uuid\" \u00bb\n\n  verify-ballots : (ballots : JSArray JSValue) \u2192 Bool\n  verify-ballots ballots =\n      and $ decodeJSArray ballots \u03bb ix ballot \u2192\n        trace \"verify ballot \" ix \u03bb _ \u2192\n        verify-ballot ballot\n\n-- Many checks are still missing!\nverify-helios-election : (arg : JSValue) \u2192 Bool\nverify-helios-election arg = trace \"res=\" res id\n where\n    ed = arg \u00b7\u00ab \"election_data\" \u00bb\n    bs = arg \u00b7\u00ab \"ballots\" \u00bbA\n    pk = ed  \u00b7\u00ab \"public_key\" \u00bb\n    g  = bigdec (pk \u00b7\u00ab \"g\" \u00bb)\n    p  = bigdec (pk \u00b7\u00ab \"p\" \u00bb)\n    q  = bigdec (pk \u00b7\u00ab \"q\" \u00bb)\n    y  = bigdec (pk \u00b7\u00ab \"y\" \u00bb)\n    res =\n        trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" p \u03bb _ \u2192\n        trace \"q=\" q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        HeliosVerifyV3.verify-ballots g p q y bs\n\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (verify-helios-election q))\n  end\n\n-- Working around Agda.Primitive.lsuc being undefined\ncase_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\ncase x of f = f x\n\nmain : JS!\nmain =\n  Process.argv >>= \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n          server \"127.0.0.1\" \"1337\" srv >>= \u03bb uri \u2192\n          Console.log (showURI uri)\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              let opts =\n                 -- fromJSObject (fromObject ((\"encoding\" , fromString \"utf8\") \u2237 []))\n                 -- nullJS\n                    JSON-parse \"{\\\"encoding\\\":\\\"utf8\\\"}\"\n              in\n              Console.log (\"readFile=\" ++ arg) >>\n              FS.readFile arg opts >>== \u03bb err dat \u2192\n                Console.log (\"readFile: err=\" ++ JS.toString err) >>\n                Console.log (Bool\u25b9String (verify-helios-election (JSON-parse (castString dat))))\n          ; _ \u2192\n              Console.log \"usage\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.PartialHeliosVerifier where\n\nopen import Type\nopen import Function hiding (case_of_)\nopen import Data.Bool.Base\nopen import Data.Product\nopen import Data.List.Base using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base  using (String)\n\nopen import FFI.JS as JS hiding (_+_; _\/_; _*_; join)\nopen import FFI.JS.BigI as BigI\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\nopen import FFI.JS.SHA1\nopen import FFI.JS.Proc\nopen import Control.Process.Type\n\njoin : String \u2192 List String \u2192 String\njoin sep []       = \"\"\njoin sep (x \u2237 xs) = x ++ foldr (\u03bb y z \u2192 sep ++ y ++ z) \"\" xs\n\n{-\ninstance\n  showBigI = mk \u03bb _ \u2192 showString \u2218 BigI.toString\n-}\n\nG  = BigI\n\u2124q = BigI\n\nBigI\u25b9G : BigI \u2192 G\nBigI\u25b9G = id\n\nBigI\u25b9\u2124q : BigI \u2192 \u2124q\nBigI\u25b9\u2124q = id\n\nnon0I : BigI \u2192 BigI\nnon0I x with equals x 0I\n... | true  = throw \"Should be non zero!\" 0I\n... | false = x\n\nmodule BigICG (p q : BigI) where\n  _^_  : G  \u2192 \u2124q \u2192 G\n  _\u00b7_  : G  \u2192 G  \u2192 G\n  _\/_  : G  \u2192 G  \u2192 G\n  _+_  : \u2124q \u2192 \u2124q \u2192 \u2124q\n  _*_  : \u2124q \u2192 \u2124q \u2192 \u2124q\n  _==_ : (x y : G) \u2192 Bool\n\n  _^_ = \u03bb x y \u2192 modPow x y p\n  _\u00b7_ = \u03bb x y \u2192 mod (multiply x y) p\n  _\/_ = \u03bb x y \u2192 mod (multiply x (modInv (non0I y) p)) p\n  _+_ = \u03bb x y \u2192 mod (add x y) q\n  _*_ = \u03bb x y \u2192 mod (multiply x y) q\n  _==_ = equals\n\n  sumI : List BigI \u2192 BigI\n  sumI = foldr _+_ 0I\n\nbignum : Number \u2192 BigI\nbignum n = bigI (Number\u25b9String n) \"10\"\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n{-\nprint : {A : Set}{{_ : Show A}} \u2192 A \u2192 Callback0\nprint = Console.log \u2218 show\n-}\n\nPubKey         = G\nEncRnd         = \u2124q {- randomness used for encryption of ct -}\nMessage        = G {- plain text message -}\nChallenge      = \u2124q\nResponse       = \u2124q\nCommitmentPart = G\nCipherTextPart = G\n\nmodule HeliosVerifyV3 (g : G) p q (y : PubKey) where\n  open BigICG p q\n\n  verify-chaum-pedersen : (\u03b1 \u03b2 : CipherTextPart)(M : Message)(A B : G)(c : Challenge)(s : Response) \u2192 Bool\n  verify-chaum-pedersen \u03b1 \u03b2 M A B c s\n      = trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n        (g ^ s) == (A \u00b7 (\u03b1 ^ c))\n      \u2227 (y ^ s) == (B \u00b7 ((\u03b2 \/ M) ^ c))\n\n  verify-individual-proof : (\u03b1 \u03b2 : CipherTextPart)(ix : Number)(\u03c0 : JSValue) \u2192 Bool\n  verify-individual-proof \u03b1 \u03b2 ix \u03c0\n      = trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"commitmentA=\" A \u03bb _ \u2192\n        trace \"commitmentB=\" B \u03bb _ \u2192\n        trace \"challenge=\" c \u03bb _ \u2192\n        trace \"response=\" s \u03bb _ \u2192\n        res\n      where\n        m  = bignum ix\n        M  = g ^ m\n        A  = bigdec (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"A\" \u00bb)\n        B  = bigdec (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"B\" \u00bb)\n        c  = bigdec (\u03c0 \u00b7\u00ab \"challenge\" \u00bb)\n        s  = bigdec (\u03c0 \u00b7\u00ab \"response\" \u00bb)\n        res = verify-chaum-pedersen \u03b1 \u03b2 M A B c s\n\n  -- Conform to HeliosV3 but it is too weak.\n  -- One should follow a \"Strong Fiat Shamir\" transformation\n  -- from interactive ZK proof to non-interactive ZK proofs.\n  -- Namely one should hash the statement as well.\n  --\n  -- SHA1(A0 + \",\" + B0 + \",\" + A1 + \",\" + B1 + ... + \"Amax\" + \",\" + Bmax)\n  hash-commitments : JSArray JSValue \u2192 BigI\n  hash-commitments \u03c0s =\n    fromHex $\n    trace-call \"SHA1(commitments)=\" SHA1       $\n    trace-call \"commitments=\"       (join \",\") $\n    decodeJSArray \u03c0s \u03bb _ \u03c0 \u2192\n      (castString (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"A\" \u00bb))\n      ++ \",\" ++\n      (castString (\u03c0 \u00b7\u00ab \"commitment\" \u00bb \u00b7\u00ab \"B\" \u00bb))\n\n  sum-challenges : JSArray JSValue \u2192 BigI\n  sum-challenges \u03c0s =\n    sumI $ decodeJSArray \u03c0s \u03bb _ \u03c0 \u2192 bigdec (\u03c0 \u00b7\u00ab \"challenge\" \u00bb)\n\n  verify-challenges : JSArray JSValue \u2192 Bool\n  verify-challenges \u03c0s =\n      trace \"hash(commitments)=\" h \u03bb _ \u2192\n      trace \"sum(challenge)=\"    c \u03bb _ \u2192\n      h == c\n    where\n      h = hash-commitments \u03c0s\n      c = sum-challenges   \u03c0s\n\n  verify-choice : (v : JSValue)(\u03c0s : JSArray JSValue) \u2192 Bool\n  verify-choice v \u03c0s =\n      trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n      trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n      verify-challenges \u03c0s\n      \u2227\n      and (decodeJSArray \u03c0s $ verify-individual-proof \u03b1 \u03b2)\n    where\n      \u03b1 = bigdec (v \u00b7\u00ab \"alpha\" \u00bb)\n      \u03b2 = bigdec (v \u00b7\u00ab \"beta\"  \u00bb)\n\n  verify-choices : (choices \u03c0s : JSArray JSValue) \u2192 Bool\n  verify-choices choices \u03c0s =\n    trace \"TODO: check the array size of choices \u03c0s together election data\" \"\" \u03bb _ \u2192\n    and $ decodeJSArray choices \u03bb ix choice \u2192\n      trace \"verify choice \" ix \u03bb _ \u2192\n      verify-choice choice (castJSArray (\u03c0s Array[ ix ]))\n\n  verify-answer : (answer : JSValue) \u2192 Bool\n  verify-answer answer =\n      trace \"TODO: CHECK the overall_proof\" \"\" \u03bb _ \u2192\n      verify-choices choices individual-proofs\n    where\n      choices           = answer \u00b7\u00ab \"choices\" \u00bbA\n      individual-proofs = answer \u00b7\u00ab \"individual_proofs\" \u00bbA\n      overall-proof     = answer \u00b7\u00ab \"overall_proof\" \u00bbA\n\n  verify-answers : (answers : JSArray JSValue) \u2192 Bool\n  verify-answers answers =\n    and $ decodeJSArray answers \u03bb ix a \u2192\n      trace \"verify answer \" ix \u03bb _ \u2192\n      verify-answer a\n\n  {-\n  verify-election-hash =\n  \"\"\n  -- notice the whitespaces and the alphabetical order of keys\n  -- {\"email\": [\"ben@adida.net\", \"ben@mit.edu\"], \"first_name\": \"Ben\", \"last_name\": \"Adida\"}\n    computed_hash = base64.b64encode(hash.new(election.toJSON()).digest())[:-1]\n    computed_hash == vote.election_hash:\n  \"\"\n  -}\n\n  -- module _ {-(election : JSValue)-} where\n  verify-ballot : (ballot : JSValue) \u2192 Bool\n  verify-ballot ballot =\n        trace \"TODO: CHECK the election_hash: \" election_hash \u03bb _ \u2192\n        trace \"TODO: CHECK the election_uuid: \" election_uuid \u03bb _ \u2192\n        verify-answers answers\n      where\n        answers       = ballot \u00b7\u00ab \"answers\" \u00bbA\n        election_hash = ballot \u00b7\u00ab \"election_hash\" \u00bb\n        election_uuid = ballot \u00b7\u00ab \"election_uuid\" \u00bb\n\n  verify-ballots : (ballots : JSArray JSValue) \u2192 Bool\n  verify-ballots ballots =\n      and $ decodeJSArray ballots \u03bb ix ballot \u2192\n        trace \"verify ballot \" ix \u03bb _ \u2192\n        verify-ballot ballot\n\n-- Many checks are still missing!\nverify-helios-election : (arg : JSValue) \u2192 Bool\nverify-helios-election arg = trace \"res=\" res id\n where\n    ed = arg \u00b7\u00ab \"election_data\" \u00bb\n    bs = arg \u00b7\u00ab \"ballots\" \u00bbA\n    pk = ed  \u00b7\u00ab \"public_key\" \u00bb\n    g  = bigdec (pk \u00b7\u00ab \"g\" \u00bb)\n    p  = bigdec (pk \u00b7\u00ab \"p\" \u00bb)\n    q  = bigdec (pk \u00b7\u00ab \"q\" \u00bb)\n    y  = bigdec (pk \u00b7\u00ab \"y\" \u00bb)\n    res =\n        trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" p \u03bb _ \u2192\n        trace \"q=\" q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        HeliosVerifyV3.verify-ballots g p q y bs\n\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (verify-helios-election q))\n  end\n\n-- Working around Agda.Primitive.lsuc being undefined\ncase_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\ncase x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n          server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n          Console.log (showURI uri)\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              let opts =\n                 -- fromJSObject (fromObject ((\"encoding\" , fromString \"utf8\") \u2237 []))\n                 -- nullJS\n                    JSON-parse \"{\\\"encoding\\\":\\\"utf8\\\"}\"\n              in\n              Console.log (\"readFile=\" ++ arg) >>\n              FS.readFile arg opts !\u2082 \u03bb err dat \u2192\n                Console.log (\"readFile: err=\" ++ JS.toString err) >>\n                Console.log (Bool\u25b9String (verify-helios-election (JSON-parse (castString dat))))\n          ; _ \u2192\n              Console.log \"usage\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d65ed0113d28c841876548e54828896460cd3411","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 280c70464ba8834c70dba89505eaa3e7\n\ndarcs-hash:20110211172355-3bd4e-da68e0330b95249a06ca8fa24c3ba199be3697d1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/MirrorStdLib.agda","new_file":"Draft\/Mirror\/MirrorStdLib.agda","new_contents":"-- Tested with the darcs version of the standard library on 11 February 2011.\n\nmodule MirrorStdLib where\n\nopen import Algebra\nopen import Data.List as List hiding ( reverse )\nopen import Data.List.Properties\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\nmodule LM {A : Set} = Monoid (List.monoid A)\n\n------------------------------------------------------------------------------\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n++-rightIdentity : {A : Set}(xs : List A) \u2192 xs ++ [] \u2261 xs\n++-rightIdentity []       = refl\n++-rightIdentity (x \u2237 xs) = cong (_\u2237_ x) (++-rightIdentity xs)\n\nreverse-++ : {A : Set}(xs ys : List A) \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ [] ys       = sym (++-rightIdentity (reverse ys))\nreverse-++ (x \u2237 xs) [] = cong (\u03bb x' \u2192 reverse x' ++ x \u2237 []) (++-rightIdentity xs)\nreverse-++ (x \u2237 xs) (y \u2237 ys) =\n  begin\n    reverse (xs ++ y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 cong (\u03bb x' \u2192 x' ++ x \u2237 []) (reverse-++ xs (y \u2237 ys)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 LM.assoc (reverse (y \u2237 ys)) (reverse xs) (x \u2237 []) \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\ndata Tree (A : Set) : Set where\n  treeT : A \u2192 List (Tree A) \u2192 Tree A\n\nmirror : {A : Set} \u2192 Tree A \u2192 Tree A\nmirror (treeT a ts) = treeT a (reverse (map mirror ts))\n\nmutual\n  mirror\u00b2 : {A : Set} \u2192 (t : Tree A) \u2192 mirror (mirror t) \u2261 t\n  mirror\u00b2 (treeT a [])       = refl\n  mirror\u00b2 (treeT a (t \u2237 ts)) =\n    begin\n      treeT a (reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts) ++\n                                                      mirror t \u2237 []))) \u2261\n                        treeT a x)\n           (aux (t \u2237 ts))\n           refl\n        \u27e9\n      treeT a (t \u2237 ts)\n    \u220e\n\n  aux : {A : Set} \u2192 (ts : List (Tree A)) \u2192\n        reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  aux []       = refl\n  aux (t \u2237 ts) =\n    begin\n      reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 []))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror (reverse (map mirror ts)) (mirror t \u2237 []))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++ (map mirror (reverse (map mirror ts)))\n                             (map mirror (mirror t \u2237 [])))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n              \u2261\u27e8 refl \u27e9\n      mirror (mirror t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror (mirror t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 t)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (aux ts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n","old_contents":"module MirrorStdLib where\n\nopen import Algebra\nopen import Data.List as List hiding ( reverse )\nopen import Data.List.Properties\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\nmodule LM {A : Set} = Monoid (List.monoid A)\n\n------------------------------------------------------------------------------\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n++-rightIdentity : {A : Set}(xs : List A) \u2192 xs ++ [] \u2261 xs\n++-rightIdentity []       = refl\n++-rightIdentity (x \u2237 xs) = cong (_\u2237_ x) (++-rightIdentity xs)\n\nreverse-++ : {A : Set}(xs ys : List A) \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ [] ys       = sym (++-rightIdentity (reverse ys))\nreverse-++ (x \u2237 xs) [] = cong (\u03bb x' \u2192 reverse x' ++ x \u2237 []) (++-rightIdentity xs)\nreverse-++ (x \u2237 xs) (y \u2237 ys) =\n  begin\n    reverse (xs ++ y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 cong (\u03bb x' \u2192 x' ++ x \u2237 []) (reverse-++ xs (y \u2237 ys)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 LM.assoc (reverse (y \u2237 ys)) (reverse xs) (x \u2237 []) \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\ndata Tree (A : Set) : Set where\n  treeT : A \u2192 List (Tree A) \u2192 Tree A\n\nmirror : {A : Set} \u2192 Tree A \u2192 Tree A\nmirror (treeT a ts) = treeT a (reverse (map mirror ts))\n\nmutual\n  mirror\u00b2 : {A : Set} \u2192 (t : Tree A) \u2192 mirror (mirror t) \u2261 t\n  mirror\u00b2 (treeT a [])       = refl\n  mirror\u00b2 (treeT a (t \u2237 ts)) =\n    begin\n      treeT a (reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts) ++\n                                                      mirror t \u2237 []))) \u2261\n                        treeT a x)\n           (aux (t \u2237 ts))\n           refl\n        \u27e9\n      treeT a (t \u2237 ts)\n    \u220e\n\n  aux : {A : Set} \u2192 (ts : List (Tree A)) \u2192\n        reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  aux []       = refl\n  aux (t \u2237 ts) =\n    begin\n      reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 []))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror (reverse (map mirror ts)) (mirror t \u2237 []))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n                          (map mirror (mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++ (map mirror (reverse (map mirror ts)))\n                             (map mirror (mirror t \u2237 [])))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror t \u2237 [])) ++\n              reverse (map mirror (reverse (map mirror ts)))\n              \u2261\u27e8 refl \u27e9\n      mirror (mirror t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror (mirror t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 t)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (aux ts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"15d68ccb1266c0c853b2906c9cef75d1daf94953","subject":"removing some redundant lemmas and moving things around in disjointness","message":"removing some redundant lemmas and moving things around in disjointness\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  -- collect up the hole names of a term as the indices of a trivial contex\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  -- the above judgement has mode (\u2200,\u2203). this doesn't prove uniqueness; any\n  -- contex that extends the one computed here will be indistinguishable\n  -- but we'll treat this one as canonical\n  find-holes : (e : hexp) \u2192 \u03a3[ H \u2208 \u22a4 ctx ](holes e H)\n  find-holes c = \u2205 , HConst\n  find-holes (e \u00b7: x) with find-holes e\n  ... | (h , d)= h , (HAsc d)\n  find-holes (X x) = \u2205 , HVar\n  find-holes (\u00b7\u03bb x e) with find-holes e\n  ... | (h , d) = h , HLam1 d\n  find-holes (\u00b7\u03bb x [ x\u2081 ] e) with find-holes e\n  ... | (h , d) = h , HLam2 d\n  find-holes \u2987\u2988[ x ] = (\u25a0 (x , <>)) , HEHole\n  find-holes \u2987 e \u2988[ x ] with find-holes e\n  ... | (h , d) = h ,, (x , <>) , HNEHole d\n  find-holes (e1 \u2218 e2) with find-holes e1 | find-holes e2\n  ... | (h1 , d1) | (h2 , d2)  = (h1 \u222a h2 ) , (HAp d1 d2)\n\n  -- two contexts that may contain different mappings have the same domain\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  -- the empty context has the same domain as itself\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  -- the singleton contexts formed with any contents but the same index has\n  -- the same domain\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  lem-dom-union-apt1 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03941 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03942 x == Some y)\n  lem-dom-union-apt1 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt1 apt xin | Some x\u2081 = abort (somenotnone apt)\n  lem-dom-union-apt1 apt xin | None = xin\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  lem-dom-union-apt2 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03942 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03941 x == Some y)\n  lem-dom-union-apt2 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt2 apt xin | Some x\u2081 = xin\n  lem-dom-union-apt2 apt xin | None = abort (somenotnone (! xin \u00b7 apt))\n\n  -- if two disjoint sets each share a domain with two other sets, those\n  -- are also disjoint.\n  dom-eq-disj : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n              H1 ## H2 \u2192\n              dom-eq \u03941 H1 \u2192\n              dom-eq \u03942 H2 \u2192\n              \u03941 ## \u03942\n  dom-eq-disj {A} {B} {\u03941} {\u03942} {H1} {H2} (d1 , d2) (de1 , de2) (de3 , de4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192 dom \u03941 n \u2192 n # \u03942\n      guts1 n dom1 with ctxindirect H2 n\n      guts1 n dom1 | Inl x = abort (somenotnone (! (\u03c02 x) \u00b7 d1 n (de1 n dom1)))\n      guts1 n dom1 | Inr x with ctxindirect \u03942 n\n      guts1 n dom1 | Inr x\u2081 | Inl x = abort (somenotnone (! (\u03c02 (de3 n x)) \u00b7 x\u2081))\n      guts1 n dom1 | Inr x\u2081 | Inr x = x\n\n      guts2 : (n : Nat) \u2192 dom \u03942 n \u2192 n # \u03941\n      guts2 n dom2 with ctxindirect H1 n\n      guts2 n dom2 | Inl x = abort (somenotnone (! (\u03c02 x) \u00b7 d2 n (de3 n dom2)))\n      guts2 n dom2 | Inr x with ctxindirect \u03941 n\n      guts2 n dom2 | Inr x\u2081 | Inl x = abort (somenotnone (! (\u03c02 (de1 n x)) \u00b7 x\u2081))\n      guts2 n dom2 | Inr x\u2081 | Inr x = x\n\n  -- if two sets share a domain with disjoint sets, then their union shares\n  -- a domain with the union\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n                                     H1 ## H2 \u2192\n                                     dom-eq \u03941 H1 \u2192\n                                     dom-eq \u03942 H2 \u2192\n                                     dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union {A} {B} {\u03941} {\u03942} {H1} {H2} disj (p1 , p2) (p3 , p4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y)\n      guts1 n (x , eq) with ctxindirect \u03941 n\n      guts1 n (x\u2081 , eq) | Inl x with p1 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al H1 H2 n q1 q2\n      guts1 n (x\u2081 , eq) | Inr x with p3 n (_ , lem-dom-union-apt1 {\u03941 = \u03941} {\u03942 = \u03942} x eq)\n      ... | q1 , q2 =  q1 , x\u2208\u222ar H1 H2 n q1 q2 (##-comm disj)\n\n      guts2 : (n : Nat) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x)\n      guts2 n (x , eq) with ctxindirect H1 n\n      guts2 n (x\u2081 , eq) | Inl x with p2 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al \u03941 \u03942 n q1 q2\n      guts2 n (x\u2081 , eq) | Inr x with p4 n (_ , lem-dom-union-apt2 {\u03941 = H2} {\u03942 = H1} x (tr (\u03bb qq \u2192 qq n == Some x\u2081) (\u222acomm H1 H2 disj) eq))\n      ... | q1 , q2 = q1 , x\u2208\u222ar \u03941 \u03942 n q1 q2 (##-comm (dom-eq-disj disj (p1 , p2) (p3 , p4)))\n\n  -- if a hole name is new then it's apart from the collection of hole\n  -- names\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  lem-dom-apt : {A : Set} {G : A ctx} {x y : Nat} \u2192 x # G \u2192 dom G y \u2192 x \u2260 y\n  lem-dom-apt {x = x} {y = y} apt dom with natEQ x y\n  lem-dom-apt apt dom | Inl refl = abort (somenotnone (! (\u03c02 dom) \u00b7 apt))\n  lem-dom-apt apt dom | Inr x\u2081 = x\u2081\n\n  -- if the holes of two expressions are disjoint, so are their collections\n  -- of hole names\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-sing-disj (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-sing-disj (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  -- the holes of an expression have the same domain as \u0394; that is, we\n  -- don't add any extra junk as we expand\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) with (holes-delta-synth h x\u2081)\n    ... | ih = dom-union {!!} ih (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union {!!} (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) = dom-union (holes-disjoint-disjoint h h\u2081 x) (holes-delta-ana h x\u2084) (holes-delta-ana h\u2081 x\u2085)\n\n  -- if you expand two hole-disjoint expressions analytically, the \u0394s\n  -- produces are disjoint. note that this is likely true for synthetic\n  -- expansions in much the same way, but we only prove \"half\" of the usual\n  -- pair here. the proof technique is *not* structurally inductive on the\n  -- expansion judgement, because of the missing weakness property, so this\n  -- proof is somewhat unusual compared to the rest in this development.\n  expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n  expand-ana-disjoint {e1} {e2} hd ana1 ana2\n    with find-holes e1 | find-holes e2\n  ... | (_ , he1) | (_ , he2) = dom-eq-disj (holes-disjoint-disjoint he1 he2 hd)\n                                            (holes-delta-ana he1 ana1)\n                                            (holes-delta-ana he2 ana2)\n\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  -- collect up the hole names of a term as the indices of a trivial contex\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  -- the above judgement has mode (\u2200,\u2203). this doesn't prove uniqueness; any\n  -- contex that extends the one computed here will be indistinguishable\n  -- but we'll treat this one as canonical\n  find-holes : (e : hexp) \u2192 \u03a3[ H \u2208 \u22a4 ctx ](holes e H)\n  find-holes c = \u2205 , HConst\n  find-holes (e \u00b7: x) with find-holes e\n  ... | (h , d)= h , (HAsc d)\n  find-holes (X x) = \u2205 , HVar\n  find-holes (\u00b7\u03bb x e) with find-holes e\n  ... | (h , d) = h , HLam1 d\n  find-holes (\u00b7\u03bb x [ x\u2081 ] e) with find-holes e\n  ... | (h , d) = h , HLam2 d\n  find-holes \u2987\u2988[ x ] = (\u25a0 (x , <>)) , HEHole\n  find-holes \u2987 e \u2988[ x ] with find-holes e\n  ... | (h , d) = h ,, (x , <>) , HNEHole d\n  find-holes (e1 \u2218 e2) with find-holes e1 | find-holes e2\n  ... | (h1 , d1) | (h2 , d2)  = (h1 \u222a h2 ) , (HAp d1 d2)\n\n  -- two contexts that may contain different mappings have the same domain\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  -- the empty context has the same domain as itself\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  -- the singleton contexts formed with any contents but the same index has\n  -- the same domain\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  lem-dom-union-apt1 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03941 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03942 x == Some y)\n  lem-dom-union-apt1 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt1 apt xin | Some x\u2081 = abort (somenotnone apt)\n  lem-dom-union-apt1 apt xin | None = xin\n\n  lem-dom-union-apt2 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03942 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03941 x == Some y)\n  lem-dom-union-apt2 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt2 apt xin | Some x\u2081 = xin\n  lem-dom-union-apt2 apt xin | None = abort (somenotnone (! xin \u00b7 apt))\n\n  -- if two disjoint sets each share a domain with two other sets, those\n  -- are also disjoint.\n  dom-eq-disj : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n              H1 ## H2 \u2192\n              dom-eq \u03941 H1 \u2192\n              dom-eq \u03942 H2 \u2192\n              \u03941 ## \u03942\n  dom-eq-disj {A} {B} {\u03941} {\u03942} {H1} {H2} (d1 , d2) (de1 , de2) (de3 , de4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192 dom \u03941 n \u2192 n # \u03942\n      guts1 n dom1 with ctxindirect H2 n\n      guts1 n dom1 | Inl x = abort (somenotnone (! (\u03c02 x) \u00b7 d1 n (de1 n dom1)))\n      guts1 n dom1 | Inr x with ctxindirect \u03942 n\n      guts1 n dom1 | Inr x\u2081 | Inl x = abort (somenotnone (! (\u03c02 (de3 n x)) \u00b7 x\u2081))\n      guts1 n dom1 | Inr x\u2081 | Inr x = x\n\n      guts2 : (n : Nat) \u2192 dom \u03942 n \u2192 n # \u03941\n      guts2 n dom2 with ctxindirect H1 n\n      guts2 n dom2 | Inl x = abort (somenotnone (! (\u03c02 x) \u00b7 d2 n (de3 n dom2)))\n      guts2 n dom2 | Inr x with ctxindirect \u03941 n\n      guts2 n dom2 | Inr x\u2081 | Inl x = abort (somenotnone (! (\u03c02 (de1 n x)) \u00b7 x\u2081))\n      guts2 n dom2 | Inr x\u2081 | Inr x = x\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n                                     H1 ## H2 \u2192\n                                     dom-eq \u03941 H1 \u2192\n                                     dom-eq \u03942 H2 \u2192\n                                     dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union {A} {B} {\u03941} {\u03942} {H1} {H2} disj (p1 , p2) (p3 , p4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y)\n      guts1 n (x , eq) with ctxindirect \u03941 n\n      guts1 n (x\u2081 , eq) | Inl x with p1 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al H1 H2 n q1 q2\n      guts1 n (x\u2081 , eq) | Inr x with p3 n (_ , lem-dom-union-apt1 {\u03941 = \u03941} {\u03942 = \u03942} x eq)\n      ... | q1 , q2 =  q1 , x\u2208\u222ar H1 H2 n q1 q2 (##-comm disj)\n\n      guts2 : (n : Nat) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x)\n      guts2 n (x , eq) with ctxindirect H1 n\n      guts2 n (x\u2081 , eq) | Inl x with p2 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al \u03941 \u03942 n q1 q2\n      guts2 n (x\u2081 , eq) | Inr x with p4 n (_ , lem-dom-union-apt2 {\u03941 = H2} {\u03942 = H1} x (tr (\u03bb qq \u2192 qq n == Some x\u2081) (\u222acomm H1 H2 disj) eq))\n      ... | q1 , q2 = q1 , x\u2208\u222ar \u03941 \u03942 n q1 q2 (##-comm (dom-eq-disj disj (p1 , p2) (p3 , p4)))\n\n\n\n  -- the holes of an expression have the same domain as \u0394; that is, we\n  -- don't add any extra junk as we expand\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union {!!} (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union {!!} (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) = dom-union {!!} (holes-delta-ana h x\u2084) (holes-delta-ana h\u2081 x\u2085)\n\n  -- if a hole name is new then it's apart from the holes\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  -- todo: lemmas file?\n  lem-dom-apt : {A : Set} {G : A ctx} {x y : Nat} \u2192 x # G \u2192 dom G y \u2192 x \u2260 y\n  lem-dom-apt {x = x} {y = y} apt dom with natEQ x y\n  lem-dom-apt apt dom | Inl refl = abort (somenotnone (! (\u03c02 dom) \u00b7 apt))\n  lem-dom-apt apt dom | Inr x\u2081 = x\u2081\n\n  -- if the holes of two expressions are disjoint, so are their collections\n  -- of hole names\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-sing-disj (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-sing-disj (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  -- if two contexsts are disjoint and each share a domain with another\n  -- context, those other two contexts are also disjoint\n  domeq-disj : {A B : Set} {H1 H2 : A ctx} {\u03941 \u03942 : B ctx} \u2192\n               H1 ## H2 \u2192\n               dom-eq \u03941 H1 \u2192\n               dom-eq \u03942 H2 \u2192\n               \u03941 ## \u03942\n  domeq-disj (\u03c01 , \u03c02) (\u03c03 , \u03c04) (\u03c05 , \u03c06) =\n                   (\u03bb n x \u2192 {!(\u03c03 n x)!}) ,\n                   (\u03bb n x \u2192 {!!})\n\n  -- if you expand two hole-disjoint expressions analytically, the \u0394s\n  -- produces are disjoint\n  expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n  expand-ana-disjoint {e1} {e2} hd ana1 ana2\n    with find-holes e1 | find-holes e2\n  ... | (_ , he1) | (_ , he2) = domeq-disj (holes-disjoint-disjoint he1 he2 hd)\n                                           (holes-delta-ana he1 ana1)\n                                           (holes-delta-ana he2 ana2)\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1c2120b65288219351b535ddd885c9200ce2e138","subject":"Use shorter name for accent constructors.","message":"Use shorter name for accent constructors.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  add-circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\n\u03b1-circumflex : \u03b1\u2032 accent\n\u03b1-circumflex = add-circumflex \u03b1-long-vowel\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth add-smooth-lower-vowel\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough add-rough-vowel\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth add-smooth-lower-vowel\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough add-rough-vowel\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\n\u03b7-circumflex : \u03b7\u2032 accent\n\u03b7-circumflex = add-circumflex \u03b7-long-vowel\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth add-smooth-lower-vowel\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough add-rough-vowel\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\n\u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\n\u03b9-circumflex : \u03b9\u2032 accent\n\u03b9-circumflex = add-circumflex \u03b9-long-vowel\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth add-smooth-lower-vowel\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing \u03b1-circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing \u03b1-circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing \u03b1-circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing \u03b1-circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota \u03b1-circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota \u03b1-circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota \u03b1-circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota \u03b1-circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent \u03b1-circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota \u03b1-circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent (add-circumflex x) = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  add-acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  add-grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  add-circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\n\u03b1-acute : \u03b1\u2032 accent\n\u03b1-acute = add-acute\n\n\u03b1-grave : \u03b1\u2032 accent\n\u03b1-grave = add-grave\n\n\u03b1-circumflex : \u03b1\u2032 accent\n\u03b1-circumflex = add-circumflex \u03b1-long-vowel\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth add-smooth-lower-vowel\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough add-rough-vowel\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\n\u03b5-acute : \u03b5\u2032 accent\n\u03b5-acute = add-acute\n\n\u03b5-grave : \u03b5\u2032 accent\n\u03b5-grave = add-grave\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth add-smooth-lower-vowel\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough add-rough-vowel\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\n\u03b7-acute : \u03b7\u2032 accent\n\u03b7-acute = add-acute\n\n\u03b7-grave : \u03b7\u2032 accent\n\u03b7-grave = add-grave\n\n\u03b7-circumflex : \u03b7\u2032 accent\n\u03b7-circumflex = add-circumflex \u03b7-long-vowel\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth add-smooth-lower-vowel\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough add-rough-vowel\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\n\u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\n\u03b9-acute : \u03b9\u2032 accent\n\u03b9-acute = add-acute\n\n\u03b9-grave : \u03b9\u2032 accent\n\u03b9-grave = add-grave\n\n\u03b9-circumflex : \u03b9\u2032 accent\n\u03b9-circumflex = add-circumflex \u03b9-long-vowel\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth add-smooth-lower-vowel\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing \u03b1-grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing \u03b1-grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing \u03b1-acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing \u03b1-acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing \u03b1-circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing \u03b1-circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing \u03b1-grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing \u03b1-grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing \u03b1-acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing \u03b1-acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing \u03b1-circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing \u03b1-circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent \u03b1-grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent \u03b1-acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota \u03b1-grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota \u03b1-grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota \u03b1-acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota \u03b1-acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota \u03b1-circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota \u03b1-circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota \u03b1-grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota \u03b1-grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota \u03b1-acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota \u03b1-acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota \u03b1-circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota \u03b1-circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota \u03b1-grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota \u03b1-acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent \u03b1-circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota \u03b1-circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent \u03b1-grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent \u03b1-acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent add-acute = acute-mark\nletter-to-accent add-grave = grave-mark\nletter-to-accent (add-circumflex x) = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"96cc380d8bec495647aa617138f89a24bf3a3266","subject":"Added a simpler way of constructing syntax for the linear solver.","message":"Added a simpler way of constructing syntax for the linear solver.\n","repos":"crypto-agda\/crypto-agda","old_file":"linear-solver.agda","new_file":"linear-solver.agda","new_contents":"module linear-solver where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'    : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'   : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n  infix 4 _\u21db_\n\n  record Eq : Set where\n    constructor _\u21db_\n    field\n      LHS RHS : Syn\n\n  open import Data.Vec.N-ary using (N-ary ; _$\u207f_)\n  open import Data.Vec using (allFin) renaming (map to vmap)\n\n  rewire' : (f : N-ary nrVars Syn Eq) \u2192 let (S\u2081 \u21db S\u2082) = f $\u207f (vmap Syn.var (allFin nrVars))\n          in CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire' f eq = let S \u21db S' = f $\u207f vmap Syn.var (allFin  nrVars)\n         in rewire S S' eq\n \nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire' (\u03bb a b c \u2192 (a , b) , c \u21db (b , a) , c) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n","old_contents":"module linear-solver where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'    : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'   : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n\nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"86469b61012b492b0ff97b1f013785d45df79fac","subject":"Avoiding bug \"undefined\", changing Zq to BigI","message":"Avoiding bug \"undefined\", changing Zq to BigI\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker.agda","new_file":"ZK\/JSChecker.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\n  hiding (check)\n  renaming (_*_ to _*Number_)\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FiniteField.JS as \ud835\udd3d\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\n-- trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n-- trace _ inp f = f inp\n\nbignum : Number \u2192 BigI\nbignum n = bigI (Number\u25b9String n) \"10\"\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- TODO bug (undefined)!\nrecord ZK-chaum-pedersen-pok-elgamal-rnd {--(\u2124q \u2124p\u2605 : Set)--} : Set where\n  field\n    m c s : BigI {--\u2124q--}\n    g p q y \u03b1 \u03b2 A B : BigI --\u2124p\u2605\n\n-- TODO dynamise me\nt : Number\nt = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\ncheck : (title  : String)\n        (pred   : Bool)\n        (errmsg : \ud835\udfd9 \u2192 String)\n     \u2192 JS!\ncheck title true  errmsg = Console.log (title ++ \": PASS\")\ncheck title false errmsg = Console.log (title ++ \": FAIL [\" ++ errmsg _ ++ \"]\")\n\ncheck-size : Number \u2192 String \u2192 BigI \u2192 JS!\ncheck-size min-bits name value =\n  check (\"check size of \" ++ name)\n        (len \u2265Number min-bits)\n        (\u03bb _ \u2192 name ++ \" is not a necessarily a safe prime: \"\n             ++ BigI.toString value    ++ \" has \"\n             ++ Number\u25b9String len      ++ \" bits which is less than \"\n             ++ Number\u25b9String min-bits ++ \" bits\")\n  module Check-size where\n    len = BigI.byteLength value *Number 8N\n\ncheck-pq-relation : (p q : BigI) \u2192 JS!\ncheck-pq-relation p q =\n  check (\"check p and q relation p-1\/q=\" ++ BigI.toString s)\n        (equals r 0I)\n        (\u03bb _ \u2192 \"Not necessarily a safe group: (p-1) mod q != 0\\np=\"\n             ++ BigI.toString p\n             ++ \", q=\" ++ BigI.toString q)\n  module Check-pq-relation where\n    open BigI\n    p-1 = subtract p 1I\n    r   = mod    p-1 q\n    s   = divide p-1 q\n\ncheck-primality : String \u2192 BigI \u2192 JS!\ncheck-primality name value =\n  check (\"check primality of \" ++ name)\n        (BigI.isProbablePrime value t)\n        (\u03bb _ \u2192 \"Not a prime number: \" ++ BigI.toString value)\n\ncheck-generator-group-order : (g q p : BigI) \u2192 JS!\ncheck-generator-group-order g q p =\n  check \"check generator & group order\"\n        (BigI.equals (BigI.modPow g q p) BigI.1I)\n        (\u03bb _ \u2192 \"Not a generator of a group of order q: modPow \"\n             ++ BigI.toString g ++ \" \" ++ BigI.toString q ++ \" \"\n             ++ BigI.toString p)\n\nmodule [\u2124q]\u2124p\u2605 (qI pI gI : BigI) where\n\n  checks : JS!\n  checks =\n    check-pq-relation      pI qI >>\n    check-size min-bits-q \"q\" qI >>\n    check-size min-bits-p \"p\" pI >>\n    check-primality       \"q\" qI >>\n    check-primality       \"p\" pI >>\n    check-generator-group-order gI qI pI\n\n  module \u2124q = \ud835\udd3d qI\n    using (0#; 1#; _+_; _\u2212_; _*_; _\/_)\n    renaming (\ud835\udd3d to \u2124q; fromBigI to BigI\u25b9\u2124q; repr to \u2124q-repr)\n  module \u2124p\u2605 = \ud835\udd3d pI\n    using (_==_)\n    renaming ( fromBigI to BigI\u25b9\u2124p\u2605; \ud835\udd3d to \u2124p\u2605; _*_ to _\u00b7_\n             ; repr to \u2124p\u2605-repr; _\/_ to _\u00b7\/_)\n\n  open \u2124q  -- public -- <- BUG\n  open \u2124p\u2605 public\n\n  g : \u2124p\u2605\n  g = BigI\u25b9\u2124p\u2605 gI\n\n  _^_ : \u2124p\u2605 \u2192 \u2124q \u2192 \u2124p\u2605\n  b ^ e = BigI\u25b9\u2124p\u2605 (BigI.modPow (\u2124p\u2605-repr b) (\u2124q-repr e) pI)\n\nzk-check-chaum-pedersen-pok-elgamal-rnd : ZK-chaum-pedersen-pok-elgamal-rnd {-BigI BigI-} \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks\n      >> check \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    open module IM = [\u2124q]\u2124p\u2605 I.q I.p I.g\n    open \u2124q\n--  open \u2124p\u2605 -- <- BUG\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check : JSValue \u2192 JS!\nzk-check arg =\n    check \"type of statement\" (typ === fromString \"chaum-pedersen-pok-elgamal-rnd\")\n                              (\u03bb _ \u2192 \"\")\n >> zk-check-chaum-pedersen-pok-elgamal-rnd pok\n module Zk-check where\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check (JSON-parse (castString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\n  hiding (check; trace)\n  renaming (_*_ to _*Number_)\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FiniteField.JS as \ud835\udd3d\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\ntrace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\ntrace _ inp f = f inp\n\nbignum : Number \u2192 BigI\nbignum n = bigI (Number\u25b9String n) \"10\"\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\nrecord ZK-chaum-pedersen-pok-elgamal-rnd (\u2124q \u2124p\u2605 : Set) : Set where\n  field\n    m c s : \u2124q\n    g p q y \u03b1 \u03b2 A B : \u2124p\u2605\n\n-- TODO dynamise me\nt : Number\nt = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\ncheck : (title  : String)\n        (pred   : Bool)\n        (errmsg : \ud835\udfd9 \u2192 String)\n     \u2192 JS!\ncheck title true  errmsg = Console.log (title ++ \": PASS\")\ncheck title false errmsg = Console.log (title ++ \": FAIL [\" ++ errmsg _ ++ \"]\")\n\ncheck-size : Number \u2192 String \u2192 BigI \u2192 JS!\ncheck-size min-bits name value =\n  check (\"check size of \" ++ name)\n        (len \u2265Number min-bits)\n        (\u03bb _ \u2192 name ++ \" is not a necessarily a safe prime: \"\n             ++ BigI.toString value    ++ \" has \"\n             ++ Number\u25b9String len      ++ \" bits which is less than \"\n             ++ Number\u25b9String min-bits ++ \" bits\")\n  module Check-size where\n    len = BigI.byteLength value *Number 8N\n\ncheck-pq-relation : (p q : BigI) \u2192 JS!\ncheck-pq-relation p q =\n  check (\"check p and q relation p-1\/q=\" ++ BigI.toString s)\n        (equals r 0I)\n        (\u03bb _ \u2192 \"Not necessarily a safe group: (p-1) mod q != 0\\np=\"\n             ++ BigI.toString p\n             ++ \", q=\" ++ BigI.toString q)\n  module Check-pq-relation where\n    open BigI\n    p-1 = subtract p 1I\n    r   = mod    p-1 q\n    s   = divide p-1 q\n\ncheck-primality : String \u2192 BigI \u2192 JS!\ncheck-primality name value =\n  check (\"check primality of \" ++ name)\n        (BigI.isProbablePrime value t)\n        (\u03bb _ \u2192 \"Not a prime number: \" ++ BigI.toString value)\n\ncheck-generator-group-order : (g q p : BigI) \u2192 JS!\ncheck-generator-group-order g q p =\n  check \"check generator & group order\"\n        (BigI.equals (BigI.modPow g q p) BigI.1I)\n        (\u03bb _ \u2192 \"Not a generator of a group of order q: modPow \"\n             ++ BigI.toString g ++ \" \" ++ BigI.toString q ++ \" \"\n             ++ BigI.toString p)\n\nmodule [\u2124q]\u2124p\u2605 (qI pI gI : BigI) where\n\n  checks : JS!\n  checks =\n    check-pq-relation      pI qI >>\n    check-size min-bits-q \"q\" qI >>\n    check-size min-bits-p \"p\" pI >>\n    check-primality       \"q\" qI >>\n    check-primality       \"p\" pI >>\n    check-generator-group-order gI qI pI\n\n  module \u2124q = \ud835\udd3d qI\n    using (0#; 1#; _+_; _\u2212_; _*_; _\/_)\n    renaming (\ud835\udd3d to \u2124q; fromBigI to BigI\u25b9\u2124q; repr to \u2124q-repr)\n  module \u2124p\u2605 = \ud835\udd3d pI\n    using (_==_)\n    renaming ( fromBigI to BigI\u25b9\u2124p\u2605; \ud835\udd3d to \u2124p\u2605; _*_ to _\u00b7_\n             ; repr to \u2124p\u2605-repr; _\/_ to _\u00b7\/_)\n\n  open \u2124q  -- public -- <- BUG\n  open \u2124p\u2605 public\n\n  g : \u2124p\u2605\n  g = BigI\u25b9\u2124p\u2605 gI\n\n  _^_ : \u2124p\u2605 \u2192 \u2124q \u2192 \u2124p\u2605\n  b ^ e = BigI\u25b9\u2124p\u2605 (BigI.modPow (\u2124p\u2605-repr b) (\u2124q-repr e) pI)\n\nzk-check-chaum-pedersen-pok-elgamal-rnd : ZK-chaum-pedersen-pok-elgamal-rnd BigI BigI \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks\n      >> check \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module Zk-check-chaume-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    open [\u2124q]\u2124p\u2605 I.q I.p I.g\n    open \u2124q\n--  open \u2124p\u2605 -- <- BUG\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check : JSValue \u2192 JS!\nzk-check arg =\n    check \"type of statement\" (typ === fromString \"chaum-pedersen-pok-elgamal-rnd\")\n                              (\u03bb _ \u2192 \"\")\n >> zk-check-chaum-pedersen-pok-elgamal-rnd pok\n module Zk-check where\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check (JSON-parse (castString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9025fe6c69d96bf8ff35600a5e95baeaadf809d6","subject":"Updated the mirror draft.","message":"Updated the mirror draft.\n\nIgnore-this: 775599b409b5d3ddd96b6542a5d61e1\n\ndarcs-hash:20110211172236-3bd4e-5593c52c7622802e3890bc8c25540f9493e5dd86.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/PropertiesI.agda","new_file":"Draft\/Mirror\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.PropertiesI where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesI\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesI using ( reverse-[x]\u2261[x] )\nopen import Draft.Mirror.ListTree.PropertiesI\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove the postulate\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\n-- mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n-- mirror-Tree (treeT d nilLT) =\n--   subst Tree (sym (mirror-eq d [])) (treeT d aux\u2082)\n--     where\n--       aux\u2081 : rev (map mirror []) [] \u2261 []\n--       aux\u2081 =\n--         begin\n--           rev (map mirror []) []\n--           \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n--                    (map-[] mirror)\n--                    refl\n--           \u27e9\n--           rev [] []\n--             \u2261\u27e8 rev-[] [] \u27e9\n--           []\n--         \u220e\n\n--       aux\u2082 : ListTree (rev (map mirror []) [])\n--       aux\u2082 = subst ListTree (sym aux\u2081) nilLT\n\n-- mirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n--   subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d aux)\n\n--   where\n--     aux : ListTree (reverse (map mirror (t \u2237 ts)))\n--     aux = rev-ListTree (map-ListTree mirror {!!} (consLT Tt LTts)) nilLT\n\nmutual\n  mirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\n  mirror\u00b2 (treeT d nilLT) =\n    begin\n      mirror \u00b7 (mirror \u00b7 (node d []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 (node d [])) \u2261 mirror \u00b7 x )\n                 (mirror-eq d [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror [])) \u2261\n                        mirror \u00b7 node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse []) \u2261 mirror \u00b7 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d []\n        \u2261\u27e8 mirror-eq d [] \u27e9\n      node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror [])) \u2261\n                        node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse []) \u2261 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      node d []\n    \u220e\n\n  mirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n    begin\n      mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 mirror \u00b7 x)\n                 (mirror-eq d (t \u2237 ts))\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror (t \u2237 ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror (t \u2237 ts))) \u2261\n                        x)\n                 (mirror-eq d (reverse (map mirror (t \u2237 ts))))\n                 refl\n        \u27e9\n      node d (reverse (map mirror (reverse (map mirror (t \u2237 ts)))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (reverse (map mirror (t \u2237 ts))))) \u2261\n                      node d x)\n               (aux (consLT Tt LTts))\n               refl\n      \u27e9\n      node d (t \u2237 ts)\n    \u220e\n\n  aux : \u2200 {ts} \u2192 ListTree ts \u2192\n        reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  aux nilLT =\n    begin\n      reverse (map mirror (reverse (map mirror [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror []))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse []))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse [])) \u2261\n                        reverse (map mirror x))\n                 (rev-[] [])\n                 refl\n        \u27e9\n      reverse (map mirror [])\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror []) \u2261\n                        reverse x)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse []\n        \u2261\u27e8 rev-[] [] \u27e9\n      []\n    \u220e\n\n  aux (consLT {t} {ts} Tt LTts) =\n    begin\n      reverse (map mirror (reverse (map mirror (t \u2237 ts))))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror (t \u2237 ts)))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-\u2237 mirror t ts)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts))) \u2261\n                        reverse (map mirror x))\n                 (reverse-\u2237 (mirror \u00b7 t) (map-ListTree mirror mirror-Tree LTts))\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror \u00b7 t \u2237 []))) \u2261\n                        reverse x)\n           (map-++ mirror\n                   mirror-Tree\n                   (rev-ListTree (map-ListTree mirror mirror-Tree LTts) nilLT)\n                   (consLT (mirror-Tree Tt) nilLT))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror \u00b7 t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++ (map-ListTree mirror\n                                           mirror-Tree\n                                           (rev-ListTree (map-ListTree mirror\n                                                                       mirror-Tree\n                                                                       LTts)\n                                                         nilLT))\n                             (map-ListTree mirror\n                                           mirror-Tree\n                                           (consLT (mirror-Tree Tt) nilLT)))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (mirror \u00b7 t \u2237 [])) ++ n\u2081 \u2261\n                        reverse x ++ n\u2081)\n                 (map-\u2237 mirror (mirror \u00b7 t) [])\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081 \u2261\n                        reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 x) ++ n\u2081)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081 \u2261\n                        x ++ n\u2081)\n                 (reverse-[x]\u2261[x] (mirror \u00b7 (mirror \u00b7 t)))\n                 refl\n        \u27e9\n      (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 ++-\u2237 (mirror \u00b7 (mirror \u00b7 t)) [] n\u2081 \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192  mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081 \u2261\n                         mirror \u00b7 (mirror \u00b7 t) \u2237 x)\n                 (++-leftIdentity LTn\u2081)\n                 refl\n        \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror \u00b7 (mirror \u00b7 t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 Tt)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (aux LTts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n      where\n        n\u2081 : D\n        n\u2081 = reverse (map mirror (reverse (map mirror ts)))\n\n        LTn\u2081 : ListTree n\u2081\n        LTn\u2081 = rev-ListTree (map-ListTree mirror\n                                          mirror-Tree\n                                          (rev-ListTree (map-ListTree mirror\n                                                                      mirror-Tree\n                                                                      LTts)\n                                                        nilLT))\n                            nilLT\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.PropertiesI where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesI\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesI using ( reverse-[x]\u2261[x] )\nopen import Draft.Mirror.ListTree.PropertiesI\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\n-- mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n-- mirror-Tree (treeT d nilLT) =\n--   subst Tree (sym (mirror-eq d [])) (treeT d aux\u2082)\n--     where\n--       aux\u2081 : rev (map mirror []) [] \u2261 []\n--       aux\u2081 =\n--         begin\n--           rev (map mirror []) []\n--           \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n--                    (map-[] mirror)\n--                    refl\n--           \u27e9\n--           rev [] []\n--             \u2261\u27e8 rev-[] [] \u27e9\n--           []\n--         \u220e\n\n--       aux\u2082 : ListTree (rev (map mirror []) [])\n--       aux\u2082 = subst ListTree (sym aux\u2081) nilLT\n\n-- mirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n--   subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d aux)\n\n--   where\n--     aux : ListTree (reverse (map mirror (t \u2237 ts)))\n--     aux = rev-ListTree (map-ListTree mirror {!!} (consLT Tt LTts)) nilLT\n\nmirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\nmirror\u00b2 (treeT d nilLT) =\n  begin\n    mirror \u00b7 (mirror \u00b7 (node d []))\n      \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 (node d [])) \u2261 mirror \u00b7 x )\n               (mirror-eq d [])\n               refl\n      \u27e9\n    mirror \u00b7 node d (reverse (map mirror []))\n      \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror [])) \u2261\n                      mirror \u00b7 node d (reverse x))\n               (map-[] mirror)\n               refl\n      \u27e9\n    mirror \u00b7 node d (reverse [])\n      \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse []) \u2261 mirror \u00b7 node d x)\n               (rev-[] [])\n               refl\n      \u27e9\n    mirror \u00b7 node d []\n      \u2261\u27e8 mirror-eq d [] \u27e9\n    node d (reverse (map mirror []))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror [])) \u2261\n                      node d (reverse x))\n               (map-[] mirror)\n               refl\n      \u27e9\n    node d (reverse [])\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse []) \u2261 node d x)\n               (rev-[] [])\n               refl\n      \u27e9\n    node d []\n  \u220e\n\nmirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n  begin\n    mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts))\n      \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 mirror \u00b7 x)\n               (mirror-eq d (t \u2237 ts))\n               refl\n      \u27e9\n    mirror \u00b7 node d (reverse (map mirror (t \u2237 ts)))\n      \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror (t \u2237 ts))) \u2261\n                      mirror \u00b7 node d (reverse x))\n               (map-\u2237 mirror t ts)\n               refl\n      \u27e9\n    mirror \u00b7 node d (reverse (mirror \u00b7 t \u2237 map mirror ts))\n      \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (mirror \u00b7 t \u2237 map mirror ts)) \u2261\n                      mirror \u00b7 node d x)\n               (reverse-\u2237 (mirror \u00b7 t) (map-ListTree mirror mirror-Tree LTts))\n               refl\n      \u27e9\n    mirror \u00b7 node d (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 []))\n      \u2261\u27e8 mirror-eq d (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 [])) \u27e9\n    node d (reverse (map mirror (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 []))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (reverse (map mirror ts) ++\n                                      (mirror \u00b7 t \u2237 [])))) \u2261\n                      node d (reverse x))\n               (map-++ mirror\n                       mirror-Tree\n                       (rev-ListTree (map-ListTree mirror mirror-Tree LTts) nilLT)\n                       (consLT (mirror-Tree Tt) nilLT))\n               refl\n      \u27e9\n    node d (reverse ((map mirror (reverse (map mirror ts))) ++\n                      map mirror (mirror \u00b7 t \u2237 [])))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse ((map mirror (reverse (map mirror ts))) ++\n                      map mirror (mirror \u00b7 t \u2237 []))) \u2261\n                      node d x)\n               (reverse-++ (map-ListTree mirror\n                                         mirror-Tree\n                                         (rev-ListTree (map-ListTree mirror\n                                                                     mirror-Tree\n                                                                     LTts)\n                                                       nilLT))\n                           (map-ListTree mirror\n                                         mirror-Tree\n                                         (consLT (mirror-Tree Tt) nilLT)))\n               refl\n      \u27e9\n    node d (reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n            reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n                             reverse (map mirror (reverse (map mirror ts)))) \u2261\n                      node d (reverse x ++\n                             reverse (map mirror (reverse (map mirror ts)))))\n               (map-\u2237 mirror (mirror \u00b7 t) [])\n               refl\n      \u27e9\n    node d (reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++\n            reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++\n                              reverse (map mirror (reverse (map mirror ts)))) \u2261\n                      node d (reverse (x \u2237 map mirror []) ++\n                             reverse (map mirror (reverse (map mirror ts)))))\n               (mirror\u00b2 Tt)  -- IH.\n               refl\n      \u27e9\n    node d (reverse (t \u2237 map mirror []) ++\n            reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (t \u2237 map mirror []) ++\n                              reverse (map mirror (reverse (map mirror ts)))) \u2261\n                      node d (reverse (t \u2237 x) ++\n                              reverse (map mirror (reverse (map mirror ts)))))\n               (map-[] mirror)\n               refl\n      \u27e9\n    node d (reverse (t \u2237 []) ++\n            reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (t \u2237 []) ++\n                              reverse (map mirror (reverse (map mirror ts)))) \u2261\n                      node d (x ++\n                              reverse (map mirror (reverse (map mirror ts)))))\n               (reverse-[x]\u2261[x] t)\n               refl\n      \u27e9\n    node d ((t \u2237 []) ++\n            reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d ((t \u2237 []) ++\n                              reverse (map mirror (reverse (map mirror ts)))) \u2261\n                      node d ((t \u2237 []) ++ x))\n               (prf (mirror\u00b2 {!!}))\n               refl\n      \u27e9\n    node d ((t \u2237 []) ++ ts)\n      \u2261\u27e8 subst (\u03bb x \u2192 node d ((t \u2237 []) ++ ts) \u2261 node d x)\n               (++-\u2237 t [] ts)\n               refl\n      \u27e9\n    node d (t \u2237 [] ++ ts)\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (t \u2237 [] ++ ts) \u2261 node d (t \u2237 x))\n               (++-leftIdentity LTts)\n               refl\n      \u27e9\n    node d (t \u2237 ts)\n  \u220e\n  where\n    postulate prf : mirror \u00b7 (mirror \u00b7 ts) \u2261 ts \u2192  -- IH\n                    rev (map mirror (rev (map mirror ts) [])) [] \u2261 ts\n    {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e9c4182f96dc269533561617c43f24f501a3411a","subject":"finished the sub lemma","message":"finished the sub lemma\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping.agda","new_file":"Agda\/ITyping.agda","new_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\ndata IType : Set\ndata IType\u1d62 : Set\n\ndata IType where\n  o : IType\n  _~>_ : IType\u1d62 -> IType\u1d62 -> IType\n\ndata IType\u1d62 where\n  \u2229 : List IType -> IType\u1d62\n\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\u1d62\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\u2229-\u2237-\u2261 : \u2200 {x y : IType} {xs ys} -> x \u2261 y -> \u2229 xs \u2261 \u2229 ys -> \u2229 (x \u2237 xs) \u2261 \u2229 (y \u2237 ys)\n\u2229-\u2237-\u2261 refl refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u1d62_ : Decidable {A = IType\u1d62} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u1d62 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u1d62 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n\u2229 [] \u225fTI\u1d62 \u2229 [] = yes refl\n\u2229 [] \u225fTI\u1d62 \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI\u1d62 (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 (IType\u1d62 \u00d7 Type))\n\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u1d62_ : IType\u1d62 -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u1d62 A -> \u03c4 \u2237'\u1d62 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {A \u0393 x \u03c4} ->\n    (x\u2209 : x \u2209 dom \u0393) -> \u03c4 \u2237'\u1d62 A -> Wf-ICtxt \u0393 ->\n    --------------------------------------------\n              Wf-ICtxt ((x , (\u03c4 , A)) \u2237 \u0393)\n\ndata _\u2237'\u1d62_ where\n  nil : \u2200 {A} ->  \u03c9 \u2237'\u1d62 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237'\u1d62 A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u1d62 A\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u1d62[_]_ : IType\u1d62 -> Type -> IType\u1d62 -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u1d62[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u1d62[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u1d62[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u1d62 A ->\n    -----------\n    \u03c9 \u2286\u1d62[ A ] \u03c4\n  -- \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229' \u03c4) \u2286[ A ] (\u2229 \u03c4\u1d62)\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> (\u2229 \u03c4'\u1d62) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62) ->\n    -------------------------------------------------------------\n                  (\u2229 (\u03c4' \u2237 \u03c4'\u1d62)) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62)\n\n-- data _\u2264\u2229_ : IType -> IType -> Set where\n--   base : o \u2264\u2229 o\n--   arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n--   \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n--   \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n--   \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\u2237'\u1d62-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A -> \u03c4 \u2237' A\n\u2237'\u1d62-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u1d62-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u1d62-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u1d62-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u1d62-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u1d62-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u1d62 A -> \u03c4 \u2286\u1d62[ A ] \u03c4\n\u2286\u1d62-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> \u2229 \u03c4\u1d62 \u2286\u1d62[ A ] \u2229 \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u1d62-refl \u03c4\u2237\u1d62A) (\u2286\u1d62-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u1d62-refl {\u03c4 = \u2229 []} nil = nil nil\n\u2286\u1d62-refl {A} {\u2229 (\u03c4 \u2237 \u03c4\u1d62)} \u03c4\u03c4\u1d62\u2237A = \u2286\u1d62-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u1d62-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u1d62-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n\n\u2286\u1d62-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u2229 \u03c4\u2081 \u2286\u1d62[ A ] \u2229 \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n\u2286\u1d62-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n\u2286\u1d62-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u1d62-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n\n\n\u2286\u1d62-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u2229 (\u03c4 \u2237 \u03c4\u1d62)) \u2286\u1d62[ A ] \u03c9 -> \u22a5\n\u2286\u1d62-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286-\u2237'-l : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4 \u2237' A\n\u2286-\u2237'-l base = base\n\u2286-\u2237'-l (arr _ _ x _) = x\n\n\u2286\u1d62-\u2237'\u1d62-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u1d62-\u2237'\u1d62-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\u2286\u1d62-\u2237'\u1d62-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u1d62 \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-l (nil x) = nil\n\u2286\u1d62-\u2237'\u1d62-l (cons (_ , _ , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) = cons (\u2286-\u2237'-l \u03c4'\u2286\u03c4) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\n\n\u2286\u1d62-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n  \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u1d62[ A ] \u03c4\u2096 ->\n  ---------------------------------\n            \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2096\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2c7c\u2286\u03c4\u2096)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u1d62-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} {\u2229 \u03c4\u2c7c} {\u2229 \u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n  cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u1d62-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n    where\n    \u03c4'' = proj\u2081 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n    \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n    \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} {\u2229 (_ \u2237 \u03c4\u2c7c)} {\u2229 \u03c4\u2096} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) (cons x \u03c4\u2c7c\u2286\u03c4\u2096) = cons (\u03c4 , ({!   !} , {!   !})) {!   !}\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 = {!   !}\n\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u1d62-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u1d62-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n\n\n\u2286->\u2286\u1d62 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u1d62[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u1d62 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app s t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y _) = Y \u03c4\n\n-- \u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n-- \u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u1d62_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType\u1d62 -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : IType\u1d62} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u03c4\u1d62 , A)) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u1d62[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u1d62 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u1d62[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u1d62 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , (\u03c4 , A)) \u2237 \u0393) \u22a9\u1d62 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> (\u03c4 ~> \u03c4') \u2237' (A \u27f6 B) ->\n    --------------------------------------------------------------------------------------------------------------\n                                                \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2081) \u00d7 ((\u03c4 ~> \u03c4) \u2286[ A \u27f6 A ] \u03c4')) -> (\u2229 \u03c4\u2081) \u2237'\u1d62 (A \u27f6 A) -> \u03c4\u2082 \u2286\u1d62[ A ] \u03c4 -> --  \u03c4\u2081 \u2286[ (A \u27f6 A) \u27f6 A ]((\u2229' (\u03c4 ~> \u03c4)) ~> \u03c4)\n    -----------------------------------------------------------------------------------------\n                                    \u0393 \u22a9 Y A \u2236 ((\u2229 \u03c4\u2081) ~> \u03c4\u2082)\n\ndata _\u22a9\u1d62_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 \u03c9\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 \u03c4\u1d62) ->\n    --------------------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n\ndata _\u2286\u0393_ : ICtxt -> ICtxt -> Set where\n  nil : \u2200 {\u0393} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          [] \u2286\u0393 \u0393\n  cons : \u2200 {A x \u03c4' \u0393 \u0393'} ->\n    \u2203(\u03bb \u03c4 -> ((x , (\u03c4 , A)) \u2208 \u0393') \u00d7 (\u03c4' \u2286\u1d62[ A ] \u03c4)) -> \u0393 \u2286\u0393 \u0393' ->\n    ------------------------------------------------------------\n                      ((x , (\u03c4' , A)) \u2237 \u0393) \u2286\u0393 \u0393'\n\n\n\u2286\u0393-wf\u0393' : \u2200 {\u0393 \u0393'} -> \u0393 \u2286\u0393 \u0393' -> Wf-ICtxt \u0393'\n\u2286\u0393-wf\u0393' (nil wf-\u0393') = wf-\u0393'\n\u2286\u0393-wf\u0393' (cons _ \u0393\u2286\u0393') = \u2286\u0393-wf\u0393' \u0393\u2286\u0393'\n\n\n\n\u22a9-\u2237' : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4 \u2237' A\n\u22a9\u1d62-\u2237'\u1d62 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u03c4 \u2237'\u1d62 A\n\n\u22a9-\u2237' (var _ _ \u03c4\u2286\u03c4\u1d62) = \u2237'\u1d62-\u2208 (here refl) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (app _ _ \u03c4\u2286\u03c4\u1d62 _) = \u2237'\u1d62-\u2208 (here refl) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (abs _ _ x) = x\n\u22a9-\u2237' (Y _ _ \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) = arr \u03c4\u2081\u2237A\u27f6A (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2082\u2286\u03c4)\n\n\u22a9\u1d62-\u2237'\u1d62 (nil _) = nil\n\u22a9\u1d62-\u2237'\u1d62 (cons \u0393\u22a9m\u2236\u03c4 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62) = cons (\u22a9-\u2237' \u0393\u22a9m\u2236\u03c4) (\u22a9\u1d62-\u2237'\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62)\n\n\n\u22a9\u1d62-\u2208-\u22a9 : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4\u1d62} -> \u0393 \u22a9\u1d62 m \u2236 \u2229 \u03c4\u1d62 -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9\u1d62-\u2208-\u22a9 (nil _) ()\n\u22a9\u1d62-\u2208-\u22a9 (cons \u0393\u22a9m\u2236\u03c4 x) (here refl) = \u0393\u22a9m\u2236\u03c4\n\u22a9\u1d62-\u2208-\u22a9 (cons _ \u0393\u22a9\u1d62m\u2236\u03c4\u1d62) (there \u03c4\u2208\u03c4\u1d62) = \u22a9\u1d62-\u2208-\u22a9 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n\u2208-\u2286\u0393-trans : \u2200 {A x \u03c4\u1d62} {\u0393 \u0393'} -> (x , (\u03c4\u1d62 , A)) \u2208 \u0393 -> \u0393 \u2286\u0393 \u0393' -> \u2203(\u03bb \u03c4\u1d62' -> ((x , (\u03c4\u1d62' , A)) \u2208 \u0393') \u00d7 \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u1d62')\n\u2208-\u2286\u0393-trans (here refl) (cons x _) = x\n\u2208-\u2286\u0393-trans (there x\u2208L) (cons _ L\u2286L') = \u2208-\u2286\u0393-trans x\u2208L L\u2286L'\n\n\n\u2286\u0393-\u2237 : \u2200 {A x \u03c4\u1d62 \u0393 \u0393'} -> x \u2209 dom \u0393' -> \u03c4\u1d62 \u2237'\u1d62 A -> \u0393 \u2286\u0393 \u0393' -> \u0393 \u2286\u0393 ((x , \u03c4\u1d62 , A) \u2237 \u0393')\n\u2286\u0393-\u2237 {\u0393 = []} x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393' = nil (cons x\u2209\u0393' \u03c4\u1d62\u2237A (\u2286\u0393-wf\u0393' \u0393\u2286\u0393'))\n\u2286\u0393-\u2237 {\u0393 = (x , \u03c4\u1d62 , A) \u2237 \u0393} x\u2209\u0393' \u03c4\u1d62\u2237A (cons (proj\u2081 , proj\u2082 , proj\u2083) \u0393\u2286\u0393') =\n  cons (proj\u2081 , ((there proj\u2082) , proj\u2083)) (\u2286\u0393-\u2237 x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393')\n\n\nsub\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4\nsub\u1d62\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u1d62 m \u2236 \u03c4\n\nsub\u0393 (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u0393\u2286\u0393' = var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u1d62-trans \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub\u0393 (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u0393\u2286\u0393' = app (sub\u0393 \u0393\u22a9m\u2236\u03c4 \u0393\u2286\u0393') (sub\u1d62\u0393 x \u0393\u2286\u0393') x\u2081 x\u2082\nsub\u0393 {\u0393' = \u0393'} (abs {\u03c4 = \u03c4} L cf (arr \u03c4\u2237A \u03c4'\u2237B)) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\u0393\n    (cf (\u2209-cons-l _ _ x\u2209))\n    (cons\n      (\u03c4 , (here refl) , (\u2286\u1d62-refl \u03c4\u2237A))\n      (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2237A \u03c4'\u2237B)\nsub\u0393 (Y x x\u2081 x\u2082 x\u2083) \u0393\u2286\u0393' = Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') x\u2081 x\u2082 x\u2083\n\nsub\u1d62\u0393 (nil wf-\u0393) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62\u0393 (cons x \u0393\u22a9\u1d62m\u2236\u03c4) \u0393\u2286\u0393' = cons (sub\u0393 x \u0393\u2286\u0393') (sub\u1d62\u0393 \u0393\u22a9\u1d62m\u2236\u03c4 \u0393\u2286\u0393')\n\n\n\u2208-\u2286\u1d62-trans : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229 \u03c4\u1d62) \u2286\u1d62[ A ] (\u2229 \u03c4\u2c7c) -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2c7c) \u00d7 \u03c4 \u2286[ A ] \u03c4')\n\u2208-\u2286\u1d62-trans (here refl) (cons x _) = x\n\u2208-\u2286\u1d62-trans (there \u03c4\u2208\u03c4\u1d62) (cons _ \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2208-\u2286\u1d62-trans \u03c4\u2208\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\nsub : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4'\nsub\u1d62 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u03c4' \u2286\u1d62[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u1d62 m \u2236 \u03c4'\n\nsub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u1d62-trans (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62) \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub (app \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u22a9\u1d62t\u2236\u03c4\u2081 \u03c4\u2286\u03c4\u2082 \u03c4\u2237A) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = app\n  (sub\u0393 \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u2286\u0393')\n  (sub\u1d62\u0393 \u0393\u22a9\u1d62t\u2236\u03c4\u2081 \u0393\u2286\u0393')\n  (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u2082)\n  \u03c4\u2237A\nsub {_} {\u0393} {\u0393'} (abs {\u03c4 = \u03c4} {\u03c4'} L {t} cf \u03c4~>\u03c4'\u2237A\u27f6B) (arr {A} {\u03c4\u2081\u2081 = \u03c4\u2081\u2081} \u03c4\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4' (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B) x\u2083) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\n    (cf (\u2209-cons-l _ _ x\u2209))\n    \u03c4\u2081\u2082\u2286\u03c4'\n    (cons (\u03c4\u2081\u2081 , (here refl) , \u03c4\u2286\u03c4\u2081\u2081) (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2081\u2081\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B)\nsub (Y wf-\u0393 (\u03c4' , \u03c4'\u2208\u03c4\u2081 , \u03c4~>\u03c4\u2286\u03c4') \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) (arr {\u03c4\u2081\u2081 = \u2229 \u03c4\u2081'} \u03c4\u2081\u2286\u03c4\u2081' \u03c4\u2082'\u2286\u03c4\u2082 (arr \u2229\u03c4\u2081'\u2237A\u27f6A \u03c4\u2082'\u2237A) x\u2084) \u0393\u2286\u0393' =\n  Y ((\u2286\u0393-wf\u0393' \u0393\u2286\u0393')) (\u03c4'' , (\u03c4''\u2208\u03c4\u2081' , \u2286-trans \u03c4~>\u03c4\u2286\u03c4' \u03c4'\u2286\u03c4'')) \u2229\u03c4\u2081'\u2237A\u27f6A (\u2286\u1d62-trans \u03c4\u2082'\u2286\u03c4\u2082 \u03c4\u2082\u2286\u03c4)\n  where\n  \u03c4'' = proj\u2081 (\u2208-\u2286\u1d62-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081')\n  \u03c4''\u2208\u03c4\u2081' = proj\u2081 (proj\u2082 (\u2208-\u2286\u1d62-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n  \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2208-\u2286\u1d62-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n\nsub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4 (nil x) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4'\u1d62\u2286\u03c4\u1d62) \u0393\u2286\u0393' with \u22a9\u1d62-\u2208-\u22a9 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n... | \u0393\u22a9m\u2236\u03c4 = cons (sub \u0393\u22a9m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393') (sub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4'\u1d62\u2286\u03c4\u1d62 \u0393\u2286\u0393')\n\n\n\n\n-- \u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u1d62Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","old_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\ndata IType : Set\ndata IType\u1d62 : Set\n\ndata IType where\n  o : IType\n  _~>_ : IType\u1d62 -> IType\u1d62 -> IType\n\ndata IType\u1d62 where\n  \u2229 : List IType -> IType\u1d62\n\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\u1d62\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\u2229-\u2237-\u2261 : \u2200 {x y : IType} {xs ys} -> x \u2261 y -> \u2229 xs \u2261 \u2229 ys -> \u2229 (x \u2237 xs) \u2261 \u2229 (y \u2237 ys)\n\u2229-\u2237-\u2261 refl refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u1d62_ : Decidable {A = IType\u1d62} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u1d62 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u1d62 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n\u2229 [] \u225fTI\u1d62 \u2229 [] = yes refl\n\u2229 [] \u225fTI\u1d62 \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI\u1d62 (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 IType\u1d62)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u1d62_ : IType\u1d62 -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u1d62 A -> \u03c4 \u2237'\u1d62 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\ndata _\u2237'\u1d62_ where\n  nil : \u2200 {A} ->  \u03c9 \u2237'\u1d62 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237'\u1d62 A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u1d62 A\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u1d62[_]_ : IType\u1d62 -> Type -> IType\u1d62 -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u1d62[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u1d62[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u1d62[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u1d62 A ->\n    -----------\n    \u03c9 \u2286\u1d62[ A ] \u03c4\n  -- \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229' \u03c4) \u2286[ A ] (\u2229 \u03c4\u1d62)\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> (\u2229 \u03c4'\u1d62) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62) ->\n    -------------------------------------------------------------\n                  (\u2229 (\u03c4' \u2237 \u03c4'\u1d62)) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62)\n\n-- data _\u2264\u2229_ : IType -> IType -> Set where\n--   base : o \u2264\u2229 o\n--   arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n--   \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n--   \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n--   \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\u2237'\u1d62-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A -> \u03c4 \u2237' A\n\u2237'\u1d62-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u1d62-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u1d62-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u1d62-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u1d62-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u1d62-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u1d62 A -> \u03c4 \u2286\u1d62[ A ] \u03c4\n\u2286\u1d62-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> \u2229 \u03c4\u1d62 \u2286\u1d62[ A ] \u2229 \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u1d62-refl \u03c4\u2237\u1d62A) (\u2286\u1d62-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u1d62-refl {\u03c4 = \u2229 []} nil = nil nil\n\u2286\u1d62-refl {A} {\u2229 (\u03c4 \u2237 \u03c4\u1d62)} \u03c4\u03c4\u1d62\u2237A = \u2286\u1d62-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u1d62-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u1d62-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n\n\u2286\u1d62-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u2229 \u03c4\u2081 \u2286\u1d62[ A ] \u2229 \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n\u2286\u1d62-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n\u2286\u1d62-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u1d62-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n\n\n\u2286\u1d62-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u2229 (\u03c4 \u2237 \u03c4\u1d62)) \u2286\u1d62[ A ] \u03c9 -> \u22a5\n\u2286\u1d62-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286-\u2237'-l : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4 \u2237' A\n\u2286-\u2237'-l base = base\n\u2286-\u2237'-l (arr _ _ x _) = x\n\n\u2286\u1d62-\u2237'\u1d62-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u1d62-\u2237'\u1d62-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\u2286\u1d62-\u2237'\u1d62-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u1d62 \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-l (nil x) = nil\n\u2286\u1d62-\u2237'\u1d62-l (cons (_ , _ , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) = cons (\u2286-\u2237'-l \u03c4'\u2286\u03c4) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\n\n\u2286\u1d62-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n  \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u1d62[ A ] \u03c4\u2096 ->\n  ---------------------------------\n            \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2096\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2c7c\u2286\u03c4\u2096)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u1d62-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} {\u2229 \u03c4\u2c7c} {\u2229 \u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n  cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u1d62-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n    where\n    \u03c4'' = proj\u2081 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n    \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n    \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} {\u2229 (_ \u2237 \u03c4\u2c7c)} {\u2229 \u03c4\u2096} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) (cons x \u03c4\u2c7c\u2286\u03c4\u2096) = cons (\u03c4 , ({!   !} , {!   !})) {!   !}\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 = {!   !}\n\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u1d62-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u1d62-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n\n\n\u2286->\u2286\u1d62 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u1d62[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u1d62 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app s t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y _) = Y \u03c4\n\n-- \u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n-- \u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\n\n-- ~>' : IType\u1d62 \u00d7 IType\u1d62 -> IType\n-- ~>' (a , b) = a ~> b\n--\n-- \u2237-++-\u2261 : \u2200 {a} {A : Set a} {x : A} {xs ys : List A} -> (x \u2237 xs) ++ ys \u2261 x \u2237 (xs ++ ys)\n-- \u2237-++-\u2261 {a} {A} {x} {xs} {ys} = refl\n--\n--\n-- \u2237\u1d62-++ : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> \u2229 (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2237'\u1d62 A\n-- \u2237\u1d62-++ nil \u03c4\u2c7c\u2237A = \u03c4\u2c7c\u2237A\n-- \u2237\u1d62-++ (cons x \u03c4\u1d62\u2237A) \u03c4\u2c7c\u2237A = cons x (\u2237\u1d62-++ \u03c4\u1d62\u2237A \u03c4\u2c7c\u2237A)\n--\n-- \u2286\u1d62-++ : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62) -> \u03c4 \u2286\u1d62[ A ] (\u2229 \u03c4\u2c7c) -> \u03c4 \u2286\u1d62[ A ] \u2229 (\u03c4\u1d62 ++ \u03c4\u2c7c)\n-- \u2286\u1d62-++ {\u03c4 = \u2229 []} \u03c4\u2286\u03c4\u1d62 \u03c4\u2286\u03c4\u2c7c = nil (\u2237\u1d62-++ (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2286\u03c4\u1d62) (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2286\u03c4\u2c7c))\n-- \u2286\u1d62-++ {\u03c4 = \u2229 (x \u2237 x\u1d62)} (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , x\u2286\u03c4) x\u2081\u2286\u03c4\u1d62) (cons \u2203\u03c4\u2082 x\u2081\u2286\u03c4\u2c7c) =\n--   cons (\u03c4 , ((\u2208-cons-l _ \u03c4\u2208\u03c4\u1d62) , x\u2286\u03c4)) (\u2286\u1d62-++ {\u03c4 = \u2229 x\u1d62} x\u2081\u2286\u03c4\u1d62 x\u2081\u2286\u03c4\u2c7c)\n--\n-- concat\u1d62 : List (IType\u1d62) -> IType\u1d62\n-- concat\u1d62 [] = \u03c9\n-- concat\u1d62 ((\u2229 x) \u2237 xs) with concat\u1d62 xs\n-- ... | \u2229 xs' = \u2229 (x ++ xs')\n--\n-- \u2229\u2081 : List (IType\u1d62 \u00d7 IType\u1d62) -> IType\u1d62\n-- \u2229\u2081 xs\u00d7ys = concat\u1d62 (Data.List.map proj\u2081 xs\u00d7ys)\n--\n-- \u2229\u2082 : List (IType\u1d62 \u00d7 IType\u1d62) -> IType\u1d62\n-- \u2229\u2082 xs\u00d7ys = concat\u1d62 (Data.List.map proj\u2082 xs\u00d7ys)\n--\n--\n-- inv-\u2286\u1d62-\u2229~>-r : \u2200 {A B \u03c4} -> \u03c4 \u2237'\u1d62 (A \u27f6 B) ->\n--   \u2203(\u03bb (\u03c4\u1d62\u00d7\u03c4\u2c7c : List (IType\u1d62 \u00d7 IType\u1d62)) -> \u03c4 \u2261 \u2229 (Data.List.map ~>' \u03c4\u1d62\u00d7\u03c4\u2c7c))\n-- inv-\u2286\u1d62-\u2229~>-r nil = [] , refl\n-- inv-\u2286\u1d62-\u2229~>-r (cons {\u03c4\u1d62 = \u03c4\u1d62} (arr {\u03b4 = \u03c4} {\u03c4'} _ _) \u03c4\u1d62\u2237A\u27f6B) = ((\u03c4 , \u03c4') \u2237 \u03c4\u1d62\u00d7\u03c4\u2c7c) , \u2229-\u2237-\u2261 refl \u03c4\u1d62\u2261\u2229\u03c4\u1d62\u00d7\u03c4\u2c7c\n--   where\n--   ih : \u2203 (\u03bb \u03c4\u1d62\u00d7\u03c4\u2c7c \u2192 \u2229 \u03c4\u1d62 \u2261 \u2229 (Data.List.map ~>' \u03c4\u1d62\u00d7\u03c4\u2c7c))\n--   ih = inv-\u2286\u1d62-\u2229~>-r \u03c4\u1d62\u2237A\u27f6B\n--\n--   \u03c4\u1d62\u00d7\u03c4\u2c7c = proj\u2081 ih\n--   \u03c4\u1d62\u2261\u2229\u03c4\u1d62\u00d7\u03c4\u2c7c = proj\u2082 ih\n\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u1d62_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType\u1d62 -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : IType\u1d62} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , \u03c4\u1d62) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u1d62[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u1d62 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u1d62[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u1d62 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u03c4) \u2237 \u0393) \u22a9\u1d62 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> (\u03c4 ~> \u03c4') \u2237' (A \u27f6 B) ->\n    ---------------------------------------------------------------------------------------------------------\n                                        \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  -- Y : \u2200 {\u0393 A \u03c4\u2082} {\u03c4\u00d7\u03c4\u2081 : List (IType\u1d62 \u00d7 IType\u1d62)} ->\n  --   Wf-ICtxt \u0393 -> \u03c4\u2082 \u2286\u1d62[ A ] (\u2229\u2082 \u03c4\u00d7\u03c4\u2081) -> (\u2229\u2082 \u03c4\u00d7\u03c4\u2081) \u2286\u1d62[ A ] (\u2229\u2081 \u03c4\u00d7\u03c4\u2081) ->\n  --   --------------------------------------------------------------------\n  --           \u0393 \u22a9 Y A \u2236 ((\u2229 (Data.List.map ~>' \u03c4\u00d7\u03c4\u2081)) ~> \u03c4\u2082)\n\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n    Wf-ICtxt \u0393 -> \u03c4\u2082 \u2286\u1d62[ A ] \u03c4\u2081 -> \u03c4 \u2286\u1d62[ A ] \u03c4\u2081 -> \u03c4\u2083 \u2286[ (A \u27f6 A) \u27f6 A ]((\u2229' (\u03c4 ~> \u03c4\u2081)) ~> \u03c4\u2082) ->\n    -------------------------------------------------------------------------------------------\n                                          \u0393 \u22a9 Y A \u2236 \u03c4\u2083\n\ndata _\u22a9\u1d62_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 \u03c9\n\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 \u03c4\u1d62) ->\n    --------------------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n\ndata _\u2286\u0393_ : ICtxt -> ICtxt -> Set where\n  nil : \u2200 {\u0393} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          [] \u2286\u0393 \u0393\n  cons : \u2200 {x \u03c4' \u0393 \u0393'} ->\n    \u2203(\u03bb \u03c4A -> ((x , (proj\u2081 \u03c4A)) \u2208 \u0393') \u00d7 (\u03c4' \u2286\u1d62[ (proj\u2082 \u03c4A) ] (proj\u2081 \u03c4A))) -> \u0393 \u2286\u0393 \u0393' ->\n    -----------------------------------------------------------------------------------\n                                  ((x , \u03c4') \u2237 \u0393) \u2286\u0393 \u0393'\n\n\n\u2286\u0393-wf\u0393' : \u2200 {\u0393 \u0393'} -> \u0393 \u2286\u0393 \u0393' -> Wf-ICtxt \u0393'\n\u2286\u0393-wf\u0393' (nil wf-\u0393') = wf-\u0393'\n\u2286\u0393-wf\u0393' (cons _ \u0393\u2286\u0393') = \u2286\u0393-wf\u0393' \u0393\u2286\u0393'\n\n\u22a9-\u2237' : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4 \u2237' A\n\u22a9\u1d62-\u2237'\u1d62 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u03c4 \u2237'\u1d62 A\n\n\u22a9-\u2237' (var _ _ \u03c4\u2286\u03c4\u1d62) = \u2237'\u1d62-\u2208 (here refl) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (app _ _ \u03c4\u2286\u03c4\u1d62 _) = \u2237'\u1d62-\u2208 (here refl) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (abs _ _ x) = x\n\u22a9-\u2237' (Y _ _ _ x) = \u2286-\u2237'-l x\n\n\u22a9\u1d62-\u2237'\u1d62 (nil _) = nil\n\u22a9\u1d62-\u2237'\u1d62 (cons \u0393\u22a9m\u2236\u03c4 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62) = cons (\u22a9-\u2237' \u0393\u22a9m\u2236\u03c4) (\u22a9\u1d62-\u2237'\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62)\n\n\n\u22a9\u1d62-\u2208-\u22a9 : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4\u1d62} -> \u0393 \u22a9\u1d62 m \u2236 \u2229 \u03c4\u1d62 -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9\u1d62-\u2208-\u22a9 (nil _) ()\n\u22a9\u1d62-\u2208-\u22a9 (cons \u0393\u22a9m\u2236\u03c4 x) (here refl) = \u0393\u22a9m\u2236\u03c4\n\u22a9\u1d62-\u2208-\u22a9 (cons _ \u0393\u22a9\u1d62m\u2236\u03c4\u1d62) (there \u03c4\u2208\u03c4\u1d62) = \u22a9\u1d62-\u2208-\u22a9 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\nsub\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4\nsub\u1d62\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u1d62 m \u2236 \u03c4\n\nsub\u0393 (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u0393\u2286\u0393' = var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') {!   !} \u03c4\u2286\u03c4\u1d62\nsub\u0393 (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u0393\u2286\u0393' = app (sub\u0393 \u0393\u22a9m\u2236\u03c4 \u0393\u2286\u0393') (sub\u1d62\u0393 x \u0393\u2286\u0393') x\u2081 x\u2082\nsub\u0393 (abs L cf x) \u0393\u2286\u0393' = abs L (\u03bb x\u2209L \u2192 sub\u1d62\u0393 (cf x\u2209L) {!   !}) x\nsub\u0393 (Y x x\u2081 x\u2082 x\u2083) \u0393\u2286\u0393' = Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') x\u2081 x\u2082 x\u2083\n\nsub\u1d62\u0393 (nil wf-\u0393) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62\u0393 (cons x \u0393\u22a9\u1d62m\u2236\u03c4) \u0393\u2286\u0393' = cons (sub\u0393 x \u0393\u2286\u0393') (sub\u1d62\u0393 \u0393\u22a9\u1d62m\u2236\u03c4 \u0393\u2286\u0393')\n\n\nsub : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4'\nsub\u1d62 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u03c4' \u2286\u1d62[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u1d62 m \u2236 \u03c4'\n\nsub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') {!   !} (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62)\nsub (app \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u22a9\u1d62t\u2236\u03c4\u2081 \u03c4\u2286\u03c4\u2082 \u03c4\u2237A) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = app\n  (sub\u0393 \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u2286\u0393')\n  (sub\u1d62\u0393 \u0393\u22a9\u1d62t\u2236\u03c4\u2081 \u0393\u2286\u0393')\n  (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u2082)\n  \u03c4\u2237A\nsub {_} {\u0393} (abs {\u03c4 = \u03c4} {\u03c4'} L {t} cf \u03c4~>\u03c4'\u2237A\u27f6B) (arr {A} {\u03c4\u2081\u2081 = \u03c4\u2081\u2081} \u03c4\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4' \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B x\u2083) \u0393\u2286\u0393' =\n  abs L (\u03bb x\u2209L \u2192 sub\u1d62 (cf x\u2209L) \u03c4\u2081\u2082\u2286\u03c4' (cons ((\u03c4\u2081\u2081 , A) , (here refl , \u03c4\u2286\u03c4\u2081\u2081)) {!   !}) ) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B\nsub {_} {\u0393} (Y {\u03c4 = \u03c4} {\u03c4\u2081} {\u03c4\u2082} {\u03c4\u2083} wf-\u0393 \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2286\u03c4\u2081 \u03c4\u2083\u2286) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2286\u03c4\u2081 (\u2286-trans \u03c4'\u2286\u03c4 \u03c4\u2083\u2286)\n\n\nsub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4 (nil x) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4'\u1d62\u2286\u03c4\u1d62) \u0393\u2286\u0393' with \u22a9\u1d62-\u2208-\u22a9 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n... | \u0393\u22a9m\u2236\u03c4 = cons (sub \u0393\u22a9m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393') (sub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4'\u1d62\u2286\u03c4\u1d62 \u0393\u2286\u0393')\n\n\n-- -- weakening-\u03c4 {A} {\u0393} {_} {\u03c4} {\u03c4'} {_} {\u03c4'\u209b\u209b} (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u1d62\u2264\u2229\u03c4 \u03c4\u2237A) \u03c4\u2264\u03c4' \u03c4'\u2237A = var {A} {\u0393} {_} {\u03c4'} {\u03c4'\u209b\u209b} wf-\u0393 \u03c4\u1d62\u2208\u0393 (\u2229-trans \u03c4\u1d62\u2264\u2229\u03c4 \u03c4\u2264\u03c4') \u03c4'\u2237A\n-- -- weakening-\u03c4 (app \u0393\u22a9m\u2236\u03c4\u209b\u209b \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081 x) \u03c4\u2264\u03c4' \u03c4'\u2237A =\n-- --   app (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b (arr \u03c4\u2264\u03c4' \u2264\u2229-refl) (arr {!   !} \u03c4'\u2237A)) (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081 \u2264\u2229-refl {!   !}) (arr {!   !} \u03c4'\u2237A)\n-- --   -- (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b (arr \u03c4\u2264\u03c4' \u2264\u2229-refl) ?) (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081 \u2264\u2229-refl ?) ?)\n-- -- weakening-\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) \u03c4\u2264\u03c4' \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m\u2236\u03c4\u209b\u209b \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081) \u03c4\u2264\u03c4' \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (abs L cf x x\u2081) \u03c4\u2264\u03c4' \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) (arr \u03c4\u2264\u03c4' \u03c4\u2264\u03c4'') \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) \u2229-nil \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) (\u2229-cons \u03c4\u2264\u03c4' \u03c4\u2264\u03c4'') \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) (\u2229-trans \u03c4\u2264\u03c4' \u03c4\u2264\u03c4'') \u03c4'\u2237A = {!   !}\n\n\n\n-- \u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u1d62Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8a55eb1f41371571c5a4c27630bf3b599688b07e","subject":"IDesc stratified model: silly code indentation","message":"IDesc stratified model: silly code indentation\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set (suc l)) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set (suc l)} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set (suc l)}(R : I -> IDesc {l = l} I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set (suc l)}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set (suc l) -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\nIDescD : {l : Level}(I : Set (suc l)) -> IDesc {l = suc l} Unit\nIDescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : {l : Level}(I : Set (suc l)) -> Unit -> Set (suc l)\nIDescl0 {x} I = IMu {l = suc x} (\\_ -> IDescD {l = x} I)\n\nIDescl : {l : Level}(I : Set (suc l)) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set (suc l)}(i : I) -> IDescl I\nvarl {x} i = con (lvar {l = suc x} , i) \n\nconstl : {l : Level}{I : Set (suc l)}(X : Set l) -> IDescl I\nconstl {x} X = con (lconst {l = suc x} , X)\n\nprodl : {l : Level}{I : Set (suc l)}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lprod {l = suc x} , (D , D'))\n\n\npil : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lpi {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n\nsigmal : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (lsigma {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n           \n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set (suc l)}\n        (xs : desc (IDescD I) (IMu (\u03bb _ -> IDescD I)))\n        (hs : desc (box (IDescD I) (IMu (\u03bb _ -> IDescD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s) )\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set (suc l)} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> IDescD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set (suc l)} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set (suc l)} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-psi-phi : {l : Level}(I : Set (suc l)) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> IDescD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n    where P : Sigma Unit (IMu (\\ x -> IDescD I)) -> Set (suc x)\n          P ( Void , D ) = psi (phi D) == D\n          proof-psi-phi-cases : (i : Unit)\n                                (xs : desc (IDescD I) (IDescl0 I))\n                                (hs : desc (box (IDescD I) (IDescl0 I) xs) P)\n                                -> P (i , con xs)\n          proof-psi-phi-cases Void (lvar , i) hs = refl\n          proof-psi-phi-cases Void (lconst , x) hs = refl\n          proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n          proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                               (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                               (\u03bb s \u2192 psi (phi (T (s))))\n                                                                               (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                           (proof-lift-unlift-eq s)))\n                                                                       (reflFun (\\s -> psi (phi (T s))) \n                                                                                T \n                                                                                hs)) \n          proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                  (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                  (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                  (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                              (proof-lift-unlift-eq s)))\n                                                                         (reflFun (\\s -> psi (phi (T s))) \n                                                                                  T \n                                                                                  hs))","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set (suc l)) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set (suc l)} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set (suc l)}(R : I -> IDesc {l = l} I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set (suc l)}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set (suc l) -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\nIDescD : {l : Level}(I : Set (suc l)) -> IDesc {l = suc l} Unit\nIDescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : {l : Level}(I : Set (suc l)) -> Unit -> Set (suc l)\nIDescl0 {x} I = IMu {l = suc x} (\\_ -> IDescD {l = x} I)\n\nIDescl : {l : Level}(I : Set (suc l)) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set (suc l)}(i : I) -> IDescl I\nvarl {x} i = con (lvar {l = suc x} , i) \n\nconstl : {l : Level}{I : Set (suc l)}(X : Set l) -> IDescl I\nconstl {x} X = con (lconst {l = suc x} , X)\n\nprodl : {l : Level}{I : Set (suc l)}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lprod {l = suc x} , (D , D'))\n\n\npil : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lpi {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n\nsigmal : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (lsigma {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n           \n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set (suc l)}\n        (xs : desc (IDescD I) (IMu (\u03bb _ -> IDescD I)))\n        (hs : desc (box (IDescD I) (IMu (\u03bb _ -> IDescD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s) )\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set (suc l)} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> IDescD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set (suc l)} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set (suc l)} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-psi-phi : {l : Level}(I : Set (suc l)) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> IDescD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> IDescD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (IDescD I) (IDescl0 I))\n                                            (hs : desc (box (IDescD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                                           (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                           (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                           (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                                       (proof-lift-unlift-eq s)))\n                                                                                  (reflFun (\\s -> psi (phi (T s))) \n                                                                                           T \n                                                                                           hs)) \n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                              (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                              (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                              (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                                          (proof-lift-unlift-eq s)))\n                                                                                     (reflFun (\\s -> psi (phi (T s))) \n                                                                                              T \n                                                                                              hs))","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24e0a0189105b8f3f2766a604a1ad5423b51b77e","subject":"Document P.C.Derive.","message":"Document P.C.Derive.\n\nOld-commit-hash: 962425cf9bb190e1b8fb2c400bd9765180081dd0\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Derive.agda","new_file":"Parametric\/Change\/Derive.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Incrementalization as term-to-term transformation (Fig. 4g).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Derive\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\n-- Extension point: Incrementalization of fully applied primitives.\nStructure : Set\nStructure = \u2200 {\u0393 \u03a3 \u03c4} \u2192\n  Const \u03a3 \u03c4 \u2192\n  Terms (\u0394Context \u0393) \u03a3 \u2192\n  Terms (\u0394Context \u0393) (mapContext \u0394Type \u03a3) \u2192\n  Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\nmodule Structure (deriveConst : Structure) where\n  fit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\n  fit = weaken \u0393\u227c\u0394\u0393\n\n  fit-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) \u03a3\n  fit-terms = weaken-terms \u0393\u227c\u0394\u0393\n\n  -- In the paper, we transform \"x\" to \"dx\". Here, we work with\n  -- de Bruijn indices, so we have to manipulate the indices to\n  -- account for a bigger context after transformation.\n  deriveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394Context \u0393) (\u0394Type \u03c4)\n  deriveVar this = this\n  deriveVar (that x) = that (that (deriveVar x))\n\n  derive-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) (mapContext \u0394Type \u03a3)\n  derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\n  derive-terms {\u2205}    \u2205 = \u2205\n  derive-terms {\u03c4 \u2022 \u03a3} (t \u2022 ts) = derive t \u2022 derive-terms ts\n\n  -- We provide: Incrementalization of arbitrary terms.\n  derive (var x) = var (deriveVar x)\n  derive (app s t) = app (app (derive s) (fit t)) (derive t)\n  derive (abs t) = abs (abs (derive t))\n  derive (const c ts) = deriveConst c (fit-terms ts) (derive-terms ts)\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Derive\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nStructure : Set\nStructure = \u2200 {\u0393 \u03a3 \u03c4} \u2192\n  Const \u03a3 \u03c4 \u2192\n  Terms (\u0394Context \u0393) \u03a3 \u2192\n  Terms (\u0394Context \u0393) (mapContext \u0394Type \u03a3) \u2192\n  Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\nmodule Structure (deriveConst : Structure) where\n  fit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\n  fit = weaken \u0393\u227c\u0394\u0393\n\n  fit-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) \u03a3\n  fit-terms = weaken-terms \u0393\u227c\u0394\u0393\n\n  deriveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394Context \u0393) (\u0394Type \u03c4)\n  deriveVar this = this\n  deriveVar (that x) = that (that (deriveVar x))\n\n  derive-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) (mapContext \u0394Type \u03a3)\n  derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\n  derive-terms {\u2205}    \u2205 = \u2205\n  derive-terms {\u03c4 \u2022 \u03a3} (t \u2022 ts) = derive t \u2022 derive-terms ts\n\n  derive (var x) = var (deriveVar x)\n  derive (app s t) = app (app (derive s) (fit t)) (derive t)\n  derive (abs t) = abs (abs (derive t))\n  derive (const c ts) = deriveConst c (fit-terms ts) (derive-terms ts)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2785cd34222fed5f734189c03ce46b310d2fd27e","subject":"Desc stratified model: descD implicit.","message":"Desc stratified model: descD implicit.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f215632e880e24517f026a58d55c3637663b1642","subject":"Simplied some proofs.","message":"Simplied some proofs.\n\nIgnore-this: 1aa23e74e5fb6656bef1c63cc2c1e84b\n\ndarcs-hash:20110218170953-3bd4e-ab204ca4c90c40c3fdcd1bd8c40e01b6bb106b5d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_file":"Draft\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Properties.AuxiliaryATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.ArithmeticATP\nopen import Draft.McCarthy91.McCarthy91\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 x\u223810-N #-}\n{-# ATP prove x<mc91x+11>100 x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 1+x-N 11+x-N\n              x\u223810-N +-comm #-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n              Nmc91>100 N111 N101\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 N91 #-}\n-- {-# ATP prove mc91<mc91+11 mc91\u226191 n102 100<102 #-}\n\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 mc91-eq\u2081 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 mc91-eq\u2081 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 mc91-eq\u2081 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 mc91-eq\u2081 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 mc91-eq\u2081 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 mc91-eq\u2081 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 mc91-eq\u2081 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 mc91-eq\u2081 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 mc91-eq\u2081 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 mc91-eq\u2081 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","old_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Properties.AuxiliaryATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.ArithmeticATP\nopen import Draft.McCarthy91.McCarthy91\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 x\u223810-N #-}\n{-# ATP prove x<mc91x+11>100 x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 1+x-N 11+x-N\n              x\u223810-N +-comm #-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n              Nmc91>100 N111 N101\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 N91 #-}\n-- {-# ATP prove mc91<mc91+11 mc91\u226191 n102 100<102 #-}\n\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 mc91-eq\u2081 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 mc91-eq\u2081 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 mc91-eq\u2081 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 mc91-eq\u2081 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 mc91-eq\u2081 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 mc91-eq\u2081 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 mc91-eq\u2081 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 mc91-eq\u2081 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 mc91-eq\u2081 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 mc91-eq\u2081 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\npostulate\n  mc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\n  mc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\n  mc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\n  mc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\n  mc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\n  mc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\n  mc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\n  mc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\n  mc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\n  mc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\n{-# ATP prove mc91-res-99' mc91-res-99 #-}\n{-# ATP prove mc91-res-98' mc91-res-98 #-}\n{-# ATP prove mc91-res-97' mc91-res-97 #-}\n{-# ATP prove mc91-res-96' mc91-res-96 #-}\n{-# ATP prove mc91-res-95' mc91-res-95 #-}\n{-# ATP prove mc91-res-94' mc91-res-94 #-}\n{-# ATP prove mc91-res-93' mc91-res-93 #-}\n{-# ATP prove mc91-res-92' mc91-res-92 #-}\n{-# ATP prove mc91-res-91' mc91-res-91 #-}\n{-# ATP prove mc91-res-90' mc91-res-90 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c2336ec78c0eecde7d41e8928d17e8c38a3f2ed9","subject":"Two: +twist","message":"Two: +twist\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Two.agda","new_file":"lib\/Data\/Two.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two where\n\nopen import Data.Bool public hiding (if_then_else_) renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\nopen import Data.Bool.Properties\n  public\n  using (isCommutativeSemiring-\u2228-\u2227\n        ;commutativeSemiring-\u2228-\u2227\n        ;module RingSolver\n        ;isCommutativeSemiring-\u2227-\u2228\n        ;commutativeSemiring-\u2227-\u2228\n        ;isBooleanAlgebra\n        ;booleanAlgebra\n        ;commutativeRing-xor-\u2227\n        ;module XorRingSolver\n        ;not-involutive\n        ;not-\u00ac\n        ;\u00ac-not\n        ;\u21d4\u2192\u2261\n        ;proof-irrelevance)\n  renaming\n        (T-\u2261 to \u2713-\u2261\n        ;T-\u2227 to \u2713-\u2227\n        ;T-\u2228 to \u2713-\u2228)\n\nopen import Type using (\u2605_)\n\nopen import Algebra                    using (module CommutativeRing; module CommutativeSemiring)\nopen import Algebra.FunctionProperties using (Op\u2081; Op\u2082)\n\nopen import Data.Nat     using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\nopen import Data.Zero    using (\ud835\udfd8-elim)\nopen import Data.One     using (\ud835\udfd9)\nopen import Data.Product using (proj\u2081; proj\u2082; uncurry; _\u00d7_; _,_)\nopen import Data.Sum     using (_\u228e_; inj\u2081; inj\u2082)\n\nopen import Function.Equivalence using (module Equivalence)\nopen import Function.Equality    using (_\u27e8$\u27e9_)\nopen import Function.NP          using (id; _\u2218_; _\u27e8_\u27e9\u00b0_; flip)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; refl; idp; _\u2219_; !_; coe; coe!; J; J-orig)\nopen import Relation.Nullary                         using (\u00ac_; Dec; yes; no)\n\nopen import HoTT\nopen Equivalences\n\n\nopen Equivalence using (to; from)\n\nmodule Xor\u00b0 = CommutativeRing     commutativeRing-xor-\u2227\nmodule \ud835\udfda\u00b0   = CommutativeSemiring commutativeSemiring-\u2227-\u2228\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: proj\u2081 1: proj\u2082 ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\ntwist-equiv : \ud835\udfda \u2243 \ud835\udfda\ntwist-equiv = self-inv-equiv not not-involutive\n\nmodule _ {{_ : UA}} where\n    twist : \ud835\udfda \u2261 \ud835\udfda\n    twist = ua twist-equiv\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 p q = _\u27e8$\u27e9_ (from \u2713-\u2227) (p , q)\n\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to \u2713-\u2227)\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (\u2713-\u2227 {b\u2081}))\n\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {b\u2081} = _\u27e8$\u27e9_ (to (\u2713-\u2228 {b\u2081}))\n\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 = _\u27e8$\u27e9_ (from \u2713-\u2228) \u2218 inj\u2081\n\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (\u2713-\u2228 {b\u2081})) \u2218 inj\u2082\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not 0\u2082 y = not-involutive y\nxor-not-not 1\u2082 _ = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 (x xor y) \u2261 (x xor z) \u2192 y \u2261 z\nxor-inj\u2081 0\u2082 = id\nxor-inj\u2081 1\u2082 = not-inj\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 (y xor x) \u2261 (z xor x) \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} p = xor-inj\u2081 x (Xor\u00b0.+-comm x y \u2219 p \u2219 Xor\u00b0.+-comm z x)\n\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \ud835\udfd9\ncheck = _\n\nmodule Indexed {a} {A : \u2605 a} where\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 \ud835\udfda)\n    not\u00b0 f = not \u2218 f\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two where\n\nopen import Data.Bool public hiding (if_then_else_) renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\nopen import Data.Bool.Properties\n  public\n  using (isCommutativeSemiring-\u2228-\u2227\n        ;commutativeSemiring-\u2228-\u2227\n        ;module RingSolver\n        ;isCommutativeSemiring-\u2227-\u2228\n        ;commutativeSemiring-\u2227-\u2228\n        ;isBooleanAlgebra\n        ;booleanAlgebra\n        ;commutativeRing-xor-\u2227\n        ;module XorRingSolver\n        ;not-involutive\n        ;not-\u00ac\n        ;\u00ac-not\n        ;\u21d4\u2192\u2261\n        ;proof-irrelevance)\n  renaming\n        (T-\u2261 to \u2713-\u2261\n        ;T-\u2227 to \u2713-\u2227\n        ;T-\u2228 to \u2713-\u2228)\n\nopen import Type using (\u2605_)\n\nopen import Algebra                    using (module CommutativeRing; module CommutativeSemiring)\nopen import Algebra.FunctionProperties using (Op\u2081; Op\u2082)\n\nopen import Data.Nat     using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\nopen import Data.Zero    using (\ud835\udfd8-elim)\nopen import Data.One     using (\ud835\udfd9)\nopen import Data.Product using (proj\u2081; proj\u2082; uncurry; _\u00d7_; _,_)\nopen import Data.Sum     using (_\u228e_; inj\u2081; inj\u2082)\n\nopen import Function.Equivalence using (module Equivalence)\nopen import Function.Equality    using (_\u27e8$\u27e9_)\nopen import Function.NP          using (id; _\u2218_; _\u27e8_\u27e9\u00b0_)\n\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; refl; idp; _\u2219_; !_)\nopen import Relation.Nullary                         using (\u00ac_; Dec; yes; no)\n\nopen Equivalence using (to; from)\n\nmodule Xor\u00b0 = CommutativeRing     commutativeRing-xor-\u2227\nmodule \ud835\udfda\u00b0   = CommutativeSemiring commutativeSemiring-\u2227-\u2228\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: proj\u2081 1: proj\u2082 ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 p q = _\u27e8$\u27e9_ (from \u2713-\u2227) (p , q)\n\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to \u2713-\u2227)\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (\u2713-\u2227 {b\u2081}))\n\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {b\u2081} = _\u27e8$\u27e9_ (to (\u2713-\u2228 {b\u2081}))\n\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 = _\u27e8$\u27e9_ (from \u2713-\u2228) \u2218 inj\u2081\n\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (\u2713-\u2228 {b\u2081})) \u2218 inj\u2082\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not 0\u2082 y = not-involutive y\nxor-not-not 1\u2082 _ = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 (x xor y) \u2261 (x xor z) \u2192 y \u2261 z\nxor-inj\u2081 0\u2082 = id\nxor-inj\u2081 1\u2082 = not-inj\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 (y xor x) \u2261 (z xor x) \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} p = xor-inj\u2081 x (Xor\u00b0.+-comm x y \u2219 p \u2219 Xor\u00b0.+-comm z x)\n\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \ud835\udfd9\ncheck = _\n\nmodule Indexed {a} {A : \u2605 a} where\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 \ud835\udfda)\n    not\u00b0 f = not \u2218 f\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8d4599155dc64db203782bdd39087df1a8014331","subject":"Move (apply|diff)-base to the top of the module.","message":"Move (apply|diff)-base to the top of the module.\n\nThis commit separates the POPL14-specific code from the\nlanguage-independent code in this module. The following commits will\nremove the language-independent code.\n\nOld-commit-hash: 245a7b618b2ac811d28d702089482e9aed8b8425\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Change\/Term.agda","new_file":"Popl14\/Change\/Term.agda","new_contents":"module Popl14.Change.Term where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Data.Integer\n\nopen import Popl14.Syntax.Type public\nopen import Popl14.Syntax.Term public\nopen import Popl14.Change.Type public\n\nimport Parametric.Change.Term Const \u0394Base as ChangeTerm\n\ndiff-base : ChangeTerm.DiffStructure\ndiff-base {base-int} = abs\u2082 (\u03bb x y \u2192 add x (minus y))\ndiff-base {base-bag} = abs\u2082 (\u03bb x y \u2192 union x (negate y))\n\napply-base : ChangeTerm.ApplyStructure\napply-base {base-int} = abs\u2082 (\u03bb \u0394x x \u2192 add x \u0394x)\napply-base {base-bag} = abs\u2082 (\u03bb \u0394x x \u2192 union x \u0394x)\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-term {base \u03b9} = apply-base {\u03b9}\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app f x \u2295 app (app \u0394f x) (x \u229d x))))\n\ndiff-term {base \u03b9} = diff-base {\u03b9}\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app g (x \u2295 \u0394x) \u229d app f x))))\n","old_contents":"module Popl14.Change.Term where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Data.Integer\n\nopen import Popl14.Syntax.Type public\nopen import Popl14.Syntax.Term public\nopen import Popl14.Change.Type public\n\nimport Parametric.Change.Term Const \u0394Base as ChangeTerm\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-base : ChangeTerm.ApplyStructure\napply-base {base-int} =\n  abs\u2082 (\u03bb \u0394x x \u2192 add x \u0394x)\napply-base {base-bag} =\n  abs\u2082 (\u03bb \u0394x x \u2192 union x \u0394x)\n\napply-term {base \u03b9} = apply-base {\u03b9}\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app f x \u2295 app (app \u0394f x) (x \u229d x))))\n\ndiff-base : ChangeTerm.DiffStructure\ndiff-base {base-int} =\n  abs\u2082 (\u03bb x y \u2192 add x (minus y))\ndiff-base {base-bag} =\n  abs\u2082 (\u03bb x y \u2192 union x (negate y))\n\ndiff-term {base \u03b9} = diff-base {\u03b9}\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app g (x \u2295 \u0394x) \u229d app f x))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cf2254421026fbe05ddc7be9b1c07c8af3c76b19","subject":"whitespace and alpha renaming #3","message":"whitespace and alpha renaming #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | V x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | V y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | V y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = {!!} -- I {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (_ , Step x a q) = S (_ , Step {!!} (ITCastFail y z w) {!!} )\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | V x = I (IFailedCast (FBoxed x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | V x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | V y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | V y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = {!!} -- I {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (_ , Step x a q) = S (_ , Step {!!} (ITCastFail y z w) {!!} )\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | V x = I (IFailedCast (FBoxed x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5e0522a26266d87d99cd7169d3df306935bdf8f4","subject":"Removed unused code.","message":"Removed unused code.\n\nIgnore-this: bb2cba0cc416e210cbca1ac97da677e5\n\ndarcs-hash:20100505162046-3bd4e-f69bdae78351e9271235c7a8d96d8aea25255f18.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities.agda","new_file":"LTC\/Relation\/Inequalities.agda","new_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities where\n\nopen import LTC.Minimal\nopen import MyStdLib.Data.Product\n\n------------------------------------------------------------------------------\n\npostulate\n  lt : D \u2192 D \u2192 D\n\n  lt-00 : lt zero zero                     \u2261 false\n\n  lt-0S : (d : D) \u2192 lt zero (succ d)       \u2261 true\n\n  lt-S0 : (d : D) \u2192 lt (succ d) zero       \u2261 false\n\n  lt-SS : (d e : D) \u2192 lt (succ d) (succ e) \u2261 lt d e\n\n{-# ATP axiom lt-00 #-}\n{-# ATP axiom lt-0S #-}\n{-# ATP axiom lt-S0 #-}\n{-# ATP axiom lt-SS #-}\n\ngt : D \u2192 D \u2192 D\ngt d e = lt e d\n{-# ATP definition gt #-}\n\n------------------------------------------------------------------------\n-- The data types\n\n-- infix 4 _\u2264_ _<_ _\u2265_ _>_\n\nGT : D \u2192 D \u2192 Set\nGT d e = gt d e \u2261 true\n{-# ATP definition GT #-}\n\nLT : D \u2192 D  \u2192 Set\nLT d e = lt d e \u2261 true\n{-# ATP definition LT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = gt d e \u2261 false\n{-# ATP definition LE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = lt d e \u2261 false\n{-# ATP definition GE #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities where\n\nopen import LTC.Minimal\nopen import MyStdLib.Data.Product\n\n------------------------------------------------------------------------------\n\npostulate\n  lt : D \u2192 D \u2192 D\n\n  lt-00 : lt zero zero                     \u2261 false\n\n  lt-0S : (d : D) \u2192 lt zero (succ d)       \u2261 true\n\n  lt-S0 : (d : D) \u2192 lt (succ d) zero       \u2261 false\n\n  lt-SS : (d e : D) \u2192 lt (succ d) (succ e) \u2261 lt d e\n\n{-# ATP axiom lt-00 #-}\n{-# ATP axiom lt-0S #-}\n{-# ATP axiom lt-S0 #-}\n{-# ATP axiom lt-SS #-}\n\ngt : D \u2192 D \u2192 D\ngt d e = lt e d\n{-# ATP definition gt #-}\n\n------------------------------------------------------------------------\n-- The data types\n\n-- infix 4 _\u2264_ _<_ _\u2265_ _>_\n\nGT : D \u2192 D \u2192 Set\nGT d e = gt d e \u2261 true\n{-# ATP definition GT #-}\n\nLT : D \u2192 D  \u2192 Set\nLT d e = lt d e \u2261 true\n{-# ATP definition LT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = gt d e \u2261 false\n{-# ATP definition LE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = lt d e \u2261 false\n{-# ATP definition GE #-}\n\n-- ------------------------------------------------------------------------\n-- -- Lexicographical order on D\n-- data LT\u2082 : D \u00d7 D \u2192 D \u00d7 D \u2192 Set where\n--   left  : {x\u2081 x\u2082 y\u2081 y\u2082 : D} \u2192 LT x\u2081 x\u2082 \u2192 LT\u2082 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n--   right : {x y\u2081 y\u2082 : D} \u2192  LT y\u2081 y\u2082 \u2192 LT\u2082 (x , y\u2081) (x , y\u2082)\n-- -- {-# ATP hint left #-}\n-- -- {-# ATP hint right #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c19f3333c5cdfd8103fff65fb873090cb0e21f88","subject":"making complete expansion a little bit stronger by also observing that the resultant hole context must be empty","message":"making complete expansion a little bit stronger by also observing that the resultant hole context must be empty\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-expansion.agda","new_file":"complete-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import typed-expansion\nopen import lemmas-gcomplete\n\nmodule complete-expansion where\n  -- this might be derivable from things below and a fact about => and ::\n  -- that we seem to have not proven\n  complete-ta : \u2200{\u0393 \u0394 d \u03c4} \u2192 (\u0393 gcomplete) \u2192 (\u0394 , \u0393 \u22a2 d :: \u03c4) \u2192 d dcomplete \u2192 \u03c4 tcomplete\n  complete-ta gc TAConst comp = TCBase\n  complete-ta gc (TAVar x\u2081) DCVar = gc _ _ x\u2081\n  complete-ta gc (TALam a wt) (DCLam comp x\u2081) = TCArr x\u2081 (complete-ta (gcomp-extend gc x\u2081 a ) wt comp)\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) with complete-ta gc wt comp\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) | TCArr qq qq\u2081 = qq\u2081\n  complete-ta gc (TAEHole x x\u2081) ()\n  complete-ta gc (TANEHole x wt x\u2081) ()\n  complete-ta gc (TACast wt x) (DCCast comp x\u2081 x\u2082) = x\u2082\n  complete-ta gc (TAFailedCast wt x x\u2081 x\u2082) ()\n\n  comp-synth : \u2200{\u0393 e \u03c4} \u2192\n                   \u0393 gcomplete \u2192\n                   e ecomplete \u2192\n                   \u0393 \u22a2 e => \u03c4 \u2192\n                   \u03c4 tcomplete\n  comp-synth gc ec SConst = TCBase\n  comp-synth gc (ECAsc x ec) (SAsc x\u2081) = x\n  comp-synth gc ec (SVar x) = gc _ _ x\n  comp-synth gc (ECAp ec ec\u2081) (SAp _ wt MAHole x\u2081) with comp-synth gc ec wt\n  ... | ()\n  comp-synth gc (ECAp ec ec\u2081) (SAp _ wt MAArr x\u2081) with comp-synth gc ec wt\n  comp-synth gc (ECAp ec ec\u2081) (SAp _ wt MAArr x\u2081) | TCArr qq qq\u2081 = qq\u2081\n  comp-synth gc () SEHole\n  comp-synth gc () (SNEHole _ wt)\n  comp-synth gc (ECLam2 ec x\u2081) (SLam x\u2082 wt) = TCArr x\u2081 (comp-synth (gcomp-extend gc x\u2081 x\u2082) ec wt)\n\n  mutual\n    complete-expansion-synth : \u2200{e \u03c4 \u0393 \u0394 d} \u2192\n                               \u0393 gcomplete \u2192\n                               e ecomplete \u2192\n                               \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                               (d dcomplete \u00d7 \u03c4 tcomplete \u00d7 \u0394 == \u2205)\n    complete-expansion-synth gc ec ESConst = DCConst , TCBase , refl\n    complete-expansion-synth gc ec (ESVar x\u2081) = DCVar , gc _ _ x\u2081 , refl\n    complete-expansion-synth gc (ECLam2 ec x\u2081) (ESLam x\u2082 exp) with complete-expansion-synth (gcomp-extend gc x\u2081 x\u2082) ec exp\n    ... | ih1 , ih2 , ih3 = DCLam ih1 x\u2081 , TCArr x\u2081 ih2 , ih3\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp _ _ x MAHole x\u2082 x\u2083) with comp-synth gc ec x\n    ... | ()\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp {\u03941 = \u03941} {\u03942 = \u03942} _ _ x MAArr x\u2082 x\u2083)\n      with comp-synth gc ec x\n    ... | TCArr t1 t2\n      with complete-expansion-ana gc ec (TCArr t1 t2) x\u2082 | complete-expansion-ana gc ec\u2081 t1 x\u2083\n    ... | ih1 , ih4 | ih2 , ih3 = DCAp (DCCast ih1 (comp-ana gc x\u2082 ih1) (TCArr t1 t2)) (DCCast ih2 (comp-ana gc x\u2083 ih2) t1) ,\n                                  t2 ,\n                                  tr (\u03bb qq \u2192 (qq \u222a \u03942) == \u2205) (! ih4) (tr (\u03bb qq \u2192 (\u2205 \u222a qq) == \u2205) (! ih3) refl)\n    complete-expansion-synth gc () ESEHole\n    complete-expansion-synth gc () (ESNEHole _ exp)\n    complete-expansion-synth gc (ECAsc x ec) (ESAsc x\u2081)\n      with complete-expansion-ana gc ec x x\u2081\n    ... | ih1 , ih2 = DCCast ih1 (comp-ana gc x\u2081 ih1) x , x , ih2\n\n    complete-expansion-ana : \u2200{e \u03c4 \u03c4' \u0393 \u0394 d} \u2192\n                             \u0393 gcomplete \u2192\n                             e ecomplete \u2192\n                             \u03c4 tcomplete \u2192\n                             \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                             (d dcomplete \u00d7 \u0394 == \u2205)\n    complete-expansion-ana gc (ECLam1 ec) () (EALam x\u2081 MAHole exp)\n    complete-expansion-ana gc (ECLam1 ec) (TCArr t1 t2)  (EALam x\u2081 MAArr exp)\n      with complete-expansion-ana (gcomp-extend gc t1 x\u2081) ec t2 exp\n    ... | ih , ih2 = DCLam ih t1 , ih2\n    complete-expansion-ana gc ec tc (EASubsume x x\u2081 x\u2082 x\u2083)\n      with complete-expansion-synth gc ec x\u2082\n    ... | ih1 , ih2 , ih3 = ih1 , ih3\n\n    --this is just a convenience since it shows up a few times above\n    comp-ana : \u2200{\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n       \u0393 gcomplete \u2192\n       \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n       d dcomplete \u2192\n       \u03c4' tcomplete\n    comp-ana gc ex dc = complete-ta gc (\u03c02 (typed-expansion-ana ex)) dc\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import typed-expansion\nopen import lemmas-gcomplete\n\nmodule complete-expansion where\n  -- this might be derivable from things below and a fact about => and ::\n  -- that we seem to have not proven\n  complete-ta : \u2200{\u0393 \u0394 d \u03c4} \u2192 (\u0393 gcomplete) \u2192 (\u0394 , \u0393 \u22a2 d :: \u03c4) \u2192 d dcomplete \u2192 \u03c4 tcomplete\n  complete-ta gc TAConst comp = TCBase\n  complete-ta gc (TAVar x\u2081) DCVar = gc _ _ x\u2081\n  complete-ta gc (TALam a wt) (DCLam comp x\u2081) = TCArr x\u2081 (complete-ta (gcomp-extend gc x\u2081 a ) wt comp)\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) with complete-ta gc wt comp\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) | TCArr qq qq\u2081 = qq\u2081\n  complete-ta gc (TAEHole x x\u2081) ()\n  complete-ta gc (TANEHole x wt x\u2081) ()\n  complete-ta gc (TACast wt x) (DCCast comp x\u2081 x\u2082) = x\u2082\n  complete-ta gc (TAFailedCast wt x x\u2081 x\u2082) ()\n\n  comp-synth : \u2200{\u0393 e \u03c4} \u2192\n                   \u0393 gcomplete \u2192\n                   e ecomplete \u2192\n                   \u0393 \u22a2 e => \u03c4 \u2192\n                   \u03c4 tcomplete\n  comp-synth gc ec SConst = TCBase\n  comp-synth gc (ECAsc x ec) (SAsc x\u2081) = x\n  comp-synth gc ec (SVar x) = gc _ _ x\n  comp-synth gc (ECAp ec ec\u2081) (SAp _ wt MAHole x\u2081) with comp-synth gc ec wt\n  ... | ()\n  comp-synth gc (ECAp ec ec\u2081) (SAp _ wt MAArr x\u2081) with comp-synth gc ec wt\n  comp-synth gc (ECAp ec ec\u2081) (SAp _ wt MAArr x\u2081) | TCArr qq qq\u2081 = qq\u2081\n  comp-synth gc () SEHole\n  comp-synth gc () (SNEHole _ wt)\n  comp-synth gc (ECLam2 ec x\u2081) (SLam x\u2082 wt) = TCArr x\u2081 (comp-synth (gcomp-extend gc x\u2081 x\u2082) ec wt)\n\n  mutual\n    complete-expansion-synth : \u2200{e \u03c4 \u0393 \u0394 d} \u2192\n                               \u0393 gcomplete \u2192\n                               e ecomplete \u2192\n                               \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                               (d dcomplete \u00d7 \u03c4 tcomplete)\n    complete-expansion-synth gc ec ESConst = DCConst , TCBase\n    complete-expansion-synth gc ec (ESVar x\u2081) = DCVar , gc _ _ x\u2081\n    complete-expansion-synth gc (ECLam2 ec x\u2081) (ESLam x\u2082 exp) with complete-expansion-synth (gcomp-extend gc x\u2081 x\u2082) ec exp\n    ... | ih1 , ih2 = DCLam ih1 x\u2081 , TCArr x\u2081 ih2\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp _ _ x MAHole x\u2082 x\u2083) with comp-synth gc ec x\n    ... | ()\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp _ _ x MAArr x\u2082 x\u2083)\n      with comp-synth gc ec x\n    ... | TCArr t1 t2\n      with complete-expansion-ana gc ec (TCArr t1 t2) x\u2082 | complete-expansion-ana gc ec\u2081 t1 x\u2083\n    ... | ih1 | ih2 = DCAp (DCCast ih1 (comp-ana gc x\u2082 ih1) (TCArr t1 t2)) (DCCast ih2 (comp-ana gc x\u2083 ih2) t1) , t2\n\n    complete-expansion-synth gc () ESEHole\n    complete-expansion-synth gc () (ESNEHole _ exp)\n    complete-expansion-synth gc (ECAsc x ec) (ESAsc x\u2081)\n      with complete-expansion-ana gc ec x x\u2081\n    ... | ih = DCCast ih (comp-ana gc x\u2081 ih) x , x\n\n    complete-expansion-ana : \u2200{e \u03c4 \u03c4' \u0393 \u0394 d} \u2192\n                             \u0393 gcomplete \u2192\n                             e ecomplete \u2192\n                             \u03c4 tcomplete \u2192\n                             \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                             d dcomplete\n    complete-expansion-ana gc (ECLam1 ec) () (EALam x\u2081 MAHole exp)\n    complete-expansion-ana gc (ECLam1 ec) (TCArr t1 t2)  (EALam x\u2081 MAArr exp)\n      with complete-expansion-ana (gcomp-extend gc t1 x\u2081) ec t2 exp\n    ... | ih = DCLam ih t1\n    complete-expansion-ana gc ec tc (EASubsume x x\u2081 x\u2082 x\u2083) = \u03c01(complete-expansion-synth gc ec x\u2082)\n\n    --this is just a convenience since it shows up a few times above\n    comp-ana : \u2200{\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n       \u0393 gcomplete \u2192\n       \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n       d dcomplete \u2192\n       \u03c4' tcomplete\n    comp-ana gc ex dc = complete-ta gc (\u03c02 (typed-expansion-ana ex)) dc\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c68a9b8870e9eaca8c1a746f2ec67e2c84d97fd5","subject":"CyclicGroup: remove useless import","message":"CyclicGroup: remove useless import\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/JS\/BigI\/CyclicGroup.agda","new_file":"Crypto\/JS\/BigI\/CyclicGroup.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import FFI.JS using (JS[_]; return; _++_; _>>_)\nopen import FFI.JS.Check using (check!)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Raw\nopen import Algebra.Group\n\n-- TODO carry on a primality proof of p\nmodule Crypto.JS.BigI.CyclicGroup (p : BigI) where\n\nabstract\n  \u2124[_]\u2605 : Set\n  \u2124[_]\u2605 = BigI\n\n  private\n    \u2124p\u2605 : Set\n    \u2124p\u2605 = BigI\n\n    mod-p : BigI \u2192 \u2124p\u2605\n    mod-p x = mod x p\n\n  -- There are two ways to go from BigI to \u2124p\u2605: check and mod-p\n  -- Use check for untrusted input data and mod-p for internal\n  -- computation.\n  BigI\u25b9\u2124[_]\u2605 : BigI \u2192 JS[ \u2124p\u2605 ]\n  BigI\u25b9\u2124[_]\u2605 x =\n    -- Console.log \"BigI\u25b9\u2124[_]\u2605\" >>\n    check! \"below modulus\" (x <I p)\n           (\u03bb _ \u2192 \"Not below the modulus: p:\" ++\n                  toString p ++\n                  \" is less than x:\" ++\n                  toString x) >>\n    check! \"strictcly positive\" (x >I 0I)\n           (\u03bb _ \u2192 \"Should be strictly positive: \" ++\n                  toString x ++\n                  \" <= 0\") >>\n    return x\n\n  repr : \u2124p\u2605 \u2192 BigI\n  repr x = x\n\n  1# : \u2124p\u2605\n  1# = 1I\n\n  1\/_ : \u2124p\u2605 \u2192 \u2124p\u2605\n  1\/ x = modInv x p\n\n  _^_ : \u2124p\u2605 \u2192 BigI \u2192 \u2124p\u2605\n  x ^ y = modPow x y p\n\n_*_ _\/_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \u2124p\u2605\n\nx * y = mod-p (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\ninstance\n  \u2124[_]\u2605-Eq? : Eq? \u2124p\u2605\n  \u2124[_]\u2605-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\nprod : List \u2124p\u2605 \u2192 \u2124p\u2605\nprod = foldr _*_ 1#\n\nmon-ops : Monoid-Ops \u2124p\u2605\nmon-ops = _*_ , 1#\n\ngrp-ops : Group-Ops \u2124p\u2605\ngrp-ops = mon-ops , 1\/_\n\npostulate\n  grp-struct : Group-Struct grp-ops\n\ngrp : Group \u2124p\u2605\ngrp = grp-ops , grp-struct\n\nmodule grp = Group grp\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import FFI.JS using (JS[_]; return; Bool; _++_; _>>_)\nopen import FFI.JS.Check using (check!)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Raw\nopen import Algebra.Group\n\n-- TODO carry on a primality proof of p\nmodule Crypto.JS.BigI.CyclicGroup (p : BigI) where\n\nabstract\n  \u2124[_]\u2605 : Set\n  \u2124[_]\u2605 = BigI\n\n  private\n    \u2124p\u2605 : Set\n    \u2124p\u2605 = BigI\n\n    mod-p : BigI \u2192 \u2124p\u2605\n    mod-p x = mod x p\n\n  -- There are two ways to go from BigI to \u2124p\u2605: check and mod-p\n  -- Use check for untrusted input data and mod-p for internal\n  -- computation.\n  BigI\u25b9\u2124[_]\u2605 : BigI \u2192 JS[ \u2124p\u2605 ]\n  BigI\u25b9\u2124[_]\u2605 x =\n    -- Console.log \"BigI\u25b9\u2124[_]\u2605\" >>\n    check! \"below modulus\" (x <I p)\n           (\u03bb _ \u2192 \"Not below the modulus: p:\" ++\n                  toString p ++\n                  \" is less than x:\" ++\n                  toString x) >>\n    check! \"strictcly positive\" (x >I 0I)\n           (\u03bb _ \u2192 \"Should be strictly positive: \" ++\n                  toString x ++\n                  \" <= 0\") >>\n    return x\n\n  repr : \u2124p\u2605 \u2192 BigI\n  repr x = x\n\n  1# : \u2124p\u2605\n  1# = 1I\n\n  1\/_ : \u2124p\u2605 \u2192 \u2124p\u2605\n  1\/ x = modInv x p\n\n  _^_ : \u2124p\u2605 \u2192 BigI \u2192 \u2124p\u2605\n  x ^ y = modPow x y p\n\n_*_ _\/_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \u2124p\u2605\n\nx * y = mod-p (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\ninstance\n  \u2124[_]\u2605-Eq? : Eq? \u2124p\u2605\n  \u2124[_]\u2605-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\nprod : List \u2124p\u2605 \u2192 \u2124p\u2605\nprod = foldr _*_ 1#\n\nmon-ops : Monoid-Ops \u2124p\u2605\nmon-ops = _*_ , 1#\n\ngrp-ops : Group-Ops \u2124p\u2605\ngrp-ops = mon-ops , 1\/_\n\npostulate\n  grp-struct : Group-Struct grp-ops\n\ngrp : Group \u2124p\u2605\ngrp = grp-ops , grp-struct\n\nmodule grp = Group grp\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ef672acfb72215f4d1bfad30178223e7de83d7d3","subject":"or-elimiation proved","message":"or-elimiation proved\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/IPL.agda","new_file":"learyouanagda\/IPL.agda","new_contents":"module IPL where\n\n\ndata _\u2227_ (P : Set) (Q : Set) : Set where\n  \u2227-intro : P \u2192 Q \u2192 (P \u2227 Q)\n\n\nproof\u2081 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 P\nproof\u2081 (\u2227-intro p q) = p\n\nproof\u2082 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 Q\nproof\u2082 (\u2227-intro p q) = q\n\n\n_\u21d4_ : (P : Set) \u2192 (Q : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\n\u2227-comm\u2032 : (P Q : Set) \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm\u2032 _ _ (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm : (P Q : Set) \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm P Q = \u2227-intro (\u2227-comm\u2032 P Q) (\u2227-comm\u2032 Q P)\n\n\n\u2227-comm1\u2032 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm1\u2032 (\u2227-intro p q) = \u2227-intro q p\n\n\n\ndata _\u2228_ (P Q : Set) : Set where\n   \u2228-intro\u2081 : P \u2192 P \u2228 Q\n   \u2228-intro\u2082 : Q \u2192 P \u2228 Q\n\n\n\u2228-elim : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 (A \u2228 B) \u2192 C\n\u2228-elim ac bc (\u2228-intro\u2081 a) = ac a\n\u2228-elim ac bc (\u2228-intro\u2082 b) = bc b\n","old_contents":"module IPL where\n\n\ndata _\u2227_ (P : Set) (Q : Set) : Set where\n  \u2227-intro : P \u2192 Q \u2192 (P \u2227 Q)\n\n\nproof\u2081 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 P\nproof\u2081 (\u2227-intro p q) = p\n\nproof\u2082 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 Q\nproof\u2082 (\u2227-intro p q) = q\n\n\n_\u21d4_ : (P : Set) \u2192 (Q : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\n\u2227-comm\u2032 : (P Q : Set) \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm\u2032 _ _ (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm : (P Q : Set) \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm P Q = \u2227-intro (\u2227-comm\u2032 P Q) (\u2227-comm\u2032 Q P)\n\n\n\u2227-comm1\u2032 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm1\u2032 (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm1 : {P Q : Set} \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm1 = \u2227-intro \u2227-comm1\u2032 \u2227-comm1\u2032\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9620d29a7588874f2cd63c3fdb4e0d447c3b59aa","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 22c69a21ac99143a432e5eaf91e798a\n\ndarcs-hash:20110305125148-3bd4e-eaf2fcfcc0c50fbe2e040203577ac1ce04f9e8d4.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/List\/LT-Length\/Induction\/Acc\/WellFoundedInduction.agda","new_file":"src\/FOTC\/Data\/List\/LT-Length\/Induction\/Acc\/WellFoundedInduction.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation LTL\n------------------------------------------------------------------------------\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\nopen import FOTC.Data.List.LT-Length\nopen import FOTC.Data.List.Type\n\nmodule FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInduction\n  (xs<[]\u2192\u22a5                   : \u2200 {xs} \u2192 List xs \u2192 \u00ac (LTL xs []))\n  (x\u2237xs<y\u2237ys\u2192xs<ys           : \u2200 {x xs y ys} \u2192 List xs \u2192 List ys \u2192\n                               LTL (x \u2237 xs) (y \u2237 ys) \u2192 LTL xs ys)\n  (<-trans                   : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n                               LTL xs ys \u2192 LTL ys zs \u2192 LTL xs zs)\n  (lg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs   : \u2200 {xs ys zs} \u2192 length xs \u2261 length ys \u2192\n                               LTL ys zs \u2192 LTL xs zs)\n  (xs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys : \u2200 {xs y ys} \u2192 List xs \u2192 List ys \u2192\n                               LTL xs (y \u2237 ys) \u2192\n                               LTL xs ys \u2228 length xs \u2261 length ys)\n  where\n\nopen import FOTC.Data.List.Induction.Acc.WellFounded\n\n------------------------------------------------------------------------------\n-- The relation LTL is well-founded.\n-- Adapted from FOTC.Data.Nat.Induction.Acc.WellFoundedInduction.WF\u2081-LT.\nwf-LTL : WellFounded LTL\nwf-LTL Lxs = acc Lxs (helper Lxs)\n  where\n    helper : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 LTL ys xs \u2192 Acc LTL ys\n    helper nilL Lys ys<[] = \u22a5-elim (xs<[]\u2192\u22a5 Lys ys<[])\n    helper (consL x {xs} Lxs) nilL []<x\u2237xs =\n      acc nilL (\u03bb Lys ys<[] \u2192 \u22a5-elim (xs<[]\u2192\u22a5 Lys ys<[]))\n    helper (consL x {xs} Lxs) (consL y {ys} Lys) y\u2237ys<x\u2237xs = acc (consL y Lys)\n      (\u03bb {zs} Lzs zs<y\u2237ys \u2192\n          let ys<xs : LTL ys xs\n              ys<xs = x\u2237xs<y\u2237ys\u2192xs<ys Lys Lxs y\u2237ys<x\u2237xs\n\n              zs<xs : LTL zs xs\n              zs<xs = [ (\u03bb zs<ys \u2192 <-trans Lzs Lys Lxs zs<ys ys<xs)\n                      , (\u03bb h \u2192 lg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs h ys<xs)\n                      ] (xs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys Lzs Lys zs<y\u2237ys)\n\n          in  helper Lxs Lzs zs<xs\n        )\n\n-- Well-founded induction on the relation LTL.\nwfInd-LTL : (P : D \u2192 Set) \u2192\n            (\u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 LTL ys xs \u2192 P ys) \u2192 P xs) \u2192\n            \u2200 {xs} \u2192 List xs \u2192 P xs\nwfInd-LTL P = WellFoundedInduction wf-LTL\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation LTL\n------------------------------------------------------------------------------\n\nopen import FOTC.Base\nopen import FOTC.Data.List\nopen import FOTC.Data.List.Type\n\nopen import FOTC.Data.List.LT-Length\n\n  module FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInduction\n  (xs<[]\u2192\u22a5                   : \u2200 {xs} \u2192 List xs \u2192 \u00ac (LTL xs []))\n  (x\u2237xs<y\u2237ys\u2192xs<ys           : \u2200 {x xs y ys} \u2192 List xs \u2192 List ys \u2192\n                               LTL (x \u2237 xs) (y \u2237 ys) \u2192 LTL xs ys)\n  (<-trans                   : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n                               LTL xs ys \u2192 LTL ys zs \u2192 LTL xs zs)\n  (lg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs   : \u2200 {xs ys zs} \u2192 length xs \u2261 length ys \u2192\n                               LTL ys zs \u2192 LTL xs zs)\n  (xs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys : \u2200 {xs y ys} \u2192 List xs \u2192 List ys \u2192\n                               LTL xs (y \u2237 ys) \u2192\n                               LTL xs ys \u2228 length xs \u2261 length ys)\n  where\n\nopen import FOTC.Data.List.Induction.Acc.WellFounded\n\n------------------------------------------------------------------------------\n-- The relation LTL is well-founded.\n-- Adapted from FOTC.Data.Nat.Induction.Acc.WellFoundedInduction.WF\u2081-LT\nwf-LTL : WellFounded LTL\nwf-LTL Lxs = acc Lxs (helper Lxs)\n  where\n    helper : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 LTL ys xs \u2192 Acc LTL ys\n    helper nilL Lys ys<[] = \u22a5-elim (xs<[]\u2192\u22a5 Lys ys<[])\n    helper (consL x {xs} Lxs) nilL []<x\u2237xs =\n      acc nilL (\u03bb Lys ys<[] \u2192 \u22a5-elim (xs<[]\u2192\u22a5 Lys ys<[]))\n    helper (consL x {xs} Lxs) (consL y {ys} Lys) y\u2237ys<x\u2237xs = acc (consL y Lys)\n      (\u03bb {zs} Lzs zs<y\u2237ys \u2192\n          let ys<xs : LTL ys xs\n              ys<xs = x\u2237xs<y\u2237ys\u2192xs<ys Lys Lxs y\u2237ys<x\u2237xs\n\n              zs<xs : LTL zs xs\n              zs<xs = [ (\u03bb zs<ys \u2192 <-trans Lzs Lys Lxs zs<ys ys<xs)\n                      , (\u03bb h \u2192 lg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs h ys<xs)\n                      ] (xs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys Lzs Lys zs<y\u2237ys)\n\n          in  helper Lxs Lzs zs<xs\n        )\n\n-- Well-founded induction on the relation LTL.\nwfInd-LTL : (P : D \u2192 Set) \u2192\n            (\u2200 {xs} \u2192 List xs \u2192 (\u2200 {ys} \u2192 List ys \u2192 LTL ys xs \u2192 P ys) \u2192 P xs) \u2192\n            \u2200 {xs} \u2192 List xs \u2192 P xs\nwfInd-LTL P = WellFoundedInduction wf-LTL\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"123b8b8d10845929beedb4c2f3e744a8d8962af8","subject":"otp: permutations","message":"otp: permutations\n","repos":"crypto-agda\/crypto-agda","old_file":"single-bit-one-time-pad.agda","new_file":"single-bit-one-time-pad.agda","new_contents":"module single-bit-one-time-pad where\n\nopen import Function.NP\nopen import Data.Bool.NP as Bool\nopen import Data.Product renaming (map to <_\u00d7_>)\nopen import Data.Nat.NP\nimport Data.Vec.NP as V\nopen V using (Vec; take; drop; drop\u2032; take\u2032; _++_) renaming (swap to vswap)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) where\n    open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n    kont\u2080-not : \u2200 b k \u2192 kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n    kont\u2080-not b k rewrite xor-not-not b k = \u2261.refl\n\n    open \u2261-Reasoning\n\n    lem\u2082 : \u2200 b \u2192 count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem\u2082 b = count\u21ba (runA b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n          \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not b 0b) (kont\u2080-not b 1b) \u27e9\n            count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n          \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n            count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (runA (not b)) \u220e\n\n    lem\u2083 : Safe\u2141? runA\n    lem\u2083 = lem\u2082 0b\n\n    -- A specialized version of lem\u2082 (\u2248lem\u2083)\n    lem\u2084 : Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\n    lem\u2084    = count\u21ba (runA 0b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n            \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b 0b) (kont\u2080-not 0b 1b) \u27e9\n              count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n            \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n              count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (runA 1b) \u220e\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = \u2261.refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\n\u2047 : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- \u2047 = random\n\u2047 = mk id\n\n\nlem'' : \u2200 {k} (f : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 2* #\u27e8 f \u27e9\nlem'' f = \u2261.refl\n\nlem' : \u2200 {k} (f g : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 #\u27e8 g \u2218 tail \u27e9 \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\nlem' f g pf = 2*-inj (\u2261.trans (lem'' f) (\u2261.trans pf (\u2261.sym (lem'' g))))\n\ndrop-tail : \u2200 k {n a} {A : Set a} \u2192 drop (suc k) {n} \u2257 drop k \u2218 tail {A = A}\ndrop-tail k (x \u2237 xs) = V.drop-\u2237 k x xs\n\nlemdrop\u2032 : \u2200 {k n} (f : Bits n \u2192 Bit) \u2192 #\u27e8 f \u2218 drop\u2032 k \u27e9 \u2261 \u27e82^ k * #\u27e8 f \u27e9 \u27e9\nlemdrop\u2032 {zero} f = \u2261.refl\nlemdrop\u2032 {suc k} f = #\u27e8 f \u2218 drop\u2032 k \u2218 tail \u27e9\n                   \u2261\u27e8 lem'' (f \u2218 drop\u2032 k) \u27e9\n                     2* #\u27e8 f \u2218 drop\u2032 k \u27e9\n                   \u2261\u27e8 \u2261.cong 2*_ (lemdrop\u2032 {k} f) \u27e9\n                     2* \u27e82^ k * #\u27e8 f \u27e9 \u27e9 \u220e\n                    where open \u2261-Reasoning\n\n\n\n\n\n\n\n-- exchange to independant statements\nlem-flip$-\u2295 : \u2200 {m n a} {A : Set a} (f : \u21ba m (A \u2192 Bit)) (x : \u21ba n A) \u2192\n               count\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb \u2261 count\u21ba (f \u229b x)\nlem-flip$-\u2295 {m} {n} f x = \u2261.sym (\n               count\u21ba fx\n              \u2261\u27e8 #-+ {m} {n} (run\u21ba fx) \u27e9\n               sum {m} (\u03bb xs \u2192 #\u27e8_\u27e9 {n} (\u03bb ys \u2192 run\u21ba fx (xs ++ ys)))\n              \u2261\u27e8 sum-sum {m} {n} (Bool.to\u2115 \u2218 run\u21ba fx) \u27e9\n               sum {n} (\u03bb ys \u2192 #\u27e8_\u27e9 {m} (\u03bb xs \u2192 run\u21ba fx (xs ++ ys)))\n              \u2261\u27e8 sum-\u2257\u2082 (\u03bb ys xs \u2192 Bool.to\u2115 (run\u21ba fx (xs ++ ys)))\n                        (\u03bb ys xs \u2192 Bool.to\u2115 (run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb (ys ++ xs)))\n                        (\u03bb ys xs \u2192 \u2261.cong Bool.to\u2115 (lem\u2081 xs ys)) \u27e9\n               sum {n} (\u03bb ys \u2192 #\u27e8_\u27e9 {m} (\u03bb xs \u2192 run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb (ys ++ xs)))\n              \u2261\u27e8 \u2261.sym (#-+ {n} {m} (run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb)) \u27e9\n               count\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb\n              \u220e )\n    where open \u2261-Reasoning\n          fx = f \u229b x\n          lem\u2081 : \u2200 xs ys \u2192 run\u21ba f (take m (xs ++ ys)) (run\u21ba x (drop m (xs ++ ys)))\n                         \u2261 run\u21ba f (drop n (ys ++ xs)) (run\u21ba x (take n (ys ++ xs)))\n          lem\u2081 xs ys rewrite V.take-++ m xs ys | V.drop-++ m xs ys\n                           | V.take-++ n ys xs | V.drop-++ n ys xs = \u2261.refl\n\n\u2248\u1d2c\u2032-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c\u2032 toss\n\u2248\u1d2c\u2032-toss true Adv = \u2115\u00b0.+-comm (count\u21ba (Adv true)) _\n\u2248\u1d2c\u2032-toss false Adv = \u2261.refl\n\n\u2248\u1d2c-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c toss\n\u2248\u1d2c-toss b Adv = \u2248\u1d2c\u2032-toss b (return\u1d30 \u2218 Adv)\n\n-- should be equivalent to #-comm if \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 x were convertible to \u27ea _\u2295_ m \u00b7 x \u27eb\n\u2248\u1d2c-\u2047 : \u2200 {k} (m : Bits k) \u2192 \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047\n\u2248\u1d2c-\u2047 {zero}  _       _ = \u2261.refl\n\u2248\u1d2c-\u2047 {suc k} (h \u2237 m) Adv\n  rewrite \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 0b))\n        | \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 1b))\n        = \u2248\u1d2c\u2032-toss h (\u03bb x \u2192 \u27ea Adv \u2218 _\u2237_ x \u00b7 \u2047 \u27eb)\n\nopen OperationSyntax\n\n_\u27e8\u2219\u27e9_ : Op \u2192 \u2200 {m n} \u2192 Endo (\u21ba m (Bits n))\nf \u27e8\u2219\u27e9 g = \u27ea _\u2219_ f \u00b7 g \u27eb\n\n\u2248\u1d2c-op-\u2047 : \u2200 {k} f \u2192 f \u27e8\u2219\u27e9 \u2047 \u2248\u1d2c \u2047 {k}\n\u2248\u1d2c-op-\u2047 = flip #-op\n\n\u2248\u1d2c-\u2047\u2082 : \u2200 {k} (m\u2080 m\u2081 : Bits k) \u2192 \u27ea m\u2080 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m\u2081 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2082 {k} m\u2080 m\u2081 = \u2248\u1d2c.trans {k} (\u2248\u1d2c-\u2047 m\u2080) (\u2248\u1d2c.sym {k} (\u2248\u1d2c-\u2047 m\u2081))\n\n\u2248\u1d2c-\u2047\u2083 : \u2200 {k} (m : Bit \u2192 Bits k) (b : Bit) \u2192 \u27ea m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2083 m b = \u2248\u1d2c-\u2047\u2082 (m b) (m (not b))\n\n\u2248\u1d2c-\u2047\u2084 : \u2200 {k} (m : Bits k \u00d7 Bits k) (b : Bit) \u2192 \u27ea proj m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea proj m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2084 = \u2248\u1d2c-\u2047\u2083 \u2218 proj\n","old_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP as Bool\nopen import Data.Product renaming (map to <_\u00d7_>)\nopen import Data.Nat.NP\nimport Data.Vec.NP as V\nopen V using (Vec; take; drop; drop\u2032; take\u2032; _++_) renaming (swap to vswap)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) where\n    open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n    kont\u2080-not : \u2200 b k \u2192 kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n    kont\u2080-not b k rewrite xor-not-not b k = \u2261.refl\n\n    open \u2261-Reasoning\n\n    lem\u2082 : \u2200 b \u2192 count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem\u2082 b = count\u21ba (runA b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n          \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not b 0b) (kont\u2080-not b 1b) \u27e9\n            count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n          \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n            count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (runA (not b)) \u220e\n\n    lem\u2083 : Safe\u2141? runA\n    lem\u2083 = lem\u2082 0b\n\n    -- A specialized version of lem\u2082 (\u2248lem\u2083)\n    lem\u2084 : Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\n    lem\u2084    = count\u21ba (runA 0b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n            \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b 0b) (kont\u2080-not 0b 1b) \u27e9\n              count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n            \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n              count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (runA 1b) \u220e\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = \u2261.refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\n\u2047 : \u2200 {n} \u2192 \u21ba n (Bits n)\n\u2047 = random\n\n\nlem'' : \u2200 {k} (f : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 2* #\u27e8 f \u27e9\nlem'' f = \u2261.refl\n\nlem' : \u2200 {k} (f g : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 #\u27e8 g \u2218 tail \u27e9 \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\nlem' f g pf = 2*-inj (\u2261.trans (lem'' f) (\u2261.trans pf (\u2261.sym (lem'' g))))\n\ndrop-tail : \u2200 k {n a} {A : Set a} \u2192 drop (suc k) {n} \u2257 drop k \u2218 tail {A = A}\ndrop-tail k (x \u2237 xs) = V.drop-\u2237 k x xs\n\nlemdrop\u2032 : \u2200 {k n} (f : Bits n \u2192 Bit) \u2192 #\u27e8 f \u2218 drop\u2032 k \u27e9 \u2261 \u27e82^ k * #\u27e8 f \u27e9 \u27e9\nlemdrop\u2032 {zero} f = \u2261.refl\nlemdrop\u2032 {suc k} f = #\u27e8 f \u2218 drop\u2032 k \u2218 tail \u27e9\n                   \u2261\u27e8 lem'' (f \u2218 drop\u2032 k) \u27e9\n                     2* #\u27e8 f \u2218 drop\u2032 k \u27e9\n                   \u2261\u27e8 \u2261.cong 2*_ (lemdrop\u2032 {k} f) \u27e9\n                     2* \u27e82^ k * #\u27e8 f \u27e9 \u27e9 \u220e\n                    where open \u2261-Reasoning\n\n\n\n\n\n\n\n-- exchange to independant statements\nlem-flip$-\u2295 : \u2200 {m n a} {A : Set a} (f : \u21ba m (A \u2192 Bit)) (x : \u21ba n A) \u2192\n               count\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb \u2261 count\u21ba (f \u229b x)\nlem-flip$-\u2295 {m} {n} f x = \u2261.sym (\n               count\u21ba fx\n              \u2261\u27e8 #-+ {m} {n} (run\u21ba fx) \u27e9\n               sum {m} (\u03bb xs \u2192 #\u27e8_\u27e9 {n} (\u03bb ys \u2192 run\u21ba fx (xs ++ ys)))\n              \u2261\u27e8 sum-sum {m} {n} (Bool.to\u2115 \u2218 run\u21ba fx) \u27e9\n               sum {n} (\u03bb ys \u2192 #\u27e8_\u27e9 {m} (\u03bb xs \u2192 run\u21ba fx (xs ++ ys)))\n              \u2261\u27e8 sum-\u2257\u2082 (\u03bb ys xs \u2192 Bool.to\u2115 (run\u21ba fx (xs ++ ys)))\n                        (\u03bb ys xs \u2192 Bool.to\u2115 (run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb (ys ++ xs)))\n                        (\u03bb ys xs \u2192 \u2261.cong Bool.to\u2115 (lem\u2081 xs ys)) \u27e9\n               sum {n} (\u03bb ys \u2192 #\u27e8_\u27e9 {m} (\u03bb xs \u2192 run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb (ys ++ xs)))\n              \u2261\u27e8 \u2261.sym (#-+ {n} {m} (run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb)) \u27e9\n               count\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb\n              \u220e )\n    where open \u2261-Reasoning\n          fx = f \u229b x\n          lem\u2081 : \u2200 xs ys \u2192 run\u21ba f (take m (xs ++ ys)) (run\u21ba x (drop m (xs ++ ys)))\n                         \u2261 run\u21ba f (drop n (ys ++ xs)) (run\u21ba x (take n (ys ++ xs)))\n          lem\u2081 xs ys rewrite V.take-++ m xs ys | V.drop-++ m xs ys\n                           | V.take-++ n ys xs | V.drop-++ n ys xs = \u2261.refl\n\n\u2248\u1d2c\u2032-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c\u2032 toss\n\u2248\u1d2c\u2032-toss true Adv = \u2115\u00b0.+-comm (count\u21ba (Adv true)) _\n\u2248\u1d2c\u2032-toss false Adv = \u2261.refl\n\n\u2248\u1d2c-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c toss\n\u2248\u1d2c-toss b Adv = \u2248\u1d2c\u2032-toss b (return\u1d30 \u2218 Adv)\n\n-- should be equivalent to #-comm if \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 x were convertible to \u27ea _\u2295_ m \u00b7 x \u27eb\n\u2248\u1d2c-\u2047 : \u2200 {k} (m : Bits k) \u2192 \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047\n\u2248\u1d2c-\u2047 {zero}  _       _ = \u2261.refl\n\u2248\u1d2c-\u2047 {suc k} (h \u2237 m) Adv\n  rewrite \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 0b))\n        | \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 1b))\n        = \u2248\u1d2c\u2032-toss h (\u03bb x \u2192 \u27ea Adv \u2218 _\u2237_ x \u00b7 \u2047 \u27eb)\n\n\u2248\u1d2c-\u2047\u2082 : \u2200 {k} (m\u2080 m\u2081 : Bits k) \u2192 \u27ea m\u2080 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m\u2081 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2082 {k} m\u2080 m\u2081 = \u2248\u1d2c.trans {k} (\u2248\u1d2c-\u2047 m\u2080) (\u2248\u1d2c.sym {k} (\u2248\u1d2c-\u2047 m\u2081))\n\n\u2248\u1d2c-\u2047\u2083 : \u2200 {k} (m : Bit \u2192 Bits k) (b : Bit) \u2192 \u27ea m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2083 m b = \u2248\u1d2c-\u2047\u2082 (m b) (m (not b))\n\n\u2248\u1d2c-\u2047\u2084 : \u2200 {k} (m : Bits k \u00d7 Bits k) (b : Bit) \u2192 \u27ea proj m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea proj m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2084 = \u2248\u1d2c-\u2047\u2083 \u2218 proj\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c8a0b87298dd1bfc0bfdd36ebc05d9f2b14d82a1","subject":"Typo.","message":"Typo.\n\nIgnore-this: 7ff4279cea3751561b0a45c24a588967\n\ndarcs-hash:20101130074907-3bd4e-2ced29f128dcfb342686615a9d6c60073dd9ec60.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/PA\/Base.agda","new_file":"Draft\/PA\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule Draft.PA.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\n-- N.B. The following module is exported by this module.\nopen import Common.Universe public renaming ( D to PA )\n\n-- Logical constants\n-- N.B. The module is exported by this module.\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : PA\n  succ : PA \u2192 PA\n  _+_  : PA \u2192 PA \u2192 PA\n  _*_  : PA \u2192 PA \u2192 PA\n\n-- The PA equality\n-- The PA equality is the propositional identity on the PA universe.\n-- N.B. The following modules are exported by this module.\n-- N.B. We are not using the refl and sym properties because they are\n-- not stated in the proper axioms (see below).\nopen import Common.Relation.Binary.PropositionalEquality public\n  using ( _\u2261_ )\nopen import Common.Relation.Binary.PropositionalEquality.Properties public\n  using ( trans )\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- S\u2081. x\u2081 = x\u2082 \u2192 x\u2081 = x\u2083 \u2192 x\u2082 = x\u2083\n-- S\u2082. x\u2081 = x\u2082 \u2192 succ x\u2081 = succ x\u2082\n-- S\u2083. 0 \u2260 succ x\n-- S\u2084. succ x\u2081 = succ x\u2082 \u2192 x\u2081 = x\u2082\n-- S\u2085. x\u2081 + 0 = x\u2081\n-- S\u2086. x\u2081 + succ x\u2082 = succ (x\u2081 + x\u2082)\n-- S\u2087. x\u2081 * 0 = 0\n-- S\u2088. x\u2081 * succ x\u2082 = (x\u2081 * x\u2082) + x\u2081\n-- S\u2089. P(0) \u2192 (\u2200x.P(x) \u2192 P(succ x)) \u2192 \u2200x.P(x)\n\npostulate\n  -- S\u2081: This axiom is the transitivity property imported from\n  --     Common.Relation.Binary.PropositionalEquality.Properties.\n\n  -- S\u2082: This axiom is not required by the ATPs.\n\n  S\u2083 : \u2200 {x} \u2192 \u00ac (zero \u2261 succ x)\n\n  S\u2084 : \u2200 {x\u2081 x\u2082} \u2192 succ x\u2081 \u2261 succ x\u2082 \u2192 x\u2081 \u2261 x\u2082\n\n  S\u2085 : \u2200 x \u2192     zero    + x  \u2261 x\n  S\u2086 : \u2200 x\u2081 x\u2082 \u2192 succ x\u2081 + x\u2082 \u2261 succ (x\u2081 + x\u2082)\n\n  S\u2087 : \u2200 x \u2192     zero    * x  \u2261 zero\n  S\u2088 : \u2200 x\u2081 x\u2082 \u2192 succ x\u2081 * x\u2082 \u2261 x\u2082 + x\u2081 * x\u2082\n\n  -- S\u2089: Instead of the induction schema we will use the inductive predicate N\n  --     (see Draft.PA.Type).\n\n{-# ATP axiom S\u2083 #-}\n{-# ATP axiom S\u2084 #-}\n{-# ATP axiom S\u2085 #-}\n{-# ATP axiom S\u2086 #-}\n{-# ATP axiom S\u2087 #-}\n{-# ATP axiom S\u2088 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule Draft.PA.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\n-- N.B. The following module is exported by this module.\nopen import Common.Universe public renaming ( D to PA )\n\n-- Logical constants\n-- N.B. The module is exported by this module.\nopen import Common.LogicalConstants public\n\n-- Non-logical constants of PA.\npostulate\n  zero : PA\n  succ : PA \u2192 PA\n  _+_  : PA \u2192 PA \u2192 PA\n  _*_  : PA \u2192 PA \u2192 PA\n\n-- The PA equality\n-- The PA equality is the propositional identity on the PA universe.\n-- N.B. The following modules are exported by this module.\n-- N.B. We are not using the refl and sym properties because they are\n-- not stated in the proper axioms (see below).\nopen import Common.Relation.Binary.PropositionalEquality public\n  using ( _\u2261_ )\nopen import Common.Relation.Binary.PropositionalEquality.Properties public\n  using ( trans )\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- S\u2081. x\u2081 = x\u2082 \u2192 x\u2081 = x\u2083 \u2192 x\u2082 = x\u2083\n-- S\u2082. x\u2081 = x\u2082 \u2192 succ x\u2081 = succ x\u2082\n-- S\u2083. 0 \u2260 succ x\n-- S\u2084. succ x\u2081 = succ x\u2082 \u2192 x\u2081 = x\u2082\n-- S\u2085. x\u2081 + 0 = x\u2081\n-- S\u2086. x\u2081 + succ x\u2082 = succ (x\u2081 + x\u2082)\n-- S\u2087. x\u2081 * 0 = 0\n-- S\u2088. x\u2081 * succ x\u2082 = (x\u2081 * x\u2082) + x\u2081\n-- S\u2089. P(0) \u2192 (\u2200x.P(x) \u2192 P(succ x)) \u2192 \u2200x.P(x)\n\npostulate\n  -- S\u2081: This axiom is the transitivity property imported from\n  --     Common.Relation.Binary.PropositionalEquality.Properties.\n\n  -- S\u2082: This axiom is not required by the ATPs.\n\n  S\u2083 : \u2200 {x} \u2192 \u00ac (zero \u2261 succ x)\n\n  S\u2084 : \u2200 {x\u2081 x\u2082} \u2192 succ x\u2081 \u2261 succ x\u2082 \u2192 x\u2081 \u2261 x\u2082\n\n  S\u2085 : \u2200 x \u2192     zero    + x  \u2261 x\n  S\u2086 : \u2200 x\u2081 x\u2082 \u2192 succ x\u2081 + x\u2082 \u2261 succ (x\u2081 + x\u2082)\n\n  S\u2087 : \u2200 x \u2192     zero    * x  \u2261 zero\n  S\u2088 : \u2200 x\u2081 x\u2082 \u2192 succ x\u2081 * x\u2082 \u2261 x\u2082 + x\u2081 * x\u2082\n\n  -- S\u2089: Instead of the induction schema we will use the inductive predicate N\n  --     see (Draft.PA.Type)\n\n{-# ATP axiom S\u2083 #-}\n{-# ATP axiom S\u2084 #-}\n{-# ATP axiom S\u2085 #-}\n{-# ATP axiom S\u2086 #-}\n{-# ATP axiom S\u2087 #-}\n{-# ATP axiom S\u2088 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3a1143b9db99b9ec7e1431f08aab91f27cf9bd7a","subject":"Document Postulate.Extensionality regularly.","message":"Document Postulate.Extensionality regularly.\n\nOld-commit-hash: 35222e92088dbd2313ec7252e50cd68e85b340bf\n","repos":"inc-lc\/ilc-agda","old_file":"Postulate\/Extensionality.agda","new_file":"Postulate\/Extensionality.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Postulate extensionality of functions.\n--\n-- Justification on Agda mailing list:\n-- http:\/\/permalink.gmane.org\/gmane.comp.lang.agda\/2343\n------------------------------------------------------------------------\n\nmodule Postulate.Extensionality where\n\n\nopen import Relation.Binary.PropositionalEquality\n\npostulate ext : \u2200 {a b} \u2192 Extensionality a b\n\n-- Convenience of using extensionality 3 times in a row\n-- (using it twice in a row is moderately tolerable)\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n","old_contents":"module Postulate.Extensionality where\n\n-- POSTULATE EXTENSIONALITY\n--\n-- Justification on Agda mailing list:\n-- http:\/\/permalink.gmane.org\/gmane.comp.lang.agda\/2343\n\nopen import Relation.Binary.PropositionalEquality\n\npostulate ext : \u2200 {a b} \u2192 Extensionality a b\n\n-- Convenience of using extensionality 3 times in a row\n-- (using it twice in a row is moderately tolerable)\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e7d0ffc6976dc3270e58e80326cf78dc7e9dcb2e","subject":"flat-funs: put the type interface in the universe module","message":"flat-funs: put the type interface in the universe module\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat\nopen import Data.Bits using (Bits)\nimport Level as L\nimport Function as F\nopen import composable\nopen import vcomp\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n\n    _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    constBits : \u2200 {n} \u2192 Bits n \u2192 `\u22a4 `\u2192 `Bits n\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Product renaming (zip to \u00d7-zip; map to \u00d7-map)\nopen import Function\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id (\u03bb f g x \u2192 g (f x)) (\u03bb f g \u2192 \u00d7-map f g) _&&&_ proj\u2081 proj\u2082 const\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id (\u03bb f g \u2192 g \u2218 f) (VComposable._***_ bitsFunVComp) _&&&_ (\u03bb {A} \u2192 take A)\n                    (\u03bb {A} \u2192 drop A) (\u03bb xs _ \u2192 xs ++ [] {- ugly? -})\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe\n                           (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                        (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                        (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4 = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                         (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                         (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ _\u2294_ _\u2294_ 0 0 (const 0)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ _+_ _+_ 0 0 (\u03bb {n} _ \u2192 n)\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n  open Composable isComposable\n  open VComposable isVComposable\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk \u22a4 Bit _\u00d7_ Vec (\u03bb A B \u2192 A \u2192 B)\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk 0 1 _+_ _*_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_ proj\u2081 proj\u2082\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id bitsFunComp bitsFunVComp _&&&_ (\u03bb {A} \u2192 take A) (\u03bb {A} \u2192 drop A)\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = ?\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = ?\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"77d92178ad638a41ae32099024888c6bf121bb0b","subject":"Desc model: generic proof that map id = id.","message":"Desc model: generic proof that map id = id.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A : Set}{B : A -> Set}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"785dddce82b3dc294f8d52a3a8a96b64366b1ac4","subject":"Remove more","message":"Remove more\n\nNo Structure in Base.\n\nOld-commit-hash: 81cee315c1ade9b842f0400fbc2bd215a189e8dc\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule Structure where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely.\n    --\n    -- * That should be be true for\n    --   functions using changes parametrically.\n    --\n    -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n    --   this is proved below on both contexts at once by fun-change-respects.\n    --\n    -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n    --   it by definition, and \u229f doesn't take change arguments - we will only\n    --   need a proof for compose, when we define it.\n    --\n    -- Stating the general result, though, seems hard, we should\n    -- rather have lemmas proving that certain classes of functions respect this\n    -- equivalence.\n\n  module _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n    {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n    module FC = FunctionChanges A B {{CA}} {{CB}}\n    open FC using (changeAlgebra; incrementalization)\n    open FC.FunctionChange\n\n    fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n      df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n    fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n      where\n        open \u2261-Reasoning\n        -- This type signature just expands the goal manually a bit.\n        lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n        -- Informally: use incrementalization on both sides and then apply\n        -- congruence.\n        lemma =\n          begin\n            f x \u229e apply df\u2081 x dx\u2081\n          \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2081)\n          \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2082)\n          \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n            (f \u229e df\u2082) (x \u229e dx\u2082)\n          \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n            f x \u229e apply df\u2082 x dx\u2082\n          \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely.\n    --\n    -- * That should be be true for\n    --   functions using changes parametrically.\n    --\n    -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n    --   this is proved below on both contexts at once by fun-change-respects.\n    --\n    -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n    --   it by definition, and \u229f doesn't take change arguments - we will only\n    --   need a proof for compose, when we define it.\n    --\n    -- Stating the general result, though, seems hard, we should\n    -- rather have lemmas proving that certain classes of functions respect this\n    -- equivalence.\n\n  module _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n    {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n    module FC = FunctionChanges A B {{CA}} {{CB}}\n    open FC using (changeAlgebra; incrementalization)\n    open FC.FunctionChange\n\n    fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n      df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n    fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n      where\n        open \u2261-Reasoning\n        open import Postulate.Extensionality\n        -- This type signature just expands the goal manually a bit.\n        lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n        -- Informally: use incrementalization on both sides and then apply\n        -- congruence.\n        lemma =\n          begin\n            f x \u229e apply df\u2081 x dx\u2081\n          \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2081)\n          \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2082)\n          \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n            (f \u229e df\u2082) (x \u229e dx\u2082)\n          \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n            f x \u229e apply df\u2082 x dx\u2082\n          \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2aa331ecc397f402c621e7b4b53886eb5ca53e91","subject":"Use HOAS-enabled helpers for nested abstractions.","message":"Use HOAS-enabled helpers for nested abstractions.\n\nOld-commit-hash: f5006b448dcc30a8ff103cf5de501e945e14faa1\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x)\n      ,\n      abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24ce7b62d29582982f0f0c2694ebd10a473113aa","subject":"IND-CCA2: Hide the Rat code","message":"IND-CCA2: Hide the Rat code\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/IND-CCA2.agda","new_file":"Game\/IND-CCA2.agda","new_contents":"\n{-# OPTIONS --without-K #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Unit\n\nopen import Data.Nat.NP\n--open import Rat\n\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Relation.Binary.PropositionalEquality\n\nmodule Game.IND-CCA2\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n\nwhere\nopen import Game.CCA-Common Message CipherText\n                         \nAdv : \u2605\nAdv = R\u2090 \u2192 PubKey \u2192 Strategy ((Message \u00d7 Message) \u00d7 (CipherText \u2192 Strategy Bit))\n\n{-\nValid-Adv : Adv \u2192 Set\nValid-Adv adv = \u2200 {r\u2090 r\u2093 pk c} \u2192 Valid (\u03bb x \u2192 x \u2262 c) {!!}\n-}\n\nR : \u2605\nR = R\u2090 \u00d7 R\u2096 \u00d7 R\u2091\n\nGame : \u2605\nGame = Adv \u2192 R \u2192 Bit\n\n\n\u2141 : Bit \u2192 Game\n\u2141 b m (r\u2090 , r\u2096 , r\u2091) with KeyGen r\u2096\n... | pk , sk = b\u2032 where\n  open Eval Dec sk\n  \n  ev = eval (m r\u2090 pk)\n  mb = proj (proj\u2081 ev) b\n  d = proj\u2082 ev\n\n  c  = Enc pk mb r\u2091\n  b\u2032 = eval (d c)\n\n  \nmodule Advantage\n  (\u03bc\u2091 : Explore\u2080 R\u2091)\n  (\u03bc\u2096 : Explore\u2080 R\u2096)\n  (\u03bc\u2090 : Explore\u2080 R\u2090)\n  where\n  \u03bcR : Explore\u2080 R\n  \u03bcR = \u03bc\u2090 \u00d7\u1d49 \u03bc\u2096 \u00d7\u1d49 \u03bc\u2091\n  \n  module \u03bcR = FromExplore\u2080 \u03bcR\n  \n  run : Bit \u2192 Adv \u2192 \u2115\n  run b adv = \u03bcR.count (\u2141 b adv)\n  \n  Advantage : Adv \u2192 \u2115\n  Advantage adv = dist (run 0b adv) (run 1b adv)\n    \n  --Advantage\u211a : Adv \u2192 \u211a\n  --Advantage\u211a adv = Advantage adv \/ \u03bcR.Card\n  \n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n{-# OPTIONS --without-K #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Unit\n\nopen import Data.Nat.NP\nopen import Rat\n\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Relation.Binary.PropositionalEquality\n\nmodule Game.IND-CCA2\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n\nwhere\nopen import Game.CCA-Common Message CipherText\n                         \nAdv : \u2605\nAdv = R\u2090 \u2192 PubKey \u2192 Strategy ((Message \u00d7 Message) \u00d7 (CipherText \u2192 Strategy Bit))\n\n{-\nValid-Adv : Adv \u2192 Set\nValid-Adv adv = \u2200 {r\u2090 r\u2093 pk c} \u2192 Valid (\u03bb x \u2192 x \u2262 c) {!!}\n-}\n\nR : \u2605\nR = R\u2090 \u00d7 R\u2096 \u00d7 R\u2091\n\nGame : \u2605\nGame = Adv \u2192 R \u2192 Bit\n\n\n\u2141 : Bit \u2192 Game\n\u2141 b m (r\u2090 , r\u2096 , r\u2091) with KeyGen r\u2096\n... | pk , sk = b\u2032 where\n  open Eval Dec sk\n  \n  ev = eval (m r\u2090 pk)\n  mb = proj (proj\u2081 ev) b\n  d = proj\u2082 ev\n\n  c  = Enc pk mb r\u2091\n  b\u2032 = eval (d c)\n\n  \nmodule Advantage\n  (\u03bc\u2091 : Explore\u2080 R\u2091)\n  (\u03bc\u2096 : Explore\u2080 R\u2096)\n  (\u03bc\u2090 : Explore\u2080 R\u2090)\n  where\n  \u03bcR : Explore\u2080 R\n  \u03bcR = \u03bc\u2090 \u00d7\u1d49 \u03bc\u2096 \u00d7\u1d49 \u03bc\u2091\n  \n  module \u03bcR = FromExplore\u2080 \u03bcR\n  \n  run : Bit \u2192 Adv \u2192 \u2115\n  run b adv = \u03bcR.count (\u2141 b adv)\n  \n  Advantage : Adv \u2192 \u2115\n  Advantage adv = dist (run 0b adv) (run 1b adv)\n    \n  Advantage\u211a : Adv \u2192 \u211a\n  Advantage\u211a adv = Advantage adv \/ \u03bcR.Card\n  \n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"43f907923a585a467516163bdb20896c50287af0","subject":"removing dep","message":"removing dep\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"cast-inert.agda","new_file":"cast-inert.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import typed-expansion\nopen import lemmas-gcomplete\nopen import lemmas-complete\nopen import progress-checks\nopen import finality\n\nmodule cast-inert where\n  -- if a term is compelete and well typed, then the casts inside are all\n  -- identity casts and there are no failed casts\n  cast-inert : \u2200{\u0394 \u0393 d \u03c4} \u2192\n                  \u0393 gcomplete \u2192\n                  d dcomplete \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                  cast-id d -- all casts are id, and there are no failed casts\n  cast-inert gc dc TAConst = CIConst\n  cast-inert gc dc (TAVar x\u2081) = CIVar\n  cast-inert gc (DCLam dc x\u2081) (TALam x\u2082 wt) = CILam (cast-inert (gcomp-extend gc x\u2081 x\u2082) dc wt)\n  cast-inert gc (DCAp dc dc\u2081) (TAAp wt wt\u2081) = CIAp (cast-inert gc dc wt)\n                                                   (cast-inert gc dc\u2081 wt\u2081)\n  cast-inert gc () (TAEHole x x\u2081)\n  cast-inert gc () (TANEHole x wt x\u2081)\n  cast-inert gc (DCCast dc x x\u2081) (TACast wt x\u2082)\n    with eq-complete-consist x x\u2081 x\u2082\n  ... | refl = CICast (cast-inert gc dc wt)\n  cast-inert gc () (TAFailedCast wt x x\u2081 x\u2082)\n\n  -- relates expressions to the same thing with all identity casts removed\n  data no-id-casts : dhexp \u2192 dhexp \u2192 Set where\n    NICConst  : no-id-casts c c\n    NICVar    : \u2200{x} \u2192 no-id-casts (X x) (X x)\n    NICLam    : \u2200{x \u03c4 d d'} \u2192 no-id-casts d d' \u2192 no-id-casts (\u00b7\u03bb x [ \u03c4 ] d) (\u00b7\u03bb x [ \u03c4 ] d')\n    NICHole   : \u2200{u} \u2192 no-id-casts (\u2987\u2988\u27e8 u \u27e9) (\u2987\u2988\u27e8 u \u27e9)\n    NICNEHole : \u2200{d d' u} \u2192 no-id-casts d d' \u2192 no-id-casts (\u2987 d \u2988\u27e8 u \u27e9) (\u2987 d' \u2988\u27e8 u \u27e9)\n    NICAp     : \u2200{d1 d2 d1' d2'} \u2192 no-id-casts d1 d1' \u2192 no-id-casts d2 d2' \u2192 no-id-casts (d1 \u2218 d2) (d1' \u2218 d2')\n    NICCast   : \u2200{d d' \u03c4} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) d'\n    NICFailed : \u2200{d d' \u03c41 \u03c42} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- removing identity casts doesn't change the type\n  no-id-casts-type : \u2200{\u0393 \u0394 d \u03c4 d' } \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                                      no-id-casts d d' \u2192\n                                      \u0394 , \u0393 \u22a2 d' :: \u03c4\n  no-id-casts-type TAConst NICConst = TAConst\n  no-id-casts-type (TAVar x\u2081) NICVar = TAVar x\u2081\n  no-id-casts-type (TALam x\u2081 wt) (NICLam nic) = TALam x\u2081 (no-id-casts-type wt nic)\n  no-id-casts-type (TAAp wt wt\u2081) (NICAp nic nic\u2081) = TAAp (no-id-casts-type wt nic) (no-id-casts-type wt\u2081 nic\u2081)\n  no-id-casts-type (TAEHole x x\u2081) NICHole = TAEHole x x\u2081\n  no-id-casts-type (TANEHole x wt x\u2081) (NICNEHole nic) = TANEHole x (no-id-casts-type wt nic) x\u2081\n  no-id-casts-type (TACast wt x) (NICCast nic) = no-id-casts-type wt nic\n  no-id-casts-type (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) = TAFailedCast (no-id-casts-type wt nic) x x\u2081 x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import typed-expansion\nopen import lemmas-gcomplete\nopen import lemmas-complete\nopen import progress-checks\nopen import finality\nopen import preservation\n\nmodule cast-inert where\n  -- if a term is compelete and well typed, then the casts inside are all\n  -- identity casts and there are no failed casts\n  cast-inert : \u2200{\u0394 \u0393 d \u03c4} \u2192\n                  \u0393 gcomplete \u2192\n                  d dcomplete \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                  cast-id d -- all casts are id, and there are no failed casts\n  cast-inert gc dc TAConst = CIConst\n  cast-inert gc dc (TAVar x\u2081) = CIVar\n  cast-inert gc (DCLam dc x\u2081) (TALam x\u2082 wt) = CILam (cast-inert (gcomp-extend gc x\u2081 x\u2082) dc wt)\n  cast-inert gc (DCAp dc dc\u2081) (TAAp wt wt\u2081) = CIAp (cast-inert gc dc wt)\n                                                   (cast-inert gc dc\u2081 wt\u2081)\n  cast-inert gc () (TAEHole x x\u2081)\n  cast-inert gc () (TANEHole x wt x\u2081)\n  cast-inert gc (DCCast dc x x\u2081) (TACast wt x\u2082)\n    with eq-complete-consist x x\u2081 x\u2082\n  ... | refl = CICast (cast-inert gc dc wt)\n  cast-inert gc () (TAFailedCast wt x x\u2081 x\u2082)\n\n  -- relates expressions to the same thing with all identity casts removed\n  data no-id-casts : dhexp \u2192 dhexp \u2192 Set where\n    NICConst  : no-id-casts c c\n    NICVar    : \u2200{x} \u2192 no-id-casts (X x) (X x)\n    NICLam    : \u2200{x \u03c4 d d'} \u2192 no-id-casts d d' \u2192 no-id-casts (\u00b7\u03bb x [ \u03c4 ] d) (\u00b7\u03bb x [ \u03c4 ] d')\n    NICHole   : \u2200{u} \u2192 no-id-casts (\u2987\u2988\u27e8 u \u27e9) (\u2987\u2988\u27e8 u \u27e9)\n    NICNEHole : \u2200{d d' u} \u2192 no-id-casts d d' \u2192 no-id-casts (\u2987 d \u2988\u27e8 u \u27e9) (\u2987 d' \u2988\u27e8 u \u27e9)\n    NICAp     : \u2200{d1 d2 d1' d2'} \u2192 no-id-casts d1 d1' \u2192 no-id-casts d2 d2' \u2192 no-id-casts (d1 \u2218 d2) (d1' \u2218 d2')\n    NICCast   : \u2200{d d' \u03c4} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) d'\n    NICFailed : \u2200{d d' \u03c41 \u03c42} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- removing identity casts doesn't change the type\n  no-id-casts-type : \u2200{\u0393 \u0394 d \u03c4 d' } \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                                      no-id-casts d d' \u2192\n                                      \u0394 , \u0393 \u22a2 d' :: \u03c4\n  no-id-casts-type TAConst NICConst = TAConst\n  no-id-casts-type (TAVar x\u2081) NICVar = TAVar x\u2081\n  no-id-casts-type (TALam x\u2081 wt) (NICLam nic) = TALam x\u2081 (no-id-casts-type wt nic)\n  no-id-casts-type (TAAp wt wt\u2081) (NICAp nic nic\u2081) = TAAp (no-id-casts-type wt nic) (no-id-casts-type wt\u2081 nic\u2081)\n  no-id-casts-type (TAEHole x x\u2081) NICHole = TAEHole x x\u2081\n  no-id-casts-type (TANEHole x wt x\u2081) (NICNEHole nic) = TANEHole x (no-id-casts-type wt nic) x\u2081\n  no-id-casts-type (TACast wt x) (NICCast nic) = no-id-casts-type wt nic\n  no-id-casts-type (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) = TAFailedCast (no-id-casts-type wt nic) x x\u2081 x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"01f4ca20dd3c0c79513911d02d3bdc3c76f870d7","subject":"Abstraction and application of stability of terms (i. e., the derivative of a stable term is its nil change in any honest environment)","message":"Abstraction and application of stability of terms\n(i. e., the derivative of a stable term is its nil change in any\nhonest environment)\n\nCaution: Validity assumption is introduced into the close-enough\nrelation. An optimized derivative expecting volatility will NOT be\npropositionally equal to the original derivative. They will only\nact identically on consistent \u0394-environments and valid future\nargument pairs.\n\nOld-commit-hash: b34946276b4d9fed70d1a900dfef8c90df0bb7ad\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-- Useful statements about Boolean\n\u2227-proj\u2081 : \u2200 {a b} \u2192 a \u2227 b \u2261 true \u2192 a \u2261 true\n\u2227-proj\u2081 {false} {_} ()\n\u2227-proj\u2081 {true} {false} ()\n\u2227-proj\u2081 {true} {true} eq = refl\n\n\u2227-proj\u2082 : \u2200 {a b} \u2192 a \u2227 b \u2261 true \u2192 b \u2261 true\n\u2227-proj\u2082 {false} {_} ()\n\u2227-proj\u2082 {true} {false} ()\n\u2227-proj\u2082 {true} {true} eq = refl\n\n-- Extending correctness-on-closed-terms\nextended-correctness : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192\n  (f : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (x : Term \u0393 \u03c4\u2081)\n    {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n    \u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 x \u27e7 \u03c1)\n    \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 x \u27e7 \u03c1) (\u27e6 derive x \u27e7 \u03c1)\n  \u2261 (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u03c1)\n    (\u27e6 weaken \u0393\u227c\u0394\u0393 x \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive x \u27e7 \u03c1)\n\nextended-correctness {\u03c4\u2081} {\u03c4\u2082} {\u0393} f x {\u03c1} {consistency} = sym (proj\u2082\n  (validity-of-derive\u2032 \u03c1 {consistency} f\n    (\u27e6 weaken \u0393\u227c\u0394\u0393 x \u27e7 \u03c1) (\u27e6 derive x \u27e7 \u03c1)\n    (validity-of-derive\u2032 \u03c1 {consistency} x)))\n\n-- Debug tool\nabsurd! : \u2200 {A B : Set} \u2192 B \u2192 A \u2192 A \u2192 B\nabsurd! b _ _ = b\n\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} {R[x,dx] : valid-\u0394 x dx}\n  \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} {validity : valid-\u0394 x dx}\n  \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\ndf\u2248df : \u2200 {\u03c4} {df : \u27e6 \u0394-Type \u03c4 \u27e7} {args : Args \u03c4} \u2192 df \u2248 df WRT args\ndf\u2248df {nats} = refl\ndf\u2248df {bags} = refl\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {abide args} =\n  \u03bb {x} {dx} _ \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {alter args} =\n  \u03bb {x} {dx}   \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {\u03c4}{_} {args} {df} {dg} {\u03c1} df=dg\n  rewrite df=dg = df\u2248df {\u03c4} {\u27e6 dg \u27e7 \u03c1} {args}\n\n-- A variable does not change if its value is unchanging.\nstabilityVar : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stableVar x vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 deriveVar x \u27e7 \u03c1 \u2261 \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1\n\nstabilityVar this (alter vars) () (alter honesty)\nstabilityVar this (abide vars) refl (abide proof honesty) = proof\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (abide vars) truth (abide _ honesty)\n  = stabilityVar x vars truth honesty\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (alter vars) truth (alter honesty)\n  = stabilityVar x vars truth honesty\n\n-- A term does not change if its free variables are unchanging.\nstability : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stable t vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {_ : Consistent-\u0394env \u03c1} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\n\nstability (nat n) vars truth {\u03c1} _ = refl\n\nstability (bag b) vars truth {\u03c1} _ = b++\u2205=b\n\nstability (var x) vars truth {\u03c1} honesty =\n  stabilityVar x vars truth honesty\n\nstability (abs {\u03c4\u2081} {\u03c4\u2082} {\u0393} t) vars truth {\u03c1} {consistency} honesty =\n  extensionality (\u03bb v \u2192 let\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6derive\u27e7 v\n    mutual-weakening : \u27e6 weaken (keep \u03c4\u2081 \u2022 \u0393\u227c\u0394\u0393) t \u27e7 (v \u2022 \u03c1)\n                     \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 (dv \u2022 v \u2022 \u03c1)\n    mutual-weakening =\n      trans (weaken-sound t (v \u2022 \u03c1))\n            (trans (cong (\u03bb hole \u2192 \u27e6 t \u27e7 hole)\n                     {x = \u27e6 keep \u03c4\u2081 \u2022 \u0393\u227c\u0394\u0393 \u27e7 (v \u2022 \u03c1)}\n                     {y = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 (dv \u2022 v \u2022 \u03c1)}\n                     refl)\n                   (sym (weaken-sound t (dv \u2022 v \u2022 \u03c1))))\n  in\n    begin\n      \u27e6 weaken (keep \u03c4\u2081 \u2022 \u0393\u227c\u0394\u0393) t \u27e7 (v \u2022 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1))\n            mutual-weakening \u27e9\n      \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 (dv \u2022 v \u2022 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n    \u2261\u27e8 stability t (abide vars) truth\n         {dv \u2022 v \u2022 \u03c1}\n         {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] v) consistency}\n         (abide (f\u2295\u0394f=f v) honesty) \u27e9\n      \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 (dv \u2022 v \u2022 \u03c1)\n    \u2261\u27e8 sym mutual-weakening \u27e9\n      \u27e6 weaken (keep \u03c4\u2081 \u2022 \u0393\u227c\u0394\u0393) t \u27e7 (v \u2022 \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\nstability (app s t) vars truth {\u03c1} {consistency} honesty =\n  let\n    f  = \u27e6 weaken \u0393\u227c\u0394\u0393 s \u27e7 \u03c1\n    x  = \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\n    df = \u27e6 derive s \u27e7 \u03c1\n    dx = \u27e6 derive t \u27e7 \u03c1\n  in\n    begin\n      f x \u27e6\u2295\u27e7 df x dx\n    \u2261\u27e8 extended-correctness s t {\u03c1} {consistency} \u27e9\n      (f \u27e6\u2295\u27e7 df) (x \u27e6\u2295\u27e7 dx)\n    \u2261\u27e8 stability s vars (\u2227-proj\u2081 truth) {\u03c1} {consistency} honesty \u27e8$\u27e9\n       stability t vars\n         (\u2227-proj\u2082 {stable s vars} truth)\n         {\u03c1} {consistency} honesty \u27e9\n      f x\n    \u220e where open \u2261-Reasoning\n\nstability (add s t) vars truth {\u03c1} honesty = {!!}\nstability (map s t) vars truth {\u03c1} honesty = {!!}\nstability (diff s t) vars truth {\u03c1} honesty = {!!}\nstability (union s t) vars truth {\u03c1} honesty = {!!}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\n-- If both the environment and the future arguments are honest\n-- about nil changes, then the optimized derivation delivers\n-- the same result as the original derivation.\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env \u03c1} \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy (app f s) args vars \u03c1 {consistency} honesty\n  with stable s vars | inspect (stable s) vars\n... | true  | [ truth ] = {!!}\n-- Task after stability\n-- --------------------\n-- To use close-enough on f, must have validity of optimized derivative\n-- of s. Validity should follow from constant expectation of volatility.\n--\n--  absurd! {!!} (stability s vars truth {\u03c1}) {!!}\n\n... | false | [ falsehood ] = {!!}\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) (abide args)\n  vars \u03c1 {consistency} honesty =\n  \u03bb {x} {dx} {R[x,dx]} x\u2295dx=x \u2192 honesty-is-the-best-policy\n    t args (abide vars)\n    (dx \u2022 x \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[x,dx] consistency}\n    (abide x\u2295dx=x honesty)\n\nhonesty-is-the-best-policy (abs t) (alter args)\n  vars \u03c1 {consistency} honesty =\n  \u03bb {x} {dx} {R[x,dx]} \u2192 honesty-is-the-best-policy\n    t args (alter vars)\n    (dx \u2022 x \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[x,dx] consistency}\n    (alter honesty)\n\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-- Debug tool\nabsurd! : \u2200 {A B : Set} \u2192 B \u2192 A \u2192 A \u2192 B\nabsurd! b _ _ = b\n\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\n{-\n-- Reformulating stableness as a decidable relation\n-- Not sure if it is necessary or not.\ndata StableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  abide-this : \u2200 {\u03c4 \u0393} \u2192 {vars : Vars \u0393} \u2192 StableVar this (abide {\u03c4} vars)\n  abide-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (abide vars)\n  alter-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (alter vars)\n\ndata Stable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  stable-nat : \u2200 {\u0393 n vars} \u2192 Stable {nats} {\u0393} (nat n) vars\n  stable-bag : \u2200 {\u0393 b vars} \u2192 Stable {bags} {\u0393} (bag b) vars\n  stable-var : \u2200 {\u03c4 \u0393 x vars} \u2192\n               StableVar {\u03c4} {\u0393} x vars \u2192 Stable (var x) vars\n  stable-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n               Stable t (abide vars) \u2192 Stable (abs t) vars\n  -- TODO app add map diff union\n-}\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\nvolatility\u21d2identity :\n  \u2200 {\u03c4} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg WRT (expect-volatility {\u03c4}) \u2192 df \u2261 dg\n\nvolatility\u21d2identity {nats} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {bags} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {dg} df\u2248dg =\n  extensionality (\u03bb x \u2192 extensionality (\u03bb dx \u2192\n    volatility\u21d2identity {\u03c4\u2082} (df\u2248dg {x} {dx})))\n\ndf\u2248df : \u2200 {\u03c4} {df : \u27e6 \u0394-Type \u03c4 \u27e7} {args : Args \u03c4} \u2192 df \u2248 df WRT args\ndf\u2248df {nats} = refl\ndf\u2248df {bags} = refl\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {abide args} =\n  \u03bb {x} {dx} _ \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {alter args} =\n  \u03bb {x} {dx}   \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {\u03c4}{_} {args} {df} {dg} {\u03c1} df=dg\n  rewrite df=dg = df\u2248df {\u03c4} {\u27e6 dg \u27e7 \u03c1} {args}\n\n-- A variable does not change if its value is unchanging.\nstabilityVar : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stableVar x vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 deriveVar x \u27e7 \u03c1 \u2261 \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1\n\nstabilityVar this (alter vars) () (alter honesty)\nstabilityVar this (abide vars) refl (abide proof honesty) = proof\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (abide vars) truth (abide _ honesty)\n  = stabilityVar x vars truth honesty\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (alter vars) truth (alter honesty)\n  = stabilityVar x vars truth honesty\n\n-- A term does not change if its free variables are unchanging.\nstability : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stable t vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\n\nstability (nat n) vars truth {\u03c1} _ = refl\n\nstability (bag b) vars truth {\u03c1} _ = b++\u2205=b\n\nstability (var x) vars truth {\u03c1} honesty =\n  stabilityVar x vars truth honesty\n\nstability (abs t) vars truth {\u03c1} honesty = {!!}\nstability (app t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (add t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (map t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (diff t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (union t t\u2081) vars truth {\u03c1} honesty = {!!}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\n-- If both the environment and the future arguments are honest\n-- about nil changes, then the optimized derivation delivers\n-- the same result as the original derivation.\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy (app f s) args vars \u03c1 honesty\n  with stable s vars | inspect (stable s) vars\n... | true  | [ truth ] = {!!}\n--  absurd! {!!} (stability s vars truth {\u03c1}) {!!}\n\n... | false | [ falsehood ] = {!!}\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) (abide args) vars \u03c1 honesty =\n  \u03bb {x} {dx} x\u2295dx=x \u2192 honesty-is-the-best-policy\n    t args (abide vars) (dx \u2022 x \u2022 \u03c1) (abide x\u2295dx=x honesty)\n\nhonesty-is-the-best-policy (abs t) (alter args) vars \u03c1 honesty =\n  \u03bb {x} {dx} \u2192 honesty-is-the-best-policy\n    t args (alter vars) (dx \u2022 x \u2022 \u03c1) (alter honesty)\n\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5ff2bafd8672cbc1ed595de165ca062a999676f7","subject":"Cleanup","message":"Cleanup\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\nchangeSemVar : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSemVar t \u03c1 d\u03c1 = \u27e6 deriveVar t \u27e7Var (alternate \u03c1 d\u03c1)\n\nchangeSem : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSem t \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1))\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (v : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 v \u27e7Var (changeSemVar v)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1))\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term (changeSem t)\n\nchangeSemConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\nchangeSemConst c = \u27e6 deriveConst c \u27e7Term \u2205\n\nchangeSemConst-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 changeSem (const c) \u03c1 d\u03c1 \u2261 changeSemConst c\nchangeSemConst-rewrite c \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6 deriveConst c \u27e7Term \u2205)\nvalidDeriveConst ()\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite changeSemConst-rewrite c \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 changeSem t \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (changeSem t \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 changeSem t \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295\n       \u27e6 derive t \u27e7Term (alternate \u03c11 d\u03c11)\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term (a \u2022 \u03c1) \u2295 \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite changeSemConst-rewrite c \u03c1 d\u03c1 = validDeriveConst c\n","old_contents":"module New.Correctness where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\nchangeSemVar : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSemVar t \u03c1 d\u03c1 = \u27e6 deriveVar t \u27e7Var (alternate \u03c1 d\u03c1)\n\nchangeSem : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSem t \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1))\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (v : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 v \u27e7Var (changeSemVar v)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1))\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term (changeSem t)\n\nsemConst-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 changeSem (const c) \u03c1 d\u03c1 \u2261 \u27e6 deriveConst c \u27e7Term \u2205\nsemConst-rewrite c \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 \u27e6 deriveConst c \u27e7Term \u2205\ncorrectDeriveConst ()\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6 deriveConst c \u27e7Term \u2205)\nvalidDeriveConst ()\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite semConst-rewrite c \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 changeSem t \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (changeSem t \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 changeSem t \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295\n       \u27e6 derive t \u27e7Term (alternate \u03c11 d\u03c11)\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term (a \u2022 \u03c1) \u2295 \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite semConst-rewrite c \u03c1 d\u03c1 = validDeriveConst c\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a297da86562d06d588cd10f4dcc0e310624bc4f4","subject":"agda : wouter \/ PT","message":"agda : wouter \/ PT\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenWP : \u2115 -> Set\n    isEvenWP = wp0 (_+ 2) isEven\n\n    _        : isEvenWP \u2261 isEven \u2218 (_+ 2)\n    _        = refl\n\n    _        : Set\n    _        = isEvenWP 5\n\n    _        : isEvenWP 5 \u2261 isEven 7\n    _        = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_; s\u2264s; z\u2264n)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    _ : Expr\n    _ = Val 3\n    _ : Expr\n    _ = Div (Val 3) (Val 0)\n\n    _ : Val 0 \u21d3 0\n    _ = \u21d3Base\n\n    _ : Val 3 \u21d3 3\n    _ = \u21d3Base\n\n    _ : Div (Val 3) (Val 3) \u21d3 1\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Val 3) \u21d3 3\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    _ : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    _     : evv \u2261 Pure 3\n    _     = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    _     : evd \u2261 Pure 1\n    _     = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    _     : evd0 \u2261 Step Abort (\u03bb ())\n    _     = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    _ : Expr -> Nat -> Set\n    _ = _\u21d3_\n\n    _ : Set\n    _ = Val 1 \u21d3 1\n\n    _ : Expr -> Partial Nat -> Set\n    _ = mustPT _\u21d3_\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    _ : mustPT _\u21d3_ (Val 1) (Pure 1)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- a value of this type cannot be constructed\n    _ : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    _ = {!!}\n    -}\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    _ : Expr -> Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    _ = refl\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    _ = \u21d3Base\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    _ = {!!}\n    -}\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    _ : SafeDiv (Val 3) \u2261 \u22a4\n    _ = refl\n    _ : SafeDiv (Val 0) \u2261 \u22a4\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Val 0)\n\n    _ : SafeDiv (Div (Val 3) (Val 3))\n                 \u2261 Pair ((x : Val 3 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 3))\n\n    _ : SafeDiv (Div (Val 3) (Val 0))\n                 \u2261 Pair ((x : Val 0 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    _ = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- domain is well-defined Exprs (i.e., no div-by-0)\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module DomTest where\n    _ : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    _ = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    _ : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    _ = {!!}\n    -}\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n            -> (wpPartial f \u2291 wpPartial g) <-> (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement\n  - to relate Kleisli morphisms\n  - to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre and post condition\n  - to its implementation\n\n  add top two elements; fails if stack has too few elements\n\n  show how to prove definition meets its specification\n  -}\n\n  -- define specification in terms of a pre\/post condition\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre  :      a         -> Set l -- precondition on 'a'\n      post : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                     -- and the corresponding outputs\n\n  -- [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  -- for non-dependent examples (e.g., type b does not depend on x : a)\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function (to discard unused arg of type a)\n\n  -- describes desired postcondition for addition function\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  -- spec for addition function\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n  --            pre                     post\n  {-\n  need to relate spec to an implementation\n  'wpPartial' assigns predicate transformer semantics to functions\n  'wpSpec'    assigns predicate transformer semantics to specifications\n  -}\n  -- given a spec, Spec a b\n  -- computes weakest precondition that satisfies an arbitrary postcondition P\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\a -> (pre a)        -- i.e., spec\u2019s precondition should hold and\n                                  \u2227 (post a \u2286 P a) --       spec's postcondition must imply P\n\n  -- using 'wpSpec' can now find a program 'add' that \"refines\" 'addSpec'\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop      Nil  = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs0 =\n    pop xs0 >>= \\{(x1 , xs1) ->\n    pop xs1 >>= \\{(x2 , xs2) ->\n    return ((x1 + x2) :: xs2)}}\n\n  -- verify correct\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd _        (_ :: Nil) (s\u2264s        () , _)\n  correctnessAdd P a@(x :: y :: xs ) (s\u2264s (s\u2264s z\u2264n) , post_addSpec_a_\u2286P_a)\n                                                        -- wpPartial add P (x :: y :: xs)\n                                                         = post_addSpec_a_\u2286P_a (x + y :: xs) AddStep\n  -- paper version has \"extra\" 'Nil\" case\n--correctnessAdd P              Nil  (           () , _)\n--correctnessAdd P        (_ :: Nil) (s\u2264s        () , _)\n--correctnessAdd P   (x :: y :: xs ) (            _ , H) = H                   (x + y :: xs) AddStep\n\n  {-\n  repeat: this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition\n  - to its implementation\n\n  compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n   where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\n{-\n------------------------------------------------------------------------------\n4 Mutable State\n\npredicate transformer semantics for mutable state\ngiving rise to Hoare logic\n\nfollowing assumes a fixed type s : Set (the type of the state)\n-}\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n\n  -- smart constructor\n  get : State s\n  get = Step Get return\n\n  -- smart constructor\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  -- map free monad to the state monad\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x ,       s)\n  run (Step  Get    k) s = run (k s)  s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  -- predicate transformer : for every stateful computation\n  -- maps a postcondition on b \u00d7 s\n  -- to the required precondition on s:\n  statePT : forall {l l'} -> {b : Set l}\n         -> (b \u00d7 s -> Set l')\n         -> State b\n         -> (    s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x ,   s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  -- generalise statePT\n  -- Sometimes describe postconditions as a relation between inputs and outputs.\n  -- For stateful computations, mean enabling the postcondition to also refer to the initial state:\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s          -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  -- weakest precondition semantics for Kleisli morphisms a \u2192 State b\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (    a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (    a \u00d7 s          -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  -- Given predicate P relating input, initial state, final state and result,\n  -- 'wpState' computes the weakest precondition required of input and initial state\n  -- to ensure P holds upon completing the computation.\n  -- The definition amounts to composing 'wp' and 'statePT' functions.\n\n  -- prove soundness of this semantics with respect to the run function\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n   where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\n{-\nExample showing how to reason about stateful programs using weakest precondition semantics.\n\nVerification problem proposed by Hutton and Fulger [2008]\n- input : binary tree\n- relabel tree so each leaf has a unique number associated with it\n\nTypical solution uses state monad to keep track of next unused label.\n\nHutton and Fulger challenge : reason about the program, without expanding definition of monadic operations.\n\nExample shows how properties of refinement relation can be used to reason about arbitrary effects.\n-}\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  -- Specification.\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  --               precondition true regardless of input tree and initial state\n  --               v\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n   --                     ^\n   --                     postcondition is conjunction\n   where\n    relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n    relabelPost (t , s) (t' , s')\n      = (flatten t' == (seq (s) (size t))) -- flattening result t\u2019 produces sequence of numbers from s to s + size t\n      \u2227 (s + size t == s')                 -- output state 's should be precisely size t larger than input state s\n\n  -- increment current state; return value (before incr)\n  fresh : State Nat\n  fresh = get          >>= \\n ->\n          put (Succ n) >>\n          return n\n\n  -- relabelling function\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  module RelableTest where\n    _ : Free C R (Tree Nat)\n    _ = relabel (Node (Leaf 10) (Leaf 20))\n    _ : relabel (Node (Leaf 10) (Leaf 20))\n        \u2261 Step  Get (\u03bb n1\n        \u2192 Step (Put (Succ n1)) (\u03bb _\n        \u2192 Step  Get (\u03bb n2\n        \u2192 Step (Put (Succ n2)) (\u03bb _\n        \u2192 Pure (Node (Leaf n1) (Leaf n2))))))\n    _ = refl\n\n    _ : Pair (Tree Nat) Nat\n    _ = run (relabel (Node (Leaf 10) (Leaf 20))) 1\n    _ : run (relabel (Node (Leaf 10) (Leaf 20))) 1 \u2261 (Node (Leaf 1) (Leaf 2) , 3)\n    _ = refl\n\n  -- Show the definition satisfies the specification,\n  -- via using 'wpState' to compute weakest precondition semantics of 'relabel'\n  -- and formulating desired correctness property:\n  correctnessRelabel : forall {a : Set}\n                    -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set}\n                  -> (c : State a)\n                     (f : a -> State b)\n                  -> \u2200 i P\n                  -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure      x)    f i P = refl\n  compositionality (Step Get  k)    f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  -- Proof by induction on input tree.\n  -- Proof of Node constructor case is proving\n  --      statePT (relabel l >>= (\u03bb l\u2019 \u2192 relabel r >>= (\u03bb r\u2019 \u2192 Pure (Node l\u2019 r\u2019)))) (P (Node l r , i)) i\n  -- Not obvious how apply induction hypothesis (IH states that P holds for l and r)\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    -- Simplify proofs of refining a specification, by first proving one side of the bind, then the second.\n    -- This is essentially the first law of consequence, specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b}\n                 (                           mx : State a)\n                 (                f : a -> State b) P i\n              -> statePT (wpState f \\_ -> P) mx        i\n              -> statePT                  P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b}\n                      (mx : State a)\n                      (f : a -> State b)\n                      spec\n                   -> \u2200 P i\n                   -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q  mx        i)\n                   ->         Spec.pre spec i \u00d7 (Spec.post spec i \u2286       wpState f (\\_ -> P))\n                   ->                                                     statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s}\n             -> SpecK {zero} (a \u00d7 s) (b \u00d7 s)\n             ->               a\n             -> SpecK     s          (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a  Zero    c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a)\n                                   (trans (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n                                          (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr\n                 -> Pair       (fl \u2261 seq s  sl)       (s  +  sl       \u2261 s')\n                 -> Pair       (fr \u2261 seq s'      sr)  (s' +       sr  \u2261 s'')\n                 -> Pair (fl ++ fr \u2261 seq s (sl + sr)) (s  + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s\n          -> wpSpec relabelSpec P (t , s)\n          -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf   x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set}\n      -> ((x : a) -> Free C R (b x))\n      -> ((x : a) -> b x -> Set)\n      -> (a              -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set}\n                  -> (c : Free C R a) (f : a -> Free C R b)\n                  -> \u2200 P\n                  -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure   x) f P = refl\n  compositionality (Step c x) f P = cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R}\n       -> (a -> Free C R b)\n       -> (b -> Free C R c)\n       ->  a\n       ->       Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set}\n                       -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c)\n                       -> wpCR f1 \u2291 wpCR f2\n                       -> wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n          | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenWP : \u2115 -> Set\n    isEvenWP = wp0 (_+ 2) isEven\n\n    _        : isEvenWP \u2261 isEven \u2218 (_+ 2)\n    _        = refl\n\n    _        : Set\n    _        = isEvenWP 5\n\n    _        : isEvenWP 5 \u2261 isEven 7\n    _        = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_; s\u2264s; z\u2264n)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    _ : Expr\n    _ = Val 3\n    _ : Expr\n    _ = Div (Val 3) (Val 0)\n\n    _ : Val 0 \u21d3 0\n    _ = \u21d3Base\n\n    _ : Val 3 \u21d3 3\n    _ = \u21d3Base\n\n    _ : Div (Val 3) (Val 3) \u21d3 1\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Val 3) \u21d3 3\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    _ : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    _     : evv \u2261 Pure 3\n    _     = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    _     : evd \u2261 Pure 1\n    _     = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    _     : evd0 \u2261 Step Abort (\u03bb ())\n    _     = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    _ : Expr -> Nat -> Set\n    _ = _\u21d3_\n\n    _ : Set\n    _ = Val 1 \u21d3 1\n\n    _ : Expr -> Partial Nat -> Set\n    _ = mustPT _\u21d3_\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    _ : mustPT _\u21d3_ (Val 1) (Pure 1)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- a value of this type cannot be constructed\n    _ : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    _ = {!!}\n    -}\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    _ : Expr -> Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    _ = refl\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    _ = \u21d3Base\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    _ = {!!}\n    -}\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    _ : SafeDiv (Val 3) \u2261 \u22a4\n    _ = refl\n    _ : SafeDiv (Val 0) \u2261 \u22a4\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Val 0)\n\n    _ : SafeDiv (Div (Val 3) (Val 3))\n                 \u2261 Pair ((x : Val 3 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 3))\n\n    _ : SafeDiv (Div (Val 3) (Val 0))\n                 \u2261 Pair ((x : Val 0 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    _ = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- domain is well-defined Exprs (i.e., no div-by-0)\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module DomTest where\n    _ : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    _ = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    _ : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    _ = {!!}\n    -}\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n            -> (wpPartial f \u2291 wpPartial g) <-> (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement\n  - to relate Kleisli morphisms\n  - to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre and post condition\n  - to its implementation\n\n  add top two elements; fails if stack has too few elements\n\n  show how to prove definition meets its specification\n  -}\n\n  -- define specification in terms of a pre\/post condition\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre  :      a         -> Set l -- precondition on 'a'\n      post : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                     -- and the corresponding outputs\n\n  -- [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  -- for non-dependent examples (e.g., type b does not depend on x : a)\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function (to discard unused arg of type a)\n\n  -- describes desired postcondition for addition function\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  -- spec for addition function\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n  --            pre                     post\n  {-\n  need to relate spec to an implementation\n  'wpPartial' assigns predicate transformer semantics to functions\n  'wpSpec'    assigns predicate transformer semantics to specifications\n  -}\n  -- given a spec, Spec a b\n  -- computes weakest precondition that satisfies an arbitrary postcondition P\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\a -> (pre a)        -- i.e., spec\u2019s precondition should hold and\n                                  \u2227 (post a \u2286 P a) --       spec's postcondition must imply P\n\n  -- using 'wpSpec' can now find a program 'add' that \"refines\" 'addSpec'\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop      Nil  = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs0 =\n    pop xs0 >>= \\{(x1 , xs1) ->\n    pop xs1 >>= \\{(x2 , xs2) ->\n    return ((x1 + x2) :: xs2)}}\n\n  -- verify correct\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd _        (_ :: Nil) (s\u2264s        () , _)\n  correctnessAdd P a@(x :: y :: xs ) (s\u2264s (s\u2264s z\u2264n) , post_addSpec_a_\u2286P_a)\n                                                        -- wpPartial add P (x :: y :: xs)\n                                                         = post_addSpec_a_\u2286P_a (x + y :: xs) AddStep\n  -- paper version has \"extra\" 'Nil\" case\n--correctnessAdd P              Nil  (           () , _)\n--correctnessAdd P        (_ :: Nil) (s\u2264s        () , _)\n--correctnessAdd P   (x :: y :: xs ) (            _ , H) = H                   (x + y :: xs) AddStep\n\n  {-\n  repeat: this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition\n  - to its implementation\n\n  compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n   where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s          -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (    a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (    a \u00d7 s          -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n   where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"dac9f1ba9752874315b02702e10ed439c6530929","subject":"Updated setoids note.","message":"Updated setoids note.\n\nIgnore-this: 207219e84550b3773e01821a2f860d8f\n\ndarcs-hash:20120513172543-3bd4e-799455fba458d48009025fccd8ec6c7a3139f4d2.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 13 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- 13 May 2012. It seems we cannot define the identity elimination\n  -- using the setoid equality.\n  --\n  -- subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- The identity type\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  -- 13 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2261_\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n-- Tested with the development version of Agda on 23 March 2012.\n\n{-\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor = : The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n-}\n\nmodule FOTC where\n\n------------------------------------------------------------------------------\n\nmodule PeterEquality where\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  infix 7 _\u2261_\n\n  data _\u2261_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2261 x\n    sym   : \u2200 {x y} \u2192 x \u2261 y \u2192                           y \u2261 x\n    trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192                 x \u2261 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2261 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2261 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- It seems we cannot define the identity elimination using the setoid\n  -- equality.\n  --\n  -- subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\nmodule PeterD where\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n  data D : Set where\n    zero succ true false : D\n    _\u00b7_             : D \u2192 D \u2192 D\n    loop            : D\n\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A refl Ax = Ax\n\n  data N : D \u2192 Set where\n    zN :               N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  -- 2012-03-23: Why the inductive structure makes 0 + 0 different\n  -- from 0? How to define _+_ ?\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n  --   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n  --   2003\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2261_\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} x\u2261y A Ay = x\u2261y (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z x\u2261y y\u2261z A Ax = y\u2261z A (x\u2261y A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- but it seems we cannot prove the congruency\n\n  -- cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  -- cong x\u2081\u2261x\u2082 y\u2081\u2261y\u2082 A Ax\u2081y\u2081 = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"16f8a95227efdacdb93599ca7b8ddfd847c2e47a","subject":"Document P.D.Value.","message":"Document P.D.Value.\n\nOld-commit-hash: bfaab88fbd6365ff0b9046dd999ccd1362c526d6\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Value.agda","new_file":"Parametric\/Denotation\/Value.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for standard evaluation (Def. 3.1 and 3.2, Fig. 4c and 4f).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\n\nmodule Parametric.Denotation.Value\n    (Base : Type.Structure)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\n\n-- Extension point: Values for base types.\nStructure : Set\u2081\nStructure = Base \u2192 Set\n\nmodule Structure (\u27e6_\u27e7Base : Structure) where\n  -- We provide: Values for arbitrary types.\n  \u27e6_\u27e7Type : Type -> Set\n  \u27e6 base \u03b9 \u27e7Type = \u27e6 \u03b9 \u27e7Base\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\n  -- This means: Overload \u27e6_\u27e7 to mean \u27e6_\u27e7Type.\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n  -- We also provide: Environments of such values.\n  open import Base.Denotation.Environment Type \u27e6_\u27e7Type public\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Value domains for languages described in Plotkin style\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\n\nmodule Parametric.Denotation.Value\n    (Base : Type.Structure)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\n\nStructure : Set\u2081\nStructure = Base \u2192 Set\n\nmodule Structure (\u27e6_\u27e7Base : Structure) where\n  \u27e6_\u27e7Type : Type -> Set\n  \u27e6 base \u03b9 \u27e7Type = \u27e6 \u03b9 \u27e7Base\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n  open import Base.Denotation.Environment Type \u27e6_\u27e7Type public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a2f7958b6a6e7245442d1e150092aacdc7c8707d","subject":"Add a hole in the examples that's strictly unnecessary","message":"Add a hole in the examples that's strictly unnecessary\n\n... but potentially confusing if it isn't there\n","repos":"bch29\/agda-holes","old_file":"examples\/Propositional.agda","new_file":"examples\/Propositional.agda","new_contents":"module Propositional where\n\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\n\nopen import Holes.Term using (\u231e_\u231f)\nopen import Holes.Cong.Propositional\n\nopen \u2261-Reasoning\n\n--------------------------------------------------------------------------------\n--  Some easy lemmas\n--------------------------------------------------------------------------------\n\n+-zero : \u2200 a \u2192 a + zero \u2261 a\n+-zero zero = refl\n+-zero (suc a) rewrite +-zero a = refl\n\n+-suc : \u2200 a b \u2192 a + suc b \u2261 suc (a + b)\n+-suc zero b = refl\n+-suc (suc a) b rewrite +-suc a b = refl\n\n+-assoc : \u2200 a b c \u2192 a + b + c \u2261 a + (b + c)\n+-assoc zero b c = refl\n+-assoc (suc a) b c rewrite +-assoc a b c = refl\n\n--------------------------------------------------------------------------------\n--  Proofs by equational reasoning\n--------------------------------------------------------------------------------\n\n-- commutativity of addition proved traditionally\n\n+-comm\u2081 : \u2200 a b \u2192 a + b \u2261 b + a\n+-comm\u2081 zero b = sym (+-zero b)\n+-comm\u2081 (suc a) b =\n  suc \u231e a + b \u231f   \u2261\u27e8 cong suc (+-comm\u2081 a b) \u27e9\n  suc (b + a)     \u2261\u27e8 sym (+-suc b a) \u27e9\n  b + suc a       \u220e\n\n-- commutativity of addition proved with holes\n\n+-comm : \u2200 a b \u2192 a + b \u2261 b + a\n+-comm zero b = sym (+-zero b)\n+-comm (suc a) b =\n  suc \u231e a + b \u231f   \u2261\u27e8 cong! (+-comm a b) \u27e9\n  suc (b + a)     \u2261\u27e8 cong! (+-suc b a) \u27e9\n  b + suc a       \u220e\n\n-- distributivity of multiplication over addition proved traditionally\n\n*-distrib-+\u2081 : \u2200 a b c \u2192 a * (b + c) \u2261 a * b + a * c\n*-distrib-+\u2081 zero b c = refl\n*-distrib-+\u2081 (suc a) b c =\n    b + c + a * (b + c)          \u2261\u27e8 cong (\u03bb h \u2192 b + c + h) (*-distrib-+\u2081 a b c) \u27e9\n    (b + c) + (a * b + a * c)    \u2261\u27e8 sym (+-assoc (b + c) (a * b) (a * c)) \u27e9\n    ((b + c) + a * b) + a * c    \u2261\u27e8 cong (\u03bb h \u2192 h + a * c) (+-assoc b c (a * b)) \u27e9\n    (b + (c + a * b)) + a * c    \u2261\u27e8 cong (\u03bb h \u2192 (b + h) + a * c) (+-comm c (a * b)) \u27e9\n    (b + (a * b + c)) + a * c    \u2261\u27e8 cong (\u03bb h \u2192 h + a * c) (sym (+-assoc b (a * b) c)) \u27e9\n    ((b + a * b) + c) + a * c    \u2261\u27e8 +-assoc (b + a * b) c (a * c) \u27e9\n    (b + a * b) + (c + a * c)\n  \u220e\n\n-- distributivity of multiplication over addition proved with holes\n\n*-distrib-+ : \u2200 a b c \u2192 a * (b + c) \u2261 a * b + a * c\n*-distrib-+ zero b c = refl\n*-distrib-+ (suc a) b c =\n  b + c + \u231e a * (b + c) \u231f         \u2261\u27e8 cong! (*-distrib-+ a b c) \u27e9\n  \u231e (b + c) + (a * b + a * c) \u231f   \u2261\u27e8 cong! (+-assoc (b + c) (a * b) (a * c)) \u27e9\n  \u231e (b + c) + a * b \u231f + a * c     \u2261\u27e8 cong! (+-assoc b c (a * b)) \u27e9\n  (b + \u231e c + a * b \u231f) + a * c     \u2261\u27e8 cong! (+-comm c (a * b)) \u27e9\n  \u231e b + (a * b + c) \u231f + a * c     \u2261\u27e8 cong! (+-assoc b (a * b) c) \u27e9\n  \u231e ((b + a * b) + c) + a * c \u231f   \u2261\u27e8 cong! (+-assoc (b + a * b) c (a * c)) \u27e9\n  (b + a * b) + (c + a * c)\n  \u220e\n","old_contents":"module Propositional where\n\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\n\nopen import Holes.Term using (\u231e_\u231f)\nopen import Holes.Cong.Propositional\n\nopen \u2261-Reasoning\n\n--------------------------------------------------------------------------------\n--  Some easy lemmas\n--------------------------------------------------------------------------------\n\n+-zero : \u2200 a \u2192 a + zero \u2261 a\n+-zero zero = refl\n+-zero (suc a) rewrite +-zero a = refl\n\n+-suc : \u2200 a b \u2192 a + suc b \u2261 suc (a + b)\n+-suc zero b = refl\n+-suc (suc a) b rewrite +-suc a b = refl\n\n+-assoc : \u2200 a b c \u2192 a + b + c \u2261 a + (b + c)\n+-assoc zero b c = refl\n+-assoc (suc a) b c rewrite +-assoc a b c = refl\n\n--------------------------------------------------------------------------------\n--  Proofs by equational reasoning\n--------------------------------------------------------------------------------\n\n-- commutativity of addition proved traditionally\n\n+-comm\u2081 : \u2200 a b \u2192 a + b \u2261 b + a\n+-comm\u2081 zero b = sym (+-zero b)\n+-comm\u2081 (suc a) b =\n  suc \u231e a + b \u231f   \u2261\u27e8 cong suc (+-comm\u2081 a b) \u27e9\n  suc (b + a)     \u2261\u27e8 sym (+-suc b a) \u27e9\n  b + suc a       \u220e\n\n-- commutativity of addition proved with holes\n\n+-comm : \u2200 a b \u2192 a + b \u2261 b + a\n+-comm zero b = sym (+-zero b)\n+-comm (suc a) b =\n  suc \u231e a + b \u231f   \u2261\u27e8 cong! (+-comm a b) \u27e9\n  suc (b + a)     \u2261\u27e8 cong! (+-suc b a) \u27e9\n  b + suc a       \u220e\n\n-- distributivity of multiplication over addition proved traditionally\n\n*-distrib-+\u2081 : \u2200 a b c \u2192 a * (b + c) \u2261 a * b + a * c\n*-distrib-+\u2081 zero b c = refl\n*-distrib-+\u2081 (suc a) b c =\n    b + c + a * (b + c)          \u2261\u27e8 cong (\u03bb h \u2192 b + c + h) (*-distrib-+\u2081 a b c) \u27e9\n    (b + c) + (a * b + a * c)    \u2261\u27e8 sym (+-assoc (b + c) (a * b) (a * c)) \u27e9\n    ((b + c) + a * b) + a * c    \u2261\u27e8 cong (\u03bb h \u2192 h + a * c) (+-assoc b c (a * b)) \u27e9\n    (b + (c + a * b)) + a * c    \u2261\u27e8 cong (\u03bb h \u2192 (b + h) + a * c) (+-comm c (a * b)) \u27e9\n    (b + (a * b + c)) + a * c    \u2261\u27e8 cong (\u03bb h \u2192 h + a * c) (sym (+-assoc b (a * b) c)) \u27e9\n    ((b + a * b) + c) + a * c    \u2261\u27e8 +-assoc (b + a * b) c (a * c) \u27e9\n    (b + a * b) + (c + a * c)\n  \u220e\n\n-- distributivity of multiplication over addition proved with holes\n\n*-distrib-+ : \u2200 a b c \u2192 a * (b + c) \u2261 a * b + a * c\n*-distrib-+ zero b c = refl\n*-distrib-+ (suc a) b c =\n  b + c + \u231e a * (b + c) \u231f         \u2261\u27e8 cong! (*-distrib-+ a b c) \u27e9\n  (b + c) + (a * b + a * c)       \u2261\u27e8 cong! (+-assoc (b + c) (a * b) (a * c)) \u27e9\n  \u231e (b + c) + a * b \u231f + a * c     \u2261\u27e8 cong! (+-assoc b c (a * b)) \u27e9\n  (b + \u231e c + a * b \u231f) + a * c     \u2261\u27e8 cong! (+-comm c (a * b)) \u27e9\n  \u231e b + (a * b + c) \u231f + a * c     \u2261\u27e8 cong! (+-assoc b (a * b) c) \u27e9\n  \u231e ((b + a * b) + c) + a * c \u231f   \u2261\u27e8 cong! (+-assoc (b + a * b) c (a * c)) \u27e9\n  (b + a * b) + (c + a * c)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3fc6dc8eead442499245bb1f54a5e9a8a2cd012e","subject":"Workaround broken pattern synonyms","message":"Workaround broken pattern synonyms\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0c9f9226d80652b033e57f60c46d5d84ca504cef","subject":"Added ATP pragma for conversion rules.","message":"Added ATP pragma for conversion rules.\n\nIgnore-this: 23c2214308d27af148d99c602cff2018\n\ndarcs-hash:20100528214442-3bd4e-a2214a104d369a5b78575a474973e4c3968b0683.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Minimal.agda","new_file":"LTC\/Minimal.agda","new_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\n------------------------------------------------------------------------------\n\nmodule LTC.Minimal where\n\n-- Standard library imports\n-- open import Data.Fin using ( Fin )\n-- open import Data.Nat using ( \u2115 )\n-- open import Data.String\n-- open import Relation.Binary.PropositionalEquality\n-- open \u2261-Reasoning\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain\n------------------------------------------------------------------------------\n\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language\n------------------------------------------------------------------------------\n\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n------------------------------------------------------------------------------\n-- Equality: identity type\n------------------------------------------------------------------------------\n\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n------------------------------------------------------------------------------\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Product public\nopen import MyStdLib.Data.Empty public\nopen import MyStdLib.Data.Product public\nopen import MyStdLib.Data.Sum public\nopen import MyStdLib.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n------------------------------------------------------------------------------\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom cB\u2081 #-}\n{-# ATP axiom cB\u2082 #-}\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom cZ\u2081 #-}\n{-# ATP axiom cZ\u2082 #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom cBeta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom cFix #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n------------------------------------------------------------------------------\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac ( zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\n------------------------------------------------------------------------------\n\nmodule LTC.Minimal where\n\n-- Standard library imports\n-- open import Data.Fin using ( Fin )\n-- open import Data.Nat using ( \u2115 )\n-- open import Data.String\n-- open import Relation.Binary.PropositionalEquality\n-- open \u2261-Reasoning\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain\n------------------------------------------------------------------------------\n\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language\n------------------------------------------------------------------------------\n\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  fixFO  : D\n\n------------------------------------------------------------------------------\n-- Equality: identity type\n------------------------------------------------------------------------------\n\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n------------------------------------------------------------------------------\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Product public\nopen import MyStdLib.Data.Empty public\nopen import MyStdLib.Data.Product public\nopen import MyStdLib.Data.Sum public\nopen import MyStdLib.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n------------------------------------------------------------------------------\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n------------------------------------------------------------------------------\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac ( zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f20333e283720fccdcb8f0a6ed25587480c9d577","subject":"Start defining ops on dependently-typed changes","message":"Start defining ops on dependently-typed changes\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/Changes.agda","new_file":"Thesis\/Changes.agda","new_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  valid : \u2200 (a : A) (da : ChA) \u2192 Set\n  valid a da = ch da from a to (a \u2295 da)\n  \u0394 : (a : A) \u2192 Set\n  \u0394 a = \u03a3[ da \u2208 ChA ] valid a da\n  \u0394\u2082 : (a1 : A) (a2 : A) \u2192 Set\n  \u0394\u2082 a1 a2 = \u03a3[ da \u2208 ChA ] ch da from a1 to a2\n\n  _\u2295'_ : (a1 : A) -> {a2 : A} -> (da : \u0394\u2082 a1 a2) -> A\n  a1 \u2295' (da , daa) = a1 \u2295 da\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\ninstance\n  unitCS : ChangeStructure \u22a4\n\n  unitCS = record\n    { Ch = \u22a4\n    ; ch_from_to_ = \u03bb dv v1 v2 \u2192 \u22a4\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = \u03bb _ _ \u2192 tt\n      ; fromto\u2192\u2295 = \u03bb { _ _ tt _ \u2192 refl }\n      ; _\u229d_ = \u03bb _ _ \u2192 tt\n      ; \u229d-fromto = \u03bb a b \u2192 tt\n      })\n    }\n","old_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\ninstance\n  unitCS : ChangeStructure \u22a4\n\n  unitCS = record\n    { Ch = \u22a4\n    ; ch_from_to_ = \u03bb dv v1 v2 \u2192 \u22a4\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = \u03bb _ _ \u2192 tt\n      ; fromto\u2192\u2295 = \u03bb { _ _ tt _ \u2192 refl }\n      ; _\u229d_ = \u03bb _ _ \u2192 tt\n      ; \u229d-fromto = \u03bb a b \u2192 tt\n      })\n    }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"48d1be2bb4bb61634b15fbd0cff064c02be1f4f9","subject":"Comment on handling of contexts","message":"Comment on handling of contexts\n\nOld-commit-hash: 0bfc0dfdc024921016f8df0d8252aa20f5f0f8d6\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Syntax\/Context.agda","new_file":"Base\/Syntax\/Context.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n--\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\nimport Data.List as List\nopen List public\n  using ()\n  renaming\n    ( [] to \u2205 ; _\u2237_ to _\u2022_\n    ; map to mapContext\n    )\n\nContext : Set\nContext = List.List Type\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\n-- This handling of contexts is recommended by [this\n-- email](https:\/\/lists.chalmers.se\/pipermail\/agda\/2011\/003423.html) and\n-- attributed to Conor McBride.\n--\n-- The associated thread discusses a few alternatives solutions, including one\n-- where beta-reduction can handle associativity of ++.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n--\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\nimport Data.List as List\nopen List public\n  using ()\n  renaming\n    ( [] to \u2205 ; _\u2237_ to _\u2022_\n    ; map to mapContext\n    )\n\nContext : Set\nContext = List.List Type\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"675a834b612e0975d9d476442347b95f20a88acc","subject":"Added missing files.","message":"Added missing files.\n\nIgnore-this: c9a956d8948883f8e4a748a6f4128d94\n\ndarcs-hash:20110211225550-3bd4e-789909a82466fe6d270219ddba2fa3b7f95424d0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download agda2atp tool (described in above paper) using darcs\n-- with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains interactive proofs\n-- that are used by the combined proofs.\n\n------------------------------------------------------------------------------\n-- Distributive laws on a binary operation\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- Group theory\n\n-- We formalize the theory of groups using Agda postulates for\n-- the group axioms.\n\n-- Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Logic\n\n-- Propositional logic\nopen import Logic.Propositional.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n-- Predicate logic\nopen import Logic.Predicate.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n------------------------------------------------------------------------------\n-- LTC\n\n-- Formalization of (a version of) Azcel's Logical Theory of constructions.\n\n-- LTC base\nopen import LTC.Base.Properties\nopen import LTC.Base.PropertiesATP\nopen import LTC.Base.PropertiesI\n\n-- Booleans\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Bool.PropertiesI\n\n-- Lists\nopen import LTC.Data.List.PropertiesATP\nopen import LTC.Data.List.PropertiesI\n\n-- Naturals numbers: Common properties\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.PropertiesI\n\nopen import LTC.Data.Nat.PropertiesByInductionATP\nopen import LTC.Data.Nat.PropertiesByInductionI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC.Data.Nat.Divisibility.PropertiesATP\nopen import LTC.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC.Data.Nat.Induction.LexicographicATP\nopen import LTC.Data.Nat.Induction.LexicographicI\nopen import LTC.Data.Nat.Induction.WellFoundedATP\nopen import LTC.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: List\nopen import LTC.Data.Nat.List.PropertiesATP\nopen import LTC.Data.Nat.List.PropertiesI\n\n-- Naturals numbers: Unary numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n-- The GCD algorithm\nopen import LTC.Program.GCD.ProofSpecificationATP\nopen import LTC.Program.GCD.ProofSpecificationI\n\n-- Burstall's sort list algorithm\nopen import LTC.Program.SortList.ProofSpecificationATP\nopen import LTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- LTC-PCF\n\n-- Formalization of a version of Azcel's Logical Theory of constructions.\n\n-- Naturals numbers: Common properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC-PCF.Data.Nat.Induction.LexicographicATP\nopen import LTC-PCF.Data.Nat.Induction.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedATP\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- The division algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- The GCD algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- Axiomatic PA\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- Inductive PA\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, so see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download agda2atp tool (described in above paper) using darcs\n-- with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains interactive proofs\n-- that are used by the combined proofs.\n\n------------------------------------------------------------------------------\n-- Distributive laws on a binary operation\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- Group theory\n\n-- We formalize the theory of groups using Agda postulates for\n-- the group axioms.\n\n-- Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Logic\n\n-- Propositional logic\nopen import Logic.Propositional.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n-- Predicate logic\nopen import Logic.Predicate.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n------------------------------------------------------------------------------\n-- LTC\n\n-- Formalization of (a version of) Azcel's Logical Theory of constructions.\n\n-- LTC base\nopen import LTC.Base.Properties\nopen import LTC.Base.PropertiesATP\nopen import LTC.Base.PropertiesI\n\n-- Booleans\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Bool.PropertiesI\n\n-- Lists\nopen import LTC.Data.List.PropertiesATP\nopen import LTC.Data.List.PropertiesI\n\n-- Naturals numbers: Common properties\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.PropertiesI\n\nopen import LTC.Data.Nat.PropertiesByInductionATP\nopen import LTC.Data.Nat.PropertiesByInductionI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC.Data.Nat.Divisibility.PropertiesATP\nopen import LTC.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC.Data.Nat.Induction.LexicographicATP\nopen import LTC.Data.Nat.Induction.LexicographicI\nopen import LTC.Data.Nat.Induction.WellFoundedATP\nopen import LTC.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: List\nopen import LTC.Data.Nat.List.PropertiesATP\nopen import LTC.Data.Nat.List.PropertiesI\n\n-- The GCD algorithm\nopen import LTC.Program.GCD.ProofSpecificationATP\nopen import LTC.Program.GCD.ProofSpecificationI\n\n-- Burstall's sort list algorithm\nopen import LTC.Program.SortList.ProofSpecificationATP\nopen import LTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- LTC-PCF\n\n-- Formalization of a version of Azcel's Logical Theory of constructions.\n\n-- Naturals numbers: Common properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC-PCF.Data.Nat.Induction.LexicographicATP\nopen import LTC-PCF.Data.Nat.Induction.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedATP\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- The division algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- The GCD algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- Axiomatic PA\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- Inductive PA\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, so see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5e711a8f559b2a518b2b9fa467bd3c15c34b1dc3","subject":"Add \u22a2\u02e2 the server, its \u0394 is \u2297-like","message":"Add \u22a2\u02e2 the server, its \u0394 is \u2297-like\n","repos":"crypto-agda\/protocols","old_file":"js-experiment\/Terms.agda","new_file":"js-experiment\/Terms.agda","new_contents":"open import proto\nopen import Types\nopen import prelude\nopen import uri\n\nmodule Terms where\n\ninfix 2 \u22a2\u02e2_ \u22a2_ \u22a2\u1d9c\u1da0_\n\ndata \u22a2_ : (\u0394 : Env) \u2192 Set\u2081 where\n  end : \u2200{\u0394}{e : EndedEnv \u0394}\n       ------------------\n      \u2192 \u22a2 \u0394\n\n  output : \u2200 {\u0394 d M P}{{_ : SER M}}\n             (l : d \u21a6 send P \u2208 \u0394)(m : M)\n             (p : \u22a2 \u0394 [ l \u2254 m ])\n             -------------------\n               \u2192 \u22a2 \u0394\n\n  input : \u2200 {\u0394 d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2 \u0394 [ l \u2254 m ])\n                 ----------------------\n                     \u2192 \u22a2 \u0394\n\n  mix : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n          (p : \u22a2 \u0394\u2080) (q : \u22a2 \u0394\u2081)\n          --------------------\n        \u2192 \u22a2 \u0394\n\n  cut : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) {P d}\n          (p : \u22a2 (\u0394\u2080 , d \u21a6 dual P))\n          (q : \u22a2 (\u0394\u2081 , d \u21a6 P))\n          ---------------------\n        \u2192 \u22a2 \u0394\n\n  fwd : \u2200 c d {P} \u2192 \u22a2 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\n\n  exch-last :\n      \u2200 {\u0394 c d P Q} \u2192\n      (p : \u22a2 \u0394 , c \u21a6 P , d \u21a6 Q)\n      \u2192 \u22a2 \u0394 , d \u21a6 Q , c \u21a6 P\n\n  wk-last : \u2200 {\u0394 d}\n              (p : \u22a2 \u0394)\n              -----------------------\n                \u2192 \u22a2 (\u0394 , d \u21a6 end)\n\n  end-last : \u2200 {\u0394 d}\n               (p : \u22a2 (\u0394 , d \u21a6 end))\n               ----------------------\n                 \u2192 \u22a2 \u0394\n\ndata \u22a2\u1d9c\u1da0_ (\u0394 : Env) : Set\u2081 where\n  end : {e : EndedEnv \u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  output : \u2200 {d M P}{{_ : SER M}}\n            (l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M)\n            (p : \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            --------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  input : \u2200 {d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            ----------------------------\n                     \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n-- The \u0394 for the server contains the view point of the clients\n-- The point is that the meaning of _,_ in \u0394 is \u2297 while it\n-- is \u214b in \u22a2\u1d9c\u1da0\nrecord \u22a2\u02e2_ (\u0394 : Env) : Set\u2081 where\n  coinductive\n  field\n    server-output :\n      \u2200 {d M}{{_ : SER M}}{P : M \u2192 Proto}\n        (l : d \u21a6 recv P \u2208 \u0394) \u2192\n        \u03a3 M \u03bb m \u2192 \u22a2\u02e2 \u0394 [ l \u2254 m ]\n    server-input :\n      \u2200 {d M}{{_ : SER M}}{P : M \u2192 Proto}\n        (l : d \u21a6 send P \u2208 \u0394)\n        (m : M) \u2192 \u22a2\u02e2 \u0394 [ l \u2254 m ]\nopen \u22a2\u02e2_ public\n\n-- This is just to confirm that we have enough cases\ntelecom' : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u02e2 \u0394 \u2192 \ud835\udfd9\ntelecom' end q = _\ntelecom' (output l m p) q\n  = telecom' p (server-input q l m)\ntelecom' (input l p) q\n  = case server-output q l of \u03bb { (m , s) \u2192\n      telecom' (p m) s }\n\nembed\u1d9c\u1da0 : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2 \u0394\nembed\u1d9c\u1da0 (end {e = e}) = end {e = e}\nembed\u1d9c\u1da0 (output l m p) = output l m (embed\u1d9c\u1da0 p)\nembed\u1d9c\u1da0 (input l p) = input l \u03bb m \u2192 embed\u1d9c\u1da0 (p m)\n\nmix\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n         (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n         (q : \u22a2\u1d9c\u1da0 \u0394\u2081)\n         -------------\n           \u2192 \u22a2\u1d9c\u1da0 \u0394\nmix\u1d9c\u1da0 \u0394\u209b end q = tr \u22a2\u1d9c\u1da0_ (Zip-identity  \u0394\u209b) q\nmix\u1d9c\u1da0 \u0394\u209b (output l m p) q\n  = output (Zip-com\u2208\u2080 \u0394\u209b l) m (mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) p q)\nmix\u1d9c\u1da0 \u0394\u209b (input l p) q\n  = input (Zip-com\u2208\u2080 \u0394\u209b l) \u03bb m \u2192 mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) (p m) q\n\ncut\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081}\n         (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) d P\n         (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n         (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n         ---------------------------\n           \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\ncut\u1d9c\u1da0 \u0394\u209b d end p q = mix\u1d9c\u1da0 (\u0394\u209b , d \u21a6\u2080 end) p q\n\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output here m p) (input here q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input here p) (output here m q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) (p m) q\n\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\n\ncut\u1d9c\u1da0 _ _ (com _ _) (end {e = _ , ()}) _\ncut\u1d9c\u1da0 _ _ (com _ _) _ (end {e = _ , ()})\n\nend-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n              (p : \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end))\n              -----------------------\n                \u2192 \u22a2\u1d9c\u1da0 \u0394\nend-last\u1d9c\u1da0 (end {e = e , _}) = end {e = e}\nend-last\u1d9c\u1da0 (output (there l) m p) = output l m (end-last\u1d9c\u1da0 p)\nend-last\u1d9c\u1da0 (input (there l) p) = input l \u03bb m \u2192 end-last\u1d9c\u1da0 (p m)\n\nwk-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n             (p : \u22a2\u1d9c\u1da0 \u0394)\n             -----------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\nwk-last\u1d9c\u1da0 end = end {e = \u2026 , _}\nwk-last\u1d9c\u1da0 (output l m p) = output (there l) m (wk-last\u1d9c\u1da0 p)\nwk-last\u1d9c\u1da0 (input l p) = input (there l) \u03bb m \u2192 wk-last\u1d9c\u1da0 (p m)\n\nwk-,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 \u2192 EndedEnv \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u03b5} p E = p\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u0394\u2081 , d \u21a6 P} p (E , e) rewrite Ended-\u2261end e\n  = wk-last\u1d9c\u1da0 (wk-,,\u1d9c\u1da0 p E)\n\nmodule _ {d P \u0394\u2080} where\n  pre-wk-\u2208 : \u2200 {\u0394\u2081} \u2192 d \u21a6 P \u2208 \u0394\u2081 \u2192 d \u21a6 P \u2208 (\u0394\u2080 ,, \u0394\u2081)\n  pre-wk-\u2208 here = here\n  pre-wk-\u2208 (there l) = there (pre-wk-\u2208 l)\n\n{-\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 E\u0394\u2080 end = end {e = {!!}}\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E\u0394\u2080 (output l m p) =\n  output (pre-wk-\u2208 l) m (pre-wk\u1d9c\u1da0 {\u0394\u2080} {{!\u0394\u2081!}} E\u0394\u2080 {!!})\npre-wk\u1d9c\u1da0 E\u0394\u2080 (input l p) = {!!}\n-}\n\nfwd-mix\u1d9c\u1da0 : \u2200 {\u0394 c d} P \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , d \u21a6 dual P)\nfwd-mix\u1d9c\u1da0 end p = wk-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 p)\nfwd-mix\u1d9c\u1da0 (recv P) p = input (there here) \u03bb m \u2192 output here m (fwd-mix\u1d9c\u1da0 (P m) p)\nfwd-mix\u1d9c\u1da0 (send P) p = input here \u03bb m \u2192 output (there here) m (fwd-mix\u1d9c\u1da0 (P m) p)\n\nfwd\u1d9c\u1da0 : \u2200 c d {P} \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\nfwd\u1d9c\u1da0 _ _ {P} = fwd-mix\u1d9c\u1da0 {\u03b5} P end\n\n\u03b5,, : \u2200 \u0394 \u2192 \u03b5 ,, \u0394 \u2261 \u0394\n\u03b5,, \u03b5 = refl\n\u03b5,, (\u0394 , d \u21a6 P) rewrite \u03b5,, \u0394 = refl\n\npostulate\n exch\u1d9c\u1da0 :\n  \u2200 \u0394\u2080 \u0394\u2081 \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2081 ,, \u0394\u2080)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n  {-\nexch\u1d9c\u1da0 \u03b5 \u0394\u2081 p rewrite \u03b5,, \u0394\u2081 = p\nexch\u1d9c\u1da0 \u0394\u2080 \u03b5 p rewrite \u03b5,, \u0394\u2080 = p\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) end = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 ._) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output here m p) = {!exch\u1d9c\u1da0!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (input l p) = {!!}\n-}\n\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E p = exch\u1d9c\u1da0 \u0394\u2080 \u0394\u2081 (wk-,,\u1d9c\u1da0 p E)\n\nend-of : Env \u2192 Env\nend-of \u03b5 = \u03b5\nend-of (\u0394 , d \u21a6 P) = end-of \u0394 , d \u21a6 end\n\nend-of-Ended : \u2200 \u0394 \u2192 EndedEnv (end-of \u0394)\nend-of-Ended \u03b5 = _\nend-of-Ended (\u0394 , d \u21a6 P) = end-of-Ended \u0394 , _\n\nend-of-\u22ce : \u2200 \u0394 \u2192 [ \u0394 is \u0394 \u22ce end-of \u0394 ]\nend-of-\u22ce \u03b5 = \u03b5\nend-of-\u22ce (\u0394 , d \u21a6 P) = end-of-\u22ce \u0394 , d \u21a6\u2080 P\n\nend-of-,,-\u22ce : \u2200 \u0394\u2080 \u0394\u2081 \u2192 [ \u0394\u2080 ,, \u0394\u2081 is \u0394\u2080 ,, end-of \u0394\u2081 \u22ce end-of \u0394\u2080 ,, \u0394\u2081 ]\nend-of-,,-\u22ce \u0394\u2080 \u03b5 = end-of-\u22ce \u0394\u2080\nend-of-,,-\u22ce \u0394\u2080 (\u0394\u2081 , d \u21a6 P) = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081 , d \u21a6\u2081 P\n\n,,-assoc : \u2200 {\u0394\u2080 \u0394\u2081 \u0394\u2082} \u2192 \u0394\u2080 ,, (\u0394\u2081 ,, \u0394\u2082) \u2261 (\u0394\u2080 ,, \u0394\u2081) ,, \u0394\u2082\n,,-assoc {\u0394\u2082 = \u03b5} = refl\n,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082 , d \u21a6 P} rewrite ,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082} = refl\n\ncut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\ncut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q =\n  end-last\u1d9c\u1da0\n    (cut\u1d9c\u1da0 \u0394\u209b d P\n       (exch\u1d9c\u1da0 (\u0394\u2080 ,, end-of \u0394\u2081) (\u03b5 , d \u21a6 dual P)\n              (tr \u22a2\u1d9c\u1da0_ (! (,,-assoc {\u03b5 , d \u21a6 dual P} {\u0394\u2080} {end-of \u0394\u2081}))\n                 (wk-,,\u1d9c\u1da0\n                   (exch\u1d9c\u1da0 (\u03b5 , d \u21a6 dual P) _ p) (end-of-Ended _))))\n       (pre-wk\u1d9c\u1da0 (end-of-Ended _) q))\n  where \u0394\u209b = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081\n\npostulate\n  !cut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 dual P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n-- !cut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q = {!!}\n\n-- only the last two are exchanged, some more has to be done\nexch-last\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q , c \u21a6 P\nexch-last\u1d9c\u1da0 (end {e = (a , b) , c}) = end {e = (a , c) , b}\nexch-last\u1d9c\u1da0 (output here m p) = output (there here) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there here) m p) = output here m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there (there l)) m p) = output (there (there l)) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (input here p) = input (there here) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there here) p) = input here \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there (there l)) p) = input (there (there l)) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p  m)\n\n{-\ndata Relabel : Env \u2192 Env \u2192 Set where\n  \u03b5 : Relabel \u03b5 \u03b5\n  _,_\u21a6_ : \u2200 {\u0394\u2080 \u0394\u2081 c d P} \u2192 Relabel \u0394\u2080 \u0394\u2081 \u2192 Relabel (\u0394\u2080 , c \u21a6 P) (\u0394\u2081 , d \u21a6 P)\n\nmodule _ where\n    rebalel-\u2208 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n                  (l : d \u21a6 P \u2208 \u0394\u2080)\n                    \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n            (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel end = {!end!}\n    relabel (output l m p) = output {!l!} {!!} {!!}\n    relabel (input l p) = {!!}\n\npar\u1d9c\u1da0 : \u2200 {\u0394 c} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , c \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 (P \u214b' Q))\n-- TODO only one channel name!!!\n-- TODO empty context\n-- TODO try to match on 'p' first\nbroken-par\u1d9c\u1da0 : \u2200 {c d e} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , e \u21a6 (P \u214b' Q))\nbroken-par\u1d9c\u1da0 end Q p = {!end-last\u1d9c\u1da0 (exch-last\u1d9c\u1da0 p)!}\nbroken-par\u1d9c\u1da0 (com x P) end p = end-last\u1d9c\u1da0 {!p!}\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (end {e = _ , ()})\n\nbroken-par\u1d9c\u1da0 (com x P) (com .OUT Q) (output here m p)\n  = output here R (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com x P) (com .IN Q) (input here p)\n  = output here R (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com .OUT P) (com y Q) (output (there here) m p)\n  = output here L (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com .IN P) (com y Q) (input (there here) p)\n  = output here L (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (output (there (there ())) m p)\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (input (there (there ())) p)\n-}\n\nmodule _ {c d cd} where\n    bi-fwd : \u2200 P Q \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , cd \u21a6 P \u2297 Q , c \u21a6 dual P , d \u21a6 dual Q)\n\n    private\n      module _ {M} {{_ : SER M}} {P : M \u2192 Proto} {Q} where\n        goL : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 P m \u2297 Q)\n                          , c  \u21a6 dual (com x P)\n                          , d  \u21a6 dual Q\n\n        goL IN  = input (there (there here)) \u03bb m \u2192 output (there here) m (bi-fwd _ _)\n        goL OUT = input (there here) \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n        goR : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 Q \u2297 P m)\n                          , c  \u21a6 dual Q\n                          , d  \u21a6 dual (com x P)\n        goR IN  = input (there (there here)) \u03bb m \u2192 output here m (bi-fwd _ _)\n        goR OUT = input here \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n    bi-fwd end Q = exch-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _))\n    bi-fwd (com x P) end = wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _)\n    bi-fwd (com x P) (com y Q) = input (there (there here)) [L: goL x ,R: goR y ]\n\n    module _ {\u0394\u2080 \u0394\u2081 P Q} where\n        \u2297\u1d9c\u1da0 : (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , c \u21a6 P))\n             (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 Q))\n             ----------------------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394\u2080 ,, \u0394\u2081 , cd \u21a6 (P \u2297 Q))\n        \u2297\u1d9c\u1da0 p q = !cut,,\u1d9c\u1da0 _ _ p (!cut,,\u1d9c\u1da0 _ _ q (bi-fwd P Q))\n\n  {-\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (l : c \u21a6 P \u2208 \u0394)\n  (p : \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 d \u21a6 Q ] , c \u21a6 P\nexch\u1d9c\u1da0 here p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (there l) p = {!!}\n-}\n\n{-\nrot\u1d9c\u1da0 : \u2200 \u0394 {c P} \u2192\n         (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P)\n      \u2192 \u22a2\u1d9c\u1da0 \u03b5 , c \u21a6 P ,, \u0394\nrot\u1d9c\u1da0 \u03b5 p = p\nrot\u1d9c\u1da0 (\u0394 , d \u21a6 P) p = {!rot\u1d9c\u1da0 \u0394 p!}\n\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394\u2080} \u0394\u2081 {c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2080 , c \u21a6 P , d \u21a6 Q ,, \u0394\u2081)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 , d \u21a6 Q , c \u21a6 P ,, \u0394\u2081\nexch\u1d9c\u1da0 \u03b5 p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (end e) = end {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d\u2081 \u21a6 ._) (output here m p) = output here m ({!exch\u1d9c\u1da0 \u0394\u2081 p!})\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (input l p) = {!!}\n-}\n\n_\u2286_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n_\u2287_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n\n\u0394\u2080 \u2286 \u0394\u2081 = \u2200 {d P} \u2192 d \u21a6 P \u2208 \u0394\u2080 \u2192 d \u21a6 P \u2208 \u0394\u2081\n\u0394\u2080 \u2287 \u0394\u2081 = \u0394\u2081 \u2286 \u0394\u2080\n\nget-put : \u2200 {d P \u0394 c Q} \u2192\n        (l : d \u21a6 P \u2208 \u0394) \u2192 c \u21a6 Q \u2208 (\u0394 [ l \u2254 c \u21a6 Q ])\nget-put here = here\nget-put (there l) = there (get-put l)\n\n{-\n\u2286_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (f : \u0394\u2080 \u2286 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2286 (\u0394\u2081 [ f l \u2254 c \u21a6 Q ])\n\u2286 f [ l \u2254 c \u21a6 Q ] {d'} {P'} l' = {!!}\n\n(l : d \u21a6 P \u2208 \u0394)\n\u2192 \u0394 [ l \u2254 ]\n\nrecord _\u2248_ (\u0394\u2080 \u0394\u2081 : Env) : Set\u2081 where\n  constructor _,_\n  field\n    \u2248\u2286 : \u0394\u2080 \u2286 \u0394\u2081\n    \u2248\u2287 : \u0394\u2080 \u2287 \u0394\u2081\nopen _\u2248_ public\n\n\u2248_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2248 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ])\n\u2248 \u0394\u2091 [ here \u2254 m ] = {!!}\n\u2248 \u0394\u2091 [ there l \u2254 m ] = {!!}\n\n{-(\u03bb l' \u2192 {!\u2248\u2286 \u0394\u2091!}) , from\n  where\n    from : \u2200 {\u0394\u2080 \u0394\u2081 d io M} {P : M \u2192 Proto} {ser : SER M}\n             {\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081} {l : d \u21a6 com io P \u2208 \u0394\u2080} {m : M} {d\u2081} {P\u2081} \u2192\n           d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ]) \u2192 d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2080 [ l \u2254 m ])\n    from = {!!}\n\n\u2248\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081}\n       (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n       (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n       -----------\n         \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n\u2248\u1d9c\u1da0 \u0394\u2091 (end {e = e}) = end {e = {!!}}\n\u2248\u1d9c\u1da0 \u0394\u2091 (output l m p) = output (\u2248\u2286 \u0394\u2091 l) m (\u2248\u1d9c\u1da0 (\u2248 \u0394\u2091 [ l \u2254 m ]) p)\n\u2248\u1d9c\u1da0 \u0394\u2091 (input l p) = {!!}\n-}\n-}\n\ncut-elim : \u2200 {\u0394} (p : \u22a2 \u0394)\n                 ------------\n                   \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut-elim (end {e = e}) = end {e = e}\ncut-elim (output l m p) = output l m (cut-elim p)\ncut-elim (input l p) = input l (\u03bb m \u2192 cut-elim (p m))\ncut-elim (mix \u0394\u209b p q) = mix\u1d9c\u1da0 \u0394\u209b (cut-elim p) (cut-elim q)\ncut-elim (cut \u0394\u209b {P} {d} p q) = end-last\u1d9c\u1da0 (cut\u1d9c\u1da0 \u0394\u209b d P (cut-elim p) (cut-elim q))\ncut-elim (end-last p) = end-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (wk-last p) = wk-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (fwd c d) = fwd\u1d9c\u1da0 c d\ncut-elim (exch-last p) = exch-last\u1d9c\u1da0 (cut-elim p)\n\n{-\n\nstart : \u2200 {\u0394} P\n       \u2192 \u22a2 [ clientURI \u21a6 dual P ]\n       \u2192 (\u2200 d \u2192 \u22a2 (\u0394 , d \u21a6 P))\n       \u2192 \u22a2 \u0394\nstart P p q = cut {!!} (... p) (q {!!})\n-}\n\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 : \u2200 {P d} \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ] \u2192 \u27e6 P \u27e7\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {end} end = _\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {com x P} (end {e = _ , ()})\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output here m der) = m , \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 der\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output (there ()) m der)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input here x\u2081) m = \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (x\u2081 m)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input (there ()) x\u2081)\n\nSatisfy : \u2200 {p d} P \u2192 (R : \u27e6 log P \u27e7 \u2192 Set p) \u2192 \u22a2 [ d \u21a6 P ] \u2192 Set p\nSatisfy P Rel d = (d' : \u27e6 dual P \u27e7) \u2192 Rel (telecom P (\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (cut-elim d)) d')\n\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {end} p = end\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 (p m))\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 p)\n\n\u27e6\u27e7\u2192\u22a2 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2 {end} p = end\n\u27e6\u27e7\u2192\u22a2 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2 (p m))\n\u27e6\u27e7\u2192\u22a2 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2 p)\n\n{-\n\u22a2toProc : \u2200 {\u0394} \u2192 \u22a2 \u0394 \u2192 JSProc\n\u22a2toProc end = end\n\u22a2toProc (output {d = d} l m prg) = output d (serialize m) (\u22a2toProc prg)\n\u22a2toProc (input {d = d} l prg) = input d ([succeed: (\u03bb m \u2192 \u22a2toProc (prg m)) ,fail: error ] \u2218 parse)\n\u22a2toProc (start P prg x) = start (\u22a2toProc prg) (\u03bb d \u2192 \u22a2toProc (x d))\n\n\n\u22a2toProc-WT : \u2200 {\u0394} (der : \u22a2 \u0394) \u2192 \u0394 \u22a2 \u22a2toProc der\n\u22a2toProc-WT (end {x}) = end {_} {x}\n\u22a2toProc-WT (output {{x}} l m der) = output l (sym (rinv m)) (\u22a2toProc-WT der)\n\u22a2toProc-WT (input {{x}} l x\u2081) = input l \u03bb s m x \u2192\n  subst (\u03bb X \u2192 _ [ l \u2254 m ] \u22a2 [succeed: (\u22a2toProc \u2218 x\u2081) ,fail: error ] X) x (\u22a2toProc-WT (x\u2081 m))\n\u22a2toProc-WT (start P der x) = start P (\u22a2toProc-WT der) (\u03bb d \u2192 \u22a2toProc-WT (x d))\n-}\n\n\u27e6_\u27e7E : Env \u2192 Set\n\u27e6 \u03b5 \u27e7E = \ud835\udfd9\n\u27e6 \u0394 , d \u21a6 P \u27e7E = \u27e6 \u0394 \u27e7E \u00d7 \u27e6 P \u27e7\n\nmapEnv : (Proto \u2192 Proto) \u2192 Env \u2192 Env\nmapEnv f \u03b5 = \u03b5\nmapEnv f (\u0394 , d \u21a6 P) = mapEnv f \u0394 , d \u21a6 f P\n\nmapEnv-all : \u2200 {P Q : URI \u2192 Proto \u2192 Set}{\u0394}{f : Proto \u2192 Proto}\n  \u2192 (\u2200 d x \u2192 P d x \u2192 Q d (f x))\n  \u2192 AllEnv P \u0394 \u2192 AllEnv Q (mapEnv f \u0394)\nmapEnv-all {\u0394 = \u03b5} f\u2081 \u2200\u0394 = \u2200\u0394\nmapEnv-all {\u0394 = \u0394 , d \u21a6 P\u2081} f\u2081 (\u2200\u0394 , P) = (mapEnv-all f\u2081 \u2200\u0394) , f\u2081 d P\u2081 P\n\nmapEnv-Ended : \u2200 {f : Proto \u2192 Proto}{\u0394} \u2192 Ended (f end)\n  \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 AllEnv (\u03bb _ \u2192 Ended) (mapEnv f \u0394)\nmapEnv-Ended eq = mapEnv-all (\u03bb { d end _ \u2192 eq ; d (send P) () ; d (recv P) () })\n\nmapEnv-\u2208 : \u2200 {d P f \u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 d \u21a6 f P \u2208 mapEnv f \u0394\nmapEnv-\u2208 here = here\nmapEnv-\u2208 (there der) = there (mapEnv-\u2208 der)\n\nmodule _ {d c M cf}{m : M}{{_ : M \u2243? SERIAL}}{p} where\n  subst-lemma-one-point-four : \u2200 {\u0394}( l : d \u21a6 com c p \u2208 \u0394 ) \u2192\n    let f = mapProto cf\n    in (mapEnv f (\u0394 [ l \u2254 m ])) \u2261 (_[_\u2254_]{c = cf c} (mapEnv f \u0394) (mapEnv-\u2208 l) m)\n  subst-lemma-one-point-four here = refl\n  subst-lemma-one-point-four (there {d' = d'}{P'} l) = ap (\u03bb X \u2192 X , d' \u21a6 mapProto cf P') (subst-lemma-one-point-four l)\n\nmodule _ {d P} where\n  project : \u2200 {\u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 \u27e6 \u0394 \u27e7E \u2192 \u27e6 P \u27e7\n  project here env = snd env\n  project (there mem) env = project mem (fst env)\n\nempty : \u2200 {\u0394} \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 \u27e6 \u0394 \u27e7E\nempty {\u03b5} <> = _\nempty {\u0394 , d \u21a6 end} (fst , <>) = empty fst , _\nempty {\u0394 , d \u21a6 com x P} (fst , ())\n\nnoRecvInLog : \u2200 {d M}{{_ : M \u2243? SERIAL}}{P : M \u2192 _}{\u0394} \u2192 d \u21a6 recv P \u2208 mapEnv log \u0394 \u2192 \ud835\udfd8\nnoRecvInLog {\u0394 = \u03b5} ()\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 end} (there l) = noRecvInLog l\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 com x\u2081 P\u2081} (there l) = noRecvInLog l\n\nmodule _ {d M P}{{_ : M \u2243? SERIAL}} where\n  lookup : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 \u03a3 M \u03bb m \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  lookup here (env , (m , p)) = m , (env , p)\n  lookup (there l) (env , P') = let (m , env') = lookup l env in m , (env' , P')\n\n  set : \u2200 {\u0394}(l : d \u21a6 recv P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  set here (env , f) m = env , f m\n  set (there l) (env , P') m = set l env m , P'\n\n  set\u03a3 : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E \u2192 \u27e6 \u0394 \u27e7E\n  set\u03a3 here m env = fst env , (m , snd env)\n  set\u03a3 (there l) m env = set\u03a3 l m (fst env) , snd env\n\n  {-\nforgetConc : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 mapEnv log \u0394 \u2192 \u27e6 mapEnv log \u0394 \u27e7E\nforgetConc (end e) = empty \u2026\nforgetConc {\u0394} (output l m der) = set\u03a3 l m (forgetConc {{!set\u03a3 l m!}} der) -- (forgetConc der)\nforgetConc (input l x\u2081) with noRecvInLog l\n... | ()\n-}\n\n\u22a2telecom : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u27e6 mapEnv dual \u0394 \u27e7E \u2192 \u22a2 mapEnv log \u0394\n\u22a2telecom end env = end {e = mapEnv-Ended _ \u2026}\n\u22a2telecom (output l m der) env = output (mapEnv-\u2208 l) m (subst (\u22a2_)\n  (subst-lemma-one-point-four l) (\u22a2telecom der (subst \u27e6_\u27e7E (sym (subst-lemma-one-point-four l)) (set (mapEnv-\u2208 l) env m))))\n\u22a2telecom (input l x\u2081) env = let (m , env') = lookup (mapEnv-\u2208 l) env\n                                hyp = \u22a2telecom (x\u2081 m) (subst (\u27e6_\u27e7E) (sym (subst-lemma-one-point-four l)) env')\n                             in output (mapEnv-\u2208 l) m\n                             (subst (\u22a2_) (subst-lemma-one-point-four l) hyp)\n\n-- old version\n{-\ncut\u1d9c\u1da0 : \u2200 {\u0394 d P} \u2192 \u27e6 dual P \u27e7 \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 P \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut\u1d9c\u1da0 D (end {allEnded = \u0394E , PE }) = end {allEnded = \u0394E}\ncut\u1d9c\u1da0 D (output here m E) = cut\u1d9c\u1da0 (D m) E\ncut\u1d9c\u1da0 D (output (there l) m E) = output l m (cut\u1d9c\u1da0 D E)\ncut\u1d9c\u1da0 (m , D) (input here x\u2081) = cut\u1d9c\u1da0 D (x\u2081 m)\ncut\u1d9c\u1da0 D (input (there l) x\u2081) = input l (\u03bb m \u2192 cut\u1d9c\u1da0 D (x\u2081 m))\n\ncut : \u2200 {\u0394 \u0394' \u0393 \u0393' d P} \u2192 \u22a2 \u0394 , clientURI \u21a6 dual P +++ \u0394' \u2192 \u22a2 \u0393 , d \u21a6 P +++ \u0393' \u2192 \u22a2 (\u0394 +++ \u0394') +++ (\u0393 +++ \u0393')\ncut D E = {!!}\n-}\n\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import proto\nopen import Types\nopen import prelude\nopen import uri\n\nmodule Terms where\n\ninfix 2 \u22a2_ \u22a2\u1d9c\u1da0_\n\ndata \u22a2_ : (\u0394 : Env) \u2192 Set\u2081 where\n  end : \u2200{\u0394}{e : EndedEnv \u0394}\n       ------------------\n      \u2192 \u22a2 \u0394\n\n  output : \u2200 {\u0394 d M P}{{_ : SER M}}\n             (l : d \u21a6 send P \u2208 \u0394)(m : M)\n             (p : \u22a2 \u0394 [ l \u2254 m ])\n             -------------------\n               \u2192 \u22a2 \u0394\n\n  input : \u2200 {\u0394 d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2 \u0394 [ l \u2254 m ])\n                 ----------------------\n                     \u2192 \u22a2 \u0394\n\n  mix : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n          (p : \u22a2 \u0394\u2080) (q : \u22a2 \u0394\u2081)\n          --------------------\n        \u2192 \u22a2 \u0394\n\n  cut : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) {P d}\n          (p : \u22a2 (\u0394\u2080 , d \u21a6 dual P))\n          (q : \u22a2 (\u0394\u2081 , d \u21a6 P))\n          ---------------------\n        \u2192 \u22a2 \u0394\n\n  fwd : \u2200 c d {P} \u2192 \u22a2 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\n\n  exch-last :\n      \u2200 {\u0394 c d P Q} \u2192\n      (p : \u22a2 \u0394 , c \u21a6 P , d \u21a6 Q)\n      \u2192 \u22a2 \u0394 , d \u21a6 Q , c \u21a6 P\n\n  wk-last : \u2200 {\u0394 d}\n              (p : \u22a2 \u0394)\n              -----------------------\n                \u2192 \u22a2 (\u0394 , d \u21a6 end)\n\n  end-last : \u2200 {\u0394 d}\n               (p : \u22a2 (\u0394 , d \u21a6 end))\n               ----------------------\n                 \u2192 \u22a2 \u0394\n\ndata \u22a2\u1d9c\u1da0_ (\u0394 : Env) : Set\u2081 where\n  end : {e : EndedEnv \u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  output : \u2200 {d M P}{{_ : SER M}}\n            (l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M)\n            (p : \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            --------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  input : \u2200 {d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            ----------------------------\n                     \u2192 \u22a2\u1d9c\u1da0 \u0394\n\nembed\u1d9c\u1da0 : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2 \u0394\nembed\u1d9c\u1da0 (end {e = e}) = end {e = e}\nembed\u1d9c\u1da0 (output l m p) = output l m (embed\u1d9c\u1da0 p)\nembed\u1d9c\u1da0 (input l p) = input l \u03bb m \u2192 embed\u1d9c\u1da0 (p m)\n\nmix\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n         (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n         (q : \u22a2\u1d9c\u1da0 \u0394\u2081)\n         -------------\n           \u2192 \u22a2\u1d9c\u1da0 \u0394\nmix\u1d9c\u1da0 \u0394\u209b end q = tr \u22a2\u1d9c\u1da0_ (Zip-identity  \u0394\u209b) q\nmix\u1d9c\u1da0 \u0394\u209b (output l m p) q\n  = output (Zip-com\u2208\u2080 \u0394\u209b l) m (mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) p q)\nmix\u1d9c\u1da0 \u0394\u209b (input l p) q\n  = input (Zip-com\u2208\u2080 \u0394\u209b l) \u03bb m \u2192 mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) (p m) q\n\ncut\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081}\n         (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) d P\n         (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n         (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n         ---------------------------\n           \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\ncut\u1d9c\u1da0 \u0394\u209b d end p q = mix\u1d9c\u1da0 (\u0394\u209b , d \u21a6\u2080 end) p q\n\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output here m p) (input here q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input here p) (output here m q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) (p m) q\n\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\n\ncut\u1d9c\u1da0 _ _ (com _ _) (end {e = _ , ()}) _\ncut\u1d9c\u1da0 _ _ (com _ _) _ (end {e = _ , ()})\n\nend-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n              (p : \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end))\n              -----------------------\n                \u2192 \u22a2\u1d9c\u1da0 \u0394\nend-last\u1d9c\u1da0 (end {e = e , _}) = end {e = e}\nend-last\u1d9c\u1da0 (output (there l) m p) = output l m (end-last\u1d9c\u1da0 p)\nend-last\u1d9c\u1da0 (input (there l) p) = input l \u03bb m \u2192 end-last\u1d9c\u1da0 (p m)\n\nwk-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n             (p : \u22a2\u1d9c\u1da0 \u0394)\n             -----------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\nwk-last\u1d9c\u1da0 end = end {e = \u2026 , _}\nwk-last\u1d9c\u1da0 (output l m p) = output (there l) m (wk-last\u1d9c\u1da0 p)\nwk-last\u1d9c\u1da0 (input l p) = input (there l) \u03bb m \u2192 wk-last\u1d9c\u1da0 (p m)\n\nwk-,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 \u2192 EndedEnv \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u03b5} p E = p\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u0394\u2081 , d \u21a6 P} p (E , e) rewrite Ended-\u2261end e\n  = wk-last\u1d9c\u1da0 (wk-,,\u1d9c\u1da0 p E)\n\nmodule _ {d P \u0394\u2080} where\n  pre-wk-\u2208 : \u2200 {\u0394\u2081} \u2192 d \u21a6 P \u2208 \u0394\u2081 \u2192 d \u21a6 P \u2208 (\u0394\u2080 ,, \u0394\u2081)\n  pre-wk-\u2208 here = here\n  pre-wk-\u2208 (there l) = there (pre-wk-\u2208 l)\n\n{-\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 E\u0394\u2080 end = end {e = {!!}}\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E\u0394\u2080 (output l m p) =\n  output (pre-wk-\u2208 l) m (pre-wk\u1d9c\u1da0 {\u0394\u2080} {{!\u0394\u2081!}} E\u0394\u2080 {!!})\npre-wk\u1d9c\u1da0 E\u0394\u2080 (input l p) = {!!}\n-}\n\nfwd-mix\u1d9c\u1da0 : \u2200 {\u0394 c d} P \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , d \u21a6 dual P)\nfwd-mix\u1d9c\u1da0 end p = wk-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 p)\nfwd-mix\u1d9c\u1da0 (recv P) p = input (there here) \u03bb m \u2192 output here m (fwd-mix\u1d9c\u1da0 (P m) p)\nfwd-mix\u1d9c\u1da0 (send P) p = input here \u03bb m \u2192 output (there here) m (fwd-mix\u1d9c\u1da0 (P m) p)\n\nfwd\u1d9c\u1da0 : \u2200 c d {P} \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\nfwd\u1d9c\u1da0 _ _ {P} = fwd-mix\u1d9c\u1da0 {\u03b5} P end\n\n\u03b5,, : \u2200 \u0394 \u2192 \u03b5 ,, \u0394 \u2261 \u0394\n\u03b5,, \u03b5 = refl\n\u03b5,, (\u0394 , d \u21a6 P) rewrite \u03b5,, \u0394 = refl\n\npostulate\n exch\u1d9c\u1da0 :\n  \u2200 \u0394\u2080 \u0394\u2081 \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2081 ,, \u0394\u2080)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n  {-\nexch\u1d9c\u1da0 \u03b5 \u0394\u2081 p rewrite \u03b5,, \u0394\u2081 = p\nexch\u1d9c\u1da0 \u0394\u2080 \u03b5 p rewrite \u03b5,, \u0394\u2080 = p\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) end = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 ._) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output here m p) = {!exch\u1d9c\u1da0!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (input l p) = {!!}\n-}\n\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E p = exch\u1d9c\u1da0 \u0394\u2080 \u0394\u2081 (wk-,,\u1d9c\u1da0 p E)\n\nend-of : Env \u2192 Env\nend-of \u03b5 = \u03b5\nend-of (\u0394 , d \u21a6 P) = end-of \u0394 , d \u21a6 end\n\nend-of-Ended : \u2200 \u0394 \u2192 EndedEnv (end-of \u0394)\nend-of-Ended \u03b5 = _\nend-of-Ended (\u0394 , d \u21a6 P) = end-of-Ended \u0394 , _\n\nend-of-\u22ce : \u2200 \u0394 \u2192 [ \u0394 is \u0394 \u22ce end-of \u0394 ]\nend-of-\u22ce \u03b5 = \u03b5\nend-of-\u22ce (\u0394 , d \u21a6 P) = end-of-\u22ce \u0394 , d \u21a6\u2080 P\n\nend-of-,,-\u22ce : \u2200 \u0394\u2080 \u0394\u2081 \u2192 [ \u0394\u2080 ,, \u0394\u2081 is \u0394\u2080 ,, end-of \u0394\u2081 \u22ce end-of \u0394\u2080 ,, \u0394\u2081 ]\nend-of-,,-\u22ce \u0394\u2080 \u03b5 = end-of-\u22ce \u0394\u2080\nend-of-,,-\u22ce \u0394\u2080 (\u0394\u2081 , d \u21a6 P) = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081 , d \u21a6\u2081 P\n\n,,-assoc : \u2200 {\u0394\u2080 \u0394\u2081 \u0394\u2082} \u2192 \u0394\u2080 ,, (\u0394\u2081 ,, \u0394\u2082) \u2261 (\u0394\u2080 ,, \u0394\u2081) ,, \u0394\u2082\n,,-assoc {\u0394\u2082 = \u03b5} = refl\n,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082 , d \u21a6 P} rewrite ,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082} = refl\n\ncut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\ncut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q =\n  end-last\u1d9c\u1da0\n    (cut\u1d9c\u1da0 \u0394\u209b d P\n       (exch\u1d9c\u1da0 (\u0394\u2080 ,, end-of \u0394\u2081) (\u03b5 , d \u21a6 dual P)\n              (tr \u22a2\u1d9c\u1da0_ (! (,,-assoc {\u03b5 , d \u21a6 dual P} {\u0394\u2080} {end-of \u0394\u2081}))\n                 (wk-,,\u1d9c\u1da0\n                   (exch\u1d9c\u1da0 (\u03b5 , d \u21a6 dual P) _ p) (end-of-Ended _))))\n       (pre-wk\u1d9c\u1da0 (end-of-Ended _) q))\n  where \u0394\u209b = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081\n\npostulate\n  !cut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 dual P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n-- !cut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q = {!!}\n\n-- only the last two are exchanged, some more has to be done\nexch-last\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q , c \u21a6 P\nexch-last\u1d9c\u1da0 (end {e = (a , b) , c}) = end {e = (a , c) , b}\nexch-last\u1d9c\u1da0 (output here m p) = output (there here) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there here) m p) = output here m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there (there l)) m p) = output (there (there l)) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (input here p) = input (there here) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there here) p) = input here \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there (there l)) p) = input (there (there l)) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p  m)\n\n{-\ndata Relabel : Env \u2192 Env \u2192 Set where\n  \u03b5 : Relabel \u03b5 \u03b5\n  _,_\u21a6_ : \u2200 {\u0394\u2080 \u0394\u2081 c d P} \u2192 Relabel \u0394\u2080 \u0394\u2081 \u2192 Relabel (\u0394\u2080 , c \u21a6 P) (\u0394\u2081 , d \u21a6 P)\n\nmodule _ where\n    rebalel-\u2208 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n                  (l : d \u21a6 P \u2208 \u0394\u2080)\n                    \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n            (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel end = {!end!}\n    relabel (output l m p) = output {!l!} {!!} {!!}\n    relabel (input l p) = {!!}\n\npar\u1d9c\u1da0 : \u2200 {\u0394 c} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , c \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 (P \u214b' Q))\n-- TODO only one channel name!!!\n-- TODO empty context\n-- TODO try to match on 'p' first\nbroken-par\u1d9c\u1da0 : \u2200 {c d e} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , e \u21a6 (P \u214b' Q))\nbroken-par\u1d9c\u1da0 end Q p = {!end-last\u1d9c\u1da0 (exch-last\u1d9c\u1da0 p)!}\nbroken-par\u1d9c\u1da0 (com x P) end p = end-last\u1d9c\u1da0 {!p!}\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (end {e = _ , ()})\n\nbroken-par\u1d9c\u1da0 (com x P) (com .OUT Q) (output here m p)\n  = output here R (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com x P) (com .IN Q) (input here p)\n  = output here R (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com .OUT P) (com y Q) (output (there here) m p)\n  = output here L (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com .IN P) (com y Q) (input (there here) p)\n  = output here L (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (output (there (there ())) m p)\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (input (there (there ())) p)\n-}\n\nmodule _ {c d cd} where\n    bi-fwd : \u2200 P Q \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , cd \u21a6 P \u2297 Q , c \u21a6 dual P , d \u21a6 dual Q)\n\n    private\n      module _ {M} {{_ : SER M}} {P : M \u2192 Proto} {Q} where\n        goL : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 P m \u2297 Q)\n                          , c  \u21a6 dual (com x P)\n                          , d  \u21a6 dual Q\n\n        goL IN  = input (there (there here)) \u03bb m \u2192 output (there here) m (bi-fwd _ _)\n        goL OUT = input (there here) \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n        goR : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 Q \u2297 P m)\n                          , c  \u21a6 dual Q\n                          , d  \u21a6 dual (com x P)\n        goR IN  = input (there (there here)) \u03bb m \u2192 output here m (bi-fwd _ _)\n        goR OUT = input here \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n    bi-fwd end Q = exch-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _))\n    bi-fwd (com x P) end = wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _)\n    bi-fwd (com x P) (com y Q) = input (there (there here)) [L: goL x ,R: goR y ]\n\n    module _ {\u0394\u2080 \u0394\u2081 P Q} where\n        \u2297\u1d9c\u1da0 : (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , c \u21a6 P))\n             (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 Q))\n             ----------------------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394\u2080 ,, \u0394\u2081 , cd \u21a6 (P \u2297 Q))\n        \u2297\u1d9c\u1da0 p q = !cut,,\u1d9c\u1da0 _ _ p (!cut,,\u1d9c\u1da0 _ _ q (bi-fwd P Q))\n\n  {-\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (l : c \u21a6 P \u2208 \u0394)\n  (p : \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 d \u21a6 Q ] , c \u21a6 P\nexch\u1d9c\u1da0 here p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (there l) p = {!!}\n-}\n\n{-\nrot\u1d9c\u1da0 : \u2200 \u0394 {c P} \u2192\n         (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P)\n      \u2192 \u22a2\u1d9c\u1da0 \u03b5 , c \u21a6 P ,, \u0394\nrot\u1d9c\u1da0 \u03b5 p = p\nrot\u1d9c\u1da0 (\u0394 , d \u21a6 P) p = {!rot\u1d9c\u1da0 \u0394 p!}\n\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394\u2080} \u0394\u2081 {c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2080 , c \u21a6 P , d \u21a6 Q ,, \u0394\u2081)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 , d \u21a6 Q , c \u21a6 P ,, \u0394\u2081\nexch\u1d9c\u1da0 \u03b5 p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (end e) = end {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d\u2081 \u21a6 ._) (output here m p) = output here m ({!exch\u1d9c\u1da0 \u0394\u2081 p!})\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (input l p) = {!!}\n-}\n\n_\u2286_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n_\u2287_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n\n\u0394\u2080 \u2286 \u0394\u2081 = \u2200 {d P} \u2192 d \u21a6 P \u2208 \u0394\u2080 \u2192 d \u21a6 P \u2208 \u0394\u2081\n\u0394\u2080 \u2287 \u0394\u2081 = \u0394\u2081 \u2286 \u0394\u2080\n\nget-put : \u2200 {d P \u0394 c Q} \u2192\n        (l : d \u21a6 P \u2208 \u0394) \u2192 c \u21a6 Q \u2208 (\u0394 [ l \u2254 c \u21a6 Q ])\nget-put here = here\nget-put (there l) = there (get-put l)\n\n{-\n\u2286_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (f : \u0394\u2080 \u2286 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2286 (\u0394\u2081 [ f l \u2254 c \u21a6 Q ])\n\u2286 f [ l \u2254 c \u21a6 Q ] {d'} {P'} l' = {!!}\n\n(l : d \u21a6 P \u2208 \u0394)\n\u2192 \u0394 [ l \u2254 ]\n\nrecord _\u2248_ (\u0394\u2080 \u0394\u2081 : Env) : Set\u2081 where\n  constructor _,_\n  field\n    \u2248\u2286 : \u0394\u2080 \u2286 \u0394\u2081\n    \u2248\u2287 : \u0394\u2080 \u2287 \u0394\u2081\nopen _\u2248_ public\n\n\u2248_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2248 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ])\n\u2248 \u0394\u2091 [ here \u2254 m ] = {!!}\n\u2248 \u0394\u2091 [ there l \u2254 m ] = {!!}\n\n{-(\u03bb l' \u2192 {!\u2248\u2286 \u0394\u2091!}) , from\n  where\n    from : \u2200 {\u0394\u2080 \u0394\u2081 d io M} {P : M \u2192 Proto} {ser : SER M}\n             {\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081} {l : d \u21a6 com io P \u2208 \u0394\u2080} {m : M} {d\u2081} {P\u2081} \u2192\n           d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ]) \u2192 d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2080 [ l \u2254 m ])\n    from = {!!}\n\n\u2248\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081}\n       (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n       (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n       -----------\n         \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n\u2248\u1d9c\u1da0 \u0394\u2091 (end {e = e}) = end {e = {!!}}\n\u2248\u1d9c\u1da0 \u0394\u2091 (output l m p) = output (\u2248\u2286 \u0394\u2091 l) m (\u2248\u1d9c\u1da0 (\u2248 \u0394\u2091 [ l \u2254 m ]) p)\n\u2248\u1d9c\u1da0 \u0394\u2091 (input l p) = {!!}\n-}\n-}\n\ncut-elim : \u2200 {\u0394} (p : \u22a2 \u0394)\n                 ------------\n                   \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut-elim (end {e = e}) = end {e = e}\ncut-elim (output l m p) = output l m (cut-elim p)\ncut-elim (input l p) = input l (\u03bb m \u2192 cut-elim (p m))\ncut-elim (mix \u0394\u209b p q) = mix\u1d9c\u1da0 \u0394\u209b (cut-elim p) (cut-elim q)\ncut-elim (cut \u0394\u209b {P} {d} p q) = end-last\u1d9c\u1da0 (cut\u1d9c\u1da0 \u0394\u209b d P (cut-elim p) (cut-elim q))\ncut-elim (end-last p) = end-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (wk-last p) = wk-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (fwd c d) = fwd\u1d9c\u1da0 c d\ncut-elim (exch-last p) = exch-last\u1d9c\u1da0 (cut-elim p)\n\n{-\n\nstart : \u2200 {\u0394} P\n       \u2192 \u22a2 [ clientURI \u21a6 dual P ]\n       \u2192 (\u2200 d \u2192 \u22a2 (\u0394 , d \u21a6 P))\n       \u2192 \u22a2 \u0394\nstart P p q = cut {!!} (... p) (q {!!})\n-}\n\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 : \u2200 {P d} \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ] \u2192 \u27e6 P \u27e7\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {end} end = _\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {com x P} (end {e = _ , ()})\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output here m der) = m , \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 der\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output (there ()) m der)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input here x\u2081) m = \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (x\u2081 m)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input (there ()) x\u2081)\n\nSatisfy : \u2200 {p d} P \u2192 (R : \u27e6 log P \u27e7 \u2192 Set p) \u2192 \u22a2 [ d \u21a6 P ] \u2192 Set p\nSatisfy P Rel d = (d' : \u27e6 dual P \u27e7) \u2192 Rel (telecom P (\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (cut-elim d)) d')\n\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {end} p = end\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 (p m))\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 p)\n\n\u27e6\u27e7\u2192\u22a2 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2 {end} p = end\n\u27e6\u27e7\u2192\u22a2 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2 (p m))\n\u27e6\u27e7\u2192\u22a2 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2 p)\n\n{-\n\u22a2toProc : \u2200 {\u0394} \u2192 \u22a2 \u0394 \u2192 JSProc\n\u22a2toProc end = end\n\u22a2toProc (output {d = d} l m prg) = output d (serialize m) (\u22a2toProc prg)\n\u22a2toProc (input {d = d} l prg) = input d ([succeed: (\u03bb m \u2192 \u22a2toProc (prg m)) ,fail: error ] \u2218 parse)\n\u22a2toProc (start P prg x) = start (\u22a2toProc prg) (\u03bb d \u2192 \u22a2toProc (x d))\n\n\n\u22a2toProc-WT : \u2200 {\u0394} (der : \u22a2 \u0394) \u2192 \u0394 \u22a2 \u22a2toProc der\n\u22a2toProc-WT (end {x}) = end {_} {x}\n\u22a2toProc-WT (output {{x}} l m der) = output l (sym (rinv m)) (\u22a2toProc-WT der)\n\u22a2toProc-WT (input {{x}} l x\u2081) = input l \u03bb s m x \u2192\n  subst (\u03bb X \u2192 _ [ l \u2254 m ] \u22a2 [succeed: (\u22a2toProc \u2218 x\u2081) ,fail: error ] X) x (\u22a2toProc-WT (x\u2081 m))\n\u22a2toProc-WT (start P der x) = start P (\u22a2toProc-WT der) (\u03bb d \u2192 \u22a2toProc-WT (x d))\n-}\n\n\u27e6_\u27e7E : Env \u2192 Set\n\u27e6 \u03b5 \u27e7E = \ud835\udfd9\n\u27e6 \u0394 , d \u21a6 P \u27e7E = \u27e6 \u0394 \u27e7E \u00d7 \u27e6 P \u27e7\n\nmapEnv : (Proto \u2192 Proto) \u2192 Env \u2192 Env\nmapEnv f \u03b5 = \u03b5\nmapEnv f (\u0394 , d \u21a6 P) = mapEnv f \u0394 , d \u21a6 f P\n\nmapEnv-all : \u2200 {P Q : URI \u2192 Proto \u2192 Set}{\u0394}{f : Proto \u2192 Proto}\n  \u2192 (\u2200 d x \u2192 P d x \u2192 Q d (f x))\n  \u2192 AllEnv P \u0394 \u2192 AllEnv Q (mapEnv f \u0394)\nmapEnv-all {\u0394 = \u03b5} f\u2081 \u2200\u0394 = \u2200\u0394\nmapEnv-all {\u0394 = \u0394 , d \u21a6 P\u2081} f\u2081 (\u2200\u0394 , P) = (mapEnv-all f\u2081 \u2200\u0394) , f\u2081 d P\u2081 P\n\nmapEnv-Ended : \u2200 {f : Proto \u2192 Proto}{\u0394} \u2192 Ended (f end)\n  \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 AllEnv (\u03bb _ \u2192 Ended) (mapEnv f \u0394)\nmapEnv-Ended eq = mapEnv-all (\u03bb { d end _ \u2192 eq ; d (send P) () ; d (recv P) () })\n\nmapEnv-\u2208 : \u2200 {d P f \u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 d \u21a6 f P \u2208 mapEnv f \u0394\nmapEnv-\u2208 here = here\nmapEnv-\u2208 (there der) = there (mapEnv-\u2208 der)\n\nmodule _ {d c M cf}{m : M}{{_ : M \u2243? SERIAL}}{p} where\n  subst-lemma-one-point-four : \u2200 {\u0394}( l : d \u21a6 com c p \u2208 \u0394 ) \u2192\n    let f = mapProto cf\n    in (mapEnv f (\u0394 [ l \u2254 m ])) \u2261 (_[_\u2254_]{c = cf c} (mapEnv f \u0394) (mapEnv-\u2208 l) m)\n  subst-lemma-one-point-four here = refl\n  subst-lemma-one-point-four (there {d' = d'}{P'} l) = ap (\u03bb X \u2192 X , d' \u21a6 mapProto cf P') (subst-lemma-one-point-four l)\n\nmodule _ {d P} where\n  project : \u2200 {\u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 \u27e6 \u0394 \u27e7E \u2192 \u27e6 P \u27e7\n  project here env = snd env\n  project (there mem) env = project mem (fst env)\n\nempty : \u2200 {\u0394} \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 \u27e6 \u0394 \u27e7E\nempty {\u03b5} <> = _\nempty {\u0394 , d \u21a6 end} (fst , <>) = empty fst , _\nempty {\u0394 , d \u21a6 com x P} (fst , ())\n\nnoRecvInLog : \u2200 {d M}{{_ : M \u2243? SERIAL}}{P : M \u2192 _}{\u0394} \u2192 d \u21a6 recv P \u2208 mapEnv log \u0394 \u2192 \ud835\udfd8\nnoRecvInLog {\u0394 = \u03b5} ()\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 end} (there l) = noRecvInLog l\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 com x\u2081 P\u2081} (there l) = noRecvInLog l\n\nmodule _ {d M P}{{_ : M \u2243? SERIAL}} where\n  lookup : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 \u03a3 M \u03bb m \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  lookup here (env , (m , p)) = m , (env , p)\n  lookup (there l) (env , P') = let (m , env') = lookup l env in m , (env' , P')\n\n  set : \u2200 {\u0394}(l : d \u21a6 recv P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  set here (env , f) m = env , f m\n  set (there l) (env , P') m = set l env m , P'\n\n  set\u03a3 : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E \u2192 \u27e6 \u0394 \u27e7E\n  set\u03a3 here m env = fst env , (m , snd env)\n  set\u03a3 (there l) m env = set\u03a3 l m (fst env) , snd env\n\n  {-\nforgetConc : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 mapEnv log \u0394 \u2192 \u27e6 mapEnv log \u0394 \u27e7E\nforgetConc (end e) = empty \u2026\nforgetConc {\u0394} (output l m der) = set\u03a3 l m (forgetConc {{!set\u03a3 l m!}} der) -- (forgetConc der)\nforgetConc (input l x\u2081) with noRecvInLog l\n... | ()\n-}\n\n\u22a2telecom : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u27e6 mapEnv dual \u0394 \u27e7E \u2192 \u22a2 mapEnv log \u0394\n\u22a2telecom end env = end {e = mapEnv-Ended _ \u2026}\n\u22a2telecom (output l m der) env = output (mapEnv-\u2208 l) m (subst (\u22a2_)\n  (subst-lemma-one-point-four l) (\u22a2telecom der (subst \u27e6_\u27e7E (sym (subst-lemma-one-point-four l)) (set (mapEnv-\u2208 l) env m))))\n\u22a2telecom (input l x\u2081) env = let (m , env') = lookup (mapEnv-\u2208 l) env\n                                hyp = \u22a2telecom (x\u2081 m) (subst (\u27e6_\u27e7E) (sym (subst-lemma-one-point-four l)) env')\n                             in output (mapEnv-\u2208 l) m\n                             (subst (\u22a2_) (subst-lemma-one-point-four l) hyp)\n\n-- old version\n{-\ncut\u1d9c\u1da0 : \u2200 {\u0394 d P} \u2192 \u27e6 dual P \u27e7 \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 P \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut\u1d9c\u1da0 D (end {allEnded = \u0394E , PE }) = end {allEnded = \u0394E}\ncut\u1d9c\u1da0 D (output here m E) = cut\u1d9c\u1da0 (D m) E\ncut\u1d9c\u1da0 D (output (there l) m E) = output l m (cut\u1d9c\u1da0 D E)\ncut\u1d9c\u1da0 (m , D) (input here x\u2081) = cut\u1d9c\u1da0 D (x\u2081 m)\ncut\u1d9c\u1da0 D (input (there l) x\u2081) = input l (\u03bb m \u2192 cut\u1d9c\u1da0 D (x\u2081 m))\n\ncut : \u2200 {\u0394 \u0394' \u0393 \u0393' d P} \u2192 \u22a2 \u0394 , clientURI \u21a6 dual P +++ \u0394' \u2192 \u22a2 \u0393 , d \u21a6 P +++ \u0393' \u2192 \u22a2 (\u0394 +++ \u0394') +++ (\u0393 +++ \u0393')\ncut D E = {!!}\n-}\n\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c797d7eb5e42bb8c41377fd0cda132082dd7438a","subject":"Desc stratified model: IDescl implicit.","message":"Desc stratified model: IDescl implicit.","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl I) -> IDescl I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7ec8bed818d248b395b4b0b2c2d2f21361f940bc","subject":"Add lemmas I present in thesis","message":"Add lemmas I present in thesis\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/LangOps.agda","new_file":"Thesis\/LangOps.agda","new_contents":"module Thesis.LangOps where\n\nopen import Thesis.Lang\nopen import Thesis.Changes\nopen import Thesis.LangChanges\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\noplus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4 \u21d2 \u03c4)\nominus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394t \u03c4)\n\nonil\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4)\nonil\u03c4o \u03c4 = abs (app\u2082 (ominus\u03c4o \u03c4) (var this) (var this))\n\n-- Do NOT try to read this, such terms are write-only. But the behavior is\n-- specified to be oplus\u03c4-equiv and ominus\u03c4-equiv.\noplus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs\n  (app\u2082 (oplus\u03c4o \u03c4)\n    (app (var (that (that this))) (var this))\n    (app\u2082 (var (that this)) (var this) (app (onil\u03c4o \u03c3) (var this))))))\noplus\u03c4o unit = abs (abs (const unit))\noplus\u03c4o int = const plus\noplus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (oplus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (oplus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\noplus\u03c4o (sum \u03c3 \u03c4) = abs (abs (app\u2083 (const match) (var (that this))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app\u2083 (const match) (var this)\n      (abs (app (const linj) (app\u2082 (oplus\u03c4o \u03c3) (var (that (that this))) (var this))))\n      (abs (app (const linj) (var (that (that this)))))))\n    (abs (var this))))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app\u2083 (const match) (var this)\n      (abs (app (const rinj) (var (that (that this)))))\n      (abs (app (const rinj) (app\u2082 (oplus\u03c4o \u03c4) (var (that (that this))) (var this))))))\n    (abs (var this))))))\n\nominus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs (abs (app\u2082 (ominus\u03c4o \u03c4)\n  (app (var (that (that (that this)))) (app\u2082 (oplus\u03c4o \u03c3) (var (that this)) (var this)))\n  (app (var (that (that this))) (var (that this)))))))\nominus\u03c4o unit = abs (abs (const unit))\nominus\u03c4o int = const minus\nominus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (ominus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (ominus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\nominus\u03c4o (sum \u03c3 \u03c4) = abs (abs (app\u2083 (const match) (var (that this))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app (const linj) (app (const linj) (app\u2082 (ominus\u03c4o \u03c3) (var (that this)) (var this)))))\n    (abs (app (const rinj) (var (that (that (that this))))))))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app (const rinj) (var (that (that (that this))))))\n    (abs (app (const linj) (app (const rinj) (app\u2082 (ominus\u03c4o \u03c4) (var (that this)) (var this)))))))))\n\noplus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 a da \u2192 \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1 a da \u2261 a \u2295 da\nominus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 b a \u2192 \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u03c1 b a \u2261 b \u229d a\n\noplus\u03c4-equiv-ext : \u2200 \u03c4 \u0393 \u2192 \u27e6 oplus\u03c4o {\u0393} \u03c4 \u27e7Term \u2261 \u03bb \u03c1 \u2192 _\u2295_\noplus\u03c4-equiv-ext \u03c4 _ = ext\u00b3 (\u03bb \u03c1 a da \u2192 oplus\u03c4-equiv _ \u03c1 \u03c4 a da)\nominus\u03c4-equiv-ext : \u2200 \u03c4 \u0393 \u2192 \u27e6 ominus\u03c4o {\u0393} \u03c4 \u27e7Term \u2261 \u03bb \u03c1 \u2192 _\u229d_\nominus\u03c4-equiv-ext \u03c4 _ = ext\u00b3 (\u03bb \u03c1 a da \u2192 ominus\u03c4-equiv _ \u03c1 \u03c4 a da)\n\noplus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) f df = ext (\u03bb a \u2192 lemma a)\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) where\n      \u03c1\u2032 = a \u2022 df \u2022 f \u2022 \u03c1\n      \u03c1\u2032\u2032 = a \u2022 \u03c1\u2032\n      lemma : \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1\u2032 (f a)\n         (df a (\u27e6 ominus\u03c4o \u03c3 \u27e7Term \u03c1\u2032\u2032 a a))\n         \u2261 f a \u2295 df a (nil a)\n      lemma\n        rewrite ominus\u03c4-equiv _ \u03c1\u2032\u2032 \u03c3 a a\n        | oplus\u03c4-equiv _ \u03c1\u2032 \u03c4 (f a) (df a (nil a))\n        = refl\noplus\u03c4-equiv \u0393 \u03c1 unit tt tt = refl\noplus\u03c4-equiv \u0393 \u03c1 int a da = refl\noplus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a , b) (da , db)\n  rewrite oplus\u03c4-equiv _ ((da , db) \u2022 (a , b) \u2022 \u03c1) \u03c3 a da\n  | oplus\u03c4-equiv _ ((da , db) \u2022 (a , b) \u2022 \u03c1) \u03c4 b db\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2081 (inj\u2081 dx))\n  rewrite oplus\u03c4-equiv-ext \u03c3 (\u0394t \u03c3 \u2022 sum (\u0394t \u03c3) (\u0394t \u03c4) \u2022 \u03c3 \u2022 \u0394t (sum \u03c3 \u03c4) \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2081 (inj\u2082 dy)) = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2082 y) = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2081 (inj\u2081 dx)) = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2081 (inj\u2082 dy))\n  rewrite oplus\u03c4-equiv-ext \u03c4 (\u0394t \u03c4 \u2022 sum (\u0394t \u03c3) (\u0394t \u03c4) \u2022 \u03c4 \u2022 \u0394t (sum \u03c3 \u03c4) \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2082 y\u2081) = refl\n\nominus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) g f = ext (\u03bb a \u2192 ext (lemma a))\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) (da : Ch\u03c4 \u03c3) where\n      \u03c1\u2032 = da \u2022 a \u2022 f \u2022 g \u2022 \u03c1\n      lemma : \u27e6 ominus\u03c4o \u03c4 \u27e7Term (da \u2022 a \u2022 f \u2022 g \u2022 \u03c1)\n        (g (\u27e6 oplus\u03c4o \u03c3 \u27e7Term (da \u2022 a \u2022 f \u2022 g \u2022 \u03c1) a da)) (f a)\n        \u2261 g (a \u2295 da) \u229d f a\n      lemma\n        rewrite oplus\u03c4-equiv _ \u03c1\u2032 \u03c3 a da\n        | ominus\u03c4-equiv _ \u03c1\u2032 \u03c4 (g (a \u2295 da)) (f a) = refl\nominus\u03c4-equiv \u0393 \u03c1 unit tt tt = refl\nominus\u03c4-equiv \u0393 \u03c1 int b a = refl\nominus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a2 , b2) (a1 , b1)\n  rewrite ominus\u03c4-equiv _ ((a1 , b1) \u2022 (a2 , b2) \u2022 \u03c1) \u03c3 a2 a1\n  | ominus\u03c4-equiv _ ((a1 , b1) \u2022 (a2 , b2) \u2022 \u03c1) \u03c4 b2 b1\n  = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2081 x\u2081)\n  rewrite ominus\u03c4-equiv-ext \u03c3 (\u03c3 \u2022 \u03c3 \u2022 sum \u03c3 \u03c4 \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2082 y) = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2081 x) = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2082 y\u2081)\n  rewrite ominus\u03c4-equiv-ext \u03c4 (\u03c4 \u2022 \u03c4 \u2022 sum \u03c3 \u03c4 \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\n\n-- Proved these lemmas to show them in thesis.\noplus\u03c4-equiv-term : \u2200 d\u0393 (d\u03c1 : \u27e6 d\u0393 \u27e7Context) \u03c4 t dt  \u2192\n  \u27e6 app\u2082 (oplus\u03c4o \u03c4) t dt \u27e7Term d\u03c1 \u2261 \u27e6 t \u27e7Term d\u03c1 \u2295 \u27e6 dt \u27e7Term d\u03c1\noplus\u03c4-equiv-term d\u0393 d\u03c1 \u03c4 t dt = oplus\u03c4-equiv d\u0393 d\u03c1 \u03c4 (\u27e6 t \u27e7Term d\u03c1) (\u27e6 dt \u27e7Term d\u03c1)\nominus\u03c4-equiv-term : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 t2 t1  \u2192\n  \u27e6 app\u2082 (ominus\u03c4o \u03c4) t2 t1 \u27e7Term \u03c1 \u2261 \u27e6 t2 \u27e7Term \u03c1 \u229d \u27e6 t1 \u27e7Term \u03c1\nominus\u03c4-equiv-term \u0393 \u03c1 \u03c4 t2 t1 = ominus\u03c4-equiv \u0393 \u03c1 \u03c4 (\u27e6 t2 \u27e7Term \u03c1) (\u27e6 t1 \u27e7Term \u03c1)\n","old_contents":"module Thesis.LangOps where\n\nopen import Thesis.Lang\nopen import Thesis.Changes\nopen import Thesis.LangChanges\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\noplus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4 \u21d2 \u03c4)\nominus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394t \u03c4)\n\nonil\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4)\nonil\u03c4o \u03c4 = abs (app\u2082 (ominus\u03c4o \u03c4) (var this) (var this))\n\n-- Do NOT try to read this, such terms are write-only. But the behavior is\n-- specified to be oplus\u03c4-equiv and ominus\u03c4-equiv.\noplus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs\n  (app\u2082 (oplus\u03c4o \u03c4)\n    (app (var (that (that this))) (var this))\n    (app\u2082 (var (that this)) (var this) (app (onil\u03c4o \u03c3) (var this))))))\noplus\u03c4o unit = abs (abs (const unit))\noplus\u03c4o int = const plus\noplus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (oplus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (oplus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\noplus\u03c4o (sum \u03c3 \u03c4) = abs (abs (app\u2083 (const match) (var (that this))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app\u2083 (const match) (var this)\n      (abs (app (const linj) (app\u2082 (oplus\u03c4o \u03c3) (var (that (that this))) (var this))))\n      (abs (app (const linj) (var (that (that this)))))))\n    (abs (var this))))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app\u2083 (const match) (var this)\n      (abs (app (const rinj) (var (that (that this)))))\n      (abs (app (const rinj) (app\u2082 (oplus\u03c4o \u03c4) (var (that (that this))) (var this))))))\n    (abs (var this))))))\n\nominus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs (abs (app\u2082 (ominus\u03c4o \u03c4)\n  (app (var (that (that (that this)))) (app\u2082 (oplus\u03c4o \u03c3) (var (that this)) (var this)))\n  (app (var (that (that this))) (var (that this)))))))\nominus\u03c4o unit = abs (abs (const unit))\nominus\u03c4o int = const minus\nominus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (ominus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (ominus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\nominus\u03c4o (sum \u03c3 \u03c4) = abs (abs (app\u2083 (const match) (var (that this))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app (const linj) (app (const linj) (app\u2082 (ominus\u03c4o \u03c3) (var (that this)) (var this)))))\n    (abs (app (const rinj) (var (that (that (that this))))))))\n  (abs (app\u2083 (const match) (var (that this))\n    (abs (app (const rinj) (var (that (that (that this))))))\n    (abs (app (const linj) (app (const rinj) (app\u2082 (ominus\u03c4o \u03c4) (var (that this)) (var this)))))))))\n\noplus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 a da \u2192 \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1 a da \u2261 a \u2295 da\nominus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 b a \u2192 \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u03c1 b a \u2261 b \u229d a\n\noplus\u03c4-equiv-ext : \u2200 \u03c4 \u0393 \u2192 \u27e6 oplus\u03c4o {\u0393} \u03c4 \u27e7Term \u2261 \u03bb \u03c1 \u2192 _\u2295_\noplus\u03c4-equiv-ext \u03c4 _ = ext\u00b3 (\u03bb \u03c1 a da \u2192 oplus\u03c4-equiv _ \u03c1 \u03c4 a da)\nominus\u03c4-equiv-ext : \u2200 \u03c4 \u0393 \u2192 \u27e6 ominus\u03c4o {\u0393} \u03c4 \u27e7Term \u2261 \u03bb \u03c1 \u2192 _\u229d_\nominus\u03c4-equiv-ext \u03c4 _ = ext\u00b3 (\u03bb \u03c1 a da \u2192 ominus\u03c4-equiv _ \u03c1 \u03c4 a da)\n\noplus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) f df = ext (\u03bb a \u2192 lemma a)\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) where\n      \u03c1\u2032 = a \u2022 df \u2022 f \u2022 \u03c1\n      \u03c1\u2032\u2032 = a \u2022 \u03c1\u2032\n      lemma : \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1\u2032 (f a)\n         (df a (\u27e6 ominus\u03c4o \u03c3 \u27e7Term \u03c1\u2032\u2032 a a))\n         \u2261 f a \u2295 df a (nil a)\n      lemma\n        rewrite ominus\u03c4-equiv _ \u03c1\u2032\u2032 \u03c3 a a\n        | oplus\u03c4-equiv _ \u03c1\u2032 \u03c4 (f a) (df a (nil a))\n        = refl\noplus\u03c4-equiv \u0393 \u03c1 unit tt tt = refl\noplus\u03c4-equiv \u0393 \u03c1 int a da = refl\noplus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a , b) (da , db)\n  rewrite oplus\u03c4-equiv _ ((da , db) \u2022 (a , b) \u2022 \u03c1) \u03c3 a da\n  | oplus\u03c4-equiv _ ((da , db) \u2022 (a , b) \u2022 \u03c1) \u03c4 b db\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2081 (inj\u2081 dx))\n  rewrite oplus\u03c4-equiv-ext \u03c3 (\u0394t \u03c3 \u2022 sum (\u0394t \u03c3) (\u0394t \u03c4) \u2022 \u03c3 \u2022 \u0394t (sum \u03c3 \u03c4) \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2081 (inj\u2082 dy)) = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2082 y) = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2081 (inj\u2081 dx)) = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2081 (inj\u2082 dy))\n  rewrite oplus\u03c4-equiv-ext \u03c4 (\u0394t \u03c4 \u2022 sum (\u0394t \u03c3) (\u0394t \u03c4) \u2022 \u03c4 \u2022 \u0394t (sum \u03c3 \u03c4) \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2082 y\u2081) = refl\n\nominus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) g f = ext (\u03bb a \u2192 ext (lemma a))\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) (da : Ch\u03c4 \u03c3) where\n      \u03c1\u2032 = da \u2022 a \u2022 f \u2022 g \u2022 \u03c1\n      lemma : \u27e6 ominus\u03c4o \u03c4 \u27e7Term (da \u2022 a \u2022 f \u2022 g \u2022 \u03c1)\n        (g (\u27e6 oplus\u03c4o \u03c3 \u27e7Term (da \u2022 a \u2022 f \u2022 g \u2022 \u03c1) a da)) (f a)\n        \u2261 g (a \u2295 da) \u229d f a\n      lemma\n        rewrite oplus\u03c4-equiv _ \u03c1\u2032 \u03c3 a da\n        | ominus\u03c4-equiv _ \u03c1\u2032 \u03c4 (g (a \u2295 da)) (f a) = refl\nominus\u03c4-equiv \u0393 \u03c1 unit tt tt = refl\nominus\u03c4-equiv \u0393 \u03c1 int b a = refl\nominus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a2 , b2) (a1 , b1)\n  rewrite ominus\u03c4-equiv _ ((a1 , b1) \u2022 (a2 , b2) \u2022 \u03c1) \u03c3 a2 a1\n  | ominus\u03c4-equiv _ ((a1 , b1) \u2022 (a2 , b2) \u2022 \u03c1) \u03c4 b2 b1\n  = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2081 x\u2081)\n  rewrite ominus\u03c4-equiv-ext \u03c3 (\u03c3 \u2022 \u03c3 \u2022 sum \u03c3 \u03c4 \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2081 x) (inj\u2082 y) = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2081 x) = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) (inj\u2082 y) (inj\u2082 y\u2081)\n  rewrite ominus\u03c4-equiv-ext \u03c4 (\u03c4 \u2022 \u03c4 \u2022 sum \u03c3 \u03c4 \u2022 sum \u03c3 \u03c4 \u2022 \u0393)\n  = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"965133f11d3863769cb8e636eff326a49bc7321b","subject":"CyclicGroup: rename \ud835\udd3e as \u2124[ p ]\u2605","message":"CyclicGroup: rename \ud835\udd3e as \u2124[ p ]\u2605\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/JS\/BigI\/CyclicGroup.agda","new_file":"Crypto\/JS\/BigI\/CyclicGroup.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import FFI.JS using (Bool; trace-call; _++_)\nopen import FFI.JS.Check\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Raw\nopen import Algebra.Group\n\n-- TODO carry on a primality proof of p\nmodule Crypto.JS.BigI.CyclicGroup (p : BigI) where\n\nabstract\n  \u2124[_]\u2605 : Set\n  \u2124[_]\u2605 = BigI\n\n  private\n    \u2124p\u2605 : Set\n    \u2124p\u2605 = BigI\n\n    mod-p : BigI \u2192 \u2124p\u2605\n    mod-p x = mod x p\n\n  -- There are two ways to go from BigI to \u2124p\u2605: check and mod-p\n  -- Use check for untrusted input data and mod-p for internal\n  -- computation.\n  BigI\u25b9\u2124[_]\u2605 : BigI \u2192 \u2124p\u2605\n  BigI\u25b9\u2124[_]\u2605 = -- trace-call \"BigI\u25b9\u2124[_]\u2605 \"\n    \u03bb x \u2192\n      (check? (x <I p)\n         (\u03bb _ \u2192 \"Not below the modulus: p:\" ++ toString p ++ \" is less than x:\" ++ toString x)\n         (check? (x >I 0I)\n            (\u03bb _ \u2192 \"Should be strictly positive: \" ++ toString x ++ \" <= 0\") x))\n\n  check-non-zero : \u2124p\u2605 \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \u2124p\u2605 \u2192 BigI\n  repr x = x\n\n  1# : \u2124p\u2605\n  1# = 1I\n\n  1\/_ : \u2124p\u2605 \u2192 \u2124p\u2605\n  1\/ x = modInv (check-non-zero x) p\n\n  _^_ : \u2124p\u2605 \u2192 BigI \u2192 \u2124p\u2605\n  x ^ y = modPow x y p\n\n_*_ _\/_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \u2124p\u2605\n\nx * y = mod-p (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\ninstance\n  \u2124[_]\u2605-Eq? : Eq? \u2124p\u2605\n  \u2124[_]\u2605-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\nprod : List \u2124p\u2605 \u2192 \u2124p\u2605\nprod = foldr _*_ 1#\n\nmon-ops : Monoid-Ops \u2124p\u2605\nmon-ops = _*_ , 1#\n\ngrp-ops : Group-Ops \u2124p\u2605\ngrp-ops = mon-ops , 1\/_\n\npostulate\n  grp-struct : Group-Struct grp-ops\n\ngrp : Group \u2124p\u2605\ngrp = grp-ops , grp-struct\n\nmodule grp = Group grp\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import FFI.JS using (Bool; trace-call; _++_)\nopen import FFI.JS.Check\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Raw\nopen import Algebra.Group\n\n-- TODO carry on a primality proof of p\nmodule Crypto.JS.BigI.CyclicGroup (p : BigI) where\n\nabstract\n  -- \u2124p*\n  \ud835\udd3e : Set\n  \ud835\udd3e = BigI\n\n  private\n    mod-p : BigI \u2192 \ud835\udd3e\n    mod-p x = mod x p\n\n  -- There are two ways to go from BigI to \ud835\udd3e: check and mod-p\n  -- Use check for untrusted input data and mod-p for internal\n  -- computation.\n  BigI\u25b9\ud835\udd3e : BigI \u2192 \ud835\udd3e\n  BigI\u25b9\ud835\udd3e = -- trace-call \"BigI\u25b9\ud835\udd3e \"\n    \u03bb x \u2192\n      (check? (x <I p)\n         (\u03bb _ \u2192 \"Not below the modulus: p:\" ++ toString p ++ \" is less than x:\" ++ toString x)\n         (check? (x >I 0I)\n            (\u03bb _ \u2192 \"Should be strictly positive: \" ++ toString x ++ \" <= 0\") x))\n\n  check-non-zero : \ud835\udd3e \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \ud835\udd3e \u2192 BigI\n  repr x = x\n\n  1# : \ud835\udd3e\n  1# = 1I\n\n  1\/_ : \ud835\udd3e \u2192 \ud835\udd3e\n  1\/ x = modInv (check-non-zero x) p\n\n  _^_ : \ud835\udd3e \u2192 BigI \u2192 \ud835\udd3e\n  x ^ y = modPow x y p\n\n_*_ _\/_ : \ud835\udd3e \u2192 \ud835\udd3e \u2192 \ud835\udd3e\n\nx * y = mod-p (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\ninstance\n  \ud835\udd3e-Eq? : Eq? \ud835\udd3e\n  \ud835\udd3e-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \ud835\udd3e \u2192 \ud835\udd3e \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\nprod : List \ud835\udd3e \u2192 \ud835\udd3e\nprod = foldr _*_ 1#\n\nmon-ops : Monoid-Ops \ud835\udd3e\nmon-ops = _*_ , 1#\n\ngrp-ops : Group-Ops \ud835\udd3e\ngrp-ops = mon-ops , 1\/_\n\npostulate\n  grp-struct : Group-Struct grp-ops\n\ngrp : Group \ud835\udd3e\ngrp = grp-ops , grp-struct\n\nmodule grp = Group grp\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fcda492aa46e222289a8d537bd357e0a62b63c45","subject":"Working in a draft about the existential elimination.","message":"Working in a draft about the existential elimination.\n\nIgnore-this: 1d4d9bccdab653e3734d917852116238\n\ndarcs-hash:20120220031726-3bd4e-400bf695e96bc4680bfccab55da676c918628e5d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Common\/Data\/Existential.agda","new_file":"Draft\/Common\/Data\/Existential.agda","new_contents":"-----------------------------------------------------------------------------\n-- Existential elimination\n-----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 19 February 2012.\n\nmodule Existential where\n\n-----------------------------------------------------------------------------\n\npostulate\n  D : Set\n\nmodule \u2203\u2081 where\n  -- Type theoretical version\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 7 _,_\n\n  -- The existential quantifier type on D.\n  data \u2203 (P : D \u2192 Set) : Set where\n    _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n  -- Sugar syntax for the existential quantifier.\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  -- The existential elimination.\n  \u2203-proj\u2081 : \u2200 {P} \u2192 \u2203 P \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {P} \u2192 (p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n  \u2203-proj\u2082 (_ , Px) = Px\n\n  -- Some examples\n\n  -- The order of quantifiers of the same sort is irrelevant.\n  \u2203-ord : {P\u00b2 : D \u2192 D \u2192 Set} \u2192 (\u2203[ x ] \u2203[ y ] P\u00b2 x y) \u2192 (\u2203[ y ] \u2203[ x ] P\u00b2 x y)\n  \u2203-ord (x , y , h) = y , x , h\n\n-----------------------------------------------------------------------------\n\nmodule \u2203\u2082 where\n  -- FOL version\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 7 _,_\n\n  -- The existential quantifier type on D.\n  data \u2203 (P : D \u2192 Set) : Set where\n    _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n  -- Sugar syntax for the existential quantifier.\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  -- FOL existential elimination\n  --     \u2203x.P(x)   P(x) \u2192 Q\n  --  ------------------------\n  --             Q\n  \u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n  \u2203-elim (x , p) h = h x p\n\n  -- Some examples\n\n  -- The order of quantifiers of the same sort is irrelevant.\n  \u2203-ord : {P\u00b2 : D \u2192 D \u2192 Set} \u2192 (\u2203[ x ] \u2203[ y ] P\u00b2 x y) \u2192 (\u2203[ y ] \u2203[ x ] P\u00b2 x y)\n  \u2203-ord h = \u2203-elim h (\u03bb x h\u2081 \u2192 \u2203-elim h\u2081 (\u03bb y prf \u2192 y , x , prf))\n\n  -- A proof non-FOL valid\n  non-FOL : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n  non-FOL h = \u2203-elim h (\u03bb x _ \u2192 x)\n\n-----------------------------------------------------------------------------\n\nmodule \u2203\u2083 where\n  -- A different version from the FOL existential introduction\n  --      P(x)\n  --  ------------\n  --     \u2203x.P(x)\n\n  -- The existential quantifier type on D.\n  data \u2203 (P : D \u2192 Set) : Set where\n    \u2203-intro : ((x : D) \u2192 P x) \u2192 \u2203 P\n\n  -- Sugar syntax for the existential quantifier.\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  postulate d : D\n\n  -- FOL existential elimination.\n  -- NB. It is neccesary that D \u2260 \u2205.\n  \u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n  \u2203-elim (\u2203-intro h\u2081) h\u2082 = h\u2082 d (h\u2081 d)\n\n  -- Some examples\n\n  -- Impossible\n  -- thm : {P : D \u2192 Set} \u2192 \u2203[ x ] P x \u2192 \u2203[ y ] P y\n  -- thm h = \u2203-elim h (\u03bb x prf \u2192 {!!})\n","old_contents":"-----------------------------------------------------------------------------\n-- Existential elimination\n-----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 16 February 2012.\n\nmodule Existential where\n\n-----------------------------------------------------------------------------\n\npostulate\n  D : Set\n\nmodule \u2203\u2081 where\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 7 _,_\n\n  -- The existential quantifier type on D.\n  data \u2203 (P : D \u2192 Set) : Set where\n    _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n  -- FOL existential elimination\n  --     \u2203x.P(x)   P(x) \u2192 Q\n  --  ------------------------\n  --             Q\n  \u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n  \u2203-elim (x , p) h = h x p\n\n  -- Type theory existential elimination.\n  \u2203-proj\u2081 : \u2200 {P} \u2192 \u2203 P \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {P} \u2192 (p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n  \u2203-proj\u2082 (_ , Px) = Px\n\n-----------------------------------------------------------------------------\n\nmodule \u2203\u2082 where\n  -- A different version from the FOL existential introduction\n  --     P(x)\n  --  ------------\n  --     \u2203x.P(x)\n\n  -- The existential quantifier type on D.\n  data \u2203 (P : D \u2192 Set) : Set where\n    \u2203-intro : ((x : D) \u2192 P x) \u2192 \u2203 P\n\n  postulate d : D\n\n  -- FOL existential elimination.\n  -- NB. It is neccesary that D \u2260 \u2205.\n  \u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n  \u2203-elim (\u2203-intro h\u2081) h\u2082 = h\u2082 d (h\u2081 d)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6448a4f7d90d0941e9df46bfb68a3553176b9ee2","subject":"Desc@ICFP: update model, adding just Variable from the thing containing Values.","message":"Desc@ICFP: update model, adding just Variable from the thing containing Values.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             Var context ty ->\n             IMu closeTerm' ty\ndischarge {n} {c} ty variable = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      Var context ty ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con ((ESu EZe) , true )) \n              (vcons (nat , con ((ESu EZe) , su (su ze)) ) \n              (vcons (pair bool nat , con ((ESu EZe) , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con ((ESu (ESu (ESu EZe))) , (con ((ESu EZe) , (su ze)) , con ( EZe , (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , (fze , refl)) ,\n                             (con (EZe , (fsu fze , refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n-- Fix menu:\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),                        -- Val t\n                              (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                               Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             (Val ty + Var context ty) ->\n             IMu closeTerm' ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      (Val ty + Var context ty) ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"da7ed128a0220380c3109913af5f58760ad241d3","subject":"Improve commentary in Base.Denotation.Notation.","message":"Improve commentary in Base.Denotation.Notation.\n\nOld-commit-hash: f51d5958c73259a9120fc8a2ad20854bb12f8870\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Denotation\/Notation.agda","new_file":"Base\/Denotation\/Notation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Overloading \u27e6_\u27e7 notation\n--\n-- This module defines a general mechanism for overloading the\n-- \u27e6_\u27e7 notation, using Agda\u2019s instance arguments.\n------------------------------------------------------------------------\n\nmodule Base.Denotation.Notation where\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\nopen Meaning public\n  using (\u27e8_\u27e9\u27e6_\u27e7)\n","old_contents":"module Base.Denotation.Notation where\n\n-- OVERLOADING \u27e6_\u27e7 NOTATION\n--\n-- This module defines a general mechanism for overloading the\n-- \u27e6_\u27e7 notation, using Agda\u2019s instance arguments.\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\nopen Meaning public\n  using (\u27e8_\u27e9\u27e6_\u27e7)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"66a643a5678be6f131477803e19ed0cb52159556","subject":"use the standard terminology refl.","message":"use the standard terminology refl.\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Equality.agda","new_file":"agda\/Equality.agda","new_contents":"-- This is an implementation of the equality type for Sets. Agda's\n-- standard equality is more powerful. The main idea here is to\n-- illustrate the equality type.\nmodule Equality where\n\nopen import Level\n-- The equality of two elements of type A. The type a \u2261 b is a family\n-- of types which captures the statement of equality in A. Not all of\n-- the types are inhabited as in general not all elements are equal\n-- here. The only type that are inhabited are a \u2261 a by the element\n-- that we call definition. This is called refl (for reflection in\n-- agda).\ndata _\u2261_  {\u2113} {A : Set \u2113} : (a b : A) \u2192 Set \u2113 where\n  refl : {x : A} \u2192 x \u2261 x\n\n-- \u2261 is a symmetric relation.\nsym : \u2200{\u2113} {A : Set \u2113} {a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\nsym refl = refl\n\n-- \u2261 is a transitive relation.\ntrans : \u2200 {\u2113} {A : Set \u2113} {a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\ntrans pAB refl = pAB\n-- trans refl pBC = pBC  -- alternate proof of transitivity.\n\n-- Congruence. If we apply f to equals the result are also equal.\ncong : \u2200{\u2113\u2080 \u2113\u2081} {A : Set \u2113\u2080} {B : Set \u2113\u2081}\n       {a\u2080 a\u2081 : A} \u2192 (f : A \u2192 B) \u2192 a\u2080 \u2261 a\u2081 \u2192 f a\u2080 \u2261 f a\u2081\ncong f refl = refl\n\n-- Pretty way of doing equational reasoning.  If we want to prove a\u2080 \u2261\n-- b through an intermediate set of equations use this. The general\n-- form will look like.\n--\n-- begin a  \u2248 a\u2080 by p\u2080\n--          \u2248 a\u2081 by p\u2081\n--          ...\n--          \u2248 b  by p\n-- \u220e\n\nbegin : \u2200{\u2113} {A : Set \u2113} (a : A) \u2192  a \u2261 a\n_\u2248_by_ : \u2200{\u2113} {A : Set \u2113} {a b : A} \u2192 a \u2261 b \u2192 (c : A) \u2192 b \u2261 c \u2192 a \u2261 c\n_\u220e : \u2200{\u2113} {A : Set \u2113} (a : A) \u2192 A\n\nbegin a = refl\naEb \u2248 c by bEc = trans aEb bEc\nx \u220e = x\n\ninfixl 1 _\u2248_by_\ninfixl 1 _\u2261_\n","old_contents":"-- This is an implementation of the equality type for Sets. Agda's\n-- standard equality is more powerful. The main idea here is to\n-- illustrate the equality type.\nmodule Equality where\n\nopen import Level\n-- The equality of two elements of type A. The type a \u2261 b is a family\n-- of types which captures the statement of equality in A. Not all of\n-- the types are inhabited as in general not all elements are equal\n-- here. The only type that are inhabited are a \u2261 a by the element\n-- that we call definition. This is called refl (for reflection in\n-- agda).\ndata _\u2261_  {\u2113} {A : Set \u2113} : (a b : A) \u2192 Set \u2113 where\n  definition : {x : A} \u2192 x \u2261 x\n\n-- \u2261 is a symmetric relation.\nsym : \u2200{\u2113} {A : Set \u2113} {a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\nsym definition = definition\n\n-- \u2261 is a transitive relation.\ntrans : \u2200 {\u2113} {A : Set \u2113} {a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\ntrans pAB definition = pAB\n-- trans definition pBC = pBC  -- alternate proof of transitivity.\n\n-- Congruence. If we apply f to equals the result are also equal.\ncong : \u2200{\u2113\u2080 \u2113\u2081} {A : Set \u2113\u2080} {B : Set \u2113\u2081}\n       {a\u2080 a\u2081 : A} \u2192 (f : A \u2192 B) \u2192 a\u2080 \u2261 a\u2081 \u2192 f a\u2080 \u2261 f a\u2081\ncong f definition = definition\n\n-- Pretty way of doing equational reasoning.  If we want to prove a\u2080 \u2261\n-- b through an intermediate set of equations use this. The general\n-- form will look like.\n--\n-- begin a  \u2248 a\u2080 by p\u2080\n--          \u2248 a\u2081 by p\u2081\n--          ...\n--          \u2248 b  by p\n-- \u220e\n\nbegin : \u2200{\u2113} {A : Set \u2113} (a : A) \u2192  a \u2261 a\n_\u2248_by_ : \u2200{\u2113} {A : Set \u2113} {a b : A} \u2192 a \u2261 b \u2192 (c : A) \u2192 b \u2261 c \u2192 a \u2261 c\n_\u220e : \u2200{\u2113} {A : Set \u2113} (a : A) \u2192 A\n\nbegin a = definition\naEb \u2248 c by bEc = trans aEb bEc\nx \u220e = x\n\ninfixl 1 _\u2248_by_\ninfixl 1 _\u2261_\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"c16588fa8c31fbc05fa4fccc81543dfcc150709d","subject":"Desc model: iso2 implicit.","message":"Desc model: iso2 implicit.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c3bd14c398dacc413acb6e18cf83eca8483f5e2c","subject":"Defined the internal hom.","message":"Defined the internal hom.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\n{-\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n{-       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 (U \u2192 V) \u00d7 (Y \u2192 X) \u2192 U \u00d7 Y \u2192 Set\n\u22b8-cond \u03b1 \u03b2 (f , g) (u , y) = \u03b1 u (g y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (Y \u2192 X)) , (U \u00d7 Y) , \u22b8-cond \u03b1 \u03b2\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , p\u2083\n  where\n   h : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 T \u2192 Z)\n   h (h\u2081 , h\u2082) = (\u03bb w \u2192 g (h\u2081 (f w))) , (\u03bb t \u2192 F (h\u2082 (G t)))\n   H : \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n   H (w , t) = f w , G t\n   p\u2083 : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond \u03b1 \u03b2 u (H y) \u2192 \u22b8-cond \u03b3 \u03b4 (h u) y\n   p\u2083 {h\u2081 , h\u2082}{w , t} c c' = p\u2082 (c (p\u2081 c'))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d2a91e9f9db12e5bb647ea91f7f5cb8b8d3dc42d","subject":"Still in progress of getting Finable to work under +","message":"Still in progress of getting Finable to work under +\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-properties.agda","new_file":"sum-properties.agda","new_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Function.NP\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nrecord Finable {A : Set}(\u03bcA : SumProp A) : Set where\n  constructor mk\n  field\n    FinCard  : \u2115\n    toFin    : A \u2192 Fin FinCard\n    fromFin  : Fin FinCard \u2192 A\n    to-inj   : Injective toFin\n    from-inj : Injective fromFin\n    iso      : fromFin \u2218 toFin \u2257 id\n    sums-ok  : \u2200 f \u2192 sum \u03bcA f \u2261 sumFin FinCard (f \u2218 fromFin)\n\npostulate\n  Fin-Stab : \u2200 n \u2192 StableUnderInjection (\u03bcFin n)\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = mk 1 (\u03bb x \u2192 zero) (\u03bb x \u2192 _) (\u03bb x\u2081 \u2192 \u2261.refl) (\u03bb x\u2081 \u2192 help) (\u03bb x \u2192 \u2261.refl) (\u03bb f \u2192 ?)\n  where help : {x y : Fin 1} \u2192 x \u2261 y\n        help {zero} {zero} = \u2261.refl\n        help {zero} {suc ()}\n        help {suc ()}\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable {A}{B} \u03bcA \u03bcB finA finB = mk FinCard toFin fromFin to-inj from-inj iso sums-ok where\n    open import Data.Sum\n    open import Data.Empty\n\n    |A| = Finable.FinCard finA\n    |B| = Finable.FinCard finB\n\n    Fsuc-injective : \u2200 {n} {i j : Fin n} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\n    Fsuc-injective \u2261.refl = \u2261.refl\n    \n    inj\u2081-inj : \u2200 {A B : Set} {x y : A} \u2192 inj\u2081 {B = B} x \u2261 inj\u2081 y \u2192 x \u2261 y\n    inj\u2081-inj \u2261.refl = \u2261.refl\n\n    inj\u2082-inj : \u2200 {A B : Set} {x y : B} \u2192 inj\u2082 {A = A} x \u2261 inj\u2082 y \u2192 x \u2261 y\n    inj\u2082-inj \u2261.refl = \u2261.refl\n\n    fin\u2081 : \u2200 n {m} \u2192 Fin n \u2192 Fin (n + m)\n    fin\u2081 zero ()\n    fin\u2081 (suc n) zero = zero\n    fin\u2081 (suc n) (suc i) = suc (fin\u2081 n i)\n\n    fin\u2082 : \u2200 n {m} \u2192 Fin m \u2192 Fin (n + m)\n    fin\u2082 zero i = i\n    fin\u2082 (suc n) i = suc (fin\u2082 n i)\n\n    toFin' : \u2200 {n} {m} \u2192 Fin n \u228e Fin m \u2192 Fin (n + m)\n    toFin' {n} (inj\u2081 x) = fin\u2081 n x\n    toFin' {n} (inj\u2082 y) = fin\u2082 n y\n\n    fin\u2081-inj : \u2200 n {m} \u2192 Injective (fin\u2081 n {m})\n    fin\u2081-inj .(suc n) {m} {zero {n}} {zero} x\u2261y = \u2261.refl\n    fin\u2081-inj .(suc n) {m} {zero {n}} {suc y} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {zero} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {suc y} x\u2261y = \u2261.cong suc (fin\u2081-inj n (Fsuc-injective x\u2261y))\n\n    fin\u2082-inj : \u2200 n {m} \u2192 Injective (fin\u2082 n {m})\n    fin\u2082-inj zero eq = eq\n    fin\u2082-inj (suc n) eq = fin\u2082-inj n (Fsuc-injective eq)\n\n    fin\u2081\u2260fin\u2082 : \u2200 n {m} x y \u2192 (fin\u2081 n {m} x \u2261 fin\u2082 n y) \u2192 \u22a5\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) zero ()\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) (suc y) ()\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) zero eq = fin\u2081\u2260fin\u2082 n x zero (Fsuc-injective eq)\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) (suc y) eq = fin\u2081\u2260fin\u2082 n x (suc y) (Fsuc-injective eq)\n\n    fromFin' : \u2200 n {m} \u2192 Fin (n + m) \u2192 Fin n \u228e Fin m\n    fromFin' zero k = inj\u2082 k\n    fromFin' (suc n) zero = inj\u2081 zero\n    fromFin' (suc n) (suc k) with fromFin' n k\n    ... | inj\u2081 x = inj\u2081 (suc x)\n    ... | inj\u2082 y = inj\u2082 y\n\n    fromFin'-inj : \u2200 n {m} x y \u2192 fromFin' n {m} x \u2261 fromFin' n y \u2192 x \u2261 y\n    fromFin'-inj zero x y eq = inj\u2082-inj eq\n    fromFin'-inj (suc n) zero zero eq = \u2261.refl\n    fromFin'-inj (suc n) zero (suc y) eq = {!!}\n    fromFin'-inj (suc n) (suc x) zero eq = {!!}\n    fromFin'-inj (suc n) (suc x) (suc y) eq = {!fromFin'-inj n x y!}\n\n    fromFin'-inj\u2081 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2081 n x) \u2261 inj\u2081 x\n    fromFin'-inj\u2081 zero ()\n    fromFin'-inj\u2081 (suc n) zero = \u2261.refl\n    fromFin'-inj\u2081 (suc n) {m} (suc x) rewrite fromFin'-inj\u2081 n {m} x  = \u2261.refl\n\n    fromFin'-inj\u2082 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2082 n x) \u2261 inj\u2082 x\n    fromFin'-inj\u2082 zero x = \u2261.refl\n    fromFin'-inj\u2082 (suc n) x rewrite fromFin'-inj\u2082 n x = \u2261.refl\n\n    FinCard  : \u2115\n    FinCard = Finable.FinCard finA + Finable.FinCard finB\n\n    toFin    : A \u228e B \u2192 Fin FinCard\n    toFin (inj\u2081 x) = toFin' (inj\u2081 (Finable.toFin finA x))\n    toFin (inj\u2082 y) = toFin' {n = |A|} (inj\u2082 (Finable.toFin finB y))\n\n    fromFin  : Fin FinCard \u2192 A \u228e B\n    fromFin x+y with fromFin' |A| x+y\n    ... | inj\u2081 x = inj\u2081 (Finable.fromFin finA x)\n    ... | inj\u2082 y = inj\u2082 (Finable.fromFin finB y)\n\n    to-inj   : Injective toFin\n    to-inj {inj\u2081 x} {inj\u2081 x\u2081} tx\u2261ty = \u2261.cong inj\u2081 (Finable.to-inj finA (fin\u2081-inj |A| tx\u2261ty))\n    to-inj {inj\u2081 x} {inj\u2082 y} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ tx\u2261ty)\n    to-inj {inj\u2082 y} {inj\u2081 x} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ (\u2261.sym tx\u2261ty))\n    to-inj {inj\u2082 y} {inj\u2082 y\u2081} tx\u2261ty = \u2261.cong inj\u2082 (Finable.to-inj finB (fin\u2082-inj |A| tx\u2261ty))\n\n    from-inj : Injective fromFin\n    from-inj {x} {y} fx\u2261fy = {!!}\n\n    iso      : fromFin \u2218 toFin \u2257 id\n    iso (inj\u2081 x) rewrite fromFin'-inj\u2081 |A| {|B|} (Finable.toFin finA x) | Finable.iso finA x = \u2261.refl\n    iso (inj\u2082 y) rewrite fromFin'-inj\u2082 |A| {|B|} (Finable.toFin finB y) | Finable.iso finB y = \u2261.refl\n\n    fin-proof : \u2200 n m (f : Fin n \u228e Fin m \u2192 \u2115) \u2192 sumFin n (f \u2218 inj\u2081) + sumFin m (f \u2218 inj\u2082) \u2261 sumFin (n + m) (f \u2218 fromFin' n)\n    fin-proof zero    m f = \u2261.refl\n    fin-proof (suc n) m f = sumFin (suc n) (f \u2218 inj\u2081) + sumFin m (f \u2218 inj\u2082)\n                          \u2261\u27e8 \u2115\u00b0.+-assoc (f (inj\u2081 zero)) (sumFin n (f' f \u2218 inj\u2081))\n                               (sumFin m (f \u2218 inj\u2082)) \u27e9\n                            f (inj\u2081 zero) + (sumFin n (f' f \u2218 inj\u2081) + sumFin m (f \u2218 inj\u2082))\n                          \u2261\u27e8 \u2261.cong (_+_ (f (inj\u2081 zero))) (fin-proof n m (f' f)) \u27e9\n                            f (inj\u2081 zero) + (sumFin (n + m) (f' f \u2218 fromFin' n))\n                          \u2261\u27e8 \u2261.cong (\u03bb p \u2192 f (inj\u2081 zero) + vsum p)\n                               (tabulate\u2257 (\u03bb x \u2192 \u2261.sym (f'-proof n x f))) \u27e9\n                            sumFin (suc n + m) (f \u2218 fromFin' (suc n)) \n                          \u220e\n     where open \u2261.\u2261-Reasoning\n           open import Data.Vec renaming (sum to vsum)\n\n           tabulate\u2257 : \u2200 {n} {A : Set}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n           tabulate\u2257 {zero} eq = \u2261.refl\n           tabulate\u2257 {suc n\u2081} eq rewrite eq zero | tabulate\u2257 (eq \u2218 suc) = \u2261.refl\n\n           f' : \u2200 {n m} \u2192 (Fin (suc n) \u228e Fin m \u2192 \u2115) \u2192 Fin n \u228e Fin m \u2192 \u2115\n           f' f (inj\u2081 x) = f (inj\u2081 (suc x))\n           f' f (inj\u2082 x) = f (inj\u2082 x)\n\n           f'-proof : \u2200 n {m} x f \u2192 f (fromFin' (suc n) (suc x)) \u2261 f' {m = m} f (fromFin' n x)\n           f'-proof n x f with fromFin' n x\n           ... | inj\u2081 p = \u2261.refl\n           ... | inj\u2082 p = \u2261.refl\n\n    sums-ok  : \u2200 f \u2192 sum (\u03bcA +\u03bc \u03bcB) f \u2261 sumFin FinCard (f \u2218 fromFin)\n    sums-ok f =\n        sum (\u03bcA +\u03bc \u03bcB) f\n      \u2261\u27e8 \u2261.cong\u2082 _+_ (Finable.sums-ok finA (f \u2218 inj\u2081))\n           (Finable.sums-ok finB (f \u2218 inj\u2082)) \u27e9\n        sumFin (Finable.FinCard finA) (f \u2218 inj\u2081 \u2218 Finable.fromFin finA)\n      + sumFin (Finable.FinCard finB) (f \u2218 inj\u2082 \u2218 Finable.fromFin finB)\n      \u2261\u27e8 {!!} \u27e9\n        sumFin FinCard (f \u2218 fromFin)\n      \u220e\n      where open \u2261.\u2261-Reasoning\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA fin f p p-inj\n  = sum \u03bcA f\n  \u2261\u27e8 sums-ok f \u27e9\n    sumFin FinCard (f \u2218 fromFin)\n  \u2261\u27e8 Fin-Stab FinCard (f \u2218 fromFin) (toFin \u2218 p \u2218 fromFin) (from-inj \u2218 p-inj \u2218 to-inj) \u27e9\n    sumFin FinCard (f \u2218 fromFin \u2218 toFin \u2218 p \u2218 fromFin)\n  \u2261\u27e8 sum-ext (\u03bcFin FinCard) (\u03bb x \u2192 \u2261.cong f (iso (p (fromFin x)))) \u27e9\n    sumFin FinCard (f \u2218 p \u2218 fromFin)\n  \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 p)) \u27e9\n    sum \u03bcA (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open Finable fin\n\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Vec.NP\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFin n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = {!sumFin-spec n (f \u2218 p)!} -- {!sumFinSUI (suc n) f p p-inj!}\n\n","old_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Function.NP\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nrecord Finable {A : Set}(\u03bcA : SumProp A) : Set where\n  constructor mk\n  field\n    FinCard  : \u2115\n    toFin    : A \u2192 Fin (suc FinCard)\n    fromFin  : Fin (suc FinCard) \u2192 A\n    to-inj   : Injective toFin\n    from-inj : Injective fromFin\n    iso      : fromFin \u2218 toFin \u2257 id\n    sums-ok  : \u2200 f \u2192 sum \u03bcA f \u2261 sum (\u03bcFin FinCard) (f \u2218 fromFin)\n\npostulate\n  Fin-Stab : \u2200 n \u2192 StableUnderInjection (\u03bcFin n)\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = mk 0 (\u03bb x \u2192 zero) (\u03bb x \u2192 _) (\u03bb x\u2081 \u2192 \u2261.refl) (\u03bb x\u2081 \u2192 help) (\u03bb x \u2192 \u2261.refl) (\u03bb f \u2192 \u2261.refl) \n  where help : {x y : Fin 1} \u2192 x \u2261 y\n        help {zero} {zero} = \u2261.refl\n        help {zero} {suc ()}\n        help {suc ()}\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable {A}{B} \u03bcA \u03bcB finA finB = mk FinCard toFin fromFin to-inj from-inj iso sums-ok where\n    open import Data.Sum\n    open import Data.Empty\n\n    |A| = suc (Finable.FinCard finA)\n    |B| = suc (Finable.FinCard finB)\n\n    Fsuc-injective : \u2200 {n} {i j : Fin n} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\n    Fsuc-injective \u2261.refl = \u2261.refl\n    \n    inj\u2081-inj : \u2200 {A B : Set} {x y : A} \u2192 inj\u2081 {B = B} x \u2261 inj\u2081 y \u2192 x \u2261 y\n    inj\u2081-inj \u2261.refl = \u2261.refl\n\n    inj\u2082-inj : \u2200 {A B : Set} {x y : B} \u2192 inj\u2082 {A = A} x \u2261 inj\u2082 y \u2192 x \u2261 y\n    inj\u2082-inj \u2261.refl = \u2261.refl\n\n    fin\u2081 : \u2200 n {m} \u2192 Fin n \u2192 Fin (n + m)\n    fin\u2081 zero ()\n    fin\u2081 (suc n) zero = zero\n    fin\u2081 (suc n) (suc i) = suc (fin\u2081 n i)\n\n    fin\u2082 : \u2200 n {m} \u2192 Fin m \u2192 Fin (n + m)\n    fin\u2082 zero i = i\n    fin\u2082 (suc n) i = suc (fin\u2082 n i)\n\n    toFin' : \u2200 {n} {m} \u2192 Fin n \u228e Fin m \u2192 Fin (n + m)\n    toFin' {n} (inj\u2081 x) = fin\u2081 n x\n    toFin' {n} (inj\u2082 y) = fin\u2082 n y\n\n    fin\u2081-inj : \u2200 n {m} \u2192 Injective (fin\u2081 n {m})\n    fin\u2081-inj .(suc n) {m} {zero {n}} {zero} x\u2261y = \u2261.refl\n    fin\u2081-inj .(suc n) {m} {zero {n}} {suc y} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {zero} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {suc y} x\u2261y = \u2261.cong suc (fin\u2081-inj n (Fsuc-injective x\u2261y))\n\n    fin\u2082-inj : \u2200 n {m} \u2192 Injective (fin\u2082 n {m})\n    fin\u2082-inj zero eq = eq\n    fin\u2082-inj (suc n) eq = fin\u2082-inj n (Fsuc-injective eq)\n\n    fin\u2081\u2260fin\u2082 : \u2200 n {m} x y \u2192 (fin\u2081 n {m} x \u2261 fin\u2082 n y) \u2192 \u22a5\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) zero ()\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) (suc y) ()\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) zero eq = fin\u2081\u2260fin\u2082 n x zero (Fsuc-injective eq)\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) (suc y) eq = fin\u2081\u2260fin\u2082 n x (suc y) (Fsuc-injective eq)\n\n    fromFin' : \u2200 n {m} \u2192 Fin (n + m) \u2192 Fin n \u228e Fin m\n    fromFin' zero k = inj\u2082 k\n    fromFin' (suc n) zero = inj\u2081 zero\n    fromFin' (suc n) (suc k) with fromFin' n k\n    ... | inj\u2081 x = inj\u2081 (suc x)\n    ... | inj\u2082 y = inj\u2082 y\n\n    fromFin'-inj : \u2200 n {m} x y \u2192 fromFin' n {m} x \u2261 fromFin' n y \u2192 x \u2261 y\n    fromFin'-inj zero x y eq = inj\u2082-inj eq\n    fromFin'-inj (suc n) zero zero eq = \u2261.refl\n    fromFin'-inj (suc n) zero (suc y) eq = {!!}\n    fromFin'-inj (suc n) (suc x) zero eq = {!!}\n    fromFin'-inj (suc n) (suc x) (suc y) eq = {!fromFin'-inj n x y!}\n\n    fromFin'-inj\u2081 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2081 n x) \u2261 inj\u2081 x\n    fromFin'-inj\u2081 zero ()\n    fromFin'-inj\u2081 (suc n) zero = \u2261.refl\n    fromFin'-inj\u2081 (suc n) {m} (suc x) rewrite fromFin'-inj\u2081 n {m} x  = \u2261.refl\n\n    fromFin'-inj\u2082 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2082 n x) \u2261 inj\u2082 x\n    fromFin'-inj\u2082 zero x = \u2261.refl\n    fromFin'-inj\u2082 (suc n) x rewrite fromFin'-inj\u2082 n x = \u2261.refl\n\n    FinCard  : \u2115\n    FinCard = Finable.FinCard finA + suc (Finable.FinCard finB)\n\n    toFin    : A \u228e B \u2192 Fin (suc FinCard)\n    toFin (inj\u2081 x) = toFin' (inj\u2081 (Finable.toFin finA x))\n    toFin (inj\u2082 y) = toFin' {n = |A|} (inj\u2082 (Finable.toFin finB y))\n\n    fromFin  : Fin (suc FinCard) \u2192 A \u228e B\n    fromFin x+y with fromFin' |A| x+y\n    ... | inj\u2081 x = inj\u2081 (Finable.fromFin finA x)\n    ... | inj\u2082 y = inj\u2082 (Finable.fromFin finB y)\n\n    to-inj   : Injective toFin\n    to-inj {inj\u2081 x} {inj\u2081 x\u2081} tx\u2261ty = \u2261.cong inj\u2081 (Finable.to-inj finA (fin\u2081-inj |A| tx\u2261ty))\n    to-inj {inj\u2081 x} {inj\u2082 y} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ tx\u2261ty)\n    to-inj {inj\u2082 y} {inj\u2081 x} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ (\u2261.sym tx\u2261ty))\n    to-inj {inj\u2082 y} {inj\u2082 y\u2081} tx\u2261ty = \u2261.cong inj\u2082 (Finable.to-inj finB (fin\u2082-inj |A| tx\u2261ty))\n\n    from-inj : Injective fromFin\n    from-inj {x} {y} fx\u2261fy = {!!}\n\n    iso      : fromFin \u2218 toFin \u2257 id\n    iso (inj\u2081 x) rewrite fromFin'-inj\u2081 |A| {|B|} (Finable.toFin finA x) | Finable.iso finA x = \u2261.refl\n    iso (inj\u2082 y) rewrite fromFin'-inj\u2082 |A| {|B|} (Finable.toFin finB y) | Finable.iso finB y = \u2261.refl\n\n    sums-ok  : \u2200 f \u2192 sum (\u03bcA +\u03bc \u03bcB) f \u2261 sum (\u03bcFin FinCard) (f \u2218 fromFin)\n    sums-ok f =\n        sum (\u03bcA +\u03bc \u03bcB) f\n      \u2261\u27e8 \u2261.cong\u2082 _+_ (Finable.sums-ok finA (f \u2218 inj\u2081))\n           (Finable.sums-ok finB (f \u2218 inj\u2082)) \u27e9\n        sum (\u03bcFin (Finable.FinCard finA)) (f \u2218 inj\u2081 \u2218 Finable.fromFin finA)\n      + sum (\u03bcFin (Finable.FinCard finB)) (f \u2218 inj\u2082 \u2218 Finable.fromFin finB)\n      \u2261\u27e8 {!!} \u27e9\n        sum (\u03bcFin FinCard) (f \u2218 fromFin)\n      \u220e\n      where open \u2261.\u2261-Reasoning\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA fin f p p-inj\n  = sum \u03bcA f\n  \u2261\u27e8 sums-ok f \u27e9\n    sum (\u03bcFin FinCard) (f \u2218 fromFin)\n  \u2261\u27e8 Fin-Stab FinCard (f \u2218 fromFin) (toFin \u2218 p \u2218 fromFin) (from-inj \u2218 p-inj \u2218 to-inj) \u27e9\n    sum (\u03bcFin FinCard) (f \u2218 fromFin \u2218 toFin \u2218 p \u2218 fromFin)\n  \u2261\u27e8 sum-ext (\u03bcFin FinCard) (\u03bb x \u2192 \u2261.cong f (iso (p (fromFin x)))) \u27e9\n    sum (\u03bcFin FinCard) (f \u2218 p \u2218 fromFin)\n  \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 p)) \u27e9\n    sum \u03bcA (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open Finable fin\n\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Vec.NP\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFin n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b2b91ed59added92ff1db3f16b2ed90b51a8fbf","subject":"Desc stratified model: painful implicitification of IDescl constructors","message":"Desc stratified model: painful implicitification of IDescl constructors","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl I) -> IDescl I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6b984ce04f570d4e3d656e2a27f2ebd3ff9e3d3f","subject":"Added doc about the conversion rules.","message":"Added doc about the conversion rules.\n\nIgnore-this: c2bffa7d9902f27b0a6892bded45551a\n\ndarcs-hash:20111005211320-3bd4e-56b7013f3a6d7a01bf747a11474b16d00af46d6d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Base.agda","new_file":"src\/FOTC\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- The first order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-\nFOTC                                  Agda\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductive predicates                * Inductive families\n-}\n\nmodule FOTC.Base where\n\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfixr 8 _\u2237_  -- We add 3 to the fixities of the standard library.\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- The universal domain.\nopen import Common.Universe public\n\n-- The FOTC equality\n-- The FOTC equality is the propositional identity on the universal domain.\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.NonInductive public\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC correspond to the PCF terms.\n\n--   t ::= x    | t t    |\n--      | true  | false  | if t then t else t\n--      | 0     | succ t | pred t             | isZero t\n--      | []    | _\u2237_    | null               | head     | tail\n--      | loop\n\npostulate\n\n  -- FOTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n\n  -- FOTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n\n  -- FOTC application.\n  _\u00b7_ : D \u2192 D \u2192 D\n\n  -- FOTC lists.\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n  null : D \u2192 D\n  head : D \u2192 D\n  tail : D \u2192 D\n\n  -- FOTC looping programs.\n  loop : D\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- Note: The conversion relation _conv_ satifies (Aczel 1977. The\n-- strength of Martin-L\u00f6f's intuitionistic type theory with one\n-- universe, p. 8).\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as FOL non-logical\n-- axioms.\n\n-- Conversion rules for booleans.\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true then d\u2081 else d\u2082  \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true #-}\n{-# ATP axiom if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred zero     \u2261 zero\n  pred-S : \u2200 d \u2192 pred (succ d) \u2261 d\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for isZero.\npostulate\n  isZero-0 :       isZero zero     \u2261 true\n  isZero-S : \u2200 d \u2192 isZero (succ d) \u2261 false\n{-# ATP axiom isZero-0 #-}\n{-# ATP axiom isZero-S #-}\n\n-- Conversion rules for null.\npostulate\n  null-[] :          null []       \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\npostulate head-\u2237 : \u2200 x xs \u2192 head (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\npostulate tail-\u2237 : \u2200 x xs \u2192 tail (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n-- Conversion rule for loop.\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a FOL axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : \u2200 {d} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The first order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-\nFOTC                                  Agda\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductive predicates                * Inductive families\n-}\n\nmodule FOTC.Base where\n\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfixr 8 _\u2237_  -- We add 3 to the fixities of the standard library.\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- The universal domain.\nopen import Common.Universe public\n\n-- The FOTC equality\n-- The FOTC equality is the propositional identity on the universal domain.\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.NonInductive public\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC correspond to the PCF terms.\n\n--   t ::= x    | t t    |\n--      | true  | false  | if t then t else t\n--      | 0     | succ t | pred t             | isZero t\n--      | []    | _\u2237_    | null               | head     | tail\n--      | loop\n\npostulate\n\n  -- FOTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n\n  -- FOTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n\n  -- FOTC application.\n  _\u00b7_ : D \u2192 D \u2192 D\n\n  -- FOTC lists.\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n  null : D \u2192 D\n  head : D \u2192 D\n  tail : D \u2192 D\n\n  -- FOTC looping programs.\n  loop : D\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- Conversion rules for booleans.\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true then d\u2081 else d\u2082  \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true #-}\n{-# ATP axiom if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred zero     \u2261 zero\n  pred-S : \u2200 d \u2192 pred (succ d) \u2261 d\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for isZero.\npostulate\n  isZero-0 :       isZero zero     \u2261 true\n  isZero-S : \u2200 d \u2192 isZero (succ d) \u2261 false\n{-# ATP axiom isZero-0 #-}\n{-# ATP axiom isZero-S #-}\n\n-- Conversion rules for null.\npostulate\n  null-[] :          null []       \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\npostulate head-\u2237 : \u2200 x xs \u2192 head (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\npostulate tail-\u2237 : \u2200 x xs \u2192 tail (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n-- Conversion rule for loop.\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a FOL axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : \u2200 {d} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"22450b7d49eb76b754ec41272ab5f116d5c52d5c","subject":"A variable does not change in an honest environment that says so.","message":"A variable does not change in an honest environment that says so.\n\nOld-commit-hash: 58a997c1f5014000ba469de18ca3018c0d55d9fd\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-- Debug tool\nabsurd! : \u2200 {A B : Set} \u2192 B \u2192 A \u2192 A \u2192 B\nabsurd! b _ _ = b\n\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\n{-\n-- Reformulating stableness as a decidable relation\n-- Not sure if it is necessary or not.\ndata StableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  abide-this : \u2200 {\u03c4 \u0393} \u2192 {vars : Vars \u0393} \u2192 StableVar this (abide {\u03c4} vars)\n  abide-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (abide vars)\n  alter-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (alter vars)\n\ndata Stable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  stable-nat : \u2200 {\u0393 n vars} \u2192 Stable {nats} {\u0393} (nat n) vars\n  stable-bag : \u2200 {\u0393 b vars} \u2192 Stable {bags} {\u0393} (bag b) vars\n  stable-var : \u2200 {\u03c4 \u0393 x vars} \u2192\n               StableVar {\u03c4} {\u0393} x vars \u2192 Stable (var x) vars\n  stable-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n               Stable t (abide vars) \u2192 Stable (abs t) vars\n  -- TODO app add map diff union\n-}\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\nvolatility\u21d2identity :\n  \u2200 {\u03c4} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg WRT (expect-volatility {\u03c4}) \u2192 df \u2261 dg\n\nvolatility\u21d2identity {nats} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {bags} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {dg} df\u2248dg =\n  extensionality (\u03bb x \u2192 extensionality (\u03bb dx \u2192\n    volatility\u21d2identity {\u03c4\u2082} (df\u2248dg {x} {dx})))\n\ndf\u2248df : \u2200 {\u03c4} {df : \u27e6 \u0394-Type \u03c4 \u27e7} {args : Args \u03c4} \u2192 df \u2248 df WRT args\ndf\u2248df {nats} = refl\ndf\u2248df {bags} = refl\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {abide args} =\n  \u03bb {x} {dx} _ \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {alter args} =\n  \u03bb {x} {dx}   \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {\u03c4}{_} {args} {df} {dg} {\u03c1} df=dg\n  rewrite df=dg = df\u2248df {\u03c4} {\u27e6 dg \u27e7 \u03c1} {args}\n\n-- A variable does not change if its value is unchanging.\nstabilityVar : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stableVar x vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 deriveVar x \u27e7 \u03c1 \u2261 \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1\n\nstabilityVar this (alter vars) () (alter honesty)\nstabilityVar this (abide vars) refl (abide proof honesty) = proof\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (abide vars) truth (abide _ honesty) =\n  stabilityVar x vars (trans eq2 truth) honesty\n  where\n    eq2 : stableVar x vars \u2261 stableVar (that {\u03c4} {\u03c4\u2032} {\u0393} x) (abide vars)\n    eq2 = refl\n\nstabilityVar (that x) (alter vars) truth honesty = {!ditto!}\n\n-- A term does not change if its free variables are unchanging.\nstability : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stable t vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\n\nstability (nat n) vars truth {\u03c1} _ = refl\n\nstability (bag b) vars truth {\u03c1} _ = b++\u2205=b\n\nstability (var x) vars truth {\u03c1} honesty =\n  {!!}\n  where\n    eq1  : stableVar x vars \u2261 stable (var x) vars\n    eq1  = refl\n    tVar : stableVar x vars \u2261 true\n    tVar = trans eq1 truth\n\nstability (abs t) vars truth {\u03c1} honesty = {!!}\nstability (app t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (add t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (map t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (diff t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (union t t\u2081) vars truth {\u03c1} honesty = {!!}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\n-- If both the environment and the future arguments are honest\n-- about nil changes, then the optimized derivation delivers\n-- the same result as the original derivation.\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy (app f s) args vars \u03c1 honesty\n  with stable s vars | inspect (stable s) vars\n... | true  | [ truth ] = {!!}\n--  absurd! {!!} (stability s vars truth {\u03c1}) {!!}\n\n... | false | [ falsehood ] = {!!}\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) (abide args) vars \u03c1 honesty =\n  \u03bb {x} {dx} x\u2295dx=x \u2192 honesty-is-the-best-policy\n    t args (abide vars) (dx \u2022 x \u2022 \u03c1) (abide x\u2295dx=x honesty)\n\nhonesty-is-the-best-policy (abs t) (alter args) vars \u03c1 honesty =\n  \u03bb {x} {dx} \u2192 honesty-is-the-best-policy\n    t args (alter vars) (dx \u2022 x \u2022 \u03c1) (alter honesty)\n\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\n{-\n-- Reformulating stableness as a decidable relation\n-- Not sure if it is necessary or not.\ndata StableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  abide-this : \u2200 {\u03c4 \u0393} \u2192 {vars : Vars \u0393} \u2192 StableVar this (abide {\u03c4} vars)\n  abide-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (abide vars)\n  alter-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (alter vars)\n\ndata Stable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  stable-nat : \u2200 {\u0393 n vars} \u2192 Stable {nats} {\u0393} (nat n) vars\n  stable-bag : \u2200 {\u0393 b vars} \u2192 Stable {bags} {\u0393} (bag b) vars\n  stable-var : \u2200 {\u03c4 \u0393 x vars} \u2192\n               StableVar {\u03c4} {\u0393} x vars \u2192 Stable (var x) vars\n  stable-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n               Stable t (abide vars) \u2192 Stable (abs t) vars\n  -- TODO app add map diff union\n-}\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\ndf\u2248df : \u2200 {\u03c4} {df : \u27e6 \u0394-Type \u03c4 \u27e7} {args : Args \u03c4} \u2192 df \u2248 df WRT args\ndf\u2248df {nats} = refl\ndf\u2248df {bags} = refl\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {abide args} =\n  \u03bb {x} {dx} _ \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {alter args} =\n  \u03bb {x} {dx}   \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {nats}{_}{_} {df} {dg} {\u03c1} df=dg =\n  cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) df=dg\ntoo-close {bags}{_}{_} {df} {dg} {\u03c1} df=dg =\n  cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) df=dg\ntoo-close {\u03c4\u2081 \u21d2 \u03c4\u2082}{_} {args} {df} {dg} {\u03c1} df=dg\n  rewrite df=dg = df\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u27e6 dg \u27e7 \u03c1} {args}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\n-- A term does not change if its free variables are unchanging.\nstability : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  T (stable t vars) \u2192\n  \u2200 {A : Set} \u2192 A\n-- Conclusion is absurdity for now\n-- to test usability inside case t-app.\n-- Using T (stable t vars) as a condition is not enough.\nstability t vars truth = {!!}\n\nhonesty-is-the-best-policy (app f s) args vars \u03c1 honesty\n  with stable s vars\n... | true = stability s vars {!tt!}\n... | false = {!!}\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) (abide args) vars \u03c1 honesty =\n  \u03bb {x} {dx} x\u2295dx=x \u2192 honesty-is-the-best-policy\n    t args (abide vars) (dx \u2022 x \u2022 \u03c1) (abide x\u2295dx=x honesty)\n\nhonesty-is-the-best-policy (abs t) (alter args) vars \u03c1 honesty =\n  \u03bb {x} {dx} \u2192 honesty-is-the-best-policy\n    t args (alter vars) (dx \u2022 x \u2022 \u03c1) (alter honesty)\n\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5303d7901af55944be2ec4695ee17bbd01624291","subject":"Document Theorem.CongApp regularly.","message":"Document Theorem.CongApp regularly.\n\nOld-commit-hash: f16f3e3d0b8e59230bd86a0c223af263a8db75e8\n","repos":"inc-lc\/ilc-agda","old_file":"Theorem\/CongApp.agda","new_file":"Theorem\/CongApp.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Congruence of application.\n--\n-- If f \u2261 g and x \u2261 y, then (f x) \u2261 (g y).\n------------------------------------------------------------------------\n\nmodule Theorem.CongApp where\n\nopen import Relation.Binary.PropositionalEquality public\n\ninfixl 0 _\u27e8$\u27e9_\n\n_\u27e8$\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b}\n  {f g : A \u2192 B} {x y : A} \u2192\n  f \u2261 g \u2192 x \u2261 y \u2192 f x \u2261 g y\n\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n","old_contents":"module Theorem.CongApp where\n\n-- Theorem CongApp.\n-- If f \u2261 g and x \u2261 y, then (f x) \u2261 (g y).\n\nopen import Relation.Binary.PropositionalEquality public\n\ninfixl 0 _\u27e8$\u27e9_\n\n_\u27e8$\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b}\n  {f g : A \u2192 B} {x y : A} \u2192\n  f \u2261 g \u2192 x \u2261 y \u2192 f x \u2261 g y\n\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"45091fb39397b33a08cf731e29b25b603d1c525b","subject":"Removed unnecessary implicit argument.","message":"Removed unnecessary implicit argument.\n\nIgnore-this: d00e8c938cb9a7fa50eccab6b862327b\n\ndarcs-hash:20120314030126-3bd4e-92953574fef902c48980a129990a5a2edd427c8e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Inductive\/Properties.agda","new_file":"src\/PA\/Inductive\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Inductive PA properties commons to the interactive and automatic proofs\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Inductive.Properties where\n\nopen import PA.Inductive.Base\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nsucc-cong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsucc-cong refl = refl\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n------------------------------------------------------------------------------\n-- Peano's third axiom.\nP\u2083 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\nP\u2083 refl = refl\n\n-- Peano's fourth axiom.\nP\u2084 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\nP\u2084 ()\n\nx\u2262Sx : \u2200 {n} \u2192 \u00ac (n \u2261 succ n)\nx\u2262Sx {zero} ()\nx\u2262Sx {succ n} h = x\u2262Sx (P\u2083 h)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = refl\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.n+0\u2261n).\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity zero     = refl\n+-rightIdentity (succ n) = succ-cong (+-rightIdentity n)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.+-assoc).\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zero     _ _ = refl\n+-assoc (succ m) n o = succ-cong (+-assoc m n o)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.m+1+n\u22611+m+n).\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] zero     _ = refl\nx+Sy\u2261S[x+y] (succ m) n = succ-cong (x+Sy\u2261S[x+y] m n)\n","old_contents":"------------------------------------------------------------------------------\n-- Inductive PA properties commons to the interactive and automatic proofs\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Inductive.Properties where\n\nopen import PA.Inductive.Base\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nsucc-cong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsucc-cong = cong succ\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong h = cong\u2082 _+_ h refl\n\n-- TODO: It is strictly necessary the implicit argument m?\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong {m} h = cong\u2082 _+_ {m} refl h\n\n------------------------------------------------------------------------------\n-- Peano's third axiom.\nP\u2083 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\nP\u2083 refl = refl\n\n-- Peano's fourth axiom.\nP\u2084 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\nP\u2084 ()\n\nx\u2262Sx : \u2200 {n} \u2192 \u00ac (n \u2261 succ n)\nx\u2262Sx {zero} ()\nx\u2262Sx {succ n} h = x\u2262Sx (P\u2083 h)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = refl\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.n+0\u2261n).\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity zero     = refl\n+-rightIdentity (succ n) = succ-cong (+-rightIdentity n)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.+-assoc).\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zero     _ _ = refl\n+-assoc (succ m) n o = succ-cong (+-assoc m n o)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.m+1+n\u22611+m+n).\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] zero     _ = refl\nx+Sy\u2261S[x+y] (succ m) n = succ-cong (x+Sy\u2261S[x+y] m n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dfc3d8fd4f8a36bc9eac388bded7b5a7635e7af8","subject":"IDescTT model: implement the Vec and Fin examples","message":"IDescTT model: implement the Vec and Fin examples","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f184f39c2cc18ca063517afb7b4501762acaf24e","subject":"various","message":"various\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e M P) (\u03a0\u1d3e M Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 source-of (Com.P c m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Tele : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n{-\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = end\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server = dual \u2218 Client\n\n    Server' : \u2115 \u2192 Proto\n    Server' zero    = end\n    Server' (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server' n\n\n    Server\u2261Server' : \u2200 n \u2192 Server n \u2261\u1d3e Server' n\n    Server\u2261Server' zero    = end\n    Server\u2261Server' (suc n) = com refl Query (\u03bb m \u2192 com refl (Resp m) (\u03bb m' \u2192 Server\u2261Server' n))\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6a83206fb6f2245d31f9953821a63d85003599a9","subject":"progress towards lemmas after restructuring TA hole rules","message":"progress towards lemmas after restructuring TA hole rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub {\u0393 = \u0393} x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!}\n  lem-idsub x d () | None\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole {!!} {!!}\n  lem-weaken\u03941 (TANEHole D x y) = TANEHole {!!} (lem-weaken\u03941 x) {!!}\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4}  D = tr (\u03bb x \u2192 x , \u0393 \u22a2 d :: \u03c4) (funext (\u03bb x \u2192 {!!})) D\n\n  lem-weaken\u0394single : \u2200{\u0394 \u0393 d \u03c4 u \u0393' \u03c4'} \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u0394 ,, u ::[ \u0393' ] \u03c4') , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u0394single = {!!}\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth ESEHole = TAEHole {!!} lem-idsub\n    typed-expansion-synth (ESNEHole ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {!!} (lem-weaken\u0394single ih1) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana EAEHole = TCRefl , TAEHole {!!} lem-idsub\n    typed-expansion-ana (EANEHole x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {!!} (lem-weaken\u0394single ih1) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub {\u0393 = \u0393} x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!}\n  lem-idsub x d () | None\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x) = {!TAEHole!}\n  lem-weaken\u03941 (TANEHole D x) = {!!}\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-##eq : {\u03941 \u03942 : hctx} (n : Nat) \u2192 \u03941 ## \u03942 \u2192 \u03941 n == (\u03941 \u222a \u03942) n\n  lem-##eq {\u03941} {\u03942} n apart with \u03941 n\n  lem-##eq n apart | Some x = refl\n  lem-##eq {\u03941} {\u03942} n apart | None with \u03942 n\n  lem-##eq n (\u03c01 , \u03c02) | None | Some x = {!!}\n  lem-##eq n apart | None | None = refl\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4}  D = tr (\u03bb x \u2192 x , \u0393 \u22a2 d :: \u03c4) (funext (\u03bb x \u2192 {!!})) D\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth ESEHole = TAEHole lem-idsub\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana EAEHole = TCRefl , TAEHole lem-idsub\n    typed-expansion-ana (EANEHole x) = TCRefl , TANEHole (typed-expansion-synth x) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"53305597f0881c1001550f6c966e1f87162dcd74","subject":"pisearch: search _\u228e_ is \u03a3","message":"pisearch: search _\u228e_ is \u03a3\n","repos":"crypto-agda\/crypto-agda","old_file":"pisearch.agda","new_file":"pisearch.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function.NP\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nData : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nData sA B = sA _\u00d7_ B\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Data sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Data sA B\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Data s _) _,_\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ toFunA toFunB d = uncurry (toFunB \u2218 toFunA d)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ fromFunA fromFunB f = fromFunA (fromFunB \u2218 curry f)\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nToFrom : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nToFrom {A} sA toFunA fromFunA = \u2200 {B} (f : \u03a0 A B) x \u2192 toFunA (fromFunA f) x \u2261 f x\n\nFromTo : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nFromTo sA toFunA fromFunA = \u2200 {B} (d : Data sA B) \u2192 fromFunA (toFunA d) \u2261 d\n\nmodule \u03a3-props {A} {B : A \u2192 \u2605}\n                (\u03bcA : Searchable A) (\u03bcB : \u2200 {x} \u2192 Searchable (B x)) where\n    sA = search \u03bcA\n    sB : \u2200 {x} \u2192 Search (B x)\n    sB {x} = search (\u03bcB {x})\n    fromFunA : FromFun sA\n    fromFunA = fromFun-searchInd (search-ind \u03bcA)\n    fromFunB : \u2200 {x} \u2192 FromFun (sB {x})\n    fromFunB {x} = fromFun-searchInd (search-ind (\u03bcB {x}))\n    module ToFrom\n             (toFunA : ToFun sA)\n             (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n             (toFromA : ToFrom sA toFunA fromFunA)\n             (toFromB : \u2200 {x} \u2192 ToFrom (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : ToFrom (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 f (x , y) rewrite toFromA (fromFunB \u2218 curry f) x = toFromB (curry f x) y\n\n    {- we need a search-ind-ext...\n    module FromTo\n                 (toFunA : ToFun sA)\n                 (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n                 (toFromA : FromTo sA toFunA fromFunA)\n                 (toFromB : \u2200 {x} \u2192 FromTo (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : FromTo (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 t = {!toFromA!} -- {!(\u03bb x \u2192 toFromB (toFunA t x))!}\n    -}\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB toFunA toFunB = toFun-\u03a3 sA sB toFunA toFunB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB fromFunA fromFunB = fromFun-\u03a3 sA sB fromFunA fromFunB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB toFunA toFunB (x , y) = [ toFunA x , toFunB y ]\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB fromFunA fromFunB f = fromFunA (f \u2218 inj\u2081) , fromFunB (f \u2218 inj\u2082)\n\nPoint : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nPoint sA B = sA _\u228e_ B\n\nToPair : \u2200 {A} (sA : Search A) \u2192 \u2605\nToPair {A} sA = \u2200 {B} \u2192 Point sA B \u2192 \u03a3 A B\n\nFromPair : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromPair {A} sA = \u2200 {B} \u2192 \u03a3 A B \u2192 Point sA B\n\ntoPair-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToPair sA\ntoPair-searchInd indA = indA ToPair (\u03bb P0 P1 \u2192 [ P0 , P1 ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n","old_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function.NP\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nTree : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nTree sA B = sA _\u00d7_ B\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Tree sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Tree sA B\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Tree s _) _,_\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ toFunA toFunB t = uncurry (toFunB \u2218 toFunA t)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ fromFunA fromFunB f = fromFunA (fromFunB \u2218 curry f)\n\nopen import Relation.Binary.PropositionalEquality\nToFrom : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nToFrom {A} sA toFunA fromFunA = \u2200 {B} (f : \u03a0 A B) x \u2192 toFunA (fromFunA f) x \u2261 f x\n\nFromTo : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nFromTo sA toFunA fromFunA = \u2200 {B} (t : Tree sA B) \u2192 fromFunA (toFunA t) \u2261 t\n\nmodule \u03a3-props {A} {B : A \u2192 \u2605}\n                (\u03bcA : Searchable A) (\u03bcB : \u2200 {x} \u2192 Searchable (B x)) where\n    sA = search \u03bcA\n    sB : \u2200 {x} \u2192 Search (B x)\n    sB {x} = search (\u03bcB {x})\n    fromFunA : FromFun sA\n    fromFunA = fromFun-searchInd (search-ind \u03bcA)\n    fromFunB : \u2200 {x} \u2192 FromFun (sB {x})\n    fromFunB {x} = fromFun-searchInd (search-ind (\u03bcB {x}))\n    module ToFrom\n             (toFunA : ToFun sA)\n             (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n             (toFromA : ToFrom sA toFunA fromFunA)\n             (toFromB : \u2200 {x} \u2192 ToFrom (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : ToFrom (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 f (x , y) rewrite toFromA (fromFunB \u2218 curry f) x = toFromB (curry f x) y\n\n    {- we need a search-ind-ext...\n    module FromTo\n                 (toFunA : ToFun sA)\n                 (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n                 (toFromA : FromTo sA toFunA fromFunA)\n                 (toFromB : \u2200 {x} \u2192 FromTo (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : FromTo (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 t = {!toFromA!} -- {!(\u03bb x \u2192 toFromB (toFunA t x))!}\n    -}\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB toFunA toFunB = toFun-\u03a3 sA sB toFunA toFunB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB fromFunA fromFunB = fromFun-\u03a3 sA sB fromFunA fromFunB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2081 x) = toFunA t x\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2082 x) = toFunB u x\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB fromFunA fromFunB f = fromFunA (f \u2218 inj\u2081) , fromFunB (f \u2218 inj\u2082)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5cc39d986b6ffa05d7fd642b7b5dae4e8d18c155","subject":"this proves expandability, except for three premises about the disjointness of Deltas which might not be relevant","message":"this proves expandability, except for three premises about the disjointness of Deltas which might not be relevant\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = _ , _ , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | _ , _ , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-ana wt2\n    ... | d2 , \u03942 , \u03c42 , D2 with expandability-ana (ASubsume wt1 TCHole1)\n    ... | d1 , \u03941 , \u03c41 , D1 =  _ , _ , ESAp1 {!!} wt1 D1 D2\n    expandability-synth (SAp wt1 (MAArr {\u03c41 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} D1 D2 neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c41  , D2 | Inl refl = _ , _ , ESAp3 {!!} D1 D2\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081)\n      with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )              | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _  = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  -- lemma : \u2200 { \u0393  e \u03c4} \u2192\n  --         \u0393 \u22a2 e => \u2987\u2988 \u2192\n  --         \u0393 \u22a2 e <= \u03c4 ==> \u2987\u2988\n  -- lemma wt = ASubsume wt TCHole1\n\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = _ , _ , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | _ , _ , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-ana wt2\n    ... | d2 , \u03942 , \u03c42 , D2 with expandability-ana (ASubsume wt1 TCHole1)\n    ... | d1 , \u03941 , \u03c41 , D1 =  _ , _ , ESAp1 {!!} wt1 D1 D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081)\n      with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )              | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _  = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"47c8540189b6bea5f1a13bfc28f05bc9a52bf47c","subject":"[ FOL ] Added documentation.","message":"[ FOL ] Added documentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/Common\/FOL\/FOL.agda","new_file":"src\/fot\/Common\/FOL\/FOL.agda","new_contents":"------------------------------------------------------------------------------\n-- First-order logic (without equality)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- This module is re-exported by the \"base\" modules whose theories are\n-- defined on first-order logic (without equality).\n\n-- The logical connectives are hard-coded in our translation, i.e. the\n-- symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192 and \u2194 must be used.\n--\n-- N.B. For the implication we use the Agda function type.\n--\n-- N.B For the universal quantifier we use the Agda (dependent)\n-- function type.\n\nmodule Common.FOL.FOL where\n\ninfixr 4 _,_\ninfix  3 \u00ac_\ninfixr 2 _\u2227_\ninfix  2 \u2203\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\n\n----------------------------------------------------------------------------\n-- The universe of discourse\/universal domain.\npostulate D : Set\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : \u2200 {A B} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (a , _) = a\n\n\u2227-proj\u2082 : \u2200 {A B} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (_ , b) = b\n\n-----------------------------------------------------------------------------\n-- The disjunction data type.\n\n-- It is not necessary to add the data constructors inj\u2081 and inj\u2082 as\n-- axioms because the ATPs implement them.\ndata _\u2228_ (A B : Set) : Set where\n  inj\u2081 : A \u2192 A \u2228 B\n  inj\u2082 : B \u2192 A \u2228 B\n\n-- It is not strictly necessary define the eliminator `case` because\n-- the ATPs implement it. For the same reason, it is not necessary to\n-- add it as a (general\/local) hint.\ncase : \u2200 {A B} \u2192 {C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\ncase f g (inj\u2081 a) = f a\ncase f g (inj\u2082 b) = g b\n\n------------------------------------------------------------------------------\n-- The empty type.\ndata \u22a5 : Set where\n\n\u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n\u22a5-elim ()\n\n------------------------------------------------------------------------------\n-- The unit type.\n-- N.B. The name of this type is \"\\top\", not T.\ndata \u22a4 : Set where\n  tt : \u22a4\n\n------------------------------------------------------------------------------\n-- Negation.\n\n-- The underscore allows to write for example '\u00ac \u00ac A' instead of '\u00ac (\u00ac A)'.\n\n-- We do not add a definition because: i) the definition of negation\n-- is not a FOL-definition and ii) the translation of the neagation is\n-- hard-coded in Apia.\n\n\u00ac_ : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n\n------------------------------------------------------------------------------\n-- Biconditional.\n\n-- We do not add a definition because: i) the definition of the\n-- biconditional is not a FOL-definition, ii) the translation of the\n-- biconditional is hard-coded in Apia.\n\n_\u2194_ : Set \u2192 Set \u2192 Set\nA \u2194 B = (A \u2192 B) \u2227 (B \u2192 A)\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n-- 2012-03-05: We avoid to use the existential elimination or the\n-- existential projections because we use pattern matching (and the\n-- Agda's with constructor).\n\n-- The existential elimination.\n--\n-- NB. We do not use the usual type theory elimination with two\n-- projections because we are working in first-order logic where we do\n-- not need extract a witness from an existence proof.\n-- \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n-- \u2203-elim (_ , Ax) h = h Ax\n\n-- The existential proyections.\n-- \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n-- \u2203-proj\u2081 (x , _) = x\n\n-- \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n-- \u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Properties\n\n\u2192-trans : {A B C : Set} \u2192 (A \u2192 B) \u2192 (B \u2192 C) \u2192 A \u2192 C\n\u2192-trans f g a = g (f a)\n","old_contents":"------------------------------------------------------------------------------\n-- First-order logic (without equality)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- This module is re-exported by the \"base\" modules whose theories are\n-- defined on first-order logic (without equality).\n\n-- The logical connectives are hard-coded in our translation, i.e. the\n-- symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192 and \u2194 must be used.\n--\n-- N.B. For the implication we use the Agda function type.\n--\n-- N.B For the universal quantifier we use the Agda (dependent)\n-- function type.\n\nmodule Common.FOL.FOL where\n\ninfixr 4 _,_\ninfix  3 \u00ac_\ninfixr 2 _\u2227_\ninfix  2 \u2203\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\n\n----------------------------------------------------------------------------\n-- The universe of discourse\/universal domain.\npostulate D : Set\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : \u2200 {A B} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (a , _) = a\n\n\u2227-proj\u2082 : \u2200 {A B} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (_ , b) = b\n\n-----------------------------------------------------------------------------\n-- The disjunction data type.\n\n-- It is not necessary to add the data constructors inj\u2081 and inj\u2082 as\n-- axioms because the ATPs implement them.\ndata _\u2228_ (A B : Set) : Set where\n  inj\u2081 : A \u2192 A \u2228 B\n  inj\u2082 : B \u2192 A \u2228 B\n\n-- It is not strictly necessary define the eliminator `case` because\n-- the ATPs implement it. For the same reason, it is not necessary to\n-- add it as a (general\/local) hint.\ncase : \u2200 {A B} \u2192 {C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\ncase f g (inj\u2081 a) = f a\ncase f g (inj\u2082 b) = g b\n\n------------------------------------------------------------------------------\n-- The empty type.\ndata \u22a5 : Set where\n\n\u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n\u22a5-elim ()\n\n------------------------------------------------------------------------------\n-- The unit type.\n-- N.B. The name of this type is \"\\top\", not T.\ndata \u22a4 : Set where\n  tt : \u22a4\n\n------------------------------------------------------------------------------\n-- Negation.\n-- The underscore allows to write for example '\u00ac \u00ac A' instead of '\u00ac (\u00ac A)'.\n\u00ac_ : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n\n------------------------------------------------------------------------------\n-- Biconditional.\n_\u2194_ : Set \u2192 Set \u2192 Set\nA \u2194 B = (A \u2192 B) \u2227 (B \u2192 A)\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n-- 2012-03-05: We avoid to use the existential elimination or the\n-- existential projections because we use pattern matching (and the\n-- Agda's with constructor).\n\n-- The existential elimination.\n--\n-- NB. We do not use the usual type theory elimination with two\n-- projections because we are working in first-order logic where we do\n-- not need extract a witness from an existence proof.\n-- \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n-- \u2203-elim (_ , Ax) h = h Ax\n\n-- The existential proyections.\n-- \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n-- \u2203-proj\u2081 (x , _) = x\n\n-- \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n-- \u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Properties\n\n\u2192-trans : {A B C : Set} \u2192 (A \u2192 B) \u2192 (B \u2192 C) \u2192 A \u2192 C\n\u2192-trans f g a = g (f a)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bf33562cdea58a03e94ec0693c42335249daaf5a","subject":"Lift change algebras to Data.All.","message":"Lift change algebras to Data.All.\n\nOld-commit-hash: 2d6c57c5ca13b3f3e97bf60d2d09f4b1f0aaf56b\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function changes\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n-- List changes\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d _ \u208e px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\u208d _ \u208e\n        ; update = update-all\n        ; diff = diff-all\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          }\n        }\n      }\n","old_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function changes\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b87eb75ad51669de831d389ef1a78b415b31a18d","subject":"Dist: add the `symIter` from Conor and some lemmas","message":"Dist: add the `symIter` from Conor and some lemmas\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Distance.agda","new_file":"lib\/Data\/Nat\/Distance.agda","new_contents":"open import Function.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Properties.Simple\nopen import Data.Product.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Binary\nopen import Relation.Nullary\n\nmodule Data.Nat.Distance where\n\n{-\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n-}\n\n-- `symIter` from Conor McBride\n-- Since symIter is essentially `dist` with callbacks, I\n-- include it here.\nmodule Generic {a}{A : Set a}(z : \u2115 \u2192 A)(s : A \u2192 A) where\n  dist : \u2115 \u2192 \u2115 \u2192 A\n  dist zero    y       = z y\n  dist x       zero    = z x\n  dist (suc x) (suc y) = s (dist x y)\n\n  dist-comm : \u2200 x y \u2192 dist x y \u2261 dist y x\n  dist-comm zero    zero    = idp\n  dist-comm zero    (suc y) = idp\n  dist-comm (suc x) zero    = idp\n  dist-comm (suc x) (suc y) = ap s (dist-comm x y)\n\n  dist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 z x\n  dist-0\u2261id _ = idp\n\n  dist-0\u2261id\u2032 : \u2200 x \u2192 dist x 0 \u2261 z x\n  dist-0\u2261id\u2032 zero    = idp\n  dist-0\u2261id\u2032 (suc _) = idp\n\nopen Generic id id public\n\nmodule Add where\n  open Generic id (suc \u2218 suc)\n    public\n    renaming ( dist to _+'_\n             ; dist-0\u2261id to +'0-identity\n             ; dist-0\u2261id\u2032 to 0+'-identity\n             ; dist-comm to +'-comm\n             )\n\n  +-spec : \u2200 x y \u2192 x +' y \u2261 x + y\n  +-spec zero    _       = idp\n  +-spec (suc x) zero    = ap suc (\u2115\u00b0.+-comm 0 x)\n  +-spec (suc x) (suc y) = ap suc (ap suc (+-spec x y) \u2219 ! +-suc x y)\n\nmodule AddMult where\n  open Generic (\u03bb x \u2192 x , 0)\n               (\u03bb { (s , p) \u2192 (2 + s , 1 + s + p) })\n    public\n    renaming ( dist to _+*_\n             ; dist-0\u2261id to +*0-identity\n             ; dist-0\u2261id\u2032 to 0+*-identity\n             ; dist-comm to +*-comm\n             )\n\n  _+'_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  x +' y = fst (x +* y)\n\n  _*'_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  x *' y = snd (x +* y)\n\n  +*-spec : \u2200 x y \u2192 (x +' y \u2261 x + y) \u00d7 (x *' y \u2261 x * y)\n  +*-spec zero    _       = idp , idp\n  +*-spec (suc x) zero    = ap suc (\u2115\u00b0.+-comm 0 x) , \u2115\u00b0.*-comm 0 x\n  +*-spec (suc x) (suc y) = ap suc p+ , ap suc p*\n    where p = +*-spec x y\n          p+ = ap suc (fst p) \u2219 ! +-suc x y\n          p* = ap\u2082 _+_ (fst p) (snd p)\n             \u2219 +-assoc x y (x * y)\n             \u2219 ap (\u03bb z \u2192 x + (y + z)) (\u2115\u00b0.*-comm x y)\n             \u2219 ! +-assoc-comm y x (y * x)\n             \u2219 ap (_+_ y) (\u2115\u00b0.*-comm (suc y) x)\n\nmodule Min where\n  open Generic (const zero) suc\n    public\n    renaming ( dist to _\u2293'_\n             ; dist-0\u2261id to \u2293'0-zero\n             ; dist-0\u2261id\u2032 to 0\u2293'-zero\n             ; dist-comm to \u2293'-comm\n             )\n\n  \u2293'-spec : \u2200 x y \u2192 x \u2293' y \u2261 x \u2293 y\n  \u2293'-spec zero    zero    = idp\n  \u2293'-spec zero    (suc y) = idp\n  \u2293'-spec (suc x) zero    = idp\n  \u2293'-spec (suc x) (suc y) = ap suc (\u2293'-spec x y)\n\nmodule Max where\n  open Generic id suc\n    public\n    renaming ( dist to _\u2294'_\n             ; dist-0\u2261id to \u2294'0-identity\n             ; dist-0\u2261id\u2032 to 0\u2294'-identity\n             ; dist-comm to \u2294'-comm\n             )\n\n  \u2294'-spec : \u2200 x y \u2192 x \u2294' y \u2261 x \u2294 y\n  \u2294'-spec zero    zero    = idp\n  \u2294'-spec zero    (suc y) = idp\n  \u2294'-spec (suc x) zero    = idp\n  \u2294'-spec (suc x) (suc y) = ap suc (\u2294'-spec x y)\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-comm (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-comm (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite fst \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-comm x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-comm x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\ndist\u22610 : \u2200 x y \u2192 dist x y \u2261 0 \u2192 x \u2261 y\ndist\u22610 zero zero d\u22610 = idp\ndist\u22610 zero (suc y) ()\ndist\u22610 (suc x) zero ()\ndist\u22610 (suc x) (suc y) d\u22610 = ap suc (dist\u22610 x y d\u22610)\n\n\u2238-\u2264-dist : \u2200 x y \u2192 x \u2238 y \u2264 dist x y\n\u2238-\u2264-dist zero zero       = z\u2264n\n\u2238-\u2264-dist zero (suc y)    = z\u2264n\n\u2238-\u2264-dist (suc x) zero    = \u2115\u2264.refl\n\u2238-\u2264-dist (suc x) (suc y) = \u2238-\u2264-dist x y\n\n\u2238-dist : \u2200 x y \u2192 y \u2264 x \u2192 x \u2238 y \u2261 dist x y\n\u2238-dist x .0  z\u2264n       = ! dist-comm x 0\n\u2238-dist ._ ._ (s\u2264s y\u2264x) = \u2238-dist _ _ y\u2264x\n\ndist-min-max : \u2200 x y \u2192 dist x y \u2261 (x \u2294 y) \u2238 (x \u2293 y)\ndist-min-max zero    zero    = idp\ndist-min-max zero    (suc y) = idp\ndist-min-max (suc x) zero    = idp\ndist-min-max (suc x) (suc y) = dist-min-max x y\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-comm : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-comm zero zero = idp\ndist-comm zero (suc y) = idp\ndist-comm (suc x) zero = idp\ndist-comm (suc x) (suc y) = dist-comm x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-comm (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-comm (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-comm x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-comm x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\ndist\u22610 : \u2200 x y \u2192 dist x y \u2261 0 \u2192 x \u2261 y\ndist\u22610 zero zero d\u22610 = refl\ndist\u22610 zero (suc y) ()\ndist\u22610 (suc x) zero ()\ndist\u22610 (suc x) (suc y) d\u22610 = ap suc (dist\u22610 x y d\u22610)\n\n\u2238-dist : \u2200 x y \u2192 y \u2264 x \u2192 x \u2238 y \u2261 dist x y\n\u2238-dist x .0 z\u2264n = ! dist-comm x 0\n\u2238-dist ._ ._ (s\u2264s y\u2264x) = \u2238-dist _ _ y\u2264x\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1dd4fe7a8a920498ea83e46ad2707edca7f6f4bf","subject":"Agda: move import to top of file in Syntax.Type.Plotkin","message":"Agda: move import to top of file in Syntax.Type.Plotkin\n\nOld-commit-hash: 0d789d2318a5215ac4ba2f68286e7db5826593ec\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Type\/Plotkin.agda","new_file":"Syntax\/Type\/Plotkin.agda","new_contents":"module Syntax.Type.Plotkin where\n\n-- Types for language description \u00e0 la Plotkin (LCF as PL)\n--\n-- Given base types, we build function types.\n\nopen import Function\n\ninfixr 5 _\u21d2_\n\ndata Type (B : Set {- of base types -}) : Set where\n  base : (\u03b9 : B) \u2192 Type B\n  _\u21d2_ : (\u03c3 : Type B) \u2192 (\u03c4 : Type B) \u2192 Type B\n\n-- Lift (\u0394 : B \u2192 Type B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift-\u0394type : \u2200 {B} \u2192 (B \u2192 Type B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type f (base \u03b9) = f \u03b9\nlift-\u0394type f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift-\u0394type f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n\n-- Note: the above is *not* a monadic bind.\n--\n-- Proof. base` is the most straightforward `return` of the\n-- functor `Type`.\n--\n--     return : B \u2192 Type B\n--     return = base\n--\n-- Let\n--\n--     m : Type B\n--     m = base \u03ba \u21d2 base \u03b9\n--\n-- then\n--\n--     m >>= return = lift-\u0394type return m\n--                  = base \u03ba \u21d2 base \u03ba \u21d2 base \u03b9\n--\n-- violating the second monadic law, m >>= return \u2261 m. \u220e\n\n-- Variant of lift-\u0394type for (\u0394 : B \u2192 B).\nlift-\u0394type\u2080 : \u2200 {B} \u2192 (B \u2192 B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type\u2080 f = lift-\u0394type $ base \u2218 f\n-- This has a similar type to the type of `fmap`,\n-- and `base` has a similar type to the type of `return`.\n--\n-- Similarly, for collections map can be defined from flatMap,\n-- like lift-\u0394type\u2080 can be defined in terms of lift-\u0394type.\n","old_contents":"module Syntax.Type.Plotkin where\n\n-- Types for language description \u00e0 la Plotkin (LCF as PL)\n--\n-- Given base types, we build function types.\n\ninfixr 5 _\u21d2_\n\ndata Type (B : Set {- of base types -}) : Set where\n  base : (\u03b9 : B) \u2192 Type B\n  _\u21d2_ : (\u03c3 : Type B) \u2192 (\u03c4 : Type B) \u2192 Type B\n\n-- Lift (\u0394 : B \u2192 Type B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift-\u0394type : \u2200 {B} \u2192 (B \u2192 Type B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type f (base \u03b9) = f \u03b9\nlift-\u0394type f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift-\u0394type f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n\n-- Note: the above is *not* a monadic bind.\n--\n-- Proof. base` is the most straightforward `return` of the\n-- functor `Type`.\n--\n--     return : B \u2192 Type B\n--     return = base\n--\n-- Let\n--\n--     m : Type B\n--     m = base \u03ba \u21d2 base \u03b9\n--\n-- then\n--\n--     m >>= return = lift-\u0394type return m\n--                  = base \u03ba \u21d2 base \u03ba \u21d2 base \u03b9\n--\n-- violating the second monadic law, m >>= return \u2261 m. \u220e\n\nopen import Function\n\n-- Variant of lift-\u0394type for (\u0394 : B \u2192 B).\nlift-\u0394type\u2080 : \u2200 {B} \u2192 (B \u2192 B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type\u2080 f = lift-\u0394type $ base \u2218 f\n-- This has a similar type to the type of `fmap`,\n-- and `base` has a similar type to the type of `return`.\n--\n-- Similarly, for collections map can be defined from flatMap,\n-- like lift-\u0394type\u2080 can be defined in terms of lift-\u0394type.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fa1e5e6ba14f3f7038a0a46052c8f81d12ed2e9d","subject":"Clarify Agda comment","message":"Clarify Agda comment\n\nOld-commit-hash: e00f9f0bb80cb86372669d674900a9abd43b7ef4\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence\/Base.agda","new_file":"Base\/Change\/Equivalence\/Base.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence - base definitions\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence.Base where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record (a Haskell \"newtype\")\n  -- instead of a \"type synonym\".\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence - base definitions\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence.Base where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"279231bbe708c42f5ac74a904e54cdc43421d2e0","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Nat\/Type.agda","new_file":"src\/fot\/FOTC\/Data\/Nat\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC natural numbers type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- N.B. This module is re-exported by FOTC.Data.Nat.\n\nmodule FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type (inductive predicate for the total\n-- natural numbers).\ndata N : D \u2192 Set where\n  nzero :               N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n{-# ATP axiom nzero nsucc #-}\n\n-- Induction principle.\nN-ind : (A : D \u2192 Set) \u2192\n        A zero \u2192\n        (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n        \u2200 {n} \u2192 N n \u2192 A n\nN-ind A A0 h nzero      = A0\nN-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- The FOTC natural numbers type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- N.B. This module is re-exported by FOTC.Data.Nat.\n\nmodule FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type (inductive predicate for the total\n-- natural numbers).\ndata N : D \u2192 Set where\n  nzero :               N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n{-# ATP axiom nzero nsucc #-}\n\n-- Induction principle.\nN-ind : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nN-ind A A0 h nzero      = A0\nN-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"697c75829b2c797ef4e527479a655a98ef5fc934","subject":"Added _\u00ab_.","message":"Added _\u00ab_.\n\nIgnore-this: ca495454edf50adffc0277133e7c4df3\n\ndarcs-hash:20110215150831-3bd4e-6b7a74d6063f5dd4275e974f1b96a807a6688f76.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/McCarthy91.agda","new_file":"Draft\/McCarthy91\/McCarthy91.agda","new_contents":"------------------------------------------------------------------------------\n-- The McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.McCarthy91 where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.ArithmeticATP\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Unary.Numbers\n\n------------------------------------------------------------------------------\n\n-- The McCarthy 91 function\npostulate\n  mc91     : D \u2192 D\n  mc91-eq\u2081 : \u2200 n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\n  mc91-eq\u2082 : \u2200 n \u2192 LE n one-hundred \u2192 mc91 n \u2261 mc91 (mc91 (n + eleven))\n{-# ATP axiom mc91-eq\u2081 #-}\n{-# ATP axiom mc91-eq\u2082 #-}\n\n-- Relation use by the properties of the McCarthy 91 function\n_\u00ab_ : D \u2192 D \u2192 D\nm \u00ab n = (hundred-one \u2238 m) < (hundred-one \u2238 n)\n{-# ATP definition _\u00ab_ #-}\n\nMCR : D \u2192 D \u2192 Set\nMCR m n = m \u00ab n \u2261 true\n{-# ATP definition MCR #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.McCarthy91 where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.ArithmeticATP\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Unary.Numbers\n\n------------------------------------------------------------------------------\n\n-- The McCarthy 91 function\npostulate\n  mc91     : D \u2192 D\n  mc91-eq\u2081 : \u2200 n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\n  mc91-eq\u2082 : \u2200 n \u2192 LE n one-hundred \u2192 mc91 n \u2261 mc91 (mc91 (n + eleven))\n{-# ATP axiom mc91-eq\u2081 #-}\n{-# ATP axiom mc91-eq\u2082 #-}\n\n-- Relation use by the properties of the McCarthy 91 function\nMCR : D \u2192 D \u2192 Set\nMCR m n = LT (hundred-one \u2238 m) (hundred-one \u2238 n)\n-- NB. The ATP pragma is not necessary at the moment.\n-- {-# ATP definition MCR #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ee3e58322cada11dac8048f080e1f4b548cf4bbf","subject":"FunUniverse.Core: adder","message":"FunUniverse.Core: adder\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Core.agda","new_file":"FunUniverse\/Core.agda","new_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nrecord FunUniverse {t} (T : Set t) : Set (L.suc t) where\n  constructor _,_\n  field\n    universe : Universe T\n    _`\u2192_     : T \u2192 T \u2192 \u2605\n\n  infix 0 _`\u2192_\n  open Universe universe public\n\n  _`\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\n  i `\u2192\u1d47 o = `Bits i `\u2192 `Bits o\n\n  `Endo : T \u2192 \u2605\n  `Endo A = A `\u2192 A\n\nmodule OpFunU {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  opFunU : FunUniverse T\n  opFunU = universe , flip _`\u2192_\n\nmodule Defaults\u27e8first-part\u27e9 {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultSecondFromFirstSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n  module Default<\u00d7>FromFirstSecond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n  module DefaultFstFromSndSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module DefaultsGroup1\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open Default<\u00d7>FromFirstSecond _\u2218_ first second public\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    open DefaultFstFromSndSwap _\u2218_ snd swap public\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\n    (_\u2218_   : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    open CompositionNotations _\u2218_\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u204f fst , < fst \u204f snd , snd > >\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCondFromFork\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultForkFromCond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\n  module DefaultForkFromBijFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B)\n    (snd  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = bijFork f g \u204f snd\n\n  module DefaultBijForkFromCond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n    bijFork f g = second < f , g > \u204f < fst , cond >\n\n  module DefaultBijForkFromFork\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B) where\n\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n    bijFork f g = < fst , fork f g >\n\n  module DefaultXor\n     (id  : \u2200 {A} \u2192 A `\u2192 A)\n     (not : `Bit `\u2192 `Bit)\n     (<_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n) where\n\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n    xor xs = < V.map (B.cond not id) xs \u229b>\n\n  module DefaultRewire\n    (rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o) where\n\n    rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewire fun = rewireTbl (V.tabulate fun)\n\n  module DefaultRewireTbl\n    (rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o\n    rewireTbl tbl = rewire (flip V.lookup tbl)\n\n  module LinDefaults\n    (_\u2218_   : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))) where\n\n    open CompositionNotations _\u2218_ public\n    open <\u00d7>Default _\u2218_ first swap public\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\nrecord LinRewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  infixr 9 _\u2218_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    tt\u2192[]  : \u2200 {A} \u2192 `\u22a4 `\u2192 `Vec A 0\n    []\u2192tt  : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\u22a4\n    <\u2237>    : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open CompositionNotations _\u2218_ public\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  infixr 3 _***_\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  <id,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\u22a4\n  <id,tt> = <tt,id> \u204f swap\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  fst<,tt> : \u2200 {A} \u2192 A `\u00d7 `\u22a4 `\u2192 A\n  fst<,tt> = swap \u204f snd<tt,>\n\n  fst<,[]> : \u2200 {A B} \u2192 A `\u00d7 `Vec B 0 `\u2192 A\n  fst<,[]> = second []\u2192tt \u204f fst<,tt>\n\n  snd<[],> : \u2200 {A B} \u2192 `Vec A 0 `\u00d7 B `\u2192 B\n  snd<[],> = first []\u2192tt \u204f snd<tt,>\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  swap-top : \u2200 {A B C} \u2192 A `\u00d7 B `\u00d7 C `\u2192 B `\u00d7 A `\u00d7 C\n  swap-top = assoc-first swap\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_\u00d7\u2081_> : \u2200 {A B C D E F} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 (C `\u2192 F) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 F\n  < f \u00d7\u2081 g > = assoc\u2032 \u204f < f \u00d7 g > \u204f assoc\n\n  <_\u00d7\u2082_> : \u2200 {A B C D E F} \u2192 (A `\u2192 D) \u2192 (B `\u00d7 C `\u2192 E `\u00d7 F) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (D `\u00d7 E) `\u00d7 F\n  < f \u00d7\u2082 g > = assoc \u204f < f \u00d7 g > \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f <\u2237>\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f <\u2237>\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f <\u2237>\n\n  <_\u2237[]> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237[]> = < f \u2237\u2032tt\u204f tt\u2192[] >\n\n  <\u2237[]> : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  <\u2237[]> = < id \u2237[]>\n\n  <[],_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <[], f > = <tt\u204f tt\u2192[] , f >\n\n  <_,[]> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,[]> = < f ,tt\u204f tt\u2192[] >\n\n  head<\u2237> : \u2200 {A} \u2192 `Vec A 1 `\u2192 A\n  head<\u2237> = uncons \u204f fst<,[]>\n\n  constVec\u22a4 : \u2200 {n a} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 `\u22a4 `\u2192 `Vec B n\n  constVec\u22a4 f [] = tt\u2192[]\n  constVec\u22a4 f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec\u22a4 f xs >\n\n  []\u2192[] : \u2200 {A B} \u2192 `Vec A 0 `\u2192 `Vec B 0\n  []\u2192[] = []\u2192tt \u204f tt\u2192[]\n\n  <[],[]>\u2192[] : \u2200 {A B C} \u2192 (`Vec A 0 `\u00d7 `Vec B 0) `\u2192 `Vec C 0\n  <[],[]>\u2192[] = fst<,[]> \u204f []\u2192[]\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = []\u2192[]\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst<,[]>\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd<[],>\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = <[],[]>\u2192[]\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f <\u2237>\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd<[],> \u2237[]>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f <\u2237>\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = id\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd<[],>\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f <\u2237>\n\n  <_++_> : \u2200 {m n A} \u2192 (`\u22a4 `\u2192 `Vec A m) \u2192 (`\u22a4 `\u2192 `Vec A n) \u2192\n                         `\u22a4 `\u2192 `Vec A (m + n)\n  < f ++ g > = <tt\u204f f , g > \u204f append\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <[], id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first <\u2237>\n\n  rot-left : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A (n + m)\n  rot-left m = splitAt m \u204f swap \u204f append\n\n  rot-right : \u2200 {m} n {A} \u2192 `Vec A (m + n) `\u2192 `Vec A (n + m)\n  rot-right {m} _ = rot-left m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head<\u2237>\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = []\u2192[]\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = []\u2192[]\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f <\u2237>\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate\u22a4 : \u2200 n \u2192 `\u22a4 `\u2192 `Vec `\u22a4 n\n  replicate\u22a4 _ = constVec\u22a4 (\u03bb _ \u2192 id) (V.replicate {A = \u22a4} _)\n\n  loop : \u2200 {A} \u2192 \u2115 \u2192 (A `\u2192 A) \u2192 (A `\u2192 A)\n  loop zero    _ = id\n  loop (suc n) f = f \u204f loop n f\n  -- or based on fold:\n  -- loop n f = < id ,tt\u204f replicate\u22a4 n > \u204f foldl (fst<,tt> \u204f f)\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <0b> <1b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","old_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nrecord FunUniverse {t} (T : Set t) : Set (L.suc t) where\n  constructor _,_\n  field\n    universe : Universe T\n    _`\u2192_     : T \u2192 T \u2192 \u2605\n\n  infix 0 _`\u2192_\n  open Universe universe public\n\n  _`\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\n  i `\u2192\u1d47 o = `Bits i `\u2192 `Bits o\n\n  `Endo : T \u2192 \u2605\n  `Endo A = A `\u2192 A\n\nmodule OpFunU {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  opFunU : FunUniverse T\n  opFunU = universe , flip _`\u2192_\n\nmodule Defaults\u27e8first-part\u27e9 {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultSecondFromFirstSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n  module Default<\u00d7>FromFirstSecond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n  module DefaultFstFromSndSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module DefaultsGroup1\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open Default<\u00d7>FromFirstSecond _\u2218_ first second public\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    open DefaultFstFromSndSwap _\u2218_ snd swap public\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\n    (_\u2218_   : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    open CompositionNotations _\u2218_\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u204f fst , < fst \u204f snd , snd > >\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCondFromFork\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultForkFromCond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\n  module DefaultForkFromBijFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B)\n    (snd  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = bijFork f g \u204f snd\n\n  module DefaultBijForkFromCond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n    bijFork f g = second < f , g > \u204f < fst , cond >\n\n  module DefaultBijForkFromFork\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst  : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B) where\n\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n    bijFork f g = < fst , fork f g >\n\n  module DefaultXor\n     (id  : \u2200 {A} \u2192 A `\u2192 A)\n     (not : `Bit `\u2192 `Bit)\n     (<_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n) where\n\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n    xor xs = < V.map (B.cond not id) xs \u229b>\n\n  module DefaultRewire\n    (rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o) where\n\n    rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewire fun = rewireTbl (V.tabulate fun)\n\n  module DefaultRewireTbl\n    (rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o\n    rewireTbl tbl = rewire (flip V.lookup tbl)\n\n  module LinDefaults\n    (_\u2218_   : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))) where\n\n    open CompositionNotations _\u2218_ public\n    open <\u00d7>Default _\u2218_ first swap public\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\nrecord LinRewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  infixr 9 _\u2218_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    tt\u2192[]  : \u2200 {A} \u2192 `\u22a4 `\u2192 `Vec A 0\n    []\u2192tt  : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\u22a4\n    <\u2237>    : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open CompositionNotations _\u2218_ public\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  infixr 3 _***_\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  <id,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\u22a4\n  <id,tt> = <tt,id> \u204f swap\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  fst<,tt> : \u2200 {A} \u2192 A `\u00d7 `\u22a4 `\u2192 A\n  fst<,tt> = swap \u204f snd<tt,>\n\n  fst<,[]> : \u2200 {A B} \u2192 A `\u00d7 `Vec B 0 `\u2192 A\n  fst<,[]> = second []\u2192tt \u204f fst<,tt>\n\n  snd<[],> : \u2200 {A B} \u2192 `Vec A 0 `\u00d7 B `\u2192 B\n  snd<[],> = first []\u2192tt \u204f snd<tt,>\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  swap-top : \u2200 {A B C} \u2192 A `\u00d7 B `\u00d7 C `\u2192 B `\u00d7 A `\u00d7 C\n  swap-top = assoc-first swap\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_\u00d7\u2081_> : \u2200 {A B C D E F} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 (C `\u2192 F) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 F\n  < f \u00d7\u2081 g > = assoc\u2032 \u204f < f \u00d7 g > \u204f assoc\n\n  <_\u00d7\u2082_> : \u2200 {A B C D E F} \u2192 (A `\u2192 D) \u2192 (B `\u00d7 C `\u2192 E `\u00d7 F) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (D `\u00d7 E) `\u00d7 F\n  < f \u00d7\u2082 g > = assoc \u204f < f \u00d7 g > \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f <\u2237>\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f <\u2237>\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f <\u2237>\n\n  <_\u2237[]> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237[]> = < f \u2237\u2032tt\u204f tt\u2192[] >\n\n  <\u2237[]> : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  <\u2237[]> = < id \u2237[]>\n\n  <[],_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <[], f > = <tt\u204f tt\u2192[] , f >\n\n  <_,[]> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,[]> = < f ,tt\u204f tt\u2192[] >\n\n  head<\u2237> : \u2200 {A} \u2192 `Vec A 1 `\u2192 A\n  head<\u2237> = uncons \u204f fst<,[]>\n\n  constVec\u22a4 : \u2200 {n a} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 `\u22a4 `\u2192 `Vec B n\n  constVec\u22a4 f [] = tt\u2192[]\n  constVec\u22a4 f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec\u22a4 f xs >\n\n  []\u2192[] : \u2200 {A B} \u2192 `Vec A 0 `\u2192 `Vec B 0\n  []\u2192[] = []\u2192tt \u204f tt\u2192[]\n\n  <[],[]>\u2192[] : \u2200 {A B C} \u2192 (`Vec A 0 `\u00d7 `Vec B 0) `\u2192 `Vec C 0\n  <[],[]>\u2192[] = fst<,[]> \u204f []\u2192[]\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = []\u2192[]\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst<,[]>\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd<[],>\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = <[],[]>\u2192[]\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f <\u2237>\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd<[],> \u2237[]>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f <\u2237>\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = id\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd<[],>\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f <\u2237>\n\n  <_++_> : \u2200 {m n A} \u2192 (`\u22a4 `\u2192 `Vec A m) \u2192 (`\u22a4 `\u2192 `Vec A n) \u2192\n                         `\u22a4 `\u2192 `Vec A (m + n)\n  < f ++ g > = <tt\u204f f , g > \u204f append\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <[], id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first <\u2237>\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head<\u2237>\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = []\u2192[]\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = []\u2192[]\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f <\u2237>\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate\u22a4 : \u2200 n \u2192 `\u22a4 `\u2192 `Vec `\u22a4 n\n  replicate\u22a4 _ = constVec\u22a4 (\u03bb _ \u2192 id) (V.replicate {A = \u22a4} _)\n\n  loop : \u2200 {A} \u2192 \u2115 \u2192 (A `\u2192 A) \u2192 (A `\u2192 A)\n  loop zero    _ = id\n  loop (suc n) f = f \u204f loop n f\n  -- or based on fold:\n  -- loop n f = < id ,tt\u204f replicate\u22a4 n > \u204f foldl (fst<,tt> \u204f f)\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <0b> <1b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"06c0a79346fc5f1d818b819a394fd296aac8ce52","subject":"Refine logical relation for integers","message":"Refine logical relation for integers\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV nat (intV v1) (intV v2) n = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV nat v1 v2 n = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat v1 v2 vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8d00761c933d85cbaf7c91e2a9f2d3a00108afb9","subject":"game.agda","message":"game.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"game.agda","new_file":"game.agda","new_contents":"module game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Bool -- hiding (_\u225f_)\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R\n  ask : {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R) \u2192 Strategy \u2610_ R\n\nrun : \u2200 {R} \u2192 Strategy id R \u2192 R\nrun (say r)   = r\nrun (ask a f) = run (f a)\n\nopen import Data.Bits\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nmodule CryptoGames where\n\n  module CipherGames (M C : Set) where\n    SemSecAdv : Set\n    SemSecAdv = C \u2192 Bit\n\n    CPAAdv : Set\u2081\n    CPAAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit\n\n    data CPAWinner (enc : M \u2192 C) b : Strategy id Bit \u2192 Set\u2081 where\n      win : CPAWinner enc b (say b)\n      ask : \u2200 {m f} \u2192 CPAWinner enc b (f (enc m)) \u2192 CPAWinner enc b (ask (enc m) f)\n\n  module MACGames (M T : Set) where\n    MacAdv : Set\u2081\n    MacAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 T) \u2192 Strategy \u2610_ (M \u00d7 T)\n\n    data MACWinner (mac : M \u2192 T) : Strategy id (M \u00d7 T) \u2192 Set\u2081 where\n      win : \u2200 {m} \u2192 MACWinner mac (say (m , mac m))\n      ask : \u2200 {m f} \u2192 MACWinner mac (f (mac m)) \u2192 MACWinner mac (ask (mac m) f)\n","old_contents":"module game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Bool -- hiding (_\u225f_)\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R\n  ask : {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R) \u2192 Strategy \u2610_ R\n\nrun : \u2200 {R} \u2192 Strategy id R \u2192 R\nrun (say r)   = r\nrun (ask a f) = run (f a)\n\nopen import Data.Bits\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nmodule CryptoGames where\n\n  module CipherGames (M C : Set) where\n    SemSecAdv : Set\n    SemSecAdv = C \u2192 Bit\n\n    CPAAdv : Set\n    CPAAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit\n\n    data CPAWinner b : Strategy \u2610_ Bit \u2192 Set where\n      win : CPAWinner b (say b)\n      ask : CPAWinner b s \u2192 CPAWinner b (ask ? (\u03bb ? \u2192 ))\n\n  module MACGames (M T : Set) where\n    MacAdv : Set\n    MacAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 T) \u2192 Strategy \u2610_ (M \u00d7 T)\n\n    data MACWinner (mac : M \u2192 T) : Strategy id (M \u00d7 T) \u2192 Set where\n      win : \u2200 {m} \u2192 MACWinner mac (say (m , mac m))\n      ask : \u2200 {m f} \u2192 MACWinner mac (f (mac m)) \u2192 MACWinner mac (ask (mac m) f)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e2d49c81f464ab9449818f964e0078660f4315da","subject":"Agda: add lemma about syntactic contexts","message":"Agda: add lemma about syntactic contexts\n\nOld-commit-hash: 3613bba9a32778eaaa8f9d22419c23a8505be346\n","repos":"inc-lc\/ilc-agda","old_file":"Syntactic\/Contexts.agda","new_file":"Syntactic\/Contexts.agda","new_contents":"module Syntactic.Contexts\n    (Type : Set)\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Specialized congruence rules\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 8 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  lift (keep \u03c4 \u2022 \u227c\u2081) this = this\n  lift (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (lift \u227c\u2081 x)\n  lift (drop \u03c4 \u2022 \u227c\u2081) x = that (lift \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","old_contents":"module Syntactic.Contexts\n    (Type : Set)\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Specialized congruence rules\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 8 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  lift (keep \u03c4 \u2022 \u227c\u2081) this = this\n  lift (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (lift \u227c\u2081 x)\n  lift (drop \u03c4 \u2022 \u227c\u2081) x = that (lift \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2c88234563182184ea1c4db69a4b3fc0742b0371","subject":"PropEq: reexport \"reflexive\"","message":"PropEq: reexport \"reflexive\"\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\nprivate\n  module Dummy {a} {A : Set a} where\n    open IsEquivalence (isEquivalence {a} {A}) public hiding (refl; sym; trans)\nopen Dummy public\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2261\u2261_ : \u2200 {a} {A : Set a} {i j : A}\n         (i\u2261j\u2081 : i \u2261 j) (i\u2261j\u2082 : i \u2261 j) \u2192 i\u2261j\u2081 \u2261 i\u2261j\u2082\n_\u2261\u2261_ refl refl = refl\n\n_\u225f\u2261_ : \u2200 {a} {A : Set a} {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n_\u225f\u2261_ refl refl = yes refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} (f g : A \u2192 B \u2192 C) \u2192 Set _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\ninjective : \u2200 {a} {A : Set a} \u2192 InjectiveRel A _\u2261_\ninjective refl refl = refl\n\nsurjective : \u2200 {a} {A : Set a} \u2192 SurjectiveRel A _\u2261_\nsurjective refl refl = refl\n\nbijective : \u2200 {a} {A : Set a} \u2192 BijectiveRel A _\u2261_\nbijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\nmodule \u2261-Reasoning {a} {A : Set a} where\n  open Setoid-Reasoning (setoid A) public renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : Set a} {B : Set b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 \u2605\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2261\u2261_ : \u2200 {a} {A : Set a} {i j : A}\n         (i\u2261j\u2081 : i \u2261 j) (i\u2261j\u2082 : i \u2261 j) \u2192 i\u2261j\u2081 \u2261 i\u2261j\u2082\n_\u2261\u2261_ refl refl = refl\n\n_\u225f\u2261_ : \u2200 {a} {A : Set a} {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n_\u225f\u2261_ refl refl = yes refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} (f g : A \u2192 B \u2192 C) \u2192 Set _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\ninjective : \u2200 {a} {A : Set a} \u2192 InjectiveRel A _\u2261_\ninjective refl refl = refl\n\nsurjective : \u2200 {a} {A : Set a} \u2192 SurjectiveRel A _\u2261_\nsurjective refl refl = refl\n\nbijective : \u2200 {a} {A : Set a} \u2192 BijectiveRel A _\u2261_\nbijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\nmodule \u2261-Reasoning {a} {A : Set a} where\n  open Setoid-Reasoning (setoid A) public renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : Set a} {B : Set b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 \u2605\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9b9ca55078054bf601d204c004b44563204271a9","subject":"few more cases of #14, including an elided exercise in loop unrolling","message":"few more cases of #14, including an elided exercise in loop unrolling\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time: that boxed values, indeterminates, cast errors, and\n-- expressions that step are pairwise disjoint.\n--\n-- note that as a consequence of currying and comutativity of products,\n-- this means that there are six theorems to prove. in addition to those,\n-- we also prove several convenince forms that combine theorems about\n-- indeterminate and boxed value forms into the same statement about final\n-- forms, which mirrors the mutual definition of indeterminate and final\n-- and saves some redundant argumentation.\nmodule progress-checks where\n  -- boxed values are not indeterminates\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values are not errors\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values don't step\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates are not errors\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  mutual\n    -- indeterminates don't step\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = final-sub-not-trans x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    -- final expressions don't step\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors don't step\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail cases\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x () x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITCastID x\u2084) (FHCast FHOuter))\n  err-not-step (CECastFail x () x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITCastSucceed x\u2084 x\u2085) (FHCast FHOuter))\n  err-not-step (CECastFail x GHole x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITGround x\u2084 x\u2085) (FHCast FHOuter)) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast (FHCast x\u2084)) x\u2085 (FHCast (FHCast x\u2086))) = final-not-step x (_ , Step x\u2084 x\u2085 x\u2086)\n\n    -- congruence cases\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!} -- final-not-err x\u2081 (CECong {!!} ce) -- proably case on x, but get incomplete pattern garbage\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!} -- cyrus: possibly another counter example\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter () FHOuter)\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!} -- cyrus: only obvious thing to do is case on x\u2082, doesn't seem to get anywhere\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- cyrus: this is a counter example, d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = final-not-err x\u2081 (ce-out-cast ce x)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = err-not-step (ce-out-cast ce x\u2081) (_ , Step x\u2083 x\u2084 x\u2086)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values and errors are disjoint\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  mutual\n    -- indeterminates and expressions that step are disjoint\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = final-sub-not-trans x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    -- final expressions don't step\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors and expressions that step are disjoint\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail cases\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x () x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITCastID x\u2084) (FHCast FHOuter))\n  err-not-step (CECastFail x () x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITCastSucceed x\u2084 x\u2085) (FHCast FHOuter))\n  err-not-step (CECastFail x GHole x\u2082 x\u2083) (_ , Step (FHCast FHOuter) (ITGround x\u2084 x\u2085) (FHCast FHOuter)) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast (FHCast x\u2084)) x\u2085 (FHCast (FHCast x\u2086))) = final-not-step x (_ , Step x\u2084 x\u2085 x\u2086)\n\n    -- congruence cases\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- cyrus this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dabd8bc14552ef33e1791e95dba210bfae26d7a0","subject":"Typo.","message":"Typo.\n\nIgnore-this: 59c6c0863d6e6aaabad15e16de6dd83\n\ndarcs-hash:20120223161854-3bd4e-069e05e6ab9b3c8e18f2242c558fee7bc0105f2b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Axiomatic\/Standard\/Base.agda","new_file":"src\/PA\/Axiomatic\/Standard\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Standard.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe.\n-- We chose the symbol M because there are non-standard models of\n-- Peano Arithmetic, where the domain is not the set of natural\n-- numbers.\nopen import FOL.Universe public renaming ( D to M )\n\n-- FOL with equality.\nopen import FOL.FOL-Eq public\n\n-- Non-logical constants\npostulate\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- Proper axioms\n-- From [Machover, 1977, p. 263] and [H\u00e1jeck and Pudl\u00e1k, 1998, p. 28]\n--\n-- * Mosh\u00e9 Machover. Set theory, logic and their\n--   limitations. Cambridge University Press, 1996.\n--\n-- * Petr H\u00e1jeck and Pavel Pudl\u00e1k. Metamathematics of First-Order\n--   Arithmetic. Springer, 1998. 2nd printing.\n-- )\n\n-- A\u2081. 0 \u2260 succ n\n-- A\u2082. succ m = succ n \u2192 m = n\n-- A\u2083. 0 + n = n\n-- A\u2084. succ m + n = succ (m + n)\n-- A\u2085. 0 * n = 0\n-- A\u2086. succ m * n = (m * n) + n\n-- A\u2087. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  A\u2081 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\n  A\u2082 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n  A\u2083 : \u2200 n \u2192 zero + n \u2261 n\n  A\u2084 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  A\u2085 : \u2200 n \u2192 zero * n \u2261 zero\n  A\u2086 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n{-# ATP axiom A\u2081 A\u2082 A\u2083 A\u2084 A\u2085 A\u2086 #-}\n\n-- A\u2087 is an axiom schema, therefore we do not translate it to TPTP.\npostulate A\u2087 : (P : M \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Standard.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe.\n-- We chose the symbol M because there are non-standard models of\n-- Peano Arithmetic, where the domain is not the set of natural\n-- numbers.\nopen import FOL.Universe public renaming ( D to M )\n\n-- FOL with equality.\nopen import FOL.FOL-Eq public\n\n-- Non-logical constants\npostulate\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- Proper axioms (see [Machover, 1977, p. 263], [H\u00e1jeck and Pudl\u00e1k,\n  1998, p. 28])\n--\n-- * Mosh\u00e9 Machover. Set theory, logic and their\n--   limitations. Cambridge University Press, 1996.\n--\n-- * Petr H\u00e1jeck and Pavel Pudl\u00e1k. Metamathematics of First-Order\n--   Arithmetic. Springer, 1998. 2nd printing.\n\n-- A\u2081. 0 \u2260 succ n\n-- A\u2082. succ m = succ n \u2192 m = n\n-- A\u2083. 0 + n = n\n-- A\u2084. succ m + n = succ (m + n)\n-- A\u2085. 0 * n = 0\n-- A\u2086. succ m * n = (m * n) + n\n-- A\u2087. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  A\u2081 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\n  A\u2082 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n  A\u2083 : \u2200 n \u2192 zero + n \u2261 n\n  A\u2084 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  A\u2085 : \u2200 n \u2192 zero * n \u2261 zero\n  A\u2086 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n{-# ATP axiom A\u2081 A\u2082 A\u2083 A\u2084 A\u2085 A\u2086 #-}\n\n-- A\u2087 is an axiom schema, therefore we do not translate it to TPTP.\npostulate A\u2087 : (P : M \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"082fca7202356a2254bdefad74bb4680042edbdb","subject":"Made a module private.","message":"Made a module private.\n\nIgnore-this: c8dd74d9a93a0ee413fd3a59fffc49f4\n\ndarcs-hash:20120305232020-3bd4e-b6b668cdb5a1c4ff19d99a001ea15a451288921a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/Equality.agda","new_file":"src\/FOTC\/Data\/Stream\/Equality.agda","new_contents":"------------------------------------------------------------------------------\n-- Equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The equality on streams.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The relation _\u2248_ is the greatest post-fixed point (by \u2248-gfp\u2081 and\n-- \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The relation _\u2248_ is a post-fixed point of the bisimulation functional\n-- (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed point, the relation\n-- _\u2248_ is also a pre-fixed point of the bisimulation functional (see\n-- below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n\n  -- as (R :: R') bs' (p. 310)\n\n  -- for the bisimulation functional.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimulation functional (Bart Jacobs and Jan\n  -- Rutten. (Co)algebras and (co)induction. EATCS Bulletin,\n  -- 62:222\u2013259, 1997).\n  Bisimulation : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  Bisimulation _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The relation _\u2248_ is a post-fixed point of Bisimulation.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Bisimulation _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 Bisimulation _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- The relation _\u2248_ is a pre-fixed point of Bisimulation.\n  pre-fp : \u2200 {xs ys} \u2192 Bisimulation _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","old_contents":"------------------------------------------------------------------------------\n-- Equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The equality on streams.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The relation _\u2248_ is the greatest post-fixed point (by \u2248-gfp\u2081 and\n-- \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The relation _\u2248_ is a post-fixed point of the bisimulation functional\n-- (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed point, the relation\n-- _\u2248_ is also a pre-fixed point of the bisimulation functional (see\n-- below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nmodule Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n\n  -- as (R :: R') bs' (p. 310)\n\n  -- for the bisimulation functional.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimulation functional (Bart Jacobs and Jan\n  -- Rutten. (Co)algebras and (co)induction. EATCS Bulletin,\n  -- 62:222\u2013259, 1997).\n  Bisimulation : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  Bisimulation _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The relation _\u2248_ is a post-fixed point of Bisimulation.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Bisimulation _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 Bisimulation _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- The relation _\u2248_ is a pre-fixed point of Bisimulation.\n  pre-fp : \u2200 {xs ys} \u2192 Bisimulation _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93eab697da9ba8131347a9dab0758cc9925c3605","subject":"Control\/Protocol\/Choreography.agda","message":"Control\/Protocol\/Choreography.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndualInOut : InOut \u2192 InOut\ndualInOut In  = Out\ndualInOut Out = In\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  com  : (q : InOut)(M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com q M P \u2261\u1d3e com q M Q\n\npattern \u03a0' M P = com In  M P\npattern \u03a3' M P = com Out M P\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0S' M P = \u03a0' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3S' M P = \u03a3' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u27e6_\u27e7\u03a0\u03a3 : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u03a0\u03a3 In  = \u03a0\n\u27e6_\u27e7\u03a0\u03a3 Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com q M P \u27e7 = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 \u27e6 P x \u27e7\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\n_-[_]\u2192\u00f8\u204f_ : \u2200 {I}(A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\nA -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  I\u03a3D : \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  I\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n--  I\u03a0S : \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 \u03a0' (\u2610 M) \u2102C ]\n  \u2605\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  \u2605\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n  --\u00f8\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ A \u2261 \u03a3' S M \u2102A ]\n  --\u00f8\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ B \u2261 \u03a0' S M \u2102B ]\n  end : \u2200 {A} \u2192 [ end \/ A \u2261 end ]\n\nTrace : Proto \u2192 Proto\nTrace end          = end\nTrace (com _ A B) = \u03a3' A \u03bb m \u2192 Trace (B m)\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (com q A B) = com (dualInOut q) A (\u03bb x \u2192 dual (B x))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com q M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com q M P) = com q M (\u03bb m \u2192 \u2261\u1d3e-refl (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end = end\nTrace-idempotent (com q M P) = \u03a3' M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end = end\nTrace-dual-oblivious (com q M P) = \u03a3' M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\ncom \u03c0 A B >>\u2261 Q = com \u03c0 A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>\u2261 Q)\n++Tele end         _        ys = ys\n++Tele (com q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\nright-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261\u1d3e P\nright-unit end = end\nright-unit (com q M P) = com q M \u03bb m \u2192 right-unit (P m)\n\nassoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>\u2261 Q) >>\u2261 R)\nassoc end         Q R = \u2261\u1d3e-refl (Q _ >>\u2261 R)\nassoc (com q M P) Q R = com q M \u03bb m \u2192 assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndata DualInOut : InOut \u2192 InOut \u2192 \u2605 where\n  DInOut : DualInOut In  Out\n  DOutIn : DualInOut Out In\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' A B) (\u03a3' A B')\n  \u03a3\u00b7\u03a0 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' A B) (\u03a0' A B')\n\ndata Sing {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  sing : \u2200 x \u2192 Sing x\n\n-- Commit = \u03bb M R \u2192 \u03a3' D (\u2610 M) \u03bb { [ m ] \u2192 \u03a0' D R \u03bb r \u2192 \u03a3' D (Sing m) (\u03bb _ \u2192 end) }\nCommit = \u03bb M R \u2192 \u03a3S' M \u03bb m \u2192 \u03a0' R \u03bb r \u2192 \u03a3' (Sing m) (\u03bb _ \u2192 end)\n\ncommit : \u2200 M R (m : M)  \u2192 \u27e6 Commit M R \u27e7\ncommit M R m = [ m ] , (\u03bb x \u2192 (sing m) , _)\n\ndecommit : \u2200 M R (r : R) \u2192 \u27e6 dual (Commit M R) \u27e7\ndecommit M R r = \u03bb { [ m ] \u2192 r , (\u03bb x \u2192 0\u2081) }\n\ndata [_&_\u2261_]InOut : InOut \u2192 InOut \u2192 InOut \u2192 \u2605\u2081 where\n  \u03a0XX : \u2200 {X} \u2192 [ In & X  \u2261 X ]InOut\n  X\u03a0X : \u2200 {X} \u2192 [ X  & In \u2261 X ]InOut\n\n&InOut-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]InOut \u2192 [ Q & P \u2261 R ]InOut\n&InOut-comm \u03a0XX = X\u03a0X\n&InOut-comm X\u03a0X = \u03a0XX\n\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end& : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  &end : \u2200 {P} \u2192 [ P & end \u2261 P ]\n  D&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q m     \u2261 R m ]) \u2192 [ com qP    M  P & com qQ    M  Q \u2261 com qR M R ]\n  S&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P [ m ] & Q m     \u2261 R m ]) \u2192 [ com qP (\u2610 M) P & com qQ    M  Q \u2261 com qR M R ]\n  D&S  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q [ m ] \u2261 R m ]) \u2192 [ com qP    M  P & com qQ (\u2610 M) Q \u2261 com qR M R ]\n\n&-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n&-comm end& = &end\n&-comm &end = end&\n&-comm (D&D q P&) = D&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (S&D q P&) = D&S (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (D&S q P&) = S&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n\nDualInOut-sym : \u2200 {P Q} \u2192 DualInOut P Q \u2192 DualInOut Q P\nDualInOut-sym DInOut = DOutIn\nDualInOut-sym DOutIn = DInOut\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDualInOut-spec : \u2200 P \u2192 DualInOut P (dualInOut P)\nDualInOut-spec In  = DInOut\nDualInOut-spec Out = DOutIn\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (com In M P) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-spec (P x))\nDual-spec (com Out M P) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-spec (P x))\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com q M P) X = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>\u2261 Q) \u2192 \u2605} \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  bind-spec end         = refl\n  bind-spec (com q M P) = cong (\u27e6 q \u27e7\u03a0\u03a3 M) (funExt \u03bb m \u2192 bind-spec (P m))\n\n\nmodule _ {A B} where\n  run-com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  run-com end      a       b       = a , b\n  run-com (\u03a0' M P) p       (m , q) = run-com (P m) (p m) q\n  run-com (\u03a3' M P) (m , p) q       = run-com (P m) p (q m)\n\ncom-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\ncom-cont end p q = (_ , p)  , (_ , q)\ncom-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\ncom-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0' (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3' (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q M P Q} \u2192 SimL (com q M P) Q \u2192 Sim (com q M P) Q\n    right : \u2200 {P q M Q} \u2192 SimR P (com q M Q) \u2192 Sim P (com q M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recv PQA)   (send m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (send m PQB) = send m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (send m PQA) (recv PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (send x\u2081 x\u2082) (recv x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = send m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = ? -- send m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb x \u2192 left (send x (sim-id (B x)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb x \u2192 right (send x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u00b7\u03a3 x\u2081) (recv x) (send x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u00b7\u03a0 x)  (send x\u2081 x\u2082) (recv x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recv PQ) QR = recv (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recv P)) = do (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (send m P)) = do (send m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (send x x\u2082)) (left (recv x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (right (recv x)) (left (send x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (right (send x x\u2082)) (left (recv x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u00b7\u03a0 x) Q-Q' (right (send x\u2081 x\u2082)) (left (recv x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' (\u03a0\u00b7\u03a3 x\u2081) (right \u00f8P) (right (recv x)) (left (send x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u00b7\u03a0 x) (right \u00f8P) (right (send x\u2081 x\u2082)) (left (recv x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recv x\u2081))\n    = cong (right \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (send x\u2081 x\u2082))\n    = cong (right \u2218 send x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recv x))    = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (send x x\u2081)) = cong (left \u2218 send x) (sim-!! x\u2081)\n  sim-!! (right (recv x))    = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (send x x\u2081)) = cong (right \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (right (recv x)) (left (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (right (send x\u2081 x\u2082)) (left (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recv x\u2081))\n    = cong (left \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (send x\u2081 x\u2082))\n    = cong (left \u2218 send x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\ndata InOut : \u2605 where\n  -- \u03a0' \u03a3' : InOut\n  In Out : InOut\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  com  : (q : InOut)(M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\npattern \u03a0' M P = com In  M P\npattern \u03a3' M P = com Out M P\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0S' M P = \u03a0' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3S' M P = \u03a3' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end         \u27e7 = \ud835\udfd9\n\u27e6 \u03a0' M P \u27e7 = \u03a0  M \u03bb x \u2192 \u27e6 P x \u27e7\n\u27e6 \u03a3' M P \u27e7 = \u03a3  M \u03bb x \u2192 \u27e6 P x \u27e7\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\n_-[_]\u2192\u00f8\u204f_ : \u2200 {I}(A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\nA -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  I\u03a3D : \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  I\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n--  I\u03a0S : \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 \u03a0' (\u2610 M) \u2102C ]\n  \u2605\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  \u2605\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n  --\u00f8\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ A \u2261 \u03a3' S M \u2102A ]\n  --\u00f8\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ B \u2261 \u03a0' S M \u2102B ]\n  end : \u2200 {A} \u2192 [ end \/ A \u2261 end ]\n\nTrace : Proto \u2192 Proto\nTrace end          = end\nTrace (com _ A B) = \u03a3' A \u03bb m \u2192 Trace (B m)\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    Trace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261 Trace P\n    Trace-idempotent end = refl\n    Trace-idempotent (com q M P) = cong (\u03a3' M) (funExt \u03bb m \u2192 Trace-idempotent (P m))\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\ncom \u03c0 A B >>\u2261 Q = com \u03c0 A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndualInOut : InOut \u2192 InOut\ndualInOut In  = Out\ndualInOut Out = In\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (com q A B) = com (dualInOut q) A (\u03bb x \u2192 dual (B x))\n\ndata DualInOut : InOut \u2192 InOut \u2192 \u2605 where\n  DInOut : DualInOut In  Out\n  DOutIn : DualInOut Out In\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' A B) (\u03a3' A B')\n  \u03a3\u00b7\u03a0 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' A B) (\u03a0' A B')\n\ndata Sing {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  sing : \u2200 x \u2192 Sing x\n\n-- Commit = \u03bb M R \u2192 \u03a3' D (\u2610 M) \u03bb { [ m ] \u2192 \u03a0' D R \u03bb r \u2192 \u03a3' D (Sing m) (\u03bb _ \u2192 end) }\nCommit = \u03bb M R \u2192 \u03a3S' M \u03bb m \u2192 \u03a0' R \u03bb r \u2192 \u03a3' (Sing m) (\u03bb _ \u2192 end)\n\ncommit : \u2200 M R (m : M)  \u2192 \u27e6 Commit M R \u27e7\ncommit M R m = [ m ] , (\u03bb x \u2192 (sing m) , _)\n\ndecommit : \u2200 M R (r : R) \u2192 \u27e6 dual (Commit M R) \u27e7\ndecommit M R r = \u03bb { [ m ] \u2192 r , (\u03bb x \u2192 {!!}) }\n\ndata [_&_\u2261_]InOut : InOut \u2192 InOut \u2192 InOut \u2192 \u2605\u2081 where\n  \u03a0XX : \u2200 {X} \u2192 [ In & X  \u2261 X ]InOut\n  X\u03a0X : \u2200 {X} \u2192 [ X  & In \u2261 X ]InOut\n\n&InOut-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]InOut \u2192 [ Q & P \u2261 R ]InOut\n&InOut-comm \u03a0XX = X\u03a0X\n&InOut-comm X\u03a0X = \u03a0XX\n\n  {-\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end& : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  &end : \u2200 {P} \u2192 [ P & end \u2261 P ]\n  XXX  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ com qP    M  P & com qQ    M  Q \u2261 com qR M R ]\n  -- SDD  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ com qP (\u2610 M) P & com qQ    M  Q \u2261 com qR M R ]\n  DSD  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ com qP    M  P & com qQ (\u2610 M) Q \u2261 com qR M R ]\n\n&-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n&-comm end& = &end\n&-comm &end = end&\n&-comm (XXX q P&) = XXX (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (SDD q P&) = DSD (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (DSD q P&) = SDD (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n-}\n\nDualInOut-sym : \u2200 {P Q} \u2192 DualInOut P Q \u2192 DualInOut Q P\nDualInOut-sym DInOut = DOutIn\nDualInOut-sym DOutIn = DInOut\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDualInOut-spec : \u2200 P \u2192 DualInOut P (dualInOut P)\nDualInOut-spec In  = DInOut\nDualInOut-spec Out = DOutIn\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (com In M P) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-spec (P x))\nDual-spec (com Out M P) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-spec (P x))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n\n{-\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M P} \u2192 (..(m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' (\u2610 M) P)\nrecvS = ?\nsendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' S M P))\nsendS = ?\n-}\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q M P Q} \u2192 SimL (com q M P) Q \u2192 Sim (com q M P) Q\n    right : \u2200 {P q M Q} \u2192 SimR P (com q M Q) \u2192 Sim P (com q M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recv PQA)   (send m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (send m PQB) = send m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (send m PQA) (recv PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (send x\u2081 x\u2082) (recv x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = send m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = ? -- send m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb x \u2192 left (send x (sim-id (B x)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb x \u2192 right (send x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u00b7\u03a3 x\u2081) (recv x) (send x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u00b7\u03a0 x)  (send x\u2081 x\u2082) (recv x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recv PQ) QR = recv (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recv P)) = do (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (send m P)) = do (send m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (send x x\u2082)) (left (recv x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (right (recv x)) (left (send x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (right (send x x\u2082)) (left (recv x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u00b7\u03a0 x) Q-Q' (right (send x\u2081 x\u2082)) (left (recv x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' (\u03a0\u00b7\u03a3 x\u2081) (right \u00f8P) (right (recv x)) (left (send x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u00b7\u03a0 x) (right \u00f8P) (right (send x\u2081 x\u2082)) (left (recv x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recv x\u2081))\n    = cong (right \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (send x\u2081 x\u2082))\n    = cong (right \u2218 send x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recv x))    = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (send x x\u2081)) = cong (left \u2218 send x) (sim-!! x\u2081)\n  sim-!! (right (recv x))    = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (send x x\u2081)) = cong (right \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (right (recv x)) (left (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (right (send x\u2081 x\u2082)) (left (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recv x\u2081))\n    = cong (left \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (send x\u2081 x\u2082))\n    = cong (left \u2218 send x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"18227b35dbe96a254f64b4a6adfb56b2f345d4ff","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: b556b22f21ca044a6c7ddf7e1b7f84c5\n\ndarcs-hash:20120105125316-3bd4e-49c60ec209442b07dc5649ba80697225c37662ac.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/MainATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/MainATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.Properties.MainATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.LT2MCR-ATP\nopen import FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Program.McCarthy91.Properties.AuxiliaryATP\n\n------------------------------------------------------------------------------\n\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 N (mc91 n) \u2227 LT n (mc91 n + eleven)\nmc91-N-ineq = wfInd-MCR P mc91-N-ineq-aux\n  where\n  P : D \u2192 Set\n  P d = N (mc91 d) \u2227 LT d (mc91 d + eleven)\n\n  mc91-N-ineq-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-N-ineq-aux {m} Nm f with x>y\u2228x\u2264y Nm 100-N\n  ... | inj\u2081 m>100 = ( Nmc91>100 Nm m>100 , x<mc91x+11>100 Nm m>100 )\n  ... | inj\u2082 m\u2264100 =\n    let Nm+11 : N (m + eleven)\n        Nm+11 = x+11-N Nm\n\n        ih1 : P (m + eleven)\n        ih1 = f Nm+11 (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n\n        Nih1 : N (mc91 (m + eleven))\n        Nih1 = \u2227-proj\u2081 ih1\n\n        LTih1 : LT (m + eleven) (mc91 (m + eleven) + eleven)\n        LTih1 = \u2227-proj\u2082 ih1\n\n        m<mc91m+11 : LT m (mc91 (m + eleven))\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm Nih1 11-N LTih1\n\n        ih2 : P (mc91 (m + eleven))\n        ih2 = f Nih1 (LT2MCR Nih1 Nm m\u2264100 m<mc91m+11)\n\n        m\u226f100 : NGT m one-hundred\n        m\u226f100 = x\u2264y\u2192x\u226fy Nm 100-N m\u2264100\n\n        Nmc91\u2264100 : N (mc91 m)\n        Nmc91\u2264100 = Nmc91\u226f100 m m\u226f100 (\u2227-proj\u2081 ih2)\n\n    in ( Nmc91\u2264100 ,\n         <-trans Nm Nih1 (x+11-N Nmc91\u2264100)\n                 m<mc91m+11\n                 (mc91x+11<mc91x+11 m m\u226f100 (\u2227-proj\u2082 ih2)))\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (GT n one-hundred \u2227 mc91 n \u2261 n \u2238 ten) \u2228\n                         (NGT n one-hundred \u2227 mc91 n \u2261 ninety-one)\nmc91-res = wfInd-MCR P mc91-res-aux\n  where\n  P : D \u2192 Set\n  P d = (GT d one-hundred \u2227 mc91 d \u2261 d \u2238 ten) \u2228\n        (NGT d one-hundred \u2227 mc91 d \u2261 ninety-one)\n\n  mc91-res-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-res-aux {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq-aux m m>100 )\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u226f100 , mc91-res-100' m\u2261100 )\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = inj\u2082 ( m\u226f100 , mc91-res-99' m\u226199 )\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = inj\u2082 ( m\u226f100 , mc91-res-98' m\u226198 )\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = inj\u2082 ( m\u226f100 , mc91-res-97' m\u226197 )\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = inj\u2082 ( m\u226f100 , mc91-res-96' m\u226196 )\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = inj\u2082 ( m\u226f100 , mc91-res-95' m\u226195 )\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = inj\u2082 ( m\u226f100 , mc91-res-94' m\u226194 )\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = inj\u2082 ( m\u226f100 , mc91-res-93' m\u226193 )\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = inj\u2082 ( m\u226f100 , mc91-res-92' m\u226192 )\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = inj\u2082 ( m\u226f100 , mc91-res-91' m\u226191 )\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = inj\u2082 ( m\u226f100 , mc91-res-90' m\u226190 )\n  ... | inj\u2081 m\u226f89 = inj\u2082 ( m\u226f100 , mc91-res-m\u226f89 )\n    where\n    m\u2264100 : LE m one-hundred\n    m\u2264100 = x\u226fy\u2192x\u2264y Nm 100-N m\u226f100\n\n    m\u226489 : LE m eighty-nine\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : mc91 (m + eleven) \u2261 ninety-one\n    mc91-res-m+11 with f (x+11-N Nm) (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n    ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm m\u226489 m+11>100)\n    ... | inj\u2082 ( _ , res ) = res\n\n    mc91-res-m\u226f89 : mc91 m \u2261 ninety-one\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m ninety-one m\u226f100 mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn = \u2227-proj\u2081 (mc91-N-ineq Nn)\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 LT n (mc91 n + eleven)\nmc91-ineq Nn = \u2227-proj\u2082 (mc91-N-ineq Nn)\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\nmc91-res>100 Nn n>100 with mc91-res Nn\n... | inj\u2081 ( _     , res ) = res\n... | inj\u2082 ( n\u226f100 , _ )   = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 NGT n one-hundred \u2192 mc91 n \u2261 ninety-one\nmc91-res\u226f100 Nn n\u226f100 with mc91-res Nn\n... | inj\u2081 ( n>100 , _   ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n... | inj\u2082 ( _     , res ) = res\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.Properties.MainATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.LT2MCR-ATP\nopen import FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Program.McCarthy91.Properties.AuxiliaryATP\n\n------------------------------------------------------------------------------\n\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 N (mc91 n) \u2227 LT n (mc91 n + eleven)\nmc91-N-ineq = wfInd-MCR P mc91-N-ineq-aux\n  where\n  P : D \u2192 Set\n  P d = N (mc91 d) \u2227 LT d (mc91 d + eleven)\n\n  mc91-N-ineq-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-N-ineq-aux {m} Nm f with x>y\u2228x\u2264y Nm 100-N\n  ... | inj\u2081 m>100 = ( Nmc91>100 Nm m>100 , x<mc91x+11>100 Nm m>100 )\n  ... | inj\u2082 m\u2264100 =\n    let Nm+11 : N (m + eleven)\n        Nm+11 = x+11-N Nm\n\n        ih1 : P (m + eleven)\n        ih1 = f Nm+11 (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n\n        Nih1 : N (mc91 (m + eleven))\n        Nih1 = \u2227-proj\u2081 ih1\n\n        LTih1 : LT (m + eleven) (mc91 (m + eleven) + eleven)\n        LTih1 = \u2227-proj\u2082 ih1\n\n        m<mc91m+11 : LT m (mc91 (m + eleven))\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm Nih1 11-N LTih1\n\n        ih2 : P (mc91 (m + eleven))\n        ih2 = f Nih1 (LT2MCR Nih1 Nm m\u2264100 m<mc91m+11)\n\n        m\u226f100 : NGT m one-hundred\n        m\u226f100 = x\u2264y\u2192x\u226fy Nm 100-N m\u2264100\n\n        Nmc91\u2264100 : N (mc91 m)\n        Nmc91\u2264100 = Nmc91\u226f100 m m\u226f100 (\u2227-proj\u2081 ih2)\n\n    in ( Nmc91\u2264100 ,\n         <-trans Nm Nih1 (x+11-N Nmc91\u2264100)\n                 m<mc91m+11\n                 (mc91x+11<mc91x+11 m m\u226f100 (\u2227-proj\u2082 ih2)))\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (GT n one-hundred \u2227 mc91 n \u2261 n \u2238 ten) \u2228\n                         (NGT n one-hundred \u2227 mc91 n \u2261 ninety-one)\nmc91-res = wfInd-MCR P mc91-res-aux\n  where\n  P : D \u2192 Set\n  P d = (GT d one-hundred \u2227 mc91 d \u2261 d \u2238 ten) \u2228\n        (NGT d one-hundred \u2227 mc91 d \u2261 ninety-one)\n\n  mc91-res-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-res-aux {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq-aux m m>100 )\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u226f100 , mc91-res-100' m\u2261100 )\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = inj\u2082 ( m\u226f100 , mc91-res-99' m\u226199 )\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = inj\u2082 ( m\u226f100 , mc91-res-98' m\u226198 )\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = inj\u2082 ( m\u226f100 , mc91-res-97' m\u226197 )\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = inj\u2082 ( m\u226f100 , mc91-res-96' m\u226196 )\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = inj\u2082 ( m\u226f100 , mc91-res-95' m\u226195 )\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = inj\u2082 ( m\u226f100 , mc91-res-94' m\u226194 )\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = inj\u2082 ( m\u226f100 , mc91-res-93' m\u226193 )\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = inj\u2082 ( m\u226f100 , mc91-res-92' m\u226192 )\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = inj\u2082 ( m\u226f100 , mc91-res-91' m\u226191 )\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = inj\u2082 ( m\u226f100 , mc91-res-90' m\u226190 )\n  ... | inj\u2081 m\u226f89 = inj\u2082 ( m\u226f100 , mc91-res-m\u226f89 )\n\n    where\n      m\u2264100 : LE m one-hundred\n      m\u2264100 = x\u226fy\u2192x\u2264y Nm 100-N m\u226f100\n\n      m\u226489 : LE m eighty-nine\n      m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n      mc91-res-m+11 : mc91 (m + eleven) \u2261 ninety-one\n      mc91-res-m+11 with f (x+11-N Nm) (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n      ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm m\u226489 m+11>100)\n      ... | inj\u2082 ( _ , res ) = res\n\n      mc91-res-m\u226f89 : mc91 m \u2261 ninety-one\n      mc91-res-m\u226f89 = mc91x-res\u226f100 m ninety-one m\u226f100 mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn = \u2227-proj\u2081 (mc91-N-ineq Nn)\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 LT n (mc91 n + eleven)\nmc91-ineq Nn = \u2227-proj\u2082 (mc91-N-ineq Nn)\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\nmc91-res>100 Nn n>100 with mc91-res Nn\n... | inj\u2081 ( _     , res ) = res\n... | inj\u2082 ( n\u226f100 , _ )   = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 NGT n one-hundred \u2192 mc91 n \u2261 ninety-one\nmc91-res\u226f100 Nn n\u226f100 with mc91-res Nn\n... | inj\u2081 ( n>100 , _   ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n... | inj\u2082 ( _     , res ) = res\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db67836522d8449804ce9b344f24877546eafb80","subject":"Only white space.","message":"Only white space.\n\nIgnore-this: e576b6173e37484c807272422296be50\n\ndarcs-hash:20100505225855-3bd4e-3eb9725e590920b79243fbc6d8a973b84fcfe044.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Function.Arithmetic.PropertiesER\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sx\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sx\u22640\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m + n) (succ m)   \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m + n) (succ m) \u2261\n                                               lt t (succ m))\n                                        (+-Sx m n)\n                                        refl\n                               \u27e9\n    lt (succ (m + n)) (succ m) \u2261\u27e8 lt-SS (m + n) m \u27e9\n    lt (m + n) m               \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    false\n  \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS m n)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Function.Arithmetic.PropertiesER\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sx\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sx\u22640\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m + n) (succ m)   \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m + n) (succ m) \u2261\n                                               lt t (succ m))\n                                        (+-Sx m n)\n                                        refl\n                               \u27e9\n    lt (succ (m + n)) (succ m) \u2261\u27e8 lt-SS (m + n) m \u27e9\n    lt (m + n) m               \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    false\n  \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 =\n  trans (+-rightIdentity (minus-N (sN Nm) zN)) (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS m n)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"886648677722c9cb94f5a4a0191634f396112553","subject":"[ Agda ] Agda's issue 1342 was fixed","message":"[ Agda ] Agda's issue 1342 was fixed\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GFPs\/Conat.agda","new_file":"notes\/fixed-points\/GFPs\/Conat.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas     #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule GFPs.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-out-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e.\n  --\n  -- \u2200 A. A \u2264 NatF A \u21d2 A \u2264 Conat.\n  Conat-coind :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial predicate A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification\/proof for\n  -- this principle.\n  Conat-stronger-coind\u2081 :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n  -- Other stronger co-induction principle\n  --\n  -- Adapted from (Paulson, 1997. p. 16).\n  Conat-stronger-coind\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 A n \u2192 (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2228 Conat n) \u2192\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-out and Conat-out-ho are equivalents\n\nConat-out-fo : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out-fo = Conat-out-ho\n\nConat-out-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-out-ho' = Conat-out\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind-fo :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-fo = Conat-coind-ho\n\nConat-coind-ho' :\n  (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e.\n--\n-- NatF Conat \u2264 Conat.\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in h = Conat-coind A h' h\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  h' : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h' (inj\u2081 n\u22610)              = inj\u2081 n\u22610\n  h' (inj\u2082 (n' , prf , Cn')) = inj\u2082 (n' , prf , Conat-out Cn')\n\n-- The higher-order version.\nConat-in-ho : \u2200 {n} \u2192 NatF Conat n \u2192 Conat n\nConat-in-ho = Conat-in\n\n-- A different definition.\nConat-in' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n            \u2200 {n} \u2192 Conat n\nConat-in' h = Conat-coind (\u03bb m \u2192 m \u2261 m) (h' h) refl\n  where\n  h' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n       \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' h'' {m} _ with (h'' {m})\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\nConat-in-ho' : (\u2200 {n} \u2192 NatF Conat n) \u2192 \u2200 {n} \u2192 Conat n\nConat-in-ho' = Conat-in'\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-stronger-coind\u2081 to Conat-stronger-coind\u2081\/Conat-coind\n\nConat-coind'' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-stronger-coind\u2081 A h An\n\n-- 22 December 2013: We couln't prove Conat-stronger-coind\u2081 using\n-- Conat-coind.\nConat-stronger-coind\u2081' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind\u2081' A {n} h An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\n------------------------------------------------------------------------------\n-- From Conat-stronger-coind\u2082 to Conat-stronger-coind\u2081\n\n-- 13 January 2014: We couln't prove Conat-stronger-coind\u2081 using\n-- Conat-stronger-coind\u2082.\nConat-stronger-coind\u2081'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind\u2081'' A h An = Conat-stronger-coind\u2082 A {!!} An\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","old_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas     #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule GFPs.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-out-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e.\n  --\n  -- \u2200 A. A \u2264 NatF A \u21d2 A \u2264 Conat.\n  Conat-coind :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial predicate A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification\/proof for\n  -- this principle.\n  Conat-stronger-coind\u2081 :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n  -- Other stronger co-induction principle\n  --\n  -- Adapted from (Paulson, 1997. p. 16).\n  Conat-stronger-coind\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 A n \u2192 (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2228 Conat n) \u2192\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-out and Conat-out-ho are equivalents\n\nConat-out-fo : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out-fo = Conat-out-ho\n\nConat-out-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-out-ho' = Conat-out\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind-fo :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-fo = Conat-coind-ho\n\nConat-coind-ho' :\n  (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e.\n--\n-- NatF Conat \u2264 Conat.\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in h = Conat-coind A h' h\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  h' : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h' (inj\u2081 n\u22610)              = inj\u2081 n\u22610\n  h' (inj\u2082 (n' , prf , Cn')) = inj\u2082 (n' , prf , Conat-out Cn')\n\n-- The higher-order version.\nConat-in-ho : \u2200 {n} \u2192 NatF Conat n \u2192 Conat n\nConat-in-ho = Conat-in\n\n-- A different definition.\nConat-in' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n            \u2200 {n} \u2192 Conat n\nConat-in' h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\nConat-in-ho' : (\u2200 {n} \u2192 NatF Conat n) \u2192 \u2200 {n} \u2192 Conat n\nConat-in-ho' = Conat-in'\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-stronger-coind\u2081 to Conat-stronger-coind\u2081\/Conat-coind\n\nConat-coind'' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-stronger-coind\u2081 A h An\n\n-- 22 December 2013: We couln't prove Conat-stronger-coind\u2081 using\n-- Conat-coind.\nConat-stronger-coind\u2081' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind\u2081' A {n} h An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\n------------------------------------------------------------------------------\n-- From Conat-stronger-coind\u2082 to Conat-stronger-coind\u2081\n\n-- 13 January 2014: We couln't prove Conat-stronger-coind\u2081 using\n-- Conat-stronger-coind\u2082.\nConat-stronger-coind\u2081'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind\u2081'' A h An = Conat-stronger-coind\u2082 A {!!} An\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b65f846ed1d44f640b5958eabb65b2f70f230ac3","subject":"Prove correctness of apply-term and diff-term (see #25).","message":"Prove correctness of apply-term and diff-term (see #25).\n\nOld-commit-hash: 30f988d1ecc240c3f9e95082b6b7506d53d58724\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Evaluation\/Total.agda","new_file":"Denotational\/Evaluation\/Total.agda","new_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\n{-\n\nHere is an example to understand the semantics of \u0394. I will use a\nnamed variable representation for the task.\n\nConsider the typing judgment:\n\n   x: T |- x: T\n\nThus, we have that:\n\n   dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nThanks to weakening, we also have:\n\n   y : S, dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nIn the formalization, we need a proof \u0393\u2032 that the context \u0393\u2081 = dx : \u0394\nT, x: T is a subcontext of \u0393\u2082 = y : S, dx : \u0394 T, x: T. Thus, \u0393\u2032 has\ntype \u0393\u2081 \u227c \u0393\u2082.\n\nNow take the environment:\n\n   \u03c1 = y \u21a6 w, dx \u21a6 dv, x \u21a6 v\n\nSince the semantics of \u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082 is a function from environments\nfor \u0393\u2082 to environments for \u0393\u2081, we have that:\n\n   \u27e6 \u0393\u2032 \u27e7 \u03c1 = dx \u21a6 dv, x \u21a6 v\n\nFrom the definitions of update and ignore, it follows that:\n\n   update (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 dv \u2295 v\n   ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 v\n\nHence, finally, we have that:\n\n   diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nis simply diff (dv \u2295 v) v (or (dv \u2295 v) \u229d v). If dv is a valid change,\nthat's just dv, that is \u27e6 dx \u27e7 \u03c1. In other words, if dv is a valid\nchange then \u27e6 \u0394 {{\u0393\u2032}} x \u27e7 \u03c1 \u2261 \u27e6 dx \u27e7 \u03c1 \u2261 \u27e6 derive-term x \u27e7 \u03c1. This\nfact, generalized for arbitrary terms, is proven formally by\nderive-term-correct.\n\n-}\n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n\n-- Simplification rules for weakening\n\n\u2261-weaken\u2070 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 weaken\u2070 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 \u03c1\n\u2261-weaken\u2070 t \u03c1 =\n  begin\n    \u27e6 weaken\u2070 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-refl \u27e7 \u03c1)\n  \u2261\u27e8 cong \u27e6 t \u27e7 (\u27e6\u27e7-\u227c-refl \u03c1) \u27e9\n    \u27e6 t \u27e7 \u03c1\n  \u220e where open \u2261-Reasoning\n\n\u2261-weaken\u00b9 : \u2200 {\u0393 \u03c4} {\u03c4\u2081 : Type} (t : Term \u0393 \u03c4) \u2192\n  \u2200 (v\u2081 : \u27e6 \u03c4\u2081 \u27e7) (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 weaken\u00b9 t \u27e7 (v\u2081 \u2022 \u03c1) \u2261 \u27e6 t \u27e7 \u03c1\n\u2261-weaken\u00b9 t v\u2081 \u03c1 =\n  begin\n    \u27e6 weaken\u00b9 t \u27e7 (v\u2081 \u2022 \u03c1)\n  \u2261\u27e8 weaken-sound t (v\u2081 \u2022 \u03c1) \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-refl \u27e7 \u03c1)\n  \u2261\u27e8 cong \u27e6 t \u27e7 (\u27e6\u27e7-\u227c-refl \u03c1) \u27e9\n    \u27e6 t \u27e7 \u03c1\n  \u220e where open \u2261-Reasoning\n\n\u2261-weaken\u00b2 : \u2200 {\u0393 \u03c4} {\u03c4\u2081 \u03c4\u2082 : Type} (t : Term \u0393 \u03c4) \u2192\n  \u2200 (v\u2081 : \u27e6 \u03c4\u2081 \u27e7) (v\u2082 : \u27e6 \u03c4\u2082 \u27e7) (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 weaken\u00b2 t \u27e7 (v\u2081 \u2022 v\u2082 \u2022 \u03c1) \u2261 \u27e6 t \u27e7 \u03c1\n\u2261-weaken\u00b2 t v\u2081 v\u2082 \u03c1 =\n  begin\n    \u27e6 weaken\u00b2 t \u27e7 (v\u2081 \u2022 v\u2082 \u2022 \u03c1)\n  \u2261\u27e8 weaken-sound t (v\u2081 \u2022 v\u2082 \u2022 \u03c1) \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-refl \u27e7 \u03c1)\n  \u2261\u27e8 cong \u27e6 t \u27e7 (\u27e6\u27e7-\u227c-refl \u03c1) \u27e9\n    \u27e6 t \u27e7 \u03c1\n  \u220e where open \u2261-Reasoning\n\n-- CORRECTNESS of NAMED TERMS\n\nxor-term-correct : \u2200 {\u0393} (t\u2081 t\u2082 : Term \u0393 bool) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 t\u2081 xor-term t\u2082 \u27e7 \u03c1 \u2261 \u27e6 t\u2081 \u27e7 \u03c1 xor \u27e6 t\u2082 \u27e7 \u03c1\nxor-term-correct t\u2081 t\u2082 \u03c1\n  with \u27e6 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2082 \u27e7 \u03c1\n... | true | true = refl\n... | true | false = refl\n... | false | true = refl\n... | false | false = refl\n\ndiff-term-correct : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 diff-term t\u2081 t\u2082 \u27e7 \u03c1 \u2261 diff (\u27e6 t\u2081 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 \u03c1)\n\napply-term-correct : \u2200 {\u03c4 \u0393} {t\u2081 : Term \u0393 (\u0394-Type \u03c4)} {t\u2082 : Term \u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 apply-term t\u2081 t\u2082 \u27e7 \u03c1 \u2261 apply (\u27e6 t\u2081 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 \u03c1)\n\ndiff-term-correct {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {f\u2081} {f\u2082} \u03c1 = ext (\u03bb dv \u2192 ext (\u03bb v \u2192\n  begin\n    (\u27e6 diff-term f\u2081 f\u2082 \u27e7 \u03c1) dv v\n  \u2261\u27e8\u27e9\n    \u27e6 diff-term\n       (app (weaken\u00b2 f\u2081) (apply-term (var this) (var (that this))))\n       (app (weaken\u00b2 f\u2082) (var (that this))) \u27e7 (v \u2022 dv \u2022 \u03c1)\n  \u2261\u27e8 diff-term-correct (v \u2022 dv \u2022 \u03c1) \u27e9\n    diff\n      (\u27e6 app (weaken\u00b2 f\u2081) (apply-term (var this) (var (that this))) \u27e7 (v \u2022 dv \u2022 \u03c1))\n      (\u27e6 app (weaken\u00b2 f\u2082) (var (that this)) \u27e7 (v \u2022 dv \u2022 \u03c1))\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 weaken\u00b2 f\u2081 \u27e7 (v \u2022 dv \u2022 \u03c1) (\u27e6 apply-term (var this) (var (that this)) \u27e7 (v \u2022 dv \u2022 \u03c1)))\n      (\u27e6 weaken\u00b2 f\u2082 \u27e7 (v \u2022 dv \u2022 \u03c1) dv)\n  \u2261\u27e8 \u2261-diff\n       (\u2261-app (\u2261-weaken\u00b2 f\u2081 v dv \u03c1) (apply-term-correct (v \u2022 dv \u2022 \u03c1)))\n       (\u2261-app (\u2261-weaken\u00b2 f\u2082 v dv \u03c1) \u2261-refl) \u27e9\n    diff\n      (\u27e6 f\u2081 \u27e7 \u03c1 (apply (\u27e6 var {_ \u2022 _ \u2022 _} this \u27e7 (v \u2022 dv \u2022 \u03c1)) (\u27e6 var (that this) \u27e7 (v \u2022 dv \u2022 \u03c1))))\n      (\u27e6 f\u2082 \u27e7 \u03c1 dv)\n  \u2261\u27e8\u27e9\n    diff (\u27e6 f\u2081 \u27e7 \u03c1 (apply v dv)) (\u27e6 f\u2082 \u27e7Term \u03c1 dv)\n  \u2261\u27e8\u27e9\n    diff (\u27e6 f\u2081 \u27e7 \u03c1) (\u27e6 f\u2082 \u27e7 \u03c1) dv v\n  \u220e)) where open \u2261-Reasoning\n\ndiff-term-correct {bool} {\u0393} {b\u2081} {b\u2082} \u03c1 =\n  begin\n    \u27e6 diff-term b\u2081 b\u2082 \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 b\u2081 xor-term b\u2082 \u27e7 \u03c1\n  \u2261\u27e8 xor-term-correct b\u2081 b\u2082 \u03c1 \u27e9\n    \u27e6 b\u2081 \u27e7 \u03c1 xor \u27e6 b\u2082 \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff (\u27e6 b\u2081 \u27e7 \u03c1) (\u27e6 b\u2082 \u27e7 \u03c1)\n  \u220e where open \u2261-Reasoning\n\napply-term-correct {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {df} {f} \u03c1 = ext (\u03bb v \u2192\n  begin\n    \u27e6 apply-term df f \u27e7 \u03c1 v\n  \u2261\u27e8\u27e9\n    \u27e6 abs (apply-term\n            (app (app (weaken\u00b9 df) (var this))\n                 (diff-term (var this) (var this)))\n            (app (weaken\u00b9 f) (var this))) \u27e7 \u03c1 v\n  \u2261\u27e8\u27e9\n    \u27e6 apply-term\n        (app (app (weaken\u00b9 df) (var this))\n             (diff-term (var this) (var this)))\n        (app (weaken\u00b9 f) (var this)) \u27e7 (v \u2022 \u03c1)\n  \u2261\u27e8 apply-term-correct (v \u2022 \u03c1) \u27e9\n    apply\n      (\u27e6 app (app (weaken\u00b9 df) (var this))\n             (diff-term (var this) (var this)) \u27e7 (v \u2022 \u03c1))\n      (\u27e6 app (weaken\u00b9 f) (var this) \u27e7 (v \u2022 \u03c1))\n  \u2261\u27e8\u27e9\n    apply\n      (\u27e6 weaken\u00b9 df \u27e7 (v \u2022 \u03c1) v\n        (\u27e6 diff-term (var this) (var this) \u27e7 (v \u2022 \u03c1)))\n      (\u27e6 weaken\u00b9 f \u27e7 (v \u2022 \u03c1) v)\n  \u2261\u27e8 \u2261-apply\n       (\u2261-app (\u2261-app (\u2261-weaken\u00b9 df v \u03c1) \u2261-refl)\n              (diff-term-correct (v \u2022 \u03c1)))\n       (\u2261-app (\u2261-weaken\u00b9 f v \u03c1) \u2261-refl) \u27e9\n    apply\n      (\u27e6 df \u27e7 \u03c1 v\n        (diff (\u27e6 var this \u27e7 (v \u2022 \u03c1)) (\u27e6 var this \u27e7 (v \u2022 \u03c1))))\n      (\u27e6 f \u27e7 \u03c1 v)\n  \u2261\u27e8\u27e9\n    apply (\u27e6 df \u27e7 \u03c1 v (diff v v)) (\u27e6 f \u27e7 \u03c1 v)\n  \u2261\u27e8 cong (\u03bb X \u2192 apply (\u27e6 df \u27e7 \u03c1 v X) (\u27e6 f \u27e7 \u03c1 v)) (diff-derive v) \u27e9\n    apply (\u27e6 df \u27e7 \u03c1 v (derive v)) (\u27e6 f \u27e7 \u03c1 v)\n  \u2261\u27e8\u27e9\n    apply (\u27e6 df \u27e7 \u03c1) (\u27e6 f \u27e7 \u03c1) v\n  \u220e) where open \u2261-Reasoning\n\napply-term-correct {bool} {\u0393} {db} {b} \u03c1 =\n  begin\n    \u27e6 apply-term db b \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 db xor-term b \u27e7 \u03c1\n  \u2261\u27e8 xor-term-correct db b \u03c1 \u27e9\n    \u27e6 db \u27e7 \u03c1 xor \u27e6 b \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    apply (\u27e6 db \u27e7 \u03c1) (\u27e6 b \u27e7 \u03c1)\n  \u220e where open \u2261-Reasoning\n","old_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\n{-\n\nHere is an example to understand the semantics of \u0394. I will use a\nnamed variable representation for the task.\n\nConsider the typing judgment:\n\n   x: T |- x: T\n\nThus, we have that:\n\n   dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nThanks to weakening, we also have:\n\n   y : S, dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nIn the formalization, we need a proof \u0393\u2032 that the context \u0393\u2081 = dx : \u0394\nT, x: T is a subcontext of \u0393\u2082 = y : S, dx : \u0394 T, x: T. Thus, \u0393\u2032 has\ntype \u0393\u2081 \u227c \u0393\u2082.\n\nNow take the environment:\n\n   \u03c1 = y \u21a6 w, dx \u21a6 dv, x \u21a6 v\n\nSince the semantics of \u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082 is a function from environments\nfor \u0393\u2082 to environments for \u0393\u2081, we have that:\n\n   \u27e6 \u0393\u2032 \u27e7 \u03c1 = dx \u21a6 dv, x \u21a6 v\n\nFrom the definitions of update and ignore, it follows that:\n\n   update (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 dv \u2295 v\n   ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 v\n\nHence, finally, we have that:\n\n   diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nis simply diff (dv \u2295 v) v (or (dv \u2295 v) \u229d v). If dv is a valid change,\nthat's just dv, that is \u27e6 dx \u27e7 \u03c1. In other words, if dv is a valid\nchange then \u27e6 \u0394 {{\u0393\u2032}} x \u27e7 \u03c1 \u2261 \u27e6 dx \u27e7 \u03c1 \u2261 \u27e6 derive-term x \u27e7 \u03c1. This\nfact, generalized for arbitrary terms, is proven formally by\nderive-term-correct.\n\n-}\n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n\n-- Simplification rules for weakening\n\n\u2261-weaken\u2070 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 weaken\u2070 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 \u03c1\n\u2261-weaken\u2070 t \u03c1 =\n  begin\n    \u27e6 weaken\u2070 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-refl \u27e7 \u03c1)\n  \u2261\u27e8 cong \u27e6 t \u27e7 (\u27e6\u27e7-\u227c-refl \u03c1) \u27e9\n    \u27e6 t \u27e7 \u03c1\n  \u220e where open \u2261-Reasoning\n\n\u2261-weaken\u00b9 : \u2200 {\u0393 \u03c4} {\u03c4\u2081 : Type} (t : Term \u0393 \u03c4) \u2192\n  \u2200 (v\u2081 : \u27e6 \u03c4\u2081 \u27e7) (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 weaken\u00b9 t \u27e7 (v\u2081 \u2022 \u03c1) \u2261 \u27e6 t \u27e7 \u03c1\n\u2261-weaken\u00b9 t v\u2081 \u03c1 =\n  begin\n    \u27e6 weaken\u00b9 t \u27e7 (v\u2081 \u2022 \u03c1)\n  \u2261\u27e8 weaken-sound t (v\u2081 \u2022 \u03c1) \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-refl \u27e7 \u03c1)\n  \u2261\u27e8 cong \u27e6 t \u27e7 (\u27e6\u27e7-\u227c-refl \u03c1) \u27e9\n    \u27e6 t \u27e7 \u03c1\n  \u220e where open \u2261-Reasoning\n\n\u2261-weaken\u00b2 : \u2200 {\u0393 \u03c4} {\u03c4\u2081 \u03c4\u2082 : Type} (t : Term \u0393 \u03c4) \u2192\n  \u2200 (v\u2081 : \u27e6 \u03c4\u2081 \u27e7) (v\u2082 : \u27e6 \u03c4\u2082 \u27e7) (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 weaken\u00b2 t \u27e7 (v\u2081 \u2022 v\u2082 \u2022 \u03c1) \u2261 \u27e6 t \u27e7 \u03c1\n\u2261-weaken\u00b2 t v\u2081 v\u2082 \u03c1 =\n  begin\n    \u27e6 weaken\u00b2 t \u27e7 (v\u2081 \u2022 v\u2082 \u2022 \u03c1)\n  \u2261\u27e8 weaken-sound t (v\u2081 \u2022 v\u2082 \u2022 \u03c1) \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-refl \u27e7 \u03c1)\n  \u2261\u27e8 cong \u27e6 t \u27e7 (\u27e6\u27e7-\u227c-refl \u03c1) \u27e9\n    \u27e6 t \u27e7 \u03c1\n  \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6942f545a655646a6bfe7b3fd9b3dce0fbbb8e64","subject":"massaging notation","message":"massaging notation\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes anywhere\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those expressions without holes anywhere\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only know about complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- application of a substution to a term\n  postulate\n    apply : subst \u2192 dhexp \u2192 dhexp\n  -- apply \u03c3 c = c\n  -- apply \u03c3 (X x) with \u03c3 x\n  -- apply \u03c3 (X x) | Some d' = d'\n  -- apply \u03c3 (X x) | None = X x\n  -- apply \u03c3 (\u00b7\u03bb x [ \u03c4 ] d) = (\u00b7\u03bb x [ \u03c4 ] (apply \u03c3 d))\n  -- apply \u03c3 \u2987\u2988\u27e8 u , \u03c3' \u27e9 =  \u2987\u2988\u27e8 u , ((\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))) \u27e9 -- (\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))\n  -- apply \u03c3 \u2987 d \u2988\u27e8 u , \u03c3' \u27e9 = \u2987 apply \u03c3 d \u2988\u27e8 u ,{!!} \u27e9\n  -- apply \u03c3 (d1 \u2218 d2) = ((apply \u03c3 d1) \u2218 (apply \u03c3 d2))\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  --hole instantiation; todo: judgemental or functional?\n  \u27e6_\/_\u27e7_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  \u27e6 d \/ u \u27e7 c = c\n  \u27e6 d \/ u \u27e7 X x = X x\n  \u27e6 d \/ u \u27e7 (\u00b7\u03bb x [ \u03c4 ] d') = \u00b7\u03bb x [ \u03c4 ] (\u27e6 d \/ u \u27e7 d')\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9  | Inr x = \u2987\u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9 | Inr x = \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 (d1 \u2218 d2) = (\u27e6 d \/ u \u27e7 d1) \u2218 (\u27e6 d \/ u \u27e7 d2)\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes anywhere\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those expressions without holes anywhere\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only know about complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (t : htyp) \u2192 (\u0393 x) == Some t \u2192 t tcomplete\n\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- application of a substution to a term\n  postulate\n    apply : subst \u2192 dhexp \u2192 dhexp\n  -- apply \u03c3 c = c\n  -- apply \u03c3 (X x) with \u03c3 x\n  -- apply \u03c3 (X x) | Some d' = d'\n  -- apply \u03c3 (X x) | None = X x\n  -- apply \u03c3 (\u00b7\u03bb x [ \u03c4 ] d) = (\u00b7\u03bb x [ \u03c4 ] (apply \u03c3 d))\n  -- apply \u03c3 \u2987\u2988\u27e8 u , \u03c3' \u27e9 =  \u2987\u2988\u27e8 u , ((\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))) \u27e9 -- (\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))\n  -- apply \u03c3 \u2987 d \u2988\u27e8 u , \u03c3' \u27e9 = \u2987 apply \u03c3 d \u2988\u27e8 u ,{!!} \u27e9\n  -- apply \u03c3 (d1 \u2218 d2) = ((apply \u03c3 d1) \u2218 (apply \u03c3 d2))\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  --hole instantiation; todo: judgemental or functional?\n  \u27e6_\/_\u27e7_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  \u27e6 d \/ u \u27e7 c = c\n  \u27e6 d \/ u \u27e7 X x = X x\n  \u27e6 d \/ u \u27e7 (\u00b7\u03bb x [ \u03c4 ] d') = \u00b7\u03bb x [ \u03c4 ] (\u27e6 d \/ u \u27e7 d')\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9  | Inr x = \u2987\u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9 | Inr x = \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 (d1 \u2218 d2) = (\u27e6 d \/ u \u27e7 d1) \u2218 (\u27e6 d \/ u \u27e7 d2)\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0ecf1536814a880e8cdbe60b6eef1ff1106504bf","subject":"make main theorem correspond to the one stated in the paper","message":"make main theorem correspond to the one stated in the paper\n\nOld-commit-hash: 84979b2067c54a685110f1bad744ba106a962a0f\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Correctness.agda","new_file":"Parametric\/Change\/Correctness.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  derive-correct-closed : \u2200 {\u03c4} (t : Term \u2205 \u03c4) \u2192\n    \u27e6 t \u27e7\u0394 \u2205 \u2205 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 \u2205\n\n  derive-correct-closed t = derive-correct t \u2205 \u2205 \u2205 \u2205\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {s : Term \u2205 \u03c3} {ds : Term \u2205 (\u0394Type \u03c3)} \u2192\n    {dv : \u0394\u208d \u03c3 \u208e (\u27e6 s \u27e7 \u2205)} {erasure : dv \u2248\u208d \u03c3 \u208e (\u27e6 ds \u27e7 \u2205)} \u2192\n\n    \u27e6 app f (s \u2295\u208d \u03c3 \u208e ds) \u27e7 \u2261 \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {s} {ds} {dv} {erasure} =\n    let\n      g  = \u27e6 f \u27e7 \u2205\n      \u0394g = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394g\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 s \u27e7 \u2205\n      dv\u2032 = \u27e6 ds \u27e7 \u2205\n      u  = \u27e6 s \u2295\u208d \u03c3 \u208e ds \u27e7 \u2205\n      -- \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        g u\n      \u2261\u27e8 cong g (sym (meaning-\u2295 {t = s} {\u0394t = ds})) \u27e9\n        g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 dv\u2032)\n      \u2261\u27e8 cong g (sym (carry-over {\u03c3} dv erasure)) \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v dv \u27e9\n        g v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394g v dv\n      \u2261\u27e8 carry-over {\u03c4} (call-change {\u03c3} {\u03c4} \u0394g v dv)\n           (derive-correct f \u2205 \u2205 \u2205 \u2205 v dv dv\u2032 erasure) \u27e9\n        g v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394g\u2032 v dv\u2032\n      \u2261\u27e8 meaning-\u2295 {t = app f s} {\u0394t = app (app (derive f) s) ds} \u27e9\n        \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  -- Our main theorem, as we used to state it in the paper.\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {y : Term \u2205 \u03c3}\n    \u2192 \u27e6 app f y \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {x} {y} =\n    let\n      h  = \u27e6 f \u27e7 \u2205\n      \u0394h = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394h\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 x \u27e7 \u2205\n      u  = \u27e6 y \u27e7 \u2205\n      \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        h u\n      \u2261\u27e8 cong h (sym (update-diff\u208d \u03c3 \u208e u v)) \u27e9\n        h (v \u229e\u208d \u03c3 \u208e (u \u229f\u208d \u03c3 \u208e v))\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v (u \u229f\u208d \u03c3 \u208e v) \u27e9\n        h v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v)\n      \u2261\u27e8 carry-over {\u03c4}\n         (call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v))\n         (derive-correct f\n           \u2205 \u2205 \u2205 \u2205 v (u \u229f\u208d \u03c3 \u208e v) (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) (u\u229fv\u2248u\u229dv {\u03c3} {u} {v})) \u27e9\n         h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 trans\n         (cong (\u03bb hole \u2192 h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v hole) (meaning-\u229d {\u03c3} {s = y} {x}))\n         (meaning-\u2295 {t = app f x} {\u0394t = \u0394output-term}) \u27e9\n         \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n\n  -- A corollary, closer to what we state in the paper.\n  main-theorem-coroll : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {dx : Term \u2205 (\u0394Type \u03c3)}\n    \u2192 \u27e6 app f (x \u2295\u208d \u03c3 \u208e dx) \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) ((x \u2295\u208d \u03c3 \u208e dx) \u229d x) \u27e7\n  main-theorem-coroll {\u03c3} {\u03c4} {f} {x} {dx} = main-theorem {\u03c3} {\u03c4} {f} {x} {x \u2295\u208d \u03c3 \u208e dx}\n\n  -- For the statement in the paper, we'd need to talk about valid changes in\n  -- the lambda calculus. In fact we can, thanks to the `implements` relation;\n  -- but I guess the required proof must be done directly from derive-correct,\n  -- not from main-theorem.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"56284e892af435d1f8e9968c5f9e7aaa7fa61706","subject":"Agda: start reflecting back changes to the main theorem in the paper","message":"Agda: start reflecting back changes to the main theorem in the paper\n\nWe simplified the statement in the paper by talking about valid changes.\nThough this is a minor change, this should be reflected in Agda, also to\nget an exact statement for the paper.\n\nThis commit discusses what to do; I didn't try writing the real\nstatement.\n\nOld-commit-hash: 0d78ce6f1fab80aa61430ee3241ca2c7080028ba\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Correctness.agda","new_file":"Parametric\/Change\/Correctness.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  -- Our main theorem, as we used to state it in the paper.\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {y : Term \u2205 \u03c3}\n    \u2192 \u27e6 app f y \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {x} {y} =\n    let\n      h  = \u27e6 f \u27e7 \u2205\n      \u0394h = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394h\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 x \u27e7 \u2205\n      u  = \u27e6 y \u27e7 \u2205\n      \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        h u\n      \u2261\u27e8 cong h (sym (update-diff\u208d \u03c3 \u208e u v)) \u27e9\n        h (v \u229e\u208d \u03c3 \u208e (u \u229f\u208d \u03c3 \u208e v))\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v (u \u229f\u208d \u03c3 \u208e v) \u27e9\n        h v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v)\n      \u2261\u27e8 carry-over {\u03c4}\n         (call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v))\n         (derive-correct f\n           \u2205 \u2205 \u2205 \u2205 v (u \u229f\u208d \u03c3 \u208e v) (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) (u\u229fv\u2248u\u229dv {\u03c3} {u} {v})) \u27e9\n         h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 trans\n         (cong (\u03bb hole \u2192 h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v hole) (meaning-\u229d {\u03c3} {s = y} {x}))\n         (meaning-\u2295 {t = app f x} {\u0394t = \u0394output-term}) \u27e9\n         \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n\n  -- A corollary, closer to what we state in the paper.\n  main-theorem-coroll : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {dx : Term \u2205 (\u0394Type \u03c3)}\n    \u2192 \u27e6 app f (x \u2295\u208d \u03c3 \u208e dx) \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) ((x \u2295\u208d \u03c3 \u208e dx) \u229d x) \u27e7\n  main-theorem-coroll {\u03c3} {\u03c4} {f} {x} {dx} = main-theorem {\u03c3} {\u03c4} {f} {x} {x \u2295\u208d \u03c3 \u208e dx}\n\n  -- For the statement in the paper, we'd need to talk about valid changes in\n  -- the lambda calculus. In fact we can, thanks to the `implements` relation;\n  -- but I guess the required proof must be done directly from derive-correct,\n  -- not from main-theorem.\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {y : Term \u2205 \u03c3}\n    \u2192 \u27e6 app f y \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {x} {y} =\n    let\n      h  = \u27e6 f \u27e7 \u2205\n      \u0394h = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394h\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 x \u27e7 \u2205\n      u  = \u27e6 y \u27e7 \u2205\n      \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        h u\n      \u2261\u27e8 cong h (sym (update-diff\u208d \u03c3 \u208e u v)) \u27e9\n        h (v \u229e\u208d \u03c3 \u208e (u \u229f\u208d \u03c3 \u208e v))\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v (u \u229f\u208d \u03c3 \u208e v) \u27e9\n        h v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v)\n      \u2261\u27e8 carry-over {\u03c4}\n         (call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v))\n         (derive-correct f\n           \u2205 \u2205 \u2205 \u2205 v (u \u229f\u208d \u03c3 \u208e v) (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) (u\u229fv\u2248u\u229dv {\u03c3} {u} {v})) \u27e9\n         h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 trans\n         (cong (\u03bb hole \u2192 h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v hole) (meaning-\u229d {\u03c3} {s = y} {x}))\n         (meaning-\u2295 {t = app f x} {\u0394t = \u0394output-term}) \u27e9\n         \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9de1d2be11ad3758770500baab5298c68874ca56","subject":"uploading all","message":"uploading all\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/WellFormedLF.agda","new_file":"agda\/WellFormedLF.agda","new_contents":"module WellFormedLF where\n\nopen import Common\nopen import LinLogic\nopen import SetLL\n\n--data IndexLF : \u2200{u} \u2192 {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set where\n--  \u2193    : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF elf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  _\u2282\u2192_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF lf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  tr   : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u ll orll rll} \u2192 {{ltr : LLTr orll ll}} \u2192 {lf : LFun {u} {i} {j} {rll} {orll}}\n--         \u2192 IndexLF lf \u2192 IndexLF (tr {{ltr = ltr}} lf) \n--\n\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n\n--                                       \u2193 probably the subtrees that contain all the inputs. \n-- We need to keep truck of all the latest subtrees that are outputs of coms. We can then check whether a transformation permutates them. If so , the tr is acceptable.\n-- transformations inside a subtree are acceptable.\n-- tranformations between two subtrees are only acceptable if the next com depends on both of them.\n-- If more than one subtree depends on a specific subtree, but the two do not depend on each other, we first separate the elements for the first subtree and then for the other.\n-- Since these coms can be independently executed, there can be tranformations of one subtree that can not be done while the others are ready. --> We remove the separation transformation and consider that\n-- the unexecuted transformations are done on those separated elements.\n\n-- > The input tree contains subtrees that represent exactly the input of the coms it can execute.\n\n-- 2 TWO\n-- We need three structures.\n-- THe first is used to identify inputs from the same com. When a tranformation splits these coms, we also split the set into two sets. Next we track all the inputs that are part of the tranformation, be it sets of inputs of a specific com or individual inputs. From all these sets, at least one item from each set needs to be the input of the next com.\n-- The set of coms that are used to allow for commutation of inputs.\n\nmodule _ where\n\n  open import Data.Vec\n  \n  data Ancestor : Set where\n    orig  : Ancestor\n    anc   : \u2115 \u2192 Ancestor \u2192 Ancestor\n    manc  : \u2115 \u2192 \u2200{n} \u2192 Vec Ancestor n \u2192 Ancestor \u2192 Ancestor\n\n\n  \n  data SetLLD {i : Size} {u} : LinLogic i {u} \u2192 Set (lsuc u) where\n    \u2193     : \u2200{ll} \u2192  Ancestor               \u2192 SetLLD ll\n    _\u2190\u2227   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2227 rs)\n    \u2227\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2227 rs)\n    _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2227 rs)\n    _\u2190\u2228   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2228 rs)\n    \u2228\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2228 rs)\n    _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2228 rs)\n    _\u2190\u2202   : \u2200{rs ls} \u2192 SetLLD ls            \u2192 SetLLD (ls \u2202 rs)\n    \u2202\u2192_   : \u2200{rs ls} \u2192 SetLLD rs            \u2192 SetLLD (ls \u2202 rs)\n    _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLLD ls \u2192 SetLLD rs \u2192 SetLLD (ls \u2202 rs)\n    \n  \n  \n  data MSetLLD {i : Size} {u} : LinLogic i {u} \u2192 Set (lsuc u) where\n    \u2205   : \u2200{ll}             \u2192 MSetLLD ll\n    \u00ac\u2205  : \u2200{ll} \u2192 SetLLD ll \u2192 MSetLLD ll\n  \n  -- TODO We shouldn't need this. When issue agda #2409 is resolved, remove this.\n  drsize : \u2200{i u ll} \u2192 {j : Size< \u2191 i} \u2192 SetLLD {i} {u} ll \u2192 SetLLD {j} ll\n  drsize (\u2193 mm)          = (\u2193 mm)\n  drsize (x \u2190\u2227)     = (drsize x) \u2190\u2227\n  drsize (\u2227\u2192 x)     = \u2227\u2192 (drsize x)\n  drsize (x \u2190\u2227\u2192 x\u2081) = (drsize x \u2190\u2227\u2192 drsize x\u2081)\n  drsize (x \u2190\u2228)     = (drsize x) \u2190\u2228\n  drsize (\u2228\u2192 x)     = \u2228\u2192 (drsize x)\n  drsize (x \u2190\u2228\u2192 x\u2081) = (drsize x \u2190\u2228\u2192 drsize x\u2081)\n  drsize (x \u2190\u2202)     = (drsize x) \u2190\u2202\n  drsize (\u2202\u2192 x)     = \u2202\u2192 (drsize x)\n  drsize (x \u2190\u2202\u2192 x\u2081) = (drsize x \u2190\u2202\u2192 drsize x\u2081)\n  \n  fillAllLowerD : \u2200{i u} \u2192 \u2200 ll \u2192 SetLLD {i} {u} ll\n  fillAllLowerD ll = {!!}\n  \n  \n  \n","old_contents":"module WellFormedLF where\n\nopen import Common\nopen import LinLogic\nimport SetLL\n\n--data IndexLF : \u2200{u} \u2192 {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set where\n--  \u2193    : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF elf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  _\u2282\u2192_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{u rll pll ell ll ind elf prf lf}\n--         \u2192 IndexLF lf\n--         \u2192 IndexLF (_\u2282_ {u} {i} {j} {k} {pll} {ll} {ell} {rll} {ind} elf {{prf}} lf)\n--  tr   : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u ll orll rll} \u2192 {{ltr : LLTr orll ll}} \u2192 {lf : LFun {u} {i} {j} {rll} {orll}}\n--         \u2192 IndexLF lf \u2192 IndexLF (tr {{ltr = ltr}} lf) \n--\n\n--  _\u2190\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 IndexLF lf\n\n--                                       \u2193 probably the subtrees that contain all the inputs. \n-- We need to keep truck of all the latest subtrees that are outputs of coms. We can then check whether a transformation permutates them. If so , the tr is acceptable.\n-- transformations inside a subtree are acceptable.\n-- tranformations between two subtrees are only acceptable if the next com depends on both of them.\n-- If more than one subtree depends on a specific subtree, but the two do not depend on each other, we first separate the elements for the first subtree and then for the other.\n-- Since these coms can be independently executed, there can be tranformations of one subtree that can not be done while the others are ready. --> We remove the separation transformation and consider that\n-- the unexecuted transformations are done on those separated elements.\n\n-- > The input tree contains subtrees that represent exactly the input of the coms it can execute.\n\n-- 2 TWO\n-- We need three structures.\n-- THe first is used to identify inputs from the same com. When a tranformation splits these coms, we also split the set into two sets. Next we track all the inputs that are part of the tranformation, be it sets of inputs of a specific com or individual inputs. From all these sets, at least one item from each set needs to be the input of the next com.\n-- The set of coms that are used to allow for commutation of inputs.\n\n\n\n-- A non-empty set of nodes in a Linear Logic tree that also can also tag intermediary nodes.\ndata SetLLInter {i : Size} {u} : LinLogic i {u} \u2192 Set where\n  \u2193     : \u2200{ll}                                    \u2192 SetLLInter ll\n  \u2193+    : \u2200{ll}    \u2192 SetLLInter ll                 \u2192 SetLLInter ll\n  _\u2190\u2227   : \u2200{rs ls} \u2192 SetLLInter ls                 \u2192 SetLLInter (ls \u2227 rs)\n  \u2227\u2192_   : \u2200{rs ls} \u2192 SetLLInter rs                 \u2192 SetLLInter (ls \u2227 rs)\n  _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLLInter ls \u2192 SetLLInter rs \u2192 SetLLInter (ls \u2227 rs)\n  _\u2190\u2228   : \u2200{rs ls} \u2192 SetLLInter ls                 \u2192 SetLLInter (ls \u2228 rs)\n  \u2228\u2192_   : \u2200{rs ls} \u2192 SetLLInter rs                 \u2192 SetLLInter (ls \u2228 rs)\n  _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLLInter ls \u2192 SetLLInter rs \u2192 SetLLInter (ls \u2228 rs)\n  _\u2190\u2202   : \u2200{rs ls} \u2192 SetLLInter ls                 \u2192 SetLLInter (ls \u2202 rs)\n  \u2202\u2192_   : \u2200{rs ls} \u2192 SetLLInter rs                 \u2192 SetLLInter (ls \u2202 rs)\n  _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLLInter ls \u2192 SetLLInter rs \u2192 SetLLInter (ls \u2202 rs)\n\n\ndata MSetLLInter {i : Size} {u} : LinLogic i {u} \u2192 Set where\n  \u2205   : \u2200{ll}                 \u2192 MSetLLInter ll\n  \u00ac\u2205  : \u2200{ll} \u2192 SetLLInter ll \u2192 MSetLLInter ll\n\n\n\u2205-add : \u2200{i u ll rll} \u2192 {j : Size< \u2191 i} \u2192 (ind : IndexLL {i} {u} rll ll) \u2192 (nrll : LinLogic j )\n        \u2192 SetLLInter (replLL ll ind nrll)\n\u2205-add \u2193 nrll = \u2193\n\u2205-add (ind \u2190\u2227) nrll = (\u2205-add ind nrll) \u2190\u2227\n\u2205-add (\u2227\u2192 ind) nrll = \u2227\u2192 (\u2205-add ind nrll)\n\u2205-add (ind \u2190\u2228) nrll = (\u2205-add ind nrll) \u2190\u2228\n\u2205-add (\u2228\u2192 ind) nrll = \u2228\u2192 (\u2205-add ind nrll)\n\u2205-add (ind \u2190\u2202) nrll = (\u2205-add ind nrll) \u2190\u2202\n\u2205-add (\u2202\u2192 ind) nrll = \u2202\u2192 (\u2205-add ind nrll)\n\n-- TODO We shouldn't need this. When issue agda #2409 is resolved, remove this.\ndsize : \u2200{i u ll} \u2192 {j : Size< \u2191 i} \u2192 SetLLInter {i} {u} ll \u2192 SetLLInter {j} ll\ndsize \u2193          = \u2193\ndsize (\u2193+ s)     = (\u2193+ $ dsize s)\ndsize (x \u2190\u2227)     = (dsize x) \u2190\u2227\ndsize (\u2227\u2192 x)     = \u2227\u2192 (dsize x)\ndsize (x \u2190\u2227\u2192 x\u2081) = (dsize x \u2190\u2227\u2192 dsize x\u2081)\ndsize (x \u2190\u2228)     = (dsize x) \u2190\u2228\ndsize (\u2228\u2192 x)     = \u2228\u2192 (dsize x)\ndsize (x \u2190\u2228\u2192 x\u2081) = (dsize x \u2190\u2228\u2192 dsize x\u2081)\ndsize (x \u2190\u2202)     = (dsize x) \u2190\u2202\ndsize (\u2202\u2192 x)     = \u2202\u2192 (dsize x)\ndsize (x \u2190\u2202\u2192 x\u2081) = (dsize x \u2190\u2202\u2192 dsize x\u2081)\n\n-- Lower level intermediary are removed when adding a new index.\nadd : \u2200{i u ll q} \u2192 {j : Size< \u2191 i} \u2192 SetLLInter ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 (rll : LinLogic j)\n      \u2192 SetLLInter (replLL ll ind rll)\nadd \u2193 \u2193 rll                 = \u2193\nadd \u2193 ind rll               = \u2193+ (\u2205-add ind rll)\nadd (\u2193+ s) ind rll          = \u2193+ (add s ind rll)\nadd (s \u2190\u2227) \u2193 rll            = \u2193\nadd (s \u2190\u2227) (ind \u2190\u2227) rll     = (add s ind rll) \u2190\u2227\nadd (s \u2190\u2227) (\u2227\u2192 ind) rll     = dsize s \u2190\u2227\u2192 (\u2205-add ind rll)\nadd (\u2227\u2192 s) \u2193 rll            = \u2193\nadd (\u2227\u2192 s) (ind \u2190\u2227) rll     = (\u2205-add ind rll) \u2190\u2227\u2192 dsize s\nadd (\u2227\u2192 s) (\u2227\u2192 ind) rll     = \u2227\u2192 add s ind rll\nadd (s \u2190\u2227\u2192 s\u2081) \u2193 rll        = \u2193\nadd (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll = (add s ind rll) \u2190\u2227\u2192 dsize s\u2081\nadd (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll = dsize s \u2190\u2227\u2192 (add s\u2081 ind rll)\nadd (s \u2190\u2228) \u2193 rll            = \u2193\nadd (s \u2190\u2228) (ind \u2190\u2228) rll     = (add s ind rll) \u2190\u2228\nadd (s \u2190\u2228) (\u2228\u2192 ind) rll     = dsize s \u2190\u2228\u2192 (\u2205-add ind rll)\nadd (\u2228\u2192 s) \u2193 rll            = \u2193\nadd (\u2228\u2192 s) (ind \u2190\u2228) rll     = (\u2205-add ind rll) \u2190\u2228\u2192 dsize s\nadd (\u2228\u2192 s) (\u2228\u2192 ind) rll     = \u2228\u2192 add s ind rll\nadd (s \u2190\u2228\u2192 s\u2081) \u2193 rll        = \u2193\nadd (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll = (add s ind rll) \u2190\u2228\u2192 dsize s\u2081\nadd (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll = dsize s \u2190\u2228\u2192 (add s\u2081 ind rll)\nadd (s \u2190\u2202) \u2193 rll            = \u2193\nadd (s \u2190\u2202) (ind \u2190\u2202) rll     = (add s ind rll) \u2190\u2202\nadd (s \u2190\u2202) (\u2202\u2192 ind) rll     = dsize s \u2190\u2202\u2192 (\u2205-add ind rll)\nadd (\u2202\u2192 s) \u2193 rll            = \u2193\nadd (\u2202\u2192 s) (ind \u2190\u2202) rll     = (\u2205-add ind rll) \u2190\u2202\u2192 dsize s\nadd (\u2202\u2192 s) (\u2202\u2192 ind) rll     = \u2202\u2192 add s ind rll\nadd (s \u2190\u2202\u2192 s\u2081) \u2193 rll        = \u2193\nadd (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll = (add s ind rll) \u2190\u2202\u2192 dsize s\u2081\nadd (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll = dsize s \u2190\u2202\u2192 (add s\u2081 ind rll)\n\n\nmadd : \u2200{i u ll q} \u2192 {j : Size< \u2191 i} \u2192 MSetLLInter ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 (rll : LinLogic j)\n      \u2192 MSetLLInter (replLL ll ind rll)\nmadd \u2205 ind rll = \u00ac\u2205 (\u2205-add ind rll)\nmadd (\u00ac\u2205 x) ind rll = \u00ac\u2205 (add x ind rll)\n\ndel : \u2200{i u ll q} \u2192 {j : Size< \u2191 i} \u2192 SetLLInter ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 (rll : LinLogic j)\n      \u2192 MSetLLInter (replLL ll ind rll)\ndel \u2193 \u2193 rll = \u2205\ndel (\u2193+ s) \u2193 rll = {!\u00ac\u2205 s!}\ndel (s \u2190\u2227) \u2193 rll = {!!}\ndel (\u2227\u2192 s) \u2193 rll = {!!}\ndel (s \u2190\u2227\u2192 s\u2081) \u2193 rll = {!!}\ndel (s \u2190\u2228) \u2193 rll = {!!}\ndel (\u2228\u2192 s) \u2193 rll = {!!}\ndel (s \u2190\u2228\u2192 s\u2081) \u2193 rll = {!!}\ndel (s \u2190\u2202) \u2193 rll = {!!}\ndel (\u2202\u2192 s) \u2193 rll = {!!}\ndel (s \u2190\u2202\u2192 s\u2081) \u2193 rll = {!!}\ndel \u2193 (ind \u2190\u2227) rll with (del \u2193 ind rll)\ndel \u2193 (ind \u2190\u2227) rll | \u2205 = \u00ac\u2205 (\u2227\u2192 \u2193)\ndel \u2193 (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227\u2192 \u2193)\ndel (s \u2190\u2227) (ind \u2190\u2227) rll with (del s ind rll)\ndel (s \u2190\u2227) (ind \u2190\u2227) rll | \u2205 = \u2205\ndel (s \u2190\u2227) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227)\ndel (\u2227\u2192 s) (ind \u2190\u2227) rll = \u00ac\u2205 (\u2227\u2192 (dsize s))\ndel (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll with (del s ind rll)\ndel (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u2205 = \u00ac\u2205 (\u2227\u2192 (dsize s\u2081))\ndel (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227\u2192 (dsize s\u2081))\ndel \u2193 (\u2227\u2192 ind) rll with (del \u2193 ind rll)\ndel \u2193 (\u2227\u2192 ind) rll | \u2205 = \u00ac\u2205 (\u2193 \u2190\u2227)\ndel \u2193 (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2193 \u2190\u2227\u2192 x)\ndel (s \u2190\u2227) (\u2227\u2192 ind) rll = \u00ac\u2205 ((dsize s) \u2190\u2227)\ndel (\u2227\u2192 s) (\u2227\u2192 ind) rll with (del s ind rll)\ndel (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u2205 = \u2205\ndel (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2227\u2192 x)\ndel (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll with (del s\u2081 ind rll)\ndel (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u2205 = \u00ac\u2205 ((dsize s) \u2190\u2227)\ndel (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((dsize s) \u2190\u2227\u2192 x)\ndel \u2193 (ind \u2190\u2228) rll with (del \u2193 ind rll)\ndel \u2193 (ind \u2190\u2228) rll | \u2205 = \u00ac\u2205 (\u2228\u2192 \u2193)\ndel \u2193 (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228\u2192 \u2193)\ndel (s \u2190\u2228) (ind \u2190\u2228) rll with (del s ind rll)\ndel (s \u2190\u2228) (ind \u2190\u2228) rll | \u2205 = \u2205\ndel (s \u2190\u2228) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228)\ndel (\u2228\u2192 s) (ind \u2190\u2228) rll = \u00ac\u2205 (\u2228\u2192 (dsize s))\ndel (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll with (del s ind rll)\ndel (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u2205 = \u00ac\u2205 (\u2228\u2192 (dsize s\u2081))\ndel (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228\u2192 (dsize s\u2081))\ndel \u2193 (\u2228\u2192 ind) rll with (del \u2193 ind rll)\ndel \u2193 (\u2228\u2192 ind) rll | \u2205 = \u00ac\u2205 (\u2193 \u2190\u2228)\ndel \u2193 (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2193 \u2190\u2228\u2192 x)\ndel (s \u2190\u2228) (\u2228\u2192 ind) rll = \u00ac\u2205 ((dsize s) \u2190\u2228)\ndel (\u2228\u2192 s) (\u2228\u2192 ind) rll with (del s ind rll)\ndel (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u2205 = \u2205\ndel (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2228\u2192 x)\ndel (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll with (del s\u2081 ind rll)\ndel (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u2205 = \u00ac\u2205 ((dsize s) \u2190\u2228)\ndel (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((dsize s) \u2190\u2228\u2192 x)\ndel \u2193 (ind \u2190\u2202) rll with (del \u2193 ind rll)\ndel \u2193 (ind \u2190\u2202) rll | \u2205 = \u00ac\u2205 (\u2202\u2192 \u2193)\ndel \u2193 (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202\u2192 \u2193)\ndel (s \u2190\u2202) (ind \u2190\u2202) rll with (del s ind rll)\ndel (s \u2190\u2202) (ind \u2190\u2202) rll | \u2205 = \u2205\ndel (s \u2190\u2202) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202)\ndel (\u2202\u2192 s) (ind \u2190\u2202) rll = \u00ac\u2205 (\u2202\u2192 (dsize s))\ndel (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll with (del s ind rll)\ndel (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u2205 = \u00ac\u2205 (\u2202\u2192 (dsize s\u2081))\ndel (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202\u2192 (dsize s\u2081))\ndel \u2193 (\u2202\u2192 ind) rll with (del \u2193 ind rll)\ndel \u2193 (\u2202\u2192 ind) rll | \u2205 = \u00ac\u2205 (\u2193 \u2190\u2202)\ndel \u2193 (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2193 \u2190\u2202\u2192 x)\ndel (s \u2190\u2202) (\u2202\u2192 ind) rll = \u00ac\u2205 ((dsize s) \u2190\u2202)\ndel (\u2202\u2192 s) (\u2202\u2192 ind) rll with (del s ind rll)\ndel (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u2205 = \u2205\ndel (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2202\u2192 x)\ndel (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll with (del s\u2081 ind rll)\ndel (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u2205 = \u00ac\u2205 ((dsize s) \u2190\u2202)\ndel (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((dsize s) \u2190\u2202\u2192 x)\n\n\n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"ec77d4b02f6afb3d807d66f7e51f3b56de0a84e0","subject":"GroupIsomorphism: is now indexed over the function","message":"GroupIsomorphism: is now indexed over the function\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Isomorphism.agda","new_file":"lib\/Algebra\/Group\/Isomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type_)\nopen import Level.NP\nopen import Function.NP\nopen import Data.Product using (_,_)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nmodule Algebra.Group.Isomorphism where\n\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n  hiding (module Stability; module Stability-Minimal)\n\nrecord GroupIsomorphism\n         {a}{A : Type a}\n         {b}{B : Type b}\n         (G+   : Group A)\n         (G*   : Group B)\n         (\u03c6    : A \u2192 B) : Set(a \u2294 b) where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n\n  field\n    \u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y\n    \u03c6\u207b\u00b9   : B \u2192 A\n    \u03c6\u207b\u00b9-\u03c6 : \u2200 {x} \u2192 \u03c6\u207b\u00b9 (\u03c6 x) \u2261 x\n    \u03c6-\u03c6\u207b\u00b9 : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x\n\n  \u03c6-hom : GroupHomomorphism G+ G* \u03c6\n  \u03c6-hom = mk \u03c6-+-*\n\n  open GroupHomomorphism \u03c6-hom public\n\n{- How this proof can be used for crypto, in particular ElGamal to DDH\n\n  the Group A is \u2124q with modular addition as operation\n  the Group B is the cyclic group with order q\n\n  \u03c6 is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  F is the summation operator and is required to be stable\n  under addition of A.\n\n  F should respects extensionality\n\n  This proof adds \u03c6\u207b\u00b9 k, because adding a constant is stable under\n  the big op F, this addition can then be pulled homomorphically\n  through f, to become a, multiplication by k.\n-}\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (\u03c6\u207b\u00b9   : B \u2192 A)\n  (\u03c6-\u03c6\u207b\u00b9 : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  where\n\n  module _ (F\u03c6+ : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)) where\n\n    *-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6)\n    *-stable {k} =\n      F \u03c6                  \u2261\u27e8 F\u03c6+ \u27e9\n      F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 k))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 k + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                         \u03c6 (\u03c6\u207b\u00b9 k) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-\u03c6\u207b\u00b9 idp \u27e9\n                                         k * \u03c6 x         \u220e) \u27e9\n      F (_*_ k \u2218 \u03c6)        \u220e\n\n  {- The reverse direction comes from the homomorphism -}\n  open Algebra.Group.Homomorphism.Stability-Minimal\n    \u03c6 _+_ _*_ \u03c6-+-* F F=\n\n  stability : (\u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)) \u2194 (\u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  stability = *-stable , +-stable\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6     : A \u2192 B)\n  (\u03c6-iso : GroupIsomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupIsomorphism \u03c6-iso\n\n  open Stability-Minimal \u03c6 \u03c6\u207b\u00b9 \u03c6-\u03c6\u207b\u00b9 _+_ _*_ \u03c6-+-* public\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type_)\nopen import Level.NP\nopen import Function.NP\nopen import Data.Product using (_,_)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nmodule Algebra.Group.Isomorphism where\n\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n  hiding (module Stability; module Stability-Minimal)\n\nrecord GroupIsomorphism\n         {a}{A : Type a}\n         {b}{B : Type b}\n         (G+   : Group A)\n         (G*   : Group B) : Set(a \u2294 b) where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n\n  field\n    \u03c6 : A \u2192 B -- TODO\n    \u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y\n    \u03c6\u207b\u00b9   : B \u2192 A\n    \u03c6\u207b\u00b9-\u03c6 : \u2200 {x} \u2192 \u03c6\u207b\u00b9 (\u03c6 x) \u2261 x\n    \u03c6-\u03c6\u207b\u00b9 : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x\n\n  \u03c6-hom : GroupHomomorphism G+ G* \u03c6\n  \u03c6-hom = mk \u03c6-+-*\n\n  open GroupHomomorphism \u03c6-hom public\n\n{- How this proof can be used for crypto, in particular ElGamal to DDH\n\n  the Group A is \u2124q with modular addition as operation\n  the Group B is the cyclic group with order q\n\n  \u03c6 is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  F is the summation operator and is required to be stable\n  under addition of A.\n\n  F should respects extensionality\n\n  This proof adds \u03c6\u207b\u00b9 k, because adding a constant is stable under\n  the big op F, this addition can then be pulled homomorphically\n  through f, to become a, multiplication by k.\n-}\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (\u03c6\u207b\u00b9   : B \u2192 A)\n  (\u03c6-\u03c6\u207b\u00b9 : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  where\n\n  module _ (F\u03c6+ : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)) where\n\n    *-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6)\n    *-stable {k} =\n      F \u03c6                  \u2261\u27e8 F\u03c6+ \u27e9\n      F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 k))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 k + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                         \u03c6 (\u03c6\u207b\u00b9 k) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-\u03c6\u207b\u00b9 idp \u27e9\n                                         k * \u03c6 x         \u220e) \u27e9\n      F (_*_ k \u2218 \u03c6)        \u220e\n\n  {- The reverse direction comes from the homomorphism -}\n  open Algebra.Group.Homomorphism.Stability-Minimal\n    \u03c6 _+_ _*_ \u03c6-+-* F F=\n\n  stability : (\u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)) \u2194 (\u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  stability = *-stable , +-stable\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6-iso : GroupIsomorphism G+ G*)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupIsomorphism \u03c6-iso\n\n  open Stability-Minimal \u03c6 \u03c6\u207b\u00b9 \u03c6-\u03c6\u207b\u00b9 _+_ _*_ \u03c6-+-* public\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"272abae5605c8fd8c5385376ffe4f4d4c26d2a9d","subject":"Add and use a congruence lemma","message":"Add and use a congruence lemma\n\nTo reduce explicit wrapping\/unwrapping of proofs of \u2259.\n\nOld-commit-hash: 2bc64b5b9f3b074a70918e11f66512232fb6796b\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) $ proof dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) $ proof df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"aa599ff81b5977aba6526f15272f7c5cd110b5ca","subject":"Minor changes to a note.","message":"Minor changes to a note.\n\nIgnore-this: e9399eaae7068336c59aed864a0878a0\n\ndarcs-hash:20120615190941-3bd4e-d47bd9ffb7bbc540837cedc9e4f06e1fba047622.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Predicates.agda","new_file":"notes\/fixed-points\/Predicates.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC types without use data, i.e. using Agda as a logical framework\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Predicates where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- The existential proyections.\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using postulates.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N  : D \u2192 Set\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    nilL  : List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    nilLN  : ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    nilL  :                      List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    nilLN  :                             ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined types.\n    --\n    -- N.B. We cannot write LFP in first-order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n-- The greatest fixed-point operator.\n\n-- N.B. At the moment, the definitions of LFP and GFP are the same.\n\nmodule GFP where\n\n  postulate\n    -- Greatest fixed-points correspond to coinductively defined types.\n\n    GFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    GFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 GFP f d \u2192 f (GFP f) d\n    GFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (GFP f) d \u2192 GFP f d\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero     = \u22a4\n  -- NatF X (succ\u2081 n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  zN : N zero\n  zN = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  sN : {n : D} \u2192 N n \u2192 N (succ\u2081 n)\n  sN {n} Nn = LFP\u2082 NatF (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = sN zN\n\n------------------------------------------------------------------------------\n-- The FOTC list type as the least fixed-point of a functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  nilL : List []\n  nilL = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  consL x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  nilLN : ListN []\n  nilLN = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  consLN {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n-- The FOTC Colist type as the greatest fixed-point of a functor.\n\nmodule CoList where\n\n  open GFP\n\n  -- Functor for the FOTC Colists type (the same functor that for the\n  -- List type).\n  ColistF : (D \u2192 Set) \u2192 D \u2192 Set\n  ColistF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC Colist type using GFP.\n  Colist : D \u2192 Set\n  Colist = GFP ColistF\n\n  -- The data constructors of Colist.\n  nilCL : Colist []\n  nilCL = GFP\u2082 ColistF [] (inj\u2081 refl)\n\n  consCL : \u2200 x xs \u2192 Colist xs \u2192 Colist (x \u2237 xs)\n  consCL x xs CLxs = GFP\u2082 ColistF (x \u2237 xs) (inj\u2082 (x , xs , refl , CLxs))\n\n  -- Example (finite colist).\n  l : Colist (zero \u2237 true \u2237 [])\n  l = consCL zero (true \u2237 []) (consCL true [] nilCL)\n\n  -- TODO: Example (infinite colist).\n  -- zerosCL : Colist {!!}\n  -- zerosCL = consCL zero {!!} zerosCL\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2081 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 X ds\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- Using StreamF we cannot define this data constructor\n  -- consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  -- consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) {!!}\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2082 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} S = GFP\u2082 StreamF xs (x , x \u2237 xs , S)\n\n  -- The functor StreamF does not link together the parts of the\n  -- stream, so I can get a stream from any stream.\n  bad : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys\n  bad {xs} {ys} S = GFP\u2082 StreamF ys (ys , xs , S)\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2083 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , refl , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} Sx\u2237xs = Sxs\n    where\n    unfoldS : StreamF (GFP StreamF) (x \u2237 xs)\n    unfoldS = GFP\u2081 StreamF (x \u2237 xs) Sx\u2237xs\n\n    e : D\n    e = \u2203-proj\u2081 unfoldS\n\n    Pe : \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pe = \u2203-proj\u2082 unfoldS\n\n    es : D\n    es = \u2203-proj\u2081 Pe\n\n    Pes : x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pes = \u2203-proj\u2082 Pe\n\n    xs\u2261es : xs \u2261 es\n    xs\u2261es = \u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2081 Pes))\n\n    Sxs : GFP StreamF xs\n    Sxs = subst (GFP StreamF) (sym xs\u2261es) (\u2227-proj\u2082 Pes)\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Predicates where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- The existential proyections.\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n-- The FOTC types without use data, i.e. using Agda as a logical framework.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N  : D \u2192 Set\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    nilL  : List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    nilLN  : ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    nilL  :                      List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    nilLN  :                             ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined types.\n    --\n    -- N.B. We cannot write LFP in first-order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n\n-- The greatest fixed-point operator.\n\n-- N.B. At the moment, the definitions of LFP and GFP are the same.\n\nmodule GFP where\n\n  postulate\n    -- Greatest fixed-points correspond to coinductively defined types.\n\n    GFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    GFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 GFP f d \u2192 f (GFP f) d\n    GFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (GFP f) d \u2192 GFP f d\n\n------------------------------------------------------------------------------\n\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero     = \u22a4\n  -- NatF X (succ\u2081 n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  zN : N zero\n  zN = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  sN : {n : D} \u2192 N n \u2192 N (succ\u2081 n)\n  sN {n} Nn = LFP\u2082 NatF (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = sN zN\n\n------------------------------------------------------------------------------\n\n-- The FOTC list type as the least fixed-point of a functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  nilL : List []\n  nilL = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  consL x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n------------------------------------------------------------------------------\n\n-- The FOTC list of natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  nilLN : ListN []\n  nilLN = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  consLN {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The FOTC Colist type as the greatest fixed-point of a functor.\n\nmodule CoList where\n\n  open GFP\n\n  -- Functor for the FOTC Colists type (the same functor that for the\n  -- List type).\n  ColistF : (D \u2192 Set) \u2192 D \u2192 Set\n  ColistF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC Colist type using GFP.\n  Colist : D \u2192 Set\n  Colist = GFP ColistF\n\n  -- The data constructors of Colist.\n  nilCL : Colist []\n  nilCL = GFP\u2082 ColistF [] (inj\u2081 refl)\n\n  consCL : \u2200 x xs \u2192 Colist xs \u2192 Colist (x \u2237 xs)\n  consCL x xs CLxs = GFP\u2082 ColistF (x \u2237 xs) (inj\u2082 (x , xs , refl , CLxs))\n\n  -- Example (finite colist).\n  l : Colist (zero \u2237 true \u2237 [])\n  l = consCL zero (true \u2237 []) (consCL true [] nilCL)\n\n  -- TODO: Example (infinite colist).\n  -- zerosCL : Colist {!!}\n  -- zerosCL = consCL zero {!!} zerosCL\n\n------------------------------------------------------------------------------\n\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2081 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 X ds\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- Using StreamF we cannot define this data constructor\n  -- consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  -- consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) {!!}\n\n------------------------------------------------------------------------------\n\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2082 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} S = GFP\u2082 StreamF xs (x , x \u2237 xs , S)\n\n  -- The functor StreamF does not link together the parts of the\n  -- stream, so I can get a stream from any stream.\n  bad : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys\n  bad {xs} {ys} S = GFP\u2082 StreamF ys (ys , xs , S)\n\n------------------------------------------------------------------------------\n\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2083 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , refl , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} Sx\u2237xs = Sxs\n    where\n    unfoldS : StreamF (GFP StreamF) (x \u2237 xs)\n    unfoldS = GFP\u2081 StreamF (x \u2237 xs) Sx\u2237xs\n\n    e : D\n    e = \u2203-proj\u2081 unfoldS\n\n    Pe : \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pe = \u2203-proj\u2082 unfoldS\n\n    es : D\n    es = \u2203-proj\u2081 Pe\n\n    Pes : x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pes = \u2203-proj\u2082 Pe\n\n    xs\u2261es : xs \u2261 es\n    xs\u2261es = \u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2081 Pes))\n\n    Sxs : GFP StreamF xs\n    Sxs = subst (GFP StreamF) (sym xs\u2261es) (\u2227-proj\u2082 Pes)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"30c4508e8343a2319a8a365ad65b0c8f13a6c8c5","subject":"More bintree fun","message":"More bintree fun\n","repos":"crypto-agda\/crypto-agda","old_file":"bintree.agda","new_file":"bintree.agda","new_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} \u2192 (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then left else right) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nopen import Relation.Binary\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","old_contents":"module bintree\n  ([0,1] : Set)\nwhere\n\nopen import Data.Nat\nopen import Data.Bits\n\ndata T : \u2115 \u2192 Set where\n  leaf : \u2200 {n} \u2192 Bits n \u2192 T n\n  fork : \u2200 {n} (proba : [0,1]) (left right : T n) \u2192 T (1 + n)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"85b48bf78ebbcaed626a162fe5616b7890cc7e8a","subject":"validity-of-derive on bags (toward #51)","message":"validity-of-derive on bags (toward #51)\n\nOld-commit-hash: 02f7fd17338ba8df5698d20442c1956021f53f0a\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/NatBag.agda","new_file":"experimental\/NatBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","old_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag)\n\nopen import Data.NatBag.Properties using\n  (b\\\\b=\u2205)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 = cong\u2082 mapBag (weaken-sound f \u03c1)\n                                        (weaken-sound b \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = {!!}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = {!!}\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n{-\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = {!!}\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = {!!}\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\nderive _ = {!!}\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {bags} b db valid-b-db = {!!}\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = {!!}\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bc8f73c86b96f119aff939fd38aa12d023ebc02e","subject":"Checkpoint: To merge alter\/abide to one constructor with Bool tag. Reason: Otherwise impossible to express \"nonempty \u0394-context\" as a type.","message":"Checkpoint: To merge alter\/abide to one constructor with Bool tag.\nReason: Otherwise impossible to express \"nonempty \u0394-context\" as\na type.\n\nOld-commit-hash: 67e1406b2769044d22b7ac779f8da7e8eea1b5f2\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag renaming\n--   (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {b : Bag} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  nats : \u0394-Type\n  bags : \u0394-Type\n  alter_\u21d2_ : \u0394-Type \u2192 \u0394-Type \u2192 \u0394-Type\n  abide_\u21d2_ : \u0394-Type \u2192 \u0394-Type \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u2205 : \u0394-Context\n  alter_\u2022_ : \u0394-Type \u2192 \u0394-Context \u2192 \u0394-Context\n  abide_\u2022_ : \u0394-Type \u2192 \u0394-Context \u2192 \u0394-Context\n\nerase : \u0394-Type \u2192 Type\nerase nats = nats\nerase bags = bags\nerase (alter \u03c4\u2081 \u21d2 \u03c4\u2082) = erase \u03c4\u2081 \u21d2 erase \u03c4\u2082\nerase (abide \u03c4\u2081 \u21d2 \u03c4\u2082) = erase \u03c4\u2081 \u21d2 erase \u03c4\u2082\n\nforget : \u0394-Context \u2192 Context\nforget \u2205 = \u2205\nforget (alter \u03c4 \u2022 \u03c1) = erase \u03c4 \u2022 forget \u03c1\nforget (abide \u03c4 \u2022 \u03c1) = erase \u03c4 \u2022 forget \u03c1\n\n-- Convert a type\/context to a \u0394-type\/\u0394-context without any\n-- assumption about arguments\n\n\u0394-type : Type \u2192 \u0394-Type\n\u0394-type nats = nats\n\u0394-type bags = bags\n\u0394-type (\u03c4\u2081 \u21d2 \u03c4\u2082) = alter \u0394-type \u03c4\u2081 \u21d2 \u0394-type \u03c4\u2082\n\n\u0394-context : Context \u2192 \u0394-Context\n\u0394-context \u2205 = \u2205\n\u0394-context (\u03c4 \u2022 \u0393) = alter \u0394-type \u03c4 \u2022 \u0394-context \u0393\n\ndata \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\ndata \u27e6_\u27e7\u0394Context : \u0394-Context \u2192 Set\n\n\u27e6_\u27e7\u0394Type : \u0394-Type \u2192 Set\n\u27e6_\u27e7\u0394Term : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7\u0394Context \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 \u27e6 erase \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type\nvalid : \u2200 {\u03c4} \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4} {u v : \u27e6 erase \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4} {u v : \u27e6 erase \u03c4 \u27e7} \u2192 v \u2295 (_\u229d_ {\u03c4} u v) \u2261 u\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7\u0394Context \u2192 \u27e6 forget \u0393 \u27e7\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7\u0394Context \u2192 \u27e6 forget \u0393 \u27e7\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var (forget \u0393) (erase \u03c4)) \u2192 \u0394-Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  -- There are two kinds of those: One who expects the argument\n  -- to change, and one who does not.\n  \u0394abs\u2080 : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393}\n    (t : \u0394-Term (abide \u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n    \u0394-Term \u0393 (abide \u03c4\u2081 \u21d2 \u03c4\u2082)\n  \u0394abs\u2081 : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393}\n    (t : \u0394-Term (alter \u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n    \u0394-Term \u0393 (alter \u03c4\u2081 \u21d2 \u03c4\u2082)\n{-\n  \u0394app\u2080 : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393}\n    (ds : \u0394-Term \u0393 (abide \u03c4\u2081 \u21d2 \u03c4\u2082))\n    (t : Term (forget \u0393) (erase \u03c4\u2081))\n    (dt : \u0394-Term \u0393 \u03c4\u2081)\n    {R[t,dt] : \u2200 {\u03c1 : \u27e6\n-}\n    \n{-\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars}\n    (ds : \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars})\n    ( t :   Term \u0393 \u03c4\u2081)\n    (dt : \u0394-Term \u0393 \u03c4\u2081 {vars})\n    (R[t,dt] : {\u03c1 : \u0394-Env \u0393 {-vars-}} \u2192\n      valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1)) \u2192\n    \u0394-Term \u0393 \u03c4\u2082 {vars}\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393 vars}\n    (ds : \u0394-Term \u0393 nats {vars})\n    (dt : \u0394-Term \u0393 nats {vars}) \u2192\n    \u0394-Term \u0393 nats {vars}\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats))\n    (df : \u0394-Term \u0393 (nats \u21d2 nats) {vars})\n    ( b :   Term \u0393 bags)\n    (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n  \u0394map\u2081 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats)) (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n-}\n\n-- \u27e6_\u27e7\u0394Type : \u0394-Type \u2192 Set\n\u27e6 nats \u27e7\u0394Type = \u2115 \u00d7 \u2115\n\u27e6 bags \u27e7\u0394Type = Bag\n\u27e6 alter \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\u0394Type =\n  (v : \u27e6 erase \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u03c4\u2081 \u27e7\u0394Type) \u2192\n  valid {\u03c4\u2081} v dv \u2192\n  \u27e6 \u03c4\u2082 \u27e7\u0394Type\n\u27e6 abide \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\u0394Type =\n  (v : \u27e6 erase \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u03c4\u2081 \u27e7\u0394Type) \u2192\n  valid {\u03c4\u2081} v dv \u2192 _\u2295_ {\u03c4\u2081} v dv \u2261 v \u2192\n  \u27e6 \u03c4\u2082 \u27e7\u0394Type\n\nmeaning-\u0394Type : Meaning \u0394-Type\nmeaning-\u0394Type = meaning \u27e6_\u27e7\u0394Type\n\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 \u27e6 erase \u03c4 \u27e7\n_\u2295_ = {!!}\n{-\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n-}\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n_\u229d_ = {!!}\n{-\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n-}\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 erase \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nvalid _ _ = \u22a4\n{-\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u03c4\u2081 \u27e7\u0394Type) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n-}\n\n-- v\u2295[u\u229dv]=u : \u2200 {\u03c4} {u v : \u27e6 erase \u03c4 \u27e7} \u2192 v \u2295 (_\u229d_ {\u03c4} u v) \u2261 u\nv\u2295[u\u229dv]=u = {!!}\n{-\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n-}\n\n-- R[v,u\u229dv] : \u2200 {\u03c4} {u v : \u27e6 erase \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] = tt\n{-\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n-}\n\n\n-- The type of environments ensures their consistency and honesty.\n-- data \u27e6_\u27e7\u0394Context : \u0394-Context \u2192 Set\ndata \u27e6_\u27e7\u0394Context where\n  \u2205 : \u27e6 \u2205 \u27e7\u0394Context\n  quad : \u2200 {\u03c4 \u0393}\n    (v : \u27e6 erase \u03c4 \u27e7)\n    (dv : \u27e6 \u03c4 \u27e7\u0394Type)\n    (R[v,dv] : valid {\u03c4} v dv)\n    (\u03c1 : \u27e6 \u0393 \u27e7\u0394Context) \u2192\n    \u27e6 alter \u03c4 \u2022 \u0393 \u27e7\u0394Context\n  quint : \u2200 {\u03c4 \u0393}\n    (v : \u27e6 erase \u03c4 \u27e7)\n    (dv : \u27e6 \u03c4 \u27e7\u0394Type)\n    (R[v,dv] : valid {\u03c4} v dv)\n    (v\u2295dv=v : _\u2295_ {\u03c4} v dv \u2261 v)\n    (\u03c1 : \u27e6 \u0393 \u27e7\u0394Context) \u2192\n    \u27e6 abide \u03c4 \u2022 \u0393 \u27e7\u0394Context\n\nmeaning-\u0394Context : Meaning \u0394-Context\nmeaning-\u0394Context = meaning \u27e6_\u27e7\u0394Context\n\n\n-- ignore : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7\u0394Context \u2192 \u27e6 forget \u0393 \u27e7\nignore \u2205 = \u2205\nignore (quad v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\nignore (quint v dv R[v,dv] v\u2295dv=v \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7\u0394Context \u2192 \u27e6 forget \u0393 \u27e7\nupdate \u2205 = \u2205\nupdate (quad {\u03c4} v dv R[v,dv] \u03c1) = (_\u2295_ {\u03c4} v dv) \u2022 update \u03c1\nupdate (quint {\u03c4} v dv R[v,dv] v\u2295dv=v \u03c1) = (_\u2295_ {\u03c4} v dv) \u2022 update \u03c1\n\n--\u27e6_\u27e7\u0394Var : \u2200 {\u03c4} {\u0393} \u2192 Var (forget \u0393) (erase \u03c4) \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n{-\n\u27e6_\u27e7\u0394Var {alter \u03c4 \u2022 _} {.\u03c4} this (quad _ dv _ _) = dv\n\u27e6_\u27e7\u0394Var {abide \u03c4 \u2022 _} {.\u03c4} this (quint _ dv _ _ _) = dv\n\u27e6_\u27e7\u0394Var {alter _ \u2022 _} (that x) (quad _ _ _ \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6_\u27e7\u0394Var {abide _ \u2022 _} (that x) (quint _ _ _ _ \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n-}\n\n\n-- \u27e6_\u27e7\u0394Term : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7\u0394Context \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6_\u27e7\u0394Term _ _ = {!!}\n{-\n\u27e6 \u0394nat old new \u27e7\u0394Term \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394Term \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394Term \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv]          \u2192 \u27e6 t \u27e7\u0394Term (alter v dv R[v,dv] \u03c1)\n\u27e6 \u0394abs\u2080 t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv] {-v\u2295dv=v-} \u2192 \u27e6 t \u27e7\u0394Term (abide v dv R[v,dv] {!v\u2295dv=v!} \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394Term \u03c1 =\n  \u27e6 ds \u27e7\u0394Term \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1) R[dt,t]\n  --(R[dt,t] {\u03c1} {honesty = {!!}})\n\u27e6 \u0394add ds dt \u27e7\u0394Term \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394Term \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394Term \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394Term \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394Term \u03c1\n    dh = \u27e6 df \u27e7\u0394Term \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394Term \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394Term \u03c1)\n-}\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term \u0393 \u03c4)\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394Term\n\n\n\n{-\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7 \u03c1\n    db = \u27e6 derive t \u27e7 \u03c1\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394-Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7 \u03c1\u2032\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n-}\n\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag renaming\n--   (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {b : Bag} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  nats : \u0394-Type\n  bags : \u0394-Type\n  alter_\u21d2_ : \u0394-Type \u2192 \u0394-Type \u2192 \u0394-Type\n  abide_\u21d2_ : \u0394-Type \u2192 \u0394-Type \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u2205 : \u0394-Context\n  alter_\u2022_ : \u0394-Type \u2192 \u0394-Context \u2192 \u0394-Context\n  abide_\u2022_ : \u0394-Type \u2192 \u0394-Context \u2192 \u0394-Context\n\n-- Convert a type\/context to a \u0394-type\/\u0394-context without any\n-- assumption about arguments\n\n\u0394-type : Type \u2192 \u0394-Type\n\u0394-type nats = nats\n\u0394-type bags = bags\n\u0394-type (\u03c4\u2081 \u21d2 \u03c4\u2082) = alter \u0394-type \u03c4\u2081 \u21d2 \u0394-type \u03c4\u2082\n\n\u0394-context : Context \u2192 \u0394-Context\n\u0394-context \u2205 = \u2205\n\u0394-context (\u03c4 \u2022 \u0393) = alter \u0394-type \u03c4 \u2022 \u0394-context \u0393\n\n{-\n\n\u27e6_\u27e7\u0394Type : \u2200 {\u03c4 args} \u2192 \u0394-Type \u03c4 {args} \u2192 Set\n\ndata \u0394-Env : (\u0393 : Context) \u2192 {vars : Vars \u0393} \u2192 Set\ndata \u0394-Term : (\u0393 : Context) \u2192 Type \u2192 {vars : Vars \u0393} \u2192 Set\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u03c4 : Type} {\u0393 : Context} {vars}\n    \u2192 \u0394-Term \u0393 \u03c4 {vars} \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} {vars} \u2192 (\u03c1 : \u0394-Env \u0393 {vars}) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} {vars} \u2192 (\u03c1 : \u0394-Env \u0393 {vars}) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393 vars} \u2192\n    (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term \u0393 nats {vars}\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393 vars} \u2192\n    (db : Bag) \u2192 \u0394-Term \u0393 bags {vars}\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393 vars} \u2192\n    (x : Var \u0393 \u03c4) \u2192 \u0394-Term \u0393 \u03c4 {vars}\n  -- changes to abstractions are binders of x and dx\n  -- There are two kinds of those: One who expects the argument\n  -- to change, and one who does not.\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} \u2192 (t : \u0394-Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082 {alter vars}) \u2192\n         \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars}\n  \u0394abs\u2080 : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} \u2192 (t : \u0394-Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082 {abide vars}) \u2192\n          \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars}\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars}\n    (ds : \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars})\n    ( t :   Term \u0393 \u03c4\u2081)\n    (dt : \u0394-Term \u0393 \u03c4\u2081 {vars})\n    (R[t,dt] : {\u03c1 : \u0394-Env \u0393 {-vars-}} \u2192\n      valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1)) \u2192\n    \u0394-Term \u0393 \u03c4\u2082 {vars}\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393 vars}\n    (ds : \u0394-Term \u0393 nats {vars})\n    (dt : \u0394-Term \u0393 nats {vars}) \u2192\n    \u0394-Term \u0393 nats {vars}\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats))\n    (df : \u0394-Term \u0393 (nats \u21d2 nats) {vars})\n    ( b :   Term \u0393 bags)\n    (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n  \u0394map\u2081 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats)) (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n\n\u27e6 \u0394 nats \u27e7\u0394Type = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394Type = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) {alter args} \u27e7\u0394Type =\n  \u2200 {args\u2081} \u2192\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 {args\u2081} \u27e7\u0394Type) \u2192 valid v dv \u2192\n  \u27e6 \u0394 \u03c4\u2082 {args} \u27e7\u0394Type\n\nmeaning-\u0394Type : Meaning\u0394 Type\nmeaning-\u0394Type = meaning\u0394 \u27e6_\u27e7\u0394Type\n\nrecord Meaning\u0394\n  (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n    constructor\n      meaning\u0394\n    field\n      {\u0394-Semantics} : Set \u2113\n      \u27e8_\u27e9\u27e6_\u27e7\u0394 : Syntax \u2192 \u0394-Semantics\n\nopen Meaning\u0394 {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7\u0394 to \u27e6_\u27e7\u0394)\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u03c4\u2081 \u27e7\u0394Type) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency and honesty.\n-- \u0394-Env : (\u0393 : Context) \u2192 {vars : Vars \u0393} \u2192 Set\ndata \u0394-Env where\n  \u2205 : \u0394-Env \u2205 {\u2205}\n  abide : \u2200 {\u03c4 \u0393 vars} \u2192\n    (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u03c4 \u27e7\u0394) \u2192 valid v dv \u2192 v \u2295 dv \u2261 v \u2192\n    \u0394-Env \u0393 {vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {abide vars}\n  alter : \u2200 {\u03c4 \u0393 vars} \u2192\n    (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u03c4 \u27e7\u0394) \u2192 valid v dv \u2192\n    \u0394-Env \u0393 {vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {alter vars}\n\n\u2016_\u2016 : \u2200 {\u03c4 \u0393 vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {vars} \u2192\n  Quadruple \u27e6 \u03c4 \u27e7 (\u03bb _ \u2192 \u27e6 \u03c4 \u27e7\u0394) (\u03bb v dv \u2192 valid v dv)\n    (\u03bb _ _ _ \u2192 \u0394-Env \u0393 {cdr vars})\n\n\u2016 abide v dv R[v,dv] _ \u03c1 \u2016 = cons v dv R[v,dv] \u03c1\n\u2016 alter v dv R[v,dv]   \u03c1 \u2016 = cons v dv R[v,dv] \u03c1\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393 vars} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6 this   \u27e7\u0394Var \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\nmeaning-\u0394Var : \u2200 {\u03c4 \u0393} {vars : Vars \u0393} \u2192 Meaning\u0394 (Var \u0393 \u03c4)\nmeaning-\u0394Var {\u03c4} {\u0393} {vars} = meaning\u0394 (\u27e6_\u27e7\u0394Var {\u03c4} {\u0393} {vars})\n\n-- \u27e6_\u27e7\u0394Term : \u2200 {\u03c4 \u0393 vars} \u2192\n--   \u0394-Term \u0393 \u03c4 {vars} \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6 \u0394nat old new \u27e7\u0394Term \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394Term \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394Term \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv]          \u2192 \u27e6 t \u27e7\u0394Term (alter v dv R[v,dv] \u03c1)\n\u27e6 \u0394abs\u2080 t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv] {-v\u2295dv=v-} \u2192 \u27e6 t \u27e7\u0394Term (abide v dv R[v,dv] {!v\u2295dv=v!} \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394Term \u03c1 =\n  \u27e6 ds \u27e7\u0394Term \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1) R[dt,t]\n  --(R[dt,t] {\u03c1} {honesty = {!!}})\n\u27e6 \u0394add ds dt \u27e7\u0394Term \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394Term \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394Term \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394Term \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394Term \u03c1\n    dh = \u27e6 df \u27e7\u0394Term \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394Term \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394Term \u03c1)\n\nmeaning-\u0394Term-0 : \u2200 {\u03c4 \u0393 vars} \u2192 Meaning (\u0394-Term \u0393 \u03c4 {vars})\nmeaning-\u0394Term-0 = meaning \u27e6_\u27e7\u0394Term\n\nmeaning-\u0394Term-1 : \u2200 {\u03c4 \u0393 vars} \u2192 Meaning\u0394 (\u0394-Term \u0393 \u03c4 {vars})\nmeaning-\u0394Term-1 = meaning\u0394 \u27e6_\u27e7\u0394Term\n\n{-\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7 \u03c1\n    db = \u27e6 derive t \u27e7 \u03c1\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394-Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7 \u03c1\u2032\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n-}\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"28811f00933ec13ba9e4a8c9ad81998ae70b7eb0","subject":"Alternative definition of function composition","message":"Alternative definition of function composition\n\nThis is the second definition taken from New\/Changes.agda. As usual, its\ncorrectness proofs are much simpler (after we've rephrased the\nstatements cleverly).\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/Changes.agda","new_file":"Thesis\/Changes.agda","new_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\nopen import Data.Integer\nopen import Data.Unit\nopen import Theorem.Groups-Nehemiah\n\nprivate\n  intCh = \u2124\ninstance\n  intCS : ChangeStructure \u2124\nintCS = record\n  { Ch = \u2124\n  ; ch_from_to_ = \u03bb dv v1 v2 \u2192 v1 + dv \u2261 v2\n  ; isCompChangeStructure = record\n    { isChangeStructure = record\n      { _\u2295_ = _+_\n      ; fromto\u2192\u2295 = \u03bb dv v1 v2 v2\u2261v1+dv \u2192 v2\u2261v1+dv\n      ; _\u229d_ = _-_\n      ; \u229d-fromto = \u03bb a b \u2192 n+[m-n]=m {a} {b}\n      }\n    ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 da1 + da2\n    ; \u229a-fromto = i\u229a-fromto\n    }\n  }\n  where\n    i\u229a-fromto : (a1 a2 a3 : \u2124) (da1 da2 : intCh) \u2192\n      a1 + da1 \u2261 a2 \u2192 a2 + da2 \u2261 a3 \u2192 a1 + (da1 + da2) \u2261 a3\n    i\u229a-fromto a1 a2 a3 da1 da2 a1+da1\u2261a2 a2+da2\u2261a3\n      rewrite sym (associative-int a1 da1 da2) | a1+da1\u2261a2 = a2+da2\u2261a3\n","old_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa = \u229a-fromto (f1 a1) (f2 a1) (f3 a2)  (df1 a1 (nil a1)) (df2 a1 da) (dff1 (nil a1) a1 a1 (nil-fromto a1)) (dff2 da a1 a2 daa)\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\nopen import Data.Integer\nopen import Data.Unit\nopen import Theorem.Groups-Nehemiah\n\nprivate\n  intCh = \u2124\ninstance\n  intCS : ChangeStructure \u2124\nintCS = record\n  { Ch = \u2124\n  ; ch_from_to_ = \u03bb dv v1 v2 \u2192 v1 + dv \u2261 v2\n  ; isCompChangeStructure = record\n    { isChangeStructure = record\n      { _\u2295_ = _+_\n      ; fromto\u2192\u2295 = \u03bb dv v1 v2 v2\u2261v1+dv \u2192 v2\u2261v1+dv\n      ; _\u229d_ = _-_\n      ; \u229d-fromto = \u03bb a b \u2192 n+[m-n]=m {a} {b}\n      }\n    ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 da1 + da2\n    ; \u229a-fromto = i\u229a-fromto\n    }\n  }\n  where\n    i\u229a-fromto : (a1 a2 a3 : \u2124) (da1 da2 : intCh) \u2192\n      a1 + da1 \u2261 a2 \u2192 a2 + da2 \u2261 a3 \u2192 a1 + (da1 + da2) \u2261 a3\n    i\u229a-fromto a1 a2 a3 da1 da2 a1+da1\u2261a2 a2+da2\u2261a3\n      rewrite sym (associative-int a1 da1 da2) | a1+da1\u2261a2 = a2+da2\u2261a3\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57dd579828aa8a7c7ff8b0546dd51520584eb18e","subject":"agda: add extended comment on semantics of \u0394","message":"agda: add extended comment on semantics of \u0394\n\nDuring my walk-throughs on the code, explaining this line was a bit\nhard and required a small example. After writing it here during the\nwalkthrough, I decided to extend it and save it.\n\nOld-commit-hash: 279b6047512b8c934cae49bda5977fa104077e12\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Evaluation\/Total.agda","new_file":"Denotational\/Evaluation\/Total.agda","new_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\n{-\n\nHere is an example to understand the semantics of \u0394. I will use a\nnamed variable representation for the task.\n\nConsider the typing judgment:\n\n   x: T |- x: T\n\nThus, we have that:\n\n   dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nThanks to weakening, we also have:\n\n   y : S, dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nIn the formalization, we need a proof \u0393\u2032 that the context \u0393\u2081 = dx : \u0394\nT, x: T is a subcontext of \u0393\u2082 = y : S, dx : \u0394 T, x: T. Thus, \u0393\u2032 has\ntype \u0393\u2081 \u227c \u0393\u2082.\n\nNow take the environment:\n\n   \u03c1 = y \u21a6 w, dx \u21a6 dv, x \u21a6 v\n\nSince the semantics of \u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082 is a function from environments\nfor \u0393\u2082 to environments for \u0393\u2081, we have that:\n\n   \u27e6 \u0393\u2032 \u27e7 \u03c1 = dx \u21a6 dv, x \u21a6 v\n\nFrom the definitions of update and ignore, it follows that:\n\n   update (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 dv \u2295 v\n   ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 v\n\nHence, finally, we have that:\n\n   diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nis simply diff (dv \u2295 v) v (or (dv \u2295 v) \u229d v). If dv is a valid change,\nthat's just dv, that is \u27e6 dx \u27e7 \u03c1. In other words\n\n-}\n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n","old_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bd2e108c643b5ec0da84a5746e9e2a4410b98970","subject":"elgamal cleanups and improvments","message":"elgamal cleanups and improvments\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\n\nmodule elgamal where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\nmodule Sum where\n    Sum : \u2605 \u2192 \u2605\n    Sum A = (A \u2192 \u2115) \u2192 \u2115 \n\n    Count : \u2605 \u2192 \u2605\n    Count A = (A \u2192 Bit) \u2192 \u2115 \n\n    SumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\n    SumExt sumA = \u2200 {f g} \u2192 f \u2257 g \u2192 sumA f \u2261 sumA g\n\n    sumToCount : \u2200 {A} \u2192 Sum A \u2192 Count A\n    sumToCount sumA f = sumA (Bool.to\u2115 \u2218 f)\n\n    sum\u22a4 : Sum \u22a4\n    sum\u22a4 f = f _\n\n    sum\u22a4-ext : SumExt sum\u22a4\n    sum\u22a4-ext f\u2257g = f\u2257g _\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    sumBit-ext : SumExt sumBit\n    sumBit-ext f\u2257g rewrite f\u2257g 0b | f\u2257g 1b = refl\n\n    -- liftM2 _,_ in the continuation monad\n    sumProd : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n    sumProd sumA sumB f = sumA (\u03bb x\u2080 \u2192\n                           sumB (\u03bb x\u2081 \u2192\n                             f (x\u2080 , x\u2081)))\n\n    sumProd-ext : \u2200 {A B} {sumA : Sum A} {sumB : Sum B} \u2192\n                  SumExt sumA \u2192 SumExt sumB \u2192 SumExt (sumProd sumA sumB)\n    sumProd-ext sumA-ext sumB-ext f\u2257g = sumA-ext (\u03bb x \u2192 sumB-ext (\u03bb y \u2192 f\u2257g (x , y)))\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum.Sum \u2124q)\n  (sum\u2124q-ext : Sum.SumExt sum\u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  where\n\n  open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  sum : \u2200 u \u2192 Sum (El u)\n  sum `\u22a4         = sum\u22a4\n  sum `X         = sum\u2124q\n  sum (u\u2080 `\u00d7 u\u2081) = sumProd (sum u\u2080) (sum u\u2081)\n\n  sum-ext : \u2200 u \u2192 SumExt (sum u)\n  sum-ext `\u22a4         = sum\u22a4-ext\n  sum-ext `X         = sum\u2124q-ext\n  sum-ext (u\u2080 `\u00d7 u\u2081) = sumProd-ext (sum-ext u\u2080) (sum-ext u\u2081)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum u (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = sumToCount sum\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _ \n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047) \n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum\u2124q (f \u2218 [suc]) \u2261 sum\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n  vmap-ext : \u2200 {A B : \u2605} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 vmap f \u2257 vmap {n = n} g\n  vmap-ext f\u2257g [] = refl\n  vmap-ext f\u2257g (x \u2237 xs) rewrite f\u2257g x | vmap-ext f\u2257g xs = refl\n\n  sum\u2124q-ext : SumExt sum\u2124q\n  sum\u2124q-ext f\u2257g rewrite vmap-ext f\u2257g all\u2124q = refl\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum.Sum \u2124q)\n  (sum\u2124q-ext : Sum.SumExt sum\u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open Sum\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-ext sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 D (r , x , y , _) = D r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 D (r , x , y , z) = D r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba D) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (D b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2295_ : G \u2192 Message \u2192 Message)\n  (_\u2295\u207b\u00b9_ : G \u2192 Message \u2192 Message)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2295 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = (g\u02b8 ^ x) \u2295\u207b\u00b9 \u03b6\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R = (R \u2192 PubKey \u2192 Bit \u2192 M) \u00d7 (R \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R _I : \u2605} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , D) b (r , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      D r pk (Enc pk (m r pk b) y)\n\n    Game : (i : Bit) \u2192 \u2200 {R} \u2192 EncAdv R \u2192 (Bit \u00d7 R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , D) (b , r , x , y , z) =\n      let g\u02e3 = g^ x\n          g\u02b8 = g^ y\n          \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n          \u03b6  = \u03b4 \u2295 m r g\u02e3 b\n      in b ==\u1d47 D r g\u02e3 (g\u02b8 , \u03b6)\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R} \u2192 Game 0b \u2261 Game0 {R}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    -- Game0 \u2248 Game 0b\n    -- Game1 = Game 1b\n\n    -- Game0 \u2264 \u03b5\n    -- Game1 \u2261 0\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M D (r , x , y , z) = D r g\u02e3 g\u02b8 (g\u1dbb \u2295 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R} \u2192 Bit \u2192 EncAdv R \u2192 DDHAdv R\n    TrA b (m , D) r g\u02e3 g\u02b8 g\u02e3\u02b8 = D r g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2295 m r g\u02e3 b)\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u207b\u00b9 : G \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n    where\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_ (flip _\/_) public\n\n    like-SS\u2141 : \u2200 {R _I : \u2605} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , D) b (r , x , y , _z) =\n      D r g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    open Sum\n    module WithSumRAnd\u2124qProps\n        (R\u2090 : \u2605)\n        (sumR\u2090 : Sum R\u2090)\n        (sumR\u2090-ext : SumExt sumR\u2090)\n        (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n        (sum\u2124q : Sum \u2124q)\n        (sum\u2124q-ext : SumExt sum\u2124q)\n        where\n\n        module SumR where\n          R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n          sumR : Sum R\n          sumR = sumProd sumR\u2090 (sumProd sum\u2124q (sumProd sum\u2124q sum\u2124q))\n          #R_ : Count R\n          #R_ = sumToCount sumR\n          sumR-ext : SumExt sumR\n          sumR-ext = sumProd-ext sumR\u2090-ext (sumProd-ext sum\u2124q-ext (sumProd-ext sum\u2124q-ext sum\u2124q-ext))\n        private\n            #q_ : Count \u2124q\n            #q_ = sumToCount sum\u2124q\n\n        open SumR\n\n        _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n        f \u2248q g = #q f \u2261 #q g\n\n        _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n        f \u2248R g = #R f \u2261 #R g\n\n        module EvenMoreProof\n            (ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n            (otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n            where\n\n                otp-lem' : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248R OTP\u2141 M\u2081 D\n                otp-lem' D M\u2080 M\u2081 = sumR\u2090-ext (\u03bb r \u2192\n                                     sum\u2124q-ext (\u03bb x \u2192\n                                       sum\u2124q-ext (\u03bb y \u2192\n                                         pf r x y)))\n                  where\n                  f0 = \u03bb M r x y z \u2192 OTP\u2141 M D (r , x , y , z)\n                  f1 = \u03bb M r x y \u2192 sum\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n                  f2 = \u03bb M r x \u2192 sum\u2124q (f1 M r x)\n                  f3 = \u03bb M r \u2192 sum\u2124q (f2 M r)\n                  pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n                  pf r x y = pf'\n                    where g\u02e3 = g^ x\n                          g\u02b8 = g^ y\n                          m\u2080 = M\u2080 r g\u02e3\n                          m\u2081 = M\u2081 r g\u02e3\n                          f5 = \u03bb M z \u2192 D r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                          pf' : f5 m\u2080 \u2248q f5 m\u2081\n                          pf' rewrite otp-lem (D r g\u02e3 g\u02b8) m\u2080\n                                    | otp-lem (D r g\u02e3 g\u02b8) m\u2081 = refl\n\n                projM : EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 G\n                projM (m , _) b r g\u02e3 = m r g\u02e3 b\n\n                projD : EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 G \u2192 Bit\n                projD (_ , D) r g\u02e3 g\u02b8 g\u1dbb\u2219M = D r g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n                module WithAdversary (A : EncAdv R\u2090) b where\n                    A\u1d47 = TrA b A\n                    A\u00ac\u1d47 = TrA (not b) A\n\n                    pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n                    pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n                    pf1 = sumR-ext (cong Bool.to\u2115 \u2218 pf0,5)\n\n                    pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n                    pf2 = ddh-hyp A\u1d47\n\n                    pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n                    pf2,5 = refl\n\n                    pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n                    pf3 = otp-lem' (projD A) (projM A b) (projM A (not b))\n\n                    pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n                    pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n                    pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n                    pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n                    pf5 = sumR-ext (cong Bool.to\u2115 \u2218 pf4,5)\n\n                    final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n                    final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|) where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u207b\u00b9 _\u2219_\n  open EB hiding (g^_)\n\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-ext sum\u2124q-\u229e-lem\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum `R\u2090\n  sumR\u2090-ext = sum-ext `R\u2090\n\n  open WithSumRAnd\u2124qProps R\u2090 sumR\u2090 sumR\u2090-ext dist-^-\u22a0 sum\u2124q sum\u2124q-ext\n\n  open SumR\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n  open EvenMoreProof ddh-hyp otp-lem\n\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = WithAdversary.final A 0b\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\n--import Data.Vec.Properties as Vec\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nmodule elgamal where\n\n\u2605\u2081 : Set\u2082\n\u2605\u2081 = Set\u2081\n\n\u2605 : Set\u2081\n\u2605 = Set\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sumA = \u2200 f g \u2192 f \u2257 g \u2192 sumA f \u2261 sumA g\n\nsumToCount : \u2200 {A} \u2192 Sum A \u2192 Count A\nsumToCount sumA f = sumA (Bool.to\u2115 \u2218 f)\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\n-- liftM2 _,_ in the continuation monad\nsumProd : \u2200 {A B : Set} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsumProd sumA sumB f = sumA (\u03bb x\u2080 \u2192\n                       sumB (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\nsumProd-ext : \u2200 {A B : Set} {sumA : Sum A} {sumB : Sum B} \u2192\n              SumExt sumA \u2192 SumExt sumB \u2192 SumExt (sumProd sumA sumB)\nsumProd-ext sumA-ext sumB-ext f g f\u2257g = sumA-ext _ _ (\u03bb x \u2192 sumB-ext _ _ (\u03bb y \u2192 f\u2257g (x , y)))\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum \u2124q)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  where\n\n  open Univ \u2124q\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  sum : (u : `\u2605) \u2192 Sum (El u)\n  sum `\u22a4         f = f _\n  sum `X         f = sum\u2124q f\n  sum (u\u2080 `\u00d7 u\u2081) f = sum u\u2080 (\u03bb x\u2080 \u2192\n                       sum u\u2081 (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum u (Bool.to\u2115 \u2218 run\u21ba f)\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _ \n  lem x Adv = sum\u2124q-lem (Bool.to\u2115 \u2218 Adv) x\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047) \n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum\u2124q (f \u2218 [suc]) \u2261 sum\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  postulate\n    g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum \u2124q)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  where\n\n  open Univ \u2124q\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g ^ x\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 D (r , x , y , _) = D r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 D (r , x , y , z) = D r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- \u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- \u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba D) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (D b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2295_ : G \u2192 Message \u2192 Message)\n  (_\u2295\u207b\u00b9_ : G \u2192 Message \u2192 Message)\n  where\n\n    -- \u03b1 is the pk\n    -- \u03b1 = g ^ x\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g ^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc \u03b1 m y = \u03b2 , \u03b6 where\n      \u03b2 = g ^ y\n      \u03b4 = \u03b1 ^ y\n      \u03b6 = \u03b4 \u2295 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba \u03b1 m = mk (Enc \u03b1 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (\u03b2 , \u03b6) = (\u03b2 ^ x) \u2295\u207b\u00b9 \u03b6\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R = PubKey \u2192 R \u2192 (Bit \u2192 M) \u00d7 (C \u2192 Bit)\n\n    {-\n    Game0 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bit\n    Game0 A (b , x , y , z , r) =\n      let (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n\n    Game : (i : Bit) \u2192 \u2200 {R} \u2192 EncAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    Game i A (b , x , y , z , r) =\n      let (\u03b1 , sk) = KeyGen x\n          (m , D) = A \u03b1 r\n          \u03b2 = g ^ y\n          \u03b4 = \u03b1 ^ case i 0\u2192 y 1\u2192 z\n          \u03b6 = \u03b4 \u2295 m b\n      in {-b ==\u1d47-} D (\u03b2 , \u03b6)\n\n    Game-0b\u2261Game0 : \u2200 {R} \u2192 Game 0b \u2261 Game0 {R}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    SS\u2141 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 A b (r , x , y , z) =\n      let -- (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n         where pk = g ^ x\n\n    open DDH \u2124q _\u22a0_ G (_^_ g) public\n\n    -- Game0 \u2248 Game 0b\n    -- Game1 = Game 1b\n\n    -- Game0 \u2264 \u03b5\n    -- Game1 \u2261 0\n\n    -- \u2047 \u2295 x \u2248 \u2047\n\n    -- g ^ \u2047 \u2219 x \u2248 g ^ \u2047\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u207b\u00b9 : G \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n    where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_ (flip _\/_) public\n\n    TrA : \u2200 {R} \u2192 Bit \u2192 EncAdv R \u2192 DDHAdv R\n    TrA b A r g\u02e3 g\u02b8 g\u02e3\u02b8 = d (g\u02b8 , g\u02e3\u02b8 \u2219 m b)\n       where m,d = A g\u02e3 r\n             m = proj\u2081 m,d\n             d = proj\u2082 m,d\n\n    like-SS\u2141 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 A b (r , x , y , _z) =\n      let -- g\u02e3 = g ^ x\n          -- g\u02b8 = g ^ y\n          (m , D) = A g\u02e3 r in\n      D (g\u02b8 , (g\u02e3 ^ y) \u2219 m b)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    OTP\u2141 : \u2200 {R : Set} \u2192 (R \u2192 G \u2192 G) \u2192 (R \u2192 G \u2192 G \u2192 G \u2192 Bit) \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M D (r , x , y , z) = D r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n            g\u1dbb = g ^ z\n\n    module With\u2124qProps\n        (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      -- (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n        (sum\u2124q : Sum \u2124q)\n        (sum\u2124q-ext : SumExt sum\u2124q) where\n        #q_ : Count \u2124q\n        #q_ = sumToCount sum\u2124q\n\n        module SumU\n          (R : Set)\n          (sumR : Sum R)\n          (sumR-ext : SumExt sumR)\n          where\n          U = R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n          sumU : Sum U\n          sumU = sumProd sumR (sumProd sum\u2124q (sumProd sum\u2124q sum\u2124q))\n          #U_ : Count U\n          #U_ = sumToCount sumU\n          sumU-ext : SumExt sumU\n          sumU-ext = sumProd-ext sumR-ext (sumProd-ext sum\u2124q-ext (sumProd-ext sum\u2124q-ext sum\u2124q-ext))\n\n        module WithSumR\n            (R : Set)\n            (sumR : Sum R)\n            (sumR-ext : SumExt sumR)\n           where\n           open SumU R sumR sumR-ext\n\n           module EvenMoreProof\n            (ddh-hyp : (A : DDHAdv R) \u2192 #U (DDH\u2141 A 0b) \u2261 #U (DDH\u2141 A 1b))\n\n            (otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 #q(\u03bb x \u2192 A (g^ x \u2219 m)) \u2261 #q(\u03bb x \u2192 A (g^ x)))\n            where\n\n                _\u2248U_ : (f g : U \u2192 Bit) \u2192 Set\n                f \u2248U g = #U f \u2261 #U g\n\n                otp-lem' : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248U OTP\u2141 M\u2081 D\n                otp-lem' D M\u2080 M\u2081 = sumR-ext (f3 M\u2080) (f3 M\u2081) (\u03bb r \u2192\n                                     sum\u2124q-ext (f2 M\u2080 r) (f2 M\u2081 r) (\u03bb x \u2192\n                                       sum\u2124q-ext (f1 M\u2080 r x) (f1 M\u2081 r x) (\u03bb y \u2192\n                                         pf r x y)))\n                  where\n                  f0 = \u03bb M r x y z \u2192 OTP\u2141 M D (r , x , y , z)\n                  f1 = \u03bb M r x y \u2192 sum\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n                  f2 = \u03bb M r x \u2192 sum\u2124q (f1 M r x)\n                  f3 = \u03bb M r \u2192 sum\u2124q (f2 M r)\n                  pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n                  pf r x y = pf'\n                    where g\u02e3 = g^ x\n                          g\u02b8 = g^ y\n                          m0 = M\u2080 r g\u02e3\n                          m1 = M\u2081 r g\u02e3\n                          f5 = \u03bb M z \u2192 D r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                          pf' : #q(f5 m0) \u2261 #q(f5 m1)\n                          pf' rewrite otp-lem (D r g\u02e3 g\u02b8) m0\n                                    | otp-lem (D r g\u02e3 g\u02b8) m1 = refl\n\n      {-\n        #-proj\u2081 : \u2200 {A B : Set} {f g : A \u2192 Bit} \u2192 # f \u2261 # g\n                   \u2192 #_ {A \u00d7 B} (f \u2218 proj\u2081) \u2261 #_ {A \u00d7 B} (g \u2218 proj\u2081)\n        #-proj\u2081 = {!!}\n\n        otp-lem'' : \u2200 (A : G \u2192 Bit) m\n           \u2192 #(\u03bb { (x , y) \u2192 A (g^ x \u2219 m) }) \u2261 #(\u03bb { (x , y) \u2192 A (g ^ x) })\n        otp-lem'' = {!!}\n        -}\n                projM : EncAdv R \u2192 Bit \u2192 R \u2192 G \u2192 G\n                projM A b r g\u02e3 = proj\u2081 (A g\u02e3 r) b\n\n                projD : EncAdv R \u2192 R \u2192 G \u2192 G \u2192 G \u2192 Bit\n                projD A r g\u02e3 g\u02b8 g\u1dbb\u2219M = proj\u2082 (A g\u02e3 r) (g\u02b8 , g\u1dbb\u2219M)\n\n\n                module WithAdversary (A : EncAdv R) b where\n                    A\u1d47 = TrA b A\n                    A\u00ac\u1d47 = TrA (not b) A\n\n                    pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n                    pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf1 : #U(SS\u2141 A b) \u2261 #U(DDH\u2141 A\u1d47 0b)\n                    pf1 = sumU-ext _ _ (cong Bool.to\u2115 \u2218 pf0,5)\n\n                    pf2 : DDH\u2141 A\u1d47 0b \u2248U DDH\u2141 A\u1d47 1b\n                    pf2 = ddh-hyp A\u1d47\n\n                    pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n                    pf2,5 = refl\n\n                    pf3 : DDH\u2141 A\u1d47 1b \u2248U DDH\u2141 A\u00ac\u1d47 1b\n                    pf3 = otp-lem' (projD A) (projM A b) (projM A (not b))\n\n                    pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248U DDH\u2141 A\u00ac\u1d47 0b\n                    pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n                    pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n                    pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf5 : #U(SS\u2141 A (not b)) \u2261 #U(DDH\u2141 A\u00ac\u1d47 0b)\n                    pf5 = sumU-ext _ _ (cong Bool.to\u2115 \u2218 pf4,5)\n\n                    final : #U(SS\u2141 A b) \u2261 #U(SS\u2141 A (not b))\n                    final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\n\n      {-\n        pf1 : \u2200 {R} A r \u2192 Game 0b {R} A r \u2261 \u2141\u2032 (TrA 1b A) r\n        pf1 A (true , x , y , z , r) rewrite dist-^-\u22a0 g x y = refl\n        pf1 A (false , x , y , z , r) = {!!}\n\n        pf2 : \u2200 {R} A r \u2192 Game 1b {R} A r \u2261 \u2141\u2032 (TrA 0b A) r\n        pf2 A (true , x , y , z , r)  = {!!}\n        pf2 A (false , x , y , z , r) = {!!}\n-}\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|) where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_\n\n    module Proof\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      -- (^-lem : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n      -- (x \u2219 x \u207b\u00b9 \u2261 0)\n      (\u211a : \u2605)\n      (dist : \u211a \u2192 \u211a \u2192 \u211a)\n      (1\/2 : \u211a)\n      (_\u2264_ : \u211a \u2192 \u211a \u2192 \u2605)\n      (\u03b5-DDH : \u211a)\n      where\n\n        Pr[S_] : Bool \u2192 \u211a\n        Pr[S b ] = {!!}\n\n        -- SS\n        -- SS = dist Pr[\n\n        Pr[S\u2080] = Pr[S 0b ]\n        Pr[S\u2081] = Pr[S 1b ]\n\n        pf1 : Pr[S\u2081] \u2261 1\/2\n        pf1 = {!!}\n\n        pf2 : dist Pr[S\u2080] 1\/2 \u2261 dist Pr[S\u2080] Pr[S\u2081]\n        pf2 = {!!}\n\n        pf3 : dist Pr[S\u2080] Pr[S\u2081] \u2264 \u03b5-DDH\n        pf3 = {!!}\n\n        pf4 : dist Pr[S\u2080] 1\/2 \u2264 \u03b5-DDH\n        pf4 = {!!}\n\nmodule G11 = G-implem 11 10 (## 2) (## 0) (## 1) (## 0) (## 1)\nopen G11\nmodule E11 = El-Gamal-Base _ _\u22a0_ G g _^_ {!!} _\u2219_\nopen E11\nopen With\u2124qProps ?\n-- open Proof {!!}\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c11e644e4393f792d6cc2dc45eb85a1e4d25a878","subject":" demand way","message":" demand way\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"canonical-indeterminate-forms.agda","new_file":"canonical-indeterminate-forms.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n  data cif-base : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    CIFBEHole : \u2200 { \u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        ((u ::[ \u0393' ] b) \u2208 \u0394))\n       \u2192 cif-base \u0394 d\n    CIFBNEHole : \u2200 { \u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        (d' final) \u00d7\n        (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n        ((u ::[ \u0393' ] b) \u2208 \u0394))\n        \u2192 cif-base \u0394 d\n    CIFBAp : \u2200 { \u0394 d} \u2192\n      \u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n        (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n        (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n        (d1 indet) \u00d7\n        (d2 final) \u00d7\n        ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n        \u2192 cif-base \u0394 d\n    CIFBCast : \u2200 { \u0394 d} \u2192\n      \u03a3[ d' \u2208 dhexp ]\n        ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n        (d' indet) \u00d7\n        ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))\n        \u2192 cif-base \u0394 d\n    CIFBFailedCast : \u2200 { \u0394 d} \u2192\n      \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == d' \u27e8 \u03c4' \u21d2\u2987\u2988\u21d2\u0338 b \u27e9) \u00d7\n        (\u03c4' ground) \u00d7\n        (\u03c4' \u2260 b) \u00d7\n        (\u0394 , \u2205 \u22a2 d' :: \u03c4'))\n       \u2192 cif-base \u0394 d\n\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192 cif-base \u0394 d\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFBAp (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = CIFBEHole (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFBNEHole (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x)\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = CIFBCast (_ , refl , ind , x\u2081)\n  canonical-indeterminate-forms-base (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = CIFBFailedCast (_ , _ , refl , x\u2085 , x\u2087 , x)\n\n  data cif-arr : (\u0394 : hctx) (d : dhexp) (\u03c41 \u03c42 : htyp) \u2192 Set where\n    CIFAEHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394)))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFANEHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        (d' final) \u00d7\n        (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n        ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394)))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFAAp : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n        (\u0394 , \u2205 \u22a2 d1 :: \u03c4 ==> (\u03c41' ==> \u03c42')) \u00d7\n        (\u0394 , \u2205 \u22a2 d2 :: \u03c4) \u00d7\n        (d1 indet) \u00d7\n        (d2 final) \u00d7\n        ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFACast : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c43 \u2208 htyp ] \u03a3[ \u03c44 \u2208 htyp ]\n        ((d == d' \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9) \u00d7\n          (d' indet) \u00d7\n          ((\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44))))\n       \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFACastHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d' \u2208 dhexp ]\n        ((\u03c41 == \u2987\u2988) \u00d7\n          (\u03c42 == \u2987\u2988) \u00d7\n          (d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n          (d' indet) \u00d7\n          ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFAFailedCast : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ]\n          ((d == (d' \u27e8 \u03c4 \u21d2\u2987\u2988\u21d2\u0338 \u2987\u2988 ==> \u2987\u2988 \u27e9) ) \u00d7\n            (\u03c4 ground) \u00d7\n            (\u03c4 \u2260 (\u2987\u2988 ==> \u2987\u2988)) \u00d7\n            (\u0394 , \u2205 \u22a2 d' :: \u03c4)))\n          \u2192 cif-arr \u0394 d \u03c41 \u03c42\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       cif-arr \u0394 d \u03c41 \u03c42\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFAAp (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = CIFAEHole (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFANEHole (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x)\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = CIFACast (_ , _ , _ , _ , _ , refl , ind , x\u2081)\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = CIFACastHole (_ , refl , refl , refl , ind , x\u2081)\n  canonical-indeterminate-forms-arr (TAFailedCast x x\u2081 GHole x\u2083) (IFailedCast x\u2084 x\u2085 GHole x\u2087) = CIFAFailedCast (_ , _ , refl , x\u2085 , x\u2087 , x)\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n                                          (\u03c4' ground) \u00d7\n                                          (d' indet)))\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = Inr (Inr (Inr (_ , _ , refl , x\u2081 , ind)))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind ())\n  canonical-indeterminate-forms-hole (TAFailedCast x x\u2081 () x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087)\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam wt) () nb na nh\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAEHole x x\u2081) IEHole nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAEHole x x\u2081) IEHole nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAEHole x x\u2081) IEHole nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb z _ _\u2081 \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ _\u2081 z \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ z _\u2081 \u2192 z \u03c4 \u03c4\u2081 refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       \u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n                                         +\n                                       (\u03a3[ d' \u2208 dhexp ]\n                                         ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n                                          (d' indet) \u00d7\n                                          ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == d' \u27e8 \u03c4' \u21d2\u2987\u2988\u21d2\u0338 b \u27e9) \u00d7\n                                          (\u03c4' ground) \u00d7\n                                          (\u03c4' \u2260 b) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4')))\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x))\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (Inl (_ , refl , ind , x\u2081))))\n  canonical-indeterminate-forms-base (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = Inr (Inr (Inr (Inr (_ , _  , refl , x\u2085 , x\u2087 , x))))\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c4 ==> (\u03c41' ==> \u03c42')) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c4) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c43 \u2208 htyp ] \u03a3[ \u03c44 \u2208 htyp ]\n                                          ((d == d' \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9) \u00d7\n                                           (d' indet) \u00d7\n                                           ((\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44)))) +\n                                       (\u03a3[ d' \u2208 dhexp ]\n                                          ((\u03c41 == \u2987\u2988) \u00d7\n                                           (\u03c42 == \u2987\u2988) \u00d7\n                                           (d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n                                           (d' indet) \u00d7\n                                           ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))) +\n                                        (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ]\n                                           ((d == (d' \u27e8 \u03c4 \u21d2\u2987\u2988\u21d2\u0338 \u2987\u2988 ==> \u2987\u2988 \u27e9) ) \u00d7\n                                             (\u03c4 ground) \u00d7\n                                             (\u03c4 \u2260 (\u2987\u2988 ==> \u2987\u2988)) \u00d7\n                                             (\u0394 , \u2205 \u22a2 d' :: \u03c4)))\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = Inr (Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , ind , x\u2081))))\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = Inr (Inr (Inr (Inr (Inl (_ , refl , refl , refl , ind , x\u2081)))))\n  canonical-indeterminate-forms-arr (TAFailedCast x x\u2081 GHole x\u2083) (IFailedCast x\u2084 x\u2085 GHole x\u2087) = Inr (Inr (Inr (Inr (Inr (_ , _ , refl , x\u2085 , x\u2083 , x)))))\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n                                          (\u03c4' ground) \u00d7\n                                          (d' indet)))\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = Inr (Inr (Inr (_ , _ , refl , x\u2081 , ind)))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind ())\n  canonical-indeterminate-forms-hole (TAFailedCast x x\u2081 () x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087)\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam wt) () nb na nh\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAEHole x x\u2081) IEHole nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAEHole x x\u2081) IEHole nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAEHole x x\u2081) IEHole nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb z _ _\u2081 \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ _\u2081 z \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ z _\u2081 \u2192 z \u03c4 \u03c4\u2081 refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64cba9367b5f3fea04faf0eb4ea064124283c935","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 5c709c40f6f324f1a0056bad771ebef4\n\ndarcs-hash:20110324133307-3bd4e-465681788611ef6934d13ab3f7bebf4d0ae71f41.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n","old_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192\n                N (succ n + x) ) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1ade0ad7a853c7c488eb2b25e66563058db1a3d6","subject":"Bool: +cond","message":"Bool: +cond\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"beadc4c6bf5a0654b270878f150131cb18ada078","subject":"Data.Bool.NP: add ring modules Xor\u00b0, Bool\u00b0","message":"Data.Bool.NP: add ring modules Xor\u00b0, Bool\u00b0\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties\nopen import Data.Unit using (\u22a4)\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing Data.Bool.Properties.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring Data.Bool.Properties.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 {true}   {true}  _   _  = _\nT\u2227 {false}  {_}     ()  _\nT\u2227 {true}   {false} _   ()\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 {true} {true} _ = _\nT\u2227\u2081 {false} {_}   ()\nT\u2227\u2081 {true} {false}   ()\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {true} {true} _ = _\nT\u2227\u2082 {false} {_}   ()\nT\u2227\u2082 {true} {false}   ()\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {true} _ = inj\u2081 _\nT\u2228'\u228e {false} {true} _ = inj\u2082 _\nT\u2228'\u228e {false} {false} ()\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 {true} _ = _\nT\u2228\u2081 {false} {true} _ = _\nT\u2228\u2081 {false} {false} ()\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {true} _ = _\nT\u2228\u2082 {false} {true} _ = _\nT\u2228\u2082 {false} {false} ()\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nopen import Data.Unit using (\u22a4)\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 {true}   {true}  _   _  = _\nT\u2227 {false}  {_}     ()  _\nT\u2227 {true}   {false} _   ()\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 {true} {true} _ = _\nT\u2227\u2081 {false} {_}   ()\nT\u2227\u2081 {true} {false}   ()\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {true} {true} _ = _\nT\u2227\u2082 {false} {_}   ()\nT\u2227\u2082 {true} {false}   ()\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {true} _ = inj\u2081 _\nT\u2228'\u228e {false} {true} _ = inj\u2082 _\nT\u2228'\u228e {false} {false} ()\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 {true} _ = _\nT\u2228\u2081 {false} {true} _ = _\nT\u2228\u2081 {false} {false} ()\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {true} _ = _\nT\u2228\u2082 {false} {true} _ = _\nT\u2228\u2082 {false} {false} ()\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"09aa7263bcfccc0537afdaba6875c6d31a8fa0e1","subject":"update Group to include left and right inverse","message":"update Group to include left and right inverse\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group.agda","new_file":"lib\/Algebra\/Group.agda","new_contents":"{-# OPTIONS --without-K #-}\n  -- TODO\n  -- If you are looking for a proof of:\n  --   f (\u03a3(x\u1d62\u2208A) g(x\u2081)) \u2261 \u03a0(x\u1d62\u2208A) (f(g(x\u1d62)))\n  -- Have a look to:\n  --   https:\/\/github.com\/crypto-agda\/explore\/blob\/master\/lib\/Explore\/GroupHomomorphism.agda\nopen import Type using (Type_)\nopen import Data.Product.NP using (_,_;fst;snd)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\n\nmodule Algebra.Group where\n\nrecord Group-Ops {\u2113} (G : Type \u2113) : Type \u2113 where\n  constructor _,_\n\n  field\n    mon-ops : Monoid-Ops G\n    _\u207b\u00b9     : G \u2192 G\n\n  open Monoid-Ops mon-ops public\n  open From-Group-Ops \u03b5 _\u2219_ _\u207b\u00b9  public\n\nrecord Group-Struct {\u2113} {G : Type \u2113} (grp-ops : Group-Ops G) : Type \u2113 where\n  constructor _,_\n  open Group-Ops grp-ops\n\n  -- laws\n  field\n    mon-struct : Monoid-Struct mon-ops\n    inverse    : Inverse \u03b5 _\u207b\u00b9 _\u2219_\n\n  mon : Monoid G\n  mon = mon-ops , mon-struct\n\n  \u207b\u00b9\u2219-inverse : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n  \u207b\u00b9\u2219-inverse = fst inverse\n\n  \u2219\u207b\u00b9-inverse : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n  \u2219\u207b\u00b9-inverse = snd inverse\n\n  open Monoid-Struct mon-struct                             public\n  open From-Assoc-Identities-Inverse assoc identity inverse public\n\n-- TODO Monoid+LeftInverse \u2192 Group\n\nrecord Group {\u2113}(G : Type \u2113) : Type \u2113 where\n  constructor _,_\n  field\n    grp-ops    : Group-Ops G\n    grp-struct : Group-Struct grp-ops\n  open Group-Ops    grp-ops    public\n  open Group-Struct grp-struct public\n\n-- A renaming of Group-Ops with additive notation\nmodule Additive-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) where\n  private\n   module M = Group-Ops grp\n    using    ()\n    renaming ( _\u207b\u00b9 to 0\u2212_\n             ; _\/_ to _\u2212_\n             ; _^\u207b_ to _\u2297\u207b_\n             ; _^_ to _\u2297_\n             ; mon-ops to +-mon-ops\n             ; \/= to \u2212=)\n  open M public using (0\u2212_; +-mon-ops; \u2212=)\n  open Additive-Monoid-Ops +-mon-ops public\n  infixl 6 _\u2212_\n  infixl 7 _\u2297\u207b_ _\u2297_\n  _\u2212_   = M._\u2212_\n  _\u2297\u207b_  = M._\u2297\u207b_\n  _\u2297_   = M._\u2297_\n\n-- A renaming of Group-Struct with additive notation\nmodule Additive-Group-Struct {\u2113}{G : Type \u2113}{grp-ops : Group-Ops G}\n                             (grp-struct : Group-Struct grp-ops)\n    = Group-Struct grp-struct\n    using    ()\n    renaming ( mon-struct to +-mon-struct\n             ; mon to +-mon\n             ; assoc to +-assoc\n             ; identity to +-identity\n             ; \u03b5\u2219-identity to 0+-identity\n             ; \u2219\u03b5-identity to +0-identity\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             ; inverse to 0\u2212-inverse\n             ; \u2219-\/ to +-\u2212; \/-\u2219 to \u2212-+\n             ; unique-\u03b5-left to unique-0-left\n             ; unique-\u03b5-right to unique-0-right\n             ; x\/y\u2261\u03b5\u2192x\u2261y to x\u2212y\u22610\u2192x\u2261y\n             ; x\/y\u2262\u03b5 to x\u2212y\u22620\n             ; is-\u03b5-left to is-0-left\n             ; is-\u03b5-right to is-0-right\n             ; unique-\u207b\u00b9 to unique-0\u2212\n             ; cancels-\u2219-left to cancels-+-left\n             ; cancels-\u2219-right to cancels-+-right\n             ; elim-\u2219-right-\/ to elim-+-right-\u2212\n             ; elim-assoc= to elim-+-assoc=\n             ; elim-!assoc= to elim-+-!assoc=\n             ; elim-inner= to elim-+-inner=\n             ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n             ; \u207b\u00b9-inj to 0\u2212-inj\n             ; \u207b\u00b9-involutive to 0\u2212-involutive\n             ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n             )\n\n-- A renaming of Group with additive notation\nmodule Additive-Group {\u2113}{G : Type \u2113}(mon : Group G) where\n  open Additive-Group-Ops    (Group.grp-ops    mon) public\n  open Additive-Group-Struct (Group.grp-struct mon) public\n\n-- A renaming of Group-Ops with multiplicative notation\nmodule Multiplicative-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) = Group-Ops grp\n    using    ( _\u207b\u00b9; _\/_; \/=; _^\u207a_ ; _^\u207b_; _^_; _\u00b2; _\u00b3; _\u2074 )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1#; mon-ops to *-mon-ops; \u2219= to *= )\n\n-- A renaming of Group-Struct with multiplicative notation\nmodule Multiplicative-Group-Struct {\u2113}{G : Type \u2113}{grp-ops : Group-Ops G}\n                                   (grp-struct : Group-Struct grp-ops)\n  = Group-Struct grp-struct\n    using    ( unique-\u207b\u00b9\n             ; \u207b\u00b9-hom\u2032\n             ; \u207b\u00b9-inj\n             ; \u207b\u00b9-involutive\n             )\n    renaming ( assoc to *-assoc\n             ; identity to *-identity\n             ; \u03b5\u2219-identity to 1*-identity\n             ; \u2219\u03b5-identity to *1-identity\n             ; inverse to \u207b\u00b9-inverse\n             ; \u2219-\/ to *-\/; \/-\u2219 to \/-*\n             ; mon-struct to *-mon-struct\n             ; mon to *-mon\n             ; unique-\u03b5-left to unique-1-left\n             ; unique-\u03b5-right to unique-1-right\n             ; x\/y\u2261\u03b5\u2192x\u2261y to x\/y\u22611\u2192x\u2261y\n             ; x\/y\u2262\u03b5 to x\/y\u22621\n             ; is-\u03b5-left to is-1-left\n             ; is-\u03b5-right to is-1-right\n             ; cancels-\u2219-left to cancels-*-left\n             ; cancels-\u2219-right to cancels-*-right\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             ; elim-\u2219-right-\/ to elim-*-right-\/\n             ; elim-assoc= to elim-*-assoc=\n             ; elim-!assoc= to elim-*-!assoc=\n             ; elim-inner= to elim-*-inner=\n             ; \u03b5\u207b\u00b9\u2261\u03b5 to 1\u207b\u00b9\u22611\n             )\n\n-- A renaming of Group with multiplicative notation\nmodule Multiplicative-Group {\u2113}{G : Type \u2113}(mon : Group G) where\n  open Multiplicative-Group-Ops    (Group.grp-ops    mon) public\n  open Multiplicative-Group-Struct (Group.grp-struct mon) public\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n  -- TODO\n  -- If you are looking for a proof of:\n  --   f (\u03a3(x\u1d62\u2208A) g(x\u2081)) \u2261 \u03a0(x\u1d62\u2208A) (f(g(x\u1d62)))\n  -- Have a look to:\n  --   https:\/\/github.com\/crypto-agda\/explore\/blob\/master\/lib\/Explore\/GroupHomomorphism.agda\nopen import Type using (Type_)\nopen import Data.Product.NP using (_,_)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\n\nmodule Algebra.Group where\n\nrecord Group-Ops {\u2113} (G : Type \u2113) : Type \u2113 where\n  constructor _,_\n\n  field\n    mon-ops : Monoid-Ops G\n    _\u207b\u00b9     : G \u2192 G\n\n  open Monoid-Ops mon-ops public\n  open From-Group-Ops \u03b5 _\u2219_ _\u207b\u00b9  public\n\nrecord Group-Struct {\u2113} {G : Type \u2113} (grp-ops : Group-Ops G) : Type \u2113 where\n  constructor _,_\n  open Group-Ops grp-ops\n\n  -- laws\n  field\n    mon-struct : Monoid-Struct mon-ops\n    inverse    : Inverse \u03b5 _\u207b\u00b9 _\u2219_\n\n  mon : Monoid G\n  mon = mon-ops , mon-struct\n\n  open Monoid-Struct mon-struct                             public\n  open From-Assoc-Identities-Inverse assoc identity inverse public\n\n-- TODO Monoid+LeftInverse \u2192 Group\n\nrecord Group {\u2113}(G : Type \u2113) : Type \u2113 where\n  constructor _,_\n  field\n    grp-ops    : Group-Ops G\n    grp-struct : Group-Struct grp-ops\n  open Group-Ops    grp-ops    public\n  open Group-Struct grp-struct public\n\n-- A renaming of Group-Ops with additive notation\nmodule Additive-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) where\n  private\n   module M = Group-Ops grp\n    using    ()\n    renaming ( _\u207b\u00b9 to 0\u2212_\n             ; _\/_ to _\u2212_\n             ; _^\u207b_ to _\u2297\u207b_\n             ; _^_ to _\u2297_\n             ; mon-ops to +-mon-ops\n             ; \/= to \u2212=)\n  open M public using (0\u2212_; +-mon-ops; \u2212=)\n  open Additive-Monoid-Ops +-mon-ops public\n  infixl 6 _\u2212_\n  infixl 7 _\u2297\u207b_ _\u2297_\n  _\u2212_   = M._\u2212_\n  _\u2297\u207b_  = M._\u2297\u207b_\n  _\u2297_   = M._\u2297_\n\n-- A renaming of Group-Struct with additive notation\nmodule Additive-Group-Struct {\u2113}{G : Type \u2113}{grp-ops : Group-Ops G}\n                             (grp-struct : Group-Struct grp-ops)\n    = Group-Struct grp-struct\n    using    ()\n    renaming ( mon-struct to +-mon-struct\n             ; mon to +-mon\n             ; assoc to +-assoc\n             ; identity to +-identity\n             ; \u03b5\u2219-identity to 0+-identity\n             ; \u2219\u03b5-identity to +0-identity\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             ; inverse to 0\u2212-inverse\n             ; \u2219-\/ to +-\u2212; \/-\u2219 to \u2212-+\n             ; unique-\u03b5-left to unique-0-left\n             ; unique-\u03b5-right to unique-0-right\n             ; x\/y\u2261\u03b5\u2192x\u2261y to x\u2212y\u22610\u2192x\u2261y\n             ; x\/y\u2262\u03b5 to x\u2212y\u22620\n             ; is-\u03b5-left to is-0-left\n             ; is-\u03b5-right to is-0-right\n             ; unique-\u207b\u00b9 to unique-0\u2212\n             ; cancels-\u2219-left to cancels-+-left\n             ; cancels-\u2219-right to cancels-+-right\n             ; elim-\u2219-right-\/ to elim-+-right-\u2212\n             ; elim-assoc= to elim-+-assoc=\n             ; elim-!assoc= to elim-+-!assoc=\n             ; elim-inner= to elim-+-inner=\n             ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n             ; \u207b\u00b9-inj to 0\u2212-inj\n             ; \u207b\u00b9-involutive to 0\u2212-involutive\n             ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n             )\n\n-- A renaming of Group with additive notation\nmodule Additive-Group {\u2113}{G : Type \u2113}(mon : Group G) where\n  open Additive-Group-Ops    (Group.grp-ops    mon) public\n  open Additive-Group-Struct (Group.grp-struct mon) public\n\n-- A renaming of Group-Ops with multiplicative notation\nmodule Multiplicative-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) = Group-Ops grp\n    using    ( _\u207b\u00b9; _\/_; \/=; _^\u207a_ ; _^\u207b_; _^_; _\u00b2; _\u00b3; _\u2074 )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1#; mon-ops to *-mon-ops; \u2219= to *= )\n\n-- A renaming of Group-Struct with multiplicative notation\nmodule Multiplicative-Group-Struct {\u2113}{G : Type \u2113}{grp-ops : Group-Ops G}\n                                   (grp-struct : Group-Struct grp-ops)\n  = Group-Struct grp-struct\n    using    ( unique-\u207b\u00b9\n             ; \u207b\u00b9-hom\u2032\n             ; \u207b\u00b9-inj\n             ; \u207b\u00b9-involutive\n             )\n    renaming ( assoc to *-assoc\n             ; identity to *-identity\n             ; \u03b5\u2219-identity to 1*-identity\n             ; \u2219\u03b5-identity to *1-identity\n             ; inverse to \u207b\u00b9-inverse\n             ; \u2219-\/ to *-\/; \/-\u2219 to \/-*\n             ; mon-struct to *-mon-struct\n             ; mon to *-mon\n             ; unique-\u03b5-left to unique-1-left\n             ; unique-\u03b5-right to unique-1-right\n             ; x\/y\u2261\u03b5\u2192x\u2261y to x\/y\u22611\u2192x\u2261y\n             ; x\/y\u2262\u03b5 to x\/y\u22621\n             ; is-\u03b5-left to is-1-left\n             ; is-\u03b5-right to is-1-right\n             ; cancels-\u2219-left to cancels-*-left\n             ; cancels-\u2219-right to cancels-*-right\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             ; elim-\u2219-right-\/ to elim-*-right-\/\n             ; elim-assoc= to elim-*-assoc=\n             ; elim-!assoc= to elim-*-!assoc=\n             ; elim-inner= to elim-*-inner=\n             ; \u03b5\u207b\u00b9\u2261\u03b5 to 1\u207b\u00b9\u22611\n             )\n\n-- A renaming of Group with multiplicative notation\nmodule Multiplicative-Group {\u2113}{G : Type \u2113}(mon : Group G) where\n  open Multiplicative-Group-Ops    (Group.grp-ops    mon) public\n  open Multiplicative-Group-Struct (Group.grp-struct mon) public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ef4cad43e7a3e2241f5561276449a1d159f04d4d","subject":"Maybe: just^, Maybe^-\u2218-+","message":"Maybe: just^, Maybe^-\u2218-+\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {A} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 m n A \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc)\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"84190aeeefe0df40d05f1276e5972c27c14ee9db","subject":"HoTTify grouphomomorphism","message":"HoTTify grouphomomorphism\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/GroupHomomorphism.agda","new_file":"lib\/Explore\/GroupHomomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.GroupHomomorphism where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP hiding (_\u2219_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Sum\n\n{-\n I had some problems with using the standard library definiton of Groups\n so I rolled my own, therefor I need some boring proofs first\n\n-}\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc    : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n  -- derived property\n  help : \u2200 x y \u2192 x \u2261 (x \u2219 y) \u2219 - y\n  help x y = x\n           \u2261\u27e8 ! proj\u2082 identity x \u27e9\n             x \u2219 \u03b5\n           \u2261\u27e8 ap (_\u2219_ x) (! proj\u2082 inverse y) \u27e9\n             x \u2219 (y \u2219 - y)\n           \u2261\u27e8 ! assoc x y (- y) \u27e9\n             (x \u2219 y) \u2219 (- y)\n           \u220e\n    where open \u2261-Reasoning\n\n  unique-1g : \u2200 x y \u2192 x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x \u2219 y) \u2219 - y\n                   \u2261\u27e8 ap (flip _\u2219_ (- y)) eq \u27e9\n                     y \u2219 - y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     \u03b5\n                   \u220e\n    where open \u2261-Reasoning\n\n  unique-\/ : \u2200 x y \u2192 x \u2219 y \u2261 \u03b5 \u2192 x \u2261 - y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x \u2219 y) \u2219 - y\n                  \u2261\u27e8 ap (flip _\u2219_ (- y)) eq \u27e9\n                    \u03b5 \u2219 - y\n                  \u2261\u27e8 proj\u2081 identity (- y) \u27e9\n                    - y\n                  \u220e\n    where open \u2261-Reasoning\n\nmodule _ {A B : Set}(GA : Group A)(GB : Group B) where\n  open Group GA using (-_;inverse;identity) renaming (_\u2219_ to _+_; \u03b5 to 0g)\n  open Group GB using (unique-1g ; unique-\/) renaming (_\u2219_ to _*_; \u03b5 to 1g; -_ to 1\/_)\n\n  GroupHomomorphism : (A \u2192 B) \u2192 Set\n  GroupHomomorphism f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n\n  module GroupHomomorphismProp {f}(f-homo : GroupHomomorphism f) where\n    f-pres-\u03b5 : f 0g \u2261 1g\n    f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n      where open \u2261-Reasoning\n            part = f 0g * f 0g\n                 \u2261\u27e8 ! f-homo 0g 0g \u27e9\n                   f (0g + 0g)\n                 \u2261\u27e8 ap f (proj\u2081 identity 0g) \u27e9\n                   f 0g\n                 \u220e\n\n    f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n    f-pres-inv x = unique-\/ (f (- x)) (f x) part\n      where open \u2261-Reasoning\n            part = f (- x) * f x\n                 \u2261\u27e8 ! f-homo (- x) x \u27e9\n                   f (- x + x)\n                 \u2261\u27e8 ap f (proj\u2081 inverse x) \u27e9\n                   f 0g\n                 \u2261\u27e8 f-pres-\u03b5 \u27e9\n                   1g\n                 \u220e\n\nmodule _ {A B}(GA : Group A)(GB : Group B)\n         (f : A \u2192 B)\n         (exploreA : Explore zero A)(f-homo : GroupHomomorphism GA GB f)\n         ([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n         (explore-ext : ExploreExt exploreA)\n         where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n  open GroupHomomorphismProp GA GB f-homo\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n  -}\n\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n  module _ {X}(z : X)(op : X \u2192 X \u2192 X)\n           (sui : \u2200 k \u2192 StableUnder' exploreA z op (flip (Group._\u2219_ GA) k))\n    where\n  -- this proof isn't actually any hard..\n    thm : \u2200 (O : B \u2192 X) m\u2080 m\u2081 \u2192 exploreA z op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 exploreA z op (\u03bb x \u2192 O (f x * m\u2081))\n    thm O m\u2080 m\u2081 = explore (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 ap O (lemma1 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) (O \u2218 f) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 ap O (lemma2 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n     where\n      open \u2261-Reasoning\n      explore = exploreA z op\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.GroupHomomorphism where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP hiding (_\u2219_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Sum\n\n{-\n I had some problems with using the standard library definiton of Groups\n so I rolled my own, therefor I need some boring proofs first\n\n-}\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc    : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n  -- derived property\n  help : \u2200 x y \u2192 x \u2261 (x \u2219 y) \u2219 - y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x \u2219 \u03b5\n           \u2261\u27e8 cong (_\u2219_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x \u2219 (y \u2219 - y)\n           \u2261\u27e8 sym (assoc x y (- y)) \u27e9\n             (x \u2219 y) \u2219 (- y)\n           \u220e\n    where open \u2261-Reasoning\n\n  unique-1g : \u2200 x y \u2192 x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x \u2219 y) \u2219 - y\n                   \u2261\u27e8 cong (flip _\u2219_ (- y)) eq \u27e9\n                     y \u2219 - y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     \u03b5\n                   \u220e\n    where open \u2261-Reasoning\n\n  unique-\/ : \u2200 x y \u2192 x \u2219 y \u2261 \u03b5 \u2192 x \u2261 - y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x \u2219 y) \u2219 - y\n                  \u2261\u27e8 cong (flip _\u2219_ (- y)) eq \u27e9\n                    \u03b5 \u2219 - y\n                  \u2261\u27e8 proj\u2081 identity (- y) \u27e9\n                    - y\n                  \u220e\n    where open \u2261-Reasoning\n\nmodule _ {A B : Set}(GA : Group A)(GB : Group B) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_; \u03b5 to 0g)\n  open Group GB using (unique-1g ; unique-\/) renaming (_\u2219_ to _*_; \u03b5 to 1g; -_ to 1\/_)\n\n  GroupHomomorphism : (A \u2192 B) \u2192 Set\n  GroupHomomorphism f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n\n  module GroupHomomorphismProp {f}(f-homo : GroupHomomorphism f) where\n    f-pres-\u03b5 : f 0g \u2261 1g\n    f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n      where open \u2261-Reasoning\n            open Group GA using (identity)\n            part = f 0g * f 0g\n                 \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                   f (0g + 0g)\n                 \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                   f 0g\n                 \u220e\n\n    f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n    f-pres-inv x = unique-\/ (f (- x)) (f x) part\n      where open \u2261-Reasoning\n            open Group GA using (inverse)\n            part = f (- x) * f x\n                 \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                   f (- x + x)\n                 \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                   f 0g\n                 \u2261\u27e8 f-pres-\u03b5 \u27e9\n                   1g\n                 \u220e\n\nmodule _ {A B}(GA : Group A)(GB : Group B)\n         (f : A \u2192 B)\n         (exploreA : Explore zero A)(f-homo : GroupHomomorphism GA GB f)\n         ([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n         (explore-ext : ExploreExt exploreA)\n         where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n  open GroupHomomorphismProp GA GB f-homo\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n  -}\n\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n  module _ {X}(z : X)(op : X \u2192 X \u2192 X)\n           (sui : \u2200 k \u2192 StableUnder' exploreA z op (flip (Group._\u2219_ GA) k))\n    where\n  -- this proof isn't actually any hard..\n    thm : \u2200 (O : B \u2192 X) m\u2080 m\u2081 \u2192 exploreA z op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 exploreA z op (\u03bb x \u2192 O (f x * m\u2081))\n    thm O m\u2080 m\u2081 = explore (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma1 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) (O \u2218 f) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma2 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n     where\n      open \u2261-Reasoning\n      explore = exploreA z op\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1ed3194fa9dfc175ebb7232172842e35a201ecc1","subject":"Updated a note on the existential elimination.","message":"Updated a note on the existential elimination.\n\nIgnore-this: fc0913e5a6f5eb20a7f038ae272d00fe\n\ndarcs-hash:20120226141843-3bd4e-3ceaba139b8cf1afb4fe19775c5d1e65df77d8ec.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Existential.agda","new_file":"notes\/Existential.agda","new_contents":"-- Tested with the development version of Agda on 26 February 2012.\n\nmodule Existential where\n\ninfixr 7 _,_\n\npostulate\n  D   : Set\n  P Q : D \u2192 Set\n  P\u2192Q : \u2200 {x} \u2192 P x \u2192 Q x\n\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 (\u2200 x \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n\n-- A proof using the existential elimination.\nprf\u2081 : \u2203 P \u2192 \u2203 Q\nprf\u2081 h = \u2203-elim h (\u03bb x Px \u2192 x , P\u2192Q Px)\n\n-- A proof using the existential elimination with a helper function.\nprf\u2082 : \u2203 P \u2192 \u2203 Q\nprf\u2082 h = \u2203-elim h helper\n  where\n  helper : \u2200 x \u2192 P x \u2192 \u2203 Q\n  helper x Px = x , P\u2192Q Px\n\n-- A proof using pattern matching.\nprf\u2083 : \u2203 P \u2192 \u2203 Q\nprf\u2083 (x , Px) = x , P\u2192Q Px\n","old_contents":"-- Tested with the development version of Agda on 25 February 2012.\n\nmodule Existential where\n\ninfixr 7 _,_\n\npostulate\n  D   : Set\n  P Q : D \u2192 Set\n  P\u2192Q : \u2200 {x} \u2192 P x \u2192 Q x\n\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 (\u2200 x \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n\nthm\u2081 : \u2203 P \u2192 \u2203 Q\nthm\u2081 h = \u2203-elim h (\u03bb x Px \u2192 x , P\u2192Q Px)\n\nthm\u2082 : \u2203 P \u2192 \u2203 Q\nthm\u2082 h = \u2203-elim h helper\n  where\n  helper : \u2200 x \u2192 P x \u2192 \u2203 Q\n  helper x Px = x , P\u2192Q Px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"039ad45a017c28bce761d79c39ee6ce685fde2a5","subject":"Added Nx+11.","message":"Added Nx+11.\n\nIgnore-this: 70697fba4fa0352e7230a83d70878b15\n\ndarcs-hash:20110211212344-3bd4e-9c970f14fcec789dcb7e01b02ea6f379dae2dc5a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  N1+n  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  N11+n : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  Nn-10 : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  Nx+11 : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove N1+n #-}\n{-# ATP prove N11+n #-}\n{-# ATP prove Nn-10 \u2238-N N10 #-}\n{-# ATP prove Nx+11 N11+n +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  N1+n  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  N11+n : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  Nn-10 : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n{-# ATP prove N1+n #-}\n{-# ATP prove N11+n #-}\n{-# ATP prove Nn-10 \u2238-N N10 #-}\n\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7d9fac76bdbab958ba63ed54cada87a0186888a3","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 1e957aefcbd418b95c65022fab6e539f\n\ndarcs-hash:20110401125940-3bd4e-6fedb07830363ae4a9d8f466812a51604b999298.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 P. app\u2081(P,zero) \u2192\n--      (\u2200 x. app1(N,x) \u2192 app\u2081(P,x) \u2192 app\u2081(P,succ x)) \u2192   -- indN\n--      (\u2200 x. app\u2081(N,x) \u2192 app\u2081(P,x))\n------------------------------------------------------------------\n-- \u2200 x y. app\u2081(N,x) \u2192 app\u2081(N,y) \u2192 app\u2081(N, x + y)          -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","old_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"77ca7303f67051396173f312a7b72369abb053c0","subject":"Added x\u00abSy\u2192x\u00aby.","message":"Added x\u00abSy\u2192x\u00aby.\n\nIgnore-this: 3603ab0adb279ea9e60ffaf28464b82c\n\ndarcs-hash:20110216152234-3bd4e-c26e56e210fb50398beb7c90652dc1f185a328f3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/RelationATP.agda","new_file":"Draft\/McCarthy91\/RelationATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for some relations\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.RelationATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\nopen import Draft.McCarthy91.ArithmeticATP\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\n0\u00abx\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 \u00ac (MCR zero n)\n0\u00abx\u2192\u22a5 zN 0\u00abn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n0\u00abx\u2192\u22a5 (sN Nn) 0\u00abSn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x\u2238y<Sx x<y\u2192y<x\u2192\u22a5 #-}\n\n\u00ab-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 MCR m n \u2192 MCR n o \u2192 MCR m o\n\u00ab-trans Nm Nn No m\u00abn n\u00abo =\n  <-trans (\u2238-N N101 Nm) (\u2238-N N101 Nn) (\u2238-N N101 No) m\u00abn n\u00abo\n\nSx\u00abSy\u2192x\u00aby : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR (succ m) (succ n) \u2192 MCR m n\nSx\u00abSy\u2192x\u00aby zN zN S0\u00abS0 = prf\n  where\n    postulate prf : MCR zero zero\n    {-# ATP prove prf #-}\n\nSx\u00abSy\u2192x\u00aby zN (sN {n} Nn) S0\u00abSSn = prf\n  where\n    postulate prf : MCR zero (succ n)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nSx\u00abSy\u2192x\u00aby (sN {m} Nm) zN SSm\u00abS0 = prf\n  where\n    postulate prf : MCR (succ m) zero\n    {-# ATP prove prf \u2238-N x\u2238y<Sx #-}\n\nSx\u00abSy\u2192x\u00aby (sN {m} Nm) (sN {n} Nn) SSm\u00abSSn = prf\n  where\n    postulate prf : MCR (succ m) (succ n)\n    {-# ATP prove prf x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z #-}\n\nx\u00abSy\u2192x\u00aby : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR m (succ n) \u2192 MCR m n\nx\u00abSy\u2192x\u00aby {n = n} zN Nn 0\u00abSn = prf\n  where\n    postulate prf : MCR zero n\n    {-# ATP prove prf 0\u00abx\u2192\u22a5 #-}\n\nx\u00abSy\u2192x\u00aby (sN {m} Nm) zN Sm\u00abS0 = prf\n   where\n     postulate prf : MCR (succ m) zero\n     {-# ATP prove prf x\u2238y<Sx #-}\n\nx\u00abSy\u2192x\u00aby (sN {m} Nm) (sN {n} Nn) Sm\u00abSSn = prf\n  where\n    postulate prf : MCR (succ m) (succ n)\n    {-# ATP prove prf \u2238-N x<y\u2192y\u2264z\u2192x<z x\u2238Sy\u2264x\u2238y #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for some relations\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.RelationATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\nopen import Draft.McCarthy91.ArithmeticATP\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\n0\u00abx\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 \u00ac (MCR zero n)\n0\u00abx\u2192\u22a5 zN 0\u00abn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n0\u00abx\u2192\u22a5 (sN Nn) 0\u00abSn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x\u2238y<Sx x<y\u2192y<x\u2192\u22a5 #-}\n\n\u00ab-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 MCR m n \u2192 MCR n o \u2192 MCR m o\n\u00ab-trans Nm Nn No m\u00abn n\u00abo =\n  <-trans (\u2238-N N101 Nm) (\u2238-N N101 Nn) (\u2238-N N101 No) m\u00abn n\u00abo\n\nSx\u00abSy\u2192x\u00aby : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR (succ m) (succ n) \u2192 MCR m n\nSx\u00abSy\u2192x\u00aby zN zN S0\u00abS0 = prf\n  where\n    postulate prf : MCR zero zero\n    {-# ATP prove prf #-}\n\nSx\u00abSy\u2192x\u00aby zN (sN {n} Nn) S0\u00abSSn = prf\n  where\n    postulate prf : MCR zero (succ n)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nSx\u00abSy\u2192x\u00aby (sN {m} Nm) zN SSm\u00abS0 = prf\n  where\n    postulate prf : MCR (succ m) zero\n    {-# ATP prove prf \u2238-N x\u2238y<Sx #-}\n\nSx\u00abSy\u2192x\u00aby (sN {m} Nm) (sN {n} Nn) SSm\u00abSSn = prf\n  where\n    postulate prf : MCR (succ m) (succ n)\n    {-# ATP prove prf x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"42334cb4d0c8b03e4046bd2ce19a4db8a6512ded","subject":"-powered-flat-funs.agda","message":"-powered-flat-funs.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"powered-flat-funs.agda","new_file":"powered-flat-funs.agda","new_contents":"","old_contents":"module powered-flat-funs where\n\nopen import Data.Nat\nopen import Data.Unit using (\u22a4)\nimport Level as L\nopen import flat-funs\nopen import composable\nopen import vcomp\nopen import ann\n\nrecord PowerOps (Power : Set) : Set where\n  constructor mk\n  infixr 1 _>>>\u1d56_\n  infixr 3 _***\u1d56_\n  field\n    id\u1d56 : Power\n    _>>>\u1d56_ _***\u1d56_ : Power \u2192 Power \u2192 Power\n\nconstPowerOps : PowerOps \u22a4\nconstPowerOps = _\n\nrecord PoweredFlatFuns (Power : Set) {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits   : \u2115 \u2192 T\n    `Bit    : T\n    _`\u00d7_    : T \u2192 T \u2192 T\n    \u27e8_\u27e9_\u219d_  : Power \u2192 T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 \u27e8_\u27e9_\u219d_\n\n\u22a4-poweredFlatFuns : \u2200 {t} {T : Set t} \u2192 FlatFuns T \u2192 PoweredFlatFuns \u22a4 T\n\u22a4-poweredFlatFuns ff = mk `Bits `Bit _`\u00d7_ (\u03bb _ A B \u2192 A `\u2192 B) where open FlatFuns ff\n\nrecord PoweredFlatFunsOps {P t} {T : Set t} (powerOps : PowerOps P) (\u266dFuns : PoweredFlatFuns P T) : Set t where\n  constructor mk\n  open PoweredFlatFuns \u266dFuns\n  open PowerOps powerOps\n  field\n    idO : \u2200 {A p} \u2192 \u27e8 p \u27e9 A \u219d A\n    isIComposable  : IComposable  {I = \u22a4} _>>>\u1d56_ \u27e8_\u27e9_\u219d_\n    isVIComposable : VIComposable {I = \u22a4} _***\u1d56_ _`\u00d7_ \u27e8_\u27e9_\u219d_\n  open PoweredFlatFuns \u266dFuns public\n  open PowerOps powerOps public\n  open IComposable isIComposable public\n  open VIComposable isVIComposable public\n\n\u22a4-poweredFlatFunsOps : \u2200 {t} {T : Set t} {\u266dFuns : FlatFuns T}\n                 \u2192 FlatFunsOps \u266dFuns \u2192 PoweredFlatFunsOps _ (\u22a4-poweredFlatFuns \u266dFuns)\n\u22a4-poweredFlatFunsOps ops = mk idO isComposable isVComposable where open FlatFunsOps ops\n\npoweredFlatFuns : \u2200 {t} {T : Set t} {P}\n                   \u2192 FlatFuns T \u2192 PoweredFlatFuns P T\npoweredFlatFuns ff = mk `Bits `Bit _`\u00d7_ (\u03bb p A B \u2192 Ann p (A `\u2192 B)) where open FlatFuns ff\n\npoweredFlatFunsOps : \u2200 {t} {T : Set t} {P} (funPowerOps : PowerOps P)\n                    {\u266dFuns : FlatFuns T}\n                    \u2192 FlatFunsOps \u266dFuns \u2192 PoweredFlatFunsOps funPowerOps (poweredFlatFuns \u266dFuns)\npoweredFlatFunsOps funPowerOps {\u266dFuns} (mk idO (mk _>>>_) (mk _***_) _)\n   = mk (mk idO) (mk (\u03bb x y \u2192 mk (get x >>> get y))) (mk (\u03bb x y \u2192 mk (get x *** get y)))\n      where open Ann\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b2aaa4d843f2cd1f54bf4cd24fadc6827fbbb5b4","subject":"agda: prove most of diff-apply (see #33)","message":"agda: prove most of diff-apply (see #33)\n\nIssue #33 discusses the remaining holes.\n\nOld-commit-hash: a9aeee0162bb9af15ba0d4a4c8b85e53211bbb36\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid {bool} v = tt\nderive-is-valid {\u03c4\u2081 \u21d2 \u03c4\u2082} v =\n  \u03bb s ds valid-\u0394-s-ds \u2192 {!!} , (\n    begin\n      (apply (derive v) v) (apply ds s)\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 x (apply ds s)) (apply-derive v) \u27e9\n      v (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v (apply ds s))) \u27e9\n      apply (derive v s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\n-- This is a postulate elsewhere, but here I want to start developing a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (derive-is-valid (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"21ea7cbcf3428ce9d579b019c1be987a27c706e9","subject":"Step-indexed evaluation: simplify back","message":"Step-indexed evaluation: simplify back\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a (suc n) = eval t (a \u2022 \u03c1) n\napply (closure t \u03c1) a zero = TimeOut\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\n-- eval (app s t) \u03c1 = eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)\neval (app s t) \u03c1 n0 with eval s \u03c1 n0\n... | Error = Error\n... | TimeOut = TimeOut\n... | Done sv n1 with eval t \u03c1 n1\n... | Done tv n2 = apply sv tv n2\n... | Error = Error\n... | TimeOut = TimeOut\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) (n0 : \u2115) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 (n\u2264 : n1 \u2264 n0) \u2192 ErrVal \u03c4 n0\n  Error : ErrVal \u03c4 n0\n  TimeOut : ErrVal \u03c4 n0\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4 n\n\nup : \u2200 {\u03c4 n0 n1} \u2192 ErrVal \u03c4 n1 \u2192 (n1 \u2264 n0) \u2192 ErrVal \u03c4 n0\nup (Done v n2 n2\u2264n1) n1\u2264n0 = Done v n2 (\u2264-trans n2\u2264n1 n1\u2264n0)\nup Error n\u2264 = Error\nup TimeOut n\u2264 = TimeOut\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 n\u2264 = up (t v n1) n\u2264\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n (\u2264-step \u2264-refl)\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a (suc n) = up (eval t (a \u2022 \u03c1) n) (\u2264-step \u2264-refl)\napply (closure t \u03c1) a zero = TimeOut\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n \u2264-refl\neval (abs t) \u03c1 n = Done (closure t \u03c1) n \u2264-refl\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\n-- eval (app s t) \u03c1 = eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)\neval (app s t) \u03c1 n0 with eval s \u03c1 n0\n... | Error = Error\n... | TimeOut = TimeOut\n... | Done sv n1 n1\u2264n0 with eval t \u03c1 n1\n... | Done tv n2 n2\u2264n1 = up (apply sv tv n2) (\u2264-trans n2\u2264n1 n1\u2264n0)\n... | Error = Error\n... | TimeOut = TimeOut\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4886cd0de64e23f77fd7608edbc35ba22f9ea2f2","subject":"tried to analyze a slightly larger example, but it is too big already","message":"tried to analyze a slightly larger example, but it is too big already\n\nOld-commit-hash: 3e1af00454de058acf5ca90a5d63198a5b3aca11\n","repos":"inc-lc\/ilc-agda","old_file":"Examples\/Examples1.agda","new_file":"Examples\/Examples1.agda","new_contents":"module Examples.Examples1 where\n\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\nopen import Syntactic.Closures\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\nopen import Natural.Evaluation\nopen import Relation.Binary.PropositionalEquality\n\n-- only some finger exercises to get used to the definitions\n \nbool-identity : Term \u2205 (bool \u21d2 bool)\nbool-identity = abs (var this)\n\nterm1 : Term \u2205 bool\nterm1 = app bool-identity true\n\nres : \u27e6 bool \u27e7\nres = \u27e6 term1 \u27e7 \u2205\n\n-- test the denotational semantics\n\n-- is this a good way to write unit tests in Agda?\ntest1 : res \u2261 true\ntest1 = refl\n\n-- test the natural semantics\nopen import Natural.Lookup\ntest2 : \u2205 \u22a2 term1 \u2193 vtrue\ntest2 = app abs e-true (var this)\n\nterm2 : Term (bool \u2022 bool \u2022 bool \u2022 \u2205) bool\nterm2 = if (var this) (if (var (that this)) false true) (if (var (that (that this))) false true)\n\nterm3 : Term \u2205 (bool \u21d2 bool \u21d2 bool \u21d2 bool)\nterm3 = abs (abs (abs term2))\n\nopen import SymbolicDerivation\n-- presumably there is some shorter way to do this?\n-- I don't understand instance arguments sufficiently\nterm4 = derive-term {\u2205} {\u2205} {bool \u21d2 bool \u21d2 bool \u21d2 bool} {{\u2205 }} term3\n\n-- it turns out that term4 is already way too complicated to be analyzed by hand\n","old_contents":"module Examples.Examples1 where\n\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\nopen import Syntactic.Closures\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\nopen import Natural.Evaluation\nopen import Relation.Binary.PropositionalEquality\n\n-- only some finger exercises to get used to the definitions\n \nbool-identity : Term \u2205 (bool \u21d2 bool)\nbool-identity = abs (var this)\n\nterm1 : Term \u2205 bool\nterm1 = app bool-identity true\n\nres : \u27e6 bool \u27e7\nres = \u27e6 term1 \u27e7 \u2205\n\n-- test the denotational semantics\n\n-- is this a good way to write unit tests in Agda?\ntest1 : res \u2261 true\ntest1 = refl\n\n-- test the natural semantics\nopen import Natural.Lookup\ntest2 : \u2205 \u22a2 term1 \u2193 vtrue\ntest2 = app abs e-true (var this)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bd4721cbfeecd02bd82f2eb2bb19570e836dac3a","subject":"Change argument order of carry-over.","message":"Change argument order of carry-over.\n\nThis commit prepares for uncurrying cary-over.\n\nOld-commit-hash: e1a21bdcbe7ee2b217917bd6ac9dfc576487102a\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Implementation\/Popl14.agda","new_file":"Denotation\/Implementation\/Popl14.agda","new_contents":"module Denotation.Implementation.Popl14 where\n\n-- Notions of programs being implementations of specifications\n-- for Calculus Popl14\n\nopen import Denotation.Specification.Canon-Popl14 public\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Derive\nopen import Popl14.Change.Value\nopen import Popl14.Change.Validity\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Tuples\nopen import Structure.Bag.Popl14\nopen import Postulate.Extensionality\n\ninfix 4 implements\nsyntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\nimplements : \u2200 (\u03c4 : Type) \u2192 Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\nu \u2248\u208d int \u208e v = u \u2261 v\nu \u2248\u208d bag \u208e v = u \u2261 v\nu \u2248\u208d \u03c3 \u21d2 \u03c4 \u208e v =\n  (w : \u27e6 \u03c3 \u27e7) (\u0394w : Change \u03c3) (R[w,\u0394w] : valid {\u03c3} w \u0394w)\n  (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 \u0394w \u0394w\u2032) \u2192\n  implements \u03c4 (u (cons w \u0394w R[w,\u0394w])) (v w \u0394w\u2032)\n\ninfix 4 _\u2248_\n_\u2248_ : \u2200 {\u03c4} \u2192 Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {\u03c4} = implements \u03c4\n\nmodule Disambiguation (\u03c4 : Type) where\n  _\u271a_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a_ _\u2212_\n  _\u2243_ : Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243_\n\nmodule FunctionDisambiguation (\u03c3 : Type) (\u03c4 : Type) where\n  open Disambiguation (\u03c3 \u21d2 \u03c4) public\n  _\u271a\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212\u2081_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a\u2081_ _\u2212\u2081_\n  _\u2243\u2081_ : Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243\u2081_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243\u2081_\n  _\u271a\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7\n  _\u271a\u2080_ = _\u27e6\u2295\u27e7_ {\u03c3}\n  _\u2212\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7\n  _\u2212\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n  infixl 6 _\u271a\u2080_ _\u2212\u2080_\n  _\u2243\u2080_ : Change \u03c3 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 Set\n  _\u2243\u2080_ = _\u2248_ {\u03c3}\n  infix 4 _\u2243\u2080_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = \u22a4\ncompatible {\u03c4 \u2022 \u0393} (cons v \u0394v _ \u2022 \u03c1) (\u0394v\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  Triple (v \u2261 v\u2032) (\u03bb _ \u2192 \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) (\u03bb _ _ \u2192 compatible \u03c1 \u03c1\u2032)\n\n-- If a program implements a specification, then certain things\n-- proven about the specification carry over to the programs.\ncarry-over : \u2200 {\u03c4}\n  {v : \u27e6 \u03c4 \u27e7} {\u0394v : Change \u03c4} (R[v,\u0394v] : valid {\u03c4} v \u0394v)\n  {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7} (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) \u2192\n let open Disambiguation \u03c4 in\n   v \u229e\u208d \u03c4 \u208e \u0394v \u2261 v \u271a \u0394v\u2032\n\nu\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let open Disambiguation \u03c4 in\n    u \u229f\u208d \u03c4 \u208e v \u2243 u \u2212 v\nu\u229fv\u2248u\u229dv {base base-int} = refl\nu\u229fv\u2248u\u229dv {base base-bag} = refl\nu\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n  open FunctionDisambiguation \u03c3 \u03c4\n  result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : Change \u03c3) \u2192 valid {\u03c3} w \u0394w \u2192\n    (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192 \u0394w \u2248\u208d \u03c3 \u208e \u0394w\u2032 \u2192\n    g (w \u229e\u208d \u03c3 \u208e \u0394w) \u229f\u208d \u03c4 \u208e f w \u2243\u2081 g (w \u271a\u2080 \u0394w\u2032) \u2212\u2081 f w\n  result w \u0394w R[w,\u0394w] \u0394w\u2032 \u0394w\u2248\u0394w\u2032\n    rewrite carry-over {\u03c3} {w} R[w,\u0394w] \u0394w\u2248\u0394w\u2032 =\n    u\u229fv\u2248u\u229dv {\u03c4} {g (w \u271a\u2080 \u0394w\u2032)} {f w}\ncarry-over {base base-int} {v} _ \u0394v\u2248\u0394v\u2032 = cong (_+_  v) \u0394v\u2248\u0394v\u2032\ncarry-over {base base-bag} {v} _ \u0394v\u2248\u0394v\u2032 = cong (_++_ v) \u0394v\u2248\u0394v\u2032\ncarry-over {\u03c3 \u21d2 \u03c4} {f} {\u0394f} R[f,\u0394f] {\u0394f\u2032} \u0394f\u2248\u0394f\u2032 =\n  ext (\u03bb v \u2192\n  let\n    open FunctionDisambiguation \u03c3 \u03c4\n    V = R[v,u-v] {\u03c3} {v} {v}\n    S = u\u229fv\u2248u\u229dv {\u03c3} {v} {v}\n  in\n    carry-over {\u03c4} {f v}\n      {\u0394f (nil-valid-change \u03c3 v)} (proj\u2081 (R[f,\u0394f] (nil-valid-change \u03c3 v)))\n      {\u0394f\u2032 v (v \u2212\u2080 v)}\n      (\u0394f\u2248\u0394f\u2032 v (v \u229f\u208d \u03c3 \u208e v) V (v \u2212\u2080 v) S))\n\n-- A property relating `ignore` and the subcontext relation \u0393\u227c\u0394\u0393\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} _ = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u03c4 \u2022 \u0393} {cons v dv _ \u2022 \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032}\n  (cons v\u2261v\u2032 _ C) = cong\u2082 _\u2022_ v\u2261v\u2032 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)\n\n-- A specialization of the soundness of weakening\n\u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 fit t \u27e7 \u03c1\u2032\n\u27e6fit\u27e7 t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)) (sym (weaken-sound t \u03c1\u2032))\n","old_contents":"module Denotation.Implementation.Popl14 where\n\n-- Notions of programs being implementations of specifications\n-- for Calculus Popl14\n\nopen import Denotation.Specification.Canon-Popl14 public\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Derive\nopen import Popl14.Change.Value\nopen import Popl14.Change.Validity\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Tuples\nopen import Structure.Bag.Popl14\nopen import Postulate.Extensionality\n\ninfix 4 implements\nsyntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\nimplements : \u2200 (\u03c4 : Type) \u2192 Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\nu \u2248\u208d int \u208e v = u \u2261 v\nu \u2248\u208d bag \u208e v = u \u2261 v\nu \u2248\u208d \u03c3 \u21d2 \u03c4 \u208e v =\n  (w : \u27e6 \u03c3 \u27e7) (\u0394w : Change \u03c3) (R[w,\u0394w] : valid {\u03c3} w \u0394w)\n  (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 \u0394w \u0394w\u2032) \u2192\n  implements \u03c4 (u (cons w \u0394w R[w,\u0394w])) (v w \u0394w\u2032)\n\ninfix 4 _\u2248_\n_\u2248_ : \u2200 {\u03c4} \u2192 Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {\u03c4} = implements \u03c4\n\nmodule Disambiguation (\u03c4 : Type) where\n  _\u271a_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a_ _\u2212_\n  _\u2243_ : Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243_\n\nmodule FunctionDisambiguation (\u03c3 : Type) (\u03c4 : Type) where\n  open Disambiguation (\u03c3 \u21d2 \u03c4) public\n  _\u271a\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212\u2081_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a\u2081_ _\u2212\u2081_\n  _\u2243\u2081_ : Change \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243\u2081_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243\u2081_\n  _\u271a\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7\n  _\u271a\u2080_ = _\u27e6\u2295\u27e7_ {\u03c3}\n  _\u2212\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7\n  _\u2212\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n  infixl 6 _\u271a\u2080_ _\u2212\u2080_\n  _\u2243\u2080_ : Change \u03c3 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 Set\n  _\u2243\u2080_ = _\u2248_ {\u03c3}\n  infix 4 _\u2243\u2080_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = \u22a4\ncompatible {\u03c4 \u2022 \u0393} (cons v \u0394v _ \u2022 \u03c1) (\u0394v\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  Triple (v \u2261 v\u2032) (\u03bb _ \u2192 \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) (\u03bb _ _ \u2192 compatible \u03c1 \u03c1\u2032)\n\n-- If a program implements a specification, then certain things\n-- proven about the specification carry over to the programs.\ncarry-over : \u2200 {\u03c4}\n  {v : \u27e6 \u03c4 \u27e7} {\u0394v : Change \u03c4} {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7}\n (R[v,\u0394v] : valid {\u03c4} v \u0394v) (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) \u2192\n let open Disambiguation \u03c4 in\n   v \u229e\u208d \u03c4 \u208e \u0394v \u2261 v \u271a \u0394v\u2032\n\nu\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let open Disambiguation \u03c4 in\n    u \u229f\u208d \u03c4 \u208e v \u2243 u \u2212 v\nu\u229fv\u2248u\u229dv {base base-int} = refl\nu\u229fv\u2248u\u229dv {base base-bag} = refl\nu\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n  open FunctionDisambiguation \u03c3 \u03c4\n  result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : Change \u03c3) \u2192 valid {\u03c3} w \u0394w \u2192\n    (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192 \u0394w \u2248\u208d \u03c3 \u208e \u0394w\u2032 \u2192\n    g (w \u229e\u208d \u03c3 \u208e \u0394w) \u229f\u208d \u03c4 \u208e f w \u2243\u2081 g (w \u271a\u2080 \u0394w\u2032) \u2212\u2081 f w\n  result w \u0394w R[w,\u0394w] \u0394w\u2032 \u0394w\u2248\u0394w\u2032\n    rewrite carry-over {\u03c3} {w} R[w,\u0394w] \u0394w\u2248\u0394w\u2032 =\n    u\u229fv\u2248u\u229dv {\u03c4} {g (w \u271a\u2080 \u0394w\u2032)} {f w}\ncarry-over {base base-int} {v} _ \u0394v\u2248\u0394v\u2032 = cong (_+_  v) \u0394v\u2248\u0394v\u2032\ncarry-over {base base-bag} {v} _ \u0394v\u2248\u0394v\u2032 = cong (_++_ v) \u0394v\u2248\u0394v\u2032\ncarry-over {\u03c3 \u21d2 \u03c4} {f} {\u0394f} {\u0394f\u2032} R[f,\u0394f] \u0394f\u2248\u0394f\u2032 =\n  ext (\u03bb v \u2192\n  let\n    open FunctionDisambiguation \u03c3 \u03c4\n    V = R[v,u-v] {\u03c3} {v} {v}\n    S = u\u229fv\u2248u\u229dv {\u03c3} {v} {v}\n  in\n    carry-over {\u03c4} {f v}\n      {\u0394f (nil-valid-change \u03c3 v)} {\u0394f\u2032 v (v \u2212\u2080 v)}\n      (proj\u2081 (R[f,\u0394f] (nil-valid-change \u03c3 v)))\n      (\u0394f\u2248\u0394f\u2032 v (v \u229f\u208d \u03c3 \u208e v) V (v \u2212\u2080 v) S))\n\n-- A property relating `ignore` and the subcontext relation \u0393\u227c\u0394\u0393\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} _ = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u03c4 \u2022 \u0393} {cons v dv _ \u2022 \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032}\n  (cons v\u2261v\u2032 _ C) = cong\u2082 _\u2022_ v\u2261v\u2032 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)\n\n-- A specialization of the soundness of weakening\n\u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 fit t \u27e7 \u03c1\u2032\n\u27e6fit\u27e7 t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)) (sym (weaken-sound t \u03c1\u2032))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9028d1ebd71caa01d4b8a57cbb884b32e787ebe1","subject":"Revert \"agda\/...: misc (to revert)\"","message":"Revert \"agda\/...: misc (to revert)\"\n\nThis reverts commit ae3a9db6fb87ddc01573ae236f8850b322bec8db.\n\nOld-commit-hash: fa2ae9742c4282275b8f242a197b35ef1acf5932\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : --\u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-var-3 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context (\u0394-Context \u0393)) \u03c4\nlift-var-3 this = that (that (that this))\nlift-var-3 (that x) = that (that (that (that (lift-var-3 x))))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : --\u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\n{-\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u2205 \u2205 t\u2081 = \u0394 t\u2081\n    weakenMore2 (\u03c4\u2081 \u2022 \u0393\u2081) \u2205 t\u2081 =\n      weakenOne \u2205 (\u0394-Type (\u0394-Type \u03c4\u2081))\n        (weakenOne (\u0394-Type \u03c4\u2081 \u2022 \u2205) (\u0394-Type \u03c4\u2081) {!\n          weakenMore2\n            (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} {\u03c4} t\u2081)!})\n          --(weakenMore2 {\u03c4\u2081 \u2022 \u0393\u2081} (\u0394-Type \u03c4\u2081 \u2022 \u03c4\u2081 \u2022 \u2205) t\u2081 ))\n\n--        (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} {\u03c4} t\u2081)))\n\n    weakenMore2 (\u03c4\u2081 \u2022 \u0393\u2081) (.(\u0394-Type \u03c4\u2081) \u2022 \u0393\u2032\u2081) t\u2081 = {!!}\n-}\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"322bfa9ea5da1ad8271fefe5c6188aaa3736ee9d","subject":"Desc model: proof-phi-psi implicit.","message":"Desc model: proof-phi-psi implicit.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : (I : Set) -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi I (var i) = refl\nproof-phi-psi I (const x) = refl\nproof-phi-psi I (prod D D') with proof-phi-psi I D | proof-phi-psi I D'\n...                             | p | q = cong2 prod p q\nproof-phi-psi I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi I (T s)))\nproof-phi-psi I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                              T\n                                              (\\s -> proof-phi-psi I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a6016a23a99d566ccb1cf3fc64f026e867f0dc5a","subject":"Universe: \u03a0\u2192\u03a0\u1d41","message":"Universe: \u03a0\u2192\u03a0\u1d41\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Universe.agda","new_file":"lib\/Explore\/Universe.agda","new_contents":"open import Level.NP\nopen import Type\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Product.NP\nopen import Data.Sum.NP\nopen import Data.Sum.Logical\nopen import Data.Nat\nopen import Data.Fin\nopen import Relation.Nullary.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Binary.Logical\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Zero\nopen import Explore.One\nopen import Explore.Two\nopen import Explore.Product\nopen import Explore.Sum\nopen import Explore.Explorable\nimport Explore.Isomorphism\n\nmodule Explore.Universe where\n\nopen Operators\n\ninfixr 2 _\u00d7\u1d41_\n\ndata U : \u2605\nEl : U \u2192 \u2605\n\ndata U where\n  \ud835\udfd8\u1d41 \ud835\udfd9\u1d41 \ud835\udfda\u1d41 : U\n  _\u00d7\u1d41_ _\u228e\u1d41_ : U \u2192 U \u2192 U\n  \u03a3\u1d41 : (t : U) (f : El t \u2192 U) \u2192 U\n\nEl \ud835\udfd8\u1d41 = \ud835\udfd8\nEl \ud835\udfd9\u1d41 = \ud835\udfd9\nEl \ud835\udfda\u1d41 = \ud835\udfda\nEl (s \u00d7\u1d41 t) = El s \u00d7 El t\nEl (s \u228e\u1d41 t) = El s \u228e El t\nEl (\u03a3\u1d41 t f) = \u03a3 (El t) \u03bb x \u2192 El (f x)\n\ndata \u27e6U\u27e7 : \u27e6\u2605\u2080\u27e7 U U\n\u27e6El\u27e7 : (\u27e6U\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) El El\n\ndata \u27e6U\u27e7 where\n  \u27e6\ud835\udfd8\u1d41\u27e7 : \u27e6U\u27e7 \ud835\udfd8\u1d41 \ud835\udfd8\u1d41\n  \u27e6\ud835\udfd9\u1d41\u27e7 : \u27e6U\u27e7 \ud835\udfd9\u1d41 \ud835\udfd9\u1d41\n  \u27e6\ud835\udfda\u1d41\u27e7 : \u27e6U\u27e7 \ud835\udfda\u1d41 \ud835\udfda\u1d41\n  _\u27e6\u00d7\u1d41\u27e7_ : \u27e6Op\u2082\u27e7 \u27e6U\u27e7 _\u00d7\u1d41_ _\u00d7\u1d41_\n  _\u27e6\u228e\u1d41\u27e7_ : \u27e6Op\u2082\u27e7 \u27e6U\u27e7 _\u228e\u1d41_ _\u228e\u1d41_\n  \u27e6\u03a3\u1d41\u27e7 : (\u27e8 t \u2236 \u27e6U\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6El\u27e7 t \u27e6\u2192\u27e7 \u27e6U\u27e7) \u27e6\u2192\u27e7 \u27e6U\u27e7) \u03a3\u1d41 \u03a3\u1d41\n\n\u27e6El\u27e7 \u27e6\ud835\udfd8\u1d41\u27e7 = _\u2261_\n\u27e6El\u27e7 \u27e6\ud835\udfd9\u1d41\u27e7 = _\u2261_\n\u27e6El\u27e7 \u27e6\ud835\udfda\u1d41\u27e7 = _\u2261_\n\u27e6El\u27e7 (s \u27e6\u00d7\u1d41\u27e7 t) = \u27e6El\u27e7 s \u27e6\u00d7\u27e7 \u27e6El\u27e7 t\n\u27e6El\u27e7 (s \u27e6\u228e\u1d41\u27e7 t) = \u27e6El\u27e7 s \u27e6\u228e\u27e7 \u27e6El\u27e7 t\n\u27e6El\u27e7 (\u27e6\u03a3\u1d41\u27e7 t f) = \u27e6\u03a3\u27e7 (\u27e6El\u27e7 t) \u03bb x \u2192 \u27e6El\u27e7 (f x)\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb x\u2081 p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) refl p\n\n    {-\n\u27e6U\u27e7-sound : \u2200 {{_ : FunExt}} {x y} \u2192 \u27e6U\u27e7 x y \u2192 x \u2261 y\n\u27e6U\u27e7-refl : \u2200 x \u2192 \u27e6U\u27e7 x x\n\n{-\n\u27e6El\u27e7-refl : \u2200 x \u2192 {!\u27e6El\u27e7 x x!}\n\u27e6El\u27e7-refl = {!!}\n-}\n\n\u27e6U\u27e7-sound \u27e6\ud835\udfd8\u1d41\u27e7 = refl\n\u27e6U\u27e7-sound \u27e6\ud835\udfd9\u1d41\u27e7 = refl\n\u27e6U\u27e7-sound \u27e6\ud835\udfda\u1d41\u27e7 = refl\n\u27e6U\u27e7-sound (u \u27e6\u00d7\u1d41\u27e7 u\u2081) = ap\u2082 _\u00d7\u1d41_ (\u27e6U\u27e7-sound u) (\u27e6U\u27e7-sound u\u2081)\n\u27e6U\u27e7-sound (u \u27e6\u228e\u1d41\u27e7 u\u2081) = ap\u2082 _\u228e\u1d41_ (\u27e6U\u27e7-sound u) (\u27e6U\u27e7-sound u\u2081)\n\u27e6U\u27e7-sound (\u27e6\u03a3\u1d41\u27e7 {u\u2080} {u\u2081} u {f\u2080} {f\u2081} f\u1d63) = apd\u2082 \u03a3\u1d41 (\u27e6U\u27e7-sound u) (tr-\u2192 El (const U) (\u27e6U\u27e7-sound u) f\u2080 \u2219 \u03bb= (\u03bb A \u2192 ap (\u03bb z \u2192 z (f\u2080 (tr El (! \u27e6U\u27e7-sound u) A))) (tr-const (\u27e6U\u27e7-sound u)) \u2219 \u27e6U\u27e7-sound (f\u1d63 {!!}))) -- (\u03bb= (\u03bb y \u2192 let foo = x\u1d63 {{!!}} {y} {!x\u1d63!} in {!tr-\u2192 El (const U) (\u27e6U\u27e7-sound u)!}))\n\n\u27e6U\u27e7-refl \ud835\udfd8\u1d41 = \u27e6\ud835\udfd8\u1d41\u27e7\n\u27e6U\u27e7-refl \ud835\udfd9\u1d41 = \u27e6\ud835\udfd9\u1d41\u27e7\n\u27e6U\u27e7-refl \ud835\udfda\u1d41 = \u27e6\ud835\udfda\u1d41\u27e7\n\u27e6U\u27e7-refl (x \u00d7\u1d41 x\u2081) = \u27e6U\u27e7-refl x \u27e6\u00d7\u1d41\u27e7 \u27e6U\u27e7-refl x\u2081\n\u27e6U\u27e7-refl (x \u228e\u1d41 x\u2081) = \u27e6U\u27e7-refl x \u27e6\u228e\u1d41\u27e7 \u27e6U\u27e7-refl x\u2081\n\u27e6U\u27e7-refl (\u03a3\u1d41 x f) = \u27e6\u03a3\u1d41\u27e7 (\u27e6U\u27e7-refl x) (\u03bb y \u2192 {!\u27e6U\u27e7-refl ?!})\n-}\n\ninfix  8 _^\u1d41_\n_^\u1d41_ : U \u2192 \u2115 \u2192 U\nt ^\u1d41 zero  = t\nt ^\u1d41 suc n = t \u00d7\u1d41 t ^\u1d41 n\n\nmodule _ {\u2113} where\n\n    explore : \u2200 t \u2192 Explore \u2113 (El t)\n    explore \ud835\udfd8\u1d41 = \ud835\udfd8\u1d49\n    explore \ud835\udfd9\u1d41 = \ud835\udfd9\u1d49\n    explore \ud835\udfda\u1d41 = \ud835\udfda\u1d49\n    explore (s \u00d7\u1d41 t) = explore s \u00d7\u1d49 explore t\n    explore (s \u228e\u1d41 t) = explore s \u228e\u1d49 explore t\n    explore (\u03a3\u1d41 t f) = explore\u03a3 (explore t) \u03bb {x} \u2192 explore (f x)\n\n    exploreU-ind : \u2200 {p} t \u2192 ExploreInd p (explore t)\n    exploreU-ind \ud835\udfd8\u1d41 = \ud835\udfd8\u2071\n    exploreU-ind \ud835\udfd9\u1d41 = \ud835\udfd9\u2071\n    exploreU-ind \ud835\udfda\u1d41 = \ud835\udfda\u2071\n    exploreU-ind (s \u00d7\u1d41 t) = exploreU-ind s \u00d7\u2071 exploreU-ind t\n    exploreU-ind (s \u228e\u1d41 t) = exploreU-ind s \u228e\u2071 exploreU-ind t\n    exploreU-ind (\u03a3\u1d41 t f) = explore\u03a3-ind (exploreU-ind t) \u03bb {x} \u2192 exploreU-ind (f x)\n\nmodule _ {\u2113\u2080 \u2113\u2081 \u2113\u1d63} where\n    \u27e6explore\u27e7 : \u2200 {t\u2080 t\u2081} (t : \u27e6U\u27e7 t\u2080 t\u2081) \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 (\u27e6El\u27e7 t) (explore t\u2080) (explore t\u2081)\n    \u27e6explore\u27e7 \u27e6\ud835\udfd8\u1d41\u27e7        = \u27e6\ud835\udfd8\u1d49\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63}\n    \u27e6explore\u27e7 \u27e6\ud835\udfd9\u1d41\u27e7        = \u27e6\ud835\udfd9\u1d49\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} {_\u2261_} {refl}\n    \u27e6explore\u27e7 \u27e6\ud835\udfda\u1d41\u27e7        = \u27e6\ud835\udfda\u1d49\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} {_\u2261_} {refl} {refl}\n    \u27e6explore\u27e7 (t \u27e6\u00d7\u1d41\u27e7 t\u2081) = \u27e6explore\u00d7\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} (\u27e6explore\u27e7 t) (\u27e6explore\u27e7 t\u2081)\n    \u27e6explore\u27e7 (t \u27e6\u228e\u1d41\u27e7 t\u2081) = \u27e6explore\u228e\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} (\u27e6explore\u27e7 t) (\u27e6explore\u27e7 t\u2081)\n    \u27e6explore\u27e7 (\u27e6\u03a3\u1d41\u27e7 t f)  = \u27e6explore\u03a3\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} (\u27e6explore\u27e7 t) (\u27e6explore\u27e7 \u2218 f)\n\nmodule _ (t : U) where\n  private\n    t\u1d49 : \u2200 {\u2113} \u2192 Explore \u2113 (El t)\n    t\u1d49 = explore t\n    t\u2071 : \u2200 {\u2113 p} \u2192 ExploreInd {\u2113} p t\u1d49\n    t\u2071 = exploreU-ind t\n\n  open FromExploreInd t\u2071 public hiding (\u27e6explore\u27e7)\n  {-\n  open From\u27e6Explore\u27e7 (\u03bb {\u2113\u2081} {\u2113\u2082} {\u2113\u1d63} \u2192 \u27e6explore\u27e7' {\u2113\u2081} {\u2113\u2082} {\u2113\u1d63} t) public\n  -}\n\nadequate-sumU : \u2200 {{_ : UA}}{{_ : FunExt}} t \u2192 Adequate-sum (sum t)\nadequate-sumU \ud835\udfd8\u1d41       = \ud835\udfd8\u02e2-ok\nadequate-sumU \ud835\udfd9\u1d41       = \ud835\udfd9\u02e2-ok\nadequate-sumU \ud835\udfda\u1d41       = \ud835\udfda\u02e2-ok\nadequate-sumU (s \u00d7\u1d41 t) = adequate-sum\u03a3 (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (s \u228e\u1d41 t) = adequate-sum\u228e (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (\u03a3\u1d41 t f) = adequate-sum\u03a3 (adequate-sumU t) (\u03bb {x} \u2192 adequate-sumU (f x))\n\nmodule _ {\u2113} where\n    lookupU : \u2200 t \u2192 Lookup {\u2113} (explore t)\n    lookupU \ud835\udfd8\u1d41 = \ud835\udfd8\u02e1\n    lookupU \ud835\udfd9\u1d41 = \ud835\udfd9\u02e1\n    lookupU \ud835\udfda\u1d41 = \ud835\udfda\u02e1\n    lookupU (s \u00d7\u1d41 t) = lookup\u00d7 {e\u1d2c = explore s} {e\u1d2e = explore t} (lookupU s) (lookupU t)\n    lookupU (s \u228e\u1d41 t) = lookup\u228e {e\u1d2c = explore s} {e\u1d2e = explore t} (lookupU s) (lookupU t)\n    lookupU (\u03a3\u1d41 t f) = lookup\u03a3 {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (f x)} (lookupU t) (\u03bb {x} \u2192 lookupU (f x))\n\n    focusU : \u2200 t \u2192 Focus {\u2113} (explore t)\n    focusU \ud835\udfd8\u1d41 = \ud835\udfd8\u1da0\n    focusU \ud835\udfd9\u1d41 = \ud835\udfd9\u1da0\n    focusU \ud835\udfda\u1d41 = \ud835\udfda\u1da0\n    focusU (s \u00d7\u1d41 t) = focus\u00d7 {e\u1d2c = explore s} {e\u1d2e = explore t} (focusU s) (focusU t)\n    focusU (s \u228e\u1d41 t) = focus\u228e {e\u1d2c = explore s} {e\u1d2e = explore t} (focusU s) (focusU t)\n    focusU (\u03a3\u1d41 t f) = focus\u03a3 {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (f x)} (focusU t) (\u03bb {x} \u2192 focusU (f x))\n\n    \u03a3\u1d49U : \u2200 A {\u2113} \u2192 (El A \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n    \u03a3\u1d49U = \u03bb A \u2192 \u03a3\u1d49 (explore A)\n    \u03a0\u1d49U : \u2200 A {\u2113} \u2192 (El A \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n    \u03a0\u1d49U = \u03bb A \u2192 \u03a0\u1d49 (explore A)\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03a3\u1d49U-ok : \u2200 t \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49U t)\n        \u03a3\u1d49U-ok \ud835\udfd8\u1d41       = \u03a3\u1d49\ud835\udfd8-ok\n        \u03a3\u1d49U-ok \ud835\udfd9\u1d41       = \u03a3\u1d49\ud835\udfd9-ok\n        \u03a3\u1d49U-ok \ud835\udfda\u1d41       = \u03a3\u1d49\ud835\udfda-ok\n        \u03a3\u1d49U-ok (t \u00d7\u1d41 u) = \u03a3\u1d49\u00d7-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a3\u1d49U-ok t) (\u03a3\u1d49U-ok u)\n        \u03a3\u1d49U-ok (t \u228e\u1d41 u) = \u03a3\u1d49\u228e-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a3\u1d49U-ok t) (\u03a3\u1d49U-ok u)\n        \u03a3\u1d49U-ok (\u03a3\u1d41 t u) = \u03a3\u1d49\u03a3-ok {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (u x)} (\u03a3\u1d49U-ok t) (\u03bb {x} \u2192 \u03a3\u1d49U-ok (u x))\n\n        \u03a0\u1d49U-ok : \u2200 t \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49U t)\n        \u03a0\u1d49U-ok \ud835\udfd8\u1d41       = \u03a0\u1d49\ud835\udfd8-ok\n        \u03a0\u1d49U-ok \ud835\udfd9\u1d41       = \u03a0\u1d49\ud835\udfd9-ok\n        \u03a0\u1d49U-ok \ud835\udfda\u1d41       = \u03a0\u1d49\ud835\udfda-ok\n        \u03a0\u1d49U-ok (t \u00d7\u1d41 u) = \u03a0\u1d49\u00d7-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a0\u1d49U-ok t) (\u03a0\u1d49U-ok u)\n        \u03a0\u1d49U-ok (t \u228e\u1d41 u) = \u03a0\u1d49\u228e-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a0\u1d49U-ok t) (\u03a0\u1d49U-ok u)\n        \u03a0\u1d49U-ok (\u03a3\u1d41 t u) = \u03a0\u1d49\u03a3-ok {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (u x)} (\u03a0\u1d49U-ok t) (\u03bb {x} \u2192 \u03a0\u1d49U-ok (u x))\n\n{-\nmodule _ (t : U) {{_ : UA}} {{_ : FunExt}} where\n  open FromAdequate-\u03a3\u1d49 t (\u03a3\u1d49U-ok t) public\n  open FromAdequate-\u03a0\u1d49 t (\u03a0\u1d49U-ok t) public\n-}\n\nmodule _ (A : U) (P : El A \u2192 \u2605\u2080) where\n    Dec-\u03a3 : \u03a0 (El A) (Dec \u2218 P) \u2192 Dec (\u03a3 (El A) P)\n    Dec-\u03a3 = map-Dec (unfocus A) (focusU A) \u2218 lift-Dec A P \u2218 reify A\n\n-- See Explore.Fin for an example of the use of this module\nmodule Isomorphism {A : \u2605\u2080} u (e : El u \u2243 A) where\n  open Explore.Isomorphism e\n\n  module _ {\u2113} where\n    iso\u1d49 : Explore \u2113 A\n    iso\u1d49 = explore-iso (explore u)\n\n    module _ {p} where\n      iso\u2071 : ExploreInd p iso\u1d49\n      iso\u2071 = explore-iso-ind (exploreU-ind u)\n\n  module _ {\u2113} where\n  {-\n    iso\u02e1 : Lookup {\u2113} iso\u1d49\n    iso\u02e1 = lookup-iso {\u2113} {exploreU u} (lookupU u)\n\n    iso\u1da0 : Focus {\u2113} iso\u1d49\n    iso\u1da0 = focus-iso {\u2113} {exploreU u} (focusU u)\n    -}\n\n    iso\u02b3 : Reify {\u2113} iso\u1d49\n    iso\u02b3 = FromExploreInd-iso.reify (exploreU-ind u)\n\n    iso\u1d58 : Unfocus {\u2113} iso\u1d49\n    iso\u1d58 = FromExploreInd-iso.unfocus (exploreU-ind u)\n\n  iso\u02e2 : Sum A\n  iso\u02e2 = sum-iso (sum u)\n\n  iso\u1d56 : Product A\n  iso\u1d56 = product-iso (sum u)\n\n  module _ {{_ : UA}}{{_ : FunExt}} where\n    iso\u02e2-ok : Adequate-sum iso\u02e2\n    iso\u02e2-ok = sum-iso-ok (adequate-sumU u)\n\n    open Adequate-sum\u2080 iso\u02e2-ok iso\u02e2-ok public renaming (sumStableUnder to iso\u02e2-stableUnder)\n\nFin\u1d41 : \u2115 \u2192 U\nFin\u1d41 zero    = \ud835\udfd8\u1d41\nFin\u1d41 (suc n) = \ud835\udfd9\u1d41 \u228e\u1d41 Fin\u1d41 n\n\nFin\u1d41' : \u2115 \u2192 U\nFin\u1d41' zero          = \ud835\udfd8\u1d41\nFin\u1d41' (suc zero)    = \ud835\udfd9\u1d41\nFin\u1d41' (suc (suc n)) = \ud835\udfd9\u1d41 \u228e\u1d41 Fin\u1d41' (suc n)\n\nFin\u1d41-Fin : \u2200 n \u2192 El (Fin\u1d41 n) \u2243 Fin n\nFin\u1d41-Fin zero    = \u2243-! Fin0\u2243\ud835\udfd8\nFin\u1d41-Fin (suc n) = \u228e\u2243 \u2243-refl (Fin\u1d41-Fin n) \u2243-\u2219 \u2243-! Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin\u1d41'-Fin : \u2200 n \u2192 El (Fin\u1d41' n) \u2243 Fin n\nFin\u1d41'-Fin zero          = \u2243-! Fin0\u2243\ud835\udfd8\nFin\u1d41'-Fin (suc zero)    = \u2243-! Fin1\u2243\ud835\udfd9\nFin\u1d41'-Fin (suc (suc n)) = \u228e\u2243 \u2243-refl (Fin\u1d41'-Fin (suc n)) \u2243-\u2219 \u2243-! Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\n\u03a0\u1d41 : (u : U) (v : El u \u2192 U) \u2192 U\n\u03a0\u1d41 \ud835\udfd8\u1d41        v = \ud835\udfd9\u1d41\n\u03a0\u1d41 \ud835\udfd9\u1d41        v = v _\n\u03a0\u1d41 \ud835\udfda\u1d41        v = v 0\u2082 \u00d7\u1d41 v 1\u2082\n\u03a0\u1d41 (u \u00d7\u1d41 u\u2081) v = \u03a0\u1d41 u \u03bb x \u2192 \u03a0\u1d41 u\u2081 \u03bb y \u2192 v (x , y)\n\u03a0\u1d41 (u \u228e\u1d41 u\u2081) v = (\u03a0\u1d41 u (v \u2218 inl)) \u00d7\u1d41 (\u03a0\u1d41 u\u2081 (v \u2218 inr))\n\u03a0\u1d41 (\u03a3\u1d41 u f)  v = \u03a0\u1d41 u \u03bb x \u2192 \u03a0\u1d41 (f x) (v \u2218 _,_ x)\n\n_\u2192\u1d41_ : (u : U) (v : U) \u2192 U\nu \u2192\u1d41 v = \u03a0\u1d41 u (const v)\n\n{-\n\ud835\udfdb\u1d41 : U\n\ud835\udfdb\u1d41 = \ud835\udfd9\u1d41 \u228e\u1d41 \ud835\udfda\u1d41\n\nlist22 = list (\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41)\nlist33 = list (\ud835\udfdb\u1d41 \u2192\u1d41 \ud835\udfdb\u1d41)\n-}\n\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n    \u03a0\u1d41-\u03a0 : \u2200 u v \u2192 El (\u03a0\u1d41 u v) \u2261 \u03a0 (El u) (El \u2218 v)\n    \u03a0\u1d41-\u03a0 \ud835\udfd8\u1d41        v = ! \u03a0\ud835\udfd8-uniq\u2080 _\n    \u03a0\u1d41-\u03a0 \ud835\udfd9\u1d41        v = ! \u03a0\ud835\udfd9-uniq _\n    \u03a0\u1d41-\u03a0 \ud835\udfda\u1d41        v = ! \u03a0\ud835\udfda-\u00d7\n    \u03a0\u1d41-\u03a0 (u \u00d7\u1d41 u\u2081) v = \u03a0\u1d41-\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 u\u2081 (v \u2218 _,_ x)) \u2219 \u03a0=\u2032 _ (\u03bb _ \u2192 \u03a0\u1d41-\u03a0 u\u2081 _) \u2219 ! \u03a0\u03a3-curry\n    \u03a0\u1d41-\u03a0 (u \u228e\u1d41 u\u2081) v = \u00d7= (\u03a0\u1d41-\u03a0 u (v \u2218 inl)) (\u03a0\u1d41-\u03a0 u\u2081 (v \u2218 inr)) \u2219 ! dist-\u00d7-\u03a0\n    \u03a0\u1d41-\u03a0 (\u03a3\u1d41 u f)  v = \u03a0\u1d41-\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 (f _) (v \u2218 _,_ x)) \u2219 \u03a0=\u2032 _ (\u03bb _ \u2192 \u03a0\u1d41-\u03a0 (f _) _) \u2219 ! \u03a0\u03a3-curry\n\n    \u2192\u1d41-\u2192 : \u2200 u v \u2192 El (u \u2192\u1d41 v) \u2261 (El u \u2192 El v)\n    \u2192\u1d41-\u2192 u v = \u03a0\u1d41-\u03a0 u (const v)\n\n    \u03a0\u1d41\u2192\u03a0 : \u2200 u v \u2192 El (\u03a0\u1d41 u v) \u2192 \u03a0 (El u) (El \u2218 v)\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfd8\u1d41 v x\u2082 ()\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfd9\u1d41 v x\u2082 x\u2083 = x\u2082\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfda\u1d41 v (x , y) 0\u2082 = x\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfda\u1d41 v (x , y) 1\u2082 = y\n    \u03a0\u1d41\u2192\u03a0 (u \u00d7\u1d41 u\u2081) v x (z , t) = \u03a0\u1d41\u2192\u03a0 u\u2081 (v \u2218 _,_ z) (\u03a0\u1d41\u2192\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 u\u2081 (v \u2218 _,_ x)) x z) t\n    \u03a0\u1d41\u2192\u03a0 (u \u228e\u1d41 _) v (x , _) (inl y) = \u03a0\u1d41\u2192\u03a0 u (v \u2218 inl) x y\n    \u03a0\u1d41\u2192\u03a0 (_ \u228e\u1d41 u) v (_ , x) (inr y) = \u03a0\u1d41\u2192\u03a0 u (v \u2218 inr) x y\n    \u03a0\u1d41\u2192\u03a0 (\u03a3\u1d41 u f) v x       (y , z) = \u03a0\u1d41\u2192\u03a0 (f _) (v \u2218 _,_ y) (\u03a0\u1d41\u2192\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 (f _) (v \u2218 _,_ x)) x y) z\n\n    \u03a0\u2192\u03a0\u1d41 : \u2200 u v \u2192 \u03a0 (El u) (El \u2218 v) \u2192 El (\u03a0\u1d41 u v)\n    \u03a0\u2192\u03a0\u1d41 \ud835\udfd8\u1d41 v f = 0\u2081\n    \u03a0\u2192\u03a0\u1d41 \ud835\udfd9\u1d41 v f = f 0\u2081\n    \u03a0\u2192\u03a0\u1d41 \ud835\udfda\u1d41 v f = f 0\u2082 , f 1\u2082\n    \u03a0\u2192\u03a0\u1d41 (u \u00d7\u1d41 u\u2081) v f = \u03a0\u2192\u03a0\u1d41 u (\u03bb x \u2192 \u03a0\u1d41 u\u2081 (v \u2218 _,_ x))\n                           (\u03bb x \u2192 \u03a0\u2192\u03a0\u1d41 u\u2081 (v \u2218 _,_ x) (f \u2218 _,_ x))\n    \u03a0\u2192\u03a0\u1d41 (u \u228e\u1d41 u\u2081) v f = \u03a0\u2192\u03a0\u1d41 u (v \u2218 inl) (f \u2218 inl) ,\n                           \u03a0\u2192\u03a0\u1d41 u\u2081 (v \u2218 inr) (f \u2218 inr)\n    \u03a0\u2192\u03a0\u1d41 (\u03a3\u1d41 u F) v f = \u03a0\u2192\u03a0\u1d41 u (\u03bb x \u2192 \u03a0\u1d41 (F x) (v \u2218 _,_ x))\n                          (\u03bb x \u2192 \u03a0\u2192\u03a0\u1d41 (F x) (v \u2218 _,_ x) (f \u2218 _,_ x))\n\n    \u2192\u1d41\u2192\u2192 : \u2200 u v \u2192 El (u \u2192\u1d41 v) \u2192 (El u \u2192 El v)\n    \u2192\u1d41\u2192\u2192 u v = \u03a0\u1d41\u2192\u03a0 u (const v)\n\n    \u2192\u2192\u2192\u1d41 : \u2200 u v \u2192 (El u \u2192 El v) \u2192 El (u \u2192\u1d41 v)\n    \u2192\u2192\u2192\u1d41 u v = \u03a0\u2192\u03a0\u1d41 u (const v)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Level.NP\nopen import Type\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Product.NP\nopen import Data.Sum.NP\nopen import Data.Sum.Logical\nopen import Data.Nat\nopen import Data.Fin\nopen import Relation.Nullary.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Binary.Logical\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Zero\nopen import Explore.One\nopen import Explore.Two\nopen import Explore.Product\nopen import Explore.Sum\nopen import Explore.Explorable\nimport Explore.Isomorphism\n\nmodule Explore.Universe where\n\nopen Operators\n\ninfixr 2 _\u00d7\u1d41_\n\ndata U : \u2605\nEl : U \u2192 \u2605\n\ndata U where\n  \ud835\udfd8\u1d41 \ud835\udfd9\u1d41 \ud835\udfda\u1d41 : U\n  _\u00d7\u1d41_ _\u228e\u1d41_ : U \u2192 U \u2192 U\n  \u03a3\u1d41 : (t : U) (f : El t \u2192 U) \u2192 U\n\nEl \ud835\udfd8\u1d41 = \ud835\udfd8\nEl \ud835\udfd9\u1d41 = \ud835\udfd9\nEl \ud835\udfda\u1d41 = \ud835\udfda\nEl (s \u00d7\u1d41 t) = El s \u00d7 El t\nEl (s \u228e\u1d41 t) = El s \u228e El t\nEl (\u03a3\u1d41 t f) = \u03a3 (El t) \u03bb x \u2192 El (f x)\n\ndata \u27e6U\u27e7 : \u27e6\u2605\u2080\u27e7 U U\n\u27e6El\u27e7 : (\u27e6U\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) El El\n\ndata \u27e6U\u27e7 where\n  \u27e6\ud835\udfd8\u1d41\u27e7 : \u27e6U\u27e7 \ud835\udfd8\u1d41 \ud835\udfd8\u1d41\n  \u27e6\ud835\udfd9\u1d41\u27e7 : \u27e6U\u27e7 \ud835\udfd9\u1d41 \ud835\udfd9\u1d41\n  \u27e6\ud835\udfda\u1d41\u27e7 : \u27e6U\u27e7 \ud835\udfda\u1d41 \ud835\udfda\u1d41\n  _\u27e6\u00d7\u1d41\u27e7_ : \u27e6Op\u2082\u27e7 \u27e6U\u27e7 _\u00d7\u1d41_ _\u00d7\u1d41_\n  _\u27e6\u228e\u1d41\u27e7_ : \u27e6Op\u2082\u27e7 \u27e6U\u27e7 _\u228e\u1d41_ _\u228e\u1d41_\n  \u27e6\u03a3\u1d41\u27e7 : (\u27e8 t \u2236 \u27e6U\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6El\u27e7 t \u27e6\u2192\u27e7 \u27e6U\u27e7) \u27e6\u2192\u27e7 \u27e6U\u27e7) \u03a3\u1d41 \u03a3\u1d41\n\n\u27e6El\u27e7 \u27e6\ud835\udfd8\u1d41\u27e7 = _\u2261_\n\u27e6El\u27e7 \u27e6\ud835\udfd9\u1d41\u27e7 = _\u2261_\n\u27e6El\u27e7 \u27e6\ud835\udfda\u1d41\u27e7 = _\u2261_\n\u27e6El\u27e7 (s \u27e6\u00d7\u1d41\u27e7 t) = \u27e6El\u27e7 s \u27e6\u00d7\u27e7 \u27e6El\u27e7 t\n\u27e6El\u27e7 (s \u27e6\u228e\u1d41\u27e7 t) = \u27e6El\u27e7 s \u27e6\u228e\u27e7 \u27e6El\u27e7 t\n\u27e6El\u27e7 (\u27e6\u03a3\u1d41\u27e7 t f) = \u27e6\u03a3\u27e7 (\u27e6El\u27e7 t) \u03bb x \u2192 \u27e6El\u27e7 (f x)\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb x\u2081 p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) refl p\n\n    {-\n\u27e6U\u27e7-sound : \u2200 {{_ : FunExt}} {x y} \u2192 \u27e6U\u27e7 x y \u2192 x \u2261 y\n\u27e6U\u27e7-refl : \u2200 x \u2192 \u27e6U\u27e7 x x\n\n{-\n\u27e6El\u27e7-refl : \u2200 x \u2192 {!\u27e6El\u27e7 x x!}\n\u27e6El\u27e7-refl = {!!}\n-}\n\n\u27e6U\u27e7-sound \u27e6\ud835\udfd8\u1d41\u27e7 = refl\n\u27e6U\u27e7-sound \u27e6\ud835\udfd9\u1d41\u27e7 = refl\n\u27e6U\u27e7-sound \u27e6\ud835\udfda\u1d41\u27e7 = refl\n\u27e6U\u27e7-sound (u \u27e6\u00d7\u1d41\u27e7 u\u2081) = ap\u2082 _\u00d7\u1d41_ (\u27e6U\u27e7-sound u) (\u27e6U\u27e7-sound u\u2081)\n\u27e6U\u27e7-sound (u \u27e6\u228e\u1d41\u27e7 u\u2081) = ap\u2082 _\u228e\u1d41_ (\u27e6U\u27e7-sound u) (\u27e6U\u27e7-sound u\u2081)\n\u27e6U\u27e7-sound (\u27e6\u03a3\u1d41\u27e7 {u\u2080} {u\u2081} u {f\u2080} {f\u2081} f\u1d63) = apd\u2082 \u03a3\u1d41 (\u27e6U\u27e7-sound u) (tr-\u2192 El (const U) (\u27e6U\u27e7-sound u) f\u2080 \u2219 \u03bb= (\u03bb A \u2192 ap (\u03bb z \u2192 z (f\u2080 (tr El (! \u27e6U\u27e7-sound u) A))) (tr-const (\u27e6U\u27e7-sound u)) \u2219 \u27e6U\u27e7-sound (f\u1d63 {!!}))) -- (\u03bb= (\u03bb y \u2192 let foo = x\u1d63 {{!!}} {y} {!x\u1d63!} in {!tr-\u2192 El (const U) (\u27e6U\u27e7-sound u)!}))\n\n\u27e6U\u27e7-refl \ud835\udfd8\u1d41 = \u27e6\ud835\udfd8\u1d41\u27e7\n\u27e6U\u27e7-refl \ud835\udfd9\u1d41 = \u27e6\ud835\udfd9\u1d41\u27e7\n\u27e6U\u27e7-refl \ud835\udfda\u1d41 = \u27e6\ud835\udfda\u1d41\u27e7\n\u27e6U\u27e7-refl (x \u00d7\u1d41 x\u2081) = \u27e6U\u27e7-refl x \u27e6\u00d7\u1d41\u27e7 \u27e6U\u27e7-refl x\u2081\n\u27e6U\u27e7-refl (x \u228e\u1d41 x\u2081) = \u27e6U\u27e7-refl x \u27e6\u228e\u1d41\u27e7 \u27e6U\u27e7-refl x\u2081\n\u27e6U\u27e7-refl (\u03a3\u1d41 x f) = \u27e6\u03a3\u1d41\u27e7 (\u27e6U\u27e7-refl x) (\u03bb y \u2192 {!\u27e6U\u27e7-refl ?!})\n-}\n\ninfix  8 _^\u1d41_\n_^\u1d41_ : U \u2192 \u2115 \u2192 U\nt ^\u1d41 zero  = t\nt ^\u1d41 suc n = t \u00d7\u1d41 t ^\u1d41 n\n\nmodule _ {\u2113} where\n\n    explore : \u2200 t \u2192 Explore \u2113 (El t)\n    explore \ud835\udfd8\u1d41 = \ud835\udfd8\u1d49\n    explore \ud835\udfd9\u1d41 = \ud835\udfd9\u1d49\n    explore \ud835\udfda\u1d41 = \ud835\udfda\u1d49\n    explore (s \u00d7\u1d41 t) = explore s \u00d7\u1d49 explore t\n    explore (s \u228e\u1d41 t) = explore s \u228e\u1d49 explore t\n    explore (\u03a3\u1d41 t f) = explore\u03a3 (explore t) \u03bb {x} \u2192 explore (f x)\n\n    exploreU-ind : \u2200 {p} t \u2192 ExploreInd p (explore t)\n    exploreU-ind \ud835\udfd8\u1d41 = \ud835\udfd8\u2071\n    exploreU-ind \ud835\udfd9\u1d41 = \ud835\udfd9\u2071\n    exploreU-ind \ud835\udfda\u1d41 = \ud835\udfda\u2071\n    exploreU-ind (s \u00d7\u1d41 t) = exploreU-ind s \u00d7\u2071 exploreU-ind t\n    exploreU-ind (s \u228e\u1d41 t) = exploreU-ind s \u228e\u2071 exploreU-ind t\n    exploreU-ind (\u03a3\u1d41 t f) = explore\u03a3-ind (exploreU-ind t) \u03bb {x} \u2192 exploreU-ind (f x)\n\nmodule _ {\u2113\u2080 \u2113\u2081 \u2113\u1d63} where\n    \u27e6explore\u27e7 : \u2200 {t\u2080 t\u2081} (t : \u27e6U\u27e7 t\u2080 t\u2081) \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 (\u27e6El\u27e7 t) (explore t\u2080) (explore t\u2081)\n    \u27e6explore\u27e7 \u27e6\ud835\udfd8\u1d41\u27e7        = \u27e6\ud835\udfd8\u1d49\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63}\n    \u27e6explore\u27e7 \u27e6\ud835\udfd9\u1d41\u27e7        = \u27e6\ud835\udfd9\u1d49\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} {_\u2261_} {refl}\n    \u27e6explore\u27e7 \u27e6\ud835\udfda\u1d41\u27e7        = \u27e6\ud835\udfda\u1d49\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} {_\u2261_} {refl} {refl}\n    \u27e6explore\u27e7 (t \u27e6\u00d7\u1d41\u27e7 t\u2081) = \u27e6explore\u00d7\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} (\u27e6explore\u27e7 t) (\u27e6explore\u27e7 t\u2081)\n    \u27e6explore\u27e7 (t \u27e6\u228e\u1d41\u27e7 t\u2081) = \u27e6explore\u228e\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} (\u27e6explore\u27e7 t) (\u27e6explore\u27e7 t\u2081)\n    \u27e6explore\u27e7 (\u27e6\u03a3\u1d41\u27e7 t f)  = \u27e6explore\u03a3\u27e7 {\u2113\u2080} {\u2113\u2081} {\u2113\u1d63} (\u27e6explore\u27e7 t) (\u27e6explore\u27e7 \u2218 f)\n\nmodule _ (t : U) where\n  private\n    t\u1d49 : \u2200 {\u2113} \u2192 Explore \u2113 (El t)\n    t\u1d49 = explore t\n    t\u2071 : \u2200 {\u2113 p} \u2192 ExploreInd {\u2113} p t\u1d49\n    t\u2071 = exploreU-ind t\n\n  open FromExploreInd t\u2071 public hiding (\u27e6explore\u27e7)\n  {-\n  open From\u27e6Explore\u27e7 (\u03bb {\u2113\u2081} {\u2113\u2082} {\u2113\u1d63} \u2192 \u27e6explore\u27e7' {\u2113\u2081} {\u2113\u2082} {\u2113\u1d63} t) public\n  -}\n\nadequate-sumU : \u2200 {{_ : UA}}{{_ : FunExt}} t \u2192 Adequate-sum (sum t)\nadequate-sumU \ud835\udfd8\u1d41       = \ud835\udfd8\u02e2-ok\nadequate-sumU \ud835\udfd9\u1d41       = \ud835\udfd9\u02e2-ok\nadequate-sumU \ud835\udfda\u1d41       = \ud835\udfda\u02e2-ok\nadequate-sumU (s \u00d7\u1d41 t) = adequate-sum\u03a3 (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (s \u228e\u1d41 t) = adequate-sum\u228e (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (\u03a3\u1d41 t f) = adequate-sum\u03a3 (adequate-sumU t) (\u03bb {x} \u2192 adequate-sumU (f x))\n\nmodule _ {\u2113} where\n    lookupU : \u2200 t \u2192 Lookup {\u2113} (explore t)\n    lookupU \ud835\udfd8\u1d41 = \ud835\udfd8\u02e1\n    lookupU \ud835\udfd9\u1d41 = \ud835\udfd9\u02e1\n    lookupU \ud835\udfda\u1d41 = \ud835\udfda\u02e1\n    lookupU (s \u00d7\u1d41 t) = lookup\u00d7 {e\u1d2c = explore s} {e\u1d2e = explore t} (lookupU s) (lookupU t)\n    lookupU (s \u228e\u1d41 t) = lookup\u228e {e\u1d2c = explore s} {e\u1d2e = explore t} (lookupU s) (lookupU t)\n    lookupU (\u03a3\u1d41 t f) = lookup\u03a3 {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (f x)} (lookupU t) (\u03bb {x} \u2192 lookupU (f x))\n\n    focusU : \u2200 t \u2192 Focus {\u2113} (explore t)\n    focusU \ud835\udfd8\u1d41 = \ud835\udfd8\u1da0\n    focusU \ud835\udfd9\u1d41 = \ud835\udfd9\u1da0\n    focusU \ud835\udfda\u1d41 = \ud835\udfda\u1da0\n    focusU (s \u00d7\u1d41 t) = focus\u00d7 {e\u1d2c = explore s} {e\u1d2e = explore t} (focusU s) (focusU t)\n    focusU (s \u228e\u1d41 t) = focus\u228e {e\u1d2c = explore s} {e\u1d2e = explore t} (focusU s) (focusU t)\n    focusU (\u03a3\u1d41 t f) = focus\u03a3 {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (f x)} (focusU t) (\u03bb {x} \u2192 focusU (f x))\n\n    \u03a3\u1d49U : \u2200 A {\u2113} \u2192 (El A \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n    \u03a3\u1d49U = \u03bb A \u2192 \u03a3\u1d49 (explore A)\n    \u03a0\u1d49U : \u2200 A {\u2113} \u2192 (El A \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n    \u03a0\u1d49U = \u03bb A \u2192 \u03a0\u1d49 (explore A)\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03a3\u1d49U-ok : \u2200 t \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49U t)\n        \u03a3\u1d49U-ok \ud835\udfd8\u1d41       = \u03a3\u1d49\ud835\udfd8-ok\n        \u03a3\u1d49U-ok \ud835\udfd9\u1d41       = \u03a3\u1d49\ud835\udfd9-ok\n        \u03a3\u1d49U-ok \ud835\udfda\u1d41       = \u03a3\u1d49\ud835\udfda-ok\n        \u03a3\u1d49U-ok (t \u00d7\u1d41 u) = \u03a3\u1d49\u00d7-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a3\u1d49U-ok t) (\u03a3\u1d49U-ok u)\n        \u03a3\u1d49U-ok (t \u228e\u1d41 u) = \u03a3\u1d49\u228e-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a3\u1d49U-ok t) (\u03a3\u1d49U-ok u)\n        \u03a3\u1d49U-ok (\u03a3\u1d41 t u) = \u03a3\u1d49\u03a3-ok {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (u x)} (\u03a3\u1d49U-ok t) (\u03bb {x} \u2192 \u03a3\u1d49U-ok (u x))\n\n        \u03a0\u1d49U-ok : \u2200 t \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49U t)\n        \u03a0\u1d49U-ok \ud835\udfd8\u1d41       = \u03a0\u1d49\ud835\udfd8-ok\n        \u03a0\u1d49U-ok \ud835\udfd9\u1d41       = \u03a0\u1d49\ud835\udfd9-ok\n        \u03a0\u1d49U-ok \ud835\udfda\u1d41       = \u03a0\u1d49\ud835\udfda-ok\n        \u03a0\u1d49U-ok (t \u00d7\u1d41 u) = \u03a0\u1d49\u00d7-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a0\u1d49U-ok t) (\u03a0\u1d49U-ok u)\n        \u03a0\u1d49U-ok (t \u228e\u1d41 u) = \u03a0\u1d49\u228e-ok {e\u1d2c = explore t} {e\u1d2e = explore u} (\u03a0\u1d49U-ok t) (\u03a0\u1d49U-ok u)\n        \u03a0\u1d49U-ok (\u03a3\u1d41 t u) = \u03a0\u1d49\u03a3-ok {e\u1d2c = explore t} {e\u1d2e = \u03bb {x} \u2192 explore (u x)} (\u03a0\u1d49U-ok t) (\u03bb {x} \u2192 \u03a0\u1d49U-ok (u x))\n\n{-\nmodule _ (t : U) {{_ : UA}} {{_ : FunExt}} where\n  open FromAdequate-\u03a3\u1d49 t (\u03a3\u1d49U-ok t) public\n  open FromAdequate-\u03a0\u1d49 t (\u03a0\u1d49U-ok t) public\n-}\n\nmodule _ (A : U) (P : El A \u2192 \u2605\u2080) where\n    Dec-\u03a3 : \u03a0 (El A) (Dec \u2218 P) \u2192 Dec (\u03a3 (El A) P)\n    Dec-\u03a3 = map-Dec (unfocus A) (focusU A) \u2218 lift-Dec A P \u2218 reify A\n\n-- See Explore.Fin for an example of the use of this module\nmodule Isomorphism {A : \u2605\u2080} u (e : El u \u2243 A) where\n  open Explore.Isomorphism e\n\n  module _ {\u2113} where\n    iso\u1d49 : Explore \u2113 A\n    iso\u1d49 = explore-iso (explore u)\n\n    module _ {p} where\n      iso\u2071 : ExploreInd p iso\u1d49\n      iso\u2071 = explore-iso-ind (exploreU-ind u)\n\n  module _ {\u2113} where\n  {-\n    iso\u02e1 : Lookup {\u2113} iso\u1d49\n    iso\u02e1 = lookup-iso {\u2113} {exploreU u} (lookupU u)\n\n    iso\u1da0 : Focus {\u2113} iso\u1d49\n    iso\u1da0 = focus-iso {\u2113} {exploreU u} (focusU u)\n    -}\n\n    iso\u02b3 : Reify {\u2113} iso\u1d49\n    iso\u02b3 = FromExploreInd-iso.reify (exploreU-ind u)\n\n    iso\u1d58 : Unfocus {\u2113} iso\u1d49\n    iso\u1d58 = FromExploreInd-iso.unfocus (exploreU-ind u)\n\n  iso\u02e2 : Sum A\n  iso\u02e2 = sum-iso (sum u)\n\n  iso\u1d56 : Product A\n  iso\u1d56 = product-iso (sum u)\n\n  module _ {{_ : UA}}{{_ : FunExt}} where\n    iso\u02e2-ok : Adequate-sum iso\u02e2\n    iso\u02e2-ok = sum-iso-ok (adequate-sumU u)\n\n    open Adequate-sum\u2080 iso\u02e2-ok iso\u02e2-ok public renaming (sumStableUnder to iso\u02e2-stableUnder)\n\nFin\u1d41 : \u2115 \u2192 U\nFin\u1d41 zero    = \ud835\udfd8\u1d41\nFin\u1d41 (suc n) = \ud835\udfd9\u1d41 \u228e\u1d41 Fin\u1d41 n\n\nFin\u1d41' : \u2115 \u2192 U\nFin\u1d41' zero          = \ud835\udfd8\u1d41\nFin\u1d41' (suc zero)    = \ud835\udfd9\u1d41\nFin\u1d41' (suc (suc n)) = \ud835\udfd9\u1d41 \u228e\u1d41 Fin\u1d41' (suc n)\n\nFin\u1d41-Fin : \u2200 n \u2192 El (Fin\u1d41 n) \u2243 Fin n\nFin\u1d41-Fin zero    = \u2243-! Fin0\u2243\ud835\udfd8\nFin\u1d41-Fin (suc n) = \u228e\u2243 \u2243-refl (Fin\u1d41-Fin n) \u2243-\u2219 \u2243-! Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin\u1d41'-Fin : \u2200 n \u2192 El (Fin\u1d41' n) \u2243 Fin n\nFin\u1d41'-Fin zero          = \u2243-! Fin0\u2243\ud835\udfd8\nFin\u1d41'-Fin (suc zero)    = \u2243-! Fin1\u2243\ud835\udfd9\nFin\u1d41'-Fin (suc (suc n)) = \u228e\u2243 \u2243-refl (Fin\u1d41'-Fin (suc n)) \u2243-\u2219 \u2243-! Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\n\u03a0\u1d41 : (u : U) (v : El u \u2192 U) \u2192 U\n\u03a0\u1d41 \ud835\udfd8\u1d41        v = \ud835\udfd9\u1d41\n\u03a0\u1d41 \ud835\udfd9\u1d41        v = v _\n\u03a0\u1d41 \ud835\udfda\u1d41        v = v 0\u2082 \u00d7\u1d41 v 1\u2082\n\u03a0\u1d41 (u \u00d7\u1d41 u\u2081) v = \u03a0\u1d41 u \u03bb x \u2192 \u03a0\u1d41 u\u2081 \u03bb y \u2192 v (x , y)\n\u03a0\u1d41 (u \u228e\u1d41 u\u2081) v = (\u03a0\u1d41 u (v \u2218 inl)) \u00d7\u1d41 (\u03a0\u1d41 u\u2081 (v \u2218 inr))\n\u03a0\u1d41 (\u03a3\u1d41 u f)  v = \u03a0\u1d41 u \u03bb x \u2192 \u03a0\u1d41 (f x) (v \u2218 _,_ x)\n\n_\u2192\u1d41_ : (u : U) (v : U) \u2192 U\nu \u2192\u1d41 v = \u03a0\u1d41 u (const v)\n\n{-\n\ud835\udfdb\u1d41 : U\n\ud835\udfdb\u1d41 = \ud835\udfd9\u1d41 \u228e\u1d41 \ud835\udfda\u1d41\n\nlist22 = list (\ud835\udfda\u1d41 \u2192\u1d41 \ud835\udfda\u1d41)\nlist33 = list (\ud835\udfdb\u1d41 \u2192\u1d41 \ud835\udfdb\u1d41)\n-}\n\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n    \u03a0\u1d41-\u03a0 : \u2200 u v \u2192 El (\u03a0\u1d41 u v) \u2261 \u03a0 (El u) (El \u2218 v)\n    \u03a0\u1d41-\u03a0 \ud835\udfd8\u1d41        v = ! \u03a0\ud835\udfd8-uniq\u2080 _\n    \u03a0\u1d41-\u03a0 \ud835\udfd9\u1d41        v = ! \u03a0\ud835\udfd9-uniq _\n    \u03a0\u1d41-\u03a0 \ud835\udfda\u1d41        v = ! \u03a0\ud835\udfda-\u00d7\n    \u03a0\u1d41-\u03a0 (u \u00d7\u1d41 u\u2081) v = \u03a0\u1d41-\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 u\u2081 (v \u2218 _,_ x)) \u2219 \u03a0=\u2032 _ (\u03bb _ \u2192 \u03a0\u1d41-\u03a0 u\u2081 _) \u2219 ! \u03a0\u03a3-curry\n    \u03a0\u1d41-\u03a0 (u \u228e\u1d41 u\u2081) v = \u00d7= (\u03a0\u1d41-\u03a0 u (v \u2218 inl)) (\u03a0\u1d41-\u03a0 u\u2081 (v \u2218 inr)) \u2219 ! dist-\u00d7-\u03a0\n    \u03a0\u1d41-\u03a0 (\u03a3\u1d41 u f)  v = \u03a0\u1d41-\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 (f _) (v \u2218 _,_ x)) \u2219 \u03a0=\u2032 _ (\u03bb _ \u2192 \u03a0\u1d41-\u03a0 (f _) _) \u2219 ! \u03a0\u03a3-curry\n\n    \u2192\u1d41-\u2192 : \u2200 u v \u2192 El (u \u2192\u1d41 v) \u2261 (El u \u2192 El v)\n    \u2192\u1d41-\u2192 u v = \u03a0\u1d41-\u03a0 u (const v)\n\n    \u03a0\u1d41\u2192\u03a0 : \u2200 u v \u2192 El (\u03a0\u1d41 u v) \u2192 \u03a0 (El u) (El \u2218 v)\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfd8\u1d41 v x\u2082 ()\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfd9\u1d41 v x\u2082 x\u2083 = x\u2082\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfda\u1d41 v (x , y) 0\u2082 = x\n    \u03a0\u1d41\u2192\u03a0 \ud835\udfda\u1d41 v (x , y) 1\u2082 = y\n    \u03a0\u1d41\u2192\u03a0 (u \u00d7\u1d41 u\u2081) v x (z , t) = \u03a0\u1d41\u2192\u03a0 u\u2081 (v \u2218 _,_ z) (\u03a0\u1d41\u2192\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 u\u2081 (v \u2218 _,_ x)) x z) t\n    \u03a0\u1d41\u2192\u03a0 (u \u228e\u1d41 _) v (x , _) (inl y) = \u03a0\u1d41\u2192\u03a0 u (v \u2218 inl) x y\n    \u03a0\u1d41\u2192\u03a0 (_ \u228e\u1d41 u) v (_ , x) (inr y) = \u03a0\u1d41\u2192\u03a0 u (v \u2218 inr) x y\n    \u03a0\u1d41\u2192\u03a0 (\u03a3\u1d41 u f) v x       (y , z) = \u03a0\u1d41\u2192\u03a0 (f _) (v \u2218 _,_ y) (\u03a0\u1d41\u2192\u03a0 u (\u03bb x \u2192 \u03a0\u1d41 (f _) (v \u2218 _,_ x)) x y) z\n\n    \u2192\u1d41\u2192\u2192 : \u2200 u v \u2192 El (u \u2192\u1d41 v) \u2192 (El u \u2192 El v)\n    \u2192\u1d41\u2192\u2192 u v = \u03a0\u1d41\u2192\u03a0 u (const v)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2e98871b50826699f6f16a2718f8f2361a559c4d","subject":"Write down real step-indexed logical relation for untyped terms","message":"Write down real step-indexed logical relation for untyped terms\n\nSadly, Agda does not recognize this is terminating, because recursing on k with\nk < n is not a structurally recursive call on a subterm. SAD!\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalUntyped.agda","new_file":"Thesis\/ANormalUntyped.agda","new_contents":"module Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Product\nopen import Data.Nat.Base\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {t : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs t \u2193 dclosure t \u03c1 d\u03c1\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\n{-# TERMINATING #-} -- Dear lord. Why on Earth.\nmutual\n  -- Single context? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3 : \u2200 {\u0393} (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val Uni) \u2192\n    \u2200 j n2 (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 \u2115 \u2192 Set\n  rrelV3 (intV v1) (dintV dv) (intV v2) n = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n  rrelV3 _ _ _ n = \u22a5\n\n-- -- [_]\u03c4  from v1 to v2) :\n-- -- [ \u03c4 ]\u03c4 dv from v1 to v2) =\n\n-- -- data [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set where\n-- --   v\u2205 : [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205\n-- --   _v\u2022_ : \u2200 {\u03c4 \u0394 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n-- --     (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n-- --     (d\u03c1\u03c1 : [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12) \u2192\n-- --     [ \u03c4 \u2022 \u0394 ]\u0394 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n-- data ErrVal : Set where\n--   Done : (v : Val Uni) \u2192 ErrVal\n--   Error : ErrVal\n--   TimeOut : ErrVal\n\n-- _>>=_ : ErrVal \u2192 (Val Uni \u2192 ErrVal) \u2192 ErrVal\n-- Done v >>= f = f v\n-- Error >>= f = Error\n-- TimeOut >>= f = TimeOut\n\n-- Res : Set\n-- Res = \u2115 \u2192 ErrVal\n\n-- -- eval-fun : \u2200 {\u0393} \u2192 Fun \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n-- -- eval-term : \u2200 {\u0393} \u2192 Term \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n\n-- -- apply : Val Uni \u2192 Val Uni \u2192 Res\n-- -- apply (closure f \u03c1) a n = eval-fun f (a \u2022 \u03c1) n\n-- -- apply (intV _) a n = Error\n\n-- -- eval-term t \u03c1 zero = TimeOut\n-- -- eval-term (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\n-- -- eval-term (lett f x t) \u03c1 (suc n) = apply (\u27e6 f \u27e7Var \u03c1) (\u27e6 x \u27e7Var \u03c1) n >>= (\u03bb v \u2192 eval-term t (v \u2022 \u03c1) n)\n\n-- -- eval-fun (term t) \u03c1 n = eval-term t \u03c1 n\n-- -- eval-fun (abs f) \u03c1 n = Done (closure f \u03c1)\n\n-- -- -- Erasure from typed to untyped values.\n-- -- import Thesis.ANormalBigStep as T\n\n-- -- erase-type : T.Type \u2192 Type\n-- -- erase-type _ = Uni\n\n-- -- erase-val : \u2200 {\u03c4} \u2192 T.Val \u03c4 \u2192 Val (erase-type \u03c4)\n\n-- -- erase-errVal : \u2200 {\u03c4} \u2192 T.ErrVal \u03c4 \u2192 ErrVal\n-- -- erase-errVal (T.Done v) = Done (erase-val v)\n-- -- erase-errVal T.Error = Error\n-- -- erase-errVal T.TimeOut = TimeOut\n\n-- -- erase-res : \u2200 {\u03c4} \u2192 T.Res \u03c4 \u2192 Res\n-- -- erase-res r n = erase-errVal (r n)\n\n-- -- erase-ctx : T.Context \u2192 Context\n-- -- erase-ctx \u2205 = \u2205\n-- -- erase-ctx (\u03c4 \u2022 \u0393) = erase-type \u03c4 \u2022 (erase-ctx \u0393)\n\n-- -- erase-env : \u2200 {\u0393} \u2192 T.Op.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 erase-ctx \u0393 \u27e7Context\n-- -- erase-env \u2205 = \u2205\n-- -- erase-env (v \u2022 \u03c1) = erase-val v \u2022 erase-env \u03c1\n\n-- -- erase-var : \u2200 {\u0393 \u03c4} \u2192 T.Var \u0393 \u03c4 \u2192 Var (erase-ctx \u0393) (erase-type \u03c4)\n-- -- erase-var T.this = this\n-- -- erase-var (T.that x) = that (erase-var x)\n\n-- -- erase-term : \u2200 {\u0393 \u03c4} \u2192 T.Term \u0393 \u03c4 \u2192 Term (erase-ctx \u0393)\n-- -- erase-term (T.var x) = var (erase-var x)\n-- -- erase-term (T.lett f x t) = lett (erase-var f) (erase-var x) (erase-term t)\n\n-- -- erase-fun : \u2200 {\u0393 \u03c4} \u2192 T.Fun \u0393 \u03c4 \u2192 Fun (erase-ctx \u0393)\n-- -- erase-fun (T.term x) = term (erase-term x)\n-- -- erase-fun (T.abs f) = abs (erase-fun f)\n\n-- -- erase-val (T.closure t \u03c1) = closure (erase-fun t) (erase-env \u03c1)\n-- -- erase-val (T.intV n) = intV n\n\n-- -- -- Different erasures commute.\n-- -- erasure-commute-var : \u2200 {\u0393 \u03c4} (x : T.Var \u0393 \u03c4) \u03c1 \u2192\n-- --   erase-val (T.Op.\u27e6 x \u27e7Var \u03c1) \u2261 \u27e6 erase-var x \u27e7Var (erase-env \u03c1)\n-- -- erasure-commute-var T.this (v \u2022 \u03c1) = refl\n-- -- erasure-commute-var (T.that x) (v \u2022 \u03c1) = erasure-commute-var x \u03c1\n\n-- -- erase-bind : \u2200 {\u03c3 \u03c4 \u0393} a (t : T.Term (\u03c3 \u2022 \u0393) \u03c4) \u03c1 n \u2192 erase-errVal (a T.>>= (\u03bb v \u2192 T.eval-term t (v \u2022 \u03c1) n)) \u2261 erase-errVal a >>= (\u03bb v \u2192 eval-term (erase-term t) (v \u2022 erase-env \u03c1) n)\n\n-- -- erasure-commute-fun : \u2200 {\u0393 \u03c4} (t : T.Fun \u0393 \u03c4) \u03c1 n \u2192\n-- --   erase-errVal (T.eval-fun t \u03c1 n) \u2261 eval-fun (erase-fun t) (erase-env \u03c1) n\n\n-- -- erasure-commute-apply : \u2200 {\u03c3 \u03c4} (f : T.Val (\u03c3 T.\u21d2 \u03c4)) a n \u2192 erase-errVal (T.apply f a n) \u2261 apply (erase-val f) (erase-val a) n\n-- -- erasure-commute-apply {\u03c3} (T.closure t \u03c1) a n = erasure-commute-fun t (a \u2022 \u03c1) n\n\n-- -- erasure-commute-term : \u2200 {\u0393 \u03c4} (t : T.Term \u0393 \u03c4) \u03c1 n \u2192\n-- --   erase-errVal (T.eval-term t \u03c1 n) \u2261 eval-term (erase-term t) (erase-env \u03c1) n\n\n-- -- erasure-commute-fun (T.term t) \u03c1 n = erasure-commute-term t \u03c1 n\n-- -- erasure-commute-fun (T.abs t) \u03c1 n = refl\n\n-- -- erasure-commute-term t \u03c1 zero = refl\n-- -- erasure-commute-term (T.var x) \u03c1 (\u2115.suc n) = cong Done (erasure-commute-var x \u03c1)\n-- -- erasure-commute-term (T.lett f x t) \u03c1 (\u2115.suc n) rewrite erase-bind (T.apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n) t \u03c1 n | erasure-commute-apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n | erasure-commute-var f \u03c1 | erasure-commute-var x \u03c1 = refl\n\n-- -- erase-bind (T.Done v) t \u03c1 n = erasure-commute-term t (v \u2022 \u03c1) n\n-- -- erase-bind T.Error t \u03c1 n = refl\n-- -- erase-bind T.TimeOut t \u03c1 n = refl\n","old_contents":"module Thesis.ANormalUntyped where\n\nopen import Data.Nat.Base\nopen import Data.Integer.Base\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nopen import Base.Syntax.Context Type public\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2124) \u2192 Val Uni\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\ndata ErrVal : Set where\n  Done : (v : Val Uni) \u2192 ErrVal\n  Error : ErrVal\n  TimeOut : ErrVal\n\n_>>=_ : ErrVal \u2192 (Val Uni \u2192 ErrVal) \u2192 ErrVal\nDone v >>= f = f v\nError >>= f = Error\nTimeOut >>= f = TimeOut\n\nRes : Set\nRes = \u2115 \u2192 ErrVal\n\neval-fun : \u2200 {\u0393} \u2192 Fun \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\neval-term : \u2200 {\u0393} \u2192 Term \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n\napply : Val Uni \u2192 Val Uni \u2192 Res\napply (closure f \u03c1) a n = eval-fun f (a \u2022 \u03c1) n\napply (intV _) a n = Error\n\neval-term t \u03c1 zero = TimeOut\neval-term (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval-term (lett f x t) \u03c1 (suc n) = apply (\u27e6 f \u27e7Var \u03c1) (\u27e6 x \u27e7Var \u03c1) n >>= (\u03bb v \u2192 eval-term t (v \u2022 \u03c1) n)\n\neval-fun (term t) \u03c1 n = eval-term t \u03c1 n\neval-fun (abs f) \u03c1 n = Done (closure f \u03c1)\n\n-- Erasure from typed to untyped values.\nimport Thesis.ANormalBigStep as T\n\nerase-type : T.Type \u2192 Type\nerase-type _ = Uni\n\nerase-val : \u2200 {\u03c4} \u2192 T.Val \u03c4 \u2192 Val (erase-type \u03c4)\n\nerase-errVal : \u2200 {\u03c4} \u2192 T.ErrVal \u03c4 \u2192 ErrVal\nerase-errVal (T.Done v) = Done (erase-val v)\nerase-errVal T.Error = Error\nerase-errVal T.TimeOut = TimeOut\n\nerase-res : \u2200 {\u03c4} \u2192 T.Res \u03c4 \u2192 Res\nerase-res r n = erase-errVal (r n)\n\nerase-ctx : T.Context \u2192 Context\nerase-ctx \u2205 = \u2205\nerase-ctx (\u03c4 \u2022 \u0393) = erase-type \u03c4 \u2022 (erase-ctx \u0393)\n\nerase-env : \u2200 {\u0393} \u2192 T.Op.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 erase-ctx \u0393 \u27e7Context\nerase-env \u2205 = \u2205\nerase-env (v \u2022 \u03c1) = erase-val v \u2022 erase-env \u03c1\n\nerase-var : \u2200 {\u0393 \u03c4} \u2192 T.Var \u0393 \u03c4 \u2192 Var (erase-ctx \u0393) (erase-type \u03c4)\nerase-var T.this = this\nerase-var (T.that x) = that (erase-var x)\n\nerase-term : \u2200 {\u0393 \u03c4} \u2192 T.Term \u0393 \u03c4 \u2192 Term (erase-ctx \u0393)\nerase-term (T.var x) = var (erase-var x)\nerase-term (T.lett f x t) = lett (erase-var f) (erase-var x) (erase-term t)\n\nerase-fun : \u2200 {\u0393 \u03c4} \u2192 T.Fun \u0393 \u03c4 \u2192 Fun (erase-ctx \u0393)\nerase-fun (T.term x) = term (erase-term x)\nerase-fun (T.abs f) = abs (erase-fun f)\n\nerase-val (T.closure t \u03c1) = closure (erase-fun t) (erase-env \u03c1)\nerase-val (T.intV n) = intV n\n\n-- Different erasures commute.\nerasure-commute-var : \u2200 {\u0393 \u03c4} (x : T.Var \u0393 \u03c4) \u03c1 \u2192\n  erase-val (T.Op.\u27e6 x \u27e7Var \u03c1) \u2261 \u27e6 erase-var x \u27e7Var (erase-env \u03c1)\nerasure-commute-var T.this (v \u2022 \u03c1) = refl\nerasure-commute-var (T.that x) (v \u2022 \u03c1) = erasure-commute-var x \u03c1\n\nerase-bind : \u2200 {\u03c3 \u03c4 \u0393} a (t : T.Term (\u03c3 \u2022 \u0393) \u03c4) \u03c1 n \u2192 erase-errVal (a T.>>= (\u03bb v \u2192 T.eval-term t (v \u2022 \u03c1) n)) \u2261 erase-errVal a >>= (\u03bb v \u2192 eval-term (erase-term t) (v \u2022 erase-env \u03c1) n)\n\nerasure-commute-fun : \u2200 {\u0393 \u03c4} (t : T.Fun \u0393 \u03c4) \u03c1 n \u2192\n  erase-errVal (T.eval-fun t \u03c1 n) \u2261 eval-fun (erase-fun t) (erase-env \u03c1) n\n\nerasure-commute-apply : \u2200 {\u03c3 \u03c4} (f : T.Val (\u03c3 T.\u21d2 \u03c4)) a n \u2192 erase-errVal (T.apply f a n) \u2261 apply (erase-val f) (erase-val a) n\nerasure-commute-apply {\u03c3} (T.closure t \u03c1) a n = erasure-commute-fun t (a \u2022 \u03c1) n\n\nerasure-commute-term : \u2200 {\u0393 \u03c4} (t : T.Term \u0393 \u03c4) \u03c1 n \u2192\n  erase-errVal (T.eval-term t \u03c1 n) \u2261 eval-term (erase-term t) (erase-env \u03c1) n\n\nerasure-commute-fun (T.term t) \u03c1 n = erasure-commute-term t \u03c1 n\nerasure-commute-fun (T.abs t) \u03c1 n = refl\n\nerasure-commute-term t \u03c1 zero = refl\nerasure-commute-term (T.var x) \u03c1 (\u2115.suc n) = cong Done (erasure-commute-var x \u03c1)\nerasure-commute-term (T.lett f x t) \u03c1 (\u2115.suc n) rewrite erase-bind (T.apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n) t \u03c1 n | erasure-commute-apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n | erasure-commute-var f \u03c1 | erasure-commute-var x \u03c1 = refl\n\nerase-bind (T.Done v) t \u03c1 n = erasure-commute-term t (v \u2022 \u03c1) n\nerase-bind T.Error t \u03c1 n = refl\nerase-bind T.TimeOut t \u03c1 n = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a66b08fcfbf8ac4b747cc3153338fe176a361f01","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expansion-unicity.agda","new_file":"expansion-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule expansion-unicity where\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp1 x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp2 x\u2084 d5 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp3 x\u2084 d5 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp1 x\u2083 x\u2084 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp2 x\u2083 d6 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp3 x\u2083 d6 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp1 x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp2 x\u2082 d6 x\u2083 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp3 x\u2082 d6 x\u2083) = {!!}\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083) = {!!}\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc2 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081) = {!!}\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c41' \u03c42 \u03c42' : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n                          \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03c41' == \u03c42' \u00d7 \u03941 == \u03942\n    expansion-unicity-ana (EALam apt1 d1) (EALam apt2 d2) = {!!}\n    expansion-unicity-ana (EALam apt1 d1) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = {!!}\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) (EALam apt2 d2) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    expansion-unicity-ana EAEHole EAEHole = {!!}\n    expansion-unicity-ana (EANEHole x x\u2081) (EASubsume x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana (EANEHole x x\u2081) (EANEHole x\u2082 x\u2083) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule expansion-unicity where\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam d1) (ESLam d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , refl\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp1 x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp2 x\u2084 d5 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp3 x\u2084 d5 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp1 x\u2083 x\u2084 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp2 x\u2083 d6 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp3 x\u2083 d6 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp1 x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp2 x\u2082 d6 x\u2083 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp3 x\u2082 d6 x\u2083) = {!!}\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083) = {!!}\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc2 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081) = {!!}\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c41' \u03c42 \u03c42' : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n                          \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03c41' == \u03c42' \u00d7 \u03941 == \u03942\n    expansion-unicity-ana (EALam d1) (EALam d2) = {!!}\n    expansion-unicity-ana (EALam d1) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = {!!}\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) (EALam d2) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    expansion-unicity-ana EAEHole EAEHole = {!!}\n    expansion-unicity-ana (EANEHole x x\u2081) (EASubsume x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana (EANEHole x x\u2081) (EANEHole x\u2082 x\u2083) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5423d9e6f476f8d4fea29421fe3397bfa9cfa786","subject":"Avoid local function (fix #1).","message":"Avoid local function (fix #1).\n\nThis fixes an error about hidden vs. visible types in the type of a\nlocal function. The error was introduced by switching to Agda 2.4.0. I\ndon't understand the error. It seems that the local function was\nintroduced to be able to use rewrite, so I rewrote the proof with subst\ninstead. This avoids the local function, and thus the error.\n\nThe ilc-agda project now type checks with current Agda. Tested versions:\n\nagda\/agda-stdlib@517d842ed43d0684e31b2e3e1f45ec9dda703761\n\nagda\/agda@b21f32a35f1a1b584538e3aaec3f62c9b4463aff (2_4_0)\nagda\/agda@70c8a575c46f6a568c7518150a1a64fcd03aa437\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Implementation.agda","new_file":"Parametric\/Change\/Implementation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Logical relation for erasure (Def. 3.8 and Lemma 3.9)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Derive as Derive\n\nmodule Parametric.Change.Implementation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n       Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (derive-const : Derive.Structure Const \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen Derive.Structure Const \u0394Base derive-const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nrecord Structure : Set\u2081 where\n\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    -- Extension point 1: Logical relation on base types.\n    --\n    -- In the paper, we assume that the logical relation is equality on base types\n    -- (see Def. 3.8a). Here, we only require that plugins define what the logical\n    -- relation is on base types, and provide proofs for the two extension points\n    -- below.\n    implements-base : \u2200 \u03b9 {v : \u27e6 \u03b9 \u27e7Base} \u2192 \u0394\u208d \u03b9 \u208e v \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 Set\n\n    -- Extension point 2: Differences on base types are logically related.\n    u\u229fv\u2248u\u229dv-base : \u2200 \u03b9 {u v : \u27e6 \u03b9 \u27e7Base} \u2192\n      implements-base \u03b9 (u \u229f\u208d \u03b9 \u208e v) (\u27e6diff-base\u27e7 \u03b9 u v)\n\n    -- Extension point 3: Lemma 3.1 for base types.\n    carry-over-base : \u2200 {\u03b9}\n      {v : \u27e6 \u03b9 \u27e7Base}\n      (\u0394v : \u0394\u208d \u03b9 \u208e v)\n      {\u0394v\u2032 : \u27e6 \u0394Base \u03b9 \u27e7Base} (\u0394v\u2248\u0394v\u2032 : implements-base \u03b9 \u0394v \u0394v\u2032) \u2192\n        v \u229e\u208d base \u03b9 \u208e \u0394v \u2261 v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v\u2032\n\n  ------------------------\n  -- Logical relation \u2248 --\n  ------------------------\n\n  infix 4 implements\n  syntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\n  implements : \u2200 \u03c4 {v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\n  implements (base \u03b9) \u0394f \u0394f\u2032 = implements-base \u03b9 \u0394f \u0394f\u2032\n  implements (\u03c3 \u21d2 \u03c4) {f} \u0394f \u0394f\u2032 =\n    (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w)\n    (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 {w} \u0394w \u0394w\u2032) \u2192\n    implements \u03c4 {f w} (call-change {\u03c3} {\u03c4} \u0394f w \u0394w) (\u0394f\u2032 w \u0394w\u2032)\n\n  infix 4 _\u2248_\n  _\u2248_ : \u2200 {\u03c4 v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2248_ {\u03c4} {v} = implements \u03c4 {v}\n\n  data implements-env : \u2200 \u0393 \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 Set where\n    \u2205 : implements-env \u2205 {\u2205} \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1 dv d\u03c1 v\u2032 \u03c1\u2032} \u2192\n      (dv\u2248v\u2032 : implements \u03c4 {v} dv v\u2032) \u2192\n      (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 {\u03c1} d\u03c1 \u03c1\u2032) \u2192\n      implements-env (\u03c4 \u2022 \u0393) {v \u2022 \u03c1} (dv \u2022 d\u03c1) (v\u2032 \u2022 \u03c1\u2032)\n\n  ----------------\n  -- carry-over --\n  ----------------\n\n  -- This is lemma 3.10.\n  carry-over : \u2200 {\u03c4}\n    {v : \u27e6 \u03c4 \u27e7}\n    (\u0394v : \u0394\u208d \u03c4 \u208e v)\n    {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7} (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) \u2192\n      after\u208d \u03c4 \u208e \u0394v \u2261 before\u208d \u03c4 \u208e \u0394v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394v\u2032\n\n  u\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    u \u229f\u208d \u03c4 \u208e v \u2248\u208d \u03c4 \u208e u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  u\u229fv\u2248u\u229dv {base \u03b9} {u} {v} = u\u229fv\u2248u\u229dv-base \u03b9 {u} {v}\n  u\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032 =\n    subst\n      (\u03bb \u25a1 \u2192 (g \u25a1 \u229f\u208d \u03c4 \u208e f (before\u208d \u03c3 \u208e \u0394w)) \u2248\u208d \u03c4 \u208e g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 f (before\u208d \u03c3 \u208e \u0394w))\n      (sym (carry-over {\u03c3} \u0394w \u0394w\u2248\u0394w\u2032))\n      (u\u229fv\u2248u\u229dv {\u03c4} {g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032)} {f (before\u208d \u03c3 \u208e \u0394w)})\n\n  carry-over {base \u03b9} \u0394v \u0394v\u2248\u0394v\u2032 = carry-over-base \u0394v \u0394v\u2248\u0394v\u2032\n  carry-over {\u03c3 \u21d2 \u03c4} {f} \u0394f {\u0394f\u2032} \u0394f\u2248\u0394f\u2032 =\n    ext (\u03bb v \u2192\n      carry-over {\u03c4} {f v} (call-change {\u03c3} {\u03c4} \u0394f v (nil\u208d \u03c3 \u208e v))\n        {\u0394f\u2032 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)}\n        (\u0394f\u2248\u0394f\u2032 v (nil\u208d \u03c3 \u208e v) (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) ( u\u229fv\u2248u\u229dv {\u03c3} {v} {v})))\n\n  -- A property relating `alternate` and the subcontext relation \u0393\u227c\u0394\u0393\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\n  -- A specialization of the soundness of weakening\n  \u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 fit t \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6fit\u27e7 t \u03c1 d\u03c1 =\n    trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)) (sym (weaken-sound t _))\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Logical relation for erasure (Def. 3.8 and Lemma 3.9)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Derive as Derive\n\nmodule Parametric.Change.Implementation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n       Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (derive-const : Derive.Structure Const \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen Derive.Structure Const \u0394Base derive-const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nrecord Structure : Set\u2081 where\n\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    -- Extension point 1: Logical relation on base types.\n    --\n    -- In the paper, we assume that the logical relation is equality on base types\n    -- (see Def. 3.8a). Here, we only require that plugins define what the logical\n    -- relation is on base types, and provide proofs for the two extension points\n    -- below.\n    implements-base : \u2200 \u03b9 {v : \u27e6 \u03b9 \u27e7Base} \u2192 \u0394\u208d \u03b9 \u208e v \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 Set\n\n    -- Extension point 2: Differences on base types are logically related.\n    u\u229fv\u2248u\u229dv-base : \u2200 \u03b9 {u v : \u27e6 \u03b9 \u27e7Base} \u2192\n      implements-base \u03b9 (u \u229f\u208d \u03b9 \u208e v) (\u27e6diff-base\u27e7 \u03b9 u v)\n\n    -- Extension point 3: Lemma 3.1 for base types.\n    carry-over-base : \u2200 {\u03b9}\n      {v : \u27e6 \u03b9 \u27e7Base}\n      (\u0394v : \u0394\u208d \u03b9 \u208e v)\n      {\u0394v\u2032 : \u27e6 \u0394Base \u03b9 \u27e7Base} (\u0394v\u2248\u0394v\u2032 : implements-base \u03b9 \u0394v \u0394v\u2032) \u2192\n        v \u229e\u208d base \u03b9 \u208e \u0394v \u2261 v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v\u2032\n\n  ------------------------\n  -- Logical relation \u2248 --\n  ------------------------\n\n  infix 4 implements\n  syntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\n  implements : \u2200 \u03c4 {v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\n  implements (base \u03b9) \u0394f \u0394f\u2032 = implements-base \u03b9 \u0394f \u0394f\u2032\n  implements (\u03c3 \u21d2 \u03c4) {f} \u0394f \u0394f\u2032 =\n    (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w)\n    (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 {w} \u0394w \u0394w\u2032) \u2192\n    implements \u03c4 {f w} (call-change {\u03c3} {\u03c4} \u0394f w \u0394w) (\u0394f\u2032 w \u0394w\u2032)\n\n  infix 4 _\u2248_\n  _\u2248_ : \u2200 {\u03c4 v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2248_ {\u03c4} {v} = implements \u03c4 {v}\n\n  data implements-env : \u2200 \u0393 \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 Set where\n    \u2205 : implements-env \u2205 {\u2205} \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1 dv d\u03c1 v\u2032 \u03c1\u2032} \u2192\n      (dv\u2248v\u2032 : implements \u03c4 {v} dv v\u2032) \u2192\n      (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 {\u03c1} d\u03c1 \u03c1\u2032) \u2192\n      implements-env (\u03c4 \u2022 \u0393) {v \u2022 \u03c1} (dv \u2022 d\u03c1) (v\u2032 \u2022 \u03c1\u2032)\n\n  ----------------\n  -- carry-over --\n  ----------------\n\n  -- This is lemma 3.10.\n  carry-over : \u2200 {\u03c4}\n    {v : \u27e6 \u03c4 \u27e7}\n    (\u0394v : \u0394\u208d \u03c4 \u208e v)\n    {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7} (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) \u2192\n      after\u208d \u03c4 \u208e \u0394v \u2261 before\u208d \u03c4 \u208e \u0394v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394v\u2032\n\n  u\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    u \u229f\u208d \u03c4 \u208e v \u2248\u208d \u03c4 \u208e u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  u\u229fv\u2248u\u229dv {base \u03b9} {u} {v} = u\u229fv\u2248u\u229dv-base \u03b9 {u} {v}\n  u\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n    result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w) \u2192\n      (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192 \u0394w \u2248\u208d \u03c3 \u208e \u0394w\u2032 \u2192\n        (g (after\u208d \u03c3 \u208e \u0394w) \u229f\u208d \u03c4 \u208e f (before\u208d \u03c3 \u208e \u0394w)) \u2248\u208d \u03c4 \u208e g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 f (before\u208d \u03c3 \u208e \u0394w)\n    result w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032\n      rewrite carry-over {\u03c3} \u0394w \u0394w\u2248\u0394w\u2032 =\n      u\u229fv\u2248u\u229dv {\u03c4} {g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032)} {f (before\u208d \u03c3 \u208e \u0394w)}\n\n  carry-over {base \u03b9} \u0394v \u0394v\u2248\u0394v\u2032 = carry-over-base \u0394v \u0394v\u2248\u0394v\u2032\n  carry-over {\u03c3 \u21d2 \u03c4} {f} \u0394f {\u0394f\u2032} \u0394f\u2248\u0394f\u2032 =\n    ext (\u03bb v \u2192\n      carry-over {\u03c4} {f v} (call-change {\u03c3} {\u03c4} \u0394f v (nil\u208d \u03c3 \u208e v))\n        {\u0394f\u2032 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)}\n        (\u0394f\u2248\u0394f\u2032 v (nil\u208d \u03c3 \u208e v) (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) ( u\u229fv\u2248u\u229dv {\u03c3} {v} {v})))\n\n  -- A property relating `alternate` and the subcontext relation \u0393\u227c\u0394\u0393\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\n  -- A specialization of the soundness of weakening\n  \u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 fit t \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6fit\u27e7 t \u03c1 d\u03c1 =\n    trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)) (sym (weaken-sound t _))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e4292c039f3f26444126c66ebb060a83ab1e914d","subject":"Added tailS.","message":"Added tailS.\n\nIgnore-this: f8c3a1c756aff5692726c70d02433824\n\ndarcs-hash:20110730172619-3bd4e-8ac339efc4e78dfcdd270f2cd26b2497b39c4bf3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/Stream\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesATP\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\npostulate tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n{-# ATP prove tailS #-}\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n  {-# ATP definition P\u2081 #-}\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n  {-# ATP definition P\u2082 #-}\n\n  postulate\n    helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n              \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 P\u2081 ws'))\n  {-# ATP prove helper\u2081 #-}\n\n  postulate\n    helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192\n              \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  {-# ATP prove helper\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesATP\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n  {-# ATP definition P\u2081 #-}\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n  {-# ATP definition P\u2082 #-}\n\n  postulate\n    helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n              \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 P\u2081 ws'))\n  {-# ATP prove helper\u2081 #-}\n\n  postulate\n    helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192\n              \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  {-# ATP prove helper\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"826c3786890b50ee2d861e13539405b2aac40bbf","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: 50b6461ccb5bd7b956e03cae0ed535da\n\ndarcs-hash:20110402162139-3bd4e-30f7fb20d91dca52965210f6fa7687e4370efcb5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","old_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(n,x) \u2192 app\u2081(n,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"55886cdd77fb42b777e571c6b0625cc4e67c7c4e","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: 4cb7c429d7aabe9ac4be5cc0acb31552\n\ndarcs-hash:20120519135448-3bd4e-301a535b796fb0130b2cc51ef323c1879532647f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC-PCF\/Data\/Nat\/PropertiesI.agda","new_file":"src\/LTC-PCF\/Data\/Nat\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n+-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n+-Sx m n =\n  rec (succ\u2081 m) n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)))\n    \u2261\u27e8 rec-S m n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u00b7 m \u00b7 (m + n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m \u00b7 (m + n)\n    \u2261\u27e8 refl \u27e9\n  lam succ\u2081 \u00b7 (m + n)\n    \u2261\u27e8 beta succ\u2081 (m + n) \u27e9\n  succ\u2081 (m + n) \u220e\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S zN =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 cong pred\u2081 (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN      = \u2238-x0 zero\n\u2238-0x (sN Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ zN =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym $ \u2238-x0 m \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS zN (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS zN Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0S Nn \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (sN {m} Nm) (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS (sN Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym $ beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 refl \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym $ beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym $ rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = rec zero zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u2261\u27e8 rec-0 zero \u27e9\n         zero                                          \u220e\n\n*-Sx : \u2200 m n \u2192 succ\u2081 m * n \u2261 n + m * n\n*-Sx m n =\n  rec (succ\u2081 m) zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)))\n    \u2261\u27e8 rec-S m zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u27e9\n  (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u00b7 m \u00b7 (m * n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m) \u27e9\n  (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m \u00b7 (m * n)\n    \u2261\u27e8 refl \u27e9\n  lam (\u03bb y \u2192 n + y) \u00b7 (m * n)\n    \u2261\u27e8 beta (\u03bb y \u2192 n + y) (m * n) \u27e9\n  (\u03bb y \u2192 n + y) (m * n)\n    \u2261\u27e8 refl \u27e9\n  n + (m * n) \u220e\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm       zN     = subst N (sym $ \u2238-x0 m) Nm\n\u2238-N     zN      (sN Nn) = subst N (sym $ \u2238-0S Nn) zN\n\u2238-N     (sN Nm) (sN Nn) = subst N (sym $ \u2238-SS Nm Nn) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t) (+-rightIdentity Nn) refl)\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym $ +-leftIdentity n) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym $ +-Sx m n) (sN (+-N Nm Nn))\n\n+-assoc : \u2200 {m} \u2192 N m \u2192  \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zN n o =\n  zero + n + o   \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o          \u2261\u27e8 sym $ +-leftIdentity (n + o) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (sN {m} Nm) n o =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm n o) refl \u27e9\n  succ\u2081 (m + (n + o))\n    \u2261\u27e8 sym $ +-Sx m (n + o) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] zN n =\n  zero + succ\u2081 n \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym $ +-leftIdentity n) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (sN {m} Nm) n =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] Nm n) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym $ +-Sx m n) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n    n \u2238 (zero + o)\n      \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n    n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym $ +-rightIdentity Nn \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym $ x+Sy\u2261S[x+y] Nn m \u27e9\n  n + succ\u2081 m   \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN          _  = subst N (sym $ *-leftZero n) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym $ *-Sx m n) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero n) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n              \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = sym\n  ( zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero n) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym $ *-leftZero (succ\u2081 n) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym $ +-assoc Nn m (m * n))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n             refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym $ +-Sx m (n + m * n) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym $ *-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero n) (sym $ *-rightZero Nn)\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n  \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m    \u2261\u27e8 sym $ x*Sy\u2261x+xy Nn Nm \u27e9\n  n * succ\u2081 m  \u220e\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o            \u2261\u27e8 sym $ \u2238-x0 (m * o) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym $ *-0x o) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S Nn) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) No) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o)\n             (sym $ *-0x o)\n             refl\n    \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) zN) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym $ *-0x (succ\u2081 m))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (\u2238-SS Nm Nn)\n             refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n    \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym $ [x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No)) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym $ *-Sx n (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n  zero                 \u2261\u27e8 sym $ *-0x m \u27e9\n  zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero             \u2261\u27e8 sym $ +-rightIdentity (*-N Nm zN) \u27e9\n  m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                (trans (sym $ *-0x n) (*-comm zN Nn))\n                                  refl\n                       \u27e9\n  m * zero + n * zero  \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym $ +-leftIdentity (n * succ\u2081 o) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym $ *-0x (succ\u2081 o))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym $ +-rightIdentity (*-N (sN Nm) (sN No)) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym $ *-leftZero (succ\u2081 o))\n             refl\n    \u27e9\n    succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym $ +-assoc (sN No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n \u220e\n\n+-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n+-Sx m n =\n  rec (succ\u2081 m) n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)))\n    \u2261\u27e8 rec-S m n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u00b7 m \u00b7 (m + n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m \u00b7 (m + n)\n    \u2261\u27e8 refl \u27e9\n  lam succ\u2081 \u00b7 (m + n)\n    \u2261\u27e8 beta succ\u2081 (m + n) \u27e9\n  succ\u2081 (m + n) \u220e\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S zN =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 cong pred\u2081 (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN      = \u2238-x0 zero\n\u2238-0x (sN Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ zN =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym $ \u2238-x0 m \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS zN (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS zN Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0S Nn \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (sN {m} Nm) (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS (sN Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym $ beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 refl \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym $ beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym $ rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = rec zero zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u2261\u27e8 rec-0 zero \u27e9\n         zero                                          \u220e\n\n*-Sx : \u2200 m n \u2192 succ\u2081 m * n \u2261 n + m * n\n*-Sx m n =\n  rec (succ\u2081 m) zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)))\n    \u2261\u27e8 rec-S m zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u27e9\n  (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u00b7 m \u00b7 (m * n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m) \u27e9\n  (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m \u00b7 (m * n)\n    \u2261\u27e8 refl \u27e9\n  lam (\u03bb y \u2192 n + y) \u00b7 (m * n)\n    \u2261\u27e8 beta (\u03bb y \u2192 n + y) (m * n) \u27e9\n  (\u03bb y \u2192 n + y) (m * n)\n    \u2261\u27e8 refl \u27e9\n  n + (m * n) \u220e\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm       zN     = subst N (sym $ \u2238-x0 m) Nm\n\u2238-N     zN      (sN Nn) = subst N (sym $ \u2238-0S Nn) zN\n\u2238-N     (sN Nm) (sN Nn) = subst N (sym $ \u2238-SS Nm Nn) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t) (+-rightIdentity Nn) refl)\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym $ +-leftIdentity n) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym $ +-Sx m n) (sN (+-N Nm Nn))\n\n+-assoc : \u2200 {m} \u2192 N m \u2192  \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zN n o =\n  zero + n + o   \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o          \u2261\u27e8 sym $ +-leftIdentity (n + o) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (sN {m} Nm) n o =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm n o) refl \u27e9\n  succ\u2081 (m + (n + o))\n    \u2261\u27e8 sym $ +-Sx m (n + o) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] zN n =\n  zero + succ\u2081 n \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym $ +-leftIdentity n) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (sN {m} Nm) n =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] Nm n) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym $ +-Sx m n) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n    n \u2238 (zero + o)\n      \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n    n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym $ +-rightIdentity Nn \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym $ x+Sy\u2261S[x+y] Nn m \u27e9\n  n + succ\u2081 m   \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN          _  = subst N (sym $ *-leftZero n) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym $ *-Sx m n) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero n) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n              \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = sym\n  ( zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero n) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym $ *-leftZero (succ\u2081 n) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym $ +-assoc Nn m (m * n))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n             refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym $ +-Sx m (n + m * n) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym $ *-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero n) (sym $ *-rightZero Nn)\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n  \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m    \u2261\u27e8 sym $ x*Sy\u2261x+xy Nn Nm \u27e9\n  n * succ\u2081 m  \u220e\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o            \u2261\u27e8 sym $ \u2238-x0 (m * o) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym $ *-0x o) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S Nn) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) No) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o)\n             (sym $ *-0x o)\n             refl\n    \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) zN) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym $ *-0x (succ\u2081 m))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (\u2238-SS Nm Nn)\n             refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n    \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym $ [x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No)) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym $ *-Sx n (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n  zero                 \u2261\u27e8 sym $ *-0x m \u27e9\n  zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero             \u2261\u27e8 sym $ +-rightIdentity (*-N Nm zN) \u27e9\n  m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                (trans (sym $ *-0x n) (*-comm zN Nn))\n                                  refl\n                       \u27e9\n  m * zero + n * zero  \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym $ +-leftIdentity (n * succ\u2081 o) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym $ *-0x (succ\u2081 o))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym $ +-rightIdentity (*-N (sN Nm) (sN No)) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym $ *-leftZero (succ\u2081 o))\n             refl\n    \u27e9\n    succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym $ +-assoc (sN No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25e15c0f23c0b4a2589c85d5401ff2f55c2d7735","subject":"Defined uncurry.","message":"Defined uncurry.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = h , (H , cond)\n  where\n  h : \u03a3 U (\u03bb x \u2192 V) \u2192 W\n  h (u , v) with f u\n  ... | i , I = i v\n  H : \u03a3 U (\u03bb x \u2192 V) \u2192 Z \u2192 \u03a3 X (\u03bb x \u2192 Y)\n  H (u , v) z with f u\n  ... | i , I = F u (v , z) , I v z\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {z : Z} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (H u z) \u2192 \u03b3 (h u) z\n  cond {u , v}{z} r with (p\u2081 {u}{v , z})\n  ... | p\u2084 with f u\n  ... | i , I with r\n  ... | p\u2082 , p\u2083 = p\u2084 p\u2082 p\u2083\n  \n{-\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n{-\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7849a93dd727b32c2606d958173b08212260367d","subject":"update the validity for the simulator","message":"update the validity for the simulator\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/Transformation\/ReceiptFreeness-CCA2d\/Valid.agda","new_file":"Game\/Transformation\/ReceiptFreeness-CCA2d\/Valid.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.One\nopen import Data.Two\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Nat\nopen import Data.Vec hiding (_>>=_ ; _\u2208_)\nopen import Data.List as L\nopen import Data.Fin as Fin using (Fin)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\nopen import Control.Strategy renaming (map to mapS)\nopen import Control.Strategy.Utils\nopen import Game.Challenge\nimport Game.ReceiptFreeness\nimport Game.IND-CCA2-dagger\nimport Game.IND-CCA2-dagger.Valid\nimport Game.IND-CPA-utils\n\nimport Data.List.Any\nopen Data.List.Any using (here; there)\nopen Data.List.Any.Membership-\u2261 using (_\u2208_ ; _\u2209_)\n\nimport Game.ReceiptFreeness.Adversary\nimport Game.ReceiptFreeness.Valid\nimport Game.Transformation.ReceiptFreeness-CCA2d.Simulator\n\nmodule Game.Transformation.ReceiptFreeness-CCA2d.Valid\n  (PubKey    : \u2605)\n  (CipherText : \u2605)\n\n  (SerialNumber : \u2605)\n  (Receipt : \u2605)\n  (MarkedReceipt? : \u2605)\n  (Ballot : \u2605)\n  (Tally : \u2605)\n--  (BB    : \u2605)\n--  ([]    : BB)\n--  (_\u2237_ : Receipt \u2192 BB \u2192 BB)\n  (Rgb   : \u2605)\n  (genBallot : PubKey \u2192 Rgb \u2192 Ballot)\n  (tallyMarkedReceipt? : let CO = \ud835\udfda in CO \u2192 MarkedReceipt? \u2192 Tally)\n  (0,0   : Tally)\n  (1,1   : Tally)\n  (_+,+_ : Tally \u2192 Tally \u2192 Tally)\n  (receipts : SerialNumber \u00b2 \u2192 CipherText \u00b2 \u2192 Receipt \u00b2)\n  (enc-co : Receipt \u2192 CipherText)\n  (r-sn   : Receipt \u2192 SerialNumber)\n  (m?     : Receipt \u2192 MarkedReceipt?)\n  (b-sn   : Ballot \u2192 SerialNumber)\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2090 : \u2605)\n  (#q : \u2115) (max#q : Fin #q)\n  (Check    : let BB = List Receipt in BB \u2192 Receipt \u2192 \ud835\udfda)\n  where\n\n_\u00b2' : \u2605 \u2192 \u2605\nX \u00b2' = X \u00d7 X\n\nCO = \ud835\udfda\nBB = List Receipt\nall-sn : BB \u2192 List SerialNumber\nall-sn = L.map r-sn\n\nmodule RF  = Game.ReceiptFreeness.Adversary PubKey (SerialNumber \u00b2) R\u2090 Receipt Ballot Tally CO BB\nmodule RFV = Game.ReceiptFreeness.Valid     PubKey SerialNumber R\u2090 Receipt Ballot Tally CO BB\n                                            CipherText enc-co r-sn b-sn\n\nopen Game.Transformation.ReceiptFreeness-CCA2d.Simulator\n  PubKey CipherText (SerialNumber \u00b2) Receipt MarkedReceipt? Ballot Tally BB [] _\u2237_ Rgb genBallot\n  tallyMarkedReceipt? 0,0 1,1 _+,+_ receipts enc-co m? R\u2090 #q max#q Check\n\nmodule CCA2\u2020V = Game.IND-CCA2-dagger.Valid PubKey Message CipherText R\u2090\u2020\n\n\nmodule Simulator-Valid (RFA : RF.Adversary)(RFA-Valid : RFV.Valid-Adversary RFA)\n  (WRONG-HYP : \u2200 r r' \u2192 enc-co r \u2261 enc-co r' \u2192 r-sn r \u2261 r-sn r')\n  (CheckMem : \u2200 bb r \u2192 \u2713 (Check bb r) \u2192 r-sn r \u2209 all-sn bb)\n  where\n  valid : CCA2\u2020V.Valid-Adversary (Simulator.A\u2020 RFA)\n  valid (r\u2090 , rgb) pk = Phase1 _ (RFA-Valid r\u2090 pk) where\n     open CCA2\u2020V.Valid-Adversary (r\u2090 , rgb) pk\n     module RFVA = RFV.Valid-Adversary r\u2090 pk\n     open RF\n     open Simulator RFA\n     open AdversaryParts rgb pk r\u2090\n\n     -- could refine r more\n     -- {-\n     Phase2 : \u2200 RF {bb i ta r} \u2192 r-sn (r 0\u2082) \u2208 all-sn bb \u2192 r-sn (r 1\u2082) \u2208 all-sn bb \u2192 RFVA.Phase2-Valid r RF\n            \u2192 Phase2-Valid (enc-co \u2218 r) (mapS proj\u2081 (mitm-to-client-trans (MITM-phase 1\u2082 i bb ta) RF))\n     Phase2 (ask REB cont) r0 r1 RF-valid = Phase2 (cont _) r0 r1 (RF-valid _)\n     Phase2 (ask RBB cont) r0 r1 RF-valid = Phase2 (cont _) r0 r1 (RF-valid _)\n     Phase2 (ask RTally cont) r0 r1 RF-valid = Phase2 (cont _) r0 r1 (RF-valid _)\n     Phase2 (ask (RCO x) cont) r0 r1 ((r\u2080 , r\u2081) , RF-valid) = r\u2080 , r\u2081 , (\u03bb r \u2192 Phase2 (cont _) r0 r1 (RF-valid _))\n     Phase2 (ask (Vote x) cont) {bb} r0 r1 RF-valid with Check bb x | CheckMem bb x\n     ... | 0\u2082 | _ = Phase2 (cont _) r0 r1 (RF-valid _)\n     ... | 1\u2082 | not-in-bb = (\u03bb eq \u2192 not-in-bb _ (subst (\u03bb x\u2081 \u2192 x\u2081 \u2208 all-sn bb) (WRONG-HYP _ _  eq) r0))\n                            , (\u03bb eq \u2192 not-in-bb _ (subst (\u03bb x\u2081 \u2192 x\u2081 \u2208 all-sn bb) (WRONG-HYP _ _ eq) r1))\n                            , \u03bb r \u2192 Phase2 (cont _) (there r0) (there r1) (RF-valid _)\n                            --Phase2 (cont _) (there r0) (there r1) (RF-valid _)\n     Phase2 (done x) r0 r1 RF-valid = RF-valid\n\n     Phase1 : \u2200 RF {sn i bb ta} \u2192 RFVA.Phase1-Valid sn RF\n            \u2192 Phase1-Valid (mapS A\u20202 (mitm-to-client-trans (MITM-phase 0\u2082 i bb ta) RF))\n     Phase1 (ask REB cont) RF-valid = Phase1 _ (RF-valid _)\n     Phase1 (ask RBB cont) RF-valid = Phase1 _ (RF-valid _)\n     Phase1 (ask RTally cont) RF-valid = Phase1 _ (RF-valid _)\n     Phase1 (ask (RCO x) cont) RF-valid r = Phase1 _ (RF-valid _)\n     Phase1 (ask (Vote x) cont) {bb = bb} RF-valid with Check bb x\n     Phase1 (ask (Vote x) cont) RF-valid | 1\u2082 = \u03bb r \u2192 Phase1 _ (RF-valid _)\n     Phase1 (ask (Vote x) cont) RF-valid | 0\u2082 = Phase1 _ (RF-valid _)\n     Phase1 (done x) {bb = bb'}{ta} (sn\u2080\u2209sn , sn\u2081\u2209sn , RF-valid) cs\n       = {!Phase2 (put-resp x (proj\u2082 (put-resp (hack-challenge x) cs))) (here refl) (there (here refl)) (RF-valid _)!}\n       -- Phase2 (put-resp x (proj\u2082 (put-resp (hack-challenge x) cs) ))\n       --     ? ? (RF-valid _)\n    -- -}\n\n-- -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.One\nopen import Data.Two\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Nat\nopen import Data.Vec hiding (_>>=_ ; _\u2208_)\nopen import Data.List as L\nopen import Data.Fin as Fin using (Fin)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\nopen import Control.Strategy renaming (map to mapS)\nopen import Game.Challenge\nimport Game.ReceiptFreeness\nimport Game.IND-CCA2-dagger\nimport Game.IND-CPA-utils\n\nimport Data.List.Any\nopen Data.List.Any using (here; there)\nopen Data.List.Any.Membership-\u2261 using (_\u2208_ ; _\u2209_)\n\nmodule Game.Transformation.ReceiptFreeness-CCA2d.Valid\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (#q : \u2115) (max#q : Fin #q)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CipherText \u2192 Message)\n  (Check    : let open Game.ReceiptFreeness PubKey SecKey CipherText SerialNumber R\u2091 R\u2096 R\u2090 #q max#q KeyGen Enc Dec\n               in BB \u2192 Receipt \u2192 \ud835\udfda)\n  (CheckMem : \u2200 bb r \u2192 \u2713 (Check bb r) \u2192 proj\u2081 (proj\u2082 r) \u2209 L.map (proj\u2081 \u2218 proj\u2082) bb)\n  -- (CheckEnc : \u2200 pk m r\u2091 \u2192 Check (Enc pk m r\u2091) \u2261 1\u2082)\n  where\n\n_\u00b2' : \u2605 \u2192 \u2605\nX \u00b2' = X \u00d7 X\n\n\nr-sn : Receipt \u2192 SerialNumber\nr-sn (_ , sn , _) = sn\n\nmodule Simulator-Valid (RFA : RF.Adversary)(RFA-Valid : RF.Valid-Adversary RFA)\n  (WRONG-HYP : \u2200 r r' \u2192 r-sn r \u2261 r-sn r' \u2192 enc-co r \u2261 enc-co r')\n  where\n  valid : CCA2\u2020.Valid-Adversary (Simulator.A\u2020 RFA)\n  valid (r\u2090 , rgb) pk = Phase1 _ (RFA-Valid r\u2090 pk) where\n     open CCA2\u2020.Valid-Adversary (r\u2090 , rgb) pk\n     module RFA = RF.Valid-Adversary r\u2090 pk\n     open Simulator RFA\n     open AdversaryParts rgb pk r\u2090\n\n     -- could refine r more\n     -- {-\n     Phase2 : \u2200 RF {bb i taA taB r} \u2192 r-sn (r 0\u2082) \u2208 L.map r-sn bb \u2192 r-sn (r 1\u2082) \u2208 L.map r-sn bb \u2192 RFA.Phase2-Valid r RF\n            \u2192 Phase2-Valid (proj\u2082 \u2218 proj\u2082 \u2218 r) (mapS proj\u2081 (mitm-to-client-trans (MITM-phase 1\u2082 i bb (taA , taB)) RF))\n     Phase2 (ask REB cont) r0 r1 RF-valid = Phase2 (cont _) r0 r1 (RF-valid _)\n     Phase2 (ask RBB cont) r0 r1 RF-valid = Phase2 (cont _) r0 r1 (RF-valid _)\n     Phase2 (ask RTally cont) r0 r1 RF-valid = Phase2 (cont _) r0 r1 (RF-valid _)\n     Phase2 (ask (RCO x) cont) r0 r1 ((r\u2080 , r\u2081) , RF-valid) = r\u2080 , r\u2081 , (\u03bb r \u2192 Phase2 (cont _) r0 r1 (RF-valid _))\n     Phase2 (ask (Vote x) cont) {bb} r0 r1 RF-valid with Check bb x | CheckMem bb x\n     ... | 0\u2082 | _ = Phase2 (cont _) r0 r1 (RF-valid _)\n     ... | 1\u2082 | not-in-bb = (\u03bb eq \u2192 not-in-bb _ {!subst (\u03bb x \u2192 x \u2208 L.map r-sn bb) eq!})\n                            , {!!},\n                            , \u03bb r \u2192 Phase2 (cont _) (there r0) (there r1) (RF-valid _)\n     Phase2 (done x) r0 r1 RF-valid = RF-valid\n\n     Phase1 : \u2200 RF {sn i bb taA taB} \u2192 RFA.Phase1-Valid sn RF\n            \u2192 Phase1-Valid (mapS A\u20202 (mitm-to-client-trans (MITM-phase 0\u2082 i bb (taA , taB)) RF))\n     Phase1 (ask REB cont) RF-valid = Phase1 _ (RF-valid _)\n     Phase1 (ask RBB cont) RF-valid = Phase1 _ (RF-valid _)\n     Phase1 (ask RTally cont) RF-valid = Phase1 _ (RF-valid _)\n     Phase1 (ask (RCO x) cont) RF-valid r = Phase1 _ (RF-valid _)\n     Phase1 (ask (Vote x) cont) {bb = bb} RF-valid with Check bb x\n     Phase1 (ask (Vote x) cont) RF-valid | 1\u2082 = \u03bb r \u2192 Phase1 _ (RF-valid _)\n     Phase1 (ask (Vote x) cont) RF-valid | 0\u2082 = Phase1 _ (RF-valid _)\n     Phase1 (done x) (sn\u2080\u2209sn , sn\u2081\u2209sn , RF-valid) cs = Phase2 (put-resp x (proj\u2082 (put-resp (hack-challenge x) cs) ))\n            (here refl) (there (here refl)) (RF-valid _)\n    -- -}\n\n-- -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2685982a354df08144518cd09a76727e9556b673","subject":"Hutton's razor: shave it with Conor(tm)","message":"Hutton's razor: shave it with Conor(tm)\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/RevisedHutton.agda","new_file":"models\/RevisedHutton.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ctxt env ty (l value) = con (EZe , value)\ndischarge ctxt env ty (r variable) = con (EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (l (r refl)) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (l (l (r refl)))) ,\n                       (con (EZe , r (l (r refl))) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( r refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\n-- Open term: holes are either values or variables in a context\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = \n    subst (cong (IMu closeTerm) (snd variable)) \n          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"de8917e4d69350f1eb3c42d6c86c64eeb18e3786","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: ecda4978e184c6c467a610bfa8971fb6\n\ndarcs-hash:20120306162449-3bd4e-ca9ba6a7b8b6177d3150a5d3ab44428f7a3496c8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/Type.agda","new_file":"src\/FOTC\/Data\/Stream\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC stream type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Functional for the FOTC Stream type.\n-- StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n-- StreamF P xs = \u2203[ x' ] \u2203[ xs' ] P xs' \u2227 xs \u2261 x' \u2237 xs'\n\n-- Stream is the greatest fixed-point of StreamF (by Stream-gfp\u2081 and\n-- Stream-gfp\u2082).\n\npostulate\n  Stream : D \u2192 Set\n\npostulate\n-- Stream is a post-fixed point of StreamF, i.e.\n--\n-- Stream \u2264 StreamF Stream.\n  Stream-gfp\u2081 : \u2200 {xs} \u2192 Stream xs \u2192\n                \u2203[ x' ] \u2203[ xs' ] Stream xs' \u2227 xs \u2261 x' \u2237 xs'\n{-# ATP axiom Stream-gfp\u2081 #-}\n\n-- Stream is the greatest post-fixed point of StreamF, i.e\n--\n-- \u2200 P. P \u2264 StreamF P \u21d2 P \u2264 Stream.\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  Stream-gfp\u2082 : (P : D \u2192 Set) \u2192\n                -- P is post-fixed point of StreamF.\n                (\u2200 {xs} \u2192 P xs \u2192 \u2203[ x' ] \u2203[ xs' ] P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n                -- Stream is greater than P.\n                \u2200 {xs} \u2192 P xs \u2192 Stream xs\n\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functor StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream.\nStream-gfp\u2083 : \u2200 {xs} \u2192\n              (\u2203[ x' ] \u2203[ xs' ] Stream xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n              Stream xs\nStream-gfp\u2083 h = Stream-gfp\u2082 P helper h\n  where\n  P : D \u2192 Set\n  P ws = \u2203[ w' ] \u2203[ ws' ] Stream ws' \u2227 ws \u2261 w' \u2237 ws'\n\n  helper : \u2200 {xs} \u2192 P xs \u2192 \u2203[ x' ] \u2203[ xs' ] P xs' \u2227 xs \u2261 x' \u2237 xs'\n  helper (_ , _ , Sxs' , xs\u2261x'\u2237xs') = _ , _ , Stream-gfp\u2081 Sxs' , xs\u2261x'\u2237xs'\n","old_contents":"------------------------------------------------------------------------------\n-- The FOTC stream type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Functional for the FOTC Stream type.\n-- StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n-- StreamF P ds = \u2203[ e ] \u2203[ es ] P es \u2227 ds \u2261 e \u2237 es\n\n-- Stream is the greatest fixed-point of StreamF (by Stream-gfp\u2081 and\n-- Stream-gfp\u2082).\n\npostulate\n  Stream : D \u2192 Set\n\npostulate\n-- Stream is a post-fixed point of StreamF, i.e.\n--\n-- Stream \u2264 StreamF Stream.\n  Stream-gfp\u2081 : \u2200 {xs} \u2192 Stream xs \u2192\n                \u2203[ x' ] \u2203[ xs' ] Stream xs' \u2227 xs \u2261 x' \u2237 xs'\n{-# ATP axiom Stream-gfp\u2081 #-}\n\n-- Stream is the greatest post-fixed point of StreamF, i.e\n--\n-- \u2200 P. P \u2264 StreamF P \u21d2 P \u2264 Stream.\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  Stream-gfp\u2082 : (P : D \u2192 Set) \u2192\n                -- P is post-fixed point of StreamF.\n                (\u2200 {xs} \u2192 P xs \u2192 \u2203[ x' ] \u2203[ xs' ] P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n                -- Stream is greater than P.\n                \u2200 {xs} \u2192 P xs \u2192 Stream xs\n\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functor StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream.\nStream-gfp\u2083 : \u2200 {xs} \u2192\n              (\u2203[ x' ] \u2203[ xs' ] Stream xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n              Stream xs\nStream-gfp\u2083 h = Stream-gfp\u2082 P helper h\n  where\n  P : D \u2192 Set\n  P ws = \u2203[ w' ] \u2203[ ws' ] Stream ws' \u2227 ws \u2261 w' \u2237 ws'\n\n  helper : \u2200 {xs} \u2192 P xs \u2192 \u2203[ x' ] \u2203[ xs' ] P xs' \u2227 xs \u2261 x' \u2237 xs'\n  helper (_ , _ , Sxs' , xs\u2261x'\u2237xs') = _ , _ , Stream-gfp\u2081 Sxs' , xs\u2261x'\u2237xs'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"41898751fdcc62758217e161ad60e93a20db6f51","subject":"Monoid: use nest and not fold","message":"Monoid: use nest and not fold\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Monoid.agda","new_file":"lib\/Algebra\/Monoid.agda","new_contents":"open import Function.NP\nopen import Data.Product.NP\nopen import Data.Nat\n  using    (\u2115; zero)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nopen import Data.Integer\n  using    (\u2124; +_; -[1+_]; _\u2296_)\n  renaming (_+_ to _+\u2124_; _*_ to _*\u2124_)\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nopen import Algebra.FunctionProperties.Eq\nopen \u2261-Reasoning\n\nmodule Algebra.Monoid where\n\nrecord Monoid-Ops {\u2113} (M : Set \u2113) : Set \u2113 where\n  constructor mk\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : M \u2192 M \u2192 M\n    \u03b5   : M\n\n  _^\u207a_ : M \u2192 \u2115 \u2192 M\n  x ^\u207a n = nest n (_\u2219_ x) \u03b5\n\n  module FromInverseOp\n         (_\u207b\u00b9   : Op\u2081 M)\n         where\n    infixl 7 _\/_\n\n    _\/_ : M \u2192 M \u2192 M\n    x \/ y = x \u2219 y \u207b\u00b9\n\n    open FromOp\u2082 _\/_ public renaming (op= to \/=)\n\n    _^\u207b_ _^\u207b\u2032_ : M \u2192 \u2115 \u2192 M\n    x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n    x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n    _^_ : M \u2192 \u2124 \u2192 M\n    x ^ -[1+ n ] = x ^\u207b(1+ n)\n    x ^ (+ n)    = x ^\u207a n\n\nrecord Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n\n  -- laws\n  field\n    assoc    : Associative _\u2219_\n    identity : Identity  \u03b5 _\u2219_\n\n  open FromOp\u2082 _\u2219_ public renaming (op= to \u2219=)\n  open FromAssoc _\u2219_ assoc public\n\n  module _ {b} where\n    ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n    ^\u207a0-\u03b5 = idp\n\n    ^\u207a1-id : b ^\u207a 1 \u2261 b\n    ^\u207a1-id = snd identity\n\n    ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n    ^\u207a2-\u2219 = ap (_\u2219_ _) ^\u207a1-id\n\n    ^\u207a-+ : \u2200 m {n} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n    ^\u207a-+ 0      = ! fst identity\n    ^\u207a-+ (1+ m) = ap (_\u2219_ b) (^\u207a-+ m) \u2666 ! assoc\n\n    ^\u207a-* : \u2200 m {n} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n    ^\u207a-* 0 = idp\n    ^\u207a-* (1+ m) {n}\n      = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n        b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n        b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n  comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n  comm-\u03b5 = snd identity \u2666 ! fst identity\n\n  module _ {c x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n    elim-assoc= = assoc \u2666 \u2219= idp e \u2666 snd identity\n\n    elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n    elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 fst identity\n\n  module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n    elim-inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module FromLeftInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-left : LeftCancel _\u2219_\n    cancels-\u2219-left {c} {x} {y} e\n      = x              \u2261\u27e8 ! fst identity \u27e9\n        \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inv-l) idp \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inv-l idp \u27e9\n        \u03b5 \u2219 y          \u2261\u27e8 fst identity \u27e9\n        y \u220e\n\n    inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-r = cancels-\u2219-left (! assoc \u2666 \u2219= inv-l idp \u2666 ! comm-\u03b5)\n\n    \/-\u2219 : \u2200 {x y} \u2192 x \u2261 (x \/ y) \u2219 y\n    \/-\u2219 {x} {y}\n      = x               \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inv-l) \u27e9\n        x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n        (x \/ y) \u2219 y     \u220e\n\n  module FromRightInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-right : RightCancel _\u2219_\n    cancels-\u2219-right {c} {x} {y} e\n      = x              \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inv-r) \u27e9\n        x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n        y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inv-r \u27e9\n        y \u2219 \u03b5          \u2261\u27e8 snd identity \u27e9\n        y \u220e\n\n    inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-l = cancels-\u2219-right (assoc \u2666 \u2219= idp inv-r \u2666 comm-\u03b5)\n\n    module _ {x y} where\n      is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n      is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 fst identity\n\n      is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n      is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 snd identity\n\n      \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n      \u2219-\/\n        = x            \u2261\u27e8 ! snd identity \u27e9\n          x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inv-r) \u27e9\n          x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n          (x \u2219 y) \/ y  \u220e\n\n    module _ {x y} where\n      unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n      unique-\u03b5-left eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          y \/ y        \u2261\u27e8 inv-r \u27e9\n          \u03b5            \u220e\n\n      unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n      unique-\u03b5-right eq\n        = y               \u2261\u27e8 ! is-\u03b5-left inv-l \u27e9\n          x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n          x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n          x \u207b\u00b9 \u2219 x        \u2261\u27e8 inv-l \u27e9\n          \u03b5               \u220e\n\n      unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n      unique-\u207b\u00b9 eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          \u03b5 \/ y        \u2261\u27e8 fst identity \u27e9\n          y \u207b\u00b9         \u220e\n\n    open FromLeftInverse _\u207b\u00b9 inv-l hiding (inv-r)\n\n    \u03b5\u207b\u00b9-\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n    \u03b5\u207b\u00b9-\u03b5 = unique-\u03b5-left inv-l\n\n    involutive : Involutive _\u207b\u00b9\n    involutive {x}\n      = cancels-\u2219-right\n        (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inv-l \u27e9\n         \u03b5              \u2261\u27e8 ! inv-r \u27e9\n         x \u2219 x \u207b\u00b9 \u220e)\n\n    \u207b\u00b9-hom\u2032 : \u2200 {x y} \u2192 (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n    \u207b\u00b9-hom\u2032 {x} {y} = cancels-\u2219-left {x \u2219 y}\n       ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 inv-r \u27e9\n        \u03b5                       \u2261\u27e8 ! inv-r \u27e9\n        x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! fst identity) \u27e9\n        x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! inv-r) idp) \u27e9\n        x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n        x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n        (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n    elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n    elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n      = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n        x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-inner= inv-l \u27e9\n        x \u207b\u00b9 \u2219 y               \u220e\n\n    elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n    elim-\u2219-right-\/ {c} {x} {y}\n      = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n        x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-inner= inv-r \u27e9\n        x \/ y \u220e \n\n    module _ {b} where\n      ^\u207asuc : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n      ^\u207asuc 0 = comm-\u03b5\n      ^\u207asuc (1+ n) = ap (_\u2219_ b) (^\u207asuc n) \u2666 ! assoc\n\n      ^\u207a-comm : \u2200 n \u2192 b \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b\n      ^\u207a-comm = ^\u207asuc\n\n      ^\u207bsuc : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n      ^\u207bsuc n = ap _\u207b\u00b9 (^\u207asuc n) \u2666 \u207b\u00b9-hom\u2032\n\n      ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n      ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9-\u03b5\n      ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                      \u2666 ! \u207b\u00b9-hom\u2032\n                      \u2666 ap _\u207b\u00b9 (! ^\u207asuc n)\n\n      ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n      ^\u207b\u20321-id = snd identity\n\n      ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n      ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n      ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n      ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\nrecord Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-struct : Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Monoid-Struct mon-struct public\n\nrecord Commutative-Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n  field\n    mon-struct : Monoid-Struct mon-ops\n    comm : Commutative _\u2219_\n  open Monoid-Struct mon-struct public\n  open FromAssocComm _\u2219_ assoc comm public\n    hiding (!assoc=; assoc=; inner=)\n\nrecord Commutative-Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-comm   : Commutative-Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Commutative-Monoid-Struct mon-comm public\n  mon : Monoid M\n  mon = record { mon-struct = mon-struct }\n\n-- A renaming of Monoid with additive notation\nmodule Additive-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             )\n\n-- A renaming of Monoid with multiplicative notation\nmodule Multiplicative-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             )\n\nmodule Additive-Commutative-Monoid {M} (mon-comm : Commutative-Monoid M)\n  = Commutative-Monoid mon-comm\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             ; assoc-comm to +-assoc-comm\n             ; interchange to +-interchange\n             ; outer= to +-outer=\n             )\n\nmodule Multiplicative-Commutative-Monoid {M} (mon : Commutative-Monoid M) = Commutative-Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             ; assoc-comm to *-assoc-comm\n             ; interchange to *-interchange\n             ; outer= to *-outer=\n             )\n\nrecord MonoidHomomorphism {A B : Set}\n                          (monA0+ : Monoid A)\n                          (monB1* : Monoid B)\n                          (f : A \u2192 B) : Set where\n  open Additive-Monoid monA0+\n  open Multiplicative-Monoid monB1*\n  field\n    0-hom-1 : f 0\u1d50 \u2261 1\u1d50\n    +-hom-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Function.NP\nopen import Data.Product.NP\nopen import Data.Nat\n  using    (\u2115; fold; zero)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nopen import Data.Integer\n  using    (\u2124; +_; -[1+_]; _\u2296_)\n  renaming (_+_ to _+\u2124_; _*_ to _*\u2124_)\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nopen import Algebra.FunctionProperties.Eq\nopen \u2261-Reasoning\n\nmodule Algebra.Monoid where\n\nrecord Monoid-Ops {\u2113} (M : Set \u2113) : Set \u2113 where\n  constructor mk\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : M \u2192 M \u2192 M\n    \u03b5   : M\n\n  _^\u207a_ : M \u2192 \u2115 \u2192 M\n  x ^\u207a n = fold \u03b5 (_\u2219_ x) n\n\n  module FromInverseOp\n         (_\u207b\u00b9   : Op\u2081 M)\n         where\n    infixl 7 _\/_\n\n    _\/_ : M \u2192 M \u2192 M\n    x \/ y = x \u2219 y \u207b\u00b9\n\n    open FromOp\u2082 _\/_ public renaming (op= to \/=)\n\n    _^\u207b_ _^\u207b\u2032_ : M \u2192 \u2115 \u2192 M\n    x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n    x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n    _^_ : M \u2192 \u2124 \u2192 M\n    x ^ -[1+ n ] = x ^\u207b(1+ n)\n    x ^ (+ n)    = x ^\u207a n\n\nrecord Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n\n  -- laws\n  field\n    assoc    : Associative _\u2219_\n    identity : Identity  \u03b5 _\u2219_\n\n  open FromOp\u2082 _\u2219_ public renaming (op= to \u2219=)\n  open FromAssoc _\u2219_ assoc public\n\n  module _ {b} where\n    ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n    ^\u207a0-\u03b5 = idp\n\n    ^\u207a1-id : b ^\u207a 1 \u2261 b\n    ^\u207a1-id = snd identity\n\n    ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n    ^\u207a2-\u2219 = ap (_\u2219_ _) ^\u207a1-id\n\n    ^\u207a-+ : \u2200 m {n} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n    ^\u207a-+ 0      = ! fst identity\n    ^\u207a-+ (1+ m) = ap (_\u2219_ b) (^\u207a-+ m) \u2666 ! assoc\n\n    ^\u207a-* : \u2200 m {n} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n    ^\u207a-* 0 = idp\n    ^\u207a-* (1+ m) {n}\n      = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n        b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n        b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n  comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n  comm-\u03b5 = snd identity \u2666 ! fst identity\n\n  module _ {c x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n    elim-assoc= = assoc \u2666 \u2219= idp e \u2666 snd identity\n\n    elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n    elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 fst identity\n\n  module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n    elim-inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module FromLeftInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-left : LeftCancel _\u2219_\n    cancels-\u2219-left {c} {x} {y} e\n      = x              \u2261\u27e8 ! fst identity \u27e9\n        \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inv-l) idp \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inv-l idp \u27e9\n        \u03b5 \u2219 y          \u2261\u27e8 fst identity \u27e9\n        y \u220e\n\n    inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-r = cancels-\u2219-left (! assoc \u2666 \u2219= inv-l idp \u2666 ! comm-\u03b5)\n\n    \/-\u2219 : \u2200 {x y} \u2192 x \u2261 (x \/ y) \u2219 y\n    \/-\u2219 {x} {y}\n      = x               \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inv-l) \u27e9\n        x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n        (x \/ y) \u2219 y     \u220e\n\n  module FromRightInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-right : RightCancel _\u2219_\n    cancels-\u2219-right {c} {x} {y} e\n      = x              \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inv-r) \u27e9\n        x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n        y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inv-r \u27e9\n        y \u2219 \u03b5          \u2261\u27e8 snd identity \u27e9\n        y \u220e\n\n    inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-l = cancels-\u2219-right (assoc \u2666 \u2219= idp inv-r \u2666 comm-\u03b5)\n\n    module _ {x y} where\n      is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n      is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 fst identity\n\n      is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n      is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 snd identity\n\n      \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n      \u2219-\/\n        = x            \u2261\u27e8 ! snd identity \u27e9\n          x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inv-r) \u27e9\n          x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n          (x \u2219 y) \/ y  \u220e\n\n    module _ {x y} where\n      unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n      unique-\u03b5-left eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          y \/ y        \u2261\u27e8 inv-r \u27e9\n          \u03b5            \u220e\n\n      unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n      unique-\u03b5-right eq\n        = y               \u2261\u27e8 ! is-\u03b5-left inv-l \u27e9\n          x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n          x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n          x \u207b\u00b9 \u2219 x        \u2261\u27e8 inv-l \u27e9\n          \u03b5               \u220e\n\n      unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n      unique-\u207b\u00b9 eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          \u03b5 \/ y        \u2261\u27e8 fst identity \u27e9\n          y \u207b\u00b9         \u220e\n\n    open FromLeftInverse _\u207b\u00b9 inv-l hiding (inv-r)\n\n    \u03b5\u207b\u00b9-\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n    \u03b5\u207b\u00b9-\u03b5 = unique-\u03b5-left inv-l\n\n    involutive : Involutive _\u207b\u00b9\n    involutive {x}\n      = cancels-\u2219-right\n        (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inv-l \u27e9\n         \u03b5              \u2261\u27e8 ! inv-r \u27e9\n         x \u2219 x \u207b\u00b9 \u220e)\n\n    \u207b\u00b9-hom\u2032 : \u2200 {x y} \u2192 (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n    \u207b\u00b9-hom\u2032 {x} {y} = cancels-\u2219-left {x \u2219 y}\n       ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 inv-r \u27e9\n        \u03b5                       \u2261\u27e8 ! inv-r \u27e9\n        x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! fst identity) \u27e9\n        x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! inv-r) idp) \u27e9\n        x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n        x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n        (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n    elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n    elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n      = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n        x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-inner= inv-l \u27e9\n        x \u207b\u00b9 \u2219 y               \u220e\n\n    elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n    elim-\u2219-right-\/ {c} {x} {y}\n      = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n        x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-inner= inv-r \u27e9\n        x \/ y \u220e \n\n    module _ {b} where\n      ^\u207asuc : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n      ^\u207asuc 0 = comm-\u03b5\n      ^\u207asuc (1+ n) = ap (_\u2219_ b) (^\u207asuc n) \u2666 ! assoc\n\n      ^\u207a-comm : \u2200 n \u2192 b \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b\n      ^\u207a-comm = ^\u207asuc\n\n      ^\u207bsuc : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n      ^\u207bsuc n = ap _\u207b\u00b9 (^\u207asuc n) \u2666 \u207b\u00b9-hom\u2032\n\n      ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n      ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9-\u03b5\n      ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                      \u2666 ! \u207b\u00b9-hom\u2032\n                      \u2666 ap _\u207b\u00b9 (! ^\u207asuc n)\n\n      ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n      ^\u207b\u20321-id = snd identity\n\n      ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n      ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n      ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n      ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\nrecord Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-struct : Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Monoid-Struct mon-struct public\n\nrecord Commutative-Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n  field\n    mon-struct : Monoid-Struct mon-ops\n    comm : Commutative _\u2219_\n  open Monoid-Struct mon-struct public\n  open FromAssocComm _\u2219_ assoc comm public\n    hiding (!assoc=; assoc=; inner=)\n\nrecord Commutative-Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-comm   : Commutative-Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Commutative-Monoid-Struct mon-comm public\n  mon : Monoid M\n  mon = record { mon-struct = mon-struct }\n\n-- A renaming of Monoid with additive notation\nmodule Additive-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             )\n\n-- A renaming of Monoid with multiplicative notation\nmodule Multiplicative-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             )\n\nmodule Additive-Commutative-Monoid {M} (mon-comm : Commutative-Monoid M)\n  = Commutative-Monoid mon-comm\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             ; assoc-comm to +-assoc-comm\n             ; interchange to +-interchange\n             ; outer= to +-outer=\n             )\n\nmodule Multiplicative-Commutative-Monoid {M} (mon : Commutative-Monoid M) = Commutative-Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             ; assoc-comm to *-assoc-comm\n             ; interchange to *-interchange\n             ; outer= to *-outer=\n             )\n\nrecord MonoidHomomorphism {A B : Set}\n                          (monA0+ : Monoid A)\n                          (monB1* : Monoid B)\n                          (f : A \u2192 B) : Set where\n  open Additive-Monoid monA0+\n  open Multiplicative-Monoid monB1*\n  field\n    0-hom-1 : f 0\u1d50 \u2261 1\u1d50\n    +-hom-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"66e4306a6d9260a51adfb53ab8e82a87b0e8ff9c","subject":"Prove that nil changes are derivatives.","message":"Prove that nil changes are derivatives.\n\nThe point of these simple definitions is to rephrase correct in more\nhigh-level terms. In the definition of correct, we inline update and\ndiff for function changes to avoid mutual recursion.\n\nOld-commit-hash: a2aff54dae7f67a166745a2df2acd581140bc7b9\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function changes\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n","old_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function changes\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"058d6bf14a3aa315b9ba2d681bb621847c3aa089","subject":"Simplified some proofs.","message":"Simplified some proofs.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/Properties.agda","new_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.Properties where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Data.Nat.Inequalities.EliminationProperties\nopen import LTC-PCF.Data.Nat.Inequalities.ConversionRules public\nopen import LTC-PCF.Data.Nat.Properties\nopen import LTC-PCF.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- N.B. The elimination properties are in the module\n-- LTC.Data.Nat.Inequalities.EliminationProperties.\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nltLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 lt m o \u2261 lt n o\nltLeftCong refl = refl\n\nltRightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 lt m n \u2261 lt m o\nltRightCong refl = refl\n\n------------------------------------------------------------------------------\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 zero \u226f n\n0\u226fx nzero          = lt-00\n0\u226fx (nsucc {n} Nn) = lt-S0 n\n\nx\u226ex : \u2200 {n} \u2192 N n \u2192 n \u226e n\nx\u226ex nzero          = lt-00\nx\u226ex (nsucc {n} Nn) = trans (lt-SS n n) (x\u226ex Nn)\n\nSx\u22700 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2270 zero\nSx\u22700 nzero          = x\u226ex (nsucc nzero)\nSx\u22700 (nsucc {n} Nn) = trans (lt-SS (succ\u2081 n) zero) (lt-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 n < succ\u2081 n\nx<Sx nzero          = lt-0S zero\nx<Sx (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x<Sx Nn)\n\nx<y\u2192Sx<Sy : \u2200 {m n} \u2192 m < n \u2192 succ\u2081 m < succ\u2081 n\nx<y\u2192Sx<Sy {m} {n} m<n = trans (lt-SS m n) m<n\n\nSx<Sy\u2192x<y : \u2200 {m n} \u2192 succ\u2081 m < succ\u2081 n \u2192 m < n\nSx<Sy\u2192x<y {m} {n} Sm<Sn = trans (sym (lt-SS m n)) Sm<Sn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 n \u2264 n\nx\u2264x nzero          = lt-0S zero\nx\u2264x (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 m \u2264 n \u2192 succ\u2081 m \u2264 succ\u2081 n\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (lt-SS m (succ\u2081 n)) m\u2264n\n\nSx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 succ\u2081 m \u2264 succ\u2081 n \u2192 m \u2264 n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 m \u226e n\nx\u2265y\u2192x\u226ey nzero          nzero          _     = x\u226ex nzero\nx\u2265y\u2192x\u226ey nzero          (nsucc Nn)     0\u2265Sn  = \u22a5-elim (0\u2265S\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) nzero          _     = lt-S0 m\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn =\n  trans (lt-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym (lt-SS n (succ\u2081 m))) Sm\u2265Sn))\n\nx\u226ey\u2192x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226e n \u2192 m \u2265 n\nx\u226ey\u2192x\u2265y nzero nzero 0\u226e0 = x\u2264x nzero\nx\u226ey\u2192x\u2265y nzero (nsucc {n} Nn) 0\u226eSn = \u22a5-elim (t\u2262f (trans (sym (lt-0S n)) 0\u226eSn))\nx\u226ey\u2192x\u2265y (nsucc {m} Nm) nzero Sm\u226en = lt-0S (succ\u2081 m)\nx\u226ey\u2192x\u2265y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226eSn =\n  trans (lt-SS n (succ\u2081 m)) (x\u226ey\u2192x\u2265y Nm Nn (trans (sym (lt-SS m n)) Sm\u226eSn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u2264 n\nx>y\u2228x\u2264y {n = n} nzero Nn = inj\u2082 (lt-0S n)\nx>y\u2228x\u2264y (nsucc {m} Nm) nzero = inj\u2081 (lt-0S m)\nx>y\u2228x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb m>n \u2192 inj\u2081 (trans (lt-SS n m) m>n))\n       (\u03bb m\u2264n \u2192 inj\u2082 (trans (lt-SS m (succ\u2081 n)) m\u2264n))\n       (x>y\u2228x\u2264y Nm Nn)\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m \u226f n\nx\u2264y\u2192x\u226fy nzero          Nn             _    = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (nsucc Nm)     nzero          Sm\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn =\n  trans (lt-SS n m) (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn))\n\nx\u226fy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226f n \u2192 m \u2264 n\nx\u226fy\u2192x\u2264y {n = n} nzero Nn _ = lt-0S n\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) nzero Sm\u226f0 = \u22a5-elim (t\u2262f (trans (sym (lt-0S m)) Sm\u226f0))\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226fSn =\n  trans (lt-SS m (succ\u2081 n)) (x\u226fy\u2192x\u2264y Nm Nn (trans (sym (lt-SS n m)) Sm\u226fSn))\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u226f n\nx>y\u2228x\u226fy nzero          Nn             = inj\u2082 (0\u226fx Nn)\nx>y\u2228x\u226fy (nsucc {m} Nm) nzero          = inj\u2081 (lt-0S m)\nx>y\u2228x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb h \u2192 inj\u2081 (trans (lt-SS n m) h))\n       (\u03bb h \u2192 inj\u2082 (trans (lt-SS n m) h))\n       (x>y\u2228x\u226fy Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u2265 n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx<y\u2228x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u226e n\nx<y\u2228x\u226ey Nm Nn = case (\u03bb m<n \u2192 inj\u2081 m<n)\n                     (\u03bb m\u2265n \u2192 inj\u2082 (x\u2265y\u2192x\u226ey Nm Nn m\u2265n))\n                     (x<y\u2228x\u2265y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 m \u2264 n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nn\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m \u2264 n\nx<y\u2192x\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x\u2264y nzero          (nsucc {n} Nn) _     = lt-0S (succ\u2081 n)\nx<y\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m \u2264 n\nx<Sy\u2192x\u2264y {n = n} nzero      Nn 0<Sn  = lt-0S n\nx<Sy\u2192x\u2264y         (nsucc Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 m \u2264 succ\u2081 m\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (nsucc Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 succ\u2081 m \u2264 n\nx<y\u2192Sx\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192Sx\u2264y nzero          (nsucc {n} Nn) _     = x\u2264y\u2192Sx\u2264Sy (lt-0S n)\nx<y\u2192Sx\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  trans (lt-SS (succ\u2081 m) (succ\u2081 n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2264 n \u2192 m < n\nSx\u2264y\u2192x<y Nm             nzero          Sm\u22640   = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nSx\u2264y\u2192x<y nzero          (nsucc {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (nsucc {m} Nm) (nsucc {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m < n \u2192 n < o \u2192 m < o\n<-trans nzero          nzero           _             0<0   _    = \u22a5-elim (0<0\u2192\u22a5 0<0)\n<-trans nzero          (nsucc Nn)     nzero          _     Sn<0 = \u22a5-elim (S<0\u2192\u22a5 Sn<0)\n<-trans nzero          (nsucc Nn)     (nsucc {o} No) _     _    = lt-0S o\n<-trans (nsucc Nm)     Nn             nzero          _     n<0  = \u22a5-elim (x<0\u2192\u22a5 Nn n<0)\n<-trans (nsucc Nm)     nzero          (nsucc No)     Sm<0  _    = \u22a5-elim (S<0\u2192\u22a5 Sm<0)\n<-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy (<-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So))\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2264 n \u2192 n \u2264 o \u2192 m \u2264 o\n\u2264-trans {o = o} nzero Nn No _ _ = lt-0S o\n\u2264-trans (nsucc Nm) nzero No Sm\u22640 _ = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\n\u2264-trans (nsucc Nm) (nsucc Nn) nzero _ Sn\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nn Sn\u22640)\n\u2264-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 m + n\nx\u2264x+y {n = n} nzero Nn = lt-0S (zero + n)\nx\u2264x+y {n = n} (nsucc {m} Nm) Nn =\n  lt (succ\u2081 m) (succ\u2081 (succ\u2081 m + n)) \u2261\u27e8 lt-SS m (succ\u2081 m + n) \u27e9\n  lt m (succ\u2081 m + n)                 \u2261\u27e8 ltRightCong (+-Sx m n) \u27e9\n  lt m (succ\u2081 (m + n))               \u2261\u27e8 refl \u27e9\n  le m (m + n)                       \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n  true                               \u220e\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2238 n < succ\u2081 m\nx\u2238y<Sx {m} Nm nzero =\n  lt (m \u2238 zero) (succ\u2081 m) \u2261\u27e8 ltLeftCong (\u2238-x0 m) \u27e9\n  lt m (succ\u2081 m)          \u2261\u27e8 x<Sx Nm \u27e9\n  true                    \u220e\n\nx\u2238y<Sx nzero (nsucc {n} Nn) =\n  lt( zero \u2238 succ\u2081 n) [1] \u2261\u27e8 ltLeftCong (0\u2238x (nsucc Nn)) \u27e9\n  lt zero [1]             \u2261\u27e8 lt-0S zero \u27e9\n  true                    \u220e\n\nx\u2238y<Sx (nsucc {m} Nm) (nsucc {n} Nn) =\n  lt (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 (succ\u2081 m))\n    \u2261\u27e8 ltLeftCong (S\u2238S Nm Nn) \u27e9\n  lt (m \u2238 n) (succ\u2081 (succ\u2081 m))\n     \u2261\u27e8 <-trans (\u2238-N Nm Nn) (nsucc Nm) (nsucc (nsucc Nm))\n                (x\u2238y<Sx Nm Nn) (x<Sx (nsucc Nm))\n     \u27e9\n  true \u220e\n\nSx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n < succ\u2081 m\nSx\u2238Sy<Sx {m} {n} Nm Nn =\n  lt (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 m) \u2261\u27e8 ltLeftCong (S\u2238S Nm Nn) \u27e9\n  lt (m \u2238 n) (succ\u2081 m)             \u2261\u27e8 x\u2238y<Sx Nm Nn \u27e9\n  true                             \u220e\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x nzero Nn 0>n = \u22a5-elim (0>x\u2192\u22a5 Nn 0>n)\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) nzero Sm>0 =\n  trans (+-rightIdentity (\u2238-N (nsucc Nm) nzero)) (\u2238-x0 (succ\u2081 m))\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn =\n  (succ\u2081 m \u2238 succ\u2081 n) + succ\u2081 n\n    \u2261\u27e8 +-leftCong (S\u2238S Nm Nn)  \u27e9\n  (m \u2238 n) + succ\u2081 n\n     \u2261\u27e8 +-comm (\u2238-N Nm Nn) (nsucc Nn) \u27e9\n  succ\u2081 n + (m \u2238 n)\n    \u2261\u27e8 +-Sx n (m \u2238 n) \u27e9\n  succ\u2081 (n + (m \u2238 n))\n    \u2261\u27e8 succCong (+-comm Nn (\u2238-N Nm Nn)) \u27e9\n  succ\u2081 ((m \u2238 n) + n)\n    \u2261\u27e8 succCong (x>y\u2192x\u2238y+y\u2261x Nm Nn (trans (sym (lt-SS n m)) Sm>Sn)) \u27e9\n  succ\u2081 m \u220e\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} nzero Nn 0\u2264n =\n  trans (+-rightIdentity (\u2238-N Nn nzero)) (\u2238-x0 n)\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc Nm) nzero Sm\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn =\n  (succ\u2081 n \u2238 succ\u2081 m) + succ\u2081 m\n    \u2261\u27e8 +-leftCong (S\u2238S Nn Nm) \u27e9\n  (n \u2238 m) + succ\u2081 m\n     \u2261\u27e8 +-comm (\u2238-N Nn Nm) (nsucc Nm) \u27e9\n  succ\u2081 m + (n \u2238 m)\n    \u2261\u27e8 +-Sx m (n \u2238 m) \u27e9\n  succ\u2081 (m + (n \u2238 m))\n    \u2261\u27e8 succCong (+-comm Nm (\u2238-N Nn Nm)) \u27e9\n  succ\u2081 ((n \u2238 m) + m)\n    \u2261\u27e8 succCong  (x\u2264y\u2192y\u2238x+x\u2261y Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn)) \u27e9\n  succ\u2081 n \u220e\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m < succ\u2081 n\nx<y\u2192x<Sy Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x<Sy nzero          (nsucc {n} Nn) 0<Sn  = lt-0S (succ\u2081 n)\nx<y\u2192x<Sy (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m < n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y nzero nzero 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y nzero (nsucc {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) nzero Sm<S0 =\n  \u22a5-elim (x<0\u2192\u22a5 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm<SSn =\n  case (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n       (\u03bb m\u2261n \u2192 inj\u2082 (succCong m\u2261n))\n       m<n\u2228m\u2261n\n  where\n  m<n\u2228m\u2261n : m < n \u2228 m \u2261 n\n  m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m < n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\nx<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 m < n \u2192 n \u2261 o \u2192 m < o\nx<y\u2192y\u2261z\u2192x<z m<n refl = m<n\n\nx\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 n < o \u2192 m < o\nx\u2261y\u2192y<z\u2192x<z refl n<o = n<o\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 n > zero \u2192 m \u2238 n < m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm             nzero          _     0>0  = \u22a5-elim (x>x\u2192\u22a5 nzero 0>0)\nx\u2265y\u2192y>0\u2192x\u2238y<x nzero          (nsucc Nn)     0\u2265Sn  _    = \u22a5-elim (S\u22640\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192y>0\u2192x\u2238y<x (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn Sn>0 =\n  lt (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 m)\n    \u2261\u27e8 ltLeftCong (S\u2238S Nm Nn) \u27e9\n  lt (m \u2238 n) (succ\u2081 m)\n     \u2261\u27e8 x\u2238y<Sx Nm Nn \u27e9\n  true \u220e\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\nxy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (Lexi m n zero zero)\nxy<00\u2192\u22a5 Nm Nn mn<00 = case (\u03bb m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n                           (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n                           mn<00\n\n0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 \u00ac (Lexi zero (succ\u2081 m) zero zero)\n0Sx<00\u2192\u22a5 0Sm<00 = case 0<0\u2192\u22a5\n                       (\u03bb 0\u22610\u2227Sm<0 \u2192 S<0\u2192\u22a5 (\u2227-proj\u2082 0\u22610\u2227Sm<0))\n                       0Sm<00\n\nSxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 \u00ac (Lexi (succ\u2081 m) n\u2081 zero n\u2082)\nSxy\u2081<0y\u2082\u2192\u22a5 Smn\u2081<0n\u2082 =\n  case S<0\u2192\u22a5\n       (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2262S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n       Smn\u2081<0n\u2082\n\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n} \u2192 N n \u2192 \u2200 {m\u2082} \u2192 Lexi m\u2081 n m\u2082 zero \u2192 m\u2081 < m\u2082\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 Nn m\u2081n<m\u20820 =\n  case (\u03bb m\u2081<n\u2081 \u2192 m\u2081<n\u2081)\n       (\u03bb m\u2081\u2261n\u2081\u2227n<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nn (\u2227-proj\u2082 m\u2081\u2261n\u2081\u2227n<0)))\n       m\u2081n<m\u20820\n\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m} \u2192 N m \u2192 \u2200 {n\u2081 n\u2082} \u2192 Lexi m n\u2081 zero n\u2082 \u2192\n                    m \u2261 zero \u2227 n\u2081 < n\u2082\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 Nm mn\u2081<0n\u2082 = case (\u03bb m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n                                    (\u03bb m\u22610\u2227n\u2081<n\u2082 \u2192 m\u22610\u2227n\u2081<n\u2082)\n                                    mn\u2081<0n\u2082\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) (succ\u2081 m) (succ\u2081 n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] Nm Nn = inj\u2081 (Sx\u2238Sy<Sx Nm Nn)\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m) (succ\u2081 n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] Nm Nn = inj\u2082 (refl , Sx\u2238Sy<Sx Nn Nm)\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.Properties where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Data.Nat.Inequalities.EliminationProperties\nopen import LTC-PCF.Data.Nat.Inequalities.ConversionRules public\nopen import LTC-PCF.Data.Nat.Properties\nopen import LTC-PCF.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- N.B. The elimination properties are in the module\n-- LTC.Data.Nat.Inequalities.EliminationProperties.\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nltLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 lt m o \u2261 lt n o\nltLeftCong refl = refl\n\nltRightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 lt m n \u2261 lt m o\nltRightCong refl = refl\n\n------------------------------------------------------------------------------\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 zero \u226f n\n0\u226fx nzero          = lt-00\n0\u226fx (nsucc {n} Nn) = lt-S0 n\n\nx\u226ex : \u2200 {n} \u2192 N n \u2192 n \u226e n\nx\u226ex nzero          = lt-00\nx\u226ex (nsucc {n} Nn) = trans (lt-SS n n) (x\u226ex Nn)\n\nSx\u22700 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2270 zero\nSx\u22700 nzero          = x\u226ex (nsucc nzero)\nSx\u22700 (nsucc {n} Nn) = trans (lt-SS (succ\u2081 n) zero) (lt-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 n < succ\u2081 n\nx<Sx nzero          = lt-0S zero\nx<Sx (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x<Sx Nn)\n\nx<y\u2192Sx<Sy : \u2200 {m n} \u2192 m < n \u2192 succ\u2081 m < succ\u2081 n\nx<y\u2192Sx<Sy {m} {n} m<n = trans (lt-SS m n) m<n\n\nSx<Sy\u2192x<y : \u2200 {m n} \u2192 succ\u2081 m < succ\u2081 n \u2192 m < n\nSx<Sy\u2192x<y {m} {n} Sm<Sn = trans (sym (lt-SS m n)) Sm<Sn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 n \u2264 n\nx\u2264x nzero          = lt-0S zero\nx\u2264x (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 m \u2264 n \u2192 succ\u2081 m \u2264 succ\u2081 n\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (lt-SS m (succ\u2081 n)) m\u2264n\n\nSx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 succ\u2081 m \u2264 succ\u2081 n \u2192 m \u2264 n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 m \u226e n\nx\u2265y\u2192x\u226ey nzero          nzero          _     = x\u226ex nzero\nx\u2265y\u2192x\u226ey nzero          (nsucc Nn)     0\u2265Sn  = \u22a5-elim (0\u2265S\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) nzero          _     = lt-S0 m\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn =\n  trans (lt-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym (lt-SS n (succ\u2081 m))) Sm\u2265Sn))\n\nx\u226ey\u2192x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226e n \u2192 m \u2265 n\nx\u226ey\u2192x\u2265y nzero nzero 0\u226e0 = x\u2264x nzero\nx\u226ey\u2192x\u2265y nzero (nsucc {n} Nn) 0\u226eSn = \u22a5-elim (t\u2262f (trans (sym (lt-0S n)) 0\u226eSn))\nx\u226ey\u2192x\u2265y (nsucc {m} Nm) nzero Sm\u226en = lt-0S (succ\u2081 m)\nx\u226ey\u2192x\u2265y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226eSn =\n  trans (lt-SS n (succ\u2081 m)) (x\u226ey\u2192x\u2265y Nm Nn (trans (sym (lt-SS m n)) Sm\u226eSn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u2264 n\nx>y\u2228x\u2264y {n = n} nzero Nn = inj\u2082 (lt-0S n)\nx>y\u2228x\u2264y (nsucc {m} Nm) nzero = inj\u2081 (lt-0S m)\nx>y\u2228x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb m>n \u2192 inj\u2081 (trans (lt-SS n m) m>n))\n       (\u03bb m\u2264n \u2192 inj\u2082 (trans (lt-SS m (succ\u2081 n)) m\u2264n))\n       (x>y\u2228x\u2264y Nm Nn)\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m \u226f n\nx\u2264y\u2192x\u226fy nzero          Nn             _    = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (nsucc Nm)     nzero          Sm\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn =\n  trans (lt-SS n m) (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn))\n\nx\u226fy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226f n \u2192 m \u2264 n\nx\u226fy\u2192x\u2264y {n = n} nzero Nn _ = lt-0S n\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) nzero Sm\u226f0 = \u22a5-elim (t\u2262f (trans (sym (lt-0S m)) Sm\u226f0))\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226fSn =\n  trans (lt-SS m (succ\u2081 n)) (x\u226fy\u2192x\u2264y Nm Nn (trans (sym (lt-SS n m)) Sm\u226fSn))\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u226f n\nx>y\u2228x\u226fy nzero          Nn             = inj\u2082 (0\u226fx Nn)\nx>y\u2228x\u226fy (nsucc {m} Nm) nzero          = inj\u2081 (lt-0S m)\nx>y\u2228x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb h \u2192 inj\u2081 (trans (lt-SS n m) h))\n       (\u03bb h \u2192 inj\u2082 (trans (lt-SS n m) h))\n       (x>y\u2228x\u226fy Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u2265 n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx<y\u2228x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u226e n\nx<y\u2228x\u226ey Nm Nn = case (\u03bb m<n \u2192 inj\u2081 m<n)\n                     (\u03bb m\u2265n \u2192 inj\u2082 (x\u2265y\u2192x\u226ey Nm Nn m\u2265n))\n                     (x<y\u2228x\u2265y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 m \u2264 n\nx\u2261y\u2192x\u2264y {n = n} Nm Nn m\u2261n = subst (\u03bb m' \u2192 m' \u2264 n) (sym m\u2261n) (x\u2264x Nn)\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m \u2264 n\nx<y\u2192x\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x\u2264y nzero          (nsucc {n} Nn) _     = lt-0S (succ\u2081 n)\nx<y\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m \u2264 n\nx<Sy\u2192x\u2264y {n = n} nzero      Nn 0<Sn  = lt-0S n\nx<Sy\u2192x\u2264y         (nsucc Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 m \u2264 succ\u2081 m\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (nsucc Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 succ\u2081 m \u2264 n\nx<y\u2192Sx\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192Sx\u2264y nzero          (nsucc {n} Nn) _     = x\u2264y\u2192Sx\u2264Sy (lt-0S n)\nx<y\u2192Sx\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  trans (lt-SS (succ\u2081 m) (succ\u2081 n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2264 n \u2192 m < n\nSx\u2264y\u2192x<y Nm             nzero          Sm\u22640   = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nSx\u2264y\u2192x<y nzero          (nsucc {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (nsucc {m} Nm) (nsucc {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m < n \u2192 n < o \u2192 m < o\n<-trans nzero          nzero           _             0<0   _    = \u22a5-elim (0<0\u2192\u22a5 0<0)\n<-trans nzero          (nsucc Nn)     nzero          _     Sn<0 = \u22a5-elim (S<0\u2192\u22a5 Sn<0)\n<-trans nzero          (nsucc Nn)     (nsucc {o} No) _     _    = lt-0S o\n<-trans (nsucc Nm)     Nn             nzero          _     n<0  = \u22a5-elim (x<0\u2192\u22a5 Nn n<0)\n<-trans (nsucc Nm)     nzero          (nsucc No)     Sm<0  _    = \u22a5-elim (S<0\u2192\u22a5 Sm<0)\n<-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy (<-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So))\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2264 n \u2192 n \u2264 o \u2192 m \u2264 o\n\u2264-trans {o = o} nzero Nn No _ _ = lt-0S o\n\u2264-trans (nsucc Nm) nzero No Sm\u22640 _ = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\n\u2264-trans (nsucc Nm) (nsucc Nn) nzero _ Sn\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nn Sn\u22640)\n\u2264-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 m + n\nx\u2264x+y {n = n} nzero Nn = lt-0S (zero + n)\nx\u2264x+y {n = n} (nsucc {m} Nm) Nn =\n  lt (succ\u2081 m) (succ\u2081 (succ\u2081 m + n)) \u2261\u27e8 lt-SS m (succ\u2081 m + n) \u27e9\n  lt m (succ\u2081 m + n)                 \u2261\u27e8 ltRightCong (+-Sx m n) \u27e9\n  lt m (succ\u2081 (m + n))               \u2261\u27e8 refl \u27e9\n  le m (m + n)                       \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n  true                               \u220e\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2238 n < succ\u2081 m\nx\u2238y<Sx {m} Nm nzero =\n  lt (m \u2238 zero) (succ\u2081 m) \u2261\u27e8 ltLeftCong (\u2238-x0 m) \u27e9\n  lt m (succ\u2081 m)          \u2261\u27e8 x<Sx Nm \u27e9\n  true                    \u220e\n\nx\u2238y<Sx nzero (nsucc {n} Nn) =\n  lt( zero \u2238 succ\u2081 n) [1] \u2261\u27e8 ltLeftCong (0\u2238x (nsucc Nn)) \u27e9\n  lt zero [1]             \u2261\u27e8 lt-0S zero \u27e9\n  true                    \u220e\n\nx\u2238y<Sx (nsucc {m} Nm) (nsucc {n} Nn) =\n  lt (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 (succ\u2081 m))\n    \u2261\u27e8 ltLeftCong (S\u2238S Nm Nn) \u27e9\n  lt (m \u2238 n) (succ\u2081 (succ\u2081 m))\n     \u2261\u27e8 <-trans (\u2238-N Nm Nn) (nsucc Nm) (nsucc (nsucc Nm))\n                (x\u2238y<Sx Nm Nn) (x<Sx (nsucc Nm))\n     \u27e9\n  true \u220e\n\nSx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n < succ\u2081 m\nSx\u2238Sy<Sx {m} {n} Nm Nn =\n  lt (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 m) \u2261\u27e8 ltLeftCong (S\u2238S Nm Nn) \u27e9\n  lt (m \u2238 n) (succ\u2081 m)             \u2261\u27e8 x\u2238y<Sx Nm Nn \u27e9\n  true                             \u220e\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x nzero Nn 0>n = \u22a5-elim (0>x\u2192\u22a5 Nn 0>n)\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) nzero Sm>0 =\n  trans (+-rightIdentity (\u2238-N (nsucc Nm) nzero)) (\u2238-x0 (succ\u2081 m))\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn =\n  (succ\u2081 m \u2238 succ\u2081 n) + succ\u2081 n\n    \u2261\u27e8 +-leftCong (S\u2238S Nm Nn)  \u27e9\n  (m \u2238 n) + succ\u2081 n\n     \u2261\u27e8 +-comm (\u2238-N Nm Nn) (nsucc Nn) \u27e9\n  succ\u2081 n + (m \u2238 n)\n    \u2261\u27e8 +-Sx n (m \u2238 n) \u27e9\n  succ\u2081 (n + (m \u2238 n))\n    \u2261\u27e8 succCong (+-comm Nn (\u2238-N Nm Nn)) \u27e9\n  succ\u2081 ((m \u2238 n) + n)\n    \u2261\u27e8 succCong (x>y\u2192x\u2238y+y\u2261x Nm Nn (trans (sym (lt-SS n m)) Sm>Sn)) \u27e9\n  succ\u2081 m \u220e\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} nzero Nn 0\u2264n =\n  trans (+-rightIdentity (\u2238-N Nn nzero)) (\u2238-x0 n)\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc Nm) nzero Sm\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn =\n  (succ\u2081 n \u2238 succ\u2081 m) + succ\u2081 m\n    \u2261\u27e8 +-leftCong (S\u2238S Nn Nm) \u27e9\n  (n \u2238 m) + succ\u2081 m\n     \u2261\u27e8 +-comm (\u2238-N Nn Nm) (nsucc Nm) \u27e9\n  succ\u2081 m + (n \u2238 m)\n    \u2261\u27e8 +-Sx m (n \u2238 m) \u27e9\n  succ\u2081 (m + (n \u2238 m))\n    \u2261\u27e8 succCong (+-comm Nm (\u2238-N Nn Nm)) \u27e9\n  succ\u2081 ((n \u2238 m) + m)\n    \u2261\u27e8 succCong  (x\u2264y\u2192y\u2238x+x\u2261y Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn)) \u27e9\n  succ\u2081 n \u220e\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m < succ\u2081 n\nx<y\u2192x<Sy Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x<Sy nzero          (nsucc {n} Nn) 0<Sn  = lt-0S (succ\u2081 n)\nx<y\u2192x<Sy (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m < n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y nzero nzero 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y nzero (nsucc {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) nzero Sm<S0 =\n  \u22a5-elim (x<0\u2192\u22a5 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm<SSn =\n  case (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n       (\u03bb m\u2261n \u2192 inj\u2082 (succCong m\u2261n))\n       m<n\u2228m\u2261n\n  where\n  m<n\u2228m\u2261n : m < n \u2228 m \u2261 n\n  m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m < n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\nx<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 m < n \u2192 n \u2261 o \u2192 m < o\nx<y\u2192y\u2261z\u2192x<z {m} m<n n\u2261o = subst (\u03bb n' \u2192 m < n') n\u2261o m<n\n\nx\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 n < o \u2192 m < o\nx\u2261y\u2192y<z\u2192x<z {o = o} m\u2261n n<o = subst (\u03bb m' \u2192 m' < o) (sym m\u2261n) n<o\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 n > zero \u2192 m \u2238 n < m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm             nzero          _     0>0  = \u22a5-elim (x>x\u2192\u22a5 nzero 0>0)\nx\u2265y\u2192y>0\u2192x\u2238y<x nzero          (nsucc Nn)     0\u2265Sn  _    = \u22a5-elim (S\u22640\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192y>0\u2192x\u2238y<x (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn Sn>0 =\n  lt (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 m)\n    \u2261\u27e8 ltLeftCong (S\u2238S Nm Nn) \u27e9\n  lt (m \u2238 n) (succ\u2081 m)\n     \u2261\u27e8 x\u2238y<Sx Nm Nn \u27e9\n  true \u220e\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\nxy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (Lexi m n zero zero)\nxy<00\u2192\u22a5 Nm Nn mn<00 = case (\u03bb m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n                           (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n                           mn<00\n\n0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 \u00ac (Lexi zero (succ\u2081 m) zero zero)\n0Sx<00\u2192\u22a5 0Sm<00 = case 0<0\u2192\u22a5\n                       (\u03bb 0\u22610\u2227Sm<0 \u2192 S<0\u2192\u22a5 (\u2227-proj\u2082 0\u22610\u2227Sm<0))\n                       0Sm<00\n\nSxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 \u00ac (Lexi (succ\u2081 m) n\u2081 zero n\u2082)\nSxy\u2081<0y\u2082\u2192\u22a5 Smn\u2081<0n\u2082 =\n  case S<0\u2192\u22a5\n       (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2262S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n       Smn\u2081<0n\u2082\n\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n} \u2192 N n \u2192 \u2200 {m\u2082} \u2192 Lexi m\u2081 n m\u2082 zero \u2192 m\u2081 < m\u2082\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 Nn m\u2081n<m\u20820 =\n  case (\u03bb m\u2081<n\u2081 \u2192 m\u2081<n\u2081)\n       (\u03bb m\u2081\u2261n\u2081\u2227n<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nn (\u2227-proj\u2082 m\u2081\u2261n\u2081\u2227n<0)))\n       m\u2081n<m\u20820\n\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m} \u2192 N m \u2192 \u2200 {n\u2081 n\u2082} \u2192 Lexi m n\u2081 zero n\u2082 \u2192\n                    m \u2261 zero \u2227 n\u2081 < n\u2082\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 Nm mn\u2081<0n\u2082 = case (\u03bb m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n                                    (\u03bb m\u22610\u2227n\u2081<n\u2082 \u2192 m\u22610\u2227n\u2081<n\u2082)\n                                    mn\u2081<0n\u2082\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) (succ\u2081 m) (succ\u2081 n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] Nm Nn = inj\u2081 (Sx\u2238Sy<Sx Nm Nn)\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m) (succ\u2081 n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] Nm Nn = inj\u2082 (refl , Sx\u2238Sy<Sx Nn Nm)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7826abee6b67c7b55aa70968ca7fd6b637d30857","subject":"Drop Prefixes","message":"Drop Prefixes\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Syntax\/Context.agda","new_file":"Base\/Syntax\/Context.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts and subcontexts,\n-- together with variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- Typing Contexts\n-- ===============\n\nimport Data.List as List\nopen import Base.Data.ContextList public\n\nContext : Set\nContext = List.List Type\n\n-- Variables\n-- =========\n--\n-- Here it is clear that we are using de Bruijn indices,\n-- encoded as natural numbers, more or less.\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- Weakening\n-- =========\n--\n-- We define weakening based on subcontext relationship.\n\n-- Subcontexts\n-- -----------\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\n-- This handling of contexts is recommended by [this\n-- email](https:\/\/lists.chalmers.se\/pipermail\/agda\/2011\/003423.html) and\n-- attributed to Conor McBride.\n--\n-- The associated thread discusses a few alternatives solutions, including one\n-- where beta-reduction can handle associativity of ++.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation.\n\nopen Subcontexts public\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- Typing Contexts\n-- ===============\n\nimport Data.List as List\nopen import Base.Data.ContextList public\n\nContext : Set\nContext = List.List Type\n\n-- Variables\n-- =========\n--\n-- Here it is clear that we are using de Bruijn indices,\n-- encoded as natural numbers, more or less.\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- Weakening\n-- =========\n--\n-- We provide two approaches for weakening: context prefixes and\n-- and subcontext relationship. In this formalization, we mostly\n-- (maybe exclusively?) use the latter.\n\n-- Context Prefixes\n-- ----------------\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  _\u22ce_ = List._++_\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Subcontexts\n-- -----------\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\n-- This handling of contexts is recommended by [this\n-- email](https:\/\/lists.chalmers.se\/pipermail\/agda\/2011\/003423.html) and\n-- attributed to Conor McBride.\n--\n-- The associated thread discusses a few alternatives solutions, including one\n-- where beta-reduction can handle associativity of ++.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"316fec2f0021d774510b2f899a6e6a4c328d2241","subject":"Working on ConatSL.agda.","message":"Working on ConatSL.agda.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.ConatSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf cozero          = inj\u2081 refl\nConat-unf (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h {n} An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n'  \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\npostulate\n  inf    : D\n  inf-eq : inf \u2261 succ\u2081 inf\n{-# ATP axiom inf-eq #-}\n\n{-# NO_TERMINATION_CHECK #-}\ninf-Conat : Conat inf\ninf-Conat = subst Conat (sym inf-eq) (cosucc (\u266f inf-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.ConatSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')\nConat-unf cozero          = inj\u2081 refl\nConat-unf (cosucc {n} Cn) = inj\u2082 (n , \u266d Cn , refl)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n') \u2192\n           Conat n\nConat-in (inj\u2081 h) = subst Conat (sym h) cozero\nConat-in (inj\u2082 (n , Cn , h)) = subst Conat (sym h) (cosucc (\u266f Cn))\n\nConat-coind : \u2200 (A : D \u2192 Set) {n} \u2192\n              (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')) \u2192\n              A n \u2192 Conat n\nConat-coind A h An = {!!}\n\npostulate\n  inf    : D\n  inf-eq : inf \u2261 succ\u2081 inf\n{-# ATP axiom inf-eq #-}\n\n{-# NO_TERMINATION_CHECK #-}\ninf-Conat : Conat inf\ninf-Conat = subst Conat (sym inf-eq) (cosucc (\u266f inf-Conat))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d4d74883fdfb5330e27bb48434b046f8291b5f3d","subject":"Improve documentation of \u0394Val.","message":"Improve documentation of \u0394Val.\n\nOld-commit-hash: fb71a0ea6af97c62b01a9d8b7d3089e5cd3139ed\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Change\/Popl14.agda","new_file":"Denotation\/Change\/Popl14.agda","new_contents":"module Denotation.Change.Popl14 where\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Popl14.Denotation.Value\nopen import Theorem.Groups-Popl14\nopen import Postulate.Extensionality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\n\n---------------\n-- Interface --\n---------------\n\n\u0394Val : Type \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\ninfixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n-- apply \u0394v v ::= v \u229e \u0394v\n_\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n--   diff u v ::= u \u229f v\n_\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n\n-- Lemma diff-is-valid\nR[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f v)\n\n-- Lemma apply-diff\nv+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let\n    _+_ = _\u229e_ {\u03c4}\n    _-_ = _\u229f_ {\u03c4}\n  in\n  v + (u - v) \u2261 u\n\n--------------------\n-- Implementation --\n--------------------\n\n-- (\u0394Val \u03c4) is the set of changes of type \u03c4. This set is\n-- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n-- particular, (\u0394Val (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n-- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n-- changes.\n--\n-- \u0394Val \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n-- Detailed motivation:\n--\n-- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n--\n-- \u0394Val : Type \u2192 Set\n\u0394Val (base base-int) = \u2124\n\u0394Val (base base-bag) = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229e_ {base base-int} n \u0394n = n + \u0394n\n_\u229e_ {base base-bag} b \u0394b = b ++ \u0394b\n_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192 f v \u229e \u0394f v (v \u229f v) (R[v,u-v] {\u03c3})\n\n-- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229f_ {base base-int} m n = m - n\n_\u229f_ {base base-bag} d b = d \\\\ b\n_\u229f_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e \u0394v) \u229f f v\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {base base-int} n \u0394n = \u22a4\nvalid {base base-bag} b \u0394b = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f \u0394f =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : \u0394Val \u03c3) (R[v,\u0394v] : valid v \u0394v)\n  \u2192 valid (f v) (\u0394f v \u0394v R[v,\u0394v])\n  -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n  \u00d7 (_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n\n-- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\nv+[u-v]=u {base base-int} {u} {v} = n+[m-n]=m {v} {u}\nv+[u-v]=u {base base-bag} {u} {v} = a++[b\\\\a]=b {v} {u}\nv+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n  let\n    _+_ = _\u229e_ {\u03c3 \u21d2 \u03c4}\n    _-_ = _\u229f_ {\u03c3 \u21d2 \u03c4}\n    _+\u2080_ = _\u229e_ {\u03c3}\n    _-\u2080_ = _\u229f_ {\u03c3}\n    _+\u2081_ = _\u229e_ {\u03c4}\n    _-\u2081_ = _\u229f_ {\u03c4}\n  in\n    ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n      begin\n        (v + (u - v)) w\n      \u2261\u27e8 refl \u27e9\n        v w +\u2081 (u (w +\u2080 (w -\u2080 w)) -\u2081 v w)\n      \u2261\u27e8 cong (\u03bb hole \u2192 v w +\u2081 (u hole -\u2081 v w)) (v+[u-v]=u {\u03c3}) \u27e9\n        v w +\u2081 (u w -\u2081 v w)\n      \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n        u w\n      \u220e) where\n      open \u2261-Reasoning\n\nR[v,u-v] {base base-int} {u} {v} = tt\nR[v,u-v] {base base-bag} {u} {v} = tt\nR[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n  let\n    w\u2032 = w \u229e \u0394w\n    _+_ = _\u229e_ {\u03c3 \u21d2 \u03c4}\n    _-_ = _\u229f_ {\u03c3 \u21d2 \u03c4}\n    _+\u2081_ = _\u229e_ {\u03c4}\n    _-\u2081_ = _\u229f_ {\u03c4}\n  in\n    R[v,u-v] {\u03c4}\n    ,\n    (begin\n      (v + (u - v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n      v w +\u2081 (u - v) w \u0394w R[w,\u0394w]\n    \u220e) where open \u2261-Reasoning\n\n-- `diff` and `apply`, without validity proofs\ninfixl 6 _\u27e6\u2295\u27e7_ _\u27e6\u229d\u27e7_\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n_\u27e6\u2295\u27e7_ {base base-int}  n \u0394n = n +  \u0394n\n_\u27e6\u2295\u27e7_ {base base-bag}  b \u0394b = b ++ \u0394b\n_\u27e6\u2295\u27e7_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192\n  let\n    _-\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n    _+\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  in\n    f v +\u2081 \u0394f v (v -\u2080 v)\n\n_\u27e6\u229d\u27e7_ {base base-int}  m n = m -  n\n_\u27e6\u229d\u27e7_ {base base-bag}  a b = a \\\\ b\n_\u27e6\u229d\u27e7_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v \u2192 _\u27e6\u229d\u27e7_ {\u03c4} (g (_\u27e6\u2295\u27e7_ {\u03c3} v \u0394v)) (f v)\n","old_contents":"module Denotation.Change.Popl14 where\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Popl14.Denotation.Value\nopen import Theorem.Groups-Popl14\nopen import Postulate.Extensionality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\n\n---------------\n-- Interface --\n---------------\n\n\u0394Val : Type \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\ninfixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n-- apply \u0394v v ::= v \u229e \u0394v\n_\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n--   diff u v ::= u \u229f v\n_\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n\n-- Lemma diff-is-valid\nR[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f v)\n\n-- Lemma apply-diff\nv+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let\n    _+_ = _\u229e_ {\u03c4}\n    _-_ = _\u229f_ {\u03c4}\n  in\n  v + (u - v) \u2261 u\n\n--------------------\n-- Implementation --\n--------------------\n\n-- \u0394Val \u03c4 is intended to be the semantic domain for changes of values\n-- of type \u03c4.\n--\n-- \u0394Val \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n-- Detailed motivation:\n--\n-- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n--\n-- \u0394Val : Type \u2192 Set\n\u0394Val (base base-int) = \u2124\n\u0394Val (base base-bag) = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229e_ {base base-int} n \u0394n = n + \u0394n\n_\u229e_ {base base-bag} b \u0394b = b ++ \u0394b\n_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192 f v \u229e \u0394f v (v \u229f v) (R[v,u-v] {\u03c3})\n\n-- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229f_ {base base-int} m n = m - n\n_\u229f_ {base base-bag} d b = d \\\\ b\n_\u229f_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e \u0394v) \u229f f v\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {base base-int} n \u0394n = \u22a4\nvalid {base base-bag} b \u0394b = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f \u0394f =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : \u0394Val \u03c3) (R[v,\u0394v] : valid v \u0394v)\n  \u2192 valid (f v) (\u0394f v \u0394v R[v,\u0394v])\n  -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n  \u00d7 (_\u229e_ {\u03c3 \u21d2 \u03c4} f \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n\n-- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\nv+[u-v]=u {base base-int} {u} {v} = n+[m-n]=m {v} {u}\nv+[u-v]=u {base base-bag} {u} {v} = a++[b\\\\a]=b {v} {u}\nv+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n  let\n    _+_ = _\u229e_ {\u03c3 \u21d2 \u03c4}\n    _-_ = _\u229f_ {\u03c3 \u21d2 \u03c4}\n    _+\u2080_ = _\u229e_ {\u03c3}\n    _-\u2080_ = _\u229f_ {\u03c3}\n    _+\u2081_ = _\u229e_ {\u03c4}\n    _-\u2081_ = _\u229f_ {\u03c4}\n  in\n    ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n      begin\n        (v + (u - v)) w\n      \u2261\u27e8 refl \u27e9\n        v w +\u2081 (u (w +\u2080 (w -\u2080 w)) -\u2081 v w)\n      \u2261\u27e8 cong (\u03bb hole \u2192 v w +\u2081 (u hole -\u2081 v w)) (v+[u-v]=u {\u03c3}) \u27e9\n        v w +\u2081 (u w -\u2081 v w)\n      \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n        u w\n      \u220e) where\n      open \u2261-Reasoning\n\nR[v,u-v] {base base-int} {u} {v} = tt\nR[v,u-v] {base base-bag} {u} {v} = tt\nR[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n  let\n    w\u2032 = w \u229e \u0394w\n    _+_ = _\u229e_ {\u03c3 \u21d2 \u03c4}\n    _-_ = _\u229f_ {\u03c3 \u21d2 \u03c4}\n    _+\u2081_ = _\u229e_ {\u03c4}\n    _-\u2081_ = _\u229f_ {\u03c4}\n  in\n    R[v,u-v] {\u03c4}\n    ,\n    (begin\n      (v + (u - v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n      v w +\u2081 (u - v) w \u0394w R[w,\u0394w]\n    \u220e) where open \u2261-Reasoning\n\n-- `diff` and `apply`, without validity proofs\ninfixl 6 _\u27e6\u2295\u27e7_ _\u27e6\u229d\u27e7_\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n_\u27e6\u2295\u27e7_ {base base-int}  n \u0394n = n +  \u0394n\n_\u27e6\u2295\u27e7_ {base base-bag}  b \u0394b = b ++ \u0394b\n_\u27e6\u2295\u27e7_ {\u03c3 \u21d2 \u03c4} f \u0394f = \u03bb v \u2192\n  let\n    _-\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n    _+\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  in\n    f v +\u2081 \u0394f v (v -\u2080 v)\n\n_\u27e6\u229d\u27e7_ {base base-int}  m n = m -  n\n_\u27e6\u229d\u27e7_ {base base-bag}  a b = a \\\\ b\n_\u27e6\u229d\u27e7_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v \u0394v \u2192 _\u27e6\u229d\u27e7_ {\u03c4} (g (_\u27e6\u2295\u27e7_ {\u03c3} v \u0394v)) (f v)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"903d0a73b1e1de747f98b06521826197786929a6","subject":"Agda: weakening for Plotkin-style terms","message":"Agda: weakening for Plotkin-style terms\n\nOld-commit-hash: e2005584bd69f4ae0b7dac1a633a1a447dd64307\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"module Syntax.Term.Plotkin where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen import Syntax.Type.Plotkin\nopen import Syntax.Context\n\ndata Term\n  {B : Set {- of base types -}}\n  {C : Set {- of constants -}}\n  {type-of : C \u2192 Type B}\n  (\u0393 : Context {Type B}) :\n  (\u03c4 : Type B) \u2192 Set\n  where\n  const : (c : C) \u2192 Term \u0393 (type-of c)\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term {B} {C} {type-of} \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term {B} {C} {type-of} \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term {B} {C} {type-of} (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {B C \u22a2 \u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term {B} {C} {\u22a2} \u0393\u2081 \u03c4 \u2192\n  Term {B} {C} {\u22a2} \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {B C \u22a2 \u0393 \u03c3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n","old_contents":"module Syntax.Term.Plotkin where\n\n-- Terms of languages described in Plotkin style\n\nopen import Syntax.Type.Plotkin\nopen import Syntax.Context\n\ndata Term\n  {B : Set {- of base types -}}\n  {C : Set {- of constants -}}\n  {type-of : C \u2192 Type B}\n  (\u0393 : Context {Type B}) :\n  (\u03c4 : Type B) \u2192 Set\n  where\n  const : (c : C) \u2192 Term \u0393 (type-of c)\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term {B} {C} {type-of} \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term {B} {C} {type-of} \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term {B} {C} {type-of} (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a17155f4ec639d1e46a267df227e5a5c2b969a8a","subject":"Little tactic to take `n` steps of evaluation.","message":"Little tactic to take `n` steps of evaluation.\n","repos":"spire\/spire","old_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\nAn alternative to the Arith codes is to create a *view* of the\nnatural numbers along with a collection of functions over them.\n-}\n\ndata ArithView : \u2115 \u2192 Set where\n  `nat : (n : \u2115) \u2192 ArithView n\n  _`+_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 + n\u2082)\n  _`*_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nHere is the ArithView specialized into the Fin type.\n-}\n\ndata Fin\u2083 : \u2115 \u2192 Set where\n  fin : (n : \u2115) (i : Fin n) \u2192 Fin\u2083 n\n  fin+ : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 + n\u2082)) \u2192 Fin\u2083 (n\u2081 + n\u2082)\n  fin* : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 * n\u2082)) \u2192 Fin\u2083 (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `ArithView `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `Fin\u2083 : (n : \u2115) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `ArithView n \u27e7 = ArithView n\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `Fin\u2083 n \u27e7 = Fin\u2083 n\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin\u2082 : Tactic\nrm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin\u2082 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin\u2082 : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin\u2082 a (fin i) = proj\u2082\n  (rm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic that simplifies an ArithView index.\nTo target Fin, ideally you want to match on something like:\n`\u03a3 `\u2115 (\u03bb n \u2192 `ArithView n `\u00d7 Fin n)\nBut this requires matching on a \u03bb, and a non-linear occurrence\nof n.\n\nThe example after this one accomplishes the same thing via the\nspecialized Fin\u2083 type instead.\n-}\n\nrm-plus0-ArithView : Tactic\nrm-plus0-ArithView (`ArithView .(n + 0) , (n `+ 0)) =\n  `ArithView n , `nat n \nrm-plus0-ArithView x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify to simplify the \"view\"-based Fin type Fin\u2083.\nThis is most similar to rm-plus0-Fin\u2082, except Fin\u2083 is view-based\nwhile Fin\u2082 is universe-based (where the universe is the codes of\npossible operations).\n\nOn one hand this Fin\u2083 approach is simpler than Fin\u2082, because you do\nnot need to write an eval function or tell it to reduce definitional\nequalities explicitly (like you would to get Arith to compute).\n\nOn the other hand, the Fin\u2082 approach is more powerful because it\nallows you to control when things reduce. This is more like Coq\ntactics, where you might not want something to reduce and you use\n\"simpl\" explicitly when you do. The \"simpl\" tactic is defined below\nvia the eval\u2082 function.\n-}\n\nrm-plus0-Fin\u2083 : Tactic\nrm-plus0-Fin\u2083 (`Fin\u2083 .(n + 0) , fin+ n 0 i) =\n  `Fin\u2083 n\n  ,\n  fin n (subst Fin (sym (plusrident n)) i)\nrm-plus0-Fin\u2083 x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\nsimp-step-n-plus0-Fin : \u2115 \u2192 Tactic\nsimp-step-n-plus0-Fin zero x = x\nsimp-step-n-plus0-Fin (suc n) x =\n  simp-step-n-plus0-Fin n (simp-step-plus0-Fin x)\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","old_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\nAn alternative to the Arith codes is to create a *view* of the\nnatural numbers along with a collection of functions over them.\n-}\n\ndata ArithView : \u2115 \u2192 Set where\n  `nat : (n : \u2115) \u2192 ArithView n\n  _`+_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 + n\u2082)\n  _`*_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nHere is the ArithView specialized into the Fin type.\n-}\n\ndata Fin\u2083 : \u2115 \u2192 Set where\n  fin : (n : \u2115) (i : Fin n) \u2192 Fin\u2083 n\n  fin+ : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 + n\u2082)) \u2192 Fin\u2083 (n\u2081 + n\u2082)\n  fin* : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 * n\u2082)) \u2192 Fin\u2083 (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `ArithView `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `Fin\u2083 : (n : \u2115) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `ArithView n \u27e7 = ArithView n\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `Fin\u2083 n \u27e7 = Fin\u2083 n\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin\u2082 : Tactic\nrm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin\u2082 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin\u2082 : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin\u2082 a (fin i) = proj\u2082\n  (rm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic that simplifies an ArithView index.\nTo target Fin, ideally you want to match on something like:\n`\u03a3 `\u2115 (\u03bb n \u2192 `ArithView n `\u00d7 Fin n)\nBut this requires matching on a \u03bb, and a non-linear occurrence\nof n.\n\nThe example after this one accomplishes the same thing via the\nspecialized Fin\u2083 type instead.\n-}\n\nrm-plus0-ArithView : Tactic\nrm-plus0-ArithView (`ArithView .(n + 0) , (n `+ 0)) =\n  `ArithView n , `nat n \nrm-plus0-ArithView x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify to simplify the \"view\"-based Fin type Fin\u2083.\nThis is most similar to rm-plus0-Fin\u2082, except Fin\u2083 is view-based\nwhile Fin\u2082 is universe-based (where the universe is the codes of\npossible operations).\n\nOn one hand this Fin\u2083 approach is simpler than Fin\u2082, because you do\nnot need to write an eval function or tell it to reduce definitional\nequalities explicitly (like you would to get Arith to compute).\n\nOn the other hand, the Fin\u2082 approach is more powerful because it\nallows you to control when things reduce. This is more like Coq\ntactics, where you might not want something to reduce and you use\n\"simpl\" explicitly when you do. The \"simpl\" tactic is defined below\nvia the eval\u2082 function.\n-}\n\nrm-plus0-Fin\u2083 : Tactic\nrm-plus0-Fin\u2083 (`Fin\u2083 .(n + 0) , fin+ n 0 i) =\n  `Fin\u2083 n\n  ,\n  fin n (subst Fin (sym (plusrident n)) i)\nrm-plus0-Fin\u2083 x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"46c906bc94e121c0380141190eebe006016dd5c5","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 2030b2f08b32be7107650d70f647ccbb\n\ndarcs-hash:20110519155111-3bd4e-273a84eb0d1491f18ef33a7d80f755f6d022f1f9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/Collatz\/PropertiesI.agda","new_file":"src\/FOTC\/Program\/Collatz\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.Collatz.PropertiesI where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityI\n\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesI\nopen import FOTC.Program.Collatz.EquationsI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 \u2203 (\u03bb k \u2192 N k \u2227 n \u2261 two ^ k) \u2192 collatz n \u2261 one\ncollatz-2^x zN _ = collatz-0\ncollatz-2^x (sN {n} Nn) (.zero , zN , Sn\u22612^0) =\n  subst (\u03bb t \u2192 collatz t \u2261 one)\n        (cong succ (sym (Sx\u22612^0\u2192x\u22610 Nn Sn\u22612^0)))\n        collatz-1\ncollatz-2^x (sN {n} Nn) (.(succ k) , sN {k} Nk , Sn\u22612^k+1) =\n  begin\n    collatz (succ n)\n      \u2261\u27e8 cong collatz Sn\u22612^k+1 \u27e9\n    collatz (two ^ (succ k))\n      \u2261\u27e8 cong collatz prf \u27e9\n    collatz (succ (succ ((two ^ (succ k)) \u2238 two)))\n      \u2261\u27e8 collatz-even (x-Even\u2192SSx-Even (\u2238-N (^-N 2-N (sN Nk)) 2-N)\n                      (\u2238-Even (^-N 2-N (sN Nk)) 2-N (2^[x+1]-Even Nk)\n                              (x-Even\u2192SSx-Even zN even-0)))\n      \u27e9\n    collatz ((succ (succ ((two ^ (succ k)) \u2238 two))) \/ two)\n      \u2261\u27e8 cong collatz\n              (subst (\u03bb t \u2192 succ (succ (two ^ succ k \u2238 two)) \/ two \u2261 t \/ two)\n                     (sym prf)\n                     refl)\n              \u27e9\n    collatz ((two ^ (succ k)) \/ two)\n      \u2261\u27e8 cong collatz (2^[x+1]\/2\u22612^x Nk) \u27e9\n    collatz (two ^ k)\n      \u2261\u27e8 collatz-2^x (^-N 2-N Nk) (k , Nk , refl) \u27e9\n    one\n  \u220e\n  where\n    prf : two ^ succ k \u2261 succ (succ (two ^ succ k \u2238 two))\n    prf = (+\u22382 (^-N 2-N (sN Nk)) (2^x\u22600 (sN Nk)) (2^[x+1]\u22601 Nk))\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.Collatz.PropertiesI where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityI\n\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesI\nopen import FOTC.Program.Collatz.EquationsI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 \u2203 (\u03bb k \u2192 N k \u2227 n \u2261 two ^ k) \u2192 collatz n \u2261 one\ncollatz-2^x zN _ = collatz-0\ncollatz-2^x (sN {n} Nn) (.zero , zN , Sn\u22612^0) =\n  subst (\u03bb t \u2192 collatz t \u2261 one)\n        (cong succ (sym (Sx\u22612^0\u2192x\u22610 Nn Sn\u22612^0)))\n        collatz-1\ncollatz-2^x (sN {n} Nn) (.(succ k) , sN {k} Nk , Sn\u22612^k+1) =\n  begin\n    collatz (succ n)\n      \u2261\u27e8 cong collatz Sn\u22612^k+1 \u27e9\n    collatz (two ^ (succ k))\n      \u2261\u27e8 cong collatz prf \u27e9\n    collatz (succ (succ ((two ^ (succ k)) \u2238 two)))\n      \u2261\u27e8 collatz-even (x-Even\u2192SSx-Even (\u2238-N (^-N 2-N (sN Nk)) 2-N)\n                      (\u2238-Even (^-N 2-N (sN Nk)) 2-N (2^[x+1]-Even Nk)\n                              (x-Even\u2192SSx-Even zN even-0)))\n      \u27e9\n    collatz ((succ (succ ((two ^ (succ k)) \u2238 two))) \/ two)\n      \u2261\u27e8 cong collatz\n              (subst (\u03bb t \u2192 succ (succ (succ (succ zero) ^ succ k \u2238 succ (succ zero))) \/\n                            two \u2261\n                            t \/ two)\n                     (sym prf)\n                     refl)\n              \u27e9\n    collatz ((two ^ (succ k)) \/ two)\n      \u2261\u27e8 cong collatz (2^[x+1]\/2\u22612^x Nk) \u27e9\n    collatz (two ^ k)\n      \u2261\u27e8 collatz-2^x (^-N 2-N Nk) (k , Nk , refl) \u27e9\n    one\n  \u220e\n  where\n    prf : two ^ succ k \u2261 succ (succ (two ^ succ k \u2238 two))\n    prf = (+\u22382 (^-N 2-N (sN Nk)) (2^x\u22600 (sN Nk)) (2^[x+1]\u22601 Nk))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"785f4ac2d55acdb149d62e892c64d89106eeb669","subject":"Some progress on Strong-Fiat-Shamir","message":"Some progress on Strong-Fiat-Shamir\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Strong-Fiat-Shamir.agda","new_file":"ZK\/Strong-Fiat-Shamir.agda","new_contents":"open import Type\nopen import Data.Maybe\nopen import Data.Two\nopen import Data.List.NP\nopen import Data.Product.NP\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Strategy\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}{Prf : \u039b \u2192 \u2605}\n  {RP RV RS : \u2605}{Q : \u2605}{Resp :    \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (Q? : Decidable (_\u2261_ {A = Q}))\n  where\n\nRandom-Oracle : \u2605\nRandom-Oracle = Q \u2192 Resp\n\nState : (S A : \u2605) \u2192 \u2605\nState S A = S \u2192 A \u00d7 S\n\ndata Adversary-Query : \u2605 where\n  query-H : (q : Q) \u2192 Adversary-Query\n  query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\nChallenger-Resp : Adversary-Query \u2192 \u2605\nChallenger-Resp (query-H s) = Resp\nChallenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\nTranscript = List (\u03a3 Adversary-Query Challenger-Resp)\n\nrecord Proof-System : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : RV \u2192 (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\nmodule H-def (ro : Random-Oracle)(t : Transcript)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n\nrecord Simulator (PF : Proof-System) : \u2605 where\n  open Proof-System PF\n\n  field\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : (rs : RS)(rv : RV)(t : Transcript)(Y : \u039b) \u2192\n                      let \u03c0 = Simulate rs t Y\n                      in Verify rv Y \u03c0 \u2261 1\u2082\n\nmodule Is-Zero-Knowledge\n  (PF : Proof-System)\n  (sim : Simulator PF)\n  where\n\n  open Simulator sim\n\n  Adversary : \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp \ud835\udfda\n\n  module _ (ro : Random-Oracle) where\n\n    open H-def ro\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H t q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    simulated-Challenger rs t (query-create-proof w Y) | yes p = just (Simulate rs t Y)\n    simulated-Challenger rs t (query-create-proof w Y) | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    Experiment : \ud835\udfda \u2192 RS \u2192 Adversary \u2192 \ud835\udfda \u00d7 Transcript\n    Experiment \u03b2 rs adv = runT (challenger \u03b2 rs) adv []\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {R\u03a3P : \u2605}\n  (PF : Proof-System)\n  where\n  open Proof-System PF\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  (v \u21c4 p) r Y c = \u03a3-Protocol.\u03a3-game (v , p) r Y c\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct (v , p) = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (mk (get-A r Y) c (get-f r Y c)) \u2261 1\u2082\n\n  module Special-Honest-Verifier-Zero-Knowledge where\n\n    record SHVZK (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n      open \u03a3-Protocol \u03a3-proto\n      field\n        Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n        Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n        -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n        -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  module Special-Soundness where\n\n    record SpSo : \u2605 where\n      field\n        Extract : (Y : \u039b)(t\u2080 t\u2081 : \u03a3-Transcript Y) \u2192 W\n        Extract-ok : \u2200 Y A c\u2080 c\u2081 f\u2080 f\u2081 \u2192 (c\u2080 \u2262 c\u2081) \u2192 L (Extract Y (mk A c\u2080 f\u2080) (mk A c\u2081 f\u2081)) Y\n\n  module Simulation-Sound-Extractability\n    (sim : Simulator PF)\n    (open Is-Zero-Knowledge PF sim)\n    {RA : \u2605}\n    (Adv : Adversary)\n    (ro : Random-Oracle)\n    where\n\n    module Initial-run where\n        K-winning-Transcript : Transcript \u2192 \ud835\udfda\n        K-winning-Transcript []            = 1\u2082\n        K-winning-Transcript ((query-H q , r) \u2237 t) = K-winning-Transcript t -- nothing else to check right?\n        K-winning-Transcript ((query-create-proof w Y , nothing) \u2237 t) = ?\n        K-winning-Transcript ((query-create-proof w Y , just \u03c0)  \u2237 t) = ?\n        {-with L? w Y\n        ... | yes p = {!!}\n        ... | no \u00acp = {!!}-}\n\n        game : (w : RA)(\u03b2 : \ud835\udfda)(rs : RS) \u2192 {!!}\n        game w \u03b2 rs = let (\u03b2' , t) = Experiment ro \u03b2 rs Adv in {!!}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Data.Maybe\nopen import Data.Two\nopen import Data.Product.NP\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Strategy\n\nmodule ZK.Strong-Fiat-Shamir\n  {W S : \u2605}{L : W \u2192 S \u2192 \u2605}{Prf : S \u2192 \u2605}\n  {RP RV RS : \u2605}{SimState : \u2605}{Q : \u2605}{Resp :    \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))(L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  where\n\nRandom-Oracle : \u2605\nRandom-Oracle = Q \u2192 Resp\n\nState : (S A : \u2605) \u2192 \u2605\nState S A = S \u2192 A \u00d7 S\n\nrecord Proof-System : \u2605 where\n  field\n    Prove : RP \u2192  (w : W)(Y : S) \u2192 Prf Y\n    Verify : RV \u2192 (Y : S)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\nrecord Simulator (PF : Proof-System) : \u2605 where\n  open Proof-System PF\n  field\n    init : RS \u2192 SimState\n    H : Q \u2192 State SimState Resp\n    Simulate : (Y : S) \u2192 State SimState (Prf Y)\n    verify-sim-spec : (rv : RV)(Y : S)(s : SimState) \u2192 let \u03c0 = proj\u2081 (Simulate Y s)\n                     in Verify rv Y \u03c0 \u2261 1\u2082\n\nmodule Is-Zero-Knowledge\n  (PF : Proof-System)\n  (sim : Simulator PF)\n  where\n\n  open Simulator sim\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : S) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp \ud835\udfda\n\n  simulated-Challenger : \u2200 q \u2192 State SimState (Challenger-Resp q)\n  simulated-Challenger (query-H q) s = H q s\n  simulated-Challenger (query-create-proof w Y) s with L? w Y\n  simulated-Challenger (query-create-proof w Y) s | yes p = first just (Simulate Y s)\n  simulated-Challenger (query-create-proof w Y) s | no \u00acp = nothing , s\n\n  open StatefulRun\n  simulated : SimState \u2192 Adversary \u2192 \ud835\udfda\n  simulated s adv = evalS simulated-Challenger adv s\n\n  module _ (ro : Random-Oracle) where\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    real : Adversary \u2192 \ud835\udfda\n    real adv = run real-Challenger adv\n\n    Experiment : \ud835\udfda \u2192 RS \u2192 Adversary \u2192 \ud835\udfda\n    Experiment 0\u2082 rs = real\n    Experiment 1\u2082 rs = simulated (init rs)\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : S \u2192 \u2605)\n  {R\u03a3P : \u2605}\n  (PF : Proof-System)\n  where\n  open Proof-System PF\n\n  record \u03a3-Prover : \u2605 where\n    field\n      A : R\u03a3P \u2192 (Y : S) \u2192 Commitment\n      f : R\u03a3P \u2192 (Y : S) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : S)(A : Commitment)(c : Challenge)\n             \u2192 \u03a3-Prf Y \u2192 \ud835\udfda\n\n  Correct : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 \u2605\n  Correct v p = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (A r Y) c (f r Y c) \u2261 1\u2082\n\n  module Special-Honest-Verifier-Zero-Knowledge\n    where\n\n    record shvzk : \u2605 where\n\n\nmodule Simulation-Sound-Extractability\n  (PF : Proof-System)\n  (sim : Simulator PF)\n  where\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3713713aaeefaa5279573ace5c7427141c6396c3","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: c9cba6e77a2ca004a4b55da3fc2e1bad\n\ndarcs-hash:20120620015414-3bd4e-a37101bccd1026e4ec2ea1720b6a25dd66be0c33.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/FOL\/VariableNamesClash.agda","new_file":"Test\/Succeed\/FOL\/VariableNamesClash.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test.Succeed.FOL.VariableNamesClash where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  a b : D\n\npostulate a\u2261b : a \u2261 b\n{-# ATP axiom a\u2261b #-}\n\nfoo : (n : D) \u2192 a \u2261 b\nfoo n = prf n\n  where\n  -- The translation of this postulate must use two diferents\n  -- quantified variables names e.g. \u2200 x. \u2200 y. a \u2261 b.\n  postulate prf : (n : D) \u2192 a \u2261 b\n  {-# ATP prove prf #-}\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test.Succeed.FOL.VariableNamesClash where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  a b : D\n\npostulate a\u2261b : a \u2261 b\n{-# ATP axiom a\u2261b #-}\n\nfoo : (n : D) \u2192 a \u2261 b\nfoo n = prf n\n  where\n  -- The translation of this postulate must use two diferents\n  -- quantified variables names e.g. \u2200 x. \u2200 y. x \u2261 y.\n  postulate prf : (n : D) \u2192 a \u2261 b\n  {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f00026118fec0ae9c017486c92bb18c9d151799f","subject":"Upgrade the some cost models","message":"Upgrade the some cost models\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Cost.agda","new_file":"FunUniverse\/Cost.agda","new_contents":"module FunUniverse.Cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nopen import Data.One\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Level.NP hiding (_\u2294_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_; _\u2192\u1d47_)\n\nopen import FunUniverse.Core\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Const\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\n{-\nseqTimeOpsD : FunOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            <[]> = 0#; <\u2237> = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FunOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n-}\n\nTime = \u2115\nTimeCost = constFuns Time\nSpace = \u2115\nSpaceCost = constFuns Space\n\nseqTimeCat : Category {\u2080} {\u2080} {\ud835\udfd9} (\u03bb _ _ \u2192 Time)\nseqTimeCat = 0 , _+_\n\nseqTimeLin : LinRewiring TimeCost\nseqTimeLin =\n  record {\n    cat = seqTimeCat;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nseqTimeRewiring : Rewiring TimeCost\nseqTimeRewiring =\n  record {\n    linRewiring = seqTimeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _+_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\nseqTimeFork : HasFork TimeCost\nseqTimeFork = record { cond = 1; fork = 1+_+_ }\n\nseqTimeOps : FunOps TimeCost\nseqTimeOps = record { rewiring = seqTimeRewiring; hasFork = seqTimeFork;\n                      <0b> = 0; <1b> = 0 }\n\nseqTimeBij : Bijective TimeCost\nseqTimeBij = FunOps.bijective seqTimeOps\n\ntimeCat : Category (\u03bb _ _ \u2192 Time)\ntimeCat = seqTimeCat\n\ntimeLin : LinRewiring TimeCost\ntimeLin =\n  record {\n    cat = timeCat;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _\u2294_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\ntimeRewiring : Rewiring TimeCost\ntimeRewiring =\n  record {\n    linRewiring = timeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _\u2294_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\ntimeFork : HasFork TimeCost\ntimeFork = record { cond = 1; fork = 1+_\u2294_ }\n\ntimeOps : FunOps TimeCost\ntimeOps = record { rewiring = timeRewiring; hasFork = timeFork;\n                   <0b> = 0; <1b> = 0 }\n\n{-\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                rewiring = record seqTimeRewiring {\n                                             linRewiring = record seqTimeLin { <_\u00d7_> = _\u2294_ };\n                                             <_,_> = _\u2294_};\n                                fork = 1+_\u2294_ }\n                                {-;\n                                <\u2237> = 0; uncons = 0 } -- Without <\u2237> = 0... this definition makes\n                                                       -- the FunOps record def yellow\n                                                       -}\ntimeOps\u2261seqTimeOps = \u2261.refl\n-}\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FunOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            <[]> = con 0; <\u2237> = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FunOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  open FunOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u22a4\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec\u22a4 {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u22a4\u2261maximum f [] = \u2261.refl\n  constVec\u22a4\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec\u22a4 f bs) 0\n                                              | constVec\u22a4\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\n                                {-\nspaceLin : LinRewiring SpaceCost\nspaceLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nspaceLin\u2261seqTimeLin : spaceLin \u2261 seqTimeLin\nspaceLin\u2261seqTimeLin = \u2261.refl\n\nspaceRewiring : Rewiring TimeCost\nspaceRewiring =\n  record {\n    linRewiring = spaceLin;\n    tt = 0;\n    dup = 1;\n    <[]> = 0;\n    <_,_> = 1+_+_;\n    fst = 0;\n    snd = 0;\n    rewire = \u03bb {_} {o} _ \u2192 o;\n    rewireTbl = \u03bb {_} {o} _ \u2192 o }\n\nspaceFork : HasFork SpaceCost\nspaceFork = record { cond = 1; fork = 1+_+_ }\n\nspaceOps : FunOps SpaceCost\nspaceOps = record { rewiring = spaceRewiring; hasFork = spaceFork;\n                    <0b> = 1; <1b> = 1 }\n\n             {-\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; <\u2237> = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n-}\n\nmodule SpaceOps where\n  open FunOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u22a4\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec\u22a4 {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u22a4\u2261sum f [] = \u2261.refl\n  constVec\u22a4\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec\u22a4 f bs) 0\n                                          | constVec\u22a4\u2261sum f bs = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f xs rewrite constVec\u22a4\u2261sum f xs | \u2115\u00b0.+-comm (V.sum (V.map f xs)) 0 = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\n{-\ntime\u00d7spaceOps : FunOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-Ops timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FunOps time\u00d7spaceOps\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module FunUniverse.Cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_; _\u2192\u1d47_)\n\nopen import FunUniverse.Core\nopen import FunUniverse.Const\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\n{-\nseqTimeOpsD : FunOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            <[]> = 0#; <\u2237> = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FunOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n-}\n\nTime = \u2115\nTimeCost = constFuns Time\nSpace = \u2115\nSpaceCost = constFuns Space\n\nseqTimeLin : LinRewiring TimeCost\nseqTimeLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nseqTimeRewiring : Rewiring TimeCost\nseqTimeRewiring =\n  record {\n    linRewiring = seqTimeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _+_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\nseqTimeFork : HasFork TimeCost\nseqTimeFork = record { cond = 1; fork = 1+_+_ }\n\nseqTimeOps : FunOps TimeCost\nseqTimeOps = record { rewiring = seqTimeRewiring; hasFork = seqTimeFork;\n                      <0b> = 0; <1b> = 0 }\n\nseqTimeBij : Bijective TimeCost\nseqTimeBij = FunOps.bijective seqTimeOps\n\ntimeLin : LinRewiring TimeCost\ntimeLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _\u2294_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\ntimeRewiring : Rewiring TimeCost\ntimeRewiring =\n  record {\n    linRewiring = timeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _\u2294_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\ntimeFork : HasFork TimeCost\ntimeFork = record { cond = 1; fork = 1+_\u2294_ }\n\ntimeOps : FunOps TimeCost\ntimeOps = record { rewiring = timeRewiring; hasFork = timeFork;\n                   <0b> = 0; <1b> = 0 }\n\n{-\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                rewiring = record seqTimeRewiring {\n                                             linRewiring = record seqTimeLin { <_\u00d7_> = _\u2294_ };\n                                             <_,_> = _\u2294_};\n                                fork = 1+_\u2294_ }\n                                {-;\n                                <\u2237> = 0; uncons = 0 } -- Without <\u2237> = 0... this definition makes\n                                                       -- the FunOps record def yellow\n                                                       -}\ntimeOps\u2261seqTimeOps = \u2261.refl\n-}\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FunOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            <[]> = con 0; <\u2237> = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FunOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  open FunOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u22a4\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec\u22a4 {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u22a4\u2261maximum f [] = \u2261.refl\n  constVec\u22a4\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec\u22a4 f bs) 0\n                                              | constVec\u22a4\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\nspaceLin : LinRewiring SpaceCost\nspaceLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nspaceLin\u2261seqTimeLin : spaceLin \u2261 seqTimeLin\nspaceLin\u2261seqTimeLin = \u2261.refl\n\nspaceRewiring : Rewiring TimeCost\nspaceRewiring =\n  record {\n    linRewiring = spaceLin;\n    tt = 0;\n    dup = 1;\n    <[]> = 0;\n    <_,_> = 1+_+_;\n    fst = 0;\n    snd = 0;\n    rewire = \u03bb {_} {o} _ \u2192 o;\n    rewireTbl = \u03bb {_} {o} _ \u2192 o }\n\nspaceFork : HasFork SpaceCost\nspaceFork = record { cond = 1; fork = 1+_+_ }\n\nspaceOps : FunOps SpaceCost\nspaceOps = record { rewiring = spaceRewiring; hasFork = spaceFork;\n                    <0b> = 1; <1b> = 1 }\n\n             {-\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; <\u2237> = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n-}\n\nmodule SpaceOps where\n  open FunOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u22a4\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec\u22a4 {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u22a4\u2261sum f [] = \u2261.refl\n  constVec\u22a4\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec\u22a4 f bs) 0\n                                          | constVec\u22a4\u2261sum f bs = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f xs rewrite constVec\u22a4\u2261sum f xs | \u2115\u00b0.+-comm (V.sum (V.map f xs)) 0 = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\n{-\ntime\u00d7spaceOps : FunOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-Ops timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FunOps time\u00d7spaceOps\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a0c0e331a5839a6f7cd2921fe50b2acfdf977244","subject":"Resolve implicit arguments.","message":"Resolve implicit arguments.\n\nWhen I filled the last hole in total.agda in the previous commit,\nAgda told me to 'resolve implicit argument', so I added explicit\nimplicit arguments until Agda was satisfied. I don't know whether\nor why it matters to do this.\n\nOld-commit-hash: ed138bd06659ab2caac47a670b7d3d60c4d97f3a\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 {{\u0393\u2032}} (abs t)\n  \u2248\u27e8  \u0394-abs {{\u0393\u2032}} t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  \u2261\u27e8\u27e9\n     derive-term {{\u0393\u2032}} (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app {{\u0393\u2032}} t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083) =\n  begin\n    \u0394 (if t\u2081 t\u2082 t\u2083)\n  \u2248\u27e8 \u0394-if {{\u0393\u2032}} t\u2081 t\u2082 t\u2083 \u27e9\n    if (\u0394 t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (\u0394 {{\u0393\u2032}} t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (\u0394 {{\u0393\u2032}} t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (\u0394 {{\u0393\u2032}} t\u2082)\n           (\u0394 {{\u0393\u2032}} t\u2083))\n  \u2248\u27e8 \u2248-if (derive-term-correct {{\u0393\u2032}} t\u2081)\n          (\u2248-if (\u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct {{\u0393\u2032}} t\u2083) \u2248-refl) \u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct {{\u0393\u2032}} t\u2082) \u2248-refl) \u2248-refl))\n          (\u2248-if (\u2248-refl)\n                (derive-term-correct {{\u0393\u2032}} t\u2082)\n                (derive-term-correct {{\u0393\u2032}} t\u2083)) \u27e9\n    if (derive-term {{\u0393\u2032}} t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (derive-term {{\u0393\u2032}} t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (derive-term {{\u0393\u2032}} t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (derive-term {{\u0393\u2032}} t\u2082)\n           (derive-term {{\u0393\u2032}} t\u2083))\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083)\n  \u220e where open \u2248-Reasoning\n\nderive-term-correct {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u2248-\u0394 {{\u0393\u2032}} (derive-term-correct {{\u0393\u2033}} t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083) =\n  begin\n    \u0394 (if t\u2081 t\u2082 t\u2083)\n  \u2248\u27e8 \u0394-if {{\u0393\u2032}} t\u2081 t\u2082 t\u2083 \u27e9\n    if (\u0394 t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (\u0394 t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (\u0394 t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (\u0394 t\u2082)\n           (\u0394 t\u2083))\n  \u2248\u27e8 \u2248-if (derive-term-correct t\u2081)\n          (\u2248-if (\u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct t\u2083) \u2248-refl) \u2248-refl)\n                (\u2248-diff-term (\u2248-apply-term (derive-term-correct t\u2082) \u2248-refl) \u2248-refl))\n          (\u2248-if (\u2248-refl)\n                (derive-term-correct t\u2082)\n                (derive-term-correct t\u2083)) \u27e9\n    if (derive-term t\u2081)\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (diff-term (apply-term (derive-term t\u2083) (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n           (diff-term (apply-term (derive-term t\u2082) (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083)))\n       (if (lift-term {{\u0393\u2032}} t\u2081)\n           (derive-term t\u2082)\n           (derive-term t\u2083))\n  \u2261\u27e8\u27e9\n    derive-term (if t\u2081 t\u2082 t\u2083)\n  \u220e where open \u2248-Reasoning\n\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"37dbe4e367fef7b52ae7dff6fe572cc48ada1378","subject":"Define step-indexed logical relation we want (almost)","message":"Define step-indexed logical relation we want (almost)\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStepSILR2.agda","new_file":"Thesis\/BigStepSILR2.agda","new_contents":"module Thesis.BigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\nopen import Thesis.FunBigStepSILR2\n\n-- Standard relational big-step semantics, with step-indexes matching a small-step semantics.\n-- Protip: doing this on ANF terms would be much easier.\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u2115 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  app : \u2200 n1 n2 n3 {\u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193[ n1 ] closure t\u2032 \u03c1\u2032 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193[ n2 ] v\u2082 \u2192\n    (v\u2082 \u2022 \u03c1\u2032) \u22a2 t\u2032 \u2193[ n3 ] v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193[ suc n1 + n2 + n3 ] v\u2032\n  var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lit : \u2200 n \u2192\n    \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- Silly lemmas for eval-dec-sound\nmodule _ where\n  Done-inj : \u2200 {\u03c4} m n {v1 v2 : Val \u03c4} \u2192 Done v1 m \u2261 Done v2 n \u2192 m \u2261 n\n  Done-inj _ _ refl = refl\n\n  lem1 : \u2200 n d \u2192 d \u2262 suc (n + d)\n  lem1 n d eq rewrite +-comm n d = m\u22621+m+n d eq\n\n  subd : \u2200 n d \u2192 n + d \u2261 d \u2192 n \u2261 0\n  subd zero d eq = refl\n  subd (suc n) d eq = \u22a5-elim (lem1 n d (sym eq))\n\n  comp\u2238 : \u2200 a b \u2192 b \u2264 a \u2192 suc a \u2238 b \u2261 suc (a \u2238 b)\n  comp\u2238 a zero le = refl\n  comp\u2238 zero (suc b) ()\n  comp\u2238 (suc a) (suc b) (s\u2264s le) = comp\u2238 a b le\n\n  rearr\u2238 : \u2200 a b c \u2192 c \u2264 b \u2192 a + (b \u2238 c) \u2261 (a + b) \u2238 c\n  rearr\u2238 a b zero c\u2264b = refl\n  rearr\u2238 a zero (suc c) ()\n  rearr\u2238 a (suc b) (suc c) (s\u2264s c\u2264b) rewrite +-suc a b = rearr\u2238 a b c c\u2264b\n\n  cancel\u2238 : \u2200 a b \u2192 b \u2264 a \u2192 a \u2238 b + b \u2261 a\n  cancel\u2238 a zero b\u2264a = +-right-identity a\n  cancel\u2238 zero (suc b) ()\n  cancel\u2238 (suc a) (suc b) (s\u2264s b\u2264a) rewrite +-suc (a \u2238 b) b = cong suc (cancel\u2238 a b b\u2264a)\n\n  lemR : \u2200 {\u03c4} n m {v1 v2 : Val \u03c4} \u2192 Done v1 m \u2261 Done v2 n \u2192 m \u2261 n\n  lemR n .n refl = refl\n\n  lem2 : \u2200 n n3 r \u2192 n3 \u2264 n \u2192 n + n3 \u2261 suc r \u2192 \u2203 \u03bb s \u2192 (n \u2261 suc s)\n  lem2 zero .0 r z\u2264n ()\n  lem2 (suc n) n3 .(n + n3) le refl = n , refl\n\n{-# TERMINATING #-}\neval-dec-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v m n \u2192 eval t \u03c1 m \u2261 Done v n \u2192 \u03c1 \u22a2 t \u2193[ m \u2238 n ] v\neval-dec-sound (const (lit x)) \u03c1 (intV v) m n eq with lemR n m eq\neval-dec-sound (const (lit x)) \u03c1 (intV .x) m .m refl | refl rewrite n\u2238n\u22610 m = lit x\neval-dec-sound (var x) \u03c1 v m n eq with lemR n m eq\neval-dec-sound (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) m .m refl | refl rewrite n\u2238n\u22610 m  = var x\neval-dec-sound (abs t) \u03c1 v m n eq with lemR n m eq\neval-dec-sound (abs t) \u03c1 .(closure t \u03c1) m .m refl | refl rewrite n\u2238n\u22610 m  = abs\neval-dec-sound (app s t) \u03c1 v zero n ()\neval-dec-sound (app s t) \u03c1 v (suc r) n eq with eval s \u03c1 r | inspect (eval s \u03c1) r\neval-dec-sound (app s t) \u03c1 v (suc r) n eq | (Done sv n1) | [ seq ] with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-dec-sound (app s t) \u03c1 v (suc r) n eq | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] with eval st (tv \u2022 s\u03c1) n2 | inspect (eval st (tv \u2022 s\u03c1)) n2\neval-dec-sound (app s t) \u03c1 .v (suc r) .n3 refl | (Done sv@(closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | (Done v n3) | [ veq ] = body\n  where\n    n1\u2264r : n1 \u2264 r\n    n1\u2264r = eval-dec s \u03c1 sv r n1 seq\n    n2\u2264n1 : n2 \u2264 n1\n    n2\u2264n1 = eval-dec t \u03c1 tv n1 n2 teq\n    n3\u2264n2 : n3 \u2264 n2\n    n3\u2264n2 = eval-dec st (tv \u2022 s\u03c1) v n2 n3 veq\n    eval1 = eval-dec-sound s \u03c1 sv r n1 seq\n    eval2 = eval-dec-sound t \u03c1 tv n1 n2 teq\n    eval3 = eval-dec-sound st (tv \u2022 s\u03c1) v n2 n3 veq\n    l1 : suc r \u2238 n3 \u2261 suc (r \u2238 n3)\n    l1 = comp\u2238 r n3 (\u2264-trans n3\u2264n2 (\u2264-trans n2\u2264n1 n1\u2264r))\n    l2 : suc r \u2238 n3 \u2261 suc (((r \u2238 n1) + (n1 \u2238 n2)) + (n2 \u2238 n3))\n    l2\n      rewrite rearr\u2238 (r \u2238 n1) n1 n2 n2\u2264n1 | cancel\u2238 r n1 n1\u2264r\n      | rearr\u2238 (r \u2238 n2) n2 n3 n3\u2264n2 | cancel\u2238 r n2 (\u2264-trans n2\u2264n1 n1\u2264r) = l1\n    foo : \u03c1 \u22a2 app s t \u2193[ suc (r \u2238 n1 + (n1 \u2238 n2) + (n2 \u2238 n3)) ] v\n    foo = app (r \u2238 n1) (n1 \u2238 n2) (n2 \u2238 n3) {t\u2081 = s} {t\u2082 = t} {t\u2032 = st} eval1 eval2 eval3\n    body : \u03c1 \u22a2 app s t \u2193[ suc r \u2238 n3 ] v\n    body rewrite l2 = foo\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | Error | [ veq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | TimeOut | [ veq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done sv n1) | [ seq ] | Error | [ teq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done sv n1) | [ seq ] | TimeOut | [ teq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | Error | [ seq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | TimeOut | [ seq ]\n-- \u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n--   ((Den.\u27e6 x \u27e7Var) (\u27e6 \u03c1 \u27e7Context)) \u2261 (\u27e6 (\u27e6 x \u27e7Var) \u03c1 \u27e7Val)\n-- \u21a6-sound (px \u2022 \u03c1) this = refl\n-- \u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n-- \u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n--   \u03c1 \u22a2 t \u2193 v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n-- \u2193-sound abs = refl\n-- \u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n-- \u2193-sound (var x) = \u21a6-sound _ x\n-- \u2193-sound (lit n) = refl\n\nimport Data.Integer as I\nopen I using (\u2124)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393 \u0394\u0393} (t1 : Term \u0393 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j n2 (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ]\n    d\u03c1 \u22a2 dt \u2193[ dn ] dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n\n  -- XXX What we want for rrelV3:\n  -- rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure dt d\u03c1) (closure {\u03932} t2 \u03c12) n =\n  --     \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n  --       \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n  --       rrelV3 \u03c3 v1 dv v2 k \u2192\n  --       rrelT3\n  --         t1\n  --         dt\n  --         t2\n  --         (v1 \u2022 \u03c11)\n  --         (dv \u2022 v1 \u2022 d\u03c1)\n  --         (v2 \u2022 \u03c12) k\n  --     }\n\n  -- However, we don't have separate change values, so we write:\n\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure dt' d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      -- Require a proof that the two contexts match:\n      \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          -- XXX The next expression is wrong.\n          -- rrelV3 should require dv to be a change closure,\n          -- (\u03bb x dx . dt , d\u03c1) or (dclosure dt d\u03c1).\n          -- Then, here we could write as conclusion.\n          -- rrelT3 t1 dt t2 (v1 \u2022 \u03c11) (dv \u2022 v1 \u2022 d\u03c1) (v2 \u2022 \u03c12) k\n\n          -- Instead, with this syntax I can just match dv as a normal closure,\n          -- closure dt' d\u03c1 or (\u03bb x . dt', d\u03c1), where we hope that dt' evaluates\n          -- to \u03bb dx. dt. So, instead of writing dt, I must write dt' dx where\n          -- dx is a newly bound variable (hence, var this), and dt' must be\n          -- weakened once. Hence, we write instead:\n          (app (weaken (drop (\u0394\u03c4 \u03c3) \u2022 \u227c-refl) dt') (var this))\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 v1 \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\n\n-- The extra \"r\" stands for \"relational\", because unlike relT and relV, rrelV\n-- and rrelT are based on a *relational* big-step semantics.\nmutual\n  rrelT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT {\u03c4} t1 t2 \u03c11 \u03c12 k =\n    (v1 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ j ] v1) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03c12 \u22a2 t2 \u2193[ n2 ] v2 \u00d7 rrelV \u03c4 v1 v2 (k \u2238 j)\n\n  rrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k<n : k < n) v1 v2 \u2192\n      (vv : rrelV \u03c3 v1 v2 k) \u2192\n      rrelT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n    }\n\nrrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = rrelV \u03c4 v1 v2 n \u00d7 rrel\u03c1 \u0393 \u03c11 \u03c12 n\n\n\u27e6_\u27e7RelVar : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192\n  rrel\u03c1 \u0393 \u03c11 \u03c12 n \u2192\n  rrelV \u03c4 (\u27e6 x \u27e7Var \u03c11) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar \u03c1\u03c1\n\nrrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 rrelV \u03c4 v1 v2 n \u2192 rrelV \u03c4 v1 v2 m\nrrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rrel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rrel\u03c1 \u0393 \u03c11 \u03c12 m\nrrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV-mono m n m\u2264n _ v1 v2 vv , rrel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rrel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 rrelT (var x) (var x) \u03c11 \u03c12 n\nrfundamentalV x n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .0 j<n (var .x) = \u27e6 x \u27e7Var \u03c12 , 0 , (var x) , \u27e6 x \u27e7RelVar \u03c1\u03c1\n\nbar : \u2200 m {n o} \u2192 o \u2264 n \u2192 m \u2264 (m + n) \u2238 o\nbar m {n} {o} o\u2264n rewrite +-\u2238-assoc m o\u2264n = m\u2264m+n m (n \u2238 o)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nm\u2261m\u22381+1 : \u2200 m {n} \u2192 n < m \u2192 m \u2261 suc (m \u2238 1)\nm\u2261m\u22381+1 (suc m) (s\u2264s n<m) = refl\n\nm\u2238[n+1]<m : \u2200 m n \u2192 n < m \u2192 m \u2238 suc n < m\nm\u2238[n+1]<m (suc m) zero (s\u2264s n<m) = s\u2264s \u2264-refl\nm\u2238[n+1]<m (suc m) (suc n) (s\u2264s n<m) rewrite sym (suc\u2238suc m n n<m) = \u2264-step (m\u2238[n+1]<m m n n<m)\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\nrfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rrel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 rrelT t t \u03c11 \u03c12 n\nrfundamental (var x) n \u03c11 \u03c12 \u03c1\u03c1 = rfundamentalV x n \u03c11 \u03c12 \u03c1\u03c1\nrfundamental (const (lit x)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV x) .0 j<n (lit .x) = intV x , 0 , lit x , I.+ 0 , refl\nrfundamental (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .0 j<n abs = closure t \u03c12 , 0 , abs , refl , (\u03bb k k<n v1 v2 vv v3 j j<k \u03c11\u22a2t1\u2193[j]v1 \u2192 rfundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rrel\u03c1-mono k n (lt1 k<n) _ _ _ \u03c1\u03c1) v3 j j<k \u03c11\u22a2t1\u2193[j]v1 )\nrfundamental (app s t) n \u03c11 \u03c12 \u03c1\u03c1 v1 .(suc (n1 + n2 + n3)) 1+n1+n2+n3<n (app n1 n2 n3 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3) = body\n  where\n    open \u2264-Reasoning\n    n1\u2264sum : n1 \u2264 n1 + n2 + n3\n    n1\u2264sum rewrite +-assoc n1 n2 n3 = m\u2264m+n n1 (n2 + n3)\n    n1<n : n1 < n\n    n1<n = \u2264-trans (s\u2264s (\u2264-step n1\u2264sum)) 1+n1+n2+n3<n\n\n    n2\u2264sum : n2 \u2264 n1 + n2 + n3\n    n2\u2264sum rewrite +-assoc n1 n2 n3 = \u2264-steps {n2} {n2 + n3} n1 (m\u2264m+n n2 n3)\n    n2<n : n2 < n\n    n2<n = \u2264-trans (s\u2264s (\u2264-step n2\u2264sum)) 1+n1+n2+n3<n\n    n1+n2<n : n1 + n2 < n\n    n1+n2<n = \u2264-trans (s\u2264s (\u2264-step (m\u2264m+n (n1 + n2) n3))) 1+n1+n2+n3<n\n    n2+n1<n : n2 + n1 < n\n    n2+n1<n rewrite +-comm n2 n1 = n1+n2<n\n\n    n1+n3\u2264sum : n1 + n3 \u2264 (n1 + n2) + n3\n    n1+n3\u2264sum = m\u2264m+n n1 n2 +-mono (\u2264-refl {n3})\n    foo : suc n2 \u2261 suc (n1 + n2 + n3) \u2238 (n1 + n3)\n    foo rewrite cong suc (sym (m+n\u2238n\u2261m n2 (n1 + n3))) | sym (+-assoc n2 n1 n3) | +-comm n2 n1 = suc\u2238 (n1 + n2 + n3) (n1 + n3) n1+n3\u2264sum\n    n2<n\u2238n1 : n2 < n \u2238 n1\n    n2<n\u2238n1 rewrite foo = \u2238-mono {suc (n1 + n2 + n3)} {n} {n1 + n3} {n1} (lt1 1+n1+n2+n3<n) (m\u2264m+n n1 n3)\n    n-[1+n1+n2]<n-n1 : n \u2238 n1 \u2238 suc n2 < n \u2238 n1\n    n-[1+n1+n2]<n-n1 = m\u2238[n+1]<m (n \u2238 n1) n2 n2<n\u2238n1\n\n    n-[1+n1+n2]<n-n2 : n \u2238 n1 \u2238 suc n2 < n \u2238 n2\n    n-[1+n1+n2]<n-n2 rewrite \u2238-+-assoc n n1 (suc n2) | +-suc n1 n2 | +-comm n1 n2 | suc\u2238suc n (n2 + n1) n2+n1<n | sym (\u2238-+-assoc n n2 n1) = n\u2238m\u2264n n1 (n \u2238 n2)\n\n    baz : n1 + suc (n2 + suc n3) \u2264 n\n    baz rewrite +-suc n2 n3 | +-suc n1 (suc (n2 + n3)) | +-suc n1 (n2 + n3) | +-assoc n1 n2 n3 = 1+n1+n2+n3<n\n    n3<n-n1-[1+n2] : n3 < n \u2238 n1 \u2238 suc n2\n    n3<n-n1-[1+n2] = sub\u2238 (suc n2) (suc n3) (n \u2238 n1) (sub\u2238 n1 (suc (n2 + suc n3)) n baz)\n\n    l1 : suc (n1 + n2 + n3) \u2261 n1 + suc n2 + n3\n    l1 = cong (_+ n3) (sym (+-suc n1 n2))\n    n-1+sum\u2261alt : n \u2238 suc (n1 + n2 + n3) \u2261 n \u2238 n1 \u2238 suc n2 \u2238 n3\n    n-1+sum\u2261alt rewrite l1 | sym (\u2238-+-assoc n (n1 + suc n2) n3) | sym (\u2238-+-assoc n n1 (suc n2)) = refl\n\n    body : \u03a3[ v2 \u2208 Val _ ] \u03a3[ tn \u2208 \u2115 ] \u03c12 \u22a2 app s t \u2193[ tn ] v2 \u00d7 rrelV _ v1 v2 (n \u2238 suc (n1 + n2 + n3))\n    body with rfundamental s n \u03c11 \u03c12 \u03c1\u03c1 _ n1 n1<n \u03c11\u22a2t1\u2193[j]v1 | rfundamental t n \u03c11 \u03c12 \u03c1\u03c1 _ n2 n2<n \u03c11\u22a2t1\u2193[j]v2\n    ... | sv2@(closure st2 s\u03c12) , sn2 , \u03c12\u22a2s\u2193 , refl , sv1v2 | tv2 , tn2 , \u03c12\u22a2t\u2193 , tv1v2 with sv1v2 (n \u2238 n1 \u2238 suc n2) n-[1+n1+n2]<n-n1 _ tv2 (rrelV-mono (n \u2238 n1 \u2238 suc n2) (n \u2238 n2) (lt1 n-[1+n1+n2]<n-n2) _ _ tv2 tv1v2) v1 n3 n3<n-n1-[1+n2]  \u03c11\u22a2t1\u2193[j]v3\n    ... | v2 , stn , \u03c12\u22a2st2\u2193 , vv rewrite n-1+sum\u2261alt = v2 , suc (sn2 + tn2 + stn) , app _ _ _ \u03c12\u22a2s\u2193 \u03c12\u22a2t\u2193  \u03c12\u22a2st2\u2193 , vv\n","old_contents":"module Thesis.BigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\nopen import Thesis.FunBigStepSILR2\n\n-- Standard relational big-step semantics, with step-indexes matching a small-step semantics.\n-- Protip: doing this on ANF terms would be much easier.\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u2115 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  app : \u2200 n1 n2 n3 {\u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193[ n1 ] closure t\u2032 \u03c1\u2032 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193[ n2 ] v\u2082 \u2192\n    (v\u2082 \u2022 \u03c1\u2032) \u22a2 t\u2032 \u2193[ n3 ] v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193[ suc n1 + n2 + n3 ] v\u2032\n  var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lit : \u2200 n \u2192\n    \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- Silly lemmas for eval-dec-sound\nmodule _ where\n  Done-inj : \u2200 {\u03c4} m n {v1 v2 : Val \u03c4} \u2192 Done v1 m \u2261 Done v2 n \u2192 m \u2261 n\n  Done-inj _ _ refl = refl\n\n  lem1 : \u2200 n d \u2192 d \u2262 suc (n + d)\n  lem1 n d eq rewrite +-comm n d = m\u22621+m+n d eq\n\n  subd : \u2200 n d \u2192 n + d \u2261 d \u2192 n \u2261 0\n  subd zero d eq = refl\n  subd (suc n) d eq = \u22a5-elim (lem1 n d (sym eq))\n\n  comp\u2238 : \u2200 a b \u2192 b \u2264 a \u2192 suc a \u2238 b \u2261 suc (a \u2238 b)\n  comp\u2238 a zero le = refl\n  comp\u2238 zero (suc b) ()\n  comp\u2238 (suc a) (suc b) (s\u2264s le) = comp\u2238 a b le\n\n  rearr\u2238 : \u2200 a b c \u2192 c \u2264 b \u2192 a + (b \u2238 c) \u2261 (a + b) \u2238 c\n  rearr\u2238 a b zero c\u2264b = refl\n  rearr\u2238 a zero (suc c) ()\n  rearr\u2238 a (suc b) (suc c) (s\u2264s c\u2264b) rewrite +-suc a b = rearr\u2238 a b c c\u2264b\n\n  cancel\u2238 : \u2200 a b \u2192 b \u2264 a \u2192 a \u2238 b + b \u2261 a\n  cancel\u2238 a zero b\u2264a = +-right-identity a\n  cancel\u2238 zero (suc b) ()\n  cancel\u2238 (suc a) (suc b) (s\u2264s b\u2264a) rewrite +-suc (a \u2238 b) b = cong suc (cancel\u2238 a b b\u2264a)\n\n  lemR : \u2200 {\u03c4} n m {v1 v2 : Val \u03c4} \u2192 Done v1 m \u2261 Done v2 n \u2192 m \u2261 n\n  lemR n .n refl = refl\n\n  lem2 : \u2200 n n3 r \u2192 n3 \u2264 n \u2192 n + n3 \u2261 suc r \u2192 \u2203 \u03bb s \u2192 (n \u2261 suc s)\n  lem2 zero .0 r z\u2264n ()\n  lem2 (suc n) n3 .(n + n3) le refl = n , refl\n\n{-# TERMINATING #-}\neval-dec-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v m n \u2192 eval t \u03c1 m \u2261 Done v n \u2192 \u03c1 \u22a2 t \u2193[ m \u2238 n ] v\neval-dec-sound (const (lit x)) \u03c1 (intV v) m n eq with lemR n m eq\neval-dec-sound (const (lit x)) \u03c1 (intV .x) m .m refl | refl rewrite n\u2238n\u22610 m = lit x\neval-dec-sound (var x) \u03c1 v m n eq with lemR n m eq\neval-dec-sound (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) m .m refl | refl rewrite n\u2238n\u22610 m  = var x\neval-dec-sound (abs t) \u03c1 v m n eq with lemR n m eq\neval-dec-sound (abs t) \u03c1 .(closure t \u03c1) m .m refl | refl rewrite n\u2238n\u22610 m  = abs\neval-dec-sound (app s t) \u03c1 v zero n ()\neval-dec-sound (app s t) \u03c1 v (suc r) n eq with eval s \u03c1 r | inspect (eval s \u03c1) r\neval-dec-sound (app s t) \u03c1 v (suc r) n eq | (Done sv n1) | [ seq ] with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-dec-sound (app s t) \u03c1 v (suc r) n eq | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] with eval st (tv \u2022 s\u03c1) n2 | inspect (eval st (tv \u2022 s\u03c1)) n2\neval-dec-sound (app s t) \u03c1 .v (suc r) .n3 refl | (Done sv@(closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | (Done v n3) | [ veq ] = body\n  where\n    n1\u2264r : n1 \u2264 r\n    n1\u2264r = eval-dec s \u03c1 sv r n1 seq\n    n2\u2264n1 : n2 \u2264 n1\n    n2\u2264n1 = eval-dec t \u03c1 tv n1 n2 teq\n    n3\u2264n2 : n3 \u2264 n2\n    n3\u2264n2 = eval-dec st (tv \u2022 s\u03c1) v n2 n3 veq\n    eval1 = eval-dec-sound s \u03c1 sv r n1 seq\n    eval2 = eval-dec-sound t \u03c1 tv n1 n2 teq\n    eval3 = eval-dec-sound st (tv \u2022 s\u03c1) v n2 n3 veq\n    l1 : suc r \u2238 n3 \u2261 suc (r \u2238 n3)\n    l1 = comp\u2238 r n3 (\u2264-trans n3\u2264n2 (\u2264-trans n2\u2264n1 n1\u2264r))\n    l2 : suc r \u2238 n3 \u2261 suc (((r \u2238 n1) + (n1 \u2238 n2)) + (n2 \u2238 n3))\n    l2\n      rewrite rearr\u2238 (r \u2238 n1) n1 n2 n2\u2264n1 | cancel\u2238 r n1 n1\u2264r\n      | rearr\u2238 (r \u2238 n2) n2 n3 n3\u2264n2 | cancel\u2238 r n2 (\u2264-trans n2\u2264n1 n1\u2264r) = l1\n    foo : \u03c1 \u22a2 app s t \u2193[ suc (r \u2238 n1 + (n1 \u2238 n2) + (n2 \u2238 n3)) ] v\n    foo = app (r \u2238 n1) (n1 \u2238 n2) (n2 \u2238 n3) {t\u2081 = s} {t\u2082 = t} {t\u2032 = st} eval1 eval2 eval3\n    body : \u03c1 \u22a2 app s t \u2193[ suc r \u2238 n3 ] v\n    body rewrite l2 = foo\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | Error | [ veq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | TimeOut | [ veq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done sv n1) | [ seq ] | Error | [ teq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done sv n1) | [ seq ] | TimeOut | [ teq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | Error | [ seq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | TimeOut | [ seq ]\n-- \u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n--   ((Den.\u27e6 x \u27e7Var) (\u27e6 \u03c1 \u27e7Context)) \u2261 (\u27e6 (\u27e6 x \u27e7Var) \u03c1 \u27e7Val)\n-- \u21a6-sound (px \u2022 \u03c1) this = refl\n-- \u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n-- \u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n--   \u03c1 \u22a2 t \u2193 v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n-- \u2193-sound abs = refl\n-- \u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n-- \u2193-sound (var x) = \u21a6-sound _ x\n-- \u2193-sound (lit n) = refl\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- The extra \"r\" stands for \"relational\", because unlike relT and relV, rrelV\n-- and rrelT are based on a *relational* big-step semantics.\nmutual\n  rrelT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT {\u03c4} t1 t2 \u03c11 \u03c12 k =\n    (v1 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ j ] v1) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03c12 \u22a2 t2 \u2193[ n2 ] v2 \u00d7 rrelV \u03c4 v1 v2 (k \u2238 j)\n\n  rrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k<n : k < n) v1 v2 \u2192\n      (vv : rrelV \u03c3 v1 v2 k) \u2192\n      rrelT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n    }\n\nrrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = rrelV \u03c4 v1 v2 n \u00d7 rrel\u03c1 \u0393 \u03c11 \u03c12 n\n\n\u27e6_\u27e7RelVar : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192\n  rrel\u03c1 \u0393 \u03c11 \u03c12 n \u2192\n  rrelV \u03c4 (\u27e6 x \u27e7Var \u03c11) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar \u03c1\u03c1\n\nrrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 rrelV \u03c4 v1 v2 n \u2192 rrelV \u03c4 v1 v2 m\nrrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rrel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rrel\u03c1 \u0393 \u03c11 \u03c12 m\nrrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV-mono m n m\u2264n _ v1 v2 vv , rrel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rrel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 rrelT (var x) (var x) \u03c11 \u03c12 n\nrfundamentalV x n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .0 j<n (var .x) = \u27e6 x \u27e7Var \u03c12 , 0 , (var x) , \u27e6 x \u27e7RelVar \u03c1\u03c1\n\nbar : \u2200 m {n o} \u2192 o \u2264 n \u2192 m \u2264 (m + n) \u2238 o\nbar m {n} {o} o\u2264n rewrite +-\u2238-assoc m o\u2264n = m\u2264m+n m (n \u2238 o)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nm\u2261m\u22381+1 : \u2200 m {n} \u2192 n < m \u2192 m \u2261 suc (m \u2238 1)\nm\u2261m\u22381+1 (suc m) (s\u2264s n<m) = refl\n\nm\u2238[n+1]<m : \u2200 m n \u2192 n < m \u2192 m \u2238 suc n < m\nm\u2238[n+1]<m (suc m) zero (s\u2264s n<m) = s\u2264s \u2264-refl\nm\u2238[n+1]<m (suc m) (suc n) (s\u2264s n<m) rewrite sym (suc\u2238suc m n n<m) = \u2264-step (m\u2238[n+1]<m m n n<m)\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\nrfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rrel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 rrelT t t \u03c11 \u03c12 n\nrfundamental (var x) n \u03c11 \u03c12 \u03c1\u03c1 = rfundamentalV x n \u03c11 \u03c12 \u03c1\u03c1\nrfundamental (const (lit x)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV x) .0 j<n (lit .x) = intV x , 0 , lit x , I.+ 0 , refl\nrfundamental (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .0 j<n abs = closure t \u03c12 , 0 , abs , refl , (\u03bb k k<n v1 v2 vv v3 j j<k \u03c11\u22a2t1\u2193[j]v1 \u2192 rfundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rrel\u03c1-mono k n (lt1 k<n) _ _ _ \u03c1\u03c1) v3 j j<k \u03c11\u22a2t1\u2193[j]v1 )\nrfundamental (app s t) n \u03c11 \u03c12 \u03c1\u03c1 v1 .(suc (n1 + n2 + n3)) 1+n1+n2+n3<n (app n1 n2 n3 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3) = body\n  where\n    open \u2264-Reasoning\n    n1\u2264sum : n1 \u2264 n1 + n2 + n3\n    n1\u2264sum rewrite +-assoc n1 n2 n3 = m\u2264m+n n1 (n2 + n3)\n    n1<n : n1 < n\n    n1<n = \u2264-trans (s\u2264s (\u2264-step n1\u2264sum)) 1+n1+n2+n3<n\n\n    n2\u2264sum : n2 \u2264 n1 + n2 + n3\n    n2\u2264sum rewrite +-assoc n1 n2 n3 = \u2264-steps {n2} {n2 + n3} n1 (m\u2264m+n n2 n3)\n    n2<n : n2 < n\n    n2<n = \u2264-trans (s\u2264s (\u2264-step n2\u2264sum)) 1+n1+n2+n3<n\n    n1+n2<n : n1 + n2 < n\n    n1+n2<n = \u2264-trans (s\u2264s (\u2264-step (m\u2264m+n (n1 + n2) n3))) 1+n1+n2+n3<n\n    n2+n1<n : n2 + n1 < n\n    n2+n1<n rewrite +-comm n2 n1 = n1+n2<n\n\n    n1+n3\u2264sum : n1 + n3 \u2264 (n1 + n2) + n3\n    n1+n3\u2264sum = m\u2264m+n n1 n2 +-mono (\u2264-refl {n3})\n    foo : suc n2 \u2261 suc (n1 + n2 + n3) \u2238 (n1 + n3)\n    foo rewrite cong suc (sym (m+n\u2238n\u2261m n2 (n1 + n3))) | sym (+-assoc n2 n1 n3) | +-comm n2 n1 = suc\u2238 (n1 + n2 + n3) (n1 + n3) n1+n3\u2264sum\n    n2<n\u2238n1 : n2 < n \u2238 n1\n    n2<n\u2238n1 rewrite foo = \u2238-mono {suc (n1 + n2 + n3)} {n} {n1 + n3} {n1} (lt1 1+n1+n2+n3<n) (m\u2264m+n n1 n3)\n    n-[1+n1+n2]<n-n1 : n \u2238 n1 \u2238 suc n2 < n \u2238 n1\n    n-[1+n1+n2]<n-n1 = m\u2238[n+1]<m (n \u2238 n1) n2 n2<n\u2238n1\n\n    n-[1+n1+n2]<n-n2 : n \u2238 n1 \u2238 suc n2 < n \u2238 n2\n    n-[1+n1+n2]<n-n2 rewrite \u2238-+-assoc n n1 (suc n2) | +-suc n1 n2 | +-comm n1 n2 | suc\u2238suc n (n2 + n1) n2+n1<n | sym (\u2238-+-assoc n n2 n1) = n\u2238m\u2264n n1 (n \u2238 n2)\n\n    baz : n1 + suc (n2 + suc n3) \u2264 n\n    baz rewrite +-suc n2 n3 | +-suc n1 (suc (n2 + n3)) | +-suc n1 (n2 + n3) | +-assoc n1 n2 n3 = 1+n1+n2+n3<n\n    n3<n-n1-[1+n2] : n3 < n \u2238 n1 \u2238 suc n2\n    n3<n-n1-[1+n2] = sub\u2238 (suc n2) (suc n3) (n \u2238 n1) (sub\u2238 n1 (suc (n2 + suc n3)) n baz)\n\n    l1 : suc (n1 + n2 + n3) \u2261 n1 + suc n2 + n3\n    l1 = cong (_+ n3) (sym (+-suc n1 n2))\n    n-1+sum\u2261alt : n \u2238 suc (n1 + n2 + n3) \u2261 n \u2238 n1 \u2238 suc n2 \u2238 n3\n    n-1+sum\u2261alt rewrite l1 | sym (\u2238-+-assoc n (n1 + suc n2) n3) | sym (\u2238-+-assoc n n1 (suc n2)) = refl\n\n    body : \u03a3[ v2 \u2208 Val _ ] \u03a3[ tn \u2208 \u2115 ] \u03c12 \u22a2 app s t \u2193[ tn ] v2 \u00d7 rrelV _ v1 v2 (n \u2238 suc (n1 + n2 + n3))\n    body with rfundamental s n \u03c11 \u03c12 \u03c1\u03c1 _ n1 n1<n \u03c11\u22a2t1\u2193[j]v1 | rfundamental t n \u03c11 \u03c12 \u03c1\u03c1 _ n2 n2<n \u03c11\u22a2t1\u2193[j]v2\n    ... | sv2@(closure st2 s\u03c12) , sn2 , \u03c12\u22a2s\u2193 , refl , sv1v2 | tv2 , tn2 , \u03c12\u22a2t\u2193 , tv1v2 with sv1v2 (n \u2238 n1 \u2238 suc n2) n-[1+n1+n2]<n-n1 _ tv2 (rrelV-mono (n \u2238 n1 \u2238 suc n2) (n \u2238 n2) (lt1 n-[1+n1+n2]<n-n2) _ _ tv2 tv1v2) v1 n3 n3<n-n1-[1+n2]  \u03c11\u22a2t1\u2193[j]v3\n    ... | v2 , stn , \u03c12\u22a2st2\u2193 , vv rewrite n-1+sum\u2261alt = v2 , suc (sn2 + tn2 + stn) , app _ _ _ \u03c12\u22a2s\u2193 \u03c12\u22a2t\u2193  \u03c12\u22a2st2\u2193 , vv\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4c2ed8317d73a9ee1544b06f5c885c5790e313e9","subject":"change par again","message":"change par again\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : proj\u2081 x \u2261 proj\u2081 y) \u2192 subst B p (proj\u2082 x) \u2261 proj\u2082 y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (proj\u2081 (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n    send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n    recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n    end  : ViewProc end _\n\n  view-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\n  view-proc end      _       = end\n  view-proc (\u03a0\u1d3e M P) p       = recv _ _ p\n  view-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\n  data ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n    send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n    recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\n  view-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\n  view-com (\u03a0\u1d9c M P) p       = recv _ p\n  view-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M} (P : M \u2192 Proto) Q (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (com Q) (`L , m , p)\n    sendR' : \u2200 {M} P (Q : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7) \u2192 View-\u214b (com P) (\u03a3\u1d3e M Q) (`R , m , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) (com Q) (`L , p)\n    recvR' : \u2200 {M} P (Q : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)) \u2192 View-\u214b (com P) (\u03a0\u1d3e M Q) (`R , p)\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 P (p : \u27e6 com P \u27e7) \u2192 View-\u214b (com P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end      Q       p                = endL _ _\n  view-\u214b (com x)  end     p                = endR _ _\n  view-\u214b (\u03a0\u1d3e M P) (com Q) (`L , p)         = recvL' _ _ _\n  view-\u214b (\u03a3\u1d3e M P) (com Q) (`L , (m , p))   = sendL' _ _ _ _\n  view-\u214b (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) (`R , p)       = recvR' _ _ _\n  view-\u214b (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) (`R , (m , p)) = sendR' _ _ _ _\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) (`R , p)       = recvR' _ _ _\n  view-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (`R , (m , p)) = sendR' _ _ _ _\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : proj\u2081 x \u2261 proj\u2081 y) \u2192 subst B p (proj\u2082 x) \u2261 proj\u2082 y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ (\u03a0-ext : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : \u2200 x \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (\u03a0-ext \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (\u03a0-ext \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (\u03a0-ext (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (\u03a0-ext (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr) = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr) = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q) = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q) = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr) = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr) = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr) = {!\u214b\u1d3e-apply {?} {?} pq qr!}\n\n    {-\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map P Q R S f g p with view-\u214b P R p\n  \u214b\u1d3e-map .(com (mk Out M P)) Q\u2081 .(com Q) S f g .(`L , m , p) | sendL' M P Q m p = {!!}\n  \u214b\u1d3e-map .(com P) Q\u2081 .(com (mk Out M Q)) S f g .(`R , m , p) | sendR' P M Q m p = {!!}\n  \u214b\u1d3e-map .(com (mk In M P)) Q\u2081 .(com Q) S f g .(`L , p) | recvL' M P Q p = {!!}\n  \u214b\u1d3e-map .(com P) Q\u2081 .(com (mk In M Q)) S f g .(`R , p) | recvR' P M Q p = {!!}\n  \u214b\u1d3e-map .end Q R S f g p | endL .R .p = {!!}\n  \u214b\u1d3e-map .(com P) Q .end S f g p | endR P .p = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  switchL' : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R p\u214bq r with view-\u214b P Q p\u214bq\n  switchL' ._ ._ R .(`L , m , p) r | sendL' M P Q m p = \u214b\u1d3e-sendL (com Q ox\u1d3e R) m (switchL' (P m) (com Q) R p r)\n  switchL' ._ ._ R .(`R , m , p) r | sendR' P M Q m p = \u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id\n                                                          (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id)\n                                                             (switchL' (com P) (Q m) R p r)\n  switchL' ._ ._ R .(`L , p) r | recvL' M P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n  switchL' ._ ._ R .(`R , p) r | recvR' (\u03a0\u1d9c M' P) M Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n  switchL' ._ ._ R .(`R , p) r | recvR' (\u03a3\u1d9c M' P) M Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n  switchL' ._ ._ R p\u214bq r | endL Q .p\u214bq = comma\u1d3e Q R p\u214bq r\n  switchL' .(com P) .end R p\u214bq r | endR P .p\u214bq = par (com P) R p\u214bq r\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"45831718134fddb82017042f777ad924faa14f70","subject":"par id defined, and send{L,R} and recv{L,R} for par","message":"par id defined, and send{L,R} and recv{L,R} for par\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _ where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    \u214b\u1d3e Q      = Q\n    P      \u214b\u1d3e end    = P\n    com P\u1d38 \u214b\u1d3e com P\u1d3f = \u03a3\u1d3e LR (P\u1d38 \u214b\u1d9c P\u1d3f)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  Equiv : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 \u2605\n  Equiv {A} {B} f = \u2200 y \u2192 \u03a3 A (_\u2261_ y \u2218 f)\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com x) = id\n\n  \u214b\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR {P = end} m p = m , p\n  \u214b\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7 \u2192 \u27e6 com' Out M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-sendL {P = end}{Q} m p = m , \u214b\u1d3e-rend (Q m) p\n  \u214b\u1d3e-sendL {P = com x} m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR {P = end}   f = f\n  \u214b\u1d3e-recvR {P = com x} f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7) \u2192 \u27e6 com' In M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-recvL {P = end}{Q} f x = \u214b\u1d3e-rend (Q x) (f x)\n  \u214b\u1d3e-recvL {P = com x} f = `L , f\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL {P = P m} m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR {P = dual (P m)} m (\u214b\u1d3e-id (P m))\n\n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end     Q   p q = q\n  comma\u1d3e (com x) end p q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  comma\u1d3e-equiv : \u2200 P Q \u2192 Equiv (comma\u1d3e P Q)\n  comma\u1d3e-equiv P Q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _ where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    \u214b\u1d3e Q      = Q\n    P      \u214b\u1d3e end    = P\n    com P\u1d38 \u214b\u1d3e com P\u1d3f = \u03a3\u1d3e LR (P\u1d38 \u214b\u1d9c P\u1d3f)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  Equiv : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 \u2605\n  Equiv {A} {B} f = \u2200 y \u2192 \u03a3 A (_\u2261_ y \u2218 f)\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id = {!!}\n  \n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end     Q   p q = q\n  comma\u1d3e (com x) end p q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  comma\u1d3e-equiv : \u2200 P Q \u2192 Equiv (comma\u1d3e P Q)\n  comma\u1d3e-equiv P Q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"db5307718bfbf262aa80ac0a0dd4b8da3799363b","subject":"Change where-clause to let-expr so that agda 2.3.2 typechecks README","message":"Change where-clause to let-expr so that agda 2.3.2 typechecks README\n\nOld-commit-hash: 6ffa74c100a7ee9d51e15d960db14c801f06e27c\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/WeakValidChanges.agda","new_file":"Denotational\/WeakValidChanges.agda","new_contents":"module Denotational.WeakValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.ChangeTypes.ChangesAreDerivatives\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Changes\nopen import Denotational.EqualityLemmas\n\n-- DEFINITION of weakly valid changes via a logical relation\n\n-- Strong validity:\nopen import Denotational.ValidChanges\n\n-- Weak validity, defined through a function producing a type.\n--\n-- Since this is a logical relation, this couldn't be defined as a\n-- datatype, since it is not strictly positive.\n\nWeak-Valid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nWeak-Valid-\u0394 {bool} v dv = \u22a4\nWeak-Valid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : Weak-Valid-\u0394 s ds) \u2192\n    Weak-Valid-\u0394 (f s) (df s ds) \u00d7\n    df is-correct-for f on s and ds\n\nstrong-to-weak-validity : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv} \u2192 Valid-\u0394 v dv \u2192 Weak-Valid-\u0394 v dv\nstrong-to-weak-validity {bool} _ = tt\nstrong-to-weak-validity {\u03c4 \u21d2 \u03c4\u2081} {v} {dv} s-valid-v-dv = \u03bb s ds _ \u2192 \n  let\n    proofs = s-valid-v-dv s ds\n    dv-s-ds-valid-on-v-s = proj\u2081 proofs\n    dv-is-correct-for-v-on-s-and-ds = proj\u2082 proofs\n  in\n    strong-to-weak-validity dv-s-ds-valid-on-v-s , dv-is-correct-for-v-on-s-and-ds\n\n{-\n-- This proof doesn't go through: the desired equivalence is too\n-- strong, and we can't fill the holes because dv is arbitrary.\n\ndiff-apply-proof-weak : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Weak-Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof-weak {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (strong-to-weak-validity (derive-is-valid (apply dv v)))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof-weak {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof-weak {bool} db b _ = xor-cancellative b db\n\n-- That'd be the core of the proof of diff-apply-weak, if we relax the\n-- equivalence in the goal to something more correct. We need a proof\n-- of validity only for the first argument, somehow: maybe because\n-- diff itself produces strongly valid changes?\n--\n-- Ahah! Not really: below we use the unprovable\n-- diff-apply-proof-weak. To be able to use induction, we need to use\n-- this weaker lemma again. Hence, we'll only be able to prove the\n-- same equivalence, which will needs another proof of validity to\n-- unfold further until the base case.\n\ndiff-apply-proof-weak-f : \u2200 {\u03c4\u2081 \u03c4\u2082} (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192\n    (Weak-Valid-\u0394 f df) \u2192 \u2200 dv v \u2192 (Weak-Valid-\u0394 v dv) \u2192 (diff (apply df f) f) v dv \u2261 df v dv\n\ndiff-apply-proof-weak-f {\u03c4\u2081} {\u03c4\u2082} df f df-valid dv v dv-valid =\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (strong-to-weak-validity (derive-is-valid (apply dv v)))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv dv-valid)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof-weak {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv dv-valid)) \u27e9\n    df v dv\n  \u220e where open \u2261-Reasoning\n-}\n\n-- A stronger delta equivalence. Note this isn't a congruence; it\n-- should be possible to define some congruence rules, but those will\n-- be more complex.\n\ndata _\u2248_ : {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set where\n  --base : \u2200 (v : \u27e6 bool \u27e7) \u2192 v \u2248 v\n  base : \u2200 (v1 v2 : \u27e6 bool \u27e7) \u2192 (\u2261 : v1 \u2261 v2) \u2192 v1 \u2248 v2\n  fun : \u2200 {\u03c3 \u03c4} (f1 f2 : \u27e6 \u0394-Type (\u03c3 \u21d2 \u03c4) \u27e7) \u2192\n    (t\u2248 : \u2200 s ds (valid : Weak-Valid-\u0394 s ds) \u2192 f1 s ds \u2248 f2 s ds) \u2192\n    f1 \u2248 f2\n\n\u2261\u2192\u2248 : \u2200 {\u03c4} {v1 v2 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192 v1 \u2261 v2 \u2192 v1 \u2248 v2\n\u2261\u2192\u2248 {bool} {v1} {.v1} refl = base v1 v1 refl\n\u2261\u2192\u2248 {\u03c4\u2081 \u21d2 \u03c4\u2082} {v1} {.v1} refl = fun v1 v1 (\u03bb s ds _ \u2192 \u2261\u2192\u2248 refl)\n\nopen import Relation.Binary hiding (_\u21d2_)\n\n\u2248-refl : \u2200 {\u03c4} \u2192 Reflexive (_\u2248_ {\u03c4})\n\u2248-refl = \u2261\u2192\u2248 refl\n\n\u2248-sym : \u2200 {\u03c4} \u2192 Symmetric (_\u2248_ {\u03c4})\n\u2248-sym (base v1 v2 \u2261) = base _ _ (sym \u2261)\n\u2248-sym {(\u03c3 \u21d2 \u03c4)} (fun f1 f2 t\u2248) = fun _ _  (\u03bb s ds valid \u2192 \u2248-sym (t\u2248 s ds valid))\n\n\u2248-trans : \u2200 {\u03c4} \u2192 Transitive (_\u2248_ {\u03c4})\n\u2248-trans {.bool} {.k} {.k} {k} (base .k .k refl) (base .k .k refl) = base k k refl\n\u2248-trans {\u03c3 \u21d2 \u03c4} {.f1} {.f2} {k} (fun f1 .f2 \u2248\u2081) (fun f2 .k \u2248\u2082) = fun f1 k (\u03bb s ds valid \u2192 \u2248-trans (\u2248\u2081 _ _ valid) (\u2248\u2082 _ _ valid))\n\n\u2248-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2248_ {\u03c4})\n\u2248-isEquivalence = record\n  { refl  = \u2248-refl\n  ; sym   = \u2248-sym\n  ; trans = \u2248-trans\n  }\n\n\u2248-setoid : Type \u2192 Setoid _ _\n\u2248-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u0394-Type \u03c4 \u27e7\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\nmodule \u2248-Reasoning where\n  module _ {\u03c4 : Type} where\n    open import Relation.Binary.EqReasoning (\u2248-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n-- Correct proof, to refactor using \u2248-Reasoning\n\ndiff-apply-proof-weak-real : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Weak-Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2248 dv\ndiff-apply-proof-weak-real {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = fun _ _ (\u03bb v dv dv-valid \u2192 \u2248-trans (\u2261\u2192\u2248 (\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (strong-to-weak-validity (derive-is-valid (apply dv v)))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv dv-valid)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u220e)) (diff-apply-proof-weak-real {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv dv-valid))))\n  --\u2248\u27e8 fill in the above proof \u27e9\n    --df v dv\n\n  where open \u2261-Reasoning\n\ndiff-apply-proof-weak-real {bool} db b _ = \u2261\u2192\u2248 (xor-cancellative b db)\n","old_contents":"module Denotational.WeakValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.ChangeTypes.ChangesAreDerivatives\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Changes\nopen import Denotational.EqualityLemmas\n\n-- DEFINITION of weakly valid changes via a logical relation\n\n-- Strong validity:\nopen import Denotational.ValidChanges\n\n-- Weak validity, defined through a function producing a type.\n--\n-- Since this is a logical relation, this couldn't be defined as a\n-- datatype, since it is not strictly positive.\n\nWeak-Valid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nWeak-Valid-\u0394 {bool} v dv = \u22a4\nWeak-Valid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : Weak-Valid-\u0394 s ds) \u2192\n    Weak-Valid-\u0394 (f s) (df s ds) \u00d7\n    df is-correct-for f on s and ds\n\nstrong-to-weak-validity : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv} \u2192 Valid-\u0394 v dv \u2192 Weak-Valid-\u0394 v dv\nstrong-to-weak-validity {bool} _ = tt\nstrong-to-weak-validity {\u03c4 \u21d2 \u03c4\u2081} {v} {dv} s-valid-v-dv s ds _ = strong-to-weak-validity dv-s-ds-valid-on-v-s , dv-is-correct-for-v-on-s-and-ds\n  where\n    proofs = s-valid-v-dv s ds\n    dv-s-ds-valid-on-v-s = proj\u2081 proofs\n    dv-is-correct-for-v-on-s-and-ds = proj\u2082 proofs\n\n{-\n-- This proof doesn't go through: the desired equivalence is too\n-- strong, and we can't fill the holes because dv is arbitrary.\n\ndiff-apply-proof-weak : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Weak-Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof-weak {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (strong-to-weak-validity (derive-is-valid (apply dv v)))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof-weak {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof-weak {bool} db b _ = xor-cancellative b db\n\n-- That'd be the core of the proof of diff-apply-weak, if we relax the\n-- equivalence in the goal to something more correct. We need a proof\n-- of validity only for the first argument, somehow: maybe because\n-- diff itself produces strongly valid changes?\n--\n-- Ahah! Not really: below we use the unprovable\n-- diff-apply-proof-weak. To be able to use induction, we need to use\n-- this weaker lemma again. Hence, we'll only be able to prove the\n-- same equivalence, which will needs another proof of validity to\n-- unfold further until the base case.\n\ndiff-apply-proof-weak-f : \u2200 {\u03c4\u2081 \u03c4\u2082} (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192\n    (Weak-Valid-\u0394 f df) \u2192 \u2200 dv v \u2192 (Weak-Valid-\u0394 v dv) \u2192 (diff (apply df f) f) v dv \u2261 df v dv\n\ndiff-apply-proof-weak-f {\u03c4\u2081} {\u03c4\u2082} df f df-valid dv v dv-valid =\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (strong-to-weak-validity (derive-is-valid (apply dv v)))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv dv-valid)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof-weak {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv dv-valid)) \u27e9\n    df v dv\n  \u220e where open \u2261-Reasoning\n-}\n\n-- A stronger delta equivalence. Note this isn't a congruence; it\n-- should be possible to define some congruence rules, but those will\n-- be more complex.\n\ndata _\u2248_ : {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set where\n  --base : \u2200 (v : \u27e6 bool \u27e7) \u2192 v \u2248 v\n  base : \u2200 (v1 v2 : \u27e6 bool \u27e7) \u2192 (\u2261 : v1 \u2261 v2) \u2192 v1 \u2248 v2\n  fun : \u2200 {\u03c3 \u03c4} (f1 f2 : \u27e6 \u0394-Type (\u03c3 \u21d2 \u03c4) \u27e7) \u2192\n    (t\u2248 : \u2200 s ds (valid : Weak-Valid-\u0394 s ds) \u2192 f1 s ds \u2248 f2 s ds) \u2192\n    f1 \u2248 f2\n\n\u2261\u2192\u2248 : \u2200 {\u03c4} {v1 v2 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192 v1 \u2261 v2 \u2192 v1 \u2248 v2\n\u2261\u2192\u2248 {bool} {v1} {.v1} refl = base v1 v1 refl\n\u2261\u2192\u2248 {\u03c4\u2081 \u21d2 \u03c4\u2082} {v1} {.v1} refl = fun v1 v1 (\u03bb s ds _ \u2192 \u2261\u2192\u2248 refl)\n\nopen import Relation.Binary hiding (_\u21d2_)\n\n\u2248-refl : \u2200 {\u03c4} \u2192 Reflexive (_\u2248_ {\u03c4})\n\u2248-refl = \u2261\u2192\u2248 refl\n\n\u2248-sym : \u2200 {\u03c4} \u2192 Symmetric (_\u2248_ {\u03c4})\n\u2248-sym (base v1 v2 \u2261) = base _ _ (sym \u2261)\n\u2248-sym {(\u03c3 \u21d2 \u03c4)} (fun f1 f2 t\u2248) = fun _ _  (\u03bb s ds valid \u2192 \u2248-sym (t\u2248 s ds valid))\n\n\u2248-trans : \u2200 {\u03c4} \u2192 Transitive (_\u2248_ {\u03c4})\n\u2248-trans {.bool} {.k} {.k} {k} (base .k .k refl) (base .k .k refl) = base k k refl\n\u2248-trans {\u03c3 \u21d2 \u03c4} {.f1} {.f2} {k} (fun f1 .f2 \u2248\u2081) (fun f2 .k \u2248\u2082) = fun f1 k (\u03bb s ds valid \u2192 \u2248-trans (\u2248\u2081 _ _ valid) (\u2248\u2082 _ _ valid))\n\n\u2248-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2248_ {\u03c4})\n\u2248-isEquivalence = record\n  { refl  = \u2248-refl\n  ; sym   = \u2248-sym\n  ; trans = \u2248-trans\n  }\n\n\u2248-setoid : Type \u2192 Setoid _ _\n\u2248-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u0394-Type \u03c4 \u27e7\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\nmodule \u2248-Reasoning where\n  module _ {\u03c4 : Type} where\n    open import Relation.Binary.EqReasoning (\u2248-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n-- Correct proof, to refactor using \u2248-Reasoning\n\ndiff-apply-proof-weak-real : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Weak-Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2248 dv\ndiff-apply-proof-weak-real {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = fun _ _ (\u03bb v dv dv-valid \u2192 \u2248-trans (\u2261\u2192\u2248 (\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (strong-to-weak-validity (derive-is-valid (apply dv v)))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv dv-valid)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u220e)) (diff-apply-proof-weak-real {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv dv-valid))))\n  --\u2248\u27e8 fill in the above proof \u27e9\n    --df v dv\n\n  where open \u2261-Reasoning\n\ndiff-apply-proof-weak-real {bool} db b _ = \u2261\u2192\u2248 (xor-cancellative b db)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c358fe1ba64f44c5f57f72267b2aa9d0ffc61be4","subject":"Pseudocode for","message":"Pseudocode for\n\nOld-commit-hash: 3d42c268469649df0f436aa927eb3587fa8d09db\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/Extension-00.agda","new_file":"nats\/Extension-00.agda","new_contents":"{-\n\nChecklist of stuff to add when adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\n\nThis file explores introducing folds over natural numbers\ninto the object language.\n\n-}\n\nmodule Extension-00 where\n\nopen import Data.Nat using\n  ( \u2115\n  ; suc\n  ; fold\n  ; _+_\n  ; _\u2238_ -- x \u2238 y = max(0, x - y)\n  )\n\nopen import Data.Product\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Level using (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  foldNat : \u2200 {\u0393 \u03c4} \u2192 (z : Term \u0393 \u03c4) \u2192 (f : Term \u0393 (\u03c4 \u21d2 \u03c4))\n                    \u2192 (n : Term \u0393 nats)\n                    \u2192 Term \u0393 \u03c4\n  _plus_ : \u2200 {\u0393} \u2192 (m : Term \u0393 nats) \u2192 (n : Term \u0393 nats) \u2192 Term \u0393 nats\n  _minus_ : \u2200 {\u0393} \u2192 (m : Term \u0393 nats) \u2192 (n : Term \u0393 nats) \u2192 Term \u0393 nats\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 = \u227c-reflexive refl\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nweaken subctx (foldNat z f n) = \n  foldNat (weaken subctx z) (weaken subctx f) (weaken subctx n)\nweaken subctx (m plus n) = (weaken subctx m) plus (weaken subctx n)\nweaken subctx (m minus n) = (weaken subctx m) minus (weaken subctx n)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\n\u27e6 foldNat z f n \u27e7Term \u03c1 with \u27e6 n \u27e7Term \u03c1\n... | i = fold (\u27e6 z \u27e7Term \u03c1) (\u27e6 f \u27e7Term \u03c1) i\n\u27e6 m plus  n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 m minus n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 \u2238 \u27e6 n \u27e7Term \u03c1\n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (foldNat n f z) \u03c1 = {!!}\nweaken-sound (m plus n) \u03c1 = {!!}\nweaken-sound (m minus n) \u03c1 = {!!}\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n-- Syntactic sugars\n-- * if0-then-else\n-- * change application\n-- * term difference\n-- * replacement pairs\n\ninfix 4 if0_then_else_\n\nif0_then_else_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 nats \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nif0 condition then then-branch else else-branch =\n  foldNat then-branch (abs (weaken (drop _ \u2022 \u0393\u227c\u0393) else-branch)) condition\n\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n\nreplace_by_ : \u2200 {\u03c4 \u0393} \u2192 (old : Term \u0393 \u03c4) (new : Term \u0393 \u03c4)\n                      \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- Term difference is replacement because symbolic\n-- derivation isn't avaliable at object level yet\ns \u229d t = replace s by t\n\n-- n \u2295 dn = dn snd\n-- f \u2295 df = \u03bb x . f x \u2295 df x (x \u229d x)\n--\n-- Replacing same by same is used to get the nil change,\n-- because symbolic derivation is not part of the object\n-- language.\n_\u2295_ {nats} t dt = app dt snd\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} t dt = abs ((app t\u2032 x) \u2295 (app (app dt\u2032 x) (x \u229d x)))\n  where\n    drop-it = drop \u03c4\u2081 \u2022 \u0393\u227c\u0393\n    x = var this\n    t\u2032 = weaken drop-it t\n    dt\u2032 = weaken drop-it dt\n\n-- replace n by m = \u03bb f . f n m\n-- replace f by g = \u03bb x . \u03bb dx . replace (f x) by (g (x \u2295 dx))\n--\n-- Amazingly, weakening is identical to arbitrary\n-- legal adjustment of de-Bruijn indices.\nreplace_by_ {nats} {\u0393} old new =\n  abs (app (app (var this) (weaken drop-f old)) (weaken drop-f new))\n  where drop-f = drop _ \u2022 \u0393\u227c\u0393\n-- Think: the new must compute upon changes...\nreplace_by_ {\u03c4\u2081 \u21d2 \u03c4\u2082} old new =\n  abs (abs (replace (app (weaken drop! old) x)\n                 by (app (weaken drop! new) (x \u2295 dx))))\n  where\n    drop! = drop (\u0394-Type \u03c4\u2081) \u2022 drop \u03c4\u2081 \u2022 \u0393\u227c\u0393\n    dx = var this\n    x = var (that this)\n\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive (foldNat z f n) =\n--   let\n--     dz = derive z\n--     df = derive f\n--     dn = derive n\n--     n\u2032 = dn snd\n--   in\n--     if (n = n\u2032) -- if0 (n\u2032 \u2238 n) + (n \u2238 n\u2032)\n--       core\u2082\n--     if (n < n\u2032) -- else if0 (n - n\u2032)\n--       (replace (\u03bb x . x) by (\u03bb x . foldNat x f (n\u2032 \u2238 n))) core\u2081 core\u2082\n--     if (n > n\u2032) -- else\n--       (replace (\u03bb x . foldNat x f (n \u2238 n\u2032)) by (\u03bb x . x)) core\u2081 core\u2082\n--   where\n--     min[n,n\u2032] = if (n < n\u2032) then n else n\u2032\n--     -- core is a Church pair (core\u2081, core\u2082)\n--     -- where core\u2081 is the result of applying f to z min[n,n\u2032] times\n--     -- and core\u2082 is the change to core\u2081 caused by changes to f and z.\n--     core = foldNat (\u03bb g . g z \u0394z)\n--                    (\u03bb p . \u03bb g . g (f (p fst)) (df (p fst) (p snd)))\n--                    min[n,n\u2032]\n--     core\u2081 = core fst\n--     core\u2082 = core snd\nderive (foldNat z f n) =  {!!}\n\n-- derive (m plus n) = ?\nderive (m plus n) = {!!}\n\n-- derive (m minus n) = ?\nderive (m minus n) = {!!}\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} (foldNat z f n) = {!!}\nvalidity-of-derive \u03c1 {consistency} (m plus n) = {!!}\nvalidity-of-derive \u03c1 {consistency} (m minus n) = {!!}\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive {\u0393} {\u03c4}\n  \u03c1 {consistency} (foldNat z f n) = {!!}\n\ncorrectness-of-derive {\u0393} {nats}\n  \u03c1 {consistency} (m plus n) = {!!}\n\ncorrectness-of-derive {\u0393} {nats}\n  \u03c1 {consistency} (m minus n) = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","old_contents":"{-\n\nChecklist of stuff to add when adding syntactic constructs\n\n- derive (symbolic derivation; most important!)\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- validity-of-derive\n- correctness-of-derive\n\n-}\n\nmodule Extension-00 where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- ad-hoc extensions\n  foldNat : \u2200 {\u0393 \u03c4} \u2192 (n : Term \u0393 nats) \u2192\n                      (f : Term \u0393 (\u03c4 \u21d2 \u03c4)) \u2192 (z : Term \u0393 \u03c4)\n                    \u2192 Term \u0393 \u03c4\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 = \u227c-reflexive refl\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nweaken subctx (foldNat n f z) = {!!}\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\n\u27e6 foldNat n f z \u27e7Term \u03c1 = {!!}\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (foldNat n f z) \u03c1 = {!!}\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- Combination as type-indexed family of terms\n-- _\u2295_ : \u2200 {\u03c4 \u0393} \u2192  TODO: IMPLEMENT ME!! {!!}\n\n-- Replacement-pairs on all types as syntactic sugar\n\nreplace_by_ : \u2200 {\u03c4 \u0393} \u2192 (old : Term \u0393 \u03c4) (new : Term \u0393 \u03c4)\n                      \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- replace n by m = \u03bb f . f n m\n-- replace f by g = \u03bb x . \u03bb dx . replace (f x) by (g (x \u2295 dx))\n--\n-- Remark. Amazingly, weakening is identical to arbitrary\n-- legal adjustment of de-Bruijn indices.\nreplace_by_ {nats} {\u0393} old new =\n  abs (app (app (var this) (weaken drop-f old)) (weaken drop-f new))\n  where drop-f = drop _ \u2022 \u0393\u227c\u0393\n-- Think: the new must compute upon changes...\nreplace_by_ {\u03c4\u2081 \u21d2 \u03c4\u2082} old new =\n  abs (abs (replace (app (weaken drop! old) (var (that this)))\n                 by (app (weaken drop! new) {!!})))\n  where drop! = drop (\u0394-Type \u03c4\u2081) \u2022 drop \u03c4\u2081 \u2022 \u0393\u227c\u0393\n\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive (foldNat n f z) = ?\nderive (foldNat n f z) =  {!!}\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} (foldNat n f z) = {!!}\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive {\u0393} {\u03c4}\n  \u03c1 {consistency} (foldNat n f z) = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4b53dd9e6c407b8e8ee939aee74a311a390cb3b6","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Paper, prerequisites, tested versions of the ATPS, and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fossacs-2012\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites, tested versions of the ATPS, and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fossacs-2012\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cad090cf1face3e9ee7c8e4575440a427c74e66a","subject":"Add more exploration functions about binary trees","message":"Add more exploration functions about binary trees\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/BinTree.agda","new_file":"lib\/Explore\/BinTree.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.BinTree where\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Data.Tree.Binary\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import HoTT\nopen Equivalences\nopen import Type.Identities\nopen import Function.Extensionality\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Zero\nopen import Explore.Sum\nopen import Explore.Isomorphism\n\nfromBinTree : \u2200 {m} {A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree empty      = empty-explore\nfromBinTree (leaf x)   = point-explore x\nfromBinTree (fork \u2113 r) = merge-explore (fromBinTree \u2113) (fromBinTree r)\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind empty      = empty-explore-ind\nfromBinTree-ind (leaf x)   = point-explore-ind x\nfromBinTree-ind (fork \u2113 r) = merge-explore-ind (fromBinTree-ind \u2113)\n                                               (fromBinTree-ind r)\n{-\nfromBinTree : \u2200 {m A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree (leaf x)   _ _   f = f x\nfromBinTree (fork \u2113 r) \u03b5 _\u2219_ f = fromBinTree \u2113 \u03b5 _\u2219_ f \u2219 fromBinTree r \u03b5 _\u2219_ f\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind (leaf x)   P _  P\u2219 Pf = Pf x\nfromBinTree-ind (fork \u2113 r) P P\u03b5 P\u2219 Pf = P\u2219 (fromBinTree-ind \u2113 P P\u03b5 P\u2219 Pf)\n                                           (fromBinTree-ind r P P\u03b5 P\u2219 Pf)\n-}\n\nAnyP\u2261\u03a3fromBinTree : \u2200 {p}{A : \u2605 _}{P : A \u2192 \u2605 p}(xs : BinTree A) \u2192 Any P xs \u2261 \u03a3\u1d49 (fromBinTree xs) P\nAnyP\u2261\u03a3fromBinTree empty    = idp\nAnyP\u2261\u03a3fromBinTree (leaf x) = idp\nAnyP\u2261\u03a3fromBinTree (fork xs xs\u2081) = \u228e= (AnyP\u2261\u03a3fromBinTree xs) (AnyP\u2261\u03a3fromBinTree xs\u2081)\n\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{A : \u2605 \u2080} where\n\n\n  explore\u03a3\u2208 : \u2200 {m} xs \u2192 Explore m (\u03a3 A \u03bb x \u2192 Any (_\u2261_ x) xs)\n  explore\u03a3\u2208 empty = explore-iso (coe-equiv (Lift\u2261id \u2219 ! \u00d7\ud835\udfd8-snd  \u2219 \u00d7= idp (! Lift\u2261id))) Lift\ud835\udfd8\u1d49\n  explore\u03a3\u2208 (leaf x) = point-explore (x , idp)\n  explore\u03a3\u2208 (fork xs xs\u2081) = explore-iso (coe-equiv (! \u03a3\u228e-split)) (explore\u03a3\u2208 xs \u228e\u1d49 explore\u03a3\u2208 xs\u2081)\n\n  \u03a3\u1d49-adq-explore\u03a3\u2208 : \u2200 {m} xs \u2192 Adequate-\u03a3\u1d49 {m} (explore\u03a3\u2208 xs)\n  \u03a3\u1d49-adq-explore\u03a3\u2208 empty = \u03a3-iso-ok (coe-equiv (Lift\u2261id \u2219 ! \u00d7\ud835\udfd8-snd \u2219 \u00d7= idp (! Lift\u2261id)))\n    {A\u1d49 = Lift\ud835\udfd8\u1d49} \u03a3\u1d49Lift\ud835\udfd8-ok\n  \u03a3\u1d49-adq-explore\u03a3\u2208 (leaf x\u2081) F = ! \u03a3\ud835\udfd9-snd \u2219 \u03a3-fst\u2243 (\u2243-sym (\u03a3x\u2261\u2243\ud835\udfd9 x\u2081)) F\n  \u03a3\u1d49-adq-explore\u03a3\u2208 (fork xs xs\u2081) = \u03a3-iso-ok (coe-equiv (! \u03a3\u228e-split)) {A\u1d49 = explore\u03a3\u2208 xs \u228e\u1d49 explore\u03a3\u2208 xs\u2081}\n               (\u03a3\u1d49\u228e-ok {e\u1d2c = explore\u03a3\u2208 xs}{e\u1d2e = explore\u03a3\u2208 xs\u2081} (\u03a3\u1d49-adq-explore\u03a3\u2208 xs) (\u03a3\u1d49-adq-explore\u03a3\u2208 xs\u2081))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{A : \u2605 \u2080}{P : A \u2192 \u2605 _}(explore-P : \u2200 {m} x \u2192 Explore m (P x)) where\n  open import Explore.Zero\n  open import Explore.Sum\n  open import Explore.Isomorphism\n\n  exploreAny : \u2200 {m} xs \u2192 Explore m (Any P xs)\n  exploreAny empty         = Lift\ud835\udfd8\u1d49\n  exploreAny (leaf x)      = explore-P x\n  exploreAny (fork xs xs\u2081) = exploreAny xs \u228e\u1d49 exploreAny xs\u2081\n\n  module _ (\u03a3\u1d49-adq-explore-P : \u2200 {m} x \u2192 Adequate-\u03a3\u1d49 {m} (explore-P x)) where\n    \u03a3\u1d49-adq-exploreAny : \u2200 {m} xs \u2192 Adequate-\u03a3\u1d49 {m} (exploreAny xs)\n    \u03a3\u1d49-adq-exploreAny empty F         = ! \u03a3\ud835\udfd8-lift\u2218fst \u2219 \u03a3-fst= (! Lift\u2261id) _\n    \u03a3\u1d49-adq-exploreAny (leaf x\u2081) F     = \u03a3\u1d49-adq-explore-P x\u2081 F\n    \u03a3\u1d49-adq-exploreAny (fork xs xs\u2081) F = \u228e= (\u03a3\u1d49-adq-exploreAny xs _) (\u03a3\u1d49-adq-exploreAny xs\u2081 _) \u2219 ! dist-\u228e-\u03a3\n\n  explore\u03a3\u1d49 : \u2200 {m} xs \u2192 Explore m (\u03a3\u1d49 (fromBinTree xs) P)\n  explore\u03a3\u1d49 {m} xs = fromBinTree-ind xs (\u03bb e \u2192 Explore m (\u03a3\u1d49 e P)) Lift\ud835\udfd8\u1d49 _\u228e\u1d49_ explore-P\n\n  module _ (\u03a3\u1d49-adq-explore-P : \u2200 {m} x \u2192 Adequate-\u03a3\u1d49 {m} (explore-P x)) where\n    \u03a3\u1d49-adq-explore\u03a3\u1d49 : \u2200 {m} xs \u2192 Adequate-\u03a3\u1d49 {m} (explore\u03a3\u1d49 xs)\n    \u03a3\u1d49-adq-explore\u03a3\u1d49 empty F         = ! \u03a3\ud835\udfd8-lift\u2218fst \u2219 \u03a3-fst= (! Lift\u2261id) _\n    \u03a3\u1d49-adq-explore\u03a3\u1d49 (leaf x\u2081) F     = \u03a3\u1d49-adq-explore-P x\u2081 F\n    \u03a3\u1d49-adq-explore\u03a3\u1d49 (fork xs xs\u2081) F = \u228e= (\u03a3\u1d49-adq-explore\u03a3\u1d49 xs _) (\u03a3\u1d49-adq-explore\u03a3\u1d49 xs\u2081 _) \u2219 ! dist-\u228e-\u03a3\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.BinTree where\n\nopen import Data.Tree.Binary\n\nopen import Explore.Core\nopen import Explore.Properties\n\nfromBinTree : \u2200 {m} {A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree empty      = empty-explore\nfromBinTree (leaf x)   = point-explore x\nfromBinTree (fork \u2113 r) = merge-explore (fromBinTree \u2113) (fromBinTree r)\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind empty      = empty-explore-ind\nfromBinTree-ind (leaf x)   = point-explore-ind x\nfromBinTree-ind (fork \u2113 r) = merge-explore-ind (fromBinTree-ind \u2113)\n                                               (fromBinTree-ind r)\n{-\nfromBinTree : \u2200 {m A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree (leaf x)   _ _   f = f x\nfromBinTree (fork \u2113 r) \u03b5 _\u2219_ f = fromBinTree \u2113 \u03b5 _\u2219_ f \u2219 fromBinTree r \u03b5 _\u2219_ f\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind (leaf x)   P _  P\u2219 Pf = Pf x\nfromBinTree-ind (fork \u2113 r) P P\u03b5 P\u2219 Pf = P\u2219 (fromBinTree-ind \u2113 P P\u03b5 P\u2219 Pf)\n                                           (fromBinTree-ind r P P\u03b5 P\u2219 Pf)\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a51b1493ba6e471baa5091f2b629e860cc75ca83","subject":"Bits: propertis about search and count","message":"Bits: propertis about search and count\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {zero}  _   f = f []\nsearch {suc n} _\u00b7_ f = search _\u00b7_ (f \u2218 0\u2237_) \u00b7 search _\u00b7_ (f \u2218 1\u2237_)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = search _+_ (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\nsearch-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\nsearch-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n  where\n    go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n    go zero = refl\n    go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^_ zero = refl\n#always\u22612^_ (suc n) = cong\u2082 _+_ pf pf where pf = #always\u22612^ n\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op {-(f |\u2228| g) (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))-} (|de-morgan| f g)\n\nsearch-comm :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-comm {zero} _+_ _*_ f p hom = refl\nsearch-comm {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-comm {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-comm _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n[0\u2194_] : \u2200 {n} \u2192 Fin n \u2192 Bits n \u2192 Bits n\n[0\u2194_] {zero}  i xs = xs\n[0\u2194_] {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n[0\u21941] : Bits 2 \u2192 Bits 2\n[0\u21941] = [0\u2194 suc zero ]\n\n[0\u21941]-spec : [0\u21941] \u2257 (\u03bb { (x \u2237 y \u2237 []) \u2192 y \u2237 x \u2237 [] })\n[0\u21941]-spec (x \u2237 y \u2237 []) = refl\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {zero}  _   f = f []\nsearch {suc n} _\u00b7_ f = search _\u00b7_ (f \u2218 0\u2237_) \u00b7 search _\u00b7_ (f \u2218 1\u2237_)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = search _+_ (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"15cd0eafd613a991286818c276496c93db0ada5d","subject":"Drop \u27e6_\u27e7*TypeHidCacheWrong","message":"Drop \u27e6_\u27e7*TypeHidCacheWrong\n\nThis wrong definition should not be counted among the failures before\nCBPV, since it was created after and it is due to a separate limitation.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingMValue.agda","new_file":"Parametric\/Denotation\/CachingMValue.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for caching evaluation of MTerm\n------------------------------------------------------------------------\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.MType as MType\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.MValue as MValue\n\nmodule Parametric.Denotation.CachingMValue\n    (Base : Type.Structure)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\nopen MValue.Structure Base \u27e6_\u27e7Base\n\nopen import Data.Product hiding (map)\nopen import Data.Sum hiding (map)\nopen import Data.Unit\nopen import Level\nopen import Function hiding (const)\n\nmodule Structure where\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  --\n  -- XXX: This line is the only change, up to now, for the caching semantics,\n  -- the rest is copied. Inheritance would handle this precisely; without\n  -- inheritance, we might want to use one of the standard encodings of related\n  -- features (delegation?).\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for caching evaluation of MTerm\n------------------------------------------------------------------------\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.MType as MType\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.MValue as MValue\n\nmodule Parametric.Denotation.CachingMValue\n    (Base : Type.Structure)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\nopen MValue.Structure Base \u27e6_\u27e7Base\n\nopen import Data.Product hiding (map)\nopen import Data.Sum hiding (map)\nopen import Data.Unit\nopen import Level\nopen import Function hiding (const)\n\nmodule Structure where\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCacheWrong )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a5ce35c64b2c723b065d127b6ae2c8a7e2fba61c","subject":"Embed62.agda:410: main theorem as stated in slides for Hessen Workshop (fix #68)","message":"Embed62.agda:410: main theorem as stated in slides for Hessen Workshop\n(fix #68)\n\nOld-commit-hash: 7254a11f4c494091307a31436b4d31d567d52b12\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/Embed62.agda","new_file":"experimental\/Embed62.agda","new_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import ExplicitNil using (ext\u00b3)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Data.Nat\n\nnatPairVisitor : Type\nnatPairVisitor = nats \u21d2 nats \u21d2 nats\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = natPairVisitor \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd! : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd! _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\noldFrom newFrom : \u2200 {\u0393} \u2192 Term \u0393 (\u0394-Type nats) \u2192 Term \u0393 nats\noldFrom d = app d fst\nnewFrom d = app d snd\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) t)) (weaken (drop _ \u2022 \u0393\u227c\u0393) s))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nnatPair : \u2200 {\u0393} \u2192 (old new : Term (natPairVisitor \u2022 \u0393) nats) \u2192 Term \u0393 (\u0394-Type nats)\nnatPair old new = abs (app (app (var this) old) new)\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = natPair (nat old) (nat new)\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = natPair\n  (add (oldFrom (embedWeaken ds))\n       (oldFrom (embedWeaken dt)))\n  (add (newFrom (embedWeaken ds))\n       (newFrom (embedWeaken dt)))\n  where\n    embedWeaken : \u2200 {\u0393 \u03c4} \u2192 (d : \u0394Term \u0393 nats) \u2192\n      Term (\u03c4 \u2022 \u0394-Context \u0393) (natPairVisitor \u21d2 nats)\n    embedWeaken d = (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed d))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {nats} (old , new) v = old \u2261 v \u27e6fst\u27e7 \u00d7 new \u2261 v \u27e6snd\u27e7\n_\u2248_ {bags} u v = u \u2261 v\n_\u2248_ {\u03c3 \u21d2 \u03c4} u v = \n  (w : \u27e6 \u03c3 \u27e7) (dw : \u0394Val \u03c3) (dw\u2032 : \u27e6 \u0394-Type \u03c3 \u27e7)\n  (R[w,dw] : valid w dw) (eq : dw \u2248 dw\u2032) \u2192\n  u w dw R[w,dw] \u2248 v w dw\u2032\n\ninfix 4 _\u2248_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = EmptySet\ncompatible {\u03c4 \u2022 \u0393} (cons v dv _ \u03c1) (dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  triple (v \u2261 v\u2032) (dv \u2248 dv\u2032) (compatible \u03c1 \u03c1\u2032)\n\nderiveVar-correct : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 x \u27e7\u0394Var \u03c1 \u2248 \u27e6 deriveVar x \u27e7 \u03c1\u2032\n\nderiveVar-correct {x = this} -- pattern-matching on \u03c1, \u03c1\u2032 NOT optional\n  {cons _ _ _ _} {_ \u2022 _ \u2022 _} {cons _ dv\u2248dv\u2032 _ _} = dv\u2248dv\u2032\nderiveVar-correct {x = that y}\n  {cons _ _ _ \u03c1} {_ \u2022 _ \u2022 \u03c1\u2032} {cons _ _ C _} =\n  deriveVar-correct {x = y} {\u03c1} {\u03c1\u2032} {C}\n\nweak-id : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nweak-id {\u2205} {\u2205} = refl\nweak-id {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} = cong\u2082 _\u2022_ {x = v} refl (weak-id {\u0393} {\u03c1})\n\nweak-eq : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\nweak-eq {\u2205} {\u2205} {\u2205} _ = refl\nweak-eq {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032} (cons v\u2261v\u2032 _ C _) =\n  cong\u2082 _\u2022_ v\u2261v\u2032 (weak-eq C)\n\nweak-eq-term : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\u2032\n\nweak-eq-term t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (weak-eq C)) (sym (weaken-sound t \u03c1\u2032))\n\nweaken-once : \u2200 {\u03c4 \u0393 \u03c3} {t : Term \u0393 \u03c4} {v : \u27e6 \u03c3 \u27e7} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) t \u27e7 (v \u2022 \u03c1)\n\nweaken-once {t = t} {v} {\u03c1} = trans\n  (cong \u27e6 t \u27e7 (sym weak-id))\n  (sym (weaken-sound t (v \u2022 \u03c1)))\n\nweaken-twice : \u2200 {\u03c4 \u0393 \u03c3\u2080 \u03c3\u2081} {t : Term \u0393 \u03c4}\n  {u : \u27e6 \u03c3\u2080 \u27e7} {v : \u27e6 \u03c3\u2081 \u27e7} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 weaken (drop \u03c3\u2080 \u2022 drop \u03c3\u2081 \u2022 \u0393\u227c\u0393) t \u27e7 (u \u2022 v \u2022 \u03c1)\n\nweaken-twice {t = t} {u} {v} {\u03c1} = trans\n  (cong \u27e6 t \u27e7 (sym weak-id))\n  (sym (weaken-sound t (u \u2022 v \u2022 \u03c1)))\n\nembed-correct : \u2200 {\u03c4 \u0393}\n  {dt : \u0394Term \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {V : dt is-valid-for \u03c1}\n  {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 dt \u27e7\u0394 \u03c1 V \u2248 \u27e6 embed dt \u27e7 \u03c1\u2032\n\nembed-correct {dt = \u0394nat old new} = refl , refl\n\nembed-correct {dt = \u0394bag db} = refl\n\nembed-correct {dt = \u0394var x} {\u03c1} {\u03c1\u2032 = \u03c1\u2032} {C} =\n  deriveVar-correct {x = x} {\u03c1} {\u03c1\u2032} {C}\n\nembed-correct {dt = \u0394add ds dt} {\u03c1} {V} {\u03c1\u2032} {C} =\n  let\n    s00 , s01 = \u27e6 ds \u27e7\u0394 \u03c1 (car V)\n    t00 , t01 = \u27e6 dt \u27e7\u0394 \u03c1 (cadr V)\n    s10 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    s11 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    t10 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    t11 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    rec-s = embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n    rec-t = embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C}\n  in\n    \n    (begin\n      s00 + t00\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2081 rec-s) (proj\u2081 rec-t) \u27e9\n      s10 + t10\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6fst\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6fst\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed ds) (\u27e6fst\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed dt) (\u27e6fst\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7\n    \u220e)\n    ,\n    (begin\n      s01 + t01\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2082 rec-s) (proj\u2082 rec-t) \u27e9\n      s11 + t11\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6snd\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6snd\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed ds) (\u27e6snd\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed dt) (\u27e6snd\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7\n    \u220e)\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394union ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _++_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394diff ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2081 s dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 mapBag\n  (weak-eq-term s C)\n  (embed-correct {dt = dt} {\u03c1} {car V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2080 s ds t dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (cong\u2082 mapBag\n    (extensionality (\u03bb v \u2192\n    begin\n      proj\u2082 (\u27e6 ds \u27e7\u0394 \u03c1 (car V) v (v , v) refl)\n    \u2261\u27e8 proj\u2082 (embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n        v (v , v) (\u03bb f \u2192 f v v) refl (refl , refl)) \u27e9\n      \u27e6 embed ds \u27e7 \u03c1\u2032 v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole v (\u03bb f \u2192 f v v) \u27e6snd\u27e7)\n        (weaken-once {t = embed ds} {v} {\u03c1\u2032}) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (v \u2022 \u03c1\u2032) v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u220e))\n    (cong\u2082 _++_\n      (weak-eq-term t C)\n      (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})))\n  (cong\u2082 mapBag (weak-eq-term s C) (weak-eq-term t C))\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394app ds t dt} {\u03c1} {cons vds vdt R[t,dt] _} {\u03c1\u2032} {C}\n  rewrite sym (weak-eq-term t C) =\n  embed-correct {dt = ds} {\u03c1} {vds} {\u03c1\u2032} {C}\n    (\u27e6 t \u27e7Term (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 vdt) (\u27e6 embed dt \u27e7Term \u03c1\u2032)\n    R[t,dt] (embed-correct {dt = dt} {\u03c1} {vdt} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394abs dt} {\u03c1} {V} {\u03c1\u2032} {C} = \u03bb w dw dw\u2032 R[w,dw] dw\u2248dw\u2032 \u2192\n  embed-correct {dt = dt}\n    {cons w dw R[w,dw] \u03c1} {V w dw R[w,dw]}\n    {dw\u2032 \u2022 w \u2022 \u03c1\u2032} {cons refl dw\u2248dw\u2032 C tt}\n\n_\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ninfixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n_\u229e_ {nats}  n dn = dn \u27e6snd\u27e7\n_\u229e_ {bags}  b db = b ++ db\n_\u229e_ {\u03c3 \u21d2 \u03c4} f df = \u03bb v \u2192 f v \u229e df v (v \u229f v)\n\n_\u229f_ {nats} m n = \u03bb f \u2192 f n m\n_\u229f_ {bags} d b = d \\\\ b\n_\u229f_ {\u03c3 \u21d2 \u03c4} g f = \u03bb v dv \u2192 g (v \u229e dv) \u229f f v\n\nu\u229dv\u2248u\u229fv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 u \u229d v \u2248 u \u229f v\n\ncarry-over : \u2200 {\u03c4}\n  {dv : \u0394Val \u03c4} {dv\u2032 : \u27e6 \u0394-Type \u03c4 \u27e7} (dv\u2248dv\u2032 : dv \u2248 dv\u2032)\n  {v : \u27e6 \u03c4 \u27e7} (R[v,dv] : valid v dv) \u2192\n  v \u2295 dv \u2261 v \u229e dv\u2032\n\nu\u229dv\u2248u\u229fv {nats} = refl , refl\nu\u229dv\u2248u\u229fv {bags} = refl\nu\u229dv\u2248u\u229fv {\u03c3 \u21d2 \u03c4} {g} {f} = result\n  where\n  result : \u2200 (w : \u27e6 \u03c3 \u27e7) (dw : \u0394Val \u03c3) (dw\u2032 : \u27e6 \u0394-Type \u03c3 \u27e7)\n    (R[w,dw] : valid w dw) (dw\u2248dw\u2032 : dw \u2248 dw\u2032) \u2192\n    g (w \u2295 dw) \u229d f w \u2248 g (w \u229e dw\u2032) \u229f f w\n  result w dw dw\u2032 R[w,dw] dw\u2248dw\u2032\n    rewrite carry-over dw\u2248dw\u2032 R[w,dw] = u\u229dv\u2248u\u229fv\n  -- Question: is there anyway not to write the signature\n  -- of u\u229dv\u2248u\u229fv twice just so as to carry out the rewrite?\n\ncarry-over {nats} dv\u2248dv\u2032 R[v,dv] = proj\u2082 dv\u2248dv\u2032\ncarry-over {bags} dv\u2248dv\u2032 {v} R[v,dv] = cong (_++_ v) dv\u2248dv\u2032\ncarry-over {\u03c3 \u21d2 \u03c4} {df} {df\u2032} df\u2248df\u2032 {f} R[f,df] =\n  extensionality (\u03bb v \u2192 carry-over {\u03c4}\n    {df v (v \u229d v) R[v,u\u229dv]} {df\u2032 v (v \u229f v)}\n    (df\u2248df\u2032 v (v \u229d v) (v \u229f v) R[v,u\u229dv] u\u229dv\u2248u\u229fv)\n    {f v}\n    (proj\u2081 (R[f,df] v (v \u229d v) R[v,u\u229dv])))\n\nignore\u2032 : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 {\u2205} \u2205 = \u2205\nignore\u2032 {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore\u2032 \u03c1\n\nupdate\u2032 : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 {\u2205} \u2205 = \u2205\nupdate\u2032 {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = (v \u229e dv) \u2022 update\u2032 \u03c1\n\nignorance : \u2200 {\u0393}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  (ignore \u03c1) \u2261 (ignore\u2032 \u03c1\u2032)\n\nignorance {\u2205} {\u2205} {\u2205} _ = refl\nignorance {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032}\n  (cons v\u2261v\u2032 _ C _) = cong\u2082 _\u2022_ v\u2261v\u2032 (ignorance C)\n\n-- ... because \"updatedness\" sounds horrible\nactualit\u00e9 : \u2200 {\u0393}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  (update \u03c1) \u2261 (update\u2032 \u03c1\u2032)\n\nactualit\u00e9 {\u2205} {\u2205} {\u2205} _ = refl\nactualit\u00e9 {\u03c4 \u2022 \u0393} {cons v dv R[v,dv] \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032}\n  (cons v\u2261v\u2032 dv\u2248dv\u2032 C _) rewrite v\u2261v\u2032 =\n  cong\u2082 _\u2022_ (carry-over dv\u2248dv\u2032 R[v,dv]) (actualit\u00e9 C)\n\nignore-both : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 t \u27e7 (ignore\u2032 \u03c1\u2032)\n\nignore-both t C rewrite ignorance C = refl\n\nupdate-both : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (update \u03c1) \u2261 \u27e6 t \u27e7 (update\u2032 \u03c1\u2032)\n\nupdate-both t C rewrite actualit\u00e9 C = refl\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  {v : \u27e6 \u03c3 \u27e7} {dv : \u0394Val \u03c3} {R[v,dv] : valid v dv}\n  \u2192 \u27e6 t \u27e7 \u2205 (v \u2295 dv)\n  \u2261 \u27e6 t \u27e7 \u2205 v \u2295 \u27e6 derive t \u27e7\u0394 \u2205 (unrestricted t) v dv R[v,dv]\n\ncorollary-closed {t = t} {v} {dv} {R[v,dv]} =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    df = \u27e6 derive t \u27e7\u0394 \u2205 (unrestricted t)\n  in\n    begin\n      f (v \u2295 dv)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (v \u2295 dv)) (sym (correctness {t = t})) \u27e9\n      (f \u2295 df) (v \u2295 dv)\n    \u2261\u27e8 proj\u2082 (validity {t = t} v dv R[v,dv]) \u27e9\n      f v \u2295 df v dv R[v,dv]\n    \u220e where open \u2261-Reasoning\n\n\u27e6apply\u27e7 : \u2200 {\u03c4 \u0393}\n  {t : Term \u0393 \u03c4} {dt : Term \u0393 (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u229e \u27e6 dt \u27e7 \u03c1 \u2261 \u27e6 apply dt t \u27e7 \u03c1\n\n\u27e6difff\u27e7 : \u2200 {\u03c4 \u0393}\n  {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u27e7 \u03c1 \u229f \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 difff s t \u27e7 \u03c1\n\n\u27e6apply\u27e7 {nats} {\u0393} {t} {dt} {\u03c1} = refl\n\u27e6apply\u27e7 {bags} {\u0393} {t} {dt} {\u03c1} = refl\n\u27e6apply\u27e7 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {dt} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u27e6 \u03c3 \u2022 \u0393 \u27e7Context\n    \u03c1\u2032 = v \u2022 \u03c1\n    dv = \u27e6 difff (var this) (var this) \u27e7 \u03c1\u2032\n    t-v = app (weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) t) (var this)\n    dt-v-dv = app (app\n      (weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) dt) (var this)) (difff (var this) (var this))\n  in\n    begin\n      \u27e6 t \u27e7 \u03c1 v \u229e \u27e6 dt \u27e7 \u03c1 v (v \u229f v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v \u229e \u27e6 dt \u27e7 \u03c1 v hole)\n        (\u27e6difff\u27e7 {s = var this} {var this} {\u03c1\u2032}) \u27e9\n      \u27e6 t \u27e7 \u03c1 v \u229e \u27e6 dt \u27e7 \u03c1 v dv\n    \u2261\u27e8 cong\u2082 _\u229e_\n        (cong (\u03bb hole \u2192 hole v   ) (weaken-once {t = t}))\n        (cong (\u03bb hole \u2192 hole v dv) (weaken-once {t = dt})) \u27e9\n      \u27e6 t-v \u27e7 \u03c1\u2032 \u229e \u27e6 dt-v-dv \u27e7 \u03c1\u2032\n    \u2261\u27e8 \u27e6apply\u27e7 \u27e9\n      \u27e6 apply dt-v-dv t-v \u27e7 \u03c1\u2032\n    \u220e) where\n    open \u2261-Reasoning\n\n\u27e6difff\u27e7 {nats} {\u0393} {s} {t} {\u03c1} = extensionality result\n  where\n  result : (f : \u27e6 nats \u21d2 nats \u21d2 nats \u27e7)\n    \u2192 f (\u27e6 t \u27e7 \u03c1) (\u27e6 s \u27e7 \u03c1)\n    \u2261 f (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) t \u27e7 (f \u2022 \u03c1))\n        (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) s \u27e7 (f \u2022 \u03c1))\n  result f rewrite weaken-once {t = s} {f} {\u03c1}\n                 | weaken-once {t = t} {f} {\u03c1} = refl\n\u27e6difff\u27e7 {bags} {\u0393} {s} {t} {\u03c1} = refl\n\u27e6difff\u27e7 {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} = \n  extensionality (\u03bb v \u2192 extensionality (\u03bb dv \u2192\n  let\n    \u03c1\u2032 : \u27e6 (\u0394-Type \u03c3) \u2022 \u03c3 \u2022 \u0393 \u27e7Context\n    \u03c1\u2032 = dv \u2022 v \u2022 \u03c1\n    f = weaken (drop \u0394-Type \u03c3 \u2022 (drop \u03c3 \u2022 \u0393\u227c\u0393)) s\n    t-x = app (weaken (drop \u0394-Type \u03c3 \u2022 (drop \u03c3 \u2022 \u0393\u227c\u0393)) t)\n              (var (that this))\n    x\u229edx = apply (var this) (var (that this))\n  in\n    begin\n      \u27e6 s \u27e7 \u03c1 (v \u229e dv) \u229f \u27e6 t \u27e7 \u03c1 v\n    \u2261\u27e8 cong\u2082 _\u229f_\n        (cong\u2082 (\u03bb f x \u2192 f x)\n          (weaken-twice {t = s} {dv} {v} {\u03c1}) \u27e6apply\u27e7)\n        (cong (\u03bb hole \u2192 hole v)\n          (weaken-twice {t = t} {dv} {v} {\u03c1})) \u27e9\n      \u27e6 f \u27e7 \u03c1\u2032 (\u27e6 x\u229edx \u27e7 \u03c1\u2032) \u229f \u27e6 t-x \u27e7 \u03c1\u2032\n    \u2261\u27e8 \u27e6difff\u27e7 {s = app f x\u229edx} {t-x} \u27e9\n      \u27e6 difff (app f x\u229edx) t-x \u27e7 \u03c1\u2032\n    \u220e)) where open \u2261-Reasoning\n\nmain-theorem : \u2200 {\u03c3 \u03c4}\n  {s : Term \u2205 (\u03c3 \u21d2 \u03c4)} {t\u2080 : Term \u2205 \u03c3} {t\u2081 : Term \u2205 \u03c3}\n  \u2192 \u27e6 app s t\u2081 \u27e7 \u2205\n  \u2261 \u27e6 apply (app (app (embed (derive s)) t\u2080) (difff t\u2081 t\u2080)) (app s t\u2080) \u27e7 \u2205\n\nmain-theorem {s = s} {t\u2080} {t\u2081} =\n  let\n    f  = \u27e6 s \u27e7 \u2205\n    df = \u27e6 derive s \u27e7\u0394 \u2205 (unrestricted s)\n    df\u2032 = \u27e6 embed (derive s) \u27e7 \u2205\n    v  = \u27e6 t\u2080 \u27e7 \u2205\n    u  = \u27e6 t\u2081 \u27e7 \u2205\n  in\n    begin\n      f u\n    \u2261\u27e8 cong (\u03bb hole \u2192 f hole) (sym v\u2295[u\u229dv]=u) \u27e9\n      f (v \u2295 (u \u229d v))\n    \u2261\u27e8 corollary-closed {t = s} {v} {u \u229d v} {R[v,u\u229dv]} \u27e9\n      f v \u2295 df v (u \u229d v) R[v,u\u229dv]\n    \u2261\u27e8 carry-over\n        (embed-correct {\u0393 = \u2205} {dt = derive s}\n          {\u2205} {unrestricted s} {\u2205} {\u2205}\n          v (u \u229d v) (u \u229f v) R[v,u\u229dv] u\u229dv\u2248u\u229fv)\n        (proj\u2081 (validity {\u0393 = \u2205} {s} v (u \u229d v) R[v,u\u229dv])) \u27e9\n      f v \u229e df\u2032 v (u \u229f v)\n    \u2261\u27e8 trans (cong (\u03bb hole \u2192 f v \u229e df\u2032 v hole)\n        (\u27e6difff\u27e7 {s = t\u2081} {t\u2080})) \u27e6apply\u27e7 \u27e9\n      \u27e6 apply (app (app (embed (derive s)) t\u2080) (difff t\u2081 t\u2080))\n              (app s t\u2080) \u27e7 \u2205\n    \u220e where open \u2261-Reasoning\n","old_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import ExplicitNil using (ext\u00b3)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Data.Nat\n\nnatPairVisitor : Type\nnatPairVisitor = nats \u21d2 nats \u21d2 nats\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = natPairVisitor \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd! : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd! _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\noldFrom newFrom : \u2200 {\u0393} \u2192 Term \u0393 (\u0394-Type nats) \u2192 Term \u0393 nats\noldFrom d = app d fst\nnewFrom d = app d snd\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) s)) (weaken (drop _ \u2022 \u0393\u227c\u0393) t))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nnatPair : \u2200 {\u0393} \u2192 (old new : Term (natPairVisitor \u2022 \u0393) nats) \u2192 Term \u0393 (\u0394-Type nats)\nnatPair old new = abs (app (app (var this) old) new)\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = natPair (nat old) (nat new)\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = natPair\n  (add (oldFrom (embedWeaken ds))\n       (oldFrom (embedWeaken dt)))\n  (add (newFrom (embedWeaken ds))\n       (newFrom (embedWeaken dt)))\n  where\n    embedWeaken : \u2200 {\u0393 \u03c4} \u2192 (d : \u0394Term \u0393 nats) \u2192\n      Term (\u03c4 \u2022 \u0394-Context \u0393) (natPairVisitor \u21d2 nats)\n    embedWeaken d = (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed d))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {nats} (old , new) v = old \u2261 v \u27e6fst\u27e7 \u00d7 new \u2261 v \u27e6snd\u27e7\n_\u2248_ {bags} u v = u \u2261 v\n_\u2248_ {\u03c3 \u21d2 \u03c4} u v = \n  (w : \u27e6 \u03c3 \u27e7) (dw : \u0394Val \u03c3) (dw\u2032 : \u27e6 \u0394-Type \u03c3 \u27e7)\n  (R[w,dw] : valid w dw) (eq : dw \u2248 dw\u2032) \u2192\n  u w dw R[w,dw] \u2248 v w dw\u2032\n\ninfix 4 _\u2248_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = EmptySet\ncompatible {\u03c4 \u2022 \u0393} (cons v dv _ \u03c1) (dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  triple (v \u2261 v\u2032) (dv \u2248 dv\u2032) (compatible \u03c1 \u03c1\u2032)\n\nderiveVar-correct : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 x \u27e7\u0394Var \u03c1 \u2248 \u27e6 deriveVar x \u27e7 \u03c1\u2032\n\nderiveVar-correct {x = this} -- pattern-matching on \u03c1, \u03c1\u2032 NOT optional\n  {cons _ _ _ _} {_ \u2022 _ \u2022 _} {cons _ dv\u2248dv\u2032 _ _} = dv\u2248dv\u2032\nderiveVar-correct {x = that y}\n  {cons _ _ _ \u03c1} {_ \u2022 _ \u2022 \u03c1\u2032} {cons _ _ C _} =\n  deriveVar-correct {x = y} {\u03c1} {\u03c1\u2032} {C}\n\nweak-id : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nweak-id {\u2205} {\u2205} = refl\nweak-id {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} = cong\u2082 _\u2022_ {x = v} refl (weak-id {\u0393} {\u03c1})\n\nweak-eq : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\nweak-eq {\u2205} {\u2205} {\u2205} _ = refl\nweak-eq {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032} (cons v\u2261v\u2032 _ C _) =\n  cong\u2082 _\u2022_ v\u2261v\u2032 (weak-eq C)\n\nweak-eq-term : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\u2032\n\nweak-eq-term t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (weak-eq C)) (sym (weaken-sound t \u03c1\u2032))\n\nweaken-once : \u2200 {\u03c4 \u0393 \u03c3} {t : Term \u0393 \u03c4} {v : \u27e6 \u03c3 \u27e7} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) t \u27e7 (v \u2022 \u03c1)\n\nweaken-once {t = t} {v} {\u03c1} = trans\n  (cong \u27e6 t \u27e7 (sym weak-id))\n  (sym (weaken-sound t (v \u2022 \u03c1)))\n\nembed-correct : \u2200 {\u03c4 \u0393}\n  {dt : \u0394Term \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {V : dt is-valid-for \u03c1}\n  {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 dt \u27e7\u0394 \u03c1 V \u2248 \u27e6 embed dt \u27e7 \u03c1\u2032\n\nembed-correct {dt = \u0394nat old new} = refl , refl\n\nembed-correct {dt = \u0394bag db} = refl\n\nembed-correct {dt = \u0394var x} {\u03c1} {\u03c1\u2032 = \u03c1\u2032} {C} =\n  deriveVar-correct {x = x} {\u03c1} {\u03c1\u2032} {C}\n\nembed-correct {dt = \u0394add ds dt} {\u03c1} {V} {\u03c1\u2032} {C} =\n  let\n    s00 , s01 = \u27e6 ds \u27e7\u0394 \u03c1 (car V)\n    t00 , t01 = \u27e6 dt \u27e7\u0394 \u03c1 (cadr V)\n    s10 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    s11 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    t10 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    t11 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    rec-s = embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n    rec-t = embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C}\n  in\n    \n    (begin\n      s00 + t00\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2081 rec-s) (proj\u2081 rec-t) \u27e9\n      s10 + t10\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6fst\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6fst\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed ds) (\u27e6fst\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed dt) (\u27e6fst\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7\n    \u220e)\n    ,\n    (begin\n      s01 + t01\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2082 rec-s) (proj\u2082 rec-t) \u27e9\n      s11 + t11\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6snd\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6snd\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed ds) (\u27e6snd\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed dt) (\u27e6snd\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7\n    \u220e)\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394union ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _++_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394diff ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2081 s dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 mapBag\n  (weak-eq-term s C)\n  (embed-correct {dt = dt} {\u03c1} {car V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2080 s ds t dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (cong\u2082 mapBag\n    (extensionality (\u03bb v \u2192\n    begin\n      proj\u2082 (\u27e6 ds \u27e7\u0394 \u03c1 (car V) v (v , v) refl)\n    \u2261\u27e8 proj\u2082 (embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n        v (v , v) (\u03bb f \u2192 f v v) refl (refl , refl)) \u27e9\n      \u27e6 embed ds \u27e7 \u03c1\u2032 v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole v (\u03bb f \u2192 f v v) \u27e6snd\u27e7)\n        (weaken-once {t = embed ds} {v} {\u03c1\u2032}) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (v \u2022 \u03c1\u2032) v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u220e))\n    (cong\u2082 _++_\n      (weak-eq-term t C)\n      (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})))\n  (cong\u2082 mapBag (weak-eq-term s C) (weak-eq-term t C))\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394app ds t dt} {\u03c1} {cons vds vdt R[t,dt] _} {\u03c1\u2032} {C}\n  rewrite sym (weak-eq-term t C) =\n  embed-correct {dt = ds} {\u03c1} {vds} {\u03c1\u2032} {C}\n    (\u27e6 t \u27e7Term (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 vdt) (\u27e6 embed dt \u27e7Term \u03c1\u2032)\n    R[t,dt] (embed-correct {dt = dt} {\u03c1} {vdt} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394abs dt} {\u03c1} {V} {\u03c1\u2032} {C} = \u03bb w dw dw\u2032 R[w,dw] dw\u2248dw\u2032 \u2192\n  embed-correct {dt = dt}\n    {cons w dw R[w,dw] \u03c1} {V w dw R[w,dw]}\n    {dw\u2032 \u2022 w \u2022 \u03c1\u2032} {cons refl dw\u2248dw\u2032 C tt}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e7b75259e34cf9de11a2696f75d1384a4dc62f8d","subject":"allow the use of more natural notation in Nat","message":"allow the use of more natural notation in Nat\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat.agda","new_file":"agda\/Nat.agda","new_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115 #-}\n{-# BUILTIN ZERO zero #-}\n{-# BUILTIN SUC  succ #-}\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\nimport Relation.Binary.EqReasoning as EqR\nopen module EqNat = EqR (setoid \u2115)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b) =\n  begin\n    a + succ b\n      \u2248\u27e8 lemma a b \u27e9\n    succ (a + b)\n      \u2248\u27e8 cong succ (+-is-commutative a b) \u27e9\n    succ (b + a)\n      \u2248\u27e8 refl \u27e9\n    succ b + a\n  \u220e\n","old_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\nimport Relation.Binary.EqReasoning as EqR\nopen module EqNat = EqR (setoid \u2115)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b) =\n  begin\n    a + succ b\n      \u2248\u27e8 lemma a b \u27e9\n    succ (a + b)\n      \u2248\u27e8 cong succ (+-is-commutative a b) \u27e9\n    succ (b + a)\n      \u2248\u27e8 refl \u27e9\n    succ b + a\n  \u220e\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"ff2b28f9855d9d8d0f36b8784508cda5ca3bb006","subject":"FLABloM: more lifting of seminearring","message":"FLABloM: more lifting of seminearring\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  lift<+> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  lift<+> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  lift<+> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    lift<+> L c p q , lift<+> L c\u2081 p\u2081 q\u2081\n  lift<+> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    lift<+> r L p q , lift<+> r\u2081 L p\u2081 q\u2081\n  lift<+> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    lift<+> r c p q , lift<+> r c\u2081 p\u2081 q\u2081 ,\n    lift<+> r\u2081 c p\u2082 q\u2082 , lift<+> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = lift<+> shape shape }\n\n\n  liftIdent\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  liftIdent\u02e1 L L (One x) = identity\u02e1\u209b x\n  liftIdent\u02e1 L (B c c\u2081) (Row x x\u2081) = liftIdent\u02e1 L c x , liftIdent\u02e1 L c\u2081 x\u2081\n  liftIdent\u02e1 (B r r\u2081) L (Col x x\u2081) = liftIdent\u02e1 r L x , liftIdent\u02e1 r\u2081 L x\u2081\n  liftIdent\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    liftIdent\u02e1 r c x , liftIdent\u02e1 r c\u2081 x\u2081 ,\n    liftIdent\u02e1 r\u2081 c x\u2082 , liftIdent\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = liftIdent\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  -- TODO: can I use \u2243S to 'rewrite' types?\n  liftZero\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    let 0\u02e1 = 0S a b\n        0\u02b3 = 0S a c\n    in (0\u02e1 \u2219S x) \u2243S' 0\u02b3\n  liftZero\u02e1 L L L (One x) = zero\u02e1 x\n  liftZero\u02e1 L L (B c c\u2081) (Row x x\u2081) = (liftZero\u02e1 L L c x) , (liftZero\u02e1 L L c\u2081 x\u2081)\n  liftZero\u02e1 L (B b b\u2081) L (Col x x\u2081) = {!!}\n  liftZero\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) = {!!} , {!!}\n  liftZero\u02e1 (B a a\u2081) L L (One x) = liftZero\u02e1 a L L (One x) , liftZero\u02e1 a\u2081 L L (One x)\n  liftZero\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    liftZero\u02e1 a L c x , liftZero\u02e1 a L c\u2081 x\u2081 ,\n    liftZero\u02e1 a\u2081 L c x , liftZero\u02e1 a\u2081 L c\u2081 x\u2081\n  liftZero\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) = {!!} , {!!}\n  liftZero\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    {!liftZero\u02e1 a b c x!} , ({!!} , ({!!} , {!!}))\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = liftZero\u02e1 shape shape shape\n      ; zero\u02b3 = {!!}\n      ; _<\u2219>_ = {!!}\n      }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  lift0 : (r c : Shape) \u2192 M s r c\n  lift0 L L = One 0\u209b\n  lift0 L (B s s\u2081) = Row (lift0 L s) (lift0 L s\u2081)\n  lift0 (B r r\u2081) L = Col (lift0 r L) (lift0 r\u2081 L)\n  lift0 (B r r\u2081) (B s s\u2081) =\n      Q (lift0 r s) (lift0 r s\u2081)\n        (lift0 r\u2081 s) (lift0 r\u2081 s\u2081)\n\n  0S = lift0 shape shape\n\n  lift\u2243 : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  lift\u2243 L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  lift\u2243 L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = lift\u2243 L c\u2081 m n \u00d7 lift\u2243 L c\u2082 m\u2081 n\u2081\n  lift\u2243 (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = lift\u2243 r\u2081 L m n \u00d7 lift\u2243 r\u2082 L m\u2081 n\u2081\n  lift\u2243 (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    lift\u2243 r\u2081 c\u2081 m00 n00 \u00d7 lift\u2243 r\u2081 c\u2082 m01 n01 \u00d7\n    lift\u2243 r\u2082 c\u2081 m10 n10 \u00d7 lift\u2243 r\u2082 c\u2082 m11 n11\n\n  _\u2243S_ = lift\u2243 shape shape\n\n  liftRefl : (r c : Shape) \u2192\n    {X : M s r c} \u2192 lift\u2243 r c X X\n  liftRefl L L {X = One x} = refl\u209b {x}\n  liftRefl L (B c\u2081 c\u2082) {X = Row X Y} = liftRefl L c\u2081 , liftRefl L c\u2082\n  liftRefl (B r\u2081 r\u2082) L {X = Col X Y} = liftRefl r\u2081 L , liftRefl r\u2082 L\n  liftRefl (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    liftRefl r\u2081 c\u2081 , liftRefl r\u2081 c\u2082 ,\n    liftRefl r\u2082 c\u2081 , liftRefl r\u2082 c\u2082\n\n  reflS = liftRefl shape shape\n\n  liftSym : (r c : Shape) \u2192\n    let R' = lift\u2243 r c\n    in {i j : M s r c} \u2192 R' i j \u2192 R' j i\n  liftSym L L {One x} {One x\u2081} p = sym\u209b p\n  liftSym L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = liftSym L c\u2081 p , liftSym L c\u2082 q\n  liftSym (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = liftSym r\u2081 L p , liftSym r\u2082 L q\n  liftSym (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    liftSym r c\u2081 p , liftSym r c\u2082 q ,\n    liftSym r\u2082 c\u2081 x , liftSym r\u2082 c\u2082 y\n\n  symS : {i j : M s shape shape} \u2192 i \u2243S j \u2192 j \u2243S i\n  symS p = liftSym shape shape p\n\n  liftTrans : (r c : Shape) \u2192\n    let R' = lift\u2243 r c\n    in {i j k : M s r c} \u2192 R' i j \u2192 R' j k \u2192 R' i k\n  liftTrans L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  liftTrans L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    liftTrans L c\u2081 p p' , liftTrans L c\u2082 q q'\n  liftTrans (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    liftTrans r\u2081 L p p' , liftTrans r\u2082 L q q'\n  liftTrans (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    liftTrans r\u2081 c\u2081 p p' , liftTrans r\u2081 c\u2082 q q' ,\n    liftTrans r\u2082 c\u2081 x x' , liftTrans r\u2082 c\u2082 y y'\n\n  transS : {i j k : M s shape shape} \u2192 i \u2243S j \u2192 j \u2243S k \u2192 i \u2243S k\n  transS p q = liftTrans shape shape p q\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 lift\u2243 r c ((x +S y) +S z) (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  assocS : (x y z : M s shape shape) \u2192 ((x +S y) +S z) \u2243S (x +S (y +S z))\n  assocS x y z = liftAssoc shape shape x y z\n\n  isEquivS =\n    record\n      { refl = reflS\n      ; sym = symS\n      ; trans = transS }\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = assocS\n      ; \u2219-cong = {!!} }\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = {!!}\n      ; comm = {!!} }\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = _\u2243S_\n      ; 0\u209b = 0S\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = {!!}\n      ; zero\u02b3 = {!!}\n      ; _<\u2219>_ = {!!}\n      }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"710f1568c8d76da54a538508f3b3ed6a43af57f4","subject":"README","message":"README\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- A module for bit-vectors which is used almost\n-- everywhere in this development.\nopen import Data.Bits\n\nopen import elgamal\nopen import schnorr\nopen import generic-zero-knowledge-interactive\nopen import sum-properties\n\n-- Randomized programs\nopen import flipbased\nopen import flipbased-counting\nopen import flipbased-running\nopen import flipbased-implem\nopen import flipbased-tree\n\n-- A distance between randomized programs\nopen import program-distance\n\n-- Pure guessing game, a game in which no strategy can\n-- consistently win or consistently lose.\nopen import bit-guessing-game\n\n-- Cryptographic pseudo-random generator game.\nopen import prg\n\n-- \u201cOne time Semantic Security\u201d of the \u201cOne time pad\u201d cipher\n-- on one bit messages. In other words, \u201cxor\u201ding any bit with\n-- a random bit will look random as well.\nopen import single-bit-one-time-pad\n\n-- Ciphers, the \u201cone time Semantic Security\u201d game.\nopen import one-time-semantic-security\n\n-- A simple reduction on ciphers.\nopen import composition-sem-sec-reduction\n\n-- Tracking space and time through a restricted universe\n-- of functions.\nopen import data-universe\nopen import fun-universe\nopen import agda-fun-universe\nopen import bits-fun-universe\nopen import fin-fun-universe\nopen import const-fun-universe\nopen import cost-fun-universe\nopen import inverse-fun-universe\nopen import circuit-fun-universe\nopen import syntax-fun-universe\n\n-- TODO: Fix & restore the product of universes and ops\n\n-- Draft modules of previous attempts.\n-- There is still some code to rescue.\nopen import circuit\nopen import flipbased-tree-probas\nopen import flipbased-no-split\nopen import flipbased-product-implem\n","old_contents":"module README where\n\n-- A module for bit-vectors which is used almost\n-- everywhere in this development.\nopen import Data.Bits\n\n-- Randomized programs\nopen import flipbased\nopen import flipbased-counting\nopen import flipbased-running\nopen import flipbased-implem\nopen import flipbased-tree\n\n-- A distance between randomized programs\nopen import program-distance\n\n-- Pure guessing game, a game in which no strategy can\n-- consistently win or consistently lose.\nopen import bit-guessing-game\n\n-- Cryptographic pseudo-random generator game.\nopen import prg\n\n-- \u201cOne time Semantic Security\u201d of the \u201cOne time pad\u201d cipher\n-- on one bit messages. In other words, \u201cxor\u201ding any bit with\n-- a random bit will look random as well.\nopen import single-bit-one-time-pad\n\n-- Ciphers, the \u201cone time Semantic Security\u201d game.\nopen import one-time-semantic-security\n\n-- A simple reduction on ciphers.\nopen import composition-sem-sec-reduction\n\n-- Tracking space and time through a restricted universe\n-- of functions.\nopen import data-universe\nopen import fun-universe\nopen import agda-fun-universe\nopen import bits-fun-universe\nopen import fin-fun-universe\nopen import const-fun-universe\nopen import cost-fun-universe\nopen import inverse-fun-universe\nopen import circuit-fun-universe\nopen import syntax-fun-universe\n\n-- TODO: Fix & restore the product of universes and ops\n\n-- Draft modules of previous attempts.\n-- There is still some code to rescue.\nopen import circuit\nopen import flipbased-tree-probas\nopen import flipbased-no-split\nopen import flipbased-product-implem\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8e1932027422fbfd5254b377ea26f80df138163c","subject":"Move parameter to module level.","message":"Move parameter to module level.\n\nAll definitions in this module take the same parameter, so moving that\nparameter to the module level makes it easier to instantiate all of the\ndefinitions at once.\n\nOld-commit-hash: c9a73b24cd02aed1ee75b46416c1893c8c37b130\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -})\n    {\u0394base : Base \u2192 Type.Type Base}\n  where\n\n-- Terms that operate on changes\n\nopen Type Base\nopen Context Type\n\nopen import Parametric.Change.Type \u0394base\nopen import Parametric.Syntax.Term Constant\n\nopen import Data.Product\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply :\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff :\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply :\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -})\n  where\n\n-- Terms that operate on changes\n\nopen Type Base\nopen Context Type\n\nopen import Parametric.Change.Type \u0394base\nopen import Parametric.Syntax.Term Constant\n\nopen import Data.Product\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"917d3d21d2860202db684184e85baa981642c3dc","subject":"Type.Identities: some more identities","message":"Type.Identities: some more identities\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin.NP as Fin using (Fin; suc; zero; [zero:_,suc:_])\nopen import Data.Vec as Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\n\ncoe\u00d7= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081){x y}\n      \u2192 coe (\u00d7= A= B=) (x , y) \u2261 (coe A= x , coe B= y)\ncoe\u00d7= idp idp = idp\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n    private\n      module A\u2243 = Equiv A\u2243\n      A\u2192 = A\u2243.\u00b7\u2192\n      A\u2190 = A\u2243.\u00b7\u2190\n      module B\u2243 = Equiv B\u2243\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n\n    {-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 A\u2192 B\u2192) (map\u00d7 A\u2190 B\u2190)\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-r x) ({!coe-\u03b2!} \u2219 B\u2243.\u00b7\u2190-inv-r y) })\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-l x) {!!} })\n    -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e A\u2192 B\u2192) (map\u228e A\u2190 B\u2190)\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-r x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-r ]\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-l x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-l ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 B\u2192 \u2218 f \u2218 A\u2190)\n               (\u03bb f \u2192 B\u2190 \u2218 f \u2218 A\u2192)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-r _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-r x))) \n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-l _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-l x)))\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n  !\u03a3=\u2032 : ! (\u03a3=\u2032 A B) \u2261 \u03a3=\u2032 A (!_ \u2218 B)\n  !\u03a3=\u2032 = !-ap _ (\u03bb= B) \u2219 ap (ap (\u03a3 A)) (!-\u03bb= B)\n\ncoe\u03a3=\u2032-aux : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (ap (\u03bb f \u2192 f x) (\u03bb= B)) y)\ncoe\u03a3=\u2032-aux A B with \u03bb= B\ncoe\u03a3=\u2032-aux A B | idp = idp\n\ncoe\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (B x) y)\ncoe\u03a3=\u2032 A B = coe\u03a3=\u2032-aux A B \u2219 ap (_,_ _) (coe-same (happly (happly-\u03bb= B) _) _)\n\ncoe!\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe! (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe! (B x) y)\ncoe!\u03a3=\u2032 A B {x}{y} = coe-same (!\u03a3=\u2032 A B) _ \u2219 coe\u03a3=\u2032 A (!_ \u2218 B)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2243Lift\ud835\udfd9\u228e : Maybe A \u2243 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n  Maybe\u2243Lift\ud835\udfd9\u228e = equiv (maybe inr (inl _))\n                        [inl: const nothing ,inr: just ]\n                        [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                        (maybe (\u03bb _ \u2192 idp) idp)\n\n  Vec0\u2243\ud835\udfd9 : Vec A 0 \u2243 \ud835\udfd9\n  Vec0\u2243\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec0\u2243Lift\ud835\udfd9 : Vec A 0 \u2243 Lift {\u2113 = a} \ud835\udfd9\n  Vec0\u2243Lift\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec\u2218suc\u2243\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2243 (A \u00d7 Vec A n)\n  Vec\u2218suc\u2243\u00d7 = equiv (\u03bb { (x \u2237 xs) \u2192 x , xs }) (\u03bb { (x , xs) \u2192 x \u2237 xs })\n                    (\u03bb { (x , xs) \u2192 idp }) (\u03bb { (x \u2237 xs) \u2192 idp })\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua Maybe\u2243Lift\ud835\udfd9\u228e\n\n    Vec0\u2261Lift\ud835\udfd9 : Vec A 0 \u2261 Lift {\u2113 = a} \ud835\udfd9\n    Vec0\u2261Lift\ud835\udfd9 = ua Vec0\u2243Lift\ud835\udfd9\n\n    Vec\u2218suc\u2261\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2261 (A \u00d7 Vec A n)\n    Vec\u2218suc\u2261\u00d7 = ua Vec\u2218suc\u2243\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}} where\n    Vec0\u2261\ud835\udfd9 : Vec A 0 \u2261 \ud835\udfd9\n    Vec0\u2261\ud835\udfd9 = ua Vec0\u2243\ud835\udfd9\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nFin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv [zero: inl _ ,suc: inr ] [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n  [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n  [zero: idp ,suc: (\u03bb _ \u2192 idp) ]\n\nFin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin.NP as Fin using (Fin; suc; zero; [zero:_,suc:_])\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\n\ncoe\u00d7= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081){x y}\n      \u2192 coe (\u00d7= A= B=) (x , y) \u2261 (coe A= x , coe B= y)\ncoe\u00d7= idp idp = idp\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n{-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 (\u2013> A\u2243) (\u2013> B\u2243)) (map\u00d7 (<\u2013 A\u2243) (<\u2013 B\u2243))\n               (\u03bb y \u2192 pair= (<\u2013-inv-r A\u2243 (fst y)) ({!!} \u2219 <\u2013-inv-r B\u2243 (snd y)))\n               {!!}\n               -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e (\u2013> A\u2243) (\u2013> B\u2243)) (map\u228e (<\u2013 A\u2243) (<\u2013 B\u2243))\n               [inl: (\u03bb x \u2192 ap inl (<\u2013-inv-r A\u2243 x)) ,inr: ap inr \u2218 <\u2013-inv-r B\u2243 ]\n               [inl: (\u03bb x \u2192 ap inl (<\u2013-inv-l A\u2243 x)) ,inr: ap inr \u2218 <\u2013-inv-l B\u2243 ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 \u2013> B\u2243 \u2218 f \u2218 <\u2013 A\u2243)\n               (\u03bb f \u2192 <\u2013 B\u2243 \u2218 f \u2218 \u2013> A\u2243)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-r B\u2243 _ \u2219 ap f (<\u2013-inv-r A\u2243 x)))\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B\u2243 _ \u2219 ap f (<\u2013-inv-l A\u2243 x)))\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n  !\u03a3=\u2032 : ! (\u03a3=\u2032 A B) \u2261 \u03a3=\u2032 A (!_ \u2218 B)\n  !\u03a3=\u2032 = !-ap _ (\u03bb= B) \u2219 ap (ap (\u03a3 A)) (!-\u03bb= B)\n\ncoe\u03a3=\u2032-aux : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (ap (\u03bb f \u2192 f x) (\u03bb= B)) y)\ncoe\u03a3=\u2032-aux A B with \u03bb= B\ncoe\u03a3=\u2032-aux A B | idp = idp\n\ncoe\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (B x) y)\ncoe\u03a3=\u2032 A B = coe\u03a3=\u2032-aux A B \u2219 ap (_,_ _) (coe-same (happly (happly-\u03bb= B) _) _)\n\ncoe!\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe! (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe! (B x) y)\ncoe!\u03a3=\u2032 A B {x}{y} = coe-same (!\u03a3=\u2032 A B) _ \u2219 coe\u03a3=\u2032 A (!_ \u2218 B)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua (equiv (maybe inr (inl _))\n                      [inl: const nothing ,inr: just ]\n                      [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                      (maybe (\u03bb _ \u2192 idp) idp))\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nFin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv [zero: inl _ ,suc: inr ] [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n  [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n  [zero: idp ,suc: (\u03bb _ \u2192 idp) ]\n\nFin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"887d992f22f6e65e6ca4201f28f73ab449fdfb41","subject":"progress progress #3","message":"progress progress #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import Agda.Primitive using (Level; lzero; lsuc) renaming (_\u2294_ to lmax)\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus (had an idea for a lemma here but it's false; see bottom)\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus (see bottom)\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FIndet i)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FBoxed v\u2082)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n    where\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter (ITCastID (FIndet x)) FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter (ITCastID (FBoxed x)) FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n\n  -- this would fill two above, but it's false: \u2987\u2988 indet but its type, \u2987\u2988, is not ground\n  -- lem-groundindet : \u2200{ \u0394 d \u03c4} \u2192 \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 d indet \u2192 \u03c4 ground\n\n\n  counter : \u03a3[ d \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0394 \u2208 hctx ]\n               ((d indet) \u00d7 (\u0394 , \u2205 \u22a2 d :: \u03c4 ==> \u03c4'))\n  counter = (\u2987\u2988\u27e8 Z , \u2205 \u27e9 \u27e8 b ==> b \u21d2 b ==> \u2987\u2988 \u27e9) ,\n            b , \u2987\u2988 ,  \u25a0 (Z , \u2205 , b ==> b ) ,\n            ICastArr (\u03bb ()) IEHole , TACast (TAEHole refl (\u03bb x d \u2192 \u03bb ())) (TCArr TCRefl TCHole1)\n\n  postulate\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n\n  problem : \u22a5\n  problem = lem (\u03c02 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter)))))\n                (\u03c01 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter))))) b b b \u2987\u2988 \u2987\u2988\u27e8 Z , \u2205 \u27e9 refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-boxed-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus\n    where\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n    lem (TACast wt TCRefl) (ICastArr x\u2083 ind) \u03c41 \u03c42 .\u03c41 .\u03c42 d refl = x\u2083 refl\n    lem (TACast wt (TCArr x\u2082 x\u2083)) (ICastArr x\u2084 ind) \u03c41 \u03c42 \u03c43 \u03c4' d refl = {!!}\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i = I (IAp {!!} {!!} (FIndet i)) -- cyrus (issue from above, and also i think missing a rule for indet applications)\n  progress (TAAp wt1 wt2) | V v | V v\u2082 = {!!} -- {!canonical-boxed-forms-arr wt1 x !}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter (ITCastID (FIndet x)) FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter (ITCastID (FBoxed x)) FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n\n  -- this would fill two above, but it's false: \u2987\u2988 indet but its type, \u2987\u2988, is not ground\n  -- lem-groundindet : \u2200{ \u0394 d \u03c4} \u2192 \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 d indet \u2192 \u03c4 ground\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6936c62b46d40ca7430caebbc9225fd24549ad9a","subject":"P{\u22a4,0} do not need FunExt,UA","message":"P{\u22a4,0} do not need FunExt,UA\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Additive.agda","new_file":"Control\/Protocol\/Additive.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (_\u00d7_; _,_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Additive where\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto) where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    module _ {{_ : FunExt}}{{_ : UA}} where\n        P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n        P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n        P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n        P\u22a4-uniq = \u03a0\ud835\udfd8-uniq _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} {P Q} where\n    dual-\u2295 : dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} P Q where\n    &-comm : P & Q \u2261 Q & P\n    &-comm = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm = send\u2243 LR!-equiv [L: refl R: refl ]\n\n    -- additive mix (left-biased)\n    amixL : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixL pq = (`L , pq `L)\n\n    amixR : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixR pq = (`R , pq `R)\n\nmodule _ {P Q R S}(f : \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7)(g : \u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) where\n    \u2295-map : \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map (`L , pr) = `L , f pr\n    \u2295-map (`R , pr) = `R , g pr\n\n    &-map : \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map p `L = f (p `L)\n    &-map p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 = [inl: _,_ `L ,inr: _,_ `R ]\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = equiv \u2295\u2192\u228e \u228e\u2192\u2295 [inl: (\u03bb _ \u2192 refl) ,inr: (\u03bb _ \u2192 refl) ] \u228e\u2192\u2295\u2192\u228e\n      where\n        \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n        \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n        \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = &\u2192\u00d7 , record { linv = \u00d7\u2192& ; is-linv = \u00d7\u2192&\u2192\u00d7 ; rinv = \u00d7\u2192& ; is-rinv = &\u2192\u00d7\u2192& }\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ P {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\nmodule _ P Q R {{_ : FunExt}}{{_ : UA}} where\n    &-assoc : \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (_\u00d7_; _,_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Additive where\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto){{_ : FunExt}}{{_ : UA}} where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n    P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n    P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n    P\u22a4-uniq = \u03a0\ud835\udfd8-uniq _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} {P Q} where\n    dual-\u2295 : dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} P Q where\n    &-comm : P & Q \u2261 Q & P\n    &-comm = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm = send\u2243 LR!-equiv [L: refl R: refl ]\n\n    -- additive mix (left-biased)\n    amixL : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixL pq = (`L , pq `L)\n\n    amixR : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixR pq = (`R , pq `R)\n\nmodule _ {P Q R S}(f : \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7)(g : \u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) where\n    \u2295-map : \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map (`L , pr) = `L , f pr\n    \u2295-map (`R , pr) = `R , g pr\n\n    &-map : \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map p `L = f (p `L)\n    &-map p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 = [inl: _,_ `L ,inr: _,_ `R ]\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = equiv \u2295\u2192\u228e \u228e\u2192\u2295 [inl: (\u03bb _ \u2192 refl) ,inr: (\u03bb _ \u2192 refl) ] \u228e\u2192\u2295\u2192\u228e\n      where\n        \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n        \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n        \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = &\u2192\u00d7 , record { linv = \u00d7\u2192& ; is-linv = \u00d7\u2192&\u2192\u00d7 ; rinv = \u00d7\u2192& ; is-rinv = &\u2192\u00d7\u2192& }\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ P {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\nmodule _ P Q R {{_ : FunExt}}{{_ : UA}} where\n    &-assoc : \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4f7fa7c66950d9794f12eff0ea347f1fd2e7c52f","subject":"program-disantce: upgrade","message":"program-disantce: upgrade\n","repos":"crypto-agda\/crypto-agda","old_file":"program-distance.agda","new_file":"program-distance.agda","new_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n  ]-[-antisym {n} f f]-[g rewrite dist-refl (count\u21ba f) with \u2115\u2264.trans (1\u22642^ (n \u2238 k)) f]-[g\n  ... | ()\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist \u27e82^ n * count\u21ba f \u27e9 \u27e82^ m * count\u21ba g \u27e9 \u2265 2^((m + n) \u2238 k)\n\n  ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n  ]-[-antisym {n} f f]-[g rewrite dist-refl \u27e82^ n * count\u21ba f \u27e9 with \u2115\u2264.trans (1\u22642^ (n + n \u2238 k)) f]-[g\n  ... | ()\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym \u27e82^ n * count\u21ba f \u27e9 \u27e82^ m * count\u21ba g \u27e9 | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 m n o k \u2192 m + ((n + o) \u2238 k) \u2261 n + ((m + o) \u2238 k)\n      helper\u2032 : \u2200 m n o k \u2192 \u27e82^ m * (2^((n + o) \u2238 k))\u27e9 \u2261 \u27e82^ n * (2^((m + o) \u2238 k))\u27e9\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n           -- 2\u1d50(dist 2\u1d52#g 2\u207f#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n  ... | q rewrite sym (dist-2^* m \u27e82^ o * count\u21ba g \u27e9 \u27e82^ n * count\u21ba h \u27e9)\n           -- dist 2\u1d502\u1d52#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m o (count\u21ba g)\n           -- dist 2\u1d522\u1d50#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | sym f\u2248g\n           -- dist 2\u1d522\u207f#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm o n (count\u21ba f)\n           -- dist 2\u207f2\u1d52#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m n (count\u21ba h)\n           -- dist 2\u207f2\u1d52#f 2\u207f2\u1d50#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | dist-2^* n \u27e82^ o * count\u21ba f \u27e9 \u27e82^ m * count\u21ba h \u27e9\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-+ m (n + o \u2238 k) 1\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d50\u207a\u207f\u207a\u1d52\u207b\u1d4f\n                | helper m n o k\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f\u207a\u1d50\u207a\u1d52\u207b\u1d4f\n                | sym (2^-+ n (m + o \u2238 k) 1)\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f2\u1d50\u207a\u1d52\u207b\u1d4f\n                = 2^*-mono\u2032 n q\n           -- dist 2\u1d52#f 2\u1d50#h \u2264 2\u1d50\u207a\u1d52\u207b\u1d4f\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_\n               ]-[-antisym\n               (\u03bb {m n f g} \u2192 ]-[-sym {f = f} {g})\n               (\u03bb {m n o f g h} \u2192 ]-[-cong-left-\u2248\u21ba {f = f} {g} {h})\n","old_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nprivate\n  2^_\u22651 : \u2200 n \u2192 2^ n \u2265 1\n  2^ zero \u22651  = s\u2264s z\u2264n\n  2^ suc n \u22651 = z\u2264n {2^ n} +-mono 2^ n \u22651\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n  ]-[-antisym {n} f f]-[g rewrite dist-refl (count\u21ba f) with \u2115\u2264.trans (2^ (n \u2238 k) \u22651) f]-[g\n  ... | ()\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist \u27e82^ n * count\u21ba f \u27e9 \u27e82^ m * count\u21ba g \u27e9 \u2265 2^((m + n) \u2238 k)\n\n  ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n  ]-[-antisym {n} f f]-[g rewrite dist-refl \u27e82^ n * count\u21ba f \u27e9 with \u2115\u2264.trans (2^ (n + n \u2238 k) \u22651) f]-[g\n  ... | ()\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym \u27e82^ n * count\u21ba f \u27e9 \u27e82^ m * count\u21ba g \u27e9 | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 m n o k \u2192 m + ((n + o) \u2238 k) \u2261 n + ((m + o) \u2238 k)\n      helper\u2032 : \u2200 m n o k \u2192 \u27e82^ m * (2^((n + o) \u2238 k))\u27e9 \u2261 \u27e82^ n * (2^((m + o) \u2238 k))\u27e9\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n           -- 2\u1d50(dist 2\u1d52#g 2\u207f#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n  ... | q rewrite sym (dist-2^* m \u27e82^ o * count\u21ba g \u27e9 \u27e82^ n * count\u21ba h \u27e9)\n           -- dist 2\u1d502\u1d52#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m o (count\u21ba g)\n           -- dist 2\u1d522\u1d50#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | sym f\u2248g\n           -- dist 2\u1d522\u207f#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm o n (count\u21ba f)\n           -- dist 2\u207f2\u1d52#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m n (count\u21ba h)\n           -- dist 2\u207f2\u1d52#f 2\u207f2\u1d50#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | dist-2^* n \u27e82^ o * count\u21ba f \u27e9 \u27e82^ m * count\u21ba h \u27e9\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-+ m (n + o \u2238 k) 1\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d50\u207a\u207f\u207a\u1d52\u207b\u1d4f\n                | helper m n o k\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f\u207a\u1d50\u207a\u1d52\u207b\u1d4f\n                | sym (2^-+ n (m + o \u2238 k) 1)\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f2\u1d50\u207a\u1d52\u207b\u1d4f\n                = 2^*-mono\u2032 n q\n           -- dist 2\u1d52#f 2\u1d50#h \u2264 2\u1d50\u207a\u1d52\u207b\u1d4f\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_\n               ]-[-antisym\n               (\u03bb {m n f g} \u2192 ]-[-sym {f = f} {g})\n               (\u03bb {m n o f g h} \u2192 ]-[-cong-left-\u2248\u21ba {f = f} {g} {h})\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"87661ddb46c3c4fe87733c8324a74ae28bd65966","subject":"Describe the purpose of binding.adga (see #28).","message":"Describe the purpose of binding.adga (see #28).\n\nThere are two comments because I intend to split this file.\n\nOld-commit-hash: 4156d12adc58a4ac0856145a0353e54b14c9a865\n","repos":"inc-lc\/ilc-agda","old_file":"binding.agda","new_file":"binding.agda","new_contents":"module binding\n    (Type : Set)\n    (\u27e6_\u27e7Type : Type \u2192 Set)\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\n-- ENVIRONMENTS\n--\n-- This module defines the meaning of contexts, that is,\n-- the type of environments that fit a context, together\n-- with operations and properties of these operations.\n--\n-- This module is parametric in the syntax and semantics\n-- of types, so it can be reused for different calculi\n-- and models.\n\nopen import meaning\n\nprivate\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Denotational Semantics : Contexts Represent Environments\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = Empty\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n-- WEAKENING\n\n-- Prefix of a context\n\ndata Prefix : Context \u2192 Set where\n  \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n  _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\ntake : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\ntake \u0393 \u2205 = \u2205\ntake (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\ndrop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\ndrop \u0393 \u2205 = \u0393\ndrop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Remove a variable from an environment\n\nweakenEnv : \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083} \u2192 \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7 \u2192 \u27e6 \u0393\u2081 \u22ce \u0393\u2083 \u27e7\nweakenEnv \u2205 \u03c4\u2082 (v \u2022 \u03c1) = \u03c1\nweakenEnv (\u03c4 \u2022 \u0393\u2081) \u03c4\u2082 (v \u2022 \u03c1) = v \u2022 weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1\n\nopen import Relation.Binary.PropositionalEquality\n\ntake-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\ntake-drop \u2205 \u2205 = refl\ntake-drop (\u03c4 \u2022 \u0393) \u2205 = refl\ntake-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\nor-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\nor-unit \u2205 = refl\nor-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\nmove-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n  \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\nmove-prefix \u2205 \u03c4 \u0393\u2032 = refl\nmove-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\nmove-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n","old_contents":"module binding\n    (Type : Set)\n    (\u27e6_\u27e7Type : Type \u2192 Set)\n  where\n\nopen import meaning\n\nprivate\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Denotational Semantics : Contexts Represent Environments\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = Empty\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n-- WEAKENING\n\n-- Prefix of a context\n\ndata Prefix : Context \u2192 Set where\n  \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n  _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\ntake : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\ntake \u0393 \u2205 = \u2205\ntake (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\ndrop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\ndrop \u0393 \u2205 = \u0393\ndrop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Remove a variable from an environment\n\nweakenEnv : \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083} \u2192 \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7 \u2192 \u27e6 \u0393\u2081 \u22ce \u0393\u2083 \u27e7\nweakenEnv \u2205 \u03c4\u2082 (v \u2022 \u03c1) = \u03c1\nweakenEnv (\u03c4 \u2022 \u0393\u2081) \u03c4\u2082 (v \u2022 \u03c1) = v \u2022 weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1\n\nopen import Relation.Binary.PropositionalEquality\n\ntake-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\ntake-drop \u2205 \u2205 = refl\ntake-drop (\u03c4 \u2022 \u0393) \u2205 = refl\ntake-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\nor-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\nor-unit \u2205 = refl\nor-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\nmove-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n  \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\nmove-prefix \u2205 \u03c4 \u0393\u2032 = refl\nmove-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\nmove-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"568f209dbe43ea336be8078951c585c924fb4d9d","subject":"bintree: toFun\/fromFun","message":"bintree: toFun\/fromFun\n","repos":"crypto-agda\/crypto-agda","old_file":"bintree.agda","new_file":"bintree.agda","new_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (0b \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (1b \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nleaf\u207f : \u2200 {n a} {A : Set a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : Set a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x rewrite fromConst\u2261leaf\u207f {n} x = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x) = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","old_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} \u2192 (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then left else right) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nopen import Relation.Binary\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"24ca737e8a2713939592b800f904f7abb2cd7816","subject":"Added minus-0x.","message":"Added minus-0x.\n\nIgnore-this: 184c72f3970e6a8e48a91ed79f94646c\n\ndarcs-hash:20100429154806-3bd4e-a8caa80a418cb33160bf13afca11eb6ff2a25c1f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/Properties.agda","new_file":"LTC\/Function\/Arithmetic\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties on total natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\n\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN  = prf\n  where\n  postulate prf : N (pred zero)\n  {-# ATP prove prf zN #-}\n\npred-N (sN {n} Nn) = prf\n  where\n  -- TODO: The postulate N (pred $ succ n) is not proved by the ATP.\n  postulate prf : N (pred (succ n))\n  {-# ATP prove prf #-}\n-- {-# ATP hint pred-N #-}\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN = prf\n  where\n  postulate prf : N (m - zero)\n  {-# ATP prove prf #-}\n\nminus-N zN (sN {n} Nn) = prf\n  where\n  postulate prf : N (zero - succ n)\n  {-# ATP prove prf zN #-}\n\nminus-N (sN {m} Nm) (sN {n} Nn) = prf $ minus-N Nm Nn\n  where\n  postulate prf : N (m - n) \u2192 N (succ m - succ n)\n  {-# ATP prove prf #-}\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero + n)\n  {-# ATP prove prf #-}\n+-N {n = n} (sN {m} Nm ) Nn = prf (+-N Nm Nn)\n  where\n  postulate prf : N (m + n) \u2192 N (succ m + n)\n  {-# ATP prove prf sN #-}\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero * n)\n  {-# ATP prove prf zN #-}\n*-N {n = n} (sN {m} Nm) Nn = prf (*-N Nm Nn)\n  where\n  postulate prf : N (m * n) \u2192 N (succ m * n)\n  {-# ATP prove prf +-N #-}\n\n------------------------------------------------------------------------------\n\n-- Some proofs are based on the proofs in the standard library\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) = prf $ +-rightIdentity Nn\n   where\n   postulate prf : n + zero \u2261 n \u2192 succ n + zero \u2261 succ n\n   {-# ATP prove prf #-}\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No = prf\n  where\n  postulate prf : zero + n + o \u2261 zero + (n + o)\n  {-# ATP prove prf #-}\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No = prf $ +-assoc Nm Nn No\n  where\n  postulate prf : m + n + o \u2261 m + (n + o) \u2192\n                  succ m + n + o \u2261 succ m + (n + o)\n  {-# ATP prove prf #-}\n\nx+1+y\u22611+x+y : {m n : D} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y {n = n} zN Nn = prf\n  where\n  postulate prf : zero + succ n \u2261 succ (zero + n)\n  {-# ATP prove prf #-}\nx+1+y\u22611+x+y {n = n} (sN {m} Nm) Nn = prf (x+1+y\u22611+x+y Nm Nn)\n  where\n  postulate prf : m + succ n \u2261 succ (m + n) \u2192\n                  succ m + succ n \u2261 succ (succ m + n)\n  {-# ATP prove prf #-}\n\n+-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero + n \u2261 n + zero\n  {-# ATP prove prf +-rightIdentity #-}\n+-comm {n = n} (sN {m} Nm) Nn = prf (+-comm Nm Nn)\n  where\n  postulate prf : m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+1+y\u22611+x+y #-}\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) = prf (*-rightZero Nn)\n  where\n  postulate prf : n * zero \u2261 zero \u2192 succ n * zero \u2261 zero\n  {-# ATP prove prf #-}\n\npostulate *-leftIdentity : {n : D} \u2192 N n \u2192 succ zero * n \u2261 n\n{-# ATP prove *-leftIdentity +-rightIdentity #-}\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = prf\n  where\n  postulate prf : zero * succ n \u2261 zero + zero * n\n  {-# ATP prove prf #-}\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn = prf (x*1+y\u2261x+xy Nm Nn)\n                                        (+-assoc Nn Nm (*-N Nm Nn))\n                                        (+-assoc Nm Nn (*-N Nm Nn))\n  where\n  -- N.B. We had to feed the ATP with the instances of the associate law\n  postulate prf :  m * succ n \u2261 m + m * n \u2192 -- IH\n                   (n + m) + (m * n) \u2261 n + (m + (m * n)) \u2192 -- Associative law\n                   (m + n) + (m * n) \u2261 m + (n + (m * n)) \u2192 -- Associateve law\n                   succ m * succ n \u2261 succ m + succ m * n\n  {-# ATP prove prf +-comm #-}\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero * n \u2261 n * zero\n  {-# ATP prove prf *-rightZero #-}\n*-comm {n = n} (sN {m} Nm) Nn = prf (*-comm Nm Nn)\n  where\n  postulate prf : m * n \u2261 n * m \u2192\n                  succ m * n \u2261 n * succ m\n  {-# ATP prove prf x*1+y\u2261x+xy #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties on total natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\n\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN  = prf\n  where\n  postulate prf : N (pred zero)\n  {-# ATP prove prf zN #-}\n\npred-N (sN {n} Nn) = prf\n  where\n  -- TODO: The postulate N (pred $ succ n) is not proved by the ATP.\n  postulate prf : N (pred (succ n))\n  {-# ATP prove prf #-}\n-- {-# ATP hint pred-N #-}\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN = prf\n  where\n  postulate prf : N (m - zero)\n  {-# ATP prove prf #-}\n\nminus-N zN (sN {n} Nn) = prf\n  where\n  postulate prf : N (zero - succ n)\n  {-# ATP prove prf zN #-}\n\nminus-N (sN {m} Nm) (sN {n} Nn) = prf $ minus-N Nm Nn\n  where\n  postulate prf : N (m - n) \u2192 N (succ m - succ n)\n  {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero + n)\n  {-# ATP prove prf #-}\n+-N {n = n} (sN {m} Nm ) Nn = prf (+-N Nm Nn)\n  where\n  postulate prf : N (m + n) \u2192 N (succ m + n)\n  {-# ATP prove prf sN #-}\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero * n)\n  {-# ATP prove prf zN #-}\n*-N {n = n} (sN {m} Nm) Nn = prf (*-N Nm Nn)\n  where\n  postulate prf : N (m * n) \u2192 N (succ m * n)\n  {-# ATP prove prf +-N #-}\n\n------------------------------------------------------------------------------\n\n-- The proofs are based on the proofs in the standard library\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) = prf $ +-rightIdentity Nn\n   where\n   postulate prf : n + zero \u2261 n \u2192 succ n + zero \u2261 succ n\n   {-# ATP prove prf #-}\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No = prf\n  where\n  postulate prf : zero + n + o \u2261 zero + (n + o)\n  {-# ATP prove prf #-}\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No = prf $ +-assoc Nm Nn No\n  where\n  postulate prf : m + n + o \u2261 m + (n + o) \u2192\n                  succ m + n + o \u2261 succ m + (n + o)\n  {-# ATP prove prf #-}\n\nx+1+y\u22611+x+y : {m n : D} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y {n = n} zN Nn = prf\n  where\n  postulate prf : zero + succ n \u2261 succ (zero + n)\n  {-# ATP prove prf #-}\nx+1+y\u22611+x+y {n = n} (sN {m} Nm) Nn = prf (x+1+y\u22611+x+y Nm Nn)\n  where\n  postulate prf : m + succ n \u2261 succ (m + n) \u2192\n                  succ m + succ n \u2261 succ (succ m + n)\n  {-# ATP prove prf #-}\n\n+-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero + n \u2261 n + zero\n  {-# ATP prove prf +-rightIdentity #-}\n+-comm {n = n} (sN {m} Nm) Nn = prf (+-comm Nm Nn)\n  where\n  postulate prf : m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+1+y\u22611+x+y #-}\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) = prf (*-rightZero Nn)\n  where\n  postulate prf : n * zero \u2261 zero \u2192 succ n * zero \u2261 zero\n  {-# ATP prove prf #-}\n\npostulate *-leftIdentity : {n : D} \u2192 N n \u2192 succ zero * n \u2261 n\n{-# ATP prove *-leftIdentity +-rightIdentity #-}\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = prf\n  where\n  postulate prf : zero * succ n \u2261 zero + zero * n\n  {-# ATP prove prf #-}\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn = prf (x*1+y\u2261x+xy Nm Nn)\n                                        (+-assoc Nn Nm (*-N Nm Nn))\n                                        (+-assoc Nm Nn (*-N Nm Nn))\n  where\n  -- N.B. We had to feed the ATP with the instances of the associate law\n  postulate prf :  m * succ n \u2261 m + m * n \u2192 -- IH\n                   (n + m) + (m * n) \u2261 n + (m + (m * n)) \u2192 -- Associative law\n                   (m + n) + (m * n) \u2261 m + (n + (m * n)) \u2192 -- Associateve law\n                   succ m * succ n \u2261 succ m + succ m * n\n  {-# ATP prove prf +-comm #-}\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero * n \u2261 n * zero\n  {-# ATP prove prf *-rightZero #-}\n*-comm {n = n} (sN {m} Nm) Nn = prf (*-comm Nm Nn)\n  where\n  postulate prf : m * n \u2261 n * m \u2192\n                  succ m * n \u2261 n * succ m\n  {-# ATP prove prf x*1+y\u2261x+xy #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e1d9192fe2f0abe2c2a369db5b7083d63331c9a5","subject":"Describe the purpose of Changes.agda (see #28).","message":"Describe the purpose of Changes.agda (see #28).\n\nOld-commit-hash: cbd11ef58e011e2b5d8af09524e6a0c07ee91098\n","repos":"inc-lc\/ilc-agda","old_file":"Changes.agda","new_file":"Changes.agda","new_contents":"module Changes where\n\n-- CHANGES\n--\n-- This module defines the representation of changes, the\n-- available operations on changes, and properties of these\n-- operations.\n\nopen import meaning\nopen import Model\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} b c = b xor c\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} b c = b xor c\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} b c = b xor c\n\n-- CONGRUENCE rules for change operations\n\nopen import Relation.Binary.PropositionalEquality\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\n\ndiff-derive {bool} b = a-xor-a-false b\n\n-- XXX: as given, this is false!\npostulate\n  diff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    diff (apply dv v) v \u2261 dv\n\n{-\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n-}\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} a b = xor-cancellative a b\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} a = a-xor-false-a a\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} a b c = xor-associative a b c\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} a b c = \u2261-sym (xor-associative c a b)\n","old_contents":"module Changes where\n\nopen import meaning\nopen import Model\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} b c = b xor c\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} b c = b xor c\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} b c = b xor c\n\n-- CONGRUENCE rules for change operations\n\nopen import Relation.Binary.PropositionalEquality\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\n\ndiff-derive {bool} b = a-xor-a-false b\n\n-- XXX: as given, this is false!\npostulate\n  diff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    diff (apply dv v) v \u2261 dv\n\n{-\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n-}\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} a b = xor-cancellative a b\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} a = a-xor-false-a a\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} a b c = xor-associative a b c\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} a b c = \u2261-sym (xor-associative c a b)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2374278020e2b6ffe37d215e7ea4fffd7a448576","subject":"extended natural semantics proof with ifthenelse but failed to finish it","message":"extended natural semantics proof with ifthenelse but failed to finish it\n\nOld-commit-hash: 542159c766c7879ae32fd9c8be27fc977bf99b16\n","repos":"inc-lc\/ilc-agda","old_file":"Natural\/Soundness.agda","new_file":"Natural\/Soundness.agda","new_contents":"module Natural.Soundness where\n\n-- SOUNDNESS of NATURAL SEMANTICS\n--\n-- This module proves consistency of the natural semantics with\n-- respect to the denotational semantics.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Closures\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\n-- Syntactic lookup is consistent with semantic lookup.\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n-- Syntactic evaluation is consistent with semantic evaluation.\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound e-true = \u2261-refl \n\u2193-sound e-false = \u2261-refl \n-- Agda does not recognize automagically that\n-- \u27e6if c t e\u27e7 \u2261 \u27e6t\u27e7, which I would need to finish \n-- the proof. Maybe I need some \"with\" magic in\n-- the pattern match here? Help!\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) = \u2261-trans {!!} (\u2193-sound \u2193\u2082)\n\u2193-sound (if-false \u2193\u2081 \u2193\u2082) = {!!}\n","old_contents":"module Natural.Soundness where\n\n-- SOUNDNESS of NATURAL SEMANTICS\n--\n-- This module proves consistency of the natural semantics with\n-- respect to the denotational semantics.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Closures\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\n-- Syntactic lookup is consistent with semantic lookup.\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n-- Syntactic evaluation is consistent with semantic evaluation.\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound e-true = \u2261-refl \n\u2193-sound e-false = \u2261-refl \n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) \u2261-trans {!!} (\u2193-sound \u2193\u2082)\n\u2193-sound (if-false \u2193\u2081 \u2193\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8ca23b07a65ea0c63d74b53511d03868a5e1ccbf","subject":"better circuits","message":"better circuits\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup : \u2200 o \u2192 C 1 o\n  dup _ = rewire (const zero)\n-- dup-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2082 : C 1 2\n  dup\u2082 = dup 2\n-- dup\u2082-spec : (b \u2237 []) =[ dup\u2082 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2261 i\u2081 \u2192 o\u2080 \u2261 o\u2081 \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081\n  coerce refl refl = id\n\n  dup\u207f : \u2200 {n} k \u2192 C n (k * n)\n  dup\u207f {n} k = coerce (proj\u2082 \u2115\u00b0.*-identity n) (\u2115\u00b0.*-comm n k) (vcat {n = n} (replicate (dup _)))\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  takeC : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) o\n  takeC {k} {_} {o} c = coerce refl (\u2115\u00b0.+-comm o 0) (c *** sink k)\n\n  takeC' : \u2200 {i} k \u2192 C (i + k) i\n  takeC' k = takeC {k} idC\n\n  dropC : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) o\n  dropC {k} c = sink k *** c\n\n  dropC' : \u2200 {i} k \u2192 C (k + i) i\n  dropC' k = dropC {k} idC\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = idC {1} *** sink _\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC' 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder public\n\n  bit : Bit \u2192 C 0 1\n  bit b = arr (const (b \u2237 []))\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2082 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5c05f76f8895526eefdbaa88f665fd4534bd4a1f","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Conat\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Conat\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Conat properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e,\n--\n-- NatF Conat \u2264 Conat (see FOTC.Data.Conat.Type).\nConat-pre-fixed : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n                  \u2200 {n} \u2192 Conat n\nConat-pre-fixed h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\n0-Conat : Conat zero\n0-Conat = Conat-coind A h refl\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero\n\n  h : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h An = inj\u2081 An\n\n-- Adapted from (Sander 1992, p. 57).\n\u221e-Conat : Conat \u221e\n\u221e-Conat = Conat-coind A h refl\n  where\n  A : D \u2192 Set\n  A n = n \u2261 \u221e\n\n  h : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h An = inj\u2082 (\u221e , trans An \u221e-eq , refl)\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat Nn = Conat-coind N h Nn\n  where\n  h : \u2200 {m} \u2192 N m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n  h nzero          = inj\u2081 refl\n  h (nsucc {m} Nm) = inj\u2082 (m , refl , Nm)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Sander, Herbert P. (1992). A Logic of Functional Programs with an\n-- Application to Concurrency. PhD thesis. Department of Computer\n-- Sciences: Chalmers University of Technology and University of\n-- Gothenburg.\n","old_contents":"------------------------------------------------------------------------------\n-- Conat properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e,\n--\n-- NatF Conat \u2264 Conat (see FOTC.Data.Conat.Type).\nConat-pre-fixed : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n                  \u2200 {n} \u2192 Conat n\nConat-pre-fixed h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610              = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\n0-Conat : Conat zero\n0-Conat = Conat-coind A h refl\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero\n\n  h : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h An = inj\u2081 An\n\n-- Adapted from (Sander 1992, p. 57).\n\u221e-Conat : Conat \u221e\n\u221e-Conat = Conat-coind A h refl\n  where\n  A : D \u2192 Set\n  A n = n \u2261 \u221e\n\n  h : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h An = inj\u2082 (\u221e , trans An \u221e-eq , refl)\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat Nn = Conat-coind N h Nn\n  where\n  h : \u2200 {m} \u2192 N m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n  h nzero          = inj\u2081 refl\n  h (nsucc {m} Nm) = inj\u2082 (m , refl , Nm)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Sander, Herbert P. (1992). A Logic of Functional Programs with an\n-- Application to Concurrency. PhD thesis. Department of Computer\n-- Sciences: Chalmers University of Technology and University of\n-- Gothenburg.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0d92eb4d13b0cf810bec100dd2fe9b96ff861d7e","subject":"[ agda ] The --no-coverage-check option was removed.","message":"[ agda ] The --no-coverage-check option was removed.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/report\/AgdaIntroduction.agda","new_file":"notes\/thesis\/report\/AgdaIntroduction.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas     #-}\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule AgdaIntroduction where\n\n-- We add 3 to the fixities of the Agda standard library\u00a00.8.1 (see\n-- Data\/Nat.agda).\ninfixl 10 _*_\ninfixl 9  _+_\ninfixr 5  _\u2237_ _++_\ninfix  4  _\u2261_\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A zero\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\nf : \u2115 \u2192 \u2115\nf zero = zero\n{-# CATCHALL #-}\nf _    = succ zero\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin zero \u2192 A\nmagic ()\n\n-- The with constructor\n\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven zero     = true\neven (succ n) = odd n\n\nodd zero   = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Normalisation\n\ndata _\u2261_ {A : Set} : A \u2192 A \u2192 Set where\n  refl : {a : A} \u2192 a \u2261 a\n\nlength : {A : Set} \u2192 List A \u2192 \u2115\nlength []       = zero\nlength (x \u2237 xs) = succ zero + length xs\n\n_++_ : {A : Set} \u2192 List A \u2192 List A \u2192 List A\n[]       ++ ys = ys\n(x \u2237 xs) ++ ys = x \u2237 xs ++ ys\n\nsuccCong : {m n : \u2115} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsuccCong refl = refl\n\nlength-++ : {A : Set}(xs ys : List A) \u2192\n            length (xs ++ ys) \u2261 length xs + length ys\nlength-++ [] ys       = refl\nlength-++ (x \u2237 xs) ys = succCong (length-++ xs ys)\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead []       = ?\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero     n        = succ n\nack (succ m) zero     = ack m (succ zero)\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n-- Combinators for equational reasoning\n\npostulate\n  sym   : {A : Set}{a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\n  trans : {A : Set}{a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\n  subst : {A : Set}(P : A \u2192 Set){a b : A} \u2192 a \u2261 b \u2192 P a \u2192 P b\n\npostulate\n  _*_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  *-comm          : \u2200 m n \u2192 m * n \u2261 n * m\n  *-rightIdentity : \u2200 n \u2192 n * succ zero \u2261 n\n\n*-leftIdentity : \u2200 n \u2192 succ zero * n \u2261 n\n*-leftIdentity n =\n  trans {\u2115} {succ zero * n} {n * succ zero} {n} (*-comm (succ zero) n) (*-rightIdentity n)\n\nmodule ER\n  {A     : Set}\n  (_\u223c_   : A \u2192 A \u2192 Set)\n  (\u223c-refl  : \u2200 {x} \u2192 x \u223c x)\n  (\u223c-trans : \u2200 {x y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z)\n  where\n\n  infixr 5 _\u223c\u27e8_\u27e9_\n  infix  6 _\u220e\n\n  _\u223c\u27e8_\u27e9_ : \u2200 x {y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z\n  _ \u223c\u27e8 x\u223cy \u27e9 y\u223cz = \u223c-trans x\u223cy y\u223cz\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _\u220e _ = \u223c-refl\n\nopen module \u2261-Reasoning = ER _\u2261_ (refl {\u2115}) (trans {\u2115})\n  renaming ( _\u223c\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_ )\n\n*-leftIdentity' : \u2200 n \u2192 succ zero * n \u2261 n\n*-leftIdentity' n =\n  succ zero * n \u2261\u27e8 *-comm (succ zero) n \u27e9\n  n * succ zero \u2261\u27e8 *-rightIdentity n \u27e9\n  n             \u220e\n","old_contents":"{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-coverage-check        #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule AgdaIntroduction where\n\n-- We add 3 to the fixities of the Agda standard library\u00a00.8.1 (see\n-- Data\/Nat.agda).\ninfixl 10 _*_\ninfixl 9  _+_\ninfixr 5  _\u2237_ _++_\ninfix  4  _\u2261_\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A zero\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\nf : \u2115 \u2192 \u2115\nf zero = zero\n{-# CATCHALL #-}\nf _    = succ zero\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin zero \u2192 A\nmagic ()\n\n-- The with constructor\n\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven zero     = true\neven (succ n) = odd n\n\nodd zero   = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Normalisation\n\ndata _\u2261_ {A : Set} : A \u2192 A \u2192 Set where\n  refl : {a : A} \u2192 a \u2261 a\n\nlength : {A : Set} \u2192 List A \u2192 \u2115\nlength []       = zero\nlength (x \u2237 xs) = succ zero + length xs\n\n_++_ : {A : Set} \u2192 List A \u2192 List A \u2192 List A\n[]       ++ ys = ys\n(x \u2237 xs) ++ ys = x \u2237 xs ++ ys\n\nsuccCong : {m n : \u2115} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsuccCong refl = refl\n\nlength-++ : {A : Set}(xs ys : List A) \u2192\n            length (xs ++ ys) \u2261 length xs + length ys\nlength-++ [] ys       = refl\nlength-++ (x \u2237 xs) ys = succCong (length-++ xs ys)\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero     n        = succ n\nack (succ m) zero     = ack m (succ zero)\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n-- Combinators for equational reasoning\n\npostulate\n  sym   : {A : Set}{a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\n  trans : {A : Set}{a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\n  subst : {A : Set}(P : A \u2192 Set){a b : A} \u2192 a \u2261 b \u2192 P a \u2192 P b\n\npostulate\n  _*_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  *-comm          : \u2200 m n \u2192 m * n \u2261 n * m\n  *-rightIdentity : \u2200 n \u2192 n * succ zero \u2261 n\n\n*-leftIdentity : \u2200 n \u2192 succ zero * n \u2261 n\n*-leftIdentity n =\n  trans {\u2115} {succ zero * n} {n * succ zero} {n} (*-comm (succ zero) n) (*-rightIdentity n)\n\nmodule ER\n  {A     : Set}\n  (_\u223c_   : A \u2192 A \u2192 Set)\n  (\u223c-refl  : \u2200 {x} \u2192 x \u223c x)\n  (\u223c-trans : \u2200 {x y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z)\n  where\n\n  infixr 5 _\u223c\u27e8_\u27e9_\n  infix  6 _\u220e\n\n  _\u223c\u27e8_\u27e9_ : \u2200 x {y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z\n  _ \u223c\u27e8 x\u223cy \u27e9 y\u223cz = \u223c-trans x\u223cy y\u223cz\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _\u220e _ = \u223c-refl\n\nopen module \u2261-Reasoning = ER _\u2261_ (refl {\u2115}) (trans {\u2115})\n  renaming ( _\u223c\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_ )\n\n*-leftIdentity' : \u2200 n \u2192 succ zero * n \u2261 n\n*-leftIdentity' n =\n  succ zero * n \u2261\u27e8 *-comm (succ zero) n \u27e9\n  n * succ zero \u2261\u27e8 *-rightIdentity n \u27e9\n  n             \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"83736713e89c34dbd05375de260cf1ac934c06ab","subject":"Try more","message":"Try more\n","repos":"louisswarren\/hieretikz","old_file":"hierarchy.agda","new_file":"hierarchy.agda","new_contents":"module hierarchy where\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n\n_and_ : Bool \u2192 Bool \u2192 Bool\ntrue and true = true\ntrue and false = false\nfalse and true = false\nfalse and false = false\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or true = true\ntrue or false = true\nfalse or true = true\nfalse or false = false\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A \u2192 List A \u2192 List A\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []      = false\nx \u2208 (y :: ys) with x == y\n...              | true  = true\n...              | false = x \u2208 ys\n\n_\u2282_ : List \u2115 \u2192 List \u2115 \u2192 Bool\n[] \u2282 ys      = true\n(x :: xs) \u2282 ys with x \u2208 ys\n...               | true  = xs \u2282 ys\n...               | false = false\n\n\n_++_ : {A : Set} \u2192 List A \u2192 List A \u2192 List A\n[] ++ ys        = ys\n(x :: xs) ++ ys = x :: (xs ++ ys)\n\nmap : {A : Set} \u2192 (A \u2192 A) \u2192 List A \u2192 List A\nmap f []        = []\nmap f (x :: xs) = (f x) :: (map f xs)\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []        = false\nany f (x :: xs) = (f x) or (any f xs)\n\nall : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nall _ []        = true\nall f (x :: xs) = (f x) and (all f xs)\n\n\ndata Arrow : Set where\n  _\u21d2_ : List \u2115 \u2192 \u2115 \u2192 Arrow\n\n_\u21d4_ : Arrow \u2192 Arrow \u2192 Bool\n(at \u21d2 ah) \u21d4 (bt \u21d2 bh) = ((at \u2282 bt) and (bt \u2282 at)) and (ah == bh)\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []      = false\nx \u2208\u2208 (y :: ys) with x \u21d4 y\n...               | true  = true\n...               | false = x \u2208\u2208 ys\n\n_\u2282\u2282_ : List Arrow \u2192 List Arrow \u2192 Bool\n[] \u2282\u2282 ys      = true\n(x :: xs) \u2282\u2282 ys with x \u2208\u2208 ys\n...                | true  = xs \u2282\u2282 ys\n...                | false = false\n\n\nArrowSearch : List Arrow \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nArrowSearch [] ds g = false\nArrowSearch (([] \u21d2 h) :: cs) ds g with (h == g)\n...         | true  = true\n...         | false = ArrowSearch cs (([] \u21d2 h) :: ds) g\nArrowSearch ((ts \u21d2 h) :: cs) ds g with (h == g)\n...         | true  = (all (ArrowSearch (cs ++ ds) []) ts)\n                  or (ArrowSearch cs ((ts \u21d2 h) :: ds) g)\n...         | false = ArrowSearch cs ((ts \u21d2 h) :: ds) g\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 ((t :: ts) \u21d2 h) = (([] \u21d2 t) :: cs) \u22a2 (ts \u21d2 h)\ncs \u22a2 ([] \u21d2 h)        = ArrowSearch cs [] h\n","old_contents":"module hierarchy where\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n\n_and_ : Bool \u2192 Bool \u2192 Bool\ntrue and true = true\ntrue and false = false\nfalse and true = false\nfalse and false = false\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A \u2192 List A \u2192 List A\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []      = false\nx \u2208 (y :: ys) with x == y\n...              | true  = true\n...              | false = x \u2208 ys\n\n_\u2282_ : List \u2115 \u2192 List \u2115 \u2192 Bool\n[] \u2282 ys      = true\n(x :: xs) \u2282 ys with x \u2208 ys\n...               | true  = xs \u2282 ys\n...               | false = false\n\n\n_++_ : List \u2115 \u2192 List \u2115 \u2192 List \u2115\n[] ++ ys        = ys\n(x :: xs) ++ ys with (x \u2208 ys)\n...               | true  = xs ++ ys\n...               | false = x :: (xs ++ ys)\n\nmap : {A : Set} \u2192 (A \u2192 A) \u2192 List A \u2192 List A\nmap f []        = []\nmap f (x :: xs) = (f x) :: (map f xs)\n\n\ndata Arrow : Set where\n  _\u21d2_ : List \u2115 \u2192 \u2115 \u2192 Arrow\n\n_\u21d4_ : Arrow \u2192 Arrow \u2192 Bool\n(at \u21d2 ah) \u21d4 (bt \u21d2 bh) = ((at \u2282 bt) and (bt \u2282 at)) and (ah == bh)\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []      = false\nx \u2208\u2208 (y :: ys) with x \u21d4 y\n...               | true  = true\n...               | false = x \u2208\u2208 ys\n\n_\u2282\u2282_ : List Arrow \u2192 List Arrow \u2192 Bool\n[] \u2282\u2282 ys      = true\n(x :: xs) \u2282\u2282 ys with x \u2208\u2208 ys\n...                | true  = xs \u2282\u2282 ys\n...                | false = false\n\n\n-- SingleClosure : Arrow \u2192 List \u2115 \u2192 List \u2115\n-- SingleClosure (ts \u21d2 h) ps with ts \u2282 ps\n-- ...                     | true  = h :: ps\n-- ...                     | false = ps\n--\n-- Closure' : List Arrow \u2192 List \u2115 \u2192 List \u2115\n-- Closure' [] roots = roots\n-- Closure' (c :: cs) roots = (SingleClosure c roots) ++ (Closure' cs roots)\n--\n-- Closure : List Arrow \u2192 List \u2115 \u2192 List \u2115\n-- Closure arrows roots with (Closure' arrows roots) \u2282 roots\n-- ...                     | true = roots\n-- ...                     | false = Closure arrows (Closure' arrows roots)\n--\n--\n-- _,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\n-- _ , [] \u22a2 _ = false\n-- thms , p :: premises \u22a2 conc = true\n--\n\nArrowIntro : \u2115 \u2192 Arrow \u2192 Arrow\nArrowIntro n (ts \u21d2 h) = (n :: ts) \u21d2 h\n\nArrowElim : \u2115 \u2192 Arrow \u2192 Arrow\nArrowElim _ ([] \u21d2 h) = [] \u21d2 h\nArrowElim n ((t :: ts) \u21d2 h) with (t == n)\n...                            | true  = ArrowElim  n (ts \u21d2 h)\n...                            | false = ArrowIntro t (ArrowElim n (ts \u21d2 h))\n\nReduce : List Arrow \u2192 List \u2115 \u2192 List Arrow\nReduce cs [] = cs\nReduce cs (n :: ns) with (map (ArrowElim n) cs) \u2282\u2282 cs\n...                    | true  = cs\n...                    | false = Reduce (map (ArrowElim n) cs) (n :: ns)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"666809a94d7b89e1d7e137009b28f6c4db550b60","subject":"CachingEvaluation: add TODO","message":"CachingEvaluation: add TODO\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\u2081\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u03c4\u2081 )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\u2081\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values (where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things)!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n\n  -- XXX: here we do have intermediate results, so we should save them.\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\u2081\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u03c4\u2081 )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\u2081\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values (where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things)!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e1edb95d9ae37f9ff8d1a6fdf36c68c75f7b3c33","subject":"Simplify","message":"Simplify\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product hiding (swap)\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  infixr 60 _\u2219S_\n  infixr 50 _+S_\n\n\n  _+S_ : \u2200 {r c} \u2192 M s r c \u2192 M s r c \u2192 M s r c\n  One x     +S One x\u2081    = One (x +\u209b x\u2081)\n  Row m m\u2081  +S Row n n\u2081  = Row (m +S n) (m\u2081 +S n\u2081)\n  Col m m\u2081  +S Col n n\u2081  = Col (m +S n) (m\u2081 +S n\u2081)\n  Q m00 m01\n    m10 m11 +S Q n00 n01\n                n10 n11 = Q (m00 +S n00) (m01 +S n01)\n                            (m10 +S n10) (m11 +S n11)\n\n  _\u2219S_ : \u2200 {r m c} \u2192 M s r m \u2192 M s m c \u2192 M s r c\n  One x     \u2219S One x\u2081    = One (x \u2219\u209b x\u2081)\n  One x     \u2219S Row n n\u2081  = Row (One x \u2219S n) (One x \u2219S n\u2081)\n  Row m m\u2081  \u2219S Col n n\u2081  = m \u2219S n +S m\u2081 \u2219S n\u2081\n  Row m m\u2081  \u2219S Q n00 n01\n                n10 n11 = Row (m \u2219S n00 +S m\u2081 \u2219S n10) (m \u2219S n01 +S m\u2081 \u2219S n11)\n  Col m m\u2081  \u2219S One x     = Col (m \u2219S One x) (m\u2081 \u2219S One x)\n  Col m m\u2081  \u2219S Row n n\u2081  = Q (m \u2219S n)   (m \u2219S n\u2081)\n                            (m\u2081 \u2219S n)  (m\u2081 \u2219S n\u2081)\n  Q m00 m01\n    m10 m11 \u2219S Col n n\u2081  = Col (m00 \u2219S n +S m01 \u2219S n\u2081) (m10 \u2219S n +S m11 \u2219S n\u2081)\n  Q m00 m01\n    m10 m11 \u2219S Q n00 n01\n                n10 n11 = Q (m00 \u2219S n00 +S m01 \u2219S n10) (m00 \u2219S n01 +S m01 \u2219S n11)\n                            (m10 \u2219S n00 +S m11 \u2219S n10) (m10 \u2219S n01 +S m11 \u2219S n11)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  _\u2243S_ : {r c : Shape} \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  _\u2243S_ {L} {L} (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  _\u2243S_ {L} {(B c\u2081 c\u2082)} (Row m m\u2081) (Row n n\u2081) =\n    _\u2243S_ m n \u00d7 _\u2243S_ m\u2081 n\u2081\n  _\u2243S_ {(B r\u2081 r\u2082)} {L} (Col m m\u2081) (Col n n\u2081) =\n    _\u2243S_ m n \u00d7 _\u2243S_ m\u2081 n\u2081\n  _\u2243S_ {(B r\u2081 r\u2082)} {(B c\u2081 c\u2082)} (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    _\u2243S_ m00 n00 \u00d7\n    _\u2243S_ m01 n01 \u00d7\n    _\u2243S_ m10 n10 \u00d7\n    _\u2243S_ m11 n11\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S j \u2192 j \u2243S i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S j \u2192 j \u2243S k \u2192 i \u2243S k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  assocS : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S (x +S (y +S z))\n  assocS L L (One x) (One y) (One z) = assoc\u209b x y z\n  assocS L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    assocS L c x y z  , assocS L c\u2081 x\u2081 y\u2081 z\u2081\n  assocS (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    assocS r L x y z , assocS r\u2081 L x\u2081 y\u2081 z\u2081\n  assocS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (assocS r c x y z) , (assocS r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (assocS r\u2081 c x\u2082 y\u2082 z\u2082) , (assocS r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S y \u2192 u \u2243S v \u2192 (x +S u) \u2243S (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = assocS shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  commS : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S (y +S x)\n  commS L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  commS L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (commS L c x y) , (commS L c\u2081 x\u2081 y\u2081)\n  commS (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (commS r L x y) , (commS r\u2081 L x\u2081 y\u2081)\n  commS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    commS r c x y , commS r c\u2081 x\u2081 y\u2081 ,\n    commS r\u2081 c x\u2082 y\u2082 , commS r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = commS shape shape }\n\n\n  -- used to make proofs nicer later on\n  setoidS : {r c : Shape} \u2192 Setoid _ _\n  setoidS {r} {c} =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = _\u2243S_ {r} {c}\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n  zerolHelp : \u2200 (r : Shape) {m m' c : Shape}\n    (x : M s m c)\n    (y : M s m' c) \u2192\n    0S r m \u2219S x \u2243S 0S r c \u2192\n    0S r m' \u2219S y \u2243S 0S r c \u2192\n    0S r m \u2219S x +S 0S r m' \u2219S y\n    \u2243S 0S r c\n  zerolHelp r {m} {m'} {c} x y p q =\n    let open EqReasoning setoidS\n    in begin\n      0S r m \u2219S x +S 0S r m' \u2219S y\n    \u2248\u27e8 <+S> _ _ {0S r m \u2219S x} {0S r c} {0S r m' \u2219S y} {0S r c} p q \u27e9\n      0S r c +S 0S r c\n    \u2248\u27e8 identS\u02e1 _ _ (0S r c) \u27e9\n     0S r c\n    \u220e\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    zerolHelp L x x\u2081 (zeroS\u02e1 L b L x) (zeroS\u02e1 L b\u2081 L x\u2081)\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    zerolHelp L x x\u2082 (zeroS\u02e1 L b c x) (zeroS\u02e1 L b\u2081 c x\u2082) ,\n    zerolHelp L x\u2081 x\u2083 (zeroS\u02e1 L b c\u2081 x\u2081) (zeroS\u02e1 L b\u2081 c\u2081 x\u2083)\n  zeroS\u02e1 (B a a\u2081) L L (One x) =\n    zeroS\u02e1 a L L (One x) ,\n    zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x ,\n    zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x ,\n    zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    zerolHelp a x x\u2081 (zeroS\u02e1 a b L x) (zeroS\u02e1 a b\u2081 L x\u2081) ,\n    zerolHelp a\u2081 x x\u2081 (zeroS\u02e1 a\u2081 b L x) (zeroS\u02e1 a\u2081 b\u2081 L x\u2081)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    zerolHelp a x x\u2082 (zeroS\u02e1 a b c x) (zeroS\u02e1 a b\u2081 c x\u2082) ,\n    zerolHelp a x\u2081 x\u2083 (zeroS\u02e1 a b c\u2081 x\u2081) (zeroS\u02e1 a b\u2081 c\u2081 x\u2083) ,\n    zerolHelp a\u2081 x x\u2082 (zeroS\u02e1 a\u2081 b c x) (zeroS\u02e1 a\u2081 b\u2081 c x\u2082) ,\n    zerolHelp a\u2081 x\u2081 x\u2083 (zeroS\u02e1 a\u2081 b c\u2081 x\u2081) (zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083)\n\n  zerorHelp : \u2200 r {m m' c}\n    (x : M s r m)\n    (x\u2081 : M s r m') \u2192\n    x \u2219S 0S m c \u2243S 0S r c \u2192\n    x\u2081 \u2219S 0S m' c \u2243S 0S r c \u2192\n    x \u2219S 0S m c +S x\u2081 \u2219S 0S m' c\n    \u2243S 0S r c\n  zerorHelp r {m} {m'} {c} x x\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      x \u2219S 0S m c +S x\u2081 \u2219S 0S m' c\n    \u2248\u27e8 <+S> _ _ {x \u2219S 0S m c} {0S r c} {x\u2081 \u2219S 0S m' c} {0S r c} p q \u27e9\n      0S r c +S 0S r c\n    \u2248\u27e8 identS\u02e1 r c (0S r c) \u27e9\n      0S r c\n    \u220e\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) ,\n    (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    zerorHelp L {c = L} x x\u2081 (zeroS\u02b3 L b L x) (zeroS\u02b3 L b\u2081 L x\u2081)\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    zerorHelp L {c = c} x x\u2081 (zeroS\u02b3 L b c x) (zeroS\u02b3 L b\u2081 c x\u2081) ,\n    zerorHelp L {c = c\u2081} x x\u2081 (zeroS\u02b3 L b c\u2081 x) (zeroS\u02b3 L b\u2081 c\u2081 x\u2081)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) =\n    zeroS\u02b3 a L L x ,\n    zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    zerorHelp a x x\u2081 (zeroS\u02b3 a b L x) (zeroS\u02b3 a b\u2081 L x\u2081) ,\n    zerorHelp a\u2081 x\u2082 x\u2083 (zeroS\u02b3 a\u2081 b L x\u2082) (zeroS\u02b3 a\u2081 b\u2081 L x\u2083)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    zerorHelp a x x\u2081 (zeroS\u02b3 a b c x) (zeroS\u02b3 a b\u2081 c x\u2081) ,\n    zerorHelp a x x\u2081 (zeroS\u02b3 a b c\u2081 x) (zeroS\u02b3 a b\u2081 c\u2081 x\u2081) ,\n    zerorHelp a\u2081 x\u2082 x\u2083 (zeroS\u02b3 a\u2081 b c x\u2082) (zeroS\u02b3 a\u2081 b\u2081 c x\u2083) ,\n    zerorHelp a\u2081 x\u2082 x\u2083 (zeroS\u02b3 a\u2081 b c\u2081 x\u2082) (zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083)\n\n  <\u2219S> : (a b c : Shape) {x y : M s a b} {u v : M s b c} \u2192\n    x \u2243S y \u2192 u \u2243S v \u2192 (x \u2219S u) \u2243S (y \u2219S v)\n  <\u2219S> L L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <\u2219> q\n  <\u2219S> L L (B c c\u2081) {One x} {One x\u2081} {Row u u\u2081} {Row v v\u2081} p (q , q\u2081) =\n    (<\u2219S> L L c {One x} {One x\u2081} {u} {v} p q) ,\n    <\u2219S> L L c\u2081 {One x} {One x\u2081} {u\u2081} {v\u2081} p q\u2081\n  <\u2219S> L (B b b\u2081) L {Row x x\u2081} {Row y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    -- Row x x\u2081 \u2219S Col u u\u2081 \u2243S Row y y\u2081 \u2219S Col v v\u2081\n    let\n      open EqReasoning setoidS\n      ih = <\u2219S> _ _ _ {x} {y} {u} {v} p q\n      ih\u2081 = <\u2219S> _ _ _ {x\u2081} {y\u2081} {u\u2081} {v\u2081} p\u2081 q\u2081\n    in begin\n      Row x x\u2081 \u2219S Col u u\u2081\n    \u2261\u27e8 refl-\u2261 \u27e9\n      x \u2219S u +S x\u2081 \u2219S u\u2081\n    \u2248\u27e8 <+S> L L {x \u2219S u} {y \u2219S v} {x\u2081 \u2219S u\u2081} {y\u2081 \u2219S v\u2081} ih ih\u2081 \u27e9\n      y \u2219S v +S y\u2081 \u2219S v\u2081\n    \u220e\n  <\u2219S> L (B b b\u2081) (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081) (q , q\u2081 , q\u2082 , q\u2083) =\n    (let\n      ih = <\u2219S> L b c p q\n      ih\u2081 = <\u2219S> L b\u2081 c p\u2081 q\u2082\n    in <+S> L c ih ih\u2081) ,\n    <+S> L c\u2081 (<\u2219S> L b c\u2081 p q\u2081) (<\u2219S> L b\u2081 c\u2081 p\u2081 q\u2083)\n  <\u2219S> (B a a\u2081) L L {Col x x\u2081} {Col y y\u2081} {One x\u2082} {One x\u2083} (p , p\u2081) q =\n    <\u2219S> a L L p q ,\n    <\u2219S> a\u2081 L L p\u2081 q\n  <\u2219S> (B a a\u2081) L (B c c\u2081) {Col x x\u2081} {Col y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <\u2219S> a L c p q ,\n    <\u2219S> a L c\u2081 p q\u2081  ,\n    <\u2219S> a\u2081 L c p\u2081 q ,\n    <\u2219S> a\u2081 L c\u2081 p\u2081 q\u2081\n  <\u2219S> (B a a\u2081) (B b b\u2081) L {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Col u u\u2081} {Col v v\u2081} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081) =\n    <+S> a L (<\u2219S> a b L p q) (<\u2219S> a b\u2081 L p\u2081 q\u2081) ,\n    <+S> a\u2081 L (<\u2219S> a\u2081 b L p\u2082 q) (<\u2219S> a\u2081 b\u2081 L p\u2083 q\u2081 )\n  <\u2219S> (B a a\u2081) (B b b\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> a c (<\u2219S> a b c p q) (<\u2219S> a b\u2081 c p\u2081 q\u2082) ,\n    <+S> a c\u2081 (<\u2219S> a b c\u2081 p q\u2081) (<\u2219S> a b\u2081 c\u2081 p\u2081 q\u2083) ,\n    <+S> a\u2081 c (<\u2219S> a\u2081 b c p\u2082 q) (<\u2219S> a\u2081 b\u2081 c p\u2083 q\u2082) ,\n    <+S> a\u2081 c\u2081 (<\u2219S> a\u2081 b c\u2081 p\u2082 q\u2081) (<\u2219S> a\u2081 b\u2081 c\u2081 p\u2083 q\u2083)\n\n  idemS : (r c : Shape) (x : M s r c) \u2192 x +S x \u2243S x\n  idemS L L (One x) = idem x\n  idemS L (B c c\u2081) (Row x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) L (Col x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (idemS _ _ x) , (idemS _ _ x\u2081 , (idemS _ _ x\u2082  , idemS _ _ x\u2083))\n\n  swapMid : {r c : Shape} (x y z w : M s r c) \u2192\n    (x +S y) +S (z +S w) \u2243S (x +S z) +S (y +S w)\n  swapMid {r} {c} x y z w =\n    let open EqReasoning setoidS\n    in begin\n      (x +S y) +S (z +S w)\n    \u2248\u27e8 assocS _ _ x y (z +S w) \u27e9\n      x +S y +S z +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (symS r c (assocS r c y z w)) \u27e9\n      x +S (y +S z) +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (<+S> r c (commS r c y z) (reflS r c)) \u27e9\n      x +S (z +S y) +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (assocS r c z y w) \u27e9\n      x +S z +S y +S w\n    \u2248\u27e8 symS r c (assocS r c x z (y +S w)) \u27e9\n      (x +S z) +S (y +S w)\n    \u220e\n\n  distlHelp : \u2200 {a b b\u2081 c\u2081}\n              (x : M s a b)\n              (y z : M s b c\u2081)\n              (x\u2081 : M s a b\u2081)\n              (y\u2081 z\u2081 : M s b\u2081 c\u2081) \u2192\n            (x \u2219S (y +S z)) \u2243S (x \u2219S y +S x \u2219S z) \u2192\n            (x\u2081 \u2219S (y\u2081 +S z\u2081)) \u2243S (x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081) \u2192\n            (x \u2219S (y +S z) +S x\u2081 \u2219S (y\u2081 +S z\u2081))\n            \u2243S ((x \u2219S y +S x\u2081 \u2219S y\u2081) +S x \u2219S z +S x\u2081 \u2219S z\u2081)\n  distlHelp x y z x\u2081 y\u2081 z\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      x \u2219S (y +S z) +S x\u2081 \u2219S (y\u2081 +S z\u2081)\n    \u2248\u27e8 <+S> _ _ {x \u2219S (y +S z)} {x \u2219S y +S x \u2219S z}\n                {x\u2081 \u2219S (y\u2081 +S z\u2081)} {x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081} p q \u27e9\n      (x \u2219S y +S x \u2219S z) +S x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081\n    \u2248\u27e8 swapMid (x \u2219S y) (x \u2219S z) (x\u2081 \u2219S y\u2081) (x\u2081 \u2219S z\u2081) \u27e9\n      (x \u2219S y +S x\u2081 \u2219S y\u2081) +S x \u2219S z +S x\u2081 \u2219S z\u2081\n    \u220e\n\n  distlS : {a b c : Shape} (x : M s a b) (y z : M s b c) \u2192\n    (x \u2219S (y +S z)) \u2243S ((x \u2219S y) +S (x \u2219S z))\n  distlS {L} {L} {L} (One x) (One y) (One z) = distl x y z\n  distlS {L} {L} {B c c\u2081} (One x) (Row y y\u2081) (Row z z\u2081) =\n    distlS (One x) y z ,\n    distlS (One x) y\u2081 z\u2081\n  distlS {L} {(B b b\u2081)} {L} (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    distlHelp x y z x\u2081 y\u2081 z\u2081 (distlS x y z) (distlS x\u2081 y\u2081 z\u2081)\n  distlS {L} {(B b b\u2081)} {(B c c\u2081)} (Row x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distlHelp x y z x\u2081 y\u2082 z\u2082 (distlS x y z) (distlS x\u2081 y\u2082 z\u2082) ,\n    distlHelp x y\u2081 z\u2081 x\u2081 y\u2083 z\u2083  (distlS x y\u2081 z\u2081) (distlS x\u2081 y\u2083 z\u2083)\n  distlS {(B a a\u2081)} {L} {L} (Col x x\u2081) (One x\u2082) (One x\u2083) =\n    distlS x (One x\u2082) (One x\u2083) ,\n    distlS x\u2081 (One x\u2082) (One x\u2083)\n  distlS {(B a a\u2081)} {L} {(B c c\u2081)} (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    distlS x y z ,\n    distlS x y\u2081 z\u2081 ,\n    distlS x\u2081 y z ,\n    distlS x\u2081 y\u2081 z\u2081\n  distlS {(B a a\u2081)} {(B b b\u2081)} {L} (Q x x\u2081 x\u2082 x\u2083) (Col y y\u2081) (Col z z\u2081) =\n    distlHelp x y z x\u2081 y\u2081 z\u2081 (distlS x y z) (distlS x\u2081 y\u2081 z\u2081) ,\n    distlHelp x\u2082 y z x\u2083 y\u2081 z\u2081 (distlS x\u2082 y z) (distlS x\u2083 y\u2081 z\u2081)\n  distlS {(B a a\u2081)} {(B b b\u2081)} {(B c c\u2081)} (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distlHelp x y z x\u2081 y\u2082 z\u2082 (distlS x y z) (distlS x\u2081 y\u2082 z\u2082) ,\n    distlHelp x y\u2081 z\u2081 x\u2081 y\u2083 z\u2083 (distlS x y\u2081 z\u2081) (distlS x\u2081 y\u2083 z\u2083) ,\n    distlHelp x\u2082 y z x\u2083 y\u2082 z\u2082 (distlS x\u2082 y z) (distlS x\u2083 y\u2082 z\u2082)  ,\n    distlHelp x\u2082 y\u2081 z\u2081 x\u2083 y\u2083 z\u2083 (distlS x\u2082 y\u2081 z\u2081) (distlS x\u2083 y\u2083 z\u2083)\n\n  distrHelp : \u2200 {r m m\u2081 c : Shape}\n              (x : M s m c)\n              (y z : M s r m)\n              (x\u2081 : M s m\u2081 c)\n              (y\u2081 z\u2081 : M s r m\u2081) \u2192\n            ((y +S z) \u2219S x) \u2243S (y \u2219S x +S z \u2219S x) \u2192\n            ((y\u2081 +S z\u2081) \u2219S x\u2081) \u2243S (y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081) \u2192\n            ((y +S z) \u2219S x +S (y\u2081 +S z\u2081) \u2219S x\u2081)\n            \u2243S ((y \u2219S x +S y\u2081 \u2219S x\u2081) +S z \u2219S x +S z\u2081 \u2219S x\u2081)\n  distrHelp x y z x\u2081 y\u2081 z\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      (y +S z) \u2219S x +S (y\u2081 +S z\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> _ _ {(y +S z) \u2219S x} {y \u2219S x +S z \u2219S x}\n                {(y\u2081 +S z\u2081) \u2219S x\u2081} {y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081} p q \u27e9\n      (y \u2219S x +S z \u2219S x) +S y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081\n    \u2248\u27e8 swapMid (y \u2219S x) (z \u2219S x) (y\u2081 \u2219S x\u2081) (z\u2081 \u2219S x\u2081) \u27e9\n      (y \u2219S x +S y\u2081 \u2219S x\u2081) +S z \u2219S x +S z\u2081 \u2219S x\u2081\n    \u220e\n\n  distrS : {r m c : Shape} (x : M s m c) (y z : M s r m) \u2192\n    ((y +S z) \u2219S x) \u2243S ((y \u2219S x) +S (z \u2219S x))\n  distrS {L} {L} {L} (One x) (One y) (One z) =\n    distr x y z\n  distrS {L} {L} {B c c\u2081} (Row x x\u2081) (One x\u2082) (One x\u2083) =\n    (distrS x (One x\u2082) (One x\u2083)) ,\n    (distrS x\u2081 (One x\u2082) (One x\u2083))\n  distrS {L} {B m m\u2081} {L} (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    distrHelp x y z x\u2081 y\u2081 z\u2081 (distrS x y z) (distrS x\u2081 y\u2081 z\u2081)\n  distrS {L} {B m m\u2081} {B c c\u2081} (Q x x\u2081 x\u2082 x\u2083) (Row y y\u2081) (Row z z\u2081) =\n    (distrHelp x y z x\u2082 y\u2081 z\u2081 (distrS x y z) (distrS x\u2082 y\u2081 z\u2081)) ,\n    (distrHelp x\u2081 y z x\u2083 y\u2081 z\u2081 (distrS x\u2081 y z) (distrS x\u2083 y\u2081 z\u2081))\n  distrS {B r r\u2081} {L} {L} (One x) (Col y y\u2081) (Col z z\u2081) =\n    distrS (One x) y z ,\n    distrS (One x) y\u2081 z\u2081\n  distrS {B r r\u2081} {L} {B c c\u2081} (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    (distrS x y z) ,\n    (distrS x\u2081 y z) ,\n    (distrS x y\u2081 z\u2081) ,\n    (distrS x\u2081 y\u2081 z\u2081)\n  distrS {B r r\u2081} {B m m\u2081} {L} (Col x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (distrHelp x y z x\u2081 y\u2081 z\u2081 (distrS x y z) (distrS x\u2081 y\u2081 z\u2081)) ,\n    (distrHelp x y\u2082 z\u2082 x\u2081 y\u2083 z\u2083 (distrS x y\u2082 z\u2082) (distrS x\u2081 y\u2083 z\u2083))\n  distrS {B r r\u2081} {B m m\u2081} {B c c\u2081} (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distrHelp x y z x\u2082 y\u2081 z\u2081 (distrS x y z) (distrS x\u2082 y\u2081 z\u2081) ,\n    distrHelp x\u2081 y z x\u2083 y\u2081 z\u2081 (distrS x\u2081 y z) (distrS x\u2083 y\u2081 z\u2081) ,\n    distrHelp x y\u2082 z\u2082 x\u2082 y\u2083 z\u2083 (distrS x y\u2082 z\u2082) (distrS x\u2082 y\u2083 z\u2083) ,\n    distrHelp x\u2081 y\u2082 z\u2082 x\u2083 y\u2083 z\u2083 (distrS x\u2081 y\u2082 z\u2082) (distrS x\u2083 y\u2083 z\u2083)\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = _\u2243S_ {shape} {shape}\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = <\u2219S> shape shape shape\n      ; idem = idemS shape shape\n      ; distl = distlS {shape} {shape}\n      ; distr = distrS {shape} {shape}\n      }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product hiding (swap)\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  infixr 60 _\u2219S_\n  infixr 50 _+S_\n\n\n  _+S_ : \u2200 {r c} \u2192 M s r c \u2192 M s r c \u2192 M s r c\n  One x     +S One x\u2081    = One (x +\u209b x\u2081)\n  Row m m\u2081  +S Row n n\u2081  = Row (m +S n) (m\u2081 +S n\u2081)\n  Col m m\u2081  +S Col n n\u2081  = Col (m +S n) (m\u2081 +S n\u2081)\n  Q m00 m01\n    m10 m11 +S Q n00 n01\n                n10 n11 = Q (m00 +S n00) (m01 +S n01)\n                            (m10 +S n10) (m11 +S n11)\n\n  _\u2219S_ : \u2200 {r m c} \u2192 M s r m \u2192 M s m c \u2192 M s r c\n  One x     \u2219S One x\u2081    = One (x \u2219\u209b x\u2081)\n  One x     \u2219S Row n n\u2081  = Row (One x \u2219S n) (One x \u2219S n\u2081)\n  Row m m\u2081  \u2219S Col n n\u2081  = m \u2219S n +S m\u2081 \u2219S n\u2081\n  Row m m\u2081  \u2219S Q n00 n01\n                n10 n11 = Row (m \u2219S n00 +S m\u2081 \u2219S n10) (m \u2219S n01 +S m\u2081 \u2219S n11)\n  Col m m\u2081  \u2219S One x     = Col (m \u2219S One x) (m\u2081 \u2219S One x)\n  Col m m\u2081  \u2219S Row n n\u2081  = Q (m \u2219S n)   (m \u2219S n\u2081)\n                            (m\u2081 \u2219S n)  (m\u2081 \u2219S n\u2081)\n  Q m00 m01\n    m10 m11 \u2219S Col n n\u2081  = Col (m00 \u2219S n +S m01 \u2219S n\u2081) (m10 \u2219S n +S m11 \u2219S n\u2081)\n  Q m00 m01\n    m10 m11 \u2219S Q n00 n01\n                n10 n11 = Q (m00 \u2219S n00 +S m01 \u2219S n10) (m00 \u2219S n01 +S m01 \u2219S n11)\n                            (m10 \u2219S n00 +S m11 \u2219S n10) (m10 \u2219S n01 +S m11 \u2219S n11)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  _\u2243S_ : {r c : Shape} \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  _\u2243S_ {L} {L} (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  _\u2243S_ {L} {(B c\u2081 c\u2082)} (Row m m\u2081) (Row n n\u2081) =\n    _\u2243S_ m n \u00d7 _\u2243S_ m\u2081 n\u2081\n  _\u2243S_ {(B r\u2081 r\u2082)} {L} (Col m m\u2081) (Col n n\u2081) =\n    _\u2243S_ m n \u00d7 _\u2243S_ m\u2081 n\u2081\n  _\u2243S_ {(B r\u2081 r\u2082)} {(B c\u2081 c\u2082)} (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    _\u2243S_ m00 n00 \u00d7\n    _\u2243S_ m01 n01 \u00d7\n    _\u2243S_ m10 n10 \u00d7\n    _\u2243S_ m11 n11\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S j \u2192 j \u2243S i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S j \u2192 j \u2243S k \u2192 i \u2243S k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  assocS : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S (x +S (y +S z))\n  assocS L L (One x) (One y) (One z) = assoc\u209b x y z\n  assocS L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    assocS L c x y z  , assocS L c\u2081 x\u2081 y\u2081 z\u2081\n  assocS (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    assocS r L x y z , assocS r\u2081 L x\u2081 y\u2081 z\u2081\n  assocS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (assocS r c x y z) , (assocS r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (assocS r\u2081 c x\u2082 y\u2082 z\u2082) , (assocS r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S y \u2192 u \u2243S v \u2192 (x +S u) \u2243S (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = assocS shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  commS : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S (y +S x)\n  commS L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  commS L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (commS L c x y) , (commS L c\u2081 x\u2081 y\u2081)\n  commS (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (commS r L x y) , (commS r\u2081 L x\u2081 y\u2081)\n  commS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    commS r c x y , commS r c\u2081 x\u2081 y\u2081 ,\n    commS r\u2081 c x\u2082 y\u2082 , commS r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = commS shape shape }\n\n\n  setoidS : {r c : Shape} \u2192 Setoid _ _\n  setoidS {r} {c} =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = _\u2243S_ {r} {c}\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) , (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 L b L x\n      ih\u2081 = zeroS\u02b3 L b\u2081 L x\u2081\n    in begin\n      Row x x\u2081 \u2219S Col (0S b L) (0S b\u2081 L)\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (x \u2219S 0S b L) +S (x\u2081 \u2219S 0S b\u2081 L)\n    \u2248\u27e8 <+S> L L {x \u2219S 0S b L} {0S L L} {x\u2081 \u2219S 0S b\u2081 L} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 L b c x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n      0S L c +S 0S L c\n    \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n      0S L c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 L b c\u2081 x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 (0S L c\u2081) \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) = zeroS\u02b3 a L L x , zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a b L x\n      ih\u2081 = zeroS\u02b3 a b\u2081 L x\u2081\n    in begin\n      x \u2219S 0S b L +S x\u2081 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L (0S _ _) \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a\u2081 b L x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 L x\u2083\n    in begin\n      x\u2082 \u2219S 0S b L +S x\u2083 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 L (0S _ _) \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a b c x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c (0S _ _) \u27e9\n      0S _ _\n    \u220e) ,\n    (let\n      ih = zeroS\u02b3 a b c\u2081 x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c\u2081 x\u2081\n    in transS a c\u2081 (<+S> a c\u2081 ih ih\u2081) (identS\u02e1 a c\u2081 (0S _ _))\n    ) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a\u2081 b c x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c +S x\u2083 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c (0S _ _) \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a\u2081 b c\u2081 x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c\u2081 +S x\u2083 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 (0S _ _) \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  <\u2219S> : (a b c : Shape) {x y : M s a b} {u v : M s b c} \u2192\n    x \u2243S y \u2192 u \u2243S v \u2192 (x \u2219S u) \u2243S (y \u2219S v)\n  <\u2219S> L L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <\u2219> q\n  <\u2219S> L L (B c c\u2081) {One x} {One x\u2081} {Row u u\u2081} {Row v v\u2081} p (q , q\u2081) =\n    (<\u2219S> L L c {One x} {One x\u2081} {u} {v} p q) ,\n    <\u2219S> L L c\u2081 {One x} {One x\u2081} {u\u2081} {v\u2081} p q\u2081\n  <\u2219S> L (B b b\u2081) L {Row x x\u2081} {Row y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    -- Row x x\u2081 \u2219S Col u u\u2081 \u2243S Row y y\u2081 \u2219S Col v v\u2081\n    let\n      open EqReasoning setoidS\n      ih = <\u2219S> _ _ _ {x} {y} {u} {v} p q\n      ih\u2081 = <\u2219S> _ _ _ {x\u2081} {y\u2081} {u\u2081} {v\u2081} p\u2081 q\u2081\n    in begin\n      Row x x\u2081 \u2219S Col u u\u2081\n    \u2261\u27e8 refl-\u2261 \u27e9\n      x \u2219S u +S x\u2081 \u2219S u\u2081\n    \u2248\u27e8 <+S> L L {x \u2219S u} {y \u2219S v} {x\u2081 \u2219S u\u2081} {y\u2081 \u2219S v\u2081} ih ih\u2081 \u27e9\n      y \u2219S v +S y\u2081 \u2219S v\u2081\n    \u220e\n  <\u2219S> L (B b b\u2081) (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081) (q , q\u2081 , q\u2082 , q\u2083) =\n    (let\n      ih = <\u2219S> L b c p q\n      ih\u2081 = <\u2219S> L b\u2081 c p\u2081 q\u2082\n    in <+S> L c ih ih\u2081) ,\n    <+S> L c\u2081 (<\u2219S> L b c\u2081 p q\u2081) (<\u2219S> L b\u2081 c\u2081 p\u2081 q\u2083)\n  <\u2219S> (B a a\u2081) L L {Col x x\u2081} {Col y y\u2081} {One x\u2082} {One x\u2083} (p , p\u2081) q =\n    <\u2219S> a L L p q ,\n    <\u2219S> a\u2081 L L p\u2081 q\n  <\u2219S> (B a a\u2081) L (B c c\u2081) {Col x x\u2081} {Col y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <\u2219S> a L c p q ,\n    <\u2219S> a L c\u2081 p q\u2081  ,\n    <\u2219S> a\u2081 L c p\u2081 q ,\n    <\u2219S> a\u2081 L c\u2081 p\u2081 q\u2081\n  <\u2219S> (B a a\u2081) (B b b\u2081) L {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Col u u\u2081} {Col v v\u2081} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081) =\n    <+S> a L (<\u2219S> a b L p q) (<\u2219S> a b\u2081 L p\u2081 q\u2081) ,\n    <+S> a\u2081 L (<\u2219S> a\u2081 b L p\u2082 q) (<\u2219S> a\u2081 b\u2081 L p\u2083 q\u2081 )\n  <\u2219S> (B a a\u2081) (B b b\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> a c (<\u2219S> a b c p q) (<\u2219S> a b\u2081 c p\u2081 q\u2082) ,\n    <+S> a c\u2081 (<\u2219S> a b c\u2081 p q\u2081) (<\u2219S> a b\u2081 c\u2081 p\u2081 q\u2083) ,\n    <+S> a\u2081 c (<\u2219S> a\u2081 b c p\u2082 q) (<\u2219S> a\u2081 b\u2081 c p\u2083 q\u2082) ,\n    <+S> a\u2081 c\u2081 (<\u2219S> a\u2081 b c\u2081 p\u2082 q\u2081) (<\u2219S> a\u2081 b\u2081 c\u2081 p\u2083 q\u2083)\n\n  idemS : (r c : Shape) (x : M s r c) \u2192 x +S x \u2243S x\n  idemS L L (One x) = idem x\n  idemS L (B c c\u2081) (Row x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) L (Col x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (idemS _ _ x) , (idemS _ _ x\u2081 , (idemS _ _ x\u2082  , idemS _ _ x\u2083))\n\n  swapMid : {r c : Shape} (x y z w : M s r c) \u2192\n    (x +S y) +S (z +S w) \u2243S (x +S z) +S (y +S w)\n  swapMid {r} {c} x y z w =\n    let open EqReasoning setoidS\n    in begin\n      (x +S y) +S (z +S w)\n    \u2248\u27e8 assocS _ _ x y (z +S w) \u27e9\n      x +S y +S z +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (symS r c (assocS r c y z w)) \u27e9\n      x +S (y +S z) +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (<+S> r c (commS r c y z) (reflS r c)) \u27e9\n      x +S (z +S y) +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (assocS r c z y w) \u27e9\n      x +S z +S y +S w\n    \u2248\u27e8 symS r c (assocS r c x z (y +S w)) \u27e9\n      (x +S z) +S (y +S w)\n    \u220e\n\n  distlHelp : \u2200 {a b b\u2081 c\u2081}\n              (x : M s a b)\n              (y z : M s b c\u2081)\n              (x\u2081 : M s a b\u2081)\n              (y\u2081 z\u2081 : M s b\u2081 c\u2081) \u2192\n            (x \u2219S (y +S z)) \u2243S (x \u2219S y +S x \u2219S z) \u2192\n            (x\u2081 \u2219S (y\u2081 +S z\u2081)) \u2243S (x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081) \u2192\n            (x \u2219S (y +S z) +S x\u2081 \u2219S (y\u2081 +S z\u2081))\n            \u2243S ((x \u2219S y +S x\u2081 \u2219S y\u2081) +S x \u2219S z +S x\u2081 \u2219S z\u2081)\n  distlHelp x y z x\u2081 y\u2081 z\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      x \u2219S (y +S z) +S x\u2081 \u2219S (y\u2081 +S z\u2081)\n    \u2248\u27e8 <+S> _ _ {x \u2219S (y +S z)} {x \u2219S y +S x \u2219S z}\n                {x\u2081 \u2219S (y\u2081 +S z\u2081)} {x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081} p q \u27e9\n      (x \u2219S y +S x \u2219S z) +S x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081\n    \u2248\u27e8 swapMid (x \u2219S y) (x \u2219S z) (x\u2081 \u2219S y\u2081) (x\u2081 \u2219S z\u2081) \u27e9\n      (x \u2219S y +S x\u2081 \u2219S y\u2081) +S x \u2219S z +S x\u2081 \u2219S z\u2081\n    \u220e\n\n  distlS : {a b c : Shape} (x : M s a b) (y z : M s b c) \u2192\n    (x \u2219S (y +S z)) \u2243S ((x \u2219S y) +S (x \u2219S z))\n  distlS {L} {L} {L} (One x) (One y) (One z) = distl x y z\n  distlS {L} {L} {B c c\u2081} (One x) (Row y y\u2081) (Row z z\u2081) =\n    distlS (One x) y z ,\n    distlS (One x) y\u2081 z\u2081\n  distlS {L} {(B b b\u2081)} {L} (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    distlHelp x y z x\u2081 y\u2081 z\u2081 (distlS x y z) (distlS x\u2081 y\u2081 z\u2081)\n  distlS {L} {(B b b\u2081)} {(B c c\u2081)} (Row x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distlHelp x y z x\u2081 y\u2082 z\u2082 (distlS x y z) (distlS x\u2081 y\u2082 z\u2082) ,\n    distlHelp x y\u2081 z\u2081 x\u2081 y\u2083 z\u2083  (distlS x y\u2081 z\u2081) (distlS x\u2081 y\u2083 z\u2083)\n  distlS {(B a a\u2081)} {L} {L} (Col x x\u2081) (One x\u2082) (One x\u2083) =\n    distlS x (One x\u2082) (One x\u2083) ,\n    distlS x\u2081 (One x\u2082) (One x\u2083)\n  distlS {(B a a\u2081)} {L} {(B c c\u2081)} (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    distlS x y z ,\n    distlS x y\u2081 z\u2081 ,\n    distlS x\u2081 y z ,\n    distlS x\u2081 y\u2081 z\u2081\n  distlS {(B a a\u2081)} {(B b b\u2081)} {L} (Q x x\u2081 x\u2082 x\u2083) (Col y y\u2081) (Col z z\u2081) =\n    distlHelp x y z x\u2081 y\u2081 z\u2081 (distlS x y z) (distlS x\u2081 y\u2081 z\u2081) ,\n    distlHelp x\u2082 y z x\u2083 y\u2081 z\u2081 (distlS x\u2082 y z) (distlS x\u2083 y\u2081 z\u2081)\n  distlS {(B a a\u2081)} {(B b b\u2081)} {(B c c\u2081)} (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distlHelp x y z x\u2081 y\u2082 z\u2082 (distlS x y z) (distlS x\u2081 y\u2082 z\u2082) ,\n    distlHelp x y\u2081 z\u2081 x\u2081 y\u2083 z\u2083 (distlS x y\u2081 z\u2081) (distlS x\u2081 y\u2083 z\u2083) ,\n    distlHelp x\u2082 y z x\u2083 y\u2082 z\u2082 (distlS x\u2082 y z) (distlS x\u2083 y\u2082 z\u2082)  ,\n    distlHelp x\u2082 y\u2081 z\u2081 x\u2083 y\u2083 z\u2083 (distlS x\u2082 y\u2081 z\u2081) (distlS x\u2083 y\u2083 z\u2083)\n\n  distrHelp : \u2200 {r m m\u2081 c : Shape}\n              (x : M s m c)\n              (y z : M s r m)\n              (x\u2081 : M s m\u2081 c)\n              (y\u2081 z\u2081 : M s r m\u2081) \u2192\n            ((y +S z) \u2219S x) \u2243S (y \u2219S x +S z \u2219S x) \u2192\n            ((y\u2081 +S z\u2081) \u2219S x\u2081) \u2243S (y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081) \u2192\n            ((y +S z) \u2219S x +S (y\u2081 +S z\u2081) \u2219S x\u2081)\n            \u2243S ((y \u2219S x +S y\u2081 \u2219S x\u2081) +S z \u2219S x +S z\u2081 \u2219S x\u2081)\n  distrHelp x y z x\u2081 y\u2081 z\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      (y +S z) \u2219S x +S (y\u2081 +S z\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> _ _ {(y +S z) \u2219S x} {y \u2219S x +S z \u2219S x}\n                {(y\u2081 +S z\u2081) \u2219S x\u2081} {y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081} p q \u27e9\n      (y \u2219S x +S z \u2219S x) +S y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081\n    \u2248\u27e8 swapMid (y \u2219S x) (z \u2219S x) (y\u2081 \u2219S x\u2081) (z\u2081 \u2219S x\u2081) \u27e9\n      (y \u2219S x +S y\u2081 \u2219S x\u2081) +S z \u2219S x +S z\u2081 \u2219S x\u2081\n    \u220e\n\n  distrS : {r m c : Shape} (x : M s m c) (y z : M s r m) \u2192\n    ((y +S z) \u2219S x) \u2243S ((y \u2219S x) +S (z \u2219S x))\n  distrS {L} {L} {L} (One x) (One y) (One z) =\n    distr x y z\n  distrS {L} {L} {B c c\u2081} (Row x x\u2081) (One x\u2082) (One x\u2083) =\n    (distrS x (One x\u2082) (One x\u2083)) ,\n    (distrS x\u2081 (One x\u2082) (One x\u2083))\n  distrS {L} {B m m\u2081} {L} (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    distrHelp x y z x\u2081 y\u2081 z\u2081 (distrS x y z) (distrS x\u2081 y\u2081 z\u2081)\n  distrS {L} {B m m\u2081} {B c c\u2081} (Q x x\u2081 x\u2082 x\u2083) (Row y y\u2081) (Row z z\u2081) =\n    (distrHelp x y z x\u2082 y\u2081 z\u2081 (distrS x y z) (distrS x\u2082 y\u2081 z\u2081)) ,\n    (distrHelp x\u2081 y z x\u2083 y\u2081 z\u2081 (distrS x\u2081 y z) (distrS x\u2083 y\u2081 z\u2081))\n  distrS {B r r\u2081} {L} {L} (One x) (Col y y\u2081) (Col z z\u2081) =\n    distrS (One x) y z ,\n    distrS (One x) y\u2081 z\u2081\n  distrS {B r r\u2081} {L} {B c c\u2081} (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    (distrS x y z) ,\n    (distrS x\u2081 y z) ,\n    (distrS x y\u2081 z\u2081) ,\n    (distrS x\u2081 y\u2081 z\u2081)\n  distrS {B r r\u2081} {B m m\u2081} {L} (Col x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (distrHelp x y z x\u2081 y\u2081 z\u2081 (distrS x y z) (distrS x\u2081 y\u2081 z\u2081)) ,\n    (distrHelp x y\u2082 z\u2082 x\u2081 y\u2083 z\u2083 (distrS x y\u2082 z\u2082) (distrS x\u2081 y\u2083 z\u2083))\n  distrS {B r r\u2081} {B m m\u2081} {B c c\u2081} (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distrHelp x y z x\u2082 y\u2081 z\u2081 (distrS x y z) (distrS x\u2082 y\u2081 z\u2081) ,\n    distrHelp x\u2081 y z x\u2083 y\u2081 z\u2081 (distrS x\u2081 y z) (distrS x\u2083 y\u2081 z\u2081) ,\n    distrHelp x y\u2082 z\u2082 x\u2082 y\u2083 z\u2083 (distrS x y\u2082 z\u2082) (distrS x\u2082 y\u2083 z\u2083) ,\n    distrHelp x\u2081 y\u2082 z\u2082 x\u2083 y\u2083 z\u2083 (distrS x\u2081 y\u2082 z\u2082) (distrS x\u2083 y\u2083 z\u2083)\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = _\u2243S_ {shape} {shape}\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = <\u2219S> shape shape shape\n      ; idem = idemS shape shape\n      ; distl = distlS {shape} {shape}\n      ; distr = distrS {shape} {shape}\n      }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7573d7f994c5821b13d69bce4942f9b5bb61dfc6","subject":"fix elgamal","message":"fix elgamal\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Unit\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; Fin\u25b9\u2115) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_; _==_)\nopen import Data.Bit\nopen import Data.Bits hiding (_==_)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\nimport Game.DDH\nimport Game.IND-CPA\nimport Cipher.ElGamal.Generic\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to Explorable)\nopen import Search.Searchable.Product\nopen import Search.Searchable.Fin\nopen import Relation.Binary.NP\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : Explorable X \u2192 \u2200 u \u2192 Explorable (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7-\u03bc \u03bcU \u03bcX u\u2081\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = count (\u03bcU \u03bc\u2124q u) (run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sym (\u229e-stable x (Bit\u25b9\u2115 \u2218 Adv))\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc]; sucmod-inj to [suc]-inj)\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q-1 : \u2115) ([0]' [1]' : Fin (suc q-1)) where\n  -- open Sum\n  q : \u2115\n  q = suc q-1\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  \u03bc\u2124q : Explorable \u2124q\n  \u03bc\u2124q = \u03bcFinSuc q-1\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q = sum \u03bc\u2124q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  [suc]-stable : SumStableUnder (sum \u03bc\u2124q) [suc]\n  [suc]-stable = \u03bcFinSUI [suc] [suc]-inj\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = [suc]-inj (\u2115\u229e-inj n eq)\n\n  \u2115\u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u2115\u229e_ m)\n  \u2115\u229e-stable m = \u03bcFinSUI (_\u2115\u229e_ m) (\u2115\u229e-inj m)\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin\u25b9\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin\u25b9\u2115 m)\n\n  \u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ m)\n  \u229e-stable m = \u03bcFinSUI (_\u229e_ m) (\u229e-inj m)\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin\u25b9\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin\u25b9\u2115 n)\n\nmodule G-implem (p-1 q-1 : \u2115) (g' 0[p] 1[p] : Fin (suc p-1)) (0[q] 1[q] : Fin (suc q-1)) where\n  open \u2124q-implem q-1 0[q] 1[q] public\n  open \u2124q-implem p-1 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin\u25b9\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule El-Gamal-Generic\n  (\u2124q       : \u2605)\n  (_\u22a0_      : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G        : \u2605)\n  (g        : G)\n  (_^_      : G \u2192 \u2124q \u2192 G)\n  (Message  : \u2605)\n  (_\u2219_      : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_      : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219      : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q      : Explorable \u2124q)\n  (R\u2090       : \u2605)\n  (\u03bcR\u2090      : Explorable R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    R\u2093 : \u2605\n    R\u2093 = \u2124q\n\n    open Cipher.ElGamal.Generic Message \u2124q G g _^_ _\u2219_ _\/_\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\n    open IND-CPA using (R)\n\n    UnusedGame : (i : Bit) \u2192 IND-CPA.Adv \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    UnusedGame i (m , d) (b , r\u2090 , x , y , z) = b == d r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    module DDH = Game.DDH \u2124q G g _^_ R\u2090\n\n    OTP\u2141 : (R\u2090 \u2192 G \u2192 Message) \u2192 (R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit) \u2192 R \u2192 Bit\n    OTP\u2141 M d (r , x , y , z) = d r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : Bit \u2192 IND-CPA.Adv \u2192 DDH.Adv\n    TrA b (m , d) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = d r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : IND-CPA.Adv \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : IND-CPA.Adv \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , d) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    module Unused where\n        like-\u2141 : Bit \u2192 IND-CPA.Game\n        like-\u2141 b (m , d) (r\u2090 , x , y , _z) =\n          d r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n          where g\u02e3 = g^ x\n                g\u02b8 = g^ y\n\n        IND-CPA-\u2141\u2261like-\u2141 : IND-CPA.\u2141 \u2261 like-\u2141\n        IND-CPA-\u2141\u2261like-\u2141 = refl\n\n    -- R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : Explorable R\n    \u03bcR = \u03bcR\u2090 \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q\n\n    #\u1d3f_ : Count R\n    #\u1d3f_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    Re = (f g : R \u2192 Bit) \u2192 \u2605\n    record Tra (_\u2248\u2080_ _\u2248\u2081_ : Re) (f g : R \u2192 Bit) : \u2605 where\n      field\n        h : R \u2192 Bit\n        f\u2248\u2080h : f \u2248\u2080 h\n        h\u2248\u2081g : h \u2248\u2081 g\n\n    record _\u2248\u1d3f_ (f g : R \u2192 Bit) : \u2605 where\n      constructor mk\n      field\n        un-\u2248\u1d3f : #\u1d3f f \u2261 #\u1d3f g\n    open _\u2248\u1d3f_ public\n\n    \u2248\u1d3f-trans : Transitive _\u2248\u1d3f_\n    \u2248\u1d3f-trans (mk p) (mk q) = mk (\u2261.trans p q)\n\n    module \u2248\u1d3f-Reasoning where\n      open Trans-Reasoning _\u2248\u1d3f_ \u2248\u1d3f-trans public using () renaming (_\u2248\u27e8_\u27e9_ to _\u2248\u1d3f\u27e8_\u27e9_)\n      infix  2 _\u220e\n\n      _\u220e : \u2200 x \u2192 x \u2248\u1d3f x\n      _ \u220e = mk refl\n\n    module Proof\n        (ddh-hyp : \u2200 A \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A)\n        (otp-lem : \u2200 A m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : IND-CPA.Adv) (b : Bit)\n      where\n        OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n        OTP\u2141-lem d M\u2080 M\u2081 = mk (\n                           sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y))))\n          where\n          pf : \u2200 r x y \u2192 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2080 d (r , x , y , z))\n                       \u2261 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2081 d (r , x , y , z))\n          pf r x y rewrite otp-lem (d r (g^ x) (g^ y)) (M\u2080 r (g^ x)) (M\u2081 r (g^ x)) = refl\n\n        -- moving this definition above OTP\u2141-lem breaks type-checking: ???\n        \u00acb : Bit\n        \u00acb = not b\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA \u00acb A\n\n        pf0,5 : IND-CPA.\u2141 b A \u2257 DDH.\u2141\u2080 A\u1d47\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf2,5 : DDH.\u2141\u2081 A\u1d47 \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf4,5 : IND-CPA.\u2141 \u00acb A \u2257 DDH.\u2141\u2080 A\u00ac\u1d47\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        open \u2248\u1d3f-Reasoning\n\n        final : IND-CPA.\u2141 b A \u2248\u1d3f IND-CPA.\u2141 \u00acb A\n        final = IND-CPA.\u2141 b A\n              \u2248\u1d3f\u27e8 mk (sum-ext \u03bcR (cong Bit\u25b9\u2115 \u2218 pf0,5)) \u27e9\n                DDH.\u2141\u2080 A\u1d47\n              \u2248\u1d3f\u27e8 ddh-hyp A\u1d47 \u27e9\n                DDH.\u2141\u2081 A\u1d47\n              \u2248\u1d3f\u27e8 OTP\u2141-lem (projD A) (projM A b) (projM A \u00acb) \u27e9\n                DDH.\u2141\u2081 A\u00ac\u1d47\n              \u2248\u1d3f\u27e8 mk (\u2261.sym (un-\u2248\u1d3f (ddh-hyp A\u00ac\u1d47))) \u27e9\n                DDH.\u2141\u2080 A\u00ac\u1d47\n              \u2248\u1d3f\u27e8 mk (\u2261.sym (sum-ext \u03bcR (cong Bit\u25b9\u2115 \u2218 pf4,5))) \u27e9\n                IND-CPA.\u2141 \u00acb A\n              \u220e\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ m \u03b4 = \u210b \u03b4 \u2295 m\n{-\n\n    \/-\u2219 : \u2200 x y \u2192 \u210b\u27e8 x \u27e9\u2295 y \/ x \u2261 y\n    \/-\u2219 x y = {!!}\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n\n  _\/_ : G \u2192 G \u2192 G\n  x \/ y = x \u2219 (y \u207b\u00b9)\n\n  postulate\n    \/-\u2022 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n    \u22a0-comm : \u2200 x y \u2192 x \u22a0 y \u2261 y \u22a0 x\n\n  comm-^ : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n  comm-^ \u03b1 x y = (\u03b1 ^ x)^ y\n               \u2261\u27e8 sym (dist-^-\u22a0 \u03b1 x y) \u27e9\n                  \u03b1 ^ (x \u22a0 y)\n               \u2261\u27e8 cong (_^_ \u03b1) (\u22a0-comm x y) \u27e9\n                  \u03b1 ^ (y \u22a0 x)\n               \u2261\u27e8 dist-^-\u22a0 \u03b1 y x \u27e9\n                  (\u03b1 ^ y)^ x\n               \u220e\n    where open \u2261-Reasoning\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u03bcR\u2090 : Explorable (El `R\u2090)\n  \u03bcR\u2090 = \u03bcU \u03bc\u2124q `R\u2090\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum \u03bcR\u2090\n  sumR\u2090-ext = sum-ext \u03bcR\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\/_ \/-\u2022 comm-^ dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090\n  open EB hiding (g^_)\n\n  otp-base-lem : \u2200 (A : G \u2192 Bit) m \u2192 (A \u2218 g^_) \u2248q (A \u2218 g^_ \u2218 _\u229e_ m)\n  otp-base-lem A m = \u229e-stable m (Bit\u25b9\u2115 \u2218 A \u2218 g^_)\n\n  postulate\n    ddh-hyp : (A : DDH.Adv) \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 IND-CPA.\u2141 A 0b \u2248\u1d3f IND-CPA.\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Unit\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_; to\u2115 to Fin\u25b9\u2115)\nopen import Data.Nat.NP hiding (_^_; _==_)\nopen import Data.Bit\nopen import Data.Bits hiding (_==_)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\nimport Game.DDH\nimport Game.IND-CPA\nimport Cipher.ElGamal.Generic\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to Explorable)\nopen import Search.Searchable.Product\nopen import Search.Searchable.Fin\nopen import Relation.Binary.NP\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : Explorable X \u2192 \u2200 u \u2192 Explorable (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7-\u03bc \u03bcU \u03bcX u\u2081\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = count (\u03bcU \u03bc\u2124q u) (run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sym (\u229e-stable x (Bit\u25b9\u2115 \u2218 Adv))\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc]; sucmod-inj to [suc]-inj)\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q-1 : \u2115) ([0]' [1]' : Fin (suc q-1)) where\n  -- open Sum\n  q : \u2115\n  q = suc q-1\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  \u03bc\u2124q : Explorable \u2124q\n  \u03bc\u2124q = \u03bcFinSuc q-1\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q = sum \u03bc\u2124q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  [suc]-stable : SumStableUnder (sum \u03bc\u2124q) [suc]\n  [suc]-stable = \u03bcFinSUI [suc] [suc]-inj\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = [suc]-inj (\u2115\u229e-inj n eq)\n\n  \u2115\u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u2115\u229e_ m)\n  \u2115\u229e-stable m = \u03bcFinSUI (_\u2115\u229e_ m) (\u2115\u229e-inj m)\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin\u25b9\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin\u25b9\u2115 m)\n\n  \u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ m)\n  \u229e-stable m = \u03bcFinSUI (_\u229e_ m) (\u229e-inj m)\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin\u25b9\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin\u25b9\u2115 n)\n\nmodule G-implem (p-1 q-1 : \u2115) (g' 0[p] 1[p] : Fin (suc p-1)) (0[q] 1[q] : Fin (suc q-1)) where\n  open \u2124q-implem q-1 0[q] 1[q] public\n  open \u2124q-implem p-1 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin\u25b9\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule El-Gamal-Generic\n  (\u2124q       : \u2605)\n  (_\u22a0_      : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G        : \u2605)\n  (g        : G)\n  (_^_      : G \u2192 \u2124q \u2192 G)\n  (Message  : \u2605)\n  (_\u2219_      : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_      : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219      : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q      : Explorable \u2124q)\n  (R\u2090       : \u2605)\n  (\u03bcR\u2090      : Explorable R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    R\u2093 : \u2605\n    R\u2093 = \u2124q\n\n    open Cipher.ElGamal.Generic Message \u2124q G g _^_ _\u2219_ _\/_\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\n    open IND-CPA using (R)\n\n    UnusedGame : (i : Bit) \u2192 IND-CPA.Adv \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    UnusedGame i (m , d) (b , r\u2090 , x , y , z) = b == d r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    module DDH = Game.DDH \u2124q G g _^_ R\u2090\n\n    OTP\u2141 : (R\u2090 \u2192 G \u2192 Message) \u2192 (R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit) \u2192 R \u2192 Bit\n    OTP\u2141 M d (r , x , y , z) = d r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : Bit \u2192 IND-CPA.Adv \u2192 DDH.Adv\n    TrA b (m , d) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = d r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : IND-CPA.Adv \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : IND-CPA.Adv \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , d) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    module Unused where\n        like-\u2141 : Bit \u2192 IND-CPA.Game\n        like-\u2141 b (m , d) (r\u2090 , x , y , _z) =\n          d r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n          where g\u02e3 = g^ x\n                g\u02b8 = g^ y\n\n        IND-CPA-\u2141\u2261like-\u2141 : IND-CPA.\u2141 \u2261 like-\u2141\n        IND-CPA-\u2141\u2261like-\u2141 = refl\n\n    -- R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : Explorable R\n    \u03bcR = \u03bcR\u2090 \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q\n\n    #\u1d3f_ : Count R\n    #\u1d3f_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    Re = (f g : R \u2192 Bit) \u2192 \u2605\n    record Tra (_\u2248\u2080_ _\u2248\u2081_ : Re) (f g : R \u2192 Bit) : \u2605 where\n      field\n        h : R \u2192 Bit\n        f\u2248\u2080h : f \u2248\u2080 h\n        h\u2248\u2081g : h \u2248\u2081 g\n\n    record _\u2248\u1d3f_ (f g : R \u2192 Bit) : \u2605 where\n      constructor mk\n      field\n        un-\u2248\u1d3f : #\u1d3f f \u2261 #\u1d3f g\n    open _\u2248\u1d3f_ public\n\n    \u2248\u1d3f-trans : Transitive _\u2248\u1d3f_\n    \u2248\u1d3f-trans (mk p) (mk q) = mk (\u2261.trans p q)\n\n    module \u2248\u1d3f-Reasoning where\n      open Trans-Reasoning _\u2248\u1d3f_ \u2248\u1d3f-trans public using () renaming (_\u2248\u27e8_\u27e9_ to _\u2248\u1d3f\u27e8_\u27e9_)\n      infix  2 _\u220e\n\n      _\u220e : \u2200 x \u2192 x \u2248\u1d3f x\n      _ \u220e = mk refl\n\n    module Proof\n        (ddh-hyp : \u2200 A \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A)\n        (otp-lem : \u2200 A m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : IND-CPA.Adv) (b : Bit)\n      where\n        OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n        OTP\u2141-lem d M\u2080 M\u2081 = mk (\n                           sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y))))\n          where\n          pf : \u2200 r x y \u2192 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2080 d (r , x , y , z))\n                       \u2261 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2081 d (r , x , y , z))\n          pf r x y rewrite otp-lem (d r (g^ x) (g^ y)) (M\u2080 r (g^ x)) (M\u2081 r (g^ x)) = refl\n\n        -- moving this definition above OTP\u2141-lem breaks type-checking: ???\n        \u00acb : Bit\n        \u00acb = not b\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA \u00acb A\n\n        pf0,5 : IND-CPA.\u2141 b A \u2257 DDH.\u2141\u2080 A\u1d47\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf2,5 : DDH.\u2141\u2081 A\u1d47 \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf4,5 : IND-CPA.\u2141 \u00acb A \u2257 DDH.\u2141\u2080 A\u00ac\u1d47\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        open \u2248\u1d3f-Reasoning\n\n        final : IND-CPA.\u2141 b A \u2248\u1d3f IND-CPA.\u2141 \u00acb A\n        final = IND-CPA.\u2141 b A\n              \u2248\u1d3f\u27e8 mk (sum-ext \u03bcR (cong Bit\u25b9\u2115 \u2218 pf0,5)) \u27e9\n                DDH.\u2141\u2080 A\u1d47\n              \u2248\u1d3f\u27e8 ddh-hyp A\u1d47 \u27e9\n                DDH.\u2141\u2081 A\u1d47\n              \u2248\u1d3f\u27e8 OTP\u2141-lem (projD A) (projM A b) (projM A \u00acb) \u27e9\n                DDH.\u2141\u2081 A\u00ac\u1d47\n              \u2248\u1d3f\u27e8 mk (\u2261.sym (un-\u2248\u1d3f (ddh-hyp A\u00ac\u1d47))) \u27e9\n                DDH.\u2141\u2080 A\u00ac\u1d47\n              \u2248\u1d3f\u27e8 mk (\u2261.sym (sum-ext \u03bcR (cong Bit\u25b9\u2115 \u2218 pf4,5))) \u27e9\n                IND-CPA.\u2141 \u00acb A\n              \u220e\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ m \u03b4 = \u210b \u03b4 \u2295 m\n{-\n\n    \/-\u2219 : \u2200 x y \u2192 \u210b\u27e8 x \u27e9\u2295 y \/ x \u2261 y\n    \/-\u2219 x y = {!!}\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n\n  _\/_ : G \u2192 G \u2192 G\n  x \/ y = x \u2219 (y \u207b\u00b9)\n\n  postulate\n    \/-\u2022 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n    \u22a0-comm : \u2200 x y \u2192 x \u22a0 y \u2261 y \u22a0 x\n\n  comm-^ : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n  comm-^ \u03b1 x y = (\u03b1 ^ x)^ y\n               \u2261\u27e8 sym (dist-^-\u22a0 \u03b1 x y) \u27e9\n                  \u03b1 ^ (x \u22a0 y)\n               \u2261\u27e8 cong (_^_ \u03b1) (\u22a0-comm x y) \u27e9\n                  \u03b1 ^ (y \u22a0 x)\n               \u2261\u27e8 dist-^-\u22a0 \u03b1 y x \u27e9\n                  (\u03b1 ^ y)^ x\n               \u220e\n    where open \u2261-Reasoning\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u03bcR\u2090 : Explorable (El `R\u2090)\n  \u03bcR\u2090 = \u03bcU \u03bc\u2124q `R\u2090\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum \u03bcR\u2090\n  sumR\u2090-ext = sum-ext \u03bcR\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\/_ \/-\u2022 comm-^ dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090\n  open EB hiding (g^_)\n\n  otp-base-lem : \u2200 (A : G \u2192 Bit) m \u2192 (A \u2218 g^_) \u2248q (A \u2218 g^_ \u2218 _\u229e_ m)\n  otp-base-lem A m = \u229e-stable m (Bit\u25b9\u2115 \u2218 A \u2218 g^_)\n\n  postulate\n    ddh-hyp : (A : DDH.Adv) \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 IND-CPA.\u2141 A 0b \u2248\u1d3f IND-CPA.\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d4894223fb978a68bdda86d285ca1023e8d45235","subject":"Removed unnecessary import.","message":"Removed unnecessary import.\n\nIgnore-this: e6495fab2648ecac4ecc85685b6b158a\n\ndarcs-hash:20100604032100-3bd4e-e89550df44e4c9bb127867dfcd4e3a8f4e6a9bc8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Division\/IsDIV-ER.agda","new_file":"Examples\/Division\/IsDIV-ER.agda","new_contents":"------------------------------------------------------------------------------\n-- The division satisfies the specification\n------------------------------------------------------------------------------\n\n-- This module formalize the division algorithm following [1].\n-- [1] Peter Dybjer. Program verification in a logical theory of\n-- constructions. In Jean-Pierre Jouannaud, editor. Functional\n-- Programming Languages and Computer Architecture, volume 201 of\n-- LNCS, 1985, pages 334-349. Appears in revised form as Programming\n-- Methodology Group Report 26, June 1986.\n\nmodule Examples.Division.IsDIV-ER where\n\nopen import LTC.Minimal\n\nopen import Examples.Division\nopen import Examples.Division.EquationsER\nopen import Examples.Division.IsCorrectER\nopen import Examples.Division.IsN-ER\nopen import Examples.Division.Specification\n\nopen import LTC.Data.N\nopen import LTC.Data.N.Induction.WellFoundedPCF\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF-ER\nopen import LTC.Relation.InequalitiesPCF\nopen import LTC.Relation.Inequalities.PropertiesPCF-ER\n\n------------------------------------------------------------------------------\n\n-- The division result satifies the specification DIV\n-- when the dividend is less than the divisor\ndiv-x<y-DIV : {i j : D} \u2192 N i \u2192 N j -> LT i j \u2192 DIV i j (div i j)\ndiv-x<y-DIV Ni Nj i<j = div-x<y-N i<j , div-x<y-correct Ni Nj i<j\n\n-- The division result satisfies the specification DIV when the\n-- dividend is greater or equal than the divisor.\ndiv-x\u2265y-DIV : {i j : D} \u2192 N i -> N j \u2192\n              ({i' : D} \u2192 N i' \u2192 LT i' i \u2192 DIV i' j (div i' j)) \u2192\n              GT j zero \u2192\n              GE i j \u2192\n              DIV i j (div i j)\ndiv-x\u2265y-DIV {i} {j} Ni Nj accH j>0 i\u2265j =\n  (div-x\u2265y-N ih i\u2265j) , div-x\u2265y-correct Ni Nj ih i\u2265j\n\n    where\n    -- The inductive hypothesis on 'i - j'.\n    ih : DIV (i - j) j (div (i - j) j)\n    ih = accH {i - j}\n              (minus-N Ni Nj )\n              (x\u2265y\u2192y>0\u2192x-y<x Ni Nj i\u2265j j>0)\n\n------------------------------------------------------------------------------\n-- The division satisfies the specification\n\n-- We do the well-founded induction on 'i' and we keep 'j' fixed.\ndiv-DIV : {i j : D} \u2192 N i \u2192 N j \u2192 GT j zero \u2192 DIV i j (div i j)\ndiv-DIV {j = j} Ni Nj j>0 = wfIndN-LT P iStep Ni\n\n   where\n     P : D \u2192 Set\n     P d = DIV d j (div d j)\n\n     -- The inductive step doesn't use the variable 'i'\n     -- (nor 'Ni'). To make this\n     -- clear we write down the inductive step using the variables\n     -- 'm' and 'n'.\n     iStep : {n : D} \u2192 N n \u2192\n             (accH : {m : D} \u2192 N m \u2192 LT m n \u2192 P m) \u2192\n             P n\n     iStep {n} Nn accH =\n       [ div-x<y-DIV Nn Nj , div-x\u2265y-DIV Nn Nj accH j>0 ] (x<y\u2228x\u2265y Nn Nj)\n","old_contents":"------------------------------------------------------------------------------\n-- The division satisfies the specification\n------------------------------------------------------------------------------\n\n-- This module formalize the division algorithm following [1].\n-- [1] Peter Dybjer. Program verification in a logical theory of\n-- constructions. In Jean-Pierre Jouannaud, editor. Functional\n-- Programming Languages and Computer Architecture, volume 201 of\n-- LNCS, 1985, pages 334-349. Appears in revised form as Programming\n-- Methodology Group Report 26, June 1986.\n\nmodule Examples.Division.IsDIV-ER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import Examples.Division.EquationsER\nopen import Examples.Division.IsCorrectER\nopen import Examples.Division.IsN-ER\nopen import Examples.Division.Specification\n\nopen import LTC.Data.N\nopen import LTC.Data.N.Induction.WellFoundedPCF\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF-ER\nopen import LTC.Relation.InequalitiesPCF\nopen import LTC.Relation.Inequalities.PropertiesPCF-ER\n\n------------------------------------------------------------------------------\n\n-- The division result satifies the specification DIV\n-- when the dividend is less than the divisor\ndiv-x<y-DIV : {i j : D} \u2192 N i \u2192 N j -> LT i j \u2192 DIV i j (div i j)\ndiv-x<y-DIV Ni Nj i<j = div-x<y-N i<j , div-x<y-correct Ni Nj i<j\n\n-- The division result satisfies the specification DIV when the\n-- dividend is greater or equal than the divisor.\ndiv-x\u2265y-DIV : {i j : D} \u2192 N i -> N j \u2192\n              ({i' : D} \u2192 N i' \u2192 LT i' i \u2192 DIV i' j (div i' j)) \u2192\n              GT j zero \u2192\n              GE i j \u2192\n              DIV i j (div i j)\ndiv-x\u2265y-DIV {i} {j} Ni Nj accH j>0 i\u2265j =\n  (div-x\u2265y-N ih i\u2265j) , div-x\u2265y-correct Ni Nj ih i\u2265j\n\n    where\n    -- The inductive hypothesis on 'i - j'.\n    ih : DIV (i - j) j (div (i - j) j)\n    ih = accH {i - j}\n              (minus-N Ni Nj )\n              (x\u2265y\u2192y>0\u2192x-y<x Ni Nj i\u2265j j>0)\n\n------------------------------------------------------------------------------\n-- The division satisfies the specification\n\n-- We do the well-founded induction on 'i' and we keep 'j' fixed.\ndiv-DIV : {i j : D} \u2192 N i \u2192 N j \u2192 GT j zero \u2192 DIV i j (div i j)\ndiv-DIV {j = j} Ni Nj j>0 = wfIndN-LT P iStep Ni\n\n   where\n     P : D \u2192 Set\n     P d = DIV d j (div d j)\n\n     -- The inductive step doesn't use the variable 'i'\n     -- (nor 'Ni'). To make this\n     -- clear we write down the inductive step using the variables\n     -- 'm' and 'n'.\n     iStep : {n : D} \u2192 N n \u2192\n             (accH : {m : D} \u2192 N m \u2192 LT m n \u2192 P m) \u2192\n             P n\n     iStep {n} Nn accH =\n       [ div-x<y-DIV Nn Nj , div-x\u2265y-DIV Nn Nj accH j>0 ] (x<y\u2228x\u2265y Nn Nj)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"625631634e57a74cffb0442c92207c7b8667b622","subject":"Only indentation.","message":"Only indentation.\n\nIgnore-this: c2263467480f89c699d375fd42dad3e5\n\ndarcs-hash:20100429202055-3bd4e-d360ccd24a43231bdc90120f8657214fc1c8535d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\n  using ( +-leftIdentity ; *-leftZero ; *-comm ; minus-0x )\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN          = subst (\u03bb t \u2192 N t) (sym cP\u2081) zN\npred-N (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (cP\u2082 n)) Nn\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN                   = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN (sN {n} Nn)          = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst ((\u03bb t \u2192 N t)) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n------------------------------------------------------------------------------\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\n  using ( +-leftIdentity ; *-leftZero ; *-comm ; minus-0x )\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN          = subst (\u03bb t \u2192 N t) (sym cP\u2081) zN\npred-N (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (cP\u2082 n)) Nn\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN                   = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN (sN {n} Nn)          = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst ((\u03bb t \u2192 N t)) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n------------------------------------------------------------------------------\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                 succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                        (minus-x0 m)\n                        refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                              t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bdfd1c903bebb00f241b98797e6544d8970bea49","subject":"\u2124q: Use warn-check since check is monadic now...","message":"\u2124q: Use warn-check since check is monadic now...\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/JS\/BigI\/FiniteField.agda","new_file":"Crypto\/JS\/BigI\/FiniteField.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq using (_==_; Eq?; \u2261\u21d2==; ==\u21d2\u2261)\nopen import Data.Two.Base using (\ud835\udfda; \u2713)\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; refl; ap)\nopen import FFI.JS using (JS[_]; _++_; return; _>>_)\nopen import FFI.JS.Check\n  using    (check!)\n--renaming (check      to check?)\n  renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nopen import Algebra.Field\n\n-- TODO carry on a primality proof of q\nmodule Crypto.JS.BigI.FiniteField (q : BigI) where\n\n-- The constructor mk is not exported.\nprivate\n  module Internals where\n    data \u2124[_] : Set where\n      mk : BigI \u2192 \u2124[_]\n\n    mk-inj : \u2200 {x y : BigI} \u2192 \u2124[_].mk x \u2261 mk y \u2192 x \u2261 y\n    mk-inj refl = refl\n\nopen Internals public using (\u2124[_])\nopen Internals\n\n\u2124q : Set\n\u2124q = \u2124[_]\n\nprivate\n  mod-q : BigI \u2192 \u2124q\n  mod-q x = mk (mod x q)\n\n-- There are two ways to go from BigI to \u2124q: BigI\u25b9\u2124[ q ] and mod-q\n-- Use BigI\u25b9\u2124[ q ] for untrusted input data and mod-q for internal\n-- computation.\nBigI\u25b9\u2124[_] : BigI \u2192 JS[ \u2124q ]\nBigI\u25b9\u2124[_] x =\n-- Console.log \"BigI\u25b9\u2124[_]\"\n  check! \"below the modulus\" (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++\n                \" is less than x:\" ++ toString x) >>\n  check! \"positivity\" (x \u2265I 0I)\n         (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") >>\n  return (mk x)\n\ncheck-non-zero : \u2124q \u2192 BigI\ncheck-non-zero (mk x) = -- trace-call \"check-non-zero \"\n  check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n\u2124[_]\u25b9BigI : \u2124q \u2192 BigI\n\u2124[_]\u25b9BigI (mk x) = x\n\n0# 1# : \u2124q\n0# = mk 0I\n1# = mk 1I\n\n-- One could choose here to return a dummy value when 0 is given.\n-- Instead we throw an exception which in some circumstances could\n-- be bad if the runtime semantics is too eager.\n1\/_ : \u2124q \u2192 \u2124q\n1\/ x = mk (modInv (check-non-zero x) q)\n\n_^I_ : \u2124q \u2192 BigI \u2192 \u2124q\nmk x ^I y = mk (modPow x y q)\n\n\u2124q\u25b9BigI = \u2124[_]\u25b9BigI\nBigI\u25b9\u2124q = BigI\u25b9\u2124[_]\n\n_\u2297I_ : \u2124q \u2192 BigI \u2192 \u2124q\nx \u2297I y = mod-q (multiply (\u2124q\u25b9BigI x) y)\n\n_+_ _\u2212_ _*_ _\/_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n\nx + y = mod-q (add      (\u2124q\u25b9BigI x) (\u2124q\u25b9BigI y))\nx \u2212 y = mod-q (subtract (\u2124q\u25b9BigI x) (\u2124q\u25b9BigI y))\nx * y = x \u2297I \u2124q\u25b9BigI y\nx \/ y = x * 1\/ y\n\n*-def : _*_ \u2261 (\u03bb x y \u2192 x \u2297I \u2124q\u25b9BigI y)\n*-def = refl\n\n0\u2212_ : \u2124q \u2192 \u2124q\n0\u2212 x = mod-q (negate (\u2124q\u25b9BigI x))\n\nsum prod : List \u2124q \u2192 \u2124q\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\ninstance\n  \u2124[_]-Eq? : Eq? \u2124q\n  \u2124[_]-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124q \u2192 \u2124q \u2192 \ud835\udfda\n      mk x ==' mk y = x == y\n      \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n      \u2261\u21d2==' {mk x} {mk y} p = \u2261\u21d2== (mk-inj p)\n      ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n      ==\u21d2\u2261' {mk x} {mk y} p = ap mk (==\u21d2\u2261 p)\n\n\u2124q-Eq?  = \u2124[_]-Eq?\n\n+-mon-ops : Monoid-Ops \u2124q\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \u2124q\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \u2124q\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \u2124q\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \u2124q\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate fld-struct : Field-Struct fld-ops\n\nfld : Field \u2124q\nfld = fld-ops , fld-struct\n\nmodule fld = Field fld\n\n-- open fld using (+-grp) public\n\n\u2124[_]+-grp : Group \u2124q\n\u2124[_]+-grp = fld.+-grp\n\n\u2124q+-grp : Group \u2124q\n\u2124q+-grp = fld.+-grp\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq using (_==_; Eq?; \u2261\u21d2==; ==\u21d2\u2261)\nopen import Data.Two.Base using (\ud835\udfda; \u2713)\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; refl; ap)\nopen import FFI.JS using (JS[_]; _++_; return; _>>_)\nopen import FFI.JS.Check\n  using    (check!)\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nopen import Algebra.Field\n\n-- TODO carry on a primality proof of q\nmodule Crypto.JS.BigI.FiniteField (q : BigI) where\n\n-- The constructor mk is not exported.\nprivate\n  module Internals where\n    data \u2124[_] : Set where\n      mk : BigI \u2192 \u2124[_]\n\n    mk-inj : \u2200 {x y : BigI} \u2192 \u2124[_].mk x \u2261 mk y \u2192 x \u2261 y\n    mk-inj refl = refl\n\nopen Internals public using (\u2124[_])\nopen Internals\n\n\u2124q : Set\n\u2124q = \u2124[_]\n\nprivate\n  mod-q : BigI \u2192 \u2124q\n  mod-q x = mk (mod x q)\n\n-- There are two ways to go from BigI to \u2124q: BigI\u25b9\u2124[ q ] and mod-q\n-- Use BigI\u25b9\u2124[ q ] for untrusted input data and mod-q for internal\n-- computation.\nBigI\u25b9\u2124[_] : BigI \u2192 JS[ \u2124q ]\nBigI\u25b9\u2124[_] x =\n-- Console.log \"BigI\u25b9\u2124[_]\"\n  check! \"below the modulus\" (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++\n                \" is less than x:\" ++ toString x) >>\n  check! \"positivity\" (x \u2265I 0I)\n         (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") >>\n  return (mk x)\n\ncheck-non-zero : \u2124q \u2192 BigI\ncheck-non-zero (mk x) = -- trace-call \"check-non-zero \"\n  check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n\u2124[_]\u25b9BigI : \u2124q \u2192 BigI\n\u2124[_]\u25b9BigI (mk x) = x\n\n0# 1# : \u2124q\n0# = mk 0I\n1# = mk 1I\n\n-- One could choose here to return a dummy value when 0 is given.\n-- Instead we throw an exception which in some circumstances could\n-- be bad if the runtime semantics is too eager.\n1\/_ : \u2124q \u2192 \u2124q\n1\/ x = mk (modInv (check-non-zero x) q)\n\n_^I_ : \u2124q \u2192 BigI \u2192 \u2124q\nmk x ^I y = mk (modPow x y q)\n\n\u2124q\u25b9BigI = \u2124[_]\u25b9BigI\nBigI\u25b9\u2124q = BigI\u25b9\u2124[_]\n\n_\u2297I_ : \u2124q \u2192 BigI \u2192 \u2124q\nx \u2297I y = mod-q (multiply (\u2124q\u25b9BigI x) y)\n\n_+_ _\u2212_ _*_ _\/_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n\nx + y = mod-q (add      (\u2124q\u25b9BigI x) (\u2124q\u25b9BigI y))\nx \u2212 y = mod-q (subtract (\u2124q\u25b9BigI x) (\u2124q\u25b9BigI y))\nx * y = x \u2297I \u2124q\u25b9BigI y\nx \/ y = x * 1\/ y\n\n*-def : _*_ \u2261 (\u03bb x y \u2192 x \u2297I \u2124q\u25b9BigI y)\n*-def = refl\n\n0\u2212_ : \u2124q \u2192 \u2124q\n0\u2212 x = mod-q (negate (\u2124q\u25b9BigI x))\n\nsum prod : List \u2124q \u2192 \u2124q\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\ninstance\n  \u2124[_]-Eq? : Eq? \u2124q\n  \u2124[_]-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124q \u2192 \u2124q \u2192 \ud835\udfda\n      mk x ==' mk y = x == y\n      \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n      \u2261\u21d2==' {mk x} {mk y} p = \u2261\u21d2== (mk-inj p)\n      ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n      ==\u21d2\u2261' {mk x} {mk y} p = ap mk (==\u21d2\u2261 p)\n\n\u2124q-Eq?  = \u2124[_]-Eq?\n\n+-mon-ops : Monoid-Ops \u2124q\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \u2124q\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \u2124q\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \u2124q\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \u2124q\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate fld-struct : Field-Struct fld-ops\n\nfld : Field \u2124q\nfld = fld-ops , fld-struct\n\nmodule fld = Field fld\n\n-- open fld using (+-grp) public\n\n\u2124[_]+-grp : Group \u2124q\n\u2124[_]+-grp = fld.+-grp\n\n\u2124q+-grp : Group \u2124q\n\u2124q+-grp = fld.+-grp\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7710ed80e465665babad11564d7f05f8642d6a90","subject":"elgamal","message":"elgamal\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Data.Unit\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; Fin\u25b9\u2115) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_; _==_)\nopen import Data.Nat.Distance\nopen import Data.Bit\nopen import Data.Zero\nopen import Data.Two\nopen import Relation.Binary.NP\nopen import Data.Bits hiding (_==_)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261 hiding (_\u2219_)\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\nopen import Explore.Sum\nopen import Explore.Product\nimport Explore.GroupHomomorphism as GH\n\nimport Game.DDH\nimport Game.IND-CPA\nimport Cipher.ElGamal.Generic\n\nmodule elgamal where\n\n{-\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : Explorable X \u2192 \u2200 u \u2192 Explorable (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7-\u03bc \u03bcU \u03bcX u\u2081\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = count (\u03bcU \u03bc\u2124q u) (run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (fst rs) (run\u21ba ax (snd rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sym (\u229e-stable x (Bit\u25b9\u2115 \u2218 Adv))\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n-}\n\nmodule El-Gamal-Generic\n  (\u2124q\u1d41 : U)\n  (G        : \u2605)\n  (g        : G)\n  (_^_      : G \u2192 El \u2124q\u1d41 \u2192 G)\n  (Message  : \u2605)\n  (_\u2219_      : G \u2192 Message \u2192 Message)\n  (_\/_      : Message \u2192 G \u2192 Message)\n  (R\u2090\u1d41       : U)\n  where\n    \u2124q = El \u2124q\u1d41\n    R\u2090  = El R\u2090\u1d41\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    R\u2093 : \u2605\n    R\u2093 = \u2124q\n\n    open Cipher.ElGamal.Generic Message Message \u2124q G g _^_ _\u2219_ _\/_\n\n    module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\n    open IND-CPA renaming (R to R')\n\n    -- R' = (R\u2090 \u00d7 R\u2096 \u00d7 R\u2091 \u00d7 R\u2093)\n    R'\u1d41 = R\u2090\u1d41 \u00d7\u1d41 \u2124q\u1d41 \u00d7\u1d41 \u2124q\u1d41 \u00d7\u1d41 \u2124q\u1d41\n    -- R' = El R'\u1d41\n    sumR' = sum R'\u1d41\n    R = \ud835\udfda \u00d7 R'\n    R\u1d41 = \ud835\udfda\u1d41 \u00d7\u1d41 R'\u1d41\n    sumR = sum R\u1d41\n    \n    sumExtR\u2090 = sum-ext R\u2090\u1d41\n    sumExt\u2124q = sum-ext \u2124q\u1d41\n    sumHomR' = sum-hom R'\u1d41\n    sumExtR' = sum-ext R'\u1d41\n    \n    IND-CPA-\u2141 : IND-CPA.Adversary \u2192 R \u2192 \ud835\udfda\n    IND-CPA-\u2141 = IND-CPA.game\n    \n    module DDH = Game.DDH \u2124q G g _^_ (\ud835\udfda \u00d7 R\u2090)\n\n    DDH-\u2141\u2080 DDH-\u2141\u2081 : DDH.Adv \u2192 R \u2192 \ud835\udfda\n    DDH-\u2141\u2080 A (b , r\u2090 , g\u02e3 , g\u02b8 , g\u1dbb) = DDH.\u2141\u2080 A ((b , r\u2090) , g\u02e3 , g\u02b8 , g\u1dbb)\n    DDH-\u2141\u2081 A (b , r\u2090 , g\u02e3 , g\u02b8 , g\u1dbb) = DDH.\u2141\u2081 A ((b , r\u2090) , g\u02e3 , g\u02b8 , g\u1dbb)\n  \n    transformAdv : IND-CPA.Adversary \u2192 DDH.Adv\n    transformAdv (m , d) (b , r\u2090) g\u02e3 g\u02b8 g\u1dbb = b == b\u2032\n                 where mb  = m r\u2090 g\u02e3 b\n                       c   = (g\u02b8 , g\u1dbb \u2219 mb)\n                       b\u2032  = d r\u2090 g\u02e3 c\n\n    #q_ : Count \u2124q\n    #q_ = count \u2124q\u1d41\n\n    _\u2248q_ : (f g : \u2124q \u2192 \u2115) \u2192 \u2605\n    f \u2248q g = sum \u2124q\u1d41 f \u2261 sum \u2124q\u1d41 g\n\n    OTP-LEM = \u2200 (O : Message \u2192 \u2115) m\u2080 m\u2081 \u2192\n                              (\u03bb x \u2192 O((g ^ x) \u2219 m\u2080)) \u2248q (\u03bb x \u2192 O((g ^ x) \u2219 m\u2081))\n\n    1\/2 : R \u2192 \ud835\udfda\n    1\/2 (b , _) = b\n\n    module _ {S} where \n      _\u2248\u1d3f_ : (f g : R \u2192 S) \u2192 \u2605\n      f \u2248\u1d3f g = \u2200 (X : S \u2192 \u2115) \u2192 sumR (X \u2218 f) \u2261 sumR (X \u2218 g) \n\n    Dist : (f g : R \u2192 \ud835\udfda) \u2192 \u2115\n    Dist f g = dist (count R\u1d41 f) (count R\u1d41 g)\n\n    dist-cong : \u2200  {f g h i} \u2192 f \u2248\u1d3f g \u2192 h \u2248\u1d3f i \u2192 Dist f h \u2261 Dist g i\n    dist-cong {f}{g}{h}{i} f\u2248g h\u2248i = ap\u2082 dist (f\u2248g \ud835\udfda\u25b9\u2115) (h\u2248i \ud835\udfda\u25b9\u2115)\n\n    OTP-\u2141 : IND-CPA.Adversary \u2192 R \u2192 \ud835\udfda\n    OTP-\u2141 (m , d) (b , r\u2090 , x , y , z) = b == d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n            g\u1dbb = g ^ z\n\n    module Proof (otp-lem : OTP-LEM)\n\n                 (A : IND-CPA.Adversary) where\n\n        module A = IND-CPA.Adversary A\n        A\u2032 = transformAdv A\n\n        goal4 : 1\/2 \u2248\u1d3f OTP-\u2141 A\n        goal4 X = sumR' (\u03bb _ \u2192 X 0b) + sumR' (\u03bb _ \u2192 X 1b)\n                \u2261\u27e8 sym (sumHomR' _  _) \u27e9\n                  sumR' (\u03bb _ \u2192 X 0b + X 1b)\n                \u2261\u27e8 sumExtR' (lemma \u2218 B 0b) \u27e9\n                  sumR' (Y 0b 0b +\u00b0 Y 1b 0b)\n                \u2261\u27e8 sumHomR' _ _ \u27e9\n                  sumR' (Y 0b 0b) + sumR' (Y 1b 0b)\n                \u2261\u27e8 cong (_+_ (sumR' (Y 0b 0b))) lemma2 \u27e9\n                  sumR' (Y 0b 0b) + sumR' (Y 1b 1b)\n                \u220e\n          where\n            open \u2261-Reasoning\n\n            B : \ud835\udfda \u2192 R' \u2192 \ud835\udfda\n            B b (r\u2090 , x , y , z) = A.b\u2032 r\u2090 (g ^ x) (g ^ y , (g ^ z) \u2219 A.m r\u2090 (g ^ x) b)\n\n            Y = \u03bb bb bbb r \u2192 X (bb == B bbb r)\n\n            lemma : \u2200 b \u2192 X 0b + X 1b \u2261  X (0b == b) + X (1b == b)\n            lemma 1b = refl\n            lemma 0b = \u2115\u00b0.+-comm (X 0b) _\n\n            lemma2 : sumR' (Y 1b 0b) \u2261 sumR' (Y 1b 1b)\n            lemma2 = sumExtR\u2090 \u03bb r\u2090 \u2192\n                     sumExt\u2124q \u03bb x \u2192\n                     sumExt\u2124q \u03bb y \u2192\n                       otp-lem (\u03bb m \u2192 X (A.b\u2032 r\u2090 (g ^ x) (g ^ y , m))) (A.m r\u2090 (g ^ x) 0\u2082) (A.m r\u2090 (g ^ x) 1\u2082)\n\n                     {-\n                      otp-lem (\u03bb m \u2192 snd A r\u2090 (g ^ x) (g ^ y , m))\n                              (fst A r\u2090 (g ^ x) 1b)\n                              (fst A r\u2090 (g ^ x) 0b)\n                              (\u03bb c \u2192 X (1b == c))\n    -}\n\n        module absDist {DIST : \u2605}(Dist : (f g : R \u2192 \ud835\udfda) \u2192 DIST)\n          (dist-cong : \u2200 {f h i} \u2192 h \u2248\u1d3f i \u2192 Dist f h \u2261 Dist f i) where\n          goal : Dist (IND-CPA-\u2141 A) 1\/2 \u2261 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n          goal = Dist (IND-CPA-\u2141 A) 1\/2\n               \u2261\u27e8 refl \u27e9\n                 Dist (DDH-\u2141\u2080 A\u2032) 1\/2\n               \u2261\u27e8 dist-cong goal4 \u27e9\n                 Dist (DDH-\u2141\u2080 A\u2032) (OTP-\u2141 A)\n               \u2261\u27e8 refl \u27e9\n                 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n               \u220e\n            where open \u2261-Reasoning\n\n        open absDist Dist (\u03bb {f}{h}{i} \u2192 dist-cong {f}{f}{h}{i} (\u03bb _ \u2192 refl)) public\n\nmodule El-Gamal-Base\n    (\u2124q\u1d41 : U)\n    (\u2124q-grp : GH.Group (El \u2124q\u1d41))\n    (G : \u2605)\n    (G-grp : GH.Group G)\n    (g : G)\n    (_^_ : G \u2192 El \u2124q\u1d41 \u2192 G)\n    (^-gh : GH.GroupHomomorphism \u2124q-grp G-grp (_^_ g))\n    (dlog : (b y : G) \u2192 El \u2124q\u1d41)\n    (dlog-ok : (b y : G) \u2192 b ^ dlog b y \u2261 y)\n    (R\u2090\u1d41 : U)\n    (R\u2090 : El R\u2090\u1d41)\n    {{_ : FunExt}}\n    {{_ : UA}}\n    (open GH.Group \u2124q-grp renaming (_\u2219_ to _\u229e_))\n    (\u229e-is-equiv : \u2200 k \u2192 Is-equiv (flip _\u229e_ k))\n    where\n\n    open GH.Group G-grp using (_\u2219_) renaming (-_ to _\u207b\u00b9)\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n\n    open El-Gamal-Generic \u2124q\u1d41 G g _^_ G _\u2219_ _\/_ R\u2090\u1d41 public\n\n    otp-lem : \u2200 (O : G \u2192 \u2115) m\u2080 m\u2081 \u2192\n        (\u03bb x \u2192 O((g ^ x) \u2219 m\u2080)) \u2248q (\u03bb x \u2192 O((g ^ x) \u2219 m\u2081))\n    otp-lem = GH.thm \u2124q-grp G-grp (_^_ g) (explore \u2124q\u1d41)\n                     ^-gh (dlog g) (dlog-ok g)\n                     (explore-ext \u2124q\u1d41) 0 _+_\n                     (\u03bb k f \u2192 ! sumStableUnder \u2124q\u1d41 (_ , \u229e-is-equiv k) f)\n    open Proof otp-lem\n\n    thm : \u2200 A \u2192\n          let A\u2032 = transformAdv A in\n          Dist (IND-CPA-\u2141 A) 1\/2 \u2261 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n    thm = goal\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Data.Unit\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; Fin\u25b9\u2115) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_; _==_)\nopen import Data.Nat.Distance\nopen import Data.Bit\nopen import Data.Two\nopen import Relation.Binary.NP\nopen import Data.Bits hiding (_==_)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261 hiding (_\u2219_)\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Universe.Base\nopen import Explore.Sum -- renaming (\u03bcBit to \u03bc\ud835\udfda)\nopen import Explore.Product\nimport Explore.GroupHomomorphism as GH\n--open import Explore.Fin\n\nimport Game.DDH\nimport Game.IND-CPA\nimport Cipher.ElGamal.Generic\n\nmodule elgamal where\n\n{-\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : Explorable X \u2192 \u2200 u \u2192 Explorable (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7-\u03bc \u03bcU \u03bcX u\u2081\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = count (\u03bcU \u03bc\u2124q u) (run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (fst rs) (run\u21ba ax (snd rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sym (\u229e-stable x (Bit\u25b9\u2115 \u2218 Adv))\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n-}\n\nmodule El-Gamal-Generic\n  (\u2124q\u1d41 : U)\n  (G        : \u2605)\n  (g        : G)\n  (_^_      : G \u2192 El \u2124q\u1d41 \u2192 G)\n  (Message  : \u2605)\n  (_\u2219_      : G \u2192 Message \u2192 Message)\n  (_\/_      : Message \u2192 G \u2192 Message)\n  (R\u2090\u1d41       : U)\n  where\n    \u2124q = El \u2124q\u1d41\n    R\u2090  = El R\u2090\u1d41\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    R\u2093 : \u2605\n    R\u2093 = \u2124q\n\n    open Cipher.ElGamal.Generic Message Message \u2124q G g _^_ _\u2219_ _\/_\n\n    module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\n    open IND-CPA renaming (R to R')\n\n    -- R' = (R\u2090 \u00d7 R\u2096 \u00d7 R\u2091 \u00d7 R\u2093)\n    R'\u1d41 = R\u2090\u1d41 \u00d7\u1d41 \u2124q\u1d41 \u00d7\u1d41 \u2124q\u1d41 \u00d7\u1d41 \u2124q\u1d41\n    -- R' = El R'\u1d41\n    sumR' = sum R'\u1d41\n    R = \ud835\udfda \u00d7 R'\n    R\u1d41 = \ud835\udfda\u1d41 \u00d7\u1d41 R'\u1d41\n    sumR = sum R\u1d41\n    \n    sumExtR\u2090 = sum-ext R\u2090\u1d41\n    sumExt\u2124q = sum-ext \u2124q\u1d41\n    sumHomR' = sum-hom R'\u1d41\n    sumExtR' = sum-ext R'\u1d41\n    \n    IND-CPA-\u2141 : IND-CPA.Adversary \u2192 R \u2192 \ud835\udfda\n    IND-CPA-\u2141 = IND-CPA.game\n    \n    module DDH = Game.DDH \u2124q G g _^_ (\ud835\udfda \u00d7 R\u2090)\n\n    DDH-\u2141\u2080 DDH-\u2141\u2081 : DDH.Adv \u2192 R \u2192 \ud835\udfda\n    DDH-\u2141\u2080 A (b , r\u2090 , g\u02e3 , g\u02b8 , g\u1dbb) = DDH.\u2141\u2080 A ((b , r\u2090) , g\u02e3 , g\u02b8 , g\u1dbb)\n    DDH-\u2141\u2081 A (b , r\u2090 , g\u02e3 , g\u02b8 , g\u1dbb) = DDH.\u2141\u2081 A ((b , r\u2090) , g\u02e3 , g\u02b8 , g\u1dbb)\n  \n    transformAdv : IND-CPA.Adversary \u2192 DDH.Adv\n    transformAdv (m , d) (b , r\u2090) g\u02e3 g\u02b8 g\u1dbb = b == b\u2032\n                 where mb  = m r\u2090 g\u02e3 b\n                       c   = (g\u02b8 , g\u1dbb \u2219 mb)\n                       b\u2032  = d r\u2090 g\u02e3 c\n\n    #q_ : Count \u2124q\n    #q_ = count \u2124q\u1d41\n\n    _\u2248q_ : (f g : \u2124q \u2192 \u2115) \u2192 \u2605\n    f \u2248q g = sum \u2124q\u1d41 f \u2261 sum \u2124q\u1d41 g\n\n    OTP-LEM = \u2200 (O : Message \u2192 \u2115) m\u2080 m\u2081 \u2192\n                              (\u03bb x \u2192 O((g ^ x) \u2219 m\u2080)) \u2248q (\u03bb x \u2192 O((g ^ x) \u2219 m\u2081))\n\n    1\/2 : R \u2192 \ud835\udfda\n    1\/2 (b , _) = b\n\n    module _ {S} where \n      _\u2248\u1d3f_ : (f g : R \u2192 S) \u2192 \u2605\n      f \u2248\u1d3f g = \u2200 (X : S \u2192 \u2115) \u2192 sumR (X \u2218 f) \u2261 sumR (X \u2218 g) \n\n    Dist : (f g : R \u2192 \ud835\udfda) \u2192 \u2115\n    Dist f g = dist (count R\u1d41 f) (count R\u1d41 g)\n\n    dist-cong : \u2200  {f g h i} \u2192 f \u2248\u1d3f g \u2192 h \u2248\u1d3f i \u2192 Dist f h \u2261 Dist g i\n    dist-cong {f}{g}{h}{i} f\u2248g h\u2248i = ap\u2082 dist (f\u2248g \ud835\udfda\u25b9\u2115) (h\u2248i \ud835\udfda\u25b9\u2115)\n\n    OTP-\u2141 : IND-CPA.Adversary \u2192 R \u2192 \ud835\udfda\n    OTP-\u2141 (m , d) (b , r\u2090 , x , y , z) = b == d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n            g\u1dbb = g ^ z\n\n    module Proof (otp-lem : OTP-LEM)\n\n                 (A : IND-CPA.Adversary) where\n\n        module A = IND-CPA.Adversary A\n        A\u2032 = transformAdv A\n\n        goal4 : 1\/2 \u2248\u1d3f OTP-\u2141 A\n        goal4 X = sumR' (\u03bb _ \u2192 X 0b) + sumR' (\u03bb _ \u2192 X 1b)\n                \u2261\u27e8 sym (sumHomR' _  _) \u27e9\n                  sumR' (\u03bb _ \u2192 X 0b + X 1b)\n                \u2261\u27e8 sumExtR' (lemma \u2218 B 0b) \u27e9\n                  sumR' (Y 0b 0b +\u00b0 Y 1b 0b)\n                \u2261\u27e8 sumHomR' _ _ \u27e9\n                  sumR' (Y 0b 0b) + sumR' (Y 1b 0b)\n                \u2261\u27e8 cong (_+_ (sumR' (Y 0b 0b))) lemma2 \u27e9\n                  sumR' (Y 0b 0b) + sumR' (Y 1b 1b)\n                \u220e\n          where\n            open \u2261-Reasoning\n\n            B : \ud835\udfda \u2192 R' \u2192 \ud835\udfda\n            B b (r\u2090 , x , y , z) = A.b\u2032 r\u2090 (g ^ x) (g ^ y , (g ^ z) \u2219 A.m r\u2090 (g ^ x) b)\n\n            Y = \u03bb bb bbb r \u2192 X (bb == B bbb r)\n\n            lemma : \u2200 b \u2192 X 0b + X 1b \u2261  X (0b == b) + X (1b == b)\n            lemma 1b = refl\n            lemma 0b = \u2115\u00b0.+-comm (X 0b) _\n\n            lemma2 : sumR' (Y 1b 0b) \u2261 sumR' (Y 1b 1b)\n            lemma2 = sumExtR\u2090 \u03bb r\u2090 \u2192\n                     sumExt\u2124q \u03bb x \u2192\n                     sumExt\u2124q \u03bb y \u2192\n                       otp-lem (\u03bb m \u2192 X (A.b\u2032 r\u2090 (g ^ x) (g ^ y , m))) (A.m r\u2090 (g ^ x) 0\u2082) (A.m r\u2090 (g ^ x) 1\u2082)\n\n                     {-\n                      otp-lem (\u03bb m \u2192 snd A r\u2090 (g ^ x) (g ^ y , m))\n                              (fst A r\u2090 (g ^ x) 1b)\n                              (fst A r\u2090 (g ^ x) 0b)\n                              (\u03bb c \u2192 X (1b == c))\n    -}\n\n        module absDist {DIST : \u2605}(Dist : (f g : R \u2192 \ud835\udfda) \u2192 DIST)\n          (dist-cong : \u2200 {f h i} \u2192 h \u2248\u1d3f i \u2192 Dist f h \u2261 Dist f i) where\n          goal : Dist (IND-CPA-\u2141 A) 1\/2 \u2261 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n          goal = Dist (IND-CPA-\u2141 A) 1\/2\n               \u2261\u27e8 refl \u27e9\n                 Dist (DDH-\u2141\u2080 A\u2032) 1\/2\n               \u2261\u27e8 dist-cong goal4 \u27e9\n                 Dist (DDH-\u2141\u2080 A\u2032) (OTP-\u2141 A)\n               \u2261\u27e8 refl \u27e9\n                 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n               \u220e\n            where open \u2261-Reasoning\n\n        open absDist Dist (\u03bb {f}{h}{i} \u2192 dist-cong {f}{f}{h}{i} (\u03bb _ \u2192 refl)) public\n\nmodule El-Gamal-Base\n    (\u2124q\u1d41 : U)\n    (\u2124q-grp : GH.Group (El \u2124q\u1d41))\n    (G : \u2605)\n    (G-grp : GH.Group G)\n    (g : G)\n    (_^_ : G \u2192 El \u2124q\u1d41 \u2192 G)\n    (^-gh : GH.GroupHomomorphism \u2124q-grp G-grp (_^_ g))\n    (dlog : (b y : G) \u2192 El \u2124q\u1d41)\n    (dlog-ok : (b y : G) \u2192 b ^ dlog b y \u2261 y)\n    (R\u2090\u1d41 : U)\n    (R\u2090 : El R\u2090\u1d41)\n    {{_ : FunExt}}\n    {{_ : UA}}\n    (open GH.Group \u2124q-grp renaming (_\u2219_ to _\u229e_))\n    (\u229e-is-equiv : \u2200 k \u2192 Is-equiv (flip _\u229e_ k))\n    where\n\n    open GH.Group G-grp using (_\u2219_) renaming (-_ to _\u207b\u00b9)\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n\n    open El-Gamal-Generic \u2124q\u1d41 G g _^_ G _\u2219_ _\/_ R\u2090\u1d41 public\n\n    otp-lem : \u2200 (O : G \u2192 \u2115) m\u2080 m\u2081 \u2192\n        (\u03bb x \u2192 O((g ^ x) \u2219 m\u2080)) \u2248q (\u03bb x \u2192 O((g ^ x) \u2219 m\u2081))\n    otp-lem = GH.thm \u2124q-grp G-grp (_^_ g) (explore \u2124q\u1d41)\n                     ^-gh (dlog g) (dlog-ok g)\n                     (explore-ext \u2124q\u1d41) 0 _+_\n                     (\u03bb k f \u2192 ! sumStableUnder \u2124q\u1d41 (_ , \u229e-is-equiv k) f)\n    open Proof otp-lem\n\n    thm : \u2200 A \u2192\n          let A\u2032 = transformAdv A in\n          Dist (IND-CPA-\u2141 A) 1\/2 \u2261 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n    thm = goal\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ac531c140ff1e48d76d49f86e8052d8213243ff4","subject":"Fixup old comments","message":"Fixup old comments\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Evaluation.agda","new_file":"Parametric\/Change\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Connecting Parametric.Change.Term and Parametric.Change.Value.\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\n\nmodule Parametric.Change.Evaluation\n    {Base : Type.Structure}\n    {Const : Term.Structure Base}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\nopen import Postulate.Extensionality\n\n-- Extension point 1: Relating \u2295 and its value on base types\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 {\u0393} \u2192\n  {t : Term \u0393 (base \u03b9)} {\u0394t : Term \u0393 (\u0394Type (base \u03b9))} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d base \u03b9 \u208e \u0394t \u27e7 \u03c1\n\n-- Extension point 2: Relating \u229d and its value on base types\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 {\u0393} \u2192\n  {s : Term \u0393 (base \u03b9)} {t : Term \u0393 (base \u03b9)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d base \u03b9 \u208e t \u27e7 \u03c1\n\nmodule Structure\n    (meaning-\u2295-base : ApplyStructure)\n    (meaning-\u229d-base : DiffStructure)\n  where\n\n  -- unique names with unambiguous types\n  -- to help type inference figure things out\n  private\n    module Disambiguation where\n      infixr 9 _\u22c6_\n      _\u22c6_ : Type \u2192 Context \u2192 Context\n      _\u22c6_ = _\u2022_\n\n  -- We provide: Relating \u2295 and \u229d and their values on arbitrary types.\n  meaning-\u2295 : \u2200 {\u03c4 \u0393}\n    {t : Term \u0393 \u03c4} {\u0394t : Term \u0393 (\u0394Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d \u03c4 \u208e \u0394t \u27e7 \u03c1\n\n  meaning-\u229d : \u2200 {\u03c4 \u0393}\n    {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d \u03c4 \u208e t \u27e7 \u03c1\n\n  meaning-\u2295 {base \u03b9} {\u0393} {\u03c4} {\u0394t} {\u03c1} = meaning-\u2295-base \u03b9 {\u0393} {\u03c4} {\u0394t} {\u03c1}\n  meaning-\u2295 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {\u0394t} {\u03c1} = ext (\u03bb v \u2192\n    let\n      \u0393\u2032 = \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0394Type (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7\n      \u03c1\u2032 = v \u2022 (\u27e6 t \u27e7 \u03c1) \u2022 (\u27e6 \u0394t \u27e7 \u03c1) \u2022 \u03c1\n      x  : Term \u0393\u2032 \u03c3\n      x  = var this\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that this)\n      \u0394f : Term \u0393\u2032 (\u0394Type (\u03c3 \u21d2 \u03c4))\n      \u0394f = var (that (that this))\n      y  = app f x\n      \u0394y = app (app \u0394f x) (x \u229d x)\n    in\n      begin\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v hole)\n           (meaning-\u229d {s = x} {x} {\u03c1\u2032}) \u27e9\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (\u27e6 x \u229d x \u27e7 \u03c1\u2032)\n      \u2261\u27e8 meaning-\u2295 {t = y} {\u0394t = \u0394y} {\u03c1\u2032} \u27e9\n        \u27e6 y \u2295\u208d \u03c4 \u208e \u0394y \u27e7 \u03c1\u2032\n      \u220e)\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n\n  meaning-\u229d {base \u03b9} {\u0393} {s} {t} {\u03c1} = meaning-\u229d-base \u03b9 {\u0393} {s} {t} {\u03c1}\n  meaning-\u229d {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} =\n    ext (\u03bb v \u2192 ext (\u03bb \u0394v \u2192\n    let\n      \u0393\u2032 = \u0394Type \u03c3 \u22c6 \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7Context\n      \u03c1\u2032 = \u0394v \u2022 v \u2022 \u27e6 t \u27e7Term \u03c1 \u2022 \u27e6 s \u27e7Term \u03c1 \u2022 \u03c1\n      \u0394x : Term \u0393\u2032 (\u0394Type \u03c3)\n      \u0394x = var this\n      x  : Term \u0393\u2032 \u03c3\n      x  = var (that this)\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that (that this))\n      g  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      g  = var (that (that (that this)))\n      y  = app f x\n      y\u2032 = app g (x \u2295\u208d \u03c3 \u208e \u0394x)\n    in\n      begin\n        \u27e6 s \u27e7 \u03c1 (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 s \u27e7 \u03c1 hole \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v)\n           (meaning-\u2295 {t = x} {\u0394t = \u0394x} {\u03c1\u2032}) \u27e9\n        \u27e6 s \u27e7 \u03c1 (\u27e6 x \u2295\u208d \u03c3 \u208e \u0394x \u27e7 \u03c1\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 meaning-\u229d {s = y\u2032} {y} {\u03c1\u2032} \u27e9\n        \u27e6 y\u2032 \u229d y \u27e7 \u03c1\u2032\n      \u220e))\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Connecting Parametric.Change.Term and Parametric.Change.Value.\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\n\nmodule Parametric.Change.Evaluation\n    {Base : Type.Structure}\n    {Const : Term.Structure Base}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\nopen import Postulate.Extensionality\n\n-- Extension point 1: Relating \u229d and its value on base types\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 {\u0393} \u2192\n  {t : Term \u0393 (base \u03b9)} {\u0394t : Term \u0393 (\u0394Type (base \u03b9))} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d base \u03b9 \u208e \u0394t \u27e7 \u03c1\n\n-- Extension point 2: Relating \u2295 and its value on base types\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 {\u0393} \u2192\n  {s : Term \u0393 (base \u03b9)} {t : Term \u0393 (base \u03b9)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d base \u03b9 \u208e t \u27e7 \u03c1\n\nmodule Structure\n    (meaning-\u2295-base : ApplyStructure)\n    (meaning-\u229d-base : DiffStructure)\n  where\n\n  -- unique names with unambiguous types\n  -- to help type inference figure things out\n  private\n    module Disambiguation where\n      infixr 9 _\u22c6_\n      _\u22c6_ : Type \u2192 Context \u2192 Context\n      _\u22c6_ = _\u2022_\n\n  -- We provide: Relating \u2295 and \u229d and their values on arbitrary types.\n  meaning-\u2295 : \u2200 {\u03c4 \u0393}\n    {t : Term \u0393 \u03c4} {\u0394t : Term \u0393 (\u0394Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d \u03c4 \u208e \u0394t \u27e7 \u03c1\n\n  meaning-\u229d : \u2200 {\u03c4 \u0393}\n    {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d \u03c4 \u208e t \u27e7 \u03c1\n\n  meaning-\u2295 {base \u03b9} {\u0393} {\u03c4} {\u0394t} {\u03c1} = meaning-\u2295-base \u03b9 {\u0393} {\u03c4} {\u0394t} {\u03c1}\n  meaning-\u2295 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {\u0394t} {\u03c1} = ext (\u03bb v \u2192\n    let\n      \u0393\u2032 = \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0394Type (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7\n      \u03c1\u2032 = v \u2022 (\u27e6 t \u27e7 \u03c1) \u2022 (\u27e6 \u0394t \u27e7 \u03c1) \u2022 \u03c1\n      x  : Term \u0393\u2032 \u03c3\n      x  = var this\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that this)\n      \u0394f : Term \u0393\u2032 (\u0394Type (\u03c3 \u21d2 \u03c4))\n      \u0394f = var (that (that this))\n      y  = app f x\n      \u0394y = app (app \u0394f x) (x \u229d x)\n    in\n      begin\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v hole)\n           (meaning-\u229d {s = x} {x} {\u03c1\u2032}) \u27e9\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (\u27e6 x \u229d x \u27e7 \u03c1\u2032)\n      \u2261\u27e8 meaning-\u2295 {t = y} {\u0394t = \u0394y} {\u03c1\u2032} \u27e9\n        \u27e6 y \u2295\u208d \u03c4 \u208e \u0394y \u27e7 \u03c1\u2032\n      \u220e)\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n\n  meaning-\u229d {base \u03b9} {\u0393} {s} {t} {\u03c1} = meaning-\u229d-base \u03b9 {\u0393} {s} {t} {\u03c1}\n  meaning-\u229d {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} =\n    ext (\u03bb v \u2192 ext (\u03bb \u0394v \u2192\n    let\n      \u0393\u2032 = \u0394Type \u03c3 \u22c6 \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7Context\n      \u03c1\u2032 = \u0394v \u2022 v \u2022 \u27e6 t \u27e7Term \u03c1 \u2022 \u27e6 s \u27e7Term \u03c1 \u2022 \u03c1\n      \u0394x : Term \u0393\u2032 (\u0394Type \u03c3)\n      \u0394x = var this\n      x  : Term \u0393\u2032 \u03c3\n      x  = var (that this)\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that (that this))\n      g  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      g  = var (that (that (that this)))\n      y  = app f x\n      y\u2032 = app g (x \u2295\u208d \u03c3 \u208e \u0394x)\n    in\n      begin\n        \u27e6 s \u27e7 \u03c1 (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 s \u27e7 \u03c1 hole \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v)\n           (meaning-\u2295 {t = x} {\u0394t = \u0394x} {\u03c1\u2032}) \u27e9\n        \u27e6 s \u27e7 \u03c1 (\u27e6 x \u2295\u208d \u03c3 \u208e \u0394x \u27e7 \u03c1\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 meaning-\u229d {s = y\u2032} {y} {\u03c1\u2032} \u27e9\n        \u27e6 y\u2032 \u229d y \u27e7 \u03c1\u2032\n      \u220e))\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"89ca2e37b8eb2d8ee3b6d8c9bf51b952925c6c45","subject":"Added x\u22650 (ER version).","message":"Added x\u22650 (ER version).\n\nIgnore-this: 156a203f7f01a3daa606860ed4c0c3ff\n\ndarcs-hash:20100502224943-3bd4e-e29c673de3e2161e48fc06c1830f38eee3d7f0f5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sx\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sx\u22640\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Relation.Inequalities\nopen import LTC.Relation.Inequalities.Properties using ( x\u22650 )\nopen import LTC.Data.N\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sx\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sx\u22640\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3be2477097d1bfe6056a5d1fbcef0fb70636e4dc","subject":"add \u2257\u2082","message":"add \u2257\u2082\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2261\u2261_ : \u2200 {a} {A : Set a} {i j : A}\n         (i\u2261j\u2081 : i \u2261 j) (i\u2261j\u2082 : i \u2261 j) \u2192 i\u2261j\u2081 \u2261 i\u2261j\u2082\n_\u2261\u2261_ refl refl = refl\n\n_\u225f\u2261_ : \u2200 {a} {A : Set a} {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n_\u225f\u2261_ refl refl = yes refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} (f g : A \u2192 B \u2192 C) \u2192 Set _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\ninjective : \u2200 {a} {A : Set a} \u2192 InjectiveRel A _\u2261_\ninjective refl refl = refl\n\nsurjective : \u2200 {a} {A : Set a} \u2192 SurjectiveRel A _\u2261_\nsurjective refl refl = refl\n\nbijective : \u2200 {a} {A : Set a} \u2192 BijectiveRel A _\u2261_\nbijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\nmodule \u2261-Reasoning {a} {A : Set a} where\n  open Setoid-Reasoning (setoid A) public renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : Set a} {B : Set b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 Set\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2261\u2261_ : \u2200 {a} {A : Set a} {i j : A}\n         (i\u2261j\u2081 : i \u2261 j) (i\u2261j\u2082 : i \u2261 j) \u2192 i\u2261j\u2081 \u2261 i\u2261j\u2082\n_\u2261\u2261_ refl refl = refl\n\n_\u225f\u2261_ : \u2200 {a} {A : Set a} {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n_\u225f\u2261_ refl refl = yes refl\n\ninjective : \u2200 {a} {A : Set a} \u2192 InjectiveRel A _\u2261_\ninjective refl refl = refl\n\nsurjective : \u2200 {a} {A : Set a} \u2192 SurjectiveRel A _\u2261_\nsurjective refl refl = refl\n\nbijective : \u2200 {a} {A : Set a} \u2192 BijectiveRel A _\u2261_\nbijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\nmodule \u2261-Reasoning {a} {A : Set a} where\n  open Setoid-Reasoning (setoid A) public renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : Set a} {B : Set b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 Set\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bc58f8eb9575320021db74fbb9d62cfccaf9f18c","subject":"Desc@ICFP: really update the model this time","message":"Desc@ICFP: really update the model this time\n\nI confused myself in the previous patch. I had been editing the\nsemi-old code, while the good one was in RevisedHutton. I've now\nmerged RevisedHutton into IDescTT.\n\n","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            Var context ty ->\n            IMu closeTerm' ty\ndischarge ctxt env ty variable = con (ESu EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     Var context ty ->\n                     IMu closeTerm' ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm' ty\nsubstExpr {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval' (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu (ESu EZe)) , (con (ESu EZe , su ze) , con ( EZe , l (r refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , l (l (r refl))) ,\n                             (con (EZe , l (r refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe ,  r refl  )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             Var context ty ->\n             IMu closeTerm' ty\ndischarge {n} {c} ty variable = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      Var context ty ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con ((ESu EZe) , true )) \n              (vcons (nat , con ((ESu EZe) , su (su ze)) ) \n              (vcons (pair bool nat , con ((ESu EZe) , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con ((ESu (ESu (ESu EZe))) , (con ((ESu EZe) , (su ze)) , con ( EZe , (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , (fze , refl)) ,\n                             (con (EZe , (fsu fze , refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"385637be7ada0ab6d5a80eee57eb20d67e57decf","subject":"FunBigStepSILR.agda: progress till I get stuck","message":"FunBigStepSILR.agda: progress till I get stuck\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 with svv n \u2264-refl tv1 tv2 {! tvv !} v1 {! eqv1 !}\n... | v2 , eqv2 , final-v = v2 , {! eqv2 !} , {! final-v !}\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"A verified framework for\n-- higher-order uncurrying optimizations\" (and a bit by \"Functional Big-Step\n-- Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n\n-- TODO: match sv2 before matching on fundamental s.\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n  where\n    v2 : Val \u03c4\n    v2 = {!!}\n    -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n    -- t\u03c12\u2193v2 = ?\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"89da077b4d098d6e0939d860c48061f66c5d3369","subject":"Add elimBool to operational semantics.","message":"Add elimBool to operational semantics.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Operational.agda","new_file":"formalization\/agda\/Spire\/Operational.agda","new_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\n\n----------------------------------------------------------------------\n\n","old_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b3753a80c591aeb5fa7f6af0bf11039337e9cc40","subject":"Redid model in terms of just leq, not leq and lt.","message":"Redid model in terms of just leq, not leq and lt.\n","repos":"agda\/agda-frp-js,agda\/agda-frp-js","old_file":"src\/agda\/FRP\/JS\/Model.agda","new_file":"src\/agda\/FRP\/JS\/Model.agda","new_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u2264_ ; _<_ )\nopen import FRP.JS.Bool using ( Bool ; true ; false ; not ; _\u225f_ ) \nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\nsym : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 (a \u2261 b) \u2192 (b \u2261 a)\nsym refl = refl\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\ndata _\u2264?_ (t u : Time) : Set where\n  leq : True (t \u2264 u) \u2192 (t \u2264? u)\n  geq : True (u \u2264 t) \u2192 (t \u2264? u)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  \u2264-asym : \u2200 t u \u2192 True (t \u2264 u) \u2192 True (u \u2264 t) \u2192 (t \u2261 u)\n  \u2264-total : \u2200 t u \u2192 (t \u2264? u) \n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Case on booleans\n\ndata Case (c : Bool) : Set where\n  _,_ : \u2200 b \u2192 True (b \u225f c) \u2192 Case c\n\nswitch : \u2200 b \u2192 Case b\nswitch true  = (true , tt)\nswitch false = (false , tt)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\nstruct-sym : \u2200 K {A B C D \u2111 \u211c} \u2192 (A\u2261B : A \u2261 B) \u2192 (C\u2261D : C \u2261 D) \u2192\n  (\u2111 \u2261 struct K A\u2261B \u211c C\u2261D) \u2192 \n    (\u211c \u2261 struct K (sym A\u2261B) \u2111 (sym C\u2261D))\nstruct-sym K refl refl refl = refl\n\nstruct-trans : \u2200 K {A B C D E F}\n  (A\u2261B : A \u2261 B) (B\u2261C : B \u2261 C) (\u211c : K \u220b C \u2194 F) (E\u2261F : E \u2261 F) (D\u2261E : D \u2261 E) \u2192\n    struct K A\u2261B (struct K B\u2261C \u211c E\u2261F) D\u2261E \u2261\n      struct K (trans A\u2261B B\u2261C) \u211c (trans D\u2261E E\u2261F)\nstruct-trans K refl refl \u211c refl refl = refl\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Concatenation of type contexts\n\n_++_ : Kinds \u2192 Kinds \u2192 Kinds\n[]      ++ \u03a5 = \u03a5\n(K \u2237 \u03a3) ++ \u03a5 = K \u2237 (\u03a3 ++ \u03a5)\n\n_\u220b_++_\u220b_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u2200 \u03a5 \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 ++ \u03a5 \u27e7\n[]      \u220b tt       ++ \u03a5 \u220b Bs = Bs\n(K \u2237 \u03a3) \u220b (A , As) ++ \u03a5 \u220b Bs = (A , (\u03a3 \u220b As ++ \u03a5 \u220b Bs))\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03a3 {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 \u2200 \u03a5 {Cs Ds} \u2192 (\u03a5 \u220b Cs \u2194* Ds) \u2192 \n  ((\u03a3 ++ \u03a5) \u220b (\u03a3 \u220b As ++ \u03a5 \u220b Cs) \u2194* (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds))\n[]      \u220b tt       ++\u00b2 \u03a5 \u220b \u2111s = \u2111s\n(K \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s = (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\ninterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\ninterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227c tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\n_[_,_] : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set \u03b1)\nT [ t , u ] = app (app (app interval T) t) u\n\nconstreq : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nconstreq {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (\u03a0 time \n  ((tvar\u2081 \u227c tvar\u2080) \u22b8 ((tvar\u2083 [ tvar\u2081 , tvar\u2080 ]) \u22b8 app tvar\u2082 tvar\u2080)))))\n\n_\u22b5_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\nT \u22b5 U = app\u2082 constreq T U\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Susbtitution on type variables under a context\n\n\u03c4substn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 TVar K (\u03a3 ++ (L \u2237 \u03a5)) \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\n\u03c4substn+ []      zero    U = U\n\u03c4substn+ []      (suc \u03c4) U = var \u03c4\n\u03c4substn+ (K \u2237 \u03a3) zero    U = var zero\n\u03c4substn+ (K \u2237 \u03a3) (suc \u03c4) U = weaken (skip K id) (\u03c4substn+ \u03a3 \u03c4 U)\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7 (A , As) Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Bs = trans \n  (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs) \n  (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip K id) (A , (\u03a3 \u220b As ++ _ \u220b Bs)))\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ (J \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s = refl\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 (J \u2237 \u03a3) {\u03a5} {L} {K} (suc \u03c4) U {A , As} {B , Bs} {Cs} {Ds} (\u211c , \u211cs) \u2111s = \n  begin\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) X (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (skip J id) (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n      (struct K\n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs))) \n        (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))) \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 struct-trans K \n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs)))\n       (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)))\n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))\n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 \u03c4substn+ (J \u2237 \u03a3) (suc \u03c4) U \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) )\n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (B , Bs) Ds)    \n  \u220e\n\n-- Substitution on types under a context\n\nsubstn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 Typ (\u03a3 ++ (L \u2237 \u03a5)) K \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\nsubstn+ \u03a3 (const C) U = const C\nsubstn+ \u03a3 (abs K T) U = abs K (substn+ (K \u2237 \u03a3) T U)\nsubstn+ \u03a3 (app S T) U = app (substn+ \u03a3 S U) (substn+ \u03a3 T U)\nsubstn+ \u03a3 (var \u03c4)   U = \u03c4substn+ \u03a3 \u03c4 U\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    T\u27e6 T \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 substn+ \u03a3 T U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7 As Bs = refl\nsubstn+ \u03a3 \u27e6 abs K T \u27e7\u27e6 U \u27e7 As Bs = ext (\u03bb A \u2192 substn+ K \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Bs)\nsubstn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Bs = apply (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Bs) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Bs)\nsubstn+ \u03a3 \u27e6 var \u03c4   \u27e7\u27e6 U \u27e7 As Bs = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = refl\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (abs J {K} T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs J T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u211c\n  \u2261\u27e8 substn+ (J \u2237 \u03a3) \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s \u27e9\n    struct K \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds)\n  \u2261\u27e8 struct-ext J K \n       (\u03bb A \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n       (\u03bb \u211c \u2192 T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b \u211c , \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (\u03bb B \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds) \u211c \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 (abs J T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 Bs Ds) \u211c\n  \u220e)))\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (app {J} {K} S T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  begin\n    T\u27e6 app S T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 cong (T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct J \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 struct-apply J K \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n       (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 substn+ \u03a3 (app S T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u220e\nsubstn+ \u03a3 \u27e6 var \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s\n\n-- Substitution on types\n\nsubstn : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 \u03a3) K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 K\nsubstn = substn+ []\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) (As : \u03a3\u27e6 \u03a3 \u27e7)\u2192\n  T\u27e6 T \u27e7 (T\u27e6 U \u27e7 As , As) \u2261 T\u27e6 substn T U \u27e7 As\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7 tt\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192\n  T\u27e6 T \u27e7\u00b2 (T\u27e6 U \u27e7\u00b2 \u211cs , \u211cs) \u2261 \n    struct K (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 Bs)\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 tt\n\n-- Eta-beta equivalence on types\n\ndata _\u220b_\u2263_ {\u03a3} : \u2200 K \u2192 Typ \u03a3 K \u2192 Typ \u03a3 K \u2192 Set where\n  abs : \u2200 K {L T U} \u2192 (L \u220b T \u2263 U) \u2192 ((K \u21d2 L) \u220b abs K T \u2263 abs K U)\n  app : \u2200 {K L F G T U} \u2192 ((K \u21d2 L) \u220b F \u2263 G) \u2192 (K \u220b T \u2263 U) \u2192 (L \u220b app F T \u2263 app G U)\n  beta : \u2200 {K L} T U \u2192 (L \u220b app (abs K T) U \u2263 substn T U)\n  eta : \u2200 {K L} T \u2192 ((K \u21d2 L) \u220b T \u2263 abs K (app (weaken (skip K id) T) tvar\u2080))\n  \u2263-refl : \u2200 {K T} \u2192 (K \u220b T \u2263 T)\n  \u2263-sym : \u2200 {K T U} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 T)\n  \u2263-trans : \u2200 {K T U V} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 V) \u2192 (K \u220b T \u2263 V)\n\n\u2263\u27e6_\u27e7 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} \u2192 (K \u220b T \u2263 U) \u2192 \u2200 As \u2192 T\u27e6 T \u27e7 As \u2261 T\u27e6 U \u27e7 As\n\u2263\u27e6 abs K T\u2263U \u27e7       As = ext (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As))\n\u2263\u27e6 app F\u2263G T\u2263U \u27e7     As = apply (\u2263\u27e6 F\u2263G \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 beta T U \u27e7        As = substn\u27e6 T \u27e7\u27e6 U \u27e7 As\n\u2263\u27e6 eta {K} T \u27e7       As = ext (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n\u2263\u27e6 \u2263-refl \u27e7          As = refl\n\u2263\u27e6 \u2263-sym T\u2263U \u27e7       As = sym (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As = trans (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As)\n\n\u2263\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} (T\u2263U : K \u220b T \u2263 U) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 \u211cs \u2261 struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n\u2263\u27e6 abs K {L} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 (\u211c , \u211cs) \u27e9\n    struct L (\u2263\u27e6 T\u2263U \u27e7 (A , As)) (T\u27e6 U \u27e7\u00b2 (\u211c , \u211cs)) (\u2263\u27e6 T\u2263U \u27e7 (B , Bs))\n  \u2261\u27e8 struct-ext K L (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As)) (\u03bb \u211c' \u2192 T\u27e6 U \u27e7\u00b2 (\u211c' , \u211cs)) (\u03bb B \u2192 \u2263\u27e6 T\u2263U \u27e7 (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 abs K T\u2263U \u27e7 As) (T\u27e6 abs K U \u27e7\u00b2 \u211cs) (\u2263\u27e6 abs K T\u2263U \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 app {K} {L} {F} {G} {T} {U} F\u2263G T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app F T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 cong (T\u27e6 F \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) \u27e9\n    T\u27e6 F \u27e7\u00b2 \u211cs (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)))\n       (\u2263\u27e6 F\u2263G \u27e7\u00b2 \u211cs) \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n      (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 struct-apply K L\n       (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n       (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct L (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 As) (T\u27e6 app G U \u27e7\u00b2 \u211cs) (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 Bs)\n  \u220e\n\u2263\u27e6 beta T U \u27e7\u00b2 \u211cs = substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs\n\u2263\u27e6 eta {K} {L} T \u27e7\u00b2 {As} {Bs} \u211cs = iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 \n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs \u211c\n  \u2261\u27e8 cong (\u03bb X \u2192 X \u211c) (weaken\u27e6 T \u27e7\u00b2 (skip K id) (\u211c , \u211cs)) \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 T \u27e7 (skip K id) (A , As))\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs))\n      (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) \u211c\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 (skip K id) (A , As)) \n       (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl \u211c refl \u27e9\n    struct L \n      (apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c)\n      (apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl)\n  \u2261\u27e8 struct-ext K L\n       (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c) \n       (\u03bb B \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl) \u211c \u27e9\n    struct (K \u21d2 L) \n      (\u2263\u27e6 eta T \u27e7 As)\n      (T\u27e6 abs K (app (weaken (skip K id) T) (var zero)) \u27e7\u00b2 \u211cs)\n      (\u2263\u27e6 eta T \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 \u2263-refl \u27e7\u00b2 \u211cs = refl\n\u2263\u27e6 \u2263-sym {K} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  struct-sym K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs)\n\u2263\u27e6 \u2263-trans {K} {T} {U} {V} T\u2263U U\u2263V \u27e7\u00b2 {As} {Bs} \u211cs =\n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u2263\u27e6 T\u2263U \u27e7 As) X (\u2263\u27e6 T\u2263U \u27e7 Bs)) (\u2263\u27e6 U\u2263V \u27e7\u00b2 \u211cs) \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (struct K (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs)) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 struct-trans K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct K (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 Bs)\n  \u220e\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  \u227c-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  \u227c-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n  \u227c-antisym : \u2200 {\u03b1} \u2192 Const (\u03a0 (rset \u03b1) (\u03a0 time (\u03a0 time ((tvar\u2081 \u227c tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 (app tvar\u2082 tvar\u2081 \u22b8 app tvar\u2082 tvar\u2080))))))\n  \u227c-case : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time (((tvar\u2081 \u227c tvar\u2080) \u22b8 tvar\u2082) \u22b8 (((tvar\u2080 \u227c tvar\u2081) \u22b8 tvar\u2082) \u22b8 tvar\u2082)))))\n\n\u2264-antisym : \u2200 {\u03b1} (A : RSet \u03b1) t u \u2192 True (t \u2264 u) \u2192 True (u \u2264 t) \u2192 A t \u2192 A u\n\u2264-antisym A t u t\u2264u u\u2264t a with \u2264-asym t u t\u2264u u\u2264t\n\u2264-antisym A t .t _   _  a | refl = a\n\n\u2264-case\u2032 : \u2200 {\u03b1} {A : Set \u03b1} {t u} \u2192 (t \u2264? u) \u2192 (True (t \u2264 u) \u2192 A) \u2192 (True (u \u2264 t) \u2192 A) \u2192 A\n\u2264-case\u2032 (leq t\u2264u) f g = f t\u2264u\n\u2264-case\u2032 (geq u\u2264t) f g = g u\u2264t\n\n\u2264-case : \u2200 {\u03b1} (A : Set \u03b1) t u \u2192 (True (t \u2264 u) \u2192 A) \u2192 (True (u \u2264 t) \u2192 A) \u2192 A\n\u2264-case A t u = \u2264-case\u2032 (\u2264-total t u)\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7      As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7       As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7       As = \u03bb A B \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7    As = \u2264-refl\nc\u27e6 \u227c-trans \u27e7   As = \u2264-trans\nc\u27e6 \u227c-antisym \u27e7 As = \u2264-antisym\nc\u27e6 \u227c-case \u27e7    As = \u2264-case\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2          \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2           \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2           \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7\u00b2        \u211cs = _\nc\u27e6 \u227c-trans \u27e7\u00b2       \u211cs = _\nc\u27e6 \u227c-antisym {\u03b1} \u27e7\u00b2 \u211cs = lemma where\n  lemma : \u2200 {\u03b1} {A B : RSet \u03b1} (\u211c : rset \u03b1 \u220b A \u2194 B) \u2192 \n    {t u : Time} \u2192 (t\u2261u : t \u2261 u) \u2192 {v w : Time} \u2192 (v\u2261w : v \u2261 w) \u2192\n    {t\u2264v : True (t \u2264 v)} {u\u2264w : True (u \u2264 w)} \u2192 \u22a4 \u2192\n    {v\u2264t : True (v \u2264 t)} {w\u2264u : True (w \u2264 u)} \u2192 \u22a4 \u2192\n    {a : A t} {b : B u} \u2192 \u211c t\u2261u a b \u2192\n    \u211c v\u2261w (\u2264-antisym A t v t\u2264v v\u2264t a) (\u2264-antisym B u w u\u2264w w\u2264u b)\n  lemma \u211c {t} refl {v} refl {t\u2264v} {u\u2264w} tt {v\u2264t} {w\u2264u} tt a\u211cb \n    with irrel (t \u2264 v) t\u2264v u\u2264w | irrel (v \u2264 t) v\u2264t w\u2264u\n  lemma \u211c {t} refl {v} refl {t\u2264v} tt {v\u2264t} tt a\u211cb\n    | refl | refl with \u2264-asym t v t\u2264v v\u2264t\n  lemma \u211c refl refl tt tt a\u211cb\n    | refl | refl | refl = a\u211cb\nc\u27e6 \u227c-case {\u03b1} \u27e7\u00b2    \u211cs = lemma where\n  lemma : \u2200 {\u03b1} {A B : Set \u03b1} (\u211c : set \u03b1 \u220b A \u2194 B) \u2192\n    \u2200 {t u : Time} \u2192 (t\u2261u : t \u2261 u) \u2192 \u2200 {v w : Time} \u2192 (v\u2261w : v \u2261 w) \u2192 \n    \u2200 {f g} \u2192 (\u2200 {t\u2264v} {u\u2264w} \u2192 \u22a4 \u2192 \u211c (f t\u2264v) (g u\u2264w)) \u2192\n    \u2200 {h i} \u2192 (\u2200 {v\u2264t} {w\u2264u} \u2192 \u22a4 \u2192 \u211c (h v\u2264t) (i w\u2264u)) \u2192\n    \u211c (\u2264-case A t v f h) (\u2264-case B u w g i)\n  lemma \u211c {t} refl {v} refl {f} {g} f\u211cg {h} {i} h\u211ci = lemma\u2032 (\u2264-total t v) where\n    lemma\u2032 : \u2200 t\u2264?v \u2192 \u211c (\u2264-case\u2032 t\u2264?v f h) (\u2264-case\u2032 t\u2264?v g i)\n    lemma\u2032 (leq t\u2264v) = f\u211cg {t\u2264v} {t\u2264v} tt\n    lemma\u2032 (geq v\u2264t) = h\u211ci {v\u2264t} {v\u2264t} tt\n  \n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n  tapp : \u2200 {K \u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} \u2192 Exp \u0393 (\u03a0 K T) \u2192 \u2200 U \u2192 Exp \u0393 (substn T U)\n  eq : \u2200 {\u03b1 T U} \u2192 (set \u03b1 \u220b T \u2263 U) \u2192 (Exp \u0393 T) \u2192 (Exp \u0393 U)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\nM\u27e6 tapp {T = T} M U \u27e7 As as = \n  cast (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (M\u27e6 M \u27e7 As as (T\u27e6 U \u27e7 As))\nM\u27e6 eq T\u2263U M \u27e7 As as = cast (\u2263\u27e6 T\u2263U \u27e7 As) (M\u27e6 M \u27e7 As as)\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\nM\u27e6 tapp {T = T} M U \u27e7\u00b2 \u211cs as\u211cbs = \n  struct-cast (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _)\n    (cast\u00b2 (substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 U \u27e7\u00b2 \u211cs)))\nM\u27e6 eq {\u03b1} {T} {U} T\u2263U M \u27e7\u00b2 {As} {Bs} \u211cs as\u211cbs = \n  struct-cast (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (cast\u00b2 (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs))\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time \n  (app (world {time \u2237 rset \u03b1 \u2237 \u03a3}) tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080))\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app (taut {[]}) \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar\u2080 \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsig : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsig T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsig\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsig\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsig-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (sig T s (sig\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsig-iso T s \u03c3 = refl\n\nsig-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (sig\u207b\u00b9 T s (sig T s \u03c3) \u2261 \u03c3)\nsig-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = sig U s (f (sig\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = sig\u207b\u00b9 U s (f (sig T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = (\u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _))\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 True (s \u2264 u) \u2192 (T at s \u22a8 \u03c3 s _ \u2248[ u ] \u03c4 s _)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (set \u03b1)) (U : STyp (set \u03b2)) \u2192 T\u27ea T \u21a6 U \u27eb \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U f = \u2200 u \u03c3 \u03c4 \u2192 \n  (T \u22a8 \u03c3 \u2248[ u ] \u03c4) \u2192 (U \u22a8 f \u03c3 \u2248[ u ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s u \u2192 True (s \u2264 u) \u2192 \u2200 a b \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u s\u2264u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u s\u2264u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u s\u2264u a c a\u211cc , \u211c-impl-\u2248 U at s u s\u2264u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u s\u2264u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u s\u2264u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u s\u2264u a b a\u2248b))\n  \u211c-impl-\u2248_at (\u25a1 T) s u s\u2264u \u03c3 \u03c4 \u03c3\u211c\u03c4 = \u03bb t s\u2264t t\u2264u \u2192 \n    \u211c-impl-\u2248 T at t u t\u2264u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n      (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s u \u2192 True (s \u2264 u) \u2192 \u2200 a b \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u s\u2264u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u s\u2264u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u s\u2264u a c a\u2248c , \u2248-impl-\u211c U at s u s\u2264u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u s\u2264u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u s\u2264u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u s\u2264u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u s\u2264u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt k\u211ck\u2032 with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt k\u211ck\u2032 | refl\n      = \u2248-impl-\u211c T at t u t\u2264u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) (\u03c3\u2248\u03c4 t s\u2264t t\u2264u) where\n        t\u2264u = k\u211ck\u2032 {t} refl {s\u2264t} {s\u2264t} tt {\u2264-refl t} {\u2264-refl t} tt\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s s\u2264u \u2192 \u211c-impl-\u2248 T at s u s\u2264u (\u03c3 s _) (\u03c4 s _) (\u03c3\u211c\u03c4 refl s\u2264u)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl s\u2264u = \u2248-impl-\u211c T at s u s\u2264u (\u03c3 s _) (\u03c4 s _) (\u03c3\u2248\u03c4 s s\u2264u)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (set \u03b1)) (U : STyp (set \u03b2)) (M : Exp [] \u27ea T \u21a6 U \u27eb) \u2192 \n  Causal T U (M\u27e6 M \u27e7 (World , tt) tt)\ncausality T U M u\n  = \u211c-impl-\u2248 (T \u21a6 U) u\n      (M\u27e6 M \u27e7 (World , tt) tt) \n      (M\u27e6 M \u27e7 (World , tt) tt) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ u ] , _) tt)\n","old_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u2264_ ; _<_ )\nopen import FRP.JS.Bool using ( Bool ; true ; false ; not ; _\u225f_ ) \nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\nsym : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 (a \u2261 b) \u2192 (b \u2261 a)\nsym refl = refl\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Case on booleans\n\ndata Case (c : Bool) : Set where\n  _,_ : \u2200 b \u2192 True (b \u225f c) \u2192 Case c\n\nswitch : \u2200 b \u2192 Case b\nswitch true  = (true , tt)\nswitch false = (false , tt)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\nstruct-sym : \u2200 K {A B C D \u2111 \u211c} \u2192 (A\u2261B : A \u2261 B) \u2192 (C\u2261D : C \u2261 D) \u2192\n  (\u2111 \u2261 struct K A\u2261B \u211c C\u2261D) \u2192 \n    (\u211c \u2261 struct K (sym A\u2261B) \u2111 (sym C\u2261D))\nstruct-sym K refl refl refl = refl\n\nstruct-trans : \u2200 K {A B C D E F}\n  (A\u2261B : A \u2261 B) (B\u2261C : B \u2261 C) (\u211c : K \u220b C \u2194 F) (E\u2261F : E \u2261 F) (D\u2261E : D \u2261 E) \u2192\n    struct K A\u2261B (struct K B\u2261C \u211c E\u2261F) D\u2261E \u2261\n      struct K (trans A\u2261B B\u2261C) \u211c (trans D\u2261E E\u2261F)\nstruct-trans K refl refl \u211c refl refl = refl\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Concatenation of type contexts\n\n_++_ : Kinds \u2192 Kinds \u2192 Kinds\n[]      ++ \u03a5 = \u03a5\n(K \u2237 \u03a3) ++ \u03a5 = K \u2237 (\u03a3 ++ \u03a5)\n\n_\u220b_++_\u220b_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u2200 \u03a5 \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 ++ \u03a5 \u27e7\n[]      \u220b tt       ++ \u03a5 \u220b Bs = Bs\n(K \u2237 \u03a3) \u220b (A , As) ++ \u03a5 \u220b Bs = (A , (\u03a3 \u220b As ++ \u03a5 \u220b Bs))\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03a3 {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 \u2200 \u03a5 {Cs Ds} \u2192 (\u03a5 \u220b Cs \u2194* Ds) \u2192 \n  ((\u03a3 ++ \u03a5) \u220b (\u03a3 \u220b As ++ \u03a5 \u220b Cs) \u2194* (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds))\n[]      \u220b tt       ++\u00b2 \u03a5 \u220b \u2111s = \u2111s\n(K \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s = (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq lt : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 lt \u27e7     = \u03bb t u \u2192 True (t < u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 lt \u27e7\u00b2     = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n_\u227a_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227a_ = capp\u2082 lt\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\n\u00bdinterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\n\u00bdinterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227a tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\n_\u27e8_,_] : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set \u03b1)\nT \u27e8 t , u ] = app (app (app \u00bdinterval T) t) u\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Susbtitution on type variables under a context\n\n\u03c4substn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 TVar K (\u03a3 ++ (L \u2237 \u03a5)) \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\n\u03c4substn+ []      zero    U = U\n\u03c4substn+ []      (suc \u03c4) U = var \u03c4\n\u03c4substn+ (K \u2237 \u03a3) zero    U = var zero\n\u03c4substn+ (K \u2237 \u03a3) (suc \u03c4) U = weaken (skip K id) (\u03c4substn+ \u03a3 \u03c4 U)\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7 (A , As) Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Bs = trans \n  (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs) \n  (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip K id) (A , (\u03a3 \u220b As ++ _ \u220b Bs)))\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ (J \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s = refl\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 (J \u2237 \u03a3) {\u03a5} {L} {K} (suc \u03c4) U {A , As} {B , Bs} {Cs} {Ds} (\u211c , \u211cs) \u2111s = \n  begin\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) X (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (skip J id) (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n      (struct K\n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs))) \n        (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))) \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 struct-trans K \n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs)))\n       (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)))\n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))\n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 \u03c4substn+ (J \u2237 \u03a3) (suc \u03c4) U \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) )\n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (B , Bs) Ds)    \n  \u220e\n\n-- Substitution on types under a context\n\nsubstn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 Typ (\u03a3 ++ (L \u2237 \u03a5)) K \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\nsubstn+ \u03a3 (const C) U = const C\nsubstn+ \u03a3 (abs K T) U = abs K (substn+ (K \u2237 \u03a3) T U)\nsubstn+ \u03a3 (app S T) U = app (substn+ \u03a3 S U) (substn+ \u03a3 T U)\nsubstn+ \u03a3 (var \u03c4)   U = \u03c4substn+ \u03a3 \u03c4 U\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    T\u27e6 T \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 substn+ \u03a3 T U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7 As Bs = refl\nsubstn+ \u03a3 \u27e6 abs K T \u27e7\u27e6 U \u27e7 As Bs = ext (\u03bb A \u2192 substn+ K \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Bs)\nsubstn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Bs = apply (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Bs) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Bs)\nsubstn+ \u03a3 \u27e6 var \u03c4   \u27e7\u27e6 U \u27e7 As Bs = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = refl\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (abs J {K} T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs J T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u211c\n  \u2261\u27e8 substn+ (J \u2237 \u03a3) \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s \u27e9\n    struct K \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds)\n  \u2261\u27e8 struct-ext J K \n       (\u03bb A \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n       (\u03bb \u211c \u2192 T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b \u211c , \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (\u03bb B \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds) \u211c \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 (abs J T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 Bs Ds) \u211c\n  \u220e)))\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (app {J} {K} S T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  begin\n    T\u27e6 app S T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 cong (T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct J \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 struct-apply J K \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n       (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 substn+ \u03a3 (app S T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u220e\nsubstn+ \u03a3 \u27e6 var \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s\n\n-- Substitution on types\n\nsubstn : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 \u03a3) K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 K\nsubstn = substn+ []\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) (As : \u03a3\u27e6 \u03a3 \u27e7)\u2192\n  T\u27e6 T \u27e7 (T\u27e6 U \u27e7 As , As) \u2261 T\u27e6 substn T U \u27e7 As\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7 tt\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192\n  T\u27e6 T \u27e7\u00b2 (T\u27e6 U \u27e7\u00b2 \u211cs , \u211cs) \u2261 \n    struct K (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 Bs)\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 tt\n\n-- Eta-beta equivalence on types\n\ndata _\u220b_\u2263_ {\u03a3} : \u2200 K \u2192 Typ \u03a3 K \u2192 Typ \u03a3 K \u2192 Set where\n  abs : \u2200 K {L T U} \u2192 (L \u220b T \u2263 U) \u2192 ((K \u21d2 L) \u220b abs K T \u2263 abs K U)\n  app : \u2200 {K L F G T U} \u2192 ((K \u21d2 L) \u220b F \u2263 G) \u2192 (K \u220b T \u2263 U) \u2192 (L \u220b app F T \u2263 app G U)\n  beta : \u2200 {K L} T U \u2192 (L \u220b app (abs K T) U \u2263 substn T U)\n  eta : \u2200 {K L} T \u2192 ((K \u21d2 L) \u220b T \u2263 abs K (app (weaken (skip K id) T) tvar\u2080))\n  \u2263-refl : \u2200 {K T} \u2192 (K \u220b T \u2263 T)\n  \u2263-sym : \u2200 {K T U} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 T)\n  \u2263-trans : \u2200 {K T U V} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 V) \u2192 (K \u220b T \u2263 V)\n\n\u2263\u27e6_\u27e7 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} \u2192 (K \u220b T \u2263 U) \u2192 \u2200 As \u2192 T\u27e6 T \u27e7 As \u2261 T\u27e6 U \u27e7 As\n\u2263\u27e6 abs K T\u2263U \u27e7       As = ext (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As))\n\u2263\u27e6 app F\u2263G T\u2263U \u27e7     As = apply (\u2263\u27e6 F\u2263G \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 beta T U \u27e7        As = substn\u27e6 T \u27e7\u27e6 U \u27e7 As\n\u2263\u27e6 eta {K} T \u27e7       As = ext (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n\u2263\u27e6 \u2263-refl \u27e7          As = refl\n\u2263\u27e6 \u2263-sym T\u2263U \u27e7       As = sym (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As = trans (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As)\n\n\u2263\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} (T\u2263U : K \u220b T \u2263 U) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 \u211cs \u2261 struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n\u2263\u27e6 abs K {L} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 (\u211c , \u211cs) \u27e9\n    struct L (\u2263\u27e6 T\u2263U \u27e7 (A , As)) (T\u27e6 U \u27e7\u00b2 (\u211c , \u211cs)) (\u2263\u27e6 T\u2263U \u27e7 (B , Bs))\n  \u2261\u27e8 struct-ext K L (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As)) (\u03bb \u211c' \u2192 T\u27e6 U \u27e7\u00b2 (\u211c' , \u211cs)) (\u03bb B \u2192 \u2263\u27e6 T\u2263U \u27e7 (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 abs K T\u2263U \u27e7 As) (T\u27e6 abs K U \u27e7\u00b2 \u211cs) (\u2263\u27e6 abs K T\u2263U \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 app {K} {L} {F} {G} {T} {U} F\u2263G T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app F T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 cong (T\u27e6 F \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) \u27e9\n    T\u27e6 F \u27e7\u00b2 \u211cs (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)))\n       (\u2263\u27e6 F\u2263G \u27e7\u00b2 \u211cs) \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n      (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 struct-apply K L\n       (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n       (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct L (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 As) (T\u27e6 app G U \u27e7\u00b2 \u211cs) (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 Bs)\n  \u220e\n\u2263\u27e6 beta T U \u27e7\u00b2 \u211cs = substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs\n\u2263\u27e6 eta {K} {L} T \u27e7\u00b2 {As} {Bs} \u211cs = iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 \n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs \u211c\n  \u2261\u27e8 cong (\u03bb X \u2192 X \u211c) (weaken\u27e6 T \u27e7\u00b2 (skip K id) (\u211c , \u211cs)) \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 T \u27e7 (skip K id) (A , As))\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs))\n      (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) \u211c\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 (skip K id) (A , As)) \n       (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl \u211c refl \u27e9\n    struct L \n      (apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c)\n      (apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl)\n  \u2261\u27e8 struct-ext K L\n       (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c) \n       (\u03bb B \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl) \u211c \u27e9\n    struct (K \u21d2 L) \n      (\u2263\u27e6 eta T \u27e7 As)\n      (T\u27e6 abs K (app (weaken (skip K id) T) (var zero)) \u27e7\u00b2 \u211cs)\n      (\u2263\u27e6 eta T \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 \u2263-refl \u27e7\u00b2 \u211cs = refl\n\u2263\u27e6 \u2263-sym {K} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  struct-sym K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs)\n\u2263\u27e6 \u2263-trans {K} {T} {U} {V} T\u2263U U\u2263V \u27e7\u00b2 {As} {Bs} \u211cs =\n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u2263\u27e6 T\u2263U \u27e7 As) X (\u2263\u27e6 T\u2263U \u27e7 Bs)) (\u2263\u27e6 U\u2263V \u27e7\u00b2 \u211cs) \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (struct K (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs)) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 struct-trans K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct K (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 Bs)\n  \u220e\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  \u227c-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  \u227c-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n  \u227c-absurd : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time ((tvar\u2081 \u227a tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 tvar\u2082)))))\n  \u227c-case : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time (((tvar\u2081 \u227c tvar\u2080) \u22b8 tvar\u2082) \u22b8 (((tvar\u2080 \u227a tvar\u2081) \u22b8 tvar\u2082) \u22b8 tvar\u2082)))))\n\nabsurd : \u2200 {\u03b1} {A : Set \u03b1} b \u2192 True b \u2192 True (not b) \u2192 A\nabsurd true  tt ()\nabsurd false () tt\n\ncond : \u2200 {\u03b1} {A : Set \u03b1} b \u2192 (True (not b) \u2192 A) \u2192 (True b \u2192 A) \u2192 A\ncond true  f g = g tt\ncond false f g = f tt\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7     As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7      As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7      As = \u03bb A B \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7   As = \u2264-refl\nc\u27e6 \u227c-trans \u27e7  As = \u2264-trans\nc\u27e6 \u227c-absurd \u27e7 As = \u03bb A t u \u2192 absurd (t < u)\nc\u27e6 \u227c-case \u27e7   As = \u03bb A t u \u2192 cond (u < t)\n\ncond\u00b2 : \u2200 {\u03b1} {A A\u2032 : Set \u03b1} (\u211c : A \u2192 A\u2032 \u2192 Set \u03b1) b \u2192\n  \u2200 {f f\u2032} \u2192 (\u2200 b\u00d7 \u2192 \u211c (f b\u00d7) (f\u2032 b\u00d7)) \u2192\n    \u2200 {g g\u2032} \u2192 (\u2200 b\u2713 \u2192 \u211c (g b\u2713) (g\u2032 b\u2713)) \u2192\n      \u211c (cond b f g) (cond b f\u2032 g\u2032)\ncond\u00b2 \u211c true  f\u211cf\u2032 g\u211cg\u2032 = g\u211cg\u2032 tt\ncond\u00b2 \u211c false f\u211cf\u2032 g\u211cg\u2032 = f\u211cf\u2032 tt\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2        \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2        \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7\u00b2     \u211cs = _\nc\u27e6 \u227c-trans \u27e7\u00b2    \u211cs = _\nc\u27e6 \u227c-absurd \u27e7\u00b2   \u211cs = \u03bb \u211c {t} t\u2261t {u} u\u2261u {t<u} tt {u\u2264t} tt \u2192 absurd (t < u) t<u u\u2264t\nc\u27e6 \u227c-case {\u03b1} \u27e7\u00b2 \u211cs = lemma where\n  lemma : \u2200 {A A\u2032 : Set \u03b1} (\u211c : A \u2192 A\u2032 \u2192 Set \u03b1) \u2192\n    \u2200 {t t\u2032} \u2192 (t \u2261 t\u2032) \u2192 \u2200 {u u\u2032} \u2192 (u \u2261 u\u2032) \u2192\n      \u2200 {f f\u2032} \u2192 (\u2200 {t\u2264u t\u2032\u2264u\u2032} \u2192 \u22a4 \u2192 \u211c (f t\u2264u) (f\u2032 t\u2032\u2264u\u2032)) \u2192\n        \u2200 {g g\u2032} \u2192 (\u2200 {u<t u\u2032<t\u2032} \u2192 \u22a4 \u2192 \u211c (g u<t) (g\u2032 u\u2032<t\u2032)) \u2192\n          \u211c (cond (u < t) f g) (cond (u\u2032 < t\u2032) f\u2032 g\u2032)\n  lemma \u211c {t} refl {u} refl f\u211cf\u2032 g\u211cg\u2032 = \n    cond\u00b2 \u211c (u < t) (\u03bb t\u2264u \u2192 f\u211cf\u2032 {t\u2264u} {t\u2264u} tt) (\u03bb u<t \u2192 g\u211cg\u2032 {u<t} {u<t} tt)\n  \n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n  tapp : \u2200 {K \u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} \u2192 Exp \u0393 (\u03a0 K T) \u2192 \u2200 U \u2192 Exp \u0393 (substn T U)\n  eq : \u2200 {\u03b1 T U} \u2192 (set \u03b1 \u220b T \u2263 U) \u2192 (Exp \u0393 T) \u2192 (Exp \u0393 U)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\nM\u27e6 tapp {T = T} M U \u27e7 As as = \n  cast (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (M\u27e6 M \u27e7 As as (T\u27e6 U \u27e7 As))\nM\u27e6 eq T\u2263U M \u27e7 As as = cast (\u2263\u27e6 T\u2263U \u27e7 As) (M\u27e6 M \u27e7 As as)\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\nM\u27e6 tapp {T = T} M U \u27e7\u00b2 \u211cs as\u211cbs = \n  struct-cast (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _)\n    (cast\u00b2 (substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 U \u27e7\u00b2 \u211cs)))\nM\u27e6 eq {\u03b1} {T} {U} T\u2263U M \u27e7\u00b2 {As} {Bs} \u211cs as\u211cbs = \n  struct-cast (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (cast\u00b2 (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs))\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time \n  (app (world {time \u2237 rset \u03b1 \u2237 \u03a3}) tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080))\n\nsignal : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1 \u21d2 rset \u03b1)\nsignal {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (\u03a0 time ((tvar\u2081 \u227c tvar\u2080) \u22b8 \n  ((world {time \u2237 time \u2237 rset \u03b1 \u2237 \u03a3} \u27e8 tvar\u2081 , tvar\u2080 ]) \u22b8 app tvar\u2082 tvar\u2080))))\n\n\u25a7 : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1) \u2192 Typ+ \u03a3 (rset \u03b1)\n\u25a7 {\u03a3} = app (signal {\u03a3})\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app (taut {[]}) \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = \u25a7 {[]} \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsig : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsig T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsig\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsig\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsig-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (sig T s (sig\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsig-iso T s \u03c3 = refl\n\nsig-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (sig\u207b\u00b9 T s (sig T s \u03c3) \u2261 \u03c3)\nsig-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = sig U s (f (sig\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = sig\u207b\u00b9 U s (f (sig T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = (\u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _))\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 True (s \u2264 u) \u2192 (T at s \u22a8 \u03c3 s _ \u2248[ u ] \u03c4 s _)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (set \u03b1)) (U : STyp (set \u03b2)) \u2192 T\u27ea T \u21a6 U \u27eb \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U f = \u2200 u \u03c3 \u03c4 \u2192 \n  (T \u22a8 \u03c3 \u2248[ u ] \u03c4) \u2192 (U \u22a8 f \u03c3 \u2248[ u ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s u \u2192 True (s \u2264 u) \u2192 \u2200 a b \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u s\u2264u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u s\u2264u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u s\u2264u a c a\u211cc , \u211c-impl-\u2248 U at s u s\u2264u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u s\u2264u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u s\u2264u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u s\u2264u a b a\u2248b))\n  \u211c-impl-\u2248_at (\u25a1 T) s u s\u2264u \u03c3 \u03c4 \u03c3\u211c\u03c4 = \u03bb t s\u2264t t\u2264u \u2192 \n    \u211c-impl-\u2248 T at t u t\u2264u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n      (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s u \u2192 True (s \u2264 u) \u2192 \u2200 a b \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u s\u2264u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u s\u2264u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u s\u2264u a c a\u2248c , \u2248-impl-\u211c U at s u s\u2264u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u s\u2264u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u s\u2264u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u s\u2264u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u s\u2264u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt k\u211ck\u2032 with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt k\u211ck\u2032 | refl\n      = \u2248-impl-\u211c T at t u t\u2264u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) (\u03c3\u2248\u03c4 t s\u2264t t\u2264u) where\n        t\u2264u : True (t \u2264 u)\n        t\u2264u with switch (s < t)\n        t\u2264u | (true , s<t)  = k\u211ck\u2032 {t} refl {s<t} {s<t} tt {\u2264-refl t} {\u2264-refl t} tt \n        t\u2264u | (false , t\u2264s) = \u2264-trans t s u t\u2264s s\u2264u\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s s\u2264u \u2192 \u211c-impl-\u2248 T at s u s\u2264u (\u03c3 s _) (\u03c4 s _) (\u03c3\u211c\u03c4 refl s\u2264u)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl s\u2264u = \u2248-impl-\u211c T at s u s\u2264u (\u03c3 s _) (\u03c4 s _) (\u03c3\u2248\u03c4 s s\u2264u)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (set \u03b1)) (U : STyp (set \u03b2)) (M : Exp [] \u27ea T \u21a6 U \u27eb) \u2192 \n  Causal T U (M\u27e6 M \u27e7 (World , tt) tt)\ncausality T U M u\n  = \u211c-impl-\u2248 (T \u21a6 U) u\n      (M\u27e6 M \u27e7 (World , tt) tt) \n      (M\u27e6 M \u27e7 (World , tt) tt) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ u ] , _) tt)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f48e8a266b458979ac6b0f3bb45639852cb2fe6a","subject":"Added failed automatic proof.","message":"Added failed automatic proof.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Exponentiation.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Exponentiation.agda","new_contents":"------------------------------------------------------------------------------\n-- Some proofs related to exponentiation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Exponentiation where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192 n ^ zero      \u2261 [1]\n  ^-S : \u2200 m n \u2192 m ^ succ\u2081 n \u2261 m * m ^ n\n{-# ATP axiom ^-0 ^-S #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 [5] \u2264 n \u2192 n ^ [5] \u2264 [5] ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ [5] \u2264 [5] ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn)\n  where postulate prf : ([5] \u2264 n \u2192 n ^ [5] \u2264 [5] ^ n) \u2192\n                        (succ\u2081 n) ^ [5] \u2264 [5] ^ (succ\u2081 n)\n        -- 10 May 2013: The ATPs could not prove the theorem (240 sec).\n        -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       (([2] ^ n) \u2238 [1]) + [1] + (([2] ^ n) \u2238 [1]) \u2261 [2] ^ (n + [1]) \u2238 [1]\nthm\u2082 nzero = prf\n  where\n  postulate prf : (([2] ^ zero) \u2238 [1]) + [1] + (([2] ^ zero) \u2238 [1]) \u2261\n                  [2] ^ (zero + [1]) \u2238 [1]\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : (([2] ^ n) \u2238 [1]) + [1] + (([2] ^ n) \u2238 [1]) \u2261\n                  [2] ^ (n + [1]) \u2238 [1] \u2192\n                  (([2] ^ succ\u2081 n) \u2238 [1]) + [1] + (([2] ^ succ\u2081 n) \u2238 [1]) \u2261\n                  [2] ^ (succ\u2081 n + [1]) \u2238 [1]\n  -- 20 May 2013: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some proofs related to exponentiation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Exponentiation where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192 n ^ zero      \u2261 [1]\n  ^-S : \u2200 m n \u2192 m ^ succ\u2081 n \u2261 m * m ^ n\n{-# ATP axiom ^-0 ^-S #-}\n\nthm : \u2200 {n} \u2192 N n \u2192 [5] \u2264 n \u2192 n ^ [5] \u2264 [5] ^ n\nthm nzero h = prf\n  where postulate prf : zero ^ [5] \u2264 [5] ^ zero\n        {-# ATP prove prf #-}\nthm (nsucc {n} Nn) h = prf (thm Nn)\n  where postulate prf : ([5] \u2264 n \u2192 n ^ [5] \u2264 [5] ^ n) \u2192\n                        (succ\u2081 n) ^ [5] \u2264 [5] ^ (succ\u2081 n)\n        -- 10 May 2013: The ATPs could not prove the theorem (240 sec).\n        -- {-# ATP prove prf 5-N #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"73e561e8189cec85d6f3d5bff5ffc1f08f7779eb","subject":"Updated a note.","message":"Updated a note.\n\nIgnore-this: 870cacddedfafd0c43021b2947d29dd8\n\ndarcs-hash:20120223210907-3bd4e-85ee68e07b61a0274a515584e5e6e4fa1a881174.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/NoWhere.agda","new_file":"notes\/thesis\/logical-framework\/NoWhere.agda","new_contents":"------------------------------------------------------------------------------\n-- A Peano arithmetic proof without using a where clause\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 23 February 2012.\n\n-- See the original proof in\n-- PA.Axiomatic.Relation.Binary.PropositionalEqualityI where\n\nmodule NoWhere where\n\nopen import FOL.Relation.Binary.EqReasoning\n\nopen import PA.Axiomatic.Standard.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  x+Sy\u2261S[x+y]     : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\n  succ-cong       : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n\nP : M \u2192 M \u2192 Set\nP n i = i + n \u2261 n + i\n\nP0 : \u2200 n \u2192 P n zero\nP0 n = zero + n   \u2261\u27e8 A\u2083 n \u27e9\n       n          \u2261\u27e8 sym (+-rightIdentity n) \u27e9\n       n + zero \u220e\n\nis : \u2200 n i \u2192 P n i \u2192 P n (succ i)\nis n i Pi = succ i + n   \u2261\u27e8 A\u2084 i n \u27e9\n            succ (i + n) \u2261\u27e8 succ-cong Pi \u27e9\n            succ (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n i) \u27e9\n            n + succ i \u220e\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm m n = A\u2087 (P n) (P0 n) (is n) m\n","old_contents":"------------------------------------------------------------------------------\n-- A Peano arithmetic proof without using a where clause\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 09 February 2012.\n\n-- See the original proof in\n-- PA.Axiomatic.Relation.Binary.PropositionalEqualityI where\n\nmodule NoWhere where\n\nopen import PA.Axiomatic.Base\nopen import PA.Axiomatic.Relation.Binary.EqReasoning\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI using ( \u2250-sym )\n\n------------------------------------------------------------------------------\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2250 n\n  x+Sy\u2250S[x+y]     : \u2200 m n \u2192 m + succ n \u2250 succ (m + n)\n\nP : \u2115 \u2192 \u2115 \u2192 Set\nP n i = i + n \u2250 n + i\n\nP0 : \u2200 n \u2192 P n zero\nP0 n = zero + n   \u2250\u27e8 S\u2085 n \u27e9\n       n          \u2250\u27e8 \u2250-sym (+-rightIdentity n) \u27e9\n       n + zero \u220e\n\nis : \u2200 n i \u2192 P n i \u2192 P n (succ i)\nis n i Pi = succ i + n   \u2250\u27e8 S\u2086 i n \u27e9\n            succ (i + n) \u2250\u27e8 S\u2082 Pi \u27e9\n            succ (n + i) \u2250\u27e8 \u2250-sym (x+Sy\u2250S[x+y] n i) \u27e9\n            n + succ i \u220e\n\n+-comm : \u2200 m n \u2192 m + n \u2250 n + m\n+-comm m n = S\u2089 (P n) (P0 n) (is n) m\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a003974adb6be493178b47ce10f67d3536c512d4","subject":"Universe: +FinU' +FinU'\u2194Fin","message":"Universe: +FinU' +FinU'\u2194Fin\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Universe.agda","new_file":"lib\/Explore\/Universe.agda","new_contents":"open import Level.NP\nopen import Type\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat\nopen import Data.Fin\nopen import Function.Inverse.NP using (_\u2194_)\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Zero\nopen import Explore.One\nopen import Explore.Two\nopen import Explore.Product\nopen import Explore.Sum\nopen import Explore.Explorable\nimport Explore.Isomorphism\n\nmodule Explore.Universe where\n\nopen Operators\n\ninfixr 2 _\u00d7\u2032_\ninfix  8 _^\u2032_\n\ndata U : \u2605\nEl : U \u2192 \u2605\n\ndata U where\n  \ud835\udfd8\u2032 \ud835\udfd9\u2032 \ud835\udfda\u2032 : U\n  _\u00d7\u2032_ _\u228e\u2032_ : U \u2192 U \u2192 U\n  \u03a3\u2032 : (t : U) \u2192 (El t \u2192 U) \u2192 U\n\nEl \ud835\udfd8\u2032 = \ud835\udfd8\nEl \ud835\udfd9\u2032 = \ud835\udfd9\nEl \ud835\udfda\u2032 = \ud835\udfda\nEl (s \u00d7\u2032 t) = El s \u00d7 El t\nEl (s \u228e\u2032 t) = El s \u228e El t\nEl (\u03a3\u2032 t f) = \u03a3 (El t) \u03bb x \u2192 El (f x)\n\n_^\u2032_ : U \u2192 \u2115 \u2192 U\nt ^\u2032 zero  = t\nt ^\u2032 suc n = t \u00d7\u2032 t ^\u2032 n\n\nmodule _ {\u2113} where\n\n    exploreU : \u2200 t \u2192 Explore \u2113 (El t)\n    exploreU \ud835\udfd8\u2032 = \ud835\udfd8\u1d49\n    exploreU \ud835\udfd9\u2032 = \ud835\udfd9\u1d49\n    exploreU \ud835\udfda\u2032 = \ud835\udfda\u1d49\n    exploreU (s \u00d7\u2032 t) = exploreU s \u00d7\u1d49 exploreU t\n    exploreU (s \u228e\u2032 t) = exploreU s \u228e\u1d49 exploreU t\n    exploreU (\u03a3\u2032 t f) = explore\u03a3 (exploreU t) \u03bb {x} \u2192 exploreU (f x)\n\n    exploreU-ind : \u2200 {p} t \u2192 ExploreInd p (exploreU t)\n    exploreU-ind \ud835\udfd8\u2032 = \ud835\udfd8\u2071\n    exploreU-ind \ud835\udfd9\u2032 = \ud835\udfd9\u2071\n    exploreU-ind \ud835\udfda\u2032 = \ud835\udfda\u2071\n    exploreU-ind (s \u00d7\u2032 t) = exploreU-ind s \u00d7\u2071 exploreU-ind t\n    exploreU-ind (s \u228e\u2032 t) = exploreU-ind s \u228e\u2071 exploreU-ind t\n    exploreU-ind (\u03a3\u2032 t f) = explore\u03a3-ind (exploreU-ind t) \u03bb {x} \u2192 exploreU-ind (f x)\n\nmodule _ (t : U) where\n  private\n    t\u1d49 : \u2200 {\u2113} \u2192 Explore \u2113 (El t)\n    t\u1d49 = exploreU t\n    t\u2071 : \u2200 {\u2113 p} \u2192 ExploreInd p {\u2113} t\u1d49\n    t\u2071 = exploreU-ind t\n  open Explorable\u2080  t\u2071 public using () renaming (sum     to sumU; product to productU)\n  open Explorable\u2081\u2080 t\u2071 public using () renaming (reify   to reifyU)\n  open Explorable\u2081\u2081 t\u2071 public using () renaming (unfocus to unfocusU)\n\nadequate-sumU : \u2200 t \u2192 AdequateSum (sumU t)\nadequate-sumU \ud835\udfd8\u2032       = \ud835\udfd8\u02e2-ok\nadequate-sumU \ud835\udfd9\u2032       = \ud835\udfd9\u02e2-ok\nadequate-sumU \ud835\udfda\u2032       = \ud835\udfda\u02e2-ok\nadequate-sumU (s \u00d7\u2032 t) = adequate-sum\u03a3 (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (s \u228e\u2032 t) = adequate-sum\u228e (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (\u03a3\u2032 t f) = adequate-sum\u03a3 (adequate-sumU t) (\u03bb {x} \u2192 adequate-sumU (f x))\n\nmodule _ {\u2113} where\n    lookupU : \u2200 t \u2192 Lookup {\u2113} (exploreU t)\n    lookupU \ud835\udfd8\u2032 = \ud835\udfd8\u02e1\n    lookupU \ud835\udfd9\u2032 = \ud835\udfd9\u02e1\n    lookupU \ud835\udfda\u2032 = \ud835\udfda\u02e1\n    lookupU (s \u00d7\u2032 t) = lookup\u00d7 {_} {_} {_} {exploreU s} {exploreU t} (lookupU s) (lookupU t)\n    lookupU (s \u228e\u2032 t) = lookup\u228e {_} {_} {_} {exploreU s} {exploreU t} (lookupU s) (lookupU t)\n    lookupU (\u03a3\u2032 t f) = lookup\u03a3 {_} {_} {_} {exploreU t} {\u03bb {x} \u2192 exploreU (f x)} (lookupU t) (\u03bb {x} \u2192 lookupU (f x))\n\n    focusU : \u2200 t \u2192 Focus {\u2113} (exploreU t)\n    focusU \ud835\udfd8\u2032 = \ud835\udfd8\u1da0\n    focusU \ud835\udfd9\u2032 = \ud835\udfd9\u1da0\n    focusU \ud835\udfda\u2032 = \ud835\udfda\u1da0\n    focusU (s \u00d7\u2032 t) = focus\u00d7 {_} {_} {_} {exploreU s} {exploreU t} (focusU s) (focusU t)\n    focusU (s \u228e\u2032 t) = focus\u228e {_} {_} {_} {exploreU s} {exploreU t} (focusU s) (focusU t)\n    focusU (\u03a3\u2032 t f) = focus\u03a3 {_} {_} {_} {exploreU t} {\u03bb {x} \u2192 exploreU (f x)} (focusU t) (\u03bb {x} \u2192 focusU (f x))\n\n-- See Explore.Fin for an example of the use of this module\nmodule Isomorphism {A : \u2605\u2080} u (u\u2194A : El u \u2194 A) where\n  open Explore.Isomorphism u\u2194A\n\n  module _ {\u2113} where\n    iso\u1d49 : Explore \u2113 A\n    iso\u1d49 = explore-iso (exploreU u)\n\n    module _ {p} where\n      iso\u2071 : ExploreInd p iso\u1d49\n      iso\u2071 = explore-iso-ind (exploreU-ind u)\n\n  module _ {\u2113} where\n    iso\u02e1 : Lookup {\u2113} iso\u1d49\n    iso\u02e1 = lookup-iso {\u2113} {exploreU u} (lookupU u)\n\n    iso\u1da0 : Focus {\u2113} iso\u1d49\n    iso\u1da0 = focus-iso {\u2113} {exploreU u} (focusU u)\n\n  iso\u02e2 : Sum A\n  iso\u02e2 = sum-iso (sumU u)\n\n  iso\u1d56 : Product A\n  iso\u1d56 = product-iso (sumU u)\n\n  iso\u02b3 : Reify iso\u1d49\n  iso\u02b3 = reify-iso (exploreU-ind u)\n\n  iso\u1d58 : Unfocus iso\u1d49\n  iso\u1d58 = unfocus-iso (exploreU-ind u)\n\n  iso\u02e2-ok : AdequateSum iso\u02e2\n  iso\u02e2-ok = sum-iso-ok (adequate-sumU u)\n\n  -- SUI\n  open AdequateSum\u2080 iso\u02e2-ok iso\u02e2-ok public renaming (sumStableUnder to iso\u02e2-stableUnder)\n\n  \u03bciso : Explorable A\n  \u03bciso = mk _ iso\u2071 iso\u02e2-ok\n\nFinU : \u2115 \u2192 U\nFinU zero    = \ud835\udfd8\u2032\nFinU (suc n) = \ud835\udfd9\u2032 \u228e\u2032 FinU n\n\nFinU\u2194Fin : \u2200 n \u2192 El (FinU n) \u2194 Fin n\nFinU\u2194Fin zero    = Inv.sym Fin0\u2194\ud835\udfd8\nFinU\u2194Fin (suc n) = Inv.sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin Inv.\u2218 Inv.id \u228e-cong FinU\u2194Fin n\n\nFinU' : \u2115 \u2192 U\nFinU' zero          = \ud835\udfd8\u2032\nFinU' (suc zero)    = \ud835\udfd9\u2032\nFinU' (suc (suc n)) = \ud835\udfd9\u2032 \u228e\u2032 FinU' (suc n)\n\nFinU'\u2194Fin : \u2200 n \u2192 El (FinU' n) \u2194 Fin n\nFinU'\u2194Fin zero          = Inv.sym Fin0\u2194\ud835\udfd8\nFinU'\u2194Fin (suc zero)    = Inv.sym Fin1\u2194\ud835\udfd9\nFinU'\u2194Fin (suc (suc n)) = Inv.sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin Inv.\u2218 Inv.id \u228e-cong FinU'\u2194Fin (suc n)\n","old_contents":"open import Level.NP\nopen import Type\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat\nopen import Data.Fin\nopen import Function.Inverse.NP using (_\u2194_)\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Zero\nopen import Explore.One\nopen import Explore.Two\nopen import Explore.Product\nopen import Explore.Sum\nopen import Explore.Explorable\nimport Explore.Isomorphism\n\nmodule Explore.Universe where\n\nopen Operators\n\ninfixr 2 _\u00d7\u2032_\ninfix  8 _^\u2032_\n\ndata U : \u2605\nEl : U \u2192 \u2605\n\ndata U where\n  \ud835\udfd8\u2032 \ud835\udfd9\u2032 \ud835\udfda\u2032 : U\n  _\u00d7\u2032_ _\u228e\u2032_ : U \u2192 U \u2192 U\n  \u03a3\u2032 : (t : U) \u2192 (El t \u2192 U) \u2192 U\n\nEl \ud835\udfd8\u2032 = \ud835\udfd8\nEl \ud835\udfd9\u2032 = \ud835\udfd9\nEl \ud835\udfda\u2032 = \ud835\udfda\nEl (s \u00d7\u2032 t) = El s \u00d7 El t\nEl (s \u228e\u2032 t) = El s \u228e El t\nEl (\u03a3\u2032 t f) = \u03a3 (El t) \u03bb x \u2192 El (f x)\n\n_^\u2032_ : U \u2192 \u2115 \u2192 U\nt ^\u2032 zero  = t\nt ^\u2032 suc n = t \u00d7\u2032 t ^\u2032 n\n\nmodule _ {\u2113} where\n\n    exploreU : \u2200 t \u2192 Explore \u2113 (El t)\n    exploreU \ud835\udfd8\u2032 = \ud835\udfd8\u1d49\n    exploreU \ud835\udfd9\u2032 = \ud835\udfd9\u1d49\n    exploreU \ud835\udfda\u2032 = \ud835\udfda\u1d49\n    exploreU (s \u00d7\u2032 t) = exploreU s \u00d7\u1d49 exploreU t\n    exploreU (s \u228e\u2032 t) = exploreU s \u228e\u1d49 exploreU t\n    exploreU (\u03a3\u2032 t f) = explore\u03a3 (exploreU t) \u03bb {x} \u2192 exploreU (f x)\n\n    exploreU-ind : \u2200 {p} t \u2192 ExploreInd p (exploreU t)\n    exploreU-ind \ud835\udfd8\u2032 = \ud835\udfd8\u2071\n    exploreU-ind \ud835\udfd9\u2032 = \ud835\udfd9\u2071\n    exploreU-ind \ud835\udfda\u2032 = \ud835\udfda\u2071\n    exploreU-ind (s \u00d7\u2032 t) = exploreU-ind s \u00d7\u2071 exploreU-ind t\n    exploreU-ind (s \u228e\u2032 t) = exploreU-ind s \u228e\u2071 exploreU-ind t\n    exploreU-ind (\u03a3\u2032 t f) = explore\u03a3-ind (exploreU-ind t) \u03bb {x} \u2192 exploreU-ind (f x)\n\nmodule _ (t : U) where\n  private\n    t\u1d49 : \u2200 {\u2113} \u2192 Explore \u2113 (El t)\n    t\u1d49 = exploreU t\n    t\u2071 : \u2200 {\u2113 p} \u2192 ExploreInd p {\u2113} t\u1d49\n    t\u2071 = exploreU-ind t\n  open Explorable\u2080  t\u2071 public using () renaming (sum     to sumU; product to productU)\n  open Explorable\u2081\u2080 t\u2071 public using () renaming (reify   to reifyU)\n  open Explorable\u2081\u2081 t\u2071 public using () renaming (unfocus to unfocusU)\n\nadequate-sumU : \u2200 t \u2192 AdequateSum (sumU t)\nadequate-sumU \ud835\udfd8\u2032       = \ud835\udfd8\u02e2-ok\nadequate-sumU \ud835\udfd9\u2032       = \ud835\udfd9\u02e2-ok\nadequate-sumU \ud835\udfda\u2032       = \ud835\udfda\u02e2-ok\nadequate-sumU (s \u00d7\u2032 t) = adequate-sum\u03a3 (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (s \u228e\u2032 t) = adequate-sum\u228e (adequate-sumU s) (adequate-sumU t)\nadequate-sumU (\u03a3\u2032 t f) = adequate-sum\u03a3 (adequate-sumU t) (\u03bb {x} \u2192 adequate-sumU (f x))\n\nmodule _ {\u2113} where\n    lookupU : \u2200 t \u2192 Lookup {\u2113} (exploreU t)\n    lookupU \ud835\udfd8\u2032 = \ud835\udfd8\u02e1\n    lookupU \ud835\udfd9\u2032 = \ud835\udfd9\u02e1\n    lookupU \ud835\udfda\u2032 = \ud835\udfda\u02e1\n    lookupU (s \u00d7\u2032 t) = lookup\u00d7 {_} {_} {_} {exploreU s} {exploreU t} (lookupU s) (lookupU t)\n    lookupU (s \u228e\u2032 t) = lookup\u228e {_} {_} {_} {exploreU s} {exploreU t} (lookupU s) (lookupU t)\n    lookupU (\u03a3\u2032 t f) = lookup\u03a3 {_} {_} {_} {exploreU t} {\u03bb {x} \u2192 exploreU (f x)} (lookupU t) (\u03bb {x} \u2192 lookupU (f x))\n\n    focusU : \u2200 t \u2192 Focus {\u2113} (exploreU t)\n    focusU \ud835\udfd8\u2032 = \ud835\udfd8\u1da0\n    focusU \ud835\udfd9\u2032 = \ud835\udfd9\u1da0\n    focusU \ud835\udfda\u2032 = \ud835\udfda\u1da0\n    focusU (s \u00d7\u2032 t) = focus\u00d7 {_} {_} {_} {exploreU s} {exploreU t} (focusU s) (focusU t)\n    focusU (s \u228e\u2032 t) = focus\u228e {_} {_} {_} {exploreU s} {exploreU t} (focusU s) (focusU t)\n    focusU (\u03a3\u2032 t f) = focus\u03a3 {_} {_} {_} {exploreU t} {\u03bb {x} \u2192 exploreU (f x)} (focusU t) (\u03bb {x} \u2192 focusU (f x))\n\n-- See Explore.Fin for an example of the use of this module\nmodule Isomorphism {A : \u2605\u2080} u (u\u2194A : El u \u2194 A) where\n  open Explore.Isomorphism u\u2194A\n\n  module _ {\u2113} where\n    iso\u1d49 : Explore \u2113 A\n    iso\u1d49 = explore-iso (exploreU u)\n\n    module _ {p} where\n      iso\u2071 : ExploreInd p iso\u1d49\n      iso\u2071 = explore-iso-ind (exploreU-ind u)\n\n  module _ {\u2113} where\n    iso\u02e1 : Lookup {\u2113} iso\u1d49\n    iso\u02e1 = lookup-iso {\u2113} {exploreU u} (lookupU u)\n\n    iso\u1da0 : Focus {\u2113} iso\u1d49\n    iso\u1da0 = focus-iso {\u2113} {exploreU u} (focusU u)\n\n  iso\u02e2 : Sum A\n  iso\u02e2 = sum-iso (sumU u)\n\n  iso\u1d56 : Product A\n  iso\u1d56 = product-iso (sumU u)\n\n  iso\u02b3 : Reify iso\u1d49\n  iso\u02b3 = reify-iso (exploreU-ind u)\n\n  iso\u1d58 : Unfocus iso\u1d49\n  iso\u1d58 = unfocus-iso (exploreU-ind u)\n\n  iso\u02e2-ok : AdequateSum iso\u02e2\n  iso\u02e2-ok = sum-iso-ok (adequate-sumU u)\n\n  -- SUI\n  open AdequateSum\u2080 iso\u02e2-ok iso\u02e2-ok public renaming (sumStableUnder to iso\u02e2-stableUnder)\n\n  \u03bciso : Explorable A\n  \u03bciso = mk _ iso\u2071 iso\u02e2-ok\n\nFinU : \u2115 \u2192 U\nFinU zero    = \ud835\udfd8\u2032\nFinU (suc n) = \ud835\udfd9\u2032 \u228e\u2032 FinU n\n\nFinU\u2194Fin : \u2200 n \u2192 El (FinU n) \u2194 Fin n\nFinU\u2194Fin zero    = Inv.sym Fin0\u2194\ud835\udfd8\nFinU\u2194Fin (suc n) = Inv.sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin Inv.\u2218 Inv.id \u228e-cong FinU\u2194Fin n\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"82f90de0921c70e4d5be5bdacaff5ffff38da256","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 26b21111344d5c4756767dc9b1b16467\n\ndarcs-hash:20101222132815-3bd4e-45eefb53fbd97ea42604e9e008fb8a71bcdc0e55.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/AxiomaticPA\/Base.agda","new_file":"src\/AxiomaticPA\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfix  7  _\u2263_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the tool agda2atp as the ATPs equality.\npostulate\n  _\u2263_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. n + 0 = n\n-- S\u2086. m + succ n = succ (m + n)\n-- S\u2087. n * 0 = 0\n-- S\u2088. m * succ n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2263 n \u2192 m \u2263 o \u2192 n \u2263 o\n{-# ATP axiom S\u2081 #-}\n\npostulate\n  S\u2082 : \u2200 {m n} \u2192 m \u2263 n \u2192 succ m \u2263 succ n\n{-# ATP axiom S\u2082 #-}\n\npostulate\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2263 succ n)\n{-# ATP axiom S\u2083 #-}\n\npostulate\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2263 succ n \u2192 m \u2263 n\n{-# ATP axiom S\u2084 #-}\n\n-- N.B. We make the recursion in the first argument.\npostulate\n  S\u2085 : \u2200 n \u2192   zero   + n \u2263 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2263 succ (m + n)\n{-# ATP axiom S\u2085 #-}\n{-# ATP axiom S\u2086 #-}\n\n-- N.B. We make the recursion on the first argument.\npostulate\n  S\u2087 : \u2200 n \u2192   zero   * n \u2263 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2263 n + m * n\n{-# ATP axiom S\u2087 #-}\n{-# ATP axiom S\u2088 #-}\n\npostulate\n  -- The axiom S\u2089 is a higher-order one, therefore we do not translate it\n  -- as an ATP axiom.\n  S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfix  7  _\u2263_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\npostulate\n  _\u2263_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. n + 0 = n\n-- S\u2086. m + succ n = succ (m + n)\n-- S\u2087. n * 0 = 0\n-- S\u2088. m * succ n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2263 n \u2192 m \u2263 o \u2192 n \u2263 o\n{-# ATP axiom S\u2081 #-}\n\npostulate\n  S\u2082 : \u2200 {m n} \u2192 m \u2263 n \u2192 succ m \u2263 succ n\n{-# ATP axiom S\u2082 #-}\n\npostulate\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2263 succ n)\n{-# ATP axiom S\u2083 #-}\n\npostulate\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2263 succ n \u2192 m \u2263 n\n{-# ATP axiom S\u2084 #-}\n\n-- N.B. We make the recursion in the first argument.\npostulate\n  S\u2085 : \u2200 n \u2192   zero   + n \u2263 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2263 succ (m + n)\n{-# ATP axiom S\u2085 #-}\n{-# ATP axiom S\u2086 #-}\n\n-- N.B. We make the recursion on the first argument.\npostulate\n  S\u2087 : \u2200 n \u2192   zero   * n \u2263 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2263 n + m * n\n{-# ATP axiom S\u2087 #-}\n{-# ATP axiom S\u2088 #-}\n\npostulate\n  -- The axiom S\u2089 is a higher-order one, therefore we do not translate it\n  -- as an ATP axiom.\n  S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d28029e6a6b72c675115285102a2d423b1c0b3f9","subject":"prg: fix argument order","message":"prg: fix argument order\n","repos":"crypto-agda\/crypto-agda","old_file":"prg.agda","new_file":"prg.agda","new_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search _\u2228_ {k} ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search _\u2227_ {k} (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = search-hom _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n","old_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search {k} _\u2228_ ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search {k} _\u2227_ (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = search-comm _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8704f60b97e9adc489cdacbb1e2c9a7d762fbf96","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --    (a : \u2115)    (a \u2264 1)    (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _  = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _) = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    xxx : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    xxx = \u21d3Base\n\n    yyy : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    yyy = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    zzz : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    zzz = {!!}\n    -}\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val _) _ = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"5748a712a365f87abbea3ba2e246ee805f07d463","subject":"progress progress #3","message":"progress progress #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFACast (d' , \u03c41 , \u03c42 , \u03c43 , \u03c44 , refl , d'' , ne ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAEHole (_ , _ , _ , refl , _)           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFANEHole (_ , _ , _ , _ , _ , refl , _)  = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAAp (_ , _ , _ , _ , _ , refl , _)      = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFACastHole (_ , refl , refl , refl , _ ) = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAFailedCast (_ , _ , refl , _ )         = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | V x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFACast (d' , \u03c41 , \u03c42 , \u03c43 , \u03c44 , refl , d'' , ne ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAEHole (_ , _ , _ , refl , _)           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFANEHole (_ , _ , _ , _ , _ , refl , _)  = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAAp (_ , _ , _ , _ , _ , refl , _)      = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFACastHole (_ , refl , refl , refl , _ ) = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAFailedCast (_ , _ , refl , _ )         = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = {!!} -- I {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (_ , Step x a q) = S (_ , Step {!!} (ITCastFail y z w) {!!} )\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | V x = I (IFailedCast (FBoxed x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFACast (d' , \u03c41 , \u03c42 , \u03c43 , \u03c44 , refl , d'' , ne ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAEHole (_ , _ , _ , refl , _)           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFANEHole (_ , _ , _ , _ , _ , refl , _)  = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAAp (_ , _ , _ , _ , _ , refl , _)      = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFACastHole (_ , refl , refl , refl , _ ) = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAFailedCast (_ , _ , refl , _ )         = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | V x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFACast (d' , \u03c41 , \u03c42 , \u03c43 , \u03c44 , refl , d'' , ne ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAEHole (_ , _ , _ , refl , _)           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFANEHole (_ , _ , _ , _ , _ , refl , _)  = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAAp (_ , _ , _ , _ , _ , refl , _)      = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFACastHole (_ , refl , refl , refl , _ ) = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAFailedCast (_ , _ , refl , _ )         = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (_ , Step x a q) = S (_ , Step {!!} (ITCastFail y z w) {!!} )\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | V x = I (IFailedCast (FBoxed x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"343a2d1d11e329ca7267e1e3098547a18992bcc5","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/Collatz\/Collatz.agda","new_file":"src\/fot\/FOTC\/Program\/Collatz\/Collatz.agda","new_contents":"------------------------------------------------------------------------------\n-- The Collatz function: A function without a termination proof\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.Collatz where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n-- The Collatz function.\npostulate\n  collatz         : D \u2192 D\n  collatz-0       : collatz [0] \u2261 [1]\n  collatz-1       : collatz [1] \u2261 [1]\n  collatz-even    : \u2200 {n} \u2192 Even (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz (div (succ\u2081 (succ\u2081 n)) [2])\n  collatz-noteven : \u2200 {n} \u2192 NotEven (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz ([3] * (succ\u2081 (succ\u2081 n)) + [1])\n{-# ATP axiom collatz-0 collatz-1 collatz-even collatz-noteven #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The Collatz function: A function without a termination proof\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.Collatz where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n-- The Collatz function.\npostulate\n  collatz         : D \u2192 D\n  collatz-0       : collatz [0] \u2261 [1]\n  collatz-1       : collatz [1]  \u2261 [1]\n  collatz-even    : \u2200 {n} \u2192 Even (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz (div (succ\u2081 (succ\u2081 n)) [2])\n  collatz-noteven : \u2200 {n} \u2192 NotEven (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz ([3] * (succ\u2081 (succ\u2081 n)) + [1])\n{-# ATP axiom collatz-0 collatz-1 collatz-even collatz-noteven #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"61d871d15a9844e258a74c2b3e37da6993ed116f","subject":"Instance arguments changed...","message":"Instance arguments changed...\n","repos":"np\/NomPa","old_file":"lib\/NomPa\/Examples\/LC\/SymanticsTy.agda","new_file":"lib\/NomPa\/Examples\/LC\/SymanticsTy.agda","new_contents":"open import Relation.Binary  using (Reflexive; Transitive)\nopen import Function\nopen import Function.InstanceArguments\nopen import Data.Nat\nopen import Data.Unit using (\u22a4)\n-- open import Reflection.NP using (\u03b7\u207f)\nopen import Data.Product.NP\nopen import Category.Functor\n            renaming (RawFunctor to Functor; module RawFunctor to Functor)\nopen import Category.Applicative\n            renaming (RawApplicative to Applicative; module RawApplicative to Applicative)\n\nmodule NomPa.Examples.LC.SymanticsTy\n  where\n\nrecord Box {a} (A : Set a) : Set a where\n  constructor mk\n  field get : A\n\nmodule DataFunctorReader {E} where\n  Reader : Set \u2192 Set\n  Reader A = E \u2192 A\n  functor : Functor Reader\n  functor = record { _<$>_ = \u03bb f g \u2192 f \u2218 g }\n  applicative : Applicative Reader\n  applicative = record { pure = const; _\u229b_ = _\u02e2_ }\n\nmodule BasicSyms (Repr : Set \u2192 Set) where\n  record BaseArithSym : Set where\n    constructor mk\n    field\n      nat : \u2115 \u2192 Repr \u2115\n      add : Repr (\u2115 \u2192 \u2115 \u2192 \u2115)\n      mul : Repr (\u2115 \u2192 \u2115 \u2192 \u2115)\n\n  AppSym' : Set\u2081\n  AppSym' = \u2200 {A B : Set} \u2192 Repr (A \u2192 B) \u2192 Repr A \u2192 Repr B\n\n  AppSym : Set\u2081\n  AppSym = Box AppSym'\n\n  record SimpleSym : Set\u2081 where\n    constructor mk\n    infixl 2 _$$_\n    field\n      baseArithSym : BaseArithSym\n      appSym       : AppSym\n    open BaseArithSym baseArithSym public\n    _$$_ : AppSym'\n    _$$_ = Box.get appSym\n    _+:_ : Repr \u2115 \u2192 Repr \u2115 \u2192 Repr \u2115\n    x +: y = add $$ x $$ y\n    _*:_ : Repr \u2115 \u2192 Repr \u2115 \u2192 Repr \u2115\n    x *: y = mul $$ x $$ y\n\n  -- lamS in Staged Haskell\n  LamPure : Set\u2081\n  LamPure = Box (\u2200 {A B} \u2192 (Repr A \u2192 Repr B) \u2192 Repr (A \u2192 B))\n\n  AssertPos : Set\u2081\n  AssertPos = \u2200 {A} \u2192 Repr \u2115 \u2192 Repr A \u2192 Repr A\n\nmodule IdSyms where\n  open BasicSyms id\n  baseArithSym : BaseArithSym\n  baseArithSym = record { nat = id; add = _+_; mul = _*_ }\n  _$$_ : AppSym\n  _$$_ = mk id\n  simpleSym : SimpleSym\n  simpleSym = record { baseArithSym = baseArithSym; appSym = _$$_ }\n  \u019b\u1d3e : LamPure\n  \u019b\u1d3e = mk id\n\nmodule ApplicativeSyms {M Repr : Set \u2192 Set} (M-app : Applicative M) where\n  open BasicSyms\n  open Applicative M-app\n\n  instance\n    baseArithSym : {{sym : BaseArithSym Repr}} \u2192 BaseArithSym (M \u2218 Repr)\n    baseArithSym = record { nat = pure \u2218 nat; add = pure add; mul = pure mul } where\n      open BaseArithSym _ \u2026\n\n  appSym : {{sym : AppSym Repr}} \u2192 AppSym (M \u2218 Repr)\n  appSym {{mk appSym'}} = mk \u03bb x y \u2192 pure appSym' \u229b x \u229b y\n\n  simpleSym : {{sym : SimpleSym Repr}} \u2192 SimpleSym (M \u2218 Repr)\n  simpleSym {{mk _ _}} = mk \u2026 appSym\n\n  assertPos : {{_ : AssertPos Repr}} \u2192 AssertPos (M \u2218 Repr)\n  assertPos {{assertPos\u2032}} n\u1d39 a\u1d39 = pure assertPos\u2032 \u229b n\u1d39 \u229b a\u1d39\n\nopen BasicSyms\n\nJ : (M Repr : Set \u2192 Set) (A : Set) \u2192 Set\nJ M Repr A = M (Repr A)\n\nHV : (H : Set) (Repr : Set \u2192 Set) (A : Set) \u2192 Set\n-- HV H = J (\u03bb B \u2192 H \u2192 B)\nHV H Repr A = H \u2192 Repr A\n\nmodule HVSyms {H Repr} = ApplicativeSyms {\u03bb B \u2192 H \u2192 B} {Repr} DataFunctorReader.applicative\n\nhmap : \u2200 {A H\u2081 H\u2082 Repr} \u2192 (H\u2082 \u2192 H\u2081) \u2192 HV H\u2081 Repr A \u2192 HV H\u2082 Repr A\nhmap f g = g \u2218 f\n\n-- called runH in Staged Haskell\nrunHV : \u2200 {Repr A} \u2192 HV \u22a4 Repr A \u2192 Repr A\nrunHV m = m _\n\nrecord Lib : Set\u2081 where\n  constructor mk\n  field\n    HA : (Repr : Set \u2192 Set) (S A : Set) \u2192 Set\n\n  LamM : Set\u2081\n  LamM = \u2200 {M Repr : Set \u2192 Set}{A B H} {{funM : Functor M}} {{sSym : SimpleSym Repr}}\n              {{\u019b\u1d3e : LamPure Repr}} \u2192 (\u2200 {S} \u2192 HV (HA Repr S A \u00d7 H) Repr A \u2192 M (HV (HA Repr S A \u00d7 H) Repr B))\n            \u2192 M (HV H Repr (A \u2192 B))\n\n  field\n    lam : LamM\n    href : \u2200 {Repr S A H} \u2192 HV (HA Repr S A \u00d7 H) Repr A\n    var : \u2200 {H M Repr A} {{M-app : Applicative M}} \u2192 HV H Repr A \u2192 M (HV H Repr A)\n    weaken : \u2200 {H H\u2032 A : Set} {M Repr : Set \u2192 Set} {{M-app : Functor M}} \u2192 M (HV H Repr A) \u2192 M (HV (H\u2032 \u00d7 H) Repr A)\n\nmodule M2\n  (Cst : Set)\n  (Env : Set)\n  -- (_\u21911 : Env \u2192 Env)\n  (Binder : Set)\n  (Ty : Set)\n  (_,\u27e8_\u2236_\u27e9 : Env \u2192 Binder \u2192 Ty \u2192 Env)\n  (_#_ : Binder \u2192 Env \u2192 Set)\n  (_\u21911\u27e8_\u27e9 : Env \u2192 Ty \u2192 Env)\n  (_\u2286_ : Env \u2192 Env \u2192 Set)\n  (_\u21d2_ : Ty \u2192 Ty \u2192 Ty)\n  (\u03b5 : Env)\n  (\u2286-\u03b5     : \u2200 {\u0393} \u2192 \u03b5 \u2286 \u0393)\n  (\u2286-# : \u2200 {\u0393 b \u03c4} \u2192 b # \u0393 \u2192 \u0393 \u2286 \u0393 ,\u27e8 b \u2236 \u03c4 \u27e9)\n  (\u2286-cong-,\u1d30 : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u2286 \u0393\u2082 \u2192 \u0393\u2081 \u21911\u27e8 \u03c4 \u27e9 \u2286 \u0393\u2082 \u21911\u27e8 \u03c4 \u27e9)\n  (\u2286-trans : Transitive _\u2286_)\n  (_+1 : Env \u2192 Env)\n  (\u2286-+1\u21911 : \u2200 {\u0393 \u03c4} \u2192 \u0393 +1 \u2286 \u0393 \u21911\u27e8 \u03c4 \u27e9)\n  (_\u22a2_ : Env \u2192 Ty \u2192 Set) where\n\n record Sym : Set where\n  infixl 6 _\u00b7_\n\n  field\n   _\u00b7_  : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : \u0393 \u22a2 (\u03c3 \u21d2 \u03c4)) (u : \u0393 \u22a2 \u03c3) \u2192 \u0393 \u22a2 \u03c4\n   \u019b\u1d3a    : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (f : \u2200 {b} \u2192 b # \u0393 \u2192 \u0393 ,\u27e8 b \u2236 \u03c3 \u27e9 \u22a2 \u03c3 \u2192 \u0393 ,\u27e8 b \u2236 \u03c3 \u27e9 \u22a2 \u03c4) \u2192 \u0393 \u22a2 (\u03c3 \u21d2 \u03c4)\n   \u019b\u1d30    : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (f : \u0393 \u21911\u27e8 \u03c3 \u27e9 \u22a2 \u03c3 \u2192 \u0393 \u21911\u27e8 \u03c3 \u27e9 \u22a2 \u03c4) \u2192 \u0393 \u22a2 (\u03c3 \u21d2 \u03c4)\n   -- Let  : \u2200 {\u03b1} \u2192 (t : Tm \u03b1) (u : Tm (\u03b1 \u21911) \u2192 Tm (\u03b1 \u21911)) \u2192 Tm \u03b1\n   -- \u019b    : \u2200 {\u03b1} \u2192 (t : \u2200 {b} \u2192 Tm (b \u25c5 \u03b1) \u2192 Tm (b \u25c5 \u03b1)) \u2192 Tm \u03b1\n   -- Let  : \u2200 {\u03b1} \u2192 (t : Tm \u03b1) (u : \u2200 {b} \u2192 Tm (b \u25c5 \u03b1) \u2192 Tm (b \u25c5 \u03b1)) \u2192 Tm \u03b1\n   -- `_   : \u2200 {\u03b1} \u2192 (\u03ba : Cst) \u2192 Tm \u03b1\n\n   coerce\u2122 : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u2286 \u0393\u2082 \u2192 \u0393\u2081 \u22a2 \u03c4 \u2192 \u0393\u2082 \u22a2 \u03c4\n   shift\u2122 : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 +1 \u2286 \u0393\u2082 \u2192 \u0393\u2081 \u22a2 \u03c4 \u2192 \u0393\u2082 \u22a2 \u03c4\n\n  weaken\u2122 : \u2200 {\u0393 \u03c3 \u03c4} \u2192 \u0393 \u22a2 \u03c4 \u2192 \u0393 \u21911\u27e8 \u03c3 \u27e9 \u22a2 \u03c4\n  weaken\u2122 = shift\u2122 \u2286-+1\u21911\n\n\n{-\n   \u2286-cong-\u21911 : \u2200 {\u03b1 \u03b2}\n                  (\u03b1\u2286\u03b2 : \u03b1 \u2286 \u03b2)\n                \u2192 \u03b1 \u21911 \u2286 \u03b2 \u21911\n-}\n\n module Examples\u1d3a (sym : Sym) where\n  open Sym sym\n\n  id\u2122 : \u2200 {\u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 \u03c4)\n  id\u2122 = \u019b\u1d3a (\u03bb _ x \u2192 x)\n\n  true\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 (\u03c3 \u21d2 \u03c4))\n  true\u2122 = \u019b\u1d3a (\u03bb _ x \u2192 \u019b\u1d3a (\u03bb y# y \u2192 coerce\u2122 (\u2286-# y#) x))\n\n  false\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c3 \u21d2 (\u03c4 \u21d2 \u03c4))\n  false\u2122 = \u019b\u1d3a (\u03bb _ x \u2192 \u019b\u1d3a (\u03bb _ y \u2192 y))\n\n module Examples\u1d30 (sym : Sym) where\n  open Sym sym\n\n  id\u2122 : \u2200 {\u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 \u03c4)\n  id\u2122 = \u019b\u1d30 (\u03bb x \u2192 x)\n\n  true\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 (\u03c3 \u21d2 \u03c4))\n  true\u2122 = \u019b\u1d30 (\u03bb x \u2192 \u019b\u1d30 (\u03bb y \u2192 weaken\u2122 x))\n\n  false\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c3 \u21d2 (\u03c4 \u21d2 \u03c4))\n  false\u2122 = \u019b\u1d30 (\u03bb x \u2192 \u019b\u1d30 (\u03bb y \u2192 y))\n\nmodule M3 where\n  -- open module F {M : Set \u2192 Set} {{x}} = Functor {_} {M} x\n  -- open module A {M : Set \u2192 Set} {{x}} = Applicative {_} {M} x\n\n  record HA (Repr : Set \u2192 Set) (S A : Set) : Set where\n    constructor mk\n    field\n      ha : Repr A\n\n  href : \u2200 {Repr S A H} \u2192 HV (HA Repr S A \u00d7 H) Repr A \n  href (mk x , h) = x \n\n  -- dup\n  LamM : Set\u2081\n  LamM = \u2200 {M Repr : Set \u2192 Set}{A B H} {{M-fun : Functor M}} {{sSym : SimpleSym Repr}}\n              {{\u019b\u1d3e : LamPure Repr}} \u2192 (\u2200 {S} \u2192 HV (HA Repr S A \u00d7 H) Repr A \u2192 M (HV (HA Repr S A \u00d7 H) Repr B))\n            \u2192 M (HV H Repr (A \u2192 B))\n\n  lam : LamM\n  lam {{M-fun}} {{sym}} {{mk \u019b\u1d3e}} f = _<$>_ (\u03bb body h \u2192 \u019b\u1d3e (\u03bb x \u2192 body (mk x , h))) (f {\u22a4} href) where\n    open SimpleSym _ sym\n    open Functor M-fun\n\n  var : \u2200 {H M Repr A} {{M-app : Applicative M}} \u2192 HV H Repr A \u2192 M (HV H Repr A)\n  var = Applicative.pure \u2026\n\n  weaken : \u2200 {H H\u2032 A : Set} {M Repr : Set \u2192 Set} {{M-app : Functor M}} \u2192 M (HV H Repr A) \u2192 M (HV (H\u2032 \u00d7 H) Repr A)\n  weaken = Functor._<$>_ \u2026 (\u03bb g \u2192 g \u2218 snd) -- hmap goes yellow\n\n  lib : Lib\n  lib = mk HA lam href\n           (\u03bb {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084} \u2192 var {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084}) -- FIXME: unquote (\u03b7\u207f (quote var))\n           (\u03bb {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084} {\u03b7\u2085} \u2192 weaken {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084} {\u03b7\u2085})\n","old_contents":"open import Relation.Binary  using (Reflexive; Transitive)\nopen import Function\nopen import Function.InstanceArguments\nopen import Data.Nat\nopen import Data.Unit using (\u22a4)\n-- open import Reflection.NP using (\u03b7\u207f)\nopen import Data.Product.NP\nopen import Category.Functor\n            renaming (RawFunctor to Functor; module RawFunctor to Functor)\nopen import Category.Applicative\n            renaming (RawApplicative to Applicative; module RawApplicative to Applicative)\n\nmodule NomPa.Examples.LC.SymanticsTy\n  where\n\nmodule DataFunctorReader {E} where\n  Reader : Set \u2192 Set\n  Reader A = E \u2192 A\n  functor : Functor Reader\n  functor = record { _<$>_ = \u03bb f g \u2192 f \u2218 g }\n  applicative : Applicative Reader\n  applicative = record { pure = const; _\u229b_ = _\u02e2_ }\n\nmodule BasicSyms (Repr : Set \u2192 Set) where\n  record BaseArithSym : Set where\n    constructor mk\n    field\n      nat : \u2115 \u2192 Repr \u2115\n      add : Repr (\u2115 \u2192 \u2115 \u2192 \u2115)\n      mul : Repr (\u2115 \u2192 \u2115 \u2192 \u2115)\n\n  AppSym : Set\u2081\n  AppSym = \u2200 {A B : Set} \u2192 Repr (A \u2192 B) \u2192 Repr A \u2192 Repr B\n\n  record SimpleSym : Set\u2081 where\n    constructor mk\n    infixl 2 _$$_\n    field\n      baseArithSym : BaseArithSym\n      _$$_         : AppSym\n    open BaseArithSym baseArithSym public\n    _+:_ : Repr \u2115 \u2192 Repr \u2115 \u2192 Repr \u2115\n    x +: y = add $$ x $$ y\n    _*:_ : Repr \u2115 \u2192 Repr \u2115 \u2192 Repr \u2115\n    x *: y = mul $$ x $$ y\n\n  -- lamS in Staged Haskell\n  LamPure : Set\u2081\n  LamPure = \u2200 {A B} \u2192 (Repr A \u2192 Repr B) \u2192 Repr (A \u2192 B)\n\n  AssertPos : Set\u2081\n  AssertPos = \u2200 {A} \u2192 Repr \u2115 \u2192 Repr A \u2192 Repr A\n\nmodule IdSyms where\n  open BasicSyms id\n  baseArithSym : BaseArithSym\n  baseArithSym = record { nat = id; add = _+_; mul = _*_ }\n  _$$_ : AppSym\n  _$$_ = id\n  simpleSym : SimpleSym\n  simpleSym = record { baseArithSym = baseArithSym; _$$_ = _$$_ }\n  \u019b\u1d3e : LamPure\n  \u019b\u1d3e = id\n\nmodule ApplicativeSyms {M Repr : Set \u2192 Set} (M-app : Applicative M) where\n  open BasicSyms\n  open Applicative M-app\n\n  baseArithSym : {{sym : BaseArithSym Repr}} \u2192 BaseArithSym (M \u2218 Repr)\n  baseArithSym = record { nat = pure \u2218 nat; add = pure add; mul = pure mul } where\n    open BaseArithSym _ \u2026\n\n  _$$_ : {{sym : AppSym Repr}} \u2192 AppSym (M \u2218 Repr)\n  _$$_ {{_$$\u2032_}} x y = pure _$$\u2032_ \u229b x \u229b y\n\n  simpleSym : {{sym : SimpleSym Repr}} \u2192 SimpleSym (M \u2218 Repr)\n  simpleSym {{mk _ _}} = mk baseArithSym _$$_\n\n  assertPos : {{_ : AssertPos Repr}} \u2192 AssertPos (M \u2218 Repr)\n  assertPos {{assertPos\u2032}} n\u1d39 a\u1d39 = pure assertPos\u2032 \u229b n\u1d39 \u229b a\u1d39\n\nopen BasicSyms\n\nJ : (M Repr : Set \u2192 Set) (A : Set) \u2192 Set\nJ M Repr A = M (Repr A)\n\nHV : (H : Set) (Repr : Set \u2192 Set) (A : Set) \u2192 Set\n-- HV H = J (\u03bb B \u2192 H \u2192 B)\nHV H Repr A = H \u2192 Repr A\n\nmodule HVSyms {H Repr} = ApplicativeSyms {\u03bb B \u2192 H \u2192 B} {Repr} DataFunctorReader.applicative\n\nhmap : \u2200 {A H\u2081 H\u2082 Repr} \u2192 (H\u2082 \u2192 H\u2081) \u2192 HV H\u2081 Repr A \u2192 HV H\u2082 Repr A\nhmap f g = g \u2218 f\n\n-- called runH in Staged Haskell\nrunHV : \u2200 {Repr A} \u2192 HV \u22a4 Repr A \u2192 Repr A\nrunHV m = m _\n\nrecord Lib : Set\u2081 where\n  constructor mk\n  field\n    HA : (Repr : Set \u2192 Set) (S A : Set) \u2192 Set\n\n  LamM : Set\u2081\n  LamM = \u2200 {M Repr : Set \u2192 Set}{A B H} {{funM : Functor M}} {{sSym : SimpleSym Repr}}\n              {{\u019b\u1d3e : LamPure Repr}} \u2192 (\u2200 {S} \u2192 HV (HA Repr S A \u00d7 H) Repr A \u2192 M (HV (HA Repr S A \u00d7 H) Repr B))\n            \u2192 M (HV H Repr (A \u2192 B))\n\n  field\n    lam : LamM\n    href : \u2200 {Repr S A H} \u2192 HV (HA Repr S A \u00d7 H) Repr A\n    var : \u2200 {H M Repr A} {{M-app : Applicative M}} \u2192 HV H Repr A \u2192 M (HV H Repr A)\n    weaken : \u2200 {H H\u2032 A : Set} {M Repr : Set \u2192 Set} {{M-app : Functor M}} \u2192 M (HV H Repr A) \u2192 M (HV (H\u2032 \u00d7 H) Repr A)\n\nmodule M2\n  (Cst : Set)\n  (Env : Set)\n  -- (_\u21911 : Env \u2192 Env)\n  (Binder : Set)\n  (Ty : Set)\n  (_,\u27e8_\u2236_\u27e9 : Env \u2192 Binder \u2192 Ty \u2192 Env)\n  (_#_ : Binder \u2192 Env \u2192 Set)\n  (_\u21911\u27e8_\u27e9 : Env \u2192 Ty \u2192 Env)\n  (_\u2286_ : Env \u2192 Env \u2192 Set)\n  (_\u21d2_ : Ty \u2192 Ty \u2192 Ty)\n  (\u03b5 : Env)\n  (\u2286-\u03b5     : \u2200 {\u0393} \u2192 \u03b5 \u2286 \u0393)\n  (\u2286-# : \u2200 {\u0393 b \u03c4} \u2192 b # \u0393 \u2192 \u0393 \u2286 \u0393 ,\u27e8 b \u2236 \u03c4 \u27e9)\n  (\u2286-cong-,\u1d30 : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u2286 \u0393\u2082 \u2192 \u0393\u2081 \u21911\u27e8 \u03c4 \u27e9 \u2286 \u0393\u2082 \u21911\u27e8 \u03c4 \u27e9)\n  (\u2286-trans : Transitive _\u2286_)\n  (_+1 : Env \u2192 Env)\n  (\u2286-+1\u21911 : \u2200 {\u0393 \u03c4} \u2192 \u0393 +1 \u2286 \u0393 \u21911\u27e8 \u03c4 \u27e9)\n  (_\u22a2_ : Env \u2192 Ty \u2192 Set) where\n\n record Sym : Set where\n  infixl 6 _\u00b7_\n\n  field\n   _\u00b7_  : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : \u0393 \u22a2 (\u03c3 \u21d2 \u03c4)) (u : \u0393 \u22a2 \u03c3) \u2192 \u0393 \u22a2 \u03c4\n   \u019b\u1d3a    : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (f : \u2200 {b} \u2192 b # \u0393 \u2192 \u0393 ,\u27e8 b \u2236 \u03c3 \u27e9 \u22a2 \u03c3 \u2192 \u0393 ,\u27e8 b \u2236 \u03c3 \u27e9 \u22a2 \u03c4) \u2192 \u0393 \u22a2 (\u03c3 \u21d2 \u03c4)\n   \u019b\u1d30    : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (f : \u0393 \u21911\u27e8 \u03c3 \u27e9 \u22a2 \u03c3 \u2192 \u0393 \u21911\u27e8 \u03c3 \u27e9 \u22a2 \u03c4) \u2192 \u0393 \u22a2 (\u03c3 \u21d2 \u03c4)\n   -- Let  : \u2200 {\u03b1} \u2192 (t : Tm \u03b1) (u : Tm (\u03b1 \u21911) \u2192 Tm (\u03b1 \u21911)) \u2192 Tm \u03b1\n   -- \u019b    : \u2200 {\u03b1} \u2192 (t : \u2200 {b} \u2192 Tm (b \u25c5 \u03b1) \u2192 Tm (b \u25c5 \u03b1)) \u2192 Tm \u03b1\n   -- Let  : \u2200 {\u03b1} \u2192 (t : Tm \u03b1) (u : \u2200 {b} \u2192 Tm (b \u25c5 \u03b1) \u2192 Tm (b \u25c5 \u03b1)) \u2192 Tm \u03b1\n   -- `_   : \u2200 {\u03b1} \u2192 (\u03ba : Cst) \u2192 Tm \u03b1\n\n   coerce\u2122 : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u2286 \u0393\u2082 \u2192 \u0393\u2081 \u22a2 \u03c4 \u2192 \u0393\u2082 \u22a2 \u03c4\n   shift\u2122 : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 +1 \u2286 \u0393\u2082 \u2192 \u0393\u2081 \u22a2 \u03c4 \u2192 \u0393\u2082 \u22a2 \u03c4\n\n  weaken\u2122 : \u2200 {\u0393 \u03c3 \u03c4} \u2192 \u0393 \u22a2 \u03c4 \u2192 \u0393 \u21911\u27e8 \u03c3 \u27e9 \u22a2 \u03c4\n  weaken\u2122 = shift\u2122 \u2286-+1\u21911\n\n\n{-\n   \u2286-cong-\u21911 : \u2200 {\u03b1 \u03b2}\n                  (\u03b1\u2286\u03b2 : \u03b1 \u2286 \u03b2)\n                \u2192 \u03b1 \u21911 \u2286 \u03b2 \u21911\n-}\n\n module Examples\u1d3a (sym : Sym) where\n  open Sym sym\n\n  id\u2122 : \u2200 {\u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 \u03c4)\n  id\u2122 = \u019b\u1d3a (\u03bb _ x \u2192 x)\n\n  true\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 (\u03c3 \u21d2 \u03c4))\n  true\u2122 = \u019b\u1d3a (\u03bb _ x \u2192 \u019b\u1d3a (\u03bb y# y \u2192 coerce\u2122 (\u2286-# y#) x))\n\n  false\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c3 \u21d2 (\u03c4 \u21d2 \u03c4))\n  false\u2122 = \u019b\u1d3a (\u03bb _ x \u2192 \u019b\u1d3a (\u03bb _ y \u2192 y))\n\n module Examples\u1d30 (sym : Sym) where\n  open Sym sym\n\n  id\u2122 : \u2200 {\u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 \u03c4)\n  id\u2122 = \u019b\u1d30 (\u03bb x \u2192 x)\n\n  true\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c4 \u21d2 (\u03c3 \u21d2 \u03c4))\n  true\u2122 = \u019b\u1d30 (\u03bb x \u2192 \u019b\u1d30 (\u03bb y \u2192 weaken\u2122 x))\n\n  false\u2122 : \u2200 {\u03c3 \u03c4} \u2192 \u03b5 \u22a2 (\u03c3 \u21d2 (\u03c4 \u21d2 \u03c4))\n  false\u2122 = \u019b\u1d30 (\u03bb x \u2192 \u019b\u1d30 (\u03bb y \u2192 y))\n\nmodule M3 where\n  -- open module F {M : Set \u2192 Set} {{x}} = Functor {_} {M} x\n  -- open module A {M : Set \u2192 Set} {{x}} = Applicative {_} {M} x\n\n  record HA (Repr : Set \u2192 Set) (S A : Set) : Set where\n    constructor mk\n    field\n      ha : Repr A\n\n  href : \u2200 {Repr S A H} \u2192 HV (HA Repr S A \u00d7 H) Repr A \n  href (mk x , h) = x \n\n  -- dup\n  LamM : Set\u2081\n  LamM = \u2200 {M Repr : Set \u2192 Set}{A B H} {{M-fun : Functor M}} {{sSym : SimpleSym Repr}}\n              {{\u019b\u1d3e : LamPure Repr}} \u2192 (\u2200 {S} \u2192 HV (HA Repr S A \u00d7 H) Repr A \u2192 M (HV (HA Repr S A \u00d7 H) Repr B))\n            \u2192 M (HV H Repr (A \u2192 B))\n\n  lam : LamM\n  lam {{M-fun}} {{sym}} {{\u019b\u1d3e}} f = _<$>_ (\u03bb body h \u2192 \u019b\u1d3e (\u03bb x \u2192 body (mk x , h))) (f {\u22a4} href) where\n    open SimpleSym _ sym\n    open Functor M-fun\n\n  var : \u2200 {H M Repr A} {{M-app : Applicative M}} \u2192 HV H Repr A \u2192 M (HV H Repr A)\n  var = Applicative.pure \u2026\n\n  weaken : \u2200 {H H\u2032 A : Set} {M Repr : Set \u2192 Set} {{M-app : Functor M}} \u2192 M (HV H Repr A) \u2192 M (HV (H\u2032 \u00d7 H) Repr A)\n  weaken = Functor._<$>_ \u2026 (\u03bb g \u2192 g \u2218 snd) -- hmap goes yellow\n\n  lib : Lib\n  lib = mk HA lam href\n           (\u03bb {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084} \u2192 var {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084}) -- FIXME: unquote (\u03b7\u207f (quote var))\n           (\u03bb {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084} {\u03b7\u2085} \u2192 weaken {\u03b7\u2081} {\u03b7\u2082} {\u03b7\u2083} {\u03b7\u2084} {\u03b7\u2085})\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"27d30eeac98b6cb21e267cddc3cc7b991fb3a144","subject":"pisearch: \u03a0 is in Function.NP","message":"pisearch: \u03a0 is in Function.NP\n","repos":"crypto-agda\/crypto-agda","old_file":"pisearch.agda","new_file":"pisearch.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function.NP\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nTree : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nTree sA B = sA _\u00d7_ B\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Tree sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Tree sA B\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Tree s _) _,_\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ toFunA toFunB t = uncurry (toFunB \u2218 toFunA t)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ fromFunA fromFunB f = fromFunA (fromFunB \u2218 curry f)\n\nopen import Relation.Binary.PropositionalEquality\nToFrom : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nToFrom {A} sA toFunA fromFunA = \u2200 {B} (f : \u03a0 A B) x \u2192 toFunA (fromFunA f) x \u2261 f x\n\nFromTo : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nFromTo sA toFunA fromFunA = \u2200 {B} (t : Tree sA B) \u2192 fromFunA (toFunA t) \u2261 t\n\nmodule \u03a3-props {A} {B : A \u2192 \u2605}\n                (\u03bcA : Searchable A) (\u03bcB : \u2200 {x} \u2192 Searchable (B x)) where\n    sA = search \u03bcA\n    sB : \u2200 {x} \u2192 Search (B x)\n    sB {x} = search (\u03bcB {x})\n    fromFunA : FromFun sA\n    fromFunA = fromFun-searchInd (search-ind \u03bcA)\n    fromFunB : \u2200 {x} \u2192 FromFun (sB {x})\n    fromFunB {x} = fromFun-searchInd (search-ind (\u03bcB {x}))\n    module ToFrom\n             (toFunA : ToFun sA)\n             (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n             (toFromA : ToFrom sA toFunA fromFunA)\n             (toFromB : \u2200 {x} \u2192 ToFrom (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : ToFrom (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 f (x , y) rewrite toFromA (fromFunB \u2218 curry f) x = toFromB (curry f x) y\n\n    {- we need a search-ind-ext...\n    module FromTo\n                 (toFunA : ToFun sA)\n                 (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n                 (toFromA : FromTo sA toFunA fromFunA)\n                 (toFromB : \u2200 {x} \u2192 FromTo (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : FromTo (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 t = {!toFromA!} -- {!(\u03bb x \u2192 toFromB (toFunA t x))!}\n    -}\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB toFunA toFunB = toFun-\u03a3 sA sB toFunA toFunB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB fromFunA fromFunB = fromFun-\u03a3 sA sB fromFunA fromFunB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2081 x) = toFunA t x\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2082 x) = toFunB u x\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB fromFunA fromFunB f = fromFunA (f \u2218 inj\u2081) , fromFunB (f \u2218 inj\u2082)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n","old_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nTree : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nTree sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605) \u2192 (B : A \u2192 \u2605) \u2192 \u2605\n\u03a0 A B = (x : A) \u2192 B x\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Tree sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Tree sA B\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Tree s _) _,_\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ toFunA toFunB t = uncurry (toFunB \u2218 toFunA t)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ fromFunA fromFunB f = fromFunA (fromFunB \u2218 curry f)\n\nopen import Relation.Binary.PropositionalEquality\nToFrom : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nToFrom {A} sA toFunA fromFunA = \u2200 {B} (f : \u03a0 A B) x \u2192 toFunA (fromFunA f) x \u2261 f x\n\nFromTo : \u2200 {A} (sA : Search A)\n           (toFunA : ToFun sA)\n           (fromFunA : FromFun sA)\n         \u2192 \u2605\nFromTo sA toFunA fromFunA = \u2200 {B} (t : Tree sA B) \u2192 fromFunA (toFunA t) \u2261 t\n\nmodule \u03a3-props {A} {B : A \u2192 \u2605}\n                (\u03bcA : Searchable A) (\u03bcB : \u2200 {x} \u2192 Searchable (B x)) where\n    sA = search \u03bcA\n    sB : \u2200 {x} \u2192 Search (B x)\n    sB {x} = search (\u03bcB {x})\n    fromFunA : FromFun sA\n    fromFunA = fromFun-searchInd (search-ind \u03bcA)\n    fromFunB : \u2200 {x} \u2192 FromFun (sB {x})\n    fromFunB {x} = fromFun-searchInd (search-ind (\u03bcB {x}))\n    module ToFrom\n             (toFunA : ToFun sA)\n             (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n             (toFromA : ToFrom sA toFunA fromFunA)\n             (toFromB : \u2200 {x} \u2192 ToFrom (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : ToFrom (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 f (x , y) rewrite toFromA (fromFunB \u2218 curry f) x = toFromB (curry f x) y\n\n    {- we need a search-ind-ext...\n    module FromTo\n                 (toFunA : ToFun sA)\n                 (toFunB : \u2200 {x} \u2192 ToFun (sB {x}))\n                 (toFromA : FromTo sA toFunA fromFunA)\n                 (toFromB : \u2200 {x} \u2192 FromTo (sB {x}) toFunB fromFunB) where\n        toFrom-\u03a3 : FromTo (\u03a3Search sA (\u03bb {x} \u2192 sB {x})) (toFun-\u03a3 sA sB toFunA toFunB) (fromFun-\u03a3 sA sB fromFunA fromFunB)\n        toFrom-\u03a3 t = {!toFromA!} -- {!(\u03bb x \u2192 toFromB (toFunA t x))!}\n    -}\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB toFunA toFunB = toFun-\u03a3 sA sB toFunA toFunB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB fromFunA fromFunB = fromFun-\u03a3 sA sB fromFunA fromFunB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2081 x) = toFunA t x\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2082 x) = toFunB u x\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB fromFunA fromFunB f = fromFunA (f \u2218 inj\u2081) , fromFunB (f \u2218 inj\u2082)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fcd66503bf65118ad5fda83327897a1b2b6fd85e","subject":"Minor changes to a test.","message":"Minor changes to a test.\n\nIgnore-this: 7b6fca5f08d62b3a275c60659cb5385b\n\ndarcs-hash:20100730043722-3bd4e-d33f283f3cb4b7f6d96c7b651cffce8783b5b2cc.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/Definitions.agda","new_file":"Test\/Succeed\/Definitions.agda","new_contents":"module Test.Succeed.Definitions where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\nP : D \u2192 Set\nP i = zero + i \u2261 i\n{-# ATP definition P #-}\n\npostulate\n  P0 : P zero\n{-# ATP prove P0 #-}\n\npostulate\n  iStep : {i : D} \u2192 N i \u2192 P i \u2192 P (succ i)\n{-# ATP prove iStep #-}\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity = indN (\u03bb i \u2192 P i) P0 iStep\n","old_contents":"module Test.Succeed.Definitions where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  add-x0 : (n : D) \u2192 n + zero     \u2261 n\n  add-xS : (m n : D) \u2192 m + succ n \u2261 succ (m + n)\n{-# ATP axiom add-x0 #-}\n{-# ATP axiom add-xS #-}\n\nP : D \u2192 Set\nP i = zero + i \u2261 i\n{-# ATP definition P #-}\n\npostulate\n  P0 : P zero\n{-# ATP prove P0 #-}\n\npostulate\n  iStep : {i : D} \u2192 N i \u2192 P i \u2192 P (succ i)\n{-# ATP prove iStep #-}\n\naddLeftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\naddLeftIdentity = indN (\u03bb i \u2192 P i) P0 iStep\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0a73b4bf215c087bbfe6a7f0207b9e8472ce9a05","subject":"ElGamal: functional correctness","message":"ElGamal: functional correctness\n","repos":"crypto-agda\/crypto-agda","old_file":"Cipher\/ElGamal\/Generic.agda","new_file":"Cipher\/ElGamal\/Generic.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Cipher.ElGamal.Generic\n  (Message : \u2605)\n  (\u2124q      : \u2605)\n  (G       : \u2605)\n  (g       : G)\n  (_^_     : G \u2192 \u2124q \u2192 G)\n\n  -- Required for encryption\n  (_\u2666_     : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_     : Message \u2192 G \u2192 Message)\nwhere\n\nPubKey     = G\nSecKey     = \u2124q\nCipherText = G \u00d7 Message\nR\u2096         = \u2124q\nR\u2091         = \u2124q\n\nPubKeyGen : R\u2096 \u2192 PubKey\nPubKeyGen x = g ^ x\n\nKeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey\nKeyGen x = (g ^ x , x)\n\nEnc : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\nEnc g\u02e3 m y = g\u02b8 , \u03b6 where\n  g\u02b8 = g  ^ y\n  \u03b4  = g\u02e3 ^ y\n  \u03b6  = \u03b4  \u2666 m\n\nDec : SecKey \u2192 CipherText \u2192 Message\nDec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n\nmodule FunctionalCorrectness\n    (\/-\u2666    : \u2200 {x y} \u2192 (x \u2666 y) \/ x \u2261 y)\n    (comm-^ : \u2200 {\u03b1 x y} \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    where\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g ^ x) m y) \u2261 m\n    functional-correctness x y m = ap Ctx comm-^ \u2219 \/-\u2666\n      where Ctx = \u03bb z \u2192 (z \u2666 m) \/ ((g ^ y)^ x)\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Product\n\nmodule Cipher.ElGamal.Generic\n  (Message : \u2605)\n  (\u2124q      : \u2605)\n  (G       : \u2605)\n  (g       : G)\n  (_^_     : G \u2192 \u2124q \u2192 G)\n\n  -- Required for encryption\n  (_\u2219_     : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_     : Message \u2192 G \u2192 Message)\nwhere\n\nPubKey     = G\nSecKey     = \u2124q\nCipherText = G \u00d7 Message\nR\u2096         = \u2124q\nR\u2091         = \u2124q\n\nKeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey\nKeyGen x = (g ^ x , x)\n\nEnc : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\nEnc g\u02e3 m y = g\u02b8 , \u03b6 where\n  g\u02b8 = g  ^ y\n  \u03b4  = g\u02e3 ^ y\n  \u03b6  = \u03b4  \u2219 m\n\nDec : SecKey \u2192 CipherText \u2192 Message\nDec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"37bd60875400557d90e8376abde43fd1c27e2afe","subject":"Adapt Data.Bits.OperationSyntax","message":"Adapt Data.Bits.OperationSyntax\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits\/OperationSyntax.agda","new_file":"lib\/Data\/Bits\/OperationSyntax.agda","new_contents":"-- NOTE with-K\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Two hiding (_==_)\n\n-- TODO get rid of IFs\nopen import Data.Bool using (if_then_else_)\n\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Vec.NP\nopen import Data.Vec.Bijection\nopen import Data.Vec.Permutation\n\nopen import Function.Bijection.SyntaxKit\nopen import Function.NP\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule Data.Bits.OperationSyntax where\n\nmodule \ud835\udfdaBij = \ud835\udfdaBijection\nopen \ud835\udfdaBij public using (`id; \ud835\udfdaBij) renaming (bool-bijKit to bitBijKit; `not to `not\u1d2e)\nopen BijectionSyntax Bit \ud835\udfdaBij public\nopen BijectionSemantics bitBijKit public\n\n{-\n  id    \u03b5          id\n  0\u21941   swp-inners interchange\n  not   swap       comm\n  when0 first      ...\n  when1 second     ...\n-}\n`not : \u2200 {n} \u2192 Bij (1 + n)\n`not = \ud835\udfdaBij.`not `\u2237 const `id\n\n`xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n`xor b = \ud835\udfdaBij.`xor b `\u2237 const `id\n\n`[0:_1:_] : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n`[0: f 1: g ] = \ud835\udfdaBij.`id `\u2237 [0: f 1: g ]\n\n`when0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n`when0 f = `[0: f    1: `id ]\n\n`when1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n`when1 f = `[0: `id  1:   f ]\n\n-- law: `when0 f `\u204f `when1 g \u2261 `when1 g `; `when0 f\n\non-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\non-firsts f = `0\u21941 `\u204f `when0 f `\u204f `0\u21941\n\n--   (a \u00b7 b) \u00b7 (c \u00b7 d)\n-- \u2261 right swap\n--   (a \u00b7 b) \u00b7 (d \u00b7 c)\n-- \u2261 interchange\n--   (a \u00b7 d) \u00b7 (b \u00b7 c)\n-- \u2261 right swap\n--   (a \u00b7 d) \u00b7 (c \u00b7 b)\nswp-seconds : \u2200 {n} \u2192 Bij (2 + n)\nswp-seconds = `when1 `not `\u204f `0\u21941 `\u204f `when1 `not\n\non-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n-- on-extremes f = swp-seconds `\u204f `when0 f `\u204f swp-seconds\n\n-- (a \u00b7 b) \u00b7 (c \u00b7 d)\n-- \u2261 right swap \u2261 when1 not\n-- (a \u00b7 b) \u00b7 (d \u00b7 c)\n-- \u2261 interchange \u2261 0\u21941\n-- (a \u00b7 d) \u00b7 (b \u00b7 c)\n-- \u2261 left f \u2261 when0 f\n-- (A \u00b7 D) \u00b7 (b \u00b7 c)\n--     where A \u00b7 D = f (a \u00b7 d)\n-- \u2261 interchange \u2261 0\u21941\n-- (A \u00b7 b) \u00b7 (D \u00b7 c)\n-- \u2261 right swap \u2261 when1 not\n-- (A \u00b7 b) \u00b7 (c \u00b7 D)\non-extremes f = `when1 `not `\u204f `0\u21941 `\u204f `when0 f `\u204f `0\u21941 `\u204f `when1 `not\n\nmap-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\nmap-inner f = `when1 `not `\u204f `0\u21941 `\u204f `when1 f `\u204f `0\u21941 `\u204f `when1 `not\n\nmap-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\nmap-outer = `[0:_1:_]\n\n0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n0\u21941\u2237 [] = `not\n0\u21941\u2237 (0\u2082 \u2237 p) = on-firsts   (0\u21941\u2237 p)\n0\u21941\u2237 (1\u2082 \u2237 p) = on-extremes (0\u21941\u2237 p)\n\n0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n0\u2194 [] = `id\n0\u2194 (0\u2082 \u2237 p) = `when0 (0\u2194 p)\n0\u2194 (1\u2082 \u2237 p) = 0\u21941\u2237 p\n\n\u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n\u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\nif\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\nif\u2237 1\u2082 x xs ys = refl\nif\u2237 0\u2082 x xs ys = refl\n\nif-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (0\u2082 \u2237 xs) else (1\u2082 \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\nif-not\u2237 1\u2082 xs ys = refl\nif-not\u2237 0\u2082 xs ys = refl\n\nif\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (1\u2082 \u2237 xs) else (0\u2082 \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\nif\u2237\u2032 1\u2082 xs ys = refl\nif\u2237\u2032 0\u2082 xs ys = refl\n\n\u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u00b7 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n\u27e80\u21941\u2237_\u27e9-spec [] (1\u2082 \u2237 []) = refl\n\u27e80\u21941\u2237_\u27e9-spec [] (0\u2082 \u2237 []) = refl\n\u27e80\u21941\u2237_\u27e9-spec (1\u2082 \u2237 ps) (1\u2082 \u2237 1\u2082 \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n     with ps == xs\n... | 1\u2082 = refl\n... | 0\u2082 = refl\n\u27e80\u21941\u2237_\u27e9-spec (1\u2082 \u2237 ps) (1\u2082 \u2237 0\u2082 \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (1\u2082 \u2237 ps) (0\u2082 \u2237 1\u2082 \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (1\u2082 \u2237 ps) (0\u2082 \u2237 0\u2082 \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n     with 0\u207f == xs\n... | 1\u2082 = refl\n... | 0\u2082 = refl\n\u27e80\u21941\u2237_\u27e9-spec (0\u2082 \u2237 ps) (1\u2082 \u2237 1\u2082 \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (0\u2082 \u2237 ps) (1\u2082 \u2237 0\u2082 \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n     with ps == xs\n... | 1\u2082 = refl\n... | 0\u2082 = refl\n\u27e80\u21941\u2237_\u27e9-spec (0\u2082 \u2237 ps) (0\u2082 \u2237 1\u2082 \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (0\u2082 \u2237 ps) (0\u2082 \u2237 0\u2082 \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n     with 0\u207f == xs\n... | 1\u2082 = refl\n... | 0\u2082 = refl\n\n\u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u00b7 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n\u27e80\u2194_\u27e9-spec [] [] = refl\n\u27e80\u2194_\u27e9-spec (0\u2082 \u2237 ps) (1\u2082 \u2237 xs) = refl\n\u27e80\u2194_\u27e9-spec (0\u2082 \u2237 ps) (0\u2082 \u2237 xs)\n  rewrite \u27e80\u2194 ps \u27e9-spec xs\n          | if\u2237 (ps == xs) 0b 0\u207f xs\n          | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n          = refl\n\u27e80\u2194_\u27e9-spec (1\u2082 \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\nprivate\n  module P where\n    open PermutationSyntax public\n    open PermutationSemantics public\nopen P using (Perm; `id; `0\u21941; _`\u204f_)\n\n`\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n`\u27e80\u21941+ zero  \u27e9 = `0\u21941\n`\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n`\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u00b7 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n`\u27e80\u21941+ zero  \u27e9-spec xs = refl\n`\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n`\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n`\u27e80\u2194 zero  \u27e9 = `id\n`\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n`\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u00b7 xs \u2261 \u27e80\u2194 i \u27e9 xs\n`\u27e80\u2194 zero  \u27e9-spec xs = refl\n`\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n{-\n`\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n`\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n`\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n`\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n`\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u00b7 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n`\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n`\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n`\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n-}\n\n`xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n`xor-head = [0: `id   1: `not ]\n\n`xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u00b7 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n`xor-head-spec 1\u2082 x xs  = refl\n`xor-head-spec 0\u2082 x xs = refl\n\n`\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n`\u27e8 []     \u2295\u27e9 = `id\n`\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n`\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u00b7 ys \u2261 xs \u2295 ys\n`\u27e8 []         \u2295\u27e9-spec []       = refl\n`\u27e8 1\u2082  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n`\u27e8 0\u2082 \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n\u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n\u2295-dist-0\u21941 _           []          = refl\n\u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bool.NP hiding (_==_)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Vec.NP\n\nopen import Function.Bijection.SyntaxKit\nopen import Function.NP\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule Data.Bits.OperationSyntax where\n\nmodule BitBij = BoolBijection\nopen BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\nopen BijectionSyntax Bit BitBij public\nopen BijectionSemantics bitBijKit public\n\n{-\n  id   \u03b5          id\n  0\u21941  swp-inners interchange\n  not  swap       comm\n  if0  first      ...\n  if1  second     ...\n-}\n`not : \u2200 {n} \u2192 Bij (1 + n)\n`not = BitBij.`not `\u2237 const `id\n\n`xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n`xor b = BitBij.`xor b `\u2237 const `id\n\n`if : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n`if f g = BitBij.`id `\u2237 cond f g\n\n`if0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n`if0 f = `if `id f\n\n`if1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n`if1 f = `if f `id\n\n-- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\non-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\non-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n--   (a \u2219 b) \u2219 (c \u2219 d)\n-- \u2261 right swap\n--   (a \u2219 b) \u2219 (d \u2219 c)\n-- \u2261 interchange\n--   (a \u2219 d) \u2219 (b \u2219 c)\n-- \u2261 right swap\n--   (a \u2219 d) \u2219 (c \u2219 b)\nswp-seconds : \u2200 {n} \u2192 Bij (2 + n)\nswp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\non-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n-- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n-- (a \u2219 b) \u2219 (c \u2219 d)\n-- \u2261 right swap \u2261 if1 not\n-- (a \u2219 b) \u2219 (d \u2219 c)\n-- \u2261 interchange \u2261 0\u21941\n-- (a \u2219 d) \u2219 (b \u2219 c)\n-- \u2261 left f \u2261 if0 f\n-- (A \u2219 D) \u2219 (b \u2219 c)\n--     where A \u2219 D = f (a \u2219 d)\n-- \u2261 interchange \u2261 0\u21941\n-- (A \u2219 b) \u2219 (D \u2219 c)\n-- \u2261 right swap \u2261 if1 not\n-- (A \u2219 b) \u2219 (c \u2219 D)\non-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\nmap-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\nmap-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\nmap-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\nmap-outer f g = `if g f\n\n0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n0\u21941\u2237 [] = `not\n0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n0\u2194 [] = `id\n0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n\u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n\u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\nif\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\nif\u2237 true x xs ys = refl\nif\u2237 false x xs ys = refl\n\nif-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\nif-not\u2237 true xs ys = refl\nif-not\u2237 false xs ys = refl\n\nif\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\nif\u2237\u2032 true xs ys = refl\nif\u2237\u2032 false xs ys = refl\n\n\u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n\u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n\u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n\u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n     with ps == xs\n... | true = refl\n... | false = refl\n\u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n     with 0\u207f == xs\n... | true = refl\n... | false = refl\n\u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n     with ps == xs\n... | true = refl\n... | false = refl\n\u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n\u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n   rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n     with 0\u207f == xs\n... | true = refl\n... | false = refl\n\n\u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n\u27e80\u2194_\u27e9-spec [] [] = refl\n\u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n\u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n  rewrite \u27e80\u2194 ps \u27e9-spec xs\n          | if\u2237 (ps == xs) 0b 0\u207f xs\n          | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n          = refl\n\u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\nprivate\n  module P where\n    open PermutationSyntax public\n    open PermutationSemantics public\nopen P using (Perm; `id; `0\u21941; _`\u204f_)\n\n`\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n`\u27e80\u21941+ zero  \u27e9 = `0\u21941\n`\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n`\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n`\u27e80\u21941+ zero  \u27e9-spec xs = refl\n`\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n`\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n`\u27e80\u2194 zero  \u27e9 = `id\n`\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n`\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n`\u27e80\u2194 zero  \u27e9-spec xs = refl\n`\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n{-\n`\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n`\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n`\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n`\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n`\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n`\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n`\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n`\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n-}\n\n`xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n`xor-head b = if b then `not else `id\n\n`xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n`xor-head-spec true x xs  = refl\n`xor-head-spec false x xs = refl\n\n`\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n`\u27e8 []     \u2295\u27e9 = `id\n`\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n`\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n`\u27e8 []         \u2295\u27e9-spec []       = refl\n`\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n`\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n\u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n\u2295-dist-0\u21941 _           []          = refl\n\u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4949f9058a6fb480bcce92b2259a9d154a7abc47","subject":"filling in ealamhole rule","message":"filling in ealamhole rule\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- not a hole\n\n  data _not-hole : dhexp \u2192 Set where\n    CNotHole : c not-hole\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- not a hole\n\n  data _not-hole : dhexp \u2192 Set where\n    CNotHole : c not-hole\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x  e d \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              {!!} \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ {!!}  ] d :: {!!} \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d135e593fa01a535210210ad2c6a558688023599","subject":"Nat: renaming","message":"Nat: renaming\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k \u27e9* m \u2261 \u27e82^ k \u27e9* n \u2192 m \u2261 n\n2^-inj zero    eq = eq\n2^-inj (suc k) eq = 2^-inj k (2*-inj eq)\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n\u27e82^_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ zero  \u27e9* = id\n\u27e82^ suc k \u27e9* = 2*_ \u2218 \u27e82^ k \u27e9*\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k \u27e9* (x + y) \u2261 \u27e82^ k \u27e9* x + \u27e82^ k \u27e9* y\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib (\u27e82^ k \u27e9* x) (\u27e82^ k \u27e9* y))\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k \u27e9* (2* x) \u2261 2* (\u27e82^ k \u27e9* x)\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist (\u27e82^ x \u27e9* y) (\u27e82^ x \u27e9* z) \u2261 \u27e82^ x \u27e9* (dist y z)\ndist-2^* x y z = dist-sym-wlog \u27e82^ x \u27e9* (pf x) y z\n  where pf : \u2200 x a k \u2192 dist (\u27e82^ x \u27e9* a) (\u27e82^ x \u27e9* (a + k)) \u2261 \u27e82^ x \u27e9* k\n        pf x a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y (\u27e82^ x \u27e9* a) (\u27e82^ x \u27e9* k)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k \u27e9* a \u2264 \u27e82^ k \u27e9* b\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k \u27e9* a \u2264 \u27e82^ k \u27e9* b \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x \u27e9* (\u27e82^ y \u27e9* z) \u2261 \u27e82^ y \u27e9* (\u27e82^ x \u27e9* z)\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y (\u27e82^ x \u27e9* z))\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x \u27e9* (\u27e82^ y \u27e9* z) \u2261 \u27e82^ x + y \u27e9* z\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k \u27e9* m \u2261 \u27e82^ k \u27e9* n \u2192 m \u2261 n\n2^-inj zero    eq = eq\n2^-inj (suc k) eq = 2^-inj k (2*-inj eq)\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n \u27e9* 1\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e9edc47e1e4dfd23330e7436b76e5a8cd65dc3c3","subject":"horsing around with 312 ABT style approach to proving substitution, where the renamings happen at the last second when you need them and sort of under the hood. i dont want to spend a lot of time on this because it seems like it might not actually go through, but if it does it wouldnt change the type of preservation so that would be good","message":"horsing around with 312 ABT style approach to proving substitution, where the renamings happen at the last second when you need them and sort of under the hood. i dont want to spend a lot of time on this because it seems like it might not actually go through, but if it does it wouldnt change the type of preservation so that would be good\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  -- there is always a natural number that's fresh for any expression,\n  -- possibly avoiding one in particular\n  exists-fresh : (d : dhexp) \u2192 \u03a3[ y \u2208 Nat ] (fresh y d)\n  exists-fresh = {!!}\n\n  -- rename all x's into y's in d to produce d'\n  data rename : (x : Nat) \u2192 (y : Nat) \u2192 (d : dhexp) (d' : dhexp) \u2192 Set where\n    RConst       : \u2200{x y} \u2192 rename x y c c\n    RVar1        : \u2200{x y z} \u2192 x \u2260 z \u2192 rename x y (X z) (X z)\n    RVar2        : \u2200{x y} \u2192 rename x y (X x) (X y)\n  --  RLam         : \u2200{x y \u03c4 e} \u2192 rename x y ? ?\n    -- REHole\n    -- RNEHole\n    -- RAp\n    -- RCast\n    -- RFailedCast\n\n  -- have to do something about the context in the typing judgement as well\n  -- and probably in the \u03c3s. bleh.\n\n  exists-rename : (x y : Nat) (d : dhexp) \u2192 \u03a3[ d' \u2208 dhexp ](rename x y d d')\n  exists-rename z y c = c , RConst\n  exists-rename z y (X x) with natEQ z x\n  exists-rename z y (X .z) | Inl refl = X y , RVar2\n  exists-rename z y (X x) | Inr x\u2081 = X x , RVar1 x\u2081\n  exists-rename z y (\u00b7\u03bb x [ x\u2081 ] d) = {!!}\n  exists-rename z y \u2987\u2988\u27e8 x \u27e9 = {!!}\n  exists-rename z y \u2987 d \u2988\u27e8 x \u27e9 = {!!}\n  exists-rename z y (d \u2218 d\u2081) = {!!}\n  exists-rename z y (d \u27e8 x \u21d2 x\u2081 \u27e9) = {!!}\n  exists-rename z y (d \u27e8 x \u21d2\u2987\u2988\u21d2\u0338 x\u2081 \u27e9) = {!!}\n\n  rename-ctx : {A : Set} \u2192 (x y : Nat) \u2192 (\u0393 : A ctx) \u2192 A ctx\n  rename-ctx x y \u0393 q with natEQ y q -- if you're asking about the new variable y,  it's now gonna be  whatever x was\n  rename-ctx x y \u0393 q | Inl x\u2081 = \u0393 x\n  rename-ctx x y \u0393 q | Inr x\u2081 = \u0393 q\n\n  -- is this the right direction? try to use it below; need that y # \u0393, likely\n  rename-preserve : \u2200{x y d d' \u0394 \u0393 \u03c4} \u2192 (f : fresh y d) \u2192 rename x y d d' \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192 \u0394 , (rename-ctx x y \u0393) \u22a2 d' :: \u03c4\n  rename-preserve FConst RConst TAConst = TAConst\n  rename-preserve (FVar x\u2082) (RVar1 x\u2081) (TAVar x\u2083) = TAVar {!!}\n  rename-preserve (FVar x\u2082) RVar2 (TAVar x\u2083) = TAVar {!!}\n  rename-preserve (FLam x\u2082 f) re (TALam x\u2083 wt) = {!!}\n  rename-preserve (FHole x\u2082) re (TAEHole x\u2083 x\u2084) = {!!}\n  rename-preserve (FNEHole x\u2082 f) re (TANEHole x\u2083 wt x\u2084) = {!!}\n  rename-preserve (FAp f f\u2081) re (TAAp wt wt\u2081) = {!!}\n  rename-preserve (FCast f) re (TACast wt x\u2082) = {!!}\n  rename-preserve (FFailedCast f) re (TAFailedCast wt x\u2082 x\u2083 x\u2084) = {!!}\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _ with exists-fresh d2\n  ... | z , neq = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n\n  -- possible fix: add a premise from a new judgement like holes disjoint\n  -- that classifies pairs of dhexps that share no variable names\n  -- whatsoever. that should imply freshness here. then propagate that\n  -- change to preservation. this is morally what \u03b1-equiv lets us do in a\n  -- real setting.\n\n\n  -- lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n  --   with lem-union-none {\u0393 = \u0393} x\u2082\n  -- ... |  x\u2260y , y#\u0393 with natEQ y x\n  -- ... | Inl eq = abort (x\u2260y (! eq))\n  -- ... | Inr _ with exists-fresh d2 None\n  -- ... | z , neq = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _ = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n\n  -- possible fix: add a premise from a new judgement like holes disjoint\n  -- that classifies pairs of dhexps that share no variable names\n  -- whatsoever. that should imply freshness here. then propagate that\n  -- change to preservation. this is morally what \u03b1-equiv lets us do in a\n  -- real setting.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f3003ac296a5a819510afb4fff39156c8247f0b5","subject":"Desc model: implicit refl, cong, cong2, reflFun, desc, and IMu","message":"Desc model: implicit refl, cong, cong2, reflFun, desc, and IMu","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box I D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box Unit (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : (A : Set)(B : Set)(f : A -> B)(x y : A) -> x == y -> f x == f y\ncong A B f x .x refl = refl\n\ncong2 : (A B C : Set)(f : A -> B -> C)(x y : A)(z t : B) -> \n        x == y -> z == t -> f x z == f y t\ncong2 A B C f x .x z .z refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : (A : Set)(B : Set)(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc Unit (descD I) (IMu Unit (\u03bb _ -> descD I)))\n        (hs : desc (Sigma Unit (IMu Unit (\u03bb _ -> descD I)))\n              (box Unit (descD I) (IMu Unit (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 (IDesc I) \n                                                (IDesc I)\n                                                (IDesc I) \n                                                prod \n                                                (iso1 I (iso2 I D))\n                                                D \n                                                (iso1 I (iso2 I D'))\n                                                D' p q\nproof-iso1-iso2 I (pi S T) = cong (S \u2192 IDesc I)\n                                  (IDesc I)\n                                  (pi S) \n                                  (\\x -> iso1 I (iso2 I (T x)))\n                                  T \n                                  (reflFun S (IDesc I)\n                                             (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                             T\n                                             (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (S \u2192 IDesc I)\n                                     (IDesc I)\n                                     (sigma S) \n                                     (\\x -> iso1 I (iso2 I (T x)))\n                                     T \n                                     (reflFun S \n                                              (IDesc I)\n                                              (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- induction : (I : Set)\n--             (R : I -> IDesc I)\n--             (P : Sigma I (IMu I R) -> Set)\n--             (m : (i : I)\n--                  (xs : desc I (R i) (IMu I R))\n--                  (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n--                  P ( i , con xs)) ->\n--             (i : I)(x : IMu I R i) -> P ( i , x )\n\n\nP : (I : Set) -> Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (IDescl I) \n                                          (IDescl I) \n                                          (IDescl I)\n                                          (prodl I) \n                                          (iso2 I (iso1 I D))\n                                          D \n                                          (iso2 I (iso1 I D'))\n                                          D' p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                          (IDescl I)\n                                                                          (pil I S) \n                                                                          (\\x -> iso2 I (iso1 I (T x)))\n                                                                          T \n                                                                          (reflFun S \n                                                                                   (IDescl I)\n                                                                                   (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                             (IDescl I)\n                                                                             (sigmal I S) \n                                                                             (\\x -> iso2 I (iso1 I (T x)))\n                                                                             T \n                                                                             (reflFun S \n                                                                                      (IDescl I)\n                                                                                      (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"21354267de027c28eef0e195f783de60ff623a81","subject":"Use instance argument for iota-subscript, diaeresis and final form.","message":"Use instance argument for iota-subscript, diaeresis and final form.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-final \u2984 \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\ninstance \u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota smooth\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota rough\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave smooth\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave rough\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute smooth\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute rough\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex smooth\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex rough\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota smooth\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota rough\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave smooth\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave rough\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute smooth\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute rough\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex smooth\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex rough\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a) = just (letter-to-accent a)\nget-accent (with-accent-iota a) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis with-diaeresis = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form final = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4282673d8da7771de7f995705e4999074f98cbad","subject":"Added x+11\u223810\u2261Sx.","message":"Added x+11\u223810\u2261Sx.\n\nIgnore-this: e605e7c3f021e9be583d6affc58f9aec\n\ndarcs-hash:20110218165317-3bd4e-885efb1503edb19ed466ae8fb96f4a60342db5e6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  102>100  : GT hundred-two one-hundred\n  103>100  : GT hundred-three one-hundred\n  104>100  : GT hundred-four one-hundred\n  105>100  : GT hundred-five one-hundred\n  106>100  : GT hundred-six one-hundred\n  107>100  : GT hundred-seven one-hundred\n  108>100  : GT hundred-eight one-hundred\n  109>100  : GT hundred-nine one-hundred\n  110>100  : GT hundred-ten one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 102>100 #-}\n{-# ATP prove 103>100 #-}\n{-# ATP prove 104>100 #-}\n{-# ATP prove 105>100 #-}\n{-# ATP prove 106>100 #-}\n{-# ATP prove 107>100 #-}\n{-# ATP prove 108>100 #-}\n{-# ATP prove 109>100 #-}\n{-# ATP prove 110>100 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven) one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven) one-hundred\n  95+11>100 : GT (ninety-five + eleven) one-hundred\n  94+11>100 : GT (ninety-four + eleven) one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven) one-hundred\n  91+11>100 : GT (ninety-one + eleven) one-hundred\n  90+11>100 : GT (ninety + eleven) one-hundred\n{-# ATP prove 99+11>100 110>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101>100' 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\nx+11\u223810\u2261Sx : \u2200 {n} \u2192 N n \u2192 (n + eleven) \u2238 ten \u2261 succ n\nx+11\u223810\u2261Sx Nn = [x+Sy]\u2238y\u2261Sx Nn N10\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  102>100  : GT hundred-two one-hundred\n  103>100  : GT hundred-three one-hundred\n  104>100  : GT hundred-four one-hundred\n  105>100  : GT hundred-five one-hundred\n  106>100  : GT hundred-six one-hundred\n  107>100  : GT hundred-seven one-hundred\n  108>100  : GT hundred-eight one-hundred\n  109>100  : GT hundred-nine one-hundred\n  110>100  : GT hundred-ten one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 102>100 #-}\n{-# ATP prove 103>100 #-}\n{-# ATP prove 104>100 #-}\n{-# ATP prove 105>100 #-}\n{-# ATP prove 106>100 #-}\n{-# ATP prove 107>100 #-}\n{-# ATP prove 108>100 #-}\n{-# ATP prove 109>100 #-}\n{-# ATP prove 110>100 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven) one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven) one-hundred\n  95+11>100 : GT (ninety-five + eleven) one-hundred\n  94+11>100 : GT (ninety-four + eleven) one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven) one-hundred\n  91+11>100 : GT (ninety-one + eleven) one-hundred\n  90+11>100 : GT (ninety + eleven) one-hundred\n{-# ATP prove 99+11>100 110>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101>100' 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"82b8cfb3b9f32510182670a10fd20292bf25b946","subject":"More flat-funs combinators","message":"More flat-funs combinators\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nopen import Data.Bool using (if_then_else_)\nopen import Function using (_\u2218\u2032_)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bits using (Bit; Bits)\n\nimport bintree as Tree\nopen Tree using (Tree)\nopen import data-universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nmodule Defaults {t} {T : Set t} (\u266dFuns : FlatFuns T) where\n  open FlatFuns \u266dFuns\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultsGroup1\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    where\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n\n  infixr 9 _\u2218_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults \u266dFuns\n  open CompositionNotations _\u2218_ public\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f cons\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f cons\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f cons\n\n  <_\u2237nil> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237nil> = < f \u2237\u2032tt\u204f nil >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 `\u22a4 `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f [] = nil\n  constVec f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec f xs >\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = nil\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <nil,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <nil, f > = <tt\u204f nil , f >\n\n  <_,nil> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,nil> = < f ,tt\u204f nil >\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd \u2237nil>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <nil, id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  replicate\u22a4 : \u2200 {n _A} \u2192 _A `\u2192 `Vec `\u22a4 n\n  replicate\u22a4 {zero}  = nil\n  replicate\u22a4 {suc n} = <tt\u204f id \u2237\u2032 replicate\u22a4 >\n\n  loop : \u2200 {A} n \u2192 (A `\u2192 A) \u2192 (A `\u2192 A)\n  loop n f = < id ,tt\u204f replicate\u22a4 {n} > \u204f foldl (fst \u204f f)\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = nil \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 `Bits i `\u2192 `Bits o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n","old_contents":"module flat-funs where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nopen import Data.Bool using (if_then_else_)\nopen import Function using (_\u2218\u2032_)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bits using (Bit; Bits)\n\nimport bintree as Tree\nopen Tree using (Tree)\nopen import data-universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nmodule Defaults {t} {T : Set t} (\u266dFuns : FlatFuns T) where\n  open FlatFuns \u266dFuns\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultsGroup1\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    where\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n\n  infixr 9 _\u2218_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults \u266dFuns\n  open CompositionNotations _\u2218_ public\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f cons\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f cons\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f cons\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 `\u22a4 `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f [] = nil\n  constVec f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec f xs >\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = nil\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <nil,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <nil, f > = <tt\u204f nil , f >\n\n  <_,nil> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,nil> = < f ,tt\u204f nil >\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id ,nil> \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <nil, id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = nil \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 `Bits i `\u2192 `Bits o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = <tt\u204f fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9ef129c5aad704a019f61d4c083b35985bf81762","subject":"Typo.","message":"Typo.\n\nIgnore-this: 931e5c7dc1533eed7aa34972dc910a62\n\ndarcs-hash:20110406162735-3bd4e-7baa4b3acc9f8747cd7dcdec04e615cf9c2fee2a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/LT2MCR-ATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/LT2MCR-ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Property LT2MCR which proves that the recursive calls of the\n-- McCarthy 91 function are on smaller arguments.\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.MCR.LT2MCR-ATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\n\nopen import FOTC.Program.McCarthy91.MCR\n\n------------------------------------------------------------------------------\n\nLT2MCR-helper : \u2200 {n m k} \u2192 N n \u2192 N m \u2192 N k \u2192\n                LT m n \u2192 LT (succ n) k \u2192 LT (succ m) k \u2192\n                LT (k \u2238 n) (k \u2238 m) \u2192 LT (k \u2238 succ n) (k \u2238 succ m)\nLT2MCR-helper zN Nm Nk p qn qm h = \u22a5-elim (x<0\u2192\u22a5 Nm p)\nLT2MCR-helper (sN Nn) Nm zN p qn qm h = \u22a5-elim (x<0\u2192\u22a5 (sN Nm) qm)\nLT2MCR-helper (sN {n} Nn) zN (sN {k} Nk) p qn qm h = prfS0S\n  where\n    postulate k\u2265Sn   : GE k (succ n)\n              k\u2238Sn<k : LT (k \u2238 (succ n)) k\n              prfS0S : LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ zero)\n    {-# ATP prove k\u2265Sn x<y\u2192x\u2264y #-}\n    {-# ATP prove k\u2238Sn<k k\u2265Sn x\u2265y\u2192y>0\u2192x\u2238y<x #-}\n    {-# ATP prove prfS0S k\u2238Sn<k #-}\n\nLT2MCR-helper (sN {n} Nn) (sN {m} Nm) (sN {k} Nk) p qn qm h =\n  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm (LT2MCR-helper Nn Nm Nk m<n Sn<k Sm<k k\u2238n<k\u2238m)\n  where\n    postulate\n      k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm : LT (k \u2238 succ n) (k \u2238 succ m) \u2192\n                                LT (succ k \u2238 succ (succ n))\n                                   (succ k \u2238 succ (succ m))\n    {-# ATP prove  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm #-}\n\n    postulate\n      m<n     : LT m n\n      Sn<k    : LT (succ n) k\n      Sm<k    : LT (succ m) k\n      k\u2238n<k\u2238m : LT (k \u2238 n) (k \u2238 m)\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn<k #-}\n    {-# ATP prove Sm<k #-}\n    {-# ATP prove k\u2238n<k\u2238m #-}\n\nLT2MCR : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE m one-hundred \u2192 LT m n \u2192 MCR n m\nLT2MCR zN Nm p h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nLT2MCR (sN {n} Nn) zN p h = prfS0\n  where\n    postulate prfS0 : MCR (succ n) zero\n    {-# ATP prove prfS0 x\u2238y<Sx #-}\n\nLT2MCR (sN {n} Nn) (sN {m} Nm) p h with x<y\u2228x\u2265y Nn 100-N\n... | inj\u2081 n<100 = LT2MCR-helper Nn Nm 101-N m<n Sn\u2264101 Sm\u2264101\n                                 (LT2MCR Nn Nm m\u2264100 m<n)\n  where\n    postulate m\u2264100  : LE m one-hundred\n              m<n    : LT m n\n              Sn\u2264101 : LT (succ n) hundred-one\n              Sm\u2264101 : LT (succ m) hundred-one\n    {-# ATP prove m\u2264100 x<y\u2192x\u2264y #-}\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn\u2264101 #-}\n    {-# ATP prove Sm\u2264101 #-}\n... | inj\u2082 n\u2265100 = prf-n\u2265100\n  where\n    postulate\n      101\u2238Sn\u22610  : hundred-one \u2238 succ n \u2261 zero\n      0<101\u2238Sm  : LT zero (hundred-one \u2238 succ m)\n      prf-n\u2265100 : MCR (succ n) (succ m)\n    {-# ATP prove 101\u2238Sn\u22610 x\u2264y\u2192x-y\u22610 #-}\n    {-# ATP prove 0<101\u2238Sm x<y\u21920<x\u2238y #-}\n    {-# ATP prove prf-n\u2265100 101\u2238Sn\u22610 0<101\u2238Sm #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Property LT2MCR which proves that the recursive calls of the\n-- McCarthy 91 function are on smaller arguments.\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.MCR.LT2MCR-ATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\n\nopen import FOTC.Program.McCarthy91.MCR\n\n------------------------------------------------------------------------------\n\nLT2MCR-helper : \u2200 {n m k} \u2192 N n \u2192 N m \u2192 N k \u2192\n                LT m n \u2192 LT (succ n) k \u2192 LT (succ m) k \u2192\n                LT (k \u2238 n) (k \u2238 m) \u2192 LT (k \u2238 succ n) (k \u2238 succ m)\nLT2MCR-helper zN Nm Nk p qn qm h = \u22a5-elim (x<0\u2192\u22a5 Nm p)\nLT2MCR-helper (sN Nn) Nm zN p qn qm h = \u22a5-elim (x<0\u2192\u22a5 (sN Nm) qm)\nLT2MCR-helper (sN {n} Nn) zN (sN {k} Nk) p qn qm h = prfS0S\n  where\n    postulate k\u2265Sn   : GE k (succ n)\n              k\u2238Sn<k : LT (k \u2238 (succ n)) k\n              prfS0S : LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ zero)\n    {-# ATP prove k\u2265Sn x<y\u2192x\u2264y #-}\n    {-# ATP prove k\u2238Sn<k k\u2265Sn x\u2265y\u2192y>0\u2192x\u2238y<x #-}\n    {-# ATP prove prfS0S k\u2238Sn<k #-}\n\nLT2MCR-helper (sN {n} Nn) (sN {m} Nm) (sN {k} Nk) p qn qm h =\n  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm (LT2MCR-helper Nn Nm Nk m<n Sn<k Sm<k k\u2238n<k\u2238m)\n  where\n    postulate\n      k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm : LT (k \u2238 succ n) (k \u2238 succ m) \u2192\n                                LT (succ k \u2238 succ (succ n))\n                                   (succ k \u2238 succ (succ m))\n    {-# ATP prove  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm #-}\n\n    postulate\n      m<n     : LT m n\n      Sn<k    : LT (succ n) k\n      Sm<k    : LT (succ m) k\n      k\u2238n<k\u2238m : LT (k \u2238 n) (k \u2238 m)\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn<k #-}\n    {-# ATP prove Sm<k #-}\n    {-# ATP prove k\u2238n<k\u2238m #-}\n\nLT2MCR : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE m one-hundred \u2192 LT m n \u2192 MCR n m\nLT2MCR zN Nm p h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nLT2MCR (sN {n} Nn) zN p h = prfS0\n  where\n    postulate prfS0 : MCR (succ n) zero\n    {-# ATP prove prfS0 x\u2238y<Sx #-}\n\nLT2MCR (sN {n} Nn) (sN {m} Nm) p h with x<y\u2228x\u2265y Nn 100-N\n... | inj\u2081 n<100 = LT2MCR-helper Nn Nm 101-N m<n Sn\u2264101 Sm\u2264101\n                                 (LT2MCR Nn Nm m\u2264100 m<n)\n  where\n    postulate m\u2264100  : LE m one-hundred\n              m<n    : LT m n\n              Sn\u2264101 : LT (succ n) hundred-one\n              Sm\u2264101 : LT (succ m) hundred-one\n    {-# ATP prove m\u2264100 x<y\u2192x\u2264y #-}\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn\u2264101 #-}\n    {-# ATP prove Sm\u2264101 #-}\n... | inj\u2082 n\u2265199 = prf-n\u2265100\n  where\n    postulate\n      101\u2238Sn\u22610  : hundred-one \u2238 succ n \u2261 zero\n      0<101\u2238Sm  : LT zero (hundred-one \u2238 succ m)\n      prf-n\u2265100 : MCR (succ n) (succ m)\n    {-# ATP prove 101\u2238Sn\u22610 x\u2264y\u2192x-y\u22610 #-}\n    {-# ATP prove 0<101\u2238Sm x<y\u21920<x\u2238y #-}\n    {-# ATP prove prf-n\u2265100 101\u2238Sn\u22610 0<101\u2238Sm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3792d48761a9ba0ccb7eefb70eafcefafde37790","subject":"Removed unnecessary explicit argument.","message":"Removed unnecessary explicit argument.\n\nIgnore-this: cd4237c3168eb4fe94c17e20426561bc\n\ndarcs-hash:20111129025124-3bd4e-cb83546017e9cd6a21e83bbd52d677884bf31ba5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Equality.agda","new_file":"notes\/thesis\/logical-framework\/Equality.agda","new_contents":"module Equality where\n\npostulate\n  D     : Set\n  _\u2261_   : D \u2192 D \u2192 Set\n  refl  : \u2200 {x} \u2192 x \u2261 x\n  subst : \u2200 (P : D \u2192 Set) {x y} \u2192 x \u2261 y \u2192 P x \u2192 P y\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym {x} h = subst (\u03bb t \u2192 t \u2261 x) h refl\n\ntrans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans {z = z} h\u2081 h\u2082 = subst (\u03bb t \u2192 t \u2261 z) (sym h\u2081) h\u2082\n","old_contents":"module Equality where\n\npostulate\n  D     : Set\n  _\u2261_   : D \u2192 D \u2192 Set\n  refl  : (x : D) \u2192 x \u2261 x\n  subst : (P : D \u2192 Set) {x y : D} \u2192 x \u2261 y \u2192 P x \u2192 P y\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym {x} h = subst (\u03bb t \u2192 t \u2261 x) h (refl x)\n\ntrans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans {z = z} h\u2081 h\u2082 = subst (\u03bb t \u2192 t \u2261 z) (sym h\u2081) h\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c35fb9496c6b965675cc671026c262ccccdc2e3a","subject":"IDesc model: substitution ok.","message":"IDesc model: substitution ok.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (domNat domBool : EnumU)\n            (gammaNat : spi domNat (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi domBool (\\_ -> IMu closeTerm' bool))\n            (sigma : (domNat domBool : EnumU)\n                     (gammaNat : spi domNat (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi domBool (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var (chooseDom ty domNat domBool) ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (toIDesc Type (exprFree ** (\\ty -> Val ty + Var (chooseDom ty domNat domBool) ty))) ty ->\n            IMu closeTerm' ty\nsubstExpr domNat domBool gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var (chooseDom ty domNat domBool) ty)\n           Val\n           exprFree\n           (sig domNat domBool gammaNat gammaBool)\n           ty\n           term\n\n\n-- sigmaExpr : (domNat domBool : EnumU)\n--             (gammaNat : spi domNat (\\_ -> IMu closeTerm' nat))\n--             (gammaBool : spi domBool (\\_ -> IMu closeTerm' bool))\n--             (ty : Type) ->\n--             (Val ty + Var (chooseDom ty domNat domBool) ty) ->\n--             IMu closeTerm' ty\n-- sigmaExpr domNat domBool gammaNat gammaBool ty v = \n--     discharge' ty\n--                (chooseDom ty domNat domBool)\n--                (chooseGamma ty (chooseDom ty domNat domBool) gammaNat gammaBool)\n--                v\n\n-- discharge' : (ty : Type)\n--             (vars : EnumU)\n--             (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n--             (Val ty + Var vars ty) -> \n--             IMu closeTerm' ty\n\n-- chooseGamma : (ty : Type)\n--               (dom : EnumU)\n--               (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n--               (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n--               spi dom (\\_ -> IMu closeTerm' ty)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f274de0be6b9f871bb446ee6381969cac1a7ad8a","subject":"Fix order of returned caches","message":"Fix order of returned caches\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c4\u2081 v\u00d7 \u03c4\u2082 v\u00d7 \u03c3) , (r\u2082 ,\u2032 (c\u2081 ,\u2032 c\u2082  ,\u2032 r\u2081))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n\n  \u27e6_\u27e7TermCacheCBV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 fromCBVCtx \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 cbvToCompType \u03c4 \u27e7CompTypeHidCache\n  \u27e6 t \u27e7TermCacheCBV = \u27e6 fromCBV t \u27e7CompTermCache\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n\n  \u27e6_\u27e7TermCacheCBV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 fromCBVCtx \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 cbvToCompType \u03c4 \u27e7CompTypeHidCache\n  \u27e6 t \u27e7TermCacheCBV = \u27e6 fromCBV t \u27e7CompTermCache\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a127cbe0afe5735ec22de57b718586c54a38c33b","subject":"Fix testd","message":"Fix testd\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 7))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"50aed72e5cdc9e329d9804c724535dfc771ad1b6","subject":"Some progress on FunBigStepSILR2","message":"Some progress on FunBigStepSILR2\n\nHowever, relV is still slightly wrong. Fixed version in Alt. Gotta change a few\nlemmas now!\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". However that doesn't help termination either,\n-- since Agda doesn't see that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) zero n1 ()\neval-const-dec (lit v) (suc n0) .n0 refl = \u2264-step \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) zero n1 ()\neval-const-mono (lit v) (suc n0) .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\nmodule Alt where\n  mutual\n    relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n    relT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n      (v1 : Val \u03c4) \u2192\n      \u2200 n-j (n-j\u2264n : n-j < n) \u2192\n      (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n      \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n      -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n    relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n    -- Seems the proof for abs would go through even if here we do not step down.\n    -- However, that only works as long as we use a typed language; not stepping\n    -- down here, in an untyped language, gives a non-well-founded definition.\n    relV nat v1 v2 n = v1 \u2261 v2\n    relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nmutual\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV \u03c4 v1 v2 zero = \u22a4\n  -- Seems the proof for abs would go through even if here we do not step down.\n  -- However, that only works as long as we use a typed language; not stepping\n  -- down here, in an untyped language, gives a non-well-founded definition.\n  relV nat v1 v2 (suc n) = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n    \u2200 (k : \u2115) (k\u2264n : k \u2264 n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV  x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- relT (app s t) (app s t)\n\nrelV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n  n1 sv1 sv2\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n  n2 tv1 tv2\n  (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n  (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n  -- (tvv)\n  -- (eqv1)\n  \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\nrelV-apply s t v1 \u03c1 n-j n1 sv1 sv2 svv n2 tv1 tv2 tvv eq = {!!}\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (const (lit v)) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , suc zero , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , n , n , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | Done tv1 n2 | [ t1eq ] with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s (eval-dec s \u03c11 _ n n1 s1eq)) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq) | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s (eval-dec s \u03c11 _ n n1 s1eq))) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s (eval-dec t \u03c11 _ n1 n2 t1eq)) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , sn4 , s2eq , svv | tv2 , tn3 , tn4 , t2eq , tvv = {! relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq!}\n--\n-- {!eval s \u03c12 !}\n-- fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) ? ?\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) zero n1 ()\neval-const-mono (lit v) (suc n0) .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) v1 v2 \u2192\n  relV \u03c3 v1 v2 k \u2192\n  relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n  (v1 : Val \u03c4) \u2192\n  \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n  (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV  x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- relT (app s t) (app s t)\n\nrelV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\nrelV-apply = {!!}\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (const (lit v)) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , suc zero , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , n , n , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | Done tv1 n2 | [ t1eq ] with fundamental s _ \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s {!!}) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq) | fundamental t _ \u03c11 \u03c12 \u03c1\u03c1 tv1 (suc n2) {!!} {!eval-mono t \u03c11 tv1 n1 n2 t1eq!}\n... | closure st2 s\u03c12 , sn3 , sn4 , s2eq , svv | tv2 , tn3 , tn4 , t2eq , tvv = {!!}\n--\n-- {!eval s \u03c12 !}\n-- fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) ? ?\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"442691e90525bfa889ed6b3aa1507ccd8fddc7df","subject":"Prove extensionality of d.o.e. (#332, #333)","message":"Prove extensionality of d.o.e. (#332, #333)\n\nOld-commit-hash: ae48f8fe6bde1aac0a03f2feb7177328be4ab5df\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : FC.FunctionChange f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} dfxdx\u2259dgxdx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      open import Postulate.Extensionality\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = dfxdx\u2259dgxdx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"475c999cc7f09cce353c6e8be0c4e04f7b3db93f","subject":"adding a few more exchange properties","message":"adding a few more exchange properties\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"exchange.agda","new_file":"exchange.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nmodule exchange where\n  -- exchanging just two disequal elements produces the same context\n  swap-little : {A : Set} {x y : Nat} {\u03c41 \u03c42 : A} \u2192 (x \u2260 y) \u2192\n    ((\u25a0 (x , \u03c41)) ,, (y , \u03c42)) == ((\u25a0 (y , \u03c42)) ,, (x , \u03c41))\n  swap-little {A} {x} {y} {\u03c41} {\u03c42} neq = \u222acomm (\u25a0 (x , \u03c41))\n                                                (\u25a0 (y , \u03c42))\n                                                (disjoint-singles neq)\n\n  -- really the core of all the exchange arguments: contexts with two\n  -- disequal elements exchanged are the same. we reassociate the unions,\n  -- swap as above, and then associate them back.\n  --\n  -- note that this is generic in the contents of the context. the proofs\n  -- below show the exchange properties that we actually need in the\n  -- various other proofs; the remaning exchange properties for both \u0394 and\n  -- \u0393 positions for all the other hypothetical judgements are exactly in\n  -- this pattern.\n  swap : {A : Set} {x y : Nat} {\u03c41 \u03c42 : A} (\u0393 : A ctx) (x\u2260y : x == y \u2192 \u22a5) \u2192\n         ((\u0393 ,, (x , \u03c41)) ,, (y , \u03c42)) == ((\u0393 ,, (y , \u03c42)) ,, (x , \u03c41))\n  swap {A} {x} {y} {\u03c41} {\u03c42} \u0393 neq =\n                        (\u222aassoc \u0393 (\u25a0 (x , \u03c41)) (\u25a0 (y , \u03c42)) (disjoint-singles neq) ) \u00b7\n                        (ap1 (\u03bb qq \u2192 \u0393 \u222a qq) (swap-little neq) \u00b7\n                        ! (\u222aassoc \u0393 (\u25a0 (y , \u03c42)) (\u25a0 (x , \u03c41)) (disjoint-singles (flip neq))))\n\n  -- the above exchange principle used via transport in the judgements we needed\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y =\n    tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (swap \u0393 x\u2260y)\n\n  exchange-synth : \u2200{\u0393 x y \u03c4 \u03c41 \u03c42 e}\n                       \u2192 x \u2260 y\n                       \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 e => \u03c4\n                       \u2192 (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 e => \u03c4\n  exchange-synth {\u0393} {x} {y} {\u03c4} {\u03c41} {\u03c42} {e} neq  =\n    tr (\u03bb qq \u2192 qq \u22a2 e => \u03c4) (swap \u0393 neq)\n\n  exchange-ana : \u2200{\u0393 x y \u03c4 \u03c41 \u03c42 e}\n                       \u2192 x \u2260 y\n                       \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 e <= \u03c4\n                       \u2192 (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 e <= \u03c4\n  exchange-ana {\u0393} {x} {y} {\u03c4} {\u03c41} {\u03c42} {e} neq  =\n    tr (\u03bb qq \u2192 qq \u22a2 e <= \u03c4) (swap \u0393 neq)\n\n  exchange-expand-synth : \u2200{\u0393 x y \u03c41 \u03c42 e \u03c4 d \u0394} \u2192\n                        x \u2260 y \u2192\n                        (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                        (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394\n  exchange-expand-synth {\u0393 = \u0393} {e = e} {\u03c4 = \u03c4} {d = d } {\u0394 = \u0394} neq =\n    tr (\u03bb qq \u2192 qq \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394) (swap \u0393 neq)\n\n  exchange-expand-ana : \u2200 {\u0393 x y \u03c41 \u03c42 \u03c4 \u03c4' d e \u0394} \u2192\n                      x \u2260 y \u2192\n                      (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                      (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n  exchange-expand-ana {\u0393 = \u0393} {\u03c4 = \u03c4} {\u03c4' = \u03c4'} {d = d} {e = e} {\u0394 = \u0394} neq =\n    tr (\u03bb qq \u2192 qq \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394) (swap \u0393 neq)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nmodule exchange where\n  -- exchanging just two disequal elements produces the same context\n  swap-little : {A : Set} {x y : Nat} {\u03c41 \u03c42 : A} \u2192 (x \u2260 y) \u2192\n    ((\u25a0 (x , \u03c41)) ,, (y , \u03c42)) == ((\u25a0 (y , \u03c42)) ,, (x , \u03c41))\n  swap-little {A} {x} {y} {\u03c41} {\u03c42} neq = \u222acomm (\u25a0 (x , \u03c41))\n                                                (\u25a0 (y , \u03c42))\n                                                (disjoint-singles neq)\n\n  -- really the core of all the exchange arguments: contexts with two\n  -- disequal elements exchanged are the same. we reassociate the unions,\n  -- swap as above, and then associate them back.\n  --\n  -- note that this is generic in the contents of the context. the proofs\n  -- below show the exchange properties that we actually need in the\n  -- various other proofs; the remaning exchange properties for both \u0394 and\n  -- \u0393 positions for all the other hypothetical judgements are exactly in\n  -- this pattern.\n  swap : {A : Set} {x y : Nat} {\u03c41 \u03c42 : A} (\u0393 : A ctx) (x\u2260y : x == y \u2192 \u22a5) \u2192\n         ((\u0393 ,, (x , \u03c41)) ,, (y , \u03c42)) == ((\u0393 ,, (y , \u03c42)) ,, (x , \u03c41))\n  swap {A} {x} {y} {\u03c41} {\u03c42} \u0393 neq =\n                        (\u222aassoc \u0393 (\u25a0 (x , \u03c41)) (\u25a0 (y , \u03c42)) (disjoint-singles neq) ) \u00b7\n                        (ap1 (\u03bb qq \u2192 \u0393 \u222a qq) (swap-little neq) \u00b7\n                        ! (\u222aassoc \u0393 (\u25a0 (y , \u03c42)) (\u25a0 (x , \u03c41)) (disjoint-singles (flip neq))))\n\n  -- the above exchange principle used via transport in the judgements we needed\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy =\n    tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (swap \u0393 x\u2260y) xy\n\n  exchange-synth : \u2200{\u0393 x y \u03c4 \u03c41 \u03c42 e}\n                       \u2192 x \u2260 y\n                       \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 e => \u03c4\n                       \u2192 (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 e => \u03c4\n  exchange-synth {\u0393} {x} {y} {\u03c4} {\u03c41} {\u03c42} {e} neq synth =\n    tr (\u03bb qq \u2192 qq \u22a2 e => \u03c4) (swap \u0393 neq) synth\n\n  exchange-ana : \u2200{\u0393 x y \u03c4 \u03c41 \u03c42 e}\n                       \u2192 x \u2260 y\n                       \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 e <= \u03c4\n                       \u2192 (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 e <= \u03c4\n  exchange-ana {\u0393} {x} {y} {\u03c4} {\u03c41} {\u03c42} {e} neq ana =\n    tr (\u03bb qq \u2192 qq \u22a2 e <= \u03c4) (swap \u0393 neq) ana\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"62397bbcc510a68f9c365f02e17fc7ec42140abc","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Rec\/ConversionRules.agda","new_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Rec\/ConversionRules.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion rules for the recursive operator rec\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Rec.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Rec\n\n----------------------------------------------------------------------------\n\nprivate\n  -- We follow the same methodology used in\n  -- LTC-PCF.Program.Division.EquationsI (see it for the\n  -- documentation).\n\n  ----------------------------------------------------------------------------\n  -- The steps\n\n  -- Initially, the conversion rule fix-eq is applied.\n  rec-s\u2081 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2081 n a f = rech (fix rech) \u00b7 n \u00b7 a \u00b7 f\n\n  -- First argument application.\n  rec-s\u2082 : D \u2192 D\n  rec-s\u2082 n = lam (\u03bb a \u2192 lam (\u03bb f \u2192\n                    (if (iszero\u2081 n)\n                        then a\n                        else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f))))\n\n  -- Second argument application.\n  rec-s\u2083 : D \u2192 D \u2192 D\n  rec-s\u2083 n a = lam (\u03bb f \u2192\n                   (if (iszero\u2081 n)\n                       then a\n                       else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)))\n\n  -- Third argument application.\n  rec-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2084 n a f = if (iszero\u2081 n)\n                     then a\n                     else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  rec-s\u2085 : D \u2192 D \u2192 D \u2192 D \u2192 D\n  rec-s\u2085 n a f b = if b\n                      then a\n                      else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)\n\n  -- Reduction of iszero\u2081 n \u2261 true\n  --\n  -- It should be\n  -- rec-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  -- rec-s\u2086 n a f = a\n  --\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  rec-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2086 n a f = f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  rec-s\u2087 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2087 n a f = f \u00b7 n \u00b7 (fix rech \u00b7 n \u00b7 a \u00b7 f)\n\n  ----------------------------------------------------------------------------\n  -- The execution steps\n\n  -- We follow the same methodology used in\n  -- LTC-PCF.Program.Division.EquationsI (see it for the\n  -- documentation).\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 n a f \u2192 fix rech \u00b7 n \u00b7 a \u00b7 f \u2261 rec-s\u2081 n a f\n  proof\u2080\u208b\u2081 n a f = subst (\u03bb x \u2192 x \u00b7 n \u00b7 a \u00b7 f \u2261\n                                rech (fix rech) \u00b7 n \u00b7 a \u00b7 f )\n                         (sym (fix-eq rech))\n                         refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 n a f \u2192 rec-s\u2081 n a f \u2261 rec-s\u2082 n \u00b7 a \u00b7 f\n  proof\u2081\u208b\u2082 n a f = subst (\u03bb x \u2192 x \u00b7 a \u00b7 f \u2261 rec-s\u2082 n \u00b7 a \u00b7 f)\n                         (sym (beta rec-s\u2082 n))\n                         refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 n a f \u2192 rec-s\u2082 n \u00b7 a \u00b7 f \u2261 rec-s\u2083 n a \u00b7 f\n  proof\u2082\u208b\u2083 n a f = subst (\u03bb x \u2192 x \u00b7 f \u2261 rec-s\u2083 n a \u00b7 f)\n                         (sym (beta (rec-s\u2083 n) a))\n                         refl\n\n  -- Application of the third argument.\n  proof\u2083\u208b\u2084 : \u2200 n a f \u2192 rec-s\u2083 n a \u00b7 f \u2261 rec-s\u2084 n a f\n  proof\u2083\u208b\u2084 n a f = beta (rec-s\u2084 n a) f\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n  --\n  -- Cases iszero\u2081 n \u2261 b using that proof.\n  proof\u2084\u208b\u2085 : \u2200 n a f b \u2192 iszero\u2081 n \u2261 b \u2192 rec-s\u2084 n a f \u2261 rec-s\u2085 n a f b\n  proof\u2084\u208b\u2085 n a f b h = subst (\u03bb x \u2192 rec-s\u2085 n a f x \u2261 rec-s\u2085 n a f b)\n                             (sym h)\n                             refl\n\n  -- Reduction of if true ... using the conversion rule if-true.\n  proof\u2085\u208a : \u2200 n a f \u2192 rec-s\u2085 n a f true \u2261 a\n  proof\u2085\u208a n a f = if-true a\n\n   -- Reduction of if false ... using the conversion rule if-false.\n  proof\u2085\u208b\u2086 : \u2200 n a f \u2192 rec-s\u2085 n a f false \u2261 rec-s\u2086 n a f\n  proof\u2085\u208b\u2086 n a f = if-false (rec-s\u2086 n a f)\n\n  -- Reduction pred\u2081 (succ n) \u2261 n using the conversion rule pred\u2081-S.\n  proof\u2086\u208b\u2087 : \u2200 n a f \u2192 rec-s\u2086 (succ\u2081 n) a f \u2261 rec-s\u2087 n a f\n  proof\u2086\u208b\u2087 n a f = subst (\u03bb x \u2192 rec-s\u2087 x a f \u2261 rec-s\u2087 n a f)\n                         (sym (pred-S n))\n                         refl\n\n------------------------------------------------------------------------------\n-- The conversion rules for rec.\n\nrec-0 : \u2200 a {f} \u2192 rec zero a f \u2261 a\nrec-0 a {f} =\n  fix rech \u00b7 zero \u00b7 a \u00b7 f  \u2261\u27e8 proof\u2080\u208b\u2081 zero a f \u27e9\n  rec-s\u2081 zero a f          \u2261\u27e8 proof\u2081\u208b\u2082 zero a f \u27e9\n  rec-s\u2082 zero \u00b7 a \u00b7 f      \u2261\u27e8 proof\u2082\u208b\u2083 zero a f \u27e9\n  rec-s\u2083 zero a \u00b7 f        \u2261\u27e8 proof\u2083\u208b\u2084 zero a f \u27e9\n  rec-s\u2084 zero a f          \u2261\u27e8 proof\u2084\u208b\u2085 zero a f true iszero-0 \u27e9\n  rec-s\u2085 zero a f true     \u2261\u27e8 proof\u2085\u208a zero a f \u27e9\n  a                        \u220e\n\nrec-S : \u2200 n a f \u2192 rec (succ\u2081 n) a f \u2261 f \u00b7 n \u00b7 (rec n a f)\nrec-S n a f =\n  fix rech \u00b7 (succ\u2081 n) \u00b7 a \u00b7 f  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 n) a f \u27e9\n  rec-s\u2081 (succ\u2081 n) a f          \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 n) a f \u27e9\n  rec-s\u2082 (succ\u2081 n) \u00b7 a \u00b7 f      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 n) a f \u27e9\n  rec-s\u2083 (succ\u2081 n) a \u00b7 f        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 n) a f \u27e9\n  rec-s\u2084 (succ\u2081 n) a f          \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 n) a f false (iszero-S n) \u27e9\n  rec-s\u2085 (succ\u2081 n) a f false    \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 n) a f \u27e9\n  rec-s\u2086 (succ\u2081 n) a f          \u2261\u27e8 proof\u2086\u208b\u2087 n a f \u27e9\n  rec-s\u2087 n a f                  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Conversion rules for the recursive operator rec\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Rec.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Rec\n\n----------------------------------------------------------------------------\n\nprivate\n  -- We follow the same methodology used in\n  -- LTC-PCF.Program.Division.EquationsI (see it for the\n  -- documentation).\n\n  ----------------------------------------------------------------------------\n  -- The steps\n\n  -- Initially, the conversion rule fix-eq is applied.\n  rec-s\u2081 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2081 n a f = rech (fix rech) \u00b7 n \u00b7 a \u00b7 f\n\n  -- First argument application.\n  rec-s\u2082 : D \u2192 D\n  rec-s\u2082 n = lam (\u03bb a \u2192 lam (\u03bb f \u2192\n                    (if (iszero\u2081 n)\n                        then a\n                        else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f))))\n\n  -- Second argument application.\n  rec-s\u2083 : D \u2192 D \u2192 D\n  rec-s\u2083 n a = lam (\u03bb f \u2192\n                   (if (iszero\u2081 n)\n                       then a\n                       else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)))\n\n  -- Third argument application.\n  rec-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2084 n a f = if (iszero\u2081 n)\n                     then a\n                     else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  rec-s\u2085 : D \u2192 D \u2192 D \u2192 D \u2192 D\n  rec-s\u2085 n a f b = if b\n                      then a\n                      else f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)\n\n  -- Reduction of iszero\u2081 n \u2261 true\n  --\n  -- It should be\n  -- rec-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  -- rec-s\u2086 n a f = a\n  --\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  rec-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2086 n a f = f \u00b7 pred\u2081 n \u00b7 (fix rech \u00b7 pred\u2081 n \u00b7 a \u00b7 f)\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  rec-s\u2087 : D \u2192 D \u2192 D \u2192 D\n  rec-s\u2087 n a f = f \u00b7 n \u00b7 (fix rech \u00b7 n \u00b7 a \u00b7 f)\n\n  ----------------------------------------------------------------------------\n  -- The execution steps\n\n  -- We follow the same methodology used in\n  -- LTC-PCF.Program.Division.EquationsI (see it for the\n  -- documentation).\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 n a f \u2192 fix rech \u00b7 n \u00b7 a \u00b7 f \u2261 rec-s\u2081 n a f\n  proof\u2080\u208b\u2081 n a f = subst (\u03bb x \u2192 x \u00b7 n \u00b7 a \u00b7 f \u2261 rech (fix rech) \u00b7 n \u00b7 a \u00b7 f )\n                         (sym (fix-eq rech))\n                         refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 n a f \u2192 rec-s\u2081 n a f \u2261 rec-s\u2082 n \u00b7 a \u00b7 f\n  proof\u2081\u208b\u2082 n a f = subst (\u03bb x \u2192 x \u00b7 a \u00b7 f \u2261 rec-s\u2082 n \u00b7 a \u00b7 f)\n                         (sym (beta rec-s\u2082 n))\n                         refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 n a f \u2192 rec-s\u2082 n \u00b7 a \u00b7 f \u2261 rec-s\u2083 n a \u00b7 f\n  proof\u2082\u208b\u2083 n a f = subst (\u03bb x \u2192 x \u00b7 f \u2261 rec-s\u2083 n a \u00b7 f)\n                         (sym (beta (rec-s\u2083 n) a))\n                         refl\n\n  -- Application of the third argument.\n  proof\u2083\u208b\u2084 : \u2200 n a f \u2192 rec-s\u2083 n a \u00b7 f \u2261 rec-s\u2084 n a f\n  proof\u2083\u208b\u2084 n a f = beta (rec-s\u2084 n a) f\n\n  -- Cases iszero\u2081 n \u2261 b using that proof.\n  proof\u2084\u208b\u2085 : \u2200 n a f b \u2192 iszero\u2081 n \u2261 b \u2192 rec-s\u2084 n a f \u2261 rec-s\u2085 n a f b\n  proof\u2084\u208b\u2085 n a f b h = subst (\u03bb x \u2192 rec-s\u2085 n a f x \u2261 rec-s\u2085 n a f b)\n                             (sym h)\n                             refl\n\n  -- Reduction of if true ... using the conversion rule if-true.\n  proof\u2085\u208a : \u2200 n a f \u2192 rec-s\u2085 n a f true \u2261 a\n  proof\u2085\u208a n a f = if-true a\n\n   -- Reduction of if false ... using the conversion rule if-false.\n  proof\u2085\u208b\u2086 : \u2200 n a f \u2192 rec-s\u2085 n a f false \u2261 rec-s\u2086 n a f\n  proof\u2085\u208b\u2086 n a f = if-false (rec-s\u2086 n a f)\n\n  -- Reduction pred\u2081 (succ n) \u2261 n using the conversion rule pred\u2081-S.\n  proof\u2086\u208b\u2087 : \u2200 n a f \u2192 rec-s\u2086 (succ\u2081 n) a f \u2261 rec-s\u2087 n a f\n  proof\u2086\u208b\u2087 n a f = subst (\u03bb x \u2192 rec-s\u2087 x a f \u2261 rec-s\u2087 n a f)\n                         (sym (pred-S n))\n                         refl\n\n------------------------------------------------------------------------------\n-- The conversion rules for rec.\n\nrec-0 : \u2200 a {f} \u2192 rec zero a f \u2261 a\nrec-0 a {f} =\n  fix rech \u00b7 zero \u00b7 a \u00b7 f  \u2261\u27e8 proof\u2080\u208b\u2081 zero a f \u27e9\n  rec-s\u2081 zero a f          \u2261\u27e8 proof\u2081\u208b\u2082 zero a f \u27e9\n  rec-s\u2082 zero \u00b7 a \u00b7 f      \u2261\u27e8 proof\u2082\u208b\u2083 zero a f \u27e9\n  rec-s\u2083 zero a \u00b7 f        \u2261\u27e8 proof\u2083\u208b\u2084 zero a f \u27e9\n  rec-s\u2084 zero a f          \u2261\u27e8 proof\u2084\u208b\u2085 zero a f true iszero-0 \u27e9\n  rec-s\u2085 zero a f true     \u2261\u27e8 proof\u2085\u208a zero a f \u27e9\n  a                        \u220e\n\nrec-S : \u2200 n a f \u2192 rec (succ\u2081 n) a f \u2261 f \u00b7 n \u00b7 (rec n a f)\nrec-S n a f =\n  fix rech \u00b7 (succ\u2081 n) \u00b7 a \u00b7 f  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 n) a f \u27e9\n  rec-s\u2081 (succ\u2081 n) a f          \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 n) a f \u27e9\n  rec-s\u2082 (succ\u2081 n) \u00b7 a \u00b7 f      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 n) a f \u27e9\n  rec-s\u2083 (succ\u2081 n) a \u00b7 f        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 n) a f \u27e9\n  rec-s\u2084 (succ\u2081 n) a f          \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 n) a f false (iszero-S n) \u27e9\n  rec-s\u2085 (succ\u2081 n) a f false    \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 n) a f \u27e9\n  rec-s\u2086 (succ\u2081 n) a f          \u2261\u27e8 proof\u2086\u208b\u2087 n a f \u27e9\n  rec-s\u2087 n a f                  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"921eab0bc4910e71e779f2cb9286cf901b189466","subject":"Updated an issue.","message":"Updated an issue.\n\nIgnore-this: 6271a60d2dbe3a0f1d5d53e46e39ae9b\n\ndarcs-hash:20110820132419-3bd4e-4a602e1125c80e4134202edb8306da90d9c4f9ad.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/ProofTerm.agda","new_file":"Issues\/ProofTerm.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the erasure of proof terms\n------------------------------------------------------------------------------\n\n-- An internal error has occurred. Please report this as a bug.\n-- Location of the error: src\/AgdaLib\/Syntax\/DeBruijn.hs:417\n\nmodule Issues.ProofTerm where\n\npostulate\n  D   : Set\n  N   : D \u2192 Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata ListN : D \u2192 Set where\n  consLN : \u2200 {n ns} \u2192 N n \u2192 (LNns : ListN ns) \u2192 ListN (n \u2237 ns)\n{-# ATP axiom consLN #-}\n\n-- We need to have at least one conjecture to generate a TPTP file.\npostulate refl : \u2200 d \u2192 d \u2261 d\n{-# ATP prove refl #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the erasure of proof terms\n------------------------------------------------------------------------------\n\nmodule Issues.Tmp.ProofTerm where\n\npostulate\n  D   : Set\n  N   : D \u2192 Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata ListN : D \u2192 Set where\n  consLN : \u2200 {n ns} \u2192 N n \u2192 (LNns : ListN ns) \u2192 ListN (n \u2237 ns)\n{-# ATP axiom consLN #-}\n\n-- We need to have at least one conjecture to generate a TPTP file.\npostulate refl : \u2200 d \u2192 d \u2261 d\n{-# ATP prove refl #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7d43e03aba3b449b2ebcdd9117c92952093ab3bb","subject":"some syntax to match the paper, more rules","message":"some syntax to match the paper, more rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_&_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_&_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< e2' > t2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u & id \u0393 ] \u22a3 (\u2205 ,, (u ::[ \u0393 ] \u2987\u2988))\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u & id \u0393 ] \u22a3 (\u0394 ,, (u ::[ \u0393 ] \u2987\u2988))\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< e' > t) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      -- EALam :\n      -- EASubsume :\n      -- EAEHole :\n      -- EANEHole :\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where -- todo not a context\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u & \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u & \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_&_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_&_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 (e1 \u2218 e2) \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      -- ESAp2\n      -- ESAp3\n      -- ESEHole\n      -- ESNEHole\n      -- ESAsc1\n      -- ESAsc2\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      -- EALam\n      -- EASubsume\n      -- EAEHole\n      -- EANEHole\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n\n  -- todo: ugh\n  postulate\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where -- todo not a context\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u & \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u & \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5bd5879c8a87ce091f5d83193ee97bd3d644101e","subject":"FunUniverse: not was broken","message":"FunUniverse: not was broken\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Core.agda","new_file":"FunUniverse\/Core.agda","new_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nopen import FunUniverse.Types public\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\nopen import FunUniverse.Rewiring.Linear\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    -- bijFork f\u2080 f\u2081 (0' , x) = 0' , f\u2080 x\n    -- bijFork f\u2080 f\u2081 (1' , x) = 1' , f\u2081 x\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  -- bijFork\u2032 f\u2080 f\u2081 (0' , x) = 0' , f\u2080 0' x\n  -- bijFork\u2032 f\u2080 f\u2081 (1' , x) = 1' , f\u2081 1' x\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- cond (0' , x\u2080 , x\u2081) = x\u2080\n    -- cond (1' , x\u2080 , x\u2081) = x\u2081\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- fork f\u2080 f\u2081 (0' , x) = f\u2080 x\n    -- fork f\u2080 f\u2081 (1' , x) = f\u2081 x\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  -- Bad idea\u2122\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <1b> <0b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","old_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nopen import FunUniverse.Types public\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\nopen import FunUniverse.Rewiring.Linear\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    -- bijFork f\u2080 f\u2081 (0' , x) = 0' , f\u2080 x\n    -- bijFork f\u2080 f\u2081 (1' , x) = 1' , f\u2081 x\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  -- bijFork\u2032 f\u2080 f\u2081 (0' , x) = 0' , f\u2080 0' x\n  -- bijFork\u2032 f\u2080 f\u2081 (1' , x) = 1' , f\u2081 1' x\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- cond (0' , x\u2080 , x\u2081) = x\u2080\n    -- cond (1' , x\u2080 , x\u2081) = x\u2081\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- fork f\u2080 f\u2081 (0' , x) = f\u2080 x\n    -- fork f\u2080 f\u2081 (1' , x) = f\u2081 x\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  -- Bad idea\u2122\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <0b> <1b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3682d42d35e0734a6a4d6a4ee62cbad3965309b9","subject":"Make _\u2295_ more coherent with _\u229d_","message":"Make _\u2295_ more coherent with _\u229d_\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Terms that operate on changes (Fig. 3).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\n-- Extension point 1: A term for \u229d on base types.\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\n-- Extension point 2: A term for \u2295 on base types.\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  -- We provide: terms for \u2295 and \u229d on arbitrary types.\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       absV 4 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       absV 3 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff _ = app\u2082 diff-term\n\n  apply : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply _ = app\u2082 apply-term\n\n  infixl 6 apply diff\n  syntax apply \u03c4 x \u0394x = \u0394x \u2295\u208d \u03c4 \u208e x\n  syntax diff \u03c4 x y = x \u229d\u208d \u03c4 \u208e y\n\n  infixl 6 _\u2295_ _\u229d_\n  _\u229d_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  _\u229d_ {\u03c4} = diff \u03c4\n\n  _\u2295_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  _\u2295_ {\u03c4} = apply \u03c4\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Terms that operate on changes (Fig. 3).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\n-- Extension point 1: A term for \u229d on base types.\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\n-- Extension point 2: A term for \u2295 on base types.\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  -- We provide: terms for \u2295 and \u229d on arbitrary types.\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       absV 4 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       absV 3 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff _ = app\u2082 diff-term\n\n  apply : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply _ = app\u2082 apply-term\n\n  infixl 6 apply diff\n  syntax apply \u03c4 x \u0394x = \u0394x \u2295\u208d \u03c4 \u208e x\n  syntax diff \u03c4 x y = x \u229d\u208d \u03c4 \u208e y\n\n  infixl 6 _\u2295_ _\u229d_\n  _\u229d_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  _\u229d_ {\u03c4} = diff \u03c4\n\n  _\u2295_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  _\u2295_ {\u03c4} = app\u2082 apply-term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ef4dba4a6f0ef00966897ba95d15854fd8cc756c","subject":"Fixed doc.","message":"Fixed doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Conat\/Type.agda","new_file":"src\/fot\/FOTC\/Data\/Conat\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC co-inductive natural numbers type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- N.B. This module is re-exported by FOTC.Data.Conat.\n\n-- Adapted from (Sander 1992).\n\n-- References:\n--\n-- \u2022 Herbert P. Sander. A logic of functional programs with an\n--   application to concurrency. PhD thesis, Chalmers University of\n--   Technology and University of Gothenburg, Department of Computer\n--   Sciences, 1992.\n\nmodule FOTC.Data.Conat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- The FOTC co-inductive natural numbers type (co-inductive predicate\n-- for total co-inductive natural)\n\n-- Functional for the NatF predicate.\n-- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n-- NatF P n = n = zero \u2228 (\u2203[ n' ] P n' \u2227 n = succ n')\n\n-- Conat is the greatest fixed-point of NatF (by Conat-unf and\n-- Conat-coind).\n\npostulate Conat : D \u2192 Set\n\n-- Conat is a post-fixed point of NatF, i.e.\n--\n-- Conat \u2264 NatF Conat.\npostulate\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')\n{-# ATP axiom Conat-unf #-}\n\n-- Conat is the greatest post-fixed point of NatF, i.e\n--\n-- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  Conat-coind : \u2200 (A : D \u2192 Set) {n} \u2192\n                -- A is post-fixed point of NatF.\n                (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                -- Conat is greater than A.\n                A n \u2192 Conat n\n\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e,\n--\n-- NatF Conat \u2264 Conat.\nConat-pre-fixed : \u2200 {n} \u2192\n                  (n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                  Conat n\nConat-pre-fixed {n} h = Conat-coind A prf h\n  where\n  A : D \u2192 Set\n  A m = m \u2261 zero \u2228 (\u2203[ m' ] Conat m' \u2227 m \u2261 succ\u2081 m')\n\n  prf : A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')\n  prf (inj\u2081 n\u22610) = inj\u2081 n\u22610\n  prf (inj\u2082 (n' , CNn' , n\u2261Sn')) = inj\u2082 (n' , Conat-unf CNn' , n\u2261Sn')\n","old_contents":"------------------------------------------------------------------------------\n-- The FOTC co-inductive natural numbers type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- N.B. This module is re-exported by FOTC.Data.Conat.\n\n-- Adapted from (Sander 1992).\n\n-- References:\n--\n-- \u2022 Herbert P. Sander. A logic of functional programs with an\n--   application to concurrency. PhD thesis, Chalmers University of\n--   Technology and University of Gothenburg, Department of Computer\n--   Sciences, 1992.\n\nmodule FOTC.Data.Conat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- The FOTC co-inductive natural numbers type (co-inductive predicate\n-- for total co-inductive natural)\n\n-- Functional for the ConatF predicate.\n-- ConatF : (D \u2192 Set) \u2192 D \u2192 Set\n-- ConatF P n = n = zero \u2228 (\u2203[ n' ] P n' \u2227 n = succ n')\n\n-- Conat is the greatest fixed-point of ConatF (by Conat-unf and\n-- Conat-coind).\n\npostulate Conat : D \u2192 Set\n\n-- Conat is a post-fixed point of ConatF, i.e.\n--\n-- Conat \u2264 ConatF Stream.\npostulate\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')\n{-# ATP axiom Conat-unf #-}\n\n-- Conat is the greatest post-fixed point of ConatF, i.e\n--\n-- \u2200 P. P \u2264 ConatF P \u21d2 P \u2264 Conat.\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  Conat-coind : \u2200 (A : D \u2192 Set) {n} \u2192\n                -- A is post-fixed point of ConatF.\n                (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                -- Conat is greater than A.\n                A n \u2192 Conat n\n\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional ConatF,\n-- i.e,\n--\n-- ConatF Conat \u2264 Conat.\nConat-pre-fixed : \u2200 {n} \u2192\n                  (n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                  Conat n\nConat-pre-fixed {n} h = Conat-coind A prf h\n  where\n  A : D \u2192 Set\n  A m = m \u2261 zero \u2228 (\u2203[ m' ] Conat m' \u2227 m \u2261 succ\u2081 m')\n\n  prf : A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')\n  prf (inj\u2081 n\u22610) = inj\u2081 n\u22610\n  prf (inj\u2082 (n' , CNn' , n\u2261Sn')) = inj\u2082 (n' , Conat-unf CNn' , n\u2261Sn')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"de127be14bbacaa2cfe0253a1a3caa890e694607","subject":"Add comment","message":"Add comment\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  -- this equals UncurriedEl I D (\u03bc I D)\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 UncurriedCases E\n    (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) \n    ((i : I) (x : \u03bc I D i) \u2192 P i x)\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\nelim2 :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n        X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in CurriedCases E Q X\nelim2 I E Cs P =\n  let D = `Arg (Tag E) Cs\n      Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n      X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in curryCases E Q X (elim I E Cs P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) VecT (VecC A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) VecT (VecC (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 UncurriedCases E\n    (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) \n    ((i : I) (x : \u03bc I D i) \u2192 P i x)\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\nelim2 :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n        X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in CurriedCases E Q X\nelim2 I E Cs P =\n  let D = `Arg (Tag E) Cs\n      Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n      X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in curryCases E Q X (elim I E Cs P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) VecT (VecC A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) VecT (VecC (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"91edc677d2001791306e00aacdc57508ae030ed2","subject":"Binary: +Equivalence-Reasoning","message":"Binary: +Equivalence-Reasoning\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/NP.agda","new_file":"lib\/Relation\/Binary\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Equivalence-Reasoning isEquivalence public renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_)\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Preorder-Reasoning (Setoid.preorder s) public renaming (_\u223c\u27e8_\u27e9_ to _\u2248\u27e8_\u27e9_)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Trans-Reasoning _\u223c_ trans public hiding (finally) renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _ \u220e = refl\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Preorder-Reasoning (Setoid.preorder s) public renaming (_\u223c\u27e8_\u27e9_ to _\u2248\u27e8_\u27e9_)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5cd94161fed3259ba24dc7c2099fc439a5fdadb1","subject":"Reorder definitions","message":"Reorder definitions\n\nMove definitions about change contexts next to their uses.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Derive.agda","new_file":"New\/Derive.agda","new_contents":"module New.Derive where\nopen import New.Lang\nopen import New.Changes\nopen import New.LangOps\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\n-- dplus = \u03bb m dm n dn \u2192 (m + dm) + (n + dn) - (m + n) = dm + dn\nderiveConst plus = abs (abs (abs (abs (app\u2082 (const plus) (var (that (that this))) (var this)))))\n-- minus = \u03bb m n \u2192 m - n\n-- dminus = \u03bb m dm n dn \u2192 (m + dm) - (n + dn) - (m - n) = dm - dn\nderiveConst minus = abs (abs (abs (abs (app\u2082 (const minus) (var (that (that this))) (var this)))))\nderiveConst cons = abs (abs (abs (abs (app (app (const cons) (var (that (that this)))) (var this)))))\nderiveConst fst = abs (abs (app (const fst) (var this)))\nderiveConst snd = abs (abs (app (const snd) (var this)))\n-- deriveConst linj = abs (abs (app (const linj) (app (const linj) (var this))))\n-- deriveConst rinj = abs (abs (app (const linj) (app (const rinj) (var this))))\n-- deriveConst (match {t1} {t2} {t3}) =\n--   -- \u03bb s ds f df g dg \u2192\n--   abs (abs (abs (abs (abs (abs\n--     -- match ds\n--     (app\u2083 (const match) (var (that (that (that (that this)))))\n--       -- \u03bb ds\u2081 \u2192 match ds\u2081\n--       (abs (app\u2083 (const match) (var this)\n--         -- case inj\u2081 da \u2192 absV 1 (\u03bb da \u2192 match s\n--         (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n--           -- \u03bb a \u2192 app\u2082 df a da\n--           (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (var (that this))))\n--           -- absurd: \u03bb b \u2192 dg b (nil b)\n--           (abs (app\u2082 (var (that (that (that this)))) (var this) (app (onil\u03c4o t2) (var this))))))\n--         -- case inj\u2082 db \u2192 absV 1 (\u03bb db \u2192 match s\n--         (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n--           -- absurd: \u03bb a \u2192 df a (nil a)\n--           (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (app (onil\u03c4o t1) (var this))))\n--           -- \u03bb b \u2192 app\u2082 dg b db\n--           (abs (app\u2082 (var (that (that (that this)))) (var this) (var (that this))))))))\n--       -- recomputation branch:\n--       -- \u03bb s2 \u2192 ominus (match s2 (f \u2295 df) (g \u2295 dg)) (match s f g)\n--       (abs (app\u2082 (ominus\u03c4o t3)\n--         -- (match s2 (f \u2295 df) (g \u2295 dg))\n--         (app\u2083 (const match)\n--           (var this)\n--           (app\u2082 (oplus\u03c4o (t1 \u21d2 t3))\n--             (var (that (that (that (that this)))))\n--             (var (that (that (that this)))))\n--           (app\u2082 (oplus\u03c4o (t2 \u21d2 t3))\n--             (var (that (that this)))\n--             (var (that this))))\n--         -- (match s f g)\n--         (app\u2083 (const match)\n--           (var (that (that (that (that (that (that this)))))))\n--           (var (that (that (that (that this)))))\n--           (var (that (that this))))))))))))\n\n-- {-\n-- derive (\u03bb s f g \u2192 match s f g) =\n-- \u03bb s ds f df g dg \u2192\n-- case ds of\n--  ch1 da \u2192\n--    case s of\n--      inj1 a \u2192 df a da\n--      inj2 b \u2192 {- absurd -} dg b (nil b)\n--   ch2 db \u2192\n--    case s of\n--      inj2 b \u2192 dg b db\n--      inj1 a \u2192 {- absurd -} df a (nil a)\n--   rp s2 \u2192\n--     match (f \u2295 df) (g \u2295 dg) s2 \u229d\n--     match f g s\n-- -}\n\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\nopen import New.LangChanges\n\n-- Lemmas needed to reason about derivation, for any correctness proof\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n","old_contents":"module New.Derive where\nopen import New.Lang\nopen import New.Changes\nopen import New.LangOps\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\n-- dplus = \u03bb m dm n dn \u2192 (m + dm) + (n + dn) - (m + n) = dm + dn\nderiveConst plus = abs (abs (abs (abs (app\u2082 (const plus) (var (that (that this))) (var this)))))\n-- minus = \u03bb m n \u2192 m - n\n-- dminus = \u03bb m dm n dn \u2192 (m + dm) - (n + dn) - (m - n) = dm - dn\nderiveConst minus = abs (abs (abs (abs (app\u2082 (const minus) (var (that (that this))) (var this)))))\nderiveConst cons = abs (abs (abs (abs (app (app (const cons) (var (that (that this)))) (var this)))))\nderiveConst fst = abs (abs (app (const fst) (var this)))\nderiveConst snd = abs (abs (app (const snd) (var this)))\n-- deriveConst linj = abs (abs (app (const linj) (app (const linj) (var this))))\n-- deriveConst rinj = abs (abs (app (const linj) (app (const rinj) (var this))))\n-- deriveConst (match {t1} {t2} {t3}) =\n--   -- \u03bb s ds f df g dg \u2192\n--   abs (abs (abs (abs (abs (abs\n--     -- match ds\n--     (app\u2083 (const match) (var (that (that (that (that this)))))\n--       -- \u03bb ds\u2081 \u2192 match ds\u2081\n--       (abs (app\u2083 (const match) (var this)\n--         -- case inj\u2081 da \u2192 absV 1 (\u03bb da \u2192 match s\n--         (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n--           -- \u03bb a \u2192 app\u2082 df a da\n--           (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (var (that this))))\n--           -- absurd: \u03bb b \u2192 dg b (nil b)\n--           (abs (app\u2082 (var (that (that (that this)))) (var this) (app (onil\u03c4o t2) (var this))))))\n--         -- case inj\u2082 db \u2192 absV 1 (\u03bb db \u2192 match s\n--         (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n--           -- absurd: \u03bb a \u2192 df a (nil a)\n--           (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (app (onil\u03c4o t1) (var this))))\n--           -- \u03bb b \u2192 app\u2082 dg b db\n--           (abs (app\u2082 (var (that (that (that this)))) (var this) (var (that this))))))))\n--       -- recomputation branch:\n--       -- \u03bb s2 \u2192 ominus (match s2 (f \u2295 df) (g \u2295 dg)) (match s f g)\n--       (abs (app\u2082 (ominus\u03c4o t3)\n--         -- (match s2 (f \u2295 df) (g \u2295 dg))\n--         (app\u2083 (const match)\n--           (var this)\n--           (app\u2082 (oplus\u03c4o (t1 \u21d2 t3))\n--             (var (that (that (that (that this)))))\n--             (var (that (that (that this)))))\n--           (app\u2082 (oplus\u03c4o (t2 \u21d2 t3))\n--             (var (that (that this)))\n--             (var (that this))))\n--         -- (match s f g)\n--         (app\u2083 (const match)\n--           (var (that (that (that (that (that (that this)))))))\n--           (var (that (that (that (that this)))))\n--           (var (that (that this))))))))))))\n\n-- {-\n-- derive (\u03bb s f g \u2192 match s f g) =\n-- \u03bb s ds f df g dg \u2192\n-- case ds of\n--  ch1 da \u2192\n--    case s of\n--      inj1 a \u2192 df a da\n--      inj2 b \u2192 {- absurd -} dg b (nil b)\n--   ch2 db \u2192\n--    case s of\n--      inj2 b \u2192 dg b db\n--      inj1 a \u2192 {- absurd -} df a (nil a)\n--   rp s2 \u2192\n--     match (f \u2295 df) (g \u2295 dg) s2 \u229d\n--     match f g s\n-- -}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\nopen import New.LangChanges\n\n-- Lemmas needed to reason about derivation, for any correctness proof\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"27936513218bb472408225b9883e149a0ed927b2","subject":"Control\/Protocol\/Choreography.agda","message":"Control\/Protocol\/Choreography.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V1 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end Q = Q\n    Sim\u1d3e P end = P\n    Sim\u1d3e (com P\u1d38) (com P\u1d3f) = \u03a3\u1d3e LR (Sim\u1d9c P\u1d38 P\u1d3f)\n\n    Sim\u1d9c : Com \u2192 Com \u2192 LR \u2192 Proto\n    Sim\u1d9c P\u1d38 P\u1d3f `L = Sim\u1d9cL P\u1d38 P\u1d3f\n    Sim\u1d9c P\u1d38 P\u1d3f `R = Sim\u1d9cR P\u1d38 P\u1d3f\n\n    Sim\u1d9cL : Com \u2192 Com \u2192 Proto\n    Sim\u1d9cL (mk q\u1d38 M\u1d38 P\u1d38) Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 Sim\u1d3e (P\u1d38 m) (com Q))\n\n    Sim\u1d9cR : Com \u2192 Com \u2192 Proto\n    Sim\u1d9cR P (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 Sim\u1d3e (com P) (P\u1d3f m))\n\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule V3 where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    P \u214b\u1d3e Q = \u03a3\u1d3e LR (Par\u1d3e P Q)\n\n    Par\u1d3e : Proto \u2192 Proto \u2192 LR \u2192 Proto\n    Par\u1d3e end          Q `L = Q\n    Par\u1d3e (com' q M P) Q `L = com' q M \u03bb m \u2192 P m \u214b\u1d3e Q\n    Par\u1d3e P end          `R = P\n    Par\u1d3e P (com' q M Q) `R = com' q M \u03bb m \u2192 P \u214b\u1d3e Q m\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    P ox\u1d3e Q = \u03a0\u1d3e LR (Ten\u1d3e P Q)\n\n    Ten\u1d3e : Proto \u2192 Proto \u2192 LR \u2192 Proto\n    Ten\u1d3e end          Q `L = Q\n    Ten\u1d3e (com' q M P) Q `L = com' q M \u03bb m \u2192 P m ox\u1d3e Q\n    Ten\u1d3e P end          `R = P\n    Ten\u1d3e P (com' q M Q) `R = com' q M \u03bb m \u2192 P ox\u1d3e Q m\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = {!!}\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id = {!!}\n  \n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e P Q p q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndualInOut : InOut \u2192 InOut\ndualInOut In  = Out\ndualInOut Out = In\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  com  : (q : InOut)(M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com q M P \u2261\u1d3e com q M Q\n\npattern \u03a0' M P = com In  M P\npattern \u03a3' M P = com Out M P\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0S' M P = \u03a0' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3S' M P = \u03a3' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u27e6_\u27e7\u03a0\u03a3 : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u03a0\u03a3 In  = \u03a0\n\u27e6_\u27e7\u03a0\u03a3 Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com q M P \u27e7 = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 \u27e6 P x \u27e7\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\n_-[_]\u2192\u00f8\u204f_ : \u2200 {I}(A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\nA -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  I\u03a3D : \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  I\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n--  I\u03a0S : \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 \u03a0' (\u2610 M) \u2102C ]\n  \u2605\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  \u2605\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n  --\u00f8\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ A \u2261 \u03a3' S M \u2102A ]\n  --\u00f8\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ B \u2261 \u03a0' S M \u2102B ]\n  end : \u2200 {A} \u2192 [ end \/ A \u2261 end ]\n\nTrace : Proto \u2192 Proto\nTrace end          = end\nTrace (com _ A B) = \u03a3' A \u03bb m \u2192 Trace (B m)\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (com q A B) = com (dualInOut q) A (\u03bb x \u2192 dual (B x))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com q M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com q M P) = com q M (\u03bb m \u2192 \u2261\u1d3e-refl (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end = end\nTrace-idempotent (com q M P) = \u03a3' M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end = end\nTrace-dual-oblivious (com q M P) = \u03a3' M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\ncom \u03c0 A B >>\u2261 Q = com \u03c0 A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>\u2261 Q)\n++Tele end         _        ys = ys\n++Tele (com q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\nright-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261\u1d3e P\nright-unit end = end\nright-unit (com q M P) = com q M \u03bb m \u2192 right-unit (P m)\n\nassoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>\u2261 Q) >>\u2261 R)\nassoc end         Q R = \u2261\u1d3e-refl (Q _ >>\u2261 R)\nassoc (com q M P) Q R = com q M \u03bb m \u2192 assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndata DualInOut : InOut \u2192 InOut \u2192 \u2605 where\n  DInOut : DualInOut In  Out\n  DOutIn : DualInOut Out In\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' A B) (\u03a3' A B')\n  \u03a3\u00b7\u03a0 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' A B) (\u03a0' A B')\n\ndata Sing {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  sing : \u2200 x \u2192 Sing x\n\n-- Commit = \u03bb M R \u2192 \u03a3' D (\u2610 M) \u03bb { [ m ] \u2192 \u03a0' D R \u03bb r \u2192 \u03a3' D (Sing m) (\u03bb _ \u2192 end) }\nCommit = \u03bb M R \u2192 \u03a3S' M \u03bb m \u2192 \u03a0' R \u03bb r \u2192 \u03a3' (Sing m) (\u03bb _ \u2192 end)\n\ncommit : \u2200 M R (m : M)  \u2192 \u27e6 Commit M R \u27e7\ncommit M R m = [ m ] , (\u03bb x \u2192 (sing m) , _)\n\ndecommit : \u2200 M R (r : R) \u2192 \u27e6 dual (Commit M R) \u27e7\ndecommit M R r = \u03bb { [ m ] \u2192 r , (\u03bb x \u2192 0\u2081) }\n\ndata [_&_\u2261_]InOut : InOut \u2192 InOut \u2192 InOut \u2192 \u2605\u2081 where\n  \u03a0XX : \u2200 {X} \u2192 [ In & X  \u2261 X ]InOut\n  X\u03a0X : \u2200 {X} \u2192 [ X  & In \u2261 X ]InOut\n\n&InOut-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]InOut \u2192 [ Q & P \u2261 R ]InOut\n&InOut-comm \u03a0XX = X\u03a0X\n&InOut-comm X\u03a0X = \u03a0XX\n\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end& : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  &end : \u2200 {P} \u2192 [ P & end \u2261 P ]\n  D&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q m     \u2261 R m ]) \u2192 [ com qP    M  P & com qQ    M  Q \u2261 com qR M R ]\n  S&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P [ m ] & Q m     \u2261 R m ]) \u2192 [ com qP (\u2610 M) P & com qQ    M  Q \u2261 com qR M R ]\n  D&S  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q [ m ] \u2261 R m ]) \u2192 [ com qP    M  P & com qQ (\u2610 M) Q \u2261 com qR M R ]\n\n&-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n&-comm end& = &end\n&-comm &end = end&\n&-comm (D&D q P&) = D&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (S&D q P&) = D&S (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (D&S q P&) = S&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n\nDualInOut-sym : \u2200 {P Q} \u2192 DualInOut P Q \u2192 DualInOut Q P\nDualInOut-sym DInOut = DOutIn\nDualInOut-sym DOutIn = DInOut\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDualInOut-spec : \u2200 P \u2192 DualInOut P (dualInOut P)\nDualInOut-spec In  = DInOut\nDualInOut-spec Out = DOutIn\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (com In M P) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-spec (P x))\nDual-spec (com Out M P) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-spec (P x))\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com q M P) X = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>\u2261 Q) \u2192 \u2605} \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  bind-spec end         = refl\n  bind-spec (com q M P) = cong (\u27e6 q \u27e7\u03a0\u03a3 M) (funExt \u03bb m \u2192 bind-spec (P m))\n\n\nmodule _ {A B} where\n  run-com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  run-com end      a       b       = a , b\n  run-com (\u03a0' M P) p       (m , q) = run-com (P m) (p m) q\n  run-com (\u03a3' M P) (m , p) q       = run-com (P m) p (q m)\n\ncom-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\ncom-cont end p q = (_ , p)  , (_ , q)\ncom-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\ncom-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0' (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3' (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q M P Q} \u2192 SimL (com q M P) Q \u2192 Sim (com q M P) Q\n    right : \u2200 {P q M Q} \u2192 SimR P (com q M Q) \u2192 Sim P (com q M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recv PQA)   (send m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (send m PQB) = send m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (send m PQA) (recv PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (send x\u2081 x\u2082) (recv x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = send m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = ? -- send m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb x \u2192 left (send x (sim-id (B x)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb x \u2192 right (send x (sim-id (B x)))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = left (recv (\u03bb m \u2192 right (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = right (recv (\u03bb m \u2192 left (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u00b7\u03a3 x\u2081) (recv x) (send x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u00b7\u03a0 x)  (send x\u2081 x\u2082) (recv x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recv PQ) QR = recv (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recv P)) = do (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (send m P)) = do (send m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule 3-way-trace where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (send x x\u2082)) (left (recv x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (right (recv x)) (left (send x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (right (send x x\u2082)) (left (recv x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end} {end} end = _\n  trace {end} {com q M P} (right (send m x)) = mod\u2082 (_,_ m) (trace x)\n  trace {com q M P} {end} (left (send m x)) = mod\u2081 (_,_ m) (trace x)\n  trace {com q M P} {com q\u2081 M\u2081 P\u2081} (left (send m x)) = mod\u2081 (_,_ m) (trace {_} {com (q\u2081) _ _} x)\n  trace {com q M P} {com q\u2081 M\u2081 P\u2081} (right (send m x)) = mod\u2082 (_,_ m) (trace {com q _ _} x)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (left (recv x)) PQ QR = cong (left \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (left (send m x)) PQ QR = cong (left \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (recv x)) (left (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (send m x)) (left (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (right x) (right (recv x\u2081)) (left (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (right x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (right x) (right (send m x\u2081)) (left (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (right x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) (right (recv x\u2082)) = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) (right (send m x\u2082)) = cong (right \u2218 send m) (sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ A where\n    Test : Proto\n    Test = com Out A (const end)\n\n    s : A \u2192 Sim Test Test\n    s m = right (send m (left (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = right (recv (\u03bb m \u2192 left (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (left (recv x)) = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (left (send m x)) = cong (left \u2218 send m) (sim-comp-id x)\n  sim-comp-id (right (recv x)) = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (right (send m x)) = {!cong (right \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recv x))    = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (send x x\u2081)) = cong (left \u2218 send x) (sim-!! x\u2081)\n  sim-!! (right (recv x))    = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (send x x\u2081)) = cong (right \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (right (recv x)) (left (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (right (send x\u2081 x\u2082)) (left (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recv x\u2081))\n    = cong (left \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (send x\u2081 x\u2082))\n    = cong (left \u2218 send x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end (right (recv x)) = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (right (send m x)) = cong (left \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a1609f07a9b1a099aa3b7854089b59bde5f5e80f","subject":"TypeIsos: a few more isos","message":"TypeIsos: a few more isos\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{-\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n{-\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3b9d65fd14f8a676b56dbf7c5ad3ab9ebb8f558c","subject":"Describe correctly relation between function changes","message":"Describe correctly relation between function changes\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/RelateToValidity.agda","new_file":"Thesis\/RelateToValidity.agda","new_contents":"module Thesis.RelateToValidity where\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\n\nopen import Thesis.Changes\nopen import Thesis.Lang\n\nmodule _\n  {A : Set} {B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 \u2200 a da \u2192 Set\n  WellDefinedFunChangePoint f df a da = (f \u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangeFromTo\u2032 : \u2200 (f1 : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 Set\n  WellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 ch da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\n  open \u2261-Reasoning\n\n  fromto\u2192WellDefined\u2032 : \u2200 {f1 f2 df} \u2192 ch df from f1 to f2 \u2192\n    WellDefinedFunChangeFromTo\u2032 f1 df\n  fromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n    begin\n      (f1 \u2295 df) (a \u2295 da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n      f2 (a \u2295 da)\n    \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n      f1 a \u2295 df a da\n    \u220e\n\n  -- This is similar (not equivalent) to the old definition of function changes.\n  -- However, such a function need not be defined on invalid changes, unlike the\n  -- functions.\n  --\n  -- Hence this is sort of bigger than \u0394 f, though not really.\n  f\u0394 : (A \u2192 B) \u2192 Set\n  f\u0394 f = (a : A) \u2192 \u0394 a \u2192 \u0394 (f a)\n\n  -- We can indeed map \u0394 f into f\u0394 f, via valid-functions-map-\u0394. If this mapping\n  -- were an injection, we could say that f\u0394 f is bigger than \u0394 f. But we'd need\n  -- first to quotient \u0394 f to turn valid-functions-map-\u0394 into an injection:\n  --\n  -- 1. We need to quotient \u0394 f by extensional equivalence of functions.\n  --    Luckily, we can just postulate it.\n  --\n  -- 2. We need to quotient \u0394 f by identifying functions that only differ by\n  --    behavior on invalid changes; such functions can't be distinguished after\n  --    being injected into f\u0394 f.\n  valid-functions-map-\u0394 : \u2200 (f : A \u2192 B) (df : \u0394 f) \u2192 f\u0394 f\n  valid-functions-map-\u0394 f (df , dff) a (da , daa) = df a da , valid-res\n    where\n      valid-res : ch df a da from f a to (f a \u2295 df a da)\n      valid-res rewrite sym (fromto\u2192WellDefined\u2032 dff da a daa) = dff da a (a \u2295 da) daa\n\n  fromto-functions-map-\u0394 : \u2200 (f1 f2 : A \u2192 B) (df : Ch (A \u2192 B)) \u2192 ch df from f1 to f2 \u2192 f\u0394 f1\n  fromto-functions-map-\u0394 f1 f2 df dff a (da , daa) = valid-functions-map-\u0394 f1 (df , dff\u2032) a (da , daa)\n    where\n      dff\u2032 : ch df from f1 to (f1 \u2295 df)\n      dff\u2032 = subst (ch df from f1 to_) (sym (fromto\u2192\u2295 df f1 f2 dff)) dff\n\nopen import Thesis.LangChanges\n\nfromto\u2192WellDefined\u2032Lang : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032Lang {f1 = f1} {f2} {df} dff da a daa =\n  fromto\u2192WellDefined\u2032 dff da a daa\n","old_contents":"module Thesis.RelateToValidity where\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\n\nopen import Thesis.Changes\nopen import Thesis.Lang\n\nmodule _\n  {A : Set} {B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 \u2200 a da \u2192 Set\n  WellDefinedFunChangePoint f df a da = (f \u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\nopen import Thesis.LangChanges\n\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Ch\u03c4 (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ]\u03c4 da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nopen \u2261-Reasoning\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\n-- TODO: prove the converse of the above statement. Which is a bit harder, since\n-- you first need to state it.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"04013ce818e48ce926f5431eb540f47288f75584","subject":"trailing whitespaces","message":"trailing whitespaces\n","repos":"crypto-agda\/crypto-agda","old_file":"single-bit-one-time-pad.agda","new_file":"single-bit-one-time-pad.agda","new_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  not\u21ba : \u2200 {n} \u2192 EXP n \u2192 EXP n\n  not\u21ba = map\u21ba not\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) (b k : Bit) where\n  open Run\u2141\u2082 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = refl\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nopen \u2261-Reasoning\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\nlem\u2082 : \u2200 {ca} (A : Adv\u2082 ca) b \u2192 count\u21ba (Run\u2141\u2082.run\u2141\u2082 A b) \u2261 count\u21ba (Run\u2141\u2082.run\u2141\u2082 A (not b))\nlem\u2082 A b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b) (kont\u2080-not 1b) \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n  where open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n        open Run\u2141\u2082-Properties A b\n\nlem\u2083 : \u2200 {ca} (A : Adv\u2082 ca) \u2192 Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\nlem\u2083 A = lem\u2082 A 0b\n\n-- A specialized version of lem\u2082\nlem\u2084 : \u2200 {ca} (A : Adv\u2082 ca) \u2192 Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\nlem\u2084 A  = count\u21ba (runA 0b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b) (kont\u2080-not 1b) \u27e9\n          count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n          count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA 1b) \u220e\n  where open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n        open Run\u2141\u2082-Properties A 0b\n","old_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n \nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n \nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n \nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k) \n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  not\u21ba : \u2200 {n} \u2192 EXP n \u2192 EXP n\n  not\u21ba = map\u21ba not\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) (b k : Bit) where\n  open Run\u2141\u2082 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k) \n  kont\u2080-not rewrite xor-not-not b k = refl\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nopen \u2261-Reasoning\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\nlem\u2082 : \u2200 {ca} (A : Adv\u2082 ca) b \u2192 count\u21ba (Run\u2141\u2082.run\u2141\u2082 A b) \u2261 count\u21ba (Run\u2141\u2082.run\u2141\u2082 A (not b))\nlem\u2082 A b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b) (kont\u2080-not 1b) \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n  where open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n        open Run\u2141\u2082-Properties A b\n\nlem\u2083 : \u2200 {ca} (A : Adv\u2082 ca) \u2192 Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\nlem\u2083 A = lem\u2082 A 0b\n\n-- A specialized version of lem\u2082\nlem\u2084 : \u2200 {ca} (A : Adv\u2082 ca) \u2192 Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\nlem\u2084 A  = count\u21ba (runA 0b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b) (kont\u2080-not 1b) \u27e9\n          count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n          count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA 1b) \u220e\n  where open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n        open Run\u2141\u2082-Properties A 0b\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"472f1d2065bc28c6100e66b94daab1293925a60e","subject":"Control\/Protocol\/Choreography.agda","message":"Control\/Protocol\/Choreography.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _ where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    \u214b\u1d3e Q      = Q\n    P      \u214b\u1d3e end    = P\n    com P\u1d38 \u214b\u1d3e com P\u1d3f = \u03a3\u1d3e LR (P\u1d38 \u214b\u1d9c P\u1d3f)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = {!!}\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id = {!!}\n  \n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e P Q p q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V1 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end Q = Q\n    Sim\u1d3e P end = P\n    Sim\u1d3e (com P\u1d38) (com P\u1d3f) = \u03a3\u1d3e LR (Sim\u1d9c P\u1d38 P\u1d3f)\n\n    Sim\u1d9c : Com \u2192 Com \u2192 LR \u2192 Proto\n    Sim\u1d9c P\u1d38 P\u1d3f `L = Sim\u1d9cL P\u1d38 P\u1d3f\n    Sim\u1d9c P\u1d38 P\u1d3f `R = Sim\u1d9cR P\u1d38 P\u1d3f\n\n    Sim\u1d9cL : Com \u2192 Com \u2192 Proto\n    Sim\u1d9cL (mk q\u1d38 M\u1d38 P\u1d38) Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 Sim\u1d3e (P\u1d38 m) (com Q))\n\n    Sim\u1d9cR : Com \u2192 Com \u2192 Proto\n    Sim\u1d9cR P (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 Sim\u1d3e (com P) (P\u1d3f m))\n\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule V3 where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    P \u214b\u1d3e Q = \u03a3\u1d3e LR (Par\u1d3e P Q)\n\n    Par\u1d3e : Proto \u2192 Proto \u2192 LR \u2192 Proto\n    Par\u1d3e end          Q `L = Q\n    Par\u1d3e (com' q M P) Q `L = com' q M \u03bb m \u2192 P m \u214b\u1d3e Q\n    Par\u1d3e P end          `R = P\n    Par\u1d3e P (com' q M Q) `R = com' q M \u03bb m \u2192 P \u214b\u1d3e Q m\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    P ox\u1d3e Q = \u03a0\u1d3e LR (Ten\u1d3e P Q)\n\n    Ten\u1d3e : Proto \u2192 Proto \u2192 LR \u2192 Proto\n    Ten\u1d3e end          Q `L = Q\n    Ten\u1d3e (com' q M P) Q `L = com' q M \u03bb m \u2192 P m ox\u1d3e Q\n    Ten\u1d3e P end          `R = P\n    Ten\u1d3e P (com' q M Q) `R = com' q M \u03bb m \u2192 P ox\u1d3e Q m\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = {!!}\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id = {!!}\n  \n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e P Q p q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fb908ee6dff3ee32438dc93dca5f6c3a00df821f","subject":"Fixed a missing cosmetic change.","message":"Fixed a missing cosmetic change.\n\nIgnore-this: e48756f412daa48aab9a7214ed4a8ac2\n\ndarcs-hash:20110316115851-3bd4e-4fdfe28c58205b26d16932f3d4e1d16208a01338.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/Bisimilarity\/PropertiesI.agda","new_file":"src\/FOTC\/Data\/Stream\/Bisimilarity\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Bisimilarity properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.Bisimilarity.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI using ( \u2237-injective )\n\nopen import FOTC.Data.Stream.Bisimilarity\n\n------------------------------------------------------------------------------\n\nx\u2237xs\u2248x\u2237ys\u2192xs\u2248ys : \u2200 {x xs ys} \u2192 x \u2237 xs \u2248 x \u2237 ys \u2192 xs \u2248 ys\nx\u2237xs\u2248x\u2237ys\u2192xs\u2248ys {x} {xs} {ys} x\u2237xs\u2248x\u2237ys = xs\u2248ys\n  where\n    x' : D\n    x' = \u2203-proj\u2081 (\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys)\n\n    xs' : D\n    xs' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys))\n\n    ys' : D\n    ys' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys)))\n\n    helper : xs' \u2248 ys' \u2227 x \u2237 xs \u2261 x' \u2237 xs' \u2227 x \u2237 ys \u2261 x' \u2237 ys'\n    helper = \u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys)))\n\n    xs'\u2248ys' : xs' \u2248 ys'\n    xs'\u2248ys' = \u2227-proj\u2081 helper\n\n    x\u2237xs\u2261x'\u2237xs' : x \u2237 xs \u2261 x' \u2237 xs'\n    x\u2237xs\u2261x'\u2237xs' = \u2227-proj\u2081 (\u2227-proj\u2082 helper)\n\n    x\u2237ys\u2261x'\u2237ys' : x \u2237 ys \u2261 x' \u2237 ys'\n    x\u2237ys\u2261x'\u2237ys' = \u2227-proj\u2082 (\u2227-proj\u2082 helper)\n\n    xs\u2261xs' : xs \u2261 xs'\n    xs\u2261xs' = \u2227-proj\u2082 (\u2237-injective x\u2237xs\u2261x'\u2237xs')\n\n    ys\u2261ys' : ys \u2261 ys'\n    ys\u2261ys' = \u2227-proj\u2082 (\u2237-injective x\u2237ys\u2261x'\u2237ys')\n\n    xs\u2248ys : xs \u2248 ys\n    xs\u2248ys = subst (\u03bb t \u2192 t \u2248 ys)\n                  (sym xs\u2261xs')\n                  (subst (\u03bb t \u2192 xs' \u2248 t) (sym ys\u2261ys') xs'\u2248ys')\n","old_contents":"------------------------------------------------------------------------------\n-- Bisimilarity properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.Bisimilarity.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI using ( \u2237-injective )\n\nopen import FOTC.Data.Stream.Bisimilarity using ( _\u2248_ ; -\u2248-gfp\u2081 )\n\n------------------------------------------------------------------------------\n\nx\u2237xs\u2248x\u2237ys\u2192xs\u2248ys : \u2200 {x xs ys} \u2192 x \u2237 xs \u2248 x \u2237 ys \u2192 xs \u2248 ys\nx\u2237xs\u2248x\u2237ys\u2192xs\u2248ys {x} {xs} {ys} x\u2237xs\u2248x\u2237ys = xs\u2248ys\n  where\n    x' : D\n    x' = \u2203-proj\u2081 (-\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys)\n\n    xs' : D\n    xs' = \u2203-proj\u2081 (\u2203-proj\u2082 (-\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys))\n\n    ys' : D\n    ys' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 (-\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys)))\n\n    helper : xs' \u2248 ys' \u2227 x \u2237 xs \u2261 x' \u2237 xs' \u2227 x \u2237 ys \u2261 x' \u2237 ys'\n    helper = \u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 (-\u2248-gfp\u2081 x\u2237xs\u2248x\u2237ys)))\n\n    xs'\u2248ys' : xs' \u2248 ys'\n    xs'\u2248ys' = \u2227-proj\u2081 helper\n\n    x\u2237xs\u2261x'\u2237xs' : x \u2237 xs \u2261 x' \u2237 xs'\n    x\u2237xs\u2261x'\u2237xs' = \u2227-proj\u2081 (\u2227-proj\u2082 helper)\n\n    x\u2237ys\u2261x'\u2237ys' : x \u2237 ys \u2261 x' \u2237 ys'\n    x\u2237ys\u2261x'\u2237ys' = \u2227-proj\u2082 (\u2227-proj\u2082 helper)\n\n    xs\u2261xs' : xs \u2261 xs'\n    xs\u2261xs' = \u2227-proj\u2082 (\u2237-injective x\u2237xs\u2261x'\u2237xs')\n\n    ys\u2261ys' : ys \u2261 ys'\n    ys\u2261ys' = \u2227-proj\u2082 (\u2237-injective x\u2237ys\u2261x'\u2237ys')\n\n    xs\u2248ys : xs \u2248 ys\n    xs\u2248ys = subst (\u03bb t \u2192 t \u2248 ys)\n                  (sym xs\u2261xs')\n                  (subst (\u03bb t \u2192 xs' \u2248 t) (sym ys\u2261ys') xs'\u2248ys')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"42316d32e1b90cea16ece1ad29627cb678610bc6","subject":"Clarify that change algebra operations trivially preserve d.o.e.","message":"Clarify that change algebra operations trivially preserve d.o.e.\n\nOld-commit-hash: c11fb0c1abae938f0c7ec3dacff96f112f6bbee4\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely.\n    --\n    -- * That should be be true for\n    --   functions using changes parametrically.\n    --\n    -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n    --   this is proved below on both contexts at once by fun-change-respects.\n    --\n    -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n    --   it by definition, and \u229f doesn't take change arguments - we will only\n    --   need a proof for compose, when we define it.\n    --\n    -- Stating the general result, though, seems hard, we should\n    -- rather have lemmas proving that certain classes of functions respect this\n    -- equivalence.\n\n  module _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n    {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n    module FC = FunctionChanges A B {{CA}} {{CB}}\n    open FC using (changeAlgebra; incrementalization)\n    open FC.FunctionChange\n\n    fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n      df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n    fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n      where\n        open \u2261-Reasoning\n        open import Postulate.Extensionality\n        -- This type signature just expands the goal manually a bit.\n        lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n        -- Informally: use incrementalization on both sides and then apply\n        -- congruence.\n        lemma =\n          begin\n            f x \u229e apply df\u2081 x dx\u2081\n          \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2081)\n          \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2082)\n          \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n            (f \u229e df\u2082) (x \u229e dx\u2082)\n          \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n            f x \u229e apply df\u2082 x dx\u2082\n          \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely.\n    --\n    -- * That should be be true for\n    --   functions using changes parametrically.\n    --\n    -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n    --   this is proved below on both contexts at once by fun-change-respects.\n    --\n    -- * Finally, change algebra operations should respect d.o.e.\n    --\n    -- Stating the general result, though, seems hard, we should\n    -- rather have lemmas proving that certain classes of functions respect this\n    -- equivalence.\n\n  module _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n    {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n    module FC = FunctionChanges A B {{CA}} {{CB}}\n    open FC using (changeAlgebra; incrementalization)\n    open FC.FunctionChange\n\n    fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n      df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n    fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n      where\n        open \u2261-Reasoning\n        open import Postulate.Extensionality\n        -- This type signature just expands the goal manually a bit.\n        lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n        -- Informally: use incrementalization on both sides and then apply\n        -- congruence.\n        lemma =\n          begin\n            f x \u229e apply df\u2081 x dx\u2081\n          \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2081)\n          \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2082)\n          \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n            (f \u229e df\u2082) (x \u229e dx\u2082)\n          \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n            f x \u229e apply df\u2082 x dx\u2082\n          \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0d3012e5695fb8ab9a9f6923e17ada2d99a1b7b6","subject":"Updated a note about the existential quantifier.","message":"Updated a note about the existential quantifier.\n\nIgnore-this: 8ad5316e644cba628b19f41d007d533\n\ndarcs-hash:20111130164653-3bd4e-9e00b7c21e8e910b34ab599f2d86be3c0e7b3570.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Existential.agda","new_file":"notes\/thesis\/logical-framework\/Existential.agda","new_contents":"-- Tested on 30 November 2011.\n\nmodule Existential where\n\npostulate\n  D       : Set\n  \u2203       : (P : D \u2192 Set) \u2192 Set\n  _,_     : {P : D \u2192 Set}(d : D) \u2192 P d \u2192 \u2203 P\n  \u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n  \u2203-proj\u2082 : {P : D \u2192 Set}(p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\npostulate P : D \u2192 D \u2192 Set\n\n\u2203\u2200 : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n\u2203\u2200 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n","old_contents":"module Existential where\n\npostulate\n  D       : Set\n  \u2203       : (P : D \u2192 Set) \u2192 Set\n  _,_     : {P : D \u2192 Set}(d : D) \u2192 P d \u2192 \u2203 P\n  \u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n  \u2203-proj\u2082 : {P : D \u2192 Set}(p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n\npostulate P : D \u2192 D \u2192 Set\n\n\u2203\u2200 : (\u2203 \u03bb x \u2192 \u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203 \u03bb x \u2192 P x y\n\u2203\u2200 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"36716645788ba47dbfa8cbb7912747ea4e52a921","subject":"Improved some proofs.","message":"Improved some proofs.\n\nIgnore-this: e7e0889f7060eb9f3adde2de878ad1d3\n\ndarcs-hash:20100524023850-3bd4e-59dcde27415395528e46b6323c5698f04b5ddac0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\npostulate\n  \u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x x\u22650 #-}\n\npostulate\n  \u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n{-# ATP prove \u00acS\u22640 #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n    -- trans (lt-SS (succ n) zero) (lt-S0 n)\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  prf (\u2264-trans Nm Nn No m\u2264n n\u2264o)\n    where\n      postulate m\u2264n : LE m n\n      {-# ATP prove m\u2264n #-}\n\n      postulate n\u2264o : LE n o\n      {-# ATP prove n\u2264o #-}\n\n      postulate prf : LE m o \u2192 LE (succ m) (succ o)\n      {-# ATP prove prf #-}\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sd\u22640 = \u22a5-elim prf\n  where\n    -- The proof uses the axiom true\u2260false\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n    -- trans (lt-SS (succ n) zero) (lt-S0 n)\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  prf (\u2264-trans Nm Nn No m\u2264n n\u2264o)\n    where\n      postulate m\u2264n : LE m n\n      {-# ATP prove m\u2264n #-}\n\n      postulate n\u2264o : LE n o\n      {-# ATP prove n\u2264o #-}\n\n      postulate prf : LE m o \u2192 LE (succ m) (succ o)\n      {-# ATP prove prf #-}\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b3c29af8a0e31056b3efe8dd76723fe0d834058a","subject":"Add Pullback","message":"Add Pullback\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Type hiding (\u2605)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product using (\u03a3; _,_)\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\nprivate\n  module Dummy {a} {A : \u2605 a} where\n    open IsEquivalence (isEquivalence {a} {A}) public hiding (refl; sym; trans)\nopen Dummy public\n\nPathOver : \u2200 {i j} {A : \u2605 i} (B : A \u2192 \u2605 j)\n  {x y : A} (p : x \u2261 y) (u : B x) (v : B y) \u2192 \u2605 j\nPathOver B refl u v = (u \u2261 v)\n\nsyntax PathOver B p u v =\n  u \u2261 v [ B \u2193 p ]\n\n-- Some definitions with the names from Agda-HoTT\n-- this is only a temporary state of affairs...\nmodule _ {a} {A : \u2605 a} where\n\n    idp : {x : A} \u2192 x \u2261 x\n    idp = refl\n\n    infixr 8 _\u2219_ _\u2219'_\n\n    _\u2219_ _\u2219'_ : {x y z : A} \u2192 (x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n\n    refl \u2219 q  = q\n    q \u2219' refl = q\n\n    ! : {x y : A} \u2192 (x \u2261 y \u2192 y \u2261 x)\n    ! refl = refl\n\nap\u2193 : \u2200 {i j k} {A : \u2605 i} {B : A \u2192 \u2605 j} {C : A \u2192 \u2605 k}\n  (g : {a : A} \u2192 B a \u2192 C a) {x y : A} {p : x \u2261 y}\n  {u : B x} {v : B y}\n  \u2192 (u \u2261 v [ B \u2193 p ] \u2192 g u \u2261 g v [ C \u2193 p ])\nap\u2193 g {p = refl} refl = refl\n\nap : \u2200 {i j} {A : \u2605 i} {B : \u2605 j} (f : A \u2192 B) {x y : A}\n  \u2192 (x \u2261 y \u2192 f x \u2261 f y)\nap f p = ap\u2193 {A = \ud835\udfd9} f {p = refl} p\n\napd : \u2200 {i j} {A : \u2605 i} {B : A \u2192 \u2605 j} (f : (a : A) \u2192 B a) {x y : A}\n  \u2192 (p : x \u2261 y) \u2192 f x \u2261 f y [ B \u2193 p ]\napd f refl = refl\n\ncoe : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 A \u2192 B\ncoe refl x = x\n\ncoe! : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 B \u2192 A\ncoe! refl x = x\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} (f g : A \u2192 B \u2192 C) \u2192 \u2605 _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\nmodule _ {a} {A : \u2605 a} where\n  injective : InjectiveRel A _\u2261_\n  injective p q = p \u2219 ! q\n\n  surjective : SurjectiveRel A _\u2261_\n  surjective p q = ! p \u2219 q\n\n  bijective : BijectiveRel A _\u2261_\n  bijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\n  Equalizer : \u2200 {b} {B : A \u2192 \u2605 b} (f g : (x : A) \u2192 B x) \u2192 \u2605 _\n  Equalizer f g = \u03a3 A (\u03bb x \u2192 f x \u2261 g x)\n  {- In a category theory context this type would the object 'E'\n     and 'proj\u2081' would be the morphism 'eq : E \u2192 A' such that\n     given any object O, and morphism 'm : O \u2192 A', there exists\n     a unique morphism 'u : O \u2192 E' such that 'eq \u2218 u \u2261 m'.\n  -}\n\n  Pullback : \u2200 {b c} {B : \u2605 b} {C : \u2605 c} (f : A \u2192 C) (g : B \u2192 C) \u2192 \u2605 _\n  Pullback {B = B} f g = \u03a3 A (\u03bb x \u2192 \u03a3 B (\u03bb y \u2192 f x \u2261 g y))\n\nmodule \u2261-Reasoning {a} {A : \u2605 a} where\n  open Setoid-Reasoning (setoid A) public renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : \u2605 a} {B : \u2605 b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 \u2605\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","old_contents":"{-# OPTIONS --without-K #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Type hiding (\u2605)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product using (\u03a3; _,_)\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\nprivate\n  module Dummy {a} {A : \u2605 a} where\n    open IsEquivalence (isEquivalence {a} {A}) public hiding (refl; sym; trans)\nopen Dummy public\n\nPathOver : \u2200 {i j} {A : \u2605 i} (B : A \u2192 \u2605 j)\n  {x y : A} (p : x \u2261 y) (u : B x) (v : B y) \u2192 \u2605 j\nPathOver B refl u v = (u \u2261 v)\n\nsyntax PathOver B p u v =\n  u \u2261 v [ B \u2193 p ]\n\n-- Some definitions with the names from Agda-HoTT\n-- this is only a temporary state of affairs...\nmodule _ {a} {A : \u2605 a} where\n\n    idp : {x : A} \u2192 x \u2261 x\n    idp = refl\n\n    infixr 8 _\u2219_ _\u2219'_\n\n    _\u2219_ _\u2219'_ : {x y z : A} \u2192 (x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n\n    refl \u2219 q  = q\n    q \u2219' refl = q\n\n    ! : {x y : A} \u2192 (x \u2261 y \u2192 y \u2261 x)\n    ! refl = refl\n\nap\u2193 : \u2200 {i j k} {A : \u2605 i} {B : A \u2192 \u2605 j} {C : A \u2192 \u2605 k}\n  (g : {a : A} \u2192 B a \u2192 C a) {x y : A} {p : x \u2261 y}\n  {u : B x} {v : B y}\n  \u2192 (u \u2261 v [ B \u2193 p ] \u2192 g u \u2261 g v [ C \u2193 p ])\nap\u2193 g {p = refl} refl = refl\n\nap : \u2200 {i j} {A : \u2605 i} {B : \u2605 j} (f : A \u2192 B) {x y : A}\n  \u2192 (x \u2261 y \u2192 f x \u2261 f y)\nap f p = ap\u2193 {A = \ud835\udfd9} f {p = refl} p\n\napd : \u2200 {i j} {A : \u2605 i} {B : A \u2192 \u2605 j} (f : (a : A) \u2192 B a) {x y : A}\n  \u2192 (p : x \u2261 y) \u2192 f x \u2261 f y [ B \u2193 p ]\napd f refl = refl\n\ncoe : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 A \u2192 B\ncoe refl x = x\n\ncoe! : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 B \u2192 A\ncoe! refl x = x\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} (f g : A \u2192 B \u2192 C) \u2192 \u2605 _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\nmodule _ {a} {A : \u2605 a} where\n  injective : InjectiveRel A _\u2261_\n  injective p q = p \u2219 ! q\n\n  surjective : SurjectiveRel A _\u2261_\n  surjective p q = ! p \u2219 q\n\n  bijective : BijectiveRel A _\u2261_\n  bijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\n  Equalizer : \u2200 {b} {B : A \u2192 \u2605 b} (f g : (x : A) \u2192 B x) \u2192 \u2605 _\n  Equalizer f g = \u03a3 A (\u03bb x \u2192 f x \u2261 g x)\n  {- In a category theory context this type would the object 'E'\n     and 'proj\u2081' would be the morphism 'eq : E \u2192 A' such that\n     given any object O, and morphism 'm : O \u2192 A', there exists\n     a unique morphism 'u : O \u2192 E' such that 'eq \u2218 u \u2261 m'.\n  -}\n\nmodule \u2261-Reasoning {a} {A : \u2605 a} where\n  open Setoid-Reasoning (setoid A) public renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : \u2605 a} {B : \u2605 b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 \u2605\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0b268926b3f7accc03c3dc681afd337d913759cd","subject":"Added k+x<k+y\u2192x<y, x+k<y+k\u2192x<y, x\u2264y\u2192k+x\u2264k+y, and x\u2264y\u2192x+k\u2264y+k (by Ana Bove).","message":"Added k+x<k+y\u2192x<y, x+k<y+k\u2192x<y, x\u2264y\u2192k+x\u2264k+y, and x\u2264y\u2192x+k\u2264y+k (by Ana Bove).\n\nIgnore-this: 19939ad70cf6f4e2b6769c003e6cac94\n\ndarcs-hash:20110211223354-3bd4e-20a00dcb2be3438bc3743b5a7c5fb70375b9d550.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ \u00ac0<0 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ \u00acS<0 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ \u00acS<0 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x-y\u21920<Sx-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x-y\u21920<Sx-y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x-y\u21920<Sx-y zN (sN {n} Nn) h = \u22a5-elim (\u00acx<0 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x-y\u21920<Sx-y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x-y\u21920<Sx-y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ \u00ac0<0 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ \u00acS<0 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ \u00acS<0 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x-y\u21920<Sx-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x-y\u21920<Sx-y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x-y\u21920<Sx-y zN (sN {n} Nn) h = \u22a5-elim (\u00acx<0 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x-y\u21920<Sx-y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x-y\u21920<Sx-y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9e5c8303ebcc88a2fe9454cf905b889947390eca","subject":"Added _,\u2032_.","message":"Added _,\u2032_.\n\nIgnore-this: 34840168d583b411434a67ebf23718cd\n\ndarcs-hash:20110527133955-3bd4e-2f779c560176c046f7fa6fe1f150539215ad6c34.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Data\/Product.agda","new_file":"src\/Common\/Data\/Product.agda","new_contents":"------------------------------------------------------------------------------\n-- Products\n------------------------------------------------------------------------------\n\nmodule Common.Data.Product where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_ _,\u2032_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n\n_,\u2032_ : \u2200 {A B : Set} \u2192 A \u2192 B \u2192 A \u2227 B\n_,\u2032_ = _,_\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (witness : D) \u2192 P witness \u2192 \u2203 P\n\n\u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : {P : D \u2192 Set}(\u2203p : \u2203 P) \u2192 P (\u2203-proj\u2081 \u2203p)\n\u2203-proj\u2082 (_ , px) = px\n","old_contents":"------------------------------------------------------------------------------\n-- Products\n------------------------------------------------------------------------------\n\nmodule Common.Data.Product where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (witness : D) \u2192 P witness \u2192 \u2203 P\n\n\u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : {P : D \u2192 Set}(\u2203p : \u2203 P) \u2192 P (\u2203-proj\u2081 \u2203p)\n\u2203-proj\u2082 (_ , px) = px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e58fdcf2bd3f99745f653c3ee049cb1244bcb092","subject":"Rename by-definition combinator for eq-reasoning","message":"Rename by-definition combinator for eq-reasoning\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/NP.agda","new_file":"lib\/Relation\/Binary\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Refl-Trans-Reasoning\n         {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113)\n         (refl : Reflexive _\u2248_)\n         (trans : Transitive _\u2248_) where\n\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\n  infixr 2 _\u2248\u27e8by-definition\u27e9_\n\n  _\u2248\u27e8by-definition\u27e9_ : \u2200 x {y : A} \u2192 x \u2248 y \u2192 x \u2248 y\n  _ \u2248\u27e8by-definition\u27e9 x\u2248y = x\u2248y\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Refl-Trans-Reasoning _\u2248_ refl trans public\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Refl-Trans-Reasoning _\u223c_ refl trans public\n     renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_; _\u2248\u27e8by-definition\u27e9_ to _\u223c\u27e8by-definition\u27e9_)\n  open Equivalence-Reasoning isEquivalence public renaming (_\u220e to _\u2610)\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Equivalence-Reasoning (Setoid.isEquivalence s) public\n","old_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Refl-Trans-Reasoning\n         {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113)\n         (refl : Reflexive _\u2248_)\n         (trans : Transitive _\u2248_) where\n\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\n  infixr 2 _\u2248\u27e8-by-definition-\u27e9_\n\n  _\u2248\u27e8-by-definition-\u27e9_ : \u2200 x {y : A} \u2192 x \u2248 y \u2192 x \u2248 y\n  _ \u2248\u27e8-by-definition-\u27e9 x\u2248y = x\u2248y\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Refl-Trans-Reasoning _\u2248_ refl trans public\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Refl-Trans-Reasoning _\u223c_ refl trans public\n     renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_; _\u2248\u27e8-by-definition-\u27e9_ to _\u223c\u27e8-by-definition-\u27e9_)\n  open Equivalence-Reasoning isEquivalence public renaming (_\u220e to _\u2610)\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Equivalence-Reasoning (Setoid.isEquivalence s) public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4994e30047dd8c471fb426bcca46a63570b95e9d","subject":"Relation.Binary.Refl-Trans-Reasoning","message":"Relation.Binary.Refl-Trans-Reasoning\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/NP.agda","new_file":"lib\/Relation\/Binary\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Refl-Trans-Reasoning\n         {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113)\n         (refl : Reflexive _\u2248_)\n         (trans : Transitive _\u2248_) where\n\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Refl-Trans-Reasoning _\u2248_ refl trans public\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Refl-Trans-Reasoning _\u223c_ refl trans public renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_)\n  open Equivalence-Reasoning isEquivalence public renaming (_\u220e to _\u2610)\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Equivalence-Reasoning (Setoid.isEquivalence s) public\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Trans-Reasoning _\u223c_ trans public hiding (finally) renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_)\n  open Equivalence-Reasoning isEquivalence public renaming (_\u220e to _\u2610)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _ \u220e = refl\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Equivalence-Reasoning (Setoid.isEquivalence s) public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a0cb2129d6925e76e7c69250fbe510b3a711b245","subject":"Correct code in comment","message":"Correct code in comment\n\nmap-IVT does not exist currently (not sure it ever did, I did not\ninvestigate); what we want and does not work is simply map. Hence, fix\nthe comment to clarify it.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Evaluation.agda","new_file":"Parametric\/Denotation\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map-IVT (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"caab0f0a9ebc53432d697d8dfef61fb6cae09b67","subject":"Equivalence: Prepare for splitting","message":"Equivalence: Prepare for splitting\n\nThe file has become too big.\n\nSplitting the anonymous modules requires more work out of type inference\nand introduces one type inference failure, which is fixed in this commmit.\n\nOld-commit-hash: 32f42f80b60cd3dd45e879d7a367b4fe933bb9d8\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR (\u2259-setoid {x = x}) public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fcd30d55c5986f78d2f7059910fd4cf1194fd76d","subject":"reexport _\u00d7-cong_ _\u228e-cong_","message":"reexport _\u00d7-cong_ _\u228e-cong_\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise public using (_\u00d7-cong_)\nopen import Relation.Binary.Sum public using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n-- {-\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n{-\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n-- {-\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n{-\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4e1289551ff82bc8329c3c1caf7e5cb6ef47ade7","subject":"Only indentation.","message":"Only indentation.\n\nIgnore-this: f64e1fdc84de57eb5a1dda281b14b019\n\ndarcs-hash:20110209033859-3bd4e-244c58fa7b5bd53e5342b1b247aff5012816d9bd.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/Numbers.agda","new_file":"Draft\/McCarthy91\/Numbers.agda","new_contents":"------------------------------------------------------------------------------\n-- Some naturales numbers\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Numbers where\n\nopen import LTC.Base\n\n------------------------------------------------------------------------------\n\none    = succ zero\ntwo    = succ one\nthree  = succ two\nfour   = succ three\nfive   = succ four\nsix    = succ five\nseven  = succ six\neight  = succ seven\nnine   = succ eight\nten    = succ nine\n\n{-# ATP definition one #-}\n{-# ATP definition two #-}\n{-# ATP definition three #-}\n{-# ATP definition four #-}\n{-# ATP definition five #-}\n{-# ATP definition six #-}\n{-# ATP definition seven #-}\n{-# ATP definition eight #-}\n{-# ATP definition nine #-}\n{-# ATP definition ten #-}\n\neleven    = succ ten\ntwelve    = succ eleven\nthirteen  = succ twelve\nfourteen  = succ thirteen\nfifteen   = succ fourteen\nsixteen   = succ fifteen\nseventeen = succ sixteen\neighteen  = succ seventeen\nnineteen  = succ eighteen\ntwenty    = succ nineteen\n\n{-# ATP definition eleven #-}\n{-# ATP definition twelve #-}\n{-# ATP definition thirteen #-}\n{-# ATP definition fourteen #-}\n{-# ATP definition fifteen #-}\n{-# ATP definition sixteen #-}\n{-# ATP definition seventeen #-}\n{-# ATP definition eighteen #-}\n{-# ATP definition nineteen #-}\n{-# ATP definition twenty #-}\n\ntwenty-one   = succ twenty\ntwenty-two   = succ twenty-one\ntwenty-three = succ twenty-two\ntwenty-four  = succ twenty-three\ntwenty-five  = succ twenty-four\ntwenty-six   = succ twenty-five\ntwenty-seven = succ twenty-six\ntwenty-eight = succ twenty-seven\ntwenty-nine  = succ twenty-eight\nthirty       = succ twenty-nine\n\n{-# ATP definition twenty-one #-}\n{-# ATP definition twenty-two #-}\n{-# ATP definition twenty-three #-}\n{-# ATP definition twenty-four #-}\n{-# ATP definition twenty-five #-}\n{-# ATP definition twenty-six #-}\n{-# ATP definition twenty-seven #-}\n{-# ATP definition twenty-eight #-}\n{-# ATP definition twenty-nine #-}\n{-# ATP definition thirty #-}\n\nthirty-one   = succ thirty\nthirty-two   = succ thirty-one\nthirty-three = succ thirty-two\nthirty-four  = succ thirty-three\nthirty-five  = succ thirty-four\nthirty-six   = succ thirty-five\nthirty-seven = succ thirty-six\nthirty-eight = succ thirty-seven\nthirty-nine  = succ thirty-eight\nforty        = succ thirty-nine\n\n{-# ATP definition thirty-one #-}\n{-# ATP definition thirty-two #-}\n{-# ATP definition thirty-three #-}\n{-# ATP definition thirty-four #-}\n{-# ATP definition thirty-five #-}\n{-# ATP definition thirty-six #-}\n{-# ATP definition thirty-seven #-}\n{-# ATP definition thirty-eight #-}\n{-# ATP definition thirty-nine #-}\n{-# ATP definition forty #-}\n\nforty-one   = succ forty\nforty-two   = succ forty-one\nforty-three = succ forty-two\nforty-four  = succ forty-three\nforty-five  = succ forty-four\nforty-six   = succ forty-five\nforty-seven = succ forty-six\nforty-eight = succ forty-seven\nforty-nine  = succ forty-eight\nfifty       = succ forty-nine\n\n{-# ATP definition forty-one #-}\n{-# ATP definition forty-two #-}\n{-# ATP definition forty-three #-}\n{-# ATP definition forty-four #-}\n{-# ATP definition forty-five #-}\n{-# ATP definition forty-six #-}\n{-# ATP definition forty-seven #-}\n{-# ATP definition forty-eight #-}\n{-# ATP definition forty-nine #-}\n{-# ATP definition fifty #-}\n\nfifty-one   = succ fifty\nfifty-two   = succ fifty-one\nfifty-three = succ fifty-two\nfifty-four  = succ fifty-three\nfifty-five  = succ fifty-four\nfifty-six   = succ fifty-five\nfifty-seven = succ fifty-six\nfifty-eight = succ fifty-seven\nfifty-nine  = succ fifty-eight\nsixty       = succ fifty-nine\n\n{-# ATP definition fifty-one #-}\n{-# ATP definition fifty-two #-}\n{-# ATP definition fifty-three #-}\n{-# ATP definition fifty-four #-}\n{-# ATP definition fifty-five #-}\n{-# ATP definition fifty-six #-}\n{-# ATP definition fifty-seven #-}\n{-# ATP definition fifty-eight #-}\n{-# ATP definition fifty-nine #-}\n{-# ATP definition sixty #-}\n\nsixty-one   = succ sixty\nsixty-two   = succ sixty-one\nsixty-three = succ sixty-two\nsixty-four  = succ sixty-three\nsixty-five  = succ sixty-four\nsixty-six   = succ sixty-five\nsixty-seven = succ sixty-six\nsixty-eight = succ sixty-seven\nsixty-nine  = succ sixty-eight\nseventy     = succ sixty-nine\n\n{-# ATP definition sixty-one #-}\n{-# ATP definition sixty-two #-}\n{-# ATP definition sixty-three #-}\n{-# ATP definition sixty-four #-}\n{-# ATP definition sixty-five #-}\n{-# ATP definition sixty-six #-}\n{-# ATP definition sixty-seven #-}\n{-# ATP definition sixty-eight #-}\n{-# ATP definition sixty-nine #-}\n{-# ATP definition seventy #-}\n\nseventy-one   = succ seventy\nseventy-two   = succ seventy-one\nseventy-three = succ seventy-two\nseventy-four  = succ seventy-three\nseventy-five  = succ seventy-four\nseventy-six   = succ seventy-five\nseventy-seven = succ seventy-six\nseventy-eight = succ seventy-seven\nseventy-nine  = succ seventy-eight\neighty        = succ seventy-nine\n\n{-# ATP definition seventy-one #-}\n{-# ATP definition seventy-two #-}\n{-# ATP definition seventy-three #-}\n{-# ATP definition seventy-four #-}\n{-# ATP definition seventy-five #-}\n{-# ATP definition seventy-six #-}\n{-# ATP definition seventy-seven #-}\n{-# ATP definition seventy-eight #-}\n{-# ATP definition seventy-nine #-}\n{-# ATP definition eighty #-}\n\neighty-one   = succ eighty\neighty-two   = succ eighty-one\neighty-three = succ eighty-two\neighty-four  = succ eighty-three\neighty-five  = succ eighty-four\neighty-six   = succ eighty-five\neighty-seven = succ eighty-six\neighty-eight = succ eighty-seven\neighty-nine  = succ eighty-eight\nninety       = succ eighty-nine\n\n{-# ATP definition eighty-one #-}\n{-# ATP definition eighty-two #-}\n{-# ATP definition eighty-three #-}\n{-# ATP definition eighty-four #-}\n{-# ATP definition eighty-five #-}\n{-# ATP definition eighty-six #-}\n{-# ATP definition eighty-seven #-}\n{-# ATP definition eighty-eight #-}\n{-# ATP definition eighty-nine #-}\n{-# ATP definition ninety #-}\n\nninety-one   = succ ninety\nninety-two   = succ ninety-one\nninety-three = succ ninety-two\nninety-four  = succ ninety-three\nninety-five  = succ ninety-four\nninety-six   = succ ninety-five\nninety-seven = succ ninety-six\nninety-eight = succ ninety-seven\nninety-nine  = succ ninety-eight\none-hundred  = succ ninety-nine\n\n{-# ATP definition ninety-one #-}\n{-# ATP definition ninety-two #-}\n{-# ATP definition ninety-three #-}\n{-# ATP definition ninety-four #-}\n{-# ATP definition ninety-five #-}\n{-# ATP definition ninety-six #-}\n{-# ATP definition ninety-seven #-}\n{-# ATP definition ninety-eight #-}\n{-# ATP definition ninety-nine #-}\n{-# ATP definition one-hundred #-}\n\nhundred-one   = succ one-hundred\nhundred-two   = succ hundred-one\nhundred-three = succ hundred-two\nhundred-four  = succ hundred-three\nhundred-five  = succ hundred-four\nhundred-six   = succ hundred-five\nhundred-seven = succ hundred-six\nhundred-eight = succ hundred-seven\nhundred-nine  = succ hundred-eight\nhundred-ten   = succ hundred-nine\n\n{-# ATP definition hundred-one #-}\n{-# ATP definition hundred-two #-}\n{-# ATP definition hundred-three #-}\n{-# ATP definition hundred-four #-}\n{-# ATP definition hundred-five #-}\n{-# ATP definition hundred-six #-}\n{-# ATP definition hundred-seven #-}\n{-# ATP definition hundred-eight #-}\n{-# ATP definition hundred-nine #-}\n{-# ATP definition hundred-ten #-}\n\nhundred-eleven = succ hundred-ten\n{-# ATP definition hundred-eleven #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some naturales numbers\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Numbers where\n\nopen import LTC.Base\n\n------------------------------------------------------------------------------\n\none    = succ zero\ntwo    = succ one\nthree  = succ two\nfour   = succ three\nfive   = succ four\nsix    = succ five\nseven  = succ six\neight  = succ seven\nnine   = succ eight\nten    = succ nine\n\n{-# ATP definition one #-}\n{-# ATP definition two #-}\n{-# ATP definition three #-}\n{-# ATP definition four #-}\n{-# ATP definition five #-}\n{-# ATP definition six #-}\n{-# ATP definition seven #-}\n{-# ATP definition eight #-}\n{-# ATP definition nine #-}\n{-# ATP definition ten #-}\n\neleven    = succ ten\ntwelve    = succ eleven\nthirteen  = succ twelve\nfourteen  = succ thirteen\nfifteen   = succ fourteen\nsixteen   = succ fifteen\nseventeen = succ sixteen\neighteen  = succ seventeen\nnineteen  = succ eighteen\ntwenty    = succ nineteen\n\n{-# ATP definition eleven #-}\n{-# ATP definition twelve #-}\n{-# ATP definition thirteen #-}\n{-# ATP definition fourteen #-}\n{-# ATP definition fifteen #-}\n{-# ATP definition sixteen #-}\n{-# ATP definition seventeen #-}\n{-# ATP definition eighteen #-}\n{-# ATP definition nineteen #-}\n{-# ATP definition twenty #-}\n\ntwenty-one   = succ twenty\ntwenty-two   = succ twenty-one\ntwenty-three = succ twenty-two\ntwenty-four  = succ twenty-three\ntwenty-five  = succ twenty-four\ntwenty-six   = succ twenty-five\ntwenty-seven = succ twenty-six\ntwenty-eight = succ twenty-seven\ntwenty-nine  = succ twenty-eight\nthirty       = succ twenty-nine\n\n{-# ATP definition twenty-one #-}\n{-# ATP definition twenty-two #-}\n{-# ATP definition twenty-three #-}\n{-# ATP definition twenty-four #-}\n{-# ATP definition twenty-five #-}\n{-# ATP definition twenty-six #-}\n{-# ATP definition twenty-seven #-}\n{-# ATP definition twenty-eight #-}\n{-# ATP definition twenty-nine #-}\n{-# ATP definition thirty #-}\n\nthirty-one   = succ thirty\nthirty-two   = succ thirty-one\nthirty-three = succ thirty-two\nthirty-four  = succ thirty-three\nthirty-five  = succ thirty-four\nthirty-six   = succ thirty-five\nthirty-seven = succ thirty-six\nthirty-eight = succ thirty-seven\nthirty-nine  = succ thirty-eight\nforty        = succ thirty-nine\n\n{-# ATP definition thirty-one #-}\n{-# ATP definition thirty-two #-}\n{-# ATP definition thirty-three #-}\n{-# ATP definition thirty-four #-}\n{-# ATP definition thirty-five #-}\n{-# ATP definition thirty-six #-}\n{-# ATP definition thirty-seven #-}\n{-# ATP definition thirty-eight #-}\n{-# ATP definition thirty-nine #-}\n{-# ATP definition forty #-}\n\nforty-one   = succ forty\nforty-two   = succ forty-one\nforty-three = succ forty-two\nforty-four  = succ forty-three\nforty-five  = succ forty-four\nforty-six   = succ forty-five\nforty-seven = succ forty-six\nforty-eight = succ forty-seven\nforty-nine  = succ forty-eight\nfifty       = succ forty-nine\n\n{-# ATP definition forty-one #-}\n{-# ATP definition forty-two #-}\n{-# ATP definition forty-three #-}\n{-# ATP definition forty-four #-}\n{-# ATP definition forty-five #-}\n{-# ATP definition forty-six #-}\n{-# ATP definition forty-seven #-}\n{-# ATP definition forty-eight #-}\n{-# ATP definition forty-nine #-}\n{-# ATP definition fifty #-}\n\nfifty-one   = succ fifty\nfifty-two   = succ fifty-one\nfifty-three = succ fifty-two\nfifty-four  = succ fifty-three\nfifty-five  = succ fifty-four\nfifty-six   = succ fifty-five\nfifty-seven = succ fifty-six\nfifty-eight = succ fifty-seven\nfifty-nine  = succ fifty-eight\nsixty       = succ fifty-nine\n\n{-# ATP definition fifty-one #-}\n{-# ATP definition fifty-two #-}\n{-# ATP definition fifty-three #-}\n{-# ATP definition fifty-four #-}\n{-# ATP definition fifty-five #-}\n{-# ATP definition fifty-six #-}\n{-# ATP definition fifty-seven #-}\n{-# ATP definition fifty-eight #-}\n{-# ATP definition fifty-nine #-}\n{-# ATP definition sixty #-}\n\nsixty-one   = succ sixty\nsixty-two   = succ sixty-one\nsixty-three = succ sixty-two\nsixty-four  = succ sixty-three\nsixty-five  = succ sixty-four\nsixty-six   = succ sixty-five\nsixty-seven = succ sixty-six\nsixty-eight = succ sixty-seven\nsixty-nine  = succ sixty-eight\nseventy     = succ sixty-nine\n\n{-# ATP definition sixty-one #-}\n{-# ATP definition sixty-two #-}\n{-# ATP definition sixty-three #-}\n{-# ATP definition sixty-four #-}\n{-# ATP definition sixty-five #-}\n{-# ATP definition sixty-six #-}\n{-# ATP definition sixty-seven #-}\n{-# ATP definition sixty-eight #-}\n{-# ATP definition sixty-nine #-}\n{-# ATP definition seventy #-}\n\nseventy-one   = succ seventy\nseventy-two   = succ seventy-one\nseventy-three = succ seventy-two\nseventy-four  = succ seventy-three\nseventy-five  = succ seventy-four\nseventy-six   = succ seventy-five\nseventy-seven = succ seventy-six\nseventy-eight = succ seventy-seven\nseventy-nine  = succ seventy-eight\neighty       = succ seventy-nine\n\n{-# ATP definition seventy-one #-}\n{-# ATP definition seventy-two #-}\n{-# ATP definition seventy-three #-}\n{-# ATP definition seventy-four #-}\n{-# ATP definition seventy-five #-}\n{-# ATP definition seventy-six #-}\n{-# ATP definition seventy-seven #-}\n{-# ATP definition seventy-eight #-}\n{-# ATP definition seventy-nine #-}\n{-# ATP definition eighty #-}\n\neighty-one   = succ eighty\neighty-two   = succ eighty-one\neighty-three = succ eighty-two\neighty-four  = succ eighty-three\neighty-five  = succ eighty-four\neighty-six   = succ eighty-five\neighty-seven = succ eighty-six\neighty-eight = succ eighty-seven\neighty-nine  = succ eighty-eight\nninety       = succ eighty-nine\n\n{-# ATP definition eighty-one #-}\n{-# ATP definition eighty-two #-}\n{-# ATP definition eighty-three #-}\n{-# ATP definition eighty-four #-}\n{-# ATP definition eighty-five #-}\n{-# ATP definition eighty-six #-}\n{-# ATP definition eighty-seven #-}\n{-# ATP definition eighty-eight #-}\n{-# ATP definition eighty-nine #-}\n{-# ATP definition ninety #-}\n\nninety-one   = succ ninety\nninety-two   = succ ninety-one\nninety-three = succ ninety-two\nninety-four  = succ ninety-three\nninety-five  = succ ninety-four\nninety-six   = succ ninety-five\nninety-seven = succ ninety-six\nninety-eight = succ ninety-seven\nninety-nine  = succ ninety-eight\none-hundred  = succ ninety-nine\n\n{-# ATP definition ninety-one #-}\n{-# ATP definition ninety-two #-}\n{-# ATP definition ninety-three #-}\n{-# ATP definition ninety-four #-}\n{-# ATP definition ninety-five #-}\n{-# ATP definition ninety-six #-}\n{-# ATP definition ninety-seven #-}\n{-# ATP definition ninety-eight #-}\n{-# ATP definition ninety-nine #-}\n{-# ATP definition one-hundred #-}\n\nhundred-one   = succ one-hundred\nhundred-two   = succ hundred-one\nhundred-three = succ hundred-two\nhundred-four  = succ hundred-three\nhundred-five  = succ hundred-four\nhundred-six   = succ hundred-five\nhundred-seven = succ hundred-six\nhundred-eight = succ hundred-seven\nhundred-nine  = succ hundred-eight\nhundred-ten   = succ hundred-nine\n\n{-# ATP definition hundred-one #-}\n{-# ATP definition hundred-two #-}\n{-# ATP definition hundred-three #-}\n{-# ATP definition hundred-four #-}\n{-# ATP definition hundred-five #-}\n{-# ATP definition hundred-six #-}\n{-# ATP definition hundred-seven #-}\n{-# ATP definition hundred-eight #-}\n{-# ATP definition hundred-nine #-}\n{-# ATP definition hundred-ten #-}\n\nhundred-eleven = succ hundred-ten\n{-# ATP definition hundred-eleven #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"929d2eace2cfc7a66c7c65a54ee19685ca37f159","subject":"Updated note on the existential quantifier.","message":"Updated note on the existential quantifier.\n\nIgnore-this: 42acf883a18689f1628a5fa5c479f987\n\ndarcs-hash:20120423162449-3bd4e-35b3d060e1280821b05f7c7b5e3c08ab39f89d37.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Existential.agda","new_file":"notes\/thesis\/logical-framework\/Existential.agda","new_contents":"-- Tested with FOT on 23 April 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Existential where\n\nmodule LF where\n  postulate\n    D       : Set\n    \u2203       : (A : D \u2192 Set) \u2192 Set\n    _,_     : {A : D \u2192 Set}(x : D) \u2192 A x \u2192 \u2203 A\n    \u2203-proj\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    \u2203-proj\u2082 : {A : D \u2192 Set}(a : \u2203 A) \u2192 A (\u2203-proj\u2081 a)\n    \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  module FOL-Example where\n    postulate A : D \u2192 D \u2192 Set\n\n    -- Using the projections.\n    \u2203\u2200\u2081 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    -- Using the elimination\n    \u2203\u2200\u2082 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} h\u2081 \u2192 x , h\u2081 y)\n\n  module NonFOL-Example where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\nmodule Inductive where\n\n  open import Common.FOL.FOL\n\n  -- The existential proyections.\n  \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n  \u2203-proj\u2082 (_ , Ax) = Ax\n\n  -- The existential elimination.\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (_ , Ax) h = h Ax\n\n  module FOL-Example where\n    postulate A : D \u2192 D \u2192 Set\n\n    -- Using the projections.\n    \u2203\u2200\u2081 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    -- Using the elimination.\n    \u2203\u2200\u2082 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} h\u2081 \u2192 x , h\u2081 y)\n\n    -- Using pattern matching.\n    \u2203\u2200\u2083 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2083 (x , Ax) y = x , Ax y\n\n  module NonFOL-Example where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\n    -- Using the pattern matching.\n    non-FOL\u2083 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2083 (x , _) = x\n","old_contents":"-- Tested with FOT on 16 February 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Existential where\n\nmodule LF where\n  postulate\n    D       : Set\n    \u2203       : (A : D \u2192 Set) \u2192 Set\n    _,_     : {A : D \u2192 Set}(x : D) \u2192 A x \u2192 \u2203 A\n    \u2203-proj\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    \u2203-proj\u2082 : {A : D \u2192 Set}(a : \u2203 A) \u2192 A (\u2203-proj\u2081 p)\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  postulate A : D \u2192 D \u2192 Set\n\n  \u2203\u2200 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n  \u2203\u2200 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\nmodule Inductive where\n\n  open import Common.Universe\n  open import Common.Data.Product\n\n  postulate A : D \u2192 D \u2192 Set\n\n  \u2203\u2200-el : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n  \u2203\u2200-el h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n  \u2203\u2200 : \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n  \u2203\u2200 (x , Ax) y = x , Ax y\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93b21c38564d9ad524fbd8dd5f62c5be86166146","subject":"ThreeBallot.agda","message":"ThreeBallot.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"ThreeBallot.agda","new_file":"ThreeBallot.agda","new_contents":"module ThreeBallot\n   -- ()\n   where\n\nopen import Data.Fin using (Fin ; zero ; suc )\nopen import Data.Nat\nopen import Data.Bool.NP using (to\u2115) renaming (false to 0b ; true to 1b)\n\nopen import Data.List using (List ; [] ; _\u2237_ ; length)\nopen import Data.Bits hiding (to\u2115 ; 0b ; 1b)\n\nopen import Data.Vec.NP renaming (map to vmap)\nopen import Data.Product\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; refl)\n\ndata Cand : Set where\n  Nice'Guy Bad'Guy : Cand\n\nVote = Bits 2\nTag  = \u2115\n\nrecord Ballot : Set where\n  constructor mk\n  field\n    vote : Vote\n    tag  : Tag\n\nopen Ballot\n\nrecord 3'_ (A : Set) : Set where\n  constructor [_][_][_]\n  field\n    1t 2t 3t : A\n\n3Vote = 3' Vote\n3Ballot = 3' Ballot\n\n[_][_] : Bit \u2192 Bit \u2192 Bits 2\n[ x ][ y ] = x \u2237 y \u2237 []\n\nsvote : Vote\nsvote = [ 0b ][ 1b ]\n\nexample : 3Vote\nexample = [ [ 1b ][ 0b ] ][ [ 0b ][ 1b ] ][ [ 0b ][ 1b ] ]\n\nrow-const : Bit \u2192 3' Bit \u2192 Set\nrow-const b [ 1t ][ 2t ][ 3t ] = 1 + to\u2115 b \u2261 to\u2115 1t + to\u2115 2t + to\u2115 3t\n\nvoteFor : Vote \u2192 3Vote \u2192 Set\nvoteFor (n \u2237 b \u2237 []) [ x \u2237 x\u2081 \u2237 [] ][ x\u2082 \u2237 x\u2083 \u2237 [] ][ x\u2084 \u2237 x\u2085 \u2237 [] ] = row-const n [ x ][ x\u2082 ][ x\u2084 ]\n                                                                     \u00d7 row-const b [ x\u2081 ][ x\u2083 ][ x\u2085 ]\n\nvote-example : voteFor svote example\nvote-example = refl , refl\n\ncast : (_ _ : Vote) \u2192 Vote\ncast i r = vnot r\n\n\nvote-any : (intent receipt : Vote) \u2192 voteFor intent [ intent ][ receipt ][ cast intent receipt ]\nvote-any (true \u2237 true \u2237 []) (true \u2237 true \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (true \u2237 false \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (false \u2237 true \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (false \u2237 false \u2237 []) = refl , refl\nvote-any (true \u2237 false \u2237 []) (true \u2237 true \u2237 []) = refl , refl\nvote-any (true \u2237 false \u2237 []) (true \u2237 false \u2237 []) = refl , refl\nvote-any (true \u2237 false \u2237 []) (false \u2237 true \u2237 []) = refl , refl\nvote-any (true \u2237 false \u2237 []) (false \u2237 false \u2237 []) = refl , refl\nvote-any (false \u2237 true \u2237 []) (true \u2237 true \u2237 []) = refl , refl\nvote-any (false \u2237 true \u2237 []) (true \u2237 false \u2237 []) = refl , refl\nvote-any (false \u2237 true \u2237 []) (false \u2237 true \u2237 []) = refl , refl\nvote-any (false \u2237 true \u2237 []) (false \u2237 false \u2237 []) = refl , refl\nvote-any (false \u2237 false \u2237 []) (true \u2237 true \u2237 []) = refl , refl\nvote-any (false \u2237 false \u2237 []) (true \u2237 false \u2237 []) = refl , refl\nvote-any (false \u2237 false \u2237 []) (false \u2237 true \u2237 []) = refl , refl\nvote-any (false \u2237 false \u2237 []) (false \u2237 false \u2237 []) = refl , refl\n\n{-\n\nNP :\n\n  (R?\u00d7V)^n \u2192 (3V)^n \u2192 (3B)^n \u2192 B^3n -\u27e8 shuffle \u27e9\u2192 b^3n \u2192 T\n\n-}\n\nPerm : \u2115 \u2192 Set\nPerm n = Fin n \u2194 Fin n\n\nshuffle : \u2200 {n}{A : Set} \u2192 Perm n \u2192 Vec A n \u2192 Vec A n\nshuffle p xs = tabulate (\u03bb i \u2192 lookup (to p i) xs)\n\nmodule _\n  (tally : \u2200 {n} \u2192 Vec Vote n \u2192 \u2115)\n  (tally-stable : \u2200 {n} vs \u03c0 \u2192 tally {n} vs \u2261 tally (shuffle \u03c0 vs))\n  where\n\n  3-tally : \u2200 {n} \u2192 Vec Ballot (3 * n) \u2192 \u2115\n  3-tally {n} bs = tally (vmap vote bs) \u2238 n\n\n\n  lookup-map : \u2200 {n}{A B : Set}(f : A \u2192 B)(i : Fin n)(xs : Vec A n)\n             \u2192 f (lookup i xs) \u2261 lookup i (vmap f xs)\n  lookup-map f zero (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  -- map g (tabulate f) = tabulate (g \u2218 f)\n  -- f (lookup i xs) \u2261 lookup i (map f xs)\n  3-tally-stable : \u2200 {n} bs \u03c0 \u2192 3-tally {n} bs \u2261 3-tally {n} (shuffle \u03c0 bs)\n  3-tally-stable {n} bs p rewrite tally-stable (vmap vote bs) p\n                           | \u2261.sym (tabulate-ext (\u03bb i \u2192 lookup-map vote (to p i) bs))\n                           = \u2261.cong (\u03bb p\u2081 \u2192 tally p\u2081 \u2238 n) (tabulate-\u2218 _ _)\n\n{-\n  Ballot privacy \u263a\n-}\n\nmodule BallotPrivacy\n  (RA : Set)\n  where\n\n  data Msg : Set where\n    vote! : (intent receipt : Vote) \u2192 Msg\n    stop! : Msg\n\n  Resp = List Ballot\n\n  Adv\u2080 = RA \u2192 Vote\n  Adv\u2081 = RA \u2192 Resp \u2192 Msg\n  Adv\u2082 = RA \u2192 (l : Resp) \u2192 Vec Ballot (3 * length l) \u2192 Bit\n\n  Adv = Adv\u2080 \u00d7 Adv\u2081 \u00d7 Adv\u2082\n\n\n  G : Adv \u2192 (\u2200 {n} \u2192 Perm n) \u2192 RA \u2192 Bit \u2192  Bit\n  G (A\u2080 , A\u2081 , A\u2082) \u03c0 Ra b = {!!} where\n    \u03b5 = A\u2080 Ra\n\n    go : (r : Resp) \u2192 Vec Ballot (3 * length r) \u2192 Bit\n    go r bb with A\u2081 Ra r | b\n    go r bb | vote! intent receipt | 1b = {!!}\n    go r bb | vote! intent receipt | 0b = {!!}\n    go r bb | stop!                | _ = A\u2082 Ra r (shuffle \u03c0 bb)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n3<4 : 3 < 4\n3<4 =  s\u2264s (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n\n","old_contents":"module ThreeBallot\n   -- ()\n   where\n\nopen import Data.Fin using (Fin ; zero ; suc )\nopen import Data.Nat\nopen import Data.Bool.NP using (to\u2115) renaming (false to 0b ; true to 1b)\n\nopen import Data.List using (List ; [] ; _\u2237_ ; length)\nopen import Data.Bits hiding (to\u2115 ; 0b ; 1b)\n\nopen import Data.Vec.NP renaming (map to vmap)\nopen import Data.Product\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; refl)\n\ndata Cand : Set where\n  Nice'Guy Bad'Guy : Cand\n\nVote = Bits 2\nTag  = \u2115 \n\nrecord Ballot : Set where\n  constructor mk\n  field\n    vote : Vote\n    tag  : Tag\n\nopen Ballot\n    \nrecord 3'_ (A : Set) : Set where\n  constructor [_][_][_]\n  field\n    1t 2t 3t : A\n\n3Vote = 3' Vote\n3Ballot = 3' Ballot\n    \n[_][_] : Bit \u2192 Bit \u2192 Bits 2\n[ x ][ y ] = x \u2237 y \u2237 []\n\nsvote : Vote\nsvote = [ 0b ][ 1b ]\n\nexample : 3Vote\nexample = [ [ 1b ][ 0b ] ][ [ 0b ][ 1b ] ][ [ 0b ][ 1b ] ] \n\nrow-const : Bit \u2192 3' Bit \u2192 Set\nrow-const b [ 1t ][ 2t ][ 3t ] = 1 + to\u2115 b \u2261 to\u2115 1t + to\u2115 2t + to\u2115 3t\n\nvoteFor : Vote \u2192 3Vote \u2192 Set\nvoteFor (n \u2237 b \u2237 []) [ x \u2237 x\u2081 \u2237 [] ][ x\u2082 \u2237 x\u2083 \u2237 [] ][ x\u2084 \u2237 x\u2085 \u2237 [] ] = row-const n [ x ][ x\u2082 ][ x\u2084 ]\n                                                                     \u00d7 row-const b [ x\u2081 ][ x\u2083 ][ x\u2085 ]\n\nvote-example : voteFor svote example\nvote-example = refl , refl\n\ncast : (_ _ : Vote) \u2192 Vote\ncast i r = vnot r\n\n\nvote-any : (intent receipt : Vote) \u2192 voteFor intent [ intent ][ receipt ][ cast intent receipt ]\nvote-any (true \u2237 true \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (true \u2237 true \u2237 []) (true \u2237 false \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (false \u2237 true \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (false \u2237 false \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (true \u2237 false \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (false \u2237 true \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (false \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (true \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (false \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (false \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (true \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (false \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (false \u2237 false \u2237 []) = refl , refl   \n\n{-\n\nNP :\n\n  (R?\u00d7V)^n \u2192 (3V)^n \u2192 (3B)^n \u2192 B^3n -\u27e8 shuffle \u27e9\u2192 b^3n \u2192 T\n\n-}\n\nPerm : \u2115 \u2192 Set\nPerm n = Fin n \u2194 Fin n\n\nshuffle : \u2200 {n}{A : Set} \u2192 Perm n \u2192 Vec A n \u2192 Vec A n\nshuffle p xs = tabulate (\u03bb i \u2192 lookup (to p i) xs)\n\nmodule _\n  (tally : \u2200 {n} \u2192 Vec Vote n \u2192 \u2115)\n  (tally-stable : \u2200 {n} vs \u03c0 \u2192 tally {n} vs \u2261 tally (shuffle \u03c0 vs)) \n  where\n\n  3-tally : \u2200 {n} \u2192 Vec Ballot (3 * n) \u2192 \u2115\n  3-tally {n} bs = tally (vmap vote bs) \u2238 n\n\n\n  lookup-map : \u2200 {n}{A B : Set}(f : A \u2192 B)(i : Fin n)(xs : Vec A n)\n             \u2192 f (lookup i xs) \u2261 lookup i (vmap f xs)\n  lookup-map f zero (x \u2237 xs) = refl \n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n  \n  -- map g (tabulate f) = tabulate (g \u2218 f)\n  -- f (lookup i xs) \u2261 lookup i (map f xs)\n  3-tally-stable : \u2200 {n} bs \u03c0 \u2192 3-tally {n} bs \u2261 3-tally {n} (shuffle \u03c0 bs)\n  3-tally-stable {n} bs p rewrite tally-stable (vmap vote bs) p\n                           | \u2261.sym (tabulate-ext (\u03bb i \u2192 lookup-map vote (to p i) bs))\n                           = \u2261.cong (\u03bb p\u2081 \u2192 tally p\u2081 \u2238 n) (tabulate-\u2218 _ _)\n                           \n{-\n  Ballot privacy \u263a\n-}\n\nmodule BallotPrivacy\n  (RA : Set)\n  where\n\n  data Msg : Set where\n    vote! : (intent receipt : Vote) \u2192 Msg\n    stop! : Msg\n  \n  Resp = List Ballot\n    \n  Adv\u2080 = RA \u2192 Vote\n  Adv\u2081 = RA \u2192 Resp \u2192 Msg\n  Adv\u2082 = RA \u2192 (l : Resp) \u2192 Vec Ballot (3 * length l) \u2192 Bit\n  \n  Adv = Adv\u2080 \u00d7 Adv\u2081 \u00d7 Adv\u2082\n  \n  \n  G : Adv \u2192 (\u2200 {n} \u2192 Perm n) \u2192 RA \u2192 Bit \u2192  Bit\n  G (A\u2080 , A\u2081 , A\u2082) \u03c0 Ra b = {!!} where\n    \u03b5 = A\u2080 Ra\n\n    go : (r : Resp) \u2192 Vec Ballot (3 * length r) \u2192 Bit\n    go r bb with A\u2081 Ra r | b\n    go r bb | vote! intent receipt | 1b = {!!}\n    go r bb | vote! intent receipt | 0b = {!!}\n    go r bb | stop!                | _ = A\u2082 Ra r (shuffle \u03c0 bb) \n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n3<4 : 3 < 4\n3<4 =  s\u2264s (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3328b0f0320cd70ea6918039e6d504a93c44d273","subject":"Added missing error message.","message":"Added missing error message.\n\nIgnore-this: 5fd5052a9a7dfad2534f325dc4dc1154\n\ndarcs-hash:20110322153008-3bd4e-28bb0c94f0a5761321053f1e67ab85c652a3039d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/NotErasedProofTerm.agda","new_file":"Test\/Fail\/NotErasedProofTerm.agda","new_contents":"------------------------------------------------------------------------------\n-- We do not erase of the proofs terms in the translation\n------------------------------------------------------------------------------\n\nmodule Test.Fail.NotErasedProofTerm where\n\n-- Error message:\n-- agda2atp: It is necessary to erase a proof term, but we do know how do it. The internal representation of term to be erased is:\n-- Pi r(El (Type (Lit (LitLevel  0))) (Def Test.Fail.NotErasedProofTerm.D [])) (Abs \"k\" El (Type (Lit (LitLevel  0))) (Def Test.Fail.NotErasedProofTerm._\u2264_ [r(Var 0 []),r(Var 0 [])]))\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  _\u2264_  : D \u2192 D \u2192 Set\n  zero : D\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nthm : \u2200 n \u2192 N n \u2192 (\u2200 k \u2192 k \u2264 k) \u2192 n \u2261 n\nthm n Nn h = prf\n  where\n\n    -- The internal type of prf is\n\n    --  \u2200 (n : D) (Nn : N n) (h : \u2200 k \u2192 k \u2264 k) \u2192 ...\n\n    -- The agda2atp tool can erase the proof term Nn, but it cannot erase the\n    -- proof term h.\n\n    postulate prf : n \u2261 n\n    {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- We do not erase of the proofs terms in the translation\n------------------------------------------------------------------------------\n\nmodule Test.Fail.NotErasedProofTerm where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  _\u2264_  : D \u2192 D \u2192 Set\n  zero : D\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nthm : \u2200 n \u2192 N n \u2192 (\u2200 k \u2192 k \u2264 k) \u2192 n \u2261 n\nthm n Nn h = prf\n  where\n\n    -- The internal type of prf is\n\n    --  \u2200 (n : D) (Nn : N n) (h : \u2200 k \u2192 k \u2264 k) \u2192 ...\n\n    -- The agda2atp tool can erase the proof term Nn, but it cannot erase the\n    -- proof term h.\n\n    postulate prf : n \u2261 n\n    {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"26feb31d8f07b57839d5ad020a366816d0ae21fb","subject":"insert missing space (cosmetic)","message":"insert missing space (cosmetic)\n\nOld-commit-hash: 4d559497f68fabe2674c6d523508f715c78256c8\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Evaluation.agda","new_file":"Parametric\/Denotation\/Evaluation.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Syntax.Context Type\nopen import Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen import Base.Denotation.Notation\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Syntax.Context Type\nopen import Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen import Base.Denotation.Notation\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure(\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b444ae81fd6c852a6409937d2ea0c38c95693b1e","subject":"Bits: upgrading Op to Bij, to\u2115-inj, some 2 ^ n turned into 2^ n","message":"Bits: upgrading Op to Bij, to\u2115-inj, some 2 ^ n turned into 2^ n\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} (f : Bits n \u2192 A) (g : Bij) \u2192 search {n} (f \u2218 eval g) \u2261 search {n} f\n        search-bij f `id     = refl\n        search-bij f `0\u21941    = search-0\u21941 f\n        search-bij f (g `\u204f h)\n          rewrite search-bij (f \u2218 eval h) g = search-bij f h\n        search-bij {zero}  f (_     `\u2237 _) = refl\n        search-bij {suc n} f (`id   `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 0b)\n                | search-bij (f \u2218 1\u2237_) (g 1b) = refl\n        search-bij {suc n} f (`not\u1d2e `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 1b)\n                | search-bij (f \u2218 1\u2237_) (g 0b) = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} (f : Bits n \u2192 Bit) (g : Bij) \u2192 #\u27e8 f \u2218 eval g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-bij f = sum-bij (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p rewrite \u2115\u00b0.+-comm (2^ n) (to\u2115 ys) = \u22a5-elim (<\u2192\u2262 (\u2264-steps (to\u2115 ys) (to\u2115-bound xs)) p)\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p rewrite \u2115\u00b0.+-comm (2^ n) (to\u2115 xs) = \u22a5-elim (<\u2192\u2262 (\u2264-steps (to\u2115 xs) (to\u2115-bound ys)) (\u2261.sym p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6a3bcddd66afcd08d8384d14a3adb3c2367319a6","subject":"Bits: fx","message":"Bits: fx\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {zero}  (_     `\u2237 _) _ = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n <= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) <= (to\u2115 xs) | \u2261.inspect (_<=_ (2^ n)) (to\u2115 xs)\n... | true  | [ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |   _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x \u2115< 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | [ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (\u2115<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | [ p ] = to\u2115\u2218from\u2115 {n} x (\u2115<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x \u2115< 2^ n \u2192 y \u2115< 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {zero}  (_     `\u2237 _) _ = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | [ p ] = 2\u207f\u2270to\u2115 xs (\u2115<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |   _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x \u2115< 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | [ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (\u2115<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | [ p ] = to\u2115\u2218from\u2115 {n} x (\u2115<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x \u2115< 2^ n \u2192 y \u2115< 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9ab34b569625320145abfa47ec7362c8cf8de49a","subject":"Add --allow-unsolved-metas to avoid CI failures","message":"Add --allow-unsolved-metas to avoid CI failures\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\nopen import Data.Nat.Base using (_<_)\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nopen import Relation.Binary hiding (_\u21d2_)\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\nopen import Data.Nat.Base using (_<_)\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nopen import Relation.Binary hiding (_\u21d2_)\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5ccaccd4756fd71b8825521605e7c23bb18e9a52","subject":"Fixe doc.","message":"Fixe doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/Collatz\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/Collatz\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesATP using ( \u2238-N )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP using ( 2-N )\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\nhelper : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ succ\u2081 n) \u2261 collatz (2' ^ n)\nhelper nzero = prf\n  where postulate prf : collatz (2' ^ succ\u2081 zero) \u2261 collatz (2' ^ zero)\n        {-# ATP prove prf #-}\nhelper (nsucc {n} Nn) = prf\n  where postulate prf : collatz (2' ^ succ\u2081 (succ\u2081 n)) \u2261 collatz (2' ^ succ\u2081 n)\n        {-# ATP prove prf +\u22382 ^-N 2-N 2^x\u22620 2^[x+1]\u22621 div-2^[x+1]-2\u22612^x x-Even\u2192SSx-Even \u2238-N \u2238-Even 2^[x+1]-Even 2-Even #-}\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ n) \u2261 1'\ncollatz-2^x nzero = prf\n  where postulate prf : collatz (2' ^ 0') \u2261 1'\n        {-# ATP prove prf #-}\n\ncollatz-2^x (nsucc {n} Nn) = prf (collatz-2^x Nn)\n  where postulate prf : collatz (2' ^ n) \u2261 1' \u2192 collatz (2' ^ succ\u2081 n) \u2261 1'\n        {-# ATP prove prf helper #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesATP using ( \u2238-N )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP using ( 2-N )\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\nhelper : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ succ\u2081 n) \u2261 collatz (2' ^ n)\nhelper nzero = prf\n  where postulate prf : collatz (2' ^ succ\u2081 zero) \u2261 collatz (2' ^ zero)\n        {-# ATP prove prf #-}\nhelper (nsucc {n} Nn) = prf\n  -- 18 December 2012: The ATPs could not prove the theorem (240 sec).\n  where postulate prf : collatz (2' ^ succ\u2081 (succ\u2081 n)) \u2261 collatz (2' ^ succ\u2081 n)\n        {-# ATP prove prf +\u22382 ^-N 2-N 2^x\u22620 2^[x+1]\u22621 div-2^[x+1]-2\u22612^x x-Even\u2192SSx-Even \u2238-N \u2238-Even 2^[x+1]-Even 2-Even #-}\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ n) \u2261 1'\ncollatz-2^x nzero = prf\n  where postulate prf : collatz (2' ^ 0') \u2261 1'\n        {-# ATP prove prf #-}\n\ncollatz-2^x (nsucc {n} Nn) = prf (collatz-2^x Nn)\n  where postulate prf : collatz (2' ^ n) \u2261 1' \u2192 collatz (2' ^ succ\u2081 n) \u2261 1'\n        {-# ATP prove prf helper #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"321249014d14436b6f3f84e01fdeb73f0d5faf7b","subject":"update crypto-agda.agda","message":"update crypto-agda.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"crypto-agda.agda","new_file":"crypto-agda.agda","new_contents":"module crypto-agda where\nimport Attack.Compression\nimport Attack.Reencryption\nimport Attack.Reencryption.OneBitMessage\nimport Cipher.ElGamal.Generic\nimport Cipher.ElGamal.Homomorphic\nimport Composition.Forkable\nimport Composition.Horizontal\nimport Composition.Vertical\nimport Crypto.Schemes\nimport FiniteField.FinImplem\nimport FunUniverse.Agda\nimport FunUniverse.BinTree\nimport FunUniverse.Bits\nimport FunUniverse.Category\nimport FunUniverse.Category.Op\nimport FunUniverse.Circuit\nimport FunUniverse.Const\nimport FunUniverse.Core\nimport FunUniverse.Cost\nimport FunUniverse.Data\nimport FunUniverse.Defaults.FirstPart\nimport FunUniverse.ExnArrow\nimport FunUniverse.Fin\nimport FunUniverse.Fin.Op\nimport FunUniverse.Fin.Op.Abstract\nimport FunUniverse.FlatFunsProd\nimport FunUniverse.Interface.Bits\nimport FunUniverse.Interface.Two\nimport FunUniverse.Interface.Vec\nimport FunUniverse.Inverse\nimport FunUniverse.Loop\nimport FunUniverse.Nand\nimport FunUniverse.Nand.Function\nimport FunUniverse.Nand.Properties\nimport FunUniverse.README\nimport FunUniverse.Rewiring.Linear\nimport FunUniverse.State\nimport FunUniverse.Syntax\nimport FunUniverse.Types\nimport Game.DDH\nimport Game.EntropySmoothing\nimport Game.EntropySmoothing.WithKey\nimport Game.IND-CCA\nimport Game.IND-CCA2-dagger\nimport Game.IND-CCA2\nimport Game.IND-CPA\nimport Game.Transformation.CCA-CPA\nimport Game.Transformation.CCA2-CCA\nimport Game.Transformation.CCA2-CCA2d\nimport Game.Transformation.CCA2d-CCA2\nimport Game.Transformation.InternalExternal\nimport Language.Simple.Abstract\nimport Language.Simple.Interface\nimport Language.Simple.Two.Mux\nimport Language.Simple.Two.Mux.Normalise\nimport Language.Simple.Two.Nand\nimport README\nimport Solver.AddMax\nimport Solver.Linear\nimport Solver.Linear.Examples\nimport Solver.Linear.Parser\nimport Solver.Linear.Syntax\nimport adder\nimport alea.cpo\nimport bijection-syntax.Bijection-Fin\nimport bijection-syntax.Bijection\nimport bijection-syntax.README\nimport circuits.bytecode\nimport circuits.circuit\nimport elgamal\nimport sha1\n","old_contents":"module crypto-agda where\nimport Cipher.ElGamal.Generic\nimport Composition.Forkable\nimport Composition.Horizontal\nimport Composition.Vertical\nimport Crypto.Schemes\n--import FiniteField.FinImplem\nimport FunUniverse.Agda\nimport FunUniverse.BinTree\nimport FunUniverse.Bits\nimport FunUniverse.Category\nimport FunUniverse.Category.Op\nimport FunUniverse.Circuit\nimport FunUniverse.Const\nimport FunUniverse.Core\nimport FunUniverse.Cost\nimport FunUniverse.Data\nimport FunUniverse.Defaults.FirstPart\n--import FunUniverse.ExnArrow\nimport FunUniverse.Fin\nimport FunUniverse.Fin.Op\nimport FunUniverse.Fin.Op.Abstract\n--import FunUniverse.FlatFunsProd\n--import FunUniverse.Interface.Bits\nimport FunUniverse.Interface.Two\nimport FunUniverse.Interface.Vec\nimport FunUniverse.Inverse\n--import FunUniverse.Loop\nimport FunUniverse.Nand\nimport FunUniverse.Nand.Function\nimport FunUniverse.Nand.Properties\nimport FunUniverse.README\nimport FunUniverse.Rewiring.Linear\n--import FunUniverse.State\nimport FunUniverse.Syntax\nimport FunUniverse.Types\nimport Game.DDH\nimport Game.EntropySmoothing\nimport Game.EntropySmoothing.WithKey\nimport Game.IND-CPA\nimport Game.Transformation.InternalExternal\nimport Language.Simple.Abstract\nimport Language.Simple.Interface\nimport Language.Simple.Two.Mux\nimport Language.Simple.Two.Mux.Normalise\nimport Language.Simple.Two.Nand\nimport README\nimport Solver.AddMax\nimport Solver.Linear\n--import Solver.Linear.Examples\nimport Solver.Linear.Parser\nimport Solver.Linear.Syntax\nimport adder\nimport alea.cpo\nimport bijection-syntax.Bijection-Fin\nimport bijection-syntax.Bijection\nimport bijection-syntax.README\n--import circuits.bytecode\nimport circuits.circuit\n--import elgamal\n--import sha1\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"56e74c3f39ed20d8bab724915e525881e26a8390","subject":"SyntaxKit renamings","message":"SyntaxKit renamings\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Bijection\/SyntaxKit.agda","new_file":"lib\/Function\/Bijection\/SyntaxKit.agda","new_contents":"module Function.Bijection.SyntaxKit where\n\nopen import Type hiding (\u2605)\nopen import Level.NP using (\u2080; \u209b; _\u2294_)\nopen import Function.NP\nopen import Data.One\nopen import Data.Two\nopen import Data.Bool.Properties\nopen import Relation.Binary.PropositionalEquality\n\nrecord BijKit b {a} (A : \u2605 a) : \u2605 (\u209b b \u2294 a) where\n  constructor mk\n  field\n    Bij   : \u2605 b\n    eval  : Bij \u2192 Endo A\n    _\u207b\u00b9   : Endo Bij\n    idBij : Bij\n    _\u204fBij_ : Bij \u2192 Bij \u2192 Bij\n\n  eval\u207b\u00b9 : Bij \u2192 Endo A\n  eval\u207b\u00b9 f = eval (f \u207b\u00b9)\n\n  _\u2257Bij_ : Bij \u2192 Bij \u2192 \u2605 _\n  _\u2257Bij_ = \u03bb f g \u2192 \u2200 (x : A) \u2192 eval f x \u2261 eval g x\n\n  field\n    id-spec : eval idBij \u2257 id\n    _\u204f-spec_ : \u2200 f g \u2192 eval (f \u204fBij g) \u2257 eval g \u2218 eval f\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 eval (f \u207b\u00b9) \u2218 eval f \u2257 id\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257Bij f\n\n  _\u207b\u00b9-inverseBij : \u2200 f \u2192 (f \u204fBij f \u207b\u00b9) \u2257Bij idBij\n  (f \u207b\u00b9-inverseBij) x rewrite (f \u204f-spec (f \u207b\u00b9)) x\n                            | id-spec x\n                            | (f \u207b\u00b9-inverse)\n                            x = refl\n\n  _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (eval f \u2218 eval (f \u207b\u00b9)) \u2257 id\n  (f \u207b\u00b9-inverse\u2032) x with ((f \u207b\u00b9) \u207b\u00b9-inverse) x\n  ... | p rewrite (f \u207b\u00b9-involutive) (eval (f \u207b\u00b9) x) = p\n\n  _\u207b\u00b9-inverseBij\u2032 : \u2200 f \u2192 (f \u207b\u00b9 \u204fBij f) \u2257Bij idBij\n  (f \u207b\u00b9-inverseBij\u2032) x rewrite (f \u207b\u00b9 \u204f-spec f) x\n                             | id-spec x\n                             | (f \u207b\u00b9-inverse\u2032)\n                            x = refl\n\nmodule \ud835\udfdaBijection where\n  data \ud835\udfdaBij : \u2605\u2080 where\n    `id `not : \ud835\udfdaBij\n\n  bool-bijKit : BijKit \u2080 \ud835\udfda\n  bool-bijKit = mk \ud835\udfdaBij eval id `id _\u204fBij_ (\u03bb _ \u2192 refl) _\u204f-spec_ _\u207b\u00b9-inverse (\u03bb _ _ \u2192 refl)\n   module BBK where\n    eval  : \ud835\udfdaBij \u2192 Endo \ud835\udfda\n    eval `id = id\n    eval `not = not\n\n    _\u204fBij_ : \ud835\udfdaBij \u2192 \ud835\udfdaBij \u2192 \ud835\udfdaBij\n    `id  \u204fBij g    = g\n    f    \u204fBij `id  = f\n    `not \u204fBij `not = `id\n\n    _\u204f-spec_ : \u2200 f g \u2192 eval (f \u204fBij g) \u2257 eval g \u2218 eval f\n    `id  \u204f-spec g    = \u03bb _ \u2192 refl\n    `not \u204f-spec `id  = \u03bb _ \u2192 refl\n    `not \u204f-spec `not = sym \u2218 not-involutive\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 eval f \u2218 eval f \u2257 id\n    (`id  \u207b\u00b9-inverse) = \u03bb _ \u2192 refl\n    (`not \u207b\u00b9-inverse) = not-involutive\n\n  module \ud835\udfdaBijKit = BijKit bool-bijKit\n\n  -- `xor can be seen as a from\ud835\udfda function\n  `xor : \ud835\udfda \u2192 \ud835\udfdaBij\n  `xor 0\u2082 = `id\n  `xor 1\u2082 = `not\n\n  -- `not? can be seen as a to\ud835\udfda function\n  `not? : \ud835\udfdaBij \u2192 \ud835\udfda\n  `not? `id  = 0\u2082\n  `not? `not = 1\u2082\n\n  `xor\u2218`not? : `xor \u2218 `not? \u2257 id\n  `xor\u2218`not? `id = refl\n  `xor\u2218`not? `not = refl\n\n  `not?\u2218`xor : `not? \u2218 `xor \u2257 id\n  `not?\u2218`xor 1\u2082 = refl\n  `not?\u2218`xor 0\u2082 = refl\n\nmodule 1-Bijection {a} (A : \u2605 a) where\n    1-bij : BijKit \u2080 A\n    1-bij = mk \ud835\udfd9 (\u03bb _ \u2192 id) _ _ _ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl) (\u03bb _ _ \u2192 refl) (\u03bb _ _ \u2192 refl)\n\n    module 1-BijKit = BijKit 1-bij\n\nmodule \ud835\udfd9Bijection where\n    open 1-Bijection\n    \ud835\udfd9-bijKit : BijKit \u2080 \ud835\udfd9\n    \ud835\udfd9-bijKit = 1-bij \ud835\udfd9\n\n    module \ud835\udfd9BijKit = 1-BijKit \ud835\udfd9\n","old_contents":"module Function.Bijection.SyntaxKit where\n\nopen import Type hiding (\u2605)\nopen import Level.NP using (\u2080; \u209b; _\u2294_)\nopen import Function.NP\nopen import Data.One\nopen import Data.Two\nopen import Data.Bool.Properties\nopen import Relation.Binary.PropositionalEquality\n\nrecord BijKit b {a} (A : \u2605 a) : \u2605 (\u209b b \u2294 a) where\n  constructor mk\n  field\n    Bij   : \u2605 b\n    eval  : Bij \u2192 Endo A\n    _\u207b\u00b9   : Endo Bij\n    idBij : Bij\n    _\u204fBij_ : Bij \u2192 Bij \u2192 Bij\n\n  eval\u207b\u00b9 : Bij \u2192 Endo A\n  eval\u207b\u00b9 f = eval (f \u207b\u00b9)\n\n  _\u2257Bij_ : Bij \u2192 Bij \u2192 \u2605 _\n  _\u2257Bij_ = \u03bb f g \u2192 \u2200 (x : A) \u2192 eval f x \u2261 eval g x\n\n  field\n    id-spec : eval idBij \u2257 id\n    _\u204f-spec_ : \u2200 f g \u2192 eval (f \u204fBij g) \u2257 eval g \u2218 eval f\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 eval (f \u207b\u00b9) \u2218 eval f \u2257 id\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257Bij f\n\n  _\u207b\u00b9-inverseBij : \u2200 f \u2192 (f \u204fBij f \u207b\u00b9) \u2257Bij idBij\n  (f \u207b\u00b9-inverseBij) x rewrite (f \u204f-spec (f \u207b\u00b9)) x\n                            | id-spec x\n                            | (f \u207b\u00b9-inverse)\n                            x = refl\n\n  _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (eval f \u2218 eval (f \u207b\u00b9)) \u2257 id\n  (f \u207b\u00b9-inverse\u2032) x with ((f \u207b\u00b9) \u207b\u00b9-inverse) x\n  ... | p rewrite (f \u207b\u00b9-involutive) (eval (f \u207b\u00b9) x) = p\n\n  _\u207b\u00b9-inverseBij\u2032 : \u2200 f \u2192 (f \u207b\u00b9 \u204fBij f) \u2257Bij idBij\n  (f \u207b\u00b9-inverseBij\u2032) x rewrite (f \u207b\u00b9 \u204f-spec f) x\n                             | id-spec x\n                             | (f \u207b\u00b9-inverse\u2032)\n                            x = refl\n\nmodule BoolBijection where\n  data BoolBij : \u2605\u2080 where\n    `id `not : BoolBij\n\n  bool-bijKit : BijKit \u2080 Bool\n  bool-bijKit = mk BoolBij eval id `id _\u204fBij_ (\u03bb _ \u2192 refl) _\u204f-spec_ _\u207b\u00b9-inverse (\u03bb _ _ \u2192 refl)\n   module BBK where\n    eval  : BoolBij \u2192 Endo Bool\n    eval `id = id\n    eval `not = not\n\n    _\u204fBij_ : BoolBij \u2192 BoolBij \u2192 BoolBij\n    `id  \u204fBij g    = g\n    f    \u204fBij `id  = f\n    `not \u204fBij `not = `id\n\n    _\u204f-spec_ : \u2200 f g \u2192 eval (f \u204fBij g) \u2257 eval g \u2218 eval f\n    `id  \u204f-spec g    = \u03bb _ \u2192 refl\n    `not \u204f-spec `id  = \u03bb _ \u2192 refl\n    `not \u204f-spec `not = sym \u2218 not-involutive\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 eval f \u2218 eval f \u2257 id\n    (`id  \u207b\u00b9-inverse) = \u03bb _ \u2192 refl\n    (`not \u207b\u00b9-inverse) = not-involutive\n\n  module BoolBijKit = BijKit bool-bijKit\n\n  -- `xor can be seen as a fromBool function\n  `xor : Bool \u2192 BoolBij\n  `xor 0\u2082 = `id\n  `xor 1\u2082 = `not\n\n  -- `not? can be seen as a toBool function\n  `not? : BoolBij \u2192 Bool\n  `not? `id  = 0\u2082\n  `not? `not = 1\u2082\n\n  `xor\u2218`not? : `xor \u2218 `not? \u2257 id\n  `xor\u2218`not? `id = refl\n  `xor\u2218`not? `not = refl\n\n  `not?\u2218`xor : `not? \u2218 `xor \u2257 id\n  `not?\u2218`xor 1\u2082 = refl\n  `not?\u2218`xor 0\u2082 = refl\n\nmodule 1-Bijection {a} (A : \u2605 a) where\n    1-bij : BijKit \u2080 A\n    1-bij = mk \ud835\udfd9 (\u03bb _ \u2192 id) _ _ _ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl) (\u03bb _ _ \u2192 refl) (\u03bb _ _ \u2192 refl)\n\n    module 1-BijKit = BijKit 1-bij\n\nmodule \ud835\udfd9Bijection where\n    open 1-Bijection\n    \ud835\udfd9-bijKit : BijKit \u2080 \ud835\udfd9\n    \ud835\udfd9-bijKit = 1-bij \ud835\udfd9\n\n    module \ud835\udfd9BijKit = 1-BijKit \ud835\udfd9\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e711bc38471d0346486631547f7942ec85c0a788","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: a4b2d89612d6d3630ba66d2f23fcfcc7\n\ndarcs-hash:20111018164755-3bd4e-a5aa09a1c15b871f8c449eba4c10548ad9ed2cab.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/ABP\/Fair.agda","new_file":"src\/FOTC\/Program\/ABP\/Fair.agda","new_contents":"------------------------------------------------------------------------------\n-- Fairness of the ABP channels\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.ABP.Fair where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- The Fair co-inductive predicate\n\n-- From the paper: al : O*L if al is a list of zero or more O's\n-- followed by a final L.\ndata O*L : D \u2192 Set where\n  nilO*L  :                 O*L (L \u2237 [])\n  consO*L : \u2200 ol \u2192 O*L ol \u2192 O*L (O \u2237 ol)\n\n-- Functor for the Fair type.\n-- FairF : (D \u2192 Set) \u2192 D \u2192 Set\n-- FairF X os = \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 X os' \u2227 os \u2261 ol ++ os'\n\npostulate Fair : D \u2192 Set\n\n-- Fair is post-fixed point of FairF (d \u2264 f d).\npostulate\n  Fair-gfp\u2081 : \u2200 {os} \u2192 Fair os \u2192\n              \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 Fair os' \u2227 os \u2261 ol ++ os'\n{-# ATP axiom Fair-gfp\u2081 #-}\n\n-- Fair is the greatest post-fixed of FairF.\n--\n-- (If \u2200 e. e \u2264 f e => e \u2264 d, then d is the greatest post-fixed point\n-- of f).\n\n-- N.B. This is a second-order axiom. In the automatic proofs, we\n-- *must* use an instance. Therefore, we do not add this postulate as\n-- an ATP axiom.\npostulate\n  Fair-gfp\u2082 : (P : D \u2192 Set) \u2192\n              -- P is post-fixed point of FairF.\n              (\u2200 {os} \u2192 P os \u2192\n               \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 P os' \u2227 os \u2261 ol ++ os') \u2192\n              -- Fair is greater than P.\n              \u2200 {os} \u2192 P os \u2192 Fair os\n\n-- Because a greatest post-fixed point is a fixed point, then the Fair\n-- predicate is also a pre-fixed point of the functor FairF (f d \u2264 d).\nFair-gfp\u2083 : \u2200 {os} \u2192\n            (\u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 Fair os' \u2227 os \u2261 ol ++ os') \u2192\n            Fair os\nFair-gfp\u2083 h = Fair-gfp\u2082 P helper h\n  where\n  P : D \u2192 Set\n  P ws = \u2203 \u03bb wl \u2192 \u2203 \u03bb ws' \u2192 O*L wl \u2227 Fair ws' \u2227 ws \u2261 wl ++ ws'\n\n  helper : {os : D} \u2192 P os \u2192 \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 P os' \u2227 os \u2261 ol ++ os'\n  helper (ol , os' , OLol , Fos' , h) = ol , os' , OLol , Fair-gfp\u2081 Fos' , h\n","old_contents":"------------------------------------------------------------------------------\n-- Fairness of the ABP channels\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.ABP.Fair where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- The Fair co-inductive predicate\n\n-- From the paper: al : O*L if al is a list of zero or more O's\n-- followed by a final L.\ndata O*L : D \u2192 Set where\n  nilO*L  :                 O*L (L \u2237 [])\n  consO*L : \u2200 ol \u2192 O*L ol \u2192 O*L (O \u2237 ol)\n\n-- Functor for the Fair type.\n-- FairF : (D \u2192 Set) \u2192 D \u2192 Set\n-- FairF X os = \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 X os' \u2227 os \u2261 ol ++ os'\n\npostulate Fair : D \u2192 Set\n\n-- Fair is post-fixed point of FairF (d \u2264 f d).\npostulate\n  Fair-gfp\u2081 : \u2200 {os} \u2192 Fair os \u2192\n              \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 Fair os' \u2227 os \u2261 ol ++ os'\n{-# ATP axiom Fair-gfp\u2081 #-}\n\n-- Fair is the greatest post-fixed of FairF.\n--\n-- (If \u2200 e. e \u2264 f e => e \u2264 d, then d is the greatest post-fixed point\n-- of f).\n\n-- N.B. This is a second-order axiom. In the automatic proofs, we\n-- *must* use an instance. Therefore, we do not add this postulate as\n-- an ATP axiom.\npostulate\n  Fair-gfp\u2082 : (P : D \u2192 Set) \u2192\n              -- P is post-fixed point of FairF.\n              ( \u2200 {os} \u2192 P os \u2192\n                \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 P os' \u2227 os \u2261 ol ++ os' ) \u2192\n              -- Fair is greater than P.\n              \u2200 {os} \u2192 P os \u2192 Fair os\n\n-- Because a greatest post-fixed point is a fixed point, then the Fair\n-- predicate is also a pre-fixed point of the functor FairF (f d \u2264 d).\nFair-gfp\u2083 : \u2200 {os} \u2192\n            (\u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 Fair os' \u2227 os \u2261 ol ++ os') \u2192\n            Fair os\nFair-gfp\u2083 h = Fair-gfp\u2082 P helper h\n  where\n  P : D \u2192 Set\n  P ws = \u2203 \u03bb wl \u2192 \u2203 \u03bb ws' \u2192 O*L wl \u2227 Fair ws' \u2227 ws \u2261 wl ++ ws'\n\n  helper : {os : D} \u2192 P os \u2192 \u2203 \u03bb ol \u2192 \u2203 \u03bb os' \u2192 O*L ol \u2227 P os' \u2227 os \u2261 ol ++ os'\n  helper (ol , os' , OLol , Fos' , h) = ol , os' , OLol , Fair-gfp\u2081 Fos' , h\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"68ba92194de15faf012a5b3ec096b43ef5b7dfc4","subject":"We cannot prove Colist-coind-stronger-ho from Colist-coind-ho.","message":"We cannot prove Colist-coind-stronger-ho from Colist-coind-ho.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GreatestFixedPoints\/Colist.agda","new_file":"notes\/fixed-points\/GreatestFixedPoints\/Colist.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Colist where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Colist is a greatest fixed-point of a functor\n\n-- The functor.\nColistF : (D \u2192 Set) \u2192 D \u2192 Set\nColistF A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n\n-- Colist is the greatest fixed-point of ColistF.\npostulate\n  Colist : D \u2192 Set\n\n  -- Colist is a post-fixed point of ColistF, i.e.\n  --\n  -- Colist \u2264 ColistF Colist.\n  Colist-out-ho : \u2200 {n} \u2192 Colist n \u2192 ColistF Colist n\n\n  -- Colist is the greatest post-fixed point of ColistF, i.e.\n  --\n  -- \u2200 A. A \u2264 ColistF A \u21d2 A \u2264 Colist.\n  Colist-coind-ho :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ColistF.\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n    -- Colist is greater than A.\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n------------------------------------------------------------------------------\n-- First-order versions\n\nColist-out : \u2200 {xs} \u2192\n             Colist xs \u2192\n             xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\nColist-out = Colist-out-ho\n\nColist-coind :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind = Colist-coind-ho\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Colist predicate is also a pre-fixed point of the functional\n-- ColistF, i.e.\n--\n-- ColistF Colist \u2264 Colist.\nColist-in-ho : \u2200 {xs} \u2192 ColistF Colist xs \u2192 Colist xs\nColist-in-ho h = Colist-coind-ho A h' h\n  where\n  A : D \u2192 Set\n  A xs = ColistF Colist xs\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 ColistF A xs\n  h' (inj\u2081 xs\u22610) = inj\u2081 xs\u22610\n  h' (inj\u2082 (x' , xs' , prf , CLxs' )) =\n    inj\u2082 (x' , xs' , prf , Colist-out CLxs')\n\n-- The first-order version.\nColist-in : \u2200 {xs} \u2192\n            xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs') \u2192\n            Colist xs\nColist-in = Colist-in-ho\n\n------------------------------------------------------------------------------\n-- A stronger co-induction principle\n--\n-- From (Paulson, 1997. p. 16).\n\npostulate\n  Colist-coind-stronger-ho :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs \u2228 Colist xs) \u2192\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\nColist-coind-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-ho' A h Axs =\n  Colist-coind-stronger-ho A (\u03bb Ays \u2192 inj\u2081 (h Ays)) Axs\n\n-- The first-order version.\nColist-coind-stronger :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192\n    (xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'))\n    \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger = Colist-coind-stronger-ho\n\n-- 13 January 2014. As expected, we cannot prove\n-- Colist-coind-stronger-ho from Colist-coind-stronger.\nColist-coind-stronger-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger-ho' A h {xs} Axs = case prf (\u03bb h' \u2192 h') (h Axs)\n  where\n  prf : ColistF A xs \u2192 Colist xs\n  prf h' = Colist-coind-ho A {!!} Axs\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. In: Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","old_contents":"------------------------------------------------------------------------------\n-- Co-lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Colist where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Colist is a greatest fixed-point of a functor\n\n-- The functor.\nColistF : (D \u2192 Set) \u2192 D \u2192 Set\nColistF A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n\n-- Colist is the greatest fixed-point of ColistF.\npostulate\n  Colist : D \u2192 Set\n\n  -- Colist is a post-fixed point of ColistF, i.e.\n  --\n  -- Colist \u2264 ColistF Colist.\n  Colist-out-ho : \u2200 {n} \u2192 Colist n \u2192 ColistF Colist n\n\n  -- Colist is the greatest post-fixed point of ColistF, i.e.\n  --\n  -- \u2200 A. A \u2264 ColistF A \u21d2 A \u2264 Colist.\n  Colist-coind-ho :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ColistF.\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n    -- Colist is greater than A.\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n------------------------------------------------------------------------------\n-- First-order versions\n\nColist-out : \u2200 {xs} \u2192\n             Colist xs \u2192\n             xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\nColist-out = Colist-out-ho\n\nColist-coind :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind = Colist-coind-ho\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Colist predicate is also a pre-fixed point of the functional\n-- ColistF, i.e.\n--\n-- ColistF Colist \u2264 Colist.\nColist-in-ho : \u2200 {xs} \u2192 ColistF Colist xs \u2192 Colist xs\nColist-in-ho h = Colist-coind-ho A h' h\n  where\n  A : D \u2192 Set\n  A xs = ColistF Colist xs\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 ColistF A xs\n  h' (inj\u2081 xs\u22610) = inj\u2081 xs\u22610\n  h' (inj\u2082 (x' , xs' , prf , CLxs' )) =\n    inj\u2082 (x' , xs' , prf , Colist-out CLxs')\n\n-- The first-order version.\nColist-in : \u2200 {xs} \u2192\n            xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs') \u2192\n            Colist xs\nColist-in = Colist-in-ho\n\n------------------------------------------------------------------------------\n-- A stronger co-induction principle\n--\n-- From (Paulson, 1997. p. 16).\n\npostulate\n  Colist-coind-stronger-ho :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs \u2228 Colist xs) \u2192\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\nColist-coind-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-ho' A h Axs =\n  Colist-coind-stronger-ho A (\u03bb Ays \u2192 inj\u2081 (h Ays)) Axs\n\n-- The first-order version.\nColist-coind-stronger :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192\n    (xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'))\n    \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger = Colist-coind-stronger-ho\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. In: Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9354d1da9a790382d98d46653573abc5a5b8bb56","subject":"finished stating and proving all but one checksum for #19","message":"finished stating and proving all but one checksum for #19\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"canonical-indeterminate-forms.agda","new_file":"canonical-indeterminate-forms.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       \u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n                                         +\n                                       (\u03a3[ d' \u2208 dhexp ]\n                                         ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n                                          (d' indet) \u00d7\n                                          ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))))\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x))\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (_ , refl , ind , x\u2081)))\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       ((\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                        (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                        (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c4 ==> (\u03c41' ==> \u03c42')) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c4) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                        (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c43 \u2208 htyp ] \u03a3[ \u03c44 \u2208 htyp ]\n                                          ((d == d' \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9) \u00d7\n                                           (d' indet) \u00d7\n                                           ((\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44)))) +\n                                        (\u03a3[ d' \u2208 dhexp ]\n                                          ((\u03c41 == \u2987\u2988) \u00d7\n                                           (\u03c42 == \u2987\u2988) \u00d7\n                                           (d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n                                           (d' indet) \u00d7\n                                           ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))))\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = Inr (Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , ind , x\u2081))))\n  -- nb: this is the only one that required pattern matching (or equivalently a lemma i didn't bother to state) on premises\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = Inr (Inr (Inr (Inr (_ , refl , refl , refl , ind , x\u2081) )))\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n                                          (\u03c4' ground) \u00d7\n                                          (d indet))) --todo: this is\n                                                      --interesting; it's\n                                                      --the only clause\n                                                      --that's filled by\n                                                      --multiple patterns\n                                                      --below; maybe we\n                                                      --could say something\n                                                      --more specific?\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = Inr (Inr (Inr (_ , _ , refl , x\u2081 , ICastGroundHole x\u2081 ind)))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (_ , _ , refl , x\u2082 , ICastGroundHole x\u2082 ind)))\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam wt) () nb na nh\n  canonical-indeterminate-forms-coverage (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = {!!}\n  canonical-indeterminate-forms-coverage (TAEHole x x\u2081) IEHole nb na nh = {!!}\n  canonical-indeterminate-forms-coverage (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = {!!}\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       \u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n                                         +\n                                       (\u03a3[ d' \u2208 dhexp ]\n                                         ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n                                          (d' indet) \u00d7\n                                          ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))))\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x))\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (_ , refl , ind , x\u2081)))\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       {!!} +\n                                       {!!} +\n                                       {!!} +\n                                       {!!} +\n                                       {!!}\n  canonical-indeterminate-forms-arr wt ind = {!!}\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       {!!}\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = {!!}\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = {!!}\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam wt) () nb na nh\n  canonical-indeterminate-forms-coverage (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = {!!}\n  canonical-indeterminate-forms-coverage (TAEHole x x\u2081) IEHole nb na nh = {!!}\n  canonical-indeterminate-forms-coverage (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = {!!}\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c85f33967d277406efd4e4db2010a9b1b64173b1","subject":"Drop dead code","message":"Drop dead code\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 c \u2294 d)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b \u2294 c)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (c \u2294 d \u2294 p)\nRawChange\u208d_,_\u208e x y f = \u2200 px (dpx : \u0394\u208d_\u208e x px) \u2192 \u0394\u208d_\u208e y (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q \u2294 c)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            _\u229e_ {{CB}} (f (_\u229e_ {{CA}} a da)) (g (_\u229e_ {{CA}} (a \u229e da) (nil (a \u229e da))) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 _\u229e_ {{CB}} (f (a \u229e da)) (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil {{CA}} (a \u229e da)) \u27e9\n            _\u229e_ {{CB}} (f (a \u229e da)) (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff {{CB}} (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff {{CB}} (g (a \u229e da)) (f a)) \u27e9\n            _\u229e_ {{CB}} (f a) (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n    funNil = \u03bb f \u2192 funDiff f f\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n\n      changeAlgebra = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          _\u229e_ {{CB}} (f a) (g (_\u229e_ {{CA}} a (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 _\u229e_ {{CB}} (f a) (g \u25a1 \u229f f a)) (update-nil {{CA}} a) \u27e9\n          _\u229e_ {{CB}} (f a) (g a \u229f f a)\n        \u2261\u27e8 update-diff {{CB}}  (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil {{changeAlgebra}} f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil {{changeAlgebra}} f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192  All\u2032 (\u0394\u208d_\u208e {{C}} _) pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d_\u208e {{C}} _) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d_\u208e {{C}} _ px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 (\u0394\u208d_\u208e {{C}} _)\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  {-\n  instance\n    change-algebra-extractor : \u2200 {x} \u2192 ChangeAlgebra \u2113 (P x)\n    change-algebra-extractor {x} = change-algebra x\n  -}\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 c \u2294 d)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b \u2294 c)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (c \u2294 d \u2294 p)\nRawChange\u208d_,_\u208e x y f = \u2200 px (dpx : \u0394\u208d_\u208e x px) \u2192 \u0394\u208d_\u208e y (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q \u2294 c)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            _\u229e_ {{CB}} (f (_\u229e_ {{CA}} a da)) (g (_\u229e_ {{CA}} (a \u229e da) (nil (a \u229e da))) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 _\u229e_ {{CB}} (f (a \u229e da)) (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil {{CA}} (a \u229e da)) \u27e9\n            _\u229e_ {{CB}} (f (a \u229e da)) (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff {{CB}} (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff {{CB}} (g (a \u229e da)) (f a)) \u27e9\n            _\u229e_ {{CB}} (f a) (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n    funNil = \u03bb f \u2192 funDiff f f\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n\n      changeAlgebra = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          _\u229e_ {{CB}} (f a) (g (_\u229e_ {{CA}} a (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 _\u229e_ {{CB}} (f a) (g \u25a1 \u229f f a)) (update-nil {{CA}} a) \u27e9\n          _\u229e_ {{CB}} (f a) (g a \u229f f a)\n        \u2261\u27e8 update-diff {{CB}}  (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil {{changeAlgebra}} f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil {{changeAlgebra}} f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192  All\u2032 (\u0394\u208d_\u208e {{C}} _) pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d_\u208e {{C}} _) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d_\u208e {{C}} _ px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 (\u0394\u208d_\u208e {{C}} _)\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93fbc39f356356b47218530ff8eaa341187b6d0a","subject":"moar curry = moar sugar","message":"moar curry = moar sugar\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 UncurriedCases E\n    (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) \n    ((i : I) (x : \u03bc I D i) \u2192 P i x)\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\nelim2 :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n        X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in CurriedCases E Q X\nelim2 I E Cs P =\n  let D = `Arg (Tag E) Cs\n      Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n      X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in curryCases E Q X (elim I E Cs P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) VecT (VecC A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) VecT (VecC (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (cs : Cases E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim \u22a4 \u2115T \u2115C _\n      ( (\u03bb n \u2192 n)\n      , (\u03bb m ih n \u2192 suc (ih n))\n      , tt\n      )\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim \u22a4 \u2115T \u2115C _\n      ( (\u03bb n \u2192 zero)\n      , (\u03bb m ih n \u2192 add n (ih n))\n      , tt\n      )\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim (\u2115 tt) VecT (VecC A) _\n      ( (\u03bb n ys \u2192 ys)\n      , (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n      , tt\n      )\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim (\u2115 tt) VecT (VecC (Vec A m)) _\n      ( (nil A)\n      , (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n      , tt\n      )\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0b0d9846a69f8b188de90f0c401104f21d016ae3","subject":"Add back generic eliminators in Darkwing, now improved with generic motives.","message":"Add back generic eliminators in Darkwing, now improved with generic motives.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  C : (A : El\u1d40 P) \u2192 Tag E \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg (Tag E) (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 CurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\nuncurryEl\u1d40 End X x tt = x\nuncurryEl\u1d40 (Arg A B) X f (a , b) = uncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (f a) b\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El\u1d30 D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R p t = curryEl\u1d30 (Data.C R p t)\n  (\u03bc (Data.D R p))\n  (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) M init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\nindCurried D M f i x =\n  ind D M (uncurryHyps D (\u03bc D) M init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  CurriedHyps (Data.C R p t) (\u03bc (Data.D R p)) M (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : UncurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 UncurriedBranches (Data.E R)\n    (SumCurriedHyps R p M)\n    (\u2200 i (x : \u03bc D i) \u2192 M i x)\nelimUncurried R p M cs i x =\n  indCurried (Data.D R p) M\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p unM)\n     (\u2200 i (x : \u03bc D i) \u2192 unM i x)\nelim R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in\n    (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n    \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p unM)\n       (\u2200 i (x : \u03bc D i) \u2192 unM i x))\n  (\u03bb p M \u2192 curryBranches\n    (elimUncurried R p\n      (uncurryEl\u1d40 (Data.I R p)\n        ((\u03bb i \u2192 \u03bc (Data.D R p) i \u2192 Set)) M)))\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n  tt\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n  tt\n\nappend : {A : Set} (m : \u2115) (xs : Vec A m) (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} m xs = elim VecR (\u03bb m xs \u2192 (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons (add m n) x (ih n ys))\n  (m , tt) xs\n\nconcat : {A : Set} (m n : \u2115) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} m n xss = elim VecR (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb n xs xss ih \u2192 append m xs (mult n m) ih)\n  (n , tt) xss\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  C : (A : El\u1d40 P) \u2192 Tag E \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg (Tag E) (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El\u1d30 D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R p t = curryEl\u1d30 (Data.C R p t)\n  (\u03bc (Data.D R p))\n  (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) M init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\nindCurried D M f i x =\n  ind D M (uncurryHyps D (\u03bc D) M init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  CurriedHyps (Data.C R p t) (\u03bc (Data.D R p)) M (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 UncurriedBranches (Data.E R)\n    (SumCurriedHyps R p M)\n    (\u2200 i (x : \u03bc D i) \u2192 M i x)\nelimUncurried R p M cs i x =\n  indCurried (Data.D R p) M\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 CurriedBranches (Data.E R)\n    (SumCurriedHyps R p M)\n    (\u2200 i (x : \u03bc D i) \u2192 M i x)\nelim R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in\n    (M : \u2200 i \u2192 \u03bc D i \u2192 Set) \u2192\n    CurriedBranches (Data.E R)\n    (SumCurriedHyps R p M)\n    (\u2200 i (x : \u03bc D i) \u2192 M i x))\n  (\u03bb p M \u2192 curryBranches (elimUncurried R p M))\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R (\u03bb u n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n  tt\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R (\u03bb u n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n  tt\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bc0f697e345522fad65844eed07cba984158e3b7","subject":"Parametric.Change.Validity: Comment out bug-triggering code","message":"Parametric.Change.Validity: Comment out bug-triggering code\n\nThis code is exported and used elsewhere :-(\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  {-\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d4e4f011631cfbb914d7d51c617806ac6e74cd50","subject":"Typo.","message":"Typo.\n\nIgnore-this: 68af14a7e65f38003c3779ccf4921770\n\ndarcs-hash:20110930182805-3bd4e-97d00b34b1a4af679e44fc8768ed579a9589a669.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/DataPostulate.agda","new_file":"Draft\/DataPostulate.agda","new_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we chose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulated one: Using the induction\n-- principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulated one: Using pattern\n-- matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulated predicate to the data one.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {_} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {_} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulated one: Using the induction\n-- principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulated one: Using pattern\n-- matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulated predicate to the data one.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {_} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {_} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ac25a046e34905a5fd630275f7f19cf585a2f9ab","subject":"Remove no longer relevant TODO","message":"Remove no longer relevant TODO\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker.agda","new_file":"ZK\/JSChecker.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.ZqZp as ZqZp\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- TODO bug (undefined)!\nrecord ZK-chaum-pedersen-pok-elgamal-rnd {--(\u2124q \u2124p\u2605 : Set)--} : Set where\n  field\n    m c s : BigI {--\u2124q--}\n    g p q y \u03b1 \u03b2 A B : BigI --\u2124p\u2605\n\nzk-check-chaum-pedersen-pok-elgamal-rnd! : ZK-chaum-pedersen-pok-elgamal-rnd {-BigI BigI-} \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd! pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks!\n      >> check! \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check! \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    params = record\n      { primality-test-probability-bound = primality-test-probability-bound\n      ; min-bits-q = min-bits-q\n      ; min-bits-p = min-bits-p\n      ; qI = I.q\n      ; pI = I.p\n      ; gI = I.g\n      }\n    open module [\u2124q]\u2124p\u2605 = ZqZp params\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n    check! \"type of statement\" (typ === fromString cpt)\n                               (\u03bb _ \u2192 \"Expected type of statement: \" ++ cpt ++ \" not \" ++ toString typ)\n >> zk-check-chaum-pedersen-pok-elgamal-rnd! pok\n module Zk-check where\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.ZqZp as ZqZp\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- TODO bug (undefined)!\nrecord ZK-chaum-pedersen-pok-elgamal-rnd {--(\u2124q \u2124p\u2605 : Set)--} : Set where\n  field\n    m c s : BigI {--\u2124q--}\n    g p q y \u03b1 \u03b2 A B : BigI --\u2124p\u2605\n\nzk-check-chaum-pedersen-pok-elgamal-rnd! : ZK-chaum-pedersen-pok-elgamal-rnd {-BigI BigI-} \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd! pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks!\n      >> check! \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check! \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    params = record\n      { primality-test-probability-bound = primality-test-probability-bound\n      ; min-bits-q = min-bits-q\n      ; min-bits-p = min-bits-p\n      ; qI = I.q\n      ; pI = I.p\n      ; gI = I.g\n      }\n    open module [\u2124q]\u2124p\u2605 = ZqZp params\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n    check! \"type of statement\" (typ === fromString cpt)\n                               (\u03bb _ \u2192 \"Expected type of statement: \" ++ cpt ++ \" not \" ++ toString typ)\n >> zk-check-chaum-pedersen-pok-elgamal-rnd! pok\n module Zk-check where\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a6dfe1f553ba79964ad12503ce1a9095fe79ffe2","subject":"Add test","message":"Add test\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 7))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\napply : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\napply _ \u2218        = \u2218\napply f (x \u2237 xs) = (f x) \u2237 (apply f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"255f75653eee710cb4309c5fe5ec0c5a42aea891","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  -- todo: add a premise from a new judgement like holes disjoint that\n  -- classifies pairs of dhexps that share no variable names\n  -- whatsoever. that should imply freshness here. then propagate that\n  -- change to preservation. this is morally what \u03b1-equiv lets us do in a\n  -- real setting.\n\n  -- the variable name x does not appear in the term d\n  data var-name-new : (x : Nat) (d : dhexp) \u2192 Set where\n    VNNConst : \u2200{x} \u2192 var-name-new x c\n    VNNVar : \u2200{x y} \u2192 x \u2260 y \u2192 var-name-new x (X y)\n    VNNLam2 : \u2200{x d y \u03c4} \u2192 x \u2260 y\n                         \u2192 var-name-new x d\n                         \u2192 var-name-new x (\u00b7\u03bb_[_]_ y \u03c4 d)\n    VNNHole : \u2200{x u \u03c3} \u2192 var-name-new x (\u2987\u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNNNEHole : \u2200{x u \u03c3 d } \u2192\n                var-name-new x d \u2192\n                var-name-new x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNNAp : \u2200{ x d1 d2 } \u2192\n           var-name-new x d1 \u2192\n           var-name-new x d2 \u2192\n           var-name-new x (d1 \u2218 d2)\n    VNNCast : \u2200{x d \u03c41 \u03c42} \u2192 var-name-new x d \u2192 var-name-new x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n    VNNFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 var-name-new x d \u2192 var-name-new x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- two terms that do not share any hole names\n  data var-names-disjoint : (d1 : dhexp) \u2192 (d2 : dhexp) \u2192 Set where\n    VNDConst : \u2200{d} \u2192 var-names-disjoint c d\n    VNDVar : \u2200{x d} \u2192 var-names-disjoint (X x) d\n    VNDLam : \u2200{x \u03c4 d1 d2} \u2192 var-names-disjoint d1 d2\n                          \u2192 var-names-disjoint (\u00b7\u03bb_[_]_ x \u03c4 d1) d2\n    VNDHole : \u2200{u \u03c3 d2} \u2192 var-name-new u d2\n                        \u2192 var-names-disjoint (\u2987\u2988\u27e8 u , \u03c3 \u27e9) d2 -- todo something about \u03c3?\n    VNDNEHole : \u2200{u \u03c3 d1 d2} \u2192 var-name-new u d2\n                             \u2192 var-names-disjoint d1 d2\n                             \u2192 var-names-disjoint (\u2987 d1 \u2988\u27e8 u , \u03c3 \u27e9) d2 -- todo something about \u03c3?\n    VNDAp :  \u2200{d1 d2 d3} \u2192 var-names-disjoint d1 d3\n                         \u2192 var-names-disjoint d2 d3\n                         \u2192 var-names-disjoint (d1 \u2218 d2) d3\n\n  -- all the variable names in the term are unique\n  data var-names-unique : dhexp \u2192 Set where\n    VNUHole : var-names-unique c\n    VNUVar : \u2200{x} \u2192 var-names-unique (X x)\n    VNULam : {x : Nat} {\u03c4 : htyp} {d : dhexp} \u2192 var-names-unique d\n                                              \u2192 var-name-new x d\n                                              \u2192 var-names-unique (\u00b7\u03bb_[_]_ x \u03c4 d)\n    VNUEHole : \u2200{u \u03c3} \u2192 var-names-unique (\u2987\u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNUNEHole : \u2200{u \u03c3 d} \u2192 var-names-unique d\n                         \u2192 var-names-unique (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNUAp : \u2200{d1 d2} \u2192 var-names-unique d1\n                     \u2192 var-names-unique d2\n                     \u2192 var-names-disjoint d1 d2\n                     \u2192 var-names-unique (d1 \u2218 d2)\n    VNUCast : \u2200{d \u03c41 \u03c42} \u2192 var-names-unique d\n                         \u2192 var-names-unique (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n    VNUFailedCast : \u2200{d \u03c41 \u03c42} \u2192 var-names-unique d\n                               \u2192 var-names-unique (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  unique-fresh : \u2200{\u0394 \u0393 d \u03c4 y} \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192 y # \u0393 \u2192 var-names-unique d \u2192 fresh y d\n  unique-fresh TAConst apt VNUHole = FConst\n  unique-fresh (TAVar x\u2081) apt VNUVar = {!!}\n  unique-fresh (TALam x\u2081 wt) apt (VNULam unq x\u2082) = {!!}\n  unique-fresh (TAAp wt wt\u2081) apt (VNUAp unq unq\u2081 x) = {!!}\n  unique-fresh (TAEHole x x\u2081) apt VNUEHole = {!!}\n  unique-fresh (TANEHole x wt x\u2081) apt (VNUNEHole unq) = {!!}\n  unique-fresh (TACast wt x) apt (VNUCast unq) = {!!}\n  unique-fresh (TAFailedCast wt x x\u2081 x\u2082) apt (VNUFailedCast unq) = {!!}\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  -- todo: add a premise from a new judgement like holes disjoint that\n  -- classifies pairs of dhexps that share no variable names\n  -- whatsoever. that should imply freshness here. then propagate that\n  -- change to preservation. this is morally what \u03b1-equiv lets us do in a\n  -- real setting.\n\n  -- the variable name x does not appear in the term d\n  data var-name-new : (x : Nat) (d : dhexp) \u2192 Set where\n    VNNConst : \u2200{x} \u2192 var-name-new x c\n    VNNVar : \u2200{x y} \u2192 x \u2260 y \u2192 var-name-new x (X y)\n    VNNLam2 : \u2200{x d y \u03c4} \u2192 x \u2260 y\n                         \u2192 var-name-new x d\n                         \u2192 var-name-new x (\u00b7\u03bb_[_]_ y \u03c4 d)\n    VNNHole : \u2200{x u \u03c3} \u2192 var-name-new x (\u2987\u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNNNEHole : \u2200{x u \u03c3 d } \u2192\n                var-name-new x d \u2192\n                var-name-new x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNNAp : \u2200{ x d1 d2 } \u2192\n           var-name-new x d1 \u2192\n           var-name-new x d2 \u2192\n           var-name-new x (d1 \u2218 d2)\n    VNNCast : \u2200{x d \u03c41 \u03c42} \u2192 var-name-new x d \u2192 var-name-new x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n    VNNFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 var-name-new x d \u2192 var-name-new x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- two terms that do not share any hole names\n  data var-names-disjoint : (d1 : dhexp) \u2192 (d2 : dhexp) \u2192 Set where\n    VNDConst : \u2200{d} \u2192 var-names-disjoint c d\n    VNDVar : \u2200{x d} \u2192 var-names-disjoint (X x) d\n    VNDLam : \u2200{x \u03c4 d1 d2} \u2192 var-names-disjoint d1 d2\n                          \u2192 var-names-disjoint (\u00b7\u03bb_[_]_ x \u03c4 d1) d2\n    VNDHole : \u2200{u \u03c3 d2} \u2192 var-name-new u d2\n                        \u2192 var-names-disjoint (\u2987\u2988\u27e8 u , \u03c3 \u27e9) d2 -- todo something about \u03c3?\n    VNDNEHole : \u2200{u \u03c3 d1 d2} \u2192 var-name-new u d2\n                             \u2192 var-names-disjoint d1 d2\n                             \u2192 var-names-disjoint (\u2987 d1 \u2988\u27e8 u , \u03c3 \u27e9) d2 -- todo something about \u03c3?\n    VNDAp :  \u2200{d1 d2 d3} \u2192 var-names-disjoint d1 d3\n                         \u2192 var-names-disjoint d2 d3\n                         \u2192 var-names-disjoint (d1 \u2218 d2) d3\n\n  -- all the variable names in the term are unique\n  data var-names-unique : dhexp \u2192 Set where\n    VNUHole : var-names-unique c\n    VNUVar : \u2200{x} \u2192 var-names-unique (X x)\n    VNULam : {x : Nat} {\u03c4 : htyp} {d : dhexp} \u2192 var-names-unique d\n                                              \u2192 var-name-new x d\n                                              \u2192 var-names-unique (\u00b7\u03bb_[_]_ x \u03c4 d)\n    VNUEHole : \u2200{u \u03c3} \u2192 var-names-unique (\u2987\u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNUNEHole : \u2200{u \u03c3 d} \u2192 var-names-unique d\n                         \u2192 var-names-unique (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNUAp : \u2200{d1 d2} \u2192 var-names-unique d1\n                     \u2192 var-names-unique d2\n                     \u2192 var-names-disjoint d1 d2\n                     \u2192 var-names-unique (d1 \u2218 d2)\n    VNUCast : \u2200{d \u03c41 \u03c42} \u2192 var-names-unique d\n                         \u2192 var-names-unique (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n    VNUFailedCast : \u2200{d \u03c41 \u03c42} \u2192 var-names-unique d\n                               \u2192 var-names-unique (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"13d78b19995ed646b53097d73e796ce8c886e695","subject":"Callback0 -> JS!","message":"Callback0 -> JS!\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS\/Proc.agda","new_file":"lib\/FFI\/JS\/Proc.agda","new_contents":"module FFI.JS.Proc where\n\nopen import FFI.JS\nopen import Control.Process.Type\n\nabstract\n  URI = String\n  showURI : URI \u2192 String\n  showURI x = x\n  readURI : String \u2192 URI\n  readURI x = x\n\nJSProc = Proc URI JSValue\n\n{-\npostulate server : (ip port  : String)\n                   (proc     : URI \u2192 JSProc)\n                   (callback : URI \u2192 JSCmd) \u2192 JSCmd\n{-# COMPILED_JS server require(\"proc\").serverCurry #-}\n\npostulate client : JSProc \u2192 JSCmd \u2192 JSCmd\n{-# COMPILED_JS client require(\"proc\").clientCurry #-}\n-}\n\npostulate server : (ip port  : String)\n                   (proc     : URI \u2192 JSProc)\n                 \u2192 Callback1 URI\n{-# COMPILED_JS server require(\"proc\").serverCurry #-}\n\npostulate client : JSProc \u2192 JS!\n{-# COMPILED_JS client require(\"proc\").clientCurry #-}\n","old_contents":"module FFI.JS.Proc where\n\nopen import FFI.JS\nopen import Control.Process.Type\n\nabstract\n  URI = String\n  showURI : URI \u2192 String\n  showURI x = x\n  readURI : String \u2192 URI\n  readURI x = x\n\nJSProc = Proc URI JSValue\n\n{-\npostulate server : (ip port  : String)\n                   (proc     : URI \u2192 JSProc)\n                   (callback : URI \u2192 JSCmd) \u2192 JSCmd\n{-# COMPILED_JS server require(\"proc\").serverCurry #-}\n\npostulate client : JSProc \u2192 JSCmd \u2192 JSCmd\n{-# COMPILED_JS client require(\"proc\").clientCurry #-}\n-}\n\npostulate server : (ip port  : String)\n                   (proc     : URI \u2192 JSProc)\n                 \u2192 Callback1 URI\n{-# COMPILED_JS server require(\"proc\").serverCurry #-}\n\npostulate client : JSProc \u2192 Callback0\n{-# COMPILED_JS client require(\"proc\").clientCurry #-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9bf7b311f51bcaf7be6e3f874a7f24f1f1ad99d9","subject":"Testing automatic proofs with streams.","message":"Testing automatic proofs with streams.\n\nIgnore-this: 19fe2f88ffe2f0b524f957d35eb3978e\n\ndarcs-hash:20110725165341-3bd4e-ffb75f484bb59247ff637ca3f5013c1e9fad2044.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/PropertiesATP.agda","new_file":"Draft\/FOTC\/Data\/Stream\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Stream.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesATP\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\npostulate\n  helper\u2081 : \u2200 {ws} \u2192 (\u2203 \u03bb zs \u2192 ws \u2248 zs) \u2192\n            \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 (\u2203 \u03bb zs \u2192 ws' \u2248 zs)))\n{-# ATP prove helper\u2081 #-}\n\npostulate\n  helper\u2082 : \u2200 {zs} \u2192 (\u2203 \u03bb ws \u2192 ws \u2248 zs) \u2192\n            \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 (\u2203 \u03bb ws \u2192 ws \u2248 zs')))\n{-# ATP prove helper\u2082 #-}\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Stream.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesATP\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective x\u2237xs\u2261e\u2237es))) Ses\n  where\n  unfold : \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 Stream es\n  unfold = Stream-gfp\u2081 h\n\n  e : D\n  e = \u2203-proj\u2081 unfold\n\n  es : D\n  es = \u2203-proj\u2081 (\u2203-proj\u2082 unfold)\n\n  x\u2237xs\u2261e\u2237es : x \u2237 xs \u2261 e \u2237 es\n  x\u2237xs\u2261e\u2237es = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold))\n\n  Ses : Stream es\n  Ses = \u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold))\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n\n  postulate\n    helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n              \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 P\u2081 ws'))\n  {-# ATP prove helper\u2081 #-}\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n\n  postulate\n    helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  {-# ATP prove helper\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"16e36c67b2fcfd4a473120e797a5f65761648085","subject":"Added Stream-build.","message":"Added Stream-build.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Functors.agda","new_file":"notes\/fixed-points\/Functors.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conat type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- The pred function is the conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n-- From (Leclerc and Paulin-Mohring 1994, p. 195).\n{-# NO_TERMINATION_CHECK #-}\nStream-build :\n  {A X : Set} \u2192\n  (X \u2192 StreamF A X) \u2192\n  X \u2192 Stream A\nStream-build f x with f x\n... | a , x' = Wrap (a , Stream-build f x')\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Leclerc, F. and Paulin-Mohring, C. (1994). Programming with Streams\n-- in Coq. A case study : the Sieve of Eratosthenes. In: Types for\n-- Proofs and Programs (TYPES \u201993). Ed. by Barendregt, H. and Nipkow,\n-- T. Vol. 806. LNCS. Springer, pp. 191\u2013212.\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conat type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- The pred function is the conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bf41bcaafc48c3940d90b646ceaad93f43a83882","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/DistributiveLaws\/PropertiesI.agda","new_file":"src\/fot\/DistributiveLaws\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Distributive laws properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule DistributiveLaws.PropertiesI where\n\nopen import DistributiveLaws.Base\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n-- The propositional equality is compatible with the binary operation.\n\n-- \u00b7-cong : \u2200 {a b c d} \u2192 a \u2261 b \u2192 c \u2261 d \u2192 a \u00b7 c \u2261 b \u00b7 d\n-- \u00b7-cong = cong\u2082 _\u00b7_\n\n\u00b7-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a \u00b7 c \u2261 b \u00b7 c\n\u00b7-leftCong refl = refl\n\n\u00b7-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a \u00b7 b \u2261 a \u00b7 c\n\u00b7-rightCong refl = refl\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule DistributiveLaws.PropertiesI where\n\nopen import DistributiveLaws.Base\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n-- The propositional equality is compatible with the binary operation.\n\n-- \u00b7-cong : \u2200 {a b c d} \u2192 a \u2261 b \u2192 c \u2261 d \u2192 a \u00b7 c \u2261 b \u00b7 d\n-- \u00b7-cong = cong\u2082 _\u00b7_\n\n\u00b7-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a \u00b7 c \u2261 b \u00b7 c\n\u00b7-leftCong refl = refl\n\n\u00b7-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a \u00b7 b \u2261 a \u00b7 c\n\u00b7-rightCong refl = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d849c7f1f712c85e23c46f6ec0ec2adec592154a","subject":"Added missing modules to README.agda","message":"Added missing modules to README.agda\n\nIgnore-this: fb16a8a7c5d2c7e33af18e9d9645d7a7\n\ndarcs-hash:20110517124334-3bd4e-4e668149cae7521fd5188937e01c04544df7bcb8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"709fb9f2b23ad42da6f0fbb124a140b01bf8bc3a","subject":"Drop (unused) System L","message":"Drop (unused) System L\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/LType.agda","new_file":"Parametric\/Syntax\/LType.agda","new_contents":"","old_contents":"import Parametric.Syntax.Type as Type\n\n{--\n-- Encoding types of polarized System L, as described in \"A dissection of L\".\n-- However, I formalized contexts in a wrong way, hence there are problems.\n--}\n\nmodule Parametric.Syntax.LType where\n\n-- The treatment of base type seems questionable. We might want instead to\n-- distinguish base value types and base computation types. Or not, I'm not\n-- sure.\n\nmodule Structure (Base : Type.Structure) where\n  open Type.Structure Base\n\n  mutual\n    -- Polarized L.\n    --  Values (positive types)\n    data VType : Set where\n      vB : (\u03b9 : Base) \u2192 VType\n\n      _\u2297_ : (\u03c4\u2081 : VType) \u2192 (\u03c4\u2082 : VType) \u2192 VType\n      vUnit : VType\n\n      _\u2295_ : (\u03c4\u2081 : VType) \u2192 (\u03c4\u2082 : VType) \u2192 VType\n      vZero : VType\n\n      ! : VType \u2192 VType\n      \u2193 : (c : CType) \u2192 VType\n\n    data CType : Set where\n      cB : (\u03b9 : Base) \u2192 CType\n\n      _\u214b_ : CType \u2192 CType \u2192 CType\n      c\u22a5 : CType\n\n      _&_ : CType \u2192 CType \u2192 CType\n      c\u22a4 : CType\n\n      \u00bf : CType \u2192 CType\n      \u2191 : VType \u2192 CType\n\n  _\u2020v : VType \u2192 CType\n  _\u2020c : CType \u2192 VType\n\n  vB \u03b9 \u2020v = cB \u03b9\n  (v\u2081 \u2297 v\u2082) \u2020v = (v\u2081 \u2020v) \u214b (v\u2082 \u2020v)\n  vUnit \u2020v = c\u22a5\n  (v\u2081 \u2295 v\u2082) \u2020v = (v\u2081 \u2020v) & (v\u2082 \u2020v)\n  vZero \u2020v = c\u22a4\n  ! v \u2020v = \u00bf (v \u2020v)\n  \u2193 c \u2020v = \u2191 (c \u2020c)\n\n  cB \u03b9 \u2020c = vB \u03b9\n  (c\u2081 \u214b c\u2082) \u2020c = c\u2081 \u2020c \u2297 c\u2082 \u2020c\n  c\u22a5 \u2020c = vUnit\n  (c\u2081 & c\u2082) \u2020c = (c\u2081 \u2020c) \u2295 (c\u2082 \u2020c)\n  c\u22a4 \u2020c = vZero\n  \u00bf c \u2020c = ! (c \u2020c)\n  \u2191 v \u2020c = \u2193 (v \u2020v)\n\n  open import Relation.Binary.PropositionalEquality\n\n  -- This proof really calls for tactics, doesn't it?\n  invV : \u2200 {v} \u2192 v \u2261 v \u2020v \u2020c\n  invC : \u2200 {c} \u2192 c \u2261 c \u2020c \u2020v\n\n  invV {vB \u03b9} = refl\n  invV {v\u2081 \u2297 v\u2082} = cong\u2082 _\u2297_ invV invV\n  invV {vUnit} = refl\n  invV {v\u2081 \u2295 v\u2082} = cong\u2082 _\u2295_ invV invV\n  invV {vZero} = refl\n  invV { ! v} = cong ! invV\n  invV {\u2193 c} = cong \u2193 invC\n\n  invC {cB \u03b9} = refl\n  invC {c\u2081 \u214b c\u2082} = cong\u2082 _\u214b_ invC invC\n  invC {c\u22a5} = refl\n  invC {c\u2081 & c\u2082} = cong\u2082 _&_ invC invC\n  invC {c\u22a4} = refl\n  invC {\u00bf c} = cong \u00bf invC\n  invC {\u2191 x} = cong \u2191 invV\n\n  open import Base.Syntax.Context VType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; _\u22ce_ to _\u22ce\u22ce_\n      ; mapContext to mapVCtx\n      ; Var to VVar\n      ; Context to VContext\n      ; this to vThis; that to vThat)\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 VType) \u2192 Var \u0393 \u03c4 \u2192 VVar (mapVCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n\n  data VTerm : (\u039e : VContext) \u2192 (\u0393 : VContext) \u2192 VType \u2192 Set\n  data CTerm : (\u039e : VContext) \u2192 (\u0393 : VContext) \u2192 CType \u2192 Set\n\n  -- Each constructor of the following types encodes a rule from the paper (Fig.\n  -- 4); the original names are given in comments at the end of each row.\n\n  data Cmd : (\u039e : VContext) \u2192 (\u0393 : VContext) \u2192 Set where\n    -- I swapped the operands compared to the cut rule in the paper (Fig. 4),\n    -- but this should be purely cosmetic.\n    \u27e8_\u2223_\u27e9 : \u2200 {\u039e \u0393 \u0394 \u03c4} \u2192 (v : CTerm \u039e \u0394 (\u03c4 \u2020v)) \u2192 (u : VTerm \u039e \u0393 \u03c4) \u2192 Cmd \u039e (\u0393 \u22ce\u22ce \u0394) -- cut\n\n  data VTerm where\n    lvar : \u2200 {\u039e A} \u2192 VTerm \u039e (A \u2022\u2022 \u2205\u2205) A        -- id\n    vvar : \u2200 {\u039e A} \u2192 VVar \u039e A \u2192 VTerm \u039e \u2205\u2205 A   -- id'\n    \u03bc\u21d1_ : \u2200 {\u039e \u0393 N} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 \u0393) \u2192 VTerm \u039e \u0393 (\u2193 N) -- \u2193\n\n    -- Multiplicative fragment\n    _,_ : \u2200 {\u039e \u0393 \u0394 A B} \u2192 VTerm \u039e \u0393 A \u2192 VTerm \u039e \u0394 B \u2192 VTerm \u039e (\u0393 \u22ce\u22ce \u0394) (A \u2297 B) -- \u2297\n    \u27e8\u27e9 : \u2200 {\u039e} \u2192 VTerm \u039e \u2205\u2205 vUnit --1\n\n    -- Additive fragment\n    _\u2081as_ : \u2200 {\u039e \u0393 A} \u2192 VTerm \u039e \u0393 A \u2192 \u2200 B \u2192 VTerm \u039e \u0393 (A \u2295 B) -- \u2295L\n    _\u2082as_ : \u2200 {\u039e \u0393 B} \u2192 VTerm \u039e \u0393 B \u2192 \u2200 A \u2192 VTerm \u039e \u0393 (A \u2295 B) -- \u2295R\n\n    -- No rule for vZero\n\n    -- Exponential fragment (in the sense of linear logic)\n\n    \u2983\u2984_ : \u2200 {\u039e A} \u2192 VTerm \u039e \u2205\u2205 A \u2192 VTerm \u039e \u2205\u2205 (! A) -- !\n    -- : \u2200 {\u039e \u0393 A} \u2192\n\n  -- Compared to the paper, we avoid using \u2020 in indexes of the result, and\n  -- exploit the fact that \u2020 is involutive. See for instance \u03bc\n  data CTerm where\n    \u03bc : \u2200 {\u039e \u0393 N} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 \u0393) \u2192 CTerm \u039e \u0393 N\n    \u21d1 : \u2200 {\u039e \u0393 A} \u2192 VTerm \u039e \u0393 A \u2192 CTerm \u039e \u0393 (\u2191 A) -- \u2191\n\n    -- Multiplicative fragment\n    \u03bc, : \u2200 {\u039e \u0393 N M} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 M \u2020c \u2022\u2022 \u0393) \u2192 CTerm \u039e \u0393 (N \u214b M) -- \u214b\n    \u03bc\u27e8\u27e9 : \u2200 {\u039e \u0393} \u2192 Cmd \u039e \u0393 \u2192 CTerm \u039e \u0393 c\u22a5 -- \u22a5\n\n    -- Additive fragment\n    \u03bc\u2081_\u03bc\u2082_ : \u2200 {\u039e \u0393 N M} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 \u0393) \u2192 Cmd \u039e (M \u2020c \u2022\u2022 \u0393) \u2192 CTerm \u039e \u0393 (N & M)\n\n    -- Exponential fragment (in the sense of linear logic)\n    \u03bc\u2983\u2984 : \u2200 {\u039e \u0393 N} \u2192 Cmd (N \u2020c \u2022\u2022 \u039e) \u0393 \u2192 CTerm \u039e \u0393 (\u00bf N) -- ?\n\n    -- : \u2200 {\u039e \u0393 \u03c4} \u2192\n\n  cast-invV : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 N \u2192 VTerm \u039e \u0393 ((N \u2020v) \u2020c)\n  cast-invV {\u039e} {\u0393} t = subst (VTerm \u039e \u0393) invV t\n\n  cast-invV2 : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 ((N \u2020v) \u2020c) \u2192 VTerm \u039e \u0393 N\n  cast-invV2 {\u039e} {\u0393} t = subst (VTerm \u039e \u0393) (sym invV) t\n\n  cast-invC : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 ((N \u2020c) \u2020v)\n  cast-invC {\u039e} {\u0393} t = subst (CTerm \u039e \u0393) invC t\n\n  -- Check that type rules match by doing eta-expansion.\n  -- (Is this checking harmony?)\n  \u03b7-\u03bc : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 N\n  \u03b7-\u03bc t = \u03bc \u27e8 cast-invC t \u2223 lvar \u27e9\n\n  maybe-\u03b7-\u03bc\u21d1 : \u2200 {\u039e \u0393 A} \u2192 VTerm \u039e \u0393 A \u2192 VTerm \u039e \u0393 (\u2193 (\u2191 A))\n  maybe-\u03b7-\u03bc\u21d1 t = \u03bc\u21d1 \u27e8 cast-invC (\u21d1 t) \u2223 lvar \u27e9\n\n  -- A \"thunk\" combinator (from paper)\n  \u21d3 : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 N \u2192 VTerm \u039e \u0393 (\u2193 N)\n  \u21d3 t = \u03bc\u21d1 \u27e8 cast-invC t \u2223 lvar \u27e9\n\n  open import UNDEFINED\n\n  -- from paper\n  force : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 (\u2193 N) \u2192 CTerm \u039e \u0393 N\n  force t = reveal UNDEFINED -- {! \u03bc \u27e8 \u21d1 lvar \u2223 t \u27e9 !}\n\n  -- XXX Here we \"just\" need to reorder variables in contexts. OMG, I screwed up\n  -- (which is bad), or they use implicit exchange (which would be worse).\n  \u03b7-\u03bc\u21d1 : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 (\u2193 N) \u2192 VTerm \u039e \u0393 (\u2193 N)\n  \u03b7-\u03bc\u21d1 t = \u03bc\u21d1 (reveal UNDEFINED) -- {! \u27e8 \u21d1 lvar \u2223 t \u27e9 !}\n\n  n-\u03bc, : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 (N \u214b M) \u2192 CTerm \u039e \u0393 (N \u214b M)\n  n-\u03bc, t = \u03bc, \u27e8 cast-invC t \u2223 lvar , lvar \u27e9 -- The two lvars are in different contexts, so they have different types!\n\n  \u03b7-\u03bc\u27e8\u27e9 : \u2200 {\u039e \u0393} \u2192 CTerm \u039e \u0393 c\u22a5 \u2192 CTerm \u039e \u0393 c\u22a5\n  \u03b7-\u03bc\u27e8\u27e9 t = \u03bc\u27e8\u27e9 \u27e8 t \u2223 \u27e8\u27e9 \u27e9\n\n  \u03b7-\u03bc\u2983\u2984 : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 (\u00bf N) \u2192 CTerm \u039e \u0393 (\u00bf N)\n  \u03b7-\u03bc\u2983\u2984 t = \u03bc\u2983\u2984 \u27e8  cast-invC (reveal UNDEFINED) {- weaken t -} \u2223 \u2983\u2984 (vvar vThis) \u27e9\n\n  \u03b2-expand-\u03bc\u27e8\u27e9 : \u2200 {\u039e \u0393} \u2192 Cmd \u039e \u0393 \u2192 Cmd \u039e \u0393\n  \u03b2-expand-\u03bc\u27e8\u27e9 c = \u27e8 \u03bc\u27e8\u27e9 c \u2223 \u27e8\u27e9 \u27e9\n\n  -- \u03b7-rules for sums look different.\n  maybe-\u03b7-\u03bc\u2081 : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 M \u2192 CTerm \u039e \u0393 (N & M)\n  maybe-\u03b7-\u03bc\u2081 t\u2081 t\u2082 = \u03bc\u2081 \u27e8 cast-invC t\u2081 \u2223 lvar \u27e9 \u03bc\u2082  \u27e8 cast-invC t\u2082 \u2223 lvar \u27e9\n\n  \u03b7-\u03bc\u2081 : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 M \u2192 CTerm \u039e \u0393 N\n  \u03b7-\u03bc\u2081 {M = M} t\u2081 t\u2082 =\n    \u03bc \u27e8\n      \u03bc\u2081 \u27e8 cast-invC (cast-invC t\u2081) \u2223 lvar \u27e9  \u03bc\u2082 \u27e8 cast-invC (cast-invC t\u2082) \u2223 lvar \u27e9\n    \u2223\n      lvar \u2081as (M \u2020c)\n    \u27e9\n\n  \u03b7-\u03bc\u2082 : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 M \u2192 CTerm \u039e \u0393 M\n  \u03b7-\u03bc\u2082 {N = N} t\u2081 t\u2082 =\n    \u03bc \u27e8\n      \u03bc\u2081 \u27e8 cast-invC (cast-invC t\u2081) \u2223 lvar \u27e9  \u03bc\u2082 \u27e8 cast-invC (cast-invC t\u2082) \u2223 lvar \u27e9\n    \u2223\n      lvar \u2082as (N \u2020c)\n    \u27e9\n\n  -- Some syntactic sugar.\n  _\u22b8_ : VType \u2192 CType \u2192 CType\n  \u03c3 \u22b8 \u03c4 = (\u03c3 \u2020v) \u214b \u03c4\n\n  _\u21db_ : VType \u2192 CType \u2192 CType\n  \u03c3 \u21db \u03c4 = ! \u03c3 \u22b8 \u03c4\n\n\n  -- Not checked again yet. This code comes from CBPV and was refactored for\n  -- renamings done to get polarized L, so it might or might not work.\n\n  cbnToCType : Type \u2192 CType\n  cbnToCType (base \u03b9) = \u2191 (vB \u03b9)\n  cbnToCType (\u03c3 \u21d2 \u03c4) = \u2193 (cbnToCType \u03c3) \u21db cbnToCType \u03c4\n\n  cbvToVType : Type \u2192 VType\n  cbvToVType (base \u03b9) = vB \u03b9\n  cbvToVType (\u03c3 \u21d2 \u03c4) = \u2193 (cbvToVType \u03c3 \u21db \u2191 (cbvToVType \u03c4))\n\n  cbnToVType : Type \u2192 VType\n  cbnToVType \u03c4 = \u2193 (cbnToCType \u03c4)\n\n  cbvToCType : Type \u2192 CType\n  cbvToCType \u03c4 = \u2191 (cbvToVType \u03c4)\n\n  fromCBNCtx : Context \u2192 VContext\n  fromCBNCtx \u0393 = mapVCtx cbnToVType \u0393\n\n  fromCBVCtx : Context \u2192 VContext\n  fromCBVCtx \u0393 = mapVCtx cbvToVType \u0393\n\n  open import Data.List\n  open Data.List using (List) public\n  fromCBVToCompList : Context \u2192 List CType\n  fromCBVToCompList \u0393 = mapVCtx cbvToCType \u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"51432b971511d6e3b28904b1d33fc55d32888906","subject":"FunExt: add a version with implicit arguments","message":"FunExt: add a version with implicit arguments\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Extensionality.agda","new_file":"lib\/Function\/Extensionality.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP  -- renaming (subst to tr)\n\nhapply : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}\n  \u2192 f \u2261 g \u2192 (x : A) \u2192 f x \u2261 g x\nhapply p x = ap (\u03bb f \u2192 f x) p\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n           (f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n\n    happly-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 happly (\u03bb= fg) \u2261 fg\n\n    \u03bb=-happly : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (\u03b1 : f \u2261 g) \u2192 \u03bb= (happly \u03b1) \u2261 \u03b1\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}{{fe : FunExt}}(fg : (x : A) \u2192 f x \u2261 g x)\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n\n\n!-\u03b1-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (\u03b1 : f \u2261 g) \u2192 ! \u03b1 \u2261 \u03bb= (!_ \u2218 happly \u03b1)\n!-\u03b1-\u03bb= refl = ! \u03bb=-happly refl\n\n!-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 ! (\u03bb= fg) \u2261 \u03bb= (!_ \u2218 fg)\n!-\u03bb= fg = !-\u03b1-\u03bb= (\u03bb= fg) \u2219 ap \u03bb= (\u03bb= (\u03bb x \u2192 ap !_ (happly (happly-\u03bb= fg) x)))\n\nmodule _ {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}} where\n  \u03bb=\u2071 : (f= : \u2200 {x} \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n  \u03bb=\u2071 f= = \u03bb= \u03bb x \u2192 f= {x}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP  -- renaming (subst to tr)\n\nhapply : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}\n  \u2192 f \u2261 g \u2192 (x : A) \u2192 f x \u2261 g x\nhapply p x = ap (\u03bb f \u2192 f x) p\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n           (f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n\n    happly-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 happly (\u03bb= fg) \u2261 fg\n\n    \u03bb=-happly : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (\u03b1 : f \u2261 g) \u2192 \u03bb= (happly \u03b1) \u2261 \u03b1\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}{{fe : FunExt}}(fg : (x : A) \u2192 f x \u2261 g x)\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n\n\n!-\u03b1-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (\u03b1 : f \u2261 g) \u2192 ! \u03b1 \u2261 \u03bb= (!_ \u2218 happly \u03b1)\n!-\u03b1-\u03bb= refl = ! \u03bb=-happly refl\n\n!-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 ! (\u03bb= fg) \u2261 \u03bb= (!_ \u2218 fg)\n!-\u03bb= fg = !-\u03b1-\u03bb= (\u03bb= fg) \u2219 ap \u03bb= (\u03bb= (\u03bb x \u2192 ap !_ (happly (happly-\u03bb= fg) x)))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3ed5e78d2041cfe6ec4072cfe37bc4f1950cd73f","subject":"[ notes ] Commented out an unprovable conjecture","message":"[ notes ] Commented out an unprovable conjecture\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Pow.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Pow.agda","new_contents":"------------------------------------------------------------------------------\n-- Some proofs related to the power function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.Nat.Pow where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192 n ^ zero      \u2261 1'\n  ^-S : \u2200 m n \u2192 m ^ succ\u2081 n \u2261 m * m ^ n\n{-# ATP axiom ^-0 ^-S #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ 5' \u2264 5' ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn)\n  where postulate prf : (5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n) \u2192\n                        (succ\u2081 n) ^ 5' \u2264 5' ^ (succ\u2081 n)\n        -- 09 December 2014: The ATPs could not prove the theorem (240 sec).\n        -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261 2' ^ (n + 1') \u2238 1'\nthm\u2082 nzero = prf\n  where\n  postulate prf : ((2' ^ zero) \u2238 1') + 1' + ((2' ^ zero) \u2238 1') \u2261\n                  2' ^ (zero + 1') \u2238 1'\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261\n                  2' ^ (n + 1') \u2238 1' \u2192\n                  ((2' ^ succ\u2081 n) \u2238 1') + 1' + ((2' ^ succ\u2081 n) \u2238 1') \u2261\n                  2' ^ (succ\u2081 n + 1') \u2238 1'\n  -- 09 December 2014: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some proofs related to the power function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.Nat.Pow where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192 n ^ zero      \u2261 1'\n  ^-S : \u2200 m n \u2192 m ^ succ\u2081 n \u2261 m * m ^ n\n{-# ATP axiom ^-0 ^-S #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ 5' \u2264 5' ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn)\n  where postulate prf : (5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n) \u2192\n                        (succ\u2081 n) ^ 5' \u2264 5' ^ (succ\u2081 n)\n        -- 09 December 2014: The ATPs could not prove the theorem (240 sec).\n        -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261 2' ^ (n + 1') \u2238 1'\nthm\u2082 nzero = prf\n  where\n  postulate prf : ((2' ^ zero) \u2238 1') + 1' + ((2' ^ zero) \u2238 1') \u2261\n                  2' ^ (zero + 1') \u2238 1'\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261\n                  2' ^ (n + 1') \u2238 1' \u2192\n                  ((2' ^ succ\u2081 n) \u2238 1') + 1' + ((2' ^ succ\u2081 n) \u2238 1') \u2261\n                  2' ^ (succ\u2081 n + 1') \u2238 1'\n  -- 09 December 2014: The ATPs could not prove the theorem (240 sec).\n  {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f313f949b97f138dd32728e2129de3db80950f64","subject":"Fin: _=?_, iterate","message":"Fin: _=?_, iterate\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"-- NOTE with-K\nmodule Data.Fin.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Zero\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Fin public renaming (to\u2115 to Fin\u25b9\u2115)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; [0:_1:_]; case_0:_1:_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081; tabulate; foldr) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n-- The isomorphisms about Fin, \ud835\udfd8, \ud835\udfd9, \ud835\udfda are in Function.Related.TypeIsomorphisms.NP\n\nFin\u25b9\ud835\udfd8 : Fin 0 \u2192 \ud835\udfd8\nFin\u25b9\ud835\udfd8 ()\n\n\ud835\udfd8\u25b9Fin : \ud835\udfd8 \u2192 Fin 0\n\ud835\udfd8\u25b9Fin ()\n\nFin\u25b9\ud835\udfd9 : Fin 1 \u2192 \ud835\udfd9\nFin\u25b9\ud835\udfd9 _ = _\n\n\ud835\udfd9\u25b9Fin : \ud835\udfd9 \u2192 Fin 1\n\ud835\udfd9\u25b9Fin _ = zero\n\nFin\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\nFin\u25b9\ud835\udfda zero    = 0\u2082\nFin\u25b9\ud835\udfda (suc _) = 1\u2082\n\n\ud835\udfda\u25b9Fin : \ud835\udfda \u2192 Fin 2\n\ud835\udfda\u25b9Fin = [0: # 0 1: # 1 ]\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_\u225f_ : \u2200 {n} (i j : Fin n) \u2192 Dec (i \u2261 j)\nzero \u225f zero = yes refl\nzero \u225f suc j = no (\u03bb())\nsuc i \u225f zero = no (\u03bb())\nsuc i \u225f suc j with i \u225f j\nsuc i \u225f suc j | yes p = yes (cong suc p)\nsuc i \u225f suc j | no \u00acp = no (\u00acp \u2218 suc-injective)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 \ud835\udfda\nx == y = \u230a x \u225f y \u230b\n{-helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 \ud835\udfda\n  helper (equal _) = 1\u2082\n  helper _         = 0\u2082-}\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = case x == z 0: (case y == z 0: z 1: x) 1: y\n\nmodule _ {a} {A : \u2605 a}\n         (B : \u2115 \u2192 \u2605\u2080)\n         (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n         (\u03b5 : B zero) where\n  iterate : \u2200 {n} (f : Fin n \u2192 A) \u2192 B n\n  iterate {zero}  f = \u03b5\n  iterate {suc n} f = f zero \u25c5 iterate (f \u2218 suc)\n\n  iterate-foldr\u2218tabulate :\n    \u2200 {n} (f : Fin n \u2192 A) \u2192 iterate f \u2261 foldr B _\u25c5_ \u03b5 (tabulate f)\n  iterate-foldr\u2218tabulate {zero} f = refl\n  iterate-foldr\u2218tabulate {suc n} f = cong (_\u25c5_ (f zero)) (iterate-foldr\u2218tabulate (f \u2218 suc))\n\nmodule _ {a} {A : \u2605 a} (B : \u2605\u2080)\n         (_\u25c5_ : A \u2192 B \u2192 B)\n         (\u03b5 : B) where\n  iterate\u2032 : \u2200 {n} (f : Fin n \u2192 A) \u2192 B\n  iterate\u2032 f = iterate _ _\u25c5_ \u03b5 f\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 \u2605\u2080 where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\n{-\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n-}\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nFin\u25b9\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 Fin\u25b9\u2115 (n \u2115- x) \u2261 n \u2238 Fin\u25b9\u2115 x\nFin\u25b9\u2115-\u2115-lem zero = to-from _\nFin\u25b9\u2115-\u2115-lem {zero} (suc ())\nFin\u25b9\u2115-\u2115-lem {suc n} (suc x) = Fin\u25b9\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse-old x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (Fin\u25b9\u2115 x) (suc n)) = Fin\u25b9\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 \u2605\u2080 where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \ud835\udfd8-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \ud835\udfd8-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : \u2605 a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : \u2605 a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","old_contents":"-- NOTE with-K\nmodule Data.Fin.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Zero\nopen import Data.One\nopen import Data.Fin public renaming (to\u2115 to Fin\u25b9\u2115)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Two hiding (_==_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n-- The isomorphisms about Fin, \ud835\udfd8, \ud835\udfd9, \ud835\udfda are in Function.Related.TypeIsomorphisms.NP\n\nFin\u25b9\ud835\udfd8 : Fin 0 \u2192 \ud835\udfd8\nFin\u25b9\ud835\udfd8 ()\n\n\ud835\udfd8\u25b9Fin : \ud835\udfd8 \u2192 Fin 0\n\ud835\udfd8\u25b9Fin ()\n\nFin\u25b9\ud835\udfd9 : Fin 1 \u2192 \ud835\udfd9\nFin\u25b9\ud835\udfd9 _ = _\n\n\ud835\udfd9\u25b9Fin : \ud835\udfd9 \u2192 Fin 1\n\ud835\udfd9\u25b9Fin _ = zero\n\nFin\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\nFin\u25b9\ud835\udfda zero    = 0\u2082\nFin\u25b9\ud835\udfda (suc _) = 1\u2082\n\n\ud835\udfda\u25b9Fin : \ud835\udfda \u2192 Fin 2\n\ud835\udfda\u25b9Fin = [0: # 0 1: # 1 ]\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 \ud835\udfda\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 \ud835\udfda\n  helper (equal _) = 1\u2082\n  helper _         = 0\u2082\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = case x == z 0: (case y == z 0: z 1: x) 1: y\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 \u2605\u2080 where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\n{-\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n-}\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nFin\u25b9\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 Fin\u25b9\u2115 (n \u2115- x) \u2261 n \u2238 Fin\u25b9\u2115 x\nFin\u25b9\u2115-\u2115-lem zero = to-from _\nFin\u25b9\u2115-\u2115-lem {zero} (suc ())\nFin\u25b9\u2115-\u2115-lem {suc n} (suc x) = Fin\u25b9\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse-old x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (Fin\u25b9\u2115 x) (suc n)) = Fin\u25b9\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 \u2605\u2080 where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \ud835\udfd8-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \ud835\udfd8-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : \u2605 a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : \u2605 a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b531cfe0eaf8c265a52d179884b4b7d3964e4f81","subject":"Vec: unconvincing experiment with reverse","message":"Vec: unconvincing experiment with reverse\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product hiding (map; zip)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c12d0f122a68f42ab685380279a40c4bed632639","subject":"Added thesis' examples.","message":"Added thesis' examples.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/LogicalFramework\/AdequacyTheorems.agda","new_file":"notes\/thesis\/LogicalFramework\/AdequacyTheorems.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.AdequacyTheorems where\n\nmodule Example5 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081    : A \u2192 C\n    f\u2082    : B \u2192 C\n\n  g : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  g f\u2081 f\u2082 (inj\u2081 a) = f\u2081 a\n  g f\u2081 f\u2082 (inj\u2082 b) = f\u2082 b\n\n  g' : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  g' f\u2081 f\u2082 x = case f\u2081 f\u2082 x\n\nmodule Example7 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    C  : D \u2192 D \u2192 Set\n    d  : \u2200 {a} \u2192 C a a\n\n  g : \u2200 {a b} \u2192 a \u2261 b \u2192 C a b\n  g refl = d\n\n  g' : \u2200 {a b} \u2192 a \u2261 b \u2192 C a b\n  g' {a} h = subst (\u03bb x \u2192 C a x) h d\n\nmodule Example10 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081 : A \u2192 C\n    f\u2082 : B \u2192 C\n\n  f : A \u2228 B \u2192 C\n  f (inj\u2081 a) = f\u2081 a\n  f (inj\u2082 b) = f\u2082 b\n\n  f' : A \u2228 B \u2192 C\n  f' = case f\u2081 f\u2082\n\nmodule Example20 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  f : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f {A} {t} {.t} refl At = d At\n    where\n    postulate d : A t \u2192 A t\n\n  f' : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f' {A} h At = subst A h At\n\nmodule Example30 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C E : Set\n    f\u2081 : A \u2192 E\n    f\u2082 : B \u2192 E\n    f\u2083 : C \u2192 E\n\n  g : (A \u2228 B) \u2228 C \u2192 E\n  g (inj\u2081 (inj\u2081 a)) = f\u2081 a\n  g (inj\u2081 (inj\u2082 b)) = f\u2082 b\n  g (inj\u2082 c)        = f\u2083 c\n\n  g' : (A \u2228 B) \u2228 C \u2192 E\n  g' = case (case f\u2081 f\u2082) f\u2083\n\nmodule Example40 where\n\n  infixl 9  _+_ _+'_\n  infix  7  _\u2261_\n\n  data M : Set where\n    zero :     M\n    succ : M \u2192 M\n\n  PA-ind : (A : M \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n  PA-ind A A0 h zero     = A0\n  PA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n  data _\u2261_ (x : M) : M \u2192 Set where\n    refl : x \u2261 x\n\n  subst : (A : M \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A refl Ax = Ax\n\n  _+_ : M \u2192 M \u2192 M\n  zero   + n = n\n  succ m + n = succ (m + n)\n\n  _+'_ : M \u2192 M \u2192 M\n  m +' n = PA-ind (\u03bb _ \u2192 M) n (\u03bb x y \u2192 succ y) m\n\n  -- Properties using pattern matching.\n  succCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong refl = refl\n\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  +-rightIdentity zero     = refl\n  +-rightIdentity (succ n) = succCong (+-rightIdentity n)\n\n  -- Properties using the basic inductive constants.\n  succCong' : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong' {m} h = subst (\u03bb x \u2192 succ m \u2261 succ x) h refl\n\n  +'-leftIdentity : \u2200 n \u2192 zero +' n \u2261 n\n  +'-leftIdentity n = refl\n\n  +'-rightIdentity : \u2200 n \u2192 n +' zero \u2261 n\n  +'-rightIdentity = PA-ind A A0 is\n    where\n    A : M \u2192 Set\n    A n = n +' zero \u2261 n\n\n    A0 : A zero\n    A0 = refl\n\n    is : \u2200 n \u2192 A n \u2192 A (succ n)\n    is n ih = succCong' ih\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.AdequacyTheorems where\n\nmodule Example10 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081 : A \u2192 C\n    f\u2082 : B \u2192 C\n\n  f : A \u2228 B \u2192 C\n  f (inj\u2081 a) = f\u2081 a\n  f (inj\u2082 b) = f\u2082 b\n\n  f' : A \u2228 B \u2192 C\n  f' = case f\u2081 f\u2082\n\nmodule Example20 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  f : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f {A} {t} {.t} refl At = d At\n    where\n    postulate d : A t \u2192 A t\n\n  f' : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f' {A} h At = subst A h At\n\nmodule Example30 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C E : Set\n    f\u2081 : A \u2192 E\n    f\u2082 : B \u2192 E\n    f\u2083 : C \u2192 E\n\n  f : (A \u2228 B) \u2228 C \u2192 E\n  f (inj\u2081 (inj\u2081 a)) = f\u2081 a\n  f (inj\u2081 (inj\u2082 b)) = f\u2082 b\n  f (inj\u2082 c)        = f\u2083 c\n\n  f' : (A \u2228 B) \u2228 C \u2192 E\n  f' = case (case f\u2081 f\u2082) f\u2083\n\nmodule Example40 where\n\n  infixl 9  _+_ _+'_\n  infix  7  _\u2261_\n\n  data M : Set where\n    zero :     M\n    succ : M \u2192 M\n\n  PA-ind : (A : M \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n  PA-ind A A0 h zero     = A0\n  PA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n  data _\u2261_ (x : M) : M \u2192 Set where\n    refl : x \u2261 x\n\n  subst : (A : M \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A refl Ax = Ax\n\n  _+_ : M \u2192 M \u2192 M\n  zero   + n = n\n  succ m + n = succ (m + n)\n\n  _+'_ : M \u2192 M \u2192 M\n  m +' n = PA-ind (\u03bb _ \u2192 M) n (\u03bb x y \u2192 succ y) m\n\n  -- Properties using pattern matching.\n  succCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong refl = refl\n\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  +-rightIdentity zero     = refl\n  +-rightIdentity (succ n) = succCong (+-rightIdentity n)\n\n  -- Properties using the basic inductive constants.\n  succCong' : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong' {m} h = subst (\u03bb x \u2192 succ m \u2261 succ x) h refl\n\n  +'-leftIdentity : \u2200 n \u2192 zero +' n \u2261 n\n  +'-leftIdentity n = refl\n\n  +'-rightIdentity : \u2200 n \u2192 n +' zero \u2261 n\n  +'-rightIdentity = PA-ind A A0 is\n    where\n    A : M \u2192 Set\n    A n = n +' zero \u2261 n\n\n    A0 : A zero\n    A0 = refl\n\n    is : \u2200 n \u2192 A n \u2192 A (succ n)\n    is n ih = succCong' ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64aeff24b29bf417d3c1ddff59acd073ff749e88","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     envfresh x \u03c3 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 {\u0393 = \u0393} (EFId x\u2081) (STAId x\u2082) = STAId (\u03bb x \u03c4 x\u2083 \u2192 x\u2208\u222al \u0393 _ x \u03c4 (x\u2082 x \u03c4 x\u2083) )\n    weaken-subst-\u0393 {x = x} {\u0393 = \u0393} (EFSubst x\u2081 efrsh x\u2082) (STASubst {y = y} {\u03c4 = \u03c4'} subst x\u2083) =\n      STASubst (exchange-subst-\u0393 {\u0393 = \u0393} (flip x\u2082) (weaken-subst-\u0393 {\u0393 = \u0393 ,, (y , \u03c4')} efrsh subst))\n               (weaken-ta x\u2081 x\u2083)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192  -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081)))\n                                                                              {!weaken-ta {\u0393 = \u0393 ,, (y , ?)} x\u2083 wt!}\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 -- = {!weaken-ta ? (TALam x\u2082 wt1)!}\n    with natEQ y x\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = abort ((\u03c01(lem-union-none {\u0393 = \u0393} x\u2083)) refl)\n  lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n    with lem-union-none {\u0393 = \u0393} x\u2083 -- | lem-subst apt wt1  -- probably not the straight IH; need to weaken\n  ... | neq , r =  {!!} --  TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     envfresh x \u03c3 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 {\u0393 = \u0393} (EFId x\u2081) (STAId x\u2082) = STAId (\u03bb x \u03c4 x\u2083 \u2192 x\u2208\u222al \u0393 _ x \u03c4 (x\u2082 x \u03c4 x\u2083) )\n    weaken-subst-\u0393 {x = x} {\u0393 = \u0393} (EFSubst x\u2081 efrsh x\u2082) (STASubst {y = y} {\u03c4 = \u03c4'} subst x\u2083) =\n      STASubst (exchange-subst-\u0393 {\u0393 = \u0393} (flip x\u2082) (weaken-subst-\u0393 {\u0393 = \u0393 ,, (y , \u03c4')} efrsh subst))\n               (weaken-ta x\u2081 x\u2083)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192  -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081))) (weaken-ta {\u0393 = {!\u25a0 (y , ? )!} \u222a \u0393} x\u2083 wt)\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 -- = {!weaken-ta ? (TALam x\u2082 wt1)!}\n    with natEQ y x\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = abort ((\u03c01(lem-union-none {\u0393 = \u0393} x\u2083)) refl)\n  lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n    with lem-union-none {\u0393 = \u0393} x\u2083 -- | lem-subst apt wt1  -- probably not the straight IH; need to weaken\n  ... | neq , r =  {!!} --  TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"aa4bab642a60975cadb113e2eddba536d1671fc3","subject":"Add doc.","message":"Add doc.\n\nIgnore-this: 88c8569ac71a4dadfb6bb8be36321c8f\n\ndarcs-hash:20110102201203-3bd4e-b02ac257d8d12f84b6f4a78bf73e3b3c41da75c1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/SortList\/Properties\/Closures\/OrdList\/FlattenI.agda","new_file":"src\/LTC\/Program\/SortList\/Properties\/Closures\/OrdList\/FlattenI.agda","new_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to OrdList (flatten-OrdList-aux)\n------------------------------------------------------------------------------\n\n-- Agda bug? The recursive calls in flatten-OrdList-aux are\n-- structurally smaller, so it is not clear why we need the option\n-- --no-termination-check. We tested the option --termination-depth=N\n-- with some values, but it had no effect.\n{-# OPTIONS --no-termination-check #-}\n\nmodule LTC.Program.SortList.Properties.Closures.OrdList.FlattenI where\n\nopen import LTC.Base\n\n-- open import Common.Function\n\nopen import LTC.Data.Bool\nopen import LTC.Data.Bool.PropertiesI\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesI\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.List\n\nopen import LTC.Program.SortList.Properties.Closures.BoolI\nopen import LTC.Program.SortList.Properties.Closures.ListN-I\nopen import LTC.Program.SortList.Properties.Closures.OrdTreeI\nopen import LTC.Program.SortList.Properties.MiscellaneousI\nopen import LTC.Program.SortList.SortList\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nflatten-OrdList-aux : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192\n                      OrdTree (node t\u2081 i t\u2082) \u2192\n                      LE-Lists (flatten t\u2081) (flatten t\u2082)\n\nflatten-OrdList-aux {t\u2082 = t\u2082} nilT Ni Tt\u2082 OTt =\n  subst (\u03bb t \u2192 LE-Lists t (flatten t\u2082))\n        (sym (flatten-nilTree))\n        (\u2264-Lists-[] (flatten t\u2082))\n\nflatten-OrdList-aux (tipT {i\u2081} Ni\u2081) _ nilT OTt =\n  begin\n    \u2264-Lists (flatten (tip i\u2081)) (flatten nilTree)\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 \u2264-Lists (flatten (tip i\u2081)) (flatten nilTree) \u2261\n                           \u2264-Lists x\u2081 x\u2082)\n                (flatten-tip i\u2081)\n                flatten-nilTree\n                refl\n      \u27e9\n    \u2264-Lists (i\u2081 \u2237 []) []\n      \u2261\u27e8 \u2264-Lists-\u2237 i\u2081 [] [] \u27e9\n    \u2264-ItemList i\u2081 [] && \u2264-Lists [] []\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 \u2264-ItemList i\u2081 [] && \u2264-Lists [] [] \u2261 x\u2081 && x\u2082 )\n                (\u2264-ItemList-[] i\u2081)\n                (\u2264-Lists-[] [])\n                refl\n      \u27e9\n    true && true\n      \u2261\u27e8 &&-tt \u27e9\n    true\n  \u220e\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni (tipT {i\u2082} Ni\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (tip i\u2081)) (flatten (tip i\u2082))\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 \u2264-Lists (flatten (tip i\u2081)) (flatten (tip i\u2082)) \u2261\n                           \u2264-Lists x\u2081 x\u2082)\n                (flatten-tip i\u2081)\n                (flatten-tip i\u2082)\n                refl\n      \u27e9\n    \u2264-Lists (i\u2081 \u2237 []) (i\u2082 \u2237 [])\n      \u2261\u27e8 \u2264-Lists-\u2237 i\u2081 [] (i\u2082 \u2237 []) \u27e9\n    \u2264-ItemList i\u2081 (i\u2082 \u2237 []) && \u2264-Lists [] (i\u2082 \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 \u2264-ItemList i\u2081 (i\u2082 \u2237 []) && \u2264-Lists [] (i\u2082 \u2237 []) \u2261\n                      t && \u2264-Lists [] (i\u2082 \u2237 []))\n               (\u2264-ItemList-\u2237 i\u2081 i\u2082 [])\n               refl\n      \u27e9\n    (i\u2081 \u2264 i\u2082 && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 (i\u2081 \u2264 i\u2082  && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 []) \u2261\n                      (t        && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 []))\n               lemma\n               refl\n      \u27e9\n    (true && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 [])\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 (true && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 []) \u2261\n                           (true && x\u2081)               && x\u2082)\n                (\u2264-ItemList-[] i\u2081)\n                (\u2264-Lists-[] (i\u2082 \u2237 []))\n                refl\n      \u27e9\n    (true && true) && true\n      \u2261\u27e8 &&-assoc tB tB tB \u27e9\n    true && true && true\n      \u2261\u27e8 &&-true\u2083 \u27e9\n    true\n  \u220e\n  where\n    aux\u2081 : Bool (ordTree (tip i\u2081))\n    aux\u2081 = ordTree-Bool (tipT Ni\u2081)\n\n    aux\u2082 : Bool (ordTree (tip i\u2082))\n    aux\u2082 = ordTree-Bool (tipT Ni\u2082)\n\n    aux\u2083 : Bool (\u2264-TreeItem (tip i\u2081) i)\n    aux\u2083 = \u2264-TreeItem-Bool (tipT Ni\u2081) Ni\n\n    aux\u2084 : Bool (\u2264-ItemTree i (tip i\u2082))\n    aux\u2084 = \u2264-ItemTree-Bool Ni (tipT Ni\u2082)\n\n    aux\u2085 : ordTree (tip i\u2081) &&\n           ordTree (tip i\u2082) &&\n           \u2264-TreeItem (tip i\u2081) i &&\n           \u2264-ItemTree i (tip i\u2082) \u2261 true\n    aux\u2085 = trans (sym (ordTree-node (tip i\u2081) i (tip i\u2082))) OTt\n\n    lemma : LE i\u2081 i\u2082\n    lemma = \u2264-trans Ni\u2081 Ni Ni\u2082 i\u2081\u2264i i\u2264i\u2082\n     where\n       i\u2081\u2264i : LE i\u2081 i\n       i\u2081\u2264i = trans (sym (\u2264-TreeItem-tip i\u2081 i))\n                    (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n       i\u2264i\u2082 : LE i i\u2082\n       i\u2264i\u2082 = trans (sym (\u2264-ItemTree-tip i i\u2082))\n                    (&&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (tip i\u2081)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (tip i\u2081)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082)) \u2261\n                      \u2264-Lists (flatten (tip i\u2081)) x)\n               (flatten-node t\u2082\u2081 i\u2082 t\u2082\u2082)\n               refl\n      \u27e9\n    \u2264-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2081 ++ flatten t\u2082\u2082)\n      \u2261\u27e8 xs\u2264ys\u2192xs\u2264zs\u2192xs\u2264ys++zs (flatten-ListN (tipT Ni\u2081))\n                               (flatten-ListN Tt\u2082\u2081)\n                               (flatten-ListN Tt\u2082\u2082)\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (tipT Ni\u2081)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (tipT Ni\u2081) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (tip i\u2081) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the fourth conjunct of OTt.\n    aux\u2086 = \u2264-ItemTree-Bool Ni Tt\u2082\u2081\n    aux\u2087 = \u2264-ItemTree-Bool Ni Tt\u2082\u2082\n    aux\u2088 = trans (sym (\u2264-ItemTree-node i t\u2082\u2081 i\u2082 t\u2082\u2082))\n                 (&&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    OrdTree-tip-i\u2081 : OrdTree (tip i\u2081)\n    OrdTree-tip-i\u2081 = &&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-TreeItem-tip-i\u2081-i : LE-TreeItem (tip i\u2081) i\n    LE-TreeItem-tip-i\u2081-i = &&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2081)\n    lemma\u2081 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2081 OT  -- IH.\n      where\n        OrdTree-t\u2082\u2081 : OrdTree t\u2082\u2081\n        OrdTree-t\u2082\u2081 = leftSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                          (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2081 : LE-ItemTree i t\u2082\u2081\n        LE-ItemTree-i-t\u2082\u2081 = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node (tip i\u2081) i t\u2082\u2081)\n        OT =\n          begin\n            ordTree (node (tip i\u2081) i t\u2082\u2081)\n              \u2261\u27e8 ordTree-node (tip i\u2081) i t\u2082\u2081 \u27e9\n            ordTree (tip i\u2081) &&\n            ordTree t\u2082\u2081 &&\n            \u2264-TreeItem (tip i\u2081) i &&\n            \u2264-ItemTree i t\u2082\u2081\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree (tip i\u2081) &&\n                                     ordTree t\u2082\u2081 &&\n                                     \u2264-TreeItem (tip i\u2081) i &&\n                                     \u2264-ItemTree i t\u2082\u2081 \u2261\n                                     w && x && y && z)\n                        OrdTree-tip-i\u2081\n                        OrdTree-t\u2082\u2081\n                        LE-TreeItem-tip-i\u2081-i\n                        LE-ItemTree-i-t\u2082\u2081\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2082)\n    lemma\u2082 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2082 OT  -- IH.\n      where\n        OrdTree-t\u2082\u2082 : OrdTree t\u2082\u2082\n        OrdTree-t\u2082\u2082 = rightSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                           (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2082 : LE-ItemTree i t\u2082\u2082\n        LE-ItemTree-i-t\u2082\u2082 = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node (tip i\u2081) i t\u2082\u2082)\n        OT =\n          begin\n            ordTree (node (tip i\u2081) i t\u2082\u2082)\n              \u2261\u27e8 ordTree-node (tip i\u2081) i t\u2082\u2082 \u27e9\n            ordTree (tip i\u2081) &&\n            ordTree t\u2082\u2082 &&\n            \u2264-TreeItem (tip i\u2081) i &&\n            \u2264-ItemTree i t\u2082\u2082\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree (tip i\u2081) &&\n                                     ordTree t\u2082\u2082 &&\n                                     \u2264-TreeItem (tip i\u2081) i &&\n                                     \u2264-ItemTree i t\u2082\u2082 \u2261\n                                     w && x && y && z)\n                        OrdTree-tip-i\u2081\n                        OrdTree-t\u2082\u2082\n                        LE-TreeItem-tip-i\u2081-i\n                        LE-ItemTree-i-t\u2082\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni nilT OTt =\n  begin\n    \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten nilTree)\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten nilTree) \u2261\n                      \u2264-Lists x                           (flatten nilTree))\n               (flatten-node t\u2081\u2081 i\u2081 t\u2081\u2082 )\n               refl\n      \u27e9\n    \u2264-Lists (flatten t\u2081\u2081 ++  flatten t\u2081\u2082) (flatten nilTree)\n      \u2261\u27e8 xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs (flatten-ListN Tt\u2081\u2081)\n                               (flatten-ListN Tt\u2081\u2082)\n                               (flatten-ListN nilT)\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool nilT\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni nilT\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i nilTree )) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    LE-ItemTree-i-niltree : LE-ItemTree i nilTree\n    LE-ItemTree-i-niltree = &&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten nilTree)\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni nilT OT  -- IH.\n      where\n        OrdTree-t\u2081\u2081 : OrdTree t\u2081\u2081\n        OrdTree-t\u2081\u2081 =\n          leftSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082\n                              (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2081-i : LE-TreeItem t\u2081\u2081 i\n        LE-TreeItem-t\u2081\u2081-i = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2081 i nilTree)\n        OT =\n          begin\n            ordTree (node t\u2081\u2081 i nilTree )\n              \u2261\u27e8 ordTree-node t\u2081\u2081 i nilTree \u27e9\n            ordTree t\u2081\u2081 &&\n            ordTree nilTree &&\n            \u2264-TreeItem t\u2081\u2081 i &&\n            \u2264-ItemTree i nilTree\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2081 &&\n                                     ordTree nilTree &&\n                                     \u2264-TreeItem t\u2081\u2081 i &&\n                                     \u2264-ItemTree i nilTree \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2081\n                        ordTree-nilTree\n                        LE-TreeItem-t\u2081\u2081-i\n                        LE-ItemTree-i-niltree\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten nilTree)\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni nilT OT  -- IH.\n      where\n        OrdTree-t\u2081\u2082 : OrdTree t\u2081\u2082\n        OrdTree-t\u2081\u2082 =\n          rightSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2082-i : LE-TreeItem t\u2081\u2082 i\n        LE-TreeItem-t\u2081\u2082-i = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2082 i nilTree)\n        OT =\n          begin\n            ordTree (node t\u2081\u2082 i nilTree )\n              \u2261\u27e8 ordTree-node t\u2081\u2082 i nilTree \u27e9\n            ordTree t\u2081\u2082 &&\n            ordTree nilTree &&\n            \u2264-TreeItem t\u2081\u2082 i &&\n            \u2264-ItemTree i nilTree\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2082 &&\n                                     ordTree nilTree &&\n                                     \u2264-TreeItem t\u2081\u2082 i &&\n                                     \u2264-ItemTree i nilTree \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2082\n                        ordTree-nilTree\n                        LE-TreeItem-t\u2081\u2082-i\n                        LE-ItemTree-i-niltree\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (tipT {i\u2082} Ni\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (tip i\u2082))\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (tip i\u2082)) \u2261\n                      \u2264-Lists x                           (flatten (tip i\u2082)))\n               (flatten-node t\u2081\u2081 i\u2081 t\u2081\u2082 )\n               refl\n      \u27e9\n    \u2264-Lists (flatten t\u2081\u2081 ++  flatten t\u2081\u2082) (flatten (tip i\u2082))\n      \u2261\u27e8 xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs (flatten-ListN Tt\u2081\u2081)\n                               (flatten-ListN Tt\u2081\u2082)\n                               (flatten-ListN (tipT Ni\u2082))\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (tipT Ni\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (tipT Ni\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (tip i\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    OrdTree-tip-i\u2082 : OrdTree (tip i\u2082)\n    OrdTree-tip-i\u2082 = &&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-ItemTree-i-tip-i\u2082 : LE-ItemTree i (tip i\u2082)\n    LE-ItemTree-i-tip-i\u2082 = &&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (tip i\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2081 : OrdTree t\u2081\u2081\n        OrdTree-t\u2081\u2081 =\n          leftSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2081-i : LE-TreeItem t\u2081\u2081 i\n        LE-TreeItem-t\u2081\u2081-i = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2081 i (tip i\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2081 i (tip i\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2081 i (tip i\u2082) \u27e9\n            ordTree t\u2081\u2081 &&\n            ordTree (tip i\u2082) &&\n            \u2264-TreeItem t\u2081\u2081 i &&\n            \u2264-ItemTree i (tip i\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2081 &&\n                                     ordTree (tip i\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2081 i &&\n                                     \u2264-ItemTree i (tip i\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2081\n                        OrdTree-tip-i\u2082\n                        LE-TreeItem-t\u2081\u2081-i\n                        LE-ItemTree-i-tip-i\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (tip i\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2082 : OrdTree t\u2081\u2082\n        OrdTree-t\u2081\u2082 =\n          rightSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2082-i : LE-TreeItem t\u2081\u2082 i\n        LE-TreeItem-t\u2081\u2082-i = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2082 i (tip i\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2082 i (tip i\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2082 i (tip i\u2082) \u27e9\n            ordTree t\u2081\u2082 &&\n            ordTree (tip i\u2082) &&\n            \u2264-TreeItem t\u2081\u2082 i &&\n            \u2264-ItemTree i (tip i\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2082 &&\n                                     ordTree (tip i\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2082 i &&\n                                     \u2264-ItemTree i (tip i\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2082\n                        OrdTree-tip-i\u2082\n                        LE-TreeItem-t\u2081\u2082-i\n                        LE-ItemTree-i-tip-i\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082))\n                              (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082)) \u2261\n                      \u2264-Lists x (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082)))\n               (flatten-node t\u2081\u2081 i\u2081 t\u2081\u2082 )\n               refl\n      \u27e9\n    \u2264-Lists (flatten t\u2081\u2081 ++  flatten t\u2081\u2082) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      \u2261\u27e8 xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs (flatten-ListN Tt\u2081\u2081)\n                               (flatten-ListN Tt\u2081\u2082)\n                               (flatten-ListN (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082))\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082 : OrdTree (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n    OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082 = &&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082 : LE-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n    LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082 = &&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2081 : OrdTree t\u2081\u2081\n        OrdTree-t\u2081\u2081 =\n          leftSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2081-i : LE-TreeItem t\u2081\u2081 i\n        LE-TreeItem-t\u2081\u2081-i = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u27e9\n            ordTree t\u2081\u2081 &&\n            ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n            \u2264-TreeItem t\u2081\u2081 i &&\n            \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2081 &&\n                                     ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2081 i &&\n                                     \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2081\n                        OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        LE-TreeItem-t\u2081\u2081-i\n                        LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2082 : OrdTree t\u2081\u2082\n        OrdTree-t\u2081\u2082 =\n          rightSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2082-i : LE-TreeItem t\u2081\u2082 i\n        LE-TreeItem-t\u2081\u2082-i = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u27e9\n            ordTree t\u2081\u2082 &&\n            ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n            \u2264-TreeItem t\u2081\u2082 i &&\n            \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2082 &&\n                                     ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2082 i &&\n                                     \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2082\n                        OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        LE-TreeItem-t\u2081\u2082-i\n                        LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to OrdList (flatten-OrdList-aux)\n------------------------------------------------------------------------------\n\n-- TODO\n{-# OPTIONS --no-termination-check #-}\n\nmodule LTC.Program.SortList.Properties.Closures.OrdList.FlattenI where\n\nopen import LTC.Base\n\n-- open import Common.Function\n\nopen import LTC.Data.Bool\nopen import LTC.Data.Bool.PropertiesI\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesI\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.List\n\nopen import LTC.Program.SortList.Properties.Closures.BoolI\nopen import LTC.Program.SortList.Properties.Closures.ListN-I\nopen import LTC.Program.SortList.Properties.Closures.OrdTreeI\nopen import LTC.Program.SortList.Properties.MiscellaneousI\nopen import LTC.Program.SortList.SortList\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nflatten-OrdList-aux : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192\n                      OrdTree (node t\u2081 i t\u2082) \u2192\n                      LE-Lists (flatten t\u2081) (flatten t\u2082)\n\nflatten-OrdList-aux {t\u2082 = t\u2082} nilT Ni Tt\u2082 OTt =\n  subst (\u03bb t \u2192 LE-Lists t (flatten t\u2082))\n        (sym (flatten-nilTree))\n        (\u2264-Lists-[] (flatten t\u2082))\n\nflatten-OrdList-aux (tipT {i\u2081} Ni\u2081) _ nilT OTt =\n  begin\n    \u2264-Lists (flatten (tip i\u2081)) (flatten nilTree)\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 \u2264-Lists (flatten (tip i\u2081)) (flatten nilTree) \u2261\n                           \u2264-Lists x\u2081 x\u2082)\n                (flatten-tip i\u2081)\n                flatten-nilTree\n                refl\n      \u27e9\n    \u2264-Lists (i\u2081 \u2237 []) []\n      \u2261\u27e8 \u2264-Lists-\u2237 i\u2081 [] [] \u27e9\n    \u2264-ItemList i\u2081 [] && \u2264-Lists [] []\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 \u2264-ItemList i\u2081 [] && \u2264-Lists [] [] \u2261 x\u2081 && x\u2082 )\n                (\u2264-ItemList-[] i\u2081)\n                (\u2264-Lists-[] [])\n                refl\n      \u27e9\n    true && true\n      \u2261\u27e8 &&-tt \u27e9\n    true\n  \u220e\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni (tipT {i\u2082} Ni\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (tip i\u2081)) (flatten (tip i\u2082))\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 \u2264-Lists (flatten (tip i\u2081)) (flatten (tip i\u2082)) \u2261\n                           \u2264-Lists x\u2081 x\u2082)\n                (flatten-tip i\u2081)\n                (flatten-tip i\u2082)\n                refl\n      \u27e9\n    \u2264-Lists (i\u2081 \u2237 []) (i\u2082 \u2237 [])\n      \u2261\u27e8 \u2264-Lists-\u2237 i\u2081 [] (i\u2082 \u2237 []) \u27e9\n    \u2264-ItemList i\u2081 (i\u2082 \u2237 []) && \u2264-Lists [] (i\u2082 \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 \u2264-ItemList i\u2081 (i\u2082 \u2237 []) && \u2264-Lists [] (i\u2082 \u2237 []) \u2261\n                      t && \u2264-Lists [] (i\u2082 \u2237 []))\n               (\u2264-ItemList-\u2237 i\u2081 i\u2082 [])\n               refl\n      \u27e9\n    (i\u2081 \u2264 i\u2082 && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 (i\u2081 \u2264 i\u2082  && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 []) \u2261\n                      (t        && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 []))\n               lemma\n               refl\n      \u27e9\n    (true && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 [])\n      \u2261\u27e8 subst\u2082 (\u03bb x\u2081 x\u2082 \u2192 (true && \u2264-ItemList i\u2081 []) && \u2264-Lists [] (i\u2082 \u2237 []) \u2261\n                           (true && x\u2081)               && x\u2082)\n                (\u2264-ItemList-[] i\u2081)\n                (\u2264-Lists-[] (i\u2082 \u2237 []))\n                refl\n      \u27e9\n    (true && true) && true\n      \u2261\u27e8 &&-assoc tB tB tB \u27e9\n    true && true && true\n      \u2261\u27e8 &&-true\u2083 \u27e9\n    true\n  \u220e\n  where\n    aux\u2081 : Bool (ordTree (tip i\u2081))\n    aux\u2081 = ordTree-Bool (tipT Ni\u2081)\n\n    aux\u2082 : Bool (ordTree (tip i\u2082))\n    aux\u2082 = ordTree-Bool (tipT Ni\u2082)\n\n    aux\u2083 : Bool (\u2264-TreeItem (tip i\u2081) i)\n    aux\u2083 = \u2264-TreeItem-Bool (tipT Ni\u2081) Ni\n\n    aux\u2084 : Bool (\u2264-ItemTree i (tip i\u2082))\n    aux\u2084 = \u2264-ItemTree-Bool Ni (tipT Ni\u2082)\n\n    aux\u2085 : ordTree (tip i\u2081) &&\n           ordTree (tip i\u2082) &&\n           \u2264-TreeItem (tip i\u2081) i &&\n           \u2264-ItemTree i (tip i\u2082) \u2261 true\n    aux\u2085 = trans (sym (ordTree-node (tip i\u2081) i (tip i\u2082))) OTt\n\n    lemma : LE i\u2081 i\u2082\n    lemma = \u2264-trans Ni\u2081 Ni Ni\u2082 i\u2081\u2264i i\u2264i\u2082\n     where\n       i\u2081\u2264i : LE i\u2081 i\n       i\u2081\u2264i = trans (sym (\u2264-TreeItem-tip i\u2081 i))\n                    (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n       i\u2264i\u2082 : LE i i\u2082\n       i\u2264i\u2082 = trans (sym (\u2264-ItemTree-tip i i\u2082))\n                    (&&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (tip i\u2081)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (tip i\u2081)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082)) \u2261\n                      \u2264-Lists (flatten (tip i\u2081)) x)\n               (flatten-node t\u2082\u2081 i\u2082 t\u2082\u2082)\n               refl\n      \u27e9\n    \u2264-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2081 ++ flatten t\u2082\u2082)\n      \u2261\u27e8 xs\u2264ys\u2192xs\u2264zs\u2192xs\u2264ys++zs (flatten-ListN (tipT Ni\u2081))\n                               (flatten-ListN Tt\u2082\u2081)\n                               (flatten-ListN Tt\u2082\u2082)\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (tipT Ni\u2081)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (tipT Ni\u2081) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (tip i\u2081) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the fourth conjunct of OTt.\n    aux\u2086 = \u2264-ItemTree-Bool Ni Tt\u2082\u2081\n    aux\u2087 = \u2264-ItemTree-Bool Ni Tt\u2082\u2082\n    aux\u2088 = trans (sym (\u2264-ItemTree-node i t\u2082\u2081 i\u2082 t\u2082\u2082))\n                 (&&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    OrdTree-tip-i\u2081 : OrdTree (tip i\u2081)\n    OrdTree-tip-i\u2081 = &&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-TreeItem-tip-i\u2081-i : LE-TreeItem (tip i\u2081) i\n    LE-TreeItem-tip-i\u2081-i = &&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2081)\n    lemma\u2081 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2081 OT  -- IH.\n      where\n        OrdTree-t\u2082\u2081 : OrdTree t\u2082\u2081\n        OrdTree-t\u2082\u2081 = leftSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                          (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2081 : LE-ItemTree i t\u2082\u2081\n        LE-ItemTree-i-t\u2082\u2081 = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node (tip i\u2081) i t\u2082\u2081)\n        OT =\n          begin\n            ordTree (node (tip i\u2081) i t\u2082\u2081)\n              \u2261\u27e8 ordTree-node (tip i\u2081) i t\u2082\u2081 \u27e9\n            ordTree (tip i\u2081) &&\n            ordTree t\u2082\u2081 &&\n            \u2264-TreeItem (tip i\u2081) i &&\n            \u2264-ItemTree i t\u2082\u2081\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree (tip i\u2081) &&\n                                     ordTree t\u2082\u2081 &&\n                                     \u2264-TreeItem (tip i\u2081) i &&\n                                     \u2264-ItemTree i t\u2082\u2081 \u2261\n                                     w && x && y && z)\n                        OrdTree-tip-i\u2081\n                        OrdTree-t\u2082\u2081\n                        LE-TreeItem-tip-i\u2081-i\n                        LE-ItemTree-i-t\u2082\u2081\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2082)\n    lemma\u2082 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2082 OT  -- IH.\n      where\n        OrdTree-t\u2082\u2082 : OrdTree t\u2082\u2082\n        OrdTree-t\u2082\u2082 = rightSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                           (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2082 : LE-ItemTree i t\u2082\u2082\n        LE-ItemTree-i-t\u2082\u2082 = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node (tip i\u2081) i t\u2082\u2082)\n        OT =\n          begin\n            ordTree (node (tip i\u2081) i t\u2082\u2082)\n              \u2261\u27e8 ordTree-node (tip i\u2081) i t\u2082\u2082 \u27e9\n            ordTree (tip i\u2081) &&\n            ordTree t\u2082\u2082 &&\n            \u2264-TreeItem (tip i\u2081) i &&\n            \u2264-ItemTree i t\u2082\u2082\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree (tip i\u2081) &&\n                                     ordTree t\u2082\u2082 &&\n                                     \u2264-TreeItem (tip i\u2081) i &&\n                                     \u2264-ItemTree i t\u2082\u2082 \u2261\n                                     w && x && y && z)\n                        OrdTree-tip-i\u2081\n                        OrdTree-t\u2082\u2082\n                        LE-TreeItem-tip-i\u2081-i\n                        LE-ItemTree-i-t\u2082\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni nilT OTt =\n  begin\n    \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten nilTree)\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten nilTree) \u2261\n                      \u2264-Lists x                           (flatten nilTree))\n               (flatten-node t\u2081\u2081 i\u2081 t\u2081\u2082 )\n               refl\n      \u27e9\n    \u2264-Lists (flatten t\u2081\u2081 ++  flatten t\u2081\u2082) (flatten nilTree)\n      \u2261\u27e8 xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs (flatten-ListN Tt\u2081\u2081)\n                               (flatten-ListN Tt\u2081\u2082)\n                               (flatten-ListN nilT)\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool nilT\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni nilT\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i nilTree )) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    LE-ItemTree-i-niltree : LE-ItemTree i nilTree\n    LE-ItemTree-i-niltree = &&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten nilTree)\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni nilT OT  -- IH.\n      where\n        OrdTree-t\u2081\u2081 : OrdTree t\u2081\u2081\n        OrdTree-t\u2081\u2081 =\n          leftSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082\n                              (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2081-i : LE-TreeItem t\u2081\u2081 i\n        LE-TreeItem-t\u2081\u2081-i = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2081 i nilTree)\n        OT =\n          begin\n            ordTree (node t\u2081\u2081 i nilTree )\n              \u2261\u27e8 ordTree-node t\u2081\u2081 i nilTree \u27e9\n            ordTree t\u2081\u2081 &&\n            ordTree nilTree &&\n            \u2264-TreeItem t\u2081\u2081 i &&\n            \u2264-ItemTree i nilTree\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2081 &&\n                                     ordTree nilTree &&\n                                     \u2264-TreeItem t\u2081\u2081 i &&\n                                     \u2264-ItemTree i nilTree \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2081\n                        ordTree-nilTree\n                        LE-TreeItem-t\u2081\u2081-i\n                        LE-ItemTree-i-niltree\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten nilTree)\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni nilT OT  -- IH.\n      where\n        OrdTree-t\u2081\u2082 : OrdTree t\u2081\u2082\n        OrdTree-t\u2081\u2082 =\n          rightSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2082-i : LE-TreeItem t\u2081\u2082 i\n        LE-TreeItem-t\u2081\u2082-i = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2082 i nilTree)\n        OT =\n          begin\n            ordTree (node t\u2081\u2082 i nilTree )\n              \u2261\u27e8 ordTree-node t\u2081\u2082 i nilTree \u27e9\n            ordTree t\u2081\u2082 &&\n            ordTree nilTree &&\n            \u2264-TreeItem t\u2081\u2082 i &&\n            \u2264-ItemTree i nilTree\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2082 &&\n                                     ordTree nilTree &&\n                                     \u2264-TreeItem t\u2081\u2082 i &&\n                                     \u2264-ItemTree i nilTree \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2082\n                        ordTree-nilTree\n                        LE-TreeItem-t\u2081\u2082-i\n                        LE-ItemTree-i-niltree\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (tipT {i\u2082} Ni\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (tip i\u2082))\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (tip i\u2082)) \u2261\n                      \u2264-Lists x                           (flatten (tip i\u2082)))\n               (flatten-node t\u2081\u2081 i\u2081 t\u2081\u2082 )\n               refl\n      \u27e9\n    \u2264-Lists (flatten t\u2081\u2081 ++  flatten t\u2081\u2082) (flatten (tip i\u2082))\n      \u2261\u27e8 xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs (flatten-ListN Tt\u2081\u2081)\n                               (flatten-ListN Tt\u2081\u2082)\n                               (flatten-ListN (tipT Ni\u2082))\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (tipT Ni\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (tipT Ni\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (tip i\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    OrdTree-tip-i\u2082 : OrdTree (tip i\u2082)\n    OrdTree-tip-i\u2082 = &&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-ItemTree-i-tip-i\u2082 : LE-ItemTree i (tip i\u2082)\n    LE-ItemTree-i-tip-i\u2082 = &&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (tip i\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2081 : OrdTree t\u2081\u2081\n        OrdTree-t\u2081\u2081 =\n          leftSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2081-i : LE-TreeItem t\u2081\u2081 i\n        LE-TreeItem-t\u2081\u2081-i = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2081 i (tip i\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2081 i (tip i\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2081 i (tip i\u2082) \u27e9\n            ordTree t\u2081\u2081 &&\n            ordTree (tip i\u2082) &&\n            \u2264-TreeItem t\u2081\u2081 i &&\n            \u2264-ItemTree i (tip i\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2081 &&\n                                     ordTree (tip i\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2081 i &&\n                                     \u2264-ItemTree i (tip i\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2081\n                        OrdTree-tip-i\u2082\n                        LE-TreeItem-t\u2081\u2081-i\n                        LE-ItemTree-i-tip-i\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (tip i\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2082 : OrdTree t\u2081\u2082\n        OrdTree-t\u2081\u2082 =\n          rightSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2082-i : LE-TreeItem t\u2081\u2082 i\n        LE-TreeItem-t\u2081\u2082-i = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2082 i (tip i\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2082 i (tip i\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2082 i (tip i\u2082) \u27e9\n            ordTree t\u2081\u2082 &&\n            ordTree (tip i\u2082) &&\n            \u2264-TreeItem t\u2081\u2082 i &&\n            \u2264-ItemTree i (tip i\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2082 &&\n                                     ordTree (tip i\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2082 i &&\n                                     \u2264-ItemTree i (tip i\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2082\n                        OrdTree-tip-i\u2082\n                        LE-TreeItem-t\u2081\u2082-i\n                        LE-ItemTree-i-tip-i\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt =\n  begin\n    \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      \u2261\u27e8 subst (\u03bb x \u2192 \u2264-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082))\n                              (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082)) \u2261\n                      \u2264-Lists x (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082)))\n               (flatten-node t\u2081\u2081 i\u2081 t\u2081\u2082 )\n               refl\n      \u27e9\n    \u2264-Lists (flatten t\u2081\u2081 ++  flatten t\u2081\u2082) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      \u2261\u27e8 xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs (flatten-ListN Tt\u2081\u2081)\n                               (flatten-ListN Tt\u2081\u2082)\n                               (flatten-ListN (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082))\n                               lemma\u2081\n                               lemma\u2082\n      \u27e9\n    true\n  \u220e\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082 : OrdTree (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n    OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082 = &&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082 : LE-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n    LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082 = &&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2081 : OrdTree t\u2081\u2081\n        OrdTree-t\u2081\u2081 =\n          leftSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2081-i : LE-TreeItem t\u2081\u2081 i\n        LE-TreeItem-t\u2081\u2081-i = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u27e9\n            ordTree t\u2081\u2081 &&\n            ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n            \u2264-TreeItem t\u2081\u2081 i &&\n            \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2081 &&\n                                     ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2081 i &&\n                                     \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2081\n                        OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        LE-TreeItem-t\u2081\u2081-i\n                        LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        OrdTree-t\u2081\u2082 : OrdTree t\u2081\u2082\n        OrdTree-t\u2081\u2082 =\n          rightSubTree-OrdTree Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082 (&&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-TreeItem-t\u2081\u2082-i : LE-TreeItem t\u2081\u2082 i\n        LE-TreeItem-t\u2081\u2082-i = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        OT : OrdTree (node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        OT =\n          begin\n            ordTree (node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) )\n              \u2261\u27e8 ordTree-node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u27e9\n            ordTree t\u2081\u2082 &&\n            ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n            \u2264-TreeItem t\u2081\u2082 i &&\n            \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082)\n              \u2261\u27e8 subst\u2084 (\u03bb w x y z \u2192 ordTree t\u2081\u2082 &&\n                                     ordTree (node t\u2082\u2081 i\u2082 t\u2082\u2082) &&\n                                     \u2264-TreeItem t\u2081\u2082 i &&\n                                     \u2264-ItemTree i (node t\u2082\u2081 i\u2082 t\u2082\u2082) \u2261\n                                     w && x && y && z)\n                        OrdTree-t\u2081\u2082\n                        OrdTree-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        LE-TreeItem-t\u2081\u2082-i\n                        LE-ItemTree-i-node-t\u2082\u2081-i\u2082-t\u2082\u2082\n                        refl\n            \u27e9\n            true && true && true && true\n              \u2261\u27e8 &&-true\u2084 \u27e9\n            true\n          \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3ae029b724ce3c3f266068195aa755f9d65a415e","subject":"Boring.","message":"Boring.\n\nIgnore-this: c17974f2bac758a84b125f426d4798a0\n\ndarcs-hash:20100410164802-3bd4e-10acc80d0acab3f763cace7626e83817333a8176.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/UsingPredicatesWithConjectures.agda","new_file":"Test\/Fail\/UsingPredicatesWithConjectures.agda","new_contents":"","old_contents":"module Test.Fail.UsingPredicatesWithConjectures where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  add-x0 : (n : D) \u2192 n + zero     \u2261 n\n  add-xS : (m n : D) \u2192 m + succ n \u2261 succ (m + n)\n\n{-# ATP axiom add-x0 #-}\n{-# ATP axiom add-xS #-}\n\nP : D \u2192 Set\nP i = zero + i \u2261 i\n\npostulate\n  P0\u2081 : zero + zero \u2261 zero\n{-# ATP prove P0\u2081 #-}\n\npostulate\n  iStep\u2081 : {i : D} \u2192 N i \u2192 zero + i \u2261 i \u2192 zero + (succ i) \u2261 succ i\n{-# ATP prove iStep\u2081 #-}\n\naddLeftIdentity\u2081 : {n : D} \u2192 N n \u2192 zero + n \u2261 n\naddLeftIdentity\u2081 = indN (\u03bb i \u2192 P i) P0\u2081 iStep\u2081\n\n-- Equinox cannot prove the postulates P0\u2082 and iStep\u2082 because he does\n-- not the definition of the predicate P. It seems we should use the\n-- TPTP definitions.\npostulate\n  P0\u2082 : P zero\n-- {-# ATP prove P0\u2082 #-}\n\npostulate\n  iStep\u2082 : {i : D} \u2192 N i \u2192 P i \u2192 P (succ i)\n-- {-# ATP prove iStep\u2082 #-}\n\naddLeftIdentity\u2082 : {n : D} \u2192 N n \u2192 zero + n \u2261 n\naddLeftIdentity\u2082 = indN (\u03bb i \u2192 P i) P0\u2082 iStep\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"702fcd37b6983eaf128ea29fca185751c2444c69","subject":"Fix in Relation.Binary.Permutation","message":"Fix in Relation.Binary.Permutation\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Permutation.agda","new_file":"lib\/Relation\/Binary\/Permutation.agda","new_contents":"--TODO {-# OPTIONS --without-K #-}\nmodule Relation.Binary.Permutation where\n\nopen import Level\nopen import Data.Product.NP\nopen import Data.Zero\nopen import Data.One\nopen import Data.Sum\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_;_\u2262_)\n\nprivate\n  _\u21d4_ : \u2200 {a b \u2113\u2081 \u2113\u2082} {A : Set a} {B : Set b} (R\u2081 : REL A B \u2113\u2081) (R\u2082 : REL A B \u2113\u2082) \u2192 Set _\n  R\u2081 \u21d4 R\u2082 = R\u2081 \u21d2 R\u2082 \u00d7 R\u2082 \u21d2 R\u2081\n\n  infix 2 _\u21d4_\n\n\u27e8_,_\u27e9\u2208_ : \u2200 {\u2113 a b} {A : Set a} {B : Set b} (x : A) (y : B) \u2192 REL A B \u2113 \u2192 Set \u2113\n\u27e8_,_\u27e9\u2208_ x y R = R x y\n\ndata _[_\u2194_] {a} {A : Set a} (R : Rel A a) (x y : A) : Rel A a where\n\n  here\u2081  : \u2200 {j}\n             (yRj : \u27e8 y , j \u27e9\u2208 R    )\n           \u2192 ----------------------\n             \u27e8 x , j \u27e9\u2208 R [ x \u2194 y ]\n\n  here\u2082  : \u2200 {j}\n             (xRj : \u27e8 x , j \u27e9\u2208 R   )\n           \u2192 ----------------------\n             \u27e8 y , j \u27e9\u2208 R [ x \u2194 y ]\n\n  there  : \u2200 {i j}\n             (x\u2262i   : x \u2262 i         )\n             (y\u2262i   : y \u2262 i         )\n             (iRj   : \u27e8 i , j \u27e9\u2208 R  )\n           \u2192 -----------------------\n             \u27e8 i , j \u27e9\u2208 R [ x \u2194 y ]\n\nmodule PermComm {a} {A : Set a} {R : Rel A a} where\n  \u27f9 : \u2200 {x y} \u2192 R [ x \u2194 y ] \u21d2 R [ y \u2194 x ]\n  \u27f9 (here\u2081 yRj)          = here\u2082 yRj\n  \u27f9 (here\u2082 xRj)          = here\u2081 xRj\n  \u27f9 (there x\u2262i x\u2262j iRj)  = there x\u2262j x\u2262i iRj\n\n  lem : \u2200 {x y} \u2192 R [ x \u2194 y ] \u21d4 R [ y \u2194 x ]\n  lem = (\u03bb {_} \u2192 \u27f9) , (\u03bb {_} \u2192 \u27f9)\n\nmodule PermIdem {a} {A : Set a} (_\u225f_ : Decidable {A = A} _\u2261_) {R : Rel A a} where\n  \u21d0 : \u2200 {x y} \u2192 R [ x \u2194 y ] [ x \u2194 y ] \u21d2 R\n  \u21d0 (here\u2081 (here\u2081 yRj))          = yRj\n  \u21d0 (here\u2081 (here\u2082 xRj))          = xRj\n  \u21d0 (here\u2081 (there _ y\u2262y _))      = \ud835\udfd8-elim (y\u2262y \u2261.refl)\n  \u21d0 (here\u2082 (here\u2081 yRj))          = yRj\n  \u21d0 (here\u2082 (here\u2082 xRj))          = xRj\n  \u21d0 (here\u2082 (there x\u2262x _ _))      = \ud835\udfd8-elim (x\u2262x \u2261.refl)\n  \u21d0 (there x\u2262x _ (here\u2081 _))      = \ud835\udfd8-elim (x\u2262x \u2261.refl)\n  \u21d0 (there _ y\u2262y (here\u2082 _))      = \ud835\udfd8-elim (y\u2262y \u2261.refl)\n  \u21d0 (there _ _ (there _ _ iRj))  = iRj\n\n  \u27f9 : \u2200 {x y} \u2192 R \u21d2 R [ x \u2194 y ] [ x \u2194 y ]\n  \u27f9 {x} {y} {i} {j} R\n   with x \u225f i     | y \u225f i\n  ... | yes x\u2261i   | _       rewrite x\u2261i = here\u2081 (here\u2082 R)\n  ... | _         | yes y\u2261i rewrite y\u2261i = here\u2082 (here\u2081 R)\n  ... | no x\u2262i    | no y\u2262i              = there x\u2262i y\u2262i (there x\u2262i y\u2262i R)\n\n  lem : \u2200 {x y} \u2192 R \u21d4 R [ x \u2194 y ] [ x \u2194 y ]\n  lem = (\u03bb {_} \u2192 \u27f9) , \u03bb {_} \u2192 \u21d0\n\nPermutation : \u2200 {a} \u2192 Set a \u2192 Set a\nPermutation A = List (A \u00d7 A)\n\npermRel : \u2200 {a} {A : Set a} \u2192 (\u03c0 : Permutation A) \u2192 Rel A a \u2192 Rel A a\npermRel \u03c0 R = foldr (\u03bb p r \u2192 r [ fst p \u2194 snd p ]) R \u03c0\n\ntoRel : \u2200 {a} {A : Set a} \u2192 (\u03c0 : Permutation A) \u2192 Rel A a\ntoRel \u03c0 = permRel \u03c0 (\u03bb _ _ \u2192 Lift \ud835\udfd9)\n\n{-\n  _\u27e8$\u27e9\u2081_ : \u2200 {a} {A : Set a} \u2192 Permutation A \u2192 A \u2192 Maybe A\n  []       \u27e8$\u27e9\u2081 y = nothing\n  (x \u2237 xs) \u27e8$\u27e9\u2081 y = if \u230a x \u225fA y \u230b then ? else\n-}\n","old_contents":"--TODO {-# OPTIONS --without-K #-}\nmodule Relation.Binary.Permutation where\n\nopen import Level\nopen import Data.Product.NP\nopen import Data.Zero\nopen import Data.One\nopen import Data.Sum\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_;_\u2262_)\n\nprivate\n  _\u21d4_ : \u2200 {a b \u2113\u2081 \u2113\u2082} {A : Set a} {B : Set b} (R\u2081 : REL A B \u2113\u2081) (R\u2082 : REL A B \u2113\u2082) \u2192 Set _\n  R\u2081 \u21d4 R\u2082 = R\u2081 \u21d2 R\u2082 \u00d7 R\u2082 \u21d2 R\u2081\n\n  infix 2 _\u21d4_\n\n\u27e8_,_\u27e9\u2208_ : \u2200 {\u2113 a b} {A : Set a} {B : Set b} (x : A) (y : B) \u2192 REL A B \u2113 \u2192 Set \u2113\n\u27e8_,_\u27e9\u2208_ x y R = R x y\n\ndata _[_\u2194_] {a} {A : Set a} (R : Rel A a) (x y : A) : Rel A a where\n\n  here\u2081  : \u2200 {j}\n             (yRj : \u27e8 y , j \u27e9\u2208 R    )\n           \u2192 ----------------------\n             \u27e8 x , j \u27e9\u2208 R [ x \u2194 y ]\n\n  here\u2082  : \u2200 {j}\n             (xRj : \u27e8 x , j \u27e9\u2208 R   )\n           \u2192 ----------------------\n             \u27e8 y , j \u27e9\u2208 R [ x \u2194 y ]\n\n  there  : \u2200 {i j}\n             (x\u2262i   : x \u2262 i         )\n             (y\u2262i   : y \u2262 i         )\n             (iRj   : \u27e8 i , j \u27e9\u2208 R  )\n           \u2192 -----------------------\n             \u27e8 i , j \u27e9\u2208 R [ x \u2194 y ]\n\nmodule PermComm {a} {A : Set a} {R : Rel A a} where\n  \u27f9 : \u2200 {x y} \u2192 R [ x \u2194 y ] \u21d2 R [ y \u2194 x ]\n  \u27f9 (here\u2081 yRj)          = here\u2082 yRj\n  \u27f9 (here\u2082 xRj)          = here\u2081 xRj\n  \u27f9 (there x\u2262i x\u2262j iRj)  = there x\u2262j x\u2262i iRj\n\n  lem : \u2200 {x y} \u2192 R [ x \u2194 y ] \u21d4 R [ y \u2194 x ]\n  lem = (\u03bb {_} \u2192 \u27f9) , (\u03bb {_} \u2192 \u27f9)\n\nmodule PermIdem {a} {A : Set a} (_\u225f_ : Decidable {A = A} _\u2261_) {x y : A} {R : Rel A a} where\n  \u21d0 : \u2200 {x y} \u2192 R [ x \u2194 y ] [ x \u2194 y ] \u21d2 R\n  \u21d0 (here\u2081 (here\u2081 yRj))          = yRj\n  \u21d0 (here\u2081 (here\u2082 xRj))          = xRj\n  \u21d0 (here\u2081 (there _ y\u2262y _))      = \ud835\udfd8-elim (y\u2262y \u2261.refl)\n  \u21d0 (here\u2082 (here\u2081 yRj))          = yRj\n  \u21d0 (here\u2082 (here\u2082 xRj))          = xRj\n  \u21d0 (here\u2082 (there x\u2262x _ _))      = \ud835\udfd8-elim (x\u2262x \u2261.refl)\n  \u21d0 (there x\u2262x _ (here\u2081 _))      = \ud835\udfd8-elim (x\u2262x \u2261.refl)\n  \u21d0 (there _ y\u2262y (here\u2082 _))      = \ud835\udfd8-elim (y\u2262y \u2261.refl)\n  \u21d0 (there _ _ (there _ _ iRj))  = iRj\n\n  \u27f9 : R \u21d2 R [ x \u2194 y ] [ x \u2194 y ]\n  \u27f9 {i} {j} R\n   with x \u225f i     | y \u225f i\n  ... | yes x\u2261i   | _       rewrite x\u2261i = here\u2081 (here\u2082 R)\n  ... | _         | yes y\u2261i rewrite y\u2261i = here\u2082 (here\u2081 R)\n  ... | no x\u2262i    | no y\u2262i              = there x\u2262i y\u2262i (there x\u2262i y\u2262i R)\n\n  lem : R \u21d4 R [ x \u2194 y ] [ x \u2194 y ]\n  lem = (\u03bb {_} \u2192 \u27f9) , \u03bb {_} \u2192 \u21d0\n\nPermutation : \u2200 {a} \u2192 Set a \u2192 Set a\nPermutation A = List (A \u00d7 A)\n\npermRel : \u2200 {a} {A : Set a} \u2192 (\u03c0 : Permutation A) \u2192 Rel A a \u2192 Rel A a\npermRel \u03c0 R = foldr (\u03bb p r \u2192 r [ fst p \u2194 snd p ]) R \u03c0\n\ntoRel : \u2200 {a} {A : Set a} \u2192 (\u03c0 : Permutation A) \u2192 Rel A a\ntoRel \u03c0 = permRel \u03c0 (\u03bb _ _ \u2192 Lift \ud835\udfd9)\n\n{-\n  _\u27e8$\u27e9\u2081_ : \u2200 {a} {A : Set a} \u2192 Permutation A \u2192 A \u2192 Maybe A\n  []       \u27e8$\u27e9\u2081 y = nothing\n  (x \u2237 xs) \u27e8$\u27e9\u2081 y = if \u230a x \u225fA y \u230b then ? else\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"42981be31978bf60f777fb69f84e05a34f8bee99","subject":"More protocols","message":"More protocols\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n    send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n    recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n    end  : ViewProc end _\n\n  view-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\n  view-proc end      _       = end\n  view-proc (\u03a0\u1d3e M P) p       = recv _ _ p\n  view-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\n  data ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n    send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n    recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\n  view-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\n  view-com (\u03a0\u1d9c M P) p       = recv _ p\n  view-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M} (P : M \u2192 Proto) Q (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (com Q) (`L , m , p)\n    sendR' : \u2200 {M} P (Q : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7) \u2192 View-\u214b (com P) (\u03a3\u1d3e M Q) (`R , m , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) (com Q) (`L , p)\n    recvR' : \u2200 {M} P (Q : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)) \u2192 View-\u214b (com P) (\u03a0\u1d3e M Q) (`R , p)\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 P (p : \u27e6 com P \u27e7) \u2192 View-\u214b (com P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end      Q       p                = endL _ _\n  view-\u214b (com x)  end     p                = endR _ _\n  view-\u214b (\u03a0\u1d3e M P) (com Q) (`L , p)         = recvL' _ _ _\n  view-\u214b (\u03a3\u1d3e M P) (com Q) (`L , (m , p))   = sendL' _ _ _ _\n  view-\u214b (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) (`R , p)       = recvR' _ _ _\n  view-\u214b (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) (`R , (m , p)) = sendR' _ _ _ _\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) (`R , p)       = recvR' _ _ _\n  view-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (`R , (m , p)) = sendR' _ _ _ _\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr) = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr) = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q) = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q) = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr) = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr) = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr) = {!\u214b\u1d3e-apply {?} {?} pq qr!}\n\n    {-\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map P Q R S f g p with view-\u214b P R p\n  \u214b\u1d3e-map .(com (mk Out M P)) Q\u2081 .(com Q) S f g .(`L , m , p) | sendL' M P Q m p = {!!}\n  \u214b\u1d3e-map .(com P) Q\u2081 .(com (mk Out M Q)) S f g .(`R , m , p) | sendR' P M Q m p = {!!}\n  \u214b\u1d3e-map .(com (mk In M P)) Q\u2081 .(com Q) S f g .(`L , p) | recvL' M P Q p = {!!}\n  \u214b\u1d3e-map .(com P) Q\u2081 .(com (mk In M Q)) S f g .(`R , p) | recvR' P M Q p = {!!}\n  \u214b\u1d3e-map .end Q R S f g p | endL .R .p = {!!}\n  \u214b\u1d3e-map .(com P) Q .end S f g p | endR P .p = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  switchL' : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R p\u214bq r with view-\u214b P Q p\u214bq\n  switchL' ._ ._ R .(`L , m , p) r | sendL' M P Q m p = \u214b\u1d3e-sendL (com Q ox\u1d3e R) m (switchL' (P m) (com Q) R p r)\n  switchL' ._ ._ R .(`R , m , p) r | sendR' P M Q m p = \u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id\n                                                          (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id)\n                                                             (switchL' (com P) (Q m) R p r)\n  switchL' ._ ._ R .(`L , p) r | recvL' M P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n  switchL' ._ ._ R .(`R , p) r | recvR' (\u03a0\u1d9c M' P) M Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n  switchL' ._ ._ R .(`R , p) r | recvR' (\u03a3\u1d9c M' P) M Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n  switchL' ._ ._ R p\u214bq r | endL Q .p\u214bq = comma\u1d3e Q R p\u214bq r\n  switchL' .(com P) .end R p\u214bq r | endR P .p\u214bq = par (com P) R p\u214bq r\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _ where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    \u214b\u1d3e Q      = Q\n    P      \u214b\u1d3e end    = P\n    com P\u1d38 \u214b\u1d3e com P\u1d3f = \u03a3\u1d3e LR (P\u1d38 \u214b\u1d9c P\u1d3f)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com x) = id\n\n  \u214b\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR {P = end} m p = m , p\n  \u214b\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7 \u2192 \u27e6 com' Out M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-sendL {P = end}{Q} m p = m , \u214b\u1d3e-rend (Q m) p\n  \u214b\u1d3e-sendL {P = com x} m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR {P = end}   f = f\n  \u214b\u1d3e-recvR {P = com x} f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7) \u2192 \u27e6 com' In M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-recvL {P = end}{Q} f x = \u214b\u1d3e-rend (Q x) (f x)\n  \u214b\u1d3e-recvL {P = com x} f = `L , f\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL {P = P m} m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR {P = dual (P m)} m (\u214b\u1d3e-id (P m))\n\n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end     Q   p q = q\n  comma\u1d3e (com x) end p q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  comma\u1d3e-equiv : \u2200 P Q \u2192 Equiv (comma\u1d3e P Q)\n  comma\u1d3e-equiv P Q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6382a6d1e7dc0c527b8890748438731dc69f3134","subject":"Agda upgrade","message":"Agda upgrade\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Type.agda","new_file":"formalization\/agda\/Spire\/Type.agda","new_contents":"module Spire.Type where\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\nrecord \u22a4 : Set where constructor tt\ndata Bool : Set where true false : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115    #-}\n\nrecord \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\nopen \u03a3 public\n\n----------------------------------------------------------------------\n\nconst : {A : Set\u2081} {B : Set} \u2192 A \u2192 B \u2192 A\nconst a _ = a\n\nuncurry : {A : Set} {B : A \u2192 Set} {C : \u03a3 A B \u2192 Set} \u2192\n  ((a : A) \u2192 (b : B a) \u2192 C (a , b)) \u2192\n  ((p : \u03a3 A B) \u2192 C p)\nuncurry f (a , b) = f a b\n\n_\u2218_ : {A B C : Set\u2081} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n_\u2218_ g f x = g (f x)\n\n----------------------------------------------------------------------\n\n\nelim\u22a5 : {A : Set} \u2192 \u22a5 \u2192 A\nelim\u22a5 ()\n\nelimBool : (P : Bool \u2192 Set)\n  (pt : P true)\n  (pf : P false)\n  (b : Bool) \u2192 P b\nelimBool P pt pf true = pt\nelimBool P pt pf false = pf\n\nif_then_else_ : {C : Set} \u2192 Bool \u2192 C \u2192 C \u2192 C\nif b then c\u2081 else c\u2082 = elimBool _ c\u2081 c\u2082 b\n\nelim\u2115 : (P : \u2115 \u2192 Set)\n  (pz : P zero)\n  (ps : (n : \u2115) \u2192 P n \u2192 P (suc n))\n  (n : \u2115) \u2192 P n\nelim\u2115 P pz ps zero = pz\nelim\u2115 P pz ps (suc n) = ps n (elim\u2115 P pz ps n)\n\n----------------------------------------------------------------------\n\nrecord Universe : Set\u2081 where\n  field\n    Codes : Set\n    Meaning : Codes \u2192 Set\n\n----------------------------------------------------------------------\n\ndata DescForm (U : Universe) : Set where\n  `\u22a4 `X : DescForm U\n  `\u03a0 `\u03a3 : (A : Universe.Codes U)\n    (D : Universe.Meaning U A \u2192 DescForm U)\n    \u2192 DescForm U\n\n\u27e6_\/_\u27e7\u1d48 : (U : Universe) \u2192 DescForm U \u2192 Set \u2192 Set\n\u27e6 U \/ `\u22a4 \u27e7\u1d48 X = \u22a4\n\u27e6 U \/ `X \u27e7\u1d48 X = X\n\u27e6 U \/ `\u03a0 A D \u27e7\u1d48 X =\n  (a : Universe.Meaning U A) \u2192 \u27e6 U \/ D a \u27e7\u1d48 X\n\u27e6 U \/ `\u03a3 A D \u27e7\u1d48 X =\n  \u03a3 (Universe.Meaning U A) (\u03bb a \u2192 \u27e6 U \/ D a \u27e7\u1d48 X)\n\ndata \u03bc {U : Universe} (D : DescForm U) : Set where\n  con : \u27e6 U \/ D \u27e7\u1d48 (\u03bc D) \u2192 \u03bc D\n\n----------------------------------------------------------------------\n\ndata TypeForm (U : Universe) : Set\n\u27e6_\/_\u27e7 : (U : Universe) \u2192 TypeForm U \u2192 Set\n\ndata TypeForm U where\n  `\u22a5 `\u22a4 `Bool `\u2115 `Desc `Type : TypeForm U\n  `\u03a0 `\u03a3 : (A : TypeForm U)\n    (B : \u27e6 U \/ A \u27e7 \u2192 TypeForm U)\n    \u2192 TypeForm U\n  `\u27e6_\u27e7 : Universe.Codes U \u2192 TypeForm U\n  `\u27e6_\u27e7\u1d48 : DescForm U \u2192 TypeForm U \u2192 TypeForm U\n  `\u03bc : DescForm U \u2192 TypeForm U\n\n\u27e6 U \/ `\u22a5 \u27e7 = \u22a5\n\u27e6 U \/ `\u22a4 \u27e7 = \u22a4\n\u27e6 U \/ `Bool \u27e7 = Bool\n\u27e6 U \/ `\u2115 \u27e7 = \u2115\n\u27e6 U \/ `\u03a0 A B \u27e7 = (a : \u27e6 U \/ A \u27e7) \u2192 \u27e6 U \/ B a \u27e7\n\u27e6 U \/ `\u03a3 A B \u27e7 = \u03a3 \u27e6 U \/ A \u27e7 (\u03bb a \u2192 \u27e6 U \/ B a \u27e7)\n\u27e6 U \/ `Type \u27e7 = Universe.Codes U\n\u27e6 U \/ `\u27e6 A \u27e7 \u27e7 = Universe.Meaning U A\n\u27e6 U \/ `Desc \u27e7 = DescForm U\n\u27e6 U \/ `\u27e6 D \u27e7\u1d48 X \u27e7 = \u27e6 U \/ D \u27e7\u1d48 \u27e6 U \/ X \u27e7\n\u27e6 U \/ `\u03bc D \u27e7 = \u03bc D\n\n----------------------------------------------------------------------\n\n_`\u2192_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u2192 B = `\u03a0 A (\u03bb _ \u2192 B)\n\nLevel : (\u2113 : \u2115) \u2192 Universe\nLevel zero = record { Codes = \u22a5 ; Meaning = \u03bb() }\nLevel (suc \u2113) = record { Codes = TypeForm (Level \u2113)\n                       ; Meaning = \u27e6_\/_\u27e7 (Level \u2113) }\n\nType : \u2115 \u2192 Set\nType \u2113 = TypeForm (Level \u2113)\n\nDesc : \u2115 \u2192 Set\nDesc \u2113 = DescForm (Level \u2113)\n\ninfix 0 \u27e6_\u2223_\u27e7\n\u27e6_\u2223_\u27e7 : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set\n\u27e6 \u2113 \u2223 A \u27e7 = \u27e6 Level \u2113 \/ A \u27e7\n\n\u27e6_\u2223_\u27e7\u1d48 : (\u2113 : \u2115) \u2192 Desc \u2113 \u2192 Set \u2192 Set\n\u27e6 \u2113 \u2223 D \u27e7\u1d48 X = \u27e6 Level \u2113 \/ D \u27e7\u1d48 X\n\n----------------------------------------------------------------------\n\nelimDesc : (P : (\u2113 : \u2115) \u2192 Desc \u2113 \u2192 Set)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u22a4)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `X)\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (D : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Desc (suc \u2113))\n    (rec : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P (suc \u2113) (D a))\n    \u2192 P (suc \u2113) (`\u03a0 A D))\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (D : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Desc (suc \u2113))\n    (rec : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P (suc \u2113) (D a))\n    \u2192 P (suc \u2113) (`\u03a3 A D))\n  \u2192 (\u2113 : \u2115) (D : Desc \u2113) \u2192 P \u2113 D\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 \u2113 `\u22a4 = p\u22a4 \u2113\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 \u2113 `X = pX \u2113\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 zero (`\u03a0 () D)\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113) (`\u03a0 A D) =\n  let f = elimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113)\n  in p\u03a0 \u2113 A D (\u03bb a \u2192 f (D a))\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 zero (`\u03a3 () D)\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113) (`\u03a3 A D) =\n  let f = elimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113)\n  in p\u03a3 \u2113 A D (\u03bb a \u2192 f (D a))\n\ndes : \u2200{\u2113} {D : Desc \u2113} \u2192 \u03bc D \u2192 \u27e6 \u2113 \u2223 D \u27e7\u1d48 (\u03bc D)\ndes (con x) = x\n\n----------------------------------------------------------------------\n\nelimType : (P : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u22a5)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u22a4)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `Bool)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u2115)\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (B : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Type \u2113)\n    (rec\u2081 : P \u2113 A)\n    (rec\u2082 : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P \u2113 (B a))\n    \u2192 P \u2113 (`\u03a0 A B))\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (B : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Type \u2113)\n    (rec\u2081 : P \u2113 A)\n    (rec\u2082 : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P \u2113 (B a))\n    \u2192 P \u2113 (`\u03a3 A B))\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `Desc)\n  \u2192 ((\u2113 : \u2115) (D : Desc \u2113) (X : Type \u2113) (rec : P \u2113 X) \u2192 P \u2113 (`\u27e6 D \u27e7\u1d48 X))\n  \u2192 ((\u2113 : \u2115) (D : Desc \u2113) \u2192 P \u2113 (`\u03bc D))\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `Type)\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (rec : P \u2113 A) \u2192 P (suc \u2113) `\u27e6 A \u27e7)\n  \u2192 (\u2113 : \u2115) (A : Type \u2113) \u2192 P \u2113 A\n\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `\u22a5 = p\u22a5 \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `\u22a4 = p\u22a4 \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `Bool = pBool \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `\u2115 = p\u2115 \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u03a0 A B) =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u03a0 \u2113 A B (f A) (\u03bb a \u2192 f (B a))\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u03a3 A B) =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u03a3 \u2113 A B (f A) (\u03bb a \u2192 f (B a))\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `Type = pType \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `Desc = pDesc \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u27e6 D \u27e7\u1d48 X) =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u27e6D\u27e7\u1d48 \u2113 D X (f X)\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u03bc D) = p\u03bc \u2113 D\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  zero `\u27e6 () \u27e7\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  (suc \u2113) `\u27e6 A \u27e7 =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u27e6A\u27e7 \u2113 A (f A)\n\n----------------------------------------------------------------------\n","old_contents":"module Spire.Type where\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\nrecord \u22a4 : Set where constructor tt\ndata Bool : Set where true false : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115    #-}\n{-# BUILTIN ZERO    zero #-}\n{-# BUILTIN SUC     suc  #-}\n\nrecord \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\nopen \u03a3 public\n\n----------------------------------------------------------------------\n\nconst : {A : Set\u2081} {B : Set} \u2192 A \u2192 B \u2192 A\nconst a _ = a\n\nuncurry : {A : Set} {B : A \u2192 Set} {C : \u03a3 A B \u2192 Set} \u2192\n  ((a : A) \u2192 (b : B a) \u2192 C (a , b)) \u2192\n  ((p : \u03a3 A B) \u2192 C p)\nuncurry f (a , b) = f a b\n\n_\u2218_ : {A B C : Set\u2081} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n_\u2218_ g f x = g (f x)\n\n----------------------------------------------------------------------\n\n\nelim\u22a5 : {A : Set} \u2192 \u22a5 \u2192 A\nelim\u22a5 ()\n\nelimBool : (P : Bool \u2192 Set)\n  (pt : P true)\n  (pf : P false)\n  (b : Bool) \u2192 P b\nelimBool P pt pf true = pt\nelimBool P pt pf false = pf\n\nif_then_else_ : {C : Set} \u2192 Bool \u2192 C \u2192 C \u2192 C\nif b then c\u2081 else c\u2082 = elimBool _ c\u2081 c\u2082 b\n\nelim\u2115 : (P : \u2115 \u2192 Set)\n  (pz : P zero)\n  (ps : (n : \u2115) \u2192 P n \u2192 P (suc n))\n  (n : \u2115) \u2192 P n\nelim\u2115 P pz ps zero = pz\nelim\u2115 P pz ps (suc n) = ps n (elim\u2115 P pz ps n)\n\n----------------------------------------------------------------------\n\nrecord Universe : Set\u2081 where\n  field\n    Codes : Set\n    Meaning : Codes \u2192 Set\n\n----------------------------------------------------------------------\n\ndata DescForm (U : Universe) : Set where\n  `\u22a4 `X : DescForm U\n  `\u03a0 `\u03a3 : (A : Universe.Codes U)\n    (D : Universe.Meaning U A \u2192 DescForm U)\n    \u2192 DescForm U\n\n\u27e6_\/_\u27e7\u1d48 : (U : Universe) \u2192 DescForm U \u2192 Set \u2192 Set\n\u27e6 U \/ `\u22a4 \u27e7\u1d48 X = \u22a4\n\u27e6 U \/ `X \u27e7\u1d48 X = X\n\u27e6 U \/ `\u03a0 A D \u27e7\u1d48 X =\n  (a : Universe.Meaning U A) \u2192 \u27e6 U \/ D a \u27e7\u1d48 X\n\u27e6 U \/ `\u03a3 A D \u27e7\u1d48 X =\n  \u03a3 (Universe.Meaning U A) (\u03bb a \u2192 \u27e6 U \/ D a \u27e7\u1d48 X)\n\ndata \u03bc {U : Universe} (D : DescForm U) : Set where\n  con : \u27e6 U \/ D \u27e7\u1d48 (\u03bc D) \u2192 \u03bc D\n\n----------------------------------------------------------------------\n\ndata TypeForm (U : Universe) : Set\n\u27e6_\/_\u27e7 : (U : Universe) \u2192 TypeForm U \u2192 Set\n\ndata TypeForm U where\n  `\u22a5 `\u22a4 `Bool `\u2115 `Desc `Type : TypeForm U\n  `\u03a0 `\u03a3 : (A : TypeForm U)\n    (B : \u27e6 U \/ A \u27e7 \u2192 TypeForm U)\n    \u2192 TypeForm U\n  `\u27e6_\u27e7 : Universe.Codes U \u2192 TypeForm U\n  `\u27e6_\u27e7\u1d48 : DescForm U \u2192 TypeForm U \u2192 TypeForm U\n  `\u03bc : DescForm U \u2192 TypeForm U\n\n\u27e6 U \/ `\u22a5 \u27e7 = \u22a5\n\u27e6 U \/ `\u22a4 \u27e7 = \u22a4\n\u27e6 U \/ `Bool \u27e7 = Bool\n\u27e6 U \/ `\u2115 \u27e7 = \u2115\n\u27e6 U \/ `\u03a0 A B \u27e7 = (a : \u27e6 U \/ A \u27e7) \u2192 \u27e6 U \/ B a \u27e7\n\u27e6 U \/ `\u03a3 A B \u27e7 = \u03a3 \u27e6 U \/ A \u27e7 (\u03bb a \u2192 \u27e6 U \/ B a \u27e7)\n\u27e6 U \/ `Type \u27e7 = Universe.Codes U\n\u27e6 U \/ `\u27e6 A \u27e7 \u27e7 = Universe.Meaning U A\n\u27e6 U \/ `Desc \u27e7 = DescForm U\n\u27e6 U \/ `\u27e6 D \u27e7\u1d48 X \u27e7 = \u27e6 U \/ D \u27e7\u1d48 \u27e6 U \/ X \u27e7\n\u27e6 U \/ `\u03bc D \u27e7 = \u03bc D\n\n----------------------------------------------------------------------\n\n_`\u2192_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u2192 B = `\u03a0 A (\u03bb _ \u2192 B)\n\nLevel : (\u2113 : \u2115) \u2192 Universe\nLevel zero = record { Codes = \u22a5 ; Meaning = \u03bb() }\nLevel (suc \u2113) = record { Codes = TypeForm (Level \u2113)\n                       ; Meaning = \u27e6_\/_\u27e7 (Level \u2113) }\n\nType : \u2115 \u2192 Set\nType \u2113 = TypeForm (Level \u2113)\n\nDesc : \u2115 \u2192 Set\nDesc \u2113 = DescForm (Level \u2113)\n\ninfix 0 \u27e6_\u2223_\u27e7\n\u27e6_\u2223_\u27e7 : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set\n\u27e6 \u2113 \u2223 A \u27e7 = \u27e6 Level \u2113 \/ A \u27e7\n\n\u27e6_\u2223_\u27e7\u1d48 : (\u2113 : \u2115) \u2192 Desc \u2113 \u2192 Set \u2192 Set\n\u27e6 \u2113 \u2223 D \u27e7\u1d48 X = \u27e6 Level \u2113 \/ D \u27e7\u1d48 X\n\n----------------------------------------------------------------------\n\nelimDesc : (P : (\u2113 : \u2115) \u2192 Desc \u2113 \u2192 Set)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u22a4)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `X)\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (D : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Desc (suc \u2113))\n    (rec : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P (suc \u2113) (D a))\n    \u2192 P (suc \u2113) (`\u03a0 A D))\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (D : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Desc (suc \u2113))\n    (rec : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P (suc \u2113) (D a))\n    \u2192 P (suc \u2113) (`\u03a3 A D))\n  \u2192 (\u2113 : \u2115) (D : Desc \u2113) \u2192 P \u2113 D\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 \u2113 `\u22a4 = p\u22a4 \u2113\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 \u2113 `X = pX \u2113\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 zero (`\u03a0 () D)\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113) (`\u03a0 A D) =\n  let f = elimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113)\n  in p\u03a0 \u2113 A D (\u03bb a \u2192 f (D a))\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 zero (`\u03a3 () D)\nelimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113) (`\u03a3 A D) =\n  let f = elimDesc P p\u22a4 pX p\u03a0 p\u03a3 (suc \u2113)\n  in p\u03a3 \u2113 A D (\u03bb a \u2192 f (D a))\n\ndes : \u2200{\u2113} {D : Desc \u2113} \u2192 \u03bc D \u2192 \u27e6 \u2113 \u2223 D \u27e7\u1d48 (\u03bc D)\ndes (con x) = x\n\n----------------------------------------------------------------------\n\nelimType : (P : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u22a5)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u22a4)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `Bool)\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `\u2115)\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (B : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Type \u2113)\n    (rec\u2081 : P \u2113 A)\n    (rec\u2082 : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P \u2113 (B a))\n    \u2192 P \u2113 (`\u03a0 A B))\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (B : \u27e6 \u2113 \u2223 A \u27e7 \u2192 Type \u2113)\n    (rec\u2081 : P \u2113 A)\n    (rec\u2082 : (a : \u27e6 \u2113 \u2223 A \u27e7) \u2192 P \u2113 (B a))\n    \u2192 P \u2113 (`\u03a3 A B))\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `Desc)\n  \u2192 ((\u2113 : \u2115) (D : Desc \u2113) (X : Type \u2113) (rec : P \u2113 X) \u2192 P \u2113 (`\u27e6 D \u27e7\u1d48 X))\n  \u2192 ((\u2113 : \u2115) (D : Desc \u2113) \u2192 P \u2113 (`\u03bc D))\n  \u2192 ((\u2113 : \u2115) \u2192 P \u2113 `Type)\n  \u2192 ((\u2113 : \u2115) (A : Type \u2113) (rec : P \u2113 A) \u2192 P (suc \u2113) `\u27e6 A \u27e7)\n  \u2192 (\u2113 : \u2115) (A : Type \u2113) \u2192 P \u2113 A\n\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `\u22a5 = p\u22a5 \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `\u22a4 = p\u22a4 \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `Bool = pBool \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `\u2115 = p\u2115 \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u03a0 A B) =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u03a0 \u2113 A B (f A) (\u03bb a \u2192 f (B a))\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u03a3 A B) =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u03a3 \u2113 A B (f A) (\u03bb a \u2192 f (B a))\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `Type = pType \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 `Desc = pDesc \u2113\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u27e6 D \u27e7\u1d48 X) =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u27e6D\u27e7\u1d48 \u2113 D X (f X)\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  \u2113 (`\u03bc D) = p\u03bc \u2113 D\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  zero `\u27e6 () \u27e7\nelimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7\n  (suc \u2113) `\u27e6 A \u27e7 =\n  let f = elimType P p\u22a5 p\u22a4 pBool p\u2115 p\u03a0 p\u03a3 pDesc p\u27e6D\u27e7\u1d48 p\u03bc pType p\u27e6A\u27e7 \u2113\n  in p\u27e6A\u27e7 \u2113 A (f A)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3822d110c060a75d914e9d0e4fc105f00e99980a","subject":"Atlas: fix bug in `union` by postulating conjunction","message":"Atlas: fix bug in `union` by postulating conjunction\n\nOld-commit-hash: 69ed9a83b2b162709e5dfcf541f9d0bf18c7e970\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n--\n-- TODO: write this and call it Syntax.Term.Plotkin.lift-term\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\n\nzip! f m\u2081 m\u2082 = app (app (app (const zip) f) m\u2081) m\u2082\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\n\nlookup! = {!!} -- TODO: add constant `lookup`\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff s t = app (app (lift-diff Atlas-diff Atlas-apply) s) t\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply s t = app (app (lift-apply Atlas-diff Atlas-apply) s) t\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\n\nupdate! k v m = app (app (app (const update) k) v) m\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- The binary operator such that\n-- union t t \u2261 t\n\n-- To implement union, we need for each non-map base-type\n-- an operator such that `op v v = v` on all values `v`.\n-- for Booleans, conjunction is good enough.\n--\n-- TODO (later): support conjunction, probably by Boolean\n-- elimination form if-then-else\n\npostulate\n  and! : \u2200 {\u0393} \u2192\n    Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n    Atlas-term \u0393 (base Bool)\n\nunion : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base \u03b9)\nunion {Bool} s t = and! s t\nunion {Map \u03ba \u03b9} s t =\n  let\n    union-term = abs (abs (union (var (that this)) (var this)))\n  in\n    zip! (abs union-term) s t\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    v\u2081 = var (that this)\n    v\u2082 = var this\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    k\u2081\u2082 = var (that (that this))\n    k\u2083\u2084 = var (that (that this))\n    f\u2081\u2082 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2081\u2082) v\u2081) v\u2082)\n      (lookup! k\u2081\u2082 (weaken\u2083 m\u2083))) (lookup! k\u2081\u2082 (weaken\u2083 m\u2084)))))\n    f\u2083\u2084 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2083\u2084)\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2081)))\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2082))) v\u2083) v\u2084)))\n  in\n    -- A correct but inefficient implementation.\n    -- May want to speed it up after constants are finalized.\n    union (zip! f\u2081\u2082 m\u2081 m\u2082) (zip! f\u2083\u2084 m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app (app (weaken\u2081 \u0394f) (var this)) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n--\n-- TODO: write this and call it Syntax.Term.Plotkin.lift-term\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\n\nzip! f m\u2081 m\u2082 = app (app (app (const zip) f) m\u2081) m\u2082\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\n\nlookup! = {!!} -- TODO: add constant `lookup`\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff s t = app (app (lift-diff Atlas-diff Atlas-apply) s) t\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply s t = app (app (lift-apply Atlas-diff Atlas-apply) s) t\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\n\nupdate! k v m = app (app (app (const update) k) v) m\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- The binary operator with which all base-type values\n-- form a group\n\nunion : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base \u03b9)\nunion {Bool} s t = app (app (const xor) s) t\nunion {Map \u03ba \u03b9} s t =\n  let\n    union-term = abs (abs (union (var (that this)) (var this)))\n  in\n    zip! (abs union-term) s t\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    v\u2081 = var (that this)\n    v\u2082 = var this\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    k\u2081\u2082 = var (that (that this))\n    k\u2083\u2084 = var (that (that this))\n    f\u2081\u2082 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2081\u2082) v\u2081) v\u2082)\n      (lookup! k\u2081\u2082 (weaken\u2083 m\u2083))) (lookup! k\u2081\u2082 (weaken\u2083 m\u2084)))))\n    f\u2083\u2084 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2083\u2084)\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2081)))\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2082))) v\u2083) v\u2084)))\n  in\n    -- A correct but inefficient implementation.\n    -- May want to speed it up after constants are finalized.\n    union (zip! f\u2081\u2082 m\u2081 m\u2082) (zip! f\u2083\u2084 m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app (app (weaken\u2081 \u0394f) (var this)) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ced202e3f87365cfeb07406d81b0480bc81320bf","subject":"Sum: untag","message":"Sum: untag\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Sum\/NP.agda","new_file":"lib\/Data\/Sum\/NP.agda","new_contents":"-- NOTE with-K\nmodule Data.Sum.NP where\n\nopen import Data.Sum public renaming (inj\u2081 to inl; inj\u2082 to inr)\n\nopen import Type hiding (\u2605)\nopen import Level.NP\nopen import Function\nopen import Data.Nat using (\u2115; zero; suc)\nopen import Data.Zero\nopen import Data.One\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_;_\u2262_;_\u2257_)\nopen import Data.Two hiding (twist)\nopen \u2261 using (\u2192-to-\u27f6)\n\n[inl:_,inr:_] = [_,_]\n\ninl-inj : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {x y : A} \u2192 inl {B = B} x \u2261 inl y \u2192 x \u2261 y\ninl-inj \u2261.refl = \u2261.refl\n\ninr-inj : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {x y : B} \u2192 inr {A = A} x \u2261 inr y \u2192 x \u2261 y\ninr-inj \u2261.refl = \u2261.refl\n\nmodule _ {a} {A : \u2605 a} where\n    untag : A \u228e A \u2192 A\n    untag = [inl: id ,inr: id ]\n\nmodule _ {a\u2081 a\u2082 b\u2081 b\u2082}\n         {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n         {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n         {c} {C : \u2605 c} (f : A\u2081 \u228e B\u2081 \u2192 A\u2082 \u228e B\u2082 \u2192 C) where\n    on-inl = \u03bb i j \u2192 f (inl i) (inl j)\n    on-inr = \u03bb i j \u2192 f (inr i) (inr j)\n\n[,]-assoc : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 c} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n              {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} {C : \u2605 c}\n              {f\u2081 : B\u2081 \u2192 C} {g\u2081 : A\u2081 \u2192 B\u2081} {f\u2082 : B\u2082 \u2192 C} {g\u2082 : A\u2082 \u2192 B\u2082} \u2192\n              [ f\u2081 , f\u2082 ] \u2218\u2032 map g\u2081 g\u2082 \u2257 [ f\u2081 \u2218 g\u2081 , f\u2082 \u2218 g\u2082 ]\n[,]-assoc (inl _) = \u2261.refl\n[,]-assoc (inr _) = \u2261.refl\n\n[,]-factor : \u2200 {a\u2081 a\u2082 b c} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} {B : \u2605 b} {C : \u2605 c}\n              {f : B \u2192 C} {g\u2081 : A\u2081 \u2192 B} {g\u2082 : A\u2082 \u2192 B} \u2192\n              [ f \u2218 g\u2081 , f \u2218 g\u2082 ] \u2257 f \u2218 [ g\u2081 , g\u2082 ]\n[,]-factor (inl _) = \u2261.refl\n[,]-factor (inr _) = \u2261.refl\n\nmap-assoc : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n              {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} {C\u2081 : \u2605 c\u2081} {C\u2082 : \u2605 c\u2082}\n              {f\u2081 : B\u2081 \u2192 C\u2081} {g\u2081 : A\u2081 \u2192 B\u2081} {f\u2082 : B\u2082 \u2192 C\u2082} {g\u2082 : A\u2082 \u2192 B\u2082} \u2192\n              map f\u2081 f\u2082 \u2218\u2032 map g\u2081 g\u2082 \u2257 map (f\u2081 \u2218 g\u2081) (f\u2082 \u2218 g\u2082)\nmap-assoc = [,]-assoc\n\nopen import Data.Product\nopen import Function.Inverse\nopen import Function.LeftInverse\n\n{- bad names\n\u228e-fst : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u228e B \u2192 \ud835\udfda\n\u228e-fst (inl _) = 0\u2082\n\u228e-fst (inr _) = 1\u2082\n\n\u228e-snd : \u2200 {\u2113} {A B : \u2605 \u2113} (x : A \u228e B) \u2192 case \u228e-fst x 0: A 1: B\n\u228e-snd (inl x) = x\n\u228e-snd (inr x) = x\n-}\n\n-- Function.Related.TypeIsomorphisms.NP for the A \u228e B, \u03a3 \ud835\udfda [0: A 1: B ] iso.\n\n\ud835\udfd9\u228e^ : \u2115 \u2192 \u2605\u2080\n\ud835\udfd9\u228e^ zero    = \ud835\udfd8\n\ud835\udfd9\u228e^ (suc n) = \ud835\udfd9 \u228e \ud835\udfd9\u228e^ n\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} where\n    twist : A \u228e B \u2192 B \u228e A\n    twist = [inl: inr ,inr: inl ]\n","old_contents":"-- NOTE with-K\nmodule Data.Sum.NP where\n\nopen import Data.Sum public renaming (inj\u2081 to inl; inj\u2082 to inr)\n\nopen import Type hiding (\u2605)\nopen import Level.NP\nopen import Function\nopen import Data.Nat using (\u2115; zero; suc)\nopen import Data.Zero\nopen import Data.One\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_;_\u2262_;_\u2257_)\nopen import Data.Two hiding (twist)\nopen \u2261 using (\u2192-to-\u27f6)\n\n[inl:_,inr:_] = [_,_]\n\ninl-inj : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {x y : A} \u2192 inl {B = B} x \u2261 inl y \u2192 x \u2261 y\ninl-inj \u2261.refl = \u2261.refl\n\ninr-inj : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {x y : B} \u2192 inr {A = A} x \u2261 inr y \u2192 x \u2261 y\ninr-inj \u2261.refl = \u2261.refl\n\nmodule _ {a\u2081 a\u2082 b\u2081 b\u2082}\n         {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n         {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n         {c} {C : \u2605 c} (f : A\u2081 \u228e B\u2081 \u2192 A\u2082 \u228e B\u2082 \u2192 C) where\n    on-inl = \u03bb i j \u2192 f (inl i) (inl j)\n    on-inr = \u03bb i j \u2192 f (inr i) (inr j)\n\n[,]-assoc : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 c} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n              {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} {C : \u2605 c}\n              {f\u2081 : B\u2081 \u2192 C} {g\u2081 : A\u2081 \u2192 B\u2081} {f\u2082 : B\u2082 \u2192 C} {g\u2082 : A\u2082 \u2192 B\u2082} \u2192\n              [ f\u2081 , f\u2082 ] \u2218\u2032 map g\u2081 g\u2082 \u2257 [ f\u2081 \u2218 g\u2081 , f\u2082 \u2218 g\u2082 ]\n[,]-assoc (inl _) = \u2261.refl\n[,]-assoc (inr _) = \u2261.refl\n\n[,]-factor : \u2200 {a\u2081 a\u2082 b c} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} {B : \u2605 b} {C : \u2605 c}\n              {f : B \u2192 C} {g\u2081 : A\u2081 \u2192 B} {g\u2082 : A\u2082 \u2192 B} \u2192\n              [ f \u2218 g\u2081 , f \u2218 g\u2082 ] \u2257 f \u2218 [ g\u2081 , g\u2082 ]\n[,]-factor (inl _) = \u2261.refl\n[,]-factor (inr _) = \u2261.refl\n\nmap-assoc : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n              {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} {C\u2081 : \u2605 c\u2081} {C\u2082 : \u2605 c\u2082}\n              {f\u2081 : B\u2081 \u2192 C\u2081} {g\u2081 : A\u2081 \u2192 B\u2081} {f\u2082 : B\u2082 \u2192 C\u2082} {g\u2082 : A\u2082 \u2192 B\u2082} \u2192\n              map f\u2081 f\u2082 \u2218\u2032 map g\u2081 g\u2082 \u2257 map (f\u2081 \u2218 g\u2081) (f\u2082 \u2218 g\u2082)\nmap-assoc = [,]-assoc\n\nopen import Data.Product\nopen import Function.Inverse\nopen import Function.LeftInverse\n\n{- bad names\n\u228e-fst : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u228e B \u2192 \ud835\udfda\n\u228e-fst (inl _) = 0\u2082\n\u228e-fst (inr _) = 1\u2082\n\n\u228e-snd : \u2200 {\u2113} {A B : \u2605 \u2113} (x : A \u228e B) \u2192 case \u228e-fst x 0: A 1: B\n\u228e-snd (inl x) = x\n\u228e-snd (inr x) = x\n-}\n\n-- Function.Related.TypeIsomorphisms.NP for the A \u228e B, \u03a3 \ud835\udfda [0: A 1: B ] iso.\n\n\ud835\udfd9\u228e^ : \u2115 \u2192 \u2605\u2080\n\ud835\udfd9\u228e^ zero    = \ud835\udfd8\n\ud835\udfd9\u228e^ (suc n) = \ud835\udfd9 \u228e \ud835\udfd9\u228e^ n\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} where\n    twist : A \u228e B \u2192 B \u228e A\n    twist = [inl: inr ,inr: inl ]\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"23cf45d8db2b7852d5a8c6726d7534b076de6c92","subject":"Iso attepmt for fin","message":"Iso attepmt for fin\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Fin.agda","new_file":"lib\/Explore\/Fin.agda","new_contents":"{-# OPTIONS --without-K #-}\n\nopen import Level.NP\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary.NP\nopen import Explore.Type\nopen import Explore.Explorable\nimport Explore.Isomorphism as Iso\n--open import Explore.Explorable.Maybe\n\nmodule Explore.Fin where\n\n\ud835\udfd9\u228e^ : \u2115 \u2192 \u2605\u2080\n\ud835\udfd9\u228e^ zero    = \ud835\udfd8\n\ud835\udfd9\u228e^ (suc n) = \ud835\udfd9 \u228e \ud835\udfd9\u228e^ n\n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 : \u2200 n \u2192 Fin n \u2194 \ud835\udfd9\u228e^ n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 zero    = Fin0\u2194\ud835\udfd8\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 (suc n) = {!{!Inv.id!} \u228e-cong ?!} Inv.\u2218 Fin\u2218suc\u2194\ud835\udfd9\u228eFin\n\n{-\nMaybe^\ud835\udfd8\u2194Fin\nopen Iso (Fin\n\n{-\nmodule _ {\u2113} where\n    Fin\u1d49 : \u2200 n \u2192 Explore \u2113 (Fin n)\n    Fin\u1d49 zero    z _\u2219_ f = z\n    Fin\u1d49 (suc n) z _\u2219_ f = f zero \u2219 Fin\u1d49 n z _\u2219_ (f \u2218 suc)\n\n    Fin\u2071 : \u2200 {p} n \u2192 ExploreInd p (Fin\u1d49 n)\n    Fin\u2071 zero    P z _\u2219_ f = z\n    Fin\u2071 (suc n) P z _\u2219_ f = f zero \u2219 Fin\u2071 n Psuc z _\u2219_ (f \u2218 suc)\n      where Psuc = \u03bb e \u2192 P (\u03bb z op f \u2192 e z op (f \u2218 suc))\n\nmodule _ {\u2113} where\n    Fin\u02e1 : \u2200 n \u2192 Lookup {\u2113} (Fin\u1d49 n)\n    Fin\u02e1 (suc _) (b , _)  zero    = b\n    Fin\u02e1 (suc n) (_ , xs) (suc x) = Fin\u02e1 n xs x\n\n    Fin\u1da0 : \u2200 n \u2192 Focus {\u2113} (Fin\u1d49 n)\n    Fin\u1da0 (suc n) (zero   , b) = inj\u2081 b\n    Fin\u1da0 (suc n) (suc x  , b) = inj\u2082 (Fin\u1da0 n (x , b))\n\nmodule _ n where\n    open Explorable\u2080  (Fin\u2071 n) public using () renaming (sum     to Fin\u02e2; product to Fin\u1d56)\n    open Explorable\u2081\u2080 (Fin\u2071 n) public using () renaming (reify   to Fin\u02b3)\n    open Explorable\u2081\u2081 (Fin\u2071 n) public using () renaming (unfocus to Fin\u1d58)\n\npostulate\n  Postulate-Fin\u02e2-ok : \u2605\n  Postulate-FinFun\u02e2-ok : \u2605\n\n  Fin\u02e2-ok : \u2200 {{_ : Postulate-Fin\u02e2-ok}} n \u2192 AdequateSum (Fin\u02e2 n)\n\n{-\nFin\u1d56-ok : \u2200 n \u2192 AdequateProduct (Fin\u1d56 n)\n-}\n\nmodule _ {A : \u2605}(\u03bcA : Explorable A) where\n\n  e\u1d2c = explore \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  FinFun\u1d49 : \u2200 n \u2192 Explore _ (Fin n \u2192 A)\n  FinFun\u1d49 zero    z op f = f \u00acFin0\n  FinFun\u1d49 (suc n) z op f = e\u1d2c z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))\n\n  FinFun\u2071 : \u2200 n \u2192 ExploreInd _ (FinFun\u1d49 n)\n  FinFun\u2071 zero    P Pz P\u2219 Pf = Pf _\n  FinFun\u2071 (suc n) P Pz P\u2219 Pf =\n    explore-ind \u03bcA (\u03bb sa \u2192 P (\u03bb z op f \u2192 sa z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))))\n      Pz P\u2219\n      (\u03bb x \u2192 FinFun\u2071 n (\u03bb sf \u2192 P (\u03bb z op f \u2192 sf z op (f \u2218 extend x)))\n        Pz P\u2219 (Pf \u2218 extend x))\n\n  FinFun\u02e2 : \u2200 n \u2192 Sum (Fin n \u2192 A)\n  FinFun\u02e2 n = FinFun\u1d49 n 0 _+_\n\n  postulate\n    FinFun\u02e2-ok : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} n \u2192 AdequateSum (FinFun\u02e2 n)\n\n  \u03bcFinFun : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} {n} \u2192 Explorable (Fin n \u2192 A)\n  \u03bcFinFun = mk _ (FinFun\u2071 _) (FinFun\u02e2-ok _)\n\n\u03bcFin : \u2200 {{_ : Postulate-Fin\u02e2-ok}} n \u2192 Explorable (Fin n)\n\u03bcFin n = mk _ (Fin\u2071 n) (Fin\u02e2-ok n)\n\n{-\n\u03bcFinSUI : \u2200 {n} \u2192 SumStableUnderInjection (sum (\u03bcFin n))\n-}\n\nmodule BigDistr\n  {{_ : Postulate-Fin\u02e2-ok}}\n  {{_ : Postulate-FinFun\u02e2-ok}}\n  {A : \u2605}(\u03bcA : Explorable A)\n  (cm           : CommutativeMonoid \u2080 \u2080)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_          : let open CMon cm in C \u2192 C \u2192 C)\n  (\u03b5-\u25ce          : let open CMon cm in Zero _\u2248_ \u03b5 _\u25ce_)\n  (distrib      : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_     : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm renaming (sym to \u2248-sym)\n\n  \u03bcF\u2192A = \u03bcFinFun \u03bcA\n\n  -- Sum over A\n  \u03a3\u1d2c = explore \u03bcA \u03b5 _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin I \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = explore \u03bcF\u2192A \u03b5 _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 explore (\u03bcFin I) \u03b5 _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))\n    \u2248\u27e8 \u2248-sym (explore-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))) (proj\u2081 \u03b5-\u25ce _) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)))\n    \u2248\u27e8 explore-mon-ext \u03bcA monoid (\u03bb j \u2192 \u2248-sym (explore-lin\u02e1 \u03bcF\u2192A monoid _\u25ce_ (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)) (F zero j) (proj\u2082 \u03b5-\u25ce _) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I ((F \u2218 suc) \u02e2 f))\n    \u220e\n\nmodule _\n  {{_ : Postulate-Fin\u02e2-ok}}\n  {{_ : Postulate-FinFun\u02e2-ok}} where\n\n  FinDist : \u2200 {n} \u2192 DistFun (\u03bcFin n) (\u03bb \u03bcX \u2192 \u03bcFinFun \u03bcX)\n  FinDist \u03bcB c \u25ce distrib \u25ce-cong f = BigDistr.bigDistr \u03bcB c \u25ce distrib \u25ce-cong f _\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n\nopen import Level.NP\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary.NP\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Isomorphism\n--open import Explore.Explorable.Maybe\n\nmodule Explore.Fin where\n\nmodule _ {\u2113} where\n    Fin\u1d49 : \u2200 n \u2192 Explore \u2113 (Fin n)\n    Fin\u1d49 zero    z _\u2219_ f = z\n    Fin\u1d49 (suc n) z _\u2219_ f = f zero \u2219 Fin\u1d49 n z _\u2219_ (f \u2218 suc)\n\n    Fin\u2071 : \u2200 {p} n \u2192 ExploreInd p (Fin\u1d49 n)\n    Fin\u2071 zero    P z _\u2219_ f = z\n    Fin\u2071 (suc n) P z _\u2219_ f = f zero \u2219 Fin\u2071 n Psuc z _\u2219_ (f \u2218 suc)\n      where Psuc = \u03bb e \u2192 P (\u03bb z op f \u2192 e z op (f \u2218 suc))\n\nmodule _ {\u2113} where\n    Fin\u02e1 : \u2200 n \u2192 Lookup {\u2113} (Fin\u1d49 n)\n    Fin\u02e1 (suc _) (b , _)  zero    = b\n    Fin\u02e1 (suc n) (_ , xs) (suc x) = Fin\u02e1 n xs x\n\n    Fin\u1da0 : \u2200 n \u2192 Focus {\u2113} (Fin\u1d49 n)\n    Fin\u1da0 (suc n) (zero   , b) = inj\u2081 b\n    Fin\u1da0 (suc n) (suc x  , b) = inj\u2082 (Fin\u1da0 n (x , b))\n\nmodule _ n where\n    open Explorable\u2080  (Fin\u2071 n) public using () renaming (sum     to Fin\u02e2; product to Fin\u1d56)\n    open Explorable\u2081\u2080 (Fin\u2071 n) public using () renaming (reify   to Fin\u02b3)\n    open Explorable\u2081\u2081 (Fin\u2071 n) public using () renaming (unfocus to Fin\u1d58)\n\npostulate\n  Postulate-Fin\u02e2-ok : \u2605\n  Postulate-FinFun\u02e2-ok : \u2605\n\n  Fin\u02e2-ok : \u2200 {{_ : Postulate-Fin\u02e2-ok}} n \u2192 AdequateSum (Fin\u02e2 n)\n\n{-\nFin\u1d56-ok : \u2200 n \u2192 AdequateProduct (Fin\u1d56 n)\n-}\n\nmodule _ {A : \u2605}(\u03bcA : Explorable A) where\n\n  e\u1d2c = explore \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  FinFun\u1d49 : \u2200 n \u2192 Explore _ (Fin n \u2192 A)\n  FinFun\u1d49 zero    z op f = f \u00acFin0\n  FinFun\u1d49 (suc n) z op f = e\u1d2c z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))\n\n  FinFun\u2071 : \u2200 n \u2192 ExploreInd _ (FinFun\u1d49 n)\n  FinFun\u2071 zero    P Pz P\u2219 Pf = Pf _\n  FinFun\u2071 (suc n) P Pz P\u2219 Pf =\n    explore-ind \u03bcA (\u03bb sa \u2192 P (\u03bb z op f \u2192 sa z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))))\n      Pz P\u2219\n      (\u03bb x \u2192 FinFun\u2071 n (\u03bb sf \u2192 P (\u03bb z op f \u2192 sf z op (f \u2218 extend x)))\n        Pz P\u2219 (Pf \u2218 extend x))\n\n  FinFun\u02e2 : \u2200 n \u2192 Sum (Fin n \u2192 A)\n  FinFun\u02e2 n = FinFun\u1d49 n 0 _+_\n\n  postulate\n    FinFun\u02e2-ok : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} n \u2192 AdequateSum (FinFun\u02e2 n)\n\n  \u03bcFinFun : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} {n} \u2192 Explorable (Fin n \u2192 A)\n  \u03bcFinFun = mk _ (FinFun\u2071 _) (FinFun\u02e2-ok _)\n\n\u03bcFin : \u2200 {{_ : Postulate-Fin\u02e2-ok}} n \u2192 Explorable (Fin n)\n\u03bcFin n = mk _ (Fin\u2071 n) (Fin\u02e2-ok n)\n\n{-\n\u03bcFinSUI : \u2200 {n} \u2192 SumStableUnderInjection (sum (\u03bcFin n))\n-}\n\nmodule BigDistr\n  {{_ : Postulate-Fin\u02e2-ok}}\n  {{_ : Postulate-FinFun\u02e2-ok}}\n  {A : \u2605}(\u03bcA : Explorable A)\n  (cm           : CommutativeMonoid \u2080 \u2080)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_          : let open CMon cm in C \u2192 C \u2192 C)\n  (\u03b5-\u25ce          : let open CMon cm in Zero _\u2248_ \u03b5 _\u25ce_)\n  (distrib      : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_     : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm renaming (sym to \u2248-sym)\n\n  \u03bcF\u2192A = \u03bcFinFun \u03bcA\n\n  -- Sum over A\n  \u03a3\u1d2c = explore \u03bcA \u03b5 _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin I \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = explore \u03bcF\u2192A \u03b5 _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 explore (\u03bcFin I) \u03b5 _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))\n    \u2248\u27e8 \u2248-sym (explore-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))) (proj\u2081 \u03b5-\u25ce _) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)))\n    \u2248\u27e8 explore-mon-ext \u03bcA monoid (\u03bb j \u2192 \u2248-sym (explore-lin\u02e1 \u03bcF\u2192A monoid _\u25ce_ (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)) (F zero j) (proj\u2082 \u03b5-\u25ce _) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I ((F \u2218 suc) \u02e2 f))\n    \u220e\n\nmodule _\n  {{_ : Postulate-Fin\u02e2-ok}}\n  {{_ : Postulate-FinFun\u02e2-ok}} where\n\n  FinDist : \u2200 {n} \u2192 DistFun (\u03bcFin n) (\u03bb \u03bcX \u2192 \u03bcFinFun \u03bcX)\n  FinDist \u03bcB c \u25ce distrib \u25ce-cong f = BigDistr.bigDistr \u03bcB c \u25ce distrib \u25ce-cong f _\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"12687bc02ef41455749e8da3f343fe679cd80336","subject":"adding new files to the \"\"\"cm file\"\"\"","message":"adding new files to the \"\"\"cm file\"\"\"\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import core\n-- open import correspondence\n-- open import expandability\n-- open import expansion-unicity\n-- open import preservation\n-- open import progress\n-- open import type-assignment-unicity\n-- open import typed-expansion\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"87c6c3bc67648f23a50338540793cd6b1bd51f0c","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import lemmas-gcomplete\n\nopen import lemmas-disjointness\n--open import disjointness -- working here\n\nopen import structural-assumptions\n\nopen import focus-formation\nopen import ground-decidable\nopen import matched-ground-invariant\nopen import finality\n\n--open import lemmas-subst-ta -- working here\nopen import lemmas-ground\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\n--open import expansion-generality -- depend on disjointness\n--open import expandability -- ditto\nopen import expansion-unicity\nopen import type-assignment-unicity\nopen import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import lemmas-progress-checks\nopen import progress-checks\nopen import progress\nopen import preservation\n\nopen import lemmas-complete\nopen import cast-inert\nopen import complete-preservation\nopen import complete-progress\nopen import complete-expansion\n\nopen import weakening\nopen import exchange\nopen import lemmas-freshness\nopen import contraction\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import lemmas-gcomplete\n\nopen import lemmas-disjointness\nopen import disjointness -- work here\n\nopen import structural-assumptions\n\nopen import focus-formation\nopen import ground-decidable\nopen import matched-ground-invariant\nopen import finality\n\nopen import lemmas-subst-ta -- work here\nopen import lemmas-ground\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\nopen import expansion-generality\nopen import expandability\nopen import expansion-unicity\nopen import type-assignment-unicity\nopen import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import lemmas-progress-checks\nopen import progress-checks\nopen import progress\nopen import preservation\n\nopen import lemmas-complete\nopen import cast-inert\nopen import complete-preservation\nopen import complete-progress\nopen import complete-expansion\n\nopen import weakening\nopen import exchange\nopen import lemmas-freshness\nopen import contraction\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"344eedbe0e9625bd6a43619ab9c60b3aaa817443","subject":"agda : wouter : predicate transformer","message":"agda : wouter : predicate transformer\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/x.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/x.agda","new_contents":"{-# OPTIONS --type-in-type   #-} -- NOT SOUND!\n\nopen import Data.Empty      using (\u22a5)\nopen import Data.Nat        renaming (\u2115 to Nat)\nopen import Data.Nat.DivMod\nopen import Data.Product    hiding (map)\nopen import Data.Unit       using (\u22a4)\n------------------------------------------------------------------------------\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl)\n\nmodule x where\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\nabstract\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nINTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to\n    - use this refinement relation to show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n2 BACKGROUND\n\nFree monads\n-}\n\n-- C          : type of commands\n-- Free C R   : returns an 'a' or issues command c : C\n-- For each c : C, there is a set of responses R c\n-- 2nd arg of Step is continuation : how to proceed after receiving response R c\ndata Free (C : Set) (R : C \u2192 Set) (a : Set) : Set where\n  Pure :                           a  \u2192 Free C R a\n  Step : (c : C) \u2192 (R c \u2192 Free C R a) \u2192 Free C R a\n\n-- show that 'Free' is a monad:\n\nmap : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 (a \u2192 b) \u2192 Free C R a \u2192 Free C R b\nmap f (Pure x)   = Pure (f x)\nmap f (Step c k) = Step c (\u03bb r \u2192 map f (k r))\n\nreturn : \u2200 {a C : Set} {R : C \u2192 Set} \u2192 a \u2192 Free C R a\nreturn = Pure\n\n_>>=_ : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 Free C R a \u2192 (a \u2192 Free C R b) \u2192 Free C R b\nPure   x >>= f = f x\nStep c x >>= f = Step c (\u03bb r \u2192 x r >>= f)\n\n{-\ndifferent effects choose C and R differently, depending on their ops\n\nWeakest precondition semantics\n\nidea of associating weakest precondition semantics with imperative programs\ndates to Dijkstra\u2019s Guarded Command Language [1975]\n\nways to specify behaviour of function f : a \u2192 b\n- reference implementation\n- define a relation R : a \u2192 b \u2192 Set\n- write contracts and test cases\n- PT semantics\n\ncall values of type a \u2192 Set : predicate on type a\n\nPTs are functions between predicates\ne.g., weakest precondition:\n-}\n\n-- \"maps\"\n-- function f : a \u2192 b and\n-- desired postcondition on the function\u2019s output, b \u2192 Set\n-- to weakest precondition a \u2192 Set on function\u2019s input that ensures postcondition satisfied\n--\n-- note: definition is just reverse function composition\n-- wp0 : \u2200 {a b : Set} \u2192 (f : a \u2192 b) \u2192 (b \u2192 Set) \u2192 (a \u2192 Set)\nwp0 : \u2200 {a : Set} {b :     Set} (f :      a  \u2192 b)   \u2192           (b   \u2192 Set) \u2192 (a \u2192 Set)\nwp0 f P = \u03bb x \u2192 P   (f x)\n{-\nabove wp semantics is sometimes too restrictive\n- no way to specify that output is related to input\n- fix via making f dependent:\n-}\nwp  : \u2200 {a : Set} {b : a \u2192 Set} (f : (x : a) \u2192 b x) \u2192 ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)\nwp  f P = \u03bb x \u2192 P x (f x)\n\n-- shorthand for working with predicates and predicates transformers\n_\u2286_ : \u2200 {a : Set} \u2192 (a \u2192 Set) \u2192 (a \u2192 Set) \u2192 Set\nP \u2286 Q = \u2200 x \u2192 P x \u2192 Q x\n\n-- refinement relation defined between PTs\n_\u2291_ : \u2200 {a : Set} {b : a \u2192 Set} \u2192 (pt1 pt2 : ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)) \u2192 Set\u2081\npt1 \u2291 pt2 = \u2200 P \u2192 pt1 P \u2286 pt2 P\n\n{-\nuse refinement relation\n- to relate PT semantics between programs and specifications\n- to show a program satisfies its specification; or\n- to show that one program is somehow \u2018better\u2019 than another,\n  where \u2018better\u2019 is defined by choice of PT semantics\n\nin pure setting, this refinement relation is not interesting:\nthe refinement relation corresponds to extensional equality between functions:\n\nlemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\nrefinement : \u2200 (f g : a \u2192 b) \u2192 (wp f \u2291 wp g) \u2194 (\u2200 x \u2192 f x \u2261 g x)\n\nthis paper defines PT semantics for Kleisli arrows of form\n\n    a \u2192 Free C R b\n\ncould use 'wp' to assign semantics to these computations directly,\nbut typically not interested in syntactic equality between free monads\n\nrather want to study semantics of effectful programs they represent\n\nto define a PT semantics for effects\ndefine a function with form:\n\n  -- how to lift a predicate on 'a' over effectful computation returning 'a'\n  pt : (a \u2192 Set) \u2192 Free C R a \u2192 Set\n\n'pt' def depends on semantics desired for a particulr free monad\n\nCrucially,\nchoice of pt and weakest precondition semantics, wp, together\ngive a way to assign weakest precondition semantics to Kleisli arrows\nrepresenting effectful computations\n\n3 PARTIALITY\n\nPartial computations : i.e., 'Maybe'\n\nmake choices for commands C and responses R\n-}\n\ndata C : Set where\n  Abort : C -- no continuation\n\nR : C \u2192 Set\nR Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\nPartial : Set \u2192 Set\nPartial = Free C R\n\n-- smart constructor for failure:\nabort : \u2200 {a : Set} \u2192 Partial a\nabort = Step Abort (\u03bb ())\n\n{-\ncomputation of type Partial a will either\n- return a value of type a or\n- fail, issuing abort command\n\nNote that responses to Abort command are empty;\nabort smart constructor abort uses this to discharge the continuation\nin the second argument of the Step constructor\n\nExample: division\n\nexpression language, closed under division and natural numbers:\n-}\n\ndata Expr : Set where\n  Val : Nat  \u2192 Expr\n  Div : Expr \u2192 Expr \u2192 Expr\n\nexv : Expr\nexv = Val 3\nexd : Expr\nexd = Div (Val 3) (Val 3)\n\n-- semantics specified using inductively defined RELATION:\n-- def rules out erroneous results by requiring the divisor evaluates to non-zero\ndata _\u21d3_ : Expr \u2192 Nat \u2192 Set where\n  Base : \u2200 {x : Nat}\n       \u2192 Val x \u21d3 x\n  Step : \u2200 {l r : Expr} {v1 v2 : Nat}\n       \u2192     l   \u21d3  v1\n       \u2192       r \u21d3         (suc v2)\n       \u2192 Div l r \u21d3 (v1 div (suc v2))\n\nexb : Val 3 \u21d3 3\nexb = Base\n\nexs : Div (Val 3) (Val 3) \u21d3 1\nexs = Step Base Base\n\n-- Alternatively\n-- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n-- op used by \u27e6_\u27e7 interpreter\n_\u00f7_ : Nat \u2192 Nat \u2192 Partial Nat\nn \u00f7 zero    = abort\nn \u00f7 (suc k) = return (n div (suc k))\n\n\u27e6_\u27e7 : Expr \u2192 Partial Nat\n\u27e6 Val x \u27e7     = return x\n\u27e6 Div e1 e2 \u27e7 = \u27e6 e1 \u27e7 >>= \u03bb v1 \u2192 \u27e6 e2 \u27e7 >>= \u03bb v2 \u2192 v1 \u00f7 v2\n\nevv   : Free C R Nat\nevv   = \u27e6 Val 3 \u27e7\nevv'  : evv \u2261 Pure 3\nevv'  = refl\n\nevd   : Free C R Nat\nevd   = \u27e6 Div (Val 3) (Val 3) \u27e7\nevd'  : evd \u2261 Pure 1\nevd'  = refl\n\nevd0  : Free C R Nat\nevd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\nevd0' : evd0 \u2261 Step Abort (\u03bb ())\nevd0' = refl\n\n{-\nHow to relate two definitions:\n- std lib 'div' requires implicit proof that divisor is non-zero\n  - \u21d3 relation generates via pattern matching\n  - _\u00f7_ does explicit check\n- interpreter uses _\u00f7_\n  - fails explicitly with abort when divisor is zero\n\nAssign a weakest precondition semantics to Kleisli arrows of the form\n\n    a \u2192 Partial b\n-}\n\nwpPartial : \u2200 {a : Set} {b : a \u2192 Set}\n          \u2192 (f : (x : a) \u2192 Partial (b x))\n          \u2192 (P : (x : a) \u2192          b x \u2192 Set)\n          \u2192 (a \u2192 Set)\nwpPartial f P = wp f (mustPT P)\n where\n  mustPT : \u2200 {a : Set} {b : a \u2192 Set}\n         \u2192 (P : (x : a) \u2192 b x \u2192 Set)\n         \u2192      (x : a)\n         \u2192 Partial       (b x) \u2192 Set\n  mustPT P _ (Pure y)       = P _ y\n  mustPT P _ (Step Abort _) = \u22a5\n{-\nTo call 'wp', must show how to transform\n-      predicate                  P :         b \u2192 Set\n- to a predicate on partial results : Partial b \u2192 Set\nDone via proposition 'mustPT P c'\n- holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\nparticular PT semantics of partial computations determined by def mustPT\nhere: rule out failure entirely\n- so case Abort returns empty type\n\nGiven this PT semantics for Kleisli arrows in general,\ncan now study semantics of above monadic interpreter\nvia passing\n- interpreter: \u27e6_\u27e7\n- desired postcondition : _\u21d3_\nas arguments to wpPartial:\n\n  wpPartial \u27e6_\u27e7 _\u21d3_ : Expr \u2192 Set\n\nresulting in a predicate on expressions\n\nfor all expressions satisfying this predicate,\nthe monadic interpreter and the relational specification, _\u21d3_,\nmust agree on the result of evaluation\n\nWhat does this say about correctness of interpreter?\nTo understand the predicate better, consider defining this predicate on expressions:\n-}\n\nSafeDiv : Expr \u2192 Set\nSafeDiv (Val x)     = \u22a4\nSafeDiv (Div e1 e2) = (e2 \u21d3 zero \u2192 \u22a5) {-\u2227-} \u00d7 SafeDiv e1 {-\u2227-} \u00d7 SafeDiv e2\n\n{-\nExpect : any expr e for which SafeDiv e holds\ncan be evaluated without division-by-zero\n\ncan prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n-- lemma relates the two semantics\n-- expressed as a relation and an evaluator\n-- for those expressions that satisfy the SafeDiv property\ncorrect : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n\nInstead of manually defining SafeDiv, define more general predicate\ncharacterising the domain of a partial function:\n-}\n\ndom : \u2200 {a : Set} {b : a \u2192 Set}\n    \u2192 ((x : a)\n    \u2192 Partial (b x))\n    \u2192 (a \u2192 Set)\ndom f = wpPartial f (\u03bb _ _ \u2192 \u22a4)\n\n{-\ncan show that the two semantics agree precisely on the domain of the interpreter:\n\nsound    : dom       \u27e6_\u27e7 \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\ncomplete : wpPartial \u27e6_\u27e7 _\u21d3_ \u2286 dom \u27e6_\u27e7\n\nboth proofs proceed by induction on the argument expression\n\nRefinement\n\nweakest precondition semantics on partial computations give rise\nto a refinement relation on Kleisli arrows of the form a \u2192 Partial b\n\ncan characterise this relation by proving:\n\nrefinement : (f g : a \u2192 Maybe b)\n           \u2192 (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x \u2192 (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\nuse refinement to relate Kleisli morphisms,\nand to relate a program to a specification given by a pre- and postcondition\n\n\nExample: Add (interpreter for stack machine)\n\nadd top two elements; can fail fail if stack has too few elements\n\nbelow shows how to prove the definition meets its specification\n\nDefine specification in terms of a pre\/post condition.\nThe specification of a function of type (x : a) \u2192 b x consists of\n-}\nrecord Spec (a : Set) (b : a \u2192 Set) : Set where\n  constructor [_,_]\n  field\n    pre  : a \u2192 Set             -- a precondition on a, and\n    post : (x : a) \u2192 b x \u2192 Set -- a postcondition relating inputs that satisfy this precondition\n                               -- and the corresponding outputs\n","old_contents":"{-# OPTIONS --type-in-type   #-} -- NOT SOUND!\n\nopen import Data.Empty      using (\u22a5)\nopen import Data.Nat        renaming (\u2115 to Nat)\nopen import Data.Nat.DivMod\nopen import Data.Product    hiding (map)\nopen import Data.Unit       using (\u22a4)\n\nmodule x where\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\nabstract\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nINTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising\n  from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to\n    - use this refinement relation to show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n2 BACKGROUND\n\nFree monads\n-}\n\n-- C          : type of commands\n-- Free C R   : returns an 'a' or issues command c : C\n-- For each c : C, there is a set of responses R c\n-- 2nd arg of Step is continuation : how to proceed after receiving response R c\ndata Free (C : Set) (R : C \u2192 Set) (a : Set) : Set where\n  Pure :                           a  \u2192 Free C R a\n  Step : (c : C) \u2192 (R c \u2192 Free C R a) \u2192 Free C R a\n\n-- show that 'Free' is a monad:\n\nmap : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 (a \u2192 b) \u2192 Free C R a \u2192 Free C R b\nmap f (Pure x)   = Pure (f x)\nmap f (Step c k) = Step c (\u03bb r \u2192 map f (k r))\n\nreturn : \u2200 {a C : Set} {R : C \u2192 Set} \u2192 a \u2192 Free C R a\nreturn = Pure\n\n_>>=_ : \u2200 {a b C : Set} {R : C \u2192 Set} \u2192 Free C R a \u2192 (a \u2192 Free C R b) \u2192 Free C R b\nPure   x >>= f = f x\nStep c x >>= f = Step c (\u03bb r \u2192 x r >>= f)\n\n{-\ndifferent effects choose C and R differently, depending on their ops\n\nWeakest precondition semantics\n\nidea of associating weakest precondition semantics with imperative programs\ndates to Dijkstra\u2019s Guarded Command Language [1975]\n\nways to specify behaviour of function f : a \u2192 b\n- reference implementation\n- define a relation R : a \u2192 b \u2192 Set\n- write contracts and test cases\n- PT semantics\n\ncall values of type a \u2192 Set : predicate on type a\n\nPTs are functions between predicates\ne.g., weakest precondition:\n-}\n\n-- \"maps\"\n-- function f : a \u2192 b and\n-- desired postcondition on the function\u2019s output, b \u2192 Set\n-- to weakest precondition a \u2192 Set on function\u2019s input that ensures postcondition satisfied\n--\n-- note: definition is just reverse function composition\n-- wp0 : \u2200 {a b : Set} \u2192 (f : a \u2192 b) \u2192 (b \u2192 Set) \u2192 (a \u2192 Set)\nwp0 : \u2200 {a : Set} {b :     Set} (f :       a \u2192 b)   \u2192           (b   \u2192 Set) \u2192 (a \u2192 Set)\nwp0 f P = \u03bb x \u2192 P   (f x)\n{-\nabove wp semantics is sometimes too restrictive\n- no way to specify that output is related to input\n- fix via making f dependent:\n-}\nwp  : \u2200 {a : Set} {b : a \u2192 Set} (f : (x : a) \u2192 b x) \u2192 ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)\nwp  f P = \u03bb x \u2192 P x (f x)\n\n-- shorthand for working with predicates and predicates transformers\n_\u2286_ : \u2200 {a : Set} \u2192 (a \u2192 Set) \u2192 (a \u2192 Set) \u2192 Set\nP \u2286 Q = \u2200 x \u2192 P x \u2192 Q x\n\n-- refinement relation defined between PTs\n_\u2291_ : \u2200 {a : Set} {b : a \u2192 Set} \u2192 (pt1 pt2 : ((x : a) \u2192 b x \u2192 Set) \u2192 (a \u2192 Set)) \u2192 Set\u2081\npt1 \u2291 pt2 = \u2200 P \u2192 pt1 P \u2286 pt2 P\n\n{-\nuse refinement relation\n- to relate PT semantics between programs and specifications\n- to show a program satisfies its specification; or\n- to show that one program is somehow \u2018better\u2019 than another,\n  where \u2018better\u2019 is defined by choice of PT semantics\n\nin pure setting, this refinement relation is not interesting:\nthe refinement relation corresponds to extensional equality between functions:\n\nlemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\nrefinement : \u2200 (f g : a \u2192 b) \u2192 (wp f \u2291 wp g) \u2194 (\u2200 x \u2192 f x \u2261 g x)\n\nthis paper defines PT semantics for Kleisli arrows of form\n\n    a \u2192 Free C R b\n\ncould use 'wp' to assign semantics to these computations directly,\nbut typically not interested in syntactic equality between free monads\n\nrather want to study semantics of effectful programs they represent\n\nto define a PT semantics for effects\ndefine a function with form:\n\n  -- how to lift a predicate on 'a' over effectful computation returning 'a'\n  pt : (a \u2192 Set) \u2192 Free C R a \u2192 Set\n\n'pt' def depends on semantics desired for a particulr free monad\n\nCrucially,\nchoice of pt and weakest precondition semantics, wp, together\ngive a way to assign weakest precondition semantics to Kleisli arrows\nrepresenting effectful computations\n\n3 PARTIALITY\n\nPartial computations : i.e., 'Maybe'\n\nmake choices for commands C and responses R\n-}\n\ndata C : Set where\n  Abort : C -- no continuation\n\nR : C \u2192 Set\nR Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\nPartial : Set \u2192 Set\nPartial = Free C R\n\n-- smart constructor for failure:\nabort : \u2200 {a : Set} \u2192 Partial a\nabort = Step Abort (\u03bb ())\n\n{-\ncomputation of type Partial a will either\n- return a value of type a or\n- fail, issuing abort command\n\nNote that responses to Abort command are empty;\nabort smart constructor abort uses this to discharge the continuation\nin the second argument of the Step constructor\n\nExample: division\n\nexpression language, closed under division and natural numbers:\n-}\n\ndata Expr : Set where\n  Val : Nat  \u2192 Expr\n  Div : Expr \u2192 Expr \u2192 Expr\n\n-- semantics specified using inductively defined RELATION:\n-- def rules out erroneous results by requiring the divisor evaluates to non-zero\ndata _\u21d3_ : Expr \u2192 Nat \u2192 Set where\n  Base : \u2200 {x : Nat}\n       \u2192 Val x \u21d3 x\n  Step : \u2200 {l r : Expr} {v1 v2 : Nat}\n       \u2192       l \u21d3  v1\n       \u2192       r \u21d3         (suc v2)\n       \u2192 Div l r \u21d3 (v1 div (suc v2))\n\n-- Alternatively\n-- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n-- op used by \u27e6_\u27e7 interpreter\n_\u00f7_ : Nat \u2192 Nat \u2192 Partial Nat\nn \u00f7 zero    = abort\nn \u00f7 (suc k) = return (n div (suc k))\n\n\u27e6_\u27e7 : Expr \u2192 Partial Nat\n\u27e6 Val x \u27e7     = return x\n\u27e6 Div e1 e2 \u27e7 = \u27e6 e1 \u27e7 >>= \u03bb v1 \u2192 \u27e6 e2 \u27e7 >>= \u03bb v2 \u2192 v1 \u00f7 v2\n\n{-\nstd lib 'div' requires implicit proof that divisor is non-zero\n\nwhen divisor is zero, interpreter fail explicitly with abort\n\nHow to relate these two definitions?\n\nAssign a weakest precondition semantics to Kleisli arrows of the form\n\n    a \u2192 Partial b\n-}\n\nwpPartial : \u2200 {a : Set} {b : a \u2192 Set}\n          \u2192 (f : (x : a) \u2192 Partial (b x))\n          \u2192 (P : (x : a) \u2192          b x \u2192 Set)\n          \u2192 (a \u2192 Set)\nwpPartial f P = wp f (mustPT P)\n where\n  mustPT : \u2200 {a : Set} {b : a \u2192 Set}\n         \u2192 (P : (x : a) \u2192 b x \u2192 Set)\n         \u2192      (x : a)\n         \u2192 Partial       (b x) \u2192 Set\n  mustPT P _ (Pure y)       = P _ y\n  mustPT P _ (Step Abort _) = \u22a5\n{-\nTo call 'wp', must show how to transform\n-      predicate                  P :         b \u2192 Set\n- to a predicate on partial results : Partial b \u2192 Set\nDone via proposition 'mustPT P c'\n- holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\nparticular PT semantics of partial computations determined by def mustPT\nhere: rule out failure entirely\n- so case Abort returns empty type\n\nGiven this PT semantics for Kleisli arrows in general,\ncan now study semantics of above monadic interpreter\nvia passing\n- interpreter: \u27e6_\u27e7\n- desired postcondition : _\u21d3_\nas arguments to wpPartial:\n\n  wpPartial \u27e6_\u27e7 _\u21d3_ : Expr \u2192 Set\n\nresulting in a predicate on expressions\n\nfor all expressions satisfying this predicate,\nthe monadic interpreter and the relational specification, _\u21d3_,\nmust agree on the result of evaluation\n\nWhat does this say about correctness of interpreter?\nTo understand the predicate better, consider defining this predicate on expressions:\n-}\n\nSafeDiv : Expr \u2192 Set\nSafeDiv (Val x)     = \u22a4\nSafeDiv (Div e1 e2) = (e2 \u21d3 zero \u2192 \u22a5) {-\u2227-} \u00d7 SafeDiv e1 {-\u2227-} \u00d7 SafeDiv e2\n\n{-\nExpect : any expr e for which SafeDiv e holds\ncan be evaluated without division-by-zero\n\ncan prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n-- lemma relates the two semantics\n-- expressed as a relation and an evaluator\n-- for those expressions that satisfy the SafeDiv property\ncorrect : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n\nInstead of manually defining SafeDiv, define more general predicate\ncharacterising the domain of a partial function:\n-}\n\ndom : \u2200 {a : Set} {b : a \u2192 Set}\n    \u2192 ((x : a)\n    \u2192 Partial (b x))\n    \u2192 (a \u2192 Set)\ndom f = wpPartial f (\u03bb _ _ \u2192 \u22a4)\n\n{-\ncan show that the two semantics agree precisely on the domain of the interpreter:\n\nsound    : dom       \u27e6_\u27e7 \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\ncomplete : wpPartial \u27e6_\u27e7 _\u21d3_ \u2286 dom \u27e6_\u27e7\n\nboth proofs proceed by induction on the argument expression\n\nRefinement\n\nweakest precondition semantics on partial computations give rise\nto a refinement relation on Kleisli arrows of the form a \u2192 Partial b\n\ncan characterise this relation by proving:\n\nrefinement : (f g : a \u2192 Maybe b)\n           \u2192 (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x \u2192 (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\nuse refinement to relate Kleisli morphisms,\nand to relate a program to a specification given by a pre- and postcondition\n\n\nExample: Add (interpreter for stack machine)\n\nadd top two elements; can fail fail if stack has too few elements\n\nbelow shows how to prove the definition meets its specification\n\nDefine specification in terms of a pre\/post condition.\nThe specification of a function of type (x : a) \u2192 b x consists of\n-}\nrecord Spec (a : Set) (b : a \u2192 Set) : Set where\n  constructor [_,_]\n  field\n    pre  : a \u2192 Set             -- a precondition on a, and\n    post : (x : a) \u2192 b x \u2192 Set -- a postcondition relating inputs that satisfy this precondition\n                               -- and the corresponding outputs\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"ab67f882ab3e8b9fa88063a7fe823a848af25c4c","subject":"Only white spaces.","message":"Only white spaces.\n\nIgnore-this: 6ca18b898d016f181623e766c9dcebeb\n\ndarcs-hash:20100727155339-3bd4e-b1c40a641d81f05c0e0291cedb65fa69721cfd7d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Nat.agda","new_file":"LTC\/Data\/Nat.agda","new_contents":"------------------------------------------------------------------------------\n-- LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat where\n\nopen import LTC.Minimal\n\ninfixl 7 _*_\ninfixl 6 _+_ _-_\n\n------------------------------------------------------------------------------\n-- The LTC natural numbers type.\nopen import LTC.Data.Nat.Type public\n\n------------------------------------------------------------------------------\n-- Arithmetic operations\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\npostulate\n  _-_      : D \u2192 D \u2192 D\n  minus-x0 : (d : D) \u2192 d - zero          \u2261 d\n  minus-0S : (d : D) \u2192 zero - succ d     \u2261 zero\n  minus-SS : (d e : D) \u2192 succ d - succ e \u2261 d - e\n{-# ATP axiom minus-x0 #-}\n{-# ATP axiom minus-0S #-}\n{-# ATP axiom minus-SS #-}\n\npostulate\n  _*_ : D \u2192 D \u2192 D\n  *-0x : (d : D)   \u2192 zero * d   \u2261 zero\n  *-Sx : (d e : D) \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x #-}\n{-# ATP axiom *-Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat where\n\nopen import LTC.Minimal\n\ninfixl 7 _*_\ninfixl 6 _+_ _-_\n\n------------------------------------------------------------------------------\n-- The LTC natural numbers type.\nopen import LTC.Data.Nat.Type public\n\n------------------------------------------------------------------------------\n-- Arithmetic operations\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192   zero   + d \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\npostulate\n  _-_      : D \u2192 D \u2192 D\n  minus-x0 : (d : D) \u2192   d      - zero   \u2261 d\n  minus-0S : (d : D) \u2192   zero   - succ d \u2261 zero\n  minus-SS : (d e : D) \u2192 succ d - succ e \u2261 d - e\n{-# ATP axiom minus-x0 #-}\n{-# ATP axiom minus-0S #-}\n{-# ATP axiom minus-SS #-}\n\npostulate\n  _*_ : D \u2192 D \u2192 D\n  *-0x : (d : D)   \u2192 zero   * d \u2261 zero\n  *-Sx : (d e : D) \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x #-}\n{-# ATP axiom *-Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"117ba614c0592b120afb1203485a93d78bc6bfbc","subject":"Data.Bits: \u2192\u1d47","message":"Data.Bits: \u2192\u1d47\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"835301426a8c885216f233fec0a8c592e4daf8f6","subject":"Updated installation instructions.","message":"Updated installation instructions.\n\nIgnore-this: efdef7acbf124dd693f607adbc492261\n\ndarcs-hash:20110507165243-3bd4e-964560d62ba745ad1c29061ecd3f5ccbc02b505f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Reasoning about Functional Programs\" by Ana Bove, Peter Dybjer, and\n-- Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n------------------------------------------------------------------------------\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 The law of the excluded middle\nopen import PredicateLogic.ClassicalATP\n\n-- 1.3 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.6 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.2 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.3 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.PropertiesI\n\n-- 6.3.4 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.5 Inequalites\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.6 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.3 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.4 Lists of natural numbers\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Programs\n\n-- 6.5.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.5.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.ProofSpecificationATP\nopen import FOTC.Program.GCD.ProofSpecificationI\n\n-- 6.5.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.5.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.5.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\n-- This module was imported in the Stanovsk\u00fd example\n-- open import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n\n------------------------------------------------------------------------------\n-- Other theories\n------------------------------------------------------------------------------\n\n-- 1. LTC-PCF\n-- Formalization of a version of Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 1.1. The axioms\nopen import LTC-PCF.Base\n\n-- 1.2 Natural numberes\n\n-- 1.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 1.2.2 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 1.2.3 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- 1.2.4 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 1.2.5 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 1.2.6 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 1.3 Programs\n\n-- 1.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 1.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Theorem Proving for Reasoning about Functional Programs\" by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download the agda2atp tool (described in above paper) using\n-- darcs with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 The law of the excluded middle\nopen import PredicateLogic.ClassicalATP\n\n-- 1.3 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.6 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.2 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.3 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.PropertiesI\n\n-- 6.3.4 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.5 Inequalites\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.6 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.3 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.4 Lists of natural numbers\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Programs\n\n-- 6.5.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.5.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.ProofSpecificationATP\nopen import FOTC.Program.GCD.ProofSpecificationI\n\n-- 6.5.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.5.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.5.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\n-- This module was imported in the Stanovsk\u00fd example\n-- open import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n\n------------------------------------------------------------------------------\n-- Other theories\n------------------------------------------------------------------------------\n\n-- 1. LTC-PCF\n-- Formalization of a version of Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 1.1. The axioms\nopen import LTC-PCF.Base\n\n-- 1.2 Natural numberes\n\n-- 1.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 1.2.2 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 1.2.3 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- 1.2.4 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 1.2.5 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 1.2.6 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 1.3 Programs\n\n-- 1.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 1.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b9f8c7e62100cb6c18942ebd737297d5edfa9267","subject":"IDesc model: implement lists","message":"IDesc model: implement lists\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- List\n--********************************************\n\ndata ListDConst (l : Level) : Set l where\n  cnil : ListDConst l\n  ccons : ListDConst l\n\nlistDChoice : (l : Level) -> Set l -> ListDConst l -> IDesc l Unit\nlistDChoice x X cnil = const Unit\nlistDChoice x X ccons = sigma X (\\_ -> var Void)\n\nlistD : (l : Level) -> Set l -> IDesc l Unit\nlistD x X = sigma (ListDConst x) (listDChoice x X)\n\nlist : (l : Level) -> Set l -> Set l\nlist x X = IMu x Unit (\\_ -> listD x X) Void\n\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7a558dcc2540370cdaf9ebd714d4a564bda778c0","subject":"\u2200 P \u2192 Rel P","message":"\u2200 P \u2192 Rel P\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : proj\u2081 x \u2261 proj\u2081 y) \u2192 subst B p (proj\u2082 x) \u2261 proj\u2082 y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (proj\u2081 (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3191b7e15dd3762e428eec4635cc26f606ccab0f","subject":"Desc stratified model: descD implicit.","message":"Desc stratified model: descD implicit.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ff2adf91d069f86e9db39f1138dcc36e94b4be7f","subject":"close #30","message":"close #30\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"canonical-indeterminate-forms.agda","new_file":"canonical-indeterminate-forms.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       \u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n                                         +\n                                       (\u03a3[ d' \u2208 dhexp ]\n                                         ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n                                          (d' indet) \u00d7\n                                          ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))))\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x))\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (_ , refl , ind , x\u2081)))\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       ((\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                        (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                        (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c4 ==> (\u03c41' ==> \u03c42')) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c4) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                        (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c43 \u2208 htyp ] \u03a3[ \u03c44 \u2208 htyp ]\n                                          ((d == d' \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9) \u00d7\n                                           (d' indet) \u00d7\n                                           ((\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44)))) +\n                                        (\u03a3[ d' \u2208 dhexp ]\n                                          ((\u03c41 == \u2987\u2988) \u00d7\n                                           (\u03c42 == \u2987\u2988) \u00d7\n                                           (d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n                                           (d' indet) \u00d7\n                                           ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))))\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = Inr (Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , ind , x\u2081))))\n  -- todo \/ cyrus: this is the only one that required pattern matching (or equivalently a lemma i didn't bother to state) on premises\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = Inr (Inr (Inr (Inr (_ , refl , refl , refl , ind , x\u2081))))\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n                                          (\u03c4' ground) \u00d7\n                                          (d' indet)))\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = Inr (Inr (Inr (_ , _ , refl , x\u2081 , ind)))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind ())\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam wt) () nb na nh\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAEHole x x\u2081) IEHole nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAEHole x x\u2081) IEHole nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAEHole x x\u2081) IEHole nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] b) \u2208 \u0394))) +\n                                       \u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n                                         +\n                                       (\u03a3[ d' \u2208 dhexp ]\n                                         ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n                                          (d' indet) \u00d7\n                                          ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))))\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x))\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (_ , refl , ind , x\u2081)))\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       ((\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                        (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394))) +\n                                        (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: \u03c4 ==> (\u03c41' ==> \u03c42')) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c4) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                        (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c43 \u2208 htyp ] \u03a3[ \u03c44 \u2208 htyp ]\n                                          ((d == d' \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9) \u00d7\n                                           (d' indet) \u00d7\n                                           ((\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44)))) +\n                                        (\u03a3[ d' \u2208 dhexp ]\n                                          ((\u03c41 == \u2987\u2988) \u00d7\n                                           (\u03c42 == \u2987\u2988) \u00d7\n                                           (d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n                                           (d' indet) \u00d7\n                                           ((d'' : dhexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))))\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = Inr (Inr (Inr (Inl (_ , _ , _ , _ , _ , refl , ind , x\u2081))))\n  -- todo \/ cyrus: this is the only one that required pattern matching (or equivalently a lemma i didn't bother to state) on premises\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = Inr (Inr (Inr (Inr (_ , refl , refl , refl , ind , x\u2081))))\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 subst ] \u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n                                         ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n                                          (d' final) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n                                          ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394))) +\n                                       (\u03a3[ d1 \u2208 dhexp ] \u03a3[ d2 \u2208 dhexp ] \u03a3[ \u03c42 \u2208 htyp ]\n                                         ((d == d1 \u2218 d2) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n                                          (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n                                          (d1 indet) \u00d7\n                                          (d2 final) \u00d7\n                                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))) +\n                                       (\u03a3[ d' \u2208 dhexp ] \u03a3[ \u03c4' \u2208 htyp ]\n                                         ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n                                          (\u03c4' ground) \u00d7\n                                          (d indet))) --todo \/ cyrus: this is\n                                                      --interesting; it's\n                                                      --the only clause\n                                                      --that's filled by\n                                                      --multiple patterns\n                                                      --below; maybe we\n                                                      --could say something\n                                                      --more specific?\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = Inr (Inr (Inl (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)))\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = Inl (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = Inr (Inl (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x ))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = Inr (Inr (Inr (_ , _ , refl , x\u2081 , ICastGroundHole x\u2081 ind)))\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = Inr (Inr (Inr (_ , _ , refl , x\u2082 , ICastGroundHole x\u2082 ind)))\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam wt) () nb na nh\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAEHole x x\u2081) IEHole nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAEHole x x\u2081) IEHole nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAEHole x x\u2081) IEHole nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"14b2f16af804d20fd6a8890ced468fa56fc999d4","subject":"lecture 1","message":"lecture 1\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/Boolean.agda","new_file":"livecode\/Boolean.agda","new_contents":"module Boolean where\n\ndata Bool : Set where \n  true : Bool\n  false : Bool\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\nand : Bool -> (Bool -> Bool)\nand true true = true\nand false _ = false\nand true false = false\n","old_contents":"module Boolean where\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3e0d9a0e494b869a18e730c5ce93e481a4161646","subject":"Yet more fiddling","message":"Yet more fiddling\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.BigStep where\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_)\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!fromtoDerive _ (suc n) s d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!LIE!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!LIE!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\nopen import Data.Nat.Base using (_<_)\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nopen import Relation.Binary hiding (_\u21d2_)\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6115015065a0334d84d5cfc5f45b76faa010f32a","subject":"Swierstra_A_Predicate_Transformer_for_Effects","message":"Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val _) _ = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Znr = magic (ernz er\u21d3Znr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val _) _ = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Znr = magic (ernz er\u21d3Znr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"ebfc5b9adb1c6f827ad5ba98efb7170f25fd251b","subject":"agda: Document \u0394Val","message":"agda: Document \u0394Val\n\nOld-commit-hash: 0db0b8be91a20a06651cdeb565c7e6d63de28db6\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  map-over-++ : \u2200 {f b d} \u2192\n    mapBag f (b ++ d) \u2261 mapBag f b ++ mapBag f d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n--\n-- \u0394Val \u03c4 is intended to be the semantic domain for changes of values of type\n-- \u03c4, which was obtained by \u27e6 \u0394-Type \u03c4 \u27e7 in other formalizations.\n--\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- Used to signal free variables of a term.\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n\n-- Declare everything in \u0393 to be volatile.\nselect-none : {\u0393 : Context} \u2192 Vars \u0393\nselect-none {\u2205} = \u2205\nselect-none {\u03c4 \u2022 \u0393} = alter (select-none {\u0393})\n\nselect-just : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393\nselect-just {\u0393 = \u03c4 \u2022 \u0393\u2080} this = abide select-none\nselect-just (that x) = alter (select-just x)\n\n-- De-facto union of free variables\nFV-union : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 Vars \u0393 \u2192 Vars \u0393\nFV-union \u2205 \u2205 = \u2205\nFV-union (alter us) (alter vs) = alter (FV-union us vs)\nFV-union (alter us) (abide vs) = abide (FV-union us vs)\nFV-union (abide us) (alter vs) = abide (FV-union us vs)\nFV-union (abide us) (abide vs) = abide (FV-union us vs)\n\ntail : \u2200 {\u03c4 \u0393} \u2192 Vars (\u03c4 \u2022 \u0393) \u2192 Vars \u0393\ntail (abide vars) = vars\ntail (alter vars) = vars\n\n-- Free variables of a term.\n-- Free variables are marked as abiding, bound variables altering.\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV {\u0393 = \u0393} (nat n) = select-none\nFV {\u0393 = \u0393} (bag b) = select-none\nFV (var x) = select-just x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV-union (FV s) (FV t)\nFV (add s t) = FV-union (FV s) (FV t)\nFV (map s t) = FV-union (FV s) (FV t)\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 Vars \u0393 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (alter vars)\n  abide : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          v \u2295 dv \u2261 v \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (abide vars)\n\n_is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-for \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-for \u03c1 = \u22a4\n\u0394bag db is-valid-for \u03c1 = \u22a4\n\u0394var x is-valid-for \u03c1 = \u22a4\n\n_is-valid-for_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-for_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-for \u03c1 = Quadruple\n  (ds is-valid-for \u03c1)\n  (\u03bb _ \u2192 dt is-valid-for \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2081 s db is-valid-for \u03c1 = couple\n  (db is-valid-for \u03c1)\n  (Honest \u03c1 (FV s))\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 (cons v-dt honesty _ _) =\n  mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-for \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-for \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- Corollary: (f \u2295 df) (v \u2295 dv) = f v \u2295 df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore \u03c1) \u2295 \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s))\n    (\u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n  \u2261  \u27e6 s \u27e7 (ignore \u03c1) (\u27e6 t \u27e7 (ignore \u03c1)) \u2295\n     \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s) (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t})\n\ncorollary s t {\u03c1} = proj\u2082\n  (validity {t = s} (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t}))\n\nunrestricted (nat n) = tt\nunrestricted (bag b) = tt\nunrestricted (var x) {\u03c1} = tt\nunrestricted (abs t) {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted t)\nunrestricted (app s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t))\n  (validity {t = t}) tt\nunrestricted (add s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\nunrestricted (map s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted t)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  map-over-++ : \u2200 {f b d} \u2192\n    mapBag f (b ++ d) \u2261 mapBag f b ++ mapBag f d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- Used to signal free variables of a term.\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n\n-- Declare everything in \u0393 to be volatile.\nselect-none : {\u0393 : Context} \u2192 Vars \u0393\nselect-none {\u2205} = \u2205\nselect-none {\u03c4 \u2022 \u0393} = alter (select-none {\u0393})\n\nselect-just : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393\nselect-just {\u0393 = \u03c4 \u2022 \u0393\u2080} this = abide select-none\nselect-just (that x) = alter (select-just x)\n\n-- De-facto union of free variables\nFV-union : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 Vars \u0393 \u2192 Vars \u0393\nFV-union \u2205 \u2205 = \u2205\nFV-union (alter us) (alter vs) = alter (FV-union us vs)\nFV-union (alter us) (abide vs) = abide (FV-union us vs)\nFV-union (abide us) (alter vs) = abide (FV-union us vs)\nFV-union (abide us) (abide vs) = abide (FV-union us vs)\n\ntail : \u2200 {\u03c4 \u0393} \u2192 Vars (\u03c4 \u2022 \u0393) \u2192 Vars \u0393\ntail (abide vars) = vars\ntail (alter vars) = vars\n\n-- Free variables of a term.\n-- Free variables are marked as abiding, bound variables altering.\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV {\u0393 = \u0393} (nat n) = select-none\nFV {\u0393 = \u0393} (bag b) = select-none\nFV (var x) = select-just x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV-union (FV s) (FV t)\nFV (add s t) = FV-union (FV s) (FV t)\nFV (map s t) = FV-union (FV s) (FV t)\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 Vars \u0393 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (alter vars)\n  abide : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          v \u2295 dv \u2261 v \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (abide vars)\n\n_is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-for \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-for \u03c1 = \u22a4\n\u0394bag db is-valid-for \u03c1 = \u22a4\n\u0394var x is-valid-for \u03c1 = \u22a4\n\n_is-valid-for_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-for_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-for \u03c1 = Quadruple\n  (ds is-valid-for \u03c1)\n  (\u03bb _ \u2192 dt is-valid-for \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2081 s db is-valid-for \u03c1 = couple\n  (db is-valid-for \u03c1)\n  (Honest \u03c1 (FV s))\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 (cons v-dt honesty _ _) =\n  mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-for \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-for \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- Corollary: (f \u2295 df) (v \u2295 dv) = f v \u2295 df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore \u03c1) \u2295 \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s))\n    (\u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n  \u2261  \u27e6 s \u27e7 (ignore \u03c1) (\u27e6 t \u27e7 (ignore \u03c1)) \u2295\n     \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s) (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t})\n\ncorollary s t {\u03c1} = proj\u2082\n  (validity {t = s} (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t}))\n\nunrestricted (nat n) = tt\nunrestricted (bag b) = tt\nunrestricted (var x) {\u03c1} = tt\nunrestricted (abs t) {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted t)\nunrestricted (app s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t))\n  (validity {t = t}) tt\nunrestricted (add s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\nunrestricted (map s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted t)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0a23f43324de7047f601450cf6e3c023a7f6ce68","subject":"Updated doc for an unproven conjecture.","message":"Updated doc for an unproven conjecture.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Properties\/UnprovedATP.agda","new_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Properties\/UnprovedATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Unproven PA properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Mendelson.Properties.UnprovedATP where\n\nopen import PA.Axiomatic.Mendelson.Base\nopen import PA.Axiomatic.Mendelson.PropertiesATP\n\n------------------------------------------------------------------------------\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2248 m + (n + o)\n+-asocc m n o = S\u2089 A A0 is m\n  where\n  A : \u2115 \u2192 Set\n  A i = i + n + o \u2248 i + (n + o)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-leftCong #-}\n\n  -- 25 November 2013: Vampire 0.6 proves the theorem using a time out\n  -- of (300 sec).\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is +-leftCong #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Unproven PA properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Mendelson.Properties.UnprovedATP where\n\nopen import PA.Axiomatic.Mendelson.Base\nopen import PA.Axiomatic.Mendelson.PropertiesATP\n\n------------------------------------------------------------------------------\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2248 m + (n + o)\n+-asocc m n o = S\u2089 A A0 is m\n  where\n  A : \u2115 \u2192 Set\n  A i = i + n + o \u2248 i + (n + o)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-leftCong #-}\n\n  -- 31 July 2013: The ATPs could not prove the theorem (240 sec).\n  --\n  -- After the addition of the inequality _\u2249_, no ATP proves the\n  -- theorem. Before it, only Equinox 5.0alpha (2010-06-29) had proved\n  -- the theorem.\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is +-leftCong #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"61e9334f7220a5789eb6f91603f319d08db77ec7","subject":"Agda 2.4.2 regression? (see issue 1264)","message":"Agda 2.4.2 regression? (see issue 1264)\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/ABP-SL.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/ABP-SL.agda","new_contents":"------------------------------------------------------------------------------\n-- The ABP using Agda standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.ABP-SL where\n\nopen import Coinduction\nopen import Data.Bool\nopen import Data.Product\nopen import Data.Stream\nopen import Relation.Nullary\n\n------------------------------------------------------------------------------\n\nBit : Set\nBit = Bool\n\n-- Data type used to model the fair unreliable transmission channel.\ndata Err (A : Set) : Set where\n  ok    : (x : A) \u2192 Err A\n  error : Err A\n\n-- The mutual sender functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nsendA  : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bool)\nawaitA : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bit)\n\nsendA b (i \u2237 is) ds = (i , b) \u2237 \u266f awaitA b (i \u2237 is) ds\n\nawaitA b (i \u2237 is) (ok b' \u2237 ds) with b \u225f b'\n... | yes p = sendA (not b) (\u266d is) (\u266d ds)\n... | no \u00acp = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\nawaitA b (i \u2237 is) (error \u2237 ds) = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\n\n-- The receiver functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nackA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream Bit\nackA b (ok (_ , b') \u2237 bs) with b \u225f b'\n... | yes p = b \u2237 \u266f (ackA (not b) (\u266d bs))\n... | no \u00acp = not b \u2237 \u266f (ackA b (\u266d bs))\nackA b (error \u2237 bs) = not b \u2237 \u266f (ackA b (\u266d bs))\n\n{-# NO_TERMINATION_CHECK #-}\noutA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream A\noutA b (ok (i , b') \u2237 bs) with b \u225f b'\n... | yes p = i \u2237 \u266f (outA (not b) (\u266d bs))\n... | no \u00acp = outA b (\u266d bs)\noutA b (error \u2237 bs) = outA b (\u266d bs)\n\n-- Model the fair unreliable tranmission channel.\n--\n-- 29 August 2014 Requires the non-termination flag when using\n-- --without-K after the release of Agda 2.4.2. See issue 1264.\n{-# NO_TERMINATION_CHECK #-}\ncorruptA : {A : Set} \u2192 Stream Bit \u2192 Stream A \u2192 Stream (Err A)\ncorruptA (true \u2237 os)  (_ \u2237 xs) = error \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\ncorruptA (false \u2237 os) (x \u2237 xs) = ok x \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\n\n-- The ABP transfer function.\n{-# NO_TERMINATION_CHECK #-}\nabpTransA : {A : Set} \u2192 Bit \u2192 Stream Bit \u2192 Stream Bit \u2192 Stream A \u2192 Stream A\nabpTransA {A} b os\u2081 os\u2082 is = outA b bs\n  where\n  as : Stream (A \u00d7 Bit)\n  bs : Stream (Err (A \u00d7 Bit))\n  cs : Stream Bit\n  ds : Stream (Err Bit)\n\n  as = sendA b is ds\n  bs = corruptA os\u2081 as\n  cs = ackA b bs\n  ds = corruptA os\u2082 cs\n","old_contents":"------------------------------------------------------------------------------\n-- The ABP using Agda standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.ABP-SL where\n\nopen import Coinduction\nopen import Data.Bool\nopen import Data.Product\nopen import Data.Stream\nopen import Relation.Nullary\n\n------------------------------------------------------------------------------\n\nBit : Set\nBit = Bool\n\n-- Data type used to model the fair unreliable transmission channel.\ndata Err (A : Set) : Set where\n  ok    : (x : A) \u2192 Err A\n  error : Err A\n\n-- The mutual sender functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nsendA  : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bool)\nawaitA : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bit)\n\nsendA b (i \u2237 is) ds = (i , b) \u2237 \u266f awaitA b (i \u2237 is) ds\n\nawaitA b (i \u2237 is) (ok b' \u2237 ds) with b \u225f b'\n... | yes p = sendA (not b) (\u266d is) (\u266d ds)\n... | no \u00acp = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\nawaitA b (i \u2237 is) (error \u2237 ds) = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\n\n-- The receiver functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nackA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream Bit\nackA b (ok (_ , b') \u2237 bs) with b \u225f b'\n... | yes p = b \u2237 \u266f (ackA (not b) (\u266d bs))\n... | no \u00acp = not b \u2237 \u266f (ackA b (\u266d bs))\nackA b (error \u2237 bs) = not b \u2237 \u266f (ackA b (\u266d bs))\n\n{-# NO_TERMINATION_CHECK #-}\noutA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream A\noutA b (ok (i , b') \u2237 bs) with b \u225f b'\n... | yes p = i \u2237 \u266f (outA (not b) (\u266d bs))\n... | no \u00acp = outA b (\u266d bs)\noutA b (error \u2237 bs) = outA b (\u266d bs)\n\n-- Model the fair unreliable tranmission channel.\ncorruptA : {A : Set} \u2192 Stream Bit \u2192 Stream A \u2192 Stream (Err A)\ncorruptA (true \u2237 os)  (_ \u2237 xs) = error \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\ncorruptA (false \u2237 os) (x \u2237 xs) = ok x \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\n\n-- The ABP transfer function.\n{-# NO_TERMINATION_CHECK #-}\nabpTransA : {A : Set} \u2192 Bit \u2192 Stream Bit \u2192 Stream Bit \u2192 Stream A \u2192 Stream A\nabpTransA {A} b os\u2081 os\u2082 is = outA b bs\n  where\n  as : Stream (A \u00d7 Bit)\n  bs : Stream (Err (A \u00d7 Bit))\n  cs : Stream Bit\n  ds : Stream (Err Bit)\n\n  as = sendA b is ds\n  bs = corruptA os\u2081 as\n  cs = ackA b bs\n  ds = corruptA os\u2082 cs\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"faa65e48031abe5b7a8ce7be25f4151cfb5f80a8","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 2c3aec9f499291f0b940d407c92b0b57\n\ndarcs-hash:20110404171438-3bd4e-6e04071ad7513f0de4e7893d81c5d1aec08974a5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/old\/LTC.agda","new_file":"Draft\/old\/LTC.agda","new_contents":"module LTC where\n\nmodule Core where\n\n{-\nAgda as a logical framework for LTC\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\n-- Fixity declarations (precedence level and associativity)\n-- Agda default: infix 20\n\ninfixl 80 _`_\ninfixl 60 _\u2227_\ninfixl 50 _\u2228_\ninfix  40 if#_then_else_\ninfix  30 _==_\n\n-------------------------------------------------------------------------\n-- Equality: identity type\n-------------------------------------------------------------------------\n\n-- The identity type\ndata _==_ {A : Set} : A -> A -> Set where\n  ==-refl : {x : A} -> x == x\n\n==-subst : {A : Set}(P : A -> Set){x y : A} -> x == y -> P x -> P y\n==-subst P ==-refl px = px\n\n==-sym : {A : Set} {x y : A} -> x == y ->  y == x\n==-sym ==-refl = ==-refl\n\n==-trans : {A : Set} {x y z : A} -> x == y -> y == z -> x == z\n==-trans ==-refl y==z  = y==z\n\n-------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n-------------------------------------------------------------------------\n\n-- The false type\ndata \u22a5 : Set where\n\n\u22a5-elim : {A : Set} -> \u22a5 -> A\n\u22a5-elim ()\n\n\u00ac : Set -> Set\n\u00ac A = A -> \u22a5\n\n-- The disjunction type ('\\vee', not 'v')\ndata _\u2228_ (A B : Set) : Set where\n  \u2228-il : A -> A \u2228 B\n  \u2228-ir : B -> A \u2228 B\n\n\u2228-elim : {A B C : Set} -> (A -> C) -> (B -> C) -> A \u2228 B -> C\n\u2228-elim f g (\u2228-il a) = f a\n\u2228-elim f g (\u2228-ir b) = g b\n\n-- The conjunction type\ndata _\u2227_ (A B : Set) : Set where\n  \u2227-i : A -> B -> A \u2227 B\n\n\u2227-fst : {A B : Set} -> A \u2227 B -> A\n\u2227-fst (\u2227-i a b) = a\n\n\u2227-snd : {A B : Set} -> A \u2227 B -> B\n\u2227-snd (\u2227-i a b) = b\n\n-- The existential quantifier type\ndata \u2203 (A : Set)(P : A -> Set) : Set where\n  \u2203-i : (witness : A)  -> P witness -> \u2203 A P\n\n\u2203-fst : {A : Set}{P : A -> Set} -> \u2203 A P -> A\n\u2203-fst (\u2203-i x px) = x\n\n\u2203-snd : {A : Set}{P : A -> Set} -> (x-px : \u2203 A P) -> P (\u2203-fst x-px)\n\u2203-snd (\u2203-i x px) = px\n\n-- The implication and the universal quantifier\n-- We use Agda (dependent) function type\n\n-------------------------------------------------------------------------\n--  The term language: postulates\n-------------------------------------------------------------------------\n\n{-\nD : a universal domain of terms\n\nPre-types (or weak types): Agda's simple type lambda calculus on 'D'\n\n              T ::= D | T -> T\n                t := x | \\x -> t | t t | c (constants)\n\n-}\n\n-------------------------\n-- The universal domain\n\npostulate D : Set\n\n--------------------\n-- Constants terms\n\npostulate\n\n  -- LTC booleans\n  true#          : D\n  false#         : D\n  if#_then_else_ : D -> D -> D -> D\n\n  -- LTC natural numbers\n  -- We will refer to these partial numbers as 'N#'\n  zero# : D\n  suc#  : D -> D\n  rec#  : D -> D -> (D -> D -> D) -> D\n\n  -- LTC abstraction and application\n  \u03bb   : (D -> D) -> D\n  -- Left associative aplication operator\n  _`_ : D -> D -> D\n\n----------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans\n  CB1 : (a : D){b : D} -> if# true# then a else b == a\n  CB2 : {a : D}(b : D) -> if# false# then a else b == b\n\n  -- Conversion rules for natural numbers\n  CN1 : (a : D){f : D -> D -> D} -> rec# zero# a f == a\n  CN2 : (a n : D)(f : D -> D -> D) ->\n        rec# (suc# n) a f == f n (rec# n a f)\n\n  -- Conversion rule for the abstraction and the application\n  beta : (f : D -> D)(a : D) -> (\u03bb f) ` a == f a\n\n--------------------------------------------------------------\n-- Inductive predicate for natural numbers : Inductive family\n--------------------------------------------------------------\n-- The inductive predicate 'N' represents the type of the natural\n-- numbers. They are a subset of 'D'.\n\n-- The natural numbers type\ndata N : D -> Set where\n  N-zero : N zero#\n  N-suc : {n : D} -> N n -> N (suc# n)\n\n-- Induction principle on 'N'  (elimination rule)\nN-ind : (P : D -> Set) ->\n        P zero# ->\n        ({n : D} -> N n -> P n -> P (suc#  n)) ->\n        {n : D} -> N n -> P n\nN-ind P p0 h N-zero     = p0\nN-ind P p0 h (N-suc Nn) = h Nn (N-ind P  p0  h Nn)\n\n\n\nmodule Recursive-Functions-LTC where\n\n  -- We define some primitive recursive functions via LTC terms\n\n  open Core\n\n  import Equality-Reasoning\n  open module ER-Recursive-Functions-LTC =\n              Equality-Reasoning (_==_ {D}) ==-refl ==-trans\n\n   ------------------------------------------------------------------------\n  -- Remark: We are using Agda's definitional equality '=' as\n  -- LTC's definitional equality\n\n  -- Fixity declarations (precedence level and associativity)\n  -- Agda default: infix 20\n\n  infixl 70 _*_\n  infixl 60 _+_ _-_\n\n  _+_ : D -> D -> D\n  m + n = rec# n m (\\x y -> suc# y)\n\n  -- recursion on the second argument\n  _*_ : D -> D -> D\n  m * n = rec# n zero# (\\x y -> y + n)\n\n  pred : D -> D\n  pred n = rec# n zero# (\\x y -> x)\n\n  -- m - 0        = m\n  -- m - (succ n) = pred (m - n)\n  _-_ : D -> D -> D\n  m - n = rec# n m (\\x y -> pred y)\n\n  or : D -> D -> D\n  or a b = if#  a then true# else b\n\n  isZero : D -> D\n  isZero n = rec# n true# (\\x y -> false#)\n\n  -- The function 'equi n' return a function which establish if an\n  -- argument 'm' is equal to 'n'\n  -- equi zero     = \\m -> isZero m\n  -- equi (succ n) = \\m -> case m of\n  --                         zero -> false\n  --                         (succ m') -> (equi n) m\n\n  equi : D -> D\n  equi n = rec# n  (\u03bb (\\m -> isZero m))\n               (\\x y -> \u03bb (\\m -> rec# m  false#  (\\m' z -> y ` m')))\n\n  -- equality on N#\n  eq : D -> D -> D\n  eq m n = (equi n) ` m\n\n  -- inequality on N#\n  -- lt m 0        = false#\n  -- lt m (succ n) = (lt m n) \u2228 (m = n)\n  lt : D -> D -> D\n  lt m n = rec# n false# (\\x y -> or y  (eq m  x))\n\n  -- inequality on N#\n  gt : D -> D -> D\n  gt m n = lt n m\n","old_contents":"module LTC where\n\n\nmodule Core where\n\n{-\nAgda as a logical framework for LTC\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\n\n-- Fixity declarations (precedence level and associativity)\n-- Agda default: infix 20\n\ninfixl 80 _`_\ninfixl 60 _\u2227_\ninfixl 50 _\u2228_\ninfix  40 if#_then_else_\ninfix  30 _==_\n\n-------------------------------------------------------------------------\n-- Equality: identity type\n-------------------------------------------------------------------------\n\n-- The identity type\ndata _==_ {A : Set} : A -> A -> Set where\n  ==-refl : {x : A} -> x == x\n\n==-subst : {A : Set}(P : A -> Set){x y : A} -> x == y -> P x -> P y\n==-subst P ==-refl px = px\n\n==-sym : {A : Set} {x y : A} -> x == y ->  y == x\n==-sym ==-refl = ==-refl\n\n==-trans : {A : Set} {x y z : A} -> x == y -> y == z -> x == z\n==-trans ==-refl y==z  = y==z\n\n-------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n-------------------------------------------------------------------------\n\n-- The false type\ndata \u22a5 : Set where\n\n\u22a5-elim : {A : Set} -> \u22a5 -> A\n\u22a5-elim ()\n\n\u00ac : Set -> Set\n\u00ac A = A -> \u22a5\n\n-- The disjunction type ('\\vee', not 'v')\ndata _\u2228_ (A B : Set) : Set where\n  \u2228-il : A -> A \u2228 B\n  \u2228-ir : B -> A \u2228 B\n\n\u2228-elim : {A B C : Set} -> (A -> C) -> (B -> C) -> A \u2228 B -> C\n\u2228-elim f g (\u2228-il a) = f a\n\u2228-elim f g (\u2228-ir b) = g b\n\n-- The conjunction type\ndata _\u2227_ (A B : Set) : Set where\n  \u2227-i : A -> B -> A \u2227 B\n\n\u2227-fst : {A B : Set} -> A \u2227 B -> A\n\u2227-fst (\u2227-i a b) = a\n\n\u2227-snd : {A B : Set} -> A \u2227 B -> B\n\u2227-snd (\u2227-i a b) = b\n\n-- The existential quantifier type\ndata \u2203 (A : Set)(P : A -> Set) : Set where\n  \u2203-i : (witness : A)  -> P witness -> \u2203 A P\n\n\u2203-fst : {A : Set}{P : A -> Set} -> \u2203 A P -> A\n\u2203-fst (\u2203-i x px) = x\n\n\u2203-snd : {A : Set}{P : A -> Set} -> (x-px : \u2203 A P) -> P (\u2203-fst x-px)\n\u2203-snd (\u2203-i x px) = px\n\n-- The implication and the universal quantifier\n-- We use Agda (dependent) function type\n\n-------------------------------------------------------------------------\n--  The term language: postulates\n-------------------------------------------------------------------------\n\n{-\nD : a universal domain of terms\n\nPre-types (or weak types): Agda's simple type lambda calculus on 'D'\n\n              T ::= D | T -> T\n                t := x | \\x -> t | t t | c (constants)\n\n-}\n\n-------------------------\n-- The universal domain\n\npostulate D : Set\n\n--------------------\n-- Constants terms\n\npostulate\n\n  -- LTC booleans\n  true#          : D\n  false#         : D\n  if#_then_else_ : D -> D -> D -> D\n\n  -- LTC natural numbers\n  -- We will refer to these partial numbers as 'N#'\n  zero# : D\n  suc#  : D -> D\n  rec#  : D -> D -> (D -> D -> D) -> D\n\n  -- LTC abstraction and application\n  \u03bb   : (D -> D) -> D\n  -- Left associative aplication operator\n  _`_ : D -> D -> D\n\n----------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans\n  CB1 : (a : D){b : D} -> if# true# then a else b == a\n  CB2 : {a : D}(b : D) -> if# false# then a else b == b\n\n  -- Conversion rules for natural numbers\n  CN1 : (a : D){f : D -> D -> D} -> rec# zero# a f == a\n  CN2 : (a n : D)(f : D -> D -> D) ->\n        rec# (suc# n) a f == f n (rec# n a f)\n\n  -- Conversion rule for the abstraction and the application\n  beta : (f : D -> D)(a : D) -> (\u03bb f) ` a == f a\n\n\n--------------------------------------------------------------\n-- Inductive predicate for natural numbers : Inductive family\n--------------------------------------------------------------\n-- The inductive predicate 'N' represents the type of the natural\n-- numbers. They are a subset of 'D'.\n\n-- The natural numbers type\ndata N : D -> Set where\n  N-zero : N zero#\n  N-suc : {n : D} -> N n -> N (suc# n)\n\n-- Induction principle on 'N'  (elimination rule)\nN-ind : (P : D -> Set) ->\n        P zero# ->\n        ({n : D} -> N n -> P n -> P (suc#  n)) ->\n        {n : D} -> N n -> P n\nN-ind P p0 h N-zero     = p0\nN-ind P p0 h (N-suc Nn) = h Nn (N-ind P  p0  h Nn)\n\n\n\nmodule Recursive-Functions-LTC where\n\n  -- We define some primitive recursive functions via LTC terms\n\n  open Core\n\n  import Equality-Reasoning\n  open module ER-Recursive-Functions-LTC =\n              Equality-Reasoning (_==_ {D}) ==-refl ==-trans\n\n\n   ------------------------------------------------------------------------\n  -- Remark: We are using Agda's definitional equality '=' as\n  -- LTC's definitional equality\n\n  -- Fixity declarations (precedence level and associativity)\n  -- Agda default: infix 20\n\n  infixl 70 _*_\n  infixl 60 _+_ _-_\n\n\n  _+_ : D -> D -> D\n  m + n = rec# n m (\\x y -> suc# y)\n\n  -- recursion on the second argument\n  _*_ : D -> D -> D\n  m * n = rec# n zero# (\\x y -> y + n)\n\n  pred : D -> D\n  pred n = rec# n zero# (\\x y -> x)\n\n\n  -- m - 0        = m\n  -- m - (succ n) = pred (m - n)\n  _-_ : D -> D -> D\n  m - n = rec# n m (\\x y -> pred y)\n\n  or : D -> D -> D\n  or a b = if#  a then true# else b\n\n  isZero : D -> D\n  isZero n = rec# n true# (\\x y -> false#)\n\n\n  -- The function 'equi n' return a function which establish if an\n  -- argument 'm' is equal to 'n'\n  -- equi zero     = \\m -> isZero m\n  -- equi (succ n) = \\m -> case m of\n  --                         zero -> false\n  --                         (succ m') -> (equi n) m\n\n  equi : D -> D\n  equi n = rec# n  (\u03bb (\\m -> isZero m))\n               (\\x y -> \u03bb (\\m -> rec# m  false#  (\\m' z -> y ` m')))\n\n\n  -- equality on N#\n  eq : D -> D -> D\n  eq m n = (equi n) ` m\n\n  -- inequality on N#\n  -- lt m 0        = false#\n  -- lt m (succ n) = (lt m n) \u2228 (m = n)\n  lt : D -> D -> D\n  lt m n = rec# n false# (\\x y -> or y  (eq m  x))\n\n  -- inequality on N#\n  gt : D -> D -> D\n  gt m n = lt n m\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e48ae532437abaf69855a88c32dbfa753f8dcb92","subject":"ZK.JSChecker: better error message while checking the type of the statement","message":"ZK.JSChecker: better error message while checking the type of the statement\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker.agda","new_file":"ZK\/JSChecker.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.ZqZp as ZqZp\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- TODO bug (undefined)!\nrecord ZK-chaum-pedersen-pok-elgamal-rnd {--(\u2124q \u2124p\u2605 : Set)--} : Set where\n  field\n    m c s : BigI {--\u2124q--}\n    g p q y \u03b1 \u03b2 A B : BigI --\u2124p\u2605\n\nzk-check-chaum-pedersen-pok-elgamal-rnd! : ZK-chaum-pedersen-pok-elgamal-rnd {-BigI BigI-} \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd! pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks!\n      >> check! \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check! \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    params = record\n      { primality-test-probability-bound = primality-test-probability-bound\n      ; min-bits-q = min-bits-q\n      ; min-bits-p = min-bits-p\n      ; qI = I.q\n      ; pI = I.p\n      ; gI = I.g\n      }\n    open module [\u2124q]\u2124p\u2605 = ZqZp params\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n    check! \"type of statement\" (typ === fromString cpt)\n                               (\u03bb _ \u2192 \"Expected type of statement: \" ++ cpt ++ \" not \" ++ toString typ)\n >> zk-check-chaum-pedersen-pok-elgamal-rnd! pok\n module Zk-check where\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.ZqZp as ZqZp\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- TODO bug (undefined)!\nrecord ZK-chaum-pedersen-pok-elgamal-rnd {--(\u2124q \u2124p\u2605 : Set)--} : Set where\n  field\n    m c s : BigI {--\u2124q--}\n    g p q y \u03b1 \u03b2 A B : BigI --\u2124p\u2605\n\nzk-check-chaum-pedersen-pok-elgamal-rnd! : ZK-chaum-pedersen-pok-elgamal-rnd {-BigI BigI-} \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd! pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks!\n      >> check! \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check! \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    params = record\n      { primality-test-probability-bound = primality-test-probability-bound\n      ; min-bits-q = min-bits-q\n      ; min-bits-p = min-bits-p\n      ; qI = I.q\n      ; pI = I.p\n      ; gI = I.g\n      }\n    open module [\u2124q]\u2124p\u2605 = ZqZp params\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n    check! \"type of statement\" (typ === fromString \"chaum-pedersen-pok-elgamal-rnd\")\n                               (\u03bb _ \u2192 \"\")\n >> zk-check-chaum-pedersen-pok-elgamal-rnd! pok\n module Zk-check where\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"080a08a480530a0bfbe7ee42fb486407caea4a5e","subject":"Remove extra imports","message":"Remove extra imports\n\nOld-commit-hash: 3a9775f6afd97585f6b425390b5431b43807b03e\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Equivalence.agda","new_file":"Parametric\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Parametric.Change.Equivalence where\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {\u2113} {A} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely. That should be true for functions\n    -- using changes parametrically, for derivatives and function changes, and\n    -- for functions using only the interface to changes (including the fact\n    -- that function changes are functions). Stating the general result, though,\n    -- seems hard, we should rather have lemmas proving that certain classes of\n    -- functions respect this equivalence.\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Equivalence\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {\u2113} {A} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely. That should be true for functions\n    -- using changes parametrically, for derivatives and function changes, and\n    -- for functions using only the interface to changes (including the fact\n    -- that function changes are functions). Stating the general result, though,\n    -- seems hard, we should rather have lemmas proving that certain classes of\n    -- functions respect this equivalence.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3409935502fb668aa913b86adee0b90b81efb211","subject":"Use standard syntax","message":"Use standard syntax\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\ntrue \u2228 _  = true\nfalse \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nfalse \u2227 _ = false\ntrue \u2227 b  = b\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[_] : {A : Set} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x == y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q == s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p == r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs ([ n ]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 []\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 [])) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (5 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (6 \u2237 3 \u2237 [])) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (6 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 []) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 []) (3 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 [])) \u2237\n  (model (10 \u2237 9 \u2237 []) (1 \u2237 [])) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 []) (4 \u2237 11 \u2237 8 \u2237 [])) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 []) (11 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 []) (11 \u2237 3 \u2237 [])) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 []) ([])) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 []) (12 \u2237 [])) \u2237 []\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"19fcdd904a770324b30cfe4d9f92a08b16860ac6","subject":"Bits: +and +\u2713-and +\u2713-and'","message":"Bits: +and +\u2713-and +\u2713-and'\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Bits where\n\nopen import Algebra\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Fin.NP using (Fin; zero; suc; inject\u2081; inject+; raise; Fin\u25b9\u2115)\nopen import Data.Vec.NP\nopen import Function.NP\nimport Data.List.NP as L\n\n-- Re-export some vector functions, maybe they should be given\n-- less generic types.\nopen Data.Vec.NP public using ([]; _\u2237_; _++_; head; tail; map; replicate; RewireTbl; rewire; rewireTbl; on\u1d62)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Notice that all empty vectors are the same, hence 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\npattern 0\u2237_ xs = 0\u2082 \u2237 xs\npattern 1\u2237_ xs = 1\u2082 \u2237 xs\n{-\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n-}\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n-- see Data.Bits.Properties\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n\u2295-group : \u2115 \u2192 Group \u2080 \u2080\n\u2295-group = LiftGroup.group Xor\u00b0.+-group\n\nmodule \u2295-Group  (n : \u2115) = Group (\u2295-group n)\nmodule \u2295-Monoid (n : \u2115) = Monoid (\u2295-Group.monoid n)\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 k {n} \u2192 Bits (n + k) \u2192 Bits k\nlsb _ {n} = drop n\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 map not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2237 1\u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = map 0\u2237_ bs ++ map 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\n\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_\u2228\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2228\u00b0_ f g x = f x \u2228 g x\n\n_\u2227\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2227\u00b0_ f g x = f x \u2227 g x\n\nnot\u00b0 : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\nnot\u00b0 f = not \u2218 f\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []      = 0\u2237 []\nsucBCarry (0\u2237 xs) = 0\u2237 sucBCarry xs\nsucBCarry (1\u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2237 xs) = 1\u2237 xs\n  sucRB (1\u2237 xs) = 0\u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []      = zero\ntoFin         (0\u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []      = zero\nBits\u25b9\u2115         (0\u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []       = head\nlookupTbl         (0\u2237 key) = lookupTbl key \u2218 take _\nlookupTbl {suc n} (1\u2237 key) = lookupTbl key \u2218 drop (2^ n)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_)\n                    ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nand : \u2200 {n} \u2192 Bits n \u2192 \ud835\udfda\nand = foldr _ _\u2227_ 1\u2082\n\n\u2713-and : \u2200 {n}{xs : Bits n} \u2192 (\u2200 l \u2192 \u2713(xs \u203c l)) \u2192 \u2713(and xs)\n\u2713-and {xs = []}     p = _\n\u2713-and {xs = x \u2237 xs} p = \u2713\u2227 (p zero) (\u2713-and (p \u2218 suc))\n\n\u2713-and' : \u2200 {n}{xs : Bits n} \u2192 \u2713(and xs) \u2192 \u2200 l \u2192 \u2713(xs \u203c l)\n\u2713-and' {xs = x \u2237 xs} e zero = \u2713\u2227\u2081 {x} e\n\u2713-and' {xs = x \u2237 xs} e (suc l) = \u2713-and' (\u2713\u2227\u2082 {x} e) l\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Bits where\n\nopen import Algebra\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Fin.NP using (Fin; zero; suc; inject\u2081; inject+; raise; Fin\u25b9\u2115)\nopen import Data.Vec.NP\nopen import Function.NP\nimport Data.List.NP as L\n\n-- Re-export some vector functions, maybe they should be given\n-- less generic types.\nopen Data.Vec.NP public using ([]; _\u2237_; _++_; head; tail; map; replicate; RewireTbl; rewire; rewireTbl; on\u1d62)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Notice that all empty vectors are the same, hence 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\npattern 0\u2237_ xs = 0\u2082 \u2237 xs\npattern 1\u2237_ xs = 1\u2082 \u2237 xs\n{-\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n-}\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n-- see Data.Bits.Properties\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n\u2295-group : \u2115 \u2192 Group \u2080 \u2080\n\u2295-group = LiftGroup.group Xor\u00b0.+-group\n\nmodule \u2295-Group  (n : \u2115) = Group (\u2295-group n)\nmodule \u2295-Monoid (n : \u2115) = Monoid (\u2295-Group.monoid n)\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 k {n} \u2192 Bits (n + k) \u2192 Bits k\nlsb _ {n} = drop n\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 map not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2237 1\u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = map 0\u2237_ bs ++ map 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\n\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_\u2228\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2228\u00b0_ f g x = f x \u2228 g x\n\n_\u2227\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2227\u00b0_ f g x = f x \u2227 g x\n\nnot\u00b0 : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\nnot\u00b0 f = not \u2218 f\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []      = 0\u2237 []\nsucBCarry (0\u2237 xs) = 0\u2237 sucBCarry xs\nsucBCarry (1\u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2237 xs) = 1\u2237 xs\n  sucRB (1\u2237 xs) = 0\u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []      = zero\ntoFin         (0\u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []      = zero\nBits\u25b9\u2115         (0\u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []       = head\nlookupTbl         (0\u2237 key) = lookupTbl key \u2218 take _\nlookupTbl {suc n} (1\u2237 key) = lookupTbl key \u2218 drop (2^ n)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_)\n                    ++ tblFromFun {n} (f \u2218 1\u2237_)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3bbe7c4df0212e945a979a2c61dc0def85743b6e","subject":"Nicer statement for fromtoDeriveConst","message":"Nicer statement for fromtoDeriveConst\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/DeriveCorrect.agda","new_file":"Thesis\/DeriveCorrect.agda","new_contents":"module Thesis.DeriveCorrect where\n\nopen import Thesis.Lang\nopen import Thesis.Changes\nopen import Thesis.LangChanges\nopen import Thesis.Derive\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Theorem.Groups-Nehemiah\n\nfromtoDeriveConst : \u2200 {\u03c4 : Type} (c : Const \u03c4) \u2192\n  ch \u27e6 c \u27e7\u0394Const from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = right-id-int n\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db})\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb | sym (-m\u00b7-n=-mn {b1} {db}) = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db})\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = sft\u2081 daa\nfromtoDeriveConst rinj db b1 b2 dbb = sft\u2082 dbb\nfromtoDeriveConst match .(inj\u2081 (inj\u2081 _)) .(inj\u2081 _) .(inj\u2081 _) (sft\u2081 daa) df f1 f2 dff dg g1 g2 dgg = dff _ _ _ daa\nfromtoDeriveConst match .(inj\u2081 (inj\u2082 _)) .(inj\u2082 _) .(inj\u2082 _) (sft\u2082 dbb) df f1 f2 dff dg g1 g2 dgg = dgg _ _ _ dbb\nfromtoDeriveConst match .(inj\u2082 (inj\u2082 b2)) .(inj\u2081 a1) .(inj\u2082 b2) (sftrp\u2081 a1 b2) df f1 f2 dff dg g1 g2 dgg\n rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2\n | sym (fromto\u2192\u2295 dg g1 g2 dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match .(inj\u2082 (inj\u2081 a2)) .(inj\u2082 b1) .(inj\u2081 a2) (sftrp\u2082 b1 a2) df f1 f2 dff dg g1 g2 dgg\n  rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2\n  | sym (fromto\u2192\u2295 df f1 f2 dff)\n   = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Getting to the original equation 1 from PLDI'14.\n\ncorrectDeriveOplus : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 t \u27e7Term \u03c11) \u2295 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus \u03c4 t d\u03c1\u03c1 = fromto\u2192\u2295 _ _ _ (fromtoDerive \u03c4 t d\u03c1\u03c1)\n\nopen import Thesis.LangOps\n\ncorrectDeriveOplus\u03c4 : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4)\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit t) (derive t) \u27e7Term d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus\u03c4 \u03c4 t {d\u03c1 = d\u03c1} {\u03c11 = \u03c11} d\u03c1\u03c1\n  rewrite oplus\u03c4-equiv _ d\u03c1 _ (\u27e6 fit t \u27e7Term d\u03c1) (\u27e6 derive t \u27e7Term d\u03c1)\n  | sym (fit-sound t d\u03c1\u03c1)\n  = correctDeriveOplus \u03c4 t d\u03c1\u03c1\n\nderiveGivesDerivative : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3)\u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app f a \u27e7Term \u03c11) \u2295 (\u27e6 app f a \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus \u03c4 (app f a) d\u03c1\u03c1\n\nderiveGivesDerivative\u2082 : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) (derive a)) \u27e7Term d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative\u2082 \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus\u03c4 \u03c4 (app f a) d\u03c1\u03c1\n\n-- Proof of the original equation 1 from PLDI'14. The original was restricted to\n-- closed terms. This is a generalization, because it holds also for open terms,\n-- *as long as* the environment change is a nil change.\neq1 : \u2200 {\u0393} \u03c3 \u03c4 \u2192\n  {nil\u03c1 : Ch\u0393 \u0393} {\u03c1 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 nil\u03c1 from \u03c1 to \u03c1 \u2192\n  \u2200 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) (da : Term (\u0394\u0393 \u0393) (\u0394t \u03c3)) \u2192\n  (daa : [ \u03c3 ]\u03c4 (\u27e6 da \u27e7Term nil\u03c1) from (\u27e6 a \u27e7Term \u03c1) to (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1)) \u2192\n    \u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1 \u2261 \u27e6 app (fit f) (app\u2082 (oplus\u03c4o \u03c3) (fit a) da) \u27e7Term nil\u03c1\neq1 \u03c3 \u03c4 {nil\u03c1} {\u03c1} d\u03c1\u03c1 f a da daa\n  rewrite\n    oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit (app f a) \u27e7Term nil\u03c1) (\u27e6 (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1)\n  | sym (fit-sound f d\u03c1\u03c1)\n  | oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit a \u27e7Term nil\u03c1) (\u27e6 da \u27e7Term nil\u03c1)\n  | sym (fit-sound a d\u03c1\u03c1)\n  = fromto\u2192\u2295 (\u27e6 f \u27e7\u0394Term \u03c1 nil\u03c1 (\u27e6 a \u27e7Term \u03c1) (\u27e6 da \u27e7Term nil\u03c1)) _ _\n    (fromtoDerive _ f d\u03c1\u03c1 (\u27e6 da \u27e7Term nil\u03c1) (\u27e6 a \u27e7Term \u03c1)\n      (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1) daa)\n","old_contents":"module Thesis.DeriveCorrect where\n\nopen import Thesis.Lang\nopen import Thesis.Changes\nopen import Thesis.LangChanges\nopen import Thesis.Derive\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Theorem.Groups-Nehemiah\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ]\u03c4 \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = right-id-int n\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db})\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb | sym (-m\u00b7-n=-mn {b1} {db}) = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db})\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = sft\u2081 daa\nfromtoDeriveConst rinj db b1 b2 dbb = sft\u2082 dbb\nfromtoDeriveConst match .(inj\u2081 (inj\u2081 _)) .(inj\u2081 _) .(inj\u2081 _) (sft\u2081 daa) df f1 f2 dff dg g1 g2 dgg = dff _ _ _ daa\nfromtoDeriveConst match .(inj\u2081 (inj\u2082 _)) .(inj\u2082 _) .(inj\u2082 _) (sft\u2082 dbb) df f1 f2 dff dg g1 g2 dgg = dgg _ _ _ dbb\nfromtoDeriveConst match .(inj\u2082 (inj\u2082 b2)) .(inj\u2081 a1) .(inj\u2082 b2) (sftrp\u2081 a1 b2) df f1 f2 dff dg g1 g2 dgg\n rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2\n | sym (fromto\u2192\u2295 dg g1 g2 dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match .(inj\u2082 (inj\u2081 a2)) .(inj\u2082 b1) .(inj\u2081 a2) (sftrp\u2082 b1 a2) df f1 f2 dff dg g1 g2 dgg\n  rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2\n  | sym (fromto\u2192\u2295 df f1 f2 dff)\n   = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Getting to the original equation 1 from PLDI'14.\n\ncorrectDeriveOplus : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 t \u27e7Term \u03c11) \u2295 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus \u03c4 t d\u03c1\u03c1 = fromto\u2192\u2295 _ _ _ (fromtoDerive \u03c4 t d\u03c1\u03c1)\n\nopen import Thesis.LangOps\n\ncorrectDeriveOplus\u03c4 : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4)\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit t) (derive t) \u27e7Term d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus\u03c4 \u03c4 t {d\u03c1 = d\u03c1} {\u03c11 = \u03c11} d\u03c1\u03c1\n  rewrite oplus\u03c4-equiv _ d\u03c1 _ (\u27e6 fit t \u27e7Term d\u03c1) (\u27e6 derive t \u27e7Term d\u03c1)\n  | sym (fit-sound t d\u03c1\u03c1)\n  = correctDeriveOplus \u03c4 t d\u03c1\u03c1\n\nderiveGivesDerivative : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3)\u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app f a \u27e7Term \u03c11) \u2295 (\u27e6 app f a \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus \u03c4 (app f a) d\u03c1\u03c1\n\nderiveGivesDerivative\u2082 : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) (derive a)) \u27e7Term d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative\u2082 \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus\u03c4 \u03c4 (app f a) d\u03c1\u03c1\n\n-- Proof of the original equation 1 from PLDI'14. The original was restricted to\n-- closed terms. This is a generalization, because it holds also for open terms,\n-- *as long as* the environment change is a nil change.\neq1 : \u2200 {\u0393} \u03c3 \u03c4 \u2192\n  {nil\u03c1 : Ch\u0393 \u0393} {\u03c1 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 nil\u03c1 from \u03c1 to \u03c1 \u2192\n  \u2200 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) (da : Term (\u0394\u0393 \u0393) (\u0394t \u03c3)) \u2192\n  (daa : [ \u03c3 ]\u03c4 (\u27e6 da \u27e7Term nil\u03c1) from (\u27e6 a \u27e7Term \u03c1) to (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1)) \u2192\n    \u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1 \u2261 \u27e6 app (fit f) (app\u2082 (oplus\u03c4o \u03c3) (fit a) da) \u27e7Term nil\u03c1\neq1 \u03c3 \u03c4 {nil\u03c1} {\u03c1} d\u03c1\u03c1 f a da daa\n  rewrite\n    oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit (app f a) \u27e7Term nil\u03c1) (\u27e6 (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1)\n  | sym (fit-sound f d\u03c1\u03c1)\n  | oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit a \u27e7Term nil\u03c1) (\u27e6 da \u27e7Term nil\u03c1)\n  | sym (fit-sound a d\u03c1\u03c1)\n  = fromto\u2192\u2295 (\u27e6 f \u27e7\u0394Term \u03c1 nil\u03c1 (\u27e6 a \u27e7Term \u03c1) (\u27e6 da \u27e7Term nil\u03c1)) _ _\n    (fromtoDerive _ f d\u03c1\u03c1 (\u27e6 da \u27e7Term nil\u03c1) (\u27e6 a \u27e7Term \u03c1)\n      (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1) daa)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"66053c0d43dccf3dd648c77092869da45b9a0486","subject":"lam case","message":"lam case\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , {!!} -- ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc x\u2081) x\u2082) = {!!}\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EASubsume (\u03bb u \u2192 \u03bb ()) (\u03bb e' u \u2192 \u03bb ()) (ESLam x\u2082 D) x\u2083\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAHole x\u2081) x\u2082) = {!!}\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' = {!!} , {!!} , {!!} , {!!}\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , {!!} -- ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc x\u2081) x\u2082) = {!!}\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083) = _ , _ , _ , EASubsume {!!} {!!} (ESLam x\u2082 {!!}) x\u2083\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAHole x\u2081) x\u2082) = {!!}\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D'  = {!!}\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6d3a8e3bd26e9797ee4be36b510dac7615f13f71","subject":"Bits: to\u2115\/from\u2115","message":"Bits: to\u2115\/from\u2115\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} (f : Bits n \u2192 A) (g : Bij) \u2192 search {n} (f \u2218 eval g) \u2261 search {n} f\n        search-bij f `id     = refl\n        search-bij f `0\u21941    = search-0\u21941 f\n        search-bij f (g `\u204f h)\n          rewrite search-bij (f \u2218 eval h) g = search-bij f h\n        search-bij {zero}  f (_     `\u2237 _) = refl\n        search-bij {suc n} f (`id   `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 0b)\n                | search-bij (f \u2218 1\u2237_) (g 1b) = refl\n        search-bij {suc n} f (`not\u1d2e `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 1b)\n                | search-bij (f \u2218 1\u2237_) (g 0b) = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} (f : Bits n \u2192 Bit) (g : Bij) \u2192 #\u27e8 f \u2218 eval g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-bij f = sum-bij (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} (f : Bits n \u2192 A) (g : Bij) \u2192 search {n} (f \u2218 eval g) \u2261 search {n} f\n        search-bij f `id     = refl\n        search-bij f `0\u21941    = search-0\u21941 f\n        search-bij f (g `\u204f h)\n          rewrite search-bij (f \u2218 eval h) g = search-bij f h\n        search-bij {zero}  f (_     `\u2237 _) = refl\n        search-bij {suc n} f (`id   `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 0b)\n                | search-bij (f \u2218 1\u2237_) (g 1b) = refl\n        search-bij {suc n} f (`not\u1d2e `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 1b)\n                | search-bij (f \u2218 1\u2237_) (g 0b) = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} (f : Bits n \u2192 Bit) (g : Bij) \u2192 #\u27e8 f \u2218 eval g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-bij f = sum-bij (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p rewrite \u2115\u00b0.+-comm (2^ n) (to\u2115 ys) = \u22a5-elim (<\u2192\u2262 (\u2264-steps (to\u2115 ys) (to\u2115-bound xs)) p)\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p rewrite \u2115\u00b0.+-comm (2^ n) (to\u2115 xs) = \u22a5-elim (<\u2192\u2262 (\u2264-steps (to\u2115 xs) (to\u2115-bound ys)) (\u2261.sym p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"72b8ff768aa116dde7ff401c43f25629968483ed","subject":"Group.Homomorphism: typo, should have been commented","message":"Group.Homomorphism: typo, should have been commented\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_; id)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n\nopen GroupHomomorphism\n\nmodule Identity\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n\n  id-hom : GroupHomomorphism \ud835\udd38 \ud835\udd38 id\n  id-hom = mk refl\n\nmodule Compose\n  {a}{A : Type a}\n  {b}{B : Type b}\n  {c}{C : Type c}\n  (\ud835\udd38 : Group A)\n  (\ud835\udd39 : Group B)\n  (\u2102 : Group C)\n  (\u03c8 : A \u2192 B)\n  (\u03c8-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39 \u03c8)\n  (\u03c6 : B \u2192 C)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd39 \u2102 \u03c6)\n  where\n\n  \u2218-hom : GroupHomomorphism \ud835\udd38 \u2102 (\u03c6 \u2218 \u03c8)\n  \u2218-hom = mk (ap \u03c6 (hom \u03c8-hom) \u2219 hom \u03c6-hom)\n\nmodule Delta\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n  open Algebra.Group.Constructions.Product\n\n  \u0394-hom : GroupHomomorphism \ud835\udd38 (\u00d7-grp \ud835\udd38 \ud835\udd38) (\u03bb x \u2192 x , x)\n  \u0394-hom = mk refl\n\nmodule Zip\n  {a\u2080}{A\u2080 : Type a\u2080}\n  {a\u2081}{A\u2081 : Type a\u2081}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38\u2080 : Group A\u2080)\n  (\ud835\udd38\u2081 : Group A\u2081)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A\u2080 \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38\u2080 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A\u2081 \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38\u2081 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  open Algebra.Group.Constructions.Product\n\n  zip-hom : GroupHomomorphism (\u00d7-grp \ud835\udd38\u2080 \ud835\udd38\u2081) (\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) (map \u03c6\u2080 \u03c6\u2081)\n  zip-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n\nmodule Pair\n  {a}{A   : Type a}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38  : Group A)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n\n  -- pair = zip \u2218 \u0394\n  pair-hom : GroupHomomorphism \ud835\udd38 (Product.\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) < \u03c6\u2080 , \u03c6\u2081 >\n  pair-hom = Compose.\u2218-hom _ _ _\n               _ (Delta.\u0394-hom \ud835\udd38)\n               _ (Zip.zip-hom _ _ _ _ _ \u03c6\u2080-hom _ \u03c6\u2081-hom)\n  -- OR:\n  -- pair-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_; id)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n\nopen GroupHomomorphism\n\nmodule Identity\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n\n  id-hom : GroupHomomorphism \ud835\udd38 \ud835\udd38 id\n  id-hom = mk refl\n\nmodule Compose\n  {a}{A : Type a}\n  {b}{B : Type b}\n  {c}{C : Type c}\n  (\ud835\udd38 : Group A)\n  (\ud835\udd39 : Group B)\n  (\u2102 : Group C)\n  (\u03c8 : A \u2192 B)\n  (\u03c8-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39 \u03c8)\n  (\u03c6 : B \u2192 C)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd39 \u2102 \u03c6)\n  where\n\n  \u2218-hom : GroupHomomorphism \ud835\udd38 \u2102 (\u03c6 \u2218 \u03c8)\n  \u2218-hom = mk (ap \u03c6 (hom \u03c8-hom) \u2219 hom \u03c6-hom)\n\nmodule Delta\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n  open Algebra.Group.Constructions.Product\n\n  \u0394-hom : GroupHomomorphism \ud835\udd38 (\u00d7-grp \ud835\udd38 \ud835\udd38) (\u03bb x \u2192 x , x)\n  \u0394-hom = mk refl\n\nmodule Zip\n  {a\u2080}{A\u2080 : Type a\u2080}\n  {a\u2081}{A\u2081 : Type a\u2081}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38\u2080 : Group A\u2080)\n  (\ud835\udd38\u2081 : Group A\u2081)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A\u2080 \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38\u2080 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A\u2081 \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38\u2081 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  open Algebra.Group.Constructions.Product\n\n  zip-hom : GroupHomomorphism (\u00d7-grp \ud835\udd38\u2080 \ud835\udd38\u2081) (\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) (map \u03c6\u2080 \u03c6\u2081)\n  zip-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n\nmodule Pair\n  {a}{A   : Type a}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38  : Group A)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n\n  -- pair = zip \u2218 \u0394\n  pair-hom : GroupHomomorphism \ud835\udd38 (Product.\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) < \u03c6\u2080 , \u03c6\u2081 >\n  pair-hom = Compose.\u2218-hom _ _ _\n               _ (Delta.\u0394-hom \ud835\udd38)\n               _ (Zip.zip-hom _ _ _ _ _ \u03c6\u2080-hom _ \u03c6\u2081-hom)\n  -- OR:\n  pair-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"48e8cfa77c1c8a68a6beee0c3263161084c49025","subject":"Bits: lift the xor group structure to vectors","message":"Bits: lift the xor group structure to vectors\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Bits where\n\nopen import Algebra\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Fin.NP using (Fin; zero; suc; inject\u2081; inject+; raise; Fin\u25b9\u2115)\nopen import Data.Vec.NP\nopen import Function.NP\nimport Data.List.NP as L\n\n-- Re-export some vector functions, maybe they should be given\n-- less generic types.\nopen Data.Vec.NP public using ([]; _\u2237_; _++_; head; tail; map; replicate; RewireTbl; rewire; rewireTbl; on\u1d62)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Notice that all empty vectors are the same, hence 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n-- see Data.Bits.Properties\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n\u2295-group : \u2115 \u2192 Group \u2080 \u2080\n\u2295-group = LiftGroup.group Xor\u00b0.+-group\n\nmodule \u2295-Group  (n : \u2115) = Group (\u2295-group n)\nmodule \u2295-Monoid (n : \u2115) = Monoid (\u2295-Group.monoid n)\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 k {n} \u2192 Bits (n + k) \u2192 Bits k\nlsb _ {n} = drop n\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2082 \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2082 \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 map not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2082 \u2237 1\u2082 \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = map 0\u2237_ bs ++ map 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\n\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_\u2228\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2228\u00b0_ f g x = f x \u2228 g x\n\n_\u2227\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2227\u00b0_ f g x = f x \u2227 g x\n\nnot\u00b0 : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\nnot\u00b0 f = not \u2218 f\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0\u2082 \u2237 []\nsucBCarry (0\u2082 \u2237 xs) = 0\u2082 \u2237 sucBCarry xs\nsucBCarry (1\u2082 \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2082 \u2237 xs) = 1\u2082 \u2237 xs\n  sucRB (1\u2082 \u2237 xs) = 0\u2082 \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0\u2082 \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2082 \u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []        = zero\nBits\u25b9\u2115         (0\u2082 \u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0\u2082 \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1\u2082 \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Bits where\n\nopen import Type hiding (\u2605)\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Fin.NP using (Fin; zero; suc; inject\u2081; inject+; raise; Fin\u25b9\u2115)\nopen import Data.Vec.NP\nopen import Function.NP\nimport Data.List.NP as L\n\n-- Re-export some vector functions, maybe they should be given\n-- less generic types.\nopen Data.Vec.NP public using ([]; _\u2237_; _++_; head; tail; map; replicate; RewireTbl; rewire; rewireTbl; on\u1d62)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Notice that all empty vectors are the same, hence 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n-- see Data.Bits.Properties\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 k {n} \u2192 Bits (n + k) \u2192 Bits k\nlsb _ {n} = drop n\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2082 \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2082 \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 map not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2082 \u2237 1\u2082 \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = map 0\u2237_ bs ++ map 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\n\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_\u2228\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2228\u00b0_ f g x = f x \u2228 g x\n\n_\u2227\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2227\u00b0_ f g x = f x \u2227 g x\n\nnot\u00b0 : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\nnot\u00b0 f = not \u2218 f\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0\u2082 \u2237 []\nsucBCarry (0\u2082 \u2237 xs) = 0\u2082 \u2237 sucBCarry xs\nsucBCarry (1\u2082 \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2082 \u2237 xs) = 1\u2082 \u2237 xs\n  sucRB (1\u2082 \u2237 xs) = 0\u2082 \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0\u2082 \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2082 \u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []        = zero\nBits\u25b9\u2115         (0\u2082 \u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0\u2082 \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1\u2082 \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6b9e09214fc24fa6f8d233d537942f27cc0bb168","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Base.agda","new_file":"src\/fot\/FOTC\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\n-- Common definitions.\nopen import Common.DefinitionsATP public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | loop\n\npostulate\n  _\u00b7_                    : D \u2192 D \u2192 D  -- FOTC application.\n  true false if          : D          -- FOTC partial Booleans.\n  zero succ pred iszero  : D          -- FOTC partial natural numbers.\n  loop                   : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 n = succ \u00b7 n\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 n = pred \u00b7 n\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 n = iszero \u00b7 n\n  -- {-# ATP definition iszero\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as non-logical axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for Booleans.\n-- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n-- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\n-- N.B. We don't need this equation.\n-- pred-0 :       pred \u00b7 zero     \u2261 zero\n-- pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\npostulate\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\n-- iszero-0 :       iszero \u00b7 zero       \u2261 true\n-- iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\npostulate\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 n \u2192 iszero\u2081 (succ\u2081 n) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rule for loop.\n--\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a first-order logic axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\n--  0\u2262S : \u2200 {n} \u2192 zero \u2262 succ \u00b7 n\npostulate\n  true\u2262false : true \u2262 false\n  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ\u2081 n\n{-# ATP axiom true\u2262false 0\u2262S #-}\n\n------------------------------------------------------------------------------\n-- FOTC combinators for lists, colists, streams, etc.\n\nmodule BList where\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 8 _\u2237_\n\n  -- List constants.\n  postulate\n    [] cons head tail null : D  -- FOTC lists.\n\n  -- Definitions\n  abstract\n    _\u2237_  : D \u2192 D \u2192 D\n    x \u2237 xs = cons \u00b7 x \u00b7 xs\n    -- {-# ATP definition _\u2237_ #-}\n\n    head\u2081 : D \u2192 D\n    head\u2081 xs = head \u00b7 xs\n    -- {-# ATP definition head\u2081 #-}\n\n    tail\u2081 : D \u2192 D\n    tail\u2081 xs = tail \u00b7 xs\n    -- {-# ATP definition tail\u2081 #-}\n\n    null\u2081 : D \u2192 D\n    null\u2081 xs = null \u00b7 xs\n    -- {-# ATP definition null\u2081 #-}\n\n  -- Conversion rules for null.\n  -- null-[] :          null \u00b7 nil             \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  postulate\n    null-[] :          null\u2081 []       \u2261 true\n    null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n  -- Conversion rule for head.\n  -- head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  postulate head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n  {-# ATP axiom head-\u2237 #-}\n\n  -- Conversion rule for tail.\n  -- tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  postulate tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n  {-# ATP axiom tail-\u2237 #-}\n\n  -- Discrimination rules\n  -- postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 cons \u00b7 x \u00b7 xs\n  postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 x \u2237 xs\n","old_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\n-- Common definitions.\nopen import Common.DefinitionsATP public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | loop\n\npostulate\n  _\u00b7_                    : D \u2192 D \u2192 D  -- FOTC application.\n  true false if          : D          -- FOTC partial Booleans.\n  zero succ pred iszero  : D          -- FOTC partial natural numbers.\n  loop                   : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 n = succ \u00b7 n\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 n = pred \u00b7 n\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 n = iszero \u00b7 n\n  -- {-# ATP definition iszero\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as non-logical axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for Booleans.\npostulate\n  -- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n  -- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred \u00b7 zero     \u2261 zero\n  -- pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\npostulate\n  -- iszero-0 :       iszero \u00b7 zero       \u2261 true\n  -- iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 n \u2192 iszero\u2081 (succ\u2081 n) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rule for loop.\n--\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a first-order logic axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2262false : true \u2262 false\n--  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ \u00b7 n\n  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ\u2081 n\n{-# ATP axiom true\u2262false 0\u2262S #-}\n\n------------------------------------------------------------------------------\n-- FOTC combinators for lists, colists, streams, etc.\n\nmodule BList where\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 8 _\u2237_\n\n  -- List constants.\n  postulate\n    [] cons head tail null : D  -- FOTC lists.\n\n  -- Definitions\n  abstract\n    _\u2237_  : D \u2192 D \u2192 D\n    x \u2237 xs = cons \u00b7 x \u00b7 xs\n    -- {-# ATP definition _\u2237_ #-}\n\n    head\u2081 : D \u2192 D\n    head\u2081 xs = head \u00b7 xs\n    -- {-# ATP definition head\u2081 #-}\n\n    tail\u2081 : D \u2192 D\n    tail\u2081 xs = tail \u00b7 xs\n    -- {-# ATP definition tail\u2081 #-}\n\n    null\u2081 : D \u2192 D\n    null\u2081 xs = null \u00b7 xs\n    -- {-# ATP definition null\u2081 #-}\n\n  -- Conversion rules for null.\n  postulate\n    null-[] :          null\u2081 []       \u2261 true\n    null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n  -- Conversion rule for head.\n  postulate head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n  {-# ATP axiom head-\u2237 #-}\n\n  -- Conversion rule for tail.\n  postulate tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n  {-# ATP axiom tail-\u2237 #-}\n\n  -- Discrimination rules\n  postulate []\u2262cons    : \u2200 {x xs} \u2192 [] \u2262 x \u2237 xs\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cb42e120fdc89dd9482591952c126c8f09c37f02","subject":"formal paramater madness","message":"formal paramater madness\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\nopen import lemmas-matching\nopen import disjointness\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = _ , _ , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | _ , _ , \u03c4' , D  = _ , _ , ESAsc D\n    expandability-synth (SVar x) = _ , _ , ESVar x\n    expandability-synth (SAp dis wt1 m wt2)\n      with expandability-ana (ASubsume wt1 (match-consist m)) | expandability-ana wt2\n    ... | _ , _ , _ , D1 | _ , _ , _ , D2 = _ , _ , ESAp dis (expand-ana-disjoint dis D1 D2) wt1 m D1 D2\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole new wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole (expand-new-disjoint-synth new wt') wt'\n    expandability-synth (SLam x\u2081 wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081)\n      with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )                   | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {\u0393} {\u2987 e \u2988[ x ]} (ASubsume (SNEHole new wt) x\u2082) | _ , _ , ESNEHole x\u2081 D' with expandability-synth wt\n    ... | w , y , z =  _ , _ , _ , EANEHole (expand-new-disjoint-synth new z) z\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 m wt)\n      with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALam x\u2081 m D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\nopen import lemmas-matching\nopen import disjointness\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = _ , _ , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | _ , _ , \u03c4' , D  = _ , _ , ESAsc D\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp dis wt1 m wt2)\n      with expandability-ana (ASubsume wt1 (match-consist m)) | expandability-ana wt2\n    ... | _ , _ , _ , D1 | _ , _ , _ , D2 = _ , _ , ESAp dis (expand-ana-disjoint dis D1 D2) wt1 m D1 D2\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole new wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole (expand-new-disjoint-synth new wt') wt'\n    expandability-synth (SLam x\u2081 wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081)\n      with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )                   | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {\u0393} {\u2987 e \u2988[ x ]} (ASubsume (SNEHole new wt) x\u2082) | _ , _ , ESNEHole x\u2081 D' with expandability-synth wt\n    ... | w , y , z =  _ , _ , _ , EANEHole (expand-new-disjoint-synth new z) z\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 m wt)\n      with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALam x\u2081 m D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2691816104bfb57b0b4e5094a85c36377cf2cc33","subject":"now uses equational reasoning","message":"now uses equational reasoning\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat.agda","new_file":"agda\/Nat.agda","new_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\nimport Relation.Binary.EqReasoning as EqR\nopen module EqNat = EqR (setoid \u2115)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b) =\n  begin\n    a + succ b\n      \u2248\u27e8 lemma a b \u27e9\n    succ (a + b)\n      \u2248\u27e8 cong succ (+-is-commutative a b) \u27e9\n    succ (b + a)\n      \u2248\u27e8 refl \u27e9\n    succ b + a\n  \u220e\n","old_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b)\n  = trans (lemma a b) (cong succ (+-is-commutative a b))\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"e9b6ce771b34dca21826aef620fdab37447e6d80","subject":"Renamed one function.","message":"Renamed one function.\n\nIgnore-this: 9c78acc0d4b0f6406274ce4cd5708f45\n\ndarcs-hash:20110402162531-3bd4e-972702749229997e6b55c5c45134d470e56c6257.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Bool\/AndTotality.agda","new_file":"Draft\/FOTC\/Data\/Bool\/AndTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Bool.AndTotality where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Bool.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  thm : \u2200 {b}(P : D \u2192 Set) \u2192 (Bool b \u2227 P true \u2227 P false) \u2192 P b\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm #-}\n\npostulate\n  thm\u2081 : \u2200 {P : D \u2192 Set}{x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192 P (if x then y else z)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm\u2081 #-}\n\n-- Typing of the if-then-else.\nif-T : \u2200 (P : D \u2192 Set){x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192\n       P (if x then y else z)\nif-T P {y = y} tB Py Pz = subst P (sym (if-true y)) Py\nif-T P {z = z} fB Py Pz = subst P (sym (if-false z)) Pz\n\n_&&_ : D \u2192 D \u2192 D\nx && y = if x then y else false\n{-# ATP definition _&&_ #-}\n\npostulate\n  &&-Bool : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n{-# ATP prove &&-Bool if-T #-}\n\n&&-Bool\u2081 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2081 {y = y} tB By = prf\n  where\n    postulate prf : Bool (true && y)\n    {-# ATP prove prf if-T #-}\n&&-Bool\u2081 {y = y} fB By = prf\n  where\n    postulate prf : Bool (false && y)\n    {-# ATP prove prf if-T #-}\n\n&&-Bool\u2082 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2082 tB By = if-T Bool tB By fB\n&&-Bool\u2082 fB By = if-T Bool fB By fB\n","old_contents":"------------------------------------------------------------------------------\n-- Testing\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Bool.AndTotality where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Bool.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  thm : \u2200 {b}(P : D \u2192 Set) \u2192 (Bool b \u2227 P true \u2227 P false) \u2192 P b\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm #-}\n\npostulate\n  thm\u2081 : \u2200 {P : D \u2192 Set}{x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192 P (if x then y else z)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm\u2081 #-}\n\nBool-elim : \u2200 (P : D \u2192 Set){x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192\n            P (if x then y else z)\nBool-elim P {y = y} tB Py Pz = subst P (sym (if-true y)) Py\nBool-elim P {z = z} fB Py Pz = subst P (sym (if-false z)) Pz\n\n_&&_ : D \u2192 D \u2192 D\nx && y = if x then y else false\n{-# ATP definition _&&_ #-}\n\npostulate\n  &&-Bool : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n{-# ATP prove &&-Bool Bool-elim #-}\n\n&&-Bool\u2081 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2081 {y = y} tB By = prf\n  where\n    postulate prf : Bool (true && y)\n    {-# ATP prove prf Bool-elim #-}\n&&-Bool\u2081 {y = y} fB By = prf\n  where\n    postulate prf : Bool (false && y)\n    {-# ATP prove prf Bool-elim #-}\n\n&&-Bool\u2082 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2082 tB By = Bool-elim Bool tB By fB\n&&-Bool\u2082 fB By = Bool-elim Bool fB By fB\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"eb62ebad0cbcd277118fcfd907c7981c71725bdc","subject":"Minor changes.","message":"Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (n > [100] \u2227 mc91 n \u2261 n \u2238 [10]) \u2228\n                         (n \u226f [100] \u2227 mc91 n \u2261 [91])\nmc91-res = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = (d > [100] \u2227 mc91 d \u2261 d \u2238 [10]) \u2228\n        (d \u226f [100] \u2227 mc91 d \u2261 [91])\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq-aux m m>100 )\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-100 m\u2261100 )\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-99 m\u226199 )\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-98 m\u226198 )\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-97 m\u226197 )\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-96 m\u226196 )\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-95 m\u226195 )\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-94 m\u226194 )\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-93 m\u226193 )\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-92 m\u226192 )\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-91 m\u226191 )\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-90 m\u226190 )\n  ... | inj\u2081 m\u226f89 = inj\u2082 ( m\u226f100 , mc91-res-m\u226f89 )\n    where\n    m\u226489 : m \u2264 [89]\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : mc91 (m + [11]) \u2261 [91]\n    mc91-res-m+11 with f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n    ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm m\u226489 m+11>100)\n    ... | inj\u2082 ( _ , res ) = res\n\n    mc91-res-m\u226f89 : mc91 m \u2261 [91]\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m [91] m\u226f100 mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 n > [100] \u2192 mc91 n \u2261 n \u2238 [10]\nmc91-res>100 Nn n>100 with mc91-res Nn\n... | inj\u2081 ( _     , res ) = res\n... | inj\u2082 ( n\u226f100 , _ )   = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f [100] \u2192 mc91 n \u2261 [91]\nmc91-res\u226f100 Nn n\u226f100 with mc91-res Nn\n... | inj\u2081 ( n>100 , _   ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n... | inj\u2082 ( _     , res ) = res\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (mc91-res>100 Nn n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226e100 = subst N (sym (mc91-res\u226f100 Nn n\u226e100)) 91-N\n\n-- For all n, n < mc91 n + 11.\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 n < mc91 n + [11]\nmc91-N-ineq = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < mc91 d + [11]\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x<mc91x+11>100 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let mc91-m+11-N : N (mc91 (m + [11]))\n        mc91-m+11-N = mc91-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + [11])\n        h\u2081 = f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<mc91m+11 : m < mc91 (m + [11])\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm mc91-m+11-N 11-N h\u2081\n\n        h\u2082 : A (mc91 (m + [11]))\n        h\u2082 = f mc91-m+11-N (<\u2192\u226a mc91-m+11-N Nm m\u226f100 m<mc91m+11)\n\n    in ( <-trans Nm mc91-m+11-N (x+11-N (mc91-N Nm))\n                 m<mc91m+11\n                 (mc91x+11<mc91x+11 m m\u226f100 h\u2082))\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (n > [100] \u2227 mc91 n \u2261 n \u2238 [10]) \u2228\n                         (n \u226f [100] \u2227 mc91 n \u2261 [91])\nmc91-res = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = (d > [100] \u2227 mc91 d \u2261 d \u2238 [10]) \u2228\n        (d \u226f [100] \u2227 mc91 d \u2261 [91])\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq-aux m m>100 )\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-100 m\u2261100 )\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-99 m\u226199 )\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-98 m\u226198 )\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-97 m\u226197 )\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-96 m\u226196 )\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-95 m\u226195 )\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-94 m\u226194 )\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-93 m\u226193 )\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-92 m\u226192 )\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-91 m\u226191 )\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = inj\u2082 ( m\u226f100 , mc91-res-aux mc91-res-90 m\u226190 )\n  ... | inj\u2081 m\u226f89 = inj\u2082 ( m\u226f100 , mc91-res-m\u226f89 )\n    where\n    m\u226489 : m \u2264 [89]\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : mc91 (m + [11]) \u2261 [91]\n    mc91-res-m+11 with f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n    ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm m\u226489 m+11>100)\n    ... | inj\u2082 ( _ , res ) = res\n\n    mc91-res-m\u226f89 : mc91 m \u2261 [91]\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m [91] m\u226f100 mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 n > [100] \u2192 mc91 n \u2261 n \u2238 [10]\nmc91-res>100 Nn n>100 with mc91-res Nn\n... | inj\u2081 ( _     , res ) = res\n... | inj\u2082 ( n\u226f100 , _ )   = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f [100] \u2192 mc91 n \u2261 [91]\nmc91-res\u226f100 Nn n\u226f100 with mc91-res Nn\n... | inj\u2081 ( n>100 , _   ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n... | inj\u2082 ( _     , res ) = res\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (mc91-res>100 Nn n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226e100 = subst N (sym (mc91-res\u226f100 Nn n\u226e100)) 91-N\n\n-- For all n, n < mc91 n + 11.\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 n < mc91 n + [11]\nmc91-N-ineq = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < mc91 d + [11]\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x<mc91x+11>100 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let mc91-m-11-N : N (mc91 (m + [11]))\n        mc91-m-11-N = mc91-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + [11])\n        h\u2081 = f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<mc91m+11 : m < mc91 (m + [11])\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm mc91-m-11-N 11-N h\u2081\n\n        h\u2082 : A (mc91 (m + [11]))\n        h\u2082 = f mc91-m-11-N (<\u2192\u226a mc91-m-11-N Nm m\u226f100 m<mc91m+11)\n\n    in ( <-trans Nm mc91-m-11-N (x+11-N (mc91-N Nm))\n                 m<mc91m+11\n                 (mc91x+11<mc91x+11 m m\u226f100 h\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8b57da7d3b2e33729211adcd26f10bf967cdb24f","subject":"Lamport: more lemmas","message":"Lamport: more lemmas\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/Lamport.agda","new_file":"Crypto\/Sig\/Lamport.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH  = hash-secret\nH\u00b2 = map2* #secret #digest H\n\n#signkey1   = OTS1.#signkey\n#verifkey1  = OTS1.#verifkey\n#signature1 = OTS1.#signature\n#signkey    = #message * #signkey1\n#verifkey   = #message * #verifkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #verifkey1 vk\n\n  concat-vk1s : concat vk1s \u2261 vk\n  concat-vk1s = concat-group #message #verifkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #signkey1 sk\n  sc1s = map OTS1.signkey.skP sk1s\n  vk   = verif-key sk\n  open verifkey vk public\n  sc1s-sk1s : map (uncurry _++_) sc1s \u2261 sk1s\n  sc1s-sk1s = ! map-\u2218 (uncurry _++_) OTS1.signkey.skP sk1s\n            \u2219 map-id= (take-drop-lem #secret) sk1s\n  H-sc1s-sk1s : map (uncurry _++_ \u2218 \u00d7-map H H) sc1s \u2261 map H\u00b2 sk1s\n  H-sc1s-sk1s = ! map-\u2218= (map2*-++ H) sc1s \u2219 ap (map H\u00b2) sc1s-sk1s\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\n  concat-sig1s : concat sig1s \u2261 sig\n  concat-sig1s = concat-group #message #secret sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\n    group-sig : group #message #signature1 sig \u2261 sig1s\n    group-sig = group-concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and checks\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n    checks = map OTS1.verify vk1s \u229b m \u229b sig1s\n\nprivate\n  module lemma {#m} (sk : Bits (suc #m * #signkey1))\n                    (b : Bit) (m : Bits #m) where\n    skL  = take #signkey1 sk\n    skH  = drop #signkey1 sk\n    vkL  = OTS1.verif-key skL\n    vkH  = map* #m OTS1.verif-key skH\n    sigL = OTS1.sign skL b\n    sigH = concat (map OTS1.sign (group #m #signkey1 skH) \u229b m)\n\nverifkey-is-hash-sig : \u2200 sk m (open signkey sk)\n                       \u2192 map (take2* _) m \u229b vk1s\n                       \u2261 map H (map OTS1.sign sk1s \u229b m)\nverifkey-is-hash-sig sk m = lemma sk m\n  module verifkey-is-hash-sig where\n    lemma : \u2200 {#m} sk m \u2192 map (take2* _) m \u229b group #m #verifkey1 (map* #m OTS1.verif-key sk)\n                        \u2261 map hash-secret (map OTS1.sign (group #m #signkey1 sk) \u229b m)\n    lemma sk [] = refl\n    lemma {suc #m} sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signkey1 skL skH)\n            | (let open lemma sk b m in drop-++ #signkey1 skL skH)\n            | (let open lemma sk b m in lemma skH m)\n            | (let open lemma sk b m in OTS1.verifkey-is-hash-sig skL b)\n      = refl\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig sk m = lemma sk m\n  module verify-correct-sig where\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #verifkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                              (concat (map OTS1.sign (group #m #signkey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH = hash-secret\n\n#signkey1   = OTS1.#signkey\n#verifkey1  = OTS1.#verifkey\n#signature1 = OTS1.#signature\n#signkey    = #message * #signkey1\n#verifkey   = #message * #verifkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #verifkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #signkey1 sk\n  vk   = verif-key sk\n  open verifkey vk public\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and (map OTS1.verify vk1s \u229b m \u229b sig1s)\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig = lemma\n  where\n    module lemma {#m} sk b m where\n      skL  = take #signkey1 sk\n      skH  = drop #signkey1 sk\n      vkL  = OTS1.verif-key skL\n      vkH  = map* #m OTS1.verif-key skH\n      sigL = OTS1.sign skL b\n      sigH = concat (map OTS1.sign (group #m #signkey1 skH) \u229b m)\n\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #verifkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                               (concat (map OTS1.sign (group #m #signkey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b58eec073a5ee089dca5655b5c7fee8a6d586935","subject":"Removed unnecessary pragma and parenthesis.","message":"Removed unnecessary pragma and parenthesis.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 \u221e (Conat n) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\nConat-coind-helper :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (\u2200 {m} \u2192 A m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m')) \u2192\n  A n \u2192 Conat n\nConat-coind-helper A h An with h An\n... | inj\u2081 n\u22610 = subst Conat (sym n\u22610) cozero\n... | inj\u2082 (n' , prf , An') =\n  subst Conat (sym prf) (cosucc (\u266f (Conat-coind-helper A h An')))\n\n-- 07 January 2014. Agda bug? We couldn't directly prove Conat-coind\n-- because Agda's termination checker doesn't accept the proof, but it\n-- accepts the proof of Conat-coind-helper. The only difference is the\n-- *position* of the universal quantifier \u2200 {n}.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h An = Conat-coind-helper A h An\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221e.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\nConat-coind-helper :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (\u2200 {m} \u2192 A m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m')) \u2192\n  A n \u2192 Conat n\nConat-coind-helper A h An with h An\n... | inj\u2081 n\u22610 = subst Conat (sym n\u22610) cozero\n... | inj\u2082 (n' , prf , An') =\n  subst Conat (sym prf) (cosucc (\u266f (Conat-coind-helper A h An')))\n\n-- 07 January 2014. Agda bug? We couldn't directly prove Conat-coind\n-- because Agda's termination checker doesn't accept the proof, but it\n-- accepts the proof of Conat-coind-helper. The only difference is the\n-- *position* of the universal quantifier \u2200 {n}.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h An = Conat-coind-helper A h An\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221e.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2b90ef2d011bbca7a36332769057d69f72556b77","subject":"Added x\u2260Sx to inductive PA.","message":"Added x\u2260Sx to inductive PA.\n\nIgnore-this: 88c8bc1743b92688e1e57667a5472580\n\ndarcs-hash:20120303040024-3bd4e-93ad7b3dc9f315fc7d519e8f39c779b33c089493.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Inductive\/Properties.agda","new_file":"src\/PA\/Inductive\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Inductive PA properties commons to the interactive and automatic proofs\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Inductive.Properties where\n\nopen import PA.Inductive.Base\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nsucc-cong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsucc-cong = cong succ\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong h = cong\u2082 _+_ h refl\n\n-- TODO: It is strictly necessary the implicit argument m?\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong {m} h = cong\u2082 _+_ {m} refl h\n\n------------------------------------------------------------------------------\n-- Peano's third axiom.\nP\u2083 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\nP\u2083 refl = refl\n\n-- Peano's fourth axiom.\nP\u2084 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\nP\u2084 ()\n\nx\u2260Sx : \u2200 {n} \u2192 \u00ac (n \u2261 succ n)\nx\u2260Sx {zero} ()\nx\u2260Sx {succ n} h = x\u2260Sx (P\u2083 h)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = refl\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.n+0\u2261n).\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity zero     = refl\n+-rightIdentity (succ n) = succ-cong (+-rightIdentity n)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.+-assoc).\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zero     _ _ = refl\n+-assoc (succ m) n o = succ-cong (+-assoc m n o)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.m+1+n\u22611+m+n).\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] zero     _ = refl\nx+Sy\u2261S[x+y] (succ m) n = succ-cong (x+Sy\u2261S[x+y] m n)\n","old_contents":"------------------------------------------------------------------------------\n-- Inductive PA properties commons to the interactive and automatic proofs\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Inductive.Properties where\n\nopen import PA.Inductive.Base\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nsucc-cong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsucc-cong = cong succ\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong h = cong\u2082 _+_ h refl\n\n-- TODO: It is strictly necessary the implicit argument m?\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong {m} h = cong\u2082 _+_ {m} refl h\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\n-- Peano's third axiom.\nP\u2083 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\nP\u2083 refl = refl\n\n-- Peano's fourth axiom.\nP\u2084 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\nP\u2084 ()\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = refl\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.n+0\u2261n).\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity zero     = refl\n+-rightIdentity (succ n) = succ-cong (+-rightIdentity n)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.+-assoc).\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zero     _ _ = refl\n+-assoc (succ m) n o = succ-cong (+-assoc m n o)\n\n-- Adapted from the Agda standard library v0.6 (see\n-- Data.Nat.Properties.m+1+n\u22611+m+n).\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] zero     _ = refl\nx+Sy\u2261S[x+y] (succ m) n = succ-cong (x+Sy\u2261S[x+y] m n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"496f9d578fddd936fb140db38194fa4db26b8a90","subject":"Fixed a draft.","message":"Fixed a draft.\n\nIgnore-this: bf3af2a57414652c34cac96c640f6436\n\ndarcs-hash:20120224154216-3bd4e-beeeb1cbd8ae38e8c2f1b9ca78eff7e80ad53f12.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n-- Tested with magda and agda2atp on 24 February 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P i = N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n    ih {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : \u2200 x \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N (n + x) \u2192 N (succ\u2081 n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = N-ind (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 indN #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n-- Tested with magda and agda2atp on 15 December 2011.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n    ih {i} Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : \u2200 x \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N (n + x) \u2192 N (succ\u2081 n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 indN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f9a6f36f367c9feb3f563dd6a2d58f02b2a16ad3","subject":"Updated documentation related to the --no-internal-equality Apia option.","message":"Updated documentation related to the --no-internal-equality Apia option.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Base.agda","new_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- Mendelson's axioms for first-order Peano arithmetic [Mendelson,\n-- 1977, p. 155].\n\n-- Elliott Mendelson. Introduction to mathematical logic. Chapman &\n-- Hall, 4th edition, 1997, p. 155.\n\n-- NB. These axioms formalize the propositional equality.\n\nmodule PA.Axiomatic.Mendelson.Base where\n\ninfixl 7 _*_\ninfixl 6 _+_\ninfix  4 _\u2248_ _\u2249_\n\n------------------------------------------------------------------------------\n-- First-order logic (without equality)\nopen import Common.FOL.FOL public renaming ( D to \u2115 )\n\n-- Non-logical constants\npostulate\n  zero    : \u2115\n  succ    : \u2115 \u2192 \u2115\n  _+_ _*_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n--\n-- N.B. We use the symbol _\u2248_ for avoiding to use the\n-- --no-internal-equality Apia option.\npostulate _\u2248_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Inequality.\n_\u2249_ : \u2115 \u2192 \u2115 \u2192 Set\nx \u2249 y = \u00ac x \u2248 y\n{-# ATP definition _\u2249_ #-}\n\n-- Proper axioms\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + n\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2248 n \u2192 m \u2248 o \u2192 n \u2248 o\n  S\u2082 : \u2200 {m n} \u2192 m \u2248 n \u2192 succ m \u2248 succ n\n  S\u2083 : \u2200 {n} \u2192 zero \u2249 succ n\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2248 succ n \u2192 m \u2248 n\n  S\u2085 : \u2200 n \u2192 zero + n \u2248 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2248 succ (m + n)\n  S\u2087 : \u2200 n \u2192 zero * n \u2248 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2248 n + m * n\n{-# ATP axiom S\u2081 S\u2082 S\u2083 S\u2084 S\u2085 S\u2086 S\u2087 S\u2088 #-}\n\n-- S\u2089 is an axiom schema, therefore we do not translate it to TPTP.\npostulate S\u2089 : (A : \u2115 \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n","old_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- Mendelson's axioms for first-order Peano arithmetic [Mendelson,\n-- 1977, p. 155].\n\n-- Elliott Mendelson. Introduction to mathematical logic. Chapman &\n-- Hall, 4th edition, 1997, p. 155.\n\n-- NB. These axioms formalize the propositional equality.\n\nmodule PA.Axiomatic.Mendelson.Base where\n\ninfixl 7 _*_\ninfixl 6 _+_\ninfix  4 _\u2248_ _\u2249_\n\n------------------------------------------------------------------------------\n-- First-order logic (without equality)\nopen import Common.FOL.FOL public renaming ( D to \u2115 )\n\n-- Non-logical constants\npostulate\n  zero    : \u2115\n  succ    : \u2115 \u2192 \u2115\n  _+_ _*_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n--\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- Apia as the ATPs equality.\npostulate _\u2248_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Inequality.\n_\u2249_ : \u2115 \u2192 \u2115 \u2192 Set\nx \u2249 y = \u00ac x \u2248 y\n{-# ATP definition _\u2249_ #-}\n\n-- Proper axioms\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + n\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2248 n \u2192 m \u2248 o \u2192 n \u2248 o\n  S\u2082 : \u2200 {m n} \u2192 m \u2248 n \u2192 succ m \u2248 succ n\n  S\u2083 : \u2200 {n} \u2192 zero \u2249 succ n\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2248 succ n \u2192 m \u2248 n\n  S\u2085 : \u2200 n \u2192 zero + n \u2248 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2248 succ (m + n)\n  S\u2087 : \u2200 n \u2192 zero * n \u2248 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2248 n + m * n\n{-# ATP axiom S\u2081 S\u2082 S\u2083 S\u2084 S\u2085 S\u2086 S\u2087 S\u2088 #-}\n\n-- S\u2089 is an axiom schema, therefore we do not translate it to TPTP.\npostulate S\u2089 : (A : \u2115 \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f6ddd0ffec6818d212d8ab73605906eb5d61c997","subject":"Fixed a module name.","message":"Fixed a module name.\n\nIgnore-this: 2f8d900a7574dc6e4dc23bbdcfde511f\n\ndarcs-hash:20101011230518-3bd4e-e59e5b05c7009146e06969a61f639da8f6f7dc73.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Notes\/FixedPoints\/Predicates.agda","new_file":"Notes\/FixedPoints\/Predicates.agda","new_contents":"module Notes.FixedPoints.Predicates where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\n-- The pure (i.e. non-inductive) LTC natural numbers.\nmodule Pure where\n  postulate\n    N  : D \u2192 Set\n    zN : N zero\n    sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- The (inductive) LTC natural numbers.\nmodule Inductive where\n\n  data N : D \u2192 Set where\n    zN : N zero\n    sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- The (fixed-point) LTC natural numbers.\nmodule FixedPoint where\n\n  -- Because N is an inductive data type, we can defined it as the least\n  -- fixed-point (Fix) of an appropriate functional.\n  postulate\n    Fix : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n    -- In the first-order version of LTC we cannot use the equality on\n    -- predicates\n    --\n    -- (i.e. _\u2263_ : (D \u2192 Set) \u2192 (D \u2192 Set) \u2192 Set),\n    --\n    -- therefore we postulate both directions of the conversion rule\n    -- Fix f = f (Fix f).\n    cFix\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 Fix f d \u2192 f (Fix f) d\n    cFix\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 f (Fix f) d \u2192 Fix f d\n\n  -- From Peter: FN if D was an inductive type\n  -- FN : (D \u2192 Set) \u2192 D \u2192 Set\n  -- FN X zero     = \u22a4\n  -- FN X (succ n) = X n\n\n  -- From Peter: FN in pure predicate logic\n  FN : (D \u2192 Set) \u2192 D \u2192 Set\n  FN X n = n \u2261 zero \u2228 (\u2203D (\u03bb m \u2192 n \u2261 succ m \u2227 X m))\n\n  -- The LTC natural numbers using Fix.\n  N : D \u2192 Set\n  N = Fix FN\n\n  -- The data constructors of the inductive version via the\n  -- fixed-point version.\n  zN : N zero\n  zN = cFix\u2082 FN zero (inj\u2081 refl)\n\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n  sN {n} Nn = cFix\u2082 FN (succ n) (inj\u2082 (n , (refl , Nn)))\n","old_contents":"module Others.FixedPoints.Predicates where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\n-- The pure (i.e. non-inductive) LTC natural numbers.\nmodule Pure where\n  postulate\n    N  : D \u2192 Set\n    zN : N zero\n    sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- The (inductive) LTC natural numbers.\nmodule Inductive where\n\n  data N : D \u2192 Set where\n    zN : N zero\n    sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- The (fixed-point) LTC natural numbers.\nmodule FixedPoint where\n\n  -- Because N is an inductive data type, we can defined it as the least\n  -- fixed-point (Fix) of an appropriate functional.\n  postulate\n    Fix : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n    -- In the first-order version of LTC we cannot use the equality on\n    -- predicates\n    --\n    -- (i.e. _\u2263_ : (D \u2192 Set) \u2192 (D \u2192 Set) \u2192 Set),\n    --\n    -- therefore we postulate both directions of the conversion rule\n    -- Fix f = f (Fix f).\n    cFix\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 Fix f d \u2192 f (Fix f) d\n    cFix\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 f (Fix f) d \u2192 Fix f d\n\n  -- From Peter: FN if D was an inductive type\n  -- FN : (D \u2192 Set) \u2192 D \u2192 Set\n  -- FN X zero     = \u22a4\n  -- FN X (succ n) = X n\n\n  -- From Peter: FN in pure predicate logic\n  FN : (D \u2192 Set) \u2192 D \u2192 Set\n  FN X n = n \u2261 zero \u2228 (\u2203D (\u03bb m \u2192 n \u2261 succ m \u2227 X m))\n\n  -- The LTC natural numbers using Fix.\n  N : D \u2192 Set\n  N = Fix FN\n\n  -- The data constructors of the inductive version via the\n  -- fixed-point version.\n  zN : N zero\n  zN = cFix\u2082 FN zero (inj\u2081 refl)\n\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n  sN {n} Nn = cFix\u2082 FN (succ n) (inj\u2082 (n , (refl , Nn)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7165b065c092271f76909280a5ee53f26556195b","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import Coinduction\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 \u221e (Conat n) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\n-- 08 January 2014. Fails with --without-K.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h An with h An\n... | inj\u2081 refl = cozero\n... | inj\u2082 (n' , refl , An') = cosucc (\u266f (Conat-coind A h An'))\n\n-- TODO (07 January 2014): We couldn't prove Conat-stronger-coind.\nConat-stronger-coind :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind A h An with h An\n... | inj\u2081 n\u22610 = subst Conat (sym n\u22610) cozero\n... | inj\u2082 (n' , prf , An') =\n  subst Conat (sym prf) (cosucc (\u266f (Conat-coind A {!!} An')))\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221eD.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import Coinduction\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 \u221e (Conat n) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\n-- 08 January 2014. Fails with --without-K.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h An with h An\n... | inj\u2081 refl = cozero\n... | inj\u2082 (n' , refl , An') = cosucc (\u266f (Conat-coind A h An'))\n\n-- TODO (07 January 2014): We couldn't prove Conat-stronger-coind.\nConat-stronger-coind :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind A h An with h An\n... | inj\u2081 n\u22610 = subst Conat (sym n\u22610) cozero\n... | inj\u2082 (n' , prf , An') =\n  subst Conat (sym prf) (cosucc (\u266f (Conat-coind A {!!} An')))\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221e.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a9f0ccd8518ec95836f6e7191621b75a3cdfa623","subject":"Fin: Fin\u25b9\u2115<","message":"Fin: Fin\u25b9\u2115<\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"-- NOTE with-K\nmodule Data.Fin.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Zero\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Fin public renaming (to\u2115 to Fin\u25b9\u2115; from\u2115 to \u2115\u25b9Fin)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0; z\u2264n; s\u2264s) renaming (_+_ to _+\u2115_; _<_ to _<\u2115_)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; [0:_1:_]; case_0:_1:_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081; tabulate; foldr) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\npattern #0 = zero\npattern #1 = suc #0\npattern #2 = suc #1\npattern #3 = suc #2\npattern #4 = suc #3\npattern #5 = suc #4\npattern #6 = suc #5\npattern #7 = suc #6\npattern #8 = suc #7\npattern #9 = suc #8\npattern #A = suc #9\npattern #B = suc #A\npattern #C = suc #B\npattern #D = suc #C\npattern #E = suc #D\npattern #F = suc #E\n\npattern #1+ x = suc x\npattern #2+ x = suc (#1+ x)\npattern #3+ x = suc (#2+ x)\npattern #4+ x = suc (#3+ x)\npattern #5+ x = suc (#4+ x)\npattern #6+ x = suc (#5+ x)\npattern #7+ x = suc (#6+ x)\npattern #8+ x = suc (#7+ x)\npattern #9+ x = suc (#8+ x)\npattern #A+ x = suc (#9+ x)\npattern #B+ x = suc (#A+ x)\npattern #C+ x = suc (#B+ x)\npattern #D+ x = suc (#C+ x)\npattern #E+ x = suc (#D+ x)\npattern #F+ x = suc (#E+ x)\n\n-- The isomorphisms about Fin, \ud835\udfd8, \ud835\udfd9, \ud835\udfda are in Function.Related.TypeIsomorphisms.NP\n\nFin\u25b9\ud835\udfd8 : Fin 0 \u2192 \ud835\udfd8\nFin\u25b9\ud835\udfd8 ()\n\n\ud835\udfd8\u25b9Fin : \ud835\udfd8 \u2192 Fin 0\n\ud835\udfd8\u25b9Fin ()\n\nFin\u25b9\ud835\udfd9 : Fin 1 \u2192 \ud835\udfd9\nFin\u25b9\ud835\udfd9 _ = _\n\n\ud835\udfd9\u25b9Fin : \ud835\udfd9 \u2192 Fin 1\n\ud835\udfd9\u25b9Fin _ = zero\n\nFin\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\nFin\u25b9\ud835\udfda zero    = 0\u2082\nFin\u25b9\ud835\udfda (suc _) = 1\u2082\n\n\ud835\udfda\u25b9Fin : \ud835\udfda \u2192 Fin 2\n\ud835\udfda\u25b9Fin = [0: # 0 1: # 1 ]\n\n[zero:_,suc:_] : \u2200 {n a}{A : Fin (suc n) \u2192 Set a}\n                   (z : A zero)(s : (x : Fin n) \u2192 A (suc x))\n                   (x : Fin (suc n)) \u2192 A x\n[zero: z ,suc: s ] zero    = z\n[zero: z ,suc: s ] (suc x) = s x\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_\u225f_ : \u2200 {n} (i j : Fin n) \u2192 Dec (i \u2261 j)\nzero \u225f zero = yes refl\nzero \u225f suc j = no (\u03bb())\nsuc i \u225f zero = no (\u03bb())\nsuc i \u225f suc j with i \u225f j\nsuc i \u225f suc j | yes p = yes (cong suc p)\nsuc i \u225f suc j | no \u00acp = no (\u00acp \u2218 suc-injective)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 \ud835\udfda\nx == y = \u230a x \u225f y \u230b\n{-helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 \ud835\udfda\n  helper (equal _) = 1\u2082\n  helper _         = 0\u2082-}\n\nFin\u25b9\u2115< : \u2200{y}(x : Fin y) \u2192 Fin\u25b9\u2115 x <\u2115 y\nFin\u25b9\u2115< zero = s\u2264s z\u2264n\nFin\u25b9\u2115< (suc i) = s\u2264s (Fin\u25b9\u2115< i)\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = case x == z 0: (case y == z 0: z 1: x) 1: y\n\nmodule _ {a} {A : \u2605 a}\n         (B : \u2115 \u2192 \u2605\u2080)\n         (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n         (\u03b5 : B zero) where\n  iterate : \u2200 {n} (f : Fin n \u2192 A) \u2192 B n\n  iterate {zero}  f = \u03b5\n  iterate {suc n} f = f zero \u25c5 iterate (f \u2218 suc)\n\n  iterate-foldr\u2218tabulate :\n    \u2200 {n} (f : Fin n \u2192 A) \u2192 iterate f \u2261 foldr B _\u25c5_ \u03b5 (tabulate f)\n  iterate-foldr\u2218tabulate {zero} f = refl\n  iterate-foldr\u2218tabulate {suc n} f = cong (_\u25c5_ (f zero)) (iterate-foldr\u2218tabulate (f \u2218 suc))\n\nmodule _ {a} {A : \u2605 a} (B : \u2605\u2080)\n         (_\u25c5_ : A \u2192 B \u2192 B)\n         (\u03b5 : B) where\n  iterate\u2032 : \u2200 {n} (f : Fin n \u2192 A) \u2192 B\n  iterate\u2032 f = iterate _ _\u25c5_ \u03b5 f\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 \u2605\u2080 where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = \u2115\u25b9Fin\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = \u2115\u25b9Fin n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Properties renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (\u2115\u25b9Fin n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\n{-\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n-}\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nFin\u25b9\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 Fin\u25b9\u2115 (n \u2115- x) \u2261 n \u2238 Fin\u25b9\u2115 x\nFin\u25b9\u2115-\u2115-lem zero = to-from _\nFin\u25b9\u2115-\u2115-lem {zero} (suc ())\nFin\u25b9\u2115-\u2115-lem {suc n} (suc x) = Fin\u25b9\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse-old x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (Fin\u25b9\u2115 x) (suc n)) = Fin\u25b9\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 \u2605\u2080 where\n  `from\u2115   : FinView (\u2115\u25b9Fin n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \ud835\udfd8-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \ud835\udfd8-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (\u2115\u25b9Fin q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (\u2115\u25b9Fin q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : \u2605 a} (xs : Vec A n) x \u2192 lookup (\u2115\u25b9Fin n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(\u2115\u25b9Fin n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : \u2605 a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","old_contents":"-- NOTE with-K\nmodule Data.Fin.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Zero\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Fin public renaming (to\u2115 to Fin\u25b9\u2115; from\u2115 to \u2115\u25b9Fin)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; [0:_1:_]; case_0:_1:_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081; tabulate; foldr) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n-- The isomorphisms about Fin, \ud835\udfd8, \ud835\udfd9, \ud835\udfda are in Function.Related.TypeIsomorphisms.NP\n\nFin\u25b9\ud835\udfd8 : Fin 0 \u2192 \ud835\udfd8\nFin\u25b9\ud835\udfd8 ()\n\n\ud835\udfd8\u25b9Fin : \ud835\udfd8 \u2192 Fin 0\n\ud835\udfd8\u25b9Fin ()\n\nFin\u25b9\ud835\udfd9 : Fin 1 \u2192 \ud835\udfd9\nFin\u25b9\ud835\udfd9 _ = _\n\n\ud835\udfd9\u25b9Fin : \ud835\udfd9 \u2192 Fin 1\n\ud835\udfd9\u25b9Fin _ = zero\n\nFin\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\nFin\u25b9\ud835\udfda zero    = 0\u2082\nFin\u25b9\ud835\udfda (suc _) = 1\u2082\n\n\ud835\udfda\u25b9Fin : \ud835\udfda \u2192 Fin 2\n\ud835\udfda\u25b9Fin = [0: # 0 1: # 1 ]\n\n[zero:_,suc:_] : \u2200 {n a}{A : Fin (suc n) \u2192 Set a}\n                   (z : A zero)(s : (x : Fin n) \u2192 A (suc x))\n                   (x : Fin (suc n)) \u2192 A x\n[zero: z ,suc: s ] zero    = z\n[zero: z ,suc: s ] (suc x) = s x\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_\u225f_ : \u2200 {n} (i j : Fin n) \u2192 Dec (i \u2261 j)\nzero \u225f zero = yes refl\nzero \u225f suc j = no (\u03bb())\nsuc i \u225f zero = no (\u03bb())\nsuc i \u225f suc j with i \u225f j\nsuc i \u225f suc j | yes p = yes (cong suc p)\nsuc i \u225f suc j | no \u00acp = no (\u00acp \u2218 suc-injective)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 \ud835\udfda\nx == y = \u230a x \u225f y \u230b\n{-helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 \ud835\udfda\n  helper (equal _) = 1\u2082\n  helper _         = 0\u2082-}\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = case x == z 0: (case y == z 0: z 1: x) 1: y\n\nmodule _ {a} {A : \u2605 a}\n         (B : \u2115 \u2192 \u2605\u2080)\n         (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n         (\u03b5 : B zero) where\n  iterate : \u2200 {n} (f : Fin n \u2192 A) \u2192 B n\n  iterate {zero}  f = \u03b5\n  iterate {suc n} f = f zero \u25c5 iterate (f \u2218 suc)\n\n  iterate-foldr\u2218tabulate :\n    \u2200 {n} (f : Fin n \u2192 A) \u2192 iterate f \u2261 foldr B _\u25c5_ \u03b5 (tabulate f)\n  iterate-foldr\u2218tabulate {zero} f = refl\n  iterate-foldr\u2218tabulate {suc n} f = cong (_\u25c5_ (f zero)) (iterate-foldr\u2218tabulate (f \u2218 suc))\n\nmodule _ {a} {A : \u2605 a} (B : \u2605\u2080)\n         (_\u25c5_ : A \u2192 B \u2192 B)\n         (\u03b5 : B) where\n  iterate\u2032 : \u2200 {n} (f : Fin n \u2192 A) \u2192 B\n  iterate\u2032 f = iterate _ _\u25c5_ \u03b5 f\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 \u2605\u2080 where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = \u2115\u25b9Fin\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = \u2115\u25b9Fin n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Properties renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (\u2115\u25b9Fin n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\n{-\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n-}\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nFin\u25b9\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 Fin\u25b9\u2115 (n \u2115- x) \u2261 n \u2238 Fin\u25b9\u2115 x\nFin\u25b9\u2115-\u2115-lem zero = to-from _\nFin\u25b9\u2115-\u2115-lem {zero} (suc ())\nFin\u25b9\u2115-\u2115-lem {suc n} (suc x) = Fin\u25b9\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse-old x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (Fin\u25b9\u2115 x) (suc n)) = Fin\u25b9\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 \u2605\u2080 where\n  `from\u2115   : FinView (\u2115\u25b9Fin n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \ud835\udfd8-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \ud835\udfd8-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (\u2115\u25b9Fin q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (\u2115\u25b9Fin q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : \u2605 a} (xs : Vec A n) x \u2192 lookup (\u2115\u25b9Fin n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(\u2115\u25b9Fin n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : \u2605 a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d2e584bb6a1be2e3d988468e83cb20e883a8bcd9","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationI.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationI.agda","new_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ProofSpecificationI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1I\nopen import FOTC.Program.ABP.Lemma2I\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 abpTransfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n         \u2203[ i' ] \u2203[ is' ] \u2203[ js' ] is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  prf\u2081 {is} {js} (b , os\u2080 , os\u2081 , as , bs , cs , ds , Sis , Bb , Fos\u2080 , Fos\u2081 , h)\n     with Stream-unf Sis\n  ... | (i' , is' , is\u2261i'\u2237is , Sis') =\n    i' , is' , js' , is\u2261i'\u2237is , js\u2261i'\u2237js' , Bis'js'\n    where\n    ABP-helper : is \u2261 i' \u2237 is' \u2192\n                 ABP b is os\u2080 os\u2081 as bs cs ds js \u2192\n                 ABP b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js\n    ABP-helper h\u2081 h\u2082 = subst (\u03bb t \u2192 ABP b t os\u2080 os\u2081 as bs cs ds js) h\u2081 h\u2082\n\n    ABP'-lemma\u2081 : \u2203[ os\u2080' ] \u2203[ os\u2081' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n                    Fair os\u2080'\n                    \u2227 Fair os\u2081'\n                    \u2227 ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                    \u2227 js \u2261 i' \u2237 js'\n    ABP'-lemma\u2081 = lemma\u2081 Bb Fos\u2080 Fos\u2081 (ABP-helper is\u2261i'\u2237is h)\n\n    -- Following Martin Escardo advice (see Agda mailing list, heap\n    -- mistery) we use pattern matching instead of \u2203 eliminators to\n    -- project the elements of the existentials.\n\n    -- 2011-08-25 update: It does not seems strictly necessary because\n    -- the Agda issue 415 was fixed.\n\n    js' : D\n    js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , js' , _ = js'\n\n    js\u2261i'\u2237js' : js \u2261 i' \u2237 js'\n    js\u2261i'\u2237js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , _ , _ , _ , _ , h = h\n\n    ABP-lemma\u2082 : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n                   Fair os\u2080''\n                   \u2227 Fair os\u2081''\n                   \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n    ABP-lemma\u2082 with ABP'-lemma\u2081\n    ABP-lemma\u2082 | _ , _ , _ , _ , _ , _ , _ , Fos\u2080' , Fos\u2081' , abp' , _ =\n      lemma\u2082 Bb Fos\u2080' Fos\u2081' abp'\n\n    Bis'js' : B is' js'\n    Bis'js' with ABP-lemma\u2082\n    ... | os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds'' , Fos\u2080'' , Fos\u2081'' , abp =\n      not b , os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds''\n      , Sis' , not-Bool Bb , Fos\u2080'' , Fos\u2081'' , abp\n\n  prf\u2082 : B is (abpTransfer b os\u2080 os\u2081 is)\n  prf\u2082 = b\n       , os\u2080\n       , os\u2081\n       , has a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , Sis\n       , Bb\n       , Fos\u2080\n       , Fos\u2081\n       , has-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , trans (abpTransfer-eq b os\u2080 os\u2081 is) (transfer-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is)\n    where\n    a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 : D\n    a\u2081 = send \u00b7 b\n    a\u2082 = ack \u00b7 b\n    a\u2083 = out \u00b7 b\n    a\u2084 = corrupt \u00b7 os\u2080\n    a\u2085 = corrupt \u00b7 os\u2081\n","old_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ProofSpecificationI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1I\nopen import FOTC.Program.ABP.Lemma2I\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 abpTransfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n       \u2203[ i' ] \u2203[ is' ] \u2203[ js' ] is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  prf\u2081 {is} {js} (b , os\u2080 , os\u2081 , as , bs , cs , ds , Sis , Bb , Fos\u2080 , Fos\u2081 , h)\n     with Stream-unf Sis\n  ... | (i' , is' , is\u2261i'\u2237is , Sis') =\n    i' , is' , js' , is\u2261i'\u2237is , js\u2261i'\u2237js' , Bis'js'\n    where\n    ABP-helper : is \u2261 i' \u2237 is' \u2192\n                 ABP b is os\u2080 os\u2081 as bs cs ds js \u2192\n                 ABP b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js\n    ABP-helper h\u2081 h\u2082 = subst (\u03bb t \u2192 ABP b t os\u2080 os\u2081 as bs cs ds js) h\u2081 h\u2082\n\n    ABP'-lemma\u2081 : \u2203[ os\u2080' ] \u2203[ os\u2081' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n                    Fair os\u2080'\n                    \u2227 Fair os\u2081'\n                    \u2227 ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                    \u2227 js \u2261 i' \u2237 js'\n    ABP'-lemma\u2081 = lemma\u2081 Bb Fos\u2080 Fos\u2081 (ABP-helper is\u2261i'\u2237is h)\n\n    -- Following Martin Escardo advice (see Agda mailing list, heap\n    -- mistery) we use pattern matching instead of \u2203 eliminators to\n    -- project the elements of the existentials.\n\n    -- 2011-08-25 update: It does not seems strictly necessary because\n    -- the Agda issue 415 was fixed.\n\n    js' : D\n    js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , js' , _ = js'\n\n    js\u2261i'\u2237js' : js \u2261 i' \u2237 js'\n    js\u2261i'\u2237js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , _ , _ , _ , _ , h = h\n\n    ABP-lemma\u2082 : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n                   Fair os\u2080''\n                   \u2227 Fair os\u2081''\n                   \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n    ABP-lemma\u2082 with ABP'-lemma\u2081\n    ABP-lemma\u2082 | _ , _ , _ , _ , _ , _ , _ , Fos\u2080' , Fos\u2081' , abp' , _ =\n      lemma\u2082 Bb Fos\u2080' Fos\u2081' abp'\n\n    Bis'js' : B is' js'\n    Bis'js' with ABP-lemma\u2082\n    ... | os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds'' , Fos\u2080'' , Fos\u2081'' , abp =\n      not b , os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds''\n      , Sis' , not-Bool Bb , Fos\u2080'' , Fos\u2081'' , abp\n\n  prf\u2082 : B is (abpTransfer b os\u2080 os\u2081 is)\n  prf\u2082 = b\n       , os\u2080\n       , os\u2081\n       , has a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , Sis\n       , Bb\n       , Fos\u2080\n       , Fos\u2081\n       , has-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , trans (abpTransfer-eq b os\u2080 os\u2081 is) (transfer-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is)\n    where\n    a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 : D\n    a\u2081 = send \u00b7 b\n    a\u2082 = ack \u00b7 b\n    a\u2083 = out \u00b7 b\n    a\u2084 = corrupt \u00b7 os\u2080\n    a\u2085 = corrupt \u00b7 os\u2081\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4dc49fe2dd738663976b5a2553906ab00fe6c248","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: d1ac86c67f59bd56c296570dbc89bce5\n\ndarcs-hash:20110920064521-3bd4e-9025b902b0445ad3dcd8f94f075ac1242bf906ef.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/MapIterate\/MapIterateATP.agda","new_file":"src\/FOTC\/Program\/MapIterate\/MapIterateATP.agda","new_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using coinduction\n------------------------------------------------------------------------------\n\n-- The map-iterate property [1]:\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\n-- [1] Jeremy Gibbons and Graham Hutton. Proof methods for corecursive\n-- programs. Fundamenta Informaticae, XX:1\u201314, 2005.\n\nmodule FOTC.Program.MapIterate.MapIterateATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream.Equality\n\n------------------------------------------------------------------------------\n-- The map-iterate property.\n\n\u2248-map-iterate : \u2200 f x \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-gfp\u2082 R helper (x , refl , refl)\n  where\n  postulate\n    unfoldMap : \u2200 f y \u2192\n                map f (iterate f y) \u2261 f \u00b7 y \u2237 map f (iterate f (f \u00b7 y))\n  {-# ATP prove unfoldMap #-}\n\n  postulate\n    unfoldIterate : \u2200 f y \u2192\n                    iterate f (f \u00b7 y) \u2261 f \u00b7 y \u2237 iterate f (f \u00b7 (f \u00b7 y))\n  {-# ATP prove unfoldIterate #-}\n\n  -- The relation R was based on the relation used by Eduardo Gim\u00e9nez\n  -- and Pierre Cast\u00e9ran. A Tutorial on [Co-]Inductive Types in\n  -- Coq. May 1998 -- August 17, 2007.\n  R : D \u2192 D \u2192 Set\n  R xs ys = \u2203 \u03bb y \u2192 xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y)\n\n  helper : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n           R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper {xs} {ys} h = f \u00b7 y\n                     , map f (iterate f (f \u00b7 y))\n                     , iterate f (f \u00b7 (f \u00b7 y))\n                     , ((f \u00b7 y) , refl , refl)\n                     , trans xs\u2261map (unfoldMap f y)\n                     , trans ys\u2261iterate (unfoldIterate f y)\n    where\n    y : D\n    y = \u2203-proj\u2081 h\n\n    xs\u2261map : xs \u2261 map f (iterate f y)\n    xs\u2261map = \u2227-proj\u2081 (\u2203-proj\u2082 h)\n\n    ys\u2261iterate : ys \u2261 iterate f (f \u00b7 y)\n    ys\u2261iterate = \u2227-proj\u2082 (\u2203-proj\u2082 h)\n","old_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using coinduction\n------------------------------------------------------------------------------\n\n-- The map-iterate property [1]:\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\n-- [1] Jeremy Gibbons and Graham Hutton. Proof methods for corecursive\n-- programs. Fundamenta Informaticae, XX:1\u201314, 2005.\n\nmodule FOTC.Program.MapIterate.MapIterateATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream.Equality\n\n------------------------------------------------------------------------------\n-- The map-iterate property.\n\n\u2248-map-iterate : (f x : D) \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-gfp\u2082 R (\u03bb {xs} {ys} \u2192 helper xs ys) (x , refl , refl)\n  where\n  postulate\n    unfoldMap : \u2200 f y \u2192\n                map f (iterate f y) \u2261 f \u00b7 y \u2237 map f (iterate f (f \u00b7 y))\n  {-# ATP prove unfoldMap #-}\n\n  postulate\n    unfoldIterate : \u2200 f y \u2192\n                    iterate f (f \u00b7 y) \u2261 f \u00b7 y \u2237 iterate f (f \u00b7 (f \u00b7 y))\n  {-# ATP prove unfoldIterate #-}\n\n  -- The relation R was based on the relation used by Eduardo Gim\u00e9nez\n  -- and Pierre Cast\u00e9ran. A Tutorial on [Co-]Inductive Types in\n  -- Coq. May 1998 -- August 17, 2007.\n  R : D \u2192 D \u2192 Set\n  R xs ys = \u2203 (\u03bb y \u2192 xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y))\n\n  helper : \u2200 xs ys \u2192 R xs ys \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n             R xs' ys' \u2227\n             xs \u2261 x' \u2237 xs' \u2227\n             ys \u2261 x' \u2237 ys'\n  helper xs ys h = f \u00b7 y\n                   , map f (iterate f (f \u00b7 y))\n                   , iterate f (f \u00b7 (f \u00b7 y))\n                   , ((f \u00b7 y) , refl , refl)\n                   , trans xs\u2261map (unfoldMap f y)\n                   , trans ys\u2261iterate (unfoldIterate f y)\n    where\n    y : D\n    y = \u2203-proj\u2081 h\n\n    xs\u2261map : xs \u2261 map f (iterate f y)\n    xs\u2261map = \u2227-proj\u2081 (\u2203-proj\u2082 h)\n\n    ys\u2261iterate : ys \u2261 iterate f (f \u00b7 y)\n    ys\u2261iterate = \u2227-proj\u2082 (\u2203-proj\u2082 h)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"90124748011256fb6d0bc20d8e6a6e21ffd15850","subject":"agda : Wouter-Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Wouter-Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven'  : isEvenPT \u2261 isEven \u2218 (_+ 2)\n    isEven'  = refl\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    xxx  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    xxx  = \u21d3Base\n\n    xxx' : Set\n    xxx' = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    yyy  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    yyy  = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    zzz  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    zzz  = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module DomTest where\n    domIs   : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    domIs   = refl\n    domIs'  : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    domIs'  = refl\n    domIs'' : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    domIs'' = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven'  : isEvenPT \u2261 isEven \u2218 (_+ 2)\n    isEven'  = refl\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    xxx  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    xxx  = \u21d3Base\n\n    xxx' : Set\n    xxx' = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    yyy  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    yyy  = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    zzz  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    zzz  = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"442388d90f9fc78a0d4b2abcc7aa451bbf8b52e9","subject":"Add +1 primitive but not really :-|","message":"Add +1 primitive but not really :-|\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- Adding this changes nothing without changes to the semantics.\n  succ : Const (nat \u21d2 nat)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5f55e3a3f33179e9ee5003810973b807581a35cc","subject":"Closure!","message":"Closure!\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or true   = true\ntrue or false  = true\nfalse or true  = true\nfalse or false = false\n\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n--------------------\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n--------------------\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218 = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n--------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\nmodusponens : \u2115 \u2192 Arrow \u2192 Arrow\nmodusponens n (\u21d2 q)   = (\u21d2 q)\nmodusponens n (p \u21d2 q) with (n \u2261 p)\n...                      | true  = modusponens n q\n...                      | false = p \u21d2 (modusponens n q)\n\n\nreduce : \u2115 \u2192 List Arrow \u2192 List Arrow\nreduce n \u2218 = \u2218\nreduce n (arr \u2237 rst) = (modusponens n arr) \u2237 (reduce n rst)\n\n\nsearch : List Arrow \u2192 List \u2115\nsearch \u2218 = \u2218\nsearch ((\u21d2 n) \u2237 rst)   = n \u2237 (search (reduce n rst))\nsearch ((n \u21d2 q) \u2237 rst) with (n \u2208 (search rst))\n...                       | true  = search (q \u2237 rst)\n...                       | false = search rst\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q) = q \u2208 (search cs)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e676b483cb67814cc3678f3c0a17a1e4ce27b25c","subject":"next en back constructors in paths kunnen de kleur niet impliciet vinden","message":"next en back constructors in paths kunnen de kleur niet impliciet vinden\n","repos":"toonn\/popartt","old_file":"koopa.agda","new_file":"koopa.agda","new_contents":"\n{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_)\n  open import Data.List\n  open import Data.Vec renaming (map to vmap; lookup to vlookup)\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa Color : Set where\n    _KT : Color \u2192 KoopaTroopa Color\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  data Clearance : Set where\n    Low  : Clearance\n    High : Clearance\n    God  : Clearance\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n      clr : Clearance\n\n  data _c>_ : Color \u2192 Clearance \u2192 Set where\n    <red>   : \u2200 {c} \u2192 c c> Low\n    <green> : Green c> High\n\n  data _follows_ : Position \u2192 Position \u2192 Set where\n    stay  : \u2200 {x y} \u2192 pos x y gas Low follows pos x y gas Low\n    next  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos (suc x) y gas Low follows pos x y gas cl\n    back  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos x y gas Low follows pos (suc x) y gas cl\n    -- jump  : \u2200 {x y} \u2192 pos x (suc y) gas High follows pos x y gas Low\n    fall  : \u2200 {cl x y} \u2192 pos x y gas cl follows pos x (suc y) gas High\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 gas Low) (pos 0 0 gas Low)\n  ex_path = pos 0 0 gas Low \u21a0\u27e8 next {Red} \u27e9\n            pos 1 0 gas Low \u21a0\u27e8 back \u27e9\n            pos 0 0 gas Low \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192\n                   Vec Position n\n  matToPosVec []           []        _ _ = []\n  matToPosVec (mat \u2237 mats) (under \u2237 unders) x y =\n    pos x y mat cl \u2237 matToPosVec mats unders (x + 1) y\n      where\n        clearance : Material \u2192 Material \u2192 Clearance\n        clearance gas   gas   = High\n        clearance gas   solid = Low\n        clearance solid _     = God\n        cl = clearance mat under\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs []                   = []\n  matToPosVecs (_\u2237_ {y} mats matss) = matToPosVec mats mats 0 y \u2237 matToPosVecs matss\n\n  matsToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matsToMat matss = Mat (matToPosVecs matss)\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = matsToMat (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 [])\n","old_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_)\n  open import Data.List\n  open import Data.Vec renaming (map to vmap; lookup to vlookup)\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa Color : Set where\n    _KT : Color \u2192 KoopaTroopa Color\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n\n  data _follows_ : Position \u2192 Position \u2192 Set where\n    stay  : \u2200 {p} \u2192 p follows p\n    next  : \u2200 {x y mat} \u2192 pos (suc x) y mat follows pos x y mat\n    back  : \u2200 {x y mat} \u2192 pos x y mat follows pos (suc x) y mat\n    -- jump  : \u2200 {x y mat} \u2192 pos x (suc y) mat follows pos x y mat\n    fall  : \u2200 {x y mat} \u2192 pos x y mat follows pos x (suc y) mat\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 solid) (pos 0 0 solid)\n  ex_path = pos 0 0 solid \u21a0\u27e8 next \u27e9\n            pos 1 0 solid \u21a0\u27e8 back \u27e9\n            pos 0 0 solid \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192 Vec Position n\n  matToPosVec []           _ _ = []\n  matToPosVec (mat \u2237 mats) x y = pos x y mat \u2237 matToPosVec mats (x + 1) y\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs []                   = []\n  matToPosVecs (_\u2237_ {y} mats matss) = matToPosVec mats 0 y \u2237 matToPosVecs matss\n\n  matToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matToMat matss = Mat (matToPosVecs matss)\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = Mat (matToPosVecs (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 []))\n\n","returncode":0,"stderr":"","license":"bsd-2-clause","lang":"Agda"}
{"commit":"20a1abbafdf38b144be117be78c688be2786fbee","subject":"Update to Neglible","message":"Update to Neglible\n","repos":"crypto-agda\/crypto-agda","old_file":"Neglible.agda","new_file":"Neglible.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (fN k) (gD k) (f'D k) (g'D k) \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange (gN k) (fD k) (g'D k) (f'D k))\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (f'N k) (fD k) (g'D k) (gD k))\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange (g'N k) (gD k) (f'D k) (fD k)\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\n\nmodule _ (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"398fc8e688c6a28e9c5c3e7f5691b7219c54a35f","subject":"more cases for #5","message":"more cases for #5\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i (FVal x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = vs x (_ , Step p q r)\n  is (IAp i (FIndet x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n  -- final expressions are not errors (not one of the 6 cases for progress)\n  fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  fe (FVal x) er = ve x er\n  fe (FIndet x) er = ie x er\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (d' , Step FHNEHoleEvaled () x\u2082)\n  es (ENEHole er) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i (FVal x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = vs x (_ , Step p q r)\n  is (IAp i (FIndet x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n  -- final expressions are not errors (not one of the 6 cases for progress)\n  fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  fe (FVal x) er = ve x er\n  fe (FIndet x) er = ie x er\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 er) (d' , Step (FHAp2 x) x\u2081 x\u2082) = {!!}\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole er) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"56068be9d5f2204763d86c0e8e47cf811b5878f7","subject":"Removed unnecesary hint.","message":"Removed unnecesary hint.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/Collatz\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/Collatz\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\nhelper-helper : \u2200 {n} \u2192 N n \u2192 collatz ([2] ^ succ\u2081 n) \u2261\n                        collatz (div ([2] ^ succ\u2081 n) [2])\nhelper-helper nzero = prf\n  where postulate prf : collatz ([2] ^ succ\u2081 zero) \u2261\n                        collatz (div ([2] ^ succ\u2081 zero) [2])\n        {-# ATP prove prf #-}\n\nhelper-helper (nsucc {n} Nn) = prf\n  where postulate prf : collatz ([2] ^ succ\u2081 (succ\u2081 n)) \u2261\n                        collatz (div ([2] ^ succ\u2081 (succ\u2081 n)) [2])\n        {-# ATP prove prf +\u22382 ^-N 2-N 2^x\u22620 2^[x+1]\u22621\n                      x-Even\u2192SSx-Even \u2238-N \u2238-Even 2^[x+1]-Even 2-Even\n        #-}\n\n-- We help the ATPs proving the helper-helper property first.\npostulate helper : \u2200 {n} \u2192 N n \u2192 collatz ([2] ^ succ\u2081 n) \u2261 collatz ([2] ^ n)\n{-# ATP prove helper helper-helper div-2^[x+1]-2\u22612^x #-}\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 collatz ([2] ^ n) \u2261 [1]\ncollatz-2^x nzero = prf\n  where postulate prf : collatz ([2] ^ [0]) \u2261 [1]\n        {-# ATP prove prf #-}\n\ncollatz-2^x (nsucc {n} Nn) = prf (collatz-2^x Nn)\n  where postulate prf : collatz ([2] ^ n) \u2261 [1] \u2192 collatz ([2] ^ succ\u2081 n) \u2261 [1]\n        {-# ATP prove prf helper #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\nhelper-helper : \u2200 {n} \u2192 N n \u2192 collatz ([2] ^ succ\u2081 n) \u2261\n                        collatz (div ([2] ^ succ\u2081 n) [2])\nhelper-helper nzero = prf\n  where postulate prf : collatz ([2] ^ succ\u2081 zero) \u2261\n                        collatz (div ([2] ^ succ\u2081 zero) [2])\n        {-# ATP prove prf #-}\n\nhelper-helper (nsucc {n} Nn) = prf\n  where postulate prf : collatz ([2] ^ succ\u2081 (succ\u2081 n)) \u2261\n                        collatz (div ([2] ^ succ\u2081 (succ\u2081 n)) [2])\n        {-# ATP prove prf +\u22382 ^-N 2-N 2^x\u22620 2^[x+1]\u22621\n                      collatz-even x-Even\u2192SSx-Even \u2238-N \u2238-Even 2^[x+1]-Even\n                      2-Even\n        #-}\n\n-- We help the ATPs proving the helper-helper property first.\npostulate helper : \u2200 {n} \u2192 N n \u2192 collatz ([2] ^ succ\u2081 n) \u2261 collatz ([2] ^ n)\n{-# ATP prove helper helper-helper div-2^[x+1]-2\u22612^x #-}\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 collatz ([2] ^ n) \u2261 [1]\ncollatz-2^x nzero = prf\n  where postulate prf : collatz ([2] ^ [0]) \u2261 [1]\n        {-# ATP prove prf #-}\n\ncollatz-2^x (nsucc {n} Nn) = prf (collatz-2^x Nn)\n  where postulate prf : collatz ([2] ^ n) \u2261 [1] \u2192 collatz ([2] ^ succ\u2081 n) \u2261 [1]\n        {-# ATP prove prf helper #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6d4a90d69f37d2853048dcfc6038e4e79f9c1db6","subject":"agda\/total.agda: Sketch symbolic derivation for if","message":"agda\/total.agda: Sketch symbolic derivation for if\n\nI left some holes for helper functions, and didn't prove correctness\nyet.\n\nOld-commit-hash: 4cbf58539c1fd77a425abf1e609db4a03e1d8c0f\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n--\u2261-cong {bool} {bool} {v\u2081} f bool = bool\n\u2261-cong {bool} {bool} f bool = bool\n\u2261-cong {bool} {\u03c4\u2083 \u21d2 \u03c4\u2084} {v\u2081} {v\u2082} f (ext \u2261\u2081) = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} {bool} {bool} f bool bool = bool\n\u2261-cong\u2082 {bool} {bool} {\u03c4\u2082 \u21d2 \u03c4\u2083} f bool (ext \u2261\u2082) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {bool} f (ext \u2261\u2081) (bool) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u03c4\u2083 \u21d2 \u03c4\u2084} f (ext \u2261\u2081) (ext \u2261\u2082) = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = ?\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = ?\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n--\u2261-cong {bool} {bool} {v\u2081} f bool = bool\n\u2261-cong {bool} {bool} f bool = bool\n\u2261-cong {bool} {\u03c4\u2083 \u21d2 \u03c4\u2084} {v\u2081} {v\u2082} f (ext \u2261\u2081) = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} {bool} {bool} f bool bool = bool\n\u2261-cong\u2082 {bool} {bool} {\u03c4\u2082 \u21d2 \u03c4\u2083} f bool (ext \u2261\u2082) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {bool} f (ext \u2261\u2081) (bool) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u03c4\u2083 \u21d2 \u03c4\u2084} f (ext \u2261\u2081) (ext \u2261\u2082) = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2fa68e863455907e4dbafa0d9e4c28fae5211452","subject":"sum: SearchInd","message":"sum: SearchInd\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nsearchInd' : \u2200 {A} {s\u1d2c : Search A} \u2192 SearchInd s\u1d2c \u2192 SearchInd' s\u1d2c\nsearchInd' Ps\u1d2c P P\u2219 {f} Pf = Ps\u1d2c (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nsearch-mono' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchMono s\u1d2c\nsearch-mono' s\u1d2c Ps\u1d2c _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nsearch-\u03b5' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 Search\u03b5 (searchMonFromSearch s\u1d2c)\nsearch-\u03b5' s\u1d2c Ps\u1d2c m = searchInd' Ps\u1d2c Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n  where open Mon m\n\nsearch-ext' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchExt s\u1d2c\nsearch-ext' s\u1d2c Ps\u1d2c sg {f} {g} f\u2248\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n  where open Sgrp sg\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = {!!}\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = ?\n{-\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk srch ext mono swp eps hom\n  where\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n-}\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e {C} P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) = ?\n{-\n   = mk srch ext mono swp eps hom\n   where\n     srch : Search _ -- (A \u00d7 B)\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n-}\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n  \nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x \n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = ?\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n(\u03bcA +\u03bc \u03bcB) = mk srch ext mono swp eps hom\n  where\n    s\u1d2c = search \u03bcA\n    s\u1d2e = search \u03bcB\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e \n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB)\n   = mk srch ext mono swp eps hom\n   where\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k \nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"697a6af2b714cdc17629bc600252cb44b2721351","subject":"Agda factorized Arg.","message":"Agda factorized Arg.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2I.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2I.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2I where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.Loop\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- Helper function for the ABP lemma 2\n\nmodule Helper where\n\n  helper : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n           Bit b \u2192\n           Fair os\u2080' \u2192\n           ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n           \u2200 ft\u2081 os\u2081'' \u2192 F*T ft\u2081 \u2192 Fair os\u2081'' \u2192 os\u2081' \u2261 ft\u2081 ++ os\u2081'' \u2192\n           \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2080''\n           \u2227 Fair os\u2081''\n           \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(T \u2237 []) os\u2081'' f*tnil Fos\u2081'' os\u2081'-eq =\n         os\u2080' , os\u2081'' , as'' , bs'' , cs'' , ds''\n         , Fos\u2080' , Fos\u2081''\n         , as''-eq , bs''-eq ,  cs''-eq , refl , js'-eq\n\n    where\n    os'\u2081-eq-helper : os\u2081' \u2261 T \u2237 os\u2081''\n    os'\u2081-eq-helper =\n      os\u2081'                 \u2261\u27e8 os\u2081'-eq \u27e9\n      (true \u2237 []) ++ os\u2081'' \u2261\u27e8 ++-\u2237 true [] os\u2081'' \u27e9\n      true \u2237 [] ++ os\u2081''   \u2261\u27e8 \u2237-rightCong (++-leftIdentity os\u2081'') \u27e9\n      true \u2237 os\u2081''         \u220e\n\n    ds'' : D\n    ds'' = corrupt \u00b7 os\u2081'' \u00b7 cs'\n\n    ds'-eq : ds' \u2261 ok b \u2237 ds''\n    ds'-eq =\n      ds'\n        \u2261\u27e8 ds'ABP' \u27e9\n      corrupt \u00b7 os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (\u00b7-rightCong os'\u2081-eq-helper) \u27e9\n      corrupt \u00b7 (T \u2237 os\u2081'') \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-T os\u2081'' b cs' \u27e9\n      ok b \u2237 corrupt \u00b7 os\u2081'' \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      ok b \u2237 ds'' \u220e\n\n    as'' : D\n    as'' = as'\n\n    as''-eq : as'' \u2261 send \u00b7 not b \u00b7 is' \u00b7 ds''\n    as''-eq =\n      as''                         \u2261\u27e8 as'ABP \u27e9\n      await b i' is' ds'           \u2261\u27e8 cong (await b i' is') ds'-eq \u27e9\n      await b i' is' (ok b \u2237 ds'') \u2261\u27e8 await-ok\u2261 b b i' is' ds'' refl \u27e9\n      send \u00b7 not b \u00b7 is' \u00b7 ds''    \u220e\n\n    bs'' : D\n    bs'' = bs'\n\n    bs''-eq : bs'' \u2261 corrupt \u00b7 os\u2080' \u00b7 as'\n    bs''-eq = bs'ABP'\n\n    cs'' : D\n    cs'' = cs'\n\n    cs''-eq : cs'' \u2261 ack \u00b7 not b \u00b7 bs'\n    cs''-eq = cs'ABP'\n\n    js'-eq : js' \u2261 out \u00b7 not b \u00b7 bs''\n    js'-eq = js'ABP'\n\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(F \u2237 ft\u2081) os\u2081'' (f*tcons {ft\u2081} FTft\u2081) Fos\u2081'' os\u2081'-eq\n         = helper Bb (tail-Fair Fos\u2080') ABP'IH ft\u2081 os\u2081'' FTft\u2081 Fos\u2081'' refl\n\n    where\n    os\u2080\u2075 : D\n    os\u2080\u2075 = tail\u2081 os\u2080'\n\n    os\u2081\u2075 : D\n    os\u2081\u2075 = ft\u2081 ++ os\u2081''\n\n    os\u2081'-eq-helper : os\u2081' \u2261 F \u2237 os\u2081\u2075\n    os\u2081'-eq-helper = os\u2081'               \u2261\u27e8 os\u2081'-eq \u27e9\n                     (F \u2237 ft\u2081) ++ os\u2081'' \u2261\u27e8 ++-\u2237 _ _ _ \u27e9\n                     F \u2237 ft\u2081 ++ os\u2081''   \u2261\u27e8 refl \u27e9\n                     F \u2237 os\u2081\u2075           \u220e\n\n    ds\u2075 : D\n    ds\u2075 = corrupt \u00b7 os\u2081\u2075 \u00b7 cs'\n\n    ds'-eq : ds' \u2261 error \u2237 ds\u2075\n    ds'-eq =\n      ds'\n        \u2261\u27e8 ds'ABP' \u27e9\n      corrupt \u00b7 os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (\u00b7-rightCong os\u2081'-eq-helper) \u27e9\n      corrupt \u00b7 (F \u2237 os\u2081\u2075) \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt \u00b7 os\u2081\u2075 \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      error \u2237 ds\u2075 \u220e\n\n    as\u2075 : D\n    as\u2075 = await b i' is' ds\u2075\n\n    as'-eq : as' \u2261 < i' , b > \u2237 as\u2075\n    as'-eq = as'                             \u2261\u27e8 as'ABP \u27e9\n             await b i' is' ds'              \u2261\u27e8 cong (await b i' is') ds'-eq \u27e9\n             await b i' is' (error \u2237 ds\u2075)    \u2261\u27e8 await-error _ _ _ _ \u27e9\n             < i' , b > \u2237 await b i' is' ds\u2075 \u2261\u27e8 refl \u27e9\n             < i' , b > \u2237 as\u2075                \u220e\n\n    bs\u2075 : D\n    bs\u2075 = corrupt \u00b7 os\u2080\u2075 \u00b7 as\u2075\n\n    bs'-eq-helper\u2081 : os\u2080' \u2261 T \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 ok < i' , b > \u2237 bs\u2075\n    bs'-eq-helper\u2081 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt \u00b7 os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 corrupt \u00b7 os\u2080' \u00b7 as' \u2261 corrupt \u00b7 t\u2081 \u00b7 t\u2082)\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt \u00b7 (T \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as\u2075)\n        \u2261\u27e8 corrupt-T _ _ _ \u27e9\n      ok < i' , b > \u2237 corrupt \u00b7 (tail\u2081 os\u2080') \u00b7 as\u2075\n        \u2261\u27e8 refl \u27e9\n      ok < i' , b > \u2237 bs\u2075 \u220e\n\n    bs'-eq-helper\u2082 : os\u2080' \u2261 F \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 error \u2237 bs\u2075\n    bs'-eq-helper\u2082 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt \u00b7 os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 corrupt \u00b7 os\u2080' \u00b7 as' \u2261 corrupt \u00b7 t\u2081 \u00b7 t\u2082)\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt \u00b7 (F \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as\u2075)\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt \u00b7 (tail\u2081 os\u2080') \u00b7 as\u2075\n        \u2261\u27e8 refl \u27e9\n      error \u2237 bs\u2075 \u220e\n\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs\u2075 \u2228 bs' \u2261 error \u2237 bs\u2075\n    bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2080')\n\n    cs\u2075 : D\n    cs\u2075 = ack \u00b7 not b \u00b7 bs\u2075\n\n    cs'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 \u2192 cs' \u2261 b \u2237 cs\u2075\n    cs'-eq-helper\u2081 h =\n      cs'\n      \u2261\u27e8 cs'ABP' \u27e9\n      ack \u00b7 not b \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack \u00b7 not b \u00b7 (ok < i' , b > \u2237 bs\u2075)\n        \u2261\u27e8 ack-ok\u2262 _ _ _ _ (not-x\u2262x Bb) \u27e9\n      not (not b) \u2237 ack \u00b7 not b \u00b7 bs\u2075\n        \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack \u00b7 not b \u00b7 bs\u2075\n        \u2261\u27e8 refl \u27e9\n      b \u2237 cs\u2075 \u220e\n\n    cs'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 \u2192 cs' \u2261 b \u2237 cs\u2075\n    cs'-eq-helper\u2082 h =\n      cs'                             \u2261\u27e8 cs'ABP' \u27e9\n      ack \u00b7 not b \u00b7 bs'               \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack \u00b7 not b \u00b7 (error \u2237 bs\u2075)     \u2261\u27e8 ack-error _ _ \u27e9\n      not (not b) \u2237 ack \u00b7 not b \u00b7 bs\u2075 \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack \u00b7 not b \u00b7 bs\u2075           \u2261\u27e8 refl \u27e9\n      b \u2237 cs\u2075                         \u220e\n\n    cs'-eq : cs' \u2261 b \u2237 cs\u2075\n    cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n    js'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 \u2192 js' \u2261 out \u00b7 not b \u00b7 bs\u2075\n    js'-eq-helper\u2081 h  =\n      js'\n        \u2261\u27e8 js'ABP' \u27e9\n      out \u00b7 not b \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out \u00b7 not b \u00b7 (ok < i' , b > \u2237 bs\u2075)\n        \u2261\u27e8 out-ok\u2262 (not b) b i' bs\u2075 (not-x\u2262x Bb) \u27e9\n      out \u00b7 not b \u00b7 bs\u2075 \u220e\n\n    js'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 \u2192 js' \u2261 out \u00b7 not b \u00b7 bs\u2075\n    js'-eq-helper\u2082 h  =\n      js'                         \u2261\u27e8 js'ABP' \u27e9\n      out \u00b7 not b \u00b7 bs'           \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out \u00b7 not b \u00b7 (error \u2237 bs\u2075) \u2261\u27e8 out-error (not b) bs\u2075 \u27e9\n      out \u00b7 not b \u00b7 bs\u2075           \u220e\n\n    js'-eq : js' \u2261 out \u00b7 not b \u00b7 bs\u2075\n    js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n    ds\u2075-eq : ds\u2075 \u2261 corrupt \u00b7 os\u2081\u2075 \u00b7 (b \u2237 cs\u2075)\n    ds\u2075-eq = \u00b7-rightCong cs'-eq\n\n    ABP'IH : ABP' b i' is' os\u2080\u2075 os\u2081\u2075 as\u2075 bs\u2075 cs\u2075 ds\u2075 js'\n    ABP'IH = ds\u2075-eq , refl , refl , refl , js'-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081 \u2208\n-- Fair, and hence by unfolding Fair we conclude that there are ft\u2081 :\n-- F*T and os\u2081'' : Fair, such that os\u2081' = ft\u2081 ++ os\u2081''.\n--\n-- We proceed by induction on ft\u2081 : F*T using helper.\n\nopen Helper\nlemma\u2082 : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2080' \u2192\n         Fair os\u2081' \u2192\n         ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n         Fair os\u2080''\n         \u2227 Fair os\u2081''\n         \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2080' Fos\u2081' abp' with Fair-unf Fos\u2081'\n... | ft , os\u2080'' , FTft , Fos\u2080'' , h =\n  helper Bb Fos\u2080' abp' ft os\u2080'' FTft Fos\u2080'' h\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2I where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.Loop\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- Helper function for the ABP lemma 2\n\nmodule Helper where\n\n  helper : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n           Bit b \u2192\n           Fair os\u2080' \u2192\n           ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n           \u2200 ft\u2081 os\u2081'' \u2192 F*T ft\u2081 \u2192 Fair os\u2081'' \u2192 os\u2081' \u2261 ft\u2081 ++ os\u2081'' \u2192\n           \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2080''\n           \u2227 Fair os\u2081''\n           \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(T \u2237 []) os\u2081'' f*tnil Fos\u2081'' os\u2081'-eq =\n         os\u2080' , os\u2081'' , as'' , bs'' , cs'' , ds''\n         , Fos\u2080' , Fos\u2081''\n         , as''-eq , bs''-eq ,  cs''-eq , refl , js'-eq\n\n    where\n    os'\u2081-eq-helper : os\u2081' \u2261 T \u2237 os\u2081''\n    os'\u2081-eq-helper =\n      os\u2081'                 \u2261\u27e8 os\u2081'-eq \u27e9\n      (true \u2237 []) ++ os\u2081'' \u2261\u27e8 ++-\u2237 true [] os\u2081'' \u27e9\n      true \u2237 [] ++ os\u2081''   \u2261\u27e8 \u2237-rightCong (++-leftIdentity os\u2081'') \u27e9\n      true \u2237 os\u2081''         \u220e\n\n    ds'' : D\n    ds'' = corrupt \u00b7 os\u2081'' \u00b7 cs'\n\n    ds'-eq : ds' \u2261 ok b \u2237 ds''\n    ds'-eq =\n      ds'\n        \u2261\u27e8 ds'ABP' \u27e9\n      corrupt \u00b7 os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (\u00b7-rightCong os'\u2081-eq-helper) \u27e9\n      corrupt \u00b7 (T \u2237 os\u2081'') \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-T os\u2081'' b cs' \u27e9\n      ok b \u2237 corrupt \u00b7 os\u2081'' \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      ok b \u2237 ds'' \u220e\n\n    as'' : D\n    as'' = as'\n\n    as''-eq : as'' \u2261 send \u00b7 not b \u00b7 is' \u00b7 ds''\n    as''-eq =\n      as''                         \u2261\u27e8 as'ABP \u27e9\n      await b i' is' ds'           \u2261\u27e8 cong (await b i' is') ds'-eq \u27e9\n      await b i' is' (ok b \u2237 ds'') \u2261\u27e8 await-ok\u2261 b b i' is' ds'' refl \u27e9\n      send \u00b7 not b \u00b7 is' \u00b7 ds''    \u220e\n\n    bs'' : D\n    bs'' = bs'\n\n    bs''-eq : bs'' \u2261 corrupt \u00b7 os\u2080' \u00b7 as'\n    bs''-eq = bs'ABP'\n\n    cs'' : D\n    cs'' = cs'\n\n    cs''-eq : cs'' \u2261 ack \u00b7 not b \u00b7 bs'\n    cs''-eq = cs'ABP'\n\n    js'-eq : js' \u2261 out \u00b7 not b \u00b7 bs''\n    js'-eq = js'ABP'\n\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(F \u2237 ft\u2081) os\u2081'' (f*tcons {ft\u2081} FTft\u2081) Fos\u2081'' os\u2081'-eq\n         = helper Bb (tail-Fair Fos\u2080') ABP'IH ft\u2081 os\u2081'' FTft\u2081 Fos\u2081'' refl\n\n    where\n    os\u2080\u2075 : D\n    os\u2080\u2075 = tail\u2081 os\u2080'\n\n    os\u2081\u2075 : D\n    os\u2081\u2075 = ft\u2081 ++ os\u2081''\n\n    os\u2081'-eq-helper : os\u2081' \u2261 F \u2237 os\u2081\u2075\n    os\u2081'-eq-helper = os\u2081'               \u2261\u27e8 os\u2081'-eq \u27e9\n                     (F \u2237 ft\u2081) ++ os\u2081'' \u2261\u27e8 ++-\u2237 _ _ _ \u27e9\n                     F \u2237 ft\u2081 ++ os\u2081''   \u2261\u27e8 refl \u27e9\n                     F \u2237 os\u2081\u2075           \u220e\n\n    ds\u2075 : D\n    ds\u2075 = corrupt \u00b7 os\u2081\u2075 \u00b7 cs'\n\n    ds'-eq : ds' \u2261 error \u2237 ds\u2075\n    ds'-eq =\n      ds'\n        \u2261\u27e8 ds'ABP' \u27e9\n      corrupt \u00b7 os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (\u00b7-rightCong os\u2081'-eq-helper) \u27e9\n      corrupt \u00b7 (F \u2237 os\u2081\u2075) \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt \u00b7 os\u2081\u2075 \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      error \u2237 ds\u2075 \u220e\n\n    as\u2075 : D\n    as\u2075 = await b i' is' ds\u2075\n\n    as'-eq : as' \u2261 < i' , b > \u2237 as\u2075\n    as'-eq = as'                             \u2261\u27e8 as'ABP \u27e9\n             await b i' is' ds'              \u2261\u27e8 cong (await b i' is') ds'-eq \u27e9\n             await b i' is' (error \u2237 ds\u2075)    \u2261\u27e8 await-error _ _ _ _ \u27e9\n             < i' , b > \u2237 await b i' is' ds\u2075 \u2261\u27e8 refl \u27e9\n             < i' , b > \u2237 as\u2075                \u220e\n\n    bs\u2075 : D\n    bs\u2075 = corrupt \u00b7 os\u2080\u2075 \u00b7 as\u2075\n\n    bs'-eq-helper\u2081 : os\u2080' \u2261 T \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 ok < i' , b > \u2237 bs\u2075\n    bs'-eq-helper\u2081 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt \u00b7 os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 corrupt \u00b7 os\u2080' \u00b7 as' \u2261 corrupt \u00b7 t\u2081 \u00b7 t\u2082)\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt \u00b7 (T \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as\u2075)\n        \u2261\u27e8 corrupt-T _ _ _ \u27e9\n      ok < i' , b > \u2237 corrupt \u00b7 (tail\u2081 os\u2080') \u00b7 as\u2075\n        \u2261\u27e8 refl \u27e9\n      ok < i' , b > \u2237 bs\u2075 \u220e\n\n    bs'-eq-helper\u2082 : os\u2080' \u2261 F \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 error \u2237 bs\u2075\n    bs'-eq-helper\u2082 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt \u00b7 os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 corrupt \u00b7 os\u2080' \u00b7 as' \u2261 corrupt \u00b7 t\u2081 \u00b7 t\u2082)\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt \u00b7 (F \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as\u2075)\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt \u00b7 (tail\u2081 os\u2080') \u00b7 as\u2075\n        \u2261\u27e8 refl \u27e9\n      error \u2237 bs\u2075 \u220e\n\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs\u2075 \u2228 bs' \u2261 error \u2237 bs\u2075\n    bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2080')\n\n    cs\u2075 : D\n    cs\u2075 = ack \u00b7 not b \u00b7 bs\u2075\n\n    cs'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 \u2192 cs' \u2261 b \u2237 cs\u2075\n    cs'-eq-helper\u2081 h =\n      cs'\n      \u2261\u27e8 cs'ABP' \u27e9\n      ack \u00b7 not b \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack \u00b7 not b \u00b7 (ok < i' , b > \u2237 bs\u2075)\n        \u2261\u27e8 ack-ok\u2262 _ _ _ _ (not-x\u2262x Bb) \u27e9\n      not (not b) \u2237 ack \u00b7 not b \u00b7 bs\u2075\n        \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack \u00b7 not b \u00b7 bs\u2075\n        \u2261\u27e8 refl \u27e9\n      b \u2237 cs\u2075 \u220e\n\n    cs'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 \u2192 cs' \u2261 b \u2237 cs\u2075\n    cs'-eq-helper\u2082 h =\n      cs'                             \u2261\u27e8 cs'ABP' \u27e9\n      ack \u00b7 not b \u00b7 bs'               \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack \u00b7 not b \u00b7 (error \u2237 bs\u2075)     \u2261\u27e8 ack-error _ _ \u27e9\n      not (not b) \u2237 ack \u00b7 not b \u00b7 bs\u2075 \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack \u00b7 not b \u00b7 bs\u2075           \u2261\u27e8 refl \u27e9\n      b \u2237 cs\u2075                         \u220e\n\n    cs'-eq : cs' \u2261 b \u2237 cs\u2075\n    cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n    js'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 \u2192 js' \u2261 out \u00b7 not b \u00b7 bs\u2075\n    js'-eq-helper\u2081 h  =\n      js'\n        \u2261\u27e8 js'ABP' \u27e9\n      out \u00b7 not b \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out \u00b7 not b \u00b7 (ok < i' , b > \u2237 bs\u2075)\n        \u2261\u27e8 out-ok\u2262 (not b) b i' bs\u2075 (not-x\u2262x Bb) \u27e9\n      out \u00b7 not b \u00b7 bs\u2075 \u220e\n\n    js'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 \u2192 js' \u2261 out \u00b7 not b \u00b7 bs\u2075\n    js'-eq-helper\u2082 h  =\n      js'                         \u2261\u27e8 js'ABP' \u27e9\n      out \u00b7 not b \u00b7 bs'           \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out \u00b7 not b \u00b7 (error \u2237 bs\u2075) \u2261\u27e8 out-error (not b) bs\u2075 \u27e9\n      out \u00b7 not b \u00b7 bs\u2075           \u220e\n\n    js'-eq : js' \u2261 out \u00b7 not b \u00b7 bs\u2075\n    js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n    ds\u2075-eq : ds\u2075 \u2261 corrupt \u00b7 os\u2081\u2075 \u00b7 (b \u2237 cs\u2075)\n    ds\u2075-eq = \u00b7-rightCong cs'-eq\n\n    ABP'IH : ABP' b i' is' os\u2080\u2075 os\u2081\u2075 as\u2075 bs\u2075 cs\u2075 ds\u2075 js'\n    ABP'IH = ds\u2075-eq , refl , refl , refl , js'-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081 \u2208\n-- Fair, and hence by unfolding Fair we conclude that there are ft\u2081 :\n-- F*T and os\u2081'' : Fair, such that os\u2081' = ft\u2081 ++ os\u2081''.\n--\n-- We proceed by induction on ft\u2081 : F*T using helper.\n\nopen Helper\nlemma\u2082 : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2080' \u2192\n         Fair os\u2081' \u2192\n         ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n         Fair os\u2080''\n         \u2227 Fair os\u2081''\n         \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2080' Fos\u2081' abp' with Fair-unf Fos\u2081'\n... | ft , os\u2080'' , FTft , Fos\u2080'' , h =\n  helper Bb Fos\u2080' abp' ft os\u2080'' FTft Fos\u2080'' h\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b099c4796003a90ed3a5269d9137f3a8e04c3d42","subject":"pisearch: renaming P* to {To,From}Fun","message":"pisearch: renaming P* to {To,From}Fun\n","repos":"crypto-agda\/crypto-agda","old_file":"pisearch.agda","new_file":"pisearch.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nTree : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nTree sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605) \u2192 (B : A \u2192 \u2605) \u2192 \u2605\n\u03a0 A B = (x : A) \u2192 B x\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Tree sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Tree sA B\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ PA PB t = uncurry (PB \u2218 PA t)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ PA PB f = PA (PB \u2218 curry f)\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB PA PB = toFun-\u03a3 sA sB PA PB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB PA PB = fromFun-\u03a3 sA sB PA PB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB PA PB (t , u) (inj\u2081 x) = PA t x\ntoFun-\u228e sA sB PA PB (t , u) (inj\u2082 x) = PB u x\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB PA PB f = PA (f \u2218 inj\u2081) , PB (f \u2218 inj\u2082)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Tree s _) _,_\n","old_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\n\u03a0' : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\n\u03a0' sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605\u2080) \u2192 (B : A \u2192 \u2605\u2080) \u2192 \u2605\u2080\n\u03a0 A B = (x : A) \u2192 B x\n\nP\u2192 : \u2200 {A} (sA : Search A) \u2192 \u2605\u2080\nP\u2192 {A} sA = \u2200 {B} \u2192 \u03a0' sA B \u2192 \u03a0 A B\n\nP\u2190 : \u2200 {A} (sA : Search A) \u2192 \u2605\u2080\nP\u2190 {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 \u03a0' sA B\n\nP\u2192\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n     \u2192 P\u2192 sA\n     \u2192 (\u2200 {x} \u2192 P\u2192 (sB {x}))\n     \u2192 P\u2192 (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nP\u2192\u03a3 _ _ PA PB t = uncurry (PB \u2218 PA t)\n\nP\u2190\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n      \u2192 P\u2190 sA\n      \u2192 (\u2200 {x} \u2192 P\u2190 (sB {x}))\n      \u2192 P\u2190 (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nP\u2190\u03a3 _ _ PA PB f = PA (PB \u2218 curry f)\n\nP\u2192\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2192 sA \u2192 P\u2192 sB \u2192 P\u2192 (sA \u00d7Search sB)\nP\u2192\u00d7 sA sB PA PB = P\u2192\u03a3 sA sB PA PB\n\nP\u2190\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2190 sA \u2192 P\u2190 sB \u2192 P\u2190 (sA \u00d7Search sB)\nP\u2190\u00d7 sA sB PA PB = P\u2190\u03a3 sA sB PA PB\n\nP\u2192Bit : P\u2192 (search \u03bcBit)\nP\u2192Bit (f , t) false = f\nP\u2192Bit (f , t) true  = t\n\nP\u2190Bit : P\u2190 (search \u03bcBit)\nP\u2190Bit f = f false , f true\n\nP\u2192\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2192 sA \u2192 P\u2192 sB \u2192 P\u2192 (sA +Search sB)\nP\u2192\u228e sA sB PA PB (t , u) (inj\u2081 x) = PA t x\nP\u2192\u228e sA sB PA PB (t , u) (inj\u2082 x) = PB u x\n\nP\u2190\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2190 sA \u2192 P\u2190 sB \u2192 P\u2190 (sA +Search sB)\nP\u2190\u228e sA sB PA PB f = PA (f \u2218 inj\u2081) , PB (f \u2218 inj\u2082)\n\n-- P\u2192Ind : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 P\u2192 sA\n-- P\u2192Ind {A} {sA} indA {B} t = ?\n\nP\u2190Ind : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 P\u2190 sA\nP\u2190Ind indA = indA (\u03bb s \u2192 \u03a0' s _) _,_\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4ce8f99bf6219694ddca34b04eec4ee51fddd10e","subject":"ECC (generic modular exp actually)","message":"ECC (generic modular exp actually)\n","repos":"crypto-agda\/crypto-agda","old_file":"ECC\/ecc.agda","new_file":"ECC\/ecc.agda","new_contents":"--open import prelude renaming (Bool to \ud835\udfda; true to 1\u2082; false to 0\u2082)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Data.Two.Base\nopen import Data.List\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen Implicits\nopen import Algebra.Raw\nopen import Algebra.Field\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Group\n\nmodule ecc\n  (\u2119 : Set)\n  (\u2119-monoid : Commutative-Monoid \u2119)\n  (Number : Set)\n  (+-comm-mon : Commutative-Monoid Number)\n  (*-mon : Monoid Number)\n  (open Additive-Commutative-Monoid +-comm-mon)\n  (open Multiplicative-Monoid *-mon)\n  (+-*-distr : \u2200 {x y z} \u2192 (x + y) * z \u2261 x * z + y * z)\n  (*-+-distr : \u2200 {x y z} \u2192 x * (y + z) \u2261 x * y + x * z)\n  (0*-zero : \u2200 {x} \u2192 0# * x \u2261 0#)\n  (*0-zero : \u2200 {x} \u2192 x * 0# \u2261 0#)\n   --(modinv-*-distr : \u2200 {x y} \u2192 modinv (x * y) \u2261 modinv x * modinv y)\n   --(modinv-modinv : \u2200 {x} \u2192 modinv (modinv x) \u2261 x)\n   --(*-assoc : \u2200 {x y z} \u2192 (x * y) * z \u2261 x * (y * z))\n   --(*-comm : \u2200 {x y} \u2192 x * y \u2261 y * x)\n   --(modinv-cancel : \u2200 {x y} \u2192 x * modinv x * y \u2261 y)\n   --(2*1\u2099 : 2* 1# \u2261 2\u2099)\n   --(2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u2099 * n)\n  where\n\nmodule \u2295 = Commutative-Monoid \u2119-monoid\nopen \u2295\n  renaming\n    ( _\u2219_ to _\u2295_\n    ; \u2219= to \u2295=\n    ; \u03b5\u2219-identity to \u03b5\u2295-identity\n    ; \u2219\u03b5-identity to \u2295\u03b5-identity\n    )\n\n2\u00b7_ : \u2119 \u2192 \u2119\n2\u00b7 P = P \u2295 P\n\n2\u00b7-\u2295-distr : \u2200 {P Q} \u2192 2\u00b7 (P \u2295 Q) \u2261 2\u00b7 P \u2295 2\u00b7 Q\n2\u00b7-\u2295-distr = \u2295.interchange\n\n2\u00b7-\u2295 : \u2200 {P Q R} \u2192 2\u00b7 P \u2295 (Q \u2295 R) \u2261 (P \u2295 Q) \u2295 (P \u2295 R)\n2\u00b7-\u2295 = \u2295.interchange\n\n{-\nec-multiply-bin : List \ud835\udfda \u2192 \u2119 \u2192 \u2119\nec-multiply-bin scalar P = go scalar\n  where\n    go : List \ud835\udfda \u2192 \u2119\n    go []       = P\n    go (b \u2237 bs) = [0: x\u2080 1: x\u2081 ] b\n      where x\u2080 = 2\u00b7 go bs\n            x\u2081 = P \u2295 x\u2080\n\nec-multiply : Number \u2192 \u2119 \u2192 \u2119\nec-multiply scalar P =\n  -- if scalar == 0 or scalar >= N: raise Exception(\"Invalid Scalar\/Private Key\")\n    ec-multiply-bin (bin scalar) P\n\n_\u00b7_ = ec-multiply\ninfixr 8 _\u00b7_\n-}\n\n\nopen From-Op\u2082.From-Assoc-Comm _+_ +-assoc +-comm\n  renaming ( on-sides to +-on-sides)\n\ninfixl 6 1+_\ninfixl 7 2*_ 1+2*_\n1+_ = \u03bb x \u2192 1# + x\n2*_ = \u03bb x \u2192 x + x\n1+2*_ = \u03bb x \u2192 1+ 2* x\n\ndata Parity-View : Number \u2192 Set where\n  zero\u27e8_\u27e9    : \u2200 {n} \u2192 n \u2261 0# \u2192 Parity-View n\n  even_by\u27e8_\u27e9 : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 2* m    \u2192 Parity-View n\n  odd_by\u27e8_\u27e9  : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 1+ 2* m \u2192 Parity-View n\n\ncast_by\u27e8_\u27e9 : \u2200 {x y} \u2192 Parity-View x \u2192 y \u2261 x \u2192 Parity-View y\ncast zero\u27e8 x\u2091 \u27e9       by\u27e8 y\u2091 \u27e9 = zero\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast even x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = even x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast odd  x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = odd  x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\n\ninfixr 8 _\u00b7\u209a_\n_\u00b7\u209a_ : \u2200 {n} (p : Parity-View n) \u2192 \u2119 \u2192 \u2119\nzero\u27e8 e \u27e9      \u00b7\u209a P = \u03b5\neven p by\u27e8 e \u27e9 \u00b7\u209a P = 2\u00b7 (p \u00b7\u209a P)\nodd  p by\u27e8 e \u27e9 \u00b7\u209a P = P \u2295 (2\u00b7 (p \u00b7\u209a P))\n\n_+2*_ : \ud835\udfda \u2192 Number \u2192 Number\n0\u2082 +2* m =   2* m\n1\u2082 +2* m = 1+2* m\n\n{-\npostulate\n    bin-2* : \u2200 {n} \u2192 bin (2* n) \u2261 0\u2082 \u2237 bin n\n    bin-1+2* : \u2200 {n} \u2192 bin (1+2* n) \u2261 1\u2082 \u2237 bin n\n\nbin-+2* : (b : \ud835\udfda)(n : Number) \u2192 bin (b +2* n) \u2261 b \u2237 bin n\nbin-+2* 1\u2082 n = bin-1+2*\nbin-+2* 0\u2082 n = bin-2*\n-}\n\n-- (msb) most significant bit first\nbin\u209a : \u2200 {n} \u2192 Parity-View n \u2192 List \ud835\udfda\nbin\u209a zero\u27e8 e \u27e9      = []\nbin\u209a even p by\u27e8 e \u27e9 = 0\u2082 \u2237 bin\u209a p\nbin\u209a odd  p by\u27e8 e \u27e9 = 1\u2082 \u2237 bin\u209a p\n\nhalf : \u2200 {n} \u2192 Parity-View n \u2192 Number\nhalf zero\u27e8 _ \u27e9            = 0#\nhalf (even_by\u27e8_\u27e9 {m} _ _) = m\nhalf (odd_by\u27e8_\u27e9  {m} _ _) = m\n\n{-\nbin-parity : \u2200 {n} (p : ParityView n) \u2192 bin n \u2261 parity p \u2237 bin (half p)\nbin-parity (even n) = bin-2*\nbin-parity (odd n)  = bin-1+2*\n-}\n\ninfixl 6 1+\u209a_ _+\u209a_\n1+\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (1+ x)\n1+\u209a zero\u27e8 e \u27e9      = odd zero\u27e8 refl \u27e9 by\u27e8 ap 1+_ (e \u2219 ! 0+-identity) \u27e9\n1+\u209a even p by\u27e8 e \u27e9 = odd p      by\u27e8 ap 1+_ e \u27e9\n1+\u209a odd  p by\u27e8 e \u27e9 = even 1+\u209a p by\u27e8 ap 1+_ e \u2219 ! +-assoc \u2219 +-interchange \u27e9\n\n_+\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x + y)\nzero\u27e8 x\u2091 \u27e9       +\u209a y\u209a        = cast y\u209a by\u27e8 ap (\u03bb z \u2192 z + _) x\u2091 \u2219 0+-identity \u27e9\nx\u209a              +\u209a zero\u27e8 y\u2091 \u27e9 = cast x\u209a by\u27e8 ap (_+_ _) y\u2091 \u2219 +-comm \u2219 0+-identity \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even     x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-interchange \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-comm \u2219 +-assoc \u2219 ap 1+_ (+-comm \u2219 +-interchange) \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-assoc \u2219 ap 1+_ +-interchange \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = even 1+\u209a (x\u209a +\u209a y\u209a) by\u27e8 += x\u2091 y\u2091 \u2219 +-on-sides refl +-interchange \u27e9\n\ninfixl 7 2*\u209a_\n2*\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (2* x)\n2*\u209a x\u209a = x\u209a +\u209a x\u209a\n\nopen \u2261-Reasoning\n\nmodule _ {P Q} where\n    \u00b7\u209a-\u2295-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a (P \u2295 Q) \u2261 x\u209a \u00b7\u209a P \u2295 x\u209a \u00b7\u209a Q\n    \u00b7\u209a-\u2295-distr zero\u27e8 x\u2091 \u27e9       = ! \u03b5\u2295-identity\n    \u00b7\u209a-\u2295-distr even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u2295-distr x\u209a) \u2219 2\u00b7-\u2295-distr\n    \u00b7\u209a-\u2295-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap (\u03bb z \u2192 P \u2295 Q \u2295 2\u00b7 z) (\u00b7\u209a-\u2295-distr x\u209a)\n                               \u2219 ap (\u03bb z \u2192 P \u2295 Q \u2295 z) (! 2\u00b7-\u2295)\n                               \u2219 \u2295.interchange\n\nmodule _ {P} where\n    cast-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u2091 : y \u2261 x) \u2192 cast x\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P\n    cast-\u00b7\u209a-distr zero\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n\n    1+\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (1+\u209a x\u209a) \u00b7\u209a P \u2261 P \u2295 x\u209a \u00b7\u209a P\n    1+\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9       = ap (_\u2295_ P) \u03b5\u2295-identity\n    1+\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 = refl\n    1+\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (1+\u209a-\u00b7\u209a-distr x\u209a) \u2219 \u2295.interchange \u2219 \u2295.assoc\n\n    +\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y)\n                 \u2192 (x\u209a +\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P\n    +\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = cast-\u00b7\u209a-distr y\u209a _ \u2219 ! \u03b5\u2295-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 \u2295.assoc-comm\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 ! \u2295.assoc\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9\n       = (odd x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd y\u209a by\u27e8 y\u2091 \u27e9) \u00b7\u209a P\n       \u2261\u27e8by-definition\u27e9\n         2\u00b7((1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P)\n       \u2261\u27e8 ap 2\u00b7_ helper \u27e9\n         2\u00b7(P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 ap (_\u2295_ (2\u00b7 P)) 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P) \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295 \u27e9\n         P \u2295 2\u00b7(x\u209a \u00b7\u209a P) \u2295 (P \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8by-definition\u27e9\n         odd x\u209a by\u27e8 x\u2091 \u27e9 \u00b7\u209a P \u2295 odd y\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P\n       \u220e\n        where helper = (1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P\n                     \u2261\u27e8 1+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) \u27e9\n                       P \u2295 ((x\u209a +\u209a y\u209a) \u00b7\u209a P)\n                     \u2261\u27e8 ap (_\u2295_ P) (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u27e9\n                       P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P)\n                     \u220e\n\n*-1+-distr : \u2200 {x y} \u2192 x * (1+ y) \u2261 x + x * y\n*-1+-distr = *-+-distr \u2219 += *1-identity refl\n\n1+-*-distr : \u2200 {x y} \u2192 (1+ x) * y \u2261 y + x * y\n1+-*-distr = +-*-distr \u2219 += 1*-identity refl\n\ninfixl 7 _*\u209a_\n_*\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x * y)\nzero\u27e8 x\u2091 \u27e9       *\u209a y\u209a              = zero\u27e8 *= x\u2091 refl \u2219 0*-zero \u27e9\nx\u209a              *\u209a zero\u27e8 y\u2091 \u27e9       = zero\u27e8 *= refl y\u2091 \u2219 *0-zero \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 *\u209a y\u209a              = even (x\u209a *\u209a y\u209a) by\u27e8 *= x\u2091 refl \u2219 +-*-distr \u27e9\nx\u209a              *\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even (x\u209a *\u209a y\u209a) by\u27e8 *= refl y\u2091 \u2219 *-+-distr \u27e9\nodd x\u209a by\u27e8 x\u2091 \u27e9  *\u209a odd y\u209a by\u27e8 y\u2091 \u27e9  = odd  (x\u209a +\u209a y\u209a +\u209a 2*\u209a (x\u209a *\u209a y\u209a)) by\u27e8 *= x\u2091 y\u2091 \u2219 helper \u27e9\n   where\n     x = _\n     y = _\n     helper = (1+2* x)*(1+2* y)\n            \u2261\u27e8 1+-*-distr \u27e9\n              1+2* y + 2* x * 1+2* y\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2* y + z)\n                 (2* x * 1+2* y\n                 \u2261\u27e8 *-1+-distr \u27e9\n                 (2* x + 2* x * 2* y)\n                 \u2261\u27e8 += refl *-+-distr \u2219 +-interchange \u27e9\n                 (2* (x + 2* x * y))\n                 \u220e) \u27e9\n              1+2* y + 2* (x + 2* x * y)\n            \u2261\u27e8 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n              1+2*(y + (x + 2* x * y))\n            \u2261\u27e8 ap 1+2*_ (! +-assoc \u2219 += +-comm refl) \u27e9\n              1+2*(x + y + 2* x * y)\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2*(x + y + z)) +-*-distr \u27e9\n              1+2*(x + y + 2* (x * y))\n            \u220e\n\nmodule _ {P} where\n    2\u00b7-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 2\u00b7(x\u209a \u00b7\u209a P) \u2261 x\u209a \u00b7\u209a 2\u00b7 P\n    2\u00b7-\u00b7\u209a-distr x\u209a = ! \u00b7\u209a-\u2295-distr x\u209a\n\n    2*\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (2*\u209a x\u209a) \u00b7\u209a P \u2261 2\u00b7(x\u209a \u00b7\u209a P)\n    2*\u209a-\u00b7\u209a-distr x\u209a = +\u209a-\u00b7\u209a-distr x\u209a x\u209a\n\n\u00b7\u209a-\u03b5 : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a \u03b5 \u2261 \u03b5\n\u00b7\u209a-\u03b5 zero\u27e8 x\u2091 \u27e9       = refl\n\u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\u00b7\u209a-\u03b5 odd x\u209a by\u27e8 x\u2091 \u27e9  = \u03b5\u2295-identity \u2219 ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\n*\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y) {P} \u2192 (x\u209a *\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a y\u209a \u00b7\u209a P\n*\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = refl\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 odd  x\u209a by\u27e8 x\u2091 \u27e9\n\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a even y\u209a by\u27e8 y\u2091 \u27e9)\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9)\n\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2091 \u27e9 y\u209a) \u2219 2\u00b7-\u2295-distr \u2219 ap (\u03bb z \u2192 2\u00b7 (y\u209a \u00b7\u209a P) \u2295 2\u00b7 z) (2\u00b7-\u00b7\u209a-distr x\u209a)\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap (_\u2295_ P)\n     (ap 2\u00b7_\n        (+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) (2*\u209a (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) (2*\u209a-\u00b7\u209a-distr (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= \u2295.comm (ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u00b7\u209a-distr x\u209a) \u2219 \u2295.assoc\n        \u2219 \u2295= refl (! \u00b7\u209a-\u2295-distr x\u209a) ) \u2219 2\u00b7-\u2295-distr)\n     \u2219 ! \u2295.assoc\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"--open import prelude renaming (Bool to \ud835\udfda; true to 1\u2082; false to 0\u2082)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Data.Two.Base\nopen import Data.List\nopen import Function\nopen import Algebra.FunctionProperties.Eq\n\nmodule _ where\n\ninfixl 6 _+_ -- _-_\ninfixl 7 _*_ -- _%_\ninfixl 6 _+P_\n\npostulate\n  Number : Set\n  _+_ _*_ : Number \u2192 Number \u2192 Number\n -- _-_ _\/_ _%_ : Number \u2192 Number \u2192 Number\n  -- Pcurve Acurve : Number\n  0\u2099 1\u2099 {-2\u2099 3\u2099-} : Number\n  -- bin : Number \u2192 List \ud835\udfda\n\n  Point : Set\n  _+P_ : Point \u2192 Point \u2192 Point\n  +P-assoc : \u2200 {P Q R} \u2192 (P +P Q) +P R \u2261 P +P (Q +P R)\n  +P-comm : \u2200 {P Q} \u2192 P +P Q \u2261 Q +P P\n\n  0P : Point\n  0P+-cancel : \u2200 {x} \u2192 0P +P x \u2261 x\n\n+0P-cancel : \u2200 {x} \u2192 x +P 0P \u2261 x\n+0P-cancel = +P-comm \u2219 0P+-cancel\n\n2\u00b7_ : Point \u2192 Point\n2\u00b7 P = P +P P\n\n+P= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x +P y \u2261 x' +P y'\n+P= {x} {y' = y'} p q = ap (_+P_ x) q \u2219 ap (\u03bb z \u2192 z +P y') p\n\n+P-interchange : \u2200 {x y z t} \u2192 (x +P y) +P (z +P t) \u2261 (x +P z) +P (y +P t)\n+P-interchange = InterchangeFromAssocComm.\u00b7-interchange _+P_ +P-assoc +P-comm\n\n2\u00b7-+P-2\u00b7 : \u2200 {P Q} \u2192 2\u00b7 (P +P Q) \u2261 2\u00b7 P +P 2\u00b7 Q\n2\u00b7-+P-2\u00b7 = +P-interchange\n\n+P-comm-2of3 : \u2200 {P Q R} \u2192 P +P (Q +P R) \u2261 Q +P (P +P R)\n+P-comm-2of3 = ! +P-assoc \u2219 +P= +P-comm refl \u2219 +P-assoc\n\n2\u00b7-+P : \u2200 {P Q R} \u2192 2\u00b7 P +P (Q +P R) \u2261 (P +P Q) +P (P +P R)\n2\u00b7-+P = +P-interchange\n \n{-\nec-multiply-bin : List \ud835\udfda \u2192 Point \u2192 Point\nec-multiply-bin scalar P = go scalar\n  where\n    go : List \ud835\udfda \u2192 Point\n    go []       = P\n    go (b \u2237 bs) = [0: x\u2080 1: x\u2081 ] b\n      where x\u2080 = 2\u00b7 go bs\n            x\u2081 = P +P x\u2080\n\nec-multiply : Number \u2192 Point \u2192 Point\nec-multiply scalar P =\n  -- if scalar == 0 or scalar >= N: raise Exception(\"Invalid Scalar\/Private Key\")\n    ec-multiply-bin (bin scalar) P\n\n_\u00b7_ = ec-multiply\ninfixr 8 _\u00b7_\n-}\n\ninfixl 6 1+_\ninfixl 7 2*_ 1+2*_\n1+_ = \u03bb x \u2192 1\u2099 + x\n2*_ = \u03bb x \u2192 x + x\n1+2*_ = \u03bb x \u2192 1+ 2* x\n\ndata Parity-View : Number \u2192 Set where\n  zero\u27e8_\u27e9    : \u2200 {n} \u2192 n \u2261 0\u2099 \u2192 Parity-View n\n  even_by\u27e8_\u27e9 : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 2* m    \u2192 Parity-View n\n  odd_by\u27e8_\u27e9  : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 1+ 2* m \u2192 Parity-View n\n\ncast_by\u27e8_\u27e9 : \u2200 {x y} \u2192 Parity-View x \u2192 y \u2261 x \u2192 Parity-View y\ncast zero\u27e8 x\u2091 \u27e9       by\u27e8 y\u2091 \u27e9 = zero\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast even x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = even x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast odd  x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = odd  x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\n\ninfixr 8 _\u00b7\u209a_\n_\u00b7\u209a_ : \u2200 {n} (p : Parity-View n) \u2192 Point \u2192 Point\nzero\u27e8 e \u27e9      \u00b7\u209a P = 0P\neven p by\u27e8 e \u27e9 \u00b7\u209a P = 2\u00b7 (p \u00b7\u209a P)\nodd  p by\u27e8 e \u27e9 \u00b7\u209a P = P +P (2\u00b7 (p \u00b7\u209a P))\n\n+= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n+= {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n*= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n*= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\nmodule Add\u209a\n       (+-assoc : \u2200 {x y z} \u2192 (x + y) + z \u2261 x + (y + z))\n       (+-comm : \u2200 {x y} \u2192 x + y \u2261 y + x)\n       (0+-cancel : \u2200 {x} \u2192 0\u2099 + x \u2261 x)\n       where\n\n    +-interchange : \u2200 {x y z t} \u2192 (x + y) + (z + t) \u2261 (x + z) + (y + t)\n    +-interchange = InterchangeFromAssocComm.\u00b7-interchange _+_ +-assoc +-comm\n\n    +-on-sides : \u2200 {x x' y y' z z' t t'}\n                 \u2192 x + z \u2261 x' + z'\n                 \u2192 y + t \u2261 y' + t'\n                 \u2192 (x + y) + (z + t) \u2261 (x' + y') + (z' + t')\n    +-on-sides p q = +-interchange \u2219 += p q \u2219 +-interchange\n\n    -- only needs interchange and comm\n    {- UNUSED\n    +-inner : \u2200 {x y y' z z' t} \u2192 y + z \u2261 z' + y' \u2192 (x + y) + (z + t) \u2261 (x + y') + (z' + t)\n    +-inner p = += +-comm refl \u2219 +-on-sides (p \u2219 +-comm) refl \u2219 += +-comm refl\n    -}\n\n    _+2*_ : \ud835\udfda \u2192 Number \u2192 Number\n    0\u2082 +2* m =   2* m\n    1\u2082 +2* m = 1+2* m\n\n    {-\n    postulate\n        bin-2* : \u2200 {n} \u2192 bin (2* n) \u2261 0\u2082 \u2237 bin n\n        bin-1+2* : \u2200 {n} \u2192 bin (1+2* n) \u2261 1\u2082 \u2237 bin n\n\n    bin-+2* : (b : \ud835\udfda)(n : Number) \u2192 bin (b +2* n) \u2261 b \u2237 bin n\n    bin-+2* 1\u2082 n = bin-1+2*\n    bin-+2* 0\u2082 n = bin-2*\n    -}\n\n    -- (msb) most significant bit first\n    bin\u209a : \u2200 {n} \u2192 Parity-View n \u2192 List \ud835\udfda\n    bin\u209a zero\u27e8 e \u27e9      = []\n    bin\u209a even p by\u27e8 e \u27e9 = 0\u2082 \u2237 bin\u209a p\n    bin\u209a odd  p by\u27e8 e \u27e9 = 1\u2082 \u2237 bin\u209a p\n\n    half : \u2200 {n} \u2192 Parity-View n \u2192 Number\n    half zero\u27e8 _ \u27e9            = 0\u2099\n    half (even_by\u27e8_\u27e9 {m} _ _) = m\n    half (odd_by\u27e8_\u27e9  {m} _ _) = m\n\n    {-\n    bin-parity : \u2200 {n} (p : ParityView n) \u2192 bin n \u2261 parity p \u2237 bin (half p)\n    bin-parity (even n) = bin-2*\n    bin-parity (odd n)  = bin-1+2*\n    -}\n\n    infixl 6 1+\u209a_ _+\u209a_\n    1+\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (1+ x)\n    1+\u209a zero\u27e8 e \u27e9      = odd zero\u27e8 refl \u27e9 by\u27e8 ap 1+_ (e \u2219 ! 0+-cancel) \u27e9\n    1+\u209a even p by\u27e8 e \u27e9 = odd p      by\u27e8 ap 1+_ e \u27e9\n    1+\u209a odd  p by\u27e8 e \u27e9 = even 1+\u209a p by\u27e8 ap 1+_ e \u2219 ! +-assoc \u2219 +-interchange \u27e9\n\n    _+\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x + y)\n    zero\u27e8 x\u2091 \u27e9       +\u209a y\u209a        = cast y\u209a by\u27e8 ap (\u03bb z \u2192 z + _) x\u2091 \u2219 0+-cancel \u27e9\n    x\u209a              +\u209a zero\u27e8 y\u2091 \u27e9 = cast x\u209a by\u27e8 ap (_+_ _) y\u2091 \u2219 +-comm \u2219 0+-cancel \u27e9\n    even x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even     x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-interchange \u27e9\n    even x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-comm \u2219 +-assoc \u2219 ap 1+_ (+-comm \u2219 +-interchange) \u27e9\n    odd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n    odd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = even 1+\u209a (x\u209a +\u209a y\u209a) by\u27e8 += x\u2091 y\u2091 \u2219 +-on-sides refl +-interchange \u27e9\n\n    infixl 7 2*\u209a_\n    2*\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (2* x)\n    2*\u209a x\u209a = x\u209a +\u209a x\u209a\n\n    open \u2261-Reasoning\n\n    module _ {P Q} where\n        \u00b7\u209a-+P-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a (P +P Q) \u2261 x\u209a \u00b7\u209a P +P x\u209a \u00b7\u209a Q\n        \u00b7\u209a-+P-distr zero\u27e8 x\u2091 \u27e9       = ! 0P+-cancel\n        \u00b7\u209a-+P-distr even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-+P-distr x\u209a) \u2219 2\u00b7-+P-2\u00b7\n        \u00b7\u209a-+P-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap (\u03bb z \u2192 P +P Q +P 2\u00b7 z) (\u00b7\u209a-+P-distr x\u209a)\n                                   \u2219 ap (\u03bb z \u2192 P +P Q +P z) (! 2\u00b7-+P)\n                                   \u2219 +P-interchange\n\n    module _ {P} where\n        cast-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u2091 : y \u2261 x) \u2192 cast x\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P\n        cast-\u00b7\u209a-distr zero\u27e8 x\u2081 \u27e9 y\u2091 = refl\n        cast-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n        cast-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    \n        1+\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (1+\u209a x\u209a) \u00b7\u209a P \u2261 P +P x\u209a \u00b7\u209a P\n        1+\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9       = ap (_+P_ P) 0P+-cancel\n        1+\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 = refl\n        1+\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (1+\u209a-\u00b7\u209a-distr x\u209a) \u2219 +P-interchange \u2219 +P-assoc\n\n        +\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y)\n                     \u2192 (x\u209a +\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P +P y\u209a \u00b7\u209a P\n        +\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = cast-\u00b7\u209a-distr y\u209a _ \u2219 ! 0P+-cancel\n\n        +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! +0P-cancel\n        +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! +0P-cancel\n\n        +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-+P-2\u00b7\n        +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap (_+P_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-+P-2\u00b7) \u2219 +P-comm-2of3\n        +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap (_+P_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-+P-2\u00b7) \u2219 ! +P-assoc\n        +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9\n           = (odd x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd y\u209a by\u27e8 y\u2091 \u27e9) \u00b7\u209a P\n           \u2261\u27e8by-definition\u27e9\n             2\u00b7((1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P)\n           \u2261\u27e8 ap 2\u00b7_ helper \u27e9\n             2\u00b7(P +P (x\u209a \u00b7\u209a P +P y\u209a \u00b7\u209a P))\n           \u2261\u27e8 2\u00b7-+P-2\u00b7 \u27e9\n             2\u00b7 P +P (2\u00b7(x\u209a \u00b7\u209a P +P y\u209a \u00b7\u209a P))\n           \u2261\u27e8 ap (_+P_ (2\u00b7 P)) 2\u00b7-+P-2\u00b7 \u27e9\n             2\u00b7 P +P (2\u00b7(x\u209a \u00b7\u209a P) +P 2\u00b7(y\u209a \u00b7\u209a P))\n           \u2261\u27e8 2\u00b7-+P \u27e9\n             P +P 2\u00b7(x\u209a \u00b7\u209a P) +P (P +P 2\u00b7(y\u209a \u00b7\u209a P))\n           \u2261\u27e8by-definition\u27e9\n             odd x\u209a by\u27e8 x\u2091 \u27e9 \u00b7\u209a P +P odd y\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P\n           \u220e\n            where helper = (1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P\n                         \u2261\u27e8 1+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) \u27e9\n                           P +P ((x\u209a +\u209a y\u209a) \u00b7\u209a P)\n                         \u2261\u27e8 ap (_+P_ P) (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u27e9\n                           P +P (x\u209a \u00b7\u209a P +P y\u209a \u00b7\u209a P)\n                         \u220e\n\n    module Mult\u209a\n       (+-*-distr : \u2200 {x y z} \u2192 (x + y) * z \u2261 x * z + y * z)\n       (*-+-distr : \u2200 {x y z} \u2192 x * (y + z) \u2261 x * y + x * z)\n       (1*-cancel : \u2200 {x} \u2192 1\u2099 * x \u2261 x)\n       (*1-cancel : \u2200 {x} \u2192 x * 1\u2099 \u2261 x)\n       (0*-cancel : \u2200 {x} \u2192 0\u2099 * x \u2261 0\u2099)\n       (*0-cancel : \u2200 {x} \u2192 x * 0\u2099 \u2261 0\u2099)\n       --(modinv-*-distr : \u2200 {x y} \u2192 modinv (x * y) \u2261 modinv x * modinv y)\n       --(modinv-modinv : \u2200 {x} \u2192 modinv (modinv x) \u2261 x)\n       --(*-assoc : \u2200 {x y z} \u2192 (x * y) * z \u2261 x * (y * z))\n       --(*-comm : \u2200 {x y} \u2192 x * y \u2261 y * x)\n       --(modinv-cancel : \u2200 {x y} \u2192 x * modinv x * y \u2261 y)\n       --(2*1\u2099 : 2* 1\u2099 \u2261 2\u2099)\n       --(2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u2099 * n)\n       where\n\n        *-1+-distr : \u2200 {x y} \u2192 x * (1+ y) \u2261 x + x * y\n        *-1+-distr = *-+-distr \u2219 += *1-cancel refl\n\n        1+-*-distr : \u2200 {x y} \u2192 (1+ x) * y \u2261 y + x * y\n        1+-*-distr = +-*-distr \u2219 += 1*-cancel refl\n\n        infixl 7 _*\u209a_\n        _*\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x * y)\n        zero\u27e8 x\u2091 \u27e9       *\u209a y\u209a              = zero\u27e8 *= x\u2091 refl \u2219 0*-cancel \u27e9\n        x\u209a              *\u209a zero\u27e8 y\u2091 \u27e9       = zero\u27e8 *= refl y\u2091 \u2219 *0-cancel \u27e9\n        even x\u209a by\u27e8 x\u2091 \u27e9 *\u209a y\u209a              = even (x\u209a *\u209a y\u209a) by\u27e8 *= x\u2091 refl \u2219 +-*-distr \u27e9\n        x\u209a              *\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even (x\u209a *\u209a y\u209a) by\u27e8 *= refl y\u2091 \u2219 *-+-distr \u27e9\n        odd x\u209a by\u27e8 x\u2091 \u27e9  *\u209a odd y\u209a by\u27e8 y\u2091 \u27e9  = odd  (x\u209a +\u209a y\u209a +\u209a 2*\u209a (x\u209a *\u209a y\u209a)) by\u27e8 *= x\u2091 y\u2091 \u2219 helper \u27e9\n           where\n             x = _\n             y = _\n             helper = (1+2* x)*(1+2* y)\n                    \u2261\u27e8 1+-*-distr \u27e9\n                      1+2* y + 2* x * 1+2* y\n                    \u2261\u27e8 ap (\u03bb z \u2192 1+2* y + z)\n                         (2* x * 1+2* y\n                         \u2261\u27e8 *-1+-distr \u27e9\n                         (2* x + 2* x * 2* y)\n                         \u2261\u27e8 += refl *-+-distr \u2219 +-interchange \u27e9\n                         (2* (x + 2* x * y))\n                         \u220e) \u27e9\n                      1+2* y + 2* (x + 2* x * y)\n                    \u2261\u27e8 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n                      1+2*(y + (x + 2* x * y))\n                    \u2261\u27e8 ap 1+2*_ (! +-assoc \u2219 += +-comm refl) \u27e9\n                      1+2*(x + y + 2* x * y)\n                    \u2261\u27e8 ap (\u03bb z \u2192 1+2*(x + y + z)) +-*-distr \u27e9\n                      1+2*(x + y + 2* (x * y))\n                    \u220e\n\n        module _ {P} where\n            2\u00b7-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 2\u00b7(x\u209a \u00b7\u209a P) \u2261 x\u209a \u00b7\u209a 2\u00b7 P\n            2\u00b7-\u00b7\u209a-distr x\u209a = ! \u00b7\u209a-+P-distr x\u209a\n\n            2*\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (2*\u209a x\u209a) \u00b7\u209a P \u2261 2\u00b7(x\u209a \u00b7\u209a P)\n            2*\u209a-\u00b7\u209a-distr x\u209a = +\u209a-\u00b7\u209a-distr x\u209a x\u209a\n\n        \u00b7\u209a-0P : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a 0P \u2261 0P\n        \u00b7\u209a-0P zero\u27e8 x\u2091 \u27e9       = refl\n        \u00b7\u209a-0P even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-0P x\u209a) \u2219 0P+-cancel\n        \u00b7\u209a-0P odd x\u209a by\u27e8 x\u2091 \u27e9  = 0P+-cancel \u2219 ap 2\u00b7_ (\u00b7\u209a-0P x\u209a) \u2219 0P+-cancel\n\n        *\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y) {P} \u2192 (x\u209a *\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a y\u209a \u00b7\u209a P\n        *\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = refl\n        *\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-0P even x\u209a by\u27e8 x\u2091 \u27e9\n        *\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-0P odd  x\u209a by\u27e8 x\u2091 \u27e9\n\n        *\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a even y\u209a by\u27e8 y\u2091 \u27e9)\n        *\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9)\n\n        *\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 {P} = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2091 \u27e9 y\u209a) \u2219 2\u00b7-+P-2\u00b7 \u2219 ap (\u03bb z \u2192 2\u00b7 (y\u209a \u00b7\u209a P) +P 2\u00b7 z) (2\u00b7-\u00b7\u209a-distr x\u209a)\n        *\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n           ap (_+P_ P)\n             (ap 2\u00b7_\n                (+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) (2*\u209a (x\u209a *\u209a y\u209a))\n                \u2219 +P= (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) (2*\u209a-\u00b7\u209a-distr (x\u209a *\u209a y\u209a))\n                \u2219 +P= +P-comm (ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u00b7\u209a-distr x\u209a) \u2219 +P-assoc\n                \u2219 +P= refl (! \u00b7\u209a-+P-distr x\u209a) ) \u2219 2\u00b7-+P-2\u00b7)\n             \u2219 ! +P-assoc\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dc4c757de571dbf9decd0ec09fa3bce159f0b948","subject":"Direct proof of well-founded induction.","message":"Direct proof of well-founded induction.\n\nIgnore-this: be4f83042601bf8c031fec62e2fb0878\n\ndarcs-hash:20100525021915-3bd4e-695229f36277d68ee396016369b2744726dd84c7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/N\/Induction\/WellFounded.agda","new_file":"LTC\/Data\/N\/Induction\/WellFounded.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on N\n------------------------------------------------------------------------------\n\nmodule LTC.Data.N.Induction.WellFounded where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Relation.Inequalities\nopen import LTC.Relation.Inequalities.Properties\n\n------------------------------------------------------------------------------\n-- Well-founded induction on N\n-- Adapted from http:\/\/code.haskell.org\/~dolio\/agda-share\/induction\/.\n\nwfIndN-LT :\n   (P : D \u2192 Set) \u2192\n   ({m : D} \u2192 N m \u2192 ({n : D} \u2192 N n \u2192 LT n m \u2192 P n ) \u2192 P m ) \u2192\n   {n : D} \u2192 N n \u2192 P n\nwfIndN-LT P accH Nn = accH Nn (wfAux Nn)\n  where\n    wfAux : {m : D} \u2192 N m \u2192 {n : D} \u2192 N n \u2192 LT n m \u2192 P n\n    wfAux zN      Nn      n<0   = \u22a5-elim (\u00acx<0 Nn n<0)\n    wfAux (sN Nm) zN      0<Sm  = accH zN (\u03bb Nn' n'<0 \u2192 \u22a5-elim (\u00acx<0 Nn' n'<0))\n    wfAux (sN {m} Nm) (sN {n} Nn) Sn<Sm =\n      accH (sN Nn) (\u03bb Nn' n'<Sn \u2192\n        wfAux Nm Nn' (Sx\u2264y\u2192x<y Nn' Nm\n                            (\u2264-trans (sN Nn') (sN Nn) Nm\n                                     (x<y\u2192Sx\u2264y Nn' (sN Nn) n'<Sn)\n                                     (Sx\u2264Sy\u2192x\u2264y {succ n} {m}\n                                                (x<y\u2192Sx\u2264y (sN Nn)\n                                                          (sN Nm)\n                                                          Sn<Sm)))))\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on N\n------------------------------------------------------------------------------\n\nmodule LTC.Data.N.Induction.WellFounded where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Relation.Inequalities\nopen import LTC.Relation.Inequalities.Properties\n\n------------------------------------------------------------------------------\n-- General setting.\n-- Adapted from http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/.\n\ndata AccN (R : D \u2192 D \u2192 Set) : {x : D} \u2192 N x \u2192 Set where\n  accN : {x : D} \u2192 (Nx : N x) \u2192 ({y : D} \u2192 (Ny : N y) \u2192 R y x \u2192 AccN R Ny) \u2192\n         AccN R Nx\n\nWellFoundedN : (D \u2192 D \u2192 Set) \u2192 Set\nWellFoundedN R = {x : D} \u2192 (Nx : N x) \u2192 AccN R Nx\n\naccFoldN : (R : D \u2192 D \u2192 Set){P : D \u2192 Set} \u2192\n           ({x : D} \u2192 N x \u2192 ({y : D} \u2192 N y \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n           {x : D} \u2192 (Nx : N x) \u2192 AccN R Nx \u2192 P x\naccFoldN R f Nx (accN .Nx h) = f Nx (\u03bb Ny y<x \u2192 accFoldN R f Ny (h Ny y<x))\n\nwfIndN : {R : D \u2192 D \u2192 Set} {P : D \u2192 Set} \u2192\n         WellFoundedN R \u2192\n         ({x : D} \u2192 N x \u2192 ({y : D} \u2192 N y \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n         {x : D} \u2192 N x \u2192 P x\nwfIndN {R = R} wf f Nx = accFoldN R f Nx (wf Nx)\n\n------------------------------------------------------------------------------\n-- LT is well-founded\n\n-- Adapted from http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/ and http:\/\/code.haskell.org\/~dolio\/agda-share\/induction\/.\n\naccess : {m : D} \u2192 N m \u2192 {n : D} \u2192 (Nn : N n) \u2192 LT n m \u2192 AccN LT Nn\naccess zN      Nn      n<0   = \u22a5-elim (\u00acx<0 Nn n<0)\naccess (sN Nm) zN      0<Sm  = accN zN (\u03bb Nn' n'<0 \u2192 \u22a5-elim (\u00acx<0 Nn' n'<0))\naccess (sN {m} Nm) (sN {n} Nn) Sn<Sm =\n  accN (sN Nn) (\u03bb Nn' n'<Sn \u2192\n    access Nm Nn' (Sx\u2264y\u2192x<y Nn' Nm\n                            (\u2264-trans (sN Nn') (sN Nn) Nm\n                                     (x<y\u2192Sx\u2264y Nn' (sN Nn) n'<Sn)\n                                     (Sx\u2264Sy\u2192x\u2264y {succ n} {m} (x<y\u2192Sx\u2264y (sN Nn)\n                                                                       (sN Nm)\n                                                                       Sn<Sm)))))\n\nwf-LT : WellFoundedN LT\nwf-LT Nx = accN Nx (access Nx)\n\n------------------------------------------------------------------------------\n-- Well-founded induction on N\n\nwfIndN-LT :\n   (P : D \u2192 Set) \u2192\n   ({m : D} \u2192 N m \u2192 ({n : D} \u2192 N n \u2192 LT n m \u2192 P n ) \u2192 P m ) \u2192\n   {n : D} \u2192 N n \u2192 P n\nwfIndN-LT P accH Nn = wfIndN {LT} {\u03bb x \u2192 P x} wf-LT accH Nn\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"89ba70944bd5b8ab30191f0e7b6948d595332dd2","subject":"Added x<y\u2192Sx-y\u22610 (by Ana Bove).","message":"Added x<y\u2192Sx-y\u22610 (by Ana Bove).\n\nIgnore-this: 8b8b39c215a93e23780151e5e8ec1a90\n\ndarcs-hash:20110210164212-3bd4e-08c2d3698ac5138c67a02ecd3e408ea7122dc0b2.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9cd4d1be62855d1e21802868d457fbaebe609715","subject":"RenderToHTML.agda: Cleaning.","message":"RenderToHTML.agda: Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/RenderToHTML.agda","new_file":"notes\/RenderToHTML.agda","new_contents":"------------------------------------------------------------------------------\n-- Draft modules render to HTML (via an external Makefile)\n------------------------------------------------------------------------------\n\nmodule RenderToHTML where\n","old_contents":"------------------------------------------------------------------------------\n-- Draft modules render to HTML\n------------------------------------------------------------------------------\n\nmodule RenderToHTML where\n\nopen import Lists\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db02f917e04f74e4f64601f3fee93a2cb7e9d9ae","subject":"Map \/ fold for ind","message":"Map \/ fold for ind\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Primitive.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Primitive.agda","new_contents":"module Spire.DarkwingDuck.Primitive where\n\n----------------------------------------------------------------------\n\ninfixr 4 _,_\ninfixr 5 _\u2237_\n\n----------------------------------------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\n\nelimBot : (P : \u22a5 \u2192 Set)\n  (v : \u22a5) \u2192 P v\nelimBot P ()\n\n----------------------------------------------------------------------\n\ndata \u22a4 : Set where\n  tt : \u22a4\n\nelimUnit : (P : \u22a4 \u2192 Set)\n  (ptt : P tt)\n  (u : \u22a4) \u2192 P u\nelimUnit P ptt tt = ptt\n\n----------------------------------------------------------------------\n\ndata \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  _,_ : (a : A) (b : B a) \u2192 \u03a3 A B\n\nelimPair : {A : Set} {B : A \u2192 Set}\n  (P : \u03a3 A B \u2192 Set)\n  (ppair : (a : A) (b : B a) \u2192 P (a , b))\n  (ab : \u03a3 A B) \u2192 P ab\nelimPair P ppair (a , b) = ppair a b\n\n----------------------------------------------------------------------\n\ndata _\u2261_ {A : Set} (x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\nelimEq : {A : Set} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  (prefl : P x refl)\n  (y : A) (q : x \u2261 y) \u2192 P y q\nelimEq P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\nelimList : {A : Set} (P : List A \u2192 Set)\n  (pnil : P [])\n  (pcons : (x : A) (xs : List A) \u2192 P xs \u2192 P (x \u2237 xs))\n  (xs : List A) \u2192 P xs\nelimList P pnil pcons [] = pnil\nelimList P pnil pcons (x \u2237 xs) = pcons x xs (elimList P pnil pcons xs)\n\n----------------------------------------------------------------------\n\ndata Elem {A : Set} : List A \u2192 Set where\n  here : \u2200{x xs} \u2192 Elem (x \u2237 xs)\n  there : \u2200{x xs} \u2192 Elem xs \u2192 Elem (x \u2237 xs)\n\nelimElem : {A : Set} (P : (xs : List A) \u2192 Elem xs \u2192 Set)\n  (phere : (x : A) (xs : List A) \u2192 P (x \u2237 xs) here)\n  (pthere : (x : A) (xs : List A) (t : Elem xs) \u2192 P xs t \u2192 P (x \u2237 xs) (there t))\n  (xs : List A) (t : Elem xs) \u2192 P xs t\nelimElem P phere pthere (x \u2237 xs) here = phere x xs\nelimElem P phere pthere (x \u2237 xs) (there t) = pthere x xs t (elimElem P phere pthere xs t)\n\n----------------------------------------------------------------------\n\ndata Tel : Set\u2081 where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nelimTel : (P : Tel \u2192 Set)\n  (pend : P End)\n  (parg  : (A : Set) (B : A \u2192 Tel) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (T : Tel) \u2192 P T\nelimTel P pend parg End = pend\nelimTel P pend parg (Arg A B) = parg A B (\u03bb a \u2192 elimTel P pend parg (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec  : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 (I \u2192 Set) \u2192 I \u2192 Set\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D)  X i = \u03a3 (X j) (\u03bb _ \u2192 El\u1d30 D X i)\nEl\u1d30 (Arg A B)  X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D)  X P i (x , xs) = \u03a3 (P j x) (\u03bb _ \u2192 Hyps D X P i xs)\nHyps (Arg A B)  X P i (a , xs) = Hyps (B a) X P i xs\n\n----------------------------------------------------------------------\n\ndata \u03bc (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p)) (i : I p) : Set where\n  init : El\u1d30 D (\u03bc \u2113 P I p D) i \u2192 \u03bc \u2113 P I p D i\n\nmap : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set)\n  (p : (i : I) (x : X i) \u2192 P i x) (i : I) (xs : El\u1d30 D X i)\n  \u2192 Hyps D X P i xs\nmap (End j) X P p i q = tt\nmap (Rec j D) X P p i (x , xs) = p j x , map D X P p i xs\nmap (Arg A B) X P p i (a , xs) = map (B a) X P p i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc \u2113 P I p D) i) (ihs : Hyps D (\u03bc \u2113 P I p D) M i xs) \u2192 M i (init xs))\n  (i : I p)\n  (x : \u03bc \u2113 P I p D i)\n  \u2192 M i x\nind \u2113 P I p D M \u03b1 i (init xs) = \u03b1 i xs (map D (\u03bc \u2113 P I p D) M (ind \u2113 P I p D M \u03b1) i xs)\n\n----------------------------------------------------------------------\n","old_contents":"module Spire.DarkwingDuck.Primitive where\n\n----------------------------------------------------------------------\n\ninfixr 4 _,_\ninfixr 5 _\u2237_\n\n----------------------------------------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\n\nelimBot : (P : \u22a5 \u2192 Set)\n  (v : \u22a5) \u2192 P v\nelimBot P ()\n\n----------------------------------------------------------------------\n\ndata \u22a4 : Set where\n  tt : \u22a4\n\nelimUnit : (P : \u22a4 \u2192 Set)\n  (ptt : P tt)\n  (u : \u22a4) \u2192 P u\nelimUnit P ptt tt = ptt\n\n----------------------------------------------------------------------\n\ndata \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  _,_ : (a : A) (b : B a) \u2192 \u03a3 A B\n\nelimPair : {A : Set} {B : A \u2192 Set}\n  (P : \u03a3 A B \u2192 Set)\n  (ppair : (a : A) (b : B a) \u2192 P (a , b))\n  (ab : \u03a3 A B) \u2192 P ab\nelimPair P ppair (a , b) = ppair a b\n\n----------------------------------------------------------------------\n\ndata _\u2261_ {A : Set} (x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\nelimEq : {A : Set} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  (prefl : P x refl)\n  (y : A) (q : x \u2261 y) \u2192 P y q\nelimEq P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\nelimList : {A : Set} (P : List A \u2192 Set)\n  (pnil : P [])\n  (pcons : (x : A) (xs : List A) \u2192 P xs \u2192 P (x \u2237 xs))\n  (xs : List A) \u2192 P xs\nelimList P pnil pcons [] = pnil\nelimList P pnil pcons (x \u2237 xs) = pcons x xs (elimList P pnil pcons xs)\n\n----------------------------------------------------------------------\n\ndata Elem {A : Set} : List A \u2192 Set where\n  here : \u2200{x xs} \u2192 Elem (x \u2237 xs)\n  there : \u2200{x xs} \u2192 Elem xs \u2192 Elem (x \u2237 xs)\n\nelimElem : {A : Set} (P : (xs : List A) \u2192 Elem xs \u2192 Set)\n  (phere : (x : A) (xs : List A) \u2192 P (x \u2237 xs) here)\n  (pthere : (x : A) (xs : List A) (t : Elem xs) \u2192 P xs t \u2192 P (x \u2237 xs) (there t))\n  (xs : List A) (t : Elem xs) \u2192 P xs t\nelimElem P phere pthere (x \u2237 xs) here = phere x xs\nelimElem P phere pthere (x \u2237 xs) (there t) = pthere x xs t (elimElem P phere pthere xs t)\n\n----------------------------------------------------------------------\n\ndata Tel : Set\u2081 where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nelimTel : (P : Tel \u2192 Set)\n  (pend : P End)\n  (parg  : (A : Set) (B : A \u2192 Tel) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (T : Tel) \u2192 P T\nelimTel P pend parg End = pend\nelimTel P pend parg (Arg A B) = parg A B (\u03bb a \u2192 elimTel P pend parg (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec  : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 (I \u2192 Set) \u2192 I \u2192 Set\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D)  X i = \u03a3 (X j) (\u03bb _ \u2192 El\u1d30 D X i)\nEl\u1d30 (Arg A B)  X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D)  X P i (x , xs) = \u03a3 (P j x) (\u03bb _ \u2192 Hyps D X P i xs)\nHyps (Arg A B)  X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p)) (i : I p) : Set where\n  init : El\u1d30 D (\u03bc \u2113 P I p D) i \u2192 \u03bc \u2113 P I p D i\n\nind : {\u2113 : String} {P : Set} {I : P \u2192 Set} {p : P} (D : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc \u2113 P I p D) i) (ihs : Hyps D (\u03bc \u2113 P I p D) M i xs) \u2192 M i (init xs))\n  (i : I p)\n  (x : \u03bc \u2113 P I p D i)\n  \u2192 M i x\n\nprove : {\u2113 : String} {P : Set} {I : P \u2192 Set} {p : P} (D E : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc \u2113 P I p E) i) (ihs : Hyps E (\u03bc \u2113 P I p E) M i xs) \u2192 M i (init xs))\n  (i : (I p)) (xs : El\u1d30 D (\u03bc \u2113 P I p E) i) \u2192 Hyps D (\u03bc \u2113 P I p E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f8ab2823470050f06bdf6739063602bc13562e28","subject":"Added succCong and predCong.","message":"Added succCong and predCong.\n\nIgnore-this: b4c73cd153958f7ba9fc0d4e83461efc\n\ndarcs-hash:20120602040222-3bd4e-9bac12b25b9f6fecd7824a3f3f73118599725822.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC-PCF\/Data\/Nat\/PropertiesI.agda","new_file":"src\/LTC-PCF\/Data\/Nat\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\npredCong : \u2200 {m n} \u2192 m \u2261 n \u2192 pred\u2081 m \u2261 pred\u2081 n\npredCong refl = refl\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n+-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n+-Sx m n =\n  rec (succ\u2081 m) n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)))\n    \u2261\u27e8 rec-S m n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u00b7 m \u00b7 (m + n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m \u00b7 (m + n)\n    \u2261\u27e8 refl \u27e9\n  lam succ\u2081 \u00b7 (m + n)\n    \u2261\u27e8 beta succ\u2081 (m + n) \u27e9\n  succ\u2081 (m + n) \u220e\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S zN =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 predCong (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN      = \u2238-x0 zero\n\u2238-0x (sN Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ zN =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS zN (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS zN Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (sN {m} Nm) (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS (sN Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 refl \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n)) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = rec zero zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u2261\u27e8 rec-0 zero \u27e9\n         zero                                          \u220e\n\n*-Sx : \u2200 m n \u2192 succ\u2081 m * n \u2261 n + m * n\n*-Sx m n =\n  rec (succ\u2081 m) zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)))\n    \u2261\u27e8 rec-S m zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u27e9\n  (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u00b7 m \u00b7 (m * n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m) \u27e9\n  (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m \u00b7 (m * n)\n    \u2261\u27e8 refl \u27e9\n  lam (\u03bb y \u2192 n + y) \u00b7 (m * n)\n    \u2261\u27e8 beta (\u03bb y \u2192 n + y) (m * n) \u27e9\n  (\u03bb y \u2192 n + y) (m * n)\n    \u2261\u27e8 refl \u27e9\n  n + (m * n) \u220e\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym (+-0x n)) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm       zN     = subst N (sym (\u2238-x0 m)) Nm\n\u2238-N     zN      (sN Nn) = subst N (sym (\u2238-0S Nn)) zN\n\u2238-N     (sN Nm) (sN Nn) = subst N (sym (\u2238-SS Nm Nn)) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t) (+-rightIdentity Nn) refl)\n\n\n+-assoc : \u2200 {m} \u2192 N m \u2192  \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zN n o =\n  zero + n + o   \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (sN {m} Nm) n o =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm n o) refl \u27e9\n  succ\u2081 (m + (n + o))\n    \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] zN n =\n  zero + succ\u2081 n \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym (+-leftIdentity n)) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (sN {m} Nm) n =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] Nm n) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym (+-Sx m n)) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n    n \u2238 (zero + o)\n      \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n    n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN          _  = subst N (sym (*-0x n)) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero n) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n              \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = sym\n  ( zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero n) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym (*-leftZero (succ\u2081 n)) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym (+-assoc Nn m (m * n)))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n             refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym (*-Sx m n))\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero n) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n  \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m    \u2261\u27e8 sym (x*Sy\u2261x+xy Nn Nm) \u27e9\n  n * succ\u2081 m  \u220e\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o            \u2261\u27e8 sym (\u2238-x0 (m * o)) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym (*-0x o)) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S Nn) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0x (*-N (sN Nn) No)) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o)\n             (sym (*-0x o))\n             refl\n    \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0x (*-N (sN Nn) zN)) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym (*-0x (succ\u2081 m)))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (\u2238-SS Nm Nn)\n             refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n    \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym ([x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No))) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym (*-Sx m (succ\u2081 o)))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym (*-Sx n (succ\u2081 o)))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n  zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n  zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n  m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                       \u27e9\n  m * zero + n * zero  \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym (+-leftIdentity (n * succ\u2081 o)) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym (*-0x (succ\u2081 o)))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym (*-leftZero (succ\u2081 o)))\n             refl\n    \u27e9\n    succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym (+-assoc (sN No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym (*-Sx m (succ\u2081 o)))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n+-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n+-Sx m n =\n  rec (succ\u2081 m) n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)))\n    \u2261\u27e8 rec-S m n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u00b7 m \u00b7 (m + n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m \u00b7 (m + n)\n    \u2261\u27e8 refl \u27e9\n  lam succ\u2081 \u00b7 (m + n)\n    \u2261\u27e8 beta succ\u2081 (m + n) \u27e9\n  succ\u2081 (m + n) \u220e\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S zN =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 cong pred\u2081 (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN      = \u2238-x0 zero\n\u2238-0x (sN Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ zN =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS zN (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS zN Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (sN {m} Nm) (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS (sN Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 refl \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n)) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = rec zero zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u2261\u27e8 rec-0 zero \u27e9\n         zero                                          \u220e\n\n*-Sx : \u2200 m n \u2192 succ\u2081 m * n \u2261 n + m * n\n*-Sx m n =\n  rec (succ\u2081 m) zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)))\n    \u2261\u27e8 rec-S m zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u27e9\n  (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u00b7 m \u00b7 (m * n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m) \u27e9\n  (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m \u00b7 (m * n)\n    \u2261\u27e8 refl \u27e9\n  lam (\u03bb y \u2192 n + y) \u00b7 (m * n)\n    \u2261\u27e8 beta (\u03bb y \u2192 n + y) (m * n) \u27e9\n  (\u03bb y \u2192 n + y) (m * n)\n    \u2261\u27e8 refl \u27e9\n  n + (m * n) \u220e\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym (+-0x n)) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm       zN     = subst N (sym (\u2238-x0 m)) Nm\n\u2238-N     zN      (sN Nn) = subst N (sym (\u2238-0S Nn)) zN\n\u2238-N     (sN Nm) (sN Nn) = subst N (sym (\u2238-SS Nm Nn)) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t) (+-rightIdentity Nn) refl)\n\n\n+-assoc : \u2200 {m} \u2192 N m \u2192  \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zN n o =\n  zero + n + o   \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (sN {m} Nm) n o =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm n o) refl \u27e9\n  succ\u2081 (m + (n + o))\n    \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] zN n =\n  zero + succ\u2081 n \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym (+-leftIdentity n)) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (sN {m} Nm) n =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] Nm n) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym (+-Sx m n)) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n    n \u2238 (zero + o)\n      \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n    n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN          _  = subst N (sym (*-0x n)) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero n) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n              \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = sym\n  ( zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero n) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym (*-leftZero (succ\u2081 n)) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym (+-assoc Nn m (m * n)))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n             refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym (*-Sx m n))\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero n) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n  \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m    \u2261\u27e8 sym (x*Sy\u2261x+xy Nn Nm) \u27e9\n  n * succ\u2081 m  \u220e\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o            \u2261\u27e8 sym (\u2238-x0 (m * o)) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym (*-0x o)) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S Nn) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0x (*-N (sN Nn) No)) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o)\n             (sym (*-0x o))\n             refl\n    \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0x (*-N (sN Nn) zN)) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym (*-0x (succ\u2081 m)))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (\u2238-SS Nm Nn)\n             refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n    \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym ([x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No))) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym (*-Sx m (succ\u2081 o)))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym (*-Sx n (succ\u2081 o)))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n  zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n  zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n  m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                       \u27e9\n  m * zero + n * zero  \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym (+-leftIdentity (n * succ\u2081 o)) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym (*-0x (succ\u2081 o)))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym (*-leftZero (succ\u2081 o)))\n             refl\n    \u27e9\n    succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym (+-assoc (sN No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym (*-Sx m (succ\u2081 o)))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bf19df96ca7ae8136f7e9ad610e9e93926ed9c4c","subject":"updating constructor name","message":"updating constructor name\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2)       | V VConst | _\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2)    | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FDot (ITLam (FVal x\u2081)) FDot)\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!}) -- stuck on defining substitution\n  progress (TAAp D1 x\u2082 D2)            | _   | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2)            | E x | _    = E (EAp1 x)\n  -- progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I IEHole | V x\u2081 = I (IAp IEHole (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I (INEHole x) | V x\u2081 = I (IAp (INEHole x) (FVal x\u2081))\n  progress (TAAp D1 x\u2083 D2) | I (IAp x x\u2081) | V x\u2082 = I (IAp (IAp x x\u2081) (FVal x\u2082))\n  progress (TAAp D1 x\u2082 D2) | I (ICast x) | V x\u2081 = I (IAp (ICast x) (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , (Step {!!} (ITLam (FIndet {!!})) {!!}))\n  progress (TAAp {d2 = d2} D1 x\u2082 D2) | S (\u03c01 , \u03c02) | _ = S ((\u03c01 \u2218 d2) , {!!})\n  -- progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  -- progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {m = \u2717} x x\u2081) = S (_ , Step FDot ITEHole FDot)\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | V v = S (_ , Step FDot (ITNEHole (FVal v)) FDot)\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | I x = S (_ , Step FDot (ITNEHole (FIndet x)) FDot)\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = E EConst\n  progress (TACast TAConst TCHole2) | V VConst = E EConst\n  progress (TACast D x\u2081) | V VLam\n    with invert-lam D\n  progress (TACast D TCRefl)        | V VLam | \u03c41 , \u03c42 , refl = {!!}\n  progress (TACast D TCHole2)       | V VLam | \u03c41 , \u03c42 , refl = {!!}\n  progress (TACast D (TCArr x\u2081 x\u2082)) | V VLam | \u03c41 , \u03c42 , refl = {!!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2)       | V VConst | _\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2)    | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!}) -- stuck on defining substitution\n  progress (TAAp D1 x\u2082 D2)            | _   | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2)            | E x | _    = E (EAp1 x)\n  -- progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I IEHole | V x\u2081 = I (IAp IEHole (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I (INEHole x) | V x\u2081 = I (IAp (INEHole x) (FVal x\u2081))\n  progress (TAAp D1 x\u2083 D2) | I (IAp x x\u2081) | V x\u2082 = I (IAp (IAp x x\u2081) (FVal x\u2082))\n  progress (TAAp D1 x\u2082 D2) | I (ICast x) | V x\u2081 = I (IAp (ICast x) (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , (Step {!!} (ITLam (FIndet {!!})) {!!}))\n  progress (TAAp {d2 = d2} D1 x\u2082 D2) | S (\u03c01 , \u03c02) | _ = S ((\u03c01 \u2218 d2) , {!!})\n  -- progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  -- progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {m = \u2717} x x\u2081) = S (_ , Step FRefl ITEHole FRefl)\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | V v = S (_ , Step FRefl (ITNEHole (FVal v)) FRefl)\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | I x = S (_ , Step FRefl (ITNEHole (FIndet x)) FRefl)\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = E EConst\n  progress (TACast TAConst TCHole2) | V VConst = E EConst\n  progress (TACast D x\u2081) | V VLam\n    with invert-lam D\n  progress (TACast D TCRefl)        | V VLam | \u03c41 , \u03c42 , refl = {!!}\n  progress (TACast D TCHole2)       | V VLam | \u03c41 , \u03c42 , refl = {!!}\n  progress (TACast D (TCArr x\u2081 x\u2082)) | V VLam | \u03c41 , \u03c42 , refl = {!!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"779e8fa8a689f3359678d254a21864800c8a0179","subject":"Nat: FromAssocComm _*_, with implicit arguments","message":"Nat: FromAssocComm _*_, with implicit arguments\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.Eq\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromAssocComm _+_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.+-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.+-comm x y)\n  renaming ( assoc-comm to +-assoc-comm\n           ; interchange to +-interchange\n           )\n\nopen FromAssocComm _*_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.*-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.*-comm x y)\n  renaming ( assoc-comm to *-assoc-comm\n           ; interchange to *-interchange\n           )\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm {a} {1} {b}\n                                  | +-assoc-comm {a \u2294 b} {1} {a \u2293 b}\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm {1} {a} {b}))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm {1} {n} {n} = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange {x} {x} {y} {y}\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm {a} {1} {a}\n                                  | +-assoc-comm {b} {1} {b} = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = snd \u2115\u00b0.*-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = ! snd \u2115\u00b0.+-identity (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! \u2115\u00b0.*-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.Eq\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromAssocComm _+_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.+-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.+-comm x y)\n  renaming ( assoc-comm to +-assoc-comm\n           ; interchange to +-interchange\n           )\n\n*-assoc-comm : \u2200 x y z \u2192 x * (y * z) \u2261 y * (x * z)\n*-assoc-comm x y z = ! \u2115\u00b0.*-assoc x y z \u2219 ap (flip _*_ z) (\u2115\u00b0.*-comm x y) \u2219 \u2115\u00b0.*-assoc y x z\n\n*-interchange : Interchange _\u2261_ _*_ _*_\n*-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _*_ \u2115\u00b0.*-assoc \u2115\u00b0.*-comm (\u03bb z \u2192 ap (flip _*_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm {a} {1} {b}\n                                  | +-assoc-comm {a \u2294 b} {1} {a \u2293 b}\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm {1} {a} {b}))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm {1} {n} {n} = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange {x} {x} {y} {y}\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm {a} {1} {a}\n                                  | +-assoc-comm {b} {1} {b} = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = snd \u2115\u00b0.*-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = ! snd \u2115\u00b0.+-identity (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! \u2115\u00b0.*-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f5f267e3952095f52394a2377e2f10618505d866","subject":"Added Agsy examples.","message":"Added Agsy examples.\n\nIgnore-this: 6a5ebd515088c189d2d71d42872cd0e8\n\ndarcs-hash:20101203172202-3bd4e-06ca65a535ff142673537ff8af3ac4820edad9fb.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Agsy\/AgsyTest.agda","new_file":"Draft\/Agsy\/AgsyTest.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing Agsy using the Agda standard library\n------------------------------------------------------------------------------\n\nmodule AgsyTest where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero    n o = refl\n+-assoc (suc m) n o = cong suc (+-assoc m n o)\n\nx+1+y\u22611+x+y : \u2200 m n \u2192 m + suc n \u2261 suc (m + n)  -- via Agsy {-c}\nx+1+y\u22611+x+y zero    n = refl\nx+1+y\u22611+x+y (suc m) n = cong suc (x+1+y\u22611+x+y m n)\n\n0+n\u2261n+0 : \u2200 n \u2192 0 + n \u2261 n + 0  -- via Agsy {-c}\n0+n\u2261n+0 zero    = refl\n0+n\u2261n+0 (suc n) = cong suc (0+n\u2261n+0 n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = 0+n\u2261n+0 n\n+-comm (suc m) n =\n  begin\n    suc (m + n) \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m) \u2261\u27e8 sym (x+1+y\u22611+x+y n m) \u27e9\n    n + suc m\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Testing Agsy using the Agda standard library\n------------------------------------------------------------------------------\n\nmodule AgsyTest where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n0+m\u2261m+0 : \u2200 m \u2192 0 + m \u2261 m + 0  -- via Agsy {-c}\n0+m\u2261m+0 zero    = refl\n0+m\u2261m+0 (suc n) = cong suc (0+m\u2261m+0 n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm zero    zero    = refl -- via Agsy\n+-comm zero    (suc n) = sym (cong suc (+-comm n zero))  -- via Agsy\n+-comm (suc m) zero    = sym (cong suc (+-comm zero m))  -- via Agsy\n+-comm (suc m) (suc n) = {!-t 20!} -- Agsy: No solution found at time out (20s)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f7a0824e23aa553c512f0fbff1879941ee29f6d0","subject":"JS: add is-null","message":"JS: add is-null\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS.agda","new_file":"lib\/FFI\/JS.agda","new_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber Number #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString String #-}\n\n-- TODO dyn check of length 1?\npostulate castChar : JSValue \u2192 Char\n{-# COMPILED_JS castChar String #-}\n\n-- TODO dyn check of length 1?\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char String #-}\n\n-- TODO dyn check?\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray function(x) { return x; } #-}\n\n-- TODO dyn check?\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject function(x) { return x; } #-}\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\npostulate onJSArray : {A : Set} (f : JSArray JSValue \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onJSArray require(\"libagda\").onJSArray #-}\n\npostulate onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onString require(\"libagda\").onString #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\npostulate is-null : JSValue \u2192 Bool\n{-# COMPILED_JS is-null function(x) { return (x == null); } #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = castString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback0 : Set\nCallback0 = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 Callback0\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ncheck : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\ncheck true  errmsg x = x\ncheck false errmsg x = throw (errmsg _) x\n\nwarn-check : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\nwarn-check true  errmsg x = x\nwarn-check false errmsg x = trace (\"Warning: \" ++ errmsg _) x id\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\ndata JS! : Set\u2081 where\n  end  : JS!\n  _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n  _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n\n_>>_ : Callback0 \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber Number #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString String #-}\n\n-- TODO dyn check of length 1?\npostulate castChar : JSValue \u2192 Char\n{-# COMPILED_JS castChar String #-}\n\n-- TODO dyn check of length 1?\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char String #-}\n\n-- TODO dyn check?\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray function(x) { return x; } #-}\n\n-- TODO dyn check?\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject function(x) { return x; } #-}\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\npostulate onJSArray : {A : Set} (f : JSArray JSValue \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onJSArray require(\"libagda\").onJSArray #-}\n\npostulate onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onString require(\"libagda\").onString #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = castString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback0 : Set\nCallback0 = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 Callback0\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ncheck : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\ncheck true  errmsg x = x\ncheck false errmsg x = throw (errmsg _) x\n\nwarn-check : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\nwarn-check true  errmsg x = x\nwarn-check false errmsg x = trace (\"Warning: \" ++ errmsg _) x id\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\ndata JS! : Set\u2081 where\n  end  : JS!\n  _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n  _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n\n_>>_ : Callback0 \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6077652e83bff7230d91cf06d3df4541b171a639","subject":"elgamal is proved... almost","message":"elgamal is proved... almost\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\n--import Data.Vec.Properties as Vec\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nmodule elgamal where\n\n\u2605\u2081 : Set\u2082\n\u2605\u2081 = Set\u2081\n\n\u2605 : Set\u2081\n\u2605 = Set\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sumA = \u2200 f g \u2192 f \u2257 g \u2192 sumA f \u2261 sumA g\n\nsumToCount : \u2200 {A} \u2192 Sum A \u2192 Count A\nsumToCount sumA f = sumA (Bool.to\u2115 \u2218 f)\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\n-- liftM2 _,_ in the continuation monad\nsumProd : \u2200 {A B : Set} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsumProd sumA sumB f = sumA (\u03bb x\u2080 \u2192\n                       sumB (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\nsumProd-ext : \u2200 {A B : Set} {sumA : Sum A} {sumB : Sum B} \u2192\n              SumExt sumA \u2192 SumExt sumB \u2192 SumExt (sumProd sumA sumB)\nsumProd-ext sumA-ext sumB-ext f g f\u2257g = sumA-ext _ _ (\u03bb x \u2192 sumB-ext _ _ (\u03bb y \u2192 f\u2257g (x , y)))\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum \u2124q)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  where\n\n  open Univ \u2124q\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  sum : (u : `\u2605) \u2192 Sum (El u)\n  sum `\u22a4         f = f _\n  sum `X         f = sum\u2124q f\n  sum (u\u2080 `\u00d7 u\u2081) f = sum u\u2080 (\u03bb x\u2080 \u2192\n                       sum u\u2081 (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum u (Bool.to\u2115 \u2218 run\u21ba f)\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _ \n  lem x Adv = sum\u2124q-lem (Bool.to\u2115 \u2218 Adv) x\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047) \n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum\u2124q (f \u2218 [suc]) \u2261 sum\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  postulate\n    g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum \u2124q)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  where\n\n  open Univ \u2124q\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g ^ x\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 D (r , x , y , _) = D r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 D (r , x , y , z) = D r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- \u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- \u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba D) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (D b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2295_ : G \u2192 Message \u2192 Message)\n  (_\u2295\u207b\u00b9_ : G \u2192 Message \u2192 Message)\n  where\n\n    -- \u03b1 is the pk\n    -- \u03b1 = g ^ x\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g ^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc \u03b1 m y = \u03b2 , \u03b6 where\n      \u03b2 = g ^ y\n      \u03b4 = \u03b1 ^ y\n      \u03b6 = \u03b4 \u2295 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba \u03b1 m = mk (Enc \u03b1 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (\u03b2 , \u03b6) = (\u03b2 ^ x) \u2295\u207b\u00b9 \u03b6\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R = PubKey \u2192 R \u2192 (Bit \u2192 M) \u00d7 (C \u2192 Bit)\n\n    {-\n    Game0 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bit\n    Game0 A (b , x , y , z , r) =\n      let (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n\n    Game : (i : Bit) \u2192 \u2200 {R} \u2192 EncAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    Game i A (b , x , y , z , r) =\n      let (\u03b1 , sk) = KeyGen x\n          (m , D) = A \u03b1 r\n          \u03b2 = g ^ y\n          \u03b4 = \u03b1 ^ case i 0\u2192 y 1\u2192 z\n          \u03b6 = \u03b4 \u2295 m b\n      in {-b ==\u1d47-} D (\u03b2 , \u03b6)\n\n    Game-0b\u2261Game0 : \u2200 {R} \u2192 Game 0b \u2261 Game0 {R}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    SS\u2141 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 A b (r , x , y , z) =\n      let -- (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n         where pk = g ^ x\n\n    open DDH \u2124q _\u22a0_ G (_^_ g) public\n\n    -- Game0 \u2248 Game 0b\n    -- Game1 = Game 1b\n\n    -- Game0 \u2264 \u03b5\n    -- Game1 \u2261 0\n\n    -- \u2047 \u2295 x \u2248 \u2047\n\n    -- g ^ \u2047 \u2219 x \u2248 g ^ \u2047\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u207b\u00b9 : G \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n    where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_ (flip _\/_) public\n\n    TrA : \u2200 {R} \u2192 Bit \u2192 EncAdv R \u2192 DDHAdv R\n    TrA b A r g\u02e3 g\u02b8 g\u02e3\u02b8 = d (g\u02b8 , g\u02e3\u02b8 \u2219 m b)\n       where m,d = A g\u02e3 r\n             m = proj\u2081 m,d\n             d = proj\u2082 m,d\n\n    like-SS\u2141 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 A b (r , x , y , _z) =\n      let -- g\u02e3 = g ^ x\n          -- g\u02b8 = g ^ y\n          (m , D) = A g\u02e3 r in\n      D (g\u02b8 , (g\u02e3 ^ y) \u2219 m b)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    OTP\u2141 : \u2200 {R : Set} \u2192 (R \u2192 G \u2192 G) \u2192 (R \u2192 G \u2192 G \u2192 G \u2192 Bit) \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M D (r , x , y , z) = D r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n            g\u1dbb = g ^ z\n\n    module With\u2124qProps\n        (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      -- (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n        (sum\u2124q : Sum \u2124q)\n        (sum\u2124q-ext : SumExt sum\u2124q) where\n        #q_ : Count \u2124q\n        #q_ = sumToCount sum\u2124q\n\n        module SumU\n          (R : Set)\n          (sumR : Sum R)\n          (sumR-ext : SumExt sumR)\n          where\n          U = R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n          sumU : Sum U\n          sumU = sumProd sumR (sumProd sum\u2124q (sumProd sum\u2124q sum\u2124q))\n          #U_ : Count U\n          #U_ = sumToCount sumU\n          sumU-ext : SumExt sumU\n          sumU-ext = sumProd-ext sumR-ext (sumProd-ext sum\u2124q-ext (sumProd-ext sum\u2124q-ext sum\u2124q-ext))\n\n        module WithSumR\n            (R : Set)\n            (sumR : Sum R)\n            (sumR-ext : SumExt sumR)\n           where\n           open SumU R sumR sumR-ext\n\n           module EvenMoreProof\n            (ddh-hyp : (A : DDHAdv R) \u2192 #U (DDH\u2141 A 0b) \u2261 #U (DDH\u2141 A 1b))\n\n            (otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 #q(\u03bb x \u2192 A (g^ x \u2219 m)) \u2261 #q(\u03bb x \u2192 A (g^ x)))\n            where\n\n                _\u2248U_ : (f g : U \u2192 Bit) \u2192 Set\n                f \u2248U g = #U f \u2261 #U g\n\n                otp-lem' : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248U OTP\u2141 M\u2081 D\n                otp-lem' D M\u2080 M\u2081 = sumR-ext (f3 M\u2080) (f3 M\u2081) (\u03bb r \u2192\n                                     sum\u2124q-ext (f2 M\u2080 r) (f2 M\u2081 r) (\u03bb x \u2192\n                                       sum\u2124q-ext (f1 M\u2080 r x) (f1 M\u2081 r x) (\u03bb y \u2192\n                                         pf r x y)))\n                  where\n                  f0 = \u03bb M r x y z \u2192 OTP\u2141 M D (r , x , y , z)\n                  f1 = \u03bb M r x y \u2192 sum\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n                  f2 = \u03bb M r x \u2192 sum\u2124q (f1 M r x)\n                  f3 = \u03bb M r \u2192 sum\u2124q (f2 M r)\n                  pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n                  pf r x y = pf'\n                    where g\u02e3 = g^ x\n                          g\u02b8 = g^ y\n                          m0 = M\u2080 r g\u02e3\n                          m1 = M\u2081 r g\u02e3\n                          f5 = \u03bb M z \u2192 D r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                          pf' : #q(f5 m0) \u2261 #q(f5 m1)\n                          pf' rewrite otp-lem (D r g\u02e3 g\u02b8) m0\n                                    | otp-lem (D r g\u02e3 g\u02b8) m1 = refl\n\n      {-\n        #-proj\u2081 : \u2200 {A B : Set} {f g : A \u2192 Bit} \u2192 # f \u2261 # g\n                   \u2192 #_ {A \u00d7 B} (f \u2218 proj\u2081) \u2261 #_ {A \u00d7 B} (g \u2218 proj\u2081)\n        #-proj\u2081 = {!!}\n\n        otp-lem'' : \u2200 (A : G \u2192 Bit) m\n           \u2192 #(\u03bb { (x , y) \u2192 A (g^ x \u2219 m) }) \u2261 #(\u03bb { (x , y) \u2192 A (g ^ x) })\n        otp-lem'' = {!!}\n        -}\n                projM : EncAdv R \u2192 Bit \u2192 R \u2192 G \u2192 G\n                projM A b r g\u02e3 = proj\u2081 (A g\u02e3 r) b\n\n                projD : EncAdv R \u2192 R \u2192 G \u2192 G \u2192 G \u2192 Bit\n                projD A r g\u02e3 g\u02b8 g\u1dbb\u2219M = proj\u2082 (A g\u02e3 r) (g\u02b8 , g\u1dbb\u2219M)\n\n\n                module WithAdversary (A : EncAdv R) b where\n                    A\u1d47 = TrA b A\n                    A\u00ac\u1d47 = TrA (not b) A\n\n                    pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n                    pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf1 : #U(SS\u2141 A b) \u2261 #U(DDH\u2141 A\u1d47 0b)\n                    pf1 = sumU-ext _ _ (cong Bool.to\u2115 \u2218 pf0,5)\n\n                    pf2 : DDH\u2141 A\u1d47 0b \u2248U DDH\u2141 A\u1d47 1b\n                    pf2 = ddh-hyp A\u1d47\n\n                    pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n                    pf2,5 = refl\n\n                    pf3 : DDH\u2141 A\u1d47 1b \u2248U DDH\u2141 A\u00ac\u1d47 1b\n                    pf3 = otp-lem' (projD A) (projM A b) (projM A (not b))\n\n                    pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248U DDH\u2141 A\u00ac\u1d47 0b\n                    pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n                    pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n                    pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf5 : #U(SS\u2141 A (not b)) \u2261 #U(DDH\u2141 A\u00ac\u1d47 0b)\n                    pf5 = sumU-ext _ _ (cong Bool.to\u2115 \u2218 pf4,5)\n\n                    final : #U(SS\u2141 A b) \u2261 #U(SS\u2141 A (not b))\n                    final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\n\n      {-\n        pf1 : \u2200 {R} A r \u2192 Game 0b {R} A r \u2261 \u2141\u2032 (TrA 1b A) r\n        pf1 A (true , x , y , z , r) rewrite dist-^-\u22a0 g x y = refl\n        pf1 A (false , x , y , z , r) = {!!}\n\n        pf2 : \u2200 {R} A r \u2192 Game 1b {R} A r \u2261 \u2141\u2032 (TrA 0b A) r\n        pf2 A (true , x , y , z , r)  = {!!}\n        pf2 A (false , x , y , z , r) = {!!}\n-}\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|) where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_\n\n    module Proof\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      -- (^-lem : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n      -- (x \u2219 x \u207b\u00b9 \u2261 0)\n      (\u211a : \u2605)\n      (dist : \u211a \u2192 \u211a \u2192 \u211a)\n      (1\/2 : \u211a)\n      (_\u2264_ : \u211a \u2192 \u211a \u2192 \u2605)\n      (\u03b5-DDH : \u211a)\n      where\n\n        Pr[S_] : Bool \u2192 \u211a\n        Pr[S b ] = {!!}\n\n        -- SS\n        -- SS = dist Pr[\n\n        Pr[S\u2080] = Pr[S 0b ]\n        Pr[S\u2081] = Pr[S 1b ]\n\n        pf1 : Pr[S\u2081] \u2261 1\/2\n        pf1 = {!!}\n\n        pf2 : dist Pr[S\u2080] 1\/2 \u2261 dist Pr[S\u2080] Pr[S\u2081]\n        pf2 = {!!}\n\n        pf3 : dist Pr[S\u2080] Pr[S\u2081] \u2264 \u03b5-DDH\n        pf3 = {!!}\n\n        pf4 : dist Pr[S\u2080] 1\/2 \u2264 \u03b5-DDH\n        pf4 = {!!}\n\nmodule G11 = G-implem 11 10 (## 2) (## 0) (## 1) (## 0) (## 1)\nopen G11\nmodule E11 = El-Gamal-Base _ _\u22a0_ G g _^_ {!!} _\u2219_\nopen E11\nopen With\u2124qProps ?\n-- open Proof {!!}\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\n--import Data.Vec.Properties as Vec\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nmodule elgamal where\n\n\u2605\u2081 : Set\u2082\n\u2605\u2081 = Set\u2081\n\n\u2605 : Set\u2081\n\u2605 = Set\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bool \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\nsumBit : (Bit \u2192 \u2115) \u2192 \u2115\nsumBit f = f 0b + f 1b\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : (\u2124q \u2192 \u2115) \u2192 \u2115)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  where\n\n  open Univ \u2124q\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  sum : (u : `\u2605) \u2192 (El u \u2192 \u2115) \u2192 \u2115\n  sum `\u22a4         f = f _\n  sum `X         f = sum\u2124q f\n  sum (u\u2080 `\u00d7 u\u2081) f = sum u\u2080 (\u03bb x\u2080 \u2192\n                       sum u\u2081 (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum u (Bool.to\u2115 \u2218 run\u21ba f)\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _ \n  lem x Adv = sum\u2124q-lem (Bool.to\u2115 \u2218 Adv) x\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047) \n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  sum\u2124q : (\u2124q \u2192 \u2115) \u2192 \u2115\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum\u2124q (f \u2218 [suc]) \u2261 sum\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  postulate\n    g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : (\u2124q \u2192 \u2115) \u2192 \u2115)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  where\n\n  open Univ \u2124q\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g ^ x\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = G \u2192 G \u2192 G \u2192 R \u2192 Bool\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (\u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bool\n    DDH\u2141\u2080 D (x , y , _ , r) = D (g^ x) (g^ y) (g^ (x \u22a0 y)) r\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (\u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    DDH\u2141\u2081 D (x , y , z , r) = D (g^ x) (g^ y) (g^ z) r\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bool \u2192 (\u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- \u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bool \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    -- \u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bool\n        \u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (`\u2124q `\u00d7 `\u2124q `\u00d7 _I `\u00d7 R) Bool\n        run\u21ba (\u2141\u2080\u21ba D) = \u2141\u2080 (\u03bb a b c \u2192 run\u21ba (D a b c))\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2295_ : G \u2192 Message \u2192 Message)\n  (_\u2295\u207b\u00b9_ : G \u2192 Message \u2192 Message)\n  where\n\n    -- \u03b1 is the pk\n    -- \u03b1 = g ^ x\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g ^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc \u03b1 m y = \u03b2 , \u03b6 where\n      \u03b2 = g ^ y\n      \u03b4 = \u03b1 ^ y\n      \u03b6 = \u03b4 \u2295 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba \u03b1 m = mk (Enc \u03b1 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (\u03b2 , \u03b6) = (\u03b2 ^ x) \u2295\u207b\u00b9 \u03b6\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R = PubKey \u2192 R \u2192 (Bool \u2192 M) \u00d7 (C \u2192 Bool)\n\n    {-\n    Game0 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 (Bool \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bool\n    Game0 A (b , x , y , z , r) =\n      let (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n\n    Game : (i : Bool) \u2192 \u2200 {R} \u2192 EncAdv R \u2192 (Bool \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    Game i A (b , x , y , z , r) =\n      let (\u03b1 , sk) = KeyGen x\n          (m , D) = A \u03b1 r\n          \u03b2 = g ^ y\n          \u03b4 = \u03b1 ^ case i 0\u2192 y 1\u2192 z\n          \u03b6 = \u03b4 \u2295 m b\n      in {-b ==\u1d47-} D (\u03b2 , \u03b6)\n\n    Game-0b\u2261Game0 : \u2200 {R} \u2192 Game 0b \u2261 Game0 {R}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    SS\u2141 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 Bool \u2192 (\u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bool\n    SS\u2141 A b (x , y , z , r) =\n      let -- (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n         where pk = g ^ x\n\n    open DDH \u2124q _\u22a0_ G (_^_ g) public\n\n    -- Game0 \u2248 Game 0b\n    -- Game1 = Game 1b\n\n    -- Game0 \u2264 \u03b5\n    -- Game1 \u2261 0\n\n    -- \u2047 \u2295 x \u2248 \u2047\n\n    -- g ^ \u2047 \u2219 x \u2248 g ^ \u2047\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u207b\u00b9 : G \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n    where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_ (flip _\/_) public\n\n    TrA : \u2200 {R} \u2192 Bool \u2192 EncAdv R \u2192 DDHAdv R\n    TrA b A g\u02e3 g\u02b8 g\u02e3\u02b8 r = d (g\u02b8 , g\u02e3\u02b8 \u2219 m b)\n       where m,d = A g\u02e3 r\n             m = proj\u2081 m,d\n             d = proj\u2082 m,d\n\n    like-SS\u2141 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 Bool \u2192 (\u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bool\n    like-SS\u2141 A b (x , y , z , r) =\n      let -- g\u02e3 = g ^ x\n          -- g\u02b8 = g ^ y\n          (m , D) = A g\u02e3 r in\n      D (g\u02b8 , (g\u02e3 ^ y) \u2219 m b)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    module Proof\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      -- (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n\n      (#_ : \u2200 {R} \u2192 (R \u2192 Bit) \u2192 \u2115)\n      (ddh-hyp : \u2200 {R} (A : DDHAdv R) \u2192 # (DDH\u2141 A 0b) \u2261 # (DDH\u2141 A 1b))\n      (otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 #(\u03bb x \u2192 A (g^ x \u2219 m)) \u2261 #(\u03bb x \u2192 A (g ^ x)))\n      -- (wk\u2248 : \u2200 {R1 R2} (f : R2 \u2192 R1)(x y : R1 \u2192 Bit) (z t : R2 \u2192 Bit) \u2192 x \u2248 y \u2192 z \u2248 t)\n      where\n\n        _\u2248_ : \u2200 {R} \u2192 (f g : R \u2192 Bit) \u2192 Set\n        f \u2248 g = # f \u2261 # g\n\n        pf1 : \u2200 {R} A b r \u2192 SS\u2141 {R} A b r \u2261 DDH\u2141 (TrA b A) 0b r\n        pf1 A b (x , y , z , r) rewrite dist-^-\u22a0 g x y = refl\n\n        pf2 : \u2200 {R} A b \u2192 DDH\u2141 {R} (TrA b A) 0b \u2248 DDH\u2141 (TrA b A) 1b\n        pf2 A b = ddh-hyp (TrA b A)\n\n        pf3 : \u2200 {R} A b \u2192 DDH\u2141 {R} (TrA b A) 1b \u2248 DDH\u2141 (TrA (not b) A) 1b\n        pf3 {R} A b = pf3'\n          where\n            A\u1d47 = TrA b A\n\n            -- pf3'' : # (DDH\u2141\u2081 {R} A\u1d47) \u2248 DDH\u2141\u2081 (TrA (not b) A)\n            -- pf3'' = ?\n\n            pf3' : DDH\u2141\u2081 {R} A\u1d47 \u2248 DDH\u2141\u2081 (TrA (not b) A)\n            pf3' = {!wk\u2248 (\u03bb { (x , y , z , r) \u2192 {!!} }) _ _ _ _ (otp-lem {!!} {!!})!}\n\n      {-\n        pf1 : \u2200 {R} A r \u2192 Game 0b {R} A r \u2261 \u2141\u2032 (TrA 1b A) r\n        pf1 A (true , x , y , z , r) rewrite dist-^-\u22a0 g x y = refl\n        pf1 A (false , x , y , z , r) = {!!}\n\n        pf2 : \u2200 {R} A r \u2192 Game 1b {R} A r \u2261 \u2141\u2032 (TrA 0b A) r\n        pf2 A (true , x , y , z , r)  = {!!}\n        pf2 A (false , x , y , z , r) = {!!}\n-}\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|) where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_\n\n    module Proof\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      -- (^-lem : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n      -- (x \u2219 x \u207b\u00b9 \u2261 0)\n      (\u211a : \u2605)\n      (dist : \u211a \u2192 \u211a \u2192 \u211a)\n      (1\/2 : \u211a)\n      (_\u2264_ : \u211a \u2192 \u211a \u2192 \u2605)\n      (\u03b5-DDH : \u211a)\n      where\n\n        Pr[S_] : Bool \u2192 \u211a\n        Pr[S b ] = {!!}\n\n        -- SS\n        -- SS = dist Pr[\n\n        Pr[S\u2080] = Pr[S 0b ]\n        Pr[S\u2081] = Pr[S 1b ]\n\n        pf1 : Pr[S\u2081] \u2261 1\/2\n        pf1 = {!!}\n\n        pf2 : dist Pr[S\u2080] 1\/2 \u2261 dist Pr[S\u2080] Pr[S\u2081]\n        pf2 = {!!}\n\n        pf3 : dist Pr[S\u2080] Pr[S\u2081] \u2264 \u03b5-DDH\n        pf3 = {!!}\n\n        pf4 : dist Pr[S\u2080] 1\/2 \u2264 \u03b5-DDH\n        pf4 = {!!}\n\nmodule G11 = G-implem 11 10 (## 2) (## 0) (## 1) (## 0) (## 1)\nopen G11\nmodule E11 = El-Gamal-Base _ _\u22a0_ G g _^_ {!!} _\u2219_\nopen E11\nopen Proof {!!}\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7b2c31fcd7415cecc448653df3d386001cbebf1b","subject":"Put constructor arguments on different lines.","message":"Put constructor arguments on different lines.\n\nOld-commit-hash: 02f9c59c59ddef7050c2c035457c98ecb00464dc\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen Type B\nopen import Syntax.Context {Type}\n\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n  where\n  const : \u2200 {\u03c4} \u2192\n    (c : C \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","old_contents":"import Syntax.Type.Plotkin as Type\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen Type B\nopen import Syntax.Context {Type}\n\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n  where\n  const : \u2200 {\u03c4} \u2192 (c : C \u03c4) \u2192 Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3d7e95768705107ac7dd83dd8c8b12e1304b3b17","subject":"Cleaning.","message":"Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1WithoutHelperATP.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1WithoutHelperATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1WithoutHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n-- Helper function for the ABP lemma 1\nmodule Helper where\n  -- 30 November 2013. If we don't have the following definitions\n  -- outside the where clause, the ATPs cannot prove the theorems.\n\n  helper : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n           Bit b \u2192\n           Fair os\u2082 \u2192\n           S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n           \u2200 ft\u2081 os\u2081' \u2192 F*T ft\u2081 \u2192 Fair os\u2081' \u2192 os\u2081 \u2261 ft\u2081 ++ os\u2081' \u2192\n           \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n             Fair os\u2081'\n             \u2227 Fair os\u2082'\n             \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n             \u2227 js \u2261 i' \u2237 js'\n  helper b i' is' os\u2081 os\u2082 as bs cs ds js\n         Bb Fos\u2082 s .(T \u2237 []) os\u2081' f*tnil Fos\u2081' os\u2081-eq = prf\n    where\n    postulate\n      prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n              Fair os\u2081'\n              \u2227 Fair os\u2082'\n              \u2227 (as' \u2261 await b i' is' ds'\n                 \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n                 \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n                 \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n                 \u2227 js' \u2261 out (not b) \u00b7 bs')\n              \u2227 js \u2261 i' \u2237 js'\n    {-# ATP prove prf #-}\n\n  -- TODO (25 January 2014): Why Agda cannot refine using the helper\n  -- function? Agda bug?\n  helper b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2082 s\n         .(F \u2237 ft\u2081^) os\u2081' (f*tcons {ft\u2081^} FTft\u2081^) Fos\u2081' os\u2081-eq =\n         helper b i' is'\n                (ft\u2081^ ++ os\u2081')\n                (tail\u2081 os\u2082)\n                (await b i' is' ds)\n                (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n                (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n                (corrupt (tail\u2081 os\u2082) \u00b7\n                   (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n                js\n                Bb (tail-Fair Fos\u2082) ihS ft\u2081^ os\u2081' FTft\u2081^ Fos\u2081' refl\n    where\n    postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n    {-# ATP prove os\u2081-eq-helper #-}\n\n    postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n    {-# ATP prove as-eq #-}\n\n    postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n    {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n    postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n    {-# ATP prove cs-eq bs-eq #-}\n\n    postulate\n      ds-eq-helper\u2081 :\n        os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n        ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n    postulate\n      ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                      ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n    ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n            \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                 (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                 (head-tail-Fair Fos\u2082)\n\n    postulate\n      as^-eq-helper\u2081 :\n        ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n        as^ b i' is' ds \u2261\n          send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n    postulate\n      as^-eq-helper\u2082 :\n        ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n        as^ b i' is' ds \u2261\n          send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove as^-eq-helper\u2082 #-}\n\n    as^-eq : as^ b i' is' ds \u2261\n             send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n    postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n    {-# ATP prove js-eq bs-eq #-}\n\n    ihS : S b\n            (i' \u2237 is')\n            (os\u2081^ os\u2081' ft\u2081^)\n            (os\u2082^ os\u2082)\n            (as^ b i' is' ds)\n            (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n            (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n            (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n            js\n    ihS = as^-eq , refl , refl , refl , js-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\n-- lemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n--          Bit b \u2192\n--          Fair os\u2081 \u2192\n--          Fair os\u2082 \u2192\n--          S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n--          \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--            Fair os\u2081'\n--            \u2227 Fair os\u2082'\n--            \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n--            \u2227 js \u2261 i' \u2237 js'\n-- lemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n-- ... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n--   where\n--   postulate\n--     prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--             Fair os\u2081'\n--             \u2227 Fair os\u2082'\n--             \u2227 (as' \u2261 await b i' is' ds'\n--               \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n--               \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n--               \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n--               \u2227 js' \u2261 out (not b) \u00b7 bs')\n--             \u2227 js \u2261 i' \u2237 js'\n--   {-# ATP prove prf #-}\n\n-- ... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n--   lemma\u2081 b i' is'\n--          (ft\u2081^ ++ os\u2081')\n--          (tail\u2081 os\u2082)\n--          (await b i' is' ds)\n--          (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n--          (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n--          (corrupt (tail\u2081 os\u2082) \u00b7\n--            (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n--          js Bb ft\u2081^++-os\u2081'-Fair ((tail-Fair Fos\u2082)) ihS\n--      where\n--      ft\u2081^++-os\u2081'-Fair : Fair (ft\u2081^ ++ os\u2081')\n--      ft\u2081^++-os\u2081'-Fair = Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')\n\n--      postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n--      {-# ATP prove os\u2081-eq-helper #-}\n\n--      postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n--      {-# ATP prove as-eq #-}\n\n--      postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--      {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n--      postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n--      {-# ATP prove cs-eq bs-eq #-}\n\n--      postulate\n--        ds-eq-helper\u2081 :\n--          os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n--          ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--      {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n--      postulate\n--        ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n--                        ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--      {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n--      ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--              \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--      ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n--                   (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n--                   (head-tail-Fair Fos\u2082)\n\n--      postulate\n--        as^-eq-helper\u2081 :\n--          ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n--          as^ b i' is' ds \u2261\n--            send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--      {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n--      postulate\n--        as^-eq-helper\u2082 :\n--          ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n--          as^ b i' is' ds \u2261\n--            send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--      {-# ATP prove as^-eq-helper\u2082 #-}\n\n--      as^-eq : as^ b i' is' ds \u2261\n--               send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--      as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n--      postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n--      {-# ATP prove js-eq bs-eq #-}\n\n--      ihS : S b\n--              (i' \u2237 is')\n--              (os\u2081^ os\u2081' ft\u2081^)\n--              (os\u2082^ os\u2082)\n--              (as^ b i' is' ds)\n--              (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--              (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--              (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n--              js\n--      ihS = as^-eq , refl , refl , refl , js-eq\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1WithoutHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n-- Helper function for the ABP lemma 1\n-- module Helper where\n--   -- 30 November 2013. If we don't have the following definitions\n--   -- outside the where clause, the ATPs cannot prove the theorems.\n\n--   helper : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n--            Bit b \u2192\n--            Fair os\u2082 \u2192\n--            S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n--            \u2200 ft\u2081 os\u2081' \u2192 F*T ft\u2081 \u2192 Fair os\u2081' \u2192 os\u2081 \u2261 ft\u2081 ++ os\u2081' \u2192\n--            \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--              Fair os\u2081'\n--              \u2227 Fair os\u2082'\n--              \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n--              \u2227 js \u2261 i' \u2237 js'\n--   helper b i' is' os\u2081 os\u2082 as bs cs ds js\n--          Bb Fos\u2082 s .(T \u2237 []) os\u2081' f*tnil Fos\u2081' os\u2081-eq = prf\n--     where\n--     postulate\n--       prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--               Fair os\u2081'\n--               \u2227 Fair os\u2082'\n--               \u2227 (as' \u2261 await b i' is' ds'\n--                  \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n--                  \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n--                  \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n--                  \u2227 js' \u2261 out (not b) \u00b7 bs')\n--               \u2227 js \u2261 i' \u2237 js'\n--     {-# ATP prove prf #-}\n\n--   -- TODO (25 January 2014): Why Agda cannot refine using the helper\n--   -- function? Agda bug?\n--   helper b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2082 s\n--          .(F \u2237 ft\u2081^) os\u2081' (f*tcons {ft\u2081^} FTft\u2081^) Fos\u2081' os\u2081-eq =\n--          helper b i' is'\n--                 (ft\u2081^ ++ os\u2081')\n--                 (tail\u2081 os\u2082)\n--                 (await b i' is' ds)\n--                 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n--                 (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n--                 (corrupt (tail\u2081 os\u2082) \u00b7\n--                    (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n--                 js\n--                 Bb (tail-Fair Fos\u2082) ihS ft\u2081^ os\u2081' FTft\u2081^ Fos\u2081' refl\n--     where\n--     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n--     {-# ATP prove os\u2081-eq-helper #-}\n\n--     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n--     {-# ATP prove as-eq #-}\n\n--     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n--     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n--     {-# ATP prove cs-eq bs-eq #-}\n\n--     postulate\n--       ds-eq-helper\u2081 :\n--         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n--         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n--     postulate\n--       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n--                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n--     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n--                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n--                  (head-tail-Fair Fos\u2082)\n\n--     postulate\n--       as^-eq-helper\u2081 :\n--         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n--         as^ b i' is' ds \u2261\n--           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n--     postulate\n--       as^-eq-helper\u2082 :\n--         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n--         as^ b i' is' ds \u2261\n--           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove as^-eq-helper\u2082 #-}\n\n--     as^-eq : as^ b i' is' ds \u2261\n--              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n--     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n--     {-# ATP prove js-eq bs-eq #-}\n\n--     ihS : S b\n--             (i' \u2237 is')\n--             (os\u2081^ os\u2081' ft\u2081^)\n--             (os\u2082^ os\u2082)\n--             (as^ b i' is' ds)\n--             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n--             js\n--     ihS = as^-eq , refl , refl , refl , js-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\n-- open Helper\nlemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n         Bit b \u2192\n         Fair os\u2081 \u2192\n         Fair os\u2082 \u2192\n         S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n         \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n           Fair os\u2081'\n           \u2227 Fair os\u2082'\n           \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n           \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n            Fair os\u2081'\n            \u2227 Fair os\u2082'\n            \u2227 (as' \u2261 await b i' is' ds'\n              \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n              \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n              \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n              \u2227 js' \u2261 out (not b) \u00b7 bs')\n            \u2227 js \u2261 i' \u2237 js'\n  {-# ATP prove prf #-}\n\n... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n  lemma\u2081 b i' is'\n         (ft\u2081^ ++ os\u2081')\n         (tail\u2081 os\u2082)\n         (await b i' is' ds)\n         (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n         (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n         (corrupt (tail\u2081 os\u2082) \u00b7\n           (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n         js Bb ft\u2081^++-os\u2081'-Fair ((tail-Fair Fos\u2082)) ihS\n     where\n     ft\u2081^++-os\u2081'-Fair : Fair (ft\u2081^ ++ os\u2081')\n     ft\u2081^++-os\u2081'-Fair = Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')\n\n     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n     {-# ATP prove os\u2081-eq-helper #-}\n\n     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n     {-# ATP prove as-eq #-}\n\n     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove cs-eq bs-eq #-}\n\n     postulate\n       ds-eq-helper\u2081 :\n         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n     postulate\n       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2082)\n\n     postulate\n       as^-eq-helper\u2081 :\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n     postulate\n       as^-eq-helper\u2082 :\n         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2082 #-}\n\n     as^-eq : as^ b i' is' ds \u2261\n              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove js-eq bs-eq #-}\n\n     ihS : S b\n             (i' \u2237 is')\n             (os\u2081^ os\u2081' ft\u2081^)\n             (os\u2082^ os\u2082)\n             (as^ b i' is' ds)\n             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n             js\n     ihS = as^-eq , refl , refl , refl , js-eq\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fbbd955ba2c82259764f220c075ee5ab09f0a3c1","subject":"Update Desc model. Follow ICFP paper. With type-in-type, but termination-checker-friendly.","message":"Update Desc model. Follow ICFP paper. With type-in-type, but termination-checker-friendly.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms )) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\nrecord One : Set where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\n-- Inductive types are implemented as a Universe. Hence, in this\n-- section, we implement their code.\n\n-- We can read this code as follow (see Conor's \"Ornamental\n-- algebras\"): a description in |Desc| is a program to read one node\n-- of the described tree.\n\ndata Desc : Set where\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  -- Read a field in |X|; continue, given its value\n\n  -- Often, |X| is an |EnumT|, hence allowing to choose a constructor\n  -- among a finite set of constructors\n\n  Ind : (H : Set) -> Desc -> Desc\n  -- Read a field in H; read a recursive subnode given the field,\n  -- continue regarless of the subnode\n\n  -- Often, |H| is |1|, hence |Ind| simplifies to |Inf : Desc -> Desc|,\n  -- meaning: read a recursive subnode and continue regardless\n\n  Done : Desc\n  -- Stop reading\n\n\n--********************************************\n-- Desc decoder\n--********************************************\n\n-- Provided the type of the recursive subnodes |R|, we decode a\n-- description as a record describing the node.\n\n[|_|]_ : Desc -> Set -> Set\n[| Arg A D |] R = Sigma A (\\ a -> [| D a |] R)\n[| Ind H D |] R = (H -> R) * [| D |] R\n[| Done |] R = One\n\n\n--********************************************\n-- Functions on codes\n--********************************************\n\n-- Saying that a \"predicate\" |p| holds everywhere in |v| amounts to\n-- write something of the following type:\n\nEverywhere : (d : Desc) (D : Set) (bp : D -> Set) (V : [| d |] D) -> Set\nEverywhere (Arg A f) d p v =  Everywhere (f (fst v)) d p (snd v)\n-- It must hold for this constructor\n\nEverywhere (Ind H x) d p v =  ((y : H) -> p (fst v y)) * Everywhere x d p (snd v)\n-- It must hold for the subtrees \n\nEverywhere Done _ _ _ =  _\n-- It trivially holds at endpoints\n\n-- Then, we can build terms that inhabits this type. That is, a\n-- function that takes a \"predicate\" |bp| and makes it hold everywhere\n-- in the data-structure. It is the equivalent of a \"map\", but in a\n-- dependently-typed setting.\n\neverywhere : (d : Desc) (D : Set) (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : [| d |] D) ->\n           Everywhere d D bp v\neverywhere (Arg a f) d bp p v =  everywhere (f (fst v)) d bp p (snd v)\n-- It holds everywhere on this constructor\n\neverywhere (Ind H x) d bp p v = (\\y -> p (fst v y)) ,  everywhere x d bp p (snd v)\n-- It holds here, and down in the recursive subtrees\n\neverywhere Done _ _ _ _ =  One\n-- Nothing needs to be done on endpoints\n\n\n-- Looking at the decoder, a natural thing to do is to define its\n-- fixpoint |Mu D|, hence instantiating |R| with |Mu D| itself.\n\ndata Mu (D : Desc) : Set where\n  Con : [| D |] (Mu D) -> Mu D\n\n-- Using the \"map\" defined by |everywhere|, we can implement a \"fold\"\n-- over the |Mu| fixpoint:\n\nfoldDesc : (D : Desc) (bp : Mu D -> Set) ->\n         ((x : [| D |] (Mu D)) -> Everywhere D (Mu D) bp x -> bp (Con x)) ->\n         (v : Mu D) ->\n         bp v\nfoldDesc D bp p (Con v) =  p v (everywhere D (Mu D) bp (\\x -> foldDesc D bp p x) v) \n\n\n--********************************************\n-- Nat\n--********************************************\n\ndata NatConst : Set where\n  ZE : NatConst\n  SU : NatConst\n\nnatc : NatConst -> Desc\nnatc ZE = Done\nnatc SU = Ind One Done\n\nnatd : Desc\nnatd = Arg NatConst natc\n\nnat : Set\nnat = Mu natd\n\nzero : nat\nzero = Con ( ZE ,  _ )\n\nsuc : nat -> nat\nsuc n = Con ( SU , ( (\\_ -> n) ,  _ ) )\n\ntwo : nat\ntwo = suc (suc zero)\n\nfour : nat\nfour = suc (suc (suc (suc zero)))\n\nsum : nat -> \n      ((x : Sigma NatConst (\\ a -> [| natc a |] Mu (Arg NatConst natc))) ->\n      Everywhere (natc (fst x)) (Mu (Arg NatConst natc)) (\\ _ -> Mu (Arg NatConst natc)) (snd x) ->\n      Mu (Arg NatConst natc))\nsum n2 (ZE , _) p =  n2\nsum n2 (SU , f) p =  suc ( fst p _) \n\n\nplus : nat -> nat -> nat\nplus n1 n2 = foldDesc natd (\\_ -> nat) (sum n2) n1 \n\nx : nat\nx = plus two two\n\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"656ee7f653c5f51df1fca8401f63c98f11f269c7","subject":"Environment changes: define \u229a without \u229d","message":"Environment changes: define \u229a without \u229d\n\nDefine \u229a as in terms of \u229a for entries, not in terms of \u229d on\nenvironments.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/LangChanges.agda","new_file":"Thesis\/LangChanges.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"bb08669e1bdc2f7d2df9bb9289bda1b42f3dece8","subject":"Alegbra.FunctionProperties.Eq: add Endo.Cycle","message":"Alegbra.FunctionProperties.Eq: add Endo.Cycle\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Like Algebra.FunctionProperties but specialized to _\u2261_ and using implict arguments\n-- Moreover there is some extensions such as:\n-- * interchange\n-- * non-zero inversions\n-- * cancelation and non-zero cancelation\n\n-- These properties can (for instance) be used to define algebraic\n-- structures.\n\nopen import Level\nopen import Function.NP using (_$\u27e8_\u27e9_; flip)\nopen import Data.Nat.Base using (\u2115)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\n-- The properties are specified using the following relation as\n-- \"equality\".\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\n------------------------------------------------------------------------\n-- Unary and binary operations\n\nopen import Algebra.FunctionProperties.Core public\n\n------------------------------------------------------------------------\n-- Properties of operations\n\nAssociative : Op\u2082 A \u2192 Set _\nAssociative _\u00b7_ = \u2200 {x y z} \u2192 ((x \u00b7 y) \u00b7 z) \u2261 (x \u00b7 (y \u00b7 z))\n\nCommutative : Op\u2082 A \u2192 Set _\nCommutative _\u00b7_ = \u2200 {x y} \u2192 (x \u00b7 y) \u2261 (y \u00b7 x)\n\nLeftIdentity : A \u2192 Op\u2082 A \u2192 Set _\nLeftIdentity e _\u00b7_ = \u2200 {x} \u2192 (e \u00b7 x) \u2261 x\n\nRightIdentity : A \u2192 Op\u2082 A \u2192 Set _\nRightIdentity e _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 e) \u2261 x\n\nIdentity : A \u2192 Op\u2082 A \u2192 Set _\nIdentity e \u00b7 = LeftIdentity e \u00b7 \u00d7 RightIdentity e \u00b7\n\nLeftZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftZero z _\u00b7_ = \u2200 {x} \u2192 (z \u00b7 x) \u2261 z\n\nRightZero : A \u2192 Op\u2082 A \u2192 Set _\nRightZero z _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 z) \u2261 z\n\nZero : A \u2192 Op\u2082 A \u2192 Set _\nZero z \u00b7 = LeftZero z \u00b7 \u00d7 RightZero z \u00b7\n\nLeftInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nLeftInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nRightInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nRightInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverse e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\nInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverseNonZero zero e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\n_DistributesOver\u02e1_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02e1 _+_ =\n  \u2200 {x y z} \u2192 (x * (y + z)) \u2261 ((x * y) + (x * z))\n\n_DistributesOver\u02b3_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02b3 _+_ =\n  \u2200 {x y z} \u2192 ((y + z) * x) \u2261 ((y * x) + (z * x))\n\n_DistributesOver_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n* DistributesOver + = (* DistributesOver\u02e1 +) \u00d7 (* DistributesOver\u02b3 +)\n\n_IdempotentOn_ : Op\u2082 A \u2192 A \u2192 Set _\n_\u00b7_ IdempotentOn x = (x \u00b7 x) \u2261 x\n\nIdempotent : Op\u2082 A \u2192 Set _\nIdempotent \u00b7 = \u2200 {x} \u2192 \u00b7 IdempotentOn x\n\nIdempotentFun : Op\u2081 A \u2192 Set _\nIdempotentFun f = \u2200 {x} \u2192 f (f x) \u2261 f x\n\n_Absorbs_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_\u00b7_ Absorbs _\u2218_ = \u2200 {x y} \u2192 (x \u00b7 (x \u2218 y)) \u2261 x\n\nAbsorptive : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nAbsorptive \u00b7 \u2218 = (\u00b7 Absorbs \u2218) \u00d7 (\u2218 Absorbs \u00b7)\n\nInvolutive : Op\u2081 A \u2192 Set _\nInvolutive f = \u2200 {x} \u2192 f (f x) \u2261 x\n\nInterchange : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nInterchange _\u00b7_ _\u2218_ = \u2200 {x y z t} \u2192 ((x \u00b7 y) \u2218 (z \u00b7 t)) \u2261 ((x \u2218 z) \u00b7 (y \u2218 t))\n\nLeftCancel : Op\u2082 A \u2192 Set _\nLeftCancel _\u00b7_ = \u2200 {c x y} \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancel : Op\u2082 A \u2192 Set _\nRightCancel _\u00b7_ = \u2200 {c x y} \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nLeftCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nRightCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nmodule InterchangeFromAssocComm\n         (_\u00b7_     : Op\u2082 A)\n         (\u00b7-assoc : Associative _\u00b7_)\n         (\u00b7-comm  : Commutative _\u00b7_)\n         where\n\n    open \u2261-Reasoning\n\n    \u00b7= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 (x \u00b7 y) \u2261 (x' \u00b7 y')\n    \u00b7= refl refl = refl\n\n    \u00b7-interchange : Interchange _\u00b7_ _\u00b7_\n    \u00b7-interchange {x} {y} {z} {t}\n                = (x \u00b7 y) \u00b7 (z \u00b7 t)\n                \u2261\u27e8 \u00b7-assoc \u27e9\n                  x \u00b7 (y \u00b7 (z \u00b7 t))\n                \u2261\u27e8 \u00b7= refl (! \u00b7-assoc) \u27e9\n                  x \u00b7 ((y \u00b7 z) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl (\u00b7= \u00b7-comm refl) \u27e9\n                  x \u00b7 ((z \u00b7 y) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl \u00b7-assoc \u27e9\n                  x \u00b7 (z \u00b7 (y \u00b7 t))\n                \u2261\u27e8 ! \u00b7-assoc \u27e9\n                  (x \u00b7 z) \u00b7 (y \u00b7 t)\n                \u220e\n\nmodule _ {b} {B : Set b} (f : A \u2192 B) where\n\n    Injective : Set (b \u2294 a)\n    Injective = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n    Conflict : Set (b \u2294 a)\n    Conflict = \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 (x \u2262 y) \u00d7 f x \u2261 f y\n\nmodule _ {b} {B : Set b} {f : A \u2192 B} where\n    Injective-\u00acConflict : Injective f \u2192 \u00ac (Conflict f)\n    Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n    Conflict-\u00acInjective : Conflict f \u2192 \u00ac (Injective f)\n    Conflict-\u00acInjective = flip Injective-\u00acConflict\n\nmodule Endo {f : A \u2192 A} where\n    Cycle^ : \u2115 \u2192 Set _\n    Cycle^ n = \u2203 \u03bb x \u2192 f $\u27e8 n \u27e9 x \u2261 x\n\n    Cycle = \u2203 Cycle^\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Like Algebra.FunctionProperties but specialized to _\u2261_ and using implict arguments\n-- Moreover there is some extensions such as:\n-- * interchange\n-- * non-zero inversions\n-- * cancelation and non-zero cancelation\n\n-- These properties can (for instance) be used to define algebraic\n-- structures.\n\nopen import Level\nopen import Function using (flip)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\n-- The properties are specified using the following relation as\n-- \"equality\".\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\n------------------------------------------------------------------------\n-- Unary and binary operations\n\nopen import Algebra.FunctionProperties.Core public\n\n------------------------------------------------------------------------\n-- Properties of operations\n\nAssociative : Op\u2082 A \u2192 Set _\nAssociative _\u00b7_ = \u2200 {x y z} \u2192 ((x \u00b7 y) \u00b7 z) \u2261 (x \u00b7 (y \u00b7 z))\n\nCommutative : Op\u2082 A \u2192 Set _\nCommutative _\u00b7_ = \u2200 {x y} \u2192 (x \u00b7 y) \u2261 (y \u00b7 x)\n\nLeftIdentity : A \u2192 Op\u2082 A \u2192 Set _\nLeftIdentity e _\u00b7_ = \u2200 {x} \u2192 (e \u00b7 x) \u2261 x\n\nRightIdentity : A \u2192 Op\u2082 A \u2192 Set _\nRightIdentity e _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 e) \u2261 x\n\nIdentity : A \u2192 Op\u2082 A \u2192 Set _\nIdentity e \u00b7 = LeftIdentity e \u00b7 \u00d7 RightIdentity e \u00b7\n\nLeftZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftZero z _\u00b7_ = \u2200 {x} \u2192 (z \u00b7 x) \u2261 z\n\nRightZero : A \u2192 Op\u2082 A \u2192 Set _\nRightZero z _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 z) \u2261 z\n\nZero : A \u2192 Op\u2082 A \u2192 Set _\nZero z \u00b7 = LeftZero z \u00b7 \u00d7 RightZero z \u00b7\n\nLeftInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nLeftInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nRightInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nRightInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverse e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\nInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverseNonZero zero e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\n_DistributesOver\u02e1_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02e1 _+_ =\n  \u2200 {x y z} \u2192 (x * (y + z)) \u2261 ((x * y) + (x * z))\n\n_DistributesOver\u02b3_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02b3 _+_ =\n  \u2200 {x y z} \u2192 ((y + z) * x) \u2261 ((y * x) + (z * x))\n\n_DistributesOver_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n* DistributesOver + = (* DistributesOver\u02e1 +) \u00d7 (* DistributesOver\u02b3 +)\n\n_IdempotentOn_ : Op\u2082 A \u2192 A \u2192 Set _\n_\u00b7_ IdempotentOn x = (x \u00b7 x) \u2261 x\n\nIdempotent : Op\u2082 A \u2192 Set _\nIdempotent \u00b7 = \u2200 {x} \u2192 \u00b7 IdempotentOn x\n\nIdempotentFun : Op\u2081 A \u2192 Set _\nIdempotentFun f = \u2200 {x} \u2192 f (f x) \u2261 f x\n\n_Absorbs_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_\u00b7_ Absorbs _\u2218_ = \u2200 {x y} \u2192 (x \u00b7 (x \u2218 y)) \u2261 x\n\nAbsorptive : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nAbsorptive \u00b7 \u2218 = (\u00b7 Absorbs \u2218) \u00d7 (\u2218 Absorbs \u00b7)\n\nInvolutive : Op\u2081 A \u2192 Set _\nInvolutive f = \u2200 {x} \u2192 f (f x) \u2261 x\n\nInterchange : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nInterchange _\u00b7_ _\u2218_ = \u2200 {x y z t} \u2192 ((x \u00b7 y) \u2218 (z \u00b7 t)) \u2261 ((x \u2218 z) \u00b7 (y \u2218 t))\n\nLeftCancel : Op\u2082 A \u2192 Set _\nLeftCancel _\u00b7_ = \u2200 {c x y} \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancel : Op\u2082 A \u2192 Set _\nRightCancel _\u00b7_ = \u2200 {c x y} \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nLeftCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nRightCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nmodule InterchangeFromAssocComm\n         (_\u00b7_     : Op\u2082 A)\n         (\u00b7-assoc : Associative _\u00b7_)\n         (\u00b7-comm  : Commutative _\u00b7_)\n         where\n\n    open \u2261-Reasoning\n\n    \u00b7= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 (x \u00b7 y) \u2261 (x' \u00b7 y')\n    \u00b7= refl refl = refl\n\n    \u00b7-interchange : Interchange _\u00b7_ _\u00b7_\n    \u00b7-interchange {x} {y} {z} {t}\n                = (x \u00b7 y) \u00b7 (z \u00b7 t)\n                \u2261\u27e8 \u00b7-assoc \u27e9\n                  x \u00b7 (y \u00b7 (z \u00b7 t))\n                \u2261\u27e8 \u00b7= refl (! \u00b7-assoc) \u27e9\n                  x \u00b7 ((y \u00b7 z) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl (\u00b7= \u00b7-comm refl) \u27e9\n                  x \u00b7 ((z \u00b7 y) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl \u00b7-assoc \u27e9\n                  x \u00b7 (z \u00b7 (y \u00b7 t))\n                \u2261\u27e8 ! \u00b7-assoc \u27e9\n                  (x \u00b7 z) \u00b7 (y \u00b7 t)\n                \u220e\n\nmodule _ {b} {B : Set b} (f : A \u2192 B) where\n\n    Injective : Set (b \u2294 a)\n    Injective = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n    Conflict : Set (b \u2294 a)\n    Conflict = \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 (x \u2262 y) \u00d7 f x \u2261 f y\n\nmodule _ {b} {B : Set b} {f : A \u2192 B} where\n    Injective-\u00acConflict : Injective f \u2192 \u00ac (Conflict f)\n    Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n    Conflict-\u00acInjective : Conflict f \u2192 \u00ac (Injective f)\n    Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1101c4e828759adabf2e6d1502aa9a74f67ca917","subject":"progress cases, removing some redundant ones, adding a couple comments","message":"progress cases, removing some redundant ones, adding a couple comments\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V VConst\n\n  -- variables\n  progress (TAVar x) = abort (somenotnone (! x))\n\n  -- lambdas\n  progress (TALam D) = V VLam\n\n  -- applications\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n    -- left applicand value\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n    -- errors propagate\n  progress (TAAp D1 x\u2082 D2) | _   | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n    -- indeterminates\n  progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n    -- either applicand steps\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , {!!})\n  progress (TAAp {d2 = d2} D1 x\u2082 D2) | S (\u03c01 , \u03c02) | _ = S ((\u03c01 \u2218 d2) , {!!})\n\n  -- empty holes\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {m = \u2717} x x\u2081) = S (_ , Step FHEHole ITEHole FHEHole)\n\n  -- non-empty holes\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | V v = S ( _ , Step (FHNEHoleFinal (FVal v)) (ITNEHole (FVal v)) FHNEHoleEvaled)\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | I x = S (_ , Step (FHNEHoleFinal (FIndet x)) (ITNEHole (FIndet x)) FHNEHoleEvaled )\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole {d = d} {u = u} {\u03c3 = \u03c3} {m = m} x\u2083 D x\u2084) | S (d' , Step x x\u2081 x\u2082) = S ( \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 , {!!}) -- maybe depends on m\n\n  -- casts\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  progress (TACast D m) | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V VConst\n\n  -- variables\n  progress (TAVar x) = abort (somenotnone (! x))\n\n  -- lambdas\n  progress (TALam D) = V VLam\n\n  -- applications\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | _   | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n  progress (TAAp D1 x\u2082 D2) | I IEHole | V x\u2081 = I (IAp IEHole (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I (INEHole x) | V x\u2081 = I (IAp (INEHole x) (FVal x\u2081))\n  progress (TAAp D1 x\u2083 D2) | I (IAp x x\u2081) | V x\u2082 = I (IAp (IAp x x\u2081) (FVal x\u2082))\n  progress (TAAp D1 x\u2082 D2) | I (ICast x) | V x\u2081 = I (IAp (ICast x) (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , (Step {!!} (ITLam (FIndet {!!})) {!!}))\n  progress (TAAp {d2 = d2} D1 x\u2082 D2) | S (\u03c01 , \u03c02) | _ = S ((\u03c01 \u2218 d2) , {!!})\n\n  -- empty holes\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {m = \u2717} x x\u2081) = S (_ , Step FHEHole ITEHole FHEHole)\n\n  -- non-empty holes\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | V v = S ( _ , Step (FHNEHoleFinal (FVal v)) (ITNEHole (FVal v)) FHNEHoleEvaled)\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | I x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S {!!} --  S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n\n  -- casts\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  progress (TACast D m) | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a70fc323eaa6eec7d9c617ee3ccef5722e7d7af5","subject":"Data.Nat","message":"Data.Nat\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3e2e2af74570385b18c0b4160f86bc1c38ea5cc0","subject":"Realign member order in env. validity","message":"Realign member order in env. validity\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n  [ \u03c4 ] dv from v1 to v2 \u00d7 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dv \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app s t) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite sym (fit-sound t \u03c11 d\u03c1) =\n  let fromToF = fromtoDerive (_ \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive _ t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive _ t (daa , d\u03c1\u03c1)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n  [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u00d7 [ \u03c4 ] dv from v1 to v2\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , d\u03c1\u03c1) = d\u03c1\u03c1\nfromtoDeriveVar (that x) (dv \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 dvv\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app s t) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite sym (fit-sound t \u03c11 d\u03c1) =\n  let fromToF = fromtoDerive (_ \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive _ t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive _ t (d\u03c1\u03c1 , daa)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e7633a6efa1638c88b0e30d79e852a1320869251","subject":"tree permutations, made easy!","message":"tree permutations, made easy!\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\n0\u2194\u2032_ : \u2200 {n} \u2192 Bits n \u2192 SwpOp n\n0\u2194\u2032_ {n} rewrite cong SwpOp (sym (\u2115\u00b0.+-comm n 0)) = 0\u2194_ {n} {0}\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = 0\u2194\u2032 p \u204f 0\u2194\u2032 q\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata AC {a} {A : Set a} : \u2200 {m n} (left : Tree A m) (right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 AC t t\n\n  _\u204f_ : \u2200 {m n o} {t : Tree A m} {u : Tree A n} {v : Tree A o} \u2192 AC t u \u2192 AC u v \u2192 AC t v\n\n  first : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         AC left\u2080 left\u2081 \u2192\n         AC (fork left\u2080 right) (fork left\u2081 right)\n\n  swp : \u2200 {n} {left right : Tree A n} \u2192\n         AC (fork left right) (fork right left)\n\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         AC (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n         {-\n  zip : \u2200 {m n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A m} {u\u2080 u\u2081 : Tree A n} \u2192\n            AC (fork t\u2080\u2080 t\u2081\u2081) u\u2080 \u2192\n            AC (fork t\u2080\u2081 t\u2081\u2080) u\u2081 \u2192\n            AC (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork u\u2080 u\u2081)\n            -}\n\ninfixr 1 _\u204f_\n\nsecond-AC : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           AC right\u2080 right\u2081 \u2192\n           AC (fork left right\u2080) (fork left right\u2081)\nsecond-AC f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-AC : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           AC left\u2080 left\u2081 \u2192\n           AC right\u2080 right\u2081 \u2192\n           AC (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-AC = first f \u204f second-AC g\n\nswp\u2083-AC : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         AC (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-AC = < swp \u00d7 swp >-AC \u204f swp\u2082 \u204f < swp \u00d7 swp >-AC\n\nSwp\u21d2AC : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 AC {n = n}\nSwp\u21d2AC (left pf) = first (Swp\u21d2AC pf)\nSwp\u21d2AC (right pf) = second-AC (Swp\u21d2AC pf)\nSwp\u21d2AC swp\u2081 = swp\nSwp\u21d2AC (swp\u2082 {t\u2080\u2080 = A} {B} {C} {D}) = swp\u2082 -- zip {!!} {!!} {-first swp \u204f zip swp \u03b5 \u204f {!!}-}\n\nSwp\u2605\u21d2AC : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 AC {n = n}\nSwp\u2605\u21d2AC \u03b5         = \u03b5\nSwp\u2605\u21d2AC (x \u25c5 xs) = Swp\u21d2AC x \u204f Swp\u2605\u21d2AC xs\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0655a6d0e4899c4ba47d263fc94275319d74c339","subject":"IDesc example: rework Hutton shaver with a more traditional context","message":"IDesc example: rework Hutton shaver with a more traditional context","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\n-- Fix menu:\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),                        -- Val t\n                              (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                               Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             (Val ty + Var context ty) ->\n             IMu closeTerm' ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      (Val ty + Var context ty) ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2\ntestContext : Context _\ntestContext = vcons (bool , con (EZe , true )) (vcons (nat , con (EZe , su (su ze)) ) vnil)\n\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"09a448f0651733ec90348ae060a019caff44fa89","subject":"agda\/derivation.agda: progress on \u0394-term","message":"agda\/derivation.agda: progress on \u0394-term\n\nImplement \u0394-term on applications, assuming the availability of a nabla\nmetafunction (which we expect to be implemented using a term\nconstructor, but who knows).\n\nOld-commit-hash: 803596156f6f88a497e4f039116e047d66aedf3c\n","repos":"inc-lc\/ilc-agda","old_file":"derivation.agda","new_file":"derivation.agda","new_contents":"module derivation where\n\nopen import lambda\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\nadaptVar : \u2200  {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","old_contents":"module derivation where\n\nopen import lambda\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- CHANGE TERMS\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9843e14b03993bc42349a93ec9737bc06686d1b9","subject":"Added setoid for FOTC.","message":"Added setoid for FOTC.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-positivity-check #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\nmodule FOTC where\n\nmodule Aczel-CA where\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 {x} \u2192 x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    K-eq  : \u2200 x y \u2192 K \u00b7 x \u00b7 y \u2250 x\n    S-eq  : \u2200 x y z \u2192 S \u00b7 x \u00b7 y \u00b7 z \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- we can proof\n\n  \u2250\u2192\u2261 : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  \u2250\u2192\u2261 {x} h = subst (\u03bb z \u2192 x \u2261 z) h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule FOTC where\n\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl     : \u2200 {x} \u2192 x \u2250 x\n    sym      : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    trans    : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    if-true  : \u2200 d\u2081 d\u2082 \u2192 if \u00b7 true \u00b7 d\u2081 \u00b7 d\u2082 \u2250 d\u2081\n    if-false : \u2200 d\u2081 d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2250 d\u2082\n    pred-0   : pred \u00b7 zero \u2250 zero\n    pred-S   : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2250 n\n    iszero-0 : iszero \u00b7 zero \u2250 true\n    iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2250 false\n    beta     : \u2200 f a \u2192 lam f \u00b7 a \u2250 f a\n    fix-eq   : \u2200 f \u2192 fix f \u2250 f (fix f)\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- you can get the instance\n\n  subst-aux : \u2200 x y \u2192 x \u2250 y \u2192 x \u2261 x \u2192 x \u2261 y\n  subst-aux x y h\u2081 h\u2082 = subst A x y h\u2081 refl\n    where A : D \u2192 Set\n          A z = x \u2261 z\n\n  -- hence you can prove\n\n  thm : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  thm {x} {y} h = subst-aux x y h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"037a6a0d1770f4eb88369b436af7b960161dd9ad","subject":"Reported issue #11.","message":"Reported issue #11.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/Terms.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/Terms.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP terms\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.ABP.Terms where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\n\n------------------------------------------------------------------------------\n-- N.B. We did not define @Bit = Bool@ due to the issue #11.\nBit : D \u2192 Set\nBit b = Bool b\n{-# ATP definition Bit #-}\n\nF : D\nF = false\n{-# ATP definition F #-}\n\nT : D\nT = true\n{-# ATP definition T #-}\n\npostulate\n  <_,_> : D \u2192 D \u2192 D\n  ok    : D \u2192 D\n","old_contents":"------------------------------------------------------------------------------\n-- ABP terms\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.ABP.Terms where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\n\n------------------------------------------------------------------------------\n\nBit : D \u2192 Set\nBit b = Bool b\n{-# ATP definition Bit #-}\n\nF : D\nF = false\n{-# ATP definition F #-}\n\nT : D\nT = true\n{-# ATP definition T #-}\n\npostulate\n  <_,_> : D \u2192 D \u2192 D\n  ok    : D \u2192 D\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6dddc3bd8f8e569a803a9ed68746ad93c6653498","subject":"Desc stratified model: fresh start","message":"Desc stratified model: fresh start\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- List\n--********************************************\n\ndata ListDConst (l : Level) : Set l where\n  cnil : ListDConst l\n  ccons : ListDConst l\n\nlistDChoice : (l : Level) -> Set l -> ListDConst l -> IDesc l Unit\nlistDChoice x X cnil = const Unit\nlistDChoice x X ccons = sigma X (\\_ -> var Void)\n\nlistD : (l : Level) -> Set l -> IDesc l Unit\nlistD x X = sigma (ListDConst x) (listDChoice x X)\n\nlist : (l : Level) -> Set l -> Set l\nlist x X = IMu x Unit (\\_ -> listD x X) Void\n\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b4c7bc65da42597091f08c82956e4c5a2dfbce3a","subject":"Updated setoid note.","message":"Updated setoid note.\n\nIgnore-this: e3efd15c1aa4c5d0d3b13ce3c8b9d2c0\n\ndarcs-hash:20120514124154-3bd4e-4a0539c4fe3ebec97983e47f466c258e4221a3a0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 14 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 13 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- 13 May 2012. It seems we cannot define the identity elimination\n  -- using the setoid equality.\n  --\n  -- subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- The identity type\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  -- 13 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2261_\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"675459709ff3563cde300754c35f229c6b963f10","subject":"Rename \u0394Env-Entry to ValidChange.","message":"Rename \u0394Env-Entry to ValidChange.\n\nAlso rename operations related to \u0394Env-Entry.\n\nOld-commit-hash: 35ab981b89bab6eced9a7b20753bb140f6caf7a8\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    \u0394Val-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u0394Val-base \u03b9 \u2192 Set\n    apply-\u0394Val-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u0394Val-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-\u0394Val-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u0394Val-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-\u0394Val-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-\u0394Val-base \u03b9 v (diff-\u0394Val-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  \u0394Val : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n  apply-\u0394Val : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-\u0394Val : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n\n  infixl 6 apply-\u0394Val diff-\u0394Val -- as with + - in GHC.Num\n  syntax apply-\u0394Val \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-\u0394Val \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (\u0394Val \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (\u0394Val (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- \u0394Val \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- \u0394Val : Type \u2192 Set\n  \u0394Val (base \u03b9) = \u0394Val-base \u03b9\n  \u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-\u0394Val-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-\u0394Val-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : \u0394Val \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 \u0394Val \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  ignore-valid-change (cons v _ _) = v\n\n  update-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  update-valid-change {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-valid-change {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-valid-change {\u03c4})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    \u0394Val-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u0394Val-base \u03b9 \u2192 Set\n    apply-\u0394Val-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u0394Val-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-\u0394Val-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u0394Val-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-\u0394Val-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-\u0394Val-base \u03b9 v (diff-\u0394Val-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  \u0394Val : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n  apply-\u0394Val : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-\u0394Val : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n\n  infixl 6 apply-\u0394Val diff-\u0394Val -- as with + - in GHC.Num\n  syntax apply-\u0394Val \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-\u0394Val \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (\u0394Val \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (\u0394Val (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- \u0394Val \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- \u0394Val : Type \u2192 Set\n  \u0394Val (base \u03b9) = \u0394Val-base \u03b9\n  \u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-\u0394Val-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-\u0394Val-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : \u0394Val \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env-Entry : Type \u2192 Set\n  \u0394Env-Entry \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 \u0394Val \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList \u0394Env-Entry\n\n  ignore-entry : \u0394Env-Entry \u2286 \u27e6_\u27e7\n  ignore-entry (cons v _ _) = v\n\n  update-entry : \u0394Env-Entry \u2286 \u27e6_\u27e7\n  update-entry {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-entry {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-entry {\u03c4})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5f825cae8441f0fd6b7b1d5bdcba4517455f3cf5","subject":"lib\/Explore\/Summable.agda","message":"lib\/Explore\/Summable.agda\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Summable.agda","new_file":"lib\/Explore\/Summable.agda","new_contents":"module Explore.Summable where\n\nopen import Type\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\nopen import Explore.Type\nopen import Explore.Product\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Bool.NP renaming (Bool to \ud835\udfda; true to 1b; false to 0b; to\u2115 to \ud835\udfda\u25b9\u2115)\nopen Data.Bool.NP.Indexed\n\nmodule FromSum {A : \u2605} (sum : Sum A) where\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\nmodule FromSumInd {A : \u2605}\n                  {sum : Sum A}\n                  (sum-ind : SumInd sum) where\n  open FromSum sum public\n\n  sum-ext : SumExt sum\n  sum-ext = sum-ind (\u03bb s \u2192 s _ \u2261 s _) (\u2261.cong\u2082 _+_)\n\n  sum-zero : SumZero sum\n  sum-zero = sum-ind (\u03bb s \u2192 s (const 0) \u2261 0) (\u2261.cong\u2082 _+_) (\u03bb _ \u2192 \u2261.refl)\n\n  sum-hom : SumHom sum\n  sum-hom f g = sum-ind (\u03bb s \u2192 s (f +\u00b0 g) \u2261 s f + s g)\n                        (\u03bb {s\u2080} {s\u2081} p\u2080 p\u2081 \u2192 \u2261.trans (\u2261.cong\u2082 _+_ p\u2080 p\u2081) (+-interchange (s\u2080 _) (s\u2080 _) _ _))\n                        (\u03bb _ \u2192 \u2261.refl)\n\n  sum-mono : SumMono sum\n  sum-mono = sum-ind (\u03bb s \u2192 s _ \u2264 s _) _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  module _ (f g : A \u2192 \u2115) where\n    open \u2261.\u2261-Reasoning\n\n    sum-\u2293-\u2238 : sum f \u2261 sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g)\n    sum-\u2293-\u2238 = sum f                          \u2261\u27e8 sum-ext (f \u27e8 a\u2261a\u2293b+a\u2238b \u27e9\u00b0 g) \u27e9\n              sum ((f \u2293\u00b0 g) +\u00b0 (f \u2238\u00b0 g))     \u2261\u27e8 sum-hom (f \u2293\u00b0 g) (f \u2238\u00b0 g) \u27e9\n              sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g) \u220e\n\n    sum-\u2294-\u2293 : sum f + sum g \u2261 sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g)\n    sum-\u2294-\u2293 = sum f + sum g               \u2261\u27e8 \u2261.sym (sum-hom f g) \u27e9\n              sum (f +\u00b0 g)                \u2261\u27e8 sum-ext (f \u27e8 a+b\u2261a\u2294b+a\u2293b \u27e9\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g +\u00b0 f \u2293\u00b0 g)      \u2261\u27e8 sum-hom (f \u2294\u00b0 g) (f \u2293\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g) \u220e\n\n    sum-\u2294 : sum (f \u2294\u00b0 g) \u2264 sum f + sum g\n    sum-\u2294 = \u2115\u2264.trans (sum-mono (f \u27e8 \u2294\u2264+ \u27e9\u00b0 g)) (\u2115\u2264.reflexive (sum-hom f g))\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sum-const : \u2200 k \u2192 sum (const k) \u2261 Card * k\n  sum-const k\n      rewrite \u2115\u00b0.*-comm Card k\n            | \u2261.sym (sum-lin (const 1) k)\n            | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\n  module _ f g where\n    count-\u2227-not : count f \u2261 count (f \u2227\u00b0 g) + count (f \u2227\u00b0 not\u00b0 g)\n    count-\u2227-not rewrite sum-\u2293-\u2238 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2238 \u27e9\u00b0 g)\n                      = \u2261.refl\n\n    count-\u2228-\u2227 : count f + count g \u2261 count (f \u2228\u00b0 g) + count (f \u2227\u00b0 g)\n    count-\u2228-\u2227 rewrite sum-\u2294-\u2293 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                    = \u2261.refl\n\n    count-\u2228\u2264+ : count (f \u2228\u00b0 g) \u2264 count f + count g\n    count-\u2228\u2264+ = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext (\u2261.sym \u2218 (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g))))\n                         (sum-\u2294 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g))\n\nmodule FromSum\u00d7\n         {A B}\n         {sum\u1d2c     : Sum A}\n         (sum-ind\u1d2c : SumInd sum\u1d2c)\n         {sum\u1d2e     : Sum B}\n         (sum-ind\u1d2e : SumInd sum\u1d2e) where\n\n  module |A| = FromSumInd sum-ind\u1d2c\n  module |B| = FromSumInd sum-ind\u1d2e\n  open Operators\n\n  sum\u1d2c\u1d2e = sum\u1d2c \u00d7\u02e2 sum\u1d2e\n\n  sum-\u2218proj\u2081\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 |B|.Card * sum\u1d2c f\n  sum-\u2218proj\u2081\u2261Card* f\n      rewrite |A|.sum-ext (|B|.sum-const \u2218 f)\n            = |A|.sum-lin f |B|.Card\n\n  sum-\u2218proj\u2082\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 |A|.Card * sum\u1d2e f\n  sum-\u2218proj\u2082\u2261Card* = |A|.sum-const \u2218 sum\u1d2e\n\n  sum-\u2218proj\u2081 : \u2200 {f} {g} \u2192 sum\u1d2c f \u2261 sum\u1d2c g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2081)\n  sum-\u2218proj\u2081 {f} {g} sumf\u2261sumg\n      rewrite sum-\u2218proj\u2081\u2261Card* f\n            | sum-\u2218proj\u2081\u2261Card* g\n            | sumf\u2261sumg = \u2261.refl\n\n  sum-\u2218proj\u2082 : \u2200 {f} {g} \u2192 sum\u1d2e f \u2261 sum\u1d2e g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2082)\n  sum-\u2218proj\u2082 sumf\u2261sumg = |A|.sum-ext (const sumf\u2261sumg)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Explore.Summable where\n\nopen import Type\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\nopen import Explore.Type\nopen import Explore.Product\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Bool.NP renaming (Bool to \ud835\udfda; true to 1b; false to 0b; to\u2115 to \ud835\udfda\u25b9\u2115)\nopen Data.Bool.NP.Indexed\n\nmodule FromSum {A : \u2605} (sum : Sum A) where\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\nmodule FromSumInd {A : \u2605}\n                  {sum : Sum A}\n                  (sum-ind : SumInd sum) where\n  open FromSum sum public\n\n  sum-ext : SumExt sum\n  sum-ext = sum-ind (\u03bb s \u2192 s _ \u2261 s _) (\u2261.cong\u2082 _+_)\n\n  sum-zero : SumZero sum\n  sum-zero = sum-ind (\u03bb s \u2192 s (const 0) \u2261 0) (\u2261.cong\u2082 _+_) (\u03bb _ \u2192 \u2261.refl)\n\n  sum-hom : SumHom sum\n  sum-hom f g = sum-ind (\u03bb s \u2192 s (f +\u00b0 g) \u2261 s f + s g)\n                        (\u03bb {s\u2080} {s\u2081} p\u2080 p\u2081 \u2192 \u2261.trans (\u2261.cong\u2082 _+_ p\u2080 p\u2081) (+-interchange (s\u2080 _) (s\u2080 _) _ _))\n                        (\u03bb _ \u2192 \u2261.refl)\n\n  sum-mono : SumMono sum\n  sum-mono = sum-ind (\u03bb s \u2192 s _ \u2264 s _) _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  module _ (f g : A \u2192 \u2115) where\n    open \u2261.\u2261-Reasoning\n\n    sum-\u2293-\u2238 : sum f \u2261 sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g)\n    sum-\u2293-\u2238 = sum f                          \u2261\u27e8 sum-ext (f \u27e8 a\u2261a\u2293b+a\u2238b \u27e9\u00b0 g) \u27e9\n              sum ((f \u2293\u00b0 g) +\u00b0 (f \u2238\u00b0 g))     \u2261\u27e8 sum-hom (f \u2293\u00b0 g) (f \u2238\u00b0 g) \u27e9\n              sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g) \u220e\n\n    sum-\u2294-\u2293 : sum f + sum g \u2261 sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g)\n    sum-\u2294-\u2293 = sum f + sum g               \u2261\u27e8 \u2261.sym (sum-hom f g) \u27e9\n              sum (f +\u00b0 g)                \u2261\u27e8 sum-ext (f \u27e8 a+b\u2261a\u2294b+a\u2293b \u27e9\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g +\u00b0 f \u2293\u00b0 g)      \u2261\u27e8 sum-hom (f \u2294\u00b0 g) (f \u2293\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g) \u220e\n\n    sum-\u2294 : sum (f \u2294\u00b0 g) \u2264 sum f + sum g\n    sum-\u2294 = \u2115\u2264.trans (sum-mono (f \u27e8 \u2294\u2264+ \u27e9\u00b0 g)) (\u2115\u2264.reflexive (sum-hom f g))\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sum-const : \u2200 k \u2192 sum (const k) \u2261 Card * k\n  sum-const k\n      rewrite \u2115\u00b0.*-comm Card k\n            | \u2261.sym (sum-lin (const 1) k)\n            | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\n  module _ f g where\n    count-\u2227-not : count f \u2261 count (f \u2227\u00b0 g) + count (f \u2227\u00b0 not\u00b0 g)\n    count-\u2227-not rewrite sum-\u2293-\u2238 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2238 \u27e9\u00b0 g)\n                      = \u2261.refl\n\n    count-\u2228-\u2227 : count f + count g \u2261 count (f \u2228\u00b0 g) + count (f \u2227\u00b0 g)\n    count-\u2228-\u2227 rewrite sum-\u2294-\u2293 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                    = \u2261.refl\n\n    count-\u2228\u2264+ : count (f \u2228\u00b0 g) \u2264 count f + count g\n    count-\u2228\u2264+ = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext (\u2261.sym \u2218 (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g))))\n                         (sum-\u2294 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g))\n\nmodule FromSum\u00d7\n         {A B}\n         {sum\u1d2c     : Sum A}\n         (sum-ind\u1d2c : SumInd sum\u1d2c)\n         {sum\u1d2e     : Sum B}\n         (sum-ind\u1d2e : SumInd sum\u1d2e) where\n\n  module |A| = FromSumInd sum-ind\u1d2c\n  module |B| = FromSumInd sum-ind\u1d2e\n  open Operators\n  \n  sum\u1d2c\u1d2e = sum\u1d2c \u00d7\u02e2 sum\u1d2e\n\n  sum-\u2218proj\u2081\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 |B|.Card * sum\u1d2c f\n  sum-\u2218proj\u2081\u2261Card* f\n      rewrite |A|.sum-ext (|B|.sum-const \u2218 f)\n            = |A|.sum-lin f |B|.Card\n\n  sum-\u2218proj\u2082\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 |A|.Card * sum\u1d2e f\n  sum-\u2218proj\u2082\u2261Card* = |A|.sum-const \u2218 sum\u1d2e\n\n  sum-\u2218proj\u2081 : \u2200 {f} {g} \u2192 sum\u1d2c f \u2261 sum\u1d2c g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2081)\n  sum-\u2218proj\u2081 {f} {g} sumf\u2261sumg\n      rewrite sum-\u2218proj\u2081\u2261Card* f\n            | sum-\u2218proj\u2081\u2261Card* g\n            | sumf\u2261sumg = \u2261.refl\n\n  sum-\u2218proj\u2082 : \u2200 {f} {g} \u2192 sum\u1d2e f \u2261 sum\u1d2e g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2082)\n  sum-\u2218proj\u2082 sumf\u2261sumg = |A|.sum-ext (const sumf\u2261sumg)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bc3f375b16b82ac64c6d762e6ffd4f7483bd4984","subject":"update linear solver with isomorphisms","message":"update linear solver with isomorphisms\n","repos":"crypto-agda\/crypto-agda","old_file":"linear-solver.agda","new_file":"linear-solver.agda","new_contents":"module linear-solver where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'    : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'   : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n  infix 4 _\u21db_\n\n  record Eq : Set where\n    constructor _\u21db_\n    field\n      LHS RHS : Syn\n\n  open import Data.Vec.N-ary using (N-ary ; _$\u207f_)\n  open import Data.Vec using (allFin) renaming (map to vmap)\n\n  rewire' : (f : N-ary nrVars Syn Eq) \u2192 let (S\u2081 \u21db S\u2082) = f $\u207f (vmap Syn.var (allFin nrVars))\n          in CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire' f eq = let S \u21db S' = f $\u207f vmap Syn.var (allFin  nrVars)\n         in rewire S S' eq\n \nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire' (\u03bb a b c \u2192 (a , b) , c \u21db (b , a) , c) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2083 where\n\n  open import Data.Unit\n  open import Data.Product\n  open import Data.Vec\n\n  open import Function using (flip ; const)\n  \n  open import Function.Inverse\n  open import Function.Related.TypeIsomorphisms.NP\n\n  open \u00d7-CMon using () renaming (\u2219-cong to \u00d7-cong ; assoc to \u00d7-assoc)\n\n  module STest n M = Syntax Set _\u00d7_ \u22a4 _\u2194_ id (flip _\u2218_) A\u00d7\u22a4\u2194A (sym A\u00d7\u22a4\u2194A) (A\u00d7\u22a4\u2194A \u2218 swap-iso) (swap-iso \u2218 sym A\u00d7\u22a4\u2194A)\n                            \u00d7-cong first-iso (\u03bb x \u2192 second-iso (const x))\n                            (sym (\u00d7-assoc _ _ _)) (\u00d7-assoc _ _ _) swap-iso n M\n\n  test : \u2200 A B C \u2192 ((A \u00d7 B) \u00d7 C) \u2194 (C \u00d7 (B \u00d7 A))\n  test A B C = rewire ((# 0 , # 1) , # 2) (# 2 , (# 1 , # 0)) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n","old_contents":"module linear-solver where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'    : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'   : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n  infix 4 _\u21db_\n\n  record Eq : Set where\n    constructor _\u21db_\n    field\n      LHS RHS : Syn\n\n  open import Data.Vec.N-ary using (N-ary ; _$\u207f_)\n  open import Data.Vec using (allFin) renaming (map to vmap)\n\n  rewire' : (f : N-ary nrVars Syn Eq) \u2192 let (S\u2081 \u21db S\u2082) = f $\u207f (vmap Syn.var (allFin nrVars))\n          in CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire' f eq = let S \u21db S' = f $\u207f vmap Syn.var (allFin  nrVars)\n         in rewire S S' eq\n \nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire' (\u03bb a b c \u2192 (a , b) , c \u21db (b , a) , c) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d3b3af37e2a3db45c5989097d40b354cd10cca92","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: 56f256a7d0c9d89dea0c836aed175f19\n\ndarcs-hash:20120303031914-3bd4e-94a323682a68c7fa19333fcfea1528dc4853b741.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/README.agda","new_file":"src\/PA\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- First-order Peano arithmetic (PA)\n------------------------------------------------------------------------------\n\nmodule PA.README where\n\n-- Two formalizations of first-order Peano aritmetic using:\n\n-- \u2022 Agda postulates for the non-logical constants and the Peano's\n--   axioms, using axioms based on the propositional equality (see, for\n--   example, [Machover, 1996, p. 263], [H\u00e1jek and Pudl\u00e1k, 1998, p. 28]).\n\n-- \u2022 Agda data types and primitive recursive functions for addition and\n--   multiplication.\n\n-- References;\n\n-- Mosh\u00e9 Machover. Set theory, logic and their limitations. Cambridge\n-- University Press, 1996.\n\n-- Petr H\u00e1jek and Pavel Pudl\u00e1k. Metamathematics of First-Order\n-- Arithmetic. Springer, 1998. 2nd printing.\n\n------------------------------------------------------------------------------\n-- Axiomatic PA\n\n-- The axioms\nopen import PA.Axiomatic.Standard.Base\n\n-- Some properties\nopen import PA.Axiomatic.Standard.PropertiesATP\nopen import PA.Axiomatic.Standard.PropertiesI\n\n------------------------------------------------------------------------------\n-- Inductive PA\n\n-- Inductive definitions\nopen import PA.Inductive.Base\n\n-- Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n","old_contents":"------------------------------------------------------------------------------\n-- First-order Peano arithmetic (PA)\n------------------------------------------------------------------------------\n\nmodule PA.README where\n\n-- Two formalizations of first-order Peano aritmetic using:\n\n-- * Agda postulates for the non-logical constants and the Peano's\n--   axioms, using axioms based on the propositional equality (see, for\n--   example, [Machover, 1996, p. 263], [H\u00e1jek and Pudl\u00e1k, 1998, p. 28]).\n\n-- * Agda data types and primitive recursive functions for addition and\n--   multiplication.\n\n-- References;\n\n-- Mosh\u00e9 Machover. Set theory, logic and their limitations. Cambridge\n-- University Press, 1996.\n\n-- Petr H\u00e1jek and Pavel Pudl\u00e1k. Metamathematics of First-Order\n-- Arithmetic. Springer, 1998. 2nd printing.\n\n------------------------------------------------------------------------------\n-- Axiomatic PA\n\n-- The axioms\nopen import PA.Axiomatic.Standard.Base\n\n-- Some properties\nopen import PA.Axiomatic.Standard.PropertiesATP\nopen import PA.Axiomatic.Standard.PropertiesI\n\n------------------------------------------------------------------------------\n-- Inductive PA\n\n-- Inductive definitions\nopen import PA.Inductive.Base\n\n-- Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a1572f3d4a91267b68f0d9fae0ab866d4c0337e1","subject":"Syntactic sugar for replacement pairs","message":"Syntactic sugar for replacement pairs\n\nOld-commit-hash: 930ad8e0b6ee020c54d0c8372373e433fb184bad\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/Extension-00.agda","new_file":"nats\/Extension-00.agda","new_contents":"{-\n\nChecklist of stuff to add when adding syntactic constructs\n\n- derive (symbolic derivation; most important!)\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- validity-of-derive\n- correctness-of-derive\n\n-}\n\nmodule Extension-00 where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- ad-hoc extensions\n  foldNat : \u2200 {\u0393 \u03c4} \u2192 (n : Term \u0393 nats) \u2192\n                      (f : Term \u0393 (\u03c4 \u21d2 \u03c4)) \u2192 (z : Term \u0393 \u03c4)\n                    \u2192 Term \u0393 \u03c4\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 = \u227c-reflexive refl\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nweaken subctx (foldNat n f z) = {!!}\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\n\u27e6 foldNat n f z \u27e7Term \u03c1 = {!!}\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (foldNat n f z) \u03c1 = {!!}\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- Combination as type-indexed family of terms\n-- _\u2295_ : \u2200 {\u03c4 \u0393} \u2192  TODO: IMPLEMENT ME!! {!!}\n\n-- Replacement-pairs on all types as syntactic sugar\n\nreplace_by_ : \u2200 {\u03c4 \u0393} \u2192 (old : Term \u0393 \u03c4) (new : Term \u0393 \u03c4)\n                      \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- replace n by m = \u03bb f . f n m\n-- replace f by g = \u03bb x . \u03bb dx . replace (f x) by (g (x \u2295 dx))\n--\n-- Remark. Amazingly, weakening is identical to arbitrary\n-- legal adjustment of de-Bruijn indices.\nreplace_by_ {nats} {\u0393} old new =\n  abs (app (app (var this) (weaken drop-f old)) (weaken drop-f new))\n  where drop-f = drop _ \u2022 \u0393\u227c\u0393\n-- Think: the new must compute upon changes...\nreplace_by_ {\u03c4\u2081 \u21d2 \u03c4\u2082} old new =\n  abs (abs (replace (app (weaken drop! old) (var (that this)))\n                 by (app (weaken drop! new) {!!})))\n  where drop! = drop (\u0394-Type \u03c4\u2081) \u2022 drop \u03c4\u2081 \u2022 \u0393\u227c\u0393\n\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive (foldNat n f z) = ?\nderive (foldNat n f z) =  {!!}\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} (foldNat n f z) = {!!}\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive {\u0393} {\u03c4}\n  \u03c1 {consistency} (foldNat n f z) = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","old_contents":"{-\n\nChecklist of stuff to add when adding syntactic constructs\n\n- derive (symbolic derivation; most important!)\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- validity-of-derive\n- correctness-of-derive\n\n-}\n\nmodule Extension-00 where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- ad-hoc extensions\n  foldNat : \u2200 {\u0393 \u03c4} \u2192 (n : Term \u0393 nats) \u2192\n                      (f : Term \u0393 (\u03c4 \u21d2 \u03c4)) \u2192 (z : Term \u0393 \u03c4)\n                    \u2192 Term \u0393 \u03c4\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nweaken subctx (foldNat n f z) = {!!}\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\n\u27e6 foldNat n f z \u27e7Term \u03c1 = {!!}\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (foldNat n f z) \u03c1 = {!!}\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive (foldNat f n z) =\nderive (foldNat f n z) =  {!!}\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} (foldNat n f z) = {!!}\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive {\u0393} {\u03c4}\n  \u03c1 {consistency} (foldNat n f z) = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"12a874526963109760b8a84a9007561aa0d82700","subject":"Fixed doc.","message":"Fixed doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Interactive proof using an instance of the induction principle.\n+-N-ind : \u2200 {n} \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = +-N-ind prf\u2081 prf\u2082 Nm\n  where\n  prf\u2081 : N (zero + n)\n  prf\u2081 = subst N (sym (+-0x n)) Nn\n\n  prf\u2082 : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n  prf\u2082 {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2082 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- N-ind\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- Because the ATPs don't handle induction, them cannot prove this\n-- postulate.\n-- {-# ATP prove +-N\u2083 N-ind #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\n+-N-ind : \u2200 {n} \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = +-N-ind prf\u2081 prf\u2082 Nm\n  where\n  prf\u2081 : N (zero + n)\n  prf\u2081 = subst N (sym (+-0x n)) Nn\n\n  prf\u2082 : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n  prf\u2082 {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2082 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- N-ind\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- Because the ATPs don't handle induction, them cannot prove this\n-- postulate.\n-- {-# ATP prove +-N\u2083 N-ind #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d635357fe352bf547faec865695ca24e9607d7c9","subject":"Renamed a property.","message":"Renamed a property.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Base\/List\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Base\/List\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC list terms properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Base.List.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n\u2237-leftCong : \u2200 {x y xs} \u2192 x \u2261 y \u2192 x \u2237 xs \u2261 y \u2237 xs\n\u2237-leftCong refl = refl\n\n\u2237-rightCong : \u2200 {x xs ys} \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 x \u2237 ys\n\u2237-rightCong refl = refl\n\n\u2237-cong : \u2200 {x y xs ys} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\n\u2237-cong refl refl = refl\n\nheadCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 head\u2081 xs \u2261 head\u2081 ys\nheadCong refl = refl\n\ntailCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 tail\u2081 xs \u2261 tail\u2081 ys\ntailCong refl = refl\n\n------------------------------------------------------------------------------\n-- Injective properties\n\n\u2237-injective : \u2200 {x y xs ys} \u2192 x \u2237 xs \u2261 y \u2237 ys \u2192 x \u2261 y \u2227 xs \u2261 ys\n\u2237-injective {x} {y} {xs} {ys} h = x\u2261y , xs\u2261ys\n  where\n  x\u2261y : x \u2261 y\n  x\u2261y = x              \u2261\u27e8 sym (head-\u2237 x xs) \u27e9\n        head\u2081 (x \u2237 xs) \u2261\u27e8 headCong h \u27e9\n        head\u2081 (y \u2237 ys) \u2261\u27e8 head-\u2237 y ys \u27e9\n        y              \u220e\n\n  xs\u2261ys : xs \u2261 ys\n  xs\u2261ys = xs             \u2261\u27e8 sym (tail-\u2237 x xs) \u27e9\n          tail\u2081 (x \u2237 xs) \u2261\u27e8 tailCong h \u27e9\n          tail\u2081 (y \u2237 ys) \u2261\u27e8 tail-\u2237 y ys \u27e9\n          ys             \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC list terms properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Base.List.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n\u2237-leftCong : \u2200 {x y xs} \u2192 x \u2261 y \u2192 x \u2237 xs \u2261 y \u2237 xs\n\u2237-leftCong refl = refl\n\n\u2237-rightCong : \u2200 {x xs ys} \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 x \u2237 ys\n\u2237-rightCong refl = refl\n\n\u2237-Cong : \u2200 {x y xs ys} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\n\u2237-Cong refl refl = refl\n\nheadCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 head\u2081 xs \u2261 head\u2081 ys\nheadCong refl = refl\n\ntailCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 tail\u2081 xs \u2261 tail\u2081 ys\ntailCong refl = refl\n\n------------------------------------------------------------------------------\n-- Injective properties\n\n\u2237-injective : \u2200 {x y xs ys} \u2192 x \u2237 xs \u2261 y \u2237 ys \u2192 x \u2261 y \u2227 xs \u2261 ys\n\u2237-injective {x} {y} {xs} {ys} h = x\u2261y , xs\u2261ys\n  where\n  x\u2261y : x \u2261 y\n  x\u2261y = x              \u2261\u27e8 sym (head-\u2237 x xs) \u27e9\n        head\u2081 (x \u2237 xs) \u2261\u27e8 headCong h \u27e9\n        head\u2081 (y \u2237 ys) \u2261\u27e8 head-\u2237 y ys \u27e9\n        y              \u220e\n\n  xs\u2261ys : xs \u2261 ys\n  xs\u2261ys = xs             \u2261\u27e8 sym (tail-\u2237 x xs) \u27e9\n          tail\u2081 (x \u2237 xs) \u2261\u27e8 tailCong h \u27e9\n          tail\u2081 (y \u2237 ys) \u2261\u27e8 tail-\u2237 y ys \u27e9\n          ys             \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ed6d75a98df4fd8dc85536ae6f51df7b5b4a2ebc","subject":"Data.Two.Base: +\u2713\u2227\u00d7","message":"Data.Two.Base: +\u2713\u2227\u00d7\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Two\/Base.agda","new_file":"lib\/Data\/Two\/Base.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two.Base where\n\nopen import Data.Zero\n\nopen import Data.Bool\n  public\n  hiding (if_then_else_)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\n\nopen import Data.Product\n  renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nopen import Data.Sum\n  using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr)\n\nopen import Data.Nat.Base\n  using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\n\nopen import Type using (\u2605_)\n\nopen import Relation.Nullary using (\u00ac_; Dec; yes; no)\n\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; _\u2262_; refl)\n\n0\u22621\u2082 : 0\u2082 \u2262 1\u2082\n0\u22621\u2082 ()\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: fst 1: snd ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n-- See also \u2713-\u2227\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 {0\u2082} p q = p\n\u2713\u2227 {1\u2082} p q = q\n\n-- See also \u2713-\u2227\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 {0\u2082} ()\n\u2713\u2227\u2081 {1\u2082} = _\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {0\u2082} ()\n\u2713\u2227\u2082 {1\u2082} p = p\n\n\u2713\u2227\u00d7 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081 \u00d7 \u2713 b\u2082\n\u2713\u2227\u00d7 {0\u2082} ()\n\u2713\u2227\u00d7 {1\u2082} p = _ , p\n\n-- Similar to \u2713-\u2228\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {0\u2082} = inr\n\u2713\u2228-\u228e {1\u2082} = inl\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 {1\u2082} = _\n\u2713\u2228\u2081 {0\u2082} ()\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {0\u2082} p = p\n\u2713\u2228\u2082 {1\u2082} _ = _\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n-- This particular implementation has been\n-- chosen for computational content.\n-- Namely the proof is \"re-created\" when b is 1\u2082.\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \u2713 b\ncheck 0\u2082 {}\ncheck 1\u2082 = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two.Base where\n\nopen import Data.Zero\n\nopen import Data.Bool\n  public\n  hiding (if_then_else_)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\n\nopen import Data.Product\n  renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nopen import Data.Sum\n  using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr)\n\nopen import Data.Nat.Base\n  using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\n\nopen import Type using (\u2605_)\n\nopen import Relation.Nullary using (\u00ac_; Dec; yes; no)\n\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; _\u2262_; refl)\n\n0\u22621\u2082 : 0\u2082 \u2262 1\u2082\n0\u22621\u2082 ()\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: fst 1: snd ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n-- See also \u2713-\u2227\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 {0\u2082} p q = p\n\u2713\u2227 {1\u2082} p q = q\n\n-- See also \u2713-\u2227\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 {0\u2082} ()\n\u2713\u2227\u2081 {1\u2082} = _\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {0\u2082} ()\n\u2713\u2227\u2082 {1\u2082} p = p\n\n-- Similar to \u2713-\u2228\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {0\u2082} = inr\n\u2713\u2228-\u228e {1\u2082} = inl\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 {1\u2082} = _\n\u2713\u2228\u2081 {0\u2082} ()\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {0\u2082} p = p\n\u2713\u2228\u2082 {1\u2082} _ = _\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n-- This particular implementation has been\n-- chosen for computational content.\n-- Namely the proof is \"re-created\" when b is 1\u2082.\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \u2713 b\ncheck 0\u2082 {}\ncheck 1\u2082 = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ff3903157144200af948a03571b132a3eabed199","subject":"Weaken the \u2219-opt assumption! Adapt generic-zero-knowledge-interactive to new sums","message":"Weaken the \u2219-opt assumption! Adapt generic-zero-knowledge-interactive to new sums\n","repos":"crypto-agda\/crypto-agda","old_file":"generic-zero-knowledge-interactive.agda","new_file":"generic-zero-knowledge-interactive.agda","new_contents":"open import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\nprivate\n  \u2605 : Set\u2081\n  \u2605 = Set\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (sum\u03c0        : Sum Permutation)\n         (\u03bc\u03c0          : SumProp sum\u03c0)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (\u03bcR\u209a-xtra    : SumProp sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sum\u03c0 \u00d7Sum sumR\u209a-xtra\n\n    \u03bcR\u209a : SumProp sumR\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    sumR : Sum R\n    sumR = sumBit \u00d7Sum sumR\u209a\n\n    \u03bcR : SumProp sumR\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl  : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-true  : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (==-false : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 false \u2192 \u03b1 \u2262 \u03b2)\n            (sum\u2124q      : Sum \u2124q)\n            (\u03bc\u2124q        : SumProp sum\u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (sumR\u209a-xtra : Sum R\u209a-xtra)\n            (\u03bcR\u209a-xtra   : SumProp sumR\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s\n      with ==-false check-p-s ?\n   ... | ()\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 sum\u2124q \u03bc\u2124q R\u209a-xtra sumR\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Data.Bool.NP as Bool\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\nprivate\n  \u2605 : Set\u2081\n  \u2605 = Set\n\n-- record VerifiableProblem :\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\n\u27e80,\u27e9 : \u2200 {A} \u2192 \u21ba A \u2192 \u21ba (Bit \u00d7 A)\n\u27e80,\u27e9 (rand x) = rand (0b , x)\n\n\u27e81,\u27e9 : \u2200 {A} \u2192 \u21ba A \u2192 \u21ba (Bit \u00d7 A)\n\u27e81,\u27e9 (rand x) = rand (1b , x)\n\nmodule G (Permutation : \u2605)\n         (sum\u03c0        : Sum Permutation)\n         (sum\u03c0-ext    : SumExt sum\u03c0)\n         -- (_\u207b\u00b9         : Endo Permutation)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (sumR\u209a-xtra-ext : SumExt sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n\n         (otp-\u2219       : let otp = \u03bb A pb s \u2192 (sumToCount (sumProd sum\u03c0 sumR\u209a-xtra)) (\u03bb { (\u03c0 , _) \u2192 A (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n                        \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 (A : _ \u2192 _ \u2192 Bit) \u2192 otp A pb\u2080 s\u2080 \u2261 otp A pb\u2081 s\u2081)\n                        {-\n                        -- is this enough\n         (otp-\u2219\u2032      : let otp = \u03bb A pb s \u2192 sum\u03c0 (\u03bb \u03c0 \u2192 A (\u03c0 \u2219P pb) (\u03c0 \u2219S s)) in\n                        \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 (A : _ \u2192 _ \u2192 \u2115) \u2192 otp A pb\u2080 s\u2080 \u2261 otp A pb\u2081 s\u2081)\n                        -}\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check p s \u2261 check (\u03c0 \u2219P p) (\u03c0 \u2219S s))\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (easy-pb \u03c0) (easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sumProd sum\u03c0 sumR\u209a-xtra\n\n    #R\u209a : Count R\u209a\n    #R\u209a = sumToCount sumR\u209a\n\n    sumR\u209a-ext : SumExt sumR\u209a\n    sumR\u209a-ext = sumProd-ext sum\u03c0-ext sumR\u209a-xtra-ext\n\n    sum : Sum (Bit \u00d7 R\u209a)\n    sum = sumProd sumBit sumR\u209a\n\n    sum-ext : SumExt sum\n    sum-ext = sumProd-ext sumBit-ext sumR\u209a-ext\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = sumToCount sum (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n    {-\n    otp-\u2219 : let otp = \u03bb A pb s \u2192 #R\u209a(\u03bb { (\u03c0 , _) \u2192 A (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 (A : _ \u2192 _ \u2192 Bit) \u2192 otp A pb\u2080 s\u2080 \u2261 otp A pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 A = ?\n    -}\n\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    Prover : \u2605\n    Prover = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n        prover : ProverWC \u2192 Prover\n        prover (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0) = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , s) = check \u03c0\u2219pb s\n\n    _\u21c4\u2032_ : Problem \u2192 Prover \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 Prover\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , ans b)\n      module Honest where\n        ans : \u2200 b \u2192 Answer b\n        ans false = \u03c0\n        ans true  with solve pb\n        ...          | just sol = \u03c0 \u2219S sol\n        ...          | nothing  = easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite sym (check-\u2219 pb s \u03c0) = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081 : Problem)\n                                   (s\u2080 s\u2081   : Solution) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check | sumR\u209a-ext helper = refl\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 Prover\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = pb\u2032 b\u2032 , ans b\n      module Cheater where\n        pb\u2032 : \u2200 b\u2032 \u2192 Problem\n        pb\u2032 true  = easy-pb \u03c0\n        pb\u2032 false = \u03c0 \u2219P pb\n        ans : \u2200 b \u2192 Answer b\n        ans true  = easy-sol \u03c0\n        ans false = \u03c0\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == easy-pb \u03c0) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"09a96ec629e65a1c06ec3c1e22b81d09c9f1f854","subject":"easy cases","message":"easy cases\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expansion-unicity.agda","new_file":"expansion-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule expansion-unicity where\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam d1) (ESLam d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , refl\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp1 x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp2 x\u2084 d5 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp3 x\u2084 d5 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp1 x\u2083 x\u2084 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp2 x\u2083 d6 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp3 x\u2083 d6 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp1 x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp2 x\u2082 d6 x\u2083 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp3 x\u2082 d6 x\u2083) = {!!}\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083) = {!!}\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc2 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081) = {!!}\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c41' \u03c42 \u03c42' : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n                          \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03c41' == \u03c42' \u00d7 \u03941 == \u03942\n    expansion-unicity-ana (EALam d1) (EALam d2) = {!!}\n    expansion-unicity-ana (EALam d1) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = {!!}\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) (EALam d2) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    expansion-unicity-ana EAEHole EAEHole = {!!}\n    expansion-unicity-ana (EANEHole x x\u2081) (EASubsume x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana (EANEHole x x\u2081) (EANEHole x\u2082 x\u2083) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule expansion-unicity where\n  mutual\n    expansion-unicity-synth : (\u0393 : tctx) (e : hexp) (\u03c41 \u03c42 : htyp) (d1 d2 : dhexp) (\u03941 \u03942 : hctx) \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth = {!!}\n\n    expansion-unicity-ana : (\u0393 : tctx) (e : hexp) (\u03c41 \u03c41' \u03c42 \u03c42' : htyp) (d1 d2 : dhexp) (\u03941 \u03942 : hctx) \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n                          \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03c41' == \u03c42' \u00d7 \u03941 == \u03942\n    expansion-unicity-ana = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0e5b51a126b354cb6a2c34c080deb1fdcc75c7cb","subject":"Generalize environments to values in higher universes.","message":"Generalize environments to values in higher universes.\n\nOld-commit-hash: 4ff4102742306da9be77151d785aa608f8558977\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Environments.agda","new_file":"Denotational\/Environments.agda","new_contents":"module Denotational.Environments\n    (Type : Set)\n    {\u2113}\n    (\u27e6_\u27e7Type : Type \u2192 Set \u2113)\n  where\n\n-- ENVIRONMENTS\n--\n-- This module defines the meaning of contexts, that is,\n-- the type of environments that fit a context, together\n-- with operations and properties of these operations.\n--\n-- This module is parametric in the syntax and semantics\n-- of types, so it can be reused for different calculi\n-- and models.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Contexts Type\nopen import Denotational.Notation\n\nprivate\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS\n\n-- Denotational Semantics : Contexts Represent Environments\n\ndata Empty : Set \u2113 where\n  \u2205 : Empty\n\ndata Bind A B : Set \u2113 where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set \u2113\n\u27e6 \u2205 \u27e7Context = Empty\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n-- VARIABLES\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n-- WEAKENING\n\n-- Remove a variable from an environment\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\n-- Properties\n\n\u27e6\u27e7-\u227c-trans : \u2200 {\u0393\u2083 \u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (\u0393\u2033 : \u0393\u2082 \u227c \u0393\u2083) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2083 \u27e7) \u2192 \u27e6 \u227c-trans \u0393\u2032 \u0393\u2033 \u27e7 \u03c1 \u2261 \u27e6 \u0393\u2032 \u27e7 (\u27e6 \u0393\u2033 \u27e7 \u03c1)\n\u27e6\u27e7-\u227c-trans \u0393\u2032 \u2205 \u2205 = refl\n\u27e6\u27e7-\u227c-trans (keep \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n\u27e6\u27e7-\u227c-trans (drop \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\u27e6\u27e7-\u227c-trans \u0393\u2032 (drop \u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\n\u27e6\u27e7-\u227c-refl : \u2200 {\u0393 : Context} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 \u227c-refl \u27e7 \u03c1 \u2261 \u03c1\n\u27e6\u27e7-\u227c-refl {\u2205} \u2205 = refl\n\u27e6\u27e7-\u227c-refl {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-refl \u03c1)\n\n-- SOUNDNESS of variable lifting\n\nlift-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 lift \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nlift-sound \u2205 () \u03c1\nlift-sound (keep \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = refl\nlift-sound (keep \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = lift-sound \u0393\u2032 x \u03c1\nlift-sound (drop \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = lift-sound \u0393\u2032 this \u03c1\nlift-sound (drop \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = lift-sound \u0393\u2032 (that x) \u03c1\n","old_contents":"module Denotational.Environments\n    (Type : Set)\n    (\u27e6_\u27e7Type : Type \u2192 Set)\n  where\n\n-- ENVIRONMENTS\n--\n-- This module defines the meaning of contexts, that is,\n-- the type of environments that fit a context, together\n-- with operations and properties of these operations.\n--\n-- This module is parametric in the syntax and semantics\n-- of types, so it can be reused for different calculi\n-- and models.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Contexts Type\nopen import Denotational.Notation\n\nprivate\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS\n\n-- Denotational Semantics : Contexts Represent Environments\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = Empty\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n-- VARIABLES\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n-- WEAKENING\n\n-- Remove a variable from an environment\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\n-- Properties\n\n\u27e6\u27e7-\u227c-trans : \u2200 {\u0393\u2083 \u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (\u0393\u2033 : \u0393\u2082 \u227c \u0393\u2083) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2083 \u27e7) \u2192 \u27e6 \u227c-trans \u0393\u2032 \u0393\u2033 \u27e7 \u03c1 \u2261 \u27e6 \u0393\u2032 \u27e7 (\u27e6 \u0393\u2033 \u27e7 \u03c1)\n\u27e6\u27e7-\u227c-trans \u0393\u2032 \u2205 \u2205 = refl\n\u27e6\u27e7-\u227c-trans (keep \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n\u27e6\u27e7-\u227c-trans (drop \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\u27e6\u27e7-\u227c-trans \u0393\u2032 (drop \u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\n\u27e6\u27e7-\u227c-refl : \u2200 {\u0393 : Context} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 \u227c-refl \u27e7 \u03c1 \u2261 \u03c1\n\u27e6\u27e7-\u227c-refl {\u2205} \u2205 = refl\n\u27e6\u27e7-\u227c-refl {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-refl \u03c1)\n\n-- SOUNDNESS of variable lifting\n\nlift-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 lift \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nlift-sound \u2205 () \u03c1\nlift-sound (keep \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = refl\nlift-sound (keep \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = lift-sound \u0393\u2032 x \u03c1\nlift-sound (drop \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = lift-sound \u0393\u2032 this \u03c1\nlift-sound (drop \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = lift-sound \u0393\u2032 (that x) \u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b7aea6f9087802948420259deb04a07ff6105ffc","subject":"Working in the equivalence N as lfp and N using data.","message":"Working in the equivalence N as lfp and N using data.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      \u2200 (A : D \u2192 Set) {n} \u2192\n      (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      N n \u2192 A n\n\n    -- Higher-order version (incomplete?).\n    N-least-pre-fixed-ho :\n      \u2200 (A : D \u2192 Set) {n} \u2192\n      (NatF A n \u2192 A n) \u2192\n      N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n  N-least-pre-fixed' :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (NatF A n \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 prf)              = inj\u2081 prf\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 is {n} Nn = N-least-pre-fixed A h Nn\n    where\n    h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n    h (inj\u2081 prf)              = subst A (sym prf) A0\n    h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is helper An')\n      where\n      helper : N n'\n      helper with N-post-fixed Nn\n      ... | inj\u2081 n\u22610 = \u22a5-elim (0\u2262S (trans (sym n\u22610) prf))\n      ... | inj\u2082 (m' , prf' , Nm') =\n        subst N (succInjective (trans (sym prf') prf)) Nm'\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 is {n} = N-least-pre-fixed A h\n    where\n    h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n    h (inj\u2081 prf)              = subst A (sym prf) A0\n    h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is An')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d      \u2261 d\n    +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {m} {n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 prf) = subst N (cong (flip _+_ n) (sym prf)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , m\u2261Sm' , Am')) =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 is nzero      = A0\n  N-ind\u2081 A A0 is (nsucc Nn) = is Nn (N-ind\u2081 A A0 is Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N\u21920\u2228S = N-ind\u2082 A A0 is\n    where\n    A : D \u2192 Set\n    A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n    A0 : A zero\n    A0 = inj\u2081 refl\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = case prf\u2081 prf\u2082 Ai\n      where\n      prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n      prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n      prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n             succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n      prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\n  N-least-pre-fixed\u2082 :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed\u2082 A {n} h Nn = case prf\u2081 prf\u2082 (N\u21920\u2228S Nn)\n    where\n    prf\u2081 : n \u2261 zero \u2192 A n\n    prf\u2081 n\u22610 = h (inj\u2081 n\u22610)\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 A n\n    prf\u2082 (n' , prf , Nn') = h (inj\u2082 (n' , prf , {!!}))\n","old_contents":"------------------------------------------------------------------------------\n-- From N as the least fixed-point to N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- We want to represent the total natural numbers data type\n--\n-- data N : D \u2192 Set where\n--   nzero : N zero\n--   nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n--\n-- using the representation of N as the least fixed-point.\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n  -- The higher-order version.\n  N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n  -- N is the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-least-pre-fixed :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n\n  -- Higher-order version (incomplete?)\n  N-least-pre-fixed-ho :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (NatF A n \u2192 A n) \u2192\n    N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- From\/to N-in\/N-in-ho.\n\nN-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\nN-in\u2081 = N-in-ho\n\nN-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\nN-in-ho\u2081 = N-in\u2081\n\n------------------------------------------------------------------------------\n-- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\nN-least-pre-fixed' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n  N n \u2192 A n\nN-least-pre-fixed' = N-least-pre-fixed-ho\n\nN-least-pre-fixed-ho' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (NatF A n \u2192 A n) \u2192\n  N n \u2192 A n\nN-least-pre-fixed-ho' = N-least-pre-fixed\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nnzero : N zero\nnzero = N-in (inj\u2081 refl)\n\nnsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nnsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-in and\n-- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN-post-fixed = N-least-pre-fixed A h\n  where\n  A : D \u2192 Set\n  A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n  h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  h (inj\u2081 prf)              = inj\u2081 prf\n  h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n    where\n    helper : A m' \u2192 N m'\n    helper (inj\u2081 prf')                = subst N (sym prf') nzero\n    helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 is {n} Nn = N-least-pre-fixed A h Nn\n  where\n  h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n  h (inj\u2081 prf)              = subst A (sym prf) A0\n  h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is helper An')\n    where\n    helper : N n'\n    helper with N-post-fixed Nn\n    ... | inj\u2081 n\u22610 = \u22a5-elim (0\u2262S (trans (sym n\u22610) prf))\n    ... | inj\u2082 (m' , prf' , Nm') = subst N (succInjective (trans (sym prf') prf)) Nm'\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 is {n} = N-least-pre-fixed A h\n  where\n  h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n  h (inj\u2081 prf)              = subst A (sym prf) A0\n  h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is An')\n\n------------------------------------------------------------------------------\n-- Example: We will use N-least-pre-fixed as the induction principle on N.\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192 zero + d      \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-least-pre-fixed A h Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  h : m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  h (inj\u2081 prf) = subst N (cong (flip _+_ n) (sym prf)) A0\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n  h (inj\u2082 (m' , m\u2261Sm' , Am')) = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n    where\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n------------------------------------------------------------------------------\n-- Example: A proof using N-post-fixed\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n  where\n  h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n  h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n  h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n  h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n------------------------------------------------------------------------------\n-- From\/to N as a least fixed-point to\/from N as data type.\n\nopen import FOTC.Data.Nat.Type renaming\n  ( N to N'\n  ; nsucc to nsucc'\n  ; nzero to nzero'\n  )\n\nN'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\nN'\u2192N nzero'      = nzero\nN'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n-- Using N-ind\u2081.\nN\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\nN\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n-- Using N-ind\u2082.\nN\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\nN\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b35ce26d137fb197a7e90b212d09b5eb09dcf5dd","subject":"Swapped premises and the conclusion in the introduction rules.","message":"Swapped premises and the conclusion in the introduction rules.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n-- infixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- Instead defining the least fixed-point via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF P n = n \u2261 zero \u2228 (\u2203[ m ] P m \u2227 n \u2261 succ\u2081 m)\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081    : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] N m \u2227 n \u2261 succ\u2081 m) \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- N is a the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082    : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] A m \u2227 n \u2261 succ\u2081 m) \u2192 A n) \u2192\n              \u2200 {n} \u2192 N n \u2192 A n\n  -- N-lfp\u2082 : (A : D \u2192 Set) \u2192  -- Higher-order version\n  --          (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192\n  --          \u2200 {n} \u2192 N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN Nn = N-lfp\u2081 (inj\u2082 (_ , (Nn , refl)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 N m \u2227 n \u2261 succ\u2081 m)\nN-lfp\u2083 Nn = N-lfp\u2082 A prf Nn\n  where\n  A : D \u2192 Set\n  A x = x \u2261 zero \u2228 (\u2203 \u03bb m \u2192 N m \u2227 x \u2261 succ\u2081 m)\n\n  prf : \u2200 {n'} \u2192 n' \u2261 zero \u2228 (\u2203[ m ] A m \u2227 n' \u2261 succ\u2081 m) \u2192 A n'\n  prf {n'} h = case inj\u2081 ((\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081))) h -- case inj\u2081 prf\u2081 h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 A m \u2227 n' \u2261 succ\u2081 m) \u2192 \u2203 (\u03bb m \u2192 N m \u2227 n' \u2261 succ\u2081 m)\n    prf\u2081 (m , Am , n'=Sm) = m , prf\u2082 Am , n'=Sm\n      where\n      prf\u2082 : A m \u2192 N m\n      prf\u2082 Am = case (\u03bb ah \u2192 subst N (sym ah) zN) prf\u2083 Am\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 N m' \u2227 m \u2261 succ\u2081 m') \u2192 N m\n        prf\u2083 (_ , Nm' , m\u2261Sm' ) = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nindN\u2081 : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nindN\u2081 A A0 is Nn = N-lfp\u2082 A (case prf\u2081 prf\u2082) Nn\n  where\n  prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n  prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n  prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 A m \u2227 n' \u2261 succ\u2081 m) \u2192 A n'\n  prf\u2082 (_ , Am , n'\u2261Sm) = subst A (sym n'\u2261Sm) (is Am)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\n--\n-- 2012-03-06. We cannot proof this principle because N-lfp\u2082 does not\n-- have the hypothesis N n.\n--\n-- indN\u2082 : (A : D \u2192 Set) \u2192\n--        A zero \u2192\n--        (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n--        \u2200 {n} \u2192 N n \u2192 A n\n-- indN\u2082 A A0 is Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n--   where\n--   prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n--   prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n--   prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n--   prf\u2082 {n'} (m , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is helper Am)\n--     where\n--     helper : N m\n--     helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 {!!})\n--       where\n--       prf\u2083 : n' \u2261 zero \u2192 N m\n--       prf\u2083 n'\u22610 = \u22a5-elim (0\u2262S (trans (sym n'\u22610) n'\u2261Sm))\n\n--       prf\u2084 : \u2203 (\u03bb m' \u2192 n' \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n--       prf\u2084 (_ , n'\u2261Sm' , Nm') =\n--         subst N (succInjective (trans (sym n'\u2261Sm') n'\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-lfp\u2082 A prf Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  prf : \u2200 {m'} \u2192 m' \u2261 zero \u2228 \u2203 (\u03bb m'' \u2192 A m'' \u2227 m' \u2261 succ\u2081 m'') \u2192 A m'\n  prf h = case prf\u2081 prf\u2082 h\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : \u2200 {m} \u2192 m \u2261 zero \u2192 A m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) A0\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n    prf\u2082 : \u2200 {m} \u2192 \u2203 (\u03bb m'' \u2192 A m'' \u2227 m \u2261 succ\u2081 m'') \u2192 A m\n    prf\u2082 (_ ,  Am'' , m\u2261Sm'') =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm'')) (is Am'')\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- Instead defining the least fixed-point via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF P n = n \u2261 zero \u2228 (\u2203[ m ] n \u2261 succ\u2081 m \u2227 P m)\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081    : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] n \u2261 succ\u2081 m \u2227 N m) \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- N is a the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082    : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] n \u2261 succ\u2081 m \u2227 A m) \u2192 A n) \u2192\n              \u2200 {n} \u2192 N n \u2192 A n\n  -- N-lfp\u2082 : (A : D \u2192 Set) \u2192  -- Higher-order version\n  --          (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192\n  --          \u2200 {n} \u2192 N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN Nn = N-lfp\u2081 (inj\u2082 (_ , (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m)\nN-lfp\u2083 Nn = N-lfp\u2082 A prf Nn\n  where\n  A : D \u2192 Set\n  A x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ\u2081 m \u2227 N m\n\n  prf : \u2200 {n'} \u2192 n' \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n  prf {n'} h = case inj\u2081 (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 N m\n    prf\u2081 (m , n'=Sm , h\u2082) = m , n'=Sm , prf\u2082 h\u2082\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = case (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) prf\u2083 h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n        prf\u2083 (_ , m\u2261Sm' , Nm') = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nindN\u2081 : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nindN\u2081 A A0 is Nn = N-lfp\u2082 A (case prf\u2081 prf\u2082) Nn\n  where\n  prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n  prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n  prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n  prf\u2082 (_ , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is Am)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\n--\n-- 2012-03-06. We cannot proof this principle because N-lfp\u2082 does not\n-- have the hypothesis N n.\n--\n-- indN\u2082 : (A : D \u2192 Set) \u2192\n--        A zero \u2192\n--        (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n--        \u2200 {n} \u2192 N n \u2192 A n\n-- indN\u2082 A A0 is Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n--   where\n--   prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n--   prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n--   prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n--   prf\u2082 {n'} (m , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is helper Am)\n--     where\n--     helper : N m\n--     helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 {!!})\n--       where\n--       prf\u2083 : n' \u2261 zero \u2192 N m\n--       prf\u2083 n'\u22610 = \u22a5-elim (0\u2262S (trans (sym n'\u22610) n'\u2261Sm))\n\n--       prf\u2084 : \u2203 (\u03bb m' \u2192 n' \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n--       prf\u2084 (_ , n'\u2261Sm' , Nm') =\n--         subst N (succInjective (trans (sym n'\u2261Sm') n'\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-lfp\u2082 A prf Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  prf : \u2200 {m'} \u2192 m' \u2261 zero \u2228 \u2203 (\u03bb m'' \u2192 m' \u2261 succ\u2081 m'' \u2227 A m'') \u2192 A m'\n  prf h = case prf\u2081 prf\u2082 h\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : \u2200 {m} \u2192 m \u2261 zero \u2192 A m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) A0\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n    prf\u2082 : \u2200 {m} \u2192 \u2203 (\u03bb m'' \u2192 m \u2261 succ\u2081 m'' \u2227 A m'') \u2192 A m\n    prf\u2082 (_ , m\u2261Sm'' , Am'') =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm'')) (is Am'')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db2f77d45aa65090c8c3fb549e22ff6740c08815","subject":"Added x+1+y\u22611+x+y (ER version).","message":"Added x+1+y\u22611+x+y (ER version).\n\nIgnore-this: cc262563efb9ce925db01b6a8ada7680\n\ndarcs-hash:20100531135507-3bd4e-18c905f00f1d32eeba52f68aef14a2d237602dbb.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Relation.Equalities.PropertiesER\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm          zN          = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN          (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ (n + zero) \u2261 succ t)\n               (+-rightIdentity Nn)\n               refl\n        )\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst (\u03bb t \u2192 N t) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No =\n  begin\n    zero + n + o \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o)\n                          (+-leftIdentity Nn)\n                          refl\n                 \u27e9\n    n + o        \u2261\u27e8 sym (+-leftIdentity (+-N Nn No)) \u27e9\n    zero + (n + o)\n  \u220e\n\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    succ m + n + o     \u2261\u27e8 subst (\u03bb t \u2192 succ m + n + o \u2261 t + o)\n                                (+-Sx m n)\n                                refl\n                       \u27e9\n    succ (m + n) + o   \u2261\u27e8 +-Sx (m + n) o \u27e9\n    succ (m + n + o)   \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + o) \u2261 succ t)\n                                (+-assoc Nm Nn No)\n                                refl\n                       \u27e9\n    succ (m + (n + o)) \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n    succ m + (n + o)\n  \u220e\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (*-N Nn zN)) (*-rightZero Nn))\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = sym\n  (\n    begin\n      zero + zero * n \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t)\n                         (*-leftZero Nn)\n                         refl\n                      \u27e9\n      zero + zero     \u2261\u27e8 +-leftIdentity zN \u27e9\n      zero            \u2261\u27e8 sym (*-leftZero (sN Nn)) \u27e9\n      zero * succ n\n    \u220e\n  )\n\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * succ n        \u2261\u27e8 *-Sx m (succ n) \u27e9\n    succ n + m * succ n    \u2261\u27e8 subst (\u03bb t \u2192 succ n + m * succ n \u2261 succ n + t)\n                                    (x*1+y\u2261x+xy Nm Nn)\n                                    refl\n                           \u27e9\n    succ n + (m + m * n)   \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n    succ (n + (m + m * n)) \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m + m * n)) \u2261 succ t)\n                                    (sym (+-assoc Nn Nm (*-N Nm Nn)))\n                                    refl\n                           \u27e9\n    succ (n + m + m * n)   \u2261\u27e8 subst (\u03bb t \u2192 succ (n + m + m * n) \u2261\n                                           succ (t + m * n))\n                                    (+-comm Nn Nm)\n                                    refl\n                           \u27e9\n     succ (m + n + m * n)  \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + m * n) \u2261 succ t)\n                                    (+-assoc Nm Nn (*-N Nm Nn))\n                                    refl\n                           \u27e9\n\n    succ (m + (n + m * n)) \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n    succ m + (n + m * n)   \u2261\u27e8 subst (\u03bb t \u2192 succ m + (n + m * n) \u2261 succ m + t)\n                                    (sym (*-Sx m n))\n                                    refl\n                           \u27e9\n    succ m + succ m * n\n    \u220e\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm zN Nn = trans (*-leftZero Nn) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * n   \u2261\u27e8 *-Sx m n \u27e9\n    n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t)\n                          (*-comm Nm Nn)\n                          refl\n                  \u27e9\n    n + n * m     \u2261\u27e8 sym (x*1+y\u2261x+xy Nn Nm) \u27e9\n    n * succ m\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} {o = o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n\n[x+y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n[x+y]z\u2261xz*yz {m} {n} Nm Nn zN =\n  begin\n    (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n    zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n    zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n    zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n    m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n    m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                  (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                         \u27e9\n    m * zero + n * zero\n  \u220e\n\n[x+y]z\u2261xz*yz {n = n} zN Nn (sN {o} No) =\n  begin\n    (zero + n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ o \u2261 t * succ o)\n                                  (+-leftIdentity Nn)\n                                  refl\n                         \u27e9\n    n * succ o           \u2261\u27e8 sym (+-leftIdentity (*-N Nn (sN No))) \u27e9\n    zero + n * succ o    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ o \u2261 t +  n * succ o)\n                                (sym (*-0x (succ o)))\n                                refl\n                         \u27e9\n    zero * succ o + n * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) zN (sN {o} No) =\n begin\n    (succ m + zero) * succ o \u2261\u27e8 subst (\u03bb t \u2192 (succ m + zero) * succ o \u2261\n                                             t * succ o)\n                                      (+-rightIdentity (sN Nm))\n                                      refl\n                              \u27e9\n    succ m * succ o          \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n    succ m * succ o + zero   \u2261\u27e8 subst (\u03bb t \u2192 succ m * succ o + zero \u2261\n                                             succ m * succ o + t)\n                                      (sym (*-leftZero (sN No)))\n                                      refl\n                             \u27e9\n    succ m * succ o + zero * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m + succ n) * succ o \u2261 t * succ o)\n               (+-Sx m (succ n))\n               refl\n      \u27e9\n    succ ( m + succ n) * succ o \u2261\u27e8 *-Sx (m + succ n) (succ o) \u27e9\n    succ o + (m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + (m + succ n) * succ o \u2261 succ o + t)\n               ([x+y]z\u2261xz*yz Nm (sN Nn) (sN No))\n               refl\n      \u27e9\n    succ o + (m * succ o + succ n * succ o)\n      \u2261\u27e8 sym (+-assoc (sN No) (*-N Nm (sN No)) (*-N (sN Nn) (sN No))) \u27e9\n    succ o + m * succ o + succ n * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + m * succ o + succ n * succ o \u2261\n                      t + succ n * succ o)\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    succ m * succ o + succ n * succ o\n      \u220e","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\n  using ( x*1+y\u2261x+xy )\nopen import LTC.Relation.Equalities.PropertiesER\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm          zN          = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN          (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ (n + zero) \u2261 succ t)\n               (+-rightIdentity Nn)\n               refl\n        )\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst (\u03bb t \u2192 N t) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No =\n  begin\n    zero + n + o \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o)\n                          (+-leftIdentity Nn)\n                          refl\n                 \u27e9\n    n + o        \u2261\u27e8 sym (+-leftIdentity (+-N Nn No)) \u27e9\n    zero + (n + o)\n  \u220e\n\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    succ m + n + o     \u2261\u27e8 subst (\u03bb t \u2192 succ m + n + o \u2261 t + o)\n                                (+-Sx m n)\n                                refl\n                       \u27e9\n    succ (m + n) + o   \u2261\u27e8 +-Sx (m + n) o \u27e9\n    succ (m + n + o)   \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + o) \u2261 succ t)\n                                (+-assoc Nm Nn No)\n                                refl\n                       \u27e9\n    succ (m + (n + o)) \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n    succ m + (n + o)\n  \u220e\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (*-N Nn zN)) (*-rightZero Nn))\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm zN Nn = trans (*-leftZero Nn) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * n   \u2261\u27e8 *-Sx m n \u27e9\n    n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t)\n                          (*-comm Nm Nn)\n                          refl\n                  \u27e9\n    n + n * m     \u2261\u27e8 sym (x*1+y\u2261x+xy Nn Nm) \u27e9\n    n * succ m\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} {o = o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n\n[x+y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n[x+y]z\u2261xz*yz {m} {n} Nm Nn zN =\n  begin\n    (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n    zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n    zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n    zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n    m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n    m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                  (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                         \u27e9\n    m * zero + n * zero\n  \u220e\n\n[x+y]z\u2261xz*yz {n = n} zN Nn (sN {o} No) =\n  begin\n    (zero + n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ o \u2261 t * succ o)\n                                  (+-leftIdentity Nn)\n                                  refl\n                         \u27e9\n    n * succ o           \u2261\u27e8 sym (+-leftIdentity (*-N Nn (sN No))) \u27e9\n    zero + n * succ o    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ o \u2261 t +  n * succ o)\n                                (sym (*-0x (succ o)))\n                                refl\n                         \u27e9\n    zero * succ o + n * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) zN (sN {o} No) =\n begin\n    (succ m + zero) * succ o \u2261\u27e8 subst (\u03bb t \u2192 (succ m + zero) * succ o \u2261\n                                             t * succ o)\n                                      (+-rightIdentity (sN Nm))\n                                      refl\n                              \u27e9\n    succ m * succ o          \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n    succ m * succ o + zero   \u2261\u27e8 subst (\u03bb t \u2192 succ m * succ o + zero \u2261\n                                             succ m * succ o + t)\n                                      (sym (*-leftZero (sN No)))\n                                      refl\n                             \u27e9\n    succ m * succ o + zero * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m + succ n) * succ o \u2261 t * succ o)\n               (+-Sx m (succ n))\n               refl\n      \u27e9\n    succ ( m + succ n) * succ o \u2261\u27e8 *-Sx (m + succ n) (succ o) \u27e9\n    succ o + (m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + (m + succ n) * succ o \u2261 succ o + t)\n               ([x+y]z\u2261xz*yz Nm (sN Nn) (sN No))\n               refl\n      \u27e9\n    succ o + (m * succ o + succ n * succ o)\n      \u2261\u27e8 sym (+-assoc (sN No) (*-N Nm (sN No)) (*-N (sN Nn) (sN No))) \u27e9\n    succ o + m * succ o + succ n * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + m * succ o + succ n * succ o \u2261\n                      t + succ n * succ o)\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    succ m * succ o + succ n * succ o\n      \u220e","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1242f3ff7ac516bd4e5d6845eb083992c3527b0b","subject":"Updated agda2atp: The TPTP definitions use an unique name.","message":"Updated agda2atp: The TPTP definitions use an unique name.\n\nIgnore-this: 46560a3592efd4fdf23dc29fe3ee2d5a\n\ndarcs-hash:20101201173120-3bd4e-1c669d5b56abab12ba22ba65c0c8ed7ac50be6b8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Properties.agda","new_file":"src\/PA\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- PA properties\n------------------------------------------------------------------------------\n\nmodule PA.Properties where\n\nopen import PA.Base\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = S\u2085\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity = S\u2089 P P0 iStep\n  where\n    P : \u2115 \u2192 Set\n    P i = i + zero \u2261 i\n    {-# ATP definition P #-}\n\n    P0 : P zero\n    P0 = S\u2085 zero\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc m n o = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n + o \u2261 i + (n + o)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\nx+1+y\u22611+x+y : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + succ n \u2261 succ (i + n)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n \u2261 n + i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 +-rightIdentity #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep x+1+y\u22611+x+y #-}\n","old_contents":"------------------------------------------------------------------------------\n-- PA properties\n------------------------------------------------------------------------------\n\nmodule PA.Properties where\n\nopen import PA.Base\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = S\u2085\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity = S\u2089 P P0 iStep\n  where\n    P : \u2115 \u2192 Set\n    P i = i + zero \u2261 i\n\n    P0 : zero + zero \u2261 zero\n    P0 = S\u2085 zero\n\n    postulate\n      iStep : \u2200 i \u2192\n              i + zero \u2261 i \u2192\n              succ i + zero \u2261 succ i\n    {-# ATP prove iStep #-}\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc m n o = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n + o \u2261 i + (n + o)\n\n    postulate\n      P0 : zero + n + o \u2261 zero + (n + o)\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : \u2200 i \u2192\n              i + n + o \u2261 i + (n + o) \u2192\n              succ i + n + o \u2261 succ i + (n + o)\n    {-# ATP prove iStep #-}\n\nx+1+y\u22611+x+y : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + succ n \u2261 succ (i + n)\n\n    postulate\n      P0 : zero + succ n \u2261 succ (zero + n)\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : \u2200 i \u2192\n              i + succ n \u2261 succ (i + n) \u2192\n              succ i + succ n \u2261 succ (succ i + n)\n    {-# ATP prove iStep #-}\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n \u2261 n + i\n\n    postulate\n      P0 : zero + n \u2261 n + zero\n    {-# ATP prove P0 +-rightIdentity #-}\n\n    postulate\n      iStep : \u2200 i \u2192\n              i + n \u2261 n + i \u2192\n              succ i + n \u2261 n + succ i\n    {-# ATP prove iStep x+1+y\u22611+x+y #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b0b16a5abb26bcc3d09fb6b74b2343c21159e33c","subject":"#31 missed another file","message":"#31 missed another file\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- todo: check that it's not that form, otherwise the cast can progress maybe\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) --\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- todo: check that it's not that form, otherwise the cast can progress maybe\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) --\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FIndet i)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FBoxed v\u2082)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter (ITCastID (FIndet x)) FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter (ITCastID (FBoxed x)) FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n\n\n\n  -- this would fill two above, but it's false: \u2987\u2988 indet but its type, \u2987\u2988, is not ground\n  postulate\n    lem-groundindet : \u2200{ \u0394 d \u03c4} \u2192 \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 d indet \u2192 \u03c4 ground\n\n  -- this is also false, but tempting looking in two holes above from the ap case\n  postulate\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n\n  counter : \u03a3[ d \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0394 \u2208 hctx ]\n               ((d indet) \u00d7 (\u0394 , \u2205 \u22a2 d :: \u03c4 ==> \u03c4'))\n  counter = (\u2987\u2988\u27e8 Z , \u2205 \u27e9 \u27e8 b ==> b \u21d2 b ==> \u2987\u2988 \u27e9) ,\n            b , \u2987\u2988 ,  \u25a0 (Z , \u2205 , b ==> b ) ,\n            ICastArr (\u03bb ()) IEHole , TACast (TAEHole refl (\u03bb x d \u2192 \u03bb ())) (TCArr TCRefl TCHole1)\n\n  oops : \u22a5\n  oops = lem (\u03c02 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter)))))\n                (\u03c01 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter))))) b b b \u2987\u2988 \u2987\u2988\u27e8 Z , \u2205 \u27e9 refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0718040f19ebe9f759f025f31d845d8f2b8f381b","subject":"The agda2atp tool fixed the issue related to the eta-expansion of TPTP definitions.","message":"The agda2atp tool fixed the issue related to the eta-expansion of TPTP definitions.\n\nIgnore-this: 447b379a77cb14d71c8a7b51a2b87837\n\ndarcs-hash:20110730142709-3bd4e-d570ee27862523ca742a98082ef46b71ab657230.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/Stream\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesATP\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n  {-# ATP definition P\u2081 #-}\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n  {-# ATP definition P\u2082 #-}\n\n  postulate\n    helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n              \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 P\u2081 ws'))\n  {-# ATP prove helper\u2081 #-}\n\n  postulate\n    helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192\n              \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  {-# ATP prove helper\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesATP\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n  {-# ATP definition P\u2082 #-}\n\n  -- TODO: We don't use the predicate P\u2081 in the type of the function\n  -- helper\u2081, because at the moment the agda2atp tool doesn't handle\n  -- the eta-expansion for equations.\n  postulate\n    helper\u2081 : \u2200 {ws} \u2192 (\u2203 \u03bb zs \u2192 ws \u2248 zs) \u2192\n              \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 (\u2203 \u03bb zs \u2192 ws' \u2248 zs)))\n  {-# ATP prove helper\u2081 #-}\n\n  postulate\n    helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192\n              \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  {-# ATP prove helper\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0ec5eda09729e608ae00f5997b50db1941b4baf5","subject":"Iso: two more isos requiring funext","message":"Iso: two more isos requiring funext\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nopen import Level.NP\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum.NP renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive; [0:_1:_])\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise public using (_\u00d7-cong_)\nopen import Relation.Binary.Sum public using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid \u2080 \u2080\n\n-- requires extensionality\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c}\n         (extB : {f g : \u03a0 A B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         (extC : {f g : \u03a0 A C} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a0 A B \u2192 \u03a0 A C\n    \u21d2 g x = to (f x) (g x)\n    \u21d0 : \u03a0 A C \u2192 \u03a0 A B\n    \u21d0 g x = from (f x) (g x)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 g = extB \u03bb x \u2192 left-f x (g x)\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 g = extC \u03bb x \u2192 right-f x (g x)\n\n  fiber-iso : \u03a0 A B \u2194 \u03a0 A C\n  fiber-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n-- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c}\n         (ext : {f g : \u03a0 (A \u228e B) C} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : \u03a0 (A \u228e B) C \u2194 (\u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082))\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) (\u03bb f \u2192 ext (\u21d0\u21d2 f)) \u21d2\u21d0\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n\n-- requires extensionality\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\n-- requires extensionality\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 : \u2200 {a} (F : \ud835\udfd8 \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfd8 F \u2194 \ud835\udfd8\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 F = inverses proj\u2081 (\u03bb ()) (\u03bb { ((), _) }) (\u03bb ())\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\u228e\u21ff\u03a32 : \u2200 {\u2113} {A B : \u2605 \u2113} \u2192 (A \u228e B) \u2194 \u03a3 \ud835\udfda [0: A 1: B ]\n\u228e\u21ff\u03a32 {A = A} {B} = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : A \u228e B \u2192 _\n    \u21d2 (inj\u2081 x) = 0\u2082 , x\n    \u21d2 (inj\u2082 x) = 1\u2082 , x\n    \u21d0 : \u03a3 _ _ \u2192 A \u228e B\n    \u21d0 (0\u2082 , x) = inj\u2081 x\n    \u21d0 (1\u2082 , x) = inj\u2082 x\n    \u21d0\u21d2 : (_ : _ \u228e _) \u2192 _\n    \u21d0\u21d2 (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 x) = \u2261.refl\n    \u21d2\u21d0 : (_ : \u03a3 _ _) \u2192 _\n    \u21d2\u21d0 (0\u2082 , x) = \u2261.refl\n    \u21d2\u21d0 (1\u2082 , x) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 : \u2200 n \u2192 Fin n \u2194 \ud835\udfd9\u228e^ n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 zero    = Fin0\u2194\ud835\udfd8\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 (suc n) = Inv.id \u228e-cong Fin\u2194\ud835\udfd9\u228e^\ud835\udfd8 n Inv.\u2218 Fin\u2218suc\u2194\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nopen import Level.NP\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum.NP renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive; [0:_1:_])\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise public using (_\u00d7-cong_)\nopen import Relation.Binary.Sum public using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid \u2080 \u2080\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n\n-- requires extensionality\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\n-- requires extensionality\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 : \u2200 {a} (F : \ud835\udfd8 \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfd8 F \u2194 \ud835\udfd8\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 F = inverses proj\u2081 (\u03bb ()) (\u03bb { ((), _) }) (\u03bb ())\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\u228e\u21ff\u03a32 : \u2200 {\u2113} {A B : \u2605 \u2113} \u2192 (A \u228e B) \u2194 \u03a3 \ud835\udfda [0: A 1: B ]\n\u228e\u21ff\u03a32 {A = A} {B} = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : A \u228e B \u2192 _\n    \u21d2 (inj\u2081 x) = 0\u2082 , x\n    \u21d2 (inj\u2082 x) = 1\u2082 , x\n    \u21d0 : \u03a3 _ _ \u2192 A \u228e B\n    \u21d0 (0\u2082 , x) = inj\u2081 x\n    \u21d0 (1\u2082 , x) = inj\u2082 x\n    \u21d0\u21d2 : (_ : _ \u228e _) \u2192 _\n    \u21d0\u21d2 (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 x) = \u2261.refl\n    \u21d2\u21d0 : (_ : \u03a3 _ _) \u2192 _\n    \u21d2\u21d0 (0\u2082 , x) = \u2261.refl\n    \u21d2\u21d0 (1\u2082 , x) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 : \u2200 n \u2192 Fin n \u2194 \ud835\udfd9\u228e^ n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 zero    = Fin0\u2194\ud835\udfd8\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 (suc n) = Inv.id \u228e-cong Fin\u2194\ud835\udfd9\u228e^\ud835\udfd8 n Inv.\u2218 Fin\u2218suc\u2194\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"142187d4f24336fe4c321504b51b961590896d1b","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/Mirror\/Type.agda","new_file":"src\/fot\/FOTC\/Program\/Mirror\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- The types used by the mirror function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Mirror.Type where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Data.List\n\n------------------------------------------------------------------------------\n-- Tree terms.\npostulate node : D \u2192 D \u2192 D\n\n-- The mutually totality predicates\n\ndata Forest : D \u2192 Set  -- The list of rose trees (called forest).\ndata Tree   : D \u2192 Set  -- The rose tree type.\n\ndata Forest where\n  fnil  :                                 Forest []\n  fcons : \u2200 {t ts} \u2192 Tree t \u2192 Forest ts \u2192 Forest (t \u2237 ts)\n{-# ATP axiom fnil fcons #-}\n\ndata Tree where\n  tree : \u2200 d {ts} \u2192 Forest ts \u2192 Tree (node d ts)\n{-# ATP axiom tree #-}\n\n------------------------------------------------------------------------------\n-- Mutual induction for Tree and Forest\n\n-- Adapted from the mutual induction principles generate from Coq 8.4\n-- using the command:\n--\n-- Scheme Tree_mutual_ind :=\n--   Minimality for Tree Sort Prop\n-- with Forest_mutual_ind :=\n--   Minimality for Forest Sort Prop.\n\nTree-ind :\n  {A B : D \u2192 Set} \u2192\n  (\u2200 d {ts} \u2192 Forest ts \u2192 B ts \u2192 A (node d ts)) \u2192\n  B [] \u2192\n  (\u2200 {t ts} \u2192 Tree t \u2192 A t \u2192 Forest ts \u2192 B ts \u2192 B (t \u2237 ts)) \u2192\n  \u2200 {t} \u2192 Tree t \u2192 A t\n\nForest-ind :\n  {P B : D \u2192 Set} \u2192\n  (\u2200 d {ts} \u2192 Forest ts \u2192 B ts \u2192 P (node d ts)) \u2192\n  B [] \u2192\n  (\u2200 {t ts} \u2192 Tree t \u2192 P t \u2192 Forest ts \u2192 B ts \u2192 B (t \u2237 ts)) \u2192\n  \u2200 {ts} \u2192 Forest ts \u2192 B ts\n\nTree-ind ihA B[] _   (tree d fnil)           = ihA d fnil B[]\nTree-ind ihA B[] ihB (tree d (fcons Tt Fts)) =\n  ihA d (fcons Tt Fts) (ihB Tt (Tree-ind ihA B[] ihB Tt)\n                              Fts (Forest-ind ihA B[] ihB Fts))\n\nForest-ind _   B[] _   fnil           = B[]\nForest-ind ihP B[] ihB (fcons Tt Fts) =\n  ihB Tt (Tree-ind ihP B[] ihB Tt) Fts (Forest-ind ihP B[] ihB Fts)\n","old_contents":"------------------------------------------------------------------------------\n-- The types used by the mirror function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Mirror.Type where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Data.List\n\n------------------------------------------------------------------------------\n-- Tree terms.\npostulate\n  node : D \u2192 D \u2192 D\n\n-- The mutually totality predicates\n\ndata Forest : D \u2192 Set  -- The list of rose trees (called forest).\ndata Tree   : D \u2192 Set  -- The rose tree type.\n\ndata Forest where\n  fnil  :                                 Forest []\n  fcons : \u2200 {t ts} \u2192 Tree t \u2192 Forest ts \u2192 Forest (t \u2237 ts)\n{-# ATP axiom fnil fcons #-}\n\ndata Tree where\n  tree : \u2200 d {ts} \u2192 Forest ts \u2192 Tree (node d ts)\n{-# ATP axiom tree #-}\n\n------------------------------------------------------------------------------\n-- Mutual induction for Tree and Forest\n\n-- Adapted from the mutual induction principles generate from Coq 8.4\n-- using the command:\n--\n-- Scheme Tree_mutual_ind :=\n--   Minimality for Tree Sort Prop\n-- with Forest_mutual_ind :=\n--   Minimality for Forest Sort Prop.\n\nTree-ind :\n  {A B : D \u2192 Set} \u2192\n  (\u2200 d {ts} \u2192 Forest ts \u2192 B ts \u2192 A (node d ts)) \u2192\n  B [] \u2192\n  (\u2200 {t ts} \u2192 Tree t \u2192 A t \u2192 Forest ts \u2192 B ts \u2192 B (t \u2237 ts)) \u2192\n  \u2200 {t} \u2192 Tree t \u2192 A t\n\nForest-ind :\n  {P B : D \u2192 Set} \u2192\n  (\u2200 d {ts} \u2192 Forest ts \u2192 B ts \u2192 P (node d ts)) \u2192\n  B [] \u2192\n  (\u2200 {t ts} \u2192 Tree t \u2192 P t \u2192 Forest ts \u2192 B ts \u2192 B (t \u2237 ts)) \u2192\n  \u2200 {ts} \u2192 Forest ts \u2192 B ts\n\nTree-ind ihA B[] _   (tree d fnil)           = ihA d fnil B[]\nTree-ind ihA B[] ihB (tree d (fcons Tt Fts)) =\n  ihA d (fcons Tt Fts) (ihB Tt (Tree-ind ihA B[] ihB Tt)\n                              Fts (Forest-ind ihA B[] ihB Fts))\n\nForest-ind _   B[] _   fnil           = B[]\nForest-ind ihP B[] ihB (fcons Tt Fts) =\n  ihB Tt (Tree-ind ihP B[] ihB Tt) Fts (Forest-ind ihP B[] ihB Fts)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0ab0b0012d2db6e26ec2716c2d95f3743111df4d","subject":"MType: Fix arity of _v\u00d7_","message":"MType: Fix arity of _v\u00d7_\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/MType.agda","new_file":"Parametric\/Syntax\/MType.agda","new_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Syntax.MType where\n\nmodule Structure (Base : Type.Structure) where\n  open Type.Structure Base\n\n  mutual\n    --  Derived from CBPV\n    data ValType : Set where\n      U : (c : CompType) \u2192 ValType\n      B : (\u03b9 : Base) \u2192 ValType\n      vUnit : ValType\n      _v\u00d7_ : (\u03c4\u2081 : ValType) \u2192 (\u03c4\u2082 : ValType) \u2192 ValType\n      _v+_ : (\u03c4\u2081 : ValType) \u2192 (\u03c4\u2082 : ValType) \u2192 ValType\n\n    -- Same associativity as the standard _\u00d7_\n    infixr 2 _v\u00d7_\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      -- We did not use this in CBPV, so dropped.\n      -- _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  open import Data.List\n  open Data.List using (List) public\n  fromCBVToCompList : Context \u2192 List CompType\n  fromCBVToCompList \u0393 = mapValCtx cbvToCompType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","old_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Syntax.MType where\n\nmodule Structure (Base : Type.Structure) where\n  open Type.Structure Base\n\n  mutual\n    --  Derived from CBPV\n    data ValType : Set where\n      U : (c : CompType) \u2192 ValType\n      B : (\u03b9 : Base) \u2192 ValType\n      vUnit : ValType\n      _v\u00d7_ : (\u03c4\u2081 : ValType) \u2192 (\u03c4\u2082 : ValType) \u2192 ValType\n      _v+_ : (\u03c4\u2081 : ValType) \u2192 (\u03c4\u2082 : ValType) \u2192 ValType\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      -- We did not use this in CBPV, so dropped.\n      -- _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  open import Data.List\n  open Data.List using (List) public\n  fromCBVToCompList : Context \u2192 List CompType\n  fromCBVToCompList \u0393 = mapValCtx cbvToCompType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fa5b698a20a02c32894d577d7a1ff7c70cd2b395","subject":"adding a small note about preservation","message":"adding a small note about preservation\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import binders-disjoint-checks\n\nopen import lemmas-subst-ta\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            binders-unique d \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans bd TAConst ()\n  preserve-trans bd (TAVar x\u2081) ()\n  preserve-trans bd (TALam _ ta) ()\n  preserve-trans (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2082 x\u2083)) (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt x\u2082 bd\u2081 ta ta\u2081\n  preserve-trans bd (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans bd (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans bd (TAEHole x x\u2081) ()\n  preserve-trans bd (TANEHole x ta x\u2081) ()\n  preserve-trans bd (TACast ta x) (ITCastID) = ta\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans bd (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans bd (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans bd (TAFailedCast x y z q) ()\n\n  lem-bd-\u03b51 : \u2200{ d \u03b5 d0} \u2192 d == \u03b5 \u27e6 d0 \u27e7 \u2192 binders-unique d \u2192 binders-unique d0\n  lem-bd-\u03b51 FHOuter bd = bd\n  lem-bd-\u03b51 (FHAp1 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHAp2 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\u2081\n  lem-bd-\u03b51 (FHNEHole eps) (BUNEHole bd x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHCast eps) (BUCast bd) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHFailedCast eps) (BUFailedCast bd) = lem-bd-\u03b51 eps bd\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             binders-unique d \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preservation bd D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans (lem-bd-\u03b51 x bd) wt x\u2081) x\u2082\n\n  -- note that the exact statement of preservation in the paper, where \u0393 is\n  -- empty indicating that the terms are closed, is an immediate corrolary\n  -- of the slightly more general statement above.\n  preservation' : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} \u2192\n             binders-unique d \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation' = preservation\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import binders-disjoint-checks\n\nopen import lemmas-subst-ta\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            binders-unique d \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans bd TAConst ()\n  preserve-trans bd (TAVar x\u2081) ()\n  preserve-trans bd (TALam _ ta) ()\n  preserve-trans (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2082 x\u2083)) (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt x\u2082 bd\u2081 ta ta\u2081\n  preserve-trans bd (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans bd (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans bd (TAEHole x x\u2081) ()\n  preserve-trans bd (TANEHole x ta x\u2081) ()\n  preserve-trans bd (TACast ta x) (ITCastID) = ta\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans bd (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans bd (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans bd (TAFailedCast x y z q) ()\n\n  lem-bd-\u03b51 : \u2200{ d \u03b5 d0} \u2192 d == \u03b5 \u27e6 d0 \u27e7 \u2192 binders-unique d \u2192 binders-unique d0\n  lem-bd-\u03b51 FHOuter bd = bd\n  lem-bd-\u03b51 (FHAp1 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHAp2 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\u2081\n  lem-bd-\u03b51 (FHNEHole eps) (BUNEHole bd x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHCast eps) (BUCast bd) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHFailedCast eps) (BUFailedCast bd) = lem-bd-\u03b51 eps bd\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             binders-unique d \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preservation bd D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans (lem-bd-\u03b51 x bd) wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d864c1b18db6945f3011ab052b5aa8b980202dd8","subject":"Still a few holes","message":"Still a few holes\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen L using (Lift; lift)\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Data.Fin using (Fin)\nopen import Function.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to Searchable)\nopen import Search.Searchable.Product\nopen import Search.Searchable.Sum\nopen import Search.Derived\n\nmodule sum where\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search _ A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 Searchable A \u2192 Searchable B\n\u03bc-iso {A}{B} A\u2194B \u03bcA = mk search\u1d2e ind ade\n  where\n    A\u2192B = to A\u2194B\n\n    search\u1d2e : Search _ B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    sum\u1d2e = search\u1d2e _+_\n\n    ind : SearchInd _ search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n    ade : AdequateSum sum\u1d2e\n    ade f = sym-first-iso A\u2194B FI.\u2218 adequate-sum \u03bcA (f \u2218 A\u2192B)\n\n-- I guess this could be more general\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : Searchable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 from A\u2194B)\n\u03bc-iso-preserve A\u2194B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 \u2261.sym \u2218 Inverse.left-inverse-of A\u2194B)\n\n\u03bcLift : \u2200 {A} \u2192 Searchable A \u2192 Searchable (Lift A)\n\u03bcLift = \u03bc-iso (FI.sym Lift\u2194id)\n\n\u03bc\u22a4 : Searchable \u22a4\n\u03bc\u22a4 = mk _ ind ade\n  where\n    srch : Search _ \u22a4\n    srch _ f = f _\n\n    ind : SearchInd _ srch\n    ind _ _ Pf = Pf _\n\n    ade : AdequateSum (srch _+_)\n    ade x = FI.sym \u22a4\u00d7A\u2194A\n\nsearchBit : \u2200 {m} \u2192 Search m Bit\nsearchBit _\u2219_ f = f 0b \u2219 f 1b\n\nsearchBit-ind : \u2200 {m p} \u2192 SearchInd p {m} searchBit\nsearchBit-ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\n\u03bcBit : Searchable Bit\n\u03bcBit = \u03bc-iso (FI.sym Bit\u2194\u22a4\u228e\u22a4) (\u03bc\u22a4 \u228e-\u03bc \u03bc\u22a4)\n\nfocusBit : \u2200 {a} \u2192 Focus {a} searchBit\nfocusBit (false , x) = inj\u2081 x\nfocusBit (true ,  x) = inj\u2082 x\n\nfocusedBit : Focused {L.zero} searchBit\nfocusedBit {B} = inverses focusBit unfocus (\u21d2) (\u21d0)\n  where open Searchable\u2081\u2081 searchBit-ind\n        \u21d2 : (x : \u03a3 Bit B) \u2192 _\n        \u21d2 (false , x) = \u2261.refl\n        \u21d2 (true  , x) = \u2261.refl\n        \u21d0 : (x : B 0b \u228e B 1b) \u2192 _\n        \u21d0 (inj\u2081 x) = \u2261.refl\n        \u21d0 (inj\u2082 x) = \u2261.refl\n\nlookupBit : \u2200 {a} \u2192 Lookup {a} searchBit\nlookupBit = proj\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nprivate -- unused\n    S\u03a0\u03a3\u207b : \u2200 {m A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n           \u2192 Search m ((x : A) (y : B x) \u2192 C (x , y))\n           \u2192 Search m (\u03a0 (\u03a3 A B) C)\n    S\u03a0\u03a3\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry)\n\n    S\u03a0\u03a3\u207b-ind : \u2200 {m p A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n               \u2192 {s : Search m ((x : A) (y : B x) \u2192 C (x , y))}\n               \u2192 SearchInd p s\n               \u2192 SearchInd p (S\u03a0\u03a3\u207b s)\n    S\u03a0\u03a3\u207b-ind ind P P\u2219 Pf = ind (P \u2218 S\u03a0\u03a3\u207b) P\u2219 (Pf \u2218 uncurry)\n\n    S\u00d7\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 B \u2192 C) \u2192 Search m (A \u00d7 B \u2192 C)\n    S\u00d7\u207b = S\u03a0\u03a3\u207b\n\n    S\u00d7\u207b-ind : \u2200 {m p A B C}\n              \u2192 {s : Search m (A \u2192 B \u2192 C)}\n              \u2192 SearchInd p s\n              \u2192 SearchInd p (S\u00d7\u207b s)\n    S\u00d7\u207b-ind = S\u03a0\u03a3\u207b-ind\n\n    S\u03a0\u228e\u207b : \u2200 {m A B} {C : A \u228e B \u2192 \u2605 _}\n           \u2192 Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n           \u2192 Search m (\u03a0 (A \u228e B) C)\n    S\u03a0\u228e\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry [_,_])\n\n    S\u03a0\u228e\u207b-ind : \u2200 {m p A B} {C : A \u228e B \u2192 \u2605 _}\n                 {s : Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))}\n                 (i : SearchInd p s)\n               \u2192 SearchInd p (S\u03a0\u228e\u207b {C = C} s) -- A sB)\n    S\u03a0\u228e\u207b-ind i P P\u2219 Pf = i (P \u2218 S\u03a0\u228e\u207b) P\u2219 (Pf \u2218 uncurry [_,_])\n\n    {- For each A\u2192C function\n       and each B\u2192C function\n       an A\u228eB\u2192C function is yield\n     -}\n    S\u228e\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 C) \u2192 Search m (B \u2192 C)\n                      \u2192 Search m (A \u228e B \u2192 C)\n    S\u228e\u207b sA sB =  S\u03a0\u228e\u207b (sA \u00d7-search sB)\n\n\u03bc\u03a0\u03a3\u207b : \u2200 {A B}{C : \u03a3 A B \u2192 \u2605\u2080} \u2192 Searchable ((x : A)(y : B x) \u2192 C (x , y)) \u2192 Searchable (\u03a0 (\u03a3 A B) C)\n\u03bc\u03a0\u03a3\u207b = \u03bc-iso (FI.sym curried)\n\n\u03a3-Fun : \u2200 {A B} \u2192 Funable A \u2192 Funable B \u2192 Funable (A \u00d7 B)\n\u03a3-Fun (\u03bcA , \u03bcA\u2192) FB  = \u03bc\u03a3 \u03bcA (searchable FB) , (\u03bb x \u2192 \u03bc\u03a0\u03a3\u207b (\u03bcA\u2192 (negative FB x)))\n  where open Funable\n\n\u03bc\u03a0\u228e\u207b : \u2200 {A B}{C : A \u228e B \u2192 \u2605 _} \u2192 Searchable (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n     \u2192 Searchable (\u03a0 (A \u228e B) C)\n\u03bc\u03a0\u228e\u207b = \u03bc-iso {!!}\n\n_\u228e-Fun_ : \u2200 {A B} \u2192 Funable A \u2192 Funable B \u2192 Funable (A \u228e B)\n_\u228e-Fun_ (\u03bcA , \u03bcA\u2192) (\u03bcB , \u03bcB\u2192) = (\u03bcA \u228e-\u03bc \u03bcB) , (\u03bb X \u2192 \u03bc\u03a0\u228e\u207b (\u03bcA\u2192 X \u00d7-\u03bc \u03bcB\u2192 X))\n\nS\u22a4 : \u2200 {m A} \u2192 Search m A \u2192 Search m (\u22a4 \u2192 A)\nS\u22a4 sA _\u2219_ f = sA _\u2219_ (f \u2218 const)\n\nS\u03a0Bit : \u2200 {m A} \u2192 Search m (A 0b) \u2192 Search m (A 1b)\n                \u2192 Search m (\u03a0 Bit A)\nS\u03a0Bit sA\u2080 sA\u2081 _\u2219_ f = sA\u2080 _\u2219_ \u03bb x \u2192 sA\u2081 _\u2219_ \u03bb y \u2192 f \u03bb {true \u2192 y; false \u2192 x}\n\nsum-const : \u2200 {A} (\u03bcA : Searchable A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : Searchable A)\n                 (\u03bcB : Searchable B)\n                 f \u2192\n                 sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : Searchable A)\n                 (\u03bcB : Searchable B)\n                 f \u2192\n                 sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : Searchable A) (\u03bcB : Searchable B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7-\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : Searchable A) (\u03bcB : Searchable B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7-\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 {c \u2113} (m : Monoid c \u2113) {n} \u2192\n          let open Mon m in\n          Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Mon m\n\nvsgsum : \u2200 {c \u2113} (sg : Semigroup c \u2113) {n} \u2192\n           let open Sgrp sg in\n           Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 {m} n \u2192 Search m (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 Searchable A \u2192 Searchable (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 \u228e-\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 Searchable A \u2192 Searchable (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n{-\n\u03bcFinSuc : \u2200 n \u2192 Searchable (Fin (suc n))\n\u03bcFinSuc n = mk _ (ind n) {!!}\n  where ind : \u2200 n \u2192 SearchInd _ (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n-}\n\n\u03bcFinSuc : \u2200 n \u2192 Searchable (Fin (suc n))\n\u03bcFinSuc n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : Searchable A) n \u2192 Searchable (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift \u03bc\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7-\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : Searchable A) n  \u2192 Searchable (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {m A} n \u2192 Search m A \u2192 Search m (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : Searchable A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : Searchable A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : Searchable A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 Searchable (A \u00d7 B) \u2192 Searchable (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : Searchable (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : Searchable A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  ind zero    P P\u2219 Pf = Pf abs\n  ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 Searchable (Fin n \u2192 A)\n  \u03bcFun = mk _ (ind _) {!!}\n\nmodule BigDistr\n  {A}(\u03bcA : Searchable A)\n  (cm       : CommutativeMonoid L.zero L.zero)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_      : let open CMon cm in C  \u2192 C \u2192 C)\n  (distrib  : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_ : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm\n\n  \u03bcF\u2192A = \u03bcFun \u03bcA\n\n  -- Sum over A\n  \u03a3\u1d2c = search \u03bcA _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin (suc I) \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = search \u03bcF\u2192A _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 search (\u03bcFinSuc I) _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))\n    \u2248\u27e8 sym (search-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)))\n    \u2248\u27e8 search-sg-ext \u03bcA semigroup (\u03bb j \u2192 sym (search-lin\u02e1 \u03bcF\u2192A monoid _\u25ce_ (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)) (F zero j) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I ((F \u2218 suc) \u02e2 f))\n    \u220e\n\nFinDist : \u2200 {n} \u2192 DistFun (\u03bcFinSuc n) (\u03bb \u03bcX \u2192 \u03bcFun \u03bcX)\nFinDist \u03bcB c \u25ce distrib \u25ce-cong f = BigDistr.bigDistr \u03bcB c \u25ce distrib \u25ce-cong _ f\n\nsimple : \u2200 {A : Set}{P : A \u2192 A \u2192 Set} \u2192 (\u2200 x \u2192 P x x) \u2192 {x y : A} \u2192 x \u2261 y \u2192 P x y\nsimple r \u2261.refl = r _\n\n\u00d7-Dist : \u2200 {A B} FA FB \u2192 DistFunable {A} FA \u2192 DistFunable {B} FB \u2192 DistFunable (\u03a3-Fun FA FB)\n\u00d7-Dist FA FB FA-dist FB-dist \u03bcX c _\u2299_ distrib _\u2299-cong_ f\n  = \u03a0\u1d2c (\u03bb x \u2192 \u03a0\u1d2e (\u03bb y \u2192 \u03a3' (f (x , y))))\n  \u2248\u27e8 \u27e6search\u27e7 (searchable FA){_\u2261_} \u2261.refl _\u2248_ (\u03bb x y \u2192 x \u2299-cong y)\n       (\u03bb { {x} {.x} \u2261.refl \u2192 FB-dist \u03bcX c _\u2299_ distrib _\u2299-cong_ (curry f x)}) \u27e9\n    \u03a0\u1d2c (\u03bb x \u2192 \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb y \u2192 f (x , y) (fb y))))\n  \u2248\u27e8 FA-dist (negative FB \u03bcX) c _\u2299_ distrib _\u2299-cong_\n       (\u03bb x fb \u2192 search (searchable FB) _\u2299_ (\u03bb y \u2192 f (x , y) (fb y))) \u27e9\n    \u03a3\u1d2c\u1d2e (\u03bb fab \u2192 \u03a0\u1d2c (\u03bb x \u2192 \u03a0\u1d2e (\u03bb y \u2192 f (x , y) (fab x y))))\n  \u220e\n  where\n    open CMon c\n    open Funable\n\n    \u03a3' = search \u03bcX _\u2219_\n\n    \u03a0\u1d2c = search (searchable FA) _\u2299_\n    \u03a0\u1d2e = search (searchable FB) _\u2299_\n\n    \u03a3\u1d2c\u1d2e = search (negative FA (negative FB \u03bcX)) _\u2219_\n    \u03a3\u1d2e  = search (negative FB \u03bcX) _\u2219_\n\n\u228e-Dist : \u2200 {A B} FA FB \u2192 DistFunable {A} FA \u2192 DistFunable {B} FB \u2192 DistFunable (FA \u228e-Fun FB)\n\u228e-Dist FA FB FA-dist FB-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ f\n = \u03a0\u1d2c (\u03a3' \u2218 f \u2218 inj\u2081) \u25ce \u03a0\u1d2e (\u03a3' \u2218 f \u2218 inj\u2082)\n \u2248\u27e8 FA-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ (f \u2218 inj\u2081) \u25ce-cong FB-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ (f \u2218 inj\u2082) \u27e9\n   \u03a3\u1d2c (\u03bb fa \u2192 \u03a0\u1d2c (\u03bb i \u2192 f (inj\u2081 i) (fa i))) \u25ce \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb i \u2192 f (inj\u2082 i) (fb i)))\n \u2248\u27e8 sym (search-lin\u02b3 (negative FA \u03bcX) monoid _\u25ce_ _ _ (proj\u2082 distrib)) \u27e9\n   \u03a3\u1d2c (\u03bb fa \u2192 \u03a0\u1d2c (\u03bb i \u2192 f (inj\u2081 i) (fa i)) \u25ce \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb i \u2192 f (inj\u2082 i) (fb i))))\n \u2248\u27e8 search-sg-ext (negative FA \u03bcX) semigroup (\u03bb fa \u2192 sym (search-lin\u02e1 (negative FB \u03bcX) monoid _\u25ce_ _ _ (proj\u2081 distrib))) \u27e9\n   (\u03a3\u1d2c \u03bb fa \u2192 \u03a3\u1d2e \u03bb fb \u2192 \u03a0\u1d2c ((f \u2218 inj\u2081) \u02e2 fa) \u25ce \u03a0\u1d2e ((f \u2218 inj\u2082) \u02e2 fb))\n \u220e\n where\n    open CMon c\n    open Funable\n\n    \u03a3' = search \u03bcX _\u2219_\n\n    \u03a0\u1d2c = search (searchable FA) _\u25ce_\n    \u03a0\u1d2e = search (searchable FB) _\u25ce_\n\n    \u03a3\u1d2c = search (negative FA \u03bcX) _\u2219_\n    \u03a3\u1d2e = search (negative FB \u03bcX) _\u2219_\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen L using (Lift; lift)\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Data.Fin using (Fin)\nopen import Function.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to Searchable)\nopen import Search.Searchable.Product\nopen import Search.Searchable.Sum\nopen import Search.Derived\n\nmodule sum where\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search _ A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 Searchable A \u2192 Searchable B\n\u03bc-iso {A}{B} A\u2194B \u03bcA = mk search\u1d2e ind ade\n  where\n    A\u2192B = to A\u2194B\n\n    search\u1d2e : Search _ B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    sum\u1d2e = search\u1d2e _+_\n\n    ind : SearchInd _ search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n    ade : AdequateSum sum\u1d2e\n    ade f = sym-first-iso A\u2194B FI.\u2218 adequate-sum \u03bcA (f \u2218 A\u2192B)\n\n-- I guess this could be more general\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : Searchable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 from A\u2194B)\n\u03bc-iso-preserve A\u2194B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 \u2261.sym \u2218 Inverse.left-inverse-of A\u2194B)\n\n\u03bcLift : \u2200 {A} \u2192 Searchable A \u2192 Searchable (Lift A)\n\u03bcLift = \u03bc-iso (FI.sym Lift\u2194id)\n\n\u03bc\u22a4 : Searchable \u22a4\n\u03bc\u22a4 = mk _ ind ade\n  where\n    srch : Search _ \u22a4\n    srch _ f = f _\n\n    ind : SearchInd _ srch\n    ind _ _ Pf = Pf _\n\n    ade : AdequateSum (srch _+_)\n    ade x = FI.sym \u22a4\u00d7A\u2194A\n\nsearchBit : \u2200 {m} \u2192 Search m Bit\nsearchBit _\u2219_ f = f 0b \u2219 f 1b\n\nsearchBit-ind : \u2200 {m p} \u2192 SearchInd p {m} searchBit\nsearchBit-ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\n\u03bcBit : Searchable Bit\n\u03bcBit = \u03bc-iso (FI.sym Bit\u2194\u22a4\u228e\u22a4) (\u03bc\u22a4 \u228e-\u03bc \u03bc\u22a4)\n\nfocusBit : \u2200 {a} \u2192 Focus {a} searchBit\nfocusBit (false , x) = inj\u2081 x\nfocusBit (true ,  x) = inj\u2082 x\n\nfocusedBit : Focused {L.zero} searchBit\nfocusedBit {B} = inverses focusBit unfocus (\u21d2) (\u21d0)\n  where open Searchable\u2081\u2081 searchBit-ind\n        \u21d2 : (x : \u03a3 Bit B) \u2192 _\n        \u21d2 (false , x) = \u2261.refl\n        \u21d2 (true  , x) = \u2261.refl\n        \u21d0 : (x : B 0b \u228e B 1b) \u2192 _\n        \u21d0 (inj\u2081 x) = \u2261.refl\n        \u21d0 (inj\u2082 x) = \u2261.refl\n\nlookupBit : \u2200 {a} \u2192 Lookup {a} searchBit\nlookupBit = {!proj!}\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nS\u03a0\u03a3\u207b : \u2200 {m A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n       \u2192 Search m ((x : A) (y : B x) \u2192 C (x , y))\n       \u2192 Search m (\u03a0 (\u03a3 A B) C)\nS\u03a0\u03a3\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry)\n\nS\u03a0\u03a3\u207b-ind : \u2200 {m p A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n           \u2192 {s : Search m ((x : A) (y : B x) \u2192 C (x , y))}\n           \u2192 SearchInd p s\n           \u2192 SearchInd p (S\u03a0\u03a3\u207b s)\nS\u03a0\u03a3\u207b-ind ind P P\u2219 Pf = ind (P \u2218 S\u03a0\u03a3\u207b) P\u2219 (Pf \u2218 uncurry)\n\n\u03bc\u03a0\u03a3\u207b : \u2200 {A B}{C : \u03a3 A B \u2192 \u2605\u2080} \u2192 Searchable ((x : A)(y : B x) \u2192 C (x , y)) \u2192 Searchable ((p : \u03a3 A B) \u2192 C p)\n\u03bc\u03a0\u03a3\u207b \u03bc = mk (S\u03a0\u03a3\u207b (search \u03bc)) (S\u03a0\u03a3\u207b-ind (search-ind \u03bc)) {!!}\n\n\u03a3-Fun : \u2200 {A B} \u2192 Funable A \u2192 (Funable B) \u2192 Funable (A \u00d7 B)\n\u03a3-Fun (\u03bcA , \u03bcA\u2192) FB  = \u03bc\u03a3 \u03bcA (searchable FB) , (\u03bb x \u2192 \u03bc\u03a0\u03a3\u207b (\u03bcA\u2192 (negative FB x)))\n  where open Funable\n\nS\u00d7\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 B \u2192 C) \u2192 Search m (A \u00d7 B \u2192 C)\nS\u00d7\u207b = S\u03a0\u03a3\u207b\n\nS\u00d7\u207b-ind : \u2200 {m p A B C}\n          \u2192 {s : Search m (A \u2192 B \u2192 C)}\n          \u2192 SearchInd p s\n          \u2192 SearchInd p (S\u00d7\u207b s)\nS\u00d7\u207b-ind = S\u03a0\u03a3\u207b-ind\n\nS\u03a0\u228e\u207b : \u2200 {m A B} {C : A \u228e B \u2192 \u2605 _}\n    -- \u2192 Search m (\u03a0 A (C \u2218 inj\u2081)) \u2192 Search m (\u03a0 B (C \u2218 inj\u2082))\n       \u2192 Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n       \u2192 Search m (\u03a0 (A \u228e B) C)\nS\u03a0\u228e\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry [_,_])\n\nS\u03a0\u228e\u207b-ind : \u2200 {m p A B} {C : A \u228e B \u2192 \u2605 _}\n             {- {sA : Search m (\u03a0 A (C \u2218 inj\u2081))}\n             (iA : SearchInd p sA)\n             {sB : Search m (\u03a0 B (C \u2218 inj\u2082))}\n             (iB : SearchInd p sB) -}\n             {s : Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))}\n             (i : SearchInd p s)\n           \u2192 SearchInd p (S\u03a0\u228e\u207b {C = C} s) -- A sB)\nS\u03a0\u228e\u207b-ind i P P\u2219 Pf = i (P \u2218 S\u03a0\u228e\u207b) P\u2219 (Pf \u2218 uncurry [_,_])\n\n\u03bc\u03a0\u228e\u207b : \u2200 {A B}{C : A \u228e B \u2192 \u2605 _} \u2192 Searchable (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n     \u2192 Searchable (\u03a0 (A \u228e B) C)\n\u03bc\u03a0\u228e\u207b (mk s s-ind s-ade) = mk (S\u03a0\u228e\u207b s) (S\u03a0\u228e\u207b-ind s-ind) {!!}\n\n{- For each A\u2192C function\n   and each B\u2192C function\n   an A\u228eB\u2192C function is yield\n -}\nS\u228e\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 C) \u2192 Search m (B \u2192 C)\n                  \u2192 Search m (A \u228e B \u2192 C)\nS\u228e\u207b sA sB =  S\u03a0\u228e\u207b (sA \u00d7-search sB)\n\n_\u228e-Fun_ : \u2200 {A B} \u2192 Funable A \u2192 Funable B \u2192 Funable (A \u228e B)\n_\u228e-Fun_ (\u03bcA , \u03bcA\u2192) (\u03bcB , \u03bcB\u2192) = (\u03bcA \u228e-\u03bc \u03bcB) , (\u03bb X \u2192 \u03bc\u03a0\u228e\u207b (\u03bcA\u2192 X \u00d7-\u03bc \u03bcB\u2192 X))\n\nS\u22a4 : \u2200 {m A} \u2192 Search m A \u2192 Search m (\u22a4 \u2192 A)\nS\u22a4 sA _\u2219_ f = sA _\u2219_ (f \u2218 const)\n\nS\u03a0Bit : \u2200 {m A} \u2192 Search m (A 0b) \u2192 Search m (A 1b)\n                \u2192 Search m (\u03a0 Bit A)\nS\u03a0Bit sA\u2080 sA\u2081 _\u2219_ f = sA\u2080 _\u2219_ \u03bb x \u2192 sA\u2081 _\u2219_ \u03bb y \u2192 f \u03bb {true \u2192 y; false \u2192 x}\n\nsum-const : \u2200 {A} (\u03bcA : Searchable A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : Searchable A)\n                 (\u03bcB : Searchable B)\n                 f \u2192\n                 sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : Searchable A)\n                 (\u03bcB : Searchable B)\n                 f \u2192\n                 sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : Searchable A) (\u03bcB : Searchable B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7-\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : Searchable A) (\u03bcB : Searchable B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7-\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7-\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 {c \u2113} (m : Monoid c \u2113) {n} \u2192\n          let open Mon m in\n          Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Mon m\n\nvsgsum : \u2200 {c \u2113} (sg : Semigroup c \u2113) {n} \u2192\n           let open Sgrp sg in\n           Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 {m} n \u2192 Search m (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 Searchable A \u2192 Searchable (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 \u228e-\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 Searchable A \u2192 Searchable (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 Searchable (Fin (suc n))\n\u03bcFinSuc n = mk _ (ind n) {!!}\n  where ind : \u2200 n \u2192 SearchInd _ (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 Searchable (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : Searchable A) n \u2192 Searchable (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift \u03bc\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7-\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : Searchable A) n  \u2192 Searchable (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {m A} n \u2192 Search m A \u2192 Search m (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : Searchable A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : Searchable A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : Searchable A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 Searchable (A \u00d7 B) \u2192 Searchable (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : Searchable (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : Searchable A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  ind zero    P P\u2219 Pf = Pf abs\n  ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 Searchable (Fin n \u2192 A)\n  \u03bcFun = mk _ (ind _) {!!}\n\nmodule BigDistr\n  {A}(\u03bcA : Searchable A)\n  (cm       : CommutativeMonoid L.zero L.zero)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_      : let open CMon cm in C  \u2192 C \u2192 C)\n  (distrib  : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_ : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm\n\n  \u03bcF\u2192A = \u03bcFun \u03bcA\n\n  -- Sum over A\n  \u03a3\u1d2c = search \u03bcA _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin (suc I) \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = search \u03bcF\u2192A _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 search (\u03bcFinSuc I) _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))\n    \u2248\u27e8 sym (search-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)))\n    \u2248\u27e8 search-sg-ext \u03bcA semigroup (\u03bb j \u2192 sym (search-lin\u02e1 \u03bcF\u2192A monoid _\u25ce_ (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)) (F zero j) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I ((F \u2218 suc) \u02e2 f))\n    \u220e\n\nFinDist : \u2200 {n} \u2192 DistFun (\u03bcFinSuc n) (\u03bb \u03bcX \u2192 \u03bcFun \u03bcX)\nFinDist \u03bcB c \u25ce distrib \u25ce-cong f = BigDistr.bigDistr \u03bcB c \u25ce distrib \u25ce-cong _ f\n\nsimple : \u2200 {A : Set}{P : A \u2192 A \u2192 Set} \u2192 (\u2200 x \u2192 P x x) \u2192 {x y : A} \u2192 x \u2261 y \u2192 P x y\nsimple r \u2261.refl = r _\n\n\u00d7-Dist : \u2200 {A B} FA FB \u2192 DistFunable {A} FA \u2192 DistFunable {B} FB \u2192 DistFunable (\u03a3-Fun FA FB)\n\u00d7-Dist FA FB FA-dist FB-dist \u03bcX c _\u2299_ distrib _\u2299-cong_ f\n  = \u03a0\u1d2c (\u03bb x \u2192 \u03a0\u1d2e (\u03bb y \u2192 \u03a3' (f (x , y))))\n  \u2248\u27e8 \u27e6search\u27e7 (searchable FA){_\u2261_} \u2261.refl _\u2248_ (\u03bb x y \u2192 x \u2299-cong y)\n       (\u03bb { {x} {.x} \u2261.refl \u2192 FB-dist \u03bcX c _\u2299_ distrib _\u2299-cong_ (curry f x)}) \u27e9\n    \u03a0\u1d2c (\u03bb x \u2192 \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb y \u2192 f (x , y) (fb y))))\n  \u2248\u27e8 FA-dist (negative FB \u03bcX) c _\u2299_ distrib _\u2299-cong_\n       (\u03bb x fb \u2192 search (searchable FB) _\u2299_ (\u03bb y \u2192 f (x , y) (fb y))) \u27e9\n    \u03a3\u1d2c\u1d2e (\u03bb fab \u2192 \u03a0\u1d2c (\u03bb x \u2192 \u03a0\u1d2e (\u03bb y \u2192 f (x , y) (fab x y))))\n  \u220e\n  where\n    open CMon c\n    open Funable\n\n    \u03a3' = search \u03bcX _\u2219_\n\n    \u03a0\u1d2c = search (searchable FA) _\u2299_\n    \u03a0\u1d2e = search (searchable FB) _\u2299_\n\n    \u03a3\u1d2c\u1d2e = search (negative FA (negative FB \u03bcX)) _\u2219_\n    \u03a3\u1d2e  = search (negative FB \u03bcX) _\u2219_\n\n\u228e-Dist : \u2200 {A B} FA FB \u2192 DistFunable {A} FA \u2192 DistFunable {B} FB \u2192 DistFunable (FA \u228e-Fun FB)\n\u228e-Dist FA FB FA-dist FB-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ f\n = \u03a0\u1d2c (\u03a3' \u2218 f \u2218 inj\u2081) \u25ce \u03a0\u1d2e (\u03a3' \u2218 f \u2218 inj\u2082)\n \u2248\u27e8 FA-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ (f \u2218 inj\u2081) \u25ce-cong FB-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ (f \u2218 inj\u2082) \u27e9\n   \u03a3\u1d2c (\u03bb fa \u2192 \u03a0\u1d2c (\u03bb i \u2192 f (inj\u2081 i) (fa i))) \u25ce \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb i \u2192 f (inj\u2082 i) (fb i)))\n \u2248\u27e8 sym (search-lin\u02b3 (negative FA \u03bcX) monoid _\u25ce_ _ _ (proj\u2082 distrib)) \u27e9\n   \u03a3\u1d2c (\u03bb fa \u2192 \u03a0\u1d2c (\u03bb i \u2192 f (inj\u2081 i) (fa i)) \u25ce \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb i \u2192 f (inj\u2082 i) (fb i))))\n \u2248\u27e8 search-sg-ext (negative FA \u03bcX) semigroup (\u03bb fa \u2192 sym (search-lin\u02e1 (negative FB \u03bcX) monoid _\u25ce_ _ _ (proj\u2081 distrib))) \u27e9\n   (\u03a3\u1d2c \u03bb fa \u2192 \u03a3\u1d2e \u03bb fb \u2192 \u03a0\u1d2c ((f \u2218 inj\u2081) \u02e2 fa) \u25ce \u03a0\u1d2e ((f \u2218 inj\u2082) \u02e2 fb))\n \u220e\n where\n    open CMon c\n    open Funable\n\n    \u03a3' = search \u03bcX _\u2219_\n\n    \u03a0\u1d2c = search (searchable FA) _\u25ce_\n    \u03a0\u1d2e = search (searchable FB) _\u25ce_\n\n    \u03a3\u1d2c = search (negative FA \u03bcX) _\u2219_\n    \u03a3\u1d2e = search (negative FB \u03bcX) _\u2219_\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5625dccd05d45bee2c7eab37e8e5842f11ca1173","subject":"Desc model: simplify proof-psi-phi","message":"Desc model: simplify proof-psi-phi","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-casesW\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (xs : Sigma DescDConst \n                                                  (\u03bb s \u2192 desc (descDChoice I s)\n                                                  (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                            (hs : desc \n                                                  (box (sigma DescDConst (descDChoice I))\n                                                  (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                  P)\n                                            \u2192 P (Void , con xs)\n                      proof-psi-phi-cases (lvar , i) hs = refl\n                      proof-psi-phi-cases (lconst , x) hs = refl\n                      proof-psi-phi-cases (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (i : Unit)\n                                             (xs : Sigma DescDConst\n                                                   (\u03bb s \u2192 desc (descDChoice I s)\n                                                   (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                             (hs : desc \n                                                   (box (sigma DescDConst (descDChoice I))\n                                                   (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                   P)\n                                              \u2192  P (i , con xs)\n                      proof-psi-phi-casesW Void = proof-psi-phi-cases\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f8c71f27e3ad30e3903adbc4857985df75a686bc","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n-----------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h with (Stream-unf h)\n... | x' , xs' , Sxs' , h\u2081 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2081))) Sxs'\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n\n-- Requires K.\nlengthStream : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nlengthStream {xs} Sxs = \u2248N-coind _R_ h\u2081 h\u2082\n  where\n  _R_ : D \u2192 D \u2192 Set\n  m R n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 m R n \u2192\n       m \u2261 zero \u2227 n \u2261 zero \u2228\n       (\u2203[ m' ] \u2203[ n' ] m' R n' \u2227 m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n')\n  h\u2081 (_ , Sxs' , _ ) with Stream-unf Sxs'\n  h\u2081 {m} {n} (.(x'' \u2237 xs'') , Sxs' , aux , n\u2261\u221e) | x'' , xs'' , Sxs'' , refl =\n    inj\u2082 (length xs'' , \u221e , (xs'' , Sxs'' , refl , refl) , helper\u2081 , helper\u2082)\n    where\n    helper\u2081 : m \u2261 succ\u2081 (length xs'')\n    helper\u2081 = trans aux (length-\u2237 x'' xs'')\n\n    helper\u2082 : n \u2261 succ\u2081 \u221e\n    helper\u2082 = trans n\u2261\u221e \u221e-eq\n\n  h\u2082 : length xs R \u221e\n  h\u2082 = xs , Sxs , refl , refl\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n-----------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h with (Stream-unf h)\n... | x' , xs' , Sxs' , h\u2081 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2081))) Sxs'\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n\n-- Requires K.\nlengthStream : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nlengthStream {xs} Sxs = \u2248N-coind _R_ h\u2081 h\u2082\n  where\n  _R_ : D \u2192 D \u2192 Set\n  m R n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 m R n \u2192\n       m \u2261 zero \u2227 n \u2261 zero \u2228\n       (\u2203[ m' ] \u2203[ n' ] m' R n' \u2227 m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n')\n  h\u2081 (_ , Sxs' , _ ) with Stream-unf Sxs'\n  h\u2081 (.(x'' \u2237 xs'') , Sxs' , m\u2261length-x''\u2237xs'' , n\u2261\u221e)\n     | x'' , xs'' , Sxs'' , refl =\n    inj\u2082 (length xs'' , \u221e , (xs'' , Sxs'' , refl , refl)\n         , trans m\u2261length-x''\u2237xs'' (length-\u2237 x'' xs'') , trans n\u2261\u221e \u221e-eq\n         )\n\n  h\u2082 : length xs R \u221e\n  h\u2082 = xs , Sxs , refl , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24244df1568a8f505be675e6d92c0427c9e962cb","subject":"IDesc in IDesc, stratified, with a 'lift'ing axiom","message":"IDesc in IDesc, stratified, with a 'lift'ing axiom","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc zero I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc zero I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc zero I)(P : I -> Set) -> desc I D P -> IDesc zero (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc zero I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc zero I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc zero I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set1 where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc (suc zero) Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc (suc zero) Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"948451ac92f2f059429dba6effbfd6f9aad6ebea","subject":"Working on the transpose map.","message":"Working on the transpose map.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DCBSets.agda","new_file":"dialectica-cats\/DCBSets.agda","new_contents":"module DCBSets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ m \u2208 (\u22a4 \u2192 U) ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (U \u2192 X \u2192 Set))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , _ , \u03b1) (V , Y , _ , _ , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , m\u2081 , n\u2081 , \u03b1)} {(V , Y , m\u2082 , n\u2082 , \u03b2)} {(W , Z , m\u2083 , n\u2083 , \u03b3)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , m , n , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , _ , \u03b1}\n         {V , Y , _ , _ , \u03b2}\n         {W , Z , _ , _ , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{W , Z , _ , _ , \u03b3}{S , T , _ , _ , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 m\u2081 m\u2082  , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)}{(W , Z , _ , _ , \u03b3)}{(S , T , _ , _ , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (U , X , m\u2081 , n\u2081 , \u03b1)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (V , Y , m\u2082 , n\u2082 , \u03b2)\n\u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n\ncart-ar : {U X V Y W Z : Set}\n  \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n  \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n  \u2192 {m\u2081 : \u22a4 \u2192 U}\n  \u2192 {n\u2081 : \u22a4 \u2192 X}\n  \u2192 {m\u2082 : \u22a4 \u2192 V}\n  \u2192 {n\u2082 : \u22a4 \u2192 Y}\n  \u2192 {m\u2083 : \u22a4 \u2192 W}\n  \u2192 {n\u2083 : \u22a4 \u2192 Z}      \n  \u2192 Hom (W , Z , m\u2083 , n\u2083 , \u03b3) (U , X , m\u2081 , n\u2081 , \u03b1)\n  \u2192 Hom (W , Z , m\u2083 , n\u2083 , \u03b3) (V , Y , m\u2082 , n\u2082 , \u03b2)\n  \u2192 Hom (W , Z , m\u2083 , n\u2083 , \u03b3) ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2))\ncart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3}{m\u2081}{n\u2081}{m\u2082}{n\u2082}{m\u2083}{n\u2083} (f , F) (g , G) = trans-\u00d7 f g , {!!} , {!!}\n","old_contents":"module DCBSets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ m \u2208 (\u22a4 \u2192 U) ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (U \u2192 X \u2192 Set))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , _ , \u03b1) (V , Y , _ , _ , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , m\u2081 , n\u2081 , \u03b1)} {(V , Y , m\u2082 , n\u2082 , \u03b2)} {(W , Z , m\u2083 , n\u2083 , \u03b3)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , m , n , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , _ , \u03b1}\n         {V , Y , _ , _ , \u03b2}\n         {W , Z , _ , _ , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{W , Z , _ , _ , \u03b3}{S , T , _ , _ , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 m\u2081 m\u2082  , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)}{(W , Z , _ , _ , \u03b3)}{(S , T , _ , _ , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (U , X , m\u2081 , n\u2081 , \u03b1)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (V , Y , m\u2082 , n\u2082 , \u03b2)\n\u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"609618e2cc51a1aa0090d436f7f82aa096f048d3","subject":"bit-guessing-game...","message":"bit-guessing-game...\n","repos":"crypto-agda\/crypto-agda","old_file":"bit-guessing-game.agda","new_file":"bit-guessing-game.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\nopen import Function\nopen import Data.Bool.NP\n\nopen import Data.Bits\n\nopen import flipbased-implem using (Coins; \u21ba; \u2141) -- ; #\u27e8_\u27e9) renaming (count\u21ba to #\u27e8_\u27e9)\nopen import program-distance using (PrgDist; module PrgDist)\nopen import Relation.Binary.PropositionalEquality\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n\nopen PrgDist prgDist\n\nGuessAdv : Coins \u2192 Set\nGuessAdv = \u2141\n\nrunGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u2141 ca\nrunGuess\u2141 A _ = A\n\n-- An oracle: an adversary who can break the guessing game.\nOracle : Coins \u2192 Set\nOracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n-- Any adversary cannot do better than a random guess.\nGuessSec : Coins \u2192 Set\nGuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\n#\u1d47 : Bool \u2192 \u2115\n#\u1d47 b = if b then 1 else 0\n\n#\u1d47\u22641 : \u2200 b \u2192 #\u1d47 b \u2264 1\n#\u1d47\u22641 true = s\u2264s z\u2264n\n#\u1d47\u22641 false = z\u2264n\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = #\u1d47\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot zero f = refl\n#-\u2218vnot (suc c) f\n  rewrite #-\u2218vnot c (f \u2218 0\u2237_)\n        | #-\u2218vnot c (f \u2218 1\u2237_) = \u2115\u00b0.+-comm #\u27e8 f \u2218 0\u2237_ \u2218 vnot \u27e9 _\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true = refl\n... | false = refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = helper c #\u27e8 not \u2218 f \u2218 0\u2237_ \u27e9 _ (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n  where helper : \u2200 c x y \u2192 x \u2264 2^ c \u2192 y \u2264 2^ c \u2192 (2^ c \u2238 x) + (2^ c \u2238 y) \u2261 2^(1 + c) \u2238 (x + y)\n        helper zero x y x\u2264 y\u2264 = {!!}\n        helper (suc c\u2081) x y x\u2264 y\u2264 = {!helper c\u2081 x y ? ?!}\n","old_contents":"open import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Data.Bits using (Bit)\n\nopen import flipbased-implem using (Coins; \u21ba)\nopen import program-distance using (PrgDist; module PrgDist)\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0528ae0789f2799e87305842a858d9c55b1f63da","subject":"Proved the induction principles from\/to data to\/from postulates.","message":"Proved the induction principles from\/to data to\/from postulates.\n\nIgnore-this: ebc245f9526e42279cdf22228edf7ff1\n\ndarcs-hash:20110930172638-3bd4e-1edcece07d36236fa14fd2108032d8a4e808ce4e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/DataPostulate.agda","new_file":"Draft\/DataPostulate.agda","new_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulate predicate: Using the\n-- induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulate predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulate predicate to the data predicate\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulate inductive principle from the data inductive principle.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {n} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulate predicate.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {n} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- From data to postulates\n\n-- The predicate: Using the induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- The predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n------------------------------------------------------------------------------\n-- From postulates to data\n\n-- The predicate.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"49294beab1d87f1535f454d4d3fed622b78f0da0","subject":"Vec: refactor","message":"Vec: refactor\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : \u2605 a}\n    (_\u2248_ : A \u2192 A \u2192 \u2605 \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 \u2605 \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : \u2605 a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= \u2261.refl \u2261.refl = \u2261.refl\n\nmodule With\u2261 {a}{A : \u2605 a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | 1b rewrite count-\u2218 f pred xs = idp\n... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : \u2605 a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : \u2605 a} where\n  count-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_)\nimport Data.Fin.Properties as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = \u2261.ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : \u2605 a}\n    (_\u2248_ : A \u2192 A \u2192 \u2605 \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 \u2605 \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : \u2605 a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= \u2261.refl \u2261.refl = \u2261.refl\n\nmodule With\u2261 {a}{A : \u2605 a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = idp\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = idp\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\ninfixl 2 _\u203c_\n_\u203c_ : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_\u203c_ = flip lookup\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 _\u203c_\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\n\nmodule Here {a} {A : \u2605 a} where\n  open Data.Vec.Equality.Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 eq = fst (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = \u2261.ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (_\u203c_ v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (_\u203c_ v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = idp\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen Algebra.FunctionProperties.Eq.Implicits\n\nmodule _ {a} {A : \u2605 a} where\n    dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n    dup xs = xs ++ xs\n\n    dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n    dup-inj = ++-inj\u2081\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bb40a5ccf2b5b8b4d1ef8c518f9c6a7c60094079","subject":"cleaning up some names and easy todos","message":"cleaning up some names and easy todos\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam ta) FHOuter = _ , TALam ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- replacing a variable in a term with contents of the appropriate type\n  -- preserves type and contracts the context\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst TAConst D2 = TAConst\n  lem-subst {\u0393 = \u0393} {x = x'} (TAVar {x = x} x\u2082) D2 with natEQ x x'\n  lem-subst {\u0393 = \u0393} {x = x} {\u03c41 = \u03c41} (TAVar x\u2083) D2 | Inl refl = {!!}\n  lem-subst {\u0393 = \u0393} {x = x} {\u03c41 = \u03c41} (TAVar {x = x'} x\u2083) D2 | Inr x\u2082 = {!x\u2208sing!}\n  -- ... | qq = TAVar {!!}\n  --   with \u0393 x\n  -- lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | Some x\u2081 = {!!}\n  --   with natEQ x' x\n  -- lem-subst (TAVar xin) D2 | Some x\u2083 | Inl refl = {!!}\n  -- lem-subst (TAVar refl) D2 | Some x\u2083 | Inr x\u2082 = {!!}\n  -- lem-subst {x = x} (TAVar {x = x'} x\u2082) D2 | None with natEQ x' x\n  --lem-subst {x = x} (TAVar x\u2083) D2 | None | Inl refl with natEQ x x\n  -- lem-subst (TAVar refl) D2 | None | Inl refl | Inl refl = D2\n  -- lem-subst (TAVar x\u2083) D2 | None | Inl refl | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  --lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | None | Inr x\u2082 with natEQ x x'\n  -- lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inl x\u2082 = abort ((flip x\u2083) x\u2082)\n  -- lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inr x\u2082 = abort (somenotnone (! x\u2084))\n  lem-subst {\u0393 = \u0393} {x = x} (TALam {x = x'} D1) D2 = TALam {!!}\n  lem-subst (TAAp D1 D2) D3 = TAAp (lem-subst D1 D3) (lem-subst D2 D3)\n  lem-subst (TAEHole x\u2081 x\u2082) D2 = TAEHole x\u2081 {!!}\n  lem-subst (TANEHole x\u2081 D1 x\u2082) D2 = TANEHole x\u2081 (lem-subst D1 D2) {!!}\n  lem-subst (TACast D1 x\u2081) D2 = TACast (lem-subst D1 D2) x\u2081\n  lem-subst (TAFailedCast D1 x\u2081 x\u2082 x\u2083) D2 = TAFailedCast (lem-subst D1 D2) x\u2081 x\u2082 x\u2083\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans TAConst ()\n  preserve-trans (TAVar x\u2081) ()\n  preserve-trans (TALam ta) ()\n  preserve-trans (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  preserve-trans (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans (TAEHole x x\u2081) ()\n  preserve-trans (TANEHole x ta x\u2081) ()\n  preserve-trans (TACast ta x) (ITCastID) = ta\n  preserve-trans (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans (TAFailedCast x y z q) ()\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import htype-decidable\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (pres-lem eps D1 D3 D4 D5) D2\n  pres-lem (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (pres-lem eps D2 D3 D4 D5)\n  pres-lem (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (pres-lem eps D1 D2 D3 D4) x\u2081\n  pres-lem (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (pres-lem eps D1 D2 D3 D4) x\n  pres-lem (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (pres-lem x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = pres-lem2 x y\n\n  -- this is the literal contents of the hole in lem3; it might not go\n  -- through exactly like this.\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst TAConst D2 = TAConst\n  lem-subst {\u0393 = \u0393} {x = x'} (TAVar {x = x} x\u2082) D2 with natEQ x x'\n  lem-subst {\u0393 = \u0393} {x = x} {\u03c41 = \u03c41} (TAVar x\u2083) D2 | Inl refl = {!!}\n  lem-subst {\u0393 = \u0393} {x = x} {\u03c41 = \u03c41} (TAVar {x = x'} x\u2083) D2 | Inr x\u2082 = {!x\u2208sing!}\n  -- ... | qq = TAVar {!!}\n  --   with \u0393 x\n  -- lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | Some x\u2081 = {!!}\n  --   with natEQ x' x\n  -- lem-subst (TAVar xin) D2 | Some x\u2083 | Inl refl = {!!}\n  -- lem-subst (TAVar refl) D2 | Some x\u2083 | Inr x\u2082 = {!!}\n  -- lem-subst {x = x} (TAVar {x = x'} x\u2082) D2 | None with natEQ x' x\n  --lem-subst {x = x} (TAVar x\u2083) D2 | None | Inl refl with natEQ x x\n  -- lem-subst (TAVar refl) D2 | None | Inl refl | Inl refl = D2\n  -- lem-subst (TAVar x\u2083) D2 | None | Inl refl | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  --lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | None | Inr x\u2082 with natEQ x x'\n  -- lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inl x\u2082 = abort ((flip x\u2083) x\u2082)\n  -- lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inr x\u2082 = abort (somenotnone (! x\u2084))\n  lem-subst {\u0393 = \u0393} {x = x} (TALam {x = x'} D1) D2 = TALam {!lem-subst D1!}\n  lem-subst (TAAp D1 D2) D3 = TAAp (lem-subst D1 D3) (lem-subst D2 D3)\n  lem-subst (TAEHole x\u2081 x\u2082) D2 = TAEHole x\u2081 {!!}\n  lem-subst (TANEHole x\u2081 D1 x\u2082) D2 = TANEHole x\u2081 (lem-subst D1 D2) {!!}\n  lem-subst (TACast D1 x\u2081) D2 = TACast (lem-subst D1 D2) x\u2081\n  lem-subst (TAFailedCast D1 x\u2081 x\u2082 x\u2083) D2 = TAFailedCast (lem-subst D1 D2) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dd73b5a36235448b400cd9d02d7983b1155617bc","subject":"Reorder definitions of Term and Terms.","message":"Reorder definitions of Term and Terms.\n\nThe order of definitions should follow the order of declaration,\nand this code seems easier to understand when one looks at Term\nfirst.\n\nOld-commit-hash: 517547b8eebe1b8803d9f30fa3945f02685b6cc5\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 TermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 TermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8d03eb08a3d8dc237fc96cbdf430c73571ead581","subject":"Add new lemmas on change equivalence","message":"Add new lemmas on change equivalence\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CB}} (df a da) (f (a \u229e da) \u229f f a)\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"be309218c7d7b2c11388e7721ba93b5817986e9c","subject":"Moved note to agda2atp repo.","message":"Moved note to agda2atp repo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/InternalSyntax.agda","new_file":"notes\/InternalSyntax.agda","new_contents":"","old_contents":"------------------------------------------------------------------------------\n-- Agda internal syntax\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule InternalSyntax where\n\n-- Internal types from Agda.Syntax.Internal.\n\n------------------------------------------------------------------------------\n-- Sets\n\npostulate A B C : Set\n\n-- El (Type (Max [ClosedLevel 1])) (Sort (Type (Max [])))\n\npostulate A\u2081 : Set\u2081\n\n-- El (Type (Max [ClosedLevel 2])) (Sort (Type (Max [ClosedLevel 1])))\n\n------------------------------------------------------------------------------\n-- The term is Def\n\npostulate defTerm : A\n\n-- El (Type (Max [])) (Def InternalSyntax.A [])\n\n------------------------------------------------------------------------------\n-- The term is Fun\n\npostulate funTerm\u2081 : A \u2192 B\n\n-- El (Type (Max []))\n--    (Fun r(El (Type (Max []))\n--              (Def InternalSyntax.A []))\n--         (El (Type (Max []))\n--             (Def InternalSyntax.B [])))\n\npostulate funTerm\u2082 : A \u2192 B \u2192 C\n\n-- El (Type (Max []))\n--    (Fun r(El (Type (Max []))\n--              (Def InternalSyntax.A []))\n--         (El (Type (Max []))\n--             (Fun r(El (Type (Max []))\n--                       (Def InternalSyntax.B []))\n--                  (El (Type (Max []))\n--                      (Def InternalSyntax.C [])))))\n\n\n------------------------------------------------------------------------------\n-- The term is a (fake) Pi\n\npostulate fakePiTerm : (a : A) \u2192 B\n\n-- El (Type (Max []))\n--    (Pi r(El (Type (Max []))\n--             (Def InternalSyntax.A []))\n--        (Abs \"a\" El (Type (Max []))\n--                 (Def InternalSyntax.B [])))\n\n------------------------------------------------------------------------------\n-- The term is Pi\n\npostulate P : A \u2192 Set\n\n-- El (Type (Max [ClosedLevel 1]))\n--    (Fun r(El (Type (Max []))\n--              (Def InternalSyntax.A []))\n--         (El (Type (Max [ClosedLevel 1]))\n--             (Sort (Type (Max [])))))\n\npostulate piTerm : (a : A) \u2192 P a\n\n-- El (Type (Max []))\n--    (Pi r(El (Type (Max []))\n--             (Def InternalSyntax.A []))\n--        (Abs \"a\" El (Type (Max []))\n--                    (Def InternalSyntax.P [r(Var 0 [])])))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ca0490d3e647551767333fe1b8af760d98b01dfa","subject":"agda: use \u2295 and \u229d also when defining diff-term and apply-term","message":"agda: use \u2295 and \u229d also when defining diff-term and apply-term\n\nGoal: make diff-term and apply-term themselves more readable.\n\nOld-commit-hash: fc206b02d6057172e7f49c02eb521abaaafcb1e8\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Popl14.agda","new_file":"Syntax\/Term\/Popl14.agda","new_contents":"module Syntax.Term.Popl14 where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Syntax.Context.Popl14 public\nopen import Data.Integer\n\ndata Term (\u0393 : Context) : Type -> Set where\n  int : (n : \u2124) \u2192 Term \u0393 int\n  add : (s : Term \u0393 int) (t : Term \u0393 int) \u2192 Term \u0393 int\n  minus : (t : Term \u0393 int) \u2192 Term \u0393 int\n\n  empty : Term \u0393 bag\n  insert : (s : Term \u0393 int) (t : Term \u0393 bag) \u2192 Term \u0393 bag\n  union : (s : Term \u0393 bag) \u2192 (t : Term \u0393 bag) \u2192 Term \u0393 bag\n  negate : (t : Term \u0393 bag) \u2192 Term \u0393 bag\n\n  flatmap : (s : Term \u0393 (int \u21d2 bag)) (t : Term \u0393 bag) \u2192 Term \u0393 bag\n  sum : (t : Term \u0393 bag) \u2192 Term \u0393 int\n\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (int x) = int x\nweaken \u0393\u2081\u227c\u0393\u2082 (add s t) = add (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (minus t) = minus (weaken \u0393\u2081\u227c\u0393\u2082 t)\n\nweaken \u0393\u2081\u227c\u0393\u2082 empty = empty\nweaken \u0393\u2081\u227c\u0393\u2082 (insert s t) = insert (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (union s t) = union (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (negate t) = negate (weaken \u0393\u2081\u227c\u0393\u2082 t)\n\nweaken \u0393\u2081\u227c\u0393\u2082 (flatmap s t) = flatmap (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (sum t) = sum (weaken \u0393\u2081\u227c\u0393\u2082 t)\n\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weakenVar \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-term {int} =\n  let \u0394x = var (that this)\n      x  = var this\n  in abs (abs (add x \u0394x))\napply-term {bag} =\n  let \u0394x = var (that this)\n      x  = var this\n  in abs (abs (union x \u0394x))\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app f x \u2295 app (app \u0394f x) (x \u229d x))))\n\ndiff-term {int} =\n  let x = var (that this)\n      y = var this\n  in abs (abs (add x (minus y)))\ndiff-term {bag} =\n  let x = var (that this)\n      y = var this\n  in abs (abs (union x (negate y)))\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app g (x \u2295 \u0394x) \u229d app f x))))\n","old_contents":"module Syntax.Term.Popl14 where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Syntax.Context.Popl14 public\nopen import Data.Integer\n\ndata Term (\u0393 : Context) : Type -> Set where\n  int : (n : \u2124) \u2192 Term \u0393 int\n  add : (s : Term \u0393 int) (t : Term \u0393 int) \u2192 Term \u0393 int\n  minus : (t : Term \u0393 int) \u2192 Term \u0393 int\n\n  empty : Term \u0393 bag\n  insert : (s : Term \u0393 int) (t : Term \u0393 bag) \u2192 Term \u0393 bag\n  union : (s : Term \u0393 bag) \u2192 (t : Term \u0393 bag) \u2192 Term \u0393 bag\n  negate : (t : Term \u0393 bag) \u2192 Term \u0393 bag\n\n  flatmap : (s : Term \u0393 (int \u21d2 bag)) (t : Term \u0393 bag) \u2192 Term \u0393 bag\n  sum : (t : Term \u0393 bag) \u2192 Term \u0393 int\n\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (int x) = int x\nweaken \u0393\u2081\u227c\u0393\u2082 (add s t) = add (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (minus t) = minus (weaken \u0393\u2081\u227c\u0393\u2082 t)\n\nweaken \u0393\u2081\u227c\u0393\u2082 empty = empty\nweaken \u0393\u2081\u227c\u0393\u2082 (insert s t) = insert (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (union s t) = union (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (negate t) = negate (weaken \u0393\u2081\u227c\u0393\u2082 t)\n\nweaken \u0393\u2081\u227c\u0393\u2082 (flatmap s t) = flatmap (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (sum t) = sum (weaken \u0393\u2081\u227c\u0393\u2082 t)\n\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weakenVar \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\napply-term {int} =\n  let \u0394x = var (that this)\n      x  = var this\n  in abs (abs (add x \u0394x))\napply-term {bag} =\n  let \u0394x = var (that this)\n      x  = var this\n  in abs (abs (union x \u0394x))\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app (app apply-term\n        (app (app \u0394f x) (app (app diff-term x) x)))\n        (app f x))))\n\ndiff-term {int} =\n  let x = var (that this)\n      y = var this\n  in abs (abs (add x (minus y)))\ndiff-term {bag} =\n  let x = var (that this)\n      y = var this\n  in abs (abs (union x (negate y)))\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app (app diff-term\n        (app g (app (app apply-term \u0394x) x)))\n        (app f x)))))\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f994df78d5187cf478e9bd11ea4ef31f16894f95","subject":"NewNew.agda: Hide deducible arguments","message":"NewNew.agda: Hide deducible arguments\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n-- XXX This would be more elegant as a datatype \u2014 that would avoid the need for\n-- an equality proof.\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1' \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n   [ \u03c4 ] dv from v1 to v2 \u00d7 v1 \u2261 v1' \u00d7 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} {\u2205} tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {_ \u2022 _} {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} {dv \u2022 .v1 \u2022 d\u03c1} (dvv , refl , d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {\u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive \u03c4 t (daa , refl , d\u03c1\u03c1)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1' \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n   [ \u03c4 ] dv from v1 to v2 \u00d7 v1 \u2261 v1' \u00d7 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 \u2205 tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dv \u2022 .v1 \u2022 d\u03c1) (dvv , refl , d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c11 \u03c12 d\u03c1 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t \u03c11 \u03c12 d\u03c1 d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c11 \u03c12 d\u03c1 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) {d\u03c1} {\u03c11} {\u03c12} d\u03c1\u03c1 rewrite sym (fit-sound t \u03c11 \u03c12 d\u03c1 d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive \u03c4 t (daa , refl , d\u03c1\u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"11a45f0108ff29d0e24b4ce3f2077ffe597a899f","subject":"Add 1-E_ and kill some postulates","message":"Add 1-E_ and kill some postulates\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      P : U \u2192 [0,1]\n      sumP\u22611 : sumP P allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs))\n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      P : U \u2192 [0,1]\n      sumP\u22611 : sumP P allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs))\n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0270576411d565e7daa6f53f108806d14fdc6582","subject":"Finished defining the SMC structure.","message":"Finished defining the SMC structure.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   F\u03b1-inv : (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\n   F\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n{-\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (W \u2192 \u03a3 (V \u2192 X) (\u03bb x \u2192 U \u2192 Y)) (\u03bb x \u2192 \u03a3 U (\u03bb x\u2081 \u2192 V) \u2192 Z)}\n     \u2192 ((\u03bb x \u2192 (\u03bb x\u2081 \u2192 fst (fst a x) x\u2081) , (\u03bb x\u2081 \u2192 snd (fst a x) x\u2081)) , (\u03bb x \u2192 snd a (fst x , snd x))) \u2261 a\n   aux' {j\u2081 , j\u2082} = eq-\u00d7 (ext-set aux'') (ext-set aux''')\n    where\n      aux'' : {a : W} \u2192 (fst (j\u2081 a) , snd (j\u2081 a)) \u2261 j\u2081 a\n      aux'' {w} with j\u2081 w\n      ... | h\u2081 , h\u2082 = refl\n\n      aux''' : {a : \u03a3 U (\u03bb x\u2081 \u2192 V)} \u2192 j\u2082 (fst a , snd a) \u2261 j\u2082 a\n      aux''' {u , v} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a\n       : \u03a3 (\u03a3 V (\u03bb x \u2192 W) \u2192 X) (\u03bb x \u2192 U \u2192 \u03a3 (W \u2192 Y) (\u03bb x\u2081 \u2192 V \u2192 Z))} \u2192\n      ((\u03bb p' \u2192 fst (fst (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) (snd p')) (fst p')) ,\n       (\u03bb u \u2192 (\u03bb w \u2192 snd (fst (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) w) u) , (\u03bb v \u2192 snd (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) (u , v))))\n      \u2261 a\n   aux' {j\u2081 , j\u2082} = eq-\u00d7 (ext-set aux'') (ext-set aux''')\n     where\n       aux'' : {a : \u03a3 V (\u03bb x \u2192 W)} \u2192 j\u2081 (fst a , snd a) \u2261 j\u2081 a\n       aux'' {v , w} = refl\n       aux''' : {a : U} \u2192 ((\u03bb w \u2192 fst (j\u2082 a) w) , (\u03bb v \u2192 snd (j\u2082 a) v)) \u2261 j\u2082 a\n       aux''' {u} with j\u2082 u\n       ... | h\u2081 , h\u2082 = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 (U \u2192 V) \u00d7 (Y \u2192 X) \u2192 U \u00d7 Y \u2192 Set\n\u22b8-cond \u03b1 \u03b2 (f , g) (u , y) = \u03b1 u (g y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (Y \u2192 X)) , (U \u00d7 Y) , \u22b8-cond \u03b1 \u03b2\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , p\u2083\n  where\n   h : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 T \u2192 Z)\n   h (h\u2081 , h\u2082) = (\u03bb w \u2192 g (h\u2081 (f w))) , (\u03bb t \u2192 F (h\u2082 (G t)))\n   H : \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n   H (w , t) = f w , G t\n   p\u2083 : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond \u03b1 \u03b2 u (H y) \u2192 \u22b8-cond \u03b3 \u03b4 (h u) y\n   p\u2083 {h\u2081 , h\u2082}{w , t} c c' = p\u2082 (c (p\u2081 c'))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n{-\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u22a4 \u00d7 U} {x : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u ((\u03bb _ \u2192 triv) , (\u03bb _ \u2192 x)) \u2192 \u03b1 (snd u) x\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} (triv , p-\u03b1) = p-\u03b1\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb x \u2192 (\u03bb _ \u2192 triv) , (\u03bb _ \u2192 x)) , \u03bb\u2297-p\n\n\u03bb\u207b\u00b9\u2297 : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u207b\u00b9\u2297 {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , ((\u03bb x \u2192 snd x triv) , \u03bb\u207b\u00b9\u2297-p)  \n where\n  \u03bb\u207b\u00b9\u2297-p : \u2200{U X \u03b1} \u2192 {u : U} {y : (U \u2192 \u22a4) \u00d7 (\u22a4 \u2192 X)} \u2192 \u03b1 u (snd y triv) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u207b\u00b9\u2297-p {U}{X}{\u03b1}{u}{(h\u2081 , h\u2082)} p-\u03b1 with h\u2081 u\n  ... | triv = triv , p-\u03b1\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb x \u2192 (\u03bb x\u2081 \u2192 x) , (\u03bb x\u2081 \u2192 triv)) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : U \u00d7 \u22a4}{x : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u ((\u03bb _ \u2192 x) , (\u03bb _ \u2192 triv)) \u2192 \u03b1 (fst u) x\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , triv)}{x} (p-\u03b1 , triv) = p-\u03b1\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb x \u2192 x , triv) , (\u03bb x \u2192 fst x triv) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 (\u22a4 \u2192 X) (\u03bb x \u2192 U \u2192 \u22a4)} \u2192 \u03b1 u (fst y triv) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{(f , g)} p-\u03b1 rewrite single-range {g = g}{u} = p-\u03b1 , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , twist-\u00d7 , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2} \u2192 {u : U \u00d7 V} {y : (U \u2192 Y) \u00d7 (V \u2192 X)} \u2192\n       (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{(u , v)}{(h\u2081 , h\u2082)} p-\u03b1 = twist-\u00d7 p-\u03b1\n\n-- The associator:\n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   F\u03b1-inv : (W \u2192 (V \u2192 X) \u00d7 (U \u2192 Y)) \u00d7 (U \u00d7 V \u2192 Z) \u2192 (V \u00d7 W \u2192 X) \u00d7 (U \u2192 (W \u2192 Y) \u00d7 (V \u2192 Z))\n   F\u03b1-inv = (\u03bb p \u2192 (\u03bb p' \u2192 fst ((fst p) (snd p')) (fst p')) , (\u03bb u \u2192 (\u03bb w \u2192 snd (fst p w) u) , (\u03bb v \u2192 (snd p) (u , v))))\n   \u03b1-inv-cond : \u2200{u : U \u00d7 V \u00d7 W} {y : (W \u2192 (V \u2192 X) \u00d7 (U \u2192 Y)) \u00d7 (U \u00d7 V \u2192 Z)} \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv y) \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {(u , v , w)} {(h\u2081 , h\u2082)} (p\u2081 , p\u2082 , p\u2083) with h\u2081 w\n   ... | (a , b) = (p\u2081 , p\u2082) , p\u2083\n\nF\u03b1 : \u2200{V W X Y U V Z : Set} \u2192 \u03a3 (\u03a3 V (\u03bb x \u2192 W) \u2192 X) (\u03bb x \u2192 U \u2192 \u03a3 (W \u2192 Y) (\u03bb x\u2081 \u2192 V \u2192 Z))\n       \u2192 \u03a3 (W \u2192 \u03a3 (V \u2192 X) (\u03bb x \u2192 U \u2192 Y)) (\u03bb x \u2192 \u03a3 U (\u03bb x\u2081 \u2192 V) \u2192 Z)\nF\u03b1 (f ,  g) = (\u03bb x \u2192 (\u03bb x\u2081 \u2192 f ((x\u2081 , x))) , (\u03bb x\u2081 \u2192 fst (g x\u2081) x)) , (\u03bb x \u2192 snd (g (fst x)) (snd x))\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 {V} , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}{y : \u03a3 (\u03a3 V (\u03bb x \u2192 W) \u2192 X) (\u03bb x \u2192 U \u2192 \u03a3 (W \u2192 Y) (\u03bb x\u2081 \u2192 V \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 {V} y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{(f , g)} ((p\u2081 , p\u2082) , p\u2083) with g u\n  ... | (h\u2081 , h\u2082) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (W \u2192 \u03a3 (V \u2192 X) (\u03bb x \u2192 U \u2192 Y)) (\u03bb x \u2192 \u03a3 U (\u03bb x\u2081 \u2192 V) \u2192 Z)}\n     \u2192 ((\u03bb x \u2192 (\u03bb x\u2081 \u2192 fst (fst a x) x\u2081) , (\u03bb x\u2081 \u2192 snd (fst a x) x\u2081)) , (\u03bb x \u2192 snd a (fst x , snd x))) \u2261 a\n   aux' {j\u2081 , j\u2082} = eq-\u00d7 (ext-set aux'') (ext-set aux''')\n    where\n      aux'' : {a : W} \u2192 (fst (j\u2081 a) , snd (j\u2081 a)) \u2261 j\u2081 a\n      aux'' {w} with j\u2081 w\n      ... | h\u2081 , h\u2082 = refl\n\n      aux''' : {a : \u03a3 U (\u03bb x\u2081 \u2192 V)} \u2192 j\u2082 (fst a , snd a) \u2261 j\u2082 a\n      aux''' {u , v} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a\n       : \u03a3 (\u03a3 V (\u03bb x \u2192 W) \u2192 X) (\u03bb x \u2192 U \u2192 \u03a3 (W \u2192 Y) (\u03bb x\u2081 \u2192 V \u2192 Z))} \u2192\n      ((\u03bb p' \u2192 fst (fst (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) (snd p')) (fst p')) ,\n       (\u03bb u \u2192 (\u03bb w \u2192 snd (fst (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) w) u) , (\u03bb v \u2192 snd (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) (u , v))))\n      \u2261 a\n   aux' {j\u2081 , j\u2082} = eq-\u00d7 (ext-set aux'') (ext-set aux''')\n     where\n       aux'' : {a : \u03a3 V (\u03bb x \u2192 W)} \u2192 j\u2081 (fst a , snd a) \u2261 j\u2081 a\n       aux'' {v , w} = refl\n       aux''' : {a : U} \u2192 ((\u03bb w \u2192 fst (j\u2082 a) w) , (\u03bb v \u2192 snd (j\u2082 a) v)) \u2261 j\u2082 a\n       aux''' {u} with j\u2082 u\n       ... | h\u2081 , h\u2082 = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 (U \u2192 V) \u00d7 (Y \u2192 X) \u2192 U \u00d7 Y \u2192 Set\n\u22b8-cond \u03b1 \u03b2 (f , g) (u , y) = \u03b1 u (g y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (Y \u2192 X)) , (U \u00d7 Y) , \u22b8-cond \u03b1 \u03b2\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , p\u2083\n  where\n   h : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 T \u2192 Z)\n   h (h\u2081 , h\u2082) = (\u03bb w \u2192 g (h\u2081 (f w))) , (\u03bb t \u2192 F (h\u2082 (G t)))\n   H : \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n   H (w , t) = f w , G t\n   p\u2083 : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond \u03b1 \u03b2 u (H y) \u2192 \u22b8-cond \u03b3 \u03b4 (h u) y\n   p\u2083 {h\u2081 , h\u2082}{w , t} c c' = p\u2082 (c (p\u2081 c'))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1ae2c016233ad8fa076a9b194e880387b889b8b0","subject":"Searchable.Sum: add bit","message":"Searchable.Sum: add bit\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Sum.agda","new_file":"Search\/Searchable\/Sum.agda","new_contents":"open import Function.NP\nopen import Data.Nat using (_+_)\nimport Level as L\nimport Function.Inverse.NP as FI\nimport Function.Related as FR\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Product.NP\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Fin using (Fin)\nopen import Search.Type\nopen import Search.Searchable\n\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; module \u2261-Reasoning; cong)\n\nmodule Search.Searchable.Sum where\n\ninfixr 4 _\u228e-search_\n\n_\u228e-search_ : \u2200 {m A B} \u2192 Search m A \u2192 Search m B \u2192 Search m (A \u228e B)\n(search\u1d2c \u228e-search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_\u228e-search-ind_ : \u2200 {m p A B} {s\u1d2c : Search m A} {s\u1d2e : Search m B}\n                 \u2192 SearchInd p s\u1d2c \u2192 SearchInd p s\u1d2e \u2192 SearchInd p (s\u1d2c \u228e-search s\u1d2e)\n(Ps\u1d2c \u228e-search-ind Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _\u228e-sum_\n\n_\u228e-sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c \u228e-sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\nmodule _ {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} where\n\n    sum\u1d2c\u1d2e = sum\u1d2c \u228e-sum sum\u1d2e\n\n    _\u228e-adequate-sum_ : AdequateSum sum\u1d2c \u2192 AdequateSum sum\u1d2e \u2192 AdequateSum sum\u1d2c\u1d2e\n    (asum\u1d2c \u228e-adequate-sum asum\u1d2e) f = (Fin (sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 FI.sym (Fin-\u228e-+ _ _) \u27e9\n                                     (Fin (sum\u1d2c (f \u2218 inj\u2081)) \u228e Fin (sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 asum\u1d2c (f \u2218 inj\u2081) \u228e-cong asum\u1d2e (f \u2218 inj\u2082) \u27e9\n                                     (\u03a3 A (Fin \u2218 f \u2218 inj\u2081) \u228e \u03a3 B (Fin \u2218 f \u2218 inj\u2082))\n                                   \u2194\u27e8 FI.sym \u03a3\u228e-distrib \u27e9\n                                     \u03a3 (A \u228e B) (Fin \u2218 f)\n                                   \u220e\n      where open FR.EquationalReasoning\n\n_\u228e-\u03bc_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u228e B)\n\u03bcA \u228e-\u03bc \u03bcB = mk _ (search-ind \u03bcA \u228e-search-ind search-ind \u03bcB)\n                 (adequate-sum \u03bcA \u228e-adequate-sum adequate-sum \u03bcB)\n\nmodule _ {A B} {s\u1d2c : Search\u2081 A} {s\u1d2e : Search\u2081 B} where\n  s\u1d2c\u207a\u1d2e = s\u1d2c \u228e-search s\u1d2e\n  _\u228e-focus_ : Focus s\u1d2c \u2192 Focus s\u1d2e \u2192 Focus s\u1d2c\u207a\u1d2e\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2081 x , y) = inj\u2081 (f\u1d2c (x , y))\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2082 x , y) = inj\u2082 (f\u1d2e (x , y))\n\n  _\u228e-unfocus_ : Unfocus s\u1d2c \u2192 Unfocus s\u1d2e \u2192 Unfocus s\u1d2c\u207a\u1d2e\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2081 x) = first inj\u2081 (f\u1d2c x)\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2082 y) = first inj\u2082 (f\u1d2e y)\n\n  {-\n  _\u228e-focused_ : Focused s\u1d2c \u2192 Focused s\u1d2e \u2192 Focused {L.zero} s\u1d2c\u207a\u1d2e\n  _\u228e-focused_ f\u1d2c f\u1d2e {B} = inverses (to f\u1d2c \u228e-focus to f\u1d2e) (from f\u1d2c \u228e-unfocus from f\u1d2e) (\u21d2) (\u21d0)\n      where\n        \u21d2 : (x : \u03a3 (A \u228e {!!}) {!!}) \u2192 _\n        \u21d2 (x , y) = {!!}\n        \u21d0 : (x : s\u1d2c _\u228e_ (B \u2218 inj\u2081) \u228e s\u1d2e _\u228e_ (B \u2218 inj\u2082)) \u2192 _\n        \u21d0 (inj\u2081 x) = cong inj\u2081 {!!}\n        \u21d0 (inj\u2082 x) = cong inj\u2082 {!!}\n  -}\n\n  _\u228e-lookup_ : Lookup s\u1d2c \u2192 Lookup s\u1d2e \u2192 Lookup (s\u1d2c \u228e-search s\u1d2e)\n  (lookup\u1d2c \u228e-lookup lookup\u1d2e) (x , y) = [ lookup\u1d2c x , lookup\u1d2e y ]\n\n  _\u228e-reify_ : Reify s\u1d2c \u2192 Reify s\u1d2e \u2192 Reify (s\u1d2c \u228e-search s\u1d2e)\n  (reify\u1d2c \u228e-reify reify\u1d2e) f = (reify\u1d2c (f \u2218 inj\u2081)) , (reify\u1d2e (f \u2218 inj\u2082))\n\nsearchBit : \u2200 {m} \u2192 Search m Bit\nsearchBit _\u2219_ f = f 0b \u2219 f 1b\n\nsearchBit-ind : \u2200 {m p} \u2192 SearchInd p {m} searchBit\nsearchBit-ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\n\u03bcBit : Searchable Bit\n\u03bcBit = \u03bc-iso (FI.sym Bit\u2194\u22a4\u228e\u22a4) (\u03bc\u22a4 \u228e-\u03bc \u03bc\u22a4)\n\nfocusBit : \u2200 {a} \u2192 Focus {a} searchBit\nfocusBit (false , x) = inj\u2081 x\nfocusBit (true ,  x) = inj\u2082 x\n\nfocusedBit : Focused {L.zero} searchBit\nfocusedBit {B} = inverses focusBit unfocus (\u21d2) (\u21d0)\n  where open Searchable\u2081\u2081 searchBit-ind\n        \u21d2 : (x : \u03a3 Bit B) \u2192 _\n        \u21d2 (false , x) = \u2261.refl\n        \u21d2 (true  , x) = \u2261.refl\n        \u21d0 : (x : B 0b \u228e B 1b) \u2192 _\n        \u21d0 (inj\u2081 x) = \u2261.refl\n        \u21d0 (inj\u2082 x) = \u2261.refl\n\nlookupBit : \u2200 {a} \u2192 Lookup {a} searchBit\nlookupBit = proj\n -- -}\n -- -}\n -- -}\n","old_contents":"open import Function.NP\nopen import Data.Nat using (_+_)\nimport Level as L\nimport Function.Inverse.NP as FI\nimport Function.Related as FR\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Product.NP\nopen import Data.Sum\nopen import Data.Fin using (Fin)\nopen import Search.Type\nopen import Search.Searchable\n\nopen import Relation.Binary.Sum\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_ ; module \u2261-Reasoning; cong)\n\nmodule Search.Searchable.Sum where\n\ninfixr 4 _\u228e-search_\n\n_\u228e-search_ : \u2200 {m A B} \u2192 Search m A \u2192 Search m B \u2192 Search m (A \u228e B)\n(search\u1d2c \u228e-search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_\u228e-search-ind_ : \u2200 {m p A B} {s\u1d2c : Search m A} {s\u1d2e : Search m B}\n                 \u2192 SearchInd p s\u1d2c \u2192 SearchInd p s\u1d2e \u2192 SearchInd p (s\u1d2c \u228e-search s\u1d2e)\n(Ps\u1d2c \u228e-search-ind Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _\u228e-sum_\n\n_\u228e-sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c \u228e-sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\nmodule _ {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} where\n\n    sum\u1d2c\u1d2e = sum\u1d2c \u228e-sum sum\u1d2e\n\n    _\u228e-adequate-sum_ : AdequateSum sum\u1d2c \u2192 AdequateSum sum\u1d2e \u2192 AdequateSum sum\u1d2c\u1d2e\n    (asum\u1d2c \u228e-adequate-sum asum\u1d2e) f = (Fin (sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 FI.sym (Fin-\u228e-+ _ _) \u27e9\n                                     (Fin (sum\u1d2c (f \u2218 inj\u2081)) \u228e Fin (sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 asum\u1d2c (f \u2218 inj\u2081) \u228e-cong asum\u1d2e (f \u2218 inj\u2082) \u27e9\n                                     (\u03a3 A (Fin \u2218 f \u2218 inj\u2081) \u228e \u03a3 B (Fin \u2218 f \u2218 inj\u2082))\n                                   \u2194\u27e8 FI.sym \u03a3\u228e-distrib \u27e9\n                                     \u03a3 (A \u228e B) (Fin \u2218 f)\n                                   \u220e\n      where open FR.EquationalReasoning\n\n_\u228e-\u03bc_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u228e B)\n\u03bcA \u228e-\u03bc \u03bcB = mk _ (search-ind \u03bcA \u228e-search-ind search-ind \u03bcB)\n                 (adequate-sum \u03bcA \u228e-adequate-sum adequate-sum \u03bcB)\n\nmodule _ {A B} {s\u1d2c : Search\u2081 A} {s\u1d2e : Search\u2081 B} where\n  s\u1d2c\u207a\u1d2e = s\u1d2c \u228e-search s\u1d2e\n  _\u228e-focus_ : Focus s\u1d2c \u2192 Focus s\u1d2e \u2192 Focus s\u1d2c\u207a\u1d2e\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2081 x , y) = inj\u2081 (f\u1d2c (x , y))\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2082 x , y) = inj\u2082 (f\u1d2e (x , y))\n\n  _\u228e-unfocus_ : Unfocus s\u1d2c \u2192 Unfocus s\u1d2e \u2192 Unfocus s\u1d2c\u207a\u1d2e\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2081 x) = first inj\u2081 (f\u1d2c x)\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2082 y) = first inj\u2082 (f\u1d2e y)\n\n  {-\n  _\u228e-focused_ : Focused s\u1d2c \u2192 Focused s\u1d2e \u2192 Focused {L.zero} s\u1d2c\u207a\u1d2e\n  _\u228e-focused_ f\u1d2c f\u1d2e {B} = inverses (to f\u1d2c \u228e-focus to f\u1d2e) (from f\u1d2c \u228e-unfocus from f\u1d2e) (\u21d2) (\u21d0)\n      where\n        \u21d2 : (x : \u03a3 (A \u228e {!!}) {!!}) \u2192 _\n        \u21d2 (x , y) = {!!}\n        \u21d0 : (x : s\u1d2c _\u228e_ (B \u2218 inj\u2081) \u228e s\u1d2e _\u228e_ (B \u2218 inj\u2082)) \u2192 _\n        \u21d0 (inj\u2081 x) = cong inj\u2081 {!!}\n        \u21d0 (inj\u2082 x) = cong inj\u2082 {!!}\n  -}\n\n  _\u228e-lookup_ : Lookup s\u1d2c \u2192 Lookup s\u1d2e \u2192 Lookup (s\u1d2c \u228e-search s\u1d2e)\n  (lookup\u1d2c \u228e-lookup lookup\u1d2e) (x , y) = [ lookup\u1d2c x , lookup\u1d2e y ]\n\n  _\u228e-reify_ : Reify s\u1d2c \u2192 Reify s\u1d2e \u2192 Reify (s\u1d2c \u228e-search s\u1d2e)\n  (reify\u1d2c \u228e-reify reify\u1d2e) f = (reify\u1d2c (f \u2218 inj\u2081)) , (reify\u1d2e (f \u2218 inj\u2082))\n\n -- -}\n -- -}\n -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d3c86de1ab251011a4ee5b6de0be901bedf0c9fe","subject":"Type Isos","message":"Type Isos\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Bits using (Bit; 0b; 1b; false; true; proj)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_)\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \u22a5-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \u22a5-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \u22a5-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \u22a5-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\u22a4 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \u22a4\n\u03a3\u2261\u2194\u22a4 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : S \u2194 T\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\u22a4\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \u22a4 \u228e A)\nMaybe\u2194Lift\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\u22a4 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \u22a4\nVec0\u2194Lift\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\u22a4 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses (\u03bb { (x \u2237 xs) \u2192 x , xs }) (uncurry _\u2237_)\n             (\u03bb { (x \u2237 xs) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\u22a4\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \u22a4) _ _ \u2218 (Maybe\u2194Lift\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nBit\u2194\u22a4\u228e\u22a4 : Bit \u2194 (\u22a4 \u228e \u22a4)\nBit\u2194\u22a4\u228e\u22a4 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0b , F.const 1b ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : Bit) \u2192 _\n  \u21d0\u21d2 true = \u2261.refl\n  \u21d0\u21d2 false = \u2261.refl\n  \u21d2\u21d0 : (_ : \u22a4 \u228e \u22a4) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9\n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n-- -}\n","old_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Bits using (Bit; 0b; 1b; false; true; proj)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_)\n\nmodule CancelMaybe where\n  private\n    module _ {A : Set} where\n      just-inj : {x y : A} \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\n      just-inj \u2261.refl = \u2261.refl\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 Inverse.to f \u27e8$\u27e9 just x \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 Inverse.from f \u27e8$\u27e9 just x \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n          to = \u03bb x \u2192 Inverse.to f \u27e8$\u27e9 x\n          from = \u03bb x \u2192 Inverse.from f \u27e8$\u27e9 x\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to (just x) | tof-tot x\n          \u21d2 x | just x\u2081 | _ = x\u2081\n          \u21d2 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from (just x) | fof-tot x\n          \u21d0 x | just x\u2081 | p = x\u2081\n          \u21d0 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to (just x)\n                  | tof-tot x\n                  | from (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | Inverse.left-inverse-of f (just x)\n                  | Inverse.right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-inj (\u2261.trans (\u2261.sym (Inverse.left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong from c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \u22a5-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \u22a5-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from (just x)\n                  | fof-tot x\n                  | to (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | Inverse.right-inverse-of f (just x)\n                  | Inverse.left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-inj (\u2261.trans (\u2261.sym (Inverse.right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong to c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \u22a5-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \u22a5-elim (p \u2261.refl)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : Inverse.to f \u27e8$\u27e9 nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 Inverse.to f \u27e8$\u27e9 just x \u2262 nothing\n      tof-tot x eq with Inverse.injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : Inverse.from f \u27e8$\u27e9 nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (\u03bb x \u2192 Inverse.from f \u27e8$\u27e9 x) f-empty)) (Inverse.left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 Inverse.from f \u27e8$\u27e9 just x \u2262 nothing\n      fof-tot x eq with Inverse.injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n          to = \u03bb x \u2192 Inverse.to f \u27e8$\u27e9 x\n          from = \u03bb x \u2192 Inverse.from f \u27e8$\u27e9 x\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to (just x)\n          ... | nothing = to nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from (just x)\n          ... | nothing = from nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to (just x) | Inverse.left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to nothing | Inverse.left-inverse-of f nothing\n          \u21d0\u21d2 (just x\u2081) | nothing | p | just x | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from (just x) | Inverse.right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from nothing | Inverse.right-inverse-of f nothing\n          \u21d2\u21d0 (just x\u2081) | nothing | p | just x | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : Inverse.to g \u27e8$\u27e9 nothing \u2261 nothing\n      g-empty = \u2261.refl\n\n      iso' : A \u2194 B\n      iso' = iso g g-empty\n  CancelMaybe : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\n  CancelMaybe f = iso' f\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\u22a4 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \u22a4\n\u03a3\u2261\u2194\u22a4 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inv.Inverse.left-inverse-of f\n    right-f = Inv.Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inv.Inverse.left-inverse-of F.\u2218 f\n    right-f = Inv.Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : S \u2194 T\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\u22a4\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \u22a4 \u228e A)\nMaybe\u2194Lift\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\u22a4 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \u22a4\nVec0\u2194Lift\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\u22a4 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses (\u03bb { (x \u2237 xs) \u2192 x , xs }) (uncurry _\u2237_)\n             (\u03bb { (x \u2237 xs) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\u22a4\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \u22a4) _ _ \u2218 (Maybe\u2194Lift\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nBit\u2194\u22a4\u228e\u22a4 : Bit \u2194 (\u22a4 \u228e \u22a4)\nBit\u2194\u22a4\u228e\u22a4 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0b , F.const 1b ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : Bit) \u2192 _\n  \u21d0\u21d2 true = \u2261.refl\n  \u21d0\u21d2 false = \u2261.refl\n  \u21d2\u21d0 : (_ : \u22a4 \u228e \u22a4) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9\n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c97b0136bc51d35c39ad49db8bc8aaec3f77f724","subject":"Prove that application preserves change equivalence","message":"Prove that application preserves change equivalence\n\nAs demanded by Tillmann in\nhttps:\/\/github.com\/ps-mr\/ilc\/commit\/46789e08b7a8bbd95738f9bf14b3ff8644151772#commitcomment-5742787.\n\nOld-commit-hash: c6ca7c1a871e61d203684d8c4f36a91e24a0222d\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Equivalence.agda","new_file":"Parametric\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Parametric.Change.Equivalence where\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely.\n    --\n    -- * That should be be true for\n    --   functions using changes parametrically.\n    --\n    -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n    --   this is proved below on both contexts at once by fun-change-respects.\n    --\n    -- * Finally, change algebra operations should respect d.o.e.\n    --\n    -- Stating the general result, though, seems hard, we should\n    -- rather have lemmas proving that certain classes of functions respect this\n    -- equivalence.\n\n  module _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n    {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n    module FC = FunctionChanges A B {{CA}} {{CB}}\n    open FC using (changeAlgebra; incrementalization)\n    open FC.FunctionChange\n\n    fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n      df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n    fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n      where\n        open \u2261-Reasoning\n        open import Postulate.Extensionality\n        -- This type signature just expands the goal manually a bit.\n        lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n        -- Informally: use incrementalization on both sides and then apply\n        -- congruence.\n        lemma =\n          begin\n            f x \u229e apply df\u2081 x dx\u2081\n          \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2081)\n          \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2082)\n          \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n            (f \u229e df\u2082) (x \u229e dx\u2082)\n          \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n            f x \u229e apply df\u2082 x dx\u2082\n          \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Parametric.Change.Equivalence where\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely. That should be true for functions\n    -- using changes parametrically, for derivatives and function changes, and\n    -- for functions using only the interface to changes (including the fact\n    -- that function changes are functions). Stating the general result, though,\n    -- seems hard, we should rather have lemmas proving that certain classes of\n    -- functions respect this equivalence.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1ceac6499ba8089373bde5f40cdf695d3b78b462","subject":"Removed unused code.","message":"Removed unused code.\n\nIgnore-this: 3d26992a8909f6f5c83b3145ec136c54\n\ndarcs-hash:20100719033523-3bd4e-9c455816cf14148a738f6c8f20ee26a9b593bc8a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  -- N.B. It is not necessary to define neither the data constructor\n  -- _,_, nor the projections -- because the ATPs implement it.\n  data _\u2227_ (A B : Set) : Set where\n\n  -- Testing the data constructor and the projections.\n  postulate\n    A B     : Set\n    _,_     : A \u2192 B \u2192 A \u2227 B\n    \u2227-proj\u2081 : A \u2227 B \u2192 A\n    \u2227-proj\u2082 : A \u2227 B \u2192 B\n  {-# ATP prove _,_ #-}\n  {-# ATP prove \u2227-proj\u2081 #-}\n  {-# ATP prove \u2227-proj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to define neither the data constructors\n  -- inj\u2081 and inj\u2082 nor the disyunction elimination because the ATP\n  -- implements them.\n  data _\u2228_ (A B : Set) : Set where\n\n  -- Testing the data constructors and the elimination.\n  postulate\n    A B C : Set\n    inj\u2081  : A \u2192 A \u2228 B\n    inj\u2082  : B \u2192 A \u2228 B\n    [_,_] : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  {-# ATP prove inj\u2081 #-}\n  {-# ATP prove inj\u2082 #-}\n  {-# ATP prove [_,_] #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\nmodule ExistentialQuantifier where\n\n  data \u2203D (P : D \u2192 Set) : Set where\n\n  postulate\n    test : (d : D) \u2192 \u2203D (\u03bb e \u2192 e \u2261 d)\n  {-# ATP prove test #-}\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  -- N.B. It is not necessary to define neither the data constructor\n  -- _,_, nor the projections -- because the ATPs implement it.\n  data _\u2227_ (A B : Set) : Set where\n\n  -- Testing the data constructor and the projections.\n  postulate\n    A B     : Set\n    d e     : D\n    _,_     : A \u2192 B \u2192 A \u2227 B\n    \u2227-proj\u2081 : A \u2227 B \u2192 A\n    \u2227-proj\u2082 : A \u2227 B \u2192 B\n  {-# ATP prove _,_ #-}\n  {-# ATP prove \u2227-proj\u2081 #-}\n  {-# ATP prove \u2227-proj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to define neither the data constructors\n  -- inj\u2081 and inj\u2082 nor the disyunction elimination because the ATP\n  -- implements them.\n  data _\u2228_ (A B : Set) : Set where\n\n  -- Testing the data constructors and the elimination.\n  postulate\n    A B C : Set\n    inj\u2081  : A \u2192 A \u2228 B\n    inj\u2082  : B \u2192 A \u2228 B\n    [_,_] : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  {-# ATP prove inj\u2081 #-}\n  {-# ATP prove inj\u2082 #-}\n  {-# ATP prove [_,_] #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\nmodule ExistentialQuantifier where\n\n  data \u2203D (P : D \u2192 Set) : Set where\n\n  postulate\n    test : (d : D) \u2192 \u2203D (\u03bb e \u2192 e \u2261 d)\n  {-# ATP prove test #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d99326e90618c5ea852a5fad09f240f5d2d6675d","subject":"Only renaming.","message":"Only renaming.\n\nIgnore-this: c0e0e712590c149fdb1dd4ebdd4efff0\n\ndarcs-hash:20100804165205-3bd4e-9ff2eb215b46180dcd0688fb7cf46f72a2769e14.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Logic\/Predicate.agda","new_file":"Examples\/Logic\/Predicate.agda","new_contents":"------------------------------------------------------------------------------\n-- Predicate logic examples\n------------------------------------------------------------------------------\n\n-- This module contains some examples showing the use of the ATPs to\n-- prove theorems from predicate logic.\n\nmodule Examples.Logic.Predicate where\n\nopen import Examples.Logic.Constants\n\n------------------------------------------------------------------------------\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\npostulate\n  \u2200DI : ((x : D) \u2192 P\u00b9 x) \u2192 \u2200D P\u00b9\n  \u2200DE : \u2200D P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- This elimination rule cannot prove in Coq because in Coq we can\n  -- have empty domains. We do not have this problem because the ATPs\n  -- assume a non-empty domain.\n  \u2203DI : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203D P\u00b9\n  -- TODO: \u2203E : (x : D) \u2192 \u2203D P\u00b9 \u2192 (P\u00b9 x \u2192 Q\u00b9 x) \u2192 Q\u00b9 x\n{-# ATP prove \u2200DI #-}\n{-# ATP prove \u2200DE #-}\n{-# ATP prove \u2203DI #-}\n-- {-# ATP prove \u2203DE #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200D P\u00b9) \u2194 \u2203D (\u03bb x \u2192 \u00ac (P\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203D P\u00b9) \u2194 \u2200D (\u03bb x  \u2192 \u00ac (P\u00b9 x))\n  gDM\u2083 : \u2200D P\u00b9     \u2194 \u00ac (\u2203D (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n  gDM\u2084 : \u2203D P\u00b9     \u2194 \u00ac (\u2200D (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  ord\u2081 : \u2200D (\u03bb x \u2192 \u2200D (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u2200D (\u03bb y \u2192 \u2200D (\u03bb x \u2192 P\u00b2 x y))\n  ord\u2082 : \u2203D (\u03bb x \u2192 \u2203D (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u2203D (\u03bb y \u2192 \u2203D (\u03bb x \u2192 P\u00b2 x y))\n{-# ATP prove ord\u2081 #-}\n{-# ATP prove ord\u2082 #-}\n\n-- Quantification over a variable that does not occur can be delete.\npostulate\n  erase\u2081 : \u2200D (\u03bb _ \u2192 P\u2070) \u2194 P\u2070\n  erase\u2082 : \u2203D (\u03bb _ \u2192 P\u2070) \u2194 P\u2070\n{-# ATP prove erase\u2081 #-}\n{-# ATP prove erase\u2082 #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  dist\u2081 : \u2200D (\u03bb x \u2192 P\u00b9 x \u2227 Q\u00b9 x) \u2194 (\u2200D P\u00b9 \u2227 \u2200D Q\u00b9)\n  dist\u2082 : \u2203D (\u03bb x \u2192 P\u00b9 x \u2228 Q\u00b9 x) \u2194 (\u2203D P\u00b9 \u2228 \u2203D Q\u00b9)\n{-# ATP prove dist\u2081 #-}\n{-# ATP prove dist\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Predicate logic examples\n------------------------------------------------------------------------------\n\n-- This module contains some examples showing the use of the ATPs to\n-- prove theorems from predicate logic.\n\nmodule Examples.Logic.Predicate where\n\nopen import Examples.Logic.Constants\n\n------------------------------------------------------------------------------\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 \u2200D P\u00b9\n  \u2200E : \u2200D P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- This elimination rule cannot prove in Coq because in Coq we can\n  -- have empty domains. We do not have this problem because the ATPs\n  -- assume a non-empty domain.\n  \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203D P\u00b9\n  -- TODO: \u2203E : (x : D) \u2192 \u2203D P\u00b9 \u2192 (P\u00b9 x \u2192 Q\u00b9 x) \u2192 Q\u00b9 x\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203I #-}\n-- {-# ATP prove \u2203E #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200D P\u00b9) \u2194 \u2203D (\u03bb x \u2192 \u00ac (P\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203D P\u00b9) \u2194 \u2200D (\u03bb x  \u2192 \u00ac (P\u00b9 x))\n  gDM\u2083 : \u2200D P\u00b9     \u2194 \u00ac (\u2203D (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n  gDM\u2084 : \u2203D P\u00b9     \u2194 \u00ac (\u2200D (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  ord\u2081 : \u2200D (\u03bb x \u2192 \u2200D (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u2200D (\u03bb y \u2192 \u2200D (\u03bb x \u2192 P\u00b2 x y))\n  ord\u2082 : \u2203D (\u03bb x \u2192 \u2203D (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u2203D (\u03bb y \u2192 \u2203D (\u03bb x \u2192 P\u00b2 x y))\n{-# ATP prove ord\u2081 #-}\n{-# ATP prove ord\u2082 #-}\n\n-- Quantification over a variable that does not occur can be delete.\npostulate\n  erase\u2081 : \u2200D (\u03bb _ \u2192 P\u2070) \u2194 P\u2070\n  erase\u2082 : \u2203D (\u03bb _ \u2192 P\u2070) \u2194 P\u2070\n{-# ATP prove erase\u2081 #-}\n{-# ATP prove erase\u2082 #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  dist\u2081 : \u2200D (\u03bb x \u2192 P\u00b9 x \u2227 Q\u00b9 x) \u2194 (\u2200D P\u00b9 \u2227 \u2200D Q\u00b9)\n  dist\u2082 : \u2203D (\u03bb x \u2192 P\u00b9 x \u2228 Q\u00b9 x) \u2194 (\u2203D P\u00b9 \u2228 \u2203D Q\u00b9)\n{-# ATP prove dist\u2081 #-}\n{-# ATP prove dist\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"de31830f9c7597f3c590f9fcf6dde952b54dd49a","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 3f66058c406361406bb3a5ee91414be3\n\ndarcs-hash:20110720040502-3bd4e-75c92a13a9ca594309249684d6305b39fd088f13.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesI\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective x\u2237xs\u2261e\u2237es))) Ses\n  where\n  unfold : \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 Stream es\n  unfold = Stream-gfp\u2081 h\n\n  e : D\n  e = \u2203-proj\u2081 unfold\n\n  es : D\n  es = \u2203-proj\u2081 (\u2203-proj\u2082 unfold)\n\n  x\u2237xs\u2261e\u2237es : x \u2237 xs \u2261 e \u2237 es\n  x\u2237xs\u2261e\u2237es = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold))\n\n  Ses : Stream es\n  Ses = \u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold))\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n\n  helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n            \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 P\u2081 ws'))\n  helper\u2081 {ws} (zs , ws\u2248zs) = w' , ws' , ws\u2261w'\u2237ws' , (zs' , ws'\u2248zs')\n    where\n    unfold-\u2248 : \u2203 \u03bb w' \u2192 \u2203 \u03bb ws' \u2192 \u2203 \u03bb zs' \u2192\n               ws' \u2248 zs' \u2227 ws \u2261 w' \u2237 ws' \u2227 zs \u2261 w' \u2237 zs'\n\n    unfold-\u2248 = \u2248-gfp\u2081 ws\u2248zs\n\n    w' : D\n    w' = \u2203-proj\u2081 unfold-\u2248\n\n    ws' : D\n    ws' = \u2203-proj\u2081 (\u2203-proj\u2082 unfold-\u2248)\n\n    zs' : D\n    zs' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))\n\n    ws'\u2248zs' : ws' \u2248 zs'\n    ws'\u2248zs' = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248)))\n\n    ws\u2261w'\u2237ws' : ws \u2261 w' \u2237 ws'\n    ws\u2261w'\u2237ws' = \u2227-proj\u2081 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))))\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n\n  helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  helper\u2082   {zs} (ws , ws\u2248zs) = w' , zs' , zs\u2261w'\u2237zs' , ws' , ws'\u2248zs'\n    where\n    unfold-\u2248 : \u2203 \u03bb w' \u2192 \u2203 \u03bb ws' \u2192 \u2203 \u03bb zs' \u2192\n               ws' \u2248 zs' \u2227 ws \u2261 w' \u2237 ws' \u2227 zs \u2261 w' \u2237 zs'\n\n    unfold-\u2248 = \u2248-gfp\u2081 ws\u2248zs\n\n    w' : D\n    w' = \u2203-proj\u2081 unfold-\u2248\n\n    ws' : D\n    ws' = \u2203-proj\u2081 (\u2203-proj\u2082 unfold-\u2248)\n\n    zs' : D\n    zs' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))\n\n    ws'\u2248zs' : ws' \u2248 zs'\n    ws'\u2248zs' = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248)))\n\n    zs\u2261w'\u2237zs' : zs \u2261 w' \u2237 zs'\n    zs\u2261w'\u2237zs' = \u2227-proj\u2082 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))))\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesI\n\nopen import FOTC.Data.Stream.Type\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective x\u2237xs\u2261e\u2237es))) Ses\n  where\n  unfold : \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 Stream es\n  unfold = Stream-gfp\u2081 h\n\n  e : D\n  e = \u2203-proj\u2081 unfold\n\n  es : D\n  es = \u2203-proj\u2081 (\u2203-proj\u2082 unfold)\n\n  x\u2237xs\u2261e\u2237es : x \u2237 xs \u2261 e \u2237 es\n  x\u2237xs\u2261e\u2237es = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold))\n\n  Ses : Stream es\n  Ses = \u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold))\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203 \u03bb zs \u2192 ws \u2248 zs\n\n  helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n           \u2203 (\u03bb w' \u2192 \u2203 (\u03bb ws' \u2192 ws \u2261 w' \u2237 ws' \u2227 P\u2081 ws'))\n  helper\u2081 {ws} (zs , ws\u2248zs) = w' , ws' , ws\u2261w'\u2237ws' , (zs' , ws'\u2248zs')\n    where\n    unfold-\u2248 : \u2203 \u03bb w' \u2192 \u2203 \u03bb ws' \u2192 \u2203 \u03bb zs' \u2192\n               ws' \u2248 zs' \u2227 ws \u2261 w' \u2237 ws' \u2227 zs \u2261 w' \u2237 zs'\n\n    unfold-\u2248 = \u2248-gfp\u2081 ws\u2248zs\n\n    w' : D\n    w' = \u2203-proj\u2081 unfold-\u2248\n\n    ws' : D\n    ws' = \u2203-proj\u2081 (\u2203-proj\u2082 unfold-\u2248)\n\n    zs' : D\n    zs' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))\n\n    ws'\u2248zs' : ws' \u2248 zs'\n    ws'\u2248zs' = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248)))\n\n    ws\u2261w'\u2237ws' : ws \u2261 w' \u2237 ws'\n    ws\u2261w'\u2237ws' = \u2227-proj\u2081 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))))\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203 \u03bb ws \u2192 ws \u2248 zs\n\n  helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203 (\u03bb z' \u2192 \u2203 (\u03bb zs' \u2192 zs \u2261 z' \u2237 zs' \u2227 P\u2082 zs'))\n  helper\u2082 {zs} (ws , ws\u2248zs) = w' , zs' , zs\u2261w'\u2237zs' , ws' , ws'\u2248zs'\n    where\n    unfold-\u2248 : \u2203 \u03bb w' \u2192 \u2203 \u03bb ws' \u2192 \u2203 \u03bb zs' \u2192\n               ws' \u2248 zs' \u2227 ws \u2261 w' \u2237 ws' \u2227 zs \u2261 w' \u2237 zs'\n\n    unfold-\u2248 = \u2248-gfp\u2081 ws\u2248zs\n\n    w' : D\n    w' = \u2203-proj\u2081 unfold-\u2248\n\n    ws' : D\n    ws' = \u2203-proj\u2081 (\u2203-proj\u2082 unfold-\u2248)\n\n    zs' : D\n    zs' = \u2203-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))\n\n    ws'\u2248zs' : ws' \u2248 zs'\n    ws'\u2248zs' = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248)))\n\n    zs\u2261w'\u2237zs' : zs \u2261 w' \u2237 zs'\n    zs\u2261w'\u2237zs' = \u2227-proj\u2082 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-\u2248))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7540b79d7380f6d1c7ca7db983d990a2aab84b00","subject":"refactor grouphomomorphism","message":"refactor grouphomomorphism\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/GroupHomomorphism.agda","new_file":"lib\/Explore\/GroupHomomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.GroupHomomorphism where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP hiding (_\u2219_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Sum\n\n{-\n I had some problems with using the standard library definiton of Groups\n so I rolled my own, therefor I need some boring proofs first\n\n-}\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc    : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n  -- derived property\n  help : \u2200 x y \u2192 x \u2261 (x \u2219 y) \u2219 - y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x \u2219 \u03b5\n           \u2261\u27e8 cong (_\u2219_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x \u2219 (y \u2219 - y)\n           \u2261\u27e8 sym (assoc x y (- y)) \u27e9\n             (x \u2219 y) \u2219 (- y)\n           \u220e\n    where open \u2261-Reasoning\n\n  unique-1g : \u2200 x y \u2192 x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x \u2219 y) \u2219 - y\n                   \u2261\u27e8 cong (flip _\u2219_ (- y)) eq \u27e9\n                     y \u2219 - y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     \u03b5\n                   \u220e\n    where open \u2261-Reasoning\n\n  unique-\/ : \u2200 x y \u2192 x \u2219 y \u2261 \u03b5 \u2192 x \u2261 - y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x \u2219 y) \u2219 - y\n                  \u2261\u27e8 cong (flip _\u2219_ (- y)) eq \u27e9\n                    \u03b5 \u2219 - y\n                  \u2261\u27e8 proj\u2081 identity (- y) \u27e9\n                    - y\n                  \u220e\n    where open \u2261-Reasoning\n\nmodule _ {A B : Set}(GA : Group A)(GB : Group B) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_; \u03b5 to 0g)\n  open Group GB using (unique-1g ; unique-\/) renaming (_\u2219_ to _*_; \u03b5 to 1g; -_ to 1\/_)\n\n  GroupHomomorphism : (A \u2192 B) \u2192 Set\n  GroupHomomorphism f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n\n  module GroupHomomorphismProp {f}(f-homo : GroupHomomorphism f) where\n    f-pres-\u03b5 : f 0g \u2261 1g\n    f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n      where open \u2261-Reasoning\n            open Group GA using (identity)\n            part = f 0g * f 0g\n                 \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                   f (0g + 0g)\n                 \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                   f 0g\n                 \u220e\n\n    f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n    f-pres-inv x = unique-\/ (f (- x)) (f x) part\n      where open \u2261-Reasoning\n            open Group GA using (inverse)\n            part = f (- x) * f x\n                 \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                   f (- x + x)\n                 \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                   f 0g\n                 \u2261\u27e8 f-pres-\u03b5 \u27e9\n                   1g\n                 \u220e\n\nmodule _ {A B}(GA : Group A)(GB : Group B)\n         (f : A \u2192 B)\n         (exploreA : Explore zero A)(f-homo : GroupHomomorphism GA GB f)\n         ([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n         (explore-ext : ExploreExt exploreA)\n         where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n  open GroupHomomorphismProp GA GB f-homo\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n  -}\n\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n  module _ {X}(z : X)(op : X \u2192 X \u2192 X)\n           (sui : \u2200 k \u2192 StableUnder' exploreA z op (flip (Group._\u2219_ GA) k))\n    where\n  -- this proof isn't actually any hard..\n    thm : \u2200 (O : B \u2192 X) m\u2080 m\u2081 \u2192 exploreA z op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 exploreA z op (\u03bb x \u2192 O (f x * m\u2081))\n    thm O m\u2080 m\u2081 = explore (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma1 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) (O \u2218 f) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma2 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n     where\n      open \u2261-Reasoning\n      explore = exploreA z op\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.GroupHomomorphism where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP hiding (_\u2219_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc    : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)\n         (f : A \u2192 B)\n         (exploreA : Explore zero A)(f-homo : GroupHomomorphism GA GB f)\n         ([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n         (explore-ext : ExploreExt exploreA)\n         where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n  -}\n\n  {-\n   I had some problems with using the standard library definiton of Groups\n   so I rolled my own, therefor I need some boring proofs first\n\n  -}\n\n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n  module _ {X}(z : X)(op : X \u2192 X \u2192 X)\n           (sui : \u2200 k \u2192 StableUnder' exploreA z op (flip (Group._\u2219_ GA) k))\n    where\n  -- this proof isn't actually any hard..\n    thm : \u2200 (O : B \u2192 X) m\u2080 m\u2081 \u2192 exploreA z op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 exploreA z op (\u03bb x \u2192 O (f x * m\u2081))\n    thm O m\u2080 m\u2081 = explore (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma1 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) (O \u2218 f) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma2 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n     where\n      open \u2261-Reasoning\n      explore = exploreA z op\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0a11660ef086b316b40c304a2804311fade09a32","subject":"Minor changes.","message":"Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/IJK.agda","new_file":"notes\/FOT\/FOTC\/IJK.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion functions i, j, and k.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.IJK where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n\ndata \u2115 : Set where\n  z : \u2115\n  s : \u2115 \u2192 \u2115\n\n-- Conversion functions from\/to \u2115 and N.\ni : \u2115 \u2192 D\ni z     = zero\ni (s n) = succ\u2081 (i n)\n\nj : (n : \u2115) \u2192 N (i n)\nj z     = nzero\nj (s n) = nsucc (j n)\n\nk : {n : D} \u2192 N n \u2192 \u2115\nk nzero      = z\nk (nsucc Nn) = s (k Nn)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nz   + n = n\ns m + n = s (m + n)\n","old_contents":"------------------------------------------------------------------------------\n-- Conversion functions i, j, and k.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.IJK where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n\n-- The inductive natural numbers.\ndata \u2115 : Set where\n  z : \u2115\n  s : \u2115 \u2192 \u2115\n\n-- Conversion functions\ni : \u2115 \u2192 D\ni z     = zero\ni (s n) = succ\u2081 (i n)\n\nj : (n : \u2115) \u2192 N (i n)\nj z     = nzero\nj (s n) = nsucc (j n)\n\nk : {n : D} \u2192 N n \u2192 \u2115\nk nzero      = z\nk (nsucc Nn) = s (k Nn)\n\n-- Addition for \u2115\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nz   + n = n\ns m + n = s (m + n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7fde932b68b4d631c9007721d6ea856d68865033","subject":"ExplicitNil: extend correctness of optimized derivation to support bag union and difference","message":"ExplicitNil: extend correctness of optimized derivation\nto support bag union and difference\n\nOld-commit-hash: 40f74c6acb6b271d0654bbc62228aba5fbfc8edf\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import TaggedDeltaTypes\n\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nopen import Relation.Binary.Core using (Decidable)\nopen import Relation.Nullary.Core using (yes ; no)\n\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n  where ext = extensionality\n\nproj-H : \u2200 {\u0393 : Context} {\u03c1 : \u0394Env \u0393} {us vs} \u2192\n  Honest \u03c1 (FV-union us vs) \u2192 Honest \u03c1 us \u00d7 Honest \u03c1 vs\nproj-H {\u2205} {us = \u2205} {vs = \u2205} clearly = clearly , clearly\nproj-H {us = alter us} {alter vs} (alter H) =\n  let uss , vss = proj-H H in alter uss , alter vss\nproj-H {us = alter us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  alter uss , abide eq vss\nproj-H {us = abide us} {alter vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , alter vss\nproj-H {us = abide us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , abide eq vss\n\n-- Equivalence proofs are unique\nequivalence-unique : \u2200 {A : Set} {a b : A} \u2192 \u2200 {p q : a \u2261 b} \u2192 p \u2261 q\nequivalence-unique {p = refl} {refl} = refl\n\n-- Product of singletons are singletons (for example)\nproduct-unique : \u2200 {A : Set} {B : A \u2192 Set} \u2192\n  (\u2200 {a b : A} \u2192 a \u2261 b) \u2192\n  (\u2200 {a : A} {c d : B a} \u2192 c \u2261 d) \u2192\n  (\u2200 {p q : \u03a3 A B} \u2192 p \u2261 q)\nproduct-unique {A} {B} lhs rhs {a , c} {b , d}\n  rewrite lhs {a} {b} = cong (_,_ b) rhs\n\n-- Validity proofs are (extensionally) unique (as functions)\nvalidity-unique : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} {dv : \u0394Val \u03c4} \u2192\n  \u2200 {p q : valid v dv} \u2192 p \u2261 q\nvalidity-unique {nats} = equivalence-unique\nvalidity-unique {bags} = refl\nvalidity-unique {\u03c3 \u21d2 \u03c4} =  ext\u00b3 (\u03bb v dv R[v,dv] \u2192\n  product-unique validity-unique equivalence-unique)\n\nstabilityVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (select-just x)) \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\n\nstabilityVar {x = this} (abide proof _) = proof\nstabilityVar {x = that y} (alter H) = stabilityVar {x = y} H\n\nstability : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV t)) \u2192\n    \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Boilerplate begins\nstabilityAbs : \u2200 {\u03c3 \u03c4 \u0393} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV (abs t))) \u2192\n  (v : \u27e6 \u03c3 \u27e7) \u2192\n    \u27e6 t \u27e7 (v \u2022 ignore \u03c1) \u2295\n    \u27e6 derive t \u27e7\u0394 (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (v \u2022 ignore \u03c1)\nstabilityAbs {t = t} {\u03c1} H v with FV t | inspect FV t\n... | abide vars | [ case0 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case0 = abide v\u2295[u\u229dv]=u H\n... | alter vars | [ case1 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case1 = alter H\n-- Boilerplate ends\n\nstability {t = nat n} H = refl\nstability {t = bag b} H = b++\u2205=b\nstability {t = union s t} {\u03c1} H =\n  let\n    a = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    da = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n  in\n    begin\n      (a ++ b) ++ (da ++ db)\n    \u2261\u27e8 [a++b]++[c++d]=[a++c]++[b++d] \u27e9\n      (a ++ da) ++ (b ++ db)\n    \u2261\u27e8 cong\u2082 _++_ (stability {t = s} Hs) (stability {t = t} Ht) \u27e9\n      a ++ b\n    \u220e where open \u2261-Reasoning\nstability {t = diff s t} {\u03c1} H =\n  let\n    a = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    da = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n  in\n    begin\n      (a \\\\ b) ++ (da \\\\ db)\n    \u2261\u27e8 [a\\\\b]++[c\\\\d]=[a++c]\\\\[b++d] \u27e9\n      (a ++ da) \\\\ (b ++ db)\n    \u2261\u27e8 cong\u2082 _\\\\_ (stability {t = s} Hs) (stability {t = t} Ht) \u27e9\n      a \\\\ b\n    \u220e where open \u2261-Reasoning\nstability {t = var x} H = stabilityVar H\nstability {t = abs t} {\u03c1} H = extensionality (stabilityAbs {t = t} H)\nstability {t = app s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n  in\n    begin\n      f v \u2295 df v dv (validity {t = t})\n    \u2261\u27e8 sym (corollary s t) \u27e9\n      (f \u2295 df) (v \u2295 dv)\n    \u2261\u27e8 stability {t = s} Hs \u27e8$\u27e9 stability {t = t} Ht \u27e9\n      f v\n    \u220e where open \u2261-Reasoning\nstability {t = add s t} H =\n  let Hs , Ht = proj-H H\n  in cong\u2082 _+_ (stability {t = s} Hs) (stability {t = t} Ht)\nstability {t = map s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n    map = mapBag\n  in\n  begin\n    map f b \u2295 (map (f \u2295 df) (b \u2295 db) \u229d map f b)\n  \u2261\u27e8 b++[d\\\\b]=d \u27e9\n    map (f \u2295 df) (b \u2295 db)\n  \u2261\u27e8 cong\u2082 map (stability {t = s} Hs) (stability {t = t} Ht) \u27e9\n    map f b\n  \u220e where open \u2261-Reasoning\n\neq-map : \u2200 {\u0393}\n  (s    : Term \u0393 (nats \u21d2 nats))\n  (t    : Term \u0393 bags)\n  (\u03c1    : \u0394Env \u0393)\n  (H    : Honest \u03c1 (FV s)) \u2192\n    \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n  \u2261 \u27e6 \u0394map\u2080 s (derive s) t (derive t) \u27e7\u0394 \u03c1 (unrestricted (map s t))\n\neq-map s t \u03c1 H =\n  let\n    ds = derive s\n    dt = derive t\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 (unrestricted t)\n\n    eq1 : \u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (unrestricted (map s t))\n        \u2261 mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n    eq1 = refl\n\n    eq2 : mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n        \u2261 mapBag f dv\n    eq2 = trans\n      (cong (\u03bb hole \u2192 hole \u229d mapBag f v) (trans\n        (cong (\u03bb hole \u2192 mapBag hole (v \u2295 dv)) (stability {t = s} H))\n        map-over-++))\n      [b++d]\\\\b=d\n\n    eq3 : mapBag f dv\n        \u2261 \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n    eq3 = refl\n\n  in sym (trans eq1 (trans eq2 eq3))\n\n-- Vars test\nnone-selected? : \u2200 {\u0393} \u2192 (vs : Vars \u0393) \u2192 (vs \u2261 select-none) \u228e \u22a4\nnone-selected? \u2205 = inj\u2081 refl\nnone-selected? (abide vs) = inj\u2082 tt\nnone-selected? (alter vs) with none-selected? vs\n... | inj\u2081 vs=\u2205 rewrite vs=\u2205 = inj\u2081 refl\n... | inj\u2082 _ = inj\u2082 tt\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 select-none) \u228e \u22a4\nclosed? t = none-selected? (FV t)\n\nvacuous-honesty : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} \u2192 Honest \u03c1 select-none\nvacuous-honesty {\u2205} {\u2205} = clearly\nvacuous-honesty {\u03c4 \u2022 \u0393} {cons _ _ _ \u03c1} = alter (vacuous-honesty {\u03c1 = \u03c1})\n\n-- Immunity of closed terms to dishonest environments\nimmune : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  (FV t \u2261 select-none) \u2192 \u2200 {\u03c1} \u2192 Honest \u03c1 (FV t)\nimmune {t = t} eq rewrite eq = vacuous-honesty\n\n-- Ineffectual first optimization step\nderive1 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive1 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive t)\n... | inj\u2082 tt = derive (map s t)\nderive1 others = derive others\n\nvalid1 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive1 t is-valid-for \u03c1\nvalid1 (nat n) = tt\nvalid1 (bag b) = tt\nvalid1 (union s t) = cons (unrestricted s) (unrestricted t) tt tt\nvalid1 (diff s t) = cons (unrestricted s) (unrestricted t) tt tt\nvalid1 (var x) = tt\nvalid1 (abs t) = \u03bb _ _ _ \u2192 unrestricted t\nvalid1 (app s t) =\n  cons (unrestricted s) (unrestricted t) (validity {t = t}) tt\nvalid1 (add s t) =\n  cons (unrestricted s) (unrestricted t) tt tt\nvalid1 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (unrestricted s) (unrestricted t) tt tt\n... | inj\u2081 if-closed =\n  cons (unrestricted t) (immune {t = s} if-closed) tt tt\n\ncorrect1 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive1 t \u27e7\u0394 \u03c1 (valid1 t) \u2261 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrect1 {t = nat n} = refl\ncorrect1 {t = bag b} = refl\ncorrect1 {t = var x} = refl\ncorrect1 {t = abs t} = refl\ncorrect1 {t = app s t} = refl\ncorrect1 {t = union s t} = refl\ncorrect1 {t = diff s t} = refl\ncorrect1 {t = add s t} = refl\ncorrect1 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt = refl\n... | inj\u2081 if-closed = eq-map s t \u03c1 (immune {t = s} if-closed)\n\n-- derive2 = derive1 + congruence\nderive2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive2 (abs t) = \u0394abs (derive2 t)\nderive2 (app s t) = \u0394app (derive2 s) t (derive2 t)\nderive2 (add s t) = \u0394add (derive2 s) (derive2 t)\nderive2 (union s t) = \u0394union (derive2 s) (derive2 t)\nderive2 (diff s t) = \u0394diff (derive2 s) (derive2 t)\nderive2 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive2 t)\n... | inj\u2082 tt = \u0394map\u2080 s (derive2 s) t (derive2 t)\nderive2 others = derive others\n\nvalid2 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive2 t is-valid-for \u03c1\ncorrect2 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t) \u2261 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\nvalid2 (nat n) = tt\nvalid2 (bag b) = tt\nvalid2 (var x) = tt\nvalid2 (abs t) = \u03bb _ _ _ \u2192 valid2 t\nvalid2 (app s t) {\u03c1} = cons (valid2 s) (valid2 t) V tt\n  where\n  V : valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t))\n  V rewrite correct2 {t = t} {\u03c1} = validity {t = t} {\u03c1}\nvalid2 (add s t) = cons (valid2 s) (valid2 t) tt tt\nvalid2 (union s t) = cons (valid2 s) (valid2 t) tt tt\nvalid2 (diff s t) = cons (valid2 s) (valid2 t) tt tt\nvalid2 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (valid2 s) (valid2 t) tt tt\n... | inj\u2081 if-closed =\n  cons (valid2 t) (immune {t = s} if-closed) tt tt\n\ncorrect2 {t = nat n} = refl\ncorrect2 {t = bag b} = refl\ncorrect2 {t = var x} = refl\ncorrect2 {t = abs t} = ext\u00b3 (\u03bb _ _ _ \u2192 correct2 {t = t})\ncorrect2 {t = app {\u03c3} {\u03c4} s t} {\u03c1} = \u2245-to-\u2261 eq-all where\n  open import Relation.Binary.HeterogeneousEquality hiding (cong\u2082)\n  import Relation.Binary.HeterogeneousEquality as HET\n  df0 : \u0394Val (\u03c3 \u21d2 \u03c4)\n  df0 = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n  df2 : \u0394Val (\u03c3 \u21d2 \u03c4)\n  df2 = \u27e6 derive2 s \u27e7\u0394 \u03c1 (valid2 s)\n  dv0 : \u0394Val \u03c3\n  dv0 = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  dv2 : \u0394Val \u03c3\n  dv2 = \u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t)\n  v   : \u27e6 \u03c3 \u27e7\n  v   = \u27e6 t \u27e7 (ignore \u03c1)\n  eq-df : df2 \u2261 df0\n  eq-df = correct2 {t = s}\n  eq-dv : dv2 \u2261 dv0\n  eq-dv = correct2 {t = t}\n  eq-all : df2 v dv2 (caddr (valid2 (app s t) {\u03c1}))\n         \u2245 df0 v dv0 (validity {t = t} {\u03c1})\n  eq-all rewrite eq-dv = HET.cong\n    (\u03bb hole \u2192 hole v dv0 (validity {t = t} {\u03c1})) (\u2261-to-\u2245 eq-df)\n    -- found without intuition by trial-and-error\ncorrect2 {t = add s t} {\u03c1} =\n  let\n    cs = correct2 {t = s} {\u03c1}\n    ct = correct2 {t = t} {\u03c1}\n  in cong\u2082 _,_ (cong\u2082 _+_ (cong proj\u2081 cs) (cong proj\u2081 ct))\n               (cong\u2082 _+_ (cong proj\u2082 cs) (cong proj\u2082 ct))\ncorrect2 {t = union s t} {\u03c1} =\n  cong\u2082 _++_ (correct2 {t = s} {\u03c1}) (correct2 {t = t} {\u03c1})\ncorrect2 {t = diff s t} {\u03c1} =\n  cong\u2082 _\\\\_ (correct2 {t = s} {\u03c1}) (correct2 {t = t} {\u03c1})\ncorrect2 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n  in\n    cong\u2082 (\u03bb h1 h2 \u2192 mapBag (f \u2295 h1) (b \u2295 h2) \\\\ mapBag f b)\n    (correct2 {t = s}) (correct2 {t = t})\n... | inj\u2081 if-closed = trans\n  (cong (mapBag (\u27e6 s \u27e7 (ignore \u03c1))) (correct2 {t = t}))\n  (eq-map s t \u03c1 (immune {t = s} if-closed))\n\n---------------\n-- Example 3 --\n---------------\n\n-- [+1] = \u03bb x \u2192 x + 1\n[+1] : \u2200 {\u0393} \u2192 Term \u0393 (nats \u21d2 nats)\n[+1] = (abs (add (var this) (nat 1)))\n\n-- inc = \u03bb bag \u2192 map [+1] bag\ninc : Term \u2205 (bags \u21d2 bags)\ninc = abs (map [+1] (var this))\n\n-- Program transformation in example 3\n-- Sadly, `inc` is not executable, for `Bag` is an abstract type\n-- whose existence we postulated.\n--\n-- derive2 inc = \u03bb bag dbag \u2192 map [+1] dbag\nexample-3 : derive2 inc \u2261 \u0394abs (\u0394map\u2081 [+1] (\u0394var this))\nexample-3 = refl\n\n-- Correctness of optimized derivation on `inc`\n--\n-- \u27e6 derive2 inc \u27e7 <embedded-proof> = \u27e6 derive inc \u27e7 <another-proof>\nexample-3-correct :\n  \u27e6 derive2 inc \u27e7\u0394 \u2205 (valid2 inc) \u2261 \u27e6 derive inc \u27e7\u0394 \u2205 (unrestricted inc)\nexample-3-correct = correct2 {t = inc} {\u2205}\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import TaggedDeltaTypes\n\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nopen import Relation.Binary.Core using (Decidable)\nopen import Relation.Nullary.Core using (yes ; no)\n\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n  where ext = extensionality\n\nproj-H : \u2200 {\u0393 : Context} {\u03c1 : \u0394Env \u0393} {us vs} \u2192\n  Honest \u03c1 (FV-union us vs) \u2192 Honest \u03c1 us \u00d7 Honest \u03c1 vs\nproj-H {\u2205} {us = \u2205} {vs = \u2205} clearly = clearly , clearly\nproj-H {us = alter us} {alter vs} (alter H) =\n  let uss , vss = proj-H H in alter uss , alter vss\nproj-H {us = alter us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  alter uss , abide eq vss\nproj-H {us = abide us} {alter vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , alter vss\nproj-H {us = abide us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , abide eq vss\n\n-- Equivalence proofs are unique\nequivalence-unique : \u2200 {A : Set} {a b : A} \u2192 \u2200 {p q : a \u2261 b} \u2192 p \u2261 q\nequivalence-unique {p = refl} {refl} = refl\n\n-- Product of singletons are singletons (for example)\nproduct-unique : \u2200 {A : Set} {B : A \u2192 Set} \u2192\n  (\u2200 {a b : A} \u2192 a \u2261 b) \u2192\n  (\u2200 {a : A} {c d : B a} \u2192 c \u2261 d) \u2192\n  (\u2200 {p q : \u03a3 A B} \u2192 p \u2261 q)\nproduct-unique {A} {B} lhs rhs {a , c} {b , d}\n  rewrite lhs {a} {b} = cong (_,_ b) rhs\n\n-- Validity proofs are (extensionally) unique (as functions)\nvalidity-unique : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} {dv : \u0394Val \u03c4} \u2192\n  \u2200 {p q : valid v dv} \u2192 p \u2261 q\nvalidity-unique {nats} = equivalence-unique\nvalidity-unique {bags} = refl\nvalidity-unique {\u03c3 \u21d2 \u03c4} =  ext\u00b3 (\u03bb v dv R[v,dv] \u2192\n  product-unique validity-unique equivalence-unique)\n\nstabilityVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (select-just x)) \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\n\nstabilityVar {x = this} (abide proof _) = proof\nstabilityVar {x = that y} (alter H) = stabilityVar {x = y} H\n\nstability : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV t)) \u2192\n    \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Boilerplate begins\nstabilityAbs : \u2200 {\u03c3 \u03c4 \u0393} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV (abs t))) \u2192\n  (v : \u27e6 \u03c3 \u27e7) \u2192\n    \u27e6 t \u27e7 (v \u2022 ignore \u03c1) \u2295\n    \u27e6 derive t \u27e7\u0394 (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (v \u2022 ignore \u03c1)\nstabilityAbs {t = t} {\u03c1} H v with FV t | inspect FV t\n... | abide vars | [ case0 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case0 = abide v\u2295[u\u229dv]=u H\n... | alter vars | [ case1 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case1 = alter H\n-- Boilerplate ends\n\nstability {t = nat n} H = refl\nstability {t = bag b} H = b++\u2205=b\nstability {t = var x} H = stabilityVar H\nstability {t = abs t} {\u03c1} H = extensionality (stabilityAbs {t = t} H)\nstability {t = app s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n  in\n    begin\n      f v \u2295 df v dv (validity {t = t})\n    \u2261\u27e8 sym (corollary s t) \u27e9\n      (f \u2295 df) (v \u2295 dv)\n    \u2261\u27e8 stability {t = s} Hs \u27e8$\u27e9 stability {t = t} Ht \u27e9\n      f v\n    \u220e where open \u2261-Reasoning\nstability {t = add s t} H =\n  let Hs , Ht = proj-H H\n  in cong\u2082 _+_ (stability {t = s} Hs) (stability {t = t} Ht)\nstability {t = map s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n    map = mapBag\n  in\n  begin\n    map f b \u2295 (map (f \u2295 df) (b \u2295 db) \u229d map f b)\n  \u2261\u27e8 b++[d\\\\b]=d \u27e9\n    map (f \u2295 df) (b \u2295 db)\n  \u2261\u27e8 cong\u2082 map (stability {t = s} Hs) (stability {t = t} Ht) \u27e9\n    map f b\n  \u220e where open \u2261-Reasoning\n\neq-map : \u2200 {\u0393}\n  (s    : Term \u0393 (nats \u21d2 nats))\n  (t    : Term \u0393 bags)\n  (\u03c1    : \u0394Env \u0393)\n  (H    : Honest \u03c1 (FV s)) \u2192\n    \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n  \u2261 \u27e6 \u0394map\u2080 s (derive s) t (derive t) \u27e7\u0394 \u03c1 (unrestricted (map s t))\n\neq-map s t \u03c1 H =\n  let\n    ds = derive s\n    dt = derive t\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 (unrestricted t)\n\n    eq1 : \u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (unrestricted (map s t))\n        \u2261 mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n    eq1 = refl\n\n    eq2 : mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n        \u2261 mapBag f dv\n    eq2 = trans\n      (cong (\u03bb hole \u2192 hole \u229d mapBag f v) (trans\n        (cong (\u03bb hole \u2192 mapBag hole (v \u2295 dv)) (stability {t = s} H))\n        map-over-++))\n      [b++d]\\\\b=d\n\n    eq3 : mapBag f dv\n        \u2261 \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n    eq3 = refl\n\n  in sym (trans eq1 (trans eq2 eq3))\n\n-- Vars test\nnone-selected? : \u2200 {\u0393} \u2192 (vs : Vars \u0393) \u2192 (vs \u2261 select-none) \u228e \u22a4\nnone-selected? \u2205 = inj\u2081 refl\nnone-selected? (abide vs) = inj\u2082 tt\nnone-selected? (alter vs) with none-selected? vs\n... | inj\u2081 vs=\u2205 rewrite vs=\u2205 = inj\u2081 refl\n... | inj\u2082 _ = inj\u2082 tt\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 select-none) \u228e \u22a4\nclosed? t = none-selected? (FV t)\n\nvacuous-honesty : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} \u2192 Honest \u03c1 select-none\nvacuous-honesty {\u2205} {\u2205} = clearly\nvacuous-honesty {\u03c4 \u2022 \u0393} {cons _ _ _ \u03c1} = alter (vacuous-honesty {\u03c1 = \u03c1})\n\n-- Immunity of closed terms to dishonest environments\nimmune : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  (FV t \u2261 select-none) \u2192 \u2200 {\u03c1} \u2192 Honest \u03c1 (FV t)\nimmune {t = t} eq rewrite eq = vacuous-honesty\n\n-- Ineffectual first optimization step\nderive1 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive1 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive t)\n... | inj\u2082 tt = derive (map s t)\nderive1 others = derive others\n\nvalid1 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive1 t is-valid-for \u03c1\nvalid1 (nat n) = tt\nvalid1 (bag b) = tt\nvalid1 (var x) = tt\nvalid1 (abs t) = \u03bb _ _ _ \u2192 unrestricted t\nvalid1 (app s t) =\n  cons (unrestricted s) (unrestricted t) (validity {t = t}) tt\nvalid1 (add s t) =\n  cons (unrestricted s) (unrestricted t) tt tt\nvalid1 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (unrestricted s) (unrestricted t) tt tt\n... | inj\u2081 if-closed =\n  cons (unrestricted t) (immune {t = s} if-closed) tt tt\n\ncorrect1 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive1 t \u27e7\u0394 \u03c1 (valid1 t) \u2261 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrect1 {t = nat n} = refl\ncorrect1 {t = bag b} = refl\ncorrect1 {t = var x} = refl\ncorrect1 {t = abs t} = refl\ncorrect1 {t = app s t} = refl\ncorrect1 {t = add s t} = refl\ncorrect1 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt = refl\n... | inj\u2081 if-closed = eq-map s t \u03c1 (immune {t = s} if-closed)\n\n-- derive2 = derive1 + congruence\nderive2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive2 (abs t) = \u0394abs (derive2 t)\nderive2 (app s t) = \u0394app (derive2 s) t (derive2 t)\nderive2 (add s t) = \u0394add (derive2 s) (derive2 t)\nderive2 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive2 t)\n... | inj\u2082 tt = \u0394map\u2080 s (derive2 s) t (derive2 t)\nderive2 others = derive others\n\nvalid2 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive2 t is-valid-for \u03c1\ncorrect2 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t) \u2261 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\nvalid2 (nat n) = tt\nvalid2 (bag b) = tt\nvalid2 (var x) = tt\nvalid2 (abs t) = \u03bb _ _ _ \u2192 valid2 t\nvalid2 (app s t) {\u03c1} = cons (valid2 s) (valid2 t) V tt\n  where\n  V : valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t))\n  V rewrite correct2 {t = t} {\u03c1} = validity {t = t} {\u03c1}\nvalid2 (add s t) = cons (valid2 s) (valid2 t) tt tt\nvalid2 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (valid2 s) (valid2 t) tt tt\n... | inj\u2081 if-closed =\n  cons (valid2 t) (immune {t = s} if-closed) tt tt\n\ncorrect2 {t = nat n} = refl\ncorrect2 {t = bag b} = refl\ncorrect2 {t = var x} = refl\ncorrect2 {t = abs t} = ext\u00b3 (\u03bb _ _ _ \u2192 correct2 {t = t})\ncorrect2 {t = app {\u03c3} {\u03c4} s t} {\u03c1} = \u2245-to-\u2261 eq-all where\n  open import Relation.Binary.HeterogeneousEquality hiding (cong\u2082)\n  import Relation.Binary.HeterogeneousEquality as HET\n  df0 : \u0394Val (\u03c3 \u21d2 \u03c4)\n  df0 = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n  df2 : \u0394Val (\u03c3 \u21d2 \u03c4)\n  df2 = \u27e6 derive2 s \u27e7\u0394 \u03c1 (valid2 s)\n  dv0 : \u0394Val \u03c3\n  dv0 = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  dv2 : \u0394Val \u03c3\n  dv2 = \u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t)\n  v   : \u27e6 \u03c3 \u27e7\n  v   = \u27e6 t \u27e7 (ignore \u03c1)\n  eq-df : df2 \u2261 df0\n  eq-df = correct2 {t = s}\n  eq-dv : dv2 \u2261 dv0\n  eq-dv = correct2 {t = t}\n  eq-all : df2 v dv2 (caddr (valid2 (app s t) {\u03c1}))\n         \u2245 df0 v dv0 (validity {t = t} {\u03c1})\n  eq-all rewrite eq-dv = HET.cong\n    (\u03bb hole \u2192 hole v dv0 (validity {t = t} {\u03c1})) (\u2261-to-\u2245 eq-df)\n    -- found without intuition by trial-and-error\ncorrect2 {t = add s t} {\u03c1} =\n  let\n    cs = correct2 {t = s} {\u03c1}\n    ct = correct2 {t = t} {\u03c1}\n  in cong\u2082 _,_ (cong\u2082 _+_ (cong proj\u2081 cs) (cong proj\u2081 ct))\n               (cong\u2082 _+_ (cong proj\u2082 cs) (cong proj\u2082 ct))\ncorrect2 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt = \n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n  in\n    cong\u2082 (\u03bb h1 h2 \u2192 mapBag (f \u2295 h1) (b \u2295 h2) \\\\ mapBag f b)\n    (correct2 {t = s}) (correct2 {t = t})\n... | inj\u2081 if-closed = trans\n  (cong (mapBag (\u27e6 s \u27e7 (ignore \u03c1))) (correct2 {t = t}))\n  (eq-map s t \u03c1 (immune {t = s} if-closed))\n\n---------------\n-- Example 3 --\n---------------\n\n-- [+1] = \u03bb x \u2192 x + 1\n[+1] : \u2200 {\u0393} \u2192 Term \u0393 (nats \u21d2 nats)\n[+1] = (abs (add (var this) (nat 1)))\n\n-- inc = \u03bb bag \u2192 map [+1] bag\ninc : Term \u2205 (bags \u21d2 bags)\ninc = abs (map [+1] (var this))\n\n-- Program transformation in example 3\n-- Sadly, `inc` is not executable, for `Bag` is an abstract type\n-- whose existence we postulated.\n--\n-- derive2 inc = \u03bb bag dbag \u2192 map [+1] dbag\nexample-3 : derive2 inc \u2261 \u0394abs (\u0394map\u2081 [+1] (\u0394var this))\nexample-3 = refl\n\n-- Correctness of optimized derivation on `inc`\n--\n-- \u27e6 derive2 inc \u27e7 <embedded-proof> = \u27e6 derive inc \u27e7 <another-proof>\nexample-3-correct :\n  \u27e6 derive2 inc \u27e7\u0394 \u2205 (valid2 inc) \u2261 \u27e6 derive inc \u27e7\u0394 \u2205 (unrestricted inc)\nexample-3-correct = correct2 {t = inc} {\u2205}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d3f497c90ef2cbc4c034e571f3eb1595410f1084","subject":"fix spaces","message":"fix spaces\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      P : U \u2192 [0,1]\n      sumP\u22611 : sumP P allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs))\n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field     \n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248 \n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n  \n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[ \n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[ \n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n \n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  > \n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      P : U \u2192 [0,1]\n      sumP\u22611 : sumP P allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e4762e8af8591526fcb5395b9c63888ed89cadd2","subject":"agda: Remove code duplication","message":"agda: Remove code duplication\n\nThe current code defines two overloads of the same function independently, with\nduplicated documentation. However, one can be defined in terms of the other.\n\nNote: I could not fully test this code, since it is not yet clear which file is\nthe main one. I typechecked Syntax.Language.Atlas, I hope this was enough.\n\nI couldn't test this change because of #41: without fixing it, either I risk\nbreaking code, or I can't fix issues I spot.\n\nOld-commit-hash: b481a293cbc77ca4b2981761c15e7771d95216af\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Type\/Plotkin.agda","new_file":"Syntax\/Type\/Plotkin.agda","new_contents":"module Syntax.Type.Plotkin where\n\n-- Types for language description \u00e0 la Plotkin (LCF as PL)\n--\n-- Given base types, we build function types.\n\ninfixr 5 _\u21d2_\n\ndata Type (B : Set {- of base types -}) : Set where\n  base : (\u03b9 : B) \u2192 Type B\n  _\u21d2_ : (\u03c3 : Type B) \u2192 (\u03c4 : Type B) \u2192 Type B\n\n-- Lift (\u0394 : B \u2192 Type B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift\u2081 : \u2200 {B} \u2192 (B \u2192 Type B) \u2192 (Type B \u2192 Type B)\nlift\u2081 f (base \u03b9) = f \u03b9\nlift\u2081 f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift\u2081 f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n\n-- Note: the above is monadic bind with a different argument order.\n\nopen import Function\n\n-- Variant of lift\u2081 for (\u0394 : B \u2192 B).\nlift\u2080 : \u2200 {B} \u2192 (B \u2192 B) \u2192 (Type B \u2192 Type B)\nlift\u2080 f = lift\u2081 $ base \u2218 f\n-- If lift\u2081 is a monadic bind, this is fmap,\n-- and base is return.\n--\n-- Similarly, for collections map can be defined from flatMap, like lift\u2080 can be\n-- defined in terms of lift\u2081.\n","old_contents":"module Syntax.Type.Plotkin where\n\n-- Types for language description \u00e0 la Plotkin (LCF as PL)\n--\n-- Given base types, we build function types.\n\ninfixr 5 _\u21d2_\n\ndata Type (B : Set {- of base types -}) : Set where\n  base : (\u03b9 : B) \u2192 Type B\n  _\u21d2_ : (\u03c3 : Type B) \u2192 (\u03c4 : Type B) \u2192 Type B\n\n-- Lift (\u0394 : B \u2192 B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift\u2080 : \u2200 {B} \u2192 (B \u2192 B) \u2192 (Type B \u2192 Type B)\nlift\u2080 f (base \u03b9) = base (f \u03b9)\nlift\u2080 f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift\u2080 f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n\n-- Lift (\u0394 : B \u2192 Type B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift\u2081 : \u2200 {B} \u2192 (B \u2192 Type B) \u2192 (Type B \u2192 Type B)\nlift\u2081 f (base \u03b9) = f \u03b9\nlift\u2081 f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift\u2081 f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5cd1142501fcdb87d0821cc80e8dfa92f1d8618d","subject":"If distance between x y are zero, then x = y","message":"If distance between x y are zero, then x = y\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Distance.agda","new_file":"lib\/Data\/Nat\/Distance.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm {x} {1} {x} | +-assoc-comm {y} {1} {y} = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm {x} {1} {k} | q | ! +-assoc-comm {x} {1} {k} | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm {1} {y} {k} | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\ndist\u22610 : \u2200 x y \u2192 dist x y \u2261 0 \u2192 x \u2261 y\ndist\u22610 zero zero d\u22610 = refl\ndist\u22610 zero (suc y) ()\ndist\u22610 (suc x) zero ()\ndist\u22610 (suc x) (suc y) d\u22610 = ap suc (dist\u22610 x y d\u22610)\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm {x} {1} {x} | +-assoc-comm {y} {1} {y} = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm {x} {1} {k} | q | ! +-assoc-comm {x} {1} {k} | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm {1} {y} {k} | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b2e8244277ed8fc4aa948d2ec5cd7ec553e600b8","subject":"Mention third claim of the Agda code (for #275).","message":"Mention third claim of the Agda code (for #275).\n\nOld-commit-hash: 3506fbc56e5ce8c777e14da362a2aa440bcd62bb\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) and formally specifies the interface between our incrementalization\n--       framework and potential plugins.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"23339dd5a9e56c3c31d42252c4020b716ec77725","subject":"Update references to paper","message":"Update references to paper\n\nDuring revision, I added Lemma 2.5 to the paper, changing the numbering.\nThis commits updates the numbering in the formalization.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  -- In the paper, this is Def. 2.2.\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  -- In the paper, this is Lemma 2.3.\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of Derivative for change algebra families.\nDerivative\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : (px : P x) (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)) \u2192\n  Set (p \u2294 q \u2294 c)\nDerivative\u208d x , y \u208e f df = Derivative f df where\n  CPx = change-algebra\u208d x \u208e\n  CQy = change-algebra\u208d y \u208e\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        -- Definition 2.6a\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n        -- in the paper, update and diff below are in Def. 2.7\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n          -- the proof of update-diff is the second half of what\n          -- we have to prove for Theorem 2.8.\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d _ \u208e px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\u208d _ \u208e\n        ; update = update-all\n        ; diff = diff-all\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          }\n        }\n      }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  -- In the paper, this is Def. 2.2.\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  -- In the paper, this is Lemma 2.3.\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\n  -- abbrevitations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of Derivative for change algebra families.\nDerivative\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : (px : P x) (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)) \u2192\n  Set (p \u2294 q \u2294 c)\nDerivative\u208d x , y \u208e f df = Derivative f df where\n  CPx = change-algebra\u208d x \u208e\n  CQy = change-algebra\u208d y \u208e\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    -- This corresponds to Definition 2.5 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        -- Definition 2.5a\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n\n        -- Definition 2.5b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n        -- in the paper, update and diff below are in Def. 2.6\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n          -- the proof of update-diff is the second half of what\n          -- we have to prove for Theorem 2.7.\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n\n    -- This is Lemma 2.8 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.9 in the paper.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d _ \u208e px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\u208d _ \u208e\n        ; update = update-all\n        ; diff = diff-all\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          }\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3ee50b10e841b511ed0ac49f411f32075c8abc00","subject":"boolean and function composition","message":"boolean and function composition\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"85137cba040f5becff87d5865659e66d93a6216c","subject":"[ doc ] Added reference for non-redundant group theory axioms.","message":"[ doc ] Added reference for non-redundant group theory axioms.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/GroupTheory\/Base.agda","new_file":"src\/fot\/GroupTheory\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Group theory base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfix  11 _\u207b\u00b9\ninfixl 10 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public renaming ( D to G )\n\n-- Group theory axioms\npostulate\n  \u03b5   : G          -- The identity element.\n  _\u00b7_ : G \u2192 G \u2192 G  -- The binary operation.\n  _\u207b\u00b9 : G \u2192 G      -- The inverse function.\n\n  -- We choose a non-redundant set of axioms. See for example (Mac\n  -- Lane and Garret 1999, exercises 5-7, p. 50-51, or Hodges 1993,\n  -- p. 37).\n  assoc         : \u2200 a b c \u2192 a \u00b7 b \u00b7 c \u2261 a \u00b7 (b \u00b7 c)\n  leftIdentity  : \u2200 a \u2192 \u03b5 \u00b7 a         \u2261 a\n  leftInverse   : \u2200 a \u2192 a \u207b\u00b9 \u00b7 a      \u2261 \u03b5\n{-# ATP axiom assoc leftIdentity leftInverse #-}\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Hodges, W. (1993). Model Theory. Vol. 42. Encyclopedia of\n-- Mathematics and its Applications. Cambridge University Press.\n--\n-- Mac Lane, S. and Birkhof, G. (1999). Algebra. 3rd ed. AMS Chelsea\n-- Publishing.\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfix  11 _\u207b\u00b9\ninfixl 10 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public renaming ( D to G )\n\n-- Group theory axioms\npostulate\n  \u03b5   : G          -- The identity element.\n  _\u00b7_ : G \u2192 G \u2192 G  -- The binary operation.\n  _\u207b\u00b9 : G \u2192 G      -- The inverse function.\n\n  -- We choose a mininal set of axioms. See for example Saunders Mac\n  -- Lane and Garret Birkhoff. Algebra. AMS Chelsea Publishing, 3rd\n  -- edition, 1999. exercises 5-7, p. 50-51.\n  assoc         : \u2200 a b c \u2192 a \u00b7 b \u00b7 c \u2261 a \u00b7 (b \u00b7 c)\n  leftIdentity  : \u2200 a \u2192 \u03b5 \u00b7 a         \u2261 a\n  leftInverse   : \u2200 a \u2192 a \u207b\u00b9 \u00b7 a      \u2261 \u03b5\n{-# ATP axiom assoc leftIdentity leftInverse #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"75b597ce205190597a67bec4c3fe0525c78d9c71","subject":"Type: cleanup","message":"Type: cleanup\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Type.agda","new_file":"lib\/Explore\/Type.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- The core types behind exploration functions\nmodule Explore.Type where\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Function.Inverse using (_\u2194_)\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Bit\nopen import Data.Bits\nopen import Data.Indexed\nopen import Algebra\nopen import Relation.Binary.NP\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Two using (\ud835\udfda; \u2713)\nopen import Data.Maybe.NP using (_\u2192?_)\nopen import Data.Fin using (Fin)\nimport Algebra.FunctionProperties.NP as FP\nopen FP using (Op\u2082)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule SgrpExtra {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605 _\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 \u2605 _\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule RawMon {c} {C : \u2605 c} (rawMon : C \u00d7 Op\u2082 C) where\n  \u03b5    = proj\u2081 rawMon\n  _\u2219_  = proj\u2082 rawMon\n\nmodule Mon {c \u2113} (m : Monoid c \u2113) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  RawMon = C \u00d7 Op\u2082 C\n  rawMon : RawMon\n  rawMon = \u03b5 , _\u2219_\n\nmodule CMon {c \u2113} (cm : CommutativeMonoid c \u2113) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n  open FP _\u2248_\n\n  \u2219-interchange : Interchange _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nExplore : \u2200 \u2113 \u2192 \u2605\u2080 \u2192 \u2605 \u209b \u2113\nExplore \u2113 A = \u2200 {M : \u2605 \u2113} \u2192 (_\u2219_ : M \u2192 M \u2192 M) \u2192 (A \u2192 M) \u2192 M\n\nExplore\u2080 : \u2605\u2080 \u2192 \u2605\u2081\nExplore\u2080 = Explore _\n\nExplore\u2081 : \u2605\u2080 \u2192 \u2605\u2082\nExplore\u2081 = Explore _\n\n[Explore] : ([\u2605\u2080] [\u2192] [\u2605\u2081]) (Explore _)\n[Explore] A\u209a = \u2200\u27e8 M\u209a \u2236 [\u2605\u2080] \u27e9[\u2192] [Op\u2082] M\u209a [\u2192] (A\u209a [\u2192] M\u209a) [\u2192] M\u209a\n\n\u27e6Explore\u27e7 : (\u27e6\u2605\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u2081\u27e7) (Explore _) (Explore _)\n\u27e6Explore\u27e7 A\u1d63 = \u2200\u27e8 M\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Op\u2082\u27e7 M\u1d63 \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 M\u1d63) \u27e6\u2192\u27e7 M\u1d63\n\n\u27e6Explore\u27e7\u1d64 : \u2200 {\u2113} \u2192 (\u27e6\u2605\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 (\u209b \u2113)) (Explore \u2113) (Explore \u2113)\n\u27e6Explore\u27e7\u1d64 {\u2113} A\u1d63 = \u2200\u27e8 M\u1d63 \u2236 \u27e6\u2605\u27e7 \u2113 \u27e9\u27e6\u2192\u27e7 \u27e6Op\u2082\u27e7 M\u1d63 \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 M\u1d63) \u27e6\u2192\u27e7 M\u1d63\n\n-- Trimmed down version of \u27e6Explore\u27e7\n\u27e6Explore\u27e7\u2081 : \u2200 {A : \u2605_ _} (A\u1d63 : A \u2192 A \u2192 \u2605_ _) \u2192 Explore _ A \u2192 \u2605\u2081\n\u27e6Explore\u27e7\u2081 A\u1d63 s = \u27e6Explore\u27e7 A\u1d63 s s\n\n_\u2219-explore_ : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 Explore \u2113 A \u2192 Explore \u2113 A\n(s\u2080 \u2219-explore s\u2081) _\u2219_ f = s\u2080 _\u2219_ f \u2219 s\u2081 _\u2219_ f\n\nconst-explore : \u2200 {\u2113 A} \u2192 A \u2192 Explore \u2113 A\nconst-explore x _ f = f x\n\nExploreInd : \u2200 p {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreInd p {\u2113} {A} srch =\n  \u2200 (P  : Explore \u2113 A \u2192 \u2605 p)\n    (P\u2219 : \u2200 {s\u2080 s\u2081 : Explore \u2113 A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb _\u2219_ f \u2192 s\u2080 _\u2219_ f \u2219 s\u2081 _\u2219_ f))\n    (Pf : \u2200 x \u2192 P (\u03bb _ f \u2192 f x))\n  \u2192 P srch\n\nconst-explore-ind : \u2200 {\u2113 p A} (x : A) \u2192 ExploreInd p (const-explore {\u2113} x)\nconst-explore-ind x _ _ Pf = Pf x\n\n{-\n_\u2219ExploreInd_ : \u2200 {\u2113 p A} {s\u2080 s\u2081 : Explore \u2113 A}\n               \u2192 ExploreInd p s\u2080 \u2192 ExploreInd p s\u2081\n               \u2192 ExploreInd p (s\u2080 \u2219Explore s\u2081)\n_\u2219ExploreInd_ {s\u2080 = s\u2080} {s\u2081} Ps\u2080 Ps\u2081 P P\u2219 Pf = Ps\u2080 (\u03bb s \u2192 P s\u2081 \u2192 P (s \u2219Explore s\u2081)) (\u03bb {s\u2082} {s\u2083} Ps\u2082 Ps\u2083 Ps\u2081' \u2192 {!!}) (P\u2219 \u2218 Pf) (Ps\u2081 P P\u2219 Pf)\n-}\n\nrecord ExploreIndKit p {\u2113 A} (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (\u209b \u2113 \u2294 p) where\n  constructor _,_\n  field\n    P\u2219 : \u2200 {s\u2080 s\u2081 : Explore \u2113 A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (s\u2080 \u2219-explore s\u2081)\n    Pf : \u2200 x \u2192 P (const-explore x)\n\n_$kit_ : \u2200 {p \u2113 A} {P : Explore \u2113 A \u2192 \u2605 p} {s : Explore \u2113 A}\n         \u2192 ExploreInd p s \u2192 ExploreIndKit p P \u2192 P s\n_$kit_ {P = P} ind (P\u2219 , Pf) = ind P P\u2219 Pf\n\nExploreInd\u2080 : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreInd\u2080 = ExploreInd \u2080\n\nExploreInd\u2081 : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreInd\u2081 = ExploreInd \u2081\n\nExploreMon : \u2200 {c \u2113} \u2192 Monoid c \u2113 \u2192 \u2605\u2080 \u2192 \u2605 _\nExploreMon M A = (A \u2192 C) \u2192 C\n  where open Mon M\n\nExploreMonInd : \u2200 p {c \u2113} {A} (M : Monoid c \u2113) \u2192 ExploreMon M A \u2192 \u2605 _\nExploreMonInd p {c} {\u2113} {A} M srch =\n  \u2200 (P  : ExploreMon M A \u2192 \u2605 p)\n    (P\u2219 : \u2200 {s\u2080 s\u2081 : ExploreMon M A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n    (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n    (P\u2248 : \u2200 {s s'} \u2192 s \u2248\u00b0 s' \u2192 P s \u2192 P s')\n  \u2192 P srch\n  where open Mon M\n\nexplore\u2218FromExplore : \u2200 {m A} \u2192 Explore m A\n                    \u2192 \u2200 {M : \u2605 m} \u2192 (M \u2192 M \u2192 M) \u2192 (A \u2192 M) \u2192 (M \u2192 M)\nexplore\u2218FromExplore explore op f = explore _\u2218\u2032_ (op \u2218 f)\n\nExplorePlug : \u2200 {m \u2113 A} (M : Monoid m \u2113) (s : Explore _ A) \u2192 \u2605 _\nExplorePlug M s = \u2200 f x \u2192 s\u2218 _\u2219_ f \u03b5 \u2219 x \u2248 s\u2218 _\u2219_ f x\n   where open Mon M\n         s\u2218 = explore\u2218FromExplore s\n\n  {-\nExploreMon : \u2200 m \u2192 \u2605\u2080 \u2192 \u2605 _\nExploreMon m A = \u2200 {M : \u2605 m} \u2192 M \u00d7 Op\u2082 M \u2192 (A \u2192 M) \u2192 M\n\nExploreMonInd : \u2200 p {\u2113} {A} \u2192 ExploreMon \u2113 A \u2192 \u2605 _\nExploreMonInd p {\u2113} {A} srch =\n  \u2200 (P  : ExploreMon _ A \u2192 \u2605 p)\n    (P\u2219 : \u2200 {s\u2080 s\u2081 : ExploreMon _ A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb M f \u2192 let _\u2219_ = proj\u2082 M in\n                                                               s\u2080 M f \u2219 s\u2081 M f))\n    (Pf : \u2200 x \u2192 P (\u03bb _ f \u2192 f x))\n  \u2192 P srch\n\nexploreMonFromExplore : \u2200 {\u2113 A}\n                      \u2192 Explore \u2113 A \u2192 ExploreMon \u2113 A\nexploreMonFromExplore s = s \u2218 proj\u2082\n  -}\n\nexploreMonFromExplore : \u2200 {c \u2113 A}\n                      \u2192 Explore c A \u2192 (M : Monoid c \u2113) \u2192 ExploreMon M A\nexploreMonFromExplore s M = s _\u2219_ where open Mon M\n\nSum : \u2605\u2080 \u2192 \u2605\u2080\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nProduct : \u2605\u2080 \u2192 \u2605\u2080\nProduct A = (A \u2192 \u2115) \u2192 \u2115\n\nAdequateSum : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2080\nAdequateSum {A} sum\u1d2c = \u2200 f \u2192 Fin (sum\u1d2c f) \u2194 \u03a3 A (Fin \u2218 f)\n\nAdequateProduct : \u2200 {A} \u2192 Product A \u2192 \u2605\u2080\nAdequateProduct {A} product\u1d2c = \u2200 f \u2192 Fin (product\u1d2c f) \u2194 \u03a0 A (Fin \u2218 f)\n\nCount : \u2605\u2080 \u2192 \u2605\u2080\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nFind? : \u2605\u2080 \u2192 \u2605\u2081\nFind? A = \u2200 {B : \u2605\u2080} \u2192 (A \u2192? B) \u2192? B\n\nFindKey : \u2605\u2080 \u2192 \u2605\u2080\nFindKey A = (A \u2192 \ud835\udfda) \u2192? A\n\n_,-kit_ : \u2200 {m p A} {P : Explore m A \u2192 \u2605 p}{Q : Explore m A \u2192 \u2605 p}\n          \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\nPk ,-kit Qk = (\u03bb x y \u2192 P\u2219 Pk (proj\u2081 x) (proj\u2081 y) , P\u2219 Qk (proj\u2082 x) (proj\u2082 y))\n            , (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n             where open ExploreIndKit\n\nExploreInd-Extra : \u2200 p {m A} \u2192 Explore m A \u2192 \u2605 _\nExploreInd-Extra p {m} {A} srch =\n  \u2200 (Q     : Explore m A \u2192 \u2605 p)\n    (Q-kit : ExploreIndKit p Q)\n    (P     : Explore m A \u2192 \u2605 p)\n    (P\u2219    : \u2200 {s\u2080 s\u2081 : Explore m A} \u2192 Q s\u2080 \u2192 Q s\u2081 \u2192 P s\u2080 \u2192 P s\u2081\n             \u2192 P (s\u2080 \u2219-explore s\u2081))\n    (Pf    : \u2200 x \u2192 P (const-explore x))\n  \u2192 P srch\n\nto-extra : \u2200 {p m A} {s : Explore m A} \u2192 ExploreInd p s \u2192 ExploreInd-Extra p s\nto-extra s-ind Q Q-kit P P\u2219 Pf =\n proj\u2082 (s-ind (Q \u00d7\u00b0 P)\n         (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n         (\u03bb x \u2192 Qf x , Pf x))\n where open ExploreIndKit Q-kit renaming (P\u2219 to Q\u2219; Pf to Qf)\n\nStableUnder : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 (A \u2192 A) \u2192 \u2605 _\nStableUnder explore p = \u2200 {M} op (f : _ \u2192 M) \u2192 explore op f \u2261 explore op (f \u2218 p)\n\nSumStableUnder : \u2200 {A} \u2192 Sum A \u2192 (A \u2192 A) \u2192 \u2605 _\nSumStableUnder sum p = \u2200 f \u2192 sum f \u2261 sum (f \u2218 p)\n\nCountStableUnder : \u2200 {A} \u2192 Count A \u2192 (A \u2192 A) \u2192 \u2605 _\nCountStableUnder count p = \u2200 f \u2192 count f \u2261 count (f \u2218 p)\n\n-- TODO: remove the hard-wired \u2261\nInjective : \u2200 {a b}{A : \u2605 a}{B : \u2605 b}(f : A \u2192 B) \u2192 \u2605 _\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nSumStableUnderInjection : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumStableUnderInjection sum = \u2200 p \u2192 Injective p \u2192 SumStableUnder sum p\n\nSumInd : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2081\nSumInd {A} sum = \u2200 (P  : Sum A \u2192 \u2605\u2080)\n                   (P+ : \u2200 {s\u2080 s\u2081 : Sum A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f))\n                   (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n                \u2192  P sum\n\nExploreMono : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreMono r s\u1d2c = \u2200 {C} (_\u2286_ : C \u2192 C \u2192 \u2605 r)\n                    {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                    {f g} \u2192\n                    (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\n\nExploreExtFun : \u2200 {A B} \u2192 Explore _ (A \u2192 B) \u2192 \u2605\u2081\nExploreExtFun {A}{B} s\u1d2c\u1d2e = \u2200 {M} op {f g : (A \u2192 B) \u2192 M} \u2192 (\u2200 {\u03c6 \u03c8} \u2192 \u03c6 \u2257 \u03c8 \u2192 f \u03c6 \u2261 g \u03c8) \u2192 s\u1d2c\u1d2e op f \u2261 s\u1d2c\u1d2e op g\n\nExploreSgExt : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreSgExt r {\u2113} s\u1d2c = \u2200 (sg : Semigroup \u2113 r) {f g}\n                       \u2192 let open Sgrp sg in\n                         f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nExploreExt : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreExt {\u2113} {A} s\u1d2c = \u2200 {M} op {f g : A \u2192 M} \u2192 f \u2257 g \u2192 s\u1d2c op f \u2261 s\u1d2c op g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605 _\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nExplore\u03b5 : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExplore\u03b5 \u2113 r s\u1d2c = \u2200 (m : Monoid \u2113 r) \u2192\n                       let open Mon m in\n                       s\u1d2c _\u2219_ (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nExploreLin\u02e1 : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExploreLin\u02e1 \u2113 r s\u1d2c = \u2200 m _\u25ce_ f k \u2192\n                     let open Mon {\u2113} {r} m\n                         open FP _\u2248_ in\n                     _\u25ce_ DistributesOver\u02e1 _\u2219_ \u2192\n                     s\u1d2c _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce s\u1d2c _\u2219_ f\n\nExploreLin\u02b3 : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExploreLin\u02b3 \u2113 r s\u1d2c =\n  \u2200 m _\u25ce_ f k \u2192\n    let open Mon {\u2113} {r} m\n        open FP _\u2248_ in\n    _\u25ce_ DistributesOver\u02b3 _\u2219_ \u2192\n    s\u1d2c _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 s\u1d2c _\u2219_ f \u25ce k\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nExploreHom : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreHom r s\u1d2c = \u2200 sg f g \u2192 let open Sgrp {_} {r} sg in\n                            s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nExploreMonHom : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExploreMonHom \u2113 r s\u1d2c =\n  \u2200 cm f g \u2192\n    let open CMon {\u2113} {r} cm in\n    s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumMono sum\u1d2c = \u2200 {f g} \u2192 (\u2200 x \u2192 f x \u2264 g x) \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nExploreSwap : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreSwap r {\u2113} {A} s\u1d2c = \u2200 {B : \u2605\u2080} sg f \u2192\n                            let open Sgrp {_} {r} sg in\n                          \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                          \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                          \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nUnique : \u2200 {A} \u2192 Cmp A \u2192 Count A \u2192 \u2605 _\nUnique cmp count = \u2200 x \u2192 count (cmp x) \u2261 1\n\nmodule _ {\u2113 A} (A\u1d49 : Explore (\u209b \u2113) A) where\n    Data\u03a0 : (A \u2192 \u2605 \u2113) \u2192 \u2605 \u2113\n    Data\u03a0 = A\u1d49 _\u00d7_\n\n    \u03a3Point : (A \u2192 \u2605 \u2113) \u2192 \u2605 \u2113\n    \u03a3Point = A\u1d49 _\u228e_\n\nmodule _ {\u2113 A} (A\u1d49 : Explore (\u209b \u2113) A) where\n    Lookup : \u2605 (\u209b \u2113)\n    Lookup = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 Data\u03a0 A\u1d49 P \u2192 \u03a0 A P\n\n    Reify : \u2605 (\u209b \u2113)\n    Reify = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a0 A P \u2192 Data\u03a0 A\u1d49 P\n\n    Reified : \u2605 (\u209b \u2113)\n    Reified = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a0 A P \u2194 Data\u03a0 A\u1d49 P\n\n    Unfocus : \u2605 (\u209b \u2113)\n    Unfocus = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3Point A\u1d49 P \u2192 \u03a3 A P\n\n    Focus : \u2605 (\u209b \u2113)\n    Focus = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3 A P \u2192 \u03a3Point A\u1d49 P\n\n    Focused : \u2605 (\u209b \u2113)\n    Focused = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3 A P \u2194 \u03a3Point A\u1d49 P\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- The core types behind exploration functions\nmodule Explore.Type where\n\nopen import Level using (_\u2294_) renaming (zero to \u2080; suc to \u209b)\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Function.Inverse using (_\u2194_)\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Bit\nopen import Data.Bits\nopen import Data.Indexed\nopen import Algebra\nopen import Relation.Binary.NP\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Two using (\ud835\udfda; \u2713)\nopen import Data.Maybe.NP using (_\u2192?_)\nopen import Data.Fin using (Fin)\nimport Algebra.FunctionProperties.NP as FP\nopen FP using (Op\u2082)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule SgrpExtra {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605 _\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 \u2605 _\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule RawMon {c} {C : \u2605 c} (rawMon : C \u00d7 Op\u2082 C) where\n  \u03b5    = proj\u2081 rawMon\n  _\u2219_  = proj\u2082 rawMon\n\nmodule Mon {c \u2113} (m : Monoid c \u2113) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  RawMon = C \u00d7 Op\u2082 C\n  rawMon : RawMon\n  rawMon = \u03b5 , _\u2219_\n\nmodule CMon {c \u2113} (cm : CommutativeMonoid c \u2113) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n  open FP _\u2248_\n\n  \u2219-interchange : Interchange _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nExplore : \u2200 \u2113 \u2192 \u2605\u2080 \u2192 \u2605 \u209b \u2113\nExplore \u2113 A = \u2200 {M : \u2605 \u2113} \u2192 (_\u2219_ : M \u2192 M \u2192 M) \u2192 (A \u2192 M) \u2192 M\n\nExplore\u2080 : \u2605\u2080 \u2192 \u2605\u2081\nExplore\u2080 = Explore _\n\nExplore\u2081 : \u2605\u2080 \u2192 \u2605\u2082\nExplore\u2081 = Explore _\n\n[Explore] : ([\u2605\u2080] [\u2192] [\u2605\u2081]) (Explore _)\n[Explore] A\u209a = \u2200\u27e8 M\u209a \u2236 [\u2605\u2080] \u27e9[\u2192] [Op\u2082] M\u209a [\u2192] (A\u209a [\u2192] M\u209a) [\u2192] M\u209a\n\n\u27e6Explore\u27e7 : (\u27e6\u2605\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u2081\u27e7) (Explore _) (Explore _)\n\u27e6Explore\u27e7 A\u1d63 = \u2200\u27e8 M\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Op\u2082\u27e7 M\u1d63 \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 M\u1d63) \u27e6\u2192\u27e7 M\u1d63\n\n\u27e6Explore\u27e7\u1d64 : \u2200 {\u2113} \u2192 (\u27e6\u2605\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 (\u209b \u2113)) (Explore \u2113) (Explore \u2113)\n\u27e6Explore\u27e7\u1d64 {\u2113} A\u1d63 = \u2200\u27e8 M\u1d63 \u2236 \u27e6\u2605\u27e7 \u2113 \u27e9\u27e6\u2192\u27e7 \u27e6Op\u2082\u27e7 M\u1d63 \u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 M\u1d63) \u27e6\u2192\u27e7 M\u1d63\n\n-- Trimmed down version of \u27e6Explore\u27e7\n\u27e6Explore\u27e7\u2081 : \u2200 {A : \u2605_ _} (A\u1d63 : A \u2192 A \u2192 \u2605_ _) \u2192 Explore _ A \u2192 \u2605\u2081\n\u27e6Explore\u27e7\u2081 A\u1d63 s = \u27e6Explore\u27e7 A\u1d63 s s\n\n_\u2219-explore_ : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 Explore \u2113 A \u2192 Explore \u2113 A\n(s\u2080 \u2219-explore s\u2081) _\u2219_ f = s\u2080 _\u2219_ f \u2219 s\u2081 _\u2219_ f\n\nconst-explore : \u2200 {\u2113 A} \u2192 A \u2192 Explore \u2113 A\nconst-explore x _ f = f x\n\nExploreInd : \u2200 p {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreInd p {\u2113} {A} srch =\n  \u2200 (P  : Explore \u2113 A \u2192 \u2605 p)\n    (P\u2219 : \u2200 {s\u2080 s\u2081 : Explore \u2113 A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb _\u2219_ f \u2192 s\u2080 _\u2219_ f \u2219 s\u2081 _\u2219_ f))\n    (Pf : \u2200 x \u2192 P (\u03bb _ f \u2192 f x))\n  \u2192 P srch\n\nconst-explore-ind : \u2200 {\u2113 p A} (x : A) \u2192 ExploreInd p (const-explore {\u2113} x)\nconst-explore-ind x _ _ Pf = Pf x\n\n{-\n_\u2219ExploreInd_ : \u2200 {\u2113 p A} {s\u2080 s\u2081 : Explore \u2113 A}\n               \u2192 ExploreInd p s\u2080 \u2192 ExploreInd p s\u2081\n               \u2192 ExploreInd p (s\u2080 \u2219Explore s\u2081)\n_\u2219ExploreInd_ {s\u2080 = s\u2080} {s\u2081} Ps\u2080 Ps\u2081 P P\u2219 Pf = Ps\u2080 (\u03bb s \u2192 P s\u2081 \u2192 P (s \u2219Explore s\u2081)) (\u03bb {s\u2082} {s\u2083} Ps\u2082 Ps\u2083 Ps\u2081' \u2192 {!!}) (P\u2219 \u2218 Pf) (Ps\u2081 P P\u2219 Pf)\n-}\n\nrecord ExploreIndKit p {\u2113 A} (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (\u209b \u2113 \u2294 p) where\n  constructor _,_\n  field\n    P\u2219 : \u2200 {s\u2080 s\u2081 : Explore \u2113 A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (s\u2080 \u2219-explore s\u2081)\n    Pf : \u2200 x \u2192 P (const-explore x)\n\n_$kit_ : \u2200 {p \u2113 A} {P : Explore \u2113 A \u2192 \u2605 p} {s : Explore \u2113 A}\n         \u2192 ExploreInd p s \u2192 ExploreIndKit p P \u2192 P s\n_$kit_ {P = P} ind (P\u2219 , Pf) = ind P P\u2219 Pf\n\nExploreInd\u2080 : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreInd\u2080 = ExploreInd \u2080\n\nExploreMon : \u2200 {c \u2113} \u2192 Monoid c \u2113 \u2192 \u2605\u2080 \u2192 \u2605 _\nExploreMon M A = (A \u2192 C) \u2192 C\n  where open Mon M\n\nExploreMonInd : \u2200 p {c \u2113} {A} (M : Monoid c \u2113) \u2192 ExploreMon M A \u2192 \u2605 _\nExploreMonInd p {c} {\u2113} {A} M srch =\n  \u2200 (P  : ExploreMon M A \u2192 \u2605 p)\n    (P\u2219 : \u2200 {s\u2080 s\u2081 : ExploreMon M A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n    (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n    (P\u2248 : \u2200 {s s'} \u2192 s \u2248\u00b0 s' \u2192 P s \u2192 P s')\n  \u2192 P srch\n  where open Mon M\n\nexplore\u2218FromExplore : \u2200 {m A} \u2192 Explore m A\n                    \u2192 \u2200 {M : \u2605 m} \u2192 (M \u2192 M \u2192 M) \u2192 (A \u2192 M) \u2192 (M \u2192 M)\nexplore\u2218FromExplore explore op f = explore _\u2218\u2032_ (op \u2218 f)\n\nExplorePlug : \u2200 {m \u2113 A} (M : Monoid m \u2113) (s : Explore _ A) \u2192 \u2605 _\nExplorePlug M s = \u2200 f x \u2192 s\u2218 _\u2219_ f \u03b5 \u2219 x \u2248 s\u2218 _\u2219_ f x\n   where open Mon M\n         s\u2218 = explore\u2218FromExplore s\n\n  {-\nExploreMon : \u2200 m \u2192 \u2605\u2080 \u2192 \u2605 _\nExploreMon m A = \u2200 {M : \u2605 m} \u2192 M \u00d7 Op\u2082 M \u2192 (A \u2192 M) \u2192 M\n\nExploreMonInd : \u2200 p {\u2113} {A} \u2192 ExploreMon \u2113 A \u2192 \u2605 _\nExploreMonInd p {\u2113} {A} srch =\n  \u2200 (P  : ExploreMon _ A \u2192 \u2605 p)\n    (P\u2219 : \u2200 {s\u2080 s\u2081 : ExploreMon _ A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb M f \u2192 let _\u2219_ = proj\u2082 M in\n                                                               s\u2080 M f \u2219 s\u2081 M f))\n    (Pf : \u2200 x \u2192 P (\u03bb _ f \u2192 f x))\n  \u2192 P srch\n\nexploreMonFromExplore : \u2200 {\u2113 A}\n                      \u2192 Explore \u2113 A \u2192 ExploreMon \u2113 A\nexploreMonFromExplore s = s \u2218 proj\u2082\n  -}\n\nexploreMonFromExplore : \u2200 {c \u2113 A}\n                      \u2192 Explore c A \u2192 (M : Monoid c \u2113) \u2192 ExploreMon M A\nexploreMonFromExplore s M = s _\u2219_ where open Mon M\n\nSum : \u2605\u2080 \u2192 \u2605\u2080\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nProduct : \u2605\u2080 \u2192 \u2605\u2080\nProduct A = (A \u2192 \u2115) \u2192 \u2115\n\nAdequateSum : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2080\nAdequateSum {A} sum\u1d2c = \u2200 f \u2192 Fin (sum\u1d2c f) \u2194 \u03a3 A (Fin \u2218 f)\n\nAdequateProduct : \u2200 {A} \u2192 Product A \u2192 \u2605\u2080\nAdequateProduct {A} product\u1d2c = \u2200 f \u2192 Fin (product\u1d2c f) \u2194 \u03a0 A (Fin \u2218 f)\n\nCount : \u2605\u2080 \u2192 \u2605\u2080\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nFind? : \u2605\u2080 \u2192 \u2605\u2081\nFind? A = \u2200 {B : \u2605\u2080} \u2192 (A \u2192? B) \u2192? B\n\nFindKey : \u2605\u2080 \u2192 \u2605\u2080\nFindKey A = (A \u2192 \ud835\udfda) \u2192? A\n\n_,-kit_ : \u2200 {m p A} {P : Explore m A \u2192 \u2605 p}{Q : Explore m A \u2192 \u2605 p}\n          \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\nPk ,-kit Qk = (\u03bb x y \u2192 P\u2219 Pk (proj\u2081 x) (proj\u2081 y) , P\u2219 Qk (proj\u2082 x) (proj\u2082 y))\n            , (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n             where open ExploreIndKit\n\nExploreInd-Extra : \u2200 p {m A} \u2192 Explore m A \u2192 \u2605 _\nExploreInd-Extra p {m} {A} srch =\n  \u2200 (Q     : Explore m A \u2192 \u2605 p)\n    (Q-kit : ExploreIndKit p Q)\n    (P     : Explore m A \u2192 \u2605 p)\n    (P\u2219    : \u2200 {s\u2080 s\u2081 : Explore m A} \u2192 Q s\u2080 \u2192 Q s\u2081 \u2192 P s\u2080 \u2192 P s\u2081\n             \u2192 P (s\u2080 \u2219-explore s\u2081))\n    (Pf    : \u2200 x \u2192 P (const-explore x))\n  \u2192 P srch\n\nto-extra : \u2200 {p m A} {s : Explore m A} \u2192 ExploreInd p s \u2192 ExploreInd-Extra p s\nto-extra s-ind Q Q-kit P P\u2219 Pf =\n proj\u2082 (s-ind (Q \u00d7\u00b0 P)\n         (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n         (\u03bb x \u2192 Qf x , Pf x))\n where open ExploreIndKit Q-kit renaming (P\u2219 to Q\u2219; Pf to Qf)\n\nStableUnder : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 (A \u2192 A) \u2192 \u2605 _\nStableUnder explore p = \u2200 {M} op (f : _ \u2192 M) \u2192 explore op f \u2261 explore op (f \u2218 p)\n\nSumStableUnder : \u2200 {A} \u2192 Sum A \u2192 (A \u2192 A) \u2192 \u2605 _\nSumStableUnder sum p = \u2200 f \u2192 sum f \u2261 sum (f \u2218 p)\n\nCountStableUnder : \u2200 {A} \u2192 Count A \u2192 (A \u2192 A) \u2192 \u2605 _\nCountStableUnder count p = \u2200 f \u2192 count f \u2261 count (f \u2218 p)\n\n-- TODO: remove the hard-wired \u2261\nInjective : \u2200 {a b}{A : \u2605 a}{B : \u2605 b}(f : A \u2192 B) \u2192 \u2605 _\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nSumStableUnderInjection : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumStableUnderInjection sum = \u2200 p \u2192 Injective p \u2192 SumStableUnder sum p\n\nSumInd : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2081\nSumInd {A} sum = \u2200 (P  : Sum A \u2192 \u2605\u2080)\n                   (P+ : \u2200 {s\u2080 s\u2081 : Sum A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f))\n                   (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n                \u2192  P sum\n\nExploreMono : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreMono r s\u1d2c = \u2200 {C} (_\u2286_ : C \u2192 C \u2192 \u2605 r)\n                    {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                    {f g} \u2192\n                    (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\n\nExploreExtFun : \u2200 {A B} \u2192 Explore _ (A \u2192 B) \u2192 \u2605\u2081\nExploreExtFun {A}{B} s\u1d2c\u1d2e = \u2200 {M} op {f g : (A \u2192 B) \u2192 M} \u2192 (\u2200 {\u03c6 \u03c8} \u2192 \u03c6 \u2257 \u03c8 \u2192 f \u03c6 \u2261 g \u03c8) \u2192 s\u1d2c\u1d2e op f \u2261 s\u1d2c\u1d2e op g\n\nExploreSgExt : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreSgExt r {\u2113} s\u1d2c = \u2200 (sg : Semigroup \u2113 r) {f g}\n                       \u2192 let open Sgrp sg in\n                         f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nExploreExt : \u2200 {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreExt {\u2113} {A} s\u1d2c = \u2200 {M} op {f g : A \u2192 M} \u2192 f \u2257 g \u2192 s\u1d2c op f \u2261 s\u1d2c op g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605 _\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nExplore\u03b5 : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExplore\u03b5 \u2113 r s\u1d2c = \u2200 (m : Monoid \u2113 r) \u2192\n                       let open Mon m in\n                       s\u1d2c _\u2219_ (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nExploreLin\u02e1 : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExploreLin\u02e1 \u2113 r s\u1d2c = \u2200 m _\u25ce_ f k \u2192\n                     let open Mon {\u2113} {r} m\n                         open FP _\u2248_ in\n                     _\u25ce_ DistributesOver\u02e1 _\u2219_ \u2192\n                     s\u1d2c _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce s\u1d2c _\u2219_ f\n\nExploreLin\u02b3 : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExploreLin\u02b3 \u2113 r s\u1d2c =\n  \u2200 m _\u25ce_ f k \u2192\n    let open Mon {\u2113} {r} m\n        open FP _\u2248_ in\n    _\u25ce_ DistributesOver\u02b3 _\u2219_ \u2192\n    s\u1d2c _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 s\u1d2c _\u2219_ f \u25ce k\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nExploreHom : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreHom r s\u1d2c = \u2200 sg f g \u2192 let open Sgrp {_} {r} sg in\n                            s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nExploreMonHom : \u2200 \u2113 r {A} \u2192 Explore _ A \u2192 \u2605 _\nExploreMonHom \u2113 r s\u1d2c =\n  \u2200 cm f g \u2192\n    let open CMon {\u2113} {r} cm in\n    s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605 _\nSumMono sum\u1d2c = \u2200 {f g} \u2192 (\u2200 x \u2192 f x \u2264 g x) \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nExploreSwap : \u2200 r {\u2113 A} \u2192 Explore \u2113 A \u2192 \u2605 _\nExploreSwap r {\u2113} {A} s\u1d2c = \u2200 {B : \u2605\u2080} sg f \u2192\n                            let open Sgrp {_} {r} sg in\n                          \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                          \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                          \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nUnique : \u2200 {A} \u2192 Cmp A \u2192 Count A \u2192 \u2605 _\nUnique cmp count = \u2200 x \u2192 count (cmp x) \u2261 1\n\nData\u03a0 : \u2200 {b A} \u2192 Explore _ A \u2192 (A \u2192 \u2605 b) \u2192 \u2605 b\nData\u03a0 sA = sA _\u00d7_\n\nLookup : \u2200 {b A} \u2192 Explore _ A \u2192 \u2605 _\nLookup {b} {A} sA = \u2200 {B : A \u2192 \u2605 b} \u2192 Data\u03a0 sA B \u2192 \u03a0 A B\n\nReify : \u2200 {b A} \u2192 Explore _ A \u2192 \u2605 _\nReify {b} {A} sA = \u2200 {B : A \u2192 \u2605 b} \u2192 \u03a0 A B \u2192 Data\u03a0 sA B\n\nReified : \u2200 {b A} \u2192 Explore _ A \u2192 \u2605 _\nReified {b} {A} sA = \u2200 {B : A \u2192 \u2605 b} \u2192 \u03a0 A B \u2194 Data\u03a0 sA B\n\n\u03a3Point : \u2200 {b A} \u2192 Explore _ A \u2192 (A \u2192 \u2605 b) \u2192 \u2605 b\n\u03a3Point sA = sA _\u228e_\n\nUnfocus : \u2200 {b A} \u2192 Explore _ A \u2192 \u2605 _\nUnfocus {b} {A} sA = \u2200 {B : A \u2192 \u2605 b} \u2192 \u03a3Point sA B \u2192 \u03a3 A B\n\nFocus : \u2200 {b A} \u2192 Explore _ A \u2192 \u2605 _\nFocus {b} {A} sA = \u2200 {B : A \u2192 \u2605 b} \u2192 \u03a3 A B \u2192 \u03a3Point sA B\n\nFocused : \u2200 {b A} \u2192 Explore _ A \u2192 \u2605 _\nFocused {b} {A} sA = \u2200 {B : A \u2192 \u2605 b} \u2192 \u03a3 A B \u2194 \u03a3Point sA B\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7462fd6d4414a70a0c18898feef13e9b44d95589","subject":"CCA2\u2020 more intermediate names","message":"CCA2\u2020 more intermediate names\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/IND-CCA2-dagger.agda","new_file":"Game\/IND-CCA2-dagger.agda","new_contents":"\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\n\nopen import Data.Nat.NP\n--open import Rat\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Control.Strategy renaming (run to runStrategy)\nimport Game.IND-CPA-utils\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Game.IND-CCA2-dagger\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n\nwhere\n\nopen Game.IND-CPA-utils Message CipherText\nopen CPAAdversary\n\nAdversary : \u2605\nAdversary = R\u2090 \u2192 PubKey \u2192\n                   DecRound              -- first round of decryption queries\n                     (CPAAdversary       -- choosen plaintext attack\n                       (CipherText \u2192     -- in which a second ciphertext is provided\n                          DecRound Bit)) -- second round of decryption queries\n\n{-\nValid-Adv : Adv \u2192 Set\nValid-Adv (m , d) = \u2200 {r\u2090 r\u2093 pk c c'} \u2192 Valid (\u03bb x \u2192 x \u2262 c \u00d7 x \u2262 c') (d r\u2090 r\u2093 pk c c')\n-}\n\nR : \u2605\nR = R\u2090 \u00d7 R\u2096 \u00d7 R\u2091 \u00d7 R\u2091\n\nExperiment : \u2605\nExperiment = Adversary \u2192 R \u2192 Bit\n\nmodule EXP (b : Bit) (A : Adversary) (r\u2090 : R\u2090) (pk : PubKey) (sk : SecKey) (r\u2091\u2080 r\u2091\u2081 : R\u2091) where\n  decRound = runStrategy (Dec sk)\n  A1       = A r\u2090 pk\n  cpaA     = decRound A1\n  mb       = proj\u2032 (get-m cpaA)\n  c\u2080       = Enc pk (mb b)       r\u2091\u2080\n  c\u2081       = Enc pk (mb (not b)) r\u2091\u2081\n  A2       = put-c cpaA c\u2080 c\u2081\n  b\u2032       = decRound A2\n\n  ct       = proj\u2032 (c\u2080 , c\u2081)\n\nEXP : Bit \u2192 Experiment\nEXP b A (r\u2090 , r\u2096 , r\u2091\u2080 , r\u2091\u2081) with KeyGen r\u2096\n... | pk , sk = EXP.b\u2032 b A r\u2090 pk sk r\u2091\u2080 r\u2091\u2081\n\nmodule Advantage\n  (\u03bc\u2091 : Explore\u2080 R\u2091)\n  (\u03bc\u2096 : Explore\u2080 R\u2096)\n  (\u03bc\u2090 : Explore\u2080 R\u2090)\n  where\n  \u03bcR : Explore\u2080 R\n  \u03bcR = \u03bc\u2090 \u00d7\u1d49 \u03bc\u2096 \u00d7\u1d49 \u03bc\u2091 \u00d7\u1d49 \u03bc\u2091\n  \n  module \u03bcR = FromExplore\u2080 \u03bcR\n  \n  run : Bit \u2192 Adversary \u2192 \u2115\n  run b adv = \u03bcR.count (EXP b adv)\n    \n  {-\n  Advantage : Adv \u2192 \u211a\n  Advantage adv = dist (run 0b adv) (run 1b adv) \/ \u03bcR.Card\n  -}\n","old_contents":"\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\n\nopen import Data.Nat.NP\n--open import Rat\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Control.Strategy renaming (run to runStrategy)\nimport Game.IND-CPA-utils\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Game.IND-CCA2-dagger\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n\nwhere\n\nopen Game.IND-CPA-utils Message CipherText\nopen CPAAdversary\n\nAdversary : \u2605\nAdversary = R\u2090 \u2192 PubKey \u2192\n                   DecRound              -- first round of decryption queries\n                     (CPAAdversary       -- choosen plaintext attack\n                       (CipherText \u2192     -- in which a second ciphertext is provided\n                          DecRound Bit)) -- second round of decryption queries\n\n{-\nValid-Adv : Adv \u2192 Set\nValid-Adv (m , d) = \u2200 {r\u2090 r\u2093 pk c c'} \u2192 Valid (\u03bb x \u2192 x \u2262 c \u00d7 x \u2262 c') (d r\u2090 r\u2093 pk c c')\n-}\n\nR : \u2605\nR = R\u2090 \u00d7 R\u2096 \u00d7 R\u2091 \u00d7 R\u2091\n\nExperiment : \u2605\nExperiment = Adversary \u2192 R \u2192 Bit\n\nmodule EXP (b : Bit) (A : Adversary) (r\u2090 : R\u2090) (pk : PubKey) (sk : SecKey) (r\u2091\u2080 r\u2091\u2081 : R\u2091) where\n  decRound = runStrategy (Dec sk)\n  cpaA     = decRound (A r\u2090 pk)\n  mb       = proj\u2032 (get-m cpaA)\n  c\u2080       = Enc pk (mb b)       r\u2091\u2080\n  c\u2081       = Enc pk (mb (not b)) r\u2091\u2081\n  b\u2032       = decRound (put-c cpaA c\u2080 c\u2081)\n\nEXP : Bit \u2192 Experiment\nEXP b A (r\u2090 , r\u2096 , r\u2091\u2080 , r\u2091\u2081) with KeyGen r\u2096\n... | pk , sk = EXP.b\u2032 b A r\u2090 pk sk r\u2091\u2080 r\u2091\u2081\n\nmodule Advantage\n  (\u03bc\u2091 : Explore\u2080 R\u2091)\n  (\u03bc\u2096 : Explore\u2080 R\u2096)\n  (\u03bc\u2090 : Explore\u2080 R\u2090)\n  where\n  \u03bcR : Explore\u2080 R\n  \u03bcR = \u03bc\u2090 \u00d7\u1d49 \u03bc\u2096 \u00d7\u1d49 \u03bc\u2091 \u00d7\u1d49 \u03bc\u2091\n  \n  module \u03bcR = FromExplore\u2080 \u03bcR\n  \n  run : Bit \u2192 Adversary \u2192 \u2115\n  run b adv = \u03bcR.count (EXP b adv)\n    \n  {-\n  Advantage : Adv \u2192 \u211a\n  Advantage adv = dist (run 0b adv) (run 1b adv) \/ \u03bcR.Card\n  -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2f337fa6c912b4b71f8a6539911608241ebc8524","subject":"Use helpers for nested applications.","message":"Use helpers for nested applications.\n\nOld-commit-hash: e4c7562ec14ff04d00f64f84b95725baa11a85ca\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app\u2082 diff\u03c3 s t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app\u2082 diff\u03c4 s t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app\u2082 apply\u03c3 \u0394t t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app\u2082 apply\u03c4 \u0394t t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"49a7a00a3832689c0045cb3cae94640175267a08","subject":"Removed unnecessary arguments.","message":"Removed unnecessary arguments.\n\nIgnore-this: a39d6ebeef4c07ea14c5405619bc8d59\n\ndarcs-hash:20111218163729-3bd4e-c8488225fefcfde5c7622ca2471e2ae95fba08c7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/ABP\/Fair\/PropertiesATP.agda","new_file":"src\/FOTC\/Program\/ABP\/Fair\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Fair properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.ABP.Fair.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nhead-tail-Fair-helper : \u2200 {os ol os'} \u2192 O*L ol \u2192 os \u2261 ol ++ os' \u2192\n                        os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\nhead-tail-Fair-helper {os} nilO*L h = prf\n  where\n  postulate prf : os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\n  {-# ATP prove prf #-}\n\nhead-tail-Fair-helper {os} (consO*L OLol) h = prf\n  where\n  postulate prf : os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\n  {-# ATP prove prf #-}\n\nhead-tail-Fair : \u2200 {os} \u2192 Fair os \u2192 os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\nhead-tail-Fair {os} Fos = prf\n  where\n  postulate prf : os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\n  {-# ATP prove prf head-tail-Fair-helper #-}\n\ntail-Fair-helper : \u2200 {os ol os'} \u2192 O*L ol \u2192 os \u2261 ol ++ os' \u2192 Fair os' \u2192\n                   Fair (tail\u2081 os)\ntail-Fair-helper {os} nilO*L h Fos' = prf\n  where\n  postulate prf : Fair (tail\u2081 os)\n  {-# ATP prove prf #-}\n\ntail-Fair-helper {os} (consO*L OLol) h Fos' = prf\n  where\n  postulate prf : Fair (tail\u2081 os)\n  {-# ATP prove prf Fair-gfp\u2083 #-}\n\ntail-Fair : \u2200 {os} \u2192 Fair os \u2192 Fair (tail\u2081 os)\ntail-Fair {os} Fos = prf\n  where\n  postulate prf : Fair (tail\u2081 os)\n  {-# ATP prove prf tail-Fair-helper #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Fair properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.ABP.Fair.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nhead-tail-Fair-helper : \u2200 {os ol os'} \u2192 O*L ol \u2192 os \u2261 ol ++ os' \u2192\n                        os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\nhead-tail-Fair-helper {os} {os' = os'} nilO*L h = prf\n  where\n  postulate prf : os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\n  {-# ATP prove prf #-}\n\nhead-tail-Fair-helper {os} {os' = os'} (consO*L OLol) h = prf\n  where\n  postulate prf : os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\n  {-# ATP prove prf #-}\n\nhead-tail-Fair : \u2200 {os} \u2192 Fair os \u2192 os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\nhead-tail-Fair {os} Fos = prf\n  where\n  postulate prf : os \u2261 L \u2237 tail\u2081 os \u2228 os \u2261 O \u2237 tail\u2081 os\n  {-# ATP prove prf head-tail-Fair-helper #-}\n\ntail-Fair-helper : \u2200 {os ol os'} \u2192 O*L ol \u2192 os \u2261 ol ++ os' \u2192 Fair os' \u2192\n                   Fair (tail\u2081 os)\ntail-Fair-helper {os} {os' = os'} nilO*L h Fos' = prf\n  where\n  postulate prf : Fair (tail\u2081 os)\n  {-# ATP prove prf #-}\n\ntail-Fair-helper {os} {os' = os'} (consO*L OLol) h Fos' = prf\n  where\n  postulate prf : Fair (tail\u2081 os)\n  {-# ATP prove prf Fair-gfp\u2083 #-}\n\ntail-Fair : \u2200 {os} \u2192 Fair os \u2192 Fair (tail\u2081 os)\ntail-Fair {os} Fos = prf\n  where\n  postulate prf : Fair (tail\u2081 os)\n  {-# ATP prove prf tail-Fair-helper #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"561e54d6fa752c3e41e0261fcb95b5fe132461b5","subject":"Rename lift-\u03b7-const-rec to curryTermConstructor.","message":"Rename lift-\u03b7-const-rec to curryTermConstructor.\n\nAlso delete the frivolous comment, which is no longer necessary\nnow that curryTermConstructor has a descriptive name and the\nmatching type signature.\n\nOld-commit-hash: bcd82c9b32ab89ebe5c1badbec2d5714d0fce9bf\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = curryTermConstructor (const constant)\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"00497b63c37149f1b209984e58341824b3174de3","subject":"Complete proof (up to termination)","message":"Complete proof (up to termination)\n\nThis completes the proof of the fundamental lemma for this step-indexed relation\nup to a few missing termination proofs, which are left external.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) zero n1 ()\neval-const-dec (lit v) (suc n0) .n0 refl = \u2264-step \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) zero n1 ()\neval-const-mono (lit v) (suc n0) .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) zero n1 ()\neval-const-strengthen (lit v) (suc n0) .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV nat v1 v2 n = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat v1 v2 vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (const (lit v)) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , suc zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". However that doesn't help termination either,\n-- since Agda doesn't see that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) zero n1 ()\neval-const-dec (lit v) (suc n0) .n0 refl = \u2264-step \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) zero n1 ()\neval-const-mono (lit v) (suc n0) .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\nmodule Alt where\n  mutual\n    relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n    relT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n      (v1 : Val \u03c4) \u2192\n      \u2200 n-j (n-j\u2264n : n-j < n) \u2192\n      (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n      \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n      -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n    relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n    -- Seems the proof for abs would go through even if here we do not step down.\n    -- However, that only works as long as we use a typed language; not stepping\n    -- down here, in an untyped language, gives a non-well-founded definition.\n    relV nat v1 v2 n = v1 \u2261 v2\n    relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nmutual\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV \u03c4 v1 v2 zero = \u22a4\n  -- Seems the proof for abs would go through even if here we do not step down.\n  -- However, that only works as long as we use a typed language; not stepping\n  -- down here, in an untyped language, gives a non-well-founded definition.\n  relV nat v1 v2 (suc n) = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n    \u2200 (k : \u2115) (k\u2264n : k \u2264 n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV  x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- relT (app s t) (app s t)\n\nrelV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n  n1 sv1 sv2\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n  n2 tv1 tv2\n  (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n  (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n  -- (tvv)\n  -- (eqv1)\n  \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\nrelV-apply s t v1 \u03c1 n-j n1 sv1 sv2 svv n2 tv1 tv2 tvv eq = {!!}\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (const (lit v)) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , suc zero , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , n , n , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | Done tv1 n2 | [ t1eq ] with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s (eval-dec s \u03c11 _ n n1 s1eq)) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq) | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s (eval-dec s \u03c11 _ n n1 s1eq))) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s (eval-dec t \u03c11 _ n1 n2 t1eq)) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , sn4 , s2eq , svv | tv2 , tn3 , tn4 , t2eq , tvv = {! relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq!}\n--\n-- {!eval s \u03c12 !}\n-- fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) ? ?\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"55154fffeaa270b4adfd6d4cc6cb42a3f47df6a6","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationI.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationI.agda","new_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ProofSpecificationI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1I\nopen import FOTC.Program.ABP.Lemma2I\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 transfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n       \u2203[ i' ] \u2203[ is' ] \u2203[ js' ] is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  prf\u2081 {is} {js} (b , os\u2080 , os\u2081 , as , bs , cs , ds , Sis , Bb , Fos\u2080 , Fos\u2081 , h)\n     with Stream-unf Sis\n  ... | (i' , is' , is\u2261i'\u2237is , Sis') =\n    i' , is' , js' , is\u2261i'\u2237is , js\u2261i'\u2237js' , Bis'js'\n    where\n    ABP-helper : is \u2261 i' \u2237 is' \u2192\n                 ABP b is os\u2080 os\u2081 as bs cs ds js \u2192\n                 ABP b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js\n    ABP-helper h\u2081 h\u2082 = subst (\u03bb t \u2192 ABP b t os\u2080 os\u2081 as bs cs ds js) h\u2081 h\u2082\n\n    ABP'-lemma\u2081 : \u2203[ os\u2080' ] \u2203[ os\u2081' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n                    Fair os\u2080'\n                    \u2227 Fair os\u2081'\n                    \u2227 ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                    \u2227 js \u2261 i' \u2237 js'\n    ABP'-lemma\u2081 = lemma\u2081 Bb Fos\u2080 Fos\u2081 (ABP-helper is\u2261i'\u2237is h)\n\n    -- Following Martin Escardo advice (see Agda mailing list, heap\n    -- mistery) we use pattern matching instead of \u2203 eliminators to\n    -- project the elements of the existentials.\n\n    -- 2011-08-25 update: It does not seems strictly necessary because\n    -- the Agda issue 415 was fixed.\n\n    js' : D\n    js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , js' , _ = js'\n\n    js\u2261i'\u2237js' : js \u2261 i' \u2237 js'\n    js\u2261i'\u2237js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , _ , _ , _ , _ , h = h\n\n    ABP-lemma\u2082 : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n                   Fair os\u2080''\n                   \u2227 Fair os\u2081''\n                   \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n    ABP-lemma\u2082 with ABP'-lemma\u2081\n    ABP-lemma\u2082 | _ , _ , _ , _ , _ , _ , _ , Fos\u2080' , Fos\u2081' , abp' , _ =\n      lemma\u2082 Bb Fos\u2080' Fos\u2081' abp'\n\n    Bis'js' : B is' js'\n    Bis'js' with ABP-lemma\u2082\n    ... | os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds'' , Fos\u2080'' , Fos\u2081'' , abp =\n      not b , os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds''\n      , Sis' , not-Bool Bb , Fos\u2080'' , Fos\u2081'' , abp\n\n  prf\u2082 : B is (transfer b os\u2080 os\u2081 is)\n  prf\u2082 = b\n       , os\u2080\n       , os\u2081\n       , has a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , Sis\n       , Bb\n       , Fos\u2080\n       , Fos\u2081\n       , has-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , trans (transfer-eq b os\u2080 os\u2081 is) (genTransfer-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is)\n    where\n    a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 : D\n    a\u2081 = send \u00b7 b\n    a\u2082 = ack \u00b7 b\n    a\u2083 = out \u00b7 b\n    a\u2084 = corrupt \u00b7 os\u2080\n    a\u2085 = corrupt \u00b7 os\u2081\n","old_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ProofSpecificationI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1I\nopen import FOTC.Program.ABP.Lemma2I\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 transfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n       \u2203[ i' ] \u2203[ is' ] \u2203[ js' ] is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  prf\u2081 {is} {js} (b , os\u2080 , os\u2081 , as , bs , cs , ds , Sis , Bb , Fos\u2080 , Fos\u2081 , h)\n     with Stream-unf Sis\n  ... | (i' , is' , is\u2261i'\u2237is , Sis') =\n    i' , is' , js' , is\u2261i'\u2237is , js\u2261i'\u2237js' , Bis'js'\n    where\n    ABP-helper : is \u2261 i' \u2237 is' \u2192\n                 ABP b is os\u2080 os\u2081 as bs cs ds js \u2192\n                 ABP b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js\n    ABP-helper h\u2081 h\u2082 = subst (\u03bb t \u2192 ABP b t os\u2080 os\u2081 as bs cs ds js) h\u2081 h\u2082\n\n    ABP'-lemma\u2081 : \u2203[ os\u2080' ] \u2203[ os\u2081' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n                  Fair os\u2080'\n                  \u2227 Fair os\u2081'\n                  \u2227 ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                  \u2227 js \u2261 i' \u2237 js'\n    ABP'-lemma\u2081 = lemma\u2081 Bb Fos\u2080 Fos\u2081 (ABP-helper is\u2261i'\u2237is h)\n\n    -- Following Martin Escardo advice (see Agda mailing list, heap\n    -- mistery) we use pattern matching instead of \u2203 eliminators to\n    -- project the elements of the existentials.\n\n    -- 2011-08-25 update: It does not seems strictly necessary because\n    -- the Agda issue 415 was fixed.\n\n    js' : D\n    js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , js' , _ = js'\n\n    js\u2261i'\u2237js' : js \u2261 i' \u2237 js'\n    js\u2261i'\u2237js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , _ , _ , _ , _ , h = h\n\n    ABP-lemma\u2082 : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n                 Fair os\u2080''\n                 \u2227 Fair os\u2081''\n                 \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n    ABP-lemma\u2082 with ABP'-lemma\u2081\n    ABP-lemma\u2082 | _ , _ , _ , _ , _ , _ , _ , Fos\u2080' , Fos\u2081' , abp' , _ =\n      lemma\u2082 Bb Fos\u2080' Fos\u2081' abp'\n\n    Bis'js' : B is' js'\n    Bis'js' with ABP-lemma\u2082\n    ... | os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds'' , Fos\u2080'' , Fos\u2081'' , abp =\n      not b , os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds''\n      , Sis' , not-Bool Bb , Fos\u2080'' , Fos\u2081'' , abp\n\n  prf\u2082 : B is (transfer b os\u2080 os\u2081 is)\n  prf\u2082 = b\n       , os\u2080\n       , os\u2081\n       , has a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , Sis\n       , Bb\n       , Fos\u2080\n       , Fos\u2081\n       , has-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , trans (transfer-eq b os\u2080 os\u2081 is) (genTransfer-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is)\n    where\n    a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 : D\n    a\u2081 = send \u00b7 b\n    a\u2082 = ack \u00b7 b\n    a\u2083 = out \u00b7 b\n    a\u2084 = corrupt \u00b7 os\u2080\n    a\u2085 = corrupt \u00b7 os\u2081\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d99b8dbc1d8b11506fe41c1f369efa0c13dc01e3","subject":"Adapt correctness framework to validity-proof embedded \u0394-terms.","message":"Adapt correctness framework to validity-proof embedded \u0394-terms.\n\nOld-commit-hash: 3a38c2651dcf938367d84ffa6dee36838a4b4fb8\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\ndata \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms,\n-- (because it embeds validity proofs)\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192(ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                                 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {_} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {_} {u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car    : A\n    cadr   : B car\n    caddr  : C car cadr\n    cadddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n  \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394 \u03c1\n    dh = \u27e6 df \u27e7\u0394 \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\n\u0394-equiv : \u2200 {\u03c4 : Type} \u2192 (du : \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4) (dv : \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4) \u2192 Set\n\u0394-equiv {\u03c4} du dv =\n  \u2200 {v : \u27e6 \u03c4 \u27e7} (R[v,du] : valid v du) (R[v,dv] : valid v dv) \u2192\n    v \u2295 du \u2261 v \u2295 dv\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u0394-equiv (\u27e6 derive t \u27e7 \u03c1) (\u27e6 t \u27e7 (update \u03c1) \u229d \u27e6 t \u27e7 (ignore \u03c1))\n\ncorollary : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\ncorollary {\u03c4} {\u0393} {t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    v\u2032 = \u27e6 t \u27e7 (update \u03c1)\n  in\n    begin\n      v \u2295 \u27e6 derive t \u27e7 \u03c1\n    \u2261\u27e8 correctness {\u03c4} {\u0393} {t} {\u03c1} {v} validity R[v,u\u229dv] \u27e9\n      v \u2295 (v\u2032 \u229d v)\n    \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n      v\u2032\n    \u220e where open \u2261-Reasoning\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag b\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {nats} {\u0393} {nat n} = refl\nvalidity {bags} {\u0393} {bag b} = tt\nvalidity {\u03c4} {\u0393} {var x} = validity-var x\nvalidity {nats} {\u0393} {add s t} = cong\u2082 _+_ R[s,ds] R[t,dt]\n  where R[s,ds] = validity {nats} {\u0393} {s}\n        R[t,dt] = validity {nats} {\u0393} {t}\nvalidity {bags} {\u0393} {map f b} = tt\n\nvalidity {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v  dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {_} {_} {t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 corollary {_} {_} {t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 {!!} \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\n\nvalidity {\u03c4} {\u0393} {app s t} = {!!}\n\ncorrectness = {!!}\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\nprivate data \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms,\n-- (because it embeds validity proofs)\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\nprivate\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\nabstract\n data \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192(ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                                 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {_} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {_} {u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car    : A\n    cadr   : B car\n    caddr  : C car cadr\n    cadddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nabstract\n \u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n \u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n \u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n \u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n \u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n   \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n \u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n   let\n     (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n     (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n   in\n     (old-s + old-t , new-s + new-t)\n \u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n   let\n     v  = \u27e6 b \u27e7 (ignore \u03c1)\n     h  = \u27e6 f \u27e7 (ignore \u03c1)\n     dv = \u27e6 db \u27e7\u0394 \u03c1\n     dh = \u27e6 df \u27e7\u0394 \u03c1\n   in\n     mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n \u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag b\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {nats} {\u0393} {nat n} {\u03c1} = {!!}\n\nvalidity {bags} {\u0393} {bag b} = {!!}\nvalidity {\u03c4} {\u0393} {var x} = {!!}\nvalidity {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} = {!!}\nvalidity {\u03c4} {\u0393} {app t t\u2081} = {!!}\nvalidity {nats} {\u0393} {add t t\u2081} = {!!}\nvalidity {bags} {\u0393} {map t t\u2081} = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8a0a4851fc2641c8c2555a37e8c04c5ba5d85659","subject":"Regenerate README.agda","message":"Regenerate README.agda\n\nOld-commit-hash: fcba64e4846b84014cac184f5b0c1db45cd742bb\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\nimport Syntax.Constant.Popl14\nimport Syntax.Context.Plotkin\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.DeltaContext\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\nimport Syntax.Language.Popl14\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Atlas\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6e6a81d4cd5e6747138da920db56cf5c044c974c","subject":"agda: rename parameter","message":"agda: rename parameter\n\nAs suggested by Tillmann.\n\nOld-commit-hash: 3b2f1506d9fc2595f06fb0d8e57a526bc30b03ac\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {T : Type} \u2192 \u27e6 T \u27e7 \u2192 \u27e6 \u0394-Type T \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7602ff1910998465653413456eb858d7f47a4949","subject":"flipbased-tree: Ongoing xor proofs...","message":"flipbased-tree: Ongoing xor proofs...\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-tree.agda","new_file":"flipbased-tree.agda","new_contents":"module flipbased-tree where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import bintree\nopen import Data.Product\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\nreturn\u21ba : \u2200 {c a} {A : Set a} \u2192 A \u2192 \u21ba c A\nreturn\u21ba = leaf\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (leaf x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (leaf 0b) (leaf 1b)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (leaf x) = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 c {m a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ c = weaken\u2264 (m\u2264n+m _ c)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (leaf x) = leaf (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (leaf x)      = weaken+ c x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 leaf map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _*\/_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  1-_ : [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n  -- \u00b7\/1+_ : \u2115 \u2192 Set\n  -- \/+\/ : \u2115 \u2192 [0,1] \u2192 [0,1] \u2192 [0,1]\n\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n  _\/2+_\/2-comm : \u2200 x y \u2192 x \/2+ y \/2 \u2261 y \/2+ x \/2\n  1*q\u2261q : \u2200 q \u2192 1\/1 *\/ q \u2261 q\n  0*q\u2261q : \u2200 q \u2192 0\/1 *\/ q \u2261 0\/1\n  distr-*-\/2+\/2 : \u2200 x y z \u2192 (x *\/ z) \/2+ (y *\/ z) \/2 \u2261 (x \/2+ y \/2) *\/ z\n  distr-\/2-* : \u2200 p q \u2192 (p \/2) *\/ (q \/2) \u2261 ((p *\/ q) \/2) \/2\n  0\/2\u22610 : 0\/1 \/2 \u2261 0\/1\n  1-_\/2+1-_\/2 : \u2200 p q \u2192 (1- p) \/2+ (1- q) \/2 \u2261 1- (p \/2+ q \/2)\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\ndata Pr[return\u21ba_\u2261_]\u2261_ {a} {A : Set a} (x y : A) : [0,1] \u2192 Set a where\n  Pr-\u2261 : x \u2261 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 1\/1\n  Pr-\u2262 : x \u2262 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 0\/1\n\ninfix 2 Pr[_\u2261_]\u2261_\ndata Pr[_\u2261_]\u2261_ {a} {A : Set a} : \u2200 {c} \u2192 \u21ba c A \u2192 A \u2192 [0,1] \u2192 Set a where\n  Pr-return : \u2200 {c x y pr} (pf : Pr[return\u21ba x \u2261 y ]\u2261 pr) \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 pr\n\n  Pr-fork : \u2200 {c} {left right : \u21ba c A} {x p q r}\n            (eq : p \/2+ q \/2 \u2261 r)\n            (pf\u2080 : Pr[ left \u2261 x ]\u2261 p)\n            (pf\u2081 : Pr[ right \u2261 x ]\u2261 q)\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 r\n\nPr-fork\u2032 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x p q}\n            \u2192 Pr[ left \u2261 x ]\u2261 p\n            \u2192 Pr[ right \u2261 x ]\u2261 q\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 (p \/2+ q \/2)\nPr-fork\u2032 = Pr-fork refl\n\nPr-return-\u2261 : \u2200 {c a} {A : Set a} {x : A} \u2192 Pr[ return\u21ba {c = c} x \u2261 x ]\u2261 1\/1\nPr-return-\u2261 = Pr-return (Pr-\u2261 refl)\n\nPr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 0\/1\nPr-return-\u2262 = Pr-return \u2218 Pr-\u2262\n\nimport Function.Equality as F\u2261\nimport Function.Equivalence as F\u2248\n\n_\u2248\u21d2_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248\u21d2 p\u2082 = \u2200 {x pr} \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr \u2192 Pr[ p\u2082 \u2261 x ]\u2261 pr\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 {x pr} \u2192 (Pr[ p\u2081 \u2261 x ]\u2261 pr) F\u2248.\u21d4 (Pr[ p\u2082 \u2261 x ]\u2261 pr)\n\n\u2248-refl : \u2200 {c a} {A : Set a} \u2192 Reflexive {A = \u21ba c A} _\u2248_\n\u2248-refl = F\u2248.id\n\n\u2248-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 p\u2081 \u2248 p\u2082 \u2192 p\u2082 \u2248 p\u2081\n\u2248-sym \u03b7 = F\u2248.sym \u03b7\n\n\u2248-trans : \u2200 {c a} {A : Set a} \u2192 Transitive {A = \u21ba c A} _\u2248_\n\u2248-trans f g = g F\u2248.\u2218 f\n\nfork-sym\u21d2 : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248\u21d2 fork p\u2082 p\u2081\nfork-sym\u21d2 (Pr-fork {p = p} {q} refl pf\u2081 pf\u2080) rewrite p \/2+ q \/2-comm = Pr-fork\u2032 pf\u2080 pf\u2081\n\nfork-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\nfork-sym = F\u2248.equivalence fork-sym\u21d2 fork-sym\u21d2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ]\u2261 0\/1\n            \u2192 Pr[ right \u2261 x ]\u2261 p\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork\u2032 eq\u2081 eq\u2082\n\nex\u2081 : \u2200 x \u2192 Pr[ toss \u2261 x ]\u2261 1\/2\nex\u2081 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261\nex\u2081 0b = F\u2248.Equivalence.to fork-sym F\u2261.\u27e8$\u27e9 (Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261)\n\nPr-map : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {x pr} {f : A \u2192 B} \u2192\n  (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n  Pr[ Alg \u2261 x ]\u2261 pr \u2192\n  Pr[ \u27ea f \u00b7 Alg \u27eb \u2261 f x ]\u2261 pr\nPr-map f-inj (Pr-return (Pr-\u2261 refl)) = Pr-return (Pr-\u2261 refl)\nPr-map f-inj (Pr-return (Pr-\u2262 x\u2262y)) = Pr-return (Pr-\u2262 (x\u2262y \u2218 f-inj))\nPr-map f-inj (Pr-fork eq pf\u2080 pf\u2081) = Pr-fork eq (Pr-map f-inj pf\u2080) (Pr-map f-inj pf\u2081)\n\nPr-same : \u2200 {c a} {A : Set a} {Alg : \u21ba c A} {x pr\u2080 pr\u2081} \u2192\n    pr\u2080 \u2261 pr\u2081 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2080 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2081\nPr-same refl = id\n\nPr-weaken\u2264 : \u2200 {c\u2080 c\u2081 a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    (c\u2080\u2264c\u2081 : c\u2080 \u2264 c\u2081) \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken\u2264 c\u2080\u2264c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken\u2264 p (Pr-return pf) = Pr-return pf\nPr-weaken\u2264 (s\u2264s c\u2080\u2264c\u2081) (Pr-fork eq pf\u2080 pf\u2081)\n  = Pr-fork eq (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2080) (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2081)\n\nPr-weaken+ : \u2200 {c\u2080} c\u2081 {a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken+ c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken+ c\u2081 = Pr-weaken\u2264 (m\u2264n+m _ c\u2081)\n\nPr-map-0 : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {f : A \u2192 B} {x} \u2192 (\u2200 y \u2192 f y \u2262 x)\n           \u2192 Pr[ map\u21ba f Alg \u2261 x ]\u2261 0\/1\nPr-map-0 {Alg = leaf x} f-prop = Pr-return (Pr-\u2262 (f-prop x))\nPr-map-0 {Alg = fork Alg Alg\u2081} f-prop = Pr-fork (trans (0\/2+p\/2\u2261p\/2 0\/1) 0\/2\u22610)\n                                                (Pr-map-0 f-prop) (Pr-map-0 f-prop)\n\nPr-zip : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} {Alg\u2080 : \u21ba c\u2080 A} {Alg\u2081 : \u21ba c\u2081 B} {x y pr\u2081 pr\u2082} \u2192\n  Pr[ Alg\u2080 \u2261 x ]\u2261 pr\u2081 \u2192\n  Pr[ Alg\u2081 \u2261 y ]\u2261 pr\u2082 \u2192\n  Pr[ zip\u21ba Alg\u2080 Alg\u2081 \u2261 (x , y) ]\u2261 (pr\u2081 *\/ pr\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2261 refl)) pf\u2082\n  rewrite 1*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map {f = _,_ x} (cong proj\u2082) pf\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2262 pf)) pf\u2082\n  rewrite 0*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map-0 (\u03bb y x\u2081 \u2192 pf (cong proj\u2081 x\u2081)))\nPr-zip (Pr-fork refl pf\u2081 pf\u2082) pf\u2083 = Pr-fork (distr-*-\/2+\/2 _ _ _) (Pr-zip pf\u2081 pf\u2083) (Pr-zip pf\u2082 pf\u2083)\n\nex\u2082 : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ]\u2261 1\/4\nex\u2082 x y = Pr-same (trans (distr-\/2-* _ _) (cong (_\/2 \u2218 _\/2) (1*q\u2261q _)))\n                  (Pr-zip {Alg\u2080 = toss} {Alg\u2081 = toss} (ex\u2081 x) (ex\u2081 y))\n\npr-choose : \u2200 {c a} {A : Set a} {p\u2080 p\u2081 : \u21ba c A} {pr\u2080 pr\u2081 x}\n             \u2192 Pr[ p\u2080 \u2261 x ]\u2261 pr\u2080\n             \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr\u2081\n             \u2192 Pr[ choose p\u2080 p\u2081 \u2261 x ]\u2261 pr\u2080 \/2+ pr\u2081 \/2\npr-choose {c} {pr\u2080 = pr\u2080} {pr\u2081} pf\u2080 pf\u2081 =\n  Pr-fork (pr\u2081 \/2+ pr\u2080 \/2-comm) (Pr-weaken+ 0 pf\u2081) (Pr-weaken+ 0 pf\u2080)\n\npostulate\n  pr-ret-xor : \u2200 {x y pr} b\n         \u2192 Pr[return\u21ba x \u2261 y ]\u2261 pr\n         \u2192 Pr[return\u21ba x \u2261 b xor y ]\u2261 pr\n-- pr-ret-xor b pf = {!!}\n\npr-xor'' : \u2200 {c} {Alg : \u21ba c Bit} {x pr} b\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ Alg \u2261 b xor x ]\u2261 pr\npr-xor'' b (Pr-return pf) = Pr-return (pr-ret-xor b pf)\npr-xor'' b (Pr-fork refl pf\u2080 pf\u2081) = Pr-fork refl (pr-xor'' b pf\u2080) (pr-xor'' b pf\u2081)\n\npr-xor' : \u2200 {c} {Alg : \u21ba c Bit} {x pr b}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ \u27ea _xor_ b \u00b7 Alg \u27eb \u2261 b xor x ]\u2261 pr\npr-xor' {b = b} = Pr-map (xor-inj {b})\n  where\n    -- move it!\n    not-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\n    not-inj {true} {true} = \u03bb _ \u2192 refl\n    not-inj {true} {false} = \u03bb ()\n    not-inj {false} {true} = \u03bb ()\n    not-inj {false} {false} = \u03bb _ \u2192 refl\n    xor-inj : \u2200 {b x y} \u2192 b xor x \u2261 b xor y \u2192 x \u2261 y\n    xor-inj {true} = not-inj\n    xor-inj {false} = id\n\npostulate\n  pr-ret-not : \u2200 {x y pr}\n         \u2192 Pr[return\u21ba x \u2261 y ]\u2261 pr\n         \u2192 Pr[return\u21ba x \u2261 not y ]\u2261 1- pr\n-- pr-ret-not = {!!}\n\npr-not : \u2200 {c} {Alg : \u21ba c Bit} {x pr}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ Alg \u2261 not x ]\u2261 1- pr\npr-not (Pr-return pf) = Pr-return (pr-ret-not pf)\npr-not (Pr-fork refl pf pf\u2081) = Pr-fork (1- _ \/2+1- _ \/2) (pr-not pf) (pr-not pf\u2081)\n\npostulate\n  pr-xor : \u2200 {c} {Alg : \u21ba c Bit} {x pr b}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ \u27ea _xor_ b \u00b7 Alg \u27eb \u2261 x ]\u2261 pr\n-- pr-xor {b = b} = {!pr-not!}\n\npostulate\n  pr-toss-xor-toss : \u2200 x \u2192 Pr[ toss \u27e8xor\u27e9 toss \u2261 x ]\u2261 1\/2\n-- pr-toss-xor-toss x = {!!}\n\npostulate\n  ex\u2083 : \u2200 {n} (x : Bits n) \u2192 Pr[ random \u2261 x ]\u2261 1\/2^ n\n","old_contents":"module flipbased-tree where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import bintree\nopen import Data.Product\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\nreturn\u21ba : \u2200 {c a} {A : Set a} \u2192 A \u2192 \u21ba c A\nreturn\u21ba = leaf\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (leaf x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (leaf 0b) (leaf 1b)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (leaf x) = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 c {m a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ c = weaken\u2264 (m\u2264n+m _ c)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (leaf x) = leaf (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (leaf x)      = weaken+ c x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 leaf map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _*\/_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n  -- \u00b7\/1+_ : \u2115 \u2192 Set\n  -- \/+\/ : \u2115 \u2192 [0,1] \u2192 [0,1] \u2192 [0,1]\n\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n  _\/2+_\/2-comm : \u2200 x y \u2192 x \/2+ y \/2 \u2261 y \/2+ x \/2\n  1*q\u2261q : \u2200 q \u2192 1\/1 *\/ q \u2261 q\n  0*q\u2261q : \u2200 q \u2192 0\/1 *\/ q \u2261 0\/1\n  distr-*-\/2+\/2 : \u2200 x y z \u2192 (x *\/ z) \/2+ (y *\/ z) \/2 \u2261 (x \/2+ y \/2) *\/ z\n  distr-\/2-* : \u2200 p q \u2192 (p \/2) *\/ (q \/2) \u2261 ((p *\/ q) \/2) \/2\n  0\/2\u22610 : 0\/1 \/2 \u2261 0\/1\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\ndata Pr[return\u21ba_\u2261_]\u2261_ {a} {A : Set a} (x y : A) : [0,1] \u2192 Set a where\n  Pr-\u2261 : x \u2261 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 1\/1\n  Pr-\u2262 : x \u2262 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 0\/1\n\ninfix 2 Pr[_\u2261_]\u2261_\ndata Pr[_\u2261_]\u2261_ {a} {A : Set a} : \u2200 {c} \u2192 \u21ba c A \u2192 A \u2192 [0,1] \u2192 Set a where\n  Pr-return : \u2200 {c x y pr} (pf : Pr[return\u21ba x \u2261 y ]\u2261 pr) \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 pr\n\n  Pr-fork : \u2200 {c} {left right : \u21ba c A} {x p q r}\n            (eq : p \/2+ q \/2 \u2261 r)\n            (pf\u2080 : Pr[ left \u2261 x ]\u2261 p)\n            (pf\u2081 : Pr[ right \u2261 x ]\u2261 q)\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 r\n\nPr-fork\u2032 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x p q}\n            \u2192 Pr[ left \u2261 x ]\u2261 p\n            \u2192 Pr[ right \u2261 x ]\u2261 q\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 (p \/2+ q \/2)\nPr-fork\u2032 = Pr-fork refl\n\nPr-return-\u2261 : \u2200 {c a} {A : Set a} {x : A} \u2192 Pr[ return\u21ba {c = c} x \u2261 x ]\u2261 1\/1\nPr-return-\u2261 = Pr-return (Pr-\u2261 refl)\n\nPr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 0\/1\nPr-return-\u2262 = Pr-return \u2218 Pr-\u2262\n\nimport Function.Equality as F\u2261\nimport Function.Equivalence as F\u2248\n\n_\u2248\u21d2_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248\u21d2 p\u2082 = \u2200 {x pr} \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr \u2192 Pr[ p\u2082 \u2261 x ]\u2261 pr\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 {x pr} \u2192 (Pr[ p\u2081 \u2261 x ]\u2261 pr) F\u2248.\u21d4 (Pr[ p\u2082 \u2261 x ]\u2261 pr)\n\n\u2248-refl : \u2200 {c a} {A : Set a} \u2192 Reflexive {A = \u21ba c A} _\u2248_\n\u2248-refl = F\u2248.id\n\n\u2248-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 p\u2081 \u2248 p\u2082 \u2192 p\u2082 \u2248 p\u2081\n\u2248-sym \u03b7 = F\u2248.sym \u03b7\n\n\u2248-trans : \u2200 {c a} {A : Set a} \u2192 Transitive {A = \u21ba c A} _\u2248_\n\u2248-trans f g = g F\u2248.\u2218 f\n\nfork-sym\u21d2 : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248\u21d2 fork p\u2082 p\u2081\nfork-sym\u21d2 (Pr-fork {p = p} {q} refl pf\u2081 pf\u2080) rewrite p \/2+ q \/2-comm = Pr-fork\u2032 pf\u2080 pf\u2081\n\nfork-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\nfork-sym = F\u2248.equivalence fork-sym\u21d2 fork-sym\u21d2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ]\u2261 0\/1\n            \u2192 Pr[ right \u2261 x ]\u2261 p\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork\u2032 eq\u2081 eq\u2082\n\nex\u2081 : \u2200 x \u2192 Pr[ toss \u2261 x ]\u2261 1\/2\nex\u2081 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261\nex\u2081 0b = F\u2248.Equivalence.to fork-sym F\u2261.\u27e8$\u27e9 (Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261)\n\nPr-map : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {x pr} {f : A \u2192 B} \u2192\n  (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n  Pr[ Alg \u2261 x ]\u2261 pr \u2192\n  Pr[ \u27ea f \u00b7 Alg \u27eb \u2261 f x ]\u2261 pr\nPr-map f-inj (Pr-return (Pr-\u2261 refl)) = Pr-return (Pr-\u2261 refl)\nPr-map f-inj (Pr-return (Pr-\u2262 x\u2262y)) = Pr-return (Pr-\u2262 (x\u2262y \u2218 f-inj))\nPr-map f-inj (Pr-fork eq pf\u2080 pf\u2081) = Pr-fork eq (Pr-map f-inj pf\u2080) (Pr-map f-inj pf\u2081)\n\nPr-same : \u2200 {c a} {A : Set a} {Alg : \u21ba c A} {x pr\u2080 pr\u2081} \u2192\n    pr\u2080 \u2261 pr\u2081 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2080 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2081\nPr-same refl = id\n\nPr-weaken\u2264 : \u2200 {c\u2080 c\u2081 a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    (c\u2080\u2264c\u2081 : c\u2080 \u2264 c\u2081) \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken\u2264 c\u2080\u2264c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken\u2264 p (Pr-return pf) = Pr-return pf\nPr-weaken\u2264 (s\u2264s c\u2080\u2264c\u2081) (Pr-fork eq pf\u2080 pf\u2081)\n  = Pr-fork eq (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2080) (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2081)\n\nPr-weaken+ : \u2200 {c\u2080} c\u2081 {a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken+ c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken+ c\u2081 = Pr-weaken\u2264 (m\u2264n+m _ c\u2081)\n\nPr-map-0 : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {f : A \u2192 B} {x} \u2192 (\u2200 y \u2192 f y \u2262 x)\n           \u2192 Pr[ map\u21ba f Alg \u2261 x ]\u2261 0\/1\nPr-map-0 {Alg = leaf x} f-prop = Pr-return (Pr-\u2262 (f-prop x))\nPr-map-0 {Alg = fork Alg Alg\u2081} f-prop = Pr-fork (trans (0\/2+p\/2\u2261p\/2 0\/1) 0\/2\u22610)\n                                                (Pr-map-0 f-prop) (Pr-map-0 f-prop)\n\nPr-zip : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} {Alg\u2080 : \u21ba c\u2080 A} {Alg\u2081 : \u21ba c\u2081 B} {x y pr\u2081 pr\u2082} \u2192\n  Pr[ Alg\u2080 \u2261 x ]\u2261 pr\u2081 \u2192\n  Pr[ Alg\u2081 \u2261 y ]\u2261 pr\u2082 \u2192\n  Pr[ zip\u21ba Alg\u2080 Alg\u2081 \u2261 (x , y) ]\u2261 (pr\u2081 *\/ pr\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2261 refl)) pf\u2082\n  rewrite 1*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map {f = _,_ x} (cong proj\u2082) pf\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2262 pf)) pf\u2082\n  rewrite 0*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map-0 (\u03bb y x\u2081 \u2192 pf (cong proj\u2081 x\u2081)))\nPr-zip (Pr-fork refl pf\u2081 pf\u2082) pf\u2083 = Pr-fork (distr-*-\/2+\/2 _ _ _) (Pr-zip pf\u2081 pf\u2083) (Pr-zip pf\u2082 pf\u2083)\n\nex\u2082 : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ]\u2261 1\/4\nex\u2082 x y = Pr-same (trans (distr-\/2-* _ _) (cong (_\/2 \u2218 _\/2) (1*q\u2261q _)))\n                  (Pr-zip {Alg\u2080 = toss} {Alg\u2081 = toss} (ex\u2081 x) (ex\u2081 y))\n\npostulate\n  ex\u2083 : \u2200 {n} (x : Bits n) \u2192 Pr[ random \u2261 x ]\u2261 1\/2^ n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"acb7dfad8b6ef7aeaa2800916aa6da9df5bf1c30","subject":"Reemphasize the importance of \"import Everything\".","message":"Reemphasize the importance of \"import Everything\".\n\nOld-commit-hash: d6200d73c289f7ab4460589159b8380be47e083b\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"83d6259ce1197a7fe5da67c77747c8df330e7d7c","subject":"ToNat: + \u2264-\u2294","message":"ToNat: + \u2264-\u2294\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/ToNat.agda","new_file":"lib\/Relation\/Binary\/ToNat.agda","new_contents":"import Data.Nat.NP as \u2115\nopen \u2115 using (\u2115; zero; suc; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Function\nopen import Relation.Binary\nopen import Algebra.FunctionProperties\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Relation.Binary.ToNat {a} {A : Set a} (f : A \u2192 \u2115) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) where\n\n_<=_ : A \u2192 A \u2192 Bool\n_<=_ = \u2115._<=_ on f \n\n_\u2264_ : A \u2192 A \u2192 Set\nx \u2264 y = T (x <= y)\n\n_\u2293_ : A \u2192 A \u2192 A\nx \u2293 y = if x <= y then x else y\n\n_\u2294_ : A \u2192 A \u2192 A\nx \u2294 y = if x <= y then y else x\n\nisPreorder : IsPreorder _\u2261_ _\u2264_\nisPreorder = record { isEquivalence = \u2261.isEquivalence\n                    ; reflexive = reflexive\n                    ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n  where open \u2115.<=\n        reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 i \u2264 j\n        reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {f i})\n        trans : Transitive _\u2264_\n        trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound (f x) (f y) p) (sound (f y) (f z) q))\n\nisPartialOrder : IsPartialOrder _\u2261_ _\u2264_\nisPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n  where open \u2115.<= using (sound)\n        antisym : Antisymmetric _\u2261_ _\u2264_\n        antisym {x} {y} p q = f-inj (\u2115\u2264.antisym (sound (f x) (f y) p) (sound (f y) (f x) q))\n\nisTotalOrder : IsTotalOrder _\u2261_ _\u2264_\nisTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total (f x) (f y)) }\n  where open \u2115.<= using (complete)\n\nopen IsTotalOrder isTotalOrder\n\n\u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n\u2294-spec {x} {y} p with x <= y\n\u2294-spec _    | true = \u2261.refl\n\u2294-spec ()   | false\n\n\u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n\u2293-spec {x} {y} p with x <= y\n\u2293-spec _    | true = \u2261.refl\n\u2293-spec ()   | false\n\n\u2293-comm : Commutative _\u2261_ _\u2293_\n\u2293-comm x y with x <= y | y <= x | antisym {x} {y} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2294-comm : Commutative _\u2261_ _\u2294_\n\u2294-comm x y with x <= y | y <= x | antisym {y} {x} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n\u2293-\u2264 x y with total x y\n\u2293-\u2264 x y | inj\u2081 p rewrite \u2293-spec p = p\n\u2293-\u2264 x y | inj\u2082 p rewrite \u2293-comm x y | \u2293-spec p = refl\n\n\u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n\u2264-\u2294 x y with total x y\n\u2264-\u2294 x y | inj\u2081 p rewrite \u2294-comm y x | \u2294-spec p = p\n\u2264-\u2294 x y | inj\u2082 p rewrite \u2294-spec p = refl\n\n\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n\u2264-<_,_> {x} {y} {z} x\u2264y y\u2264z with total y z\n... | inj\u2081 p rewrite \u2293-spec p = x\u2264y\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = y\u2264z\n\n\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n\u2264-[_,_] {x} {y} {z} x\u2264z y\u2264z with total x y\n... | inj\u2081 p rewrite \u2294-spec p = y\u2264z\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = x\u2264z\n\n\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n\u2264-\u2293\u2080 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = id\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = flip trans p\n\n\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n\u2264-\u2293\u2081 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = flip trans p\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = id\n\n\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n\u2264-\u2294\u2080 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = trans p\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = id\n\n\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\u2264-\u2294\u2081 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = id\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = trans p\n","old_contents":"import Data.Nat.NP as \u2115\nopen \u2115 using (\u2115; zero; suc; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Function\nopen import Relation.Binary\nopen import Algebra.FunctionProperties\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Relation.Binary.ToNat {a} {A : Set a} (f : A \u2192 \u2115) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) where\n\n_<=_ : A \u2192 A \u2192 Bool\n_<=_ = \u2115._<=_ on f \n\n_\u2264_ : A \u2192 A \u2192 Set\nx \u2264 y = T (x <= y)\n\n_\u2293_ : A \u2192 A \u2192 A\nx \u2293 y = if x <= y then x else y\n\n_\u2294_ : A \u2192 A \u2192 A\nx \u2294 y = if x <= y then y else x\n\nisPreorder : IsPreorder _\u2261_ _\u2264_\nisPreorder = record { isEquivalence = \u2261.isEquivalence\n                    ; reflexive = reflexive\n                    ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n  where open \u2115.<=\n        reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 i \u2264 j\n        reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {f i})\n        trans : Transitive _\u2264_\n        trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound (f x) (f y) p) (sound (f y) (f z) q))\n\nisPartialOrder : IsPartialOrder _\u2261_ _\u2264_\nisPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n  where open \u2115.<= using (sound)\n        antisym : Antisymmetric _\u2261_ _\u2264_\n        antisym {x} {y} p q = f-inj (\u2115\u2264.antisym (sound (f x) (f y) p) (sound (f y) (f x) q))\n\nisTotalOrder : IsTotalOrder _\u2261_ _\u2264_\nisTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total (f x) (f y)) }\n  where open \u2115.<= using (complete)\n\nopen IsTotalOrder isTotalOrder\n\n\u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n\u2294-spec {x} {y} p with x <= y\n\u2294-spec _    | true = \u2261.refl\n\u2294-spec ()   | false\n\n\u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n\u2293-spec {x} {y} p with x <= y\n\u2293-spec _    | true = \u2261.refl\n\u2293-spec ()   | false\n\n\u2293-comm : Commutative _\u2261_ _\u2293_\n\u2293-comm x y with x <= y | y <= x | antisym {x} {y} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2294-comm : Commutative _\u2261_ _\u2294_\n\u2294-comm x y with x <= y | y <= x | antisym {y} {x} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n\u2293-\u2264 x y with total x y\n\u2293-\u2264 x y | inj\u2081 p rewrite \u2293-spec p = p\n\u2293-\u2264 x y | inj\u2082 p rewrite \u2293-comm x y | \u2293-spec p = refl\n\n\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n\u2264-<_,_> {x} {y} {z} x\u2264y y\u2264z with total y z\n... | inj\u2081 p rewrite \u2293-spec p = x\u2264y\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = y\u2264z\n\n\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n\u2264-[_,_] {x} {y} {z} x\u2264z y\u2264z with total x y\n... | inj\u2081 p rewrite \u2294-spec p = y\u2264z\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = x\u2264z\n\n\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n\u2264-\u2293\u2080 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = id\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = flip trans p\n\n\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n\u2264-\u2293\u2081 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = flip trans p\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = id\n\n\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n\u2264-\u2294\u2080 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = trans p\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = id\n\n\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\u2264-\u2294\u2081 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = id\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = trans p\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ef8b8ce4e463417ddc66f1c9d64f5f7497d0389e","subject":"Desc model: implicit box","message":"Desc model: implicit box","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box I D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box Unit (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d186c9c55549e759484d079ec26e88aa5dd4bfef","subject":"Removed general hints.","message":"Removed general hints.\n\nIgnore-this: 2f7102a575b4c2fdb80e99fde9cc97f0\n\ndarcs-hash:20100710142946-3bd4e-567a84c7ee48f426f69c53a0b41c8fb40007e53b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Bool\/Properties.agda","new_file":"LTC\/Data\/Bool\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- The booleans properties\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Bool.Properties where\n\nopen import LTC.Minimal\nopen import LTC.Data.Bool\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.Properties using ( \u2264-SS ; S\u22700 )\n\n------------------------------------------------------------------------------\n-- Basic properties.\n\n&&-Bool : {b\u2081 b\u2082 : D} \u2192 Bool b\u2081 \u2192 Bool b\u2082 \u2192 Bool (b\u2081 && b\u2082)\n&&-Bool tB tB = prf\n  where\n    postulate prf : Bool (true && true)\n    {-# ATP prove prf #-}\n&&-Bool tB fB = prf\n  where\n    postulate prf : Bool (true && false)\n    {-# ATP prove prf #-}\n&&-Bool fB tB = prf\n  where\n    postulate prf : Bool (false && true)\n    {-# ATP prove prf #-}\n&&-Bool fB fB = prf\n  where\n    postulate prf : Bool (false && false)\n    {-# ATP prove prf #-}\n\nx&&y\u2261true\u2192x\u2261true : {b\u2081 b\u2082 : D} \u2192 Bool b\u2081 \u2192 Bool b\u2082 \u2192 b\u2081 && b\u2082 \u2261 true \u2192\n                   b\u2081 \u2261 true\nx&&y\u2261true\u2192x\u2261true tB _ _    = refl\nx&&y\u2261true\u2192x\u2261true fB tB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-ft))\nx&&y\u2261true\u2192x\u2261true fB fB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-ff))\n\nx&&y\u2261true\u2192y\u2261true : {b\u2081 b\u2082 : D} \u2192 Bool b\u2081 \u2192 Bool b\u2082 \u2192 b\u2081 && b\u2082 \u2261 true \u2192\n                   b\u2082 \u2261 true\nx&&y\u2261true\u2192y\u2261true _  tB _   = refl\nx&&y\u2261true\u2192y\u2261true tB fB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-tf))\nx&&y\u2261true\u2192y\u2261true fB fB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-ff))\n\n------------------------------------------------------------------------------\n-- Properties with inequalities\n\n\u2264-Bool : {m n : D} \u2192 N m \u2192 N n \u2192 Bool (m \u2264 n)\n\u2264-Bool {n = n} zN Nn = prf\n  where\n    postulate prf : Bool (zero \u2264 n)\n    {-# ATP prove prf #-}\n\u2264-Bool (sN {m} Nm) zN = prf\n  where\n    postulate prf : Bool (succ m \u2264 zero)\n    {-# ATP prove prf S\u22700 #-}\n\u2264-Bool (sN {m} Nm) (sN {n} Nn) = prf (\u2264-Bool Nm Nn)\n  where\n    postulate prf : Bool (m \u2264 n) \u2192 Bool (succ m \u2264 succ n)\n    {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The booleans properties\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Bool.Properties where\n\nopen import LTC.Minimal\nopen import LTC.Data.Bool\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.Properties using ( \u2264-SS ; S\u22700 )\n\n------------------------------------------------------------------------------\n-- Basic properties.\n\n-- This function is a general hint.\n&&-Bool : {b\u2081 b\u2082 : D} \u2192 Bool b\u2081 \u2192 Bool b\u2082 \u2192 Bool (b\u2081 && b\u2082)\n&&-Bool tB tB = prf\n  where\n    postulate prf : Bool (true && true)\n    {-# ATP prove prf #-}\n&&-Bool tB fB = prf\n  where\n    postulate prf : Bool (true && false)\n    {-# ATP prove prf #-}\n&&-Bool fB tB = prf\n  where\n    postulate prf : Bool (false && true)\n    {-# ATP prove prf #-}\n&&-Bool fB fB = prf\n  where\n    postulate prf : Bool (false && false)\n    {-# ATP prove prf #-}\n{-# ATP hint &&-Bool #-}\n\nx&&y\u2261true\u2192x\u2261true : {b\u2081 b\u2082 : D} \u2192 Bool b\u2081 \u2192 Bool b\u2082 \u2192 b\u2081 && b\u2082 \u2261 true \u2192\n                   b\u2081 \u2261 true\nx&&y\u2261true\u2192x\u2261true tB _ _    = refl\nx&&y\u2261true\u2192x\u2261true fB tB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-ft))\nx&&y\u2261true\u2192x\u2261true fB fB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-ff))\n\nx&&y\u2261true\u2192y\u2261true : {b\u2081 b\u2082 : D} \u2192 Bool b\u2081 \u2192 Bool b\u2082 \u2192 b\u2081 && b\u2082 \u2261 true \u2192\n                   b\u2082 \u2261 true\nx&&y\u2261true\u2192y\u2261true _  tB _   = refl\nx&&y\u2261true\u2192y\u2261true tB fB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-tf))\nx&&y\u2261true\u2192y\u2261true fB fB prf = \u22a5-elim (true\u2260false (trans (sym prf) &&-ff))\n\n------------------------------------------------------------------------------\n-- Properties with inequalities\n\n-- This function is a general hint.\n\u2264-Bool : {m n : D} \u2192 N m \u2192 N n \u2192 Bool (m \u2264 n)\n\u2264-Bool {n = n} zN Nn = prf\n  where\n    postulate prf : Bool (zero \u2264 n)\n    {-# ATP prove prf #-}\n\u2264-Bool (sN {m} Nm) zN = prf\n  where\n    postulate prf : Bool (succ m \u2264 zero)\n    {-# ATP prove prf S\u22700 #-}\n\u2264-Bool (sN {m} Nm) (sN {n} Nn) = prf (\u2264-Bool Nm Nn)\n  where\n    postulate prf : Bool (m \u2264 n) \u2192 Bool (succ m \u2264 succ n)\n    {-# ATP prove prf #-}\n{-# ATP hint \u2264-Bool #-}\n\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6fe83e1d6365cf25bee0bdc823a5212aa8024ac7","subject":"Modified a test.","message":"Modified a test.\n\nIgnore-this: e571f29238bfd5b15fa8eb93e8a0edc4\n\ndarcs-hash:20100803142440-3bd4e-1d6f95a0094c35a9ec7f3ad30fa19b594c8fe390.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/DefinitionsInsideWhereClauses.agda","new_file":"Test\/Succeed\/DefinitionsInsideWhereClauses.agda","new_contents":"module Test.Succeed.DefinitionsInsideWhereClauses where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity {n} Nn = indN P P0 iStep Nn\n  where\n    P : D \u2192 Set\n    P i = i + zero \u2261 i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n","old_contents":"module Test.Succeed.DefinitionsInsideWhereClauses where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity {n} Nn = indN Q Q0 iStep Nn\n  where\n    Q : D \u2192 Set\n    Q i = i + zero \u2261 i\n    {-# ATP definition Q #-}\n\n    postulate\n      Q0 : Q zero\n    {-# ATP prove Q0 #-}\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192 Q i \u2192 Q (succ i)\n    {-# ATP prove iStep #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4b8af6242399a26626161c4b8f6ed956f115c0d6","subject":"Avoid const in favor of lift-\u03b7-const.","message":"Avoid const in favor of lift-\u03b7-const.\n\nThis commit prepares for changing the type of const.\n\nOld-commit-hash: d7ebdbeb22d70fa60c6ce08ebf4832f56797b59a\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin Atlas-type\n\ndata Atlas-const : Type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type \u2192 Type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nopen import Syntax.Context {Type}\nopen import Syntax.Term.Plotkin {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\ntrue! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool)\ntrue! = lift-\u03b7-const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool)\nfalse! = lift-\u03b7-const false\n\nxor! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool) \u2192 Term \u0393 (base Bool) \u2192\n  Term \u0393 (base Bool)\nxor! = lift-\u03b7-const xor\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base (Map \u03ba \u03b9))\nempty! = lift-\u03b7-const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Term \u0393 (base (Map \u03ba \u03b9))\nupdate! = lift-\u03b7-const update\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Term \u0393 (base \u03b9)\nlookup! = lift-\u03b7-const lookup\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Term \u0393 (base (Map \u03ba a)) \u2192 Term \u0393 (base (Map \u03ba b)) \u2192\n  Term \u0393 (base (Map \u03ba c))\nzip! = lift-\u03b7-const zip\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Term \u0393 (base b) \u2192 Term \u0393 (base (Map \u03ba a)) \u2192\n  Term \u0393 (base b)\nfold! = lift-\u03b7-const fold\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9)\nneutral-term {Bool}   = lift-\u03b7-const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = lift-\u03b7-const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = lift-\u03b7-const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = lift-\u03b7-const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Term \u0393 (base \u03b1) \u2192 Term \u0393 (base \u03b2) \u2192\n  Term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Term \u0393 (base (Map \u03ba a)) \u2192 Term \u0393 (base (Map \u03ba b)) \u2192\n  Term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = abs (abs (lift-\u03b7-const xor (var (that this)) (var this)))\nAtlas-diff {Map \u03ba \u03b9} = abs (abs (lift-\u03b7-const zip (abs Atlas-diff) (var (that this)) (var this)))\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = abs (abs (lift-\u03b7-const xor (var (that this)) (var this)))\nAtlas-apply {Map \u03ba \u03b9} = abs (abs (lift-\u03b7-const zip (abs Atlas-apply) (var (that this)) (var this)))\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Term \u0393 (base \u03b9)\n  in\n    Term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin Atlas-type\n\ndata Atlas-const : Type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type \u2192 Type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nopen import Syntax.Context {Type}\nopen import Syntax.Term.Plotkin {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\ntrue! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool)\ntrue! = lift-\u03b7-const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool)\nfalse! = lift-\u03b7-const false\n\nxor! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool) \u2192 Term \u0393 (base Bool) \u2192\n  Term \u0393 (base Bool)\nxor! = lift-\u03b7-const xor\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base (Map \u03ba \u03b9))\nempty! = lift-\u03b7-const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Term \u0393 (base (Map \u03ba \u03b9))\nupdate! = lift-\u03b7-const update\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Term \u0393 (base \u03b9)\nlookup! = lift-\u03b7-const lookup\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Term \u0393 (base (Map \u03ba a)) \u2192 Term \u0393 (base (Map \u03ba b)) \u2192\n  Term \u0393 (base (Map \u03ba c))\nzip! = lift-\u03b7-const zip\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Term \u0393 (base b) \u2192 Term \u0393 (base (Map \u03ba a)) \u2192\n  Term \u0393 (base b)\nfold! = lift-\u03b7-const fold\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Term \u0393 (base \u03b1) \u2192 Term \u0393 (base \u03b2) \u2192\n  Term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Term \u0393 (base (Map \u03ba a)) \u2192 Term \u0393 (base (Map \u03ba b)) \u2192\n  Term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Term \u0393 (base \u03b9)\n  in\n    Term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7dbb7878763fef3587923b0005e33195c0135014","subject":"Add tagged descriptions to model.","message":"Add tagged descriptions to model.\n\nThis lets elim just take a single \"Desc\" arg.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nTagDesc : (I : Set) \u2192 Set\nTagDesc I = \u03a3 Enum (\u03bb E \u2192 Cases E (\u03bb _ \u2192 Desc I))\n\ntoCase : (I : Set) (E,cs : TagDesc I) \u2192 Tag (proj\u2081 E,cs) \u2192 Desc I\ntoCase I (E , cs) = case E (\u03bb _ \u2192 Desc I) cs\n\ntoDesc : (I : Set) \u2192 TagDesc I \u2192 Desc I\ntoDesc I (E , cs) = `Arg (Tag E) (toCase I (E , cs))\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ndata \u03bc (I : Set) (D : Desc I) : I \u2192 Set where\n  con : UncurriedEl I D (\u03bc I D)\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (TD : TagDesc I)\n  \u2192 let\n    D = toDesc I TD\n    E = proj\u2081 TD\n    Cs = toCase I TD\n  in (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let\n    Q = \u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))\n    X = (i : I) (x : \u03bc I D i) \u2192 P i x\n  in UncurriedCases E Q X\nelim I TD P cs i x =\n  let\n    D = toDesc I TD\n    E = proj\u2081 TD\n    Cs = toCase I TD\n    p = case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs\n  in ind2 I D P p i x\n\nelim2 :\n  (I : Set)\n  (TD : TagDesc I)\n  \u2192 let\n    D = toDesc I TD\n    E = proj\u2081 TD\n    Cs = toCase I TD\n  in (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let\n    Q = \u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))\n    X = (i : I) (x : \u03bc I D i) \u2192 P i x\n  in CurriedCases E Q X\nelim2 I TD P =\n  let\n    D = toDesc I TD\n    E = proj\u2081 TD\n    Cs = toCase I TD\n    Q = \u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))\n    X = (i : I) (x : \u03bc I D i) \u2192 P i x\n  in curryCases E Q X (elim I TD P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n\n  \u2115TD : TagDesc \u22a4\n  \u2115TD = \u2115T\n    , `End tt\n    , `Rec tt (`End tt)\n    , tt\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = toCase \u22a4 \u2115TD\n  \n  \u2115D : Desc \u22a4\n  \u2115D = toDesc \u22a4 \u2115TD\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecTD : (A : Set) \u2192 TagDesc (\u2115 tt)\n  VecTD A = VecT\n    , `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = toCase (\u2115 tt) (VecTD A)\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = toDesc (\u2115 tt) (VecTD A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115TD _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115TD _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) (VecTD A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) (VecTD (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ndata \u03bc (I : Set) (D : Desc I) : I \u2192 Set where\n  con : UncurriedEl I D (\u03bc I D)\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 UncurriedCases E\n    (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) \n    ((i : I) (x : \u03bc I D i) \u2192 P i x)\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\nelim2 :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n        X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in CurriedCases E Q X\nelim2 I E Cs P =\n  let D = `Arg (Tag E) Cs\n      Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n      X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in curryCases E Q X (elim I E Cs P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) VecT (VecC A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) VecT (VecC (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"be764a2fe0b6c2a23e9f4f348a8a152d3071d5ae","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: 2a9374401357d23b6287d5d86a825749\n\ndarcs-hash:20120323134058-3bd4e-217b454a1a805ea13e62f9a24a8e6663c006a0c0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Relation\/Binary\/Bisimilarity.agda","new_file":"src\/FOTC\/Relation\/Binary\/Bisimilarity.agda","new_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The bisimilarity relation _\u2248_ is the greatest fixed point (by\n-- \u2248-gfp\u2081 and \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation _\u2248_ is a post-fixed point of the\n-- bisimulation functional (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n-- the bisimulation functional (see below).\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed-point, the\n-- bisimilarity relation _\u2248_ is also a pre-fixed point of the\n-- bisimulation functional (see below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- References:\n  --\n  -- \u2022 Peter Dybjer and Herbert Sander. A functional programming\n  --   approach to the specification and verification of concurrent\n  --   systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n  --\n  -- \u2022 Bart Jacobs and Jan Rutten. (Co)algebras and\n  --   (co)induction. EATCS Bulletin, 62:222\u2013259, 1997.\n\n  ----------------------------------------------------------------------------\n  -- Adapted from (Dybjer and Sander 1989, p. 310). In this paper, the\n  -- authors use the name\n\n  -- as (R :: R') bs'\n\n  -- for the bisimulation functional.\n\n  -- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The bisimulation functional (Jacobs and Rutten 1997, p. 30).\n  BisimulationF : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BisimulationF _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The bisimilarity relation _\u2248_ is a post-fixed point of\n  -- Bisimulation, i.e,\n  --\n  -- _\u2248_ \u2264 Bisimulation _\u2248_.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 BisimulationF _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  -- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation, i.e\n  --\n  -- \u2200 R. R \u2264 Bisimulation R \u21d2 R \u2264 _\u2248_.\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 BisimulationF _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- Because a greatest post-fixed point is a fixed-point, the\n  -- bisimilarity relation _\u2248_ is also a pre-fixed point of\n  -- Bisimulation, i.e.\n  --\n  -- Bisimulation _\u2248_ \u2264 _\u2248_.\n  pre-fp : \u2200 {xs ys} \u2192 BisimulationF _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","old_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The bisimilarity relation _\u2248_ is the greatest fixed point (by\n-- \u2248-gfp\u2081 and \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation _\u2248_ is a post-fixed point of the\n-- bisimulation functional (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n-- the bisimulation functional (see below).\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed-point, the\n-- bisimilarity relation _\u2248_ is also a pre-fixed point of the\n-- bisimulation functional (see below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- References:\n  --\n  -- \u2022 Peter Dybjer and Herbert Sander. A functional programming\n  --   approach to the specification and verification of concurrent\n  --   systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n  --\n  -- \u2022 Bart Jacobs and Jan Rutten. (Co)algebras and\n  --   (co)induction. EATCS Bulletin, 62:222\u2013259, 1997.\n\n  ----------------------------------------------------------------------------\n  -- Adapted from (Dybjer and Sander 1989, p. 310). In this paper, the\n  -- authors use the name\n\n  -- as (R :: R') bs'\n\n  -- for the bisimulation functional.\n\n  -- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The bisimulation functional (Jacobs and Rutten 1997, p. 30).\n  Bisimulation : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  Bisimulation _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The bisimilarity relation _\u2248_ is a post-fixed point of\n  -- Bisimulation, i.e,\n  --\n  -- _\u2248_ \u2264 Bisimulation _\u2248_.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Bisimulation _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  -- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation, i.e\n  --\n  -- \u2200 R. R \u2264 Bisimulation R \u21d2 R \u2264 _\u2248_.\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 Bisimulation _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- Because a greatest post-fixed point is a fixed-point, the\n  -- bisimilarity relation _\u2248_ is also a pre-fixed point of\n  -- Bisimulation, i.e.\n  --\n  -- Bisimulation _\u2248_ \u2264 _\u2248_.\n  pre-fp : \u2200 {xs ys} \u2192 Bisimulation _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d798e80276e0926b00be478453bef02b5735eae4","subject":"Agda's positive checker is looking for occurrences in the indexes.","message":"Agda's positive checker is looking for occurrences in the indexes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/strictly-positive-inductive-types\/StrictlyPositive.agda","new_file":"notes\/strictly-positive-inductive-types\/StrictlyPositive.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule StrictlyPositive where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n------------------------------------------------------------------------------\n-- Parametric stability constraint (Coq Art, p. 378).\n\n-- Error: B != A of type Set when checking the constructor c in the\n-- declaration of T\n\n-- data T (A : Set) : Set where\n--   c : (B : Set) \u2192 T B\n\n------------------------------------------------------------------------------\n-- Head type constraint (Coq Art, section 14.1.2.1).\n\n-- Error: T is not strictly positive, because it occurs in an index of\n-- the target type of the constructor c in the definition of T.\n-- data T : Set \u2192 Set where\n--   c : T (T \u2115)\n\n------------------------------------------------------------------------------\n-- Strictly positive constraints (Coq Art, section 14.1.2.2).\n\n-- Error: A is not strictly positive, because it occurs to the left of\n-- an arrow in the type of the constructor c in the definition of A.\n--\n-- data A : Set where\n--   c : (A \u2192 A) \u2192 A\n\n------------------------------------------------------------------------------\n-- Universe constraints (Coq Art, section 14.1.2.3).\n\nrecord R : Set\u2081 where\n  field\n    A  : Set\n    op : A \u2192 A \u2192 A\n    e  : A\n\n-- Nils (Agda mailing list): According to Voevodsky this type is\n-- incompatible with his univalent model.\ndata _\u2261_ (A : Set) : Set \u2192 Set where\n  refl : A \u2261 A\n\ndata A : Set \u2192 Set where\n  c : A \u2115\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule StrictlyPositive where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n------------------------------------------------------------------------------\n-- Parametric stability constraint (Coq Art, p. 378).\n\n-- Error: B != A of type Set when checking the constructor c in the\n-- declaration of T\n\n-- data T (A : Set) : Set where\n--   c : (B : Set) \u2192 T B\n\n------------------------------------------------------------------------------\n-- Head type constraint (Coq Art, section 14.1.2.1).\n\n-- NB. This type is not accepted by Coq.\ndata T : Set \u2192 Set where\n  c : T (T \u2115)\n\n------------------------------------------------------------------------------\n-- Strictly positive constraints (Coq Art, section 14.1.2.2).\n\n-- Error: A is not strictly positive, because it occurs to the left of\n-- an arrow in the type of the constructor c in the definition of A.\n--\n-- data A : Set where\n--   c : (A \u2192 A) \u2192 A\n\n------------------------------------------------------------------------------\n-- Universe constraints (Coq Art, section 14.1.2.3).\n\nrecord R : Set\u2081 where\n  field\n    A  : Set\n    op : A \u2192 A \u2192 A\n    e  : A\n\n-- Nils (Agda mailing list): According to Voevodsky this type is\n-- incompatible with his univalent model.\ndata _\u2261_ (A : Set) : Set \u2192 Set where\n  refl : A \u2261 A\n\ndata A : Set \u2192 Set where\n  c : A \u2115\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b50b0f77764d63042aa5a88075d4320129b3681a","subject":"Change order of implicit arguments of term constructors.","message":"Change order of implicit arguments of term constructors.\n\nThis simplifies extracting the argument type from a term\nof function type.\n\nOld-commit-hash: 2742a31a130e9d01c2283f3e7cb7fae6102a4062\n","repos":"inc-lc\/ilc-agda","old_file":"Syntactic\/Terms\/Total.agda","new_file":"Syntactic\/Terms\/Total.agda","new_contents":"module Syntactic.Terms.Total where\n\n-- TERMS with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the syntax of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Changes\nopen import ChangeContexts\n\nopen import Relation.Binary.PropositionalEquality\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n  weakenOne : \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u03c4\n\nsubstTerm : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nsubstTerm {\u03c4} {\u0393\u2081} {\u0393\u2082} \u2261\u2081 t = subst (\u03bb \u0393 \u2192 Term \u0393 \u03c4) \u2261\u2081 t\n","old_contents":"module Syntactic.Terms.Total where\n\n-- TERMS with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the syntax of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Changes\nopen import ChangeContexts\n\nopen import Relation.Binary.PropositionalEquality\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n  weakenOne : \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u03c4\n\nsubstTerm : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nsubstTerm {\u03c4} {\u0393\u2081} {\u0393\u2082} \u2261\u2081 t = subst (\u03bb \u0393 \u2192 Term \u0393 \u03c4) \u2261\u2081 t\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c16de318c104447ae09330656978df89645b8aad","subject":"IDesc example: for fun, add pair (but no projection)","message":"IDesc example: for fun, add pair (but no projection)","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n-- Fix menu:\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),                        -- Val t\n                              (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                               Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             (Val ty + Var context ty) ->\n             IMu closeTerm' ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      (Val ty + Var context ty) ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\n-- Fix menu:\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),                        -- Val t\n                              (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                               Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             (Val ty + Var context ty) ->\n             IMu closeTerm' ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      (Val ty + Var context ty) ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2\ntestContext : Context _\ntestContext = vcons (bool , con (EZe , true )) (vcons (nat , con (EZe , su (su ze)) ) vnil)\n\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ff8df0e78b5c026b9f0c8643e8b27ac0b03d8491","subject":"Cleanup README.agda (partially)","message":"Cleanup README.agda (partially)\n\nReorder modules and express a bit of the structure.\nSome other changes are proposed in comments, because I'm not fully sure\nabout them yet.\n\nOld-commit-hash: 0bf53e9e29581399c12fb7e3f5a8ff3dc6879e98\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will help with the\n-- \"generate Everything.agda\" part.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help.\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Popl14 plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Popl14\n\n-- ## Definitions\nimport Popl14.Syntax.Type\nimport Popl14.Syntax.Term\n\nimport Popl14.Denotation.Value\nimport Popl14.Denotation.Evaluation\n\nimport Popl14.Change.Term\nimport Popl14.Change.Type\n\nimport Popl14.Change.Derive\n\nimport Popl14.Change.Value\nimport Popl14.Change.Evaluation\n\n-- ## Proofs\n\nimport Popl14.Change.Validity\nimport Popl14.Change.Specification\nimport Popl14.Change.Implementation\nimport Popl14.Change.Correctness\n\n-- Some other calculus (remove from here?)\n\nimport Atlas.Syntax.Type\nimport Atlas.Syntax.Term\nimport Atlas.Change.Type\nimport Atlas.Change.Term\nimport Atlas.Change.Derive\n\n-- Import everything else.\nimport Everything\n\n-- XXX: Do we still care about these?\n\nimport Property.Uniqueness\n\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.Irrelevance-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\n\n-- Old stuff.\n-- XXX Update and integrate those descriptions where appropriate.\n\n{-\n-- Correctness theorem for canonical derivation\nimport Popl14.Change.Correctness\n\n-- Denotation-as-specification for canonical derivation\nimport Popl14.Change.Specification\n\n-- The idea of implementing a denotational specification\nimport Popl14.Change.Implementation\n-}\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will help with the\n-- \"generate Everything.agda\" part.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help.\n\n\nimport Popl14.Syntax.Term\nimport Base.Syntax.Context\nimport Base.Change.Context\nimport Popl14.Change.Derive\n{- Language definition of Calc. Atlas -}\nimport Atlas.Syntax.Term\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Parametric.Syntax.Term\nimport Popl14.Change.Term\nimport Atlas.Syntax.Type\nimport Parametric.Syntax.Type\nimport Popl14.Syntax.Type\nimport Base.Syntax.Vars\n\nimport Popl14.Change.Validity\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Popl14.Change.Correctness\nimport Theorem.Irrelevance-Popl14\nimport Base.Denotation.Environment\nimport Popl14.Denotation.Evaluation\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Popl14.Change.Implementation\nimport Base.Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Popl14.Change.Specification\nimport Popl14.Denotation.Value\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport UNDEFINED\n\nimport Everything\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"53ffa41592e8de74ad057626e193d26cc8e3b78b","subject":"Groups-Nehemiah: reorder theorems","message":"Groups-Nehemiah: reorder theorems\n","repos":"inc-lc\/ilc-agda","old_file":"Theorem\/Groups-Nehemiah.agda","new_file":"Theorem\/Groups-Nehemiah.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- About the group structure of integers and bags for Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Theorem.Groups-Nehemiah where\n\nopen import Structure.Bag.Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Structures\n\n4-way-shuffle : \u2200 {A : Set} {f} {z a b c d : A}\n  {{comm-monoid : IsCommutativeMonoid _\u2261_ f z}} \u2192\n  f (f a b) (f c d) \u2261 f (f a c) (f b d)\n4-way-shuffle {f = f} {z = z} {a} {b} {c} {d} {{comm-monoid}} =\n  let\n    assoc = associative comm-monoid\n    cmute = commutative comm-monoid\n  in\n    begin\n      f (f a b) (f c d)\n    \u2261\u27e8 assoc a b (f c d) \u27e9\n      f a (f b (f c d))\n    \u2261\u27e8 cong (f a) (sym (assoc b c d)) \u27e9\n      f a (f (f b c) d)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f a (f hole d)) (cmute b c) \u27e9\n      f a (f (f c b) d)\n    \u2261\u27e8 cong (f a) (assoc c b d) \u27e9\n      f a (f c (f b d))\n    \u2261\u27e8 sym (assoc a c (f b d)) \u27e9\n      f (f a c) (f b d)\n    \u220e where open \u2261-Reasoning\n\nopen import Data.Integer\n\nn+[m-n]=m : \u2200 {n m} \u2192 n + (m - n) \u2261 m\nn+[m-n]=m {n} {m} =\n  begin\n    n + (m - n)\n  \u2261\u27e8 cong (\u03bb hole \u2192 n + hole) (commutative-int m (- n)) \u27e9\n    n + (- n + m)\n  \u2261\u27e8 sym (associative-int n (- n) m) \u27e9\n    (n - n) + m\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole + m) (right-inv-int n) \u27e9\n    (+ 0) + m\n  \u2261\u27e8 left-id-int m \u27e9\n    m\n  \u220e where open \u2261-Reasoning\n\na++[b\\\\a]=b : \u2200 {a b} \u2192 a ++ (b \\\\ a) \u2261 b\na++[b\\\\a]=b {b} {d} = trans\n  (cong (\u03bb hole \u2192 b ++ hole) (commutative-bag d (negateBag b))) (trans\n  (sym (associative-bag b (negateBag b) d)) (trans\n  (cong (\u03bb hole \u2192 hole ++ d) (right-inv-bag b))\n  (left-id-bag d)))\n\nab\u00b7cd=ac\u00b7bd : \u2200 {a b c d : Bag} \u2192\n  (a ++ b) ++ (c ++ d) \u2261 (a ++ c) ++ (b ++ d)\nab\u00b7cd=ac\u00b7bd {a} {b} {c} {d} =\n  4-way-shuffle {a = a} {b} {c} {d} {{comm-monoid-bag}}\n\nmn\u00b7pq=mp\u00b7nq : \u2200 {m n p q : \u2124} \u2192\n  (m + n) + (p + q) \u2261 (m + p) + (n + q)\nmn\u00b7pq=mp\u00b7nq {m} {n} {p} {q} =\n  4-way-shuffle {a = m} {n} {p} {q} {{comm-monoid-int}}\n\ninverse-unique : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f a b \u2261 z \u2192 b \u2261 neg a\n\ninverse-unique {f = f} {neg} {z} {a} {b} {{abelian}} ab=z =\n  let\n    assoc = associative (IsAbelianGroup.isCommutativeMonoid abelian)\n    cmute = commutative (IsAbelianGroup.isCommutativeMonoid abelian)\n  in\n    begin\n      b\n    \u2261\u27e8 sym (left-identity abelian b) \u27e9\n      f z b\n    \u2261\u27e8 cong (\u03bb hole \u2192 f hole b) (sym (left-inverse abelian a)) \u27e9\n      f (f (neg a) a) b\n    \u2261\u27e8 assoc (neg a) a b \u27e9\n      f (neg a) (f a b)\n    \u2261\u27e8 cong (f (neg a)) ab=z \u27e9\n      f (neg a) z\n    \u2261\u27e8 right-identity abelian (neg a) \u27e9\n      neg a\n    \u220e where\n      open \u2261-Reasoning\n      eq1 : f (neg a) (f a b) \u2261 f (neg a) z\n      eq1 rewrite ab=z = cong (f (neg a)) refl\n\ndistribute-neg : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f (neg a) (neg b) \u2261 neg (f a b)\ndistribute-neg {f = f} {neg} {z} {a} {b} {{abelian}} = inverse-unique\n  {{abelian}}\n  (begin\n    f (f a b) (f (neg a) (neg b))\n  \u2261\u27e8 4-way-shuffle {{IsAbelianGroup.isCommutativeMonoid abelian}} \u27e9\n    f (f a (neg a)) (f b (neg b))\n  \u2261\u27e8 cong\u2082 f (inverse a) (inverse b) \u27e9\n    f z z\n  \u2261\u27e8 left-identity abelian z \u27e9\n    z\n  \u220e) where\n     open \u2261-Reasoning\n     inverse = right-inverse abelian\n\n-a\u00b7-b=-ab : \u2200 {a b : Bag} \u2192\n  negateBag a ++ negateBag b \u2261 negateBag (a ++ b)\n-a\u00b7-b=-ab {a} {b} = distribute-neg {a = a} {b} {{abelian-bag}}\n\n-m\u00b7-n=-mn : \u2200 {m n : \u2124} \u2192\n  (- m) + (- n) \u2261 - (m + n)\n-m\u00b7-n=-mn {m} {n} = distribute-neg {a = m} {n} {{abelian-int}}\n\n[a++b]\\\\a=b : \u2200 {a b} \u2192 (a ++ b) \\\\ a \u2261 b\n[a++b]\\\\a=b {b} {d} =\n  begin\n    (b ++ d) \\\\ b\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole \\\\ b) (commutative-bag b d) \u27e9\n    (d ++ b) \\\\ b\n  \u2261\u27e8 associative-bag d b (negateBag b) \u27e9\n    d ++ (b \\\\ b)\n  \u2261\u27e8 cong (_++_ d) (right-inv-bag b) \u27e9\n    d ++ emptyBag\n  \u2261\u27e8 right-id-bag d \u27e9\n    d\n  \u220e where open \u2261-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- About the group structure of integers and bags for Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Theorem.Groups-Nehemiah where\n\nopen import Structure.Bag.Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Integer\nopen import Algebra.Structures\n\nn+[m-n]=m : \u2200 {n m} \u2192 n + (m - n) \u2261 m\nn+[m-n]=m {n} {m} =\n  begin\n    n + (m - n)\n  \u2261\u27e8 cong (\u03bb hole \u2192 n + hole) (commutative-int m (- n)) \u27e9\n    n + (- n + m)\n  \u2261\u27e8 sym (associative-int n (- n) m) \u27e9\n    (n - n) + m\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole + m) (right-inv-int n) \u27e9\n    (+ 0) + m\n  \u2261\u27e8 left-id-int m \u27e9\n    m\n  \u220e where open \u2261-Reasoning\n\na++[b\\\\a]=b : \u2200 {a b} \u2192 a ++ (b \\\\ a) \u2261 b\na++[b\\\\a]=b {b} {d} = trans\n  (cong (\u03bb hole \u2192 b ++ hole) (commutative-bag d (negateBag b))) (trans\n  (sym (associative-bag b (negateBag b) d)) (trans\n  (cong (\u03bb hole \u2192 hole ++ d) (right-inv-bag b))\n  (left-id-bag d)))\n\n4-way-shuffle : \u2200 {A : Set} {f} {z a b c d : A}\n  {{comm-monoid : IsCommutativeMonoid _\u2261_ f z}} \u2192\n  f (f a b) (f c d) \u2261 f (f a c) (f b d)\n4-way-shuffle {f = f} {z = z} {a} {b} {c} {d} {{comm-monoid}} =\n  let\n    assoc = associative comm-monoid\n    cmute = commutative comm-monoid\n  in\n    begin\n      f (f a b) (f c d)\n    \u2261\u27e8 assoc a b (f c d) \u27e9\n      f a (f b (f c d))\n    \u2261\u27e8 cong (f a) (sym (assoc b c d)) \u27e9\n      f a (f (f b c) d)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f a (f hole d)) (cmute b c) \u27e9\n      f a (f (f c b) d)\n    \u2261\u27e8 cong (f a) (assoc c b d) \u27e9\n      f a (f c (f b d))\n    \u2261\u27e8 sym (assoc a c (f b d)) \u27e9\n      f (f a c) (f b d)\n    \u220e where open \u2261-Reasoning\n\nab\u00b7cd=ac\u00b7bd : \u2200 {a b c d : Bag} \u2192\n  (a ++ b) ++ (c ++ d) \u2261 (a ++ c) ++ (b ++ d)\nab\u00b7cd=ac\u00b7bd {a} {b} {c} {d} =\n  4-way-shuffle {a = a} {b} {c} {d} {{comm-monoid-bag}}\n\nmn\u00b7pq=mp\u00b7nq : \u2200 {m n p q : \u2124} \u2192\n  (m + n) + (p + q) \u2261 (m + p) + (n + q)\nmn\u00b7pq=mp\u00b7nq {m} {n} {p} {q} =\n  4-way-shuffle {a = m} {n} {p} {q} {{comm-monoid-int}}\n\ninverse-unique : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f a b \u2261 z \u2192 b \u2261 neg a\n\ninverse-unique {f = f} {neg} {z} {a} {b} {{abelian}} ab=z =\n  let\n    assoc = associative (IsAbelianGroup.isCommutativeMonoid abelian)\n    cmute = commutative (IsAbelianGroup.isCommutativeMonoid abelian)\n  in\n    begin\n      b\n    \u2261\u27e8 sym (left-identity abelian b) \u27e9\n      f z b\n    \u2261\u27e8 cong (\u03bb hole \u2192 f hole b) (sym (left-inverse abelian a)) \u27e9\n      f (f (neg a) a) b\n    \u2261\u27e8 assoc (neg a) a b \u27e9\n      f (neg a) (f a b)\n    \u2261\u27e8 cong (f (neg a)) ab=z \u27e9\n      f (neg a) z\n    \u2261\u27e8 right-identity abelian (neg a) \u27e9\n      neg a\n    \u220e where\n      open \u2261-Reasoning\n      eq1 : f (neg a) (f a b) \u2261 f (neg a) z\n      eq1 rewrite ab=z = cong (f (neg a)) refl\n\ndistribute-neg : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f (neg a) (neg b) \u2261 neg (f a b)\ndistribute-neg {f = f} {neg} {z} {a} {b} {{abelian}} = inverse-unique\n  {{abelian}}\n  (begin\n    f (f a b) (f (neg a) (neg b))\n  \u2261\u27e8 4-way-shuffle {{IsAbelianGroup.isCommutativeMonoid abelian}} \u27e9\n    f (f a (neg a)) (f b (neg b))\n  \u2261\u27e8 cong\u2082 f (inverse a) (inverse b) \u27e9\n    f z z\n  \u2261\u27e8 left-identity abelian z \u27e9\n    z\n  \u220e) where\n     open \u2261-Reasoning\n     inverse = right-inverse abelian\n\n-a\u00b7-b=-ab : \u2200 {a b : Bag} \u2192\n  negateBag a ++ negateBag b \u2261 negateBag (a ++ b)\n-a\u00b7-b=-ab {a} {b} = distribute-neg {a = a} {b} {{abelian-bag}}\n\n-m\u00b7-n=-mn : \u2200 {m n : \u2124} \u2192\n  (- m) + (- n) \u2261 - (m + n)\n-m\u00b7-n=-mn {m} {n} = distribute-neg {a = m} {n} {{abelian-int}}\n\n[a++b]\\\\a=b : \u2200 {a b} \u2192 (a ++ b) \\\\ a \u2261 b\n[a++b]\\\\a=b {b} {d} =\n  begin\n    (b ++ d) \\\\ b\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole \\\\ b) (commutative-bag b d) \u27e9\n    (d ++ b) \\\\ b\n  \u2261\u27e8 associative-bag d b (negateBag b) \u27e9\n    d ++ (b \\\\ b)\n  \u2261\u27e8 cong (_++_ d) (right-inv-bag b) \u27e9\n    d ++ emptyBag\n  \u2261\u27e8 right-id-bag d \u27e9\n    d\n  \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"46d4925ee9668c8d90cf14f4daa40bc575c33cbd","subject":"Added x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z.","message":"Added x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z.\n\nIgnore-this: 1f38dc2929e9ae67e230281b80da9c5f\n\ndarcs-hash:20110215150522-3bd4e-ea78a1de99a7034b356e0e0f5bc625c4098b56f9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x\u2238y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x\u2238y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x\u2238y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x\u2238y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x\u2238y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x\u2238y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d396f72bfe5b39d0add4e4e74c3e2c6a6844fced","subject":"Avoid the use of record update for definitions (robustness against changes)","message":"Avoid the use of record update for definitions (robustness against changes)\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs-cost.agda","new_file":"flat-funs-cost.agda","new_contents":"module flat-funs-cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_)\n\nopen import flat-funs\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\nseqTimeOpsD : FlatFunsOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            nil = 0#; cons = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FlatFunsOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n\nseqTimeOps : FlatFunsOps (constFuns \u2115)\nseqTimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1; fork = 1+_+_; tt = 0;\n            <_,_> = _+_; fst = 0; snd = 0;\n            dup = 0; first = F.id; swap = 0; assoc = 0;\n            <tt,id> = 0; snd<tt,> = 0;\n            <_\u00d7_> = _+_; second = F.id;\n            nil = 0; cons = 0; uncons = 0 }\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1; fork = 1+_\u2294_; tt = 0;\n            <_,_> = _\u2294_; fst = 0; snd = 0;\n            dup = 0; first = F.id; swap = 0; assoc = 0;\n            <tt,id> = 0; snd<tt,> = 0;\n            <_\u00d7_> = _\u2294_; second = F.id;\n            nil = 0; cons = 0; uncons = 0 }\n\n\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                <_,_> = _\u2294_; <_\u00d7_> = _\u2294_; fork = 1+_\u2294_\n                              ; cons = 0; uncons = 0 } -- Without cons = 0... this definition makes\n                                                       -- the FlatFunsOps record def yellow\ntimeOps\u2261seqTimeOps = \u2261.refl\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FlatFunsOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            nil = con 0; cons = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FlatFunsOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  Time = \u2115\n  open FlatFunsOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u2261maximum f [] = \u2261.refl\n  constVec\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec f bs) 0\n                                            | constVec\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 1; <1b> = 1; cond = 1; fork = 1+_+_; tt = 0;\n             <_,_> = 1+_+_; fst = 0; snd = 0;\n             dup = 1; first = F.id; swap = 0; assoc = 0;\n             <tt,id> = 0; snd<tt,> = 0;\n             <_\u00d7_> = _+_; second = F.id;\n             nil = 0; cons = 0; uncons = 0 }\n\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; cons = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n\nmodule SpaceOps where\n  Space = \u2115\n  open FlatFunsOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f [] = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec f bs) 0\n                                        | constVec\u2261sum f bs = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps","old_contents":"module flat-funs-cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_)\n\nopen import flat-funs\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\nseqTimeOpsD : FlatFunsOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            nil = 0#; cons = 0#; uncons = 0# }\n   where open D\n         1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n         1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FlatFunsOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n\nseqTimeOps : FlatFunsOps (constFuns \u2115)\nseqTimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1; fork = 1+_+_; tt = 0;\n            <_,_> = _+_; fst = 0; snd = 0;\n            dup = 0; first = F.id; swap = 0; assoc = 0;\n            <tt,id> = 0; snd<tt,> = 0;\n            <_\u00d7_> = _+_; second = F.id;\n            nil = 0; cons = 0; uncons = 0 }\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record seqTimeOps {<_,_> = _\u2294_; <_\u00d7_> = _\u2294_; fork = 1+_\u2294_; cons = 0; uncons = 0 }\n-- Without cons = 0... this definition makes the FlatFunsOps record def yellow\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FlatFunsOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            nil = con 0; cons = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FlatFunsOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  Time = \u2115\n  open FlatFunsOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u2261maximum f [] = \u2261.refl\n  constVec\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec f bs) 0\n                                            | constVec\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 1; <1b> = 1; cond = 1; fork = 1+_+_; tt = 0;\n             <_,_> = 1+_+_; fst = 0; snd = 0;\n             dup = 1; first = F.id; swap = 0; assoc = 0;\n             <tt,id> = 0; snd<tt,> = 0;\n             <_\u00d7_> = _+_; second = F.id;\n             nil = 0; cons = 0; uncons = 0 }\n\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; cons = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n\nmodule SpaceOps where\n  Space = \u2115\n  open FlatFunsOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f [] = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec f bs) 0\n                                        | constVec\u2261sum f bs = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8bb712cd1a8c9c9cc0885ef29918d9cc47588189","subject":"Document P.D.Evaluation.","message":"Document P.D.Evaluation.\n\nOld-commit-hash: a346d6ffd091c3041ecb8d087a4171aeffa63576\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Evaluation.agda","new_file":"Parametric\/Denotation\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map-IVT (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Evaluation for languages described in Plotkin style\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map-IVT (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"df3db95515c4144eec20758b26fd5ef6c4c2f54a","subject":"Desc stratified model: I hate yellow. Almost done with the proof, but stuck on yellow.","message":"Desc stratified model: I hate yellow. Almost done with the proof, but stuck on yellow.","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n\nIDescl0 : {l : Level}(I : Set l) -> Unit -> Set (suc l)\nIDescl0 I = IMu (\\_ -> descD I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set l} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\nproof-psi-phi : {l : Level}(I : Set l) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      lvar' : DescDConst {l = suc x}\n                      lvar' = lvar\n                      lpi' : DescDConst {l = suc x}\n                      lpi' = lpi\n                      lsigma' : DescDConst {l = suc x}\n                      lsigma' = lsigma\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = cong (\\t -> con (lvar' , t)) \n                                                                    (proof-lift-unlift-eq i)\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\t ->  con ((lpi' , ((S , \\ s -> t s))))) \n                                                                          (reflFun (\\s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = {!!} {- cong (\\t -> con (lsigma' , ( S , \\s -> t (unlift s) ))) \n                                                                              (reflFun (\\s -> psi (phi (T (lifter s)))) \n                                                                                       (\\s -> T (lifter s)) \n                                                                                       (\\s -> hs (lifter s))) -}","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4ed02027800055ba1e71c3b4722146c588db6259","subject":"Instantiate (Type Atlas-type) at module import.","message":"Instantiate (Type Atlas-type) at module import.\n\nOld-commit-hash: 37cd45916c0d36a09e68e7828338aab353d4f404\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin Atlas-type\n\ndata Atlas-const : Type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type \u2192 Type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nopen import Syntax.Context\n\nAtlas-context : Set\nAtlas-context = Context {Type}\n\nopen import Syntax.Term.Plotkin\n\nAtlas-term : Atlas-context \u2192 Type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n  Atlas-term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin\n\ndata Atlas-const : Type Atlas-type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nopen import Syntax.Context\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nopen import Syntax.Term.Plotkin\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n  Atlas-term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"61a5152115a91d3d1aa7aae91b383785619dd89e","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-progress.agda","new_file":"complete-progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import progress\nopen import htype-decidable\nopen import lemmas-complete\n\nmodule complete-progress where\n\n  -- todo: explain this\n  data okc : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 okc d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 okc d \u0394\n\n  -- todo: lemma file if an arrow is disequal, it disagrees in the first or second argument\n  ne-factor : \u2200{\u03c41 \u03c42 \u03c43 \u03c44} \u2192 (\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44) \u2192 (\u03c41 \u2260 \u03c43) + (\u03c42 \u2260 \u03c44)\n  ne-factor {\u03c41} {\u03c42} {\u03c43} {\u03c44} ne with htype-dec \u03c41 \u03c43 | htype-dec \u03c42 \u03c44\n  ne-factor ne | Inl refl | Inl refl = Inl (\u03bb x \u2192 ne refl)\n  ne-factor ne | Inl x | Inr x\u2081 = Inr x\u2081\n  ne-factor ne | Inr x | Inl x\u2081 = Inl x\n  ne-factor ne | Inr x | Inr x\u2081 = Inl x\n\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n                       \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                       d dcomplete \u2192\n                       okc d \u0394\n  complete-progress wt comp with progress wt\n  complete-progress wt comp | I x = abort (lem-ind-comp comp x)\n  complete-progress wt comp | S x = S x\n  complete-progress wt comp | BV (BVVal x) = V x\n  complete-progress wt (DCCast comp x\u2082 ()) | BV (BVHoleCast x x\u2081)\n  complete-progress (TACast wt x) (DCCast comp x\u2083 x\u2084) | BV (BVArrCast x\u2081 x\u2082) = abort (x\u2081 (eq-complete-consist x\u2083 x\u2084 x))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import progress\nopen import htype-decidable\nopen import lemmas-complete\n\nmodule complete-progress where\n\n  data okc : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 okc d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 okc d \u0394\n\n  -- if an arrow is disequal, it disagrees in the first or second argument\n  ne-factor : \u2200{\u03c41 \u03c42 \u03c43 \u03c44} \u2192 (\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44) \u2192 (\u03c41 \u2260 \u03c43) + (\u03c42 \u2260 \u03c44)\n  ne-factor {\u03c41} {\u03c42} {\u03c43} {\u03c44} ne with htype-dec \u03c41 \u03c43 | htype-dec \u03c42 \u03c44\n  ne-factor ne | Inl refl | Inl refl = Inl (\u03bb x \u2192 ne refl)\n  ne-factor ne | Inl x | Inr x\u2081 = Inr x\u2081\n  ne-factor ne | Inr x | Inl x\u2081 = Inl x\n  ne-factor ne | Inr x | Inr x\u2081 = Inl x\n\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n                       \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                       d dcomplete \u2192\n                       okc d \u0394\n  complete-progress wt comp with progress wt\n  complete-progress wt comp | I x = abort (lem-ind-comp comp x)\n  complete-progress wt comp | S x = S x\n  complete-progress wt comp | BV (BVVal x) = V x\n  complete-progress wt (DCCast comp x\u2082 ()) | BV (BVHoleCast x x\u2081)\n  complete-progress (TACast wt x) (DCCast comp x\u2083 x\u2084) | BV (BVArrCast x\u2081 x\u2082) = abort (x\u2081 (eq-complete-consist x\u2083 x\u2084 x))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"45b8ba0a0ef2036716a3ecc0d5ba60c89c0b5209","subject":"README: also show flipbased-tree","message":"README: also show flipbased-tree\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import composable\nopen import vcomp\nopen import forkable\n\nopen import program-distance\nopen import diff\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","old_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import composable\nopen import vcomp\nopen import forkable\n\nopen import program-distance\nopen import diff\n\nopen import flipbased\nopen import flipbased-implem\n\nopen import bit-guessing-game\n\nopen import circuit\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dab62d30d5d25c4f6b0f1826882c53b0cc5440ae","subject":"Typo.","message":"Typo.\n\nIgnore-this: 9da2ea6ef1e7200ec7366497f70e13c\n\ndarcs-hash:20110212174745-3bd4e-7324e4e35a5f0260910ab9c48669f01a570ccb03.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download the agda2atp tool (described in above paper) using\n-- darcs with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains interactive proofs\n-- that are used by the combined proofs.\n\n------------------------------------------------------------------------------\n-- Distributive laws on a binary operation\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- Group theory\n\n-- We formalize the theory of groups using Agda postulates for\n-- the group axioms.\n\n-- Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Logic\n\n-- Propositional logic\nopen import Logic.Propositional.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n-- Predicate logic\nopen import Logic.Predicate.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n------------------------------------------------------------------------------\n-- LTC\n\n-- Formalization of (a version of) Azcel's Logical Theory of constructions.\n\n-- LTC base\nopen import LTC.Base.Properties\nopen import LTC.Base.PropertiesATP\nopen import LTC.Base.PropertiesI\n\n-- Booleans\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Bool.PropertiesI\n\n-- Lists\nopen import LTC.Data.List.PropertiesATP\nopen import LTC.Data.List.PropertiesI\n\n-- Naturals numbers: Common properties\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.PropertiesI\n\nopen import LTC.Data.Nat.PropertiesByInductionATP\nopen import LTC.Data.Nat.PropertiesByInductionI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC.Data.Nat.Divisibility.PropertiesATP\nopen import LTC.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC.Data.Nat.Induction.LexicographicATP\nopen import LTC.Data.Nat.Induction.LexicographicI\nopen import LTC.Data.Nat.Induction.WellFoundedATP\nopen import LTC.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: List\nopen import LTC.Data.Nat.List.PropertiesATP\nopen import LTC.Data.Nat.List.PropertiesI\n\n-- Naturals numbers: Unary numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n-- The GCD algorithm\nopen import LTC.Program.GCD.ProofSpecificationATP\nopen import LTC.Program.GCD.ProofSpecificationI\n\n-- Burstall's sort list algorithm\nopen import LTC.Program.SortList.ProofSpecificationATP\nopen import LTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- LTC-PCF\n\n-- Formalization of a version of Azcel's Logical Theory of constructions.\n\n-- Naturals numbers: Common properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC-PCF.Data.Nat.Induction.LexicographicATP\nopen import LTC-PCF.Data.Nat.Induction.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedATP\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- The division algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- The GCD algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- Axiomatic PA\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- Inductive PA\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, so see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download agda2atp tool (described in above paper) using darcs\n-- with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains interactive proofs\n-- that are used by the combined proofs.\n\n------------------------------------------------------------------------------\n-- Distributive laws on a binary operation\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- Group theory\n\n-- We formalize the theory of groups using Agda postulates for\n-- the group axioms.\n\n-- Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Logic\n\n-- Propositional logic\nopen import Logic.Propositional.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n-- Predicate logic\nopen import Logic.Predicate.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n------------------------------------------------------------------------------\n-- LTC\n\n-- Formalization of (a version of) Azcel's Logical Theory of constructions.\n\n-- LTC base\nopen import LTC.Base.Properties\nopen import LTC.Base.PropertiesATP\nopen import LTC.Base.PropertiesI\n\n-- Booleans\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Bool.PropertiesI\n\n-- Lists\nopen import LTC.Data.List.PropertiesATP\nopen import LTC.Data.List.PropertiesI\n\n-- Naturals numbers: Common properties\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.PropertiesI\n\nopen import LTC.Data.Nat.PropertiesByInductionATP\nopen import LTC.Data.Nat.PropertiesByInductionI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC.Data.Nat.Divisibility.PropertiesATP\nopen import LTC.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC.Data.Nat.Induction.LexicographicATP\nopen import LTC.Data.Nat.Induction.LexicographicI\nopen import LTC.Data.Nat.Induction.WellFoundedATP\nopen import LTC.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: List\nopen import LTC.Data.Nat.List.PropertiesATP\nopen import LTC.Data.Nat.List.PropertiesI\n\n-- Naturals numbers: Unary numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n-- The GCD algorithm\nopen import LTC.Program.GCD.ProofSpecificationATP\nopen import LTC.Program.GCD.ProofSpecificationI\n\n-- Burstall's sort list algorithm\nopen import LTC.Program.SortList.ProofSpecificationATP\nopen import LTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- LTC-PCF\n\n-- Formalization of a version of Azcel's Logical Theory of constructions.\n\n-- Naturals numbers: Common properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC-PCF.Data.Nat.Induction.LexicographicATP\nopen import LTC-PCF.Data.Nat.Induction.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedATP\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- The division algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- The GCD algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- Axiomatic PA\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- Inductive PA\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, so see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"509e5de975b6eee17da4b6baeb32823e1688e23c","subject":"Added x+1+y\u22611+x+y (ER version).","message":"Added x+1+y\u22611+x+y (ER version).\n\nIgnore-this: e099f22302e7b85516d7b432f9309d04\n\ndarcs-hash:20100531140940-3bd4e-8dfca741032e39e7dabeb662da649d5f5a74d439.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Relation.Equalities.PropertiesER\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm          zN          = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN          (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ (n + zero) \u2261 succ t)\n               (+-rightIdentity Nn)\n               refl\n        )\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst (\u03bb t \u2192 N t) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No =\n  begin\n    zero + n + o \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o)\n                          (+-leftIdentity Nn)\n                          refl\n                 \u27e9\n    n + o        \u2261\u27e8 sym (+-leftIdentity (+-N Nn No)) \u27e9\n    zero + (n + o)\n  \u220e\n\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    succ m + n + o     \u2261\u27e8 subst (\u03bb t \u2192 succ m + n + o \u2261 t + o)\n                                (+-Sx m n)\n                                refl\n                       \u27e9\n    succ (m + n) + o   \u2261\u27e8 +-Sx (m + n) o \u27e9\n    succ (m + n + o)   \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + o) \u2261 succ t)\n                                (+-assoc Nm Nn No)\n                                refl\n                       \u27e9\n    succ (m + (n + o)) \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n    succ m + (n + o)\n  \u220e\n\nx+1+y\u22611+x+y : {m n : D} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y {n = n} zN Nn =\n  begin\n    zero + succ n \u2261\u27e8 +-0x (succ n ) \u27e9\n    succ n        \u2261\u27e8 subst (\u03bb t \u2192 succ n \u2261 succ t)\n                           (sym (+-leftIdentity Nn))\n                           refl\n                  \u27e9\n    succ (zero + n)\n  \u220e\n\nx+1+y\u22611+x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m + succ n     \u2261\u27e8 +-Sx m (succ n) \u27e9\n    succ (m + succ n)   \u2261\u27e8 subst (\u03bb t \u2192 succ (m + succ n) \u2261 succ t)\n                                 (x+1+y\u22611+x+y Nm Nn)\n                                 refl\n                        \u27e9\n    succ (succ (m + n)) \u2261\u27e8 subst (\u03bb t \u2192 succ (succ (m + n)) \u2261 succ t)\n                                 (sym (+-Sx m n))\n                                 refl\n                        \u27e9\n    succ (succ m + n)\n  \u220e\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (*-N Nn zN)) (*-rightZero Nn))\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = sym\n  (\n    begin\n      zero + zero * n \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t)\n                         (*-leftZero Nn)\n                         refl\n                      \u27e9\n      zero + zero     \u2261\u27e8 +-leftIdentity zN \u27e9\n      zero            \u2261\u27e8 sym (*-leftZero (sN Nn)) \u27e9\n      zero * succ n\n    \u220e\n  )\n\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * succ n        \u2261\u27e8 *-Sx m (succ n) \u27e9\n    succ n + m * succ n    \u2261\u27e8 subst (\u03bb t \u2192 succ n + m * succ n \u2261 succ n + t)\n                                    (x*1+y\u2261x+xy Nm Nn)\n                                    refl\n                           \u27e9\n    succ n + (m + m * n)   \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n    succ (n + (m + m * n)) \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m + m * n)) \u2261 succ t)\n                                    (sym (+-assoc Nn Nm (*-N Nm Nn)))\n                                    refl\n                           \u27e9\n    succ (n + m + m * n)   \u2261\u27e8 subst (\u03bb t \u2192 succ (n + m + m * n) \u2261\n                                           succ (t + m * n))\n                                    (+-comm Nn Nm)\n                                    refl\n                           \u27e9\n     succ (m + n + m * n)  \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + m * n) \u2261 succ t)\n                                    (+-assoc Nm Nn (*-N Nm Nn))\n                                    refl\n                           \u27e9\n\n    succ (m + (n + m * n)) \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n    succ m + (n + m * n)   \u2261\u27e8 subst (\u03bb t \u2192 succ m + (n + m * n) \u2261 succ m + t)\n                                    (sym (*-Sx m n))\n                                    refl\n                           \u27e9\n    succ m + succ m * n\n    \u220e\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm zN Nn = trans (*-leftZero Nn) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * n   \u2261\u27e8 *-Sx m n \u27e9\n    n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t)\n                          (*-comm Nm Nn)\n                          refl\n                  \u27e9\n    n + n * m     \u2261\u27e8 sym (x*1+y\u2261x+xy Nn Nm) \u27e9\n    n * succ m\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} {o = o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n\n[x+y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n[x+y]z\u2261xz*yz {m} {n} Nm Nn zN =\n  begin\n    (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n    zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n    zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n    zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n    m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n    m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                  (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                         \u27e9\n    m * zero + n * zero\n  \u220e\n\n[x+y]z\u2261xz*yz {n = n} zN Nn (sN {o} No) =\n  begin\n    (zero + n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ o \u2261 t * succ o)\n                                  (+-leftIdentity Nn)\n                                  refl\n                         \u27e9\n    n * succ o           \u2261\u27e8 sym (+-leftIdentity (*-N Nn (sN No))) \u27e9\n    zero + n * succ o    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ o \u2261 t +  n * succ o)\n                                (sym (*-0x (succ o)))\n                                refl\n                         \u27e9\n    zero * succ o + n * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) zN (sN {o} No) =\n begin\n    (succ m + zero) * succ o \u2261\u27e8 subst (\u03bb t \u2192 (succ m + zero) * succ o \u2261\n                                             t * succ o)\n                                      (+-rightIdentity (sN Nm))\n                                      refl\n                              \u27e9\n    succ m * succ o          \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n    succ m * succ o + zero   \u2261\u27e8 subst (\u03bb t \u2192 succ m * succ o + zero \u2261\n                                             succ m * succ o + t)\n                                      (sym (*-leftZero (sN No)))\n                                      refl\n                             \u27e9\n    succ m * succ o + zero * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m + succ n) * succ o \u2261 t * succ o)\n               (+-Sx m (succ n))\n               refl\n      \u27e9\n    succ ( m + succ n) * succ o \u2261\u27e8 *-Sx (m + succ n) (succ o) \u27e9\n    succ o + (m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + (m + succ n) * succ o \u2261 succ o + t)\n               ([x+y]z\u2261xz*yz Nm (sN Nn) (sN No))\n               refl\n      \u27e9\n    succ o + (m * succ o + succ n * succ o)\n      \u2261\u27e8 sym (+-assoc (sN No) (*-N Nm (sN No)) (*-N (sN Nn) (sN No))) \u27e9\n    succ o + m * succ o + succ n * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + m * succ o + succ n * succ o \u2261\n                      t + succ n * succ o)\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    succ m * succ o + succ n * succ o\n      \u220e","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Relation.Equalities.PropertiesER\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm          zN          = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN          (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ (n + zero) \u2261 succ t)\n               (+-rightIdentity Nn)\n               refl\n        )\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst (\u03bb t \u2192 N t) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No =\n  begin\n    zero + n + o \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o)\n                          (+-leftIdentity Nn)\n                          refl\n                 \u27e9\n    n + o        \u2261\u27e8 sym (+-leftIdentity (+-N Nn No)) \u27e9\n    zero + (n + o)\n  \u220e\n\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    succ m + n + o     \u2261\u27e8 subst (\u03bb t \u2192 succ m + n + o \u2261 t + o)\n                                (+-Sx m n)\n                                refl\n                       \u27e9\n    succ (m + n) + o   \u2261\u27e8 +-Sx (m + n) o \u27e9\n    succ (m + n + o)   \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + o) \u2261 succ t)\n                                (+-assoc Nm Nn No)\n                                refl\n                       \u27e9\n    succ (m + (n + o)) \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n    succ m + (n + o)\n  \u220e\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (*-N Nn zN)) (*-rightZero Nn))\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = sym\n  (\n    begin\n      zero + zero * n \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t)\n                         (*-leftZero Nn)\n                         refl\n                      \u27e9\n      zero + zero     \u2261\u27e8 +-leftIdentity zN \u27e9\n      zero            \u2261\u27e8 sym (*-leftZero (sN Nn)) \u27e9\n      zero * succ n\n    \u220e\n  )\n\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * succ n        \u2261\u27e8 *-Sx m (succ n) \u27e9\n    succ n + m * succ n    \u2261\u27e8 subst (\u03bb t \u2192 succ n + m * succ n \u2261 succ n + t)\n                                    (x*1+y\u2261x+xy Nm Nn)\n                                    refl\n                           \u27e9\n    succ n + (m + m * n)   \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n    succ (n + (m + m * n)) \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m + m * n)) \u2261 succ t)\n                                    (sym (+-assoc Nn Nm (*-N Nm Nn)))\n                                    refl\n                           \u27e9\n    succ (n + m + m * n)   \u2261\u27e8 subst (\u03bb t \u2192 succ (n + m + m * n) \u2261\n                                           succ (t + m * n))\n                                    (+-comm Nn Nm)\n                                    refl\n                           \u27e9\n     succ (m + n + m * n)  \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n + m * n) \u2261 succ t)\n                                    (+-assoc Nm Nn (*-N Nm Nn))\n                                    refl\n                           \u27e9\n\n    succ (m + (n + m * n)) \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n    succ m + (n + m * n)   \u2261\u27e8 subst (\u03bb t \u2192 succ m + (n + m * n) \u2261 succ m + t)\n                                    (sym (*-Sx m n))\n                                    refl\n                           \u27e9\n    succ m + succ m * n\n    \u220e\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm zN Nn = trans (*-leftZero Nn) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  begin\n    succ m * n   \u2261\u27e8 *-Sx m n \u27e9\n    n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t)\n                          (*-comm Nm Nn)\n                          refl\n                  \u27e9\n    n + n * m     \u2261\u27e8 sym (x*1+y\u2261x+xy Nn Nm) \u27e9\n    n * succ m\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} {o = o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n\n[x+y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n[x+y]z\u2261xz*yz {m} {n} Nm Nn zN =\n  begin\n    (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n    zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n    zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n    zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n    m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n    m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                  (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                         \u27e9\n    m * zero + n * zero\n  \u220e\n\n[x+y]z\u2261xz*yz {n = n} zN Nn (sN {o} No) =\n  begin\n    (zero + n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ o \u2261 t * succ o)\n                                  (+-leftIdentity Nn)\n                                  refl\n                         \u27e9\n    n * succ o           \u2261\u27e8 sym (+-leftIdentity (*-N Nn (sN No))) \u27e9\n    zero + n * succ o    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ o \u2261 t +  n * succ o)\n                                (sym (*-0x (succ o)))\n                                refl\n                         \u27e9\n    zero * succ o + n * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) zN (sN {o} No) =\n begin\n    (succ m + zero) * succ o \u2261\u27e8 subst (\u03bb t \u2192 (succ m + zero) * succ o \u2261\n                                             t * succ o)\n                                      (+-rightIdentity (sN Nm))\n                                      refl\n                              \u27e9\n    succ m * succ o          \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n    succ m * succ o + zero   \u2261\u27e8 subst (\u03bb t \u2192 succ m * succ o + zero \u2261\n                                             succ m * succ o + t)\n                                      (sym (*-leftZero (sN No)))\n                                      refl\n                             \u27e9\n    succ m * succ o + zero * succ o\n  \u220e\n\n[x+y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m + succ n) * succ o \u2261 t * succ o)\n               (+-Sx m (succ n))\n               refl\n      \u27e9\n    succ ( m + succ n) * succ o \u2261\u27e8 *-Sx (m + succ n) (succ o) \u27e9\n    succ o + (m + succ n) * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + (m + succ n) * succ o \u2261 succ o + t)\n               ([x+y]z\u2261xz*yz Nm (sN Nn) (sN No))\n               refl\n      \u27e9\n    succ o + (m * succ o + succ n * succ o)\n      \u2261\u27e8 sym (+-assoc (sN No) (*-N Nm (sN No)) (*-N (sN Nn) (sN No))) \u27e9\n    succ o + m * succ o + succ n * succ o\n      \u2261\u27e8 subst (\u03bb t \u2192 succ o + m * succ o + succ n * succ o \u2261\n                      t + succ n * succ o)\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    succ m * succ o + succ n * succ o\n      \u220e","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8dcb8c60821890604e6bd92f3acf8aaa9b58a6f9","subject":"Follow-up question to commit message 3367105","message":"Follow-up question to commit message 3367105\n\nOld-commit-hash: d0b179f01726529fe06b0707c242472fce315998\n","repos":"inc-lc\/ilc-agda","old_file":"Data\/NatBag\/Properties.agda","new_file":"Data\/NatBag\/Properties.agda","new_contents":"module Data.NatBag.Properties where\n\nimport Data.Nat as \u2115\nopen import Relation.Binary.PropositionalEquality\nopen import Data.NatBag\nopen import Data.Integer\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\n\n-- This import is too slow.\n-- It causes Agda 2.3.2 to use so much memory that cai's\n-- computer with 4GB RAM begins to thresh.\n--\n-- open import Data.Integer.Properties using (n\u2296n\u22610)\nn\u2296n\u22610 : \u2200 n \u2192 n \u2296 n \u2261 + 0\nn\u2296n\u22610 \u2115.zero    = refl\nn\u2296n\u22610 (\u2115.suc n) = n\u2296n\u22610 n\n\n\n----------------\n-- Statements --\n----------------\n\n-- Caution: Please convert all implicit bag argument to\n-- instance bag argument in the next iteration so that\n-- using the lemmas here incurs less typing overhead.\n-- Leave the implicit arguments on multivariate lemmas,\n-- though.\n--\n-- TODO: Convert bags from a sum to an ADT so that\n-- constructor names give hints to the bag type,\n-- and a set-theoretic development can write things\n-- like \u2200 {b} \u2192 \u2205 \u2286 b.\n\n\nb\\\\b=\u2205 : \u2200 {b} \u2192 b \\\\ b \u2261 empty\n\n\u2205++b=b : \u2200 {b} \u2192 empty ++ b \u2261 b\n\nb\\\\\u2205=b : \u2200 {b} \u2192 b \\\\ empty \u2261 b\n\nb++\u2205=b : \u2200 {{b}} \u2192 b ++ empty \u2261 b\n\nb++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\n\n[b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n------------\n-- Proofs --\n------------\n\nb++\u2205=b = {!!}\n\ni-i=0 : \u2200 {i : \u2124} \u2192 (i - i) \u2261 (+ 0)\ni-i=0 {+ \u2115.zero} = refl\ni-i=0 {+ \u2115.suc n} = n\u2296n\u22610 n\ni-i=0 { -[1+ n ]} = n\u2296n\u22610 n\n\n-- Debug tool\n-- Lets you try out inhabitance of any type anywhere\nabsurd! : \u2200 {B C : Set} \u2192 0 \u2261 1 \u2192 B \u2192 {x : B} \u2192 C\nabsurd! ()\n\n-- Specialized absurdity needed to type check.\n-- \u03bb () hasn't enough information sometimes.\nabsurd : Nonzero (+ 0) \u2192 \u2200 {A : Set} \u2192 A\nabsurd ()\n\n-- Here to please the termination checker.\nneb\\\\neb=\u2205 : \u2200 {neb : NonemptyBag} \u2192 zipNonempty _-_ neb neb \u2261 empty\nneb\\\\neb=\u2205 {singleton i i\u22600} with nonzero? (i - i)\n... | inj\u2081 _ = refl\n... | inj\u2082 0\u22600 rewrite i-i=0 {i} = absurd 0\u22600\nneb\\\\neb=\u2205 {i \u2237 neb} with nonzero? (i - i)\n... | inj\u2081 _ rewrite neb\\\\neb=\u2205 {neb} = refl\n... | inj\u2082 0\u22600 rewrite neb\\\\neb=\u2205 {neb} | i-i=0 {i} = absurd 0\u22600\n\nb\\\\b=\u2205 {inj\u2081 \u2205} = refl\nb\\\\b=\u2205 {inj\u2082 neb} = neb\\\\neb=\u2205 {neb}\n\n\u2205++b=b {b} = {!!}\n\nb\\\\\u2205=b {b} = {!!}\n\nnegate : \u2200 {i} \u2192 Nonzero i \u2192 Nonzero (- i)\nnegate (negative n) = positive n\nnegate (positive n) = negative n\n\nnegate\u2032 : \u2200 {i} \u2192 (i\u22600 : Nonzero i) \u2192 Nonzero (+ 0 - i)\nnegate\u2032 { -[1+ n ]} (negative .n) = positive n\nnegate\u2032 {+ .(\u2115.suc n)} (positive n) = negative n\n\n0-i=-i : \u2200 {i} \u2192 + 0 - i \u2261 - i\n0-i=-i { -[1+ n ]} = refl -- cases are split, for arguments to\n0-i=-i {+ \u2115.zero}  = refl -- refl are different.\n0-i=-i {+ \u2115.suc n} = refl\n\nrewrite-singleton :\n  \u2200 (i : \u2124) (0-i\u22600 : Nonzero (+ 0 - i)) ( -i\u22600 : Nonzero (- i)) \u2192\n    singleton (+ 0 - i) 0-i\u22600 \u2261 singleton (- i) -i\u22600\nrewrite-singleton (+ \u2115.zero) () ()\nrewrite-singleton (+ \u2115.suc n) (negative .n) (negative .n) = refl\nrewrite-singleton ( -[1+ n ]) (positive .n) (positive .n) = refl\n\nnegateSingleton : \u2200 {i i\u22600} \u2192\n  mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\n  inj\u2082 (singleton (- i) (negate i\u22600))\n\n-- Fun fact:\n-- Pattern-match on implicit parameters in the first two\n-- cases results in rejection by Agda.\nnegateSingleton {i} {i\u22600} with nonzero? i | nonzero? (+ 0 - i)\nnegateSingleton | inj\u2082 (negative n) | inj\u2081 ()\nnegateSingleton | inj\u2082 (positive n) | inj\u2081 ()\nnegateSingleton {_} {i\u22600} | inj\u2081 i=0 | _ rewrite i=0 = absurd i\u22600\nnegateSingleton {i} {i\u22600} | inj\u2082 _ | inj\u2082 0-i\u22600 =\n  begin -- Reasoning done in 1 step. Included for clarity only.\n    inj\u2082 (singleton (+ 0 + - i) 0-i\u22600)\n  \u2261\u27e8 cong inj\u2082 (rewrite-singleton i 0-i\u22600 (negate i\u22600)) \u27e9\n    inj\u2082 (singleton (- i) (negate i\u22600))\n  \u220e  where open \u2261-Reasoning\n\nabsurd[i-i\u22600] : \u2200 {i} \u2192 Nonzero (i - i) \u2192 \u2200 {A : Set} \u2192 A\nabsurd[i-i\u22600] {+ \u2115.zero} = absurd\nabsurd[i-i\u22600] {+ \u2115.suc n} = absurd[i-i\u22600] { -[1+ n ]}\nabsurd[i-i\u22600] { -[1+ \u2115.zero ]} = absurd\nabsurd[i-i\u22600] { -[1+ \u2115.suc n ]} = absurd[i-i\u22600] { -[1+ n ]}\n\nannihilate : \u2200 {i i\u22600} \u2192\n  inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600)) \u2261 inj\u2081 \u2205\n\nannihilate {i} with nonzero? (i - i)\n... | inj\u2081 i-i=0 = \u03bb {i\u22600} \u2192 refl\n... | inj\u2082 i-i\u22600 = absurd[i-i\u22600] {i} i-i\u22600\n{-\nFollow-up question to\nhttps:\/\/github.com\/ps-mr\/ilc\/commit\/978e3f94e70762904e077bb4d51d6b7b17695103#commitcomment-3367105\n\nMysterious error message when the case above is replaced by the one\nbelow.\n\nannihilate {i} with i - i\n... | w = ?\n-}\n\nb++[\u2205\\\\b]=\u2205 : \u2200 {b} \u2192 b ++ (empty \\\\ b) \u2261 empty\nb++[\u2205\\\\b]=\u2205 {inj\u2081 \u2205} = refl\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (singleton i i\u22600)} =\n  begin\n    inj\u2082 (singleton i i\u22600) ++\n      mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\u27e8 cong\u2082 _++_ {x = inj\u2082 (singleton i i\u22600)} refl\n                (negateSingleton {i} {i\u22600}) \u27e9\n    inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600))\n  \u2261\u27e8 annihilate {i} {i\u22600} \u27e9\n    inj\u2081 \u2205\n  \u220e where open \u2261-Reasoning\n\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (i \u2237 y)} = ?\n\nb++[d\\\\b]=d {inj\u2081 \u2205} {d} rewrite b\\\\\u2205=b {d} | \u2205++b=b {d} = refl\nb++[d\\\\b]=d {b} {inj\u2081 \u2205} = b++[\u2205\\\\b]=\u2205 {b}\nb++[d\\\\b]=d {inj\u2082 b} {inj\u2082 d} = {!!}\n\n[b++d]\\\\b=d = {!!}\n","old_contents":"module Data.NatBag.Properties where\n\nimport Data.Nat as \u2115\nopen import Relation.Binary.PropositionalEquality\nopen import Data.NatBag\nopen import Data.Integer\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\n\n-- This import is too slow.\n-- It causes Agda 2.3.2 to use so much memory that cai's\n-- computer with 4GB RAM begins to thresh.\n--\n-- open import Data.Integer.Properties using (n\u2296n\u22610)\nn\u2296n\u22610 : \u2200 n \u2192 n \u2296 n \u2261 + 0\nn\u2296n\u22610 \u2115.zero    = refl\nn\u2296n\u22610 (\u2115.suc n) = n\u2296n\u22610 n\n\n\n----------------\n-- Statements --\n----------------\n\n\nb\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 empty\n\n\u2205++b=b : \u2200 {b : Bag} \u2192 empty ++ b \u2261 b\n\nb\\\\\u2205=b : \u2200 {b : Bag} \u2192 b \\\\ empty \u2261 b\n\n\u2205\\\\b=-b : \u2200 {b : Bag} \u2192 empty \\\\ b \u2261 map\u2082 -_ b\n\nb++[d\\\\b]=d : \u2200 {b d : Bag} \u2192 b ++ (d \\\\ b) \u2261 d\n\n\n------------\n-- Proofs --\n------------\n\ni-i=0 : \u2200 {i : \u2124} \u2192 (i - i) \u2261 (+ 0)\ni-i=0 {+ \u2115.zero} = refl\ni-i=0 {+ \u2115.suc n} = n\u2296n\u22610 n\ni-i=0 { -[1+ n ]} = n\u2296n\u22610 n\n\n-- Debug tool\n-- Lets you try out inhabitance of any type anywhere\nabsurd! : \u2200 {B C : Set} \u2192 0 \u2261 1 \u2192 B \u2192 {x : B} \u2192 C\nabsurd! ()\n\n-- Specialized absurdity needed to type check.\n-- \u03bb () hasn't enough information sometimes.\nabsurd : Nonzero (+ 0) \u2192 \u2200 {A : Set} \u2192 A\nabsurd ()\n\n-- Here to please the termination checker.\nneb\\\\neb=\u2205 : \u2200 {neb : NonemptyBag} \u2192 zipNonempty _-_ neb neb \u2261 empty\nneb\\\\neb=\u2205 {singleton i i\u22600} with nonzero? (i - i)\n... | inj\u2081 _ = refl\n... | inj\u2082 0\u22600 rewrite i-i=0 {i} = absurd 0\u22600\nneb\\\\neb=\u2205 {i \u2237 neb} with nonzero? (i - i)\n... | inj\u2081 _ rewrite neb\\\\neb=\u2205 {neb} = refl\n... | inj\u2082 0\u22600 rewrite neb\\\\neb=\u2205 {neb} | i-i=0 {i} = absurd 0\u22600\n\nb\\\\b=\u2205 {inj\u2081 \u2205} = refl\nb\\\\b=\u2205 {inj\u2082 neb} = neb\\\\neb=\u2205 {neb}\n\n\u2205++b=b {b} = {!!}\n\nb\\\\\u2205=b {b} = {!!}\n\n\u2205\\\\b=-b {b} = {!!}\n\nnegate : \u2200 {i} \u2192 Nonzero i \u2192 Nonzero (- i)\nnegate (negative n) = positive n\nnegate (positive n) = negative n\n\nnegate\u2032 : \u2200 {i} \u2192 (i\u22600 : Nonzero i) \u2192 Nonzero (+ 0 - i)\nnegate\u2032 { -[1+ n ]} (negative .n) = positive n\nnegate\u2032 {+ .(\u2115.suc n)} (positive n) = negative n\n\n0-i=-i : \u2200 {i} \u2192 + 0 - i \u2261 - i\n0-i=-i { -[1+ n ]} = refl -- cases are split, for arguments to\n0-i=-i {+ \u2115.zero}  = refl -- refl are different.\n0-i=-i {+ \u2115.suc n} = refl\n\nrewrite-singleton :\n  \u2200 (i : \u2124) (0-i\u22600 : Nonzero (+ 0 - i)) ( -i\u22600 : Nonzero (- i)) \u2192\n    singleton (+ 0 - i) 0-i\u22600 \u2261 singleton (- i) -i\u22600\nrewrite-singleton (+ \u2115.zero) () ()\nrewrite-singleton (+ \u2115.suc n) (negative .n) (negative .n) = refl\nrewrite-singleton ( -[1+ n ]) (positive .n) (positive .n) = refl\n\nnegateSingleton : \u2200 {i i\u22600} \u2192\n  mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\n  inj\u2082 (singleton (- i) (negate i\u22600))\n\n-- Fun fact:\n-- Pattern-match on implicit parameters in the first two\n-- cases results in rejection by Agda.\nnegateSingleton {i} {i\u22600} with nonzero? i | nonzero? (+ 0 - i)\nnegateSingleton | inj\u2082 (negative n) | inj\u2081 ()\nnegateSingleton | inj\u2082 (positive n) | inj\u2081 ()\nnegateSingleton {_} {i\u22600} | inj\u2081 i=0 | _ rewrite i=0 = absurd i\u22600\nnegateSingleton {i} {i\u22600} | inj\u2082 _ | inj\u2082 0-i\u22600 =\n  begin -- Reasoning done in 1 step. Included for clarity only.\n    inj\u2082 (singleton (+ 0 + - i) 0-i\u22600)\n  \u2261\u27e8 cong inj\u2082 (rewrite-singleton i 0-i\u22600 (negate i\u22600)) \u27e9\n    inj\u2082 (singleton (- i) (negate i\u22600))\n  \u220e  where open \u2261-Reasoning\n\nabsurd[i-i\u22600] : \u2200 {i} \u2192 Nonzero (i - i) \u2192 \u2200 {A : Set} \u2192 A\nabsurd[i-i\u22600] {+ \u2115.zero} = absurd\nabsurd[i-i\u22600] {+ \u2115.suc n} = absurd[i-i\u22600] { -[1+ n ]}\nabsurd[i-i\u22600] { -[1+ \u2115.zero ]} = absurd\nabsurd[i-i\u22600] { -[1+ \u2115.suc n ]} = absurd[i-i\u22600] { -[1+ n ]}\n\nannihilate : \u2200 {i i\u22600} \u2192\n  inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600)) \u2261 inj\u2081 \u2205\nannihilate {i} with nonzero? (i - i)\n... | inj\u2081 i-i=0 = \u03bb {i\u22600} \u2192 refl\n... | inj\u2082 i-i\u22600 = absurd[i-i\u22600] {i} i-i\u22600\n\nb++[\u2205\\\\b]=\u2205 : \u2200 {b} \u2192 b ++ (empty \\\\ b) \u2261 empty\nb++[\u2205\\\\b]=\u2205 {inj\u2081 \u2205} = refl\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (singleton i i\u22600)} =\n  begin\n    inj\u2082 (singleton i i\u22600) ++\n      mapNonempty\u2082 (\u03bb j \u2192 + 0 - j) (singleton i i\u22600)\n  \u2261\u27e8 cong\u2082 _++_ {x = inj\u2082 (singleton i i\u22600)} refl\n                (negateSingleton {i} {i\u22600}) \u27e9\n    inj\u2082 (singleton i i\u22600) ++ inj\u2082 (singleton (- i) (negate i\u22600))\n  \u2261\u27e8 annihilate {i} {i\u22600} \u27e9\n    inj\u2081 \u2205\n  \u220e where open \u2261-Reasoning\nb++[\u2205\\\\b]=\u2205 {inj\u2082 (i \u2237 y)} = {!!}\n\nb++[d\\\\b]=d {inj\u2081 \u2205} {d} rewrite b\\\\\u2205=b {d} | \u2205++b=b {d} = refl\nb++[d\\\\b]=d {b} {inj\u2081 \u2205} = b++[\u2205\\\\b]=\u2205 {b}\nb++[d\\\\b]=d {inj\u2082 b} {inj\u2082 d} = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dda828f86a18387618d4b27b17c1325955ae135d","subject":"StepIndexedRelBigStepTypedAnfIlcCorrect: Prove SOS sound","message":"StepIndexedRelBigStepTypedAnfIlcCorrect: Prove SOS sound\n\nThat's soundness wrt. denotational semantics. No completeness though, that would\nbe tricky (and `succ` violates it right now).\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- Adding this changes nothing without changes to the semantics.\n  succ : Const (nat \u21d2 nat)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\ndata Val : Type \u2192 Set\n\nimport Base.Denotation.Environment\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 lit n \u27e7Const = n\n\u27e6 succ \u27e7Const = suc\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Den.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7SVal : \u2200 {\u0393 \u03c4} \u2192 SVal \u0393 \u03c4 \u2192 Den.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 var x \u27e7SVal \u03c1 = Den.\u27e6 x \u27e7Var \u03c1\n\u27e6 abs t \u27e7SVal \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 val sv \u27e7Term \u03c1 = \u27e6 sv \u27e7SVal \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7SVal \u03c1 (\u27e6 t \u27e7SVal \u03c1)\n\u27e6 lett s t \u27e7Term \u03c1 = \u27e6 t \u27e7Term ((\u27e6 s \u27e7Term \u03c1) \u2022 \u03c1)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 natV n \u27e7Val = n\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v hasIdx} {n : Idx hasIdx} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193[ n ] v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\u2193-sound abs = refl\n\u2193-sound (app _ _ \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n\u2193-sound (var x) = \u21a6-sound _ x\n\u2193-sound (lit n) = refl\n\u2193-sound (lett n1 n2 vsv s t \u2193 \u2193\u2081) rewrite \u2193-sound \u2193 | \u2193-sound \u2193\u2081 = refl\n-- \u2193-sound (add \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n-- \u2193-sound (minus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\n-- No proof of completeness yet: the statement does not hold here.\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- Adding this changes nothing without changes to the semantics.\n  succ : Const (nat \u21d2 nat)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7b8de230698933472e580b5ece63a74cf55998a4","subject":"Document P.C.Term.","message":"Document P.C.Term.\n\nOld-commit-hash: 0263f48a6674e209dc10d253b3ffab2671e313cd\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Terms that operate on changes (Fig. 3).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\n-- Extension point 1: A term for \u229d on base types.\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\n-- Extension point 2: A term for \u2295 on base types.\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  -- We provide: terms for \u2295 and \u229d on arbitrary types.\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff _ = app\u2082 diff-term\n\n  apply : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply _ = app\u2082 apply-term\n\n  infixl 6 apply diff\n  syntax apply \u03c4 x \u0394x = \u0394x \u2295\u208d \u03c4 \u208e x\n  syntax diff \u03c4 x y = x \u229d\u208d \u03c4 \u208e y\n\n  infixl 6 _\u2295_ _\u229d_\n  _\u229d_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  _\u229d_ {\u03c4} = diff \u03c4\n\n  _\u2295_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  _\u2295_ {\u03c4} = app\u2082 apply-term\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff _ = app\u2082 diff-term\n\n  apply : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply _ = app\u2082 apply-term\n\n  infixl 6 apply diff\n  syntax apply \u03c4 x \u0394x = \u0394x \u2295\u208d \u03c4 \u208e x\n  syntax diff \u03c4 x y = x \u229d\u208d \u03c4 \u208e y\n\n  infixl 6 _\u2295_ _\u229d_\n  _\u229d_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  _\u229d_ {\u03c4} = diff \u03c4\n\n  _\u2295_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  _\u2295_ {\u03c4} = app\u2082 apply-term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c4e3244dfb2bd811229088cb749c8d2c74ecec32","subject":"Avoid \"white smoke\" warning for fallback cases that do not hold","message":"Avoid \"white smoke\" warning for fallback cases that do not hold\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalUntyped.agda","new_file":"Thesis\/ANormalUntyped.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set\n\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  term : \u2200 {v} n (t : Term \u0393) \u2192\n    \u03c1 \u22a2 t \u2193[ n ] v \u2192\n    \u03c1 F\u22a2 term t \u2193[ n ] v\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {dt : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs dt \u2193 dclosure dt \u03c1 d\u03c1\n  dterm : \u2200 {dv} (dt : DTerm \u0393) \u2192\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 F\u22a2 dterm dt \u2193 dv\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k < n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 =  <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n =  proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n =  proj\u2082 (rrelTV3 n)\n\n  s-rrelV3 :\n    \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n    \u2200 j \u2192 j < k \u2192 rrelV3-Type (k \u2238 j)\n  s-rrelV3 k rec-rrelTV3 rrelV3-k 0 _ = rrelV3-k\n  s-rrelV3 k rec-rrelTV3 rrelV3-k (suc j) sucj<k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt j k sucj<k))\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    -- rrelV3-Type n\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n      (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    \u2200 j (j<k : j < k) n2 \u2192\n    (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 k rec-rrelTV3 rrelV3-k j j<k v1 dv v2\n    -- where\n    --   s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n    --   -- If j = 0, we can't do a recursive call on (k - j) because that's not\n    --   -- well-founded. Luckily, we don't need to, just use relV as defined at\n    --   -- the same level.\n    --   -- Yet, this will be a pain in proofs.\n    --   s-rrelV3 0 _ = rrelV3-k\n    --   s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelV3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV n\u2081) (closure t \u03c1) = \u22a5\n  rrelV3-step n rec-rrelTV3 (intV v1) (dclosure dt \u03c1 d\u03c1) c = \u22a5\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (intV n\u2081) = \u22a5\n  rrelV3-step n rec-rrelTV3 (closure t \u03c1) (dintV n\u2081) c = \u22a5\n\nopen import Postulate.Extensionality\nmutual\n  rrelT3-equiv : \u2200 k {\u0393} {t1 : Fun \u0393} {dt t2 \u03c11 d\u03c1 \u03c12} \u2192\n    rrelT3 k t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2261\n    \u2200 j (j<k : j < k) n2 \u2192\n    \u2200 (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 j v1 dv v2\n  rrelT3-equiv n = cong (\u03bb \u25a1 \u2192 \u2200 j \u2192 \u25a1 j) (ext (\u03bb j \u2192 {!!}))\n\n  rrelV3-equiv : \u2200 n {\u03931 \u03932 \u0394\u0393 t1 dt t2 \u03c11 \u03c1' d\u03c1 \u03c12} \u2192\n    rrelV3 n (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) \u2261 \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        rrelV3 k v1 dv v2 \u2192\n        rrelT3 k\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-equiv n = cong (\u03bb \u25a1 \u2192 \u03a3 (_ \u2261 _ \u00d7 _ \u2261 _) \u25a1) (ext (\u03bb { (eq1 , eq2) \u2192\n    cong\n      (\u03bb \u25a1 \u2192\n         (\u03bb { (refl , refl)\n                \u2192 \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192 \u25a1 k k<n v1 dv v2\n            })\n         eq1 eq2)\n      (ext (\u03bb x \u2192 ext {!!}))}))\n  --\n  -- cong (\u03bb \u25a1 \u2192 \u2200 k \u2192 (k<n : k <\u2032 n) \u2192 \u25a1 k k<n)\n  -- rrelV3-equiv k\n\nrrel\u03c13 : \u2115 \u2192 \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13 n \u2205 \u2205 \u2205 \u2205 = \u22a4\nrrel\u03c13 n (Uni \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) = rrelV3 n v1 dv v2 \u00d7 rrel\u03c13 n \u0393 \u03c11 d\u03c1 \u03c12\n\n--\nrfundamental3 : \u2200 {\u0393} (t : Fun \u0393) \u2192 \u2200 k \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 k \u0393 \u03c11 d\u03c1 \u03c12) \u2192 rrelT3 k t (derive-dfun t) t \u03c11 d\u03c1 \u03c12\nrfundamental3 (term x) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 j j<k n2 v1 v2 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 = {!!}\nrfundamental3 {\u0393} (abs t) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .0 j<k .0 (closure .t .\u03c11) (closure .t .\u03c12) abs abs = dclosure (derive-dfun t) \u03c11 d\u03c1 , 0 , dabs , (refl , refl) , {!body !}\n  where\n    body1 : \u2200 k\u2081 (k\u2081<k : k\u2081 <\u2032 k) \u2192 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 rrelV3 k\u2081 v1 dv v2 \u2192 rrelT3 k\u2081 t (derive-dfun t) t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12)\n    body1 = \u03bb k\u2081 k\u2081<k v1 dv v2 x v3 v4 j n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n    rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n    !}\n  -- \u03bb\n  -- { k\u2081 k<n v1 dv v2 vv v3 v4 0 n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- !}\n  -- -- (Some.wfRec-builder rrelTV3-Type rrelTV3-step k\u2081\n  --                     -- (<-well-founded\u2032 k k\u2081 k<n))\n\n  -- -- All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k k\u2081 k<n : rrelTV3-Type k\u2081\n  -- --\n  -- ; k\u2081 k<n v1 dv v2 vv v3 v4 (suc j) n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- !}\n  -- -- rfundamental3 t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 {!suc j !} n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- }\n\n  --   -- \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n  --   -- \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  --   foo : s-rrelV3 k\n  --     (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k)\n  --     (rrelV3-step k (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k))\n  --     j j<k v1 (dclosure (derive-dfun t) \u03c11 d\u03c1) v2\n  --   foo with j\n  --   ... | s = {!!}\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set\n\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  term : \u2200 {v} n (t : Term \u0393) \u2192\n    \u03c1 \u22a2 t \u2193[ n ] v \u2192\n    \u03c1 F\u22a2 term t \u2193[ n ] v\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {dt : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs dt \u2193 dclosure dt \u03c1 d\u03c1\n  dterm : \u2200 {dv} (dt : DTerm \u0393) \u2192\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 F\u22a2 dterm dt \u2193 dv\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k < n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 =  <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n =  proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n =  proj\u2082 (rrelTV3 n)\n\n  s-rrelV3 :\n    \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n    \u2200 j \u2192 j < k \u2192 rrelV3-Type (k \u2238 j)\n  s-rrelV3 k rec-rrelTV3 rrelV3-k 0 _ = rrelV3-k\n  s-rrelV3 k rec-rrelTV3 rrelV3-k (suc j) sucj<k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt j k sucj<k))\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    -- rrelV3-Type n\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n      (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    \u2200 j (j<k : j < k) n2 \u2192\n    (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 k rec-rrelTV3 rrelV3-k j j<k v1 dv v2\n    -- where\n    --   s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n    --   -- If j = 0, we can't do a recursive call on (k - j) because that's not\n    --   -- well-founded. Luckily, we don't need to, just use relV as defined at\n    --   -- the same level.\n    --   -- Yet, this will be a pain in proofs.\n    --   s-rrelV3 0 _ = rrelV3-k\n    --   s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelV3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 _ _ _ = \u22a5\n\nopen import Postulate.Extensionality\nmutual\n  rrelT3-equiv : \u2200 k {\u0393} {t1 : Fun \u0393} {dt t2 \u03c11 d\u03c1 \u03c12} \u2192\n    rrelT3 k t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2261\n    \u2200 j (j<k : j < k) n2 \u2192\n    \u2200 (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 j v1 dv v2\n  rrelT3-equiv n = cong (\u03bb \u25a1 \u2192 \u2200 j \u2192 \u25a1 j) (ext (\u03bb j \u2192 {!!}))\n\n  rrelV3-equiv : \u2200 n {\u03931 \u03932 \u0394\u0393 t1 dt t2 \u03c11 \u03c1' d\u03c1 \u03c12} \u2192\n    rrelV3 n (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) \u2261 \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        rrelV3 k v1 dv v2 \u2192\n        rrelT3 k\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-equiv n = cong (\u03bb \u25a1 \u2192 \u03a3 (_ \u2261 _ \u00d7 _ \u2261 _) \u25a1) (ext (\u03bb { (eq1 , eq2) \u2192\n    cong\n      (\u03bb \u25a1 \u2192\n         (\u03bb { (refl , refl)\n                \u2192 \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192 \u25a1 k k<n v1 dv v2\n            })\n         eq1 eq2)\n      (ext (\u03bb x \u2192 ext {!!}))}))\n  --\n  -- cong (\u03bb \u25a1 \u2192 \u2200 k \u2192 (k<n : k <\u2032 n) \u2192 \u25a1 k k<n)\n  -- rrelV3-equiv k\n\nrrel\u03c13 : \u2115 \u2192 \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13 n \u2205 \u2205 \u2205 \u2205 = \u22a4\nrrel\u03c13 n (Uni \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) = rrelV3 n v1 dv v2 \u00d7 rrel\u03c13 n \u0393 \u03c11 d\u03c1 \u03c12\n\n--\nrfundamental3 : \u2200 {\u0393} (t : Fun \u0393) \u2192 \u2200 k \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 k \u0393 \u03c11 d\u03c1 \u03c12) \u2192 rrelT3 k t (derive-dfun t) t \u03c11 d\u03c1 \u03c12\nrfundamental3 (term x) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 j j<k n2 v1 v2 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 = {!!}\nrfundamental3 {\u0393} (abs t) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .0 j<k .0 (closure .t .\u03c11) (closure .t .\u03c12) abs abs = dclosure (derive-dfun t) \u03c11 d\u03c1 , 0 , dabs , (refl , refl) , {!body !}\n  where\n    body1 : \u2200 k\u2081 (k\u2081<k : k\u2081 <\u2032 k) \u2192 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 rrelV3 k\u2081 v1 dv v2 \u2192 rrelT3 k\u2081 t (derive-dfun t) t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12)\n    body1 = \u03bb k\u2081 k\u2081<k v1 dv v2 x v3 v4 j n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n    rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n    !}\n  -- \u03bb\n  -- { k\u2081 k<n v1 dv v2 vv v3 v4 0 n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- !}\n  -- -- (Some.wfRec-builder rrelTV3-Type rrelTV3-step k\u2081\n  --                     -- (<-well-founded\u2032 k k\u2081 k<n))\n\n  -- -- All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k k\u2081 k<n : rrelTV3-Type k\u2081\n  -- --\n  -- ; k\u2081 k<n v1 dv v2 vv v3 v4 (suc j) n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- !}\n  -- -- rfundamental3 t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 {!suc j !} n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- }\n\n  --   -- \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n  --   -- \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  --   foo : s-rrelV3 k\n  --     (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k)\n  --     (rrelV3-step k (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k))\n  --     j j<k v1 (dclosure (derive-dfun t) \u03c11 d\u03c1) v2\n  --   foo with j\n  --   ... | s = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"45a1deac50158607e07a6dce216d21001bc16512","subject":"give the `Terms` parameter of `const` a default name `args` for induction","message":"give the `Terms` parameter of `const` a default name `args` for induction\n\nOld-commit-hash: 0657adc1ef2e2e51a3d863ac5d78085bd8913b08\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\n\nopen Type.Structure Base\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n\n  -- Declarations of Term and Terms to enable mutual recursion\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  data Terms\n    (\u0393 : Context) :\n    (\u03a3 : Context) \u2192 Set\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  data Terms \u0393 where\n    \u2205 : Terms \u0393 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n      Term \u0393 \u03c4 \u2192\n      Terms \u0393 \u03a3 \u2192\n      Terms \u0393 (\u03c4 \u2022 \u03a3)\n\n  infixr 9 _\u2022_\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  abs\u2081 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 (x : Term \u0393\u2032 \u03c4\u2081) \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n  abs\u2081 {\u0393} {\u03c4\u2081} =  \u03bb f \u2192 abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n\n  abs\u2082 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4))\n  abs\u2082 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2081 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2083 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4))\n  abs\u2083 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2082 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2084 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4))\n  abs\u2084 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2083 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2085 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4))\n  abs\u2085 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2084 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2086 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4\u2086 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4\u2086 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4\u2086 \u21d2 \u03c4))\n  abs\u2086 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2085 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n","old_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\n\nopen Type.Structure Base\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n\n  -- Declarations of Term and Terms to enable mutual recursion\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  data Terms\n    (\u0393 : Context) :\n    (\u03a3 : Context) \u2192 Set\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      Terms \u0393 \u03a3 \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  data Terms \u0393 where\n    \u2205 : Terms \u0393 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n      Term \u0393 \u03c4 \u2192\n      Terms \u0393 \u03a3 \u2192\n      Terms \u0393 (\u03c4 \u2022 \u03a3)\n\n  infixr 9 _\u2022_\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  abs\u2081 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 (x : Term \u0393\u2032 \u03c4\u2081) \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n  abs\u2081 {\u0393} {\u03c4\u2081} =  \u03bb f \u2192 abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n\n  abs\u2082 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4))\n  abs\u2082 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2081 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2083 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4))\n  abs\u2083 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2082 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2084 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4))\n  abs\u2084 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2083 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2085 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4))\n  abs\u2085 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2084 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2086 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4\u2086 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4\u2086 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4\u2086 \u21d2 \u03c4))\n  abs\u2086 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2085 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a8f01929b8609987c9bee122265cf4a00fba1e88","subject":"Compile fix","message":"Compile fix\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Type.agda","new_file":"Parametric\/Syntax\/Type.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of function types (Fig. 1a).\n------------------------------------------------------------------------\n\nmodule Parametric.Syntax.Type where\n\n-- This is a module in the Parametric.* hierarchy, so it defines\n-- an extension point for plugins. In this case, the plugin can\n-- choose the syntax for data types. We always provide a binding\n-- called \"Structure\" that defines the type of the value that has\n-- to be provided by the plugin. In this case, the plugin has to\n-- provide a type, so we say \"Structure = Set\".\n\nStructure : Set\u2081\nStructure = Set\n\n-- Here is the parametric module that defines the syntax of\n-- simply types in terms of the syntax of base types. In the\n-- Parametric.* hierarchy, we always call these parametric\n-- modules \"Structure\". Note that in Agda, there is a separate\n-- namespace for module names and for other bindings, so this\n-- doesn't clash with the \"Structure\" above.\n--\n-- The idea behind all these structures is that the parametric\n-- module lifts some structure of the plugin to the corresponding\n-- structure in the full language. In this case, we lift the\n-- structure of base types to the structure of simple types.\n-- Maybe not the best choice of names, but it seemed clever at\n-- the time.\n\nmodule Structure (Base : Structure) where\n  infixr 5 _\u21d2_\n\n  -- A simple type is either a base type or a function types.\n  -- Note that we can use our module parameter \"Base\" here just\n  -- like any other type.\n  data Type : Set where\n    base : (\u03b9 : Base) \u2192 Type\n    _\u21d2_ : (\u03c3 : Type) \u2192 (\u03c4 : Type) \u2192 Type\n\n  -- We also provide the definitions of contexts of the newly\n  -- defined simple types, variables as de Bruijn indices\n  -- pointing into such a context, and sets of bound variables.\n  open import Base.Syntax.Context Type public\n  open import Base.Syntax.Vars Type public\n\n  -- Internalize a context to a type.\n  --\n  -- Is there a better name for this function?\n  internalizeContext : (\u03a3 : Context) (\u03c4\u2032 : Type) \u2192 Type\n  internalizeContext \u2205 \u03c4\u2032 = \u03c4\u2032\n  internalizeContext (\u03c4 \u2022 \u03a3) \u03c4\u2032 = \u03c4 \u21d2 internalizeContext \u03a3 \u03c4\u2032\n\n\n  mutual\n    --  CBPV\n    data ValType : Set where\n      U : CompType \u2192 ValType\n      B : Base \u2192 ValType\n      Unit : ValType\n      _\u00d7_ : ValType \u2192 ValType \u2192 ValType\n      _\u22b9_ : ValType \u2192 ValType \u2192 ValType\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of function types (Fig. 1a).\n------------------------------------------------------------------------\n\nmodule Parametric.Syntax.Type where\n\n-- This is a module in the Parametric.* hierarchy, so it defines\n-- an extension point for plugins. In this case, the plugin can\n-- choose the syntax for data types. We always provide a binding\n-- called \"Structure\" that defines the type of the value that has\n-- to be provided by the plugin. In this case, the plugin has to\n-- provide a type, so we say \"Structure = Set\".\n\nStructure : Set\u2081\nStructure = Set\n\n-- Here is the parametric module that defines the syntax of\n-- simply types in terms of the syntax of base types. In the\n-- Parametric.* hierarchy, we always call these parametric\n-- modules \"Structure\". Note that in Agda, there is a separate\n-- namespace for module names and for other bindings, so this\n-- doesn't clash with the \"Structure\" above.\n--\n-- The idea behind all these structures is that the parametric\n-- module lifts some structure of the plugin to the corresponding\n-- structure in the full language. In this case, we lift the\n-- structure of base types to the structure of simple types.\n-- Maybe not the best choice of names, but it seemed clever at\n-- the time.\n\nmodule Structure (Base : Structure) where\n  infixr 5 _\u21d2_\n\n  -- A simple type is either a base type or a function types.\n  -- Note that we can use our module parameter \"Base\" here just\n  -- like any other type.\n  data Type : Set where\n    base : (\u03b9 : Base) \u2192 Type\n    _\u21d2_ : (\u03c3 : Type) \u2192 (\u03c4 : Type) \u2192 Type\n\n  -- We also provide the definitions of contexts of the newly\n  -- defined simple types, variables as de Bruijn indices\n  -- pointing into such a context, and sets of bound variables.\n  open import Base.Syntax.Context Type public\n  open import Base.Syntax.Vars Type public\n\n  -- Internalize a context to a type.\n  --\n  -- Is there a better name for this function?\n  internalizeContext : (\u03a3 : Context) (\u03c4\u2032 : Type) \u2192 Type\n  internalizeContext \u2205 \u03c4\u2032 = \u03c4\u2032\n  internalizeContext (\u03c4 \u2022 \u03a3) \u03c4\u2032 = \u03c4 \u21d2 internalizeContext \u03a3 \u03c4\u2032\n\n\n  mutual\n    --  CBPV\n    data ValType : Set where\n      U : CompType \u2192 ValType\n      B : Base \u2192 ValType\n      Unit : ValType\n      _\u00d7_ : ValType \u2192 ValType \u2192 ValType\n      _+_ : ValType \u2192 ValType \u2192 ValType\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3c5110cc5671b1357a521af48e1e4926d615bd15","subject":"Added numbers.","message":"Added numbers.\n\nIgnore-this: c59bc2222a766dd1e2d9419a5f29e709\n\ndarcs-hash:20110208163438-3bd4e-889aba18405b0b32701d88789d240233e5d75a80.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/Numbers.agda","new_file":"Draft\/McCarthy91\/Numbers.agda","new_contents":"------------------------------------------------------------------------------\n-- Some naturales numbers\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Numbers where\n\nopen import LTC.Base\n\n------------------------------------------------------------------------------\n\none    = succ zero\ntwo    = succ one\nthree  = succ two\nfour   = succ three\nfive   = succ four\nsix    = succ five\nseven  = succ six\neight  = succ seven\nnine   = succ eight\nten    = succ nine\n\n{-# ATP definition one #-}\n{-# ATP definition two #-}\n{-# ATP definition three #-}\n{-# ATP definition four #-}\n{-# ATP definition five #-}\n{-# ATP definition six #-}\n{-# ATP definition seven #-}\n{-# ATP definition eight #-}\n{-# ATP definition nine #-}\n{-# ATP definition ten #-}\n\neleven    = succ ten\ntwelve    = succ eleven\nthirteen  = succ twelve\nfourteen  = succ thirteen\nfifteen   = succ fourteen\nsixteen   = succ fifteen\nseventeen = succ sixteen\neighteen  = succ seventeen\nnineteen  = succ eighteen\ntwenty    = succ nineteen\n\n{-# ATP definition eleven #-}\n{-# ATP definition twelve #-}\n{-# ATP definition thirteen #-}\n{-# ATP definition fourteen #-}\n{-# ATP definition fifteen #-}\n{-# ATP definition sixteen #-}\n{-# ATP definition seventeen #-}\n{-# ATP definition eighteen #-}\n{-# ATP definition nineteen #-}\n{-# ATP definition twenty #-}\n\ntwenty-one   = succ twenty\ntwenty-two   = succ twenty-one\ntwenty-three = succ twenty-two\ntwenty-four  = succ twenty-three\ntwenty-five  = succ twenty-four\ntwenty-six   = succ twenty-five\ntwenty-seven = succ twenty-six\ntwenty-eight = succ twenty-seven\ntwenty-nine  = succ twenty-eight\nthirty       = succ twenty-nine\n\n{-# ATP definition twenty-one #-}\n{-# ATP definition twenty-two #-}\n{-# ATP definition twenty-three #-}\n{-# ATP definition twenty-four #-}\n{-# ATP definition twenty-five #-}\n{-# ATP definition twenty-six #-}\n{-# ATP definition twenty-seven #-}\n{-# ATP definition twenty-eight #-}\n{-# ATP definition twenty-nine #-}\n{-# ATP definition thirty #-}\n\nthirty-one   = succ thirty\nthirty-two   = succ thirty-one\nthirty-three = succ thirty-two\nthirty-four  = succ thirty-three\nthirty-five  = succ thirty-four\nthirty-six   = succ thirty-five\nthirty-seven = succ thirty-six\nthirty-eight = succ thirty-seven\nthirty-nine  = succ thirty-eight\nforty        = succ thirty-nine\n\n{-# ATP definition thirty-one #-}\n{-# ATP definition thirty-two #-}\n{-# ATP definition thirty-three #-}\n{-# ATP definition thirty-four #-}\n{-# ATP definition thirty-five #-}\n{-# ATP definition thirty-six #-}\n{-# ATP definition thirty-seven #-}\n{-# ATP definition thirty-eight #-}\n{-# ATP definition thirty-nine #-}\n{-# ATP definition forty #-}\n\nforty-one   = succ forty\nforty-two   = succ forty-one\nforty-three = succ forty-two\nforty-four  = succ forty-three\nforty-five  = succ forty-four\nforty-six   = succ forty-five\nforty-seven = succ forty-six\nforty-eight = succ forty-seven\nforty-nine  = succ forty-eight\nfifty       = succ forty-nine\n\n{-# ATP definition forty-one #-}\n{-# ATP definition forty-two #-}\n{-# ATP definition forty-three #-}\n{-# ATP definition forty-four #-}\n{-# ATP definition forty-five #-}\n{-# ATP definition forty-six #-}\n{-# ATP definition forty-seven #-}\n{-# ATP definition forty-eight #-}\n{-# ATP definition forty-nine #-}\n{-# ATP definition fifty #-}\n\nfifty-one   = succ fifty\nfifty-two   = succ fifty-one\nfifty-three = succ fifty-two\nfifty-four  = succ fifty-three\nfifty-five  = succ fifty-four\nfifty-six   = succ fifty-five\nfifty-seven = succ fifty-six\nfifty-eight = succ fifty-seven\nfifty-nine  = succ fifty-eight\nsixty       = succ fifty-nine\n\n{-# ATP definition fifty-one #-}\n{-# ATP definition fifty-two #-}\n{-# ATP definition fifty-three #-}\n{-# ATP definition fifty-four #-}\n{-# ATP definition fifty-five #-}\n{-# ATP definition fifty-six #-}\n{-# ATP definition fifty-seven #-}\n{-# ATP definition fifty-eight #-}\n{-# ATP definition fifty-nine #-}\n{-# ATP definition sixty #-}\n\nsixty-one   = succ sixty\nsixty-two   = succ sixty-one\nsixty-three = succ sixty-two\nsixty-four  = succ sixty-three\nsixty-five  = succ sixty-four\nsixty-six   = succ sixty-five\nsixty-seven = succ sixty-six\nsixty-eight = succ sixty-seven\nsixty-nine  = succ sixty-eight\nseventy     = succ sixty-nine\n\n{-# ATP definition sixty-one #-}\n{-# ATP definition sixty-two #-}\n{-# ATP definition sixty-three #-}\n{-# ATP definition sixty-four #-}\n{-# ATP definition sixty-five #-}\n{-# ATP definition sixty-six #-}\n{-# ATP definition sixty-seven #-}\n{-# ATP definition sixty-eight #-}\n{-# ATP definition sixty-nine #-}\n{-# ATP definition seventy #-}\n\nseventy-one   = succ seventy\nseventy-two   = succ seventy-one\nseventy-three = succ seventy-two\nseventy-four  = succ seventy-three\nseventy-five  = succ seventy-four\nseventy-six   = succ seventy-five\nseventy-seven = succ seventy-six\nseventy-eight = succ seventy-seven\nseventy-nine  = succ seventy-eight\neighty       = succ seventy-nine\n\n{-# ATP definition seventy-one #-}\n{-# ATP definition seventy-two #-}\n{-# ATP definition seventy-three #-}\n{-# ATP definition seventy-four #-}\n{-# ATP definition seventy-five #-}\n{-# ATP definition seventy-six #-}\n{-# ATP definition seventy-seven #-}\n{-# ATP definition seventy-eight #-}\n{-# ATP definition seventy-nine #-}\n{-# ATP definition eighty #-}\n\neighty-one   = succ eighty\neighty-two   = succ eighty-one\neighty-three = succ eighty-two\neighty-four  = succ eighty-three\neighty-five  = succ eighty-four\neighty-six   = succ eighty-five\neighty-seven = succ eighty-six\neighty-eight = succ eighty-seven\neighty-nine  = succ eighty-eight\nninety       = succ eighty-nine\n\n{-# ATP definition eighty-one #-}\n{-# ATP definition eighty-two #-}\n{-# ATP definition eighty-three #-}\n{-# ATP definition eighty-four #-}\n{-# ATP definition eighty-five #-}\n{-# ATP definition eighty-six #-}\n{-# ATP definition eighty-seven #-}\n{-# ATP definition eighty-eight #-}\n{-# ATP definition eighty-nine #-}\n{-# ATP definition ninety #-}\n\nninety-one   = succ ninety\nninety-two   = succ ninety-one\nninety-three = succ ninety-two\nninety-four  = succ ninety-three\nninety-five  = succ ninety-four\nninety-six   = succ ninety-five\nninety-seven = succ ninety-six\nninety-eight = succ ninety-seven\nninety-nine  = succ ninety-eight\none-hundred  = succ ninety-nine\n\n{-# ATP definition ninety-one #-}\n{-# ATP definition ninety-two #-}\n{-# ATP definition ninety-three #-}\n{-# ATP definition ninety-four #-}\n{-# ATP definition ninety-five #-}\n{-# ATP definition ninety-six #-}\n{-# ATP definition ninety-seven #-}\n{-# ATP definition ninety-eight #-}\n{-# ATP definition ninety-nine #-}\n{-# ATP definition one-hundred #-}\n\nhundred-one   = succ one-hundred\nhundred-two   = succ hundred-one\nhundred-three = succ hundred-two\nhundred-four  = succ hundred-three\nhundred-five  = succ hundred-four\nhundred-six   = succ hundred-five\nhundred-seven = succ hundred-six\nhundred-eight = succ hundred-seven\nhundred-nine  = succ hundred-eight\nhundred-ten   = succ hundred-nine\n\n{-# ATP definition hundred-one #-}\n{-# ATP definition hundred-two #-}\n{-# ATP definition hundred-three #-}\n{-# ATP definition hundred-four #-}\n{-# ATP definition hundred-five #-}\n{-# ATP definition hundred-six #-}\n{-# ATP definition hundred-seven #-}\n{-# ATP definition hundred-eight #-}\n{-# ATP definition hundred-nine #-}\n{-# ATP definition hundred-ten #-}\n\nhundred-eleven = succ hundred-ten\n{-# ATP definition hundred-eleven #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some naturales numbers\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Numbers where\n\nopen import LTC.Base\n\n------------------------------------------------------------------------------\n\none    = succ zero\ntwo    = succ one\nthree  = succ two\nfour   = succ three\nfive   = succ four\nsix    = succ five\nseven  = succ six\neight  = succ seven\nnine   = succ eight\nten    = succ nine\n\n{-# ATP definition one #-}\n{-# ATP definition two #-}\n{-# ATP definition three #-}\n{-# ATP definition four #-}\n{-# ATP definition five #-}\n{-# ATP definition six #-}\n{-# ATP definition seven #-}\n{-# ATP definition eight #-}\n{-# ATP definition nine #-}\n{-# ATP definition ten #-}\n\neleven    = succ ten\ntwelve    = succ eleven\nthirteen  = succ twelve\nfourteen  = succ thirteen\nfifteen   = succ fourteen\nsixteen   = succ fifteen\nseventeen = succ sixteen\neighteen  = succ seventeen\nnineteen  = succ eighteen\ntwenty    = succ nineteen\n\n{-# ATP definition eleven #-}\n{-# ATP definition twelve #-}\n{-# ATP definition thirteen #-}\n{-# ATP definition fourteen #-}\n{-# ATP definition fifteen #-}\n{-# ATP definition sixteen #-}\n{-# ATP definition seventeen #-}\n{-# ATP definition eighteen #-}\n{-# ATP definition nineteen #-}\n{-# ATP definition twenty #-}\n\ntwenty-one   = succ twenty\ntwenty-two   = succ twenty-one\ntwenty-three = succ twenty-two\ntwenty-four  = succ twenty-three\ntwenty-five  = succ twenty-four\ntwenty-six   = succ twenty-five\ntwenty-seven = succ twenty-six\ntwenty-eight = succ twenty-seven\ntwenty-nine  = succ twenty-eight\nthirty       = succ twenty-nine\n\n{-# ATP definition twenty-one #-}\n{-# ATP definition twenty-two #-}\n{-# ATP definition twenty-three #-}\n{-# ATP definition twenty-four #-}\n{-# ATP definition twenty-five #-}\n{-# ATP definition twenty-six #-}\n{-# ATP definition twenty-seven #-}\n{-# ATP definition twenty-eight #-}\n{-# ATP definition twenty-nine #-}\n{-# ATP definition thirty #-}\n\nthirty-one   = succ thirty\nthirty-two   = succ thirty-one\nthirty-three = succ thirty-two\nthirty-four  = succ thirty-three\nthirty-five  = succ thirty-four\nthirty-six   = succ thirty-five\nthirty-seven = succ thirty-six\nthirty-eight = succ thirty-seven\nthirty-nine  = succ thirty-eight\nforty        = succ thirty-nine\n\n{-# ATP definition thirty-one #-}\n{-# ATP definition thirty-two #-}\n{-# ATP definition thirty-three #-}\n{-# ATP definition thirty-four #-}\n{-# ATP definition thirty-five #-}\n{-# ATP definition thirty-six #-}\n{-# ATP definition thirty-seven #-}\n{-# ATP definition thirty-eight #-}\n{-# ATP definition thirty-nine #-}\n{-# ATP definition forty #-}\n\nforty-one   = succ forty\nforty-two   = succ forty-one\nforty-three = succ forty-two\nforty-four  = succ forty-three\nforty-five  = succ forty-four\nforty-six   = succ forty-five\nforty-seven = succ forty-six\nforty-eight = succ forty-seven\nforty-nine  = succ forty-eight\nfifty       = succ forty-nine\n\n{-# ATP definition forty-one #-}\n{-# ATP definition forty-two #-}\n{-# ATP definition forty-three #-}\n{-# ATP definition forty-four #-}\n{-# ATP definition forty-five #-}\n{-# ATP definition forty-six #-}\n{-# ATP definition forty-seven #-}\n{-# ATP definition forty-eight #-}\n{-# ATP definition forty-nine #-}\n{-# ATP definition fifty #-}\n\nfifty-one   = succ fifty\nfifty-two   = succ fifty-one\nfifty-three = succ fifty-two\nfifty-four  = succ fifty-three\nfifty-five  = succ fifty-four\nfifty-six   = succ fifty-five\nfifty-seven = succ fifty-six\nfifty-eight = succ fifty-seven\nfifty-nine  = succ fifty-eight\nsixty       = succ fifty-nine\n\n{-# ATP definition fifty-one #-}\n{-# ATP definition fifty-two #-}\n{-# ATP definition fifty-three #-}\n{-# ATP definition fifty-four #-}\n{-# ATP definition fifty-five #-}\n{-# ATP definition fifty-six #-}\n{-# ATP definition fifty-seven #-}\n{-# ATP definition fifty-eight #-}\n{-# ATP definition fifty-nine #-}\n{-# ATP definition sixty #-}\n\nsixty-one   = succ sixty\nsixty-two   = succ sixty-one\nsixty-three = succ sixty-two\nsixty-four  = succ sixty-three\nsixty-five  = succ sixty-four\nsixty-six   = succ sixty-five\nsixty-seven = succ sixty-six\nsixty-eight = succ sixty-seven\nsixty-nine  = succ sixty-eight\nseventy     = succ sixty-nine\n\n{-# ATP definition sixty-one #-}\n{-# ATP definition sixty-two #-}\n{-# ATP definition sixty-three #-}\n{-# ATP definition sixty-four #-}\n{-# ATP definition sixty-five #-}\n{-# ATP definition sixty-six #-}\n{-# ATP definition sixty-seven #-}\n{-# ATP definition sixty-eight #-}\n{-# ATP definition sixty-nine #-}\n{-# ATP definition seventy #-}\n\nseventy-one   = succ seventy\nseventy-two   = succ seventy-one\nseventy-three = succ seventy-two\nseventy-four  = succ seventy-three\nseventy-five  = succ seventy-four\nseventy-six   = succ seventy-five\nseventy-seven = succ seventy-six\nseventy-eight = succ seventy-seven\nseventy-nine  = succ seventy-eight\neighty       = succ seventy-nine\n\n{-# ATP definition seventy-one #-}\n{-# ATP definition seventy-two #-}\n{-# ATP definition seventy-three #-}\n{-# ATP definition seventy-four #-}\n{-# ATP definition seventy-five #-}\n{-# ATP definition seventy-six #-}\n{-# ATP definition seventy-seven #-}\n{-# ATP definition seventy-eight #-}\n{-# ATP definition seventy-nine #-}\n{-# ATP definition eighty #-}\n\neighty-one   = succ eighty\neighty-two   = succ eighty-one\neighty-three = succ eighty-two\neighty-four  = succ eighty-three\neighty-five  = succ eighty-four\neighty-six   = succ eighty-five\neighty-seven = succ eighty-six\neighty-eight = succ eighty-seven\neighty-nine  = succ eighty-eight\nninety       = succ eighty-nine\n\n{-# ATP definition eighty-one #-}\n{-# ATP definition eighty-two #-}\n{-# ATP definition eighty-three #-}\n{-# ATP definition eighty-four #-}\n{-# ATP definition eighty-five #-}\n{-# ATP definition eighty-six #-}\n{-# ATP definition eighty-seven #-}\n{-# ATP definition eighty-eight #-}\n{-# ATP definition eighty-nine #-}\n{-# ATP definition ninety #-}\n\nninety-one   = succ ninety\nninety-two   = succ ninety-one\nninety-three = succ ninety-two\nninety-four  = succ ninety-three\nninety-five  = succ ninety-four\nninety-six   = succ ninety-five\nninety-seven = succ ninety-six\nninety-eight = succ ninety-seven\nninety-nine  = succ ninety-eight\none-hundred  = succ ninety-nine\n\n{-# ATP definition ninety-one #-}\n{-# ATP definition ninety-two #-}\n{-# ATP definition ninety-three #-}\n{-# ATP definition ninety-four #-}\n{-# ATP definition ninety-five #-}\n{-# ATP definition ninety-six #-}\n{-# ATP definition ninety-seven #-}\n{-# ATP definition ninety-eight #-}\n{-# ATP definition ninety-nine #-}\n{-# ATP definition one-hundred #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ba174bee353125d20b6c312ebc1086368d3daf6d","subject":"Instantiate (Term ...) at module import and inline Atlas-term.","message":"Instantiate (Term ...) at module import and inline Atlas-term.\n\nOld-commit-hash: d1f5249bb1b939c52aa2da6bf3e0b54cd0be9e9e\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin Atlas-type\n\ndata Atlas-const : Type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type \u2192 Type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nopen import Syntax.Context {Type}\nopen import Syntax.Term.Plotkin {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Term \u0393 (base Bool) \u2192 Term \u0393 (base Bool) \u2192\n  Term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Term \u0393 (base (Map \u03ba a)) \u2192 Term \u0393 (base (Map \u03ba b)) \u2192\n  Term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Term \u0393 (base b) \u2192 Term \u0393 (base (Map \u03ba a)) \u2192\n  Term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Term \u0393 (base \u03b1) \u2192 Term \u0393 (base \u03b2) \u2192\n  Term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Term \u0393 (base (Map \u03ba a)) \u2192 Term \u0393 (base (Map \u03ba b)) \u2192\n  Term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Term \u0393 (base \u03b9)\n  in\n    Term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin Atlas-type\n\ndata Atlas-const : Type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type \u2192 Type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nopen import Syntax.Context {Type}\n\nopen import Syntax.Term.Plotkin\n\nAtlas-term : Context \u2192 Type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n  Atlas-term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"42ae1a35bb1ac9d1c07108e9ae8061964838aae2","subject":"Add elimDesc to formalization","message":"Add elimDesc to formalization\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (C t) (\u03bc E C)\n\ninj : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) (t : Tag E) \u2192 CurriedEl (C t) (\u03bc E C)\ninj E C t = curryEl (C t) (\u03bc E C) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i x,xs = \u03b1 j (proj\u2081 x,xs) , hyps D X P \u03b1 i (proj\u2082 x,xs)\nhyps (Arg A B) X P \u03b1 i a,xs = hyps (B (proj\u2081 a,xs)) X P \u03b1 i (proj\u2082 a,xs)\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nind E C P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (C t) (\u03bc E C) P (ind E C P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nindCurried E C P f i x = ind E C P (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E C X cn P t = CurriedHyps (C t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E C P t = Summer E C (\u03bc E C) init P t\n\nelimUncurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E C P)\n  \u2192 (i : I) (x : \u03bc E C i) \u2192 P i x\nelimUncurried E C P cs i x =\n  indCurried E C P\n    (case (SumCurriedHyps E C P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E C P)\n      ((i : I) (x : \u03bc E C i) \u2192 P i x)\nelim E C P = curryBranches (elimUncurried E C P)\n\n----------------------------------------------------------------------\n\nSoundness : Set\u2081\nSoundness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (cs : Branches E (SumCurriedHyps E C P))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb \u03b1\n  \u2192 elimUncurried E C P cs i x \u2261 ind E C P \u03b1 i x\n\nsound : Soundness\nsound E C P cs i xs =\n  let D = Arg (Tag E) C in\n  (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) cs t)) , refl\n\nCompleteness : Set\u2081\nCompleteness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb cs\n  \u2192 ind E C P \u03b1 i x \u2261 elimUncurried E C P cs i x\n\nuncurryHypsIdent : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  (\u03b1 : UncurriedHyps D X P cn)\n  (i : I) (xs : El D X i) (ihs : Hyps D X P i xs)\n  \u2192 \u03b1 i xs ihs \u2261 uncurryHyps D X P cn (curryHyps D X P cn \u03b1) i xs ihs\nuncurryHypsIdent (End .i) X P cn \u03b1 i refl tt = refl\nuncurryHypsIdent (Rec j D) X P cn \u03b1 i (x , xs) (p , ps) =\n  uncurryHypsIdent D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb k ys rs \u2192 \u03b1 k (x , ys) (p , rs)) i xs ps\nuncurryHypsIdent (Arg A B) X P cn \u03b1 i (a , xs) ps =\n  uncurryHypsIdent (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb j ys \u2192 \u03b1 j (a , ys)) i xs ps\n\npostulate\n  ext4 : {A : Set} {B : A \u2192 Set} {C : (a : A) \u2192 B a \u2192 Set}\n    {D : (a : A) (b : B a) \u2192 C a b \u2192 Set}\n    {Z : (a : A) (b : B a) (c : C a b) \u2192 D a b c \u2192 Set}\n    (f g : (a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 Z a b c d)\n    \u2192 ((a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 f a b c d \u2261 g a b c d)\n    \u2192 f \u2261 g\n\ntoBranches : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  \u2192 Branches E (Summer E C X cn P)\ntoBranches [] C X cn P \u03b1 = tt\ntoBranches (l \u2237 E) C X cn P \u03b1 =\n    curryHyps (C here) X P (\u03bb xs \u2192 cn here xs) (\u03bb i xs \u2192 \u03b1 here i xs)\n  , toBranches E (\u03bb t \u2192 C (there t)) X\n     (\u03bb t \u2192 cn (there t))\n     P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih)\n\nToBranches : {I : Set} {E : Enum} (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  (t : Tag E)\n  \u2192 let \u03b2 = toBranches E C X cn P \u03b1 in\n  case (Summer E C X cn P) \u03b2 t \u2261 curryHyps (C t) X P (cn t) (\u03b1 t)\nToBranches C X cn P \u03b1 here = refl\nToBranches C X cn P \u03b1 (there t)\n  with ToBranches (\u03bb t \u2192 C (there t)) X\n    (\u03bb t xs \u2192 cn (there t) xs)\n    P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih) t\n... | ih rewrite ih = refl\n\ncomplete\u03b1 : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (t : Tag E) (i : I) (xs : El (C t) (\u03bc E C) i) (ihs : Hyps (C t) (\u03bc E C) P i xs)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  \u03b1 t i xs ihs \u2261 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t) i xs ihs\ncomplete\u03b1 E C P \u03b1 t i xs ihs\n  with ToBranches C (\u03bc E C) init P \u03b1 t\n... | q rewrite q = uncurryHypsIdent (C t) (\u03bc E C) P (init t) (\u03b1 t) i xs ihs\n\ncomplete' : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  ind E C P \u03b1 i x \u2261 elimUncurried E C P \u03b2 i x\ncomplete' E C P \u03b1 i (init t xs) = cong\n  (\u03bb f \u2192 ind E C P f i (init t xs))\n  (ext4 \u03b1\n    (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t))\n    (complete\u03b1 E C P \u03b1)\n  )\n  where \u03b2 = toBranches E C (\u03bc E C) init P \u03b1\n\ncomplete : Completeness\ncomplete E C P \u03b1 i x =\n  let D = Arg (Tag E) C in\n    toBranches E C (\u03bc E C) init P \u03b1\n  , complete' E C P \u03b1 i x\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115C : \u2115T \u2192 Desc \u22a4\n\u2115C = caseD $\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T \u2115C\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115C\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecC : (A : Set) \u2192 VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD $\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (VecC A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecC A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecC A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecC A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecC A) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC A t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecC (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC (Vec A m) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115C _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115C _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecC A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecC (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (C t) (\u03bc E C)\n\ninj : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) (t : Tag E) \u2192 CurriedEl (C t) (\u03bc E C)\ninj E C t = curryEl (C t) (\u03bc E C) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i x,xs = \u03b1 j (proj\u2081 x,xs) , hyps D X P \u03b1 i (proj\u2082 x,xs)\nhyps (Arg A B) X P \u03b1 i a,xs = hyps (B (proj\u2081 a,xs)) X P \u03b1 i (proj\u2082 a,xs)\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nind E C P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (C t) (\u03bc E C) P (ind E C P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nindCurried E C P f i x = ind E C P (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E C X cn P t = CurriedHyps (C t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E C P t = Summer E C (\u03bc E C) init P t\n\nelimUncurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E C P)\n  \u2192 (i : I) (x : \u03bc E C i) \u2192 P i x\nelimUncurried E C P cs i x =\n  indCurried E C P\n    (case (SumCurriedHyps E C P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E C P)\n      ((i : I) (x : \u03bc E C i) \u2192 P i x)\nelim E C P = curryBranches (elimUncurried E C P)\n\n----------------------------------------------------------------------\n\nSoundness : Set\u2081\nSoundness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (cs : Branches E (SumCurriedHyps E C P))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb \u03b1\n  \u2192 elimUncurried E C P cs i x \u2261 ind E C P \u03b1 i x\n\nsound : Soundness\nsound E C P cs i xs =\n  let D = Arg (Tag E) C in\n  (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) cs t)) , refl\n\nCompleteness : Set\u2081\nCompleteness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb cs\n  \u2192 ind E C P \u03b1 i x \u2261 elimUncurried E C P cs i x\n\nuncurryHypsIdent : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  (\u03b1 : UncurriedHyps D X P cn)\n  (i : I) (xs : El D X i) (ihs : Hyps D X P i xs)\n  \u2192 \u03b1 i xs ihs \u2261 uncurryHyps D X P cn (curryHyps D X P cn \u03b1) i xs ihs\nuncurryHypsIdent (End .i) X P cn \u03b1 i refl tt = refl\nuncurryHypsIdent (Rec j D) X P cn \u03b1 i (x , xs) (p , ps) =\n  uncurryHypsIdent D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb k ys rs \u2192 \u03b1 k (x , ys) (p , rs)) i xs ps\nuncurryHypsIdent (Arg A B) X P cn \u03b1 i (a , xs) ps =\n  uncurryHypsIdent (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb j ys \u2192 \u03b1 j (a , ys)) i xs ps\n\npostulate\n  ext4 : {A : Set} {B : A \u2192 Set} {C : (a : A) \u2192 B a \u2192 Set}\n    {D : (a : A) (b : B a) \u2192 C a b \u2192 Set}\n    {Z : (a : A) (b : B a) (c : C a b) \u2192 D a b c \u2192 Set}\n    (f g : (a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 Z a b c d)\n    \u2192 ((a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 f a b c d \u2261 g a b c d)\n    \u2192 f \u2261 g\n\ntoBranches : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  \u2192 Branches E (Summer E C X cn P)\ntoBranches [] C X cn P \u03b1 = tt\ntoBranches (l \u2237 E) C X cn P \u03b1 =\n    curryHyps (C here) X P (\u03bb xs \u2192 cn here xs) (\u03bb i xs \u2192 \u03b1 here i xs)\n  , toBranches E (\u03bb t \u2192 C (there t)) X\n     (\u03bb t \u2192 cn (there t))\n     P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih)\n\nToBranches : {I : Set} {E : Enum} (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  (t : Tag E)\n  \u2192 let \u03b2 = toBranches E C X cn P \u03b1 in\n  case (Summer E C X cn P) \u03b2 t \u2261 curryHyps (C t) X P (cn t) (\u03b1 t)\nToBranches C X cn P \u03b1 here = refl\nToBranches C X cn P \u03b1 (there t)\n  with ToBranches (\u03bb t \u2192 C (there t)) X\n    (\u03bb t xs \u2192 cn (there t) xs)\n    P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih) t\n... | ih rewrite ih = refl\n\ncomplete\u03b1 : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (t : Tag E) (i : I) (xs : El (C t) (\u03bc E C) i) (ihs : Hyps (C t) (\u03bc E C) P i xs)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  \u03b1 t i xs ihs \u2261 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t) i xs ihs\ncomplete\u03b1 E C P \u03b1 t i xs ihs\n  with ToBranches C (\u03bc E C) init P \u03b1 t\n... | q rewrite q = uncurryHypsIdent (C t) (\u03bc E C) P (init t) (\u03b1 t) i xs ihs\n\ncomplete' : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  ind E C P \u03b1 i x \u2261 elimUncurried E C P \u03b2 i x\ncomplete' E C P \u03b1 i (init t xs) = cong\n  (\u03bb f \u2192 ind E C P f i (init t xs))\n  (ext4 \u03b1\n    (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t))\n    (complete\u03b1 E C P \u03b1)\n  )\n  where \u03b2 = toBranches E C (\u03bc E C) init P \u03b1\n\ncomplete : Completeness\ncomplete E C P \u03b1 i x =\n  let D = Arg (Tag E) C in\n    toBranches E C (\u03bc E C) init P \u03b1\n  , complete' E C P \u03b1 i x\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115C : \u2115T \u2192 Desc \u22a4\n\u2115C = caseD $\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T \u2115C\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115C\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecC : (A : Set) \u2192 VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD $\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (VecC A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecC A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecC A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecC A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecC A) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC A t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecC (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC (Vec A m) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115C _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115C _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecC A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecC (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d3b24ab0dab8b2f1fb47722d3ac214bc1561629b","subject":"Minor example change.","message":"Minor example change.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , IArg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} {n : \u2115} (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} {m n : \u2115} (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} {m} {n} = elim VecR\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb x xs ih ys \u2192 cons x (ih ys))\n  m\n\nconcat : {A : Set} {m n : \u2115} (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} {m} {n} = elim VecR\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb xs xss ih \u2192 append xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : \u2115\none = suc zero\n\ntwo : \u2115\ntwo = suc one\n\nthree : \u2115\nthree = suc two\n\n[1] : Vec \u2115 one\n[1] = cons one nil\n\n[2,3] : Vec \u2115 two\n[2,3] = cons two (cons three nil)\n\n[1,2,3] : Vec \u2115 three\n[1,2,3] = cons one (cons two (cons three nil))\n\ntest-append : [1,2,3] \u2261 append [1] [2,3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , IArg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} {n : \u2115} (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} {m : \u2115} (xs : Vec A m) {n : \u2115} (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} {m} = elim VecR\n  (\u03bb m xs \u2192 {n : \u2115} (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb x xs ih ys \u2192 cons x (ih ys))\n  m\n\nconcat : {A : Set} {m n : \u2115} (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} {m} {n} = elim VecR\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb xs xss ih \u2192 append xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : \u2115\none = suc zero\n\ntwo : \u2115\ntwo = suc one\n\nthree : \u2115\nthree = suc two\n\n[1] : Vec \u2115 one\n[1] = cons one nil\n\n[2,3] : Vec \u2115 two\n[2,3] = cons two (cons three nil)\n\n[1,2,3] : Vec \u2115 three\n[1,2,3] = cons one (cons two (cons three nil))\n\ntest-append : [1,2,3] \u2261 append [1] [2,3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1da8d765d3b45cbc385435bbd6c113c680b6f896","subject":"Generalize helper methods.","message":"Generalize helper methods.\n\nBefore this commit, the helper methods worked for one particular context\nthat was inferred from the helper's use sites. I want to change the\nhelper's use suite in the next commit so that the context would no\nlonger get inferred automatically, so I have to generalize the helper\nmethods for arbitrary contexts first.\n\nOld-commit-hash: 331ecddb578d8491bb6b65d1312b789d9260ff31\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app\u2082 diff\u03c3 s t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app\u2082 diff\u03c4 s t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app\u2082 apply\u03c3 \u0394t t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app\u2082 apply\u03c4 \u0394t t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"91c228f51658b4d57015ba27b291178a7cbfb4ee","subject":"Desc model: proof-phi-psi implicit.","message":"Desc model: proof-phi-psi implicit.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : (I : Set) -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi I (var i) = refl\nproof-phi-psi I (const x) = refl\nproof-phi-psi I (prod D D') with proof-phi-psi I D | proof-phi-psi I D'\n...                             | p | q = cong2 prod p q\nproof-phi-psi I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi I (T s)))\nproof-phi-psi I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                              T\n                                              (\\s -> proof-phi-psi I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"65a027f17f89f3f1babdbfeb27d53029aeef353a","subject":"Fixed notes.","message":"Fixed notes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/CombinedProofs\/AddRecursiveEquation.agda","new_file":"notes\/thesis\/CombinedProofs\/AddRecursiveEquation.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of addition using a recursive equation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule CombinedProofs.AddRecursiveEquation where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-eq : \u2200 m n \u2192 m + n \u2261 if (iszero\u2081 m) then n else succ\u2081 (pred\u2081 m + n)\n{-# ATP axiom +-eq #-}\n\npostulate\n  +-0x : \u2200 n \u2192   zero    + n \u2261 n\n  +-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n{-# ATP prove +-0x #-}\n{-# ATP prove +-Sx #-}\n\n-- $ agda2atp -i. -i ~\/fotc\/fot\/ FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation.agda\n-- Proving the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/23-43-Sx.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/23-43-Sx.tptp\n-- Proving the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/22-43-0x.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/22-43-0x.tptp\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of addition using a recursive equation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.AddRecursiveEquation where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-eq : \u2200 m n \u2192 m + n \u2261 if (iszero\u2081 m) then n else succ\u2081 (pred\u2081 m + n)\n{-# ATP axiom +-eq #-}\n\npostulate\n  +-0x : \u2200 n \u2192   zero    + n \u2261 n\n  +-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n{-# ATP prove +-0x #-}\n{-# ATP prove +-Sx #-}\n\n-- $ agda2atp -i. -i ~\/fotc\/fot\/ FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation.agda\n-- Proving the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/23-43-Sx.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/23-43-Sx.tptp\n-- Proving the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/22-43-0x.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/22-43-0x.tptp\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fa53dc9c768d14c58160f019f07c795b973f0043","subject":"bintree: sort is sorting!","message":"bintree: sort is sorting!\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\nopen Dummy public\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    -- open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\n    {-\nmodule M {n} where\n  postulate\n    _\u2264_ : {-\u2200 {n} \u2192-} Bits n \u2192 Bits n \u2192 Set\n    _\u2294_ : Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : Bits n \u2192 Bits n \u2192 Bits n\n    isPreorder : IsPreorder _\u2261_ _\u2264_\n    \u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n    \u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n    \u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n    \u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n    \u2293-comm : Commutative _\u2261_ _\u2293_\n    \u2294-comm : Commutative _\u2261_ _\u2294_\n    \u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n    \u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n    \u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n    \u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n    \u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n    \u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\nmodule BitsSorting {m} where\n    open M {m}\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2294-spec \u2293-spec \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open SortedData _\u2264_\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n  -}\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n{-\n    data _\u2264\u1d3e_ : \u2200 {x y t} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set where\n      here-here   : here \u2264\u1d3e here\n      left-left   : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 left p \u2264\u1d3e left q\n      right-right : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 right p \u2264\u1d3e right q\n      left-right  : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 left p \u2264\u1d3e right q\n      -}\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\nmodule Sorting {a} {A : Set a} -- (sort\u1d2c : A \u00d7 A \u2192 A \u00d7 A)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    {-\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\n    {-\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted' _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    postulate dec-\u2264 : \u2200 x y \u2192 Dec (x \u2264\u1d2c y)\n    postulate dec-\u2294 : \u2200 x y \u2192 x \u2294\u1d2c y \u2261 x \u228e x \u2294\u1d2c y \u2261 y\n    postulate dec-\u2293 : \u2200 x y \u2192 x \u2293\u1d2c y \u2261 x \u228e x \u2293\u1d2c y \u2261 y\n\n    merge-spec : \u2200 {n} (t u : Tree A n) \u2192\n                 Sorted t \u2192 Sorted u \u2192 Sorted (merge t u)\n    merge-spec (leaf x) (leaf y) sx sy (left here) (left here) pf = \u2264\u1d2c.refl\n    merge-spec (leaf x) (leaf y) sx sy (left here) (right here) (0-1 ._ ._) = \u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)\n    merge-spec (leaf x) (leaf y) sx sy (right p) (left q) ()\n    merge-spec (leaf x) (leaf y) sx sy (right here) (right here) pf = \u2264\u1d2c.refl\n    merge-spec (fork t\u2080 t\u2081) (fork u\u2080 u\u2081) st su p q p\u2264q\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 -- | merge-spec t\u2080 u\u2080 ? ? | merge-spec t\u2081 u\u2081 ? ?\n    ... | fork l m\u2080    | fork m\u2081 h\n      with merge m\u2080 m\u2081 -- | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 -- | fork {high_t = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with p | q | p\u2264q\n    ... | left  pp | left  qq | there ._ pp\u2264qq = {!merge-spec t\u2080 u\u2080 !}\n    ... | left  pp | right qq | 0-1 ._ ._ = {!!}\n    ... | right pp | left qq  | ()\n    ... | right pp | right qq | there ._ pp\u2264qq = {!!}\n    -}\n    -}\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf)\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = refl\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = refl\n\n             {-\n    \u2203Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    \u2203Sorted t = \u2203 \u03bb l \u2192 \u2203 \u03bb h \u2192 Sorted t l h\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = \u2203Sorted t \u00d7 \u2203Sorted u\n    -}\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = SD.Sorted t \u00d7 SD.Sorted u\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork t t\u2081) = refl\n\n    {-\n    record MergeInnerHyp {n} (t : Tree A (2 + n)) : Set (a L.\u2294 \u2113) where\n      constructor mk\n      field\n        st\u2080 : SD.Sorted (lft t)\n        st\u2081 : SD.Sorted (rght t)\n      u = interchange t\n      field\n        su\u2080 : SD.Sorted (lft u)\n        su\u2081 : SD.Sorted (rght u)\n\n    merge-inner : \u2200 {n} {t : Tree A (2 + n)} \u2192\n                 MergeInnerHyp t \u2192 SD.Sorted (map-inner merge t)\n    merge-inner {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n                (mk (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                    (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n                    (fork sv\u2080 sv\u2081 hv\u2080\u2264lv\u2081)\n                    (fork sw\u2080 sw\u2081 lw\u2080\u2264hw\u2081)) = {!!}\n                    -}\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n                -- last  (merge t u) \u2261 last  t \u2293\u1d2c last  u\n    merge-spec : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192 SD.Sorted (merge (fork t u))\n    merge-spec (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec st\u2080 su\u2080 | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec st\u2081 su\u2081\n    ... | fork v\u2080 w\u2080 | fork sv\u2080 sw\u2080 p1 | fpf1\n        | fork v\u2081 w\u2081 | fork sv\u2081 sw\u2081 p2\n      with merge (fork w\u2080 v\u2081) | merge-spec sw\u2080 sv\u2081\n    ... | fork a b | fork sa sb p3\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3\n         where\n             postulate\n                pf3 : last v\u2080 \u2264\u1d2c first t\u2081\n                pf4 : last v\u2080 \u2264\u1d2c first u\u2081\n                -- Sorted (merge t u) \u2192\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 = {!first-merge !}\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 = {!!}\n    {-\n      -}\n\n\n--      map-outer merge id \u2218 interchange\n\n      {-\n\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high_t = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high_t = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high_t = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      {-with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl-} =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ef0b3de030e69c13f6abe2f657976fd6056b39d2","subject":"Update example1","message":"Update example1\n","repos":"crypto-agda\/agda-libjs","old_file":"example1.agda","new_file":"example1.agda","new_contents":"module example1 where\n\nopen import Data.String.Base using (String)\nopen import Data.Product\nopen import Data.List.Base using (List; []; _\u2237_)\nopen import Data.Bool.Base\nopen import Function\n\nopen import Control.Process.Type\n\nopen import FFI.JS as JS\nopen import FFI.JS.Proc\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS      as FS\n\ntake-half : String \u2192 String\ntake-half s = substring s 0N (length s \/ 2N)\n\ndrop-half : String \u2192 String\ndrop-half s = substring1 s (length s \/ 2N)\n\ntest-value : Value\ntest-value = object ((\"array\"  , array (array [] \u2237 array (array [] \u2237 []) \u2237 [])) \u2237\n                     (\"object\" , array (object [] \u2237 object ((\"a\", string \"b\") \u2237 []) \u2237 [])) \u2237\n                     (\"string\" , array (string \"\" \u2237 string \"a\" \u2237 [])) \u2237\n                     (\"number\" , array (number 0N \u2237 number 1N \u2237 [])) \u2237\n                     (\"bool\"   , array (bool true \u2237 bool false \u2237 [])) \u2237\n                     (\"null\"   , array (null \u2237 [])) \u2237 [])\n\ntest =\n    fromString (JSON-stringify (fromValue test-value))\n    ===\n    fromString \"{\\\"array\\\":[[],[[]]],\\\"object\\\":[{},{\\\"a\\\":\\\"b\\\"}],\\\"string\\\":[\\\"\\\",\\\"a\\\"],\\\"number\\\":[0,1],\\\"bool\\\":[true,false],\\\"null\\\":[null]}\"\n\n\nmodule _ {A : Set} (_\u2264_ : A \u2192 A \u2192 Bool) where\n\n    merge-sort-list : (l\u2080 l\u2081 : List A) \u2192 List A\n    merge-sort-list [] l\u2081 = l\u2081\n    merge-sort-list l\u2080 [] = l\u2080\n    merge-sort-list (x\u2080 \u2237 l\u2080) (x\u2081 \u2237 l\u2081) with x\u2080 \u2264 x\u2081\n    ... | true  = x\u2080 \u2237 merge-sort-list l\u2080 (x\u2081 \u2237 l\u2081)\n    ... | false = x\u2081 \u2237 merge-sort-list (x\u2080 \u2237 l\u2080) l\u2081\n\nmerge-sort-string : String \u2192 String \u2192 String\nmerge-sort-string s\u2080 s\u2081 = List\u25b9String (merge-sort-list _\u2264Char_ (String\u25b9List s\u2080) (String\u25b9List s\u2081))\n\nmapJSArray : (JSArray String \u2192 JSArray String) \u2192 JSValue \u2192 JSValue\nmapJSArray f v = fromString (join \"\" \u2218 f \u2218 split \"\" \u2218 castString $ v)\n\nreverser : URI \u2192 JSProc\nreverser d = recv d \u03bb s \u2192 send d (mapJSArray JS.reverse s) end\n\nadder : URI \u2192 JSProc\nadder d = recv d \u03bb s\u2080 \u2192 recv d \u03bb s\u2081 \u2192 send d (s\u2080 +JS s\u2081) end\n\nadder-client : URI \u2192 JSValue \u2192 JSValue \u2192 JSProc\nadder-client d s\u2080 s\u2081 = send d s\u2080 (send d s\u2081 (recv d \u03bb _ \u2192 end))\n\nmodule _ (adder-addr reverser-addr : URI)(s : JSValue) where\n  adder-reverser-client : JSProc\n  adder-reverser-client =\n    send reverser-addr s $\n    send adder-addr s $\n    recv reverser-addr \u03bb rs \u2192\n    send adder-addr rs $\n    recv adder-addr \u03bb res \u2192\n    end\n\nstr-sorter\u2080 : URI \u2192 JSProc\nstr-sorter\u2080 d = recv d \u03bb s \u2192 send d (mapJSArray sort s) end\n\nstr-sorter-client : \u2200 {D} \u2192 D \u2192 JSValue \u2192 Proc D JSValue\nstr-sorter-client d s = send d s $ recv d \u03bb _ \u2192 end\n\nmodule _ (upstream helper\u2080 helper\u2081 : URI) where\n  str-merger : JSProc\n  str-merger =\n    recv upstream \u03bb s \u2192\n    send helper\u2080 (fromString (take-half (castString s))) $\n    send helper\u2081 (fromString (drop-half (castString s))) $\n    recv helper\u2080 \u03bb ss\u2080 \u2192\n    recv helper\u2081 \u03bb ss\u2081 \u2192\n    send upstream (fromString (merge-sort-string (castString ss\u2080) (castString ss\u2081)))\n    end\n\ndyn-merger : URI \u2192 (URI \u2192 JSProc) \u2192 JSProc\ndyn-merger upstream helper =\n  spawn helper \u03bb helper\u2080 \u2192\n  spawn helper \u03bb helper\u2081 \u2192\n  str-merger upstream helper\u2080 helper\u2081\n\nstr-sorter\u2081 : URI \u2192 JSProc\nstr-sorter\u2081 upstream = dyn-merger upstream str-sorter\u2080\n\nstr-sorter\u2082 : URI \u2192 JSProc\nstr-sorter\u2082 upstream = dyn-merger upstream str-sorter\u2081\n\nstringifier : URI \u2192 JSProc\nstringifier d = recv d \u03bb v \u2192 send d (fromString (JSON-stringify v)) end\n\nstringifier-client : \u2200 {D} \u2192 D \u2192 JSValue \u2192 Proc D JSValue\nstringifier-client d v = send d v $ recv d \u03bb _ \u2192 end\n\nmain : JS!\nmain =\n  Console.log \"Hey!\" >> assert test >>\n  Process.argv !\u2081 \u03bb argv \u2192 Console.log (\"argv=\" ++ join \" \" argv) >>\n  Console.log \"server(adder):\" >> server \"127.0.0.1\" \"1337\" adder !\u2081 \u03bb adder-uri \u2192\n  Console.log \"client(adderclient):\" >>\n  client (adder-client adder-uri (fromString \"Hello \") (fromString \"World!\")) >>\n  client (adder-client adder-uri (fromString \"Bonjour \") (fromString \"monde!\")) >>\n  Console.log \"server(reverser):\" >>\n  server \"127.0.0.1\" \"1338\" reverser !\u2081 \u03bb reverser-uri \u2192\n  Console.log \"client(adder-reverser-client):\" >>\n  client (adder-reverser-client adder-uri reverser-uri (fromString \"red\")) >>\n\n  server \"127.0.0.1\" \"1339\" str-sorter\u2080 !\u2081 \u03bb str-sorter\u2080-uri \u2192\n  Console.log \"str-sorter-client for str-sorter\u2080:\" >>\n  client (str-sorter-client str-sorter\u2080-uri (fromString \"Something to be sorted!\")) >>\n\n  server \"127.0.0.1\" \"1342\" str-sorter\u2082 !\u2081 \u03bb str-sorter\u2082-uri \u2192\n  Console.log \"str-sorter-client:\" >>\n  client (str-sorter-client str-sorter\u2082-uri (fromString \"Something to be sorted!\")) >>\n\n  server \"127.0.0.1\" \"1343\" stringifier !\u2081 \u03bb stringifier-uri \u2192\n  client (stringifier-client stringifier-uri (fromValue test-value)) >>\n  FS.readFile \"README.md\" nullJS !\u2082 \u03bb err dat \u2192\n  Console.log (\"README.md, length is \" ++ Number\u25b9String (length (toString dat))) >>\n  Console.log \"Bye!\"\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module example1 where\n\nopen import Data.String.Base using (String)\nopen import Data.Product\nopen import Data.List.Base using (List; []; _\u2237_)\nopen import Data.Bool.Base\nopen import Function\n\nopen import Control.Process.Type\n\nopen import FFI.JS as JS\nopen import FFI.JS.Proc\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS      as FS\n\npostulate take-half : String \u2192 String\n{-# COMPILED_JS take-half function(x) { return x.substring(0,x.length\/2); } #-}\npostulate drop-half : String \u2192 String\n{-# COMPILED_JS drop-half function(x) { return x.substring(x.length\/2); } #-}\n\ntest-value : Value\ntest-value = object ((\"array\"  , array (array [] \u2237 array (array [] \u2237 []) \u2237 [])) \u2237\n                     (\"object\" , array (object [] \u2237 object ((\"a\", string \"b\") \u2237 []) \u2237 [])) \u2237\n                     (\"string\" , array (string \"\" \u2237 string \"a\" \u2237 [])) \u2237\n                     (\"number\" , array (number zero \u2237 number one \u2237 [])) \u2237\n                     (\"bool\"   , array (bool true \u2237 bool false \u2237 [])) \u2237\n                     (\"null\"   , array (null \u2237 [])) \u2237 [])\n\ntest =\n    fromString (JSON-stringify (fromValue test-value))\n    ===\n    fromString \"{\\\"array\\\":[[],[[]]],\\\"object\\\":[{},{\\\"a\\\":\\\"b\\\"}],\\\"string\\\":[\\\"\\\",\\\"a\\\"],\\\"number\\\":[0,1],\\\"bool\\\":[true,false],\\\"null\\\":[null]}\"\n\n\nmodule _ {A : Set} (_\u2264_ : A \u2192 A \u2192 Bool) where\n\n    merge-sort-list : (l\u2080 l\u2081 : List A) \u2192 List A\n    merge-sort-list [] l\u2081 = l\u2081\n    merge-sort-list l\u2080 [] = l\u2080\n    merge-sort-list (x\u2080 \u2237 l\u2080) (x\u2081 \u2237 l\u2081) with x\u2080 \u2264 x\u2081\n    ... | true  = x\u2080 \u2237 merge-sort-list l\u2080 (x\u2081 \u2237 l\u2081)\n    ... | false = x\u2081 \u2237 merge-sort-list (x\u2080 \u2237 l\u2080) l\u2081\n\nmerge-sort-string : String \u2192 String \u2192 String\nmerge-sort-string s\u2080 s\u2081 = List\u25b9String (merge-sort-list _\u2264Char_ (String\u25b9List s\u2080) (String\u25b9List s\u2081))\n\nmapJSArray : (JSArray String \u2192 JSArray String) \u2192 JSValue \u2192 JSValue\nmapJSArray f v = fromString (onString (join \"\" \u2218 f \u2218 split \"\") v)\n\nreverser : URI \u2192 JSProc\nreverser d = recv d \u03bb s \u2192 send d (mapJSArray JS.reverse s) end\n\nadder : URI \u2192 JSProc\nadder d = recv d \u03bb s\u2080 \u2192 recv d \u03bb s\u2081 \u2192 send d (s\u2080 +JS s\u2081) end\n\nadder-client : URI \u2192 JSValue \u2192 JSValue \u2192 JSProc\nadder-client d s\u2080 s\u2081 = send d s\u2080 (send d s\u2081 (recv d \u03bb _ \u2192 end))\n\nmodule _ (adder-addr reverser-addr : URI)(s : JSValue) where\n  adder-reverser-client : JSProc\n  adder-reverser-client =\n    send reverser-addr s $\n    send adder-addr s $\n    recv reverser-addr \u03bb rs \u2192\n    send adder-addr rs $\n    recv adder-addr \u03bb res \u2192\n    end\n\nstr-sorter\u2080 : URI \u2192 JSProc\nstr-sorter\u2080 d = recv d \u03bb s \u2192 send d (mapJSArray sort s) end\n\nstr-sorter-client : \u2200 {D} \u2192 D \u2192 JSValue \u2192 Proc D JSValue\nstr-sorter-client d s = send d s $ recv d \u03bb _ \u2192 end\n\nmodule _ (upstream helper\u2080 helper\u2081 : URI) where\n  str-merger : JSProc\n  str-merger =\n    recv upstream \u03bb s \u2192\n    send helper\u2080 (fromString (onString take-half s)) $\n    send helper\u2081 (fromString (onString drop-half s)) $\n    recv helper\u2080 \u03bb ss\u2080 \u2192\n    recv helper\u2081 \u03bb ss\u2081 \u2192\n    send upstream (fromString (onString (onString merge-sort-string ss\u2080) ss\u2081))\n    end\n\ndyn-merger : URI \u2192 (URI \u2192 JSProc) \u2192 JSProc\ndyn-merger upstream helper =\n  spawn helper \u03bb helper\u2080 \u2192\n  spawn helper \u03bb helper\u2081 \u2192\n  str-merger upstream helper\u2080 helper\u2081\n\nstr-sorter\u2081 : URI \u2192 JSProc\nstr-sorter\u2081 upstream = dyn-merger upstream str-sorter\u2080\n\nstr-sorter\u2082 : URI \u2192 JSProc\nstr-sorter\u2082 upstream = dyn-merger upstream str-sorter\u2081\n\nstringifier : URI \u2192 JSProc\nstringifier d = recv d \u03bb v \u2192 send d (fromString (JSON-stringify v)) end\n\nstringifier-client : \u2200 {D} \u2192 D \u2192 JSValue \u2192 Proc D JSValue\nstringifier-client d v = send d v $ recv d \u03bb _ \u2192 end\n\nmain : JS!\nmain =\n  Console.log \"Hey!\" >> assert test >>\n  Process.argv !\u2081 \u03bb argv \u2192 Console.log (\"argv=\" ++ join \" \" argv) >>\n  Console.log \"server(adder):\" >> server \"127.0.0.1\" \"1337\" adder !\u2081 \u03bb adder-uri \u2192\n  Console.log \"client(adderclient):\" >>\n  client (adder-client adder-uri (fromString \"Hello \") (fromString \"World!\")) >>\n  client (adder-client adder-uri (fromString \"Bonjour \") (fromString \"monde!\")) >>\n  Console.log \"server(reverser):\" >>\n  server \"127.0.0.1\" \"1338\" reverser !\u2081 \u03bb reverser-uri \u2192\n  Console.log \"client(adder-reverser-client):\" >>\n  client (adder-reverser-client adder-uri reverser-uri (fromString \"red\")) >>\n\n  server \"127.0.0.1\" \"1342\" str-sorter\u2082 !\u2081 \u03bb str-sorter\u2082-uri \u2192\n  Console.log \"str-sorter-client:\" >>\n  client (str-sorter-client str-sorter\u2082-uri (fromString \"Something to be sorted!\")) >>\n\n  server \"127.0.0.1\" \"1343\" stringifier !\u2081 \u03bb stringifier-uri \u2192\n  client (stringifier-client stringifier-uri (fromValue test-value)) >>\n  FS.readFile \"README.md\" nullJS !\u2082 \u03bb err dat \u2192\n  Console.log (\"README.md, length is \" ++ Number\u25b9String (length (castString dat))) >>\n  Console.log \"Bye!\" >>\n  end\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fb1d9843b7b5fef2c931b969b8e78f25726b0909","subject":"IDesc: tagged indexed description","message":"IDesc: tagged indexed description","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b3651f102b65a31ef5dd4d541002915b3e5cc9fe","subject":"Improve commentary in Base.Data.DependentList.","message":"Improve commentary in Base.Data.DependentList.\n\nOld-commit-hash: 7a4d9aea2f3574b1b3f21022bced5f9ef112264c\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Data\/DependentList.agda","new_file":"Base\/Data\/DependentList.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Reexport Data.List.All from the standard library.\n--\n-- At one point, we reinvented Data.List.All from the Agda\n-- standard library, under the name dependent list. We later\n-- replaced our reinvention by this adapter module that just\n-- exports the standard library's version with partly different\n-- names.\n------------------------------------------------------------------------\n\nmodule Base.Data.DependentList where\n\nopen import Data.List.All public\n  using\n    ( head\n    ; tail\n    ; map\n    ; tabulate\n    )\n  renaming\n    ( All to DependentList\n    ; _\u2237_ to _\u2022_\n    ; [] to \u2205\n    )\n\n-- Maps a binary function over two dependent lists.\n-- Should this be in the Agda standard library?\nzipWith : \u2200 {a p q r} {A : Set a} {P : A \u2192 Set p} {Q : A \u2192 Set q} {R : A \u2192 Set r} \u2192\n  (f : {a : A} \u2192 P a \u2192 Q a \u2192 R a) \u2192\n  \u2200 {xs} \u2192 DependentList P xs \u2192 DependentList Q xs \u2192 DependentList R xs\nzipWith f \u2205 \u2205 = \u2205\nzipWith f (p \u2022 ps) (q \u2022 qs) = f p q \u2022 zipWith f ps qs\n","old_contents":"module Base.Data.DependentList where\n\nopen import Data.List.All public\n  using\n    ( head\n    ; tail\n    ; map\n    ; tabulate\n    )\n  renaming\n    ( All to DependentList\n    ; _\u2237_ to _\u2022_\n    ; [] to \u2205\n    )\n\nzipWith : \u2200 {a p q r} {A : Set a} {P : A \u2192 Set p} {Q : A \u2192 Set q} {R : A \u2192 Set r} \u2192\n  (f : {a : A} \u2192 P a \u2192 Q a \u2192 R a) \u2192\n  \u2200 {xs} \u2192 DependentList P xs \u2192 DependentList Q xs \u2192 DependentList R xs\nzipWith f \u2205 \u2205 = \u2205\nzipWith f (p \u2022 ps) (q \u2022 qs) = f p q \u2022 zipWith f ps qs\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f36c392d31fa25a5e11981c52c34ff11f7b2d2de","subject":"Erased unnecessary parenthesis.","message":"Erased unnecessary parenthesis.\n\nIgnore-this: f72b1f73b49c371bb5ae4bdf44e761c\n\ndarcs-hash:20100501002912-3bd4e-c556b26bbefb1e09c60f2decd251473a241a4c66.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/Properties.agda","new_file":"LTC\/Function\/Arithmetic\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties on total natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\n\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN  = prf\n  where\n  postulate prf : N (pred zero)\n  {-# ATP prove prf zN #-}\n\npred-N (sN {n} Nn) = prf\n  where\n  -- TODO: The postulate N (pred $ succ n) is not proved by the ATP.\n  postulate prf : N (pred (succ n))\n  {-# ATP prove prf #-}\n-- {-# ATP hint pred-N #-}\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN = prf\n  where\n  postulate prf : N (m - zero)\n  {-# ATP prove prf #-}\n\nminus-N zN (sN {n} Nn) = prf\n  where\n  postulate prf : N (zero - succ n)\n  {-# ATP prove prf zN #-}\n\nminus-N (sN {m} Nm) (sN {n} Nn) = prf $ minus-N Nm Nn\n  where\n  postulate prf : N (m - n) \u2192 N (succ m - succ n)\n  {-# ATP prove prf #-}\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero + n)\n  {-# ATP prove prf #-}\n+-N {n = n} (sN {m} Nm ) Nn = prf (+-N Nm Nn)\n  where\n  postulate prf : N (m + n) \u2192 N (succ m + n)\n  {-# ATP prove prf sN #-}\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero * n)\n  {-# ATP prove prf zN #-}\n*-N {n = n} (sN {m} Nm) Nn = prf (*-N Nm Nn)\n  where\n  postulate prf : N (m * n) \u2192 N (succ m * n)\n  {-# ATP prove prf +-N #-}\n\n------------------------------------------------------------------------------\n\n-- Some proofs are based on the proofs in the standard library\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) = prf $ +-rightIdentity Nn\n   where\n   postulate prf : n + zero \u2261 n \u2192 succ n + zero \u2261 succ n\n   {-# ATP prove prf #-}\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No = prf\n  where\n  postulate prf : zero + n + o \u2261 zero + (n + o)\n  {-# ATP prove prf #-}\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No = prf $ +-assoc Nm Nn No\n  where\n  postulate prf : m + n + o \u2261 m + (n + o) \u2192\n                  succ m + n + o \u2261 succ m + (n + o)\n  {-# ATP prove prf #-}\n\nx+1+y\u22611+x+y : {m n : D} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y {n = n} zN Nn = prf\n  where\n  postulate prf : zero + succ n \u2261 succ (zero + n)\n  {-# ATP prove prf #-}\nx+1+y\u22611+x+y {n = n} (sN {m} Nm) Nn = prf (x+1+y\u22611+x+y Nm Nn)\n  where\n  postulate prf : m + succ n \u2261 succ (m + n) \u2192\n                  succ m + succ n \u2261 succ (succ m + n)\n  {-# ATP prove prf #-}\n\n+-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero + n \u2261 n + zero\n  {-# ATP prove prf +-rightIdentity #-}\n+-comm {n = n} (sN {m} Nm) Nn = prf (+-comm Nm Nn)\n  where\n  postulate prf : m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+1+y\u22611+x+y #-}\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No = prf\n  where\n  postulate prf : (zero + n) - (zero + o) \u2261 n - o\n  {-# ATP prove prf #-}\n\n-- Nice proof by the ATP.\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  prf ([x+y]-[x+z]\u2261y-z Nm Nn No)\n  where\n  postulate prf : (m + n) - (m + o) \u2261 n - o \u2192\n                  (succ m + n) - (succ m + o) \u2261 n - o\n  {-# ATP prove prf #-}\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) = prf (*-rightZero Nn)\n  where\n  postulate prf : n * zero \u2261 zero \u2192 succ n * zero \u2261 zero\n  {-# ATP prove prf #-}\n\npostulate *-leftIdentity : {n : D} \u2192 N n \u2192 succ zero * n \u2261 n\n{-# ATP prove *-leftIdentity +-rightIdentity #-}\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = prf\n  where\n  postulate prf : zero * succ n \u2261 zero + zero * n\n  {-# ATP prove prf #-}\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn = prf (x*1+y\u2261x+xy Nm Nn)\n                                        (+-assoc Nn Nm (*-N Nm Nn))\n                                        (+-assoc Nm Nn (*-N Nm Nn))\n  where\n  -- N.B. We had to feed the ATP with the instances of the associate law\n  postulate prf :  m * succ n \u2261 m + m * n \u2192 -- IH\n                   (n + m) + (m * n) \u2261 n + (m + (m * n)) \u2192 -- Associative law\n                   (m + n) + (m * n) \u2261 m + (n + (m * n)) \u2192 -- Associateve law\n                   succ m * succ n \u2261 succ m + succ m * n\n  {-# ATP prove prf +-comm #-}\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero * n \u2261 n * zero\n  {-# ATP prove prf *-rightZero #-}\n*-comm {n = n} (sN {m} Nm) Nn = prf (*-comm Nm Nn)\n  where\n  postulate prf : m * n \u2261 n * m \u2192\n                  succ m * n \u2261 n * succ m\n  {-# ATP prove prf x*1+y\u2261x+xy #-}\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No = prf\n  where\n  postulate prf : (m - zero) * o \u2261 m * o - zero * o\n  {-# ATP prove prf #-}\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No = prf (minus-0x (*-N (sN Nn) No))\n  where\n  postulate prf : zero - succ n * o \u2261 zero \u2192\n                  (zero - succ n) * o \u2261 zero * o - succ n * o\n  {-# ATP prove prf #-}\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN = prf\n  where\n  postulate prf : (succ m - succ n) * zero \u2261 succ m * zero - succ n * zero\n  {-# ATP prove prf *-comm minus-N zN #-}\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  prf ([x-y]z\u2261xz*yz Nm Nn (sN No))\n  where\n  postulate prf : (m - n) * succ o \u2261 m * succ o - n * succ o \u2192 -- IH\n                  (succ m - succ n) * succ o \u2261\n                  succ m * succ o - succ n * succ o\n  {-# ATP prove prf sN *-N [x+y]-[x+z]\u2261y-z #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties on total natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\n\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN  = prf\n  where\n  postulate prf : N (pred zero)\n  {-# ATP prove prf zN #-}\n\npred-N (sN {n} Nn) = prf\n  where\n  -- TODO: The postulate N (pred $ succ n) is not proved by the ATP.\n  postulate prf : N (pred (succ n))\n  {-# ATP prove prf #-}\n-- {-# ATP hint pred-N #-}\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN = prf\n  where\n  postulate prf : N (m - zero)\n  {-# ATP prove prf #-}\n\nminus-N zN (sN {n} Nn) = prf\n  where\n  postulate prf : N (zero - succ n)\n  {-# ATP prove prf zN #-}\n\nminus-N (sN {m} Nm) (sN {n} Nn) = prf $ minus-N Nm Nn\n  where\n  postulate prf : N (m - n) \u2192 N (succ m - succ n)\n  {-# ATP prove prf #-}\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero + n)\n  {-# ATP prove prf #-}\n+-N {n = n} (sN {m} Nm ) Nn = prf (+-N Nm Nn)\n  where\n  postulate prf : N (m + n) \u2192 N (succ m + n)\n  {-# ATP prove prf sN #-}\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN Nn = prf\n  where\n  postulate prf : N (zero * n)\n  {-# ATP prove prf zN #-}\n*-N {n = n} (sN {m} Nm) Nn = prf (*-N Nm Nn)\n  where\n  postulate prf : N (m * n) \u2192 N (succ m * n)\n  {-# ATP prove prf +-N #-}\n\n------------------------------------------------------------------------------\n\n-- Some proofs are based on the proofs in the standard library\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) = prf $ +-rightIdentity Nn\n   where\n   postulate prf : n + zero \u2261 n \u2192 succ n + zero \u2261 succ n\n   {-# ATP prove prf #-}\n\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) + o \u2261 m + (n + o)\n+-assoc {n = n} {o = o} zN Nn No = prf\n  where\n  postulate prf : zero + n + o \u2261 zero + (n + o)\n  {-# ATP prove prf #-}\n+-assoc {n = n} {o = o} (sN {m} Nm) Nn No = prf $ +-assoc Nm Nn No\n  where\n  postulate prf : m + n + o \u2261 m + (n + o) \u2192\n                  succ m + n + o \u2261 succ m + (n + o)\n  {-# ATP prove prf #-}\n\nx+1+y\u22611+x+y : {m n : D} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+1+y\u22611+x+y {n = n} zN Nn = prf\n  where\n  postulate prf : zero + succ n \u2261 succ (zero + n)\n  {-# ATP prove prf #-}\nx+1+y\u22611+x+y {n = n} (sN {m} Nm) Nn = prf (x+1+y\u22611+x+y Nm Nn)\n  where\n  postulate prf : m + succ n \u2261 succ (m + n) \u2192\n                  succ m + succ n \u2261 succ (succ m + n)\n  {-# ATP prove prf #-}\n\n+-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero + n \u2261 n + zero\n  {-# ATP prove prf +-rightIdentity #-}\n+-comm {n = n} (sN {m} Nm) Nn = prf (+-comm Nm Nn)\n  where\n  postulate prf : m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+1+y\u22611+x+y #-}\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No = prf\n  where\n  postulate prf : (zero + n) - (zero + o) \u2261 n - o\n  {-# ATP prove prf #-}\n\n-- Nice proof by the ATP.\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  prf ([x+y]-[x+z]\u2261y-z Nm Nn No)\n  where\n  postulate prf : (m + n) - (m + o) \u2261 n - o \u2192\n                  (succ m + n) - (succ m + o) \u2261 n - o\n  {-# ATP prove prf #-}\n\n*-leftZero : {n : D} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-rightZero : {n : D} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) = prf (*-rightZero Nn)\n  where\n  postulate prf : n * zero \u2261 zero \u2192 succ n * zero \u2261 zero\n  {-# ATP prove prf #-}\n\npostulate *-leftIdentity : {n : D} \u2192 N n \u2192 succ zero * n \u2261 n\n{-# ATP prove *-leftIdentity +-rightIdentity #-}\n\nx*1+y\u2261x+xy : {m n : D} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*1+y\u2261x+xy {n = n} zN Nn = prf\n  where\n  postulate prf : zero * succ n \u2261 zero + zero * n\n  {-# ATP prove prf #-}\nx*1+y\u2261x+xy {n = n} (sN {m} Nm) Nn = prf (x*1+y\u2261x+xy Nm Nn)\n                                        (+-assoc Nn Nm (*-N Nm Nn))\n                                        (+-assoc Nm Nn (*-N Nm Nn))\n  where\n  -- N.B. We had to feed the ATP with the instances of the associate law\n  postulate prf :  m * succ n \u2261 m + m * n \u2192 -- IH\n                   (n + m) + (m * n) \u2261 n + (m + (m * n)) \u2192 -- Associative law\n                   (m + n) + (m * n) \u2261 m + (n + (m * n)) \u2192 -- Associateve law\n                   succ m * succ n \u2261 succ m + succ m * n\n  {-# ATP prove prf +-comm #-}\n\n*-comm : {m n : D} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn = prf\n  where\n  postulate prf : zero * n \u2261 n * zero\n  {-# ATP prove prf *-rightZero #-}\n*-comm {n = n} (sN {m} Nm) Nn = prf (*-comm Nm Nn)\n  where\n  postulate prf : m * n \u2261 n * m \u2192\n                  succ m * n \u2261 n * succ m\n  {-# ATP prove prf x*1+y\u2261x+xy #-}\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No = prf\n  where\n  postulate prf : (m - zero) * o \u2261 m * o - zero * o\n  {-# ATP prove prf #-}\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No = prf (minus-0x (*-N (sN Nn) No))\n  where\n  postulate prf : zero - succ n * o \u2261 zero \u2192\n                  (zero - succ n) * o \u2261 zero * o - succ n * o\n  {-# ATP prove prf #-}\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN = prf\n  where\n  postulate prf : (succ m - succ n) * zero \u2261 succ m * zero - succ n * zero\n  {-# ATP prove prf *-comm minus-N zN #-}\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  prf ([x-y]z\u2261xz*yz Nm Nn (sN No))\n  where\n  postulate prf : (m - n) * succ o \u2261 m * succ o - n * succ o \u2192 -- IH\n                  (succ m - succ n) * succ o \u2261\n                  succ m * succ o - succ n * succ o\n  {-# ATP prove prf sN *-N [x+y]-[x+z]\u2261y-z #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57b3c49f6694f8e55a607d041782d114dabf89b4","subject":"adapt bintree to composable and vcomposable","message":"adapt bintree to composable and vcomposable\n","repos":"crypto-agda\/crypto-agda","old_file":"bintree.agda","new_file":"bintree.agda","new_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; _\u2264_; s\u2264s; _+_; module \u2115\u2264; module \u2115\u00b0)\nopen import Data.Nat.Properties\nopen import Data.Bool\nopen import Data.Vec using (_++_)\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero}  f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (0b \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (1b \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nleaf\u207f : \u2200 {n a} {A : Set a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : Set a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x rewrite fromConst\u2261leaf\u207f {n} x = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x) = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 Tree A m \u2192 Tree A n\nweaken\u2264 _       (leaf x)          = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 n {m a} {A : Set a} \u2192 Tree A m \u2192 Tree A (n + m)\nweaken+ n = weaken\u2264 (m\u2264n+m _ n)\n\njoin : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 Tree (Tree A c\u2082) c\u2081 \u2192 Tree A (c\u2081 + c\u2082)\njoin {c} (leaf x)          = weaken+ c x\njoin     (fork left right) = fork (join left) (join right)\n\n_>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\nf >>> g = map (flip lookup g) f\n\n_\u2192\u1d57_ : (i o : \u2115) \u2192 Set\ni \u2192\u1d57 o = Tree (Bits o) i\n\ncomposable : Composable _\u2192\u1d57_\ncomposable = mk _>>>_\n\n_***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2192\u1d57 o\u2080 \u2192 i\u2081 \u2192\u1d57 o\u2081 \u2192 (i\u2080 + i\u2081) \u2192\u1d57 (o\u2080 + o\u2081)\n(f *** g) = join (map (\u03bb xs \u2192 map (_++_ xs) g) f)\n\nvcomposable : VComposable _+_ _\u2192\u1d57_\nvcomposable = mk _***_\n\nforkable : Forkable suc _\u2192\u1d57_\nforkable = mk fork\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","old_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (0b \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (1b \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nleaf\u207f : \u2200 {n a} {A : Set a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : Set a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x rewrite fromConst\u2261leaf\u207f {n} x = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x) = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"362f857ed3f64793c31edc314bd464b3913cc72b","subject":"updating progress checks for #13","message":"updating progress checks for #13\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (_ , Step (FHFinal _) () _)\n  vs VLam (_ , Step (FHFinal _) () _)\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    ie IEHole ()\n    ie (INEHole x) (ENEHole e) = fe x e\n    ie (IAp i x) (EAp1 e) = ie i e\n    ie (IAp i x) (EAp2 y e) = fe x e\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    fe (FVal x) er = ve x er\n    fe (FIndet x) er = ie x er\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole f) ()\n  lem2 (IAp () _) (ITLam _)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 f (FHFinal x) = x\n  lem4 (FVal ()) (FHAp1 x\u2082 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp1 x\u2083 sub) = lem4 x\u2082 sub\n  lem4 (FVal ()) (FHAp2 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp2 sub) = lem4 (FIndet x\u2081) sub\n  lem4 f FHEHole = f\n  lem4 f (FHNEHoleFinal x) = f\n  lem4 (FVal ()) (FHCast sub)\n  lem4 (FIndet ()) (FHCast sub)\n  lem4 f (FHCastFinal x) = f\n  lem4 (FVal ()) (FHNEHole y)\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole y) = lem4 x\u2081 y\n\n  lem5 : \u2200{d \u0394 d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 \u0394 \u22a2 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () _)\n  is (INEHole _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () FHEHole)\n  is (INEHole x) (d , Step (FHNEHoleFinal x\u2081) () (FHFinal x\u2083))\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHCastFinal x\u2083))\n  is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 a er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 a er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 a er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = lem5 a x x\u2081\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (_ , Step (FHNEHole a) x (FHNEHole x\u2082)) = es er (_ , Step a x x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (_ , Step (FHFinal _) () (FHFinal _))\n  vs VConst (_ , Step (FHFinal _) () FHEHole)\n  vs VConst (_ , Step (FHFinal _) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal _) () (FHNEHoleFinal _))\n  vs VConst (_ , Step (FHFinal _) () (FHCastFinal _))\n  vs VLam (_ , Step (FHFinal _) () (FHFinal _))\n  vs VLam (_ , Step (FHFinal _) () FHEHole)\n  vs VLam (_ , Step (FHFinal _) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal _) () (FHNEHoleFinal _))\n  vs VLam (_ , Step (FHFinal _) () (FHCastFinal _))\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    ie IEHole ()\n    ie (INEHole x) (ENEHole e) = fe x e\n    ie (IAp i x) (EAp1 e) = ie i e\n    ie (IAp i x) (EAp2 y e) = fe x e\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    fe (FVal x) er = ve x er\n    fe (FIndet x) er = ie x er\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole _) ()\n  lem2 (IAp () _) (ITLam _)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal _))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal _))\n  is IEHole (_ , Step FHEHole () (FHCastFinal _))\n  is (INEHole _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole _) (_ , Step FHNEHoleEvaled () (FHFinal _))\n  is (INEHole _) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole _) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole _) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal _))\n  is (INEHole _) (_ , Step FHNEHoleEvaled () (FHCastFinal _))\n  is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 f (FHFinal x) = x\n  lem4 (FVal ()) (FHAp1 x\u2082 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp1 x\u2083 sub) = lem4 x\u2082 sub\n  lem4 (FVal ()) (FHAp2 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp2 sub) = lem4 (FIndet x\u2081) sub\n  lem4 f FHEHole = f\n  lem4 f FHNEHoleEvaled = f\n  lem4 (FVal ()) (FHNEHoleInside sub)\n  lem4 (FIndet ()) (FHNEHoleInside sub)\n  lem4 f (FHNEHoleFinal x) = f\n  lem4 (FVal ()) (FHCast sub)\n  lem4 (FIndet ()) (FHCast sub)\n  lem4 f (FHCastFinal x) = f\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 a er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 a er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 a er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = lem1 (lem4 a x) x\u2081\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (d' , Step FHNEHoleEvaled () x\u2082)\n  es (ENEHole er) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"69f6068e034334d162dcf8e4f4cf795c1e52c652","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: 38884787069d54d1e3561ca859c8a953\n\ndarcs-hash:20120615045929-3bd4e-a154492ae826893a3747789a3772973a10b352f0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Inductive\/Base.agda","new_file":"src\/PA\/Inductive\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Inductive Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Inductive.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import PA.Inductive.Base.Core public\n\n-- FOL (without equality)\n--\nopen import Common.FOL.FOL public hiding ( _,_ ; \u2203 )\n-- 2012-04-24. Agda bug? Why it is necessary to use the modifier\n-- @using@ in the following importation?\nopen import PA.Inductive.Existential public using ( _,_ ; \u2203 )\n\n-- The induction principle on the PA universe\n\n-- TODO: 19 May 2012. We don't use an implicit argument for the\n-- inductive step, because it yields some unsolved meta-variables (see\n-- notes\/FOT.PA.Inductive.ImplicitArgumentInductionSL).\nPA-ind : (A : M \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\nPA-ind A A0 h zero     = A0\nPA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n-- The identity type on the PA universe\nopen import PA.Inductive.Relation.Binary.PropositionalEquality public\n\n-- PA primitive recursive functions\n_+_ : M \u2192 M \u2192 M\nzero   + n = n\nsucc m + n = succ (m + n)\n\n_*_ : M \u2192 M \u2192 M\nzero   * n = zero\nsucc m * n = n + m * n\n\n------------------------------------------------------------------------------\n-- ATPs helper\n-- We don't traslate the body of functions, only the types. Therefore\n-- we need to feed the ATPs with the functions' equations.\n\n-- Addition axioms\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = refl\n-- {-# ATP hint +-0x #-}\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\n-- Multiplication axioms\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = refl\n-- {-# ATP hint *-0x #-}\n\n*-Sx : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n*-Sx m n = refl\n-- {-# ATP hint *-Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Inductive Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Inductive.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import PA.Inductive.Base.Core public\n\n-- FOL (without equality)\n--\nopen import Common.FOL.FOL public hiding ( _,_ ; \u2203 )\n-- 2012-04-24. Agda bug? Why it is necessary to use the modifier\n-- @using@ in the following importation?\nopen import PA.Inductive.Existential public using ( _,_ ; \u2203 )\n\n-- The induction principle on the PA universe\n\n-- TODO: 19 May 2012. We don't use an implicit argument for the\n-- inductive step, because it yields some unsolved meta-variables (see\n-- Draft.PA.Inductive.ImplicitArgumentInductionSL).\nPA-ind : (A : M \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\nPA-ind A A0 h zero     = A0\nPA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n-- The identity type on the PA universe\nopen import PA.Inductive.Relation.Binary.PropositionalEquality public\n\n-- PA primitive recursive functions\n_+_ : M \u2192 M \u2192 M\nzero   + n = n\nsucc m + n = succ (m + n)\n\n_*_ : M \u2192 M \u2192 M\nzero   * n = zero\nsucc m * n = n + m * n\n\n------------------------------------------------------------------------------\n-- ATPs helper\n-- We don't traslate the body of functions, only the types. Therefore\n-- we need to feed the ATPs with the functions' equations.\n\n-- Addition axioms\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = refl\n-- {-# ATP hint +-0x #-}\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\n-- Multiplication axioms\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = refl\n-- {-# ATP hint *-0x #-}\n\n*-Sx : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n*-Sx m n = refl\n-- {-# ATP hint *-Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"eea966329ff9f953b7330cd87ef1188ffe40f0f0","subject":"Correct caching for constants (by delegation)","message":"Correct caching for constants (by delegation)\n\nThis defines the plugin interface for constants (with a type that makes\nsemantic sense), and delegates to that, while stipulating:\n* what are the appropriate types;\n* that constant arguments needn't be cached because they are values.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  open import UNDEFINED\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConstB c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstVB c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstVB2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  open import UNDEFINED\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- TODO No caching. Uh??? What about the intermediate results of the\n  -- arguments? Ah, they're values, so they aren't interesting (or they're\n  -- already cached). Yu-uh!\n\n  -- XXX We need a caching ValBase, so the caching type semantics needs to move.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = reveal UNDEFINED -- {!\u27e6 c \u27e7ValBase !}\n\n  -- The real deal, finally.\n  -- TODO Do caching. No! Delegate to caching version of the base semantics!\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED -- {!\u27e6 c \u27e7CompBase (\u27e6 args \u27e7Vals ?)!}\n\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConstB c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstVB c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstVB2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"308d57f744384b21a2811f278b455dfb952dd324","subject":"Define \u0394Const for Popl14 and fill in calculus-with","message":"Define \u0394Const for Popl14 and fill in calculus-with\n\nOld-commit-hash: c1cdbc71d45afdb85da9ac22b3207d1cb03e0bf7\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Popl14.agda","new_file":"Syntax\/Language\/Popl14.agda","new_contents":"module Syntax.Language.Popl14 where\n\nopen import Syntax.Term.Popl14\nopen import Syntax.Context.Plotkin Popl14-type\n\nopen import Data.Integer\n\nimport Syntax.Language.Calculus as Calc\n\n\u0394Const : \u2200 {\u0393 \u03a3 \u03c4} \u2192\n  Const \u03a3 \u03c4 \u2192\n  Term \u0393\n    (internalizeContext\n      (Calc.\u0394Context\u2032 (Calc.Type Popl14-type) \u0394Type \u03a3) (\u0394Type \u03c4))\n\n-- These helpers hide deBrujin indexes, providing an interface which is as\n-- comfortable as HOAS. This should be generalized and moved to Syntax.Term.Plotkin\n\n\u0394Const (intlit-c n) = intlit (+ 0)\n\u0394Const add-c = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 add \u0394x \u0394y)\n\u0394Const minus-c = abs\u2082 (\u03bb x \u0394x \u2192 minus \u0394x)\n\u0394Const empty-c = empty\n\u0394Const insert-c = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 insert (x \u2295 \u0394x) (y \u2295 \u0394y) \u229d insert x y)\n\u0394Const union-c = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 union \u0394x \u0394y)\n\u0394Const negate-c = abs\u2082 (\u03bb x \u0394x \u2192 negate \u0394x)\n\u0394Const flatmap-c = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 flatmap (x \u2295 \u0394x) (y \u2295 \u0394y) \u229d flatmap x y)\n\u0394Const sum-c = abs\u2082 (\u03bb x \u0394x \u2192 sum \u0394x)\n\nPopl14 = Calc.calculus-with\n  Popl14-type\n  Const\n  \u0394Type\n  \u0394Const\n","old_contents":"module Syntax.Language.Popl14 where\n\nopen import Syntax.Term.Popl14\nopen import Syntax.Context.Plotkin Popl14-type\n\nopen import Data.Integer\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"450dbdd83ddf95d4b2095ee7d8ea56a92b5c2302","subject":"Cleaning.","message":"Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Functors.agda","new_file":"notes\/fixed-points\/Functors.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conat type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- The pred function is the conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n-- From (Leclerc and Paulin-Mohring 1994, p. 195).\n--\n-- TODO (07 January 2014): Agda doesn't accept the definition of\n-- Stream-build.\n{-# NO_TERMINATION_CHECK #-}\nStream-build :\n  {A X : Set} \u2192\n  (X \u2192 StreamF A X) \u2192\n  X \u2192 Stream A\nStream-build h x with h x\n... | a , x' = Wrap (a , Stream-build h x')\n\n-- From (Gim\u00e9nez, 1995, p. 40).\n--\n-- TODO (07 January 2014): Agda doesn't accept the definition of\n-- Stream-corec.\n{-# NO_TERMINATION_CHECK #-}\nStream-corec :\n  {A X : Set} \u2192\n  (X \u2192 (A \u00d7 (Stream A + X))) \u2192\n  X \u2192 Stream A\nStream-corec h x with h x\n... | a , inl xs = Wrap (a , xs)\n... | a , inr x' = Wrap (a , (Stream-corec h x'))\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Gim\u00e9nez, E. (1995). Codifying guarded de\ufb01nitions with recursive\n-- schemes. In: Types for Proofs and Programs (TYPES \u201994). Ed. by\n-- Dybjer, P., Nordstr\u00f6m, B. and Smith, J. Vol. 996. LNCS. Springer,\n-- pp. 39\u201359.\n--\n-- Leclerc, F. and Paulin-Mohring, C. (1994). Programming with Streams\n-- in Coq. A case study : the Sieve of Eratosthenes. In: Types for\n-- Proofs and Programs (TYPES \u201993). Ed. by Barendregt, H. and Nipkow,\n-- T. Vol. 806. LNCS. Springer, pp. 191\u2013212.\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\nN-ind : (A : N \u2192 Set) \u2192\n        A zero \u2192\n        (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192\n        \u2200 n \u2192 A n\nN-ind A A0 h n = {!!}\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\nList-ind : {A : Set}(B : List A \u2192 Set) \u2192\n           B nil \u2192\n           (\u2200 {x} (xs : List A) \u2192 B xs \u2192 B (cons x xs)) \u2192\n           (xs : List A) \u2192 B xs\nList-ind B Bnil h xs = {!!}\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conat type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- The pred function is the conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n-- From (Leclerc and Paulin-Mohring 1994, p. 195).\n--\n-- TODO (07 January 2014): Agda doesn't accept the definition of\n-- Stream-build.\n{-# NO_TERMINATION_CHECK #-}\nStream-build :\n  {A X : Set} \u2192\n  (X \u2192 StreamF A X) \u2192\n  X \u2192 Stream A\nStream-build h x with h x\n... | a , x' = Wrap (a , Stream-build h x')\n\n-- From (Gim\u00e9nez, 1995, p. 40).\n--\n-- TODO (07 January 2014): Agda doesn't accept the definition of\n-- Stream-corec.\n{-# NO_TERMINATION_CHECK #-}\nStream-corec :\n  {A X : Set} \u2192\n  (X \u2192 (A \u00d7 (Stream A + X))) \u2192\n  X \u2192 Stream A\nStream-corec h x with h x\n... | a , inl xs = Wrap (a , xs)\n... | a , inr x' = Wrap (a , (Stream-corec h x'))\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Gim\u00e9nez, E. (1995). Codifying guarded de\ufb01nitions with recursive\n-- schemes. In: Types for Proofs and Programs (TYPES \u201994). Ed. by\n-- Dybjer, P., Nordstr\u00f6m, B. and Smith, J. Vol. 996. LNCS. Springer,\n-- pp. 39\u201359.\n--\n-- Leclerc, F. and Paulin-Mohring, C. (1994). Programming with Streams\n-- in Coq. A case study : the Sieve of Eratosthenes. In: Types for\n-- Proofs and Programs (TYPES \u201993). Ed. by Barendregt, H. and Nipkow,\n-- T. Vol. 806. LNCS. Springer, pp. 191\u2013212.\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d3a0501a305802a2bdcd4ab9568ea4a2ac6d6410","subject":"Obligatory ascii art for this next-level research project.","message":"Obligatory ascii art for this next-level research project.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_contents":"{-\n\nProject Darkwing Duck:\n\nTowards the ultimate Agda model of the Spire canonical\ntype theory and its associated derived generic operations\nlibrary.\n\n                                                      ____\n                                                     \/    `.\n                                                    \/-----.|          ____\n                                                ___\/___.---`--.__.---'    `--.\n                                  _______.-----'           __.--'             )\n                              ,--'---.______________..----'(  __         __.-'\n                                        `---.___,-.|(a (a) \/-'  )___.---'\n                                                `-.>------<__.-'\n            ______                       _____..--'      \/\/\n    __.----'      `---._                `._.--._______.-'\/))\n,--'---.__              -_                  _.-(`-.____.'\/\/ \\\n          `-._            `---.________.---'    >\\      \/<   \\\n              \\_             `--.___            \\ \\-__-\/ \/    \\\n                \\_                  `----._______\\ \\  \/ \/__    \\\n                  \\                      \/  |,-------'-'\\  `-.__\\\n                   \\                    (   ||            \\      )\n                    `\\                   \\  ||            \/\\    \/\n                      \\                   >-||  @)    @) \/\\    \/\n                      \\                  ((_||           \\ \\_.'|\n                       \\                    ||            `-'  |\n                       \\                    ||             \/   |\n                        \\                   ||            (   '|\n                        \\                   ||  @)     @)  \\   |\n                         \\                  ||              \\  )\n                          `\\_               `|__         ____\\ |\n                             \\_               | ``----'''     \\|\n                               \\_              \\    .--___    |)\n                                 `-.__          \\   |     \\   |\n                                      `----.___  \\^\/|      \\\/\\|\n                                               `--\\ \\-._  \/ | |\n                                                   \\ \\  `'  \\ \\\n                                            __...--'  )     (  `-._\n                                           (_        \/       `.    `-.__\n                                             `--.__.'          `.       )\n                                                                 `.__.-'\n-}\n\n----------------------------------------------------------------------\n\n{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  C : (A : El\u1d40 P) \u2192 Tag E \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg (Tag E) (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 CurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\nuncurryEl\u1d40 End X x tt = x\nuncurryEl\u1d40 (Arg A B) X f (a , b) = uncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (f a) b\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El\u1d30 D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R p t = curryEl\u1d30 (Data.C R p t)\n  (\u03bc (Data.D R p))\n  (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) M init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\nindCurried D M f i x =\n  ind D M (uncurryHyps D (\u03bc D) M init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc (Data.D R p) i \u2192 Set) M in\n  CurriedHyps (Data.C R p t) (\u03bc (Data.D R p)) unM (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in UncurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x))\nelimUncurried R p M cs =\n  let D = Data.D R p\n      unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x) \u03bb i x \u2192\n  indCurried (Data.D R p) unM\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x))\nelim R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in\n    (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n    \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p M)\n       (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x)))\n  (\u03bb p M \u2192 curryBranches\n    (elimUncurried R p M))\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} (m : \u2115) (xs : Vec A m) (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} = elim VecR (\u03bb m xs \u2192 (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons (add m n) x (ih n ys))\n\nconcat : {A : Set} (m n : \u2115) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} m = elim VecR (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb n xs xss ih \u2192 append m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  C : (A : El\u1d40 P) \u2192 Tag E \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg (Tag E) (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 CurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\nuncurryEl\u1d40 End X x tt = x\nuncurryEl\u1d40 (Arg A B) X f (a , b) = uncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (f a) b\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El\u1d30 D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R p t = curryEl\u1d30 (Data.C R p t)\n  (\u03bc (Data.D R p))\n  (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) M init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\nindCurried D M f i x =\n  ind D M (uncurryHyps D (\u03bc D) M init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc (Data.D R p) i \u2192 Set) M in\n  CurriedHyps (Data.C R p t) (\u03bc (Data.D R p)) unM (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in UncurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x))\nelimUncurried R p M cs =\n  let D = Data.D R p\n      unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x) \u03bb i x \u2192\n  indCurried (Data.D R p) unM\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x))\nelim R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in\n    (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n    \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p M)\n       (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x)))\n  (\u03bb p M \u2192 curryBranches\n    (elimUncurried R p M))\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} (m : \u2115) (xs : Vec A m) (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} = elim VecR (\u03bb m xs \u2192 (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons (add m n) x (ih n ys))\n\nconcat : {A : Set} (m n : \u2115) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} m = elim VecR (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb n xs xss ih \u2192 append m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5ea0220b999f1d1ef9f072ca4bedfb82dc747ca1","subject":"Use a record for doe, as suggested by Tillmann","message":"Use a record for doe, as suggested by Tillmann\n\nOld-commit-hash: f1a0e47b6fd1c171f165e2a9ba96fd258042a8ae\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) $ proof dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) $ proof df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-symm (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-symm (derivative-is-nil dg fdg)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n{-\n      lemma : nil f \u2259 dg\n      -- Goes through\n      --lemma = sym (derivative-is-nil dg fdg)\n      -- Generates tons of crazy yellow ambiguities of equality type. Apparently, unifying against \u2259 does not work so well.\n      lemma = \u2259-symm {{changeAlgebra}} {f} {dg} {nil f} (derivative-is-nil dg fdg)\n-}\n\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ed9cb7b49d88102a5aa3017140c748073cf6cd6a","subject":"Neglible: ~ invariant under equivalence","message":"Neglible: ~ invariant under equivalence\n","repos":"crypto-agda\/crypto-agda","old_file":"Neglible.agda","new_file":"Neglible.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\nopen import Function.Extensionality\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-refl : \u2200 {f} \u2192 f \u2264\u2192 f\n_\u2264\u2192_.\u2264\u2192 \u2264\u2192-refl k = OR.refl\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {fN k} {gD k} {f'D k} {g'D k} \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange {gN k} {fD k} {g'D k} {f'D k})\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {f'N k} {fD k} {g'D k} {gD k})\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange {g'N k} {gD k} {f'D k} {fD k}\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n\n  fromNeg : {f : \u2115\u2192\u211a} \u2192 Is-Neg f \u2192 NegBounded f\n  \u03b5 (fromNeg f-neg) = _\n  \u03b5-neg (fromNeg f-neg) = f-neg\n  bounded (fromNeg f-neg) = \u2264\u2192-refl\n\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\n  ~-Inv : {{_ : FunExt}}{{_ : UA}}(\u03c0 : \u2200 n \u2192 R n \u2243 R n)(f g : \u2200 x \u2192 R x \u2192 \ud835\udfda)\n          (eq : \u2200 x (r : R x) \u2192 f x r \u2261 g x (proj\u2081 (\u03c0 x) r)) \u2192 f ~ g\n  _~_.~ (~-Inv \u03c0 f g eq) = \u2264-NB lemma (fromNeg 0\u2115\u211a-neg)\n    where\n      open \u2264-Reasoning\n      lemma : ~dist f g \u2264\u2192 0\u2115\u211a\n      _\u2264\u2192_.\u2264\u2192 lemma k = dist (# (f k)) (# (g k)) * 1\n                      \u2261\u27e8 proj\u2082 prop.*-identity _ \u27e9\n                        dist (# (f k)) (# (g k))\n                      \u2261\u27e8 ap (flip dist (# (g k))) (count-ext (R\u1d41 k) (eq k)) \u27e9\n                        dist (# (g k \u2218 proj\u2081 (\u03c0 k))) (# (g k))\n                      \u2261\u27e8 ap (flip dist (# (g k))) (sumStableUnder (R\u1d41 k) (\u03c0 k) (\ud835\udfda\u25b9\u2115 \u2218 g k)) \u27e9\n                        dist (# (g k)) (# (g k))\n                      \u2261\u27e8 dist-refl (# (g k)) \u27e9\n                        0\n                      \u220e\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {fN k} {gD k} {f'D k} {g'D k} \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange {gN k} {fD k} {g'D k} {f'D k})\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {f'N k} {fD k} {g'D k} {gD k})\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange {g'N k} {gD k} {f'D k} {fD k}\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"de617d2ad76260052f9d6ecbf3c054fc7eb183ef","subject":"Agda: Remove unused imports violating architectural constraints","message":"Agda: Remove unused imports violating architectural constraints\n\nThe parametric formalization shouldn't import stuff specific to the\nPOPL14 calculus.\n\nOld-commit-hash: 30c3071e33a99422b2c878f7bca7937e8b1c216f\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Value.agda","new_file":"Parametric\/Change\/Value.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Value\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen ChangeType.Structure Base \u0394Base\n\nopen import Base.Denotation.Notation\n\n-- `diff` and `apply`, without validity proofs\n\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base\n\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base\n\nmodule Structure\n    (\u27e6apply-base\u27e7 : ApplyStructure)\n    (\u27e6diff-base\u27e7 : DiffStructure)\n  where\n\n  \u27e6apply\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  \u27e6diff\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n  infixl 6 \u27e6apply\u27e7 \u27e6diff\u27e7\n  syntax \u27e6apply\u27e7 \u03c4 v dv = v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 dv\n  syntax \u27e6diff\u27e7 \u03c4 u v = u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v = \u27e6apply-base\u27e7 \u03b9 v \u0394v\n  f \u27e6\u2295\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 \u0394f = \u03bb v \u2192 f v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394f v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n\n  u \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 v = \u27e6diff-base\u27e7 \u03b9 u v\n  g \u27e6\u229d\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 f = \u03bb v \u0394v \u2192 (g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v)) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 (f v)\n\n  _\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u27e6\u2295\u27e7_ {\u03c4} = \u27e6apply\u27e7 \u03c4\n\n  _\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u27e6\u229d\u27e7_ {\u03c4} = \u27e6diff\u27e7 \u03c4\n\n  alternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 \u27e6 \u0394Context \u0393 \u27e7\n  alternate {\u2205} \u2205 \u2205 = \u2205\n  alternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Value\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen ChangeType.Structure Base \u0394Base\n\nopen import Base.Denotation.Notation\n\nopen import Data.Integer\nopen import Structure.Bag.Popl14\n\n-- `diff` and `apply`, without validity proofs\n\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base\n\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base\n\nmodule Structure\n    (\u27e6apply-base\u27e7 : ApplyStructure)\n    (\u27e6diff-base\u27e7 : DiffStructure)\n  where\n\n  \u27e6apply\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  \u27e6diff\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n  infixl 6 \u27e6apply\u27e7 \u27e6diff\u27e7\n  syntax \u27e6apply\u27e7 \u03c4 v dv = v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 dv\n  syntax \u27e6diff\u27e7 \u03c4 u v = u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v = \u27e6apply-base\u27e7 \u03b9 v \u0394v\n  f \u27e6\u2295\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 \u0394f = \u03bb v \u2192 f v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394f v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n\n  u \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 v = \u27e6diff-base\u27e7 \u03b9 u v\n  g \u27e6\u229d\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 f = \u03bb v \u0394v \u2192 (g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v)) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 (f v)\n\n  _\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u27e6\u2295\u27e7_ {\u03c4} = \u27e6apply\u27e7 \u03c4\n\n  _\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u27e6\u229d\u27e7_ {\u03c4} = \u27e6diff\u27e7 \u03c4\n\n  alternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 \u27e6 \u0394Context \u0393 \u27e7\n  alternate {\u2205} \u2205 \u2205 = \u2205\n  alternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3eec36d839aa4ec0250dbd55c9767a1d968d0649","subject":"proof of associativity and commutativity hard way","message":"proof of associativity and commutativity hard way\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat.agda","new_file":"agda\/Nat.agda","new_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b)\n  = trans (lemma a b) (cong succ (+-is-commutative a b))\n","old_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"19985bd94f56f9d826d6eae344261fe20f4b7f4d","subject":"Removed empty line.","message":"Removed empty line.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\n\u2248-refl {xs} Sxs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = Stream xs \u2227 xs \u2261 ys\n\n  h\u2081 : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n         xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 R xs' ys'\n  h\u2081 (Sxs , refl) with Stream-unf Sxs\n  ... | x' , xs' , prf , Sxs' =\n    x' , xs' , xs' , prf , prf , (Sxs' , refl)\n\n  h\u2082 : R xs xs\n  h\u2082 = Sxs , refl\n\n\u2248-sym : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 ys \u2248 xs\n\u2248-sym {xs} {ys} xs\u2248ys = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = ys \u2248 xs\n\n  h\u2081 : R ys xs \u2192\n       \u2203[ y' ] \u2203[ ys' ] \u2203[ xs' ]\n         ys \u2261 y' \u2237 ys' \u2227 xs \u2261 y' \u2237 xs' \u2227 R ys' xs'\n  h\u2081 Rxsys with \u2248-unf Rxsys\n  ... | y' , ys' , xs' , prf\u2081 , prf\u2082 , ys'\u2248xs' =\n    y' , xs' , ys' , prf\u2082 , prf\u2081 , ys'\u2248xs'\n\n  h\u2082 : R ys xs\n  h\u2082 = xs\u2248ys\n\n\u2248-trans : \u2200 {xs ys zs} \u2192 xs \u2248 ys \u2192 ys \u2248 zs \u2192 xs \u2248 zs\n\u2248-trans {xs} {ys} {zs} xs\u2248ys ys\u2248zs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs zs = \u2203[ ys ] xs \u2248 ys \u2227 ys \u2248 zs\n\n  h\u2081 : R xs zs \u2192\n       \u2203[ x' ] \u2203[ xs' ] \u2203[ zs' ]\n         xs \u2261 x' \u2237 xs' \u2227 zs \u2261 x' \u2237 zs' \u2227 R xs' zs'\n  h\u2081 (ys , xs\u2248ys , ys\u2248zs) with \u2248-unf xs\u2248ys\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , xs'\u2248ys' with \u2248-unf ys\u2248zs\n  ... | y' , ys'' , zs' , prf\u2083 , prf\u2084 , ys''\u2248zs' =\n    x'\n    , xs'\n    , zs'\n    , prf\u2081\n    , subst (\u03bb t \u2192 zs \u2261 t \u2237 zs') y'\u2261x' prf\u2084\n    , (ys' , (xs'\u2248ys' , (subst (\u03bb t \u2192 t \u2248 zs') ys''\u2261ys' ys''\u2248zs')))\n\n    where\n    y'\u2261x' : y' \u2261 x'\n    y'\u2261x' = \u2227-proj\u2081 (\u2237-injective (trans (sym prf\u2083) prf\u2082))\n\n    ys''\u2261ys' : ys'' \u2261 ys'\n    ys''\u2261ys' = \u2227-proj\u2082 (\u2237-injective (trans (sym prf\u2083) prf\u2082))\n\n  h\u2082 : R xs zs\n  h\u2082 = ys , (xs\u2248ys , ys\u2248zs)\n\n\u2237-injective\u2248 : \u2200 {x xs ys} \u2192 x \u2237 xs \u2248 x \u2237 ys \u2192 xs \u2248 ys\n\u2237-injective\u2248 {x} {xs} {ys} h with \u2248-unf h\n... | x' , xs' , ys' , prf\u2081 , prf\u2082 , prf\u2083 = xs\u2248ys\n  where\n  xs\u2261xs' : xs \u2261 xs'\n  xs\u2261xs' = \u2227-proj\u2082 (\u2237-injective prf\u2081)\n\n  ys\u2261ys' : ys \u2261 ys'\n  ys\u2261ys' = \u2227-proj\u2082 (\u2237-injective prf\u2082)\n\n  xs\u2248ys : xs \u2248 ys\n  xs\u2248ys = subst (\u03bb t \u2192 t \u2248 ys)\n                (sym xs\u2261xs')\n                (subst (\u03bb t \u2192 xs' \u2248 t) (sym ys\u2261ys') prf\u2083)\n\n\u2237-rightCong\u2248 : \u2200 {x xs ys} \u2192 xs \u2248 ys \u2192 x \u2237 xs \u2248 x \u2237 ys\n\u2237-rightCong\u2248 {x} {xs} {ys} h = \u2248-pre-fixed (x , xs , ys , refl , refl , h)\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for the bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\n\u2248-refl {xs} Sxs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = Stream xs \u2227 xs \u2261 ys\n\n  h\u2081 : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n         xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 R xs' ys'\n  h\u2081 (Sxs , refl) with Stream-unf Sxs\n  ... | x' , xs' , prf , Sxs' =\n    x' , xs' , xs' , prf , prf , (Sxs' , refl)\n\n  h\u2082 : R xs xs\n  h\u2082 = Sxs , refl\n\n\n\u2248-sym : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 ys \u2248 xs\n\u2248-sym {xs} {ys} xs\u2248ys = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = ys \u2248 xs\n\n  h\u2081 : R ys xs \u2192\n       \u2203[ y' ] \u2203[ ys' ] \u2203[ xs' ]\n         ys \u2261 y' \u2237 ys' \u2227 xs \u2261 y' \u2237 xs' \u2227 R ys' xs'\n  h\u2081 Rxsys with \u2248-unf Rxsys\n  ... | y' , ys' , xs' , prf\u2081 , prf\u2082 , ys'\u2248xs' =\n    y' , xs' , ys' , prf\u2082 , prf\u2081 , ys'\u2248xs'\n\n  h\u2082 : R ys xs\n  h\u2082 = xs\u2248ys\n\n\u2248-trans : \u2200 {xs ys zs} \u2192 xs \u2248 ys \u2192 ys \u2248 zs \u2192 xs \u2248 zs\n\u2248-trans {xs} {ys} {zs} xs\u2248ys ys\u2248zs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs zs = \u2203[ ys ] xs \u2248 ys \u2227 ys \u2248 zs\n\n  h\u2081 : R xs zs \u2192\n       \u2203[ x' ] \u2203[ xs' ] \u2203[ zs' ]\n         xs \u2261 x' \u2237 xs' \u2227 zs \u2261 x' \u2237 zs' \u2227 R xs' zs'\n  h\u2081 (ys , xs\u2248ys , ys\u2248zs) with \u2248-unf xs\u2248ys\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , xs'\u2248ys' with \u2248-unf ys\u2248zs\n  ... | y' , ys'' , zs' , prf\u2083 , prf\u2084 , ys''\u2248zs' =\n    x'\n    , xs'\n    , zs'\n    , prf\u2081\n    , subst (\u03bb t \u2192 zs \u2261 t \u2237 zs') y'\u2261x' prf\u2084\n    , (ys' , (xs'\u2248ys' , (subst (\u03bb t \u2192 t \u2248 zs') ys''\u2261ys' ys''\u2248zs')))\n\n    where\n    y'\u2261x' : y' \u2261 x'\n    y'\u2261x' = \u2227-proj\u2081 (\u2237-injective (trans (sym prf\u2083) prf\u2082))\n\n    ys''\u2261ys' : ys'' \u2261 ys'\n    ys''\u2261ys' = \u2227-proj\u2082 (\u2237-injective (trans (sym prf\u2083) prf\u2082))\n\n  h\u2082 : R xs zs\n  h\u2082 = ys , (xs\u2248ys , ys\u2248zs)\n\n\u2237-injective\u2248 : \u2200 {x xs ys} \u2192 x \u2237 xs \u2248 x \u2237 ys \u2192 xs \u2248 ys\n\u2237-injective\u2248 {x} {xs} {ys} h with \u2248-unf h\n... | x' , xs' , ys' , prf\u2081 , prf\u2082 , prf\u2083 = xs\u2248ys\n  where\n  xs\u2261xs' : xs \u2261 xs'\n  xs\u2261xs' = \u2227-proj\u2082 (\u2237-injective prf\u2081)\n\n  ys\u2261ys' : ys \u2261 ys'\n  ys\u2261ys' = \u2227-proj\u2082 (\u2237-injective prf\u2082)\n\n  xs\u2248ys : xs \u2248 ys\n  xs\u2248ys = subst (\u03bb t \u2192 t \u2248 ys)\n                (sym xs\u2261xs')\n                (subst (\u03bb t \u2192 xs' \u2248 t) (sym ys\u2261ys') prf\u2083)\n\n\u2237-rightCong\u2248 : \u2200 {x xs ys} \u2192 xs \u2248 ys \u2192 x \u2237 xs \u2248 x \u2237 ys\n\u2237-rightCong\u2248 {x} {xs} {ys} h = \u2248-pre-fixed (x , xs , ys , refl , refl , h)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1cbcb26f8da55497f03917175c94772edb7077cd","subject":"Cosmetic change.","message":"Cosmetic change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/Collatz\/ConversionRulesATP.agda","new_file":"src\/fot\/FOTC\/Program\/Collatz\/ConversionRulesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion rules for the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.ConversionRulesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n-- Conversion rules for the Collatz function.\npostulate\n  collatz-0       : collatz zero \u2261 [1]\n  collatz-1       : collatz [1]  \u2261 [1]\n  collatz-even    : \u2200 {n} \u2192 Even (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz (div (succ\u2081 (succ\u2081 n)) [2])\n  collatz-noteven : \u2200 {n} \u2192 NotEven (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz ([3] * (succ\u2081 (succ\u2081 n)) + [1])\n{-# ATP prove collatz-0 #-}\n{-# ATP prove collatz-1 #-}\n{-# ATP prove collatz-even #-}\n{-# ATP prove collatz-noteven #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Conversion rules for the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.ConversionRulesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n-- Conversion rules for the Collatz function.\npostulate\n  collatz-0       : collatz zero \u2261 [1]\n  collatz-1       : collatz [1]  \u2261 [1]\n  collatz-even    : \u2200 {n} \u2192 Even (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261 collatz (div (succ\u2081 (succ\u2081 n)) [2])\n  collatz-noteven : \u2200 {n} \u2192 NotEven (succ\u2081 (succ\u2081 n)) \u2192\n                    collatz (succ\u2081 (succ\u2081 n)) \u2261\n                    collatz ([3] * (succ\u2081 (succ\u2081 n)) + [1])\n{-# ATP prove collatz-0 #-}\n{-# ATP prove collatz-1 #-}\n{-# ATP prove collatz-even #-}\n{-# ATP prove collatz-noteven #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b0acfde8e76ff27fb9d7b8609c3d7acf1ccf1ef2","subject":"update PLDI14-List-of-Theorems.agda for consistency with #4","message":"update PLDI14-List-of-Theorems.agda for consistency with #4\n\nCamera-ready version inserts a \"lemma 2.5\", which means math items\n2.5--2.9 should be renumbered 2.6--2.10.\n\nRelevant commits:\n23339dd5a9e56c3c31d42252c4020b716ec77725\n0c1e97fbf3b62bddf23559545a4c801b4189cb29\n","repos":"inc-lc\/ilc-agda","old_file":"PLDI14-List-of-Theorems.agda","new_file":"PLDI14-List-of-Theorems.agda","new_contents":"module PLDI14-List-of-Theorems where\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important names. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.5 (Behavior of derivatives on nil)\nopen import Base.Change.Equivalence using (deriv-zero)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.6 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.7 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.8 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.8 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.9 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.10 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- Definition 3.1 (Domains)\nimport Parametric.Denotation.Value\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Definition 3.2 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.3 (Evaluation)\nimport Parametric.Denotation.Evaluation\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Definition 3.4 (Changes)\n-- Definition 3.5 (Change environments)\nimport Parametric.Change.Validity\nopen Parametric.Change.Validity.Structure using (change-algebra)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Definition 3.6 (Change semantics)\n-- Lemma 3.7 (Change semantics is the derivative of semantics)\nimport Parametric.Change.Specification\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.8 (Erasure)\n-- Lemma 3.9 (The erased version of a change is almost the same)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.10 (\u27e6 t \u27e7\u0394 erases to Derive(t))\n-- Theorem 3.11 (Correctness of differentiation)\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","old_contents":"module PLDI14-List-of-Theorems where\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important names. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- Definition 3.1 (Domains)\nimport Parametric.Denotation.Value\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Definition 3.2 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.3 (Evaluation)\nimport Parametric.Denotation.Evaluation\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Definition 3.4 (Changes)\n-- Definition 3.5 (Change environments)\nimport Parametric.Change.Validity\nopen Parametric.Change.Validity.Structure using (change-algebra)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Definition 3.6 (Change semantics)\n-- Lemma 3.7 (Change semantics is the derivative of semantics)\nimport Parametric.Change.Specification\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.8 (Erasure)\n-- Lemma 3.9 (The erased version of a change is almost the same)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.10 (\u27e6 t \u27e7\u0394 erases to Derive(t))\n-- Theorem 3.11 (Correctness of differentiation)\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a5f10ca08fbc34a5cea8a019fefc59bf379d19a6","subject":"flipbased.agda","message":"flipbased.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"open import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule flipbased\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n where\n\nCoins = \u2115\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\n-- An experiment\nEXP : \u2115 \u2192 Set\nEXP n = \u21ba n Bit\n\n-- A guessing game\n\u2141? : \u2200 c \u2192 Set\n\u2141? c = Bit \u2192 EXP c\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = return\u21ba\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ x f = join\u21ba (map\u21ba f x)\n\n_=<<_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       (A \u2192 \u21ba n\u2081 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_=<<_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2081 n\u2082 = flip _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\n_>=>_ : \u2200 {n\u2081 n\u2082 a b c} {A : Set a} {B : Set b} {C : Set c}\n        \u2192 (A \u2192 \u21ba n\u2081 B) \u2192 (B \u2192 \u21ba n\u2082 C) \u2192 A \u2192 \u21ba (n\u2081 + n\u2082) C\n(f >=> g) x = f x >>= g\n\nweaken : \u2200 m {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken m x = return\u21ba {m} 0 >> x\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 x = x >>= return\u21ba\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 pf x with \u2264\u21d2\u2203 pf\n... | k , \u2261.refl = weaken\u2032 x\n\npure\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure\u21ba = return\u21ba\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = toss >>= return\u21ba\n\n_\u25b9\u21ba_ : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 \u21ba n A \u2192 (A \u2192 B) \u2192 \u21ba n B\nx \u25b9\u21ba f = map\u21ba f x\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\u21ba\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map\u21ba f x\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 \u27ea f \u00b7 mx \u27eb \n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map\u21ba f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\nzip\u21ba : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} \u2192 \u21ba c\u2080 A \u2192 \u21ba c\u2081 B \u2192 \u21ba (c\u2080 + c\u2081) (A \u00d7 B)\nzip\u21ba x y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\n_\u27e8,\u27e9_ = zip\u21ba\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2237\u27e9_ : \u2200 {n\u2081 n\u2082 m a} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 (Vec A m) \u2192 \u21ba (n\u2081 + n\u2082) (Vec A (suc m))\nx \u27e8\u2237\u27e9 xs = \u27ea _\u2237_ \u00b7 x \u00b7 xs \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\nT\u21ba : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\nT\u21ba p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = x \u27e8\u2237\u27e9 replicate\u21ba x\n\nnot\u21ba : \u2200 {n} \u2192 EXP n \u2192 EXP n\nnot\u21ba = map\u21ba not\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFun : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\ncostRndFun-lem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\ncostRndFun-lem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\ncostRndFun-lem (suc m) n rewrite costRndFun-lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 \u21ba (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 return\u1d30 =<< x \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map\u21ba swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"open import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule flipbased\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n where\n\nCoins = \u2115\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\n-- An experiment\nEXP : \u2115 \u2192 Set\nEXP n = \u21ba n Bit\n\n-- A guessing game\n\u2141? : \u2200 c \u2192 Set\n\u2141? c = Bit \u2192 EXP c\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = return\u21ba\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ x f = join\u21ba (map\u21ba f x)\n\n_=<<_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       (A \u2192 \u21ba n\u2081 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_=<<_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2081 n\u2082 = flip _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\n_>=>_ : \u2200 {n\u2081 n\u2082 a b c} {A : Set a} {B : Set b} {C : Set c}\n        \u2192 (A \u2192 \u21ba n\u2081 B) \u2192 (B \u2192 \u21ba n\u2082 C) \u2192 A \u2192 \u21ba (n\u2081 + n\u2082) C\n(f >=> g) x = f x >>= g\n\nweaken : \u2200 m {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken m x = return\u21ba {m} 0 >> x\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 x = x >>= return\u21ba\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 pf x with \u2264\u21d2\u2203 pf\n... | k , \u2261.refl = weaken\u2032 x\n\npure\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure\u21ba = return\u21ba\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = toss >>= return\u21ba\n\n_\u25b9\u21ba_ : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 \u21ba n A \u2192 (A \u2192 B) \u2192 \u21ba n B\nx \u25b9\u21ba f = map\u21ba f x\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\u21ba\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map\u21ba f x\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 \u27ea f \u00b7 mx \u27eb \n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map\u21ba f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\nzip\u21ba : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} \u2192 \u21ba c\u2080 A \u2192 \u21ba c\u2081 B \u2192 \u21ba (c\u2080 + c\u2081) (A \u00d7 B)\nzip\u21ba x y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\n_\u27e8,\u27e9_ = zip\u21ba\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\nT\u21ba : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\nT\u21ba p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicate\u21ba x \u27eb\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFun : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\ncostRndFun-lem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\ncostRndFun-lem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\ncostRndFun-lem (suc m) n rewrite costRndFun-lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 \u21ba (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 return\u1d30 =<< x \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map\u21ba swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5e8b1aa79d7d28b80b01e29591028ad21f2751b4","subject":"Prove new lemma from thesis","message":"Prove new lemma from thesis\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\n  -- \\begin{lemma}[Equivalence cancellation]\n  --   |v2 `ominus` v1 `doe` dv| holds if and only if |v2 = v1 `oplus`\n  --   dv|, for any |v1, v2 `elem` V| and |dv `elem` Dt ^ v1|.\n  -- \\end{lemma}\n\n  equiv-cancel-1 : \u2200 x' dx \u2192 _\u2259_ {{ca}} (x' \u229f x) dx \u2192 x' \u2261 x \u229e dx\n  equiv-cancel-1 x' dx (doe x'\u229fx\u2259dx) = trans (sym (update-diff x' x)) x'\u229fx\u2259dx\n  equiv-cancel-2 : \u2200 x' dx \u2192 x' \u2261 x \u229e dx \u2192 _\u2259_ {{ca}} (x' \u229f x) dx\n  equiv-cancel-2 _ dx refl = diff-update\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CB}} (df a da) (f (a \u229e da) \u229f f a)\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CB}} (df a da) (f (a \u229e da) \u229f f a)\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f0468dc61e6551228ac0e70f23b50c06ad9c2a14","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 1264a77944a1d117ad1b3633d2b311d2\n\ndarcs-hash:20110212194846-3bd4e-f2ad17ea1c00d9c7b31f0ff4056470b7cbe98384.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat.agda","new_file":"src\/LTC\/Data\/Nat.agda","new_contents":"------------------------------------------------------------------------------\n-- LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat where\n\nopen import LTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_ _\u2238_\n\n------------------------------------------------------------------------------\n-- The LTC natural numbers type.\nopen import LTC.Data.Nat.Type public\n\n------------------------------------------------------------------------------\n-- Arithmetic operations\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero   + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\npostulate\n  _\u2238_  : D \u2192 D \u2192 D\n  \u2238-x0 : \u2200 d \u2192   d      \u2238 zero   \u2261 d\n  \u2238-0S : \u2200 d \u2192   zero   \u2238 succ d \u2261 zero\n  \u2238-SS : \u2200 d e \u2192 succ d \u2238 succ e \u2261 d \u2238 e\n{-# ATP axiom \u2238-x0 #-}\n{-# ATP axiom \u2238-0S #-}\n{-# ATP axiom \u2238-SS #-}\n\npostulate\n  _*_  : D \u2192 D \u2192 D\n  *-0x : \u2200 d \u2192   zero   * d \u2261 zero\n  *-Sx : \u2200 d e \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x #-}\n{-# ATP axiom *-Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat where\n\nopen import LTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_ _\u2238_\n\n------------------------------------------------------------------------------\n-- The LTC natural numbers type.\nopen import LTC.Data.Nat.Type public\n\n------------------------------------------------------------------------------\n-- Arithmetic operations\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero   + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\npostulate\n  _\u2238_      : D \u2192 D \u2192 D\n  \u2238-x0 : \u2200 d \u2192   d      \u2238 zero   \u2261 d\n  \u2238-0S : \u2200 d \u2192   zero   \u2238 succ d \u2261 zero\n  \u2238-SS : \u2200 d e \u2192 succ d \u2238 succ e \u2261 d \u2238 e\n{-# ATP axiom \u2238-x0 #-}\n{-# ATP axiom \u2238-0S #-}\n{-# ATP axiom \u2238-SS #-}\n\npostulate\n  _*_ : D \u2192 D \u2192 D\n  *-0x : \u2200 d \u2192   zero   * d \u2261 zero\n  *-Sx : \u2200 d e \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x #-}\n{-# ATP axiom *-Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2525e7d2c808f4a14be0656c9bf760dc0db42f6d","subject":"The agda2atp are using the Koen's approach for the translation of predicate symbols.","message":"The agda2atp are using the Koen's approach for the translation of predicate symbols.\n\nIgnore-this: bf6dda1de53baa440b98cd82a80cd1af\n\ndarcs-hash:20110318170057-3bd4e-bebcaa3fbf61993e0433fb96ed3f5d961f78a1a1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Bool\/TestATP.agda","new_file":"Draft\/FOTC\/Data\/Bool\/TestATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Bool.TestATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Bool.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  thm : \u2200 {b}(P : D \u2192 Set) \u2192 (Bool b \u2227 P true \u2227 P false) \u2192 P b\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm #-}\n\npostulate\n  thm\u2081 : \u2200 {P : D \u2192 Set}{x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192 P (if x then y else z)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm\u2081 #-}\n\nBool-elim : \u2200 (P : D \u2192 Set){x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192\n            P (if x then y else z)\nBool-elim P {y = y} tB Py Pz = subst P (sym (if-true y)) Py\nBool-elim P {z = z} fB Py Pz = subst P (sym (if-false z)) Pz\n\n_&&_ : D \u2192 D \u2192 D\nx && y = if x then y else false\n{-# ATP definition _&&_ #-}\n\npostulate\n  &&-Bool : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n{-# ATP prove &&-Bool Bool-elim #-}\n\n&&-Bool\u2081 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2081 {y = y} tB By = prf\n  where\n    postulate prf : Bool (true && y)\n    {-# ATP prove prf Bool-elim #-}\n&&-Bool\u2081 {y = y} fB By = prf\n  where\n    postulate prf : Bool (false && y)\n    {-# ATP prove prf Bool-elim #-}\n\n&&-Bool\u2082 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2082 tB By = Bool-elim Bool tB By fB\n&&-Bool\u2082 fB By = Bool-elim Bool fB By fB\n","old_contents":"------------------------------------------------------------------------------\n-- Testing\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Bool.TestATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Bool.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  thm : \u2200 {b}(P : D \u2192 Set) \u2192 (Bool b \u2227 P true \u2227 P false) \u2192 P b\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm #-}\n\npostulate\n  thm\u2081 : \u2200 {P : D \u2192 Set}{x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192 P (if x then y else z)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm\u2081 #-}\n\nBool-elim : \u2200 (P : D \u2192 Set){x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192\n            P (if x then y else z)\nBool-elim P {y = y} tB Py Pz = subst P (sym (if-true y)) Py\nBool-elim P {z = z} fB Py Pz = subst P (sym (if-false z)) Pz\n\n_&&_ : D \u2192 D \u2192 D\nx && y = if x then y else false\n{-# ATP definition _&&_ #-}\n\npostulate\n  &&-Bool : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove &&-Bool Bool-elim #-}\n\n&&-Bool\u2081 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2081 {y = y} tB By = prf\n  where\n    postulate prf : Bool (true && y)\n    {-# ATP prove prf Bool-elim #-}\n&&-Bool\u2081 {y = y} fB By = prf\n  where\n    postulate prf : Bool (false && y)\n    {-# ATP prove prf Bool-elim #-}\n\n&&-Bool\u2082 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2082 tB By = Bool-elim Bool tB By fB\n&&-Bool\u2082 fB By = Bool-elim Bool fB By fB\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0be3dd74597a8a36c9a931238927ac17095c3522","subject":"Added draft about the FOTC booleans.","message":"Added draft about the FOTC booleans.\n\nIgnore-this: 917f7ddaa04101914e11938deb91f45b\n\ndarcs-hash:20110316145903-3bd4e-9d4543d873379bde71fa7e85c33b9b1fe709c320.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Bool\/TestATP.agda","new_file":"Draft\/FOTC\/Data\/Bool\/TestATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Bool.TestATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Bool.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  thm : \u2200 {b}(P : D \u2192 Set) \u2192 (Bool b \u2227 P true \u2227 P false) \u2192 P b\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm #-}\n\npostulate\n  thm\u2081 : \u2200 {P : D \u2192 Set}{x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192 P (if x then y else z)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm\u2081 #-}\n\nBool-elim : \u2200 (P : D \u2192 Set){x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192\n            P (if x then y else z)\nBool-elim P {y = y} tB Py Pz = subst P (sym (if-true y)) Py\nBool-elim P {z = z} fB Py Pz = subst P (sym (if-false z)) Pz\n\n_&&_ : D \u2192 D \u2192 D\nx && y = if x then y else false\n{-# ATP definition _&&_ #-}\n\npostulate\n  &&-Bool : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove &&-Bool Bool-elim #-}\n\n&&-Bool\u2081 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2081 {y = y} tB By = prf\n  where\n    postulate prf : Bool (true && y)\n    {-# ATP prove prf Bool-elim #-}\n&&-Bool\u2081 {y = y} fB By = prf\n  where\n    postulate prf : Bool (false && y)\n    {-# ATP prove prf Bool-elim #-}\n\n&&-Bool\u2082 : \u2200 {x y} \u2192 Bool x \u2192 Bool y \u2192 Bool (x && y)\n&&-Bool\u2082 tB By = Bool-elim Bool tB By fB\n&&-Bool\u2082 fB By = Bool-elim Bool fB By fB\n","old_contents":"------------------------------------------------------------------------------\n-- Testing\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Bool.TestATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Bool.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  thm : \u2200 {b}(P : D \u2192 Set) \u2192 (Bool b \u2227 P true \u2227 P false) \u2192 P b\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm #-}\n\npostulate\n  thm\u2081 : \u2200 {P : D \u2192 Set}{x y z} \u2192 Bool x \u2192 P y \u2192 P z \u2192 P (if x then y else z)\n-- The ATPs couldn't prove this postulate.\n-- {-# ATP prove thm\u2081 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"acd256774e7cff7d41fddcb59d4ee1a166592477","subject":"Removed MCR-prop","message":"Removed MCR-prop\n\nIgnore-this: ba33703ab5552046fafa46f8d50d370\n\ndarcs-hash:20110213041239-3bd4e-02db08fc7032c1d17a5c545da14dc167ebab8d9a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/RelationATP.agda","new_file":"Draft\/McCarthy91\/RelationATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for some relations\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.RelationATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\nopen import Draft.McCarthy91.ArithmeticATP\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for some relations\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.RelationATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\nopen import Draft.McCarthy91.ArithmeticATP\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\nMCR-prop : \u2200 {n} \u2192 N n \u2192 LE n one-hundred \u2192 MCR (n + eleven) n\nMCR-prop zN 0\u2264100 = prf\n  where\n    postulate\n      prf : (hundred-one \u2238 (zero + eleven)) < (hundred-one \u2238 zero) \u2261 true\n    {-# ATP prove prf #-}\nMCR-prop (sN {n} Nn) Sn\u2264100 = prf (MCR-prop Nn n\u2264100)\n  where\n    n\u2264100 : LE n one-hundred\n    n\u2264100 = \u2264-trans Nn (sN Nn) N100 (x<y\u2192x\u2264y Nn (sN Nn) (x<Sx Nn)) Sn\u2264100\n\n    postulate\n      prf : (hundred-one \u2238 (n + eleven)) < (hundred-one \u2238 n) \u2261 true \u2192  -- IH.\n            (hundred-one \u2238 (succ n + eleven)) < (hundred-one \u2238 succ n) \u2261 true\n    -- {-# ATP prove prf #-} -- Fail with all the ATPs.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8fbacc398d3d884274c0a5d0564e3802b8ff356b","subject":"Dice: use the second flavor of Fin exploration to show that it behave as expected","message":"Dice: use the second flavor of Fin exploration to show that it behave as expected\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Dice.agda","new_file":"lib\/Explore\/Dice.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Fin using (Fin; zero; suc; #_)\nimport Function.Inverse.NP as Inv\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl)\nopen Inv using (_\u2194_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Explorable\nopen import Explore.Universe\n\nmodule Explore.Dice where\n\ndata Dice : \u2605\u2080 where\n  \u2680 \u2681 \u2682 \u2683 \u2684 \u2685 : Dice\n\nmodule ByHand where\n    exploreDice : \u2200 {m} \u2192 Explore m Dice\n    exploreDice \u03b5 _\u2219_ f = f \u2680 \u2219 (f \u2681 \u2219 (f \u2682 \u2219 (f \u2683 \u2219 (f \u2684 \u2219 f \u2685))))\n\n    exploreDice-ind : \u2200 {m p} \u2192 ExploreInd p (exploreDice {m})\n    exploreDice-ind P \u03b5 _\u2219_ f = f \u2680 \u2219 (f \u2681 \u2219 (f \u2682 \u2219 (f \u2683 \u2219 (f \u2684 \u2219 f \u2685))))\n\n    open Explorable\u2080  exploreDice-ind public using () renaming (sum     to sumDice; product to productDice)\n    open Explorable\u2081\u2080 exploreDice-ind public using () renaming (reify   to reifyDice)\n    open Explorable\u2081\u2081 exploreDice-ind public using () renaming (unfocus to unfocusDice)\n\nDice\u2194Fin6 : Dice \u2194 Fin 6\nDice\u2194Fin6 = Inv.inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  module Dice\u2194Fin6 where\n    S = Dice\n    T = Fin 6\n    \u21d2 : S \u2192 T\n    \u21d2 \u2680 = # 0\n    \u21d2 \u2681 = # 1\n    \u21d2 \u2682 = # 2\n    \u21d2 \u2683 = # 3\n    \u21d2 \u2684 = # 4\n    \u21d2 \u2685 = # 5\n    \u21d0 : T \u2192 S\n    \u21d0 zero = \u2680\n    \u21d0 (suc zero) = \u2681\n    \u21d0 (suc (suc zero)) = \u2682\n    \u21d0 (suc (suc (suc zero))) = \u2683\n    \u21d0 (suc (suc (suc (suc zero)))) = \u2684\n    \u21d0 (suc (suc (suc (suc (suc zero))))) = \u2685\n    \u21d0 (suc (suc (suc (suc (suc (suc ()))))))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 \u2680 = refl\n    \u21d0\u21d2 \u2681 = refl\n    \u21d0\u21d2 \u2682 = refl\n    \u21d0\u21d2 \u2683 = refl\n    \u21d0\u21d2 \u2684 = refl\n    \u21d0\u21d2 \u2685 = refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 zero = refl\n    \u21d2\u21d0 (suc zero) = refl\n    \u21d2\u21d0 (suc (suc zero)) = refl\n    \u21d2\u21d0 (suc (suc (suc zero))) = refl\n    \u21d2\u21d0 (suc (suc (suc (suc zero)))) = refl\n    \u21d2\u21d0 (suc (suc (suc (suc (suc zero))))) = refl\n    \u21d2\u21d0 (suc (suc (suc (suc (suc (suc ()))))))\n\n-- By using FinU' instead of FinU one get a special case for Fin 1 thus avoiding\n-- a final \u03b5 in the exploration function.\nopen Explore.Universe.Isomorphism (FinU' 6) (Inv.sym Dice\u2194Fin6 Inv.\u2218 FinU'\u2194Fin 6)\n  public\n  renaming ( iso\u1d49 to Dice\u1d49\n           ; iso\u2071 to Dice\u2071\n           ; iso\u02e1 to Dice\u02e1\n           ; iso\u1da0 to Dice\u1da0\n           ; iso\u02e2 to Dice\u02e2\n           ; iso\u1d56 to Dice\u1d56\n           ; iso\u02b3 to Dice\u02b3\n           ; iso\u1d58 to Dice\u1d58\n           ; iso\u02e2-ok to Dice\u02e2-ok\n           ; iso\u02e2-stableUnder to Dice\u02e2-stableUnder\n           ; \u03bciso to \u03bcDice\n           )\n\nmodule _ {m} where\n  open ByHand\n  _\u2261\u1d49_ : (e\u2080 e\u2081 : Explore m Dice) \u2192 \u2605_ _\n  _\u2261\u1d49_ = _\u2261_\n\n  same-as-by-hand : exploreDice \u2261\u1d49 Dice\u1d49\n  same-as-by-hand = refl\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Fin using (Fin; zero; suc; #_)\nimport Function.Inverse.NP as Inv\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl)\nopen Inv using (_\u2194_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Explorable\nopen import Explore.Universe\n\nmodule Explore.Dice where\n\ndata Dice : \u2605\u2080 where\n  \u2680 \u2681 \u2682 \u2683 \u2684 \u2685 : Dice\n\nmodule ByHand where\n    exploreDice : \u2200 {m} \u2192 Explore m Dice\n    exploreDice \u03b5 _\u2219_ f = f \u2680 \u2219 (f \u2681 \u2219 (f \u2682 \u2219 (f \u2683 \u2219 (f \u2684 \u2219 f \u2685))))\n\n    exploreDice-ind : \u2200 {m p} \u2192 ExploreInd p (exploreDice {m})\n    exploreDice-ind P \u03b5 _\u2219_ f = f \u2680 \u2219 (f \u2681 \u2219 (f \u2682 \u2219 (f \u2683 \u2219 (f \u2684 \u2219 f \u2685))))\n\n    open Explorable\u2080  exploreDice-ind public using () renaming (sum     to sumDice; product to productDice)\n    open Explorable\u2081\u2080 exploreDice-ind public using () renaming (reify   to reifyDice)\n    open Explorable\u2081\u2081 exploreDice-ind public using () renaming (unfocus to unfocusDice)\n\nDice\u2194Fin6 : Dice \u2194 Fin 6\nDice\u2194Fin6 = Inv.inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  module Dice\u2194Fin6 where\n    S = Dice\n    T = Fin 6\n    \u21d2 : S \u2192 T\n    \u21d2 \u2680 = # 0\n    \u21d2 \u2681 = # 1\n    \u21d2 \u2682 = # 2\n    \u21d2 \u2683 = # 3\n    \u21d2 \u2684 = # 4\n    \u21d2 \u2685 = # 5\n    \u21d0 : T \u2192 S\n    \u21d0 zero = \u2680\n    \u21d0 (suc zero) = \u2681\n    \u21d0 (suc (suc zero)) = \u2682\n    \u21d0 (suc (suc (suc zero))) = \u2683\n    \u21d0 (suc (suc (suc (suc zero)))) = \u2684\n    \u21d0 (suc (suc (suc (suc (suc zero))))) = \u2685\n    \u21d0 (suc (suc (suc (suc (suc (suc ()))))))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 \u2680 = refl\n    \u21d0\u21d2 \u2681 = refl\n    \u21d0\u21d2 \u2682 = refl\n    \u21d0\u21d2 \u2683 = refl\n    \u21d0\u21d2 \u2684 = refl\n    \u21d0\u21d2 \u2685 = refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 zero = refl\n    \u21d2\u21d0 (suc zero) = refl\n    \u21d2\u21d0 (suc (suc zero)) = refl\n    \u21d2\u21d0 (suc (suc (suc zero))) = refl\n    \u21d2\u21d0 (suc (suc (suc (suc zero)))) = refl\n    \u21d2\u21d0 (suc (suc (suc (suc (suc zero))))) = refl\n    \u21d2\u21d0 (suc (suc (suc (suc (suc (suc ()))))))\n\nopen Explore.Universe.Isomorphism (FinU 6) (Inv.sym Dice\u2194Fin6 Inv.\u2218 FinU\u2194Fin 6)\n  public\n  renaming ( iso\u1d49 to Dice\u1d49\n           ; iso\u2071 to Dice\u2071\n           ; iso\u02e1 to Dice\u02e1\n           ; iso\u1da0 to Dice\u1da0\n           ; iso\u02e2 to Dice\u02e2\n           ; iso\u1d56 to Dice\u1d56\n           ; iso\u02b3 to Dice\u02b3\n           ; iso\u1d58 to Dice\u1d58\n           ; iso\u02e2-ok to Dice\u02e2-ok\n           ; iso\u02e2-stableUnder to Dice\u02e2-stableUnder\n           ; \u03bciso to \u03bcDice\n           )\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c691b85f8bfd7f9b4a97533c4013b53d694c2eb0","subject":"Drop more dead comments","message":"Drop more dead comments\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/MTerm.agda","new_file":"Parametric\/Syntax\/MTerm.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Syntax.MType as MType\n\nmodule Parametric.Syntax.MTerm\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n  where\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Term.Structure Base Const\n\n-- Our extension points are sets of primitives, indexed by the\n-- types of their arguments and their return type.\n\n-- We want different types of constants; some produce values, some produce\n-- computations. In all cases, arguments are assumed to be values (though\n-- individual primitives might require thunks).\n\nValConstStructure : Set\u2081\nValConstStructure = ValContext \u2192 ValType \u2192 Set\n\nCompConstStructure : Set\u2081\nCompConstStructure = ValContext \u2192 CompType \u2192 Set\n\nmodule Structure (ValConst : ValConstStructure) (CompConst : CompConstStructure) where\n  mutual\n    -- Analogues of Terms\n    Vals : ValContext \u2192 ValContext \u2192 Set\n    Vals \u0393 = DependentList (Val \u0393)\n\n    Comps : ValContext \u2192 List CompType \u2192 Set\n    Comps \u0393 = DependentList (Comp \u0393)\n\n    data Val \u0393 : (\u03c4 : ValType) \u2192 Set where\n      vVar : \u2200 {\u03c4} (x : ValVar \u0393 \u03c4) \u2192 Val \u0393 \u03c4\n      -- XXX Do we need thunks? The draft in the paper doesn't have them.\n      -- However, they will start being useful if we deal with CBN source\n      -- languages.\n      vThunk : \u2200 {\u03c4} \u2192 Comp \u0393 \u03c4 \u2192 Val \u0393 (U \u03c4)\n      vConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : ValConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Val \u0393 \u03c4\n\n    data Comp \u0393 : (\u03c4 : CompType) \u2192 Set where\n      cConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : CompConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Comp \u0393 \u03c4\n\n      cForce : \u2200 {\u03c4} \u2192 Val \u0393 (U \u03c4) \u2192 Comp \u0393 \u03c4\n      cReturn : \u2200 {\u03c4} (v : Val \u0393 \u03c4) \u2192 Comp \u0393 (F \u03c4)\n      {-\n      -- Originally, M to x in N. But here we have no names!\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 \u03c4\n      -}\n      -- The following constructor is the main difference between CBPV and this\n      -- monadic calculus. This is better for the caching transformation.\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) (F \u03c4)) \u2192\n        Comp \u0393 (F \u03c4)\n      cAbs : \u2200 {\u03c3 \u03c4} \u2192\n        (t : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 (\u03c3 \u21db \u03c4)\n      cApp : \u2200 {\u03c3 \u03c4} \u2192\n        (s : Comp \u0393 (\u03c3 \u21db \u03c4)) \u2192\n        (t : Val \u0393 \u03c3) \u2192\n        Comp \u0393 \u03c4\n\n  weaken-val : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Val \u0393\u2081 \u03c4 \u2192\n    Val \u0393\u2082 \u03c4\n\n  weaken-comp : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Comp \u0393\u2081 \u03c4 \u2192\n    Comp \u0393\u2082 \u03c4\n\n  weaken-vals : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Vals \u0393\u2081 \u03a3 \u2192\n    Vals \u0393\u2082 \u03a3\n\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vVar x) = vVar (weaken-val-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vThunk x) = vThunk (weaken-comp \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vConst c args) = vConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cConst c args) = cConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cForce x) = cForce (weaken-val \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cReturn v) = cReturn (weaken-val \u0393\u2081\u227c\u0393\u2082 v)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (_into_ {\u03c3} c c\u2081) = (weaken-comp \u0393\u2081\u227c\u0393\u2082 c) into (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c\u2081)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cAbs {\u03c3} c) = cAbs (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cApp s t) = cApp (weaken-comp \u0393\u2081\u227c\u0393\u2082 s) (weaken-val \u0393\u2081\u227c\u0393\u2082 t)\n\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 (px \u2022 ts) = (weaken-val \u0393\u2081\u227c\u0393\u2082 px) \u2022 (weaken-vals \u0393\u2081\u227c\u0393\u2082 ts)\n\n  fromCBN : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBNCtx \u0393) (cbnToCompType \u03c4)\n\n  fromCBNTerms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Vals (fromCBNCtx \u0393) (fromCBNCtx \u03a3)\n  fromCBNTerms \u2205 = \u2205\n  fromCBNTerms (px \u2022 ts) = vThunk (fromCBN px) \u2022 fromCBNTerms ts\n\n  open import UNDEFINED\n  -- This is really supposed to be part of the plugin interface.\n  cbnToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBNCtx \u03a3) (cbnToCompType \u03c4)\n  cbnToCompConst = reveal UNDEFINED\n\n  fromCBN (const c args) = cConst (cbnToCompConst c) (fromCBNTerms args)\n  fromCBN (var x) = cForce (vVar (fromVar cbnToValType x))\n  fromCBN (app s t) = cApp (fromCBN s) (vThunk (fromCBN t))\n  fromCBN (abs t) = cAbs (fromCBN t)\n\n  -- To satisfy termination checking, we'd need to inline fromCBV and weaken: fromCBV needs to produce a term in a bigger context.\n  -- But let's ignore that.\n  {-# NO_TERMINATION_CHECK #-}\n  fromCBV : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n\n  -- This is really supposed to be part of the plugin interface.\n  cbvToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBVCtx \u03a3) (cbvToCompType \u03c4)\n  cbvToCompConst = reveal UNDEFINED\n\n  cbvTermsToComps : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Comps (fromCBVCtx \u0393) (fromCBVToCompList \u03a3)\n  cbvTermsToComps \u2205 = \u2205\n  cbvTermsToComps (px \u2022 ts) = fromCBV px \u2022 cbvTermsToComps ts\n\n  module _ where\n\n    dequeValContexts : ValContext \u2192 ValContext \u2192 ValContext\n    dequeValContexts \u2205 \u0393 = \u0393\n    dequeValContexts (x \u2022 \u03a3) \u0393 = dequeValContexts \u03a3 (x \u2022 \u0393)\n\n    dequeValContexts\u227c\u227c : \u2200 \u03a3 \u0393 \u2192 \u0393 \u227c\u227c dequeValContexts \u03a3 \u0393\n    dequeValContexts\u227c\u227c \u2205 \u0393 = \u227c\u227c-refl\n    dequeValContexts\u227c\u227c (x \u2022 \u03a3) \u0393 = \u227c\u227c-trans (drop_\u2022\u2022_ x \u227c\u227c-refl) (dequeValContexts\u227c\u227c \u03a3 (x \u2022 \u0393))\n\n    dequeContexts : Context \u2192 Context \u2192 ValContext\n    dequeContexts \u03a3 \u0393 = dequeValContexts (fromCBVCtx \u03a3) (fromCBVCtx \u0393)\n\n    fromCBVArg : \u2200 {\u03c3 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 (Val (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToValType \u03c4) \u2192 Comp (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToCompType \u03c3)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c3)\n    fromCBVArg t k = (fromCBV t) into k (vVar vThis)\n\n    fromCBVArgs : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Terms \u0393 \u03a3 \u2192 (Vals (dequeContexts \u03a3 \u0393) (fromCBVCtx \u03a3) \u2192 Comp (dequeContexts \u03a3 \u0393) (cbvToCompType \u03c4)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    fromCBVArgs \u2205 k = k \u2205\n    fromCBVArgs {\u03c3 \u2022 \u03a3} {\u0393} (t \u2022 ts) k = fromCBVArg t (\u03bb v \u2192 fromCBVArgs (weaken-terms (drop_\u2022_ _ \u227c-refl) ts) (\u03bb vs \u2192 k (weaken-val (dequeValContexts\u227c\u227c (fromCBVCtx \u03a3) _) v \u2022 vs)))\n\n    -- Transform a constant application into\n    --\n    --   arg1 to x1 in (arg2 to x2 in ... (argn to xn in (cConst (cbvToCompConst c) (x1 :: x2 :: ... xn)))).\n\n    fromCBVConstCPSRoot : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 Terms \u0393 \u03a3 \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    -- pass that as a closure to compose with (_\u2022_ x) for each new variable.\n    fromCBVConstCPSRoot c ts = fromCBVArgs ts (\u03bb vs \u2192 cConst (cbvToCompConst c) vs)\n\n  fromCBV (const c args) = fromCBVConstCPSRoot c args\n  fromCBV (app s t) =\n    (fromCBV s) into\n      (fromCBV (weaken (drop _ \u2022 \u227c-refl) t) into\n        cApp (cForce (vVar (vThat vThis))) (vVar vThis))\n  -- Values\n  fromCBV (var x) = cReturn (vVar (fromVar cbvToValType x))\n  fromCBV (abs t) = cReturn (vThunk (cAbs (fromCBV t)))\n\n  -- This reflects the CBV conversion of values in TLCA '99, but we can't write\n  -- this because it's partial on the Term type. Instead, we duplicate thunking\n  -- at the call site.\n  {-\n  fromCBVToVal : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Val (fromCBVCtx \u0393) (cbvToValType \u03c4)\n  fromCBVToVal (var x) = vVar (fromVar cbvToValType x)\n  fromCBVToVal (abs t) = vThunk (cAbs (fromCBV t))\n\n  fromCBVToVal (const c args) = {!!}\n  fromCBVToVal (app t t\u2081) = {!!} -- Not a value\n  -}\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Syntax.MType as MType\n\nmodule Parametric.Syntax.MTerm\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n  where\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Term.Structure Base Const\n\n-- Our extension points are sets of primitives, indexed by the\n-- types of their arguments and their return type.\n\n-- We want different types of constants; some produce values, some produce\n-- computations. In all cases, arguments are assumed to be values (though\n-- individual primitives might require thunks).\n\nValConstStructure : Set\u2081\nValConstStructure = ValContext \u2192 ValType \u2192 Set\n\nCompConstStructure : Set\u2081\nCompConstStructure = ValContext \u2192 CompType \u2192 Set\n\nmodule Structure (ValConst : ValConstStructure) (CompConst : CompConstStructure) where\n  mutual\n    -- Analogues of Terms\n    Vals : ValContext \u2192 ValContext \u2192 Set\n    Vals \u0393 = DependentList (Val \u0393)\n\n    Comps : ValContext \u2192 List CompType \u2192 Set\n    Comps \u0393 = DependentList (Comp \u0393)\n\n    data Val \u0393 : (\u03c4 : ValType) \u2192 Set where\n      vVar : \u2200 {\u03c4} (x : ValVar \u0393 \u03c4) \u2192 Val \u0393 \u03c4\n      -- XXX Do we need thunks? The draft in the paper doesn't have them.\n      -- However, they will start being useful if we deal with CBN source\n      -- languages.\n      vThunk : \u2200 {\u03c4} \u2192 Comp \u0393 \u03c4 \u2192 Val \u0393 (U \u03c4)\n      vConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : ValConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Val \u0393 \u03c4\n\n    data Comp \u0393 : (\u03c4 : CompType) \u2192 Set where\n      cConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : CompConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Comp \u0393 \u03c4\n\n      cForce : \u2200 {\u03c4} \u2192 Val \u0393 (U \u03c4) \u2192 Comp \u0393 \u03c4\n      cReturn : \u2200 {\u03c4} (v : Val \u0393 \u03c4) \u2192 Comp \u0393 (F \u03c4)\n      {-\n      -- Originally, M to x in N. But here we have no names!\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 \u03c4\n      -}\n      -- The following constructor is the main difference between CBPV and this\n      -- monadic calculus. This is better for the caching transformation.\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) (F \u03c4)) \u2192\n        Comp \u0393 (F \u03c4)\n      cAbs : \u2200 {\u03c3 \u03c4} \u2192\n        (t : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 (\u03c3 \u21db \u03c4)\n      cApp : \u2200 {\u03c3 \u03c4} \u2192\n        (s : Comp \u0393 (\u03c3 \u21db \u03c4)) \u2192\n        (t : Val \u0393 \u03c3) \u2192\n        Comp \u0393 \u03c4\n\n  weaken-val : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Val \u0393\u2081 \u03c4 \u2192\n    Val \u0393\u2082 \u03c4\n\n  weaken-comp : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Comp \u0393\u2081 \u03c4 \u2192\n    Comp \u0393\u2082 \u03c4\n\n  weaken-vals : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Vals \u0393\u2081 \u03a3 \u2192\n    Vals \u0393\u2082 \u03a3\n\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vVar x) = vVar (weaken-val-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vThunk x) = vThunk (weaken-comp \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vConst c args) = vConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cConst c args) = cConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cForce x) = cForce (weaken-val \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cReturn v) = cReturn (weaken-val \u0393\u2081\u227c\u0393\u2082 v)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (_into_ {\u03c3} c c\u2081) = (weaken-comp \u0393\u2081\u227c\u0393\u2082 c) into (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c\u2081)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cAbs {\u03c3} c) = cAbs (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cApp s t) = cApp (weaken-comp \u0393\u2081\u227c\u0393\u2082 s) (weaken-val \u0393\u2081\u227c\u0393\u2082 t)\n\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 (px \u2022 ts) = (weaken-val \u0393\u2081\u227c\u0393\u2082 px) \u2022 (weaken-vals \u0393\u2081\u227c\u0393\u2082 ts)\n\n  fromCBN : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBNCtx \u0393) (cbnToCompType \u03c4)\n\n  fromCBNTerms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Vals (fromCBNCtx \u0393) (fromCBNCtx \u03a3)\n  fromCBNTerms \u2205 = \u2205\n  fromCBNTerms (px \u2022 ts) = vThunk (fromCBN px) \u2022 fromCBNTerms ts\n\n  open import UNDEFINED\n  -- This is really supposed to be part of the plugin interface.\n  cbnToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBNCtx \u03a3) (cbnToCompType \u03c4)\n  cbnToCompConst = reveal UNDEFINED\n\n  fromCBN (const c args) = cConst (cbnToCompConst c) (fromCBNTerms args)\n  fromCBN (var x) = cForce (vVar (fromVar cbnToValType x))\n  fromCBN (app s t) = cApp (fromCBN s) (vThunk (fromCBN t))\n  fromCBN (abs t) = cAbs (fromCBN t)\n\n  -- To satisfy termination checking, we'd need to inline fromCBV and weaken: fromCBV needs to produce a term in a bigger context.\n  -- But let's ignore that.\n  {-# NO_TERMINATION_CHECK #-}\n  fromCBV : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n\n  -- This is really supposed to be part of the plugin interface.\n  cbvToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBVCtx \u03a3) (cbvToCompType \u03c4)\n  cbvToCompConst = reveal UNDEFINED\n\n  cbvTermsToComps : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Comps (fromCBVCtx \u0393) (fromCBVToCompList \u03a3)\n  cbvTermsToComps \u2205 = \u2205\n  cbvTermsToComps (px \u2022 ts) = fromCBV px \u2022 cbvTermsToComps ts\n\n  module _ where\n\n    dequeValContexts : ValContext \u2192 ValContext \u2192 ValContext\n    dequeValContexts \u2205 \u0393 = \u0393\n    dequeValContexts (x \u2022 \u03a3) \u0393 = dequeValContexts \u03a3 (x \u2022 \u0393)\n\n    dequeValContexts\u227c\u227c : \u2200 \u03a3 \u0393 \u2192 \u0393 \u227c\u227c dequeValContexts \u03a3 \u0393\n    dequeValContexts\u227c\u227c \u2205 \u0393 = \u227c\u227c-refl\n    dequeValContexts\u227c\u227c (x \u2022 \u03a3) \u0393 = \u227c\u227c-trans (drop_\u2022\u2022_ x \u227c\u227c-refl) (dequeValContexts\u227c\u227c \u03a3 (x \u2022 \u0393)) --\n\n    dequeContexts : Context \u2192 Context \u2192 ValContext\n    dequeContexts \u03a3 \u0393 = dequeValContexts (fromCBVCtx \u03a3) (fromCBVCtx \u0393)\n\n    fromCBVArg : \u2200 {\u03c3 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 (Val (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToValType \u03c4) \u2192 Comp (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToCompType \u03c3)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c3)\n    fromCBVArg t k = (fromCBV t) into k (vVar vThis)\n\n    fromCBVArgs : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Terms \u0393 \u03a3 \u2192 (Vals (dequeContexts \u03a3 \u0393) (fromCBVCtx \u03a3) \u2192 Comp (dequeContexts \u03a3 \u0393) (cbvToCompType \u03c4)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    fromCBVArgs \u2205 k = k \u2205\n    fromCBVArgs {\u03c3 \u2022 \u03a3} {\u0393} (t \u2022 ts) k = fromCBVArg t (\u03bb v \u2192 fromCBVArgs (weaken-terms (drop_\u2022_ _ \u227c-refl) ts) (\u03bb vs \u2192 k (weaken-val (dequeValContexts\u227c\u227c (fromCBVCtx \u03a3) _) v \u2022 vs)))\n\n    -- Transform a constant application into\n    --\n    --   arg1 to x1 in (arg2 to x2 in ... (argn to xn in (cConst (cbvToCompConst c) (x1 :: x2 :: ... xn)))).\n\n    fromCBVConstCPSRoot : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 Terms \u0393 \u03a3 \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    -- pass that as a closure to compose with (_\u2022_ x) for each new variable.\n    fromCBVConstCPSRoot c ts = fromCBVArgs ts (\u03bb vs \u2192 cConst (cbvToCompConst c) vs) -- fromCBVConstCPSDo ts {!cConst (cbvToCompConst c)!}\n\n-- In the beginning, we should get a function that expects a whole set of\n-- arguments (Vals \u0393 \u03a3) for c, in the initial context \u0393.\n-- Later, each call should match:\n\n-- -  \u03a3 = \u03c4\u03a3 \u2022 \u03a3\u2032, then we recurse with \u0393\u2032 = \u03c4\u03a3 \u2022 \u0393 and \u03a3\u2032. Terms ought to be\n--    weakened. The result of f (or the arguments) also ought to be weakened! So f should weaken\n--    all arguments.\n\n-- - \u03a3 = \u2205.\n\n  fromCBV (const c args) = fromCBVConstCPSRoot c args\n  fromCBV (app s t) =\n    (fromCBV s) into\n      (fromCBV (weaken (drop _ \u2022 \u227c-refl) t) into\n        cApp (cForce (vVar (vThat vThis))) (vVar vThis))\n  -- Values\n  fromCBV (var x) = cReturn (vVar (fromVar cbvToValType x))\n  fromCBV (abs t) = cReturn (vThunk (cAbs (fromCBV t)))\n\n  -- This reflects the CBV conversion of values in TLCA '99, but we can't write\n  -- this because it's partial on the Term type. Instead, we duplicate thunking\n  -- at the call site.\n  {-\n  fromCBVToVal : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Val (fromCBVCtx \u0393) (cbvToValType \u03c4)\n  fromCBVToVal (var x) = vVar (fromVar cbvToValType x)\n  fromCBVToVal (abs t) = vThunk (cAbs (fromCBV t))\n\n  fromCBVToVal (const c args) = {!!}\n  fromCBVToVal (app t t\u2081) = {!!} -- Not a value\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c1960f9f07f5b25dbb7225149d30534a5c8f3464","subject":"Groups-Nehemiah: add variant of 4-way-shuffle for naturals","message":"Groups-Nehemiah: add variant of 4-way-shuffle for naturals\n","repos":"inc-lc\/ilc-agda","old_file":"Theorem\/Groups-Nehemiah.agda","new_file":"Theorem\/Groups-Nehemiah.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- About the group structure of integers and bags for Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Theorem.Groups-Nehemiah where\n\nopen import Structure.Bag.Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Structures\n\n4-way-shuffle : \u2200 {A : Set} {f} {z a b c d : A}\n  {{comm-monoid : IsCommutativeMonoid _\u2261_ f z}} \u2192\n  f (f a b) (f c d) \u2261 f (f a c) (f b d)\n4-way-shuffle {f = f} {z = z} {a} {b} {c} {d} {{comm-monoid}} =\n  let\n    assoc = associative comm-monoid\n    cmute = commutative comm-monoid\n  in\n    begin\n      f (f a b) (f c d)\n    \u2261\u27e8 assoc a b (f c d) \u27e9\n      f a (f b (f c d))\n    \u2261\u27e8 cong (f a) (sym (assoc b c d)) \u27e9\n      f a (f (f b c) d)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f a (f hole d)) (cmute b c) \u27e9\n      f a (f (f c b) d)\n    \u2261\u27e8 cong (f a) (assoc c b d) \u27e9\n      f a (f c (f b d))\n    \u2261\u27e8 sym (assoc a c (f b d)) \u27e9\n      f (f a c) (f b d)\n    \u220e where open \u2261-Reasoning\n\nmodule _ where\n  open import Data.Nat\n  -- Apologies for irregular name.\n  \u2115-mn\u00b7pq=mp\u00b7nq : \u2200 {m n p q : \u2115} \u2192\n    (m + n) + (p + q) \u2261 (m + p) + (n + q)\n  \u2115-mn\u00b7pq=mp\u00b7nq {m} {n} {p} {q} =\n    4-way-shuffle {a = m} {n} {p} {q} {{comm-monoid-nat}}\n\nopen import Data.Integer\n\nn+[m-n]=m : \u2200 {n m} \u2192 n + (m - n) \u2261 m\nn+[m-n]=m {n} {m} =\n  begin\n    n + (m - n)\n  \u2261\u27e8 cong (\u03bb hole \u2192 n + hole) (commutative-int m (- n)) \u27e9\n    n + (- n + m)\n  \u2261\u27e8 sym (associative-int n (- n) m) \u27e9\n    (n - n) + m\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole + m) (right-inv-int n) \u27e9\n    (+ 0) + m\n  \u2261\u27e8 left-id-int m \u27e9\n    m\n  \u220e where open \u2261-Reasoning\n\na++[b\\\\a]=b : \u2200 {a b} \u2192 a ++ (b \\\\ a) \u2261 b\na++[b\\\\a]=b {b} {d} = trans\n  (cong (\u03bb hole \u2192 b ++ hole) (commutative-bag d (negateBag b))) (trans\n  (sym (associative-bag b (negateBag b) d)) (trans\n  (cong (\u03bb hole \u2192 hole ++ d) (right-inv-bag b))\n  (left-id-bag d)))\n\nab\u00b7cd=ac\u00b7bd : \u2200 {a b c d : Bag} \u2192\n  (a ++ b) ++ (c ++ d) \u2261 (a ++ c) ++ (b ++ d)\nab\u00b7cd=ac\u00b7bd {a} {b} {c} {d} =\n  4-way-shuffle {a = a} {b} {c} {d} {{comm-monoid-bag}}\n\nmn\u00b7pq=mp\u00b7nq : \u2200 {m n p q : \u2124} \u2192\n  (m + n) + (p + q) \u2261 (m + p) + (n + q)\nmn\u00b7pq=mp\u00b7nq {m} {n} {p} {q} =\n  4-way-shuffle {a = m} {n} {p} {q} {{comm-monoid-int}}\n\ninverse-unique : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f a b \u2261 z \u2192 b \u2261 neg a\n\ninverse-unique {f = f} {neg} {z} {a} {b} {{abelian}} ab=z =\n  let\n    assoc = associative (IsAbelianGroup.isCommutativeMonoid abelian)\n    cmute = commutative (IsAbelianGroup.isCommutativeMonoid abelian)\n  in\n    begin\n      b\n    \u2261\u27e8 sym (left-identity abelian b) \u27e9\n      f z b\n    \u2261\u27e8 cong (\u03bb hole \u2192 f hole b) (sym (left-inverse abelian a)) \u27e9\n      f (f (neg a) a) b\n    \u2261\u27e8 assoc (neg a) a b \u27e9\n      f (neg a) (f a b)\n    \u2261\u27e8 cong (f (neg a)) ab=z \u27e9\n      f (neg a) z\n    \u2261\u27e8 right-identity abelian (neg a) \u27e9\n      neg a\n    \u220e where\n      open \u2261-Reasoning\n      eq1 : f (neg a) (f a b) \u2261 f (neg a) z\n      eq1 rewrite ab=z = cong (f (neg a)) refl\n\ndistribute-neg : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f (neg a) (neg b) \u2261 neg (f a b)\ndistribute-neg {f = f} {neg} {z} {a} {b} {{abelian}} = inverse-unique\n  {{abelian}}\n  (begin\n    f (f a b) (f (neg a) (neg b))\n  \u2261\u27e8 4-way-shuffle {{IsAbelianGroup.isCommutativeMonoid abelian}} \u27e9\n    f (f a (neg a)) (f b (neg b))\n  \u2261\u27e8 cong\u2082 f (inverse a) (inverse b) \u27e9\n    f z z\n  \u2261\u27e8 left-identity abelian z \u27e9\n    z\n  \u220e) where\n     open \u2261-Reasoning\n     inverse = right-inverse abelian\n\n-a\u00b7-b=-ab : \u2200 {a b : Bag} \u2192\n  negateBag a ++ negateBag b \u2261 negateBag (a ++ b)\n-a\u00b7-b=-ab {a} {b} = distribute-neg {a = a} {b} {{abelian-bag}}\n\n-m\u00b7-n=-mn : \u2200 {m n : \u2124} \u2192\n  (- m) + (- n) \u2261 - (m + n)\n-m\u00b7-n=-mn {m} {n} = distribute-neg {a = m} {n} {{abelian-int}}\n\n[a++b]\\\\a=b : \u2200 {a b} \u2192 (a ++ b) \\\\ a \u2261 b\n[a++b]\\\\a=b {b} {d} =\n  begin\n    (b ++ d) \\\\ b\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole \\\\ b) (commutative-bag b d) \u27e9\n    (d ++ b) \\\\ b\n  \u2261\u27e8 associative-bag d b (negateBag b) \u27e9\n    d ++ (b \\\\ b)\n  \u2261\u27e8 cong (_++_ d) (right-inv-bag b) \u27e9\n    d ++ emptyBag\n  \u2261\u27e8 right-id-bag d \u27e9\n    d\n  \u220e where open \u2261-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- About the group structure of integers and bags for Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Theorem.Groups-Nehemiah where\n\nopen import Structure.Bag.Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Structures\n\n4-way-shuffle : \u2200 {A : Set} {f} {z a b c d : A}\n  {{comm-monoid : IsCommutativeMonoid _\u2261_ f z}} \u2192\n  f (f a b) (f c d) \u2261 f (f a c) (f b d)\n4-way-shuffle {f = f} {z = z} {a} {b} {c} {d} {{comm-monoid}} =\n  let\n    assoc = associative comm-monoid\n    cmute = commutative comm-monoid\n  in\n    begin\n      f (f a b) (f c d)\n    \u2261\u27e8 assoc a b (f c d) \u27e9\n      f a (f b (f c d))\n    \u2261\u27e8 cong (f a) (sym (assoc b c d)) \u27e9\n      f a (f (f b c) d)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f a (f hole d)) (cmute b c) \u27e9\n      f a (f (f c b) d)\n    \u2261\u27e8 cong (f a) (assoc c b d) \u27e9\n      f a (f c (f b d))\n    \u2261\u27e8 sym (assoc a c (f b d)) \u27e9\n      f (f a c) (f b d)\n    \u220e where open \u2261-Reasoning\n\nopen import Data.Integer\n\nn+[m-n]=m : \u2200 {n m} \u2192 n + (m - n) \u2261 m\nn+[m-n]=m {n} {m} =\n  begin\n    n + (m - n)\n  \u2261\u27e8 cong (\u03bb hole \u2192 n + hole) (commutative-int m (- n)) \u27e9\n    n + (- n + m)\n  \u2261\u27e8 sym (associative-int n (- n) m) \u27e9\n    (n - n) + m\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole + m) (right-inv-int n) \u27e9\n    (+ 0) + m\n  \u2261\u27e8 left-id-int m \u27e9\n    m\n  \u220e where open \u2261-Reasoning\n\na++[b\\\\a]=b : \u2200 {a b} \u2192 a ++ (b \\\\ a) \u2261 b\na++[b\\\\a]=b {b} {d} = trans\n  (cong (\u03bb hole \u2192 b ++ hole) (commutative-bag d (negateBag b))) (trans\n  (sym (associative-bag b (negateBag b) d)) (trans\n  (cong (\u03bb hole \u2192 hole ++ d) (right-inv-bag b))\n  (left-id-bag d)))\n\nab\u00b7cd=ac\u00b7bd : \u2200 {a b c d : Bag} \u2192\n  (a ++ b) ++ (c ++ d) \u2261 (a ++ c) ++ (b ++ d)\nab\u00b7cd=ac\u00b7bd {a} {b} {c} {d} =\n  4-way-shuffle {a = a} {b} {c} {d} {{comm-monoid-bag}}\n\nmn\u00b7pq=mp\u00b7nq : \u2200 {m n p q : \u2124} \u2192\n  (m + n) + (p + q) \u2261 (m + p) + (n + q)\nmn\u00b7pq=mp\u00b7nq {m} {n} {p} {q} =\n  4-way-shuffle {a = m} {n} {p} {q} {{comm-monoid-int}}\n\ninverse-unique : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f a b \u2261 z \u2192 b \u2261 neg a\n\ninverse-unique {f = f} {neg} {z} {a} {b} {{abelian}} ab=z =\n  let\n    assoc = associative (IsAbelianGroup.isCommutativeMonoid abelian)\n    cmute = commutative (IsAbelianGroup.isCommutativeMonoid abelian)\n  in\n    begin\n      b\n    \u2261\u27e8 sym (left-identity abelian b) \u27e9\n      f z b\n    \u2261\u27e8 cong (\u03bb hole \u2192 f hole b) (sym (left-inverse abelian a)) \u27e9\n      f (f (neg a) a) b\n    \u2261\u27e8 assoc (neg a) a b \u27e9\n      f (neg a) (f a b)\n    \u2261\u27e8 cong (f (neg a)) ab=z \u27e9\n      f (neg a) z\n    \u2261\u27e8 right-identity abelian (neg a) \u27e9\n      neg a\n    \u220e where\n      open \u2261-Reasoning\n      eq1 : f (neg a) (f a b) \u2261 f (neg a) z\n      eq1 rewrite ab=z = cong (f (neg a)) refl\n\ndistribute-neg : \u2200 {A : Set} {f neg} {z a b : A}\n  {{abelian : IsAbelianGroup _\u2261_ f z neg}} \u2192\n  f (neg a) (neg b) \u2261 neg (f a b)\ndistribute-neg {f = f} {neg} {z} {a} {b} {{abelian}} = inverse-unique\n  {{abelian}}\n  (begin\n    f (f a b) (f (neg a) (neg b))\n  \u2261\u27e8 4-way-shuffle {{IsAbelianGroup.isCommutativeMonoid abelian}} \u27e9\n    f (f a (neg a)) (f b (neg b))\n  \u2261\u27e8 cong\u2082 f (inverse a) (inverse b) \u27e9\n    f z z\n  \u2261\u27e8 left-identity abelian z \u27e9\n    z\n  \u220e) where\n     open \u2261-Reasoning\n     inverse = right-inverse abelian\n\n-a\u00b7-b=-ab : \u2200 {a b : Bag} \u2192\n  negateBag a ++ negateBag b \u2261 negateBag (a ++ b)\n-a\u00b7-b=-ab {a} {b} = distribute-neg {a = a} {b} {{abelian-bag}}\n\n-m\u00b7-n=-mn : \u2200 {m n : \u2124} \u2192\n  (- m) + (- n) \u2261 - (m + n)\n-m\u00b7-n=-mn {m} {n} = distribute-neg {a = m} {n} {{abelian-int}}\n\n[a++b]\\\\a=b : \u2200 {a b} \u2192 (a ++ b) \\\\ a \u2261 b\n[a++b]\\\\a=b {b} {d} =\n  begin\n    (b ++ d) \\\\ b\n  \u2261\u27e8 cong (\u03bb hole \u2192 hole \\\\ b) (commutative-bag b d) \u27e9\n    (d ++ b) \\\\ b\n  \u2261\u27e8 associative-bag d b (negateBag b) \u27e9\n    d ++ (b \\\\ b)\n  \u2261\u27e8 cong (_++_ d) (right-inv-bag b) \u27e9\n    d ++ emptyBag\n  \u2261\u27e8 right-id-bag d \u27e9\n    d\n  \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e53fdae9c0ced28b2af7af751aaf4131cd2684a7","subject":"progress progress #3","message":"progress progress #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-boxed-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus\n    where\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n    lem (TACast wt TCRefl) (ICastArr x\u2083 ind) \u03c41 \u03c42 .\u03c41 .\u03c42 d refl = x\u2083 refl\n    lem (TACast wt (TCArr x\u2082 x\u2083)) (ICastArr x\u2084 ind) \u03c41 \u03c42 \u03c43 \u03c4' d refl = {!!}\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i = I (IAp {!!} {!!} (FIndet i)) -- cyrus (issue from above, and also i think missing a rule for indet applications)\n  progress (TAAp wt1 wt2) | V v | V v\u2082 = {!!} -- {!canonical-boxed-forms-arr wt1 x !}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  ... | I x = I {!!}\n  ... | V x = V {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n  progress (TAAp wt1 wt2) | S x | ih2 = {!!}\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n  progress (TAAp wt1 wt2) | I x | ih2 = {!!}\n  progress (TAAp wt1 wt2) | V x | ih2 = {!!}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (d' , Step x y z) = S (_ , Step {!!} {!!} {!!})\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt y)\n    with progress wt\n  ... | S x = {!!}\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  ... | I x = {!x!}\n  ... | V x = {!!}\n\n\n  -- -- applications\n  -- progress (TAAp D1 x D2)\n  --   with progress D1 | progress D2\n  --   -- left applicand value\n  -- progress (TAAp TAConst () D2) | V VConst | _\n  -- progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  -- progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n  --   -- errors propagate\n  -- progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = E (EAp2 (FVal x) x\u2081)\n  -- progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = E (EAp2 (FIndet x) x\u2081)\n  -- progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = E (EAp1 x) -- NB: could have picked the other one here, too; this is a source of non-determinism\n  -- progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = S {!!}\n  -- progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n  --   -- indeterminates\n  -- progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  -- progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  --   -- either applicand steps\n  -- progress (TAAp D1 x\u2083 D2) | S (d1' , Step x x\u2081 x\u2082)  | _ = S (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082))\n  -- progress (TAAp D1 x\u2084 D2) | _ | S (d2' , Step x\u2081 x\u2082 x\u2083) = S (_ , Step (FHAp1 {!!} x\u2081) x\u2082 (FHAp1 {!!} x\u2083))\n\n\n  -- -- non-empty holes\n  -- progress (TANEHole x D x\u2081)\n  --   with progress D\n  -- progress (TANEHole x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  -- progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  -- progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  -- progress (TANEHole x\u2083 D x\u2084) | S (_ , Step x x\u2081 x\u2082) = S (_ , Step (FHNEHole x) x\u2081 (FHNEHole x\u2082))\n\n  -- -- casts\n  -- progress (TACast D x)\n  --   with progress D\n  -- progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  -- progress (TACast D m)  | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  -- progress (TACast D x\u2081) | I x = I (ICast x)\n  -- progress (TACast D x\u2081) | E x = E (ECastProp x)\n  -- progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f8bd15f415762af0fce2117302889b7e97a25936","subject":"Rename diff to diff-base.","message":"Rename diff to diff-base.\n\nI want to move code between Parametric.Syntax.Term and\nAtlas.Syntax.Term, but I have to unify the naming conventions first.\n\nOld-commit-hash: 73a829a379ce12d2c01dbaf184ae2d380ad9abb1\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff-base apply {base \u03b9} = diff-base , apply\n  lift-diff-apply diff-base apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff-base apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n\n  lift-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff-base apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff apply {base \u03b9} = diff , apply\n  lift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n  lift-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fcfc151932c7801c0c8b94a5b4653842199a6f6c","subject":"Finish Greek Unicode 0370-03FF (partial).","message":"Finish Greek Unicode 0370-03FF (partial).\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-final \u2984 \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\ninstance \u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\ninstance \u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\ninstance \u03b9-diaeresis : \u03b9\u2032 diaeresis\n\u03b9-diaeresis = add-diaeresis here\n\n-- \u039f \u03bf\ninstance \u03bf-vowel : \u03bf\u2032 vowel\n\u03bf-vowel = is-vowel (there (there (there (there here))))\n\n\u03bf-always-short : \u03bf\u2032 always-short\n\u03bf-always-short = is-always-short (there here)\n\ninstance \u03bf-smooth : \u03bf\u2032 \u27e6 lower \u27e7-smooth\n\u03bf-smooth = add-smooth-lower-vowel\n\ninstance \u039f-smooth : \u03bf\u2032 \u27e6 upper \u27e7-smooth\n\u039f-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03bf-rough : \u03bf\u2032 with-rough\n\u03bf-rough = add-rough-vowel\n\n-- \u03a1 \u03c1\ninstance \u03c1-smooth : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n\u03c1-smooth = add-smooth-\u03c1\n\ninstance \u03c1-rough : \u03c1\u2032 with-rough\n\u03c1-rough = add-rough-\u03c1\n\n-- \u03a3 \u03c3\ninstance \u03c3-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\u03c3-final = make-final\n\n-- \u03a5 \u03c5\ninstance \u03c5-vowel : \u03c5\u2032 vowel\n\u03c5-vowel = is-vowel (there (there (there (there (there here)))))\n\n\u00ac\u03c5-always-short : \u00ac \u03c5\u2032 always-short\n\u00ac\u03c5-always-short (is-always-short (there (there ())))\n\ninstance \u03c5-long-vowel : \u03c5\u2032 long-vowel\n\u03c5-long-vowel = make-long-vowel \u00ac\u03c5-always-short\n\ninstance \u03c5-smooth : \u03c5\u2032 \u27e6 lower \u27e7-smooth\n\u03c5-smooth = add-smooth-lower-vowel\n\ninstance \u03c5-rough : \u03c5\u2032 with-rough\n\u03c5-rough = add-rough-vowel\n\ninstance \u03c5-diaeresis : \u03c5\u2032 diaeresis\n\u03c5-diaeresis = add-diaeresis (there here)\n\n-- \u03a9 \u03c9\ninstance \u03c9-vowel : \u03c9\u2032 vowel\n\u03c9-vowel = is-vowel (there (there (there (there (there (there here))))))\n\n\u00ac\u03c9-always-short : \u00ac \u03c9\u2032 always-short\n\u00ac\u03c9-always-short (is-always-short (there (there ())))\n\ninstance \u03c9-long-vowel : \u03c9\u2032 long-vowel\n\u03c9-long-vowel = make-long-vowel \u00ac\u03c9-always-short\n\ninstance \u03c9-smooth : \u03c9\u2032 \u27e6 lower \u27e7-smooth\n\u03c9-smooth = add-smooth-lower-vowel\n\ninstance \u03a9-smooth : \u03c9\u2032 \u27e6 upper \u27e7-smooth\n\u03a9-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03c9-rough : \u03c9\u2032 with-rough\n\u03c9-rough = add-rough-vowel\n\ninstance \u03c9-iota-subscript : \u03c9\u2032 iota-subscript\n\u03c9-iota-subscript = add-iota-subscript (there (there here))\n\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = with-accent-diaeresis acute\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = with-diaeresis\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = with-diaeresis\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = with-accent acute\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = with-accent acute\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = with-accent acute\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = with-accent acute\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = with-accent-diaeresis acute\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = final\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = with-diaeresis\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = with-diaeresis\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = with-accent acute\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = with-accent acute\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = with-accent acute\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota smooth\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota rough\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave smooth\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave rough\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute smooth\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute rough\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex smooth\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex rough\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota smooth\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota rough\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave smooth\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave rough\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute smooth\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute rough\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex smooth\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex rough\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a) = just (letter-to-accent a)\nget-accent (with-accent-iota a) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis with-diaeresis = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form final = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-final \u2984 \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\ninstance \u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota smooth\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota rough\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave smooth\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave rough\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute smooth\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute rough\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex smooth\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex rough\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota smooth\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota rough\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave smooth\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave rough\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute smooth\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute rough\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex smooth\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex rough\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a) = just (letter-to-accent a)\nget-accent (with-accent-iota a) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis with-diaeresis = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form final = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7b98f7162bfda73ebd29d9b3e00a96ffe8354b0c","subject":"Add the remainder of polytonic Greek Unicode.","message":"Add the remainder of polytonic Greek Unicode.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-final \u2984 \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\ninstance \u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\ninstance \u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\ninstance \u03b9-diaeresis : \u03b9\u2032 diaeresis\n\u03b9-diaeresis = add-diaeresis here\n\n-- \u039f \u03bf\ninstance \u03bf-vowel : \u03bf\u2032 vowel\n\u03bf-vowel = is-vowel (there (there (there (there here))))\n\n\u03bf-always-short : \u03bf\u2032 always-short\n\u03bf-always-short = is-always-short (there here)\n\ninstance \u03bf-smooth : \u03bf\u2032 \u27e6 lower \u27e7-smooth\n\u03bf-smooth = add-smooth-lower-vowel\n\ninstance \u039f-smooth : \u03bf\u2032 \u27e6 upper \u27e7-smooth\n\u039f-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03bf-rough : \u03bf\u2032 with-rough\n\u03bf-rough = add-rough-vowel\n\n-- \u03a1 \u03c1\ninstance \u03c1-smooth : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n\u03c1-smooth = add-smooth-\u03c1\n\ninstance \u03c1-rough : \u03c1\u2032 with-rough\n\u03c1-rough = add-rough-\u03c1\n\n-- \u03a3 \u03c3\ninstance \u03c3-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\u03c3-final = make-final\n\n-- \u03a5 \u03c5\ninstance \u03c5-vowel : \u03c5\u2032 vowel\n\u03c5-vowel = is-vowel (there (there (there (there (there here)))))\n\n\u00ac\u03c5-always-short : \u00ac \u03c5\u2032 always-short\n\u00ac\u03c5-always-short (is-always-short (there (there ())))\n\ninstance \u03c5-long-vowel : \u03c5\u2032 long-vowel\n\u03c5-long-vowel = make-long-vowel \u00ac\u03c5-always-short\n\ninstance \u03c5-smooth : \u03c5\u2032 \u27e6 lower \u27e7-smooth\n\u03c5-smooth = add-smooth-lower-vowel\n\ninstance \u03c5-rough : \u03c5\u2032 with-rough\n\u03c5-rough = add-rough-vowel\n\ninstance \u03c5-diaeresis : \u03c5\u2032 diaeresis\n\u03c5-diaeresis = add-diaeresis (there here)\n\n-- \u03a9 \u03c9\ninstance \u03c9-vowel : \u03c9\u2032 vowel\n\u03c9-vowel = is-vowel (there (there (there (there (there (there here))))))\n\n\u00ac\u03c9-always-short : \u00ac \u03c9\u2032 always-short\n\u00ac\u03c9-always-short (is-always-short (there (there ())))\n\ninstance \u03c9-long-vowel : \u03c9\u2032 long-vowel\n\u03c9-long-vowel = make-long-vowel \u00ac\u03c9-always-short\n\ninstance \u03c9-smooth : \u03c9\u2032 \u27e6 lower \u27e7-smooth\n\u03c9-smooth = add-smooth-lower-vowel\n\ninstance \u03a9-smooth : \u03c9\u2032 \u27e6 upper \u27e7-smooth\n\u03a9-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03c9-rough : \u03c9\u2032 with-rough\n\u03c9-rough = add-rough-vowel\n\ninstance \u03c9-iota-subscript : \u03c9\u2032 iota-subscript\n\u03c9-iota-subscript = add-iota-subscript (there (there here))\n\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = with-accent-diaeresis acute\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = with-diaeresis\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = with-diaeresis\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = with-accent acute\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = with-accent acute\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = with-accent acute\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = with-accent acute\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = with-accent-diaeresis acute\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = final\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = with-diaeresis\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = with-diaeresis\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = with-accent acute\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = with-accent acute\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = with-accent acute\n\n-- U+1F00 - U+1FFF\n\u1f00 : Token \u03b1\u2032 lower -- U+1F00  \u1f00 GREEK SMALL LETTER ALPHA WITH PSILI\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower -- U+1F01  \u1f01 GREEK SMALL LETTER ALPHA WITH DASIA\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower -- U+1F02  \u1f02 GREEK SMALL LETTER ALPHA WITH PSILI AND VARIA\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower -- U+1F03  \u1f03 GREEK SMALL LETTER ALPHA WITH DASIA AND VARIA\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower -- U+1F04  \u1f04 GREEK SMALL LETTER ALPHA WITH PSILI AND OXIA\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower -- U+1F05  \u1f05 GREEK SMALL LETTER ALPHA WITH DASIA AND OXIA\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower -- U+1F06  \u1f06 GREEK SMALL LETTER ALPHA WITH PSILI AND PERISPOMENI\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower -- U+1F07  \u1f07 GREEK SMALL LETTER ALPHA WITH DASIA AND PERISPOMENI\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper -- U+1F08  \u1f08 GREEK CAPITAL LETTER ALPHA WITH PSILI\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper -- U+1F09  \u1f09 GREEK CAPITAL LETTER ALPHA WITH DASIA\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper -- U+1F0A  \u1f0a GREEK CAPITAL LETTER ALPHA WITH PSILI AND VARIA\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper -- U+1F0B  \u1f0b GREEK CAPITAL LETTER ALPHA WITH DASIA AND VARIA\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper -- U+1F0C  \u1f0c GREEK CAPITAL LETTER ALPHA WITH PSILI AND OXIA\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper -- U+1F0D  \u1f0d GREEK CAPITAL LETTER ALPHA WITH DASIA AND OXIA\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper -- U+1F0E  \u1f0e GREEK CAPITAL LETTER ALPHA WITH PSILI AND PERISPOMENI\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper -- U+1F0F  \u1f0f GREEK CAPITAL LETTER ALPHA WITH DASIA AND PERISPOMENI\n\u1f0f = with-accent-breathing circumflex rough\n\u1f10 : Token \u03b5\u2032 lower -- U+1F10  \u1f10 GREEK SMALL LETTER EPSILON WITH PSILI\n\u1f10 = with-breathing smooth\n\u1f11 : Token \u03b5\u2032 lower -- U+1F11  \u1f11 GREEK SMALL LETTER EPSILON WITH DASIA\n\u1f11 = with-breathing rough\n\u1f12 : Token \u03b5\u2032 lower -- U+1F12  \u1f12 GREEK SMALL LETTER EPSILON WITH PSILI AND VARIA\n\u1f12 = with-accent-breathing grave smooth\n\u1f13 : Token \u03b5\u2032 lower -- U+1F13  \u1f13 GREEK SMALL LETTER EPSILON WITH DASIA AND VARIA\n\u1f13 = with-accent-breathing grave rough\n\u1f14 : Token \u03b5\u2032 lower -- U+1F14  \u1f14 GREEK SMALL LETTER EPSILON WITH PSILI AND OXIA\n\u1f14 = with-accent-breathing acute smooth\n\u1f15 : Token \u03b5\u2032 lower -- U+1F15  \u1f15 GREEK SMALL LETTER EPSILON WITH DASIA AND OXIA\n\u1f15 = with-accent-breathing acute rough\n-- U+1F16\n-- U+1F17\n\u1f18 : Token \u03b5\u2032 upper -- U+1F18  \u1f18 GREEK CAPITAL LETTER EPSILON WITH PSILI\n\u1f18 = with-breathing smooth\n\u1f19 : Token \u03b5\u2032 upper -- U+1F19  \u1f19 GREEK CAPITAL LETTER EPSILON WITH DASIA\n\u1f19 = with-breathing rough\n\u1f1a : Token \u03b5\u2032 upper -- U+1F1A  \u1f1a GREEK CAPITAL LETTER EPSILON WITH PSILI AND VARIA\n\u1f1a = with-accent-breathing grave smooth\n\u1f1b : Token \u03b5\u2032 upper -- U+1F1B  \u1f1b GREEK CAPITAL LETTER EPSILON WITH DASIA AND VARIA\n\u1f1b = with-accent-breathing grave rough\n\u1f1c : Token \u03b5\u2032 upper -- U+1F1C  \u1f1c GREEK CAPITAL LETTER EPSILON WITH PSILI AND OXIA\n\u1f1c = with-accent-breathing acute smooth\n\u1f1d : Token \u03b5\u2032 upper -- U+1F1D  \u1f1d GREEK CAPITAL LETTER EPSILON WITH DASIA AND OXIA\n\u1f1d = with-accent-breathing acute rough\n-- U+1F1E\n-- U+1F1F\n\u1f20 : Token \u03b7\u2032 lower -- U+1F20  \u1f20 GREEK SMALL LETTER ETA WITH PSILI\n\u1f20 = with-breathing smooth\n\u1f21 : Token \u03b7\u2032 lower -- U+1F21  \u1f21 GREEK SMALL LETTER ETA WITH DASIA\n\u1f21 = with-breathing rough\n\u1f22 : Token \u03b7\u2032 lower -- U+1F22  \u1f22 GREEK SMALL LETTER ETA WITH PSILI AND VARIA\n\u1f22 = with-accent-breathing grave smooth\n\u1f23 : Token \u03b7\u2032 lower -- U+1F23  \u1f23 GREEK SMALL LETTER ETA WITH DASIA AND VARIA\n\u1f23 = with-accent-breathing grave rough\n\u1f24 : Token \u03b7\u2032 lower -- U+1F24  \u1f24 GREEK SMALL LETTER ETA WITH PSILI AND OXIA\n\u1f24 = with-accent-breathing acute smooth\n\u1f25 : Token \u03b7\u2032 lower -- U+1F25  \u1f25 GREEK SMALL LETTER ETA WITH DASIA AND OXIA\n\u1f25 = with-accent-breathing acute rough\n\u1f26 : Token \u03b7\u2032 lower -- U+1F26  \u1f26 GREEK SMALL LETTER ETA WITH PSILI AND PERISPOMENI\n\u1f26 = with-accent-breathing circumflex smooth\n\u1f27 : Token \u03b7\u2032 lower -- U+1F27  \u1f27 GREEK SMALL LETTER ETA WITH DASIA AND PERISPOMENI\n\u1f27 = with-accent-breathing circumflex rough\n\u1f28 : Token \u03b7\u2032 upper -- U+1F28  \u1f28 GREEK CAPITAL LETTER ETA WITH PSILI\n\u1f28 = with-breathing smooth\n\u1f29 : Token \u03b7\u2032 upper -- U+1F29  \u1f29 GREEK CAPITAL LETTER ETA WITH DASIA\n\u1f29 = with-breathing rough\n\u1f2a : Token \u03b7\u2032 upper -- U+1F2A  \u1f2a GREEK CAPITAL LETTER ETA WITH PSILI AND VARIA\n\u1f2a = with-accent-breathing grave smooth\n\u1f2b : Token \u03b7\u2032 upper -- U+1F2B  \u1f2b GREEK CAPITAL LETTER ETA WITH DASIA AND VARIA\n\u1f2b = with-accent-breathing grave rough\n\u1f2c : Token \u03b7\u2032 upper -- U+1F2C  \u1f2c GREEK CAPITAL LETTER ETA WITH PSILI AND OXIA\n\u1f2c = with-accent-breathing acute smooth\n\u1f2d : Token \u03b7\u2032 upper -- U+1F2D  \u1f2d GREEK CAPITAL LETTER ETA WITH DASIA AND OXIA\n\u1f2d = with-accent-breathing acute rough\n\u1f2e : Token \u03b7\u2032 upper -- U+1F2E  \u1f2e GREEK CAPITAL LETTER ETA WITH PSILI AND PERISPOMENI\n\u1f2e = with-accent-breathing circumflex smooth\n\u1f2f : Token \u03b7\u2032 upper -- U+1F2F  \u1f2f GREEK CAPITAL LETTER ETA WITH DASIA AND PERISPOMENI\n\u1f2f = with-accent-breathing circumflex rough\n\u1f30 : Token \u03b9\u2032 lower -- U+1F30  \u1f30 GREEK SMALL LETTER IOTA WITH PSILI\n\u1f30 = with-breathing smooth\n\u1f31 : Token \u03b9\u2032 lower -- U+1F31  \u1f31 GREEK SMALL LETTER IOTA WITH DASIA\n\u1f31 = with-breathing rough\n\u1f32 : Token \u03b9\u2032 lower -- U+1F32  \u1f32 GREEK SMALL LETTER IOTA WITH PSILI AND VARIA\n\u1f32 = with-accent-breathing grave smooth\n\u1f33 : Token \u03b9\u2032 lower -- U+1F33  \u1f33 GREEK SMALL LETTER IOTA WITH DASIA AND VARIA\n\u1f33 = with-accent-breathing grave rough\n\u1f34 : Token \u03b9\u2032 lower -- U+1F34  \u1f34 GREEK SMALL LETTER IOTA WITH PSILI AND OXIA\n\u1f34 = with-accent-breathing acute smooth\n\u1f35 : Token \u03b9\u2032 lower -- U+1F35  \u1f35 GREEK SMALL LETTER IOTA WITH DASIA AND OXIA\n\u1f35 = with-accent-breathing acute rough\n\u1f36 : Token \u03b9\u2032 lower -- U+1F36  \u1f36 GREEK SMALL LETTER IOTA WITH PSILI AND PERISPOMENI\n\u1f36 = with-accent-breathing circumflex smooth\n\u1f37 : Token \u03b9\u2032 lower -- U+1F37  \u1f37 GREEK SMALL LETTER IOTA WITH DASIA AND PERISPOMENI\n\u1f37 = with-accent-breathing circumflex rough\n\u1f38 : Token \u03b9\u2032 upper -- U+1F38  \u1f38 GREEK CAPITAL LETTER IOTA WITH PSILI\n\u1f38 = with-breathing smooth\n\u1f39 : Token \u03b9\u2032 upper -- U+1F39  \u1f39 GREEK CAPITAL LETTER IOTA WITH DASIA\n\u1f39 = with-breathing rough\n\u1f3a : Token \u03b9\u2032 upper -- U+1F3A  \u1f3a GREEK CAPITAL LETTER IOTA WITH PSILI AND VARIA\n\u1f3a = with-accent-breathing grave smooth\n\u1f3b : Token \u03b9\u2032 upper -- U+1F3B  \u1f3b GREEK CAPITAL LETTER IOTA WITH DASIA AND VARIA\n\u1f3b = with-accent-breathing grave rough\n\u1f3c : Token \u03b9\u2032 upper -- U+1F3C  \u1f3c GREEK CAPITAL LETTER IOTA WITH PSILI AND OXIA\n\u1f3c = with-accent-breathing acute smooth\n\u1f3d : Token \u03b9\u2032 upper -- U+1F3D  \u1f3d GREEK CAPITAL LETTER IOTA WITH DASIA AND OXIA\n\u1f3d = with-accent-breathing acute rough\n\u1f3e : Token \u03b9\u2032 upper -- U+1F3E  \u1f3e GREEK CAPITAL LETTER IOTA WITH PSILI AND PERISPOMENI\n\u1f3e = with-accent-breathing circumflex smooth\n\u1f3f : Token \u03b9\u2032 upper -- U+1F3F  \u1f3f GREEK CAPITAL LETTER IOTA WITH DASIA AND PERISPOMENI\n\u1f3f = with-accent-breathing circumflex rough\n\u1f40 : Token \u03bf\u2032 lower -- U+1F40  \u1f40 GREEK SMALL LETTER OMICRON WITH PSILI\n\u1f40 = with-breathing smooth\n\u1f41 : Token \u03bf\u2032 lower -- U+1F41  \u1f41 GREEK SMALL LETTER OMICRON WITH DASIA\n\u1f41 = with-breathing rough\n\u1f42 : Token \u03bf\u2032 lower -- U+1F42  \u1f42 GREEK SMALL LETTER OMICRON WITH PSILI AND VARIA\n\u1f42 = with-accent-breathing grave smooth\n\u1f43 : Token \u03bf\u2032 lower -- U+1F43  \u1f43 GREEK SMALL LETTER OMICRON WITH DASIA AND VARIA\n\u1f43 = with-accent-breathing grave rough\n\u1f44 : Token \u03bf\u2032 lower -- U+1F44  \u1f44 GREEK SMALL LETTER OMICRON WITH PSILI AND OXIA\n\u1f44 = with-accent-breathing acute smooth\n\u1f45 : Token \u03bf\u2032 lower -- U+1F45  \u1f45 GREEK SMALL LETTER OMICRON WITH DASIA AND OXIA\n\u1f45 = with-accent-breathing acute rough\n-- U+1F46\n-- U+1F47\n\u1f48 : Token \u03bf\u2032 upper -- U+1F48  \u1f48 GREEK CAPITAL LETTER OMICRON WITH PSILI\n\u1f48 = with-breathing smooth\n\u1f49 : Token \u03bf\u2032 upper -- U+1F49  \u1f49 GREEK CAPITAL LETTER OMICRON WITH DASIA\n\u1f49 = with-breathing rough\n\u1f4a : Token \u03bf\u2032 upper -- U+1F4A  \u1f4a GREEK CAPITAL LETTER OMICRON WITH PSILI AND VARIA\n\u1f4a = with-accent-breathing grave smooth\n\u1f4b : Token \u03bf\u2032 upper -- U+1F4B  \u1f4b GREEK CAPITAL LETTER OMICRON WITH DASIA AND VARIA\n\u1f4b = with-accent-breathing grave rough\n\u1f4c : Token \u03bf\u2032 upper -- U+1F4C  \u1f4c GREEK CAPITAL LETTER OMICRON WITH PSILI AND OXIA\n\u1f4c = with-accent-breathing acute smooth\n\u1f4d : Token \u03bf\u2032 upper -- U+1F4D  \u1f4d GREEK CAPITAL LETTER OMICRON WITH DASIA AND OXIA\n\u1f4d = with-accent-breathing acute rough\n-- U+1F4E\n-- U+1F4F\n\u1f50 : Token \u03c5\u2032 lower -- U+1F50  \u1f50 GREEK SMALL LETTER UPSILON WITH PSILI\n\u1f50 = with-breathing smooth\n\u1f51 : Token \u03c5\u2032 lower -- U+1F51  \u1f51 GREEK SMALL LETTER UPSILON WITH DASIA\n\u1f51 = with-breathing rough\n\u1f52 : Token \u03c5\u2032 lower -- U+1F52  \u1f52 GREEK SMALL LETTER UPSILON WITH PSILI AND VARIA\n\u1f52 = with-accent-breathing grave smooth\n\u1f53 : Token \u03c5\u2032 lower -- U+1F53  \u1f53 GREEK SMALL LETTER UPSILON WITH DASIA AND VARIA\n\u1f53 = with-accent-breathing grave rough\n\u1f54 : Token \u03c5\u2032 lower -- U+1F54  \u1f54 GREEK SMALL LETTER UPSILON WITH PSILI AND OXIA\n\u1f54 = with-accent-breathing acute smooth\n\u1f55 : Token \u03c5\u2032 lower -- U+1F55  \u1f55 GREEK SMALL LETTER UPSILON WITH DASIA AND OXIA\n\u1f55 = with-accent-breathing acute rough\n\u1f56 : Token \u03c5\u2032 lower -- U+1F56  \u1f56 GREEK SMALL LETTER UPSILON WITH PSILI AND PERISPOMENI\n\u1f56 = with-accent-breathing circumflex smooth\n\u1f57 : Token \u03c5\u2032 lower -- U+1F57  \u1f57 GREEK SMALL LETTER UPSILON WITH DASIA AND PERISPOMENI\n\u1f57 = with-accent-breathing circumflex rough\n-- U+1F58\n\u1f59 : Token \u03c5\u2032 upper -- U+1F59  \u1f59 GREEK CAPITAL LETTER UPSILON WITH DASIA\n\u1f59 = with-breathing rough\n-- U+1F5A\n\u1f5b : Token \u03c5\u2032 upper -- U+1F5B  \u1f5b GREEK CAPITAL LETTER UPSILON WITH DASIA AND VARIA\n\u1f5b = with-accent-breathing grave rough\n-- U+1F5C\n\u1f5d : Token \u03c5\u2032 upper -- U+1F5D  \u1f5d GREEK CAPITAL LETTER UPSILON WITH DASIA AND OXIA\n\u1f5d = with-accent-breathing acute rough\n-- U+1F5E\n\u1f5f : Token \u03c5\u2032 upper -- U+1F5F  \u1f5f GREEK CAPITAL LETTER UPSILON WITH DASIA AND PERISPOMENI\n\u1f5f = with-accent-breathing circumflex rough\n\u1f60 : Token \u03c9\u2032 lower -- U+1F60  \u1f60 GREEK SMALL LETTER OMEGA WITH PSILI\n\u1f60 = with-breathing smooth\n\u1f61 : Token \u03c9\u2032 lower -- U+1F61  \u1f61 GREEK SMALL LETTER OMEGA WITH DASIA\n\u1f61 = with-breathing rough\n\u1f62 : Token \u03c9\u2032 lower -- U+1F62  \u1f62 GREEK SMALL LETTER OMEGA WITH PSILI AND VARIA\n\u1f62 = with-accent-breathing grave smooth\n\u1f63 : Token \u03c9\u2032 lower -- U+1F63  \u1f63 GREEK SMALL LETTER OMEGA WITH DASIA AND VARIA\n\u1f63 = with-accent-breathing grave rough\n\u1f64 : Token \u03c9\u2032 lower -- U+1F64  \u1f64 GREEK SMALL LETTER OMEGA WITH PSILI AND OXIA\n\u1f64 = with-accent-breathing acute smooth\n\u1f65 : Token \u03c9\u2032 lower -- U+1F65  \u1f65 GREEK SMALL LETTER OMEGA WITH DASIA AND OXIA\n\u1f65 = with-accent-breathing acute rough\n\u1f66 : Token \u03c9\u2032 lower -- U+1F66  \u1f66 GREEK SMALL LETTER OMEGA WITH PSILI AND PERISPOMENI\n\u1f66 = with-accent-breathing circumflex smooth\n\u1f67 : Token \u03c9\u2032 lower -- U+1F67  \u1f67 GREEK SMALL LETTER OMEGA WITH DASIA AND PERISPOMENI\n\u1f67 = with-accent-breathing circumflex rough\n\u1f68 : Token \u03c9\u2032 upper -- U+1F68  \u1f68 GREEK CAPITAL LETTER OMEGA WITH PSILI\n\u1f68 = with-breathing smooth\n\u1f69 : Token \u03c9\u2032 upper -- U+1F69  \u1f69 GREEK CAPITAL LETTER OMEGA WITH DASIA\n\u1f69 = with-breathing rough\n\u1f6a : Token \u03c9\u2032 upper -- U+1F6A  \u1f6a GREEK CAPITAL LETTER OMEGA WITH PSILI AND VARIA\n\u1f6a = with-accent-breathing grave smooth\n\u1f6b : Token \u03c9\u2032 upper -- U+1F6B  \u1f6b GREEK CAPITAL LETTER OMEGA WITH DASIA AND VARIA\n\u1f6b = with-accent-breathing grave rough\n\u1f6c : Token \u03c9\u2032 upper -- U+1F6C  \u1f6c GREEK CAPITAL LETTER OMEGA WITH PSILI AND OXIA\n\u1f6c = with-accent-breathing acute smooth\n\u1f6d : Token \u03c9\u2032 upper -- U+1F6D  \u1f6d GREEK CAPITAL LETTER OMEGA WITH DASIA AND OXIA\n\u1f6d = with-accent-breathing acute rough\n\u1f6e : Token \u03c9\u2032 upper -- U+1F6E  \u1f6e GREEK CAPITAL LETTER OMEGA WITH PSILI AND PERISPOMENI\n\u1f6e = with-accent-breathing circumflex smooth\n\u1f6f : Token \u03c9\u2032 upper -- U+1F6F  \u1f6f GREEK CAPITAL LETTER OMEGA WITH DASIA AND PERISPOMENI\n\u1f6f = with-accent-breathing circumflex rough\n\u1f70 : Token \u03b1\u2032 lower -- U+1F70  \u1f70 GREEK SMALL LETTER ALPHA WITH VARIA\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower -- U+1F71  \u1f71 GREEK SMALL LETTER ALPHA WITH OXIA\n\u1f71 = with-accent acute\n\u1f72 : Token \u03b5\u2032 lower -- U+1F72  \u1f72 GREEK SMALL LETTER EPSILON WITH VARIA\n\u1f72 = with-accent grave\n\u1f73 : Token \u03b5\u2032 lower -- U+1F73  \u1f73 GREEK SMALL LETTER EPSILON WITH OXIA\n\u1f73 = with-accent acute\n\u1f74 : Token \u03b7\u2032 lower -- U+1F74  \u1f74 GREEK SMALL LETTER ETA WITH VARIA\n\u1f74 = with-accent grave\n\u1f75 : Token \u03b7\u2032 lower -- U+1F75  \u1f75 GREEK SMALL LETTER ETA WITH OXIA\n\u1f75 = with-accent acute\n\u1f76 : Token \u03b9\u2032 lower -- U+1F76  \u1f76 GREEK SMALL LETTER IOTA WITH VARIA\n\u1f76 = with-accent grave\n\u1f77 : Token \u03b9\u2032 lower -- U+1F77  \u1f77 GREEK SMALL LETTER IOTA WITH OXIA\n\u1f77 = with-accent acute\n\u1f78 : Token \u03bf\u2032 lower -- U+1F78  \u1f78 GREEK SMALL LETTER OMICRON WITH VARIA\n\u1f78 = with-accent grave\n\u1f79 : Token \u03bf\u2032 lower -- U+1F79  \u1f79 GREEK SMALL LETTER OMICRON WITH OXIA\n\u1f79 = with-accent acute\n\u1f7a : Token \u03c5\u2032 lower -- U+1F7A  \u1f7a GREEK SMALL LETTER UPSILON WITH VARIA\n\u1f7a = with-accent grave\n\u1f7b : Token \u03c5\u2032 lower -- U+1F7B  \u1f7b GREEK SMALL LETTER UPSILON WITH OXIA\n\u1f7b = with-accent acute\n\u1f7c : Token \u03c9\u2032 lower -- U+1F7C  \u1f7c GREEK SMALL LETTER OMEGA WITH VARIA\n\u1f7c = with-accent grave\n\u1f7d : Token \u03c9\u2032 lower -- U+1F7D  \u1f7d GREEK SMALL LETTER OMEGA WITH OXIA\n\u1f7d = with-accent acute\n-- U+1F7E\n-- U+1F7F\n\u1f80 : Token \u03b1\u2032 lower -- U+1F80  \u1f80 GREEK SMALL LETTER ALPHA WITH PSILI AND YPOGEGRAMMENI\n\u1f80 = with-breathing-iota smooth\n\u1f81 : Token \u03b1\u2032 lower -- U+1F81  \u1f81 GREEK SMALL LETTER ALPHA WITH DASIA AND YPOGEGRAMMENI\n\u1f81 = with-breathing-iota rough\n\u1f82 : Token \u03b1\u2032 lower -- U+1F82  \u1f82 GREEK SMALL LETTER ALPHA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1f82 = with-accent-breathing-iota grave smooth\n\u1f83 : Token \u03b1\u2032 lower -- U+1F83  \u1f83 GREEK SMALL LETTER ALPHA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1f83 = with-accent-breathing grave rough\n\u1f84 : Token \u03b1\u2032 lower -- U+1F84  \u1f84 GREEK SMALL LETTER ALPHA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1f84 = with-accent-breathing-iota acute smooth\n\u1f85 : Token \u03b1\u2032 lower -- U+1F85  \u1f85 GREEK SMALL LETTER ALPHA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1f85 = with-accent-breathing-iota acute rough\n\u1f86 : Token \u03b1\u2032 lower -- U+1F86  \u1f86 GREEK SMALL LETTER ALPHA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f86 = with-accent-breathing-iota circumflex smooth\n\u1f87 : Token \u03b1\u2032 lower -- U+1F87  \u1f87 GREEK SMALL LETTER ALPHA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f87 = with-accent-breathing-iota circumflex rough\n\u1f88 : Token \u03b1\u2032 upper -- U+1F88  \u1f88 GREEK CAPITAL LETTER ALPHA WITH PSILI AND PROSGEGRAMMENI\n\u1f88 = with-breathing-iota smooth\n\u1f89 : Token \u03b1\u2032 upper -- U+1F89  \u1f89 GREEK CAPITAL LETTER ALPHA WITH DASIA AND PROSGEGRAMMENI\n\u1f89 = with-breathing-iota rough\n\u1f8a : Token \u03b1\u2032 upper -- U+1F8A  \u1f8a GREEK CAPITAL LETTER ALPHA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1f8a = with-accent-breathing-iota grave smooth\n\u1f8b : Token \u03b1\u2032 upper -- U+1F8B  \u1f8b GREEK CAPITAL LETTER ALPHA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1f8b = with-accent-breathing-iota grave rough\n\u1f8c : Token \u03b1\u2032 upper -- U+1F8C  \u1f8c GREEK CAPITAL LETTER ALPHA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1f8c = with-accent-breathing-iota acute smooth\n\u1f8d : Token \u03b1\u2032 upper -- U+1F8D  \u1f8d GREEK CAPITAL LETTER ALPHA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1f8d = with-accent-breathing-iota acute rough\n\u1f8e : Token \u03b1\u2032 upper -- U+1F8E  \u1f8e GREEK CAPITAL LETTER ALPHA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f8e = with-accent-breathing-iota circumflex smooth\n\u1f8f : Token \u03b1\u2032 upper -- U+1F8F  \u1f8f GREEK CAPITAL LETTER ALPHA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f8f = with-accent-breathing-iota circumflex rough\n\u1f90 : Token \u03b7\u2032 lower -- U+1F90  \u1f90 GREEK SMALL LETTER ETA WITH PSILI AND YPOGEGRAMMENI\n\u1f90 = with-breathing-iota smooth\n\u1f91 : Token \u03b7\u2032 lower -- U+1F91  \u1f91 GREEK SMALL LETTER ETA WITH DASIA AND YPOGEGRAMMENI\n\u1f91 = with-breathing-iota rough\n\u1f92 : Token \u03b7\u2032 lower -- U+1F92  \u1f92 GREEK SMALL LETTER ETA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1f92 = with-accent-breathing-iota grave smooth\n\u1f93 : Token \u03b7\u2032 lower -- U+1F93  \u1f93 GREEK SMALL LETTER ETA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1f93 = with-accent-breathing-iota grave rough\n\u1f94 : Token \u03b7\u2032 lower -- U+1F94  \u1f94 GREEK SMALL LETTER ETA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1f94 = with-accent-breathing-iota acute smooth\n\u1f95 : Token \u03b7\u2032 lower -- U+1F95  \u1f95 GREEK SMALL LETTER ETA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1f95 = with-accent-breathing-iota acute rough\n\u1f96 : Token \u03b7\u2032 lower -- U+1F96  \u1f96 GREEK SMALL LETTER ETA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f96 = with-accent-breathing-iota circumflex smooth\n\u1f97 : Token \u03b7\u2032 lower -- U+1F97  \u1f97 GREEK SMALL LETTER ETA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f97 = with-accent-breathing-iota circumflex rough\n\u1f98 : Token \u03b7\u2032 upper -- U+1F98  \u1f98 GREEK CAPITAL LETTER ETA WITH PSILI AND PROSGEGRAMMENI\n\u1f98 = with-breathing-iota smooth\n\u1f99 : Token \u03b7\u2032 upper -- U+1F99  \u1f99 GREEK CAPITAL LETTER ETA WITH DASIA AND PROSGEGRAMMENI\n\u1f99 = with-breathing-iota rough\n\u1f9a : Token \u03b7\u2032 upper -- U+1F9A  \u1f9a GREEK CAPITAL LETTER ETA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1f9a = with-accent-breathing-iota grave smooth\n\u1f9b : Token \u03b7\u2032 upper -- U+1F9B  \u1f9b GREEK CAPITAL LETTER ETA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1f9b = with-accent-breathing-iota grave rough\n\u1f9c : Token \u03b7\u2032 upper -- U+1F9C  \u1f9c GREEK CAPITAL LETTER ETA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1f9c = with-accent-breathing-iota acute smooth\n\u1f9d : Token \u03b7\u2032 upper -- U+1F9D  \u1f9d GREEK CAPITAL LETTER ETA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1f9d = with-accent-breathing-iota acute rough\n\u1f9e : Token \u03b7\u2032 upper -- U+1F9E  \u1f9e GREEK CAPITAL LETTER ETA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f9e = with-accent-breathing-iota circumflex smooth\n\u1f9f : Token \u03b7\u2032 upper -- U+1F9F  \u1f9f GREEK CAPITAL LETTER ETA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f9f = with-accent-breathing-iota circumflex rough\n\u1fa0 : Token \u03c9\u2032 lower -- U+1FA0  \u1fa0 GREEK SMALL LETTER OMEGA WITH PSILI AND YPOGEGRAMMENI\n\u1fa0 = with-breathing-iota smooth\n\u1fa1 : Token \u03c9\u2032 lower -- U+1FA1  \u1fa1 GREEK SMALL LETTER OMEGA WITH DASIA AND YPOGEGRAMMENI\n\u1fa1 = with-breathing-iota rough\n\u1fa2 : Token \u03c9\u2032 lower -- U+1FA2  \u1fa2 GREEK SMALL LETTER OMEGA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1fa2 = with-accent-breathing-iota grave smooth\n\u1fa3 : Token \u03c9\u2032 lower -- U+1FA3  \u1fa3 GREEK SMALL LETTER OMEGA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1fa3 = with-accent-breathing-iota grave rough\n\u1fa4 : Token \u03c9\u2032 lower -- U+1FA4  \u1fa4 GREEK SMALL LETTER OMEGA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1fa4 = with-accent-breathing-iota acute smooth\n\u1fa5 : Token \u03c9\u2032 lower -- U+1FA5  \u1fa5 GREEK SMALL LETTER OMEGA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1fa5 = with-accent-breathing-iota acute rough\n\u1fa6 : Token \u03c9\u2032 lower -- U+1FA6  \u1fa6 GREEK SMALL LETTER OMEGA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1fa6 = with-accent-breathing-iota circumflex smooth\n\u1fa7 : Token \u03c9\u2032 lower -- U+1FA7  \u1fa7 GREEK SMALL LETTER OMEGA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1fa7 = with-accent-breathing-iota circumflex rough\n\u1fa8 : Token \u03c9\u2032 upper -- U+1FA8  \u1fa8 GREEK CAPITAL LETTER OMEGA WITH PSILI AND PROSGEGRAMMENI\n\u1fa8 = with-breathing-iota smooth\n\u1fa9 : Token \u03c9\u2032 upper -- U+1FA9  \u1fa9 GREEK CAPITAL LETTER OMEGA WITH DASIA AND PROSGEGRAMMENI\n\u1fa9 = with-breathing-iota rough\n\u1faa : Token \u03c9\u2032 upper -- U+1FAA  \u1faa GREEK CAPITAL LETTER OMEGA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1faa = with-accent-breathing-iota grave smooth\n\u1fab : Token \u03c9\u2032 upper -- U+1FAB  \u1fab GREEK CAPITAL LETTER OMEGA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1fab = with-accent-breathing-iota grave rough\n\u1fac : Token \u03c9\u2032 upper -- U+1FAC  \u1fac GREEK CAPITAL LETTER OMEGA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1fac = with-accent-breathing-iota acute smooth\n\u1fad : Token \u03c9\u2032 upper -- U+1FAD  \u1fad GREEK CAPITAL LETTER OMEGA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1fad = with-accent-breathing-iota acute rough\n\u1fae : Token \u03c9\u2032 upper -- U+1FAE  \u1fae GREEK CAPITAL LETTER OMEGA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1fae = with-accent-breathing-iota circumflex smooth\n\u1faf : Token \u03c9\u2032 upper -- U+1FAF  \u1faf GREEK CAPITAL LETTER OMEGA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1faf = with-accent-breathing-iota circumflex rough\n-- U+1FB0 \u1fb0 GREEK SMALL LETTER ALPHA WITH VRACHY\n-- U+1FB1 \u1fb1 GREEK SMALL LETTER ALPHA WITH MACRON\n\u1fb2 : Token \u03b1\u2032 lower -- U+1FB2  \u1fb2 GREEK SMALL LETTER ALPHA WITH VARIA AND YPOGEGRAMMENI\n\u1fb2 = with-accent-iota grave\n\u1fb3 : Token \u03b1\u2032 lower -- U+1FB3  \u1fb3 GREEK SMALL LETTER ALPHA WITH YPOGEGRAMMENI\n\u1fb3 = with-iota\n\u1fb4 : Token \u03b1\u2032 lower -- U+1FB4  \u1fb4 GREEK SMALL LETTER ALPHA WITH OXIA AND YPOGEGRAMMENI\n\u1fb4 = with-accent-iota acute\n-- U+1FB5\n\u1fb6 : Token \u03b1\u2032 lower -- U+1FB6  \u1fb6 GREEK SMALL LETTER ALPHA WITH PERISPOMENI\n\u1fb6 = with-accent-iota circumflex\n\u1fb7 : Token \u03b1\u2032 lower -- U+1FB7  \u1fb7 GREEK SMALL LETTER ALPHA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1fb7 = with-accent-iota circumflex\n-- U+1FB8 \u1fb8 GREEK CAPITAL LETTER ALPHA WITH VRACHY\n-- U+1FB9 \u1fb9 GREEK CAPITAL LETTER ALPHA WITH MACRON\n\u1fba : Token \u03b1\u2032 upper -- U+1FBA  \u1fba GREEK CAPITAL LETTER ALPHA WITH VARIA\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper -- U+1FBB  \u1fbb GREEK CAPITAL LETTER ALPHA WITH OXIA\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper -- U+1FBC  \u1fbc GREEK CAPITAL LETTER ALPHA WITH PROSGEGRAMMENI\n\u1fbc = with-iota\n-- U+1FBD  \u1fbd  GREEK KORONIS\n-- U+1FBE  \u1fbe  GREEK PROSGEGRAMMENI\n-- U+1FBF  \u1fbf  GREEK PSILI\n-- U+1FC0  \u1fc0  GREEK PERISPOMENI\n-- U+1FC1  \u1fc1  GREEK DIALYTIKA AND PERISPOMENI\n\u1fc2 : Token \u03b7\u2032 lower -- U+1FC2  \u1fc2 GREEK SMALL LETTER ETA WITH VARIA AND YPOGEGRAMMENI accent circumflex\n\u1fc2 = with-accent-iota grave\n\u1fc3 : Token \u03b7\u2032 lower -- U+1FC3  \u1fc3 GREEK SMALL LETTER ETA WITH YPOGEGRAMMENI\n\u1fc3 = with-iota\n\u1fc4 : Token \u03b7\u2032 lower -- U+1FC4  \u1fc4 GREEK SMALL LETTER ETA WITH OXIA AND YPOGEGRAMMENI\n\u1fc4 = with-accent-iota acute\n-- U+1FC5\n\u1fc6 : Token \u03b7\u2032 lower -- U+1FC6  \u1fc6 GREEK SMALL LETTER ETA WITH PERISPOMENI\n\u1fc6 = with-accent circumflex\n\u1fc7 : Token \u03b7\u2032 lower -- U+1FC7  \u1fc7 GREEK SMALL LETTER ETA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1fc7 = with-accent-iota circumflex\n\u1fc8 : Token \u03b5\u2032 upper -- U+1FC8  \u1fc8 GREEK CAPITAL LETTER EPSILON WITH VARIA\n\u1fc8 = with-accent grave\n\u1fc9 : Token \u03b5\u2032 upper -- U+1FC9  \u1fc9 GREEK CAPITAL LETTER EPSILON WITH OXIA\n\u1fc9 = with-accent acute\n\u1fca : Token \u03b7\u2032 upper -- U+1FCA  \u1fca GREEK CAPITAL LETTER ETA WITH VARIA\n\u1fca = with-accent grave\n\u1fcb : Token \u03b7\u2032 upper -- U+1FCB  \u1fcb GREEK CAPITAL LETTER ETA WITH OXIA\n\u1fcb = with-accent acute\n\u1fcc : Token \u03b7\u2032 upper -- U+1FCC  \u1fcc GREEK CAPITAL LETTER ETA WITH PROSGEGRAMMENI\n\u1fcc = with-iota\n-- U+1FCD  \u1fcd GREEK PSILI AND VARIA\n-- U+1FCE  \u1fce GREEK PSILI AND OXIA\n-- U+1FCF  \u1fcf GREEK PSILI AND PERISPOMENI\n-- U+1FD0  \u1fd0 GREEK SMALL LETTER IOTA WITH VRACHY\n-- U+1FD1  \u1fd1 GREEK SMALL LETTER IOTA WITH MACRON\n\u1fd2 : Token \u03b9\u2032 lower -- U+1FD2  \u1fd2 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND VARIA\n\u1fd2 = with-accent-diaeresis grave\n\u1fd3 : Token \u03b9\u2032 lower -- U+1FD3  \u1fd3 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND OXIA\n\u1fd3 = with-accent-diaeresis acute\n-- U+1FD4\n-- U+1FD5\n\u1fd6 : Token \u03b9\u2032 lower -- U+1FD6  \u1fd6 GREEK SMALL LETTER IOTA WITH PERISPOMENI\n\u1fd6 = with-accent circumflex\n\u1fd7 : Token \u03b9\u2032 lower -- U+1FD7  \u1fd7 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND PERISPOMENI\n\u1fd7 = with-accent-diaeresis circumflex\n-- U+1FD8 \u1fd8 GREEK CAPITAL LETTER IOTA WITH VRACHY\n-- U+1FD9 \u1fd9 GREEK CAPITAL LETTER IOTA WITH MACRON\n\u1fda : Token \u03b9\u2032 upper -- U+1FDA  \u1fda GREEK CAPITAL LETTER IOTA WITH VARIA\n\u1fda = with-accent grave\n\u1fdb : Token \u03b9\u2032 upper -- U+1FDB  \u1fdb GREEK CAPITAL LETTER IOTA WITH OXIA\n\u1fdb = with-accent acute\n-- U+1FDC\n-- U+1FDD  \u1fdd GREEK DASIA AND VARIA\n-- U+1FDE  \u1fde GREEK DASIA AND OXIA\n-- U+1FDF  \u1fdf GREEK DASIA AND PERISPOMENI\n-- U+1FE0 \u1fe0 GREEK SMALL LETTER UPSILON WITH VRACHY\n-- U+1FE1 \u1fe1 GREEK SMALL LETTER UPSILON WITH MACRON\n\u1fe2 : Token \u03c5\u2032 lower -- U+1FE2  \u1fe2 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND VARIA\n\u1fe2 = with-accent-diaeresis grave\n\u1fe3 : Token \u03c5\u2032 lower -- U+1FE3  \u1fe3 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND OXIA\n\u1fe3 = with-accent-diaeresis acute\n\u1fe4 : Token \u03c1\u2032 lower -- U+1FE4 \u1fe4 GREEK SMALL LETTER RHO WITH PSILI\n\u1fe4 = with-breathing smooth\n\u1fe5 : Token \u03c1\u2032 lower -- U+1FE5 \u1fe5 GREEK SMALL LETTER RHO WITH DASIA\n\u1fe5 = with-breathing rough\n\u1fe6 : Token \u03c5\u2032 lower -- U+1FE6  \u1fe6 GREEK SMALL LETTER UPSILON WITH PERISPOMENI\n\u1fe6 = with-accent circumflex\n\u1fe7 : Token \u03c5\u2032 lower -- U+1FE7  \u1fe7 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND PERISPOMENI\n\u1fe7 = with-accent-diaeresis circumflex\n-- U+1FE8 \u1fe8 GREEK CAPITAL LETTER UPSILON WITH VRACHY\n-- U+1FE9 \u1fe9 GREEK CAPITAL LETTER UPSILON WITH MACRON\n\u1fea : Token \u03c5\u2032 upper -- U+1FEA  \u1fea GREEK CAPITAL LETTER UPSILON WITH VARIA\n\u1fea = with-accent grave\n\u1feb : Token \u03c5\u2032 upper -- U+1FEB  \u1feb GREEK CAPITAL LETTER UPSILON WITH OXIA\n\u1feb = with-accent acute\n\u1fec : Token \u03c1\u2032 upper -- U+1FEC \u1fec GREEK CAPITAL LETTER RHO WITH DASIA\n\u1fec = with-breathing rough\n-- U+1FED  \u1fed GREEK DIALYTIKA AND VARIA\n-- U+1FEE  \u1fee GREEK DIALYTIKA AND OXIA\n-- U+1FEF  \u1fef GREEK VARIA\n-- U+1FF0\n-- U+1FF1\n\u1ff2 : Token \u03c9\u2032 lower -- U+1FF2  \u1ff2 GREEK SMALL LETTER OMEGA WITH VARIA AND YPOGEGRAMMENI\n\u1ff2 = with-accent-iota grave\n\u1ff3 : Token \u03c9\u2032 lower -- U+1FF3  \u1ff3 GREEK SMALL LETTER OMEGA WITH YPOGEGRAMMENI\n\u1ff3 = with-iota\n\u1ff4 : Token \u03c9\u2032 lower -- U+1FF4  \u1ff4 GREEK SMALL LETTER OMEGA WITH OXIA AND YPOGEGRAMMENI\n\u1ff4 = with-accent-iota acute\n-- U+1FF5\n\u1ff6 : Token \u03c9\u2032 lower -- U+1FF6  \u1ff6 GREEK SMALL LETTER OMEGA WITH PERISPOMENI\n\u1ff6 = with-accent circumflex\n\u1ff7 : Token \u03c9\u2032 lower -- U+1FF7  \u1ff7 GREEK SMALL LETTER OMEGA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1ff7 = with-accent-iota circumflex\n\u1ff8 : Token \u03bf\u2032 upper -- U+1FF8  \u1ff8 GREEK CAPITAL LETTER OMICRON WITH VARIA\n\u1ff8 = with-accent grave\n\u1ff9 : Token \u03bf\u2032 upper -- U+1FF9  \u1ff9 GREEK CAPITAL LETTER OMICRON WITH OXIA\n\u1ff9 = with-accent acute\n\u1ffa : Token \u03c9\u2032 upper -- U+1FFA  \u1ffa GREEK CAPITAL LETTER OMEGA WITH VARIA\n\u1ffa = with-accent grave\n\u1ffb : Token \u03c9\u2032 upper -- U+1FFB  \u1ffb GREEK CAPITAL LETTER OMEGA WITH OXIA\n\u1ffb = with-accent acute\n\u1ffc : Token \u03c9\u2032 upper -- U+1FFC  \u1ffc GREEK CAPITAL LETTER OMEGA WITH PROSGEGRAMMENI\n\u1ffc = with-iota\n-- U+1FFD  \u1ffd GREEK OXIA\n-- U+1FFE  \u1ffe GREEK DASIA\n-- U+1FFF\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a) = just (letter-to-accent a)\nget-accent (with-accent-iota a) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis with-diaeresis = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form final = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-final \u2984 \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\ninstance \u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\ninstance \u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\ninstance \u03b9-diaeresis : \u03b9\u2032 diaeresis\n\u03b9-diaeresis = add-diaeresis here\n\n-- \u039f \u03bf\ninstance \u03bf-vowel : \u03bf\u2032 vowel\n\u03bf-vowel = is-vowel (there (there (there (there here))))\n\n\u03bf-always-short : \u03bf\u2032 always-short\n\u03bf-always-short = is-always-short (there here)\n\ninstance \u03bf-smooth : \u03bf\u2032 \u27e6 lower \u27e7-smooth\n\u03bf-smooth = add-smooth-lower-vowel\n\ninstance \u039f-smooth : \u03bf\u2032 \u27e6 upper \u27e7-smooth\n\u039f-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03bf-rough : \u03bf\u2032 with-rough\n\u03bf-rough = add-rough-vowel\n\n-- \u03a1 \u03c1\ninstance \u03c1-smooth : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n\u03c1-smooth = add-smooth-\u03c1\n\ninstance \u03c1-rough : \u03c1\u2032 with-rough\n\u03c1-rough = add-rough-\u03c1\n\n-- \u03a3 \u03c3\ninstance \u03c3-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\u03c3-final = make-final\n\n-- \u03a5 \u03c5\ninstance \u03c5-vowel : \u03c5\u2032 vowel\n\u03c5-vowel = is-vowel (there (there (there (there (there here)))))\n\n\u00ac\u03c5-always-short : \u00ac \u03c5\u2032 always-short\n\u00ac\u03c5-always-short (is-always-short (there (there ())))\n\ninstance \u03c5-long-vowel : \u03c5\u2032 long-vowel\n\u03c5-long-vowel = make-long-vowel \u00ac\u03c5-always-short\n\ninstance \u03c5-smooth : \u03c5\u2032 \u27e6 lower \u27e7-smooth\n\u03c5-smooth = add-smooth-lower-vowel\n\ninstance \u03c5-rough : \u03c5\u2032 with-rough\n\u03c5-rough = add-rough-vowel\n\ninstance \u03c5-diaeresis : \u03c5\u2032 diaeresis\n\u03c5-diaeresis = add-diaeresis (there here)\n\n-- \u03a9 \u03c9\ninstance \u03c9-vowel : \u03c9\u2032 vowel\n\u03c9-vowel = is-vowel (there (there (there (there (there (there here))))))\n\n\u00ac\u03c9-always-short : \u00ac \u03c9\u2032 always-short\n\u00ac\u03c9-always-short (is-always-short (there (there ())))\n\ninstance \u03c9-long-vowel : \u03c9\u2032 long-vowel\n\u03c9-long-vowel = make-long-vowel \u00ac\u03c9-always-short\n\ninstance \u03c9-smooth : \u03c9\u2032 \u27e6 lower \u27e7-smooth\n\u03c9-smooth = add-smooth-lower-vowel\n\ninstance \u03a9-smooth : \u03c9\u2032 \u27e6 upper \u27e7-smooth\n\u03a9-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03c9-rough : \u03c9\u2032 with-rough\n\u03c9-rough = add-rough-vowel\n\ninstance \u03c9-iota-subscript : \u03c9\u2032 iota-subscript\n\u03c9-iota-subscript = add-iota-subscript (there (there here))\n\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = with-accent-diaeresis acute\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = with-diaeresis\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = with-diaeresis\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = with-accent acute\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = with-accent acute\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = with-accent acute\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = with-accent acute\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = with-accent-diaeresis acute\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = final\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = with-diaeresis\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = with-diaeresis\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = with-accent acute\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = with-accent acute\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = with-accent acute\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota smooth\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota rough\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave smooth\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave rough\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute smooth\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute rough\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex smooth\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex rough\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota smooth\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota rough\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave smooth\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave rough\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute smooth\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute rough\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex smooth\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex rough\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a) = just (letter-to-accent a)\nget-accent (with-accent-iota a) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis with-diaeresis = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form final = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"59a112037f5ddcf24c9663b9399e5b60b19d1198","subject":"IDesc: tagged indexed description","message":"IDesc: tagged indexed description\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8062ff2e3a4db2f2405fcecd8ecd3687f9574da1","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: d943a6359f1fa72487ab808788c3bd0e\n\ndarcs-hash:20110227043610-3bd4e-0cb5a48fd55778fc07d4e4e5389bf3e2d2648b58.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/WellFoundedInductionATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/WellFoundedInductionATP.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n----------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check #-}\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.MCR.WellFoundedInductionATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\n\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.PropertiesATP\n\n----------------------------------------------------------------------------\n\n-- Adapted from FOTC.Data.Nat.Induction.WellFoundedI.wfInd-LT\u2081.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 MCR n m \u2192 P n) \u2192 P m) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P accH Nn = accH Nn (helper Nn)\n  where\n    helper : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR n m \u2192 P n\n    helper Nm zN 0\u00abm = \u22a5-elim (0\u00abx\u2192\u22a5 Nm 0\u00abm)\n\n    -- This equation does not pass the termination check.\n    helper zN (sN {n} Nn) Sn\u00ab0 = accH (sN Nn)\n      (\u03bb {n'} Nn' n'\u00abSn \u2192\n        let n'\u00ab0 : MCR n' zero\n            n'\u00ab0 = \u00ab-trans Nn' (sN Nn) zN n'\u00abSn Sn\u00ab0\n\n        in helper zN Nn' n'\u00ab0\n      )\n\n    -- Other version of the previous equation (this version neither\n    -- pass the termination check).\n    -- helper zN (sN {n} Nn) Sn\u00ab0 = accH (sN Nn)\n    --   (\u03bb {n'} Nn' n'\u00abSn \u2192\n    --     let n'\u00abn : MCR n' n\n    --         n'\u00abn = x\u00abSy\u2192x\u00aby Nn' Nn n'\u00abSn\n\n    --     in helper Nn Nn' n'\u00abn\n    --   )\n\n    -- Other version of the previous equation (this version neither\n    -- pass the termination check).\n    -- helper zN Nn n\u00ab0 = accH Nn\n    --   (\u03bb {n'} Nn' n'\u00abn \u2192\n    --     let n'\u00ab0 : MCR n' zero\n    --         n'\u00ab0 = \u00ab-trans Nn' Nn zN n'\u00abn n\u00ab0\n\n    --     in helper zN Nn' n'\u00ab0\n    --   )\n\n    -- Other version of the previous equation (this version neither\n    -- pass the termination check).\n    -- helper {n = n} zN Nn n\u00ab0 =\n    --   accH Nn (\u03bb {n'} Nn' n'\u00abn \u2192 helper Nn Nn' n'\u00abn)\n\n    helper (sN {m} Nm) (sN {n} Nn) Sn\u00abSm = helper Nm (sN Nn) Sn\u00abm\n      where\n        Sn\u00abm : MCR (succ n) m\n        Sn\u00abm = x\u00abSy\u2192x\u00aby (sN Nn) Nm Sn\u00abSm\n","old_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n----------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check #-}\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.MCR.WellFoundedInductionATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\n\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.PropertiesATP\n\n----------------------------------------------------------------------------\n\n-- Adapted from FOTC.Data.Nat.Induction.WellFoundedI.wfInd-LT\u2081.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 MCR n m \u2192 P n) \u2192 P m) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P accH Nn = accH Nn (helper Nn)\n  where\n    helper : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR n m \u2192 P n\n    helper Nm zN 0\u00abm = \u22a5-elim (0\u00abx\u2192\u22a5 Nm 0\u00abm)\n\n    -- This equation does not pass the termination check.\n    helper zN Nn n\u00ab0 = accH Nn\n      (\u03bb {n'} Nn' n'\u00abn \u2192\n        let n'\u00ab0 : MCR n' zero\n            n'\u00ab0 = \u00ab-trans Nn' Nn zN n'\u00abn n\u00ab0\n\n        in helper zN Nn' n'\u00ab0\n      )\n\n    helper (sN {m} Nm) (sN {n} Nn) Sn\u00abSm = helper Nm (sN Nn) Sn\u00abm\n      where\n        Sn\u00abm : MCR (succ n) m\n        Sn\u00abm = x\u00abSy\u2192x\u00aby (sN Nn) Nm Sn\u00abSm\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3d513b19df5421df9f336ebcdb620a5af4aa0ce4","subject":"progress but stuck...","message":"progress but stuck...\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\npostulate\n    FunExt : \u2605\n    funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 {{fe : FunExt}} \u2192 f \u2261 g\n\nContractible : \u2200 {a}{A : \u2605_ a}(x : A) \u2192 \u2605_ a\nContractible x = \u2200 y \u2192 x \u2261 y\n\nmodule Equivalences where\n\n  module _ {A B : \u2605} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  module _ {A B : \u2605}{f : A \u2192 B}(f\u1d31 : Equiv f) where\n    open Equiv f\u1d31\n    inv : B \u2192 A\n    inv = linv \u2218 f \u2218 rinv\n\n    inv-equiv : Equiv inv\n    inv-equiv = record { linv = f\n                       ; is-linv = \u03bb x \u2192 cong f (is-linv (rinv x)) \u2219 is-rinv x\n                       ; rinv = f\n                       ; is-rinv = \u03bb x \u2192 cong linv (is-rinv (f x)) \u2219 is-linv x }\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  infix 0 _\u2243_\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u2243-refl : Reflexive _\u2243_\n  \u2243-refl = _ , id\u1d31\n\n  \u2243-sym : Symmetric _\u2243_\n  \u2243-sym (_ , f\u1d31) = _ , inv-equiv f\u1d31\n\n  \u2243-trans : Transitive _\u2243_\n  \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n  \u2243-! = \u2243-sym\n  _\u2243-\u2219_ = \u2243-trans\n\n  module Contractible\u2192Equiv {A : \u2605}{x : A}(x-contr : Contractible x) where\n    const-equiv : Equiv {\ud835\udfd9} (\u03bb _ \u2192 x)\n    const-equiv = record { linv = _ ; is-linv = \u03bb _ \u2192 refl ; rinv = _ ; is-rinv = x-contr }\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-equiv\n  module Equiv\u2192Contractible {A : \u2605}(f : \ud835\udfd9 \u2192 A)(f-equiv : Equiv f) where\n    open Equiv f-equiv\n    A-contr : Contractible (f _)\n    A-contr = is-rinv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\nopen Equivalences\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com_ \u2113 : \u2605_(\u209b \u2113) where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605_ \u2113\n        P  : M \u2192 Proto_ \u2113\n\n    data Proto_ \u2113 : \u2605_(\u209b \u2113) where\n      end : Proto_ \u2113\n      com : Com_ \u2113 \u2192 Proto_ \u2113\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\nCom   : \u2605\u2081\nCom   = Com_ \u2080\n\n{-\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n-}\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e _ P) (\u03a0\u1d3e _ Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c (mk _ M P) = \u03a3\u1d9c M \u03bb m \u2192 source-of (P m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk io M P) = mk (dual\u1d35\u1d3c io) M \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113) (P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end        \u27e7 = End\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u211b\u27e6_\u27e7 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u211b\u27e6 end    \u27e7 p q = End\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7 p q = (m : M) \u2192 \u211b\u27e6 P m \u27e7 (p m) (q m)\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7 p q = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u211b\u27e6 P (fst q) \u27e7 (subst (\u27e6_\u27e7 \u2218 P) e (snd p)) (snd q)\n\n\u211b\u27e6_\u27e7-refl : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Reflexive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-refl     = end\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7-refl     = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-refl\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7-refl {x} = refl , \u211b\u27e6 P (fst x) \u27e7-refl\n\n\u211b\u27e6_\u27e7-sym : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Symmetric \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-sym p          = end\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7-sym p          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-sym (p m)\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7-sym (refl , q) = refl , \u211b\u27e6 P _ \u27e7-sym q    -- TODO HoTT\n\n\u211b\u27e6_\u27e7-trans : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Transitive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-trans p          q          = end\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7-trans p          q          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-trans (p m) (q m)\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7-trans (refl , p) (refl , q) = refl , \u211b\u27e6 P _ \u27e7-trans p q    -- TODO HoTT\n\ndata ViewProc {\u2113} : \u2200 (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u2605_(\u209b \u2113) where\n  send : \u2200 M(P : M \u2192 Proto_ \u2113)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto_ \u2113)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 {\u2113} (P : Proto_ \u2113) (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl M x) (com refl .M x\u2081) = com refl M (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\nmodule _ {{_ : FunExt}} where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end            = _\nsink (com' _ M P) x = sink (P x)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P \u2192 Contractible (sink P)\n    sink-contr end          s = refl\n    sink-contr (com' _ _ P) s = funExt \u03bb m \u2192 sink-contr (P m) (s m)\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Contractible\u2192Equiv.\ud835\udfd9\u2243 (sink-contr P)\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end         _        ys = ys\n++Log (com' q M P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n\nlift\u1d3e : \u2200 a {\u2113} \u2192 Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\nlift\u1d3e a end           = end\nlift\u1d3e a (com' io M P) = com' io (Lift {_} {a} M) \u03bb m \u2192 lift\u1d3e a (P (lower m))\n\nlift-proc : \u2200 a {\u2113} (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e a P \u27e7\nlift-proc a {\u2113} P0 p0 = lift-view (view-proc P0 p0)\n  where\n    lift-view : \u2200 {P : Proto_ \u2113}{p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 lift\u1d3e a P \u27e7\n    lift-view (send M P m p) = lift m , lift-proc _ (P m) p\n    lift-view (recv M P x)   = \u03bb { (lift m) \u2192 lift-proc _ (P m) (x m) }\n    lift-view end            = end\n\nmodule MonoMobility (P : Proto\u2080) where\n  Com\u1d3e : Proto\u2080\n  Com\u1d3e = \u03a0\u1d3e \u27e6 P  \u27e7  \u03bb p \u2192\n         \u03a0\u1d3e \u27e6 P \u22a5\u27e7  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Log P) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc p p\u22a5 = tele-com P p p\u22a5 , _\n\nmodule PolyMobility where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         lift\u1d3e \u2081 (MonoMobility.Com\u1d3e P)\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P = lift-proc \u2081 (MonoMobility.Com\u1d3e P) (MonoMobility.com-proc P)\n\nmodule PolyMobility' where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         \u03a0\u1d3e (Lift \u27e6 P  \u27e7)  \u03bb p \u2192\n         \u03a0\u1d3e (Lift \u27e6 P \u22a5\u27e7)  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Lift (Log P)) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P (lift p) (lift p\u22a5) = lift (tele-com P p p\u22a5) , _\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\n           {-\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n  \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind {{_ : FunExt}} where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Log P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = \u03a0\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n    StaticServer  = \u03a3\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ {{_ : FunExt}} where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end Q = \u2261\u1d3e-refl _\ndual->> (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\ndual->> (\u03a3\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 subst id (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (\u03a0\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (\u03a3\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _ {{_ : FunExt}} (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecv\u2610 : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecv\u2610 = recv \u2218\u2032 un\u2610\n\nsend\u2610 : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsend\u2610 m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\n{-\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n-}\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : LR \u2192 \u2605_ a} (l : A `L)(r : A `R) \u2192 (lr : LR) \u2192 A lr\n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\nmodule _ {P Q R S} where\n    \u2295\u1d3e-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n    \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n    \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n    &\u1d3e-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n    &\u1d3e-map f g p `L = f (p `L)\n    &\u1d3e-map f g p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u1d3e\u2192\u228e : \u27e6 P \u2295\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u1d3e\u2192\u228e (`L , p) = inl p\n    \u2295\u1d3e\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295\u1d3e : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295\u1d3e Q \u27e7\n    \u228e\u2192\u2295\u1d3e (inl p) = `L , p\n    \u228e\u2192\u2295\u1d3e (inr q) = `R , q\n\n    \u228e\u2192\u2295\u1d3e\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295\u1d3e (\u2295\u1d3e\u2192\u228e x) \u2261 x\n    \u228e\u2192\u2295\u1d3e\u2192\u228e (`L , _) = refl\n    \u228e\u2192\u2295\u1d3e\u2192\u228e (`R , _) = refl\n\n    \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e : \u2200 x \u2192 \u2295\u1d3e\u2192\u228e (\u228e\u2192\u2295\u1d3e x) \u2261 x\n    \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e (inl _) = refl\n    \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e (inr _) = refl\n\n    \u2295\u1d3e\u2243\u228e : \u27e6 P \u2295\u1d3e Q \u27e7 \u2243 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u1d3e\u2243\u228e = \u2295\u1d3e\u2192\u228e , record { linv = \u228e\u2192\u2295\u1d3e ; is-linv = \u228e\u2192\u2295\u1d3e\u2192\u228e ; rinv = \u228e\u2192\u2295\u1d3e ; is-rinv = \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e }\n\n    &\u1d3e\u2192\u00d7 : \u27e6 P &\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u1d3e\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192&\u1d3e : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P &\u1d3e Q \u27e7\n    \u00d7\u2192&\u1d3e (p , q) `L = p\n    \u00d7\u2192&\u1d3e (p , q) `R = q\n\n    &\u1d3e\u2192\u00d7\u2192&\u1d3e : \u2200 x \u2192 &\u1d3e\u2192\u00d7 (\u00d7\u2192&\u1d3e x) \u2261 x\n    &\u1d3e\u2192\u00d7\u2192&\u1d3e (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u1d3e\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192&\u1d3e (&\u1d3e\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u1d3e\u2192\u00d7 p = funExt \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u1d3e\u2243\u00d7 : \u27e6 P &\u1d3e Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n        &\u1d3e\u2243\u00d7 = &\u1d3e\u2192\u00d7 , record { linv = \u00d7\u2192&\u1d3e ; is-linv = \u00d7\u2192&\u1d3e\u2192\u00d7 ; rinv = \u00d7\u2192&\u1d3e ; is-rinv = &\u1d3e\u2192\u00d7\u2192&\u1d3e }\n\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP\u1d9c >>\u1d9c S = record P\u1d9c { P = \u03bb m \u2192 S (P m) }\n  where open Com_ P\u1d9c\n\nmodule _ where\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' \u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m')\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  data View-\u214b-proto : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    end-X     : \u2200 Q \u2192 View-\u214b-proto end Q\n    recv-X    : \u2200 {M}(P : M \u2192 Proto)Q \u2192 View-\u214b-proto (\u03a0\u1d3e M P) Q\n    send-send : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) \u2192 View-\u214b-proto (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q)\n    send-recv : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) \u2192 View-\u214b-proto (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q)\n    send-end  : \u2200 {M}(P : M \u2192 Proto) \u2192 View-\u214b-proto (\u03a3\u1d3e M P) end\n\n  view-\u214b-proto : \u2200 P Q \u2192 View-\u214b-proto P Q\n  view-\u214b-proto end      Q        = end-X Q\n  view-\u214b-proto (\u03a0\u1d3e _ P) Q        = recv-X P Q\n  view-\u214b-proto (\u03a3\u1d3e _ P) end      = send-end P\n  view-\u214b-proto (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) = send-recv P Q\n  view-\u214b-proto (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) = send-send P Q\n\n  data View-\u2297-proto : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    end-X     : \u2200 Q \u2192 View-\u2297-proto end Q\n    send-X    : \u2200 {M}(P : M \u2192 Proto)Q \u2192 View-\u2297-proto (\u03a3\u1d3e M P) Q\n    recv-recv : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) \u2192 View-\u2297-proto (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q)\n    recv-send : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) \u2192 View-\u2297-proto (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q)\n    recv-end  : \u2200 {M}(P : M \u2192 Proto) \u2192 View-\u2297-proto (\u03a0\u1d3e M P) end\n\n  view-\u2297-proto : \u2200 P Q \u2192 View-\u2297-proto P Q\n  view-\u2297-proto end      Q        = end-X Q\n  view-\u2297-proto (\u03a3\u1d3e _ P) Q        = send-X P Q\n  view-\u2297-proto (\u03a0\u1d3e _ P) end      = recv-end P\n  view-\u2297-proto (\u03a0\u1d3e _ P) (\u03a0\u1d3e _ Q) = recv-recv P Q\n  view-\u2297-proto (\u03a0\u1d3e _ P) (\u03a3\u1d3e _ Q) = recv-send P Q\n\n  -- the terminology used for the constructor follows the behavior of the combined process\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    send   : \u2200 {M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end (m , p)\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end (m , p) = send P m p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end      = end\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n{- Useless\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL Q f x = f x\n-}\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) R (m : M)(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a3\u1d3e _ Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (\u03a3\u1d3e _ Q) R (inl m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7))(q : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) Q R p q\n    recvR-sendR : \u2200 {M M'}P(Q : M \u2192 Proto)(R : M' \u2192 Proto)(m : M')(p : \u27e6 com P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inr m , q)\n    recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n               (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : (m : MR) \u2192 \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a0\u1d3e _ R) p q\n    sendR-recvL : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)R(m : MQ)\n                    (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : (m : MQ) \u2192 \u27e6 dual (Q m) \u214b\u1d3e R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) R (inr m , p) q\n    recvR-sendL : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (p : (m : MQ) \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(m : MQ)(q : \u27e6 dual (Q m) \u214b\u1d3e \u03a3\u1d3e _ R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inl m , q)\n    endL : \u2200 Q R\n           \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n           \u2192 View-\u2218 end Q R q qr\n    sendLM : \u2200 {MP}(P : MP \u2192 Proto)R\n               (m : MP)(p : \u27e6 P m \u27e7)(r : \u27e6 R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) end R (m , p) r\n    recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                    (m : MQ)(p : \u2200 m \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : \u27e6 dual (Q m) \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) end p (m , q)\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) _                 = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = recvR-sendR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (send ._ _ _)     = recvL-sendR _ _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (send _ _ _)     _                 = sendLM _ _ _ _ _\n\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n\n  dist-\u214b-fst : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  dist-\u214b-fst (\u03a0\u1d3e _ P) Q R p = \u03bb m \u2192 dist-\u214b-fst (P m) Q R (p m)\n  dist-\u214b-fst (\u03a3\u1d3e _ P) Q R p = p `L\n  dist-\u214b-fst end      Q R p = p `L\n\n  dist-\u214b-snd : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  dist-\u214b-snd (\u03a0\u1d3e _ P) Q R p = \u03bb m \u2192 dist-\u214b-snd (P m) Q R (p m)\n  dist-\u214b-snd (\u03a3\u1d3e _ P) Q R p = p `R\n  dist-\u214b-snd end      Q R p = p `R\n\n  dist-\u214b-\u00d7 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u00d7 \u27e6 P \u214b\u1d3e R \u27e7\n  dist-\u214b-\u00d7 P Q R p = dist-\u214b-fst P Q R p , dist-\u214b-snd P Q R p\n\n  dist-\u214b-& : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) &\u1d3e (P \u214b\u1d3e R) \u27e7\n  dist-\u214b-& P Q R p = \u00d7\u2192&\u1d3e (dist-\u214b-\u00d7 P Q R p)\n\n  factor-,-\u214b : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7\n  factor-,-\u214b end      Q R pq pr = \u00d7\u2192&\u1d3e (pq , pr)\n  factor-,-\u214b (\u03a0\u1d3e _ P) Q R pq pr = \u03bb m \u2192 factor-,-\u214b (P m) Q R (pq m) (pr m)\n  factor-,-\u214b (\u03a3\u1d3e _ P) Q R pq pr = [L: pq R: pr ]\n\n  factor-\u00d7-\u214b : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u00d7 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7\n  factor-\u00d7-\u214b P Q R (p , q) = factor-,-\u214b P Q R p q\n\n  factor-&-\u214b : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) &\u1d3e (P \u214b\u1d3e R) \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7\n  factor-&-\u214b P Q R p = factor-\u00d7-\u214b P Q R (&\u1d3e\u2192\u00d7 p)\n\n  module _ {{_ : FunExt}} where\n    dist-\u214b-fst-factor-&-, : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(pr : \u27e6 P \u214b\u1d3e R \u27e7)\n                            \u2192 dist-\u214b-fst P Q R (factor-,-\u214b P Q R pq pr) \u2261 pq\n    dist-\u214b-fst-factor-&-, (\u03a0\u1d3e _ P) Q R pq pr = funExt \u03bb m \u2192 dist-\u214b-fst-factor-&-, (P m) Q R (pq m) (pr m)\n    dist-\u214b-fst-factor-&-, (\u03a3\u1d3e _ P) Q R pq pr = refl\n    dist-\u214b-fst-factor-&-, end      Q R pq pr = refl\n\n    dist-\u214b-snd-factor-&-, : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(pr : \u27e6 P \u214b\u1d3e R \u27e7)\n                            \u2192 dist-\u214b-snd P Q R (factor-,-\u214b P Q R pq pr) \u2261 pr\n    dist-\u214b-snd-factor-&-, (\u03a0\u1d3e _ P) Q R pq pr = funExt \u03bb m \u2192 dist-\u214b-snd-factor-&-, (P m) Q R (pq m) (pr m)\n    dist-\u214b-snd-factor-&-, (\u03a3\u1d3e _ P) Q R pq pr = refl\n    dist-\u214b-snd-factor-&-, end      Q R pq pr = refl\n\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 : \u2200 P Q R \u2192 (factor-\u00d7-\u214b P Q R) LeftInverseOf (dist-\u214b-\u00d7 P Q R)\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (\u03a0\u1d3e _ P) Q R p = funExt \u03bb m \u2192 factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (P m) Q R (p m)\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (\u03a3\u1d3e _ P) Q R p = funExt \u03bb { `L \u2192 refl ; `R \u2192 refl }\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 end      Q R p = funExt \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n    module _ P Q R where\n        factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 : (factor-\u00d7-\u214b P Q R) RightInverseOf (dist-\u214b-\u00d7 P Q R)\n        factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 (x , y) = cong\u2082 _,_ (dist-\u214b-fst-factor-&-, P Q R x y) (dist-\u214b-snd-factor-&-, P Q R x y)\n\n        dist-\u214b-\u00d7-\u2243 : \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7 \u2243 \u27e6 P \u214b\u1d3e Q \u27e7 \u00d7 \u27e6 P \u214b\u1d3e R \u27e7\n        dist-\u214b-\u00d7-\u2243 = dist-\u214b-\u00d7 P Q R\n                   , record { linv = factor-\u00d7-\u214b P Q R; is-linv = factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 P Q R\n                            ; rinv = factor-\u00d7-\u214b P Q R; is-rinv = factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 }\n\n        dist-\u214b-&-\u2243 : \u27e6 P \u214b\u1d3e (Q &\u1d3e R) \u27e7 \u2243 \u27e6 (P \u214b\u1d3e Q) &\u1d3e (P \u214b\u1d3e R) \u27e7\n        dist-\u214b-&-\u2243 = dist-\u214b-\u00d7-\u2243 \u2243-\u2219 \u2243-! &\u1d3e\u2243\u00d7\n\nmodule _ {{_ : FunExt}} where\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply (dual P) Q pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound (dual-involutive P))) p)\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)      = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (\u03a3\u1d3e _ Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) Q R (p m) qr\n    \u214b\u1d3e-\u2218-view (recvR-sendR P Q R m pq q) = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (\u03a0\u1d3e _ Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (R m) pq (q m)\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) R p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)           = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (sendLM P R m pq qr)       = \u214b\u1d3e-sendL R m (par (P m) R pq qr)\n    \u214b\u1d3e-\u2218-view (recvL-sendR P Q m pq qr)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) end (pq m) (\u214b\u1d3e-rend' (dual (Q m)) qr)\n\n  mutual\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm P Q p = \u214b\u1d3e-comm-view (view-\u214b P Q p)\n\n    {-\n    \u214b\u1d3e-comm-\u03a0 : \u2200 P {N}(Q : N \u2192 Proto) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e N Q \u27e7 \u2192 \u03a0 N \u03bb n \u2192 \u27e6 Q n \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm-\u03a0 end Q p n = \u214b\u1d3e-rend' (Q n) (p n)\n    \u214b\u1d3e-comm-\u03a0 (\u03a0\u1d3e _ P) Q p n = ?\n    \u214b\u1d3e-comm-\u03a0 (\u03a3\u1d3e _ P) Q p n = \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q n) (p n)\n    -}\n\n    \u214b\u1d3e-comm-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm-view (sendL' P Q m p)  = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-comm (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-comm-view (sendR' P Q m' p) = inl m' , \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') p\n    \u214b\u1d3e-comm-view (recvL' P Q pq)   = \u214b\u1d3e-recvR Q P \u03bb m \u2192 \u214b\u1d3e-comm (P m) Q (pq m)\n    \u214b\u1d3e-comm-view (recvR' P Q pq)   = \u03bb m' \u2192 \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') (pq m')\n    \u214b\u1d3e-comm-view (endL Q pq)       = \u214b\u1d3e-rend' Q pq\n    \u214b\u1d3e-comm-view (send P m pq)     = m , pq\n\n  \u214b\u1d3e-comm-equiv : \u2200 P Q \u2192 Equiv (\u214b\u1d3e-comm P Q)\n  \u214b\u1d3e-comm-equiv P Q = record { linv = \u214b\u1d3e-comm Q P ; is-linv = {!!} ; rinv = \u214b\u1d3e-comm Q P ; is-rinv = {!!} }\n    where\n      toto : \u2200 {P Q}{pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u214b\u1d3e-comm Q P (\u214b\u1d3e-comm P Q pq) \u2261 pq\n      toto (sendL' P Q m p) = cong (_,_ (inl m)) (toto (view-\u214b (P m) (\u03a3\u1d3e _ Q) p))\n      toto (sendR' P Q m' p) = cong (_,_ (inr m')) (toto (view-\u214b (\u03a3\u1d3e _ P) (Q m') p))\n      toto (recvL' P Q pq) = funExt \u03bb m \u2192 {!MAKE A LEMMA!}\n      toto (recvR' P Q pq) = funExt \u03bb m \u2192 toto (view-\u214b (\u03a3\u1d3e _ P) (Q m) (pq m))\n      toto (endL end p) = refl\n      toto (endL (\u03a0\u1d3e _ Q) p) = funExt \u03bb m \u2192 toto (view-\u214b end (Q m) (p m))\n      toto (endL (\u03a3\u1d3e _ Q) p) = refl\n      toto (send P m p) = refl\n\n  \u214b\u1d3e-comm-\u2248 : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm-\u2248 P Q = _ , \u214b\u1d3e-comm-equiv P Q\n\n  comma\u1d3e : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n  comma\u1d3e {end}    {Q}      p q       = q\n  comma\u1d3e {\u03a3\u1d3e M P} {Q}      (m , p) q = m , comma\u1d3e {P m} p q\n  comma\u1d3e {\u03a0\u1d3e M P} {end}    p end     = p\n  comma\u1d3e {\u03a0\u1d3e M P} {\u03a3\u1d3e _ Q} p (m , q) = m , comma\u1d3e {\u03a0\u1d3e M P} {Q m} p q\n  {-\n  comma\u1d3e {\u03a0\u1d3e M P} {\u03a0\u1d3e N Q} p q       (inl m)  = comma\u1d3e (P m)    (\u03a0\u1d3e _ Q) (p m) q\n  comma\u1d3e {\u03a0\u1d3e M P} {\u03a0\u1d3e N Q} p q       (inr m') = comma\u1d3e (\u03a0\u1d3e _ P) (Q m')   p     (q m')\n  -}\n  comma\u1d3e {\u03a0\u1d3e M P} {\u03a0\u1d3e N Q} p q       = [inl: (\u03bb m  \u2192 comma\u1d3e {P m}    {\u03a0\u1d3e _ Q} (p m) q)\n                                       ,inr: (\u03bb m' \u2192 comma\u1d3e {\u03a0\u1d3e _ P} {Q m'}   p     (q m')) ]\n\n  \u2297\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297\u1d3e-fst end      Q        pq       = _\n  \u2297\u1d3e-fst (\u03a3\u1d3e M P) Q        (m , pq) = m , \u2297\u1d3e-fst (P m) Q pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) end      pq       = pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (_ , pq) = \u2297\u1d3e-fst (\u03a0\u1d3e M P) (Q _) pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-fst (P m) (\u03a0\u1d3e N Q) (pq (inl m))\n\n  \u2297\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  \u2297\u1d3e-snd end      Q        pq       = pq\n  \u2297\u1d3e-snd (\u03a3\u1d3e M P) Q        (_ , pq) = \u2297\u1d3e-snd (P _) Q pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) end      pq       = end\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (m , pq) = m , \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) (pq (inr m))\n\n  \u2297\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297\u1d3e-fst P Q (comma\u1d3e {P} {Q} p q) \u2261 p\n  \u2297\u1d3e-comma-fst end      Q        p q       = refl\n  \u2297\u1d3e-comma-fst (\u03a3\u1d3e M P) Q        (m , p) q = \u03a3-ext refl (\u2297\u1d3e-comma-fst (P m) Q p q)\n  \u2297\u1d3e-comma-fst (\u03a0\u1d3e M P) end      p q       = refl\n  \u2297\u1d3e-comma-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) p (m , q) = \u2297\u1d3e-comma-fst (\u03a0\u1d3e _ P) (Q m) p q\n  \u2297\u1d3e-comma-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) p q       = funExt \u03bb m \u2192 \u2297\u1d3e-comma-fst (P m) (\u03a0\u1d3e _ Q) (p m) q\n\n  \u2297\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297\u1d3e-snd P Q (comma\u1d3e {P} {Q} p q) \u2261 q\n  \u2297\u1d3e-comma-snd end      Q        p q       = refl\n  \u2297\u1d3e-comma-snd (\u03a3\u1d3e M P) Q        (m , p) q = \u2297\u1d3e-comma-snd (P m) Q p q\n  \u2297\u1d3e-comma-snd (\u03a0\u1d3e M P) end      p q       = refl\n  \u2297\u1d3e-comma-snd (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) p (m , q) = \u03a3-ext refl (\u2297\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  \u2297\u1d3e-comma-snd (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) p q       = funExt \u03bb m \u2192 \u2297\u1d3e-comma-snd (\u03a0\u1d3e M P) (Q m) p (q m)\n\n  module _ P Q where\n    \u2297\u2192\u00d7 : \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    \u2297\u2192\u00d7 pq = \u2297\u1d3e-fst P Q pq , \u2297\u1d3e-snd P Q pq\n\n    \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n    \u00d7\u2192\u2297 (p , q) = comma\u1d3e {P} {Q} p q\n\n  \u2297\u2192\u00d7\u2192\u2297 : \u2200 P Q \u2192 (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n  foo : \u2200 M N P Q (x : M \u228e N)(pq : \u27e6 \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e N Q \u27e7) \u2192\n            \u00d7\u2192\u2297 (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) (\u2297\u2192\u00d7 (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq) x \u2261 pq x\n  fooL : \u2200 (M : \u2605) N (P : M \u2192 Proto) Q (m : M) (pq : \u27e6 P m \u2297\u1d3e \u03a0\u1d3e N Q \u27e7) \u2192\n            \u00d7\u2192\u2297 (P m) (\u03a0\u1d3e N Q) (\u2297\u2192\u00d7 (P m) (\u03a0\u1d3e N Q) pq) \u2261 pq\n\n  \u2297\u2192\u00d7\u2192\u2297 end      Q        pq       = refl\n  \u2297\u2192\u00d7\u2192\u2297 (\u03a3\u1d3e M P) Q        (m , pq) = cong (_,_ m) (\u2297\u2192\u00d7\u2192\u2297 (P m) Q pq)\n  \u2297\u2192\u00d7\u2192\u2297 (\u03a0\u1d3e M P) end      pq       = refl\n  \u2297\u2192\u00d7\u2192\u2297 (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (m , pq) = cong (_,_ m) (\u2297\u2192\u00d7\u2192\u2297 (\u03a0\u1d3e _ P) (Q m) pq)\n  \u2297\u2192\u00d7\u2192\u2297 (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq = funExt (\u03bb m \u2192 foo M N P Q m pq) {-funExt \u03bb { (inl x) \u2192 {!!} \u2219 \u2297\u2192\u00d7\u2192\u2297 (P x) (\u03a0\u1d3e _ Q) (pq (inl x))\n                                        ; (inr x) \u2192 {!!} \u2219 \u2297\u2192\u00d7\u2192\u2297 (\u03a0\u1d3e _ P) (Q x) (pq (inr x)) }-}\n\n  fooL M N P Q m pq = \u2297\u2192\u00d7\u2192\u2297 (P m) (\u03a0\u1d3e _ Q) pq\n\n  foo M N P Q (inl m) pq = {!STUCK!} \u2219 fooL M N P Q m (pq (inl m))\n  foo M N P Q (inr m) pq = {!!}\n\n  module _ P Q where\n    \u2297\u2243\u00d7 : \u27e6 P \u2297\u1d3e Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    \u2297\u2243\u00d7 = \u2297\u2192\u00d7 P Q\n        , record { linv = \u00d7\u2192\u2297 P Q ; is-linv = \u2297\u2192\u00d7\u2192\u2297 P Q\n                 ; rinv = \u00d7\u2192\u2297 P Q ; is-rinv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (\u2297\u1d3e-comma-fst P Q x y) (\u2297\u1d3e-comma-snd P Q x y) } }\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n  -}\n\nmodule _ {{_ : FunExt}} where\n    \u03a0\ud835\udfd8-uniq : \u2200 (F G : \ud835\udfd8 \u2192 \u2605) \u2192 \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq F G = cong (\u03a0 \ud835\udfd8) (funExt (\u03bb()))\n\n{-\nA `\u2297 B 'times', context chooses how A and B are used\nA `\u214b B 'par', \"we\" chooses how A and B are used\nA `\u2295 B 'plus', select from A or B\nA `& B 'with', offer choice of A or B\n`! A   'of course!', server accept\n`? A   'why not?', client request\n`1     unit for `\u2297\n`\u22a4     unit for `\u214b\n`0     unit for `\u2295\n`\u22a5     unit for `&\n-}\ndata CLL : \u2605 where\n  `1 `\u22a4 `0 `\u22a5 : CLL\n  _`\u2297_ _`\u214b_ _`\u2295_ _`&_ : (A B : CLL) \u2192 CLL\n  -- `!_ `?_ : (A : CLL) \u2192 CLL\n\n_\u22a5 : CLL \u2192 CLL\n`1 \u22a5 = `\u22a4\n`\u22a4 \u22a5 = `1\n`0 \u22a5 = `\u22a5\n`\u22a5 \u22a5 = `0\n(A `\u2297 B)\u22a5 = A \u22a5 `\u214b B \u22a5\n(A `\u214b B)\u22a5 = A \u22a5 `\u2297 B \u22a5\n(A `\u2295 B)\u22a5 = A \u22a5 `& B \u22a5\n(A `& B)\u22a5 = A \u22a5 `\u2295 B \u22a5\n{-\n(`! A)\u22a5 = `?(A \u22a5)\n(`? A)\u22a5 = `!(A \u22a5)\n-}\n\nCLL-proto : CLL \u2192 Proto\nCLL-proto `1       = end  -- TODO\nCLL-proto `\u22a4       = end  -- TODO\nCLL-proto `0       = \u03a0\u1d3e \ud835\udfd8 \u03bb()\nCLL-proto `\u22a5       = \u03a3\u1d3e \ud835\udfd8 \u03bb()\nCLL-proto (A `\u2297 B) = CLL-proto A \u2297\u1d3e CLL-proto B\nCLL-proto (A `\u214b B) = CLL-proto A \u214b\u1d3e CLL-proto B\nCLL-proto (A `\u2295 B) = CLL-proto A \u2295\u1d3e CLL-proto B\nCLL-proto (A `& B) = CLL-proto A &\u1d3e CLL-proto B\n\n{- The point of this could be to devise a particular equivalence\n   relation for processes. It could properly deal with \u214b. -}\n\n  {-\nmodule V4 {{_ : FunExt}} where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _ {{_ : FunExt}} where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _ {{_ : FunExt}} where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _ {{_ : FunExt}}\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\npostulate\n    FunExt : \u2605\n    funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 {{fe : FunExt}} \u2192 f \u2261 g\n\nContractible : \u2200 {a}{A : \u2605_ a}(x : A) \u2192 \u2605_ a\nContractible x = \u2200 y \u2192 x \u2261 y\n\nmodule Equivalences where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  module _ {A B : \u2605}{f : A \u2192 B}(f\u1d31 : Equiv f) where\n    open Equiv f\u1d31\n    inv : B \u2192 A\n    inv = linv \u2218 f \u2218 rinv\n\n    inv-equiv : Equiv inv\n    inv-equiv = record { linv = f\n                       ; is-linv = \u03bb x \u2192 cong f (is-linv (rinv x)) \u2219 is-rinv x\n                       ; rinv = f\n                       ; is-rinv = \u03bb x \u2192 cong linv (is-rinv (f x)) \u2219 is-linv x }\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  infix 0 _\u2243_\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u2243-refl : Reflexive _\u2243_\n  \u2243-refl = _ , id\u1d31\n\n  \u2243-sym : Symmetric _\u2243_\n  \u2243-sym (_ , f\u1d31) = _ , inv-equiv f\u1d31\n\n  \u2243-trans : Transitive _\u2243_\n  \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n  module Contractible\u2192Equiv {A : \u2605}{x : A}(x-contr : Contractible x) where\n    const-equiv : Equiv {\ud835\udfd9} (\u03bb _ \u2192 x)\n    const-equiv = record { linv = _ ; is-linv = \u03bb _ \u2192 refl ; rinv = _ ; is-rinv = x-contr }\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-equiv\n  module Equiv\u2192Contractible {A : \u2605}(f : \ud835\udfd9 \u2192 A)(f-equiv : Equiv f) where\n    open Equiv f-equiv\n    A-contr : Contractible (f _)\n    A-contr = is-rinv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\nopen Equivalences\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com_ \u2113 : \u2605_(\u209b \u2113) where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605_ \u2113\n        P  : M \u2192 Proto_ \u2113\n\n    data Proto_ \u2113 : \u2605_(\u209b \u2113) where\n      end : Proto_ \u2113\n      com : Com_ \u2113 \u2192 Proto_ \u2113\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\nCom   : \u2605\u2081\nCom   = Com_ \u2080\n\n{-\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n-}\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e _ P) (\u03a0\u1d3e _ Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c (mk _ M P) = \u03a3\u1d9c M \u03bb m \u2192 source-of (P m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk io M P) = mk (dual\u1d35\u1d3c io) M \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113) (P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end        \u27e7 = End\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u211b\u27e6_\u27e7 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u211b\u27e6 end    \u27e7 p q = End\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7 p q = (m : M) \u2192 \u211b\u27e6 P m \u27e7 (p m) (q m)\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7 p q = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u211b\u27e6 P (fst q) \u27e7 (subst (\u27e6_\u27e7 \u2218 P) e (snd p)) (snd q)\n\n\u211b\u27e6_\u27e7-refl : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Reflexive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-refl     = end\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7-refl     = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-refl\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7-refl {x} = refl , \u211b\u27e6 P (fst x) \u27e7-refl\n\n\u211b\u27e6_\u27e7-sym : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Symmetric \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-sym p          = end\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7-sym p          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-sym (p m)\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7-sym (refl , q) = refl , \u211b\u27e6 P _ \u27e7-sym q    -- TODO HoTT\n\n\u211b\u27e6_\u27e7-trans : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Transitive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-trans p          q          = end\n\u211b\u27e6 \u03a0\u1d3e M P \u27e7-trans p          q          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-trans (p m) (q m)\n\u211b\u27e6 \u03a3\u1d3e M P \u27e7-trans (refl , p) (refl , q) = refl , \u211b\u27e6 P _ \u27e7-trans p q    -- TODO HoTT\n\ndata ViewProc {\u2113} : \u2200 (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u2605_(\u209b \u2113) where\n  send : \u2200 M(P : M \u2192 Proto_ \u2113)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto_ \u2113)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 {\u2113} (P : Proto_ \u2113) (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl M x) (com refl .M x\u2081) = com refl M (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\nmodule _ {{_ : FunExt}} where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end            = _\nsink (com' _ M P) x = sink (P x)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P \u2192 Contractible (sink P)\n    sink-contr end          s = refl\n    sink-contr (com' _ _ P) s = funExt \u03bb m \u2192 sink-contr (P m) (s m)\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Contractible\u2192Equiv.\ud835\udfd9\u2243 (sink-contr P)\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end         _        ys = ys\n++Log (com' q M P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n\nlift\u1d3e : \u2200 a {\u2113} \u2192 Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\nlift\u1d3e a end           = end\nlift\u1d3e a (com' io M P) = com' io (Lift {_} {a} M) \u03bb m \u2192 lift\u1d3e a (P (lower m))\n\nlift-proc : \u2200 a {\u2113} (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e a P \u27e7\nlift-proc a {\u2113} P0 p0 = lift-view (view-proc P0 p0)\n  where\n    lift-view : \u2200 {P : Proto_ \u2113}{p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 lift\u1d3e a P \u27e7\n    lift-view (send M P m p) = lift m , lift-proc _ (P m) p\n    lift-view (recv M P x)   = \u03bb { (lift m) \u2192 lift-proc _ (P m) (x m) }\n    lift-view end            = end\n\nmodule MonoMobility (P : Proto\u2080) where\n  Com\u1d3e : Proto\u2080\n  Com\u1d3e = \u03a0\u1d3e \u27e6 P  \u27e7  \u03bb p \u2192\n         \u03a0\u1d3e \u27e6 P \u22a5\u27e7  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Log P) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc p p\u22a5 = tele-com P p p\u22a5 , _\n\nmodule PolyMobility where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         lift\u1d3e \u2081 (MonoMobility.Com\u1d3e P)\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P = lift-proc \u2081 (MonoMobility.Com\u1d3e P) (MonoMobility.com-proc P)\n\nmodule PolyMobility' where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         \u03a0\u1d3e (Lift \u27e6 P  \u27e7)  \u03bb p \u2192\n         \u03a0\u1d3e (Lift \u27e6 P \u22a5\u27e7)  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Lift (Log P)) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P (lift p) (lift p\u22a5) = lift (tele-com P p p\u22a5) , _\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\n           {-\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n  \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind {{_ : FunExt}} where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Log P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = \u03a0\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n    StaticServer  = \u03a3\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ {{_ : FunExt}} where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end Q = \u2261\u1d3e-refl _\ndual->> (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\ndual->> (\u03a3\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 subst id (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (\u03a0\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (\u03a3\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _ {{_ : FunExt}} (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecv\u2610 : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecv\u2610 = recv \u2218\u2032 un\u2610\n\nsend\u2610 : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsend\u2610 m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\n{-\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n-}\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\nmodule _ {P Q R S} where\n    \u2295\u1d3e-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n    \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n    \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n    &\u1d3e-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n    &\u1d3e-map f g p `L = f (p `L)\n    &\u1d3e-map f g p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u1d3e\u2192\u228e : \u27e6 P \u2295\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u1d3e\u2192\u228e (`L , p) = inl p\n    \u2295\u1d3e\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295\u1d3e : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295\u1d3e Q \u27e7\n    \u228e\u2192\u2295\u1d3e (inl p) = `L , p\n    \u228e\u2192\u2295\u1d3e (inr q) = `R , q\n\n    \u228e\u2192\u2295\u1d3e\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295\u1d3e (\u2295\u1d3e\u2192\u228e x) \u2261 x\n    \u228e\u2192\u2295\u1d3e\u2192\u228e (`L , _) = refl\n    \u228e\u2192\u2295\u1d3e\u2192\u228e (`R , _) = refl\n\n    \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e : \u2200 x \u2192 \u2295\u1d3e\u2192\u228e (\u228e\u2192\u2295\u1d3e x) \u2261 x\n    \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e (inl _) = refl\n    \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e (inr _) = refl\n\n    \u2295\u1d3e\u2243\u228e : \u27e6 P \u2295\u1d3e Q \u27e7 \u2243 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u1d3e\u2243\u228e = \u2295\u1d3e\u2192\u228e , record { linv = \u228e\u2192\u2295\u1d3e ; is-linv = \u228e\u2192\u2295\u1d3e\u2192\u228e ; rinv = \u228e\u2192\u2295\u1d3e ; is-rinv = \u2295\u1d3e\u2192\u228e\u2192\u2295\u1d3e }\n\n    &\u1d3e\u2192\u00d7 : \u27e6 P &\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u1d3e\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192&\u1d3e : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P &\u1d3e Q \u27e7\n    \u00d7\u2192&\u1d3e (p , q) `L = p\n    \u00d7\u2192&\u1d3e (p , q) `R = q\n\n    &\u1d3e\u2192\u00d7\u2192&\u1d3e : \u2200 x \u2192 &\u1d3e\u2192\u00d7 (\u00d7\u2192&\u1d3e x) \u2261 x\n    &\u1d3e\u2192\u00d7\u2192&\u1d3e (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u1d3e\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192&\u1d3e (&\u1d3e\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u1d3e\u2192\u00d7 p = funExt \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u1d3e\u2243\u00d7 : \u27e6 P &\u1d3e Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n        &\u1d3e\u2243\u00d7 = &\u1d3e\u2192\u00d7 , record { linv = \u00d7\u2192&\u1d3e ; is-linv = \u00d7\u2192&\u1d3e\u2192\u00d7 ; rinv = \u00d7\u2192&\u1d3e ; is-rinv = &\u1d3e\u2192\u00d7\u2192&\u1d3e }\n\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP\u1d9c >>\u1d9c S = record P\u1d9c { P = \u03bb m \u2192 S (P m) }\n  where open Com_ P\u1d9c\n\nmodule _ where\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' \u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m')\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  -- the terminology used for the constructor follows the behavior of the combined process\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    send   : \u2200 {M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end (m , p)\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end (m , p) = send P m p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end      = end\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n{- Useless\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL Q f x = f x\n-}\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) R (m : M)(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a3\u1d3e _ Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (\u03a3\u1d3e _ Q) R (inl m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7))(q : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) Q R p q\n    recvR-sendR : \u2200 {M M'}P(Q : M \u2192 Proto)(R : M' \u2192 Proto)(m : M')(p : \u27e6 com P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inr m , q)\n    recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n               (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : (m : MR) \u2192 \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a0\u1d3e _ R) p q\n    sendR-recvL : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)R(m : MQ)\n                    (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : (m : MQ) \u2192 \u27e6 dual (Q m) \u214b\u1d3e R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) R (inr m , p) q\n    recvR-sendL : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (p : (m : MQ) \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(m : MQ)(q : \u27e6 dual (Q m) \u214b\u1d3e \u03a3\u1d3e _ R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inl m , q)\n    endL : \u2200 Q R\n           \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n           \u2192 View-\u2218 end Q R q qr\n    sendLM : \u2200 {MP}(P : MP \u2192 Proto)R\n               (m : MP)(p : \u27e6 P m \u27e7)(r : \u27e6 R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) end R (m , p) r\n    recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                    (m : MQ)(p : \u2200 m \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : \u27e6 dual (Q m) \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) end p (m , q)\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) _                 = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = recvR-sendR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (send ._ _ _)     = recvL-sendR _ _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (send _ _ _)     _                 = sendLM _ _ _ _ _\n\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n\nmodule _ {{_ : FunExt}} where\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply (dual P) Q pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound (dual-involutive P))) p)\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)      = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (\u03a3\u1d3e _ Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) Q R (p m) qr\n    \u214b\u1d3e-\u2218-view (recvR-sendR P Q R m pq q) = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (\u03a0\u1d3e _ Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (R m) pq (q m)\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) R p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)           = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (sendLM P R m pq qr)       = \u214b\u1d3e-sendL R m (par (P m) R pq qr)\n    \u214b\u1d3e-\u2218-view (recvL-sendR P Q m pq qr)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) end (pq m) (\u214b\u1d3e-rend' (dual (Q m)) qr)\n \n  mutual\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm P Q p = \u214b\u1d3e-comm-view (view-\u214b P Q p)\n\n    \u214b\u1d3e-comm-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm-view (sendL' P Q m p)  = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-comm (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-comm-view (sendR' P Q m' p) = inl m' , \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') p\n    \u214b\u1d3e-comm-view (recvL' P Q pq)   = \u214b\u1d3e-recvR Q P \u03bb m \u2192 \u214b\u1d3e-comm (P m) Q (pq m)\n    \u214b\u1d3e-comm-view (recvR' P Q pq)   = \u03bb m' \u2192 \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') (pq m')\n    \u214b\u1d3e-comm-view (endL Q pq)       = \u214b\u1d3e-rend' Q pq\n    \u214b\u1d3e-comm-view (send P m pq)     = m , pq\n\n  \u214b\u1d3e-comm-equiv : \u2200 P Q \u2192 Equiv (\u214b\u1d3e-comm P Q)\n  \u214b\u1d3e-comm-equiv P Q = record { linv = \u214b\u1d3e-comm Q P ; is-linv = {!!} ; rinv = \u214b\u1d3e-comm Q P ; is-rinv = {!!} }\n  {-\n    where\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 \u214b\u1d3e-comm Q P (\u214b\u1d3e-comm P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n   -}\n  \u214b\u1d3e-comm-\u2248 : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm-\u2248 P Q = _ , \u214b\u1d3e-comm-equiv P Q\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n  comma\u1d3e P Q p q with view-proc P p | view-proc Q q\n  comma\u1d3e .(com (mk Out M P)) Q .(m , p) q | send M P m p | q' = m , comma\u1d3e (P m) Q p q\n  comma\u1d3e .(com (mk In M P)) .(com (mk Out M\u2081 P\u2081)) p .(m , p\u2081) | recv M P .p | send M\u2081 P\u2081 m p\u2081 = m , comma\u1d3e (\u03a0\u1d3e M P) (P\u2081 m) p p\u2081\n  comma\u1d3e ._ ._ ._ ._ | recv M P p | recv M\u2081 P\u2081 x = [inl: (\u03bb m \u2192 comma\u1d3e (P m) (\u03a0\u1d3e _ P\u2081) (p m) x)\n                                                   ,inr: (\u03bb m' \u2192 comma\u1d3e (\u03a0\u1d3e _ P) (P\u2081 m') p (x m') ) ]\n  comma\u1d3e ._ ._ ._ ._ | recv M P x | end = x\n  comma\u1d3e .end Q .end q | end | q' = q\n\n  \u2297\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297\u1d3e-fst end      Q        pq       = _\n  \u2297\u1d3e-fst (\u03a3\u1d3e M P) Q        (m , pq) = m , \u2297\u1d3e-fst (P m) Q pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) end      pq       = pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (_ , pq) = \u2297\u1d3e-fst (\u03a0\u1d3e M P) (Q _) pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-fst (P m) (\u03a0\u1d3e N Q) (pq (inl m))\n\n  \u2297\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  \u2297\u1d3e-snd end      Q        pq       = pq\n  \u2297\u1d3e-snd (\u03a3\u1d3e M P) Q        (_ , pq) = \u2297\u1d3e-snd (P _) Q pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) end      pq       = end\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (m , pq) = m , \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) (pq (inr m))\n\n  \u00d7\u2192\u2297\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n  \u00d7\u2192\u2297\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  \u2297\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u1d3e\u2192\u00d7 P Q p = \u2297\u1d3e-fst P Q p , \u2297\u1d3e-snd P Q p\n\n  \u2297\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  \u2297\u1d3e-comma-fst end      Q        p q = refl\n  \u2297\u1d3e-comma-fst (\u03a3\u1d3e M P) Q        (m , p) q = \u03a3-ext refl (\u2297\u1d3e-comma-fst (P m) Q p q)\n  \u2297\u1d3e-comma-fst (\u03a0\u1d3e M P) end      p q = refl\n  \u2297\u1d3e-comma-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) p (m , q) = {!\u03a3-ext!}\n  \u2297\u1d3e-comma-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) p q = {!!}\n  {-\n  \u2297\u1d3e-comma-fst end      Q       p  q = refl\n  \u2297\u1d3e-comma-fst (com P)  end     p  q = refl\n  \u2297\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  \u2297\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (\u2297\u1d3e-comma-fst (P m) (com Q) p q)\n  \u2297\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 \u2297\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  \u2297\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  \u2297\u1d3e-comma-snd end     Q        p  q = refl\n  \u2297\u1d3e-comma-snd (com P) end      p  q = refl\n  \u2297\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  \u2297\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (\u2297\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  \u2297\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (\u2297\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  \u2297\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 \u2297\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  \u2297\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 \u2297\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n  -}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n  -}\n\nmodule _ {{_ : FunExt}} where\n    \u03a0\ud835\udfd8-uniq : \u2200 (F G : \ud835\udfd8 \u2192 \u2605) \u2192 \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq F G = cong (\u03a0 \ud835\udfd8) (funExt (\u03bb()))\n\n{-\nA `\u2297 B 'times', context chooses how A and B are used\nA `\u214b B 'par', \"we\" chooses how A and B are used\nA `\u2295 B 'plus', select from A or B\nA `& B 'with', offer choice of A or B\n`! A   'of course!', server accept\n`? A   'why not?', client request\n`1     unit for `\u2297\n`\u22a4     unit for `\u214b\n`0     unit for `\u2295\n`\u22a5     unit for `&\n-}\ndata CLL : \u2605 where\n  `1 `\u22a4 `0 `\u22a5 : CLL\n  _`\u2297_ _`\u214b_ _`\u2295_ _`&_ : (A B : CLL) \u2192 CLL\n  -- `!_ `?_ : (A : CLL) \u2192 CLL\n\n_\u22a5 : CLL \u2192 CLL\n`1 \u22a5 = `\u22a4\n`\u22a4 \u22a5 = `1\n`0 \u22a5 = `\u22a5\n`\u22a5 \u22a5 = `0\n(A `\u2297 B)\u22a5 = A \u22a5 `\u214b B \u22a5\n(A `\u214b B)\u22a5 = A \u22a5 `\u2297 B \u22a5\n(A `\u2295 B)\u22a5 = A \u22a5 `& B \u22a5\n(A `& B)\u22a5 = A \u22a5 `\u2295 B \u22a5\n{-\n(`! A)\u22a5 = `?(A \u22a5)\n(`? A)\u22a5 = `!(A \u22a5)\n-}\n\nCLL-proto : CLL \u2192 Proto\nCLL-proto `1       = end  -- TODO\nCLL-proto `\u22a4       = end  -- TODO\nCLL-proto `0       = \u03a0\u1d3e \ud835\udfd8 \u03bb()\nCLL-proto `\u22a5       = \u03a3\u1d3e \ud835\udfd8 \u03bb()\nCLL-proto (A `\u2297 B) = CLL-proto A \u2297\u1d3e CLL-proto B\nCLL-proto (A `\u214b B) = CLL-proto A \u214b\u1d3e CLL-proto B\nCLL-proto (A `\u2295 B) = CLL-proto A \u2295\u1d3e CLL-proto B\nCLL-proto (A `& B) = CLL-proto A &\u1d3e CLL-proto B\n\n{- The point of this could be to devise a particular equivalence\n   relation for processes. It could properly deal with \u214b. -}\n\n  {-\nmodule V4 {{_ : FunExt}} where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _ {{_ : FunExt}} where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _ {{_ : FunExt}} where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _ {{_ : FunExt}}\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c9be4e22888383844a7b653557e9463330618caa","subject":"more startx proofs","message":"more startx proofs\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata Mod : \u2605 where S D : Mod\n\n\u2192M : \u2200 {a b} \u2192 Mod \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (a \u2294 b)\n\u2192M S A B = ..(_ : A) \u2192 B\n\u2192M D A B = A \u2192 B\n\n\u03a0M : \u2200 {a b}(m : Mod) \u2192 (A : \u2605_ a) \u2192 (B : \u2192M m A (\u2605_ b)) \u2192 \u2605_ (a \u2294 b)\n\u03a0M S A B = \u03a0\u00b7 A B\n\u03a0M D A B = \u03a0 A B\n\n-- appM : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2192M m A (\u2605_ b)}(P : \u03a0M m A B)(x : A) \u2192 B\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (f : Mod)(A : \u2605)(B : \u2192M f A Proto) \u2192 Proto\n\n{-\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' D A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' D A \u03bb _ \u2192 B\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' S A B) = \u03a3' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a0' D A B) = \u03a3' D A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' S A B) = \u03a0' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' D A B) = \u03a0' D A (\u03bb x \u2192 dual (B x))\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u03a3'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' S A B) (\u03a3' S A B')\n  \u03a0\u03a3'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' D A B) (\u03a3' D A B')\n  \u03a3\u03a0'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' S A B) (\u03a0' S A B')\n  \u03a3\u03a0'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' D A B) (\u03a0' D A B')\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u03a3'S f) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a0\u03a3'D f) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'S f) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'D f) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0' S A B) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a0' D A B) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' S A B) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' D A B) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-spec (B x))\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recvD : \u2200 {M P} \u2192 (\u03a0M D M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' D M P)\n  recvS : \u2200 {M P} \u2192 (\u03a0M S M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' S M P)\n  sendD : \u2200 {M P} \u2192 \u03a0M D M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' D M P))\n  sendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' S M P))\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {P Q} \u2192 SimL P Q \u2192 Sim P Q\n    right : \u2200 {P Q} \u2192 SimR P Q \u2192 Sim P Q\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' S A B) = right (recvS (\u03bb x \u2192 left (sendS x (sim-id (B x)))))\nsim-id (\u03a0' D A B) = right (recvD (\u03bb x \u2192 left (sendD x (sim-id (B x)))))\nsim-id (\u03a3' S A B) = left (recvS (\u03bb x \u2192 right (sendS x (sim-id (B x)))))\nsim-id (\u03a3' D A B) = left (recvD (\u03bb x \u2192 right (sendD x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp end (right x) end = right x\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u03a3'S x\u2081) (recvS x) (sendS x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a0\u03a3'D x\u2081) (recvD x) (sendD x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u03a0'S x) (sendS x\u2081 x\u2082) (recvS x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\nsim-compRL (\u03a3\u03a0'D x) (sendD x\u2081 x\u2082) (recvD x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recvD PQ) QR = recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (recvS PQ) QR = recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (sendD m PQ) QR = sendD m (sim-comp Q-Q' PQ QR)\nsim-compL Q-Q' (sendS m PQ) QR = sendS m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recvD QR)   = recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (recvS QR)   = recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x))\nsim-compR Q-Q' PQ (sendD m QR) = sendD m (sim-comp Q-Q' PQ QR)\nsim-compR Q-Q' PQ (sendS m QR) = sendS m (sim-comp Q-Q' PQ QR)\n\n{-\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-comp Q-Q' (left (recvD PQ)) QR = left (recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (recvS PQ)) QR = left (recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (sendD m PQ)) QR = left (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' (left (sendS m PQ)) QR = left (sendS m (sim-comp Q-Q' PQ QR))\nsim-comp end (right ()) (left x\u2081)\nsim-comp end end QR = QR\nsim-comp end PQ end = PQ\nsim-comp (\u03a0\u03a3'S Q-Q') (right (recvS PQ)) (left (sendS m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a0\u03a3'D Q-Q') (right (recvD PQ)) (left (sendD m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a3\u03a0'S Q-Q') (right (sendS m PQ)) (left (recvS QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp (\u03a3\u03a0'D Q-Q') (right (sendD m PQ)) (left (recvD QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp Q-Q' PQ (right (recvD QR)) = right (recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m)))\nsim-comp Q-Q' PQ (right (recvS QR)) = right (recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x)))\nsim-comp Q-Q' PQ (right (sendD m QR)) = right (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' PQ (right (sendS m QR)) = right (sendS m (sim-comp Q-Q' PQ QR))\n-}\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symL (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-symR (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symR (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (recvD P)) = do (recvD (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (recvS P)) = do (recvS (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (sendD m P)) = do (sendD m (sim-unit P))\nsim-unit (right (sendS m P)) = do (sendS m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u03a3'S x) = cong \u03a0\u03a3'S (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a0\u03a3'D x) = cong \u03a0\u03a3'D (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u03a0'S x) = cong \u03a3\u03a0'S (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u03a0'D x) = cong \u03a3\u03a0'D (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end P-P' Q-Q' (left ()) PQ QR\n  sim-comp-assoc-end end Q-Q' (right ()) (left PQ) QR\n  sim-comp-assoc-end (\u03a0\u03a3'S x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a0\u03a3'D x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u03a0'S x) Q-Q' (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end (\u03a3\u03a0'D x) Q-Q' (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) (left x\u2081)\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'S x\u2081) (right \u00f8P) (right (recvS x)) (left (sendS x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'D x\u2081) (right \u00f8P) (right (recvD x)) (left (sendD x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'S x) (right \u00f8P) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'D x) (right \u00f8P) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvD x\u2081))\n    = cong (right \u2218 recvD) (funExtD (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvS x\u2081))\n    = cong (right \u2218 recvS) (funExtS (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendD x\u2081 x\u2082))\n    = cong (right \u2218 sendD x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendS x\u2081 x\u2082))\n    = cong (right \u2218 sendS x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) end\n  sim-comp-assoc-end end end (right \u00f8P) end QR = refl\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recvD x))    = cong (left \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (recvS x))    = cong (left \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (sendD x x\u2081)) = cong (left \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (left (sendS x x\u2081)) = cong (left \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! (right (recvD x))    = cong (right \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (recvS x))    = cong (right \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (sendD x x\u2081)) = cong (right \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (right (sendS x x\u2081)) = cong (right \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end Q-Q' (left ()) QR\n  sim-comp-!-end end (right ()) (left x\u2081)\n  sim-comp-!-end (\u03a0\u03a3'S x\u2081) (right (recvS x)) (left (sendS x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a0\u03a3'D x\u2081) (right (recvD x)) (left (sendD x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u03a0'S x) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end (\u03a3\u03a0'D x) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recvD x\u2081))\n    = cong (left \u2218 recvD) (funExtD (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (recvS x\u2081))\n    = cong (left \u2218 recvS) (funExtS (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (sendD x\u2081 x\u2082))\n    = cong (left \u2218 sendD x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end Q-Q' (right x) (right (sendS x\u2081 x\u2082))\n    = cong (left \u2218 sendS x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end (right x) end = refl\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExtD funExtS R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExtD funExtS Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata Mod : \u2605 where S D : Mod\n\n\u2192M : \u2200 {a b} \u2192 Mod \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (a \u2294 b)\n\u2192M S A B = ..(_ : A) \u2192 B\n\u2192M D A B = A \u2192 B\n\n\u03a0M : \u2200 {a b}(m : Mod) \u2192 (A : \u2605_ a) \u2192 (B : \u2192M m A (\u2605_ b)) \u2192 \u2605_ (a \u2294 b)\n\u03a0M S A B = \u03a0\u00b7 A B\n\u03a0M D A B = \u03a0 A B\n\n-- appM : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2192M m A (\u2605_ b)}(P : \u03a0M m A B)(x : A) \u2192 B\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (f : Mod)(A : \u2605)(B : \u2192M f A Proto) \u2192 Proto\n\n{-\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' D A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' D A \u03bb _ \u2192 B\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' S A B) = \u03a3' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a0' D A B) = \u03a3' D A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' S A B) = \u03a0' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' D A B) = \u03a0' D A (\u03bb x \u2192 dual (B x))\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u03a3'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' S A B) (\u03a3' S A B')\n  \u03a0\u03a3'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' D A B) (\u03a3' D A B')\n  \u03a3\u03a0'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' S A B) (\u03a0' S A B')\n  \u03a3\u03a0'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' D A B) (\u03a0' D A B')\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u03a3'S f) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a0\u03a3'D f) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'S f) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'D f) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0' S A B) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a0' D A B) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' S A B) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' D A B) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-spec (B x))\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recvD : \u2200 {M P} \u2192 (\u03a0M D M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' D M P)\n  recvS : \u2200 {M P} \u2192 (\u03a0M S M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' S M P)\n  sendD : \u2200 {M P} \u2192 \u03a0M D M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' D M P))\n  sendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' S M P))\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {P Q} \u2192 SimL P Q \u2192 Sim P Q\n    right : \u2200 {P Q} \u2192 SimR P Q \u2192 Sim P Q\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' S A B) = right (recvS (\u03bb x \u2192 left (sendS x (sim-id (B x)))))\nsim-id (\u03a0' D A B) = right (recvD (\u03bb x \u2192 left (sendD x (sim-id (B x)))))\nsim-id (\u03a3' S A B) = left (recvS (\u03bb x \u2192 right (sendS x (sim-id (B x)))))\nsim-id (\u03a3' D A B) = left (recvD (\u03bb x \u2192 right (sendD x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp end (right x) end = right x\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u03a3'S x\u2081) (recvS x) (sendS x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a0\u03a3'D x\u2081) (recvD x) (sendD x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u03a0'S x) (sendS x\u2081 x\u2082) (recvS x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\nsim-compRL (\u03a3\u03a0'D x) (sendD x\u2081 x\u2082) (recvD x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recvD PQ) QR = recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (recvS PQ) QR = recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (sendD m PQ) QR = sendD m (sim-comp Q-Q' PQ QR)\nsim-compL Q-Q' (sendS m PQ) QR = sendS m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recvD QR)   = recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (recvS QR)   = recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x))\nsim-compR Q-Q' PQ (sendD m QR) = sendD m (sim-comp Q-Q' PQ QR)\nsim-compR Q-Q' PQ (sendS m QR) = sendS m (sim-comp Q-Q' PQ QR)\n\n{-\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-comp Q-Q' (left (recvD PQ)) QR = left (recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (recvS PQ)) QR = left (recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (sendD m PQ)) QR = left (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' (left (sendS m PQ)) QR = left (sendS m (sim-comp Q-Q' PQ QR))\nsim-comp end (right ()) (left x\u2081)\nsim-comp end end QR = QR\nsim-comp end PQ end = PQ\nsim-comp (\u03a0\u03a3'S Q-Q') (right (recvS PQ)) (left (sendS m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a0\u03a3'D Q-Q') (right (recvD PQ)) (left (sendD m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a3\u03a0'S Q-Q') (right (sendS m PQ)) (left (recvS QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp (\u03a3\u03a0'D Q-Q') (right (sendD m PQ)) (left (recvD QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp Q-Q' PQ (right (recvD QR)) = right (recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m)))\nsim-comp Q-Q' PQ (right (recvS QR)) = right (recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x)))\nsim-comp Q-Q' PQ (right (sendD m QR)) = right (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' PQ (right (sendS m QR)) = right (sendS m (sim-comp Q-Q' PQ QR))\n-}\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\nsim-sym : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\nsim-sym (left x) = right (sim-symL x)\nsim-sym (right x) = left (sim-symR x)\nsim-sym end = end\n\nsim-symL (recvD PQ)   = recvD (\u03bb m \u2192 sim-sym (PQ m))\nsim-symL (recvS PQ)   = recvS (\u03bb m \u2192 sim-sym (PQ m))\nsim-symL (sendD m PQ) = sendD m (sim-sym PQ)\nsim-symL (sendS m PQ) = sendS m (sim-sym PQ)\n\nsim-symR (recvD PQ)   = recvD (\u03bb m \u2192 sim-sym (PQ m))\nsim-symR (recvS PQ)   = recvS (\u03bb m \u2192 sim-sym (PQ m))\nsim-symR (sendD m PQ) = sendD m (sim-sym PQ)\nsim-symR (sendS m PQ) = sendS m (sim-sym PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (recvD P)) = do (recvD (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (recvS P)) = do (recvS (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (sendD m P)) = do (sendD m (sim-unit P))\nsim-unit (right (sendS m P)) = do (sendS m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end P-P' Q-Q' (left ()) PQ QR\n  sim-comp-assoc-end end Q-Q' (right ()) (left PQ) QR\n  sim-comp-assoc-end (\u03a0\u03a3'S x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a0\u03a3'D x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u03a0'S x) Q-Q' (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end (\u03a3\u03a0'D x) Q-Q' (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) (left x\u2081)\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'S x\u2081) (right \u00f8P) (right (recvS x)) (left (sendS x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'D x\u2081) (right \u00f8P) (right (recvD x)) (left (sendD x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'S x) (right \u00f8P) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'D x) (right \u00f8P) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvD x\u2081))\n    = cong (right \u2218 recvD) (funExtD (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvS x\u2081))\n    = cong (right \u2218 recvS) (funExtS (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendD x\u2081 x\u2082))\n    = cong (right \u2218 sendD x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendS x\u2081 x\u2082))\n    = cong (right \u2218 sendS x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) end\n  sim-comp-assoc-end end end (right \u00f8P) end QR = refl\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\nsim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n  \u2192 sim-comp (Dual-sym Q-Q') (sim-sym Q'R) (sim-sym PQ) \u2261 sim-sym (sim-comp Q-Q' PQ Q'R)\nsim-comp-! Q-Q' (left (recvD x)) (left x\u2081) = {!!}\nsim-comp-! Q-Q' (left (recvS x)) (left x\u2081) = {!!}\nsim-comp-! Q-Q' (left (sendD x x\u2081)) (left x\u2082) = cong (right \u2218 sendD x) (sim-comp-! Q-Q' x\u2081 (left x\u2082))\nsim-comp-! Q-Q' (left (sendS x x\u2081)) (left x\u2082) = cong (right \u2218 sendS x) (sim-comp-! Q-Q' x\u2081 (left x\u2082))\nsim-comp-! Q-Q' (left x) (right x\u2081) = {!!}\nsim-comp-! Q-Q' (left x) end = {!!}\nsim-comp-! Q-Q' (right x) QR = {!!}\nsim-comp-! end end (left x) = refl\nsim-comp-! end end (right x) = {!!}\nsim-comp-! end end end = refl\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4dd4448f43898fbd80877ba687ffc2fbc1558a5d","subject":"Readd instance","message":"Readd instance\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d_\u208e {{change-algebra-base}} \u03b9\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n    change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n    change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d_\u208e {{environment-changes}} \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d_\u208e {{change-algebra-base}} \u03b9\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d_\u208e {{environment-changes}} \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9f49923507a898888e19681f5d20c3540f5cf834","subject":"Added foldr.","message":"Added foldr.\n\nIgnore-this: ea015214744665e73d11c17cfab71321\n\ndarcs-hash:20100702160458-3bd4e-fbf3e5957dd6548bc9fe93f1963cdf972226d6c0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List.agda","new_file":"LTC\/Data\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\n\npostulate\n  length : D \u2192 D\n  length-[] : length []                     \u2261 zero\n  length-\u2237  : ( d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D ) \u2192 map f []           \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\npostulate\n  replicate : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192 replicate zero d       \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n-- Reducing lists\n\npostulate\n  foldr    : D \u2192 D \u2192 D \u2192 D\n  foldr-[] : (f n : D) \u2192 foldr f n []            \u2261 n\n  foldr-\u2237  : (f n d ds : D) \u2192 foldr f n (d \u2237 ds) \u2261 f \u2219 d \u2219 (foldr f n ds)\n{-# ATP axiom foldr-[] #-}\n{-# ATP axiom foldr-\u2237 #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","old_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\n\npostulate\n  length : D \u2192 D\n  length-[] : length []                     \u2261 zero\n  length-\u2237  : ( d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D ) \u2192 map f []           \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\npostulate\n  replicate : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192 replicate zero d       \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"48205df178d2e57970bb42813b33f76e3c63c5ed","subject":"Minor changes.","message":"Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Functors.agda","new_file":"notes\/fixed-points\/Functors.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conaturals type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- TODO: The conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Functors where\n\n-- The carrier of the initial algebra is (up to isomorphism) a\n-- fixed-point of the functor (Vene 2000, p).\n\n------------------------------------------------------------------------------\n\ndata Bool : Set where\n  false true : Bool\n\ndata _\u228e_  (A B : Set) : Set where\n  inl : A \u2192 A \u228e B\n  inr : B \u2192 A \u228e B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata One : Set where\n   one : One\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = One \u228e X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = One \u228e (A \u00d7 X)\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl one)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl one)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conaturals type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl one)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- TODO: The conat destructor.\npred : Conat \u2192 One \u228e Conat\npred cn with out cn\n... | inl _ = inl one\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl one)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a03a22bdb93489d665eb5cd0150477b68a74b8a7","subject":"Inline Structure module","message":"Inline Structure module\n\nOld-commit-hash: a1f90eeec3bbf0ee39c738ae1cb8b8557443544c\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule Structure where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely.\n    --\n    -- * That should be be true for\n    --   functions using changes parametrically.\n    --\n    -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n    --   this is proved below on both contexts at once by fun-change-respects.\n    --\n    -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n    --   it by definition, and \u229f doesn't take change arguments - we will only\n    --   need a proof for compose, when we define it.\n    --\n    -- Stating the general result, though, seems hard, we should\n    -- rather have lemmas proving that certain classes of functions respect this\n    -- equivalence.\n\n  module _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n    {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n    module FC = FunctionChanges A B {{CA}} {{CB}}\n    open FC using (changeAlgebra; incrementalization)\n    open FC.FunctionChange\n\n    fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n      df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n    fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n      where\n        open \u2261-Reasoning\n        -- This type signature just expands the goal manually a bit.\n        lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n        -- Informally: use incrementalization on both sides and then apply\n        -- congruence.\n        lemma =\n          begin\n            f x \u229e apply df\u2081 x dx\u2081\n          \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2081)\n          \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n            (f \u229e df\u2081) (x \u229e dx\u2082)\n          \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n            (f \u229e df\u2082) (x \u229e dx\u2082)\n          \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n            f x \u229e apply df\u2082 x dx\u2082\n          \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"be6f1fa3a0adbb680ac349286c28d45c17722184","subject":"Fix some concrete non-sense about the division","message":"Fix some concrete non-sense about the division\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1]-ops (]0,1] : Set) (_\u2264E_ : ]0,1] \u2192 ]0,1] \u2192 Set) : Set where\n  constructor mk\n  field     \n    1E : ]0,1]\n    _+E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\u00b7E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\/E_<_> : (x y : ]0,1]) \u2192 x \u2264E y \u2192 ]0,1]\n\nmodule [0,1] {]0,1] _\u2264E_} (]0,1]R : ]0,1]-ops ]0,1] _\u2264E_) where\n\n  open ]0,1]-ops ]0,1]R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   _I : ]0,1] \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    E\u2264E : \u2200 {x y} \u2192 x \u2264E y \u2192 (x I) \u2264I (y I)\n\n  1I : [0,1]\n  1I = 1E I\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos (_ I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Pos x \u2192 ]0,1] \n  0I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  x I +I 0I = x I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  (x I) \u00b7I 0I = 0I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  _  \/I 0I < _ , () >\n  0I \/I _  < _ , _ > = 0I\n  (x I) \/I (y I) < E\u2264E pf , _ > = (x \/E y < pf >)I\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  postulate\n    +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n    +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n\n    \u2264I-refl : {x : [0,1]} \u2192 x \u2264I x\n\n    \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n    \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n    \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n\nmodule Univ {]0,1] _\u2264E_} (]0,1]R : ]0,1]-ops ]0,1] _\u2264E_)\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1]R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  module Prob (P : U \u2192 [0,1])\n              (sumP\u22611 : sumP P allU \u2261 1I) where\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1]-ops (]0,1] : Set) : Set where\n  field     \n    1E : ]0,1]\n    _+E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\u00b7E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    1\/E_ : ]0,1] \u2192 ]0,1]\n\nmodule [0,1] {]0,1]}(]0,1]R : ]0,1]-ops ]0,1]) where\n\n  open ]0,1]-ops ]0,1]R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   _I : ]0,1] \u2192 [0,1]\n\n  1I : [0,1]\n  1I = 1E I\n\n  Pos : [0,1] \u2192 Set\n  Pos x = \u00ac (x \u2261 0I)\n\n  _<_> : (x : [0,1]) \u2192 Pos x \u2192 ]0,1] \n  0I < pos > with pos refl\n  ... | ()\n  (x I) < pos > = x\n\n  \n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  x I +I 0I = x I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  (x I) \u00b7I 0I = 0I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  1\/I_ : ]0,1] \u2192 [0,1]\n  1\/I x = (1\/E x) I\n\n  _\/I_ : [0,1] \u2192 ]0,1] \u2192 [0,1]\n  x \/I y = x \u00b7I (1\/I y)\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  postulate\n    +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n    +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n\n    _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set\n    \u2264I-refl : {x : [0,1]} \u2192 x \u2264I x\n\n    \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n    \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n    \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n\nmodule Univ ( ]0,1] : Set)(]0,1]R : ]0,1]-ops ]0,1])\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1]R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  module Prob (P : U \u2192 [0,1])\n              (sumP\u22611 : sumP P allU \u2261 1I) where\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < pr >  \n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b) ","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d2408aa967f85d327836b63f65e967e844774910","subject":"Cleaning","message":"Cleaning\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Derived.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Derived.agda","new_contents":"{-# OPTIONS --type-in-type --no-pattern-matching #-}\nopen import Spire.DarkwingDuck.Primitive\nmodule Spire.DarkwingDuck.Derived where\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEnum : Set\nEnum = List String\n\nTag : Enum \u2192 Set\nTag xs = Elem String xs\n\nproj\u2081 : \u2200{A B} \u2192 \u03a3 A B \u2192 A\nproj\u2081 = elimPair _ (\u03bb a b \u2192 a)\n\nproj\u2082 : \u2200{A B} (ab : \u03a3 A B) \u2192 B (proj\u2081 ab)\nproj\u2082 = elimPair _ (\u03bb a b \u2192 b)\n\nBranchesM : Enum \u2192 Set\nBranchesM E = (P : Tag E \u2192 Set) \u2192 Set\n\nBranches : (E : Enum) \u2192 BranchesM E\nBranches = elimList BranchesM\n  (\u03bb P \u2192 \u22a4)\n  (\u03bb l E ih P \u2192 \u03a3 (P here) (\u03bb _ \u2192 ih (\u03bb t \u2192 P (there t))))\n\nCase : (E : Enum) \u2192 Tag E \u2192 Set\nCase E t = (P : Tag E \u2192 Set) (cs : Branches E P) \u2192 P t\n\ncase' : (E : Enum) (t : Tag E) \u2192 Case E t\ncase' = elimElem String Case\n  (\u03bb l E P c,cs \u2192 elimPair (\u03bb _ \u2192 P here) (\u03bb a b \u2192 a) c,cs)\n  (\u03bb l E t ih P c,cs \u2192 ih (\u03bb t \u2192 P (there t)) (elimPair (\u03bb _ \u2192 Branches E (\u03bb t \u2192 P (there t))) (\u03bb a b \u2192 b) c,cs))\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase E P cs t = case' E t P cs\n\nScope  : Tel \u2192 Set\nScope = elimTel (\u03bb _ \u2192 Set) \u22a4 (\u03bb A B ih \u2192 \u03a3 A ih)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranchesM : Enum \u2192 Set\nCurriedBranchesM E = (P : Tag E \u2192 Set) (X : Set) \u2192 Set\n\nCurriedBranches : (E : Enum) \u2192 CurriedBranchesM E\nCurriedBranches = elimList CurriedBranchesM\n  (\u03bb P X \u2192 X)\n  (\u03bb l E ih P X \u2192 P here \u2192 ih (\u03bb t \u2192 P (there t)) X)\n\nCurryBranches : Enum \u2192 Set\nCurryBranches E = (P : Tag E \u2192 Set) (X : Set) \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\n\ncurryBranches : (E : Enum) \u2192 CurryBranches E\ncurryBranches = elimList CurryBranches\n  (\u03bb P X f \u2192 f tt)\n  (\u03bb l E ih P X f c \u2192 ih (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs)))\n\n----------------------------------------------------------------------\n\nUncurriedScope : (T : Tel) (X : Scope T \u2192 Set) \u2192 Set\nUncurriedScope T X = (xs : Scope T) \u2192 X xs\n\nCurriedScope : (T : Tel) (X : Scope T \u2192 Set) \u2192 Set\nCurriedScope = elimTel\n  (\u03bb T \u2192 (X : Scope T \u2192 Set) \u2192 Set)\n  (\u03bb X \u2192 X tt)\n  (\u03bb A B ih X \u2192 (a : A) \u2192 ih a (\u03bb b \u2192 X (a , b)))\n\nCurryScope : Tel \u2192 Set\nCurryScope T = (X : Scope T \u2192 Set) \u2192 UncurriedScope T X \u2192 CurriedScope T X\n\ncurryScope : (T : Tel) \u2192 CurryScope T\ncurryScope = elimTel CurryScope\n  (\u03bb X f \u2192 f tt)\n  (\u03bb A B ih X f a \u2192 ih a (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b)))\n\nUncurryScope : Tel \u2192 Set\nUncurryScope T = (X : Scope T \u2192 Set) \u2192 CurriedScope T X \u2192 UncurriedScope T X\n\nuncurryScope : (T : Tel) \u2192 UncurryScope T\nuncurryScope = elimTel UncurryScope\n  (\u03bb X x \u2192 elimUnit X x)\n  (\u03bb A B ih X f \u2192 elimPair X (\u03bb a b \u2192 ih a (\u03bb b \u2192 X (a , b)) (f a) b))\n\n----------------------------------------------------------------------\n\nUncurriedFunc : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedFunc I D X = (i : I) \u2192 Func I D X i \u2192 X i\n\nCurriedFunc : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedFunc I = elimDesc\n  (\u03bb _ \u2192 (X : ISet I) \u2192 Set)\n  (\u03bb i X \u2192 X i)\n  (\u03bb i D ih X \u2192 (x : X i) \u2192 ih X )\n  (\u03bb A B ih X \u2192 (a : A) \u2192 ih a X)\n\ncurryFunc : (I : Set) (D : Desc I) (X : ISet I)\n  \u2192 UncurriedFunc I D X \u2192 CurriedFunc I D X\ncurryFunc I = elimDesc\n  (\u03bb D \u2192 (X : ISet I) \u2192 UncurriedFunc I D X \u2192 CurriedFunc I D X)\n  (\u03bb i X cn \u2192 cn i refl)\n  (\u03bb i D ih X cn x \u2192 ih X (\u03bb j xs \u2192 cn j (x , xs)))\n  (\u03bb A B ih X cn a \u2192 ih a X (\u03bb j xs \u2192 cn j (a , xs)))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedFunc I D X)\n  \u2192 Set\nUncurriedHyps I D X P cn =\n  (i : I) (xs : Func I D X i) (ihs : Hyps I D X P i xs) \u2192 P i (cn i xs)\n\nCurriedHypsM : (I : Set) (D : Desc I) \u2192 Set\nCurriedHypsM I D = (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (cn : UncurriedFunc I D X) \u2192 Set\n\nCurriedHyps : (I : Set) (D : Desc I) \u2192 CurriedHypsM I D\nCurriedHyps I = elimDesc (CurriedHypsM I)\n  (\u03bb i X P cn \u2192 P i (cn i refl))\n  (\u03bb i D ih X P cn \u2192 (x : X i) \u2192 P i x \u2192 ih X P (\u03bb j xs \u2192 cn j (x , xs)))\n  (\u03bb A B ih X P cn \u2192 (a : A) \u2192 ih a X P (\u03bb j xs \u2192 cn j (a , xs)))\n\nUncurryHyps : (I : Set) (D : Desc I) \u2192 Set\nUncurryHyps I D = (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (cn : UncurriedFunc I D X)\n  \u2192 CurriedHyps I D X P cn \u2192 UncurriedHyps I D X P cn\n\nuncurryHyps : (I : Set) (D : Desc I) \u2192 UncurryHyps I D\nuncurryHyps I = elimDesc\n  (UncurryHyps I)\n  (\u03bb j X P cn pf \u2192\n    elimEq _ (\u03bb u \u2192 pf))\n  (\u03bb j D ih X P cn pf i \u2192\n    elimPair _ (\u03bb x xs ih,ihs \u2192\n      ih X P (\u03bb j ys \u2192 cn j (x , ys)) (pf x (proj\u2081 ih,ihs)) i xs (proj\u2082 ih,ihs)))\n  (\u03bb A B ih X P cn pf i \u2192\n    elimPair _ (\u03bb a xs \u2192 ih a X P (\u03bb j ys \u2192 cn j (a , ys)) (pf a) i xs))\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    N : String\n    P : Tel\n    I : Scope P \u2192 Tel\n    E : Enum\n    B : (p : Scope P) \u2192 Branches E (\u03bb _ \u2192 Desc (Scope (I p)))\n\n  PS : Set\n  PS = Scope P\n\n  IS : PS \u2192 Set\n  IS p = Scope (I p)\n\n  C : (p : PS) \u2192 Tag E \u2192 Desc (IS p)\n  C p = case E (\u03bb _ \u2192 Desc (IS p)) (B p)\n\n  D : (p : PS) \u2192 Desc (IS p)\n  D p = Arg (Tag E) (C p)\n\n  F : (p : PS) \u2192 IS p \u2192 Set\n  F p = \u03bc N PS IS p (D p)\n\n----------------------------------------------------------------------\n\nDecl :\n  (N : String)\n  (P : Tel)\n  (I : CurriedScope P (\u03bb _ \u2192 Tel))\n  (E : Enum)\n  (B : let I = uncurryScope P (\u03bb _ \u2192 Tel) I\n      in CurriedScope P \u03bb A \u2192 Branches E (\u03bb _ \u2192 Desc (Scope (I A))))\n  \u2192 Data\nDecl N P I E B = record\n  { N = N\n  ; P = P\n  ; I = uncurryScope P _ I\n  ; E = E\n  ; B = uncurryScope P _ B\n  }\n\n----------------------------------------------------------------------\n\nEnd[_] : (I : Tel)\n  \u2192 CurriedScope I (\u03bb _ \u2192 Desc (Scope I))\nEnd[_] I = curryScope I _ End\n\nRec[_] : (I : Tel)\n  \u2192 CurriedScope I (\u03bb _ \u2192 Desc (Scope I) \u2192 Desc (Scope I))\nRec[_] I = curryScope I _ Rec\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 UncurriedScope (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried = Data.F\n\nForm : (R : Data)\n  \u2192 CurriedScope (Data.P R) \u03bb p\n  \u2192 CurriedScope (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryScope (Data.P R) (\u03bb p \u2192 CurriedScope (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryScope (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 CurriedFunc (Data.IS R p) (Data.D R p) (Data.F R p)\ninjUncurried R p t = curryFunc (Data.IS R p) (Data.C R p t)\n  (Data.F R p)\n  (\u03bb i xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 CurriedScope (Data.P R) \u03bb p\n  \u2192 CurriedFunc (Data.IS R p) (Data.D R p) (Data.F R p)\ninj R = curryScope (Data.P R)\n  (\u03bb p \u2192 CurriedFunc (Data.IS R p) (Data.D R p) (Data.F R p))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p D i \u2192 Set)\n  (f : CurriedHyps (I p) D (\u03bc \u2113 P I p D) M (\u03bb _ \u2192 init))\n  (i : I p)\n  (x : \u03bc \u2113 P I p D i)\n  \u2192 M i x\nindCurried \u2113 P I p D M f i x =\n  ind \u2113 P I p D M (uncurryHyps (I p) D (\u03bc \u2113 P I p D) M (\u03bb _ \u2192 init) f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M in\n  CurriedHyps (Data.IS R p) (Data.C R p t) (Data.F R p) unM (\u03bb i xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n  \u2192 let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n  in UncurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x))\nelimUncurried R p M cs = let\n    unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n  in\n  curryScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x) \u03bb i x \u2192\n  indCurried (Data.N R) (Data.PS R) (Data.IS R) p (Data.D R p) unM\n    (case (Data.E R) (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 CurriedScope (Data.P R) \u03bb p\n  \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n  \u2192 let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x))\nelim R = curryScope (Data.P R)\n  (\u03bb p \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n    \u2192 let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p M)\n       (CurriedScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x)))\n  (\u03bb p M \u2192 curryBranches (Data.E R) _ _\n    (elimUncurried R p M))\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type --no-pattern-matching #-}\nopen import Spire.DarkwingDuck.Primitive\nmodule Spire.DarkwingDuck.Derived where\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEnum : Set\nEnum = List String\n\nTag : Enum \u2192 Set\nTag xs = Elem String xs\n\nproj\u2081 : \u2200{A B} \u2192 \u03a3 A B \u2192 A\nproj\u2081 = elimPair _ (\u03bb a b \u2192 a)\n\nproj\u2082 : \u2200{A B} (ab : \u03a3 A B) \u2192 B (proj\u2081 ab)\nproj\u2082 = elimPair _ (\u03bb a b \u2192 b)\n\nBranches : (E : List String) (P : Tag E \u2192 Set) \u2192 Set\nBranches = elimList (\u03bb E \u2192 (P : Tag E \u2192 Set) \u2192 Set)\n  (\u03bb P \u2192 \u22a4)\n  (\u03bb l E ih P \u2192 \u03a3 (P here) (\u03bb _ \u2192 ih (\u03bb t \u2192 P (there t))))\n\ncase' : (E : List String) (t : Tag E) (P : Tag E \u2192 Set) (cs : Branches E P) \u2192 P t\ncase' = elimElem String (\u03bb E t \u2192 (P : Tag E \u2192 Set) (cs : Branches E P) \u2192 P t)\n  (\u03bb l E P c,cs \u2192 elimPair (\u03bb _ \u2192 P here) (\u03bb a b \u2192 a) c,cs)\n  (\u03bb l E t ih P c,cs \u2192 ih (\u03bb t \u2192 P (there t)) (elimPair (\u03bb _ \u2192 Branches E (\u03bb t \u2192 P (there t))) (\u03bb a b \u2192 b) c,cs))\n\ncase : {E : List String} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P cs t = case' _ t P cs\n\nScope  : Tel \u2192 Set\nScope = elimTel (\u03bb _ \u2192 Set) \u22a4 (\u03bb A B ih \u2192 \u03a3 A ih)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches = elimList\n  (\u03bb E \u2192 (P : Tag E \u2192 Set) (X : Set) \u2192 Set)\n  (\u03bb P X \u2192 X)\n  (\u03bb l E ih P X \u2192 P here \u2192 ih (\u03bb t \u2192 P (there t)) X)\n\nCurryBranches : Enum \u2192 Set\nCurryBranches E = (P : Tag E \u2192 Set) (X : Set) \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\n\ncurryBranches : (E : Enum) \u2192 CurryBranches E\ncurryBranches = elimList CurryBranches\n  (\u03bb P X f \u2192 f tt)\n  (\u03bb l E ih P X f c \u2192 ih (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs)))\n\n----------------------------------------------------------------------\n\nUncurriedScope : (T : Tel) (X : Scope T \u2192 Set) \u2192 Set\nUncurriedScope T X = (xs : Scope T) \u2192 X xs\n\nCurriedScope : (T : Tel) (X : Scope T \u2192 Set) \u2192 Set\nCurriedScope = elimTel\n  (\u03bb T \u2192 (X : Scope T \u2192 Set) \u2192 Set)\n  (\u03bb X \u2192 X tt)\n  (\u03bb A B ih X \u2192 (a : A) \u2192 ih a (\u03bb b \u2192 X (a , b)))\n\nCurryScope : Tel \u2192 Set\nCurryScope T = (X : Scope T \u2192 Set) \u2192 UncurriedScope T X \u2192 CurriedScope T X\n\ncurryScope : (T : Tel) \u2192 CurryScope T\ncurryScope = elimTel CurryScope\n  (\u03bb X f \u2192 f tt)\n  (\u03bb A B ih X f a \u2192 ih a (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b)))\n\nUncurryScope : Tel \u2192 Set\nUncurryScope T = (X : Scope T \u2192 Set) \u2192 CurriedScope T X \u2192 UncurriedScope T X\n\nuncurryScope : (T : Tel) \u2192 UncurryScope T\nuncurryScope = elimTel UncurryScope\n  (\u03bb X x \u2192 elimUnit X x)\n  (\u03bb A B ih X f \u2192 elimPair X (\u03bb a b \u2192 ih a (\u03bb b \u2192 X (a , b)) (f a) b))\n\n----------------------------------------------------------------------\n\nUncurriedFunc : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedFunc I D X = (i : I) \u2192 Func I D X i \u2192 X i\n\nCurriedFunc : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedFunc I = elimDesc\n  (\u03bb _ \u2192 (X : ISet I) \u2192 Set)\n  (\u03bb i X \u2192 X i)\n  (\u03bb i D ih X \u2192 (x : X i) \u2192 ih X )\n  (\u03bb A B ih X \u2192 (a : A) \u2192 ih a X)\n\ncurryFunc : (I : Set) (D : Desc I) (X : ISet I)\n  \u2192 UncurriedFunc I D X \u2192 CurriedFunc I D X\ncurryFunc I = elimDesc\n  (\u03bb D \u2192 (X : ISet I) \u2192 UncurriedFunc I D X \u2192 CurriedFunc I D X)\n  (\u03bb i X cn \u2192 cn i refl)\n  (\u03bb i D ih X cn x \u2192 ih X (\u03bb j xs \u2192 cn j (x , xs)))\n  (\u03bb A B ih X cn a \u2192 ih a X (\u03bb j xs \u2192 cn j (a , xs)))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedFunc I D X)\n  \u2192 Set\nUncurriedHyps I D X P cn =\n  (i : I) (xs : Func I D X i) (ihs : Hyps I D X P i xs) \u2192 P i (cn i xs)\n\nCurriedHyps : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedFunc I D X)\n  \u2192 Set\nCurriedHyps I = elimDesc\n  (\u03bb D \u2192 (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (cn : UncurriedFunc I D X) \u2192 Set)\n  (\u03bb i X P cn \u2192 P i (cn i refl))\n  (\u03bb i D ih X P cn \u2192 (x : X i) \u2192 P i x \u2192 ih X P (\u03bb j xs \u2192 cn j (x , xs)))\n  (\u03bb A B ih X P cn \u2192 (a : A) \u2192 ih a X P (\u03bb j xs \u2192 cn j (a , xs)))\n\nUncurryHyps : (I : Set) (D : Desc I) \u2192 Set\nUncurryHyps I D = (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (cn : UncurriedFunc I D X)\n  \u2192 CurriedHyps I D X P cn \u2192 UncurriedHyps I D X P cn\n\nuncurryHyps : (I : Set) (D : Desc I) \u2192 UncurryHyps I D\nuncurryHyps I = elimDesc\n  (UncurryHyps I)\n  (\u03bb j X P cn pf \u2192\n    elimEq _ (\u03bb u \u2192 pf))\n  (\u03bb j D ih X P cn pf i \u2192\n    elimPair _ (\u03bb x xs ih,ihs \u2192\n      ih X P (\u03bb j ys \u2192 cn j (x , ys)) (pf x (proj\u2081 ih,ihs)) i xs (proj\u2082 ih,ihs)))\n  (\u03bb A B ih X P cn pf i \u2192\n    elimPair _ (\u03bb a xs \u2192 ih a X P (\u03bb j ys \u2192 cn j (a , ys)) (pf a) i xs))\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    N : String\n    P : Tel\n    I : Scope P \u2192 Tel\n    E : Enum\n    B : (p : Scope P) \u2192 Branches E (\u03bb _ \u2192 Desc (Scope (I p)))\n\n  PS : Set\n  PS = Scope P\n\n  IS : PS \u2192 Set\n  IS p = Scope (I p)\n\n  C : (p : PS) \u2192 Tag E \u2192 Desc (IS p)\n  C p = case (\u03bb _ \u2192 Desc (IS p)) (B p)\n\n  D : (p : PS) \u2192 Desc (IS p)\n  D p = Arg (Tag E) (C p)\n\n  F : (p : PS) \u2192 IS p \u2192 Set\n  F p = \u03bc N PS IS p (D p)\n\n----------------------------------------------------------------------\n\nDecl :\n  (N : String)\n  (P : Tel)\n  (I : CurriedScope P (\u03bb _ \u2192 Tel))\n  (E : Enum)\n  (B : let I = uncurryScope P (\u03bb _ \u2192 Tel) I\n      in CurriedScope P \u03bb A \u2192 Branches E (\u03bb _ \u2192 Desc (Scope (I A))))\n  \u2192 Data\nDecl N P I E B = record\n  { N = N\n  ; P = P\n  ; I = uncurryScope P _ I\n  ; E = E\n  ; B = uncurryScope P _ B\n  }\n\n----------------------------------------------------------------------\n\nEnd[_] : (I : Tel)\n  \u2192 CurriedScope I (\u03bb _ \u2192 Desc (Scope I))\nEnd[_] I = curryScope I _ End\n\nRec[_] : (I : Tel)\n  \u2192 CurriedScope I (\u03bb _ \u2192 Desc (Scope I) \u2192 Desc (Scope I))\nRec[_] I = curryScope I _ Rec\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 UncurriedScope (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried = Data.F\n\nForm : (R : Data)\n  \u2192 CurriedScope (Data.P R) \u03bb p\n  \u2192 CurriedScope (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryScope (Data.P R) (\u03bb p \u2192 CurriedScope (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryScope (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 CurriedFunc (Data.IS R p) (Data.D R p) (Data.F R p)\ninjUncurried R p t = curryFunc (Data.IS R p) (Data.C R p t)\n  (Data.F R p)\n  (\u03bb i xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 CurriedScope (Data.P R) \u03bb p\n  \u2192 CurriedFunc (Data.IS R p) (Data.D R p) (Data.F R p)\ninj R = curryScope (Data.P R)\n  (\u03bb p \u2192 CurriedFunc (Data.IS R p) (Data.D R p) (Data.F R p))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p D i \u2192 Set)\n  (f : CurriedHyps (I p) D (\u03bc \u2113 P I p D) M (\u03bb _ \u2192 init))\n  (i : I p)\n  (x : \u03bc \u2113 P I p D i)\n  \u2192 M i x\nindCurried \u2113 P I p D M f i x =\n  ind \u2113 P I p D M (uncurryHyps (I p) D (\u03bc \u2113 P I p D) M (\u03bb _ \u2192 init) f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M in\n  CurriedHyps (Data.IS R p) (Data.C R p t) (Data.F R p) unM (\u03bb i xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedScope (Data.P R) \u03bb p\n  \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n  \u2192 let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n  in UncurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x))\nelimUncurried R p M cs = let\n    unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n  in\n  curryScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x) \u03bb i x \u2192\n  indCurried (Data.N R) (Data.PS R) (Data.IS R) p (Data.D R p) unM\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 CurriedScope (Data.P R) \u03bb p\n  \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n  \u2192 let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x))\nelim R = curryScope (Data.P R)\n  (\u03bb p \u2192 (M : CurriedScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set))\n    \u2192 let unM = uncurryScope (Data.I R p) (\u03bb i \u2192 Data.F R p i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p M)\n       (CurriedScope (Data.I R p) (\u03bb i \u2192 (x : Data.F R p i) \u2192 unM i x)))\n  (\u03bb p M \u2192 curryBranches (Data.E R) _ _\n    (elimUncurried R p M))\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4d607ac9e5770bdf5c6387421b0deb44c4fe3c1d","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/ABP.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/ABP.agda","new_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module define the ABP following the presentation in Dybjer and\n-- Sander (1989).\n--\n-- References:\n--\n-- \u2022 Dybjer, Peter and Sander, Herbert P. (1989). A Functional\n--   Programming Approach to the Speci\ufb01cation and Veri\ufb01cation of\n--   Concurrent Systems. In: Formal Aspects of Computing 1,\n--   pp. 303\u2013319.\n\nmodule FOTC.Program.ABP.ABP where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- ABP equations\n\npostulate\n  send ack out corrupt : D \u2192 D\n  await                : D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate send-eq : \u2200 b i is ds \u2192\n                    send b \u00b7 (i \u2237 is) \u00b7 ds \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom send-eq #-}\n\npostulate\n  await-ok\u2261 : \u2200 b b' i is ds \u2192\n              b \u2261 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 send (not b) \u00b7 is \u00b7 ds\n\n  await-ok\u2262 : \u2200 b b' i is ds \u2192\n              b \u2262 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n\n  await-error : \u2200 b i is ds \u2192\n                await b i is (error \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom await-ok\u2261 await-ok\u2262 await-error #-}\n\npostulate\n  ack-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192\n            ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 b \u2237 ack (not b) \u00b7 bs\n\n  ack-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192\n            ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n\n  ack-error : \u2200 b bs \u2192 ack b \u00b7 (error \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n{-# ATP axiom ack-ok\u2261 ack-ok\u2262 ack-error #-}\n\npostulate\n  out-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192\n            out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 i \u2237 out (not b) \u00b7 bs\n\n  out-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192\n            out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 out b \u00b7 bs\n\n  out-error : \u2200 b bs \u2192 out b \u00b7 (error \u2237 bs) \u2261 out b \u00b7 bs\n{-# ATP axiom out-ok\u2261 out-ok\u2262 out-error #-}\n\npostulate\n  corrupt-T : \u2200 os x xs \u2192\n              corrupt (T \u2237 os) \u00b7 (x \u2237 xs) \u2261 ok x \u2237 corrupt os \u00b7 xs\n  corrupt-F : \u2200 os x xs \u2192\n              corrupt (F \u2237 os) \u00b7 (x \u2237 xs) \u2261 error \u2237 corrupt os \u00b7 xs\n{-# ATP axiom corrupt-T corrupt-F #-}\n\npostulate has hbs hcs hds : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate has-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2081 \u00b7 is \u00b7 (hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom has-eq #-}\n\npostulate hbs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2081 \u00b7 (has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hbs-eq #-}\n\npostulate hcs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2082 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hcs-eq #-}\n\npostulate hds-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2082 \u00b7 (hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hds-eq #-}\n\npostulate\n  transfer    : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  transfer-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                transfer f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2083 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom transfer-eq #-}\n\npostulate\n  abpTransfer : D \u2192 D \u2192 D \u2192 D \u2192 D\n  abpTransfer-eq :\n    \u2200 b os\u2081 os\u2082 is \u2192\n      abpTransfer b os\u2081 os\u2082 is \u2261\n        transfer (send b) (ack b) (out b) (corrupt os\u2081) (corrupt os\u2082) is\n{-# ATP axiom abpTransfer-eq #-}\n\n------------------------------------------------------------------------------\n-- ABP relations\n\n-- Start state for the ABP.\nS : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nS b is os\u2081 os\u2082 as bs cs ds js =\n  as \u2261 send b \u00b7 is \u00b7 ds\n  \u2227 bs \u2261 corrupt os\u2081 \u00b7 as\n  \u2227 cs \u2261 ack b \u00b7 bs\n  \u2227 ds \u2261 corrupt os\u2082 \u00b7 cs\n  \u2227 js \u2261 out b \u00b7 bs\n{-# ATP definition S #-}\n\n-- State for the ABP.\nS' : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nS' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' =\n  as' \u2261 await b i' is' ds'  -- Typo in ds'.\n  \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n  \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n  \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n  \u2227 js' \u2261 out (not b) \u00b7 bs'\n{-# ATP definition S' #-}\n\n-- Auxiliary bisimulation.\nB : D \u2192 D \u2192 Set\nB is js = \u2203[ b ] \u2203[ os\u2081 ] \u2203[ os\u2082 ] \u2203[ as ] \u2203[ bs ] \u2203[ cs ] \u2203[ ds ]\n            Stream is\n            \u2227 Bit b\n            \u2227 Fair os\u2081\n            \u2227 Fair os\u2082\n            \u2227 S b is os\u2081 os\u2082 as bs cs ds js\n{-# ATP definition B #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module define the ABP following the presentation in Dybjer and\n-- Sander (1989).\n--\n-- References:\n--\n-- \u2022 Dybjer, Peter and Sander, Herbert P. (1989). A Functional\n--   Programming Approach to the Speci\ufb01cation and Veri\ufb01cation of\n--   Concurrent Systems. In: Formal Aspects of Computing 1,\n--   pp. 303\u2013319.\n\nmodule FOTC.Program.ABP.ABP where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- ABP equations\n\npostulate\n  send ack out corrupt : D \u2192 D\n  await                : D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate send-eq : \u2200 b i is ds \u2192\n                    send b \u00b7 (i \u2237 is) \u00b7 ds \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom send-eq #-}\n\npostulate\n  await-ok\u2261 : \u2200 b b' i is ds \u2192\n              b \u2261 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 send (not b) \u00b7 is \u00b7 ds\n\n  await-ok\u2262 : \u2200 b b' i is ds \u2192\n              b \u2262 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n\n  await-error : \u2200 b i is ds \u2192\n                await b i is (error \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom await-ok\u2261 await-ok\u2262 await-error #-}\n\npostulate\n  ack-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192\n            ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 b \u2237 ack (not b) \u00b7 bs\n\n  ack-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192\n            ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n\n  ack-error : \u2200 b bs \u2192 ack b \u00b7 (error \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n{-# ATP axiom ack-ok\u2261 ack-ok\u2262 ack-error #-}\n\npostulate\n  out-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192\n            out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 i \u2237 out (not b) \u00b7 bs\n\n  out-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192\n            out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 out b \u00b7 bs\n\n  out-error : \u2200 b bs \u2192 out b \u00b7 (error \u2237 bs) \u2261 out b \u00b7 bs\n{-# ATP axiom out-ok\u2261 out-ok\u2262 out-error #-}\n\npostulate\n  corrupt-T : \u2200 os x xs \u2192\n              corrupt (T \u2237 os) \u00b7 (x \u2237 xs) \u2261 ok x \u2237 corrupt os \u00b7 xs\n  corrupt-F : \u2200 os x xs \u2192\n              corrupt (F \u2237 os) \u00b7 (x \u2237 xs) \u2261 error \u2237 corrupt os \u00b7 xs\n{-# ATP axiom corrupt-T corrupt-F #-}\n\npostulate has hbs hcs hds : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate has-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2081 \u00b7 is \u00b7 (hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom has-eq #-}\n\npostulate hbs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2081 \u00b7 (has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hbs-eq #-}\n\npostulate hcs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2082 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hcs-eq #-}\n\npostulate hds-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2082 \u00b7 (hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hds-eq #-}\n\npostulate\n  transfer    : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  transfer-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                transfer f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2083 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom transfer-eq #-}\n\npostulate\n  abpTransfer : D \u2192 D \u2192 D \u2192 D \u2192 D\n  abpTransfer-eq :\n    \u2200 b os\u2081 os\u2082 is \u2192\n      abpTransfer b os\u2081 os\u2082 is \u2261\n      transfer (send b) (ack b) (out b) (corrupt os\u2081) (corrupt os\u2082) is\n{-# ATP axiom abpTransfer-eq #-}\n\n------------------------------------------------------------------------------\n-- ABP relations\n\n-- Start state for the ABP.\nS : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nS b is os\u2081 os\u2082 as bs cs ds js =\n  as \u2261 send b \u00b7 is \u00b7 ds\n  \u2227 bs \u2261 corrupt os\u2081 \u00b7 as\n  \u2227 cs \u2261 ack b \u00b7 bs\n  \u2227 ds \u2261 corrupt os\u2082 \u00b7 cs\n  \u2227 js \u2261 out b \u00b7 bs\n{-# ATP definition S #-}\n\n-- State for the ABP.\nS' : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nS' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' =\n  as' \u2261 await b i' is' ds'  -- Typo in ds'.\n  \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n  \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n  \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n  \u2227 js' \u2261 out (not b) \u00b7 bs'\n{-# ATP definition S' #-}\n\n-- Auxiliary bisimulation.\nB : D \u2192 D \u2192 Set\nB is js = \u2203[ b ] \u2203[ os\u2081 ] \u2203[ os\u2082 ] \u2203[ as ] \u2203[ bs ] \u2203[ cs ] \u2203[ ds ]\n            Stream is\n            \u2227 Bit b\n            \u2227 Fair os\u2081\n            \u2227 Fair os\u2082\n            \u2227 S b is os\u2081 os\u2082 as bs cs ds js\n{-# ATP definition B #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3852931ffc56b9e51f939e2c7f10aafb8889360e","subject":"updating comments","message":"updating comments\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- external expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- the type of type contexts, i.e. \u0393s in the judegments below\n  tctx : Set\n  tctx = htyp ctx\n\n  mutual\n    data env : Set where\n      Id : (\u0393 : tctx) \u2192 env\n      Subst : (d : dhexp) \u2192 (y : Nat) \u2192 env \u2192 env\n\n    -- internal expressions\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 env) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 env) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- convenient notation for chaining together two agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- the type of hole contexts, i.e. \u0394s in the judgements\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- notation for a triple to match the CMTT syntax\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- the hole name u does not appear in the term e\n  data hole-name-new : (e : hexp) (u : Nat) \u2192 Set where\n    HNConst : \u2200{u} \u2192 hole-name-new c u\n    HNAsc : \u2200{e \u03c4 u} \u2192\n            hole-name-new e u \u2192\n            hole-name-new (e \u00b7: \u03c4) u\n    HNVar : \u2200{x u} \u2192 hole-name-new (X x) u\n    HNLam1 : \u2200{x e u} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x e) u\n    HNLam2 : \u2200{x e u \u03c4} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x [ \u03c4 ] e) u\n    HNHole : \u2200{u u'} \u2192\n             u' \u2260 u \u2192\n             hole-name-new (\u2987\u2988[ u' ]) u\n    HNNEHole : \u2200{u u' e} \u2192\n               u' \u2260 u \u2192\n               hole-name-new e u \u2192\n               hole-name-new (\u2987 e \u2988[ u' ]) u\n    HNAp : \u2200{ u e1 e2 } \u2192\n           hole-name-new e1 u \u2192\n           hole-name-new e2 u \u2192\n           hole-name-new (e1 \u2218 e2) u\n\n  -- two terms that do not share any hole names\n  data holes-disjoint : (e1 : hexp) \u2192 (e2 : hexp) \u2192 Set where\n    HDConst : \u2200{e} \u2192 holes-disjoint c e\n    HDAsc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (e1 \u00b7: \u03c4) e2\n    HDVar : \u2200{x e} \u2192 holes-disjoint (X x) e\n    HDLam1 : \u2200{x e1 e2} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x e1) e2\n    HDLam2 : \u2200{x e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x [ \u03c4 ] e1) e2\n    HDHole : \u2200{u e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint (\u2987\u2988[ u ]) e2\n    HDNEHole : \u2200{u e1 e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u2987 e1 \u2988[ u ]) e2\n    HDAp :  \u2200{e1 e2 e3} \u2192 holes-disjoint e1 e3 \u2192 holes-disjoint e2 e3 \u2192 holes-disjoint (e1 \u2218 e2) e3\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {x : Nat} \u2192\n                 (x , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X x => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 holes-disjoint e1 e2 \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 hole-name-new e u \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those external expressions without holes\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  -- those internal expressions without holes\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only produce complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n  -- those internal expressions where every cast is the identity cast and\n  -- there are no failed casts\n  data cast-id : dhexp \u2192 Set where\n    CIConst  : cast-id c\n    CIVar    : \u2200{x} \u2192 cast-id (X x)\n    CILam    : \u2200{x \u03c4 d} \u2192 cast-id d \u2192 cast-id (\u00b7\u03bb x [ \u03c4 ] d)\n    CIHole   : \u2200{u} \u2192 cast-id (\u2987\u2988\u27e8 u \u27e9)\n    CINEHole : \u2200{d u} \u2192 cast-id d \u2192 cast-id (\u2987 d \u2988\u27e8 u \u27e9)\n    CIAp     : \u2200{d1 d2} \u2192 cast-id d1 \u2192 cast-id d2 \u2192 cast-id (d1 \u2218 d2)\n    CICast   : \u2200{d \u03c4} \u2192 cast-id d \u2192 cast-id (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9)\n\n  -- expansion\n  mutual\n    -- synthesis\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              holes-disjoint e1 e2 \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , Id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u2987\u2988)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    -- analysis\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u03c4)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground types\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- todo: clean up above, dom?\n    data _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 env \u2192 tctx \u2192 Set where\n      STAId : \u2200{\u0393 \u0393' \u0394} \u2192\n                  ((x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393' \u2192 (x , \u03c4) \u2208 \u0393) \u2192\n                  \u0394 , \u0393 \u22a2 Id \u0393' :s: \u0393'\n      STASubst : \u2200{\u0393 \u0394 \u03c3 y \u0393' d \u03c4 } \u2192\n               \u0394 , \u0393 ,, (y , \u03c4) \u22a2 \u03c3 :s: \u0393' \u2192\n               \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n               \u0394 , \u0393 \u22a2 Subst d y \u03c3 :s: \u0393'\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              x # \u0393 \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution\n  --\n  -- todo: if substitution lemma is hard to prove, maybe get a premise that\n  -- it's final; analagous to \"value substitution\". or define it\n  -- judgementally instead of as a function.\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d')\n    with natEQ x y\n  [ d \/ y ] (\u00b7\u03bb .y [ \u03c4 ] d') | Inl refl = \u00b7\u03bb y [ \u03c4 ] d'\n  [ d \/ y ] (\u00b7\u03bb x [ \u03c4 ] d')  | Inr x\u2081 = \u00b7\u03bb x [ \u03c4 ] ( [ d \/ y ] d')\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- applying an environment to an expression\n  apply-env : env \u2192 dhexp \u2192 dhexp\n  apply-env (Id \u0393) d = d\n  apply-env (Subst d y \u03c3) d' = [ d \/ y ] ( apply-env \u03c3 d')\n\n  -- values\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- boxed values\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate forms\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final expressions\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet    \u2192 d final\n\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 env ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n  -- note: this judgement is redundant: in the absence of the premises in\n  -- the red brackets, all syntactically well formed ectxs are valid. with\n  -- finality premises, that's not true, and that would propagate through\n  -- additions to the calculus. so we leave it here for clarity but note\n  -- that, as written, in any use case its either trival to prove or\n  -- provides no additional information\n\n   --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red brackets\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red brackets\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  -- matched ground types\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red brackets\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1)\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red brackets\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red brackets\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red brackets\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red brackets\n               -- d2 final \u2192 -- red brackets\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  -- single step (in contextual evaluation sense)\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  -- reflexive transitive closure of single steps into multi steps\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- freshness\n  mutual\n    -- ... with respect to a hole context\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    -- ... for inernal expressions\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- ... for external expressions\n  data freshh : Nat \u2192 hexp \u2192 Set where\n    FRHConst : \u2200{x} \u2192 freshh x c\n    FRHAsc   : \u2200{x e \u03c4} \u2192 freshh x e \u2192 freshh x (e \u00b7: \u03c4)\n    FRHVar   : \u2200{x y} \u2192 x \u2260 y \u2192 freshh x (X y)\n    FRHLam1  : \u2200{x y e} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y e)\n    FRHLam2  : \u2200{x \u03c4 e y} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y [ \u03c4 ] e)\n    FRHEHole : \u2200{x u} \u2192 freshh x (\u2987\u2988[ u ])\n    FRHNEHole : \u2200{x u e} \u2192 freshh x e \u2192 freshh x (\u2987 e \u2988[ u ])\n    FRHAp : \u2200{x e1 e2} \u2192 freshh x e1 \u2192 freshh x e2 \u2192 freshh x (e1 \u2218 e2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- the type of type contexts, i.e. \u0393s in the judegments below\n  tctx : Set\n  tctx = htyp ctx\n\n  mutual\n    data env : Set where\n      Id : (\u0393 : tctx) \u2192 env\n      Subst : (d : dhexp) \u2192 (y : Nat) \u2192 env \u2192 env\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 env) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 env) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- convenient notation for chaining together two agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- the type of hole contexts, i.e. \u0394s in the judgements\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- notation for a triple to match the CMTT syntax\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- the hole name u does not appear in the term e\n  data hole-name-new : (e : hexp) (u : Nat) \u2192 Set where\n    HNConst : \u2200{u} \u2192 hole-name-new c u\n    HNAsc : \u2200{e \u03c4 u} \u2192\n            hole-name-new e u \u2192\n            hole-name-new (e \u00b7: \u03c4) u\n    HNVar : \u2200{x u} \u2192 hole-name-new (X x) u\n    HNLam1 : \u2200{x e u} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x e) u\n    HNLam2 : \u2200{x e u \u03c4} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x [ \u03c4 ] e) u\n    HNHole : \u2200{u u'} \u2192\n             u' \u2260 u \u2192\n             hole-name-new (\u2987\u2988[ u' ]) u\n    HNNEHole : \u2200{u u' e} \u2192\n               u' \u2260 u \u2192\n               hole-name-new e u \u2192\n               hole-name-new (\u2987 e \u2988[ u' ]) u\n    HNAp : \u2200{ u e1 e2 } \u2192\n           hole-name-new e1 u \u2192\n           hole-name-new e2 u \u2192\n           hole-name-new (e1 \u2218 e2) u\n\n  -- two terms that do not share any hole names\n  data holes-disjoint : (e1 : hexp) \u2192 (e2 : hexp) \u2192 Set where\n    HDConst : \u2200{e} \u2192 holes-disjoint c e\n    HDAsc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (e1 \u00b7: \u03c4) e2\n    HDVar : \u2200{x e} \u2192 holes-disjoint (X x) e\n    HDLam1 : \u2200{x e1 e2} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x e1) e2\n    HDLam2 : \u2200{x e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x [ \u03c4 ] e1) e2\n    HDHole : \u2200{u e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint (\u2987\u2988[ u ]) e2\n    HDNEHole : \u2200{u e1 e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u2987 e1 \u2988[ u ]) e2\n    HDAp :  \u2200{e1 e2 e3} \u2192 holes-disjoint e1 e3 \u2192 holes-disjoint e2 e3 \u2192 holes-disjoint (e1 \u2218 e2) e3\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {x : Nat} \u2192\n                 (x , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X x => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 holes-disjoint e1 e2 \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 hole-name-new e u \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those external expressions without holes\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  -- those internal expressions without holes\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only produce complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n  -- those internal expressions where every cast is the identity cast and\n  -- there are no failed casts\n  data cast-id : dhexp \u2192 Set where\n    CIConst  : cast-id c\n    CIVar    : \u2200{x} \u2192 cast-id (X x)\n    CILam    : \u2200{x \u03c4 d} \u2192 cast-id d \u2192 cast-id (\u00b7\u03bb x [ \u03c4 ] d)\n    CIHole   : \u2200{u} \u2192 cast-id (\u2987\u2988\u27e8 u \u27e9)\n    CINEHole : \u2200{d u} \u2192 cast-id d \u2192 cast-id (\u2987 d \u2988\u27e8 u \u27e9)\n    CIAp     : \u2200{d1 d2} \u2192 cast-id d1 \u2192 cast-id d2 \u2192 cast-id (d1 \u2218 d2)\n    CICast   : \u2200{d \u03c4} \u2192 cast-id d \u2192 cast-id (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9)\n\n  -- expansion\n  mutual\n    -- synthesis\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              holes-disjoint e1 e2 \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , Id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u2987\u2988)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    -- analysis\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u03c4)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground types\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- todo: clean up above, dom?\n    data _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 env \u2192 tctx \u2192 Set where\n      STAId : \u2200{\u0393 \u0393' \u0394} \u2192\n                  ((x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393' \u2192 (x , \u03c4) \u2208 \u0393) \u2192\n                  \u0394 , \u0393 \u22a2 Id \u0393' :s: \u0393'\n      STASubst : \u2200{\u0393 \u0394 \u03c3 y \u0393' d \u03c4 } \u2192\n               \u0394 , \u0393 ,, (y , \u03c4) \u22a2 \u03c3 :s: \u0393' \u2192\n               \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n               \u0394 , \u0393 \u22a2 Subst d y \u03c3 :s: \u0393'\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              x # \u0393 \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution\n  --\n  -- todo: if substitution lemma is hard to prove, maybe get a premise that\n  -- it's final; analagous to \"value substitution\". or define it\n  -- judgementally instead of as a function.\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d')\n    with natEQ x y\n  [ d \/ y ] (\u00b7\u03bb .y [ \u03c4 ] d') | Inl refl = \u00b7\u03bb y [ \u03c4 ] d'\n  [ d \/ y ] (\u00b7\u03bb x [ \u03c4 ] d')  | Inr x\u2081 = \u00b7\u03bb x [ \u03c4 ] ( [ d \/ y ] d')\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- applying an environment to an expression\n  apply-env : env \u2192 dhexp \u2192 dhexp\n  apply-env (Id \u0393) d = d\n  apply-env (Subst d y \u03c3) d' = [ d \/ y ] ( apply-env \u03c3 d')\n\n  -- values\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- boxed values\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate forms\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final expressions\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet    \u2192 d final\n\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 env ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n  -- note: this judgement is redundant: in the absence of the premises in\n  -- the red brackets, all syntactically well formed ectxs are valid. with\n  -- finality premises, that's not true, and that would propagate through\n  -- additions to the calculus. so we leave it here for clarity but note\n  -- that, as written, in any use case its either trival to prove or\n  -- provides no additional information\n\n   --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red brackets\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red brackets\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  -- matched ground types\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red brackets\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1)\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red brackets\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red brackets\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red brackets\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red brackets\n               -- d2 final \u2192 -- red brackets\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  -- single step (in contextual evaluation sense)\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  -- reflexive transitive closure of single steps into multi steps\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- freshness\n  mutual\n    -- ... with respect to a hole context\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    -- ... for inernal expressions\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- ... for external expressions\n  data freshh : Nat \u2192 hexp \u2192 Set where\n    FRHConst : \u2200{x} \u2192 freshh x c\n    FRHAsc   : \u2200{x e \u03c4} \u2192 freshh x e \u2192 freshh x (e \u00b7: \u03c4)\n    FRHVar   : \u2200{x y} \u2192 x \u2260 y \u2192 freshh x (X y)\n    FRHLam1  : \u2200{x y e} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y e)\n    FRHLam2  : \u2200{x \u03c4 e y} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y [ \u03c4 ] e)\n    FRHEHole : \u2200{x u} \u2192 freshh x (\u2987\u2988[ u ])\n    FRHNEHole : \u2200{x u e} \u2192 freshh x e \u2192 freshh x (\u2987 e \u2988[ u ])\n    FRHAp : \u2200{x e1 e2} \u2192 freshh x e1 \u2192 freshh x e2 \u2192 freshh x (e1 \u2218 e2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4d4cc54595eef42ea3f87392acc4712358dc360b","subject":"SynGrp: updated but not used anymore so far","message":"SynGrp: updated but not used anymore so far\n","repos":"crypto-agda\/crypto-agda","old_file":"SynGrp.agda","new_file":"SynGrp.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import Function.Base\nopen import Data.Product.NP renaming (map to <_\u00d7_>)\nopen import FFI.JS.BigI\nopen import Crypto.JS.BigI.CyclicGroup\nopen import Crypto.JS.BigI.FiniteField\n  hiding (_*_; _^I_)\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Algebra.Group.Homomorphism\nopen import Relation.Binary.PropositionalEquality\n\nmodule SynGrp where\n\ndata SynGrp : Set where\n  `\u2124[_]+ : (q : BigI) \u2192 SynGrp\n  `\u2124[_]\u2605 : (p : BigI) \u2192 SynGrp\n  _`\u00d7_  : (\ud835\udd38 \ud835\udd39 : SynGrp) \u2192 SynGrp\n\nElGrp : SynGrp \u2192 Set\nElGrp `\u2124[ q ]+ = \u2124[ q ]\nElGrp `\u2124[ p ]\u2605 = \u2124[ p ]\u2605\nElGrp (`\ud835\udd38 `\u00d7 `\ud835\udd39) = ElGrp `\ud835\udd38 \u00d7 ElGrp `\ud835\udd39\n\nEl\ud835\udd3erp : \u2200 `\ud835\udd3e \u2192 Group (ElGrp `\ud835\udd3e)\nEl\ud835\udd3erp `\u2124[ q ]+ = \u2124[ q ]+-grp\nEl\ud835\udd3erp `\u2124[ p ]\u2605 = \u2124[ p ]\u2605-grp\nEl\ud835\udd3erp (`\ud835\udd38 `\u00d7 `\ud835\udd39) = Product.\u00d7-grp (El\ud835\udd3erp `\ud835\udd38) (El\ud835\udd3erp `\ud835\udd39)\n\nexp-\u00d7 : \u2200 {a b c}{A : Set a}{B : Set b}{C : Set c}\n          (expA : A \u2192 C \u2192 A)\n          (expB : B \u2192 C \u2192 B)\n        \u2192 A \u00d7 B \u2192 C \u2192 A \u00d7 B\nexp-\u00d7 expA expB (b0 , b1) e = expA b0 e , expB b1 e\n\n-- This iterate the group operation of \ud835\udd38 based on the given\n-- BigI value. Calling this operations exp(onential) makes\n-- sense when the group is \u2124p\u2605, but for \u2124q+ this corresponds\n-- to multiplication.\nexpI : \u2200 \ud835\udd38 \u2192 ElGrp \ud835\udd38 \u2192 BigI \u2192 ElGrp \ud835\udd38\nexpI `\u2124[ q ]+  = _\u2297I_ _\nexpI `\u2124[ p ]\u2605  = _^I_ _\nexpI (\ud835\udd38 `\u00d7 \ud835\udd39) = exp-\u00d7 (expI \ud835\udd38) (expI \ud835\udd39)\n\n-- See the remark on expI\nexp : \u2200 {q} \ud835\udd38 \u2192 ElGrp \ud835\udd38 \u2192 \u2124[ q ] \u2192 ElGrp \ud835\udd38\nexp \ud835\udd38 b e = expI \ud835\udd38 b (\u2124q\u25b9BigI _ e)\n\nmodule _ {q q'} where\n\n  _*_ : \u2124[ q ] \u2192 \u2124[ q' ] \u2192 \u2124[ q ]\n  x * y = _\u2297I_ _ x (\u2124q\u25b9BigI _ y)\n\n  -- TODO check on the assumptions on q,q'\n  -- x * (y +{q'} z) = x * y +{q} x * z\n  -- mod (x * (mod (y + z) q')) q = ((x * y) mod q + (x * z) mod q) mod q\n  postulate *-hom : \u2200 x \u2192 GroupHomomorphism \u2124[ q' ]+-grp \u2124[ q ]+-grp (_*_ x)\n\nopen module ^-hom-p-q {p q} = ^-hom p {q}\n\nexp-hom : \u2200 {q} \ud835\udd38 (b : ElGrp \ud835\udd38)\n           \u2192 GroupHomomorphism \u2124[ q ]+-grp (El\ud835\udd3erp \ud835\udd38) (exp \ud835\udd38 b)\nexp-hom `\u2124[ q ]+ b = *-hom b\nexp-hom `\u2124[ p ]\u2605 b = ^-hom b\nexp-hom (\ud835\udd38 `\u00d7 \ud835\udd39) (b0 , b1) = < exp-hom \ud835\udd38 b0 , exp-hom \ud835\udd39 b1 >-hom\n\ndata SynHom : (\ud835\udd38 \ud835\udd39 : SynGrp) \u2192 Set where\n  `id  : \u2200{\ud835\udd38} \u2192 SynHom \ud835\udd38 \ud835\udd38 \n  _`\u2218_ : \u2200{\ud835\udd38 \ud835\udd39 \u2102}(f : SynHom \ud835\udd39 \u2102)(g : SynHom \ud835\udd38 \ud835\udd39) \u2192 SynHom \ud835\udd38 \u2102\n  `<_\u00d7_> : \u2200{\ud835\udd38\u2080 \ud835\udd38\u2081 \ud835\udd39\u2080 \ud835\udd39\u2081}\n            (f\u2080 : SynHom \ud835\udd38\u2080 \ud835\udd39\u2080)(f\u2081 : SynHom \ud835\udd38\u2081 \ud835\udd39\u2081)\n           \u2192 SynHom (\ud835\udd38\u2080 `\u00d7 \ud835\udd38\u2081) (\ud835\udd39\u2080 `\u00d7 \ud835\udd39\u2081)\n  `\u0394 : \u2200{\ud835\udd38} \u2192 SynHom \ud835\udd38 (\ud835\udd38 `\u00d7 \ud835\udd38) \n  _`^_ : \u2200 {q \ud835\udd38} \u2192 ElGrp \ud835\udd38 \u2192 SynHom `\u2124[ q ]+ \ud835\udd38\n\n`<_,_> : \u2200{\ud835\udd38 \ud835\udd39\u2080 \ud835\udd39\u2081}\n          (f\u2080 : SynHom \ud835\udd38 \ud835\udd39\u2080)\n          (f\u2081 : SynHom \ud835\udd38 \ud835\udd39\u2081)\n         \u2192 SynHom \ud835\udd38 (\ud835\udd39\u2080 `\u00d7 \ud835\udd39\u2081)\n`< f\u2080 , f\u2081 > = `< f\u2080 \u00d7 f\u2081 > `\u2218 `\u0394\n\nElHom : \u2200{\ud835\udd38 \ud835\udd39 : SynGrp} \u2192 SynHom \ud835\udd38 \ud835\udd39 \u2192 ElGrp \ud835\udd38 \u2192 ElGrp \ud835\udd39\nElHom `id        = id\nElHom (f `\u2218 g)   = ElHom f \u2218 ElHom g\nElHom `< f \u00d7 g > = < ElHom f \u00d7 ElHom g >\nElHom `\u0394         = \u0394\nElHom (_`^_ {\ud835\udd38 = \ud835\udd38} b) = exp \ud835\udd38 b\n\nEl\u210dom : \u2200{\ud835\udd38 \ud835\udd39 : SynGrp}(\u03c6 : SynHom \ud835\udd38 \ud835\udd39) \u2192 GroupHomomorphism (El\ud835\udd3erp \ud835\udd38) (El\ud835\udd3erp \ud835\udd39) (ElHom \u03c6)\nEl\u210dom `id = Identity.id-hom _\nEl\u210dom (\u03c6 `\u2218 \u03c8) = El\u210dom \u03c6 \u2218-hom El\u210dom \u03c8\nEl\u210dom `< \u03c6 \u00d7 \u03c8 > = < El\u210dom \u03c6 \u00d7 El\u210dom \u03c8 >-hom\nEl\u210dom `\u0394 = Delta.\u0394-hom _\nEl\u210dom (_`^_ {\ud835\udd38 = \ud835\udd38} x) = exp-hom \ud835\udd38 x\n\nSynGrp-Eq? : (\ud835\udd38 : SynGrp) \u2192 Eq? (ElGrp \ud835\udd38)\nSynGrp-Eq? `\u2124[ q ]+ = \u2124[ q ]-Eq?\nSynGrp-Eq? `\u2124[ p ]\u2605 = \u2124[ p ]\u2605-Eq?\nSynGrp-Eq? (\ud835\udd38 `\u00d7 \ud835\udd39) = \u00d7-Eq? {{SynGrp-Eq? \ud835\udd38}} {{SynGrp-Eq? \ud835\udd39}}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import Function.NP\nopen import Data.Product.NP renaming (map to <_\u00d7_>)\nopen import FFI.JS.BigI\nimport      Crypto.JS.BigI.CyclicGroup as \ud835\udd3e\nimport      Crypto.JS.BigI.FiniteField as \ud835\udd3d\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Algebra.Group.Homomorphism\n-- open import Relation.Binary.PropositionalEquality\n\nmodule SynGrp where\n\ndata SynGrp : Set where\n  `\u2124[_]+ : (q : BigI) \u2192 SynGrp\n  `\u2124[_]\u2605 : (p : BigI) \u2192 SynGrp\n  _`\u00d7_  : (\ud835\udd38 \ud835\udd39 : SynGrp) \u2192 SynGrp\n\n\u2124[_]  = \ud835\udd3d.\ud835\udd3d\n\u2124[_]\u2605 = \ud835\udd3e.\ud835\udd3e\n\n\u2124[_]+-grp = \ud835\udd3d.fld.+-grp\n\u2124[_]\u2605-grp = \ud835\udd3e.grp\n\nElGrp : SynGrp \u2192 Set\nElGrp `\u2124[ q ]+ = \u2124[ q ]\nElGrp `\u2124[ p ]\u2605 = \u2124[ p ]\u2605\nElGrp (`\ud835\udd38 `\u00d7 `\ud835\udd39) = ElGrp `\ud835\udd38 \u00d7 ElGrp `\ud835\udd39\n\nEl\ud835\udd3erp : \u2200 `\ud835\udd3e \u2192 Group (ElGrp `\ud835\udd3e)\nEl\ud835\udd3erp `\u2124[ q ]+ = \u2124[ q ]+-grp\nEl\ud835\udd3erp `\u2124[ p ]\u2605 = \u2124[ p ]\u2605-grp\nEl\ud835\udd3erp (`\ud835\udd38 `\u00d7 `\ud835\udd39) = Product.\u00d7-grp (El\ud835\udd3erp `\ud835\udd38) (El\ud835\udd3erp `\ud835\udd39)\n\n-- This iterate the group operation of \ud835\udd38 based on the given\n-- BigI value. Calling this operations exp(onential) makes\n-- sense when the group is \u2124p\u2605, but for \u2124q+ this corresponds\n-- to multiplication.\nexpI : \u2200 \ud835\udd38 \u2192 ElGrp \ud835\udd38 \u2192 BigI \u2192 ElGrp \ud835\udd38\nexpI `\u2124[ q ]+ b e = \ud835\udd3d._\u2297_ _ b e\nexpI `\u2124[ p ]\u2605 b e = \ud835\udd3e._^_ _ b e\nexpI (\ud835\udd38 `\u00d7 \ud835\udd39) (b0 , b1) e = expI \ud835\udd38 b0 e , expI \ud835\udd39 b1 e\n\n-- See the remark on expI\nexp : \u2200 {q} \ud835\udd38 \u2192 ElGrp \ud835\udd38 \u2192 \u2124[ q ] \u2192 ElGrp \ud835\udd38\nexp \ud835\udd38 b e = expI \ud835\udd38 b (\ud835\udd3d.repr _ e)\n\nmodule _ {q q'} where\n\n  _*_ : \u2124[ q ] \u2192 \u2124[ q' ] \u2192 \u2124[ q ]\n  x * y = \ud835\udd3d._\u2297_ _ x (\ud835\udd3d.repr _ y)\n\n  -- TODO check on the assumptions on q,q'\n  postulate\n    *-hom : \u2200 x \u2192 GroupHomomorphism \u2124[ q' ]+-grp \u2124[ q ]+-grp (_*_ x)\n\nmodule _ {q p} where\n  _^_ : \u2124[ p ]\u2605 \u2192 \u2124[ q ] \u2192 \u2124[ p ]\u2605\n  b ^ e = \ud835\udd3e._^_ _ b (\ud835\udd3d.repr _ e)\n\n  -- TODO check on the assumptions on p,q\n  postulate\n    ^-hom : \u2200 b \u2192 GroupHomomorphism \u2124[ q ]+-grp \u2124[ p ]\u2605-grp (_^_ b)\n\n    -- ^-comm : \u2200 {\u03b1 x y} \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n\nexp-hom : \u2200 {q} \ud835\udd38 (b : ElGrp \ud835\udd38)\n           \u2192 GroupHomomorphism \u2124[ q ]+-grp (El\ud835\udd3erp \ud835\udd38) (exp \ud835\udd38 b)\nexp-hom `\u2124[ q ]+ b = *-hom b\nexp-hom `\u2124[ p ]\u2605 b = ^-hom b\nexp-hom (\ud835\udd38 `\u00d7 \ud835\udd39) (b0 , b1) = < exp-hom \ud835\udd38 b0 , exp-hom \ud835\udd39 b1 >-hom\n\ndata SynHom : (\ud835\udd38 \ud835\udd39 : SynGrp) \u2192 Set where\n  `id  : \u2200{\ud835\udd38} \u2192 SynHom \ud835\udd38 \ud835\udd38 \n  _`\u2218_ : \u2200{\ud835\udd38 \ud835\udd39 \u2102}(f : SynHom \ud835\udd39 \u2102)(g : SynHom \ud835\udd38 \ud835\udd39) \u2192 SynHom \ud835\udd38 \u2102\n  `<_\u00d7_> : \u2200{\ud835\udd38\u2080 \ud835\udd38\u2081 \ud835\udd39\u2080 \ud835\udd39\u2081}\n            (f\u2080 : SynHom \ud835\udd38\u2080 \ud835\udd39\u2080)(f\u2081 : SynHom \ud835\udd38\u2081 \ud835\udd39\u2081)\n           \u2192 SynHom (\ud835\udd38\u2080 `\u00d7 \ud835\udd38\u2081) (\ud835\udd39\u2080 `\u00d7 \ud835\udd39\u2081)\n  `\u0394 : \u2200{\ud835\udd38} \u2192 SynHom \ud835\udd38 (\ud835\udd38 `\u00d7 \ud835\udd38) \n  _`^_ : \u2200 {q \ud835\udd38} \u2192 ElGrp \ud835\udd38 \u2192 SynHom `\u2124[ q ]+ \ud835\udd38\n\n`<_,_> : \u2200{\ud835\udd38 \ud835\udd39\u2080 \ud835\udd39\u2081}\n          (f\u2080 : SynHom \ud835\udd38 \ud835\udd39\u2080)\n          (f\u2081 : SynHom \ud835\udd38 \ud835\udd39\u2081)\n         \u2192 SynHom \ud835\udd38 (\ud835\udd39\u2080 `\u00d7 \ud835\udd39\u2081)\n`< f\u2080 , f\u2081 > = `< f\u2080 \u00d7 f\u2081 > `\u2218 `\u0394\n\nElHom : \u2200{\ud835\udd38 \ud835\udd39 : SynGrp} \u2192 SynHom \ud835\udd38 \ud835\udd39 \u2192 ElGrp \ud835\udd38 \u2192 ElGrp \ud835\udd39\nElHom `id        = id\nElHom (f `\u2218 g)   = ElHom f \u2218 ElHom g\nElHom `< f \u00d7 g > = < ElHom f \u00d7 ElHom g >\nElHom `\u0394         = \u0394\nElHom (_`^_ {\ud835\udd38 = \ud835\udd38} b) = exp \ud835\udd38 b\n\nEl\u210dom : \u2200{\ud835\udd38 \ud835\udd39 : SynGrp}(\u03c6 : SynHom \ud835\udd38 \ud835\udd39) \u2192 GroupHomomorphism (El\ud835\udd3erp \ud835\udd38) (El\ud835\udd3erp \ud835\udd39) (ElHom \u03c6)\nEl\u210dom `id = Identity.id-hom _\nEl\u210dom (\u03c6 `\u2218 \u03c8) = El\u210dom \u03c6 \u2218-hom El\u210dom \u03c8\nEl\u210dom `< \u03c6 \u00d7 \u03c8 > = < El\u210dom \u03c6 \u00d7 El\u210dom \u03c8 >-hom\nEl\u210dom `\u0394 = Delta.\u0394-hom _\nEl\u210dom (_`^_ {\ud835\udd38 = \ud835\udd38} x) = exp-hom \ud835\udd38 x\n\nSynGrp-Eq? : (\ud835\udd38 : SynGrp) \u2192 Eq? (ElGrp \ud835\udd38)\nSynGrp-Eq? `\u2124[ q ]+ = \ud835\udd3d.\ud835\udd3d-Eq? q\nSynGrp-Eq? `\u2124[ p ]\u2605 = \ud835\udd3e.\ud835\udd3e-Eq? p\nSynGrp-Eq? (\ud835\udd38 `\u00d7 \ud835\udd39) = \u00d7-Eq? {{SynGrp-Eq? \ud835\udd38}} {{SynGrp-Eq? \ud835\udd39}}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"eff34c6c08c12aa06efeaea8b12668e9a2418ce1","subject":"Fixed a definition of Stream-coind.","message":"Fixed a definition of Stream-coind.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Stream\/StreamSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Stream\/StreamSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC streams using the Agda co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Stream.StreamSL where\n\nopen import Data.Product renaming ( _\u00d7_ to _\u2227_ )\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata Stream : D \u2192 Set where\n  consS : \u2200 x {xs} \u2192 \u221e (Stream xs) \u2192 Stream (x \u2237 xs)\n\nStream-unf : \u2200 {xs} \u2192 Stream xs \u2192\n             \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 Stream xs' \u2227 xs \u2261 x' \u2237 xs'\nStream-unf (consS x' {xs'} Sxs') = x' , xs' , \u266d Sxs' , refl\n\n{-# NO_TERMINATION_CHECK #-}\nStream-coind :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 xs \u2261 x' \u2237 xs' \u2227 A xs') \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Stream xs\nStream-coind A h Axs with h Axs\n... | x' , xs' , prf\u2081 , Axs' = subst Stream (sym prf\u2081) prf\u2082\n  where\n  prf\u2082 : Stream (x' \u2237 xs')\n  prf\u2082 = consS x' (\u266f Stream-coind A h Axs')\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC streams using the Agda co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Stream.StreamSL where\n\nopen import Data.Product renaming ( _\u00d7_ to _\u2227_ )\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata Stream : D \u2192 Set where\n  consS : \u2200 x {xs} \u2192 \u221e (Stream xs) \u2192 Stream (x \u2237 xs)\n\nStream-unf : \u2200 {xs} \u2192 Stream xs \u2192\n             \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 Stream xs' \u2227 xs \u2261 x' \u2237 xs'\nStream-unf (consS x' {xs'} Sxs') = x' , xs' , \u266d Sxs' , refl\n\n{-# NO_TERMINATION_CHECK #-}\nStream-coind : \u2200 (A : D \u2192 Set) {xs} \u2192\n             -- A is post-fixed point of StreamF.\n             (A xs \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 A xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n             -- Stream is greater than A.\n             A xs \u2192 Stream xs\nStream-coind A {xs} h Axs = subst Stream (sym xs\u2261x'\u2237xs') prf\n  where\n    x' : D\n    x' = proj\u2081 (h Axs)\n\n    xs' : D\n    xs' = proj\u2081 (proj\u2082 (h Axs))\n\n    Axs' : A xs'\n    Axs' = proj\u2081 (proj\u2082 (proj\u2082 (h Axs)))\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' = proj\u2082 (proj\u2082 (proj\u2082 (h Axs)))\n\n    prf : Stream (x' \u2237 xs')\n    prf = consS x' (\u266f (Stream-coind A {!!} Axs'))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4b68efdc89a5dc5bad6ad5ef71ae18bb262c1c86","subject":"Prove that equivalences don't equate everything (fix #23).","message":"Prove that equivalences don't equate everything (fix #23).\n\nOld-commit-hash: 9c2bf11b7532012cd196e4eecd4cc3a7b5b5586a\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n\u2261-cong {bool} f \u2261 = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} f \u2261\u2081 \u2261\u2082 = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n\u2261-cong {bool} f \u2261 = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} f \u2261\u2081 \u2261\u2082 = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"53075c15ffec0c11417aecdf27b9920b9b88a56c","subject":"otp: +lem-flip$-\u2295","message":"otp: +lem-flip$-\u2295\n","repos":"crypto-agda\/crypto-agda","old_file":"single-bit-one-time-pad.agda","new_file":"single-bit-one-time-pad.agda","new_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP as Bool\nopen import Data.Product renaming (map to <_\u00d7_>)\nopen import Data.Nat.NP\nimport Data.Vec.NP as V\nopen V using (Vec; take; drop; drop\u2032; take\u2032; _++_) renaming (swap to vswap)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) where\n    open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n    kont\u2080-not : \u2200 b k \u2192 kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n    kont\u2080-not b k rewrite xor-not-not b k = \u2261.refl\n\n    open \u2261-Reasoning\n\n    lem\u2082 : \u2200 b \u2192 count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem\u2082 b = count\u21ba (runA b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n          \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not b 0b) (kont\u2080-not b 1b) \u27e9\n            count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n          \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n            count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (runA (not b)) \u220e\n\n    lem\u2083 : Safe\u2141? runA\n    lem\u2083 = lem\u2082 0b\n\n    -- A specialized version of lem\u2082 (\u2248lem\u2083)\n    lem\u2084 : Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\n    lem\u2084    = count\u21ba (runA 0b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n            \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b 0b) (kont\u2080-not 0b 1b) \u27e9\n              count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n            \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n              count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (runA 1b) \u220e\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = \u2261.refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\n\u2047 : \u2200 {n} \u2192 \u21ba n (Bits n)\n\u2047 = random\n\n\nlem'' : \u2200 {k} (f : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 2* #\u27e8 f \u27e9\nlem'' f = \u2261.refl\n\nlem' : \u2200 {k} (f g : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 #\u27e8 g \u2218 tail \u27e9 \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\nlem' f g pf = 2*-inj (\u2261.trans (lem'' f) (\u2261.trans pf (\u2261.sym (lem'' g))))\n\ndrop-tail : \u2200 k {n a} {A : Set a} \u2192 drop (suc k) {n} \u2257 drop k \u2218 tail {A = A}\ndrop-tail k (x \u2237 xs) = V.drop-\u2237 k x xs\n\nlemdrop\u2032 : \u2200 {k n} (f : Bits n \u2192 Bit) \u2192 #\u27e8 f \u2218 drop\u2032 k \u27e9 \u2261 \u27e82^ k * #\u27e8 f \u27e9 \u27e9\nlemdrop\u2032 {zero} f = \u2261.refl\nlemdrop\u2032 {suc k} f = #\u27e8 f \u2218 drop\u2032 k \u2218 tail \u27e9\n                   \u2261\u27e8 lem'' (f \u2218 drop\u2032 k) \u27e9\n                     2* #\u27e8 f \u2218 drop\u2032 k \u27e9\n                   \u2261\u27e8 \u2261.cong 2*_ (lemdrop\u2032 {k} f) \u27e9\n                     2* \u27e82^ k * #\u27e8 f \u27e9 \u27e9 \u220e\n                    where open \u2261-Reasoning\n\n\n\n\n\n\n\n-- exchange to independant statements\nlem-flip$-\u2295 : \u2200 {m n a} {A : Set a} (f : \u21ba m (A \u2192 Bit)) (x : \u21ba n A) \u2192\n               count\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb \u2261 count\u21ba (f \u229b x)\nlem-flip$-\u2295 {m} {n} f x = \u2261.sym (\n               count\u21ba fx\n              \u2261\u27e8 #-+ {m} {n} (run\u21ba fx) \u27e9\n               sum {m} (\u03bb xs \u2192 #\u27e8_\u27e9 {n} (\u03bb ys \u2192 run\u21ba fx (xs ++ ys)))\n              \u2261\u27e8 sum-sum {m} {n} (Bool.to\u2115 \u2218 run\u21ba fx) \u27e9\n               sum {n} (\u03bb ys \u2192 #\u27e8_\u27e9 {m} (\u03bb xs \u2192 run\u21ba fx (xs ++ ys)))\n              \u2261\u27e8 sum-\u2257\u2082 (\u03bb ys xs \u2192 Bool.to\u2115 (run\u21ba fx (xs ++ ys)))\n                        (\u03bb ys xs \u2192 Bool.to\u2115 (run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb (ys ++ xs)))\n                        (\u03bb ys xs \u2192 \u2261.cong Bool.to\u2115 (lem\u2081 xs ys)) \u27e9\n               sum {n} (\u03bb ys \u2192 #\u27e8_\u27e9 {m} (\u03bb xs \u2192 run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb (ys ++ xs)))\n              \u2261\u27e8 \u2261.sym (#-+ {n} {m} (run\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb)) \u27e9\n               count\u21ba \u27ea flip _$_ \u00b7 x \u00b7 f \u27eb\n              \u220e )\n    where open \u2261-Reasoning\n          fx = f \u229b x\n          lem\u2081 : \u2200 xs ys \u2192 run\u21ba f (take m (xs ++ ys)) (run\u21ba x (drop m (xs ++ ys)))\n                         \u2261 run\u21ba f (drop n (ys ++ xs)) (run\u21ba x (take n (ys ++ xs)))\n          lem\u2081 xs ys rewrite V.take-++ m xs ys | V.drop-++ m xs ys\n                           | V.take-++ n ys xs | V.drop-++ n ys xs = \u2261.refl\n\n\u2248\u1d2c\u2032-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c\u2032 toss\n\u2248\u1d2c\u2032-toss true Adv = \u2115\u00b0.+-comm (count\u21ba (Adv true)) _\n\u2248\u1d2c\u2032-toss false Adv = \u2261.refl\n\n\u2248\u1d2c-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c toss\n\u2248\u1d2c-toss b Adv = \u2248\u1d2c\u2032-toss b (return\u1d30 \u2218 Adv)\n\n-- should be equivalent to #-comm if \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 x were convertible to \u27ea _\u2295_ m \u00b7 x \u27eb\n\u2248\u1d2c-\u2047 : \u2200 {k} (m : Bits k) \u2192 \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047\n\u2248\u1d2c-\u2047 {zero}  _       _ = \u2261.refl\n\u2248\u1d2c-\u2047 {suc k} (h \u2237 m) Adv\n  rewrite \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 0b))\n        | \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 1b))\n        = \u2248\u1d2c\u2032-toss h (\u03bb x \u2192 \u27ea Adv \u2218 _\u2237_ x \u00b7 \u2047 \u27eb)\n\n\u2248\u1d2c-\u2047\u2082 : \u2200 {k} (m\u2080 m\u2081 : Bits k) \u2192 \u27ea m\u2080 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m\u2081 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2082 {k} m\u2080 m\u2081 = \u2248\u1d2c.trans {k} (\u2248\u1d2c-\u2047 m\u2080) (\u2248\u1d2c.sym {k} (\u2248\u1d2c-\u2047 m\u2081))\n\n\u2248\u1d2c-\u2047\u2083 : \u2200 {k} (m : Bit \u2192 Bits k) (b : Bit) \u2192 \u27ea m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2083 m b = \u2248\u1d2c-\u2047\u2082 (m b) (m (not b))\n\n\u2248\u1d2c-\u2047\u2084 : \u2200 {k} (m : Bits k \u00d7 Bits k) (b : Bit) \u2192 \u27ea proj m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea proj m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2084 = \u2248\u1d2c-\u2047\u2083 \u2218 proj\n","old_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP\nopen import Data.Product renaming (map to <_\u00d7_>)\nopen import Data.Nat.NP\nimport Data.Vec.NP as V\nopen V using (Vec; take; drop; drop\u2032; take\u2032; _++_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) where\n    open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n    kont\u2080-not : \u2200 b k \u2192 kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n    kont\u2080-not b k rewrite xor-not-not b k = \u2261.refl\n\n    open \u2261-Reasoning\n\n    lem\u2082 : \u2200 b \u2192 count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem\u2082 b = count\u21ba (runA b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n          \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not b 0b) (kont\u2080-not b 1b) \u27e9\n            count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n          \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n            count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (runA (not b)) \u220e\n\n    lem\u2083 : Safe\u2141? runA\n    lem\u2083 = lem\u2082 0b\n\n    -- A specialized version of lem\u2082 (\u2248lem\u2083)\n    lem\u2084 : Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\n    lem\u2084    = count\u21ba (runA 0b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n            \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b 0b) (kont\u2080-not 0b 1b) \u27e9\n              count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n            \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n              count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (runA 1b) \u220e\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = \u2261.refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\n\u2047 : \u2200 {n} \u2192 \u21ba n (Bits n)\n\u2047 = random\n\n\nlem'' : \u2200 {k} (f : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 2* #\u27e8 f \u27e9\nlem'' f = \u2261.refl\n\nlem' : \u2200 {k} (f g : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 #\u27e8 g \u2218 tail \u27e9 \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\nlem' f g pf = 2*-inj (\u2261.trans (lem'' f) (\u2261.trans pf (\u2261.sym (lem'' g))))\n\ndrop-tail : \u2200 k {n a} {A : Set a} \u2192 drop (suc k) {n} \u2257 drop k \u2218 tail {A = A}\ndrop-tail k (x \u2237 xs) = V.drop-\u2237 k x xs\n\nlemdrop\u2032 : \u2200 {k n} (f : Bits n \u2192 Bit) \u2192 #\u27e8 f \u2218 drop\u2032 k \u27e9 \u2261 \u27e82^ k * #\u27e8 f \u27e9 \u27e9\nlemdrop\u2032 {zero} f = \u2261.refl\nlemdrop\u2032 {suc k} f = #\u27e8 f \u2218 drop\u2032 k \u2218 tail \u27e9\n                   \u2261\u27e8 lem'' (f \u2218 drop\u2032 k) \u27e9\n                     2* #\u27e8 f \u2218 drop\u2032 k \u27e9\n                   \u2261\u27e8 \u2261.cong 2*_ (lemdrop\u2032 {k} f) \u27e9\n                     2* \u27e82^ k * #\u27e8 f \u27e9 \u27e9 \u220e\n                    where open \u2261-Reasoning\n\nvswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nvswap m xs = drop\u2032 m xs ++ take\u2032 m xs\n\n\n\n\n\n\n\u2248\u1d2c\u2032-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c\u2032 toss\n\u2248\u1d2c\u2032-toss true Adv = \u2115\u00b0.+-comm (count\u21ba (Adv true)) _\n\u2248\u1d2c\u2032-toss false Adv = \u2261.refl\n\n\u2248\u1d2c-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c toss\n\u2248\u1d2c-toss b Adv = \u2248\u1d2c\u2032-toss b (return\u1d30 \u2218 Adv)\n\n-- should be equivalent to #-comm if \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 x were convertible to \u27ea _\u2295_ m \u00b7 x \u27eb\n\u2248\u1d2c-\u2047 : \u2200 {k} (m : Bits k) \u2192 \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047\n\u2248\u1d2c-\u2047 {zero}  _       _ = \u2261.refl\n\u2248\u1d2c-\u2047 {suc k} (h \u2237 m) Adv\n  rewrite \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 0b))\n        | \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 1b))\n        = \u2248\u1d2c\u2032-toss h (\u03bb x \u2192 \u27ea Adv \u2218 _\u2237_ x \u00b7 \u2047 \u27eb)\n\n\u2248\u1d2c-\u2047\u2082 : \u2200 {k} (m\u2080 m\u2081 : Bits k) \u2192 \u27ea m\u2080 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m\u2081 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2082 {k} m\u2080 m\u2081 = \u2248\u1d2c.trans {k} (\u2248\u1d2c-\u2047 m\u2080) (\u2248\u1d2c.sym {k} (\u2248\u1d2c-\u2047 m\u2081))\n\n\u2248\u1d2c-\u2047\u2083 : \u2200 {k} (m : Bit \u2192 Bits k) (b : Bit) \u2192 \u27ea m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2083 m b = \u2248\u1d2c-\u2047\u2082 (m b) (m (not b))\n\n\u2248\u1d2c-\u2047\u2084 : \u2200 {k} (m : Bits k \u00d7 Bits k) (b : Bit) \u2192 \u27ea proj m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea proj m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2084 = \u2248\u1d2c-\u2047\u2083 \u2218 proj\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"17f8e744461ddd3f5ba4efa2b67bc86b88e65197","subject":"ExplicitNils: stability of terms, reworked (Reminder: Stability of terms means that the value denoted by a term is unchanging if all its free variables are unchanging in an environment.)","message":"ExplicitNils: stability of terms, reworked\n(Reminder: Stability of terms means that the value denoted by\na term is unchanging if all its free variables are unchanging\nin an environment.)\n\nOld-commit-hash: 82fcad297e8d931c8f855fa0ba0c31f1db0f5d3f\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import TaggedDeltaTypes\n\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-----------------------------------\n-- Describing the lack of change --\n-----------------------------------\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\nproj-\u2227 : \u2200 {a b} \u2192 a \u2227 b \u2261 true \u2192 a \u2261 true \u00d7 b \u2261 true\nproj-\u2227 {false} {_} ()\nproj-\u2227 {true} {false} ()\nproj-\u2227 {true} {true} truth = refl , refl\n\n--------------------------\n-- Optimized derivation --\n--------------------------\n\nderive' : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 {args : Args \u03c4} \u2192 {vars : Vars \u0393} \u2192 \u0394Term \u0393 \u03c4\n\nderive' (nat n) = derive (nat n)\nderive' (bag b) = derive (bag b)\nderive' (var x) = derive (var x)\n\nderive' (add s t) {\u2205-nat} {vars} =\n  \u0394add (derive' s {\u2205-nat} {vars}) (derive' t {\u2205-nat} {vars})\n\nderive' (abs t) {alter args} {vars} = \u0394abs (derive' t {args} {alter vars})\nderive' (abs t) {abide args} {vars} = \u0394abs (derive' t {args} {abide vars})\n\nderive' (app s t) {args} {vars} =\n  if stable t vars\n  then \u0394app (derive' s {abide args} {vars})\n          t (derive' t {expect-volatility} {vars})\n  else \u0394app (derive' s {alter args} {vars})\n          t (derive' t {expect-volatility} {vars})\n\nderive' (map s t) {\u2205-bag} {vars} =\n  if stable s vars\n  then \u0394map\u2081 s (derive' t {\u2205-bag} {vars})\n  else \u0394map\u2080 s (derive' s {abide \u2205-nat} {vars})\n             t (derive' t {\u2205-bag} {vars})\n\n-----------------------------------------------------\n-- A program equivalence preserved by optimization --\n-----------------------------------------------------\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 Vars \u0393 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (alter vars)\n  abide : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          v \u2295 dv \u2261 v \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (abide vars)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\n--\n--     du \u2248 dv wrt args\n--\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u0394Val \u03c4 \u2192 \u0394Val \u03c4 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} du dv args = du \u2261 dv -- extensionally\nclose-enough {bags} du dv args = du \u2261 dv -- literally\nclose-enough {\u03c3 \u21d2 \u03c4} df dg (alter args) = \u2200 {v dv R[v,dv]} \u2192\n  close-enough (df v dv R[v,dv]) (dg v dv R[v,dv]) args\nclose-enough {\u03c3 \u21d2 \u03c4} df dg (abide args) = \u2200 {v dv R[v,dv]} \u2192\n  v \u2295 dv \u2261 v \u2192 close-enough (df v dv R[v,dv]) (dg v dv R[v,dv]) args\n\nsyntax close-enough du dv args = du \u2248 dv wrt args\n\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n  where ext = extensionality\n\n\u2261to\u2248 : \u2200 {\u03c4 args} {df dg : \u0394Val \u03c4} \u2192\n  df \u2261 dg \u2192 df \u2248 dg wrt args\n\n\u2261to\u2248 {nats} df\u2261dg = df\u2261dg\n\u2261to\u2248 {bags} df\u2261dg = df\u2261dg\n\u2261to\u2248 {\u03c3 \u21d2 \u03c4} {alter args} df\u2261dg = \u03bb {v} {dv} {R[v,dv]} \u2192\n  \u2261to\u2248 (cong (\u03bb hole \u2192 hole v dv R[v,dv]) df\u2261dg)\n\u2261to\u2248 {\u03c3 \u21d2 \u03c4} {abide args} df\u2261dg = \u03bb {v} {dv} {R[v,dv]} v\u2295dv=v \u2192\n  \u2261to\u2248 (cong (\u03bb hole \u2192 hole v dv R[v,dv]) df\u2261dg)\n\n\u2248to\u2261 : \u2200 {\u03c4} {df dg : \u0394Val \u03c4} \u2192\n  df \u2248 dg wrt (expect-volatility {\u03c4}) \u2192 df \u2261 dg\n\n\u2248to\u2261 {nats} df\u2248dg = df\u2248dg\n\u2248to\u2261 {bags} df\u2248dg = df\u2248dg\n\u2248to\u2261 {\u03c3 \u21d2 \u03c4} {df} {dg} df\u2248dg =\n  ext\u00b3 (\u03bb v dv R[v,dv] \u2192 \u2248to\u2261 {\u03c4} (df\u2248dg {v} {dv} {R[v,dv]}))\n\n------------------------\n-- Stability of terms --\n------------------------\n\n-- A variable does not change if its value is unchanging.\n\nstabilityVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {vars} \u2192\n  (S : stableVar x vars \u2261 true) \u2192 \u2200 {\u03c1 : \u0394Env \u0393}\n  (H : Honest \u03c1 vars) \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\n\nstabilityVar {x = this} {alter vars} () _\nstabilityVar {x = this} {abide vars} refl (abide proof _) = proof\nstabilityVar {x = that y} {alter vars} S (alter H) =\n  stabilityVar {x = y} {vars} S H\nstabilityVar {x = that y} {abide vars} S (abide _ H) =\n  stabilityVar {x = y} {vars} S H\n\n-- A term does not change if its free variables are unchanging.\n\nstability : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {vars} \u2192\n  (S : stable t vars \u2261 true) \u2192 \u2200 {\u03c1 : \u0394Env \u0393}\n  (H : Honest \u03c1 vars) \u2192\n    \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (ignore \u03c1)\n\nstability {t = nat n} _ _ = refl\nstability {t = bag b} _ _ = b++\u2205=b\nstability {t = var x} {vars} S H = stabilityVar {x = x} {vars} S H\n\nstability {t = abs t} {vars} S {\u03c1} H = extensionality\n  (\u03bb w \u2192 stability {t = t} {abide vars} S (abide v\u2295[u\u229dv]=u H))\n\nstability {t = app s t} {vars} S {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Ss , St = proj-\u2227 S\n  in\n    begin\n      f v \u2295 df v dv (validity {t = t})\n    \u2261\u27e8 sym (corollary s t) \u27e9\n      (f \u2295 df) (v \u2295 dv)\n    \u2261\u27e8 stability {t = s} Ss H \u27e8$\u27e9 stability {t = t} St H \u27e9\n      f v\n    \u220e where open \u2261-Reasoning\n\nstability {t = add s t} {vars} S {\u03c1} H =\n  let\n    Ss , St = proj-\u2227 S\n  in cong\u2082 _+_ (stability {t = s} Ss H) (stability {t = t} St H)\n\nstability {t = map s t} {vars} S {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Ss , St = proj-\u2227 S\n    map = mapBag\n  in\n  begin\n    map f b \u2295 (map (f \u2295 df) (b \u2295 db) \u229d map f b)\n  \u2261\u27e8 b++[d\\\\b]=d \u27e9\n    map (f \u2295 df) (b \u2295 db)\n  \u2261\u27e8 cong\u2082 map (stability {t = s} Ss H) (stability {t = t} St H) \u27e9\n    map f b\n  \u220e where open \u2261-Reasoning\n\n\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import TaggedDeltaTypes\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-- Debug tool\nabsurd! : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd! _ _ b = b\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t) bzgl. vars\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\n\n-- Optimized derivation\n\nderive' : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 {args : Args \u03c4} \u2192 {vars : Vars \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity' : \u2200 {\u03c4 \u0393 vars}\n  {t : Term \u0393 \u03c4}\n  {\u03c1 : \u0394-Env \u0393} {honesty : Honest \u03c1 vars} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive' t {fickle-args} {vars} \u27e7 \u03c1)\n\nderive' (nat n) = derive (nat n)\nderive' (bag b) = derive (bag b)\nderive' (var x) = derive (var x)\n\nderive' (add s t) {\u2205-nat} {vars} =\n  \u0394add (derive' s {\u2205-nat} {vars}) (derive' t {\u2205-nat} {vars})\n\nderive' (map s t) {\u2205-bag} {vars} =\n  if stable s vars\n  then \u0394map\u2081 s (derive' t {\u2205-bag} {vars})\n  else \u0394map\u2080 s (derive' s {abide \u2205-nat} {vars})\n             t (derive' t {\u2205-bag} {vars})\n\nderive' (app s t) {args} {vars} =\n  if stable t vars\n  then \u0394app (derive' s {abide args} {vars})\n          t (derive' t {fickle-args} {vars})\n            (\u03bb {\u03c1 vars honesty} \u2192\n               validity' {vars = vars} {t = t} {\u03c1}\n                         {honesty = ?})\n  else \u0394app (derive' s {alter args} {vars})\n          t (derive' t {fickle-args} {vars}) validity'\n\nderive' t = {!!}\n\nvalidity' = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"26d1b2d6a8dde3981ebe9617d7c3e4cf600a9368","subject":"patching type-assignment-unicity after rewrite","message":"patching type-assignment-unicity after rewrite\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"type-assignment-unicity.agda","new_file":"type-assignment-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule type-assignment-unicity where\n  type-assignment-unicity : {\u0393 : tctx} {d : dhexp} {\u03c4' \u03c4 : htyp} {\u0394 : hctx} \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                              \u03c4 == \u03c4'\n  type-assignment-unicity TAConst TAConst = refl\n  type-assignment-unicity {\u0393 = \u0393} (TAVar x\u2081) (TAVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082\n  type-assignment-unicity (TALam d1) (TALam d2)\n    with type-assignment-unicity d1 d2\n  ... | refl = refl\n  type-assignment-unicity (TAAp x x\u2081) (TAAp y y\u2081)\n    with type-assignment-unicity x y\n  ... | refl = refl\n  type-assignment-unicity (TAEHole {\u0394 = \u0394} x y) (TAEHole x\u2081 x\u2082)\n    with ctxunicity {\u0393 = \u0394} x x\u2081\n  ... | refl = refl\n  type-assignment-unicity (TANEHole {\u0394 = \u0394} x d1 y) (TANEHole x\u2081 d2 x\u2082)\n    with ctxunicity {\u0393 = \u0394} x\u2081 x\n  ... | refl = refl\n  type-assignment-unicity (TACast d1 x) (TACast d2 x\u2081)\n    with type-assignment-unicity d1 d2\n  ... | refl = refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule type-assignment-unicity where\n  type-assignment-unicity : {\u0393 : tctx} {d : dhexp} {\u03c4' \u03c4 : htyp} {\u0394 : hctx} \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                              \u03c4 == \u03c4'\n  type-assignment-unicity TAConst TAConst = refl\n  type-assignment-unicity {\u0393 = \u0393} (TAVar x\u2081) (TAVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082\n  type-assignment-unicity (TALam d1) (TALam d2)\n    with type-assignment-unicity d1 d2\n  ... | refl = refl\n  type-assignment-unicity (TAAp d3 MAHole d4) (TAAp d5 MAHole d6) = refl\n  type-assignment-unicity (TAAp d3 MAHole d4) (TAAp d5 MAArr d6)\n      with type-assignment-unicity d3 d5\n  ... | ()\n  type-assignment-unicity (TAAp d3 MAArr d4) (TAAp d5 MAHole d6)\n    with type-assignment-unicity d3 d5\n  ... | ()\n  type-assignment-unicity (TAAp d3 MAArr d4) (TAAp d5 MAArr d6)\n    with type-assignment-unicity d3 d5 | type-assignment-unicity d4 d6\n  ... | refl | refl = refl\n  type-assignment-unicity (TAEHole {\u0394 = \u0394} x y) (TAEHole x\u2081 x\u2082)\n    with ctxunicity {\u0393 = \u0394} x x\u2081\n  ... | refl = refl\n  type-assignment-unicity (TANEHole {\u0394 = \u0394} x d1 y) (TANEHole x\u2081 d2 x\u2082)\n    with ctxunicity {\u0393 = \u0394} x\u2081 x\n  ... | refl = refl\n  type-assignment-unicity (TACast d1 x) (TACast d2 x\u2081)\n    with type-assignment-unicity d1 d2\n  ... | refl = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b04d285696190406b19d3bead396447ab71f875b","subject":"removing a postulate; feels good, man","message":"removing a postulate; feels good, man\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"finality.agda","new_file":"finality.agda","new_contents":"open import Prelude\nopen import core\n\nopen import progress-checks\n\nmodule finality where\n  finality : \u2200{d d'} \u2192 d final \u2192 d \u21a6* d' \u2192 d == d'\n  finality fin MSRefl = refl\n  finality fin (MSStep x ms) = abort (final-not-step fin (_ , x))\n","old_contents":"open import Prelude\nopen import core\n\nmodule finality where\n  -- todo: this will come from progress checks, once that's all squared away\n  postulate\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n\n  finality : \u2200{d d'} \u2192 d final \u2192 d \u21a6* d' \u2192 d == d'\n  finality fin MSRefl = refl\n  finality fin (MSStep x ms) = abort (fs fin (_ , x))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"abfc56ed18fd208d29e65400cf629b952508aa35","subject":"Reordered a module.","message":"Reordered a module.\n\nIgnore-this: 56bcd35741bf120b61fd36f3a9149101\n\ndarcs-hash:20110301052416-3bd4e-83667c713d5a74ccfb53ef3ed0b22dd170580915.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Induction\/Acc\/WellFoundedInduction.agda","new_file":"src\/FOTC\/Data\/Nat\/Induction\/Acc\/WellFoundedInduction.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Induction.Acc.WellFoundedInduction where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The relation LT is well-founded.\nmodule WF-LT\u2081\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; Sx<Sy\u2192x<y    : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n  ; <-trans      : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n  ; x\u2261y\u2192y<z\u2192x<z  : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n  ; x<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\n  )\n  where\n\n  wf-LT : WellFounded LT\n  wf-LT Nn = acc Nn (helper Nn)\n    where\n      -- N.B. The helper function is the same that the function used by\n      -- FOTC.Data.Nat.Induction.NonAcc.WellFoundedInductionATP.\n      helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LT m n \u2192 Acc LT m\n      helper zN     Nm m<0  = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\n      helper (sN _) zN 0<Sn = acc zN (\u03bb Nm' m'<0 \u2192 \u22a5-elim $ x<0\u2192\u22a5 Nm' m'<0)\n\n      helper (sN {n} Nn) (sN {m} Nm) Sm<Sn = acc (sN Nm)\n        (\u03bb {m'} Nm' m'<Sm \u2192\n          let m<n : LT m n\n              m<n = Sx<Sy\u2192x<y Sm<Sn\n\n              m'<n : LT m' n\n              m'<n = [ (\u03bb m'<m \u2192 <-trans Nm' Nm Nn m'<m m<n)\n                     , (\u03bb m'\u2261m \u2192 x\u2261y\u2192y<z\u2192x<z m'\u2261m m<n)\n                     ] (x<Sy\u2192x<y\u2228x\u2261y Nm' Nm m'<Sm)\n\n          in  helper Nn Nm' m'<n\n        )\n\n------------------------------------------------------------------------------\n-- A different proof that the relation LT is well-founded.\nmodule WF-LT\u2082\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; x\u2264y\u2192x<y\u2228x\u2261y  : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\n  ; x<Sy\u2192x\u2264y     : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\n  )\n  where\n\n  wf-LT : WellFounded LT\n  wf-LT zN      = acc zN (\u03bb Nm m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\n  wf-LT (sN Nn) = acc (sN Nn)\n                      (\u03bb Nm m<Sn \u2192 helper Nm Nn (wf-LT Nn)\n                                          (x<Sy\u2192x\u2264y Nm Nn m<Sn))\n    where\n      helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 Acc LT m \u2192 LE n m \u2192 Acc LT n\n      helper {n} {m} Nn Nm (acc _ h) n\u2264m =\n        [ (\u03bb n<m \u2192 h Nn n<m)\n        , (\u03bb n\u2261m \u2192 helper\u2081 (sym n\u2261m) (acc Nm h))\n        ] (x\u2264y\u2192x<y\u2228x\u2261y Nn Nm n\u2264m)\n        where\n          helper\u2081 : \u2200 {a b} \u2192 a \u2261 b \u2192 Acc LT a \u2192 Acc LT b\n          helper\u2081 refl acc-a = acc-a\n\n------------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers.\nmodule WFInd-LT\n  ( wf-LT : WellFounded LT )\n  where\n\n  wfInd-LT : (P : D \u2192 Set) \u2192\n             (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 LT n m \u2192 P n) \u2192 P m) \u2192\n             \u2200 {n} \u2192 N n \u2192 P n\n  wfInd-LT P = WellFoundedInduction wf-LT\n\n","old_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers\n----------------------------------------------------------------------------\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.Nat.Inequalities\n\nmodule FOTC.Data.Nat.Induction.Acc.WellFoundedInduction\n  ( x<0\u2192\u22a5        : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n  ; Sx<Sy\u2192x<y    : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n  ; <-trans      : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n  ; x\u2261y\u2192y<z\u2192x<z  : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n  ; x<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\n  ; x\u2264y\u2192x<y\u2228x\u2261y  : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\n  ; x<Sy\u2192x\u2264y     : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\n  )\n  where\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded\n\n------------------------------------------------------------------------------\n-- The relation LT is well-founded.\nLT-wf : WellFounded LT\nLT-wf Nn = acc Nn (helper Nn)\n  where\n    -- N.B. The helper function is the same that the function used by\n    -- FOTC.Data.Nat.Induction.NonAcc.WellFoundedInductionATP.\n    helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LT m n \u2192 Acc LT m\n    helper zN     Nm m<0  = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\n    helper (sN _) zN 0<Sn = acc zN (\u03bb Nm' m'<0 \u2192 \u22a5-elim $ x<0\u2192\u22a5 Nm' m'<0)\n\n    helper (sN {n} Nn) (sN {m} Nm) Sm<Sn = acc (sN Nm)\n      (\u03bb {m'} Nm' m'<Sm \u2192\n        let m<n : LT m n\n            m<n = Sx<Sy\u2192x<y Sm<Sn\n\n            m'<n : LT m' n\n            m'<n = [ (\u03bb m'<m \u2192 <-trans Nm' Nm Nn m'<m m<n)\n                   , (\u03bb m'\u2261m \u2192 x\u2261y\u2192y<z\u2192x<z m'\u2261m m<n)\n                   ] (x<Sy\u2192x<y\u2228x\u2261y Nm' Nm m'<Sm)\n\n        in  helper Nn Nm' m'<n\n      )\n\n-- A different proof that the relation LT is well-founded.\nLT-wf\u2081 : WellFounded LT\nLT-wf\u2081 zN      = acc zN (\u03bb Nm m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\nLT-wf\u2081 (sN Nn) = acc (sN Nn)\n                     (\u03bb Nm m<Sn \u2192 helper Nm Nn (LT-wf\u2081 Nn)\n                                         (x<Sy\u2192x\u2264y Nm Nn m<Sn))\n  where\n    helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 Acc LT m \u2192 LE n m \u2192 Acc LT n\n    helper {n} {m} Nn Nm (acc _ h) n\u2264m =\n      [ (\u03bb n<m \u2192 h Nn n<m)\n      , (\u03bb n\u2261m \u2192 helper\u2081 (sym n\u2261m) (acc Nm h))\n      ] (x\u2264y\u2192x<y\u2228x\u2261y Nn Nm n\u2264m)\n      where\n        helper\u2081 : \u2200 {a b} \u2192 a \u2261 b \u2192 Acc LT a \u2192 Acc LT b\n        helper\u2081 refl acc-a = acc-a\n\n-- Well-founded induction on the natural numbers.\nwfInd-LT : (P : D \u2192 Set) \u2192\n           (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 LT n m \u2192 P n) \u2192 P m) \u2192\n           \u2200 {n} \u2192 N n \u2192 P n\nwfInd-LT P = WellFoundedInduction LT-wf\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3a6acf7f81f342e6bddff009ea70cdbddc46ce9e","subject":"IDesc example: rework Hutton shaver with a more traditional context","message":"IDesc example: rework Hutton shaver with a more traditional context\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\n-- Fix menu:\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),                        -- Val t\n                              (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                               Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             (Val ty + Var context ty) ->\n             IMu closeTerm' ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      (Val ty + Var context ty) ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2\ntestContext : Context _\ntestContext = vcons (bool , con (EZe , true )) (vcons (nat , con (EZe , su (su ze)) ) vnil)\n\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a0c26d1d2731a3ccca7b00c77068c7bb3491ecab","subject":"FiniteField.JS: rename \u2124q to \ud835\udd3d","message":"FiniteField.JS: rename \u2124q to \ud835\udd3d\n","repos":"crypto-agda\/crypto-agda","old_file":"FiniteField\/JS.agda","new_file":"FiniteField\/JS.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import FFI.JS using (\ud835\udfd9; Number; Bool; true; false; String; warn-check; check; trace-call; _++_)\nopen import FFI.JS.BigI\nopen import Data.List.Base hiding (sum; _++_)\n{-\nopen import Algebra.Raw\nopen import Algebra.Field\n-}\n\nmodule FiniteField.JS (q : BigI) where\n\nabstract\n  \ud835\udd3d : Set\n  \ud835\udd3d = BigI\n\n  private\n    mod-q : BigI \u2192 \ud835\udd3d\n    mod-q x = mod x q\n\n    check' : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\n    -- check' = warn-check\n    check' = check\n\n  -- There is two ways to go from BigI to \u2124q: check and mod-q\n  -- Use check for untrusted input data and mod-q for internal\n  -- computation.\n  fromBigI : BigI \u2192 \ud835\udd3d\n  fromBigI = -- trace-call \"BigI\u25b9\u2124q \"\n    \u03bb x \u2192\n      (check' (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++ \" is less than x:\" ++ toString x)\n         (check' (x \u2265I 0I)\n            (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") x))\n\n  check-non-zero : \ud835\udd3d \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \ud835\udd3d \u2192 BigI\n  repr x = x\n\n  0# 1# : \ud835\udd3d\n  0# = 0I\n  1# = 1I\n\n  1\/_ : Op\u2081 \ud835\udd3d\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : Op\u2082 \ud835\udd3d\n  x ^ y = modPow (repr x) (repr y) q\n\n_+_ _\u2212_ _*_ _\/_ : Op\u2082 \ud835\udd3d\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = mod-q (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\n0\u2212_ : Op\u2081 \ud835\udd3d\n0\u2212 x = mod-q (negate (repr x))\n\n_==_ : (x y : \ud835\udd3d) \u2192 Bool\nx == y = equals (repr x) (repr y)\n\nsum prod : List \ud835\udd3d \u2192 \ud835\udd3d\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\n{-\n+-mon-ops : Monoid-Ops \ud835\udd3d\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \ud835\udd3d\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \ud835\udd3d\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \ud835\udd3d\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \ud835\udd3d\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \ud835\udd3d\nfld = fld-ops , fld-struct\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import FFI.JS using (Number; Bool; true; false; String; warn-check; check; trace-call; _++_)\nopen import FFI.JS.BigI\nopen import Data.List.Base hiding (sum; _++_)\n{-\nopen import Algebra.Raw\nopen import Algebra.Field\n-}\n\nmodule FiniteField.JS (q : BigI) where\n\nabstract\n  \u2124q : Set\n  \u2124q = BigI\n\n  private\n    mod-q : BigI \u2192 \u2124q\n    mod-q x = mod x q\n\n  -- There is two ways to go from BigI to \u2124q: check and mod-q\n  -- Use check for untrusted input data and mod-q for internal\n  -- computation.\n  BigI\u25b9\u2124q : BigI \u2192 \u2124q\n  BigI\u25b9\u2124q = -- trace-call \"BigI\u25b9\u2124q \"\n    \u03bb x \u2192\n    mod-q\n      (warn-check (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: \" ++ toString q ++ \" < \" ++ toString x)\n         (warn-check (x \u2265I 0I)\n            (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") x))\n\n  check-non-zero : \u2124q \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \u2124q \u2192 BigI\n  repr x = x\n\n  0# 1# : \u2124q\n  0# = 0I\n  1# = 1I\n\n  1\/_ : Op\u2081 \u2124q\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : Op\u2082 \u2124q\n  x ^ y = modPow (repr x) (repr y) q\n\n_+_ _\u2212_ _*_ _\/_ : Op\u2082 \u2124q\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = mod-q (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\n0\u2212_ : Op\u2081 \u2124q\n0\u2212 x = mod-q (negate (repr x))\n\n_==_ : (x y : \u2124q) \u2192 Bool\nx == y = equals (repr x) (repr y)\n\nsum prod : List \u2124q \u2192 \u2124q\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\n{-\n+-mon-ops : Monoid-Ops \u2124q\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \u2124q\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \u2124q\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \u2124q\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \u2124q\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \u2124q\nfld = fld-ops , fld-struct\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0f09d9f4ac44a07ba808622a5206a04b696a6f76","subject":"agda\/derivation.agda: correct \u0394-term, more comments","message":"agda\/derivation.agda: correct \u0394-term, more comments\n\nIn the previous commit I forgot some comments and I forgot applying some\nchanges, something the typechecker unfortunately would not detect. Fix that.\n\nOld-commit-hash: 9a72bde5058103929f5d3c0eda97b9850a5d7c9c\n","repos":"inc-lc\/ilc-agda","old_file":"derivation.agda","new_file":"derivation.agda","new_contents":"module derivation where\n\nopen import lambda\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","old_contents":"module derivation where\n\nopen import lambda\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\nadaptVar : \u2200  {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8f895d5906dc2d743ba0dc39c4f907e17931b4f9","subject":"agda: implement proof-decorated environments (#33)","message":"agda: implement proof-decorated environments (#33)\n\nOld-commit-hash: eb0097a135a39c572cfb62c967a388632b42f839\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/ValidChanges.agda","new_file":"Denotational\/ValidChanges.agda","new_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid-\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ninvalid-changes-exist : \u00ac (\u2200 {\u03c4} v dv \u2192 Valid-\u0394 {\u03c4} v dv)\ninvalid-changes-exist k with k (\u03bb x \u2192 x) (\u03bb x dx \u2192 false) false true\n... | _ , ()\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\nopen import Syntactic.Contexts Type\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Changes\nopen import ChangeContexts\n\nProofVal : Type \u2192 Set\nProofVal \u03c4 = \u03a3[ v \u2208 \u27e6 \u03c4 \u27e7 ] (\u03a3[ dv \u2208 \u27e6 \u0394-Type \u03c4 \u27e7 ] Valid-\u0394 v dv)\n\nimport Denotational.Environments Type ProofVal as ProofEnv\n\neraseVal : \u2200 {\u03c4} \u2192 ProofVal \u03c4 \u2192 \u27e6 \u03c4 \u27e7\neraseVal (v , dv , dv-valid) = v\n\n-- Specification: eraseEnv = map eraseVal\neraseEnv : \u2200 {\u0393} \u2192 ProofEnv.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u0393 \u27e7\neraseEnv {\u2205} \u2205 = \u2205\neraseEnv {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) = eraseVal v \u2022 eraseEnv \u03c1\n\neraseProofs : \u2200 {\u0393} \u2192 ProofEnv.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u0394-Context \u0393 \u27e7\neraseProofs {\u2205} \u2205 = \u2205\neraseProofs {\u03c4 \u2022 \u0393} ((v , dv , dv-valid) \u2022 \u03c1) = dv \u2022 v \u2022 eraseProofs \u03c1\n","old_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid-\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ninvalid-changes-exist : \u00ac (\u2200 {\u03c4} v dv \u2192 Valid-\u0394 {\u03c4} v dv)\ninvalid-changes-exist k with k (\u03bb x \u2192 x) (\u03bb x dx \u2192 false) false true\n... | _ , ()\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e758586a59fdf8218ce20e2fc52ba3b20db11401","subject":"Changed base type from ]0,1] to ]0,1[ and <= to <","message":"Changed base type from ]0,1] to ]0,1[ and <= to <\n\nAlso added more proof obligations to ]0,1[-ops so that the [0,1]\nmodule doesn't contain any postulates.\n\n1 + x = 1 -- by definition now. More properties regarding 1 are\nsimpler now aswell.\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field     \n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248 \n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n  \n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[ \n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[ \n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n \n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  > \n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  module Prob (P : U \u2192 [0,1])\n              (sumP\u22611 : sumP P allU \u2261 1I) where\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1]-ops (]0,1] : Set) (_\u2264E_ : ]0,1] \u2192 ]0,1] \u2192 Set) : Set where\n  constructor mk\n  field     \n    1E : ]0,1]\n    _+E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\u00b7E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\/E_<_> : (x y : ]0,1]) \u2192 x \u2264E y \u2192 ]0,1]\n\nmodule ]0,1]-\u211a where\n  data ]0,1] : Set where\n    1+_\/1+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1]\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n  1E : ]0,1]\n  1E = 1+ 0 \/1+x+ 0\n  postulate\n    _\u2264E_ : ]0,1] \u2192 ]0,1] \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\u00b7E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\/E_<_> : (x y : ]0,1]) \u2192 x \u2264E y \u2192 ]0,1]\n  ops : ]0,1]-ops ]0,1] _\u2264E_\n  ops = mk 1E _+E_ _\u00b7E_ _\/E_<_>\n\nmodule [0,1] {]0,1] _\u2264E_} (]0,1]R : ]0,1]-ops ]0,1] _\u2264E_) where\n\n  open ]0,1]-ops ]0,1]R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   _I : ]0,1] \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    E\u2264E : \u2200 {x y} \u2192 x \u2264E y \u2192 (x I) \u2264I (y I)\n\n  1I : [0,1]\n  1I = 1E I\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos (_ I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Pos x \u2192 ]0,1] \n  0I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  x I +I 0I = x I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  (x I) \u00b7I 0I = 0I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  _  \/I 0I < _ , () >\n  0I \/I _  < _ , _ > = 0I\n  (x I) \/I (y I) < E\u2264E pf , _ > = (x \/E y < pf >)I\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  postulate\n    +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n    +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n\n    \u2264I-refl : {x : [0,1]} \u2192 x \u2264I x\n\n    \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n    \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n    \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n\nmodule Univ {]0,1] _\u2264E_} (]0,1]R : ]0,1]-ops ]0,1] _\u2264E_)\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1]R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  module Prob (P : U \u2192 [0,1])\n              (sumP\u22611 : sumP P allU \u2261 1I) where\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f40291968fc13e8adb20c12fff5d457c41d78279","subject":"Bits: Get rid of one duplicate","message":"Bits: Get rid of one duplicate\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u00b7 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule SimpleSearch {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u00b7_ public\n\n    search-\u00b7-\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-\u00b7-\u03b5\u2261\u03b5 \u03b5 \u03b5\u00b7\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module Interchange (\u00b7-interchange : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (x \u00b7 z) \u00b7 (y \u00b7 t)) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u00b7 f\u2081 x) \u2261 search f\u2080 \u00b7 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u00b7-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\nopen SimpleSearch public\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-\u2257 : \u2200 {n} (f g : Bits n \u2192 \u2115) \u2192 f \u2257 g \u2192 sum f \u2261 sum g\nsum-\u2257 = search-\u2257 _+_\n\nsum-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 \u2115) \u2192 sum f \u2261 sum (f \u2218 _\u2295_ pad)\nsum-comm = search\u2032-comm _+_ \u2115\u00b0.+-comm\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n-- #-ext\n#-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n#-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n#-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n#-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 = #-comm\n\n#-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n#-const n b = sum-const n (Bool.to\u2115 b)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = search-\u2257 _+_ _ _ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n#-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n#-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n           \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n             2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n           \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n             2^ c \u2238 #\u27e8 f \u27e9\n           \u220e\n  where open \u2261-Reasoning\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero} _+_ _*_ f p hom = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u00b7 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule SimpleSearch {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u00b7_ public\n\n    search-\u00b7-\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-\u00b7-\u03b5\u2261\u03b5 \u03b5 \u03b5\u00b7\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module Interchange (\u00b7-interchange : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (x \u00b7 z) \u00b7 (y \u00b7 t)) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u00b7 f\u2081 x) \u2261 search f\u2080 \u00b7 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u00b7-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\nopen SimpleSearch public\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-\u2257 : \u2200 {n} (f g : Bits n \u2192 \u2115) \u2192 f \u2257 g \u2192 sum f \u2261 sum g\nsum-\u2257 = search-\u2257 _+_\n\nsum-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 \u2115) \u2192 sum f \u2261 sum (f \u2218 _\u2295_ pad)\nsum-comm = search\u2032-comm _+_ \u2115\u00b0.+-comm\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n-- #-ext\n#-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n#-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n#-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n#-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 = #-comm\n\n#-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n#-const n b = sum-const n (Bool.to\u2115 b)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n#-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n#-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n           \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n             2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n           \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n             2^ c \u2238 #\u27e8 f \u27e9\n           \u220e\n  where open \u2261-Reasoning\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero} _+_ _*_ f p hom = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e1bda97e68132c63d228bd4154434843f6d59c0c","subject":"circuit: restore ipC-spec","message":"circuit: restore ipC-spec\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n-}\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7c497fbc194526d5d3f1a4335a6f74574050c633","subject":"Fix TotalNaturalSemantics.agda (fix #32).","message":"Fix TotalNaturalSemantics.agda (fix #32).\n\nOld-commit-hash: 3665667d355c2d049d92115cd27f98cc9dc4f3a2\n","repos":"inc-lc\/ilc-agda","old_file":"TotalNaturalSemantics.agda","new_file":"TotalNaturalSemantics.agda","new_contents":"module TotalNaturalSemantics where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\nopen import Model\n--open import Changes\n--open import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n","old_contents":"module TotalNaturalSemantics where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\nopen import Model\n--open import Changes\n--open import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d9a070a53dc596858137ebe6d0a29d1de2b7d461","subject":"updating statement of finality","message":"updating statement of finality\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"finality.agda","new_file":"finality.agda","new_contents":"open import Prelude\nopen import core\n\nopen import progress-checks\n\nmodule finality where\n  finality : \u03a3[ d \u2208 dhexp ] (d final \u00d7 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d'))) \u2192 \u22a5\n  finality (\u03c01 , \u03c02 , \u03c03 , \u03c04) = final-not-step \u03c02 (\u03c03 , \u03c04)\n\n  -- a slight restatement of the above, generalizing it to the\n  -- multistep judgement\n  finality* : \u2200{d d'} \u2192 d final \u2192 d \u21a6* d' \u2192 d == d'\n  finality* fin MSRefl = refl\n  finality* fin (MSStep x ms) = abort (final-not-step fin (_ , x))\n","old_contents":"open import Prelude\nopen import core\n\nopen import progress-checks\n\nmodule finality where\n  finality : \u2200{d d'} \u2192 d final \u2192 d \u21a6* d' \u2192 d == d'\n  finality fin MSRefl = refl\n  finality fin (MSStep x ms) = abort (final-not-step fin (_ , x))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3ecae75aaafb233597792507020b6d74bee0ecfe","subject":"Implement map, union, difference on bags of natural numbers (#51)","message":"Implement map, union, difference on bags of natural numbers (#51)\n\nTODO before closing the issue:\n- Introduce bags and prove the stupid derivative of map correct.\n\nOld-commit-hash: 76c8ff03739579a6839431ba637557395ccbf8b2\n","repos":"inc-lc\/ilc-agda","old_file":"Data\/NatBag.agda","new_file":"Data\/NatBag.agda","new_contents":"-- An agda type isomorphic to bags of natural numbers\n\n-- How to import this file from anywhere in the world\n--\n-- 1. M-x customize-group RET agda2 RET\n-- 2. Look for the option `Agda2 Include Dirs`\n-- 3. Insert \"$ILC\/agda\", where $ILC is path to ILC repo on your computer\n-- 4. \"Save for future sessions\"\n-- 5. open import Data.NatBag\n\nmodule Data.NatBag where\n\nopen import Data.Nat hiding (zero) renaming (_+_ to plus)\nopen import Data.Integer renaming (suc to +1 ; pred to -1)\nopen import Data.Maybe.Core\nopen import Data.Sum hiding (map)\n\n-----------\n-- Types --\n-----------\n\ndata Nonzero : \u2124 \u2192 Set where\n  negative : (n : \u2115) \u2192 Nonzero -[1+ n ]\n  positive : (n : \u2115) \u2192 Nonzero (+ (suc n))\n\ndata EmptyBag : Set where\n  \u2205 : EmptyBag\n\ndata NonemptyBag : Set where\n  singleton : (i : \u2124) \u2192 (i\u22600 : Nonzero i) \u2192 NonemptyBag\n  _\u2237_ : (i : \u2124) \u2192 (bag : NonemptyBag) \u2192 NonemptyBag\n\ninfixr 5 _\u2237_\n\nBag : Set\nBag = EmptyBag \u228e NonemptyBag\n\n-------------------\n-- Bag interface --\n-------------------\n\nempty : Bag\n\ninsert : \u2115 \u2192 Bag \u2192 Bag\n\ndelete : \u2115 \u2192 Bag \u2192 Bag\n\nlookup : \u2115 \u2192 Bag \u2192 \u2124\n\nupdate : \u2115 \u2192 \u2124 \u2192 Bag \u2192 Bag\n\nupdateWith : \u2115 \u2192 (\u2124 \u2192 \u2124) \u2192 Bag \u2192 Bag\n\nmap : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag -- map on bags, mapKeys on maps\n\nmap\u2082 : (\u2124 \u2192 \u2124) \u2192 Bag \u2192 Bag -- mapValues on maps\n\nzipWith : (\u2124 \u2192 \u2124 \u2192 \u2124) \u2192 Bag \u2192 Bag \u2192 Bag\n\n_++_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_ -- right-associativity follows Haskell, for efficiency\n\n_\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixl 9 _\\\\_ -- fixity follows Haskell's Data.Map.(\\\\)\n\n--------------------\n-- Implementation --\n--------------------\n\ninsert n bag = updateWith n +1 bag\n\ndelete n bag = updateWith n -1 bag\n\nupdate n i bag = updateWith n (\u03bb _ \u2192 i) bag\n\n_++_ = zipWith _+_\n\n_\\\\_ = zipWith _-_\n\nzero : \u2124\nzero = + 0\n\n-- It is very easy to compute a proof that an integer is\n-- nonzero whenever such a thing exists.\nnonzero? : (i : \u2124) \u2192 Maybe (Nonzero i)\nnonzero? (+ 0) = nothing\nnonzero? (+ (suc n)) = just (positive n)\nnonzero? -[1+ n ] = just (negative n)\n\nempty = inj\u2081 \u2205\n\nmakeSingleton : \u2115 \u2192 (i : \u2124) \u2192 Nonzero i \u2192 NonemptyBag\nmakeSingleton 0 i i\u22600 = singleton i i\u22600\nmakeSingleton (suc n) i i\u22600 = (+ 0) \u2237 makeSingleton n i i\u22600\n\nupdateWith n f (inj\u2081 \u2205) with nonzero? (f zero)\n... | nothing = empty\n... | just i\u22600 = inj\u2082 (makeSingleton n (f zero) i\u22600)\nupdateWith 0 f (inj\u2082 (singleton i i\u22600)) with nonzero? (f i)\n... | nothing = empty\n... | just j\u22600 = inj\u2082 (singleton (f i) j\u22600)\nupdateWith (suc n) f (inj\u2082 (singleton i i\u22600)) with nonzero? (f zero)\n... | nothing = inj\u2082 (singleton i i\u22600)\n... | just j\u22600 = inj\u2082 (i \u2237 makeSingleton n (f zero) j\u22600)\nupdateWith 0 f (inj\u2082 (i \u2237 bag)) = inj\u2082 (f i \u2237 bag)\nupdateWith (suc n) f (inj\u2082 (i \u2237 y))\n  with updateWith n f (inj\u2082 y) | nonzero? i\n... | inj\u2081 \u2205 | nothing = empty\n... | inj\u2081 \u2205 | just i\u22600 = inj\u2082 (singleton i i\u22600)\n... | inj\u2082 bag | _ = inj\u2082 (i \u2237 bag)\n\nlookupNonempty : \u2115 \u2192 NonemptyBag \u2192 \u2124\nlookupNonempty 0 (singleton i i\u22600) = i\nlookupNonempty (suc n) (singleton i i\u22600) = zero\nlookupNonempty 0 (i \u2237 bag) = i\nlookupNonempty (suc n) (i \u2237 bag) = lookupNonempty n bag\n\nlookup n (inj\u2081 \u2205) = zero\nlookup n (inj\u2082 bag) = lookupNonempty n bag\n\n-- It is possible to get empty bags by mapping over a nonempty bag:\n-- map (\u03bb _ \u2192 3) { 1 \u21d2 5 , 2 \u21d2 -5 }\nmapKeysFrom : \u2115 \u2192 (\u2115 \u2192 \u2115) \u2192 NonemptyBag \u2192 Bag\nmapKeysFrom n f (singleton i i\u22600) = inj\u2082 (makeSingleton (f n) i i\u22600)\nmapKeysFrom n f (i \u2237 bag) with nonzero? i\n... | nothing = mapKeysFrom (suc n) f bag\n... | just i\u22600 = updateWith (f n) (\u03bb j \u2192 j + i) (mapKeysFrom (suc n) f bag)\n\nmap f (inj\u2081 \u2205) = empty\nmap f (inj\u2082 bag) = mapKeysFrom 0 f bag\n\nmapNonempty\u2082 : (\u2124 \u2192 \u2124) \u2192 NonemptyBag \u2192 Bag\nmapNonempty\u2082 f (singleton i i\u22600) with nonzero? (f i)\n... | nothing = empty\n... | just j\u22600 = inj\u2082 (singleton (f i) j\u22600)\nmapNonempty\u2082 f (i \u2237 bag\u2080)\n  with mapNonempty\u2082 f bag\u2080 | nonzero? (f i)\n... | inj\u2081 \u2205   | nothing  = empty\n... | inj\u2081 \u2205   | just j\u22600 = inj\u2082 (singleton (f i) j\u22600)\n... | inj\u2082 bag | _        = inj\u2082 (f i \u2237 bag)\n\nmap\u2082 f (inj\u2081 \u2205) = empty\nmap\u2082 f (inj\u2082 b) = mapNonempty\u2082 f b\n\nzipLeft : (\u2124 \u2192 \u2124 \u2192 \u2124) \u2192 NonemptyBag \u2192 Bag\nzipLeft f b\u2081 = mapNonempty\u2082 (\u03bb x \u2192 f x zero) b\u2081\n\nzipRight : (\u2124 \u2192 \u2124 \u2192 \u2124) \u2192 NonemptyBag \u2192 Bag\nzipRight f b\u2082 = mapNonempty\u2082 (\u03bb y \u2192 f zero y) b\u2082\n\nzipNonempty : (\u2124 \u2192 \u2124 \u2192 \u2124) \u2192 NonemptyBag \u2192 NonemptyBag \u2192 Bag\nzipNonempty f (singleton i i\u22600) (singleton j j\u22600) with nonzero? (f i j)\n... | nothing = empty\n... | just k\u22600 = inj\u2082 (singleton (f i j) k\u22600)\nzipNonempty f (singleton i i\u22600) (j \u2237 b\u2082)\n  with zipRight f b\u2082 | nonzero? (f i j)\n... | inj\u2081 \u2205 | nothing = empty\n... | inj\u2081 \u2205 | just k\u22600 = inj\u2082 (singleton (f i j) k\u22600)\n... | inj\u2082 bag | _ = inj\u2082 (f i j \u2237 bag)\nzipNonempty f (i \u2237 b\u2081) (singleton j j\u22600)\n  with zipLeft f b\u2081 | nonzero? (f i j)\n... | inj\u2081 \u2205 | nothing = empty\n... | inj\u2081 \u2205 | just k\u22600 = inj\u2082 (singleton (f i j) k\u22600)\n... | inj\u2082 bag | _ = inj\u2082 (f i j \u2237 bag)\nzipNonempty f (i \u2237 b\u2081) (j \u2237 b\u2082)\n  with zipNonempty f b\u2081 b\u2082 | nonzero? (f i j)\n... | inj\u2081 \u2205 | nothing = empty\n... | inj\u2081 \u2205 | just k\u22600 = inj\u2082 (singleton (f i j) k\u22600)\n... | inj\u2082 bag | _ = inj\u2082 (f i j \u2237 bag)\n\nzipWith f (inj\u2081 \u2205) (inj\u2081 \u2205) = empty\nzipWith f (inj\u2081 \u2205) (inj\u2082 b\u2082) = zipRight f b\u2082\nzipWith f (inj\u2082 b\u2081) (inj\u2081 \u2205) = zipLeft f b\u2081\nzipWith f (inj\u2082 b\u2081) (inj\u2082 b\u2082) = zipNonempty f b\u2081 b\u2082\n","old_contents":"-- An agda type isomorphic to bags of natural numbers\n\n-- How to import this file from anywhere in the world\n--\n-- 1. M-x customize-group RET agda2 RET\n-- 2. Look for the option `Agda2 Include Dirs`\n-- 3. Insert \"$ILC\/agda\", where $ILC is path to ILC repo on your computer\n-- 4. \"Save for future sessions\"\n-- 5. open import Data.NatBag\n\nmodule Data.NatBag where\n\nopen import Data.Nat hiding (zero) renaming (_+_ to plus)\nopen import Data.Integer renaming (suc to +1 ; pred to -1)\nopen import Data.Maybe.Core\nopen import Data.Sum\n\n-----------\n-- Types --\n-----------\n\ndata Nonzero : \u2124 \u2192 Set where\n  negative : (n : \u2115) \u2192 Nonzero -[1+ n ]\n  positive : (n : \u2115) \u2192 Nonzero (+ (suc n))\n\ndata EmptyBag : Set where\n  \u2205 : EmptyBag\n\ndata NonemptyBag : Set where\n  singleton : (i : \u2124) \u2192 (i\u22600 : Nonzero i) \u2192 NonemptyBag\n  _\u2237_ : (i : \u2124) \u2192 (bag : NonemptyBag) \u2192 NonemptyBag\n\ninfixr 5 _\u2237_\n\nBag : Set\nBag = EmptyBag \u228e NonemptyBag\n\n-------------------\n-- Bag interface --\n-------------------\n\nempty : Bag\n\ninsert : \u2115 \u2192 Bag \u2192 Bag\n\ndelete : \u2115 \u2192 Bag \u2192 Bag\n\nlookup : \u2115 \u2192 Bag \u2192 \u2124\n\nupdate : \u2115 \u2192 \u2124 \u2192 Bag \u2192 Bag\n\nupdateWith : \u2115 \u2192 (\u2124 \u2192 \u2124) \u2192 Bag \u2192 Bag\n\nmap\u2081 : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag -- map on bags, mapKeys on maps\n\nmap\u2082 : (\u2124 \u2192 \u2124) \u2192 Bag \u2192 Bag -- mapValues on maps\n\n{- TODO: Wait for map.\nzipWith : (\u2124 \u2192 \u2124 \u2192 \u2124) \u2192 Bag \u2192 Bag \u2192 Bag\n\n_++_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_ -- right-associativity follows Haskell, for efficiency\n\n_\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixl 9 _\\\\_ -- fixity follows Haskell's Data.Map.(\\\\)\n-}\n\n--------------------\n-- Implementation --\n--------------------\n\ninsert n bag = updateWith n +1 bag\n\ndelete n bag = updateWith n -1 bag\n\nupdate n i bag = updateWith n (\u03bb _ \u2192 i) bag\n\nzero : \u2124\nzero = + 0\n\n-- It is very easy to compute a proof that an integer is\n-- nonzero whenever such a thing exists.\nnonzero? : (i : \u2124) \u2192 Maybe (Nonzero i)\nnonzero? (+ 0) = nothing\nnonzero? (+ (suc n)) = just (positive n)\nnonzero? -[1+ n ] = just (negative n)\n\nempty = inj\u2081 \u2205\n\nmakeSingleton : \u2115 \u2192 (i : \u2124) \u2192 Nonzero i \u2192 NonemptyBag\nmakeSingleton 0 i i\u22600 = singleton i i\u22600\nmakeSingleton (suc n) i i\u22600 = (+ 0) \u2237 makeSingleton n i i\u22600\n\nupdateWith n f (inj\u2081 \u2205) with nonzero? (f zero)\n... | nothing = empty\n... | just i\u22600 = inj\u2082 (makeSingleton n (f zero) i\u22600)\nupdateWith 0 f (inj\u2082 (singleton i i\u22600)) with nonzero? (f i)\n... | nothing = empty\n... | just j\u22600 = inj\u2082 (singleton (f i) j\u22600)\nupdateWith (suc n) f (inj\u2082 (singleton i i\u22600)) with nonzero? (f zero)\n... | nothing = inj\u2082 (singleton i i\u22600)\n... | just j\u22600 = inj\u2082 (i \u2237 makeSingleton n (f zero) j\u22600)\nupdateWith 0 f (inj\u2082 (i \u2237 bag)) = inj\u2082 (f i \u2237 bag)\nupdateWith (suc n) f (inj\u2082 (i \u2237 y))\n  with updateWith n f (inj\u2082 y) | nonzero? i\n... | inj\u2081 \u2205 | nothing = empty\n... | inj\u2081 \u2205 | just i\u22600 = inj\u2082 (singleton i i\u22600)\n... | inj\u2082 bag | _ = inj\u2082 (i \u2237 bag)\n\nlookupNonempty : \u2115 \u2192 NonemptyBag \u2192 \u2124\nlookupNonempty 0 (singleton i i\u22600) = i\nlookupNonempty (suc n) (singleton i i\u22600) = zero\nlookupNonempty 0 (i \u2237 bag) = i\nlookupNonempty (suc n) (i \u2237 bag) = lookupNonempty n bag\n\nlookup n (inj\u2081 \u2205) = zero\nlookup n (inj\u2082 bag) = lookupNonempty n bag\n\n-- It is possible to get empty bags by mapping over a nonempty bag:\n-- map\u2081 (\u03bb _ \u2192 3) { 1 \u21d2 5 , 2 \u21d2 -5 }\nmapKeysFrom : \u2115 \u2192 (\u2115 \u2192 \u2115) \u2192 NonemptyBag \u2192 Bag\nmapKeysFrom n f (singleton i i\u22600) = inj\u2082 (makeSingleton (f n) i i\u22600)\nmapKeysFrom n f (i \u2237 bag) with nonzero? i\n... | nothing = mapKeysFrom (suc n) f bag\n... | just i\u22600 = updateWith (f n) (\u03bb j \u2192 j + i) (mapKeysFrom (suc n) f bag)\n\nmap\u2081 f (inj\u2081 \u2205) = empty\nmap\u2081 f (inj\u2082 bag) = mapKeysFrom 0 f bag\n\nmap\u2082 f (inj\u2081 \u2205) = empty\nmap\u2082 f (inj\u2082 (singleton i i\u22600)) = {!!}\nmap\u2082 f (inj\u2082 (i \u2237 bag\u2080)) with map\u2082 f (inj\u2082 bag\u2080) | nonzero? i | nonzero? (f i)\n-- If an element has multiplicity 0, it should not be mapped over.\n-- Conceptually, we are mapping over the values of this infinite\n-- map not by f, but by {case 0 => 0}.orElse(f).\n... | inj\u2081 \u2205   | nothing  | _        = empty\n... | inj\u2082 bag | nothing  | _        = inj\u2082 (zero \u2237 bag)\n... | inj\u2081 \u2205   | just i\u22600 | nothing  = empty\n... | inj\u2081 \u2205   | just i\u22600 | just j\u22600 = inj\u2082 (singleton (f i) j\u22600)\n... | inj\u2082 bag | just i\u22600 | _        = inj\u2082 (f i \u2237 bag)\n\n{- TODO: Redefine in terms of zipWith, which will be defined\n         in terms of map.\n-- The union of nonempty bags (with possibly negative\n-- multiplicities) can be empty.\nunionNonempty : NonemptyBag \u2192 NonemptyBag \u2192 Bag\nunionNonempty (singleton i i\u22600) (singleton j j\u22600) with nonzero? (i + j)\n... | nothing = empty\n... | just k\u22600 = inj\u2082 (singleton (i + j) k\u22600)\nunionNonempty (singleton i i\u22600) (j \u2237 b\u2082) = inj\u2082 (i + j \u2237 b\u2082)\n\ninj\u2081 \u2205 ++ b\u2082 = b\u2082\nb\u2081 ++ inj\u2081 \u2205 = b\u2081\n(inj\u2082 b\u2081) ++ (inj\u2082 b\u2082) = (unionNonempty b\u2081 b\u2082)\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f44ff9c8dc76e1365c69137588490cc7705c8477","subject":"agda: rename datatype valid-\u0394 |-> Valid-\u0394","message":"agda: rename datatype valid-\u0394 |-> Valid-\u0394\n\nOld-commit-hash: 37b8db1e2601a886b5bcd31b170ad1560352b8a7\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/ValidChanges.agda","new_file":"Denotational\/ValidChanges.agda","new_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","old_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : valid-\u0394 s ds) -} \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9ecbe1387ef1b739fd49c32e7766fae28250aa45","subject":"Added new versions of J to the note about the identity type.","message":"Added new versions of J to the note about the identity type.\n\nIgnore-this: 762053066cff60c961b3d681934fc31d\n\ndarcs-hash:20111001144638-3bd4e-a232ee2ee9691326ad224dcdc46d20ec88ff6132.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/IdentityType.agda","new_file":"Draft\/IdentityType.agda","new_contents":"------------------------------------------------------------------------------\n-- The identity type\n------------------------------------------------------------------------------\n\n-- We can prove the properties of equality used in the formalization\n-- of FOTC, from refl and J.\n\nmodule IdentityType where\n\ninfix 7 _\u2261_\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 {x} \u2192 x \u2261 x\n\nmodule TypeTheory where\n\n  -- Using the type-theoretic eliminator for equality.\n\n  postulate\n    J : (C : \u2200 x y \u2192 x \u2261 y \u2192 Set) \u2192\n        (\u2200 x \u2192 (C x x refl)) \u2192\n        \u2200 x y \u2192 (c : x \u2261 y) \u2192 C x y c\n\n  -- From Thorsten's slides: A short history of equality.\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} {y} = J (\u03bb x' y' _ \u2192 y' \u2261 x') (\u03bb x' \u2192 refl) x y\n\n  -- From Thorsten's slides: A short history of equality.\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans {x} {y} {z} = J (\u03bb x' y' _ \u2192 y' \u2261 z \u2192 x' \u2261 z) (\u03bb x' pr \u2192 pr) x y\n\n  subst : \u2200 {x} {y} \u2192 (P : D \u2192 Set) \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst {x} {y} P x\u2261y = J (\u03bb x' y' _ \u2192 P x' \u2192 P y') (\u03bb x' pr \u2192 pr) x y x\u2261y\n\nmodule FOL where\n\n  -- Using the usual elimination schema for predicate logic.\n\n  postulate\n    J : (C : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2261 y \u2192 C x \u2192 C y\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} {y} x\u2261y = J (\u03bb y' \u2192 y' \u2261 x) x y x\u2261y refl\n\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans {x} {y} {z} x\u2261y = J (\u03bb y' \u2192 y' \u2261 z \u2192 x \u2261 z) x y x\u2261y (\u03bb pr \u2192 pr)\n\n  subst : \u2200 {x} {y} \u2192 (P : D \u2192 Set) \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst {x} {y} P x\u2261y Px = J P x y x\u2261y Px\n\nmodule ML where\n\n  -- Using Martin-L\u00f6f elimination (\"Hauptsatz ...\", 1971).\n\n  postulate\n    J  : (C : D \u2192 D \u2192 Set) \u2192\n         (\u2200 x \u2192 (C x x)) \u2192\n         \u2200 x y \u2192 x \u2261 y \u2192 C x y\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} {y} = J (\u03bb x' y' \u2192 y' \u2261 x') (\u03bb x' \u2192 refl) x y\n\n  trans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans {x} {y} {z} = J (\u03bb x' y' \u2192 y' \u2261 z \u2192 x' \u2261 z) (\u03bb x' pr \u2192 pr) x y\n\n  subst : \u2200 {x} {y} \u2192 (P : D \u2192 Set) \u2192 x \u2261 y \u2192 P x \u2192 P y\n  subst {x} {y} P x\u2261y = J (\u03bb x' y' \u2192 P x' \u2192 P y') (\u03bb x' pr \u2192 pr) x y x\u2261y\n","old_contents":"------------------------------------------------------------------------------\n-- The identity type\n------------------------------------------------------------------------------\n\n-- We can prove the properties of equality used in the formalization\n-- of FOTC, from refl and J.\n\nmodule IdentityType where\n\ninfix 7 _\u2261_\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 {x} \u2192 x \u2261 x\n  J    : (C : \u2200 x y \u2192 x \u2261 y \u2192 Set) \u2192\n         (\u2200 x \u2192 (C x x refl)) \u2192\n         \u2200 x y \u2192 (c : x \u2261 y) \u2192 C x y c\n\n-- From Thorsten's slides: A short history of equality.\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym {x} {y} = J (\u03bb x' y' _ \u2192 y' \u2261 x') (\u03bb x' \u2192 refl) x y\n\n-- From Thorsten's slides: A short history of equality.\ntrans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans {x} {y} {z} = J (\u03bb x' y' _ \u2192 y' \u2261 z \u2192 x' \u2261 z) (\u03bb x' pr \u2192 pr) x y\n\nsubst : \u2200 {x} {y} \u2192 (P : D \u2192 Set) \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst {x} {y} P x\u2261y = J (\u03bb x' y' _ \u2192 P x' \u2192 P y') (\u03bb x' pr \u2192 pr) x y x\u2261y\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8428371dc1838756182848c20525884faef5dcf9","subject":"Levitate","message":"Levitate\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Primitive.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Primitive.agda","new_contents":"{-# OPTIONS --type-in-type --no-positivity-check #-}\nmodule Spire.DarkwingDuck.Primitive where\n\n----------------------------------------------------------------------\n\ninfixr 4 _,_\ninfixr 5 _\u2237_\n\n----------------------------------------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\n\nelimBot : (P : \u22a5 \u2192 Set)\n  (v : \u22a5) \u2192 P v\nelimBot P ()\n\n----------------------------------------------------------------------\n\ndata \u22a4 : Set where\n  tt : \u22a4\n\nelimUnit : (P : \u22a4 \u2192 Set)\n  (ptt : P tt)\n  (u : \u22a4) \u2192 P u\nelimUnit P ptt tt = ptt\n\n----------------------------------------------------------------------\n\ndata \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  _,_ : (a : A) (b : B a) \u2192 \u03a3 A B\n\nelimPair : {A : Set} {B : A \u2192 Set}\n  (P : \u03a3 A B \u2192 Set)\n  (ppair : (a : A) (b : B a) \u2192 P (a , b))\n  (ab : \u03a3 A B) \u2192 P ab\nelimPair P ppair (a , b) = ppair a b\n\n----------------------------------------------------------------------\n\ndata _\u2261_ {A : Set} (x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\nelimEq : {A : Set} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  (prefl : P x refl)\n  (y : A) (q : x \u2261 y) \u2192 P y q\nelimEq P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\nelimList : {A : Set} (P : List A \u2192 Set)\n  (pnil : P [])\n  (pcons : (x : A) (xs : List A) \u2192 P xs \u2192 P (x \u2237 xs))\n  (xs : List A) \u2192 P xs\nelimList P pnil pcons [] = pnil\nelimList P pnil pcons (x \u2237 xs) = pcons x xs (elimList P pnil pcons xs)\n\n----------------------------------------------------------------------\n\ndata Elem {A : Set} : List A \u2192 Set where\n  here : \u2200{x xs} \u2192 Elem (x \u2237 xs)\n  there : \u2200{x xs} \u2192 Elem xs \u2192 Elem (x \u2237 xs)\n\nelimElem : {A : Set} (P : (xs : List A) \u2192 Elem xs \u2192 Set)\n  (phere : (x : A) (xs : List A) \u2192 P (x \u2237 xs) here)\n  (pthere : (x : A) (xs : List A) (t : Elem xs) \u2192 P xs t \u2192 P (x \u2237 xs) (there t))\n  (xs : List A) (t : Elem xs) \u2192 P xs t\nelimElem P phere pthere (x \u2237 xs) here = phere x xs\nelimElem P phere pthere (x \u2237 xs) (there t) = pthere x xs t (elimElem P phere pthere xs t)\n\n----------------------------------------------------------------------\n\ndata Tel : Set\u2081 where\n  Emp : Tel\n  Ext : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nelimTel : (P : Tel \u2192 Set)\n  (pemp : P Emp)\n  (pext  : (A : Set) (B : A \u2192 Tel) (pb : (a : A) \u2192 P (B a)) \u2192 P (Ext A B))\n  (T : Tel) \u2192 P T\nelimTel P pemp pext Emp = pemp\nelimTel P pemp pext (Ext A B) = pext A B (\u03bb a \u2192 elimTel P pemp pext (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec  : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\nFunc : (I : Set) (D : Desc I) \u2192 (I \u2192 Set) \u2192 I \u2192 Set\nFunc I = elimDesc\n  (\u03bb D \u2192 (I \u2192 Set) \u2192 I \u2192 Set)\n  (\u03bb j X i \u2192 j \u2261 i)\n  (\u03bb j D ih X i \u2192 \u03a3 (X j) (\u03bb _ \u2192 ih X i))\n  (\u03bb A B ih X i \u2192 \u03a3 A (\u03bb a \u2192 ih a X i))\n\nHyps : (I : Set) (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : Func I D X i) \u2192 Set\nHyps I = elimDesc\n  (\u03bb D \u2192 (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : Func I D X i) \u2192 Set)\n  (\u03bb j X P i q \u2192 \u22a4)\n  (\u03bb j D ih X P i x,xs \u2192 elimPair (\u03bb ab \u2192 Set) (\u03bb x xs \u2192 \u03a3 (P j x) (\u03bb _ \u2192 ih X P i xs)) x,xs)\n  (\u03bb A B ih X P i a,xs \u2192 elimPair (\u03bb ab \u2192 Set) (\u03bb a xs \u2192 ih a X P i xs) a,xs)\n\n----------------------------------------------------------------------\n\ndata \u03bc (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p)) (i : I p) : Set where\n  init : Func (I p) D (\u03bc \u2113 P I p D) i \u2192 \u03bc \u2113 P I p D i\n\nall : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set)\n  (p : (i : I) (x : X i) \u2192 P i x) (i : I) (xs : Func I D X i)\n  \u2192 Hyps I D X P i xs\nall (End j) X P p i q = tt\nall (Rec j D) X P p i (x , xs) = p j x , all D X P p i xs\nall (Arg A B) X P p i (a , xs) = all (B a) X P p i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : Func (I p) D (\u03bc \u2113 P I p D) i) (ihs : Hyps (I p) D (\u03bc \u2113 P I p D) M i xs) \u2192 M i (init xs))\n  (i : I p)\n  (x : \u03bc \u2113 P I p D i)\n  \u2192 M i x\nind \u2113 P I p D M \u03b1 i (init xs) = \u03b1 i xs (all D (\u03bc \u2113 P I p D) M (ind \u2113 P I p D M \u03b1) i xs)\n\n----------------------------------------------------------------------\n","old_contents":"module Spire.DarkwingDuck.Primitive where\n\n----------------------------------------------------------------------\n\ninfixr 4 _,_\ninfixr 5 _\u2237_\n\n----------------------------------------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\n\nelimBot : (P : \u22a5 \u2192 Set)\n  (v : \u22a5) \u2192 P v\nelimBot P ()\n\n----------------------------------------------------------------------\n\ndata \u22a4 : Set where\n  tt : \u22a4\n\nelimUnit : (P : \u22a4 \u2192 Set)\n  (ptt : P tt)\n  (u : \u22a4) \u2192 P u\nelimUnit P ptt tt = ptt\n\n----------------------------------------------------------------------\n\ndata \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  _,_ : (a : A) (b : B a) \u2192 \u03a3 A B\n\nelimPair : {A : Set} {B : A \u2192 Set}\n  (P : \u03a3 A B \u2192 Set)\n  (ppair : (a : A) (b : B a) \u2192 P (a , b))\n  (ab : \u03a3 A B) \u2192 P ab\nelimPair P ppair (a , b) = ppair a b\n\n----------------------------------------------------------------------\n\ndata _\u2261_ {A : Set} (x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\nelimEq : {A : Set} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  (prefl : P x refl)\n  (y : A) (q : x \u2261 y) \u2192 P y q\nelimEq P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\nelimList : {A : Set} (P : List A \u2192 Set)\n  (pnil : P [])\n  (pcons : (x : A) (xs : List A) \u2192 P xs \u2192 P (x \u2237 xs))\n  (xs : List A) \u2192 P xs\nelimList P pnil pcons [] = pnil\nelimList P pnil pcons (x \u2237 xs) = pcons x xs (elimList P pnil pcons xs)\n\n----------------------------------------------------------------------\n\ndata Elem {A : Set} : List A \u2192 Set where\n  here : \u2200{x xs} \u2192 Elem (x \u2237 xs)\n  there : \u2200{x xs} \u2192 Elem xs \u2192 Elem (x \u2237 xs)\n\nelimElem : {A : Set} (P : (xs : List A) \u2192 Elem xs \u2192 Set)\n  (phere : (x : A) (xs : List A) \u2192 P (x \u2237 xs) here)\n  (pthere : (x : A) (xs : List A) (t : Elem xs) \u2192 P xs t \u2192 P (x \u2237 xs) (there t))\n  (xs : List A) (t : Elem xs) \u2192 P xs t\nelimElem P phere pthere (x \u2237 xs) here = phere x xs\nelimElem P phere pthere (x \u2237 xs) (there t) = pthere x xs t (elimElem P phere pthere xs t)\n\n----------------------------------------------------------------------\n\ndata Tel : Set\u2081 where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nelimTel : (P : Tel \u2192 Set)\n  (pend : P End)\n  (parg  : (A : Set) (B : A \u2192 Tel) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (T : Tel) \u2192 P T\nelimTel P pend parg End = pend\nelimTel P pend parg (Arg A B) = parg A B (\u03bb a \u2192 elimTel P pend parg (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec  : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 (I \u2192 Set) \u2192 I \u2192 Set\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D)  X i = \u03a3 (X j) (\u03bb _ \u2192 El\u1d30 D X i)\nEl\u1d30 (Arg A B)  X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D)  X P i (x , xs) = \u03a3 (P j x) (\u03bb _ \u2192 Hyps D X P i xs)\nHyps (Arg A B)  X P i (a , xs) = Hyps (B a) X P i xs\n\n----------------------------------------------------------------------\n\ndata \u03bc (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p)) (i : I p) : Set where\n  init : El\u1d30 D (\u03bc \u2113 P I p D) i \u2192 \u03bc \u2113 P I p D i\n\nmap : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set)\n  (p : (i : I) (x : X i) \u2192 P i x) (i : I) (xs : El\u1d30 D X i)\n  \u2192 Hyps D X P i xs\nmap (End j) X P p i q = tt\nmap (Rec j D) X P p i (x , xs) = p j x , map D X P p i xs\nmap (Arg A B) X P p i (a , xs) = map (B a) X P p i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : (\u2113 : String) (P : Set) (I : P \u2192 Set) (p : P) (D : Desc (I p))\n  (M : (i : I p) \u2192 \u03bc \u2113 P I p D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc \u2113 P I p D) i) (ihs : Hyps D (\u03bc \u2113 P I p D) M i xs) \u2192 M i (init xs))\n  (i : I p)\n  (x : \u03bc \u2113 P I p D i)\n  \u2192 M i x\nind \u2113 P I p D M \u03b1 i (init xs) = \u03b1 i xs (map D (\u03bc \u2113 P I p D) M (ind \u2113 P I p D M \u03b1) i xs)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"52889b467ae1b2d3302590984af3fa49681bec28","subject":"elgamal: fixes","message":"elgamal: fixes\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\nopen import sum\nopen import sum-properties\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : SumProp X \u2192 \u2200 u \u2192 SumProp (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7\u03bc \u03bcU \u03bcX u\u2081\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum (\u03bcU \u03bc\u2124q u) (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  -- open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = sucmod-inj _ _ (\u2115\u229e-inj n eq)\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin.to\u2115 m)\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  \u03bc\u2124q : SumProp \u2124q\n  \u03bc\u2124q = \u03bcFin q\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  {-\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n  -}\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum \u03bc\u2124q (f \u2218 [suc]) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 d (r , x , y , _) = d r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 d (r , x , y , z) = d r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 d b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) d\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 d (b , x , y , z , r) = DDH\u2141 d b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba d) = DDH\u2141\u2080 (\u03bb x y z t \u2192 run\u21ba (d y z t) x)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2219_ : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_ : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (R\u2090 : \u2605)\n  (\u03bcR\u2090 : SumProp R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2219 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R\u2090 = (R\u2090 \u2192 PubKey \u2192 Bit \u2192 M)\n              \u00d7 (R\u2090 \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , d) b (r\u2090 , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      d r\u2090 pk (Enc pk (m r\u2090 pk b) y)\n\n      -- Unused\n    Game : (i : Bit) \u2192 \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , d) (b , r\u2090 , x , y , z) = b ==\u1d47 d r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R\u2090} \u2192 Game 0b \u2261 Game0 {R\u2090}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M d (r , x , y , z) = d r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R\u2090} \u2192 Bit \u2192 EncAdv R\u2090 \u2192 DDHAdv R\u2090\n    TrA b (m , d) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = d r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , d) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    like-SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , d) b (r\u2090 , x , y , _z) =\n      d r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    -- open Sum\n\n    R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : SumProp R\n    \u03bcR = \u03bcR\u2090 \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q\n\n    #R_ : Count R\n    #R_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n    f \u2248R g = #R f \u2261 #R g\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module Proof\n        (ddh-hyp : \u2200 A \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n        (otp-lem : \u2200 A m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : EncAdv R\u2090) (b : Bit)\n      where\n\n        OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248R OTP\u2141 M\u2081 d\n        OTP\u2141-lem d M\u2080 M\u2081 = sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y)))\n          where\n          pf : \u2200 r x y \u2192 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2080 d (r , x , y , z))\n                       \u2261 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2081 d (r , x , y , z))\n          pf r x y rewrite otp-lem (d r (g^ x) (g^ y)) (M\u2080 r (g^ x)) (M\u2081 r (g^ x))  = refl\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA (not b) A\n\n        pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n        pf1 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf0,5)\n\n        pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n        pf2 = ddh-hyp A\u1d47\n\n        pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n        pf3 = OTP\u2141-lem (projD A) (projM A b) (projM A (not b))\n\n        pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n        pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf5 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf4,5)\n\n        final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n        final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    \n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ m \u03b4 = \u210b \u03b4 \u2295 m\n\n \n    \/-\u2219 : \u2200 x y \u2192 \u210b\u27e8 x \u27e9\u2295 y \/ x \u2261 y\n    \/-\u2219 x y = {!!}\n    {-\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248R OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    _\/_ : G \u2192 G \u2192 G\n    \/-\u2022 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    comm-^ : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u03bcR\u2090 : SumProp (El `R\u2090)\n  \u03bcR\u2090 = \u03bcU \u03bc\u2124q `R\u2090\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum \u03bcR\u2090\n  sumR\u2090-ext = sum-ext \u03bcR\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\/_ \/-\u2022 comm-^ dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090\n  open EB hiding (g^_)\n\n  otp-base-lem : \u2200 (A : G \u2192 Bit) m \u2192 (A \u2218 g^_) \u2248q (A \u2218 g^_ \u2218 _\u229e_ m)\n  otp-base-lem A m = #-StableUnderInjection {\u03bc = \u03bcFin _} \u03bcFinSUI\n         (A \u2218 g^_) (_\u229e_ m) (\u229e-inj m)\n   -- oh hai! this is:  sym (sum\u2124q-\u229e-lem m (Bool.to\u2115 \u2218 A \u2218 g^_)) -- kthxbye\n\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\nopen import sum\nopen import sum-properties\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : SumProp X \u2192 \u2200 u \u2192 SumProp (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7\u03bc \u03bcU \u03bcX u\u2081\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum (\u03bcU \u03bc\u2124q u) (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  -- open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = sucmod-inj _ _ (\u2115\u229e-inj n eq)\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin.to\u2115 m)\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  \u03bc\u2124q : SumProp \u2124q\n  \u03bc\u2124q = \u03bcFin q\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  {-\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n  -}\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum \u03bc\u2124q (f \u2218 [suc]) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 d (r , x , y , _) = d r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 d (r , x , y , z) = d r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 d b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) d\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 d (b , x , y , z , r) = DDH\u2141 d b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba d) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (d b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2219_ : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_ : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (R\u2090 : \u2605)\n  (\u03bcR\u2090 : SumProp R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2219 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R\u2090 = (R\u2090 \u2192 PubKey \u2192 Bit \u2192 M)\n              \u00d7 (R\u2090 \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , d) b (r\u2090 , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      d r\u2090 pk (Enc pk (m r\u2090 pk b) y)\n\n      -- Unused\n    Game : (i : Bit) \u2192 \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , d) (b , r\u2090 , x , y , z) = b ==\u1d47 d r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R\u2090} \u2192 Game 0b \u2261 Game0 {R\u2090}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M d (r , x , y , z) = d r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R\u2090} \u2192 Bit \u2192 EncAdv R\u2090 \u2192 DDHAdv R\u2090\n    TrA b (m , d) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = d r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , d) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    like-SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , d) b (r\u2090 , x , y , _z) =\n      d r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    -- open Sum\n\n    R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : SumProp R\n    \u03bcR = \u03bcR\u2090 \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q\n\n    #R_ : Count R\n    #R_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n    f \u2248R g = #R f \u2261 #R g\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module Proof\n        (ddh-hyp : \u2200 A \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n        (otp-lem : \u2200 A m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : EncAdv R\u2090) (b : Bit)\n      where\n\n        OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248R OTP\u2141 M\u2081 d\n        OTP\u2141-lem d M\u2080 M\u2081 = sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y)))\n          where\n          pf : \u2200 r x y \u2192 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2080 D (r , x , y , z))\n                       \u2261 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2081 D (r , x , y , z))\n          pf r x y rewrite otp-lem (D r (g^ x) (g^ y)) (M\u2080 r (g^ x)) (M\u2081 r (g^ x))  = refl\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA (not b) A\n\n        pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n        pf1 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf0,5)\n\n        pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n        pf2 = ddh-hyp A\u1d47\n\n        pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n        pf3 = OTP\u2141-lem (projD A) (projM A b) (projM A (not b))\n\n        pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n        pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf5 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf4,5)\n\n        final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n        final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    \n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ m \u03b4 = \u210b \u03b4 \u2295 m\n\n \n    \/-\u2219 : \u2200 x y \u2192 \u210b\u27e8 x \u27e9\u2295 y \/ x \u2261 y\n    \/-\u2219 x y = {!!}\n    {-\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248R OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    _\/_ : G \u2192 G \u2192 G\n    \/-\u2022 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    comm-^ : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u03bcR\u2090 : SumProp (El `R\u2090)\n  \u03bcR\u2090 = \u03bcU \u03bc\u2124q `R\u2090\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum \u03bcR\u2090\n  sumR\u2090-ext = sum-ext \u03bcR\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\/_ \/-\u2022 comm-^ dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090\n  open EB hiding (g^_)\n\n  otp-base-lem : \u2200 (A : G \u2192 Bit) m \u2192 (A \u2218 g^_) \u2248q (A \u2218 g^_ \u2218 _\u229e_ m)\n  otp-base-lem A m = #-StableUnderInjection {\u03bc = \u03bcFin _} \u03bcFinSUI\n         (A \u2218 g^_) (_\u229e_ m) (\u229e-inj m)\n   -- oh hai! this is:  sym (sum\u2124q-\u229e-lem m (Bool.to\u2115 \u2218 A \u2218 g^_)) -- kthxbye\n\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c35a02e641512184761dc90196c26b6363980f5e","subject":"regen the crypto-agda toplevel file","message":"regen the crypto-agda toplevel file\n","repos":"crypto-agda\/crypto-agda","old_file":"crypto-agda.agda","new_file":"crypto-agda.agda","new_contents":"module crypto-agda where\nimport Attack.Compression\nimport Attack.Reencryption\nimport Crypto.Cipher.ElGamal.CPA-DDH\nimport Crypto.Cipher.ElGamal.Generic\nimport Crypto.Cipher.ElGamal.Group\nimport Crypto.Cipher.ElGamal.Homomorphic\nimport Composition.Forkable\nimport Composition.Horizontal\nimport Composition.Vertical\nimport Control.Beh\nimport Control.Protocol.BiSim\nimport Control.Protocol.CoreOld\nimport Control.Protocol.Reduction\nimport Control.Strategy\nimport Control.Strategy.Utils\nimport Crypto.Schemes\nimport FiniteField.FinImplem\nimport FiniteField.JS\nimport FunUniverse.Agda\nimport FunUniverse.BinTree\nimport FunUniverse.Bits\nimport FunUniverse.Category\nimport FunUniverse.Category.Op\nimport FunUniverse.Circuit\nimport FunUniverse.Const\nimport FunUniverse.Core\nimport FunUniverse.Cost\nimport FunUniverse.Data\nimport FunUniverse.Defaults.FirstPart\n-- import FunUniverse.ExnArrow\nimport FunUniverse.Fin\nimport FunUniverse.Fin.Op\nimport FunUniverse.Fin.Op.Abstract\n-- import FunUniverse.FlatFunsProd\nimport FunUniverse.Interface.Bits\nimport FunUniverse.Interface.Two\nimport FunUniverse.Interface.Vec\nimport FunUniverse.Inverse\nimport FunUniverse.Loop\nimport FunUniverse.Nand\nimport FunUniverse.Nand.Function\nimport FunUniverse.Nand.Properties\nimport FunUniverse.README\nimport FunUniverse.Rewiring.Linear\n-- import FunUniverse.State\nimport FunUniverse.Syntax\nimport FunUniverse.Types\nimport Game.Challenge\nimport Game.DDH\nimport Game.EntropySmoothing\nimport Game.EntropySmoothing.WithKey\nimport Game.Generic\nimport Game.IND-CCA\nimport Game.IND-CCA2-dagger\nimport Game.IND-CCA2-dagger.Adversary\nimport Game.IND-CCA2-dagger.Experiment\nimport Game.IND-CCA2-dagger.Protocol\nimport Game.IND-CCA2-dagger.ProtocolImplementation\nimport Game.IND-CCA2-dagger.Valid\nimport Game.IND-CCA2-gen.Protocol\nimport Game.IND-CCA2-gen.ProtocolImplementation\nimport Game.IND-CCA2\n-- import Game.IND-CCA2.Advantage\nimport Game.IND-CPA-alt\nimport Game.IND-CPA-dagger\nimport Game.IND-CPA-utils\nimport Game.IND-CPA\nimport Game.IND-CPA.Core\nimport Game.IND-NM-CPA\nimport Game.NCE\nimport Game.ReceiptFreeness\nimport Game.ReceiptFreeness.Adversary\nimport Game.ReceiptFreeness.CheatingAdversaries\nimport Game.ReceiptFreeness.Definitions\nimport Game.ReceiptFreeness.Definitions.Encryption\nimport Game.ReceiptFreeness.Definitions.Receipt\nimport Game.ReceiptFreeness.Definitions.Tally\nimport Game.ReceiptFreeness.Experiment\nimport Game.ReceiptFreeness.Protocol\nimport Game.ReceiptFreeness.ProtocolImplementation\nimport Game.ReceiptFreeness.Valid\nimport Game.ReceiptFreeness.ValidInst\nimport Game.Transformation.CCA-CPA\nimport Game.Transformation.CCA2-CCA\nimport Game.Transformation.CCA2-CCA2d\nimport Game.Transformation.CCA2d-CCA2\nimport Game.Transformation.CPA-CPAd\nimport Game.Transformation.CPAd-CPA\nimport Game.Transformation.Naor-Yung-proof\nimport Game.Transformation.Naor-Yung\nimport Game.Transformation.ReceiptFreeness-CCA2d\nimport Game.Transformation.ReceiptFreeness-CCA2d.Proof\nimport Game.Transformation.ReceiptFreeness-CCA2d.Protocol\n--TODO import Game.Transformation.ReceiptFreeness-CCA2d.ProtocolImplementation\nimport Game.Transformation.ReceiptFreeness-CCA2d.Simulator\nimport Game.Transformation.ReceiptFreeness-CCA2d.SimulatorInst\n--TODO import Game.Transformation.ReceiptFreeness-CCA2d.Valid\nimport Helios\nimport Language.Simple.Abstract\nimport Language.Simple.Free\nimport Language.Simple.Interface\nimport Language.Simple.Two.Mux\nimport Language.Simple.Two.Mux.Normalise\nimport Language.Simple.Two.Nand\nimport Negligible\nimport README\nimport Solver.AddMax\nimport Solver.Linear\nimport Solver.Linear.Examples\nimport Solver.Linear.Parser\nimport Solver.Linear.Syntax\nimport ZK.ChaumPedersen\n-- import ZK.Disjunctive\nimport ZK.GroupHom\nimport ZK.GroupHom.ElGamal\nimport ZK.GroupHom.FieldChal\nimport ZK.GroupHom.FieldChal2\nimport ZK.GroupHom.FieldChal3\nimport ZK.GroupHom.NatChal\nimport ZK.GroupHom.NumChal\nimport ZK.GroupHom.Types\nimport ZK.JSChecker\nimport ZK.Lemmas\nimport ZK.PartialHeliosVerifier\nimport ZK.Schnorr\nimport ZK.Schnorr.KnownStatement\nimport ZK.SigmaProtocol\nimport ZK.SigmaProtocol.KnownStatement\nimport ZK.SigmaProtocol.Map\nimport ZK.SigmaProtocol.Signature\nimport ZK.SigmaProtocol.Structure\nimport ZK.SigmaProtocol.Types\nimport ZK.Statement\n-- import ZK.Strong-Fiat-Shamir\nimport ZK.Types\nimport adder\nimport alea.cpo\nimport bijection-syntax.Bijection-Fin\nimport bijection-syntax.Bijection\nimport bijection-syntax.README\nimport circuits.bytecode\nimport circuits.circuit\nimport cycle-id\nimport cycle\nimport cycle3\nimport cyclic10\nimport forking-lemma\n-- import hash-param\n-- import misc.merkle-tree\n-- import misc.secret-sharing\n-- import misc.zk\n-- import rewind-on-success\nimport sha1\n","old_contents":"module crypto-agda where\n\n--import Control.Protocol\nimport Explore.README\n\nimport Attack.Compression\nimport Attack.Reencryption\nimport Crypto.Cipher.ElGamal.CPA-DDH\nimport Crypto.Cipher.ElGamal.Generic\nimport Crypto.Cipher.ElGamal.Group\nimport Crypto.Cipher.ElGamal.Homomorphic\nimport Composition.Forkable\nimport Composition.Horizontal\nimport Composition.Vertical\nimport Control.Beh\nimport Control.Protocol.BiSim\nimport Control.Protocol.CoreOld\nimport Control.Protocol.Reduction\nimport Control.Strategy\nimport Control.Strategy.Utils\nimport Crypto.Schemes\nimport FiniteField.FinImplem\nimport FunUniverse.Agda\nimport FunUniverse.BinTree\nimport FunUniverse.Bits\nimport FunUniverse.Category\nimport FunUniverse.Category.Op\nimport FunUniverse.Circuit\nimport FunUniverse.Const\nimport FunUniverse.Core\nimport FunUniverse.Cost\nimport FunUniverse.Data\nimport FunUniverse.Defaults.FirstPart\n--import FunUniverse.ExnArrow\nimport FunUniverse.Fin\nimport FunUniverse.Fin.Op\nimport FunUniverse.Fin.Op.Abstract\n--import FunUniverse.FlatFunsProd\nimport FunUniverse.Interface.Bits\nimport FunUniverse.Interface.Two\nimport FunUniverse.Interface.Vec\nimport FunUniverse.Inverse\nimport FunUniverse.Loop\nimport FunUniverse.Nand\nimport FunUniverse.Nand.Function\nimport FunUniverse.Nand.Properties\nimport FunUniverse.README\nimport FunUniverse.Rewiring.Linear\n--import FunUniverse.State\nimport FunUniverse.Syntax\nimport FunUniverse.Types\nimport Game.Challenge\nimport Game.DDH\nimport Game.EntropySmoothing\nimport Game.EntropySmoothing.WithKey\nimport Game.Generic\nimport Game.IND-CCA\nimport Game.IND-CCA2-dagger\nimport Game.IND-CCA2-dagger.Adversary\nimport Game.IND-CCA2-dagger.Experiment\nimport Game.IND-CCA2-dagger.Protocol\nimport Game.IND-CCA2-dagger.ProtocolImplementation\nimport Game.IND-CCA2-dagger.Valid\nimport Game.IND-CCA2-gen.Protocol\nimport Game.IND-CCA2-gen.ProtocolImplementation\nimport Game.IND-CCA2\nimport Game.IND-CPA-alt\nimport Game.IND-CPA-dagger\nimport Game.IND-CPA-utils\nimport Game.IND-CPA\nimport Game.NCE\nimport Game.ReceiptFreeness\nimport Game.ReceiptFreeness.Adversary\nimport Game.ReceiptFreeness.CheatingAdversaries\nimport Game.ReceiptFreeness.Definitions\nimport Game.ReceiptFreeness.Definitions.Encryption\nimport Game.ReceiptFreeness.Definitions.Receipt\nimport Game.ReceiptFreeness.Definitions.Tally\nimport Game.ReceiptFreeness.Experiment\nimport Game.ReceiptFreeness.Protocol\nimport Game.ReceiptFreeness.ProtocolImplementation\nimport Game.ReceiptFreeness.Valid\nimport Game.ReceiptFreeness.ValidInst\nimport Game.Transformation.CCA-CPA\nimport Game.Transformation.CCA2-CCA\n--import Game.Transformation.CCA2-CCA2d\nimport Game.Transformation.CCA2d-CCA2\nimport Game.Transformation.CPA-CPAd\nimport Game.Transformation.CPAd-CPA\nimport Game.Transformation.Naor-Yung-proof\nimport Game.Transformation.Naor-Yung\nimport Game.Transformation.ReceiptFreeness-CCA2d\nimport Game.Transformation.ReceiptFreeness-CCA2d.Proof\nimport Game.Transformation.ReceiptFreeness-CCA2d.Protocol\nimport Game.Transformation.ReceiptFreeness-CCA2d.ProtocolImplementation\nimport Game.Transformation.ReceiptFreeness-CCA2d.Simulator\nimport Game.Transformation.ReceiptFreeness-CCA2d.SimulatorInst\n-- import Game.Transformation.ReceiptFreeness-CCA2d.Valid\nimport Language.Simple.Abstract\nimport Language.Simple.Free\nimport Language.Simple.Interface\nimport Language.Simple.Two.Mux\nimport Language.Simple.Two.Mux.Normalise\nimport Language.Simple.Two.Nand\nimport README\nimport Solver.AddMax\nimport Solver.Linear\nimport Solver.Linear.Examples\nimport Solver.Linear.Parser\nimport Solver.Linear.Syntax\nimport ZK.PartialHeliosVerifier\nimport ZK.JSChecker\nimport ZK.GroupHom\nimport ZK.GroupHom.FieldChal\nimport ZK.GroupHom.FieldChal2\nimport ZK.GroupHom.FieldChal3\nimport ZK.GroupHom.ElGamal\nimport adder\nimport alea.cpo\nimport bijection-syntax.Bijection-Fin\nimport bijection-syntax.Bijection\nimport bijection-syntax.README\nimport circuits.bytecode\nimport circuits.circuit\nimport sha1\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"716297f35eafe17ce828bb82de3d5879b784645f","subject":"findB -> findKey","message":"findB -> findKey\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits\/Count.agda","new_file":"lib\/Data\/Bits\/Count.agda","new_contents":"\nmodule Data.Bits.Count where\n\nopen import Type hiding (\u2605)\nopen import Data.Bits\nopen import Data.Bits.OperationSyntax\nimport Data.Bits.Search as Search\nopen Search.SimpleSearch\nopen import Data.Bits.Sum\n\n\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\n\nopen import Data.Maybe.NP\n\nopen import Data.Nat.NP hiding (_==_) \nopen import Data.Nat.Properties\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\n\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\n\nopen import Function.NP\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n-- #-ext\n#-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n#-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n#-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n#-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n#-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n#-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 = #-comm\n\n#-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n#-const n b = sum-const n (Bool.to\u2115 b)\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = sum-const0\u22610\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n#-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n#-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n#-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                 #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n#-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n#-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n#-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n#-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n#-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n-- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n-- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g\n  rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                   ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                     (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n        | #-take-drop m n (f \u2218 0\u2237_) g\n        | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                   ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                     (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n        | #-take-drop m n (f \u2218 1\u2237_) g\n          = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                  \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n              #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n            \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n              #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n            \u2261\u27e8 #-take-drop m n g f \u27e9\n              #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n            \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n              #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n            \u220e\n      where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n               +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-LEM : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g |\u2227| f \u27e9 + #\u27e8 |not| g |\u2227| f \u27e9\n#-LEM {zero} f g with g []\n... | false = refl\n... | true  = \u2115\u00b0.+-comm 0 #\u27e8 f \u27e9\n#-LEM {suc n} f g \n  rewrite #-LEM (f \u2218 0\u2237_) (g \u2218 0\u2237_)\n        | #-LEM (f \u2218 1\u2237_) (g \u2218 1\u2237_)\n          = +-interchange #\u27e8 (g \u2218 0\u2237_) |\u2227| (f \u2218 0\u2237_) \u27e9\n                #\u27e8 |not| (g \u2218 0\u2237_) |\u2227| (f \u2218 0\u2237_) \u27e9 \n                #\u27e8 (g \u2218 1\u2237_) |\u2227| (f \u2218 1\u2237_) \u27e9\n                #\u27e8 |not| (g \u2218 1\u2237_) |\u2227| (f \u2218 1\u2237_) \u27e9\n\n\n#-\u2227-snd : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 #\u27e8 g \u27e9\n#-\u2227-snd {zero} f g with f [] | g []\n... | false | false = z\u2264n\n... | false | true  = z\u2264n\n... | true  | _     = \u2115\u2264.reflexive refl\n#-\u2227-snd {suc n} f g = #-\u2227-snd (f \u2218 0\u2237_) (g \u2218 0\u2237_) \n               +-mono #-\u2227-snd (f \u2218 1\u2237_) (g \u2218 1\u2237_)\n\n#-\u2227-fst : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 #\u27e8 f \u27e9\n#-\u2227-fst f g = \n          #\u27e8 f |\u2227| g \u27e9 \n            \u2261\u27e8 #-\u2257 (f |\u2227| g) (g |\u2227| f) (|\u2227|-comm f g) \u27e9 \n          #\u27e8 g |\u2227| f \u27e9 \n            \u2264\u27e8 #-\u2227-snd g f \u27e9 \n          #\u27e8 f \u27e9 \u220e\n      where open \u2264-Reasoning\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n      (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n              #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n              #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n      (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                 (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                            #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n#-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n#-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n            \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n              2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n            \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n              2^ c \u2238 #\u27e8 f \u27e9\n            \u220e\n      where open \u2261-Reasoning\n\n\ndifference-lemma : \u2200 {n}(A B F : Bits n \u2192 Bit) \n      \u2192 #\u27e8 |not| F |\u2227| A \u27e9 \u2261 #\u27e8 |not| F |\u2227| B \u27e9\n      \u2192 dist #\u27e8 A \u27e9 #\u27e8 B \u27e9 \u2264 #\u27e8 F \u27e9\ndifference-lemma A B F A\u2227\u00acF\u2261B\u2227\u00acF = \n  dist #\u27e8 A \u27e9 #\u27e8 B \u27e9 \n    \u2261\u27e8 cong\u2082 dist (#-LEM A F) (#-LEM B F) \u27e9\n  dist (#\u27e8 F |\u2227| A \u27e9 + #\u27e8 |not| F |\u2227| A \u27e9)\n       (#\u27e8 F |\u2227| B \u27e9 + #\u27e8 |not| F |\u2227| B \u27e9)\n    \u2261\u27e8 cong\u2082 dist (\u2115\u00b0.+-comm #\u27e8 F |\u2227| A \u27e9 #\u27e8 |not| F |\u2227| A \u27e9) \n                  (\u2115\u00b0.+-comm #\u27e8 F |\u2227| B \u27e9 #\u27e8 |not| F |\u2227| B \u27e9) \u27e9\n  dist (#\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| A \u27e9)\n       (#\u27e8 |not| F |\u2227| B \u27e9 + #\u27e8 F |\u2227| B \u27e9)\n    \u2261\u27e8 cong\u2082 dist (refl {x = #\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| A \u27e9})\n                  (cong\u2082 _+_ (sym A\u2227\u00acF\u2261B\u2227\u00acF) (refl {x = #\u27e8 F |\u2227| B \u27e9})) \u27e9\n  dist (#\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| A \u27e9)\n       (#\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| B \u27e9)\n    \u2261\u27e8 dist-x+ #\u27e8 |not| F |\u2227| A \u27e9 #\u27e8 F |\u2227| A \u27e9 #\u27e8 F |\u2227| B \u27e9 \u27e9\n  dist #\u27e8 F |\u2227| A \u27e9 #\u27e8 F |\u2227| B \u27e9\n    \u2264\u27e8 dist-bounded {#\u27e8 F |\u2227| A \u27e9} {#\u27e8 F |\u2227| B \u27e9} {#\u27e8 F \u27e9} (#-\u2227-fst F A) (#-\u2227-fst F B) \u27e9 \n  #\u27e8 F \u27e9 \u220e\n     where open \u2264-Reasoning\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindKey : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindKey pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n","old_contents":"\nmodule Data.Bits.Count where\n\nopen import Type hiding (\u2605)\nopen import Data.Bits\nopen import Data.Bits.OperationSyntax\nimport Data.Bits.Search as Search\nopen Search.SimpleSearch\nopen import Data.Bits.Sum\n\n\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\n\nopen import Data.Maybe.NP\n\nopen import Data.Nat.NP hiding (_==_) \nopen import Data.Nat.Properties\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\n\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\n\nopen import Function.NP\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n-- #-ext\n#-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n#-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n#-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n#-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n#-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n#-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 = #-comm\n\n#-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n#-const n b = sum-const n (Bool.to\u2115 b)\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = sum-const0\u22610\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n#-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n#-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n#-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                 #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n#-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n#-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n#-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n#-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n#-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n-- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n-- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g\n  rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                   ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                     (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n        | #-take-drop m n (f \u2218 0\u2237_) g\n        | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                   ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                     (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n        | #-take-drop m n (f \u2218 1\u2237_) g\n          = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                  \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n              #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n            \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n              #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n            \u2261\u27e8 #-take-drop m n g f \u27e9\n              #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n            \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n              #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n            \u220e\n      where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n               +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-LEM : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g |\u2227| f \u27e9 + #\u27e8 |not| g |\u2227| f \u27e9\n#-LEM {zero} f g with g []\n... | false = refl\n... | true  = \u2115\u00b0.+-comm 0 #\u27e8 f \u27e9\n#-LEM {suc n} f g \n  rewrite #-LEM (f \u2218 0\u2237_) (g \u2218 0\u2237_)\n        | #-LEM (f \u2218 1\u2237_) (g \u2218 1\u2237_)\n          = +-interchange #\u27e8 (g \u2218 0\u2237_) |\u2227| (f \u2218 0\u2237_) \u27e9\n                #\u27e8 |not| (g \u2218 0\u2237_) |\u2227| (f \u2218 0\u2237_) \u27e9 \n                #\u27e8 (g \u2218 1\u2237_) |\u2227| (f \u2218 1\u2237_) \u27e9\n                #\u27e8 |not| (g \u2218 1\u2237_) |\u2227| (f \u2218 1\u2237_) \u27e9\n\n\n#-\u2227-snd : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 #\u27e8 g \u27e9\n#-\u2227-snd {zero} f g with f [] | g []\n... | false | false = z\u2264n\n... | false | true  = z\u2264n\n... | true  | _     = \u2115\u2264.reflexive refl\n#-\u2227-snd {suc n} f g = #-\u2227-snd (f \u2218 0\u2237_) (g \u2218 0\u2237_) \n               +-mono #-\u2227-snd (f \u2218 1\u2237_) (g \u2218 1\u2237_)\n\n#-\u2227-fst : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 #\u27e8 f \u27e9\n#-\u2227-fst f g = \n          #\u27e8 f |\u2227| g \u27e9 \n            \u2261\u27e8 #-\u2257 (f |\u2227| g) (g |\u2227| f) (|\u2227|-comm f g) \u27e9 \n          #\u27e8 g |\u2227| f \u27e9 \n            \u2264\u27e8 #-\u2227-snd g f \u27e9 \n          #\u27e8 f \u27e9 \u220e\n      where open \u2264-Reasoning\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n      (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n              #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n              #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n      (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                 (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                            #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n#-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n#-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n            \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n              2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n            \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n              2^ c \u2238 #\u27e8 f \u27e9\n            \u220e\n      where open \u2261-Reasoning\n\n\ndifference-lemma : \u2200 {n}(A B F : Bits n \u2192 Bit) \n      \u2192 #\u27e8 |not| F |\u2227| A \u27e9 \u2261 #\u27e8 |not| F |\u2227| B \u27e9\n      \u2192 dist #\u27e8 A \u27e9 #\u27e8 B \u27e9 \u2264 #\u27e8 F \u27e9\ndifference-lemma A B F A\u2227\u00acF\u2261B\u2227\u00acF = \n  dist #\u27e8 A \u27e9 #\u27e8 B \u27e9 \n    \u2261\u27e8 cong\u2082 dist (#-LEM A F) (#-LEM B F) \u27e9\n  dist (#\u27e8 F |\u2227| A \u27e9 + #\u27e8 |not| F |\u2227| A \u27e9)\n       (#\u27e8 F |\u2227| B \u27e9 + #\u27e8 |not| F |\u2227| B \u27e9)\n    \u2261\u27e8 cong\u2082 dist (\u2115\u00b0.+-comm #\u27e8 F |\u2227| A \u27e9 #\u27e8 |not| F |\u2227| A \u27e9) \n                  (\u2115\u00b0.+-comm #\u27e8 F |\u2227| B \u27e9 #\u27e8 |not| F |\u2227| B \u27e9) \u27e9\n  dist (#\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| A \u27e9)\n       (#\u27e8 |not| F |\u2227| B \u27e9 + #\u27e8 F |\u2227| B \u27e9)\n    \u2261\u27e8 cong\u2082 dist (refl {x = #\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| A \u27e9})\n                  (cong\u2082 _+_ (sym A\u2227\u00acF\u2261B\u2227\u00acF) (refl {x = #\u27e8 F |\u2227| B \u27e9})) \u27e9\n  dist (#\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| A \u27e9)\n       (#\u27e8 |not| F |\u2227| A \u27e9 + #\u27e8 F |\u2227| B \u27e9)\n    \u2261\u27e8 dist-x+ #\u27e8 |not| F |\u2227| A \u27e9 #\u27e8 F |\u2227| A \u27e9 #\u27e8 F |\u2227| B \u27e9 \u27e9\n  dist #\u27e8 F |\u2227| A \u27e9 #\u27e8 F |\u2227| B \u27e9\n    \u2264\u27e8 dist-bounded {#\u27e8 F |\u2227| A \u27e9} {#\u27e8 F |\u2227| B \u27e9} {#\u27e8 F \u27e9} (#-\u2227-fst F A) (#-\u2227-fst F B) \u27e9 \n  #\u27e8 F \u27e9 \u220e\n     where open \u2264-Reasoning\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"825b66ffe80a00ed52000825c62d346bf51ddbc1","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3vl           | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val x) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {v : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure v\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} h    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = aux el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = aux er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (   _ , (sdel , sder))\n                                          with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  correct (Div  _  _) (ernz , (   _ ,    _)) | Pure _       | Pure Zero     | _               | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Pure (Succ _) | el\u21d3vl           | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Step Abort _  | _               | ()\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val x) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {v : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure v\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} h    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = aux el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = aux er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"2c0998148ca4d0f531c98144173c0d27ec548b8a","subject":"Update and extend ZK\/Schnorr","message":"Update and extend ZK\/Schnorr\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Schnorr.agda","new_file":"ZK\/Schnorr.agda","new_contents":"open import Type using (Type)\nopen import Data.Bool.Minimal using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; idp; ap; ap\u2082; !_; module \u2261-Reasoning)\nopen import ZK.Types using (Cyclic-group; module Cyclic-group\n                           ; Cyclic-group-properties; module Cyclic-group-properties)\nimport ZK.SigmaProtocol\n\nmodule ZK.Schnorr\n    {G \u2124q : Type}\n    (cg : Cyclic-group G \u2124q)\n    where\n  open Cyclic-group cg\n\n  Commitment = G\n  Challenge  = \u2124q\n  Response   = \u2124q\n\n  open ZK.SigmaProtocol Commitment Challenge Response\n\n  module _ (x a : \u2124q) where\n    prover-commitment : Commitment\n    prover-commitment = g ^ a\n\n    prover-response : Challenge \u2192 Response\n    prover-response c = a + (x * c)\n\n    prover : Prover\n    prover = prover-commitment , prover-response\n\n  module _ (y : G) where\n    verifier : Verifier\n    verifier (mk A c s) = (g ^ s) == (A \u00b7 (y ^ c))\n\n    -- This simulator shows why it is so important for the\n    -- challenge to be picked once the commitment is known.\n\n    -- To fake a transcript, the challenge and response can\n    -- be arbitrarily chosen. However to be indistinguishable\n    -- from a valid proof it they need to be picked at random.\n    simulator : Simulator\n    simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (g ^ s) \/ (y ^ c)\n\n    witness-extractor : Transcript\u00b2 verifier \u2192 \u2124q\n    witness-extractor t\u00b2 = x\n      module Witness-extractor where\n        open Transcript\u00b2 t\u00b2\n        fd = get-f\u2080 - get-f\u2081\n        cd = get-c\u2080 - get-c\u2081\n        x  = fd * modinv cd\n\n    extractor : \u2200 a \u2192 Extractor verifier\n    extractor a t\u00b2 = prover (witness-extractor t\u00b2) a\n\n  module Proofs (cg-props : Cyclic-group-properties cg) where\n    open Cyclic-group-properties cg-props\n\n    correct : \u2200 x a \u2192 Correct (prover x a) (let y = g ^ x in verifier y)\n    correct x a c\n      = \u2713-== (g ^(a + (x * c))\n           \u2261\u27e8 ^-+ \u27e9\n              g\u02b7 \u00b7 (g ^(x * c))\n           \u2261\u27e8 ap (\u03bb z \u2192 g\u02b7 \u00b7 z) ^-* \u27e9\n              g\u02b7 \u00b7 (y ^ c)\n           \u220e)\n      where\n        open \u2261-Reasoning\n        g\u02b7 = g ^ a\n        y  = g ^ x\n\n    module _ (y : G) where\n      shvzk : Special-Honest-Verifier-Zero-Knowledge (verifier y)\n      shvzk = record { simulator = simulator y\n                     ; correct-simulator = \u03bb _ _ \u2192 \u2713-== \/-\u00b7 }\n\n    module _ (x : \u2124q) (t\u00b2 : Transcript\u00b2 (verifier (g ^ x))) where\n      private\n        y = g ^ x\n        x' = witness-extractor y t\u00b2\n\n      open Transcript\u00b2 t\u00b2 renaming (get-A to A; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                                   ;get-f\u2080 to f\u2080; get-f\u2081 to f\u2081)\n      open Witness-extractor y t\u00b2 hiding (x)\n      open \u2261-Reasoning\n\n      g^xcd\u2261g^fd = g ^(x * cd)\n                 \u2261\u27e8 ^-* \u27e9\n                   y ^ (c\u2080 - c\u2081)\n                 \u2261\u27e8 ^-- \u27e9\n                   (y ^ c\u2080) \/ (y ^ c\u2081)\n                 \u2261\u27e8 ! cancels-\/ \u27e9\n                   (A \u00b7 (y ^ c\u2080)) \/ (A \u00b7 (y ^ c\u2081))\n                 \u2261\u27e8 ap\u2082 _\/_ (! ==-\u2713 verify\u2080) (! ==-\u2713 verify\u2081) \u27e9\n                   (g ^ f\u2080) \/ (g ^ f\u2081)\n                 \u2261\u27e8 ! ^-- \u27e9\n                   g ^ fd\n                 \u220e\n\n      -- The extracted x is correct\n      x\u2261x' : x \u2261 x'\n      x\u2261x' = left-*-to-right-\/ (^-inj g^xcd\u2261g^fd)\n\n      extractor-exact : \u2200 a \u2192 EqProver (extractor y a t\u00b2) (prover x a)\n      extractor-exact a = idp , (\u03bb c \u2192 ap (\u03bb z \u2192 _+_ a (_*_ z c)) (! x\u2261x'))\n\n      special-soundness : Special-Soundness (verifier y)\n      special-soundness = record { extractor = extractor y a\n                                 ; extractor-correct = extractor-correct }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module ZK.Schnorr where\n\nopen import Type\nopen import Data.Two\nopen import Relation.Binary.PropositionalEquality.NP\nimport ZK.Sigma-Protocol\n\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G)\n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_\/_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_==_ : (x y : G) \u2192 \ud835\udfda)\n    where\n\n  Commitment = G\n  Challenge  = \u2124q\n  Response   = \u2124q\n\n  open ZK.Sigma-Protocol Commitment Challenge Response\n\n  module _ (x w : \u2124q) where\n    prover-commitment : Commitment\n    prover-commitment = g ^ w\n\n    prover-response : Challenge \u2192 Response\n    prover-response c = w + (x * c)\n\n    prover : Prover\n    prover = prover-commitment , prover-response\n\n  module _ (y : G) where\n    verifier : Verifier\n    verifier (mk A c s) = (g ^ s) == (A \u00b7 (y ^ c))\n\n    -- This simulator shows why it is so important for the\n    -- challenge to be picked once the commitment is known.\n    \n    -- To fake a transcript, the challenge and response can\n    -- be arbitrarily chosen. However to be indistinguishable\n    -- from a valid proof it they need to be picked at random.\n    module _ (c : Challenge) (s : Response) where\n        -- Compute A, such that the verifier accepts!\n        private\n            A = (g ^ s) \/ (y ^ c)\n\n        simulate-commitment : Commitment\n        simulate-commitment = A\n\n        simulate : Transcript\n        simulate = mk simulate-commitment c s\n\n        module Correct-simulation\n           (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n           (\/-\u00b7  : \u2200 {P Q} \u2192 P \u2261 (P \/ Q) \u00b7 Q)\n          where\n          correct-simulation : \u2713(verifier simulate)\n          correct-simulation = \u2713-== \/-\u00b7\n\n  module Correctness-proof\n           (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n           (^-+  : \u2200 {b x y} \u2192 b ^(x + y) \u2261 (b ^ x) \u00b7 (b ^ y))\n           (^-*  : \u2200 {b x y} \u2192 b ^(x * y) \u2261 (b ^ x) ^ y)\n           (x r  : \u2124q) where\n    open \u2261-Reasoning\n    g\u02b3 = g ^ r\n    correctness : Correctness (prover x r) (verifier (g ^ x))\n    correctness c = \u2713-== (g ^(r + (x * c))\n                        \u2261\u27e8 ^-+ \u27e9\n                         g\u02b3 \u00b7 (g ^(x * c))\n                        \u2261\u27e8 ap (\u03bb z \u2192 g\u02b3 \u00b7 z) ^-* \u27e9\n                         g\u02b3 \u00b7 ((g ^ x) ^ c)\n                        \u220e)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c0a1b912bb72e4c332092cdb7f26de8fcb7313fb","subject":"More robust definition for \u27e6\u2286-\u00f8\u27e7","message":"More robust definition for \u27e6\u2286-\u00f8\u27e7\n","repos":"np\/NomPa","old_file":"lib\/NomPa\/Implem\/LogicalRelation\/Internals.agda","new_file":"lib\/NomPa\/Implem\/LogicalRelation\/Internals.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nimport Level as L\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Sum.NP\nopen import Data.Sum.Logical\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Nullary.Negation using (contraposition)\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP as Bin\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Equivalence as \u21d4 using (_\u21d4_; equivalence; module Equivalence)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nimport Data.Nat.NP as \u2115\nopen \u2115 renaming (_==_ to _==\u2115_)\nimport Data.Nat.Properties as \u2115\nopen import NomPa.Record using (module NomPa)\nopen import NomPa.Implem\nimport NaPa\nopen NaPa using (\u2208-uniq)\nopen import NomPa.Worlds\nimport NomPa.Derived\n\nmodule NomPa.Implem.LogicalRelation.Internals where\n\nopen \u2261 using (_\u2261_; _\u2262_; \u27e6\u2261\u27e7)\nopen \u2261.\u2261-Reasoning\nopen NomPa.Implem.Internals\nopen NomPa NomPa.Implem.nomPa using (worldSym; Name\u00f8-elim; suc\u1d3a; suc\u1d3a\u2191)\n\n-- move to record\npred\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 +1) \u2192 Name \u03b1\npred\u1d3a = subtract\u1d3a 1\n\n\u2262\u2192\u2713-not-==\u2115 : \u2200 {x y} \u2192 x \u2262 y \u2192 \u2713 (not (x ==\u2115 y))\n\u2262\u2192\u2713-not-==\u2115 \u00acp = \u2713-\u00ac-not (\u00acp \u2218 \u2115.==.sound _ _)\n\n#\u21d2\u2209 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 b \u2209 \u03b1\n#\u21d2\u2209 (_ #\u00f8)      = id\n#\u21d2\u2209 (suc#\u2237 b#\u03b1) = #\u21d2\u2209 b#\u03b1\n#\u21d2\u2209 (0# _)      = id\n\n#\u21d2\u2262 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 (x : Name \u03b1) \u2192 binder\u1d3a x \u2262 b\n#\u21d2\u2262 b# (b , b\u2208) \u2261.refl = #\u21d2\u2209 b# b\u2208\n\nbinder\u1d3a-injective : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u2192 x \u2261 y\nbinder\u1d3a-injective {\u03b1} {_ , p\u2081} {a , p\u2082} eq rewrite eq = \u2261.cong (_,_ a) (\u2208-uniq \u03b1 p\u2081 p\u2082)\n\npf-irr\u2032 : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x x\u2032 : Name \u03b1} {y y\u2032 : Name \u03b2}\n         \u2192 binder\u1d3a x \u2261 binder\u1d3a x\u2032 \u2192 binder\u1d3a y \u2261 binder\u1d3a y\u2032 \u2192 \u211b x y \u2192 \u211b x\u2032 y\u2032\npf-irr\u2032 {\u03b1} {\u03b2} \u211b {x , x\u2208} {x\u2032 , x\u2032\u2208} {y , y\u2208} {y\u2032 , y\u2032\u2208} eq\u2081 eq\u2082 x\u1d63y\n  rewrite eq\u2081 | eq\u2082 | \u2208-uniq \u03b1 x\u2208 x\u2032\u2208 | \u2208-uniq \u03b2 y\u2208 y\u2032\u2208 = x\u1d63y\n\npf-irr : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x y x\u2208 x\u2208\u2032 y\u2208 y\u2208\u2032}\n         \u2192 \u211b (x , x\u2208) (y , y\u2208) \u2192 \u211b (x , x\u2208\u2032) (y , y\u2208\u2032)\npf-irr \u211b = pf-irr\u2032 \u211b \u2261.refl \u2261.refl\n\nPreserve-\u2248 : \u2200 {a b \u2113 \u2113a \u2113b} {A : Set a} {B : Set b} \u2192 Rel A \u2113a \u2192 Rel (B) \u2113b \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2248 _\u2248a_ _\u2248b_ _\u223c_ = \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u223c x\u2082 \u2192 y\u2081 \u223c y\u2082 \u2192 (x\u2081 \u2248a y\u2081) \u21d4 (x\u2082 \u2248b y\u2082)\n\n==\u2115\u21d4\u2261 : \u2200 {x y} \u2192 \u2713 (x ==\u2115 y) \u21d4 x \u2261 y\n==\u2115\u21d4\u2261 {x} {y} = equivalence (\u2115.==.sound _ _) \u2115.==.reflexive\n\n\u2261-on-name\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u21d4 x \u2261 y\n\u2261-on-name\u21d4\u2261 {\u03b1} {x} {y} = equivalence binder\u1d3a-injective (\u2261.cong binder\u1d3a)\n\nT\u21d4T\u2192\u2261 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u21d4 \u2713 b\u2082 \u2192 b\u2081 \u2261 b\u2082\nT\u21d4T\u2192\u2261 {1\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\nT\u21d4T\u2192\u2261 {1\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.to b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.from b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\n\n==\u1d3a\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 \u2713 (x ==\u1d3a y) \u21d4 x \u2261 y\n==\u1d3a\u21d4\u2261 = \u2261-on-name\u21d4\u2261 \u27e8\u2218\u27e9 ==\u2115\u21d4\u2261 where open \u21d4 renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n-- Preserving the equalities also mean that the relation is functional and injective\nPreserve-\u2261 : \u2200 {a b \u2113} {A : Set a} {B : Set b} \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2261 _\u223c_ = Preserve-\u2248 _\u2261_ _\u2261_ _\u223c_\n\nexport\u1d3a?-nothing : \u2200 {b \u03b1} (x : Name (b \u25c5 \u03b1)) \u2192 \u2713 (binder\u1d3a x ==\u2115 b) \u2192 export\u1d3a? {b} x \u2261 nothing\nexport\u1d3a?-nothing {b} {\u03b1} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u2261.refl\n... | 0\u2082 | _ = \u22a5-elim p\n\nexport\u1d3a?-just : \u2200 {\u03b1 b} (x : Name (b \u25c5 \u03b1)) {x\u2208} \u2192 \u2713 (not (binder\u1d3a x ==\u2115 b)) \u2192 export\u1d3a? {b} x \u2261 just (binder\u1d3a x , x\u2208)\nexport\u1d3a?-just {\u03b1} {b} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u22a5-elim p\n... | 0\u2082 | _ = \u2261.cong just (binder\u1d3a-injective \u2261.refl)\n\nrecord \u27e6World\u27e7 (\u03b1\u2081 \u03b1\u2082 : World) : Set\u2081 where\n  constructor _,_\n  field\n    _\u223c_ : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set\n    \u223c-pres-\u2261 : Preserve-\u2261 _\u223c_\n\n  \u223c-inj : \u2200 {x y z} \u2192 x \u223c z \u2192 y \u223c z \u2192 x \u2261 y\n  \u223c-inj p q = Equivalence.from (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n  \u223c-fun : \u2200 {x y z} \u2192 x \u223c y \u2192 x \u223c z \u2192 y \u2261 z\n  \u223c-fun p q = Equivalence.to (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n\n  \u223c-\u2208-uniq : \u2200 {x\u2081 x\u2082} {x\u2081\u2208\u2032 x\u2082\u2208\u2032} \u2192\n        x\u2081 \u223c x\u2082 \u2261 (binder\u1d3a x\u2081 , x\u2081\u2208\u2032) \u223c (binder\u1d3a x\u2082 , x\u2082\u2208\u2032)\n  \u223c-\u2208-uniq {_ , x\u2081\u2208} {_ , x\u2082\u2208} {x\u2081\u2208\u2032} {x\u2082\u2208\u2032} = \u2261.ap\u2082 (\u03bb x\u2081\u2208 x\u2082\u2208 \u2192 (_ , x\u2081\u2208) \u223c (_ , x\u2082\u2208)) (\u2208-uniq \u03b1\u2081 x\u2081\u2208 x\u2081\u2208\u2032) (\u2208-uniq \u03b1\u2082 x\u2082\u2208 x\u2082\u2208\u2032)\n\n\u27e6sym\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 \u27e6World\u27e7 \u03b1\u2082 \u03b1\u2081\n\u27e6sym\u27e7 (\u211b , \u211b-pres-\u2261) = \u211b-sym , \u211b-sym-pres-\u2261\n  where \u211b-sym = flip \u211b\n        \u211b-sym-pres-\u2261 : Preserve-\u2261 \u211b-sym\n        \u211b-sym-pres-\u2261 p q = \u21d4.sym (\u211b-pres-\u2261 p q)\n\nrecord \u27e6Binder\u27e7 (b\u2081 b\u2082 : Binder) : Set where\n\nmodule \u27e6\u25c5\u27e7 (b\u2081 b\u2082 : Binder) {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  data _\u211b_ (x : Name (b\u2081 \u25c5 \u03b1\u2081)) (y : Name (b\u2082 \u25c5 \u03b1\u2082)) : Set where\n    here  : (x\u2261b\u2081 : binder\u1d3a x \u2261 b\u2081) (y\u2261b\u2082 : binder\u1d3a y \u2261 b\u2082) \u2192 x \u211b y\n    there : \u2200 (x\u2262b\u2081 : binder\u1d3a x \u2262 b\u2081) (y\u2262b\u2082 : binder\u1d3a y \u2262 b\u2082) {x\u2208 y\u2208} (x\u223cy : (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a y , y\u2208)) \u2192 x \u211b y\n\n  \u211b-inj : \u2200 {x y z} \u2192 x \u211b z \u2192 y \u211b z \u2192 x \u2261 y\n  \u211b-inj (here x\u2261b\u2081 _)    (here y\u2261b\u2081 _)     = binder\u1d3a-injective (\u2261.trans x\u2261b\u2081 (\u2261.sym y\u2261b\u2081))\n  \u211b-inj (here _ z\u2261b\u2082)    (there _ z\u2262b\u2082 _)  = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj (there _ z\u2262b\u2082 _) (here _ z\u2261b\u2082)     = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj {x} {y} {z} (there x\u2262b\u2081 z\u2262b\u2082 {x\u2208} {z\u2208} x\u223cz) (there y\u2262b\u2081 _ {y\u2208} {z\u2208\u2032} y\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-inj x\u223cz (\u2261.tr (\u03bb z\u2208 \u2192 (binder\u1d3a y , y\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2082 z\u2208\u2032 z\u2208) y\u223cz )))\n\n  \u211b-fun : \u2200 {x y z} \u2192 x \u211b y \u2192 x \u211b z \u2192 y \u2261 z\n  \u211b-fun (here _ y\u2261b\u2082)  (here _ z\u2261b\u2082)    = binder\u1d3a-injective (\u2261.trans y\u2261b\u2082 (\u2261.sym z\u2261b\u2082))\n  \u211b-fun (here p _)     (there \u00acp _ _)    = \u22a5-elim (\u00acp p)\n  \u211b-fun (there \u00acp _ _) (here p _)        = \u22a5-elim (\u00acp p)\n  \u211b-fun {x} {y} {z} (there x\u2262b\u2081 y\u2262b\u2082 {x\u2208} {y\u2208} x\u223cy) (there p q {x\u2208\u2032} {z\u2208} x\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-fun x\u223cy (\u2261.tr (\u03bb x\u2208 \u2192 (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2081 x\u2208\u2032 x\u2208) x\u223cz )))\n\n  \u211b-pres-\u2261 : Preserve-\u2261 _\u211b_\n  \u211b-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 x\u2081 \u211b x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b-inj x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 y\u2081 \u211b y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n  \u27e6world\u27e7 : \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n  \u27e6world\u27e7 = _\u211b_ , \u211b-pres-\u2261\n\nhere\u2032 : \u2200 {\u03b1\u2081 \u03b1\u2082 b\u2081 b\u2082 pf\u2081 pf\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} \u2192 \u27e6\u25c5\u27e7._\u211b_ b\u2081 b\u2082 \u03b1\u1d63 (b\u2081 , pf\u2081) (b\u2082 , pf\u2082)\nhere\u2032 = \u27e6\u25c5\u27e7.here \u2261.refl \u2261.refl\n\n_\u27e6\u25c5\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u25c5_ _\u25c5_\n_\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63 = \u27e6\u25c5\u27e7.\u27e6world\u27e7 b\u2081 b\u2082 \u03b1\u1d63\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 (\u211b , _) x\u2081 x\u2082 = \u211b x\u2081 x\u2082\n\n\u27e6zero\u1d2e\u27e7 : \u27e6Binder\u27e7 zero\u1d2e zero\u1d2e\n\u27e6zero\u1d2e\u27e7 = _\n\n\u27e6suc\u1d2e\u27e7 : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) suc\u1d2e suc\u1d2e\n\u27e6suc\u1d2e\u27e7 = _\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = (\u03bb _ _ \u2192 \u22a5) , (\u03bb())\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 x\u1d63 y\u1d63 =\n   \u27e6\ud835\udfda\u27e7-Props.reflexive (T\u21d4T\u2192\u2261 (sym ==\u1d3a\u21d4\u2261 \u27e8\u2218\u27e9 \u223c-pres-\u2261 x\u1d63 y\u1d63 \u27e8\u2218\u27e9 ==\u1d3a\u21d4\u2261)) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  open \u21d4 using (sym) renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n\u27e6name\u1d2e\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) name\u1d2e name\u1d2e\n\u27e6name\u1d2e\u27e7 _ _ = here\u2032\n\n\u27e6binder\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) binder\u1d3a binder\u1d3a\n\u27e6binder\u1d3a\u27e7 _ _ = _\n\n\u27e6nothing\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082}\n            \u2192 x \u2261 nothing \u2192 y \u2261 nothing \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6nothing\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6nothing\u27e7\n\n\u27e6just\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082} {x\u2032 y\u2032}\n            \u2192 x \u2261 just x\u2032 \u2192 y \u2261 just y\u2032 \u2192 A\u1d63 x\u2032 y\u2032 \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6just\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6just\u27e7\n\n\u27e6export\u1d3a?\u27e7 : (\u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) (\u03bb {b} \u2192 export\u1d3a? {b}) (\u03bb {b} \u2192 export\u1d3a? {b})\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.here e\u2081 e\u2082)\n  = \u27e6nothing\u27e7\u2032 (export\u1d3a?-nothing x\u2081 (\u2115.==.reflexive e\u2081)) (export\u1d3a?-nothing x\u2082 (\u2115.==.reflexive e\u2082))\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.there x\u2081\u2262b\u2081 x\u2082\u2262b\u2082 x\u2081\u223cx\u2082)\n  = \u27e6just\u27e7\u2032 (export\u1d3a?-just x\u2081 (\u2262\u2192\u2713-not-==\u2115 x\u2081\u2262b\u2081))\n           (export\u1d3a?-just x\u2082 (\u2262\u2192\u2713-not-==\u2115 x\u2082\u2262b\u2082)) x\u2081\u223cx\u2082\n\n_\u27e6\u2286\u27e7b_ : \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a x) (coerce\u1d3a y)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x y} \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) x y \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7b \u03b2\u1d63) x y\nun\u27e6\u2286\u27e7 (mk x) {a} {b} c = x {a} {b} c\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {mk f\u2081} {mk f\u2082} f {mk g\u2081} {mk g\u2082} g\n  = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {coerce\u1d3a (mk f\u2081) x\u2081} {coerce\u1d3a (mk f\u2082) x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 ()\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk \u03bb { {_ , ()} }\n\n_\u27e6#\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) _#_ _#_\n_\u27e6#\u27e7_ _ _ _ _ = \u22a4\n\n_\u27e6#\u00f8\u27e7 : (\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u27e6\u00f8\u27e7) _#\u00f8 _#\u00f8\n_\u27e6#\u00f8\u27e7 _ = _\n\n\u27e6suc#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 (\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6#\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) suc# suc#\n\u27e6suc#\u27e7 _ _ _ = _\n\n\u27e6\u2286-#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                  (b\u1d63 \u27e6#\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) \u2286-# \u2286-#\n\u27e6\u2286-#\u27e7 _ {b\u2081} {b\u2082} _ {b\u2081#\u03b1\u2081} {b\u2082#\u03b1\u2082} _ = mk (\u03bb {x\u2081} {x\u2082} \u2192 \u27e6\u25c5\u27e7.there (#\u21d2\u2262 b\u2081#\u03b1\u2081 x\u2081) (#\u21d2\u2262 b\u2082#\u03b1\u2082 x\u2082))\n\n\u27e6\u2286-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b2\u1d63)) \u2286-\u25c5 \u2286-\u25c5\n\u27e6\u2286-\u25c5\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {b\u2081} {b\u2082} _ {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} (mk f) = mk g where\n  open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n  g : \u2200 {x\u2081 x\u2082} \u2192 _\u211b_ \u03b1\u1d63 x\u2081 x\u2082 \u2192 _\u211b_ \u03b2\u1d63 (coerce\u1d3a (\u2286-\u25c5 b\u2081 \u03b1\u2286\u03b2\u2081) x\u2081) (coerce\u1d3a (\u2286-\u25c5 b\u2082 \u03b1\u2286\u03b2\u2082) x\u2082)\n  g (here x\u2261b\u2081 y\u2261b\u2082)       = here x\u2261b\u2081 y\u2261b\u2082\n  g (there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = there x\u2262b\u2081 y\u2262b\u2082 (f x\u223cy)\n\ncong\u2115 : \u2200 {\u03b1 \u03b2 : World} {x y : Name \u03b1} (f : \u2115 \u2192 \u2115) {fx\u2208 fy\u2208} \u2192 x \u2261 y \u2192 _\u2261_ {A = Name \u03b2} (f (binder\u1d3a x) , fx\u2208) (f (binder\u1d3a y) , fy\u2208)\ncong\u2115 f = binder\u1d3a-injective \u2218 \u2261.cong (f \u2218 binder\u1d3a)\n\nmodule \u27e6+1\u27e7 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63 renaming (_\u223c_ to \u211b; \u223c-inj to \u211b-inj; \u223c-fun to \u211b-fun)\n  \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set\n  \u211b+1 x y = \u27e6Name\u27e7 \u03b1\u1d63 (pred\u1d3a x) (pred\u1d3a y)\n{-\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x x\u2208 y y\u2208} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (suc x , ) (sucN y)\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x y} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (sucN x) (sucN y)\n-}\n\n  \u211b+1-inj : \u2200 {x y z} \u2192 \u211b+1 x z \u2192  \u211b+1 y z \u2192 x \u2261 y\n  -- \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} (suc a) (suc b) = {!!}\n  \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-inj a b)\n  \u211b+1-inj {zero , ()} _ _\n  \u211b+1-inj {_} {zero , ()} _ _\n  \u211b+1-inj {_} {_} {zero , ()} _ _\n  \u211b+1-fun : \u2200 {x y z} \u2192 \u211b+1 x y \u2192  \u211b+1 x z \u2192 y \u2261 z\n  \u211b+1-fun {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-fun a b)\n  \u211b+1-fun {zero , ()} _ _\n  \u211b+1-fun {_} {zero , ()} _ _\n  \u211b+1-fun {_} {_} {zero , ()} _ _\n\n  \u211b+1-pres-\u2261 : Preserve-\u2261 \u211b+1\n  -- \u211b+1-pres-\u2261 x y = equivalence {!\u211b+1-inj!} {!!}\n  \u211b+1-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     -- factor this move\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b+1-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 \u211b+1 x\u2081 x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b+1-inj {x\u2081} {y\u2081} {x\u2082} x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 \u211b+1 y\u2081 y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 \u03b1\u1d63 = \u211b+1 , \u211b+1-pres-\u2261 where open \u27e6+1\u27e7 \u03b1\u1d63\n\nopen WorldOps worldSym\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 \u03b1\u1d63 = \u27e6zero\u1d2e\u27e7 \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)\n\n_\u27e6\u2191\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6\u2191\u27e7 k) \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082)\n                 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 _  (\u27e6\u25c5\u27e7.here () _)\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b1\u1d63 (\u27e6\u25c5\u27e7.there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63) x\u223cy\n\n\u27e6\u00f8+1\u2286\u00f8\u27e7 : (\u27e6\u00f8\u27e7 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u00f8+1\u2286\u00f8 \u00f8+1\u2286\u00f8\n\u27e6\u00f8+1\u2286\u00f8\u27e7 = mk (\u03bb {x} _ \u2192 Name\u00f8-elim (pred\u1d3a x))\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 _ \u03b2\u1d63 \u03b1\u2286\u03b2\u1d63 = mk (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) \u2218 un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk helper where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper (here x\u2261b\u2081 y\u2261b\u2082) = here x\u2261b\u2081 y\u2261b\u2082\n  helper {x\u2081 , _} {x\u2082 , _} (there x\u2262b\u2081 y\u2262b\u2082 {p\u2081} {p\u2082} x\u223cy)\n     = there x\u2262b\u2081 y\u2262b\u2082 {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081) x\u2081 p\u2081}\n                        {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082) x\u2082 p\u2082}\n                        (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 x\u223cy))\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {add\u1d3a 1 x\u2081} {add\u1d3a 1 x\u2082}) \n\n\u22620 : \u2200 {\u03b1} (x : Name (\u03b1 +1)) \u2192 binder\u1d3a x \u2262 0\n\u22620 (0 , ()) \u2261.refl\n\u22620 (suc _ , _) ()\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 there (\u22620 x\u2081) (\u22620 x\u2082)) where open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ \u03b2\u1d63 \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b2\u1d63 (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (there (\u03bb()) (\u03bb()) x\u1d63))) where\n  open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ \u03b2\u1d63 \u03b1+\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} x\u1d63)) where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b2\u1d63 x\u2081 x\u2082\n  helper (here () _)\n  helper (there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) x\u223cy\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 \u03b1\u1d63 x\u1d63 = \u27e6coerce\u1d3a\u27e7 _ _ (\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63) (\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 x\u1d63)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _  zero x\u1d63 = x\u1d63\n\u27e6add\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) x\u1d63 = pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = pf-irr\u2032 (\u27e6Name\u27e7 \u03b1\u1d63) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2081) 1 k\u2081) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2082) 1 k\u2082) (\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {pf\u2081 : 0 \u2208 \u03b1\u2081 \u21911} {n} {pf\u2082 : suc n \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (0 , pf\u2081) (suc n , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.there p _ _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.here _ ())\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {n} {pf\u2081 : suc n \u2208 \u03b1\u2081 \u21911} {pf\u2082 : 0 \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc n , pf\u2081) (0 , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.there _ p _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.here () _)\n\n-- cmp\u1d3a-bool : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 \ud835\udfda\n-- cmp\u1d3a-bool \u2113 x = suc (binder\u1d3a x) <= \u2113\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) NaPa.cmp\u1d3a-bool NaPa.cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 _ zero _ = \u27e60\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 _ (suc _) {zero , _} {zero , _} _ = \u27e61\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) NaPa.easy-cmp\u1d3a NaPa.easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 _ zero x\u1d63 = \u27e6inr\u27e7 x\u1d63\n\u27e6easy-cmp\u1d3a\u27e7 _ (suc _) {zero , _} {zero , _} x\u1d63 = \u27e6inl\u27e7 here\u2032\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6map\u27e7 _ _ _ _ (\u27e6suc\u1d3a\u2191\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63)) (pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63))\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (NaPa.easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6\u2286-dist-+1-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) \u27e6\u2286\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))))\n                 \u2286-dist-+1-\u25c5 \u2286-dist-+1-\u25c5\n\u27e6\u2286-dist-+1-\u25c5\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083) where\n  -- open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n\n  hper :   \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (suc\u1d3a x\u2081) (suc\u1d3a x\u2082)\n  hper (\u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082) = \u27e6\u25c5\u27e7.here (\u2261.cong suc x\u2261b\u2081) (\u2261.cong suc y\u2261b\u2082)\n  hper (\u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong pred) (y\u2262b\u2082 \u2218 \u2261.cong pred) x\u223cy\n\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 = hper x\u1d63\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n\u27e6\u2286-dist-\u25c5-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) \u27e6\u2286\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)))\n                 \u2286-dist-\u25c5-+1 \u2286-dist-\u25c5-+1\n\u27e6\u2286-dist-\u25c5-+1\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk helper where\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 with x\u1d63\n  ... | \u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082 = \u27e6\u25c5\u27e7.here (\u2261.cong (\u03bb x \u2192 x \u2238 1) x\u2261b\u2081) (\u2261.cong (\u03bb x \u2192 x \u2238 1) y\u2261b\u2082)\n  ... | \u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong suc) (y\u2262b\u2082 \u2218 \u2261.cong suc) x\u223cy\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n{-\nBinder\u2261\u2115 : Binder \u2261 \u2115\nBinder\u2261\u2115 = \u2261.refl\n\npostulate\n  \u27e6Binder\u27e7\u2262\u27e6\u2115\u27e7 : \u27e6Binder\u27e7 \u2262 \u27e6\u2115\u27e7\n\n\u27e6\u2261\u27e7-lem : \u2200 {a\u2081 a\u2082 a\u1d63 A\u2081 A\u2082} {A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082} {x\u2081 x\u2082} {x\u1d63 : A\u1d63 x\u2081 x\u2082} {y\u2081 y\u2082} {y\u1d63 : A\u1d63 y\u2081 y\u2082} {pf\u2081 pf\u2082} \u2192 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 y\u1d63 pf\u2081 pf\u2082 \u2192 \u2200 (P : \u2200 {z\u2081 z\u2082} \u2192 A\u1d63 z\u2081 z\u2082 \u2192 Set) \u2192 P x\u1d63 \u2192 P y\u1d63\n\u27e6\u2261\u27e7-lem = {!!}\n\n\u00ac\u27e6Binder\u2261\u2115\u27e7 : \u00ac((\u27e6\u2261\u27e7 \u27e6Set\u2080\u27e7 \u27e6Binder\u27e7 \u27e6\u2115\u27e7) Binder\u2261\u2115 Binder\u2261\u2115)\n\u00ac\u27e6Binder\u2261\u2115\u27e7 = \u03bb pf \u2192 \u27e6\u2261\u27e7-lem pf (\u03bb x \u2192 \u22a5) {!!}\n-}\n\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 : (\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                   \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                    \u27e6\u2261\u27e7 \u27e6Binder\u27e7 (\u27e6binder\u1d3a\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) (\u27e6name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63)) b\u1d63) binder\u1d3a\u2218name\u1d2e binder\u1d3a\u2218name\u1d2e\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63 = \u2261.\u27e6refl\u27e7\n\n\u27e8_,_\u27e9\u27e6\u25c5\u27e7_ : (b\u2081 b\u2082 : Binder) \u2192 \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) \u2192 \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n\u27e8 b\u2081 , b\u2082 \u27e9\u27e6\u25c5\u27e7 \u03b1\u1d63 = _\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63\n\nmodule Perm (m n : \u2115) (m\u2262n : m \u2262 n) where\n  mn\u1d42 : World\n  mn\u1d42 = m \u25c5 n \u25c5 \u00f8\n  nm\u1d42 : World\n  nm\u1d42 = n \u25c5 m \u25c5 \u00f8\n  perm-m-n : \u27e6World\u27e7 mn\u1d42 nm\u1d42\n  perm-m-n = \u27e8 m , n \u27e9\u27e6\u25c5\u27e7 (\u27e8 n , m \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7)\n  m-n : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (m , m\u2208) (n , n\u2208)\n  m-n = here\u2032\n  n-m : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (n , n\u2208) (m , m\u2208)\n  n-m = \u27e6\u25c5\u27e7.there (m\u2262n \u2218 \u2261.sym) m\u2262n {b\u2208b\u25c5 n \u00f8} {b\u2208b\u25c5 m \u00f8} here\u2032\n\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 : \u00ac(\u27e6\ud835\udfda\u27e7 1\u2082 0\u2082)\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 ()\n\nbinder-irrelevance : \u2200 (f : Binder \u2192 \ud835\udfda)\n                     \u2192 (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f\n                     \u2192 \u2200 {b\u2081 b\u2082} \u2192 f b\u2081 \u2261 f b\u2082\nbinder-irrelevance _ f\u1d63 = \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 _)\n\ncontrab : \u2200 (f : Binder \u2192 \ud835\udfda) {b\u2081 b\u2082}\n          \u2192 f b\u2081 \u2262 f b\u2082\n          \u2192 \u00ac((\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\ncontrab f = contraposition (\u03bb f\u1d63 \u2192 binder-irrelevance f (\u03bb x\u1d63 \u2192 f\u1d63 x\u1d63))\n\nmodule Single \u03b1\u2081 \u03b1\u2082 x\u2081 x\u2082 where\n  data \u211b : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set where\n    refl : \u211b x\u2081 x\u2082\n  \u211b-pres-\u2261 : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 \u211b x\u2081 x\u2082 \u2192 \u211b y\u2081 y\u2082 \u2192 (x\u2081 \u2261 y\u2081) \u21d4 (x\u2082 \u2261 y\u2082)\n  \u211b-pres-\u2261 refl refl = equivalence (const \u2261.refl) (const \u2261.refl)\n  \u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082\n  \u03b1\u1d63 = \u211b , \u211b-pres-\u2261\n\npoly-name-uniq : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1} (x : Name \u03b1) \u2192 f x \u2261 f {\u00f8 \u21911} (0 , _)\npoly-name-uniq f f\u1d63 {\u03b1} x =\n  \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 {\u03b1} {\u00f8 \u21911} \u03b1\u1d63 {x} {0 , _} refl)\n  where open Single \u03b1 (\u00f8 \u21911) x (0 , _)\n\npoly-name-irrelevance : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1\u2081 \u03b1\u2082} (x\u2081 : Name \u03b1\u2081) (x\u2082 : Name \u03b1\u2082)\n                        \u2192 f x\u2081 \u2261 f x\u2082\npoly-name-irrelevance f f\u1d63 x\u2081 x\u2082 =\n  \u2261.trans (poly-name-uniq f f\u1d63 x\u2081) (\u2261.sym (poly-name-uniq f f\u1d63 x\u2082))\n\nmodule Broken where\n  _<=\u1d3a_ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name \u03b1 \u2192 \ud835\udfda\n  (m , _) <=\u1d3a (n , _) = \u2115._<=_ m n\n\n  \u00ac\u27e6<=\u1d3a\u27e7 : \u00ac((\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _<=\u1d3a_ _<=\u1d3a_)\n  \u00ac\u27e6<=\u1d3a\u27e7 \u27e6<=\u27e7 = \u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 (\u27e6<=\u27e7 perm-0-1 {0 , _} {1 , _} 0-1 {1 , _} {0 , _} 1-0)\n     where open Perm 0 1 (\u03bb()) renaming (perm-m-n to perm-0-1; m-n to 0-1; n-m to 1-0)\n\n  \u2286-broken : \u2200 \u03b1 b \u2192 \u03b1 \u2286 (b \u25c5 \u03b1)\n  \u2286-broken \u03b1 b = mk (\u03bb b\u2032 b\u2032\u2208\u03b1 \u2192 \u2261.tr id (\u2261.sym (\u25c5-sem \u03b1 b\u2032 b)) (If\u2032 \u2115._==_ b\u2032 b then _ else b\u2032\u2208\u03b1))\n\n  \u00ac\u27e6\u2286-broken\u27e7 : \u00ac((\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63))\n                  \u2286-broken \u2286-broken)\n  \u00ac\u27e6\u2286-broken\u27e7 \u27e6\u2286-broken\u27e7 = bot\n     where 0\u21940 : \u27e6Name\u27e7 (\u27e8 0 , 1 \u27e9\u27e6\u25c5\u27e7 \u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) (0 , _) (0 , _)\n           0\u21940 = un\u27e6\u2286\u27e7 (\u27e6\u2286-broken\u27e7 (\u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) _) {0 , _} {0 , _} here\u2032\n\n           bot : \u22a5\n           bot with 0\u21940\n           bot | \u27e6\u25c5\u27e7.here _ ()\n           bot | \u27e6\u25c5\u27e7.there 0\u22620 _ _ = 0\u22620 \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nimport Level as L\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Sum.NP\nopen import Data.Sum.Logical\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Nullary.Negation using (contraposition)\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP as Bin\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Equivalence as \u21d4 using (_\u21d4_; equivalence; module Equivalence)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nimport Data.Nat.NP as \u2115\nopen \u2115 renaming (_==_ to _==\u2115_)\nimport Data.Nat.Properties as \u2115\nopen import NomPa.Record using (module NomPa)\nopen import NomPa.Implem\nimport NaPa\nopen NaPa using (\u2208-uniq)\nopen import NomPa.Worlds\nimport NomPa.Derived\n\nmodule NomPa.Implem.LogicalRelation.Internals where\n\nopen \u2261 using (_\u2261_; _\u2262_; \u27e6\u2261\u27e7)\nopen \u2261.\u2261-Reasoning\nopen NomPa.Implem.Internals\nopen NomPa NomPa.Implem.nomPa using (worldSym; Name\u00f8-elim; suc\u1d3a; suc\u1d3a\u2191)\n\n-- move to record\npred\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 +1) \u2192 Name \u03b1\npred\u1d3a = subtract\u1d3a 1\n\n\u2262\u2192\u2713-not-==\u2115 : \u2200 {x y} \u2192 x \u2262 y \u2192 \u2713 (not (x ==\u2115 y))\n\u2262\u2192\u2713-not-==\u2115 \u00acp = \u2713-\u00ac-not (\u00acp \u2218 \u2115.==.sound _ _)\n\n#\u21d2\u2209 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 b \u2209 \u03b1\n#\u21d2\u2209 (_ #\u00f8)      = id\n#\u21d2\u2209 (suc#\u2237 b#\u03b1) = #\u21d2\u2209 b#\u03b1\n#\u21d2\u2209 (0# _)      = id\n\n#\u21d2\u2262 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 (x : Name \u03b1) \u2192 binder\u1d3a x \u2262 b\n#\u21d2\u2262 b# (b , b\u2208) \u2261.refl = #\u21d2\u2209 b# b\u2208\n\nbinder\u1d3a-injective : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u2192 x \u2261 y\nbinder\u1d3a-injective {\u03b1} {_ , p\u2081} {a , p\u2082} eq rewrite eq = \u2261.cong (_,_ a) (\u2208-uniq \u03b1 p\u2081 p\u2082)\n\npf-irr\u2032 : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x x\u2032 : Name \u03b1} {y y\u2032 : Name \u03b2}\n         \u2192 binder\u1d3a x \u2261 binder\u1d3a x\u2032 \u2192 binder\u1d3a y \u2261 binder\u1d3a y\u2032 \u2192 \u211b x y \u2192 \u211b x\u2032 y\u2032\npf-irr\u2032 {\u03b1} {\u03b2} \u211b {x , x\u2208} {x\u2032 , x\u2032\u2208} {y , y\u2208} {y\u2032 , y\u2032\u2208} eq\u2081 eq\u2082 x\u1d63y\n  rewrite eq\u2081 | eq\u2082 | \u2208-uniq \u03b1 x\u2208 x\u2032\u2208 | \u2208-uniq \u03b2 y\u2208 y\u2032\u2208 = x\u1d63y\n\npf-irr : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x y x\u2208 x\u2208\u2032 y\u2208 y\u2208\u2032}\n         \u2192 \u211b (x , x\u2208) (y , y\u2208) \u2192 \u211b (x , x\u2208\u2032) (y , y\u2208\u2032)\npf-irr \u211b = pf-irr\u2032 \u211b \u2261.refl \u2261.refl\n\nPreserve-\u2248 : \u2200 {a b \u2113 \u2113a \u2113b} {A : Set a} {B : Set b} \u2192 Rel A \u2113a \u2192 Rel (B) \u2113b \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2248 _\u2248a_ _\u2248b_ _\u223c_ = \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u223c x\u2082 \u2192 y\u2081 \u223c y\u2082 \u2192 (x\u2081 \u2248a y\u2081) \u21d4 (x\u2082 \u2248b y\u2082)\n\n==\u2115\u21d4\u2261 : \u2200 {x y} \u2192 \u2713 (x ==\u2115 y) \u21d4 x \u2261 y\n==\u2115\u21d4\u2261 {x} {y} = equivalence (\u2115.==.sound _ _) \u2115.==.reflexive\n\n\u2261-on-name\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u21d4 x \u2261 y\n\u2261-on-name\u21d4\u2261 {\u03b1} {x} {y} = equivalence binder\u1d3a-injective (\u2261.cong binder\u1d3a)\n\nT\u21d4T\u2192\u2261 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u21d4 \u2713 b\u2082 \u2192 b\u2081 \u2261 b\u2082\nT\u21d4T\u2192\u2261 {1\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\nT\u21d4T\u2192\u2261 {1\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.to b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.from b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\n\n==\u1d3a\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 \u2713 (x ==\u1d3a y) \u21d4 x \u2261 y\n==\u1d3a\u21d4\u2261 = \u2261-on-name\u21d4\u2261 \u27e8\u2218\u27e9 ==\u2115\u21d4\u2261 where open \u21d4 renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n-- Preserving the equalities also mean that the relation is functional and injective\nPreserve-\u2261 : \u2200 {a b \u2113} {A : Set a} {B : Set b} \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2261 _\u223c_ = Preserve-\u2248 _\u2261_ _\u2261_ _\u223c_\n\nexport\u1d3a?-nothing : \u2200 {b \u03b1} (x : Name (b \u25c5 \u03b1)) \u2192 \u2713 (binder\u1d3a x ==\u2115 b) \u2192 export\u1d3a? {b} x \u2261 nothing\nexport\u1d3a?-nothing {b} {\u03b1} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u2261.refl\n... | 0\u2082 | _ = \u22a5-elim p\n\nexport\u1d3a?-just : \u2200 {\u03b1 b} (x : Name (b \u25c5 \u03b1)) {x\u2208} \u2192 \u2713 (not (binder\u1d3a x ==\u2115 b)) \u2192 export\u1d3a? {b} x \u2261 just (binder\u1d3a x , x\u2208)\nexport\u1d3a?-just {\u03b1} {b} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u22a5-elim p\n... | 0\u2082 | _ = \u2261.cong just (binder\u1d3a-injective \u2261.refl)\n\nrecord \u27e6World\u27e7 (\u03b1\u2081 \u03b1\u2082 : World) : Set\u2081 where\n  constructor _,_\n  field\n    _\u223c_ : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set\n    \u223c-pres-\u2261 : Preserve-\u2261 _\u223c_\n\n  \u223c-inj : \u2200 {x y z} \u2192 x \u223c z \u2192 y \u223c z \u2192 x \u2261 y\n  \u223c-inj p q = Equivalence.from (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n  \u223c-fun : \u2200 {x y z} \u2192 x \u223c y \u2192 x \u223c z \u2192 y \u2261 z\n  \u223c-fun p q = Equivalence.to (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n\n  \u223c-\u2208-uniq : \u2200 {x\u2081 x\u2082} {x\u2081\u2208\u2032 x\u2082\u2208\u2032} \u2192\n        x\u2081 \u223c x\u2082 \u2261 (binder\u1d3a x\u2081 , x\u2081\u2208\u2032) \u223c (binder\u1d3a x\u2082 , x\u2082\u2208\u2032)\n  \u223c-\u2208-uniq {_ , x\u2081\u2208} {_ , x\u2082\u2208} {x\u2081\u2208\u2032} {x\u2082\u2208\u2032} = \u2261.ap\u2082 (\u03bb x\u2081\u2208 x\u2082\u2208 \u2192 (_ , x\u2081\u2208) \u223c (_ , x\u2082\u2208)) (\u2208-uniq \u03b1\u2081 x\u2081\u2208 x\u2081\u2208\u2032) (\u2208-uniq \u03b1\u2082 x\u2082\u2208 x\u2082\u2208\u2032)\n\n\u27e6sym\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 \u27e6World\u27e7 \u03b1\u2082 \u03b1\u2081\n\u27e6sym\u27e7 (\u211b , \u211b-pres-\u2261) = \u211b-sym , \u211b-sym-pres-\u2261\n  where \u211b-sym = flip \u211b\n        \u211b-sym-pres-\u2261 : Preserve-\u2261 \u211b-sym\n        \u211b-sym-pres-\u2261 p q = \u21d4.sym (\u211b-pres-\u2261 p q)\n\nrecord \u27e6Binder\u27e7 (b\u2081 b\u2082 : Binder) : Set where\n\nmodule \u27e6\u25c5\u27e7 (b\u2081 b\u2082 : Binder) {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  data _\u211b_ (x : Name (b\u2081 \u25c5 \u03b1\u2081)) (y : Name (b\u2082 \u25c5 \u03b1\u2082)) : Set where\n    here  : (x\u2261b\u2081 : binder\u1d3a x \u2261 b\u2081) (y\u2261b\u2082 : binder\u1d3a y \u2261 b\u2082) \u2192 x \u211b y\n    there : \u2200 (x\u2262b\u2081 : binder\u1d3a x \u2262 b\u2081) (y\u2262b\u2082 : binder\u1d3a y \u2262 b\u2082) {x\u2208 y\u2208} (x\u223cy : (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a y , y\u2208)) \u2192 x \u211b y\n\n  \u211b-inj : \u2200 {x y z} \u2192 x \u211b z \u2192 y \u211b z \u2192 x \u2261 y\n  \u211b-inj (here x\u2261b\u2081 _)    (here y\u2261b\u2081 _)     = binder\u1d3a-injective (\u2261.trans x\u2261b\u2081 (\u2261.sym y\u2261b\u2081))\n  \u211b-inj (here _ z\u2261b\u2082)    (there _ z\u2262b\u2082 _)  = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj (there _ z\u2262b\u2082 _) (here _ z\u2261b\u2082)     = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj {x} {y} {z} (there x\u2262b\u2081 z\u2262b\u2082 {x\u2208} {z\u2208} x\u223cz) (there y\u2262b\u2081 _ {y\u2208} {z\u2208\u2032} y\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-inj x\u223cz (\u2261.tr (\u03bb z\u2208 \u2192 (binder\u1d3a y , y\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2082 z\u2208\u2032 z\u2208) y\u223cz )))\n\n  \u211b-fun : \u2200 {x y z} \u2192 x \u211b y \u2192 x \u211b z \u2192 y \u2261 z\n  \u211b-fun (here _ y\u2261b\u2082)  (here _ z\u2261b\u2082)    = binder\u1d3a-injective (\u2261.trans y\u2261b\u2082 (\u2261.sym z\u2261b\u2082))\n  \u211b-fun (here p _)     (there \u00acp _ _)    = \u22a5-elim (\u00acp p)\n  \u211b-fun (there \u00acp _ _) (here p _)        = \u22a5-elim (\u00acp p)\n  \u211b-fun {x} {y} {z} (there x\u2262b\u2081 y\u2262b\u2082 {x\u2208} {y\u2208} x\u223cy) (there p q {x\u2208\u2032} {z\u2208} x\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-fun x\u223cy (\u2261.tr (\u03bb x\u2208 \u2192 (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2081 x\u2208\u2032 x\u2208) x\u223cz )))\n\n  \u211b-pres-\u2261 : Preserve-\u2261 _\u211b_\n  \u211b-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 x\u2081 \u211b x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b-inj x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 y\u2081 \u211b y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n  \u27e6world\u27e7 : \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n  \u27e6world\u27e7 = _\u211b_ , \u211b-pres-\u2261\n\nhere\u2032 : \u2200 {\u03b1\u2081 \u03b1\u2082 b\u2081 b\u2082 pf\u2081 pf\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} \u2192 \u27e6\u25c5\u27e7._\u211b_ b\u2081 b\u2082 \u03b1\u1d63 (b\u2081 , pf\u2081) (b\u2082 , pf\u2082)\nhere\u2032 = \u27e6\u25c5\u27e7.here \u2261.refl \u2261.refl\n\n_\u27e6\u25c5\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u25c5_ _\u25c5_\n_\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63 = \u27e6\u25c5\u27e7.\u27e6world\u27e7 b\u2081 b\u2082 \u03b1\u1d63\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 (\u211b , _) x\u2081 x\u2082 = \u211b x\u2081 x\u2082\n\n\u27e6zero\u1d2e\u27e7 : \u27e6Binder\u27e7 zero\u1d2e zero\u1d2e\n\u27e6zero\u1d2e\u27e7 = _\n\n\u27e6suc\u1d2e\u27e7 : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) suc\u1d2e suc\u1d2e\n\u27e6suc\u1d2e\u27e7 = _\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = (\u03bb _ _ \u2192 \u22a5) , (\u03bb())\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 x\u1d63 y\u1d63 =\n   \u27e6\ud835\udfda\u27e7-Props.reflexive (T\u21d4T\u2192\u2261 (sym ==\u1d3a\u21d4\u2261 \u27e8\u2218\u27e9 \u223c-pres-\u2261 x\u1d63 y\u1d63 \u27e8\u2218\u27e9 ==\u1d3a\u21d4\u2261)) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  open \u21d4 using (sym) renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n\u27e6name\u1d2e\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) name\u1d2e name\u1d2e\n\u27e6name\u1d2e\u27e7 _ _ = here\u2032\n\n\u27e6binder\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) binder\u1d3a binder\u1d3a\n\u27e6binder\u1d3a\u27e7 _ _ = _\n\n\u27e6nothing\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082}\n            \u2192 x \u2261 nothing \u2192 y \u2261 nothing \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6nothing\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6nothing\u27e7\n\n\u27e6just\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082} {x\u2032 y\u2032}\n            \u2192 x \u2261 just x\u2032 \u2192 y \u2261 just y\u2032 \u2192 A\u1d63 x\u2032 y\u2032 \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6just\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6just\u27e7\n\n\u27e6export\u1d3a?\u27e7 : (\u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) (\u03bb {b} \u2192 export\u1d3a? {b}) (\u03bb {b} \u2192 export\u1d3a? {b})\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.here e\u2081 e\u2082)\n  = \u27e6nothing\u27e7\u2032 (export\u1d3a?-nothing x\u2081 (\u2115.==.reflexive e\u2081)) (export\u1d3a?-nothing x\u2082 (\u2115.==.reflexive e\u2082))\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.there x\u2081\u2262b\u2081 x\u2082\u2262b\u2082 x\u2081\u223cx\u2082)\n  = \u27e6just\u27e7\u2032 (export\u1d3a?-just x\u2081 (\u2262\u2192\u2713-not-==\u2115 x\u2081\u2262b\u2081))\n           (export\u1d3a?-just x\u2082 (\u2262\u2192\u2713-not-==\u2115 x\u2082\u2262b\u2082)) x\u2081\u223cx\u2082\n\n_\u27e6\u2286\u27e7b_ : \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a x) (coerce\u1d3a y)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x y} \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) x y \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7b \u03b2\u1d63) x y\nun\u27e6\u2286\u27e7 (mk x) {a} {b} c = x {a} {b} c\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {mk f\u2081} {mk f\u2082} f {mk g\u2081} {mk g\u2082} g\n  = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {coerce\u1d3a (mk f\u2081) x\u2081} {coerce\u1d3a (mk f\u2082) x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 ()\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk helper where\n   helper : (\u27e6\u00f8\u27e7 \u27e6\u2286\u27e7b \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n   helper {_ , ()}\n\n_\u27e6#\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) _#_ _#_\n_\u27e6#\u27e7_ _ _ _ _ = \u22a4\n\n_\u27e6#\u00f8\u27e7 : (\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u27e6\u00f8\u27e7) _#\u00f8 _#\u00f8\n_\u27e6#\u00f8\u27e7 _ = _\n\n\u27e6suc#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 (\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6#\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) suc# suc#\n\u27e6suc#\u27e7 _ _ _ = _\n\n\u27e6\u2286-#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                  (b\u1d63 \u27e6#\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) \u2286-# \u2286-#\n\u27e6\u2286-#\u27e7 _ {b\u2081} {b\u2082} _ {b\u2081#\u03b1\u2081} {b\u2082#\u03b1\u2082} _ = mk (\u03bb {x\u2081} {x\u2082} \u2192 \u27e6\u25c5\u27e7.there (#\u21d2\u2262 b\u2081#\u03b1\u2081 x\u2081) (#\u21d2\u2262 b\u2082#\u03b1\u2082 x\u2082))\n\n\u27e6\u2286-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b2\u1d63)) \u2286-\u25c5 \u2286-\u25c5\n\u27e6\u2286-\u25c5\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {b\u2081} {b\u2082} _ {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} (mk f) = mk g where\n  open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n  g : \u2200 {x\u2081 x\u2082} \u2192 _\u211b_ \u03b1\u1d63 x\u2081 x\u2082 \u2192 _\u211b_ \u03b2\u1d63 (coerce\u1d3a (\u2286-\u25c5 b\u2081 \u03b1\u2286\u03b2\u2081) x\u2081) (coerce\u1d3a (\u2286-\u25c5 b\u2082 \u03b1\u2286\u03b2\u2082) x\u2082)\n  g (here x\u2261b\u2081 y\u2261b\u2082)       = here x\u2261b\u2081 y\u2261b\u2082\n  g (there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = there x\u2262b\u2081 y\u2262b\u2082 (f x\u223cy)\n\ncong\u2115 : \u2200 {\u03b1 \u03b2 : World} {x y : Name \u03b1} (f : \u2115 \u2192 \u2115) {fx\u2208 fy\u2208} \u2192 x \u2261 y \u2192 _\u2261_ {A = Name \u03b2} (f (binder\u1d3a x) , fx\u2208) (f (binder\u1d3a y) , fy\u2208)\ncong\u2115 f = binder\u1d3a-injective \u2218 \u2261.cong (f \u2218 binder\u1d3a)\n\nmodule \u27e6+1\u27e7 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63 renaming (_\u223c_ to \u211b; \u223c-inj to \u211b-inj; \u223c-fun to \u211b-fun)\n  \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set\n  \u211b+1 x y = \u27e6Name\u27e7 \u03b1\u1d63 (pred\u1d3a x) (pred\u1d3a y)\n{-\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x x\u2208 y y\u2208} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (suc x , ) (sucN y)\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x y} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (sucN x) (sucN y)\n-}\n\n  \u211b+1-inj : \u2200 {x y z} \u2192 \u211b+1 x z \u2192  \u211b+1 y z \u2192 x \u2261 y\n  -- \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} (suc a) (suc b) = {!!}\n  \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-inj a b)\n  \u211b+1-inj {zero , ()} _ _\n  \u211b+1-inj {_} {zero , ()} _ _\n  \u211b+1-inj {_} {_} {zero , ()} _ _\n  \u211b+1-fun : \u2200 {x y z} \u2192 \u211b+1 x y \u2192  \u211b+1 x z \u2192 y \u2261 z\n  \u211b+1-fun {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-fun a b)\n  \u211b+1-fun {zero , ()} _ _\n  \u211b+1-fun {_} {zero , ()} _ _\n  \u211b+1-fun {_} {_} {zero , ()} _ _\n\n  \u211b+1-pres-\u2261 : Preserve-\u2261 \u211b+1\n  -- \u211b+1-pres-\u2261 x y = equivalence {!\u211b+1-inj!} {!!}\n  \u211b+1-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     -- factor this move\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b+1-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 \u211b+1 x\u2081 x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b+1-inj {x\u2081} {y\u2081} {x\u2082} x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 \u211b+1 y\u2081 y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 \u03b1\u1d63 = \u211b+1 , \u211b+1-pres-\u2261 where open \u27e6+1\u27e7 \u03b1\u1d63\n\nopen WorldOps worldSym\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 \u03b1\u1d63 = \u27e6zero\u1d2e\u27e7 \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)\n\n_\u27e6\u2191\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6\u2191\u27e7 k) \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082)\n                 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 _  (\u27e6\u25c5\u27e7.here () _)\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b1\u1d63 (\u27e6\u25c5\u27e7.there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63) x\u223cy\n\n\u27e6\u00f8+1\u2286\u00f8\u27e7 : (\u27e6\u00f8\u27e7 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u00f8+1\u2286\u00f8 \u00f8+1\u2286\u00f8\n\u27e6\u00f8+1\u2286\u00f8\u27e7 = mk (\u03bb {x} _ \u2192 Name\u00f8-elim (pred\u1d3a x))\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 _ \u03b2\u1d63 \u03b1\u2286\u03b2\u1d63 = mk (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) \u2218 un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk helper where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper (here x\u2261b\u2081 y\u2261b\u2082) = here x\u2261b\u2081 y\u2261b\u2082\n  helper {x\u2081 , _} {x\u2082 , _} (there x\u2262b\u2081 y\u2262b\u2082 {p\u2081} {p\u2082} x\u223cy)\n     = there x\u2262b\u2081 y\u2262b\u2082 {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081) x\u2081 p\u2081}\n                        {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082) x\u2082 p\u2082}\n                        (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 x\u223cy))\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {add\u1d3a 1 x\u2081} {add\u1d3a 1 x\u2082}) \n\n\u22620 : \u2200 {\u03b1} (x : Name (\u03b1 +1)) \u2192 binder\u1d3a x \u2262 0\n\u22620 (0 , ()) \u2261.refl\n\u22620 (suc _ , _) ()\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 there (\u22620 x\u2081) (\u22620 x\u2082)) where open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ \u03b2\u1d63 \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b2\u1d63 (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (there (\u03bb()) (\u03bb()) x\u1d63))) where\n  open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ \u03b2\u1d63 \u03b1+\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} x\u1d63)) where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b2\u1d63 x\u2081 x\u2082\n  helper (here () _)\n  helper (there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) x\u223cy\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 \u03b1\u1d63 x\u1d63 = \u27e6coerce\u1d3a\u27e7 _ _ (\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63) (\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 x\u1d63)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _  zero x\u1d63 = x\u1d63\n\u27e6add\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) x\u1d63 = pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = pf-irr\u2032 (\u27e6Name\u27e7 \u03b1\u1d63) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2081) 1 k\u2081) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2082) 1 k\u2082) (\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {pf\u2081 : 0 \u2208 \u03b1\u2081 \u21911} {n} {pf\u2082 : suc n \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (0 , pf\u2081) (suc n , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.there p _ _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.here _ ())\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {n} {pf\u2081 : suc n \u2208 \u03b1\u2081 \u21911} {pf\u2082 : 0 \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc n , pf\u2081) (0 , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.there _ p _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.here () _)\n\n-- cmp\u1d3a-bool : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 \ud835\udfda\n-- cmp\u1d3a-bool \u2113 x = suc (binder\u1d3a x) <= \u2113\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) NaPa.cmp\u1d3a-bool NaPa.cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 _ zero _ = \u27e60\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 _ (suc _) {zero , _} {zero , _} _ = \u27e61\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) NaPa.easy-cmp\u1d3a NaPa.easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 _ zero x\u1d63 = \u27e6inr\u27e7 x\u1d63\n\u27e6easy-cmp\u1d3a\u27e7 _ (suc _) {zero , _} {zero , _} x\u1d63 = \u27e6inl\u27e7 here\u2032\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6map\u27e7 _ _ _ _ (\u27e6suc\u1d3a\u2191\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63)) (pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63))\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (NaPa.easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6\u2286-dist-+1-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) \u27e6\u2286\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))))\n                 \u2286-dist-+1-\u25c5 \u2286-dist-+1-\u25c5\n\u27e6\u2286-dist-+1-\u25c5\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083) where\n  -- open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n\n  hper :   \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (suc\u1d3a x\u2081) (suc\u1d3a x\u2082)\n  hper (\u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082) = \u27e6\u25c5\u27e7.here (\u2261.cong suc x\u2261b\u2081) (\u2261.cong suc y\u2261b\u2082)\n  hper (\u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong pred) (y\u2262b\u2082 \u2218 \u2261.cong pred) x\u223cy\n\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 = hper x\u1d63\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n\u27e6\u2286-dist-\u25c5-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) \u27e6\u2286\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)))\n                 \u2286-dist-\u25c5-+1 \u2286-dist-\u25c5-+1\n\u27e6\u2286-dist-\u25c5-+1\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk helper where\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 with x\u1d63\n  ... | \u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082 = \u27e6\u25c5\u27e7.here (\u2261.cong (\u03bb x \u2192 x \u2238 1) x\u2261b\u2081) (\u2261.cong (\u03bb x \u2192 x \u2238 1) y\u2261b\u2082)\n  ... | \u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong suc) (y\u2262b\u2082 \u2218 \u2261.cong suc) x\u223cy\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n{-\nBinder\u2261\u2115 : Binder \u2261 \u2115\nBinder\u2261\u2115 = \u2261.refl\n\npostulate\n  \u27e6Binder\u27e7\u2262\u27e6\u2115\u27e7 : \u27e6Binder\u27e7 \u2262 \u27e6\u2115\u27e7\n\n\u27e6\u2261\u27e7-lem : \u2200 {a\u2081 a\u2082 a\u1d63 A\u2081 A\u2082} {A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082} {x\u2081 x\u2082} {x\u1d63 : A\u1d63 x\u2081 x\u2082} {y\u2081 y\u2082} {y\u1d63 : A\u1d63 y\u2081 y\u2082} {pf\u2081 pf\u2082} \u2192 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 y\u1d63 pf\u2081 pf\u2082 \u2192 \u2200 (P : \u2200 {z\u2081 z\u2082} \u2192 A\u1d63 z\u2081 z\u2082 \u2192 Set) \u2192 P x\u1d63 \u2192 P y\u1d63\n\u27e6\u2261\u27e7-lem = {!!}\n\n\u00ac\u27e6Binder\u2261\u2115\u27e7 : \u00ac((\u27e6\u2261\u27e7 \u27e6Set\u2080\u27e7 \u27e6Binder\u27e7 \u27e6\u2115\u27e7) Binder\u2261\u2115 Binder\u2261\u2115)\n\u00ac\u27e6Binder\u2261\u2115\u27e7 = \u03bb pf \u2192 \u27e6\u2261\u27e7-lem pf (\u03bb x \u2192 \u22a5) {!!}\n-}\n\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 : (\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                   \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                    \u27e6\u2261\u27e7 \u27e6Binder\u27e7 (\u27e6binder\u1d3a\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) (\u27e6name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63)) b\u1d63) binder\u1d3a\u2218name\u1d2e binder\u1d3a\u2218name\u1d2e\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63 = \u2261.\u27e6refl\u27e7\n\n\u27e8_,_\u27e9\u27e6\u25c5\u27e7_ : (b\u2081 b\u2082 : Binder) \u2192 \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) \u2192 \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n\u27e8 b\u2081 , b\u2082 \u27e9\u27e6\u25c5\u27e7 \u03b1\u1d63 = _\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63\n\nmodule Perm (m n : \u2115) (m\u2262n : m \u2262 n) where\n  mn\u1d42 : World\n  mn\u1d42 = m \u25c5 n \u25c5 \u00f8\n  nm\u1d42 : World\n  nm\u1d42 = n \u25c5 m \u25c5 \u00f8\n  perm-m-n : \u27e6World\u27e7 mn\u1d42 nm\u1d42\n  perm-m-n = \u27e8 m , n \u27e9\u27e6\u25c5\u27e7 (\u27e8 n , m \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7)\n  m-n : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (m , m\u2208) (n , n\u2208)\n  m-n = here\u2032\n  n-m : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (n , n\u2208) (m , m\u2208)\n  n-m = \u27e6\u25c5\u27e7.there (m\u2262n \u2218 \u2261.sym) m\u2262n {b\u2208b\u25c5 n \u00f8} {b\u2208b\u25c5 m \u00f8} here\u2032\n\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 : \u00ac(\u27e6\ud835\udfda\u27e7 1\u2082 0\u2082)\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 ()\n\nbinder-irrelevance : \u2200 (f : Binder \u2192 \ud835\udfda)\n                     \u2192 (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f\n                     \u2192 \u2200 {b\u2081 b\u2082} \u2192 f b\u2081 \u2261 f b\u2082\nbinder-irrelevance _ f\u1d63 = \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 _)\n\ncontrab : \u2200 (f : Binder \u2192 \ud835\udfda) {b\u2081 b\u2082}\n          \u2192 f b\u2081 \u2262 f b\u2082\n          \u2192 \u00ac((\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\ncontrab f = contraposition (\u03bb f\u1d63 \u2192 binder-irrelevance f (\u03bb x\u1d63 \u2192 f\u1d63 x\u1d63))\n\nmodule Single \u03b1\u2081 \u03b1\u2082 x\u2081 x\u2082 where\n  data \u211b : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set where\n    refl : \u211b x\u2081 x\u2082\n  \u211b-pres-\u2261 : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 \u211b x\u2081 x\u2082 \u2192 \u211b y\u2081 y\u2082 \u2192 (x\u2081 \u2261 y\u2081) \u21d4 (x\u2082 \u2261 y\u2082)\n  \u211b-pres-\u2261 refl refl = equivalence (const \u2261.refl) (const \u2261.refl)\n  \u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082\n  \u03b1\u1d63 = \u211b , \u211b-pres-\u2261\n\npoly-name-uniq : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1} (x : Name \u03b1) \u2192 f x \u2261 f {\u00f8 \u21911} (0 , _)\npoly-name-uniq f f\u1d63 {\u03b1} x =\n  \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 {\u03b1} {\u00f8 \u21911} \u03b1\u1d63 {x} {0 , _} refl)\n  where open Single \u03b1 (\u00f8 \u21911) x (0 , _)\n\npoly-name-irrelevance : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1\u2081 \u03b1\u2082} (x\u2081 : Name \u03b1\u2081) (x\u2082 : Name \u03b1\u2082)\n                        \u2192 f x\u2081 \u2261 f x\u2082\npoly-name-irrelevance f f\u1d63 x\u2081 x\u2082 =\n  \u2261.trans (poly-name-uniq f f\u1d63 x\u2081) (\u2261.sym (poly-name-uniq f f\u1d63 x\u2082))\n\nmodule Broken where\n  _<=\u1d3a_ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name \u03b1 \u2192 \ud835\udfda\n  (m , _) <=\u1d3a (n , _) = \u2115._<=_ m n\n\n  \u00ac\u27e6<=\u1d3a\u27e7 : \u00ac((\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _<=\u1d3a_ _<=\u1d3a_)\n  \u00ac\u27e6<=\u1d3a\u27e7 \u27e6<=\u27e7 = \u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 (\u27e6<=\u27e7 perm-0-1 {0 , _} {1 , _} 0-1 {1 , _} {0 , _} 1-0)\n     where open Perm 0 1 (\u03bb()) renaming (perm-m-n to perm-0-1; m-n to 0-1; n-m to 1-0)\n\n  \u2286-broken : \u2200 \u03b1 b \u2192 \u03b1 \u2286 (b \u25c5 \u03b1)\n  \u2286-broken \u03b1 b = mk (\u03bb b\u2032 b\u2032\u2208\u03b1 \u2192 \u2261.tr id (\u2261.sym (\u25c5-sem \u03b1 b\u2032 b)) (If\u2032 \u2115._==_ b\u2032 b then _ else b\u2032\u2208\u03b1))\n\n  \u00ac\u27e6\u2286-broken\u27e7 : \u00ac((\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63))\n                  \u2286-broken \u2286-broken)\n  \u00ac\u27e6\u2286-broken\u27e7 \u27e6\u2286-broken\u27e7 = bot\n     where 0\u21940 : \u27e6Name\u27e7 (\u27e8 0 , 1 \u27e9\u27e6\u25c5\u27e7 \u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) (0 , _) (0 , _)\n           0\u21940 = un\u27e6\u2286\u27e7 (\u27e6\u2286-broken\u27e7 (\u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) _) {0 , _} {0 , _} here\u2032\n\n           bot : \u22a5\n           bot with 0\u21940\n           bot | \u27e6\u25c5\u27e7.here _ ()\n           bot | \u27e6\u25c5\u27e7.there 0\u22620 _ _ = 0\u22620 \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b5c551446ac6bd8508e98e0b22480ff263a62540","subject":"implicit arguments","message":"implicit arguments\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule typed-expansion where\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = {!!}\n    typed-expansion-synth (ESAp1 x x\u2081 x\u2082 x\u2083) = {!!}\n    typed-expansion-synth (ESAp2 x ex x\u2081 x\u2082) = {!!}\n    typed-expansion-synth (ESAp3 x ex x\u2081) = {!!}\n    typed-expansion-synth ESEHole = {!!}\n    typed-expansion-synth (ESNEHole ex) = {!!}\n    typed-expansion-synth (ESAsc1 x x\u2081) = {!!}\n    typed-expansion-synth (ESAsc2 x) = {!!}\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex) = {!!}\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    typed-expansion-ana EAEHole = {!!}\n    typed-expansion-ana (EANEHole x x\u2081) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule typed-expansion where\n  mutual\n    typed-expansion-synth : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth \u0393 .c .b .c .(\u03bb _ \u2192 None) ESConst = TAConst\n    typed-expansion-synth _ _ \u03c4 _ .(\u03bb _ \u2192 None) ESVar = TAVar\n    typed-expansion-synth \u0393 _ _ _ .(\u03bb _ \u2192 None) (ESLam D) with typed-expansion-synth _ _ _ _ _ D\n    ... | ih = TALam {!!}\n    typed-expansion-synth \u0393 _ .\u2987\u2988 _ _ (ESAp1 y x x\u2081 x\u2082) = {!!}\n    typed-expansion-synth \u0393 _ \u03c4 _ _ (ESAp2 y D x x\u2081) = {!!}\n    typed-expansion-synth \u0393 _ \u03c4 _ _ (ESAp3 y D x) = {!!}\n    typed-expansion-synth \u0393 _ .\u2987\u2988 _ _ ESEHole = TAEHole {!!}\n    typed-expansion-synth \u0393 _ .\u2987\u2988 _ _ (ESNEHole D) = TANEHole {!!} {!!}\n    typed-expansion-synth \u0393 _ \u03c4 _ \u0394 (ESAsc1 x x\u2081) = {!!}\n    typed-expansion-synth \u0393 _ \u03c4 d \u0394 (ESAsc2 x) = {!!}\n\n    typed-expansion-ana : (\u0393 : tctx) (e : hexp) (\u03c4 \u03c4' : htyp) (d : dhexp) (\u0394 : hctx) \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana \u0393 _ _ _ _ \u0394 (EALam D) = (TCArr TCRefl {!!}) , {!!}\n    typed-expansion-ana \u0393 e \u03c4 \u03c4' d \u0394 (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    typed-expansion-ana \u0393 _ \u03c4' .\u03c4' _ _ EAEHole = TCRefl , {!!}\n    typed-expansion-ana \u0393 _ \u03c4' .\u03c4' _ _ (EANEHole x x\u2081) = TCRefl , {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5dcb5ee53faebcf8147bafebc216b8e23aeb02d8","subject":"removing comments, whitespace","message":"removing comments, whitespace\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     envfresh x \u03c3 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 {\u0393 = \u0393} (EFId x\u2081) (STAId x\u2082) = STAId (\u03bb x \u03c4 x\u2083 \u2192 x\u2208\u222al \u0393 _ x \u03c4 (x\u2082 x \u03c4 x\u2083) )\n    weaken-subst-\u0393 {x = x} {\u0393 = \u0393} (EFSubst x\u2081 efrsh x\u2082) (STASubst {y = y} {\u03c4 = \u03c4'} subst x\u2083) =\n      STASubst (exchange-subst-\u0393 {\u0393 = \u0393} (flip x\u2082) (weaken-subst-\u0393 {\u0393 = \u0393 ,, (y , \u03c4')} efrsh subst))\n               (weaken-ta x\u2081 x\u2083)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192  -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081))) (weaken-ta {\u0393 = {!\u25a0 (y , ? )!} \u222a \u0393} x\u2083 wt)\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 -- = {!weaken-ta ? (TALam x\u2082 wt1)!}\n    with natEQ y x\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = abort ((\u03c01(lem-union-none {\u0393 = \u0393} x\u2083)) refl)\n  lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n    with lem-union-none {\u0393 = \u0393} x\u2083 -- | lem-subst apt wt1  -- probably not the straight IH; need to weaken\n  ... | neq , r =  {!!} --  TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y -- todo: maybe?\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     envfresh x \u03c3 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 {\u0393 = \u0393} (EFId x\u2081) (STAId x\u2082) = STAId (\u03bb x \u03c4 x\u2083 \u2192 x\u2208\u222al \u0393 _ x \u03c4 (x\u2082 x \u03c4 x\u2083) )\n    weaken-subst-\u0393 {x = x} {\u0393 = \u0393} (EFSubst x\u2081 efrsh x\u2082) (STASubst {y = y} {\u03c4 = \u03c4'} subst x\u2083) =\n      STASubst (exchange-subst-\u0393 {\u0393 = \u0393} (flip x\u2082) (weaken-subst-\u0393 {\u0393 = \u0393 ,, (y , \u03c4')} efrsh subst))\n               (weaken-ta x\u2081 x\u2083)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192\n                -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081))) (weaken-ta {\u0393 = {!\u25a0 (y , ? )!} \u222a \u0393} x\u2083 wt)\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 x\u2081 x\u2083) -- second argument rel to todo above\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 x\u2081 x\u2083) -- second argument rel to todo above\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 -- = {!weaken-ta ? (TALam x\u2082 wt1)!}\n    with natEQ y x\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = abort ((\u03c01(lem-union-none {\u0393 = \u0393} x\u2083)) refl)\n  lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n    with lem-union-none {\u0393 = \u0393} x\u2083 -- | lem-subst apt wt1  -- probably not the straight IH; need to weaken\n  ... | neq , r =  {!!} --  TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f91b15c3d4b631cfaa10d6b9f0ec06f4e556c1d3","subject":"IDesc model: cool down on universe polymorphism on Desc","message":"IDesc model: cool down on universe polymorphism on Desc\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"de962511ed25886f7e30cde9537dbe0b8146958d","subject":"Boring.","message":"Boring.\n\nIgnore-this: 29b776acd29766be249fdded6f93d79f\n\ndarcs-hash:20100414150814-3bd4e-916ed4ed8f8a00a1edd98792247972840d63fb14.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  infixr 4 _,_\n  infixr 2 _\u00d7_\n\n  -- We want to use the product type to define the conjunction type\n  record _\u00d7_ (A B : Set) : Set where\n    constructor _,_\n    field\n      proj\u2081 : A\n      proj\u2082 : B\n\n  -- N.B. It is not necessary to add the constructor _,_, nor the fields\n  -- proj\u2081, proj\u2082 as hints, because the ATP implements it.\n\n  _\u2227_ : (A B : Set) \u2192 Set\n  A \u2227 B = A \u00d7 B\n\n  postulate\n    P : D \u2192 Set\n\n  -- Testing the conjunction data constructor\n  postulate\n    testAndDataConstructor : {m n : D} \u2192 P m \u2192 P n \u2192 P m \u2227 P n\n  {-# ATP prove testAndDataConstructor #-}\n\n  -- Testing the first projection\n  postulate\n    testProj\u2081 : {m n : D} \u2192 P m \u2227 P n \u2192 P m\n  {-# ATP prove testProj\u2081 #-}\n\n  -- Testing the second projection\n  postulate\n    testProj\u2082 : {m n : D} \u2192 P m \u2227 P n \u2192 P n\n  {-# ATP prove testProj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true  : D\n    false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  infixr 4 _,_\n  infixr 2 _\u00d7_\n\n  -- We want to use the product type to define the conjunction type\n  record _\u00d7_ (A B : Set) : Set where\n    constructor _,_\n    field\n      proj\u2081 : A\n      proj\u2082 : B\n\n  -- N.B. It is not necessary to add the constructor _,_, nor the fields\n  -- proj\u2081, proj\u2082 as hints, because the ATP implements it.\n\n  _\u2227_ : (A B : Set) \u2192 Set\n  A \u2227 B = A \u00d7 B\n\n  postulate\n    P : D \u2192 Set\n\n  -- Testing the conjunction data constructor\n  postulate\n    testAndDataConstructor : {m n : D} \u2192 P m \u2192 P n \u2192 P m \u2227 P n\n  {-# ATP prove testAndDataConstructor #-}\n\n  -- Testing the first projection\n  postulate\n    testProj\u2081 : {m n : D} \u2192 P m \u2227 P n \u2192 P m\n  {-# ATP prove testProj\u2081 #-}\n\n-- Testing the second projection\n  postulate\n    testProj\u2082 : {m n : D} \u2192 P m \u2227 P n \u2192 P n\n  {-# ATP prove testProj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true  : D\n    false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"72fbb40433f647484e1e61bb7541542a115c6027","subject":"sem-sec: reductions are now abstracted from the notion of function","message":"sem-sec: reductions are now abstracted from the notion of function\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule FunctionExtra where\n  _***_ : \u2200 {A B C D : Set} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n  -- Fanout\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n  _>>>_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n            (A \u2192 B) \u2192 (B \u2192 C) \u2192 (A \u2192 C)\n  f >>> g = g \u2218 f\n  infixr 1 _>>>_\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\nmodule Guess (Power : Set) (coins : Power \u2192 Coins) (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Power \u2192 Set\n  Oracle power = \u2203 (\u03bb (A : GuessAdv (coins power)) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nrecord FlatFuns (Power : Set) {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits   : \u2115 \u2192 T\n    `Bit    : T\n    _`\u00d7_    : T \u2192 T \u2192 T\n    \u27e8_\u27e9_\u219d_  : Power \u2192 T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 \u27e8_\u27e9_\u219d_\n\nrecord PowerOps (Power : Set) : Set where\n  constructor mk\n  infixr 1 _>>>\u1d56_\n  infixr 3 _***\u1d56_\n  field\n    id\u1d56 : Power\n    _>>>\u1d56_ _***\u1d56_ : Power \u2192 Power \u2192 Power\n\nconstPowerOps : PowerOps \u22a4\nconstPowerOps = _\n\nrecord FlatFunsOps {P t} {T : Set t} (powerOps : PowerOps P) (\u266dFuns : FlatFuns P T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  open PowerOps powerOps\n  field\n    idO : \u2200 {A p} \u2192 \u27e8 p \u27e9 A \u219d A\n    isIComposable  : IComposable  {I = \u22a4} _>>>\u1d56_ \u27e8_\u27e9_\u219d_\n    isVIComposable : VIComposable {I = \u22a4} _***\u1d56_ _`\u00d7_ \u27e8_\u27e9_\u219d_\n  open FlatFuns \u266dFuns public\n  open PowerOps powerOps public\n  open IComposable isIComposable public\n  open VIComposable isVIComposable public\n\nfun\u266dFuns : FlatFuns \u22a4 Set\nfun\u266dFuns = mk Bits Bit _\u00d7_ (\u03bb _ A B \u2192 A \u2192 B)\n\nfun\u266dOps : FlatFunsOps _ fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp\n\nmodule AbsSemSec (|M| |C| : \u2115) {FunPower : Set} {t} {T : Set t}\n                 (\u266dFuns : FlatFuns FunPower T) where\n\n  record Power : Set where\n    constructor mk\n    field\n      p\u2080 p\u2081 : FunPower\n      |R| : Coins\n  coins = Power.|R|\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (p : Power) : Set where\n    constructor mk\n\n    open Power p\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Power \u2192 Power) \u2192 Set\n  SemSecReduction f = \u2200 {p} \u2192 AbsSemSecAdv p \u2192 AbsSemSecAdv (f p)\n\n-- Here we use Agda functions for FlatFuns and \u22a4 for power.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open PowerOps constPowerOps\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv |R| = AbsSemSecAdv (mk _ _ |R|)\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n    where open FunctionExtra using (_&&&_)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Power \u2192 Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {p} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {p} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {FunPower : Set} {t} {T : Set t}\n             {\u266dFuns : FlatFuns FunPower T}\n             {funPowerOps : PowerOps FunPower}\n             (\u266dops : FlatFunsOps funPowerOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red-power : FunPower \u2192 Power \u2192 Power\n    post-comp-red-power p (mk p\u2080 p\u2081 |R|) = mk p\u2080 (p ***\u1d56 id\u1d56 >>>\u1d56 p\u2081) |R|\n\n    post-comp-red : \u2200 {p} (post-E : \u27e8 p \u27e9 C \u219d C) \u2192 SemSecReduction (post-comp-red-power p)\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {p} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {p} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take (coins p) R)) b))\n                                       (drop (coins p) R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess Coins id prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\n-- open import circuit\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule FunctionExtra where\n  _***_ : \u2200 {A B C D : Set} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n  -- Fanout\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n  _>>>_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n            (A \u2192 B) \u2192 (B \u2192 C) \u2192 (A \u2192 C)\n  f >>> g = g \u2218 f\n  infixr 1 _>>>_\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\nPorts = \u2115\nSize  = \u2115\nTime  = \u2115\n\nrecord Power : Set where\n  constructor mk\n  field\n    coins : Coins\n    size  : Size\n    time  : Time\nopen Power public\n\nsame-power : Power \u2192 Power\nsame-power = id\n\n_+\u1d56_ : Power \u2192 Power \u2192 Power\n(mk c\u2080 s\u2080 t\u2080) +\u1d56 (mk c\u2081 s\u2081 t\u2081) = mk (c\u2080 + c\u2081) (s\u2080 + s\u2081) (t\u2080 + t\u2081)\n\npowerComp : Composable (ConstArr Power)\npowerComp = constComp _+\u1d56_\n\npowerVComp : VComposable _ (ConstArr Power)\npowerVComp = constVComp (\u03bb { (Power.mk c\u2080 s\u2080 t\u2080) (mk c\u2081 s\u2081 t\u2081) \u2192 mk (c\u2080 + c\u2081) (s\u2080 + s\u2081) (t\u2080 \u2294 t\u2081) })\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Power \u2192 Set\n  Oracle power = \u2203 (\u03bb (A : GuessAdv (coins power)) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nrecord FlatFuns (Power : Set) {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits   : \u2115 \u2192 T\n    `Bit    : T\n    _`\u00d7_    : T \u2192 T \u2192 T\n    \u27e8_\u27e9_\u219d_  : Power \u2192 T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 \u27e8_\u27e9_\u219d_\n\nrecord PowerOps (Power : Set) : Set where\n  constructor mk\n  field\n    _>>>\u1d56_ _***\u1d56_ : Power \u2192 Power \u2192 Power\n\nconstPowerOps : PowerOps \u22a4\nconstPowerOps = _\n\nrecord FlatFunsOps {P t} {T : Set t} (powerOps : PowerOps P) (\u266dFuns : FlatFuns P T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  open PowerOps powerOps\n  field\n    idO : \u2200 {A p} \u2192 \u27e8 p \u27e9 A \u219d A\n    isIComposable  : IComposable  {I = \u22a4} _>>>\u1d56_ \u27e8_\u27e9_\u219d_\n    isVIComposable : VIComposable {I = \u22a4} _***\u1d56_ _`\u00d7_ \u27e8_\u27e9_\u219d_\n  open IComposable isIComposable public\n  open VIComposable isVIComposable public\n\nfun\u266dFuns : FlatFuns \u22a4 Set\nfun\u266dFuns = mk Bits Bit _\u00d7_ (\u03bb _ A B \u2192 A \u2192 B)\n\nrecord AbsSemSecAdv |M| |C|\n                 {Power : Set} {t} {T : Set t} (\u266dFuns : FlatFuns Power T)\n                 (p\u2080 p\u2081 : Power) (|R| : Coins) : Set where\n  constructor _,_\n\n  open FlatFuns \u266dFuns\n\n  field\n    {|S|} : \u2115\n\n  S = `Bits |S|\n  R = `Bits |R|\n  M = `Bits |M|\n  C = `Bits |C|\n\n  field\n    step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n    step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\nmodule FunAdv (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n\n  M = Bits |M|\n  C = Bits |C|\n  -- module AbsSemSecAdv' |R| = AbsSemSecAdv {|M|} {|C|} {fun\u266dFuns} |R|\n\n  -- open AbsSemSecAdv' using (M; C)\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  record FunSemSecAdv |R| : Set where\n    constructor mk\n\n    field\n      semSecAdv : AbsSemSecAdv |M| |C| fun\u266dFuns _ _ |R|\n\n    open AbsSemSecAdv semSecAdv public hiding (M; C)\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** id\n      where open FunctionExtra\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk (A-step\u2080 , A-step\u2081)) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n    where open FunctionExtra\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc p} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv p} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {cc} {ca} {E} {A\u2081} {A\u2082} pf b R with splitAt ca R\n  change-adv {cc} {ca} {E} {A\u2081} {A\u2082} pf b ._ | pre , post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  ext-as-broken : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                  \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n  ext-as-broken same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\n  SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n  SemBroken E power = \u2203 (\u03bb (A : FunSemSecAdv (coins power)) \u2192 breaks (E \u21c4 A))\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n  --   security of E\u2080 reduces to security of E\u2081\n  --   breaking E\u2081 reduces to breaking E\u2080\n  SemSecReduction : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} {cc\u2081} f tr = \u2200 {p} {E : Enc cc\u2080} \u2192 SemSecReduction (f p) p E (tr E)\n\n  -- SemSecReductionToOracle : \u2200 (p\u2080 p\u2081 : Power) {cc} (E : Enc cc) \u2192 Set\n  -- SemSecReductionToOracle = SemBroken E p\u2080 \u2192 Oracle p\u2081\n\n  open FunctionExtra\n\n  -- post-comp : \u2200 {cc k} (post-E : C \u2192 \u21ba k C) \u2192 Tr cc (cc + k)\n  -- post-comp post-E E = E >=> post-E\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr same-power (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E (mk (A'\u2080 , A'\u2081) , A'-breaks-E') = A , A'-breaks-E'\n     where A = mk (A'\u2080 , (post-E *** id >>> A'\u2081))\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc p} {E : Enc cc}\n                            \u2192 SemSecReduction p p (post-comp post-E E) E\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {p} {E} (A , A-breaks-E)\n       = A' , A'-breaks-E'\n     where E' = post-comp post-E E\n           open FunSemSecAdv A renaming (step\u2080 to A\u2080; step\u2080F to A\u2080F; step\u2081 to A\u2081)\n           A' = mk (A\u2080 , (post-E\u207b\u00b9 *** id >>> A\u2081))\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take (coins p) R)) b))\n                                       (drop (coins p) R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = ext-as-broken same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg E = E >>> map\u21ba vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr same-power (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {p} {E} = post-comp-pres-sem-sec vnot {p} {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc p} {E : Enc cc}\n                            \u2192 SemSecReduction p p (post-neg E) E\n  post-neg-pres-sem-sec' {cc} {p} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {p} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\n{-\nmodule F''\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (Pr-ext : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 Pr[ f \u22611] \u2261 Pr[ g \u22611])\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n  (non-negligible : Proba \u2192 Set)\n\n  where\n  advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Proba\n  advantage EXP = dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = non-negligible (advantage \u2141)\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible (advantage (E \u21c4 A))\n-}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecReduction (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec = ?\n-}\n\n-- Actually Ops + Spec\nrecord FlatFunsSpec {t} {T : Set t} (\u266dFuns : FlatFuns \u22a4 T) : Set (L.suc L.zero L.\u2294 t) where\n  constructor mk\n\n  open FlatFuns \u266dFuns\n  Cp : T \u2192 T \u2192 Set\n  Cp = \u27e8_\u27e9_\u219d_ _\n\n  field\n    \u27e6_\u27e7 : T \u2192 Set\n    _\u27e6\u00d7\u27e7_ : \u2200 {i\u2080 i\u2081} \u2192 \u27e6 i\u2080 \u27e7 \u2192 \u27e6 i\u2081 \u27e7 \u2192 \u27e6 i\u2080 `\u00d7 i\u2081 \u27e7\n    \u266dops : FlatFunsOps constPowerOps \u266dFuns\n    =[]= : \u2200 {i o} \u2192 Cp i o \u2192 \u27e6 i \u27e7 \u2192 \u27e6 o \u27e7 \u2192 Set\n\n  _=[_]=_ : \u2200 {i o} \u2192 \u27e6 i \u27e7 \u2192 Cp i o \u2192 \u27e6 o \u27e7 \u2192 Set\n  _=[_]=_ = \u03bb is c os \u2192 =[]= c is os\n\n  open FlatFunsOps \u266dops\n  field\n    idC-spec : \u2200 {i} (bs : \u27e6 i \u27e7) \u2192 bs =[ idO ]= bs\n    >>>-spec : IComposable  {_\u219d\u1d62_ = Cp} _>>>_ =[]=\n    ***-spec : VIComposable {_\u219d\u1d62_ = Cp} _***_ _\u27e6\u00d7\u27e7_ =[]=\n\n    runC : \u2200 {i o} \u2192 Cp i o \u2192 \u27e6 i \u27e7 \u2192 \u27e6 o \u27e7\n    runC-spec : \u2200 {i o} (c : Cp i o) is \u2192 is =[ c ]= runC c is\n{-\n    splitAtC : \u2200 k {n} \u2192 \u27e6 `Bits (k + n) \u27e7 \u2192 (\u27e6 `Bits k \u27e7 \u00d7 \u27e6 `Bits n \u27e7)\n    splitAtC2 : \u2200 k {n} \u2192 \u27e6 `Bits (k + n) \u27e7 \u2192 (\u27e6 `Bits k `\u00d7 `Bits n \u27e7)\n    splitAtC3 : \u2200 k {n} \u2192 Cp (`Bits (k + n)) (`Bits k `\u00d7 `Bits n)\n    splitAtC4 : \u2200 k {n} \u2192 (\u27e6 `Bits k `\u00d7 `Bits n \u27e7) \u2192 (\u27e6 `Bits k \u27e7 \u00d7 \u27e6 `Bits n \u27e7)\n-}\n\n{-\nmodule CpAdv\n  {t}\n  {T : Set t}\n  (\u266dFuns : FlatFuns \u22a4 T)\n  (\u266dFunsSpec : FlatFunsSpec \u266dFuns)\n  -- (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  -- (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n\n  (prgDist : PrgDist)\n  (|M| |C| : \u2115)\n\n  where\n  open FlatFuns \u266dFuns\n  open FlatFunsSpec \u266dFunsSpec\n  open FlatFunsOps  \u266dops\n\n  open PrgDist prgDist\n  `M = `Bits |M|\n  `C = `Bits |C|\n  M = \u27e6 `M \u27e7\n  C = \u27e6 `C \u27e7\n  M\u00b2 = Bit \u2192 M\n\n  -- module FunAdv' = FunAdv prgDist |M| |C|\n  -- open FunAdv' public using (mk; {-module SplitSemSecAdv;-} FunSemSecAdv; M; C; M\u00b2; Enc; Tr; ext-as-broken; change-adv; _\u2257A_) renaming (_\u21c4_ to _\u21c4F_)\n\n  module FE = FunctionExtra\n\n  record CpSemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to |R|; size to s; time to t)\n\n    -- open FunctionExtra\n\n    field\n      semSecAdv : AbsSemSecAdv |M| |C| \u266dFuns _ _ |R|\n\n    open AbsSemSecAdv semSecAdv public hiding (M; C; R; S)\n\n    `R = `Bits |R|\n    `S = `Bits |S|\n    R = \u27e6 `R \u27e7\n    S = \u27e6 `S \u27e7\n\n    step\u2080f : R \u2192 ((M \u00d7 M) \u00d7 S)\n    step\u2080f = runC step\u2080 FE.>>> {!splitAtC4 (|M| + |M|) FE.>>> {!splitAtC |M| FE.*** id!}!}\n\n    -- step\u2080f : R \u2192 ((M \u00d7 M) \u00d7 S)\n    -- step\u2080f = runC step\u2080 FE.>>> {!splitAtC3 (|M| + |M|) FE.>>> {!splitAtC |M| FE.*** id!}!}\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080f >>> proj *** id\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081f : C \u00d7 S \u2192 Bit\n    step\u2081f (c , s) = head (runC step\u2081 (c ++ s))\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081f (c , s)\n\n--    funAdv : FunSemSecAdv |R|\n--    funAdv = mk (step\u2080f , step\u2081f) \n\n{-\n  record SemSecAdv+FunBeh power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n\n    field\n      Acp : CpSemSecAdv power\n      A\u21ba : FunSemSecAdv c\n\n    open CpSemSecAdv Acp renaming (beh\u21ba to Acp-beh\u21ba; beh to Acp-beh; beh\u2082 to Acp-beh\u2082; beh\u2081 to Acp-beh\u2081)\n    A\u21ba-beh : R \u2192 M\u00b2 \u00d7 (C \u2192 Bit)\n    A\u21ba-beh = run\u21ba A\u21ba\n\n    module SplitA\u21ba = SplitSemSecAdv A\u21ba\n    open SplitA\u21ba using () renaming (beh\u2081 to As-beh\u2081; beh\u2082 to As-beh\u2082)\n\n    field\n      coh\u2081 : Acp-beh\u2081 \u2257 As-beh\u2081\n      coh\u2082 : \u2200 R \u2192 Acp-beh\u2082 R \u2257 As-beh\u2082 R\n\n    coh : CpSemSecAdv.adv\u21ba Acp \u2257A A\u21ba\n    coh = coh\u2081 , coh\u2082\n-}\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  _\u21c4_ : \u2200 {cc p} (E : Enc cc) (A : CpSemSecAdv p) b \u2192 \u21ba (coins p + cc) Bit\n  E \u21c4 A = E \u21c4F CpSemSecAdv.funAdv A\n\n  SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n  SemBroken E power = \u2203 (\u03bb (A : CpSemSecAdv power) \u2192 breaks (E \u21c4 A))\n\n  -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n  --   security of E\u2080 reduces to security of E\u2081\n  --   breaking E\u2081 reduces to breaking E\u2080\n  SemSecReduction : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} {cc\u2081} f tr = \u2200 {p} {E : Enc cc\u2080} \u2192 SemSecReduction (f p) p E (tr E)\n\n  post-comp-cp : \u2200 {cc} post-E \u2192 Tr cc cc\n  post-comp-cp post-E E = E >>> map\u21ba (runC post-E)\n    where open FunctionExtra\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : Cp |C| |C|) \u2192 SemSecTr same-power (post-comp-cp {cc} post-E)\n  post-comp-pres-sem-sec {cc} post-E {p} {E} (A' , A'-breaks-E') = A , A-breaks-E\n     where E' : Enc cc\n           E' = post-comp-cp post-E E\n           open Power p renaming (coins to c)\n           open CpSemSecAdv A' using (R; S; |S|; step\u2080) renaming (step\u2080f to A'-step\u2080f; step\u2081f to A'-step\u2081f; step\u2081 to A'-step\u2081)\n           A-step\u2081-spec = runC post-E *** id >>> A'-step\u2081f\n             where open FunctionExtra\n           A-spec : FunSemSecAdv (coins p)\n           A-spec = mk (A'-step\u2080f , A-step\u2081-spec)\n           open FlatFunsOps \u266dops\n           A-step\u2081 = post-E *** idO {|S|} >>> A'-step\u2081\n           A : CpSemSecAdv p\n           A = mk (step\u2080 , A-step\u2081)\n           open CpSemSecAdv A using () renaming (funAdv to funA; step\u2080f to A-step\u2080f; step\u2081f to A-step\u2081f)\n\n           coh\u2081 : A-step\u2081-spec \u2257 A-step\u2081f\n           coh\u2081 (C , S) rewrite >>>-spec (post-E *** idO {|S|}) A'-step\u2081 (C ++ S)\n                         | ***-spec post-E (idO {|S|}) C {S}\n                         | idC-spec S = refl\n           pf3 : A-spec \u2257A funA\n           pf3 C R rewrite coh\u2081 (C , proj\u2082 (A-step\u2080f R)) = refl\n           same-games : (E' \u21c4 A') \u2257\u2141 (E \u21c4 A)\n           same-games = change-adv {E = E} {A-spec} {funA} pf3\n           A-breaks-E : breaks (E \u21c4 A)\n           A-breaks-E = ext-as-broken same-games A'-breaks-E'\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecReduction (_\u2218_ (_\u2295_ mask))\n-}\n\n{-\nfunBits-kit : FlatFunsSpec\nfunBits-kit = mk _\u2192\u1d47_ bitsFunCircuitBuilder id idC-spec >>>-spec ***-spec where\n  open CircuitBuilder bitsFunCircuitBuilder\n  open BitsFunExtras\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess prgDist\n  module FunAdv' = FunAdv prgDist\n  module CpAdv' = CpAdv funBits-kit prgDist\n-}\n\n{-\nmodule OTP (prgDist : PrgDist) where\n  open FunAdv prgDist 1 1\n  open Guess prgDist renaming (breaks to reads-your-mind)\n\n  -- OTP\u2081 : \u2200 {c} (k : Bit) \u2192 Enc c\n  -- OTP\u2081 k m = return\u21ba ((k xor (head m)) \u2237 [])\n\n  OTP\u2081 : Enc 1\n  OTP\u2081 m = map\u21ba (\u03bb k \u2192 (k xor (head m)) \u2237 []) toss\n\n  open FunctionExtra\n\n  -- \u2200 k \u2192 SemSecReductionToOracle no-power no-power (OTP\u2081 k)\n  foo : \u2200 p \u2192 SemBroken OTP\u2081 p \u2192 Oracle {!(mk 1 0 0 +\u1d56 p)!}\n  foo p (A , A-breaks-OTP) = O , O-reads-your-mind\n     where postulate\n             b : Bit\n           -- b = {!!}\n             k : Bit\n           -- k = {!!}\n           O : \u21ba (coins p + 1) Bit\n           O = A >>= \u03bb { (m , kont) \u2192 map\u21ba (_xor_ (head (m b)) \u2218 kont) (OTP\u2081 (m b)) } --  (k \u2237 []))) }\n           O-reads-your-mind : reads-your-mind (runGuess\u2141 O)\n           O-reads-your-mind = {!!}\n{-\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba kont (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba kont (map\u21ba (\u03bb k \u2192 (k xor (head (m b))) \u2237 []) toss) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (kont \u2218 \u03bb k \u2192 (k xor (head (m b))) \u2237 []) toss) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (\u03bb k \u2192 kont ((k xor (head (m b))) \u2237 []) toss) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (\u03bb k \u2192 kont ((k xor (head (m b))) \u2237 []) (choose (return\u21ba 0b) (return\u21ba 1b))) }\n    A >>= \u03bb { (m , kont) \u2192 choose (kont ((0b xor (head (m b))) \u2237 []))\n                                   (kont ((1b xor (head (m b))) \u2237 [])) }\n    A >>= \u03bb { (m , kont) \u2192 choose (kont (head (m b) \u2237 []))\n                                   (kont (not (head (m b)) \u2237 [])) }\n\nkont = const b\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba kont (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (const b) (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 toss >> return b }\n\nkont = const b\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (\u03bb c \u2192 c xor head (m (kont c))) (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (_xor_ (head (m b)) \u2218 const b) (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (const (head (m b) xor b)) (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 return\u21ba (head (m b) xor b) }\n\n---\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (_xor_ (head (m b)) \u2218 kont) (OTP\u2081 (m b)) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (_xor_ (head (m b)) \u2218 kont) (map\u21ba (\u03bb k \u2192 (k xor (head (m b))) \u2237 []) toss) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (_xor_ (head (m b)) \u2218 kont \u2218 \u03bb k \u2192 (k xor (head (m b))) \u2237 []) toss) }\n    A >>= \u03bb { (m , kont) \u2192 map\u21ba (\u03bb k \u2192 (head (m b)) xor (kont ((k xor (head (m b))) \u2237 [])) toss) }\n    A >>= \u03bb { (m , kont) \u2192 choose ((head (m b)) xor (kont ((0b xor (head (m b))) \u2237 [])))\n                                   ((head (m b)) xor (kont ((1b xor (head (m b))) \u2237 [])))\n            }\n-}\n-}\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"838565e1e2803a8aaf798d3dc124bda52434357b","subject":"one-time-semantic-security: plenty of minor improvments","message":"one-time-semantic-security: plenty of minor improvments\n","repos":"crypto-agda\/crypto-agda","old_file":"one-time-semantic-security.agda","new_file":"one-time-semantic-security.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a9feb3a901a7d53440a0a2887ae907ada16e3f20","subject":"Type.Identities: one more","message":"Type.Identities: one more\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin as Fin using (Fin ; suc ; zero)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; Reveal_is_ ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n{-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 (\u2013> A\u2243) (\u2013> B\u2243)) (map\u00d7 (<\u2013 A\u2243) (<\u2013 B\u2243))\n               (\u03bb y \u2192 pair= (<\u2013-inv-r A\u2243 (fst y)) ({!!} \u2219 <\u2013-inv-r B\u2243 (snd y)))\n               {!!}\n               -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e (\u2013> A\u2243) (\u2013> B\u2243)) (map\u228e (<\u2013 A\u2243) (<\u2013 B\u2243))\n               [inl: (\u03bb x \u2192 ap inl (<\u2013-inv-r A\u2243 x)) ,inr: ap inr \u2218 <\u2013-inv-r B\u2243 ]\n               [inl: (\u03bb x \u2192 ap inl (<\u2013-inv-l A\u2243 x)) ,inr: ap inr \u2218 <\u2013-inv-l B\u2243 ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 \u2013> B\u2243 \u2218 f \u2218 <\u2013 A\u2243)\n               (\u03bb f \u2192 <\u2013 B\u2243 \u2218 f \u2218 \u2013> A\u2243)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-r B\u2243 _ \u2219 ap f (<\u2013-inv-r A\u2243 x)))\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B\u2243 _ \u2219 ap f (<\u2013-inv-l A\u2243 x)))\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua (equiv (maybe inr (inl _))\n                      [inl: const nothing ,inr: just ]\n                      [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                      (maybe (\u03bb _ \u2192 idp) idp))\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nmodule _ where\n  isZero? : \u2200 {n}{A : Fin (suc n) \u2192 Set} \u2192 ((i : Fin n) \u2192 A (suc i)) \u2192 A zero\n    \u2192 (i : Fin (suc n)) \u2192 A i\n  isZero? f x zero = x\n  isZero? f x (suc i) = f i\n\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv (isZero? inr (inl _)) [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n    [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n    (isZero? (\u03bb _ \u2192 idp) idp)\n\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin as Fin using (Fin ; suc ; zero)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; Reveal_is_ ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n{-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 (\u2013> A\u2243) (\u2013> B\u2243)) (map\u00d7 (<\u2013 A\u2243) (<\u2013 B\u2243))\n               (\u03bb y \u2192 pair= (<\u2013-inv-r A\u2243 (fst y)) ({!!} \u2219 <\u2013-inv-r B\u2243 (snd y)))\n               {!!}\n               -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e (\u2013> A\u2243) (\u2013> B\u2243)) (map\u228e (<\u2013 A\u2243) (<\u2013 B\u2243))\n               [inl: (\u03bb x \u2192 ap inl (<\u2013-inv-r A\u2243 x)) ,inr: ap inr \u2218 <\u2013-inv-r B\u2243 ]\n               [inl: (\u03bb x \u2192 ap inl (<\u2013-inv-l A\u2243 x)) ,inr: ap inr \u2218 <\u2013-inv-l B\u2243 ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 \u2013> B\u2243 \u2218 f \u2218 <\u2013 A\u2243)\n               (\u03bb f \u2192 <\u2013 B\u2243 \u2218 f \u2218 \u2013> A\u2243)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-r B\u2243 _ \u2219 ap f (<\u2013-inv-r A\u2243 x)))\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B\u2243 _ \u2219 ap f (<\u2013-inv-l A\u2243 x)))\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua (equiv (maybe inr (inl _))\n                      [inl: const nothing ,inr: just ]\n                      [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                      (maybe (\u03bb _ \u2192 idp) idp))\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nmodule _ where\n  isZero? : \u2200 {n}{A : Fin (suc n) \u2192 Set} \u2192 ((i : Fin n) \u2192 A (suc i)) \u2192 A zero\n    \u2192 (i : Fin (suc n)) \u2192 A i\n  isZero? f x zero = x\n  isZero? f x (suc i) = f i\n\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv (isZero? inr (inl _)) [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n    [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n    (isZero? (\u03bb _ \u2192 idp) idp)\n\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e725104ca30e21c58d1af1ec4d1e96783d4520c0","subject":"Removed unnecessary option.","message":"Removed unnecessary option.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool.Type\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Heterogeneous lists\ndata List : D \u2192 Set where\n  lnil  :                      List []\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n-- Lists of total natural numbers\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n-- Lists of total Booleans\ndata ListB : D \u2192 Set where\n  lbnil  :                                ListB []\n  lbcons : \u2200 {b bs} \u2192 Bool b \u2192 ListB bs \u2192 ListB (b \u2237 bs)\n\n-- Polymorphic lists.\n-- NB. The data type list is in *Set\u2081*.\ndata ListP (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                               ListP A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n\nList\u2081 : D \u2192 Set\u2081\nList\u2081 = ListP (\u03bb d \u2192 d \u2261 d)\n\nListN\u2081 : D \u2192 Set\u2081\nListN\u2081 = ListP N\n\nListB\u2081 : D \u2192 Set\u2081\nListB\u2081 = ListP Bool\n","old_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --universal-quantified-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool.Type\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Heterogeneous lists\ndata List : D \u2192 Set where\n  lnil  :                      List []\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n-- Lists of total natural numbers\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n-- Lists of total Booleans\ndata ListB : D \u2192 Set where\n  lbnil  :                                ListB []\n  lbcons : \u2200 {b bs} \u2192 Bool b \u2192 ListB bs \u2192 ListB (b \u2237 bs)\n\n-- Polymorphic lists.\n-- NB. The data type list is in *Set\u2081*.\ndata ListP (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                               ListP A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n\nList\u2081 : D \u2192 Set\u2081\nList\u2081 = ListP (\u03bb d \u2192 d \u2261 d)\n\nListN\u2081 : D \u2192 Set\u2081\nListN\u2081 = ListP N\n\nListB\u2081 : D \u2192 Set\u2081\nListB\u2081 = ListP Bool\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"920d6b2f3f692db778063b92415a07047d82dee3","subject":"Minor change to doc.","message":"Minor change to doc.\n\nIgnore-this: f157c6b20376d15e0a44e561b15fc97e\n\ndarcs-hash:20120308161712-3bd4e-db631d345a42a8d25e0c71de4d4549198275a09b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/README.agda","new_file":"src\/FOTC\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- First-Order Theory of Combinators (FOTC)\n------------------------------------------------------------------------------\n\nmodule FOTC.README where\n\n-- Formalization of (a version of) Azcel's First-Order Theory of\n-- Combinators.\n\n------------------------------------------------------------------------------\n-- The axioms\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Booleans\n\n-- The axioms\nopen import FOTC.Data.Bool\n\n-- The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n------------------------------------------------------------------------------\n-- Natural numbers\n\n-- The axioms\nopen import FOTC.Data.Nat\n\n-- The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- Inequalites\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n-- Lists\n\n-- The axioms\nopen import FOTC.Data.List\n\n-- The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n------------------------------------------------------------------------------\n-- Lists of natural numbers\n\n-- The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n------------------------------------------------------------------------------\n-- Coinductive natural numbers\n\n-- The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n------------------------------------------------------------------------------\n-- Streams\n\n-- The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n------------------------------------------------------------------------------\n-- Bisimilary relation\n\n-- The axioms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n-- Properties\nopen import FOTC.Relation.Binary.Bisimilarity.PropertiesATP\nopen import FOTC.Relation.Binary.Bisimilarity.PropertiesI\n\n------------------------------------------------------------------------------\n-- Verification of programs\n\n-- The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- The map-iterate property: A property using co-induction\nopen import FOTC.Program.MapIterate.MapIterateATP\nopen import FOTC.Program.MapIterate.MapIterateI\n\n-- The alternating bit protocol: A program using induction and co-induction\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n","old_contents":"------------------------------------------------------------------------------\n-- First-Order Theory of Combinators (FOTC)\n------------------------------------------------------------------------------\n\nmodule FOTC.README where\n\n-- Formalization of (a version of) Azcel's First-Order Theory of\n-- Combinators.\n\n------------------------------------------------------------------------------\n-- The axioms\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Booleans\n\n-- The axioms\nopen import FOTC.Data.Bool\n\n-- The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n------------------------------------------------------------------------------\n-- Natural numbers\n\n-- The axioms\nopen import FOTC.Data.Nat\n\n-- The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- Inequalites\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n-- Lists\n\n-- The axioms\nopen import FOTC.Data.List\n\n-- The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n------------------------------------------------------------------------------\n-- Lists of natural numbers\n\n-- The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n------------------------------------------------------------------------------\n-- Coinductive natural numbers\n\n-- The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n------------------------------------------------------------------------------\n-- Streams\n\n-- The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n------------------------------------------------------------------------------\n-- Bisimilary relation\n\n-- The axioms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n-- Properties\nopen import FOTC.Relation.Binary.Bisimilarity.PropertiesATP\nopen import FOTC.Relation.Binary.Bisimilarity.PropertiesI\n\n------------------------------------------------------------------------------\n-- Verification of programs\n\n-- The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\nopen import FOTC.Program.MapIterate.MapIterateI\n\n-- The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"636d5a56d24486b6265526052a187e78c4538067","subject":"proof of preservation with out induction -- something is rotten in the kingdom of denmark","message":"proof of preservation with out induction -- something is rotten in the kingdom of denmark\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule preservation where\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             \u0394 \u22a2 d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation TAConst (Step _ _ _) = TAConst\n  preservation (TAVar x\u2081) step = abort (somenotnone (! x\u2081))\n  preservation (TALam wt) (Step x\u2082 (ITLam x\u2083) x\u2084) = TALam wt\n  preservation (TALam wt) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TALam wt\n  preservation (TAAp wt x\u2081 wt\u2081) (Step x\u2082 (ITLam x\u2083) x\u2084) = TAAp wt x\u2081 wt\u2081\n  preservation (TAAp wt x wt\u2081) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TAAp wt x wt\u2081\n  preservation (TAEHole x x\u2081) (Step x\u2082 x\u2083 x\u2084) = TAEHole x x\u2081\n  preservation (TANEHole x\u2081 wt x\u2082) (Step x\u2083 (ITLam x\u2084) x\u2085) = TANEHole x\u2081 wt x\u2082\n  preservation (TANEHole x wt x\u2081) (Step x\u2082 (ITCast x\u2083 x\u2084 x\u2085) x\u2086) = TANEHole x wt x\u2081\n  preservation (TACast wt x\u2081) (Step x\u2082 (ITLam x\u2083) x\u2084) = TACast wt x\u2081\n  preservation (TACast wt x) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TACast wt x\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule preservation where\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             \u0394 \u22a2 d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation TAConst (Step _ _ _) = TAConst\n  preservation (TAVar x\u2081) step = abort (somenotnone (! x\u2081))\n  preservation (TALam wt) (Step x\u2081 x\u2082 x\u2083) = {!!}\n  preservation (TAAp wt x wt\u2081) step = {!!}\n  preservation (TAEHole x x\u2081) (Step x\u2082 x\u2083 x\u2084) = {!!}\n  preservation (TANEHole x wt x\u2081) step = {!!}\n  preservation (TACast wt x) step = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1fdf8b41d9320f80c520ee0682a8fdfd595b10c2","subject":"Still unfinished, Chaum Pedersen.","message":"Still unfinished, Chaum Pedersen.\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/ChaumPedersen.agda","new_file":"ZK\/ChaumPedersen.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Zero\nopen import Data.Two.Base\nopen import Data.ShapePolymorphism\nopen import Relation.Binary.PropositionalEquality.NP\nimport ZK.SigmaProtocol as \u03a3Proto\nopen import ZK.Types\nopen import ZK.Statement\n\nmodule ZK.ChaumPedersen\n    {G \u2124q : \u2605}\n    -- (cyclic-group : Cyclic-group G \u2124q)\n    (g    : G)\n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_\/_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_==_ : (x y : G) \u2192 \ud835\udfda)\n    where\n\n  -- TODO: Re-use another module\n  module ElGamal-encryption where\n    record CipherText : \u2605 where\n      constructor _,_\n      field\n        get-\u03b1 get-\u03b2 : G\n\n    PubKey  = G\n    EncRnd  = \u2124q {- randomness used for encryption of ct -}\n    Message = G {- plain text message -}\n\n    enc : PubKey \u2192 EncRnd \u2192 Message \u2192 CipherText\n    enc y r M = \u03b1 , \u03b2 where\n      \u03b1 = g ^ r\n      \u03b2 = (y ^ r) \u00b7 M\n  open ElGamal-encryption\n\n  module Tmp where\n   module _ (y : PubKey) (M : Message) (ct : CipherText) where\n    Statement : Set\n    Statement =\n      -- Reads as follows:\n      -- A value `r` (of type `EncRnd`) is known to be\n      -- the encryption randomness used to produce the\n      -- cipher-text `c` of message `M` using public-key `y`.\n      ZKStatement EncRnd \u03bb { [ r ] \u2192 ct \u2261\u2610 enc y r M }\n\n   -- Assume the randomness `r` is known\n   module _ (y : PubKey) (M : Message) (r : EncRnd) where\n    -- Then the Statement holds\n    Statement-complete : Statement y M (enc y r M)\n    Statement-complete = [ r ] , refl\n\n  record Commitment : \u2605 where\n    constructor _,_\n    field\n      get-A get-B : G\n\n  Challenge  = \u2124q\n  Response   = \u2124q\n  Randomness = \u2124q\n  Witness    = EncRnd\n\n  record Statement : Type where\n    constructor mk\n    field\n      y  : PubKey\n      M  : Message\n      ct : CipherText\n    open CipherText ct public renaming (get-\u03b1 to \u03b1; get-\u03b2 to \u03b2)\n\n  _\u2208_ : Witness \u2192 Statement \u2192 Type\n  r \u2208 (mk y M ct) = {!!}\n\n  open \u03a3Proto Commitment Challenge Response Randomness Witness Statement _\u2208_ public\n\n{-\n  module _ (y : PubKey) (r : EncRnd) (w : \u2124q) where\n  prover : (w : \u2124q) \u2192 Prover\n  prover r w  = prover-commitment , prover-response\n    where\n    prover-commitment : Commitment\n    prover-commitment = (g ^ w) , (y ^ w)\n\n    prover-response : Challenge \u2192 Response\n    prover-response c = w + (r * c)\n\n\n  module _ (y : PubKey) (M : Message) (ct : CipherText) where\n    private\n        \u03b1 = CipherText.get-\u03b1 ct\n        \u03b2 = CipherText.get-\u03b2 ct\n        \u03b2\/M = \u03b2 \/ M\n\n    verifier : Verifier\n    verifier (mk (A , B) c s)\n      = (g ^ s) == (A \u00b7 (\u03b1 ^ c))\n      \u2227 (y ^ s) == (B \u00b7 (\u03b2\/M ^ c))\n\n    -- This simulator shows why it is so important for the\n    -- challenge to be picked once the commitment is known.\n\n    -- To fake a transcript, the challenge and response can\n    -- be arbitrarily chosen. However to be indistinguishable\n    -- from a valid proof it they need to be picked at random.\n    module _ (c : Challenge) (s : Response) where\n        -- Compute A and B, such that the verifier accepts!\n        private\n            A = (g ^ s) \/ (\u03b1 ^ c)\n            B = (y ^ s) \/ (\u03b2\/M ^ c)\n\n        simulate-commitment : Commitment\n        simulate-commitment = (A , B)\n\n        simulate : Transcript\n        simulate = mk simulate-commitment c s\n\n        module Correct-simulation\n           (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n           (\/-\u00b7  : \u2200 {P Q} \u2192 P \u2261 (P \/ Q) \u00b7 Q)\n          where\n          correct-simulation : \u2713(verifier simulate)\n          correct-simulation = \u2713\u2227 (\u2713-== \/-\u00b7) (\u2713-== \/-\u00b7)\n\n  module Correctness-proof\n           (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n           (^-+  : \u2200 {b x y} \u2192 b ^(x + y) \u2261 (b ^ x) \u00b7 (b ^ y))\n           (^-*  : \u2200 {b x y} \u2192 b ^(x * y) \u2261 (b ^ x) ^ y)\n           (\u00b7-\/  : \u2200 {P Q} \u2192 P \u2261 (P \u00b7 Q) \/ Q)\n           (y : PubKey) (r : EncRnd) (w : \u2124q) (M : Message) where\n    open \u2261-Reasoning\n\n    correctness : Correct (prover y r w , verifier y M (enc y r M))\n    correctness c = \u2713\u2227 (\u2713-== pf1) (\u2713-== pf2)\n      where\n        g\u02b7 = g ^ w\n        pf1 = g ^(w + (r * c))\n            \u2261\u27e8 ^-+ \u27e9\n              g\u02b7 \u00b7 (g ^(r * c))\n            \u2261\u27e8 ap (\u03bb z \u2192 g\u02b7 \u00b7 z) ^-* \u27e9\n              g\u02b7 \u00b7 ((g ^ r) ^ c)\n            \u220e\n        pf3 = y ^ (r * c)\n            \u2261\u27e8 ^-* \u27e9\n              (y ^ r)^ c\n            \u2261\u27e8 ap (\u03bb b \u2192 b ^ c) \u00b7-\/ \u27e9\n              (((y ^ r) \u00b7 M) \/ M) ^ c\n            \u220e\n        pf2 = y ^(w + (r * c))\n            \u2261\u27e8 ^-+ \u27e9\n              (y ^ w) \u00b7 (y ^(r * c))\n            \u2261\u27e8 ap (\u03bb z \u2192 (y ^ w) \u00b7 z) pf3 \u27e9\n             (y ^ w) \u00b7 ((((y ^ r) \u00b7 M) \/ M) ^ c)\n            \u220e\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Zero\nopen import Data.Two.Base\nopen import Data.ShapePolymorphism\nopen import Relation.Binary.PropositionalEquality.NP\nimport ZK.SigmaProtocol as \u03a3Proto\nopen import ZK.Types\nopen import ZK.Statement\n\nmodule ZK.ChaumPedersen\n    {G \u2124q : \u2605}\n    -- (cyclic-group : Cyclic-group G \u2124q)\n    (g    : G)\n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_\/_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_==_ : (x y : G) \u2192 \ud835\udfda)\n    where\n\n  -- TODO: Re-use another module\n  module ElGamal-encryption where\n    record CipherText : \u2605 where\n      constructor _,_\n      field\n        get-\u03b1 get-\u03b2 : G\n\n    PubKey  = G\n    EncRnd  = \u2124q {- randomness used for encryption of ct -}\n    Message = G {- plain text message -}\n\n    enc : PubKey \u2192 EncRnd \u2192 Message \u2192 CipherText\n    enc y r M = \u03b1 , \u03b2 where\n      \u03b1 = g ^ r\n      \u03b2 = (y ^ r) \u00b7 M\n  open ElGamal-encryption\n\n  module _ (y : PubKey) (M : Message) (ct : CipherText) where\n    Statement : Set\n    Statement =\n      -- Reads as follows:\n      -- A value `r` (of type `EncRnd`) is known to be\n      -- the encryption randomness used to produce the\n      -- cipher-text `c` of message `M` using public-key `y`.\n      ZKStatement EncRnd \u03bb { [ r ] \u2192 ct \u2261\u2610 enc y r M }\n\n  -- Assume the randomness `r` is known\n  module _ (y : PubKey) (M : Message) (r : EncRnd) where\n    -- Then the Statement holds\n    Statement-complete : Statement y M (enc y r M)\n    Statement-complete = [ r ] , refl\n\n  record Commitment : \u2605 where\n    constructor _,_\n    field\n      get-A get-B : G\n\n  Challenge  = \u2124q\n  Response   = \u2124q\n\n  open \u03a3Proto Commitment Challenge Response public\n\n  module _ (y : PubKey) (r : EncRnd) (w : \u2124q) where\n    prover-commitment : Commitment\n    prover-commitment = (g ^ w) , (y ^ w)\n\n    prover-response : Challenge \u2192 Response\n    prover-response c = w + (r * c)\n\n    prover : Prover\n    prover = prover-commitment , prover-response\n\n  module _ (y : PubKey) (M : Message) (ct : CipherText) where\n    private\n        \u03b1 = CipherText.get-\u03b1 ct\n        \u03b2 = CipherText.get-\u03b2 ct\n        \u03b2\/M = \u03b2 \/ M\n\n    verifier : Verifier\n    verifier (mk (A , B) c s)\n      = (g ^ s) == (A \u00b7 (\u03b1 ^ c))\n      \u2227 (y ^ s) == (B \u00b7 (\u03b2\/M ^ c))\n\n    -- This simulator shows why it is so important for the\n    -- challenge to be picked once the commitment is known.\n\n    -- To fake a transcript, the challenge and response can\n    -- be arbitrarily chosen. However to be indistinguishable\n    -- from a valid proof it they need to be picked at random.\n    module _ (c : Challenge) (s : Response) where\n        -- Compute A and B, such that the verifier accepts!\n        private\n            A = (g ^ s) \/ (\u03b1 ^ c)\n            B = (y ^ s) \/ (\u03b2\/M ^ c)\n\n        simulate-commitment : Commitment\n        simulate-commitment = (A , B)\n\n        simulate : Transcript\n        simulate = mk simulate-commitment c s\n\n        module Correct-simulation\n           (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n           (\/-\u00b7  : \u2200 {P Q} \u2192 P \u2261 (P \/ Q) \u00b7 Q)\n          where\n          correct-simulation : \u2713(verifier simulate)\n          correct-simulation = \u2713\u2227 (\u2713-== \/-\u00b7) (\u2713-== \/-\u00b7)\n\n  module Correctness-proof\n           (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n           (^-+  : \u2200 {b x y} \u2192 b ^(x + y) \u2261 (b ^ x) \u00b7 (b ^ y))\n           (^-*  : \u2200 {b x y} \u2192 b ^(x * y) \u2261 (b ^ x) ^ y)\n           (\u00b7-\/  : \u2200 {P Q} \u2192 P \u2261 (P \u00b7 Q) \/ Q)\n           (y : PubKey) (r : EncRnd) (w : \u2124q) (M : Message) where\n    open \u2261-Reasoning\n\n    correctness : Correct (prover y r w , verifier y M (enc y r M))\n    correctness c = \u2713\u2227 (\u2713-== pf1) (\u2713-== pf2)\n      where\n        g\u02b7 = g ^ w\n        pf1 = g ^(w + (r * c))\n            \u2261\u27e8 ^-+ \u27e9\n              g\u02b7 \u00b7 (g ^(r * c))\n            \u2261\u27e8 ap (\u03bb z \u2192 g\u02b7 \u00b7 z) ^-* \u27e9\n              g\u02b7 \u00b7 ((g ^ r) ^ c)\n            \u220e\n        pf3 = y ^ (r * c)\n            \u2261\u27e8 ^-* \u27e9\n              (y ^ r)^ c\n            \u2261\u27e8 ap (\u03bb b \u2192 b ^ c) \u00b7-\/ \u27e9\n              (((y ^ r) \u00b7 M) \/ M) ^ c\n            \u220e\n        pf2 = y ^(w + (r * c))\n            \u2261\u27e8 ^-+ \u27e9\n              (y ^ w) \u00b7 (y ^(r * c))\n            \u2261\u27e8 ap (\u03bb z \u2192 (y ^ w) \u00b7 z) pf3 \u27e9\n             (y ^ w) \u00b7 ((((y ^ r) \u00b7 M) \/ M) ^ c)\n            \u220e\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d88bde24da1e5c0823a6bebbe460684813833212","subject":"Desc model: cases implicit.","message":"Desc model: cases implicit.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"78d5c087d7e245e7995a82da881435a72bfdf850","subject":"HoTT: \u03a0-is-prop and others","message":"HoTT: \u03a0-is-prop and others\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/HoTT.agda","new_file":"lib\/HoTT.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\nopen \u2261.\u2261-Reasoning\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  -- could be derived in any groupoid\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  module _ {x y : A} where\n\n    \u2219-assoc : (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n    \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n                 (\u03bb x q r \u2192 idp)\n\n    module _ (p : x \u2261 y) where\n      -- ! is a left-inverse for _\u2219_\n      !-\u2219 : ! p \u2219 p \u2261 idp_ y\n      !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp) p\n\n      -- ! is a right-inverse for _\u2219_\n      \u2219-! : p \u2219 ! p \u2261 idp_ x\n      \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp) p\n\n      -- ! is involutive\n      !-involutive : ! (! p) \u2261 p\n      !-involutive = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp) p\n\n      !p\u2219p = !-\u2219\n      p\u2219!p = \u2219-!\n\n      ==-refl-\u2219 : {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n      ==-refl-\u2219 = ap (flip _\u2219_ p)\n\n      \u2219-==-refl : {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n      \u2219-==-refl qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  module _ {x y : A} where\n    module _ (p : x \u2261 x)(q : x \u2261 y)(e : p \u2219 q \u2261 q) where\n      unique-idp-left : p \u2261 idp\n      unique-idp-left\n        = p              \u2261\u27e8 ! \u2219-refl p \u27e9\n          p \u2219 idp        \u2261\u27e8 ap (_\u2219_ p) (! \u2219-! q) \u27e9\n          p \u2219 (q \u2219 ! q)  \u2261\u27e8 \u2219-assoc p q (! q) \u27e9\n          (p \u2219 q) \u2219 ! q  \u2261\u27e8 ap (flip _\u2219_ (! q)) e \u27e9\n          q \u2219 ! q        \u2261\u27e8 \u2219-! q \u27e9\n          idp            \u220e\n\n  module _ {x y z : A}{p\u2080 p\u2081 : x \u2261 y}{q\u2080 q\u2081 : y \u2261 z}(p : p\u2080 \u2261 p\u2081)(q : q\u2080 \u2261 q\u2081) where\n      \u2219= : p\u2080 \u2219 q\u2080 \u2261 p\u2081 \u2219 q\u2081\n      \u2219= = ap (flip _\u2219_ q\u2080) p \u2219 ap (_\u2219_ p\u2081) q\n\n  module _ {x y z : A} where\n    module _ (p : x \u2261 y)(q : y \u2261 z) where\n      \u2219-\u2219-==-refl : {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n      \u2219-\u2219-==-refl rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n      !p\u2219p\u2219q : ! p \u2219 p \u2219 q \u2261 q\n      !p\u2219p\u2219q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n    p\u2219!p\u2219q : (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n    p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n    p\u2219!q\u2219q : (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n    p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n    p\u2219q\u2219!q : (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n    p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n    \u2219-cancel : {p q : x \u2261 y}(r : y \u2261 z) \u2192 p \u2219 r \u2261 q \u2219 r \u2192 p \u2261 q\n    \u2219-cancel {p = p} {q} r e\n      = p               \u2261\u27e8 ! p\u2219q\u2219!q p r \u27e9\n         p \u2219 r  \u2219 ! r   \u2261\u27e8 \u2219-assoc p r (! r) \u27e9\n        (p \u2219 r) \u2219 ! r   \u2261\u27e8 \u2219= e idp \u27e9\n        (q \u2219 r) \u2219 ! r   \u2261\u27e8 ! \u2219-assoc q r (! r) \u27e9\n        q \u2219 (r  \u2219 ! r)  \u2261\u27e8 p\u2219q\u2219!q q r \u27e9\n        q               \u220e\n\n!-ap : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B){x y}(p : x \u2261 y)\n       \u2192 ! (ap f p) \u2261 ap f (! p)\n!-ap f idp = idp\n\nap-id : \u2200 {a}{A : Set a}{x y : A}(p : x \u2261 y) \u2192 ap id p \u2261 p\nap-id idp = idp\n\nmodule _ {a b}{A : Set a}{B : Set b}{f g : A \u2192 B}(H : \u2200 x \u2192 f x \u2261 g x) where\n  ap-nat : \u2200 {x y}(q : x \u2261 y) \u2192 ap f q \u2219 H _ \u2261 H _ \u2219 ap g q\n  ap-nat idp = ! \u2219-refl _\n\nmodule _ {a}{A : Set a}{f : A \u2192 A}(H : \u2200 x \u2192 f x \u2261 x) where\n  ap-nat-id : \u2200 x \u2192 ap f (H x) \u2261 H (f x)\n  ap-nat-id x = \u2219-cancel (H x) (ap-nat H (H x) \u2219 ap (_\u2219_ (H (f x))) (ap-id (H x)))\n\ntr-\u2218 : \u2200 {a b p}{A : Set a}{B : Set b}(P : B \u2192 Set p)(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 tr (P \u2218 f) p \u2261 tr P (ap f p)\ntr-\u2218 P f idp = idp\n\nmodule _ {a}{A : \u2605_ a} where\n    tr-\u2219\u2032 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) \u2192\n             tr P (p \u2219 q) \u223c tr P q \u2218 tr P p\n    tr-\u2219\u2032 P idp _ _ = idp\n\n    tr-\u2219 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) (pq : x \u2261 z) \u2192\n             pq \u2261 p \u2219 q \u2192\n             tr P pq \u223c tr P q \u2218 tr P p\n    tr-\u2219 P p q ._ idp = tr-\u2219\u2032 P p q\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb _ p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) idp p\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605_(a \u2294 b)\n    HAE {f} f-qinv = \u2200 x \u2192 ap f (F.inv-is-linv x) \u2261 F.inv-is-rinv (f x)\n      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      f-g' : (x : B) \u2192 f (g x) \u2261 x\n      f-g' x = ! ap (f \u2218 g) (f-g x) \u2219 ap f (g-f (g x)) \u2219 f-g x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n\n      postulate hae : HAE (qinv g f-g' g-f)\n\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g' g-f\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module Equiv {a b}{A : \u2605_ a}{B : \u2605_ b}(e : A \u2243 B) where\n    \u00b7\u2192 : A \u2192 B\n    \u00b7\u2192 = fst e\n\n    open Is-equiv (snd e) public\n      renaming (linv to \u00b7\u2190; rinv to \u00b7\u2190\u2032; is-linv to \u00b7\u2190-inv-l; is-rinv to \u00b7\u2190-inv-r)\n\n    -- Equivalences are \"injective\"\n    equiv-inj : {x y : A} \u2192 (\u00b7\u2192 x \u2261 \u00b7\u2192 y \u2192 x \u2261 y)\n    equiv-inj p = ! \u00b7\u2190-inv-l _ \u2219 ap \u00b7\u2190 p \u2219 \u00b7\u2190-inv-l _\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n    <\u2013' = \u00b7\u2190\u2032\n\n    <\u2013-inv-l = \u00b7\u2190-inv-l\n    <\u2013-inv-r = \u00b7\u2190-inv-r\n\n  open Equiv public\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n      prop-has-all-paths : is-prop A \u2192 has-all-paths A\n      prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n      all-paths-is-prop : has-all-paths A \u2192 is-prop A\n      all-paths-is-prop c x y = c x y , canon-path\n        where\n        lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n        lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n        canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n        canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                        (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n      Is-contr\u2192is-prop : Is-contr A \u2192 is-prop A\n      Is-contr\u2192is-prop (x , p) y z\n         = ! p y \u2219 p z\n         , J' (\u03bb y\u2081 z\u2081 q \u2192 ! p y\u2081 \u2219 p z\u2081 \u2261 q) (!-\u2219 \u2218 p)\n\n      {-\n      has-level-up : \u2200 {n} \u2192 has-level n A \u2192 has-level \u27e8S n \u27e9 A\n      has-level-up {\u27e8-2\u27e9} = Is-contr\u2192is-prop\n      has-level-up {\u27e8S n \u27e9} \u03c0 x y p q = {!!}\n\n      Is-contr-is-prop : is-prop (Is-contr A)\n      Is-contr-is-prop (x , p) (y , q) = ?\n\n      module _ {{_ : FunExt}} where\n        is-prop-is-prop : has-all-paths (has-all-paths A)\n        is-prop-is-prop h0 h1 = \u03bb= \u03bb x \u2192 \u03bb= \u03bb y \u2192 {!!}\n      -}\n\n\ud835\udfd8-is-prop : is-prop \ud835\udfd8\n\ud835\udfd8-is-prop () _\n\n\ud835\udfd8-has-all-paths : has-all-paths \ud835\udfd8\n\ud835\udfd8-has-all-paths () _\n\n\ud835\udfd9-is-contr : Is-contr \ud835\udfd9\n\ud835\udfd9-is-contr = _ , \u03bb _ \u2192 idp\n\n\ud835\udfd9-is-prop : is-prop \ud835\udfd9\n\ud835\udfd9-is-prop = Is-contr\u2192is-prop \ud835\udfd9-is-contr\n\n\ud835\udfd9-has-all-paths : has-all-paths \ud835\udfd9\n\ud835\udfd9-has-all-paths _ _ = idp\n\nmodule _ {{_ : FunExt}}{a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n  \u03a0-has-all-paths : (\u2200 x \u2192 has-all-paths (B x)) \u2192 has-all-paths (\u03a0 A B)\n  \u03a0-has-all-paths B-has-all-paths f g\n    = \u03bb= \u03bb _ \u2192 B-has-all-paths _ _ _\n\n  \u03a0-is-prop : (\u2200 x \u2192 is-prop (B x)) \u2192 is-prop (\u03a0 A B)\n  \u03a0-is-prop B-is-prop = all-paths-is-prop (\u03a0-has-all-paths (prop-has-all-paths \u2218 B-is-prop))\n\nmodule _ {{_ : FunExt}}{a}{A : \u2605_ a} where\n    \u00ac-has-all-paths : has-all-paths (\u00ac A)\n    \u00ac-has-all-paths = \u03a0-has-all-paths (\u03bb _ \u2192 \ud835\udfd8-has-all-paths)\n\n    \u00ac-is-prop : is-prop (\u00ac A)\n    \u00ac-is-prop = \u03a0-is-prop (\u03bb _ \u2192 \ud835\udfd8-is-prop)\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same p x = ap (\u03bb X \u2192 coe X x) p\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u00b7\u2192 e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u00b7\u2192 e a) (coe-equiv-\u03b2 e)\n\n  postulate\n    coe!-\u03b2 : (e : A \u2243 B) (b : B) \u2192 coe! (ua e) b \u2261 \u00b7\u2190 e b\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u00b7\u2192-paths-equiv : (x \u2261 y) \u2261 (\u00b7\u2192 e x \u2261 \u00b7\u2192 e y)\n    \u00b7\u2192-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n    \u2013>-paths-equiv = \u00b7\u2192-paths-equiv\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\nopen \u2261.\u2261-Reasoning\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  -- could be derived in any groupoid\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  module _ {x y : A} where\n\n    \u2219-assoc : (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n    \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n                 (\u03bb x q r \u2192 idp)\n\n    module _ (p : x \u2261 y) where\n      -- ! is a left-inverse for _\u2219_\n      !-\u2219 : ! p \u2219 p \u2261 idp_ y\n      !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp) p\n\n      -- ! is a right-inverse for _\u2219_\n      \u2219-! : p \u2219 ! p \u2261 idp_ x\n      \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp) p\n\n      -- ! is involutive\n      !-involutive : ! (! p) \u2261 p\n      !-involutive = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp) p\n\n      !p\u2219p = !-\u2219\n      p\u2219!p = \u2219-!\n\n      ==-refl-\u2219 : {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n      ==-refl-\u2219 = ap (flip _\u2219_ p)\n\n      \u2219-==-refl : {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n      \u2219-==-refl qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  module _ {x y : A} where\n    module _ (p : x \u2261 x)(q : x \u2261 y)(e : p \u2219 q \u2261 q) where\n      unique-idp-left : p \u2261 idp\n      unique-idp-left\n        = p              \u2261\u27e8 ! \u2219-refl p \u27e9\n          p \u2219 idp        \u2261\u27e8 ap (_\u2219_ p) (! \u2219-! q) \u27e9\n          p \u2219 (q \u2219 ! q)  \u2261\u27e8 \u2219-assoc p q (! q) \u27e9\n          (p \u2219 q) \u2219 ! q  \u2261\u27e8 ap (flip _\u2219_ (! q)) e \u27e9\n          q \u2219 ! q        \u2261\u27e8 \u2219-! q \u27e9\n          idp            \u220e\n\n  module _ {x y z : A}{p\u2080 p\u2081 : x \u2261 y}{q\u2080 q\u2081 : y \u2261 z}(p : p\u2080 \u2261 p\u2081)(q : q\u2080 \u2261 q\u2081) where\n      \u2219= : p\u2080 \u2219 q\u2080 \u2261 p\u2081 \u2219 q\u2081\n      \u2219= = ap (flip _\u2219_ q\u2080) p \u2219 ap (_\u2219_ p\u2081) q\n\n  module _ {x y z : A} where\n    module _ (p : x \u2261 y)(q : y \u2261 z) where\n      \u2219-\u2219-==-refl : {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n      \u2219-\u2219-==-refl rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n      !p\u2219p\u2219q : ! p \u2219 p \u2219 q \u2261 q\n      !p\u2219p\u2219q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n    p\u2219!p\u2219q : (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n    p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n    p\u2219!q\u2219q : (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n    p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n    p\u2219q\u2219!q : (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n    p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n    \u2219-cancel : {p q : x \u2261 y}(r : y \u2261 z) \u2192 p \u2219 r \u2261 q \u2219 r \u2192 p \u2261 q\n    \u2219-cancel {p = p} {q} r e\n      = p               \u2261\u27e8 ! p\u2219q\u2219!q p r \u27e9\n         p \u2219 r  \u2219 ! r   \u2261\u27e8 \u2219-assoc p r (! r) \u27e9\n        (p \u2219 r) \u2219 ! r   \u2261\u27e8 \u2219= e idp \u27e9\n        (q \u2219 r) \u2219 ! r   \u2261\u27e8 ! \u2219-assoc q r (! r) \u27e9\n        q \u2219 (r  \u2219 ! r)  \u2261\u27e8 p\u2219q\u2219!q q r \u27e9\n        q               \u220e\n\n!-ap : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B){x y}(p : x \u2261 y)\n       \u2192 ! (ap f p) \u2261 ap f (! p)\n!-ap f idp = idp\n\nap-id : \u2200 {a}{A : Set a}{x y : A}(p : x \u2261 y) \u2192 ap id p \u2261 p\nap-id idp = idp\n\nmodule _ {a b}{A : Set a}{B : Set b}{f g : A \u2192 B}(H : \u2200 x \u2192 f x \u2261 g x) where\n  ap-nat : \u2200 {x y}(q : x \u2261 y) \u2192 ap f q \u2219 H _ \u2261 H _ \u2219 ap g q\n  ap-nat idp = ! \u2219-refl _\n\nmodule _ {a}{A : Set a}{f : A \u2192 A}(H : \u2200 x \u2192 f x \u2261 x) where\n  ap-nat-id : \u2200 x \u2192 ap f (H x) \u2261 H (f x)\n  ap-nat-id x = \u2219-cancel (H x) (ap-nat H (H x) \u2219 ap (_\u2219_ (H (f x))) (ap-id (H x)))\n\ntr-\u2218 : \u2200 {a b p}{A : Set a}{B : Set b}(P : B \u2192 Set p)(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 tr (P \u2218 f) p \u2261 tr P (ap f p)\ntr-\u2218 P f idp = idp\n\nmodule _ {a}{A : \u2605_ a} where\n    tr-\u2219\u2032 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) \u2192\n             tr P (p \u2219 q) \u223c tr P q \u2218 tr P p\n    tr-\u2219\u2032 P idp _ _ = idp\n\n    tr-\u2219 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) (pq : x \u2261 z) \u2192\n             pq \u2261 p \u2219 q \u2192\n             tr P pq \u223c tr P q \u2218 tr P p\n    tr-\u2219 P p q ._ idp = tr-\u2219\u2032 P p q\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb _ p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) idp p\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605_(a \u2294 b)\n    HAE {f} f-qinv = \u2200 x \u2192 ap f (F.inv-is-linv x) \u2261 F.inv-is-rinv (f x)\n      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      f-g' : (x : B) \u2192 f (g x) \u2261 x\n      f-g' x = ! ap (f \u2218 g) (f-g x) \u2219 ap f (g-f (g x)) \u2219 f-g x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n\n      postulate hae : HAE (qinv g f-g' g-f)\n\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g' g-f\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module Equiv {a b}{A : \u2605_ a}{B : \u2605_ b}(e : A \u2243 B) where\n    \u00b7\u2192 : A \u2192 B\n    \u00b7\u2192 = fst e\n\n    open Is-equiv (snd e) public\n      renaming (linv to \u00b7\u2190; rinv to \u00b7\u2190\u2032; is-linv to \u00b7\u2190-inv-l; is-rinv to \u00b7\u2190-inv-r)\n\n    -- Equivalences are \"injective\"\n    equiv-inj : {x y : A} \u2192 (\u00b7\u2192 x \u2261 \u00b7\u2192 y \u2192 x \u2261 y)\n    equiv-inj p = ! \u00b7\u2190-inv-l _ \u2219 ap \u00b7\u2190 p \u2219 \u00b7\u2190-inv-l _\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n    <\u2013' = \u00b7\u2190\u2032\n\n    <\u2013-inv-l = \u00b7\u2190-inv-l\n    <\u2013-inv-r = \u00b7\u2190-inv-r\n\n  open Equiv public\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n        prop-has-all-paths : is-prop A \u2192 has-all-paths A\n        prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n        all-paths-is-prop : has-all-paths A \u2192 is-prop A\n        all-paths-is-prop c x y = c x y , canon-path\n          where\n          lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n          lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n          canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n          canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                          (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same p x = ap (\u03bb X \u2192 coe X x) p\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u00b7\u2192 e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u00b7\u2192 e a) (coe-equiv-\u03b2 e)\n\n  postulate\n    coe!-\u03b2 : (e : A \u2243 B) (b : B) \u2192 coe! (ua e) b \u2261 \u00b7\u2190 e b\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u00b7\u2192-paths-equiv : (x \u2261 y) \u2261 (\u00b7\u2192 e x \u2261 \u00b7\u2192 e y)\n    \u00b7\u2192-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n    \u2013>-paths-equiv = \u00b7\u2192-paths-equiv\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e583d1acadf29bc3856d3599b8082a70b3f41ca1","subject":"Add big-lift\u2227 big-lift\u2228 lift-\u2713all-\u03a0\u1d49 lift-\u2713any\u2194\u03a3\u1d49","message":"Add big-lift\u2227 big-lift\u2228 lift-\u2713all-\u03a0\u1d49 lift-\u2713any\u2194\u03a3\u1d49\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Explorable.agda","new_file":"lib\/Explore\/Explorable.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base\nopen import Data.Indexed\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin) renaming (zero to fzero)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product.NP renaming (map to \u00d7-map) hiding (first)\nopen import Data.Sum.NP renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary.NP\nopen import Relation.Binary\nopen import Relation.Binary.Sum using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise using (_\u00d7-cong_)\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen import HoTT\nopen Equivalences\nopen \u2261 using (_\u2261_)\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule _ {m a} {A : \u2605 a} where\n  open EM {a} m\n  gfilter-explore : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore m A \u2192 Explore m B\n  gfilter-explore f e\u1d2c = e\u1d2c >>= \u03bb x \u2192 maybe (\u03bb \u03b7 \u2192 point-explore \u03b7) empty-explore (f x)\n\n  filter-explore : (A \u2192 \ud835\udfda) \u2192 Explore m A \u2192 Explore m A\n  filter-explore p = gfilter-explore \u03bb x \u2192 [0: nothing 1: just x ] (p x)\n\n  -- monoidal exploration: explore A with a monoid M\n  explore-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-monoid e\u1d2c M = e\u1d2c \u03b5 _\u00b7_ where open Mon M renaming (_\u2219_ to _\u00b7_)\n\n  explore-endo : Explore m A \u2192 Explore m A\n  explore-endo e\u1d2c \u03b5 op f = e\u1d2c id _\u2218\u2032_ (op \u2218 f) \u03b5\n\n  explore-endo-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-endo-monoid = explore-monoid \u2218 explore-endo\n\n  explore-backward : Explore m A \u2192 Explore m A\n  explore-backward e\u1d2c \u03b5 _\u2219_ f = e\u1d2c \u03b5 (flip _\u2219_) f\n\n  -- explore-backward \u2218 explore-backward = id\n  -- (m : a comm monoid) \u2192 explore-backward m = explore m\n\nprivate\n  module FindForward {a} {A : \u2605 a} (explore : Explore a A) where\n    find? : Find? A\n    find? = explore nothing (M?._\u2223_ _)\n\n    first : Maybe A\n    first = find? just\n\n    findKey : FindKey A\n    findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule ExplorePlug {\u2113 a} {A : \u2605 a} where\n  record ExploreIndKit p (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (a \u2294 \u209b \u2113 \u2294 p) where\n    constructor mk\n    field\n      P\u03b5 : P empty-explore\n      P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081)\n      Pf : \u2200 x \u2192 P (point-explore x)\n\n  _$kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p} {e : Explore \u2113 A}\n           \u2192 ExploreInd p e \u2192 ExploreIndKit p P \u2192 P e\n  _$kit_ {P = P} ind (mk P\u03b5 P\u2219 Pf) = ind P P\u03b5 P\u2219 Pf\n\n  _,-kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p}{Q : Explore \u2113 A \u2192 \u2605 p}\n            \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\n  Pk ,-kit Qk = mk (P\u03b5 Pk , P\u03b5 Qk)\n                   (\u03bb x y \u2192 P\u2219 Pk (fst x) (fst y) , P\u2219 Qk (snd x) (snd y))\n                   (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n               where open ExploreIndKit\n\n  ExploreInd-Extra : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 _\n  ExploreInd-Extra p exp =\n    \u2200 (Q     : Explore \u2113 A \u2192 \u2605 p)\n      (Q-kit : ExploreIndKit p Q)\n      (P     : Explore \u2113 A \u2192 \u2605 p)\n      (P\u03b5    : P empty-explore)\n      (P\u2219    : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 Q e\u2080 \u2192 Q e\u2081 \u2192 P e\u2080 \u2192 P e\u2081\n               \u2192 P (merge-explore e\u2080 e\u2081))\n      (Pf    : \u2200 x \u2192 P (point-explore x))\n    \u2192 P exp\n\n  to-extra : \u2200 {p} {e : Explore \u2113 A} \u2192 ExploreInd p e \u2192 ExploreInd-Extra p e\n  to-extra e-ind Q Q-kit P P\u03b5 P\u2219 Pf =\n   snd (e-ind (Q \u00d7\u00b0 P)\n           (Q\u03b5 , P\u03b5)\n           (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n           (\u03bb x \u2192 Qf x , Pf x))\n   where open ExploreIndKit Q-kit renaming (P\u03b5 to Q\u03b5; P\u2219 to Q\u2219; Pf to Qf)\n\n  ExplorePlug : \u2200 {m} (M : Monoid \u2113 m) (e : Explore _ A) \u2192 \u2605 _\n  ExplorePlug M e = \u2200 f x \u2192 e\u2218 \u03b5 _\u2219_ f \u2219 x \u2248 e\u2218 x _\u2219_ f\n     where open Mon M\n           e\u2218 = explore-endo e\n\n  plugKit : \u2200 {m} (M : Monoid \u2113 m) \u2192 ExploreIndKit _ (ExplorePlug M)\n  plugKit M = mk (\u03bb _ \u2192 fst identity)\n                 (\u03bb Ps Ps' f x \u2192\n                    trans (\u2219-cong (! Ps _ _) refl)\n                          (trans (assoc _ _ _)\n                                 (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n                 (\u03bb x f _ \u2192 \u2219-cong (snd identity (f x)) refl)\n       where open Mon M\n\nmodule FromExplore\n    {a} {A : \u2605 a}\n    (explore : \u2200 {\u2113} \u2192 Explore \u2113 A) where\n\n  module _ {\u2113} where\n    with-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-monoid = explore-monoid explore\n\n    with\u2218 : Explore \u2113 A\n    with\u2218 = explore-endo explore\n\n    with-endo-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-endo-monoid = explore-endo-monoid explore\n\n    backward : Explore \u2113 A\n    backward = explore-backward explore\n\n    gfilter : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore \u2113 B\n    gfilter f = gfilter-explore f explore\n\n    filter : (A \u2192 \ud835\udfda) \u2192 Explore \u2113 A\n    filter p = filter-explore p explore\n\n  sum : Sum A\n  sum = explore 0 _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore 1 _*_\n\n  big-\u2227 big-\u2228 big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore 1\u2082 _\u2227_\n  and   = big-\u2227\n  all   = big-\u2227\n\n  big-\u2228 = explore 0\u2082 _\u2228_\n  or    = big-\u2228\n  any   = big-\u2228\n\n  big-xor = explore 0\u2082 _xor_\n\n  big-lift\u2227 big-lift\u2228 : Level \u2192 (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n  big-lift\u2227 \u2113 f = lower (explore {\u2113} (lift 1\u2082) (lift-op\u2082 _\u2227_) (lift \u2218 f))\n  big-lift\u2228 \u2113 f = lower (explore {\u2113} (lift 0\u2082) (lift-op\u2082 _\u2228_) (lift \u2218 f))\n\n  bin-tree : BinTree A\n  bin-tree = explore empty fork leaf\n\n  list : List A\n  list = explore List.[] _++_ List.[_]\n\n  module FindBackward = FindForward backward\n\n  findLast? : Find? A\n  findLast? = FindBackward.find?\n\n  last : Maybe A\n  last = FindBackward.first\n\n  findLastKey : FindKey A\n  findLastKey = FindBackward.findKey\n\n  open FindForward explore public\n\nmodule FromLookup\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (lookup : \u2200 {\u2113} \u2192 Lookup {\u2113} explore)\n    where\n\n  module CheckDec! {\u2113}{P : A \u2192 \u2605 \u2113}(decP : \u2200 x \u2192 Dec (P x)) where\n    CheckDec! : \u2605 _\n    CheckDec! = explore (Lift \ud835\udfd9) _\u00d7_ \u03bb x \u2192 \u2713 \u230a decP x \u230b\n\n    checkDec! : {p\u2713 : CheckDec!} \u2192 \u2200 x \u2192 P x\n    checkDec! {p\u2713} x = toWitness (lookup p\u2713 x)\n\nmodule FromExploreInd\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (explore-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p explore)\n    where\n\n  open FromExplore explore public\n\n  module _ {\u2113 p} where\n    explore-mon-ext : ExploreMonExt {\u2113} p explore\n    explore-mon-ext m {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ _ f \u2248 s _ _ g) refl \u2219-cong f\u2248\u00b0g\n      where open Mon m\n\n    explore-mono : ExploreMono {\u2113} p explore\n    explore-mono _\u2286_ z\u2286 _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n      explore-ind (\u03bb e \u2192 e _ _ f \u2286 e _ _ g) z\u2286 _\u2219-mono_ f\u2286\u00b0g\n\n    open ExplorePlug {\u2113} {a} {A}\n\n    explore\u2218-plug : (M : Monoid \u2113 \u2113) \u2192 ExplorePlug M explore\n    explore\u2218-plug M = explore-ind $kit plugKit M\n\n    module _ (M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = explore-ind (\u03bb e \u2192 \u2200 z \u2192 e \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo e z _\u2219_ f)\n                                                (fst identity) (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _))) (\u03bb _ _ \u2192 refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n    explore\u2218-ind : \u2200 (M : Monoid \u2113 \u2113) \u2192 BigOpMonInd \u2113 M (with-endo-monoid M)\n    explore\u2218-ind M P P\u03b5 P\u2219 Pf P\u2248 =\n      snd (explore-ind (\u03bb e \u2192 ExplorePlug M e \u00d7 P (\u03bb f \u2192 e id _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n                 (const (fst identity) , P\u03b5)\n                 (\u03bb {e} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {e} {s'} (fst Ps) (fst Ps')\n                                    , P\u2248 (\u03bb f \u2192 fst Ps f _) (P\u2219 (snd Ps) (snd Ps')))\n                 (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                      , P\u2248 (\u03bb f \u2192 ! snd identity _) (Pf x)))\n      where open Mon M\n\n    explore-swap : \u2200 {b} \u2192 ExploreSwap {\u2113} p explore {b}\n    explore-swap mon {e\u1d2e} e\u1d2e-\u03b5 pf f =\n      explore-ind (\u03bb e \u2192 e _ _ (e\u1d2e \u2218 f) \u2248 e\u1d2e (e _ _ \u2218 flip f))\n                  (! e\u1d2e-\u03b5)\n                  (\u03bb p q \u2192 trans (\u2219-cong p q) (! pf _ _))\n                  (\u03bb _ \u2192 refl)\n      where open Mon mon\n\n    explore-\u03b5 : Explore\u03b5 {\u2113} p explore\n    explore-\u03b5 M = explore-ind (\u03bb e \u2192 e \u03b5 _ (const \u03b5) \u2248 \u03b5)\n                              refl\n                              (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (fst identity \u03b5))\n                              (\u03bb _ \u2192 refl)\n      where open Mon M\n\n    explore-hom : ExploreHom {\u2113} p explore\n    explore-hom cm f g = explore-ind (\u03bb e \u2192 e _ _ (f \u2219\u00b0 g) \u2248 e _ _ f \u2219 e _ _ g)\n                                     (! fst identity \u03b5)\n                                     (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _))\n                                     (\u03bb _ \u2192 refl)\n      where open CMon cm\n\n    explore-lin\u02e1 : ExploreLin\u02e1 {\u2113} p explore\n    explore-lin\u02e1 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e \u03b5 _\u2219_ f) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    explore-lin\u02b3 : ExploreLin\u02b3 {\u2113} p explore\n    explore-lin\u02b3 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e \u03b5 _\u2219_ f \u25ce k) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    module ProductMonoid\n        {M : \u2605\u2080} (\u03b5\u2098 : M) (_\u2295\u2098_ : Op\u2082 M)\n        {N : \u2605\u2080} (\u03b5\u2099 : N) (_\u2295\u2099_ : Op\u2082 N)\n        where\n        \u03b5 = (\u03b5\u2098 , \u03b5\u2099)\n        _\u2295_ : Op\u2082 (M \u00d7 N)\n        (x\u2098 , x\u2099) \u2295 (y\u2098 , y\u2099) = (x\u2098 \u2295\u2098 y\u2098 , x\u2099 \u2295\u2099 y\u2099)\n\n        explore-product-monoid :\n          \u2200 f\u2098 f\u2099 \u2192 explore \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (explore \u03b5\u2098 _\u2295\u2098_ f\u2098 , explore \u03b5\u2099 _\u2295\u2099_ f\u2099)\n        explore-product-monoid f\u2098 f\u2099 =\n          explore-ind (\u03bb e \u2192 e \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (e \u03b5\u2098 _\u2295\u2098_ f\u2098 , e \u03b5\u2099 _\u2295\u2099_ f\u2099)) \u2261.refl (\u2261.ap\u2082 _\u2295_) (\u03bb _ \u2192 \u2261.refl)\n  {-\n  empty-explore:\n    \u03b5 \u2261 (\u03b5\u2098 , \u03b5\u2099) \u2713\n  point-explore (x , y):\n    < f\u2098 , f\u2099 > (x , y) \u2261 (f\u2098 x , f\u2099 y) \u2713\n  merge-explore e\u2080 e\u2081:\n    e\u2080 \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2295 e\u2081 \u03b5 _\u2295_ < f\u2098 , f\u2099 >\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099) \u2295 (e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 \u2295 e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099 \u2295 e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n  -}\n\n  module _ {\u2113} where\n    reify : Reify {\u2113} explore\n    reify = explore-ind (\u03bb e\u1d2c \u2192 \u03a0\u1d49 e\u1d2c _) _ _,_\n\n    unfocus : Unfocus {\u2113} explore\n    unfocus = explore-ind Unfocus (\u03bb{ (lift ()) }) (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n    module _ {\u2113\u1d63 a\u1d63} {A\u1d63 : A \u2192 A \u2192 \u2605 a\u1d63}\n             (A\u1d63-refl : Reflexive A\u1d63) where\n      \u27e6explore\u27e7 : \u27e6Explore\u27e7 \u2113\u1d63 A\u1d63 (explore {\u2113}) (explore {\u2113})\n      \u27e6explore\u27e7 M\u1d63 z\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb e \u2192 M\u1d63 (e _ _ _) (e _ _ _)) z\u1d63 (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n    explore-ext : ExploreExt {\u2113} explore\n    explore-ext \u03b5 op = explore-ind (\u03bb e \u2192 e _ _ _ \u2261 e _ _ _) \u2261.refl (\u2261.ap\u2082 op)\n\n  module LiftHom\n       {m p}\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 p)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (zero : S)\n       (_+_  : Op\u2082 S)\n       (one  : T)\n       (_*_  : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom-0-1 : f zero \u2248 one)\n       (hom-+-* : \u2200 {x y} \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore zero _+_ g) \u2248 explore one _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb e \u2192 f (e zero _+_ g) \u2248 e one _*_ (f \u2218 g))\n                               hom-0-1\n                               (\u03bb p q \u2192 \u2248-trans hom-+-* (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  module _ {\u2113} {P : A \u2192 \u2605_ \u2113} where\n    open LiftHom {S = \u2605_ \u2113} {\u2605_ \u2113} (\u03bb A B \u2192 B \u2192 A) id _\u2218\u2032_\n                 (Lift \ud835\udfd8) _\u228e_ (Lift \ud835\udfd9) _\u00d7_\n                 (\u03bb f g \u2192 \u00d7-map f g) Dec P (const (no (\u03bb{ (lift ()) })))\n                 (uncurry Dec-\u228e)\n                 public renaming (lift-hom to lift-Dec)\n\n  module FromFocus {p} (focus : Focus {p} explore) where\n    Dec-\u03a3 : \u2200 {P} \u2192 \u03a0 A (Dec \u2218 P) \u2192 Dec (\u03a3 A P)\n    Dec-\u03a3 = map-Dec unfocus focus \u2218 lift-Dec \u2218 reify\n\n  lift-hom-\u2261 :\n      \u2200 {m} {S T : \u2605 m}\n        (zero : S)\n        (_+_  : Op\u2082 S)\n        (one  : T)\n        (_*_  : Op\u2082 T)\n        (f    : S \u2192 T)\n        (g    : A \u2192 S)\n        (hom-0-1 : f zero \u2261 one)\n        (hom-+-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore zero _+_ g) \u2261 explore one _*_ (f \u2218 g)\n  lift-hom-\u2261 z _+_ o _*_ = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans z _+_ o _*_ (\u2261.ap\u2082 _*_)\n\n  -- Since so far S and T should have the same level, we get this mess of resizing\n  -- There is a later version based on \u27e6explore\u27e7.\n  module _ (f : A \u2192 \ud835\udfda) {{_ : UA}} where\n    lift-\u2713all-\u03a0\u1d49 : \u2713 (big-lift\u2227 \u2081 f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    lift-\u2713all-\u03a0\u1d49 = lift-hom-\u2261 (lift 1\u2082) (lift-op\u2082 _\u2227_) (Lift \ud835\udfd9) _\u00d7_ (\u2713 \u2218 lower) (lift \u2218 f) (\u2261.! Lift\u2261id) (\u2713-\u2227-\u00d7 _ _)\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    lift-\u2713any\u2194\u03a3\u1d49 : \u2713 (big-lift\u2228 \u2081 f) \u2194 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    lift-\u2713any\u2194\u03a3\u1d49 = LiftHom.lift-hom _\u2194_ (id , id) (zip (flip _\u2218\u2032_) _\u2218\u2032_)\n                     (lift 0\u2082) (lift-op\u2082 _\u2228_) (Lift \ud835\udfd8) _\u228e_\n                     (zip \u228e-map \u228e-map) (\u2713 \u2218 lower) (lift \u2218 f)\n                     ((\u03bb()) , \u03bb{(lift())}) (\u2713\u2228-\u228e , \u228e-\u2713\u2228)\n\n  sum-ind : SumInd sum\n  sum-ind P P0 P+ Pf = explore-ind (\u03bb e \u2192 P (e 0 _+_)) P0 P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext 0 _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ z\u2264n _+-mono_\n\n  sum-swap' : SumSwap sum\n  sum-swap' {sum\u1d2e = s\u1d2e} s\u1d2e-0 hom f =\n    sum-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2261 s\u1d2e (s \u2218 flip f))\n            (! s\u1d2e-0)\n            (\u03bb p q \u2192 (ap\u2082 _+_ p q) \u2219 (! hom _ _)) (\u03bb _ \u2192 refl)\n    where open \u2261\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.ap\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sum-const : SumConst sum\n  sum-const x = sum-ext (\u03bb _ \u2192 ! snd \u2115\u00b0.*-identity x) \u2219 sum-lin (const 1) x \u2219 \u2115\u00b0.*-comm x Card\n    where open \u2261\n\n  exploreStableUnder\u2192sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  exploreStableUnder\u2192sumStableUnder SU-p = SU-p 0 _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sumStableUnder\u2192countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  sumStableUnder\u2192countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  diff-list = with-endo-monoid (List.monoid A) List.[_]\n\n  {-\n  list\u2261diff-list : list \u2261 diff-list\n  list\u2261diff-list = {!explore-endo-monoid-spec (List.monoid A) List.[_]!}\n  -}\n\n  lift-sum : \u2200 \u2113 \u2192 Sum A\n  lift-sum \u2113 f = lower {\u2080} {\u2113} (explore (lift 0) (lift-op\u2082 _+_) (lift \u2218 f))\n\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin : \u2200 {{_ : UA}}(f : A \u2192 \u2115) \u2192 Fin (lift-sum _ f) \u2261 \u03a3\u1d49 explore (Fin \u2218 f)\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin f = lift-hom-\u2261 (lift 0) (lift-op\u2082 _+_) (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 lower) (lift \u2218 f) (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id) (! Fin-\u228e-+)\n    where open \u2261\n\nmodule FromTwoExploreInd\n    {a} {A : \u2605 a}\n    {e\u1d2c : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (e\u1d2c-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2c)\n    {b} {B : \u2605 b}\n    {e\u1d2e : \u2200 {\u2113} \u2192 Explore \u2113 B}\n    (e\u1d2e-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2e)\n    where\n\n    module A = FromExploreInd e\u1d2c-ind\n    module B = FromExploreInd e\u1d2e-ind\n\n    module _ {c \u2113}(cm : CommutativeMonoid c \u2113) where\n        open CMon cm\n\n        op\u1d2c = e\u1d2c \u03b5 _\u2219_\n        op\u1d2e = e\u1d2e \u03b5 _\u2219_\n\n        -- TODO use lift-hom\n        explore-swap' : \u2200 f \u2192 op\u1d2c (op\u1d2e \u2218 f) \u2248 op\u1d2e (op\u1d2c \u2218 flip f)\n        explore-swap' = A.explore-swap m (B.explore-\u03b5 m) (B.explore-hom cm)\n\n    sum-swap : \u2200 f \u2192 A.sum (B.sum \u2218 f) \u2261 B.sum (A.sum \u2218 flip f)\n    sum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n\nmodule FromTwoAdequate-sum\n  {{_ : UA}}{{_ : FunExt}}\n  {A}{B}\n  {sum\u1d2c : Sum A}{sum\u1d2e : Sum B}\n  (open Adequacy _\u2261_)\n  (sum\u1d2c-adq : Adequate-sum sum\u1d2c)\n  (sum\u1d2e-adq : Adequate-sum sum\u1d2e) where\n\n  open \u2261\n  sumStableUnder : (p : A \u2243 B)(f : B \u2192 \u2115)\n                 \u2192 sum\u1d2c (f \u2218 \u00b7\u2192 p) \u2261 sum\u1d2e f\n  sumStableUnder p f = Fin-injective (sum\u1d2c-adq (f \u2218 \u00b7\u2192 p)\n                                      \u2219 \u03a3-fst\u2243 p _\n                                      \u2219 ! sum\u1d2e-adq f)\n\n  sumStableUnder\u2032 : (p : A \u2243 B)(f : A \u2192 \u2115)\n                 \u2192 sum\u1d2c f \u2261 sum\u1d2e (f \u2218 <\u2013 p)\n  sumStableUnder\u2032 p f = Fin-injective (sum\u1d2c-adq f\n                                      \u2219 \u03a3-fst\u2243\u2032 p _\n                                      \u2219 ! sum\u1d2e-adq (f \u2218 <\u2013 p))\n\nmodule FromAdequate-sum\n  {A}\n  {sum : Sum A}\n  (open Adequacy _\u2261_)\n  (sum-adq : Adequate-sum sum)\n  {{_ : UA}}{{_ : FunExt}}\n  where\n\n  open FromTwoAdequate-sum sum-adq sum-adq public\n  open \u2261\n\n  sum-ext : SumExt sum\n  sum-ext = ap sum \u2218 \u03bb=\n\n  private\n    count : Count A\n    count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  private\n    module M {p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) where\n      private\n\n        P = \u03bb x \u2192 p x \u2261 1\u2082\n        Q = \u03bb x \u2192 q x \u2261 1\u2082\n        \u00acP = \u03bb x \u2192 p x \u2261 0\u2082\n        \u00acQ = \u03bb x \u2192 q x \u2261 0\u2082\n\n        \u03c0 : \u03a3 A P \u2261 \u03a3 A Q\n        \u03c0 = ! \u03a3=\u2032 _ (count-\u2261 p)\n            \u2219 ! (sum-adq (\ud835\udfda\u25b9\u2115 \u2218 p))\n            \u2219 ap Fin same-count\n            \u2219 sum-adq (\ud835\udfda\u25b9\u2115 \u2218 q)\n            \u2219 \u03a3=\u2032 _ (count-\u2261 q)\n\n        lem1 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 qx \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 (not px) * \ud835\udfda\u25b9\u2115 qx\n        lem1 1\u2082 1\u2082 = \u2261.refl\n        lem1 1\u2082 0\u2082 = \u2261.refl\n        lem1 0\u2082 1\u2082 = \u2261.refl\n        lem1 0\u2082 0\u2082 = \u2261.refl\n\n        lem2 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 px \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 px * \ud835\udfda\u25b9\u2115 (not qx)\n        lem2 1\u2082 1\u2082 = \u2261.refl\n        lem2 1\u2082 0\u2082 = \u2261.refl\n        lem2 0\u2082 1\u2082 = \u2261.refl\n        lem2 0\u2082 0\u2082 = \u2261.refl\n\n        lemma1 : \u2200 px qx \u2192 (qx \u2261 1\u2082) \u2261 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 0\u2082 \u00d7 qx \u2261 1\u2082))\n        lemma1 px qx = ! Fin-\u2261-\u22611\u2082 qx\n                     \u2219 ap Fin (lem1 px qx)\n                     \u2219 ! Fin-\u228e-+\n                     \u2219 \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22610\u2082 px) (Fin-\u2261-\u22611\u2082 qx))\n\n        lemma2 : \u2200 px qx \u2192 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 1\u2082 \u00d7 qx \u2261 0\u2082)) \u2261 (px \u2261 1\u2082)\n        lemma2 px qx = ! \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22611\u2082 px) (Fin-\u2261-\u22610\u2082 qx)) \u2219 Fin-\u228e-+ \u2219 ap Fin (! lem2 px qx) \u2219 Fin-\u2261-\u22611\u2082 px\n\n        \u03c0' : (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192  P x \u00d7 \u00acQ x))\n           \u2261 (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192 \u00acP x \u00d7  Q x))\n        \u03c0' = \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n           \u2219 ! \u03a3\u228e-split\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma2 (p x) (q x))\n           \u2219 \u03c0\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma1 (p x) (q x))\n           \u2219 \u03a3\u228e-split\n           \u2219 ! \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n\n        \u03c0'' : \u03a3 A (P \u00d7\u00b0 \u00acQ) \u2261 \u03a3 A (\u00acP \u00d7\u00b0 Q)\n        \u03c0'' = Fin\u228e-injective (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u03c0'\n\n      open EquivalentSubsets \u03c0'' public\n\n  same-count\u2192iso : \u2200{p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) \u2192 p \u2261 q \u2218 M.\u03c0 {p} {q} same-count\n  same-count\u2192iso {p} {q} sc = M.prop {p} {q} sc\n\nmodule From\u27e6Explore\u27e7\n    {-a-} {A : \u2605\u2080 {- a-}}\n    {explore   : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (\u27e6explore\u27e7 : \u2200 {\u2113\u2080 \u2113\u2081} \u2113\u1d63 \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 _\u2261_ explore explore)\n    {{_ : UA}}\n    where\n  open FromExplore explore\n\n  module AlsoInFromExploreInd\n          {\u2113}(M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = \u27e6explore\u27e7 \u2080 {C} {C \u2192 C}\n                                              (\u03bb r s \u2192 \u2200 z \u2192 r \u2219 z \u2248 s z)\n                                              (fst identity)\n                                              (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _)))\n                                              (\u03bb x\u1d63 _ \u2192 \u2219-cong (reflexive (\u2261.ap f x\u1d63)) refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n  open \u2261\n  module _ (f : A \u2192 \u2115) where\n    sum\u21d2\u03a3\u1d49 : Fin (explore 0 _+_ f) \u2261 explore (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 f)\n    sum\u21d2\u03a3\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb n X \u2192 Fin n \u2261 X)\n                (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id)\n                (\u03bb p q \u2192 ! Fin-\u228e-+ \u2219 \u228e= p q)\n                (ap (Fin \u2218 f))\n\n    product\u21d2\u03a0\u1d49 : Fin (explore 1 _*_ f) \u2261 explore (Lift \ud835\udfd9) _\u00d7_ (Fin \u2218 f)\n    product\u21d2\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                    (\u03bb n X \u2192 Fin n \u2261 X)\n                    (Fin1\u2261\ud835\udfd9 \u2219 ! Lift\u2261id)\n                    (\u03bb p q \u2192 ! Fin-\u00d7-* \u2219 \u00d7= p q)\n                    (ap (Fin \u2218 f))\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    \u2713all-\u03a0\u1d49 : \u2713 (all f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    \u2713all-\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb b X \u2192 \u2713 b \u2261 X)\n                (! Lift\u2261id)\n                (\u03bb p q \u2192 \u2713-\u2227-\u00d7 _ _ \u2219 \u00d7= p q)\n                (ap (\u2713 \u2218 f))\n\n    \u2713any\u2192\u03a3\u1d49 : \u2713 (any f) \u2192 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    \u2713any\u2192\u03a3\u1d49 p = \u27e6explore\u27e7 {\u2080} {\u209b \u2080} \u2081\n                         (\u03bb b (X : \u2605\u2080) \u2192 Lift (\u2713 b) \u2192 X)\n                         (\u03bb x \u2192 lift (lower x))\n                         (\u03bb { {0\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inr (y\u1d63 z\u1d63)\n                            ; {1\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inl (x\u1d63 _) })\n                         (\u03bb x\u1d63 x \u2192 tr (\u2713 \u2218 f) x\u1d63 (lower x)) (lift p)\n\n  module FromAdequate-\u03a3\u1d49\n           (adequate-\u03a3\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-sum : Adequate-sum _\u2261_ sum\n    adequate-sum f = sum\u21d2\u03a3\u1d49 f \u2219 adequate-\u03a3\u1d49 (Fin \u2218 f)\n\n    open FromAdequate-sum adequate-sum public\n\n    adequate-any : Adequate-any -\u2192- any\n    adequate-any f e = coe (adequate-\u03a3\u1d49 (\u2713 \u2218 f)) (\u2713any\u2192\u03a3\u1d49 f e)\n\n  module FromAdequate-\u03a0\u1d49\n           (adequate-\u03a0\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-product : Adequate-product _\u2261_ product\n    adequate-product f = product\u21d2\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 (Fin \u2218 f)\n\n    adequate-all : Adequate-all _\u2261_ all\n    adequate-all f = \u2713all-\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 _\n\n    check! : (f : A \u2192 \ud835\udfda) {pf : \u2713 (all f)} \u2192 (\u2200 x \u2192 \u2713 (f x))\n    check! f {pf} = coe (adequate-all f) pf\n\n{-\nmodule ExplorableRecord where\n    record Explorable A : \u2605\u2081 where\n      constructor mk\n      field\n        explore     : Explore\u2080 A\n        explore-ind : ExploreInd\u2080 explore\n\n      open FromExploreInd explore-ind\n      field\n        adequate-sum     : Adequate-sum sum\n    --  adequate-product : AdequateProduct product\n\n      open FromExploreInd explore-ind public\n\n    open Explorable public\n\n    ExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\n    ExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\n    record Funable A : \u2605\u2082 where\n      constructor _,_\n      field\n        explorable : Explorable A\n        negative   : ExploreForFun A\n\n    module DistFun {A} (\u03bcA : Explorable A)\n                       (\u03bcA\u2192 : ExploreForFun A)\n                       {B} (\u03bcB : Explorable B){X}\n                       (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                       (0\u2032  : X)\n                       (_+_ : X \u2192 X \u2192 X)\n                       (_*_ : X \u2192 X \u2192 X) where\n\n      \u03a3\u1d2e = explore \u03bcB 0\u2032 _+_\n      \u03a0' = explore \u03bcA 0\u2032 _*_\n      \u03a3' = explore (\u03bcA\u2192 \u03bcB) 0\u2032 _+_\n\n      DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\n    DistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\n    DistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                       \u2200 _*_ \u2192 Zero _\u2248_ \u03b5 _*_ \u2192 _DistributesOver_ _\u2248_ _*_ _\u2219_ \u2192 _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                       \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ \u03b5 _\u2219_ _*_\n\n    DistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\n    DistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03bc-iso : \u2200 {A B} \u2192 (A \u2243 B) \u2192 Explorable A \u2192 Explorable B\n        \u03bc-iso {A}{B} A\u2243B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n          where\n            open \u2261\n            A\u2192B = \u2013> A\u2243B\n            ade = \u03bb f \u2192 adequate-sum \u03bcA (f \u2218 A\u2192B) \u2219 \u03a3-fst\u2243 A\u2243B _\n\n        -- I guess this could be more general\n        \u03bc-iso-preserve : \u2200 {A B} (A\u2243B : A \u2243 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2243B \u03bcA) (f \u2218 <\u2013 A\u2243B)\n        \u03bc-iso-preserve A\u2243B f \u03bcA = sum-ext \u03bcA (\u03bb x \u2192 ap f (! (<\u2013-inv-l A\u2243B x)))\n          where open \u2261\n\n          {-\n        \u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n        \u03bcLift = \u03bc-iso {!(! Lift\u2194id)!}\n          where open \u2261\n          -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base\nopen import Data.Indexed\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin) renaming (zero to fzero)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product.NP renaming (map to \u00d7-map) hiding (first)\nopen import Data.Sum.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary.NP\nopen import Relation.Binary\nopen import Relation.Binary.Sum using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise using (_\u00d7-cong_)\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen import HoTT\nopen Equivalences\nopen \u2261 using (_\u2261_)\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule _ {m a} {A : \u2605 a} where\n  open EM {a} m\n  gfilter-explore : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore m A \u2192 Explore m B\n  gfilter-explore f e\u1d2c = e\u1d2c >>= \u03bb x \u2192 maybe (\u03bb \u03b7 \u2192 point-explore \u03b7) empty-explore (f x)\n\n  filter-explore : (A \u2192 \ud835\udfda) \u2192 Explore m A \u2192 Explore m A\n  filter-explore p = gfilter-explore \u03bb x \u2192 [0: nothing 1: just x ] (p x)\n\n  -- monoidal exploration: explore A with a monoid M\n  explore-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-monoid e\u1d2c M = e\u1d2c \u03b5 _\u00b7_ where open Mon M renaming (_\u2219_ to _\u00b7_)\n\n  explore-endo : Explore m A \u2192 Explore m A\n  explore-endo e\u1d2c \u03b5 op f = e\u1d2c id _\u2218\u2032_ (op \u2218 f) \u03b5\n\n  explore-endo-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-endo-monoid = explore-monoid \u2218 explore-endo\n\n  explore-backward : Explore m A \u2192 Explore m A\n  explore-backward e\u1d2c \u03b5 _\u2219_ f = e\u1d2c \u03b5 (flip _\u2219_) f\n\n  -- explore-backward \u2218 explore-backward = id\n  -- (m : a comm monoid) \u2192 explore-backward m = explore m\n\nprivate\n  module FindForward {a} {A : \u2605 a} (explore : Explore a A) where\n    find? : Find? A\n    find? = explore nothing (M?._\u2223_ _)\n\n    first : Maybe A\n    first = find? just\n\n    findKey : FindKey A\n    findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule ExplorePlug {\u2113 a} {A : \u2605 a} where\n  record ExploreIndKit p (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (a \u2294 \u209b \u2113 \u2294 p) where\n    constructor mk\n    field\n      P\u03b5 : P empty-explore\n      P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081)\n      Pf : \u2200 x \u2192 P (point-explore x)\n\n  _$kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p} {e : Explore \u2113 A}\n           \u2192 ExploreInd p e \u2192 ExploreIndKit p P \u2192 P e\n  _$kit_ {P = P} ind (mk P\u03b5 P\u2219 Pf) = ind P P\u03b5 P\u2219 Pf\n\n  _,-kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p}{Q : Explore \u2113 A \u2192 \u2605 p}\n            \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\n  Pk ,-kit Qk = mk (P\u03b5 Pk , P\u03b5 Qk)\n                   (\u03bb x y \u2192 P\u2219 Pk (fst x) (fst y) , P\u2219 Qk (snd x) (snd y))\n                   (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n               where open ExploreIndKit\n\n  ExploreInd-Extra : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 _\n  ExploreInd-Extra p exp =\n    \u2200 (Q     : Explore \u2113 A \u2192 \u2605 p)\n      (Q-kit : ExploreIndKit p Q)\n      (P     : Explore \u2113 A \u2192 \u2605 p)\n      (P\u03b5    : P empty-explore)\n      (P\u2219    : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 Q e\u2080 \u2192 Q e\u2081 \u2192 P e\u2080 \u2192 P e\u2081\n               \u2192 P (merge-explore e\u2080 e\u2081))\n      (Pf    : \u2200 x \u2192 P (point-explore x))\n    \u2192 P exp\n\n  to-extra : \u2200 {p} {e : Explore \u2113 A} \u2192 ExploreInd p e \u2192 ExploreInd-Extra p e\n  to-extra e-ind Q Q-kit P P\u03b5 P\u2219 Pf =\n   snd (e-ind (Q \u00d7\u00b0 P)\n           (Q\u03b5 , P\u03b5)\n           (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n           (\u03bb x \u2192 Qf x , Pf x))\n   where open ExploreIndKit Q-kit renaming (P\u03b5 to Q\u03b5; P\u2219 to Q\u2219; Pf to Qf)\n\n  ExplorePlug : \u2200 {m} (M : Monoid \u2113 m) (e : Explore _ A) \u2192 \u2605 _\n  ExplorePlug M e = \u2200 f x \u2192 e\u2218 \u03b5 _\u2219_ f \u2219 x \u2248 e\u2218 x _\u2219_ f\n     where open Mon M\n           e\u2218 = explore-endo e\n\n  plugKit : \u2200 {m} (M : Monoid \u2113 m) \u2192 ExploreIndKit _ (ExplorePlug M)\n  plugKit M = mk (\u03bb _ \u2192 fst identity)\n                 (\u03bb Ps Ps' f x \u2192\n                    trans (\u2219-cong (! Ps _ _) refl)\n                          (trans (assoc _ _ _)\n                                 (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n                 (\u03bb x f _ \u2192 \u2219-cong (snd identity (f x)) refl)\n       where open Mon M\n\nmodule FromExplore\n    {a} {A : \u2605 a}\n    (explore : \u2200 {\u2113} \u2192 Explore \u2113 A) where\n\n  module _ {\u2113} where\n    with-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-monoid = explore-monoid explore\n\n    with\u2218 : Explore \u2113 A\n    with\u2218 = explore-endo explore\n\n    with-endo-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-endo-monoid = explore-endo-monoid explore\n\n    backward : Explore \u2113 A\n    backward = explore-backward explore\n\n    gfilter : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore \u2113 B\n    gfilter f = gfilter-explore f explore\n\n    filter : (A \u2192 \ud835\udfda) \u2192 Explore \u2113 A\n    filter p = filter-explore p explore\n\n  sum : Sum A\n  sum = explore 0 _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore 1 _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore 1\u2082 _\u2227_\n  and   = big-\u2227\n  all   = big-\u2227\n\n  big-\u2228 = explore 0\u2082 _\u2228_\n  or    = big-\u2228\n  any   = big-\u2228\n\n  big-xor = explore 0\u2082 _xor_\n\n  bin-tree : BinTree A\n  bin-tree = explore empty fork leaf\n\n  list : List A\n  list = explore List.[] _++_ List.[_]\n\n  module FindBackward = FindForward backward\n\n  findLast? : Find? A\n  findLast? = FindBackward.find?\n\n  last : Maybe A\n  last = FindBackward.first\n\n  findLastKey : FindKey A\n  findLastKey = FindBackward.findKey\n\n  open FindForward explore public\n\nmodule FromLookup\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (lookup : \u2200 {\u2113} \u2192 Lookup {\u2113} explore)\n    where\n\n  module CheckDec! {\u2113}{P : A \u2192 \u2605 \u2113}(decP : \u2200 x \u2192 Dec (P x)) where\n    CheckDec! : \u2605 _\n    CheckDec! = explore (Lift \ud835\udfd9) _\u00d7_ \u03bb x \u2192 \u2713 \u230a decP x \u230b\n\n    checkDec! : {p\u2713 : CheckDec!} \u2192 \u2200 x \u2192 P x\n    checkDec! {p\u2713} x = toWitness (lookup p\u2713 x)\n\nmodule FromExploreInd\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (explore-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p explore)\n    where\n\n  open FromExplore explore public\n\n  module _ {\u2113 p} where\n    explore-mon-ext : ExploreMonExt {\u2113} p explore\n    explore-mon-ext m {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ _ f \u2248 s _ _ g) refl \u2219-cong f\u2248\u00b0g\n      where open Mon m\n\n    explore-mono : ExploreMono {\u2113} p explore\n    explore-mono _\u2286_ z\u2286 _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n      explore-ind (\u03bb e \u2192 e _ _ f \u2286 e _ _ g) z\u2286 _\u2219-mono_ f\u2286\u00b0g\n\n    open ExplorePlug {\u2113} {a} {A}\n\n    explore\u2218-plug : (M : Monoid \u2113 \u2113) \u2192 ExplorePlug M explore\n    explore\u2218-plug M = explore-ind $kit plugKit M\n\n    module _ (M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = explore-ind (\u03bb e \u2192 \u2200 z \u2192 e \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo e z _\u2219_ f)\n                                                (fst identity) (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _))) (\u03bb _ _ \u2192 refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n    explore\u2218-ind : \u2200 (M : Monoid \u2113 \u2113) \u2192 BigOpMonInd \u2113 M (with-endo-monoid M)\n    explore\u2218-ind M P P\u03b5 P\u2219 Pf P\u2248 =\n      snd (explore-ind (\u03bb e \u2192 ExplorePlug M e \u00d7 P (\u03bb f \u2192 e id _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n                 (const (fst identity) , P\u03b5)\n                 (\u03bb {e} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {e} {s'} (fst Ps) (fst Ps')\n                                    , P\u2248 (\u03bb f \u2192 fst Ps f _) (P\u2219 (snd Ps) (snd Ps')))\n                 (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                      , P\u2248 (\u03bb f \u2192 ! snd identity _) (Pf x)))\n      where open Mon M\n\n    explore-swap : \u2200 {b} \u2192 ExploreSwap {\u2113} p explore {b}\n    explore-swap mon {e\u1d2e} e\u1d2e-\u03b5 pf f =\n      explore-ind (\u03bb e \u2192 e _ _ (e\u1d2e \u2218 f) \u2248 e\u1d2e (e _ _ \u2218 flip f))\n                  (! e\u1d2e-\u03b5)\n                  (\u03bb p q \u2192 trans (\u2219-cong p q) (! pf _ _))\n                  (\u03bb _ \u2192 refl)\n      where open Mon mon\n\n    explore-\u03b5 : Explore\u03b5 {\u2113} p explore\n    explore-\u03b5 M = explore-ind (\u03bb e \u2192 e \u03b5 _ (const \u03b5) \u2248 \u03b5)\n                              refl\n                              (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (fst identity \u03b5))\n                              (\u03bb _ \u2192 refl)\n      where open Mon M\n\n    explore-hom : ExploreHom {\u2113} p explore\n    explore-hom cm f g = explore-ind (\u03bb e \u2192 e _ _ (f \u2219\u00b0 g) \u2248 e _ _ f \u2219 e _ _ g)\n                                     (! fst identity \u03b5)\n                                     (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _))\n                                     (\u03bb _ \u2192 refl)\n      where open CMon cm\n\n    explore-lin\u02e1 : ExploreLin\u02e1 {\u2113} p explore\n    explore-lin\u02e1 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e \u03b5 _\u2219_ f) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    explore-lin\u02b3 : ExploreLin\u02b3 {\u2113} p explore\n    explore-lin\u02b3 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e \u03b5 _\u2219_ f \u25ce k) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    module ProductMonoid\n        {M : \u2605\u2080} (\u03b5\u2098 : M) (_\u2295\u2098_ : Op\u2082 M)\n        {N : \u2605\u2080} (\u03b5\u2099 : N) (_\u2295\u2099_ : Op\u2082 N)\n        where\n        \u03b5 = (\u03b5\u2098 , \u03b5\u2099)\n        _\u2295_ : Op\u2082 (M \u00d7 N)\n        (x\u2098 , x\u2099) \u2295 (y\u2098 , y\u2099) = (x\u2098 \u2295\u2098 y\u2098 , x\u2099 \u2295\u2099 y\u2099)\n\n        explore-product-monoid :\n          \u2200 f\u2098 f\u2099 \u2192 explore \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (explore \u03b5\u2098 _\u2295\u2098_ f\u2098 , explore \u03b5\u2099 _\u2295\u2099_ f\u2099)\n        explore-product-monoid f\u2098 f\u2099 =\n          explore-ind (\u03bb e \u2192 e \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (e \u03b5\u2098 _\u2295\u2098_ f\u2098 , e \u03b5\u2099 _\u2295\u2099_ f\u2099)) \u2261.refl (\u2261.ap\u2082 _\u2295_) (\u03bb _ \u2192 \u2261.refl)\n  {-\n  empty-explore:\n    \u03b5 \u2261 (\u03b5\u2098 , \u03b5\u2099) \u2713\n  point-explore (x , y):\n    < f\u2098 , f\u2099 > (x , y) \u2261 (f\u2098 x , f\u2099 y) \u2713\n  merge-explore e\u2080 e\u2081:\n    e\u2080 \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2295 e\u2081 \u03b5 _\u2295_ < f\u2098 , f\u2099 >\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099) \u2295 (e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 \u2295 e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099 \u2295 e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n  -}\n\n  module _ {\u2113} where\n    reify : Reify {\u2113} explore\n    reify = explore-ind (\u03bb e\u1d2c \u2192 \u03a0\u1d49 e\u1d2c _) _ _,_\n\n    unfocus : Unfocus {\u2113} explore\n    unfocus = explore-ind Unfocus (\u03bb{ (lift ()) }) (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n    module _ {\u2113\u1d63 a\u1d63} {A\u1d63 : A \u2192 A \u2192 \u2605 a\u1d63}\n             (A\u1d63-refl : Reflexive A\u1d63) where\n      \u27e6explore\u27e7 : \u27e6Explore\u27e7 \u2113\u1d63 A\u1d63 (explore {\u2113}) (explore {\u2113})\n      \u27e6explore\u27e7 M\u1d63 z\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb e \u2192 M\u1d63 (e _ _ _) (e _ _ _)) z\u1d63 (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n    explore-ext : ExploreExt {\u2113} explore\n    explore-ext \u03b5 op = explore-ind (\u03bb e \u2192 e _ _ _ \u2261 e _ _ _) \u2261.refl (\u2261.ap\u2082 op)\n\n  module LiftHom\n       {m p}\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 p)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (zero : S)\n       (_+_  : Op\u2082 S)\n       (one  : T)\n       (_*_  : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom-0-1 : f zero \u2248 one)\n       (hom-+-* : \u2200 {x y} \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore zero _+_ g) \u2248 explore one _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb e \u2192 f (e zero _+_ g) \u2248 e one _*_ (f \u2218 g))\n                               hom-0-1\n                               (\u03bb p q \u2192 \u2248-trans hom-+-* (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  module _ {\u2113} {P : A \u2192 \u2605_ \u2113} where\n    open LiftHom {S = \u2605_ \u2113} {\u2605_ \u2113} (\u03bb A B \u2192 B \u2192 A) id _\u2218\u2032_\n                 (Lift \ud835\udfd8) _\u228e_ (Lift \ud835\udfd9) _\u00d7_\n                 (\u03bb f g \u2192 \u00d7-map f g) Dec P (const (no (\u03bb{ (lift ()) })))\n                 (uncurry Dec-\u228e)\n                 public renaming (lift-hom to lift-Dec)\n\n  module FromFocus {p} (focus : Focus {p} explore) where\n    Dec-\u03a3 : \u2200 {P} \u2192 \u03a0 A (Dec \u2218 P) \u2192 Dec (\u03a3 A P)\n    Dec-\u03a3 = map-Dec unfocus focus \u2218 lift-Dec \u2218 reify\n\n  lift-hom-\u2261 :\n      \u2200 {m} {S T : \u2605 m}\n        (zero : S)\n        (_+_  : Op\u2082 S)\n        (one  : T)\n        (_*_  : Op\u2082 T)\n        (f    : S \u2192 T)\n        (g    : A \u2192 S)\n        (hom-0-1 : f zero \u2261 one)\n        (hom-+-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore zero _+_ g) \u2261 explore one _*_ (f \u2218 g)\n  lift-hom-\u2261 z _+_ o _*_ = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans z _+_ o _*_ (\u2261.ap\u2082 _*_)\n\n  sum-ind : SumInd sum\n  sum-ind P P0 P+ Pf = explore-ind (\u03bb e \u2192 P (e 0 _+_)) P0 P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext 0 _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ z\u2264n _+-mono_\n\n  sum-swap' : SumSwap sum\n  sum-swap' {sum\u1d2e = s\u1d2e} s\u1d2e-0 hom f =\n    sum-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2261 s\u1d2e (s \u2218 flip f))\n            (! s\u1d2e-0)\n            (\u03bb p q \u2192 (ap\u2082 _+_ p q) \u2219 (! hom _ _)) (\u03bb _ \u2192 refl)\n    where open \u2261\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.ap\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sum-const : SumConst sum\n  sum-const x = sum-ext (\u03bb _ \u2192 ! snd \u2115\u00b0.*-identity x) \u2219 sum-lin (const 1) x \u2219 \u2115\u00b0.*-comm x Card\n    where open \u2261\n\n  exploreStableUnder\u2192sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  exploreStableUnder\u2192sumStableUnder SU-p = SU-p 0 _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sumStableUnder\u2192countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  sumStableUnder\u2192countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  diff-list = with-endo-monoid (List.monoid A) List.[_]\n\n  {-\n  list\u2261diff-list : list \u2261 diff-list\n  list\u2261diff-list = {!explore-endo-monoid-spec (List.monoid A) List.[_]!}\n  -}\n\n  lift-sum : \u2200 \u2113 \u2192 Sum A\n  lift-sum \u2113 f = lower {\u2080} {\u2113} (explore (lift 0) (lift-op\u2082 _+_) (lift \u2218 f))\n\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin : \u2200 {{_ : UA}}(f : A \u2192 \u2115) \u2192 Fin (lift-sum _ f) \u2261 \u03a3\u1d49 explore (Fin \u2218 f)\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin f = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans (lift 0) (lift-op\u2082 _+_) (Lift \ud835\udfd8) _\u228e_ \u228e= (Fin \u2218 lower) (lift \u2218 f) (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id) (! Fin-\u228e-+)\n    where open \u2261\n\nmodule FromTwoExploreInd\n    {a} {A : \u2605 a}\n    {e\u1d2c : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (e\u1d2c-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2c)\n    {b} {B : \u2605 b}\n    {e\u1d2e : \u2200 {\u2113} \u2192 Explore \u2113 B}\n    (e\u1d2e-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2e)\n    where\n\n    module A = FromExploreInd e\u1d2c-ind\n    module B = FromExploreInd e\u1d2e-ind\n\n    module _ {c \u2113}(cm : CommutativeMonoid c \u2113) where\n        open CMon cm\n\n        op\u1d2c = e\u1d2c \u03b5 _\u2219_\n        op\u1d2e = e\u1d2e \u03b5 _\u2219_\n\n        -- TODO use lift-hom\n        explore-swap' : \u2200 f \u2192 op\u1d2c (op\u1d2e \u2218 f) \u2248 op\u1d2e (op\u1d2c \u2218 flip f)\n        explore-swap' = A.explore-swap m (B.explore-\u03b5 m) (B.explore-hom cm)\n\n    sum-swap : \u2200 f \u2192 A.sum (B.sum \u2218 f) \u2261 B.sum (A.sum \u2218 flip f)\n    sum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n\nmodule FromTwoAdequate-sum\n  {{_ : UA}}{{_ : FunExt}}\n  {A}{B}\n  {sum\u1d2c : Sum A}{sum\u1d2e : Sum B}\n  (open Adequacy _\u2261_)\n  (sum\u1d2c-adq : Adequate-sum sum\u1d2c)\n  (sum\u1d2e-adq : Adequate-sum sum\u1d2e) where\n\n  open \u2261\n  sumStableUnder : (p : A \u2243 B)(f : B \u2192 \u2115)\n                 \u2192 sum\u1d2c (f \u2218 \u00b7\u2192 p) \u2261 sum\u1d2e f\n  sumStableUnder p f = Fin-injective (sum\u1d2c-adq (f \u2218 \u00b7\u2192 p)\n                                      \u2219 \u03a3-fst\u2243 p _\n                                      \u2219 ! sum\u1d2e-adq f)\n\n  sumStableUnder\u2032 : (p : A \u2243 B)(f : A \u2192 \u2115)\n                 \u2192 sum\u1d2c f \u2261 sum\u1d2e (f \u2218 <\u2013 p)\n  sumStableUnder\u2032 p f = Fin-injective (sum\u1d2c-adq f\n                                      \u2219 \u03a3-fst\u2243\u2032 p _\n                                      \u2219 ! sum\u1d2e-adq (f \u2218 <\u2013 p))\n\nmodule FromAdequate-sum\n  {A}\n  {sum : Sum A}\n  (open Adequacy _\u2261_)\n  (sum-adq : Adequate-sum sum)\n  {{_ : UA}}{{_ : FunExt}}\n  where\n\n  open FromTwoAdequate-sum sum-adq sum-adq public\n  open \u2261\n\n  sum-ext : SumExt sum\n  sum-ext = ap sum \u2218 \u03bb=\n\n  private\n    count : Count A\n    count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  private\n    module M {p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) where\n      private\n\n        P = \u03bb x \u2192 p x \u2261 1\u2082\n        Q = \u03bb x \u2192 q x \u2261 1\u2082\n        \u00acP = \u03bb x \u2192 p x \u2261 0\u2082\n        \u00acQ = \u03bb x \u2192 q x \u2261 0\u2082\n\n        \u03c0 : \u03a3 A P \u2261 \u03a3 A Q\n        \u03c0 = ! \u03a3=\u2032 _ (count-\u2261 p)\n            \u2219 ! (sum-adq (\ud835\udfda\u25b9\u2115 \u2218 p))\n            \u2219 ap Fin same-count\n            \u2219 sum-adq (\ud835\udfda\u25b9\u2115 \u2218 q)\n            \u2219 \u03a3=\u2032 _ (count-\u2261 q)\n\n        lem1 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 qx \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 (not px) * \ud835\udfda\u25b9\u2115 qx\n        lem1 1\u2082 1\u2082 = \u2261.refl\n        lem1 1\u2082 0\u2082 = \u2261.refl\n        lem1 0\u2082 1\u2082 = \u2261.refl\n        lem1 0\u2082 0\u2082 = \u2261.refl\n\n        lem2 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 px \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 px * \ud835\udfda\u25b9\u2115 (not qx)\n        lem2 1\u2082 1\u2082 = \u2261.refl\n        lem2 1\u2082 0\u2082 = \u2261.refl\n        lem2 0\u2082 1\u2082 = \u2261.refl\n        lem2 0\u2082 0\u2082 = \u2261.refl\n\n        lemma1 : \u2200 px qx \u2192 (qx \u2261 1\u2082) \u2261 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 0\u2082 \u00d7 qx \u2261 1\u2082))\n        lemma1 px qx = ! Fin-\u2261-\u22611\u2082 qx\n                     \u2219 ap Fin (lem1 px qx)\n                     \u2219 ! Fin-\u228e-+\n                     \u2219 \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22610\u2082 px) (Fin-\u2261-\u22611\u2082 qx))\n\n        lemma2 : \u2200 px qx \u2192 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 1\u2082 \u00d7 qx \u2261 0\u2082)) \u2261 (px \u2261 1\u2082)\n        lemma2 px qx = ! \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22611\u2082 px) (Fin-\u2261-\u22610\u2082 qx)) \u2219 Fin-\u228e-+ \u2219 ap Fin (! lem2 px qx) \u2219 Fin-\u2261-\u22611\u2082 px\n\n        \u03c0' : (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192  P x \u00d7 \u00acQ x))\n           \u2261 (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192 \u00acP x \u00d7  Q x))\n        \u03c0' = \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n           \u2219 ! \u03a3\u228e-split\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma2 (p x) (q x))\n           \u2219 \u03c0\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma1 (p x) (q x))\n           \u2219 \u03a3\u228e-split\n           \u2219 ! \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n\n        \u03c0'' : \u03a3 A (P \u00d7\u00b0 \u00acQ) \u2261 \u03a3 A (\u00acP \u00d7\u00b0 Q)\n        \u03c0'' = Fin\u228e-injective (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u03c0'\n\n      open EquivalentSubsets \u03c0'' public\n\n  same-count\u2192iso : \u2200{p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) \u2192 p \u2261 q \u2218 M.\u03c0 {p} {q} same-count\n  same-count\u2192iso {p} {q} sc = M.prop {p} {q} sc\n\nmodule From\u27e6Explore\u27e7\n    {-a-} {A : \u2605\u2080 {- a-}}\n    {explore   : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (\u27e6explore\u27e7 : \u2200 {\u2113\u2080 \u2113\u2081} \u2113\u1d63 \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 _\u2261_ explore explore)\n    {{_ : UA}}\n    where\n  open FromExplore explore\n\n  module AlsoInFromExploreInd\n          {\u2113}(M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = \u27e6explore\u27e7 \u2080 {C} {C \u2192 C}\n                                              (\u03bb r s \u2192 \u2200 z \u2192 r \u2219 z \u2248 s z)\n                                              (fst identity)\n                                              (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _)))\n                                              (\u03bb x\u1d63 _ \u2192 \u2219-cong (reflexive (\u2261.ap f x\u1d63)) refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n  open \u2261\n  module _ (f : A \u2192 \u2115) where\n    sum\u21d2\u03a3\u1d49 : Fin (explore 0 _+_ f) \u2261 explore (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 f)\n    sum\u21d2\u03a3\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb n X \u2192 Fin n \u2261 X)\n                (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id)\n                (\u03bb p q \u2192 ! Fin-\u228e-+ \u2219 \u228e= p q)\n                (ap (Fin \u2218 f))\n\n    product\u21d2\u03a0\u1d49 : Fin (explore 1 _*_ f) \u2261 explore (Lift \ud835\udfd9) _\u00d7_ (Fin \u2218 f)\n    product\u21d2\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                    (\u03bb n X \u2192 Fin n \u2261 X)\n                    (Fin1\u2261\ud835\udfd9 \u2219 ! Lift\u2261id)\n                    (\u03bb p q \u2192 ! Fin-\u00d7-* \u2219 \u00d7= p q)\n                    (ap (Fin \u2218 f))\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    \u2713all-\u03a0\u1d49 : \u2713 (all f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    \u2713all-\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb b X \u2192 \u2713 b \u2261 X)\n                (! Lift\u2261id)\n                (\u03bb p q \u2192 \u2713-\u2227-\u00d7 _ _ \u2219 \u00d7= p q)\n                (ap (\u2713 \u2218 f))\n\n    \u2713any\u2192\u03a3\u1d49 : \u2713 (any f) \u2192 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    \u2713any\u2192\u03a3\u1d49 p = \u27e6explore\u27e7 {\u2080} {\u209b \u2080} \u2081\n                         (\u03bb b (X : \u2605\u2080) \u2192 Lift (\u2713 b) \u2192 X)\n                         (\u03bb x \u2192 lift (lower x))\n                         (\u03bb { {0\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inr (y\u1d63 z\u1d63)\n                            ; {1\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inl (x\u1d63 _) })\n                         (\u03bb x\u1d63 x \u2192 tr (\u2713 \u2218 f) x\u1d63 (lower x)) (lift p)\n\n  module FromAdequate-\u03a3\u1d49\n           (adequate-\u03a3\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-sum : Adequate-sum _\u2261_ sum\n    adequate-sum f = sum\u21d2\u03a3\u1d49 f \u2219 adequate-\u03a3\u1d49 (Fin \u2218 f)\n\n    open FromAdequate-sum adequate-sum public\n\n    adequate-any : Adequate-any -\u2192- any\n    adequate-any f e = coe (adequate-\u03a3\u1d49 (\u2713 \u2218 f)) (\u2713any\u2192\u03a3\u1d49 f e)\n\n  module FromAdequate-\u03a0\u1d49\n           (adequate-\u03a0\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-product : Adequate-product _\u2261_ product\n    adequate-product f = product\u21d2\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 (Fin \u2218 f)\n\n    adequate-all : Adequate-all _\u2261_ all\n    adequate-all f = \u2713all-\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 _\n\n    check! : (f : A \u2192 \ud835\udfda) {pf : \u2713 (all f)} \u2192 (\u2200 x \u2192 \u2713 (f x))\n    check! f {pf} = coe (adequate-all f) pf\n\n{-\nmodule ExplorableRecord where\n    record Explorable A : \u2605\u2081 where\n      constructor mk\n      field\n        explore     : Explore\u2080 A\n        explore-ind : ExploreInd\u2080 explore\n\n      open FromExploreInd explore-ind\n      field\n        adequate-sum     : Adequate-sum sum\n    --  adequate-product : AdequateProduct product\n\n      open FromExploreInd explore-ind public\n\n    open Explorable public\n\n    ExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\n    ExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\n    record Funable A : \u2605\u2082 where\n      constructor _,_\n      field\n        explorable : Explorable A\n        negative   : ExploreForFun A\n\n    module DistFun {A} (\u03bcA : Explorable A)\n                       (\u03bcA\u2192 : ExploreForFun A)\n                       {B} (\u03bcB : Explorable B){X}\n                       (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                       (0\u2032  : X)\n                       (_+_ : X \u2192 X \u2192 X)\n                       (_*_ : X \u2192 X \u2192 X) where\n\n      \u03a3\u1d2e = explore \u03bcB 0\u2032 _+_\n      \u03a0' = explore \u03bcA 0\u2032 _*_\n      \u03a3' = explore (\u03bcA\u2192 \u03bcB) 0\u2032 _+_\n\n      DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\n    DistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\n    DistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                       \u2200 _*_ \u2192 Zero _\u2248_ \u03b5 _*_ \u2192 _DistributesOver_ _\u2248_ _*_ _\u2219_ \u2192 _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                       \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ \u03b5 _\u2219_ _*_\n\n    DistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\n    DistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03bc-iso : \u2200 {A B} \u2192 (A \u2243 B) \u2192 Explorable A \u2192 Explorable B\n        \u03bc-iso {A}{B} A\u2243B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n          where\n            open \u2261\n            A\u2192B = \u2013> A\u2243B\n            ade = \u03bb f \u2192 adequate-sum \u03bcA (f \u2218 A\u2192B) \u2219 \u03a3-fst\u2243 A\u2243B _\n\n        -- I guess this could be more general\n        \u03bc-iso-preserve : \u2200 {A B} (A\u2243B : A \u2243 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2243B \u03bcA) (f \u2218 <\u2013 A\u2243B)\n        \u03bc-iso-preserve A\u2243B f \u03bcA = sum-ext \u03bcA (\u03bb x \u2192 ap f (! (<\u2013-inv-l A\u2243B x)))\n          where open \u2261\n\n          {-\n        \u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n        \u03bcLift = \u03bc-iso {!(! Lift\u2194id)!}\n          where open \u2261\n          -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"518e11d3c270fa1bc9465dcf6b737a977b65500a","subject":"Proof-irrelevance on paths when the type is claimed to be Is-set","message":"Proof-irrelevance on paths when the type is claimed to be Is-set\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/K.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/K.agda","new_contents":"-- NOTE with-K\nmodule Relation.Binary.PropositionalEquality.K where\n\nopen import Type hiding (\u2605)\nopen import Relation.Binary.PropositionalEquality\n{-\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product using (\u03a3; _,_)\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\n-}\nopen import Relation.Binary.NP\nopen import Relation.Nullary\n\nmodule _ where\n  postulate\n    Is-set : \u2200 {a} \u2192 Set a \u2192 Set a\n\nproof-irrelevance : \u2200 {a} {A : Set a} {A-is-set : Is-set A} {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\nproof-irrelevance refl refl = refl\n\nmodule _ {a} {A : \u2605 a} where\n  _\u2261\u2261_ : \u2200 {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n  _\u2261\u2261_ refl refl = refl\n\n  _\u225f\u2261_ : \u2200 {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n  _\u225f\u2261_ refl refl = yes refl\n","old_contents":"-- NOTE with-K\nmodule Relation.Binary.PropositionalEquality.K where\n\nopen import Type hiding (\u2605)\nopen import Relation.Binary.PropositionalEquality\n{-\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product using (\u03a3; _,_)\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\n-}\nopen import Relation.Binary.NP\nopen import Relation.Nullary\n\nmodule _ {a} {A : \u2605 a} where\n  _\u2261\u2261_ : \u2200 {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n  _\u2261\u2261_ refl refl = refl\n\n  _\u225f\u2261_ : \u2200 {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n  _\u225f\u2261_ refl refl = yes refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"11ccfac008b3587ccf8f74ecc0e0da44907f22d5","subject":"adding a couple of rules after double checking with @cyrus-","message":"adding a couple of rules after double checking with @cyrus-\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- postulate -- todo: write this later\n  -- substitution\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 = \u2987\u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] (d1 \u2218 d2) =  ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- postulate -- todo: write this later\n  -- substitution\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 = \u2987\u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] (d1 \u2218 d2) =  ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    -- EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err -- todo: is it an error to ever cast the constant?\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 -- todo: added to make progress work\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192 -- todo: aded this to make progress work\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6344e9fed9ae231dae91c77a68e2a74aa27511be","subject":"whitespace","message":"whitespace\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFail : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFail : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d8553018dc2fe750ba11fedbda0f3db018125324","subject":"figured out the last judgement","message":"figured out the last judgement\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error -- todo\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               (\u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9) ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  -- d is result of filling the hole in \u03b5 with d'\n  data _==_[_] : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 [ d ]\n    -- FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n    --        d1 == \u03b5 [ d ] \u2192\n    --        (\u00b7\u03bb x [ \u03c4 ] d1) == ?\n    -- FAp1 :\n    -- FAp2 :\n    -- FNEHole : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n    --        d1 == \u03b5 [ d ] \u2192\n    --        (\u2987 d1 \u2988\u27e8 u , \u03c3 , m\u27e9) == \u2987 d \u2988\u27e8 u , \u03c3 , m\u27e9\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n           d1 == \u03b5 [ d ] \u2192\n           (< \u03c4 > d1) == < \u03c4 > \u03b5 [ d ]\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 [ d0 ] \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 [ d0' ] \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"26bbe5e6c76ff022da19d20c80116757a322d8e2","subject":"Control\/Protocol.agda","message":"Control\/Protocol.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol.agda","new_file":"Control\/Protocol.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_) renaming (proj\u2081 to fst)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nmodule Control.Protocol where\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual\u1d35\u1d3c-equiv : Is-equiv dual\u1d35\u1d3c\ndual\u1d35\u1d3c-equiv = self-inv-is-equiv dual\u1d35\u1d3c-involutive\n\ndual\u1d35\u1d3c-inj : \u2200 {x y} \u2192 dual\u1d35\u1d3c x \u2261 dual\u1d35\u1d3c y \u2192 x \u2261 y\ndual\u1d35\u1d3c-inj = Is-equiv.injective dual\u1d35\u1d3c-equiv\n\n{-\nmodule UniversalProtocols \u2113 {U : \u2605_(\u209b \u2113)}(U\u27e6_\u27e7 : U \u2192 \u2605_ \u2113) where\n-}\nmodule _ \u2113 where\n  U = \u2605_ \u2113\n  U\u27e6_\u27e7 = id\n  data Proto_ : \u2605_(\u209b \u2113) where\n    end : Proto_\n    com : (io : InOut){M : U}(P : U\u27e6 M \u27e7 \u2192 Proto_) \u2192 Proto_\n{-\nmodule U\u2605 \u2113 = UniversalProtocols \u2113 {\u2605_ \u2113} id\nopen U\u2605\n-}\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n    com= refl refl P= = ap (com _) (\u03bb= P=)\n\n    module _ {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n        com\u2243 = com= io= (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    module _ io {M N}(P : M \u00d7 N \u2192 Proto)\n             where\n        com\u2082 : Proto\n        com\u2082 = com io \u03bb m \u2192 com io \u03bb n \u2192 P (m , n)\n\n    {- Proving this would be awesome...\n    module _ io\n             {M\u2080 M\u2081 N\u2080 N\u2081 : \u2605}\n             (M\u00d7N\u2243 : (M\u2080 \u00d7 N\u2080) \u2243 (M\u2081 \u00d7 N\u2081))\n             {P\u2080 : M\u2080 \u00d7 N\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u00d7 N\u2081 \u2192 Proto}\n             (P= : \u2200 m,n\u2080 \u2192 P\u2080 m,n\u2080 \u2261 P\u2081 (\u2013> M\u00d7N\u2243 m,n\u2080))\n             {{_ : UA}} where\n        com\u2082\u2243 : com\u2082 io P\u2080 \u2261 com\u2082 io P\u2081\n        com\u2082\u2243 = {!com=!}\n    -}\n\n    send= = com= {Out} refl\n    send\u2243 = com\u2243 {Out} refl\n    recv= = com= {In} refl\n    recv\u2243 = com\u2243 {In} refl\n\n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= refl refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\npattern recvE M P = com In  {M} P\npattern sendE M P = com Out {M} P\n\nrecv\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nrecv\u2610 P = recv (\u03bb { [ m ] \u2192 P m })\n\nsend\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nsend\u2610 P = send (\u03bb { [ m ] \u2192 P m })\n\n{-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (send P) = recv \u03bb m \u2192 dual (P m)\ndual (recv P) = send \u03bb m \u2192 dual (P m)\n-}\n\ndual : Proto \u2192 Proto\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nsource-of : Proto \u2192 Proto\nsource-of end       = end\nsource-of (com _ P) = send \u03bb m \u2192 source-of (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end   : IsSource end\n  send' : \u2200 {M P} (PT : (m : M) \u2192 IsSource (P m)) \u2192 IsSource (send P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end   : IsSink end\n  recv' : \u2200 {M P} (PT : (m : M) \u2192 IsSink (P m)) \u2192 IsSink (recv P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 io {M P} (P\u2610 : \u2200 (m : \u2610 M) \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com io P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\n\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\nEnd-equiv : End \u2243 \ud835\udfd9\nEnd-equiv = equiv {\u2080} _ _ (\u03bb _ \u2192 refl) (\u03bb _ \u2192 refl)\n\nEnd-uniq : {{_ : UA}} \u2192 End \u2261 \ud835\udfd9\nEnd-uniq = ua End-equiv\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113)(P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end      \u27e7 = End\n\u27e6 com io P \u27e7 = \u27e6 io \u27e7\u1d35\u1d3c _ \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end         = _\nsink (com _ P) x = sink (P x)\n\ntelecom : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntelecom end      _       _       = _\ntelecom (recv P) p       (m , q) = m , telecom (P m) (p m) q\ntelecom (send P) (m , p) q       = m , telecom (P m) p (q m)\n\nsend\u2032 : \u2605 \u2192 Proto \u2192 Proto\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto \u2192 Proto\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 P \u2192 dual (dual P) \u2261 P\n    dual-involutive end        = refl\n    dual-involutive (com io P) = com= (dual\u1d35\u1d3c-involutive io) refl \u03bb m \u2192 dual-involutive (P m)\n\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n\n    source-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261 source-of P\n    source-of-idempotent end       = refl\n    source-of-idempotent (com _ P) = com= refl refl \u03bb m \u2192 source-of-idempotent (P m)\n\n    source-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261 source-of P\n    source-of-dual-oblivious end       = refl\n    source-of-dual-oblivious (com _ P) = com= refl refl \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P s \u2192 sink P \u2261 s\n    sink-contr end       s = refl\n    sink-contr (com _ P) s = \u03bb= \u03bb m \u2192 sink-contr (P m) (s m)\n\n    Sink-is-contr : \u2200 P \u2192 Is-contr (Sink P)\n    Sink-is-contr P = sink P , sink-contr P\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (Sink-is-contr P)\n\n    dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n    dual-Log = ap \u27e6_\u27e7 \u2218 source-of-dual-oblivious\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Data.LR\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; !_; _\u2219_; refl; subst; J; ap; coe; coe!; J-orig; _\u2262_)\n\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nmodule Control.Protocol where\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual\u1d35\u1d3c-equiv : Is-equiv dual\u1d35\u1d3c\ndual\u1d35\u1d3c-equiv = self-inv-is-equiv dual\u1d35\u1d3c-involutive\n\ndual\u1d35\u1d3c-inj : \u2200 {x y} \u2192 dual\u1d35\u1d3c x \u2261 dual\u1d35\u1d3c y \u2192 x \u2261 y\ndual\u1d35\u1d3c-inj = Is-equiv.injective dual\u1d35\u1d3c-equiv\n\n{-\nmodule UniversalProtocols \u2113 {U : \u2605_(\u209b \u2113)}(U\u27e6_\u27e7 : U \u2192 \u2605_ \u2113) where\n-}\nmodule _ \u2113 where\n  U = \u2605_ \u2113\n  U\u27e6_\u27e7 = id\n  data Proto_ : \u2605_(\u209b \u2113) where\n    end : Proto_\n    com : (io : InOut){M : U}(P : U\u27e6 M \u27e7 \u2192 Proto_) \u2192 Proto_\n{-\nmodule U\u2605 \u2113 = UniversalProtocols \u2113 {\u2605_ \u2113} id\nopen U\u2605\n-}\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 io {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n                {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 com io P\u2080 \u2261 com io P\u2081\n    com= io refl P= = ap (com io) (\u03bb= P=)\n\n    module _ io {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io P\u2080 \u2261 com io P\u2081\n        com\u2243 = com= io (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    module _ io {M N}(P : M \u00d7 N \u2192 Proto)\n             where\n        com\u2082 : Proto\n        com\u2082 = com io \u03bb m \u2192 com io \u03bb n \u2192 P (m , n)\n\n    {- Proving this would be awesome...\n    module _ io\n             {M\u2080 M\u2081 N\u2080 N\u2081 : \u2605}\n             (M\u00d7N\u2243 : (M\u2080 \u00d7 N\u2080) \u2243 (M\u2081 \u00d7 N\u2081))\n             {P\u2080 : M\u2080 \u00d7 N\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u00d7 N\u2081 \u2192 Proto}\n             (P= : \u2200 m,n\u2080 \u2192 P\u2080 m,n\u2080 \u2261 P\u2081 (\u2013> M\u00d7N\u2243 m,n\u2080))\n             {{_ : UA}} where\n        com\u2082\u2243 : com\u2082 io P\u2080 \u2261 com\u2082 io P\u2081\n        com\u2082\u2243 = {!com=!}\n    -}\n\n    -- send= : \u2200 {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081){P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080)) \u2192 send P\u2080 \u2261 send P\u2081\n    send= = com= Out\n    send\u2243 = com\u2243 Out\n\n    -- recv= : \u2200 {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081){P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080)) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv= = com= In\n    recv\u2243 = com\u2243 In\n\n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= io refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\npattern recvE M P = com In  {M} P\npattern sendE M P = com Out {M} P\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {io\u2080 io\u2081} (io : io\u2080 \u2248\u1d35\u1d3c io\u2081){M P Q} \u2192 (\u2200 (m : M) \u2192 P m \u2248\u1d3e Q m) \u2192 com io\u2080 P \u2248\u1d3e com io\u2081 Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : (m : M) \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (send P) (send Q) (com refl p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : (m : M) \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (recv P) (recv Q) (com refl p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _) = \u2261-\u03a3 _\n\nrecv\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nrecv\u2610 P = recv (\u03bb { [ m ] \u2192 P m })\n\nsend\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nsend\u2610 P = send (\u03bb { [ m ] \u2192 P m })\n\nsource-of : Proto \u2192 Proto\nsource-of end       = end\nsource-of (com _ P) = send \u03bb m \u2192 source-of (P m)\n\n{-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (send P) = recv \u03bb m \u2192 dual (P m)\ndual (recv P) = send \u03bb m \u2192 dual (P m)\n-}\n\ndual : Proto \u2192 Proto\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 {M P} (PT : (m : M) \u2192 IsSource (P m)) \u2192 IsSource (send P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 {M P} (PT : (m : M) \u2192 IsSink (P m)) \u2192 IsSink (recv P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q {M P} (P\u2610 : \u2200 (m : \u2610 M) \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com q P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\n\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\nmodule _ {{_ : UA}} where\n    End-uniq : End \u2261 \ud835\udfd9\n    End-uniq = ua (equiv _ _ (\u03bb _ \u2192 refl) (\u03bb _ \u2192 refl))\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113)(P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end     \u27e7 = End\n\u27e6 com q P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c _ \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u211b\u27e6_\u27e7 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u211b\u27e6 end    \u27e7 p q = End\n\u211b\u27e6 recv P \u27e7 p q = \u2200 m \u2192 \u211b\u27e6 P m \u27e7 (p m) (q m)\n\u211b\u27e6 send P \u27e7 p q = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u211b\u27e6 P (fst q) \u27e7 (subst (\u27e6_\u27e7 \u2218 P) e (snd p)) (snd q)\n\n\u211b\u27e6_\u27e7-refl : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Reflexive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-refl     = end\n\u211b\u27e6 recv P \u27e7-refl     = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-refl\n\u211b\u27e6 send P \u27e7-refl {x} = refl , \u211b\u27e6 P (fst x) \u27e7-refl\n\n\u211b\u27e6_\u27e7-sym : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Symmetric \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-sym p          = end\n\u211b\u27e6 recv P \u27e7-sym p          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-sym (p m)\n\u211b\u27e6 send P \u27e7-sym (refl , q) = refl , \u211b\u27e6 P _ \u27e7-sym q    -- TODO HoTT\n\n\u211b\u27e6_\u27e7-trans : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Transitive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-trans p          q          = end\n\u211b\u27e6 recv P \u27e7-trans p          q          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-trans (p m) (q m)\n\u211b\u27e6 send P \u27e7-trans (refl , p) (refl , q) = refl , \u211b\u27e6 P _ \u27e7-trans p q    -- TODO HoTT\n\nsend\u2032 : \u2605 \u2192 Proto \u2192 Proto\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto \u2192 Proto\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto){{_ : FunExt}}{{_ : UA}} where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n    P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n    P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n    P\u22a4-uniq = \u03a0\ud835\udfd8-uniq _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end       = end\n\u2261\u1d3e-refl (com q P) = com refl \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-sym : Symmetric _\u2261\u1d3e_\n\u2261\u1d3e-sym end = end\n\u2261\u1d3e-sym (com refl r) = com refl \u03bb m \u2192 \u2261\u1d3e-sym (r m)\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl x) (com refl x\u2081) = com refl (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n!\u1d3e = \u2261\u1d3e-sym\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end       = end\ndual-involutive (com q P) = com (dual\u1d35\u1d3c-involutive q) \u03bb m \u2192 dual-involutive (P m)\n\nmodule _ {{_ : FunExt}} where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end            = refl\n    \u2261\u1d3e-sound (com refl P\u2261Q) = ap (com _) (\u03bb= \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (ap f (\u2261\u1d3e-sound P\u2261Q))\n\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv (\u2261\u1d3e-sound \u2218 dual-involutive)\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end       = end\nsource-of-idempotent (com _ P) = com refl \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end       = end\nsource-of-dual-oblivious (com _ P) = com refl \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end         = _\nsink (com _ P) x = sink (P x)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P s \u2192 sink P \u2261 s\n    sink-contr end       s = refl\n    sink-contr (com _ P) s = \u03bb= \u03bb m \u2192 sink-contr (P m) (s m)\n\n    Sink-is-contr : \u2200 P \u2192 Is-contr (Sink P)\n    Sink-is-contr P = sink P , sink-contr P\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (Sink-is-contr P)\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end       = end\n>>=-right-unit (com q P) = com refl \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end       Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com q P) Q R = com refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    extend-send : Proto \u2192 Proto\n    extend-send end      = end\n    extend-send (send P) = send [inl: (\u03bb m \u2192 extend-send (P m)) ,inr: A\u1d3e ]\n    extend-send (recv P) = recv \u03bb m \u2192 extend-send (P m)\n\n    extend-recv : Proto \u2192 Proto\n    extend-recv end      = end\n    extend-recv (recv P) = recv [inl: (\u03bb m \u2192 extend-recv (P m)) ,inr: A\u1d3e ]\n    extend-recv (send P) = send \u03bb m \u2192 extend-recv (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-extend-send : \u2200 P \u2192 dual (extend-send A\u1d3e P) \u2261\u1d3e extend-recv (dual \u2218 A\u1d3e) (dual P)\n    dual-extend-send end      = end\n    dual-extend-send (recv P) = com refl (\u03bb m \u2192 dual-extend-send (P m))\n    dual-extend-send (send P) = com refl [inl: (\u03bb m \u2192 dual-extend-send (P m))\n                                         ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = extend-send Abort\u1d3e\n\ntelecom : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntelecom end      _       _       = _\ntelecom (recv P) p       (m , q) = m , telecom (P m) (p m) q\ntelecom (send P) (m , p) q       = m , telecom (P m) p (q m)\n\nlift\u1d3e : \u2200 a {\u2113} \u2192 Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\nlift\u1d3e a end        = end\nlift\u1d3e a (com io P) = com io \u03bb m \u2192 lift\u1d3e a (P (lower {\u2113 = a} m))\n\nlift-proc : \u2200 a {\u2113} (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e a P \u27e7\nlift-proc a {\u2113} end      end     = end\nlift-proc a {\u2113} (send P) (m , p) = lift m , lift-proc a (P m) p\nlift-proc a {\u2113} (recv P) p       = \u03bb { (lift m) \u2192 lift-proc _ (P m) (p m) }\n\nmodule MonomorphicSky (P : Proto\u2080) where\n  Cloud : Proto\u2080\n  Cloud = recv \u03bb (t   : \u27e6 P  \u27e7) \u2192\n          recv \u03bb (u   : \u27e6 P \u22a5\u27e7) \u2192\n          send \u03bb (log : Log P)  \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud t u = telecom P t u , _\n\nmodule PolySky where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P : Proto\u2080) \u2192\n          lift\u1d3e \u2081 (MonomorphicSky.Cloud P)\n  cloud : \u27e6 Cloud \u27e7\n  cloud P = lift-proc \u2081 (MonomorphicSky.Cloud P) (MonomorphicSky.cloud P)\n\nmodule PolySky' where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P   : Proto\u2080) \u2192\n          recv \u03bb (t   : Lift \u27e6 P  \u27e7)  \u2192\n          recv \u03bb (u   : Lift \u27e6 P \u22a5\u27e7)  \u2192\n          send \u03bb (log : Lift (Log P)) \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud P (lift t) (lift u) = lift (telecom P t u) , _\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I)(M : \u2605)(\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = case p A\n                               0: case p B\n                                    0: recv (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: recv (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: send (\u03bb m \u2192 \u2102 m \/\/ p)\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com (case p A 0: In 1: Out) \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (send P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (recv P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule _ {{_ : FunExt}} where\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    P2 = send\u2032 \ud835\udfda end\n\n    0\u2082\u22621\u2082 : 0\u2082 \u2262 1\u2082\n    0\u2082\u22621\u2082 ()\n\n    \ud835\udfd8\u2262 : \u2200 {A} (x : A) \u2192 \ud835\udfd8 \u2262 A\n    \ud835\udfd8\u2262 x e = coe! e x\n\n    \ud835\udfd8\u2262\ud835\udfd9 : \ud835\udfd8 \u2262 \ud835\udfd9\n    \ud835\udfd8\u2262\ud835\udfd9 = \ud835\udfd8\u2262 _\n\n    \ud835\udfd8\u2262\ud835\udfda : \ud835\udfd8 \u2262 \ud835\udfda\n    \ud835\udfd8\u2262\ud835\udfda = \ud835\udfd8\u2262 0\u2082\n\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = send \u03bb (q : Query) \u2192 recv \u03bb (r : Resp q) \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = recv \u03bb (q : Query) \u2192 send \u03bb (r : Resp q) \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = send \u03bb (q : Query) \u2192 recv \u03bb (r : Resp q) \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = recv \u03bb n \u2192\n                    Server n\n    StaticServer  = send \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ {{_ : FunExt}} where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = ap \u27e6_\u27e7 (\u2261\u1d3e-sound (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end      Q = \u2261\u1d3e-refl _\ndual->> (recv P) Q = com refl \u03bb m \u2192 dual->> (P m) Q\ndual->> (send P) Q = com refl \u03bb m \u2192 dual->> (P m) Q\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (recv M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (send M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _ {{_ : FunExt}} (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} where\n    dual-\u2295 : \u2200 {P Q} \u2192 dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : \u2200 {P Q} \u2192 dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    &-comm : \u2200 P Q \u2192 P & Q \u2261 Q & P\n    &-comm P Q = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : \u2200 P Q \u2192 P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm P Q = send\u2243 LR!-equiv [L: refl R: refl ]\n\nmodule _ {P Q R S} where\n    \u2295-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map f g (`L , pr) = `L , f pr\n    \u2295-map f g (`R , pr) = `R , g pr\n\n    &-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map f g p `L = f (p `L)\n    &-map f g p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 (inl p) = `L , p\n    \u228e\u2192\u2295 (inr q) = `R , q\n\n    \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n    \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n    \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2192\u228e\u2192\u2295 : \u2200 x \u2192 \u2295\u2192\u228e (\u228e\u2192\u2295 x) \u2261 x\n    \u2295\u2192\u228e\u2192\u2295 (inl _) = refl\n    \u2295\u2192\u228e\u2192\u2295 (inr _) = refl\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = \u2295\u2192\u228e , record { linv = \u228e\u2192\u2295 ; is-linv = \u228e\u2192\u2295\u2192\u228e ; rinv = \u228e\u2192\u2295 ; is-rinv = \u2295\u2192\u228e\u2192\u2295 }\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = &\u2192\u00d7 , record { linv = \u00d7\u2192& ; is-linv = \u00d7\u2192&\u2192\u00d7 ; rinv = \u00d7\u2192& ; is-rinv = &\u2192\u00d7\u2192& }\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u2200 P \u2192 \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& P = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u2200 P \u2192 \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 P = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\n    &-assoc : \u2200 P Q R \u2192 \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc P Q R = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u2200 P Q R \u2192 \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc P Q R = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n\nmodule _ where\n\n  _\u214b_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b Q      = Q\n  recv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\n  P      \u214b end    = P\n  P      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\n  send P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                         ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    -- absorption\n    \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n    \u22a4-\u214b P = \u03bb()\n\n  _\u2297_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297 Q      = Q\n  send P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\n  P      \u2297 end    = P\n  P      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\n  recv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                         ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\n  _-o_ : (P Q : Proto) \u2192 Proto\n  P -o Q = dual P \u214b Q\n\n  _o-o_ : (P Q : Proto) \u2192 Proto\n  P o-o Q = (P -o Q) \u2297 (Q -o P)\n\n  module _ {{_ : FunExt}} where\n    \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n    \u2297-endR end      = refl\n    \u2297-endR (recv _) = refl\n    \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n    \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n    \u214b-endR end      = refl\n    \u214b-endR (send _) = refl\n    \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n    \u2297-sendR end      Q = refl\n    \u2297-sendR (recv P) Q = refl\n    \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n    \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n    \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n    \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n    \u2297-comm (recv P) end      = refl\n    \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n    \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                               ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n    \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n    \u2297-assoc end      Q        R        = refl\n    \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n    \u2297-assoc (recv P) end      R        = refl\n    \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n    \u2297-assoc (recv P) (recv Q) end      = refl\n    \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n    \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                             \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                               ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                               ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n\n    \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n    \u214b-recvR end      Q = refl\n    \u214b-recvR (send P) Q = refl\n    \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n    \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n    \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n    \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n    \u214b-comm (send P) end      = refl\n    \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n    \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                               ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n    \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n    \u214b-assoc end      Q        R        = refl\n    \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n    \u214b-assoc (send P) end      R        = refl\n    \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n    \u214b-assoc (send P) (send Q) end      = refl\n    \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n    \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                             \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                               ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                               ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\n  module _ {P Q R}{{_ : FunExt}} where\n    dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n    dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n    dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n    dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n    module _ {{_ : UA}} where\n        dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n        dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                 \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                 \u2219 \u2295\u2261\u228e\n                 \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                 \u2219 ! \u2295\u2261\u228e\n\n        dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n        dist-\u214b-& = \u214b-comm P (Q & R)\n                 \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                 \u2219 &\u2261\u00d7\n                 \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                 \u2219 ! &\u2261\u00d7\n\n  -- P \u27e6\u2297\u27e7 Q \u2243 \u27e6 P \u2297 Q \u27e7\n  -- but potentially more convenient\n  _\u27e6\u2297\u27e7_ : Proto \u2192 Proto \u2192 \u2605\n  end    \u27e6\u2297\u27e7 Q      = \u27e6 Q \u27e7\n  send P \u27e6\u2297\u27e7 Q      = \u2203 \u03bb m \u2192 P m \u27e6\u2297\u27e7 Q\n  P      \u27e6\u2297\u27e7 end    = \u27e6 P \u27e7\n  P      \u27e6\u2297\u27e7 send Q = \u2203 \u03bb m \u2192 P \u27e6\u2297\u27e7 Q m\n  recv P \u27e6\u2297\u27e7 recv Q = (\u03a0 _ \u03bb m \u2192 P m    \u27e6\u2297\u27e7 recv Q)\n                    \u00d7 (\u03a0 _ \u03bb n \u2192 recv P \u27e6\u2297\u27e7 Q n)\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    \u27e6\u2297\u27e7-correct : \u2200 P Q \u2192 P \u27e6\u2297\u27e7 Q \u2261 \u27e6 P \u2297 Q \u27e7\n    \u27e6\u2297\u27e7-correct end      Q        = refl\n    \u27e6\u2297\u27e7-correct (send P) Q        = \u03a3=\u2032 _ \u03bb m \u2192 \u27e6\u2297\u27e7-correct (P m) Q\n    \u27e6\u2297\u27e7-correct (recv P) end      = refl\n    \u27e6\u2297\u27e7-correct (recv P) (send Q) = \u03a3=\u2032 _ \u03bb n \u2192 \u27e6\u2297\u27e7-correct (recv P) (Q n)\n    \u27e6\u2297\u27e7-correct (recv P) (recv Q) = ! dist-\u00d7-\u03a0\n                                  \u2219 \u03a0=\u2032 (_ \u228e _) \u03bb { (inl m)  \u2192 \u27e6\u2297\u27e7-correct (P m) (recv Q)\n                                                  ; (inr n) \u2192 \u27e6\u2297\u27e7-correct (recv P) (Q n) }\n\n  -- an alternative, potentially more convenient\n  _\u27e6\u214b\u27e7_ : Proto \u2192 Proto \u2192 \u2605\n  end    \u27e6\u214b\u27e7 Q      = \u27e6 Q \u27e7\n  recv P \u27e6\u214b\u27e7 Q      = \u2200 m \u2192 P m \u27e6\u214b\u27e7 Q\n  P      \u27e6\u214b\u27e7 end    = \u27e6 P \u27e7\n  P      \u27e6\u214b\u27e7 recv Q = \u2200 n \u2192 P \u27e6\u214b\u27e7 Q n\n  send P \u27e6\u214b\u27e7 send Q = (\u2203 \u03bb m \u2192 P m    \u27e6\u214b\u27e7 send Q)\n                    \u228e (\u2203 \u03bb n \u2192 send P \u27e6\u214b\u27e7 Q n)\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    \u27e6\u214b\u27e7-correct : \u2200 P Q \u2192 P \u27e6\u214b\u27e7 Q \u2261 \u27e6 P \u214b Q \u27e7\n    \u27e6\u214b\u27e7-correct end      Q        = refl\n    \u27e6\u214b\u27e7-correct (recv P) Q        = \u03a0=\u2032 _ \u03bb m \u2192 \u27e6\u214b\u27e7-correct (P m) Q\n    \u27e6\u214b\u27e7-correct (send P) end      = refl\n    \u27e6\u214b\u27e7-correct (send P) (recv Q) = \u03a0=\u2032 _ \u03bb n \u2192 \u27e6\u214b\u27e7-correct (send P) (Q n)\n    \u27e6\u214b\u27e7-correct (send P) (send Q) = ! dist-\u228e-\u03a3\n                                  \u2219 \u03a3=\u2032 (_ \u228e _) \u03bb { (inl m) \u2192 \u27e6\u214b\u27e7-correct (P m) (send Q)\n                                                  ; (inr n) \u2192 \u27e6\u214b\u27e7-correct (send P) (Q n) }\n\n                                                  {-\n    -- sends can be pulled out of tensor\n    source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n    source->>=-\u2297 end       Q R = refl\n    source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n    -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n    -- recvs can be pulled out of par\n    sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n    sink->>=-\u214b end       Q R = refl\n    sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n    -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\n  Log-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\n  Log-\u214b-\u00d7 {end}   {Q}      q           = end , q\n  Log-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\n  Log-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\n  Log-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n  Log-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\n  Log-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\n  module _ {{_ : FunExt}} where\n    \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n    \u2297\u214b-dual end Q = refl\n    \u2297\u214b-dual (recv P) Q = com=\u2032 _ \u03bb m \u2192 \u2297\u214b-dual (P m) _\n    \u2297\u214b-dual (send P) end = refl\n    \u2297\u214b-dual (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)\n    \u2297\u214b-dual (send P) (send Q) = com=\u2032 _\n      [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (send Q))\n      ,inr: (\u03bb n \u2192 \u2297\u214b-dual (send P) (Q n))\n      ]\n\n  data View-\u214b-proto : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    end-X     : \u2200 Q \u2192 View-\u214b-proto end Q\n    recv-X    : \u2200 {M}(P : M \u2192 Proto)Q \u2192 View-\u214b-proto (recv P) Q\n    send-send : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u214b-proto (send P) (send Q)\n    send-recv : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u214b-proto (send P) (recv Q)\n    send-end  : \u2200 {M}(P : M \u2192 Proto) \u2192 View-\u214b-proto (send P) end\n\n  view-\u214b-proto : \u2200 P Q \u2192 View-\u214b-proto P Q\n  view-\u214b-proto end      Q        = end-X Q\n  view-\u214b-proto (recv P) Q        = recv-X P Q\n  view-\u214b-proto (send P) end      = send-end P\n  view-\u214b-proto (send P) (recv Q) = send-recv P Q\n  view-\u214b-proto (send P) (send Q) = send-send P Q\n\n  data View-\u2297-proto : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    end-X     : \u2200 Q \u2192 View-\u2297-proto end Q\n    send-X    : \u2200 {M}(P : M \u2192 Proto)Q \u2192 View-\u2297-proto (send P) Q\n    recv-recv : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u2297-proto (recv P) (recv Q)\n    recv-send : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u2297-proto (recv P) (send Q)\n    recv-end  : \u2200 {M}(P : M \u2192 Proto) \u2192 View-\u2297-proto (recv P) end\n\n  view-\u2297-proto : \u2200 P Q \u2192 View-\u2297-proto P Q\n  view-\u2297-proto end      Q        = end-X Q\n  view-\u2297-proto (send P) Q        = send-X P Q\n  view-\u2297-proto (recv P) end      = recv-end P\n  view-\u2297-proto (recv P) (recv Q) = recv-recv P Q\n  view-\u2297-proto (recv P) (send Q) = recv-send P Q\n\n  -- the terminology used for the constructor follows the behavior of the combined process\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b send Q \u27e7) \u2192 View-\u214b (send P) (send Q) (inl m  , p)\n    sendR' : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto)(n : N)(p : \u27e6 send P \u214b Q n \u27e7) \u2192 View-\u214b (send P) (send Q) (inr n , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b Q \u27e7)) \u2192 View-\u214b (recv P) Q p\n    recvR' : \u2200 {M N} (P : M \u2192 Proto) (Q : N \u2192 Proto)(p : (n : N) \u2192 \u27e6 send P \u214b Q n \u27e7) \u2192 View-\u214b (send P) (recv Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    send'  : \u2200 {M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 View-\u214b (send P) end (m , p)\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (recv P) Q p = recvL' P Q p\n  view-\u214b (send P) end (m , p) = send' P m p\n  view-\u214b (send P) (recv Q) p = recvR' P Q p\n  view-\u214b (send P) (send Q) (inl x , p) = sendL' P Q x p\n  view-\u214b (send P) (send Q) (inr y , p) = sendR' P Q y p\n\n  {-\n  -- use coe (... \u214b-assoc P Q R)\n  \u214b-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b (Q \u214b R) \u27e7 \u2192 \u27e6 (P \u214b Q) \u214b R \u27e7\n  \u214b-assoc end      Q        R         s                 = s\n  \u214b-assoc (recv P) Q        R         s m               = \u214b-assoc (P m) _ _ (s m)\n  \u214b-assoc (send P) end      R         s                 = s\n  \u214b-assoc (send P) (recv Q) R         s m               = \u214b-assoc (send P) (Q m) _ (s m)\n  \u214b-assoc (send P) (send Q) end       s                 = s\n  \u214b-assoc (send P) (send Q) (recv R)  s m               = \u214b-assoc (send P) (send Q) (R m) (s m)\n  \u214b-assoc (send P) (send Q) (send R) (inl m , s)       = inl (inl m) , \u214b-assoc (P m) (send Q) (send R) s\n  \u214b-assoc (send P) (send Q) (send R) (inr (inl m) , s) = inl (inr m) , \u214b-assoc (send P) (Q m) (send R) s\n  \u214b-assoc (send P) (send Q) (send R) (inr (inr m) , s) = inr m       , \u214b-assoc (send P) (send Q) (R m) s\n\n  -- use coe (\u214b-endR P) instead\n  \u214b-rend : \u2200 P \u2192 \u27e6 P \u214b end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b-rend end      p = p\n  \u214b-rend (send _) p = p\n  \u214b-rend (recv P) p = \u03bb m \u2192 \u214b-rend (P m) (p m)\n\n  -- use coe! (\u214b-endR P) instead\n  \u214b-rend! : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b end \u27e7\n  \u214b-rend! end      p = p\n  \u214b-rend! (send _) p = p\n  \u214b-rend! (recv P) p = \u03bb m \u2192 \u214b-rend! (P m) (p m)\n\n  -- use coe! (\u214b-recvR P Q) instead\n  \u214b-isendR : \u2200 {N} P Q \u2192 \u27e6 P \u214b recv Q \u27e7 \u2192 (n : N) \u2192 \u27e6 P \u214b Q n \u27e7\n  \u214b-isendR end Q s n = s n\n  \u214b-isendR (recv P) Q s n = \u03bb m \u2192 \u214b-isendR (P m) Q (s m) n\n  \u214b-isendR (send P) Q s n = s n\n\n\n  -- see \u214b-recvR\n  \u214b-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b Q m \u27e7) \u2192 \u27e6 P \u214b recv Q \u27e7\n  \u214b-recvR end      Q s = s\n  \u214b-recvR (recv P) Q s = \u03bb x \u2192 \u214b-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b-recvR (send P) Q s = s\n  -}\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n\n    \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n    \u214b-sendR end      m p = m , p\n    \u214b-sendR (send P) m p = inr m , p\n    \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n    \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n    \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n    \u214b-id : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n    \u214b-id end      = end\n    \u214b-id (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (\u214b-id (P x))\n    \u214b-id (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (\u214b-id (P x))\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R (m : M)(p : \u27e6 P m \u214b send Q \u27e7)(q : \u27e6 dual (send Q) \u214b R \u27e7)\n             \u2192 View-\u2218 (send P) (send Q) R (inl m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b Q \u27e7))(q : \u27e6 dual Q \u214b R \u27e7)\n             \u2192 View-\u2218 (recv P) Q R p q\n    recvR-sendR : \u2200 {MP MQ MR}ioP(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (mR : MR)(p : \u27e6 com ioP P \u214b recv Q \u27e7)(q : \u27e6 dual (recv Q) \u214b R mR \u27e7)\n                    \u2192 View-\u2218 (com ioP P) (recv Q) (send R) p (inr mR , q)\n\n    recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n               (p : \u27e6 send P \u214b recv Q \u27e7)(q : (m : MR) \u2192 \u27e6 dual (recv Q) \u214b R m \u27e7)\n             \u2192 View-\u2218 (send P) (recv Q) (recv R) p q\n    sendR-recvL : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)R(m : MQ)\n                    (p : \u27e6 send P \u214b Q m \u27e7)(q : (m : MQ) \u2192 \u27e6 dual (Q m) \u214b R \u27e7)\n                  \u2192 View-\u2218 (send P) (send Q) R (inr m , p) q\n    recvR-sendL : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (p : (m : MQ) \u2192 \u27e6 send P \u214b Q m \u27e7)(m : MQ)(q : \u27e6 dual (Q m) \u214b send R \u27e7)\n                  \u2192 View-\u2218 (send P) (recv Q) (send R) p (inl m , q)\n    endL : \u2200 Q R\n           \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b R \u27e7)\n           \u2192 View-\u2218 end Q R q qr\n    sendLM : \u2200 {MP}(P : MP \u2192 Proto)R\n               (m : MP)(p : \u27e6 P m \u27e7)(r : \u27e6 R \u27e7)\n             \u2192 View-\u2218 (send P) end R (m , p) r\n    recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                    (m : MQ)(p : \u2200 m \u2192 \u27e6 send P \u214b Q m \u27e7)(q : \u27e6 dual (Q m) \u27e7)\n                  \u2192 View-\u2218 (send P) (recv Q) end p (m , q)\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7)(qr : \u27e6 dual Q \u214b R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b Q \u27e7}{qr : \u27e6 dual Q \u214b R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) _                 = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = recvR-sendR _ _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (send' ._ _ _)    = recvL-sendR _ _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (send' _ _ _)    _                 = sendLM _ _ _ _ _\n\n  \u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b-apply end      Q        s           p       = s\n  \u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n  \u214b-apply (send P) end      s           p       = _\n  \u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n  \u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n  \u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n  {-\n  -- see dist-\u214b-&\n  dist-\u214b-fst : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  dist-\u214b-fst (recv P) Q R p = \u03bb m \u2192 dist-\u214b-fst (P m) Q R (p m)\n  dist-\u214b-fst (send P) Q R p = p `L\n  dist-\u214b-fst end      Q R p = p `L\n\n  -- see dist-\u214b-&\n  dist-\u214b-snd : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n  dist-\u214b-snd (recv P) Q R p = \u03bb m \u2192 dist-\u214b-snd (P m) Q R (p m)\n  dist-\u214b-snd (send P) Q R p = p `R\n  dist-\u214b-snd end      Q R p = p `R\n\n  -- see dist-\u214b-&\n  dist-\u214b-\u00d7 : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 P \u214b Q \u27e7 \u00d7 \u27e6 P \u214b R \u27e7\n  dist-\u214b-\u00d7 P Q R p = dist-\u214b-fst P Q R p , dist-\u214b-snd P Q R p\n\n  -- see dist-\u214b-&\n  dist-\u214b-& : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n  dist-\u214b-& P Q R p = \u00d7\u2192& (dist-\u214b-\u00d7 P Q R p)\n\n  -- see dist-\u214b-&\n  factor-,-\u214b : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 P \u214b R \u27e7 \u2192 \u27e6 P \u214b (Q & R) \u27e7\n  factor-,-\u214b end      Q R pq pr = \u00d7\u2192& (pq , pr)\n  factor-,-\u214b (recv P) Q R pq pr = \u03bb m \u2192 factor-,-\u214b (P m) Q R (pq m) (pr m)\n  factor-,-\u214b (send P) Q R pq pr = [L: pq R: pr ]\n\n  -- see dist-\u214b-&\n  factor-\u00d7-\u214b : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u00d7 \u27e6 P \u214b R \u27e7 \u2192 \u27e6 P \u214b (Q & R) \u27e7\n  factor-\u00d7-\u214b P Q R (p , q) = factor-,-\u214b P Q R p q\n\n  -- see dist-\u214b-&\n  factor-&-\u214b : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7 \u2192 \u27e6 P \u214b (Q & R) \u27e7\n  factor-&-\u214b P Q R p = factor-\u00d7-\u214b P Q R (&\u2192\u00d7 p)\n\n  -- see dist-\u214b-&\n  module _ {{_ : FunExt}} where\n    dist-\u214b-fst-factor-&-, : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7)(pr : \u27e6 P \u214b R \u27e7)\n                            \u2192 dist-\u214b-fst P Q R (factor-,-\u214b P Q R pq pr) \u2261 pq\n    dist-\u214b-fst-factor-&-, (recv P) Q R pq pr = \u03bb= \u03bb m \u2192 dist-\u214b-fst-factor-&-, (P m) Q R (pq m) (pr m)\n    dist-\u214b-fst-factor-&-, (send P) Q R pq pr = refl\n    dist-\u214b-fst-factor-&-, end      Q R pq pr = refl\n\n    dist-\u214b-snd-factor-&-, : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7)(pr : \u27e6 P \u214b R \u27e7)\n                            \u2192 dist-\u214b-snd P Q R (factor-,-\u214b P Q R pq pr) \u2261 pr\n    dist-\u214b-snd-factor-&-, (recv P) Q R pq pr = \u03bb= \u03bb m \u2192 dist-\u214b-snd-factor-&-, (P m) Q R (pq m) (pr m)\n    dist-\u214b-snd-factor-&-, (send P) Q R pq pr = refl\n    dist-\u214b-snd-factor-&-, end      Q R pq pr = refl\n\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 : \u2200 P Q R \u2192 (factor-\u00d7-\u214b P Q R) LeftInverseOf (dist-\u214b-\u00d7 P Q R)\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (recv P) Q R p = \u03bb= \u03bb m \u2192 factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (P m) Q R (p m)\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (send P) Q R p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 end      Q R p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n    module _ P Q R where\n        factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 : (factor-\u00d7-\u214b P Q R) RightInverseOf (dist-\u214b-\u00d7 P Q R)\n        factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 (x , y) = pair\u00d7= (dist-\u214b-fst-factor-&-, P Q R x y) (dist-\u214b-snd-factor-&-, P Q R x y)\n\n        dist-\u214b-\u00d7-\u2243 : \u27e6 P \u214b (Q & R) \u27e7 \u2243 (\u27e6 P \u214b Q \u27e7 \u00d7 \u27e6 P \u214b R \u27e7)\n        dist-\u214b-\u00d7-\u2243 = dist-\u214b-\u00d7 P Q R\n                   , record { linv = factor-\u00d7-\u214b P Q R; is-linv = factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 P Q R\n                            ; rinv = factor-\u00d7-\u214b P Q R; is-rinv = factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 }\n\n        dist-\u214b-&-\u2243 : \u27e6 P \u214b (Q & R) \u27e7 \u2243 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n        dist-\u214b-&-\u2243 = dist-\u214b-\u00d7-\u2243 \u2243-\u2219 \u2243-! &\u2243\u00d7\n  -}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u2297-pair {end}    {Q}      p q       = q\n  \u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n  \u2297-pair {recv P} {end}    p end     = p\n  \u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n  \u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                       ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n  \u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297-fst end      Q        pq       = _\n  \u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n  \u2297-fst (recv P) end      pq       = pq\n  \u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n  \u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n  \u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n  \u2297-snd end      Q        pq       = pq\n  \u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n  \u2297-snd (recv P) end      pq       = end\n  \u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n  \u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\n  \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n  \u2297-pair-fst end      Q        p q       = refl\n  \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n  \u2297-pair-fst (recv P) end      p q       = refl\n  \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n  \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n  \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n  \u2297-pair-snd end      Q        p q       = refl\n  \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n  \u2297-pair-snd (recv P) end      p q       = refl\n  \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n  \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\n  module _ P Q where\n    \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n    \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n    \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n  {- WRONG\n  \u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n  \u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n  -}\n\n  -o-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  -o-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound (dual-involutive P))) p)\n\n  o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  o-o-apply P Q Po-oQ p = -o-apply {P} {Q} (\u2297-fst (P -o Q) (Q -o P) Po-oQ) p\n\n  o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n  o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n  \u214b-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n  \u214b-\u2218 P Q R pq qr = \u214b-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b Q \u27e7}{qr : \u27e6 dual Q \u214b R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218-view (sendLL P Q R m p qr)          = \u214b-sendL R m (\u214b-\u2218 (P m) (send Q) R p qr)\n    \u214b-\u2218-view (recvLL P Q R p qr)            = \u03bb m \u2192 \u214b-\u2218 (P m) Q R (p m) qr\n    \u214b-\u2218-view (recvR-sendR ioP P Q R m pq q) = \u214b-sendR (com ioP P) m (\u214b-\u2218 (com ioP P) (recv Q) (R m) pq q)\n    \u214b-\u2218-view (recvRR P Q R pq q)            = \u03bb m \u2192 \u214b-\u2218 (send P) (recv Q) (R m) pq (q m)\n    \u214b-\u2218-view (sendR-recvL P Q R m p q)      = \u214b-\u2218 (send P) (Q m) R p (q m)\n    \u214b-\u2218-view (recvR-sendL P Q R p m q)      = \u214b-\u2218 (send P) (Q m) (send R) (p m) q\n    \u214b-\u2218-view (endL Q R pq qr)               = -o-apply {Q} {R} qr pq\n    \u214b-\u2218-view (sendLM P R m pq qr)           = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218-view (recvL-sendR P Q m pq qr)      = \u214b-\u2218 (send P) (Q m) end (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n    {-\n  mutual\n    \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 Q \u214b P \u27e7\n    \u214b-comm P Q p = \u214b-comm-view (view-\u214b P Q p)\n\n    \u214b-comm-view : \u2200 {P Q} {pq : \u27e6 P \u214b Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b P \u27e7\n    \u214b-comm-view (sendL' P Q m p) = \u214b-sendR (send Q) m (\u214b-comm (P m) (send Q) p)\n    \u214b-comm-view (sendR' P Q n p) = inl n , \u214b-comm (send P) (Q n) p\n    \u214b-comm-view (recvL' P Q pq)  = \u214b-recvR Q P \u03bb m \u2192 \u214b-comm (P m) Q (pq m)\n    \u214b-comm-view (recvR' P Q pq)  = \u03bb n \u2192 \u214b-comm (send P) (Q n) (pq n)\n    \u214b-comm-view (endL Q pq)      = \u214b-rend! Q pq\n    \u214b-comm-view (send P m pq)    = m , pq\n  -}\n\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n  switchL' end      Q        R        q  r = \u2297-pair {Q} {R} q r\n  switchL' (send P) end      R        p  r = par (send P) R p r\n  switchL' (send P) (send Q) R        (inl m , pq) r = inl m , switchL' (P m) (send Q) R pq r\n  switchL' (send P) (send Q) R        (inr m , pq) r = inr m , switchL' (send P) (Q m) R pq r\n  switchL' (send P) (recv Q) end      pq r = pq\n  switchL' (send P) (recv Q) (send R) pq (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n  switchL' (send P) (recv Q) (recv R) pq r (inl m) = switchL' (send P) (Q m) (recv R) (pq m) r\n  switchL' (send P) (recv Q) (recv R) pq r (inr m) = switchL' (send P) (recv Q) (R m) pq (r m)\n  switchL' (recv P) Q R pq r = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n  -- multiplicative mix (left-biased)\n  mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\n  -- additive mix (left-biased)\n  amix : \u2200 P Q \u2192 \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n  amix P Q pq = (`L , pq `L)\n\n{-\nA `\u2297 B 'times', context chooses how A and B are used\nA `\u214b B 'par', \"we\" chooses how A and B are used\nA `\u2295 B 'plus', select from A or B\nA `& B 'with', offer choice of A or B\n`! A   'of course!', server accept\n`? A   'why not?', client request\n`1     unit for `\u2297\n`\u22a5     unit for `\u214b\n`0     unit for `\u2295\n`\u22a4     unit for `&\n-}\ndata CLL : \u2605 where\n  `1 `\u22a4 `0 `\u22a5 : CLL\n  _`\u2297_ _`\u214b_ _`\u2295_ _`&_ : (A B : CLL) \u2192 CLL\n  -- `!_ `?_ : (A : CLL) \u2192 CLL\n\n_\u22a5 : CLL \u2192 CLL\n`1 \u22a5 = `\u22a5\n`\u22a5 \u22a5 = `1\n`0 \u22a5 = `\u22a4\n`\u22a4 \u22a5 = `0\n(A `\u2297 B)\u22a5 = A \u22a5 `\u214b B \u22a5\n(A `\u214b B)\u22a5 = A \u22a5 `\u2297 B \u22a5\n(A `\u2295 B)\u22a5 = A \u22a5 `& B \u22a5\n(A `& B)\u22a5 = A \u22a5 `\u2295 B \u22a5\n{-\n(`! A)\u22a5 = `?(A \u22a5)\n(`? A)\u22a5 = `!(A \u22a5)\n-}\n\nCLL-proto : CLL \u2192 Proto\nCLL-proto `1       = end  -- TODO\nCLL-proto `\u22a5       = end  -- TODO\nCLL-proto `0       = send\u2032 \ud835\udfd8 end -- Alt: send \u03bb()\nCLL-proto `\u22a4       = recv\u2032 \ud835\udfd8 end -- Alt: recv \u03bb()\nCLL-proto (A `\u2297 B) = CLL-proto A \u2297 CLL-proto B\nCLL-proto (A `\u214b B) = CLL-proto A \u214b CLL-proto B\nCLL-proto (A `\u2295 B) = CLL-proto A \u2295 CLL-proto B\nCLL-proto (A `& B) = CLL-proto A & CLL-proto B\n\n{- The point of this could be to devise a particular equivalence\n   relation for processes. It could properly deal with \u214b. -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = send\u2610 \u03bb (s : Secret) \u2192\n             recv  \u03bb (g : Guess)  \u2192\n             send  \u03bb (_ : S\u27e8 s \u27e9) \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = send \u03bb (g\u02b3 : G)  \u2192 -- commitment\n                 recv \u03bb (c  : \u2124q) \u2192 -- challenge\n                 send \u03bb (s  : \u2124q) \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) \u2192 Proto\n        Honest-Prover x y\n          = send\u2610 \u03bb (r  : \u2124q)                \u2192 -- ideal commitment\n            send  \u03bb (g\u02b3 : S\u27e8 g ^ r \u27e9)        \u2192 -- real  commitment\n            recv  \u03bb (c  : \u2124q)                \u2192 -- challenge\n            send  \u03bb (s  : S\u27e8 r + (c * x) \u27e9)  \u2192 -- response\n            recv  \u03bb (_  : Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u2192\n            end\n\n        Honest-Verifier : ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S\u27e8 g ^ x \u27e9) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S\u27e8 g ^ x \u27e9) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"69291665aee627be442146337bc1002cde0f963f","subject":"half corrected","message":"half corrected\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"code\/GoaTest.agda","new_file":"code\/GoaTest.agda","new_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where -- finite type\n  true : Bool\n  false : Bool\n\nidBool : Bool \u2192 Bool -- lambda\nidBool x = x\n\nalwaysTrue : Bool \u2192 Bool\nalwaysTrue x = true\n\nnot : Bool \u2192 Bool -- case defn\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool -- lambda\nnotnot x = not(not(x))\n\n_&_ : Bool \u2192 Bool \u2192 Bool --curried function\ntrue & x = x\nfalse & _ = false\n\ndata \u2115 : Type where -- infinite type\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool -- recursive definition\neven zero = true\neven (succ x) = not (even x)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115 \nzero + y = y\nsucc x + y = x + succ y\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata \u2115List : Type where --list type \n  [] : \u2115List -- empty list \n  _::_ : \u2115 \u2192 \u2115List \u2192 \u2115List -- add number to head of list\n\nmylist : \u2115List \nmylist = 3 :: (4 :: (2 :: [])) -- the list [3, 4, 2]\n\ndata Vector : \u2115 \u2192 Type where -- type family\n  [] : Vector 0\n  _::_ : {n : \u2115} \u2192 \u2115 \u2192 Vector n \u2192 Vector (succ n) \n\nsum : {n : \u2115} \u2192 Vector n \u2192 \u2115\nsum [] = 0\nsum (x :: l) = x + sum l\n\n\ncountdown : (n : \u2115) \u2192 Vector n -- dependent function\ncountdown 0 = []\ncountdown (succ n) = (succ n) :: (countdown n)\n\nsumToN : \u2115 \u2192 \u2115 -- calculation\nsumToN n = sum(countdown n)\n\ndata isEven : \u2115 \u2192 Type where\n  0even : isEven 0\n  +2even : (n : \u2115) \u2192 isEven n \u2192 isEven (succ(succ(n)))\n\n4even : isEven 4\n4even = +2even _ (+2even _ 0even)\n\ndata False : Type where\n\n1odd : isEven 1 \u2192 False\n1odd ()\n\n3odd : isEven 3 \u2192 False\n3odd (+2even .1 ())\n\nhalf : (n : \u2115) \u2192 isEven n \u2192 \u2115\nhalf .0 0even = 0\nhalf .(succ (succ n)) (+2even n pf) = succ(half n pf)\n\ndouble : (n : \u2115) \u2192 \u2115\ndouble 0 = 0\ndouble (succ n) = succ(succ(double(n)))\n\nstep : (n : \u2115) \u2192 (isEven (double n)) \u2192 isEven (double(succ(n)))\nstep n pf = +2even _ pf\n\nthm : (n  : \u2115) \u2192 isEven (double n)\nthm zero = 0even\nthm (succ n) = step _ (thm n)\n\nhalfOfDouble : \u2115 \u2192 \u2115\nhalfOfDouble n = half (double n) (thm n)\n","old_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where -- finite type\n  true : Bool\n  false : Bool\n\nidBool : Bool \u2192 Bool -- lambda\nidBool x = x\n\nalwaysTrue : Bool \u2192 Bool\nalwaysTrue x = true\n\nnot : Bool \u2192 Bool -- case defn\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool -- lambda\nnotnot x = not(not(x))\n\n_&_ : Bool \u2192 Bool \u2192 Bool --curried function\ntrue & x = x\nfalse & _ = false\n\ndata \u2115 : Type where -- infinite type\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool -- recursive definition\neven zero = true\neven (succ x) = not (even x)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115 \nzero + y = y\nsucc x + y = x + succ y\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata \u2115List : Type where --list type \n  [] : \u2115List -- empty list \n  _::_ : \u2115 \u2192 \u2115List \u2192 \u2115List -- add number to head of list\n\nmylist : \u2115List \nmylist = 3 :: (4 :: (2 :: [])) -- the list [3, 4, 2]\n\ndata Vector : \u2115 \u2192 Type where -- type family\n  [] : Vector 0\n  _::_ : {n : \u2115} \u2192 \u2115 \u2192 Vector n \u2192 Vector (succ n) \n\nsum : {n : \u2115} \u2192 Vector n \u2192 \u2115\nsum [] = 0\nsum (x :: l) = x + sum l\n\n\ncountdown : (n : \u2115) \u2192 Vector n -- dependent function\ncountdown 0 = []\ncountdown (succ n) = (succ n) :: (countdown n)\n\nsumToN : \u2115 \u2192 \u2115 -- calculation\nsumToN n = sum(countdown n)\n\ndata isEven : \u2115 \u2192 Type where\n  0even : isEven 0\n  +2even : (n : \u2115) \u2192 isEven n \u2192 isEven (succ(succ(n)))\n\n4even : isEven 4\n4even = +2even _ (+2even _ 0even)\n\ndata False : Type where\n\n1odd : isEven 1 \u2192 False\n1odd ()\n\n3odd : isEven 3 \u2192 False\n3odd (+2even .1 ())\n\nhalf : (n : \u2115) \u2192 isEven n \u2192 \u2115\nhalf .0 0even = 0\nhalf .(succ (succ n)) (+2even n pf) = half n pf\n\ndouble : (n : \u2115) \u2192 \u2115\ndouble 0 = 0\ndouble (succ n) = succ(succ(double(n)))\n\nstep : (n : \u2115) \u2192 (isEven (double n)) \u2192 isEven (double(succ(n)))\nstep n pf = +2even _ pf\n\nthm : (n  : \u2115) \u2192 isEven (double n)\nthm zero = 0even\nthm (succ n) = step _ (thm n)\n\nhalfOfDouble : \u2115 \u2192 \u2115\nhalfOfDouble n = half (double n) (thm n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"40d66dfc3f2210419c40886d095f223a2cfd8315","subject":"Typos.","message":"Typos.\n\nIgnore-this: a9f125da338e4853a6c0a6f49b4f60ad\n\ndarcs-hash:20111217061246-3bd4e-453a79b478812c0a80a233f53de72d1545fce6a8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/ABP\/Lemma1ATP\/Helper.agda","new_file":"src\/FOTC\/Program\/ABP\/Lemma1ATP\/Helper.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"484bb111868a8b921157a7822aff897305838442","subject":"cleaning up progress after canonical forms change","message":"cleaning up progress after canonical forms change\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress\n  data ok : (d : ihexp) (\u0394 : hctx) \u2192 Set where\n    S  : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 ihexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I  : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : ihexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam _ wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- similar if the left is indetermiante but the right is a boxed val\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes are indeterminate\n  progress (TAEHole _ _ ) = I IEHole\n\n    -- nonempty holes step if the innards step, indet otherwise\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I  x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxedVal x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n    -- step if the innards step\n  progress (TACast wt con) | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n    -- if indet, inspect how the types in the cast are realted by consistency:\n    -- if they're the same, step by ID\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n    -- if first type is hole\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n    -- if second type is hole\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHEHole (_ , _ , _ , refl , f)           = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHNEHole (_ , _ , _ , _ , _ , refl , _ ) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHAp (_ , _ , _ , refl , _ )             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHCast (_ , \u03c4 , refl , _)\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , .b , refl , _ , grn , _) | Inl refl = S (_ , Step FHOuter (ITCastSucceed grn ) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , _ , refl , \u03c02 , grn , _)  | Inr x =    S (_ , Step FHOuter (ITCastFail grn GBase x) FHOuter)\n  progress (TACast wt (TCHole2 {\u2987\u2988}))| I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHEHole (_ , _ , _ , refl , _)          = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHNEHole (_ , _ , _ , _ , _ , refl , _) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHAp (_ , _ , _ , refl , _ )            = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , _ , GBase , _)  = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , _ , GHole , _)  = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n    -- if both are arrows\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n    -- boxed value cases, inspect how the casts are realted by consistency\n    -- step by ID if the casts are the same\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n    -- if left is hole\n  progress (TACast wt (TCHole1 {\u03c4 = \u03c4})) | BV x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole1) | BV x\u2081 | Inl g = BV (BVHoleCast g x\u2081)\n  progress (TACast wt (TCHole1 {b})) | BV x\u2081 | Inr x = abort (x GBase)\n  progress (TACast wt (TCHole1 {\u2987\u2988})) | BV x\u2081 | Inr x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2081 | Inr x\n    with (htype-dec  (\u03c41 ==> \u03c42) (\u2987\u2988 ==> \u2987\u2988))\n  progress (TACast wt (TCHole1 {.\u2987\u2988 ==> .\u2987\u2988})) | BV x\u2082 | Inr x\u2081 | Inl refl = BV (BVHoleCast GHole x\u2082)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2082 | Inr x\u2081 | Inr x = S (_ , Step FHOuter (ITGround (MGArr x)) FHOuter)\n    -- if right is hole\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4 \u03c4'\n  progress (TACast wt TCHole2) | BV x\u2081 | d' , \u03c4 , refl , gnd , wt' | Inl refl = S (_  , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | _ , _ , refl , _ , _ | Inr _\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole2) | BV x\u2082 | _ , _ , refl , gnd , _ | Inr x\u2081 | Inl x = S(_ , Step FHOuter (ITCastFail gnd x (flip x\u2081)) FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2082 | _ , _ , refl , _ , _ | Inr x\u2081 | Inr x\n    with notground x\n  progress (TACast wt TCHole2) | BV x\u2083 | _ , _ , refl , _ , _ | Inr _ | Inr _ | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2083 | _ , _ , refl , _ , _ | Inr _ | Inr x | Inr (_ , _ , refl) = S(_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter )\n    -- if both arrows\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | BV x\n    with htype-dec (\u03c41 ==> \u03c42) (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inr x = BV (BVArrCast x x\u2081)\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S (_ , Step (FHFailedCast x) a (FHFailedCast q))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxedVal x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress\n  data ok : (d : ihexp) (\u0394 : hctx) \u2192 Set where\n    S  : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 ihexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I  : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : ihexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam _ wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- similar if the left is indetermiante but the right is a boxed val\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxedVal y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes are indeterminate\n  progress (TAEHole _ _ ) = I IEHole\n\n    -- nonempty holes step if the innards step, indet otherwise\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I  x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxedVal x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n    -- step if the innards step\n  progress (TACast wt con) | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n    -- if indet, inspect how the types in the cast are realted by consistency:\n    -- if they're the same, step by ID\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n    -- if first type is hole\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n    -- if second type is hole\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHEHole (_ , _ , _ , refl , f)           = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHNEHole (_ , _ , _ , _ , _ , refl , _ ) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHAp (_ , _ , _ , refl , _ )             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHCast (_ , \u03c4 , refl , _)\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , .b , refl , \u03c02 , _) | Inl refl = S (_ , Step FHOuter (ITCastSucceed \u03c02) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , _ , refl , \u03c02 , _)  | Inr x =    S (_ , Step FHOuter (ITCastFail \u03c02 GBase x) FHOuter)\n  progress (TACast wt (TCHole2 {\u2987\u2988}))| I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHEHole (_ , _ , _ , refl , _)          = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHNEHole (_ , _ , _ , _ , _ , refl , _) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHAp (_ , _ , _ , refl , _ )            = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , GBase , _)      = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , GHole , _)      = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n    -- if both are arrows\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n    -- boxed value cases, inspect how the casts are realted by consistency\n    -- step by ID if the casts are the same\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n    -- if left is hole\n  progress (TACast wt (TCHole1 {\u03c4 = \u03c4})) | BV x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole1) | BV x\u2081 | Inl g = BV (BVHoleCast g x\u2081)\n  progress (TACast wt (TCHole1 {b})) | BV x\u2081 | Inr x = abort (x GBase)\n  progress (TACast wt (TCHole1 {\u2987\u2988})) | BV x\u2081 | Inr x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2081 | Inr x\n    with (htype-dec  (\u03c41 ==> \u03c42) (\u2987\u2988 ==> \u2987\u2988))\n  progress (TACast wt (TCHole1 {.\u2987\u2988 ==> .\u2987\u2988})) | BV x\u2082 | Inr x\u2081 | Inl refl = BV (BVHoleCast GHole x\u2082)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2082 | Inr x\u2081 | Inr x = S (_ , Step FHOuter (ITGround (MGArr x)) FHOuter)\n    -- if right is hole\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4 \u03c4'\n  progress (TACast wt TCHole2) | BV x\u2081 | d' , \u03c4 , refl , gnd , wt' | Inl refl = S (_  , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | _ , _ , refl , _ , _ | Inr _\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole2) | BV x\u2082 | _ , _ , refl , gnd , _ | Inr x\u2081 | Inl x = S(_ , Step FHOuter (ITCastFail gnd x (flip x\u2081)) FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2082 | _ , _ , refl , _ , _ | Inr x\u2081 | Inr x\n    with notground x\n  progress (TACast wt TCHole2) | BV x\u2083 | _ , _ , refl , _ , _ | Inr _ | Inr _ | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2083 | _ , _ , refl , _ , _ | Inr _ | Inr x | Inr (_ , _ , refl) = S(_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter )\n    -- if both arrows\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | BV x\n    with htype-dec (\u03c41 ==> \u03c42) (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inr x = BV (BVArrCast x x\u2081)\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S (_ , Step (FHFailedCast x) a (FHFailedCast q))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxedVal x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4c8433af18a51f6d63ab5f28270991b262ef907b","subject":"agda : Swierstra Predicate Transformer","message":"agda : Swierstra Predicate Transformer\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module _ where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenWP : \u2115 -> Set\n    isEvenWP = wp0 (_+ 2) isEven\n\n    _        : isEvenWP \u2261 isEven \u2218 (_+ 2)\n    _        = refl\n\n    _        : Set\n    _        = isEvenWP 5\n\n    _        : isEvenWP 5 \u2261 isEven 7\n    _        = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module _ where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_; s\u2264s; z\u2264n)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module _ where\n    _ : Expr\n    _ = Val 3\n    _ : Expr\n    _ = Div (Val 3) (Val 0)\n\n    _ : Val 0 \u21d3 0\n    _ = \u21d3Base\n\n    _ : Val 3 \u21d3 3\n    _ = \u21d3Base\n\n    _ : Div (Val 3) (Val 3) \u21d3 1\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Val 3) \u21d3 3\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    _ : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module _ where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    _     : evv \u2261 Pure 3\n    _     = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    _     : evd \u2261 Pure 1\n    _     = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    _     : evd0 \u2261 Step Abort (\u03bb ())\n    _     = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module _ where\n    _ : Expr -> Nat -> Set\n    _ = _\u21d3_\n\n    _ : Set\n    _ = Val 1 \u21d3 1\n\n    _ : Expr -> Partial Nat -> Set\n    _ = mustPT _\u21d3_\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    _ : mustPT _\u21d3_ (Val 1) (Pure 1)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- a value of this type cannot be constructed\n    _ : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    _ = {!!}\n    -}\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module _ where\n    _ : Expr -> Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    _ = refl\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    {- this type cannot be constructed\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    _ = {!!}\n    -}\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module _ where\n    _ : SafeDiv (Val 3) \u2261 \u22a4\n    _ = refl\n\n    _ : Set\n    _ = SafeDiv (Val 0)\n    _ : SafeDiv (Val 0) \u2261 \u22a4\n    _ = refl\n\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 3))\n    _ : SafeDiv (Div (Val 3) (Val 3))\n                 \u2261 Pair ((x : Val 3 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 0))\n    _ : SafeDiv (Div (Val 3) (Val 0))\n                 \u2261 Pair ((x : Val 0 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module _ where\n    _ :          Val 3 \u21d3 3\n    _ = correct (Val 3) tt\n\n    _ :          Div (Val 3) (Val 1) \u21d3 3\n    _ = correct (Div (Val 3) (Val 1)) ((\u03bb ()) , (tt , tt))\n\n    {- TODO\n    _  : Set\n    _  = SafeDiv (Div (Val 3) (Val 0))\n    sd : SafeDiv (Div (Val 3) (Val 0))\n    sd = {!!} , (tt , tt)\n\n    _ :          \u22a5\n    _ = correct (Div (Val 3) (Val 0)) sd\n    -}\n\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- domain is well-defined Exprs (i.e., no div-by-0)\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module _ where\n    _ : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    _ = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module _ where\n    {-\n    _ : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    _ = {!!}\n    -}\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n            -> (wpPartial f \u2291 wpPartial g) <-> (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement\n  - to relate Kleisli morphisms\n  - to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre and post condition\n  - to its implementation\n\n  add top two elements; fails if stack has too few elements\n\n  show how to prove definition meets its specification\n  -}\n\n  -- define specification in terms of a pre\/post condition\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre  :      a         -> Set l -- precondition on 'a'\n      post : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                     -- and the corresponding outputs\n\n  -- [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  -- for non-dependent examples (e.g., type b does not depend on x : a)\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function (to discard unused arg of type a)\n\n  -- describes desired postcondition for addition function\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  -- spec for addition function\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n  --            pre                     post\n  {-\n  need to relate spec to an implementation\n  'wpPartial' assigns predicate transformer semantics to functions\n  'wpSpec'    assigns predicate transformer semantics to specifications\n  -}\n  -- given a spec, Spec a b\n  -- computes weakest precondition that satisfies an arbitrary postcondition P\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\a -> (pre a)        -- i.e., spec\u2019s precondition should hold and\n                                  \u2227 (post a \u2286 P a) --       spec's postcondition must imply P\n\n  -- using 'wpSpec' can now find a program 'add' that \"refines\" 'addSpec'\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop      Nil  = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs0 =\n    pop xs0 >>= \\{(x1 , xs1) ->\n    pop xs1 >>= \\{(x2 , xs2) ->\n    return ((x1 + x2) :: xs2)}}\n\n  -- verify correct\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd _        (_ :: Nil) (s\u2264s        () , _)\n  correctnessAdd P a@(x :: y :: xs ) (s\u2264s (s\u2264s z\u2264n) , post_addSpec_a_\u2286P_a)\n                                                        -- wpPartial add P (x :: y :: xs)\n                                                         = post_addSpec_a_\u2286P_a (x + y :: xs) AddStep\n  -- paper version has \"extra\" 'Nil\" case\n--correctnessAdd P              Nil  (           () , _)\n--correctnessAdd P        (_ :: Nil) (s\u2264s        () , _)\n--correctnessAdd P   (x :: y :: xs ) (            _ , H) = H                   (x + y :: xs) AddStep\n\n  {-\n  repeat: this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition\n  - to its implementation\n\n  compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n   where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\n{-\n------------------------------------------------------------------------------\n4 Mutable State\n\npredicate transformer semantics for mutable state\ngiving rise to Hoare logic\n\nfollowing assumes a fixed type s : Set (the type of the state)\n-}\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n\n  -- smart constructor\n  get : State s\n  get = Step Get return\n\n  -- smart constructor\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  -- map free monad to the state monad\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x ,       s)\n  run (Step  Get    k) s = run (k s)  s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  -- predicate transformer : for every stateful computation\n  -- maps a postcondition on b \u00d7 s\n  -- to the required precondition on s:\n  statePT : forall {l l'} -> {b : Set l}\n         -> (b \u00d7 s -> Set l')\n         -> State b\n         -> (    s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x ,   s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  -- generalise statePT\n  -- Sometimes describe postconditions as a relation between inputs and outputs.\n  -- For stateful computations, mean enabling the postcondition to also refer to the initial state:\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s          -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  -- weakest precondition semantics for Kleisli morphisms a \u2192 State b\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (    a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (    a \u00d7 s          -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  -- Given predicate P relating input, initial state, final state and result,\n  -- 'wpState' computes the weakest precondition required of input and initial state\n  -- to ensure P holds upon completing the computation.\n  -- The definition amounts to composing 'wp' and 'statePT' functions.\n\n  -- prove soundness of this semantics with respect to the run function\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n   where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step  Get    k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\n{-\nExample showing how to reason about stateful programs using weakest precondition semantics.\n\nVerification problem proposed by Hutton and Fulger [2008]\n- input : binary tree\n- relabel tree so each leaf has a unique number associated with it\n\nTypical solution uses state monad to keep track of next unused label.\n\nHutton and Fulger challenge : reason about the program, without expanding definition of monadic operations.\n\nExample shows how properties of refinement relation can be used to reason about arbitrary effects.\n-}\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  -- Specification.\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  --               precondition true regardless of input tree and initial state\n  --               v\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n   --                     ^\n   --                     postcondition is conjunction\n   where\n    relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n    relabelPost (t , s) (t' , s')\n      = (flatten t' == (seq (s) (size t))) -- flattening result t\u2019 produces sequence of numbers from s to s + size t\n      \u2227 (s + size t == s')                 -- output state 's should be precisely size t larger than input state s\n\n  -- increment current state; return value (before incr)\n  fresh : State Nat\n  fresh = get          >>= \\n ->\n          put (Succ n) >>\n          return n\n\n  -- relabelling function\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  module _ where\n    _ : Free C R (Tree Nat)\n    _ = relabel (Node (Leaf 10) (Leaf 20))\n    _ : relabel (Node (Leaf 10) (Leaf 20))\n        \u2261 Step  Get (\u03bb n1\n        \u2192 Step (Put (Succ n1)) (\u03bb _\n        \u2192 Step  Get (\u03bb n2\n        \u2192 Step (Put (Succ n2)) (\u03bb _\n        \u2192 Pure (Node (Leaf n1) (Leaf n2))))))\n    _ = refl\n\n    _ : Pair (Tree Nat) Nat\n    _ = run (relabel (Node (Leaf 10) (Leaf 20))) 1\n    _ : run (relabel (Node (Leaf 10) (Leaf 20))) 1 \u2261 (Node (Leaf 1) (Leaf 2) , 3)\n    _ = refl\n\n  -- Show the definition satisfies the specification,\n  -- via using 'wpState' to compute weakest precondition semantics of 'relabel'\n  -- and formulating desired correctness property:\n  correctnessRelabel : forall {a : Set}\n                    -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set}\n                  -> (c : State a)\n                     (f : a -> State b)\n                  -> \u2200 i P\n                  -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure      x)    f i P = refl\n  compositionality (Step Get  k)    f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  -- Proof by induction on input tree.\n  -- Proof of Node constructor case is proving\n  --      statePT (relabel l >>= (\u03bb l\u2019 \u2192 relabel r >>= (\u03bb r\u2019 \u2192 Pure (Node l\u2019 r\u2019)))) (P (Node l r , i)) i\n  -- Not obvious how apply induction hypothesis (IH states that P holds for l and r)\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    -- Simplify proofs of refining a specification, by first proving one side of the bind, then the second.\n    -- This is essentially the first law of consequence, specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b}\n                 (                           mx : State a)\n                 (                f : a -> State b) P i\n              -> statePT (wpState f \\_ -> P) mx        i\n              -> statePT                  P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b}\n                      (mx : State a)\n                      (f : a -> State b)\n                      spec\n                   -> \u2200 P i\n                   -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q  mx        i)\n                   ->         Spec.pre spec i \u00d7 (Spec.post spec i \u2286       wpState f (\\_ -> P))\n                   ->                                                     statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s}\n             -> SpecK {zero} (a \u00d7 s) (b \u00d7 s)\n             ->               a\n             -> SpecK     s          (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a  Zero    c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a)\n                                   (trans (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n                                          (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr\n                 -> Pair       (fl \u2261 seq s  sl)       (s  +  sl       \u2261 s')\n                 -> Pair       (fr \u2261 seq s'      sr)  (s' +       sr  \u2261 s'')\n                 -> Pair (fl ++ fr \u2261 seq s (sl + sr)) (s  + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s\n          -> wpSpec relabelSpec P (t , s)\n          -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf   x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set}\n      -> ((x : a) -> Free C R (b x))\n      -> ((x : a) -> b x -> Set)\n      -> (a              -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set}\n                  -> (c : Free C R a) (f : a -> Free C R b)\n                  -> \u2200 P\n                  -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure   x) f P = refl\n  compositionality (Step c x) f P = cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R}\n       -> (a -> Free C R b)\n       -> (b -> Free C R c)\n       ->  a\n       ->       Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set}\n                       -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c)\n                       -> wpCR f1 \u2291 wpCR f2\n                       -> wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n          | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 Data.Nat.\u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n Data.Nat.\u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n Data.Nat.\u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x Data.Nat.\u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x Data.Nat.\u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n Data.Nat.\u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n Data.Nat.\u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenWP : \u2115 -> Set\n    isEvenWP = wp0 (_+ 2) isEven\n\n    _        : isEvenWP \u2261 isEven \u2218 (_+ 2)\n    _        = refl\n\n    _        : Set\n    _        = isEvenWP 5\n\n    _        : isEvenWP 5 \u2261 isEven 7\n    _        = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  module \u2286Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data _\u2264_ : \u2115 \u2192 \u2115 \u2192 Set where\n      z\u2264n : \u2200   {n : \u2115}         \u2192         0 \u2264         n\n      s\u2264s : \u2200 {m n : \u2115} \u2192 m \u2264 n \u2192 Nat.suc m \u2264 Nat.suc n\n\n    \u22641\u2286\u22642 : (_\u2264 1) \u2286 (_\u2264 2)\n    --           P \u2286 Q\n    --   \u2200 a      ->   P a    -> Q a\n    --    (a : \u2115) ->  (a \u2264 1) -> (a \u2264 2)\n    \u22641\u2286\u22642       Zero       _   = z\u2264n\n    \u22641\u2286\u22642 (Succ Zero) (s\u2264s _)  = s\u2264s z\u2264n\n\n    {- cannot be proved\n    \u22642\u2286\u22641 : (_\u2264 2) \u2286 (_\u2264 1)\n    \u22642\u2286\u22641  Zero _          = z\u2264n\n    \u22642\u2286\u22641 (Succ a) (s\u2264s p) = {!!}\n    -}\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_; s\u2264s; z\u2264n)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    _ : Expr\n    _ = Val 3\n    _ : Expr\n    _ = Div (Val 3) (Val 0)\n\n    _ : Val 0 \u21d3 0\n    _ = \u21d3Base\n\n    _ : Val 3 \u21d3 3\n    _ = \u21d3Base\n\n    _ : Div (Val 3) (Val 3) \u21d3 1\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Val 3) \u21d3 3\n    _ = \u21d3Step \u21d3Base \u21d3Base\n\n    _ : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    _ : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    _ = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    _     : evv \u2261 Pure 3\n    _     = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    _     : evd \u2261 Pure 1\n    _     = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    _     : evd0 \u2261 Step Abort (\u03bb ())\n    _     = refl\n\n  -- relate big step semantics to monadic interpreter\n\n  -- convert post-condition for a pure function  'f : (x : A) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    _ : Expr -> Nat -> Set\n    _ = _\u21d3_\n\n    _ : Set\n    _ = Val 1 \u21d3 1\n\n    _ : Expr -> Partial Nat -> Set\n    _ = mustPT _\u21d3_\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    _ : mustPT _\u21d3_ (Val 1) (Pure 1)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    {- a value of this type cannot be constructed\n    _ : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    _ = {!!}\n    -}\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    _ = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n    - done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so mustPT Abort case returns empty type\n  -}\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    _ : Expr -> Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    _ = refl\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    _ = \u21d3Base\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    _ = {!!}\n    -}\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    _ : SafeDiv (Val 3) \u2261 \u22a4\n    _ = refl\n    _ : SafeDiv (Val 0) \u2261 \u22a4\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Val 0)\n\n    _ : SafeDiv (Div (Val 3) (Val 3))\n                 \u2261 Pair ((x : Val 3 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 3))\n\n    _ : SafeDiv (Div (Val 3) (Val 0))\n                 \u2261 Pair ((x : Val 0 \u21d3 0) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    _ = refl\n    _ : Set\n    _ = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  expect : any expr : e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  --      (a : Expr)  (SafeDiv a)                                 (wpPartial \u27e6_\u27e7 _\u21d3_ a)\n  correct (Val _)     _                                         = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  module CorrectTest where\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n    _ = \u21d3Base\n\n    _ : Set\n    _ = wpPartial \u27e6_\u27e7 _\u21d3_ (Val 3)\n\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 1))\n    _ = \u21d3Step \u21d3Base \u21d3Base\n    {-\n    _ : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 3) (Val 0))\n    _ = {!!}\n    -}\n  {-\n  Generalization.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial.\n  -}\n\n  -- domain is well-defined Exprs (i.e., no div-by-0)\n  -- returns \u22a4 on Pure; \u22a5 on Step Abort ...\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  module DomTest where\n    _ : dom \u27e6_\u27e7 \u2261 \u03bb z \u2192 mustPT (\u03bb x _ \u2192 \u22a4' zero) z \u27e6 z \u27e7\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Val 0) \u2261 \u22a4' zero\n    _ = refl\n    _ : dom \u27e6_\u27e7 (Div (Val 1) (Val 0)) \u2261 \u22a5\n    _ = refl\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eqnl ]]]  | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eqnl\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with                   \u27e6 el \u27e7   | inspect \u27e6_\u27e7 el    | \u27e6 er \u27e7        | inspect \u27e6_\u27e7 er\n                                                                                      | complete el (wpPartial1 {el} {er} h)\n                                                                                      | complete er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    _ : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    _ = {!!}\n    -}\n    _ : Set\n    _ = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n            -> (wpPartial f \u2291 wpPartial g) <-> (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement\n  - to relate Kleisli morphisms\n  - to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre and post condition\n  - to its implementation\n\n  add top two elements; fails if stack has too few elements\n\n  show how to prove definition meets its specification\n  -}\n\n  -- define specification in terms of a pre\/post condition\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre  :      a         -> Set l -- precondition on 'a'\n      post : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                     -- and the corresponding outputs\n\n  -- [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  -- for non-dependent examples (e.g., type b does not depend on x : a)\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function (to discard unused arg of type a)\n\n  -- describes desired postcondition for addition function\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  -- spec for addition function\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n  --            pre                     post\n  {-\n  need to relate spec to an implementation\n  'wpPartial' assigns predicate transformer semantics to functions\n  'wpSpec'    assigns predicate transformer semantics to specifications\n  -}\n  -- given a spec, Spec a b\n  -- computes weakest precondition that satisfies an arbitrary postcondition P\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\a -> (pre a)        -- i.e., spec\u2019s precondition should hold and\n                                  \u2227 (post a \u2286 P a) --       spec's postcondition must imply P\n\n  -- using 'wpSpec' can now find a program 'add' that \"refines\" 'addSpec'\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop      Nil  = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs0 =\n    pop xs0 >>= \\{(x1 , xs1) ->\n    pop xs1 >>= \\{(x2 , xs2) ->\n    return ((x1 + x2) :: xs2)}}\n\n  -- verify correct\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd _        (_ :: Nil) (s\u2264s        () , _)\n  correctnessAdd P a@(x :: y :: xs ) (s\u2264s (s\u2264s z\u2264n) , post_addSpec_a_\u2286P_a)\n                                                        -- wpPartial add P (x :: y :: xs)\n                                                         = post_addSpec_a_\u2286P_a (x + y :: xs) AddStep\n  -- paper version has \"extra\" 'Nil\" case\n--correctnessAdd P              Nil  (           () , _)\n--correctnessAdd P        (_ :: Nil) (s\u2264s        () , _)\n--correctnessAdd P   (x :: y :: xs ) (            _ , H) = H                   (x + y :: xs) AddStep\n\n  {-\n  repeat: this example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition\n  - to its implementation\n\n  compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n   where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\n{-\n------------------------------------------------------------------------------\n4 Mutable State\n\npredicate transformer semantics for mutable state\ngiving rise to Hoare logic\n\nfollowing assumes a fixed type s : Set (the type of the state)\n-}\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n\n  -- smart constructor\n  get : State s\n  get = Step Get return\n\n  -- smart constructor\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  -- map free monad to the state monad\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x ,       s)\n  run (Step  Get    k) s = run (k s)  s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  -- predicate transformer : for every stateful computation\n  -- maps a postcondition on b \u00d7 s\n  -- to the required precondition on s:\n  statePT : forall {l l'} -> {b : Set l}\n         -> (b \u00d7 s -> Set l')\n         -> State b\n         -> (    s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x ,   s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  -- generalise statePT\n  -- Sometimes describe postconditions as a relation between inputs and outputs.\n  -- For stateful computations, mean enabling the postcondition to also refer to the initial state:\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s          -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  -- weakest precondition semantics for Kleisli morphisms a \u2192 State b\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (    a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (    a \u00d7 s          -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  -- Given predicate P relating input, initial state, final state and result,\n  -- 'wpState' computes the weakest precondition required of input and initial state\n  -- to ensure P holds upon completing the computation.\n  -- The definition amounts to composing 'wp' and 'statePT' functions.\n\n  -- prove soundness of this semantics with respect to the run function\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n   where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\n{-\nExample showing how to reason about stateful programs using weakest precondition semantics.\n\nVerification problem proposed by Hutton and Fulger [2008]\n- input : binary tree\n- relabel tree so each leaf has a unique number associated with it\n\nTypical solution uses state monad to keep track of next unused label.\n\nHutton and Fulger challenge : reason about the program, without expanding definition of monadic operations.\n\nExample shows how properties of refinement relation can be used to reason about arbitrary effects.\n-}\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  -- Specification.\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  --               precondition true regardless of input tree and initial state\n  --               v\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n   --                     ^\n   --                     postcondition is conjunction\n   where\n    relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n    relabelPost (t , s) (t' , s')\n      = (flatten t' == (seq (s) (size t))) -- flattening result t\u2019 produces sequence of numbers from s to s + size t\n      \u2227 (s + size t == s')                 -- output state 's should be precisely size t larger than input state s\n\n  -- increment current state; return value (before incr)\n  fresh : State Nat\n  fresh = get          >>= \\n ->\n          put (Succ n) >>\n          return n\n\n  -- relabelling function\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  module RelableTest where\n    _ : Free C R (Tree Nat)\n    _ = relabel (Node (Leaf 10) (Leaf 20))\n    _ : relabel (Node (Leaf 10) (Leaf 20))\n        \u2261 Step  Get (\u03bb n1\n        \u2192 Step (Put (Succ n1)) (\u03bb _\n        \u2192 Step  Get (\u03bb n2\n        \u2192 Step (Put (Succ n2)) (\u03bb _\n        \u2192 Pure (Node (Leaf n1) (Leaf n2))))))\n    _ = refl\n\n    _ : Pair (Tree Nat) Nat\n    _ = run (relabel (Node (Leaf 10) (Leaf 20))) 1\n    _ : run (relabel (Node (Leaf 10) (Leaf 20))) 1 \u2261 (Node (Leaf 1) (Leaf 2) , 3)\n    _ = refl\n\n  -- Show the definition satisfies the specification,\n  -- via using 'wpState' to compute weakest precondition semantics of 'relabel'\n  -- and formulating desired correctness property:\n  correctnessRelabel : forall {a : Set}\n                    -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set}\n                  -> (c : State a)\n                     (f : a -> State b)\n                  -> \u2200 i P\n                  -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure      x)    f i P = refl\n  compositionality (Step Get  k)    f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  -- Proof by induction on input tree.\n  -- Proof of Node constructor case is proving\n  --      statePT (relabel l >>= (\u03bb l\u2019 \u2192 relabel r >>= (\u03bb r\u2019 \u2192 Pure (Node l\u2019 r\u2019)))) (P (Node l r , i)) i\n  -- Not obvious how apply induction hypothesis (IH states that P holds for l and r)\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    -- Simplify proofs of refining a specification, by first proving one side of the bind, then the second.\n    -- This is essentially the first law of consequence, specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b}\n                 (                           mx : State a)\n                 (                f : a -> State b) P i\n              -> statePT (wpState f \\_ -> P) mx        i\n              -> statePT                  P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b}\n                      (mx : State a)\n                      (f : a -> State b)\n                      spec\n                   -> \u2200 P i\n                   -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q  mx        i)\n                   ->         Spec.pre spec i \u00d7 (Spec.post spec i \u2286       wpState f (\\_ -> P))\n                   ->                                                     statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s}\n             -> SpecK {zero} (a \u00d7 s) (b \u00d7 s)\n             ->               a\n             -> SpecK     s          (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a  Zero    c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a)\n                                   (trans (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n                                          (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr\n                 -> Pair       (fl \u2261 seq s  sl)       (s  +  sl       \u2261 s')\n                 -> Pair       (fr \u2261 seq s'      sr)  (s' +       sr  \u2261 s'')\n                 -> Pair (fl ++ fr \u2261 seq s (sl + sr)) (s  + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s\n          -> wpSpec relabelSpec P (t , s)\n          -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf   x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set}\n      -> ((x : a) -> Free C R (b x))\n      -> ((x : a) -> b x -> Set)\n      -> (a              -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set}\n                  -> (c : Free C R a) (f : a -> Free C R b)\n                  -> \u2200 P\n                  -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure   x) f P = refl\n  compositionality (Step c x) f P = cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R}\n       -> (a -> Free C R b)\n       -> (b -> Free C R c)\n       ->  a\n       ->       Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set}\n                       -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c)\n                       -> wpCR f1 \u2291 wpCR f2\n                       -> wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n          | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"147a00bc095894a35d063ba5f815e115c7798064","subject":"Working on polymorphic lists.","message":"Working on polymorphic lists.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --universal-quantified-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool.Type\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Heterogeneous lists\ndata List : D \u2192 Set where\n  lnil  :                      List []\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n-- Lists of total natural numbers\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n-- Lists of total Booleans\ndata ListB : D \u2192 Set where\n  lbnil  :                                ListB []\n  lbcons : \u2200 {b bs} \u2192 Bool b \u2192 ListB bs \u2192 ListB (b \u2237 bs)\n\n-- Polymorphic lists.\n-- NB. The data type list is in *Set\u2081*.\ndata ListP (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                               ListP A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n\nList\u2081 : D \u2192 Set\u2081\nList\u2081 = ListP (\u03bb d \u2192 d \u2261 d)\n\nListN\u2081 : D \u2192 Set\u2081\nListN\u2081 = ListP N\n\nListB\u2081 : D \u2192 Set\u2081\nListB\u2081 = ListP Bool\n","old_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --universal-quantified-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n\n-- NB. The data type list is in *Set\u2081*.\ndata List (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                              List A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 List A xs \u2192 List A (x \u2237 xs)\n{-# ATP axiom lnil lcons #-}\n\npostulate foo : \u2200 x \u2192 x \u2261 x\n{-# ATP prove foo #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ade85e632519eda34331154c3a961ce6642a9497","subject":"adding cases for new IT rules","message":"adding cases for new IT rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule preservation where\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             \u0394 \u22a2 d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation TAConst (Step _ _ _) = TAConst\n  preservation (TAAp wt x wt\u2081) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TAAp wt x wt\u2081\n  preservation (TAAp wt x\u2081 wt\u2081) (Step x\u2082 (ITLam x\u2083) x\u2084) = TAAp wt x\u2081 wt\u2081\n  preservation (TAAp x x\u2081 x\u2082) (Step x\u2083 (ITNEHole x\u2084) x\u2085) = TAAp x x\u2081 x\u2082\n  preservation (TAAp x x\u2081 x\u2082) (Step x\u2083 ITEHole x\u2085) = TAAp x x\u2081 x\u2082\n  preservation (TACast wt x) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TACast wt x\n  preservation (TACast wt x\u2081) (Step x\u2082 (ITLam x\u2083) x\u2084) = TACast wt x\u2081\n  preservation (TACast x x\u2081) (Step x\u2082 (ITNEHole x\u2083) x\u2084) = TACast x x\u2081\n  preservation (TACast x x\u2081) (Step x\u2082 ITEHole x\u2084) = TACast x x\u2081\n  preservation (TAEHole x x\u2081) (Step x\u2082 x\u2083 x\u2084) = TAEHole x x\u2081\n  preservation (TALam wt) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TALam wt\n  preservation (TALam wt) (Step x\u2082 (ITLam x\u2083) x\u2084) = TALam wt\n  preservation (TALam x\u2081) (Step x\u2082 (ITNEHole x\u2083) x\u2084) = TALam x\u2081\n  preservation (TALam x\u2081) (Step x\u2082 ITEHole x\u2084) = TALam x\u2081\n  preservation (TANEHole x wt x\u2081) (Step x\u2082 (ITCast x\u2083 x\u2084 x\u2085) x\u2086) = TANEHole x wt x\u2081\n  preservation (TANEHole x x\u2081 x\u2082) (Step x\u2083 (ITNEHole x\u2084) x\u2085) = TANEHole x x\u2081 x\u2082\n  preservation (TANEHole x x\u2081 x\u2082) (Step x\u2083 ITEHole x\u2085) = TANEHole x x\u2081 x\u2082\n  preservation (TANEHole x\u2081 wt x\u2082) (Step x\u2083 (ITLam x\u2084) x\u2085) = TANEHole x\u2081 wt x\u2082\n  preservation (TAVar x\u2081) step = abort (somenotnone (! x\u2081))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule preservation where\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             \u0394 \u22a2 d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation TAConst (Step _ _ _) = TAConst\n  preservation (TAVar x\u2081) step = abort (somenotnone (! x\u2081))\n  preservation (TALam wt) (Step x\u2082 (ITLam x\u2083) x\u2084) = TALam wt\n  preservation (TALam wt) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TALam wt\n  preservation (TAAp wt x\u2081 wt\u2081) (Step x\u2082 (ITLam x\u2083) x\u2084) = TAAp wt x\u2081 wt\u2081\n  preservation (TAAp wt x wt\u2081) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TAAp wt x wt\u2081\n  preservation (TAEHole x x\u2081) (Step x\u2082 x\u2083 x\u2084) = TAEHole x x\u2081\n  preservation (TANEHole x\u2081 wt x\u2082) (Step x\u2083 (ITLam x\u2084) x\u2085) = TANEHole x\u2081 wt x\u2082\n  preservation (TANEHole x wt x\u2081) (Step x\u2082 (ITCast x\u2083 x\u2084 x\u2085) x\u2086) = TANEHole x wt x\u2081\n  preservation (TACast wt x\u2081) (Step x\u2082 (ITLam x\u2083) x\u2084) = TACast wt x\u2081\n  preservation (TACast wt x) (Step x\u2081 (ITCast x\u2082 x\u2083 x\u2084) x\u2085) = TACast wt x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70def950c8925dc369db199a79e13cb59ff30970","subject":"adder: use mapAccum","message":"adder: use mapAccum\n","repos":"crypto-agda\/crypto-agda","old_file":"adder.agda","new_file":"adder.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule adder where\n\nmodule FunAdder\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    msb-adder : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n    msb-adder = <tt\u204f <0\u2082> , zip > \u204f mapAccum full-adder \u204f snd\n\n    -- TODO reverses all over the places... switch to lsb first?\n    -- lsb\n    adder : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n    adder = < reverse \u00d7 reverse > \u204f msb-adder \u204f reverse\n\n    open import Data.Digit\n\n    \u2115\u25b9`Bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    \u2115\u25b9`Bits \u2113 n\u2080 = constBits (V.reverse (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080)))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0\u2082\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0\u2082\n            F\u25b9\ud835\udfda (suc _) = 1\u2082\n\n{-\nopen import IO\nimport IO.Primitive\n-}\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\n--open import Coinduction\nopen import FunUniverse.Agda\n--open import Data.Nat.Show\nopen FunAdder agdaFunOps\nopen FunOps agdaFunOps\nimport FunUniverse.Cost as Cost\nmodule TimeCost = FunOps Cost.timeOps\n{-\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1\u2082 = putStr \"1\"\nputBit 0\u2082 = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\n-}\narg1   = \u2115\u25b9`Bits 8 0x0b _\narg2   = \u2115\u25b9`Bits 8 0x1f _\nresult = adder (arg1 , arg2)\nadder-cost : \u2115 \u2192 \u2115\nadder-cost n = FunAdder.adder Cost.timeOps {n}\n{-\nmainIO : IO \ud835\udfd9\nmainIO = \u266f putBits arg1 >>\n      \u266f (\u266f putStr \" + \" >>\n      \u266f (\u266f putBits arg2 >>\n      \u266f (\u266f putStr \" = \" >>\n      \u266f (\u266f putBits result >>\n      \u266f (\u266f putStr \" cost:\" >>\n         \u266f putStr (show (adder-cost 8)))))))\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run mainIO\n-}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule adder where\n\nmodule FunAdder\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 S `\u00d7 `Vec B n\n    iter {zero}  F = second <[]>\n    iter {suc n} F = second uncons\n                   \u204f assoc-first F \u204f around (iter F)\n                   \u204f second <\u2237>\n\n    msb-adder : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n    msb-adder = <tt\u204f <0b> , zip > \u204f iter full-adder \u204f snd\n\n    -- TODO reverses all over the places... switch to lsb first?\n    -- lsb\n    adder : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n    adder = < reverse \u00d7 reverse > \u204f msb-adder \u204f reverse\n\n    open import Data.Digit\n\n    \u2115\u25b9`Bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    \u2115\u25b9`Bits \u2113 n\u2080 = constBits (V.reverse (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080)))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0\u2082\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0\u2082\n            F\u25b9\ud835\udfda (suc _) = 1\u2082\n\n{-\nopen import IO\nimport IO.Primitive\n-}\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\n--open import Coinduction\nopen import FunUniverse.Agda\n--open import Data.Nat.Show\nopen FunAdder agdaFunOps\nopen FunOps agdaFunOps\nimport FunUniverse.Cost as Cost\nmodule TimeCost = FunOps Cost.timeOps\n{-\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1\u2082 = putStr \"1\"\nputBit 0\u2082 = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\n-}\narg1   = \u2115\u25b9`Bits 8 0x0b _\narg2   = \u2115\u25b9`Bits 8 0x1f _\nresult = adder (arg1 , arg2)\nadder-cost : \u2115 \u2192 \u2115\nadder-cost n = FunAdder.adder Cost.timeOps {n}\n{-\nmainIO : IO \ud835\udfd9\nmainIO = \u266f putBits arg1 >>\n      \u266f (\u266f putStr \" + \" >>\n      \u266f (\u266f putBits arg2 >>\n      \u266f (\u266f putStr \" = \" >>\n      \u266f (\u266f putBits result >>\n      \u266f (\u266f putStr \" cost:\" >>\n         \u266f putStr (show (adder-cost 8)))))))\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run mainIO\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0281d874e9b8c1ea549242d7a1ca28f18df3bf90","subject":"Added x<x\u2238y\u2192\u22a5.","message":"Added x<x\u2238y\u2192\u22a5.\n\nIgnore-this: 126888dcec026c41373c92b862bf82d3\n\ndarcs-hash:20110215034929-3bd4e-70d149030e75804cc1ac587620afa976fa5a1806.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\n-- TODO: To move to LTC.Data.Nat.Inequalities.Properties.\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x\u2238y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x\u2238y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x\u2238y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\n-- TODO: To move to LTC.Data.Nat.Inequalities.Properties.\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x\u2238y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x\u2238y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x\u2238y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1d168fbda6fabfd6a24bf393c188808d758c682f","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2I.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2I.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2I where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.Loop\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- Helper function for the ABP lemma 2\n\nmodule Helper where\n\n  helper : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n           Bit b \u2192\n           Fair os\u2080' \u2192\n           ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n           \u2200 ft\u2081 os\u2081'' \u2192 F*T ft\u2081 \u2192 Fair os\u2081'' \u2192 os\u2081' \u2261 ft\u2081 ++ os\u2081'' \u2192\n           \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2080''\n           \u2227 Fair os\u2081''\n           \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(T \u2237 []) os\u2081'' f*tnil Fos\u2081'' os\u2081'-eq =\n         os\u2080' , os\u2081'' , as'' , bs'' , cs'' , ds''\n         , Fos\u2080' , Fos\u2081''\n         , as''-eq , bs''-eq ,  cs''-eq , refl , js'-eq\n\n    where\n    os\u2081'-eq-helper : os\u2081' \u2261 T \u2237 os\u2081''\n    os\u2081'-eq-helper =\n      os\u2081'                 \u2261\u27e8 os\u2081'-eq \u27e9\n      (true \u2237 []) ++ os\u2081'' \u2261\u27e8 ++-\u2237 true [] os\u2081'' \u27e9\n      true \u2237 [] ++ os\u2081''   \u2261\u27e8 \u2237-rightCong (++-leftIdentity os\u2081'') \u27e9\n      true \u2237 os\u2081''         \u220e\n\n    ds'' : D\n    ds'' = corrupt os\u2081'' \u00b7 cs'\n\n    ds'-eq : ds' \u2261 ok b \u2237 ds''\n    ds'-eq =\n      ds' \u2261\u27e8 ds'ABP' \u27e9\n      corrupt os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (corruptCong os\u2081'-eq-helper) \u27e9\n      corrupt (T \u2237 os\u2081'') \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-T os\u2081'' b cs' \u27e9\n      ok b \u2237 corrupt os\u2081'' \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      ok b \u2237 ds'' \u220e\n\n    as'' : D\n    as'' = as'\n\n    as''-eq : as'' \u2261 send (not b) \u00b7 is' \u00b7 ds''\n    as''-eq =\n      as''                         \u2261\u27e8 as'ABP \u27e9\n      await b i' is' ds'           \u2261\u27e8 awaitCong\u2084 ds'-eq \u27e9\n      await b i' is' (ok b \u2237 ds'') \u2261\u27e8 await-ok\u2261 b b i' is' ds'' refl \u27e9\n      send (not b) \u00b7 is' \u00b7 ds''    \u220e\n\n    bs'' : D\n    bs'' = bs'\n\n    bs''-eq : bs'' \u2261 corrupt os\u2080' \u00b7 as'\n    bs''-eq = bs'ABP'\n\n    cs'' : D\n    cs'' = cs'\n\n    cs''-eq : cs'' \u2261 ack (not b) \u00b7 bs'\n    cs''-eq = cs'ABP'\n\n    js'-eq : js' \u2261 out (not b) \u00b7 bs''\n    js'-eq = js'ABP'\n\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(F \u2237 ft\u2081) os\u2081'' (f*tcons {ft\u2081} FTft\u2081) Fos\u2081'' os\u2081'-eq\n         = helper Bb (tail-Fair Fos\u2080') ABP'IH ft\u2081 os\u2081'' FTft\u2081 Fos\u2081'' refl\n\n    where\n    os\u2080^ : D\n    os\u2080^ = tail\u2081 os\u2080'\n\n    os\u2081^ : D\n    os\u2081^ = ft\u2081 ++ os\u2081''\n\n    os\u2081'-eq-helper : os\u2081' \u2261 F \u2237 os\u2081^\n    os\u2081'-eq-helper = os\u2081'               \u2261\u27e8 os\u2081'-eq \u27e9\n                     (F \u2237 ft\u2081) ++ os\u2081'' \u2261\u27e8 ++-\u2237 _ _ _ \u27e9\n                     F \u2237 ft\u2081 ++ os\u2081''   \u2261\u27e8 refl \u27e9\n                     F \u2237 os\u2081^           \u220e\n\n    ds^ : D\n    ds^ = corrupt os\u2081^ \u00b7 cs'\n\n    ds'-eq : ds' \u2261 error \u2237 ds^\n    ds'-eq =\n      ds'\n        \u2261\u27e8 ds'ABP' \u27e9\n      corrupt os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (corruptCong os\u2081'-eq-helper) \u27e9\n      corrupt (F \u2237 os\u2081^) \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt os\u2081^ \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      error \u2237 ds^ \u220e\n\n    as^ : D\n    as^ = await b i' is' ds^\n\n    as'-eq : as' \u2261 < i' , b > \u2237 as^\n    as'-eq = as'                             \u2261\u27e8 as'ABP \u27e9\n             await b i' is' ds'              \u2261\u27e8 awaitCong\u2084 ds'-eq \u27e9\n             await b i' is' (error \u2237 ds^)    \u2261\u27e8 await-error _ _ _ _ \u27e9\n             < i' , b > \u2237 await b i' is' ds^ \u2261\u27e8 refl \u27e9\n             < i' , b > \u2237 as^                \u220e\n\n    bs^ : D\n    bs^ = corrupt os\u2080^ \u00b7 as^\n\n    bs'-eq-helper\u2081 : os\u2080' \u2261 T \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 ok < i' , b > \u2237 bs^\n    bs'-eq-helper\u2081 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 corrupt os\u2080' \u00b7 as' \u2261 corrupt t \u00b7 t')\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt (T \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as^)\n        \u2261\u27e8 corrupt-T _ _ _ \u27e9\n      ok < i' , b > \u2237 corrupt (tail\u2081 os\u2080') \u00b7 as^\n        \u2261\u27e8 refl \u27e9\n      ok < i' , b > \u2237 bs^ \u220e\n\n    bs'-eq-helper\u2082 : os\u2080' \u2261 F \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 error \u2237 bs^\n    bs'-eq-helper\u2082 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 corrupt os\u2080' \u00b7 as' \u2261 corrupt t \u00b7 t')\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt (F \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as^)\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt (tail\u2081 os\u2080') \u00b7 as^\n        \u2261\u27e8 refl \u27e9\n      error \u2237 bs^ \u220e\n\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs^ \u2228 bs' \u2261 error \u2237 bs^\n    bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2080')\n\n    cs^ : D\n    cs^ = ack (not b) \u00b7 bs^\n\n    cs'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs^ \u2192 cs' \u2261 b \u2237 cs^\n    cs'-eq-helper\u2081 h =\n      cs'\n      \u2261\u27e8 cs'ABP' \u27e9\n      ack (not b) \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack (not b) \u00b7 (ok < i' , b > \u2237 bs^)\n        \u2261\u27e8 ack-ok\u2262 _ _ _ _ (not-x\u2262x Bb) \u27e9\n      not (not b) \u2237 ack (not b) \u00b7 bs^\n        \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack (not b) \u00b7 bs^\n        \u2261\u27e8 refl \u27e9\n      b \u2237 cs^ \u220e\n\n    cs'-eq-helper\u2082 : bs' \u2261 error \u2237 bs^ \u2192 cs' \u2261 b \u2237 cs^\n    cs'-eq-helper\u2082 h =\n      cs'                             \u2261\u27e8 cs'ABP' \u27e9\n      ack (not b) \u00b7 bs'               \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack (not b) \u00b7 (error \u2237 bs^)     \u2261\u27e8 ack-error _ _ \u27e9\n      not (not b) \u2237 ack (not b) \u00b7 bs^ \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack (not b) \u00b7 bs^           \u2261\u27e8 refl \u27e9\n      b \u2237 cs^                         \u220e\n\n    cs'-eq : cs' \u2261 b \u2237 cs^\n    cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n    js'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs^ \u2192 js' \u2261 out (not b) \u00b7 bs^\n    js'-eq-helper\u2081 h  =\n      js'\n        \u2261\u27e8 js'ABP' \u27e9\n      out (not b) \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out (not b) \u00b7 (ok < i' , b > \u2237 bs^)\n        \u2261\u27e8 out-ok\u2262 (not b) b i' bs^ (not-x\u2262x Bb) \u27e9\n      out (not b) \u00b7 bs^ \u220e\n\n    js'-eq-helper\u2082 : bs' \u2261 error \u2237 bs^ \u2192 js' \u2261 out (not b) \u00b7 bs^\n    js'-eq-helper\u2082 h  =\n      js'                         \u2261\u27e8 js'ABP' \u27e9\n      out (not b) \u00b7 bs'           \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out (not b) \u00b7 (error \u2237 bs^) \u2261\u27e8 out-error (not b) bs^ \u27e9\n      out (not b) \u00b7 bs^           \u220e\n\n    js'-eq : js' \u2261 out (not b) \u00b7 bs^\n    js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n    ds^-eq : ds^ \u2261 corrupt os\u2081^ \u00b7 (b \u2237 cs^)\n    ds^-eq = \u00b7-rightCong cs'-eq\n\n    ABP'IH : ABP' b i' is' os\u2080^ os\u2081^ as^ bs^ cs^ ds^ js'\n    ABP'IH = ds^-eq , refl , refl , refl , js'-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081 \u2208\n-- Fair, and hence by unfolding Fair we conclude that there are ft\u2081 :\n-- F*T and os\u2081'' : Fair, such that os\u2081' = ft\u2081 ++ os\u2081''.\n--\n-- We proceed by induction on ft\u2081 : F*T using helper.\n\nopen Helper\nlemma\u2082 : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2080' \u2192\n         Fair os\u2081' \u2192\n         ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n         Fair os\u2080''\n         \u2227 Fair os\u2081''\n         \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2080' Fos\u2081' abp' with Fair-unf Fos\u2081'\n... | ft , os\u2080'' , FTft , h , Fos\u2080'' =\n  helper Bb Fos\u2080' abp' ft os\u2080'' FTft Fos\u2080'' h\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2I where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.Loop\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- Helper function for the ABP lemma 2\n\nmodule Helper where\n\n  helper : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n           Bit b \u2192\n           Fair os\u2080' \u2192\n           ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n           \u2200 ft\u2081 os\u2081'' \u2192 F*T ft\u2081 \u2192 Fair os\u2081'' \u2192 os\u2081' \u2261 ft\u2081 ++ os\u2081'' \u2192\n           \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2080''\n           \u2227 Fair os\u2081''\n           \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(T \u2237 []) os\u2081'' f*tnil Fos\u2081'' os\u2081'-eq =\n         os\u2080' , os\u2081'' , as'' , bs'' , cs'' , ds''\n         , Fos\u2080' , Fos\u2081''\n         , as''-eq , bs''-eq ,  cs''-eq , refl , js'-eq\n\n    where\n    os'\u2081-eq-helper : os\u2081' \u2261 T \u2237 os\u2081''\n    os'\u2081-eq-helper =\n      os\u2081'                 \u2261\u27e8 os\u2081'-eq \u27e9\n      (true \u2237 []) ++ os\u2081'' \u2261\u27e8 ++-\u2237 true [] os\u2081'' \u27e9\n      true \u2237 [] ++ os\u2081''   \u2261\u27e8 \u2237-rightCong (++-leftIdentity os\u2081'') \u27e9\n      true \u2237 os\u2081''         \u220e\n\n    ds'' : D\n    ds'' = corrupt os\u2081'' \u00b7 cs'\n\n    ds'-eq : ds' \u2261 ok b \u2237 ds''\n    ds'-eq =\n      ds' \u2261\u27e8 ds'ABP' \u27e9\n      corrupt os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (corruptCong os'\u2081-eq-helper) \u27e9\n      corrupt (T \u2237 os\u2081'') \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-T os\u2081'' b cs' \u27e9\n      ok b \u2237 corrupt os\u2081'' \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      ok b \u2237 ds'' \u220e\n\n    as'' : D\n    as'' = as'\n\n    as''-eq : as'' \u2261 send (not b) \u00b7 is' \u00b7 ds''\n    as''-eq =\n      as''                         \u2261\u27e8 as'ABP \u27e9\n      await b i' is' ds'           \u2261\u27e8 awaitCong\u2084 ds'-eq \u27e9\n      await b i' is' (ok b \u2237 ds'') \u2261\u27e8 await-ok\u2261 b b i' is' ds'' refl \u27e9\n      send (not b) \u00b7 is' \u00b7 ds''    \u220e\n\n    bs'' : D\n    bs'' = bs'\n\n    bs''-eq : bs'' \u2261 corrupt os\u2080' \u00b7 as'\n    bs''-eq = bs'ABP'\n\n    cs'' : D\n    cs'' = cs'\n\n    cs''-eq : cs'' \u2261 ack (not b) \u00b7 bs'\n    cs''-eq = cs'ABP'\n\n    js'-eq : js' \u2261 out (not b) \u00b7 bs''\n    js'-eq = js'ABP'\n\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' (ds'ABP' , as'ABP , bs'ABP' , cs'ABP' , js'ABP')\n         .(F \u2237 ft\u2081) os\u2081'' (f*tcons {ft\u2081} FTft\u2081) Fos\u2081'' os\u2081'-eq\n         = helper Bb (tail-Fair Fos\u2080') ABP'IH ft\u2081 os\u2081'' FTft\u2081 Fos\u2081'' refl\n\n    where\n    os\u2080^ : D\n    os\u2080^ = tail\u2081 os\u2080'\n\n    os\u2081^ : D\n    os\u2081^ = ft\u2081 ++ os\u2081''\n\n    os\u2081'-eq-helper : os\u2081' \u2261 F \u2237 os\u2081^\n    os\u2081'-eq-helper = os\u2081'               \u2261\u27e8 os\u2081'-eq \u27e9\n                     (F \u2237 ft\u2081) ++ os\u2081'' \u2261\u27e8 ++-\u2237 _ _ _ \u27e9\n                     F \u2237 ft\u2081 ++ os\u2081''   \u2261\u27e8 refl \u27e9\n                     F \u2237 os\u2081^           \u220e\n\n    ds^ : D\n    ds^ = corrupt os\u2081^ \u00b7 cs'\n\n    ds'-eq : ds' \u2261 error \u2237 ds^\n    ds'-eq =\n      ds'\n        \u2261\u27e8 ds'ABP' \u27e9\n      corrupt os\u2081' \u00b7 (b \u2237 cs')\n        \u2261\u27e8 \u00b7-leftCong (corruptCong os\u2081'-eq-helper) \u27e9\n      corrupt (F \u2237 os\u2081^) \u00b7 (b \u2237 cs')\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt os\u2081^ \u00b7 cs'\n        \u2261\u27e8 refl \u27e9\n      error \u2237 ds^ \u220e\n\n    as^ : D\n    as^ = await b i' is' ds^\n\n    as'-eq : as' \u2261 < i' , b > \u2237 as^\n    as'-eq = as'                             \u2261\u27e8 as'ABP \u27e9\n             await b i' is' ds'              \u2261\u27e8 awaitCong\u2084 ds'-eq \u27e9\n             await b i' is' (error \u2237 ds^)    \u2261\u27e8 await-error _ _ _ _ \u27e9\n             < i' , b > \u2237 await b i' is' ds^ \u2261\u27e8 refl \u27e9\n             < i' , b > \u2237 as^                \u220e\n\n    bs^ : D\n    bs^ = corrupt os\u2080^ \u00b7 as^\n\n    bs'-eq-helper\u2081 : os\u2080' \u2261 T \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 ok < i' , b > \u2237 bs^\n    bs'-eq-helper\u2081 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 corrupt os\u2080' \u00b7 as' \u2261 corrupt t \u00b7 t')\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt (T \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as^)\n        \u2261\u27e8 corrupt-T _ _ _ \u27e9\n      ok < i' , b > \u2237 corrupt (tail\u2081 os\u2080') \u00b7 as^\n        \u2261\u27e8 refl \u27e9\n      ok < i' , b > \u2237 bs^ \u220e\n\n    bs'-eq-helper\u2082 : os\u2080' \u2261 F \u2237 tail\u2081 os\u2080' \u2192 bs' \u2261 error \u2237 bs^\n    bs'-eq-helper\u2082 h =\n      bs'\n        \u2261\u27e8 bs'ABP' \u27e9\n      corrupt os\u2080' \u00b7 as'\n        \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 corrupt os\u2080' \u00b7 as' \u2261 corrupt t \u00b7 t')\n                  h\n                  as'-eq\n                  refl\n        \u27e9\n      corrupt (F \u2237 tail\u2081 os\u2080') \u00b7 (< i' , b > \u2237 as^)\n        \u2261\u27e8 corrupt-F _ _ _ \u27e9\n      error \u2237 corrupt (tail\u2081 os\u2080') \u00b7 as^\n        \u2261\u27e8 refl \u27e9\n      error \u2237 bs^ \u220e\n\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs^ \u2228 bs' \u2261 error \u2237 bs^\n    bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2080')\n\n    cs^ : D\n    cs^ = ack (not b) \u00b7 bs^\n\n    cs'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs^ \u2192 cs' \u2261 b \u2237 cs^\n    cs'-eq-helper\u2081 h =\n      cs'\n      \u2261\u27e8 cs'ABP' \u27e9\n      ack (not b) \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack (not b) \u00b7 (ok < i' , b > \u2237 bs^)\n        \u2261\u27e8 ack-ok\u2262 _ _ _ _ (not-x\u2262x Bb) \u27e9\n      not (not b) \u2237 ack (not b) \u00b7 bs^\n        \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack (not b) \u00b7 bs^\n        \u2261\u27e8 refl \u27e9\n      b \u2237 cs^ \u220e\n\n    cs'-eq-helper\u2082 : bs' \u2261 error \u2237 bs^ \u2192 cs' \u2261 b \u2237 cs^\n    cs'-eq-helper\u2082 h =\n      cs'                             \u2261\u27e8 cs'ABP' \u27e9\n      ack (not b) \u00b7 bs'               \u2261\u27e8 \u00b7-rightCong h \u27e9\n      ack (not b) \u00b7 (error \u2237 bs^)     \u2261\u27e8 ack-error _ _ \u27e9\n      not (not b) \u2237 ack (not b) \u00b7 bs^ \u2261\u27e8 \u2237-leftCong (not-involutive Bb) \u27e9\n      b \u2237 ack (not b) \u00b7 bs^           \u2261\u27e8 refl \u27e9\n      b \u2237 cs^                         \u220e\n\n    cs'-eq : cs' \u2261 b \u2237 cs^\n    cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n    js'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs^ \u2192 js' \u2261 out (not b) \u00b7 bs^\n    js'-eq-helper\u2081 h  =\n      js'\n        \u2261\u27e8 js'ABP' \u27e9\n      out (not b) \u00b7 bs'\n        \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out (not b) \u00b7 (ok < i' , b > \u2237 bs^)\n        \u2261\u27e8 out-ok\u2262 (not b) b i' bs^ (not-x\u2262x Bb) \u27e9\n      out (not b) \u00b7 bs^ \u220e\n\n    js'-eq-helper\u2082 : bs' \u2261 error \u2237 bs^ \u2192 js' \u2261 out (not b) \u00b7 bs^\n    js'-eq-helper\u2082 h  =\n      js'                         \u2261\u27e8 js'ABP' \u27e9\n      out (not b) \u00b7 bs'           \u2261\u27e8 \u00b7-rightCong h \u27e9\n      out (not b) \u00b7 (error \u2237 bs^) \u2261\u27e8 out-error (not b) bs^ \u27e9\n      out (not b) \u00b7 bs^           \u220e\n\n    js'-eq : js' \u2261 out (not b) \u00b7 bs^\n    js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n    ds^-eq : ds^ \u2261 corrupt os\u2081^ \u00b7 (b \u2237 cs^)\n    ds^-eq = \u00b7-rightCong cs'-eq\n\n    ABP'IH : ABP' b i' is' os\u2080^ os\u2081^ as^ bs^ cs^ ds^ js'\n    ABP'IH = ds^-eq , refl , refl , refl , js'-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081 \u2208\n-- Fair, and hence by unfolding Fair we conclude that there are ft\u2081 :\n-- F*T and os\u2081'' : Fair, such that os\u2081' = ft\u2081 ++ os\u2081''.\n--\n-- We proceed by induction on ft\u2081 : F*T using helper.\n\nopen Helper\nlemma\u2082 : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2080' \u2192\n         Fair os\u2081' \u2192\n         ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n         Fair os\u2080''\n         \u2227 Fair os\u2081''\n         \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2080' Fos\u2081' abp' with Fair-unf Fos\u2081'\n... | ft , os\u2080'' , FTft , h , Fos\u2080'' =\n  helper Bb Fos\u2080' abp' ft os\u2080'' FTft Fos\u2080'' h\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"079485362fad6d34f46d04d1da4c3327a2c8b742","subject":"Uncomment","message":"Uncomment\n","repos":"spire\/spire","old_file":"proposal\/examples\/HierMatchExt.agda","new_file":"proposal\/examples\/HierMatchExt.agda","new_contents":"{-\nDemonstrating the technique of using type-changing functions\nto pattern match against types with higher-order arguments\nlike \u03a3 and \u03a0.\n\nThe technique was originally explored to compare higher-order\nsmall arguments that occur in \u03a6 types in\nTacticsMatchingFunctionCalls.agda\n\nThis file shows that the technique scales to large higher-order\nfunctions , in a closed universe with a predicative hierarchy\nof levels that supports elimType.\n\n-}\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Fin.Props\nopen import Data.Product\nopen import Data.String\nopen import Function\nopen import Relation.Binary.PropositionalEquality hiding ( inspect )\nopen Deprecated-inspect\nmodule HierMatchExt where\n\n----------------------------------------------------------------------\n\nplusrident : (n : \u2115) \u2192 n + 0 \u2261 n\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\nrecord Universe : Set\u2081 where\n  field\n    Codes : Set\n    Meaning : Codes \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Even : \u2115 \u2192 Set where\n  ezero : Even 0\n  esuc : {n : \u2115} \u2192 Even n \u2192 Even (2 + n)\n\ndata Odd : \u2115 \u2192 Set where\n  ozero : Odd 1\n  osuc : {n : \u2115} \u2192 Odd n \u2192 Odd (2 + n)\n\n----------------------------------------------------------------------\n\ndata TypeForm (U : Universe) : Set\n\u27e6_\/_\u27e7 : (U : Universe) \u2192 TypeForm U \u2192 Set\n\ndata TypeForm U where\n  `\u22a5 `\u22a4 `Bool `\u2115 `Type : TypeForm U\n  `Fin `Even `Odd : (n : \u2115) \u2192 TypeForm U\n  `\u03a0 `\u03a3 : (A : TypeForm U)\n    (B : \u27e6 U \/ A \u27e7 \u2192 TypeForm U)\n    \u2192 TypeForm U\n  `Id : (A : TypeForm U) (x y : \u27e6 U \/ A \u27e7) \u2192 TypeForm U\n  `\u27e6_\u27e7 : (A : Universe.Codes U) \u2192 TypeForm U\n\n\u27e6 U \/ `\u22a5 \u27e7 = \u22a5\n\u27e6 U \/ `\u22a4 \u27e7 = \u22a4\n\u27e6 U \/ `Bool \u27e7 = Bool\n\u27e6 U \/ `\u2115 \u27e7 = \u2115\n\u27e6 U \/ `Fin n \u27e7 = Fin n\n\u27e6 U \/ `Even n \u27e7 = Even n\n\u27e6 U \/ `Odd n \u27e7 = Odd n\n\u27e6 U \/ `\u03a0 A B \u27e7 = (a : \u27e6 U \/ A \u27e7) \u2192 \u27e6 U \/ B a \u27e7\n\u27e6 U \/ `\u03a3 A B \u27e7 = \u03a3 \u27e6 U \/ A \u27e7 (\u03bb a \u2192 \u27e6 U \/ B a \u27e7)\n\u27e6 U \/ `Id A x y \u27e7 = _\u2261_ {A = \u27e6 U \/ A \u27e7} x y\n\u27e6 U \/ `Type \u27e7 = Universe.Codes U\n\u27e6 U \/ `\u27e6 A \u27e7 \u27e7 = Universe.Meaning U A\n\n----------------------------------------------------------------------\n\n_`\u2192_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u2192 B = `\u03a0 A (\u03bb _ \u2192 B)\n\n_`\u00d7_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u00d7 B = `\u03a3 A (\u03bb _ \u2192 B)\n\nLevel : (\u2113 : \u2115) \u2192 Universe\nLevel zero = record { Codes = \u22a5 ; Meaning = \u03bb() }\nLevel (suc \u2113) = record { Codes = TypeForm (Level \u2113)\n                       ; Meaning = \u27e6_\/_\u27e7 (Level \u2113) }\n\nType : \u2115 \u2192 Set\nType \u2113 = TypeForm (Level \u2113)\n\n\u27e6_\u2223_\u27e7 : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set\n\u27e6 \u2113 \u2223 A \u27e7 = \u27e6 Level \u2113 \/ A \u27e7\n\n`Dynamic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Dynamic \u2113 = `\u03a3 `Type `\u27e6_\u27e7\n\n`Tactic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Tactic \u2113 = `Dynamic \u2113 `\u2192 `Dynamic \u2113\n\nTactic : (\u2113 : \u2115) \u2192 Set\nTactic \u2113 = \u27e6 suc \u2113 \u2223 `Tactic \u2113 \u27e7\n\n----------------------------------------------------------------------\n\nrm-plus0-Fin : (\u2113 : \u2115) \u2192 Tactic (suc \u2113)\nrm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , b))\n  with inspect (B n)\n... | `Fin m with-\u2261 p =\n  `\u03a0 `\u2115 (\u03bb x \u2192 `Id `Type (B x) (`Fin (x + 0)))\n  `\u2192\n  `\u03a3 `\u2115 `Fin\n  ,\n  \u03bb f \u2192 n , subst Fin\n    (plusrident n)\n    (subst (\u03bb A \u2192 \u27e6 \u2113 \u2223 A \u27e7) (f n) b)\n... | Bn with-\u2261 p = (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , subst (\u03bb x \u2192 \u27e6 \u2113 \u2223 B x \u27e7) refl b))\nrm-plus0-Fin \u2113 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin : (\u2113 : \u2115) \u2192\n  \u27e6 suc \u2113 \u2223 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7\n        `\u2192 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin n) \u27e7 \u27e7\neg-rm-plus0-Fin \u2113 (n , i) = proj\u2082\n  (rm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7 , (n , i)))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n","old_contents":"{-\nDemonstrating the technique of using type-changing functions\nto pattern match against types with higher-order arguments\nlike \u03a3 and \u03a0.\n\nThe technique was originally explored to compare higher-order\nsmall arguments that occur in \u03a6 types in\nTacticsMatchingFunctionCalls.agda\n\nThis file shows that the technique scales to large higher-order\nfunctions , in a closed universe with a predicative hierarchy\nof levels that supports elimType.\n\n-}\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Fin.Props\nopen import Data.Product\nopen import Data.String\nopen import Function\nopen import Relation.Binary.PropositionalEquality hiding ( inspect )\nopen Deprecated-inspect\nmodule HierMatchExt where\n\n----------------------------------------------------------------------\n\nplusrident : (n : \u2115) \u2192 n + 0 \u2261 n\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\nrecord Universe : Set\u2081 where\n  field\n    Codes : Set\n    Meaning : Codes \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Even : \u2115 \u2192 Set where\n  ezero : Even 0\n  esuc : {n : \u2115} \u2192 Even n \u2192 Even (2 + n)\n\ndata Odd : \u2115 \u2192 Set where\n  ozero : Odd 1\n  osuc : {n : \u2115} \u2192 Odd n \u2192 Odd (2 + n)\n\n----------------------------------------------------------------------\n\ndata TypeForm (U : Universe) : Set\n\u27e6_\/_\u27e7 : (U : Universe) \u2192 TypeForm U \u2192 Set\n\ndata TypeForm U where\n  `\u22a5 `\u22a4 `Bool `\u2115 `Type : TypeForm U\n  `Fin `Even `Odd : (n : \u2115) \u2192 TypeForm U\n  `\u03a0 `\u03a3 : (A : TypeForm U)\n    (B : \u27e6 U \/ A \u27e7 \u2192 TypeForm U)\n    \u2192 TypeForm U\n  `Id : (A : TypeForm U) (x y : \u27e6 U \/ A \u27e7) \u2192 TypeForm U\n  `\u27e6_\u27e7 : (A : Universe.Codes U) \u2192 TypeForm U\n\n\u27e6 U \/ `\u22a5 \u27e7 = \u22a5\n\u27e6 U \/ `\u22a4 \u27e7 = \u22a4\n\u27e6 U \/ `Bool \u27e7 = Bool\n\u27e6 U \/ `\u2115 \u27e7 = \u2115\n\u27e6 U \/ `Fin n \u27e7 = Fin n\n\u27e6 U \/ `Even n \u27e7 = Even n\n\u27e6 U \/ `Odd n \u27e7 = Odd n\n\u27e6 U \/ `\u03a0 A B \u27e7 = (a : \u27e6 U \/ A \u27e7) \u2192 \u27e6 U \/ B a \u27e7\n\u27e6 U \/ `\u03a3 A B \u27e7 = \u03a3 \u27e6 U \/ A \u27e7 (\u03bb a \u2192 \u27e6 U \/ B a \u27e7)\n\u27e6 U \/ `Id A x y \u27e7 = _\u2261_ {A = \u27e6 U \/ A \u27e7} x y\n\u27e6 U \/ `Type \u27e7 = Universe.Codes U\n\u27e6 U \/ `\u27e6 A \u27e7 \u27e7 = Universe.Meaning U A\n\n----------------------------------------------------------------------\n\n_`\u2192_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u2192 B = `\u03a0 A (\u03bb _ \u2192 B)\n\n_`\u00d7_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u00d7 B = `\u03a3 A (\u03bb _ \u2192 B)\n\nLevel : (\u2113 : \u2115) \u2192 Universe\nLevel zero = record { Codes = \u22a5 ; Meaning = \u03bb() }\nLevel (suc \u2113) = record { Codes = TypeForm (Level \u2113)\n                       ; Meaning = \u27e6_\/_\u27e7 (Level \u2113) }\n\nType : \u2115 \u2192 Set\nType \u2113 = TypeForm (Level \u2113)\n\n\u27e6_\u2223_\u27e7 : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set\n\u27e6 \u2113 \u2223 A \u27e7 = \u27e6 Level \u2113 \/ A \u27e7\n\n`Dynamic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Dynamic \u2113 = `\u03a3 `Type `\u27e6_\u27e7\n\n`Tactic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Tactic \u2113 = `Dynamic \u2113 `\u2192 `Dynamic \u2113\n\nTactic : (\u2113 : \u2115) \u2192 Set\nTactic \u2113 = \u27e6 suc \u2113 \u2223 `Tactic \u2113 \u27e7\n\n----------------------------------------------------------------------\n\nrm-plus0-Fin : (\u2113 : \u2115) \u2192 Tactic (suc \u2113)\nrm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , b))\n  with inspect (B n)\n... | `Fin m with-\u2261 p =\n  `\u03a0 `\u2115 (\u03bb x \u2192 `Id `Type (B x) (`Fin (x + 0)))\n  `\u2192\n  `\u03a3 `\u2115 `Fin\n  ,\n  \u03bb f \u2192 n , subst Fin\n    (plusrident n)\n    (subst (\u03bb A \u2192 \u27e6 \u2113 \u2223 A \u27e7) (f n) b)\n... | Bn with-\u2261 p = (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , subst (\u03bb x \u2192 \u27e6 \u2113 \u2223 B x \u27e7) refl b))\nrm-plus0-Fin \u2113 x = x\n\n----------------------------------------------------------------------\n\n-- eg-rm-plus0-Fin : (\u2113 : \u2115) \u2192\n--   \u27e6 suc \u2113 \u2223 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7\n--         `\u2192 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin n) \u27e7 \u27e7\n-- eg-rm-plus0-Fin \u2113 (n , i) = proj\u2082\n--   (rm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7 , (n , i)))\n--   (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"49783fece9819834dc0920e36e937d1a820c0909","subject":"flat-funs: more polishing work","message":"flat-funs: more polishing work\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_)\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nimport Level as L\nimport Function as F\nimport Data.Product as \u00d7\nopen V using (Vec; []; _\u2237_; _++_)\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bits using (Bits; _\u2192\u1d47_)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    El       : T \u2192 Set\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id    : \u2200 {A} \u2192 A `\u2192 A\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n    const : \u2200 {_A B} \u2192 El B \u2192 _A `\u2192 B\n\n    -- Products\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n  f \u2218 g = g >>> f\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = < fst >>> f , snd >>> g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = f *** id\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = id *** f\n\n  assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc = < fst >>> fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd >>> snd >\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = < < fst \u00d7 fst > >>> f ,\n                    < snd \u00d7 snd > >>> g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons >>> fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons >>> snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons >>> < f \u00d7 g > >>> cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > >>> cons\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                >>> assoc\u2032\n                >>> first f\n                >>> foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons >>> foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                >>> assoc\n                >>> second (foldr f)\n                >>> f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons >>> swap >>> foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map {zero}  f = nil\n  map {suc n} f = < f \u2237 map f >\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                  >>> < f `zip` (zipWith f) >\n                  >>> cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id , nil > >>> cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd >>> singleton\n  snoc {suc n} = first uncons >>> assoc >>> second snoc >>> cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons >>> swap >>> first reverse >>> snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n               >>> assoc\n               >>> second append\n               >>> cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                >>> second (splitAt m)\n                >>> assoc\u2032\n                >>> first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail >>> drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) >>> < folda n f \u00d7 folda n f > >>> f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail >>> last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons >>> second concat >>> append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k >>> second (group n k) >>> cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f >>> concat\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = uncons >>> < f \u00d7 \u229b fs > >>> cons\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > >>> cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz >>> f , tabulate (fs >>> f) > >>> cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst >>> elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail >>> lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U F.id -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U Bits _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U Fin _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id (\u03bb f g x \u2192 g (f x)) F.const \u00d7.<_,_> proj\u2081 proj\u2082\n             _ (F.const []) (uncurry _\u2237_) V.uncons\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _>>>_ const <_,_> (\u03bb {A} \u2192 V.take A)\n                    (\u03bb {A} \u2192 V.drop A) (const []) (const []) id id\n  where\n  open FlatFuns bitsFun\u266dFuns\n  open FlatFunsOps fun\u266dOps using (id; _>>>_; const)\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > x = f x ++ g x\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) \u2192 S.El A\u2080 \u00d7 T.El A\u2081 })\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A \u2192 S.El A \u00d7 T.El _)\n                            (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._>>>_ T._>>>_) < S.const \u00d7 T.const >\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._>>>_ T._>>>_) < S.const \u00d7 T.const >\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ \u2192 A) (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ (F.const 0) _\u2294_ 0 0 0 0 0 0\n\nmodule TimeOps = FlatFunsOps timeOps\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ const _+_ 0 0 0 0 0 0\n  where open FlatFuns (constFuns \u2115)\n        const : \u2200 {_A B} \u2192 El B \u2192 _A `\u2192 B\n        const {_} {_} x = x\n\nmodule SpaceOps = FlatFunsOps spaceOps\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_; _++_)\nimport Level as L\nimport Function as F\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nimport Data.Product as \u00d7\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bits using (Bits; _\u2192\u1d47_)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    El      : T \u2192 Set\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n    const : \u2200 {_A B} \u2192 El B \u2192 _A `\u2192 B\n\n    -- Products\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Vectors\n    nil : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = < fst >>> f , snd >>> g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = f *** id\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = id *** f\n\n  assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc = < fst >>> fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd >>> snd >\n\n  `\u00d7-zip : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  `\u00d7-zip f g = < < fst >>> fst , snd >>> fst > >>> f ,\n                 < fst >>> snd , snd >>> snd > >>> g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons >>> fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons >>> snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons >>> < f \u00d7 g > >>> cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > >>> cons\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                >>> assoc\u2032\n                >>> first f\n                >>> foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons >>> foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                >>> assoc\n                >>> second (foldr f)\n                >>> f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons >>> swap >>> foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map {zero}  f = nil\n  map {suc n} f = < f \u2237 map f >\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                  >>> `\u00d7-zip f (zipWith f)\n                  >>> cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id , nil > >>> cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd >>> singleton\n  snoc {suc n} = first uncons >>> assoc >>> second snoc >>> cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons >>> swap >>> first reverse >>> snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n               >>> assoc\n               >>> second append\n               >>> cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                >>> second (splitAt m)\n                >>> assoc\u2032\n                >>> first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail >>> drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) >>> < folda n f \u00d7 folda n f > >>> f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail >>> last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons >>> second concat >>> append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k >>> second (group n k) >>> cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f >>> concat\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = uncons >>> < f \u00d7 \u229b fs > >>> cons\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > >>> cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz >>> f , tabulate (fs >>> f) > >>> cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst >>> elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail >>> lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U F.id -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U Bits _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U Fin _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id (\u03bb f g x \u2192 g (f x)) F.const \u00d7.<_,_> proj\u2081 proj\u2082\n             _ (F.const []) (uncurry _\u2237_) uncons\n  where\n  uncons : \u2200 {n A} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _>>>_ const <_,_> (\u03bb {A} \u2192 V.take A)\n                    (\u03bb {A} \u2192 V.drop A) (const []) (const []) id id\n  where\n  open FlatFuns bitsFun\u266dFuns\n  open FlatFunsOps fun\u266dOps using (id; _>>>_; const)\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > x = f x ++ g x\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) \u2192 S.El A\u2080 \u00d7 T.El A\u2081 })\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A \u2192 S.El A \u00d7 T.El _)\n                            (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._>>>_ T._>>>_) < S.const \u00d7 T.const >\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._>>>_ T._>>>_) < S.const \u00d7 T.const >\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ \u2192 A) (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ (F.const 0) _\u2294_ 0 0 0 0 0 0\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ const _+_ 0 0 0 0 0 0\n  where open FlatFuns (constFuns \u2115)\n        const : \u2200 {_A B} \u2192 El B \u2192 _A `\u2192 B\n        const {_} {_} x = x\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1860f92cd71fc4cd326cb8563fd03149af8bb53c","subject":"Two.Equality: export subst and \u21d2\u2261","message":"Two.Equality: export subst and \u21d2\u2261\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Two\/Equality.agda","new_file":"lib\/Data\/Two\/Equality.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Data.Two hiding (_\u225f_; decSetoid)\nopen import Type\nopen import Relation.Binary.NP\nopen import Relation.Nullary\nopen import Function\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Data.Two.Equality where\n\nmodule \u2713-== where\n\n    _\u2248_ : (x y : \ud835\udfda) \u2192 \u2605\u2080\n    x \u2248 y = \u2713 (x == y)\n\n    subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n    subst _ {0\u2082} {0\u2082} _ = id\n    subst _ {1\u2082} {1\u2082} _ = id\n    subst _ {0\u2082} {1\u2082} ()\n    subst _ {1\u2082} {0\u2082} ()\n\n    \u21d2\u2261 : _\u2248_ \u21d2 _\u2261_\n    \u21d2\u2261 = substitutive\u21d2\u2261 subst\n\n    decSetoid : DecSetoid _ _\n    decSetoid = record { Carrier = \ud835\udfda; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n      where\n        refl : Reflexive _\u2248_\n        refl {0\u2082} = _\n        refl {1\u2082} = _\n\n        sym : Symmetric _\u2248_\n        sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n        trans : Transitive _\u2248_\n        trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n        _\u225f_ : Decidable _\u2248_\n        0\u2082 \u225f 0\u2082 = yes _\n        1\u2082 \u225f 1\u2082 = yes _\n        0\u2082 \u225f 1\u2082 = no (\u03bb())\n        1\u2082 \u225f 0\u2082 = no (\u03bb())\n\n        isEquivalence : IsEquivalence _\u2248_\n        isEquivalence = record { refl  = \u03bb {x} \u2192 refl {x}\n                               ; sym   = \u03bb {x} {y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n        isDecEquivalence : IsDecEquivalence _\u2248_\n        isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n    open DecSetoid decSetoid public hiding (_\u2248_)\n\nmodule ==-\u22611\u2082 where\n\n    _\u2248_ : (x y : \ud835\udfda) \u2192 \u2605\u2080\n    x \u2248 y = (x == y) \u2261 1\u2082\n\n    subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n    subst _ {0\u2082} {0\u2082} _ = id\n    subst _ {1\u2082} {1\u2082} _ = id\n    subst _ {0\u2082} {1\u2082} ()\n    subst _ {1\u2082} {0\u2082} ()\n\n    \u21d2\u2261 : _\u2248_ \u21d2 _\u2261_\n    \u21d2\u2261 = substitutive\u21d2\u2261 subst\n\n    decSetoid : DecSetoid _ _\n    decSetoid = record { Carrier = \ud835\udfda; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n      where\n        refl : Reflexive _\u2248_\n        refl {0\u2082} = \u2261.refl\n        refl {1\u2082} = \u2261.refl\n\n        sym : Symmetric _\u2248_\n        sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n        trans : Transitive _\u2248_\n        trans {x} x\u2248y y\u2248z = subst (_\u2248_ x) y\u2248z x\u2248y\n\n        _\u225f_ : Decidable _\u2248_\n        0\u2082 \u225f 0\u2082 = yes \u2261.refl\n        1\u2082 \u225f 1\u2082 = yes \u2261.refl\n        0\u2082 \u225f 1\u2082 = no (\u03bb())\n        1\u2082 \u225f 0\u2082 = no (\u03bb())\n\n        isEquivalence : IsEquivalence _\u2248_\n        isEquivalence = record { refl  = \u03bb {x} \u2192 refl {x}\n                               ; sym   = \u03bb {x} {y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n        isDecEquivalence : IsDecEquivalence _\u2248_\n        isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n    open DecSetoid decSetoid public hiding (_\u2248_)\n\nneg-xor : \u2200 b\u2080 b\u2081 \u2192 b\u2080 == b\u2081 \u2261 not (b\u2080 xor b\u2081)\nneg-xor 0\u2082 b = \u2261.refl\nneg-xor 1\u2082 b = \u2261.sym (not-involutive b)\n\ncomm : \u2200 b\u2080 b\u2081 \u2192 b\u2080 == b\u2081 \u2261 b\u2081 == b\u2080\ncomm 0\u2082 0\u2082 = \u2261.refl\ncomm 0\u2082 1\u2082 = \u2261.refl\ncomm 1\u2082 0\u2082 = \u2261.refl\ncomm 1\u2082 1\u2082 = \u2261.refl\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Data.Two hiding (_\u225f_; decSetoid)\nopen import Type\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Data.Two.Equality where\n\nmodule \u2713-== where\n\n    decSetoid : DecSetoid _ _\n    decSetoid = record { Carrier = \ud835\udfda; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n      where\n        _\u2248_ : (x y : \ud835\udfda) \u2192 \u2605\u2080\n        x \u2248 y = \u2713 (x == y)\n\n        refl : Reflexive _\u2248_\n        refl {0\u2082} = _\n        refl {1\u2082} = _\n\n        subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n        subst _ {0\u2082} {0\u2082} _ = id\n        subst _ {1\u2082} {1\u2082} _ = id\n        subst _ {0\u2082} {1\u2082} ()\n        subst _ {1\u2082} {0\u2082} ()\n\n        sym : Symmetric _\u2248_\n        sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n        trans : Transitive _\u2248_\n        trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n        _\u225f_ : Decidable _\u2248_\n        0\u2082 \u225f 0\u2082 = yes _\n        1\u2082 \u225f 1\u2082 = yes _\n        0\u2082 \u225f 1\u2082 = no (\u03bb())\n        1\u2082 \u225f 0\u2082 = no (\u03bb())\n\n        isEquivalence : IsEquivalence _\u2248_\n        isEquivalence = record { refl  = \u03bb {x} \u2192 refl {x}\n                               ; sym   = \u03bb {x} {y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n        isDecEquivalence : IsDecEquivalence _\u2248_\n        isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n    open DecSetoid decSetoid public hiding (_\u2248_; _\u225f_)\n\nmodule ==-\u22611\u2082 where\n\n    decSetoid : DecSetoid _ _\n    decSetoid = record { Carrier = \ud835\udfda; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n      where\n        _\u2248_ : (x y : \ud835\udfda) \u2192 \u2605\u2080\n        x \u2248 y = (x == y) \u2261 1\u2082\n\n        refl : Reflexive _\u2248_\n        refl {0\u2082} = \u2261.refl\n        refl {1\u2082} = \u2261.refl\n\n        subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n        subst _ {0\u2082} {0\u2082} _ = id\n        subst _ {1\u2082} {1\u2082} _ = id\n        subst _ {0\u2082} {1\u2082} ()\n        subst _ {1\u2082} {0\u2082} ()\n\n        sym : Symmetric _\u2248_\n        sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n        trans : Transitive _\u2248_\n        trans {x} x\u2248y y\u2248z = subst (_\u2248_ x) y\u2248z x\u2248y\n\n        _\u225f_ : Decidable _\u2248_\n        0\u2082 \u225f 0\u2082 = yes \u2261.refl\n        1\u2082 \u225f 1\u2082 = yes \u2261.refl\n        0\u2082 \u225f 1\u2082 = no (\u03bb())\n        1\u2082 \u225f 0\u2082 = no (\u03bb())\n\n        isEquivalence : IsEquivalence _\u2248_\n        isEquivalence = record { refl  = \u03bb {x} \u2192 refl {x}\n                               ; sym   = \u03bb {x} {y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n        isDecEquivalence : IsDecEquivalence _\u2248_\n        isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n    open DecSetoid decSetoid public\n\nneg-xor : \u2200 b\u2080 b\u2081 \u2192 b\u2080 == b\u2081 \u2261 not (b\u2080 xor b\u2081)\nneg-xor 0\u2082 b = \u2261.refl\nneg-xor 1\u2082 b = \u2261.sym (not-involutive b)\n\ncomm : \u2200 b\u2080 b\u2081 \u2192 b\u2080 == b\u2081 \u2261 b\u2081 == b\u2080\ncomm 0\u2082 0\u2082 = \u2261.refl\ncomm 0\u2082 1\u2082 = \u2261.refl\ncomm 1\u2082 0\u2082 = \u2261.refl\ncomm 1\u2082 1\u2082 = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6811ac8de37eac34033c36c115fc359dfa88edc5","subject":"Turn context validity into a datatype","message":"Turn context validity into a datatype\n\nThis avoids inelegant embedded equality proofs. Also revise a bit the\nclient code.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 v1 v2 dv \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb v1 v2 dv x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) (da , db) (daa , dbb) = (fromto\u2192valid  _ _ _ daa) , (fromto\u2192valid _ _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} f1 f2 df dff = \u03bb a da ada \u2192\n  fromto\u2192valid _ _ _ (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto _ _ ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid _ _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 _ _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n-- XXX This would be more elegant as a datatype \u2014 that would avoid the need for\n-- an equality proof.\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1' \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n   [ \u03c4 ] dv from v1 to v2 \u00d7 v1 \u2261 v1' \u00d7 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} {\u2205} tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {_ \u2022 _} {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} {dv \u2022 .v1 \u2022 d\u03c1} (dvv , refl , d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {\u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive \u03c4 t (daa , refl , d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 v1 v2 dv \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb v1 v2 dv x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) (da , db) (daa , dbb) = (fromto\u2192valid  _ _ _ daa) , (fromto\u2192valid _ _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} f1 f2 df dff = \u03bb a da ada \u2192\n  fromto\u2192valid _ _ _ (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto _ _ ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid _ _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 _ _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ab15e9ee2e4fab1712f707ef0f885e95995a8e73","subject":"Renamed data constructors for the Even predicate.","message":"Renamed data constructors for the Even predicate.\n\nIgnore-this: 95d8f33e5eedf116169e2910b36df9d7\n\ndarcs-hash:20120308132006-3bd4e-3429ea1e04d20b7aeb35b73858d37ff9eb1f23c8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/Even.agda","new_file":"Draft\/FOTC\/Data\/Nat\/Even.agda","new_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 08 March 2012.\n\nmodule Draft.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Even : D \u2192 Set where\n  zE :                  Even zero\n  nE : \u2200 {d} \u2192 Even d \u2192 Even (succ\u2081 (succ\u2081 d))\n\nEven-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {d} \u2192 A d \u2192 A (succ\u2081 (succ\u2081 d))) \u2192\n          \u2200 {d} \u2192 Even d \u2192 A d\nEven-ind A A0 h zE      = A0\nEven-ind A A0 h (nE Ed) = h (Even-ind A A0 h Ed)\n","old_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 02 March 2012.\n\nmodule Draft.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Even : D \u2192 Set where\n  zeroeven : Even zero\n  nexteven : \u2200 {d} \u2192 Even d \u2192 Even (succ\u2081 (succ\u2081 d))\n\nEven-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {d} \u2192 A d \u2192 A (succ\u2081 (succ\u2081 d))) \u2192\n          \u2200 {d} \u2192 Even d \u2192 A d\nEven-ind A A0 h zeroeven      = A0\nEven-ind A A0 h (nexteven Ed) = h (Even-ind A A0 h Ed)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"58d573424f0945e2a1df71e3e22bac9f751fc1e3","subject":"proved that holes-disjoint is symmetric","message":"proved that holes-disjoint is symmetric\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nopen import structural\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n\n  mutual\n    -- this looks good but may not *quite* work because of the\n    -- weakening calls in the two lambda cases\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint hd (EASubsume x x\u2081 x\u2082 x\u2083) E2 = expand-synth-disjoint hd x\u2082 E2\n    expand-ana-disjoint (HDLam1 hd) (EALam x\u2081 x\u2082 ex1) E2 = expand-ana-disjoint hd ex1 (weaken-ana-expand x\u2081 E2)\n    expand-ana-disjoint (HDHole x) EAEHole E2 = ##-comm (expand-new-disjoint-ana x E2)\n    expand-ana-disjoint (HDNEHole x hd) (EANEHole x\u2081 x\u2082) E2 = disjoint-parts (expand-synth-disjoint hd x\u2082 E2) (##-comm (expand-new-disjoint-ana x E2))\n\n    expand-synth-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d2 \u03c41 ~> e1' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-synth-disjoint HDConst ESConst ana = empty-disj _\n    expand-synth-disjoint (HDAsc hd) (ESAsc x) ana = expand-ana-disjoint hd x ana\n    expand-synth-disjoint HDVar (ESVar x\u2081) ana = empty-disj _\n    expand-synth-disjoint (HDLam1 hd) () ana\n    expand-synth-disjoint (HDLam2 hd) (ESLam x\u2081 synth) ana = expand-synth-disjoint hd synth (weaken-ana-expand x\u2081 ana)\n    expand-synth-disjoint (HDHole x) ESEHole ana = ##-comm (expand-new-disjoint-ana x ana)\n    expand-synth-disjoint (HDNEHole x hd) (ESNEHole x\u2081 synth) ana = disjoint-parts (expand-synth-disjoint hd synth ana) (##-comm (expand-new-disjoint-ana x ana))\n    expand-synth-disjoint (HDAp hd hd\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) ana = disjoint-parts (expand-ana-disjoint hd x\u2084 ana) (expand-ana-disjoint hd\u2081 x\u2085 ana)\n\n\n  -- these lemmas are all structurally recursive. morally, they\n  -- establish the properties about reduction that would be obvious \/\n  -- baked into Agda if holes-disjoint was defined as a function\n  -- rather than a judgement (datatype), or if we had defined all the\n  -- O(n^2) cases rather than relying on a little indirection to only\n  -- have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since u == u; it's not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (abeit vacuously)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nopen import structural\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n\n  mutual\n    -- this looks good but may not *quite* work because of the\n    -- weakening calls in the two lambda cases\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint hd (EASubsume x x\u2081 x\u2082 x\u2083) E2 = expand-synth-disjoint hd x\u2082 E2\n    expand-ana-disjoint (HDLam1 hd) (EALam x\u2081 x\u2082 ex1) E2 = expand-ana-disjoint hd ex1 (weaken-ana-expand x\u2081 E2)\n    expand-ana-disjoint (HDHole x) EAEHole E2 = ##-comm (expand-new-disjoint-ana x E2)\n    expand-ana-disjoint (HDNEHole x hd) (EANEHole x\u2081 x\u2082) E2 = disjoint-parts (expand-synth-disjoint hd x\u2082 E2) (##-comm (expand-new-disjoint-ana x E2))\n\n    expand-synth-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d2 \u03c41 ~> e1' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-synth-disjoint HDConst ESConst ana = empty-disj _\n    expand-synth-disjoint (HDAsc hd) (ESAsc x) ana = expand-ana-disjoint hd x ana\n    expand-synth-disjoint HDVar (ESVar x\u2081) ana = empty-disj _\n    expand-synth-disjoint (HDLam1 hd) () ana\n    expand-synth-disjoint (HDLam2 hd) (ESLam x\u2081 synth) ana = expand-synth-disjoint hd synth (weaken-ana-expand x\u2081 ana)\n    expand-synth-disjoint (HDHole x) ESEHole ana = ##-comm (expand-new-disjoint-ana x ana)\n    expand-synth-disjoint (HDNEHole x hd) (ESNEHole x\u2081 synth) ana = disjoint-parts (expand-synth-disjoint hd synth ana) (##-comm (expand-new-disjoint-ana x ana))\n    expand-synth-disjoint (HDAp hd hd\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) ana = disjoint-parts (expand-ana-disjoint hd x\u2084 ana) (expand-ana-disjoint hd\u2081 x\u2085 ana)\n\n\n\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) = HDAsc {!!}\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) = {!!}\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) = {!!}\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) = {!!}\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) = {!!}\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) = {!!}\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) = {!!}\n\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since u == u; it's not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (abeit vacuously)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c71d9bf25d860bc65cb2c8a3e4ad0f801328fb6e","subject":"Added missing ATP definitions.","message":"Added missing ATP definitions.\n\nIgnore-this: 3b3c373c2dd18ce05ccc1efd4812a2b4\n\ndarcs-hash:20110205131613-3bd4e-50691c642fd8de075a1c33c365bf096588af93a0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/Numbers.agda","new_file":"Draft\/McCarthy91\/Numbers.agda","new_contents":"------------------------------------------------------------------------------\n-- Some naturales numbers\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Numbers where\n\nopen import LTC.Base\n\n------------------------------------------------------------------------------\n\none    = succ zero\ntwo    = succ one\nthree  = succ two\nfour   = succ three\nfive   = succ four\nsix    = succ five\nseven  = succ six\neight  = succ seven\nnine   = succ eight\nten    = succ nine\n\n{-# ATP definition one #-}\n{-# ATP definition two #-}\n{-# ATP definition three #-}\n{-# ATP definition four #-}\n{-# ATP definition five #-}\n{-# ATP definition six #-}\n{-# ATP definition seven #-}\n{-# ATP definition eight #-}\n{-# ATP definition nine #-}\n{-# ATP definition ten #-}\n\neleven    = succ ten\ntwelve    = succ eleven\nthirteen  = succ twelve\nfourteen  = succ thirteen\nfifteen   = succ fourteen\nsixteen   = succ fifteen\nseventeen = succ sixteen\neighteen  = succ seventeen\nnineteen  = succ eighteen\ntwenty    = succ nineteen\n\n{-# ATP definition eleven #-}\n{-# ATP definition twelve #-}\n{-# ATP definition thirteen #-}\n{-# ATP definition fourteen #-}\n{-# ATP definition fifteen #-}\n{-# ATP definition sixteen #-}\n{-# ATP definition seventeen #-}\n{-# ATP definition eighteen #-}\n{-# ATP definition nineteen #-}\n{-# ATP definition twenty #-}\n\ntwenty-one   = succ twenty\ntwenty-two   = succ twenty-one\ntwenty-three = succ twenty-two\ntwenty-four  = succ twenty-three\ntwenty-five  = succ twenty-four\ntwenty-six   = succ twenty-five\ntwenty-seven = succ twenty-six\ntwenty-eight = succ twenty-seven\ntwenty-nine  = succ twenty-eight\nthirty       = succ twenty-nine\n\n{-# ATP definition twenty-one #-}\n{-# ATP definition twenty-two #-}\n{-# ATP definition twenty-three #-}\n{-# ATP definition twenty-four #-}\n{-# ATP definition twenty-five #-}\n{-# ATP definition twenty-six #-}\n{-# ATP definition twenty-seven #-}\n{-# ATP definition twenty-eight #-}\n{-# ATP definition twenty-nine #-}\n{-# ATP definition thirty #-}\n\nthirty-one   = succ thirty\nthirty-two   = succ thirty-one\nthirty-three = succ thirty-two\nthirty-four  = succ thirty-three\nthirty-five  = succ thirty-four\nthirty-six   = succ thirty-five\nthirty-seven = succ thirty-six\nthirty-eight = succ thirty-seven\nthirty-nine  = succ thirty-eight\nforty        = succ thirty-nine\n\n{-# ATP definition thirty-one #-}\n{-# ATP definition thirty-two #-}\n{-# ATP definition thirty-three #-}\n{-# ATP definition thirty-four #-}\n{-# ATP definition thirty-five #-}\n{-# ATP definition thirty-six #-}\n{-# ATP definition thirty-seven #-}\n{-# ATP definition thirty-eight #-}\n{-# ATP definition thirty-nine #-}\n{-# ATP definition forty #-}\n\nforty-one   = succ forty\nforty-two   = succ forty-one\nforty-three = succ forty-two\nforty-four  = succ forty-three\nforty-five  = succ forty-four\nforty-six   = succ forty-five\nforty-seven = succ forty-six\nforty-eight = succ forty-seven\nforty-nine  = succ forty-eight\nfifty       = succ forty-nine\n\n{-# ATP definition forty-one #-}\n{-# ATP definition forty-two #-}\n{-# ATP definition forty-three #-}\n{-# ATP definition forty-four #-}\n{-# ATP definition forty-five #-}\n{-# ATP definition forty-six #-}\n{-# ATP definition forty-seven #-}\n{-# ATP definition forty-eight #-}\n{-# ATP definition forty-nine #-}\n{-# ATP definition fifty #-}\n\nfifty-one   = succ fifty\nfifty-two   = succ fifty-one\nfifty-three = succ fifty-two\nfifty-four  = succ fifty-three\nfifty-five  = succ fifty-four\nfifty-six   = succ fifty-five\nfifty-seven = succ fifty-six\nfifty-eight = succ fifty-seven\nfifty-nine  = succ fifty-eight\nsixty       = succ fifty-nine\n\n{-# ATP definition fifty-one #-}\n{-# ATP definition fifty-two #-}\n{-# ATP definition fifty-three #-}\n{-# ATP definition fifty-four #-}\n{-# ATP definition fifty-five #-}\n{-# ATP definition fifty-six #-}\n{-# ATP definition fifty-seven #-}\n{-# ATP definition fifty-eight #-}\n{-# ATP definition fifty-nine #-}\n{-# ATP definition sixty #-}\n\nsixty-one   = succ sixty\nsixty-two   = succ sixty-one\nsixty-three = succ sixty-two\nsixty-four  = succ sixty-three\nsixty-five  = succ sixty-four\nsixty-six   = succ sixty-five\nsixty-seven = succ sixty-six\nsixty-eight = succ sixty-seven\nsixty-nine  = succ sixty-eight\nseventy     = succ sixty-nine\n\n{-# ATP definition sixty-one #-}\n{-# ATP definition sixty-two #-}\n{-# ATP definition sixty-three #-}\n{-# ATP definition sixty-four #-}\n{-# ATP definition sixty-five #-}\n{-# ATP definition sixty-six #-}\n{-# ATP definition sixty-seven #-}\n{-# ATP definition sixty-eight #-}\n{-# ATP definition sixty-nine #-}\n{-# ATP definition seventy #-}\n\nseventy-one   = succ seventy\nseventy-two   = succ seventy-one\nseventy-three = succ seventy-two\nseventy-four  = succ seventy-three\nseventy-five  = succ seventy-four\nseventy-six   = succ seventy-five\nseventy-seven = succ seventy-six\nseventy-eight = succ seventy-seven\nseventy-nine  = succ seventy-eight\neighty       = succ seventy-nine\n\n{-# ATP definition seventy-one #-}\n{-# ATP definition seventy-two #-}\n{-# ATP definition seventy-three #-}\n{-# ATP definition seventy-four #-}\n{-# ATP definition seventy-five #-}\n{-# ATP definition seventy-six #-}\n{-# ATP definition seventy-seven #-}\n{-# ATP definition seventy-eight #-}\n{-# ATP definition seventy-nine #-}\n{-# ATP definition eighty #-}\n\neighty-one   = succ eighty\neighty-two   = succ eighty-one\neighty-three = succ eighty-two\neighty-four  = succ eighty-three\neighty-five  = succ eighty-four\neighty-six   = succ eighty-five\neighty-seven = succ eighty-six\neighty-eight = succ eighty-seven\neighty-nine  = succ eighty-eight\nninety       = succ eighty-nine\n\n{-# ATP definition eighty-one #-}\n{-# ATP definition eighty-two #-}\n{-# ATP definition eighty-three #-}\n{-# ATP definition eighty-four #-}\n{-# ATP definition eighty-five #-}\n{-# ATP definition eighty-six #-}\n{-# ATP definition eighty-seven #-}\n{-# ATP definition eighty-eight #-}\n{-# ATP definition eighty-nine #-}\n{-# ATP definition ninety #-}\n\nninety-one   = succ ninety\nninety-two   = succ ninety-one\nninety-three = succ ninety-two\nninety-four  = succ ninety-three\nninety-five  = succ ninety-four\nninety-six   = succ ninety-five\nninety-seven = succ ninety-six\nninety-eight = succ ninety-seven\nninety-nine  = succ ninety-eight\none-hundred  = succ ninety-nine\n\n{-# ATP definition ninety-one #-}\n{-# ATP definition ninety-two #-}\n{-# ATP definition ninety-three #-}\n{-# ATP definition ninety-four #-}\n{-# ATP definition ninety-five #-}\n{-# ATP definition ninety-six #-}\n{-# ATP definition ninety-seven #-}\n{-# ATP definition ninety-eight #-}\n{-# ATP definition ninety-nine #-}\n{-# ATP definition one-hundred #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some naturales numbers\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Numbers where\n\nopen import LTC.Base\n\n------------------------------------------------------------------------------\n\none    = succ zero\ntwo    = succ one\nthree  = succ two\nfour   = succ three\nfive   = succ four\nsix    = succ five\nseven  = succ six\neight  = succ seven\nnine   = succ eight\nten    = succ nine\n\neleven    = succ ten\ntwelve    = succ eleven\nthirteen  = succ twelve\nfourteen  = succ thirteen\nfifteen   = succ fourteen\nsixteen   = succ fifteen\nseventeen = succ sixteen\neighteen  = succ seventeen\nnineteen  = succ eighteen\n\ntwenty       = succ nineteen\ntwenty-one   = succ twenty\ntwenty-two   = succ twenty-one\ntwenty-three = succ twenty-two\ntwenty-four  = succ twenty-three\ntwenty-five  = succ twenty-four\ntwenty-six   = succ twenty-five\ntwenty-seven = succ twenty-six\ntwenty-eight = succ twenty-seven\ntwenty-nine  = succ twenty-eight\n\nthirty       = succ twenty-nine\nthirty-one   = succ thirty\nthirty-two   = succ thirty-one\nthirty-three = succ thirty-two\nthirty-four  = succ thirty-three\nthirty-five  = succ thirty-four\nthirty-six   = succ thirty-five\nthirty-seven = succ thirty-six\nthirty-eight = succ thirty-seven\nthirty-nine  = succ thirty-eight\n\nforty       = succ thirty-nine\nforty-one   = succ forty\nforty-two   = succ forty-one\nforty-three = succ forty-two\nforty-four  = succ forty-three\nforty-five  = succ forty-four\nforty-six   = succ forty-five\nforty-seven = succ forty-six\nforty-eight = succ forty-seven\nforty-nine  = succ forty-eight\n\nfifty       = succ forty-nine\nfifty-one   = succ fifty\nfifty-two   = succ fifty-one\nfifty-three = succ fifty-two\nfifty-four  = succ fifty-three\nfifty-five  = succ fifty-four\nfifty-six   = succ fifty-five\nfifty-seven = succ fifty-six\nfifty-eight = succ fifty-seven\nfifty-nine  = succ fifty-eight\n\nsixty       = succ fifty-nine\nsixty-one   = succ sixty\nsixty-two   = succ sixty-one\nsixty-three = succ sixty-two\nsixty-four  = succ sixty-three\nsixty-five  = succ sixty-four\nsixty-six   = succ sixty-five\nsixty-seven = succ sixty-six\nsixty-eight = succ sixty-seven\nsixty-nine  = succ sixty-eight\n\nseventy       = succ sixty-nine\nseventy-one   = succ seventy\nseventy-two   = succ seventy-one\nseventy-three = succ seventy-two\nseventy-four  = succ seventy-three\nseventy-five  = succ seventy-four\nseventy-six   = succ seventy-five\nseventy-seven = succ seventy-six\nseventy-eight = succ seventy-seven\nseventy-nine  = succ seventy-eight\n\neighty       = succ seventy-nine\neighty-one   = succ eighty\neighty-two   = succ eighty-one\neighty-three = succ eighty-two\neighty-four  = succ eighty-three\neighty-five  = succ eighty-four\neighty-six   = succ eighty-five\neighty-seven = succ eighty-six\neighty-eight = succ eighty-seven\neighty-nine  = succ eighty-eight\n\nninety       = succ eighty-nine\nninety-one   = succ ninety\nninety-two   = succ ninety-one\nninety-three = succ ninety-two\nninety-four  = succ ninety-three\nninety-five  = succ ninety-four\nninety-six   = succ ninety-five\nninety-seven = succ ninety-six\nninety-eight = succ ninety-seven\nninety-nine  = succ ninety-eight\n\none-hundred = succ ninety-nine\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5d2800ead3ebd1b56a9d52097aa972bb180a479b","subject":"Switch to Agda's support for well-founded induction","message":"Switch to Agda's support for well-founded induction\n\nUgh, this seems as annoying as what you have in Coq :-(\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalUntyped.agda","new_file":"Thesis\/ANormalUntyped.agda","new_contents":"module Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {t : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs t \u2193 dclosure t \u03c1 d\u03c1\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k <\u2032 n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 = <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n = proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n = proj\u2082 (rrelTV3 n)\n\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    (v1 v2 : Val Uni) \u2192\n    \u2200 j n2 (j<k : j <\u2032 k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 j j<k v1 dv v2\n    where\n      s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n      -- If j = 0, we can't do a recursive call on (k - j) because that's not\n      -- well-founded. Luckily, we don't need to, just use relV as defined at\n      -- the same level.\n      -- Yet, this will be a pain in proofs.\n      s-rrelV3 0 _ = rrelV3-k\n      s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelT3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 _ _ _ = \u22a5\n\n-- -- [_]\u03c4  from v1 to v2) :\n-- -- [ \u03c4 ]\u03c4 dv from v1 to v2) =\n\n-- -- data [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set where\n-- --   v\u2205 : [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205\n-- --   _v\u2022_ : \u2200 {\u03c4 \u0394 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n-- --     (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n-- --     (d\u03c1\u03c1 : [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12) \u2192\n-- --     [ \u03c4 \u2022 \u0394 ]\u0394 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n-- data ErrVal : Set where\n--   Done : (v : Val Uni) \u2192 ErrVal\n--   Error : ErrVal\n--   TimeOut : ErrVal\n\n-- _>>=_ : ErrVal \u2192 (Val Uni \u2192 ErrVal) \u2192 ErrVal\n-- Done v >>= f = f v\n-- Error >>= f = Error\n-- TimeOut >>= f = TimeOut\n\n-- Res : Set\n-- Res = \u2115 \u2192 ErrVal\n\n-- -- eval-fun : \u2200 {\u0393} \u2192 Fun \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n-- -- eval-term : \u2200 {\u0393} \u2192 Term \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n\n-- -- apply : Val Uni \u2192 Val Uni \u2192 Res\n-- -- apply (closure f \u03c1) a n = eval-fun f (a \u2022 \u03c1) n\n-- -- apply (intV _) a n = Error\n\n-- -- eval-term t \u03c1 zero = TimeOut\n-- -- eval-term (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\n-- -- eval-term (lett f x t) \u03c1 (suc n) = apply (\u27e6 f \u27e7Var \u03c1) (\u27e6 x \u27e7Var \u03c1) n >>= (\u03bb v \u2192 eval-term t (v \u2022 \u03c1) n)\n\n-- -- eval-fun (term t) \u03c1 n = eval-term t \u03c1 n\n-- -- eval-fun (abs f) \u03c1 n = Done (closure f \u03c1)\n\n-- -- -- Erasure from typed to untyped values.\n-- -- import Thesis.ANormalBigStep as T\n\n-- -- erase-type : T.Type \u2192 Type\n-- -- erase-type _ = Uni\n\n-- -- erase-val : \u2200 {\u03c4} \u2192 T.Val \u03c4 \u2192 Val (erase-type \u03c4)\n\n-- -- erase-errVal : \u2200 {\u03c4} \u2192 T.ErrVal \u03c4 \u2192 ErrVal\n-- -- erase-errVal (T.Done v) = Done (erase-val v)\n-- -- erase-errVal T.Error = Error\n-- -- erase-errVal T.TimeOut = TimeOut\n\n-- -- erase-res : \u2200 {\u03c4} \u2192 T.Res \u03c4 \u2192 Res\n-- -- erase-res r n = erase-errVal (r n)\n\n-- -- erase-ctx : T.Context \u2192 Context\n-- -- erase-ctx \u2205 = \u2205\n-- -- erase-ctx (\u03c4 \u2022 \u0393) = erase-type \u03c4 \u2022 (erase-ctx \u0393)\n\n-- -- erase-env : \u2200 {\u0393} \u2192 T.Op.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 erase-ctx \u0393 \u27e7Context\n-- -- erase-env \u2205 = \u2205\n-- -- erase-env (v \u2022 \u03c1) = erase-val v \u2022 erase-env \u03c1\n\n-- -- erase-var : \u2200 {\u0393 \u03c4} \u2192 T.Var \u0393 \u03c4 \u2192 Var (erase-ctx \u0393) (erase-type \u03c4)\n-- -- erase-var T.this = this\n-- -- erase-var (T.that x) = that (erase-var x)\n\n-- -- erase-term : \u2200 {\u0393 \u03c4} \u2192 T.Term \u0393 \u03c4 \u2192 Term (erase-ctx \u0393)\n-- -- erase-term (T.var x) = var (erase-var x)\n-- -- erase-term (T.lett f x t) = lett (erase-var f) (erase-var x) (erase-term t)\n\n-- -- erase-fun : \u2200 {\u0393 \u03c4} \u2192 T.Fun \u0393 \u03c4 \u2192 Fun (erase-ctx \u0393)\n-- -- erase-fun (T.term x) = term (erase-term x)\n-- -- erase-fun (T.abs f) = abs (erase-fun f)\n\n-- -- erase-val (T.closure t \u03c1) = closure (erase-fun t) (erase-env \u03c1)\n-- -- erase-val (T.intV n) = intV n\n\n-- -- -- Different erasures commute.\n-- -- erasure-commute-var : \u2200 {\u0393 \u03c4} (x : T.Var \u0393 \u03c4) \u03c1 \u2192\n-- --   erase-val (T.Op.\u27e6 x \u27e7Var \u03c1) \u2261 \u27e6 erase-var x \u27e7Var (erase-env \u03c1)\n-- -- erasure-commute-var T.this (v \u2022 \u03c1) = refl\n-- -- erasure-commute-var (T.that x) (v \u2022 \u03c1) = erasure-commute-var x \u03c1\n\n-- -- erase-bind : \u2200 {\u03c3 \u03c4 \u0393} a (t : T.Term (\u03c3 \u2022 \u0393) \u03c4) \u03c1 n \u2192 erase-errVal (a T.>>= (\u03bb v \u2192 T.eval-term t (v \u2022 \u03c1) n)) \u2261 erase-errVal a >>= (\u03bb v \u2192 eval-term (erase-term t) (v \u2022 erase-env \u03c1) n)\n\n-- -- erasure-commute-fun : \u2200 {\u0393 \u03c4} (t : T.Fun \u0393 \u03c4) \u03c1 n \u2192\n-- --   erase-errVal (T.eval-fun t \u03c1 n) \u2261 eval-fun (erase-fun t) (erase-env \u03c1) n\n\n-- -- erasure-commute-apply : \u2200 {\u03c3 \u03c4} (f : T.Val (\u03c3 T.\u21d2 \u03c4)) a n \u2192 erase-errVal (T.apply f a n) \u2261 apply (erase-val f) (erase-val a) n\n-- -- erasure-commute-apply {\u03c3} (T.closure t \u03c1) a n = erasure-commute-fun t (a \u2022 \u03c1) n\n\n-- -- erasure-commute-term : \u2200 {\u0393 \u03c4} (t : T.Term \u0393 \u03c4) \u03c1 n \u2192\n-- --   erase-errVal (T.eval-term t \u03c1 n) \u2261 eval-term (erase-term t) (erase-env \u03c1) n\n\n-- -- erasure-commute-fun (T.term t) \u03c1 n = erasure-commute-term t \u03c1 n\n-- -- erasure-commute-fun (T.abs t) \u03c1 n = refl\n\n-- -- erasure-commute-term t \u03c1 zero = refl\n-- -- erasure-commute-term (T.var x) \u03c1 (\u2115.suc n) = cong Done (erasure-commute-var x \u03c1)\n-- -- erasure-commute-term (T.lett f x t) \u03c1 (\u2115.suc n) rewrite erase-bind (T.apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n) t \u03c1 n | erasure-commute-apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n | erasure-commute-var f \u03c1 | erasure-commute-var x \u03c1 = refl\n\n-- -- erase-bind (T.Done v) t \u03c1 n = erasure-commute-term t (v \u2022 \u03c1) n\n-- -- erase-bind T.Error t \u03c1 n = refl\n-- -- erase-bind T.TimeOut t \u03c1 n = refl\n","old_contents":"module Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Product\nopen import Data.Nat.Base\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {t : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs t \u2193 dclosure t \u03c1 d\u03c1\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\n{-# TERMINATING #-} -- Dear lord. Why on Earth.\nmutual\n  -- Single context? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3 : \u2200 {\u0393} (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val Uni) \u2192\n    \u2200 j n2 (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 \u2115 \u2192 Set\n  rrelV3 (intV v1) (dintV dv) (intV v2) n = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n  rrelV3 _ _ _ n = \u22a5\n\n-- -- [_]\u03c4  from v1 to v2) :\n-- -- [ \u03c4 ]\u03c4 dv from v1 to v2) =\n\n-- -- data [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set where\n-- --   v\u2205 : [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205\n-- --   _v\u2022_ : \u2200 {\u03c4 \u0394 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n-- --     (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n-- --     (d\u03c1\u03c1 : [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12) \u2192\n-- --     [ \u03c4 \u2022 \u0394 ]\u0394 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n-- data ErrVal : Set where\n--   Done : (v : Val Uni) \u2192 ErrVal\n--   Error : ErrVal\n--   TimeOut : ErrVal\n\n-- _>>=_ : ErrVal \u2192 (Val Uni \u2192 ErrVal) \u2192 ErrVal\n-- Done v >>= f = f v\n-- Error >>= f = Error\n-- TimeOut >>= f = TimeOut\n\n-- Res : Set\n-- Res = \u2115 \u2192 ErrVal\n\n-- -- eval-fun : \u2200 {\u0393} \u2192 Fun \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n-- -- eval-term : \u2200 {\u0393} \u2192 Term \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n\n-- -- apply : Val Uni \u2192 Val Uni \u2192 Res\n-- -- apply (closure f \u03c1) a n = eval-fun f (a \u2022 \u03c1) n\n-- -- apply (intV _) a n = Error\n\n-- -- eval-term t \u03c1 zero = TimeOut\n-- -- eval-term (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\n-- -- eval-term (lett f x t) \u03c1 (suc n) = apply (\u27e6 f \u27e7Var \u03c1) (\u27e6 x \u27e7Var \u03c1) n >>= (\u03bb v \u2192 eval-term t (v \u2022 \u03c1) n)\n\n-- -- eval-fun (term t) \u03c1 n = eval-term t \u03c1 n\n-- -- eval-fun (abs f) \u03c1 n = Done (closure f \u03c1)\n\n-- -- -- Erasure from typed to untyped values.\n-- -- import Thesis.ANormalBigStep as T\n\n-- -- erase-type : T.Type \u2192 Type\n-- -- erase-type _ = Uni\n\n-- -- erase-val : \u2200 {\u03c4} \u2192 T.Val \u03c4 \u2192 Val (erase-type \u03c4)\n\n-- -- erase-errVal : \u2200 {\u03c4} \u2192 T.ErrVal \u03c4 \u2192 ErrVal\n-- -- erase-errVal (T.Done v) = Done (erase-val v)\n-- -- erase-errVal T.Error = Error\n-- -- erase-errVal T.TimeOut = TimeOut\n\n-- -- erase-res : \u2200 {\u03c4} \u2192 T.Res \u03c4 \u2192 Res\n-- -- erase-res r n = erase-errVal (r n)\n\n-- -- erase-ctx : T.Context \u2192 Context\n-- -- erase-ctx \u2205 = \u2205\n-- -- erase-ctx (\u03c4 \u2022 \u0393) = erase-type \u03c4 \u2022 (erase-ctx \u0393)\n\n-- -- erase-env : \u2200 {\u0393} \u2192 T.Op.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 erase-ctx \u0393 \u27e7Context\n-- -- erase-env \u2205 = \u2205\n-- -- erase-env (v \u2022 \u03c1) = erase-val v \u2022 erase-env \u03c1\n\n-- -- erase-var : \u2200 {\u0393 \u03c4} \u2192 T.Var \u0393 \u03c4 \u2192 Var (erase-ctx \u0393) (erase-type \u03c4)\n-- -- erase-var T.this = this\n-- -- erase-var (T.that x) = that (erase-var x)\n\n-- -- erase-term : \u2200 {\u0393 \u03c4} \u2192 T.Term \u0393 \u03c4 \u2192 Term (erase-ctx \u0393)\n-- -- erase-term (T.var x) = var (erase-var x)\n-- -- erase-term (T.lett f x t) = lett (erase-var f) (erase-var x) (erase-term t)\n\n-- -- erase-fun : \u2200 {\u0393 \u03c4} \u2192 T.Fun \u0393 \u03c4 \u2192 Fun (erase-ctx \u0393)\n-- -- erase-fun (T.term x) = term (erase-term x)\n-- -- erase-fun (T.abs f) = abs (erase-fun f)\n\n-- -- erase-val (T.closure t \u03c1) = closure (erase-fun t) (erase-env \u03c1)\n-- -- erase-val (T.intV n) = intV n\n\n-- -- -- Different erasures commute.\n-- -- erasure-commute-var : \u2200 {\u0393 \u03c4} (x : T.Var \u0393 \u03c4) \u03c1 \u2192\n-- --   erase-val (T.Op.\u27e6 x \u27e7Var \u03c1) \u2261 \u27e6 erase-var x \u27e7Var (erase-env \u03c1)\n-- -- erasure-commute-var T.this (v \u2022 \u03c1) = refl\n-- -- erasure-commute-var (T.that x) (v \u2022 \u03c1) = erasure-commute-var x \u03c1\n\n-- -- erase-bind : \u2200 {\u03c3 \u03c4 \u0393} a (t : T.Term (\u03c3 \u2022 \u0393) \u03c4) \u03c1 n \u2192 erase-errVal (a T.>>= (\u03bb v \u2192 T.eval-term t (v \u2022 \u03c1) n)) \u2261 erase-errVal a >>= (\u03bb v \u2192 eval-term (erase-term t) (v \u2022 erase-env \u03c1) n)\n\n-- -- erasure-commute-fun : \u2200 {\u0393 \u03c4} (t : T.Fun \u0393 \u03c4) \u03c1 n \u2192\n-- --   erase-errVal (T.eval-fun t \u03c1 n) \u2261 eval-fun (erase-fun t) (erase-env \u03c1) n\n\n-- -- erasure-commute-apply : \u2200 {\u03c3 \u03c4} (f : T.Val (\u03c3 T.\u21d2 \u03c4)) a n \u2192 erase-errVal (T.apply f a n) \u2261 apply (erase-val f) (erase-val a) n\n-- -- erasure-commute-apply {\u03c3} (T.closure t \u03c1) a n = erasure-commute-fun t (a \u2022 \u03c1) n\n\n-- -- erasure-commute-term : \u2200 {\u0393 \u03c4} (t : T.Term \u0393 \u03c4) \u03c1 n \u2192\n-- --   erase-errVal (T.eval-term t \u03c1 n) \u2261 eval-term (erase-term t) (erase-env \u03c1) n\n\n-- -- erasure-commute-fun (T.term t) \u03c1 n = erasure-commute-term t \u03c1 n\n-- -- erasure-commute-fun (T.abs t) \u03c1 n = refl\n\n-- -- erasure-commute-term t \u03c1 zero = refl\n-- -- erasure-commute-term (T.var x) \u03c1 (\u2115.suc n) = cong Done (erasure-commute-var x \u03c1)\n-- -- erasure-commute-term (T.lett f x t) \u03c1 (\u2115.suc n) rewrite erase-bind (T.apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n) t \u03c1 n | erasure-commute-apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n | erasure-commute-var f \u03c1 | erasure-commute-var x \u03c1 = refl\n\n-- -- erase-bind (T.Done v) t \u03c1 n = erasure-commute-term t (v \u2022 \u03c1) n\n-- -- erase-bind T.Error t \u03c1 n = refl\n-- -- erase-bind T.TimeOut t \u03c1 n = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c01bc7ef92279fc0a542d633c350a718d1a000bd","subject":"Control\/Protocol\/Choreography.agda","message":"Control\/Protocol\/Choreography.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\n{-\nrecord \u03a3\u00b7 {a b} (A : Set a) (B : ..(_ : A) \u2192 Set b) : Set (a \u2294 b) where\n  constructor _,_\n  field\n    {-..-}proj\u2081\u00b7 : A\n    proj\u2082\u00b7 : B proj\u2081\u00b7\n-}\ndata \u03a3\u00b7 {a b} (A : Set a) (B : ..(_ : A) \u2192 Set b) : Set (a \u2294 b) where\n  _,_ : ..(proj\u2081\u00b7 : A) (proj\u2082\u00b7 : B proj\u2081\u00b7) \u2192 \u03a3\u00b7 A B\n\ndata Mod : \u2605 where S D : Mod\n\n\u2192M : \u2200 {a b} \u2192 Mod \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (a \u2294 b)\n\u2192M S A B = ..(_ : A) \u2192 B\n\u2192M D A B = A \u2192 B\n\n\u03a0M : \u2200 {a b}(m : Mod) \u2192 (A : \u2605_ a) \u2192 (B : \u2192M m A (\u2605_ b)) \u2192 \u2605_ (a \u2294 b)\n\u03a0M S A B = \u03a0\u00b7 A B\n\u03a0M D A B = \u03a0 A B\n\nappM' : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2605_ b}(P : \u2192M m A B) \u2192 A \u2192 B\nappM' S P x = P x\nappM' D P x = P x\n\nappM : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2192M m A (\u2605_ b)}(P : \u03a0M m A B)(x : A) \u2192 appM' m B x\nappM S P x = P x\nappM D P x = P x\n\ndata \u03a0\u03a3 : \u2605 where\n  \u03a0' \u03a3' : \u03a0\u03a3\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  mk  : (q : \u03a0\u03a3)(d : Mod)(A : \u2605)(B : \u2192M d A Proto) \u2192 Proto\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  \u00f8         : Choreo I\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  \u03a3D :  \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200 m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 mk \u03a3' D M \u2102A ]\n  \u03a0D :  \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 mk \u03a0' D M \u2102B ]\n  \u03a0S :  \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 mk \u03a0' S M \u2102C ]\n\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (mk \u03a0' D A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (mk \u03a3' D A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (mk \u03a0' S A B) = \u03a3\u00b7 A \u03bb x \u2192 Tele (B x)\nTele (mk \u03a3' S A B) = \u03a3\u00b7 A \u03bb x \u2192 Tele (B x)\n\n{-\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = mk \u03a3' D A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = mk \u03a0' D A \u03bb _ \u2192 B\n\ndual\u03a0\u03a3 : \u03a0\u03a3 \u2192 \u03a0\u03a3\ndual\u03a0\u03a3 \u03a0' = \u03a3'\ndual\u03a0\u03a3 \u03a3' = \u03a0'\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (mk q S A B) = mk (dual\u03a0\u03a3 q) S A (\u03bb x \u2192 dual (B x))\ndual (mk q D A B) = mk (dual\u03a0\u03a3 q) D A (\u03bb x \u2192 dual (B x))\n\ndata Dual\u03a0\u03a3 : \u03a0\u03a3 \u2192 \u03a0\u03a3 \u2192 \u2605 where\n  D\u03a0\u03a3 : Dual\u03a0\u03a3 \u03a0' \u03a3'\n  D\u03a3\u03a0 : Dual\u03a0\u03a3 \u03a3' \u03a0'\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  S : \u2200 {A B B' \u03c0 \u03c3} \u2192 Dual\u03a0\u03a3 \u03c0 \u03c3 \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (mk \u03c0 S A B) (mk \u03c3 S A B')\n  D : \u2200 {A B B' \u03c0 \u03c3} \u2192 Dual\u03a0\u03a3 \u03c0 \u03c3 \u2192 (\u2200   x \u2192 Dual (B x) (B' x)) \u2192 Dual (mk \u03c0 D A B) (mk \u03c3 D A B')\n\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n -- comm   : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n  end&   : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  \u03a0S-S-S : \u2200 {q M P Q R} \u2192 (\u2200 ..m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  S M Q \u2261 mk q  S M R ]\n  \u03a0S-D-D : \u2200 {q M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  D M Q \u2261 mk q  D M R ]\n  \u03a0D\u03a3D\u03a3S : \u2200   {M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' D M P & mk \u03a3' D M Q \u2261 mk \u03a3' S M R ]\n  \u03a0D\u03a0D\u03a0D : \u2200   {M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' D M P & mk \u03a0' D M Q \u2261 mk \u03a0' D M R ]\n\n  &end    : \u2200 {P} \u2192 [ P & end \u2261 P ]\n--  \u03a0S-S-S : \u2200 {q M P Q R} \u2192 (\u2200 ..m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  S M Q \u2261 mk q  S M R ]\n--  \u03a0S-D-D : \u2200 {q M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  D M Q \u2261 mk q  D M R ]\n \u00a0\u03a3D\u03a0D\u03a3S : \u2200   {M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a3' D M P & mk \u03a0' D M Q \u2261 mk \u03a3' S M R ]\n\nDual\u03a0\u03a3-sym : \u2200 {P Q} \u2192 Dual\u03a0\u03a3 P Q \u2192 Dual\u03a0\u03a3 Q P\nDual\u03a0\u03a3-sym D\u03a0\u03a3 = D\u03a3\u03a0\nDual\u03a0\u03a3-sym D\u03a3\u03a0 = D\u03a0\u03a3\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (S d f) = S (Dual\u03a0\u03a3-sym d) (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (D d f) = D (Dual\u03a0\u03a3-sym d) (\u03bb x \u2192 Dual-sym (f x))\n\nDual\u03a0\u03a3-spec : \u2200 P \u2192 Dual\u03a0\u03a3 P (dual\u03a0\u03a3 P)\nDual\u03a0\u03a3-spec \u03a0' = D\u03a0\u03a3\nDual\u03a0\u03a3-spec \u03a3' = D\u03a3\u03a0\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (mk \u03c0 S A B) = S (Dual\u03a0\u03a3-spec \u03c0) (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (mk \u03c0 D A B) = D (Dual\u03a0\u03a3-spec \u03c0) (\u03bb x \u2192 Dual-spec (B x))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recvD : \u2200 {M P} \u2192 (\u03a0M D M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (mk \u03a0' D M P)\n  recvS : \u2200 {M P} \u2192 (\u03a0M S M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (mk \u03a0' S M P)\n  sendD : \u2200 {M P} \u2192 \u03a0M D M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (mk \u03a3' D M P))\n  sendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (mk \u03a3' S M P))\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q d M P Q} \u2192 SimL (mk q d M P) Q \u2192 Sim (mk q d M P) Q\n    right : \u2200 {P q d M Q} \u2192 SimR P (mk q d M Q) \u2192 Sim P (mk q d M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recvD PQA)   (sendD m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recvD PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (sendD m PQB) = sendD m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (sendD m PQA) (recvD PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (sendD x\u2081 x\u2082) (recvD x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recvD PQA)   (sendD m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recvD PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (sendD m PQB) = sendD m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (sendD m PQA) (recvD PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recvD PQA)   (sendD m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recvD PQA)   (recvD PQB)   = ? -- recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recvD PQB)   = ? -- recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (sendD m PQB) = ? -- sendD m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (sendD m PQA) (recvD PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (mk \u03a0' S A B) = right (recvS (\u03bb x \u2192 left (sendS x (sim-id (B x)))))\nsim-id (mk \u03a0' D A B) = right (recvD (\u03bb x \u2192 left (sendD x (sim-id (B x)))))\nsim-id (mk \u03a3' S A B) = left (recvS (\u03bb x \u2192 right (sendS x (sim-id (B x)))))\nsim-id (mk \u03a3' D A B) = left (recvD (\u03bb x \u2192 right (sendD x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (S D\u03a0\u03a3 x\u2081) (recvS x) (sendS x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (D D\u03a0\u03a3 x\u2081) (recvD x) (sendD x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (S D\u03a3\u03a0 x)  (sendS x\u2081 x\u2082) (recvS x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\nsim-compRL (D D\u03a3\u03a0 x)  (sendD x\u2081 x\u2082) (recvD x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recvD PQ) QR = recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (recvS PQ) QR = recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (sendD m PQ) QR = sendD m (sim-comp Q-Q' PQ QR)\nsim-compL Q-Q' (sendS m PQ) QR = sendS m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recvD QR)   = recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (recvS QR)   = recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x))\nsim-compR Q-Q' PQ (sendD m QR) = sendD m (sim-comp Q-Q' PQ QR)\nsim-compR Q-Q' PQ (sendS m QR) = sendS m (sim-comp Q-Q' PQ QR)\n\n{-\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-comp Q-Q' (left (recvD PQ)) QR = left (recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (recvS PQ)) QR = left (recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (sendD m PQ)) QR = left (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' (left (sendS m PQ)) QR = left (sendS m (sim-comp Q-Q' PQ QR))\nsim-comp end (right ()) (left x\u2081)\nsim-comp end end QR = QR\nsim-comp end PQ end = PQ\nsim-comp (\u03a0\u03a3'S Q-Q') (right (recvS PQ)) (left (sendS m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a0\u03a3'D Q-Q') (right (recvD PQ)) (left (sendD m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a3\u03a0'S Q-Q') (right (sendS m PQ)) (left (recvS QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp (\u03a3\u03a0'D Q-Q') (right (sendD m PQ)) (left (recvD QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp Q-Q' PQ (right (recvD QR)) = right (recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m)))\nsim-comp Q-Q' PQ (right (recvS QR)) = right (recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x)))\nsim-comp Q-Q' PQ (right (sendD m QR)) = right (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' PQ (right (sendS m QR)) = right (sendS m (sim-comp Q-Q' PQ QR))\n-}\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symL (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-symR (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symR (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recvD P)) = do (recvD (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (recvS P)) = do (recvS (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (sendD m P)) = do (sendD m (sim-unit P))\nsim-unit (right (sendS m P)) = do (sendS m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (S D\u03a0\u03a3 x) = cong (S D\u03a0\u03a3) (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (D D\u03a0\u03a3 x) = cong (D D\u03a0\u03a3) (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (S D\u03a3\u03a0 x) = cong (S D\u03a3\u03a0) (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (D D\u03a3\u03a0 x) = cong (D D\u03a3\u03a0) (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end (S D\u03a0\u03a3 x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (D D\u03a0\u03a3 x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (S D\u03a3\u03a0 x) Q-Q' (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end (D D\u03a3\u03a0 x) Q-Q' (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' (S D\u03a0\u03a3 x\u2081) (right \u00f8P) (right (recvS x)) (left (sendS x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (D D\u03a0\u03a3 x\u2081) (right \u00f8P) (right (recvD x)) (left (sendD x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (S D\u03a3\u03a0 x) (right \u00f8P) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' (D D\u03a3\u03a0 x) (right \u00f8P) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvD x\u2081))\n    = cong (right \u2218 recvD) (funExtD (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvS x\u2081))\n    = cong (right \u2218 recvS) (funExtS (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendD x\u2081 x\u2082))\n    = cong (right \u2218 sendD x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendS x\u2081 x\u2082))\n    = cong (right \u2218 sendS x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recvD x))    = cong (left \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (recvS x))    = cong (left \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (sendD x x\u2081)) = cong (left \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (left (sendS x x\u2081)) = cong (left \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! (right (recvD x))    = cong (right \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (recvS x))    = cong (right \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (sendD x x\u2081)) = cong (right \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (right (sendS x x\u2081)) = cong (right \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (S D\u03a0\u03a3 x\u2081) (right (recvS x)) (left (sendS x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (D D\u03a0\u03a3 x\u2081) (right (recvD x)) (left (sendD x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (S D\u03a3\u03a0 x) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end (D D\u03a3\u03a0 x) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recvD x\u2081))\n    = cong (left \u2218 recvD) (funExtD (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (recvS x\u2081))\n    = cong (left \u2218 recvS) (funExtS (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (sendD x\u2081 x\u2082))\n    = cong (left \u2218 sendD x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end Q-Q' (right x) (right (sendS x\u2081 x\u2082))\n    = cong (left \u2218 sendS x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExtD funExtS R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExtD funExtS Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata Mod : \u2605 where S D : Mod\n\n\u2192M : \u2200 {a b} \u2192 Mod \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (a \u2294 b)\n\u2192M S A B = ..(_ : A) \u2192 B\n\u2192M D A B = A \u2192 B\n\n\u03a0M : \u2200 {a b}(m : Mod) \u2192 (A : \u2605_ a) \u2192 (B : \u2192M m A (\u2605_ b)) \u2192 \u2605_ (a \u2294 b)\n\u03a0M S A B = \u03a0\u00b7 A B\n\u03a0M D A B = \u03a0 A B\n\n-- appM : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2192M m A (\u2605_ b)}(P : \u03a0M m A B)(x : A) \u2192 B\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (f : Mod)(A : \u2605)(B : \u2192M f A Proto) \u2192 Proto\n\n{-\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' D A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' D A \u03bb _ \u2192 B\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' S A B) = \u03a3' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a0' D A B) = \u03a3' D A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' S A B) = \u03a0' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' D A B) = \u03a0' D A (\u03bb x \u2192 dual (B x))\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u03a3'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' S A B) (\u03a3' S A B')\n  \u03a0\u03a3'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' D A B) (\u03a3' D A B')\n  \u03a3\u03a0'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' S A B) (\u03a0' S A B')\n  \u03a3\u03a0'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' D A B) (\u03a0' D A B')\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u03a3'S f) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a0\u03a3'D f) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'S f) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'D f) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0' S A B) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a0' D A B) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' S A B) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' D A B) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-spec (B x))\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recvD : \u2200 {M P} \u2192 (\u03a0M D M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' D M P)\n  recvS : \u2200 {M P} \u2192 (\u03a0M S M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' S M P)\n  sendD : \u2200 {M P} \u2192 \u03a0M D M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' D M P))\n  sendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' S M P))\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {P Q} \u2192 SimL P Q \u2192 Sim P Q\n    right : \u2200 {P Q} \u2192 SimR P Q \u2192 Sim P Q\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' S A B) = right (recvS (\u03bb x \u2192 left (sendS x (sim-id (B x)))))\nsim-id (\u03a0' D A B) = right (recvD (\u03bb x \u2192 left (sendD x (sim-id (B x)))))\nsim-id (\u03a3' S A B) = left (recvS (\u03bb x \u2192 right (sendS x (sim-id (B x)))))\nsim-id (\u03a3' D A B) = left (recvD (\u03bb x \u2192 right (sendD x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp end (right x) end = right x\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u03a3'S x\u2081) (recvS x) (sendS x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a0\u03a3'D x\u2081) (recvD x) (sendD x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u03a0'S x) (sendS x\u2081 x\u2082) (recvS x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\nsim-compRL (\u03a3\u03a0'D x) (sendD x\u2081 x\u2082) (recvD x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recvD PQ) QR = recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (recvS PQ) QR = recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (sendD m PQ) QR = sendD m (sim-comp Q-Q' PQ QR)\nsim-compL Q-Q' (sendS m PQ) QR = sendS m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recvD QR)   = recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (recvS QR)   = recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x))\nsim-compR Q-Q' PQ (sendD m QR) = sendD m (sim-comp Q-Q' PQ QR)\nsim-compR Q-Q' PQ (sendS m QR) = sendS m (sim-comp Q-Q' PQ QR)\n\n{-\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-comp Q-Q' (left (recvD PQ)) QR = left (recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (recvS PQ)) QR = left (recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (sendD m PQ)) QR = left (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' (left (sendS m PQ)) QR = left (sendS m (sim-comp Q-Q' PQ QR))\nsim-comp end (right ()) (left x\u2081)\nsim-comp end end QR = QR\nsim-comp end PQ end = PQ\nsim-comp (\u03a0\u03a3'S Q-Q') (right (recvS PQ)) (left (sendS m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a0\u03a3'D Q-Q') (right (recvD PQ)) (left (sendD m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a3\u03a0'S Q-Q') (right (sendS m PQ)) (left (recvS QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp (\u03a3\u03a0'D Q-Q') (right (sendD m PQ)) (left (recvD QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp Q-Q' PQ (right (recvD QR)) = right (recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m)))\nsim-comp Q-Q' PQ (right (recvS QR)) = right (recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x)))\nsim-comp Q-Q' PQ (right (sendD m QR)) = right (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' PQ (right (sendS m QR)) = right (sendS m (sim-comp Q-Q' PQ QR))\n-}\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symL (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-symR (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symR (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (recvD P)) = do (recvD (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (recvS P)) = do (recvS (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (sendD m P)) = do (sendD m (sim-unit P))\nsim-unit (right (sendS m P)) = do (sendS m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u03a3'S x) = cong \u03a0\u03a3'S (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a0\u03a3'D x) = cong \u03a0\u03a3'D (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u03a0'S x) = cong \u03a3\u03a0'S (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u03a0'D x) = cong \u03a3\u03a0'D (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end P-P' Q-Q' (left ()) PQ QR\n  sim-comp-assoc-end end Q-Q' (right ()) (left PQ) QR\n  sim-comp-assoc-end (\u03a0\u03a3'S x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a0\u03a3'D x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u03a0'S x) Q-Q' (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end (\u03a3\u03a0'D x) Q-Q' (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) (left x\u2081)\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'S x\u2081) (right \u00f8P) (right (recvS x)) (left (sendS x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'D x\u2081) (right \u00f8P) (right (recvD x)) (left (sendD x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'S x) (right \u00f8P) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'D x) (right \u00f8P) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvD x\u2081))\n    = cong (right \u2218 recvD) (funExtD (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvS x\u2081))\n    = cong (right \u2218 recvS) (funExtS (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendD x\u2081 x\u2082))\n    = cong (right \u2218 sendD x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendS x\u2081 x\u2082))\n    = cong (right \u2218 sendS x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) end\n  sim-comp-assoc-end end end (right \u00f8P) end QR = refl\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recvD x))    = cong (left \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (recvS x))    = cong (left \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (sendD x x\u2081)) = cong (left \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (left (sendS x x\u2081)) = cong (left \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! (right (recvD x))    = cong (right \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (recvS x))    = cong (right \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (sendD x x\u2081)) = cong (right \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (right (sendS x x\u2081)) = cong (right \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end Q-Q' (left ()) QR\n  sim-comp-!-end end (right ()) (left x\u2081)\n  sim-comp-!-end (\u03a0\u03a3'S x\u2081) (right (recvS x)) (left (sendS x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a0\u03a3'D x\u2081) (right (recvD x)) (left (sendD x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u03a0'S x) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end (\u03a3\u03a0'D x) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recvD x\u2081))\n    = cong (left \u2218 recvD) (funExtD (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (recvS x\u2081))\n    = cong (left \u2218 recvS) (funExtS (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (sendD x\u2081 x\u2082))\n    = cong (left \u2218 sendD x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end Q-Q' (right x) (right (sendS x\u2081 x\u2082))\n    = cong (left \u2218 sendS x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end (right x) end = refl\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExtD funExtS R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExtD funExtS Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1b5c3931a31ac1d019f7a68f56a036201f92a46e","subject":"Fix mixed-if.","message":"Fix mixed-if.\n\nOld-commit-hash: 18ad0d9a08607bece81abf4bedc81a7763b56b9a\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n","old_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2083 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d50f72236e4009f8db49bd947117cb58b3cbfccb","subject":"Removed Agda with constructor from the proof of streamLength.","message":"Removed Agda with constructor from the proof of streamLength.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality.Type\nopen import FOTC.Data.Colist\nopen import FOTC.Data.Colist.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\n\n-----------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functional\n-- StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream (see FOTC.Data.Stream.Type).\nStream-in : \u2200 {xs} \u2192\n            \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n            Stream xs\nStream-in h = Stream-coind A h' h\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h' (x' , xs' , prf , Sxs') = x' , xs' , prf , Stream-out Sxs'\n\nfoo : \u2200 {xs} \u2192 Stream xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\nfoo Sxs = Stream-out Sxs\n\n\u2237-Stream : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream {x} {xs} h = \u2237-Stream-helper (Stream-out h)\n  where\n  \u2237-Stream-helper : \u2203[ x' ] \u2203[ xs' ] x \u2237 xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n                    Stream xs\n  \u2237-Stream-helper (x' , xs' , prf , Sxs') =\n    subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\n-- Version using Agda with constructor.\n\u2237-Stream' : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream' h with Stream-out h\n... | x' , xs' , prf , Sxs' =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\nStream\u2192Colist : \u2200 {xs} \u2192 Stream xs \u2192 Colist xs\nStream\u2192Colist {xs} Sxs = Colist-coind A h\u2081 h\u2082\n  where\n  A : D \u2192 Set\n  A ys = Stream ys\n\n  h\u2081 : \u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n  h\u2081 Axs with Stream-out Axs\n  ... | x' , xs' , prf , Sxs' = inj\u2082 (x' , xs' , prf , Sxs')\n\n  h\u2082 : A xs\n  h\u2082 = Sxs\n\n-- Adapted from (Sander 1992, p. 59).\nstreamLength : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength {xs} Sxs = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R m n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 R m n \u2192\n       m \u2261 zero \u2227 n \u2261 zero\n         \u2228 (\u2203[ m' ] \u2203[ n' ] m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n' \u2227 R m' n')\n  h\u2081 {m} {n} (xs , Sxs , m=lxs , n\u2261\u221e) = helper\u2081 (Stream-out Sxs)\n    where\n    helper\u2081 : (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs') \u2192\n              m \u2261 zero \u2227 n \u2261 zero\n                \u2228 (\u2203[ m' ] \u2203[ n' ] m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n' \u2227 R m' n')\n    helper\u2081 (x' , xs' , xs\u2261x'\u2237xs' , Sxs') =\n      inj\u2082 (length xs'\n           , \u221e\n           , helper\u2082\n           , trans n\u2261\u221e \u221e-eq\n           , (xs' , Sxs' , refl , refl))\n        where\n        helper\u2082 : m \u2261 succ\u2081 (length xs')\n        helper\u2082 = trans m=lxs (trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs'))\n\n  h\u2082 : R (length xs) \u221e\n  h\u2082 = xs , Sxs , refl , refl\n\n-- Adapted from (Sander 1992, p. 59). Version using Agda with\n-- constructor.\nstreamLength' : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength' {xs} Sxs = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R m n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 R m n \u2192\n       m \u2261 zero \u2227 n \u2261 zero\n         \u2228 (\u2203[ m' ] \u2203[ n' ] m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n' \u2227 R m' n')\n  h\u2081 {m} (xs , Sxs , m=lxs , n\u2261\u221e) with Stream-out Sxs\n  ... | x' , xs' , xs\u2261x'\u2237xs' , Sxs' =\n    inj\u2082 (length xs' , \u221e , helper , trans n\u2261\u221e \u221e-eq , (xs' , Sxs' , refl , refl))\n    where\n    helper : m \u2261 succ\u2081 (length xs')\n    helper = trans m=lxs (trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs'))\n\n  h\u2082 : R (length xs) \u221e\n  h\u2082 = xs , Sxs , refl , refl\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Sander, Herbert P. (1992). A Logic of Functional Programs with an\n-- Application to Concurrency. PhD thesis. Department of Computer\n-- Sciences: Chalmers University of Technology and University of\n-- Gothenburg.\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality.Type\nopen import FOTC.Data.Colist\nopen import FOTC.Data.Colist.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\n\n-----------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functional\n-- StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream (see FOTC.Data.Stream.Type).\nStream-in : \u2200 {xs} \u2192\n            \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n            Stream xs\nStream-in h = Stream-coind A h' h\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h' (x' , xs' , prf , Sxs') = x' , xs' , prf , Stream-out Sxs'\n\nfoo : \u2200 {xs} \u2192 Stream xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\nfoo Sxs = Stream-out Sxs\n\n\u2237-Stream : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream {x} {xs} h = \u2237-Stream-helper (Stream-out h)\n  where\n  \u2237-Stream-helper : \u2203[ x' ] \u2203[ xs' ] x \u2237 xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n                    Stream xs\n  \u2237-Stream-helper (x' , xs' , prf , Sxs') =\n    subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\n-- Version using Agda with constructor.\n\u2237-Stream' : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream' h with Stream-out h\n... | x' , xs' , prf , Sxs' =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\nStream\u2192Colist : \u2200 {xs} \u2192 Stream xs \u2192 Colist xs\nStream\u2192Colist {xs} Sxs = Colist-coind A h\u2081 h\u2082\n  where\n  A : D \u2192 Set\n  A ys = Stream ys\n\n  h\u2081 : \u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n  h\u2081 Axs with Stream-out Axs\n  ... | x' , xs' , prf , Sxs' = inj\u2082 (x' , xs' , prf , Sxs')\n\n  h\u2082 : A xs\n  h\u2082 = Sxs\n\n-- Adapted from (Sander 1992, p. 59).\nstreamLength : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength {xs} Sxs = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R m n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 R m n \u2192\n       m \u2261 zero \u2227 n \u2261 zero\n         \u2228 (\u2203[ m' ] \u2203[ n' ] m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n' \u2227 R m' n')\n  h\u2081 {m} (xs , Sxs , m=lxs , n\u2261\u221e) with Stream-out Sxs\n  ... | x' , xs' , xs\u2261x'\u2237xs' , Sxs' =\n    inj\u2082 (length xs' , \u221e , helper , trans n\u2261\u221e \u221e-eq , (xs' , Sxs' , refl , refl))\n    where\n    helper : m \u2261 succ\u2081 (length xs')\n    helper = trans m=lxs (trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs'))\n\n  h\u2082 : R (length xs) \u221e\n  h\u2082 = xs , Sxs , refl , refl\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Sander, Herbert P. (1992). A Logic of Functional Programs with an\n-- Application to Concurrency. PhD thesis. Department of Computer\n-- Sciences: Chalmers University of Technology and University of\n-- Gothenburg.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"54593ec140f1f6fbdb3f258b1affc25608b8b890","subject":"Bits: Bij is now typed","message":"Bits: Bij is now typed\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : \u2200 {n} \u2192 Bij (1 + n)\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if0 f = `if `id f\n\n    `if1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : \u2200 {n} \u2192 Bij (2 + n)\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n <= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) <= (to\u2115 xs) | \u2261.inspect (_<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n <= x | \u2261.inspect (_<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {zero}  (_     `\u2237 _) _ = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n <= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) <= (to\u2115 xs) | \u2261.inspect (_<=_ (2^ n)) (to\u2115 xs)\n... | true  | [ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |   _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x \u2115< 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | [ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (\u2115<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | [ p ] = to\u2115\u2218from\u2115 {n} x (\u2115<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x \u2115< 2^ n \u2192 y \u2115< 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0ff475d5cd044d90c54a459348a4ec5227336c0f","subject":"flipbased-tree: more","message":"flipbased-tree: more\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-tree.agda","new_file":"flipbased-tree.agda","new_contents":"module flipbased-tree where\n\nopen import Level using () renaming (_\u2294_ to _L\u2294_)\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP using (\u2115; zero; suc; _\u2264_; s\u2264s; _+_; module \u2115\u2264; module \u2115\u00b0)\nopen import Data.Nat.Properties\nopen import bintree\nopen import Data.Product\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\nreturn\u21ba : \u2200 {c a} {A : Set a} \u2192 A \u2192 \u21ba c A\nreturn\u21ba = leaf\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (leaf x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (leaf 0b) (leaf 1b)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (leaf x) = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 c {m a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ c = weaken\u2264 (m\u2264n+m _ c)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (leaf x) = leaf (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (leaf x)      = weaken+ c x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 leaf map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _*\/_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  1-_ : [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n  -- \u00b7\/1+_ : \u2115 \u2192 Set\n  -- \/+\/ : \u2115 \u2192 [0,1] \u2192 [0,1] \u2192 [0,1]\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\npostulate\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n  \/2+\/2-refl : \u2200 x \u2192 x \/2+ x \/2 \u2261 x\n  _\/2+_\/2-comm : \u2200 x y \u2192 x \/2+ y \/2 \u2261 y \/2+ x \/2\n  *\/-comm : \u2200 x y \u2192 x *\/ y \u2261 y *\/ x\n  1*q\u2261q : \u2200 q \u2192 1\/1 *\/ q \u2261 q\n  0*q\u2261q : \u2200 q \u2192 0\/1 *\/ q \u2261 0\/1\n  distr-*-\/2+\/2 : \u2200 {x y z} \u2192 (x *\/ z) \/2+ (y *\/ z) \/2 \u2261 (x \/2+ y \/2) *\/ z\n  distr-\/2-* : \u2200 {p q} \u2192 (p \/2) *\/ (q \/2) \u2261 ((p *\/ q) \/2) \/2\n  0\/2\u22610 : 0\/1 \/2 \u2261 0\/1\n  1-1\u22610 : 1- 1\/1 \u2261 0\/1\n  1-0\u22611 : 1- 0\/1 \u2261 1\/1\n  1-1\/2\u22611\/2 : 1- 1\/2 \u2261 1\/2\n  1-_\/2+1-_\/2 : \u2200 p q \u2192 (1- p) \/2+ (1- q) \/2 \u2261 1- (p \/2+ q \/2)\n\nsyntax Pr\u21ba P pr Alg = Pr[ Alg ! P ]\u2261 pr\ndata Pr\u21ba {a} {A : Set a} (P : A \u2192 [0,1] \u2192 Set a) (pr : [0,1]) : \u2200 {c} \u2192 \u21ba c A \u2192 Set a where\n  Pr-return : \u2200 {c x} (Px : P x pr) \u2192 Pr[ return\u21ba {c = c} x ! P ]\u2261 pr\n\n  Pr-fork : \u2200 {c} {left right : \u21ba c A} {p q}\n            (eq : p \/2+ q \/2 \u2261 pr)\n            (pf\u2080 : Pr[ left ! P ]\u2261 p)\n            (pf\u2081 : Pr[ right ! P ]\u2261 q)\n            \u2192 Pr[ fork left right ! P ]\u2261 pr\n\ndata Pr[return\u21ba_\u2261_]\u2261_ {a} {A : Set a} (x y : A) : [0,1] \u2192 Set a where\n  Pr-\u2261 : (x\u2261y : x \u2261 y) \u2192 Pr[return\u21ba x \u2261 y ]\u2261 1\/1\n  Pr-\u2262 : (x\u2262y : x \u2262 y) \u2192 Pr[return\u21ba x \u2261 y ]\u2261 0\/1\n\ninfix 2 Pr[_\u2261_]\u2261_\nPr[_\u2261_]\u2261_ : \u2200 {a} {A : Set a} {c} \u2192 \u21ba c A \u2192 A \u2192 [0,1] \u2192 Set a\nPr[ Alg \u2261 x ]\u2261 pr = Pr[ Alg ! flip Pr[return\u21ba_\u2261_]\u2261_ x ]\u2261 pr\n\nPr[_\u2261*]\u2261_ : \u2200 {a} {A : Set a} {c} \u2192 \u21ba c A \u2192 [0,1] \u2192 Set a\nPr[ Alg \u2261*]\u2261 pr = \u2200 {x} \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n\nPr-fork\u2032 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x p q}\n            \u2192 Pr[ left \u2261 x ]\u2261 p\n            \u2192 Pr[ right \u2261 x ]\u2261 q\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 (p \/2+ q \/2)\nPr-fork\u2032 = Pr-fork refl\n\nPr-return-\u2261 : \u2200 {c a} {A : Set a} {x : A} \u2192 Pr[ return\u21ba {c = c} x \u2261 x ]\u2261 1\/1\nPr-return-\u2261 = Pr-return (Pr-\u2261 refl)\n\nPr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 0\/1\nPr-return-\u2262 = Pr-return \u2218 Pr-\u2262\n\nimport Function.Equality as F\u2261\nimport Function.Equivalence as F\u2248\n\n_\u2248\u21d2_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248\u21d2 p\u2082 = \u2200 {x pr} \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr \u2192 Pr[ p\u2082 \u2261 x ]\u2261 pr\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 {x pr} \u2192 (Pr[ p\u2081 \u2261 x ]\u2261 pr) F\u2248.\u21d4 (Pr[ p\u2082 \u2261 x ]\u2261 pr)\n\n\u2248-refl : \u2200 {c a} {A : Set a} \u2192 Reflexive {A = \u21ba c A} _\u2248_\n\u2248-refl = F\u2248.id\n\n\u2248-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 p\u2081 \u2248 p\u2082 \u2192 p\u2082 \u2248 p\u2081\n\u2248-sym \u03b7 = F\u2248.sym \u03b7\n\n\u2248-trans : \u2200 {c a} {A : Set a} \u2192 Transitive {A = \u21ba c A} _\u2248_\n\u2248-trans f g = g F\u2248.\u2218 f\n\nfork-sym\u21d2 : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248\u21d2 fork p\u2082 p\u2081\nfork-sym\u21d2 (Pr-fork {p = p} {q} refl pf\u2081 pf\u2080) rewrite p \/2+ q \/2-comm = Pr-fork\u2032 pf\u2080 pf\u2081\n\nfork-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\nfork-sym = F\u2248.equivalence fork-sym\u21d2 fork-sym\u21d2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ]\u2261 0\/1\n            \u2192 Pr[ right \u2261 x ]\u2261 p\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork\u2032 eq\u2081 eq\u2082\n\n1-Pr : \u2200 {c a} {A : Set a} {Alg : \u21ba c A} {p P Q}\n       \u2192 (\u2200 {p x} \u2192 P x p \u2192 Q x (1- p))\n       \u2192 Pr[ Alg ! P ]\u2261 p\n       \u2192 Pr[ Alg ! Q ]\u2261 1- p\n1-Pr PQ (Pr-return pf) = Pr-return (PQ pf)\n1-Pr PQ (Pr-fork refl pf pf\u2081) = Pr-fork (1- _ \/2+1- _ \/2) (1-Pr PQ pf) (1-Pr PQ pf\u2081)\n\nPr-ret-not : \u2200 {p x y} \u2192 Pr[return\u21ba x \u2261 y ]\u2261 p \u2192 Pr[return\u21ba x \u2261 not y ]\u2261 (1- p)\nPr-ret-not (Pr-\u2261 x\u2261y) rewrite 1-1\u22610 = Pr-\u2262 (not-\u00ac x\u2261y)\nPr-ret-not (Pr-\u2262 x\u2262y) rewrite 1-0\u22611 = Pr-\u2261 (\u00ac-not x\u2262y)\n\nPr-not\u21d2 : \u2200 {c} {Alg : \u21ba c Bit} {x pr}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ Alg \u2261 not x ]\u2261 1- pr\nPr-not\u21d2 = 1-Pr Pr-ret-not\n\nPr-not\u21d0 : \u2200 {c} {Alg : \u21ba c Bit} {x pr}\n         \u2192 Pr[ Alg \u2261 not x ]\u2261 pr\n         \u2192 Pr[ Alg \u2261 x ]\u2261 1- pr\nPr-not\u21d0 {Alg = Alg} {x} {pr} pf = subst (\u03bb z \u2192 Pr[ Alg \u2261 z ]\u2261 1- pr) (not-involutive x) (Pr-not\u21d2 pf)\n\nPr-toss : \u2200 x \u2192 Pr[ toss \u2261 x ]\u2261 1\/2\nPr-toss 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261\nPr-toss 0b = F\u2248.Equivalence.to fork-sym F\u2261.\u27e8$\u27e9 (Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261)\n\nPr-map : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {P Q pr} {F : [0,1] \u2192 [0,1]} {f : A \u2192 B} \u2192\n  (R : \u2200 {pr x} \u2192 P x pr \u2192 Q (f x) (F pr)) \u2192\n  (F-R : \u2200 {p q} \u2192 F p \/2+ F q \/2 \u2261 F (p \/2+ q \/2)) \u2192\n  Pr[ Alg ! P ]\u2261 pr \u2192\n  Pr[ \u27ea f \u00b7 Alg \u27eb ! Q ]\u2261 F pr\nPr-map R F-R (Pr-return pf) = Pr-return (R pf)\nPr-map R F-R (Pr-fork refl pf\u2080 pf\u2081) = Pr-fork F-R (Pr-map R F-R pf\u2080) (Pr-map R F-R pf\u2081)\n\nPr-return-inj : \u2200 {a b} {A : Set a} {B : Set b} {x y pr} {f : A \u2192 B} \u2192\n  (f-inj-xy : f x \u2261 f y \u2192 x \u2261 y) \u2192\n  Pr[return\u21ba x \u2261 y ]\u2261 pr \u2192\n  Pr[return\u21ba f x \u2261 f y ]\u2261 pr\nPr-return-inj f-inj-xy (Pr-\u2261 refl) = Pr-\u2261 refl\nPr-return-inj f-inj-xy (Pr-\u2262 x\u2262y) = Pr-\u2262 (x\u2262y \u2218 f-inj-xy)\n\nPr-map-inj : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {x pr} {f : A \u2192 B} \u2192\n  (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n  Pr[ Alg \u2261 x ]\u2261 pr \u2192\n  Pr[ \u27ea f \u00b7 Alg \u27eb \u2261 f x ]\u2261 pr\nPr-map-inj f-inj pf = Pr-map {F = id} (Pr-return-inj f-inj) refl pf\n\nPr-same : \u2200 {c a} {A : Set a} {Alg : \u21ba c A} {x pr\u2080 pr\u2081} \u2192\n    pr\u2080 \u2261 pr\u2081 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2080 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2081\nPr-same refl = id\n\nPr-weaken\u2264 : \u2200 {c\u2080 c\u2081 a} {A : Set a} {Alg : \u21ba c\u2080 A} {P pr} \u2192\n    (c\u2080\u2264c\u2081 : c\u2080 \u2264 c\u2081) \u2192\n    Pr[ Alg ! P ]\u2261 pr \u2192\n    Pr[ weaken\u2264 c\u2080\u2264c\u2081 Alg ! P ]\u2261 pr\nPr-weaken\u2264 p (Pr-return pf) = Pr-return pf\nPr-weaken\u2264 (s\u2264s c\u2080\u2264c\u2081) (Pr-fork eq pf\u2080 pf\u2081)\n  = Pr-fork eq (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2080) (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2081)\n\nPr-weaken+ : \u2200 {c\u2080} c\u2081 {a} {A : Set a} {Alg : \u21ba c\u2080 A} {P pr} \u2192\n    Pr[ Alg ! P ]\u2261 pr \u2192\n    Pr[ weaken+ c\u2081 Alg ! P ]\u2261 pr\nPr-weaken+ c\u2081 = Pr-weaken\u2264 (m\u2264n+m _ c\u2081)\n\nPr-map-0 : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {f : A \u2192 B} {x} \u2192 (\u2200 y \u2192 f y \u2262 x)\n           \u2192 Pr[ map\u21ba f Alg \u2261 x ]\u2261 0\/1\nPr-map-0 {Alg = leaf x} f-prop = Pr-return (Pr-\u2262 (f-prop x))\nPr-map-0 {Alg = fork Alg Alg\u2081} f-prop = Pr-fork (trans (0\/2+p\/2\u2261p\/2 0\/1) 0\/2\u22610)\n                                                (Pr-map-0 f-prop) (Pr-map-0 f-prop)\n\nPR : \u2200 {a} \u2192 Set a \u2192 Set _\nPR {a} A = A \u2192 [0,1] \u2192 Set a\n\nrecord Pr\u00d7 {a b} {A : Set a} {B : Set b} (F : [0,1] \u2192 [0,1] \u2192 [0,1]) (PrA : PR A) (PrB : PR B) (xy : A \u00d7 B) pr : Set (a L\u2294 b) where\n  constructor mk\n  field\n    {pr\u2080 pr\u2081} : [0,1]\n    pr\u2080*pr\u2081\u2261pr : F pr\u2080 pr\u2081 \u2261 pr\n    prA : PrA (proj\u2081 xy) pr\u2080\n    prB : PrB (proj\u2082 xy) pr\u2081\n\nPr!-zip : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} {Alg\u2080 : \u21ba c\u2080 A} {Alg\u2081 : \u21ba c\u2081 B} {P\u2080 P\u2081 pr\u2080 pr\u2081}\n  {F}\n  (F\u2080 : \u2200 {pr\u2080 p q} \u2192 (F pr\u2080 p) \/2+ (F pr\u2080 q) \/2 \u2261 F pr\u2080 (p \/2+ q \/2))\n  (F\u2081 : \u2200 {pr\u2081 p q} \u2192 (F p pr\u2081) \/2+ (F q pr\u2081) \/2 \u2261 F (p \/2+ q \/2) pr\u2081)\n  \u2192 Pr[ Alg\u2080 ! P\u2080 ]\u2261 pr\u2080\n  \u2192 Pr[ Alg\u2081 ! P\u2081 ]\u2261 pr\u2081\n  \u2192 Pr[ zip\u21ba Alg\u2080 Alg\u2081 ! Pr\u00d7 F P\u2080 P\u2081 ]\u2261 F pr\u2080 pr\u2081\nPr!-zip {c\u2080} {pr\u2080 = pr\u2080} {F = F} F\u2080 F\u2081 (Pr-return Px) pf\u2081\n  = Pr-weaken+ c\u2080 (Pr-map {F = F pr\u2080} (mk refl Px) F\u2080 pf\u2081)\nPr!-zip F\u2080 F\u2081 (Pr-fork refl pf\u2080 pf\u2081) pf\u2082 = Pr-fork F\u2081 (Pr!-zip F\u2080 F\u2081 pf\u2080 pf\u2082) (Pr!-zip F\u2080 F\u2081 pf\u2081 pf\u2082)\n\nPr!\u21d2 : \u2200 {c a} {A : Set a} {P Q : PR A} {Alg : \u21ba c A} {pr}\n        \u2192 (\u2200 {x pr} \u2192 P x pr \u2192 Q x pr) \u2192 Pr[ Alg ! P ]\u2261 pr \u2192 Pr[ Alg ! Q ]\u2261 pr\nPr!\u21d2 P\u21d2Q (Pr-return Px) = Pr-return (P\u21d2Q Px)\nPr!\u21d2 P\u21d2Q (Pr-fork eq pf pf\u2081) = Pr-fork eq (Pr!\u21d2 P\u21d2Q pf) (Pr!\u21d2 P\u21d2Q pf\u2081)\n\nPr-zip : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} {Alg\u2080 : \u21ba c\u2080 A} {Alg\u2081 : \u21ba c\u2081 B} {x y pr\u2081 pr\u2082} \u2192\n  Pr[ Alg\u2080 \u2261 x ]\u2261 pr\u2081 \u2192\n  Pr[ Alg\u2081 \u2261 y ]\u2261 pr\u2082 \u2192\n  Pr[ zip\u21ba Alg\u2080 Alg\u2081 \u2261 (x , y) ]\u2261 (pr\u2081 *\/ pr\u2082)\nPr-zip {x = x} {y} pf\u2080 pf\u2081 = Pr!\u21d2 helper' (Pr!-zip {F = _*\/_} helper distr-*-\/2+\/2 pf\u2080 pf\u2081)\n  where\n    helper = \u03bb {pr\u2080} {p} {q} \u2192 trans (trans (cong\u2082 _\/2+_\/2 (*\/-comm _ _) (*\/-comm _ _)) distr-*-\/2+\/2) (sym (*\/-comm pr\u2080 (p \/2+ q \/2)))\n    helper' : \u2200 {xy pr} \u2192 Pr\u00d7 _*\/_ (flip Pr[return\u21ba_\u2261_]\u2261_ x) (flip Pr[return\u21ba_\u2261_]\u2261_ y) xy pr \u2192 Pr[return\u21ba xy \u2261 x , y ]\u2261 pr\n    helper' (mk {pr\u2081 = pr\u2083} refl (Pr-\u2261 refl) prB) rewrite 1*q\u2261q pr\u2083 = Pr-return-inj (cong proj\u2082) prB\n    helper' (mk {pr\u2081 = pr\u2083} refl (Pr-\u2262 x\u2262y) prB) rewrite 0*q\u2261q pr\u2083 = Pr-\u2262 (x\u2262y \u2218 cong proj\u2081)\n\nPr-zip\u21ba-toss-toss : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ]\u2261 1\/4\nPr-zip\u21ba-toss-toss x y = Pr-same (trans distr-\/2-* (cong (_\/2 \u2218 _\/2) (1*q\u2261q _)))\n                                 (Pr-zip {Alg\u2080 = toss} {Alg\u2081 = toss} (Pr-toss x) (Pr-toss y))\n\npr-choose : \u2200 {c a} {A : Set a} {p\u2080 p\u2081 : \u21ba c A} {pr\u2080 pr\u2081 x}\n             \u2192 Pr[ p\u2080 \u2261 x ]\u2261 pr\u2080\n             \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr\u2081\n             \u2192 Pr[ choose p\u2080 p\u2081 \u2261 x ]\u2261 pr\u2080 \/2+ pr\u2081 \/2\npr-choose {c} {pr\u2080 = pr\u2080} {pr\u2081} pf\u2080 pf\u2081 =\n  Pr-fork (pr\u2081 \/2+ pr\u2080 \/2-comm) (Pr-weaken+ 0 pf\u2081) (Pr-weaken+ 0 pf\u2080)\n\nPr-\u2261xor\u21d2 : \u2200 {c} {Alg : \u21ba c Bit} {x} b\n         \u2192 Pr[ Alg \u2261 x ]\u2261 1\/2\n         \u2192 Pr[ Alg \u2261 b xor x ]\u2261 1\/2\nPr-\u2261xor\u21d2 true pf = Pr-same 1-1\/2\u22611\/2 (Pr-not\u21d2 pf)\nPr-\u2261xor\u21d2 false pf = pf\n\nPr-\u2261xor\u21d0 : \u2200 {c} {Alg : \u21ba c Bit} {x} b\n         \u2192 Pr[ Alg \u2261 b xor x ]\u2261 1\/2\n         \u2192 Pr[ Alg \u2261 x ]\u2261 1\/2\nPr-\u2261xor\u21d0 {x = x} true pf  = Pr-same 1-1\/2\u22611\/2 (Pr-not\u21d0 pf)\nPr-\u2261xor\u21d0 false pf = pf\n\nPr-xor : \u2200 {c} {Alg : \u21ba c Bit} {x b}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 1\/2\n         \u2192 Pr[ \u27ea _xor_ b \u00b7 Alg \u27eb \u2261 x ]\u2261 1\/2\nPr-xor {b = b} = Pr-\u2261xor\u21d0 b \u2218 Pr-map-inj (xor-inj {b})\n  where\n    -- move it!\n    not-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\n    not-inj {true} {true} = \u03bb _ \u2192 refl\n    not-inj {true} {false} = \u03bb ()\n    not-inj {false} {true} = \u03bb ()\n    not-inj {false} {false} = \u03bb _ \u2192 refl\n    xor-inj : \u2200 {b x y} \u2192 b xor x \u2261 b xor y \u2192 x \u2261 y\n    xor-inj {true} = not-inj\n    xor-inj {false} = id\n\nPr-xor-toss : \u2200 {i} \u2192 Pr[ \u27ea _xor_ i \u00b7 toss \u27eb \u2261*]\u2261 1\/2\nPr-xor-toss {i} = Pr-xor {b = i} (Pr-toss _)\n","old_contents":"module flipbased-tree where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import bintree\nopen import Data.Product\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\nreturn\u21ba : \u2200 {c a} {A : Set a} \u2192 A \u2192 \u21ba c A\nreturn\u21ba = leaf\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (leaf x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (leaf 0b) (leaf 1b)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (leaf x) = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 c {m a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ c = weaken\u2264 (m\u2264n+m _ c)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (leaf x) = leaf (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (leaf x)      = weaken+ c x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 leaf map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _*\/_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  1-_ : [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n  -- \u00b7\/1+_ : \u2115 \u2192 Set\n  -- \/+\/ : \u2115 \u2192 [0,1] \u2192 [0,1] \u2192 [0,1]\n\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n  _\/2+_\/2-comm : \u2200 x y \u2192 x \/2+ y \/2 \u2261 y \/2+ x \/2\n  1*q\u2261q : \u2200 q \u2192 1\/1 *\/ q \u2261 q\n  0*q\u2261q : \u2200 q \u2192 0\/1 *\/ q \u2261 0\/1\n  distr-*-\/2+\/2 : \u2200 x y z \u2192 (x *\/ z) \/2+ (y *\/ z) \/2 \u2261 (x \/2+ y \/2) *\/ z\n  distr-\/2-* : \u2200 p q \u2192 (p \/2) *\/ (q \/2) \u2261 ((p *\/ q) \/2) \/2\n  0\/2\u22610 : 0\/1 \/2 \u2261 0\/1\n  1-_\/2+1-_\/2 : \u2200 p q \u2192 (1- p) \/2+ (1- q) \/2 \u2261 1- (p \/2+ q \/2)\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\ndata Pr[return\u21ba_\u2261_]\u2261_ {a} {A : Set a} (x y : A) : [0,1] \u2192 Set a where\n  Pr-\u2261 : x \u2261 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 1\/1\n  Pr-\u2262 : x \u2262 y \u2192 Pr[return\u21ba x \u2261 y ]\u2261 0\/1\n\ninfix 2 Pr[_\u2261_]\u2261_\ndata Pr[_\u2261_]\u2261_ {a} {A : Set a} : \u2200 {c} \u2192 \u21ba c A \u2192 A \u2192 [0,1] \u2192 Set a where\n  Pr-return : \u2200 {c x y pr} (pf : Pr[return\u21ba x \u2261 y ]\u2261 pr) \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 pr\n\n  Pr-fork : \u2200 {c} {left right : \u21ba c A} {x p q r}\n            (eq : p \/2+ q \/2 \u2261 r)\n            (pf\u2080 : Pr[ left \u2261 x ]\u2261 p)\n            (pf\u2081 : Pr[ right \u2261 x ]\u2261 q)\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 r\n\nPr-fork\u2032 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x p q}\n            \u2192 Pr[ left \u2261 x ]\u2261 p\n            \u2192 Pr[ right \u2261 x ]\u2261 q\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 (p \/2+ q \/2)\nPr-fork\u2032 = Pr-fork refl\n\nPr-return-\u2261 : \u2200 {c a} {A : Set a} {x : A} \u2192 Pr[ return\u21ba {c = c} x \u2261 x ]\u2261 1\/1\nPr-return-\u2261 = Pr-return (Pr-\u2261 refl)\n\nPr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ]\u2261 0\/1\nPr-return-\u2262 = Pr-return \u2218 Pr-\u2262\n\nimport Function.Equality as F\u2261\nimport Function.Equivalence as F\u2248\n\n_\u2248\u21d2_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248\u21d2 p\u2082 = \u2200 {x pr} \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr \u2192 Pr[ p\u2082 \u2261 x ]\u2261 pr\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 {x pr} \u2192 (Pr[ p\u2081 \u2261 x ]\u2261 pr) F\u2248.\u21d4 (Pr[ p\u2082 \u2261 x ]\u2261 pr)\n\n\u2248-refl : \u2200 {c a} {A : Set a} \u2192 Reflexive {A = \u21ba c A} _\u2248_\n\u2248-refl = F\u2248.id\n\n\u2248-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 p\u2081 \u2248 p\u2082 \u2192 p\u2082 \u2248 p\u2081\n\u2248-sym \u03b7 = F\u2248.sym \u03b7\n\n\u2248-trans : \u2200 {c a} {A : Set a} \u2192 Transitive {A = \u21ba c A} _\u2248_\n\u2248-trans f g = g F\u2248.\u2218 f\n\nfork-sym\u21d2 : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248\u21d2 fork p\u2082 p\u2081\nfork-sym\u21d2 (Pr-fork {p = p} {q} refl pf\u2081 pf\u2080) rewrite p \/2+ q \/2-comm = Pr-fork\u2032 pf\u2080 pf\u2081\n\nfork-sym : \u2200 {c a} {A : Set a} {p\u2081 p\u2082 : \u21ba c A} \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\nfork-sym = F\u2248.equivalence fork-sym\u21d2 fork-sym\u21d2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ]\u2261 0\/1\n            \u2192 Pr[ right \u2261 x ]\u2261 p\n            \u2192 Pr[ fork left right \u2261 x ]\u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork\u2032 eq\u2081 eq\u2082\n\nex\u2081 : \u2200 x \u2192 Pr[ toss \u2261 x ]\u2261 1\/2\nex\u2081 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261\nex\u2081 0b = F\u2248.Equivalence.to fork-sym F\u2261.\u27e8$\u27e9 (Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) Pr-return-\u2261)\n\nPr-map : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {x pr} {f : A \u2192 B} \u2192\n  (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n  Pr[ Alg \u2261 x ]\u2261 pr \u2192\n  Pr[ \u27ea f \u00b7 Alg \u27eb \u2261 f x ]\u2261 pr\nPr-map f-inj (Pr-return (Pr-\u2261 refl)) = Pr-return (Pr-\u2261 refl)\nPr-map f-inj (Pr-return (Pr-\u2262 x\u2262y)) = Pr-return (Pr-\u2262 (x\u2262y \u2218 f-inj))\nPr-map f-inj (Pr-fork eq pf\u2080 pf\u2081) = Pr-fork eq (Pr-map f-inj pf\u2080) (Pr-map f-inj pf\u2081)\n\nPr-same : \u2200 {c a} {A : Set a} {Alg : \u21ba c A} {x pr\u2080 pr\u2081} \u2192\n    pr\u2080 \u2261 pr\u2081 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2080 \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr\u2081\nPr-same refl = id\n\nPr-weaken\u2264 : \u2200 {c\u2080 c\u2081 a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    (c\u2080\u2264c\u2081 : c\u2080 \u2264 c\u2081) \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken\u2264 c\u2080\u2264c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken\u2264 p (Pr-return pf) = Pr-return pf\nPr-weaken\u2264 (s\u2264s c\u2080\u2264c\u2081) (Pr-fork eq pf\u2080 pf\u2081)\n  = Pr-fork eq (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2080) (Pr-weaken\u2264 c\u2080\u2264c\u2081 pf\u2081)\n\nPr-weaken+ : \u2200 {c\u2080} c\u2081 {a} {A : Set a} {Alg : \u21ba c\u2080 A} {x pr} \u2192\n    Pr[ Alg \u2261 x ]\u2261 pr \u2192\n    Pr[ weaken+ c\u2081 Alg \u2261 x ]\u2261 pr\nPr-weaken+ c\u2081 = Pr-weaken\u2264 (m\u2264n+m _ c\u2081)\n\nPr-map-0 : \u2200 {c a b} {A : Set a} {B : Set b} {Alg : \u21ba c A} {f : A \u2192 B} {x} \u2192 (\u2200 y \u2192 f y \u2262 x)\n           \u2192 Pr[ map\u21ba f Alg \u2261 x ]\u2261 0\/1\nPr-map-0 {Alg = leaf x} f-prop = Pr-return (Pr-\u2262 (f-prop x))\nPr-map-0 {Alg = fork Alg Alg\u2081} f-prop = Pr-fork (trans (0\/2+p\/2\u2261p\/2 0\/1) 0\/2\u22610)\n                                                (Pr-map-0 f-prop) (Pr-map-0 f-prop)\n\nPr-zip : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} {Alg\u2080 : \u21ba c\u2080 A} {Alg\u2081 : \u21ba c\u2081 B} {x y pr\u2081 pr\u2082} \u2192\n  Pr[ Alg\u2080 \u2261 x ]\u2261 pr\u2081 \u2192\n  Pr[ Alg\u2081 \u2261 y ]\u2261 pr\u2082 \u2192\n  Pr[ zip\u21ba Alg\u2080 Alg\u2081 \u2261 (x , y) ]\u2261 (pr\u2081 *\/ pr\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2261 refl)) pf\u2082\n  rewrite 1*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map {f = _,_ x} (cong proj\u2082) pf\u2082)\nPr-zip {c\u2080} {pr\u2082 = pr\u2082} (Pr-return {x = x} (Pr-\u2262 pf)) pf\u2082\n  rewrite 0*q\u2261q pr\u2082 = Pr-weaken+ c\u2080 (Pr-map-0 (\u03bb y x\u2081 \u2192 pf (cong proj\u2081 x\u2081)))\nPr-zip (Pr-fork refl pf\u2081 pf\u2082) pf\u2083 = Pr-fork (distr-*-\/2+\/2 _ _ _) (Pr-zip pf\u2081 pf\u2083) (Pr-zip pf\u2082 pf\u2083)\n\nex\u2082 : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ]\u2261 1\/4\nex\u2082 x y = Pr-same (trans (distr-\/2-* _ _) (cong (_\/2 \u2218 _\/2) (1*q\u2261q _)))\n                  (Pr-zip {Alg\u2080 = toss} {Alg\u2081 = toss} (ex\u2081 x) (ex\u2081 y))\n\npr-choose : \u2200 {c a} {A : Set a} {p\u2080 p\u2081 : \u21ba c A} {pr\u2080 pr\u2081 x}\n             \u2192 Pr[ p\u2080 \u2261 x ]\u2261 pr\u2080\n             \u2192 Pr[ p\u2081 \u2261 x ]\u2261 pr\u2081\n             \u2192 Pr[ choose p\u2080 p\u2081 \u2261 x ]\u2261 pr\u2080 \/2+ pr\u2081 \/2\npr-choose {c} {pr\u2080 = pr\u2080} {pr\u2081} pf\u2080 pf\u2081 =\n  Pr-fork (pr\u2081 \/2+ pr\u2080 \/2-comm) (Pr-weaken+ 0 pf\u2081) (Pr-weaken+ 0 pf\u2080)\n\npostulate\n  pr-ret-xor : \u2200 {x y pr} b\n         \u2192 Pr[return\u21ba x \u2261 y ]\u2261 pr\n         \u2192 Pr[return\u21ba x \u2261 b xor y ]\u2261 pr\n-- pr-ret-xor b pf = {!!}\n\npr-xor'' : \u2200 {c} {Alg : \u21ba c Bit} {x pr} b\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ Alg \u2261 b xor x ]\u2261 pr\npr-xor'' b (Pr-return pf) = Pr-return (pr-ret-xor b pf)\npr-xor'' b (Pr-fork refl pf\u2080 pf\u2081) = Pr-fork refl (pr-xor'' b pf\u2080) (pr-xor'' b pf\u2081)\n\npr-xor' : \u2200 {c} {Alg : \u21ba c Bit} {x pr b}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ \u27ea _xor_ b \u00b7 Alg \u27eb \u2261 b xor x ]\u2261 pr\npr-xor' {b = b} = Pr-map (xor-inj {b})\n  where\n    -- move it!\n    not-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\n    not-inj {true} {true} = \u03bb _ \u2192 refl\n    not-inj {true} {false} = \u03bb ()\n    not-inj {false} {true} = \u03bb ()\n    not-inj {false} {false} = \u03bb _ \u2192 refl\n    xor-inj : \u2200 {b x y} \u2192 b xor x \u2261 b xor y \u2192 x \u2261 y\n    xor-inj {true} = not-inj\n    xor-inj {false} = id\n\npostulate\n  pr-ret-not : \u2200 {x y pr}\n         \u2192 Pr[return\u21ba x \u2261 y ]\u2261 pr\n         \u2192 Pr[return\u21ba x \u2261 not y ]\u2261 1- pr\n-- pr-ret-not = {!!}\n\npr-not : \u2200 {c} {Alg : \u21ba c Bit} {x pr}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ Alg \u2261 not x ]\u2261 1- pr\npr-not (Pr-return pf) = Pr-return (pr-ret-not pf)\npr-not (Pr-fork refl pf pf\u2081) = Pr-fork (1- _ \/2+1- _ \/2) (pr-not pf) (pr-not pf\u2081)\n\npostulate\n  pr-xor : \u2200 {c} {Alg : \u21ba c Bit} {x pr b}\n         \u2192 Pr[ Alg \u2261 x ]\u2261 pr\n         \u2192 Pr[ \u27ea _xor_ b \u00b7 Alg \u27eb \u2261 x ]\u2261 pr\n-- pr-xor {b = b} = {!pr-not!}\n\npostulate\n  pr-toss-xor-toss : \u2200 x \u2192 Pr[ toss \u27e8xor\u27e9 toss \u2261 x ]\u2261 1\/2\n-- pr-toss-xor-toss x = {!!}\n\npostulate\n  ex\u2083 : \u2200 {n} (x : Bits n) \u2192 Pr[ random \u2261 x ]\u2261 1\/2^ n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c60b480ba03b7621a07b5f09841535a5d87428b9","subject":"The program agda2atp fixed an issue with the translation.","message":"The program agda2atp fixed an issue with the translation.\n\nIgnore-this: 55121fd07ed28b96fa0b7351d6d86846\n\ndarcs-hash:20120627174447-3bd4e-574390a410434b10062b759958a7a178214b45c9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/NonEmptyDomain\/TheoremsATP.agda","new_file":"src\/FOL\/NonEmptyDomain\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Theorems which require a non-empty domain\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Using the ATPs we don't have to use the postulate about a non-empty\n-- domain because the ATPs assume it.\n\n-- References:\n--\n-- \u2022 Elliott Mendelson. Introduction to mathematical logic. Chapman &\n--   Hall, 4th edition, 1997.\n\nmodule FOL.NonEmptyDomain.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A  : Set\n  A\u00b9 : D \u2192 Set\n\npostulate \u2200\u2192\u2203 : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u2203 A\u00b9\n{-# ATP prove \u2200\u2192\u2203 #-}\n\n-- Let A be a formula. If x is not free in A then \u22a2 (\u2203x)A \u2194 A\n-- (Mendelson 1997, proposition 2.18 (b), p. 70).\npostulate \u2203-erase-add\u2081 : \u2203 (\u03bb _ \u2192 A) \u2194 A\n{-# ATP prove \u2203-erase-add\u2081 #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate \u2200-erase-add : ((x : D) \u2192 A) \u2194 A\n{-# ATP prove \u2200-erase-add #-}\n\npostulate \u2203-erase-add\u2082 : (\u2203[ x ] A \u2228 A\u00b9 x) \u2194 A \u2228 (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2203-erase-add\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Theorems which require a non-empty domain\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Using the ATPs we don't have to use the postulate about a non-empty\n-- domain because the ATPs assume it.\n\n-- References:\n--\n-- \u2022 Elliott Mendelson. Introduction to mathematical logic. Chapman &\n--   Hall, 4th edition, 1997.\n\nmodule FOL.NonEmptyDomain.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A  : Set\n  A\u00b9 : D \u2192 Set\n\npostulate \u2200\u2192\u2203 : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u2203 A\u00b9\n{-# ATP prove \u2200\u2192\u2203 #-}\n\n-- Let A be a formula. If x is not free in A then \u22a2 (\u2203x)A \u2194 A\n-- (Mendelson 1997, proposition 2.18 (b), p. 70).\npostulate \u2203-erase-add\u2081 : \u2203 (\u03bb _ \u2192 A) \u2194 A\n{-# ATP prove \u2203-erase-add\u2081 #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\n-- TODO: 2012-02-03. The ATPs are not proving the theorem.\npostulate \u2200-erase-add : ((x : D) \u2192 A) \u2194 A\n-- {-# ATP prove \u2200-erase-add #-}\n\npostulate \u2203-erase-add\u2082 : (\u2203[ x ] A \u2228 A\u00b9 x) \u2194 A \u2228 (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2203-erase-add\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"60db7538f37541bf1f7be85fba27f42dd2d848b4","subject":"Boring.","message":"Boring.\n\nIgnore-this: 3c406abb2528388d742ce9a5ffb94f5\n\ndarcs-hash:20100713155035-3bd4e-ec983b1c0b74f54606c903e6f0555e58b21d4b22.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"MyStdLib\/Relation\/Binary\/EqReasoning.agda","new_file":"MyStdLib\/Relation\/Binary\/EqReasoning.agda","new_contents":"-----------------------------------------------------------------------------\n-- Equality reasoning\n-----------------------------------------------------------------------------\n\nmodule MyStdLib.Relation.Binary.EqReasoning\n  {A : Set}\n  (_\u2261_ : A \u2192 A \u2192 Set)\n  (refl : {x : A} \u2192 x \u2261 x)\n  (trans : {x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n  where\n\ninfix 4 _\u2243_\n\nprivate\n  data _\u2243_ (x y : A) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\n------------------------------------------------------------------------------\n-- A version from Ulf's thesis\nmodule Original where\n\n  infix  2 chain>_\n  infixl 2 _===_by_\n  infix  1 _qed\n\n  chain>_ : (x : A) \u2192 x \u2243 x\n  chain> x = prf (refl {x})\n\n  _===_by_ : {x y : A} \u2192 x \u2243 y \u2192 (z : A) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\n------------------------------------------------------------------------------\n-- A version from the standard library (Relation.Binary.PreorderReasoning)\nmodule StdLib where\n\n  infix  1 begin_\n  infixr 2 _\u2261\u27e8_\u27e9_\n  infix  2 _\u220e\n\n  begin_ : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : A){y z : A} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : A) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n","old_contents":"-----------------------------------------------------------------------------\n-- Equality reasoning\n-----------------------------------------------------------------------------\n\nmodule MyStdLib.Relation.Binary.EqReasoning\n  {A : Set}\n  (_\u2261_ : A \u2192 A \u2192 Set)\n  (refl : {x : A} \u2192 x \u2261 x)\n  (trans : {x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n  where\n\ninfix 4 _\u2243_\n\nprivate\n  data _\u2243_ (x y : A) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\nmodule Original where\n  ---------------------------------------------------------------------------\n  -- A version from Ulf's thesis\n\n  infix  2 chain>_\n  infixl 2 _===_by_\n  infix  1 _qed\n\n  chain>_ : (x : A) \u2192 x \u2243 x\n  chain> x = prf (refl {x})\n\n  _===_by_ : {x y : A} \u2192 x \u2243 y \u2192 (z : A) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\nmodule StdLib where\n\n  -- A version from the standard library (Relation.Binary.PreorderReasoning)\n\n  infix  1 begin_\n  infixr 2 _\u2261\u27e8_\u27e9_\n  infix  2 _\u220e\n\n  begin_ : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : A){y z : A} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : A) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e23166c1580dcc4547699a337cfc1203c6dae91b","subject":"SHA1 main","message":"SHA1 main\n","repos":"crypto-agda\/crypto-agda","old_file":"sha1.agda","new_file":"sha1.agda","new_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1 where\n\nmodule FunSHA1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n    Word : T\n    Word = `Bits 32\n\n    map\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\n    map\u00b2\u02b7 = zipWith\n\n    map\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\n    map\u02b7 = map\n\n    lift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\n    lift f g = g \u204f f\n\n    lift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n    lift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n    `not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    `not = lift (map\u02b7 not)\n\n    infixr 3 _`\u2295_\n    _`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\n    infixr 3 _`\u2227_\n    _`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\n    infixr 2 _`\u2228_\n    _`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\n    open import Solver.Linear\n\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                   \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                   \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    <\u229e> adder : Word `\u00d7 Word `\u2192 Word\n    adder = <tt\u204f <0b> , id > \u204f iter full-adder\n    <\u229e> = adder\n\n    infixl 4 _`\u229e_\n    _`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n    _`\u229e_ = lift\u2082 <\u229e>\n\n    <_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n    < f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n    <_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                                  \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                                  \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n    < f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n    <<<\u2085 : `Endo Word\n    <<<\u2085 = rot-left 5\n\n    <<<\u2083\u2080 : `Endo Word\n    <<<\u2083\u2080 = rot-left 30\n\n    Word\u00b2 = Word `\u00d7 Word\n    Word\u00b3 = Word `\u00d7 Word\u00b2\n    Word\u2074 = Word `\u00d7 Word\u00b3\n    Word\u2075 = Word `\u00d7 Word\u2074\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 =  constBits (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0'\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0'\n            F\u25b9\ud835\udfda (suc _) = 1'\n\n            {-\n    [_-_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m - n mod p ] = {!!}\n\n    [_+_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m + n mod p ] = {!!}\n    -}\n\n    #\u02b7 = bits 32\n\n    <\u229e\u2075> : Word\u2075 `\u00d7 Word\u2075 `\u2192 Word\u2075\n    <\u229e\u2075> = helper \u204f < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 <\u229e> > > > >\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {C} {D} {E} {F} {G} {H} {I} {J} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 C \u2237 D \u2237 E \u2237 F \u2237 G \u2237 H \u2237 I \u2237 J \u2237 [])\n                  (\u03bb a b c d e f g h i j \u2192\n                    ((a , b , c , d , e) , (f , g , h , i , j) \u21a6\n                     ((a , f) , (b , g) , (c , h) , (d , i) , (e , j))))\n\n    iterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\n    iterate\u207f zero    f = id\n    iterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n    _\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n    _\u00b2\u2070 = iterate\u207f 20\n\n    module _ where\n\n      K\u2080 = #\u02b7 0x5A827999\n      K\u2082 = #\u02b7 0x6ED9EBA1\n      K\u2084 = #\u02b7 0x8F1BBCDC\n      K\u2086 = #\u02b7 0xCA62C1D6\n\n      H0 = #\u02b7 0x67452301\n      H1 = #\u02b7 0xEFCDAB89\n      H2 = #\u02b7 0x98BADCFE\n      H3 = #\u02b7 0x10325476\n      H4 = #\u02b7 0xC3D2E1F0\n\n      A B C D E : Word\u2075 `\u2192 Word\n      A = fst\n      B = snd \u204f fst\n      C = snd \u204f snd \u204f fst\n      D = snd \u204f snd \u204f snd \u204f fst\n      E = snd \u204f snd \u204f snd \u204f snd\n\n      F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n      F\u2082 = B `\u2295 C `\u2295 D\n      F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n      F\u2086 = F\u2082\n\n      module _ (F : Word\u2075 `\u2192 Word)\n               (K : `\ud835\udfd9  `\u2192 Word)\n               (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n          where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n      module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n        W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n        W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n        W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n        W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n        W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n        Iteration\u2078\u2070 : `Endo Word\u2075\n        Iteration\u2078\u2070 =\n          (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n          (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n          (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n          (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n        pad0s : \u2115 \u2192 \u2115\n        pad0s zero = 512 \u2238 65\n        pad0s (suc _) = STUCK where postulate STUCK : \u2115\n        -- pad0s n = [ 512 - [ n + 65 mod 512 ] mod 512 ]\n\n        paddedLength : \u2115 \u2192 \u2115\n        paddedLength n = n + (1 + pad0s n + 64)\n\n        padding : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (paddedLength n)\n        padding {n} = < id ,tt\u204f <1\u2237 < <0\u207f> {pad0s n} ++ bits 64 n > > > \u204f append\n\n        ite : Endo (`Endo Word\u2075)\n        ite f = dup \u204f first f \u204f <\u229e\u2075>\n\n        hash-block : `Endo Word\u2075\n        hash-block = ite Iteration\u2078\u2070\n\n      ite' : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word)\n             \u2192 ((Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `Endo Word\u2075) \u2192 `Endo Word\u2075\n      ite' zero    W f = id\n      ite' (suc n) W f = f (W zero) \u204f ite' n (W \u2218 suc) f\n\n      SHA1 : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1 n W = < H0 , H1 , H2 , H3 , H4 >\n               \u204f ite' n W hash-block\n\nmodule AgdaSHA1 where\n  open import FunUniverse.Agda\n  open FunSHA1 agdaFunOps\n  open import Data.Two\n\n  SHA1-on-0s : Word\u2075\n  SHA1-on-0s = SHA1 1 (\u03bb _ _ _ \u2192 V.replicate 0') _\n\nopen import IO\nimport IO.Primitive\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Coinduction\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1' = putStr \"1\"\nputBit 0' = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\nput\u00d7 : \u2200 {A B : Set} \u2192 (A \u2192 IO \ud835\udfd9) \u2192 (B \u2192 IO \ud835\udfd9) \u2192 (A \u00d7 B) \u2192 IO \ud835\udfd9\nput\u00d7 pA pB (x , y) = \u266f pA x >> \u266f pB y\n{-\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits\n          putBits))) AgdaSHA1.SHA1-on-0s)\n-}\nfirstBit : \u2200 {A : Set} \u2192 (V.Vec \ud835\udfda 32 \u00d7 A) \u2192 \ud835\udfda\nfirstBit ((b \u2237 _) , _) = b\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (putBit (firstBit AgdaSHA1.SHA1-on-0s))\n\n-- -}\n","old_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1 where\n\nmodule FunSHA1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n    Word : T\n    Word = `Bits 32\n\n    map\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\n    map\u00b2\u02b7 = zipWith\n\n    map\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\n    map\u02b7 = map\n\n    lift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\n    lift f g = g \u204f f\n\n    lift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n    lift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n    `not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    `not = lift (map\u02b7 not)\n\n    infixr 3 _`\u2295_\n    _`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\n    infixr 3 _`\u2227_\n    _`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\n    infixr 2 _`\u2228_\n    _`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\n    open import Solver.Linear\n\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                   \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                   \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    <\u229e> adder : Word `\u00d7 Word `\u2192 Word\n    adder = <tt\u204f <0b> , id > \u204f iter full-adder\n    <\u229e> = adder\n\n    infixl 4 _`\u229e_\n    _`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n    _`\u229e_ = lift\u2082 <\u229e>\n\n    <_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n    < f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n    <_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                                  \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                                  \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n    < f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n    <<<\u2085 : `Endo Word\n    <<<\u2085 = rot-left 5\n\n    <<<\u2083\u2080 : `Endo Word\n    <<<\u2083\u2080 = rot-left 30\n\n    Word\u00b2 = Word `\u00d7 Word\n    Word\u00b3 = Word `\u00d7 Word\u00b2\n    Word\u2074 = Word `\u00d7 Word\u00b3\n    Word\u2075 = Word `\u00d7 Word\u2074\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 =  constBits (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0'\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0'\n            F\u25b9\ud835\udfda (suc _) = 1'\n\n            {-\n    [_-_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m - n mod p ] = {!!}\n\n    [_+_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m + n mod p ] = {!!}\n    -}\n\n    #\u02b7 = bits 32\n\n    <\u229e\u2075> : Word\u2075 `\u00d7 Word\u2075 `\u2192 Word\u2075\n    <\u229e\u2075> = helper \u204f < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 <\u229e> > > > >\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {C} {D} {E} {F} {G} {H} {I} {J} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 C \u2237 D \u2237 E \u2237 F \u2237 G \u2237 H \u2237 I \u2237 J \u2237 [])\n                  (\u03bb a b c d e f g h i j \u2192\n                    ((a , b , c , d , e) , (f , g , h , i , j) \u21a6\n                     ((a , f) , (b , g) , (c , h) , (d , i) , (e , j))))\n\n    iterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\n    iterate\u207f zero    f = id\n    iterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n    _\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n    _\u00b2\u2070 = iterate\u207f 20\n\n    module _ where\n\n      K\u2080 = #\u02b7 0x5A827999\n      K\u2082 = #\u02b7 0x6ED9EBA1\n      K\u2084 = #\u02b7 0x8F1BBCDC\n      K\u2086 = #\u02b7 0xCA62C1D6\n\n      H0 = #\u02b7 0x67452301\n      H1 = #\u02b7 0xEFCDAB89\n      H2 = #\u02b7 0x98BADCFE\n      H3 = #\u02b7 0x10325476\n      H4 = #\u02b7 0xC3D2E1F0\n\n      A B C D E : Word\u2075 `\u2192 Word\n      A = fst\n      B = snd \u204f fst\n      C = snd \u204f snd \u204f fst\n      D = snd \u204f snd \u204f snd \u204f fst\n      E = snd \u204f snd \u204f snd \u204f snd\n\n      F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n      F\u2082 = B `\u2295 C `\u2295 D\n      F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n      F\u2086 = F\u2082\n\n      module _ (F : Word\u2075 `\u2192 Word)\n               (K : `\ud835\udfd9  `\u2192 Word)\n               (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n          where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n      module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n        W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n        W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n        W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n        W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n        W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n        Iteration\u2078\u2070 : `Endo Word\u2075\n        Iteration\u2078\u2070 =\n          (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n          (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n          (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n          (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n        pad0s : \u2115 \u2192 \u2115\n        pad0s zero = 512 \u2238 65\n        pad0s (suc _) = STUCK where postulate STUCK : \u2115\n        -- pad0s n = [ 512 - [ n + 65 mod 512 ] mod 512 ]\n\n        paddedLength : \u2115 \u2192 \u2115\n        paddedLength n = n + (1 + pad0s n + 64)\n\n        padding : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (paddedLength n)\n        padding {n} = < id ,tt\u204f <1\u2237 < <0\u207f> {pad0s n} ++ bits 64 n > > > \u204f append\n\n        ite : Endo (`Endo Word\u2075)\n        ite f = dup \u204f first f \u204f <\u229e\u2075>\n\n        hash-block : `Endo Word\u2075\n        hash-block = ite Iteration\u2078\u2070\n\n      ite' : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word)\n             \u2192 ((Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `Endo Word\u2075) \u2192 `Endo Word\u2075\n      ite' zero    W f = id\n      ite' (suc n) W f = f (W zero) \u204f ite' n (W \u2218 suc) f\n\n      SHA1 : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1 n W = < H0 , H1 , H2 , H3 , H4 >\n               \u204f ite' n W hash-block\n\nmodule AgdaSHA1 where\n  open import FunUniverse.Agda\n  open FunSHA1 agdaFunOps\n  open import Data.Two\n\n  SHA1-on-0s : Word\u2075\n  SHA1-on-0s = SHA1 1 (\u03bb _ _ _ \u2192 V.replicate 0') _\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7c96b8369eaff904dd9270e53985e5b78739b42c","subject":"Desc@ICFP: add \\update to model","message":"Desc@ICFP: add \\update to model","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\nupdate : {ty : Type} -> IMu closeTerm ty -> IMu closeTerm' ty\nupdate {ty} tm = cata Type closeTerm (IMu closeTerm') (\\ _ tagTm -> con (ESu (fst tagTm) , (snd tagTm))) ty tm\n\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            Var context ty ->\n            IMu closeTerm' ty\ndischarge ctxt env ty variable = con (ESu EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     Var context ty ->\n                     IMu closeTerm' ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm' ty\nsubstExpr {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval' (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu (ESu EZe)) , (con (ESu EZe , su ze) , con ( EZe , l (r refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , l (l (r refl))) ,\n                             (con (EZe , l (r refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe ,  r refl  )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            Var context ty ->\n            IMu closeTerm' ty\ndischarge ctxt env ty variable = con (ESu EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     Var context ty ->\n                     IMu closeTerm' ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm' ty\nsubstExpr {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval' (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu (ESu EZe)) , (con (ESu EZe , su ze) , con ( EZe , l (r refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , l (l (r refl))) ,\n                             (con (EZe , l (r refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe ,  r refl  )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ae6b95a252d0d2c5d03ca4dfd2beb4cd1f8f683e","subject":"Cosmetics","message":"Cosmetics\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n    }\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- The original relation allows unrelated environments. However, while that is\n  -- fine as a logical relation, it's not OK if we want to prove that validity\n  -- agrees with oplus. We want a finer relation.\n  -- Also: we still need to demand the actual environments to be related, and\n  -- the bodies to match. Haven't done that yet. On the other hand, since we do want\n  -- to allow for replacement changes, that would probably complicate the proof\n  -- elsewhere.\n  relT3 : \u2200 {\u03c4 \u0393 \u0394\u0393} (t1 : Term \u0393 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  -- Weakening in this definition is going to be annoying to use. And having to\n  -- construct terms is ugly.\n\n  -- Weakening could be avoided if we use a separate language of change terms\n  -- with two environments, and with a dclosure binding two variables at once,\n  -- and so on.\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure dt d\u03c1) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n      relV3 \u03c3 v1 dv v2 k \u2192\n      relT3 t1 (app (weaken (drop (\u0394\u03c4 \u03c3) \u2022 \u227c-refl) dt) (var this)) t2 (v1 \u2022 \u03c11) (dv \u2022 v1 \u2022 d\u03c1) (v2 \u2022 \u03c12) k\n    }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k<n v1 v2 x \u2192 tt\n      ; (suc k) k<n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k<n v1 v2 x \u2192 tt\n      ; (suc k) k<n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k<n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k<n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n    }\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- The original relation allows unrelated environments. However, while that is\n  -- fine as a logical relation, it's not OK if we want to prove that validity\n  -- agrees with oplus. We want a finer relation.\n  -- Also: we still need to demand the actual environments to be related, and\n  -- the bodies to match. Haven't done that yet. On the other hand, since we do want\n  -- to allow for replacement changes, that would probably complicate the proof\n  -- elsewhere.\n  relT3 : \u2200 {\u03c4 \u0393 \u0394\u0393} (t1 : Term \u0393 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  -- Weakening in this definition is going to be annoying to use. And having to\n  -- construct terms is ugly.\n\n  -- Weakening could be avoided if we use a separate language of change terms\n  -- with two environments, and with a dclosure binding two variables at once,\n  -- and so on.\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure dt d\u03c1) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n      relV3 \u03c3 v1 dv v2 k \u2192\n      relT3 t1 (app (weaken (drop (\u0394\u03c4 \u03c3) \u2022 \u227c-refl) dt) (var this)) t2 (v1 \u2022 \u03c11) (dv \u2022 v1 \u2022 d\u03c1) (v2 \u2022 \u03c12) k\n    }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bae5c62464397c761ac1d7396e349985e33b44af","subject":"Desc stratified model: put implicits up to DescDConst. Troubles starting.","message":"Desc stratified model: put implicits up to DescDConst. Troubles starting.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : (l : Level) -> Set l -> DescDConst -> IDesc Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3a71a35e614110c17cabf82ec6bcb4b5ead5c9b8","subject":"Desc@ICFP: update model, adding just Variable from the thing containing Values.","message":"Desc@ICFP: update model, adding just Variable from the thing containing Values.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             Var context ty ->\n             IMu closeTerm' ty\ndischarge {n} {c} ty variable = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      Var context ty ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con ((ESu EZe) , true )) \n              (vcons (nat , con ((ESu EZe) , su (su ze)) ) \n              (vcons (pair bool nat , con ((ESu EZe) , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con ((ESu (ESu (ESu EZe))) , (con ((ESu EZe) , (su ze)) , con ( EZe , (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , (fze , refl)) ,\n                             (con (EZe , (fsu fze , refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n-- Fix menu:\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),                        -- Val t\n                              (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                               Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm' ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {.(su n)} (vcons x xs) (fze {n}) = fst x\ntypeAt {.(su n)} (vcons x xs) (fsu {n} y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm' (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : {n : Nat}\n             {context : Context n}\n             (ty : Type) ->\n             (Val ty + Var context ty) ->\n             IMu closeTerm' ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = subst (cong (IMu closeTerm') (snd variable)) (lookup c (fst variable))\n\nsubstExpr : {n : Nat}\n             {ty : Type}\n             (context : Context n)\n             (sigma : (ty : Type) ->\n                      (Val ty + Var context ty) ->\n                      IMu closeTerm' ty) ->\n             IMu (openTerm context) ty ->\n             IMu closeTerm' ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval' (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"37a612ebc77389e35e80565062d1ac398fe5c5fd","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 364f24408c8fdd04184a133e22e3c7a9\n\ndarcs-hash:20101102124050-3bd4e-94a179f58a55dba886d13fbc0b4c89f2d268ef4b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Stream\/Bisimilarity.agda","new_file":"src\/LTC\/Data\/Stream\/Bisimilarity.agda","new_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation on LTC streams\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Stream.Bisimilarity where\n\nopen import LTC.Minimal\n\ninfix 4 _\u2248_\n\n------------------------------------------------------------------------------\n-- Because the LTC is a first-order theory, we define a first-order\n-- version of the bisimilarity relation.\n\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation is a post-fixed point of a bisimilar\n-- relation BISI (see below).\npostulate\n  -\u2248-gfp\u2081 : {xs ys : D} \u2192 xs \u2248 ys \u2192\n            \u2203D (\u03bb x' \u2192\n            \u2203D (\u03bb xs' \u2192\n            \u2203D (\u03bb ys' \u2192 xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))\n{-# ATP axiom -\u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation is the greatest post-fixed point of a\n-- bisimilar relation BISI (see below).\n\n-- N.B. This is a second-order axiom. In the proofs, we *must* use an\n-- axiom scheme instead. Therefore, we do not add this postulate as an\n-- ATP axiom.\npostulate\n  -\u2248-gfp\u2082 : {_R_ : D \u2192 D \u2192 Set} \u2192\n            -- R is a post-fixed point of BISI.\n            ({xs ys : D} \u2192 xs R ys \u2192\n              \u2203D (\u03bb x' \u2192\n              \u2203D (\u03bb xs' \u2192\n              \u2203D (\u03bb ys' \u2192 xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))) \u2192\n            -- \u2248 is greater than R.\n           {xs ys : D} \u2192 xs R ys \u2192 xs \u2248 ys\n\nmodule Bisimulation where\n  -- In LTC we won't use the bisimilar relation BISI. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n  -- 'as (R :: R') bs' (p. 310).\n  -- N.B. This definition should work on streams.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimilar relation.\n  BISI : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BISI _R_ xs ys =\n    \u2203D (\u03bb x' \u2192\n    \u2203D (\u03bb xs' \u2192\n    \u2203D (\u03bb ys' \u2192\n       xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))\n\n  -- The bisimilarity relation is a post-fixed point of BISI.\n  -\u2248\u2192BISI\u2248 : {xs ys : D} \u2192 xs \u2248 ys \u2192 BISI _\u2248_ xs ys\n  -\u2248\u2192BISI\u2248 = -\u2248-gfp\u2081\n\n  -- The bisimilarity relation is the greatest post-fixed point of BISI.\n  R\u2192BISI-R\u2192R\u2192\u2248 : {_R_ : D \u2192 D \u2192 Set} \u2192\n                 -- R is a post-fixed point of BISI.\n                 ({xs ys : D} \u2192 xs R ys \u2192 BISI _R_ xs ys) \u2192\n                 -- \u2248 is greater than R.\n                 {xs ys : D} \u2192 xs R ys \u2192 xs \u2248 ys\n  R\u2192BISI-R\u2192R\u2192\u2248 = -\u2248-gfp\u2082\n","old_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation on LTC streams\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Stream.Bisimilarity where\n\nopen import LTC.Minimal\n\ninfix 4 _\u2248_\n\n------------------------------------------------------------------------------\n-- Because the LTC is a first-order theory, we define a first-order\n-- version of the bisimilarity relation.\n\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation is a post-fixed point of a bisimilar\n-- relation BISI (see below).\npostulate\n  -\u2248-gfp\u2081 : {xs ys : D} \u2192 xs \u2248 ys \u2192\n            \u2203D (\u03bb x' \u2192\n            \u2203D (\u03bb xs' \u2192\n            \u2203D (\u03bb ys' \u2192 xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))\n{-# ATP axiom -\u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation is the greatest post-fixed point of a\n-- bisimilar relation BISI (see below).\n\n-- N.B. This is a second-order axiom. In the proofs, we *must* use an\n-- axiom scheme instead. Therefore, we do not add this postulate as an\n-- ATP axiom.\npostulate\n  -\u2248-gfp\u2082 : {_R_ : D \u2192 D \u2192 Set} \u2192\n            -- R is a post-fixed point of BISI.\n            ({xs ys : D} \u2192 xs R ys \u2192\n              \u2203D (\u03bb x' \u2192\n              \u2203D (\u03bb xs' \u2192\n              \u2203D (\u03bb ys' \u2192 xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))) \u2192\n            -- \u2248 is the greater than R.\n           {xs ys : D} \u2192 xs R ys \u2192 xs \u2248 ys\n\nmodule Bisimulation where\n  -- In LTC we won't use the bisimilar relation BISI. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n  -- 'as (R :: R') bs' (p. 310).\n  -- N.B. This definition should work on streams.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimilar relation.\n  BISI : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BISI _R_ xs ys =\n    \u2203D (\u03bb x' \u2192\n    \u2203D (\u03bb xs' \u2192\n    \u2203D (\u03bb ys' \u2192\n       xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys')))\n\n  -- The bisimilarity relation is a post-fixed point of BISI.\n  -\u2248\u2192BISI\u2248 : {xs ys : D} \u2192 xs \u2248 ys \u2192 BISI _\u2248_ xs ys\n  -\u2248\u2192BISI\u2248 = -\u2248-gfp\u2081\n\n  -- The bisimilarity relation is the greatest post-fixed point of BISI.\n  R\u2192BISI-R\u2192R\u2192\u2248 : {_R_ : D \u2192 D \u2192 Set} \u2192\n                 ({xs ys : D} \u2192 xs R ys \u2192 BISI _R_ xs ys) \u2192\n                 {xs ys : D} \u2192 xs R ys \u2192 xs \u2248 ys\n  R\u2192BISI-R\u2192R\u2192\u2248 = -\u2248-gfp\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"909af4f0d102082a0c7a29516dca9a8436fcb3ad","subject":"Added mirror-Tree.","message":"Added mirror-Tree.\n\nIgnore-this: fa2b49ec2e9ab0b33a40ad384b5817f3\n\ndarcs-hash:20110220170905-3bd4e-4e7f837e6d45c5a8ec44cfcf227c9795a5715811.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/Mirror\/Tree\/Closures.agda","new_file":"src\/LTC\/Program\/Mirror\/Tree\/Closures.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with the closures of the tree type\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check #-}\n\nmodule LTC.Program.Mirror.Tree.Closures where\n\nopen import LTC.Base\n\nopen import LTC.Data.List\n\nopen import LTC.Program.Mirror.Mirror\nopen import LTC.Program.Mirror.ListTree.Closures\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nmirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\nmirror-Tree (treeT d nilLT) =\n  subst Tree (sym (mirror-eq d [])) (treeT d helper\u2082)\n    where\n      helper\u2081 : rev (map mirror []) [] \u2261 []\n      helper\u2081 =\n        begin\n          rev (map mirror []) []\n          \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n                   (map-[] mirror)\n                   refl\n          \u27e9\n          rev [] []\n            \u2261\u27e8 rev-[] [] \u27e9\n          []\n        \u220e\n\n      helper\u2082 : ListTree (rev (map mirror []) [])\n      helper\u2082 = subst ListTree (sym helper\u2081) nilLT\n\nmirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n  subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d helper)\n\n  where\n    helper : ListTree (reverse (map mirror (t \u2237 ts)))\n    helper = reverse-ListTree (map-ListTree mirror mirror-Tree (consLT Tt LTts))\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with the closures of the tree type\n------------------------------------------------------------------------------\n\nmodule LTC.Program.Mirror.Tree.Closures where\n\nopen import LTC.Base\n\nopen import LTC.Program.Mirror.Mirror\n\nopen import LTC.Data.List\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove the postulate\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\n-- mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n-- mirror-Tree (treeT d nilLT) =\n--   subst Tree (sym (mirror-eq d [])) (treeT d helper\u2082)\n--     where\n--       helper\u2081 : rev (map mirror []) [] \u2261 []\n--       helper\u2081 =\n--         begin\n--           rev (map mirror []) []\n--           \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n--                    (map-[] mirror)\n--                    refl\n--           \u27e9\n--           rev [] []\n--             \u2261\u27e8 rev-[] [] \u27e9\n--           []\n--         \u220e\n\n--       helper\u2082 : ListTree (rev (map mirror []) [])\n--       helper\u2082 = subst ListTree (sym helper\u2081) nilLT\n\n-- mirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n--   subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d helper)\n\n--   where\n--     helper : ListTree (reverse (map mirror (t \u2237 ts)))\n--     helper = rev-ListTree (map-ListTree mirror {!!} (consLT Tt LTts)) nilLT\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"038ac179b72512cbb57611d217eb7188f986ee3e","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: a6d436fde1c00a6bec397a3db93e5eb4\n\ndarcs-hash:20120208145259-3bd4e-53e88003a7bb73c14766634e95a9aa210713ab4a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Axiomatic\/Base.agda","new_file":"src\/PA\/Axiomatic\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\ninfix  7  _\u2250_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero    : \u2115\n  succ    : \u2115 \u2192 \u2115\n  _+_ _*_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the program agda2atp as the ATPs equality.\npostulate _\u2250_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2250 n \u2192 m \u2250 o \u2192 n \u2250 o\n  S\u2082 : \u2200 {m n} \u2192 m \u2250 n \u2192 succ m \u2250 succ n\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2250 succ n)\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2250 succ n \u2192 m \u2250 n\n  S\u2085 : \u2200 n \u2192 zero + n \u2250 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2250 succ (m + n)\n  S\u2087 : \u2200 n \u2192 zero * n \u2250 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2250 n + m * n\n{-# ATP axiom S\u2081 S\u2082 S\u2083 S\u2084 S\u2085 S\u2086 S\u2087 S\u2088 #-}\n\n-- The axiom S\u2089 is a higher-order one, therefore we do not translate\n-- it as an ATP axiom.\npostulate S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\ninfix  7  _\u2250_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the program agda2atp as the ATPs equality.\npostulate _\u2250_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2250 n \u2192 m \u2250 o \u2192 n \u2250 o\n  S\u2082 : \u2200 {m n} \u2192 m \u2250 n \u2192 succ m \u2250 succ n\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2250 succ n)\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2250 succ n \u2192 m \u2250 n\n  S\u2085 : \u2200 n \u2192 zero + n \u2250 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2250 succ (m + n)\n  S\u2087 : \u2200 n \u2192 zero * n \u2250 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2250 n + m * n\n{-# ATP axiom S\u2081 S\u2082 S\u2083 S\u2084 S\u2085 S\u2086 S\u2087 S\u2088 #-}\n\n-- The axiom S\u2089 is a higher-order one, therefore we do not translate\n-- it as an ATP axiom.\npostulate S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0e5b57e4c178ef7eb9eb2bf41c231d36dbe70f5c","subject":"Moved a bug report to the agda2atp tool.","message":"Moved a bug report to the agda2atp tool.\n\nIgnore-this: e1514c644722b889facf44f91ec27a62\n\ndarcs-hash:20110322153307-3bd4e-0b07e862cc164decb95313adbe8b811bafa0a08d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bugs\/GeneralHints.agda","new_file":"Draft\/Bugs\/GeneralHints.agda","new_contents":"","old_contents":"module Draft.Bugs.GeneralHints where\n\nopen import LTC.Base\nopen import LTC.Data.Nat\n\n-- TODO: Bug. The agda2atp tool does not translate the general hints\n\n-- {-# ATP hint zN #-}\n-- {-# ATP hint sN #-}\n\n-- because they are not defined in the file LTC.Data.Nat.Type.\n\npostulate\n N0 : N zero\n{-# ATP prove N0 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"998682f0d09f06ad7918732abe550b386afe02e2","subject":"Make Syntax.Type.Popl14 use Syntax.Type.Plotkin","message":"Make Syntax.Type.Popl14 use Syntax.Type.Plotkin\n\nOld-commit-hash: 031c97aee2022a85ee855ec3149f0f8cd160d5af\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Type\/Popl14.agda","new_file":"Syntax\/Type\/Popl14.agda","new_contents":"module Syntax.Type.Popl14 where\n\n-- Types of Calculus Popl14\n\ndata Popl14-type : Set where\n  base-int : Popl14-type\n  base-bag : Popl14-type\n\nopen import Syntax.Type.Plotkin Popl14-type public\n\npattern int = base base-int\npattern bag = base base-bag\n\nPopl14-\u0394base : Popl14-type \u2192 Popl14-type\nPopl14-\u0394base base-int = base-int\nPopl14-\u0394base base-bag = base-bag\n\n\u0394Type : Type \u2192 Type\n\u0394Type = lift-\u0394type\u2080 Popl14-\u0394base\n","old_contents":"module Syntax.Type.Popl14 where\n\n-- Types of Calculus Popl14\n\ninfixr 5 _\u21d2_\n\ndata Type : Set where\n  int : Type\n  bag : Type\n  _\u21d2_ : (\u03c3 : Type) \u2192 (\u03c4 : Type) \u2192 Type\n\n\u0394Type : Type \u2192 Type\n\u0394Type int = int\n\u0394Type bag = bag\n\u0394Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394Type \u03c3 \u21d2 \u0394Type \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8f4895d3dd6492e95fe8a6c7c34c7bcf5a727766","subject":"Lamport: +extract-signkey +extract-signkey-correct","message":"Lamport: +extract-signkey +extract-signkey-correct\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/Lamport.agda","new_file":"Crypto\/Sig\/Lamport.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nopen \u2261-Reasoning\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH  = hash-secret\nH\u00b2 = map2* #secret #digest H\n\n#signkey1   = OTS1.#signkey\n#verifkey1  = OTS1.#verifkey\n#signature1 = OTS1.#signature\n#signkey    = #message * #signkey1\n#verifkey   = #message * #verifkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #verifkey1 vk\n\n  concat-vk1s : concat vk1s \u2261 vk\n  concat-vk1s = concat-group #message #verifkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #signkey1 sk\n  sc1s = map OTS1.signkey.skP sk1s\n  vk   = verif-key sk\n  open verifkey vk public\n  sc1s-sk1s : map (uncurry _++_) sc1s \u2261 sk1s\n  sc1s-sk1s = ! map-\u2218 (uncurry _++_) OTS1.signkey.skP sk1s\n            \u2219 map-id= (take-drop-lem #secret) sk1s\n  H-sc1s-sk1s : map (uncurry _++_ \u2218 \u00d7-map H H) sc1s \u2261 map H\u00b2 sk1s\n  H-sc1s-sk1s = ! map-\u2218= (map2*-++ H) sc1s \u2219 ap (map H\u00b2) sc1s-sk1s\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\n  concat-sig1s : concat sig1s \u2261 sig\n  concat-sig1s = concat-group #message #secret sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\n    group-sig : group #message #signature1 sig \u2261 sig1s\n    group-sig = group-concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and checks\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n    checks = map OTS1.verify vk1s \u229b m \u229b sig1s\n\nprivate\n  module lemma {#m} (sk : Bits (suc #m * #signkey1))\n                    (b : Bit) (m : Bits #m) where\n    skL  = take #signkey1 sk\n    skH  = drop #signkey1 sk\n    vkL  = OTS1.verif-key skL\n    vkH  = map* #m OTS1.verif-key skH\n    sigL = OTS1.sign skL b\n    sigH = concat (map OTS1.sign (group #m #signkey1 skH) \u229b m)\n\nverifkey-is-hash-sig : \u2200 sk m (open signkey sk)\n                       \u2192 map (take2* _) m \u229b vk1s\n                       \u2261 map H (map OTS1.sign sk1s \u229b m)\nverifkey-is-hash-sig sk m = lemma sk m\n  module verifkey-is-hash-sig where\n    lemma : \u2200 {#m} sk m \u2192 map (take2* _) m \u229b group #m #verifkey1 (map* #m OTS1.verif-key sk)\n                        \u2261 map hash-secret (map OTS1.sign (group #m #signkey1 sk) \u229b m)\n    lemma sk [] = refl\n    lemma {suc #m} sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signkey1 skL skH)\n            | (let open lemma sk b m in drop-++ #signkey1 skL skH)\n            | (let open lemma sk b m in lemma skH m)\n            | (let open lemma sk b m in OTS1.verifkey-is-hash-sig skL b)\n      = refl\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig sk m = lemma sk m\n  module verify-correct-sig where\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #verifkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                              (concat (map OTS1.sign (group #m #signkey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n\nextract-signkey : (sig\u2080 sig\u2081 : Signature) \u2192 SignKey\nextract-signkey sig\u2080 sig\u2081 = sk\n  module extract-signkey where\n    module sig\u2080 = signature sig\u2080\n    module sig\u2081 = signature sig\u2081\n    sk1s = map _++_ sig\u2080.sig1s \u229b sig\u2081.sig1s\n    sk   = concat sk1s\n\nextract-signkey-correct :\n  \u2200 sk \u2192 extract-signkey (sign sk 0\u207f) (sign sk 1\u207f) \u2261 sk\nextract-signkey-correct sk = sk-lemma\n  where\n    module sk   = signkey sk\n    module sig\u2080 = sign sk 0\u207f\n    module sig\u2081 = sign sk 1\u207f\n    module sk'  = extract-signkey sig\u2080.sig sig\u2081.sig\n\n    sk1s-lemma : sk'.sk1s \u2261 sk.sk1s\n    sk1s-lemma = ap\u2082 (\u03bb x y \u2192 map _++_ x \u229b y)\n                     (sig\u2080.group-sig \u2219 \u229b-replicate _ 0b \u2219 map-\u2218= (\u03bb _ \u2192 refl) _)\n                     (sig\u2081.group-sig \u2219 \u229b-replicate _ 1b \u2219 map-\u2218= (\u03bb _ \u2192 refl) _)\n               \u2219 \u229b-take-drop-lem #secret sk.sk1s\n\n    sk-lemma : sk'.sk \u2261 sk\n    sk-lemma = ap concat sk1s-lemma\n             \u2219 concat-group #message _ sk\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH  = hash-secret\nH\u00b2 = map2* #secret #digest H\n\n#signkey1   = OTS1.#signkey\n#verifkey1  = OTS1.#verifkey\n#signature1 = OTS1.#signature\n#signkey    = #message * #signkey1\n#verifkey   = #message * #verifkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #verifkey1 vk\n\n  concat-vk1s : concat vk1s \u2261 vk\n  concat-vk1s = concat-group #message #verifkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #signkey1 sk\n  sc1s = map OTS1.signkey.skP sk1s\n  vk   = verif-key sk\n  open verifkey vk public\n  sc1s-sk1s : map (uncurry _++_) sc1s \u2261 sk1s\n  sc1s-sk1s = ! map-\u2218 (uncurry _++_) OTS1.signkey.skP sk1s\n            \u2219 map-id= (take-drop-lem #secret) sk1s\n  H-sc1s-sk1s : map (uncurry _++_ \u2218 \u00d7-map H H) sc1s \u2261 map H\u00b2 sk1s\n  H-sc1s-sk1s = ! map-\u2218= (map2*-++ H) sc1s \u2219 ap (map H\u00b2) sc1s-sk1s\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\n  concat-sig1s : concat sig1s \u2261 sig\n  concat-sig1s = concat-group #message #secret sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\n    group-sig : group #message #signature1 sig \u2261 sig1s\n    group-sig = group-concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and checks\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n    checks = map OTS1.verify vk1s \u229b m \u229b sig1s\n\nprivate\n  module lemma {#m} (sk : Bits (suc #m * #signkey1))\n                    (b : Bit) (m : Bits #m) where\n    skL  = take #signkey1 sk\n    skH  = drop #signkey1 sk\n    vkL  = OTS1.verif-key skL\n    vkH  = map* #m OTS1.verif-key skH\n    sigL = OTS1.sign skL b\n    sigH = concat (map OTS1.sign (group #m #signkey1 skH) \u229b m)\n\nverifkey-is-hash-sig : \u2200 sk m (open signkey sk)\n                       \u2192 map (take2* _) m \u229b vk1s\n                       \u2261 map H (map OTS1.sign sk1s \u229b m)\nverifkey-is-hash-sig sk m = lemma sk m\n  module verifkey-is-hash-sig where\n    lemma : \u2200 {#m} sk m \u2192 map (take2* _) m \u229b group #m #verifkey1 (map* #m OTS1.verif-key sk)\n                        \u2261 map hash-secret (map OTS1.sign (group #m #signkey1 sk) \u229b m)\n    lemma sk [] = refl\n    lemma {suc #m} sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signkey1 skL skH)\n            | (let open lemma sk b m in drop-++ #signkey1 skL skH)\n            | (let open lemma sk b m in lemma skH m)\n            | (let open lemma sk b m in OTS1.verifkey-is-hash-sig skL b)\n      = refl\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig sk m = lemma sk m\n  module verify-correct-sig where\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #verifkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                              (concat (map OTS1.sign (group #m #signkey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a11bc0494d5a53ae852712fa2ba53aa140392e9f","subject":"Updated a note.","message":"Updated a note.\n\nIgnore-this: ff92cf63d71d2a20c973e95b701a3ff3\n\ndarcs-hash:20111103145522-3bd4e-b34a41ced2551fcad2ad1814dcd1d73f59a63884.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_contents":"-- Tested with Agda 2.2.11 on 11 October 2011.\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d               -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e  -- d is the least prefixed-point of f\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d               -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e  -- d is the least prefixed-point of f\n\n-- Thm: If d is the least pre-fixed point of f, then d is a fixed\n-- point of f (TODO: source).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least pre-fixed point of a functor\n\n-- Instead defining the least pre-fixed via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\n-- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n-- NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 X m)\n\n-- The natural numbers are the least pre-fixed point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is pre-fixed point of NatF.\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081 : \u2200 n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m) \u2192 N n\n\n  -- N is the least prefixed-point of NatF.\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082 : \u2200 (P : D \u2192 Set) {n} \u2192\n           (n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n) \u2192\n           N n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 zero (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN {n} Nn = N-lfp\u2081 (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m)\nN-lfp\u2083 {n} Nn = N-lfp\u2082 P prf Nn\n  where\n  P : D \u2192 Set\n  P x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ\u2081 m \u2227 N m\n\n  prf : n \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf h = [ inj\u2081 , (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) ] h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m\n    prf\u2081 (m , n=Sm , h\u2082) = m , n=Sm , prf\u2082 h\u2082\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = [ (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) , prf\u2083 ] h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n        prf\u2083 (m' , m\u2261Sm' , Nm') = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N.\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 P n \u2192 P (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 is {n} Nn = N-lfp\u2082 P [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 P n\n  prf\u2081 n\u22610 = subst P (sym n\u22610) P0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf\u2082 (m , n\u2261Sm , Pm) = subst P (sym n\u2261Sm) (is Pm)\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-lfp\u2082 P prf Nm\n\n  where\n  P : D \u2192 Set\n  P i = N (i + n)\n\n  prf : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 P m') \u2192 P m\n  prf h = [ prf\u2081 , prf\u2082 ] h\n    where\n    P0 : P zero\n    P0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : m \u2261 zero \u2192 P m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) P0\n\n    is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n    is {i} Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n    prf\u2082 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 P m') \u2192 P m\n    prf\u2082 (m' , m\u2261Sm' , Pm') = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Pm')\n","old_contents":"-- Tested with Agda 2.2.11 on 11 October 2011.\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d               -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e  -- d is the least prefixed-point of f\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d               -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e  -- d is the least prefixed-point of f\n\n-- Thm: If d is the least pre-fixed point of f, then d is a fixed\n-- point of f (TODO: source).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least pre-fixed point of a functor\n\n-- Instead defining the least pre-fixed via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\n-- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n-- NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 X m)\n\n-- The natural numbers are the least pre-fixed point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is pre-fixed point of NatF.\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081 : \u2200 n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 N m) \u2192 N n\n\n  -- N is the least prefixed-point of NatF.\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082 : \u2200 (P : D \u2192 Set) {n} \u2192\n           (n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 P m) \u2192 P n) \u2192\n           N n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 zero (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ n)\nsN {n} Nn = N-lfp\u2081 (succ n) (inj\u2082 (n , (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 N m)\nN-lfp\u2083 {n} Nn = N-lfp\u2082 P prf Nn\n  where\n  P : D \u2192 Set\n  P x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ m \u2227 N m\n\n  prf : n \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n \u2261 succ m \u2227 P m) \u2192 P n\n  prf h = [ inj\u2081 , (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) ] h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n \u2261 succ m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n \u2261 succ m \u2227 N m\n    prf\u2081 (m , n=Sm , h\u2082) = m , n=Sm , prf\u2082 h\u2082\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = [ (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) , prf\u2083 ] h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ m' \u2227 N m') \u2192 N m\n        prf\u2083 (m' , m\u2261Sm' , Nm') = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N.\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 is {n} Nn = N-lfp\u2082 P [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 P n\n  prf\u2081 n\u22610 = subst P (sym n\u22610) P0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ m \u2227 P m) \u2192 P n\n  prf\u2082 (m , n\u2261Sm , Pm) = subst P (sym n\u2261Sm) (is helper Pm)\n    where\n    helper : N m\n    helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 Nn)\n      where\n      prf\u2083 : n \u2261 zero \u2192 N m\n      prf\u2083 n\u22610 = \u22a5-elim (0\u2260S (trans (sym n\u22610) n\u2261Sm))\n\n      prf\u2084 : \u2203 (\u03bb m' \u2192 n \u2261 succ m' \u2227 N m') \u2192 N m\n      prf\u2084 (m' , n\u2261Sm' , Nm') =\n        subst N (succInjective (trans (sym n\u2261Sm') n\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero   + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ d + e \u2261 succ (d + e)\n\n+-leftIdentity : \u2200 {n} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-lfp\u2082 P prf Nm\n\n  where\n  P : D \u2192 Set\n  P i = N (i + n)\n\n  prf : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ m' \u2227 P m') \u2192 P m\n  prf h = [ prf\u2081 , prf\u2082 ] h\n    where\n    P0 : P zero\n    P0 = subst N (sym (+-leftIdentity Nn)) Nn\n\n    prf\u2081 : m \u2261 zero \u2192 P m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) P0\n\n    is : \u2200 {i} \u2192 P i \u2192 P (succ i)\n    is {i} Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n    prf\u2082 : \u2203 (\u03bb m' \u2192 m \u2261 succ m' \u2227 P m') \u2192 P m\n    prf\u2082 (m' , m\u2261Sm' , Pm') = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Pm')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"91c27a6528221dcbcdc0cfc04817e2db672ed84a","subject":"Fixed module.","message":"Fixed module.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/NoPatternMatchingOnRefl.agda","new_file":"notes\/FOT\/FOTC\/NoPatternMatchingOnRefl.agda","new_contents":"------------------------------------------------------------------------------\n-- Proving properties without using pattern matching on refl\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.NoPatternMatchingOnRefl where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Stream\n\nopen import FOTC.Program.McCarthy91.McCarthy91\n\nopen import FOTC.Relation.Binary.Bisimilarity\nopen import FOTC.Relation.Binary.Bisimilarity.PropertiesI\n\n------------------------------------------------------------------------------\n-- From FOTC.Base.PropertiesI\n\n-- Congruence properties\n\n\u00b7-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a \u00b7 c \u2261 b \u00b7 c\n\u00b7-leftCong {a} {c = c} h = subst (\u03bb t \u2192 a \u00b7 c \u2261 t \u00b7 c) h refl\n\n\u00b7-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a \u00b7 b \u2261 a \u00b7 c\n\u00b7-rightCong {a} {b} h = subst (\u03bb t \u2192 a \u00b7 b \u2261 a \u00b7 t) h refl\n\n\u00b7-cong : \u2200 {a b c d} \u2192 a \u2261 b \u2192 c \u2261 d \u2192 a \u00b7 c \u2261 b \u00b7 d\n\u00b7-cong {a} {c = c} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 a \u00b7 c \u2261 t\u2081 \u00b7 t\u2082) h\u2081 h\u2082 refl\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong {m} h = subst (\u03bb t \u2192 succ\u2081 m \u2261 succ\u2081 t) h refl\n\npredCong : \u2200 {m n} \u2192 m \u2261 n \u2192 pred\u2081 m \u2261 pred\u2081 n\npredCong {m} h = subst (\u03bb t \u2192 pred\u2081 m \u2261 pred\u2081 t) h refl\n\nifCong\u2081 : \u2200 {b b' t t'} \u2192 b \u2261 b' \u2192 if b then t else t' \u2261 if b' then t else t'\nifCong\u2081 {b} {t = t} {t'} h =\n  subst (\u03bb x \u2192 if b then t else t' \u2261 if x then t else t') h refl\n\nifCong\u2082 : \u2200 {b t\u2081 t\u2082 t} \u2192 t\u2081 \u2261 t\u2082 \u2192 if b then t\u2081 else t \u2261 if b then t\u2082 else t\nifCong\u2082 {b} {t\u2081} {t = t} h =\n  subst (\u03bb x \u2192 if b then t\u2081 else t \u2261 if b then x else t) h refl\n\nifCong\u2083 : \u2200 {b t t\u2081 t\u2082} \u2192 t\u2081 \u2261 t\u2082 \u2192 if b then t else t\u2081 \u2261 if b then t else t\u2082\nifCong\u2083 {b} {t} {t\u2081} h =\n  subst (\u03bb x \u2192 if b then t else t\u2081 \u2261 if b then t else x) h refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Base.List.PropertiesI\n\n-- Congruence properties\n\n\u2237-leftCong : \u2200 {x y xs} \u2192 x \u2261 y \u2192 x \u2237 xs \u2261 y \u2237 xs\n\u2237-leftCong {x} {xs = xs} h = subst (\u03bb t \u2192 x \u2237 xs \u2261 t \u2237 xs) h refl\n\n\u2237-rightCong : \u2200 {x xs ys} \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 x \u2237 ys\n\u2237-rightCong {x}{xs} h = subst (\u03bb t \u2192 x \u2237 xs \u2261 x \u2237 t) h refl\n\n\u2237-Cong : \u2200 {x y xs ys} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\n\u2237-Cong {x} {xs = xs} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 x \u2237 xs \u2261 t\u2081 \u2237 t\u2082) h\u2081 h\u2082 refl\n\nheadCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 head\u2081 xs \u2261 head\u2081 ys\nheadCong {xs} h = subst (\u03bb t \u2192 head\u2081 xs \u2261 head\u2081 t) h refl\n\ntailCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 tail\u2081 xs \u2261 tail\u2081 ys\ntailCong {xs} h = subst (\u03bb t \u2192 tail\u2081 xs \u2261 tail\u2081 t) h refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Data.Bool.PropertiesI\n\n-- Congruence properties\n\n&&-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a && c \u2261 b && c\n&&-leftCong {a} {c = c} h = subst (\u03bb t \u2192 a && c \u2261 t && c) h refl\n\n&&-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a && b \u2261 a && c\n&&-rightCong {a} {b} h = subst (\u03bb t \u2192 a && b \u2261 a && t) h refl\n\n&&-cong : \u2200 {a b c d } \u2192 a \u2261 c \u2192 b \u2261 d \u2192 a && b \u2261 c && d\n&&-cong {a} {b} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 a && b \u2261 t\u2081 && t\u2082) h\u2081 h\u2082 refl\n\nnotCong : \u2200 {a b} \u2192 a \u2261 b \u2192 not a \u2261 not b\nnotCong {a} h = subst (\u03bb t \u2192 not a \u2261 not t) h refl\n\n------------------------------------------------------------------------------\n-- FOTC.Data.Conat.Equality.PropertiesI\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} Cn = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n\n  h\u2081 : \u2200 {a b} \u2192 R a b \u2192\n       a \u2261 zero \u2227 b \u2261 zero\n         \u2228 (\u2203[ a' ] \u2203[ b' ] a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b' \u2227 R a' b')\n  h\u2081 (Ca , Cb , h) with Conat-unf Ca\n  ... | inj\u2081 prf              = inj\u2081 (prf , trans (sym h) prf)\n  ... | inj\u2082 (a' , prf , Ca') =\n    inj\u2082 (a' , a' , prf , trans (sym h) prf , (Ca' , Ca' , refl))\n\n  h\u2082 : R n n\n  h\u2082 = Cn , Cn , refl\n\n\u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n\u2261\u2192\u2248N {m} Cm _ h = subst (_\u2248N_ m) h (\u2248N-refl Cm)\n\n------------------------------------------------------------------------------\n-- FOTC.Data.List.PropertiesI\n\n-- Congruence properties\n\n++-leftCong : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 xs ++ zs \u2261 ys ++ zs\n++-leftCong {xs} {zs = zs} h = subst (\u03bb t \u2192 xs ++ zs \u2261 t ++ zs) h refl\n\n++-rightCong : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 xs ++ ys \u2261 xs ++ zs\n++-rightCong {xs} {ys} h = subst (\u03bb t \u2192 xs ++ ys \u2261 xs ++ t) h refl\n\nmapCong\u2082 : \u2200 {f xs ys} \u2192 xs \u2261 ys \u2192 map f xs \u2261 map f ys\nmapCong\u2082 {f} {xs} h = subst (\u03bb t \u2192 map f xs \u2261 map f t) h refl\n\nrevCong\u2081 : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 rev xs zs \u2261 rev ys zs\nrevCong\u2081 {xs} {zs = zs} h = subst (\u03bb t \u2192 rev xs zs \u2261 rev t zs) h refl\n\nrevCong\u2082 : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 rev xs ys \u2261 rev xs zs\nrevCong\u2082 {xs} {ys} h = subst (\u03bb t \u2192 rev xs ys \u2261 rev xs t) h refl\n\nreverseCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse xs \u2261 reverse ys\nreverseCong {xs} h = subst (\u03bb t \u2192 reverse xs \u2261 reverse t) h refl\n\nlengthCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 length xs \u2261 length ys\nlengthCong {xs} h = subst (\u03bb t \u2192 length xs \u2261 length t) h refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- Congruence properties\n\nleLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 le m o \u2261 le n o\nleLeftCong {m} {o = o} h = subst (\u03bb t \u2192 le m o \u2261 le t o) h refl\n\nltLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 lt m o \u2261 lt n o\nltLeftCong {m} {o = o} h = subst (\u03bb t \u2192 lt m o \u2261 lt t o) h refl\n\nltRightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 lt m n \u2261 lt m o\nltRightCong {m} {n} h = subst (\u03bb t \u2192 lt m n \u2261 lt m t) h refl\n\nltCong : \u2200 {m\u2081 n\u2081 m\u2082 n\u2082} \u2192 m\u2081 \u2261 m\u2082 \u2192 n\u2081 \u2261 n\u2082 \u2192 lt m\u2081 n\u2081 \u2261 lt m\u2082 n\u2082\nltCong {m\u2081} {n\u2081} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 lt m\u2081 n\u2081 \u2261 lt t\u2081 t\u2082) h\u2081 h\u2082 refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Data.Nat.PropertiesI\n\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong {m} {o = o} h = subst (\u03bb t \u2192 m + o \u2261 t + o) h refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong {m} {n} h = subst (\u03bb t \u2192 m + n \u2261 m + t) h refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong {m} {o = o} h = subst (\u03bb t \u2192 m \u2238 o \u2261 t \u2238 o) h refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong {m} {n} h = subst (\u03bb t \u2192 m \u2238 n \u2261 m \u2238 t) h refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong {m} {o = o} h = subst (\u03bb t \u2192 m * o \u2261 t * o) h refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong {m} {n} h = subst (\u03bb t \u2192 m * n \u2261 m * t) h refl\n\n------------------------------------------------------------------------------\n-- From FOT.FOTC.Data.Stream.Equality.PropertiesI where\n\nstream-\u2261\u2192\u2248 : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2261 ys \u2192 xs \u2248 ys\nstream-\u2261\u2192\u2248 {xs} Sxs _ h = subst (_\u2248_ xs) h (\u2248-refl Sxs)\n\n------------------------------------------------------------------------------\n-- From FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\n\nf\u2089\u2081-x\u2261y : \u2200 {m n o} \u2192 f\u2089\u2081 m \u2261 n \u2192 o \u2261 m \u2192 f\u2089\u2081 o \u2261 n\nf\u2089\u2081-x\u2261y {n = n} h\u2081 h\u2082 = subst (\u03bb t \u2192 f\u2089\u2081 t \u2261 n) (sym h\u2082) h\u2081\n","old_contents":"------------------------------------------------------------------------------\n-- Proving properties without using pattern matching on refl\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.NoPatternMatchingOnRefl where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Stream\n\nopen import FOTC.Program.McCarthy91.McCarthy91\n\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- From FOTC.Base.PropertiesI\n\n-- Congruence properties\n\n\u00b7-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a \u00b7 c \u2261 b \u00b7 c\n\u00b7-leftCong {a} {c = c} h = subst (\u03bb t \u2192 a \u00b7 c \u2261 t \u00b7 c) h refl\n\n\u00b7-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a \u00b7 b \u2261 a \u00b7 c\n\u00b7-rightCong {a} {b} h = subst (\u03bb t \u2192 a \u00b7 b \u2261 a \u00b7 t) h refl\n\n\u00b7-cong : \u2200 {a b c d} \u2192 a \u2261 b \u2192 c \u2261 d \u2192 a \u00b7 c \u2261 b \u00b7 d\n\u00b7-cong {a} {c = c} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 a \u00b7 c \u2261 t\u2081 \u00b7 t\u2082) h\u2081 h\u2082 refl\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong {m} h = subst (\u03bb t \u2192 succ\u2081 m \u2261 succ\u2081 t) h refl\n\npredCong : \u2200 {m n} \u2192 m \u2261 n \u2192 pred\u2081 m \u2261 pred\u2081 n\npredCong {m} h = subst (\u03bb t \u2192 pred\u2081 m \u2261 pred\u2081 t) h refl\n\nifCong\u2081 : \u2200 {b b' t t'} \u2192 b \u2261 b' \u2192 if b then t else t' \u2261 if b' then t else t'\nifCong\u2081 {b} {t = t} {t'} h =\n  subst (\u03bb x \u2192 if b then t else t' \u2261 if x then t else t') h refl\n\nifCong\u2082 : \u2200 {b t\u2081 t\u2082 t} \u2192 t\u2081 \u2261 t\u2082 \u2192 if b then t\u2081 else t \u2261 if b then t\u2082 else t\nifCong\u2082 {b} {t\u2081} {t = t} h =\n  subst (\u03bb x \u2192 if b then t\u2081 else t \u2261 if b then x else t) h refl\n\nifCong\u2083 : \u2200 {b t t\u2081 t\u2082} \u2192 t\u2081 \u2261 t\u2082 \u2192 if b then t else t\u2081 \u2261 if b then t else t\u2082\nifCong\u2083 {b} {t} {t\u2081} h =\n  subst (\u03bb x \u2192 if b then t else t\u2081 \u2261 if b then t else x) h refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Base.List.PropertiesI\n\n-- Congruence properties\n\n\u2237-leftCong : \u2200 {x y xs} \u2192 x \u2261 y \u2192 x \u2237 xs \u2261 y \u2237 xs\n\u2237-leftCong {x} {xs = xs} h = subst (\u03bb t \u2192 x \u2237 xs \u2261 t \u2237 xs) h refl\n\n\u2237-rightCong : \u2200 {x xs ys} \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 x \u2237 ys\n\u2237-rightCong {x}{xs} h = subst (\u03bb t \u2192 x \u2237 xs \u2261 x \u2237 t) h refl\n\n\u2237-Cong : \u2200 {x y xs ys} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\n\u2237-Cong {x} {xs = xs} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 x \u2237 xs \u2261 t\u2081 \u2237 t\u2082) h\u2081 h\u2082 refl\n\nheadCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 head\u2081 xs \u2261 head\u2081 ys\nheadCong {xs} h = subst (\u03bb t \u2192 head\u2081 xs \u2261 head\u2081 t) h refl\n\ntailCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 tail\u2081 xs \u2261 tail\u2081 ys\ntailCong {xs} h = subst (\u03bb t \u2192 tail\u2081 xs \u2261 tail\u2081 t) h refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Data.Bool.PropertiesI\n\n-- Congruence properties\n\n&&-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a && c \u2261 b && c\n&&-leftCong {a} {c = c} h = subst (\u03bb t \u2192 a && c \u2261 t && c) h refl\n\n&&-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a && b \u2261 a && c\n&&-rightCong {a} {b} h = subst (\u03bb t \u2192 a && b \u2261 a && t) h refl\n\n&&-cong : \u2200 {a b c d } \u2192 a \u2261 c \u2192 b \u2261 d \u2192 a && b \u2261 c && d\n&&-cong {a} {b} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 a && b \u2261 t\u2081 && t\u2082) h\u2081 h\u2082 refl\n\nnotCong : \u2200 {a b} \u2192 a \u2261 b \u2192 not a \u2261 not b\nnotCong {a} h = subst (\u03bb t \u2192 not a \u2261 not t) h refl\n\n------------------------------------------------------------------------------\n-- FOTC.Data.Conat.Equality.PropertiesI\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} Cn = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n\n  h\u2081 : \u2200 {a b} \u2192 R a b \u2192\n       a \u2261 zero \u2227 b \u2261 zero\n         \u2228 (\u2203[ a' ] \u2203[ b' ] a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b' \u2227 R a' b')\n  h\u2081 (Ca , Cb , h) with Conat-unf Ca\n  ... | inj\u2081 prf              = inj\u2081 (prf , trans (sym h) prf)\n  ... | inj\u2082 (a' , prf , Ca') =\n    inj\u2082 (a' , a' , prf , trans (sym h) prf , (Ca' , Ca' , refl))\n\n  h\u2082 : R n n\n  h\u2082 = Cn , Cn , refl\n\n\u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n\u2261\u2192\u2248N {m} Cm _ h = subst (_\u2248N_ m) h (\u2248N-refl Cm)\n\n------------------------------------------------------------------------------\n-- FOTC.Data.List.PropertiesI\n\n-- Congruence properties\n\n++-leftCong : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 xs ++ zs \u2261 ys ++ zs\n++-leftCong {xs} {zs = zs} h = subst (\u03bb t \u2192 xs ++ zs \u2261 t ++ zs) h refl\n\n++-rightCong : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 xs ++ ys \u2261 xs ++ zs\n++-rightCong {xs} {ys} h = subst (\u03bb t \u2192 xs ++ ys \u2261 xs ++ t) h refl\n\nmapCong\u2082 : \u2200 {f xs ys} \u2192 xs \u2261 ys \u2192 map f xs \u2261 map f ys\nmapCong\u2082 {f} {xs} h = subst (\u03bb t \u2192 map f xs \u2261 map f t) h refl\n\nrevCong\u2081 : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 rev xs zs \u2261 rev ys zs\nrevCong\u2081 {xs} {zs = zs} h = subst (\u03bb t \u2192 rev xs zs \u2261 rev t zs) h refl\n\nrevCong\u2082 : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 rev xs ys \u2261 rev xs zs\nrevCong\u2082 {xs} {ys} h = subst (\u03bb t \u2192 rev xs ys \u2261 rev xs t) h refl\n\nreverseCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse xs \u2261 reverse ys\nreverseCong {xs} h = subst (\u03bb t \u2192 reverse xs \u2261 reverse t) h refl\n\nlengthCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 length xs \u2261 length ys\nlengthCong {xs} h = subst (\u03bb t \u2192 length xs \u2261 length t) h refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- Congruence properties\n\nleLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 le m o \u2261 le n o\nleLeftCong {m} {o = o} h = subst (\u03bb t \u2192 le m o \u2261 le t o) h refl\n\nltLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 lt m o \u2261 lt n o\nltLeftCong {m} {o = o} h = subst (\u03bb t \u2192 lt m o \u2261 lt t o) h refl\n\nltRightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 lt m n \u2261 lt m o\nltRightCong {m} {n} h = subst (\u03bb t \u2192 lt m n \u2261 lt m t) h refl\n\nltCong : \u2200 {m\u2081 n\u2081 m\u2082 n\u2082} \u2192 m\u2081 \u2261 m\u2082 \u2192 n\u2081 \u2261 n\u2082 \u2192 lt m\u2081 n\u2081 \u2261 lt m\u2082 n\u2082\nltCong {m\u2081} {n\u2081} h\u2081 h\u2082 = subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 lt m\u2081 n\u2081 \u2261 lt t\u2081 t\u2082) h\u2081 h\u2082 refl\n\n------------------------------------------------------------------------------\n-- From FOTC.Data.Nat.PropertiesI\n\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong {m} {o = o} h = subst (\u03bb t \u2192 m + o \u2261 t + o) h refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong {m} {n} h = subst (\u03bb t \u2192 m + n \u2261 m + t) h refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong {m} {o = o} h = subst (\u03bb t \u2192 m \u2238 o \u2261 t \u2238 o) h refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong {m} {n} h = subst (\u03bb t \u2192 m \u2238 n \u2261 m \u2238 t) h refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong {m} {o = o} h = subst (\u03bb t \u2192 m * o \u2261 t * o) h refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong {m} {n} h = subst (\u03bb t \u2192 m * n \u2261 m * t) h refl\n\n------------------------------------------------------------------------------\n-- From FOT.FOTC.Data.Stream.Equality.PropertiesI where\n\nstream-\u2261\u2192\u2248 : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2261 ys \u2192 xs \u2248 ys\nstream-\u2261\u2192\u2248 {xs} Sxs _ h = subst (_\u2248_ xs) h (\u2248-refl Sxs)\n\n------------------------------------------------------------------------------\n-- From FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\n\nf\u2089\u2081-x\u2261y : \u2200 {m n o} \u2192 f\u2089\u2081 m \u2261 n \u2192 o \u2261 m \u2192 f\u2089\u2081 o \u2261 n\nf\u2089\u2081-x\u2261y {n = n} h\u2081 h\u2082 = subst (\u03bb t \u2192 f\u2089\u2081 t \u2261 n) (sym h\u2082) h\u2081\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5ab3590c56f09a19fa5fe32ec9a9eacf498b4da6","subject":"One Bit->\ud835\udfda","message":"One Bit->\ud835\udfda\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Sum.agda","new_file":"lib\/Explore\/Sum.agda","new_contents":"{-\n\n  The main definitions of this module are:\n\n    * explore\u228e\n    * explore\u228e-ind\n    * adequate-sum\u228e\n\n-}\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Data.Nat using (_+_)\nimport Level as L\nimport Function.Inverse.NP as FI\nimport Function.Related as FR\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Product.NP\nopen import Data.Sum\nopen import Data.Bit\nopen import Data.Fin using (Fin)\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; module \u2261-Reasoning; cong)\n\nopen import Explore.Type\nopen import Explore.Explorable\n\nmodule Explore.Sum where\n\nmodule _ {a b u} {A : \u2605 a} {B : \u2605 b} {U : \u2605 u}\n         (_\u2219_ : U \u2192 U \u2192 U)\n         (e\u2081  : (A \u2192 U) \u2192 U)\n         (e\u2082  : (B \u2192 U) \u2192 U)\n         (f   : (A \u228e B) \u2192 U)\n  where\n      -- find a better place\/name for it\n      \u228e\u1d9c : U\n      \u228e\u1d9c = e\u2081 (f \u2218 inj\u2081) \u2219 e\u2082 (f \u2218 inj\u2082)\n\nmodule _ {m A B} where\n    explore\u228e : Explore m A \u2192 Explore m B \u2192 Explore m (A \u228e B)\n    explore\u228e explore\u1d2c explore\u1d2e _\u2219_ = \u228e\u1d9c _\u2219_ (explore\u1d2c _\u2219_) (explore\u1d2e _\u2219_)\n\n    module _ {p} {s\u1d2c : Explore m A} {s\u1d2e : Explore m B} where\n        explore\u228e-ind : ExploreInd p s\u1d2c \u2192 ExploreInd p s\u1d2e \u2192 ExploreInd p (explore\u228e s\u1d2c s\u1d2e)\n        explore\u228e-ind Ps\u1d2c Ps\u1d2e P P\u2219 Pf\n        -- TODO clean this up:\n          = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n               (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _\u228e\u1d49_ _\u228e\u2071_ _\u228e\u02e2_\n_\u228e\u1d49_ = explore\u228e\n_\u228e\u2071_ = explore\u228e-ind\n\n_\u228e\u02e2_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n_\u228e\u02e2_ = \u228e\u1d9c _+_\n\nmodule _ {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} where\n\n    adequate-sum\u228e : AdequateSum sum\u1d2c \u2192 AdequateSum sum\u1d2e \u2192 AdequateSum (sum\u1d2c \u228e\u02e2 sum\u1d2e)\n    adequate-sum\u228e asum\u1d2c asum\u1d2e f    = (Fin (sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 FI.sym (Fin-\u228e-+ _ _) \u27e9\n                                     (Fin (sum\u1d2c (f \u2218 inj\u2081)) \u228e Fin (sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 asum\u1d2c (f \u2218 inj\u2081) \u228e-cong asum\u1d2e (f \u2218 inj\u2082) \u27e9\n                                     (\u03a3 A (Fin \u2218 f \u2218 inj\u2081) \u228e \u03a3 B (Fin \u2218 f \u2218 inj\u2082))\n                                   \u2194\u27e8 FI.sym \u03a3\u228e-distrib \u27e9\n                                     \u03a3 (A \u228e B) (Fin \u2218 f)\n                                   \u220e\n      where open FR.EquationalReasoning\n\nmodule _ {A B} {s\u1d2c : Explore\u2081 A} {s\u1d2e : Explore\u2081 B} where\n  s\u1d2c\u207a\u1d2e = s\u1d2c \u228e\u1d49 s\u1d2e\n  _\u228e-focus_ : Focus s\u1d2c \u2192 Focus s\u1d2e \u2192 Focus s\u1d2c\u207a\u1d2e\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2081 x , y) = inj\u2081 (f\u1d2c (x , y))\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2082 x , y) = inj\u2082 (f\u1d2e (x , y))\n\n  _\u228e-unfocus_ : Unfocus s\u1d2c \u2192 Unfocus s\u1d2e \u2192 Unfocus s\u1d2c\u207a\u1d2e\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2081 x) = first inj\u2081 (f\u1d2c x)\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2082 y) = first inj\u2082 (f\u1d2e y)\n\n  {-\n  _\u228e-focused_ : Focused s\u1d2c \u2192 Focused s\u1d2e \u2192 Focused {L.zero} s\u1d2c\u207a\u1d2e\n  _\u228e-focused_ f\u1d2c f\u1d2e {B} = inverses (to f\u1d2c \u228e-focus to f\u1d2e) (from f\u1d2c \u228e-unfocus from f\u1d2e) (\u21d2) (\u21d0)\n      where\n        \u21d2 : (x : \u03a3 (A \u228e {!!}) {!!}) \u2192 _\n        \u21d2 (x , y) = {!!}\n        \u21d0 : (x : s\u1d2c _\u228e_ (B \u2218 inj\u2081) \u228e s\u1d2e _\u228e_ (B \u2218 inj\u2082)) \u2192 _\n        \u21d0 (inj\u2081 x) = cong inj\u2081 {!!}\n        \u21d0 (inj\u2082 x) = cong inj\u2082 {!!}\n  -}\n\n  _\u228e-lookup_ : Lookup s\u1d2c \u2192 Lookup s\u1d2e \u2192 Lookup (s\u1d2c \u228e\u1d49 s\u1d2e)\n  (lookup\u1d2c \u228e-lookup lookup\u1d2e) (x , y) = [ lookup\u1d2c x , lookup\u1d2e y ]\n\n  _\u228e-reify_ : Reify s\u1d2c \u2192 Reify s\u1d2e \u2192 Reify (s\u1d2c \u228e\u1d49 s\u1d2e)\n  (reify\u1d2c \u228e-reify reify\u1d2e) f = (reify\u1d2c (f \u2218 inj\u2081)) , (reify\u1d2e (f \u2218 inj\u2082))\n\nexploreBit : \u2200 {m} \u2192 Explore m Bit\nexploreBit _\u2219_ f = f 0b \u2219 f 1b\n\nexploreBit-ind : \u2200 {m p} \u2192 ExploreInd p {m} exploreBit\nexploreBit-ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\nfocusBit : \u2200 {a} \u2192 Focus {a} exploreBit\nfocusBit (0b , x) = inj\u2081 x\nfocusBit (1b , x) = inj\u2082 x\n\nfocusedBit : Focused {L.zero} exploreBit\nfocusedBit {B} = inverses focusBit unfocus (\u21d2) (\u21d0)\n  where open Explorable\u2081\u2081 exploreBit-ind\n        \u21d2 : (x : \u03a3 Bit B) \u2192 _\n        \u21d2 (0b , x) = \u2261.refl\n        \u21d2 (1b , x) = \u2261.refl\n        \u21d0 : (x : B 0b \u228e B 1b) \u2192 _\n        \u21d0 (inj\u2081 x) = \u2261.refl\n        \u21d0 (inj\u2082 x) = \u2261.refl\n\nlookupBit : \u2200 {a} \u2192 Lookup {a} exploreBit\nlookupBit = proj\n\n-- DEPRECATED\nmodule \u03bc where\n    _\u228e-\u03bc_ : \u2200 {A B} \u2192 Explorable A \u2192 Explorable B \u2192 Explorable (A \u228e B)\n    \u03bcA \u228e-\u03bc \u03bcB = mk _ (explore-ind \u03bcA \u228e\u2071 explore-ind \u03bcB)\n                     (adequate-sum\u228e (adequate-sum \u03bcA) (adequate-sum \u03bcB))\n\n    \u03bcBit : Explorable Bit\n    \u03bcBit = \u03bc-iso (FI.sym \ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9) (\u03bc\ud835\udfd9 \u228e-\u03bc \u03bc\ud835\udfd9)\n\n -- -}\n -- -}\n -- -}\n","old_contents":"{-\n\n  The main definitions of this module are:\n\n    * explore\u228e\n    * explore\u228e-ind\n    * adequate-sum\u228e\n\n-}\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Data.Nat using (_+_)\nimport Level as L\nimport Function.Inverse.NP as FI\nimport Function.Related as FR\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Product.NP\nopen import Data.Sum\nopen import Data.Bit\nopen import Data.Fin using (Fin)\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; module \u2261-Reasoning; cong)\n\nopen import Explore.Type\nopen import Explore.Explorable\n\nmodule Explore.Sum where\n\nmodule _ {a b u} {A : \u2605 a} {B : \u2605 b} {U : \u2605 u}\n         (_\u2219_ : U \u2192 U \u2192 U)\n         (e\u2081  : (A \u2192 U) \u2192 U)\n         (e\u2082  : (B \u2192 U) \u2192 U)\n         (f   : (A \u228e B) \u2192 U)\n  where\n      -- find a better place\/name for it\n      \u228e\u1d9c : U\n      \u228e\u1d9c = e\u2081 (f \u2218 inj\u2081) \u2219 e\u2082 (f \u2218 inj\u2082)\n\nmodule _ {m A B} where\n    explore\u228e : Explore m A \u2192 Explore m B \u2192 Explore m (A \u228e B)\n    explore\u228e explore\u1d2c explore\u1d2e _\u2219_ = \u228e\u1d9c _\u2219_ (explore\u1d2c _\u2219_) (explore\u1d2e _\u2219_)\n\n    module _ {p} {s\u1d2c : Explore m A} {s\u1d2e : Explore m B} where\n        explore\u228e-ind : ExploreInd p s\u1d2c \u2192 ExploreInd p s\u1d2e \u2192 ExploreInd p (explore\u228e s\u1d2c s\u1d2e)\n        explore\u228e-ind Ps\u1d2c Ps\u1d2e P P\u2219 Pf\n        -- TODO clean this up:\n          = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n               (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _\u228e\u1d49_ _\u228e\u2071_ _\u228e\u02e2_\n_\u228e\u1d49_ = explore\u228e\n_\u228e\u2071_ = explore\u228e-ind\n\n_\u228e\u02e2_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n_\u228e\u02e2_ = \u228e\u1d9c _+_\n\nmodule _ {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} where\n\n    adequate-sum\u228e : AdequateSum sum\u1d2c \u2192 AdequateSum sum\u1d2e \u2192 AdequateSum (sum\u1d2c \u228e\u02e2 sum\u1d2e)\n    adequate-sum\u228e asum\u1d2c asum\u1d2e f    = (Fin (sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 FI.sym (Fin-\u228e-+ _ _) \u27e9\n                                     (Fin (sum\u1d2c (f \u2218 inj\u2081)) \u228e Fin (sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 asum\u1d2c (f \u2218 inj\u2081) \u228e-cong asum\u1d2e (f \u2218 inj\u2082) \u27e9\n                                     (\u03a3 A (Fin \u2218 f \u2218 inj\u2081) \u228e \u03a3 B (Fin \u2218 f \u2218 inj\u2082))\n                                   \u2194\u27e8 FI.sym \u03a3\u228e-distrib \u27e9\n                                     \u03a3 (A \u228e B) (Fin \u2218 f)\n                                   \u220e\n      where open FR.EquationalReasoning\n\nmodule _ {A B} {s\u1d2c : Explore\u2081 A} {s\u1d2e : Explore\u2081 B} where\n  s\u1d2c\u207a\u1d2e = s\u1d2c \u228e\u1d49 s\u1d2e\n  _\u228e-focus_ : Focus s\u1d2c \u2192 Focus s\u1d2e \u2192 Focus s\u1d2c\u207a\u1d2e\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2081 x , y) = inj\u2081 (f\u1d2c (x , y))\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2082 x , y) = inj\u2082 (f\u1d2e (x , y))\n\n  _\u228e-unfocus_ : Unfocus s\u1d2c \u2192 Unfocus s\u1d2e \u2192 Unfocus s\u1d2c\u207a\u1d2e\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2081 x) = first inj\u2081 (f\u1d2c x)\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2082 y) = first inj\u2082 (f\u1d2e y)\n\n  {-\n  _\u228e-focused_ : Focused s\u1d2c \u2192 Focused s\u1d2e \u2192 Focused {L.zero} s\u1d2c\u207a\u1d2e\n  _\u228e-focused_ f\u1d2c f\u1d2e {B} = inverses (to f\u1d2c \u228e-focus to f\u1d2e) (from f\u1d2c \u228e-unfocus from f\u1d2e) (\u21d2) (\u21d0)\n      where\n        \u21d2 : (x : \u03a3 (A \u228e {!!}) {!!}) \u2192 _\n        \u21d2 (x , y) = {!!}\n        \u21d0 : (x : s\u1d2c _\u228e_ (B \u2218 inj\u2081) \u228e s\u1d2e _\u228e_ (B \u2218 inj\u2082)) \u2192 _\n        \u21d0 (inj\u2081 x) = cong inj\u2081 {!!}\n        \u21d0 (inj\u2082 x) = cong inj\u2082 {!!}\n  -}\n\n  _\u228e-lookup_ : Lookup s\u1d2c \u2192 Lookup s\u1d2e \u2192 Lookup (s\u1d2c \u228e\u1d49 s\u1d2e)\n  (lookup\u1d2c \u228e-lookup lookup\u1d2e) (x , y) = [ lookup\u1d2c x , lookup\u1d2e y ]\n\n  _\u228e-reify_ : Reify s\u1d2c \u2192 Reify s\u1d2e \u2192 Reify (s\u1d2c \u228e\u1d49 s\u1d2e)\n  (reify\u1d2c \u228e-reify reify\u1d2e) f = (reify\u1d2c (f \u2218 inj\u2081)) , (reify\u1d2e (f \u2218 inj\u2082))\n\nexploreBit : \u2200 {m} \u2192 Explore m Bit\nexploreBit _\u2219_ f = f 0b \u2219 f 1b\n\nexploreBit-ind : \u2200 {m p} \u2192 ExploreInd p {m} exploreBit\nexploreBit-ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\nfocusBit : \u2200 {a} \u2192 Focus {a} exploreBit\nfocusBit (0b , x) = inj\u2081 x\nfocusBit (1b , x) = inj\u2082 x\n\nfocusedBit : Focused {L.zero} exploreBit\nfocusedBit {B} = inverses focusBit unfocus (\u21d2) (\u21d0)\n  where open Explorable\u2081\u2081 exploreBit-ind\n        \u21d2 : (x : \u03a3 Bit B) \u2192 _\n        \u21d2 (0b , x) = \u2261.refl\n        \u21d2 (1b , x) = \u2261.refl\n        \u21d0 : (x : B 0b \u228e B 1b) \u2192 _\n        \u21d0 (inj\u2081 x) = \u2261.refl\n        \u21d0 (inj\u2082 x) = \u2261.refl\n\nlookupBit : \u2200 {a} \u2192 Lookup {a} exploreBit\nlookupBit = proj\n\n-- DEPRECATED\nmodule \u03bc where\n    _\u228e-\u03bc_ : \u2200 {A B} \u2192 Explorable A \u2192 Explorable B \u2192 Explorable (A \u228e B)\n    \u03bcA \u228e-\u03bc \u03bcB = mk _ (explore-ind \u03bcA \u228e\u2071 explore-ind \u03bcB)\n                     (adequate-sum\u228e (adequate-sum \u03bcA) (adequate-sum \u03bcB))\n\n    \u03bcBit : Explorable Bit\n    \u03bcBit = \u03bc-iso (FI.sym Bit\u2194\ud835\udfd9\u228e\ud835\udfd9) (\u03bc\ud835\udfd9 \u228e-\u03bc \u03bc\ud835\udfd9)\n\n -- -}\n -- -}\n -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6e770f559a5e16728cdbe21d12ba340aca39e6aa","subject":"disjointness lemma","message":"disjointness lemma\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  -- collect up the hole names of a term as the indices of a trivial contex\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  -- the above judgement has mode (\u2200,\u2203). this doesn't prove uniqueness; any\n  -- contex that extends the one computed here will be indistinguishable\n  -- but we'll treat this one as canonical\n  find-holes : (e : hexp) \u2192 \u03a3[ H \u2208 \u22a4 ctx ](holes e H)\n  find-holes c = \u2205 , HConst\n  find-holes (e \u00b7: x) with find-holes e\n  ... | (h , d)= h , (HAsc d)\n  find-holes (X x) = \u2205 , HVar\n  find-holes (\u00b7\u03bb x e) with find-holes e\n  ... | (h , d) = h , HLam1 d\n  find-holes (\u00b7\u03bb x [ x\u2081 ] e) with find-holes e\n  ... | (h , d) = h , HLam2 d\n  find-holes \u2987\u2988[ x ] = (\u25a0 (x , <>)) , HEHole\n  find-holes \u2987 e \u2988[ x ] with find-holes e\n  ... | (h , d) = h ,, (x , <>) , HNEHole d\n  find-holes (e1 \u2218 e2) with find-holes e1 | find-holes e2\n  ... | (h1 , d1) | (h2 , d2)  = (h1 \u222a h2 ) , (HAp d1 d2)\n\n  -- two contexts that may contain different mappings have the same domain\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  -- the empty context has the same domain as itself\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  -- the singleton contexts formed with any contents but the same index has\n  -- the same domain\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  lem-dom-union-apt1 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03941 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03942 x == Some y)\n  lem-dom-union-apt1 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt1 apt xin | Some x\u2081 = abort (somenotnone apt)\n  lem-dom-union-apt1 apt xin | None = xin\n\n  lem-dom-union-apt2 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03942 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03941 x == Some y)\n  lem-dom-union-apt2 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt2 apt xin | Some x\u2081 = xin\n  lem-dom-union-apt2 apt xin | None = abort (somenotnone (! xin \u00b7 apt))\n\n  -- if two disjoint sets each share a domain with two other sets, those\n  -- are also disjoint.\n  dom-eq-disj : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n              H1 ## H2 \u2192\n              dom-eq \u03941 H1 \u2192\n              dom-eq \u03942 H2 \u2192\n              \u03941 ## \u03942\n  dom-eq-disj {A} {B} {\u03941} {\u03942} {H1} {H2} (d1 , d2) (de1 , de2) (de3 , de4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192 dom \u03941 n \u2192 n # \u03942\n      guts1 n dom1 with ctxindirect H2 n\n      guts1 n dom1 | Inl x = abort (somenotnone (! (\u03c02 x) \u00b7 d1 n (de1 n dom1)))\n      guts1 n dom1 | Inr x with ctxindirect \u03942 n\n      guts1 n dom1 | Inr x\u2081 | Inl x = abort (somenotnone (! (\u03c02 (de3 n x)) \u00b7 x\u2081))\n      guts1 n dom1 | Inr x\u2081 | Inr x = x\n\n      guts2 : (n : Nat) \u2192 dom \u03942 n \u2192 n # \u03941\n      guts2 n dom2 with ctxindirect H1 n\n      guts2 n dom2 | Inl x = abort (somenotnone (! (\u03c02 x) \u00b7 d2 n (de3 n dom2)))\n      guts2 n dom2 | Inr x with ctxindirect \u03941 n\n      guts2 n dom2 | Inr x\u2081 | Inl x = abort (somenotnone (! (\u03c02 (de1 n x)) \u00b7 x\u2081))\n      guts2 n dom2 | Inr x\u2081 | Inr x = x\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n                                     H1 ## H2 \u2192\n                                     dom-eq \u03941 H1 \u2192\n                                     dom-eq \u03942 H2 \u2192\n                                     dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union {A} {B} {\u03941} {\u03942} {H1} {H2} disj (p1 , p2) (p3 , p4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y)\n      guts1 n (x , eq) with ctxindirect \u03941 n\n      guts1 n (x\u2081 , eq) | Inl x with p1 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al H1 H2 n q1 q2\n      guts1 n (x\u2081 , eq) | Inr x with p3 n (_ , lem-dom-union-apt1 {\u03941 = \u03941} {\u03942 = \u03942} x eq)\n      ... | q1 , q2 =  q1 , x\u2208\u222ar H1 H2 n q1 q2 (##-comm disj)\n\n      guts2 : (n : Nat) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x)\n      guts2 n (x , eq) with ctxindirect H1 n\n      guts2 n (x\u2081 , eq) | Inl x with p2 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al \u03941 \u03942 n q1 q2\n      guts2 n (x\u2081 , eq) | Inr x with p4 n (_ , lem-dom-union-apt2 {\u03941 = H2} {\u03942 = H1} x (tr (\u03bb qq \u2192 qq n == Some x\u2081) (\u222acomm H1 H2 disj) eq))\n      ... | q1 , q2 = q1 , x\u2208\u222ar \u03941 \u03942 n q1 q2 (##-comm (dom-eq-disj disj (p1 , p2) (p3 , p4)))\n\n\n\n  -- the holes of an expression have the same domain as \u0394; that is, we\n  -- don't add any extra junk as we expand\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union {!!} (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union {!!} (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) = dom-union {!!} (holes-delta-ana h x\u2084) (holes-delta-ana h\u2081 x\u2085)\n\n  -- if a hole name is new then it's apart from the holes\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  -- todo: lemmas file?\n  lem-dom-apt : {A : Set} {G : A ctx} {x y : Nat} \u2192 x # G \u2192 dom G y \u2192 x \u2260 y\n  lem-dom-apt {x = x} {y = y} apt dom with natEQ x y\n  lem-dom-apt apt dom | Inl refl = abort (somenotnone (! (\u03c02 dom) \u00b7 apt))\n  lem-dom-apt apt dom | Inr x\u2081 = x\u2081\n\n  -- if the holes of two expressions are disjoint, so are their collections\n  -- of hole names\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-sing-disj (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-sing-disj (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  -- if two contexsts are disjoint and each share a domain with another\n  -- context, those other two contexts are also disjoint\n  domeq-disj : {A B : Set} {H1 H2 : A ctx} {\u03941 \u03942 : B ctx} \u2192\n               H1 ## H2 \u2192\n               dom-eq \u03941 H1 \u2192\n               dom-eq \u03942 H2 \u2192\n               \u03941 ## \u03942\n  domeq-disj (\u03c01 , \u03c02) (\u03c03 , \u03c04) (\u03c05 , \u03c06) =\n                   (\u03bb n x \u2192 {!(\u03c03 n x)!}) ,\n                   (\u03bb n x \u2192 {!!})\n\n  -- if you expand two hole-disjoint expressions analytically, the \u0394s\n  -- produces are disjoint\n  expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n  expand-ana-disjoint {e1} {e2} hd ana1 ana2\n    with find-holes e1 | find-holes e2\n  ... | (_ , he1) | (_ , he2) = domeq-disj (holes-disjoint-disjoint he1 he2 hd)\n                                           (holes-delta-ana he1 ana1)\n                                           (holes-delta-ana he2 ana2)\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  -- proving that the above judgement has mode (\u2200,\u2203), or that it defines a\n  -- function, so we can use it in a functional modality below\n  find-holes : (e : hexp) \u2192 \u03a3[ H \u2208 \u22a4 ctx ](holes e H)\n  find-holes c = \u2205 , HConst\n  find-holes (e \u00b7: x) with find-holes e\n  ... | (h , d)= h , (HAsc d)\n  find-holes (X x) = \u2205 , HVar\n  find-holes (\u00b7\u03bb x e) with find-holes e\n  ... | (h , d) = h , HLam1 d\n  find-holes (\u00b7\u03bb x [ x\u2081 ] e) with find-holes e\n  ... | (h , d) = h , HLam2 d\n  find-holes \u2987\u2988[ x ] = (\u25a0 (x , <>)) , HEHole\n  find-holes \u2987 e \u2988[ x ] with find-holes e\n  ... | (h , d) = h ,, (x , <>) , HNEHole d\n  find-holes (e1 \u2218 e2) with find-holes e1 | find-holes e2\n  ... | (h1 , d1) | (h2 , d2)  = (h1 \u222a h2 ) , (HAp d1 d2)\n\n  -- two contexts that may contain different mappings have the same domain\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  -- the empty context has the same domain as itself\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  -- todo: this seems like i would have proven it already? otw move to lemmas\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  -- the singleton contexts formed with any contents but the same index has\n  -- the same domain\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  lem-dom-union-apt1 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03941 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03942 x == Some y)\n  lem-dom-union-apt1 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt1 apt xin | Some x\u2081 = abort (somenotnone apt)\n  lem-dom-union-apt1 apt xin | None = xin\n\n  lem-dom-union-apt2 : {A : Set} {\u03941 \u03942 : A ctx} {x : Nat} {y : A} \u2192 x # \u03942 \u2192 ((\u03941 \u222a \u03942) x == Some y) \u2192 (\u03941 x == Some y)\n  lem-dom-union-apt2 {A} {\u03941} {\u03942} {x} {y} apt xin with \u03941 x\n  lem-dom-union-apt2 apt xin | Some x\u2081 = xin\n  lem-dom-union-apt2 apt xin | None = abort (somenotnone (! xin \u00b7 apt))\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192\n                                     H1 ## H2 \u2192\n                                     dom-eq \u03941 H1 \u2192\n                                     dom-eq \u03942 H2 \u2192\n                                     dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union {A} {B} {\u03941} {\u03942} {H1} {H2} disj (p1 , p2) (p3 , p4) = guts1 , guts2\n    where\n      guts1 : (n : Nat) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y)\n      guts1 n (x , eq) with ctxindirect \u03941 n\n      guts1 n (x\u2081 , eq) | Inl x with p1 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al H1 H2 n q1 q2\n      guts1 n (x\u2081 , eq) | Inr x with p3 n (_ , lem-dom-union-apt1 {\u03941 = \u03941} {\u03942 = \u03942} x eq)\n      ... | q1 , q2 =  q1 , x\u2208\u222ar H1 H2 n q1 q2 (##-comm disj)\n\n      guts2 : (n : Nat) \u2192\n                 \u03a3[ y \u2208 B ] ((H1 \u222a H2) n == Some y) \u2192\n                 \u03a3[ x \u2208 A ] ((\u03941 \u222a \u03942) n == Some x)\n      guts2 n (x , eq) with ctxindirect H1 n\n      guts2 n (x\u2081 , eq) | Inl x with p2 n x\n      ... | q1 , q2 = q1 , x\u2208\u222al \u03941 \u03942 n q1 q2\n      guts2 n (x\u2081 , eq) | Inr x with p4 n (_ , lem-dom-union-apt2 {\u03941 = H2} {\u03942 = H1} x (tr (\u03bb qq \u2192 qq n == Some x\u2081) (\u222acomm H1 H2 disj) eq))\n      ... | q1 , q2 = q1 , x\u2208\u222ar \u03941 \u03942 n q1 q2 (##-comm {!!})\n\n\n\n  -- the holes of an expression have the same domain as \u0394; that is, we\n  -- don't add any extra junk as we expand\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union {!!} (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union {!!} (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) = dom-union {!!} (holes-delta-ana h x\u2084) (holes-delta-ana h\u2081 x\u2085)\n\n  -- if a hole name is new then it's apart from the holes\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  -- todo: lemmas file?\n  lem-dom-apt : {A : Set} {G : A ctx} {x y : Nat} \u2192 x # G \u2192 dom G y \u2192 x \u2260 y\n  lem-dom-apt {x = x} {y = y} apt dom with natEQ x y\n  lem-dom-apt apt dom | Inl refl = abort (somenotnone (! (\u03c02 dom) \u00b7 apt))\n  lem-dom-apt apt dom | Inr x\u2081 = x\u2081\n\n  -- if the holes of two expressions are disjoint, so are their collections\n  -- of hole names\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-sing-disj (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-sing-disj (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  -- if two contexsts are disjoint and each share a domain with another\n  -- context, those other two contexts are also disjoint\n  domeq-disj : {A B : Set} {H1 H2 : A ctx} {\u03941 \u03942 : B ctx} \u2192\n               H1 ## H2 \u2192\n               dom-eq \u03941 H1 \u2192\n               dom-eq \u03942 H2 \u2192\n               \u03941 ## \u03942\n  domeq-disj (\u03c01 , \u03c02) (\u03c03 , \u03c04) (\u03c05 , \u03c06) =\n                   (\u03bb n x \u2192 {!(\u03c03 n x)!}) ,\n                   (\u03bb n x \u2192 {!!})\n\n  -- if you expand two hole-disjoint expressions analytically, the \u0394s\n  -- produces are disjoint\n  expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n  expand-ana-disjoint {e1} {e2} hd ana1 ana2\n    with find-holes e1 | find-holes e2\n  ... | (_ , he1) | (_ , he2) = domeq-disj (holes-disjoint-disjoint he1 he2 hd)\n                                           (holes-delta-ana he1 ana1)\n                                           (holes-delta-ana he2 ana2)\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"36c32cf0ea2bf68af624df007475d90a3f2fba5f","subject":"Remove an extra \u2192\u1d47 def","message":"Remove an extra \u2192\u1d47 def\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec.NP as Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nRewireTbl : CircuitType\nRewireTbl i o = Vec (Fin i) o\n\nmodule Rewire where\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Rewire.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec.NP as Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nRewireTbl : CircuitType\nRewireTbl i o = Vec (Fin i) o\n\nmodule Rewire where\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Rewire.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c828ae94c2b18b988846aa0e7467ff5c68d19206","subject":"Data.Nat: + \u2115+ and \u2115+\u2032","message":"Data.Nat: + \u2115+ and \u2115+\u2032\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nopen import Type\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = \u2261.refl\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = \u2261.refl\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = \u2261.refl\na\u2261a\u2293b+a\u2238b zero (suc b) = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) zero = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite \u2261.sym (a\u2261a\u2293b+a\u2238b a b) = \u2261.refl\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 \u2605\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 \u2605\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 \u2605\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : \u2605} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 \u2605\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 \u2605) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 \u2605\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nopen import Type\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = \u2261.refl\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = \u2261.refl\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = \u2261.refl\na\u2261a\u2293b+a\u2238b zero (suc b) = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) zero = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite \u2261.sym (a\u2261a\u2293b+a\u2238b a b) = \u2261.refl\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 \u2605\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 \u2605\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 \u2605\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : \u2605} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 \u2605\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 \u2605) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 \u2605\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0ba3572a7a7523455c601da5c200dd85039dcced","subject":"Mention our claims about the Agda code (for #275).","message":"Mention our claims about the Agda code (for #275).\n\nOld-commit-hash: f20d237f6179fe5e79864e2e68d0203143182c6c\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) and formally specifies the interface between our incrementalization\n--       framework and potential plugins.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"109b8cf60463ce4ef6a46602bede3b14a658aec6","subject":"IDesc model: paranoid alpha-renaming.","message":"IDesc model: paranoid alpha-renaming.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set (suc l)) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set (suc l)} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set (suc l)}(R : I -> IDesc {l = l} I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set (suc l)}(D : IDesc I)(X : I -> Set l) -> desc D X -> IDesc (Sigma I X)\nbox (var i)     X x        = var (i , x)\nbox (const _)   X x        = const Unit\nbox (prod D D') X (d , d') = prod (box D X d) (box D' X d')\nbox (sigma S T) X (a , b)  = box (T a) X b\nbox (pi S T)    X f        = pi S (\\s -> box (T s) X (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set (suc l) -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\nIDescD : {l : Level}(I : Set (suc l)) -> IDesc {l = suc l} Unit\nIDescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : {l : Level}(I : Set (suc l)) -> Unit -> Set (suc l)\nIDescl0 {x} I = IMu {l = suc x} (\\_ -> IDescD {l = x} I)\n\nIDescl : {l : Level}(I : Set (suc l)) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set (suc l)}(i : I) -> IDescl I\nvarl {x} i = con (lvar {l = suc x} , i) \n\nconstl : {l : Level}{I : Set (suc l)}(X : Set l) -> IDescl I\nconstl {x} X = con (lconst {l = suc x} , X)\n\nprodl : {l : Level}{I : Set (suc l)}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lprod {l = suc x} , (D , D'))\n\n\npil : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lpi {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n\nsigmal : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (lsigma {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n           \n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set (suc l)}\n        (xs : desc (IDescD I) (IMu (\u03bb _ -> IDescD I)))\n        (hs : desc (box (IDescD I) (IMu (\u03bb _ -> IDescD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s) )\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set (suc l)} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> IDescD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set (suc l)} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set (suc l)} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-psi-phi : {l : Level}(I : Set (suc l)) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> IDescD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n    where P : Sigma Unit (IMu (\\ x -> IDescD I)) -> Set (suc x)\n          P ( Void , D ) = psi (phi D) == D\n          proof-psi-phi-cases : (i : Unit)\n                                (xs : desc (IDescD I) (IDescl0 I))\n                                (hs : desc (box (IDescD I) (IDescl0 I) xs) P)\n                                -> P (i , con xs)\n          proof-psi-phi-cases Void (lvar , i) hs = refl\n          proof-psi-phi-cases Void (lconst , x) hs = refl\n          proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n          proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                               (trans (reflFun (\\ s -> psi (phi (T (lifter (unlift s)))))\n                                                                               (\\ s -> psi (phi (T (s))))\n                                                                               (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                           (proof-lift-unlift-eq s)))\n                                                                       (reflFun (\\s -> psi (phi (T s))) \n                                                                                T \n                                                                                hs)) \n          proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                  (trans (reflFun (\\ s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                  (\\ s \u2192 psi (phi (T (s))))\n                                                                                  (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                              (proof-lift-unlift-eq s)))\n                                                                         (reflFun (\\s -> psi (phi (T s))) \n                                                                                  T \n                                                                                  hs))","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set (suc l)) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set (suc l)} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set (suc l)}(R : I -> IDesc {l = l} I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set (suc l)}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const Unit\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set (suc l) -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\nIDescD : {l : Level}(I : Set (suc l)) -> IDesc {l = suc l} Unit\nIDescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : {l : Level}(I : Set (suc l)) -> Unit -> Set (suc l)\nIDescl0 {x} I = IMu {l = suc x} (\\_ -> IDescD {l = x} I)\n\nIDescl : {l : Level}(I : Set (suc l)) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set (suc l)}(i : I) -> IDescl I\nvarl {x} i = con (lvar {l = suc x} , i) \n\nconstl : {l : Level}{I : Set (suc l)}(X : Set l) -> IDescl I\nconstl {x} X = con (lconst {l = suc x} , X)\n\nprodl : {l : Level}{I : Set (suc l)}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lprod {l = suc x} , (D , D'))\n\n\npil : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lpi {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n\nsigmal : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (lsigma {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n           \n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set (suc l)}\n        (xs : desc (IDescD I) (IMu (\u03bb _ -> IDescD I)))\n        (hs : desc (box (IDescD I) (IMu (\u03bb _ -> IDescD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s) )\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set (suc l)} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> IDescD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set (suc l)} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set (suc l)} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-psi-phi : {l : Level}(I : Set (suc l)) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> IDescD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n    where P : Sigma Unit (IMu (\\ x -> IDescD I)) -> Set (suc x)\n          P ( Void , D ) = psi (phi D) == D\n          proof-psi-phi-cases : (i : Unit)\n                                (xs : desc (IDescD I) (IDescl0 I))\n                                (hs : desc (box (IDescD I) (IDescl0 I) xs) P)\n                                -> P (i , con xs)\n          proof-psi-phi-cases Void (lvar , i) hs = refl\n          proof-psi-phi-cases Void (lconst , x) hs = refl\n          proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n          proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                               (trans (reflFun (\\ s -> psi (phi (T (lifter (unlift s)))))\n                                                                               (\\ s -> psi (phi (T (s))))\n                                                                               (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                           (proof-lift-unlift-eq s)))\n                                                                       (reflFun (\\s -> psi (phi (T s))) \n                                                                                T \n                                                                                hs)) \n          proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                  (trans (reflFun (\\ s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                  (\\ s \u2192 psi (phi (T (s))))\n                                                                                  (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                              (proof-lift-unlift-eq s)))\n                                                                         (reflFun (\\s -> psi (phi (T s))) \n                                                                                  T \n                                                                                  hs))","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"56113e6c31daaab6d85a46d0bc59c4b4d25cb6c9","subject":"tiny proof simplification","message":"tiny proof simplification\n\nOld-commit-hash: f98c7776e0c807aae905d00908b7eb27c17a0a99\n","repos":"inc-lc\/ilc-agda","old_file":"Natural\/Soundness.agda","new_file":"Natural\/Soundness.agda","new_contents":"module Natural.Soundness where\n\n-- SOUNDNESS of NATURAL SEMANTICS\n--\n-- This module proves consistency of the natural semantics with\n-- respect to the denotational semantics.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Closures\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\n-- Syntactic lookup is consistent with semantic lookup.\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n-- Syntactic evaluation is consistent with semantic evaluation.\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound e-true = \u2261-refl \n\u2193-sound e-false = \u2261-refl \n\u2193-sound (if-true {\u03c1 = \u03c1} {c = c} \u2193\u2081 \u2193\u2082) with \u27e6 c \u27e7 \u27e6 \u03c1 \u27e7 | \u2193-sound \u2193\u2081\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) | true | refl = \u2193-sound \u2193\u2082\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) | false | ()\n\u2193-sound (if-false {\u03c1 = \u03c1} {c = c} \u2193\u2081 \u2193\u2082) with \u27e6 c \u27e7 \u27e6 \u03c1 \u27e7 | \u2193-sound \u2193\u2081\n\u2193-sound (if-false \u2193\u2081 \u2193\u2082) | false | refl = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2082) | true | ()\n","old_contents":"module Natural.Soundness where\n\n-- SOUNDNESS of NATURAL SEMANTICS\n--\n-- This module proves consistency of the natural semantics with\n-- respect to the denotational semantics.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Closures\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\n-- Syntactic lookup is consistent with semantic lookup.\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n-- Syntactic evaluation is consistent with semantic evaluation.\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound e-true = \u2261-refl \n\u2193-sound e-false = \u2261-refl \n\u2193-sound (if-true {\u03c1 = \u03c1} {c = c} \u2193\u2081 \u2193\u2082) with \u27e6 c \u27e7 \u27e6 \u03c1 \u27e7 | \u2193-sound \u2193\u2081\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) | true | refl = \u2261-trans refl (\u2193-sound \u2193\u2082)\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) | false | ()\n\u2193-sound (if-false {\u03c1 = \u03c1} {c = c} \u2193\u2081 \u2193\u2082) with \u27e6 c \u27e7 \u27e6 \u03c1 \u27e7 | \u2193-sound \u2193\u2081\n\u2193-sound (if-false \u2193\u2081 \u2193\u2082) | false | refl = \u2261-trans refl (\u2193-sound \u2193\u2082)\n\u2193-sound (if-false \u2193\u2081 \u2193\u2082) | true | ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cb0be83f848e2a7872c46ba210d3df2d2f1d8263","subject":"Bits: remove\/comment the broken stuff","message":"Bits: remove\/comment the broken stuff\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    toPerm : Op \u2192 Perm\n    toPerm `id     = `id\n    toPerm `0\u21941    = `0\u21941\n    toPerm `not    = `id -- Important\n    toPerm (`tl f) = `tl (toPerm f)\n    toPerm (f `\u204f g) = toPerm f `\u204f toPerm g\n\n    infixr 9 _\u2219\u2032_\n    _\u2219\u2032_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    f \u2219\u2032 xs = toPerm f P.\u2219 xs\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    almost-\u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) f xs \u2192 f \u2219\u2032 pad \u2295 f \u2219 xs \u2261 f \u2219 (pad \u2295 xs)\n    almost-\u2295-dist-\u2219 pad           `id     xs = refl\n    almost-\u2295-dist-\u2219 pad           `0\u21941    xs = \u2295-dist-0\u21941 pad xs\n    almost-\u2295-dist-\u2219 []            `not    [] = refl\n    almost-\u2295-dist-\u2219 (true  \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 (false \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 []            (`tl f) [] = refl\n    almost-\u2295-dist-\u2219 (p \u2237 pad)     (`tl f) (x \u2237 xs) rewrite almost-\u2295-dist-\u2219 pad f xs = refl\n    almost-\u2295-dist-\u2219 pad           (f `\u204f g) xs rewrite almost-\u2295-dist-\u2219 (f \u2219\u2032 pad) g (f \u2219 xs)\n                                                   | almost-\u2295-dist-\u2219 pad f xs = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"41f662646924858d5d32ac199e0887cf908df64d","subject":"Added infor for an issue.","message":"Added infor for an issue.\n\nIgnore-this: 3d18d25b01fdd40e186f7f1195c43154\n\ndarcs-hash:20120619225108-3bd4e-09bf31bb63af16215eb30f7625f2746506400e2a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/BadErase.agda","new_file":"Issues\/BadErase.agda","new_contents":"------------------------------------------------------------------------------\n-- The translation is badly erasing the universal quantification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issues.BadErase where\n\npostulate\n  _\u2194_ : Set \u2192 Set \u2192 Set\n  D   : Set\n  A   : Set\n\npostulate bad\u2081 : ((x : D) \u2192 A) \u2192 A\n{-# ATP prove bad\u2081 #-}\n\n-- Before the patch\n--\n-- Wed Sep 21 04:50:43 COT 2011  ulfn@chalmers.se\n--   * got rid of the Fun constructor in internal syntax (using Pi _ (NoAbs _ _) instead)\n--\n-- the type of bad\u2081 was:\n--\n-- Type: El (Type (Max [])) (Fun r(El (Type (Max [])) (Pi r(El (Type (Max [])) (Def BadErase.D [])) (Abs \"x\" El (Type (Max [])) (Def BadErase.A [])))) (El (Type (Max [])) (Def BadErase.A [])))\n--\n--\n-- After the above patch the type of bad\u2081 was:\n--\n-- Type: El (Type (Max [])) (Pi r(El (Type (Max [])) (Pi r(El (Type (Max [])) (Def BadErase.D [])) (Abs \"x\" El (Type (Max [])) (Def BadErase.A [])))) (NoAbs \"_\" El (Type (Max [])) (Def BadErase.A [])))\n--\n--\n-- On 19 June 2012, the type of bad\u2081 is:\n--\n-- Type: El {getSort = Type (Max []), unEl = Pi r(El {getSort = Type (Max []), unEl = Pi r(El {getSort = Type (Max []), unEl = Def BadErase.D []}) (NoAbs \"x\" El {getSort = Type (Max []), unEl = Def BadErase.A []})}) (NoAbs \"_\" El {getSort = Type (Max []), unEl = Def BadErase.A []})}\n\npostulate bad\u2082 : A \u2192 ((x : D) \u2192 A)\n{-# ATP prove bad\u2082 #-}\n\npostulate bad\u2083 : ((x : D) \u2192 A) \u2194 A\n{-# ATP prove bad\u2083 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The translation is badly erasing the universal quantification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Found on 14 March 2012.\n\nmodule Issues.BadErase where\n\npostulate\n  _\u2194_ : Set \u2192 Set \u2192 Set\n  D   : Set\n  A   : Set\n\npostulate bad\u2081 : ((x : D) \u2192 A) \u2192 A\n{-# ATP prove bad\u2081 #-}\n\npostulate bad\u2082 : A \u2192 ((x : D) \u2192 A)\n{-# ATP prove bad\u2082 #-}\n\npostulate bad\u2083 : ((x : D) \u2192 A) \u2194 A\n{-# ATP prove bad\u2083 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7ebc94aba2846920a72e18a454c5be2ed6faea1d","subject":"Added arithmetic properties (by Ana Bove).","message":"Added arithmetic properties (by Ana Bove).\n\nIgnore-this: f4f694d231eb13a003e65c404d641cb2\n\ndarcs-hash:20110218163424-3bd4e-631646d7d2092969a662e91b13a18d35345a889a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  102>100  : GT hundred-two one-hundred\n  103>100  : GT hundred-three one-hundred\n  104>100  : GT hundred-four one-hundred\n  105>100  : GT hundred-five one-hundred\n  106>100  : GT hundred-six one-hundred\n  107>100  : GT hundred-seven one-hundred\n  108>100  : GT hundred-eight one-hundred\n  109>100  : GT hundred-nine one-hundred\n  110>100  : GT hundred-ten one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 102>100 #-}\n{-# ATP prove 103>100 #-}\n{-# ATP prove 104>100 #-}\n{-# ATP prove 105>100 #-}\n{-# ATP prove 106>100 #-}\n{-# ATP prove 107>100 #-}\n{-# ATP prove 108>100 #-}\n{-# ATP prove 109>100 #-}\n{-# ATP prove 110>100 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven) one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven) one-hundred\n  95+11>100 : GT (ninety-five + eleven) one-hundred\n  94+11>100 : GT (ninety-four + eleven) one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven) one-hundred\n  91+11>100 : GT (ninety-one + eleven) one-hundred\n  90+11>100 : GT (ninety + eleven) one-hundred\n{-# ATP prove 99+11>100 110>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101>100' 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"62a643cc7f1e2ea25a239431d779ca974cdeed73","subject":"Data.Vec.NP: count, count\u1da0 and properties","message":"Data.Vec.NP: count, count\u1da0 and properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin hiding (_+_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount pred [] = zero\ncount pred (x \u2237 xs) = (if pred x then suc else inject\u2081) (count pred xs)\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (to\u2115 (count pred xs))\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"47b55ae0449921322131bd6dbdab41837ccb68c8","subject":"Add support for general vector to our flat-funs abstraction","message":"Add support for general vector to our flat-funs abstraction\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n  open FlatFuns \u266dFuns public\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk \u22a4 Bit _\u00d7_ Vec (\u03bb A B \u2192 A \u2192 B)\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk 0 1 _+_ _*_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_ proj\u2081 proj\u2082\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id bitsFunComp bitsFunVComp _&&&_ (\u03bb {A} \u2192 take A) (\u03bb {A} \u2192 drop A)\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits : \u2115 \u2192 T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n  open FlatFuns \u266dFuns public\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Bits Bit _\u00d7_ (\u03bb A B \u2192 A \u2192 B)\n\nbitsfun\u266dFuns : FlatFuns \u2115\nbitsfun\u266dFuns = mk id 1 _+_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"da85f556d63b099902d17027a8e055c87c053e25","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-unf-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e\n  --\n  -- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n  Conat-coind :\n    \u2200 (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    \u2200 (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification for this\n  -- principle.\n  Conat-coind-stronger :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-unf and Conat-unf-ho are equivalents\n\nConat-unf' : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf' = Conat-unf-ho\n\nConat-unf-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-unf-ho' = Conat-unf\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind' :\n  \u2200 (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind' = Conat-coind-ho\n\nConat-coind-ho' :\n  \u2200 (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-coind-stronger to Conat-coind-stronger\/Conat-coind\n\nConat-coind'' :\n  \u2200 (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-coind-stronger A h An\n\n-- 22 December 2013: We cannot prove Conat-coind-stronger using\n-- Conat-coind.\nConat-coind-stronger'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-coind-stronger'' A h An = Conat-coind A {!!} An\n","old_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-unf-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e\n  --\n  -- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n  Conat-coind : \u2200 (A : D \u2192 Set) \u2192\n                (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho : \u2200 (A : D \u2192 Set) \u2192\n                   (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192\n                   \u2200 {n} \u2192 A n \u2192 Conat n\n\n  Conat-coind-wrong : \u2200 (A : D \u2192 Set) {n} \u2192\n                      -- A is post-fixed point of ConatF.\n                      (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                      -- Conat is greater than A.\n                      A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-unf and Conat-unf-ho are equivalents\n\nConat-unf' : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf' = Conat-unf-ho\n\nConat-unf-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-unf-ho' = Conat-unf\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind' : \u2200 (A : D \u2192 Set) \u2192\n               (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n               \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind' = Conat-coind-ho\n\nConat-coind-ho' : \u2200 (A : D \u2192 Set) \u2192\n                  (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192\n                  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-coind-wrong to Conat-coind-wrong\/Conat-coind\n\nConat-coind'' : \u2200 (A : D \u2192 Set) \u2192\n               (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n               \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-coind-wrong A h An\n\n-- 22 December 2013: We cannot prove Conat-coind-wrong'' using\n-- Conat-coind.\nConat-coind-wrong'' : \u2200 (A : D \u2192 Set) {n} \u2192\n                      (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                      A n \u2192 Conat n\nConat-coind-wrong'' A h An = Conat-coind A {!!} An\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"160cee2820de60aaa4ac5af26e55e79c762eaa39","subject":"forking-lemma: some progress","message":"forking-lemma: some progress\n","repos":"crypto-agda\/crypto-agda","old_file":"forking-lemma.agda","new_file":"forking-lemma.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Vec.NP hiding (sum)\nopen import Data.Maybe\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Fin.NP as Fin hiding (_+_; _-_; _\u2264_)\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP hiding (J)\n\nmodule forking-lemma where\n\n{-\n_\u22651 : \u2200 {n} \u2192 Fin n \u2192 \ud835\udfda\nzero  \u22651 = 0\u2082\nsuc _ \u22651 = 1\u2082\n-}\n\nreplace : \u2200 {A : Type} {q} (I : Fin q)\n            (hs hs' : Vec A q) \u2192 Vec A q\nreplace zero         hs       hs'  = hs'\nreplace (suc I) (h \u2237 hs) (_ \u2237 hs') = h \u2237 replace I hs hs'\n\ntest-replace : replace (suc zero) (40 \u2237 41 \u2237 42 \u2237 []) (60 \u2237 61 \u2237 62 \u2237 []) \u2261 40 \u2237 61 \u2237 62 \u2237 []\ntest-replace = refl\n\n\u2261-prefix : \u2200 {A : Type} {q} (I : Fin (suc q))\n             (v\u2080 v\u2081 : Vec A q) \u2192 Type\n\u2261-prefix zero    _         _         = \ud835\udfd9\n\u2261-prefix (suc I) (x\u2080 \u2237 v\u2080) (x\u2081 \u2237 v\u2081) = (x\u2080 \u2261 x\u2081) \u00d7 \u2261-prefix I v\u2080 v\u2081\n\n\u2261-prefix (suc ()) [] []\n\npostulate\n  -- Pub    : Type\n  Res    : Type -- Side output\n  {-RndIG  : Type-}\n\n  -- IG : RndIG \u2192 Pub\n\n  q     : \u2115\n\n  instance\n    q\u22651   : q \u2265 1\n\n  #h     : \u2115\n  #h\u22652   : #h \u2265 2\n\n  #\u03c1 : \u2115\n  #\u03c1\u22651 : #\u03c1 \u2265 1\n\nRndAdv : Type -- Coin\nRndAdv = Fin #\u03c1\n\nH : Type\nH = Fin #h\n\nAdv = {-(x : Pub)-}(hs : Vec H q)(\u03c1 : RndAdv) \u2192 Fin (suc q) \u00d7 Res\n\npostulate\n  A : Adv\n-- A x hs \u03c1\n-- A x hs = \u03c1 \u2190$ ...H ; A x hs \u03c1\n\n-- IO : Type \u2192 Type\n-- print : String \u2192 IO ()\n-- #r : \u2115\n-- #r\u22651 : #r \u2265 1\n-- Rnd = Fin #r\n-- random : IO Rnd\n-- run : (A : Input \u2192 Rnd \u2192 Res) \u2192 Input \u2192 IO Res\n-- run A i = do r \u2190 random\n--              return (A i r)\n\n-- Not used yet\nwell-def : Adv \u2192 Type\nwell-def A =\n  \u2200 {-x-} hs \u03c1 \u2192\n    case A {-x-} hs \u03c1 of \u03bb { (I , \u03c3) \u2192\n    \u2200 hs' \u2192 \u2261-prefix I hs hs' \u2192  A {-x-} hs' \u03c1 \u2261 (I , \u03c3) }\n\nrecord \u03a9 : Type where\n  field\n    -- rIG : RndIG\n    hs hs' : Vec H q\n    \u03c1      : RndAdv\n\nEvent = \u03a9 \u2192 \ud835\udfda\n\n#\u03a9 = ((#h ^ q) ^2) * #\u03c1\n-- \u03a9 \u2243 Fin #\u03a9\n\ninstance\n  #\u03a9\u22651 : #\u03a9 \u2265 1\n  #\u03a9\u22651 = {!!}\n\nmodule M (r : \u03a9) where\n  open \u03a9 r\n\n  F' : (I-1 : Fin q)(\u03c3 : Res) \u2192 Maybe (Res \u00d7 Res)\n  F' I-1 \u03c3 =\n    case I=I' \u2227 h=h'\n      0: nothing\n      1: just (\u03c3 , \u03c3')\n    module F' where\n      res' = A (replace I-1 hs hs') \u03c1\n      I'   = fst res'\n      \u03c3'   = snd res'\n      h    = hs  \u203c I-1\n      h'   = hs' \u203c I-1\n      I    = suc I-1\n      I=I' = I == I'\n      h=h' = h == h'\n\n  F : Maybe (Res \u00d7 Res)\n  F =\n    case I of \u03bb\n    { zero      \u2192 nothing\n    ; (suc I-1) \u2192 F' I-1 \u03c3\n    }\n    module F where\n      res  = A hs \u03c1\n      I    = fst res\n      \u03c3    = snd res\n      -- I\u22651  = not (I == zero)\n\nI\u22651\u2227_ : (f : \u03a9 \u2192 Fin q \u2192 Res \u2192 \ud835\udfda) \u2192 Event\n(I\u22651\u2227 f) r = case I of \u03bb\n             { zero \u2192 0\u2082\n             ; (suc I-1) \u2192 f r I-1 \u03c3\n             }\n  where open M.F r\n\nI\u22651 I\u22651\u2227I=I' I\u22651\u2227h=h' I\u22651\u2227I=I'\u2227h\u2262h' : Event\n\nI\u22651 = I\u22651\u2227 \u03bb _ _ _ \u2192 1\u2082\n\nI\u22651\u2227I=I'      = I\u22651\u2227 \u03bb r I-1 \u03c3 \u2192 let open M.F' r I-1 \u03c3 in I=I'\nI\u22651\u2227h=h'      = I\u22651\u2227 \u03bb r I-1 \u03c3 \u2192 let open M.F' r I-1 \u03c3 in h=h'\nI\u22651\u2227I=I'\u2227h\u2262h' = I\u22651\u2227 \u03bb r I-1 \u03c3 \u2192 let open M.F' r I-1 \u03c3 in I=I' \u2227 not h=h'\n\nI=1+_ I=I'=1+_ : (i : Fin q) \u2192 Event\n\nI=1+    i = I\u22651\u2227 \u03bb r I-1 \u03c3 \u2192 let open M.F' r I-1 \u03c3 in I == suc i\nI=I'=1+ i = I\u22651\u2227 \u03bb r I-1 \u03c3 \u2192 let open M.F' r I-1 \u03c3 in I == suc i \u2227 I=I'\n\n-- Acceptance event for A\nacc : Event\nacc = I\u22651\n\nfrk : Event\nfrk = is-just \u2218 M.F\n\ninstance\n    #h\u22651 : #h \u2265 1\n    #h\u22651 = \u2115\u2264.trans (s\u2264s z\u2264n) #h\u22652\n\ninfix 0 _\u2265'_\ninfixr 2 _\u2265\u27e8_\u27e9_ _\u2261\u27e8_\u27e9_ _\u2261\u27e8by-definition\u27e9_\ninfix 2 _\u220e\n\npostulate\n  -- [0,1] : Type\n  \u211d : Type\n  -- [0,1]\u25b9\u211d : [0,1] \u2192 \u211d\n  -- x - y requires x \u2265 y\n  -- _\u00b7_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _-_ _\u00b7_ : \u211d \u2192 \u211d \u2192 \u211d\n  -- _-_ :\n  -- _-_ : [0,1] \u2192 [0,1] \u2192 \u211d\n  -- _\u2265'_ : [0,1] \u2192 [0,1] \u2192 Type\n  _\u2265'_ : \u211d \u2192 \u211d \u2192 Type\n  1\/_ : (d : \u2115){{_ : d \u2265 1}} \u2192 \u211d\n\n  _\u2265\u27e8_\u27e9_ : \u2200 x {y} \u2192 x \u2265' y \u2192 \u2200 {z} \u2192 y \u2265' z \u2192 x \u2265' z\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y} \u2192 x \u2261  y \u2192 \u2200 {z} \u2192 y \u2265' z \u2192 x \u2265' z\n  _\u220e : \u2200 x \u2192 x \u2265' x\n\n  \u2115\u25b9\u211d : \u2115 \u2192 \u211d\n\n  sum : Event \u2192 \u2115\n  sum-ext : {A B : Event} \u2192 (\u2200 r \u2192 A r \u2261 B r) \u2192 sum A \u2261 sum B\n\n  sumFin : (q : \u2115) (f : Fin q \u2192 \u211d) \u2192 \u211d\n\nRndVar = \u03a9 \u2192 \u211d\n\n_\u00b2 : \u211d \u2192 \u211d\nx \u00b2 = x \u00b7 x\n\n_\u00b2' : RndVar \u2192 RndVar\n(X \u00b2') r = (X r)\u00b2\n\npostulate\n  E[_] : RndVar \u2192 \u211d\n\n  lemma2 : \u2200 X \u2192 E[ X \u00b2' ] \u2265' E[ X ] \u00b2\n\n_\u2261\u27e8by-definition\u27e9_ : \u2200 x {z} \u2192 x \u2265' z \u2192 x \u2265' z\nx \u2261\u27e8by-definition\u27e9 y = x \u2261\u27e8 refl \u27e9 y\n\n-- _\/_ : [0,1] \u2192 (d : \u2115){{_ : d \u2265 1}} \u2192 [0,1]\n_\/_ : \u211d \u2192 (d : \u2115){{_ : d \u2265 1}} \u2192 \u211d\nx \/ y = x \u00b7 1\/ y\n\nPr : Event \u2192 \u211d -- [0,1]\nPr A = \u2115\u25b9\u211d (sum A) \/ #\u03a9\n\nPr-ext : \u2200 {A B : Event} \u2192 (\u2200 r \u2192 A r \u2261 B r) \u2192 Pr A \u2261 Pr B\nPr-ext f = ap (\u03bb x \u2192 \u2115\u25b9\u211d x \/ #\u03a9) (sum-ext f)\n\n-- Lemma 1, equation (3)\nlemma1-3 : Pr frk \u2265' Pr acc \u00b7 ((Pr acc \/ q) - (1\/ #h))\nlemma1-3 = Pr frk\n       \u2261\u27e8 {!Pr!} \u27e9\n         Pr I\u22651\u2227I=I'\u2227h\u2262h'\n       \u2265\u27e8 {!!} \u27e9\n         Pr I\u22651\u2227I=I' - Pr I\u22651\u2227h=h'\n       \u2261\u27e8 ap (\u03bb x \u2192 Pr I\u22651\u2227I=I' - x) {!!} \u27e9\n         Pr I\u22651\u2227I=I' - (Pr I\u22651 \/ #h)\n       \u2261\u27e8by-definition\u27e9\n         Pr I\u22651\u2227I=I' - (Pr acc \/ #h)\n       \u2261\u27e8 {!!} \u27e9\n         Pr acc \u00b7 ((Pr acc \/ q) - (1\/ #h))\n       \u220e\n\nlemma1-4 : Pr I\u22651\u2227I=I' \u2265' (Pr acc)\u00b2 \/ q\nlemma1-4\n  = Pr I\u22651\u2227I=I'\n  \u2261\u27e8 {!!} \u27e9\n    sumFin q (\u03bb i \u2192 Pr (I=I'=1+ i))\n  \u2261\u27e8 {!!} \u27e9 -- Conditional probabilities\n    sumFin q (\u03bb i \u2192 Pr (I=1+ i) \u00b7 Pr (I=I'=1+ i))\n  \u2261\u27e8 {!!} \u27e9\n    (Pr acc)\u00b2 \/ q\n  \u220e\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Vec\nopen import Data.Maybe\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two.Base hiding (_==_)\nopen import Data.Fin.NP as Fin hiding (_+_; _-_; _\u2264_)\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nmodule forking-lemma where\n\n_\u22651 : \u2200 {n} \u2192 Fin n \u2192 \ud835\udfda\nzero  \u22651 = 0\u2082\nsuc _ \u22651 = 1\u2082\n\ninfix 8 _\u203c_\n_\u203c_ : \u2200 {n a}{A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_\u203c_ = flip lookup\n\nreplace : \u2200 {A : Type} {q} (I : Fin q)\n            (hs hs' : Vec A q) \u2192 Vec A q\nreplace zero    hs             hs' = hs'\nreplace (suc I) (h \u2237 hs) (_ \u2237 hs') = h \u2237 replace I hs hs'\n\ntest-replace : replace (suc zero) (40 \u2237 41 \u2237 42 \u2237 []) (60 \u2237 61 \u2237 62 \u2237 []) \u2261 40 \u2237 61 \u2237 62 \u2237 []\ntest-replace = refl\n\n\u2261-prefix : \u2200 {A : Type} {q} (I : Fin (suc q))\n             (v\u2080 v\u2081 : Vec A q) \u2192 Type\n\u2261-prefix zero    _         _         = \ud835\udfd9\n\u2261-prefix (suc I) (x\u2080 \u2237 v\u2080) (x\u2081 \u2237 v\u2081) = (x\u2080 \u2261 x\u2081) \u00d7 \u2261-prefix I v\u2080 v\u2081\n\n\u2261-prefix (suc ()) [] []\n\npostulate\n  -- Pub    : Type\n  Res    : Type -- Side output\n  RndAdv : Type -- Coin\n  {-RndIG  : Type-}\n\n  -- IG : RndIG \u2192 Pub\n\n  q     : \u2115\n\n  instance\n    q\u22651   : q \u2265 1\n\n  #h     : \u2115\n  #h\u22652   : #h \u2265 2\n\nH : Type\nH = Fin #h\n\nAdv = {-(x : Pub)-}(hs : Vec H q)(\u03c1 : RndAdv) \u2192 Fin (suc q) \u00d7 Res\n\npostulate\n  A : Adv\n-- A x hs \u03c1\n-- A x hs = \u03c1 \u2190$ ...H ; A x hs \u03c1\n\n-- IO : Type \u2192 Type\n-- print : String \u2192 IO ()\n-- #r : \u2115\n-- #r\u22651 : #r \u2265 1\n-- Rnd = Fin #r\n-- random : IO Rnd\n-- run : (A : Input \u2192 Rnd \u2192 Res) \u2192 Input \u2192 IO Res\n-- run A i = do r \u2190 random\n--              return (A i r)\n\n-- Not used yet\nwell-def : Adv \u2192 Type\nwell-def A =\n  \u2200 {-x-} hs \u03c1 \u2192\n    case A {-x-} hs \u03c1 of \u03bb { (I , \u03c3) \u2192\n    \u2200 hs' \u2192 \u2261-prefix I hs hs' \u2192  A {-x-} hs' \u03c1 \u2261 (I , \u03c3) }\n\nrecord \u03a9 : Type where\n  field\n    -- rIG : RndIG\n    hs hs' : Vec H q\n    \u03c1      : RndAdv\n\nEvent = \u03a9 \u2192 \ud835\udfda\n\nmodule M (r : \u03a9) where\n  open \u03a9 r\n\n  F' : (I-1 : Fin q)(\u03c3 : Res) \u2192 Maybe (Res \u00d7 Res)\n  F' I-1 \u03c3 =\n    case I=I' \u2227 h=h' of \u03bb\n    { 1\u2082 \u2192 just (\u03c3 , \u03c3')\n    ; 0\u2082 \u2192 nothing\n    }\n    module F' where\n      res' = A (replace I-1 hs hs') \u03c1\n      I'   = fst res'\n      \u03c3'   = snd res'\n      h    = hs  \u203c I-1\n      h'   = hs' \u203c I-1\n      I    = suc I-1\n      I=I' = I == I'\n      h=h' = h == h'\n\n  F : Maybe (Res \u00d7 Res)\n  F =\n    case I of \u03bb\n    { zero      \u2192 nothing\n    ; (suc I-1) \u2192 F' I-1 \u03c3\n    }\n    module F where\n      res  = A hs \u03c1\n      I    = fst res\n      \u03c3    = snd res\n      I\u22651  = not (I == zero)\n\n-- Acceptance event for A\nacc : Event\nacc r = J \u22651\n  where open \u03a9 r\n     -- x = IG rIG\n        J = fst (A {-x-} hs \u03c1)\n\nfrk : Event\nfrk r = case M.F r of \u03bb { (just _) \u2192 1\u2082 ; nothing \u2192 0\u2082 }\n\ninstance\n    #h\u22651 : #h \u2265 1\n    #h\u22651 = \u2115\u2264.trans (s\u2264s z\u2264n) #h\u22652\n\ninfix 0 _\u2265'_\ninfixr 2 _\u2265\u27e8_\u27e9_ _\u2261\u27e8_\u27e9_\ninfix 2 _\u220e\n\npostulate\n  [0,1] : Type\n  Pr : Event \u2192 [0,1]\n  _-_ _\u00b7_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _\u2265'_ : [0,1] \u2192 [0,1] \u2192 Type\n  1\/_ : (d : \u2115){{_ : d \u2265 1}} \u2192 [0,1]\n\n  _\u2265\u27e8_\u27e9_ : \u2200 x {y} \u2192 x \u2265' y \u2192 \u2200 {z} \u2192 y \u2265' z \u2192 x \u2265' z\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y} \u2192 x \u2261  y \u2192 \u2200 {z} \u2192 y \u2265' z \u2192 x \u2265' z\n  _\u220e : \u2200 x \u2192 x \u2265' x\n\n_\/_ : [0,1] \u2192 (d : \u2115){{_ : d \u2265 1}} \u2192 [0,1]\nx \/ y = x \u00b7 1\/ y\n\n{-\nlemma3 : Pr frk \u2265' Pr acc \u00b7 ((Pr acc \/ q) - (1\/ #h))\nlemma3 = Pr frk\n       \u2261\u27e8 {!!} \u27e9\n         Pr {!\u03bb r \u2192 I=I' r \u2227 I\u22651 r!}\n       \u2265\u27e8 {!!} \u27e9\n         Pr acc \u00b7 ((Pr acc \/ q) - (1\/ #h))\n       \u220e\n  where\n    open M.F\n    -- open M.F' I-1 \u03c3\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"51004af03df620436d3816d6e0778bba60c327f4","subject":"Vec: +map-ext +sum-distrib\u02e1","message":"Vec: +map-ext +sum-distrib\u02e1\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : Set a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = refl\n    map-id (x \u2237 xs) = \u2261.cong (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = refl\n    map-\u2218 f g (x \u2237 xs) = \u2261.cong (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : Set a} {B : Set b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g [] = refl\n    map-ext f\u2257g (x \u2237 xs) rewrite f\u2257g x | map-ext f\u2257g xs = refl\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x [] = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) rewrite sum-\u2237\u02b3 x xs | \u2115\u00b0.+-assoc x\u2081 (sum xs) x = refl\n\nrot\u2081 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : Set a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : Set} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = sym (proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 [] = refl\nsum-rot\u2081 (x \u2237 xs) rewrite sum-\u2237\u02b3 x xs = \u2115\u00b0.+-comm x _\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x [] = refl\nmap-\u2237\u02b3 f x (x\u2081 \u2237 xs) rewrite map-\u2237\u02b3 f x xs = refl\n\nsum-map-rot\u2081 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = refl\nsum-map-rot\u2081 f (x \u2237 xs) rewrite \u2115\u00b0.+-comm (f x) (sum (map f xs))\n                              | \u2261.sym (sum-\u2237\u02b3 (f x) (map f xs))\n                              | \u2261.sym (map-\u2237\u02b3 f x xs)\n                              = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; module \u2115\u00b0)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : Set a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = refl\n    map-id (x \u2237 xs) = \u2261.cong (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = refl\n    map-\u2218 f g (x \u2237 xs) = \u2261.cong (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x [] = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) rewrite sum-\u2237\u02b3 x xs | \u2115\u00b0.+-assoc x\u2081 (sum xs) x = refl\n\nrot\u2081 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : Set a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 [] = refl\nsum-rot\u2081 (x \u2237 xs) rewrite sum-\u2237\u02b3 x xs = \u2115\u00b0.+-comm x _\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x [] = refl\nmap-\u2237\u02b3 f x (x\u2081 \u2237 xs) rewrite map-\u2237\u02b3 f x xs = refl\n\nsum-map-rot\u2081 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = refl\nsum-map-rot\u2081 f (x \u2237 xs) rewrite \u2115\u00b0.+-comm (f x) (sum (map f xs))\n                              | \u2261.sym (sum-\u2237\u02b3 (f x) (map f xs))\n                              | \u2261.sym (map-\u2237\u02b3 f x xs)\n                              = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ade0337152617482eadda17198a9884581d80579","subject":"Bits: minor","message":"Bits: minor\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {zero}  (_     `\u2237 _) _ = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} (f : Bits n \u2192 A) (g : Bij) \u2192 search {n} (f \u2218 eval g) \u2261 search {n} f\n        search-bij f `id     = refl\n        search-bij f `0\u21941    = search-0\u21941 f\n        search-bij f (g `\u204f h)\n          rewrite search-bij (f \u2218 eval h) g = search-bij f h\n        search-bij {zero}  f (_     `\u2237 _) = refl\n        search-bij {suc n} f (`id   `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 0b)\n                | search-bij (f \u2218 1\u2237_) (g 1b) = refl\n        search-bij {suc n} f (`not\u1d2e `\u2237 g)\n          rewrite search-bij (f \u2218 0\u2237_) (g 1b)\n                | search-bij (f \u2218 1\u2237_) (g 0b) = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} (f : Bits n \u2192 Bit) (g : Bij) \u2192 #\u27e8 f \u2218 eval g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-bij f = sum-bij (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f70c419c4ad3b2f93a3d4359d83f44a12d4aa748","subject":"all but a lemma about envfresh","message":"all but a lemma about envfresh\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\nopen import binders-disjoint-checks\n\nmodule lemmas-subst-ta where\n  mutual\n    binders-envfresh : \u2200{\u0394 \u0393 \u0393' y \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 y # \u0393 \u2192 unbound-in-\u03c3 y \u03c3  \u2192 envfresh y \u03c3\n    binders-envfresh (STAId x) apt UB\u03c3Id = {!!} -- EFId {!!}\n    binders-envfresh (STASubst sub x\u2081) apt (UB\u03c3Subst x\u2082 ub) = {!!} --EFSubst {!!} (binders-envfresh sub (apart-extend1 _ {!!} apt) ub) {!!}\n\n    binders-fresh : \u2200{ \u0394 \u0393 d1 d2 \u03c4 y} \u2192 \u0394 , \u0393 \u22a2 d2 :: \u03c4\n                                      \u2192 binders-unique d1 -- todo: ditch?\n                                      \u2192 binders-unique d2\n                                      \u2192 binders-disjoint d1 d2\n                                      \u2192 unbound-in y d2\n                                      \u2192 \u0393 y == None\n                                      \u2192 fresh y d2\n    binders-fresh TAConst bu1 BUHole bd UBConst apt = FConst\n    binders-fresh {y = y}  (TAVar {x = x} x\u2081) bu1 BUVar bd UBVar apt with natEQ y x\n    binders-fresh (TAVar x\u2082) bu1 BUVar bd UBVar apt | Inl refl = abort (somenotnone (! x\u2082 \u00b7 apt))\n    binders-fresh (TAVar x\u2082) bu1 BUVar bd UBVar apt | Inr x\u2081 = FVar x\u2081\n    binders-fresh {y = y} (TALam {x = x} x\u2081 wt) bu1 bu2 bd ub apt  with natEQ y x\n    binders-fresh (TALam x\u2082 wt) bu1 bu2 bd (UBLam2 x\u2081 ub) apt | Inl refl = abort (x\u2081 refl)\n    binders-fresh {\u0393 = \u0393} (TALam {x = x} x\u2082 wt) bu1 (BULam bu2 x\u2083) bd (UBLam2 x\u2084 ub) apt | Inr x\u2081 =  FLam x\u2081 (binders-fresh wt bu1 bu2 (lem-bd-lam bd) ub (apart-extend1 \u0393 x\u2084 apt))\n    binders-fresh (TAAp wt wt\u2081) bu1 (BUAp bu2 bu3 x) bd (UBAp ub ub\u2081) apt = FAp (binders-fresh wt BUHole bu2 BDConst ub apt)\n                                                                                (binders-fresh wt\u2081 bu2 bu3 x ub\u2081 apt)\n    binders-fresh (TAEHole x\u2081 x\u2082) bu1 bu2 bd (UBHole x\u2083) apt = FHole {!binders-envfresh x\u2082 apt x\u2083 !}\n    binders-fresh (TANEHole x\u2081 wt x\u2082) bu1 (BUNEHole bu2 x) bd (UBNEHole x\u2083 ub) apt = FNEHole {!binders-envfresh x\u2082 apt x\u2083!} (binders-fresh wt bu1 bu2 (lem-bd-hole bd) ub apt)\n    binders-fresh (TACast wt x\u2081) bu1 (BUCast bu2) bd (UBCast ub) apt = FCast (binders-fresh wt bu1 bu2 (lem-bd-cast bd) ub apt)\n    binders-fresh (TAFailedCast wt x x\u2081 x\u2082) bu1 (BUFailedCast bu2) bd (UBFailedCast ub) apt = FFailedCast (binders-fresh wt bu1 bu2 (lem-bd-failedcast bd) ub apt)\n\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  binders-disjoint d1 d2 \u2192\n                  binders-unique d1 \u2192 -- todo: do i need both of these?\n                  binders-unique d2 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt bd bu1 bu2  TAConst wt2 = TAConst\n  lem-subst {x = x} apt bd bu1 bu2  (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2 (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (BDLam bd bd') (BULam bu1' ub) bu2 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) bd bu1' bu2 (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1)\n                                         (weaken-ta (binders-fresh wt2 (BULam {\u03c4 = \u03c42} bu1' ub) bu2 (BDLam bd bd') bd' y#\u0393) wt2))\n  lem-subst apt (BDAp bd bd\u2081) (BUAp bu1 bu2 x\u2081) bu3 (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt bd bu1 bu3 wt1 wt3) (lem-subst apt bd\u2081 bu2 bu3 wt2 wt3)\n  lem-subst apt bd bu1 bu2 (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (BDNEHole x\u2081 bd) (BUNEHole bu1 x\u2082) bu2 (TANEHole x\u2083 wt1 x\u2084) wt2 = TANEHole x\u2083 (lem-subst apt bd bu1 bu2 wt1 wt2) (STASubst x\u2084 wt2)\n  lem-subst apt (BDCast bd) (BUCast bu1) bu2 (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt bd bu1 bu2 wt1 wt2) x\u2081\n  lem-subst apt (BDFailedCast bd) (BUFailedCast bu1) bu2 (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt bd bu1 bu2 wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  lem-bd-lam :  \u2200{ d1 x \u03c41 d} \u2192 binders-disjoint d1 (\u00b7\u03bb_[_]_ x  \u03c41 d) \u2192 binders-disjoint d1 d\n  lem-bd-lam = {!!}\n\n  binders-fresh : \u2200{ \u0394 \u0393 d1 d2 \u03c4 y} \u2192 \u0394 , \u0393 \u22a2 d2 :: \u03c4\n                                    \u2192 binders-unique d1 -- todo: ditch\n                                    \u2192 binders-unique d2\n                                    \u2192 binders-disjoint d1 d2\n                                    \u2192 unbound-in y d2\n                                    \u2192 \u0393 y == None\n                                    \u2192 fresh y d2\n  binders-fresh TAConst bu1 BUHole bd UBConst apt = FConst\n  binders-fresh {y = y}  (TAVar {x = x} x\u2081) bu1 BUVar bd UBVar apt with natEQ y x\n  binders-fresh (TAVar x\u2082) bu1 BUVar bd UBVar apt | Inl refl = abort (somenotnone (! x\u2082 \u00b7 apt))\n  binders-fresh (TAVar x\u2082) bu1 BUVar bd UBVar apt | Inr x\u2081 = FVar x\u2081\n  binders-fresh {y = y} (TALam {x = x} x\u2081 wt) bu1 bu2 bd ub apt  with natEQ y x\n  binders-fresh (TALam x\u2082 wt) bu1 bu2 bd (UBLam2 x\u2081 ub) apt | Inl refl = abort (x\u2081 refl)\n  binders-fresh {\u0393 = \u0393} (TALam {x = x} x\u2082 wt) bu1 (BULam bu2 x\u2083) bd (UBLam2 x\u2084 ub) apt | Inr x\u2081 =  FLam x\u2081 (binders-fresh wt bu1 bu2 (lem-bd-lam bd) ub (apart-extend1 \u0393 x\u2084 apt))\n  binders-fresh (TAAp wt wt\u2081) bu1 bu2 bd ub apt = {!!}\n  binders-fresh (TAEHole x x\u2081) bu1 bu2 bd ub apt = {!!}\n  binders-fresh (TANEHole x wt x\u2081) bu1 bu2 bd ub apt = {!!}\n  binders-fresh (TACast wt x) bu1 bu2 bd ub apt = {!!}\n  binders-fresh (TAFailedCast wt x x\u2081 x\u2082) bu1 bu2 bd ub apt = {!!}\n\n  -- todo\n  lem : \u2200{y \u03c4 d d2} \u2192 binders-disjoint (\u00b7\u03bb_[_]_ y \u03c4 d) d2 \u2192 unbound-in y d2\n  lem (BDLam bd x\u2081) = x\u2081\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  binders-disjoint d1 d2 \u2192\n                  binders-unique d1 \u2192\n                  binders-unique d2 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt bd bu1 bu2  TAConst wt2 = TAConst\n  lem-subst {x = x} apt bd bu1 bu2  (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2 (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 bd bu1 bu2 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) {!!} {!!} {!!} (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1)\n                                         (weaken-ta (binders-fresh wt2 bu1 bu2 bd (lem bd) y#\u0393) wt2))\n  lem-subst apt bd bu1 bu2 (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt {!!} {!!} {!!} wt1 wt3) (lem-subst apt {!!} {!!} {!!} wt2 wt3)\n  lem-subst apt bd bu1 bu2 (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt bd bu1 bu2 (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt {!!} {!!} {!!} wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt bd bu1 bu2 (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt {!!} {!!} {!!} wt1 wt2) x\u2081\n  lem-subst apt bd bu1 bu2 (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt {!!} {!!} {!!} wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1c2efbbaff233ebf078e6cde7ccd4486629a1a44","subject":"Nat: some permutations","message":"Nat: some permutations\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"-- NOTE with-K so far\n-- TODO {-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_;_\u00b2)\nimport Data.Two.Equality as \ud835\udfda==\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; coe) renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z = ! \u2115\u00b0.+-assoc x y z \u2219 ap (flip _+_ z) (\u2115\u00b0.+-comm x y) \u2219 \u2115\u00b0.+-assoc y x z\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 ap (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! proj\u2082 \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- NOTE with-K so far\n-- TODO {-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nimport Data.Two.Equality as \ud835\udfda==\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap) renaming (refl to idp)\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z = ! \u2115\u00b0.+-assoc x y z \u2219 ap (flip _+_ z) (\u2115\u00b0.+-comm x y) \u2219 \u2115\u00b0.+-assoc y x z\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 ap (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! proj\u2082 \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"20096c73e1203a5dcd9f0ce97c3f22ecc30c0414","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/Nest\/NestBC-SL.agda","new_file":"notes\/FOT\/FOTC\/Program\/Nest\/NestBC-SL.agda","new_contents":"------------------------------------------------------------------------------\n-- Example of a nested recursive function using the Bove-Capretta\n-- method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: Bove, A. and Capretta, V. (2001). Nested General Recursion\n-- and Partiality in Type Theory. In: Theorem Proving in Higher Order\n-- Logics (TPHOLs 2001). Ed. by Boulton, R. J. and Jackson,\n-- P. B. Vol. 2152. LNCS. Springer, pp. 121\u2013135.\n\nmodule FOT.FOTC.Program.Nest.NestBC-SL where\n\nopen import Data.Nat renaming ( suc to succ )\nopen import Data.Nat.Properties\n\nopen import Induction\nopen import Induction.Nat\n\nopen import Relation.Binary\n\nmodule NDTO = DecTotalOrder decTotalOrder\n\n------------------------------------------------------------------------------\n-- The original non-terminating function.\n\n{-# NO_TERMINATION_CHECK #-}\nnestI : \u2115 \u2192 \u2115\nnestI 0        = 0\nnestI (succ n) = nestI (nestI n)\n\n\u2264\u2032-trans : Transitive _\u2264\u2032_\n\u2264\u2032-trans i\u2264\u2032j j\u2264\u2032k = \u2264\u21d2\u2264\u2032 (NDTO.trans (\u2264\u2032\u21d2\u2264 i\u2264\u2032j) (\u2264\u2032\u21d2\u2264 j\u2264\u2032k))\n\nmutual\n  -- The domain predicate of the nest function.\n  data NestDom : \u2115 \u2192 Set where\n    nestDom0 : NestDom 0\n    nestDomS : \u2200 {n} \u2192 (h\u2081 : NestDom n) \u2192\n               (h\u2082 : NestDom (nestD n h\u2081)) \u2192\n               NestDom (succ n)\n\n  -- The nest function by structural recursion on the domain predicate.\n  nestD : \u2200 n \u2192 NestDom n \u2192 \u2115\n  nestD .0        nestDom0             = 0\n  nestD .(succ n) (nestDomS {n} h\u2081 h\u2082) = nestD (nestD n h\u2081) h\u2082\n\nnestD-\u2264\u2032 : \u2200 n \u2192 (h : NestDom n) \u2192 nestD n h \u2264\u2032 n\nnestD-\u2264\u2032 .0        nestDom0             = \u2264\u2032-refl\nnestD-\u2264\u2032 .(succ n) (nestDomS {n} h\u2081 h\u2082) =\n  \u2264\u2032-trans (\u2264\u2032-trans (nestD-\u2264\u2032 (nestD n h\u2081) h\u2082) (nestD-\u2264\u2032 n h\u2081))\n           (\u2264\u2032-step \u2264\u2032-refl)\n\n-- The nest function is total.\nallNestDom : \u2200 n \u2192 NestDom n\nallNestDom = build <-rec-builder P ih\n  where\n  P : \u2115 \u2192 Set\n  P = NestDom\n\n  ih : \u2200 y \u2192 <-Rec P y \u2192 P y\n  ih zero     rec = nestDom0\n  ih (succ y) rec = nestDomS nd-y (rec (nestD y nd-y) (s\u2264\u2032s (nestD-\u2264\u2032 y nd-y)))\n    where\n    helper : \u2200 x \u2192 x <\u2032 y \u2192 P x\n    helper x Sx\u2264\u2032y = rec x (\u2264\u2032-step Sx\u2264\u2032y)\n\n    nd-y : NestDom y\n    nd-y = ih y helper\n\n-- The nest function.\nnest : \u2115 \u2192 \u2115\nnest n = nestD n (allNestDom n)\n","old_contents":"------------------------------------------------------------------------------\n-- Example of a nested recursive function using the Bove-Capretta\n-- method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. vol 2152 LNCS. 2001.\n\nmodule FOT.FOTC.Program.Nest.NestBC-SL where\n\nopen import Data.Nat renaming ( suc to succ )\nopen import Data.Nat.Properties\n\nopen import Induction\nopen import Induction.Nat\n\nopen import Relation.Binary\n\nmodule NDTO = DecTotalOrder decTotalOrder\n\n------------------------------------------------------------------------------\n\n-- The original non-terminating function.\n\n{-# NO_TERMINATION_CHECK #-}\nnestI : \u2115 \u2192 \u2115\nnestI 0        = 0\nnestI (succ n) = nestI (nestI n)\n\n\u2264\u2032-trans : Transitive _\u2264\u2032_\n\u2264\u2032-trans i\u2264\u2032j j\u2264\u2032k = \u2264\u21d2\u2264\u2032 (NDTO.trans (\u2264\u2032\u21d2\u2264 i\u2264\u2032j) (\u2264\u2032\u21d2\u2264 j\u2264\u2032k))\n\nmutual\n  -- The domain predicate of the nest function.\n  data NestDom : \u2115 \u2192 Set where\n    nestDom0 : NestDom 0\n    nestDomS : \u2200 {n} \u2192 (h\u2081 : NestDom n) \u2192\n               (h\u2082 : NestDom (nestD n h\u2081)) \u2192\n               NestDom (succ n)\n\n  -- The nest function by structural recursion on the domain predicate.\n  nestD : \u2200 n \u2192 NestDom n \u2192 \u2115\n  nestD .0        nestDom0             = 0\n  nestD .(succ n) (nestDomS {n} h\u2081 h\u2082) = nestD (nestD n h\u2081) h\u2082\n\nnestD-\u2264\u2032 : \u2200 n \u2192 (h : NestDom n) \u2192 nestD n h \u2264\u2032 n\nnestD-\u2264\u2032 .0        nestDom0             = \u2264\u2032-refl\nnestD-\u2264\u2032 .(succ n) (nestDomS {n} h\u2081 h\u2082) =\n  \u2264\u2032-trans (\u2264\u2032-trans (nestD-\u2264\u2032 (nestD n h\u2081) h\u2082) (nestD-\u2264\u2032 n h\u2081))\n           (\u2264\u2032-step \u2264\u2032-refl)\n\n-- The nest function is total.\nallNestDom : \u2200 n \u2192 NestDom n\nallNestDom = build <-rec-builder P ih\n  where\n  P : \u2115 \u2192 Set\n  P = NestDom\n\n  ih : \u2200 y \u2192 <-Rec P y \u2192 P y\n  ih zero     rec = nestDom0\n  ih (succ y) rec = nestDomS nd-y (rec (nestD y nd-y) (s\u2264\u2032s (nestD-\u2264\u2032 y nd-y)))\n    where\n    helper : \u2200 x \u2192 x <\u2032 y \u2192 P x\n    helper x Sx\u2264\u2032y = rec x (\u2264\u2032-step Sx\u2264\u2032y)\n\n    nd-y : NestDom y\n    nd-y = ih y helper\n\n-- The nest function.\nnest : \u2115 \u2192 \u2115\nnest n = nestD n (allNestDom n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ce69f696ace671125437eeb170328cc742ca9336","subject":"Bits: Add _<=_","message":"Bits: Add _<=_\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : \u2200 {n} \u2192 Bij (1 + n)\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if0 f = `if `id f\n\n    `if1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : \u2200 {n} \u2192 Bij (2 + n)\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : \u2200 {n} \u2192 Bij (1 + n)\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if0 f = `if `id f\n\n    `if1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : \u2200 {n} \u2192 Bij (2 + n)\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n <= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) <= (to\u2115 xs) | \u2261.inspect (_<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n <= x | \u2261.inspect (_<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"00556ffcb26ebd4a171f0d5e578e5d0601023af4","subject":"Bits: + sum-\u2257\u2082","message":"Bits: + sum-\u2257\u2082\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u00b7 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule SimpleSearch {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u00b7_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u00b7\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module Interchange (\u00b7-interchange : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (x \u00b7 z) \u00b7 (y \u00b7 t)) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u00b7 f\u2081 x) \u2261 search f\u2080 \u00b7 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u00b7-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module Interchange; search-const\u03b5\u2261\u03b5)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open Interchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-+ to sum-+;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u00b7 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule SimpleSearch {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u00b7_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u00b7\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module Interchange (\u00b7-interchange : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (x \u00b7 z) \u00b7 (y \u00b7 t)) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u00b7 f\u2081 x) \u2261 search f\u2080 \u00b7 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u00b7-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module Interchange; search-const\u03b5\u2261\u03b5)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open Interchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-+ to sum-+;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"37bbe9fa0a623adb0f646c246f1d9a408a51b023","subject":"Changed rev-ListTree by reverse-ListTree.","message":"Changed rev-ListTree by reverse-ListTree.\n\nIgnore-this: 9fa338cf5a7edf48fcacfde2c78cde76\n\ndarcs-hash:20110220160725-3bd4e-56827edae047bdc954128142183b060ef4dcd4d8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/PropertiesI.agda","new_file":"Draft\/Mirror\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.PropertiesI where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesI\nopen import Draft.Mirror.ListTree.Closures\nopen import Draft.Mirror.Tree.Closures\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesI using ( reverse-[x]\u2261[x] )\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nmutual\n  mirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\n  mirror\u00b2 (treeT d nilLT) =\n    begin\n      mirror \u00b7 (mirror \u00b7 (node d []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 (node d [])) \u2261 mirror \u00b7 x )\n                 (mirror-eq d [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror [])) \u2261\n                        mirror \u00b7 node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse []) \u2261 mirror \u00b7 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d []\n        \u2261\u27e8 mirror-eq d [] \u27e9\n      node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror [])) \u2261\n                        node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse []) \u2261 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      node d []\n    \u220e\n\n  mirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n    begin\n      mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 mirror \u00b7 x)\n                 (mirror-eq d (t \u2237 ts))\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror (t \u2237 ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror (t \u2237 ts))) \u2261\n                        x)\n                 (mirror-eq d (reverse (map mirror (t \u2237 ts))))\n                 refl\n        \u27e9\n      node d (reverse (map mirror (reverse (map mirror (t \u2237 ts)))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (reverse (map mirror (t \u2237 ts))))) \u2261\n                      node d x)\n               (helper (consLT Tt LTts))\n               refl\n      \u27e9\n      node d (t \u2237 ts)\n    \u220e\n\n  helper : \u2200 {ts} \u2192 ListTree ts \u2192\n           reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  helper nilLT =\n    begin\n      reverse (map mirror (reverse (map mirror [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror []))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse []))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse [])) \u2261\n                        reverse (map mirror x))\n                 (rev-[] [])\n                 refl\n        \u27e9\n      reverse (map mirror [])\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror []) \u2261\n                        reverse x)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse []\n        \u2261\u27e8 rev-[] [] \u27e9\n      []\n    \u220e\n\n  helper (consLT {t} {ts} Tt LTts) =\n    begin\n      reverse (map mirror (reverse (map mirror (t \u2237 ts))))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror (t \u2237 ts)))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-\u2237 mirror t ts)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts))) \u2261\n                        reverse (map mirror x))\n                 (reverse-\u2237 (mirror-Tree Tt) (map-ListTree mirror mirror-Tree LTts))\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror \u00b7 t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror\n                   mirror-Tree\n                   (reverse-ListTree (map-ListTree mirror mirror-Tree LTts))\n                   (consLT (mirror-Tree Tt) nilLT))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror \u00b7 t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++-commute\n                   (map-ListTree mirror mirror-Tree\n                                 (reverse-ListTree\n                                 (map-ListTree mirror mirror-Tree LTts)))\n                   (map-ListTree mirror mirror-Tree\n                                 (consLT (mirror-Tree Tt) nilLT)))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (mirror \u00b7 t \u2237 [])) ++ n\u2081 \u2261\n                        reverse x ++ n\u2081)\n                 (map-\u2237 mirror (mirror \u00b7 t) [])\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081 \u2261\n                        reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 x) ++ n\u2081)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081 \u2261\n                        x ++ n\u2081)\n                 (reverse-[x]\u2261[x] (mirror \u00b7 (mirror \u00b7 t)))\n                 refl\n        \u27e9\n      (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 ++-\u2237 (mirror \u00b7 (mirror \u00b7 t)) [] n\u2081 \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192  mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081 \u2261\n                         mirror \u00b7 (mirror \u00b7 t) \u2237 x)\n                 (++-leftIdentity LTn\u2081)\n                 refl\n        \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror \u00b7 (mirror \u00b7 t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 Tt)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (helper LTts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n      where\n        n\u2081 : D\n        n\u2081 = reverse (map mirror (reverse (map mirror ts)))\n\n        LTn\u2081 : ListTree n\u2081\n        LTn\u2081 = rev-ListTree\n                 (map-ListTree mirror mirror-Tree\n                               (reverse-ListTree (map-ListTree mirror mirror-Tree LTts)))\n                 nilLT\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.PropertiesI where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesI\nopen import Draft.Mirror.ListTree.Closures\nopen import Draft.Mirror.Tree.Closures\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesI using ( reverse-[x]\u2261[x] )\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nmutual\n  mirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\n  mirror\u00b2 (treeT d nilLT) =\n    begin\n      mirror \u00b7 (mirror \u00b7 (node d []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 (node d [])) \u2261 mirror \u00b7 x )\n                 (mirror-eq d [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror [])) \u2261\n                        mirror \u00b7 node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse []) \u2261 mirror \u00b7 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d []\n        \u2261\u27e8 mirror-eq d [] \u27e9\n      node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror [])) \u2261\n                        node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse []) \u2261 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      node d []\n    \u220e\n\n  mirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n    begin\n      mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 mirror \u00b7 x)\n                 (mirror-eq d (t \u2237 ts))\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror (t \u2237 ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror (t \u2237 ts))) \u2261\n                        x)\n                 (mirror-eq d (reverse (map mirror (t \u2237 ts))))\n                 refl\n        \u27e9\n      node d (reverse (map mirror (reverse (map mirror (t \u2237 ts)))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (reverse (map mirror (t \u2237 ts))))) \u2261\n                      node d x)\n               (helper (consLT Tt LTts))\n               refl\n      \u27e9\n      node d (t \u2237 ts)\n    \u220e\n\n  helper : \u2200 {ts} \u2192 ListTree ts \u2192\n           reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  helper nilLT =\n    begin\n      reverse (map mirror (reverse (map mirror [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror []))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse []))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse [])) \u2261\n                        reverse (map mirror x))\n                 (rev-[] [])\n                 refl\n        \u27e9\n      reverse (map mirror [])\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror []) \u2261\n                        reverse x)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse []\n        \u2261\u27e8 rev-[] [] \u27e9\n      []\n    \u220e\n\n  helper (consLT {t} {ts} Tt LTts) =\n    begin\n      reverse (map mirror (reverse (map mirror (t \u2237 ts))))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror (t \u2237 ts)))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-\u2237 mirror t ts)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts))) \u2261\n                        reverse (map mirror x))\n                 (reverse-\u2237 (mirror-Tree Tt) (map-ListTree mirror mirror-Tree LTts))\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror \u00b7 t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror\n                   mirror-Tree\n                   (rev-ListTree (map-ListTree mirror mirror-Tree LTts) nilLT)\n                   (consLT (mirror-Tree Tt) nilLT))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror \u00b7 t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++-commute (map-ListTree mirror\n                                           mirror-Tree\n                                           (rev-ListTree (map-ListTree mirror\n                                                                       mirror-Tree\n                                                                       LTts)\n                                                         nilLT))\n                             (map-ListTree mirror\n                                           mirror-Tree\n                                           (consLT (mirror-Tree Tt) nilLT)))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (mirror \u00b7 t \u2237 [])) ++ n\u2081 \u2261\n                        reverse x ++ n\u2081)\n                 (map-\u2237 mirror (mirror \u00b7 t) [])\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081 \u2261\n                        reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 x) ++ n\u2081)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081 \u2261\n                        x ++ n\u2081)\n                 (reverse-[x]\u2261[x] (mirror \u00b7 (mirror \u00b7 t)))\n                 refl\n        \u27e9\n      (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 ++-\u2237 (mirror \u00b7 (mirror \u00b7 t)) [] n\u2081 \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192  mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081 \u2261\n                         mirror \u00b7 (mirror \u00b7 t) \u2237 x)\n                 (++-leftIdentity LTn\u2081)\n                 refl\n        \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror \u00b7 (mirror \u00b7 t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 Tt)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (helper LTts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n      where\n        n\u2081 : D\n        n\u2081 = reverse (map mirror (reverse (map mirror ts)))\n\n        LTn\u2081 : ListTree n\u2081\n        LTn\u2081 = rev-ListTree (map-ListTree mirror\n                                          mirror-Tree\n                                          (rev-ListTree (map-ListTree mirror\n                                                                      mirror-Tree\n                                                                      LTts)\n                                                        nilLT))\n                            nilLT\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57951c517e0aa766a934a88677ff29d4808291d9","subject":"Updated the alea stub","message":"Updated the alea stub\n","repos":"crypto-agda\/crypto-agda","old_file":"alea\/cpo.agda","new_file":"alea\/cpo.agda","new_contents":"{-# OPTIONS --copatterns #-}\nmodule alea.cpo where\n\nimport Data.Nat.NP as Nat\n\nimport Relation.Binary.PropositionalEquality as \u2261\n\nrecord Order (A : Set) : Set\u2081 where\n  constructor mk\n  infix 4 _\u2261_ _\u2264_\n  field\n    _\u2264_ : A \u2192 A \u2192 Set\n    _\u2261_ : A \u2192 A \u2192 Set\n    reflexive : \u2200 {x} \u2192 x \u2264 x\n    antisym : \u2200 {x y} \u2192 x \u2264 y \u2192 y \u2264 x \u2192 x \u2261 y\n    transitive : \u2200 {x y z} \u2192 x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z\n\n  \u2261-reflexive : \u2200 {x} \u2192 x \u2261 x\n  \u2261-reflexive = antisym reflexive reflexive\n    \n\u2115 : Order Nat.\u2115\n\u2115 = mk Nat._\u2264_ \u2261._\u2261_ Nat.\u2115\u2264.refl Nat.\u2115\u2264.antisym Nat.\u2115\u2264.trans\n\nfunOrder : \u2200 {A B : Set} \u2192 Order B \u2192 Order (A \u2192 B)\nfunOrder or = mk (\u03bb f g  \u2192 \u2200 x \u2192 f x \u2264 g x) (\u03bb f g \u2192 \u2200 x \u2192 f x \u2261 g x)\n              (\u03bb x\u2081 \u2192 reflexive)\n              (\u03bb f g x \u2192 antisym (f x) (g x)) \n              (\u03bb f g x  \u2192 transitive (f x) (g x))\n  where open Order or\n\n_o\u2192_ : \u2200 A {B} \u2192 Order B \u2192 Order (A \u2192 B)\nA o\u2192 ob = funOrder {A} ob\n  \nrecord _\u2192m_ {A B}(oa : Order A)(ob : Order B) : Set where\n  coinductive\n  constructor mk\n  field\n    _$_ : A \u2192 B\n    .mon : \u2200 {x y} \u2192 Order._\u2264_ oa x y \u2192 Order._\u2264_ ob (_$_ x) (_$_ y)\nopen _\u2192m_ using (_$_ ; mon)\n\n_o\u2192m_ : \u2200 {A B}(oa : Order A)(ob : Order B) \u2192 Order (oa \u2192m ob)\n_o\u2192m_ {A} oa ob = mk (\u03bb f g \u2192 _$_ f \u2264 _$_ g)\n                     (\u03bb f g \u2192 _$_ f \u2261 _$_ g)\n                     reflexive antisym transitive\n  where open Order (A o\u2192 ob)\n\nswapm : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\nswapm f = mk (\u03bb b \u2192 mk (\u03bb a \u2192 (f $ a) $ b) (\u03bb r \u2192 mon f r b )) (\u03bb r x \u2192 mon (f $ x) r)\n\nswapm' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n_$_ (_$_ (swapm' f) b) a = (f $ a) $ b\nmon (_$_ (swapm' f) b) r = mon f r b\nmon (swapm' f) r x = mon (f $ x) r\n\n{-\nswapm'' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n(swapm'' f $ b)    $ a = (f $ a) $ b\nmon (swapm' f $ b) r   = mon f r b\nmon (swapm' f)     r x = mon (f $ x) r\n-}\n\n_\u2218m_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\nf \u2218m g = mk (\u03bb x \u2192 f $ (g $ x)) (\u03bb x \u2192 mon f (mon g x))\n\n_\u2218m'_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\n_$_ (f \u2218m' g) x = f $ (g $ x)\nmon (f \u2218m' g) x = mon f (mon g x)\n\n_$m_ : \u2200 {A B C}{oa : Order A}{oc : Order C}\n       \u2192 oa \u2192m (B o\u2192 oc) \u2192 B \u2192 oa \u2192m oc\n_$m_ {oa = oa}{oc} f b = mk (\u03bb a \u2192 (f $ a) b) (\u03bb x\u2264y \u2192 mon f x\u2264y b )\n\n_$m'_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$_ (f $m' b) a   = (f $ a) b\nmon (f $m' b) x\u2264y = mon f x\u2264y b\n\nUB : \u2200 {A} \u2192 Order A \u2192 A \u2192 Set\nUB or a = \u2200 x \u2192 x \u2264 a\n  where open Order or\n\nrecord cpo {D} (o : Order D) : Set where\n  constructor mk\n  open Order o\n\n  field\n    d0 : D\n    lub : (f : \u2115 \u2192m o) \u2192 D\n\n  -- proofs\n  field\n    .Dbot : \u2200 x \u2192 d0 \u2264 x\n    .lubUB : \u2200 (f : \u2115 \u2192m o) n \u2192 (f $ n) \u2264 lub f\n    .lubLUB : \u2200 (f : \u2115 \u2192m o) x \u2192 (\u2200 n \u2192 (f $ n) \u2264 x) \u2192 lub f \u2264 x\n\n  .lub-ext : \u2200 {f g} \u2192 Order._\u2264_ (\u2115 o\u2192m o) f g \u2192 lub f \u2264 lub g\n  lub-ext {f} {g} f\u2264g = lubLUB f (lub g) (\u03bb n \u2192 transitive (f\u2264g n) (lubUB g n))\n\n  lubm : (\u2115 o\u2192m o) \u2192m o\n  lubm = mk lub (\u03bb {a}{b} x\u2081 \u2192 lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081)\n\n  lubm' : (\u2115 o\u2192m o) \u2192m o\n  _$_ lubm' = lub\n  mon lubm' {a}{b} x\u2081 = lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081\n\nmodule _ where\n  .lub-swap : \u2200 {D}{o : Order D}(c : cpo o)\n             \u2192 let open cpo c\n                   open Order o\n               in \u2200 f \u2192 lub (lubm \u2218m f) \u2264 lub (lubm \u2218m swapm f)\n  lub-swap {D}{o} c f = lubLUB (lubm \u2218m f) (lub (lubm \u2218m swapm f))\n                     (\u03bb n \u2192 lubLUB (f $ n) (lub (lubm \u2218m swapm f))\n                     (\u03bb n\u2081 \u2192 transitive (lubUB (swapm f $ n\u2081) n) (lubUB (lubm \u2218m swapm f) n\u2081)))\n    where\n      open cpo c\n      open Order o\n\n  funcpo : \u2200 {A D}{od : Order D} \u2192 cpo od \u2192 cpo (A o\u2192 od)\n  funcpo {A}{D}{od}cpod = mk fbot flub fDbot flubUB flubLUB\n    where\n      module D = cpo cpod\n      open Nat\n      open \u2261 using (_\u2261_)\n      fbot : A \u2192 D\n      fbot _ = D.d0\n      flub : _ \u2192 _\n      flub f = \u03bb x \u2192 D.lub (f $m x)\n      .fDbot : (_ : _) \u2192 _\n      fDbot f = \u03bb x \u2192 D.Dbot (f x)\n      .flubUB : (_ : _) \u2192 _\n      flubUB f = \u03bb n x \u2192 D.lubUB (f $m x) n\n      .flubLUB : (_ : _) \u2192 _\n      flubLUB f = \u03bb x x\u2081 x\u2082 \u2192 D.lubLUB (f $m x\u2082) (x x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082)\n\n  _c\u2192m_ : \u2200 {A D}(oa : Order A){od : Order D} \u2192 cpo od \u2192 cpo (oa o\u2192m od)\n  _c\u2192m_ {A} oa {od} cpod = mk (mk (\u03bb x \u2192 d0) (\u03bb x\u2081 \u2192 Order.reflexive od))\n                          (\u03bb f \u2192 mk (cpo.lub (funcpo {A} cpod)\n                                       (mk (\u03bb n a \u2192 (f $ n) $ a) (mon f)))\n                                    (\u03bb {x}{y}x\u2264y \u2192 lubLUB _ _ (\u03bb n \u2192 transitive (mon (f $ n) x\u2264y) (lubUB (mk (\u03bb v \u2192 (f $ v) $ y) (\u03bb r \u2192 mon f r y)) n))))\n                          (\u03bb x x\u2081 \u2192 Dbot (x $ x\u2081))\n                          (\u03bb f n x \u2192 lubUB (mk (\u03bb m \u2192 (f $ m) $ x) (\u03bb x\u2082 \u2192 mon f x\u2082 x)) n)\n                          (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (mk (\u03bb n \u2192 _$_ (f $ n)) (mon f) $m x\u2082) (x $ x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n    where\n      open cpo cpod\n      open Order od renaming (_\u2264_ to _\u2264\u1d48_)\n      open Order oa using () renaming (_\u2264_ to _\u2264\u1d43_)\n\nrecord continous {A B}{oa : Order A}{ob : Order B}\n                 (c1 : cpo oa)(c2 : cpo ob)(f : oa \u2192m ob) : Set where\n  constructor mk\n  open cpo c1 renaming (lub to luba ; lubUB to lubUBa ; lubLUB to lubLUBa)\n  open cpo c2\n  open Order ob\n  field\n    .contIntro : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2264 lub (f \u2218m h)\n\n  .contIntro' : \u2200 (h : \u2115 \u2192m oa) \u2192 lub (f \u2218m h) \u2264 f $ luba h\n  contIntro' h = lubLUB (f \u2218m h) (f $ luba h) (\u03bb n \u2192 mon f (lubUBa _ n))\n\n  .contEq : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2261 lub (f \u2218m h)\n  contEq h = antisym (contIntro h) (contIntro' h)\n  \nopen continous\n\nmodule Distr\n  (Ur  : Set) -- the set [0,1]\n  (1\/_+1 : Nat.\u2115 \u2192 Ur)\n  (_+_ : Ur \u2192 Ur \u2192 Ur)\n  (_\u00d7_ : Ur \u2192 Ur \u2192 Ur)\n  (oUr : Order Ur)\n  (\u2264-cong-+ : let open Order oUr in \u2200 {x y z w} \u2192 x \u2264 z \u2192 y \u2264 w \u2192 x + y \u2264 z + w)\n  (\u2264-cong-\u00d7 : let open Order oUr in \u2200 {x y z w} \u2192 x \u2264 z \u2192 y \u2264 w \u2192 x \u00d7 y \u2264 z \u00d7 w)\n  (U   : cpo oUr) where\n\n  record distr A : Set where\n    constructor mk\n    field\n      \u03bc : (A o\u2192 oUr) \u2192m oUr\n      .muContinous : \u2200 (h : \u2115 \u2192m (A o\u2192 oUr)) \u2192 Order._\u2264_ oUr (\u03bc $ cpo.lub (funcpo U) h) (cpo.lub U (\u03bc \u2218m h)) -- continous (funcpo U) U \u03bc\n\n  open distr\n\n  module _ A where\n    module M = cpo ((A o\u2192 oUr) c\u2192m U)\n\n    distrOrder : Order (distr A)\n    distrOrder = mk (\u03bb f g \u2192 \u03bc f \u2264 \u03bc g) (\u03bb f g \u2192 \u03bc f \u2261 \u03bc g)\n      (\u03bb {x} \u2192 reflexive {\u03bc x}) (\u03bb {x} {y} \u2192 antisym {\u03bc x} {\u03bc y}) (\u03bb {x} {y} {z} \u2192 transitive {\u03bc x} {\u03bc y} {\u03bc z})\n      where open Order ((A o\u2192 oUr) o\u2192m oUr)\n\n    \u03bcm : distrOrder \u2192m ((A o\u2192 oUr) o\u2192m oUr)\n    \u03bcm = mk \u03bc (\u03bb x\u2081 x\u2082 \u2192 x\u2081 x\u2082)\n\n    distrcpo : cpo distrOrder\n    distrcpo = mk (mk M.d0 (\u03bb h \u2192 cpo.Dbot U (cpo.lub U (M.d0 \u2218m h))))\n\n                    (\u03bb f \u2192 mk (M.lub (\u03bcm \u2218m f)) (\u03bb h \u2192 transitive\n                                                       (cpo.lub-ext U (\u03bb a \u2192 muContinous (f $ a) h))\n                                                       (lub-swap U (mk (\u03bb z \u2192 mk (\u03bb z\u2081 \u2192 \u03bc (f $ z) $ (h $ z\u2081))\n                                                                                 (\u03bb x \u2192 mon (\u03bc (f $ z)) (mon h x)))\n                                                                       (\u03bb {x}{y} x\u2081 x\u2082 \u2192\n                                                                              mon f x\u2081 (\u03bb x\u2083 \u2192 (h $ x\u2082) x\u2083))))))\n                    (\u03bb x x\u2081 \u2192 cpo.Dbot U (\u03bc x $ x\u2081))\n                    (\u03bb f n x \u2192 cpo.lubUB U _ _ )\n                    (\u03bb f x x\u2081 x\u2082 \u2192 cpo.lubLUB U _ _ (\u03bb n \u2192 x\u2081 n x\u2082))\n       where open Order oUr\n\n  module FIXPOINTS {d} {D : Order d} {c : cpo D} where\n    open cpo c\n    open Order D\n\n    iter : (f : D \u2192m D) \u2192 Nat.\u2115 \u2192 d\n    iter f Nat.zero = d0\n    iter f (Nat.suc n) = f $ iter f n\n\n    iter-mon : (f : D \u2192m D) \u2192 \u2115 \u2192m D\n    iter-mon f = mk (iter f) mono\n      where\n        .mono : \u2200 {x y} \u2192 x Nat.\u2264 y \u2192 iter f x \u2264 iter f y\n        mono Nat.z\u2264n = Dbot _\n        mono (Nat.s\u2264s prf) = mon f (mono prf)\n\n    fix : (f : D \u2192m D) \u2192 d\n    fix f = lub (iter-mon f)\n\n    .fix-le : (F : D \u2192m D) \u2192 fix F \u2264 F $ (fix F)\n    fix-le F = lubLUB _ _ prf\n      where\n        .prf : (n : Nat.\u2115) \u2192 iter-mon F $ n \u2264 F $ fix F\n        prf Nat.zero = Dbot (F $ fix F)\n        prf (Nat.suc n) = mon F (lubUB (iter-mon F) n)\n\n\n\n  module _ where\n\n    open Order oUr\n    open cpo U\n    open import Data.Bool\n\n    postulate\n      EXPLODE : \u2200 {A : Set} \u2192 A\n\n\n    flip : distr Bool\n    _$_ (\u03bc flip) f = (1\/ 1 +1 \u00d7 f true) + (1\/ 1 +1 \u00d7 f false)\n    mon (\u03bc flip) r = \u2264-cong-+ (\u2264-cong-\u00d7 reflexive (r true)) (\u2264-cong-\u00d7 reflexive (r false))\n    muContinous flip h = EXPLODE\n\n    Munit : \u2200 {A} \u2192 A \u2192 distr A\n    Munit x = mk (mk (\u03bb f \u2192 f x) (\u03bb x\u2264y \u2192 x\u2264y x)) (\u03bb h \u2192 reflexive)\n\n    Mlet : \u2200 {A B} \u2192 distr A \u2192 (A \u2192 distr B) \u2192 distr B\n    Mlet m k = mk (mk (\u03bb f \u2192 \u03bc m $ (\u03bb x \u2192 \u03bc (k x) $ f)) (\u03bb x\u2264y \u2192 mon (\u03bc m) (\u03bb x \u2192 mon (\u03bc (k x)) x\u2264y)))\n       (\u03bb h \u2192  transitive (mon (\u03bc m) (\u03bb x \u2192 muContinous (k x) h))\n                   (muContinous m (mk (\u03bb z z\u2081 \u2192 \u03bc (k z\u2081) $ (h $ z)) (\u03bb x\u2081 x\u2082 \u2192 mon (\u03bc (k x\u2082)) (mon h x\u2081)))))\n\n    Munit-simpl : \u2200 {A}(q : A \u2192 Ur) x \u2192 \u03bc (Munit x) $ q \u2261 q x\n    Munit-simpl q x = \u2261-reflexive\n\n    Mlet-simpl : \u2200 {A B}(m : distr A)(M : A \u2192 distr B)(f : B \u2192 Ur)\n               \u2192 \u03bc (Mlet m M) $ f \u2261 \u03bc m $ (\u03bb x \u2192 \u03bc (M x) $ f)\n    Mlet-simpl m M f = \u2261-reflexive\n\n    .Mlet-le-compat : \u2200 {A B} (m1 m2 : distr A)(M1 M2 : A \u2192 distr B)\n      \u2192 Order._\u2264_ (distrOrder A) m1 m2\n      \u2192 Order._\u2264_ (A o\u2192 distrOrder B) M1 M2\n      \u2192 Order._\u2264_ (distrOrder B) (Mlet m1 M1) (Mlet m2 M2)\n    Mlet-le-compat m1 m2 M1 M2 m1\u2264m2 M1\u2264M2 f = transitive (mon (\u03bc m1) (\u03bb x \u2192 M1\u2264M2 x f)) (m1\u2264m2 _)\n\n    -- Monad laws\n\n    Mlet-unit : \u2200 {A B} (x : A)(m : A \u2192 distr B)\n              \u2192 Order._\u2261_ (distrOrder B) (Mlet (Munit x) m) (m x)\n    Mlet-unit x m f = \u2261-reflexive\n\n    Mlet-ext : \u2200 {A}(m : distr A) \u2192 Order._\u2261_ (distrOrder A) (Mlet m Munit) m\n    Mlet-ext m f = \u2261-reflexive\n\n    Mlet-assoc : \u2200 {A B C}(m1 : distr A)(m2 : A \u2192 distr B)(m3 : B \u2192 distr C)\n               \u2192 Order._\u2261_ (distrOrder C)\n                  (Mlet (Mlet m1 m2) m3)\n                  (Mlet m1 (\u03bb x \u2192 Mlet (m2 x) m3))\n    Mlet-assoc m1 m2 m3 f = \u2261-reflexive\n\n","old_contents":"{-# OPTIONS --copatterns #-}\nmodule alea.cpo where\n\nimport Data.Nat.NP as Nat\n\nimport Relation.Binary.PropositionalEquality as \u2261\n\nrecord Order (A : Set) : Set\u2081 where\n  constructor mk\n  infix 4 _\u2261_ _\u2264_\n  field\n    _\u2264_ : A \u2192 A \u2192 Set\n    _\u2261_ : A \u2192 A \u2192 Set\n    reflexive : \u2200 {x} \u2192 x \u2264 x\n    antisym : \u2200 {x y} \u2192 x \u2264 y \u2192 y \u2264 x \u2192 x \u2261 y\n    transitive : \u2200 {x y z} \u2192 x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z\n\n  \u2261-reflexive : \u2200 {x} \u2192 x \u2261 x\n  \u2261-reflexive = antisym reflexive reflexive\n    \n\u2115 : Order Nat.\u2115\n\u2115 = mk Nat._\u2264_ \u2261._\u2261_ Nat.\u2115\u2264.refl Nat.\u2115\u2264.antisym Nat.\u2115\u2264.trans\n\nfunOrder : \u2200 {A B : Set} \u2192 Order B \u2192 Order (A \u2192 B)\nfunOrder or = mk (\u03bb f g  \u2192 \u2200 x \u2192 f x \u2264 g x) (\u03bb f g \u2192 \u2200 x \u2192 f x \u2261 g x)\n              (\u03bb x\u2081 \u2192 reflexive)\n              (\u03bb f g x \u2192 antisym (f x) (g x)) \n              (\u03bb f g x  \u2192 transitive (f x) (g x))\n  where open Order or\n\n_o\u2192_ : \u2200 A {B} \u2192 Order B \u2192 Order (A \u2192 B)\nA o\u2192 ob = funOrder {A} ob\n  \nrecord _\u2192m_ {A B}(oa : Order A)(ob : Order B) : Set where\n  coinductive\n  constructor mk\n  field\n    _$_ : A \u2192 B\n    .mon : \u2200 {x y} \u2192 Order._\u2264_ oa x y \u2192 Order._\u2264_ ob (_$_ x) (_$_ y)\nopen _\u2192m_ using (_$_ ; mon)\n\n_o\u2192m_ : \u2200 {A B}(oa : Order A)(ob : Order B) \u2192 Order (oa \u2192m ob)\n_o\u2192m_ {A} oa ob = mk (\u03bb f g \u2192 _$_ f \u2264 _$_ g)\n                     (\u03bb f g \u2192 _$_ f \u2261 _$_ g)\n                     reflexive antisym transitive\n  where open Order (A o\u2192 ob)\n\nswapm : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\nswapm f = mk (\u03bb b \u2192 mk (\u03bb a \u2192 (f $ a) $ b) (\u03bb r \u2192 mon f r b )) (\u03bb r x \u2192 mon (f $ x) r)\n\nswapm' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n_$_ (_$_ (swapm' f) b) a = (f $ a) $ b\nmon (_$_ (swapm' f) b) r = mon f r b\nmon (swapm' f) r x = mon (f $ x) r\n\n{-\nswapm'' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n(swapm'' f $ b)    $ a = (f $ a) $ b\nmon (swapm' f $ b) r   = mon f r b\nmon (swapm' f)     r x = mon (f $ x) r\n-}\n\n_\u2218m_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\nf \u2218m g = mk (\u03bb x \u2192 f $ (g $ x)) (\u03bb x \u2192 mon f (mon g x))\n\n_\u2218m'_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\n_$_ (f \u2218m' g) x = f $ (g $ x)\nmon (f \u2218m' g) x = mon f (mon g x)\n\n_$m_ : \u2200 {A B C}{oa : Order A}{oc : Order C}\n       \u2192 oa \u2192m (B o\u2192 oc) \u2192 B \u2192 oa \u2192m oc\n_$m_ {oa = oa}{oc} f b = mk (\u03bb a \u2192 (f $ a) b) (\u03bb x\u2264y \u2192 mon f x\u2264y b )\n\n_$m'_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$_ (f $m' b) a   = (f $ a) b\nmon (f $m' b) x\u2264y = mon f x\u2264y b\n\nUB : \u2200 {A} \u2192 Order A \u2192 A \u2192 Set\nUB or a = \u2200 x \u2192 x \u2264 a\n  where open Order or\n\nrecord cpo {D} (o : Order D) : Set where\n  constructor mk\n  open Order o\n\n  field\n    d0 : D\n    lub : (f : \u2115 \u2192m o) \u2192 D\n\n  -- proofs\n  field\n    .Dbot : \u2200 x \u2192 d0 \u2264 x\n    .lubUB : \u2200 (f : \u2115 \u2192m o) n \u2192 (f $ n) \u2264 lub f\n    .lubLUB : \u2200 (f : \u2115 \u2192m o) x \u2192 (\u2200 n \u2192 (f $ n) \u2264 x) \u2192 lub f \u2264 x\n\n  .lub-ext : \u2200 {f g} \u2192 Order._\u2264_ (\u2115 o\u2192m o) f g \u2192 lub f \u2264 lub g\n  lub-ext {f} {g} f\u2264g = lubLUB f (lub g) (\u03bb n \u2192 transitive (f\u2264g n) (lubUB g n))\n\n  lubm : (\u2115 o\u2192m o) \u2192m o\n  lubm = mk lub (\u03bb {a}{b} x\u2081 \u2192 lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081)\n\n  lubm' : (\u2115 o\u2192m o) \u2192m o\n  _$_ lubm' = lub\n  mon lubm' {a}{b} x\u2081 = lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081\n\nmodule _ where\n  .lub-swap : \u2200 {D}{o : Order D}(c : cpo o)\n             \u2192 let open cpo c\n                   open Order o\n               in \u2200 f \u2192 lub (lubm \u2218m f) \u2264 lub (lubm \u2218m swapm f)\n  lub-swap {D}{o} c f = lubLUB (lubm \u2218m f) (lub (lubm \u2218m swapm f))\n                     (\u03bb n \u2192 lubLUB (f $ n) (lub (lubm \u2218m swapm f))\n                     (\u03bb n\u2081 \u2192 transitive (lubUB (swapm f $ n\u2081) n) (lubUB (lubm \u2218m swapm f) n\u2081)))\n    where\n      open cpo c\n      open Order o\n\n  funcpo : \u2200 {A D}{od : Order D} \u2192 cpo od \u2192 cpo (A o\u2192 od)\n  funcpo cpod = mk (\u03bb x \u2192 d0) (\u03bb f x \u2192 lub (f $m x)) (\u03bb x x\u2081 \u2192 Dbot (x x\u2081)) (\u03bb f n x \u2192 lubUB (f $m x) n)\n    (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (f $m x\u2082) (x x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n    where\n      open cpo cpod\n      open Nat\n      open \u2261 using (_\u2261_)\n\n  _c\u2192m_ : \u2200 {A D}(oa : Order A){od : Order D} \u2192 cpo od \u2192 cpo (oa o\u2192m od)\n  _c\u2192m_ {A} oa {od} cpod = mk (mk (\u03bb x \u2192 d0) (\u03bb x\u2081 \u2192 Order.reflexive od))\n                          (\u03bb f \u2192 mk (cpo.lub (funcpo {A} cpod)\n                                       (mk (\u03bb n a \u2192 (f $ n) $ a) (mon f)))\n                                    (\u03bb {x}{y}x\u2264y \u2192 lubLUB _ _ (\u03bb n \u2192 transitive (mon (f $ n) x\u2264y) (lubUB (mk (\u03bb v \u2192 (f $ v) $ y) (\u03bb r \u2192 mon f r y)) n))))\n                          (\u03bb x x\u2081 \u2192 Dbot (x $ x\u2081))\n                          (\u03bb f n x \u2192 lubUB (mk (\u03bb m \u2192 (f $ m) $ x) (\u03bb x\u2082 \u2192 mon f x\u2082 x)) n)\n                          (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (mk (\u03bb n \u2192 _$_ (f $ n)) (mon f) $m x\u2082) (x $ x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n    where\n      open cpo cpod\n      open Order od renaming (_\u2264_ to _\u2264\u1d48_)\n      open Order oa using () renaming (_\u2264_ to _\u2264\u1d43_)\n\nrecord continous {A B}{oa : Order A}{ob : Order B}\n                 (c1 : cpo oa)(c2 : cpo ob)(f : oa \u2192m ob) : Set where\n  constructor mk\n  open cpo c1 renaming (lub to luba ; lubUB to lubUBa ; lubLUB to lubLUBa)\n  open cpo c2\n  open Order ob\n  field\n    .contIntro : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2264 lub (f \u2218m h)\n\n  .contIntro' : \u2200 (h : \u2115 \u2192m oa) \u2192 lub (f \u2218m h) \u2264 f $ luba h\n  contIntro' h = lubLUB (f \u2218m h) (f $ luba h) (\u03bb n \u2192 mon f (lubUBa _ n))\n\n  .contEq : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2261 lub (f \u2218m h)\n  contEq h = antisym (contIntro h) (contIntro' h)\n  \nopen continous\n\nmodule Distr\n  (Ur  : Set) -- the set [0,1]\n  (1\/_+1 : Nat.\u2115 \u2192 Ur)\n  (_+_ : Ur \u2192 Ur \u2192 Ur)\n  (_\u00d7_ : Ur \u2192 Ur \u2192 Ur)\n  (oUr : Order Ur)\n  (\u2264-cong-+ : let open Order oUr in \u2200 {x y z w} \u2192 x \u2264 z \u2192 y \u2264 w \u2192 x + y \u2264 z + w)\n  (\u2264-cong-\u00d7 : let open Order oUr in \u2200 {x y z w} \u2192 x \u2264 z \u2192 y \u2264 w \u2192 x \u00d7 y \u2264 z \u00d7 w)\n  (U   : cpo oUr) where\n\n  record distr A : Set where\n    constructor mk\n    field\n      \u03bc : (A o\u2192 oUr) \u2192m oUr\n      .muContinous : \u2200 (h : \u2115 \u2192m (A o\u2192 oUr)) \u2192 Order._\u2264_ oUr (\u03bc $ cpo.lub (funcpo U) h) (cpo.lub U (\u03bc \u2218m h)) -- continous (funcpo U) U \u03bc\n\n  open distr\n\n  module _ A where\n    module M = cpo ((A o\u2192 oUr) c\u2192m U)\n\n    distrOrder : Order (distr A)\n    distrOrder = mk (\u03bb f g \u2192 \u03bc f \u2264 \u03bc g) (\u03bb f g \u2192 \u03bc f \u2261 \u03bc g)\n      (\u03bb {x} \u2192 reflexive {\u03bc x}) (\u03bb {x} {y} \u2192 antisym {\u03bc x} {\u03bc y}) (\u03bb {x} {y} {z} \u2192 transitive {\u03bc x} {\u03bc y} {\u03bc z})\n      where open Order ((A o\u2192 oUr) o\u2192m oUr)\n\n    \u03bcm : distrOrder \u2192m ((A o\u2192 oUr) o\u2192m oUr)\n    \u03bcm = mk \u03bc (\u03bb x\u2081 x\u2082 \u2192 x\u2081 x\u2082)\n\n    distrcpo : cpo distrOrder\n    distrcpo = mk (mk M.d0 (\u03bb h \u2192 cpo.Dbot U (cpo.lub U (M.d0 \u2218m h))))\n\n                    (\u03bb f \u2192 mk (M.lub (\u03bcm \u2218m f)) (\u03bb h \u2192 transitive\n                                                       (cpo.lub-ext U (\u03bb a \u2192 muContinous (f $ a) h))\n                                                       (lub-swap U (mk (\u03bb z \u2192 mk (\u03bb z\u2081 \u2192 \u03bc (f $ z) $ (h $ z\u2081))\n                                                                                 (\u03bb x \u2192 mon (\u03bc (f $ z)) (mon h x)))\n                                                                       (\u03bb {x}{y} x\u2081 x\u2082 \u2192\n                                                                              mon f x\u2081 (\u03bb x\u2083 \u2192 (h $ x\u2082) x\u2083))))))\n                    (\u03bb x x\u2081 \u2192 cpo.Dbot U (\u03bc x $ x\u2081))\n                    (\u03bb f n x \u2192 cpo.lubUB U _ _ )\n                    (\u03bb f x x\u2081 x\u2082 \u2192 cpo.lubLUB U _ _ (\u03bb n \u2192 x\u2081 n x\u2082))\n       where open Order oUr\n\n  module FIXPOINTS {d} {D : Order d} {c : cpo D} where\n    open cpo c\n    open Order D\n\n    iter : (f : D \u2192m D) \u2192 Nat.\u2115 \u2192 d\n    iter f Nat.zero = d0\n    iter f (Nat.suc n) = f $ iter f n\n\n    iter-mon : (f : D \u2192m D) \u2192 \u2115 \u2192m D\n    iter-mon f = mk (iter f) mono\n      where\n        .mono : \u2200 {x y} \u2192 x Nat.\u2264 y \u2192 iter f x \u2264 iter f y\n        mono Nat.z\u2264n = Dbot _\n        mono (Nat.s\u2264s prf) = mon f (mono prf)\n\n    fix : (f : D \u2192m D) \u2192 d\n    fix f = lub (iter-mon f)\n\n    .fix-le : (F : D \u2192m D) \u2192 fix F \u2264 F $ (fix F)\n    fix-le F = lubLUB _ _ prf\n      where\n        .prf : (n : Nat.\u2115) \u2192 iter-mon F $ n \u2264 F $ fix F\n        prf Nat.zero = Dbot (F $ fix F)\n        prf (Nat.suc n) = mon F (lubUB (iter-mon F) n)\n\n\n\n  module _ where\n\n    open Order oUr\n    open cpo U\n    open import Data.Bool\n\n    postulate\n      EXPLODE : \u2200 {A : Set} \u2192 A\n\n\n    flip : distr Bool\n    _$_ (\u03bc flip) f = (1\/ 1 +1 \u00d7 f true) + (1\/ 1 +1 \u00d7 f false)\n    mon (\u03bc flip) r = \u2264-cong-+ (\u2264-cong-\u00d7 reflexive (r true)) (\u2264-cong-\u00d7 reflexive (r false))\n    muContinous flip h = EXPLODE\n\n    Munit : \u2200 {A} \u2192 A \u2192 distr A\n    Munit x = mk (mk (\u03bb f \u2192 f x) (\u03bb x\u2264y \u2192 x\u2264y x)) (\u03bb h \u2192 reflexive)\n\n    Mlet : \u2200 {A B} \u2192 distr A \u2192 (A \u2192 distr B) \u2192 distr B\n    Mlet m k = mk (mk (\u03bb f \u2192 \u03bc m $ (\u03bb x \u2192 \u03bc (k x) $ f)) (\u03bb x\u2264y \u2192 mon (\u03bc m) (\u03bb x \u2192 mon (\u03bc (k x)) x\u2264y)))\n       (\u03bb h \u2192  transitive (mon (\u03bc m) (\u03bb x \u2192 muContinous (k x) h))\n                   (muContinous m (mk (\u03bb z z\u2081 \u2192 \u03bc (k z\u2081) $ (h $ z)) (\u03bb x\u2081 x\u2082 \u2192 mon (\u03bc (k x\u2082)) (mon h x\u2081)))))\n\n    Munit-simpl : \u2200 {A}(q : A \u2192 Ur) x \u2192 \u03bc (Munit x) $ q \u2261 q x\n    Munit-simpl q x = \u2261-reflexive\n\n    Mlet-simpl : \u2200 {A B}(m : distr A)(M : A \u2192 distr B)(f : B \u2192 Ur)\n               \u2192 \u03bc (Mlet m M) $ f \u2261 \u03bc m $ (\u03bb x \u2192 \u03bc (M x) $ f)\n    Mlet-simpl m M f = \u2261-reflexive\n\n    .Mlet-le-compat : \u2200 {A B} (m1 m2 : distr A)(M1 M2 : A \u2192 distr B)\n      \u2192 Order._\u2264_ (distrOrder A) m1 m2\n      \u2192 Order._\u2264_ (A o\u2192 distrOrder B) M1 M2\n      \u2192 Order._\u2264_ (distrOrder B) (Mlet m1 M1) (Mlet m2 M2)\n    Mlet-le-compat m1 m2 M1 M2 m1\u2264m2 M1\u2264M2 f = transitive (mon (\u03bc m1) (\u03bb x \u2192 M1\u2264M2 x f)) (m1\u2264m2 _)\n\n    -- Monad laws\n\n    Mlet-unit : \u2200 {A B} (x : A)(m : A \u2192 distr B)\n              \u2192 Order._\u2261_ (distrOrder B) (Mlet (Munit x) m) (m x)\n    Mlet-unit x m f = \u2261-reflexive\n\n    Mlet-ext : \u2200 {A}(m : distr A) \u2192 Order._\u2261_ (distrOrder A) (Mlet m Munit) m\n    Mlet-ext m f = \u2261-reflexive\n\n    Mlet-assoc : \u2200 {A B C}(m1 : distr A)(m2 : A \u2192 distr B)(m3 : B \u2192 distr C)\n               \u2192 Order._\u2261_ (distrOrder C)\n                  (Mlet (Mlet m1 m2) m3)\n                  (Mlet m1 (\u03bb x \u2192 Mlet (m2 x) m3))\n    Mlet-assoc m1 m2 m3 f = \u2261-reflexive\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"51434bce443c5e1a7fb2c7f5f7ea559edc53dbb0","subject":"bintree: SwpOp preserve membership","message":"bintree: SwpOp preserve membership\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\n0\u2194\u2032_ : \u2200 {n} \u2192 Bits n \u2192 SwpOp n\n0\u2194\u2032_ {n} rewrite cong SwpOp (sym (\u2115\u00b0.+-comm n 0)) = 0\u2194_ {n} {0}\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = 0\u2194\u2032 p \u204f 0\u2194\u2032 q\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\n0\u2194\u2032_ : \u2200 {n} \u2192 Bits n \u2192 SwpOp n\n0\u2194\u2032_ {n} rewrite cong SwpOp (sym (\u2115\u00b0.+-comm n 0)) = 0\u2194_ {n} {0}\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = 0\u2194\u2032 p \u204f 0\u2194\u2032 q\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9999ee97cc4e6910cd88265a1a3e71cac16b407f","subject":"bintree,sorting: cleanup","message":"bintree,sorting: cleanup\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree-sorting.agda","new_file":"prefect-bintree-sorting.agda","new_contents":"module prefect-bintree-sorting where\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; _\u2238_; module \u2115\u00b0; module \u2115\u2264; +-\u2264-inj; \u00acn+\u2264y<n; sucx\u2270x; \u00acn\u2264x<n; module \u2115cmp) renaming (module <= to \u2115<=; _<=_ to _\u2115<=_)\nopen import Data.Bool.NP hiding (to\u2115)\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse; shallow-\u03b7)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning; [_])\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen import prefect-bintree\nopen import Data.Nat.Properties\nopen Nat using (z\u2264n; s\u2264s; \u2264-pred; 2*_; 2*\u2032_; 2*-spec; 2*\u2032-spec; suc-injective) renaming (_\u2264_ to _\u2115\u2264_; _<_ to _\u2115<_)\n\nmodule MM where\n open Nat using (_\u2264_; _<_)\n split-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\n split-\u2264 {zero} {zero} p = inj\u2081 \u2261.refl\n split-\u2264 {zero} {suc y} p = inj\u2082 (s\u2264s z\u2264n)\n split-\u2264 {suc x} {zero} ()\n split-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n ... | inj\u2081 q rewrite q = inj\u2081 \u2261.refl\n ... | inj\u2082 q = inj\u2082 (s\u2264s q)\n\n <\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n <\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\n Monotone : \u2115 \u2192 Endo \u2115 \u2192 Set\n Monotone ub f = \u2200 {x y} \u2192 x \u2264 y \u2192 y < ub \u2192 f x \u2264 f y\n\n IsInj : \u2115 \u2192 Endo \u2115 \u2192 Set\n IsInj ub f = \u2200 {x y} \u2192 x < ub \u2192 y < ub \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n Bounded : \u2115 \u2192 Endo \u2115 \u2192 Set\n Bounded ub f = \u2200 x \u2192 x < ub \u2192 f x < ub\n\n module M (f : \u2115 \u2192 \u2115) {ub}\n          (f-mono : Monotone ub f)\n          (f-inj : IsInj ub f)\n          (f-bounded : Bounded ub f) where\n\n  f-mono-< : \u2200 {x y} \u2192 x < y \u2192 y < ub \u2192 f x < f y\n  f-mono-< {x} {y} p y<ub with split-\u2264 (f-mono {x} {y} (<\u2192\u2264 p) y<ub)\n  ... | inj\u2081 q = \u22a5-elim (sucx\u2270x y (\u2261.subst (\u03bb z \u2192 suc z \u2115\u2264 y) (f-inj {x} {y} (<-trans p y<ub) y<ub q) p))\n  ... | inj\u2082 q = q\n\n  -- f-mono-<\u2032 : \u2200 {x} \u2192 suc x < ub \u2192 f x < f (suc x)\n\n  le : \u2200 n \u2192 suc n < ub \u2192 f (suc n) \u2264 n \u2192 f n < n\n  le n 1+n<ub p = \u2115\u2264.trans (f-mono-< {n} {suc n} \u2115\u2264.refl 1+n<ub) p\n\n  bo : \u2200 b \u2192 b < ub \u2192 \u00ac(f b < b)\n  bo zero _ ()\n  bo (suc b) b<ub (s\u2264s p) = bo b (\u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) b<ub) (le b b<ub p)\n\n  fp : \u2200 b \u2192 b < ub \u2192 Bounded (suc b) f \u2192 f b \u2261 b\n  fp b b<ub bub with split-\u2264 (bub b \u2115\u2264.refl)\n  ... | inj\u2081 p = suc-injective p\n  ... | inj\u2082 p = \u22a5-elim (bo b b<ub (\u2264-pred p))\n\n  ob : \u2200 b \u2192 b \u2264 ub \u2192 Bounded b f \u2192 \u2200 x \u2192 x < b \u2192 f x \u2261 x\n  ob zero b\u2264ub bub _ ()\n  ob (suc b) b\u2264ub bub x pf with split-\u2264 pf\n  ... | inj\u2081 p rewrite suc-injective p = fp b b\u2264ub bub\n  ... | inj\u2082 (s\u2264s p) = ob b (<\u2192\u2264 b\u2264ub) ((\u03bb y y<b \u2192 \u2115\u2264.trans (f-mono-< y<b b\u2264ub) (\u2115\u2264.reflexive (fp b b\u2264ub bub)))) x p\n\n  is-id : \u2200 x \u2192 x < ub \u2192 f x \u2261 x\n  is-id = ob ub \u2115\u2264.refl f-bounded\n\nopen BitsSorting\nopen OperationSyntax\nopen BijSpec\nopen EvalTree\nopen Alternative-Reverse\n\nsortBijT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nsortBijT = bij \u2218 sort-bij\n\nsortBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij n\nsortBij = sortBijT \u2218 fromFun\n\nsortBij-sort : \u2200 {n} (t : Tree (Bits n) n) \u2192 evalTree (sortBijT t) t \u2261 sort t\nsortBij-sort = proof \u2218 sort-bij\n\nIsInj : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set\nIsInj f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nidT : \u2200 {n} \u2192 Tree (Bits n) n\nidT = fromFun id\n\ntoFun-idT : \u2200 {n} \u2192 toFun idT \u2257 id {A = Bits n}\ntoFun-idT x = toFun\u2218fromFun id x\n\nInjT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Set\nInjT = IsInj \u2218 toFun\n\n_\u207b\u00b9-inverseTree : \u2200 {n} f \u2192 evalTree (f `\u204f f \u207b\u00b9) \u2257 id {A = Tree (Bits n) n}\n(f \u207b\u00b9-inverseTree) t = evalTree (f `\u204f f \u207b\u00b9) t\n                     \u2261\u27e8 \u2261.sym (fromFun\u2218toFun (evalTree (f `\u204f f \u207b\u00b9) t)) \u27e9\n                       fromFun (toFun (evalTree (f `\u204f f \u207b\u00b9) t))\n                     \u2261\u27e8 fromFun-\u2257 (evalTree-eval\u2032 (f `\u204f f \u207b\u00b9) t) \u27e9\n                       fromFun (toFun t \u2218 eval (f \u207b\u00b9 \u207b\u00b9 `\u204f f \u207b\u00b9))\n                     \u2261\u27e8 fromFun-\u2257 (\u03bb x \u2192 \u2261.cong (toFun t) (VecBijKit._\u207b\u00b9-inverse\u2032 _ (f \u207b\u00b9) x)) \u27e9\n                       fromFun (toFun t)\n                     \u2261\u27e8 fromFun\u2218toFun t \u27e9\n                       t\n                     \u220e where open \u2261-Reasoning\n\nmodule SortedMonotonicFunctionsBits {n} where\n    open SortedMonotonicFunctions (_\u2264_ {n}) isPreorder public\nopen SortedMonotonicFunctionsBits hiding (Sorted)\n\n-- _\u2218-inj_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {f : B \u2192 C} {g : A \u2192 B} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\n_\u2218-inj_ : \u2200 {a b c} {f : Bits b \u2192 Bits c} {g : Bits a \u2192 Bits b} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\nf-inj \u2218-inj g-inj = g-inj \u2218 f-inj\n\n_\u2237-inj : \u2200 {n} b \u2192 IsInj (_\u2237_ {n = n} b)\n(_ \u2237-inj) \u2261.refl = \u2261.refl\n\nlemvec : \u2200 {n} (xs ys : Bits (suc n)) \u2192 head xs \u2261 head ys \u2192 tail xs \u2261 tail ys \u2192 xs \u2261 ys\nlemvec (x \u2237 xs) (.x \u2237 .xs) \u2261.refl \u2261.refl = \u2261.refl\n\n\u2264\u2192\u2264\u1d2e : \u2200 {n} {x y : Bits n} \u2192 x \u2264 y \u2192 x \u2264\u1d2e y\n\u2264\u2192\u2264\u1d2e {x = []}         {[]}         p = []\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {true \u2237 ys}  p = there 1b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} (\u2115<=.complete {to\u2115 xs} {to\u2115 ys} (+-\u2264-inj (2^ n) (\u2115<=.sound _ _ p))))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {false \u2237 ys} p = there 0b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} p)\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {false \u2237 ys} p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115<=.sound (2^ n + to\u2115 xs) (to\u2115 ys) p))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {true \u2237 ys}  p = 0-1 xs ys\n\n\u2264\u1d2e\u2192\u2264 : \u2200 {n} {x y : Bits n} \u2192 x \u2264\u1d2e y \u2192 x \u2264 y\n\u2264\u1d2e\u2192\u2264 [] = _\n\u2264\u1d2e\u2192\u2264 {suc n} (there true p) = \u2115<=.complete (\u2115\u2264.refl {2^ n} +-mono (\u2115<=.sound _ _ (\u2264\u1d2e\u2192\u2264 p)))\n\u2264\u1d2e\u2192\u2264 (there false p) = \u2264\u1d2e\u2192\u2264 p\n\u2264\u1d2e\u2192\u2264 (0-1 p q) = \u2115<=.complete (to\u2115\u22642\u207f+ p {to\u2115 q})\n\nMonotone' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set _\nMonotone' {i} {o} f = \u2200 {p q : Bits i} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2e f q\n\ntoEndo\u2115 : \u2200 {n} \u2192 Endo (Bits n) \u2192 Endo \u2115\ntoEndo\u2115 f = to\u2115 \u2218 f \u2218 from\u2115\n\nopen Nat using (_\u2270_)\n\nmodule ToEndo\u2115 {n} (f : Endo (Bits n)) where\n    f\u2115 = toEndo\u2115 f\n\n    from\u2115n = from\u2115 {n}\n\n    mono : Monotone f \u2192 MM.Monotone (2^ n) f\u2115\n    mono f-mono {x} {y} x\u2264y y<2\u207f\n       = \u2115<=.sound _ _\n          (f-mono\n            (\u2264\u2192\u2264\u1d2e (\u2115<=.complete {to\u2115 (from\u2115n x)} {to\u2115 (from\u2115n y)}\n                     (\u2115\u2264.trans (\u2115\u2264.reflexive (to\u2115\u2218from\u2115 {n} x (\u2115\u2264.trans (s\u2264s x\u2264y) y<2\u207f)))\n                        (\u2115\u2264.trans x\u2264y (\u2115\u2264.reflexive (\u2261.sym (to\u2115\u2218from\u2115 {n} y y<2\u207f))))))))\n\n    inj : IsInj f \u2192 MM.IsInj (2^ n) f\u2115\n    inj f-inj {x} {y} x< y< = from\u2115-inj x< y< \u2218 f-inj \u2218 to\u2115-inj _ _\n\n    bounded : MM.Bounded (2^ n) f\u2115\n    bounded x x\u22642\u207f = to\u2115-bound (f (from\u2115 x))\n\nlem : \u2200 {n} (f : Endo (Bits n)) \u2192 Monotone f \u2192 IsInj f \u2192 to\u2115 \u2218 f \u2257 to\u2115\nlem {n} f f-mono f-inj x = \u2261.trans (\u2261.cong (to\u2115 \u2218 f) (\u2261.sym (from\u2115\u2218to\u2115 x)))\n                                   (MM.M.is-id (toEndo\u2115 f) {2^ n} (mono f-mono) (inj f-inj) bounded (to\u2115 x) (to\u2115-bound x))\n  where open ToEndo\u2115 f\n\nIsInj-toFunFromFun : \u2200 {n} (f : Endo (Bits n)) \u2192 IsInj f \u2192 IsInj (toFun (fromFun f))\nIsInj-toFunFromFun f f-inj {x} {y} eq\n  rewrite toFun\u2218fromFun f x\n        | toFun\u2218fromFun f y\n        = f-inj eq\n\nsort-spec\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 Sorted (evalTree (sortBijT t) t)\nsort-spec\u2032 t rewrite sortBij-sort t = sort-spec t\n\nsort-id\u2032\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted t \u2192 toFun t \u2257 id\nsort-id\u2032\u2032 t Pt st x = to\u2115-inj _ _ (lem (toFun t) (DataSorted\u2192Sorted st) Pt x)\n\ntoFun-eval-inj : \u2200 {n} (f : Bij n) \u2192 IsInj (eval f)\ntoFun-eval-inj f {x} {y} eq = x\n                            \u2261\u27e8 \u2261.sym ((f \u207b\u00b9-inverse) x) \u27e9\n                              eval (f \u207b\u00b9) (eval f x)\n                            \u2261\u27e8 \u2261.cong (eval (f \u207b\u00b9)) eq \u27e9\n                              eval (f \u207b\u00b9) (eval f y)\n                            \u2261\u27e8 (f \u207b\u00b9-inverse) y \u27e9\n                              y\n                            \u220e where open \u2261-Reasoning\n\ntoFun-evalTree-inj : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 InjT (evalTree f t)\ntoFun-evalTree-inj f t Pt {x} {y} eq\n  rewrite evalTree-eval\u2032 f t x\n        | evalTree-eval\u2032 f t y = toFun-eval-inj (f \u207b\u00b9) (Pt eq)\n\nsort-id : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun (sort t) \u2257 id\nsort-id t t-inj rewrite \u2261.sym (sortBij-sort t)\n  = sort-id\u2032\u2032 (evalTree f t) (\u03bb {x} {y} eq \u2192 toFun-evalTree-inj f t t-inj eq) (sort-spec\u2032 t)\n  where f = sortBijT t\n\nsort-id-fun : \u2200 {n} (f : Endo (Bits n)) \u2192 IsInj f \u2192 sortFun f \u2257 id\nsort-id-fun f f-inj x = sort-id (fromFun f) (IsInj-toFunFromFun f f-inj) x\n\nthm : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) \u2192\n        eval (sortBij f) \u2257 f\nthm f f-inj x = eval (sortBij f) x\n              \u2261\u27e8 \u2261.sym (sort-id-fun f f-inj (eval (sortBij f) x)) \u27e9\n                sortFun f (eval (sortBij f) x)\n              \u2261\u27e8 \u2261.cong (\u03bb t \u2192 toFun t (eval (sortBij f) x)) (\u2261.sym (sortBij-sort (fromFun f))) \u27e9\n                toFun (evalTree (sortBij f) (fromFun f)) (eval (sortBij f) x)\n              \u2261\u27e8 evalTree-eval\u2032 (sortBij f) (fromFun f) (eval (sortBij f) x) \u27e9\n                toFun (fromFun f) (eval (sortBij f \u207b\u00b9) (eval (sortBij f) x))\n              \u2261\u27e8 toFun\u2218fromFun f _ \u27e9\n                f (eval (sortBij f \u207b\u00b9) (eval (sortBij f) x))\n              \u2261\u27e8 \u2261.cong f ((sortBij f \u207b\u00b9-inverse) x) \u27e9\n                f x\n              \u220e where open \u2261-Reasoning\n\nthm# : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) (g : Bits n \u2192 Bit) \u2192\n       #\u27e8 g \u2218 f \u27e9 \u2261 #\u27e8 g \u27e9\nthm# f f-inj g = #\u27e8 g \u2218 f \u27e9\n               \u2261\u27e8 #-\u2257 (g \u2218 f) (g \u2218 eval (sortBij f)) (\u03bb x \u2192 \u2261.cong g (\u2261.sym (thm f f-inj x))) \u27e9\n                 #\u27e8 g \u2218 eval (sortBij f) \u27e9\n               \u2261\u27e8 #-bij (sortBij f) g \u27e9\n                 #\u27e8 g \u27e9\n               \u220e where open \u2261-Reasoning\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree-sorting where\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; _\u2238_; module \u2115\u00b0; module \u2115\u2264; +-\u2264-inj; \u00acn+\u2264y<n; sucx\u2270x; \u00acn\u2264x<n; module \u2115cmp) renaming (module <= to \u2115<=; _<=_ to _\u2115<=_)\nopen import Data.Bool.NP hiding (to\u2115)\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse; shallow-\u03b7)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning; [_])\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen import prefect-bintree\nopen import Data.Nat.Properties\nopen Nat using (z\u2264n; s\u2264s; \u2264-pred; 2*_; 2*\u2032_; 2*-spec; 2*\u2032-spec; suc-injective) renaming (_\u2264_ to _\u2115\u2264_; _<_ to _\u2115<_)\n\nmodule MM where\n open Nat using (_\u2264_; _<_)\n split-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\n split-\u2264 {zero} {zero} p = inj\u2081 \u2261.refl\n split-\u2264 {zero} {suc y} p = inj\u2082 (s\u2264s z\u2264n)\n split-\u2264 {suc x} {zero} ()\n split-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n ... | inj\u2081 q rewrite q = inj\u2081 \u2261.refl\n ... | inj\u2082 q = inj\u2082 (s\u2264s q)\n\n <\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n <\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\n Monotone : \u2115 \u2192 Endo \u2115 \u2192 Set\n Monotone ub f = \u2200 {x y} \u2192 x \u2264 y \u2192 y < ub \u2192 f x \u2264 f y\n\n IsInj : \u2115 \u2192 Endo \u2115 \u2192 Set\n IsInj ub f = \u2200 {x y} \u2192 x < ub \u2192 y < ub \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n Bounded : \u2115 \u2192 Endo \u2115 \u2192 Set\n Bounded ub f = \u2200 x \u2192 x < ub \u2192 f x < ub\n\n module M (f : \u2115 \u2192 \u2115) {ub}\n          (f-mono : Monotone ub f)\n          (f-inj : IsInj ub f)\n          (f-bounded : Bounded ub f) where\n\n  f-mono-< : \u2200 {x y} \u2192 x < y \u2192 y < ub \u2192 f x < f y\n  f-mono-< {x} {y} p y<ub with split-\u2264 (f-mono {x} {y} (<\u2192\u2264 p) y<ub)\n  ... | inj\u2081 q = \u22a5-elim (sucx\u2270x y (\u2261.subst (\u03bb z \u2192 suc z \u2115\u2264 y) (f-inj {x} {y} (<-trans p y<ub) y<ub q) p))\n  ... | inj\u2082 q = q\n\n  -- f-mono-<\u2032 : \u2200 {x} \u2192 suc x < ub \u2192 f x < f (suc x)\n\n  le : \u2200 n \u2192 suc n < ub \u2192 f (suc n) \u2264 n \u2192 f n < n\n  le n 1+n<ub p = \u2115\u2264.trans (f-mono-< {n} {suc n} \u2115\u2264.refl 1+n<ub) p\n\n  bo : \u2200 b \u2192 b < ub \u2192 \u00ac(f b < b)\n  bo zero _ ()\n  bo (suc b) b<ub (s\u2264s p) = bo b (\u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) b<ub) (le b b<ub p)\n\n  fp : \u2200 b \u2192 b < ub \u2192 Bounded (suc b) f \u2192 f b \u2261 b\n  fp b b<ub bub with split-\u2264 (bub b \u2115\u2264.refl)\n  ... | inj\u2081 p = suc-injective p\n  ... | inj\u2082 p = \u22a5-elim (bo b b<ub (\u2264-pred p))\n\n  ob : \u2200 b \u2192 b \u2264 ub \u2192 Bounded b f \u2192 \u2200 x \u2192 x < b \u2192 f x \u2261 x\n  ob zero b\u2264ub bub _ ()\n  ob (suc b) b\u2264ub bub x pf with split-\u2264 pf\n  ... | inj\u2081 p rewrite suc-injective p = fp b b\u2264ub bub\n  ... | inj\u2082 (s\u2264s p) = ob b (<\u2192\u2264 b\u2264ub) ((\u03bb y y<b \u2192 \u2115\u2264.trans (f-mono-< y<b b\u2264ub) (\u2115\u2264.reflexive (fp b b\u2264ub bub)))) x p\n\n  is-id : \u2200 x \u2192 x < ub \u2192 f x \u2261 x\n  is-id = ob ub \u2115\u2264.refl f-bounded\n\nopen BitsSorting\nopen OperationSyntax\nopen BijSpec\nopen EvalTree\nopen Alternative-Reverse\nopen SPP\n\nmkBijT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nmkBijT = bij \u2218 sort-bij\n\nmkBijT\u207b\u00b9 : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nmkBijT\u207b\u00b9 f = mkBijT f \u207b\u00b9\n\nmkBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij n\nmkBij f = mkBijT (fromFun f) \u207b\u00b9\n\nmkBij-p : \u2200 {n} (t : Tree (Bits n) n) \u2192 t \u2261 evalTree (mkBijT t) (sort t)\nmkBij-p = proof \u2218 sort-bij\n\nIsInj : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set\nIsInj f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nidT : \u2200 {n} \u2192 Tree (Bits n) n\nidT = fromFun id\n\ntoFun-idT : \u2200 {n} \u2192 toFun idT \u2257 id {A = Bits n}\ntoFun-idT x = toFun\u2218fromFun id x\n\nInjT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Set\nInjT = IsInj \u2218 toFun\n\n_\u207b\u00b9-inverseTree : \u2200 {n} f \u2192 evalTree (f `\u204f f \u207b\u00b9) \u2257 id {A = Tree (Bits n) n}\n(f \u207b\u00b9-inverseTree) t = evalTree (f `\u204f f \u207b\u00b9) t\n                     \u2261\u27e8 \u2261.sym (fromFun\u2218toFun (evalTree (f `\u204f f \u207b\u00b9) t)) \u27e9\n                       fromFun (toFun (evalTree (f `\u204f f \u207b\u00b9) t))\n                     \u2261\u27e8 fromFun-\u2257 (evalTree-eval\u2032 (f `\u204f f \u207b\u00b9) t) \u27e9\n                       fromFun (toFun t \u2218 eval (f \u207b\u00b9 \u207b\u00b9 `\u204f f \u207b\u00b9))\n                     \u2261\u27e8 fromFun-\u2257 (\u03bb x \u2192 \u2261.cong (toFun t) (VecBijKit._\u207b\u00b9-inverse\u2032 _ (f \u207b\u00b9) x)) \u27e9\n                       fromFun (toFun t)\n                     \u2261\u27e8 fromFun\u2218toFun t \u27e9\n                       t\n                     \u220e where open \u2261-Reasoning\n\nmkBij-p\u207b\u00b9 : \u2200 {n} (t : Tree (Bits n) n) \u2192 evalTree (mkBijT\u207b\u00b9 t) t \u2261 sort t\nmkBij-p\u207b\u00b9 t = \u2261.trans (\u2261.cong (evalTree (mkBijT\u207b\u00b9 t)) (mkBij-p t)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t))\n\nfoo : \u2200 {n} (t : Tree (Bits n) n) \u2192 toFun t \u2257 toFun (sort t) \u2218 eval (mkBijT\u207b\u00b9 t)\nfoo t xs = toFun t xs\n         \u2261\u27e8 \u2261.cong (\u03bb t \u2192 toFun t xs) (mkBij-p t) \u27e9\n           toFun (evalTree (mkBijT t) (sort t)) xs\n         \u2261\u27e8 evalTree-eval (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t)) xs \u27e9\n           toFun (evalTree (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t))) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.refl \u27e9\n           toFun (evalTree (mkBijT t `\u204f mkBijT\u207b\u00b9 t) (sort t)) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.cong (\u03bb u \u2192 toFun u (eval (mkBijT\u207b\u00b9 t) xs)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t)) \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u220e where open \u2261-Reasoning\n\n\nmodule SortedMonotonicFunctionsBits {n} where\n    open SortedMonotonicFunctions (_\u2264_ {n}) isPreorder public\nopen SortedMonotonicFunctionsBits hiding (Sorted)\n\n-- _\u2218-inj_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {f : B \u2192 C} {g : A \u2192 B} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\n_\u2218-inj_ : \u2200 {a b c} {f : Bits b \u2192 Bits c} {g : Bits a \u2192 Bits b} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\nf-inj \u2218-inj g-inj = g-inj \u2218 f-inj\n\n_\u2237-inj : \u2200 {n} b \u2192 IsInj (_\u2237_ {n = n} b)\n(_ \u2237-inj) \u2261.refl = \u2261.refl\n\nlemvec : \u2200 {n} (xs ys : Bits (suc n)) \u2192 head xs \u2261 head ys \u2192 tail xs \u2261 tail ys \u2192 xs \u2261 ys\nlemvec (x \u2237 xs) (.x \u2237 .xs) \u2261.refl \u2261.refl = \u2261.refl\n\n\u2264\u2192\u2264\u1d2e : \u2200 {n} {x y : Bits n} \u2192 x \u2264 y \u2192 x \u2264\u1d2e y\n\u2264\u2192\u2264\u1d2e {x = []}         {[]}         p = []\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {true \u2237 ys}  p = there 1b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} (\u2115<=.complete {to\u2115 xs} {to\u2115 ys} (+-\u2264-inj (2^ n) (\u2115<=.sound _ _ p))))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {false \u2237 ys} p = there 0b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} p)\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {false \u2237 ys} p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115<=.sound (2^ n + to\u2115 xs) (to\u2115 ys) p))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {true \u2237 ys}  p = 0-1 xs ys\n\n\u2264\u1d2e\u2192\u2264 : \u2200 {n} {x y : Bits n} \u2192 x \u2264\u1d2e y \u2192 x \u2264 y\n\u2264\u1d2e\u2192\u2264 [] = _\n\u2264\u1d2e\u2192\u2264 {suc n} (there true p) = \u2115<=.complete (\u2115\u2264.refl {2^ n} +-mono (\u2115<=.sound _ _ (\u2264\u1d2e\u2192\u2264 p)))\n\u2264\u1d2e\u2192\u2264 (there false p) = \u2264\u1d2e\u2192\u2264 p\n\u2264\u1d2e\u2192\u2264 (0-1 p q) = \u2115<=.complete (to\u2115\u22642\u207f+ p {to\u2115 q})\n\nMonotone' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set _\nMonotone' {i} {o} f = \u2200 {p q : Bits i} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2e f q\n\ntoEndo\u2115 : \u2200 {n} \u2192 Endo (Bits n) \u2192 Endo \u2115\ntoEndo\u2115 f = to\u2115 \u2218 f \u2218 from\u2115\n\nopen Nat using (_\u2270_)\n\nmodule ToEndo\u2115 {n} (f : Endo (Bits n)) where\n    f\u2115 = toEndo\u2115 f\n\n    from\u2115n = from\u2115 {n}\n\n    mono : Monotone f \u2192 MM.Monotone (2^ n) f\u2115\n    mono f-mono {x} {y} x\u2264y y<2\u207f\n       = \u2115<=.sound _ _\n          (f-mono\n            (\u2264\u2192\u2264\u1d2e (\u2115<=.complete {to\u2115 (from\u2115n x)} {to\u2115 (from\u2115n y)}\n                     (\u2115\u2264.trans (\u2115\u2264.reflexive (to\u2115\u2218from\u2115 {n} x (\u2115\u2264.trans (s\u2264s x\u2264y) y<2\u207f)))\n                        (\u2115\u2264.trans x\u2264y (\u2115\u2264.reflexive (\u2261.sym (to\u2115\u2218from\u2115 {n} y y<2\u207f))))))))\n    \n    inj : IsInj f \u2192 MM.IsInj (2^ n) f\u2115\n    inj f-inj {x} {y} x< y< = from\u2115-inj x< y< \u2218 f-inj \u2218 to\u2115-inj _ _\n\n    bounded : MM.Bounded (2^ n) f\u2115\n    bounded x x\u22642\u207f = to\u2115-bound (f (from\u2115 x))\n\nlem : \u2200 {n} (f : Endo (Bits n)) \u2192 Monotone f \u2192 IsInj f \u2192 f \u2257 id\nlem {n} f f-mono f-inj x = to\u2115-inj _ _ (\u2261.trans (\u2261.cong (to\u2115 \u2218 f) (\u2261.sym (from\u2115\u2218to\u2115 x))) (MM.M.is-id (toEndo\u2115 f) {2^ n} (mono f-mono) (inj f-inj) bounded (to\u2115 x) (to\u2115-bound x)))\n  where open ToEndo\u2115 f\n\nIsInj-toFunFromFun : \u2200 {n} (f : Endo (Bits n)) \u2192 IsInj f \u2192 IsInj (toFun (fromFun f))\nIsInj-toFunFromFun f f-inj {x} {y} eq\n  rewrite toFun\u2218fromFun f x\n        | toFun\u2218fromFun f y\n        = f-inj eq\n\nsort-spec\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 Sorted (evalTree (mkBijT\u207b\u00b9 t) t)\nsort-spec\u2032 t rewrite mkBij-p\u207b\u00b9 t = sort-spec t\n\nsort-id\u2032\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted t \u2192 toFun t \u2257 id\nsort-id\u2032\u2032 t Pt st x = lem (toFun t) (DataSorted\u2192Sorted st) Pt x\n\ntoFun-eval-inj : \u2200 {n} (f : Bij n) \u2192 IsInj (eval f)\ntoFun-eval-inj f {x} {y} eq = x\n                            \u2261\u27e8 \u2261.sym ((f \u207b\u00b9-inverse) x) \u27e9\n                              eval (f \u207b\u00b9) (eval f x)\n                            \u2261\u27e8 \u2261.cong (eval (f \u207b\u00b9)) eq \u27e9\n                              eval (f \u207b\u00b9) (eval f y)\n                            \u2261\u27e8 (f \u207b\u00b9-inverse) y \u27e9\n                              y\n                            \u220e where open \u2261-Reasoning\n\ntoFun-evalTree-inj : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 InjT (evalTree f t)\ntoFun-evalTree-inj f t Pt {x} {y} eq\n  rewrite evalTree-eval\u2032 f t x\n        | evalTree-eval\u2032 f t y = toFun-eval-inj (f \u207b\u00b9) (Pt eq)\n\nsort-id\u2032 : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted (evalTree f t) \u2192 toFun (evalTree f t) \u2257 id\nsort-id\u2032 f t Pt = sort-id\u2032\u2032 (evalTree f t) (\u03bb {x} {y} eq \u2192 toFun-evalTree-inj f t Pt eq)\n\nsort-id : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun (sort t) \u2257 id\nsort-id t Pt rewrite \u2261.sym (mkBij-p\u207b\u00b9 t) = sort-id\u2032 (mkBijT\u207b\u00b9 t) t Pt (sort-spec\u2032 t)\n\nbar : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun t \u2257 eval (mkBijT\u207b\u00b9 t)\nbar t Pt xs = toFun t xs\n         \u2261\u27e8 foo t xs \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 sort-id t Pt (eval (mkBijT\u207b\u00b9 t) xs) \u27e9\n           eval (mkBijT\u207b\u00b9 t) xs\n         \u220e where open \u2261-Reasoning\n\nthm : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) \u2192\n        eval (mkBij f) \u2257 f\nthm f f-inj xs = eval (mkBij f) xs\n               \u2261\u27e8 \u2261.sym (bar (fromFun f) (IsInj-toFunFromFun f f-inj) xs) \u27e9\n                 toFun (fromFun f) xs\n               \u2261\u27e8 toFun\u2218fromFun f xs \u27e9\n                 f xs\n               \u220e where open \u2261-Reasoning\n\nthm# : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) (g : Bits n \u2192 Bit) \u2192\n       #\u27e8 g \u2218 f \u27e9 \u2261 #\u27e8 g \u27e9\nthm# f f-inj g = #\u27e8 g \u2218 f \u27e9\n               \u2261\u27e8 #-\u2257 (g \u2218 f) (g \u2218 eval (mkBij f)) (\u03bb x \u2192 \u2261.cong g (\u2261.sym (thm f f-inj x))) \u27e9\n                 #\u27e8 g \u2218 eval (mkBij f) \u27e9\n               \u2261\u27e8 #-bij (mkBij f) g \u27e9\n                 #\u27e8 g \u27e9\n               \u220e where open \u2261-Reasoning\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"066e28ecf3ffbe7b3f955e21dc1fe9ea22d4236c","subject":"Added test related to the --without-K option.","message":"Added test related to the --without-K option.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/k-rule\/Test.agda","new_file":"notes\/k-rule\/Test.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"79ac2503853e27b2c555c089b4e27fc79bf7ea1c","subject":"a few lemmas about Nats","message":"a few lemmas about Nats\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"Nat.agda","new_file":"Nat.agda","new_contents":"open import Prelude\n\nmodule Nat where\n  data Nat : Set where\n    Z : Nat\n    1+ : Nat \u2192 Nat\n\n  {-# BUILTIN NATURAL Nat #-}\n\n  -- the succ operation is injective\n  1+inj : (x y : Nat) \u2192 (1+ x == 1+ y) \u2192 x == y\n  1+inj Z .0 refl = refl\n  1+inj (1+ x) .(1+ x) refl = refl\n\n  1+notZ : (x : Nat) \u2192 1+ x \u2260 Z\n  1+notZ Z = \u03bb ()\n  1+notZ (1+ x) = \u03bb ()\n\n  1+not2+ : (x : Nat) \u2192 1+ (1+ x) \u2260 1+ x\n  1+not2+ Z ()\n  1+not2+ (1+ x) ()\n\n  -- equality of naturals is decidable. we represent this as computing a\n  -- choice of units, with inl <> meaning that the naturals are indeed the\n  -- same and inr <> that they are not.\n  natEQ : (x y : Nat) \u2192 ((x == y) + ((x == y) \u2192 \u22a5))\n  natEQ Z Z = Inl refl\n  natEQ Z (1+ y) = Inr (\u03bb ())\n  natEQ (1+ x) Z = Inr (\u03bb ())\n  natEQ (1+ x) (1+ y) with natEQ x y\n  natEQ (1+ x) (1+ .x) | Inl refl = Inl refl\n  ... | Inr b = Inr (\u03bb x\u2081 \u2192 b (1+inj x y x\u2081))\n\n  -- nat equality as a predicate. this saves some very repetative casing.\n  natEQp : (x y : Nat) \u2192 Set\n  natEQp x y with natEQ x y\n  natEQp x .x | Inl refl = \u22a5\n  natEQp x y | Inr x\u2081 = \u22a4\n\n  -- there are plenty of nats\n  different-nat : (x : Nat) \u2192 \u03a3[ y \u2208 Nat ] (y \u2260 x)\n  different-nat Z = 1+ Z , (\u03bb ())\n  different-nat (1+ x) with different-nat x\n  different-nat (1+ x) | \u03c01 , \u03c02 with natEQ \u03c01 x\n  different-nat (1+ x) | .x , \u03c02 | Inl refl = abort (\u03c02 refl)\n  different-nat (1+ x) | \u03c01 , \u03c02 | Inr x\u2081 = Z , (\u03bb ())\n\n  different-nat2 : (x y : Nat) \u2192 \u03a3[ z \u2208 Nat ](z \u2260 y \u00d7 z \u2260 x)\n  different-nat2 Z Z = 1+ Z , (\u03bb ()) , (\u03bb ())\n  different-nat2 Z (1+ y) = (1+ (1+ y)) , 1+not2+ y , 1+notZ (1+ y)\n  different-nat2 (1+ x) Z = 1+ (1+ x) , 1+notZ (1+ x) , 1+not2+ x\n  different-nat2 (1+ x) (1+ y) with different-nat2 x y\n  ... | z , \u2260x , \u2260y = 1+ z , (\u03bb x\u2081 \u2192 \u2260x (1+inj z y x\u2081)) , (\u03bb x\u2081 \u2192 \u2260y (1+inj z x x\u2081))\n","old_contents":"open import Prelude\n\nmodule Nat where\n  data Nat : Set where\n    Z : Nat\n    1+ : Nat \u2192 Nat\n\n  {-# BUILTIN NATURAL Nat #-}\n\n  -- the succ operation is injective\n  1+inj : (x y : Nat) \u2192 (1+ x == 1+ y) \u2192 x == y\n  1+inj Z .0 refl = refl\n  1+inj (1+ x) .(1+ x) refl = refl\n\n  -- equality of naturals is decidable. we represent this as computing a\n  -- choice of units, with inl <> meaning that the naturals are indeed the\n  -- same and inr <> that they are not.\n  natEQ : (x y : Nat) \u2192 ((x == y) + ((x == y) \u2192 \u22a5))\n  natEQ Z Z = Inl refl\n  natEQ Z (1+ y) = Inr (\u03bb ())\n  natEQ (1+ x) Z = Inr (\u03bb ())\n  natEQ (1+ x) (1+ y) with natEQ x y\n  natEQ (1+ x) (1+ .x) | Inl refl = Inl refl\n  ... | Inr b = Inr (\u03bb x\u2081 \u2192 b (1+inj x y x\u2081))\n\n  -- nat equality as a predicate. this saves some very repetative casing.\n  natEQp : (x y : Nat) \u2192 Set\n  natEQp x y with natEQ x y\n  natEQp x .x | Inl refl = \u22a5\n  natEQp x y | Inr x\u2081 = \u22a4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"864ac27c4d8534c751ee19fc93407a6e3fda2863","subject":"missing a file","message":"missing a file\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\nopen import correspondence\n-- open import expandability\n-- open import expansion-unicity\n-- open import preservation\n-- open import progress\n-- open import type-assignment-unicity\n-- open import typed-expansion\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\n\nopen import correspondence\n-- open import expandability\n-- open import expansion-unicity\n-- open import preservation\n-- open import progress\n-- open import type-assignment-unicity\n-- open import typed-expansion\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9911b24bcce90c2dbcecb1d4addf102a7bf6cb2e","subject":"bij, Alt-Syn: ``tail based on ``\u2218","message":"bij, Alt-Syn: ``tail based on ``\u2218\n","repos":"crypto-agda\/crypto-agda","old_file":"bijection-fin.agda","new_file":"bijection-fin.agda","new_contents":"module bijection-fin where\n\n  open import bijection\n  open import Function.NP hiding (Cmp)\n  open import Relation.Binary.PropositionalEquality\n\n  open import Data.Empty\n  open import Data.Nat.NP\n  open import Data.Fin using (Fin ; zero ; suc ; from\u2115 ; inject\u2081)\n  open import Data.Vec hiding ([_])\n\n  data `Syn : \u2115 \u2192 Set where \n    `id   : \u2200 {n} \u2192 `Syn n\n    `swap : \u2200 {n} \u2192 `Syn (2 + n)\n    `tail : \u2200 {n} \u2192 `Syn n \u2192 `Syn (1 + n)\n    _`\u2218_  : \u2200 {n} \u2192 `Syn n \u2192 `Syn n \u2192 `Syn n\n\n  `Rep = Fin\n\n  `Ix = \u2115\n\n  `Tree : Set \u2192 `Ix \u2192 Set\n  `Tree X = Vec X\n\n  `fromFun : \u2200 {i X} \u2192 (`Rep i \u2192 X) \u2192 `Tree X i\n  `fromFun = tabulate\n\n  `toFun : \u2200 {i X} \u2192 `Tree X i \u2192 (`Rep i \u2192 X)\n  `toFun T zero    = head T\n  `toFun T (suc i) = `toFun (tail T) i\n\n  `toFun\u2218fromFun : \u2200 {i X}(f : `Rep i \u2192 X) \u2192 f \u2257 `toFun (`fromFun f)\n  `toFun\u2218fromFun f zero = refl\n  `toFun\u2218fromFun f (suc i) = `toFun\u2218fromFun (f \u2218 suc) i\n\n  fin-swap : \u2200 {n} \u2192 Endo (Fin (2 + n))\n  fin-swap zero = suc zero\n  fin-swap (suc zero) = zero\n  fin-swap (suc (suc i)) = suc (suc i)\n\n  fin-tail : \u2200 {n} \u2192 Endo (Fin n) \u2192 Endo (Fin (1 + n))\n  fin-tail f zero = zero\n  fin-tail f (suc i) = suc (f i)\n\n  `evalArg : \u2200 {i} \u2192 `Syn i \u2192 Endo (`Rep i)\n  `evalArg `id       = id\n  `evalArg `swap     = fin-swap\n  `evalArg (`tail f) = fin-tail (`evalArg f)\n  `evalArg (S `\u2218 S\u2081) = `evalArg S \u2218 `evalArg S\u2081\n\n  vec-swap : \u2200 {n}{X : Set} \u2192 Endo (Vec X (2 + n))\n  vec-swap xs = head (tail xs) \u2237 head xs \u2237 tail (tail xs)\n\n  vec-tail : \u2200 {n}{X : Set} \u2192 Endo (Vec X n) \u2192 Endo (Vec X (1 + n))\n  vec-tail f xs = head xs \u2237 f (tail xs)\n\n  `evalTree : \u2200 {i X} \u2192 `Syn i \u2192 Endo (`Tree X i)\n  `evalTree `id       = id\n  `evalTree `swap     = vec-swap\n  `evalTree (`tail f) = vec-tail (`evalTree f)\n  `evalTree (S `\u2218 S\u2081) = `evalTree S \u2218 `evalTree S\u2081\n\n  `eval-proof : \u2200 {i X} S (T : `Tree X i) \u2192 `toFun T \u2257 `toFun (`evalTree S T) \u2218 `evalArg S\n  `eval-proof `id       T i = refl\n  `eval-proof `swap T zero = refl\n  `eval-proof `swap T (suc zero) = refl\n  `eval-proof `swap T (suc (suc i)) = refl\n  `eval-proof (`tail S) T zero = refl\n  `eval-proof (`tail S) T (suc i) = `eval-proof S (tail T) i\n  `eval-proof (S `\u2218 S\u2081) T i rewrite\n    `eval-proof S\u2081 T i |\n    `eval-proof S (`evalTree S\u2081 T) (`evalArg S\u2081 i) = refl\n\n  `inv : \u2200 {i} \u2192 Endo (`Syn i)\n  `inv `id       = `id\n  `inv `swap     = `swap\n  `inv (`tail S) = `tail (`inv S)\n  `inv (S `\u2218 S\u2081) = `inv S\u2081 `\u2218 `inv S\n\n  `inv-proof : \u2200 {i} \u2192 (S : `Syn i) \u2192 `evalArg S \u2218 `evalArg (`inv S) \u2257 id\n  `inv-proof `id x       = refl\n  `inv-proof `swap zero          = refl\n  `inv-proof `swap (suc zero)    = refl\n  `inv-proof `swap (suc (suc x)) = refl\n  `inv-proof (`tail S) zero = refl\n  `inv-proof (`tail S) (suc x) rewrite `inv-proof S x = refl\n  `inv-proof (S `\u2218 S\u2081) x rewrite \n    `inv-proof S\u2081 (`evalArg (`inv S) x) |\n    `inv-proof S x = refl\n\n  `RC : \u2200 {i} \u2192 Cmp (`Rep i)\n  `RC zero zero = eq\n  `RC zero (suc j) = lt\n  `RC (suc i) zero = gt\n  `RC (suc i) (suc j) = `RC i j\n\n  insert : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 Vec X (1 + n)\n  insert X-cmp x [] = x \u2237 []\n  insert X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert X-cmp x (x\u2081 \u2237 xs) | lt = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | eq = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | gt = x\u2081 \u2237 insert X-cmp x xs\n\n  `sort : \u2200 {i X} \u2192 Cmp X \u2192 Endo (`Tree X i)\n  `sort X-cmp [] = []\n  `sort X-cmp (x \u2237 xs) = insert X-cmp x (`sort X-cmp xs)\n\n  insert-syn : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 `Syn (1 + n)\n  insert-syn X-cmp x [] = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | lt = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | eq = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | gt = `tail (insert-syn X-cmp x xs) `\u2218 `swap\n\n  `sort-syn : \u2200 {i X} \u2192 Cmp X \u2192 `Tree X i \u2192 `Syn i\n  `sort-syn X-cmp []       = `id\n  `sort-syn X-cmp (x \u2237 xs) = insert-syn X-cmp x (`sort X-cmp xs) `\u2218 `tail (`sort-syn X-cmp xs)\n\n  insert-proof : \u2200 {n X}(X-cmp : Cmp X) x (T : Vec X n) \u2192 insert X-cmp x T \u2261 `evalTree (insert-syn X-cmp x T) (x \u2237 T)\n  insert-proof X-cmp x [] = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) with X-cmp x x\u2081\n  insert-proof X-cmp x (x\u2081 \u2237 T) | lt = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | eq = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | gt rewrite insert-proof X-cmp x T = refl\n\n  `sort-proof : \u2200 {i X}(X-cmp : Cmp X)(T : `Tree X i) \u2192 `sort X-cmp T \u2261 `evalTree (`sort-syn X-cmp T) T\n  `sort-proof X-cmp [] = refl\n  `sort-proof X-cmp (x \u2237 T) rewrite \n    sym (`sort-proof X-cmp T)= insert-proof X-cmp x (`sort X-cmp T)\n\n  module Alt-Syn where\n\n    data ``Syn : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 ``Syn n\n      _`\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n      `swap : \u2200 {n} m \u2192 ``Syn (m + 2 + n)\n\n    swap-fin : \u2200 {n} m \u2192 Endo (Fin (m + 2 + n))\n    swap-fin zero zero = suc zero\n    swap-fin zero (suc zero) = zero\n    swap-fin zero (suc (suc i)) = suc (suc i)\n    swap-fin (suc m) zero = zero\n    swap-fin (suc m) (suc i) = suc (swap-fin m i)\n\n    ``evalArg : \u2200 {n} \u2192 ``Syn n \u2192 Endo (`Rep n)\n    ``evalArg `id       = id\n    ``evalArg (S `\u2218 S\u2081) = ``evalArg S \u2218 ``evalArg S\u2081\n    ``evalArg (`swap m) = swap-fin m\n\n    _``\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n    `id ``\u2218 y = y\n    (x `\u2218 x\u2081) ``\u2218 `id = x `\u2218 x\u2081\n    (x `\u2218 x\u2081) ``\u2218 (y `\u2218 y\u2081) = x `\u2218 (x\u2081 `\u2218 (y `\u2218 y\u2081))\n    (x `\u2218 x\u2081) ``\u2218 `swap m = x `\u2218 (x\u2081 ``\u2218 `swap m)\n    `swap m ``\u2218 y = `swap m `\u2218 y\n\n    ``tail : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn (suc n)\n    ``tail `id = `id\n    ``tail (S `\u2218 S\u2081) = ``tail S ``\u2218 ``tail S\u2081\n    ``tail (`swap m) = `swap (suc m)\n\n    translate : \u2200 {n} \u2192 `Syn n \u2192 ``Syn n\n    translate `id = `id\n    translate `swap = `swap 0\n    translate (`tail S) = ``tail (translate S)\n    translate (S `\u2218 S\u2081) = translate S ``\u2218 translate S\u2081\n\n    ``\u2218-p : \u2200 {n}(A B : ``Syn n) \u2192 ``evalArg (A ``\u2218 B) \u2257 ``evalArg (A `\u2218 B)\n    ``\u2218-p `id B x = refl\n    ``\u2218-p (A `\u2218 A\u2081) `id x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (B `\u2218 B\u2081) x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (`swap m) x rewrite ``\u2218-p A\u2081 (`swap m) x = refl\n    ``\u2218-p (`swap m) B x = refl\n\n    ``tail-p : \u2200 {n} (S : ``Syn n) \u2192 fin-tail (``evalArg S) \u2257 ``evalArg (``tail S)\n    ``tail-p `id zero = refl\n    ``tail-p `id (suc x) = refl\n    ``tail-p (S `\u2218 S\u2081) zero rewrite ``\u2218-p (``tail S) (``tail S\u2081) zero\n                                  | sym (``tail-p S\u2081 zero) = ``tail-p S zero\n    ``tail-p (S `\u2218 S\u2081) (suc x) rewrite ``\u2218-p (``tail S) (``tail S\u2081) (suc x)\n                                     | sym (``tail-p S\u2081 (suc x)) = ``tail-p S (suc (``evalArg S\u2081 x))\n    ``tail-p (`swap m) zero = refl\n    ``tail-p (`swap m) (suc x) = refl\n\n    `eval`` : \u2200 {n} (S : `Syn n) \u2192 `evalArg S \u2257 ``evalArg (translate S)\n    `eval`` `id       x = refl\n    `eval`` `swap zero = refl\n    `eval`` `swap (suc zero) = refl\n    `eval`` `swap (suc (suc x)) = refl\n    `eval`` (`tail S) zero = ``tail-p (translate S) zero\n    `eval`` (`tail S) (suc x) rewrite `eval`` S x = ``tail-p (translate S) (suc x)\n    `eval`` (S `\u2218 S\u2081) x rewrite ``\u2218-p (translate S) (translate S\u2081) x | sym (`eval`` S\u2081 x) | `eval`` S (`evalArg S\u2081 x) = refl\n\n\n  data Fin-View : \u2200 {n} \u2192 Fin n \u2192 Set where\n    max : \u2200 {n} \u2192 Fin-View (from\u2115 n)\n    inject : \u2200 {n} \u2192 (i : Fin n) \u2192 Fin-View (inject\u2081 i)\n\n  data _\u2264F_ : \u2200 {n} \u2192 Fin n \u2192 Fin n \u2192 Set where\n    z\u2264i : {n : \u2115}{i : Fin (suc n)} \u2192 zero \u2264F i\n    s\u2264s : {n : \u2115}{i j : Fin n} \u2192 i \u2264F j \u2192 suc i \u2264F suc j\n\n  \u2264F-refl : \u2200 {n} (x : Fin n) \u2192 x \u2264F x\n  \u2264F-refl zero = z\u2264i\n  \u2264F-refl (suc i) = s\u2264s (\u2264F-refl i)\n\n  _<F_ : \u2200 {n} \u2192 Fin n \u2192 Fin n \u2192 Set\n  i <F j = suc i \u2264F inject\u2081 j\n\n  nsuc-inj : \u2200 {x y} \u2192 Data.Nat.NP.suc x \u2261 suc y \u2192 x \u2261 y\n  nsuc-inj refl = refl\n\n  suc-inj : \u2200 {n}{i j : Fin n} \u2192 Data.Fin.suc i \u2261 suc j \u2192 i \u2261 j\n  suc-inj refl = refl \n\n  data Sorted {X}(XC : Cmp X) : \u2200 {l} \u2192 Vec X l  \u2192 Set where\n    []  : Sorted XC []\n    sing : \u2200 x \u2192 Sorted XC (x \u2237 [])\n    dbl-lt  : \u2200 {l} x y {xs : Vec X l} \u2192 lt \u2261 XC x y \u2192 Sorted XC (y \u2237 xs) \u2192 Sorted XC (x \u2237 y \u2237 xs)\n    dbl-eq  : \u2200 {l} x {xs : Vec X l} \u2192 Sorted XC (x \u2237 xs) \u2192 Sorted XC (x \u2237 x \u2237 xs)\n\n  opposite : Ord \u2192 Ord\n  opposite lt = gt\n  opposite eq = eq\n  opposite gt = lt\n\n  flip-RC : \u2200 {n}(x y : Fin n) \u2192 opposite (`RC x y) \u2261 `RC y x\n  flip-RC zero zero = refl\n  flip-RC zero (suc y) = refl\n  flip-RC (suc x) zero = refl\n  flip-RC (suc x) (suc y) = flip-RC x y\n\n  eq=>\u2261 : \u2200 {i} (x y : Fin i) \u2192 eq \u2261 `RC x y \u2192 x \u2261 y\n  eq=>\u2261 zero zero p = refl\n  eq=>\u2261 zero (suc y) ()\n  eq=>\u2261 (suc x) zero ()\n  eq=>\u2261 (suc x) (suc y) p rewrite eq=>\u2261 x y p = refl\n\n  insert-Sorted : \u2200 {n l}{V : Vec (Fin n) l}(x : Fin n) \u2192 Sorted {Fin n} `RC V \u2192 Sorted {Fin n} `RC (insert `RC x V)\n  insert-Sorted x [] = sing x\n  insert-Sorted x (sing x\u2081) with `RC x x\u2081 | dbl-lt {XC = `RC} x x\u2081 {[]} | eq=>\u2261 x x\u2081 | flip-RC x x\u2081\n  insert-Sorted x (sing x\u2081) | lt | b | _ | _ = b refl (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | eq | _ | p | _ rewrite p refl = dbl-eq x\u2081 (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | gt | b | _ | l = dbl-lt x\u2081 x l (sing x)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) with `RC x y | dbl-lt {XC = `RC} x y {y' \u2237 xs} | eq=>\u2261 x y | flip-RC x y\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | lt | b | p | l\u2081 = b refl (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | eq | b | p | l\u2081 rewrite p refl = dbl-eq y (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) | gt | b | p | l\u2081 with `RC x y' | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | lt | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | eq | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | gt | xs' = dbl-lt y y' prf xs'\n  insert-Sorted x (dbl-eq y {xs} xs\u2081) with `RC x y | inspect (`RC x) y | dbl-lt {XC = `RC} x y {y \u2237 xs} | eq=>\u2261 x y | flip-RC x y | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-eq y xs\u2081) | lt | _ | b | p | l | _ = b refl (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | eq | _ | b | p | l | _ rewrite p refl = dbl-eq y (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | gt | [ prf ] | b | p | l | ss  rewrite prf = dbl-eq y ss \n\n  sort-Sorted : \u2200 {n l}(V : Vec (Fin n) l) \u2192 Sorted `RC (`sort `RC V)\n  sort-Sorted [] = []\n  sort-Sorted (x \u2237 V) = insert-Sorted x (sort-Sorted V)\n\n  RC-refl : \u2200 {i}(x : Fin i) \u2192 `RC x x \u2261 eq\n  RC-refl zero = refl\n  RC-refl (suc x) = RC-refl x\n\n  STail : \u2200 {X l}{XC : Cmp X}{xs : Vec X (suc l)} \u2192 Sorted XC xs \u2192 Sorted XC (tail xs)\n  STail (sing x) = []\n  STail (dbl-lt x y x\u2081 T) = T\n  STail (dbl-eq x T) = T\n\n  module sproof {X}(XC : Cmp X)(XC-refl : \u2200 x \u2192 XC x x \u2261 eq)\n     (eq\u2261 : \u2200 x y \u2192 XC x y \u2261 eq \u2192 x \u2261 y)\n     (lt-trans : \u2200 x y z \u2192 XC x y \u2261 lt \u2192 XC y z \u2261 lt \u2192 XC x z \u2261 lt)\n     (XC-flip : \u2200 x y \u2192 opposite (XC x y) \u2261 XC y x)\n     where\n\n    open import Data.Sum\n\n    _\u2264X_ : X \u2192 X \u2192 Set\n    x \u2264X y = XC x y \u2261 lt \u228e XC x y \u2261 eq\n\n    \u2264X-trans : \u2200 {x y z} \u2192 x \u2264X y \u2192 y \u2264X z \u2192 x \u2264X z\n    \u2264X-trans (inj\u2081 x\u2081) (inj\u2081 x\u2082) = inj\u2081 (lt-trans _ _ _ x\u2081 x\u2082)\n    \u2264X-trans {_}{y}{z}(inj\u2081 x\u2081) (inj\u2082 y\u2081) rewrite eq\u2261 y z y\u2081 = inj\u2081 x\u2081\n    \u2264X-trans {x}{y} (inj\u2082 y\u2081) y\u2264z rewrite eq\u2261 x y y\u2081 = y\u2264z\n\n    h\u2264t : \u2200 {n}{T : `Tree X (2 + n)} \u2192 Sorted XC T \u2192 head T \u2264X head (tail T)\n    h\u2264t (dbl-lt x y x\u2081 ST) = inj\u2081 (sym x\u2081)\n    h\u2264t (dbl-eq x ST) rewrite XC-refl x = inj\u2082 refl\n\n    head-p : \u2200 {n}{T : `Tree X (suc n)} i \u2192 Sorted XC T \u2192 head T \u2264X `toFun T i\n    head-p {T = T} zero ST rewrite XC-refl (head T) = inj\u2082 refl\n    head-p {zero} (suc ()) ST\n    head-p {suc n} (suc i) ST = \u2264X-trans (h\u2264t ST) (head-p i (STail ST))\n\n    toFun-p : \u2200 {n}{T : `Tree X n}{i j : Fin n} \u2192 i \u2264F j \u2192 Sorted XC T \u2192 `toFun T i \u2264X `toFun T j\n    toFun-p {j = j} z\u2264i ST = head-p j ST\n    toFun-p (s\u2264s i\u2264Fj) ST = toFun-p i\u2264Fj (STail ST)\n\n    sort-proof : \u2200 {i}{T : `Tree X i} \u2192 Sorted XC T \u2192 Is-Mono `RC XC (`toFun T)\n    sort-proof {T = T} T\u2081 zero zero rewrite XC-refl (head T) = _\n    sort-proof T\u2081 zero (suc y) with toFun-p (z\u2264i {i = suc y}) T\u2081\n    sort-proof T zero (suc y) | inj\u2081 x rewrite x = _\n    sort-proof T zero (suc y) | inj\u2082 y\u2081 rewrite y\u2081 = _\n    sort-proof {T = T} T\u2081 (suc x) zero with toFun-p (z\u2264i {i = suc x}) T\u2081 | XC-flip (head T) (`toFun (tail T) x)\n    sort-proof T (suc x) zero | inj\u2081 x\u2081 | l rewrite x\u2081 | sym l = _\n    sort-proof T (suc x) zero | inj\u2082 y | l rewrite y | sym l = _\n    sort-proof T\u2081 (suc x) (suc y) = sort-proof (STail T\u2081) x y\n\n  lt-trans-RC : \u2200 {i} (x y z : Fin i) \u2192 `RC x y \u2261 lt \u2192 `RC y z \u2261 lt \u2192 `RC x z \u2261 lt\n  lt-trans-RC zero zero zero x<y y<z = y<z\n  lt-trans-RC zero zero (suc z) x<y y<z = refl\n  lt-trans-RC zero (suc y) zero x<y ()\n  lt-trans-RC zero (suc y) (suc z) x<y y<z = refl\n  lt-trans-RC (suc x) zero zero x<y y<z = x<y\n  lt-trans-RC (suc x) zero (suc z) () y<z\n  lt-trans-RC (suc x) (suc y) zero x<y y<z = y<z\n  lt-trans-RC (suc x) (suc y) (suc z) x<y y<z = lt-trans-RC x y z x<y y<z\n\n  `sort-mono : \u2200 {i}(T : `Tree (`Rep i) i) \u2192 Is-Mono `RC `RC (`toFun (`sort `RC T))\n  `sort-mono T x y = sproof.sort-proof `RC RC-refl (\u03bb x\u2081 y\u2081 x\u2082 \u2192 eq=>\u2261 x\u2081 y\u2081 (sym x\u2082)) lt-trans-RC flip-RC (sort-Sorted T) x y\n\n\n\n  module toNat n (f : Endo (Fin (suc n)))(f-inj : Is-Inj f)(f-mono : Is-Mono `RC `RC f) where\n  \n    import prefect-bintree-sorting\n    open prefect-bintree-sorting.MM\n    open import Data.Sum\n\n    move-to-RC : \u2200 {n}{x y : Fin n} \u2192 x \u2264F y \u2192 `RC x y \u2261 lt \u228e `RC x y \u2261 eq\n    move-to-RC {y = zero} z\u2264i = inj\u2082 refl\n    move-to-RC {y = suc y} z\u2264i = inj\u2081 refl\n    move-to-RC (s\u2264s x\u2264Fy) = move-to-RC x\u2264Fy\n\n    move-from-RC : \u2200 {n}(x y : Fin n) \u2192 lt \u2261 `RC x y \u228e eq \u2261 `RC x y \u2192 x \u2264F y\n    move-from-RC zero zero prf = z\u2264i\n    move-from-RC zero (suc y) prf = z\u2264i\n    move-from-RC (suc x) zero (inj\u2081 ())\n    move-from-RC (suc x) zero (inj\u2082 ())\n    move-from-RC (suc x) (suc y) prf = s\u2264s (move-from-RC x y prf)\n\n    proper-mono : \u2200 {x y} \u2192 x \u2264F y \u2192 f x \u2264F f y\n    proper-mono {x} {y} x\u2264Fy with `RC x y | `RC (f x) (f y) | move-to-RC x\u2264Fy | f-mono x y | move-from-RC (f x) (f y)\n    proper-mono x\u2264Fy | .lt | lt | inj\u2081 refl | r4 | r5 = r5 (inj\u2081 refl)\n    proper-mono x\u2264Fy | .lt | eq | inj\u2081 refl | r4 | r5 = r5 (inj\u2082 refl)\n    proper-mono x\u2264Fy | .lt | gt | inj\u2081 refl | () | r5\n    proper-mono x\u2264Fy | .eq | lt | inj\u2082 refl | () | r5\n    proper-mono x\u2264Fy | .eq | eq | inj\u2082 refl | r4 | r5 = r5 (inj\u2082 refl)\n    proper-mono x\u2264Fy | .eq | gt | inj\u2082 refl | () | r5\n\n    getFrom : \u2200 n \u2192 \u2115 \u2192 Fin (suc n)\n    getFrom zero i = zero\n    getFrom (suc n\u2081) zero = zero\n    getFrom (suc n\u2081) (suc i) = suc (getFrom n\u2081 i)\n\n    getInj : {n x y : \u2115} \u2192 x \u2264 n \u2192 y \u2264 n \u2192 getFrom n x \u2261 getFrom n y \u2192 x \u2261 y\n    getInj z\u2264n z\u2264n prf = refl\n    getInj z\u2264n (s\u2264s y\u2264n) ()\n    getInj (s\u2264s x\u2264n) z\u2264n ()\n    getInj (s\u2264s x\u2264n) (s\u2264s y\u2264n) prf rewrite (getInj x\u2264n y\u2264n (suc-inj prf)) = refl\n\n    getMono : {n x y : \u2115} \u2192 x \u2264 y \u2192 y \u2264 n \u2192 getFrom n x \u2264F getFrom n y\n    getMono z\u2264n z\u2264n = \u2264F-refl _\n    getMono z\u2264n (s\u2264s y\u2264n) = z\u2264i\n    getMono (s\u2264s x\u2264y) (s\u2264s y\u2264n) = s\u2264s (getMono x\u2264y y\u2264n)\n\n    forget : \u2200 {n} \u2192 Fin n \u2192 \u2115\n    forget zero = zero\n    forget (suc i) = suc (forget i)\n\n    forgetInj : \u2200 {n}{i j : Fin n} \u2192 forget i \u2261 forget j \u2192 i \u2261 j\n    forgetInj {.(suc _)} {zero} {zero} prf = refl\n    forgetInj {.(suc _)} {zero} {suc j} ()\n    forgetInj {.(suc _)} {suc i} {zero} ()\n    forgetInj {.(suc _)} {suc i} {suc j} prf rewrite forgetInj (nsuc-inj prf) = refl\n\n    getForget : \u2200 {n}(i : Fin (suc n)) \u2192 getFrom n (forget i) \u2261 i\n    getForget {zero} zero = refl\n    getForget {zero} (suc ())\n    getForget {suc n\u2081} zero = refl\n    getForget {suc n\u2081} (suc i) rewrite getForget i = refl\n\n\n    forget< : \u2200 {n} \u2192 (i : Fin n) \u2192 forget i < n\n    forget< {zero} ()\n    forget< {suc n\u2081} zero = s\u2264s z\u2264n\n    forget< {suc n\u2081} (suc i) = s\u2264s (forget< i)\n\n    forget-mono : \u2200 {n}{i j : Fin n} \u2192 i \u2264F j \u2192 forget i \u2264 forget j\n    forget-mono z\u2264i = z\u2264n\n    forget-mono (s\u2264s i\u2264F) = s\u2264s (forget-mono i\u2264F)\n\n    fn : Endo \u2115\n    fn = forget \u2218 f \u2218 getFrom n\n\n    return : f \u2257 getFrom n \u2218 fn \u2218 forget\n    return x rewrite getForget x | getForget (f x) = refl\n\n    fn-monotone : Monotone (suc n) fn\n    fn-monotone {x} {y} x\u2264y (s\u2264s y\u2264n) = forget-mono (proper-mono (getMono x\u2264y y\u2264n))\n\n    fn-inj : IsInj (suc n) fn\n    fn-inj {x}{y} (s\u2264s sx\u2264sn) (s\u2264s sy\u2264sn) prf = getInj sx\u2264sn sy\u2264sn (f-inj (getFrom n x) (getFrom n y) (forgetInj prf))\n\n    fn-bounded : Bounded (suc n) fn\n    fn-bounded x _ = forget< (f (getFrom n x))\n\n    fn\u2257id : \u2200 x \u2192 x < (suc n) \u2192 fn x \u2261 x\n    fn\u2257id = M.is-id fn fn-monotone fn-inj fn-bounded \n\n    f\u2257id : f \u2257 id\n    f\u2257id x rewrite return x | fn\u2257id (forget x) (forget< x) = getForget x\n\n  fin-view : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 Fin-View i\n  fin-view {zero} zero = max\n  fin-view {zero} (suc ())\n  fin-view {suc n} zero = inject _\n  fin-view {suc n} (suc i) with fin-view i\n  fin-view {suc n} (suc .(from\u2115 n)) | max = max\n  fin-view {suc n} (suc .(inject\u2081 i)) | inject i = inject _\n\n  absurd : {X : Set} \u2192 .\u22a5 \u2192 X\n  absurd ()\n\n  drop\u2081 : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 .(i \u2262 from\u2115 n) \u2192 Fin n\n  drop\u2081 i neq with fin-view i\n  drop\u2081 {n} .(from\u2115 n) neq | max = absurd (neq refl)\n  drop\u2081 .(inject\u2081 i) neq | inject i = i\n\n  drop\u2081\u2192inject\u2081 : \u2200 {n}(i : Fin (suc n))(j : Fin n).(p : i \u2262 from\u2115 n) \u2192 drop\u2081 i p \u2261 j \u2192 i \u2261 inject\u2081 j\n  drop\u2081\u2192inject\u2081 i j p q with fin-view i\n  drop\u2081\u2192inject\u2081 {n} .(from\u2115 n) j p q | max = absurd (p refl)\n  drop\u2081\u2192inject\u2081 .(inject\u2081 i) j p q | inject i = cong inject\u2081 q\n\n\n  `mono-inj\u2192id : \u2200{i}(f : Endo (`Rep i)) \u2192 Is-Inj f \u2192 Is-Mono `RC `RC f \u2192 f \u2257 id\n  `mono-inj\u2192id {zero}  = \u03bb f x x\u2081 ()\n  `mono-inj\u2192id {suc i} = toNat.f\u2257id i \n\n\n  interface : Interface\n  interface = record \n    { Ix            = `Ix\n    ; Rep           = `Rep\n    ; Syn           = `Syn\n    ; Tree          = `Tree\n    ; fromFun       = `fromFun\n    ; toFun         = `toFun\n    ; toFun\u2218fromFun = `toFun\u2218fromFun\n    ; evalArg       = `evalArg\n    ; evalTree      = `evalTree\n    ; eval-proof    = `eval-proof\n    ; inv           = `inv\n    ; inv-proof     = `inv-proof\n    ; RC            = `RC\n    ; sort          = `sort\n    ; sort-syn      = `sort-syn\n    ; sort-proof    = `sort-proof\n    ; sort-mono     = `sort-mono\n    ; mono-inj\u2192id   = `mono-inj\u2192id\n    }\n\n  open import Data.Bool\n\n  count : \u2200 {n} \u2192 (Fin n \u2192 \u2115) \u2192 \u2115\n  count {n} f = sum (tabulate f)\n\n  #\u27e8_\u27e9 : \u2200 {n} \u2192 (Fin n \u2192 Bool) \u2192 \u2115\n  #\u27e8 f \u27e9 = count (\u03bb x \u2192 if f x then 1 else 0)\n\n  #-ext : \u2200 {n} \u2192 (f g : Fin n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n  #-ext {zero} f g f\u2257g = refl\n  #-ext {suc n} f g f\u2257g rewrite f\u2257g zero | #-ext (f \u2218 suc) (g \u2218 suc) (f\u2257g \u2218 suc) = refl\n\n  com-assoc : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n  com-assoc x y z rewrite \n    sym (\u2115\u00b0.+-assoc x y z) |\n    \u2115\u00b0.+-comm x y    |\n    \u2115\u00b0.+-assoc y x z = refl\n    \n  syn-pres : \u2200 {n}(f : Fin n \u2192 \u2115)(S : `Syn n)\n           \u2192 count f \u2261 count (f \u2218 `evalArg S)\n  syn-pres f `id = refl\n  syn-pres f `swap = com-assoc (f zero) (f (suc zero)) (count (\u03bb i \u2192 f (suc (suc i))))\n  syn-pres f (`tail S) rewrite syn-pres (f \u2218 suc) S = refl\n  syn-pres f (S `\u2218 S\u2081) rewrite syn-pres f S = syn-pres (f \u2218 `evalArg S) S\u2081\n\n  #-perm : \u2200 {n}(f : Fin n \u2192 Bool)(p : Endo (Fin n)) \u2192 Is-Inj p\n         \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 p \u27e9\n  #-perm f p p-inj = trans (syn-pres (\u03bb x \u2192 if f x then 1 else 0) (sort-bij p))\n                       (#-ext _ _ f\u2218eval\u2257f\u2218p)\n    where \n      open abs interface\n      f\u2218eval\u2257f\u2218p : f \u2218 `evalArg (sort-bij p) \u2257 f \u2218 p\n      f\u2218eval\u2257f\u2218p x rewrite thm p p-inj x = refl\n\n  test : `Syn 8\n  test = abs.sort-bij interface (\u03bb x \u2192 `evalArg (`tail `swap) x)\n","old_contents":"module bijection-fin where\n\n  open import bijection\n  open import Function.NP hiding (Cmp)\n  open import Relation.Binary.PropositionalEquality\n\n  open import Data.Empty\n  open import Data.Nat.NP\n  open import Data.Fin using (Fin ; zero ; suc ; from\u2115 ; inject\u2081)\n  open import Data.Vec hiding ([_])\n\n  data `Syn : \u2115 \u2192 Set where \n    `id   : \u2200 {n} \u2192 `Syn n\n    `swap : \u2200 {n} \u2192 `Syn (2 + n)\n    `tail : \u2200 {n} \u2192 `Syn n \u2192 `Syn (1 + n)\n    _`\u2218_  : \u2200 {n} \u2192 `Syn n \u2192 `Syn n \u2192 `Syn n\n\n  `Rep = Fin\n\n  `Ix = \u2115\n\n  `Tree : Set \u2192 `Ix \u2192 Set\n  `Tree X = Vec X\n\n  `fromFun : \u2200 {i X} \u2192 (`Rep i \u2192 X) \u2192 `Tree X i\n  `fromFun = tabulate\n\n  `toFun : \u2200 {i X} \u2192 `Tree X i \u2192 (`Rep i \u2192 X)\n  `toFun T zero    = head T\n  `toFun T (suc i) = `toFun (tail T) i\n\n  `toFun\u2218fromFun : \u2200 {i X}(f : `Rep i \u2192 X) \u2192 f \u2257 `toFun (`fromFun f)\n  `toFun\u2218fromFun f zero = refl\n  `toFun\u2218fromFun f (suc i) = `toFun\u2218fromFun (f \u2218 suc) i\n\n  fin-swap : \u2200 {n} \u2192 Endo (Fin (2 + n))\n  fin-swap zero = suc zero\n  fin-swap (suc zero) = zero\n  fin-swap (suc (suc i)) = suc (suc i)\n\n  fin-tail : \u2200 {n} \u2192 Endo (Fin n) \u2192 Endo (Fin (1 + n))\n  fin-tail f zero = zero\n  fin-tail f (suc i) = suc (f i)\n\n  `evalArg : \u2200 {i} \u2192 `Syn i \u2192 Endo (`Rep i)\n  `evalArg `id       = id\n  `evalArg `swap     = fin-swap\n  `evalArg (`tail f) = fin-tail (`evalArg f)\n  `evalArg (S `\u2218 S\u2081) = `evalArg S \u2218 `evalArg S\u2081\n\n  vec-swap : \u2200 {n}{X : Set} \u2192 Endo (Vec X (2 + n))\n  vec-swap xs = head (tail xs) \u2237 head xs \u2237 tail (tail xs)\n\n  vec-tail : \u2200 {n}{X : Set} \u2192 Endo (Vec X n) \u2192 Endo (Vec X (1 + n))\n  vec-tail f xs = head xs \u2237 f (tail xs)\n\n  `evalTree : \u2200 {i X} \u2192 `Syn i \u2192 Endo (`Tree X i)\n  `evalTree `id       = id\n  `evalTree `swap     = vec-swap\n  `evalTree (`tail f) = vec-tail (`evalTree f)\n  `evalTree (S `\u2218 S\u2081) = `evalTree S \u2218 `evalTree S\u2081\n\n  `eval-proof : \u2200 {i X} S (T : `Tree X i) \u2192 `toFun T \u2257 `toFun (`evalTree S T) \u2218 `evalArg S\n  `eval-proof `id       T i = refl\n  `eval-proof `swap T zero = refl\n  `eval-proof `swap T (suc zero) = refl\n  `eval-proof `swap T (suc (suc i)) = refl\n  `eval-proof (`tail S) T zero = refl\n  `eval-proof (`tail S) T (suc i) = `eval-proof S (tail T) i\n  `eval-proof (S `\u2218 S\u2081) T i rewrite\n    `eval-proof S\u2081 T i |\n    `eval-proof S (`evalTree S\u2081 T) (`evalArg S\u2081 i) = refl\n\n  `inv : \u2200 {i} \u2192 Endo (`Syn i)\n  `inv `id       = `id\n  `inv `swap     = `swap\n  `inv (`tail S) = `tail (`inv S)\n  `inv (S `\u2218 S\u2081) = `inv S\u2081 `\u2218 `inv S\n\n  `inv-proof : \u2200 {i} \u2192 (S : `Syn i) \u2192 `evalArg S \u2218 `evalArg (`inv S) \u2257 id\n  `inv-proof `id x       = refl\n  `inv-proof `swap zero          = refl\n  `inv-proof `swap (suc zero)    = refl\n  `inv-proof `swap (suc (suc x)) = refl\n  `inv-proof (`tail S) zero = refl\n  `inv-proof (`tail S) (suc x) rewrite `inv-proof S x = refl\n  `inv-proof (S `\u2218 S\u2081) x rewrite \n    `inv-proof S\u2081 (`evalArg (`inv S) x) |\n    `inv-proof S x = refl\n\n  `RC : \u2200 {i} \u2192 Cmp (`Rep i)\n  `RC zero zero = eq\n  `RC zero (suc j) = lt\n  `RC (suc i) zero = gt\n  `RC (suc i) (suc j) = `RC i j\n\n  insert : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 Vec X (1 + n)\n  insert X-cmp x [] = x \u2237 []\n  insert X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert X-cmp x (x\u2081 \u2237 xs) | lt = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | eq = x \u2237 x\u2081 \u2237 xs\n  insert X-cmp x (x\u2081 \u2237 xs) | gt = x\u2081 \u2237 insert X-cmp x xs\n\n  `sort : \u2200 {i X} \u2192 Cmp X \u2192 Endo (`Tree X i)\n  `sort X-cmp [] = []\n  `sort X-cmp (x \u2237 xs) = insert X-cmp x (`sort X-cmp xs)\n\n  insert-syn : \u2200 {n X} \u2192 Cmp X \u2192 X \u2192 Vec X n \u2192 `Syn (1 + n)\n  insert-syn X-cmp x [] = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) with X-cmp x x\u2081\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | lt = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | eq = `id\n  insert-syn X-cmp x (x\u2081 \u2237 xs) | gt = `tail (insert-syn X-cmp x xs) `\u2218 `swap\n\n  `sort-syn : \u2200 {i X} \u2192 Cmp X \u2192 `Tree X i \u2192 `Syn i\n  `sort-syn X-cmp []       = `id\n  `sort-syn X-cmp (x \u2237 xs) = insert-syn X-cmp x (`sort X-cmp xs) `\u2218 `tail (`sort-syn X-cmp xs)\n\n  insert-proof : \u2200 {n X}(X-cmp : Cmp X) x (T : Vec X n) \u2192 insert X-cmp x T \u2261 `evalTree (insert-syn X-cmp x T) (x \u2237 T)\n  insert-proof X-cmp x [] = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) with X-cmp x x\u2081\n  insert-proof X-cmp x (x\u2081 \u2237 T) | lt = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | eq = refl\n  insert-proof X-cmp x (x\u2081 \u2237 T) | gt rewrite insert-proof X-cmp x T = refl\n\n  `sort-proof : \u2200 {i X}(X-cmp : Cmp X)(T : `Tree X i) \u2192 `sort X-cmp T \u2261 `evalTree (`sort-syn X-cmp T) T\n  `sort-proof X-cmp [] = refl\n  `sort-proof X-cmp (x \u2237 T) rewrite \n    sym (`sort-proof X-cmp T)= insert-proof X-cmp x (`sort X-cmp T)\n\n  module Alt-Syn where\n\n    data ``Syn : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 ``Syn n\n      _`\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n      `swap : \u2200 {n} m \u2192 ``Syn (m + 2 + n)\n\n    swap-fin : \u2200 {n} m \u2192 Endo (Fin (m + 2 + n))\n    swap-fin zero zero = suc zero\n    swap-fin zero (suc zero) = zero\n    swap-fin zero (suc (suc i)) = suc (suc i)\n    swap-fin (suc m) zero = zero\n    swap-fin (suc m) (suc i) = suc (swap-fin m i)\n\n    ``evalArg : \u2200 {n} \u2192 ``Syn n \u2192 Endo (`Rep n)\n    ``evalArg `id       = id\n    ``evalArg (S `\u2218 S\u2081) = ``evalArg S \u2218 ``evalArg S\u2081\n    ``evalArg (`swap m) = swap-fin m\n    \n    ``tail : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn (suc n)\n    ``tail `id = `id\n    ``tail (S `\u2218 S\u2081) = ``tail S `\u2218 ``tail S\u2081\n    ``tail (`swap m) = `swap (suc m)\n\n    _``\u2218_ : \u2200 {n} \u2192 ``Syn n \u2192 ``Syn n \u2192 ``Syn n\n    `id ``\u2218 y = y\n    (x `\u2218 x\u2081) ``\u2218 `id = x `\u2218 x\u2081\n    (x `\u2218 x\u2081) ``\u2218 (y `\u2218 y\u2081) = x `\u2218 (x\u2081 `\u2218 (y `\u2218 y\u2081))\n    (x `\u2218 x\u2081) ``\u2218 `swap m = x `\u2218 (x\u2081 ``\u2218 `swap m)\n    `swap m ``\u2218 y = `swap m `\u2218 y\n\n    translate : \u2200 {n} \u2192 `Syn n \u2192 ``Syn n\n    translate `id = `id\n    translate `swap = `swap 0\n    translate (`tail S) = ``tail (translate S)\n    translate (S `\u2218 S\u2081) = translate S ``\u2218 translate S\u2081\n\n    ``tail-p : \u2200 {n} (S : ``Syn n) \u2192 fin-tail (``evalArg S) \u2257 ``evalArg (``tail S)\n    ``tail-p `id zero = refl\n    ``tail-p `id (suc x) = refl\n    ``tail-p (S `\u2218 S\u2081) zero rewrite sym (``tail-p S\u2081 zero)   = ``tail-p S zero\n    ``tail-p (S `\u2218 S\u2081) (suc x) rewrite sym (``tail-p S\u2081 (suc x)) = ``tail-p S (suc (``evalArg S\u2081 x))\n    ``tail-p (`swap m) zero = refl\n    ``tail-p (`swap m) (suc x) = refl\n\n    ``\u2218-p : \u2200 {n}(A B : ``Syn n) \u2192 ``evalArg (A ``\u2218 B) \u2257 ``evalArg A \u2218 ``evalArg B\n    ``\u2218-p `id B x = refl\n    ``\u2218-p (A `\u2218 A\u2081) `id x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (B `\u2218 B\u2081) x = refl\n    ``\u2218-p (A `\u2218 A\u2081) (`swap m) x rewrite ``\u2218-p A\u2081 (`swap m) x = refl\n    ``\u2218-p (`swap m) B x = refl\n\n    `eval`` : \u2200 {n} (S : `Syn n) \u2192 `evalArg S \u2257 ``evalArg (translate S)\n    `eval`` `id       x = refl\n    `eval`` `swap zero = refl\n    `eval`` `swap (suc zero) = refl\n    `eval`` `swap (suc (suc x)) = refl\n    `eval`` (`tail S) zero = ``tail-p (translate S) zero\n    `eval`` (`tail S) (suc x) rewrite `eval`` S x = ``tail-p (translate S) (suc x)\n    `eval`` (S `\u2218 S\u2081) x rewrite ``\u2218-p (translate S) (translate S\u2081) x | sym (`eval`` S\u2081 x) | `eval`` S (`evalArg S\u2081 x) = refl\n\n\n  data Fin-View : \u2200 {n} \u2192 Fin n \u2192 Set where\n    max : \u2200 {n} \u2192 Fin-View (from\u2115 n)\n    inject : \u2200 {n} \u2192 (i : Fin n) \u2192 Fin-View (inject\u2081 i)\n\n  data _\u2264F_ : \u2200 {n} \u2192 Fin n \u2192 Fin n \u2192 Set where\n    z\u2264i : {n : \u2115}{i : Fin (suc n)} \u2192 zero \u2264F i\n    s\u2264s : {n : \u2115}{i j : Fin n} \u2192 i \u2264F j \u2192 suc i \u2264F suc j\n\n  \u2264F-refl : \u2200 {n} (x : Fin n) \u2192 x \u2264F x\n  \u2264F-refl zero = z\u2264i\n  \u2264F-refl (suc i) = s\u2264s (\u2264F-refl i)\n\n  _<F_ : \u2200 {n} \u2192 Fin n \u2192 Fin n \u2192 Set\n  i <F j = suc i \u2264F inject\u2081 j\n\n  nsuc-inj : \u2200 {x y} \u2192 Data.Nat.NP.suc x \u2261 suc y \u2192 x \u2261 y\n  nsuc-inj refl = refl\n\n  suc-inj : \u2200 {n}{i j : Fin n} \u2192 Data.Fin.suc i \u2261 suc j \u2192 i \u2261 j\n  suc-inj refl = refl \n\n  data Sorted {X}(XC : Cmp X) : \u2200 {l} \u2192 Vec X l  \u2192 Set where\n    []  : Sorted XC []\n    sing : \u2200 x \u2192 Sorted XC (x \u2237 [])\n    dbl-lt  : \u2200 {l} x y {xs : Vec X l} \u2192 lt \u2261 XC x y \u2192 Sorted XC (y \u2237 xs) \u2192 Sorted XC (x \u2237 y \u2237 xs)\n    dbl-eq  : \u2200 {l} x {xs : Vec X l} \u2192 Sorted XC (x \u2237 xs) \u2192 Sorted XC (x \u2237 x \u2237 xs)\n\n  opposite : Ord \u2192 Ord\n  opposite lt = gt\n  opposite eq = eq\n  opposite gt = lt\n\n  flip-RC : \u2200 {n}(x y : Fin n) \u2192 opposite (`RC x y) \u2261 `RC y x\n  flip-RC zero zero = refl\n  flip-RC zero (suc y) = refl\n  flip-RC (suc x) zero = refl\n  flip-RC (suc x) (suc y) = flip-RC x y\n\n  eq=>\u2261 : \u2200 {i} (x y : Fin i) \u2192 eq \u2261 `RC x y \u2192 x \u2261 y\n  eq=>\u2261 zero zero p = refl\n  eq=>\u2261 zero (suc y) ()\n  eq=>\u2261 (suc x) zero ()\n  eq=>\u2261 (suc x) (suc y) p rewrite eq=>\u2261 x y p = refl\n\n  insert-Sorted : \u2200 {n l}{V : Vec (Fin n) l}(x : Fin n) \u2192 Sorted {Fin n} `RC V \u2192 Sorted {Fin n} `RC (insert `RC x V)\n  insert-Sorted x [] = sing x\n  insert-Sorted x (sing x\u2081) with `RC x x\u2081 | dbl-lt {XC = `RC} x x\u2081 {[]} | eq=>\u2261 x x\u2081 | flip-RC x x\u2081\n  insert-Sorted x (sing x\u2081) | lt | b | _ | _ = b refl (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | eq | _ | p | _ rewrite p refl = dbl-eq x\u2081 (sing x\u2081)\n  insert-Sorted x (sing x\u2081) | gt | b | _ | l = dbl-lt x\u2081 x l (sing x)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) with `RC x y | dbl-lt {XC = `RC} x y {y' \u2237 xs} | eq=>\u2261 x y | flip-RC x y\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | lt | b | p | l\u2081 = b refl (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | eq | b | p | l\u2081 rewrite p refl = dbl-eq y (dbl-lt y y' prf xs\u2081)\n  insert-Sorted x (dbl-lt y y' {xs} prf xs\u2081) | gt | b | p | l\u2081 with `RC x y' | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | lt | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | eq | xs' = dbl-lt y x l\u2081 xs'\n  insert-Sorted x (dbl-lt y y' prf xs\u2081) | gt | b | p | l\u2081 | gt | xs' = dbl-lt y y' prf xs'\n  insert-Sorted x (dbl-eq y {xs} xs\u2081) with `RC x y | inspect (`RC x) y | dbl-lt {XC = `RC} x y {y \u2237 xs} | eq=>\u2261 x y | flip-RC x y | insert-Sorted x xs\u2081\n  insert-Sorted x (dbl-eq y xs\u2081) | lt | _ | b | p | l | _ = b refl (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | eq | _ | b | p | l | _ rewrite p refl = dbl-eq y (dbl-eq y xs\u2081)\n  insert-Sorted x (dbl-eq y xs\u2081) | gt | [ prf ] | b | p | l | ss  rewrite prf = dbl-eq y ss \n\n  sort-Sorted : \u2200 {n l}(V : Vec (Fin n) l) \u2192 Sorted `RC (`sort `RC V)\n  sort-Sorted [] = []\n  sort-Sorted (x \u2237 V) = insert-Sorted x (sort-Sorted V)\n\n  RC-refl : \u2200 {i}(x : Fin i) \u2192 `RC x x \u2261 eq\n  RC-refl zero = refl\n  RC-refl (suc x) = RC-refl x\n\n  STail : \u2200 {X l}{XC : Cmp X}{xs : Vec X (suc l)} \u2192 Sorted XC xs \u2192 Sorted XC (tail xs)\n  STail (sing x) = []\n  STail (dbl-lt x y x\u2081 T) = T\n  STail (dbl-eq x T) = T\n\n  module sproof {X}(XC : Cmp X)(XC-refl : \u2200 x \u2192 XC x x \u2261 eq)\n     (eq\u2261 : \u2200 x y \u2192 XC x y \u2261 eq \u2192 x \u2261 y)\n     (lt-trans : \u2200 x y z \u2192 XC x y \u2261 lt \u2192 XC y z \u2261 lt \u2192 XC x z \u2261 lt)\n     (XC-flip : \u2200 x y \u2192 opposite (XC x y) \u2261 XC y x)\n     where\n\n    open import Data.Sum\n\n    _\u2264X_ : X \u2192 X \u2192 Set\n    x \u2264X y = XC x y \u2261 lt \u228e XC x y \u2261 eq\n\n    \u2264X-trans : \u2200 {x y z} \u2192 x \u2264X y \u2192 y \u2264X z \u2192 x \u2264X z\n    \u2264X-trans (inj\u2081 x\u2081) (inj\u2081 x\u2082) = inj\u2081 (lt-trans _ _ _ x\u2081 x\u2082)\n    \u2264X-trans {_}{y}{z}(inj\u2081 x\u2081) (inj\u2082 y\u2081) rewrite eq\u2261 y z y\u2081 = inj\u2081 x\u2081\n    \u2264X-trans {x}{y} (inj\u2082 y\u2081) y\u2264z rewrite eq\u2261 x y y\u2081 = y\u2264z\n\n    h\u2264t : \u2200 {n}{T : `Tree X (2 + n)} \u2192 Sorted XC T \u2192 head T \u2264X head (tail T)\n    h\u2264t (dbl-lt x y x\u2081 ST) = inj\u2081 (sym x\u2081)\n    h\u2264t (dbl-eq x ST) rewrite XC-refl x = inj\u2082 refl\n\n    head-p : \u2200 {n}{T : `Tree X (suc n)} i \u2192 Sorted XC T \u2192 head T \u2264X `toFun T i\n    head-p {T = T} zero ST rewrite XC-refl (head T) = inj\u2082 refl\n    head-p {zero} (suc ()) ST\n    head-p {suc n} (suc i) ST = \u2264X-trans (h\u2264t ST) (head-p i (STail ST))\n\n    toFun-p : \u2200 {n}{T : `Tree X n}{i j : Fin n} \u2192 i \u2264F j \u2192 Sorted XC T \u2192 `toFun T i \u2264X `toFun T j\n    toFun-p {j = j} z\u2264i ST = head-p j ST\n    toFun-p (s\u2264s i\u2264Fj) ST = toFun-p i\u2264Fj (STail ST)\n\n    sort-proof : \u2200 {i}{T : `Tree X i} \u2192 Sorted XC T \u2192 Is-Mono `RC XC (`toFun T)\n    sort-proof {T = T} T\u2081 zero zero rewrite XC-refl (head T) = _\n    sort-proof T\u2081 zero (suc y) with toFun-p (z\u2264i {i = suc y}) T\u2081\n    sort-proof T zero (suc y) | inj\u2081 x rewrite x = _\n    sort-proof T zero (suc y) | inj\u2082 y\u2081 rewrite y\u2081 = _\n    sort-proof {T = T} T\u2081 (suc x) zero with toFun-p (z\u2264i {i = suc x}) T\u2081 | XC-flip (head T) (`toFun (tail T) x)\n    sort-proof T (suc x) zero | inj\u2081 x\u2081 | l rewrite x\u2081 | sym l = _\n    sort-proof T (suc x) zero | inj\u2082 y | l rewrite y | sym l = _\n    sort-proof T\u2081 (suc x) (suc y) = sort-proof (STail T\u2081) x y\n\n  lt-trans-RC : \u2200 {i} (x y z : Fin i) \u2192 `RC x y \u2261 lt \u2192 `RC y z \u2261 lt \u2192 `RC x z \u2261 lt\n  lt-trans-RC zero zero zero x<y y<z = y<z\n  lt-trans-RC zero zero (suc z) x<y y<z = refl\n  lt-trans-RC zero (suc y) zero x<y ()\n  lt-trans-RC zero (suc y) (suc z) x<y y<z = refl\n  lt-trans-RC (suc x) zero zero x<y y<z = x<y\n  lt-trans-RC (suc x) zero (suc z) () y<z\n  lt-trans-RC (suc x) (suc y) zero x<y y<z = y<z\n  lt-trans-RC (suc x) (suc y) (suc z) x<y y<z = lt-trans-RC x y z x<y y<z\n\n  `sort-mono : \u2200 {i}(T : `Tree (`Rep i) i) \u2192 Is-Mono `RC `RC (`toFun (`sort `RC T))\n  `sort-mono T x y = sproof.sort-proof `RC RC-refl (\u03bb x\u2081 y\u2081 x\u2082 \u2192 eq=>\u2261 x\u2081 y\u2081 (sym x\u2082)) lt-trans-RC flip-RC (sort-Sorted T) x y\n\n\n\n  module toNat n (f : Endo (Fin (suc n)))(f-inj : Is-Inj f)(f-mono : Is-Mono `RC `RC f) where\n  \n    import prefect-bintree-sorting\n    open prefect-bintree-sorting.MM\n    open import Data.Sum\n\n    move-to-RC : \u2200 {n}{x y : Fin n} \u2192 x \u2264F y \u2192 `RC x y \u2261 lt \u228e `RC x y \u2261 eq\n    move-to-RC {y = zero} z\u2264i = inj\u2082 refl\n    move-to-RC {y = suc y} z\u2264i = inj\u2081 refl\n    move-to-RC (s\u2264s x\u2264Fy) = move-to-RC x\u2264Fy\n\n    move-from-RC : \u2200 {n}(x y : Fin n) \u2192 lt \u2261 `RC x y \u228e eq \u2261 `RC x y \u2192 x \u2264F y\n    move-from-RC zero zero prf = z\u2264i\n    move-from-RC zero (suc y) prf = z\u2264i\n    move-from-RC (suc x) zero (inj\u2081 ())\n    move-from-RC (suc x) zero (inj\u2082 ())\n    move-from-RC (suc x) (suc y) prf = s\u2264s (move-from-RC x y prf)\n\n    proper-mono : \u2200 {x y} \u2192 x \u2264F y \u2192 f x \u2264F f y\n    proper-mono {x} {y} x\u2264Fy with `RC x y | `RC (f x) (f y) | move-to-RC x\u2264Fy | f-mono x y | move-from-RC (f x) (f y)\n    proper-mono x\u2264Fy | .lt | lt | inj\u2081 refl | r4 | r5 = r5 (inj\u2081 refl)\n    proper-mono x\u2264Fy | .lt | eq | inj\u2081 refl | r4 | r5 = r5 (inj\u2082 refl)\n    proper-mono x\u2264Fy | .lt | gt | inj\u2081 refl | () | r5\n    proper-mono x\u2264Fy | .eq | lt | inj\u2082 refl | () | r5\n    proper-mono x\u2264Fy | .eq | eq | inj\u2082 refl | r4 | r5 = r5 (inj\u2082 refl)\n    proper-mono x\u2264Fy | .eq | gt | inj\u2082 refl | () | r5\n\n    getFrom : \u2200 n \u2192 \u2115 \u2192 Fin (suc n)\n    getFrom zero i = zero\n    getFrom (suc n\u2081) zero = zero\n    getFrom (suc n\u2081) (suc i) = suc (getFrom n\u2081 i)\n\n    getInj : {n x y : \u2115} \u2192 x \u2264 n \u2192 y \u2264 n \u2192 getFrom n x \u2261 getFrom n y \u2192 x \u2261 y\n    getInj z\u2264n z\u2264n prf = refl\n    getInj z\u2264n (s\u2264s y\u2264n) ()\n    getInj (s\u2264s x\u2264n) z\u2264n ()\n    getInj (s\u2264s x\u2264n) (s\u2264s y\u2264n) prf rewrite (getInj x\u2264n y\u2264n (suc-inj prf)) = refl\n\n    getMono : {n x y : \u2115} \u2192 x \u2264 y \u2192 y \u2264 n \u2192 getFrom n x \u2264F getFrom n y\n    getMono z\u2264n z\u2264n = \u2264F-refl _\n    getMono z\u2264n (s\u2264s y\u2264n) = z\u2264i\n    getMono (s\u2264s x\u2264y) (s\u2264s y\u2264n) = s\u2264s (getMono x\u2264y y\u2264n)\n\n    forget : \u2200 {n} \u2192 Fin n \u2192 \u2115\n    forget zero = zero\n    forget (suc i) = suc (forget i)\n\n    forgetInj : \u2200 {n}{i j : Fin n} \u2192 forget i \u2261 forget j \u2192 i \u2261 j\n    forgetInj {.(suc _)} {zero} {zero} prf = refl\n    forgetInj {.(suc _)} {zero} {suc j} ()\n    forgetInj {.(suc _)} {suc i} {zero} ()\n    forgetInj {.(suc _)} {suc i} {suc j} prf rewrite forgetInj (nsuc-inj prf) = refl\n\n    getForget : \u2200 {n}(i : Fin (suc n)) \u2192 getFrom n (forget i) \u2261 i\n    getForget {zero} zero = refl\n    getForget {zero} (suc ())\n    getForget {suc n\u2081} zero = refl\n    getForget {suc n\u2081} (suc i) rewrite getForget i = refl\n\n\n    forget< : \u2200 {n} \u2192 (i : Fin n) \u2192 forget i < n\n    forget< {zero} ()\n    forget< {suc n\u2081} zero = s\u2264s z\u2264n\n    forget< {suc n\u2081} (suc i) = s\u2264s (forget< i)\n\n    forget-mono : \u2200 {n}{i j : Fin n} \u2192 i \u2264F j \u2192 forget i \u2264 forget j\n    forget-mono z\u2264i = z\u2264n\n    forget-mono (s\u2264s i\u2264F) = s\u2264s (forget-mono i\u2264F)\n\n    fn : Endo \u2115\n    fn = forget \u2218 f \u2218 getFrom n\n\n    return : f \u2257 getFrom n \u2218 fn \u2218 forget\n    return x rewrite getForget x | getForget (f x) = refl\n\n    fn-monotone : Monotone (suc n) fn\n    fn-monotone {x} {y} x\u2264y (s\u2264s y\u2264n) = forget-mono (proper-mono (getMono x\u2264y y\u2264n))\n\n    fn-inj : IsInj (suc n) fn\n    fn-inj {x}{y} (s\u2264s sx\u2264sn) (s\u2264s sy\u2264sn) prf = getInj sx\u2264sn sy\u2264sn (f-inj (getFrom n x) (getFrom n y) (forgetInj prf))\n\n    fn-bounded : Bounded (suc n) fn\n    fn-bounded x _ = forget< (f (getFrom n x))\n\n    fn\u2257id : \u2200 x \u2192 x < (suc n) \u2192 fn x \u2261 x\n    fn\u2257id = M.is-id fn fn-monotone fn-inj fn-bounded \n\n    f\u2257id : f \u2257 id\n    f\u2257id x rewrite return x | fn\u2257id (forget x) (forget< x) = getForget x\n\n  fin-view : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 Fin-View i\n  fin-view {zero} zero = max\n  fin-view {zero} (suc ())\n  fin-view {suc n} zero = inject _\n  fin-view {suc n} (suc i) with fin-view i\n  fin-view {suc n} (suc .(from\u2115 n)) | max = max\n  fin-view {suc n} (suc .(inject\u2081 i)) | inject i = inject _\n\n  absurd : {X : Set} \u2192 .\u22a5 \u2192 X\n  absurd ()\n\n  drop\u2081 : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 .(i \u2262 from\u2115 n) \u2192 Fin n\n  drop\u2081 i neq with fin-view i\n  drop\u2081 {n} .(from\u2115 n) neq | max = absurd (neq refl)\n  drop\u2081 .(inject\u2081 i) neq | inject i = i\n\n  drop\u2081\u2192inject\u2081 : \u2200 {n}(i : Fin (suc n))(j : Fin n).(p : i \u2262 from\u2115 n) \u2192 drop\u2081 i p \u2261 j \u2192 i \u2261 inject\u2081 j\n  drop\u2081\u2192inject\u2081 i j p q with fin-view i\n  drop\u2081\u2192inject\u2081 {n} .(from\u2115 n) j p q | max = absurd (p refl)\n  drop\u2081\u2192inject\u2081 .(inject\u2081 i) j p q | inject i = cong inject\u2081 q\n\n\n  `mono-inj\u2192id : \u2200{i}(f : Endo (`Rep i)) \u2192 Is-Inj f \u2192 Is-Mono `RC `RC f \u2192 f \u2257 id\n  `mono-inj\u2192id {zero}  = \u03bb f x x\u2081 ()\n  `mono-inj\u2192id {suc i} = toNat.f\u2257id i \n\n\n  interface : Interface\n  interface = record \n    { Ix            = `Ix\n    ; Rep           = `Rep\n    ; Syn           = `Syn\n    ; Tree          = `Tree\n    ; fromFun       = `fromFun\n    ; toFun         = `toFun\n    ; toFun\u2218fromFun = `toFun\u2218fromFun\n    ; evalArg       = `evalArg\n    ; evalTree      = `evalTree\n    ; eval-proof    = `eval-proof\n    ; inv           = `inv\n    ; inv-proof     = `inv-proof\n    ; RC            = `RC\n    ; sort          = `sort\n    ; sort-syn      = `sort-syn\n    ; sort-proof    = `sort-proof\n    ; sort-mono     = `sort-mono\n    ; mono-inj\u2192id   = `mono-inj\u2192id\n    }\n\n  open import Data.Bool\n\n  count : \u2200 {n} \u2192 (Fin n \u2192 \u2115) \u2192 \u2115\n  count {n} f = sum (tabulate f)\n\n  #\u27e8_\u27e9 : \u2200 {n} \u2192 (Fin n \u2192 Bool) \u2192 \u2115\n  #\u27e8 f \u27e9 = count (\u03bb x \u2192 if f x then 1 else 0)\n\n  #-ext : \u2200 {n} \u2192 (f g : Fin n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n  #-ext {zero} f g f\u2257g = refl\n  #-ext {suc n} f g f\u2257g rewrite f\u2257g zero | #-ext (f \u2218 suc) (g \u2218 suc) (f\u2257g \u2218 suc) = refl\n\n  com-assoc : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n  com-assoc x y z rewrite \n    sym (\u2115\u00b0.+-assoc x y z) |\n    \u2115\u00b0.+-comm x y    |\n    \u2115\u00b0.+-assoc y x z = refl\n    \n  syn-pres : \u2200 {n}(f : Fin n \u2192 \u2115)(S : `Syn n)\n           \u2192 count f \u2261 count (f \u2218 `evalArg S)\n  syn-pres f `id = refl\n  syn-pres f `swap = com-assoc (f zero) (f (suc zero)) (count (\u03bb i \u2192 f (suc (suc i))))\n  syn-pres f (`tail S) rewrite syn-pres (f \u2218 suc) S = refl\n  syn-pres f (S `\u2218 S\u2081) rewrite syn-pres f S = syn-pres (f \u2218 `evalArg S) S\u2081\n\n  #-perm : \u2200 {n}(f : Fin n \u2192 Bool)(p : Endo (Fin n)) \u2192 Is-Inj p\n         \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 p \u27e9\n  #-perm f p p-inj = trans (syn-pres (\u03bb x \u2192 if f x then 1 else 0) (sort-bij p))\n                       (#-ext _ _ f\u2218eval\u2257f\u2218p)\n    where \n      open abs interface\n      f\u2218eval\u2257f\u2218p : f \u2218 `evalArg (sort-bij p) \u2257 f \u2218 p\n      f\u2218eval\u2257f\u2218p x rewrite thm p p-inj x = refl\n\n  test : `Syn 8\n  test = abs.sort-bij interface (\u03bb x \u2192 `evalArg (`tail `swap) x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5c27a957c2f9418fc4c228d5ab37e5d074265b89","subject":"So Bool","message":"So Bool\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\n\u2261\u2192T : \u2200 {b} \u2192 b \u2261 true \u2192 T b\n\u2261\u2192T \u2261.refl = _\n\n\u2261\u2192Tnot : \u2200 {b} \u2192 b \u2261 false \u2192 T (not b)\n\u2261\u2192Tnot \u2261.refl = _\n\nT\u2192\u2261 : \u2200 {b} \u2192 T b \u2192 b \u2261 true\nT\u2192\u2261 {true}  _  = \u2261.refl\nT\u2192\u2261 {false} ()\n\nTnot\u2192\u2261 : \u2200 {b} \u2192 T (not b) \u2192 b \u2261 false\nTnot\u2192\u2261 {false} _  = \u2261.refl\nTnot\u2192\u2261 {true}  ()\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n\ndata So : Bool \u2192 Set where\n  oh! : So true\n\nSo\u2192T : \u2200 {b} \u2192 So b \u2192 T b\nSo\u2192T oh! = _\n\nT\u2192So : \u2200 {b} \u2192 T b \u2192 So b\nT\u2192So {true}  _  = oh!\nT\u2192So {false} ()\n\nSo\u2192\u2261 : \u2200 {b} \u2192 So b \u2192 b \u2261 true\nSo\u2192\u2261 oh! = \u2261.refl\n\n\u2261\u2192So : \u2200 {b} \u2192 b \u2261 true \u2192 So b\n\u2261\u2192So \u2261.refl = oh!\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\n\u2261\u2192T : \u2200 {b} \u2192 b \u2261 true \u2192 T b\n\u2261\u2192T \u2261.refl = _\n\n\u2261\u2192Tnot : \u2200 {b} \u2192 b \u2261 false \u2192 T (not b)\n\u2261\u2192Tnot \u2261.refl = _\n\nT\u2192\u2261 : \u2200 {b} \u2192 T b \u2192 b \u2261 true\nT\u2192\u2261 {true}  _  = \u2261.refl\nT\u2192\u2261 {false} ()\n\nTnot\u2192\u2261 : \u2200 {b} \u2192 T (not b) \u2192 b \u2261 false\nTnot\u2192\u2261 {false} _  = \u2261.refl\nTnot\u2192\u2261 {true}  ()\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"82e8bb92981e2fee80ec5aaf32d899355e992d09","subject":"Made the dist proof more general","message":"Made the dist proof more general\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen L using (Lift)\nopen import Type hiding (\u2605)\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to SumProp)\nopen import Search.Searchable.Product\n\nmodule sum where\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bcLift\u22a4 : SumProp (Lift \u22a4)\n\u03bcLift\u22a4 = _ , ind\n  where\n    srch : Search (Lift \u22a4)\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search \u22a4\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nvsgsum : \u2200 sg {n} \u2192 let open Sgrp sg in\n                    Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf = \n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) \n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n\nmodule _ {A}(\u03bcA : SumProp A)\n  (cmonoid : CommutativeMonoid L.zero L.zero)\n  (_\u25ce_      : let open CMon cmonoid in Carrier  \u2192 Carrier \u2192 Carrier)\n  (distrib  : let open CMon cmonoid in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_ : let open CMon cmonoid in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cmonoid\n  s\u1d2c = search \u03bcA _\u2219_\n\n  bigDistr : \u2200 I F \u2192 search (\u03bcFinSuc I) _\u25ce_ (s\u1d2c \u2218 F)\n                   \u2248 search (\u03bcFun \u03bcA) _\u2219_ (\u03bb f \u2192 search (\u03bcFinSuc I) _\u25ce_ (\u03bb i \u2192 F i (f i)))\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a0' (suc I) (s\u1d2c \u2218 F)\n    \u2248\u27e8 refl \u27e9\n      s\u1d2c (F zero) \u25ce \u03a0' I (s\u1d2c \u2218 Fj)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I Fj \u27e9\n      (s\u1d2c \u2218 F) zero \u25ce \u03a3F I (\u03bb f \u2192 \u03a0' I (\u03bb i \u2192 F (suc i) (f i)))\n    \u2248\u27e8 sym\n      (search-lin\u02e1 (\u03bcF I) monoid _\u25ce_\n             (\u03bb f \u2192 \u03a0' I (\u03bb i \u2192 F (suc i) (f i))) (s\u1d2c (F zero)) (proj\u2081 distrib)) \u27e9\n      \u03a3F I (\u03bb f \u2192 (s\u1d2c \u2218 F) zero \u25ce \u03a0' I (\u03bb i \u2192 F (suc i) (f i)))\n    \u2248\u27e8 search-sg-ext (\u03bcF I) (Monoid.semigroup monoid)\n      (\u03bb f \u2192 sym (search-lin\u02b3 \u03bcA  monoid _\u25ce_ (F zero) (\u03a0' I (\u03bb i \u2192 F (suc i) (f i))) (proj\u2082 distrib))) \u27e9\n      \u03a3F I (\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 F zero j \u25ce (\u03a0' I \u03bb i \u2192 F  (suc i) (f i))))\n    \u2248\u27e8 search-sg-ext (\u03bcF I) (Monoid.semigroup monoid)\n      {(\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 F zero j \u25ce (\u03a0' I \u03bb i \u2192 F  (suc i) (f i))))}\n      {(\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i))))}\n      (\u03bb f \u2192 refl) \u27e9\n      \u03a3F I (\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i))))\n    \u2248\u27e8 search-swap (\u03bcF I) (Monoid.semigroup monoid) (\u03bb f j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i))) {s\u1d2e = s\u1d2c} (search-hom \u03bcA cmonoid) \u27e9\n      s\u1d2c (\u03bb j \u2192 \u03a3F I (\u03bb f \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i))))\n    \u2248\u27e8 refl \u27e9\n      \u03a3F (suc I) (\u03bb f \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (f i)))\n    \u220e\n    where\n      \u03bcF = \u03bb i \u2192 \u03bcFun \u03bcA {suc i}\n      \u03a0' = \u03bb i \u2192 search (\u03bcFinSuc i) _\u25ce_\n      \u03a3F = \u03bb i \u2192 search (\u03bcF i) _\u2219_\n      Fj = \u03bb i j \u2192 F (suc i) j\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen L using (Lift)\nopen import Type hiding (\u2605)\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to SumProp)\nopen import Search.Searchable.Product\n\nmodule sum where\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bcLift\u22a4 : SumProp (Lift \u22a4)\n\u03bcLift\u22a4 = _ , ind\n  where\n    srch : Search (Lift \u22a4)\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search \u22a4\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nvsgsum : \u2200 sg {n} \u2192 let open Sgrp sg in\n                    Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf = \n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) \n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n\nbigDistr : \u2200 I J F \u2192 search (\u03bcFinSuc I) _*_ (search (\u03bcFinSuc J) _+_ \u2218 F)\n                   \u2261.\u2261 search (\u03bcFun (\u03bcFinSuc J)) _+_ (\u03bb f \u2192 search (\u03bcFinSuc I) _*_ (\u03bb i \u2192 F i (f i)))\nbigDistr zero    _ _ = \u2261.refl\nbigDistr (suc I) J F\n  = \u03a0' (suc I) (\u03a3' J \u2218 F)\n  \u2261\u27e8 \u2261.refl \u27e9\n    (\u03a3' J \u2218 F) zero * \u03a0' I (\u03a3' J \u2218 Fj)\n  \u2261\u27e8 \u2261.cong (_*_ ((\u03a3' J \u2218 F) zero)) (bigDistr I J Fj) \u27e9\n    (\u03a3' J \u2218 F) zero * \u03a3F I J (\u03bb f \u2192 \u03a0' I (\u03bb i \u2192 F (suc i) (f i)))\n  \u2261\u27e8 \u2261.sym\n       (search-lin\u02e1 (\u03bcF I J) \u2115+.monoid _*_\n        (\u03bb f \u2192 \u03a0' I (\u03bb i \u2192 F (suc i) (f i))) (\u03a3' J (F zero)) (proj\u2081 (\u2115\u00b0.distrib)))\u27e9\n    \u03a3F I J (\u03bb f \u2192 (\u03a3' J \u2218 F) zero * \u03a0' I (\u03bb i \u2192 F (suc i) (f i)))\n  \u2261\u27e8 search-ext (\u03bcF I J) _+_\n    {\u03bb f \u2192 (\u03a3' J \u2218 F) zero * \u03a0' I (\u03bb i \u2192 F (suc i) (f i))}\n    {\u03bb f \u2192 \u03a3' J (\u03bb j \u2192 F zero j * (\u03a0' I \u03bb i \u2192 F  (suc i) (f i)))}\n    (\u03bb f \u2192 \u2261.sym (search-lin\u02b3 (\u03bc J) \u2115+.monoid _*_ (F zero) (\u03a0' I (\u03bb i \u2192 F (suc i) (f i))) (proj\u2082 \u2115\u00b0.distrib))) \u27e9\n    \u03a3F I J (\u03bb f \u2192 \u03a3' J (\u03bb j \u2192 F zero j * (\u03a0' I \u03bb i \u2192 F  (suc i) (f i))))\n  \u2261\u27e8 search-ext (\u03bcF I J) _+_\n     {(\u03bb f \u2192 \u03a3' J (\u03bb j \u2192 F zero j * (\u03a0' I \u03bb i \u2192 F  (suc i) (f i))))}\n     {(\u03bb f \u2192 \u03a3' J (\u03bb j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend (\u03bcFinSuc J) j f i))))}\n     (\u03bb f \u2192 \u2261.refl) \u27e9\n    \u03a3F I J (\u03bb f \u2192 \u03a3' J (\u03bb j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend (\u03bcFinSuc J) j f i))))\n  \u2261\u27e8 search-swap (\u03bcF I J) \u2115+.semigroup (\u03bb f j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend (\u03bc J) j f i))) {s\u1d2e = sum (\u03bc J)} (sum-hom (\u03bc J)) \u27e9\n    \u03a3' J (\u03bb j \u2192 \u03a3F I J (\u03bb f \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend (\u03bcFinSuc J) j f i))))\n  \u2261\u27e8 \u2261.refl \u27e9\n    \u03a3F (suc I) J (\u03bb f \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (f i)))\n  \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    \u03bc = \u03bcFinSuc\n    \u03bcF = \u03bb i j \u2192 \u03bcFun (\u03bc j) {suc i}\n    \u03a0' = \u03bb i \u2192 search (\u03bc i) _*_\n    \u03a3' = \u03bb i \u2192 search (\u03bc i) _+_\n    \u03a3F = \u03bb i j \u2192 search (\u03bcF i j) _+_\n    Fj = \u03bb i j \u2192 F (suc i) j\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"46c02ef522b5957f407dbf9523e47826963a24e1","subject":"TaggedDeltaTypes: Add corollary of correctness theorem\\ ... as well as rename `t is-valid-wrt \u03c1` to `t is-valid-for \u03c1`","message":"TaggedDeltaTypes: Add corollary of correctness theorem\\\n... as well as rename `t is-valid-wrt \u03c1` to `t is-valid-for \u03c1`\n\nOld-commit-hash: 575e637fabc7edce80fd8549273de2fda059f876\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n_is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-for \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-for \u03c1 = \u22a4\n\u0394bag db is-valid-for \u03c1 = \u22a4\n\u0394var x is-valid-for \u03c1 = \u22a4\n\n_is-valid-for_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-for_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-for \u03c1 = Quadruple\n  (ds is-valid-for \u03c1)\n  (\u03bb _ \u2192 dt is-valid-for \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n_is-valid-for_ (\u0394map\u2081 s db) \u03c1 = db is-valid-for \u03c1\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 v-dt = mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-for \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-for \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- Corollary: (f \u2295 df) (v \u2295 dv) = f v \u2295 df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore \u03c1) \u2295 \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s))\n    (\u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n  \u2261  \u27e6 s \u27e7 (ignore \u03c1) (\u27e6 t \u27e7 (ignore \u03c1)) \u2295\n     \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s) (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t})\n\ncorollary s t {\u03c1} = proj\u2082\n  (validity {t = s} (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t}))\n\nunrestricted (nat n) = tt\nunrestricted (bag b) = tt\nunrestricted (var x) {\u03c1} = tt\nunrestricted (abs t) {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted t)\nunrestricted (app s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t))\n  (validity {t = t}) tt\nunrestricted (add s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\nunrestricted (map s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted t)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n_is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-wrt \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-wrt \u03c1 = \u22a4\n\u0394bag db is-valid-wrt \u03c1 = \u22a4\n\u0394var x is-valid-wrt \u03c1 = \u22a4\n\n_is-valid-wrt_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-wrt_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-wrt \u03c1 = Quadruple\n  (ds is-valid-wrt \u03c1)\n  (\u03bb _ \u2192 dt is-valid-wrt \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n_is-valid-wrt_ (\u0394map\u2081 s db) \u03c1 = db is-valid-wrt \u03c1\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 v-dt = mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-wrt \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-wrt \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t}))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\nunrestricted {t = nat n} = tt\nunrestricted {t = bag b} = tt\nunrestricted {t = var x} {\u03c1} = tt\nunrestricted {t = abs t} {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted {t = t})\nunrestricted {t = app s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t})\n  (validity {t = t}) tt\nunrestricted {t = add s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\nunrestricted {t = map s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted {t = t})\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted {t = s})\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"da122aedbefd1f85da61af8842c203dbcd40a83c","subject":"ok now actually close #1","message":"ok now actually close #1\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b6ddc4ccb2f694ad05da5eae2ba3c79390c94479","subject":"adding back lemma i actually use!","message":"adding back lemma i actually use!\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import binders-disjoint-checks\n\nopen import lemmas-subst-ta\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            binders-unique d \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans bd TAConst ()\n  preserve-trans bd (TAVar x\u2081) ()\n  preserve-trans bd (TALam _ ta) ()\n  preserve-trans (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2082 x\u2083)) (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt x\u2082 bd\u2081 ta ta\u2081\n  preserve-trans bd (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans bd (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans bd (TAEHole x x\u2081) ()\n  preserve-trans bd (TANEHole x ta x\u2081) ()\n  preserve-trans bd (TACast ta x) (ITCastID) = ta\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans bd (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans bd (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans bd (TAFailedCast x y z q) ()\n\n  lem-bd-\u03b51 : \u2200{ d \u03b5 d0} \u2192 d == \u03b5 \u27e6 d0 \u27e7 \u2192 binders-unique d \u2192 binders-unique d0\n  lem-bd-\u03b51 FHOuter bd = bd\n  lem-bd-\u03b51 (FHAp1 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHAp2 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\u2081\n  lem-bd-\u03b51 (FHNEHole eps) (BUNEHole bd x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHCast eps) (BUCast bd) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHFailedCast eps) (BUFailedCast bd) = lem-bd-\u03b51 eps bd\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             binders-unique d \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preservation bd D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans (lem-bd-\u03b51 x bd) wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import binders-disjoint-checks\n\nopen import lemmas-subst-ta\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            binders-unique d \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans bd TAConst ()\n  preserve-trans bd (TAVar x\u2081) ()\n  preserve-trans bd (TALam _ ta) ()\n  preserve-trans (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2082 x\u2083)) (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt x\u2082 bd\u2081 ta ta\u2081\n  preserve-trans bd (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans bd (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans bd (TAEHole x x\u2081) ()\n  preserve-trans bd (TANEHole x ta x\u2081) ()\n  preserve-trans bd (TACast ta x) (ITCastID) = ta\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans bd (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans bd (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans bd (TAFailedCast x y z q) ()\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             binders-unique d \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preservation bd D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans (lem-bd-\u03b51 x bd) wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c31ff66e6a63f4fa648d6283d999831bc309a2b8","subject":"adding a comment","message":"adding a comment\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- notation for a chained together failing cast into hole\n  _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2\u2987\u2988\u21d2\u0338 t2 \u27e9 = d \u27e8 t1 \u21d2 \u2987\u2988 \u27e9 \u27e8\u2987\u2988\u21d2\u0338 t2 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c4 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8\u2987\u2988\u21d2\u0338 \u03c4 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : --\u2200{ d d' \u03b5 \u03c4} \u2192\n            {d d' : dhexp } {\u03c4 : htyp } { \u03b5 : ectx } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8\u2987\u2988\u21d2\u0338 \u03c4 \u27e9) == (\u03b5 \u27e8\u2987\u2988\u21d2\u0338 \u03c4 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- notation for a chained together failing cast into hole\n  _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2\u2987\u2988\u21d2\u0338 t2 \u27e9 = d \u27e8 t1 \u21d2 \u2987\u2988 \u27e9 \u27e8\u2987\u2988\u21d2\u0338 t2 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c4 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8\u2987\u2988\u21d2\u0338 \u03c4 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : --\u2200{ d d' \u03b5 \u03c4} \u2192\n            {d d' : dhexp } {\u03c4 : htyp } { \u03b5 : ectx } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8\u2987\u2988\u21d2\u0338 \u03c4 \u27e9) == (\u03b5 \u27e8\u2987\u2988\u21d2\u0338 \u03c4 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"29a2acc368333b64034b4cab4b405a5db447d918","subject":"IDescTT model: implement the Vec and Fin examples","message":"IDescTT model: implement the Vec and Fin examples\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2a67bfcd92735cccbdf442cb056b3261f69532be","subject":"Added missing file to a README.agad.","message":"Added missing file to a README.agad.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/DistributiveLaws\/README.agda","new_file":"src\/fot\/DistributiveLaws\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- Distributive laws on a binary operation (Stanovsk\u00fd example)\n------------------------------------------------------------------------------\n\nmodule DistributiveLaws.README where\n\n-- Let _\u00b7_ be a left-associative binary operation which satifies the\n-- left and right distributive axioms:\n--\n-- \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n-- \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z).\n\n-- We prove some properties of Stanovsk\u00fd (2008): Task\u00a0B, Lemma\u00a03,\n-- Lemma\u00a04, Lemma\u00a05 (Task\u00a0A) and Lemma\u00a06.\n\n------------------------------------------------------------------------------\n-- The axioms\nopen import DistributiveLaws.Base\n\n-- The interactive and combined proofs\nopen import DistributiveLaws.TaskB-AllStepsATP\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\nopen import DistributiveLaws.Lemma3-ATP\nopen import DistributiveLaws.Lemma4-ATP\nopen import DistributiveLaws.Lemma5-ATP\nopen import DistributiveLaws.Lemma6-ATP\n\n-- Unproven theorem by the ATPs\nopen import DistributiveLaws.TaskB.UnprovedATP\n\n------------------------------------------------------------------------------\n-- References:\n--\n-- Stanovsk\u00fd, David (2008). Distributive Groupoids are\n-- Symmetrical-by-Medial: An Elementary Proof. Commentations\n-- Mathematicae Universitatis Carolinae 49.4, pp. 541\u2013546.\n","old_contents":"------------------------------------------------------------------------------\n-- Distributive laws on a binary operation (Stanovsk\u00fd example)\n------------------------------------------------------------------------------\n\nmodule DistributiveLaws.README where\n\n-- Let _\u00b7_ be a left-associative binary operation which satifies the\n-- left and right distributive axioms:\n--\n-- \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n-- \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z).\n\n-- We prove some properties of Stanovsk\u00fd (2008): Task\u00a0B, Lemma\u00a03,\n-- Lemma\u00a04, Lemma\u00a05 (Task\u00a0A) and Lemma\u00a06.\n\n------------------------------------------------------------------------------\n-- The axioms\nopen import DistributiveLaws.Base\n\n-- The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\nopen import DistributiveLaws.Lemma3-ATP\nopen import DistributiveLaws.Lemma4-ATP\nopen import DistributiveLaws.Lemma5-ATP\nopen import DistributiveLaws.Lemma6-ATP\n\n-- Unproven theorem by the ATPs\nopen import DistributiveLaws.TaskB.UnprovedATP\n\n------------------------------------------------------------------------------\n-- References:\n--\n-- Stanovsk\u00fd, David (2008). Distributive Groupoids are\n-- Symmetrical-by-Medial: An Elementary Proof. Commentations\n-- Mathematicae Universitatis Carolinae 49.4, pp. 541\u2013546.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3ba9b45f84d23195411b05ae3c14e2a5859aeb61","subject":"Describe the purpose of Model.agda (see #28).","message":"Describe the purpose of Model.agda (see #28).\n\nThere are two comments because I intend to split this file.\n\nOld-commit-hash: f61a7ef89adc97a015021be9ef43504b3b124e96\n","repos":"inc-lc\/ilc-agda","old_file":"Model.agda","new_file":"Model.agda","new_contents":"module Model where\n\n-- SIMPLE TYPES\n--\n-- This module defines the syntax of simple types.\n\n\n-- VALUES\n--\n-- This module defines the model theory of simple types, that is,\n-- it defines for every type, the set of values of that type.\n--\n-- In fact, we only describe a single model here.\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n_xor_ : Bool \u2192 Bool \u2192 Bool\ntrue xor b = not b\nfalse xor b = b\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\nopen import Level using (zero)\nopen import Relation.Binary.PropositionalEquality\npostulate ext : Extensionality zero zero\n\nnot-not : \u2200 a \u2192 a \u2261 not (not a)\nnot-not true = refl\nnot-not false = refl\n\na-xor-a-false : \u2200 a \u2192 (a xor a) \u2261 false\na-xor-a-false true = refl\na-xor-a-false false = refl\n\na-xor-false-a : \u2200 a \u2192 (false xor a) \u2261 a\na-xor-false-a b = refl\n\nxor-associative : \u2200 a b c \u2192 ((b xor c) xor a) \u2261 (b xor (c xor a))\nxor-associative a true true = not-not a\nxor-associative a true false = refl\nxor-associative a false c = refl\n\na-xor-false : \u2200 a \u2192 a xor false \u2261 a\na-xor-false true = refl\na-xor-false false = refl\n\na-xor-true : \u2200 a \u2192 a xor true \u2261 not a\na-xor-true true = refl\na-xor-true false = refl\n\nxor-commutative : \u2200 a b \u2192 a xor b \u2261 b xor a\nxor-commutative true b rewrite a-xor-true b = refl\nxor-commutative false b rewrite a-xor-false b = refl\n\nxor-cancellative-2 : \u2200 a b \u2192 (b xor (a xor a)) \u2261 b\nxor-cancellative-2 a b rewrite a-xor-a-false a = a-xor-false b\n\nxor-cancellative : \u2200 a b \u2192 ((b xor a) xor a) \u2261 b\nxor-cancellative a b rewrite xor-associative a b a = xor-cancellative-2 a b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl = refl\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym = sym\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans = trans\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong f = cong f\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 f = cong\u2082 f\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4 : Set} \u2192 IsEquivalence (_\u2261_ {A = \u03c4})\n\u2261-isEquivalence = isEquivalence\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n","old_contents":"module Model where\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n_xor_ : Bool \u2192 Bool \u2192 Bool\ntrue xor b = not b\nfalse xor b = b\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\nopen import Level using (zero)\nopen import Relation.Binary.PropositionalEquality\npostulate ext : Extensionality zero zero\n\nnot-not : \u2200 a \u2192 a \u2261 not (not a)\nnot-not true = refl\nnot-not false = refl\n\na-xor-a-false : \u2200 a \u2192 (a xor a) \u2261 false\na-xor-a-false true = refl\na-xor-a-false false = refl\n\na-xor-false-a : \u2200 a \u2192 (false xor a) \u2261 a\na-xor-false-a b = refl\n\nxor-associative : \u2200 a b c \u2192 ((b xor c) xor a) \u2261 (b xor (c xor a))\nxor-associative a true true = not-not a\nxor-associative a true false = refl\nxor-associative a false c = refl\n\na-xor-false : \u2200 a \u2192 a xor false \u2261 a\na-xor-false true = refl\na-xor-false false = refl\n\na-xor-true : \u2200 a \u2192 a xor true \u2261 not a\na-xor-true true = refl\na-xor-true false = refl\n\nxor-commutative : \u2200 a b \u2192 a xor b \u2261 b xor a\nxor-commutative true b rewrite a-xor-true b = refl\nxor-commutative false b rewrite a-xor-false b = refl\n\nxor-cancellative-2 : \u2200 a b \u2192 (b xor (a xor a)) \u2261 b\nxor-cancellative-2 a b rewrite a-xor-a-false a = a-xor-false b\n\nxor-cancellative : \u2200 a b \u2192 ((b xor a) xor a) \u2261 b\nxor-cancellative a b rewrite xor-associative a b a = xor-cancellative-2 a b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl = refl\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym = sym\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans = trans\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong f = cong f\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 f = cong\u2082 f\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4 : Set} \u2192 IsEquivalence (_\u2261_ {A = \u03c4})\n\u2261-isEquivalence = isEquivalence\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d7de8ffd12566de5f900f5f5c79efb62ebdd7862","subject":"In progress adding StableUnderInjection","message":"In progress adding StableUnderInjection\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-properties.agda","new_file":"sum-properties.agda","new_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Function.NP\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nrecord Finable {A : Set}(\u03bcA : SumProp A) : Set where\n  constructor mk\n  field\n    FinCard  : \u2115\n    toFin    : A \u2192 Fin (suc FinCard)\n    fromFin  : Fin (suc FinCard) \u2192 A\n    to-inj   : Injective toFin\n    from-inj : Injective fromFin\n    iso      : fromFin \u2218 toFin \u2257 id\n    sums-ok  : \u2200 f \u2192 sum \u03bcA f \u2261 sum (\u03bcFin FinCard) (f \u2218 fromFin)\n\npostulate\n  Fin-Stab : \u2200 n \u2192 StableUnderInjection (\u03bcFin n)\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = mk 0 (\u03bb x \u2192 zero) (\u03bb x \u2192 _) (\u03bb x\u2081 \u2192 \u2261.refl) (\u03bb x\u2081 \u2192 help) (\u03bb x \u2192 \u2261.refl) (\u03bb f \u2192 \u2261.refl) \n  where help : {x y : Fin 1} \u2192 x \u2261 y\n        help {zero} {zero} = \u2261.refl\n        help {zero} {suc ()}\n        help {suc ()}\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable {A}{B} \u03bcA \u03bcB finA finB = mk FinCard toFin fromFin to-inj from-inj iso sums-ok where\n    open import Data.Sum\n    open import Data.Empty\n\n    |A| = suc (Finable.FinCard finA)\n    |B| = suc (Finable.FinCard finB)\n\n    Fsuc-injective : \u2200 {n} {i j : Fin n} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\n    Fsuc-injective \u2261.refl = \u2261.refl\n    \n    inj\u2081-inj : \u2200 {A B : Set} {x y : A} \u2192 inj\u2081 {B = B} x \u2261 inj\u2081 y \u2192 x \u2261 y\n    inj\u2081-inj \u2261.refl = \u2261.refl\n\n    inj\u2082-inj : \u2200 {A B : Set} {x y : B} \u2192 inj\u2082 {A = A} x \u2261 inj\u2082 y \u2192 x \u2261 y\n    inj\u2082-inj \u2261.refl = \u2261.refl\n\n    fin\u2081 : \u2200 n {m} \u2192 Fin n \u2192 Fin (n + m)\n    fin\u2081 zero ()\n    fin\u2081 (suc n) zero = zero\n    fin\u2081 (suc n) (suc i) = suc (fin\u2081 n i)\n\n    fin\u2082 : \u2200 n {m} \u2192 Fin m \u2192 Fin (n + m)\n    fin\u2082 zero i = i\n    fin\u2082 (suc n) i = suc (fin\u2082 n i)\n\n    toFin' : \u2200 {n} {m} \u2192 Fin n \u228e Fin m \u2192 Fin (n + m)\n    toFin' {n} (inj\u2081 x) = fin\u2081 n x\n    toFin' {n} (inj\u2082 y) = fin\u2082 n y\n\n    fin\u2081-inj : \u2200 n {m} \u2192 Injective (fin\u2081 n {m})\n    fin\u2081-inj .(suc n) {m} {zero {n}} {zero} x\u2261y = \u2261.refl\n    fin\u2081-inj .(suc n) {m} {zero {n}} {suc y} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {zero} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {suc y} x\u2261y = \u2261.cong suc (fin\u2081-inj n (Fsuc-injective x\u2261y))\n\n    fin\u2082-inj : \u2200 n {m} \u2192 Injective (fin\u2082 n {m})\n    fin\u2082-inj zero eq = eq\n    fin\u2082-inj (suc n) eq = fin\u2082-inj n (Fsuc-injective eq)\n\n    fin\u2081\u2260fin\u2082 : \u2200 n {m} x y \u2192 (fin\u2081 n {m} x \u2261 fin\u2082 n y) \u2192 \u22a5\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) zero ()\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) (suc y) ()\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) zero eq = fin\u2081\u2260fin\u2082 n x zero (Fsuc-injective eq)\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) (suc y) eq = fin\u2081\u2260fin\u2082 n x (suc y) (Fsuc-injective eq)\n\n    fromFin' : \u2200 n {m} \u2192 Fin (n + m) \u2192 Fin n \u228e Fin m\n    fromFin' zero k = inj\u2082 k\n    fromFin' (suc n) zero = inj\u2081 zero\n    fromFin' (suc n) (suc k) with fromFin' n k\n    ... | inj\u2081 x = inj\u2081 (suc x)\n    ... | inj\u2082 y = inj\u2082 y\n\n    fromFin'-inj : \u2200 n {m} x y \u2192 fromFin' n {m} x \u2261 fromFin' n y \u2192 x \u2261 y\n    fromFin'-inj zero x y eq = inj\u2082-inj eq\n    fromFin'-inj (suc n) zero zero eq = \u2261.refl\n    fromFin'-inj (suc n) zero (suc y) eq = {!!}\n    fromFin'-inj (suc n) (suc x) zero eq = {!!}\n    fromFin'-inj (suc n) (suc x) (suc y) eq = {!fromFin'-inj n x y!}\n\n    fromFin'-inj\u2081 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2081 n x) \u2261 inj\u2081 x\n    fromFin'-inj\u2081 zero ()\n    fromFin'-inj\u2081 (suc n) zero = \u2261.refl\n    fromFin'-inj\u2081 (suc n) {m} (suc x) rewrite fromFin'-inj\u2081 n {m} x  = \u2261.refl\n\n    fromFin'-inj\u2082 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2082 n x) \u2261 inj\u2082 x\n    fromFin'-inj\u2082 zero x = \u2261.refl\n    fromFin'-inj\u2082 (suc n) x rewrite fromFin'-inj\u2082 n x = \u2261.refl\n\n    FinCard  : \u2115\n    FinCard = Finable.FinCard finA + suc (Finable.FinCard finB)\n\n    toFin    : A \u228e B \u2192 Fin (suc FinCard)\n    toFin (inj\u2081 x) = toFin' (inj\u2081 (Finable.toFin finA x))\n    toFin (inj\u2082 y) = toFin' {n = |A|} (inj\u2082 (Finable.toFin finB y))\n\n    fromFin  : Fin (suc FinCard) \u2192 A \u228e B\n    fromFin x+y with fromFin' |A| x+y\n    ... | inj\u2081 x = inj\u2081 (Finable.fromFin finA x)\n    ... | inj\u2082 y = inj\u2082 (Finable.fromFin finB y)\n\n    to-inj   : Injective toFin\n    to-inj {inj\u2081 x} {inj\u2081 x\u2081} tx\u2261ty = \u2261.cong inj\u2081 (Finable.to-inj finA (fin\u2081-inj |A| tx\u2261ty))\n    to-inj {inj\u2081 x} {inj\u2082 y} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ tx\u2261ty)\n    to-inj {inj\u2082 y} {inj\u2081 x} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ (\u2261.sym tx\u2261ty))\n    to-inj {inj\u2082 y} {inj\u2082 y\u2081} tx\u2261ty = \u2261.cong inj\u2082 (Finable.to-inj finB (fin\u2082-inj |A| tx\u2261ty))\n\n    from-inj : Injective fromFin\n    from-inj {x} {y} fx\u2261fy = {!!}\n\n    iso      : fromFin \u2218 toFin \u2257 id\n    iso (inj\u2081 x) rewrite fromFin'-inj\u2081 |A| {|B|} (Finable.toFin finA x) | Finable.iso finA x = \u2261.refl\n    iso (inj\u2082 y) rewrite fromFin'-inj\u2082 |A| {|B|} (Finable.toFin finB y) | Finable.iso finB y = \u2261.refl\n\n    sums-ok  : \u2200 f \u2192 sum (\u03bcA +\u03bc \u03bcB) f \u2261 sum (\u03bcFin FinCard) (f \u2218 fromFin)\n    sums-ok f =\n        sum (\u03bcA +\u03bc \u03bcB) f\n      \u2261\u27e8 \u2261.cong\u2082 _+_ (Finable.sums-ok finA (f \u2218 inj\u2081))\n           (Finable.sums-ok finB (f \u2218 inj\u2082)) \u27e9\n        sum (\u03bcFin (Finable.FinCard finA)) (f \u2218 inj\u2081 \u2218 Finable.fromFin finA)\n      + sum (\u03bcFin (Finable.FinCard finB)) (f \u2218 inj\u2082 \u2218 Finable.fromFin finB)\n      \u2261\u27e8 {!!} \u27e9\n        sum (\u03bcFin FinCard) (f \u2218 fromFin)\n      \u220e\n      where open \u2261.\u2261-Reasoning\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA fin f p p-inj\n  = sum \u03bcA f\n  \u2261\u27e8 sums-ok f \u27e9\n    sum (\u03bcFin FinCard) (f \u2218 fromFin)\n  \u2261\u27e8 Fin-Stab FinCard (f \u2218 fromFin) (toFin \u2218 p \u2218 fromFin) (from-inj \u2218 p-inj \u2218 to-inj) \u27e9\n    sum (\u03bcFin FinCard) (f \u2218 fromFin \u2218 toFin \u2218 p \u2218 fromFin)\n  \u2261\u27e8 sum-ext (\u03bcFin FinCard) (\u03bb x \u2192 \u2261.cong f (iso (p (fromFin x)))) \u27e9\n    sum (\u03bcFin FinCard) (f \u2218 p \u2218 fromFin)\n  \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 p)) \u27e9\n    sum \u03bcA (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open Finable fin\n\n\n{-\nmodule _ where\n  open import bijection-fin\n  open import Data.Vec.NP\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFin n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (tabulate-\u2218 f id)\n                              | \u2261.sym (tabulate-\u2218 (f \u2218 p) id) = count-perm f p (\u03bb x y \u2192 p-inj)\n-}\n\n\n","old_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Function.NP\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Vec.NP\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFin n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (tabulate-\u2218 f id)\n                              | \u2261.sym (tabulate-\u2218 (f \u2218 p) id) = count-perm f p (\u03bb x y \u2192 p-inj)\n\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"74de191538e421fd0a62f67344c5e83fd18f7612","subject":"possible agda bug demonstration","message":"possible agda bug demonstration\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/IndexLF.agda","new_file":"agda\/IndexLF.agda","new_contents":"\nmodule IndexLF where\n\nopen import Common\nopen import LinLogic\n--open import LinDepT\n--open import LinT \n--open import SetLL\n--open import SetLLProp\n--open import SetLLRem\nopen import LinFun\n\nopen import Data.Product\n\n\ndata IndexLFC {i u} : \u2200{ll rll} \u2192 LFun {i} {u} ll rll \u2192 Set where\n  \u2193c    : \u2200{ll \u221erll prf \u221elf} \u2192 IndexLFC (call {i} {u} {ll} {\u221erll} {prf} \u221elf)\n  _\u2190\u2282 : \u2200{rll pll ell ll ind elf lf}\n         \u2192 IndexLFC elf\n         \u2192 IndexLFC (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  \u2282\u2192_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 IndexLFC lf\n         \u2192 IndexLFC (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  tr   : \u2200{ll orll rll} \u2192 {ltr : LLTr orll ll} \u2192 {lf : LFun {i} {u} orll rll}\n         \u2192 IndexLFC lf \u2192 IndexLFC (tr ltr lf) \n  com  : \u2200{rll ll frll prfi prfo df lf}\n         \u2192 IndexLFC lf\n         \u2192 IndexLFC (com {i} {u} {rll} {ll} {frll} {{prfi}} {{prfo}} df lf)\n\n\n\ndata SetLFC {i u oll orll} (olf : LFun {i} {u} oll orll) : \u2200{ll rll} \u2192 LFun {i} {u} ll rll \u2192 Set (lsuc u) where\n  \u2193c    : \u2200{ll \u221erll prf \u221elf} \u2192 IndexLFC olf \u2192 SetLFC olf (call {i} {u} {ll} {\u221erll} {prf} \u221elf)\n  _\u2190\u2282 : \u2200{rll pll ell ll ind elf lf}\n         \u2192 SetLFC olf elf\n         \u2192 SetLFC olf (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  \u2282\u2192_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  _\u2190\u2282\u2192_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 SetLFC olf elf\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  tr   : \u2200{ll orll rll} \u2192 {ltr : LLTr orll ll} \u2192 {lf : LFun {i} {u} orll rll}\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (tr ltr lf) \n  com  : \u2200{rll ll frll prfi prfo df lf}\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (com {i} {u} {rll} {ll} {frll} {{prfi}} {{prfo}} df lf)\n\ndata MSetLFC {i u oll orll} (olf : LFun {i} {u} oll orll) : \u2200{ll rll} \u2192 LFun {i} {u} ll rll \u2192 Set (lsuc u) where\n  \u2205   : \u2200{ll rll} \u2192 {lf : LFun {i} {u} ll rll}            \u2192 MSetLFC olf lf\n  \u00ac\u2205  : \u2200{ll rll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 SetLFC olf lf \u2192 MSetLFC olf lf\n\n\u2205-addLFC : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 IndexLFC olf \u2192 IndexLFC lf \u2192 SetLFC olf lf \n\u2205-addLFC oic \u2193c = \u2193c oic\n\u2205-addLFC oic (ic \u2190\u2282) = (\u2205-addLFC oic ic) \u2190\u2282\n\u2205-addLFC oic (\u2282\u2192 ic) = \u2282\u2192 (\u2205-addLFC oic ic)\n\u2205-addLFC oic (tr ic) = tr (\u2205-addLFC oic ic)\n\u2205-addLFC oic (com ic) = com (\u2205-addLFC oic ic)\n\naddLFC : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 SetLFC olf lf \u2192 IndexLFC olf \u2192 IndexLFC lf \u2192 SetLFC olf lf \naddLFC (\u2193c x) oic \u2193c = \u2193c oic -- replace\naddLFC (s \u2190\u2282) oic (ic \u2190\u2282) = (addLFC s oic ic) \u2190\u2282\naddLFC (\u2282\u2192 s) oic (ic \u2190\u2282) = (\u2205-addLFC oic ic) \u2190\u2282\u2192 s\naddLFC (s \u2190\u2282\u2192 s\u2081) oic (ic \u2190\u2282) =  (addLFC s oic ic) \u2190\u2282\u2192 s\u2081\naddLFC (s \u2190\u2282) oic (\u2282\u2192 ic) = s \u2190\u2282\u2192 (\u2205-addLFC oic ic)\naddLFC (\u2282\u2192 s) oic (\u2282\u2192 ic) = \u2282\u2192 (addLFC s oic ic)\naddLFC (s \u2190\u2282\u2192 s\u2081) oic (\u2282\u2192 ic) = s \u2190\u2282\u2192 (addLFC s\u2081 oic ic)\naddLFC (tr s) oic (tr ic) = tr (addLFC s oic ic)\naddLFC (com s) oic (com ic) = com (addLFC s oic ic)\n\n\nmaddLFC : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 MSetLFC olf lf \u2192 IndexLFC olf \u2192 IndexLFC lf \u2192 MSetLFC olf lf\nmaddLFC \u2205 oic ic = \u00ac\u2205 (\u2205-addLFC oic ic)\nmaddLFC (\u00ac\u2205 x) oic ic = \u00ac\u2205 (addLFC x oic ic)\n\ndata SetLFCRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2193c    : \u2200{\u221ell} \u2192 IndexLFC {i} olf                     \u2192 SetLFCRem olf (call \u221ell)\n  _\u2190\u2227   : \u2200{rs ls} \u2192 SetLFCRem olf ls                   \u2192 SetLFCRem olf (ls \u2227 rs)\n  \u2227\u2192_   : \u2200{rs ls} \u2192 SetLFCRem olf rs                   \u2192 SetLFCRem olf (ls \u2227 rs)\n  _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLFCRem olf ls \u2192 SetLFCRem olf rs \u2192 SetLFCRem olf (ls \u2227 rs)\n  _\u2190\u2228   : \u2200{rs ls} \u2192 SetLFCRem olf ls                   \u2192 SetLFCRem olf (ls \u2228 rs)\n  \u2228\u2192_   : \u2200{rs ls} \u2192 SetLFCRem olf rs                   \u2192 SetLFCRem olf (ls \u2228 rs)\n  _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLFCRem olf ls \u2192 SetLFCRem olf rs \u2192 SetLFCRem olf (ls \u2228 rs)\n  _\u2190\u2202   : \u2200{rs ls} \u2192 SetLFCRem olf ls                   \u2192 SetLFCRem olf (ls \u2202 rs)\n  \u2202\u2192_   : \u2200{rs ls} \u2192 SetLFCRem olf rs                   \u2192 SetLFCRem olf (ls \u2202 rs)\n  _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLFCRem olf ls \u2192 SetLFCRem olf rs \u2192 SetLFCRem olf (ls \u2202 rs)\n\ndata MSetLFCRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2205   : \u2200{ll}            \u2192 MSetLFCRem olf ll\n  \u00ac\u2205  : \u2200{ll} \u2192 SetLFCRem olf ll \u2192 MSetLFCRem olf ll\n\n\u2205-addLFCRem : \u2200{i u ll \u221erll oll orll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} (call \u221erll) ll) \u2192 IndexLFC olf\n        \u2192 SetLFCRem olf ll\n\u2205-addLFCRem \u2193 m = \u2193c m\n\u2205-addLFCRem (ind \u2190\u2227) m = (\u2205-addLFCRem ind m) \u2190\u2227\n\u2205-addLFCRem (\u2227\u2192 ind) m = \u2227\u2192 (\u2205-addLFCRem ind m)\n\u2205-addLFCRem (ind \u2190\u2228) m = (\u2205-addLFCRem ind m) \u2190\u2228\n\u2205-addLFCRem (\u2228\u2192 ind) m = \u2228\u2192 (\u2205-addLFCRem ind m)\n\u2205-addLFCRem (ind \u2190\u2202) m = (\u2205-addLFCRem ind m) \u2190\u2202\n\u2205-addLFCRem (\u2202\u2192 ind) m = \u2202\u2192 (\u2205-addLFCRem ind m)\n\naddLFCRem : \u2200{i u ll \u221erll oll orll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCRem olf ll \u2192 (ind : IndexLL {i} {u} (call \u221erll) ll) \u2192 IndexLFC olf\n        \u2192 SetLFCRem olf ll\naddLFCRem (\u2193c rm) ind m               = \u2193c m\naddLFCRem (s \u2190\u2227) (ind \u2190\u2227) m     = (addLFCRem s ind m) \u2190\u2227\naddLFCRem (s \u2190\u2227) (\u2227\u2192 ind) m     = s \u2190\u2227\u2192 (\u2205-addLFCRem ind m)\naddLFCRem (\u2227\u2192 s) (ind \u2190\u2227) m     = (\u2205-addLFCRem ind m) \u2190\u2227\u2192 s\naddLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) m     = \u2227\u2192 addLFCRem s ind m\naddLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) m = (addLFCRem s ind m) \u2190\u2227\u2192 s\u2081\naddLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) m = s \u2190\u2227\u2192 (addLFCRem s\u2081 ind m)\naddLFCRem (s \u2190\u2228) (ind \u2190\u2228) m     = (addLFCRem s ind m) \u2190\u2228\naddLFCRem (s \u2190\u2228) (\u2228\u2192 ind) m     = s \u2190\u2228\u2192 (\u2205-addLFCRem ind m)\naddLFCRem (\u2228\u2192 s) (ind \u2190\u2228) m     = (\u2205-addLFCRem ind m) \u2190\u2228\u2192 s\naddLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) m     = \u2228\u2192 addLFCRem s ind m\naddLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) m = (addLFCRem s ind m) \u2190\u2228\u2192 s\u2081\naddLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) m = s \u2190\u2228\u2192 (addLFCRem s\u2081 ind m)\naddLFCRem (s \u2190\u2202) (ind \u2190\u2202) m     = (addLFCRem s ind m) \u2190\u2202\naddLFCRem (s \u2190\u2202) (\u2202\u2192 ind) m     = s \u2190\u2202\u2192 (\u2205-addLFCRem ind m)\naddLFCRem (\u2202\u2192 s) (ind \u2190\u2202) m     = (\u2205-addLFCRem ind m) \u2190\u2202\u2192 s\naddLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) m     = \u2202\u2192 addLFCRem s ind m\naddLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) m = (addLFCRem s ind m) \u2190\u2202\u2192 s\u2081\naddLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) m = s \u2190\u2202\u2192 (addLFCRem s\u2081 ind m)\n\nmadd : \u2200{i u ll \u221erll oll orll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCRem olf ll \u2192 (ind : IndexLL {i} {u} (call \u221erll) ll) \u2192 IndexLFC olf\n      \u2192 MSetLFCRem olf ll\nmadd \u2205 ind m = \u00ac\u2205 (\u2205-addLFCRem ind m)\nmadd (\u00ac\u2205 x) ind m = \u00ac\u2205 (addLFCRem x ind m)\n\n\ntruncSetLFCRem : \u2200{i} \u2192 \u2200{u ll oll orll q} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCRem {i} {u} olf ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 MSetLFCRem {i} olf q\ntruncSetLFCRem \u2205 ind = \u2205\ntruncSetLFCRem (\u00ac\u2205 x) \u2193 = \u00ac\u2205 x\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227)) (ind \u2190\u2227) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (\u2227\u2192 x)) (ind \u2190\u2227) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (ind \u2190\u2227) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227)) (\u2227\u2192 ind) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (\u2227\u2192 x)) (\u2227\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (\u2227\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228)) (ind \u2190\u2228) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (\u2228\u2192 x)) (ind \u2190\u2228) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (ind \u2190\u2228) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228)) (\u2228\u2192 ind) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (\u2228\u2192 x)) (\u2228\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (\u2228\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202)) (ind \u2190\u2202) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (\u2202\u2192 x)) (ind \u2190\u2202) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (ind \u2190\u2202) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202)) (\u2202\u2192 ind) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (\u2202\u2192 x)) (\u2202\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (\u2202\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x\u2081) ind\n\ndelLFCRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n      \u2192 MSetLFCRem {i} olf (replLL ll ind rll)\ndelLFCRem s \u2193 rll = \u2205\ndelLFCRem (s \u2190\u2227) (ind \u2190\u2227) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u2205 = \u2205\ndelLFCRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227)\ndelLFCRem (\u2227\u2192 s) (ind \u2190\u2227) rll = \u00ac\u2205 (\u2227\u2192 (s))\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u2205 = \u00ac\u2205 (\u2227\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2227) (\u2227\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) rll with (delLFCRem s ind rll)\ndelLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u2205 = \u2205\ndelLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2227\u2192 x)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll with (delLFCRem s\u2081 ind rll)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2227\u2192 x)\ndelLFCRem (s \u2190\u2228) (ind \u2190\u2228) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u2205 = \u2205\ndelLFCRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228)\ndelLFCRem (\u2228\u2192 s) (ind \u2190\u2228) rll = \u00ac\u2205 (\u2228\u2192 (s))\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u2205 = \u00ac\u2205 (\u2228\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2228) (\u2228\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) rll with (delLFCRem s ind rll)\ndelLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u2205 = \u2205\ndelLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2228\u2192 x)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll with (delLFCRem s\u2081 ind rll)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2228\u2192 x)\ndelLFCRem (s \u2190\u2202) (ind \u2190\u2202) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u2205 = \u2205\ndelLFCRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202)\ndelLFCRem (\u2202\u2192 s) (ind \u2190\u2202) rll = \u00ac\u2205 (\u2202\u2192 (s))\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u2205 = \u00ac\u2205 (\u2202\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2202) (\u2202\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) rll with (delLFCRem s ind rll)\ndelLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u2205 = \u2205\ndelLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2202\u2192 x)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll with (delLFCRem s\u2081 ind rll)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2202\u2192 x)\n\nmdelLFCRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n             \u2192 MSetLFCRem {i} olf (replLL ll ind rll)\nmdelLFCRem \u2205 ind rll = \u2205\nmdelLFCRem (\u00ac\u2205 x) ind rll = delLFCRem x ind rll\n\ntranLFCRem : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCRem {i} olf ll \u2192 (tr : LLTr {i} {u} rll ll)\n       \u2192 SetLFCRem olf rll\ntranLFCRem s I                           = s\ntranLFCRem (s \u2190\u2202) (\u2202c ltr)                = tranLFCRem (\u2202\u2192 s) ltr\ntranLFCRem (\u2202\u2192 s) (\u2202c ltr)                = tranLFCRem (s \u2190\u2202) ltr\ntranLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202c ltr)            = tranLFCRem (s\u2081 \u2190\u2202\u2192 s) ltr\ntranLFCRem (s \u2190\u2228) (\u2228c ltr)                = tranLFCRem (\u2228\u2192 s) ltr\ntranLFCRem (\u2228\u2192 s) (\u2228c ltr)                = tranLFCRem (s \u2190\u2228) ltr\ntranLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228c ltr)            = tranLFCRem (s\u2081 \u2190\u2228\u2192 s) ltr\ntranLFCRem (s \u2190\u2227) (\u2227c ltr)                = tranLFCRem (\u2227\u2192 s) ltr\ntranLFCRem (\u2227\u2192 s) (\u2227c ltr)                = tranLFCRem (s \u2190\u2227) ltr\ntranLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227c ltr)            = tranLFCRem (s\u2081 \u2190\u2227\u2192 s) ltr\ntranLFCRem ((s \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCRem (s \u2190\u2227) ltr\ntranLFCRem ((\u2227\u2192 s) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCRem (\u2227\u2192 (s \u2190\u2227)) ltr\ntranLFCRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227) (\u2227\u2227d ltr)      = tranLFCRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227)) ltr\ntranLFCRem (\u2227\u2192 s) (\u2227\u2227d ltr)               = tranLFCRem (\u2227\u2192 (\u2227\u2192 s)) ltr\ntranLFCRem ((s \u2190\u2227) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCRem (s \u2190\u2227\u2192 (\u2227\u2192 s\u2081)) ltr\ntranLFCRem ((\u2227\u2192 s) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCRem (\u2227\u2192 (s \u2190\u2227\u2192 s\u2081)) ltr\ntranLFCRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227\u2192 s\u2082) (\u2227\u2227d ltr)  = tranLFCRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227\u2192 s\u2082)) ltr\ntranLFCRem (s \u2190\u2227) (\u00ac\u2227\u2227d ltr)              = tranLFCRem ((s \u2190\u2227) \u2190\u2227) ltr\ntranLFCRem (\u2227\u2192 (s \u2190\u2227)) (\u00ac\u2227\u2227d ltr)         = tranLFCRem ((\u2227\u2192 s) \u2190\u2227) ltr\ntranLFCRem (\u2227\u2192 (\u2227\u2192 s)) (\u00ac\u2227\u2227d ltr)         = tranLFCRem (\u2227\u2192 s) ltr\ntranLFCRem (\u2227\u2192 (s \u2190\u2227\u2192 s\u2081)) (\u00ac\u2227\u2227d ltr)     = tranLFCRem ((\u2227\u2192 s) \u2190\u2227\u2192 s\u2081) ltr\ntranLFCRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227)) (\u00ac\u2227\u2227d ltr)     = tranLFCRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227) ltr\ntranLFCRem (s \u2190\u2227\u2192 (\u2227\u2192 s\u2081)) (\u00ac\u2227\u2227d ltr)     = tranLFCRem ((s \u2190\u2227) \u2190\u2227\u2192 s\u2081) ltr\ntranLFCRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227\u2192 s\u2082)) (\u00ac\u2227\u2227d ltr) = tranLFCRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227\u2192 s\u2082) ltr\ntranLFCRem ((s \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCRem (s \u2190\u2228) ltr\ntranLFCRem ((\u2228\u2192 s) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCRem (\u2228\u2192 (s \u2190\u2228)) ltr\ntranLFCRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228) (\u2228\u2228d ltr)      = tranLFCRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228)) ltr\ntranLFCRem (\u2228\u2192 s) (\u2228\u2228d ltr)               = tranLFCRem (\u2228\u2192 (\u2228\u2192 s)) ltr\ntranLFCRem ((s \u2190\u2228) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCRem (s \u2190\u2228\u2192 (\u2228\u2192 s\u2081)) ltr\ntranLFCRem ((\u2228\u2192 s) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCRem (\u2228\u2192 (s \u2190\u2228\u2192 s\u2081)) ltr\ntranLFCRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228\u2192 s\u2082) (\u2228\u2228d ltr)  = tranLFCRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228\u2192 s\u2082)) ltr\ntranLFCRem (s \u2190\u2228) (\u00ac\u2228\u2228d ltr)              = tranLFCRem ((s \u2190\u2228) \u2190\u2228) ltr\ntranLFCRem (\u2228\u2192 (s \u2190\u2228)) (\u00ac\u2228\u2228d ltr)         = tranLFCRem ((\u2228\u2192 s) \u2190\u2228) ltr\ntranLFCRem (\u2228\u2192 (\u2228\u2192 s)) (\u00ac\u2228\u2228d ltr)         = tranLFCRem (\u2228\u2192 s) ltr\ntranLFCRem (\u2228\u2192 (s \u2190\u2228\u2192 s\u2081)) (\u00ac\u2228\u2228d ltr)     = tranLFCRem ((\u2228\u2192 s) \u2190\u2228\u2192 s\u2081) ltr\ntranLFCRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228)) (\u00ac\u2228\u2228d ltr)     = tranLFCRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228) ltr\ntranLFCRem (s \u2190\u2228\u2192 (\u2228\u2192 s\u2081)) (\u00ac\u2228\u2228d ltr)     = tranLFCRem ((s \u2190\u2228) \u2190\u2228\u2192 s\u2081) ltr\ntranLFCRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228\u2192 s\u2082)) (\u00ac\u2228\u2228d ltr) = tranLFCRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228\u2192 s\u2082) ltr\ntranLFCRem ((s \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCRem (s \u2190\u2202) ltr\ntranLFCRem ((\u2202\u2192 s) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCRem (\u2202\u2192 (s \u2190\u2202)) ltr\ntranLFCRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202) (\u2202\u2202d ltr)      = tranLFCRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202)) ltr\ntranLFCRem (\u2202\u2192 s) (\u2202\u2202d ltr)               = tranLFCRem (\u2202\u2192 (\u2202\u2192 s)) ltr\ntranLFCRem ((s \u2190\u2202) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCRem (s \u2190\u2202\u2192 (\u2202\u2192 s\u2081)) ltr\ntranLFCRem ((\u2202\u2192 s) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCRem (\u2202\u2192 (s \u2190\u2202\u2192 s\u2081)) ltr\ntranLFCRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202\u2192 s\u2082) (\u2202\u2202d ltr)  = tranLFCRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202\u2192 s\u2082)) ltr\ntranLFCRem (s \u2190\u2202) (\u00ac\u2202\u2202d ltr)              = tranLFCRem ((s \u2190\u2202) \u2190\u2202) ltr\ntranLFCRem (\u2202\u2192 (s \u2190\u2202)) (\u00ac\u2202\u2202d ltr)         = tranLFCRem ((\u2202\u2192 s) \u2190\u2202) ltr\ntranLFCRem (\u2202\u2192 (\u2202\u2192 s)) (\u00ac\u2202\u2202d ltr)         = tranLFCRem (\u2202\u2192 s) ltr\ntranLFCRem (\u2202\u2192 (s \u2190\u2202\u2192 s\u2081)) (\u00ac\u2202\u2202d ltr)     = tranLFCRem ((\u2202\u2192 s) \u2190\u2202\u2192 s\u2081) ltr\ntranLFCRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202)) (\u00ac\u2202\u2202d ltr)     = tranLFCRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202) ltr\ntranLFCRem (s \u2190\u2202\u2192 (\u2202\u2192 s\u2081)) (\u00ac\u2202\u2202d ltr)     = tranLFCRem ((s \u2190\u2202) \u2190\u2202\u2192 s\u2081) ltr\ntranLFCRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202\u2192 s\u2082)) (\u00ac\u2202\u2202d ltr) = tranLFCRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202\u2192 s\u2082) ltr\n\n\nextendLFCRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 IndexLL {i} {u} pll ll \u2192 SetLFCRem {i} olf pll \u2192 SetLFCRem olf ll\nextendLFCRem \u2193 sr = sr\nextendLFCRem (ind \u2190\u2227) sr = (extendLFCRem ind sr) \u2190\u2227\nextendLFCRem (\u2227\u2192 ind) sr = \u2227\u2192 (extendLFCRem ind sr)\nextendLFCRem (ind \u2190\u2228) sr = (extendLFCRem ind sr) \u2190\u2228\nextendLFCRem (\u2228\u2192 ind) sr = \u2228\u2192 (extendLFCRem ind sr)\nextendLFCRem (ind \u2190\u2202) sr = (extendLFCRem ind sr) \u2190\u2202\nextendLFCRem (\u2202\u2192 ind) sr = \u2202\u2192 (extendLFCRem ind sr)\n\nreplaceLFCRem : \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 SetLFCRem {i} olf rll \u2192 SetLFCRem olf ll \u2192 SetLFCRem olf (replLL ll ind rll)\nreplaceLFCRem \u2193 esr sr = esr\nreplaceLFCRem {rll = rll} (ind \u2190\u2227) esr (sr \u2190\u2227) = replaceLFCRem ind esr sr \u2190\u2227\nreplaceLFCRem {rll = rll} (ind \u2190\u2227) esr (\u2227\u2192 sr) = (extendLFCRem (updateIndex rll ind) esr) \u2190\u2227\u2192 sr\nreplaceLFCRem {rll = rll} (ind \u2190\u2227) esr (sr \u2190\u2227\u2192 sr\u2081) = (replaceLFCRem ind esr sr) \u2190\u2227\u2192 sr\u2081\nreplaceLFCRem {rll = rll} (\u2227\u2192 ind) esr (sr \u2190\u2227) = sr \u2190\u2227\u2192 (extendLFCRem (updateIndex rll ind) esr)\nreplaceLFCRem {rll = rll} (\u2227\u2192 ind) esr (\u2227\u2192 sr) = \u2227\u2192 replaceLFCRem ind esr sr\nreplaceLFCRem {rll = rll} (\u2227\u2192 ind) esr (sr \u2190\u2227\u2192 sr\u2081) = sr \u2190\u2227\u2192 replaceLFCRem ind esr sr\u2081\nreplaceLFCRem {rll = rll} (ind \u2190\u2228) esr (sr \u2190\u2228) = replaceLFCRem ind esr sr \u2190\u2228\nreplaceLFCRem {rll = rll} (ind \u2190\u2228) esr (\u2228\u2192 sr) = (extendLFCRem (updateIndex rll ind) esr) \u2190\u2228\u2192 sr\nreplaceLFCRem {rll = rll} (ind \u2190\u2228) esr (sr \u2190\u2228\u2192 sr\u2081) = (replaceLFCRem ind esr sr) \u2190\u2228\u2192 sr\u2081\nreplaceLFCRem {rll = rll} (\u2228\u2192 ind) esr (sr \u2190\u2228) = sr \u2190\u2228\u2192 (extendLFCRem (updateIndex rll ind) esr)\nreplaceLFCRem {rll = rll} (\u2228\u2192 ind) esr (\u2228\u2192 sr) = \u2228\u2192 replaceLFCRem ind esr sr\nreplaceLFCRem {rll = rll} (\u2228\u2192 ind) esr (sr \u2190\u2228\u2192 sr\u2081) = sr \u2190\u2228\u2192 replaceLFCRem ind esr sr\u2081\nreplaceLFCRem {rll = rll} (ind \u2190\u2202) esr (sr \u2190\u2202) = replaceLFCRem ind esr sr \u2190\u2202\nreplaceLFCRem {rll = rll} (ind \u2190\u2202) esr (\u2202\u2192 sr) = (extendLFCRem (updateIndex rll ind) esr) \u2190\u2202\u2192 sr\nreplaceLFCRem {rll = rll} (ind \u2190\u2202) esr (sr \u2190\u2202\u2192 sr\u2081) = (replaceLFCRem ind esr sr) \u2190\u2202\u2192 sr\u2081\nreplaceLFCRem {rll = rll} (\u2202\u2192 ind) esr (sr \u2190\u2202) = sr \u2190\u2202\u2192 (extendLFCRem (updateIndex rll ind) esr)\nreplaceLFCRem {rll = rll} (\u2202\u2192 ind) esr (\u2202\u2192 sr) = \u2202\u2192 replaceLFCRem ind esr sr\nreplaceLFCRem {rll = rll} (\u2202\u2192 ind) esr (sr \u2190\u2202\u2192 sr\u2081) = sr \u2190\u2202\u2192 replaceLFCRem ind esr sr\u2081\n\n\nmreplaceLFCRem :  \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 MSetLFCRem {i} olf rll \u2192 MSetLFCRem olf ll \u2192 MSetLFCRem olf (replLL ll ind rll)\nmreplaceLFCRem ind \u2205 \u2205 = \u2205\nmreplaceLFCRem {rll = rll} ind \u2205 (\u00ac\u2205 x) = delLFCRem x ind rll\nmreplaceLFCRem {rll = rll} ind (\u00ac\u2205 x) \u2205 = \u00ac\u2205 (extendLFCRem (updateIndex rll ind) x)\nmreplaceLFCRem ind (\u00ac\u2205 x) (\u00ac\u2205 x\u2081) = \u00ac\u2205 (replaceLFCRem ind x x\u2081)\n\nfindCallGraph : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (lf : LFun {i} {u} ll rll) \u2192 (IndexLFC lf \u2192 IndexLFC olf) \u2192 MSetLFCRem olf ll \u2192 MSetLFC olf olf \u2192 MSetLFCRem olf rll \u00d7 MSetLFC olf olf\nfindCallGraph I if msr ms = msr , ms\nfindCallGraph (_\u2282_ {ind = ind} lf lf\u2081) if msr ms = let emsr , ems = findCallGraph lf (\u03bb x \u2192 if (x \u2190\u2282)) (truncSetLFCRem msr ind) ms\n                                                in findCallGraph lf\u2081 (\u03bb x \u2192 if (\u2282\u2192 x)) (mreplaceLFCRem ind emsr msr) ems \nfindCallGraph (tr ltr lf) if \u2205 ms = \u2205 , ms\nfindCallGraph (tr ltr lf) if (\u00ac\u2205 x) ms = findCallGraph lf (\u03bb x \u2192 if (tr x)) (\u00ac\u2205 $ tranLFCRem x ltr) ms\nfindCallGraph (com df lf) if \u2205 ms = findCallGraph lf (\u03bb x \u2192 if (com x)) \u2205 ms\nfindCallGraph (com df lf) if (\u00ac\u2205 x) ms = IMPOSSIBLE\nfindCallGraph {ll = ll} {rll = rll} {olf = olf} (call x) if msr ms = {!!} -- where\n\n--findCallGraph {ll = ll} {rll = call .\u221erll} {olf = olf} (call {\u221erll = \u221erll} x) if msr ms = {!!} -- where\n\n--  hf : IndexLFC olf \u2192 MSetLFCRem olf ll \u2192 MSetLFC olf olf \u2192 MSetLFCRem olf rll \u00d7 MSetLFC olf olf\n--  hf oic msr ms = {!!}\n","old_contents":"\nmodule IndexLF where\n\nopen import Common\nopen import LinLogic\n--open import LinDepT\n--open import LinT \n--open import SetLL\n--open import SetLLProp\n--open import SetLLRem\nopen import LinFun\n\nopen import Data.Product\n\n\ndata IndexLFC {i u} : \u2200{ll rll} \u2192 LFun {i} {u} ll rll \u2192 Set where\n  \u2193    : \u2200{ll rll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 IndexLFC lf\n  _\u2190\u2282_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 IndexLFC elf\n         \u2192 IndexLFC (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  _\u2282\u2192_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 IndexLFC lf\n         \u2192 IndexLFC (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  tr   : \u2200{ll orll rll} \u2192 {ltr : LLTr orll ll} \u2192 {lf : LFun {i} {u} orll rll}\n         \u2192 IndexLFC lf \u2192 IndexLFC (tr ltr lf) \n  com  : \u2200{rll ll frll prfi prfo df lf}\n         \u2192 IndexLFC lf\n         \u2192 IndexLFC (com {i} {u} {rll} {ll} {frll} {{prfi}} {{prfo}} df lf)\n\n\n\ndata SetLFC {i u oll orll} (olf : LFun {i} {u} oll orll) : \u2200{ll rll} \u2192 LFun {i} {u} ll rll \u2192 Set (lsuc u) where\n  \u2193    : \u2200{ll rll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 IndexLFC olf \u2192 SetLFC olf lf\n  _\u2190\u2282_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 SetLFC olf elf\n         \u2192 SetLFC olf (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  _\u2282\u2192_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  _\u2190\u2282\u2192_ : \u2200{rll pll ell ll ind elf lf}\n         \u2192 SetLFC olf elf\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  tr   : \u2200{ll orll rll} \u2192 {ltr : LLTr orll ll} \u2192 {lf : LFun {i} {u} orll rll}\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (tr ltr lf) \n  com  : \u2200{rll ll frll prfi prfo df lf}\n         \u2192 SetLFC olf lf\n         \u2192 SetLFC olf (com {i} {u} {rll} {ll} {frll} {{prfi}} {{prfo}} df lf)\n\ndata MSetLFC {i u oll orll} (olf : LFun {i} {u} oll orll) : \u2200{ll rll} \u2192 LFun {i} {u} ll rll \u2192 Set (lsuc u) where\n  \u2205   : \u2200{ll rll} \u2192 {lf : LFun {i} {u} ll rll}            \u2192 MSetLFC olf lf\n  \u00ac\u2205  : \u2200{ll rll} \u2192 {lf : LFun {i} {u} ll rll} \u2192 SetLFC olf lf \u2192 MSetLFC olf lf\n\n\ndata SetLFCRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2193c    : \u2200{\u221ell} \u2192 IndexLFC {i} olf                     \u2192 SetLFCRem olf (call \u221ell)\n  _\u2190\u2227   : \u2200{rs ls} \u2192 SetLFCRem olf ls                   \u2192 SetLFCRem olf (ls \u2227 rs)\n  \u2227\u2192_   : \u2200{rs ls} \u2192 SetLFCRem olf rs                   \u2192 SetLFCRem olf (ls \u2227 rs)\n  _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLFCRem olf ls \u2192 SetLFCRem olf rs \u2192 SetLFCRem olf (ls \u2227 rs)\n  _\u2190\u2228   : \u2200{rs ls} \u2192 SetLFCRem olf ls                   \u2192 SetLFCRem olf (ls \u2228 rs)\n  \u2228\u2192_   : \u2200{rs ls} \u2192 SetLFCRem olf rs                   \u2192 SetLFCRem olf (ls \u2228 rs)\n  _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLFCRem olf ls \u2192 SetLFCRem olf rs \u2192 SetLFCRem olf (ls \u2228 rs)\n  _\u2190\u2202   : \u2200{rs ls} \u2192 SetLFCRem olf ls                   \u2192 SetLFCRem olf (ls \u2202 rs)\n  \u2202\u2192_   : \u2200{rs ls} \u2192 SetLFCRem olf rs                   \u2192 SetLFCRem olf (ls \u2202 rs)\n  _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLFCRem olf ls \u2192 SetLFCRem olf rs \u2192 SetLFCRem olf (ls \u2202 rs)\n\ndata MSetLFCRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2205   : \u2200{ll}            \u2192 MSetLFCRem olf ll\n  \u00ac\u2205  : \u2200{ll} \u2192 SetLFCRem olf ll \u2192 MSetLFCRem olf ll\n\n\ntruncSetLFCRem : \u2200{i} \u2192 \u2200{u ll oll orll q} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCRem {i} {u} olf ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 MSetLFCRem {i} olf q\ntruncSetLFCRem \u2205 ind = \u2205\ntruncSetLFCRem (\u00ac\u2205 x) \u2193 = \u00ac\u2205 x\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227)) (ind \u2190\u2227) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (\u2227\u2192 x)) (ind \u2190\u2227) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (ind \u2190\u2227) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227)) (\u2227\u2192 ind) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (\u2227\u2192 x)) (\u2227\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (\u2227\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228)) (ind \u2190\u2228) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (\u2228\u2192 x)) (ind \u2190\u2228) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (ind \u2190\u2228) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228)) (\u2228\u2192 ind) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (\u2228\u2192 x)) (\u2228\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (\u2228\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202)) (ind \u2190\u2202) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (\u2202\u2192 x)) (ind \u2190\u2202) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (ind \u2190\u2202) = truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202)) (\u2202\u2192 ind) = \u2205\ntruncSetLFCRem (\u00ac\u2205 (\u2202\u2192 x)) (\u2202\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x) ind\ntruncSetLFCRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (\u2202\u2192 ind) =  truncSetLFCRem (\u00ac\u2205 x\u2081) ind\n\ndelLFCRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n      \u2192 MSetLFCRem {i} olf (replLL ll ind rll)\ndelLFCRem s \u2193 rll = \u2205\ndelLFCRem (s \u2190\u2227) (ind \u2190\u2227) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u2205 = \u2205\ndelLFCRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227)\ndelLFCRem (\u2227\u2192 s) (ind \u2190\u2227) rll = \u00ac\u2205 (\u2227\u2192 (s))\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u2205 = \u00ac\u2205 (\u2227\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2227) (\u2227\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) rll with (delLFCRem s ind rll)\ndelLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u2205 = \u2205\ndelLFCRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2227\u2192 x)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll with (delLFCRem s\u2081 ind rll)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2227\u2192 x)\ndelLFCRem (s \u2190\u2228) (ind \u2190\u2228) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u2205 = \u2205\ndelLFCRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228)\ndelLFCRem (\u2228\u2192 s) (ind \u2190\u2228) rll = \u00ac\u2205 (\u2228\u2192 (s))\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u2205 = \u00ac\u2205 (\u2228\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2228) (\u2228\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) rll with (delLFCRem s ind rll)\ndelLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u2205 = \u2205\ndelLFCRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2228\u2192 x)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll with (delLFCRem s\u2081 ind rll)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2228\u2192 x)\ndelLFCRem (s \u2190\u2202) (ind \u2190\u2202) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u2205 = \u2205\ndelLFCRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202)\ndelLFCRem (\u2202\u2192 s) (ind \u2190\u2202) rll = \u00ac\u2205 (\u2202\u2192 (s))\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll with (delLFCRem s ind rll)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u2205 = \u00ac\u2205 (\u2202\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202\u2192 (s\u2081))\ndelLFCRem (s \u2190\u2202) (\u2202\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) rll with (delLFCRem s ind rll)\ndelLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u2205 = \u2205\ndelLFCRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2202\u2192 x)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll with (delLFCRem s\u2081 ind rll)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2202\u2192 x)\n\nmdelLFCRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n             \u2192 MSetLFCRem {i} olf (replLL ll ind rll)\nmdelLFCRem \u2205 ind rll = \u2205\nmdelLFCRem (\u00ac\u2205 x) ind rll = delLFCRem x ind rll\n\n\n\nmreplaceLFCRem :  \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 MSetLFCRem {i} olf rll \u2192 MSetLFCRem olf ll \u2192 MSetLFCRem olf (replLL ll ind rll)\nmreplaceLFCRem ind \u2205 \u2205 = \u2205\nmreplaceLFCRem ind \u2205 (\u00ac\u2205 x) = {!!}\nmreplaceLFCRem ind (\u00ac\u2205 x) ms = {!!}\n\n\nfindCallGraph : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (lf : LFun {i} {u} ll rll) \u2192 MSetLFCRem olf ll \u2192 MSetLFC olf olf \u2192 MSetLFCRem olf rll \u00d7 MSetLFC olf olf\nfindCallGraph I msr ms = msr , ms\nfindCallGraph (_\u2282_ {ind = ind} lf lf\u2081) msr ms = let emsr , ems = findCallGraph lf (truncSetLFCRem msr ind) ms\n                                                in {!!} \nfindCallGraph (tr ltr lf) msr ms = {!!}\nfindCallGraph (com df lf) msr ms = {!!}\nfindCallGraph (call x) msr ms = {!!}\n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"fce4641e9e8aec601f2ab48e128ad529e5ac3a98","subject":"Clarify the problem even more","message":"Clarify the problem even more\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/DeriveCorrect.agda","new_file":"Thesis\/SIRelBigStep\/DeriveCorrect.agda","new_contents":"--\n-- In this module we conclude our proof: we have shown that t, derive-dterm t\n-- and produce related values (in the sense of the fundamental property,\n-- rfundamental3), and for related values v1, dv and v2 we have v1 \u2295 dv \u2261\n-- v2 (because \u2295 agrees with validity, rrelV3\u2192\u2295).\n--\n-- We now put these facts together via derive-correct-si and derive-correct.\n-- This is immediate, even though I spend so much code on it:\n-- Indeed, all theorems are immediate corollaries of what we established (as\n-- explained above); only the statements are longer, especially because I bother\n-- expanding them.\n\nmodule Thesis.SIRelBigStep.DeriveCorrect where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\nopen import Thesis.SIRelBigStep.FundamentalProperty\n\n-- Theorem statement. This theorem still mentions step-indexes explicitly.\nderive-correct-si-type =\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4) \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3-skeleton (\u03bb v1 dv v2 _ \u2192 v1 \u2295 dv \u2261 v2) t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\n-- A verified expansion of the theorem statement.\nderive-correct-si-type-means :\n  derive-correct-si-type \u2261\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j (j<k : j < k) \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct-si-type-means = refl\n\nderive-correct-si : derive-correct-si-type\nderive-correct-si t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rrelT3\u2192\u2295 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 (rfundamental3 _ t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1)\n\n-- Infinitely related environments:\nrrel\u03c13-inf : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12 = \u2200 k \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k\n\n-- A theorem statement for infinitely-related environments. On paper, if we informally\n-- omit the arbitrary derivation step-index as usual, we get a statement without\n-- step-indexes.\nderive-correct :\n  \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j = derive-correct-si t \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 (suc j)) v1 v2 j \u2264-refl\n\nopen import Data.Unit\n\nnil\u03c1 : \u2200 {\u0393 n} (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c1 d\u03c1 \u03c1 n\nnilV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\nnilV (closure t \u03c1) = let d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dclosure (derive-dterm t) \u03c1 d\u03c1 , rfundamental3svv _ (abs t) \u03c1 d\u03c1 \u03c1 \u03c1\u03c1\nnilV (natV n\u2081) = dnatV zero , refl\nnilV (pairV a b) = let 0a , aa = nilV a; 0b , bb = nilV b in dpairV 0a 0b , aa , bb\n\nnil\u03c1 \u2205 = \u2205 , tt\nnil\u03c1 (v \u2022 \u03c1) = let dv , vv = nilV v ; d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n\n-- -- Try to prove to define validity-preserving change composition.\n-- -- If we don't store proofs that environment changes in closures are valid, we\n-- -- can't finish the proof for the closure case using the Fundamental property.\n-- -- But it seems that's not the approach we need to use: we might need\n-- -- environment validity elsewhere.\n-- -- Moreover, the proof is annoying to do unless we build a datatype to invert\n-- -- validity proofs (as suggested by Ezyang in\n-- -- http:\/\/blog.ezyang.com\/2013\/09\/induction-and-logical-relations\/).\n\n-- ocompose\u03c1 : \u2200 {\u0393} n \u03c11 d\u03c11 \u03c12 d\u03c12 \u03c13 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c11 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c12 d\u03c12 \u03c13 n \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c13 n\n\n-- ocomposev : \u2200 {\u03c4} n v1 dv1 v2 dv2 v3 \u2192 rrelV3 \u03c4 v1 dv1 v2 n \u2192 rrelV3 \u03c4 v2 dv2 v3 n \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v1 dv v3 n\n-- ocomposev n v1 dv1 v2 (bang v3) .v3 vv1 refl = bang v3 , refl\n-- ocomposev n v1 (bang x) v2 (dclosure dt \u03c1 d\u03c1) v3 vv1 vv2 = {!!}\n-- ocomposev n (closure t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c11) (closure .t .(\u03c1 \u2295\u03c1 d\u03c11)) (dclosure .(derive-dterm t) .(\u03c1 \u2295\u03c1 d\u03c11) d\u03c12) (closure .t .((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12)) ((refl , refl) , refl , refl , refl , refl , vv1) ((refl , refl) , refl , refl , refl , refl , vv2) =\n--   let d\u03c1 , \u03c1\u03c1 = ocompose\u03c1 n \u03c1 d\u03c11 (\u03c1 \u2295\u03c1 d\u03c11) d\u03c12 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12) {!!} {!!}\n--   in dclosure (derive-dterm t) \u03c1 d\u03c1 , (refl , refl) , refl , {!!} , refl , refl ,\n--     \u03bb { k (s\u2264s k\u2264n) v1 dv v2 vv \u2192\n--       let nilv2 , nilv2v = nilV v2\n--       -- vv1 k ? v1 dv v2 vv\n--       -- vv2 k ? v2 nilv2 v2 nilv2v\n--       in {!!}\n--       }\n--       -- rfundamental3 k t (v1 \u2022 \u03c1) (dv \u2022 d\u03c1) (v2 \u2022 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12)) (vv , rrel\u03c13-mono k n (\u2264-step k<n) _ \u03c1 d\u03c1 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12) \u03c1\u03c1)}\n-- -- p1 , p2 , p3 , p4 , p5\n-- ocomposev n v1 dv1 v2 (dnatV n\u2081) v3 vv1 vv2 = {!!}\n-- ocomposev n v1 dv1 v2 (dpairV dv2 dv3) v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (bang v2) .v2 dv2 v3 refl vv2 = bang (v2 \u2295 dv2) , rrelV3\u2192\u2295 v2 dv2 v3 vv2\n-- -- ocomposev n v1 (dclosure dt \u03c1 d\u03c1) v2 dv2 v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (dnatV n\u2081) v2 dv2 v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (dpairV dv1 dv2) v2 dv3 v3 vv1 vv2 = {!!}\n-- -- ocomposeV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\n\n-- ocompose\u03c1 {\u2205} n \u2205 d\u03c11 \u03c12 d\u03c12 \u2205 \u03c1\u03c11 \u03c1\u03c12 = \u2205 , tt\n-- ocompose\u03c1 {\u03c4 \u2022 \u0393} n (v1 \u2022 \u03c11) (dv1 \u2022 d\u03c11) (v2 \u2022 \u03c12) (dv2 \u2022 d\u03c12) (v3 \u2022 \u03c13) (vv1 , \u03c1\u03c11) (vv2 , \u03c1\u03c12) =\n--   let dv , vv = ocomposev n v1 dv1 v2 dv2 v3 vv1 vv2\n--       d\u03c1 , \u03c1\u03c1 = ocompose\u03c1 n \u03c11 d\u03c11 \u03c12 d\u03c12 \u03c13 \u03c1\u03c11 \u03c1\u03c12\n--   in  dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n\n-- -- But we sort of know how to store environments validity proofs in closures, you just need to use well-founded inductions, even if you're using types. Sad, yes, that's insanely tedious. Let's leave that to Coq.\n-- -- What is also interesting: if that were fixed, could we prove correct\n-- -- composition for change *terms* and rrelT3? And for *open expressions*? The\n-- -- last is the main problem Ahmed run into.\n\n-- 1: the proof of transitivity of rrelT by Ahmed is for the case where the second rrelT holds at arbitrary step counts :-)\n-- 2: that proof doesn't go through for us. If de1 : e1 -> e2 and de2 : e2 ->\n-- e3, we must show that if e1 and e3 evaluate something happens, but to use our\n-- hypothesis we need that e2 also terminates, which in our case does not\n-- follow. For Ahmed 2006, instead, e1 terminates and e1 \u2264 e2 implies that e2\n-- terminates.\n--\n-- Indeed, it seems that we can have a change from e1 to looping e2 and a change\n-- from looping e2 to e3, and I don't expect the composition of such changes to\n-- satisfy anything.\n--\n-- Indeed, \"Imperative self-adjusting computation\" does not mention any\n-- transitivity result (even in the technical report).\n","old_contents":"--\n-- In this module we conclude our proof: we have shown that t, derive-dterm t\n-- and produce related values (in the sense of the fundamental property,\n-- rfundamental3), and for related values v1, dv and v2 we have v1 \u2295 dv \u2261\n-- v2 (because \u2295 agrees with validity, rrelV3\u2192\u2295).\n--\n-- We now put these facts together via derive-correct-si and derive-correct.\n-- This is immediate, even though I spend so much code on it:\n-- Indeed, all theorems are immediate corollaries of what we established (as\n-- explained above); only the statements are longer, especially because I bother\n-- expanding them.\n\nmodule Thesis.SIRelBigStep.DeriveCorrect where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\nopen import Thesis.SIRelBigStep.FundamentalProperty\n\n-- Theorem statement. This theorem still mentions step-indexes explicitly.\nderive-correct-si-type =\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4) \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3-skeleton (\u03bb v1 dv v2 _ \u2192 v1 \u2295 dv \u2261 v2) t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\n-- A verified expansion of the theorem statement.\nderive-correct-si-type-means :\n  derive-correct-si-type \u2261\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j (j<k : j < k) \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct-si-type-means = refl\n\nderive-correct-si : derive-correct-si-type\nderive-correct-si t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rrelT3\u2192\u2295 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 (rfundamental3 _ t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1)\n\n-- Infinitely related environments:\nrrel\u03c13-inf : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12 = \u2200 k \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k\n\n-- A theorem statement for infinitely-related environments. On paper, if we informally\n-- omit the arbitrary derivation step-index as usual, we get a statement without\n-- step-indexes.\nderive-correct :\n  \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j = derive-correct-si t \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 (suc j)) v1 v2 j \u2264-refl\n\nopen import Data.Unit\n\nnil\u03c1 : \u2200 {\u0393 n} (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c1 d\u03c1 \u03c1 n\nnilV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\nnilV (closure t \u03c1) = let d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dclosure (derive-dterm t) \u03c1 d\u03c1 , rfundamental3svv _ (abs t) \u03c1 d\u03c1 \u03c1 \u03c1\u03c1\nnilV (natV n\u2081) = dnatV zero , refl\nnilV (pairV a b) = let 0a , aa = nilV a; 0b , bb = nilV b in dpairV 0a 0b , aa , bb\n\nnil\u03c1 \u2205 = \u2205 , tt\nnil\u03c1 (v \u2022 \u03c1) = let dv , vv = nilV v ; d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n\n-- Try to prove to define validity-preserving change composition.\n-- Sadly, if we don't store proofs that environment changes in closures are valid, we can't finish the proof for the closure case :-(.\n-- Moreover, the proof is annoying to do unless we build a datatype to invert\n-- validity proofs (as suggested by Ezyang in\n-- http:\/\/blog.ezyang.com\/2013\/09\/induction-and-logical-relations\/).\n\n-- ocompose\u03c1 : \u2200 {\u0393} n \u03c11 d\u03c11 \u03c12 d\u03c12 \u03c13 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c11 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c12 d\u03c12 \u03c13 n \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c13 n\n\n-- ocomposev : \u2200 {\u03c4} n v1 dv1 v2 dv2 v3 \u2192 rrelV3 \u03c4 v1 dv1 v2 n \u2192 rrelV3 \u03c4 v2 dv2 v3 n \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v1 dv v3 n\n-- ocomposev n v1 dv1 v2 (bang v3) .v3 vv1 refl = bang v3 , refl\n-- ocomposev n v1 (bang x) v2 (dclosure dt \u03c1 d\u03c1) v3 vv1 vv2 = {!!}\n-- ocomposev n (closure t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c11) (closure .t .(\u03c1 \u2295\u03c1 d\u03c11)) (dclosure .(derive-dterm t) .(\u03c1 \u2295\u03c1 d\u03c11) d\u03c12) (closure .t .((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12)) ((refl , refl) , refl , refl , refl , refl , p5) ((refl , refl) , refl , refl , refl , refl , p5q) =\n--   let d\u03c1 , \u03c1\u03c1 = ocompose\u03c1 n \u03c1 d\u03c11 (\u03c1 \u2295\u03c1 d\u03c11) d\u03c12 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12) {!!} {!!}\n--   in dclosure (derive-dterm t) \u03c1 d\u03c1 , (refl , refl) , refl , {!!} , refl , refl ,\n--     \u03bb { k (s\u2264s k<n) v1 dv v2 vv \u2192\n--       rfundamental3 k t (v1 \u2022 \u03c1) (dv \u2022 d\u03c1) (v2 \u2022 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12)) (vv , rrel\u03c13-mono k n (\u2264-step k<n) _ \u03c1 d\u03c1 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12) \u03c1\u03c1)}\n-- -- p1 , p2 , p3 , p4 , p5\n-- ocomposev n v1 dv1 v2 (dnatV n\u2081) v3 vv1 vv2 = {!!}\n-- ocomposev n v1 dv1 v2 (dpairV dv2 dv3) v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (bang v2) .v2 dv2 v3 refl vv2 = bang (v2 \u2295 dv2) , rrelV3\u2192\u2295 v2 dv2 v3 vv2\n-- -- ocomposev n v1 (dclosure dt \u03c1 d\u03c1) v2 dv2 v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (dnatV n\u2081) v2 dv2 v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (dpairV dv1 dv2) v2 dv3 v3 vv1 vv2 = {!!}\n-- -- ocomposeV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\n\n-- ocompose\u03c1 {\u2205} n \u2205 d\u03c11 \u03c12 d\u03c12 \u2205 \u03c1\u03c11 \u03c1\u03c12 = \u2205 , tt\n-- ocompose\u03c1 {\u03c4 \u2022 \u0393} n (v1 \u2022 \u03c11) (dv1 \u2022 d\u03c11) (v2 \u2022 \u03c12) (dv2 \u2022 d\u03c12) (v3 \u2022 \u03c13) (vv1 , \u03c1\u03c11) (vv2 , \u03c1\u03c12) =\n--   let dv , vv = ocomposev n v1 dv1 v2 dv2 v3 vv1 vv2\n--       d\u03c1 , \u03c1\u03c1 = ocompose\u03c1 n \u03c11 d\u03c11 \u03c12 d\u03c12 \u03c13 \u03c1\u03c11 \u03c1\u03c12\n--   in  dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n\n-- But we sort of know how to store environments validity proofs in closures, you just need to use well-founded inductions, even if you're using types. Sad, yes, that's insanely tedious. Let's leave that to Coq.\n-- What is also interesting: if that were fixed, could we prove correct\n-- composition for change *terms* and rrelT3? And for *open expressions*? The\n-- last is the main problem Ahmed run into.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4e6af19175375d6783363dec2024d4f5dd69a761","subject":"Make WellDefinedFunChangePoint more reusable","message":"Make WellDefinedFunChangePoint more reusable\n\nThe equation doesn't need to take a validity proof, that's the job of\nthe caller who adds a quantifier.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da (ada : valid a da) \u2192 WellDefinedFunChangePoint f df a da\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","old_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da (v : valid a da) \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da ada = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da ada \u2192 WellDefinedFunChangePoint f df a da ada\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da ada\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25a8c33774069a007d62b7a925de59a42ac1a524","subject":"Added missing file.","message":"Added missing file.\n\nIgnore-this: 45063ef9edde26e8f962f5fef0fdff1\n\ndarcs-hash:20120303171837-3bd4e-2248a101c8906fdd7fc50705b3dddbf0c6ad793b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Inductive\/Existential.agda","new_file":"src\/PA\/Inductive\/Existential.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"94baa2277eab970473ec537bd06b77a81c0f50d3","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: eaa80898c9e592e71c0bc7a22bc4421d\n\ndarcs-hash:20110326142536-3bd4e-f70bb98b37004c280c864c27990959289ad56034.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Agsy\/McCarthy91\/Arithmetic.agda","new_file":"src\/Agsy\/McCarthy91\/Arithmetic.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"2730b5e257ec16c740e4a258174cbad4cf935bd4","subject":"Removed unnecessary hint.","message":"Removed unnecessary hint.\n\nIgnore-this: 9ad1f3cc7186aff0022741330cc1e430\n\ndarcs-hash:20110516153044-3bd4e-137c221b12b0b62e85e9ad09440d6b8d01154138.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/McCarthy91.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/McCarthy91.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"8ac6a2540197b2687f8f8ecac3c467726a46260d","subject":"Added tabulate-ext","message":"Added tabulate-ext\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_) \nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Set a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero} f\u2257g = refl\n    tabulate-ext {suc n} f\u2257g rewrite f\u2257g zero | tabulate-ext (f\u2257g \u2218 suc) = refl\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : Set a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = refl\n    map-id (x \u2237 xs) = \u2261.cong (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = refl\n    map-\u2218 f g (x \u2237 xs) = \u2261.cong (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : Set a} {B : Set b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g [] = refl\n    map-ext f\u2257g (x \u2237 xs) rewrite f\u2257g x | map-ext f\u2257g xs = refl\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x [] = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) rewrite sum-\u2237\u02b3 x xs | \u2115\u00b0.+-assoc x\u2081 (sum xs) x = refl\n\nrot\u2081 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : Set a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : Set} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = sym (proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : Set} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = refl\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : Set} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 [] = refl\nsum-rot\u2081 (x \u2237 xs) rewrite sum-\u2237\u02b3 x xs = \u2115\u00b0.+-comm x _\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x [] = refl\nmap-\u2237\u02b3 f x (x\u2081 \u2237 xs) rewrite map-\u2237\u02b3 f x xs = refl\n\nsum-map-rot\u2081 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = refl\nsum-map-rot\u2081 f (x \u2237 xs) rewrite \u2115\u00b0.+-comm (f x) (sum (map f xs))\n                              | \u2261.sym (sum-\u2237\u02b3 (f x) (map f xs))\n                              | \u2261.sym (map-\u2237\u02b3 f x xs)\n                              = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_) \nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : Set a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = refl\n    map-id (x \u2237 xs) = \u2261.cong (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = refl\n    map-\u2218 f g (x \u2237 xs) = \u2261.cong (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : Set a} {B : Set b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g [] = refl\n    map-ext f\u2257g (x \u2237 xs) rewrite f\u2257g x | map-ext f\u2257g xs = refl\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x [] = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) rewrite sum-\u2237\u02b3 x xs | \u2115\u00b0.+-assoc x\u2081 (sum xs) x = refl\n\nrot\u2081 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : Set a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : Set} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = sym (proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : Set} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = refl\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : Set} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 [] = refl\nsum-rot\u2081 (x \u2237 xs) rewrite sum-\u2237\u02b3 x xs = \u2115\u00b0.+-comm x _\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x [] = refl\nmap-\u2237\u02b3 f x (x\u2081 \u2237 xs) rewrite map-\u2237\u02b3 f x xs = refl\n\nsum-map-rot\u2081 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = refl\nsum-map-rot\u2081 f (x \u2237 xs) rewrite \u2115\u00b0.+-comm (f x) (sum (map f xs))\n                              | \u2261.sym (sum-\u2237\u02b3 (f x) (map f xs))\n                              | \u2261.sym (map-\u2237\u02b3 f x xs)\n                              = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"859032ca05deb8289658f5a7f221093c80c4a98c","subject":"Updated draft on the existential quantifier.","message":"Updated draft on the existential quantifier.\n\nIgnore-this: 5046750c3e123601a08c81751ccf35\n\ndarcs-hash:20120423154933-3bd4e-daa1b5317e3ea83978dd71f98144270fc4cc3996.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/PA\/Inductive\/Witness.agda","new_file":"Draft\/PA\/Inductive\/Witness.agda","new_contents":"-- Tested with FOT on 23 April 2012.\n\nmodule Draft.PA.Inductive.Witness where\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n\nopen import PA.Inductive.Base\nopen import PA.Inductive.Properties\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on M\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\ndata \u2203 (A : M \u2192 Set) : Set where\n  _,_ : (x : M) \u2192 A x \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 M\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Non-intuitionistic logic theorems\n\npostulate\n  -- The principle of indirect proof (proof by contradiction).\n  \u00ac-elim : \u2200 {A} \u2192 (\u00ac A \u2192 \u22a5) \u2192 A\n\n  -- \u2203 in terms of \u2200 and \u00ac.\n  \u00ac\u2203\u00ac\u2192\u2200 : {A : M \u2192 Set} \u2192 \u00ac (\u2203[ x ] \u00ac A x) \u2192 \u2200 {x} \u2192 A x\n\n------------------------------------------------------------------------------\n\n-- Constructive proof.\nproof\u2081 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2081 = succ zero , x\u2262Sx\n\n-- Non-constructive proof.\nproof\u2082 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2082 = \u00ac-elim (\u03bb h \u2192 x\u2262Sx {succ zero} (\u00ac\u2203\u00ac\u2192\u2200 h))\n\n-- We can extract an existential witness from a constructive proof.\nwitness\u2081 : \u2203-proj\u2081 proof\u2081 \u2261 succ zero\nwitness\u2081 = refl\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n-- witness\u2082 : \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n-- witness\u2082 = {!refl!}\n\n-- Agda error:\n--\n-- \u2203-proj\u2081 (\u00ac-elim (\u03bb h \u2192 x\u2262Sx (\u00ac\u2203\u00ac\u2192\u2200 h))) != succ zero of type M\n-- when checking that the expression refl has type\n-- \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n","old_contents":"-- Tested with FOT on 12 March 2012.\n\nmodule Draft.PA.Inductive.Witness where\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n\nopen import PA.Inductive.Base\nopen import PA.Inductive.Properties\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on M\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\ndata \u2203 (A : M \u2192 Set) : Set where\n  _,_ : (x : M) \u2192 A x \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 M\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Non-intuitionistic logic theorems\n\npostulate\n  -- The principle of indirect proof (proof by contradiction).\n  \u00ac-elim : \u2200 {A} \u2192 (\u00ac A \u2192 \u22a5) \u2192 A\n\n  -- \u2203 in terms of \u2200 and \u00ac.\n  \u00ac\u2203\u00ac\u2192\u2200 : {A : M \u2192 Set} \u2192 \u00ac (\u2203[ x ] \u00ac A x) \u2192 \u2200 {x} \u2192 A x\n\n------------------------------------------------------------------------------\n\n-- Constructive proof.\nproof\u2081 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2081 = succ zero , x\u2262Sx\n\n-- Non-constructive proof.\nproof\u2082 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2082 = \u00ac-elim (\u03bb h \u2192 x\u2262Sx {succ zero} (\u00ac\u2203\u00ac\u2192\u2200 h))\n\n-- We can extract an existential witness from a constructive proof.\nwitness\u2081 : \u2203-proj\u2081 proof\u2081 \u2261 succ zero\nwitness\u2081 = refl\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n-- witness\u2082 : \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n-- witness\u2082 = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6ecbb2d5d6f0f6024afa19dfbd12fdcad3b23439","subject":"Implement natural semantics.","message":"Implement natural semantics.\n\nNatural semantics (= big-step operational semantics) came up in\na Skype chat with Paolo.\n\nOld-commit-hash: 40138450230bdc6326bbd4d47b011921433466b1\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Denotational Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Denotational Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- CHANGE TERMS\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n","old_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- CHANGE TERMS\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64e621c3d2a135c31988f71656671c97b59d8eca","subject":"Add separations","message":"Add separations\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or true   = true\ntrue or false  = true\nfalse or true  = true\nfalse or false = false\n\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\napply : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\napply _ \u2218        = \u2218\napply f (x \u2237 xs) = (f x) \u2237 (apply f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\n_,_\u22a8_ : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\n((model holds _) , cs \u22a8 n) = (cs , holds \u22a2 n)\n\n_,_\u00ac\u22a8_ : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\n((model _ fails) , cs \u00ac\u22a8 n) = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n\n\n--_,_,_\u00ac\u22a8_ : List Separation \u2192 List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\n\n\n\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or true   = true\ntrue or false  = true\nfalse or true  = true\nfalse or false = false\n\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dde3bebb67c586b0525f1ff8a908f73f37e56d97","subject":"Stratified Desc model: checkpoint Set0\/Set1","message":"Stratified Desc model: checkpoint Set0\/Set1","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : A -> Set m}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set1 where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {l : Level}(D : Desc)(X : Set)(P : X -> Set l) -> [| D |] X -> Set l\nAll id          X P x        = P x\nAll (const Z)   X P x        = Unit\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = Void\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\nproof-map-compos : (D : Desc)(X Y Z : Set)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\n-- But the termination checker is unhappy.\n-- So we write the following:\n\nmodule Elim {l : Level}\n            (D : Desc)\n            (P : Mu D -> Set l)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = Void\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst  natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsu : Nat -> Nat\nsu n = con (Su , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n\n--********************************************\n-- Finite sets\n--********************************************\n\n-- If we weren't such bug fans of levitating things, we would\n-- implement finite sets with:\n\n{-\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n-- But no, we make it fly in Desc:\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = su e \n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n-- All : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l) -> [| D |] X -> Set l\n\n{-\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD (Nat {l = zero}) (\\e -> (EnumT e -> Set1) -> Set1) xs -> \n           (EnumT (con xs) -> Set1) -> Set1\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n-}\n\n{-\ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\n-}\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (EnumT e -> Set) -> Set) xs -> \n           (EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P =  induction NatD (\\E -> (EnumT E -> Set) -> Set) casesSpi e P\n\n{-\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set1)\n                                  (b' : spi e P')\n                                  (x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set1)\n              (b' : spi (con xs) P')\n              (x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Su , n) hs P' b' EZe = fst b'\ncasesSwitch (Su , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\n\nswitch : (e : EnumU)\n         (P : EnumT e -> Set1)\n         (b : spi e P)\n         (x : EnumT e) -> P x\nswitch e P b xs =  induction NatD\n                            (\\e -> (P : EnumT e -> Set1)\n                                   (b : spi e P)\n                                   (xs : EnumT e) -> P xs) \n                            casesSwitch e P b xs \n\n\n--********************************************\n-- Tagged description\n--********************************************\n\n-- data Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j)\n-- spi : {l : Level}(e : EnumU)(P : EnumT e -> Set l) -> Set l\n\nTagDesc : Set2\nTagDesc = Sigma (EnumU {l = suc (suc zero)}) (\\e -> spi e (\\_ -> Desc {l = zero}))\n\ntoDesc : {l : Level} -> TagDesc -> Desc {l = zero}\ntoDesc {x} (B , F) = {!!} -- sigma (EnumT B) (\\e -> ?) --switch B (\\_ -> Desc {l = x}) F e)\n\ntest : (E : EnumU {l = suc (suc zero)}) -> spi E (\\_ -> Desc {l = zero}) -> EnumT E -> Desc {l = zero}\ntest B F e = {!switch B (\\_ -> Desc {l = zero}) F e!}\n\n-}\n--********************************************\n-- Catamorphism\n--********************************************\n\n{-\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n-}","old_contents":"\n {-# OPTIONS --universe-polymorphism #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : A -> Set m}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc {l : Level} : Set (suc l) where\n  id : Desc\n  const : Set l -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set l) -> (S -> Desc) -> Desc\n  pi : (S : Set l) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : {l : Level} -> Desc -> Set l -> Set l\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu {l : Level}(D : Desc {l = l}) : Set l where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l) -> [| D |] X -> Set l\nAll id          X P x        = P x\nAll (const Z)   X P x        = Unit\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = Void\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : {l : Level}(D : Desc)(X Y : Set l)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : {l : Level}(D : Desc)(X : Set l)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\nproof-map-compos : {l : Level}(D : Desc)(X Y Z : Set l)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\ninduction : {l : Set}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\n-- But the termination checker is unhappy.\n-- So we write the following:\n\nmodule Elim {l : Level}\n            (D : Desc)\n            (P : Mu D -> Set l)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = Void\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst {l : Level} : Set l where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : {l : Level} -> NatConst {l = l} -> Desc {l = l}\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : {l : Level} -> Desc {l = l}\nNatD {x} = sigma {l = x} NatConst  natCases\n\nNat : {l : Level} -> Set l\nNat = Mu NatD\n\nze : {l : Level} -> Nat {l = l}\nze {x} = con (pair {i = x} {j = x} Ze Void)\n\nsu : {l : Level} -> Nat {l = l} -> Nat {l = l}\nsu {x} n = con (pair {i = x} {j = x} Su n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n\n--********************************************\n-- Finite sets\n--********************************************\n\n-- If we weren't such bug fans of levitating things, we would\n-- implement finite sets with:\n\n{-\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n-- But no, we make it fly in Desc:\n\nEnumU : {l : Level} -> Set l\nEnumU = Nat\n\nnilE : {l : Level} -> EnumU {l = l}\nnilE {x} = ze {l = x}\n\nconsE : {l : Level} -> EnumU -> EnumU {l = l}\nconsE {x} e = su {l = x} e \n\ndata EnumT {l : Level} : (e : EnumU {l = l}) -> Set l where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n-- All : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l) -> [| D |] X -> Set l\n\n{-\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD (Nat {l = zero}) (\\e -> (EnumT e -> Set1) -> Set1) xs -> \n           (EnumT (con xs) -> Set1) -> Set1\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n-}\n\n{-\ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\n-}\n\nspi : (e : EnumU {l = zero})(P : EnumT {l = zero} e -> Set) -> Set\nspi e P =  induction NatD (\\E -> {!(EnumT {l = zero} e -> Set) -> Set!}) {!!} e  ?\n\n{-\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set1)\n                                  (b' : spi e P')\n                                  (x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set1)\n              (b' : spi (con xs) P')\n              (x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Su , n) hs P' b' EZe = fst b'\ncasesSwitch (Su , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\n\nswitch : (e : EnumU)\n         (P : EnumT e -> Set1)\n         (b : spi e P)\n         (x : EnumT e) -> P x\nswitch e P b xs =  induction NatD\n                            (\\e -> (P : EnumT e -> Set1)\n                                   (b : spi e P)\n                                   (xs : EnumT e) -> P xs) \n                            casesSwitch e P b xs \n\n\n--********************************************\n-- Tagged description\n--********************************************\n\n-- data Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j)\n-- spi : {l : Level}(e : EnumU)(P : EnumT e -> Set l) -> Set l\n\nTagDesc : Set2\nTagDesc = Sigma (EnumU {l = suc (suc zero)}) (\\e -> spi e (\\_ -> Desc {l = zero}))\n\ntoDesc : {l : Level} -> TagDesc -> Desc {l = zero}\ntoDesc {x} (B , F) = {!!} -- sigma (EnumT B) (\\e -> ?) --switch B (\\_ -> Desc {l = x}) F e)\n\ntest : (E : EnumU {l = suc (suc zero)}) -> spi E (\\_ -> Desc {l = zero}) -> EnumT E -> Desc {l = zero}\ntest B F e = {!switch B (\\_ -> Desc {l = zero}) F e!}\n\n-}\n--********************************************\n-- Catamorphism\n--********************************************\n\n{-\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n-}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2ec8aea3dc17eefe9711d6abe7b362518630d643","subject":"Only white space.","message":"Only white space.\n\nIgnore-this: 3c31ab0a948cb2060a9774b36afb0522\n\ndarcs-hash:20100505162557-3bd4e-272690a3fb1e771a2d00378d140d2867c3e088dc.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"MyStdLib\/Induction\/WellFounded.agda","new_file":"MyStdLib\/Induction\/WellFounded.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction\n------------------------------------------------------------------------------\n\nmodule MyStdLib.Induction.WellFounded where\n\n-- From: http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/\n\ndata Acc {A : Set}(R : A \u2192 A \u2192 Set) : A \u2192 Set where\n  acc : (x : A) \u2192 ((y : A) \u2192 R y x \u2192 Acc R y) \u2192 Acc R x\n\nWellFounded : {A : Set} \u2192 (A \u2192 A \u2192 Set) \u2192 Set\nWellFounded {A} R = (x : A) \u2192 Acc R x\n\naccFold : {A : Set}(R : A \u2192 A \u2192 Set){P : A \u2192 Set} \u2192\n          ((x : A) \u2192 ((y : A) \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n          (x : A) \u2192 Acc R x \u2192 P x\naccFold R f x (acc .x h) = f x (\u03bb y y<x \u2192 accFold R f y (h y y<x))\n\nwellFoundedInd : {A : Set} {R : A \u2192 A \u2192 Set} {P : A \u2192 Set} \u2192\n                 WellFounded R \u2192\n                 ((x : A) \u2192 ((y : A) \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n                 (x : A) \u2192 P x\nwellFoundedInd {R = R} wf f x = accFold R f x (wf x)\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded induction\n------------------------------------------------------------------------------\n\nmodule MyStdLib.Induction.WellFounded where\n\n-- From: http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/\n\ndata Acc {A : Set}(R : A \u2192 A \u2192 Set) : A \u2192 Set where\n  acc : (x : A) \u2192 (( y : A) \u2192  R y x \u2192 Acc R y) \u2192 Acc R x\n\nWellFounded : {A : Set} \u2192 (A \u2192 A \u2192 Set) \u2192 Set\nWellFounded {A} R = (x : A) \u2192 Acc R x\n\naccFold : {A : Set}(R : A \u2192 A \u2192 Set){P : A \u2192 Set} \u2192\n          ((x : A) \u2192 ((y : A) \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n          (x : A) \u2192 Acc R x \u2192 P x\naccFold R f x (acc .x h) = f x (\u03bb y y<x \u2192 accFold R f y (h y y<x))\n\nwellFoundedInd : {A : Set} {R : A \u2192 A \u2192 Set} {P : A \u2192 Set} \u2192\n                 WellFounded R \u2192\n                 ((x : A) \u2192 ((y : A) \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n                 (x : A) \u2192 P x\nwellFoundedInd {R = R} wf f x = accFold R f x (wf x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"af9d23c116a68842dcbe0843b111aba0220b3560","subject":"Rename Atlas-apply to apply-base.","message":"Rename Atlas-apply to apply-base.\n\nOld-commit-hash: aa571d5b9c41b6b4aa16886b55721dc3dbd9e033\n","repos":"inc-lc\/ilc-agda","old_file":"Atlas\/Change\/Term.agda","new_file":"Atlas\/Change\/Term.agda","new_contents":"module Atlas.Change.Term where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\n\n-- nil-changes\n\nnil-const : \u2200 {\u03b9 : Base} \u2192 Const  \u2205 (base (\u0394Base \u03b9))\nnil-const {\u03b9} = neutral {\u0394Base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9))\nnil-term {Bool}   = curriedConst (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = curriedConst (nil-const {Map \u03ba \u03b9})\n\n-- diff-term and apply-term\n\nopen import Parametric.Change.Term Const \u0394Base\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\ndiff-base : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))\ndiff-base {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\ndiff-base {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs diff-base) m\u2081 m\u2082)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\napply-base : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\napply-base {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\napply-base {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs apply-base) m\u2081 m\u2082)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (\u0394Type \u03c4)\ndiff = app\u2082 (lift-diff diff-base apply-base)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply diff-base apply-base)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n","old_contents":"module Atlas.Change.Term where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\n\n-- nil-changes\n\nnil-const : \u2200 {\u03b9 : Base} \u2192 Const  \u2205 (base (\u0394Base \u03b9))\nnil-const {\u03b9} = neutral {\u0394Base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9))\nnil-term {Bool}   = curriedConst (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = curriedConst (nil-const {Map \u03ba \u03b9})\n\n-- diff-term and apply-term\n\nopen import Parametric.Change.Term Const \u0394Base\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\ndiff-base : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))\ndiff-base {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\ndiff-base {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs diff-base) m\u2081 m\u2082)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\nAtlas-apply {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs Atlas-apply) m\u2081 m\u2082)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (\u0394Type \u03c4)\ndiff = app\u2082 (lift-diff diff-base Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply diff-base Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3cd27f996a67185bbbf95d3ca1036589584df0c8","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: 47e76b0e22c2d499cca454901b5785c\n\ndarcs-hash:20110301044040-3bd4e-fef74b987c4ab44e77595f538c5b01230921319d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Induction\/Acc\/WellFounded.agda","new_file":"src\/FOTC\/Data\/Nat\/Induction\/Acc\/WellFounded.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Induction.Acc.WellFounded where\n\n-- Adapted from\n-- http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/\n-- and the Agda standard library.\n\nopen import FOTC.Base\n\nopen import Common.Function\nopen import FOTC.Data.Nat.Type\n\n----------------------------------------------------------------------------\n-- The accessibility predicate: x is accessible if everything which is\n-- smaller than x is also accessible (inductively).\ndata Acc (_<_ : D \u2192 D \u2192 Set) : D \u2192 Set where\n acc : \u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m < n \u2192 Acc _<_ m) \u2192 Acc _<_ n\n\naccFold : {P : D \u2192 Set} (_<_ : D \u2192 D \u2192 Set) \u2192\n          (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m < n \u2192 P m) \u2192 P n) \u2192\n          \u2200 {n} \u2192 N n \u2192 Acc _<_ n \u2192 P n\naccFold _<_ f Nn (acc _ h) = f Nn (\u03bb Nm m<n \u2192 accFold _<_ f Nm (h Nm m<n))\n\n-- The accessibility predicate encodes what it means to be\n-- well-founded; if all elements are accessible, then _<_ is\n-- well-founded.\nWellFounded : (D \u2192 D \u2192 Set) \u2192 Set\nWellFounded _<_ = \u2200 {n} \u2192 N n \u2192 Acc _<_ n\n\nWellFoundedInduction : {P : D \u2192 Set} {_<_ : D \u2192 D \u2192 Set} \u2192\n                       WellFounded _<_ \u2192\n                       (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m < n \u2192 P m) \u2192 P n) \u2192\n                       \u2200 {n} \u2192 N n \u2192 P n\nWellFoundedInduction {_<_ = _<_} wf f Nn = accFold _<_ f Nn (wf Nn)\n\nmodule InverseImage {_<_ : D \u2192 D \u2192 Set}\n                    (f : D \u2192 D)\n                    (f-N : \u2200 {n} \u2192 N n \u2192 N (f n))\n                    where\n\n  accessible : \u2200 {n} \u2192 N n \u2192 Acc _<_ (f n) \u2192 Acc (\u03bb x y \u2192 (f x) < (f y)) n\n  accessible Nn (acc _ rs) =\n    acc Nn (\u03bb {m} Nm fm<fn \u2192 accessible Nm (rs (f-N Nm) fm<fn) )\n\n  wellFounded : WellFounded _<_ \u2192 WellFounded (\u03bb x y \u2192 (f x) < (f y))\n  wellFounded wf = \u03bb Nn \u2192 accessible Nn (wf (f-N Nn))\n","old_contents":"----------------------------------------------------------------------------\n-- Well-founded induction\n----------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Induction.Acc.WellFounded where\n\n-- Adapted from\n-- http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/ and the Agda standard library.\n\nopen import FOTC.Base\n\nopen import Common.Function\nopen import FOTC.Data.Nat.Type\n\n----------------------------------------------------------------------------\n-- The accessibility predicate: x is accessible if everything which is\n-- smaller than x is also accessible (inductively).\ndata Acc (_<_ : D \u2192 D \u2192 Set) : D \u2192 Set where\n acc : \u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m < n \u2192 Acc _<_ m) \u2192 Acc _<_ n\n\naccFold : {P : D \u2192 Set} (_<_ : D \u2192 D \u2192 Set) \u2192\n          (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m < n \u2192 P m) \u2192 P n) \u2192\n          \u2200 {n} \u2192 N n \u2192 Acc _<_ n \u2192 P n\naccFold _<_ f Nn (acc _ h) = f Nn (\u03bb Nm m<n \u2192 accFold _<_ f Nm (h Nm m<n))\n\n-- The accessibility predicate encodes what it means to be\n-- well-founded; if all elements are accessible, then _<_ is\n-- well-founded.\nWellFounded : (D \u2192 D \u2192 Set) \u2192 Set\nWellFounded _<_ = \u2200 {n} \u2192 N n \u2192 Acc _<_ n\n\nWellFoundedInduction : {P : D \u2192 Set} {_<_ : D \u2192 D \u2192 Set} \u2192\n                       WellFounded _<_ \u2192\n                       (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m < n \u2192 P m) \u2192 P n) \u2192\n                       \u2200 {n} \u2192 N n \u2192 P n\nWellFoundedInduction {_<_ = _<_} wf f Nn = accFold _<_ f Nn (wf Nn)\n\nmodule InverseImage {_<_ : D \u2192 D \u2192 Set}\n                    (f : D \u2192 D)\n                    (f-N : \u2200 {n} \u2192 N n \u2192 N (f n))\n                    where\n\n  accessible : \u2200 {n} \u2192 N n \u2192 Acc _<_ (f n) \u2192 Acc (\u03bb x y \u2192 (f x) < (f y)) n\n  accessible Nn (acc _ rs) =\n    acc Nn (\u03bb {m} Nm fm<fn \u2192 accessible Nm (rs (f-N Nm) fm<fn) )\n\n  wellFounded : WellFounded _<_ \u2192 WellFounded (\u03bb x y \u2192 (f x) < (f y))\n  wellFounded wf = \u03bb Nn \u2192 accessible Nn (wf (f-N Nn))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2155c808fb95a6bd8680325f29f4c8ff37b92c47","subject":"working on #14 after #26","message":"working on #14 after #26\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFinal x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFinal x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err -- this feels extremely strange\n    ie (INEHole x) (CECong FHOuter err) = {!!}\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err) = {!!}\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = {!!}\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFinal x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = {!!}\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFinal x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = {!!}\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFinal x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFinal x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err -- this is extremely strange\n    ie (INEHole x) (CECong x\u2081 err) = {!!} -- fe x {!!}\n    ie (IAp x indet x\u2081) (CECong x\u2082 err) = {!x\u2082!}\n    ie (ICastArr x indet) (CECong x\u2081 err) = {!!}\n    ie (ICastGroundHole x indet) (CECastFinal x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong x\u2081 err) = {!!}\n    ie (ICastHoleGround x indet x\u2081) (CECastFinal x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    ie (ICastHoleGround x indet x\u2081) (CECong x\u2082 err) = {!!}\n\n    -- ie (INEHole x) (ENEHole e) = fe x e\n    -- ie (IAp i x) (EAp1 e) = ie i e\n    -- ie (IAp i x) (EAp2 y e) = fe x e\n    -- ie (ICast i) (ECastError x x\u2081) = {!!} -- todo: this is evidence that casts are busted\n    -- ie (ICast i) (ECastProp x) = ie i x\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = {!!} -- cyrus\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 (FBoxed x) FHOuter = FBoxed x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet x) FHOuter = FIndet x\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = {!!} -- cyrus \/ maybe that error in the rule with pi-types\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!}\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter x\u2082 FHOuter) = {!!}\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1bea8799be81d6b9e2d85a6427556d95be3d529e","subject":"red counterexamples; green examples and counterexamples","message":"red counterexamples; green examples and counterexamples\n","repos":"toonn\/popartt","old_file":"koopa.agda","new_file":"koopa.agda","new_contents":"{-\n\n    Verified KoopaTroopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_; _<_ to _F<_; suc to fsuc;\n    zero to fzero)\n  open import Data.Vec renaming (map to vmap; lookup to vlookup;\n                                     replicate to vreplicate)\n  open import Data.Unit\n  open import Data.Empty\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa : Color \u2192 Set where\n    _KT : (c : Color) \u2192 KoopaTroopa c\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  data Clearance : Set where\n    Low  : Clearance\n    High : Clearance\n    God  : Clearance\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n      clr : Clearance\n\n  data _c>_ : Color \u2192 Clearance \u2192 Set where\n    <red>   : \u2200 {c} \u2192 c c> Low\n    <green> : Green c> High\n\n  data _follows_\u27e8_\u27e9 : Position \u2192 Position \u2192 Color \u2192 Set where\n    stay  : \u2200 {c x y} \u2192 pos x y gas Low follows pos x y gas Low \u27e8 c \u27e9\n    next  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos (suc x) y gas cl follows pos x y gas Low \u27e8 c \u27e9\n    back  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos x y gas cl follows pos (suc x) y gas Low \u27e8 c \u27e9\n    -- jump  : \u2200 {c x y} \u2192 pos x (suc y) gas High follows pos x y gas Low \u27e8 c \u27e9\n    fall  : \u2200 {c cl x y} \u2192 pos x y gas cl follows pos x (suc y) gas High \u27e8 c \u27e9\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path {c : Color} (Koopa : KoopaTroopa c) :\n       Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p \u27e8 c \u27e9\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 gas Low) (pos 0 0 gas Low)\n  ex_path = pos 0 0 gas Low \u21a0\u27e8 next \u27e9\n            pos 1 0 gas Low \u21a0\u27e8 back \u27e9\n            pos 0 0 gas Low \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192\n                   Vec Position n\n  matToPosVec []           []        _ _ = []\n  matToPosVec (mat \u2237 mats) (under \u2237 unders) x y =\n    pos x y mat cl \u2237 matToPosVec mats unders (x + 1) y\n      where\n        clearance : Material \u2192 Material \u2192 Clearance\n        clearance gas   gas   = High\n        clearance gas   solid = Low\n        clearance solid _     = God\n        cl = clearance mat under\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs [] = []\n  matToPosVecs (_\u2237_ {y} mats matss) =\n    matToPosVec mats (unders matss gas) 0 y \u2237 matToPosVecs matss\n    where\n      unders : \u2200 {m n \u2113}{A : Set \u2113} \u2192 Vec (Vec A m) n \u2192 A \u2192 Vec A m\n      unders [] fallback = vreplicate fallback\n      unders (us \u2237 _) _ = us\n\n  matsToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matsToMat matss = Mat (reverse (matToPosVecs matss))\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = matsToMat (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 [])\n\n  _<'_ : \u2115 \u2192 \u2115 \u2192 Set\n  m <' zero = \u22a5\n  zero <' suc n = \u22a4\n  suc m <' suc n = m <' n\n\n  fromNat : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  fromNat {zero} k {}\n  fromNat {suc n} zero = fzero\n  fromNat {suc n} (suc k) {p} = fsuc (fromNat k {p})\n \n  f : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  f = fromNat\n\n  p : (x : Fin 10) \u2192 (y : Fin 7) \u2192 Position\n  p x y = lookup y x example_level\n\n  red_path_one : Path (Red KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  red_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                 p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  red_path_two : Path (Red KT) (p (f 2) (f 1)) (p (f 3) (f 1))\n  red_path_two = p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 4) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 stay \u27e9\n                 []\n\n  -- -- Type error shows up 'late' because 'cons' is right associative\n  -- red_nopath_one : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- red_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9\n  --                  p (f 0) (f 1) \u21a0\u27e8 stay \u27e9\n  --                  []\n\n  -- -- Red KoopaTroopa can't step into a wall\n  -- red_nopath_two : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- red_nopath_two = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n\n  -- -- Red KoopaTroopa can't step into air\n  -- red_nopath_three : Path (Red KT) (p (f 4) (f 1)) (p (f 5) (f 1))\n  -- red_nopath_three = p (f 4) (f 1) \u21a0\u27e8 next \u27e9 []\n\n  -- Any path that is valid for red KoopaTroopas, is also valid for green\n  -- KoopaTroopas because we did not constrain KoopaTroopas to only turn\n  -- When there is an obstacle\n  green_path_one : Path (Green KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  green_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                   p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                   p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  green_path_two : Path (Green KT) (p (f 7) (f 6)) (p (f 5) (f 0))\n  green_path_two = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 6) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 5) (f 6) \u21a0\u27e8 fall \u27e9\n                   p (f 5) (f 5) \u21a0\u27e8 fall \u27e9\n                   p (f 5) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 4) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 3) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 2) (f 4) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 3) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 2) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                   p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 4) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 5) (f 1) \u21a0\u27e8 fall \u27e9\n                   []\n\n  -- -- Green KoopaTroopa can't step into a wall\n  -- green_nopath_one : Path (Green KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- green_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n","old_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_; _<_ to _F<_; suc to fsuc;\n    zero to fzero)\n  open import Data.Vec renaming (map to vmap; lookup to vlookup;\n                                     replicate to vreplicate)\n  open import Data.Unit\n  open import Data.Empty\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa : Color \u2192 Set where\n    _KT : (c : Color) \u2192 KoopaTroopa c\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  data Clearance : Set where\n    Low  : Clearance\n    High : Clearance\n    God  : Clearance\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n      clr : Clearance\n\n  data _c>_ : Color \u2192 Clearance \u2192 Set where\n    <red>   : \u2200 {c} \u2192 c c> Low\n    <green> : Green c> High\n\n  data _follows_\u27e8_\u27e9 : Position \u2192 Position \u2192 Color \u2192 Set where\n    stay  : \u2200 {c x y} \u2192 pos x y gas Low follows pos x y gas Low \u27e8 c \u27e9\n    next  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos (suc x) y gas cl follows pos x y gas Low \u27e8 c \u27e9\n    back  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos x y gas cl follows pos (suc x) y gas Low \u27e8 c \u27e9\n    -- jump  : \u2200 {c x y} \u2192 pos x (suc y) gas High follows pos x y gas Low \u27e8 c \u27e9\n    fall  : \u2200 {c cl x y} \u2192 pos x y gas cl follows pos x (suc y) gas High \u27e8 c \u27e9\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path {c : Color} (Koopa : KoopaTroopa c) :\n       Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p \u27e8 c \u27e9\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 gas Low) (pos 0 0 gas Low)\n  ex_path = pos 0 0 gas Low \u21a0\u27e8 next \u27e9\n            pos 1 0 gas Low \u21a0\u27e8 back \u27e9\n            pos 0 0 gas Low \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192\n                   Vec Position n\n  matToPosVec []           []        _ _ = []\n  matToPosVec (mat \u2237 mats) (under \u2237 unders) x y =\n    pos x y mat cl \u2237 matToPosVec mats unders (x + 1) y\n      where\n        clearance : Material \u2192 Material \u2192 Clearance\n        clearance gas   gas   = High\n        clearance gas   solid = Low\n        clearance solid _     = God\n        cl = clearance mat under\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs [] = []\n  matToPosVecs (_\u2237_ {y} mats matss) =\n    matToPosVec mats (unders matss gas) 0 y \u2237 matToPosVecs matss\n    where\n      unders : \u2200 {m n \u2113}{A : Set \u2113} \u2192 Vec (Vec A m) n \u2192 A \u2192 Vec A m\n      unders [] fallback = vreplicate fallback\n      unders (us \u2237 _) _ = us\n\n  matsToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matsToMat matss = Mat (reverse (matToPosVecs matss))\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = matsToMat (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 [])\n\n  _<'_ : \u2115 \u2192 \u2115 \u2192 Set\n  m <' zero = \u22a5\n  zero <' suc n = \u22a4\n  suc m <' suc n = m <' n\n\n  fromNat : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  fromNat {zero} k {}\n  fromNat {suc n} zero = fzero\n  fromNat {suc n} (suc k) {p} = fsuc (fromNat k {p})\n \n  f : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  f = fromNat\n\n  p : (x : Fin 10) \u2192 (y : Fin 7) \u2192 Position\n  p x y = lookup y x example_level\n\n  red_path_one : Path (Red KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  red_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                 p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  red_path_two : Path (Red KT) (p (f 2) (f 1)) (p (f 3) (f 1))\n  red_path_two = p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 4) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 stay \u27e9\n                 []\n\n  red_nopath_one : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  red_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9\n                   p (f 0) (f 1) \u21a0\u27e8 stay \u27e9\n                   []\n\n  red_nopath_two : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  red_nopath_two = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n","returncode":0,"stderr":"","license":"bsd-2-clause","lang":"Agda"}
{"commit":"5d9abe59d5961777849cc3f8b64da8fe85ce608b","subject":"Logical: +\u27e6\u2605\u27e7\u2080, +\u27e6\u2192\u27e7\u21d4Preserves, +\u27e6\u2192\u27e7\u00b2\u21d4Preserves\u2082","message":"Logical: +\u27e6\u2605\u27e7\u2080, +\u27e6\u2192\u27e7\u21d4Preserves, +\u27e6\u2192\u27e7\u00b2\u21d4Preserves\u2082\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Logical.agda","new_file":"lib\/Relation\/Binary\/Logical.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.Logical where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\nopen import Relation.Nullary\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\n\n\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63\n\n\u27e6\u2605\u27e7\u2080 : \u2200 {a\u2081 a\u2082} (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6\u2605\u27e7\u2080 = \u27e6\u2605\u27e7 zero\n\n\u27e6\u2605\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2080) \u2192 \u2605\u2081\n\u27e6\u2605\u2080\u27e7 = \u27e6\u2605\u27e7\u2080\n\n\u27e6\u2605\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2081) \u2192 \u2605\u2082\n\u27e6\u2605\u2081\u27e7 = \u27e6\u2605\u27e7 (suc zero)\n\n-- old name\n\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63\n\n-- old name\n\u27e6Set\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : Set) \u2192 Set\u2081\n\u27e6Set\u2080\u27e7 = \u27e6\u2605\u2080\u27e7\n\n-- old name\n\u27e6Set\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : Set\u2081) \u2192 Set\u2082\n\u27e6Set\u2081\u27e7 = \u27e6\u2605\u2081\u27e7\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\nrecord \u27e6\u22a4\u27e7 (x\u2081 x\u2082 : \u22a4) : \u2605\u2080 where\n  constructor \u27e6tt\u27e7\n\ndata \u27e6\u22a5\u27e7 (x\u2081 x\u2082 : \u22a5) : \u2605\u2080 where\n\ninfix 3 \u27e6\u00ac\u27e7_\n\n\u27e6\u00ac\u27e7_ : \u2200 {a\u2081 a\u2082 a\u209a} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u209a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) \u00ac_ \u00ac_\n\u27e6\u00ac\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u22a5\u27e7\n\n-- Products \u27e6\u03a3\u27e7, \u27e6\u2203\u27e7, \u27e6\u00d7\u27e7 are in Data.Product.NP\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n\u27e6Pred\u27e7 p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 p\u1d63\n\nprivate\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 \u2605 \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 r\u2081 r\u2082} r\u1d63 \u2192\n          (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (REL\u2032 r\u2081) (REL\u2032 r\u2082)\n  \u27e6REL\u27e7\u2032 r\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          {r\u2081 r\u2082} r\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 r\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 r\u2082) \u2192 \u2605 _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 r\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {r\u2081 r\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 r\u2081) (\u223c\u2082 : Rel A\u2082 r\u2082) \u2192 \u2605 _\n\u27e6Rel\u27e7 A\u1d63 r\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 r\u1d63\n\ndata \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : \u2605 \u2113\u2081} {P\u2082 : \u2605 \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 \u2605 \u2113\u1d63) : \u27e6\u2605\u27e7 (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) (Dec P\u2081) (Dec P\u2082) where\n  yes : {p\u2081 : P\u2081} {p\u2082 : P\u2082} (p\u1d63 : P\u1d63 p\u2081 p\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (yes p\u2081) (yes p\u2082)\n  no  : {\u00acp\u2081 : \u00ac P\u2081} {\u00acp\u2082 : \u00ac P\u2082} (\u00acp\u1d63 : (\u27e6\u00ac\u27e7 P\u1d63) \u00acp\u2081 \u00acp\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (no \u00acp\u2081) (no \u00acp\u2082)\n\nprivate\n  \u27e6Dec\u27e7' : \u2200 {p\u2081} {p\u2082} {p\u1d63} \u2192 \u27e6Pred\u27e7 {p\u2081} {p\u2082} _ (\u27e6\u2605\u27e7 p\u1d63) Dec Dec\n  \u27e6Dec\u27e7' = \u27e6Dec\u27e7\n\n\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082 \u2113\u1d63}\n              \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7\n                 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e9\u27e6\u2192\u27e7\n                   \u27e6REL\u27e7 A\u1d63 B\u1d63 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Decidable Decidable\n\u27e6Decidable\u27e7 A\u1d63 B\u1d63 _\u223c\u1d63_ = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e8 y\u1d63 \u2236 B\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Dec\u27e7 (x\u1d63 \u223c\u1d63 y\u1d63)\n\n\u27e6Op\u2081\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a) Op\u2081 Op\u2081\n\u27e6Op\u2081\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\n\u27e6Op\u2082\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a) Op\u2082 Op\u2082\n\u27e6Op\u2082\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\nopen import Function.Equivalence\nprivate\n  \u27e6\u2192\u27e7\u21d4Preserves :\n    \u2200 {a b a\u1d63 b\u1d63}\n      {A : \u2605 a} {B : \u2605 b}\n      {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A A}\n      {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B B}\n      {f}\n    \u2192 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) f f \u21d4 f Preserves A\u1d63 \u27f6 B\u1d63\n\n  \u27e6\u2192\u27e7\u21d4Preserves = equivalence (\u03bb x \u2192 x) (\u03bb x \u2192 x)\n\n  \u27e6\u2192\u27e7\u00b2\u21d4Preserves\u2082 :\n    \u2200 {a b c a\u1d63 b\u1d63 c\u1d63}\n      {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n      {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A A}\n      {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B B}\n      {C\u1d63 : \u27e6\u2605\u27e7 c\u1d63 C C}\n      {f}\n    \u2192 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 C\u1d63) f f \u21d4 f Preserves\u2082 A\u1d63 \u27f6 B\u1d63 \u27f6 C\u1d63\n\n  \u27e6\u2192\u27e7\u00b2\u21d4Preserves\u2082 = equivalence (\u03bb f {x} {y}   {_} {_} z \u2192 f {x} {y} z)\n                                (\u03bb f {x} {y} z {_} {_}   \u2192 f z)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.Logical where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\nopen import Relation.Nullary\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\n\n\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63\n\n\u27e6\u2605\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2080) \u2192 \u2605\u2081\n\u27e6\u2605\u2080\u27e7 = \u27e6\u2605\u27e7 zero\n\n\u27e6\u2605\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2081) \u2192 \u2605\u2082\n\u27e6\u2605\u2081\u27e7 = \u27e6\u2605\u27e7 (suc zero)\n\n-- old name\n\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63\n\n-- old name\n\u27e6Set\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : Set) \u2192 Set\u2081\n\u27e6Set\u2080\u27e7 = \u27e6\u2605\u2080\u27e7\n\n-- old name\n\u27e6Set\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : Set\u2081) \u2192 Set\u2082\n\u27e6Set\u2081\u27e7 = \u27e6\u2605\u2081\u27e7\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\nrecord \u27e6\u22a4\u27e7 (x\u2081 x\u2082 : \u22a4) : \u2605\u2080 where\n  constructor \u27e6tt\u27e7\n\ndata \u27e6\u22a5\u27e7 (x\u2081 x\u2082 : \u22a5) : \u2605\u2080 where\n\ninfix 3 \u27e6\u00ac\u27e7_\n\n\u27e6\u00ac\u27e7_ : \u2200 {a\u2081 a\u2082 a\u209a} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u209a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) \u00ac_ \u00ac_\n\u27e6\u00ac\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u22a5\u27e7\n\n-- Products \u27e6\u03a3\u27e7, \u27e6\u2203\u27e7, \u27e6\u00d7\u27e7 are in Data.Product.NP\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n\u27e6Pred\u27e7 p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 p\u1d63\n\nprivate\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 \u2605 \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 r\u2081 r\u2082} r\u1d63 \u2192\n          (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (REL\u2032 r\u2081) (REL\u2032 r\u2082)\n  \u27e6REL\u27e7\u2032 r\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          {r\u2081 r\u2082} r\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 r\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 r\u2082) \u2192 \u2605 _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 r\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {r\u2081 r\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 r\u2081) (\u223c\u2082 : Rel A\u2082 r\u2082) \u2192 \u2605 _\n\u27e6Rel\u27e7 A\u1d63 r\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 r\u1d63\n\ndata \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : \u2605 \u2113\u2081} {P\u2082 : \u2605 \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 \u2605 \u2113\u1d63) : \u27e6\u2605\u27e7 (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) (Dec P\u2081) (Dec P\u2082) where\n  yes : {p\u2081 : P\u2081} {p\u2082 : P\u2082} (p\u1d63 : P\u1d63 p\u2081 p\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (yes p\u2081) (yes p\u2082)\n  no  : {\u00acp\u2081 : \u00ac P\u2081} {\u00acp\u2082 : \u00ac P\u2082} (\u00acp\u1d63 : (\u27e6\u00ac\u27e7 P\u1d63) \u00acp\u2081 \u00acp\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (no \u00acp\u2081) (no \u00acp\u2082)\n\nprivate\n  \u27e6Dec\u27e7' : \u2200 {p\u2081} {p\u2082} {p\u1d63} \u2192 \u27e6Pred\u27e7 {p\u2081} {p\u2082} _ (\u27e6\u2605\u27e7 p\u1d63) Dec Dec\n  \u27e6Dec\u27e7' = \u27e6Dec\u27e7\n\n\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082 \u2113\u1d63}\n              \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7\n                 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e9\u27e6\u2192\u27e7\n                   \u27e6REL\u27e7 A\u1d63 B\u1d63 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Decidable Decidable\n\u27e6Decidable\u27e7 A\u1d63 B\u1d63 _\u223c\u1d63_ = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e8 y\u1d63 \u2236 B\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Dec\u27e7 (x\u1d63 \u223c\u1d63 y\u1d63)\n\n\u27e6Op\u2081\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a) Op\u2081 Op\u2081\n\u27e6Op\u2081\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\n\u27e6Op\u2082\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a) Op\u2082 Op\u2082\n\u27e6Op\u2082\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5a4ce333657f637b5ecb5b95c970bcacc6ba9988","subject":"Nand.Properties: renaming","message":"Nand.Properties: renaming\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Nand\/Properties.agda","new_file":"FunUniverse\/Nand\/Properties.agda","new_contents":"open import Level.NP\nopen import Type\nopen import Data.Two\nopen import Data.Product\nopen import Function.NP\nopen import Relation.Binary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\nimport Explore.Explorable\nopen import Explore.Universe\n\nopen import FunUniverse.Nand\nopen import FunUniverse.Agda\n\nmodule FunUniverse.Nand.Properties where\n\nmodule Test {B : \u2605} (_\u225f_ : Decidable {A = B} _\u2261_)\n            (A : Ty)\n            {f g : El A \u2192 B} where\n  module _ {\u2113} where\n    A\u1d49 : Explore \u2113 (El A)\n    A\u1d49 = exploreU A\n    A\u2071 : ExploreInd \u2113 A\u1d49\n    A\u2071 = exploreU-ind A\n  A\u02e1 : Lookup {\u2080} A\u1d49\n  A\u02e1 = lookupU A\n\n  Check! = A\u1d49 _\u00d7_ \u03bb x \u2192 \u2713 \u230a f x \u225f g x \u230b\n\n  check! : {p\u2713 : Check!} \u2192 f \u2257 g\n  check! {p\u2713} x = toWitness (A\u02e1 p\u2713 x)\n\n  {- Unused\n  open Explore.Explorable.Explorable\u2080 A\u2071\n  test-\u2227 = big-\u2227 \u03bb x \u2192 \u230a f x \u225f g x \u230b\n  -}\n\nmodule Test22 where\n  nand nand' : \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\n\n  nand  (x , y) = not (x \u2227 y)\n  nand' (x , y) = not x \u2228 not y\n\n  module N = FromNand funRewiring nand\n  module T = Test Data.Two._\u225f_\n\n  module UnOp where\n    open T \ud835\udfda\u2032\n\n    not-ok : N.not \u2257 not\n    not-ok = check!\n\n  module BinOp where\n    open T (\ud835\udfda\u2032 \u00d7\u2032 \ud835\udfda\u2032)\n\n    nand-ok : nand \u2257 nand'\n    nand-ok = check!\n\n    and-ok : N.and \u2257 uncurry _\u2227_\n    and-ok = check!\n\n    or-ok : N.or \u2257 uncurry _\u2228_\n    or-ok = check!\n\n    nor-ok : N.nor \u2257 (not \u2218 uncurry _\u2228_)\n    nor-ok = check!\n\n    xor-ok : N.xor \u2257 uncurry _xor_\n    xor-ok = check!\n\n    xnor-ok : N.xnor \u2257 uncurry _==_\n    xnor-ok = check!\n\n  module TriOp where\n    open T (\ud835\udfda\u2032 \u00d7\u2032 (\ud835\udfda\u2032 \u00d7\u2032 \ud835\udfda\u2032))\n\n    fork : \ud835\udfda \u00d7 \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\n    fork (c , e\u1d62) = proj e\u1d62 c\n\n    fork-ok : N.fork \u2257 fork\n    fork-ok = check!\n","old_contents":"open import Level.NP\nopen import Type\nopen import Data.Two\nopen import Data.Product\nopen import Function.NP\nopen import Relation.Binary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\nimport Explore.Explorable\nopen import Explore.Universe\n\nopen import FunUniverse.Nand\nopen import FunUniverse.Agda\n\nmodule FunUniverse.Nand.Properties where\n\nmodule Test {B : \u2605} (_\u225f_ : Decidable {A = B} _\u2261_)\n            (A : Ty)\n            {f g : El A \u2192 B} where\n  module _ {\u2113} where\n    A\u1d49 : Explore \u2113 (El A)\n    A\u1d49 = exploreU A\n    A\u2071 : ExploreInd \u2113 A\u1d49\n    A\u2071 = exploreU-ind A\n  A\u02e1 : Lookup {\u2080} A\u1d49\n  A\u02e1 = lookupU A\n\n  Check! = A\u1d49 _\u00d7_ \u03bb x \u2192 \u2713 \u230a f x \u225f g x \u230b\n\n  check! : {p\u2713 : Check!} \u2192 f \u2257 g\n  check! {p\u2713} x = toWitness (A\u02e1 p\u2713 x)\n\n  {- Unused\n  open Explore.Explorable.Explorable\u2080 A\u2071\n  test-\u2227 = big-\u2227 \u03bb x \u2192 \u230a f x \u225f g x \u230b\n  -}\n\nmodule Test22 where\n  nand nand' : \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\n\n  nand  (x , y) = not (x \u2227 y)\n  nand' (x , y) = not x \u2228 not y\n\n  module N = FromNand funRewiring nand\n  module T = Test Data.Two._\u225f_\n\n  module UnOp where\n    open T \ud835\udfda\u2032\n\n    pf-not : N.not \u2257 not\n    pf-not = check! \n\n  module BinOp where\n    open T (\ud835\udfda\u2032 \u00d7\u2032 \ud835\udfda\u2032)\n\n    pf-nand : nand \u2257 nand'\n    pf-nand = check!\n\n    pf-and : N.and \u2257 uncurry _\u2227_\n    pf-and = check!\n\n    pf-or : N.or \u2257 uncurry _\u2228_\n    pf-or = check!\n\n    pf-nor : N.nor \u2257 (not \u2218 uncurry _\u2228_)\n    pf-nor = check!\n\n    pf-xor : N.xor \u2257 uncurry _xor_\n    pf-xor = check!\n\n    pf-xnor : N.xnor \u2257 uncurry _==_\n    pf-xnor = check!\n\n  module TriOp where\n    open T (\ud835\udfda\u2032 \u00d7\u2032 (\ud835\udfda\u2032 \u00d7\u2032 \ud835\udfda\u2032))\n\n    fork : \ud835\udfda \u00d7 \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\n    fork (c , e\u1d62) = proj e\u1d62 c\n\n    pf-fork : N.fork \u2257 fork\n    pf-fork = check!\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a18ee58c296f46e92e3f6f577648c5b871e3b128","subject":"Added a proof using an instance of an induction principle.","message":"Added a proof using an instance of an induction principle.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\n+-N-ind : \u2200 {n} \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = +-N-ind prf\u2081 prf\u2082 Nm\n  where\n  prf\u2081 : N (zero + n)\n  prf\u2081 = subst N (sym (+-0x n)) Nn\n\n  prf\u2082 : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n  prf\u2082 {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2082 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2083 N-ind #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\n+-N-ind : \u2200 n \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind n = N-ind (\u03bb i \u2192 N (i + n))\n\npostulate +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 N-ind #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"aa15436d10416e76d1a5f6494f6bf9dfde53c404","subject":"Drop unused redundant case","message":"Drop unused redundant case\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/DSyntax.agda","new_file":"Thesis\/SIRelBigStep\/DSyntax.agda","new_contents":"module Thesis.SIRelBigStep.DSyntax where\n\nopen import Thesis.SIRelBigStep.Syntax public\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context\nmodule DC = Base.Syntax.Context DType\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 (pair \u03c41 \u03c42) = pair (\u0394\u03c4 \u03c41) (\u0394\u03c4 \u03c42)\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var \u0394 \u03c4) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n  dcons : \u2200 {\u03c41 \u03c42}\n    (dsv1 : DSVal \u0394 \u03c41)\n    (dsv2 : DSVal \u0394 \u03c42) \u2192\n    DSVal \u0394 (pair \u03c41 \u03c42)\n  dconst : \u2200 {\u03c4} \u2192 (dc : Const (\u0394\u03c4 \u03c4)) \u2192 DSVal \u0394 \u03c4\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  dprimapp : \u2200 {\u03c3}\n    (p : Primitive (\u03c3 \u21d2 \u03c4)) \u2192\n    (sv : SVal \u0394 \u03c3) \u2192\n    (dsv : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n","old_contents":"module Thesis.SIRelBigStep.DSyntax where\n\nopen import Thesis.SIRelBigStep.Syntax public\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context\nmodule DC = Base.Syntax.Context DType\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 (pair \u03c41 \u03c42) = pair (\u0394\u03c4 \u03c41) (\u0394\u03c4 \u03c42)\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var \u0394 \u03c4) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n  dcons : \u2200 {\u03c41 \u03c42}\n    (dsv1 : DSVal \u0394 \u03c41)\n    (dsv2 : DSVal \u0394 \u03c42) \u2192\n    DSVal \u0394 (pair \u03c41 \u03c42)\n  dconst : \u2200 {\u03c4} \u2192 (dc : Const (\u0394\u03c4 \u03c4)) \u2192 DSVal \u0394 \u03c4\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  dprimapp : \u2200 {\u03c3}\n    (p : Primitive (\u03c3 \u21d2 \u03c4)) \u2192\n    (sv : SVal \u0394 \u03c3) \u2192\n    (dsv : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7ca8c4411d108521cba992c9c85cd4eb2fdff0ed","subject":"Let the code speak for itself","message":"Let the code speak for itself\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ndata \u03bc (I : Set) (D : Desc I) : I \u2192 Set where\n  con : UncurriedEl I D (\u03bc I D)\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 UncurriedCases E\n    (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) \n    ((i : I) (x : \u03bc I D i) \u2192 P i x)\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\nelim2 :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n        X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in CurriedCases E Q X\nelim2 I E Cs P =\n  let D = `Arg (Tag E) Cs\n      Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n      X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in curryCases E Q X (elim I E Cs P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) VecT (VecC A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) VecT (VecC (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedCases E P X = Cases E P \u2192 X\n\nCurriedCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedCases [] P X = X\nCurriedCases (l \u2237 E) P X = P here \u2192 CurriedCases E (\u03bb t \u2192 P (there t)) X\n\ncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : UncurriedCases E P X) \u2192 CurriedCases E P X\ncurryCases [] P X f = f tt\ncurryCases (l \u2237 E) P X f = \u03bb c \u2192 curryCases E (\u03bb t \u2192 P (there t)) X (\u03bb cs \u2192 f (c , cs))\n\nuncurryCases : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  (f : CurriedCases E P X) \u2192 UncurriedCases E P X\nuncurryCases [] P X x tt = x\nuncurryCases (l \u2237 E) P X f (c , cs) = uncurryCases E (\u03bb t \u2192 P (there t)) X (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  -- this equals UncurriedEl I D (\u03bc I D)\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 UncurriedCases E\n    (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) \n    ((i : I) (x : \u03bc I D i) \u2192 P i x)\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\nelim2 :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  \u2192 let Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n        X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in CurriedCases E Q X\nelim2 I E Cs P =\n  let D = `Arg (Tag E) Cs\n      Q = (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs)))\n      X = ((i : I) (x : \u03bc I D i) \u2192 P i x)\n  in curryCases E Q X (elim I E Cs P)\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim2 \u22a4 \u2115T \u2115C _\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim2 (\u2115 tt) VecT (VecC A) _\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim2 (\u2115 tt) VecT (VecC (Vec A m)) _\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1131c2f7ec949c07dd71af9ef1372bf7a89cf90e","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 195af1bec8b3d065622ddc54d20fb410\n\ndarcs-hash:20110805133551-3bd4e-85248138588af499a8b16b7c2c1211186cfb3687.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Setoids\/FOTC.agda","new_file":"Draft\/Setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids to formalize the FOTC\n------------------------------------------------------------------------------\n\n{-\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor = : The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n-}\n\nmodule FOTC where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  K S : D\n  _\u00b7_ : D \u2192 D \u2192 D\n\nmodule PeterEquality where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  infix 7 _\u2250_\n\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- It seems we cannot define the identity elimination using the setoid\n  -- equality\n  -- subst : \u2200 {x y} (P : D \u2192 Set) \u2192 x \u2250 y \u2192 P x \u2192 P y\n  -- subst P x\u2250y Px = {!!}\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  -- Barthe et al. [*, p. 262] use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- [*] Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n  -- type theory. Journal of Functional Programming, 13(2):261\u2013293, 2003\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2250_\n\n  _\u2250_ : D \u2192 D \u2192 Set\u2081\n  x \u2250 y = (P : D \u2192 Set) \u2192 P x \u2192 P y\n\n  -- We can proof the setoids properties\n\n  \u2250-refl : \u2200 x \u2192 x \u2250 x\n  \u2250-refl x P Px = Px\n\n  \u2250-sym : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n  \u2250-sym {x} x\u2250y P Py = x\u2250y (\u03bb z \u2192 P z \u2192 P x) (\u03bb Px \u2192 Px) Py\n\n  \u2250-trans : \u2200 x y z \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n  \u2250-trans x y z x\u2250y y\u2250z P Px = y\u2250z P (x\u2250y P Px)\n\n  -- and the identity elimination\n\n  \u2250-subst : \u2200 (P : D \u2192 Set) {x y} \u2192 x \u2250 y \u2192 P x \u2192 P y\n  \u2250-subst P x\u2250y = x\u2250y P\n\n  -- but it seems we cannot prove the congruency\n\n  -- \u2250-cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n  -- \u2250-cong x\u2081\u2250x\u2082 y\u2081\u2250y\u2082 P Px\u2081y\u2081 = {!!}\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids to formalize the FOTC\n------------------------------------------------------------------------------\n\nmodule FOTC where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n------------------------------------------------------------------------------\n\ndata D : Set where\n  K S : D\n  _\u00b7_ : D \u2192 D \u2192 D\n\nmodule PeterEquality where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  infix 7 _\u2250_\n\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- It seems we cannot define the identity elimination using the setoid\n  -- equality\n  -- subst : \u2200 {x y} (P : D \u2192 Set) \u2192 x \u2250 y \u2192 P x \u2192 P y\n  -- subst P x\u2250y Px = {!!}\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  -- Barthe et al. [*, p. 262] use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- [*] Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n  -- type theory. Journal of Functional Programming, 13(2):261\u2013293, 2003\n\n  -- Using the Leibniz equality\n  -- (Adapted from Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  infix 7 _\u2250_\n\n  _\u2250_ : D \u2192 D \u2192 Set\u2081\n  x \u2250 y = (P : D \u2192 Set) \u2192 P x \u2192 P y\n\n  -- We can proof the setoids properties\n\n  \u2250-refl : \u2200 x \u2192 x \u2250 x\n  \u2250-refl x P Px = Px\n\n  \u2250-sym : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n  \u2250-sym {x} x\u2250y P Py = x\u2250y (\u03bb z \u2192 P z \u2192 P x) (\u03bb Px \u2192 Px) Py\n\n  \u2250-trans : \u2200 x y z \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n  \u2250-trans x y z x\u2250y y\u2250z P Px = y\u2250z P (x\u2250y P Px)\n\n  -- and the identity elimination\n\n  \u2250-subst : \u2200 (P : D \u2192 Set) {x y} \u2192 x \u2250 y \u2192 P x \u2192 P y\n  \u2250-subst P x\u2250y = x\u2250y P\n\n  -- but it seems we cannot prove the congruency\n  -- \u2250-cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n  -- \u2250-cong x\u2081\u2250x\u2082 y\u2081\u2250y\u2082 P Px\u2081y\u2081 = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"92509a078139c27bd15ff2e40e8df4061281e344","subject":"Fixed a note on least fixed-points.","message":"Fixed a note on least fixed-points.\n\nIgnore-this: 8b79d84c55313d3921e3072d6e704449\n\ndarcs-hash:20120306150936-3bd4e-79d9d5387fe2a301e75e785c41d72d50a0c0bd81.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 06 March 2012.\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least fixed point of a functor\n\n-- Instead defining the least fixed-point via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF P n = n \u2261 zero \u2228 (\u2203[ m ] n \u2261 succ\u2081 m \u2227 P m)\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081    : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] n \u2261 succ\u2081 m \u2227 N m) \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- N is a the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082    : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] n \u2261 succ\u2081 m \u2227 A m) \u2192 A n) \u2192\n              \u2200 {n} \u2192 N n \u2192 A n\n  -- N-lfp\u2082 : (A : D \u2192 Set) \u2192  -- Higher-order version\n  --          (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192\n  --          \u2200 {n} \u2192 N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN Nn = N-lfp\u2081 (inj\u2082 (_ , (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m)\nN-lfp\u2083 Nn = N-lfp\u2082 A prf Nn\n  where\n  A : D \u2192 Set\n  A x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ\u2081 m \u2227 N m\n\n  prf : \u2200 {n'} \u2192 n' \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n  prf {n'} h = [ inj\u2081 , (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) ] h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 N m\n    prf\u2081 (m , n'=Sm , h\u2082) = m , n'=Sm , prf\u2082 h\u2082\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = [ (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) , prf\u2083 ] h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n        prf\u2083 (_ , m\u2261Sm' , Nm') = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nindN\u2081 : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nindN\u2081 A A0 is Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n  prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n  prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n  prf\u2082 (_ , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is Am)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\n--\n-- 2012-03-06. We cannot proof this principle because N-lfp\u2082 does not\n-- have the hypothesis N n.\n--\n-- indN\u2082 : (A : D \u2192 Set) \u2192\n--        A zero \u2192\n--        (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n--        \u2200 {n} \u2192 N n \u2192 A n\n-- indN\u2082 A A0 is Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n--   where\n--   prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n--   prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n--   prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n--   prf\u2082 {n'} (m , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is helper Am)\n--     where\n--     helper : N m\n--     helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 {!!})\n--       where\n--       prf\u2083 : n' \u2261 zero \u2192 N m\n--       prf\u2083 n'\u22610 = \u22a5-elim (0\u2260S (trans (sym n'\u22610) n'\u2261Sm))\n\n--       prf\u2084 : \u2203 (\u03bb m' \u2192 n' \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n--       prf\u2084 (_ , n'\u2261Sm' , Nm') =\n--         subst N (succInjective (trans (sym n'\u2261Sm') n'\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-lfp\u2082 A prf Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  prf : \u2200 {m'} \u2192 m' \u2261 zero \u2228 \u2203 (\u03bb m'' \u2192 m' \u2261 succ\u2081 m'' \u2227 A m'') \u2192 A m'\n  prf h = [ prf\u2081 , prf\u2082 ] h\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : \u2200 {m} \u2192 m \u2261 zero \u2192 A m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) A0\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n    prf\u2082 : \u2200 {m} \u2192 \u2203 (\u03bb m'' \u2192 m \u2261 succ\u2081 m'' \u2227 A m'') \u2192 A m\n    prf\u2082 (_ , m\u2261Sm'' , Am'') =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm'')) (is Am'')\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 02 March 2012.\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is a fixed\n-- point of f (TODO: source).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least pre-fixed point of a functor\n\n-- Instead defining the least pre-fixed via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n-- The natural numbers are the least pre-fixed point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is pre-fixed point of NatF.\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081    : \u2200 n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m) \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082    : (A : D \u2192 Set) \u2192 \u2200 {n} \u2192\n              (n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 A m) \u2192 A n) \u2192\n              N n \u2192 A n\n  -- N-lfp\u2082 : (A : D \u2192 Set) \u2192 \u2200 {n} \u2192  -- Higher-order version\n  --          (NatF A n \u2192 A n) \u2192\n  --          N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 zero (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN {n} Nn = N-lfp\u2081 (succ\u2081 n) (inj\u2082 (\u2203-intro (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m)\nN-lfp\u2083 {n} Nn = N-lfp\u2082 A prf Nn\n  where\n  A : D \u2192 Set\n  A x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ\u2081 m \u2227 N m\n\n  prf : n \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 A m) \u2192 A n\n  prf h = [ inj\u2081 , (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) ] h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m\n    prf\u2081 (\u2203-intro {m} (n=Sm , h\u2082)) = \u2203-intro (n=Sm , prf\u2082 h\u2082)\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = [ (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) , prf\u2083 ] h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n        prf\u2083 (\u2203-intro (m\u2261Sm' , Nm')) = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nindN\u2081 : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nindN\u2081 A A0 is {n} Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 A n\n  prf\u2081 n\u22610 = subst A (sym n\u22610) A0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 A m) \u2192 A n\n  prf\u2082 (\u2203-intro (n\u2261Sm , Am)) = subst A (sym n\u2261Sm) (is Am)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\nindN\u2082 : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nindN\u2082 A A0 is {n} Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 A n\n  prf\u2081 n\u22610 = subst A (sym n\u22610) A0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 A m) \u2192 A n\n  prf\u2082 (\u2203-intro {m} (n\u2261Sm , Am)) = subst A (sym n\u2261Sm) (is helper Am)\n    where\n    helper : N m\n    helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 Nn)\n      where\n      prf\u2083 : n \u2261 zero \u2192 N m\n      prf\u2083 n\u22610 = \u22a5-elim (0\u2260S (trans (sym n\u22610) n\u2261Sm))\n\n      prf\u2084 : \u2203 (\u03bb m' \u2192 n \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2084 (\u2203-intro (n\u2261Sm' , Nm')) =\n        subst N (succInjective (trans (sym n\u2261Sm') n\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-lfp\u2082 A prf Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  prf : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n  prf h = [ prf\u2081 , prf\u2082 ] h\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : m \u2261 zero \u2192 A m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) A0\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n    prf\u2082 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    prf\u2082 (\u2203-intro (m\u2261Sm' , Am')) =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b50b779421271cc824ea5ccbbe9d9ee045719ec7","subject":"t-abs case of validity-of-derive; remove outdated bug report task","message":"t-abs case of validity-of-derive; remove outdated bug report task\n\nOld-commit-hash: 423c6e803350c77c286eca5a83687e57ecf313c9\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/A.agda","new_file":"nats\/A.agda","new_contents":"-- TODO: Include proof of\n-- \"not all correctly-typed values are valid changes\"\n\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- Question: Would it not have been better if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`?\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 (app t t\u2081) = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 {!!} \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e\n  ) where open \u2261-Reasoning\n\n","old_contents":"-- TODO: Include proof of\n-- \"not all correctly-typed values are valid changes\"\n\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- Question: Would it not have been better if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`?\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n  --Putting `df=f\u2295df\u229df v dv ?` on RHS triggers Agda bug.\n  --TODO: Report it as Agda tells us to.\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 {!!} \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)\n        \u27e6\u2295\u27e7 {!!}\n        \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n\n      \u2261\u27e8 {!!} \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 (app t t\u2081) = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 {!!} \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e\n  ) where open \u2261-Reasoning\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9e3f211e88b705e21001d83b692c5504a15a4f1a","subject":"Data.Bits: use 2^ instead of _^_ 2 in allBits and #\u27e8_\u27e9","message":"Data.Bits: use 2^ instead of _^_ 2 in allBits and #\u27e8_\u27e9\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5e52be904bd2a54d670b67f500b2a286bace1ddf","subject":"lemma churn","message":"lemma churn\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"de7e98cf756a7d615f97b80685caf879cc0c9211","subject":"cleaning up a little and documenting why the holes are puzzling #37","message":"cleaning up a little and documenting why the holes are puzzling #37\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-expansion.agda","new_file":"complete-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\n\nopen import correspondence\n\nmodule complete-expansion where\n  -- todo remove when you've proven typed expansion\n  postulate\n        typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n        typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n\n  gcomp-extend : \u2200{\u0393 \u03c4 x} \u2192 \u0393 gcomplete \u2192 \u03c4 tcomplete \u2192 (\u0393 ,, (x , \u03c4)) gcomplete\n  gcomp-extend {x = x} gc tc x\u2081 t x\u2082 with natEQ x\u2081 x\n  gcomp-extend {\u0393 = \u0393} gc tc x t x\u2083 | Inl refl with \u0393 x\n  gcomp-extend gc tc x x\u2081 refl | Inl refl | Some .x\u2081 = {!!} -- stupid context stuff again\n  gcomp-extend gc tc x t x\u2083 | Inl refl | None with natEQ x x\n  gcomp-extend gc tc x \u03c4 refl | Inl refl | None | Inl refl = tc\n  gcomp-extend gc tc x t x\u2083 | Inl refl | None | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  gcomp-extend {\u0393 = \u0393} gc tc x\u2081 t x\u2083 | Inr x\u2082 with \u0393 x\u2081\n  gcomp-extend gc tc x\u2081 t x\u2084 | Inr x\u2083 | Some x = {!tc!} -- ditto\n  gcomp-extend {x = x} gc tc x\u2081 t x\u2083 | Inr x\u2082 | None with natEQ x x\u2081\n  gcomp-extend gc tc x t refl | Inr x\u2083 | None | Inl refl = tc\n  gcomp-extend gc tc x\u2081 t x\u2084 | Inr x\u2083 | None | Inr x\u2082 = abort (somenotnone (! x\u2084))\n\n  -- this might be derivable from things below and a fact about => and ::\n  -- that we seem to have not proven\n  complete-ta : \u2200{\u0393 \u0394 d \u03c4} \u2192 (\u0393 gcomplete) \u2192 (\u0394 , \u0393 \u22a2 d :: \u03c4) \u2192 d dcomplete \u2192 \u03c4 tcomplete\n  complete-ta gc TAConst comp = TCBase\n  complete-ta gc (TAVar x\u2081) DCVar = gc _ _ x\u2081\n  complete-ta gc (TALam wt) (DCLam comp x\u2081) = TCArr x\u2081 (complete-ta (gcomp-extend gc x\u2081) wt comp)\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) with complete-ta gc wt comp\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) | TCArr qq qq\u2081 = qq\u2081\n  complete-ta gc (TAEHole x x\u2081) ()\n  complete-ta gc (TANEHole x wt x\u2081) ()\n  complete-ta gc (TACast wt x) (DCCast comp x\u2081 x\u2082) = x\u2082\n  complete-ta gc (TAFailedCast wt x x\u2081 x\u2082) ()\n\n  comp-synth : \u2200{\u0393 e \u03c4} \u2192\n                   \u0393 gcomplete \u2192\n                   e ecomplete \u2192\n                   \u0393 \u22a2 e => \u03c4 \u2192\n                   \u03c4 tcomplete\n  comp-synth gc ec SConst = TCBase\n  comp-synth gc (ECAsc x ec) (SAsc x\u2081) = x\n  comp-synth gc ec (SVar x) = gc _ _ x\n  comp-synth gc (ECAp ec ec\u2081) (SAp wt MAHole x\u2081) with comp-synth gc ec wt\n  ... | ()\n  comp-synth gc (ECAp ec ec\u2081) (SAp wt MAArr x\u2081) with comp-synth gc ec wt\n  comp-synth gc (ECAp ec ec\u2081) (SAp wt MAArr x\u2081) | TCArr qq qq\u2081 = qq\u2081\n  comp-synth gc () SEHole\n  comp-synth gc () (SNEHole wt)\n  comp-synth gc (ECLam2 ec x\u2081) (SLam x\u2082 wt) = TCArr x\u2081 (comp-synth (gcomp-extend gc x\u2081) ec wt)\n\n  mutual\n    complete-expansion-synth : \u2200{e \u03c4 \u0393 \u0394 d} \u2192\n                               \u0393 gcomplete \u2192\n                               e ecomplete \u2192\n                               \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                               d dcomplete\n    complete-expansion-synth gc ec ESConst = DCConst\n    complete-expansion-synth gc ec (ESVar x\u2081) = DCVar\n    complete-expansion-synth gc (ECLam2 ec x\u2081) (ESLam x\u2082 exp) = DCLam (complete-expansion-synth (gcomp-extend gc x\u2081) ec exp) x\u2081\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp x MAHole x\u2082 x\u2083) with comp-synth gc ec x\n    ... | ()\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp x MAArr x\u2082 x\u2083)\n      with complete-expansion-ana gc ec x\u2082 | complete-expansion-ana gc ec\u2081 x\u2083 | comp-synth gc ec x\n    ... | ih1 | ih2 | TCArr c1 c2 = DCAp (DCCast ih1 (comp-ana gc x\u2082 ih1) (TCArr c1 c2))\n                                         (DCCast ih2 (comp-ana gc x\u2083 ih2) c1)\n    complete-expansion-synth gc () ESEHole\n    complete-expansion-synth gc () (ESNEHole exp)\n    complete-expansion-synth gc (ECAsc x ec) (ESAsc x\u2081)\n      with complete-expansion-ana gc ec x\u2081\n    ... | ih = DCCast ih (comp-ana gc x\u2081 ih) x\n\n    complete-expansion-ana : \u2200{e \u03c4 \u03c4' \u0393 \u0394 d} \u2192\n                             \u0393 gcomplete \u2192\n                             e ecomplete \u2192\n                             \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                             d dcomplete\n    complete-expansion-ana gc (ECLam1 ec) (EALam x\u2081 MAHole exp) = {!!} -- not an ih because the ctx isn't gcomplete,\n    complete-expansion-ana gc (ECLam1 ec) (EALam x\u2081 MAArr exp) -- since this is the unannotated lambda form, \u03c41 comes out of the sky with no premise about completeness\n      with complete-expansion-ana (gcomp-extend gc {!!}) ec exp\n    ... | ih = DCLam ih {!!}\n    complete-expansion-ana gc ec (EASubsume x x\u2081 x\u2082 x\u2083) = complete-expansion-synth gc ec x\u2082\n\n    --this is just a convenience since it shows up a few times above\n    comp-ana : \u2200{\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n       \u0393 gcomplete \u2192\n       \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n       d dcomplete \u2192\n       \u03c4' tcomplete\n    comp-ana gc ex dc = complete-ta gc (\u03c02 (typed-expansion-ana ex)) dc\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\n\nopen import correspondence\n\nmodule complete-expansion where\n  -- todo: do you really need this? didn't in complete preservation; come back to this\n  gcomp-extend : \u2200{\u0393 \u03c4 x} \u2192 \u0393 gcomplete \u2192 \u03c4 tcomplete \u2192 (\u0393 ,, (x , \u03c4)) gcomplete\n  gcomp-extend {x = x} gc tc x\u2081 t x\u2082 with natEQ x\u2081 x\n  gcomp-extend {\u0393 = \u0393} gc tc x t x\u2083 | Inl refl with \u0393 x\n  gcomp-extend gc tc x x\u2081 refl | Inl refl | Some .x\u2081 = {!!} -- stupid context stuff again\n  gcomp-extend gc tc x t x\u2083 | Inl refl | None with natEQ x x\n  gcomp-extend gc tc x \u03c4 refl | Inl refl | None | Inl refl = tc\n  gcomp-extend gc tc x t x\u2083 | Inl refl | None | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  gcomp-extend {\u0393 = \u0393} gc tc x\u2081 t x\u2083 | Inr x\u2082 with \u0393 x\u2081\n  gcomp-extend gc tc x\u2081 t x\u2084 | Inr x\u2083 | Some x = {!tc!} -- ditto\n  gcomp-extend {x = x} gc tc x\u2081 t x\u2083 | Inr x\u2082 | None with natEQ x x\u2081\n  gcomp-extend gc tc x t refl | Inr x\u2083 | None | Inl refl = tc\n  gcomp-extend gc tc x\u2081 t x\u2084 | Inr x\u2083 | None | Inr x\u2082 = abort (somenotnone (! x\u2084))\n\n  complete-ta : \u2200{\u0393 \u0394 d \u03c4} \u2192 (\u0393 gcomplete) \u2192 (\u0394 , \u0393 \u22a2 d :: \u03c4) \u2192 d dcomplete \u2192 \u03c4 tcomplete\n  complete-ta gc TAConst comp = TCBase\n  complete-ta gc (TAVar x\u2081) DCVar = gc _ _ x\u2081\n  complete-ta gc (TALam wt) (DCLam comp x\u2081) = TCArr x\u2081 (complete-ta (gcomp-extend gc x\u2081) wt comp)\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) with complete-ta gc wt comp\n  complete-ta gc (TAAp wt wt\u2081) (DCAp comp comp\u2081) | TCArr qq qq\u2081 = qq\u2081\n  complete-ta gc (TAEHole x x\u2081) ()\n  complete-ta gc (TANEHole x wt x\u2081) ()\n  complete-ta gc (TACast wt x) (DCCast comp x\u2081 x\u2082) = x\u2082\n  complete-ta gc (TAFailedCast wt x x\u2081 x\u2082) ()\n\n  synth-hole-not-comp : \u2200{\u0393 e} \u2192  \u0393 \u22a2 e => \u2987\u2988 \u2192 \u0393 gcomplete \u2192 e ecomplete \u2192 \u22a5 -- this is not a strong enough IH\n  synth-hole-not-comp (SAsc x) gc (ECAsc () ec)\n  synth-hole-not-comp (SVar {n = n} x) gc ec with gc n _ x\n  ... | ()\n  synth-hole-not-comp (SAp wt MAHole x\u2081) gc (ECAp ec ec\u2081) = synth-hole-not-comp wt gc ec\n  synth-hole-not-comp (SAp wt MAArr x\u2081) gc (ECAp ec ec\u2081) = {!!}\n  synth-hole-not-comp SEHole gc ()\n  synth-hole-not-comp (SNEHole wt) gc ()\n\n  -- todo remove when you've proven typed expansion\n  postulate\n        typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n        typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n  mutual\n    complete-expansion-synth : \u2200{e \u03c4 \u0393 \u0394 d} \u2192\n                               \u0393 gcomplete \u2192\n                               e ecomplete \u2192\n                               \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                               d dcomplete\n    complete-expansion-synth gc ec ESConst = DCConst\n    complete-expansion-synth gc ec (ESVar x\u2081) = DCVar\n    complete-expansion-synth gc (ECLam2 ec x\u2081) (ESLam x\u2082 exp) = DCLam (complete-expansion-synth (gcomp-extend gc x\u2081) ec exp) x\u2081\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp x MAHole x\u2082 x\u2083) = abort (synth-hole-not-comp x gc ec)\n    complete-expansion-synth gc (ECAp ec ec\u2081) (ESAp x MAArr x\u2082 x\u2083) = {!!}\n    complete-expansion-synth gc () ESEHole\n    complete-expansion-synth gc () (ESNEHole exp)\n    complete-expansion-synth gc (ECAsc x ec) (ESAsc x\u2081) with complete-expansion-ana gc ec x\u2081\n    ... | ih = DCCast ih (complete-ta gc (\u03c02 (typed-expansion-ana x\u2081)) ih) x\n\n    complete-expansion-ana : \u2200{e \u03c4 \u03c4' \u0393 \u0394 d} \u2192\n                             \u0393 gcomplete \u2192\n                             e ecomplete \u2192\n                             \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                             d dcomplete\n    complete-expansion-ana gc (ECLam1 ec) (EALam x\u2081 MAHole exp) = {!!}\n    complete-expansion-ana gc (ECLam1 ec) (EALam x\u2081 MAArr exp)\n      with complete-expansion-ana (gcomp-extend gc {!!}) ec exp\n    ... | ih = DCLam ih {!!}\n    complete-expansion-ana gc ec (EASubsume x x\u2081 x\u2082 x\u2083) = complete-expansion-synth gc ec x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8df38c66c4e14c281a382410d07f8d6895927b9d","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 4effbc2cfa6aca7c6869101cc72a1849\n\ndarcs-hash:20110511171225-3bd4e-1596536d6deb7589d87a6ff01f607d89c680aaeb.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/MainATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/MainATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.Properties.MainATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.EquationsATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.LT2MCR-ATP\nopen import FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Program.McCarthy91.Properties.AuxiliaryATP\n\n------------------------------------------------------------------------------\n\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 N (mc91 n) \u2227 LT n (mc91 n + eleven)\nmc91-N-ineq = wfInd-MCR P mc91-N-ineq-aux\n  where\n    P : D \u2192 Set\n    P d = N (mc91 d) \u2227 LT d (mc91 d + eleven)\n\n    mc91-N-ineq-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n    mc91-N-ineq-aux {m} Nm f with x>y\u2228x\u2264y Nm 100-N\n    ... | inj\u2081 m>100 = ( Nmc91>100 Nm m>100 , x<mc91x+11>100 Nm m>100 )\n    ... | inj\u2082 m\u2264100 =\n      let Nm+11 : N (m + eleven)\n          Nm+11 = x+11-N Nm\n\n          ih1 : P (m + eleven)\n          ih1 = f Nm+11 (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n\n          Nih1 : N (mc91 (m + eleven))\n          Nih1 = \u2227-proj\u2081 ih1\n\n          LTih1 : LT (m + eleven) (mc91 (m + eleven) + eleven)\n          LTih1 = \u2227-proj\u2082 ih1\n\n          m<mc91m+11 : LT m (mc91 (m + eleven))\n          m<mc91m+11 = x+k<y+k\u2192x<y Nm Nih1 11-N LTih1\n\n          ih2 : P (mc91 (m + eleven))\n          ih2 = f Nih1 (LT2MCR Nih1 Nm m\u2264100 m<mc91m+11)\n\n          Nmc91\u2264100 : N (mc91 m)\n          Nmc91\u2264100 = Nmc91\u2264100' Nm m\u2264100 (\u2227-proj\u2081 ih2)\n\n      in ( Nmc91\u2264100 ,\n           <-trans Nm Nih1 (x+11-N Nmc91\u2264100)\n                   m<mc91m+11\n                   (mc91x+11<mc91x+11 Nm m\u2264100 (\u2227-proj\u2082 ih2)))\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (GT n one-hundred \u2227 mc91 n \u2261 n \u2238 ten) \u2228\n                         (LE n one-hundred \u2227 mc91 n \u2261 ninety-one)\nmc91-res = wfInd-MCR P mc91-res-aux\n  where\n    P : D \u2192 Set\n    P d = (GT d one-hundred \u2227 mc91 d \u2261 d \u2238 ten) \u2228\n          (LE d one-hundred \u2227 mc91 d \u2261 ninety-one)\n\n    mc91-res-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n    mc91-res-aux {m} Nm f with x>y\u2228x\u2264y Nm 100-N\n    ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq\u2081 m m>100 )\n    ... | inj\u2082 m\u2264100 with x<Sy\u2192x<y\u2228x\u2261y Nm 100-N m\u2264100\n    ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u2264100 , mc91-res-100' m\u2261100 )\n    ... | inj\u2081 m\u226499 with x<Sy\u2192x<y\u2228x\u2261y Nm 99-N m\u226499\n    ... | inj\u2082 m\u226199 = inj\u2082 ( m\u2264100 , mc91-res-99' m\u226199 )\n    ... | inj\u2081 m\u226498 with x<Sy\u2192x<y\u2228x\u2261y Nm 98-N m\u226498\n    ... | inj\u2082 m\u226198 = inj\u2082 ( m\u2264100 , mc91-res-98' m\u226198 )\n    ... | inj\u2081 m\u226497 with x<Sy\u2192x<y\u2228x\u2261y Nm 97-N m\u226497\n    ... | inj\u2082 m\u226197 = inj\u2082 ( m\u2264100 , mc91-res-97' m\u226197 )\n    ... | inj\u2081 m\u226496 with x<Sy\u2192x<y\u2228x\u2261y Nm 96-N m\u226496\n    ... | inj\u2082 m\u226196 = inj\u2082 ( m\u2264100 , mc91-res-96' m\u226196 )\n    ... | inj\u2081 m\u226495 with x<Sy\u2192x<y\u2228x\u2261y Nm 95-N m\u226495\n    ... | inj\u2082 m\u226195 = inj\u2082 ( m\u2264100 , mc91-res-95' m\u226195 )\n    ... | inj\u2081 m\u226494 with x<Sy\u2192x<y\u2228x\u2261y Nm 94-N m\u226494\n    ... | inj\u2082 m\u226194 = inj\u2082 ( m\u2264100 , mc91-res-94' m\u226194 )\n    ... | inj\u2081 m\u226493 with x<Sy\u2192x<y\u2228x\u2261y Nm 93-N m\u226493\n    ... | inj\u2082 m\u226193 = inj\u2082 ( m\u2264100 , mc91-res-93' m\u226193 )\n    ... | inj\u2081 m\u226492 with x<Sy\u2192x<y\u2228x\u2261y Nm 92-N m\u226492\n    ... | inj\u2082 m\u226192 = inj\u2082 ( m\u2264100 , mc91-res-92' m\u226192 )\n    ... | inj\u2081 m\u226491 with x<Sy\u2192x<y\u2228x\u2261y Nm 91-N m\u226491\n    ... | inj\u2082 m\u226191 = inj\u2082 ( m\u2264100 , mc91-res-91' m\u226191 )\n    ... | inj\u2081 m\u226490 with x<Sy\u2192x<y\u2228x\u2261y Nm 90-N m\u226490\n    ... | inj\u2082 m\u226190 = inj\u2082 ( m\u2264100 , mc91-res-90' m\u226190 )\n    ... | inj\u2081 m\u226489 = inj\u2082 ( m\u2264100 , mc91-res-m\u226489 )\n\n      where mc91-res-m+11 : mc91 (m + eleven) \u2261 ninety-one\n            mc91-res-m+11 with f (x+11-N Nm)\n                                 (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n            ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm\n                                                 m\u226489 m+11>100)\n            ... | inj\u2082 ( _ , res ) = res\n\n            mc91-res-m\u226489 : mc91 m \u2261 ninety-one\n            mc91-res-m\u226489 = mc91x-res\u2264100 Nm ninety-one m\u2264100\n                                          mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn = \u2227-proj\u2081 (mc91-N-ineq Nn)\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 LT n (mc91 n + eleven)\nmc91-ineq Nn = \u2227-proj\u2082 (mc91-N-ineq Nn)\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\nmc91-res>100 Nn n>100  with mc91-res Nn\n... | inj\u2081 ( _ , res ) = res\n... | inj\u2082 ( n\u2264100 , _ ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100 n\u2264100)\n\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u2264100 : \u2200 {n} \u2192 N n \u2192 LE n one-hundred \u2192 mc91 n \u2261 ninety-one\nmc91-res\u2264100 Nn n\u2264100  with mc91-res Nn\n... | inj\u2081 ( n>100 , _ ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100 n\u2264100)\n... | inj\u2082 ( _ , res ) = res\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.Properties.MainATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.EquationsATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.LT2MCR-ATP\nopen import FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Program.McCarthy91.Properties.AuxiliaryATP\n\n------------------------------------------------------------------------------\n\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 N (mc91 n) \u2227 LT n (mc91 n + eleven)\nmc91-N-ineq = wfInd-MCR P mc91-N-ineq-aux\n  where\n    P : D \u2192 Set\n    P d = N (mc91 d) \u2227 LT d (mc91 d + eleven)\n\n    mc91-N-ineq-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n    mc91-N-ineq-aux {m} Nm f with x>y\u2228x\u2264y Nm 100-N\n    ... | inj\u2081 m>100 = ( Nmc91>100 Nm m>100 , x<mc91x+11>100 Nm m>100 )\n    ... | inj\u2082 m\u2264100 =\n      let Nm+11 : N (m + eleven)\n          Nm+11 = x+11-N Nm\n\n          ih1 : P (m + eleven)\n          ih1 = f Nm+11 (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n\n          Nih1 : N (mc91 (m + eleven))\n          Nih1 = \u2227-proj\u2081 ih1\n\n          LTih1 : LT (m + eleven) (mc91 (m + eleven) + eleven)\n          LTih1 = \u2227-proj\u2082 ih1\n\n          m<mc91m+11 : LT m (mc91 (m + eleven))\n          m<mc91m+11 = x+k<y+k\u2192x<y Nm Nih1 11-N LTih1\n\n          ih2 : P (mc91 (m + eleven))\n          ih2 = f Nih1 (LT2MCR Nih1 Nm m\u2264100 m<mc91m+11)\n\n          Nmc91\u2264100 : N (mc91 m)\n          Nmc91\u2264100 = Nmc91\u2264100' Nm m\u2264100 (\u2227-proj\u2081 ih2)\n\n      in ( Nmc91\u2264100 ,\n           <-trans Nm Nih1 (x+11-N Nmc91\u2264100)\n                   m<mc91m+11\n                   (mc91x+11<mc91x+11 Nm m\u2264100 (\u2227-proj\u2082 ih2)))\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (GT n one-hundred \u2227 mc91 n \u2261 n \u2238 ten) \u2228\n                         (LE n one-hundred \u2227 mc91 n \u2261 ninety-one)\nmc91-res = wfInd-MCR P mc91-res-aux\n  where\n    P : D \u2192 Set\n    P d = (GT d one-hundred \u2227 mc91 d \u2261 d \u2238 ten) \u2228\n          (LE d one-hundred \u2227 mc91 d \u2261 ninety-one)\n\n    mc91-res-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n    mc91-res-aux {m} Nm f with x>y\u2228x\u2264y Nm 100-N\n    ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq\u2081 m m>100 )\n    ... | inj\u2082 m\u2264100 with x<Sy\u2192x<y\u2228x\u2261y Nm 100-N m\u2264100\n    ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u2264100 , mc91-res-100' m\u2261100 )\n    ... | inj\u2081 m\u226499 with x<Sy\u2192x<y\u2228x\u2261y Nm 99-N m\u226499\n    ... | inj\u2082 m\u226199 = inj\u2082 ( m\u2264100 , mc91-res-99' m\u226199 )\n    ... | inj\u2081 m\u226498 with x<Sy\u2192x<y\u2228x\u2261y Nm 98-N m\u226498\n    ... | inj\u2082 m\u226198 = inj\u2082 ( m\u2264100 , mc91-res-98' m\u226198 )\n    ... | inj\u2081 m\u226497 with x<Sy\u2192x<y\u2228x\u2261y Nm 97-N m\u226497\n    ... | inj\u2082 m\u226197 = inj\u2082 ( m\u2264100 , mc91-res-97' m\u226197 )\n    ... | inj\u2081 m\u226496 with x<Sy\u2192x<y\u2228x\u2261y Nm 96-N m\u226496\n    ... | inj\u2082 m\u226196 = inj\u2082 ( m\u2264100 , mc91-res-96' m\u226196 )\n    ... | inj\u2081 m\u226495 with x<Sy\u2192x<y\u2228x\u2261y Nm 95-N m\u226495\n    ... | inj\u2082 m\u226195 = inj\u2082 ( m\u2264100 , mc91-res-95' m\u226195 )\n    ... | inj\u2081 m\u226494 with x<Sy\u2192x<y\u2228x\u2261y Nm 94-N m\u226494\n    ... | inj\u2082 m\u226194 = inj\u2082 ( m\u2264100 , mc91-res-94' m\u226194 )\n    ... | inj\u2081 m\u226493 with x<Sy\u2192x<y\u2228x\u2261y Nm 93-N m\u226493\n    ... | inj\u2082 m\u226193 = inj\u2082 ( m\u2264100 , mc91-res-93' m\u226193 )\n    ... | inj\u2081 m\u226492 with x<Sy\u2192x<y\u2228x\u2261y Nm 92-N m\u226492\n    ... | inj\u2082 m\u226192 = inj\u2082 ( m\u2264100 , mc91-res-92' m\u226192 )\n    ... | inj\u2081 m\u226491 with x<Sy\u2192x<y\u2228x\u2261y Nm 91-N m\u226491\n    ... | inj\u2082 m\u226191 = inj\u2082 ( m\u2264100 , mc91-res-91' m\u226191 )\n    ... | inj\u2081 m\u226490 with x<Sy\u2192x<y\u2228x\u2261y Nm 90-N m\u226490\n    ... | inj\u2082 m\u226190 = inj\u2082 ( m\u2264100 , mc91-res-90' m\u226190 )\n    ... | inj\u2081 m\u226489 =  inj\u2082 ( m\u2264100 , mc91-res-m\u226489 )\n\n      where mc91-res-m+11 : mc91 (m + eleven) \u2261 ninety-one\n            mc91-res-m+11 with f (x+11-N Nm)\n                                 (LT2MCR (x+11-N Nm) Nm m\u2264100 (x<x+11 Nm))\n            ... | inj\u2081 ( m+11>100 , _) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm\n                                                m\u226489 m+11>100)\n            ... | inj\u2082 ( _ , res ) = res\n\n            mc91-res-m\u226489 : mc91 m \u2261 ninety-one\n            mc91-res-m\u226489 = mc91x-res\u2264100 Nm ninety-one m\u2264100\n                                          mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn = \u2227-proj\u2081 (mc91-N-ineq Nn)\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 LT n (mc91 n + eleven)\nmc91-ineq Nn = \u2227-proj\u2082 (mc91-N-ineq Nn)\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\nmc91-res>100 Nn n>100  with mc91-res Nn\n... | inj\u2081 ( _ , res ) = res\n... | inj\u2082 ( n\u2264100 , _ ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100 n\u2264100)\n\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u2264100 : \u2200 {n} \u2192 N n \u2192 LE n one-hundred \u2192 mc91 n \u2261 ninety-one\nmc91-res\u2264100 Nn n\u2264100  with mc91-res Nn\n... | inj\u2081 ( n>100 , _ ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100 n\u2264100)\n... | inj\u2082 ( _ , res ) = res\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fce2b2a119cd45e59ea06ab4bbb3f453a5d037f2","subject":"Derive during symbolic execution (see #50).","message":"Derive during symbolic execution (see #50).\n\nSurprisingly painless. Is this what we want?\n\nOld-commit-hash: af9942ac1056e270f468dc06560517ac7de2cc39\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n\n-- SYMBOLIC DERIVATION\n--\n-- compare derive with _\u27ea_\u27eb_Term\n-- and derive-var with \u27e6_\u27e7Var\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = bool\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\nxor\u2083 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nxor\u2083 t\u2081 t\u2082 t\u2083\n  = if t\u2081 (if t\u2082 t\u2083 (if t\u2083 false true)) (if t\u2082 (if t\u2083 false true) t\u2083)\n\nderive-if : \u2200 {\u03c4 \u0393\u2081} \u2192\n  (t dt : Term \u0393\u2081 bool) \u2192\n  (v\u2081 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2081 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  (v\u2082 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2082 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-if {\u03c4\u2081 \u21d2 \u03c4\u2082} t dt v\u2081 dv\u2081 v\u2082 dv\u2082 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 let \u227c\u2083 = \u227c-trans \u227c\u2081 \u227c\u2082 in\n    derive-if (weaken \u227c\u2083 t) (weaken \u227c\u2083 dt)\n              (v\u2081 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2081 \u227c\u2081 v \u227c\u2082 dv)\n              (v\u2082 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2082 \u227c\u2081 v \u227c\u2082 dv)\nderive-if {bool} t\u2081 dt\u2081 t\u2082 dt\u2082 t\u2083 dt\u2083 =\n  if dt\u2081 (if t\u2081 (xor\u2083 t\u2082 t\u2083 dt\u2083) (xor\u2083 t\u2082 t\u2083 dt\u2082)) (if t\u2081 dt\u2083 dt\u2082)\n\nderive-var : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-var this (dv SymEnv.\u2022 d\u03c1) = dv\nderive-var (that x) (dv SymEnv.\u2022 d\u03c1) = derive-var x d\u03c1\n\nderive : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive (abs t) \u03c1 d\u03c1 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 derive t (liftVal \u227c\u2082 v SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) \u03c1) (dv SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) d\u03c1)\nderive (app t\u2081 t\u2082) \u03c1 d\u03c1 =\n  derive t\u2081 \u03c1 d\u03c1 \u227c-refl (_ \u27ea t\u2082 \u27ebTerm \u03c1) \u227c-refl (derive t\u2082 \u03c1 d\u03c1)\nderive (var x) \u03c1 d\u03c1 = derive-var x d\u03c1\nderive true \u03c1 d\u03c1 = false\nderive false \u03c1 d\u03c1 = false\nderive {\u0393\u2081} (if t\u2081 t\u2082 t\u2083) \u03c1 d\u03c1 =\n  derive-if (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (derive t\u2081 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (derive t\u2082 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2083 \u27ebTerm \u03c1) (derive t\u2083 \u03c1 d\u03c1)\n","old_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8dc6fd5c821f19a87d96c7731f39280af92e1c17","subject":"Renamed some properties.","message":"Renamed some properties.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/Equality\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/Equality\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\nstream-\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\nstream-\u2248-refl {xs} Sxs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = Stream xs \u2227 Stream xs \u2227 xs \u2261 ys\n\n  h\u2081 : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n       R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  h\u2081 (Sxs , Sys , refl) with (Stream-unf Sxs)\n  ... | x' , xs' , Sxs' , prf = x' , xs' , xs' , (Sxs' , Sxs' , refl) , prf , prf\n\n  h\u2082 : R xs xs\n  h\u2082 = Sxs , Sxs , refl\n\nstream-\u2261\u2192\u2248 : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2261 ys \u2192 xs \u2248 ys\nstream-\u2261\u2192\u2248 Sxs _ refl = stream-\u2248-refl Sxs\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for the equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n\u2248-Stream-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\n\u2248-Stream-refl {xs} Sxs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = Stream xs \u2227 Stream xs \u2227 xs \u2261 ys\n\n  h\u2081 : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n       R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  h\u2081 (Sxs , Sys , refl) with (Stream-unf Sxs)\n  ... | x' , xs' , Sxs' , prf = x' , xs' , xs' , (Sxs' , Sxs' , refl) , prf , prf\n\n  h\u2082 : R xs xs\n  h\u2082 = Sxs , Sxs , refl\n\n\u2261\u2192\u2248-Stream : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2261 ys \u2192 xs \u2248 ys\n\u2261\u2192\u2248-Stream Sxs _ refl = \u2248-Stream-refl Sxs\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4dd2701282fa75b452d77d8bb170e29ea6fa6956","subject":"clean up cut elimination","message":"clean up cut elimination\n","repos":"crypto-agda\/protocols","old_file":"js-experiment\/Terms.agda","new_file":"js-experiment\/Terms.agda","new_contents":"open import proto\nopen import Types\nopen import prelude\nopen import uri\n\nmodule Terms where\n\ninfix 2 \u22a2\u02e2_ \u22a2_ \u22a2\u1d9c\u1da0_\n\ndata \u22a2_ : (\u0394 : Env) \u2192 Set\u2081 where\n  end : \u2200{\u0394}{e : EndedEnv \u0394}\n       ------------------\n      \u2192 \u22a2 \u0394\n\n  output : \u2200 {\u0394 d M P}{{_ : SER M}}\n             (l : d \u21a6 send P \u2208 \u0394)(m : M)\n             (p : \u22a2 \u0394 [ l \u2254 m ])\n             -------------------\n               \u2192 \u22a2 \u0394\n\n  input : \u2200 {\u0394 d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2 \u0394 [ l \u2254 m ])\n                 ----------------------\n                     \u2192 \u22a2 \u0394\n\n  mix : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n          (p : \u22a2 \u0394\u2080) (q : \u22a2 \u0394\u2081)\n          --------------------\n        \u2192 \u22a2 \u0394\n\n  cut : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) {P d}\n          (p : \u22a2 (\u0394\u2080 , d \u21a6 dual P))\n          (q : \u22a2 (\u0394\u2081 , d \u21a6 P))\n          ---------------------\n        \u2192 \u22a2 \u0394\n\n  fwd : \u2200 c d {P} \u2192 \u22a2 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\n\n  exch-last :\n      \u2200 {\u0394 c d P Q} \u2192\n      (p : \u22a2 \u0394 , c \u21a6 P , d \u21a6 Q)\n      \u2192 \u22a2 \u0394 , d \u21a6 Q , c \u21a6 P\n\n  wk-last : \u2200 {\u0394 d}\n              (p : \u22a2 \u0394)\n              -----------------------\n                \u2192 \u22a2 (\u0394 , d \u21a6 end)\n\n  end-last : \u2200 {\u0394 d}\n               (p : \u22a2 (\u0394 , d \u21a6 end))\n               ----------------------\n                 \u2192 \u22a2 \u0394\n\ndata \u22a2\u1d9c\u1da0_ (\u0394 : Env) : Set\u2081 where\n  end : {e : EndedEnv \u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  output : \u2200 {d M P}{{_ : SER M}}\n            (l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M)\n            (p : \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            --------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  input : \u2200 {d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            ----------------------------\n                     \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n-- The \u0394 for the server contains the view point of the clients\n-- The point is that the meaning of _,_ in \u0394 is \u2297 while it\n-- is \u214b in \u22a2\u1d9c\u1da0\nrecord \u22a2\u02e2_ (\u0394 : Env) : Set\u2081 where\n  coinductive\n  field\n    server-output :\n      \u2200 {d M}{{_ : SER M}}{P : M \u2192 Proto}\n        (l : d \u21a6 recv P \u2208 \u0394) \u2192\n        \u03a3 M \u03bb m \u2192 \u22a2\u02e2 \u0394 [ l \u2254 m ]\n    server-input :\n      \u2200 {d M}{{_ : SER M}}{P : M \u2192 Proto}\n        (l : d \u21a6 send P \u2208 \u0394)\n        (m : M) \u2192 \u22a2\u02e2 \u0394 [ l \u2254 m ]\nopen \u22a2\u02e2_ public\n\n-- This is just to confirm that we have enough cases\ntelecom' : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u02e2 \u0394 \u2192 \ud835\udfd9\ntelecom' end q = _\ntelecom' (output l m p) q\n  = telecom' p (server-input q l m)\ntelecom' (input l p) q\n  = case server-output q l of \u03bb { (m , s) \u2192\n      telecom' (p m) s }\n\nembed\u1d9c\u1da0 : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2 \u0394\nembed\u1d9c\u1da0 (end {e = e}) = end {e = e}\nembed\u1d9c\u1da0 (output l m p) = output l m (embed\u1d9c\u1da0 p)\nembed\u1d9c\u1da0 (input l p) = input l \u03bb m \u2192 embed\u1d9c\u1da0 (p m)\n\nmix\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n         (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n         (q : \u22a2\u1d9c\u1da0 \u0394\u2081)\n         -------------\n           \u2192 \u22a2\u1d9c\u1da0 \u0394\nmix\u1d9c\u1da0 \u0394\u209b end q = tr \u22a2\u1d9c\u1da0_ (Zip-identity  \u0394\u209b) q\nmix\u1d9c\u1da0 \u0394\u209b (output l m p) q\n  = output (Zip-com\u2208\u2080 \u0394\u209b l) m (mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) p q)\nmix\u1d9c\u1da0 \u0394\u209b (input l p) q\n  = input (Zip-com\u2208\u2080 \u0394\u209b l) \u03bb m \u2192 mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) (p m) q\n\ncut\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081}\n         (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) d P\n         (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n         (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n         ---------------------------\n           \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\n\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output here m p) (input here q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input here p) (output here m q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) (p m) q\n\ncut\u1d9c\u1da0 \u0394\u209b d P (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d P (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d P p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d P p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\n\ncut\u1d9c\u1da0 \u0394\u209b d end p q = mix\u1d9c\u1da0 (\u0394\u209b , d \u21a6\u2080 end) p q\ncut\u1d9c\u1da0 _ _ (com _ _) (end {e = _ , ()}) _\ncut\u1d9c\u1da0 _ _ (com _ _) _ (end {e = _ , ()})\n\n\nend-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n              (p : \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end))\n              -----------------------\n                \u2192 \u22a2\u1d9c\u1da0 \u0394\nend-last\u1d9c\u1da0 (end {e = e , _}) = end {e = e}\nend-last\u1d9c\u1da0 (output (there l) m p) = output l m (end-last\u1d9c\u1da0 p)\nend-last\u1d9c\u1da0 (input (there l) p) = input l \u03bb m \u2192 end-last\u1d9c\u1da0 (p m)\n\nwk-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n             (p : \u22a2\u1d9c\u1da0 \u0394)\n             -----------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\nwk-last\u1d9c\u1da0 end = end {e = \u2026 , _}\nwk-last\u1d9c\u1da0 (output l m p) = output (there l) m (wk-last\u1d9c\u1da0 p)\nwk-last\u1d9c\u1da0 (input l p) = input (there l) \u03bb m \u2192 wk-last\u1d9c\u1da0 (p m)\n\nwk-,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 \u2192 EndedEnv \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u03b5} p E = p\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u0394\u2081 , d \u21a6 P} p (E , e) rewrite Ended-\u2261end e\n  = wk-last\u1d9c\u1da0 (wk-,,\u1d9c\u1da0 p E)\n\nmodule _ {d P \u0394\u2080} where\n  pre-wk-\u2208 : \u2200 {\u0394\u2081} \u2192 d \u21a6 P \u2208 \u0394\u2081 \u2192 d \u21a6 P \u2208 (\u0394\u2080 ,, \u0394\u2081)\n  pre-wk-\u2208 here = here\n  pre-wk-\u2208 (there l) = there (pre-wk-\u2208 l)\n\n{-\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 E\u0394\u2080 end = end {e = {!!}}\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E\u0394\u2080 (output l m p) =\n  output (pre-wk-\u2208 l) m (pre-wk\u1d9c\u1da0 {\u0394\u2080} {{!\u0394\u2081!}} E\u0394\u2080 {!!})\npre-wk\u1d9c\u1da0 E\u0394\u2080 (input l p) = {!!}\n-}\n\nfwd-mix\u1d9c\u1da0 : \u2200 {\u0394 c d} P \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , d \u21a6 dual P)\nfwd-mix\u1d9c\u1da0 end p = wk-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 p)\nfwd-mix\u1d9c\u1da0 (recv P) p = input (there here) \u03bb m \u2192 output here m (fwd-mix\u1d9c\u1da0 (P m) p)\nfwd-mix\u1d9c\u1da0 (send P) p = input here \u03bb m \u2192 output (there here) m (fwd-mix\u1d9c\u1da0 (P m) p)\n\nfwd\u1d9c\u1da0 : \u2200 c d {P} \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\nfwd\u1d9c\u1da0 _ _ {P} = fwd-mix\u1d9c\u1da0 {\u03b5} P end\n\n\u03b5,, : \u2200 \u0394 \u2192 \u03b5 ,, \u0394 \u2261 \u0394\n\u03b5,, \u03b5 = refl\n\u03b5,, (\u0394 , d \u21a6 P) rewrite \u03b5,, \u0394 = refl\n\npostulate\n exch\u1d9c\u1da0 :\n  \u2200 \u0394\u2080 \u0394\u2081 \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2081 ,, \u0394\u2080)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n  {-\nexch\u1d9c\u1da0 \u03b5 \u0394\u2081 p rewrite \u03b5,, \u0394\u2081 = p\nexch\u1d9c\u1da0 \u0394\u2080 \u03b5 p rewrite \u03b5,, \u0394\u2080 = p\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) end = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 ._) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output here m p) = {!exch\u1d9c\u1da0!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (input l p) = {!!}\n-}\n\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E p = exch\u1d9c\u1da0 \u0394\u2080 \u0394\u2081 (wk-,,\u1d9c\u1da0 p E)\n\nend-of : Env \u2192 Env\nend-of \u03b5 = \u03b5\nend-of (\u0394 , d \u21a6 P) = end-of \u0394 , d \u21a6 end\n\nend-of-Ended : \u2200 \u0394 \u2192 EndedEnv (end-of \u0394)\nend-of-Ended \u03b5 = _\nend-of-Ended (\u0394 , d \u21a6 P) = end-of-Ended \u0394 , _\n\nend-of-\u22ce : \u2200 \u0394 \u2192 [ \u0394 is \u0394 \u22ce end-of \u0394 ]\nend-of-\u22ce \u03b5 = \u03b5\nend-of-\u22ce (\u0394 , d \u21a6 P) = end-of-\u22ce \u0394 , d \u21a6\u2080 P\n\nend-of-,,-\u22ce : \u2200 \u0394\u2080 \u0394\u2081 \u2192 [ \u0394\u2080 ,, \u0394\u2081 is \u0394\u2080 ,, end-of \u0394\u2081 \u22ce end-of \u0394\u2080 ,, \u0394\u2081 ]\nend-of-,,-\u22ce \u0394\u2080 \u03b5 = end-of-\u22ce \u0394\u2080\nend-of-,,-\u22ce \u0394\u2080 (\u0394\u2081 , d \u21a6 P) = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081 , d \u21a6\u2081 P\n\n,,-assoc : \u2200 {\u0394\u2080 \u0394\u2081 \u0394\u2082} \u2192 \u0394\u2080 ,, (\u0394\u2081 ,, \u0394\u2082) \u2261 (\u0394\u2080 ,, \u0394\u2081) ,, \u0394\u2082\n,,-assoc {\u0394\u2082 = \u03b5} = refl\n,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082 , d \u21a6 P} rewrite ,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082} = refl\n\ncut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\ncut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q =\n  end-last\u1d9c\u1da0\n    (cut\u1d9c\u1da0 \u0394\u209b d P\n       (exch\u1d9c\u1da0 (\u0394\u2080 ,, end-of \u0394\u2081) (\u03b5 , d \u21a6 dual P)\n              (tr \u22a2\u1d9c\u1da0_ (! (,,-assoc {\u03b5 , d \u21a6 dual P} {\u0394\u2080} {end-of \u0394\u2081}))\n                 (wk-,,\u1d9c\u1da0 {_} {end-of \u0394\u2081}\n                   (exch\u1d9c\u1da0 (\u03b5 , d \u21a6 dual P) _ p) (end-of-Ended _))))\n       (pre-wk\u1d9c\u1da0 (end-of-Ended _) q))\n  where \u0394\u209b = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081\n\npostulate\n  !cut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 dual P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n-- !cut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q = {!!}\n\n-- only the last two are exchanged, some more has to be done\nexch-last\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q , c \u21a6 P\nexch-last\u1d9c\u1da0 (end {e = (a , b) , c}) = end {e = (a , c) , b}\nexch-last\u1d9c\u1da0 (output here m p) = output (there here) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there here) m p) = output here m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there (there l)) m p) = output (there (there l)) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (input here p) = input (there here) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there here) p) = input here \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there (there l)) p) = input (there (there l)) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p  m)\n\n{-\ndata Relabel : Env \u2192 Env \u2192 Set where\n  \u03b5 : Relabel \u03b5 \u03b5\n  _,_\u21a6_ : \u2200 {\u0394\u2080 \u0394\u2081 c d P} \u2192 Relabel \u0394\u2080 \u0394\u2081 \u2192 Relabel (\u0394\u2080 , c \u21a6 P) (\u0394\u2081 , d \u21a6 P)\n\nmodule _ where\n    rebalel-\u2208 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n                  (l : d \u21a6 P \u2208 \u0394\u2080)\n                    \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n            (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel end = {!end!}\n    relabel (output l m p) = output {!l!} {!!} {!!}\n    relabel (input l p) = {!!}\n\npar\u1d9c\u1da0 : \u2200 {\u0394 c} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , c \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 (P \u214b' Q))\n-- TODO only one channel name!!!\n-- TODO empty context\n-- TODO try to match on 'p' first\nbroken-par\u1d9c\u1da0 : \u2200 {c d e} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , e \u21a6 (P \u214b' Q))\nbroken-par\u1d9c\u1da0 end Q p = {!end-last\u1d9c\u1da0 (exch-last\u1d9c\u1da0 p)!}\nbroken-par\u1d9c\u1da0 (com x P) end p = end-last\u1d9c\u1da0 {!p!}\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (end {e = _ , ()})\n\nbroken-par\u1d9c\u1da0 (com x P) (com .OUT Q) (output here m p)\n  = output here R (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com x P) (com .IN Q) (input here p)\n  = output here R (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com .OUT P) (com y Q) (output (there here) m p)\n  = output here L (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com .IN P) (com y Q) (input (there here) p)\n  = output here L (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (output (there (there ())) m p)\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (input (there (there ())) p)\n-}\n\nmodule _ {c d cd} where\n    bi-fwd : \u2200 P Q \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , cd \u21a6 P \u2297 Q , c \u21a6 dual P , d \u21a6 dual Q)\n\n    private\n      module _ {M} {{_ : SER M}} {P : M \u2192 Proto} {Q} where\n        goL : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 P m \u2297 Q)\n                          , c  \u21a6 dual (com x P)\n                          , d  \u21a6 dual Q\n\n        goL IN  = input (there (there here)) \u03bb m \u2192 output (there here) m (bi-fwd _ _)\n        goL OUT = input (there here) \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n        goR : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 Q \u2297 P m)\n                          , c  \u21a6 dual Q\n                          , d  \u21a6 dual (com x P)\n        goR IN  = input (there (there here)) \u03bb m \u2192 output here m (bi-fwd _ _)\n        goR OUT = input here \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n    bi-fwd end Q = exch-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _))\n    bi-fwd (com x P) end = wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _)\n    bi-fwd (com x P) (com y Q) = input (there (there here)) [L: goL x ,R: goR y ]\n\n    module _ {\u0394\u2080 \u0394\u2081 P Q} where\n        \u2297\u1d9c\u1da0 : (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , c \u21a6 P))\n             (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 Q))\n             ----------------------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394\u2080 ,, \u0394\u2081 , cd \u21a6 (P \u2297 Q))\n        \u2297\u1d9c\u1da0 p q = !cut,,\u1d9c\u1da0 _ _ p (!cut,,\u1d9c\u1da0 _ _ q (bi-fwd P Q))\n\n  {-\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (l : c \u21a6 P \u2208 \u0394)\n  (p : \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 d \u21a6 Q ] , c \u21a6 P\nexch\u1d9c\u1da0 here p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (there l) p = {!!}\n-}\n\n{-\nrot\u1d9c\u1da0 : \u2200 \u0394 {c P} \u2192\n         (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P)\n      \u2192 \u22a2\u1d9c\u1da0 \u03b5 , c \u21a6 P ,, \u0394\nrot\u1d9c\u1da0 \u03b5 p = p\nrot\u1d9c\u1da0 (\u0394 , d \u21a6 P) p = {!rot\u1d9c\u1da0 \u0394 p!}\n\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394\u2080} \u0394\u2081 {c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2080 , c \u21a6 P , d \u21a6 Q ,, \u0394\u2081)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 , d \u21a6 Q , c \u21a6 P ,, \u0394\u2081\nexch\u1d9c\u1da0 \u03b5 p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (end e) = end {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d\u2081 \u21a6 ._) (output here m p) = output here m ({!exch\u1d9c\u1da0 \u0394\u2081 p!})\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (input l p) = {!!}\n-}\n\n_\u2286_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n_\u2287_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n\n\u0394\u2080 \u2286 \u0394\u2081 = \u2200 {d P} \u2192 d \u21a6 P \u2208 \u0394\u2080 \u2192 d \u21a6 P \u2208 \u0394\u2081\n\u0394\u2080 \u2287 \u0394\u2081 = \u0394\u2081 \u2286 \u0394\u2080\n\nget-put : \u2200 {d P \u0394 c Q} \u2192\n        (l : d \u21a6 P \u2208 \u0394) \u2192 c \u21a6 Q \u2208 (\u0394 [ l \u2254 c \u21a6 Q ])\nget-put here = here\nget-put (there l) = there (get-put l)\n\n{-\n\u2286_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (f : \u0394\u2080 \u2286 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2286 (\u0394\u2081 [ f l \u2254 c \u21a6 Q ])\n\u2286 f [ l \u2254 c \u21a6 Q ] {d'} {P'} l' = {!!}\n\n(l : d \u21a6 P \u2208 \u0394)\n\u2192 \u0394 [ l \u2254 ]\n\nrecord _\u2248_ (\u0394\u2080 \u0394\u2081 : Env) : Set\u2081 where\n  constructor _,_\n  field\n    \u2248\u2286 : \u0394\u2080 \u2286 \u0394\u2081\n    \u2248\u2287 : \u0394\u2080 \u2287 \u0394\u2081\nopen _\u2248_ public\n\n\u2248_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2248 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ])\n\u2248 \u0394\u2091 [ here \u2254 m ] = {!!}\n\u2248 \u0394\u2091 [ there l \u2254 m ] = {!!}\n\n{-(\u03bb l' \u2192 {!\u2248\u2286 \u0394\u2091!}) , from\n  where\n    from : \u2200 {\u0394\u2080 \u0394\u2081 d io M} {P : M \u2192 Proto} {ser : SER M}\n             {\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081} {l : d \u21a6 com io P \u2208 \u0394\u2080} {m : M} {d\u2081} {P\u2081} \u2192\n           d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ]) \u2192 d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2080 [ l \u2254 m ])\n    from = {!!}\n\n\u2248\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081}\n       (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n       (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n       -----------\n         \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n\u2248\u1d9c\u1da0 \u0394\u2091 (end {e = e}) = end {e = {!!}}\n\u2248\u1d9c\u1da0 \u0394\u2091 (output l m p) = output (\u2248\u2286 \u0394\u2091 l) m (\u2248\u1d9c\u1da0 (\u2248 \u0394\u2091 [ l \u2254 m ]) p)\n\u2248\u1d9c\u1da0 \u0394\u2091 (input l p) = {!!}\n-}\n-}\n\ncut-elim : \u2200 {\u0394} (p : \u22a2 \u0394)\n                 ------------\n                   \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut-elim (end {e = e}) = end {e = e}\ncut-elim (output l m p) = output l m (cut-elim p)\ncut-elim (input l p) = input l (\u03bb m \u2192 cut-elim (p m))\ncut-elim (mix \u0394\u209b p q) = mix\u1d9c\u1da0 \u0394\u209b (cut-elim p) (cut-elim q)\ncut-elim (cut \u0394\u209b {P} {d} p q) = end-last\u1d9c\u1da0 (cut\u1d9c\u1da0 \u0394\u209b d P (cut-elim p) (cut-elim q))\ncut-elim (end-last p) = end-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (wk-last p) = wk-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (fwd c d) = fwd\u1d9c\u1da0 c d\ncut-elim (exch-last p) = exch-last\u1d9c\u1da0 (cut-elim p)\n\n{-\n\nstart : \u2200 {\u0394} P\n       \u2192 \u22a2 [ clientURI \u21a6 dual P ]\n       \u2192 (\u2200 d \u2192 \u22a2 (\u0394 , d \u21a6 P))\n       \u2192 \u22a2 \u0394\nstart P p q = cut {!!} (... p) (q {!!})\n-}\n\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 : \u2200 {P d} \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ] \u2192 \u27e6 P \u27e7\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {end} end = _\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {com x P} (end {e = _ , ()})\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output here m der) = m , \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 der\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output (there ()) m der)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input here x\u2081) m = \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (x\u2081 m)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input (there ()) x\u2081)\n\nSatisfy : \u2200 {p d} P \u2192 (R : \u27e6 log P \u27e7 \u2192 Set p) \u2192 \u22a2 [ d \u21a6 P ] \u2192 Set p\nSatisfy P Rel d = (d' : \u27e6 dual P \u27e7) \u2192 Rel (telecom P (\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (cut-elim d)) d')\n\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {end} p = end\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 (p m))\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 p)\n\n\u27e6\u27e7\u2192\u22a2 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2 {end} p = end\n\u27e6\u27e7\u2192\u22a2 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2 (p m))\n\u27e6\u27e7\u2192\u22a2 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2 p)\n\n{-\n\u22a2toProc : \u2200 {\u0394} \u2192 \u22a2 \u0394 \u2192 JSProc\n\u22a2toProc end = end\n\u22a2toProc (output {d = d} l m prg) = output d (serialize m) (\u22a2toProc prg)\n\u22a2toProc (input {d = d} l prg) = input d ([succeed: (\u03bb m \u2192 \u22a2toProc (prg m)) ,fail: error ] \u2218 parse)\n\u22a2toProc (start P prg x) = start (\u22a2toProc prg) (\u03bb d \u2192 \u22a2toProc (x d))\n\n\n\u22a2toProc-WT : \u2200 {\u0394} (der : \u22a2 \u0394) \u2192 \u0394 \u22a2 \u22a2toProc der\n\u22a2toProc-WT (end {x}) = end {_} {x}\n\u22a2toProc-WT (output {{x}} l m der) = output l (sym (rinv m)) (\u22a2toProc-WT der)\n\u22a2toProc-WT (input {{x}} l x\u2081) = input l \u03bb s m x \u2192\n  subst (\u03bb X \u2192 _ [ l \u2254 m ] \u22a2 [succeed: (\u22a2toProc \u2218 x\u2081) ,fail: error ] X) x (\u22a2toProc-WT (x\u2081 m))\n\u22a2toProc-WT (start P der x) = start P (\u22a2toProc-WT der) (\u03bb d \u2192 \u22a2toProc-WT (x d))\n-}\n\n\u27e6_\u27e7E : Env \u2192 Set\n\u27e6 \u03b5 \u27e7E = \ud835\udfd9\n\u27e6 \u0394 , d \u21a6 P \u27e7E = \u27e6 \u0394 \u27e7E \u00d7 \u27e6 P \u27e7\n\nmapEnv : (Proto \u2192 Proto) \u2192 Env \u2192 Env\nmapEnv f \u03b5 = \u03b5\nmapEnv f (\u0394 , d \u21a6 P) = mapEnv f \u0394 , d \u21a6 f P\n\nmapEnv-all : \u2200 {P Q : URI \u2192 Proto \u2192 Set}{\u0394}{f : Proto \u2192 Proto}\n  \u2192 (\u2200 d x \u2192 P d x \u2192 Q d (f x))\n  \u2192 AllEnv P \u0394 \u2192 AllEnv Q (mapEnv f \u0394)\nmapEnv-all {\u0394 = \u03b5} f\u2081 \u2200\u0394 = \u2200\u0394\nmapEnv-all {\u0394 = \u0394 , d \u21a6 P\u2081} f\u2081 (\u2200\u0394 , P) = (mapEnv-all f\u2081 \u2200\u0394) , f\u2081 d P\u2081 P\n\nmapEnv-Ended : \u2200 {f : Proto \u2192 Proto}{\u0394} \u2192 Ended (f end)\n  \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 AllEnv (\u03bb _ \u2192 Ended) (mapEnv f \u0394)\nmapEnv-Ended eq = mapEnv-all (\u03bb { d end _ \u2192 eq ; d (send P) () ; d (recv P) () })\n\nmapEnv-\u2208 : \u2200 {d P f \u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 d \u21a6 f P \u2208 mapEnv f \u0394\nmapEnv-\u2208 here = here\nmapEnv-\u2208 (there der) = there (mapEnv-\u2208 der)\n\nmodule _ {d c M cf}{m : M}{{_ : M \u2243? SERIAL}}{p} where\n  subst-lemma-one-point-four : \u2200 {\u0394}( l : d \u21a6 com c p \u2208 \u0394 ) \u2192\n    let f = mapProto cf\n    in (mapEnv f (\u0394 [ l \u2254 m ])) \u2261 (_[_\u2254_]{c = cf c} (mapEnv f \u0394) (mapEnv-\u2208 l) m)\n  subst-lemma-one-point-four here = refl\n  subst-lemma-one-point-four (there {d' = d'}{P'} l) = ap (\u03bb X \u2192 X , d' \u21a6 mapProto cf P') (subst-lemma-one-point-four l)\n\nmodule _ {d P} where\n  project : \u2200 {\u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 \u27e6 \u0394 \u27e7E \u2192 \u27e6 P \u27e7\n  project here env = snd env\n  project (there mem) env = project mem (fst env)\n\nempty : \u2200 {\u0394} \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 \u27e6 \u0394 \u27e7E\nempty {\u03b5} <> = _\nempty {\u0394 , d \u21a6 end} (fst , <>) = empty fst , _\nempty {\u0394 , d \u21a6 com x P} (fst , ())\n\nnoRecvInLog : \u2200 {d M}{{_ : M \u2243? SERIAL}}{P : M \u2192 _}{\u0394} \u2192 d \u21a6 recv P \u2208 mapEnv log \u0394 \u2192 \ud835\udfd8\nnoRecvInLog {\u0394 = \u03b5} ()\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 end} (there l) = noRecvInLog l\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 com x\u2081 P\u2081} (there l) = noRecvInLog l\n\nmodule _ {d M P}{{_ : M \u2243? SERIAL}} where\n  lookup : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 \u03a3 M \u03bb m \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  lookup here (env , (m , p)) = m , (env , p)\n  lookup (there l) (env , P') = let (m , env') = lookup l env in m , (env' , P')\n\n  set : \u2200 {\u0394}(l : d \u21a6 recv P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  set here (env , f) m = env , f m\n  set (there l) (env , P') m = set l env m , P'\n\n  set\u03a3 : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E \u2192 \u27e6 \u0394 \u27e7E\n  set\u03a3 here m env = fst env , (m , snd env)\n  set\u03a3 (there l) m env = set\u03a3 l m (fst env) , snd env\n\n  {-\nforgetConc : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 mapEnv log \u0394 \u2192 \u27e6 mapEnv log \u0394 \u27e7E\nforgetConc (end e) = empty \u2026\nforgetConc {\u0394} (output l m der) = set\u03a3 l m (forgetConc {{!set\u03a3 l m!}} der) -- (forgetConc der)\nforgetConc (input l x\u2081) with noRecvInLog l\n... | ()\n-}\n\n\u22a2telecom : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u27e6 mapEnv dual \u0394 \u27e7E \u2192 \u22a2 mapEnv log \u0394\n\u22a2telecom end env = end {e = mapEnv-Ended _ \u2026}\n\u22a2telecom (output l m der) env = output (mapEnv-\u2208 l) m (subst (\u22a2_)\n  (subst-lemma-one-point-four l) (\u22a2telecom der (subst \u27e6_\u27e7E (sym (subst-lemma-one-point-four l)) (set (mapEnv-\u2208 l) env m))))\n\u22a2telecom (input l x\u2081) env = let (m , env') = lookup (mapEnv-\u2208 l) env\n                                hyp = \u22a2telecom (x\u2081 m) (subst (\u27e6_\u27e7E) (sym (subst-lemma-one-point-four l)) env')\n                             in output (mapEnv-\u2208 l) m\n                             (subst (\u22a2_) (subst-lemma-one-point-four l) hyp)\n\n-- old version\n{-\ncut\u1d9c\u1da0 : \u2200 {\u0394 d P} \u2192 \u27e6 dual P \u27e7 \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 P \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut\u1d9c\u1da0 D (end {allEnded = \u0394E , PE }) = end {allEnded = \u0394E}\ncut\u1d9c\u1da0 D (output here m E) = cut\u1d9c\u1da0 (D m) E\ncut\u1d9c\u1da0 D (output (there l) m E) = output l m (cut\u1d9c\u1da0 D E)\ncut\u1d9c\u1da0 (m , D) (input here x\u2081) = cut\u1d9c\u1da0 D (x\u2081 m)\ncut\u1d9c\u1da0 D (input (there l) x\u2081) = input l (\u03bb m \u2192 cut\u1d9c\u1da0 D (x\u2081 m))\n\ncut : \u2200 {\u0394 \u0394' \u0393 \u0393' d P} \u2192 \u22a2 \u0394 , clientURI \u21a6 dual P +++ \u0394' \u2192 \u22a2 \u0393 , d \u21a6 P +++ \u0393' \u2192 \u22a2 (\u0394 +++ \u0394') +++ (\u0393 +++ \u0393')\ncut D E = {!!}\n-}\n\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import proto\nopen import Types\nopen import prelude\nopen import uri\n\nmodule Terms where\n\ninfix 2 \u22a2\u02e2_ \u22a2_ \u22a2\u1d9c\u1da0_\n\ndata \u22a2_ : (\u0394 : Env) \u2192 Set\u2081 where\n  end : \u2200{\u0394}{e : EndedEnv \u0394}\n       ------------------\n      \u2192 \u22a2 \u0394\n\n  output : \u2200 {\u0394 d M P}{{_ : SER M}}\n             (l : d \u21a6 send P \u2208 \u0394)(m : M)\n             (p : \u22a2 \u0394 [ l \u2254 m ])\n             -------------------\n               \u2192 \u22a2 \u0394\n\n  input : \u2200 {\u0394 d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2 \u0394 [ l \u2254 m ])\n                 ----------------------\n                     \u2192 \u22a2 \u0394\n\n  mix : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n          (p : \u22a2 \u0394\u2080) (q : \u22a2 \u0394\u2081)\n          --------------------\n        \u2192 \u22a2 \u0394\n\n  cut : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) {P d}\n          (p : \u22a2 (\u0394\u2080 , d \u21a6 dual P))\n          (q : \u22a2 (\u0394\u2081 , d \u21a6 P))\n          ---------------------\n        \u2192 \u22a2 \u0394\n\n  fwd : \u2200 c d {P} \u2192 \u22a2 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\n\n  exch-last :\n      \u2200 {\u0394 c d P Q} \u2192\n      (p : \u22a2 \u0394 , c \u21a6 P , d \u21a6 Q)\n      \u2192 \u22a2 \u0394 , d \u21a6 Q , c \u21a6 P\n\n  wk-last : \u2200 {\u0394 d}\n              (p : \u22a2 \u0394)\n              -----------------------\n                \u2192 \u22a2 (\u0394 , d \u21a6 end)\n\n  end-last : \u2200 {\u0394 d}\n               (p : \u22a2 (\u0394 , d \u21a6 end))\n               ----------------------\n                 \u2192 \u22a2 \u0394\n\ndata \u22a2\u1d9c\u1da0_ (\u0394 : Env) : Set\u2081 where\n  end : {e : EndedEnv \u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  output : \u2200 {d M P}{{_ : SER M}}\n            (l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M)\n            (p : \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            --------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n  input : \u2200 {d M}{P : M \u2192 _}{{_ : SER M}}\n            (l : d \u21a6 recv P \u2208 \u0394)\n            (p : \u2200 m \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 m ])\n            ----------------------------\n                     \u2192 \u22a2\u1d9c\u1da0 \u0394\n\n-- The \u0394 for the server contains the view point of the clients\n-- The point is that the meaning of _,_ in \u0394 is \u2297 while it\n-- is \u214b in \u22a2\u1d9c\u1da0\nrecord \u22a2\u02e2_ (\u0394 : Env) : Set\u2081 where\n  coinductive\n  field\n    server-output :\n      \u2200 {d M}{{_ : SER M}}{P : M \u2192 Proto}\n        (l : d \u21a6 recv P \u2208 \u0394) \u2192\n        \u03a3 M \u03bb m \u2192 \u22a2\u02e2 \u0394 [ l \u2254 m ]\n    server-input :\n      \u2200 {d M}{{_ : SER M}}{P : M \u2192 Proto}\n        (l : d \u21a6 send P \u2208 \u0394)\n        (m : M) \u2192 \u22a2\u02e2 \u0394 [ l \u2254 m ]\nopen \u22a2\u02e2_ public\n\n-- This is just to confirm that we have enough cases\ntelecom' : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u02e2 \u0394 \u2192 \ud835\udfd9\ntelecom' end q = _\ntelecom' (output l m p) q\n  = telecom' p (server-input q l m)\ntelecom' (input l p) q\n  = case server-output q l of \u03bb { (m , s) \u2192\n      telecom' (p m) s }\n\nembed\u1d9c\u1da0 : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2 \u0394\nembed\u1d9c\u1da0 (end {e = e}) = end {e = e}\nembed\u1d9c\u1da0 (output l m p) = output l m (embed\u1d9c\u1da0 p)\nembed\u1d9c\u1da0 (input l p) = input l \u03bb m \u2192 embed\u1d9c\u1da0 (p m)\n\nmix\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081} (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ])\n         (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n         (q : \u22a2\u1d9c\u1da0 \u0394\u2081)\n         -------------\n           \u2192 \u22a2\u1d9c\u1da0 \u0394\nmix\u1d9c\u1da0 \u0394\u209b end q = tr \u22a2\u1d9c\u1da0_ (Zip-identity  \u0394\u209b) q\nmix\u1d9c\u1da0 \u0394\u209b (output l m p) q\n  = output (Zip-com\u2208\u2080 \u0394\u209b l) m (mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) p q)\nmix\u1d9c\u1da0 \u0394\u209b (input l p) q\n  = input (Zip-com\u2208\u2080 \u0394\u209b l) \u03bb m \u2192 mix\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) (p m) q\n\ncut\u1d9c\u1da0 : \u2200 {\u0394 \u0394\u2080 \u0394\u2081}\n         (\u0394\u209b : [ \u0394 is \u0394\u2080 \u22ce \u0394\u2081 ]) d P\n         (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n         (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n         ---------------------------\n           \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\ncut\u1d9c\u1da0 \u0394\u209b d end p q = mix\u1d9c\u1da0 (\u0394\u209b , d \u21a6\u2080 end) p q\n\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output here m p) (input here q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input here p) (output here m q) = cut\u1d9c\u1da0 \u0394\u209b d (P m) (p m) q\n\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (output (there l) m p) q\n  = output (there (Zip-com\u2208\u2080 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d (send P) (input (there l) p) q\n  = input (there (Zip-com\u2208\u2080 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2080 l \u0394\u209b) d _ (p m) q\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) p (output (there l) m q)\n  = output (there (Zip-com\u2208\u2081 \u0394\u209b l)) m (cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p q)\ncut\u1d9c\u1da0 \u0394\u209b d (send P) p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\ncut\u1d9c\u1da0 \u0394\u209b d (recv P) p (input (there l) q)\n  = input (there (Zip-com\u2208\u2081 \u0394\u209b l)) \u03bb m \u2192 cut\u1d9c\u1da0 (Zip-\u2254\u2081 l \u0394\u209b) d _ p (q m)\n\ncut\u1d9c\u1da0 _ _ (com _ _) (end {e = _ , ()}) _\ncut\u1d9c\u1da0 _ _ (com _ _) _ (end {e = _ , ()})\n\nend-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n              (p : \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end))\n              -----------------------\n                \u2192 \u22a2\u1d9c\u1da0 \u0394\nend-last\u1d9c\u1da0 (end {e = e , _}) = end {e = e}\nend-last\u1d9c\u1da0 (output (there l) m p) = output l m (end-last\u1d9c\u1da0 p)\nend-last\u1d9c\u1da0 (input (there l) p) = input l \u03bb m \u2192 end-last\u1d9c\u1da0 (p m)\n\nwk-last\u1d9c\u1da0 : \u2200 {\u0394 d}\n             (p : \u22a2\u1d9c\u1da0 \u0394)\n             -----------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394 , d \u21a6 end)\nwk-last\u1d9c\u1da0 end = end {e = \u2026 , _}\nwk-last\u1d9c\u1da0 (output l m p) = output (there l) m (wk-last\u1d9c\u1da0 p)\nwk-last\u1d9c\u1da0 (input l p) = input (there l) \u03bb m \u2192 wk-last\u1d9c\u1da0 (p m)\n\nwk-,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 \u2192 EndedEnv \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u03b5} p E = p\nwk-,,\u1d9c\u1da0 {\u0394\u2081 = \u0394\u2081 , d \u21a6 P} p (E , e) rewrite Ended-\u2261end e\n  = wk-last\u1d9c\u1da0 (wk-,,\u1d9c\u1da0 p E)\n\nmodule _ {d P \u0394\u2080} where\n  pre-wk-\u2208 : \u2200 {\u0394\u2081} \u2192 d \u21a6 P \u2208 \u0394\u2081 \u2192 d \u21a6 P \u2208 (\u0394\u2080 ,, \u0394\u2081)\n  pre-wk-\u2208 here = here\n  pre-wk-\u2208 (there l) = there (pre-wk-\u2208 l)\n\n{-\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 E\u0394\u2080 end = end {e = {!!}}\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E\u0394\u2080 (output l m p) =\n  output (pre-wk-\u2208 l) m (pre-wk\u1d9c\u1da0 {\u0394\u2080} {{!\u0394\u2081!}} E\u0394\u2080 {!!})\npre-wk\u1d9c\u1da0 E\u0394\u2080 (input l p) = {!!}\n-}\n\nfwd-mix\u1d9c\u1da0 : \u2200 {\u0394 c d} P \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , d \u21a6 dual P)\nfwd-mix\u1d9c\u1da0 end p = wk-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 p)\nfwd-mix\u1d9c\u1da0 (recv P) p = input (there here) \u03bb m \u2192 output here m (fwd-mix\u1d9c\u1da0 (P m) p)\nfwd-mix\u1d9c\u1da0 (send P) p = input here \u03bb m \u2192 output (there here) m (fwd-mix\u1d9c\u1da0 (P m) p)\n\nfwd\u1d9c\u1da0 : \u2200 c d {P} \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 dual P)\nfwd\u1d9c\u1da0 _ _ {P} = fwd-mix\u1d9c\u1da0 {\u03b5} P end\n\n\u03b5,, : \u2200 \u0394 \u2192 \u03b5 ,, \u0394 \u2261 \u0394\n\u03b5,, \u03b5 = refl\n\u03b5,, (\u0394 , d \u21a6 P) rewrite \u03b5,, \u0394 = refl\n\npostulate\n exch\u1d9c\u1da0 :\n  \u2200 \u0394\u2080 \u0394\u2081 \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2081 ,, \u0394\u2080)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n  {-\nexch\u1d9c\u1da0 \u03b5 \u0394\u2081 p rewrite \u03b5,, \u0394\u2081 = p\nexch\u1d9c\u1da0 \u0394\u2080 \u03b5 p rewrite \u03b5,, \u0394\u2080 = p\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) end = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 ._) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output here m p) = {!exch\u1d9c\u1da0!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P) (\u0394\u2081 , d\u2081 \u21a6 P\u2081) (input l p) = {!!}\n-}\n\npre-wk\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 EndedEnv \u0394\u2080 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081 \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\npre-wk\u1d9c\u1da0 {\u0394\u2080} {\u0394\u2081} E p = exch\u1d9c\u1da0 \u0394\u2080 \u0394\u2081 (wk-,,\u1d9c\u1da0 p E)\n\nend-of : Env \u2192 Env\nend-of \u03b5 = \u03b5\nend-of (\u0394 , d \u21a6 P) = end-of \u0394 , d \u21a6 end\n\nend-of-Ended : \u2200 \u0394 \u2192 EndedEnv (end-of \u0394)\nend-of-Ended \u03b5 = _\nend-of-Ended (\u0394 , d \u21a6 P) = end-of-Ended \u0394 , _\n\nend-of-\u22ce : \u2200 \u0394 \u2192 [ \u0394 is \u0394 \u22ce end-of \u0394 ]\nend-of-\u22ce \u03b5 = \u03b5\nend-of-\u22ce (\u0394 , d \u21a6 P) = end-of-\u22ce \u0394 , d \u21a6\u2080 P\n\nend-of-,,-\u22ce : \u2200 \u0394\u2080 \u0394\u2081 \u2192 [ \u0394\u2080 ,, \u0394\u2081 is \u0394\u2080 ,, end-of \u0394\u2081 \u22ce end-of \u0394\u2080 ,, \u0394\u2081 ]\nend-of-,,-\u22ce \u0394\u2080 \u03b5 = end-of-\u22ce \u0394\u2080\nend-of-,,-\u22ce \u0394\u2080 (\u0394\u2081 , d \u21a6 P) = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081 , d \u21a6\u2081 P\n\n,,-assoc : \u2200 {\u0394\u2080 \u0394\u2081 \u0394\u2082} \u2192 \u0394\u2080 ,, (\u0394\u2081 ,, \u0394\u2082) \u2261 (\u0394\u2080 ,, \u0394\u2081) ,, \u0394\u2082\n,,-assoc {\u0394\u2082 = \u03b5} = refl\n,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082 , d \u21a6 P} rewrite ,,-assoc {\u0394\u2080} {\u0394\u2081} {\u0394\u2082} = refl\n\ncut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 dual P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\ncut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q =\n  end-last\u1d9c\u1da0\n    (cut\u1d9c\u1da0 \u0394\u209b d P\n       (exch\u1d9c\u1da0 (\u0394\u2080 ,, end-of \u0394\u2081) (\u03b5 , d \u21a6 dual P)\n              (tr \u22a2\u1d9c\u1da0_ (! (,,-assoc {\u03b5 , d \u21a6 dual P} {\u0394\u2080} {end-of \u0394\u2081}))\n                 (wk-,,\u1d9c\u1da0\n                   (exch\u1d9c\u1da0 (\u03b5 , d \u21a6 dual P) _ p) (end-of-Ended _))))\n       (pre-wk\u1d9c\u1da0 (end-of-Ended _) q))\n  where \u0394\u209b = end-of-,,-\u22ce \u0394\u2080 \u0394\u2081\n\npostulate\n  !cut,,\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081} d P\n            (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , d \u21a6 P))\n            (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 dual P))\n           ---------------------------\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 ,, \u0394\u2081\n-- !cut,,\u1d9c\u1da0 {\u0394\u2080}{\u0394\u2081} d P p q = {!!}\n\n-- only the last two are exchanged, some more has to be done\nexch-last\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q , c \u21a6 P\nexch-last\u1d9c\u1da0 (end {e = (a , b) , c}) = end {e = (a , c) , b}\nexch-last\u1d9c\u1da0 (output here m p) = output (there here) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there here) m p) = output here m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (output (there (there l)) m p) = output (there (there l)) m (exch-last\u1d9c\u1da0 p)\nexch-last\u1d9c\u1da0 (input here p) = input (there here) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there here) p) = input here \u03bb m \u2192 exch-last\u1d9c\u1da0 (p m)\nexch-last\u1d9c\u1da0 (input (there (there l)) p) = input (there (there l)) \u03bb m \u2192 exch-last\u1d9c\u1da0 (p  m)\n\n{-\ndata Relabel : Env \u2192 Env \u2192 Set where\n  \u03b5 : Relabel \u03b5 \u03b5\n  _,_\u21a6_ : \u2200 {\u0394\u2080 \u0394\u2081 c d P} \u2192 Relabel \u0394\u2080 \u0394\u2081 \u2192 Relabel (\u0394\u2080 , c \u21a6 P) (\u0394\u2081 , d \u21a6 P)\n\nmodule _ where\n    rebalel-\u2208 : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n                  (l : d \u21a6 P \u2208 \u0394\u2080)\n                    \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel : \u2200 {\u0394\u2080 \u0394\u2081} \u2192 Relabel \u0394\u2080 \u0394\u2081\n            (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n              \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n    relabel end = {!end!}\n    relabel (output l m p) = output {!l!} {!!} {!!}\n    relabel (input l p) = {!!}\n\npar\u1d9c\u1da0 : \u2200 {\u0394 c} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 P , c \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u0394 , c \u21a6 (P \u214b' Q))\n-- TODO only one channel name!!!\n-- TODO empty context\n-- TODO try to match on 'p' first\nbroken-par\u1d9c\u1da0 : \u2200 {c d e} P Q\n        (p : \u22a2\u1d9c\u1da0 (\u03b5 , c \u21a6 P , d \u21a6 Q))\n          \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , e \u21a6 (P \u214b' Q))\nbroken-par\u1d9c\u1da0 end Q p = {!end-last\u1d9c\u1da0 (exch-last\u1d9c\u1da0 p)!}\nbroken-par\u1d9c\u1da0 (com x P) end p = end-last\u1d9c\u1da0 {!p!}\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (end {e = _ , ()})\n\nbroken-par\u1d9c\u1da0 (com x P) (com .OUT Q) (output here m p)\n  = output here R (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com x P) (com .IN Q) (input here p)\n  = output here R (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com .OUT P) (com y Q) (output (there here) m p)\n  = output here L (output here m (broken-par\u1d9c\u1da0 _ _ p))\nbroken-par\u1d9c\u1da0 (com .IN P) (com y Q) (input (there here) p)\n  = output here L (input here \u03bb m \u2192 broken-par\u1d9c\u1da0 _ _ (p m))\n\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (output (there (there ())) m p)\nbroken-par\u1d9c\u1da0 (com x P) (com y Q) (input (there (there ())) p)\n-}\n\nmodule _ {c d cd} where\n    bi-fwd : \u2200 P Q \u2192 \u22a2\u1d9c\u1da0 (\u03b5 , cd \u21a6 P \u2297 Q , c \u21a6 dual P , d \u21a6 dual Q)\n\n    private\n      module _ {M} {{_ : SER M}} {P : M \u2192 Proto} {Q} where\n        goL : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 P m \u2297 Q)\n                          , c  \u21a6 dual (com x P)\n                          , d  \u21a6 dual Q\n\n        goL IN  = input (there (there here)) \u03bb m \u2192 output (there here) m (bi-fwd _ _)\n        goL OUT = input (there here) \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n        goR : \u2200 x \u2192 \u22a2\u1d9c\u1da0 \u03b5 , cd \u21a6 com x (\u03bb m \u2192 Q \u2297 P m)\n                          , c  \u21a6 dual Q\n                          , d  \u21a6 dual (com x P)\n        goR IN  = input (there (there here)) \u03bb m \u2192 output here m (bi-fwd _ _)\n        goR OUT = input here \u03bb m \u2192 output (there (there here)) m (bi-fwd _ _)\n\n    bi-fwd end Q = exch-last\u1d9c\u1da0 (wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _))\n    bi-fwd (com x P) end = wk-last\u1d9c\u1da0 (fwd\u1d9c\u1da0 _ _)\n    bi-fwd (com x P) (com y Q) = input (there (there here)) [L: goL x ,R: goR y ]\n\n    module _ {\u0394\u2080 \u0394\u2081 P Q} where\n        \u2297\u1d9c\u1da0 : (p : \u22a2\u1d9c\u1da0 (\u0394\u2080 , c \u21a6 P))\n             (q : \u22a2\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 Q))\n             ----------------------------------\n               \u2192 \u22a2\u1d9c\u1da0 (\u0394\u2080 ,, \u0394\u2081 , cd \u21a6 (P \u2297 Q))\n        \u2297\u1d9c\u1da0 p q = !cut,,\u1d9c\u1da0 _ _ p (!cut,,\u1d9c\u1da0 _ _ q (bi-fwd P Q))\n\n  {-\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394 c d P Q} \u2192\n  (l : c \u21a6 P \u2208 \u0394)\n  (p : \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 Q)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394 [ l \u2254 d \u21a6 Q ] , c \u21a6 P\nexch\u1d9c\u1da0 here p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (there l) p = {!!}\n-}\n\n{-\nrot\u1d9c\u1da0 : \u2200 \u0394 {c P} \u2192\n         (p : \u22a2\u1d9c\u1da0 \u0394 , c \u21a6 P)\n      \u2192 \u22a2\u1d9c\u1da0 \u03b5 , c \u21a6 P ,, \u0394\nrot\u1d9c\u1da0 \u03b5 p = p\nrot\u1d9c\u1da0 (\u0394 , d \u21a6 P) p = {!rot\u1d9c\u1da0 \u0394 p!}\n\nexch\u1d9c\u1da0 :\n  \u2200 {\u0394\u2080} \u0394\u2081 {c d P Q} \u2192\n  (p : \u22a2\u1d9c\u1da0 \u0394\u2080 , c \u21a6 P , d \u21a6 Q ,, \u0394\u2081)\n  \u2192 \u22a2\u1d9c\u1da0 \u0394\u2080 , d \u21a6 Q , c \u21a6 P ,, \u0394\u2081\nexch\u1d9c\u1da0 \u03b5 p = exch-last\u1d9c\u1da0 p\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (end e) = end {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d\u2081 \u21a6 ._) (output here m p) = output here m ({!exch\u1d9c\u1da0 \u0394\u2081 p!})\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (output (there l) m p) = {!!}\nexch\u1d9c\u1da0 (\u0394\u2081 , d \u21a6 P)  (input l p) = {!!}\n-}\n\n_\u2286_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n_\u2287_ : (\u0394\u2080 \u0394\u2081 : Env) \u2192 Set\u2081\n\n\u0394\u2080 \u2286 \u0394\u2081 = \u2200 {d P} \u2192 d \u21a6 P \u2208 \u0394\u2080 \u2192 d \u21a6 P \u2208 \u0394\u2081\n\u0394\u2080 \u2287 \u0394\u2081 = \u0394\u2081 \u2286 \u0394\u2080\n\nget-put : \u2200 {d P \u0394 c Q} \u2192\n        (l : d \u21a6 P \u2208 \u0394) \u2192 c \u21a6 Q \u2208 (\u0394 [ l \u2254 c \u21a6 Q ])\nget-put here = here\nget-put (there l) = there (get-put l)\n\n{-\n\u2286_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (f : \u0394\u2080 \u2286 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2286 (\u0394\u2081 [ f l \u2254 c \u21a6 Q ])\n\u2286 f [ l \u2254 c \u21a6 Q ] {d'} {P'} l' = {!!}\n\n(l : d \u21a6 P \u2208 \u0394)\n\u2192 \u0394 [ l \u2254 ]\n\nrecord _\u2248_ (\u0394\u2080 \u0394\u2081 : Env) : Set\u2081 where\n  constructor _,_\n  field\n    \u2248\u2286 : \u0394\u2080 \u2286 \u0394\u2081\n    \u2248\u2287 : \u0394\u2080 \u2287 \u0394\u2081\nopen _\u2248_ public\n\n\u2248_[_\u2254_\u21a6_] : \u2200 {\u0394\u2080 \u0394\u2081} (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n               {d P} (l : d \u21a6 P \u2208 \u0394\u2080) (c : URI) (Q : Proto)\n             \u2192 (\u0394\u2080 [ l \u2254 c \u21a6 Q ]) \u2248 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ])\n\u2248 \u0394\u2091 [ here \u2254 m ] = {!!}\n\u2248 \u0394\u2091 [ there l \u2254 m ] = {!!}\n\n{-(\u03bb l' \u2192 {!\u2248\u2286 \u0394\u2091!}) , from\n  where\n    from : \u2200 {\u0394\u2080 \u0394\u2081 d io M} {P : M \u2192 Proto} {ser : SER M}\n             {\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081} {l : d \u21a6 com io P \u2208 \u0394\u2080} {m : M} {d\u2081} {P\u2081} \u2192\n           d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2081 [ \u2248\u2286 \u0394\u2091 l \u2254 m ]) \u2192 d\u2081 \u21a6 P\u2081 \u2208 (\u0394\u2080 [ l \u2254 m ])\n    from = {!!}\n\n\u2248\u1d9c\u1da0 : \u2200 {\u0394\u2080 \u0394\u2081}\n       (\u0394\u2091 : \u0394\u2080 \u2248 \u0394\u2081)\n       (p : \u22a2\u1d9c\u1da0 \u0394\u2080)\n       -----------\n         \u2192 \u22a2\u1d9c\u1da0 \u0394\u2081\n\u2248\u1d9c\u1da0 \u0394\u2091 (end {e = e}) = end {e = {!!}}\n\u2248\u1d9c\u1da0 \u0394\u2091 (output l m p) = output (\u2248\u2286 \u0394\u2091 l) m (\u2248\u1d9c\u1da0 (\u2248 \u0394\u2091 [ l \u2254 m ]) p)\n\u2248\u1d9c\u1da0 \u0394\u2091 (input l p) = {!!}\n-}\n-}\n\ncut-elim : \u2200 {\u0394} (p : \u22a2 \u0394)\n                 ------------\n                   \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut-elim (end {e = e}) = end {e = e}\ncut-elim (output l m p) = output l m (cut-elim p)\ncut-elim (input l p) = input l (\u03bb m \u2192 cut-elim (p m))\ncut-elim (mix \u0394\u209b p q) = mix\u1d9c\u1da0 \u0394\u209b (cut-elim p) (cut-elim q)\ncut-elim (cut \u0394\u209b {P} {d} p q) = end-last\u1d9c\u1da0 (cut\u1d9c\u1da0 \u0394\u209b d P (cut-elim p) (cut-elim q))\ncut-elim (end-last p) = end-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (wk-last p) = wk-last\u1d9c\u1da0 (cut-elim p)\ncut-elim (fwd c d) = fwd\u1d9c\u1da0 c d\ncut-elim (exch-last p) = exch-last\u1d9c\u1da0 (cut-elim p)\n\n{-\n\nstart : \u2200 {\u0394} P\n       \u2192 \u22a2 [ clientURI \u21a6 dual P ]\n       \u2192 (\u2200 d \u2192 \u22a2 (\u0394 , d \u21a6 P))\n       \u2192 \u22a2 \u0394\nstart P p q = cut {!!} (... p) (q {!!})\n-}\n\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 : \u2200 {P d} \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ] \u2192 \u27e6 P \u27e7\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {end} end = _\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 {com x P} (end {e = _ , ()})\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output here m der) = m , \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 der\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (output (there ()) m der)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input here x\u2081) m = \u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (x\u2081 m)\n\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (input (there ()) x\u2081)\n\nSatisfy : \u2200 {p d} P \u2192 (R : \u27e6 log P \u27e7 \u2192 Set p) \u2192 \u22a2 [ d \u21a6 P ] \u2192 Set p\nSatisfy P Rel d = (d' : \u27e6 dual P \u27e7) \u2192 Rel (telecom P (\u22a2\u1d9c\u1da0\u2192\u27e6\u27e7 (cut-elim d)) d')\n\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2\u1d9c\u1da0 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {end} p = end\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 (p m))\n\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2\u1d9c\u1da0 p)\n\n\u27e6\u27e7\u2192\u22a2 : \u2200 {P d} \u2192 \u27e6 P \u27e7 \u2192 \u22a2 [ d \u21a6 P ]\n\u27e6\u27e7\u2192\u22a2 {end} p = end\n\u27e6\u27e7\u2192\u22a2 {recv P} p = input here (\u03bb m \u2192 \u27e6\u27e7\u2192\u22a2 (p m))\n\u27e6\u27e7\u2192\u22a2 {send P} (m , p) = output here m (\u27e6\u27e7\u2192\u22a2 p)\n\n{-\n\u22a2toProc : \u2200 {\u0394} \u2192 \u22a2 \u0394 \u2192 JSProc\n\u22a2toProc end = end\n\u22a2toProc (output {d = d} l m prg) = output d (serialize m) (\u22a2toProc prg)\n\u22a2toProc (input {d = d} l prg) = input d ([succeed: (\u03bb m \u2192 \u22a2toProc (prg m)) ,fail: error ] \u2218 parse)\n\u22a2toProc (start P prg x) = start (\u22a2toProc prg) (\u03bb d \u2192 \u22a2toProc (x d))\n\n\n\u22a2toProc-WT : \u2200 {\u0394} (der : \u22a2 \u0394) \u2192 \u0394 \u22a2 \u22a2toProc der\n\u22a2toProc-WT (end {x}) = end {_} {x}\n\u22a2toProc-WT (output {{x}} l m der) = output l (sym (rinv m)) (\u22a2toProc-WT der)\n\u22a2toProc-WT (input {{x}} l x\u2081) = input l \u03bb s m x \u2192\n  subst (\u03bb X \u2192 _ [ l \u2254 m ] \u22a2 [succeed: (\u22a2toProc \u2218 x\u2081) ,fail: error ] X) x (\u22a2toProc-WT (x\u2081 m))\n\u22a2toProc-WT (start P der x) = start P (\u22a2toProc-WT der) (\u03bb d \u2192 \u22a2toProc-WT (x d))\n-}\n\n\u27e6_\u27e7E : Env \u2192 Set\n\u27e6 \u03b5 \u27e7E = \ud835\udfd9\n\u27e6 \u0394 , d \u21a6 P \u27e7E = \u27e6 \u0394 \u27e7E \u00d7 \u27e6 P \u27e7\n\nmapEnv : (Proto \u2192 Proto) \u2192 Env \u2192 Env\nmapEnv f \u03b5 = \u03b5\nmapEnv f (\u0394 , d \u21a6 P) = mapEnv f \u0394 , d \u21a6 f P\n\nmapEnv-all : \u2200 {P Q : URI \u2192 Proto \u2192 Set}{\u0394}{f : Proto \u2192 Proto}\n  \u2192 (\u2200 d x \u2192 P d x \u2192 Q d (f x))\n  \u2192 AllEnv P \u0394 \u2192 AllEnv Q (mapEnv f \u0394)\nmapEnv-all {\u0394 = \u03b5} f\u2081 \u2200\u0394 = \u2200\u0394\nmapEnv-all {\u0394 = \u0394 , d \u21a6 P\u2081} f\u2081 (\u2200\u0394 , P) = (mapEnv-all f\u2081 \u2200\u0394) , f\u2081 d P\u2081 P\n\nmapEnv-Ended : \u2200 {f : Proto \u2192 Proto}{\u0394} \u2192 Ended (f end)\n  \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 AllEnv (\u03bb _ \u2192 Ended) (mapEnv f \u0394)\nmapEnv-Ended eq = mapEnv-all (\u03bb { d end _ \u2192 eq ; d (send P) () ; d (recv P) () })\n\nmapEnv-\u2208 : \u2200 {d P f \u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 d \u21a6 f P \u2208 mapEnv f \u0394\nmapEnv-\u2208 here = here\nmapEnv-\u2208 (there der) = there (mapEnv-\u2208 der)\n\nmodule _ {d c M cf}{m : M}{{_ : M \u2243? SERIAL}}{p} where\n  subst-lemma-one-point-four : \u2200 {\u0394}( l : d \u21a6 com c p \u2208 \u0394 ) \u2192\n    let f = mapProto cf\n    in (mapEnv f (\u0394 [ l \u2254 m ])) \u2261 (_[_\u2254_]{c = cf c} (mapEnv f \u0394) (mapEnv-\u2208 l) m)\n  subst-lemma-one-point-four here = refl\n  subst-lemma-one-point-four (there {d' = d'}{P'} l) = ap (\u03bb X \u2192 X , d' \u21a6 mapProto cf P') (subst-lemma-one-point-four l)\n\nmodule _ {d P} where\n  project : \u2200 {\u0394} \u2192 d \u21a6 P \u2208 \u0394 \u2192 \u27e6 \u0394 \u27e7E \u2192 \u27e6 P \u27e7\n  project here env = snd env\n  project (there mem) env = project mem (fst env)\n\nempty : \u2200 {\u0394} \u2192 AllEnv (\u03bb _ \u2192 Ended) \u0394 \u2192 \u27e6 \u0394 \u27e7E\nempty {\u03b5} <> = _\nempty {\u0394 , d \u21a6 end} (fst , <>) = empty fst , _\nempty {\u0394 , d \u21a6 com x P} (fst , ())\n\nnoRecvInLog : \u2200 {d M}{{_ : M \u2243? SERIAL}}{P : M \u2192 _}{\u0394} \u2192 d \u21a6 recv P \u2208 mapEnv log \u0394 \u2192 \ud835\udfd8\nnoRecvInLog {\u0394 = \u03b5} ()\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 end} (there l) = noRecvInLog l\nnoRecvInLog {\u0394 = \u0394 , d\u2081 \u21a6 com x\u2081 P\u2081} (there l) = noRecvInLog l\n\nmodule _ {d M P}{{_ : M \u2243? SERIAL}} where\n  lookup : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 \u03a3 M \u03bb m \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  lookup here (env , (m , p)) = m , (env , p)\n  lookup (there l) (env , P') = let (m , env') = lookup l env in m , (env' , P')\n\n  set : \u2200 {\u0394}(l : d \u21a6 recv P \u2208 \u0394) \u2192 \u27e6 \u0394 \u27e7E \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E\n  set here (env , f) m = env , f m\n  set (there l) (env , P') m = set l env m , P'\n\n  set\u03a3 : \u2200 {\u0394}(l : d \u21a6 send P \u2208 \u0394) \u2192 (m : M) \u2192 \u27e6 \u0394 [ l \u2254 m ] \u27e7E \u2192 \u27e6 \u0394 \u27e7E\n  set\u03a3 here m env = fst env , (m , snd env)\n  set\u03a3 (there l) m env = set\u03a3 l m (fst env) , snd env\n\n  {-\nforgetConc : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 mapEnv log \u0394 \u2192 \u27e6 mapEnv log \u0394 \u27e7E\nforgetConc (end e) = empty \u2026\nforgetConc {\u0394} (output l m der) = set\u03a3 l m (forgetConc {{!set\u03a3 l m!}} der) -- (forgetConc der)\nforgetConc (input l x\u2081) with noRecvInLog l\n... | ()\n-}\n\n\u22a2telecom : \u2200 {\u0394} \u2192 \u22a2\u1d9c\u1da0 \u0394 \u2192 \u27e6 mapEnv dual \u0394 \u27e7E \u2192 \u22a2 mapEnv log \u0394\n\u22a2telecom end env = end {e = mapEnv-Ended _ \u2026}\n\u22a2telecom (output l m der) env = output (mapEnv-\u2208 l) m (subst (\u22a2_)\n  (subst-lemma-one-point-four l) (\u22a2telecom der (subst \u27e6_\u27e7E (sym (subst-lemma-one-point-four l)) (set (mapEnv-\u2208 l) env m))))\n\u22a2telecom (input l x\u2081) env = let (m , env') = lookup (mapEnv-\u2208 l) env\n                                hyp = \u22a2telecom (x\u2081 m) (subst (\u27e6_\u27e7E) (sym (subst-lemma-one-point-four l)) env')\n                             in output (mapEnv-\u2208 l) m\n                             (subst (\u22a2_) (subst-lemma-one-point-four l) hyp)\n\n-- old version\n{-\ncut\u1d9c\u1da0 : \u2200 {\u0394 d P} \u2192 \u27e6 dual P \u27e7 \u2192 \u22a2\u1d9c\u1da0 \u0394 , d \u21a6 P \u2192 \u22a2\u1d9c\u1da0 \u0394\ncut\u1d9c\u1da0 D (end {allEnded = \u0394E , PE }) = end {allEnded = \u0394E}\ncut\u1d9c\u1da0 D (output here m E) = cut\u1d9c\u1da0 (D m) E\ncut\u1d9c\u1da0 D (output (there l) m E) = output l m (cut\u1d9c\u1da0 D E)\ncut\u1d9c\u1da0 (m , D) (input here x\u2081) = cut\u1d9c\u1da0 D (x\u2081 m)\ncut\u1d9c\u1da0 D (input (there l) x\u2081) = input l (\u03bb m \u2192 cut\u1d9c\u1da0 D (x\u2081 m))\n\ncut : \u2200 {\u0394 \u0394' \u0393 \u0393' d P} \u2192 \u22a2 \u0394 , clientURI \u21a6 dual P +++ \u0394' \u2192 \u22a2 \u0393 , d \u21a6 P +++ \u0393' \u2192 \u22a2 (\u0394 +++ \u0394') +++ (\u0393 +++ \u0393')\ncut D E = {!!}\n-}\n\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e35fc05915324dc4d1f2447d4d45b5c919b06b78","subject":"Removed files related to agda2atp.","message":"Removed files related to agda2atp.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/README\/Test.agda","new_file":"notes\/README\/Test.agda","new_contents":"","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test where\n\ndata _\u2228_ (A B : Set) : Set where\n  inj\u2081 : A \u2192 A \u2228 B\n  inj\u2082 : B \u2192 A \u2228 B\n\npostulate\n  A B    : Set\n  \u2228-comm : A \u2228 B \u2192 B \u2228 A\n{-# ATP prove \u2228-comm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"00065eb75ec8eb897e66dfd3bfb4c202d976fd57","subject":"Use booleans from Data.Bool (fix #31).","message":"Use booleans from Data.Bool (fix #31).\n\nOld-commit-hash: c2de96202cc68d0b06b45d3c977bdf935e14eec1\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Values.agda","new_file":"Denotational\/Values.agda","new_contents":"module Denotational.Values where\n\n-- VALUES\n--\n-- This module defines the model theory of simple types, that is,\n-- it defines for every type, the set of values of that type.\n--\n-- In fact, we only describe a single model here.\n\nopen import Data.Bool public\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Denotational.Notation\n\nopen import Syntactic.Types\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\nopen import Level using (zero)\nopen import Relation.Binary.PropositionalEquality\npostulate ext : Extensionality zero zero\n\nnot-not : \u2200 a \u2192 a \u2261 not (not a)\nnot-not true = refl\nnot-not false = refl\n\na-xor-a-false : \u2200 a \u2192 (a xor a) \u2261 false\na-xor-a-false true = refl\na-xor-a-false false = refl\n\na-xor-false-a : \u2200 a \u2192 (false xor a) \u2261 a\na-xor-false-a b = refl\n\nxor-associative : \u2200 a b c \u2192 ((b xor c) xor a) \u2261 (b xor (c xor a))\nxor-associative a true true = not-not a\nxor-associative a true false = refl\nxor-associative a false c = refl\n\na-xor-false : \u2200 a \u2192 a xor false \u2261 a\na-xor-false true = refl\na-xor-false false = refl\n\na-xor-true : \u2200 a \u2192 a xor true \u2261 not a\na-xor-true true = refl\na-xor-true false = refl\n\nxor-commutative : \u2200 a b \u2192 a xor b \u2261 b xor a\nxor-commutative true b rewrite a-xor-true b = refl\nxor-commutative false b rewrite a-xor-false b = refl\n\nxor-cancellative-2 : \u2200 a b \u2192 (b xor (a xor a)) \u2261 b\nxor-cancellative-2 a b rewrite a-xor-a-false a = a-xor-false b\n\nxor-cancellative : \u2200 a b \u2192 ((b xor a) xor a) \u2261 b\nxor-cancellative a b rewrite xor-associative a b a = xor-cancellative-2 a b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl = refl\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym = sym\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans = trans\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong f = cong f\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 f = cong\u2082 f\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4 : Set} \u2192 IsEquivalence (_\u2261_ {A = \u03c4})\n\u2261-isEquivalence = isEquivalence\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n","old_contents":"module Denotational.Values where\n\n-- VALUES\n--\n-- This module defines the model theory of simple types, that is,\n-- it defines for every type, the set of values of that type.\n--\n-- In fact, we only describe a single model here.\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Denotational.Notation\n\nopen import Syntactic.Types\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n_xor_ : Bool \u2192 Bool \u2192 Bool\ntrue xor b = not b\nfalse xor b = b\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\nopen import Level using (zero)\nopen import Relation.Binary.PropositionalEquality\npostulate ext : Extensionality zero zero\n\nnot-not : \u2200 a \u2192 a \u2261 not (not a)\nnot-not true = refl\nnot-not false = refl\n\na-xor-a-false : \u2200 a \u2192 (a xor a) \u2261 false\na-xor-a-false true = refl\na-xor-a-false false = refl\n\na-xor-false-a : \u2200 a \u2192 (false xor a) \u2261 a\na-xor-false-a b = refl\n\nxor-associative : \u2200 a b c \u2192 ((b xor c) xor a) \u2261 (b xor (c xor a))\nxor-associative a true true = not-not a\nxor-associative a true false = refl\nxor-associative a false c = refl\n\na-xor-false : \u2200 a \u2192 a xor false \u2261 a\na-xor-false true = refl\na-xor-false false = refl\n\na-xor-true : \u2200 a \u2192 a xor true \u2261 not a\na-xor-true true = refl\na-xor-true false = refl\n\nxor-commutative : \u2200 a b \u2192 a xor b \u2261 b xor a\nxor-commutative true b rewrite a-xor-true b = refl\nxor-commutative false b rewrite a-xor-false b = refl\n\nxor-cancellative-2 : \u2200 a b \u2192 (b xor (a xor a)) \u2261 b\nxor-cancellative-2 a b rewrite a-xor-a-false a = a-xor-false b\n\nxor-cancellative : \u2200 a b \u2192 ((b xor a) xor a) \u2261 b\nxor-cancellative a b rewrite xor-associative a b a = xor-cancellative-2 a b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl = refl\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym = sym\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans = trans\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong f = cong f\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 f = cong\u2082 f\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4 : Set} \u2192 IsEquivalence (_\u2261_ {A = \u03c4})\n\u2261-isEquivalence = isEquivalence\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ccabb7380964bd2d763e0b6231ec252a0f2de3b6","subject":"More (failed) progress on the proof","message":"More (failed) progress on the proof\n\nPart of this goes in a wrong direction, since `fundamental-aux2` appears\nto be very false (even though it has just one hole): we need a proof of\n`rel\u03c1 \u0393 \u03c11 \u03c12 (2 + n)` when we only have `rel\u03c1 \u0393 \u03c11 \u03c12 (1 + n)` in\ncontext.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n \u2192 evalConst c n \u2261 Done v \u2192 evalConst c (suc n) \u2261 Done v\neval-const-mono (lit v) n eq = eq\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n \u2192 eval t \u03c1 n \u2261 Done v \u2192 eval t \u03c1 (suc n) \u2261 Done v\neval-mono t \u03c1 v zero ()\neval-mono (const c) \u03c1 v (suc n) eq = eval-const-mono c n eq\neval-mono (var x) \u03c1 v (suc n) eq = eq\neval-mono (app s t) \u03c1 v (suc n) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-mono (app s t) \u03c1 v (suc n) eq | Done sv | [ seq ] | (Done tv) | [ teq ] with eval s \u03c1 (suc n) | eval-mono s \u03c1 sv n seq | eval t \u03c1 (suc n) | eval-mono t \u03c1 tv n teq\neval-mono (app s t) \u03c1 v (suc n) eq | Done (closure ct c\u03c1) | [ seq ] | (Done tv) | [ teq ] | .(Done (closure ct c\u03c1)) | refl | .(Done tv) | refl = eval-mono ct (tv \u2022 c\u03c1) _ n eq\neval-mono (app s t) \u03c1 v (suc n) () | Done _ | [ seq ] | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n) () | TimeOut | [ seq ] | tv | [ teq ]\n-- = {!eval-mono s \u03c1 (suc n)!}\neval-mono (abs t) \u03c1 v (suc n) eq = eq\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- XXX don't we want *RELATED* input environments? XXX seems not???\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc (suc n)))\n  (tvv : relV \u03c3 tv1 tv2 (suc (suc n)))\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n  with svv (suc n) (\u2264-step \u2264-refl) tv1 tv2 (relV-mono _ _ (\u2264-step (\u2264-step \u2264-refl)) _ _ _ tvv) v1 eqv1\n  | svv (suc (suc n)) \u2264-refl tv1 tv2 (relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv) v1 (eval-mono st1 (tv1 \u2022 \u03c11) v1 (suc n) eqv1)\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2' , eqv2' , final-v' | v2 , eqv2 , final-v with trans (sym (eval-mono st2 (tv2 \u2022 \u03c12) v2' (suc n) eqv2')) eqv2\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | .v2 , eqv2' , final-v' | v2 , eqv2 , final-v | refl = v2 , eqv2' , final-v\n\n-- eqv2'\n-- (suc n) ? tv1 tv2 (relV-mono _ _ (n\u22641+n (suc n))  _ _ _ tvv) v1 eqv1\n-- ... | v2 , eqv2 , final-v = v2 , eqv2  , final-v\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc (suc n)))\n  (tvv : relV \u03c3 tv1 tv2 (suc (suc n)))\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n--\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\n\nfundamental-aux : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n  eval t \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 eval t \u03c12 (suc n) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nfundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\nfundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = sv2 , refl , sveq , svv'\n\nfundamental-aux2 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n  eval t \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc (suc n)))\nfundamental-aux2 s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1\n... | sv2 , eq1 , eq2 , svv with fundamental-aux s (suc n) \u03c11 \u03c12 {!!} sv1 (eval-mono s \u03c11 sv1 n seq1)\n... | sv2' , eq1' , eq2' , svv' = sv2' , trans eq1 (trans (sym eq2) eq1') , svv'\n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | fundamental-aux2 s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | eval t \u03c12 n | fundamental-aux2 t n \u03c11 \u03c12 \u03c1\u03c1 tv1 teq1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | (.sv2 , refl , svv') | (Done tv2) | (.tv2 , refl , tvv') = relV-apply n svv' tvv' t\u03c11\u2193v1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = {!!}\n-- | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\n\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n \u2192 evalConst c n \u2261 Done v \u2192 evalConst c (suc n) \u2261 Done v\neval-const-mono (lit v) n eq = eq\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n \u2192 eval t \u03c1 n \u2261 Done v \u2192 eval t \u03c1 (suc n) \u2261 Done v\neval-mono t \u03c1 v zero ()\neval-mono (const c) \u03c1 v (suc n) eq = eval-const-mono c n eq\neval-mono (var x) \u03c1 v (suc n) eq = eq\neval-mono (app s t) \u03c1 v (suc n) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-mono (app s t) \u03c1 v (suc n) eq | Done sv | [ seq ] | (Done tv) | [ teq ] with eval s \u03c1 (suc n) | eval-mono s \u03c1 sv n seq | eval t \u03c1 (suc n) | eval-mono t \u03c1 tv n teq\neval-mono (app s t) \u03c1 v (suc n) eq | Done (closure ct c\u03c1) | [ seq ] | (Done tv) | [ teq ] | .(Done (closure ct c\u03c1)) | refl | .(Done tv) | refl = eval-mono ct (tv \u2022 c\u03c1) _ n eq\neval-mono (app s t) \u03c1 v (suc n) () | Done _ | [ seq ] | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n) () | TimeOut | [ seq ] | tv | [ teq ]\n-- = {!eval-mono s \u03c1 (suc n)!}\neval-mono (abs t) \u03c1 v (suc n) eq = eq\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n  (tvv : relV \u03c3 tv1 tv2 (suc n))\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 with svv (suc n) \u2264-refl tv1 tv2 {! tvv !} v1  eqv1\n... | v2 , eqv2 , final-v = v2 , eqv2  , final-v\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n  (tvv : relV \u03c3 tv1 tv2 (suc n))\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = {!!}\n-- relV-apply-go _ _ _ _ n svv tvv _ eqv1\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\n\nfundamental-aux : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n  eval t \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 eval t \u03c12 (suc n) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nfundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\nfundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = sv2 , refl , sveq , svv'\n--\n-- |\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | eval t \u03c12 n | fundamental-aux t n \u03c11 \u03c12 \u03c1\u03c1 tv1 teq1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | (.sv2 , refl , seq2' , svv') | (Done tv2) | (.tv2 , refl , teq2' , tvv') = relV-apply n svv' tvv' t\u03c11\u2193v1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = {!!}\n-- | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\n\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64a0d4678c9a3cfe69e473c9b5f335cd071c1e13","subject":"Move import to more sensible place","message":"Move import to more sensible place\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0c7f128c9e77179348618b2a7fbb141228b0b822","subject":"Extensional equivalence on bags Checkpoint: NatMap is unnecessary. Remove it.","message":"Extensional equivalence on bags\nCheckpoint: NatMap is unnecessary. Remove it.\n\nOld-commit-hash: f465114a1f29ce95ce687d717c48ad6d5bbf7693\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/MapBag.agda","new_file":"experimental\/MapBag.agda","new_contents":"{-\n\nIntroducing the MapBag, which can be considered a bag\nof \u2124 with negative multiplicities, or a map from\n\u2124 to \u2124 with default value 0.\n\nThe initial goal of this file is to make the 5th example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    -- Haskell's Data.Map.map is Scala's Map.mapValues\n    map :: (a -> b) -> Map k a -> Map k b\n    \n    incVal :: Map Integer Integer -> Map Integer Integer\n    incVal = map (+1)\n\n    old = fromAscList  [(1, 1), (2, 2) .. (n, n)]\n    res = incVal old = [(1, 2), (2, 3) .. (n, n + 1)]\n\n\nTODO\nX. Stop not getting why hole in half\u2081-WF can't be filled\nX. Prove well-foundedness of half\u2081\nX. Fix singleton\n3. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n\n4. Finish ExplicitNils\n5. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule MapBag where\n\nopen import Data.Unit using\n  (\u22a4)\nopen import Data.Nat using\n  (\u2115 ; suc ; _\u2264\u2032_ ; \u2264\u2032-refl ; \u2264\u2032-step)\nopen import Induction.Nat using\n  (<-rec)\nopen import Data.Integer using\n  (\u2124 ; +_ ; -[1+_] ; _+_ ; _-_)\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n-- Map nats to ints with default 0\ndata NatMap : Set where\n  \u2205 : NatMap\n  \u2115tree : (v : \u2124) \u2192 (left : NatMap) \u2192 (right : NatMap) \u2192 NatMap\n\n-- To understand why there must be an empty NatMap,\n-- observe the termination checker's complaint upon\n-- seeing Haskell's empty NatMap:\n--\n--     emptyMap : NatMap\n--     emptyMap = \u2115tree (+ 0) emptyMap emptyMap\n\ndata Oddity : Set where\n  odd  : Oddity\n  even : Oddity\n\noddity : \u2115 \u2192 Oddity\noddity 0 = even\noddity 1 = odd\noddity (suc (suc n)) = oddity n\n\n-- Elimination form of Oddity to please termination checker\n-- ... eventually.\n\nif-odd_then_else_ : \u2200 {A : Set} \u2192 \u2115 \u2192 A \u2192 A \u2192 A\nif-odd k then thenBranch else elseBranch with oddity k\n... | odd  = thenBranch\n... | even = elseBranch\n\n-- oddity-index k = (oddity k , (k - 1) \/ 2)\n-- Here to please termination checker so that \u2115lookup\n-- doesn't have to pass well-foundedness-proofs around\noddity-index : \u2115 \u2192 Oddity \u00d7 \u2115\noddity-index 0 = (even , 0)\noddity-index 1 = (odd  , 0)\noddity-index 2 = (even , 0)\noddity-index (suc (suc k)) = (oddity-k , suc [k-1]\/2) where\n  oddity-index-k = oddity-index k\n  oddity-k = proj\u2081 oddity-index-k\n  [k-1]\/2  = proj\u2082 oddity-index-k\n\n-- Computes next key after going down 1 level of a nat tree\nhalf\u2081 : \u2115 \u2192 \u2115\nhalf\u2081 0 = 0\nhalf\u2081 1 = 0\nhalf\u2081 2 = 0\nhalf\u2081 (suc (suc n)) = suc (half\u2081 n)\n\n-- Nonzero n holds a proof that n != 0.\ndata Nonzero : \u2115 \u2192 Set where\n  suc : (n : \u2115) \u2192 Nonzero (suc n)\n\n\u2264\u2032-suc : \u2200 {m n : \u2115} \u2192 m \u2264\u2032 n \u2192 (suc m) \u2264\u2032 (suc n)\n\u2264\u2032-suc \u2264\u2032-refl = \u2264\u2032-refl\n\u2264\u2032-suc (\u2264\u2032-step le) = \u2264\u2032-step (\u2264\u2032-suc le)\n\nhalf\u2081-WF : \u2200 (n : \u2115) \u2192 Nonzero n \u2192 suc (half\u2081 n) \u2264\u2032 n\nhalf\u2081-WF 0 ()\nhalf\u2081-WF 1 _ = \u2264\u2032-refl\nhalf\u2081-WF 2 _ = \u2264\u2032-step \u2264\u2032-refl\nhalf\u2081-WF (suc (suc (suc n))) _ =\n  \u2264\u2032-suc {suc (half\u2081 (suc n))} {suc (suc n)}\n         (\u2264\u2032-step (half\u2081-WF (suc n) (suc n)))\n\n-- Here to please the termination checker\nsingleton : \u2115 \u2192 \u2124 \u2192 NatMap\nsingleton k v = loop k where\n  loop : \u2115 \u2192 NatMap\n  loop = <-rec _ \u03bb\n    { 0 _ \u2192 \u2115tree v \u2205 \u2205\n    ; (suc n\u2080) rec \u2192\n      let\n        n = suc n\u2080\n        next = rec (half\u2081 n) (half\u2081-WF n (suc n\u2080))\n      in\n        if-odd n\n        then \u2115tree (+ 0) next \u2205\n        else \u2115tree (+ 0) \u2205 next\n    }\n\n\u2115lookup : \u2115 \u2192 NatMap \u2192 \u2124\n\u2115lookup _ \u2205 = (+ 0)\n\u2115lookup 0 (\u2115tree v _ _) = v\n\u2115lookup k (\u2115tree _ left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115lookup [k-1]\/2 left\n... | (even , [k-1]\/2) = \u2115lookup [k-1]\/2 right\n\n-- `update` in the sense of\n-- Data.Sequence.update : Int \u2192 a \u2192 Seq a \u2192 Seq a\n\u2115update : \u2115 \u2192 \u2124 \u2192 NatMap \u2192 NatMap\n\u2115update k v \u2205 = singleton k v\n\u2115update 0 v (\u2115tree v\u2080 left right) = \u2115tree v left right\n\u2115update k v (\u2115tree v\u2080 left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115tree v\u2080 (\u2115update [k-1]\/2 v left) right\n... | (even , [k-1]\/2) = \u2115tree v\u2080 left (\u2115update [k-1]\/2 v right)\n\nMapBag = NatMap \u00d7 NatMap\n\nemptyBag : MapBag\nemptyBag = (\u2205 , \u2205)\n\nlookup : \u2124 \u2192 MapBag \u2192 \u2124\nlookup -[1+ k ] (negative , nonnegative) = \u2115lookup k negative\nlookup   (+ k)  (negative , nonnegative) = \u2115lookup k nonnegative\n\nupdate : \u2124 \u2192 \u2124 \u2192 MapBag \u2192 MapBag\nupdate -[1+ k ] v (neg , pos) = (\u2115update k v neg , pos)\nupdate  ( + k ) v (neg , pos) = (neg , \u2115update k v pos)\n\n-- We implement nothing but the necessary NatMap operations to save time:\n-- union, difference, mapValues.\n\n\u2115mapValues : (\u2124 \u2192 \u2124) \u2192 NatMap \u2192 NatMap\n\u2115mapValues _ \u2205 = \u2205\n\u2115mapValues f (\u2115tree v left right) =\n  \u2115tree (f v) (\u2115mapValues f left) (\u2115mapValues f right)\n\nmapValues : (\u2124 \u2192 \u2124) \u2192 MapBag \u2192 MapBag\nmapValues f (neg , pos) = (\u2115mapValues f neg , \u2115mapValues f pos)\n\n\u2115union : NatMap \u2192 NatMap \u2192 NatMap\n\u2115union \u2205 b = b\n\u2115union b \u2205 = b\n\u2115union (\u2115tree v\u2081 left\u2081 right\u2081) (\u2115tree v\u2082 left\u2082 right\u2082) =\n  \u2115tree (v\u2081 + v\u2082) (\u2115union left\u2081 left\u2082) (\u2115union right\u2081 right\u2082)\n\n-- Prelude.(++) : [a] \u2192 [a] \u2192 [a]\n_++_ : MapBag \u2192 MapBag \u2192 MapBag\nb\u2081 ++ b\u2082 = Product.zip \u2115union \u2115union b\u2081 b\u2082\n\ninfixr 5 _++_\n\n\u2115diff : NatMap \u2192 NatMap \u2192 NatMap\n\u2115diff \u2205 b = \u2205\n\u2115diff b \u2205 = b\n\u2115diff (\u2115tree v\u2081 left\u2081 right\u2081) (\u2115tree v\u2082 left\u2082 right\u2082) =\n  \u2115tree (v\u2081 - v\u2082) (\u2115diff left\u2081 left\u2082) (\u2115diff right\u2081 right\u2082)\n\n-- Data.Map.(\\\\) : Map k a \u2192 Map k b \u2192 Map k a (where b = a = \u2124)\n_\\\\_ : MapBag \u2192 MapBag \u2192 MapBag\nb\u2081 \\\\ b\u2082 = Product.zip \u2115diff \u2115diff b\u2081 b\u2082\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n-- Bag's extensional equivalence:\n-- Remove Quinean \"identity without identity\" on bags\n-- (keys with 0 multiplicities and no subkeys are as good\n-- as not to exist).\n\ndata \u2115Empty : NatMap \u2192 Set where\n  is-empty : \u2115Empty \u2205\n  0-mult : \u2200 {left right} \u2192 \u2115Empty left \u2192 \u2115Empty right \u2192\n             \u2115Empty (\u2115tree (+ 0) left right)\n\nEmpty : MapBag \u2192 Set\nEmpty (neg , pos) = \u2115Empty neg \u00d7 \u2115Empty pos\n\npostulate\n  ext-bag : \u2200 {b\u2081 b\u2082} \u2192 Empty (b\u2081 \\\\ b\u2082) \u2192 b\u2081 \u2261 b\u2082\n\ninfixl 9 _\\\\_\n\ndata Type : Set where\n  ints : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  int : \u2200 {\u0393} \u2192 (n : \u2124) \u2192 Term \u0393 ints\n  bag : \u2200 {\u0393} \u2192 (b : MapBag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 ints) \u2192 (t\u2082 : Term \u0393 ints) \u2192 Term \u0393 ints\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (ints \u21d2 ints)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to ints = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (int x) = int x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 ints \u27e7Type = \u2124\n\u27e6 bags \u27e7Type = MapBag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 int n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapValues (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (int n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 = cong\u2082 mapValues (weaken-sound f \u03c1)\n                                           (weaken-sound b \u03c1)\n\n-- Changes to \u2124 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on mapbags are mapbags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type ints = (ints \u21d2 ints \u21d2 ints) \u21d2 ints\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\u27e6snd\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\n\u27e6derive\u27e7 {ints} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {ints} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {ints} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n{-\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {ints} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = {!!}\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {ints} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = {!!}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {ints} n = refl\nf\u2295\u0394f=f {bags} b = {!!}\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {ints} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = {!!}\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {ints} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = {!!}\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (int n) = abs (app (app (var this) (int n)) (int n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\nderive _ = {!!}\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case int: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {ints} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {bags} b db valid-b-db = {!!}\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (int n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {ints} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = {!!}\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (int n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n-}\n\n","old_contents":"{-\n\nIntroducing the MapBag, which can be considered a bag\nof \u2124 with negative multiplicities, or a map from\n\u2124 to \u2124 with default value 0.\n\nThe initial goal of this file is to make the 5th example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    -- Haskell's Data.Map.map is Scala's Map.mapValues\n    map :: (a -> b) -> Map k a -> Map k b\n    \n    incVal :: Map Integer Integer -> Map Integer Integer\n    incVal = map (+1)\n\n    old = fromAscList  [(1, 1), (2, 2) .. (n, n)]\n    res = incVal old = [(1, 2), (2, 3) .. (n, n + 1)]\n\nTODO\nX. Replace \u2115 by \u2124\n2. Introduce addition\n3. Add MapBags and map\n4. Figure out a way to communicate to a derivative that\n   certain changes are always nil (in this case, `+1`).\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule MapBag where\n\nopen import Data.Unit using\n  (\u22a4)\nopen import Data.Nat using\n  (\u2115 ; suc ; _\u2264\u2032_ ; \u2264\u2032-refl ; \u2264\u2032-step)\nopen import Induction.Nat using\n  (<-rec)\nopen import Data.Integer using\n  (\u2124 ; +_ ; -[1+_] ; _+_ ; _-_)\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Level\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n-- Map nats to ints with default 0\ndata NatMap : Set where\n  \u2205 : NatMap\n  \u2115tree : (v : \u2124) \u2192 (left : NatMap) \u2192 (right : NatMap) \u2192 NatMap\n\n{- To understand why there must be an empty NatMap,\n   observe the termination checker's complaint upon\n   seeing Haskell's empty NatMap:\n\nemptyMap : NatMap\nemptyMap = \u2115tree (+ 0) emptyMap emptyMap\n-}\n\ndata Oddity : Set where\n  odd  : Oddity\n  even : Oddity\n\noddity : \u2115 \u2192 Oddity\noddity 0 = even\noddity 1 = odd\noddity (suc (suc n)) = oddity n\n\n-- Elimination form of Oddity to please termination checker\n-- ... eventually.\n\nif-odd_then_else_ : \u2200 {A : Set} \u2192 \u2115 \u2192 A \u2192 A \u2192 A\nif-odd k then thenBranch else elseBranch with oddity k\n... | odd  = thenBranch\n... | even = elseBranch\n\n-- oddity-index k = (oddity k , (k - 1) \/ 2)\noddity-index : \u2115 \u2192 Oddity \u00d7 \u2115\noddity-index 0 = (even , 0)\noddity-index 1 = (odd  , 0)\noddity-index 2 = (even , 0)\noddity-index (suc (suc k)) = (oddity-k , suc [k-1]\/2) where\n  oddity-index-k = oddity-index k\n  oddity-k = proj\u2081 oddity-index-k\n  [k-1]\/2  = proj\u2082 oddity-index-k\n\n-- Copied from Induction.Nat.Examples (declared private)\n\nhalf\u2081 : \u2115 \u2192 \u2115\nhalf\u2081 0 = 0\nhalf\u2081 1 = 0\nhalf\u2081 2 = 0\nhalf\u2081 (suc (suc n)) = suc (half\u2081 n)\n\ndata Nonzero : \u2115 \u2192 Set where\n  suc : (n : \u2115) \u2192 Nonzero (suc n)\n\n\u2264\u2032-suc : \u2200 {m n : \u2115} \u2192 m \u2264\u2032 n \u2192 (suc m) \u2264\u2032 (suc n)\n\u2264\u2032-suc \u2264\u2032-refl = \u2264\u2032-refl\n\u2264\u2032-suc (\u2264\u2032-step le) = \u2264\u2032-step (\u2264\u2032-suc le)\n\nhalf\u2081-\u03b2\u2261 : \u2200 {n : \u2115} \u2192 Nonzero n \u2192 half\u2081 (suc (suc n)) \u2261 suc (half\u2081 n)\nhalf\u2081-\u03b2\u2261 {0} ()\nhalf\u2081-\u03b2\u2261 {suc n} nz = refl\n\nhalf\u2081-WF : \u2200 (n : \u2115) \u2192 Nonzero n \u2192 suc (half\u2081 n) \u2264\u2032 n\nhalf\u2081-WF 0 ()\nhalf\u2081-WF 1 _ = \u2264\u2032-refl\nhalf\u2081-WF 2 _ = \u2264\u2032-step \u2264\u2032-refl\nhalf\u2081-WF (suc (suc (suc n))) _ =\n  \u2264\u2032-suc {suc (half\u2081 (suc n))} {suc (suc n)} {!half\u2081-WF (suc n) ?!}\n-- TODO:\n-- 0. Stop not getting why hole#0 can't be filled\n-- 1. Prove well-foundedness of half\u2081\n-- 2. Fix singleton\n-- 3. Make sure this file has no hole\n-- 4. Finish ExplicitNils\n-- 5. Consider appending ExplicitNils\n\n-- Here to please the termination checker\nsingleton : \u2115 \u2192 \u2124 \u2192 NatMap\nsingleton k v = loop k where\n  loop : \u2115 \u2192 NatMap\n  loop = <-rec _ \u03bb\n    { 0 _ \u2192 \u2115tree v \u2205 \u2205\n    ; n rec \u2192\n      let\n        next = half\u2081 n\n      in -- TODO: Case distinction\n        rec next {!!}\n    }\n\n{-\ncRec _ \u03bb\n  { 0 _ \u2192 (0 , \u2115tree v \u2205 \u2205)\n  ; 1 _ \u2192 (100 , \u2115tree (+ 0) (\u2115tree v \u2205 \u2205) \u2205)\n  ; 2 _ \u2192 (200 , \u2115tree (+ 0) \u2205 (\u2115tree v \u2205 \u2205))\n  ; (suc (suc n)) (_ , self , _) \u2192\n      let\n        [[n+2]-1]\/2 = suc (proj\u2081 self)\n        half-map    =      proj\u2082 self\n      in if-odd n\n          then ([[n+2]-1]\/2 , \u2115tree (+ 0) half-map \u2205)\n          else ([[n+2]-1]\/2 , \u2115tree (+ 0) \u2205 half-map)\n  }\n-}\n\n\u2115lookup : \u2115 \u2192 NatMap \u2192 \u2124\n\u2115lookup _ \u2205 = (+ 0)\n\u2115lookup 0 (\u2115tree v _ _) = v\n\u2115lookup k (\u2115tree _ left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115lookup [k-1]\/2 left\n... | (even , [k-1]\/2) = \u2115lookup [k-1]\/2 right\n\n-- `\u2192` and `in` are keywords\n\u2115set_\u21d2_within_ : \u2115 \u2192 \u2124 \u2192 NatMap \u2192 NatMap\n\u2115set k \u21d2 v within \u2205 = singleton k v\n\u2115set 0 \u21d2 v within (\u2115tree v\u2080 left right) = \u2115tree v left right\n\u2115set k \u21d2 v within (\u2115tree v\u2080 left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115tree v\u2080 left (\u2115set [k-1]\/2 \u21d2 v within right)\n... | (even , [k-1]\/2) = \u2115tree v\u2080 (\u2115set [k-1]\/2 \u21d2 v within left) right\n\n-- We implement nothing but the necessary NatMap operations to save time:\n-- union, difference, mapValues.\n\n\u2115mapValues : (\u2124 \u2192 \u2124) \u2192 NatMap \u2192 NatMap\n\u2115mapValues _ \u2205 = \u2205\n\u2115mapValues f (\u2115tree v left right) =\n  \u2115tree (f v) (\u2115mapValues f left) (\u2115mapValues f right)\n\nMapBag = NatMap \u00d7 NatMap\n\nlookup : \u2124 \u2192 MapBag \u2192 \u2124\nlookup -[1+ k ] b = \u2115lookup k (proj\u2081 b)\nlookup   (+ k)  b = \u2115lookup k (proj\u2082 b)\n\nset_\u21d2_within_ : \u2124 \u2192 \u2124 \u2192 MapBag \u2192 MapBag\nset -[1+ k ] \u21d2 v within (neg , pos) = (\u2115set k \u21d2 v within neg , pos)\nset    + k   \u21d2 v within (neg , pos) = (neg , \u2115set k \u21d2 v within pos)\n\nmapValues : (\u2124 \u2192 \u2124) \u2192 MapBag \u2192 MapBag\nmapValues f (neg , pos) = (\u2115mapValues f neg , \u2115mapValues f pos)\n\n{-\n\ndata Type : Set where\n  ints : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  int : \u2200 {\u0393} \u2192 (n : \u2124) \u2192 Term \u0393 ints\n  bag : \u2200 {\u0393} \u2192 (b : MapBag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 ints) \u2192 (t\u2082 : Term \u0393 ints) \u2192 Term \u0393 ints\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (ints \u21d2 ints)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to ints = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (int x) = int x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 ints \u27e7Type = \u2124\n\u27e6 bags \u27e7Type = MapBag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 int n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapValues (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (int n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 = cong\u2082 mapValues (weaken-sound f \u03c1)\n                                           (weaken-sound b \u03c1)\n\n-- Changes to \u2124 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type ints = (ints \u21d2 ints \u21d2 ints) \u21d2 ints\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\u27e6snd\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\n\u27e6derive\u27e7 {ints} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = \u2205\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {ints} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = {!!}\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {ints} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = {!!}\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {ints} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = {!!}\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {ints} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = {!!}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {ints} n = refl\nf\u2295\u0394f=f {bags} b = {!!}\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {ints} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = {!!}\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {ints} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = {!!}\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (int n) = abs (app (app (var this) (int n)) (int n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\nderive _ = {!!}\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case int: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {ints} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {bags} b db valid-b-db = {!!}\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (int n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {ints} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = {!!}\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (int n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"49d4d79844d886e9a42816b68ac8c38d2ec28bc1","subject":"Syntax.Type.Plotkin: prove that lift-\u0394type is not a monadic bind.","message":"Syntax.Type.Plotkin: prove that lift-\u0394type is not a monadic bind.\n\nOld-commit-hash: 68f7ee53aed3f5811b2bafa2deef7f8b4349c6b3\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Type\/Plotkin.agda","new_file":"Syntax\/Type\/Plotkin.agda","new_contents":"module Syntax.Type.Plotkin where\n\n-- Types for language description \u00e0 la Plotkin (LCF as PL)\n--\n-- Given base types, we build function types.\n\ninfixr 5 _\u21d2_\n\ndata Type (B : Set {- of base types -}) : Set where\n  base : (\u03b9 : B) \u2192 Type B\n  _\u21d2_ : (\u03c3 : Type B) \u2192 (\u03c4 : Type B) \u2192 Type B\n\n-- Lift (\u0394 : B \u2192 Type B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift-\u0394type : \u2200 {B} \u2192 (B \u2192 Type B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type f (base \u03b9) = f \u03b9\nlift-\u0394type f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift-\u0394type f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n\n-- Note: the above is *not* a monadic bind.\n--\n-- Proof. base` is the most straightforward `return` of the\n-- functor `Type`.\n--\n--     return : B \u2192 Type B\n--     return = base\n--\n-- Let\n--\n--     m : Type B\n--     m = base \u03ba \u21d2 base \u03b9\n--\n-- then\n--\n--     m >>= return = lift-\u0394type return m\n--                  = base \u03ba \u21d2 base \u03ba \u21d2 base \u03b9\n--\n-- violating the second monadic law, m >>= return \u2261 m. \u220e\n\nopen import Function\n\n-- Variant of lift-\u0394type for (\u0394 : B \u2192 B).\nlift-\u0394type\u2080 : \u2200 {B} \u2192 (B \u2192 B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type\u2080 f = lift-\u0394type $ base \u2218 f\n-- This has a similar type to the type of `fmap`,\n-- and `base` has a similar type to the type of `return`.\n--\n-- Similarly, for collections map can be defined from flatMap,\n-- like lift-\u0394type\u2080 can be defined in terms of lift-\u0394type.\n","old_contents":"module Syntax.Type.Plotkin where\n\n-- Types for language description \u00e0 la Plotkin (LCF as PL)\n--\n-- Given base types, we build function types.\n\ninfixr 5 _\u21d2_\n\ndata Type (B : Set {- of base types -}) : Set where\n  base : (\u03b9 : B) \u2192 Type B\n  _\u21d2_ : (\u03c3 : Type B) \u2192 (\u03c4 : Type B) \u2192 Type B\n\n-- Lift (\u0394 : B \u2192 Type B) to (\u0394type : Type B \u2192 Type B)\n-- according to \u0394 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\nlift-\u0394type\u2081 : \u2200 {B} \u2192 (B \u2192 Type B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type\u2081 f (base \u03b9) = f \u03b9\nlift-\u0394type\u2081 f (\u03c3 \u21d2 \u03c4) = let \u0394 = lift-\u0394type\u2081 f in \u03c3 \u21d2 \u0394 \u03c3 \u21d2 \u0394 \u03c4\n\n-- Note: the above is monadic bind with a different argument order.\n\nopen import Function\n\n-- Variant of lift\u2081 for (\u0394 : B \u2192 B).\nlift-\u0394type\u2080 : \u2200 {B} \u2192 (B \u2192 B) \u2192 (Type B \u2192 Type B)\nlift-\u0394type\u2080 f = lift-\u0394type\u2081 $ base \u2218 f\n-- If lift\u2081 is a monadic bind, this is fmap,\n-- and base is return.\n--\n-- Similarly, for collections map can be defined from flatMap, like lift\u2080 can be\n-- defined in terms of lift\u2081.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6c73670338d1e3fca42f15aabc24ef33c2774bd5","subject":"messing around #10","message":"messing around #10\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"commutativity.agda","new_file":"commutativity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\n\nmodule commutativity where\n  commutativity : \u2200{d0 d1 d u} \u2192\n                  -- probably need a premise that d0 is well typed\n                  d0 \u21a6* d1 \u2192\n                  (\u27e6 d \/ u \u27e7 d0) \u21a6* (\u27e6 d \/ u \u27e7 d1)\n  commutativity MSRefl = MSRefl\n  commutativity (MSStep x stp) with commutativity stp\n  ... | ih = MSStep {!!} {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\n\nmodule commutativity where\n  commutativity : \u2200{d0 d1 d u} \u2192\n                  -- probably need a premise that d0 is well typed\n                  d0 \u21a6* d1 \u2192\n                  (\u27e6 d \/ u \u27e7 d0) \u21a6* (\u27e6 d \/ u \u27e7 d1)\n  commutativity = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5d76e4ef03b0213900ffaeb8869c559255671064","subject":"Desc model: add the map operator","message":"Desc model: add the map operator","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms )) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms )) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"eafb25706ba502727c7d1859e45b4de324d4e1f6","subject":"fix parens error in Type.Identity","message":"fix parens error in Type.Identity\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two using (\ud835\udfda ; 0\u2082 ; 1\u2082 ; [0:_1:_]; twist)\nopen import Data.Fin as Fin using (Fin ; suc ; zero)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; Reveal_is_ ; [_]; tr) renaming (refl to idp; cong\u2082 to ap\u2082; _\u2257_ to _\u223c_)\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n-- for use with ap\u2082 etc.\n_\u27f6_ : \u2200 {a b} \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (b \u2294 a)\nA \u27f6 B = A \u2192 B\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u228e B \u2192 \u2605} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : (x : A) \u2192 B x \u2192 \u2605} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {A B : \u2605} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {A B C : \u2605} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}(A : \ud835\udfd9 \u2192 \u2605) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}(A : \u2605) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}(F G : \ud835\udfd8 \u2192 \u2605) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {A : \ud835\udfd9 \u2192 \u2605} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    \u228e\ud835\udfd8-inl : {{_ : UA}} \u2192 A \u2261 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n    \ud835\udfd9\u00d7-snd : {{_ : UA}} \u2192 (\ud835\udfd9 \u00d7 A) \u2261 A\n    \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n    \u00d7\ud835\udfd9-fst : {{_ : UA}} \u2192 (A \u00d7 \ud835\udfd9) \u2261 A\n    \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n    -- old names\n    A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n    \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {A : \ud835\udfda \u2192 \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u228e-\u03a3 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u00d7-\u03a0 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  -- \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n  -- \ud835\udfda\u2192A\u2194A\u00d7A\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2261\ud835\udfd9\u228e : \u2200 {{_ : UA}}\u2192 Maybe A \u2261 (\ud835\udfd9 \u228e A)\n  Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua (equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2)\n      where \u03b2 : (_ : Fin 1) \u2192 _\n            \u03b2 zero = idp\n            \u03b2 (suc ())\n\nmodule _ where\n  isZero? : \u2200 {n}{A : Fin (suc n) \u2192 Set} \u2192 ((i : Fin n) \u2192 A (suc i)) \u2192 A zero\n    \u2192 (i : Fin (suc n)) \u2192 A i\n  isZero? f x zero = x\n  isZero? f x (suc i) = f i\n\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv (isZero? inr (inl _)) [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n    [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n    (isZero? (\u03bb _ \u2192 idp) idp)\n\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero} = ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero} = ap\u2082 _\u00d7_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = ap\u2082 _\u00d7_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero} = ap\u2082 _\u27f6_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = ap\u2082 _\u27f6_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  ap\u2082 _\u228e_ (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 ap\u2082 _\u228e_ (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nmodule _ {{_ : UA}} where\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua (equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp }))\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n{-\nTODO ?\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two using (\ud835\udfda ; 0\u2082 ; 1\u2082 ; [0:_1:_]; twist)\nopen import Data.Fin as Fin using (Fin ; suc ; zero)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; Reveal_is_ ; [_]; tr) renaming (refl to idp; cong\u2082 to ap\u2082; _\u2257_ to _\u223c_)\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n-- for use with ap\u2082 etc.\n_\u27f6_ : \u2200 {a b} \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (b \u2294 a)\nA \u27f6 B = A \u2192 B\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u228e B \u2192 \u2605} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : (x : A) \u2192 B x \u2192 \u2605} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {A B : \u2605} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {A B C : \u2605} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}(A : \ud835\udfd9 \u2192 \u2605) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}(A : \u2605) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}(F G : \ud835\udfd8 \u2192 \u2605) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {A : \ud835\udfd9 \u2192 \u2605} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    \u228e\ud835\udfd8-inl : {{_ : UA}} \u2192 A \u2261 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n    \ud835\udfd9\u00d7-snd : {{_ : UA}} \u2192 (\ud835\udfd9 \u00d7 A) \u2261 A\n    \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n    \u00d7\ud835\udfd9-fst : {{_ : UA}} \u2192 (A \u00d7 \ud835\udfd9) \u2261 A\n    \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n    -- old names\n    A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n    \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {A : \ud835\udfda \u2192 \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u228e-\u03a3 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u00d7-\u03a0 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : \ud835\udfda \u00d7 A \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n  \ud835\udfda\u2192A\u2194A\u00d7A\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2261\ud835\udfd9\u228e : \u2200 {{_ : UA}}\u2192 Maybe A \u2261 (\ud835\udfd9 \u228e A)\n  Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua (equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2)\n      where \u03b2 : (_ : Fin 1) \u2192 _\n            \u03b2 zero = idp\n            \u03b2 (suc ())\n\nmodule _ where\n  isZero? : \u2200 {n}{A : Fin (suc n) \u2192 Set} \u2192 ((i : Fin n) \u2192 A (suc i)) \u2192 A zero\n    \u2192 (i : Fin (suc n)) \u2192 A i\n  isZero? f x zero = x\n  isZero? f x (suc i) = f i\n\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv (isZero? inr (inl _)) [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n    [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n    (isZero? (\u03bb _ \u2192 idp) idp)\n\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero} = ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero} = ap\u2082 _\u00d7_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = ap\u2082 _\u00d7_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero} = ap\u2082 _\u27f6_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = ap\u2082 _\u27f6_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  ap\u2082 _\u228e_ (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 ap\u2082 _\u228e_ (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nmodule _ {{_ : UA}} where\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua (equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp }))\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n{-\nTODO ?\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"053d434b30937a1cd275add324655ea0c27625ee","subject":"induction on slightly different thing to make the ih match the premise, result of 19 july meeting","message":"induction on slightly different thing to make the ih match the premise, result of 19 july meeting\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  -- lemma : \u2200 { \u0393  e \u03c4} \u2192\n  --         \u0393 \u22a2 e => \u2987\u2988 \u2192\n  --         \u0393 \u22a2 e <= \u03c4 ==> \u2987\u2988\n  -- lemma wt = ASubsume wt TCHole1\n\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = _ , _ , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | _ , _ , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-ana wt2\n    ... | d2 , \u03942 , \u03c42 , D2 with expandability-ana (ASubsume wt1 TCHole1)\n    ... | d1 , \u03941 , \u03c41 , D1 =  _ , _ , ESAp1 {!!} wt1 D1 D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081)\n      with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )              | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _  = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = _ , _ , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | _ , _ , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt)\n      with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081)\n      with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )              | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _  = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9a897b02de3ac334ef579145aaec743a52f84ecf","subject":"circuit: Some specs","message":"circuit: Some specs\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec.NP as Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fe1b0ebc0939db24b3806bfd4606e08a57d54b68","subject":"Data.Type->Type","message":"Data.Type->Type\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Text\/Printer.agda","new_file":"lib\/Text\/Printer.agda","new_contents":"module Text.Printer where\n\nopen import Type\nopen import Data.String\nopen import Data.Nat\nopen import Data.Nat.Show\nopen import Data.Bool\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nShowS : \u2605\nShowS = String \u2192 String\n\nPr : \u2605 \u2192 \u2605\nPr A = A \u2192 ShowS\n\n`_ : String \u2192 ShowS\n(` s) tail = Data.String._++_ s tail\n\nparenBase : ShowS \u2192 ShowS\nparenBase doc = ` \"(\" \u2218 doc \u2218 ` \")\"\n\nrecord PrEnv : \u2605 where\n  constructor mk\n  field\n    level : \u2115\n\n  withLevel : \u2115 \u2192 PrEnv\n  withLevel x = record { level = x }\n\nopen PrEnv\n\nparen : PrEnv \u2192 PrEnv \u2192 ShowS \u2192 ShowS\nparen \u0393 \u0394 = if \u230a level \u0393 \u2264? level \u0394 \u230b then id else parenBase\n","old_contents":"module Text.Printer where\n\nopen import Data.Type\nopen import Data.String\nopen import Data.Nat\nopen import Data.Nat.Show\nopen import Data.Bool\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nShowS : \u2605\nShowS = String \u2192 String\n\nPr : \u2605 \u2192 \u2605\nPr A = A \u2192 ShowS\n\n`_ : String \u2192 ShowS\n(` s) tail = Data.String._++_ s tail\n\nparenBase : ShowS \u2192 ShowS\nparenBase doc = ` \"(\" \u2218 doc \u2218 ` \")\"\n\nrecord PrEnv : \u2605 where\n  constructor mk\n  field\n    level : \u2115\n\n  withLevel : \u2115 \u2192 PrEnv\n  withLevel x = record { level = x }\n\nopen PrEnv\n\nparen : PrEnv \u2192 PrEnv \u2192 ShowS \u2192 ShowS\nparen \u0393 \u0394 = if \u230a level \u0393 \u2264? level \u0394 \u230b then id else parenBase\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9f6aedf21fc179011cac644c9caaa069ed07e28c","subject":"Extract ECC\/ecc as Algebra.DoubleAndAdd in agda-nplib","message":"Extract ECC\/ecc as Algebra.DoubleAndAdd in agda-nplib\n","repos":"crypto-agda\/crypto-agda","old_file":"ECC\/ecc.agda","new_file":"ECC\/ecc.agda","new_contents":"","old_contents":"open import Relation.Binary.PropositionalEquality.NP\nopen import Data.Two.Base\nopen import Data.List\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen Implicits\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Ring\n\nmodule ecc\n  {\ud835\udd3d : Set} (\ud835\udd3d-ring : Ring \ud835\udd3d)\n  {\u2119 : Set} (\u2119-monoid : Commutative-Monoid \u2119)\n  where\n\nopen module \ud835\udd3d = Ring \ud835\udd3d-ring\n\nopen module \u2295 = Commutative-Monoid \u2119-monoid\n  renaming\n    ( _\u2219_ to _\u2295_\n    ; \u2219= to \u2295=\n    ; \u03b5\u2219-identity to \u03b5\u2295-identity\n    ; \u2219\u03b5-identity to \u2295\u03b5-identity\n    ; _\u00b2 to 2\u00b7_\n    )\n\n2\u00b7-\u2295-distr : \u2200 {P Q} \u2192 2\u00b7 (P \u2295 Q) \u2261 2\u00b7 P \u2295 2\u00b7 Q\n2\u00b7-\u2295-distr = \u2295.interchange\n\n2\u00b7-\u2295 : \u2200 {P Q R} \u2192 2\u00b7 P \u2295 (Q \u2295 R) \u2261 (P \u2295 Q) \u2295 (P \u2295 R)\n2\u00b7-\u2295 = \u2295.interchange\n\n-- NOT used yet\nmultiply-bin : List \ud835\udfda \u2192 \u2119 \u2192 \u2119\nmultiply-bin scalar P = go scalar\n  where\n    go : List \ud835\udfda \u2192 \u2119\n    go []       = P\n    go (b \u2237 bs) = [0: x\u2080 1: x\u2081 ] b\n      where x\u2080 = 2\u00b7 go bs\n            x\u2081 = P \u2295 x\u2080\n\n{-\nmultiply : \ud835\udd3d \u2192 \u2119 \u2192 \u2119\nmultiply scalar P =\n  -- if scalar == 0 or scalar >= N: raise Exception(\"Invalid Scalar\/Private Key\")\n    multiply-bin (bin scalar) P\n\n_\u00b7_ = multiply\ninfixr 8 _\u00b7_\n-}\n\ninfixl 7 1+2*_\n1+2*_ = \u03bb x \u2192 1+ 2* x\n\ndata Parity-View : \ud835\udd3d \u2192 Set where\n  zero\u27e8_\u27e9    : \u2200 {n} \u2192 n \u2261 0# \u2192 Parity-View n\n  even_by\u27e8_\u27e9 : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 2* m    \u2192 Parity-View n\n  odd_by\u27e8_\u27e9  : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 1+ 2* m \u2192 Parity-View n\n\ncast_by\u27e8_\u27e9 : \u2200 {x y} \u2192 Parity-View x \u2192 y \u2261 x \u2192 Parity-View y\ncast zero\u27e8 x\u2091 \u27e9       by\u27e8 y\u2091 \u27e9 = zero\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast even x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = even x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast odd  x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = odd  x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\n\ninfixr 8 _\u00b7\u209a_\n_\u00b7\u209a_ : \u2200 {n} (p : Parity-View n) \u2192 \u2119 \u2192 \u2119\nzero\u27e8 e \u27e9      \u00b7\u209a P = \u03b5\neven p by\u27e8 e \u27e9 \u00b7\u209a P = 2\u00b7 (p \u00b7\u209a P)\nodd  p by\u27e8 e \u27e9 \u00b7\u209a P = P \u2295 (2\u00b7 (p \u00b7\u209a P))\n\n_+2*_ : \ud835\udfda \u2192 \ud835\udd3d \u2192 \ud835\udd3d\n0\u2082 +2* m =   2* m\n1\u2082 +2* m = 1+2* m\n\n{-\npostulate\n    bin-2* : \u2200 {n} \u2192 bin (2* n) \u2261 0\u2082 \u2237 bin n\n    bin-1+2* : \u2200 {n} \u2192 bin (1+2* n) \u2261 1\u2082 \u2237 bin n\n\nbin-+2* : (b : \ud835\udfda)(n : \ud835\udd3d) \u2192 bin (b +2* n) \u2261 b \u2237 bin n\nbin-+2* 1\u2082 n = bin-1+2*\nbin-+2* 0\u2082 n = bin-2*\n-}\n\n-- (msb) most significant bit first\nbin\u209a : \u2200 {n} \u2192 Parity-View n \u2192 List \ud835\udfda\nbin\u209a zero\u27e8 e \u27e9      = []\nbin\u209a even p by\u27e8 e \u27e9 = 0\u2082 \u2237 bin\u209a p\nbin\u209a odd  p by\u27e8 e \u27e9 = 1\u2082 \u2237 bin\u209a p\n\nhalf : \u2200 {n} \u2192 Parity-View n \u2192 \ud835\udd3d\nhalf zero\u27e8 _ \u27e9            = 0#\nhalf (even_by\u27e8_\u27e9 {m} _ _) = m\nhalf (odd_by\u27e8_\u27e9  {m} _ _) = m\n\n{-\nbin-parity : \u2200 {n} (p : ParityView n) \u2192 bin n \u2261 parity p \u2237 bin (half p)\nbin-parity (even n) = bin-2*\nbin-parity (odd n)  = bin-1+2*\n-}\n\ninfixl 6 1+\u209a_ _+\u209a_\n1+\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (1+ x)\n1+\u209a zero\u27e8 e \u27e9      = odd zero\u27e8 refl \u27e9 by\u27e8 ap 1+_ (e \u2219 ! 0+-identity) \u27e9\n1+\u209a even p by\u27e8 e \u27e9 = odd p      by\u27e8 ap 1+_ e \u27e9\n1+\u209a odd  p by\u27e8 e \u27e9 = even 1+\u209a p by\u27e8 ap 1+_ e \u2219 ! +-assoc \u2219 +-interchange \u27e9\n\n_+\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x + y)\nzero\u27e8 x\u2091 \u27e9       +\u209a y\u209a        = cast y\u209a by\u27e8 ap (\u03bb z \u2192 z + _) x\u2091 \u2219 0+-identity \u27e9\nx\u209a              +\u209a zero\u27e8 y\u2091 \u27e9 = cast x\u209a by\u27e8 ap (_+_ _) y\u2091 \u2219 +-comm \u2219 0+-identity \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even     x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-interchange \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-comm \u2219 +-assoc \u2219 ap 1+_ (+-comm \u2219 +-interchange) \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-assoc \u2219 ap 1+_ +-interchange \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = even 1+\u209a (x\u209a +\u209a y\u209a) by\u27e8 += x\u2091 y\u2091 \u2219 +-on-sides refl +-interchange \u27e9\n\ninfixl 7 2*\u209a_\n2*\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (2* x)\n2*\u209a x\u209a = x\u209a +\u209a x\u209a\n\nopen \u2261-Reasoning\n\nmodule _ {P Q} where\n    \u00b7\u209a-\u2295-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a (P \u2295 Q) \u2261 x\u209a \u00b7\u209a P \u2295 x\u209a \u00b7\u209a Q\n    \u00b7\u209a-\u2295-distr zero\u27e8 x\u2091 \u27e9       = ! \u03b5\u2295-identity\n    \u00b7\u209a-\u2295-distr even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u2295-distr x\u209a) \u2219 2\u00b7-\u2295-distr\n    \u00b7\u209a-\u2295-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap (\u03bb z \u2192 P \u2295 Q \u2295 2\u00b7 z) (\u00b7\u209a-\u2295-distr x\u209a)\n                               \u2219 ap (\u03bb z \u2192 P \u2295 Q \u2295 z) (! 2\u00b7-\u2295)\n                               \u2219 \u2295.interchange\n\nmodule _ {P} where\n    cast-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u2091 : y \u2261 x) \u2192 cast x\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P\n    cast-\u00b7\u209a-distr zero\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n\n    1+\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (1+\u209a x\u209a) \u00b7\u209a P \u2261 P \u2295 x\u209a \u00b7\u209a P\n    1+\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9       = ap (_\u2295_ P) \u03b5\u2295-identity\n    1+\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 = refl\n    1+\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (1+\u209a-\u00b7\u209a-distr x\u209a) \u2219 \u2295.interchange \u2219 \u2295.assoc\n\n    +\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y)\n                 \u2192 (x\u209a +\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P\n    +\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = cast-\u00b7\u209a-distr y\u209a _ \u2219 ! \u03b5\u2295-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 \u2295.assoc-comm\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 ! \u2295.assoc\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9\n       = (odd x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd y\u209a by\u27e8 y\u2091 \u27e9) \u00b7\u209a P\n       \u2261\u27e8by-definition\u27e9\n         2\u00b7((1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P)\n       \u2261\u27e8 ap 2\u00b7_ helper \u27e9\n         2\u00b7(P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 ap (_\u2295_ (2\u00b7 P)) 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P) \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295 \u27e9\n         P \u2295 2\u00b7(x\u209a \u00b7\u209a P) \u2295 (P \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8by-definition\u27e9\n         odd x\u209a by\u27e8 x\u2091 \u27e9 \u00b7\u209a P \u2295 odd y\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P\n       \u220e\n        where helper = (1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P\n                     \u2261\u27e8 1+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) \u27e9\n                       P \u2295 ((x\u209a +\u209a y\u209a) \u00b7\u209a P)\n                     \u2261\u27e8 ap (_\u2295_ P) (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u27e9\n                       P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P)\n                     \u220e\n\n*-1+-distr : \u2200 {x y} \u2192 x * (1+ y) \u2261 x + x * y\n*-1+-distr = *-+-distr\u02e1 \u2219 += *1-identity refl\n\n1+-*-distr : \u2200 {x y} \u2192 (1+ x) * y \u2261 y + x * y\n1+-*-distr = *-+-distr\u02b3 \u2219 += 1*-identity refl\n\ninfixl 7 _*\u209a_\n_*\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x * y)\nzero\u27e8 x\u2091 \u27e9       *\u209a y\u209a              = zero\u27e8 *= x\u2091 refl \u2219 0*-zero \u27e9\nx\u209a              *\u209a zero\u27e8 y\u2091 \u27e9       = zero\u27e8 *= refl y\u2091 \u2219 *0-zero \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 *\u209a y\u209a              = even (x\u209a *\u209a y\u209a) by\u27e8 *= x\u2091 refl \u2219 *-+-distr\u02b3 \u27e9\nx\u209a              *\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even (x\u209a *\u209a y\u209a) by\u27e8 *= refl y\u2091 \u2219 *-+-distr\u02e1 \u27e9\nodd x\u209a by\u27e8 x\u2091 \u27e9  *\u209a odd y\u209a by\u27e8 y\u2091 \u27e9  = odd  (x\u209a +\u209a y\u209a +\u209a 2*\u209a (x\u209a *\u209a y\u209a)) by\u27e8 *= x\u2091 y\u2091 \u2219 helper \u27e9\n   where\n     x = _\n     y = _\n     helper = (1+2* x)*(1+2* y)\n            \u2261\u27e8 1+-*-distr \u27e9\n              1+2* y + 2* x * 1+2* y\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2* y + z)\n                 (2* x * 1+2* y\n                 \u2261\u27e8 *-1+-distr \u27e9\n                 (2* x + 2* x * 2* y)\n                 \u2261\u27e8 += refl *-+-distr\u02e1 \u2219 +-interchange \u27e9\n                 (2* (x + 2* x * y))\n                 \u220e) \u27e9\n              1+2* y + 2* (x + 2* x * y)\n            \u2261\u27e8 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n              1+2*(y + (x + 2* x * y))\n            \u2261\u27e8 ap 1+2*_ (! +-assoc \u2219 += +-comm refl) \u27e9\n              1+2*(x + y + 2* x * y)\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2*(x + y + z)) *-+-distr\u02b3 \u27e9\n              1+2*(x + y + 2* (x * y))\n            \u220e\n\nmodule _ {P} where\n    2\u00b7-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 2\u00b7(x\u209a \u00b7\u209a P) \u2261 x\u209a \u00b7\u209a 2\u00b7 P\n    2\u00b7-\u00b7\u209a-distr x\u209a = ! \u00b7\u209a-\u2295-distr x\u209a\n\n    2*\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (2*\u209a x\u209a) \u00b7\u209a P \u2261 2\u00b7(x\u209a \u00b7\u209a P)\n    2*\u209a-\u00b7\u209a-distr x\u209a = +\u209a-\u00b7\u209a-distr x\u209a x\u209a\n\n\u00b7\u209a-\u03b5 : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a \u03b5 \u2261 \u03b5\n\u00b7\u209a-\u03b5 zero\u27e8 x\u2091 \u27e9       = refl\n\u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\u00b7\u209a-\u03b5 odd x\u209a by\u27e8 x\u2091 \u27e9  = \u03b5\u2295-identity \u2219 ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\n*\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y) {P} \u2192 (x\u209a *\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a y\u209a \u00b7\u209a P\n*\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = refl\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 odd  x\u209a by\u27e8 x\u2091 \u27e9\n\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a even y\u209a by\u27e8 y\u2091 \u27e9)\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9)\n\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2091 \u27e9 y\u209a) \u2219 2\u00b7-\u2295-distr \u2219 ap (\u03bb z \u2192 2\u00b7 (y\u209a \u00b7\u209a P) \u2295 2\u00b7 z) (2\u00b7-\u00b7\u209a-distr x\u209a)\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap (_\u2295_ P)\n     (ap 2\u00b7_\n        (+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) (2*\u209a (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) (2*\u209a-\u00b7\u209a-distr (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= \u2295.comm (ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u00b7\u209a-distr x\u209a) \u2219 \u2295.assoc\n        \u2219 \u2295= refl (! \u00b7\u209a-\u2295-distr x\u209a) ) \u2219 2\u00b7-\u2295-distr)\n     \u2219 \u2295.!assoc\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5ee6d5ce31819d87d8744774d89d87955150f01e","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nltLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 lt m o \u2261 lt n o\nltLeftCong refl = refl\n\nltRightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 lt m n \u2261 lt m o\nltRightCong refl = refl\n\n------------------------------------------------------------------------------\n-- N.B. The elimination properties are in the module\n-- FOTC.Data.Nat.Inequalities.EliminationProperties.\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 n \u2265 zero\nx\u22650 nzero          = lt-0S zero\nx\u22650 (nsucc {n} Nn) = lt-0S (succ\u2081 n)\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 zero \u2264 n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 zero \u226f n\n0\u226fx nzero          = lt-00\n0\u226fx (nsucc {n} Nn) = lt-S0 n\n\nx\u226ex : \u2200 {n} \u2192 N n \u2192 n \u226e n\nx\u226ex nzero          = lt-00\nx\u226ex (nsucc {n} Nn) = trans (lt-SS n n) (x\u226ex Nn)\n\nSx\u22700 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2270 zero\nSx\u22700 nzero          = x\u226ex (nsucc nzero)\nSx\u22700 (nsucc {n} Nn) = trans (lt-SS (succ\u2081 n) zero) (lt-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 n < succ\u2081 n\nx<Sx nzero          = lt-0S zero\nx<Sx (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x<Sx Nn)\n\npostulate x<y\u2192Sx<Sy : \u2200 {m n} \u2192 m < n \u2192 succ\u2081 m < succ\u2081 n\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate Sx<Sy\u2192x<y : \u2200 {m n} \u2192 succ\u2081 m < succ\u2081 n \u2192 m < n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m < succ\u2081 n\nx<y\u2192x<Sy Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x<Sy nzero          (nsucc {n} Nn) 0<Sn  = lt-0S (succ\u2081 n)\nx<y\u2192x<Sy (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 n \u2264 n\nx\u2264x nzero          = lt-0S zero\nx\u2264x (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 n \u2265 n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 m \u2264 n \u2192 succ\u2081 m \u2264 succ\u2081 n\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (lt-SS m (succ\u2081 n)) m\u2264n\n\npostulate Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 succ\u2081 m \u2264 succ\u2081 n \u2192 m \u2264 n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nSx\u2264y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2264 n \u2192 m \u2264 n\nSx\u2264y\u2192x\u2264y {m} {n} Nm Nn Sm\u2264n = x<y\u2192x<Sy Nm Nn (trans (sym (lt-SS m n)) Sm\u2264n)\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 m \u2270 n \u2192 succ\u2081 m \u2270 succ\u2081 n\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (lt-SS m (succ\u2081 n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 n < m\nx>y\u2192y<x nzero          Nn             0>n   = \u22a5-elim (0>x\u2192\u22a5 Nn 0>n)\nx>y\u2192y<x (nsucc {m} Nm) nzero          _     = lt-0S m\nx>y\u2192y<x (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn =\n  trans (lt-SS n m) (x>y\u2192y<x Nm Nn (trans (sym (lt-SS n m)) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 m \u226e n\nx\u2265y\u2192x\u226ey nzero          nzero          _     = x\u226ex nzero\nx\u2265y\u2192x\u226ey nzero          (nsucc Nn)     0\u2265Sn  = \u22a5-elim (0\u2265S\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) nzero          _     = lt-S0 m\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn =\n  prf (x\u2265y\u2192x\u226ey Nm Nn (trans (sym (lt-SS n (succ\u2081 m))) Sm\u2265Sn))\n  where postulate prf : m \u226e n \u2192 succ\u2081 m \u226e succ\u2081 n\n        {-# ATP prove prf #-}\n\nx\u226ey\u2192x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226e n \u2192 m \u2265 n\nx\u226ey\u2192x\u2265y nzero nzero 0\u226e0 = x\u2265x nzero\nx\u226ey\u2192x\u2265y nzero (nsucc {n} Nn) 0\u226eSn =\n  \u22a5-elim (true\u2262false (trans (sym (lt-0S n)) 0\u226eSn))\nx\u226ey\u2192x\u2265y (nsucc Nm) nzero Sm\u226en = x\u22650 (nsucc Nm)\nx\u226ey\u2192x\u2265y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226eSn =\n  prf (x\u226ey\u2192x\u2265y Nm Nn (trans (sym (lt-SS m n)) Sm\u226eSn))\n  where postulate prf : m \u2265 n \u2192 succ\u2081 m \u2265 succ\u2081 n\n        {-# ATP prove prf #-}\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 m \u2270 n\nx>y\u2192x\u2270y nzero          Nn             0>m   = \u22a5-elim (0>x\u2192\u22a5 Nn 0>m)\nx>y\u2192x\u2270y (nsucc Nm)     nzero          _     = Sx\u22700 Nm\nx>y\u2192x\u2270y (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym (lt-SS n m)) Sm>Sn))\n\npostulate x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 m \u2264 n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u2264 n\nx>y\u2228x\u2264y nzero          Nn             = inj\u2082 (x\u22650 Nn)\nx>y\u2228x\u2264y (nsucc {m} Nm) nzero          = inj\u2081 (lt-0S m)\nx>y\u2228x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb m>n \u2192 inj\u2081 (trans (lt-SS n m) m>n))\n       (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n       (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u2265 n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx<y\u2228x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u226e n\nx<y\u2228x\u226ey Nm Nn = case (\u03bb m<n \u2192 inj\u2081 m<n)\n                     (\u03bb m\u2265n \u2192 inj\u2082 (x\u2265y\u2192x\u226ey Nm Nn m\u2265n))\n                     (x<y\u2228x\u2265y Nm Nn)\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2228 m \u2270 n\nx\u2264y\u2228x\u2270y nzero          Nn             = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (nsucc Nm)     nzero          = inj\u2082 (Sx\u22700 Nm)\nx\u2264y\u2228x\u2270y (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n       (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n       (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 m \u2264 n\nx\u2261y\u2192x\u2264y {n = n} Nm Nn m\u2261n = subst (\u03bb m' \u2192 m' \u2264 n) (sym m\u2261n) (x\u2264x Nn)\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m \u2264 n\nx<y\u2192x\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x\u2264y nzero          (nsucc {n} Nn) _     = lt-0S (succ\u2081 n)\nx<y\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m \u2264 n\nx<Sy\u2192x\u2264y nzero      Nn 0<Sn  = 0\u2264x Nn\nx<Sy\u2192x\u2264y (nsucc Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m  < succ\u2081 n\nx\u2264y\u2192x<Sy {n = n} nzero      Nn 0\u2264n  = lt-0S n\nx\u2264y\u2192x<Sy         (nsucc Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 m \u2264 succ\u2081 m\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (nsucc Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 succ\u2081 m \u2264 n\nx<y\u2192Sx\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192Sx\u2264y nzero          (nsucc Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn = trans (lt-SS (succ\u2081 m) (succ\u2081 n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2264 n \u2192 m < n\nSx\u2264y\u2192x<y Nm             nzero          Sm\u22640   = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nSx\u2264y\u2192x<y nzero          (nsucc {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (nsucc {m} Nm) (nsucc {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m \u226f n\nx\u2264y\u2192x\u226fy nzero          Nn             0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (nsucc Nm)     nzero          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn))\n  where postulate prf : m \u226f n \u2192 succ\u2081 m \u226f succ\u2081 n\n        {-# ATP prove prf #-}\n\nx\u226fy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226f n \u2192 m \u2264 n\nx\u226fy\u2192x\u2264y nzero Nn _ = 0\u2264x Nn\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) nzero Sm\u226f0 = \u22a5-elim (true\u2262false (trans (sym (lt-0S m)) Sm\u226f0))\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226fSn =\n  prf (x\u226fy\u2192x\u2264y Nm Nn (trans (sym (lt-SS n m)) Sm\u226fSn))\n  where postulate prf : m \u2264 n \u2192 succ\u2081 m \u2264 succ\u2081 n\n        {-# ATP prove prf #-}\n\nSx\u226fy\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u226f n \u2192 m \u226f n\nSx\u226fy\u2192x\u226fy Nm Nn Sm\u2264n = x\u2264y\u2192x\u226fy Nm Nn (Sx\u2264y\u2192x\u2264y Nm Nn (x\u226fy\u2192x\u2264y (nsucc Nm) Nn Sm\u2264n))\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u226f n\nx>y\u2228x\u226fy nzero          Nn             = inj\u2082 (0\u226fx Nn)\nx>y\u2228x\u226fy (nsucc {m} Nm) nzero          = inj\u2081 (lt-0S m)\nx>y\u2228x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb h \u2192 inj\u2081 (trans (lt-SS n m) h))\n       (\u03bb h \u2192 inj\u2082 (trans (lt-SS n m) h))\n       (x>y\u2228x\u226fy Nm Nn)\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m < n \u2192 n < o \u2192 m < o\n<-trans nzero          nzero          _              0<0   _     = \u22a5-elim (0<0\u2192\u22a5 0<0)\n<-trans nzero          (nsucc Nn)     nzero          _     Sn<0  = \u22a5-elim (S<0\u2192\u22a5 Sn<0)\n<-trans nzero          (nsucc Nn)     (nsucc {o} No) _     _     = lt-0S o\n<-trans (nsucc Nm)     Nn             nzero          _     n<0   = \u22a5-elim (x<0\u2192\u22a5 Nn n<0)\n<-trans (nsucc Nm)     nzero          (nsucc No)     Sm<0  _     = \u22a5-elim (S<0\u2192\u22a5 Sm<0)\n<-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy (<-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So))\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2264 n \u2192 n \u2264 o \u2192 m \u2264 o\n\u2264-trans nzero          Nn             No             _     _     = 0\u2264x No\n\u2264-trans (nsucc Nm)     nzero          No             Sm\u22640  _     = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\n\u2264-trans (nsucc Nm)     (nsucc Nn)     nzero          _     Sn\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nn Sn\u22640)\n\u2264-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\npred-\u2264 : \u2200 {n} \u2192 N n \u2192 pred\u2081 n \u2264 n\npred-\u2264 nzero = prf\n  where postulate prf : pred\u2081 zero \u2264 zero\n        {-# ATP prove prf #-}\n\npred-\u2264 (nsucc {n} Nn) = prf\n  where postulate prf : pred\u2081 (succ\u2081 n) \u2264 succ\u2081 n\n        {-# ATP prove prf <-trans x<Sx #-}\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 m + n\nx\u2264x+y         nzero          Nn = x\u22650 (+-N nzero Nn)\nx\u2264x+y {n = n} (nsucc {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where postulate prf : m \u2264 m + n \u2192 succ\u2081 m \u2264 succ\u2081 m + n\n        {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < m + succ\u2081 n\nx<x+Sy {n = n} nzero Nn = prf0\n  where postulate prf0 : zero < zero + succ\u2081 n\n        {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (nsucc {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where postulate prfS : m < m + succ\u2081 n \u2192 succ\u2081 m < succ\u2081 m + succ\u2081 n\n        {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 k + m < k + n \u2192 m < n\nk+x<k+y\u2192x<y {m} {n} Nm Nn nzero h = prf0\n  where postulate prf0 : m < n\n        {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (nsucc {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where postulate prfS : k + m < k + n\n        {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        m + k < n + k \u2192 m < n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 m \u2264 n \u2192 k + m \u2264 k + n\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn nzero h = prf0\n  where\n  postulate prf0 : zero + m \u2264 zero + n\n  {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (nsucc {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n  postulate prfS : k + m \u2264 k + n \u2192 succ\u2081 k + m \u2264 succ\u2081 k + n\n  {-# ATP prove prfS #-}\n\npostulate\n  x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 m \u2264 n \u2192 m + k \u2264 n + k\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 succ\u2081 m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm nzero h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 nzero (nsucc {n} Nn) h = prf0S\n  where postulate prf0S : succ\u2081 zero \u2238 succ\u2081 n \u2261 zero\n        {-# ATP prove prf0S S\u2238S 0\u2238x #-}\nx<y\u2192Sx\u2238y\u22610 (nsucc {m} Nm) (nsucc {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n  postulate m<n : m < n\n  {-# ATP prove m<n #-}\n\n  postulate prfSS : succ\u2081 m \u2238 n \u2261 zero \u2192 succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n \u2261 zero\n  {-# ATP prove prfSS S\u2238S #-}\n\npostulate x\u2264y\u2192x\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m \u2238 n \u2261 zero\n{-# ATP prove x\u2264y\u2192x\u2238y\u22610 x<y\u2192Sx\u2238y\u22610 S\u2238S #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 zero < n \u2238 m\nx<y\u21920<y\u2238x Nm nzero h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x nzero (nsucc {n} Nn) h = prf0S\n  where postulate prf0S : zero < succ\u2081 n \u2238 zero\n        {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (nsucc {m} Nm) (nsucc {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n  postulate m<n : m < n\n  {-# ATP prove m<n #-}\n\n  postulate prfSS : zero < n \u2238 m \u2192 zero < succ\u2081 n \u2238 succ\u2081 m\n  {-# ATP prove prfSS S\u2238S #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 zero < m \u2238 n \u2192 zero < succ\u2081 m \u2238 n\n0<x\u2238y\u21920<Sx\u2238y {m} Nm nzero h = prfx0\n  where\n  postulate prfx0 : zero < succ\u2081 m \u2238 zero\n  {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y nzero (nsucc {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 nzero h')\n  where postulate h' : zero < zero\n        {-# ATP prove h' 0\u2238x #-}\n\n0<x\u2238y\u21920<Sx\u2238y (nsucc {m} Nm) (nsucc {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n  postulate 0<m-n : zero < m \u2238 n\n  {-# ATP prove 0<m-n S\u2238S #-}\n\n  postulate prfSS : zero < succ\u2081 m \u2238 n \u2192 zero < succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n\n  {-# ATP prove prfSS <-trans S\u2238S #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2238 n < succ\u2081 m\nx\u2238y<Sx {m} Nm nzero = prf\n  where postulate prf : m \u2238 zero < succ\u2081 m\n        {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx nzero (nsucc {n} Nn) = prf\n  where postulate prf : zero \u2238 succ\u2081 n < succ\u2081 zero\n        {-# ATP prove prf 0\u2238x #-}\n\nx\u2238y<Sx (nsucc {m} Nm) (nsucc {n} Nn) = prf (x\u2238y<Sx Nm Nn)\n  where postulate prf : m \u2238 n < succ\u2081 m \u2192\n                        succ\u2081 m \u2238 succ\u2081 n < succ\u2081 (succ\u2081 m)\n        {-# ATP prove prf <-trans \u2238-N x<Sx S\u2238S #-}\n\npostulate Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n < succ\u2081 m\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx S\u2238S #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (m < m \u2238 n)\nx<x\u2238y\u2192\u22a5 {m} Nm nzero m<m\u22380 = prf\n  where postulate prf : \u22a5\n        {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 nzero (nsucc Nn) 0<0\u2238Sn = prf\n where postulate prf : \u22a5\n       {-# ATP prove prf x<x\u2192\u22a5 0\u2238x #-}\nx<x\u2238y\u2192\u22a5 (nsucc Nm) (nsucc Nn) Sm<Sm\u2238Sn = prf\n  where postulate prf : \u22a5\n        {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx S\u2238S #-}\n\npostulate x\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2238 succ\u2081 n \u2264 m \u2238 n\n{-# ATP prove x\u2238Sy\u2264x\u2238y pred-\u2264 \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x nzero          Nn 0>n = \u22a5-elim (0>x\u2192\u22a5 Nn 0>n)\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) nzero Sm>0 = prf\n  where postulate prf : (succ\u2081 m \u2238 zero) + zero \u2261 succ\u2081 m\n        {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn = prf (x>y\u2192x\u2238y+y\u2261x Nm Nn m>n)\n  where\n  postulate m>n : m > n\n  {-# ATP prove m>n #-}\n\n  postulate prf : (m \u2238 n) + n \u2261 m \u2192 (succ\u2081 m \u2238 succ\u2081 n) + succ\u2081 n \u2261 succ\u2081 m\n  {-# ATP prove prf +-comm \u2238-N S\u2238S #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} nzero Nn 0\u2264n = prf\n  where postulate prf : (n \u2238 zero) + zero \u2261 n\n        {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc Nm) nzero Sm\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\n\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n)\n  where\n  postulate m\u2264n : m \u2264 n\n  {-# ATP prove m\u2264n #-}\n\n  postulate prf : (n \u2238 m) + m \u2261 n \u2192 (succ\u2081 n \u2238 succ\u2081 m) + succ\u2081 m \u2261 succ\u2081 n\n  {-# ATP prove prf +-comm \u2238-N S\u2238S #-}\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m < n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y nzero nzero 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y nzero (nsucc {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) nzero Sm<S0 =\n  \u22a5-elim (x<0\u2192\u22a5 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm<SSn =\n  case (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n       (\u03bb m\u2261n \u2192 inj\u2082 (succCong m\u2261n))\n       m<n\u2228m\u2261n\n\n  where\n  m<n\u2228m\u2261n : m < n \u2228 m \u2261 n\n  m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m < n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 m < n \u2192 n \u2261 o \u2192 m < o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 n < o \u2192 m < o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u226fSy\u2192x\u226fy\u2228x\u2261Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226f succ\u2081 n \u2192 m \u226f n \u2228 m \u2261 succ\u2081 n\nx\u226fSy\u2192x\u226fy\u2228x\u2261Sy {m} {n} Nm Nn m\u226fSn =\n  case (\u03bb m<Sn \u2192 inj\u2081 (x\u2264y\u2192x\u226fy Nm Nn (x<Sy\u2192x\u2264y Nm Nn m<Sn)))\n       (\u03bb m\u2261Sn \u2192 inj\u2082 m\u2261Sn)\n       (x<Sy\u2192x<y\u2228x\u2261y Nm (nsucc Nn) (x\u2264y\u2192x<Sy Nm (nsucc Nn) (x\u226fy\u2192x\u2264y Nm (nsucc Nn) m\u226fSn)))\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 n > zero \u2192 m \u2238 n < m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm             nzero          _     0>0  = \u22a5-elim (x>x\u2192\u22a5 nzero 0>0)\nx\u2265y\u2192y>0\u2192x\u2238y<x nzero          (nsucc Nn)     0\u2265Sn  _    = \u22a5-elim (S\u22640\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192y>0\u2192x\u2238y<x (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where postulate prf : succ\u2081 m \u2238 succ\u2081 n < succ\u2081 m\n        {-# ATP prove prf x\u2238y<Sx 0\u2238x S\u2238S #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m < n \u2192 n \u2264 o \u2192 m < o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o = case (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n                                    (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n                                    (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n + (m \u2238 n)\nx\u2264y+x\u2238y {n = n} nzero Nn = prf0\n  where postulate prf0 : zero \u2264 n + (zero \u2238 n)\n        {-# ATP prove prf0 0\u2264x +-N #-}\nx\u2264y+x\u2238y (nsucc {m} Nm) nzero = prfx0\n  where postulate prfx0 : succ\u2081 m \u2264 zero + (succ\u2081 m \u2238 zero)\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (nsucc {m} Nm) (nsucc {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : m \u2264 n + (m \u2238 n) \u2192\n                          succ\u2081 m \u2264 succ\u2081 n + (succ\u2081 m \u2238 succ\u2081 n)\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N S\u2238S #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    m \u2238 n < m \u2238 o \u2192 succ\u2081 m \u2238 n < succ\u2081 m \u2238 o\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} nzero Nn No 0\u2238n<0\u2238o = prf\n  where postulate prf : succ\u2081 zero \u2238 n < succ\u2081 zero \u2238 o\n        {-# ATP prove prf 0\u2238x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (nsucc {m} Nm) nzero No Sm\u22380<Sm\u2238o = prf\n  where postulate prf : succ\u2081 (succ\u2081 m) \u2238 zero < succ\u2081 (succ\u2081 m) \u2238 o\n        {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (nsucc {m} Nm) (nsucc {n} Nn) nzero Sm\u2238Sn<Sm\u22380 = prf\n  where postulate prf : succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n < succ\u2081 (succ\u2081 m) \u2238 zero\n        {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n  postulate\n    prf : (m \u2238 n < m \u2238 o \u2192 succ\u2081 m \u2238 n < succ\u2081 m \u2238 o) \u2192\n          succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n < succ\u2081 (succ\u2081 m) \u2238 succ\u2081 o\n  {-# ATP prove prf S\u2238S #-}\n\n------------------------------------------------------------------------------\n-- Properties about the lexicographical order\n\npostulate xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (Lexi m n zero zero)\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate 0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 \u00ac (Lexi zero (succ\u2081 m) zero zero)\n{-# ATP prove 0Sx<00\u2192\u22a5 S<0\u2192\u22a5 #-}\n\npostulate Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 \u00ac (Lexi (succ\u2081 m) n\u2081 zero n\u2082)\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 #-}\n\npostulate x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n} \u2192 N n \u2192 \u2200 {m\u2082} \u2192 Lexi m\u2081 n m\u2082 zero \u2192 m\u2081 < m\u2082\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m} \u2192 N m \u2192 \u2200 {n\u2081 n\u2082} \u2192 Lexi m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 n\u2081 < n\u2082\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) (succ\u2081 m) (succ\u2081 n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n  postulate prf : Lexi (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) (succ\u2081 m) (succ\u2081 n)\n  {-# ATP prove prf x\u2238y<Sx S\u2238S #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m) (succ\u2081 n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n  postulate prf : Lexi (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m) (succ\u2081 n)\n  {-# ATP prove prf x\u2238y<Sx S\u2238S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\nltLeftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 lt m o \u2261 lt n o\nltLeftCong refl = refl\n\nltRightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 lt m n \u2261 lt m o\nltRightCong refl = refl\n\n------------------------------------------------------------------------------\n-- N.B. The elimination properties are in the module\n-- FOTC.Data.Nat.Inequalities.EliminationProperties.\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 n \u2265 zero\nx\u22650 nzero          = lt-0S zero\nx\u22650 (nsucc {n} Nn) = lt-0S (succ\u2081 n)\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 zero \u2264 n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 zero \u226f n\n0\u226fx nzero          = lt-00\n0\u226fx (nsucc {n} Nn) = lt-S0 n\n\nx\u226ex : \u2200 {n} \u2192 N n \u2192 n \u226e n\nx\u226ex nzero          = lt-00\nx\u226ex (nsucc {n} Nn) = trans (lt-SS n n) (x\u226ex Nn)\n\nSx\u22700 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2270 zero\nSx\u22700 nzero          = x\u226ex (nsucc nzero)\nSx\u22700 (nsucc {n} Nn) = trans (lt-SS (succ\u2081 n) zero) (lt-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 n < succ\u2081 n\nx<Sx nzero          = lt-0S zero\nx<Sx (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 m < n \u2192 succ\u2081 m < succ\u2081 n\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 succ\u2081 m < succ\u2081 n \u2192 m < n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m < succ\u2081 n\nx<y\u2192x<Sy Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x<Sy nzero          (nsucc {n} Nn) 0<Sn  = lt-0S (succ\u2081 n)\nx<y\u2192x<Sy (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 n \u2264 n\nx\u2264x nzero          = lt-0S zero\nx\u2264x (nsucc {n} Nn) = trans (lt-SS n (succ\u2081 n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 n \u2265 n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 m \u2264 n \u2192 succ\u2081 m \u2264 succ\u2081 n\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (lt-SS m (succ\u2081 n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 succ\u2081 m \u2264 succ\u2081 n \u2192 m \u2264 n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nSx\u2264y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2264 n \u2192 m \u2264 n\nSx\u2264y\u2192x\u2264y {m} {n} Nm Nn Sm\u2264n = x<y\u2192x<Sy Nm Nn (trans (sym (lt-SS m n)) Sm\u2264n)\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 m \u2270 n \u2192 succ\u2081 m \u2270 succ\u2081 n\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (lt-SS m (succ\u2081 n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 n < m\nx>y\u2192y<x nzero          Nn             0>n   = \u22a5-elim (0>x\u2192\u22a5 Nn 0>n)\nx>y\u2192y<x (nsucc {m} Nm) nzero          _     = lt-0S m\nx>y\u2192y<x (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn =\n  trans (lt-SS n m) (x>y\u2192y<x Nm Nn (trans (sym (lt-SS n m)) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 m \u226e n\nx\u2265y\u2192x\u226ey nzero          nzero          _     = x\u226ex nzero\nx\u2265y\u2192x\u226ey nzero          (nsucc Nn)     0\u2265Sn  = \u22a5-elim (0\u2265S\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) nzero          _     = lt-S0 m\nx\u2265y\u2192x\u226ey (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn =\n  prf (x\u2265y\u2192x\u226ey Nm Nn (trans (sym (lt-SS n (succ\u2081 m))) Sm\u2265Sn))\n  where postulate prf : m \u226e n \u2192 succ\u2081 m \u226e succ\u2081 n\n        {-# ATP prove prf #-}\n\nx\u226ey\u2192x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226e n \u2192 m \u2265 n\nx\u226ey\u2192x\u2265y nzero nzero 0\u226e0 = x\u2265x nzero\nx\u226ey\u2192x\u2265y nzero (nsucc {n} Nn) 0\u226eSn =\n  \u22a5-elim (true\u2262false (trans (sym (lt-0S n)) 0\u226eSn))\nx\u226ey\u2192x\u2265y (nsucc Nm) nzero Sm\u226en = x\u22650 (nsucc Nm)\nx\u226ey\u2192x\u2265y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226eSn =\n  prf (x\u226ey\u2192x\u2265y Nm Nn (trans (sym (lt-SS m n)) Sm\u226eSn))\n  where postulate prf : m \u2265 n \u2192 succ\u2081 m \u2265 succ\u2081 n\n        {-# ATP prove prf #-}\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 m \u2270 n\nx>y\u2192x\u2270y nzero          Nn             0>m   = \u22a5-elim (0>x\u2192\u22a5 Nn 0>m)\nx>y\u2192x\u2270y (nsucc Nm)     nzero          _     = Sx\u22700 Nm\nx>y\u2192x\u2270y (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym (lt-SS n m)) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 m \u2264 n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u2264 n\nx>y\u2228x\u2264y nzero          Nn             = inj\u2082 (x\u22650 Nn)\nx>y\u2228x\u2264y (nsucc {m} Nm) nzero          = inj\u2081 (lt-0S m)\nx>y\u2228x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb m>n \u2192 inj\u2081 (trans (lt-SS n m) m>n))\n       (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n       (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u2265 n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx<y\u2228x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2228 m \u226e n\nx<y\u2228x\u226ey Nm Nn = case (\u03bb m<n \u2192 inj\u2081 m<n)\n                     (\u03bb m\u2265n \u2192 inj\u2082 (x\u2265y\u2192x\u226ey Nm Nn m\u2265n))\n                     (x<y\u2228x\u2265y Nm Nn)\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2228 m \u2270 n\nx\u2264y\u2228x\u2270y nzero          Nn             = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (nsucc Nm)     nzero          = inj\u2082 (Sx\u22700 Nm)\nx\u2264y\u2228x\u2270y (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n       (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n       (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 m \u2264 n\nx\u2261y\u2192x\u2264y {n = n} Nm Nn m\u2261n = subst (\u03bb m' \u2192 m' \u2264 n) (sym m\u2261n) (x\u2264x Nn)\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 m \u2264 n\nx<y\u2192x\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192x\u2264y nzero          (nsucc {n} Nn) _     = lt-0S (succ\u2081 n)\nx<y\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m \u2264 n\nx<Sy\u2192x\u2264y nzero      Nn 0<Sn  = 0\u2264x Nn\nx<Sy\u2192x\u2264y (nsucc Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m  < succ\u2081 n\nx\u2264y\u2192x<Sy {n = n} nzero      Nn 0\u2264n  = lt-0S n\nx\u2264y\u2192x<Sy         (nsucc Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 m \u2264 succ\u2081 m\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (nsucc Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 succ\u2081 m \u2264 n\nx<y\u2192Sx\u2264y Nm             nzero          m<0   = \u22a5-elim (x<0\u2192\u22a5 Nm m<0)\nx<y\u2192Sx\u2264y nzero          (nsucc Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm<Sn = trans (lt-SS (succ\u2081 m) (succ\u2081 n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2264 n \u2192 m < n\nSx\u2264y\u2192x<y Nm             nzero          Sm\u22640   = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nSx\u2264y\u2192x<y nzero          (nsucc {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (nsucc {m} Nm) (nsucc {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m \u226f n\nx\u2264y\u2192x\u226fy nzero          Nn             0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (nsucc Nm)     nzero          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm\u2264Sn))\n  where postulate prf : m \u226f n \u2192 succ\u2081 m \u226f succ\u2081 n\n        {-# ATP prove prf #-}\n\nx\u226fy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226f n \u2192 m \u2264 n\nx\u226fy\u2192x\u2264y nzero Nn _ = 0\u2264x Nn\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) nzero Sm\u226f0 = \u22a5-elim (true\u2262false (trans (sym (lt-0S m)) Sm\u226f0))\nx\u226fy\u2192x\u2264y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u226fSn =\n  prf (x\u226fy\u2192x\u2264y Nm Nn (trans (sym (lt-SS n m)) Sm\u226fSn))\n  where postulate prf : m \u2264 n \u2192 succ\u2081 m \u2264 succ\u2081 n\n        {-# ATP prove prf #-}\n\nSx\u226fy\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u226f n \u2192 m \u226f n\nSx\u226fy\u2192x\u226fy Nm Nn Sm\u2264n = x\u2264y\u2192x\u226fy Nm Nn (Sx\u2264y\u2192x\u2264y Nm Nn (x\u226fy\u2192x\u2264y (nsucc Nm) Nn Sm\u2264n))\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2228 m \u226f n\nx>y\u2228x\u226fy nzero          Nn             = inj\u2082 (0\u226fx Nn)\nx>y\u2228x\u226fy (nsucc {m} Nm) nzero          = inj\u2081 (lt-0S m)\nx>y\u2228x\u226fy (nsucc {m} Nm) (nsucc {n} Nn) =\n  case (\u03bb h \u2192 inj\u2081 (trans (lt-SS n m) h))\n       (\u03bb h \u2192 inj\u2082 (trans (lt-SS n m) h))\n       (x>y\u2228x\u226fy Nm Nn)\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m < n \u2192 n < o \u2192 m < o\n<-trans nzero          nzero          _              0<0   _     = \u22a5-elim (0<0\u2192\u22a5 0<0)\n<-trans nzero          (nsucc Nn)     nzero          _     Sn<0  = \u22a5-elim (S<0\u2192\u22a5 Sn<0)\n<-trans nzero          (nsucc Nn)     (nsucc {o} No) _     _     = lt-0S o\n<-trans (nsucc Nm)     Nn             nzero          _     n<0   = \u22a5-elim (x<0\u2192\u22a5 Nn n<0)\n<-trans (nsucc Nm)     nzero          (nsucc No)     Sm<0  _     = \u22a5-elim (S<0\u2192\u22a5 Sm<0)\n<-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy (<-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So))\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2264 n \u2192 n \u2264 o \u2192 m \u2264 o\n\u2264-trans nzero          Nn             No             _     _     = 0\u2264x No\n\u2264-trans (nsucc Nm)     nzero          No             Sm\u22640  _     = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\n\u2264-trans (nsucc Nm)     (nsucc Nn)     nzero          _     Sn\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nn Sn\u22640)\n\u2264-trans (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\npred-\u2264 : \u2200 {n} \u2192 N n \u2192 pred\u2081 n \u2264 n\npred-\u2264 nzero = prf\n  where postulate prf : pred\u2081 zero \u2264 zero\n        {-# ATP prove prf #-}\n\npred-\u2264 (nsucc {n} Nn) = prf\n  where postulate prf : pred\u2081 (succ\u2081 n) \u2264 succ\u2081 n\n        {-# ATP prove prf <-trans x<Sx #-}\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 m + n\nx\u2264x+y         nzero          Nn = x\u22650 (+-N nzero Nn)\nx\u2264x+y {n = n} (nsucc {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where postulate prf : m \u2264 m + n \u2192 succ\u2081 m \u2264 succ\u2081 m + n\n        {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < m + succ\u2081 n\nx<x+Sy {n = n} nzero Nn = prf0\n  where postulate prf0 : zero < zero + succ\u2081 n\n        {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (nsucc {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where postulate prfS : m < m + succ\u2081 n \u2192 succ\u2081 m < succ\u2081 m + succ\u2081 n\n        {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 k + m < k + n \u2192 m < n\nk+x<k+y\u2192x<y {m} {n} Nm Nn nzero h = prf0\n  where postulate prf0 : m < n\n        {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (nsucc {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where postulate prfS : k + m < k + n\n        {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        m + k < n + k \u2192 m < n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 m \u2264 n \u2192 k + m \u2264 k + n\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn nzero h = prf0\n  where\n  postulate prf0 : zero + m \u2264 zero + n\n  {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (nsucc {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n  postulate prfS : k + m \u2264 k + n \u2192 succ\u2081 k + m \u2264 succ\u2081 k + n\n  {-# ATP prove prfS #-}\n\npostulate\n  x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 m \u2264 n \u2192 m + k \u2264 n + k\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 succ\u2081 m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm nzero h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 nzero (nsucc {n} Nn) h = prf0S\n  where postulate prf0S : succ\u2081 zero \u2238 succ\u2081 n \u2261 zero\n        {-# ATP prove prf0S S\u2238S 0\u2238x #-}\nx<y\u2192Sx\u2238y\u22610 (nsucc {m} Nm) (nsucc {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n  postulate m<n : m < n\n  {-# ATP prove m<n #-}\n\n  postulate prfSS : succ\u2081 m \u2238 n \u2261 zero \u2192 succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n \u2261 zero\n  {-# ATP prove prfSS S\u2238S #-}\n\npostulate x\u2264y\u2192x\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m \u2238 n \u2261 zero\n{-# ATP prove x\u2264y\u2192x\u2238y\u22610 x<y\u2192Sx\u2238y\u22610 S\u2238S #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < n \u2192 zero < n \u2238 m\nx<y\u21920<y\u2238x Nm nzero h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x nzero (nsucc {n} Nn) h = prf0S\n  where postulate prf0S : zero < succ\u2081 n \u2238 zero\n        {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (nsucc {m} Nm) (nsucc {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n  postulate m<n : m < n\n  {-# ATP prove m<n #-}\n\n  postulate prfSS : zero < n \u2238 m \u2192 zero < succ\u2081 n \u2238 succ\u2081 m\n  {-# ATP prove prfSS S\u2238S #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 zero < m \u2238 n \u2192 zero < succ\u2081 m \u2238 n\n0<x\u2238y\u21920<Sx\u2238y {m} Nm nzero h = prfx0\n  where\n  postulate prfx0 : zero < succ\u2081 m \u2238 zero\n  {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y nzero (nsucc {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 nzero h')\n  where postulate h' : zero < zero\n        {-# ATP prove h' 0\u2238x #-}\n\n0<x\u2238y\u21920<Sx\u2238y (nsucc {m} Nm) (nsucc {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n  postulate 0<m-n : zero < m \u2238 n\n  {-# ATP prove 0<m-n S\u2238S #-}\n\n  postulate prfSS : zero < succ\u2081 m \u2238 n \u2192 zero < succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n\n  {-# ATP prove prfSS <-trans S\u2238S #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2238 n < succ\u2081 m\nx\u2238y<Sx {m} Nm nzero = prf\n  where postulate prf : m \u2238 zero < succ\u2081 m\n        {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx nzero (nsucc {n} Nn) = prf\n  where postulate prf : zero \u2238 succ\u2081 n < succ\u2081 zero\n        {-# ATP prove prf 0\u2238x #-}\n\nx\u2238y<Sx (nsucc {m} Nm) (nsucc {n} Nn) = prf (x\u2238y<Sx Nm Nn)\n  where postulate prf : m \u2238 n < succ\u2081 m \u2192\n                        succ\u2081 m \u2238 succ\u2081 n < succ\u2081 (succ\u2081 m)\n        {-# ATP prove prf <-trans \u2238-N x<Sx S\u2238S #-}\n\npostulate Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n < succ\u2081 m\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx S\u2238S #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (m < m \u2238 n)\nx<x\u2238y\u2192\u22a5 {m} Nm nzero m<m\u22380 = prf\n  where postulate prf : \u22a5\n        {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 nzero (nsucc Nn) 0<0\u2238Sn = prf\n where postulate prf : \u22a5\n       {-# ATP prove prf x<x\u2192\u22a5 0\u2238x #-}\nx<x\u2238y\u2192\u22a5 (nsucc Nm) (nsucc Nn) Sm<Sm\u2238Sn = prf\n  where postulate prf : \u22a5\n        {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx S\u2238S #-}\n\npostulate x\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2238 succ\u2081 n \u2264 m \u2238 n\n{-# ATP prove x\u2238Sy\u2264x\u2238y pred-\u2264 \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m > n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x nzero          Nn 0>n = \u22a5-elim (0>x\u2192\u22a5 Nn 0>n)\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) nzero Sm>0 = prf\n  where postulate prf : (succ\u2081 m \u2238 zero) + zero \u2261 succ\u2081 m\n        {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (nsucc {m} Nm) (nsucc {n} Nn) Sm>Sn = prf (x>y\u2192x\u2238y+y\u2261x Nm Nn m>n)\n  where\n  postulate m>n : m > n\n  {-# ATP prove m>n #-}\n\n  postulate prf : (m \u2238 n) + n \u2261 m \u2192 (succ\u2081 m \u2238 succ\u2081 n) + succ\u2081 n \u2261 succ\u2081 m\n  {-# ATP prove prf +-comm \u2238-N S\u2238S #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} nzero Nn 0\u2264n = prf\n  where postulate prf : (n \u2238 zero) + zero \u2261 n\n        {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc Nm) nzero Sm\u22640 = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\n\nx\u2264y\u2192y\u2238x+x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n)\n  where\n  postulate m\u2264n : m \u2264 n\n  {-# ATP prove m\u2264n #-}\n\n  postulate prf : (n \u2238 m) + m \u2261 n \u2192 (succ\u2081 n \u2238 succ\u2081 m) + succ\u2081 m \u2261 succ\u2081 n\n  {-# ATP prove prf +-comm \u2238-N S\u2238S #-}\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m < succ\u2081 n \u2192 m < n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y nzero nzero 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y nzero (nsucc {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) nzero Sm<S0 =\n  \u22a5-elim (x<0\u2192\u22a5 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (nsucc {m} Nm) (nsucc {n} Nn) Sm<SSn =\n  case (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n       (\u03bb m\u2261n \u2192 inj\u2082 (succCong m\u2261n))\n       m<n\u2228m\u2261n\n\n  where\n  m<n\u2228m\u2261n : m < n \u2228 m \u2261 n\n  m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ\u2081 n))) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n \u2192 m < n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 m < n \u2192 n \u2261 o \u2192 m < o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 n < o \u2192 m < o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u226fSy\u2192x\u226fy\u2228x\u2261Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u226f succ\u2081 n \u2192 m \u226f n \u2228 m \u2261 succ\u2081 n\nx\u226fSy\u2192x\u226fy\u2228x\u2261Sy {m} {n} Nm Nn m\u226fSn =\n  case (\u03bb m<Sn \u2192 inj\u2081 (x\u2264y\u2192x\u226fy Nm Nn (x<Sy\u2192x\u2264y Nm Nn m<Sn)))\n       (\u03bb m\u2261Sn \u2192 inj\u2082 m\u2261Sn)\n       (x<Sy\u2192x<y\u2228x\u2261y Nm (nsucc Nn) (x\u2264y\u2192x<Sy Nm (nsucc Nn) (x\u226fy\u2192x\u2264y Nm (nsucc Nn) m\u226fSn)))\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2265 n \u2192 n > zero \u2192 m \u2238 n < m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm             nzero          _     0>0  = \u22a5-elim (x>x\u2192\u22a5 nzero 0>0)\nx\u2265y\u2192y>0\u2192x\u2238y<x nzero          (nsucc Nn)     0\u2265Sn  _    = \u22a5-elim (S\u22640\u2192\u22a5 Nn 0\u2265Sn)\nx\u2265y\u2192y>0\u2192x\u2238y<x (nsucc {m} Nm) (nsucc {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where postulate prf : succ\u2081 m \u2238 succ\u2081 n < succ\u2081 m\n        {-# ATP prove prf x\u2238y<Sx 0\u2238x S\u2238S #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m < n \u2192 n \u2264 o \u2192 m < o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o = case (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n                                    (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n                                    (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m \u2264 n + (m \u2238 n)\nx\u2264y+x\u2238y {n = n} nzero Nn = prf0\n  where postulate prf0 : zero \u2264 n + (zero \u2238 n)\n        {-# ATP prove prf0 0\u2264x +-N #-}\nx\u2264y+x\u2238y (nsucc {m} Nm) nzero = prfx0\n  where postulate prfx0 : succ\u2081 m \u2264 zero + (succ\u2081 m \u2238 zero)\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (nsucc {m} Nm) (nsucc {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : m \u2264 n + (m \u2238 n) \u2192\n                          succ\u2081 m \u2264 succ\u2081 n + (succ\u2081 m \u2238 succ\u2081 n)\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N S\u2238S #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    m \u2238 n < m \u2238 o \u2192 succ\u2081 m \u2238 n < succ\u2081 m \u2238 o\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} nzero Nn No 0\u2238n<0\u2238o = prf\n  where postulate prf : succ\u2081 zero \u2238 n < succ\u2081 zero \u2238 o\n        {-# ATP prove prf 0\u2238x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (nsucc {m} Nm) nzero No Sm\u22380<Sm\u2238o = prf\n  where postulate prf : succ\u2081 (succ\u2081 m) \u2238 zero < succ\u2081 (succ\u2081 m) \u2238 o\n        {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (nsucc {m} Nm) (nsucc {n} Nn) nzero Sm\u2238Sn<Sm\u22380 = prf\n  where postulate prf : succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n < succ\u2081 (succ\u2081 m) \u2238 zero\n        {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n  postulate\n    prf : (m \u2238 n < m \u2238 o \u2192 succ\u2081 m \u2238 n < succ\u2081 m \u2238 o) \u2192\n          succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n < succ\u2081 (succ\u2081 m) \u2238 succ\u2081 o\n  {-# ATP prove prf S\u2238S #-}\n\n------------------------------------------------------------------------------\n-- Properties about the lexicographical order\n\npostulate xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (Lexi m n zero zero)\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate 0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 \u00ac (Lexi zero (succ\u2081 m) zero zero)\n{-# ATP prove 0Sx<00\u2192\u22a5 S<0\u2192\u22a5 #-}\n\npostulate Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 \u00ac (Lexi (succ\u2081 m) n\u2081 zero n\u2082)\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 #-}\n\npostulate x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n} \u2192 N n \u2192 \u2200 {m\u2082} \u2192 Lexi m\u2081 n m\u2082 zero \u2192 m\u2081 < m\u2082\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m} \u2192 N m \u2192 \u2200 {n\u2081 n\u2082} \u2192 Lexi m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 n\u2081 < n\u2082\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) (succ\u2081 m) (succ\u2081 n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n  postulate prf : Lexi (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) (succ\u2081 m) (succ\u2081 n)\n  {-# ATP prove prf x\u2238y<Sx S\u2238S #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     Lexi (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m) (succ\u2081 n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n  postulate prf : Lexi (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m) (succ\u2081 n)\n  {-# ATP prove prf x\u2238y<Sx S\u2238S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2b39f7d1104bd54771761e91edc271195b1ca64e","subject":"Temporarily disable Explore.GroupHomomorphism","message":"Temporarily disable Explore.GroupHomomorphism\n","repos":"crypto-agda\/explore","old_file":"explore.agda","new_file":"explore.agda","new_contents":"module explore where\nimport Explore.BigDistr\nimport Explore.BinTree\nimport Explore.Core\nimport Explore.Dice\nimport Explore.Examples\nimport Explore.Explorable\nimport Explore.Fin\n-- import Explore.GroupHomomorphism\nimport Explore.GuessingGameFlipping\nimport Explore.Isomorphism\nimport Explore.Monad\nimport Explore.One\nimport Explore.Product\nimport Explore.Properties\nimport Explore.README\nimport Explore.Subset\nimport Explore.Sum\nimport Explore.Summable\nimport Explore.Two\nimport Explore.Universe\nimport Explore.Universe.Base\nimport Explore.Universe.Type\nimport Explore.Zero\n-- BROKEN import Explore.Explorable.Fun\n-- BROKEN import Explore.Function.Fin\n","old_contents":"module explore where\nimport Explore.BigDistr\nimport Explore.BinTree\nimport Explore.Core\nimport Explore.Dice\nimport Explore.Examples\nimport Explore.Explorable\nimport Explore.Fin\nimport Explore.GroupHomomorphism\nimport Explore.GuessingGameFlipping\nimport Explore.Isomorphism\nimport Explore.Monad\nimport Explore.One\nimport Explore.Product\nimport Explore.Properties\nimport Explore.README\nimport Explore.Subset\nimport Explore.Sum\nimport Explore.Summable\nimport Explore.Two\nimport Explore.Universe\nimport Explore.Universe.Base\nimport Explore.Universe.Type\nimport Explore.Zero\n-- BROKEN import Explore.Explorable.Fun\n-- BROKEN import Explore.Function.Fin\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4740a9f99cf4dec031912284adaaae0af78459d6","subject":"remove old file","message":"remove old file\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/peano.agda","new_file":"learyouanagda\/peano.agda","new_contents":"","old_contents":"module peano where\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\nzero + n     = n\n(suc n) + n\u2032 = suc (n + n\u2032)\n\n\ndata _even : \u2115 \u2192 Set where\n  ZERO : zero even\n  STEP : \u2200 x \u2192 x even \u2192 (suc (suc x)) even\n\n\n-- To prove four is even\nproof\u2081 : suc (suc (suc (suc zero))) even\nproof\u2081 = STEP (suc (suc zero)) (STEP zero ZERO)\n\nproof\u2082\u2032 : (A : Set) \u2192 A \u2192 A\nproof\u2082\u2032 _ a = a\n\nproof\u2082 : \u2115 \u2192 \u2115\nproof\u2082 = proof\u2082\u2032 \u2115\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ab8d7492ec07bab4a700a3b58ebefd97c2ed45fc","subject":"upgrade flat-funs-cost (most of it)","message":"upgrade flat-funs-cost (most of it)\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs-cost.agda","new_file":"flat-funs-cost.agda","new_contents":"module flat-funs-cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_)\n\nopen import flat-funs\nopen import flat-funs-implem\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\n{-\nseqTimeOpsD : FlatFunsOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            nil = 0#; cons = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FlatFunsOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n-}\n\nTime = \u2115\nTimeCost = constFuns Time\nSpace = \u2115\nSpaceCost = constFuns Space\n\nseqTimeLin : LinRewiring TimeCost\nseqTimeLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192nil = 0;\n    nil\u2192tt = 0;\n    cons = 0;\n    uncons = 0 }\n\nseqTimeRewiring : Rewiring TimeCost\nseqTimeRewiring =\n  record {\n    linRewiring = seqTimeLin;\n    tt = 0;\n    dup = 0;\n    nil = 0;\n    <_,_> = _+_;\n    fst = 0;\n    snd = 0 }\n\nseqTimeOps : FlatFunsOps TimeCost\nseqTimeOps = record { rewiring = seqTimeRewiring;\n                      <0b> = 0; <1b> = 0; cond = 1; fork = 1+_+_ }\n\ntimeLin : LinRewiring TimeCost\ntimeLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _\u2294_;\n    second = F.id;\n    tt\u2192nil = 0;\n    nil\u2192tt = 0;\n    cons = 0;\n    uncons = 0 }\n\ntimeRewiring : Rewiring TimeCost\ntimeRewiring =\n  record {\n    linRewiring = timeLin;\n    tt = 0;\n    dup = 0;\n    nil = 0;\n    <_,_> = _\u2294_;\n    fst = 0;\n    snd = 0 }\n\ntimeOps : FlatFunsOps TimeCost\ntimeOps = record { rewiring = timeRewiring;\n                   <0b> = 0; <1b> = 0; cond = 1; fork = 1+_\u2294_ }\n\n{-\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                rewiring = record seqTimeRewiring {\n                                             linRewiring = record seqTimeLin { <_\u00d7_> = _\u2294_ };\n                                             <_,_> = _\u2294_};\n                                fork = 1+_\u2294_ }\n                                {-;\n                                cons = 0; uncons = 0 } -- Without cons = 0... this definition makes\n                                                       -- the FlatFunsOps record def yellow\n                                                       -}\ntimeOps\u2261seqTimeOps = \u2261.refl\n-}\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FlatFunsOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            nil = con 0; cons = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FlatFunsOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  open FlatFunsOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u22a4\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec\u22a4 {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u22a4\u2261maximum f [] = \u2261.refl\n  constVec\u22a4\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec\u22a4 f bs) 0\n                                              | constVec\u22a4\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\nspaceLin : LinRewiring SpaceCost\nspaceLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192nil = 0;\n    nil\u2192tt = 0;\n    cons = 0;\n    uncons = 0 }\n\nspaceLin\u2261seqTimeLin : spaceLin \u2261 seqTimeLin\nspaceLin\u2261seqTimeLin = \u2261.refl\n\nspaceRewiring : Rewiring TimeCost\nspaceRewiring =\n  record {\n    linRewiring = spaceLin;\n    tt = 0;\n    dup = 1;\n    nil = 0;\n    <_,_> = 1+_+_;\n    fst = 0;\n    snd = 0 }\n\nspaceOps : FlatFunsOps SpaceCost\nspaceOps = record { rewiring = spaceRewiring;\n                    <0b> = 1; <1b> = 1; cond = 1; fork = 1+_+_ }\n\n             {-\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; cons = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n-}\n\nmodule SpaceOps where\n  open FlatFunsOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u22a4\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec\u22a4 {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u22a4\u2261sum f [] = \u2261.refl\n  constVec\u22a4\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec\u22a4 f bs) 0\n                                          | constVec\u22a4\u2261sum f bs = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f xs rewrite constVec\u22a4\u2261sum f xs | \u2115\u00b0.+-comm (V.sum (V.map f xs)) 0 = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\n{-\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps\n-}","old_contents":"module flat-funs-cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_)\n\nopen import flat-funs\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\nseqTimeOpsD : FlatFunsOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            nil = 0#; cons = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FlatFunsOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n\nseqTimeOps : FlatFunsOps (constFuns \u2115)\nseqTimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1; fork = 1+_+_; tt = 0;\n            <_,_> = _+_; fst = 0; snd = 0;\n            dup = 0; first = F.id; swap = 0; assoc = 0;\n            <tt,id> = 0; snd<tt,> = 0;\n            <_\u00d7_> = _+_; second = F.id;\n            nil = 0; cons = 0; uncons = 0 }\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1; fork = 1+_\u2294_; tt = 0;\n            <_,_> = _\u2294_; fst = 0; snd = 0;\n            dup = 0; first = F.id; swap = 0; assoc = 0;\n            <tt,id> = 0; snd<tt,> = 0;\n            <_\u00d7_> = _\u2294_; second = F.id;\n            nil = 0; cons = 0; uncons = 0 }\n\n\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                <_,_> = _\u2294_; <_\u00d7_> = _\u2294_; fork = 1+_\u2294_\n                              ; cons = 0; uncons = 0 } -- Without cons = 0... this definition makes\n                                                       -- the FlatFunsOps record def yellow\ntimeOps\u2261seqTimeOps = \u2261.refl\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FlatFunsOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            nil = con 0; cons = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FlatFunsOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  Time = \u2115\n  open FlatFunsOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u2261maximum f [] = \u2261.refl\n  constVec\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec f bs) 0\n                                            | constVec\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 1; <1b> = 1; cond = 1; fork = 1+_+_; tt = 0;\n             <_,_> = 1+_+_; fst = 0; snd = 0;\n             dup = 1; first = F.id; swap = 0; assoc = 0;\n             <tt,id> = 0; snd<tt,> = 0;\n             <_\u00d7_> = _+_; second = F.id;\n             nil = 0; cons = 0; uncons = 0 }\n\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; cons = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n\nmodule SpaceOps where\n  Space = \u2115\n  open FlatFunsOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f [] = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec f bs) 0\n                                        | constVec\u2261sum f bs = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : Bits i \u2192 Bits o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"417a1c46e905d1203ecabf95cc49646b7d090d33","subject":"Product: \u0394","message":"Product: \u0394\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Product\/NP.agda","new_file":"lib\/Data\/Product\/NP.agda","new_contents":"-- move this to product\n{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Product.NP where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Data.Product public hiding (\u2203)\nopen import Relation.Binary.PropositionalEquality as \u2261\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nopen import Function.Injection using (Injection; module Injection)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\n\n\u2203 : \u2200 {a b} {A : \u2605 a} \u2192 (A \u2192 \u2605 b) \u2192 \u2605 (a \u2294 b)\n\u2203 = \u03a3 _\n\nfirst : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 \u03a3 A (C \u2218 f) \u2192 \u03a3 B C\nfirst f = map f id -- f (x , y) = (f x , y)\n\n-- generalized first\u2032 but differently than first\nfirst' : \u2200 {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} \u2192\n          (f : (x : A) \u2192 B x) (p : \u03a3 A C) \u2192 B (proj\u2081 p) \u00d7 C (proj\u2081 p)\nfirst' f (x , y) = (f x , y)\n\nfirst\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 A \u00d7 C \u2192 B \u00d7 C\nfirst\u2032 = first\n\nsecond : \u2200 {a p q} {A : \u2605 a} {P : A \u2192 \u2605 p} {Q : A \u2192 \u2605 q} \u2192\n           (\u2200 {x} \u2192 P x \u2192 Q x) \u2192 \u03a3 A P \u2192 \u03a3 A Q\nsecond = map id\n\nsecond\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n           (B \u2192 C) \u2192 A \u00d7 B \u2192 A \u00d7 C\nsecond\u2032 f = second f\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\nrecord [\u03a3] {a b a\u209a b\u209a}\n           {A : \u2605 a}\n           {B : A \u2192 \u2605 b}\n           (A\u209a : A \u2192 \u2605 a\u209a)\n           (B\u209a : {x : A} (x\u209a : A\u209a x)\n                 \u2192 B x \u2192 \u2605 b\u209a)\n           (p : \u03a3 A B) : \u2605 (a\u209a \u2294 b\u209a) where\n  constructor _[,]_\n  field\n    [proj\u2081] : A\u209a (proj\u2081 p)\n    [proj\u2082] : B\u209a [proj\u2081] (proj\u2082 p)\nopen [\u03a3] public\ninfixr 4 _[,]_\n\nsyntax [\u03a3] A\u209a (\u03bb x\u209a \u2192 e) = [ x\u209a \u2236 A\u209a ][\u00d7][ e ]\n\n[\u2203] : \u2200 {a a\u209a b b\u209a} \u2192\n        (\u2200i\u27e8 A\u209a \u2236 [\u2605] {a} a\u209a \u27e9[\u2192] ((A\u209a [\u2192] [\u2605] {b} b\u209a) [\u2192] [\u2605] _)) \u2203\n[\u2203] {x\u209a = A\u209a} = [\u03a3] A\u209a\n\nsyntax [\u2203] (\u03bb A\u209a \u2192 f) = [\u2203][ A\u209a ] f\n\n_[\u00d7]_ : \u2200 {a b a\u209a b\u209a} \u2192 ([\u2605] {a} a\u209a [\u2192] [\u2605] {b} b\u209a [\u2192] [\u2605] _) _\u00d7_\n_[\u00d7]_ A\u209a B\u209a = [\u03a3] A\u209a (\u03bb _ \u2192 B\u209a)\n\nprivate\n\n  Dec\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n               (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n             \u2192 \u2605 _\n  Dec\u27e6\u2605\u27e7 A\u1d63 = \u2200 x\u2081 x\u2082 \u2192 Dec (A\u1d63 x\u2081 x\u2082)\n\n  Dec\u27e6Pred\u27e7 : \u2200 {a\u2081 a\u2082 p\u2081 p\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n                (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n                {P\u2081 : Pred p\u2081 A\u2081} {P\u2082 : Pred p\u2082 A\u2082} {p\u1d63}\n              \u2192 (P\u1d63 : \u27e6Pred\u27e7 p\u1d63 A\u1d63 P\u2081 P\u2082) \u2192 \u2605 _\n  Dec\u27e6Pred\u27e7 {A\u2081 = A\u2081} {A\u2082} A\u1d63 {P\u2081} {P\u2082} P\u1d63 = \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) y\u2081 y\u2082 \u2192 Dec (P\u1d63 x\u1d63 y\u2081 y\u2082) -- Dec\u27e6\u2605\u27e7 A\u1d63 \u21d2 Dec\u27e6\u2605\u27e7 P\u1d63\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n           {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : \u2605 (a\u1d63 \u2294 b\u1d63) where\n  constructor _,_\n  field\n    \u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082)\n    \u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _,_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u27e6\u00d7\u27e7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63}\n        {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n        {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 \u2605 _\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n          {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082) \u00d7\n                        B\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082))\n       (\u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\ndec\u27e6\u03a3\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63 A\u2081 A\u2082 B\u2081 B\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 {b\u2081} {b\u2082} b\u1d63 A\u1d63 B\u2081 B\u2082}\n       (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n       -- (substA\u1d63 : \u2200 {x y \u2113} {p q} (P : A\u1d63 x y \u2192 \u2605 \u2113) \u2192 P p \u2192 P q)\n       (uniqA\u1d63 : \u2200 {x y} (p q : A\u1d63 x y) \u2192 p \u2261 q)\n       (decB\u1d63 : Dec\u27e6Pred\u27e7 A\u1d63 {_} {_} {b\u1d63 {-BUG: with _ here Agda loops-}} B\u1d63)\n     \u2192 Dec\u27e6\u2605\u27e7 (\u27e6\u03a3\u27e7 A\u1d63 B\u1d63)\ndec\u27e6\u03a3\u27e7 {B\u1d63 = B\u1d63} decA\u1d63 uniqA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 x\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 , y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 f \u2218 \u27e6proj\u2082\u27e7)\n  where f : \u2200 {x\u1d63'} \u2192 B\u1d63 x\u1d63' y\u2081 y\u2082 \u2192 B\u1d63 x\u1d63 y\u2081 y\u2082\n        f {x\u1d63'} y\u1d63 rewrite uniqA\u1d63 x\u1d63' x\u1d63 = y\u1d63\n\nmodule Dec\u2082 where\n  map\u2082\u2032 : \u2200 {p\u2081 p\u2082 q} {P\u2081 : \u2605 p\u2081} {P\u2082 : \u2605 p\u2082} {Q : \u2605 q}\n          \u2192 (P\u2081 \u2192 P\u2082 \u2192 Q) \u2192 (Q \u2192 P\u2081) \u2192 (Q \u2192 P\u2082) \u2192 Dec P\u2081 \u2192 Dec P\u2082 \u2192 Dec Q\n  map\u2082\u2032 _   \u03c0\u2081  _   (no \u00acp\u2081)  _         = no (\u00acp\u2081 \u2218 \u03c0\u2081)\n  map\u2082\u2032 _   _   \u03c0\u2082  _         (no \u00acp\u2082)  = no (\u00acp\u2082 \u2218 \u03c0\u2082)\n  map\u2082\u2032 mk  _   _   (yes p\u2081)  (yes p\u2082)  = yes (mk p\u2081 p\u2082)\n\ndec\u27e6\u00d7\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n           {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n           {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082}\n           (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n           (decB\u1d63 : Dec\u27e6\u2605\u27e7 B\u1d63)\n         \u2192 Dec\u27e6\u2605\u27e7 (A\u1d63 \u27e6\u00d7\u27e7 B\u1d63)\ndec\u27e6\u00d7\u27e7 decA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 , y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 \u27e6proj\u2082\u27e7)\n\nmk\u03a3\u2261 : \u2200 {a b} {A : \u2605 a} {x y : A} (B : A \u2192 \u2605 b) {p : B x} {q : B y} (xy : x \u2261 y) \u2192 subst B xy p \u2261 q \u2192 (x \u03a3., p) \u2261 (y , q)\nmk\u03a3\u2261 _ xy h rewrite xy | h = \u2261.refl\n\n\u03a3,-injective\u2082 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x : A} {y z : B x} \u2192 (_,_ {B = B} x y) \u2261 (x \u03a3., z) \u2192 y \u2261 z\n\u03a3,-injective\u2082 refl = refl\n\n\u03a3,-injective\u2081 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x\u2081 x\u2082 : A} {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} \u2192 (x\u2081 \u03a3., y\u2081) \u2261 (x\u2082 , y\u2082) \u2192 x\u2081 \u2261 x\u2082\n\u03a3,-injective\u2081 refl = refl\n\nproj\u2081-injective : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x y : \u03a3 A B}\n                    (B-uniq : \u2200 {z} (p\u2081 p\u2082 : B z) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 proj\u2081 x \u2261 proj\u2081 y \u2192 x \u2261 y\nproj\u2081-injective {x = (a , p\u2081)} {y = (_ , p\u2082)} B-uniq eq rewrite sym eq\n  = cong (\u03bb p \u2192 (a , p)) (B-uniq p\u2081 p\u2082)\n\nproj\u2082-irrelevance : \u2200 {a b} {A : \u2605 a} {B C : A \u2192 \u2605 b} {x\u2081 x\u2082 : A}\n                      {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} {z\u2081 : C x\u2081} {z\u2082 : C x\u2082}\n                    \u2192 (C-uniq : \u2200 {z} (p\u2081 p\u2082 : C z) \u2192 p\u2081 \u2261 p\u2082)\n                    \u2192 (x\u2081 \u03a3., y\u2081) \u2261 (x\u2082 , y\u2082)\n                    \u2192 (x\u2081 \u03a3., z\u2081) \u2261 (x\u2082 , z\u2082)\nproj\u2082-irrelevance C-uniq = proj\u2081-injective C-uniq \u2218 \u03a3,-injective\u2081\n\n\u225f\u03a3 : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (decP : \u2200 x \u2192 Decidable {A = P x} _\u2261_)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl with decP w p\u2081 p\u2082\n\u225f\u03a3 decA decP (w  , p)  (.w , .p) | yes refl | yes refl = yes refl\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl | no  p\u2262\n    = no (p\u2262 \u2218 \u03a3,-injective\u2082)\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\n\u225f\u03a3' : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (uniqP : \u2200 {x} (p q : P x) \u2192 p \u2261 q)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3' decA uniqP (w  , p\u2081) (.w , p\u2082) | yes refl\n  = yes (cong (\u03bb p \u2192 (w , p)) (uniqP p\u2081 p\u2082))\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\nproj\u2081-Injection : \u2200 {a b} {A : \u2605 a} {B : A \u2192  \u2605 b}\n                  \u2192 (\u2200 {x} (p\u2081 p\u2082 : B x) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 Injection (setoid (\u03a3 A B))\n                              (setoid A)\nproj\u2081-Injection {B = B} B-uniq\n     = record { to        = \u2192-to-\u27f6 (proj\u2081 {B = B})\n              ; injective = proj\u2081-injective B-uniq\n              }\n\n\u0394 : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 A \u00d7 A\n\u0394 x = x , x\n","old_contents":"-- move this to product\n{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Product.NP where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Data.Product public hiding (\u2203)\nopen import Relation.Binary.PropositionalEquality as \u2261\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nopen import Function.Injection using (Injection; module Injection)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\n\n\u2203 : \u2200 {a b} {A : \u2605 a} \u2192 (A \u2192 \u2605 b) \u2192 \u2605 (a \u2294 b)\n\u2203 = \u03a3 _\n\nfirst : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 \u03a3 A (C \u2218 f) \u2192 \u03a3 B C\nfirst f = map f id -- f (x , y) = (f x , y)\n\n-- generalized first\u2032 but differently than first\nfirst' : \u2200 {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} \u2192\n          (f : (x : A) \u2192 B x) (p : \u03a3 A C) \u2192 B (proj\u2081 p) \u00d7 C (proj\u2081 p)\nfirst' f (x , y) = (f x , y)\n\nfirst\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 A \u00d7 C \u2192 B \u00d7 C\nfirst\u2032 = first\n\nsecond : \u2200 {a p q} {A : \u2605 a} {P : A \u2192 \u2605 p} {Q : A \u2192 \u2605 q} \u2192\n           (\u2200 {x} \u2192 P x \u2192 Q x) \u2192 \u03a3 A P \u2192 \u03a3 A Q\nsecond = map id\n\nsecond\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n           (B \u2192 C) \u2192 A \u00d7 B \u2192 A \u00d7 C\nsecond\u2032 f = second f\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\nrecord [\u03a3] {a b a\u209a b\u209a}\n           {A : \u2605 a}\n           {B : A \u2192 \u2605 b}\n           (A\u209a : A \u2192 \u2605 a\u209a)\n           (B\u209a : {x : A} (x\u209a : A\u209a x)\n                 \u2192 B x \u2192 \u2605 b\u209a)\n           (p : \u03a3 A B) : \u2605 (a\u209a \u2294 b\u209a) where\n  constructor _[,]_\n  field\n    [proj\u2081] : A\u209a (proj\u2081 p)\n    [proj\u2082] : B\u209a [proj\u2081] (proj\u2082 p)\nopen [\u03a3] public\ninfixr 4 _[,]_\n\nsyntax [\u03a3] A\u209a (\u03bb x\u209a \u2192 e) = [ x\u209a \u2236 A\u209a ][\u00d7][ e ]\n\n[\u2203] : \u2200 {a a\u209a b b\u209a} \u2192\n        (\u2200i\u27e8 A\u209a \u2236 [\u2605] {a} a\u209a \u27e9[\u2192] ((A\u209a [\u2192] [\u2605] {b} b\u209a) [\u2192] [\u2605] _)) \u2203\n[\u2203] {x\u209a = A\u209a} = [\u03a3] A\u209a\n\nsyntax [\u2203] (\u03bb A\u209a \u2192 f) = [\u2203][ A\u209a ] f\n\n_[\u00d7]_ : \u2200 {a b a\u209a b\u209a} \u2192 ([\u2605] {a} a\u209a [\u2192] [\u2605] {b} b\u209a [\u2192] [\u2605] _) _\u00d7_\n_[\u00d7]_ A\u209a B\u209a = [\u03a3] A\u209a (\u03bb _ \u2192 B\u209a)\n\nprivate\n\n  Dec\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n               (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n             \u2192 \u2605 _\n  Dec\u27e6\u2605\u27e7 A\u1d63 = \u2200 x\u2081 x\u2082 \u2192 Dec (A\u1d63 x\u2081 x\u2082)\n\n  Dec\u27e6Pred\u27e7 : \u2200 {a\u2081 a\u2082 p\u2081 p\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n                (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n                {P\u2081 : Pred p\u2081 A\u2081} {P\u2082 : Pred p\u2082 A\u2082} {p\u1d63}\n              \u2192 (P\u1d63 : \u27e6Pred\u27e7 p\u1d63 A\u1d63 P\u2081 P\u2082) \u2192 \u2605 _\n  Dec\u27e6Pred\u27e7 {A\u2081 = A\u2081} {A\u2082} A\u1d63 {P\u2081} {P\u2082} P\u1d63 = \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) y\u2081 y\u2082 \u2192 Dec (P\u1d63 x\u1d63 y\u2081 y\u2082) -- Dec\u27e6\u2605\u27e7 A\u1d63 \u21d2 Dec\u27e6\u2605\u27e7 P\u1d63\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n           {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : \u2605 (a\u1d63 \u2294 b\u1d63) where\n  constructor _,_\n  field\n    \u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082)\n    \u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _,_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u27e6\u00d7\u27e7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63}\n        {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n        {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 \u2605 _\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n          {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082) \u00d7\n                        B\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082))\n       (\u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\ndec\u27e6\u03a3\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63 A\u2081 A\u2082 B\u2081 B\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 {b\u2081} {b\u2082} b\u1d63 A\u1d63 B\u2081 B\u2082}\n       (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n       -- (substA\u1d63 : \u2200 {x y \u2113} {p q} (P : A\u1d63 x y \u2192 \u2605 \u2113) \u2192 P p \u2192 P q)\n       (uniqA\u1d63 : \u2200 {x y} (p q : A\u1d63 x y) \u2192 p \u2261 q)\n       (decB\u1d63 : Dec\u27e6Pred\u27e7 A\u1d63 {_} {_} {b\u1d63 {-BUG: with _ here Agda loops-}} B\u1d63)\n     \u2192 Dec\u27e6\u2605\u27e7 (\u27e6\u03a3\u27e7 A\u1d63 B\u1d63)\ndec\u27e6\u03a3\u27e7 {B\u1d63 = B\u1d63} decA\u1d63 uniqA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 x\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 , y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 f \u2218 \u27e6proj\u2082\u27e7)\n  where f : \u2200 {x\u1d63'} \u2192 B\u1d63 x\u1d63' y\u2081 y\u2082 \u2192 B\u1d63 x\u1d63 y\u2081 y\u2082\n        f {x\u1d63'} y\u1d63 rewrite uniqA\u1d63 x\u1d63' x\u1d63 = y\u1d63\n\nmodule Dec\u2082 where\n  map\u2082\u2032 : \u2200 {p\u2081 p\u2082 q} {P\u2081 : \u2605 p\u2081} {P\u2082 : \u2605 p\u2082} {Q : \u2605 q}\n          \u2192 (P\u2081 \u2192 P\u2082 \u2192 Q) \u2192 (Q \u2192 P\u2081) \u2192 (Q \u2192 P\u2082) \u2192 Dec P\u2081 \u2192 Dec P\u2082 \u2192 Dec Q\n  map\u2082\u2032 _   \u03c0\u2081  _   (no \u00acp\u2081)  _         = no (\u00acp\u2081 \u2218 \u03c0\u2081)\n  map\u2082\u2032 _   _   \u03c0\u2082  _         (no \u00acp\u2082)  = no (\u00acp\u2082 \u2218 \u03c0\u2082)\n  map\u2082\u2032 mk  _   _   (yes p\u2081)  (yes p\u2082)  = yes (mk p\u2081 p\u2082)\n\ndec\u27e6\u00d7\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n           {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n           {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082}\n           (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n           (decB\u1d63 : Dec\u27e6\u2605\u27e7 B\u1d63)\n         \u2192 Dec\u27e6\u2605\u27e7 (A\u1d63 \u27e6\u00d7\u27e7 B\u1d63)\ndec\u27e6\u00d7\u27e7 decA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 , y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 \u27e6proj\u2082\u27e7)\n\nmk\u03a3\u2261 : \u2200 {a b} {A : \u2605 a} {x y : A} (B : A \u2192 \u2605 b) {p : B x} {q : B y} (xy : x \u2261 y) \u2192 subst B xy p \u2261 q \u2192 (x \u03a3., p) \u2261 (y , q)\nmk\u03a3\u2261 _ xy h rewrite xy | h = \u2261.refl\n\n\u03a3,-injective\u2082 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x : A} {y z : B x} \u2192 (_,_ {B = B} x y) \u2261 (x \u03a3., z) \u2192 y \u2261 z\n\u03a3,-injective\u2082 refl = refl\n\n\u03a3,-injective\u2081 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x\u2081 x\u2082 : A} {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} \u2192 (x\u2081 \u03a3., y\u2081) \u2261 (x\u2082 , y\u2082) \u2192 x\u2081 \u2261 x\u2082\n\u03a3,-injective\u2081 refl = refl\n\nproj\u2081-injective : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x y : \u03a3 A B}\n                    (B-uniq : \u2200 {z} (p\u2081 p\u2082 : B z) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 proj\u2081 x \u2261 proj\u2081 y \u2192 x \u2261 y\nproj\u2081-injective {x = (a , p\u2081)} {y = (_ , p\u2082)} B-uniq eq rewrite sym eq\n  = cong (\u03bb p \u2192 (a , p)) (B-uniq p\u2081 p\u2082)\n\nproj\u2082-irrelevance : \u2200 {a b} {A : \u2605 a} {B C : A \u2192 \u2605 b} {x\u2081 x\u2082 : A}\n                      {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} {z\u2081 : C x\u2081} {z\u2082 : C x\u2082}\n                    \u2192 (C-uniq : \u2200 {z} (p\u2081 p\u2082 : C z) \u2192 p\u2081 \u2261 p\u2082)\n                    \u2192 (x\u2081 \u03a3., y\u2081) \u2261 (x\u2082 , y\u2082)\n                    \u2192 (x\u2081 \u03a3., z\u2081) \u2261 (x\u2082 , z\u2082)\nproj\u2082-irrelevance C-uniq = proj\u2081-injective C-uniq \u2218 \u03a3,-injective\u2081\n\n\u225f\u03a3 : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (decP : \u2200 x \u2192 Decidable {A = P x} _\u2261_)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl with decP w p\u2081 p\u2082\n\u225f\u03a3 decA decP (w  , p)  (.w , .p) | yes refl | yes refl = yes refl\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl | no  p\u2262\n    = no (p\u2262 \u2218 \u03a3,-injective\u2082)\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\n\u225f\u03a3' : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (uniqP : \u2200 {x} (p q : P x) \u2192 p \u2261 q)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3' decA uniqP (w  , p\u2081) (.w , p\u2082) | yes refl\n  = yes (cong (\u03bb p \u2192 (w , p)) (uniqP p\u2081 p\u2082))\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\nproj\u2081-Injection : \u2200 {a b} {A : \u2605 a} {B : A \u2192  \u2605 b}\n                  \u2192 (\u2200 {x} (p\u2081 p\u2082 : B x) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 Injection (setoid (\u03a3 A B))\n                              (setoid A)\nproj\u2081-Injection {B = B} B-uniq\n     = record { to        = \u2192-to-\u27f6 (proj\u2081 {B = B})\n              ; injective = proj\u2081-injective B-uniq\n              }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"56460e69fe9711b8fb337c5527a4b79861d48324","subject":"Redid the model -- it is now given as System F-omega plus a kind for time.","message":"Redid the model -- it is now given as System F-omega plus a kind for time.\n","repos":"agda\/agda-frp-js,agda\/agda-frp-js","old_file":"src\/agda\/FRP\/JS\/Model.agda","new_file":"src\/agda\/FRP\/JS\/Model.agda","new_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.Bool using ( true ; false )\nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq lt : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 lt \u27e7     = \u03bb t u \u2192 True (t < u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 lt \u27e7\u00b2     = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n_\u227a_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227a_ = capp\u2082 lt\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time (app tvar\u2081 tvar\u2080))\n\ninterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\ninterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227c tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\nconstreq : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nconstreq {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (\u03a0 time \n  ((tvar\u2081 \u227c tvar\u2080) \u22b8 (app (app (app interval tvar\u2083) tvar\u2081) tvar\u2080 \u22b8 app tvar\u2082 tvar\u2080)))))\n\n_\u22b5_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b5_ = app\u2082 constreq\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  leq-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  leq-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7      As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7       As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7       As = \u03bb A B \u2192 proj\u2082\nc\u27e6 leq-refl \u27e7  As = \u2264-refl\nc\u27e6 leq-trans \u27e7 As = \u2264-trans\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2      \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 leq-refl \u27e7\u00b2  \u211cs = _\nc\u27e6 leq-trans \u27e7\u00b2 \u211cs = _\n\n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app taut \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar\u2080 \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsignal : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsignal T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsignal\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsignal\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsignal-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (signal T s (signal\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsignal-iso T s \u03c3 = refl\n\nsignal-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (signal\u207b\u00b9 T s (signal T s \u03c3) \u2261 \u03c3)\nsignal-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = signal U s (f (signal\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = signal\u207b\u00b9 U s (f (signal T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _)\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 (T at s \u22a8 \u03c3 s \u2248[ u ] \u03c4 s)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192 T\u27ea T \u21d2 U \u27eb s \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U s f = \u2200 t \u03c3 \u03c4 \u2192 \n  (T at s \u22a8 \u03c3 \u2248[ t ] \u03c4) \u2192 (U at s \u22a8 f \u03c3 \u2248[ t ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u a c a\u211cc , \u211c-impl-\u2248 U at s u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u a b a\u2248b))\n  \u211c-impl-\u2248 (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb t s\u2264t t\u2264u \u2192 \u211c-impl-\u2248 T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u a c a\u2248c , \u2248-impl-\u211c U at s u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt t\u2264u with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt t\u2264u | refl = \n      \u2248-impl-\u211c T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u2248\u03c4 t s\u2264t (t\u2264u refl {s\u2264t} {s\u2264t} tt {\u2264-refl t} {\u2264-refl t} tt))\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s \u2192 \u211c-impl-\u2248 T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u211c\u03c4 refl)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl = \u2248-impl-\u211c T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u2248\u03c4 s)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) (M : Exp [] \u27ea [ T \u21d2 U ] \u27eb) s \u2192 \n  Causal T U s (M\u27e6 M \u27e7 (World , tt) tt s)\ncausality T U M s t \n  = \u211c-impl-\u2248 T \u21d2 U at s t \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ t ] , _) tt refl)\n\n","old_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.True using ( True )\nopen import FRP.JS.Nat using ( \u2115 ; zero ; suc )\n\nmodule FRP.JS.Model where\n\n-- Preliminaries\n\ninfixr 4 _+_\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n\n-- Relations on Set\n\n_\u220b_\u2194_ : \u2200 \u03b1 \u2192 Set \u03b1 \u2192 Set \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n\n-- RSets and relations on RSet\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\nRSet\u2080 = RSet o\nRSet\u2081 = RSet (\u2191 o)\n\n_\u220b_\u21d4_ : \u2200 \u03b1 \u2192 RSet \u03b1 \u2192 RSet \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u21d4 B = \u2200 t \u2192 (\u03b1 \u220b A t \u2194 B t)\n\n--- Level sequences\n\ndata Levels : Set where\n  \u03b5 : Levels\n  _,_ : \u2200 (\u03b1s : Levels) (\u03b1 : Level) \u2192 Levels\n\nmax : Levels \u2192 Level\nmax \u03b5 = o\nmax (\u03b1s , \u03b1) = max \u03b1s \u2294 \u03b1\n\n-- Sequences of Sets\n\nSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nSets \u03b5        = \u22a4\nSets (\u03b1s , \u03b1) = Sets \u03b1s \u00d7 Set \u03b1\n\n\u27e8_\u220b_\u27e9 : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Set (max \u03b1s)\n\u27e8 \u03b5 \u220b tt \u27e9 = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (As , A) \u27e9 = \u27e8 \u03b1s \u220b As \u27e9 \u00d7 A\n\n_\u220b_\u2194*_ : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Sets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u2194* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u2194* (Bs , B) = (\u03b1s \u220b As \u2194* Bs) \u00d7 (\u03b1 \u220b A \u2194 B)\n\n\u27e8_\u220b_\u27e9\u00b2 : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u2194* Bs) \u2192 (max \u03b1s \u220b \u27e8 \u03b1s \u220b As \u27e9 \u2194 \u27e8 \u03b1s \u220b Bs \u27e9)\n\u27e8 \u03b5        \u220b tt       \u27e9\u00b2 tt       tt       = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (\u211cs , \u211c) \u27e9\u00b2 (as , a) (bs , b) = (\u27e8 \u03b1s \u220b \u211cs \u27e9\u00b2 as bs) \u00d7 (\u211c a b)\n\n-- Sequences of RSets\n\nRSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nRSets \u03b5        = \u22a4\nRSets (\u03b1s , \u03b1) = RSets \u03b1s \u00d7 RSet \u03b1\n\n_\u220b_\u21d4*_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 RSets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u21d4* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u21d4* (Bs , B) = (\u03b1s \u220b As \u21d4* Bs) \u00d7 (\u03b1 \u220b A \u21d4 B)\n\n-- Concatenation of sequences\n\n_+_ : Levels \u2192 Levels \u2192 Levels\n\u03b1s + \u03b5        = \u03b1s\n\u03b1s + (\u03b2s , \u03b2) = (\u03b1s + \u03b2s) , \u03b2\n\n_\u220b_++_\u220b_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 \u2200 \u03b2s \u2192 RSets \u03b2s \u2192 RSets (\u03b1s + \u03b2s)\n\u03b1s \u220b As ++ \u03b5        \u220b tt       = As\n\u03b1s \u220b As ++ (\u03b2s , \u03b2) \u220b (Bs , B) = ((\u03b1s \u220b As ++ \u03b2s \u220b Bs) , B)\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u21d4* Bs) \u2192 \u2200 \u03b2s {Cs Ds} \u2192 (\u03b2s \u220b Cs \u21d4* Ds) \u2192 \n  ((\u03b1s + \u03b2s) \u220b (\u03b1s \u220b As ++ \u03b2s \u220b Cs) \u21d4* (\u03b1s \u220b Bs ++ \u03b2s \u220b Ds))\n\u03b1s \u220b \u211cs ++\u00b2 \u03b5        \u220b tt       = \u211cs\n\u03b1s \u220b \u211cs ++\u00b2 (\u03b2s , \u03b2) \u220b (\u2111s , \u2111) = ((\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) , \u2111)\n\n-- Intervals\n\n_[_,_] : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 Time \u2192 Time \u2192 Set \u03b1\nA [ s , u ] = \u2200 t \u2192 True (s \u2264 t) \u2192 True (t \u2264 u) \u2192 A t\n\n_[_,_]\u00b2 : \u2200 {\u03b1 A B} \u2192 (\u03b1 \u220b A \u21d4 B) \u2192 \u2200 s u \u2192 (\u03b1 \u220b A [ s , u ] \u2194 B [ s , u ])\n(\u211c [ s , u ]\u00b2) \u03c3 \u03c4 = \u2200 t s\u2264t t\u2264u \u2192 \u211c t (\u03c3 t s\u2264t t\u2264u) (\u03c4 t s\u2264t t\u2264u)\n\n-- Type variables\n\ndata TVar : Levels \u2192 Set\u2081 where\n  zero : \u2200 {\u03b1s \u03b1} \u2192 TVar (\u03b1s , \u03b1)\n  suc : \u2200 {\u03b1s \u03b1} \u2192 (\u03c4 : TVar \u03b1s) \u2192 TVar (\u03b1s , \u03b1)\n\n\u03c4level : \u2200 {\u03b1s} \u2192 TVar \u03b1s \u2192 Level\n\u03c4level (zero {\u03b1 = \u03b1}) = \u03b1\n\u03c4level (suc \u03c4) = \u03c4level \u03c4\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03b1s} (\u03c4 : TVar \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (\u03c4level \u03c4)\n\u03c4\u27e6 zero \u27e7  (As , A) = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (As , A) = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u03c4 : TVar \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (\u03c4level \u03c4 \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u21d4 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211cs , \u211c) t a b = \u211c t a b\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211cs , \u211c) t a b = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t a b\n\n-- Types\n\ndata Typ (\u03b1s : Levels) : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 Typ \u03b1s\n  _\u2227_ _\u21d2_ _\u22b5_ : (T : Typ \u03b1s) \u2192 (U : Typ \u03b1s) \u2192 Typ \u03b1s\n  tvar : (\u03c4 : TVar \u03b1s) \u2192 Typ \u03b1s\n  univ : \u2200 \u03b1 \u2192 (T : Typ (\u03b1s , \u03b1)) \u2192 Typ \u03b1s\n\ntlevel : \u2200 {\u03b1s} \u2192 Typ \u03b1s \u2192 Level\ntlevel \u27e8 A \u27e9      = o\ntlevel (T \u2227 U)    = tlevel T \u2294 tlevel U\ntlevel (T \u21d2 U)    = tlevel T \u2294 tlevel U\ntlevel (T \u22b5 U)    = tlevel T \u2294 tlevel U\ntlevel (tvar \u03c4)   = \u03c4level \u03c4\ntlevel (univ \u03b1 T) = \u2191 \u03b1 \u2294 tlevel T\n\nT\u27e6_\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (tlevel T)\nT\u27e6 \u27e8 A \u27e9 \u27e7    As t = A\nT\u27e6 T \u2227 U \u27e7    As t = T\u27e6 T \u27e7 As t \u00d7 T\u27e6 U \u27e7 As t\nT\u27e6 T \u21d2 U \u27e7    As t = T\u27e6 T \u27e7 As t \u2192 T\u27e6 U \u27e7 As t\nT\u27e6 T \u22b5 U \u27e7    As t = \u2200 u \u2192 True (t \u2264 u) \u2192 T\u27e6 T \u27e7 As [ t , u ] \u2192 T\u27e6 U \u27e7 As u\nT\u27e6 tvar \u03c4 \u27e7   As t = \u03c4\u27e6 \u03c4 \u27e7 As t\nT\u27e6 univ \u03b1 T \u27e7 As t = \u2200 (A : RSet \u03b1) \u2192 T\u27e6 T \u27e7 (As , A) t\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (T : Typ \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (tlevel T \u220b T\u27e6 T \u27e7 As \u21d4 T\u27e6 T \u27e7 Bs)\nT\u27e6 \u27e8 A \u27e9 \u27e7\u00b2    \u211cs t a       b       = a \u2261 b\nT\u27e6 T \u2227 U \u27e7\u00b2    \u211cs t (a , b) (c , d) = T\u27e6 T \u27e7\u00b2 \u211cs t a c \u00d7 T\u27e6 U \u27e7\u00b2 \u211cs t b d\nT\u27e6 T \u21d2 U \u27e7\u00b2    \u211cs t f       g       = \u2200 {a b} \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t a b \u2192 T\u27e6 U \u27e7\u00b2 \u211cs t (f a) (g b)\nT\u27e6 T \u22b5 U \u27e7\u00b2    \u211cs t f       g       = \u2200 u t\u2264u {\u03c3 \u03c4} \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs [ t , u ]\u00b2) \u03c3 \u03c4 \u2192 T\u27e6 U \u27e7\u00b2 \u211cs u (f u t\u2264u \u03c3) (g u t\u2264u \u03c4)\nT\u27e6 tvar \u03c4 \u27e7\u00b2   \u211cs t v       w       = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t v w\nT\u27e6 univ \u03b1 T \u27e7\u00b2 \u211cs t f       g       = \u2200 {A B} (\u211c : \u03b1 \u220b A \u21d4 B) \u2192 T\u27e6 T \u27e7\u00b2 (\u211cs , \u211c) t (f A) (g B)\n\n-- Contexts\n\ndata Ctxt (\u03b1s : Levels) : Set\u2081 where\n  \u03b5 : Ctxt \u03b1s\n  _,_at_ : (\u0393 : Ctxt \u03b1s) (T : Typ \u03b1s) (t : Time) \u2192 Ctxt \u03b1s\n\nclevels : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Levels\nclevels \u03b5            = \u03b5\nclevels (\u0393 , T at t) = (clevels \u0393 , tlevel T)\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03b1s} (\u0393 : Ctxt \u03b1s) \u2192 RSets \u03b1s \u2192 Sets (clevels \u0393)\n\u0393\u27e6 \u03b5 \u27e7          As = tt\n\u0393\u27e6 \u0393 , T at t \u27e7 As = (\u0393\u27e6 \u0393 \u27e7 As , T\u27e6 T \u27e7 As t)\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u0393 : Ctxt \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u2194* \u0393\u27e6 \u0393 \u27e7 Bs)\n\u0393\u27e6 \u03b5 \u27e7\u00b2          \u211cs = tt\n\u0393\u27e6 \u0393 , T at t \u27e7\u00b2 \u211cs = (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs , T\u27e6 T \u27e7\u00b2 \u211cs t)\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 TVar (\u03b1s + \u03b2s) \u2192 TVar ((\u03b1s , \u03b1) + \u03b2s)\n\u03c4weaken \u03b1 \u03b5        \u03c4       = suc \u03c4\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) zero    = zero\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) (suc \u03c4) = suc (\u03c4weaken \u03b1 \u03b2s \u03c4)\n\n\u27e6\u03c4weaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n-- Weakening of types\n\ntweaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 Typ (\u03b1s + \u03b2s) \u2192 Typ ((\u03b1s , \u03b1) + \u03b2s)\ntweaken \u03b1 \u03b2s \u27e8 A \u27e9      = \u27e8 A \u27e9\ntweaken \u03b1 \u03b2s (T \u2227 U)    = tweaken \u03b1 \u03b2s T \u2227 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (T \u21d2 U)    = tweaken \u03b1 \u03b2s T \u21d2 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (T \u22b5 U)    = tweaken \u03b1 \u03b2s T \u22b5 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (tvar \u03c4)   = tvar (\u03c4weaken \u03b1 \u03b2s \u03c4)\ntweaken \u03b1 \u03b2s (univ \u03b2 T) = univ \u03b2 (tweaken \u03b1 (\u03b2s , \u03b2) T)\n\nmutual\n\n  \u27e6tweaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u22b5 U)    As A Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (tvar \u03c4)   As A Bs t a       = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\n  \u27e6tweaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192\n    T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u22b5 U)    As A Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (tvar \u03c4)   As A Bs t a       = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u22b5 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (tvar \u03c4)   \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u22b5 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (tvar \u03c4)   \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Weakening of contexts\n\ncweaken : \u2200 {\u03b1s} \u03b1 \u2192 Ctxt \u03b1s \u2192 Ctxt (\u03b1s , \u03b1)\ncweaken \u03b1 \u03b5            = \u03b5\ncweaken \u03b1 (\u0393 , T at t) = (cweaken \u03b1 \u0393 , tweaken \u03b1 \u03b5 T at t)\n  \n\u27e6cweaken\u27e7 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) As A \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7 (As , A) \u27e9\n\u27e6cweaken\u27e7 \u03b1 \u03b5            As A tt       = tt\n\u27e6cweaken\u27e7 \u03b1 (\u0393 , T at t) As A (as , a) = (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as , \u27e6tweaken\u27e7 \u03b1 \u03b5 T As A tt t a)\n\n\u27e6cweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) {As Bs A B} \u211cs \u211c {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192\n  \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7\u00b2 (\u211cs , \u211c) \u27e9\u00b2 (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as) (\u27e6cweaken\u27e7 \u03b1 \u0393 Bs B bs)\n\u27e6cweaken\u27e7\u00b2 \u03b1 \u03b5            \u211cs \u211c tt            = tt\n\u27e6cweaken\u27e7\u00b2 \u03b1 (\u0393 , T at t) \u211cs \u211c (as\u211cbs , a\u211cb) = (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b5 T \u211cs \u211c tt t a\u211cb)\n\n-- Substitution into type variables\n\n\u03c4subst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 TVar ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\n\u03c4subst T \u03b5        zero    = T\n\u03c4subst T \u03b5        (suc \u03c4) = tvar \u03c4\n\u03c4subst T (\u03b2s , \u03b2) zero    = tvar zero\n\u03c4subst T (\u03b2s , \u03b2) (suc \u03c4) = tweaken \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4)\n\n\u27e6\u03c4subst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6tweaken\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t \n    (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u207b\u00b9\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t \n    (\u27e6tweaken\u207b\u00b9\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t a)\n\n\u27e6\u03c4subst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6tweaken\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t \n    (\u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t \n    (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t a\u211cb)\n\n-- Substitution into types\n\ntsubst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 Typ ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\ntsubst T \u03b2s \u27e8 A \u27e9      = \u27e8 A \u27e9\ntsubst T \u03b2s (U \u2227 V)    = tsubst T \u03b2s U \u2227 tsubst T \u03b2s V\ntsubst T \u03b2s (U \u21d2 V)    = tsubst T \u03b2s U \u21d2 tsubst T \u03b2s V\ntsubst T \u03b2s (U \u22b5 V)    = tsubst T \u03b2s U \u22b5 tsubst T \u03b2s V\ntsubst T \u03b2s (tvar \u03c4)   = \u03c4subst T \u03b2s \u03c4\ntsubst T \u03b2s (univ \u03b2 U) = univ \u03b2 (tsubst T (\u03b2s , \u03b2) U)\n\nmutual\n\n  \u27e6tsubst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u27e7 T \u03b2s (U \u22b5 V)    As Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tsubst\u27e7 T \u03b2s V As Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u27e7 T \u03b2s (tvar \u03c4)   As Bs t a       = \u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\n  \u27e6tsubst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u22b5 V)    As Bs t f       = \u03bb v t\u2264v \u03c3 \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs v (f v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u27e7 T \u03b2s U As Bs u (\u03c3 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (tvar \u03c4)   As Bs t a       = \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tsubst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u22b5 V)    \u211cs \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (tvar \u03c4)   \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u22b5 V)    \u211cs \u2111s t f\u211cg         = \u03bb v t\u2264v \u03c3\u211c\u03c4 \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s v (f\u211cg v t\u2264v (\u03bb u t\u2264u u\u2264v \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s u (\u03c3\u211c\u03c4 u t\u2264u u\u2264v)))\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (tvar \u03c4)   \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Variables\n\ndata Var {\u03b1s} (T : Typ \u03b1s) (t : Time) : (\u0393 : Ctxt \u03b1s) \u2192 Set\u2081 where\n  zero : \u2200 {\u0393 : Ctxt \u03b1s} \u2192 Var T t (\u0393 , T at t)\n  suc : \u2200 {\u0393 : Ctxt \u03b1s} {U : Typ \u03b1s} {u} \u2192 Var T t \u0393 \u2192 Var T t (\u0393 , U at u)\n\nv\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Var T t \u0393 \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\nv\u27e6 zero \u27e7  As (as , a) = a\nv\u27e6 suc v \u27e7 As (as , a) = v\u27e6 v \u27e7 As as\n\nv\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (\u03c4 : Var T t \u0393) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (v\u27e6 \u03c4 \u27e7 As as) (v\u27e6 \u03c4 \u27e7 Bs bs)\nv\u27e6 zero \u27e7\u00b2  \u211cs (as\u211cbs , a\u211cb) = a\u211cb\nv\u27e6 suc v \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb) = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Expressions\n\ndata Exp : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Typ \u03b1s \u2192 RSet\u2081 where\n  const : \u2200 {\u03b1s \u0393 A t} \u2192 (a : A) \u2192 Exp {\u03b1s} \u0393 \u27e8 A \u27e9 t\n  pair : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 T t) \u2192 (f : Exp \u0393 U t) \u2192 Exp {\u03b1s} \u0393 (T \u2227 U) t\n  fst : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 T t\n  snd : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 U t\n  abs : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp (\u0393 , T at t) U t) \u2192 Exp {\u03b1s} \u0393 (T \u21d2 U) t\n  app : \u2200 {\u03b1s \u0393 T U t} \u2192 (f : Exp \u0393 (T \u21d2 U) t) \u2192 (e : Exp \u0393 T t) \u2192 Exp {\u03b1s} \u0393 U t\n  var : \u2200 {\u03b1s \u0393 T t} \u2192 (v : Var T t \u0393) \u2192 Exp {\u03b1s} \u0393 T t\n  tabs : \u2200 {\u03b1s \u0393} \u03b1 {T t} \u2192 (e : Exp (cweaken \u03b1 \u0393) T t) \u2192 Exp {\u03b1s} \u0393 (univ \u03b1 T) t\n  tapp : \u2200 {\u03b1s \u0393 t} (T : Typ \u03b1s) {U} \u2192 (e : Exp \u0393 (univ (tlevel T) U) t) \u2192 Exp {\u03b1s} \u0393 (tsubst T \u03b5 U) t\n\ne\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Exp \u0393 T t \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\ne\u27e6 const a \u27e7                  As as = a\ne\u27e6 pair e f \u27e7                 As as = (e\u27e6 e \u27e7 As as , e\u27e6 f \u27e7 As as)\ne\u27e6 fst e \u27e7                    As as = proj\u2081 (e\u27e6 e \u27e7 As as)\ne\u27e6 snd e \u27e7                    As as = proj\u2082 (e\u27e6 e \u27e7 As as)\ne\u27e6 abs e \u27e7                    As as = \u03bb a \u2192 e\u27e6 e \u27e7 As (as , a)\ne\u27e6 app f e \u27e7                  As as = e\u27e6 f \u27e7 As as (e\u27e6 e \u27e7 As as)\ne\u27e6 var v \u27e7                    As as = v\u27e6 v \u27e7 As as\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7         As as = \u03bb A \u2192 e\u27e6 e \u27e7 (As , A) (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7 As as = \u27e6tsubst\u27e7 T \u03b5 U As tt t (e\u27e6 e \u27e7 As as (T\u27e6 T \u27e7 As))\n\ne\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (e : Exp \u0393 T t) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (e\u27e6 e \u27e7 As as) (e\u27e6 e \u27e7 Bs bs)\ne\u27e6 const a \u27e7\u00b2                  \u211cs as\u211cbs = refl\ne\u27e6 pair e f \u27e7\u00b2                 \u211cs as\u211cbs = (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs , e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 fst e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2081 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 snd e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2082 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 abs e \u27e7\u00b2                    \u211cs as\u211cbs = \u03bb a\u211cb \u2192 e\u27e6 e \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb)\ne\u27e6 app f e \u27e7\u00b2                  \u211cs as\u211cbs = e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 var v \u27e7\u00b2                    \u211cs as\u211cbs = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7\u00b2         \u211cs as\u211cbs = \u03bb \u211c \u2192 e\u27e6 e \u27e7\u00b2 (\u211cs , \u211c) (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7\u00b2 \u211cs as\u211cbs = \u27e6tsubst\u27e7\u00b2 T \u03b5 U \u211cs tt t (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 T \u27e7\u00b2 \u211cs))\n\n-- Surface level types\n\ndata STyp : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 STyp\n  _\u2227_ _\u21d2_ : (T : STyp) \u2192 (U : STyp) \u2192 STyp\n  \u25a1 : (T : STyp) \u2192 STyp\n\n-- Translation of surface level types into types\n\n\u27ea_\u27eb : STyp \u2192 Typ (\u03b5 , o)\n\u27ea \u27e8 A \u27e9 \u27eb = \u27e8 A \u27e9\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2227 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u21d2 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar zero \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : STyp \u2192 RSet\u2080\nT\u27ea \u27e8 A \u27e9 \u27eb t = A\nT\u27ea T \u2227 U \u27eb t = T\u27ea T \u27eb t \u00d7 T\u27ea U \u27eb t\nT\u27ea T \u21d2 U \u27eb t = T\u27ea T \u27eb t \u2192 T\u27ea U \u27eb t\nT\u27ea \u25a1 T \u27eb   t = \u2200 u \u2192 True (t \u2264 u) \u2192 T\u27ea T \u27eb u\n\nWorld : RSet\u2080\nWorld t = \u22a4\n\nmutual\n\n  trans : \u2200 T {t} \u2192 T\u27ea T \u27eb t \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) t\n  trans \u27e8 A \u27e9   a       = a\n  trans (T \u2227 U) (a , b) = (trans T a , trans U b)\n  trans (T \u21d2 U) f       = \u03bb a \u2192 trans U (f (trans\u207b\u00b9 T a))\n  trans (\u25a1 T)   \u03c3       = \u03bb u t\u2264u \u03c4 \u2192 trans T (\u03c3 u t\u2264u)\n\n  trans\u207b\u00b9 : \u2200 T {t} \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) t \u2192 T\u27ea T \u27eb t\n  trans\u207b\u00b9 \u27e8 A \u27e9   a       = a\n  trans\u207b\u00b9 (T \u2227 U) (a , b) = (trans\u207b\u00b9 T a , trans\u207b\u00b9 U b)\n  trans\u207b\u00b9 (T \u21d2 U) f       = \u03bb a \u2192 trans\u207b\u00b9 U (f (trans T a))\n  trans\u207b\u00b9 (\u25a1 T)   \u03c3       = \u03bb u t\u2264u \u2192 trans\u207b\u00b9 T (\u03c3 u t\u2264u _)\n\n-- Causality\n\n_at_\u220b_\u2248[_\u2235_]_ : \u2200 T t \u2192 T\u27ea T \u27eb t \u2192 \u2200 u \u2192 True (t \u2264 u) \u2192 T\u27ea T \u27eb t \u2192 Set\n\u27e8 A \u27e9   at t \u220b a       \u2248[ u \u2235 t\u2264u ] b       = a \u2261 b\n(T \u2227 U) at t \u220b (a , b) \u2248[ u \u2235 t\u2264u ] (c , d) = (T at t \u220b a \u2248[ u \u2235 t\u2264u ] c) \u00d7 (U at t \u220b b \u2248[ u \u2235 t\u2264u ] d)\n(T \u21d2 U) at t \u220b f       \u2248[ u \u2235 t\u2264u ] g       = \u2200 {a b} \u2192 (T at t \u220b a \u2248[ u \u2235 t\u2264u ] b) \u2192 (U at t \u220b f a \u2248[ u \u2235 t\u2264u ] g b)\n(\u25a1 T)   at s \u220b \u03c3       \u2248[ u \u2235 s\u2264u ] \u03c4       = \u2200 t s\u2264t t\u2264u \u2192 (T at t \u220b \u03c3 t s\u2264t \u2248[ u \u2235 t\u2264u ] \u03c4 t s\u2264t)\n\nCausal : \u2200 T U t \u2192 T\u27ea T \u21d2 U \u27eb t \u2192 Set\nCausal T U t f = \u2200 u t\u2264u {a b} \u2192 (T at t \u220b a \u2248[ u \u2235 t\u2264u ] b) \u2192 (U at t \u220b f a \u2248[ u \u2235 t\u2264u ] f b)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (o \u220b World \u21d4 World)\n\u211c[ u ] t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248 : \u2200 T t u t\u2264u {a b} \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) t a b \u2192\n    (T at t \u220b trans\u207b\u00b9 T a \u2248[ u \u2235 t\u2264u ] trans\u207b\u00b9 T b)\n  \u211c-impl-\u2248 \u27e8 A \u27e9   t u t\u2264u a\u211cb         = a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) t u t\u2264u (a\u211cc , b\u211cd) = (\u211c-impl-\u2248 T t u t\u2264u a\u211cc , \u211c-impl-\u2248 U t u t\u2264u b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) t u t\u2264u f\u211cg         = \u03bb a\u2248b \u2192 \u211c-impl-\u2248 U t u t\u2264u (f\u211cg (\u2248-impl-\u211c T t u t\u2264u a\u2248b))\n  \u211c-impl-\u2248 (\u25a1 T)   s v s\u2264v \u03c3\u211c\u03c4         = \u03bb u s\u2264u u\u2264v \u2192 \u211c-impl-\u2248 T u v u\u2264v (\u03c3\u211c\u03c4 u s\u2264u (\u03bb t s\u2264t t\u2264u \u2192 \u2264-trans t u v t\u2264u u\u2264v))\n\n  \u2248-impl-\u211c : \u2200 T t u t\u2264u {a b} \u2192\n    (T at t \u220b a \u2248[ u \u2235 t\u2264u ] b) \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) t (trans T a) (trans T b)\n  \u2248-impl-\u211c \u27e8 A \u27e9   t u t\u2264u a\u2248b         = a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) t u t\u2264u (a\u2248c , b\u2248d) = (\u2248-impl-\u211c T t u t\u2264u a\u2248c , \u2248-impl-\u211c U t u t\u2264u b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) t u t\u2264u f\u2248g         = \u03bb a\u211cb \u2192 \u2248-impl-\u211c U t u t\u2264u (f\u2248g (\u211c-impl-\u2248 T t u t\u2264u a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   s v s\u2264v \u03c3\u2248\u03c4         = \u03bb u s\u2264u \u03c1 \u2192 \u2248-impl-\u211c T u v (\u03c1 u s\u2264u (\u2264-refl u)) (\u03c3\u2248\u03c4 u s\u2264u (\u03c1 u s\u2264u (\u2264-refl u)))\n\n-- Every expression is causal\n\ne\u27ea_at_\u220b_\u27eb : \u2200 T t \u2192 Exp \u03b5 \u27ea T \u27eb t \u2192 T\u27ea T \u27eb t\ne\u27ea T at t \u220b e \u27eb = trans\u207b\u00b9 T (e\u27e6 e \u27e7 (tt , World) tt)\n\ncausality : \u2200 T U t f \u2192 Causal T U t e\u27ea (T \u21d2 U) at t \u220b f \u27eb \ncausality T U t f u t\u2264u = \u211c-impl-\u2248 (T \u21d2 U) t u t\u2264u (e\u27e6 f \u27e7\u00b2 (tt , \u211c[ u ]) tt)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"872efb80a4cc6d30136b2075976989235e6b3de3","subject":"Possible agda bug","message":"Possible agda bug\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/LinFunContructw.agda","new_file":"agda\/LinFunContructw.agda","new_contents":"module LinFunContructw where\n\nopen import Common\nopen import LinLogic\nopen import IndexLLProp\nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\next\u21d2\u00acho : \u2200{i u pll rll ll} \u2192 \u2200 s \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL rll ll)\n          \u2192 \u00ac Ordered\u1d62 lind ind \u2192 \u00ac hitsAtLeastOnce (extend ind s) lind\next\u21d2\u00acho s \u2193 lind \u00acord x = \u00acord (b\u2264\u1d62a \u2264\u1d62\u2193)\next\u21d2\u00acho s (ind \u2190\u2227) \u2193 \u00acord x = \u00acord (a\u2264\u1d62b \u2264\u1d62\u2193)\next\u21d2\u00acho {pll = pll} {_} {ll = li \u2227 _} s (ind \u2190\u2227) (lind \u2190\u2227) \u00acord\n      with replLL li ind pll | replLL-id li ind pll refl | extendg ind s | ext\u21d2\u00acho s ind lind hf where\n  hf : \u00ac Ordered\u1d62 lind ind\n  hf (a\u2264\u1d62b x\u2081) = \u00acord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081))\n  hf (b\u2264\u1d62a x\u2081) = \u00acord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081))\next\u21d2\u00acho {pll = pll} {_} {li \u2227 _} s (ind \u2190\u2227) (lind \u2190\u2227) \u00acord | .li | refl | t | e = {!!} where\n  hf : \u00ac hitsAtLeastOnce (t \u2190\u2227) (lind \u2190\u2227)\n  hf (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x) = {!!}\next\u21d2\u00acho {pll = pll} {_} {ll = li \u2227 _} s (ind \u2190\u2227) (\u2227\u2192 lind) \u00acord x with replLL li ind pll | replLL-id li ind pll refl | extendg ind s\next\u21d2\u00acho {_} {_} {pll} {_} {li \u2227 _} s (ind \u2190\u2227) (\u2227\u2192 lind) \u00acord () | .li | refl | t\next\u21d2\u00acho s (\u2227\u2192 ind) \u2193 \u00acord x = \u00acord (a\u2264\u1d62b \u2264\u1d62\u2193)\next\u21d2\u00acho s (\u2227\u2192 ind) (lind \u2190\u2227) \u00acord x = {!!}\next\u21d2\u00acho s (\u2227\u2192 ind) (\u2227\u2192 lind) \u00acord x = {!!}\next\u21d2\u00acho s (ind \u2190\u2228) lind \u00acord x = {!!}\next\u21d2\u00acho s (\u2228\u2192 ind) lind \u00acord x = {!!}\next\u21d2\u00acho s (ind \u2190\u2202) lind \u00acord x = {!!}\next\u21d2\u00acho s (\u2202\u2192 ind) lind \u00acord x = {!!}\n\ngoo : \u2200{i u rll pll ll tll} \u2192 (lind : IndexLL {i} {u} rll ll) \u2192 (s : SetLL ll) \u2192 \u2200{rs : SetLL tll}\n      \u2192 \u00ac (hitsAtLeastOnce s lind) \u2192 (ind : IndexLL pll ll)\n      \u2192 (nord : \u00ac Ordered\u1d62 lind ind)\n      \u2192 \u00ac (hitsAtLeastOnce (replacePartOf s to rs at ind) (\u00acord-morph lind ind tll (flipNotOrd\u1d62 nord)))\ngoo \u2193 s \u00acho ind \u00acord = \u03bb _ \u2192 \u00acord (a\u2264\u1d62b \u2264\u1d62\u2193)\ngoo (lind \u2190\u2227) \u2193 \u00acho ind \u00acord = \u03bb _ \u2192 \u00acho hitsAtLeastOnce\u2193\ngoo (lind \u2190\u2227) (s \u2190\u2227) \u00acho \u2193 \u00acord = \u03bb _ \u2192 \u00acord (b\u2264\u1d62a \u2264\u1d62\u2193)\ngoo (lind \u2190\u2227) (s \u2190\u2227) {rs} \u00acho (ind \u2190\u2227) \u00acord x\n                                 with goo lind s {rs} (\u03bb z \u2192 \u00acho (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) ind hf where\n  hf : \u00ac Ordered\u1d62 lind ind\n  hf (a\u2264\u1d62b x) = \u00acord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n  hf (b\u2264\u1d62a x) = \u00acord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x))\ngoo (lind \u2190\u2227) (s \u2190\u2227) {rs} \u00acho (ind \u2190\u2227) \u00acord (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x) | r = \u22a5-elim (r x)\ngoo (lind \u2190\u2227) (s \u2190\u2227) \u00acho (\u2227\u2192 ind) \u00acord (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x) = \u00acho (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\ngoo (lind \u2190\u2227) (\u2227\u2192 s) \u00acho \u2193 \u00acord = \u03bb _ \u2192 \u00acord (b\u2264\u1d62a \u2264\u1d62\u2193)\ngoo (lind \u2190\u2227) (\u2227\u2192 s) \u00acho (ind \u2190\u2227) \u00acord = {!!}\ngoo (lind \u2190\u2227) (\u2227\u2192 s) \u00acho (\u2227\u2192 ind) \u00acord = \u03bb ()\ngoo (lind \u2190\u2227) (s \u2190\u2227\u2192 s\u2081) \u00acho ind \u00acord = {!!}\ngoo (\u2227\u2192 lind) s \u00acho ind \u00acord = {!!}\ngoo (lind \u2190\u2228) s \u00acho ind \u00acord = {!!}\ngoo (\u2228\u2192 lind) s \u00acho ind \u00acord = {!!}\ngoo (lind \u2190\u2202) s \u00acho ind \u00acord = {!!}\ngoo (\u2202\u2192 lind) s \u00acho ind \u00acord = {!!}\n\n\n\n\ngest : \u2200{i u rll ll n dt df tind ts} \u2192 (lf : LFun ll rll)\n       \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll) \u2192 \u00ac (hitsAtLeastOnce s ind)\n       \u2192 \u00ac\u2205 tind \u2261 tranLFMIndexLL lf (\u00ac\u2205 ind) \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n       \u2192 \u00ac (hitsAtLeastOnce ts tind) \ngest I ind s \u00acho refl refl = \u00acho\ngest (_\u2282_ {ind = lind} lf lf\u2081) ind s \u00acho eqi eqs with isOrd\u1d62 lind ind\n... | no \u00acp = {!!} where\n  is = gest lf {!!} {!!} -- (\u00ac\u2205 (truncSetLL s ind))\n... | yes p = {!!}\ngest {tind = tind} {ts} (tr ltr lf) ind s \u00acho eqi eqs = gest lf (IndexLLProp.tran ind ltr ut) (SetLL.tran s ltr) \u00actho eqi eqs  where\n  ut = indLow\u21d2UpTran ind ltr \n  \u00actho = \u00actrho ltr s ind \u00acho ut\ngest (com df\u2081 lf) ind s \u00acho () eqs\ngest (call x) ind s \u00acho () eqs\n\n\n\n\nmodule _ where\n\n  \n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n-- (shrinkcms ll s cs ceqi)\n  data MLFun {i u ll rll n dt df} (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s ind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    \u2205 : MLFun ind s m\u00acho lf\n    \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf ind in\n           \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll ind s ceqi m\u00acho cll) (hf rll tind ts ceqo {!!} cll))\n           \u2192 MLFun ind s m\u00acho lf\n           -- We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n  \n  \n    -- s here does contain the ind.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test ind s lf = {!!}\n  \n  \n  \n  \n  \n  \n  \n","old_contents":"module LinFunContructw where\n\nopen import Common\nopen import LinLogic\nopen import IndexLLProp\nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\next\u21d2\u00acho : \u2200{i u pll rll ll} \u2192 \u2200 s \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL rll ll)\n          \u2192 \u00ac Ordered\u1d62 lind ind \u2192 \u00ac hitsAtLeastOnce (extend ind s) lind\next\u21d2\u00acho s \u2193 lind \u00acord x = \u00acord (b\u2264\u1d62a \u2264\u1d62\u2193)\next\u21d2\u00acho s (ind \u2190\u2227) \u2193 \u00acord x = \u00acord (a\u2264\u1d62b \u2264\u1d62\u2193)\next\u21d2\u00acho {pll = pll} {_} {ll = li \u2227 _} s (ind \u2190\u2227) (lind \u2190\u2227) \u00acord x\n      with replLL li ind pll | replLL-id li ind pll refl | extendg ind s | ext\u21d2\u00acho s ind lind hf where\n  hf : \u00ac Ordered\u1d62 lind ind\n  hf (a\u2264\u1d62b x\u2081) = \u00acord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081))\n  hf (b\u2264\u1d62a x\u2081) = \u00acord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081))\next\u21d2\u00acho {pll = pll} {_} {li \u2227 _} s (ind \u2190\u2227) (lind \u2190\u2227) \u00acord x | .li | refl | t | e = {!!}\next\u21d2\u00acho {pll = pll} {_} {ll = li \u2227 _} s (ind \u2190\u2227) (\u2227\u2192 lind) \u00acord x with replLL li ind pll | replLL-id li ind pll refl | extendg ind s\next\u21d2\u00acho {_} {_} {pll} {_} {li \u2227 _} s (ind \u2190\u2227) (\u2227\u2192 lind) \u00acord () | .li | refl | t\next\u21d2\u00acho s (\u2227\u2192 ind) \u2193 \u00acord x = \u00acord (a\u2264\u1d62b \u2264\u1d62\u2193)\next\u21d2\u00acho s (\u2227\u2192 ind) (lind \u2190\u2227) \u00acord x = {!!}\next\u21d2\u00acho s (\u2227\u2192 ind) (\u2227\u2192 lind) \u00acord x = {!!}\next\u21d2\u00acho s (ind \u2190\u2228) lind \u00acord x = {!!}\next\u21d2\u00acho s (\u2228\u2192 ind) lind \u00acord x = {!!}\next\u21d2\u00acho s (ind \u2190\u2202) lind \u00acord x = {!!}\next\u21d2\u00acho s (\u2202\u2192 ind) lind \u00acord x = {!!}\n\ngoo : \u2200{i u rll pll ll tll} \u2192 (lind : IndexLL {i} {u} rll ll) \u2192 (s : SetLL ll) \u2192 \u2200{rs : SetLL tll}\n      \u2192 \u00ac (hitsAtLeastOnce s lind) \u2192 (ind : IndexLL pll ll)\n      \u2192 (nord : \u00ac Ordered\u1d62 lind ind)\n      \u2192 \u00ac (hitsAtLeastOnce (replacePartOf s to rs at ind) (\u00acord-morph lind ind tll (flipNotOrd\u1d62 nord)))\ngoo \u2193 s \u00acho ind \u00acord = \u03bb _ \u2192 \u00acord (a\u2264\u1d62b \u2264\u1d62\u2193)\ngoo (lind \u2190\u2227) \u2193 \u00acho ind \u00acord = \u03bb _ \u2192 \u00acho hitsAtLeastOnce\u2193\ngoo (lind \u2190\u2227) (s \u2190\u2227) \u00acho \u2193 \u00acord = \u03bb _ \u2192 \u00acord (b\u2264\u1d62a \u2264\u1d62\u2193)\ngoo (lind \u2190\u2227) (s \u2190\u2227) {rs} \u00acho (ind \u2190\u2227) \u00acord x\n                                 with goo lind s {rs} (\u03bb z \u2192 \u00acho (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) ind hf where\n  hf : \u00ac Ordered\u1d62 lind ind\n  hf (a\u2264\u1d62b x) = \u00acord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n  hf (b\u2264\u1d62a x) = \u00acord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x))\ngoo (lind \u2190\u2227) (s \u2190\u2227) {rs} \u00acho (ind \u2190\u2227) \u00acord (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x) | r = \u22a5-elim (r x)\ngoo (lind \u2190\u2227) (s \u2190\u2227) \u00acho (\u2227\u2192 ind) \u00acord (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x) = \u00acho (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\ngoo (lind \u2190\u2227) (\u2227\u2192 s) \u00acho \u2193 \u00acord = \u03bb _ \u2192 \u00acord (b\u2264\u1d62a \u2264\u1d62\u2193)\ngoo (lind \u2190\u2227) (\u2227\u2192 s) \u00acho (ind \u2190\u2227) \u00acord = {!!}\ngoo (lind \u2190\u2227) (\u2227\u2192 s) \u00acho (\u2227\u2192 ind) \u00acord = \u03bb ()\ngoo (lind \u2190\u2227) (s \u2190\u2227\u2192 s\u2081) \u00acho ind \u00acord = {!!}\ngoo (\u2227\u2192 lind) s \u00acho ind \u00acord = {!!}\ngoo (lind \u2190\u2228) s \u00acho ind \u00acord = {!!}\ngoo (\u2228\u2192 lind) s \u00acho ind \u00acord = {!!}\ngoo (lind \u2190\u2202) s \u00acho ind \u00acord = {!!}\ngoo (\u2202\u2192 lind) s \u00acho ind \u00acord = {!!}\n\n\n\n\ngest : \u2200{i u rll ll n dt df tind ts} \u2192 (lf : LFun ll rll)\n       \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll) \u2192 \u00ac (hitsAtLeastOnce s ind)\n       \u2192 \u00ac\u2205 tind \u2261 tranLFMIndexLL lf (\u00ac\u2205 ind) \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n       \u2192 \u00ac (hitsAtLeastOnce ts tind) \ngest I ind s \u00acho refl refl = \u00acho\ngest (_\u2282_ {ind = lind} lf lf\u2081) ind s \u00acho eqi eqs with isOrd\u1d62 lind ind\n... | no \u00acp = {!!} where\n  is = gest lf {!!} {!!} -- (\u00ac\u2205 (truncSetLL s ind))\n... | yes p = {!!}\ngest {tind = tind} {ts} (tr ltr lf) ind s \u00acho eqi eqs = gest lf (IndexLLProp.tran ind ltr ut) (SetLL.tran s ltr) \u00actho eqi eqs  where\n  ut = indLow\u21d2UpTran ind ltr \n  \u00actho = \u00actrho ltr s ind \u00acho ut\ngest (com df\u2081 lf) ind s \u00acho () eqs\ngest (call x) ind s \u00acho () eqs\n\n\n\n\nmodule _ where\n\n  \n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n-- (shrinkcms ll s cs ceqi)\n  data MLFun {i u ll rll n dt df} (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s ind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    \u2205 : MLFun ind s m\u00acho lf\n    \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf ind in\n           \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll ind s ceqi m\u00acho cll) (hf rll tind ts ceqo {!!} cll))\n           \u2192 MLFun ind s m\u00acho lf\n           -- We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n  \n  \n    -- s here does contain the ind.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test ind s lf = {!!}\n  \n  \n  \n  \n  \n  \n  \n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"a6bbe5d2b2206e1090e8303580a73956ec707683","subject":"Add bot to primitives.","message":"Add bot to primitives.\n\nConsistency at last.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck\/Primitive.agda","new_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck\/Primitive.agda","new_contents":"module Spire.Examples.DarkwingDuck.Primitive where\n\n----------------------------------------------------------------------\n\ninfixr 4 _,_\ninfixr 5 _\u2237_\n\n----------------------------------------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\n----------------------------------------------------------------------\n\ndata \u22a5 : Set where\n\nelimBot : (P : \u22a5 \u2192 Set)\n  (v : \u22a5) \u2192 P v\nelimBot P ()\n\n----------------------------------------------------------------------\n\ndata \u22a4 : Set where\n  tt : \u22a4\n\nelimUnit : (P : \u22a4 \u2192 Set)\n  (ptt : P tt)\n  (u : \u22a4) \u2192 P u\nelimUnit P ptt tt = ptt\n\n----------------------------------------------------------------------\n\ndata \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  _,_ : (a : A) (b : B a) \u2192 \u03a3 A B\n\nelimPair : {A : Set} {B : A \u2192 Set}\n  (P : \u03a3 A B \u2192 Set)\n  (ppair : (a : A) (b : B a) \u2192 P (a , b))\n  (ab : \u03a3 A B) \u2192 P ab\nelimPair P ppair (a , b) = ppair a b\n\n----------------------------------------------------------------------\n\ndata _\u2261_ {A : Set} (x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\nelimEq : {A : Set} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  (prefl : P x refl)\n  (y : A) (q : x \u2261 y) \u2192 P y q\nelimEq P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\nelimList : {A : Set} (P : List A \u2192 Set)\n  (pnil : P [])\n  (pcons : (x : A) (xs : List A) \u2192 P xs \u2192 P (x \u2237 xs))\n  (xs : List A) \u2192 P xs\nelimList P pnil pcons [] = pnil\nelimList P pnil pcons (x \u2237 xs) = pcons x xs (elimList P pnil pcons xs)\n\n----------------------------------------------------------------------\n\ndata Point (A : Set) : List A \u2192 Set where\n  here : \u2200{x xs} \u2192 Point A (x \u2237 xs)\n  there : \u2200{x xs} \u2192 Point A xs \u2192 Point A (x \u2237 xs)\n\nelimPoint : {A : Set} (P : (xs : List A) \u2192 Point A xs \u2192 Set)\n  (phere : (x : A) (xs : List A) \u2192 P (x \u2237 xs) here)\n  (pthere : (x : A) (xs : List A) (t : Point A xs) \u2192 P xs t \u2192 P (x \u2237 xs) (there t))\n  (xs : List A) (t : Point A xs) \u2192 P xs t\nelimPoint P phere pthere (x \u2237 xs) here = phere x xs\nelimPoint P phere pthere (x \u2237 xs) (there t) = pthere x xs t (elimPoint P phere pthere xs t)\n\n----------------------------------------------------------------------\n\ndata Tel : Set\u2081 where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nelimTel : (P : Tel \u2192 Set)\n  (pend : P End)\n  (parg : (A : Set) (B : A \u2192 Tel) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (T : Tel) \u2192 P T\nelimTel P pend parg End = pend\nelimTel P pend parg (Arg A B) = parg A B (\u03bb a \u2192 elimTel P pend parg (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 (I \u2192 Set) \u2192 I \u2192 Set\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = \u03a3 (X j) (\u03bb _ \u2192 El\u1d30 D X i)\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = \u03a3 (P j x) (\u03bb _ \u2192 Hyps D X P i xs)\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n","old_contents":"module Spire.Examples.DarkwingDuck.Primitive where\n\n----------------------------------------------------------------------\n\ninfixr 4 _,_\ninfixr 5 _\u2237_\n\n----------------------------------------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\n----------------------------------------------------------------------\n\ndata \u22a4 : Set where\n  tt : \u22a4\n\nelimUnit : (P : \u22a4 \u2192 Set)\n  (ptt : P tt)\n  (u : \u22a4) \u2192 P u\nelimUnit P ptt tt = ptt\n\n----------------------------------------------------------------------\n\ndata \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n  _,_ : (a : A) (b : B a) \u2192 \u03a3 A B\n\nelimPair : {A : Set} {B : A \u2192 Set}\n  (P : \u03a3 A B \u2192 Set)\n  (ppair : (a : A) (b : B a) \u2192 P (a , b))\n  (ab : \u03a3 A B) \u2192 P ab\nelimPair P ppair (a , b) = ppair a b\n\n----------------------------------------------------------------------\n\ndata _\u2261_ {A : Set} (x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\nelimEq : {A : Set} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  (prefl : P x refl)\n  (y : A) (q : x \u2261 y) \u2192 P y q\nelimEq P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\nelimList : {A : Set} (P : List A \u2192 Set)\n  (pnil : P [])\n  (pcons : (x : A) (xs : List A) \u2192 P xs \u2192 P (x \u2237 xs))\n  (xs : List A) \u2192 P xs\nelimList P pnil pcons [] = pnil\nelimList P pnil pcons (x \u2237 xs) = pcons x xs (elimList P pnil pcons xs)\n\n----------------------------------------------------------------------\n\ndata Point (A : Set) : List A \u2192 Set where\n  here : \u2200{x xs} \u2192 Point A (x \u2237 xs)\n  there : \u2200{x xs} \u2192 Point A xs \u2192 Point A (x \u2237 xs)\n\nelimPoint : {A : Set} (P : (xs : List A) \u2192 Point A xs \u2192 Set)\n  (phere : (x : A) (xs : List A) \u2192 P (x \u2237 xs) here)\n  (pthere : (x : A) (xs : List A) (t : Point A xs) \u2192 P xs t \u2192 P (x \u2237 xs) (there t))\n  (xs : List A) (t : Point A xs) \u2192 P xs t\nelimPoint P phere pthere (x \u2237 xs) here = phere x xs\nelimPoint P phere pthere (x \u2237 xs) (there t) = pthere x xs t (elimPoint P phere pthere xs t)\n\n----------------------------------------------------------------------\n\ndata Tel : Set\u2081 where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nelimTel : (P : Tel \u2192 Set)\n  (pend : P End)\n  (parg : (A : Set) (B : A \u2192 Tel) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (T : Tel) \u2192 P T\nelimTel P pend parg End = pend\nelimTel P pend parg (Arg A B) = parg A B (\u03bb a \u2192 elimTel P pend parg (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 (I \u2192 Set) \u2192 I \u2192 Set\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = \u03a3 (X j) (\u03bb _ \u2192 El\u1d30 D X i)\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : I \u2192 Set) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = \u03a3 (P j x) (\u03bb _ \u2192 Hyps D X P i xs)\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d43fb2f89ef473dccaa4c9fcc455c8cbc9963bd5","subject":"Identify new Agda substitution bug","message":"Identify new Agda substitution bug\n","repos":"inc-lc\/ilc-agda","old_file":"Nehemiah\/Syntax\/Term.agda","new_file":"Nehemiah\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Syntax.Term where\n\nopen import Nehemiah.Syntax.Type\n\nopen import Data.Integer\n\nimport Parametric.Syntax.Term Base as Term\n\ndata Const : Term.Structure where\n  intlit-const : (n : \u2124) \u2192 Const \u2205 int\n  add-const : Const (int \u2022 int \u2022 \u2205) int\n  minus-const : Const (int \u2022 \u2205) (int)\n\n  empty-const : Const \u2205 (bag)\n  insert-const : Const (int \u2022 bag \u2022 \u2205) (bag)\n  union-const : Const (bag \u2022 bag \u2022 \u2205) (bag)\n  negate-const : Const (bag \u2022 \u2205) (bag)\n\n  flatmap-const : Const ((int \u21d2 bag) \u2022 bag \u2022 \u2205) (bag)\n  sum-const : Const (bag \u2022 \u2205) (int)\n\n\n--open Term.Structure --works\nopen Term.Structure Const --fails, even with the rest disabled.\n{-\nAn internal error has occurred. Please report this as a bug.\nLocation of the error: src\/full\/Agda\/TypeChecking\/Substitute.hs:183\n-}\n\n{-\n-- Shorthands of constants\n\npattern intlit n = const (intlit-const n) \u2205\npattern add s t = const add-const (s \u2022 t \u2022 \u2205)\npattern minus t = const minus-const (t \u2022 \u2205)\npattern empty = const empty-const \u2205\npattern insert s t = const insert-const (s \u2022 t \u2022 \u2205)\npattern union s t = const union-const (s \u2022 t \u2022 \u2205)\npattern negate t = const negate-const (t \u2022 \u2205)\npattern flatmap s t = const flatmap-const (s \u2022 t \u2022 \u2205)\npattern sum t = const sum-const (t \u2022 \u2205)\n-}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Syntax.Term where\n\nopen import Nehemiah.Syntax.Type\n\nopen import Data.Integer\n\nimport Parametric.Syntax.Term Base as Term\n\ndata Const : Term.Structure where\n  intlit-const : (n : \u2124) \u2192 Const \u2205 int\n  add-const : Const (int \u2022 int \u2022 \u2205) int\n  minus-const : Const (int \u2022 \u2205) (int)\n\n  empty-const : Const \u2205 (bag)\n  insert-const : Const (int \u2022 bag \u2022 \u2205) (bag)\n  union-const : Const (bag \u2022 bag \u2022 \u2205) (bag)\n  negate-const : Const (bag \u2022 \u2205) (bag)\n\n  flatmap-const : Const ((int \u21d2 bag) \u2022 bag \u2022 \u2205) (bag)\n  sum-const : Const (bag \u2022 \u2205) (int)\n\nopen Term.Structure Const public\n\n-- Shorthands of constants\n\npattern intlit n = const (intlit-const n) \u2205\npattern add s t = const add-const (s \u2022 t \u2022 \u2205)\npattern minus t = const minus-const (t \u2022 \u2205)\npattern empty = const empty-const \u2205\npattern insert s t = const insert-const (s \u2022 t \u2022 \u2205)\npattern union s t = const union-const (s \u2022 t \u2022 \u2205)\npattern negate t = const negate-const (t \u2022 \u2205)\npattern flatmap s t = const flatmap-const (s \u2022 t \u2022 \u2205)\npattern sum t = const sum-const (t \u2022 \u2205)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f78289d4953f689d6d6a0b6d65d36254dc5118e2","subject":"Added map-++.","message":"Added map-++.\n\nIgnore-this: 58116847ac82637c3f4b08d6251c2b1b\n\ndarcs-hash:20110206231146-3bd4e-97dc9840fff42257313c5a2a804d80c41a5d0dea.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/List\/PropertiesI.agda","new_file":"src\/LTC\/Data\/List\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List.PropertiesI where\n\nopen import LTC.Base\n\nopen import Common.Function\n\nopen import LTC.Data.List\nopen import LTC.Data.List.Type\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL               Lys = subst List (sym (++-[] ys)) Lys\n++-List {ys = ys} (consL x {xs} Lxs) Lys =\n  subst List (sym (++-\u2237 x xs ys)) (consL x (++-List Lxs Lys))\n\nmap-List : \u2200 {xs} f \u2192 List xs \u2192 List (map f xs)\nmap-List f nilL = subst List (sym (map-[] f)) nilL\nmap-List f (consL x {xs} Lxs) =\n  subst List (sym (map-\u2237 f x xs)) (consL (f \u00b7 x) (map-List f Lxs))\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} nilL          Lys = subst List (sym (rev-[] ys)) Lys\nrev-List {ys = ys} (consL x {xs} Lxs) Lys =\n  subst List (sym (rev-\u2237 x xs ys)) (rev-List Lxs (consL x Lys))\n\n++-leftIdentity : \u2200 {xs} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL               = ++-[] []\n++-rightIdentity (consL x {xs} Lxs) =\n  begin\n    (x \u2237 xs) ++ []\n      \u2261\u27e8 ++-\u2237 x xs [] \u27e9\n    x \u2237 (xs ++ [])\n      \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 (xs ++ []) \u2261 x \u2237 t)\n               (++-rightIdentity Lxs)\n               refl\n      \u27e9\n    x \u2237 xs\n  \u220e\n\n++-assoc : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n           (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc {ys = ys} {zs} nilL Lys Lzs =\n  begin\n    ([] ++ ys) ++ zs\n      \u2261\u27e8 subst (\u03bb t \u2192 ([] ++ ys) ++ zs \u2261 t ++ zs)\n               (++-[] ys)\n               refl\n      \u27e9\n    ys ++ zs\n      \u2261\u27e8 sym (++-leftIdentity (++-List Lys Lzs)) \u27e9\n    [] ++ ys ++ zs\n  \u220e\n\n++-assoc {ys = ys} {zs} (consL x {xs} Lxs) Lys Lzs =\n  begin\n    ((x \u2237 xs) ++ ys) ++ zs\n      \u2261\u27e8 subst (\u03bb t \u2192 ((x \u2237 xs) ++ ys) ++ zs \u2261 t ++ zs)\n               (++-\u2237 x xs ys)\n               refl\n      \u27e9\n    (x \u2237 (xs ++ ys)) ++ zs\n      \u2261\u27e8 ++-\u2237 x (xs ++ ys) zs \u27e9\n    x \u2237 ((xs ++ ys) ++ zs)\n      \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ((xs ++ ys) ++ zs) \u2261 x \u2237 t)\n               (++-assoc Lxs Lys Lzs) -- IH.\n               refl\n      \u27e9\n    x \u2237 (xs ++ ys ++ zs)\n      \u2261\u27e8 sym (++-\u2237 x xs (ys ++ zs)) \u27e9\n    (x \u2237 xs) ++ ys ++ zs\n  \u220e\n\nrev-++ : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++ {ys = ys} nilL Lys =\n  begin\n    rev [] ys \u2261\u27e8 rev-[] ys \u27e9\n    ys        \u2261\u27e8 sym $ ++-leftIdentity Lys \u27e9\n    [] ++ ys  \u2261\u27e8 subst (\u03bb t \u2192 [] ++ ys \u2261 t ++ ys)\n                       (sym $ rev-[] [])\n                       refl\n              \u27e9\n    rev [] [] ++ ys\n  \u220e\nrev-++ {ys = ys} (consL x {xs} Lxs) Lys =\n  begin\n    rev (x \u2237 xs) ys      \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n    rev xs (x \u2237 ys)      \u2261\u27e8 rev-++ Lxs (consL x Lys) \u27e9  -- IH.\n    rev xs [] ++ x \u2237 ys\n      \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n               (sym\n                 ( begin\n                     (x \u2237 []) ++ ys \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                     x \u2237 ([] ++ ys) \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ([] ++ ys) \u2261 x \u2237 t)\n                                             (++-leftIdentity Lys)\n                                             refl\n                                    \u27e9\n                     x \u2237 ys\n                   \u220e\n                 )\n               )\n               refl\n      \u27e9\n    rev xs [] ++ (x \u2237 []) ++ ys\n      \u2261\u27e8 sym $ ++-assoc (rev-List Lxs nilL) (consL x nilL) Lys \u27e9\n    (rev xs [] ++ (x \u2237 [])) ++ ys\n      \u2261\u27e8 subst (\u03bb t \u2192 (rev xs [] ++ (x \u2237 [])) ++ ys \u2261 t ++ ys)\n               (sym $ rev-++ Lxs (consL x nilL))  -- IH.\n               refl\n      \u27e9\n    rev xs (x \u2237 []) ++ ys\n      \u2261\u27e8 subst (\u03bb t \u2192 rev xs (x \u2237 []) ++ ys \u2261 t ++ ys)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    rev (x \u2237 xs) [] ++ ys\n  \u220e\n\nreverse-++ : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ {ys = ys} nilL Lys =\n  begin\n    reverse ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ([] ++ ys) \u2261 reverse t)\n               (++-[] ys)\n               refl\n      \u27e9\n    reverse ys\n      \u2261\u27e8 sym (++-rightIdentity (rev-List Lys nilL)) \u27e9\n    reverse ys ++ []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ys ++ [] \u2261 reverse ys ++ t)\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse ys ++ reverse []\n  \u220e\n\nreverse-++ (consL x {xs} Lxs) nilL =\n  begin\n    reverse ((x \u2237 xs) ++ [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ((x \u2237 xs) ++ []) \u2261 reverse t)\n               (++-rightIdentity (consL x Lxs))\n               refl\n      \u27e9\n    reverse (x \u2237 xs)\n      \u2261\u27e8 sym (++-[] (reverse (x \u2237 xs))) \u27e9\n    [] ++ reverse (x \u2237 xs)\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ reverse (x \u2237 xs) \u2261 t ++ reverse (x \u2237 xs))\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse [] ++ reverse (x \u2237 xs)\n  \u220e\n\nreverse-++ (consL x {xs} Lxs) (consL y {ys} Lys) =\n  begin\n    reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261\u27e8 refl \u27e9\n    rev ((x \u2237 xs) ++ y \u2237 ys) []\n      \u2261\u27e8 subst (\u03bb t \u2192 rev ((x \u2237 xs) ++ y \u2237 ys) [] \u2261 rev t [])\n               (++-\u2237 x xs (y \u2237 ys))\n               refl\n      \u27e9\n    rev (x \u2237 (xs ++ y \u2237 ys)) [] \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n    rev (xs ++ y \u2237 ys) (x \u2237 [])\n      \u2261\u27e8 rev-++ (++-List Lxs (consL y Lys)) (consL x nilL) \u27e9\n    rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 rev (xs ++ y \u2237 ys) [] ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               refl\n               refl\n      \u27e9\n    reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (xs ++ y \u2237 ys) ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               (reverse-++ Lxs (consL y Lys))  -- IH.\n               refl\n      \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 ++-assoc (rev-List (consL y Lys) nilL)\n                  (rev-List Lxs nilL)\n                  (consL x nilL)\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t ++ x \u2237 [])\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-++ Lxs (consL x nilL))\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs (x \u2237 []) \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev (x \u2237 xs) [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\nmap-++ : \u2200 f {xs ys} \u2192 List xs \u2192 List ys \u2192\n         map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++ f {ys = ys} nilL Lys =\n  begin\n    map f ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 map f ([] ++ ys) \u2261 map f t)\n               (++-[] ys)\n               refl\n      \u27e9\n    map f ys\n      \u2261\u27e8 sym (++-leftIdentity (map-List f Lys)) \u27e9\n    [] ++ map f ys\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ map f ys \u2261 t ++ map f ys)\n               (sym (map-[] f))\n               refl\n      \u27e9\n    map f [] ++ map f ys\n  \u220e\n\nmap-++ f {ys = ys} (consL x {xs} Lxs) Lys =\n  begin\n    map f ((x \u2237 xs) ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 map f ((x \u2237 xs) ++ ys) \u2261 map f t)\n               (++-\u2237 x xs ys)\n               refl\n      \u27e9\n    map f (x \u2237 xs ++ ys)\n      \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n    f \u00b7 x \u2237 map f (xs ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 f \u00b7 x \u2237 map f (xs ++ ys) \u2261 f \u00b7 x \u2237 t)\n               (map-++ f Lxs Lys)  -- IH.\n               refl\n      \u27e9\n    f \u00b7 x \u2237 (map f xs ++ map f ys)\n      \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n    (f \u00b7 x \u2237 map f xs) ++ map f ys\n      \u2261\u27e8 subst (\u03bb t \u2192 (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261 t ++ map f ys)\n               (sym (map-\u2237 f x xs))\n               refl\n      \u27e9\n    map f (x \u2237 xs) ++ map f ys\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List.PropertiesI where\n\nopen import LTC.Base\n\nopen import Common.Function\n\nopen import LTC.Data.List\nopen import LTC.Data.List.Type\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL               Lys = subst List (sym (++-[] ys)) Lys\n++-List {ys = ys} (consL x {xs} Lxs) Lys =\n  subst List (sym (++-\u2237 x xs ys)) (consL x (++-List Lxs Lys))\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} nilL          Lys = subst List (sym (rev-[] ys)) Lys\nrev-List {ys = ys} (consL x {xs} Lxs) Lys =\n  subst List (sym (rev-\u2237 x xs ys)) (rev-List Lxs (consL x Lys))\n\n++-leftIdentity : \u2200 {xs} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL               = ++-[] []\n++-rightIdentity (consL x {xs} Lxs) =\n  begin\n    (x \u2237 xs) ++ []\n      \u2261\u27e8 ++-\u2237 x xs [] \u27e9\n    x \u2237 (xs ++ [])\n      \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 (xs ++ []) \u2261 x \u2237 t)\n               (++-rightIdentity Lxs)\n               refl\n      \u27e9\n    x \u2237 xs\n  \u220e\n\n++-assoc : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n           (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc {ys = ys} {zs} nilL Lys Lzs =\n  begin\n    ([] ++ ys) ++ zs\n      \u2261\u27e8 subst (\u03bb t \u2192 ([] ++ ys) ++ zs \u2261 t ++ zs)\n               (++-[] ys)\n               refl\n      \u27e9\n    ys ++ zs\n      \u2261\u27e8 sym (++-leftIdentity (++-List Lys Lzs)) \u27e9\n    [] ++ ys ++ zs\n  \u220e\n\n++-assoc {ys = ys} {zs} (consL x {xs} Lxs) Lys Lzs =\n  begin\n    ((x \u2237 xs) ++ ys) ++ zs\n      \u2261\u27e8 subst (\u03bb t \u2192 ((x \u2237 xs) ++ ys) ++ zs \u2261 t ++ zs)\n               (++-\u2237 x xs ys)\n               refl\n      \u27e9\n    (x \u2237 (xs ++ ys)) ++ zs\n      \u2261\u27e8 ++-\u2237 x (xs ++ ys) zs \u27e9\n    x \u2237 ((xs ++ ys) ++ zs)\n      \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ((xs ++ ys) ++ zs) \u2261 x \u2237 t)\n               (++-assoc Lxs Lys Lzs) -- IH.\n               refl\n      \u27e9\n    x \u2237 (xs ++ ys ++ zs)\n      \u2261\u27e8 sym (++-\u2237 x xs (ys ++ zs)) \u27e9\n    (x \u2237 xs) ++ ys ++ zs\n  \u220e\n\nrev-++ : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++ {ys = ys} nilL Lys =\n  begin\n    rev [] ys \u2261\u27e8 rev-[] ys \u27e9\n    ys        \u2261\u27e8 sym $ ++-leftIdentity Lys \u27e9\n    [] ++ ys  \u2261\u27e8 subst (\u03bb t \u2192 [] ++ ys \u2261 t ++ ys)\n                       (sym $ rev-[] [])\n                       refl\n              \u27e9\n    rev [] [] ++ ys\n  \u220e\nrev-++ {ys = ys} (consL x {xs} Lxs) Lys =\n  begin\n    rev (x \u2237 xs) ys      \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n    rev xs (x \u2237 ys)      \u2261\u27e8 rev-++ Lxs (consL x Lys) \u27e9  -- IH.\n    rev xs [] ++ x \u2237 ys\n      \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n               (sym\n                 ( begin\n                     (x \u2237 []) ++ ys \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                     x \u2237 ([] ++ ys) \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ([] ++ ys) \u2261 x \u2237 t)\n                                             (++-leftIdentity Lys)\n                                             refl\n                                    \u27e9\n                     x \u2237 ys\n                   \u220e\n                 )\n               )\n               refl\n      \u27e9\n    rev xs [] ++ (x \u2237 []) ++ ys\n      \u2261\u27e8 sym $ ++-assoc (rev-List Lxs nilL) (consL x nilL) Lys \u27e9\n    (rev xs [] ++ (x \u2237 [])) ++ ys\n      \u2261\u27e8 subst (\u03bb t \u2192 (rev xs [] ++ (x \u2237 [])) ++ ys \u2261 t ++ ys)\n               (sym $ rev-++ Lxs (consL x nilL))  -- IH.\n               refl\n      \u27e9\n    rev xs (x \u2237 []) ++ ys\n      \u2261\u27e8 subst (\u03bb t \u2192 rev xs (x \u2237 []) ++ ys \u2261 t ++ ys)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    rev (x \u2237 xs) [] ++ ys\n  \u220e\n\nreverse-++ : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ {ys = ys} nilL Lys =\n  begin\n    reverse ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ([] ++ ys) \u2261 reverse t)\n               (++-[] ys)\n               refl\n      \u27e9\n    reverse ys\n      \u2261\u27e8 sym (++-rightIdentity (rev-List Lys nilL)) \u27e9\n    reverse ys ++ []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ys ++ [] \u2261 reverse ys ++ t)\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse ys ++ reverse []\n  \u220e\n\nreverse-++ (consL x {xs} Lxs) nilL =\n  begin\n    reverse ((x \u2237 xs) ++ [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ((x \u2237 xs) ++ []) \u2261 reverse t)\n               (++-rightIdentity (consL x Lxs))\n               refl\n      \u27e9\n    reverse (x \u2237 xs)\n      \u2261\u27e8 sym (++-[] (reverse (x \u2237 xs))) \u27e9\n    [] ++ reverse (x \u2237 xs)\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ reverse (x \u2237 xs) \u2261 t ++ reverse (x \u2237 xs))\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse [] ++ reverse (x \u2237 xs)\n  \u220e\n\nreverse-++ (consL x {xs} Lxs) (consL y {ys} Lys) =\n  begin\n    reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261\u27e8 refl \u27e9\n    rev ((x \u2237 xs) ++ y \u2237 ys) []\n      \u2261\u27e8 subst (\u03bb t \u2192 rev ((x \u2237 xs) ++ y \u2237 ys) [] \u2261 rev t [])\n               (++-\u2237 x xs (y \u2237 ys))\n               refl\n      \u27e9\n    rev (x \u2237 (xs ++ y \u2237 ys)) [] \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n    rev (xs ++ y \u2237 ys) (x \u2237 [])\n      \u2261\u27e8 rev-++ (++-List Lxs (consL y Lys)) (consL x nilL) \u27e9\n    rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 rev (xs ++ y \u2237 ys) [] ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               refl\n               refl\n      \u27e9\n    reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (xs ++ y \u2237 ys) ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               (reverse-++ Lxs (consL y Lys))  -- IH.\n               refl\n      \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 ++-assoc (rev-List (consL y Lys) nilL)\n                  (rev-List Lxs nilL)\n                  (consL x nilL)\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t ++ x \u2237 [])\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-++ Lxs (consL x nilL))\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs (x \u2237 []) \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev (x \u2237 xs) [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b69fdacecd04c4b7e4945680a7ef182b34551c79","subject":"Prob: start some distE work","message":"Prob: start some distE work\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n    distE : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  (1+ x \/2+x+ y) <E (1+ x' \/2+x+ y') = (1 + x) * (2 + x' + y') < (1 + x') * (2 + x + y)\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  -- |(a\/b)-(c\/d)| = |(ad-bc)\/bd| = |ad-bc|\/bd\n  postulate\n    distE : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  -- distE (1+ x \/2+x+ y) (1+ x' \/2+x+ y') = {!!}\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ distE (\u03bb {x} \u2192 <E-trans {x}) +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E<E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  1-I_ : [0,1] \u2192 [0,1]\n  1-I 0I    = 1I\n  1-I 1I    = 0I\n  1-I (x I) = (1-E x) I\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I  +I x    = x\n  x I +I 0I   = x I\n  1I  +I _    = 1I -- troublesome\n  x I +I 1I   = 1I -- troublesome\n  x I +I x\u2081 I = (x +E x\u2081) I -- faithful\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E<E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E<E pf) = E<E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E<E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E<E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E<E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E<E x\u2081) (E<E x\u2082) = E<E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E<E x\u2081) E\u2261E = E<E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E<E x\u2082) = E<E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E<E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E<E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E<E x\u2082) = E<E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E<E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  1-I_ : [0,1] \u2192 [0,1]\n  1-I 0I    = 1I\n  1-I 1I    = 0I\n  1-I (x I) = (1-E x) I\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I  +I x    = x\n  x I +I 0I   = x I\n  1I  +I _    = 1I -- troublesome\n  x I +I 1I   = 1I -- troublesome\n  x I +I x\u2081 I = (x +E x\u2081) I -- faithful\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E<E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E<E pf) = E<E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E<E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E<E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E<E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E<E x\u2081) (E<E x\u2082) = E<E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E<E x\u2081) E\u2261E = E<E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E<E x\u2082) = E<E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E<E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E<E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E<E x\u2082) = E<E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cce35b611f1cd9557c69b78c3ff9de51d121ee43","subject":"Fixed note.","message":"Fixed note.\n\nIgnore-this: f2e859dc6606334257adeb3012c3436b\n\ndarcs-hash:20120509041415-3bd4e-ac8e7583ced12243b8d1d1b1a5197e9de9c45c18.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Existential.agda","new_file":"notes\/thesis\/logical-framework\/Existential.agda","new_contents":"-- Tested with FOT on 08 May 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Existential where\n\nmodule LF where\n  postulate\n    D : Set\n\n    -- Disjunction.\n    _\u2228_   : Set \u2192 Set \u2192 Set\n    inj\u2081  : {A B : Set} \u2192 A \u2192 A \u2228 B\n    inj\u2082  : {A B : Set} \u2192 B \u2192 A \u2228 B\n    [_,_] : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n\n    -- The existential quantifier type on D.\n    \u2203       : (A : D \u2192 Set) \u2192 Set\n    _,_     : {A : D \u2192 Set}(t : D) \u2192 A t \u2192 \u2203 A\n    \u2203-proj\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    \u2203-proj\u2082 : {A : D \u2192 Set}(h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n    \u2203-elim  : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = [ (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n            , (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n            ] (\u2203-proj\u2082 h)\n\n    -- Using the elimination\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 [ (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                 , (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                 ] ah)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\nmodule Inductive where\n\n  open import Common.FOL.FOL\n\n  -- The existential proyections.\n  \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n  \u2203-proj\u2082 (_ , Ax) = Ax\n\n  -- The existential elimination.\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (_ , Ax) h = h Ax\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = [ (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n            , (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n            ] (\u2203-proj\u2082 h)\n\n    -- Using the elimination.\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 [ (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                 , (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                 ] ah)\n\n    -- Using pattern matching.\n    \u2203\u2200\u2083 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2083 (x , Ax) y = x , Ax y\n\n    \u2203\u2228\u2083 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2083 (x , inj\u2081 Ax) = inj\u2081 (x , Ax)\n    \u2203\u2228\u2083 (x , inj\u2082 Bx) = inj\u2082 (x , Bx)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\n    -- Using the pattern matching.\n    non-FOL\u2083 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2083 (x , _) = x\n","old_contents":"-- Tested with FOT on 08 May 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Existential where\n\nmodule LF where\n  postulate\n    D       : Set\n\n    -- Disjunction.\n    _\u2228_   : Set \u2192 Set \u2192 Set\n    inj\u2081  : {A B : Set} \u2192 A \u2192 A \u2228 B\n    inj\u2082  : {A B : Set} \u2192 B \u2192 A \u2228 B\n    [_,_] : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n\n    -- The existential quantifier type on D.\n    \u2203       : (A : D \u2192 Set) \u2192 Set\n    _,_     : {A : D \u2192 Set}(t : D) \u2192 A t \u2192 \u2203 A\n    \u2203-proj\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    \u2203-proj\u2082 : {A : D \u2192 Set}(h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n    \u2203-elim  : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = [ (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n            , (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n            ] (\u2203-proj\u2082 h)\n\n    -- Using the elimination\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 [ (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                 , (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                 ] ah)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\nmodule Inductive where\n\n  open import Common.FOL.FOL\n\n  -- The existential proyections.\n  \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n  \u2203-proj\u2082 (_ , Ax) = Ax\n\n  -- The existential elimination.\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (_ , Ax) h = h Ax\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = [ (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n            , (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n            ] (\u2203-proj\u2082 h)\n\n    -- Using the elimination.\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah\u2081 \u2192 [ (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                 , (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                 ] ah)\n\n    -- Using pattern matching.\n    \u2203\u2200\u2083 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2083 (x , Ax) y = x , Ax y\n\n    \u2203\u2228\u2083 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2083 (x , inj\u2081 Ax) = inj\u2081 (x , Ax)\n    \u2203\u2228\u2083 (x , inj\u2082 Bx) = inj\u2082 (x , Bx)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\n    -- Using the pattern matching.\n    non-FOL\u2083 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2083 (x , _) = x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"53e7d78959ce72a27c1c3a13af3ce667fbe4e935","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      \u2200 (A : D \u2192 Set) {n} \u2192\n      (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      N n \u2192 A n\n\n    -- Higher-order version (incomplete?).\n    N-least-pre-fixed-ho :\n      \u2200 (A : D \u2192 Set) {n} \u2192\n      (NatF A n \u2192 A n) \u2192\n      N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n  N-least-pre-fixed' :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (NatF A n \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 prf)              = inj\u2081 prf\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 is {n} Nn = N-least-pre-fixed A h Nn\n    where\n    h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n    h (inj\u2081 prf)              = subst A (sym prf) A0\n    h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is helper An')\n      where\n      helper : N n'\n      helper with N-post-fixed Nn\n      ... | inj\u2081 n\u22610 = \u22a5-elim (0\u2262S (trans (sym n\u22610) prf))\n      ... | inj\u2082 (m' , prf' , Nm') =\n        subst N (succInjective (trans (sym prf') prf)) Nm'\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 is {n} = N-least-pre-fixed A h\n    where\n    h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n    h (inj\u2081 prf)              = subst A (sym prf) A0\n    h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is An')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d      \u2261 d\n    +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {m} {n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 prf) = subst N (cong (flip _+_ n) (sym prf)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , m\u2261Sm' , Am')) =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 is nzero      = A0\n  N-ind\u2081 A A0 is (nsucc Nn) = is Nn (N-ind\u2081 A A0 is Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N\u21920\u2228S = N-ind\u2082 A A0 is\n    where\n    A : D \u2192 Set\n    A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n    A0 : A zero\n    A0 = inj\u2081 refl\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = case prf\u2081 prf\u2082 Ai\n      where\n      prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n      prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n      prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n             succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n      prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\n  N-least-pre-fixed\u2082 :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed\u2082 A {n} h Nn = case prf\u2081 prf\u2082 (N\u21920\u2228S Nn)\n    where\n    prf\u2081 : n \u2261 zero \u2192 A n\n    prf\u2081 n\u22610 = h (inj\u2081 n\u22610)\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 A n\n    prf\u2082 (n' , prf , Nn') = h (inj\u2082 (n' , prf , {!!}))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      \u2200 (A : D \u2192 Set) {n} \u2192\n      (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      N n \u2192 A n\n\n    -- Higher-order version (incomplete?).\n    N-least-pre-fixed-ho :\n      \u2200 (A : D \u2192 Set) {n} \u2192\n      (NatF A n \u2192 A n) \u2192\n      N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n  N-least-pre-fixed' :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (NatF A n \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 prf)              = inj\u2081 prf\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 is {n} Nn = N-least-pre-fixed A h Nn\n    where\n    h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n    h (inj\u2081 prf)              = subst A (sym prf) A0\n    h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is helper An')\n      where\n      helper : N n'\n      helper with N-post-fixed Nn\n      ... | inj\u2081 n\u22610 = \u22a5-elim (0\u2262S (trans (sym n\u22610) prf))\n      ... | inj\u2082 (m' , prf' , Nm') =\n        subst N (succInjective (trans (sym prf') prf)) Nm'\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 is {n} = N-least-pre-fixed A h\n    where\n    h : n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n\n    h (inj\u2081 prf)              = subst A (sym prf) A0\n    h (inj\u2082 (n' , prf , An')) = subst A (sym prf) (is An')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d      \u2261 d\n    +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {m} {n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 prf) = subst N (cong (flip _+_ n) (sym prf)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , m\u2261Sm' , Am')) =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 is nzero      = A0\n  N-ind\u2081 A A0 is (nsucc Nn) = is Nn (N-ind\u2081 A A0 is Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N\u21920\u2228S = N-ind\u2082 A A0 is\n    where\n    A : D \u2192 Set\n    A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n    A0 : A zero\n    A0 = inj\u2081 refl\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = case prf\u2081 prf\u2082 Ai\n      where\n      prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n      prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n      prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n             succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n      prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\n  N-least-pre-fixed\u2082 :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    N n \u2192 A n\n  N-least-pre-fixed\u2082 A {n} h Nn = case prf\u2081 prf\u2082 (N\u21920\u2228S Nn)\n    where\n    prf\u2081 : n \u2261 zero \u2192 A n\n    prf\u2081 n\u22610 = h (inj\u2081 n\u22610)\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 A n\n    prf\u2082 (n' , prf , Nn') = h (inj\u2082 (n' , prf , {!!}))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bdda2e0a702f91fb4c31af12c06276f4be5a20a5","subject":"Showed that the associator is an iso.","message":"Showed that the associator is an iso.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n{-       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 (U \u2192 V) \u00d7 (Y \u2192 X) \u2192 U \u00d7 Y \u2192 Set\n\u22b8-cond \u03b1 \u03b2 (f , g) (u , y) = \u03b1 u (g y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (Y \u2192 X)) , (U \u00d7 Y) , \u22b8-cond \u03b1 \u03b2\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , p\u2083\n  where\n   h : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 T \u2192 Z)\n   h (h\u2081 , h\u2082) = (\u03bb w \u2192 g (h\u2081 (f w))) , (\u03bb t \u2192 F (h\u2082 (G t)))\n   H : \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n   H (w , t) = f w , G t\n   p\u2083 : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond \u03b1 \u03b2 u (H y) \u2192 \u22b8-cond \u03b3 \u03b4 (h u) y\n   p\u2083 {h\u2081 , h\u2082}{w , t} c c' = p\u2082 (c (p\u2081 c'))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   F\u03b1-inv : (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\n   F\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n{-\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (W \u2192 \u03a3 (V \u2192 X) (\u03bb x \u2192 U \u2192 Y)) (\u03bb x \u2192 \u03a3 U (\u03bb x\u2081 \u2192 V) \u2192 Z)}\n     \u2192 ((\u03bb x \u2192 (\u03bb x\u2081 \u2192 fst (fst a x) x\u2081) , (\u03bb x\u2081 \u2192 snd (fst a x) x\u2081)) , (\u03bb x \u2192 snd a (fst x , snd x))) \u2261 a\n   aux' {j\u2081 , j\u2082} = eq-\u00d7 (ext-set aux'') (ext-set aux''')\n    where\n      aux'' : {a : W} \u2192 (fst (j\u2081 a) , snd (j\u2081 a)) \u2261 j\u2081 a\n      aux'' {w} with j\u2081 w\n      ... | h\u2081 , h\u2082 = refl\n\n      aux''' : {a : \u03a3 U (\u03bb x\u2081 \u2192 V)} \u2192 j\u2082 (fst a , snd a) \u2261 j\u2082 a\n      aux''' {u , v} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a\n       : \u03a3 (\u03a3 V (\u03bb x \u2192 W) \u2192 X) (\u03bb x \u2192 U \u2192 \u03a3 (W \u2192 Y) (\u03bb x\u2081 \u2192 V \u2192 Z))} \u2192\n      ((\u03bb p' \u2192 fst (fst (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) (snd p')) (fst p')) ,\n       (\u03bb u \u2192 (\u03bb w \u2192 snd (fst (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) w) u) , (\u03bb v \u2192 snd (F\u03b1 {V} {W} {X} {Y} {U} {V} {Z} a) (u , v))))\n      \u2261 a\n   aux' {j\u2081 , j\u2082} = eq-\u00d7 (ext-set aux'') (ext-set aux''')\n     where\n       aux'' : {a : \u03a3 V (\u03bb x \u2192 W)} \u2192 j\u2081 (fst a , snd a) \u2261 j\u2081 a\n       aux'' {v , w} = refl\n       aux''' : {a : U} \u2192 ((\u03bb w \u2192 fst (j\u2082 a) w) , (\u03bb v \u2192 snd (j\u2082 a) v)) \u2261 j\u2082 a\n       aux''' {u} with j\u2082 u\n       ... | h\u2081 , h\u2082 = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 (U \u2192 V) \u00d7 (Y \u2192 X) \u2192 U \u00d7 Y \u2192 Set\n\u22b8-cond \u03b1 \u03b2 (f , g) (u , y) = \u03b1 u (g y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (Y \u2192 X)) , (U \u00d7 Y) , \u22b8-cond \u03b1 \u03b2\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , p\u2083\n  where\n   h : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 T \u2192 Z)\n   h (h\u2081 , h\u2082) = (\u03bb w \u2192 g (h\u2081 (f w))) , (\u03bb t \u2192 F (h\u2082 (G t)))\n   H : \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n   H (w , t) = f w , G t\n   p\u2083 : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond \u03b1 \u03b2 u (H y) \u2192 \u22b8-cond \u03b3 \u03b4 (h u) y\n   p\u2083 {h\u2081 , h\u2082}{w , t} c c' = p\u2082 (c (p\u2081 c'))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6dbde1e52f3b14199a1e9ee7097d079561b3923e","subject":"Correct local variable name","message":"Correct local variable name\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/FundamentalProperty.agda","new_file":"Thesis\/SIRelBigStep\/FundamentalProperty.agda","new_contents":"module Thesis.SIRelBigStep.FundamentalProperty where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb j j<k v1 dv v2 vv \u2192\n      rfundamental3 j t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrelV3\u2192\u2295 vtv1 dtv vtv2 dtvv) =\n        bang v2 , bangapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"module Thesis.SIRelBigStep.FundamentalProperty where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 k (lt1 k<n) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrelV3\u2192\u2295 vtv1 dtv vtv2 dtvv) =\n        bang v2 , bangapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b4534773dd6c1842f0dacb12bcc5a4eb1d235fae","subject":"updating correspondence for removed premise in subsume rule","message":"updating correspondence for removed premise in subsume rule\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"correspondence.agda","new_file":"correspondence.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule correspondence where\n  mutual\n    correspondence-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0393 \u22a2 e => \u03c4\n    correspondence-synth ESConst = SConst\n    correspondence-synth (ESVar x\u2081) = SVar x\u2081\n    correspondence-synth (ESLam apt ex) with correspondence-synth ex\n    ... | ih = SLam apt ih\n    correspondence-synth (ESAp1 x x\u2081 x\u2082 x\u2083) = SAp x\u2081 MAHole (correspondence-ana x\u2083)\n    correspondence-synth (ESAp2 x ex x\u2081 x\u2082) = SAp (correspondence-synth ex) MAArr (correspondence-ana x\u2081)\n    correspondence-synth (ESAp3 x ex x\u2081) = SAp (correspondence-synth ex) MAArr (correspondence-ana x\u2081)\n    correspondence-synth ESEHole = SEHole\n    correspondence-synth (ESNEHole ex) = SNEHole (correspondence-synth ex)\n    correspondence-synth (ESAsc1 x _) = SAsc (correspondence-ana x)\n    correspondence-synth (ESAsc2 x) = SAsc (correspondence-ana x)\n\n    correspondence-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          \u0393 \u22a2 e <= \u03c4\n    correspondence-ana (EALam apt ex) with correspondence-ana ex\n    ... | ih = ALam apt MAArr ih\n    correspondence-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ASubsume (correspondence-synth x\u2082) x\u2083\n    correspondence-ana EAEHole = ASubsume SEHole TCHole1\n    correspondence-ana (EANEHole x) = ASubsume (SNEHole (correspondence-synth x)) TCHole1\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule correspondence where\n  mutual\n    correspondence-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0393 \u22a2 e => \u03c4\n    correspondence-synth ESConst = SConst\n    correspondence-synth (ESVar x\u2081) = SVar x\u2081\n    correspondence-synth (ESLam apt ex) with correspondence-synth ex\n    ... | ih = SLam apt ih\n    correspondence-synth (ESAp1 x x\u2081 x\u2082 x\u2083) = SAp x\u2081 MAHole (correspondence-ana x\u2083)\n    correspondence-synth (ESAp2 x ex x\u2081 x\u2082) = SAp (correspondence-synth ex) MAArr (correspondence-ana x\u2081)\n    correspondence-synth (ESAp3 x ex x\u2081) = SAp (correspondence-synth ex) MAArr (correspondence-ana x\u2081)\n    correspondence-synth ESEHole = SEHole\n    correspondence-synth (ESNEHole ex) = SNEHole (correspondence-synth ex)\n    correspondence-synth (ESAsc1 x _) = SAsc (correspondence-ana x)\n    correspondence-synth (ESAsc2 x) = SAsc (correspondence-ana x)\n\n    correspondence-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          \u0393 \u22a2 e <= \u03c4\n    correspondence-ana (EALam apt ex) with correspondence-ana ex\n    ... | ih = ALam apt MAArr ih\n    correspondence-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ASubsume (correspondence-synth x\u2082) x\u2083\n    correspondence-ana EAEHole = ASubsume SEHole TCHole1\n    correspondence-ana (EANEHole x x\u2081) = ASubsume (SNEHole (correspondence-synth x)) TCHole1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c97a711c13106d425228036a4950ce43957d9ca","subject":"Only match the higher-order argument if its image looks like what we expect.","message":"Only match the higher-order argument if its image looks like what we expect.\n","repos":"spire\/spire","old_file":"proposal\/examples\/HierMatchExt.agda","new_file":"proposal\/examples\/HierMatchExt.agda","new_contents":"{-\nDemonstrating the technique of using type-changing functions\nto pattern match against types with higher-order arguments\nlike \u03a3 and \u03a0.\n\nThe technique was originally explored to compare higher-order\nsmall arguments that occur in \u03a6 types in\nTacticsMatchingFunctionCalls.agda\n\nThis file shows that the technique scales to large higher-order\nfunctions , in a closed universe with a predicative hierarchy\nof levels that supports elimType.\n\n-}\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Fin.Props\nopen import Data.Product\nopen import Data.String\nopen import Function\nopen import Relation.Binary.PropositionalEquality hiding ( inspect )\nopen Deprecated-inspect\nmodule HierMatchExt where\n\n----------------------------------------------------------------------\n\nplusrident : (n : \u2115) \u2192 n + 0 \u2261 n\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\nrecord Universe : Set\u2081 where\n  field\n    Codes : Set\n    Meaning : Codes \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Even : \u2115 \u2192 Set where\n  ezero : Even 0\n  esuc : {n : \u2115} \u2192 Even n \u2192 Even (2 + n)\n\ndata Odd : \u2115 \u2192 Set where\n  ozero : Odd 1\n  osuc : {n : \u2115} \u2192 Odd n \u2192 Odd (2 + n)\n\n----------------------------------------------------------------------\n\ndata TypeForm (U : Universe) : Set\n\u27e6_\/_\u27e7 : (U : Universe) \u2192 TypeForm U \u2192 Set\n\ndata TypeForm U where\n  `\u22a5 `\u22a4 `Bool `\u2115 `Type : TypeForm U\n  `Fin `Even `Odd : (n : \u2115) \u2192 TypeForm U\n  `\u03a0 `\u03a3 : (A : TypeForm U)\n    (B : \u27e6 U \/ A \u27e7 \u2192 TypeForm U)\n    \u2192 TypeForm U\n  `Id : (A : TypeForm U) (x y : \u27e6 U \/ A \u27e7) \u2192 TypeForm U\n  `\u27e6_\u27e7 : (A : Universe.Codes U) \u2192 TypeForm U\n\n\u27e6 U \/ `\u22a5 \u27e7 = \u22a5\n\u27e6 U \/ `\u22a4 \u27e7 = \u22a4\n\u27e6 U \/ `Bool \u27e7 = Bool\n\u27e6 U \/ `\u2115 \u27e7 = \u2115\n\u27e6 U \/ `Fin n \u27e7 = Fin n\n\u27e6 U \/ `Even n \u27e7 = Even n\n\u27e6 U \/ `Odd n \u27e7 = Odd n\n\u27e6 U \/ `\u03a0 A B \u27e7 = (a : \u27e6 U \/ A \u27e7) \u2192 \u27e6 U \/ B a \u27e7\n\u27e6 U \/ `\u03a3 A B \u27e7 = \u03a3 \u27e6 U \/ A \u27e7 (\u03bb a \u2192 \u27e6 U \/ B a \u27e7)\n\u27e6 U \/ `Id A x y \u27e7 = _\u2261_ {A = \u27e6 U \/ A \u27e7} x y\n\u27e6 U \/ `Type \u27e7 = Universe.Codes U\n\u27e6 U \/ `\u27e6 A \u27e7 \u27e7 = Universe.Meaning U A\n\n----------------------------------------------------------------------\n\n_`\u2192_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u2192 B = `\u03a0 A (\u03bb _ \u2192 B)\n\n_`\u00d7_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u00d7 B = `\u03a3 A (\u03bb _ \u2192 B)\n\nLevel : (\u2113 : \u2115) \u2192 Universe\nLevel zero = record { Codes = \u22a5 ; Meaning = \u03bb() }\nLevel (suc \u2113) = record { Codes = TypeForm (Level \u2113)\n                       ; Meaning = \u27e6_\/_\u27e7 (Level \u2113) }\n\nType : \u2115 \u2192 Set\nType \u2113 = TypeForm (Level \u2113)\n\n\u27e6_\u2223_\u27e7 : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set\n\u27e6 \u2113 \u2223 A \u27e7 = \u27e6 Level \u2113 \/ A \u27e7\n\n`Dynamic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Dynamic \u2113 = `\u03a3 `Type `\u27e6_\u27e7\n\n`Tactic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Tactic \u2113 = `Dynamic \u2113 `\u2192 `Dynamic \u2113\n\nTactic : (\u2113 : \u2115) \u2192 Set\nTactic \u2113 = \u27e6 suc \u2113 \u2223 `Tactic \u2113 \u27e7\n\n----------------------------------------------------------------------\n\nrm-plus0-Fin : (\u2113 : \u2115) \u2192 Tactic (suc \u2113)\nrm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , b))\n  with inspect (B n)\n... | `Fin m with-\u2261 p =\n  `\u03a0 `\u2115 (\u03bb x \u2192 `Id `Type (B x) (`Fin (x + 0)))\n  `\u2192\n  `\u03a3 `\u2115 `Fin\n  ,\n  \u03bb f \u2192 n , subst Fin\n    (plusrident n)\n    (subst (\u03bb A \u2192 \u27e6 \u2113 \u2223 A \u27e7) (f n) b)\n... | Bn with-\u2261 p = (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , subst (\u03bb x \u2192 \u27e6 \u2113 \u2223 B x \u27e7) refl b))\nrm-plus0-Fin \u2113 x = x\n\n----------------------------------------------------------------------\n\n-- eg-rm-plus0-Fin : (\u2113 : \u2115) \u2192\n--   \u27e6 suc \u2113 \u2223 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7\n--         `\u2192 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin n) \u27e7 \u27e7\n-- eg-rm-plus0-Fin \u2113 (n , i) = proj\u2082\n--   (rm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7 , (n , i)))\n--   (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n","old_contents":"{-\nDemonstrating the technique of using type-changing functions\nto pattern match against types with higher-order arguments\nlike \u03a3 and \u03a0.\n\nThe technique was originally explored to compare higher-order\nsmall arguments that occur in \u03a6 types in\nTacticsMatchingFunctionCalls.agda\n\nThis file shows that the technique scales to large higher-order\nfunctions , in a closed universe with a predicative hierarchy\nof levels that supports elimType.\n\n-}\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Fin.Props\nopen import Data.Product\nopen import Data.String\nopen import Function\nopen import Relation.Binary.PropositionalEquality\nmodule HierMatchExt where\n\n----------------------------------------------------------------------\n\nplusrident : (n : \u2115) \u2192 n + 0 \u2261 n\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\nrecord Universe : Set\u2081 where\n  field\n    Codes : Set\n    Meaning : Codes \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Even : \u2115 \u2192 Set where\n  ezero : Even 0\n  esuc : {n : \u2115} \u2192 Even n \u2192 Even (2 + n)\n\ndata Odd : \u2115 \u2192 Set where\n  ozero : Odd 1\n  osuc : {n : \u2115} \u2192 Odd n \u2192 Odd (2 + n)\n\n----------------------------------------------------------------------\n\ndata TypeForm (U : Universe) : Set\n\u27e6_\/_\u27e7 : (U : Universe) \u2192 TypeForm U \u2192 Set\n\ndata TypeForm U where\n  `\u22a5 `\u22a4 `Bool `\u2115 `Type : TypeForm U\n  `Fin `Even `Odd : (n : \u2115) \u2192 TypeForm U\n  `\u03a0 `\u03a3 : (A : TypeForm U)\n    (B : \u27e6 U \/ A \u27e7 \u2192 TypeForm U)\n    \u2192 TypeForm U\n  `Id : (A : TypeForm U) (x y : \u27e6 U \/ A \u27e7) \u2192 TypeForm U\n  `\u27e6_\u27e7 : (A : Universe.Codes U) \u2192 TypeForm U\n\n\u27e6 U \/ `\u22a5 \u27e7 = \u22a5\n\u27e6 U \/ `\u22a4 \u27e7 = \u22a4\n\u27e6 U \/ `Bool \u27e7 = Bool\n\u27e6 U \/ `\u2115 \u27e7 = \u2115\n\u27e6 U \/ `Fin n \u27e7 = Fin n\n\u27e6 U \/ `Even n \u27e7 = Even n\n\u27e6 U \/ `Odd n \u27e7 = Odd n\n\u27e6 U \/ `\u03a0 A B \u27e7 = (a : \u27e6 U \/ A \u27e7) \u2192 \u27e6 U \/ B a \u27e7\n\u27e6 U \/ `\u03a3 A B \u27e7 = \u03a3 \u27e6 U \/ A \u27e7 (\u03bb a \u2192 \u27e6 U \/ B a \u27e7)\n\u27e6 U \/ `Id A x y \u27e7 = _\u2261_ {A = \u27e6 U \/ A \u27e7} x y\n\u27e6 U \/ `Type \u27e7 = Universe.Codes U\n\u27e6 U \/ `\u27e6 A \u27e7 \u27e7 = Universe.Meaning U A\n\n----------------------------------------------------------------------\n\n_`\u2192_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u2192 B = `\u03a0 A (\u03bb _ \u2192 B)\n\n_`\u00d7_ : \u2200{U} (A B : TypeForm U) \u2192 TypeForm U\nA `\u00d7 B = `\u03a3 A (\u03bb _ \u2192 B)\n\nLevel : (\u2113 : \u2115) \u2192 Universe\nLevel zero = record { Codes = \u22a5 ; Meaning = \u03bb() }\nLevel (suc \u2113) = record { Codes = TypeForm (Level \u2113)\n                       ; Meaning = \u27e6_\/_\u27e7 (Level \u2113) }\n\nType : \u2115 \u2192 Set\nType \u2113 = TypeForm (Level \u2113)\n\n\u27e6_\u2223_\u27e7 : (\u2113 : \u2115) \u2192 Type \u2113 \u2192 Set\n\u27e6 \u2113 \u2223 A \u27e7 = \u27e6 Level \u2113 \/ A \u27e7\n\n`Dynamic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Dynamic \u2113 = `\u03a3 `Type `\u27e6_\u27e7\n\n`Tactic : (\u2113 : \u2115) \u2192 Type (suc \u2113)\n`Tactic \u2113 = `Dynamic \u2113 `\u2192 `Dynamic \u2113\n\nTactic : (\u2113 : \u2115) \u2192 Set\nTactic \u2113 = \u27e6 suc \u2113 \u2223 `Tactic \u2113 \u27e7\n\n----------------------------------------------------------------------\n\nrm-plus0-Fin : (\u2113 : \u2115) \u2192 Tactic (suc \u2113)\nrm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 B \u27e7 , (n , b)) =\n  `\u03a0 `\u2115 (\u03bb x \u2192 `Id `Type (B x) (`Fin (x + 0)))\n  `\u2192\n  `\u03a3 `\u2115 `Fin\n  ,\n  \u03bb f \u2192 n , subst Fin\n    (plusrident n)\n    (subst (\u03bb A \u2192 \u27e6 \u2113 \u2223 A \u27e7) (f n) b)\nrm-plus0-Fin \u2113 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin : (\u2113 : \u2115) \u2192\n  \u27e6 suc \u2113 \u2223 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7\n        `\u2192 `\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin n) \u27e7 \u27e7\neg-rm-plus0-Fin \u2113 (n , i) = proj\u2082\n  (rm-plus0-Fin \u2113 (`\u27e6 `\u03a3 `\u2115 (\u03bb n \u2192 `Fin (n + zero)) \u27e7 , (n , i)))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1b96b34f502b64afe8b0080793725a878bdd13d9","subject":"Added N-unf-ho.","message":"Added N-unf-ho.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-ind :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-ind-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ind\/N-ind-ho\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind-ho\n\n  N-ind-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-ind-ho' = N-ind\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-unf : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-unf = N-ind A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  -- Higher-order version.\n  N-unf-ho : \u2200 {n} \u2192 N n \u2192 NatF N n\n  N-unf-ho = N-unf\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-ind.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-ind B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-ind\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-ind A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-ind as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d        \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ\u2081 d) + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-ind A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-unf.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-unf Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-ind.\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-ind :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-ind-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ind\/N-ind-ho\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind-ho\n\n  N-ind-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-ind-ho' = N-ind\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-unf : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-unf = N-ind A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-ind.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-ind B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-ind\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-ind A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-ind as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d        \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ\u2081 d) + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-ind A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-unf.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-unf Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-ind.\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93e6584220412202dbb64437dd86bca11691e722","subject":"Added x+1\u2264x\u223810+11 and x\u226489\u2192x+11>100\u2192\u22a5 (by Ana Bove).","message":"Added x+1\u2264x\u223810+11 and x\u226489\u2192x+11>100\u2192\u22a5 (by Ana Bove).\n\nIgnore-this: b1dbc3a9d82d80f5abcd37691da80221\n\ndarcs-hash:20110211225502-3bd4e-2c25dac7312cce72f06454d59caa9d09ecc9426b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n  n100' : eleven + eighty-nine \u2261 one-hundred\n  n100  : eighty-nine + eleven \u2261 one-hundred\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x-y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 n100 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n  n100' : eleven + eighty-nine \u2261 one-hundred\n  n100  : eighty-nine + eleven \u2261 one-hundred\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1c937f93f2d8a6adf90e821c16730959c7d7b238","subject":"Removed nil.","message":"Removed nil.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Base.agda","new_file":"src\/fot\/FOTC\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfixr 8 _\u2237_\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\n-- Common definitions.\nopen import Common.DefinitionsATP public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | nil  | cons  | null | head   | tail\n--      | loop\n\npostulate\n  _\u00b7_                    : D \u2192 D \u2192 D  -- FOTC application.\n  true false if          : D          -- FOTC partial Booleans.\n  zero succ pred iszero  : D          -- FOTC partial natural numbers.\n  [] cons head tail null : D          -- FOTC lists.\n  loop                   : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 n = succ \u00b7 n\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 n = pred \u00b7 n\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 n = iszero \u00b7 n\n  -- {-# ATP definition iszero\u2081 #-}\n\n  _\u2237_  : D \u2192 D \u2192 D\n  x \u2237 xs = cons \u00b7 x \u00b7 xs\n  -- {-# ATP definition _\u2237_ #-}\n\n  head\u2081 : D \u2192 D\n  head\u2081 xs = head \u00b7 xs\n  -- {-# ATP definition head\u2081 #-}\n\n  tail\u2081 : D \u2192 D\n  tail\u2081 xs = tail \u00b7 xs\n  -- {-# ATP definition tail\u2081 #-}\n\n  null\u2081 : D \u2192 D\n  null\u2081 xs = null \u00b7 xs\n  -- {-# ATP definition null\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as non-logical axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for Booleans.\npostulate\n  -- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n  -- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred \u00b7 zero     \u2261 zero\n  -- pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\npostulate\n  -- iszero-0 :       iszero \u00b7 zero       \u2261 true\n  -- iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 n \u2192 iszero\u2081 (succ\u2081 n) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rules for null.\npostulate\n  -- null-[] :          null \u00b7 nil             \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  null-[] :          null\u2081 []       \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\npostulate\n--  head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\npostulate\n--  tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n-- Conversion rule for loop.\n--\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a first-order logic axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2262false : true \u2262 false\n--  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ \u00b7 n\n  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ\u2081 n\n{-# ATP axiom true\u2262false 0\u2262S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfixr 8 _\u2237_\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\n-- Common definitions.\nopen import Common.DefinitionsATP public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | nil  | cons  | null | head   | tail\n--      | loop\n\npostulate\n  _\u00b7_                     : D \u2192 D \u2192 D  -- FOTC application.\n  true false if           : D          -- FOTC partial Booleans.\n  zero succ pred iszero   : D          -- FOTC partial natural numbers.\n  nil cons head tail null : D          -- FOTC lists.\n  loop                    : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 n = succ \u00b7 n\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 n = pred \u00b7 n\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 n = iszero \u00b7 n\n  -- {-# ATP definition iszero\u2081 #-}\n\n  [] : D\n  [] = nil\n  -- {-# ATP definition [] #-}\n\n  _\u2237_  : D \u2192 D \u2192 D\n  x \u2237 xs = cons \u00b7 x \u00b7 xs\n  -- {-# ATP definition _\u2237_ #-}\n\n  head\u2081 : D \u2192 D\n  head\u2081 xs = head \u00b7 xs\n  -- {-# ATP definition head\u2081 #-}\n\n  tail\u2081 : D \u2192 D\n  tail\u2081 xs = tail \u00b7 xs\n  -- {-# ATP definition tail\u2081 #-}\n\n  null\u2081 : D \u2192 D\n  null\u2081 xs = null \u00b7 xs\n  -- {-# ATP definition null\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as non-logical axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for Booleans.\npostulate\n  -- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n  -- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred \u00b7 zero     \u2261 zero\n  -- pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\npostulate\n  -- iszero-0 :       iszero \u00b7 zero       \u2261 true\n  -- iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 n \u2192 iszero\u2081 (succ\u2081 n) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rules for null.\npostulate\n  -- null-[] :          null \u00b7 nil             \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  null-[] :          null\u2081 []       \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\npostulate\n--  head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\npostulate\n--  tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n-- Conversion rule for loop.\n--\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a first-order logic axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2262false : true \u2262 false\n--  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ \u00b7 n\n  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ\u2081 n\n{-# ATP axiom true\u2262false 0\u2262S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b1479e21c569e836f0a8509f99b5b4c287c68821","subject":"Added missing local hint.","message":"Added missing local hint.\n\nIgnore-this: 82466ef5652a782ebf50430560b5d833\n\ndarcs-hash:20110219052705-3bd4e-13b3ac8b81002750798d46f8af10fd7bf4555c01.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/McCarthy91\/ArithmeticATP.agda","new_file":"src\/LTC\/Program\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule LTC.Program.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.UnaryNumbers\nopen import LTC.Data.Nat.UnaryNumbers.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one                    one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  102>100  : GT hundred-two                    one-hundred\n  103>100  : GT hundred-three                  one-hundred\n  104>100  : GT hundred-four                   one-hundred\n  105>100  : GT hundred-five                   one-hundred\n  106>100  : GT hundred-six                    one-hundred\n  107>100  : GT hundred-seven                  one-hundred\n  108>100  : GT hundred-eight                  one-hundred\n  109>100  : GT hundred-nine                   one-hundred\n  110>100  : GT hundred-ten                    one-hundred\n  111>100' : GT hundred-eleven                 one-hundred\n  111>100  : GT (one-hundred + eleven)         one-hundred\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 102>100 #-}\n{-# ATP prove 103>100 #-}\n{-# ATP prove 104>100 #-}\n{-# ATP prove 105>100 #-}\n{-# ATP prove 106>100 #-}\n{-# ATP prove 107>100 #-}\n{-# ATP prove 108>100 #-}\n{-# ATP prove 109>100 #-}\n{-# ATP prove 110>100 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven)  one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven)   one-hundred\n  95+11>100 : GT (ninety-five + eleven)  one-hundred\n  94+11>100 : GT (ninety-four + eleven)  one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven)   one-hundred\n  91+11>100 : GT (ninety-one + eleven)   one-hundred\n  90+11>100 : GT (ninety + eleven)       one-hundred\n{-# ATP prove 99+11>100 110>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101>100' 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\nx+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\nx+11-N Nn = +-N Nn N11\n\nx+11\u223810\u2261Sx : \u2200 {n} \u2192 N n \u2192 (n + eleven) \u2238 ten \u2261 succ n\nx+11\u223810\u2261Sx Nn = [x+Sy]\u2238y\u2261Sx Nn N10\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 N11 +-N \u2238-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule LTC.Program.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.UnaryNumbers\nopen import LTC.Data.Nat.UnaryNumbers.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one                    one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  102>100  : GT hundred-two                    one-hundred\n  103>100  : GT hundred-three                  one-hundred\n  104>100  : GT hundred-four                   one-hundred\n  105>100  : GT hundred-five                   one-hundred\n  106>100  : GT hundred-six                    one-hundred\n  107>100  : GT hundred-seven                  one-hundred\n  108>100  : GT hundred-eight                  one-hundred\n  109>100  : GT hundred-nine                   one-hundred\n  110>100  : GT hundred-ten                    one-hundred\n  111>100' : GT hundred-eleven                 one-hundred\n  111>100  : GT (one-hundred + eleven)         one-hundred\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 102>100 #-}\n{-# ATP prove 103>100 #-}\n{-# ATP prove 104>100 #-}\n{-# ATP prove 105>100 #-}\n{-# ATP prove 106>100 #-}\n{-# ATP prove 107>100 #-}\n{-# ATP prove 108>100 #-}\n{-# ATP prove 109>100 #-}\n{-# ATP prove 110>100 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven)  one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven)   one-hundred\n  95+11>100 : GT (ninety-five + eleven)  one-hundred\n  94+11>100 : GT (ninety-four + eleven)  one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven)   one-hundred\n  91+11>100 : GT (ninety-one + eleven)   one-hundred\n  90+11>100 : GT (ninety + eleven)       one-hundred\n{-# ATP prove 99+11>100 110>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101>100' 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\nx+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\nx+11-N Nn = +-N Nn N11\n\nx+11\u223810\u2261Sx : \u2200 {n} \u2192 N n \u2192 (n + eleven) \u2238 ten \u2261 succ n\nx+11\u223810\u2261Sx Nn = [x+Sy]\u2238y\u2261Sx Nn N10\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 N11 +-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 100\u226189+11 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"accc893b55bf3a37d7657a72ec9ac1e69f23636d","subject":"Internal error in agda","message":"Internal error in agda\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/LinFunContructw.agda","new_file":"agda\/LinFunContructw.agda","new_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n  \nmodule _ where\n\n  open IndexLLProp\n\n  boo : \u2200{ i u ll pll ell cs} \u2192 \u2200 s \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL ell ll) \u2192 \u2200 \u00achob \u00achoh\n        \u2192 let bind = \u00acho-shr-morph s eq ind \u00achob in\n          let hind = \u00acho-shr-morph s eq lind \u00achoh in\n          Ordered\u1d62 bind hind \u2192 Ordered\u1d62 ind lind\n  boo \u2193 () ind lind \u00achob \u00achoh ord\n\n  boo (s \u2190\u2227) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2193)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2193)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (a\u2264\u1d62b y)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (b\u2264\u1d62a y)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2193) \n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq lind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s lind)\n    \u00acnho x = \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo {ll = lll \u2227 rll} (s \u2190\u2227) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s \n  boo {ll = lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 with shrink rll (fillAllLower rll) | shr-fAL-id rll\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b x) | \u2205 | .rll | refl = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a x) | \u2205 | .rll | refl = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)\n  boo {ll = lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u00ac\u2205 x with shrink rll (fillAllLower rll) | shr-fAL-id rll\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | .rll | refl = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | .rll | refl = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)\n  boo (\u2227\u2192 s) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2227\u2192\u2193)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2227\u2192\u2193)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (a\u2264\u1d62b y)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (b\u2264\u1d62a y)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2227\u2192\u2193) \n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq lind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s lind)\n    \u00acnho x = \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s \n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 with shrink lll (fillAllLower lll) | shr-fAL-id lll\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b x) | \u2205 | .lll | refl = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a x) | \u2205 | .lll | refl = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)\n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x with shrink lll (fillAllLower lll) | shr-fAL-id lll\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | .lll | refl = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | .lll | refl = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)\n\n\n\n\n  boo (s \u2190\u2227\u2192 s\u2081) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s | complL\u209b s\u2081\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] | e = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind)) where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u2205 with boo s ieq ind lind (\u03bb x\u2081 \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x\u2081)) (\u03bb x\u2081 \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x\u2081)) ord\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u2205 | a\u2264\u1d62b y = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u2205 | b\u2264\u1d62a y = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 with boo s ieq ind lind (\u03bb x\u2081 \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x\u2081)) (\u03bb x\u2081 \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x\u2081)) (a\u2264\u1d62b y)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 | a\u2264\u1d62b z = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 z)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 | b\u2264\u1d62a z = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 z)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 with boo s ieq ind lind (\u03bb x\u2081 \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x\u2081)) (\u03bb x\u2081 \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x\u2081)) (b\u2264\u1d62a y)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 | a\u2264\u1d62b z = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 z)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 | b\u2264\u1d62a z = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 z)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s | complL\u209b s\u2081\n  ... | g | t | e = ?\n  boo (s \u2190\u2227\u2192 s\u2081) eq (\u2227\u2192 ind) lind \u00achob \u00achoh ord = {!!}\n\n  boo (s \u2190\u2228) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (\u2228\u2192 s) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2228\u2192 s\u2081) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2202) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (\u2202\u2192 s) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2202\u2192 s\u2081) eq ind lind \u00achob \u00achoh ord = {!!}\n\n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n    hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n          \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n          \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n          \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n    hf2 \u2205 mnho s lf eqs = \u2205\n    hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (\u03bb z \u2192 tranLFMIndexLL lf (\u00ac\u2205 z)) x\n    hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n    hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n\n\n  data MLFun {i u ll rll n dt df} (mind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s mind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    \u2205 : \u2200{ts} \u2192 (\u2205 \u2261 mind) \u2192 (complL\u209b s \u2261 \u2205)\n        \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun mind s m\u00acho lf\n    \u00ac\u2205\u2205 : \u2200 {ind cs} \u2192 (\u00ac\u2205 ind \u2261 mind) \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n          \u2192 (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n          \u2192 (cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) rll\n          \u2192 MLFun mind s m\u00acho lf \n    \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf mind in\n           (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 mind s m\u00acho lf eqs) cll))\n           \u2192 MLFun mind s m\u00acho lf \n           --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n           -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n    -- s here does contain the ind.\n  -- TODO IMPORTANT After test , we need to remove \u2282 that contain I in the first place.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test ind s mnho lf with complL\u209b s | inspect complL\u209b s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u2205 | [ r ] = IMPOSSIBLE\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] with complL\u209b x | inspect complL\u209b x\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u2205 | [ t ] = \u2205 refl e (sym r) t\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf | \u2205 | [ e ] = \u22a5-elim (\u00acho (compl\u2261\u2205\u21d2ho s e x)) \n\n  test \u2205 s mnho I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n  test (\u00ac\u2205 x\u2081) s (\u00ac\u2205 \u00acho) I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  ... | \u2205 indeq ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n  ... | \u00ac\u2205\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00ac\u2205\u00ac\u2205 eq tseq ceqo ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n  ... | \u2205 indeq ceq eqs ceqo with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  ... | \u00ac\u2205 mx | r with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u2205 = IMPOSSIBLE -- IMPOSSIBLE and not provable with the current information we have.\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x with compl\u2261\u00ac\u2205\u21d2replace-compl\u2261\u00ac\u2205 s lind eq teq ceq x\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , proj\u2084 with complL\u209b (replacePartOf s to x at ind)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 refl eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , () | .\u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi1 eqs1 cll1 t1\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 {ts = ts} indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 {cs} ceqi1 eqs1 ceqo1 t1\n    = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 ((\u03bb cll \u2192 subst (\u03bb z \u2192 LFun z (shrink rll _)) (shrink-repl\u2261\u2205 s lind eq teq ceq _ ceqo t) (t1 cll))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    t : complL\u209b (replacePartOf s to ts at lind) \u2261 \u00ac\u2205 cs\n    t with trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    ... | g with replacePartOf s to ts at lind\n    t | refl | e = ceqi1\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq refl ceqo | \u2205 | [ () ] | .(\u00ac\u2205 _)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind\u2081} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t\n       with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi () ceqo t | \u2205 | [ meq ] | \u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ () ] | \u00ac\u2205 _\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u2205 indeq ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl ceqo t | \u00ac\u2205 .(replacePartOf s to _ at lind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | .(\u00ac\u2205 _) with complL\u209b ts | ct where\n    m = replacePartOf s to ts at lind\n    mind = subst (\u03bb z \u2192 IndexLL z (replLL ll lind ell)) (replLL-\u2193 lind) ((a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))\n    ct = compl\u2261\u2205\u21d2compltr\u2261\u2205 m ceq1 mind (sym (tr-repl\u21d2id s lind ts))\n  test {rll = rll} {ll} {n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl () t | \u00ac\u2205 .(replacePartOf s to ts at ind) | [ refl ] | \u2205 indeq ceq1 eqs1 ceqo1 | .(\u00ac\u2205 ts) | .\u2205 | refl\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi\u2081 eqs\u2081 cll1 x\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 ceqi1 eqs1 ceqo1 t1\n      = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 (\u03bb cll \u2192 _\u2282_ {ind = hind} (t cll) (subst (\u03bb z \u2192 LFun z _) (sym (ho-shrink-repl-comm s lind eq teq ceqi ceqo w ceqi1)) (t1 cll))) where\n    w = trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    hind = ho-shr-morph s eq lind (sym teq) ceqi\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with isLTi lind ind\n  ... | no \u00acp with test {n = n} {dt} {df} nind mx (\u00ac\u2205 n\u00acho) lf\u2081 where\n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    nind = \u00ac\u2205 (\u00acord-morph ind lind ell (flipNotOrd\u1d62 nord))\n    n\u00acho = \u00acord&\u00acho-del\u21d2\u00acho ind s \u00acho lind nord (sym meq)\n  ... | \u2205 () ceq eqs ceqo\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u2205 {ind = tind} indeq ceqi eqs cll t\n    = \u00ac\u2205\u2205 refl eq tseq cll (_\u2282_ {ind = sind} lf {!best!}) where \n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    hind = \u00acho-shr-morph s eq lind ((trunc\u2261\u2205\u21d2\u00acho s lind teq))\n    bind = \u00acho-shr-morph s eq ind \u00acho\n    sind = \u00acord-morph hind bind cll {!!}\n    best = replLL-\u00acordab\u2261ba bind cll hind ell {!!}\n\n    find = \u00acord-morph ind lind ell (flipNotOrd\u1d62 nord)\n    nind = \u00acord-morph lind ind cll nord\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t = {!!}\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | yes p = {!!}\n  test (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] = {!!}\n  test ind s mnho (tr ltr lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (com df\u2081 lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (call x\u2081) | \u00ac\u2205 x | eq = {!!}\n\n\n\n\n\n--  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | \u00acI\u00ac\u2205 ceqi () ceqo x | .(\u00ac\u2205 _)\n--\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n--\n--\n---- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n----   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n----     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n---- --    is = test \u2205 {!!} \u2205 lf\u2081\n----   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n----   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n----   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n----   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n----     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n----     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n----   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n--  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n--  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n--  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n--  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n--  \n--  \n  \n  \n  \n  \n  \n","old_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n  \nmodule _ where\n\n  open IndexLLProp\n\n  boo : \u2200{ i u ll pll ell cs} \u2192 \u2200 s \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL ell ll) \u2192 \u2200 \u00achob \u00achoh\n        \u2192 let bind = \u00acho-shr-morph s eq ind \u00achob in\n          let hind = \u00acho-shr-morph s eq lind \u00achoh in\n          Ordered\u1d62 bind hind \u2192 Ordered\u1d62 ind lind\n  boo \u2193 () ind lind \u00achob \u00achoh ord\n\n  boo (s \u2190\u2227) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2193)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2193)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (a\u2264\u1d62b y)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (b\u2264\u1d62a y)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2193) \n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq lind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s lind)\n    \u00acnho x = \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo {ll = lll \u2227 rll} (s \u2190\u2227) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s \n  boo {ll = lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 with shrink rll (fillAllLower rll) | shr-fAL-id rll\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b x) | \u2205 | .rll | refl = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a x) | \u2205 | .rll | refl = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)\n  boo {ll = lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u00ac\u2205 x with shrink rll (fillAllLower rll) | shr-fAL-id rll\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | .rll | refl = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | .rll | refl = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)\n  boo (\u2227\u2192 s) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2227\u2192\u2193)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2227\u2192\u2193)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (a\u2264\u1d62b y)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (b\u2264\u1d62a y)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2227\u2192\u2193) \n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq lind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s lind)\n    \u00acnho x = \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s \n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 with shrink lll (fillAllLower lll) | shr-fAL-id lll\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b x) | \u2205 | .lll | refl = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a x) | \u2205 | .lll | refl = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)\n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x with shrink lll (fillAllLower lll) | shr-fAL-id lll\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | .lll | refl = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | .lll | refl = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)\n\n\n\n\n  boo (s \u2190\u2227\u2192 s\u2081) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s | complL\u209b s\u2081\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] | e = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind)) where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u2205 with boo s ieq ind lind {!!} {!!} ord\n  ... | g = {!!}\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 = {!!}\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2227\u2192 s\u2081) eq (\u2227\u2192 ind) lind \u00achob \u00achoh ord = {!!}\n\n  boo (s \u2190\u2228) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (\u2228\u2192 s) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2228\u2192 s\u2081) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2202) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (\u2202\u2192 s) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2202\u2192 s\u2081) eq ind lind \u00achob \u00achoh ord = {!!}\n\n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n    hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n          \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n          \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n          \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n    hf2 \u2205 mnho s lf eqs = \u2205\n    hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (\u03bb z \u2192 tranLFMIndexLL lf (\u00ac\u2205 z)) x\n    hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n    hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n\n\n  data MLFun {i u ll rll n dt df} (mind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s mind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    \u2205 : \u2200{ts} \u2192 (\u2205 \u2261 mind) \u2192 (complL\u209b s \u2261 \u2205)\n        \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun mind s m\u00acho lf\n    \u00ac\u2205\u2205 : \u2200 {ind cs} \u2192 (\u00ac\u2205 ind \u2261 mind) \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n          \u2192 (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n          \u2192 (cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) rll\n          \u2192 MLFun mind s m\u00acho lf \n    \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf mind in\n           (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 mind s m\u00acho lf eqs) cll))\n           \u2192 MLFun mind s m\u00acho lf \n           --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n           -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n    -- s here does contain the ind.\n  -- TODO IMPORTANT After test , we need to remove \u2282 that contain I in the first place.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test ind s mnho lf with complL\u209b s | inspect complL\u209b s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u2205 | [ r ] = IMPOSSIBLE\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] with complL\u209b x | inspect complL\u209b x\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u2205 | [ t ] = \u2205 refl e (sym r) t\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf | \u2205 | [ e ] = \u22a5-elim (\u00acho (compl\u2261\u2205\u21d2ho s e x)) \n\n  test \u2205 s mnho I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n  test (\u00ac\u2205 x\u2081) s (\u00ac\u2205 \u00acho) I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  ... | \u2205 indeq ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n  ... | \u00ac\u2205\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00ac\u2205\u00ac\u2205 eq tseq ceqo ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n  ... | \u2205 indeq ceq eqs ceqo with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  ... | \u00ac\u2205 mx | r with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u2205 = IMPOSSIBLE -- IMPOSSIBLE and not provable with the current information we have.\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x with compl\u2261\u00ac\u2205\u21d2replace-compl\u2261\u00ac\u2205 s lind eq teq ceq x\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , proj\u2084 with complL\u209b (replacePartOf s to x at ind)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 refl eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , () | .\u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi1 eqs1 cll1 t1\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 {ts = ts} indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 {cs} ceqi1 eqs1 ceqo1 t1\n    = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 ((\u03bb cll \u2192 subst (\u03bb z \u2192 LFun z (shrink rll _)) (shrink-repl\u2261\u2205 s lind eq teq ceq _ ceqo t) (t1 cll))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    t : complL\u209b (replacePartOf s to ts at lind) \u2261 \u00ac\u2205 cs\n    t with trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    ... | g with replacePartOf s to ts at lind\n    t | refl | e = ceqi1\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq refl ceqo | \u2205 | [ () ] | .(\u00ac\u2205 _)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind\u2081} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t\n       with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi () ceqo t | \u2205 | [ meq ] | \u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ () ] | \u00ac\u2205 _\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u2205 indeq ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl ceqo t | \u00ac\u2205 .(replacePartOf s to _ at lind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | .(\u00ac\u2205 _) with complL\u209b ts | ct where\n    m = replacePartOf s to ts at lind\n    mind = subst (\u03bb z \u2192 IndexLL z (replLL ll lind ell)) (replLL-\u2193 lind) ((a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))\n    ct = compl\u2261\u2205\u21d2compltr\u2261\u2205 m ceq1 mind (sym (tr-repl\u21d2id s lind ts))\n  test {rll = rll} {ll} {n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl () t | \u00ac\u2205 .(replacePartOf s to ts at ind) | [ refl ] | \u2205 indeq ceq1 eqs1 ceqo1 | .(\u00ac\u2205 ts) | .\u2205 | refl\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi\u2081 eqs\u2081 cll1 x\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 ceqi1 eqs1 ceqo1 t1\n      = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 (\u03bb cll \u2192 _\u2282_ {ind = hind} (t cll) (subst (\u03bb z \u2192 LFun z _) (sym (ho-shrink-repl-comm s lind eq teq ceqi ceqo w ceqi1)) (t1 cll))) where\n    w = trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    hind = ho-shr-morph s eq lind (sym teq) ceqi\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with isLTi lind ind\n  ... | no \u00acp with test {n = n} {dt} {df} nind mx (\u00ac\u2205 n\u00acho) lf\u2081 where\n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    nind = \u00ac\u2205 (\u00acord-morph ind lind ell (flipNotOrd\u1d62 nord))\n    n\u00acho = \u00acord&\u00acho-del\u21d2\u00acho ind s \u00acho lind nord (sym meq)\n  ... | \u2205 () ceq eqs ceqo\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u2205 {ind = tind} indeq ceqi eqs cll t\n    = \u00ac\u2205\u2205 refl eq tseq cll (_\u2282_ {ind = sind} lf {!best!}) where \n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    hind = \u00acho-shr-morph s eq lind ((trunc\u2261\u2205\u21d2\u00acho s lind teq))\n    bind = \u00acho-shr-morph s eq ind \u00acho\n    sind = \u00acord-morph hind bind cll {!!}\n    best = replLL-\u00acordab\u2261ba bind cll hind ell {!!}\n\n    find = \u00acord-morph ind lind ell (flipNotOrd\u1d62 nord)\n    nind = \u00acord-morph lind ind cll nord\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t = {!!}\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | yes p = {!!}\n  test (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] = {!!}\n  test ind s mnho (tr ltr lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (com df\u2081 lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (call x\u2081) | \u00ac\u2205 x | eq = {!!}\n\n\n\n\n\n--  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | \u00acI\u00ac\u2205 ceqi () ceqo x | .(\u00ac\u2205 _)\n--\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n--\n--\n---- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n----   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n----     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n---- --    is = test \u2205 {!!} \u2205 lf\u2081\n----   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n----   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n----   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n----   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n----     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n----     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n----   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n--  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n--  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n--  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n--  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n--  \n--  \n  \n  \n  \n  \n  \n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"fb73c4c68d0d9e00dac3fd35d9b93d3d01182522","subject":"Use HOAS-enabled helpers for nested abstractions.","message":"Use HOAS-enabled helpers for nested abstractions.\n\nOld-commit-hash: cffcfea69d7b1877dc5d63b49ee0ed9f24dce0d2\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Change\/Term.agda","new_file":"Popl14\/Change\/Term.agda","new_contents":"module Popl14.Change.Term where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Syntax.Context.Popl14 public\nopen import Data.Integer\n\nopen import Popl14.Syntax.Term public\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-term {int} =\n  abs\u2082 (\u03bb \u0394x x \u2192 add x \u0394x)\napply-term {bag} =\n  abs\u2082 (\u03bb \u0394x x \u2192 union x \u0394x)\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app f x \u2295 app (app \u0394f x) (x \u229d x))))\n\ndiff-term {int} =\n  abs\u2082 (\u03bb x y \u2192 add x (minus y))\ndiff-term {bag} =\n  abs\u2082 (\u03bb x y \u2192 union x (negate y))\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app g (x \u2295 \u0394x) \u229d app f x))))\n","old_contents":"module Popl14.Change.Term where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Syntax.Context.Popl14 public\nopen import Data.Integer\n\nopen import Popl14.Syntax.Term public\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-term {int} =\n  let \u0394x = var (that this)\n      x  = var this\n  in abs (abs (add x \u0394x))\napply-term {bag} =\n  let \u0394x = var (that this)\n      x  = var this\n  in abs (abs (union x \u0394x))\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app f x \u2295 app (app \u0394f x) (x \u229d x))))\n\ndiff-term {int} =\n  let x = var (that this)\n      y = var this\n  in abs (abs (add x (minus y)))\ndiff-term {bag} =\n  let x = var (that this)\n      y = var this\n  in abs (abs (union x (negate y)))\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app g (x \u2295 \u0394x) \u229d app f x))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f1f7df2dcdfa464f6aa33d7715b3bde8b94e46f9","subject":"Also curry index arguments of eliminators.","message":"Also curry index arguments of eliminators.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  C : (A : El\u1d40 P) \u2192 Tag E \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg (Tag E) (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 CurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\nuncurryEl\u1d40 End X x tt = x\nuncurryEl\u1d40 (Arg A B) X f (a , b) = uncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (f a) b\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El\u1d30 D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R p t = curryEl\u1d30 (Data.C R p t)\n  (\u03bc (Data.D R p))\n  (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) M init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\nindCurried D M f i x =\n  ind D M (uncurryHyps D (\u03bc D) M init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc (Data.D R p) i \u2192 Set) M in\n  CurriedHyps (Data.C R p t) (\u03bc (Data.D R p)) unM (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in UncurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x))\nelimUncurried R p M cs =\n  let D = Data.D R p\n      unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x) \u03bb i x \u2192\n  indCurried (Data.D R p) unM\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p M)\n     (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x))\nelim R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in\n    (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n    \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p M)\n       (CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 (x : \u03bc D i) \u2192 unM i x)))\n  (\u03bb p M \u2192 curryBranches\n    (elimUncurried R p M))\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} (m : \u2115) (xs : Vec A m) (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} = elim VecR (\u03bb m xs \u2192 (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons (add m n) x (ih n ys))\n\nconcat : {A : Set} (m n : \u2115) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} m = elim VecR (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb n xs xss ih \u2192 append m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El\u1d30 D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 D (\u03bc D) i) (ihs : Hyps D (\u03bc D) M i xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\n\nprove : {I : Set} (D E : Desc I)\n  (M : (i : I) \u2192 \u03bc E i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 E (\u03bc E) i) (ihs : Hyps E (\u03bc E) M i xs) \u2192 M i (init xs))\n  (i : I) (xs : El\u1d30 D (\u03bc E) i) \u2192 Hyps D (\u03bc E) M i xs\n\nind D M \u03b1 i (init xs) = \u03b1 i xs (prove D D M \u03b1 i xs)\n\nprove (End j) E M \u03b1 i q = tt\nprove (Rec j D) E M \u03b1 i (x , xs) = ind E M \u03b1 j x , prove D E M \u03b1 i xs\nprove (Arg A B) E M \u03b1 i (a , xs) = prove (B a) E M \u03b1 i xs\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  C : (A : El\u1d40 P) \u2192 Tag E \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg (Tag E) (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 CurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\nuncurryEl\u1d40 End X x tt = x\nuncurryEl\u1d40 (Arg A B) X f (a , b) = uncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (f a) b\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El\u1d30 D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl\u1d30 D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R p t = curryEl\u1d30 (Data.C R p t)\n  (\u03bc (Data.D R p))\n  (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n  (injUncurried R)\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (M : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) M init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 M i x\nindCurried D M f i x =\n  ind D M (uncurryHyps D (\u03bc D) M init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 Tag (Data.E R) \u2192 Set\nSumCurriedHyps R p M t =\n  CurriedHyps (Data.C R p t) (\u03bc (Data.D R p)) M (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : UncurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 UncurriedBranches (Data.E R)\n    (SumCurriedHyps R p M)\n    (\u2200 i (x : \u03bc D i) \u2192 M i x)\nelimUncurried R p M cs i x =\n  indCurried (Data.D R p) M\n    (case (SumCurriedHyps R p M) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p in\n  (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n  \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n  in CurriedBranches (Data.E R)\n     (SumCurriedHyps R p unM)\n     (\u2200 i (x : \u03bc D i) \u2192 unM i x)\nelim R = icurryEl\u1d40 (Data.P R)\n  (\u03bb p \u2192 let D = Data.D R p in\n    (M : CurriedEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set))\n    \u2192 let unM = uncurryEl\u1d40 (Data.I R p) (\u03bb i \u2192 \u03bc D i \u2192 Set) M\n    in CurriedBranches (Data.E R)\n       (SumCurriedHyps R p unM)\n       (\u2200 i (x : \u03bc D i) \u2192 unM i x))\n  (\u03bb p M \u2192 curryBranches\n    (elimUncurried R p\n      (uncurryEl\u1d40 (Data.I R p)\n        ((\u03bb i \u2192 \u03bc (Data.D R p) i \u2192 Set)) M)))\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n  tt\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n  tt\n\nappend : {A : Set} (m : \u2115) (xs : Vec A m) (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} m xs = elim VecR (\u03bb m xs \u2192 (n : \u2115) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons (add m n) x (ih n ys))\n  (m , tt) xs\n\nconcat : {A : Set} (m n : \u2115) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} m n xss = elim VecR (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb n xs xss ih \u2192 append m xs (mult n m) ih)\n  (n , tt) xss\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9d1e02a049e0eac1e796dc1d74d47064871effb3","subject":"Remove power from the guessing game","message":"Remove power from the guessing game\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\nmodule Guess (Power : Set) (coins : Power \u2192 Coins) (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Power \u2192 Set\n  Oracle power = \u2203 (\u03bb (A : GuessAdv (coins power)) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess Coins id prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"64c75b3b67cf69a5efa9a5d4f14ab7deb2f1e5ae","subject":"Fin: +Fin\u25b9\ud835\udfd8, +\ud835\udfd8\u25b9Fin, +Fin\u25b9\ud835\udfd9, +\ud835\udfd9\u25b9Fin, +Fin\u25b9\ud835\udfda, +\ud835\udfda\u25b9Fin","message":"Fin: +Fin\u25b9\ud835\udfd8, +\ud835\udfd8\u25b9Fin, +Fin\u25b9\ud835\udfd9, +\ud835\udfd9\u25b9Fin, +Fin\u25b9\ud835\udfda, +\ud835\udfda\u25b9Fin\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"-- NOTE with-K\nmodule Data.Fin.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Zero\nopen import Data.One\nopen import Data.Fin public renaming (to\u2115 to Fin\u25b9\u2115)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Two hiding (_==_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n-- The isomorphisms about Fin, \ud835\udfd8, \ud835\udfd9, \ud835\udfda are in Function.Related.TypeIsomorphisms.NP\n\nFin\u25b9\ud835\udfd8 : Fin 0 \u2192 \ud835\udfd8\nFin\u25b9\ud835\udfd8 ()\n\n\ud835\udfd8\u25b9Fin : \ud835\udfd8 \u2192 Fin 0\n\ud835\udfd8\u25b9Fin ()\n\nFin\u25b9\ud835\udfd9 : Fin 1 \u2192 \ud835\udfd9\nFin\u25b9\ud835\udfd9 _ = _\n\n\ud835\udfd9\u25b9Fin : \ud835\udfd9 \u2192 Fin 1\n\ud835\udfd9\u25b9Fin _ = zero\n\nFin\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\nFin\u25b9\ud835\udfda zero    = 0\u2082\nFin\u25b9\ud835\udfda (suc _) = 1\u2082\n\n\ud835\udfda\u25b9Fin : \ud835\udfda \u2192 Fin 2\n\ud835\udfda\u25b9Fin = [0: # 0 1: # 1 ]\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 \ud835\udfda\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 \ud835\udfda\n  helper (equal _) = 1\u2082\n  helper _         = 0\u2082\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = case x == z 0: (case y == z 0: z 1: x) 1: y\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 \u2605\u2080 where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\n{-\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n-}\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nFin\u25b9\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 Fin\u25b9\u2115 (n \u2115- x) \u2261 n \u2238 Fin\u25b9\u2115 x\nFin\u25b9\u2115-\u2115-lem zero = to-from _\nFin\u25b9\u2115-\u2115-lem {zero} (suc ())\nFin\u25b9\u2115-\u2115-lem {suc n} (suc x) = Fin\u25b9\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse-old x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (Fin\u25b9\u2115 x) (suc n)) = Fin\u25b9\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 \u2605\u2080 where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \ud835\udfd8-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \ud835\udfd8-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : \u2605 a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : \u2605 a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","old_contents":"-- NOTE with-K\nmodule Data.Fin.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Zero\nopen import Data.Fin public renaming (to\u2115 to Fin\u25b9\u2115)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 \u2605\u2080 where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\n{-\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n-}\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nFin\u25b9\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 Fin\u25b9\u2115 (n \u2115- x) \u2261 n \u2238 Fin\u25b9\u2115 x\nFin\u25b9\u2115-\u2115-lem zero = to-from _\nFin\u25b9\u2115-\u2115-lem {zero} (suc ())\nFin\u25b9\u2115-\u2115-lem {suc n} (suc x) = Fin\u25b9\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 Fin\u25b9\u2115 (reverse-old x) \u2261 n \u2238 suc (Fin\u25b9\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (Fin\u25b9\u2115 x) (suc n)) = Fin\u25b9\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 \u2605\u2080 where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \ud835\udfd8-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \ud835\udfd8-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : \u2605 a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : \u2605 a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : \u2605 a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"91b84d72680d7c2c8cecec7fa7f3c4db8507bdfc","subject":"Updated paper's title.","message":"Updated paper's title.\n\nIgnore-this: 2199f50d6284b89c6e8457cec04013a8\n\ndarcs-hash:20110430153854-3bd4e-41252e26df59f2df8a4dd02ba87fc2522d3b58c0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Theorem Proving for Reasoning about Functional Programs\" by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download the agda2atp tool (described in above paper) using\n-- darcs with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 The law of the excluded middle\nopen import PredicateLogic.ClassicalATP\n\n-- 1.3 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.6 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.2 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.3 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.PropertiesI\n\n-- 6.3.4 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.5 Inequalites\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.6 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.3 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.4 Lists of natural numbers\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Programs\n\n-- 6.5.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.5.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.ProofSpecificationATP\nopen import FOTC.Program.GCD.ProofSpecificationI\n\n-- 6.5.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.5.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.5.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\n-- This module was imported in the Stanovsk\u00fd example\n-- open import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n\n------------------------------------------------------------------------------\n-- Other theories\n------------------------------------------------------------------------------\n\n-- 1. LTC-PCF\n-- Formalization of a version of Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 1.1. The axioms\nopen import LTC-PCF.Base\n\n-- 1.2 Natural numberes\n\n-- 1.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 1.2.2 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 1.2.3 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- 1.2.4 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 1.2.5 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 1.2.6 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 1.3 Programs\n\n-- 1.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 1.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download the agda2atp tool (described in above paper) using\n-- darcs with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 The law of the excluded middle\nopen import PredicateLogic.ClassicalATP\n\n-- 1.3 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.6 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.2 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.3 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.PropertiesI\n\n-- 6.3.4 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.5 Inequalites\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.6 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.3 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.4 Lists of natural numbers\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Programs\n\n-- 6.5.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.5.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.ProofSpecificationATP\nopen import FOTC.Program.GCD.ProofSpecificationI\n\n-- 6.5.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.5.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.5.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\n-- This module was imported in the Stanovsk\u00fd example\n-- open import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n\n------------------------------------------------------------------------------\n-- Other theories\n------------------------------------------------------------------------------\n\n-- 1. LTC-PCF\n-- Formalization of a version of Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 1.1. The axioms\nopen import LTC-PCF.Base\n\n-- 1.2 Natural numberes\n\n-- 1.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 1.2.2 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 1.2.3 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- 1.2.4 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 1.2.5 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 1.2.6 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 1.3 Programs\n\n-- 1.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 1.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9ce820b684bc9a0775d9532aa64cb264a9cc8f0b","subject":"Wire together CBV\u21a6CBPV conversion and caching","message":"Wire together CBV\u21a6CBPV conversion and caching\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n\n  \u27e6_\u27e7TermCacheCBV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 fromCBVCtx \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 cbvToCompType \u03c4 \u27e7CompTypeHidCache\n  \u27e6 t \u27e7TermCacheCBV = \u27e6 fromCBV t \u27e7CompTermCache\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"242a602d070c3965f3dedd346bff306ec822c452","subject":"Group.Homomorphism: add id, \u2218, \u0394, zip, pair","message":"Group.Homomorphism: add id, \u2218, \u0394, zip, pair\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_; id)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n\nopen GroupHomomorphism\n\nmodule Identity\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n\n  id-hom : GroupHomomorphism \ud835\udd38 \ud835\udd38 id\n  id-hom = mk refl\n\nmodule Compose\n  {a}{A : Type a}\n  {b}{B : Type b}\n  {c}{C : Type c}\n  (\ud835\udd38 : Group A)\n  (\ud835\udd39 : Group B)\n  (\u2102 : Group C)\n  (\u03c8 : A \u2192 B)\n  (\u03c8-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39 \u03c8)\n  (\u03c6 : B \u2192 C)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd39 \u2102 \u03c6)\n  where\n\n  \u2218-hom : GroupHomomorphism \ud835\udd38 \u2102 (\u03c6 \u2218 \u03c8)\n  \u2218-hom = mk (ap \u03c6 (hom \u03c8-hom) \u2219 hom \u03c6-hom)\n\nmodule Delta\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n  open Algebra.Group.Constructions.Product\n\n  \u0394-hom : GroupHomomorphism \ud835\udd38 (\u00d7-grp \ud835\udd38 \ud835\udd38) (\u03bb x \u2192 x , x)\n  \u0394-hom = mk refl\n\nmodule Zip\n  {a\u2080}{A\u2080 : Type a\u2080}\n  {a\u2081}{A\u2081 : Type a\u2081}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38\u2080 : Group A\u2080)\n  (\ud835\udd38\u2081 : Group A\u2081)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A\u2080 \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38\u2080 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A\u2081 \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38\u2081 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  open Algebra.Group.Constructions.Product\n\n  zip-hom : GroupHomomorphism (\u00d7-grp \ud835\udd38\u2080 \ud835\udd38\u2081) (\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) (map \u03c6\u2080 \u03c6\u2081)\n  zip-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n\nmodule Pair\n  {a}{A   : Type a}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38  : Group A)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n\n  -- pair = zip \u2218 \u0394\n  pair-hom : GroupHomomorphism \ud835\udd38 (Product.\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) < \u03c6\u2080 , \u03c6\u2081 >\n  pair-hom = Compose.\u2218-hom _ _ _\n               _ (Delta.\u0394-hom \ud835\udd38)\n               _ (Zip.zip-hom _ _ _ _ _ \u03c6\u2080-hom _ \u03c6\u2081-hom)\n  -- OR:\n  pair-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"07ec73114140de9e379a1a389af4c43e57ad0b9d","subject":"Updated mirror draft.","message":"Updated mirror draft.\n\nIgnore-this: a5731e95fafcd4ae0b5c3a0055e1aafb\n\ndarcs-hash:20111129114129-3bd4e-320936ab5c1803f73f10c676e42599f5b831afd8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Mirror\/PropertiesListSL.agda","new_file":"Draft\/FOTC\/Program\/Mirror\/PropertiesListSL.agda","new_contents":"-- Tested with the development version of the standard library on\n-- 29 November 2011.\n\nmodule PropertiesListSL where\n\nopen import Algebra\nopen import Data.List as List hiding ( reverse )\nopen import Data.List.Properties hiding ( reverse-++-commute )\nopen import Data.Product hiding (map)\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\nmodule LM {A : Set} = Monoid (List.monoid A)\n\n------------------------------------------------------------------------------\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n++-rightIdentity : {A : Set}(xs : List A) \u2192 xs ++ [] \u2261 xs\n++-rightIdentity = proj\u2082 LM.identity\n\nreverse-++-commute : {A : Set}(xs ys : List A) \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute [] ys       = sym (++-rightIdentity (reverse ys))\nreverse-++-commute (x \u2237 xs) [] = cong (\u03bb x' \u2192 reverse x' ++ x \u2237 [])\n                                      (++-rightIdentity xs)\nreverse-++-commute (x \u2237 xs) (y \u2237 ys) =\n  begin\n    reverse (xs ++ y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 cong (\u03bb x' \u2192 x' ++ x \u2237 []) (reverse-++-commute xs (y \u2237 ys)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 LM.assoc (reverse (y \u2237 ys)) (reverse xs) (x \u2237 []) \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\ndata Tree (A : Set) : Set where\n  treeT : A \u2192 List (Tree A) \u2192 Tree A\n\nmirror : {A : Set} \u2192 Tree A \u2192 Tree A\nmirror (treeT a ts) = treeT a (reverse (map mirror ts))\n\nmutual\n  mirror\u00b2 : {A : Set} \u2192 (t : Tree A) \u2192 mirror (mirror t) \u2261 t\n  mirror\u00b2 (treeT a [])       = refl\n  mirror\u00b2 (treeT a (t \u2237 ts)) =\n    begin\n      treeT a (reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts) ++\n                                                      mirror t \u2237 []))) \u2261\n                        treeT a x)\n           (helper (t \u2237 ts))\n           refl\n        \u27e9\n      treeT a (t \u2237 ts)\n    \u220e\n\n  helper : {A : Set} \u2192 (ts : List (Tree A)) \u2192\n           reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  helper []       = refl\n  helper (t \u2237 ts) =\n    begin\n      reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 []))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror (reverse (map mirror ts)) (mirror t \u2237 []))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++-commute (map mirror (reverse (map mirror ts)))\n                                     (map mirror (mirror t \u2237 [])))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n              \u2261\u27e8 refl \u27e9\n      mirror (mirror t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror (mirror t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 t)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (helper ts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n","old_contents":"-- Tested with the development version of the standard library on\n-- 02 May 2011.\n\nmodule PropertiesListSL where\n\nopen import Algebra\nopen import Data.List as List hiding ( reverse )\nopen import Data.List.Properties hiding ( reverse-++-commute )\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\nmodule LM {A : Set} = Monoid (List.monoid A)\n\n------------------------------------------------------------------------------\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n++-rightIdentity : {A : Set}(xs : List A) \u2192 xs ++ [] \u2261 xs\n++-rightIdentity []       = refl\n++-rightIdentity (x \u2237 xs) = cong (_\u2237_ x) (++-rightIdentity xs)\n\nreverse-++-commute : {A : Set}(xs ys : List A) \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute [] ys       = sym (++-rightIdentity (reverse ys))\nreverse-++-commute (x \u2237 xs) [] = cong (\u03bb x' \u2192 reverse x' ++ x \u2237 [])\n                                      (++-rightIdentity xs)\nreverse-++-commute (x \u2237 xs) (y \u2237 ys) =\n  begin\n    reverse (xs ++ y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 cong (\u03bb x' \u2192 x' ++ x \u2237 []) (reverse-++-commute xs (y \u2237 ys)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 LM.assoc (reverse (y \u2237 ys)) (reverse xs) (x \u2237 []) \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\ndata Tree (A : Set) : Set where\n  treeT : A \u2192 List (Tree A) \u2192 Tree A\n\nmirror : {A : Set} \u2192 Tree A \u2192 Tree A\nmirror (treeT a ts) = treeT a (reverse (map mirror ts))\n\nmutual\n  mirror\u00b2 : {A : Set} \u2192 (t : Tree A) \u2192 mirror (mirror t) \u2261 t\n  mirror\u00b2 (treeT a [])       = refl\n  mirror\u00b2 (treeT a (t \u2237 ts)) =\n    begin\n      treeT a (reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts) ++\n                                                      mirror t \u2237 []))) \u2261\n                        treeT a x)\n           (helper (t \u2237 ts))\n           refl\n        \u27e9\n      treeT a (t \u2237 ts)\n    \u220e\n\n  helper : {A : Set} \u2192 (ts : List (Tree A)) \u2192\n           reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  helper []       = refl\n  helper (t \u2237 ts) =\n    begin\n      reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 []))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror (reverse (map mirror ts)) (mirror t \u2237 []))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++-commute (map mirror (reverse (map mirror ts)))\n                                     (map mirror (mirror t \u2237 [])))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n              \u2261\u27e8 refl \u27e9\n      mirror (mirror t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror (mirror t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 t)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (helper ts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39d9e13ae25672af17181a0c3d1fee2118ac33c9","subject":"Introduce and use UncurriedTermConstructor.","message":"Introduce and use UncurriedTermConstructor.\n\nOld-commit-hash: d36f33b097ddb2444e6409de52048ef9bf94b001\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f520724186899be6fbfeda9feea90380e16bd696","subject":"Removed an unproven conjecture.","message":"Removed an unproven conjecture.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Properties\/UnprovedATP.agda","new_file":"src\/fot\/PA\/Axiomatic\/Mendelson\/Properties\/UnprovedATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Unproven PA properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Mendelson.Properties.UnprovedATP where\n\nopen import PA.Axiomatic.Mendelson.Base\nopen import PA.Axiomatic.Mendelson.PropertiesATP\n\n------------------------------------------------------------------------------\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2248 m + (n + o)\n+-asocc m n o = S\u2089 A A0 is m\n  where\n  A : \u2115 \u2192 Set\n  A i = i + n + o \u2248 i + (n + o)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-leftCong #-}\n\n  -- 25 November 2013: Vampire 0.6 proves the theorem using a time out\n  -- of (300 sec).\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  -- {-# ATP prove is +-leftCong #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Unproven PA properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Mendelson.Properties.UnprovedATP where\n\nopen import PA.Axiomatic.Mendelson.Base\nopen import PA.Axiomatic.Mendelson.PropertiesATP\n\n------------------------------------------------------------------------------\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2248 m + (n + o)\n+-asocc m n o = S\u2089 A A0 is m\n  where\n  A : \u2115 \u2192 Set\n  A i = i + n + o \u2248 i + (n + o)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-leftCong #-}\n\n  -- 25 November 2013: Vampire 0.6 proves the theorem using a time out\n  -- of (300 sec).\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is +-leftCong #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"81c758c5545a825b9152e8e2ab898c688bf0d4a9","subject":"Solver.Linear.Parser: comment out the example (long to check)","message":"Solver.Linear.Parser: comment out the example (long to check)\n","repos":"crypto-agda\/crypto-agda","old_file":"Solver\/Linear\/Parser.agda","new_file":"Solver\/Linear\/Parser.agda","new_contents":"open import Data.Vec as Vec using ([]; _\u2237_)\nopen import Text.Parser\nopen import Data.One\nopen import Data.Char\nopen import Data.Nat\nopen import Data.List\nopen import Data.Product hiding (map)\nopen import Data.String as String renaming (toList to S\u25b9L; fromList to L\u25b9S)\nopen import Solver.Linear.Syntax\nopen import Data.Maybe.NP hiding (Eq)\nopen import Relation.Binary.PropositionalEquality\n\nmodule Solver.Linear.Parser where\n\nspaces : Parser \ud835\udfd9\nspaces = manyOneOf\u02e2 \" \\t\\n\\r\" *> pure _\n\npIdent : Parser String\npIdent = someNoneOf\u02e2 \"(,)\u21a6_ \\t\\n\\r\" <* spaces\n\ntok\u1d9c : Char \u2192 Parser \ud835\udfd9\ntok\u1d9c c = char c <* spaces\n\ntok\u02e2 : String \u2192 Parser \ud835\udfd9\ntok\u02e2 s = string s <* spaces\n\n--pPair : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser (A \u00d7 B)\n--pPair pA pB = tok\u1d9c '(' *> \u27ea _,_ \u00b7 pA <* tok\u1d9c ',' \u00b7 pB \u27eb <* tok\u1d9c ')'\n\nmodule _ {A} (pA : Parser A) where\n\n    pSimpleSyn pSyn : \u2115 \u2192 Parser (Syn A)\n\n    pSimpleSyn n =  pure tt <* oneOf\u02e2 \"_\ud835\udfd9\"\n                \u27e8|\u27e9 tok\u1d9c '(' *> pSyn n <* tok\u1d9c ')'\n                \u27e8|\u27e9 \u27ea var \u00b7 pA \u27eb\n\n    pSyn zero    = empty\n    pSyn (suc n) = \u27ea tuple _ \u00b7 pSimpleSyn n \u00b7 many n (tok\u1d9c ',' *> pSyn n) \u27eb\n\n    pEq : \u2115 \u2192 Parser (Eq A)\n    pEq n = \u27ea _\u21a6_ \u00b7 pSyn n <* tok\u1d9c '\u21a6' \u00b7 pSyn n \u27eb\n\npSyn\u02e2 : \u2115 \u2192 Parser (Syn String)\npSyn\u02e2 = pSyn pIdent\n\npEq\u02e2 : \u2115 \u2192 Parser (Eq String)\npEq\u02e2 = pEq pIdent\n\nparseSyn\u02e2? : String \u2192? Syn String\nparseSyn\u02e2? s = runParser (pSyn\u02e2 n <* eof) \u2113\n  where \u2113 = S\u25b9L s\n        n = length \u2113\n\nparseSyn\u02e2 : T[ parseSyn\u02e2? ]\nparseSyn\u02e2 = F[ parseSyn\u02e2? ]\n\nparseEq\u02e2? : String \u2192? Eq String\nparseEq\u02e2? s = runParser (spaces *> pEq\u02e2 (2 * n) <* eof) \u2113\n  where \u2113 = S\u25b9L s\n        n = length \u2113\n\nparseEq\u02e2 : T[ parseEq\u02e2? ]\nparseEq\u02e2 = F[ parseEq\u02e2? ]\n\nex-A\u21d2B : Eq String\nex-A\u21d2B = parseEq\u02e2 \"A\u21a6B\"\n\nex-\ud835\udfd9\u21d2\ud835\udfd9 : Eq String\nex-\ud835\udfd9\u21d2\ud835\udfd9 = parseEq\u02e2 \"\ud835\udfd9\u21a6\ud835\udfd9\"\n\nex\u2081 : Eq String\nex\u2081 = parseEq\u02e2 \" ( A , B,(\ud835\udfd9,C),(D))\u21a6 B,C , A\"\n\n{-\ntest-ex\u2081 : ex\u2081 \u2261 ((var\"A\" , (var\"B\", (tt , var\"C\") , var\"D\"))\n                \u21a6 (var\"B\" , var\"C\" , var\"A\"))\ntest-ex\u2081 = refl\n-}\n\n-- -}\n","old_contents":"open import Data.Vec as Vec using ([]; _\u2237_)\nopen import Text.Parser\nopen import Data.One\nopen import Data.Char\nopen import Data.Nat\nopen import Data.List\nopen import Data.Product hiding (map)\nopen import Data.String as String renaming (toList to S\u25b9L; fromList to L\u25b9S)\nopen import Solver.Linear.Syntax\nopen import Data.Maybe.NP hiding (Eq)\nopen import Relation.Binary.PropositionalEquality\n\nmodule Solver.Linear.Parser where\n\nspaces : Parser \ud835\udfd9\nspaces = manyOneOf\u02e2 \" \\t\\n\\r\" *> pure _\n\npIdent : Parser String\npIdent = someNoneOf\u02e2 \"(,)\u21a6_ \\t\\n\\r\" <* spaces\n\ntok\u1d9c : Char \u2192 Parser \ud835\udfd9\ntok\u1d9c c = char c <* spaces\n\ntok\u02e2 : String \u2192 Parser \ud835\udfd9\ntok\u02e2 s = string s <* spaces\n\n--pPair : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser (A \u00d7 B)\n--pPair pA pB = tok\u1d9c '(' *> \u27ea _,_ \u00b7 pA <* tok\u1d9c ',' \u00b7 pB \u27eb <* tok\u1d9c ')'\n\nmodule _ {A} (pA : Parser A) where\n\n    pSimpleSyn pSyn : \u2115 \u2192 Parser (Syn A)\n\n    pSimpleSyn n =  pure tt <* oneOf\u02e2 \"_\ud835\udfd9\"\n                \u27e8|\u27e9 tok\u1d9c '(' *> pSyn n <* tok\u1d9c ')'\n                \u27e8|\u27e9 \u27ea var \u00b7 pA \u27eb\n\n    pSyn zero    = empty\n    pSyn (suc n) = \u27ea tuple _ \u00b7 pSimpleSyn n \u00b7 many n (tok\u1d9c ',' *> pSyn n) \u27eb\n\n    pEq : \u2115 \u2192 Parser (Eq A)\n    pEq n = \u27ea _\u21a6_ \u00b7 pSyn n <* tok\u1d9c '\u21a6' \u00b7 pSyn n \u27eb\n\npSyn\u02e2 : \u2115 \u2192 Parser (Syn String)\npSyn\u02e2 = pSyn pIdent\n\npEq\u02e2 : \u2115 \u2192 Parser (Eq String)\npEq\u02e2 = pEq pIdent\n\nparseSyn\u02e2? : String \u2192? Syn String\nparseSyn\u02e2? s = runParser (pSyn\u02e2 n <* eof) \u2113\n  where \u2113 = S\u25b9L s\n        n = length \u2113\n\nparseSyn\u02e2 : T[ parseSyn\u02e2? ]\nparseSyn\u02e2 = F[ parseSyn\u02e2? ]\n\nparseEq\u02e2? : String \u2192? Eq String\nparseEq\u02e2? s = runParser (spaces *> pEq\u02e2 (2 * n) <* eof) \u2113\n  where \u2113 = S\u25b9L s\n        n = length \u2113\n\nparseEq\u02e2 : T[ parseEq\u02e2? ]\nparseEq\u02e2 = F[ parseEq\u02e2? ]\n\nex-A\u21d2B : Eq String\nex-A\u21d2B = parseEq\u02e2 \"A\u21a6B\"\n\nex-\ud835\udfd9\u21d2\ud835\udfd9 : Eq String\nex-\ud835\udfd9\u21d2\ud835\udfd9 = parseEq\u02e2 \"\ud835\udfd9\u21a6\ud835\udfd9\"\n\nex\u2081 : Eq String\nex\u2081 = parseEq\u02e2 \" ( A , B,(\ud835\udfd9,C),(D))\u21a6 B,C , A\"\n\ntest-ex\u2081 : ex\u2081 \u2261 ((var\"A\" , (var\"B\", (tt , var\"C\") , var\"D\"))\n                \u21a6 (var\"B\" , var\"C\" , var\"A\"))\ntest-ex\u2081 = refl\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e2eea833ff64ff0f23fff6c4f3d997a65c4b5ef8","subject":"'with' pattern matching","message":"'with' pattern matching\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n\n\ndata   False : Set where\nrecord True  : Set where\n\n\ntrivial : True\ntrivial = _\n\n\nisTrue : Bool -> Set\nisTrue true  = True\nisTrue false = False\n\n\n_<_ : \u2115 -> \u2115 -> Bool\n_   < O   = false\nO   < S n = true\nS m < S n = m < n\n\nlength : {A : Set} -> List A -> \u2115\nlength [] = O\nlength (x :: xs) = S (length xs)\n\nlookup : {A : Set}(xs : List A)(n : \u2115) -> isTrue (n < length xs) -> A\nlookup (x :: xs) O b = x\nlookup (x :: xs) (S n) b = lookup xs n b\nlookup [] a ()\n\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\n\ndata _\u2264_ : \u2115 -> \u2115 -> Set where\n  \u2264O : {m n : \u2115} -> m == n -> m \u2264 n\n  \u2264I : {m n : \u2115} -> m \u2264  n -> m \u2264 S n\n\n\nleq-trans : {l m n : \u2115} -> l \u2264 m -> m \u2264 n -> l \u2264 n\nleq-trans a (\u2264O refl) = a\nleq-trans a (\u2264I b)    = \u2264I (leq-trans a b)\n\n\nmin : \u2115 -> \u2115 -> \u2115\nmin a b with a < b\nmin a b | true  = a\nmin a b | false = b\n\n\nfilter : {A : Set} -> (A -> Bool) -> List A -> List A\nfilter f [] = []\nfilter f (x :: xs) with f x\n... | true  = x :: filter f xs\n... | false =      filter f xs\n\n\ndata _\u2260_ : \u2115 -> \u2115 -> Set where\n  z\u2260s : {n : \u2115}   -> O   \u2260 S n\n  s\u2260z : {n : \u2115}   -> S n \u2260 O\n  s\u2260s : {m n : \u2115} -> n   \u2260 m \u2192 S n \u2260 S m\n\n\ndata Equal? (n m : \u2115) : Set where\n  eq  : n == m -> Equal? n m\n  neq : n \u2260  m -> Equal? n m\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n\n\ndata   False : Set where\nrecord True  : Set where\n\n\ntrivial : True\ntrivial = _\n\n\nisTrue : Bool -> Set\nisTrue true  = True\nisTrue false = False\n\n\n_<_ : \u2115 -> \u2115 -> Bool\n_   < O   = false\nO   < S n = true\nS m < S n = m < n\n\nlength : {A : Set} -> List A -> \u2115\nlength [] = O\nlength (x :: xs) = S (length xs)\n\nlookup : {A : Set}(xs : List A)(n : \u2115) -> isTrue (n < length xs) -> A\nlookup (x :: xs) O b = x\nlookup (x :: xs) (S n) b = lookup xs n b\nlookup [] a ()\n\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\n\ndata _\u2264_ : \u2115 -> \u2115 -> Set where\n  \u2264O : {m n : \u2115} -> m == n -> m \u2264 n\n  \u2264I : {m n : \u2115} -> m \u2264  n -> m \u2264 S n\n\n\nleq-trans : {l m n : \u2115} -> l \u2264 m -> m \u2264 n -> l \u2264 n\nleq-trans a (\u2264O refl) = a\nleq-trans a (\u2264I b)    = \u2264I (leq-trans a b)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"90198c4b819b244f1e0cf351e13107ce74303049","subject":"\u00d7 and + are commutative monoids of setoids","message":"\u00d7 and + are commutative monoids of setoids\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\n\nimport Function as F\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\n\nInv-isEquivalence : IsEquivalence (Inv.Inverse {f\u2081 = L.zero} {f\u2082 = L.zero})\nInv-isEquivalence = record\n   { refl = Inv.id\n   ; sym = Inv.sym\n   ; trans = F.flip Inv._\u2218_ }\n\nSEToid : Set _\nSEToid = Setoid L.zero L.zero\n\nSEToid\u2081 : Setoid _ _\nSEToid\u2081 = record\n  { Carrier = Setoid L.zero L.zero\n  ; _\u2248_ = Inv.Inverse\n  ; isEquivalence = Inv-isEquivalence }\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         Carrier = SEToid ;\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv-isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B : SEToid} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         Carrier = SEToid ;\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv-isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B : SEToid} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { Carrier = SEToid\n  ; _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = from\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      from : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      from = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = (\u2081\u223c\u2081 B-rel , A-rel)\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection L.zero)\nmodule \u228e-CMon = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection L.zero)\nmodule \u00d7\u228e\u00b0    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection L.zero)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nswap-iso : \u2200 {A B} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = \u00d7-CMon.comm _ _\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = record { to = \u2192-to-\u27f6 lower\n                 ; from = \u2192-to-\u27f6 lift\n                 ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                       ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {A B : \u2605} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e = sym Maybe\u2194\u22a4\u228e \u2218 \u228e-CMon.assoc \u22a4 _ _ \u2218 (Maybe\u2194\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n","old_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nopen L using (Lift; lower; lift)\nopen import Type\nimport Function as F\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\nopen import Data.Maybe.NP\nopen import Data.Product\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nopen import Function.Inverse using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\nmodule \u00d7-CMon = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection L.zero)\nmodule \u228e-CMon = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection L.zero)\nmodule \u00d7\u228e\u00b0    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection L.zero)\n\nswap-iso : \u2200 {A B} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = \u00d7-CMon.comm _ _\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = record { to = \u2192-to-\u27f6 lower\n                 ; from = \u2192-to-\u27f6 lift\n                 ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                       ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {A B : \u2605} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e = sym Maybe\u2194\u22a4\u228e \u2218 \u228e-CMon.assoc \u22a4 _ _ \u2218 (Maybe\u2194\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"32b4e15a9837ed6e6276af1fd200e4a3dffc2223","subject":"flipbased: fix choose","message":"flipbased: fix choose\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 M n A \u2192 M n A \u2192 M (suc n) A\nchoose x y = toss\u2032 >>= \u03bb b \u2192 if b then x else y\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 M k (Bits n \u2192 A) \u2192 M (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 M (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 M n A \u2192 M n A \u2192 M (suc n) A\nchoose x y = \u27ea if_then_else_ \u00b7 toss \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 M k (Bits n \u2192 A) \u2192 M (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 M (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"58623914e9c7463bbe57441fd756041084ceb0ff","subject":"liftVec: better integration","message":"liftVec: better integration\n","repos":"crypto-agda\/agda-nplib","old_file":"experiment\/liftVec.agda","new_file":"experiment\/liftVec.agda","new_contents":"open import Function\nopen import Data.Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin using (Fin)\nopen import Data.Vec\nopen import Data.Vec.N-ary.NP\nopen import Relation.Binary.PropositionalEquality\n\nmodule liftVec where\nmodule single (A : Set)(_+_ : A \u2192 A \u2192 A)where\n\n  data Tm : Set where\n    var : Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  eval : Tm \u2192 A \u2192 A\n  eval var x = x\n  eval (cst c) x = c\n  eval (fun tm tm') x = eval tm x + eval tm' x\n\n  veval : {m : \u2115} \u2192 Tm \u2192 Vec A m \u2192 Vec A m\n  veval var xs = xs\n  veval (cst x) xs = replicate x\n  veval (fun tm tm') xs = zipWith _+_ (veval tm xs) (veval tm' xs)\n\n  record Fm : Set where\n    constructor _=='_\n    field\n      lhs : Tm\n      rhs : Tm\n\n  s[_] : Fm \u2192 Set\n  s[ lhs ==' rhs ] = (x : A) \u2192 eval lhs x \u2261 eval rhs x\n\n  v[_] : Fm \u2192 Set\n  v[ lhs ==' rhs ] = {m : \u2115}(xs : Vec A m) \u2192 veval lhs xs \u2261 veval rhs xs\n\n  lemma1 : (t : Tm) \u2192 veval t [] \u2261 []\n  lemma1 var = refl\n  lemma1 (cst x) = refl\n  lemma1 (fun t t') rewrite lemma1 t | lemma1 t' = refl\n\n  lemma2 : {m : \u2115}(x : A)(xs : Vec A m)(t : Tm) \u2192 veval t (x \u2237 xs) \u2261 eval t x \u2237 veval t xs\n  lemma2 _ _ var = refl\n  lemma2 _ _ (cst c) = refl\n  lemma2 x xs (fun t t') rewrite lemma2 x xs t | lemma2 x xs t' = refl\n\n  prf : (t : Fm) \u2192 s[ t ] \u2192 v[ t ]\n  prf (lhs ==' rhs) pr [] rewrite lemma1 lhs | lemma1 rhs = refl\n  prf (l ==' r) pr (x \u2237 xs) rewrite lemma2 x xs l | lemma2 x xs r = cong\u2082 (_\u2237_) (pr x) (prf (l ==' r) pr xs)\n\nmodule MultiDataType (A : Set) (nrVar : \u2115) where\n  data Tm : Set where\n    var : Fin nrVar \u2192 Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\nmodule multi (A : Set)(_+_ : A \u2192 A \u2192 A)(nrVar : \u2115) where\n  open MultiDataType A nrVar public\n\n  sEnv : Set\n  sEnv = Vec A nrVar\n\n  vEnv : \u2115 \u2192 Set\n  vEnv m = Vec (Vec A m) nrVar\n\n  Var : Set\n  Var = Fin nrVar\n\n  _!_ : {X : Set} \u2192 Vec X nrVar \u2192 Var \u2192 X\n  E ! v = lookup v E\n\n  eval : Tm \u2192 sEnv \u2192 A\n  eval (var x) G = G ! x\n  eval (cst c) G = c\n  eval (fun t t') G = eval t G + eval t' G\n\n  veval : {m : \u2115} \u2192 Tm \u2192 vEnv m \u2192 Vec A m\n  veval (var x) G = G ! x\n  veval (cst c) G = replicate c\n  veval (fun t t') G = zipWith _+_ (veval t G) (veval t' G)\n\n  lemma1 : {xs : Vec (Vec A 0) nrVar} (t : Tm) \u2192 veval t xs \u2261 []\n  lemma1 {xs} (var x) with xs ! x\n  ... | [] = refl\n  lemma1 (cst x) = refl\n  lemma1 {xs} (fun t t') rewrite lemma1 {xs} t | lemma1 {xs} t' = refl\n\n  lemVar : {m n : \u2115}(G : Vec (Vec A (suc m)) n)(i : Fin n) \u2192 lookup i G \u2261 lookup i (map head G) \u2237 lookup i (map tail G)\n  lemVar [] ()\n  lemVar ((x \u2237 G) \u2237 G\u2081) Data.Fin.zero = refl\n  lemVar (G \u2237 G') (Data.Fin.suc i) = lemVar G' i\n\n  lemma2 : {n : \u2115}(xs : vEnv (suc n))(t : Tm) \u2192 veval t xs \u2261 eval t (map head xs) \u2237 veval t (map tail xs)\n  lemma2 G (var x) = lemVar G x\n  lemma2 _ (cst x) = refl\n  lemma2 G (fun t t') rewrite lemma2 G t | lemma2 G t' = refl\n\n  s[_==_] : Tm \u2192 Tm \u2192 Set\n  s[ l == r ] = (G : Vec A nrVar) \u2192 eval l G \u2261 eval r G\n\n  v[_==_] : Tm \u2192 Tm \u2192 Set\n  v[ l == r ] = {m : \u2115}(G : Vec (Vec A m) nrVar) \u2192 veval l G \u2261 veval r G\n\n  prf : (l r : Tm) \u2192 s[ l == r ] \u2192 v[ l == r ]\n  prf l r pr {zero} G rewrite lemma1 {G} l | lemma1 {G} r = refl\n  prf l r pr {suc m} G rewrite lemma2 G l | lemma2 G r\n     = cong\u2082 _\u2237_ (pr (replicate head \u229b G)) (prf l r pr (map tail G))\n\nmodule Full (A : Set)(_+_ : A \u2192 A \u2192 A) (m : \u2115) n where\n  open multi A _+_ n public\n\n  solve : \u2200 (l r : Tm) \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  solve l r x = curry\u207f {n} {A = Vec A m} {B = curry\u207f\u2032 f}\n        (\u03bb xs \u2192 subst id (sym (curry-$\u207f\u2032 f xs))\n                   (prf l r (\u03bb G \u2192 subst id (curry-$\u207f\u2032 g G) (x $\u207f G)) xs))\n    where f = \u03bb G \u2192 veval {m} l G \u2261 veval r G\n          g = \u03bb G \u2192 eval l G \u2261 eval r G\n\nopen import Data.Bool\n\nmodule example where\n\n  open import Data.Fin\n  open import Data.Bool.NP\n\n  module VecBoolXorProps m n = Full Bool _xor_ m n\n  open MultiDataType\n\n  coolTheorem : {m : \u2115} \u2192 (xs ys : Vec Bool m) \u2192 zipWith _xor_ xs ys \u2261 zipWith _xor_ ys xs\n  coolTheorem {m} = VecBoolXorProps.solve m 2 l r Xor\u00b0.+-comm where\n    l = fun (var zero) (var (suc zero))\n    r = fun (var (suc zero)) (var zero)\n\ntest = example.coolTheorem (true \u2237 false \u2237 []) (false \u2237 false \u2237 [])\n","old_contents":"open import Data.Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin using (Fin)\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality\n\nmodule liftVec where \nmodule single (A : Set)(_+_ : A -> A -> A)where\n\n  data Tm : Set where\n    var : Tm\n    cst : A -> Tm\n    fun : Tm -> Tm -> Tm\n\n  eval : Tm -> A -> A\n  eval var x = x\n  eval (cst c) x = c\n  eval (fun tm tm') x = eval tm x + eval tm' x \n  \n  veval : {m : \u2115} -> Tm -> Vec A m -> Vec A m \n  veval var xs = xs\n  veval (cst x) xs = replicate x\n  veval (fun tm tm') xs = zipWith _+_ (veval tm xs) (veval tm' xs)\n\n  record Fm : Set where\n    constructor _=='_\n    field\n      lhs : Tm\n      rhs : Tm\n\n  s[_] : Fm -> Set\n  s[ lhs ==' rhs ] = (x : A) \u2192 eval lhs x \u2261 eval rhs x\n  \n  v[_] : Fm -> Set\n  v[ lhs ==' rhs ] = {m : \u2115}(xs : Vec A m) \u2192 veval lhs xs \u2261 veval rhs xs\n\n  lemma1 : (t : Tm) -> veval t [] \u2261 []\n  lemma1 var = refl\n  lemma1 (cst x) = refl\n  lemma1 (fun t t') rewrite lemma1 t | lemma1 t' = refl\n\n  lemma2 : {m : \u2115}(x : A)(xs : Vec A m)(t : Tm) -> veval t (x \u2237 xs) \u2261 eval t x \u2237 veval t xs\n  lemma2 _ _ var = refl\n  lemma2 _ _ (cst c) = refl\n  lemma2 x xs (fun t t') rewrite lemma2 x xs t | lemma2 x xs t' = refl\n\n  prf : (t : Fm) \u2192 s[ t ] \u2192 v[ t ]\n  prf (lhs ==' rhs) pr [] rewrite lemma1 lhs | lemma1 rhs = refl\n  prf (l ==' r) pr (x \u2237 xs) rewrite lemma2 x xs l | lemma2 x xs r = cong\u2082 (_\u2237_) (pr x) (prf (l ==' r) pr xs)\n\nmodule multi {nrVar : \u2115}(A : Set)(_+_ : A -> A -> A) where\n\n  sEnv : Set\n  sEnv = Vec A nrVar\n\n  vEnv : \u2115 -> Set\n  vEnv m = Vec (Vec A m) nrVar\n\n  Var : Set\n  Var = Fin nrVar\n\n  _!_ : {X : Set} -> Vec X nrVar -> Var -> X\n  E ! v = lookup v E\n\n  data Tm : Set where\n    var : Fin nrVar -> Tm\n    cst : A -> Tm\n    fun : Tm -> Tm -> Tm\n\n  eval : Tm -> sEnv -> A\n  eval (var x) G = G ! x\n  eval (cst c) G = c\n  eval (fun t t') G = eval t G + eval t' G \n\n  veval : {m : \u2115} -> Tm -> vEnv m -> Vec A m\n  veval (var x) G = G ! x\n  veval (cst c) G = replicate c\n  veval (fun t t') G = zipWith _+_ (veval t G) (veval t' G)\n\n  lemma1 : {xs : Vec (Vec A 0) nrVar} (t : Tm) -> veval t xs \u2261 []\n  lemma1 {xs} (var x) with xs ! x\n  ... | [] = refl\n  lemma1 (cst x) = refl\n  lemma1 {xs} (fun t t') rewrite lemma1 {xs} t | lemma1 {xs} t' = refl\n\n  lemVar : {m n : \u2115}(G : Vec (Vec A (suc m)) n)(i : Fin n) -> lookup i G \u2261 lookup i (map head G) \u2237 lookup i (map tail G)\n  lemVar [] ()\n  lemVar ((x \u2237 G) \u2237 G\u2081) Data.Fin.zero = refl\n  lemVar (G \u2237 G') (Data.Fin.suc i) = lemVar G' i\n\n  lemma2 : {n : \u2115}(xs : vEnv (suc n))(t : Tm) -> veval t xs \u2261 eval t (map head xs) \u2237 veval t (map tail xs)\n  lemma2 G (var x) = lemVar G x \n  lemma2 _ (cst x) = refl\n  lemma2 G (fun t t') rewrite lemma2 G t | lemma2 G t' = refl\n\n  s[_==_] : Tm -> Tm -> Set\n  s[ l == r ] = (G : Vec A nrVar) -> eval l G \u2261 eval r G\n\n  v[_==_] : Tm -> Tm -> Set\n  v[ l == r ] = {m : \u2115}(G : Vec (Vec A m) nrVar) -> veval l G \u2261 veval r G\n\n  prf : (l r : Tm) -> s[ l == r ] -> v[ l == r ]\n  prf l r pr {zero} G rewrite lemma1 {G} l | lemma1 {G} r = refl\n  prf l r pr {suc m} G rewrite lemma2 G l | lemma2 G r = cong\u2082 _\u2237_ (pr (replicate head \u229b G)) (prf l r pr (map tail G))\n\nmodule Forall (A : Set) where\n\n  module N = Data.Nat\n\n  NPred : \u2115 -> Set\u2081\n  NPred 0 = Set\n  NPred (N.suc n) = A -> NPred n \n\n  NFun : (n : \u2115) -> NPred n -> Set\n  NFun zero P = P\n  NFun (suc n) P = (x : A) \u2192 NFun n (P x)\n\n  \n  NApply : {m n : \u2115} -> Vec A n -> Vec (Fin n) m -> NPred m -> Set \n  NApply xs [] P = P\n  NApply xs (x \u2237 ys) P = NApply xs ys (P (lookup x xs))\n\n\n  NVec : {n : \u2115} -> NPred n -> Set\n  NVec {n} P = (xs : Vec A n) -> NApply xs (tabulate (\\ x -> x)) P\n\n  Build : {n : \u2115}{P : NPred n} -> NFun n P -> NVec P\n  Build {n} f xs = go (tabulate (\u03bb x \u2192 x)) f where \n    go : {m : \u2115}{P : NPred m}(ys : Vec (Fin n) m)(f : NFun m P) -> NApply xs ys P\n    go [] f = f\n    go (x \u2237 ys) f = go ys (f (lookup x xs)) \n\nmodule example where\n\n  open import Data.Bool\n  import Data.Bool.Properties\n  open Data.Fin\n\n  open import Algebra\n\n  module Xor = CommutativeRing Data.Bool.Properties.commutativeRing-xor-\u2227 \n\n  coolTheorem : {m : \u2115} -> (xs ys : Vec Bool m) -> zipWith _xor_ xs ys \u2261 zipWith _xor_ ys xs\n  coolTheorem xs ys = prf l r (lemma) (xs \u2237 (ys \u2237 [])) where\n    open multi {2} Bool _xor_ \n\n    open Forall Bool\n \n    l = (fun (var zero) (var (suc zero)))\n    r =  (fun (var (suc zero)) (var zero))\n\n    Thm : NPred 2\n    Thm = \u03bb x y \u2192 x xor y \u2261 y xor x\n\n    Prf : NFun 2 Thm\n    Prf true true = refl\n    Prf true false = refl\n    Prf false true = refl\n    Prf false false = refl\n\n    lem  : NVec Thm\n    lem = Build Prf \n \n    lemma : (G : Vec Bool 2) -> lookup zero G xor lookup (suc zero) G\n                              \u2261 lookup (suc zero) G xor lookup zero G\n    lemma (true \u2237 true \u2237 G) = refl\n    lemma (true \u2237 false \u2237 G) = refl\n    lemma (false \u2237 true \u2237 G) = refl\n    lemma (false \u2237 false \u2237 G) = refl\n\nopen import Data.Bool\nopen Forall Bool\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c087206d005020617069a7715e8b6d095680aca3","subject":"Only renaming.","message":"Only renaming.\n\nIgnore-this: 4ba18c3a1346ab5098ff19ca162a5b73\n\ndarcs-hash:20101221115850-3bd4e-6abcef31213c6163c23d468dadb1f281395dc591.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Agsy\/AgsyTest.agda","new_file":"Draft\/Agsy\/AgsyTest.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing Agsy using the Agda standard library\n------------------------------------------------------------------------------\n\nmodule AgsyTest where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero    n o = refl\n+-assoc (suc m) n o = cong suc (+-assoc m n o)\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + suc n \u2261 suc (m + n)  -- via Agsy {-c}\nx+Sy\u2261S[x+y] zero    n = refl\nx+Sy\u2261S[x+y] (suc m) n = cong suc (x+Sy\u2261S[x+y] m n)\n\n0+x\u2261x+0 : \u2200 x \u2192 0 + x \u2261 x + 0  -- via Agsy {-c}\n0+x\u2261x+0 zero    = refl\n0+x\u2261x+0 (suc n) = cong suc (0+x\u2261x+0 n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = 0+x\u2261x+0 n\n+-comm (suc m) n =\n  begin\n    suc (m + n) \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n    n + suc m\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Testing Agsy using the Agda standard library\n------------------------------------------------------------------------------\n\nmodule AgsyTest where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero    n o = refl\n+-assoc (suc m) n o = cong suc (+-assoc m n o)\n\nx+1+y\u22611+x+y : \u2200 m n \u2192 m + suc n \u2261 suc (m + n)  -- via Agsy {-c}\nx+1+y\u22611+x+y zero    n = refl\nx+1+y\u22611+x+y (suc m) n = cong suc (x+1+y\u22611+x+y m n)\n\n0+x\u2261x+0 : \u2200 x \u2192 0 + x \u2261 x + 0  -- via Agsy {-c}\n0+x\u2261x+0 zero    = refl\n0+x\u2261x+0 (suc n) = cong suc (0+x\u2261x+0 n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = 0+x\u2261x+0 n\n+-comm (suc m) n =\n  begin\n    suc (m + n) \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m) \u2261\u27e8 sym (x+1+y\u22611+x+y n m) \u27e9\n    n + suc m\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"15803dc967c439750b8fa3a9354f2a8732d4d8ca","subject":"flat-funs: more derived ops, <if_then_else_>, Maybe..., search, find?, findB","message":"flat-funs: more derived ops, <if_then_else_>, Maybe..., search, find?, findB\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nopen import Data.Bool using (if_then_else_)\nopen import Function using (_\u2218\u2032_)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bits using (Bit; Bits)\n\nimport bintree as Tree\nopen Tree using (Tree)\nopen import data-universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nmodule Defaults {t} {T : Set t} (\u266dFuns : FlatFuns T) where\n  open FlatFuns \u266dFuns\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultsGroup1\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    where\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n\n  infixr 9 _\u2218_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults \u266dFuns\n  open CompositionNotations _\u2218_ public\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f cons\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f cons\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f cons\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 `\u22a4 `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f [] = nil\n  constVec f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec f xs >\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = nil\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <nil,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <nil, f > = <tt\u204f nil , f >\n\n  <_,nil> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,nil> = < f ,tt\u204f nil >\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id ,nil> \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <nil, id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = nil \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 `Bits i `\u2192 `Bits o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = <tt\u204f fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n","old_contents":"module flat-funs where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nopen import Data.Bool using (if_then_else_)\nopen import Function using (_\u2218\u2032_)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bits using (Bit; Bits)\n\nimport bintree as Tree\nopen Tree using (Tree)\nopen import data-universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nmodule Defaults {t} {T : Set t} (\u266dFuns : FlatFuns T) where\n  open FlatFuns \u266dFuns\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultsGroup1\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    where\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n\n  infixr 9 _\u2218_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults \u266dFuns\n  open CompositionNotations _\u2218_ public\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f cons\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f cons\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f cons\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 `\u22a4 `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f [] = nil\n  constVec f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec f xs >\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = nil\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <nil,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <nil, f > = <tt\u204f nil , f >\n\n  <_,nil> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,nil> = < f ,tt\u204f nil >\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id ,nil> \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <nil, id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 `Bits i `\u2192 `Bits o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = <tt\u204f fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"306f2622f460038c9688f491c6623f15ec94b233","subject":"Generalize search, add sum, # using sum","message":"Generalize search, add sum, # using sum\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\nsearch\u2032 : \u2200 {n a} {A : \u2115 \u2192 Set a} \u2192 (\u2200 {m} \u2192 A m \u2192 A m \u2192 A (2* m)) \u2192 (Bits n \u2192 A 1) \u2192 A (2^ n)\nsearch\u2032 {n} {a} {A} op f = Search.search 1 2*_ {a} {A} (\u03bb {m} \u2192 op {m}) f\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {n} {a} {A} _\u00b7_ f = search\u2032 {n} {a} {const A} _\u00b7_ f\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\nsearch-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\nsearch-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n  where\n    go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n    go zero = refl\n    go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^_ zero = refl\n#always\u22612^_ (suc n) = cong\u2082 _+_ pf pf where pf = #always\u22612^ n\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op {-(f |\u2228| g) (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))-} (|de-morgan| f g)\n\nsearch-comm :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-comm {zero} _+_ _*_ f p hom = refl\nsearch-comm {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-comm {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-comm _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n[0\u2194_] : \u2200 {n} \u2192 Fin n \u2192 Bits n \u2192 Bits n\n[0\u2194_] {zero}  i xs = xs\n[0\u2194_] {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n[0\u21941] : Bits 2 \u2192 Bits 2\n[0\u21941] = [0\u2194 suc zero ]\n\n[0\u21941]-spec : [0\u21941] \u2257 (\u03bb { (x \u2237 y \u2237 []) \u2192 y \u2237 x \u2237 [] })\n[0\u21941]-spec (x \u2237 y \u2237 []) = refl\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {zero}  _   f = f []\nsearch {suc n} _\u00b7_ f = search _\u00b7_ (f \u2218 0\u2237_) \u00b7 search _\u00b7_ (f \u2218 1\u2237_)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = search _+_ (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\nsearch-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\nsearch-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n  where\n    go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n    go zero = refl\n    go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^_ zero = refl\n#always\u22612^_ (suc n) = cong\u2082 _+_ pf pf where pf = #always\u22612^ n\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op {-(f |\u2228| g) (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))-} (|de-morgan| f g)\n\nsearch-comm :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-comm {zero} _+_ _*_ f p hom = refl\nsearch-comm {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-comm {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-comm _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n[0\u2194_] : \u2200 {n} \u2192 Fin n \u2192 Bits n \u2192 Bits n\n[0\u2194_] {zero}  i xs = xs\n[0\u2194_] {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n[0\u21941] : Bits 2 \u2192 Bits 2\n[0\u21941] = [0\u2194 suc zero ]\n\n[0\u21941]-spec : [0\u21941] \u2257 (\u03bb { (x \u2237 y \u2237 []) \u2192 y \u2237 x \u2237 [] })\n[0\u21941]-spec (x \u2237 y \u2237 []) = refl\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5d213b30c7818d7ea1a1fd530b7ca86d27825545","subject":"+Data.Vec.NP.uncons","message":"+Data.Vec.NP.uncons\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin hiding (_+_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount pred [] = zero\ncount pred (x \u2237 xs) = (if pred x then suc else inject\u2081) (count pred xs)\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (to\u2115 (count pred xs))\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin hiding (_+_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount pred [] = zero\ncount pred (x \u2237 xs) = (if pred x then suc else inject\u2081) (count pred xs)\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (to\u2115 (count pred xs))\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a7ca6f8ee722767ec1edd8583735a23faff8e0d1","subject":"also should be *that* particular delta","message":"also should be *that* particular delta\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d err[ \u0394 ] \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | V x\u2081\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp D1 x\u2082 D2) | V x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | I x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | E x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | S x = {!!}\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = E {!!}\n  progress (TACast TAConst TCHole2) | V VConst = E {!!}\n  progress (TACast D x\u2081) | V VLam = {!!}\n  progress (TACast D x\u2081) | I x = {!!}\n  progress (TACast D x\u2081) | E x = {!!}\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) \u2192 Set where\n    V : \u2200{d} \u2192 d val \u2192 ok d\n    I : \u2200{d} \u2192 d indet \u2192 ok d\n    E : \u2200{d \u0394} \u2192 d err[ \u0394 ] \u2192 ok d\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | V x\u2081\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp D1 x\u2082 D2) | V x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | I x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | E x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | S x = {!!}\n  progress (TACast D x)\n    with progress D\n  progress (TACast D x\u2081) | V x = {!!}\n  progress (TACast D x\u2081) | I x = {!!}\n  progress (TACast D x\u2081) | E x = {!!}\n  progress (TACast D x\u2081) | S x = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4271926cfe37f975efd867eb6cbd1673f449ac89","subject":"close #4","message":"close #4\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i (FVal x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = vs x (_ , Step p q r)\n  is (IAp i (FIndet x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n--  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n\n  --es (ENEHole e) x = {!e!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step FHNEHoleEvaled x\u2082 x\u2083) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleInside x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  -- es (ENEHole e) (d' , Step (FHFinal (FVal ())) x\u2081 x\u2082)\n  -- es (ENEHole e) (d' , Step (FHFinal (FIndet (INEHole x))) () x\u2082)\n  -- es (ENEHole e) (d' , Step FHNEHoleEvaled () x\u2082)\n  -- es (ENEHole e) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FVal ())))\n  -- es (ENEHole (ECastError x x\u2081)) (_ , Step (FHNEHoleFinal x\u2082) (ITNEHole x\u2083) (FHFinal (FIndet (INEHole x\u2084)))) = {!x\u2081!}\n  -- es (ENEHole (EAp1 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (EAp2 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ENEHole e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ECastProp e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) FHNEHoleEvaled) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i x) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = {!!}\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = {!!}\n\n\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n--  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n\n  --es (ENEHole e) x = {!e!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step FHNEHoleEvaled x\u2082 x\u2083) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleInside x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  -- es (ENEHole e) (d' , Step (FHFinal (FVal ())) x\u2081 x\u2082)\n  -- es (ENEHole e) (d' , Step (FHFinal (FIndet (INEHole x))) () x\u2082)\n  -- es (ENEHole e) (d' , Step FHNEHoleEvaled () x\u2082)\n  -- es (ENEHole e) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FVal ())))\n  -- es (ENEHole (ECastError x x\u2081)) (_ , Step (FHNEHoleFinal x\u2082) (ITNEHole x\u2083) (FHFinal (FIndet (INEHole x\u2084)))) = {!x\u2081!}\n  -- es (ENEHole (EAp1 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (EAp2 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ENEHole e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ECastProp e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) FHNEHoleEvaled) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"adefe17101926c368aec9494e33f955f80ab1c53","subject":"Added rev-++-commute (ATP version).","message":"Added rev-++-commute (ATP version).\n\nIgnore-this: b06a4aa75d259c3709c1d9455b834b42\n\ndarcs-hash:20110219140841-3bd4e-622543c31fc443c146ca4ac556726eaabcad0d30.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/ListTree\/PropertiesATP.agda","new_file":"Draft\/Mirror\/ListTree\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists of trees\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.ListTree.PropertiesATP where\n\nopen import LTC.Base\n\nopen import Common.Function\n\nopen import Draft.Mirror.Mirror\n\nopen import LTC.Data.List\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n++-ListTree : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192 ListTree (xs ++ ys)\n++-ListTree {ys = ys} nilLT LTys = subst ListTree (sym (++-[] ys)) LTys\n++-ListTree {ys = ys} (consLT {x} {xs} Tx LTxs) LTys =\n  subst ListTree (sym (++-\u2237 x xs ys)) (consLT Tx (++-ListTree LTxs LTys))\n\nmap-ListTree : \u2200 {xs} f \u2192 (\u2200 {x} \u2192 Tree x \u2192 Tree (f \u00b7 x)) \u2192\n               ListTree xs \u2192 ListTree (map f xs)\nmap-ListTree f fTree nilLT = subst ListTree (sym (map-[] f)) nilLT\nmap-ListTree f fTree (consLT {x} {xs} Tx LTxs) =\n  subst ListTree\n        (sym (map-\u2237 f x xs))\n        (consLT (fTree Tx) (map-ListTree f fTree LTxs))\n\nrev-ListTree : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192 ListTree (rev xs ys)\nrev-ListTree {ys = ys} nilLT LTys = subst ListTree (sym (rev-[] ys)) LTys\nrev-ListTree {ys = ys} (consLT {x} {xs} Tx LTxs) LTys =\n  subst ListTree (sym (rev-\u2237 x xs ys)) (rev-ListTree LTxs (consLT Tx LTys))\n\n++-leftIdentity : \u2200 {xs} \u2192 ListTree xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : \u2200 {xs} \u2192 ListTree xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilLT               = ++-[] []\n++-rightIdentity (consLT {x} {xs} Tx LTxs) = prf (++-rightIdentity LTxs)\n  where\n    postulate prf :  xs ++ [] \u2261 xs \u2192\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n    {-# ATP prove prf #-}\n\n++-assoc : \u2200 {xs ys zs} \u2192 ListTree xs \u2192 ListTree ys \u2192 ListTree zs \u2192\n           (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc {ys = ys} {zs} nilLT LTys LTzs = prf\n  where\n    postulate prf : ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs\n    {-# ATP prove prf #-}\n\n++-assoc {ys = ys} {zs} (consLT {x} {xs} Tx LTxs) LTys LTzs =\n  prf (++-assoc LTxs LTys LTzs)\n  where\n    postulate prf : (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192 -- IH.\n                    ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs\n    {-# ATP prove prf #-}\n\nrev-++-commute : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192\n                 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute {ys = ys} nilLT LTys = prf\n  where\n    postulate prf : rev [] ys \u2261 rev [] [] ++ ys\n    {-# ATP prove prf #-}\n\nrev-++-commute {ys = ys} (consLT {x} {xs} Tx LTxs) LTys =\n  prf (rev-++-commute LTxs (consLT Tx LTys))\n      (rev-++-commute LTxs (consLT Tx nilLT))\n  where\n    postulate prf : rev xs (x \u2237 ys) \u2261 rev xs [] ++ x \u2237 ys \u2192  -- IH.\n                    rev xs (x \u2237 []) \u2261 rev xs [] ++ x \u2237 [] \u2192  -- IH.\n                    rev (x \u2237 xs) ys \u2261 rev (x \u2237 xs) [] ++ ys\n    {-# ATP prove prf ++-assoc rev-ListTree ++-ListTree #-}\n\npostulate\n  reverse-\u2237 : \u2200 x {ys} \u2192 ListTree ys \u2192\n              reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\n-- {-# ATP prove reverse-\u2237 #-}\n\nreverse-++-commute : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} nilLT LTys =\n  begin\n    reverse ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ([] ++ ys) \u2261 reverse t)\n               (++-[] ys)\n               refl\n      \u27e9\n    reverse ys\n      \u2261\u27e8 sym (++-rightIdentity (rev-ListTree LTys nilLT)) \u27e9\n    reverse ys ++ []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ys ++ [] \u2261 reverse ys ++ t)\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse ys ++ reverse []\n  \u220e\n\nreverse-++-commute (consLT {x} {xs} Tx LTxs) nilLT =\n  begin\n    reverse ((x \u2237 xs) ++ [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ((x \u2237 xs) ++ []) \u2261 reverse t)\n               (++-rightIdentity (consLT Tx LTxs))\n               refl\n      \u27e9\n    reverse (x \u2237 xs)\n      \u2261\u27e8 sym (++-[] (reverse (x \u2237 xs))) \u27e9\n    [] ++ reverse (x \u2237 xs)\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ reverse (x \u2237 xs) \u2261 t ++ reverse (x \u2237 xs))\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse [] ++ reverse (x \u2237 xs)\n  \u220e\n\nreverse-++-commute (consLT {x} {xs} Tx LTxs) (consLT {y} {ys} Ty LTys) =\n  begin\n    reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261\u27e8 refl \u27e9\n    rev ((x \u2237 xs) ++ y \u2237 ys) []\n      \u2261\u27e8 subst (\u03bb t \u2192 rev ((x \u2237 xs) ++ y \u2237 ys) [] \u2261 rev t [])\n               (++-\u2237 x xs (y \u2237 ys))\n               refl\n      \u27e9\n    rev (x \u2237 (xs ++ y \u2237 ys)) [] \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n    rev (xs ++ y \u2237 ys) (x \u2237 [])\n      \u2261\u27e8 rev-++-commute (++-ListTree LTxs (consLT Ty LTys)) (consLT Tx nilLT) \u27e9\n    rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 rev (xs ++ y \u2237 ys) [] ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               refl\n               refl\n      \u27e9\n    reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (xs ++ y \u2237 ys) ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               (reverse-++-commute LTxs (consLT Ty LTys))  -- IH.\n               refl\n      \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 ++-assoc (rev-ListTree (consLT Ty LTys) nilLT)\n                  (rev-ListTree LTxs nilLT)\n                  (consLT Tx nilLT)\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t ++ x \u2237 [])\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-++-commute LTxs (consLT Tx nilLT))\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs (x \u2237 []) \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev (x \u2237 xs) [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\nmap-++-commute : \u2200 f {xs ys} \u2192 (\u2200 {x} \u2192 Tree x \u2192 Tree (f \u00b7 x)) \u2192\n                 ListTree xs \u2192 ListTree ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f {ys = ys} fTree nilLT LTys =\n  begin\n    map f ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 map f ([] ++ ys) \u2261 map f t)\n               (++-[] ys)\n               refl\n      \u27e9\n    map f ys\n      \u2261\u27e8 sym (++-leftIdentity (map-ListTree f fTree LTys)) \u27e9\n    [] ++ map f ys\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ map f ys \u2261 t ++ map f ys)\n               (sym (map-[] f))\n               refl\n      \u27e9\n    map f [] ++ map f ys\n  \u220e\n\nmap-++-commute f {ys = ys} fTree (consLT {x} {xs} Tx LTxs) LTys =\n  begin\n    map f ((x \u2237 xs) ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 map f ((x \u2237 xs) ++ ys) \u2261 map f t)\n               (++-\u2237 x xs ys)\n               refl\n      \u27e9\n    map f (x \u2237 xs ++ ys)\n      \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n    f \u00b7 x \u2237 map f (xs ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 f \u00b7 x \u2237 map f (xs ++ ys) \u2261 f \u00b7 x \u2237 t)\n               (map-++-commute f fTree LTxs LTys) -- IH.\n               refl\n      \u27e9\n    f \u00b7 x \u2237 (map f xs ++ map f ys)\n      \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n    (f \u00b7 x \u2237 map f xs) ++ map f ys\n      \u2261\u27e8 subst (\u03bb t \u2192 (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261 t ++ map f ys)\n               (sym (map-\u2237 f x xs))\n               refl\n      \u27e9\n    map f (x \u2237 xs) ++ map f ys\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists of trees\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.ListTree.PropertiesATP where\n\nopen import LTC.Base\n\nopen import Common.Function\n\nopen import Draft.Mirror.Mirror\n\nopen import LTC.Data.List\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n++-ListTree : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192 ListTree (xs ++ ys)\n++-ListTree {ys = ys} nilLT LTys = subst ListTree (sym (++-[] ys)) LTys\n++-ListTree {ys = ys} (consLT {x} {xs} Tx LTxs) LTys =\n  subst ListTree (sym (++-\u2237 x xs ys)) (consLT Tx (++-ListTree LTxs LTys))\n\nmap-ListTree : \u2200 {xs} f \u2192 (\u2200 {x} \u2192 Tree x \u2192 Tree (f \u00b7 x)) \u2192\n               ListTree xs \u2192 ListTree (map f xs)\nmap-ListTree f fTree nilLT = subst ListTree (sym (map-[] f)) nilLT\nmap-ListTree f fTree (consLT {x} {xs} Tx LTxs) =\n  subst ListTree\n        (sym (map-\u2237 f x xs))\n        (consLT (fTree Tx) (map-ListTree f fTree LTxs))\n\nrev-ListTree : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192 ListTree (rev xs ys)\nrev-ListTree {ys = ys} nilLT LTys = subst ListTree (sym (rev-[] ys)) LTys\nrev-ListTree {ys = ys} (consLT {x} {xs} Tx LTxs) LTys =\n  subst ListTree (sym (rev-\u2237 x xs ys)) (rev-ListTree LTxs (consLT Tx LTys))\n\n++-leftIdentity : \u2200 {xs} \u2192 ListTree xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : \u2200 {xs} \u2192 ListTree xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilLT               = ++-[] []\n++-rightIdentity (consLT {x} {xs} Tx LTxs) = prf (++-rightIdentity LTxs)\n  where\n    postulate prf :  xs ++ [] \u2261 xs \u2192\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n    {-# ATP prove prf #-}\n\n++-assoc : \u2200 {xs ys zs} \u2192 ListTree xs \u2192 ListTree ys \u2192 ListTree zs \u2192\n           (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc {ys = ys} {zs} nilLT LTys LTzs = prf\n  where\n    postulate prf : ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs\n    {-# ATP prove prf #-}\n\n++-assoc {ys = ys} {zs} (consLT {x} {xs} Tx LTxs) LTys LTzs =\n  prf (++-assoc LTxs LTys LTzs)\n  where\n    postulate prf : (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192 -- IH.\n                    ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs\n    {-# ATP prove prf #-}\n\nrev-++-commute : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192\n                 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute {ys = ys} nilLT LTys =\n  begin\n    rev [] ys \u2261\u27e8 rev-[] ys \u27e9\n    ys        \u2261\u27e8 sym $ ++-leftIdentity LTys \u27e9\n    [] ++ ys  \u2261\u27e8 subst (\u03bb t \u2192 [] ++ ys \u2261 t ++ ys)\n                       (sym $ rev-[] [])\n                       refl\n              \u27e9\n    rev [] [] ++ ys\n  \u220e\n\nrev-++-commute {ys = ys} (consLT {x} {xs} Tx LTxs) LTys =\n  begin\n    rev (x \u2237 xs) ys      \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n    rev xs (x \u2237 ys)      \u2261\u27e8 rev-++-commute LTxs (consLT Tx LTys) \u27e9  -- IH.\n    rev xs [] ++ x \u2237 ys\n      \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n               (sym\n                 ( begin\n                     (x \u2237 []) ++ ys \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                     x \u2237 ([] ++ ys) \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ([] ++ ys) \u2261 x \u2237 t)\n                                             (++-leftIdentity LTys)\n                                             refl\n                                    \u27e9\n                     x \u2237 ys\n                   \u220e\n                 )\n               )\n               refl\n      \u27e9\n    rev xs [] ++ (x \u2237 []) ++ ys\n      \u2261\u27e8 sym $ ++-assoc (rev-ListTree LTxs nilLT) (consLT Tx nilLT) LTys \u27e9\n    (rev xs [] ++ (x \u2237 [])) ++ ys\n      \u2261\u27e8 subst (\u03bb t \u2192 (rev xs [] ++ (x \u2237 [])) ++ ys \u2261 t ++ ys)\n               (sym $ rev-++-commute LTxs (consLT Tx nilLT))  -- IH.\n               refl\n      \u27e9\n    rev xs (x \u2237 []) ++ ys\n      \u2261\u27e8 subst (\u03bb t \u2192 rev xs (x \u2237 []) ++ ys \u2261 t ++ ys)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    rev (x \u2237 xs) [] ++ ys\n  \u220e\n\npostulate\n  reverse-\u2237 : \u2200 x {ys} \u2192 ListTree ys \u2192\n              reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\n-- {-# ATP prove reverse-\u2237 #-}\n\nreverse-++-commute : \u2200 {xs ys} \u2192 ListTree xs \u2192 ListTree ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} nilLT LTys =\n  begin\n    reverse ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ([] ++ ys) \u2261 reverse t)\n               (++-[] ys)\n               refl\n      \u27e9\n    reverse ys\n      \u2261\u27e8 sym (++-rightIdentity (rev-ListTree LTys nilLT)) \u27e9\n    reverse ys ++ []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ys ++ [] \u2261 reverse ys ++ t)\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse ys ++ reverse []\n  \u220e\n\nreverse-++-commute (consLT {x} {xs} Tx LTxs) nilLT =\n  begin\n    reverse ((x \u2237 xs) ++ [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse ((x \u2237 xs) ++ []) \u2261 reverse t)\n               (++-rightIdentity (consLT Tx LTxs))\n               refl\n      \u27e9\n    reverse (x \u2237 xs)\n      \u2261\u27e8 sym (++-[] (reverse (x \u2237 xs))) \u27e9\n    [] ++ reverse (x \u2237 xs)\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ reverse (x \u2237 xs) \u2261 t ++ reverse (x \u2237 xs))\n               (sym (rev-[] []))\n               refl\n      \u27e9\n    reverse [] ++ reverse (x \u2237 xs)\n  \u220e\n\nreverse-++-commute (consLT {x} {xs} Tx LTxs) (consLT {y} {ys} Ty LTys) =\n  begin\n    reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261\u27e8 refl \u27e9\n    rev ((x \u2237 xs) ++ y \u2237 ys) []\n      \u2261\u27e8 subst (\u03bb t \u2192 rev ((x \u2237 xs) ++ y \u2237 ys) [] \u2261 rev t [])\n               (++-\u2237 x xs (y \u2237 ys))\n               refl\n      \u27e9\n    rev (x \u2237 (xs ++ y \u2237 ys)) [] \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n    rev (xs ++ y \u2237 ys) (x \u2237 [])\n      \u2261\u27e8 rev-++-commute (++-ListTree LTxs (consLT Ty LTys)) (consLT Tx nilLT) \u27e9\n    rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 rev (xs ++ y \u2237 ys) [] ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               refl\n               refl\n      \u27e9\n    reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (xs ++ y \u2237 ys) ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n               (reverse-++-commute LTxs (consLT Ty LTys))  -- IH.\n               refl\n      \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 ++-assoc (rev-ListTree (consLT Ty LTys) nilLT)\n                  (rev-ListTree LTxs nilLT)\n                  (consLT Tx nilLT)\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t ++ x \u2237 [])\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-++-commute LTxs (consLT Tx nilLT))\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs (x \u2237 []) \u2261\n                      reverse (y \u2237 ys) ++ t)\n               (sym $ rev-\u2237 x xs [])\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n      \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev (x \u2237 xs) [] \u2261\n                      reverse (y \u2237 ys) ++ t)\n               refl\n               refl\n      \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\nmap-++-commute : \u2200 f {xs ys} \u2192 (\u2200 {x} \u2192 Tree x \u2192 Tree (f \u00b7 x)) \u2192\n                 ListTree xs \u2192 ListTree ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f {ys = ys} fTree nilLT LTys =\n  begin\n    map f ([] ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 map f ([] ++ ys) \u2261 map f t)\n               (++-[] ys)\n               refl\n      \u27e9\n    map f ys\n      \u2261\u27e8 sym (++-leftIdentity (map-ListTree f fTree LTys)) \u27e9\n    [] ++ map f ys\n      \u2261\u27e8 subst (\u03bb t \u2192 [] ++ map f ys \u2261 t ++ map f ys)\n               (sym (map-[] f))\n               refl\n      \u27e9\n    map f [] ++ map f ys\n  \u220e\n\nmap-++-commute f {ys = ys} fTree (consLT {x} {xs} Tx LTxs) LTys =\n  begin\n    map f ((x \u2237 xs) ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 map f ((x \u2237 xs) ++ ys) \u2261 map f t)\n               (++-\u2237 x xs ys)\n               refl\n      \u27e9\n    map f (x \u2237 xs ++ ys)\n      \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n    f \u00b7 x \u2237 map f (xs ++ ys)\n      \u2261\u27e8 subst (\u03bb t \u2192 f \u00b7 x \u2237 map f (xs ++ ys) \u2261 f \u00b7 x \u2237 t)\n               (map-++-commute f fTree LTxs LTys) -- IH.\n               refl\n      \u27e9\n    f \u00b7 x \u2237 (map f xs ++ map f ys)\n      \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n    (f \u00b7 x \u2237 map f xs) ++ map f ys\n      \u2261\u27e8 subst (\u03bb t \u2192 (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261 t ++ map f ys)\n               (sym (map-\u2237 f x xs))\n               refl\n      \u27e9\n    map f (x \u2237 xs) ++ map f ys\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f5174554f8808bb36ec976d1d6d35f45e84062f0","subject":"universe.agda","message":"universe.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"universe.agda","new_file":"universe.agda","new_contents":"module universe where\n\nimport Level as L\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ _*_\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U\nfold-U {_} {T} u\u2080 uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U = ?\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","old_contents":"module universe where\n\nimport Level as L\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\nwidth-U : Universe \u2115\nwidth-U = mk 0 1 _+_ _*_\n\ncard-U : Universe \u2115\ncard-U = mk 1 2 _*_ _^_\n\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T\n\ndata U : Set where\n  \u22a4\u2032 Bit\u2032 : U\n  _\u00d7\u2032_    : U \u2192 U \u2192 U\n  _^\u2032_    : U \u2192 \u2115 \u2192 U\n\nsyn-U : Universe U\nsyn-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\nfold-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 U \u2192 T\nfold-U {_} {T} uni = go\n  where open Universe uni\n        go : U \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bd04af9edb07ad3112759a1a95ba54c2c0f3601b","subject":"wip cut","message":"wip cut\n","repos":"crypto-agda\/protocols","old_file":"partensor\/Shallow\/Cut.agda","new_file":"partensor\/Shallow\/Cut.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product hiding (zip)\n                         renaming (_,_ to \u27e8_,_\u27e9; proj\u2081 to fst; proj\u2082 to snd;\n                                   map to \u00d7map)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum\nopen import Data.Nat\n{-\nopen import Data.Vec\nopen import Data.Fin\n-}\n-- open import Data.List\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP hiding ([_]; J)\nopen import partensor.Shallow.Dom\nimport partensor.Shallow.Session as Session\nimport partensor.Shallow.Map as Map\nimport partensor.Shallow.Env as Env\nimport partensor.Shallow.Proto as Proto\nopen Session hiding (Ended)\nopen Env     hiding (_\/\u2080_; _\/\u2081_; _\/_; Ended)\nopen Proto   hiding ()\nopen import partensor.Shallow.Equiv\nopen import partensor.Shallow.Term\n\nmodule partensor.Shallow.Cut where\ninfixl 4 _\u2666Proto'_\n-- things we have but I want better unification\n{-\n  _\u2248'_ : \u2200 {\u03b4I \u03b4J} \u2192 Proto \u03b4I \u2192 Proto \u03b4J \u2192 Set\u2081\n  \u2248'-refl : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 I \u2248' I\n  \u2248'-sym : \u2200 {\u03b4I \u03b4J}{I : Proto \u03b4I}{J : Proto \u03b4J} \u2192 I \u2248' J \u2192 J \u2248' I\n  \u2248'-trans : \u2200 {\u03b4a \u03b4b \u03b4c}{A : Proto \u03b4a}{B : Proto \u03b4b}{C : Proto \u03b4c} \u2192 A \u2248' B \u2192 B \u2248' C \u2192 A \u2248' C\n-}\npostulate\n  _\u2666Proto'_ : \u2200 {\u03b4a \u03b4b}(A : Proto \u03b4a)(B : Proto \u03b4b) \u2192 Proto (\u03b4a \u2666Doms \u03b4b)\n\n{-\ndata DifferentVarsDoms : \u2200 {\u03b4I c d} \u2192 Doms'.[ c ]\u2208 \u03b4I \u2192 Doms'.[ d ]\u2208 \u03b4I \u2192 Set where\n  h\/t : \u2200 {a b Db l}\n   \u2192 DifferentVarsDoms {Db ,[ a ]}{a}{b} Doms'.here (Doms'.there l)\n  t\/h : \u2200 {a b Db l}\n   \u2192 DifferentVarsDoms {Db ,[ a ]}{b}{a} (Doms'.there l) Doms'.here\n  t\/t : \u2200 {a c d D l l'} \u2192 DifferentVarsDoms {D ,[ a ]}{c}{d} (Doms'.there l) (Doms'.there l')\n\n-- Need to update this, they may point to the same tensor block but different inside it...\n-- boring I know!\nDifferentVars : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B} \u2192 [ c \u21a6 A ]\u2208 I \u2192 [ d \u21a6 B ]\u2208 I \u2192 Set\nDifferentVars l l' = DifferentVarsDoms (Proto.forget ([\u21a6]\u2208.lI l)) (Proto.forget ([\u21a6]\u2208.lI l'))\n-}\npostulate\n  DifferentVars\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B} \u2192 [ c \u21a6 A \u2026]\u2208 I \u2192 [ d \u21a6 B \u2026]\u2208 I \u2192 Set\n  Diff-sym\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}{l : [ c \u21a6 A \u2026]\u2208 I}{l' : [ d \u21a6 B \u2026]\u2208 I}\n    \u2192 DifferentVars\u2026 l l' \u2192 DifferentVars\u2026 l' l\n\nrecord DifferentVars {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A ]\u2208 I)(l' : [ d \u21a6 B ]\u2208 I) : Set where\n  constructor mk\n  field\n    Diff\u2026 : DifferentVars\u2026 ([\u21a6]\u2208.l\u2026 l) ([\u21a6]\u2208.l\u2026 l')\nopen DifferentVars\n\nDiff-sym : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}{l : [ c \u21a6 A ]\u2208 I}{l' : [ d \u21a6 B ]\u2208 I}\n    \u2192 DifferentVars l l' \u2192 DifferentVars l' l\nDiff\u2026 (Diff-sym df) = Diff-sym\u2026 (Diff\u2026 df)\n\ndata SameVar? {\u03b4I}{I : Proto \u03b4I} : \u2200 {c c' A A'} \u2192 [ c \u21a6 A \u2026]\u2208 I \u2192 [ c' \u21a6 A' \u2026]\u2208 I \u2192 Set\u2081 where\n  same : \u2200 {c A}{l : [ c \u21a6 A \u2026]\u2208 I} \u2192 SameVar? l l\n  diff : \u2200 {c c' A B}{l : [ c \u21a6 A \u2026]\u2208 I}{l' : [ c' \u21a6 B \u2026]\u2208 I} \u2192 DifferentVars\u2026 l l' \u2192 SameVar? l l'\n\npostulate\n  sameVar? : \u2200 {\u03b4I}{I : Proto \u03b4I}{c c' A A'}(l : [ c \u21a6 A \u2026]\u2208 I)(l' : [ c' \u21a6 A' \u2026]\u2208 I) \u2192 SameVar? l l'\n\npostulate\n  TC-conv : \u2200 {\u03b4I \u03b4J}{I : Proto \u03b4I}{J : Proto \u03b4J}\n    \u2192 I \u2248 J \u2192 TC\u27e8 I \u27e9 \u2192 TC\u27e8 J \u27e9\n\n  \u2666-assoc : \u2200 {\u03b4a \u03b4b \u03b4c}{A : Proto \u03b4a}{B : Proto \u03b4b}{C : Proto \u03b4c} \u2192 A \u2666Proto' (B \u2666Proto' C) \u2248 (A \u2666Proto' B) \u2666Proto' C\n  \u2666-com : \u2200 {\u03b4a \u03b4b}{A : Proto \u03b4a}{B : Proto \u03b4b} \u2192 (A \u2666Proto' B) \u2248 (B \u2666Proto' A)\n  \u2666-cong\u2082 : \u2200 {\u03b4a \u03b4b \u03b4c \u03b4d}{A : Proto \u03b4a}{B : Proto \u03b4b}{C : Proto \u03b4c}{D : Proto \u03b4d}\n          \u2192 A \u2248 B \u2192 C \u2248 D \u2192 A \u2666Proto' C \u2248 B \u2666Proto' D\n  \u2666-com, : \u2200 {\u03b4a \u03b4 \u03b4b}{A : Proto \u03b4a}{B : Proto \u03b4b}{E : Env \u03b4} \u2192 (A ,[ E ]) \u2666Proto' B \u2248 (A \u2666Proto' B),[ E ]\n\n\n  \u2208\u2666\u2080\u2026 : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081} \u2192 [ c \u21a6 A \u2026]\u2208 I\u2080 \u2192 [ c \u21a6 A \u2026]\u2208 (I\u2080 \u2666Proto' I\u2081)\n  \u2208\u2666\u2081\u2026 : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081} \u2192 [ c \u21a6 A \u2026]\u2208 I\u2081 \u2192 [ c \u21a6 A \u2026]\u2208 (I\u2080 \u2666Proto' I\u2081)\n\n\n\n  move\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A \u2026]\u2208 I)(l' : [ d \u21a6 B \u2026]\u2208 I) \u2192 DifferentVars\u2026 l l'\n          \u2192 [ d \u21a6 B \u2026]\u2208 (I [\/\u2026] l)\n  move-compute\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A \u2026]\u2208 I)(l' : [ d \u21a6 B \u2026]\u2208 I)(l\/=l' : DifferentVars\u2026 l l')\n    \u2192 (I [\/\u2026] l) [\/\u2026] move\u2026 l l' l\/=l' \u2248 (I [\/\u2026] l) \/Ds Proto.forget ([\u21a6\u2026]\u2208.lI l')\n\n\u2208\u2666\u2080 : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081} \u2192 [ c \u21a6 A ]\u2208 I\u2080 \u2192 [ c \u21a6 A ]\u2208 (I\u2080 \u2666Proto' I\u2081)\n\u2208\u2666\u2080 (mk l\u2026 E\/c) = mk (\u2208\u2666\u2080\u2026 l\u2026) {!!}\n\n\u2208\u2666\u2081 : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081} \u2192 [ c \u21a6 A ]\u2208 I\u2081 \u2192 [ c \u21a6 A ]\u2208 (I\u2080 \u2666Proto' I\u2081)\n\u2208\u2666\u2081 (mk l\u2026 E\/c) = {!!}\n\n\u2208\u2666\u2080-compute : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081}(l : [ c \u21a6 A ]\u2208 I\u2080) \u2192\n          (I\u2080 \u2666Proto' I\u2081) [\/] (\u2208\u2666\u2080 l) \u2248 (I\u2080 [\/] l) \u2666Proto' I\u2081\n\u2208\u2666\u2080-compute = {!!}\n\n\u2208\u2666\u2081-compute : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081}(l : [ c \u21a6 A ]\u2208 I\u2081) \u2192\n          (I\u2080 \u2666Proto' I\u2081) [\/] (\u2208\u2666\u2081 l) \u2248 I\u2080 \u2666Proto' (I\u2081 [\/] l)\n\u2208\u2666\u2081-compute = {!!}\n\nmove : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A ]\u2208 I)(l' : [ d \u21a6 B ]\u2208 I) \u2192 DifferentVars l l'\n          \u2192 [ d \u21a6 B ]\u2208 (I [\/] l)\nmove (mk l X) (mk l' Y) df = mk (move\u2026 l l' (Diff\u2026 df)) {!!}\n\n\nTC-\u2208\u214b : \u2200 {\u03b4I \u03b4K c A B}{I : Proto \u03b4I}{K : Proto \u03b4K}(l : [ c \u21a6 A \u214b B ]\u2208 I)\n  \u2192 (\u2200 {d e \u03b4J}{J : Proto \u03b4J} (l : [ d \u21a6 A ]\u2208 J)(l' : [ e \u21a6  B ]\u2208 J) \u2192 DifferentVars l l' \u2192 TC\u27e8 J \u27e9\n     \u2192 TC\u27e8 ((J \/ [\u21a6]\u2208.lI l) \/Ds Proto.forget ([\u21a6]\u2208.lI l')) \u2666Proto' K \u27e9)\n  \u2192 TC\u27e8 I \u27e9 \u2192  TC\u27e8 I [\/] l \u2666Proto' K \u27e9\nTC-\u2208\u214b l cont (TC-\u2297-out l\u2081 \u03c3s \u03c3E A0 P\u2080 P\u2081) with sameVar? ([\u21a6]\u2208.l\u2026 l) l\u2081\n... | X = {!!}\nTC-\u2208\u214b l cont (TC-\u214b-inp l\u2081 P) with sameVar? ([\u21a6]\u2208.l\u2026 l) ([\u21a6]\u2208.l\u2026 l\u2081)\nTC-\u2208\u214b (mk l y) cont (TC-\u214b-inp (mk .l x) P) | same = TC-conv (\u2666-cong\u2082 (\u2248-trans (\u2248,[end] _) (\u2248,[end] _)) \u2248-refl) (cont (mk (mk (there here) here) _) (mk (mk here here) _) {!!} (TC-conv \u2248-refl (P c\u2080 c\u2081)))\n  where\n    postulate\n      c\u2080 c\u2081 : _\nTC-\u2208\u214b l cont (TC-\u214b-inp l\u2081 P) | diff l\/=l\u2081 = TC-\u214b-inp (\u2208\u2666\u2080 (move  l l\u2081 (mk l\/=l\u2081))) (\u03bb c\u2080 c\u2081 \u2192\n   TC-conv (\u2248-trans \u2666-com, (\u2248,[] (\u2248-trans \u2666-com, (\u2248,[] (\u2248-sym (\u2248-trans (\u2208\u2666\u2080-compute (move l l\u2081 (mk l\/=l\u2081)))\n           (\u2666-cong\u2082 (\u2248-trans (move-compute\u2026 ([\u21a6]\u2208.l\u2026 l) ([\u21a6]\u2208.l\u2026 l\u2081) l\/=l\u2081) \n           (\u2248-trans {!!}\n            (\u2248-sym (move-compute\u2026 _ _ _)))) \u2248-refl))) \u223c-refl)) \u223c-refl))\n  (TC-\u2208\u214b (there[] (there[] (move l\u2081 l (Diff-sym (mk l\/=l\u2081))))) cont (P c\u2080 c\u2081)))\nTC-\u2208\u214b l cont (TC-end E) = {!!}\nTC-\u2208\u214b l cont (TC-split \u03c3s A1 P P\u2081) = {!!}\n\n{-\nTC-\u2208\u2297 : \u2200 {\u03b4I \u03b4K c A B}{I : Proto \u03b4I}{K : Proto \u03b4K}(l : [ c \u21a6 A \u2297 B ]\u2208 I)\n  \u2192 (\u2200 {d e \u03b4J\u2080 \u03b4J\u2081}{J\u2080 : Proto \u03b4J\u2080}{J\u2081 : Proto \u03b4J\u2081}\n       (l\u2080 : [ d \u21a6 A ]\u2208 J\u2080)(l\u2081 : [ e \u21a6 B ]\u2208 J\u2081) \u2192 TC\u27e8 J\u2080 \u27e9 \u2192 TC\u27e8 J\u2081 \u27e9\n        \u2192 TC\u27e8 (J\u2080 [\/] l\u2080 \u2666Proto' J\u2081 [\/] l\u2081) \u2666Proto' K \u27e9)\n  \u2192 TC\u27e8 I \u27e9 \u2192 TC\u27e8 I [\/] l \u2666Proto' K \u27e9\nTC-\u2208\u2297 = {!!}\n\n\n{-\nTC-cut :\n    \u2200 {c\u2080 c\u2081 S\u2080 S\u2081 \u03b4\u2080 \u03b4\u2081}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (l\u2080 : [ c\u2080 \u21a6 S\u2080 ]\u2208 I\u2080)(l\u2081 : [ c\u2081 \u21a6 S\u2081 ]\u2208 I\u2081)\n      (P\u2080 : TC\u27e8 I\u2080 \u27e9) (P\u2081 : TC\u27e8 I\u2081 \u27e9)\n    \u2192 TC\u27e8 (I\u2080 [\/] l\u2080) \u2666Proto' (I\u2081 [\/] l\u2081) \u27e9\nTC-cut end \u03c3s A0 P\u2080 P\u2081 = {!TC-split \u03c3s A0 P\u2080 P\u2081!}\nTC-cut (\u2297\u214b D D\u2081 D\u2082 D\u2083) l\u2080 l\u2081 P\u2080 P\u2081 = TC-conv \u2666-com\n  (TC-\u2208\u214b l\u2081 (\u03bb d e d\/=e a'b' \u2192 TC-conv (\u2248-trans \u2666-com (\u2666-cong\u2082 (\u2248-trans (move-compute {!e!} {!d!} {!(Diff-sym d\/=e)!}) {!Proto.forget!}) \u2248-refl))\n   (TC-\u2208\u2297 l\u2080 (\u03bb d' e' a b \u2192 TC-conv (\u2248-trans (\u2666-cong\u2082 \u2248-refl\n               (\u2208\u2666\u2081-compute (move {!e!} {!d!} {!(Diff-sym d\/=e)!}))) \u2666-assoc)\n     (TC-cut  D d' (\u2208\u2666\u2081 (move {!e!} {!d!} {!(Diff-sym d\/=e)!})) a (TC-cut D\u2082 e' e b a'b')))\n   P\u2080))\n  P\u2081)\nTC-cut (\u214b\u2297 D D\u2081 D\u2082 D\u2083) l\u2080 l\u2081 P\u2080 P\u2081 = TC-conv \u2248-refl\n  (TC-\u2208\u214b l\u2080 (\u03bb {_}{_}{_}{J} d e d\/=e ab \u2192 TC-conv \u2666-com\n  (TC-\u2208\u2297 l\u2081 (\u03bb {_}{_}{_}{_}{J\u2080}{J\u2081} d' e' a b \u2192 let X = TC-cut D\u2081 d' d a ab\n       in TC-conv (\u2248-trans (\u2666-cong\u2082 \u2248-refl (\u2208\u2666\u2081-compute (move d e d\/=e)))\n               (\u2248-trans \u2666-assoc (\u2666-cong\u2082 \u2666-com (move-compute d e (mk d\/=e)))))\n          (TC-cut D\u2083 e' (\u2208\u2666\u2081 (move d e d\/=e)) b X))\n   P\u2081)) P\u2080)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product hiding (zip)\n                         renaming (_,_ to \u27e8_,_\u27e9; proj\u2081 to fst; proj\u2082 to snd;\n                                   map to \u00d7map)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum\nopen import Data.Nat\n{-\nopen import Data.Vec\nopen import Data.Fin\n-}\n-- open import Data.List\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP hiding ([_]; J)\nopen import partensor.Shallow.Dom\nimport partensor.Shallow.Session as Session\nimport partensor.Shallow.Map as Map\nimport partensor.Shallow.Env as Env\nimport partensor.Shallow.Proto as Proto\nopen Session hiding (Ended)\nopen Env     hiding (_\/\u2080_; _\/\u2081_; _\/_; Ended)\nopen Proto   hiding ()\nopen import partensor.Shallow.Equiv\nopen import partensor.Shallow.Term\n\nmodule partensor.Shallow.Cut where\ninfixl 4 _\u2666Proto'_\n-- things we have but I want better unification\n{-\n  _\u2248'_ : \u2200 {\u03b4I \u03b4J} \u2192 Proto \u03b4I \u2192 Proto \u03b4J \u2192 Set\u2081\n  \u2248'-refl : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 I \u2248' I\n  \u2248'-sym : \u2200 {\u03b4I \u03b4J}{I : Proto \u03b4I}{J : Proto \u03b4J} \u2192 I \u2248' J \u2192 J \u2248' I\n  \u2248'-trans : \u2200 {\u03b4a \u03b4b \u03b4c}{A : Proto \u03b4a}{B : Proto \u03b4b}{C : Proto \u03b4c} \u2192 A \u2248' B \u2192 B \u2248' C \u2192 A \u2248' C\n-}\npostulate\n  _\u2666Proto'_ : \u2200 {\u03b4a \u03b4b}(A : Proto \u03b4a)(B : Proto \u03b4b) \u2192 Proto (\u03b4a \u2666Doms \u03b4b)\n\n{-\ndata DifferentVarsDoms : \u2200 {\u03b4I c d} \u2192 Doms'.[ c ]\u2208 \u03b4I \u2192 Doms'.[ d ]\u2208 \u03b4I \u2192 Set where\n  h\/t : \u2200 {a b Db l}\n   \u2192 DifferentVarsDoms {Db ,[ a ]}{a}{b} Doms'.here (Doms'.there l)\n  t\/h : \u2200 {a b Db l}\n   \u2192 DifferentVarsDoms {Db ,[ a ]}{b}{a} (Doms'.there l) Doms'.here\n  t\/t : \u2200 {a c d D l l'} \u2192 DifferentVarsDoms {D ,[ a ]}{c}{d} (Doms'.there l) (Doms'.there l')\n\n-- Need to update this, they may point to the same tensor block but different inside it...\n-- boring I know!\nDifferentVars : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B} \u2192 [ c \u21a6 A ]\u2208 I \u2192 [ d \u21a6 B ]\u2208 I \u2192 Set\nDifferentVars l l' = DifferentVarsDoms (Proto.forget ([\u21a6]\u2208.lI l)) (Proto.forget ([\u21a6]\u2208.lI l'))\n-}\npostulate\n  DifferentVars\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B} \u2192 [ c \u21a6 A \u2026]\u2208 I \u2192 [ d \u21a6 B \u2026]\u2208 I \u2192 Set\n  Diff-sym\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}{l : [ c \u21a6 A \u2026]\u2208 I}{l' : [ d \u21a6 B \u2026]\u2208 I}\n    \u2192 DifferentVars\u2026 l l' \u2192 DifferentVars\u2026 l' l\n\nrecord DifferentVars {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A ]\u2208 I)(l' : [ d \u21a6 B ]\u2208 I) : Set where\n  constructor mk\n  field\n    Diff\u2026 : DifferentVars\u2026 ([\u21a6]\u2208.l\u2026 l) ([\u21a6]\u2208.l\u2026 l')\nopen DifferentVars\n\nDiff-sym : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}{l : [ c \u21a6 A ]\u2208 I}{l' : [ d \u21a6 B ]\u2208 I}\n    \u2192 DifferentVars l l' \u2192 DifferentVars l' l\nDiff\u2026 (Diff-sym df) = Diff-sym\u2026 (Diff\u2026 df)\n\ndata SameVar? {\u03b4I}{I : Proto \u03b4I} : \u2200 {c c' A A'} \u2192 [ c \u21a6 A \u2026]\u2208 I \u2192 [ c' \u21a6 A' \u2026]\u2208 I \u2192 Set\u2081 where\n  same : \u2200 {c A}{l : [ c \u21a6 A \u2026]\u2208 I} \u2192 SameVar? l l\n  diff : \u2200 {c c' A B}{l : [ c \u21a6 A \u2026]\u2208 I}{l' : [ c' \u21a6 B \u2026]\u2208 I} \u2192 DifferentVars\u2026 l l' \u2192 SameVar? l l'\n\npostulate\n  sameVar? : \u2200 {\u03b4I}{I : Proto \u03b4I}{c c' A A'}(l : [ c \u21a6 A \u2026]\u2208 I)(l' : [ c' \u21a6 A' \u2026]\u2208 I) \u2192 SameVar? l l'\n\npostulate\n  TC-conv : \u2200 {\u03b4I \u03b4J}{I : Proto \u03b4I}{J : Proto \u03b4J}\n    \u2192 I \u2248 J \u2192 TC\u27e8 I \u27e9 \u2192 TC\u27e8 J \u27e9\n\n\n  move\u2026 : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A \u2026]\u2208 I)(l' : [ d \u21a6 B \u2026]\u2208 I) \u2192 DifferentVars\u2026 l l'\n          \u2192 [ d \u21a6 B \u2026]\u2208 (I [\/\u2026] l)\n\n  \u2208\u2666\u2080 : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081} \u2192 [ c \u21a6 A ]\u2208 I\u2080 \u2192 [ c \u21a6 A ]\u2208 (I\u2080 \u2666Proto' I\u2081)\n  \u2208\u2666\u2081 : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081} \u2192 [ c \u21a6 A ]\u2208 I\u2081 \u2192 [ c \u21a6 A ]\u2208 (I\u2080 \u2666Proto' I\u2081)\n  \u2666-assoc : \u2200 {\u03b4a \u03b4b \u03b4c}{A : Proto \u03b4a}{B : Proto \u03b4b}{C : Proto \u03b4c} \u2192 A \u2666Proto' (B \u2666Proto' C) \u2248 (A \u2666Proto' B) \u2666Proto' C\n  \u2666-com : \u2200 {\u03b4a \u03b4b}{A : Proto \u03b4a}{B : Proto \u03b4b} \u2192 (A \u2666Proto' B) \u2248 (B \u2666Proto' A)\n  \u2208\u2666\u2080-compute : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081}(l : [ c \u21a6 A ]\u2208 I\u2080) \u2192\n          (I\u2080 \u2666Proto' I\u2081) [\/] (\u2208\u2666\u2080 l) \u2248 (I\u2080 [\/] l) \u2666Proto' I\u2081\n  \u2208\u2666\u2081-compute : \u2200 {\u03b4\u2080 \u03b4\u2081 c A}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081}(l : [ c \u21a6 A ]\u2208 I\u2081) \u2192\n          (I\u2080 \u2666Proto' I\u2081) [\/] (\u2208\u2666\u2081 l) \u2248 I\u2080 \u2666Proto' (I\u2081 [\/] l)\n  \u2666-cong\u2082 : \u2200 {\u03b4a \u03b4b \u03b4c \u03b4d}{A : Proto \u03b4a}{B : Proto \u03b4b}{C : Proto \u03b4c}{D : Proto \u03b4d}\n          \u2192 A \u2248 B \u2192 C \u2248 D \u2192 A \u2666Proto' C \u2248 B \u2666Proto' D\n  move-compute : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A \u2026]\u2208 I)(l' : [ d \u21a6 B \u2026]\u2208 I)(l\/=l' : DifferentVars\u2026 l l')\n    \u2192 (I [\/\u2026] l) [\/\u2026] move\u2026 l l' l\/=l' \u2248 (I [\/\u2026] l) \/Ds Proto.forget ([\u21a6\u2026]\u2208.lI l')\n\nmove : \u2200 {\u03b4I}{I : Proto \u03b4I}{c d A B}(l : [ c \u21a6 A ]\u2208 I)(l' : [ d \u21a6 B ]\u2208 I) \u2192 DifferentVars l l'\n          \u2192 [ d \u21a6 B ]\u2208 (I [\/] l)\nmove (mk l X) (mk l' Y) df = mk (move\u2026 l l' (Diff\u2026 df)) {!!}\n\n\nTC-\u2208\u214b : \u2200 {\u03b4I \u03b4K c A B}{I : Proto \u03b4I}{K : Proto \u03b4K}(l : [ c \u21a6 A \u214b B ]\u2208 I)\n  \u2192 (\u2200 {d e \u03b4J}{J : Proto \u03b4J} (l : [ d \u21a6 A ]\u2208 J)(l' : [ e \u21a6  B ]\u2208 J) \u2192 DifferentVars l l' \u2192 TC\u27e8 J \u27e9\n     \u2192 TC\u27e8 ((J \/ [\u21a6]\u2208.lI l) \/Ds Proto.forget ([\u21a6]\u2208.lI l')) \u2666Proto' K \u27e9)\n  \u2192 TC\u27e8 I \u27e9 \u2192  TC\u27e8 I [\/] l \u2666Proto' K \u27e9\nTC-\u2208\u214b l cont (TC-\u2297-out l\u2081 \u03c3s \u03c3E A0 P\u2080 P\u2081) with sameVar? ([\u21a6]\u2208.l\u2026 l) l\u2081\n... | X = {!!}\nTC-\u2208\u214b l cont (TC-\u214b-inp l\u2081 P) with sameVar? ([\u21a6]\u2208.l\u2026 l) ([\u21a6]\u2208.l\u2026 l\u2081)\nTC-\u2208\u214b (mk l y) cont (TC-\u214b-inp (mk .l x) P) | same = TC-conv (\u2666-cong\u2082 (\u2248-trans (\u2248,[end] _) (\u2248,[end] _)) \u2248-refl) (cont (mk (mk (there here) here) _) (mk (mk here here) _) {!!} (TC-conv \u2248-refl (P c\u2080 c\u2081)))\n  where\n    postulate\n      c\u2080 c\u2081 : _\nTC-\u2208\u214b l cont (TC-\u214b-inp l\u2081 P) | diff l\/=l\u2081 = TC-\u214b-inp (\u2208\u2666\u2080 (move  l l\u2081 (mk l\/=l\u2081))) (\u03bb c\u2080 c\u2081 \u2192\n   TC-conv {!there[]!} (TC-\u2208\u214b (there[] (there[] (move l\u2081 l (Diff-sym (mk l\/=l\u2081))))) cont (P c\u2080 c\u2081)))\nTC-\u2208\u214b l cont (TC-end E) = {!!}\nTC-\u2208\u214b l cont (TC-split \u03c3s A1 P P\u2081) = {!!}\n\nTC-\u2208\u2297 : \u2200 {\u03b4I \u03b4K c A B}{I : Proto \u03b4I}{K : Proto \u03b4K}(l : [ c \u21a6 A \u2297 B ]\u2208 I)\n  \u2192 (\u2200 {d e \u03b4J\u2080 \u03b4J\u2081}{J\u2080 : Proto \u03b4J\u2080}{J\u2081 : Proto \u03b4J\u2081}\n       (l\u2080 : [ d \u21a6 A ]\u2208 J\u2080)(l\u2081 : [ e \u21a6 B ]\u2208 J\u2081) \u2192 TC\u27e8 J\u2080 \u27e9 \u2192 TC\u27e8 J\u2081 \u27e9\n        \u2192 TC\u27e8 (J\u2080 [\/] l\u2080 \u2666Proto' J\u2081 [\/] l\u2081) \u2666Proto' K \u27e9)\n  \u2192 TC\u27e8 I \u27e9 \u2192 TC\u27e8 I [\/] l \u2666Proto' K \u27e9\nTC-\u2208\u2297 = {!!}\n\n\nTC-cut :\n    \u2200 {c\u2080 c\u2081 S\u2080 S\u2081 \u03b4\u2080 \u03b4\u2081}{I\u2080 : Proto \u03b4\u2080}{I\u2081 : Proto \u03b4\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (l\u2080 : [ c\u2080 \u21a6 S\u2080 ]\u2208 I\u2080)(l\u2081 : [ c\u2081 \u21a6 S\u2081 ]\u2208 I\u2081)\n      (P\u2080 : TC\u27e8 I\u2080 \u27e9) (P\u2081 : TC\u27e8 I\u2081 \u27e9)\n    \u2192 TC\u27e8 (I\u2080 [\/] l\u2080) \u2666Proto' (I\u2081 [\/] l\u2081) \u27e9\nTC-cut end \u03c3s A0 P\u2080 P\u2081 = {!TC-split \u03c3s A0 P\u2080 P\u2081!}\nTC-cut (\u2297\u214b D D\u2081 D\u2082 D\u2083) l\u2080 l\u2081 P\u2080 P\u2081 = TC-conv \u2666-com\n  (TC-\u2208\u214b l\u2081 (\u03bb d e d\/=e a'b' \u2192 TC-conv (\u2248-trans \u2666-com (\u2666-cong\u2082 (\u2248-trans (move-compute {!e!} {!d!} {!(Diff-sym d\/=e)!}) {!Proto.forget!}) \u2248-refl))\n   (TC-\u2208\u2297 l\u2080 (\u03bb d' e' a b \u2192 TC-conv (\u2248-trans (\u2666-cong\u2082 \u2248-refl\n               (\u2208\u2666\u2081-compute (move {!e!} {!d!} {!(Diff-sym d\/=e)!}))) \u2666-assoc)\n     (TC-cut  D d' (\u2208\u2666\u2081 (move {!e!} {!d!} {!(Diff-sym d\/=e)!})) a (TC-cut D\u2082 e' e b a'b')))\n   P\u2080))\n  P\u2081)\nTC-cut (\u214b\u2297 D D\u2081 D\u2082 D\u2083) l\u2080 l\u2081 P\u2080 P\u2081 = TC-conv \u2248-refl\n  (TC-\u2208\u214b l\u2080 (\u03bb {_}{_}{_}{J} d e d\/=e ab \u2192 TC-conv \u2666-com\n  (TC-\u2208\u2297 l\u2081 (\u03bb {_}{_}{_}{_}{J\u2080}{J\u2081} d' e' a b \u2192 let X = TC-cut D\u2081 d' d a ab\n       in TC-conv (\u2248-trans (\u2666-cong\u2082 \u2248-refl (\u2208\u2666\u2081-compute (move d e d\/=e)))\n               (\u2248-trans \u2666-assoc (\u2666-cong\u2082 \u2666-com (move-compute d e (mk d\/=e)))))\n          (TC-cut D\u2083 e' (\u2208\u2666\u2081 (move d e d\/=e)) b X))\n   P\u2081)) P\u2080)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ba05583275f6c5a8cd56f8736ba5d795a68c59bf","subject":"Induce change algebras from abelian groups.","message":"Induce change algebras from abelian groups.\n\nOld-commit-hash: 88f468a21264dd66570d65288e4ee1357f848428\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n","old_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1d9c622bcff742ac6217fb698eb8efe929eea1bd","subject":"At least define Var in OOO semantics","message":"At least define Var in OOO semantics\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/OperationalTypeOperationalTermOperationalValue.agda","new_file":"formalization\/agda\/Spire\/OperationalTypeOperationalTermOperationalValue.agda","new_contents":"module Spire.OperationalTypeOperationalTermOperationalValue where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\n\n----------------------------------------------------------------------\n\n","old_contents":"module Spire.OperationalTypeOperationalTermOperationalValue where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9523dacab73786f130dd16572f06f3e56e660206","subject":"Redid translation of tautology and future into System F-omega, since the previous version didn't support the comonadic structure of future.","message":"Redid translation of tautology and future into System F-omega, since the previous version didn't support the comonadic structure of future.\n","repos":"agda\/agda-frp-js,agda\/agda-frp-js","old_file":"src\/agda\/FRP\/JS\/Model.agda","new_file":"src\/agda\/FRP\/JS\/Model.agda","new_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u2264_ ; _<_ )\nopen import FRP.JS.Bool using ( Bool ; true ; false ; not ; _\u225f_ ) \nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\nsym : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 (a \u2261 b) \u2192 (b \u2261 a)\nsym refl = refl\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Case on booleans\n\ndata Case (c : Bool) : Set where\n  _,_ : \u2200 b \u2192 True (b \u225f c) \u2192 Case c\n\nswitch : \u2200 b \u2192 Case b\nswitch true  = (true , tt)\nswitch false = (false , tt)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\nstruct-sym : \u2200 K {A B C D \u2111 \u211c} \u2192 (A\u2261B : A \u2261 B) \u2192 (C\u2261D : C \u2261 D) \u2192\n  (\u2111 \u2261 struct K A\u2261B \u211c C\u2261D) \u2192 \n    (\u211c \u2261 struct K (sym A\u2261B) \u2111 (sym C\u2261D))\nstruct-sym K refl refl refl = refl\n\nstruct-trans : \u2200 K {A B C D E F}\n  (A\u2261B : A \u2261 B) (B\u2261C : B \u2261 C) (\u211c : K \u220b C \u2194 F) (E\u2261F : E \u2261 F) (D\u2261E : D \u2261 E) \u2192\n    struct K A\u2261B (struct K B\u2261C \u211c E\u2261F) D\u2261E \u2261\n      struct K (trans A\u2261B B\u2261C) \u211c (trans D\u2261E E\u2261F)\nstruct-trans K refl refl \u211c refl refl = refl\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Concatenation of type contexts\n\n_++_ : Kinds \u2192 Kinds \u2192 Kinds\n[]      ++ \u03a5 = \u03a5\n(K \u2237 \u03a3) ++ \u03a5 = K \u2237 (\u03a3 ++ \u03a5)\n\n_\u220b_++_\u220b_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u2200 \u03a5 \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 ++ \u03a5 \u27e7\n[]      \u220b tt       ++ \u03a5 \u220b Bs = Bs\n(K \u2237 \u03a3) \u220b (A , As) ++ \u03a5 \u220b Bs = (A , (\u03a3 \u220b As ++ \u03a5 \u220b Bs))\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03a3 {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 \u2200 \u03a5 {Cs Ds} \u2192 (\u03a5 \u220b Cs \u2194* Ds) \u2192 \n  ((\u03a3 ++ \u03a5) \u220b (\u03a3 \u220b As ++ \u03a5 \u220b Cs) \u2194* (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds))\n[]      \u220b tt       ++\u00b2 \u03a5 \u220b \u2111s = \u2111s\n(K \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s = (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq lt : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 lt \u27e7     = \u03bb t u \u2192 True (t < u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 lt \u27e7\u00b2     = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n_\u227a_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227a_ = capp\u2082 lt\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\n\u00bdinterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\n\u00bdinterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227a tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\n_\u27e8_,_] : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set \u03b1)\nT \u27e8 t , u ] = app (app (app \u00bdinterval T) t) u\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Susbtitution on type variables under a context\n\n\u03c4substn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 TVar K (\u03a3 ++ (L \u2237 \u03a5)) \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\n\u03c4substn+ []      zero    U = U\n\u03c4substn+ []      (suc \u03c4) U = var \u03c4\n\u03c4substn+ (K \u2237 \u03a3) zero    U = var zero\n\u03c4substn+ (K \u2237 \u03a3) (suc \u03c4) U = weaken (skip K id) (\u03c4substn+ \u03a3 \u03c4 U)\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7 (A , As) Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Bs = trans \n  (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs) \n  (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip K id) (A , (\u03a3 \u220b As ++ _ \u220b Bs)))\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ (J \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s = refl\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 (J \u2237 \u03a3) {\u03a5} {L} {K} (suc \u03c4) U {A , As} {B , Bs} {Cs} {Ds} (\u211c , \u211cs) \u2111s = \n  begin\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) X (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (skip J id) (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n      (struct K\n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs))) \n        (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))) \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 struct-trans K \n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs)))\n       (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)))\n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))\n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 \u03c4substn+ (J \u2237 \u03a3) (suc \u03c4) U \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) )\n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (B , Bs) Ds)    \n  \u220e\n\n-- Substitution on types under a context\n\nsubstn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 Typ (\u03a3 ++ (L \u2237 \u03a5)) K \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\nsubstn+ \u03a3 (const C) U = const C\nsubstn+ \u03a3 (abs K T) U = abs K (substn+ (K \u2237 \u03a3) T U)\nsubstn+ \u03a3 (app S T) U = app (substn+ \u03a3 S U) (substn+ \u03a3 T U)\nsubstn+ \u03a3 (var \u03c4)   U = \u03c4substn+ \u03a3 \u03c4 U\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    T\u27e6 T \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 substn+ \u03a3 T U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7 As Bs = refl\nsubstn+ \u03a3 \u27e6 abs K T \u27e7\u27e6 U \u27e7 As Bs = ext (\u03bb A \u2192 substn+ K \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Bs)\nsubstn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Bs = apply (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Bs) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Bs)\nsubstn+ \u03a3 \u27e6 var \u03c4   \u27e7\u27e6 U \u27e7 As Bs = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = refl\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (abs J {K} T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs J T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u211c\n  \u2261\u27e8 substn+ (J \u2237 \u03a3) \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s \u27e9\n    struct K \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds)\n  \u2261\u27e8 struct-ext J K \n       (\u03bb A \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n       (\u03bb \u211c \u2192 T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b \u211c , \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (\u03bb B \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds) \u211c \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 (abs J T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 Bs Ds) \u211c\n  \u220e)))\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (app {J} {K} S T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  begin\n    T\u27e6 app S T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 cong (T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct J \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 struct-apply J K \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n       (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 substn+ \u03a3 (app S T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u220e\nsubstn+ \u03a3 \u27e6 var \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s\n\n-- Substitution on types\n\nsubstn : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 \u03a3) K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 K\nsubstn = substn+ []\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) (As : \u03a3\u27e6 \u03a3 \u27e7)\u2192\n  T\u27e6 T \u27e7 (T\u27e6 U \u27e7 As , As) \u2261 T\u27e6 substn T U \u27e7 As\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7 tt\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192\n  T\u27e6 T \u27e7\u00b2 (T\u27e6 U \u27e7\u00b2 \u211cs , \u211cs) \u2261 \n    struct K (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 Bs)\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 tt\n\n-- Eta-beta equivalence on types\n\ndata _\u220b_\u2263_ {\u03a3} : \u2200 K \u2192 Typ \u03a3 K \u2192 Typ \u03a3 K \u2192 Set where\n  abs : \u2200 K {L T U} \u2192 (L \u220b T \u2263 U) \u2192 ((K \u21d2 L) \u220b abs K T \u2263 abs K U)\n  app : \u2200 {K L F G T U} \u2192 ((K \u21d2 L) \u220b F \u2263 G) \u2192 (K \u220b T \u2263 U) \u2192 (L \u220b app F T \u2263 app G U)\n  beta : \u2200 {K L} T U \u2192 (L \u220b app (abs K T) U \u2263 substn T U)\n  eta : \u2200 {K L} T \u2192 ((K \u21d2 L) \u220b T \u2263 abs K (app (weaken (skip K id) T) tvar\u2080))\n  \u2263-refl : \u2200 {K T} \u2192 (K \u220b T \u2263 T)\n  \u2263-sym : \u2200 {K T U} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 T)\n  \u2263-trans : \u2200 {K T U V} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 V) \u2192 (K \u220b T \u2263 V)\n\n\u2263\u27e6_\u27e7 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} \u2192 (K \u220b T \u2263 U) \u2192 \u2200 As \u2192 T\u27e6 T \u27e7 As \u2261 T\u27e6 U \u27e7 As\n\u2263\u27e6 abs K T\u2263U \u27e7       As = ext (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As))\n\u2263\u27e6 app F\u2263G T\u2263U \u27e7     As = apply (\u2263\u27e6 F\u2263G \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 beta T U \u27e7        As = substn\u27e6 T \u27e7\u27e6 U \u27e7 As\n\u2263\u27e6 eta {K} T \u27e7       As = ext (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n\u2263\u27e6 \u2263-refl \u27e7          As = refl\n\u2263\u27e6 \u2263-sym T\u2263U \u27e7       As = sym (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As = trans (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As)\n\n\u2263\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} (T\u2263U : K \u220b T \u2263 U) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 \u211cs \u2261 struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n\u2263\u27e6 abs K {L} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 (\u211c , \u211cs) \u27e9\n    struct L (\u2263\u27e6 T\u2263U \u27e7 (A , As)) (T\u27e6 U \u27e7\u00b2 (\u211c , \u211cs)) (\u2263\u27e6 T\u2263U \u27e7 (B , Bs))\n  \u2261\u27e8 struct-ext K L (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As)) (\u03bb \u211c' \u2192 T\u27e6 U \u27e7\u00b2 (\u211c' , \u211cs)) (\u03bb B \u2192 \u2263\u27e6 T\u2263U \u27e7 (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 abs K T\u2263U \u27e7 As) (T\u27e6 abs K U \u27e7\u00b2 \u211cs) (\u2263\u27e6 abs K T\u2263U \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 app {K} {L} {F} {G} {T} {U} F\u2263G T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app F T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 cong (T\u27e6 F \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) \u27e9\n    T\u27e6 F \u27e7\u00b2 \u211cs (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)))\n       (\u2263\u27e6 F\u2263G \u27e7\u00b2 \u211cs) \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n      (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 struct-apply K L\n       (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n       (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct L (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 As) (T\u27e6 app G U \u27e7\u00b2 \u211cs) (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 Bs)\n  \u220e\n\u2263\u27e6 beta T U \u27e7\u00b2 \u211cs = substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs\n\u2263\u27e6 eta {K} {L} T \u27e7\u00b2 {As} {Bs} \u211cs = iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 \n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs \u211c\n  \u2261\u27e8 cong (\u03bb X \u2192 X \u211c) (weaken\u27e6 T \u27e7\u00b2 (skip K id) (\u211c , \u211cs)) \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 T \u27e7 (skip K id) (A , As))\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs))\n      (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) \u211c\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 (skip K id) (A , As)) \n       (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl \u211c refl \u27e9\n    struct L \n      (apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c)\n      (apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl)\n  \u2261\u27e8 struct-ext K L\n       (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c) \n       (\u03bb B \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl) \u211c \u27e9\n    struct (K \u21d2 L) \n      (\u2263\u27e6 eta T \u27e7 As)\n      (T\u27e6 abs K (app (weaken (skip K id) T) (var zero)) \u27e7\u00b2 \u211cs)\n      (\u2263\u27e6 eta T \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 \u2263-refl \u27e7\u00b2 \u211cs = refl\n\u2263\u27e6 \u2263-sym {K} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  struct-sym K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs)\n\u2263\u27e6 \u2263-trans {K} {T} {U} {V} T\u2263U U\u2263V \u27e7\u00b2 {As} {Bs} \u211cs =\n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u2263\u27e6 T\u2263U \u27e7 As) X (\u2263\u27e6 T\u2263U \u27e7 Bs)) (\u2263\u27e6 U\u2263V \u27e7\u00b2 \u211cs) \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (struct K (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs)) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 struct-trans K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct K (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 Bs)\n  \u220e\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  \u227c-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  \u227c-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n  \u227c-absurd : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time ((tvar\u2081 \u227a tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 tvar\u2082)))))\n  \u227c-case : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time (((tvar\u2081 \u227c tvar\u2080) \u22b8 tvar\u2082) \u22b8 (((tvar\u2080 \u227a tvar\u2081) \u22b8 tvar\u2082) \u22b8 tvar\u2082)))))\n\nabsurd : \u2200 {\u03b1} {A : Set \u03b1} b \u2192 True b \u2192 True (not b) \u2192 A\nabsurd true  tt ()\nabsurd false () tt\n\ncond : \u2200 {\u03b1} {A : Set \u03b1} b \u2192 (True (not b) \u2192 A) \u2192 (True b \u2192 A) \u2192 A\ncond true  f g = g tt\ncond false f g = f tt\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7     As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7      As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7      As = \u03bb A B \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7   As = \u2264-refl\nc\u27e6 \u227c-trans \u27e7  As = \u2264-trans\nc\u27e6 \u227c-absurd \u27e7 As = \u03bb A t u \u2192 absurd (t < u)\nc\u27e6 \u227c-case \u27e7   As = \u03bb A t u \u2192 cond (u < t)\n\ncond\u00b2 : \u2200 {\u03b1} {A A\u2032 : Set \u03b1} (\u211c : A \u2192 A\u2032 \u2192 Set \u03b1) b \u2192\n  \u2200 {f f\u2032} \u2192 (\u2200 b\u00d7 \u2192 \u211c (f b\u00d7) (f\u2032 b\u00d7)) \u2192\n    \u2200 {g g\u2032} \u2192 (\u2200 b\u2713 \u2192 \u211c (g b\u2713) (g\u2032 b\u2713)) \u2192\n      \u211c (cond b f g) (cond b f\u2032 g\u2032)\ncond\u00b2 \u211c true  f\u211cf\u2032 g\u211cg\u2032 = g\u211cg\u2032 tt\ncond\u00b2 \u211c false f\u211cf\u2032 g\u211cg\u2032 = f\u211cf\u2032 tt\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2        \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2        \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7\u00b2     \u211cs = _\nc\u27e6 \u227c-trans \u27e7\u00b2    \u211cs = _\nc\u27e6 \u227c-absurd \u27e7\u00b2   \u211cs = \u03bb \u211c {t} t\u2261t {u} u\u2261u {t<u} tt {u\u2264t} tt \u2192 absurd (t < u) t<u u\u2264t\nc\u27e6 \u227c-case {\u03b1} \u27e7\u00b2 \u211cs = lemma where\n  lemma : \u2200 {A A\u2032 : Set \u03b1} (\u211c : A \u2192 A\u2032 \u2192 Set \u03b1) \u2192\n    \u2200 {t t\u2032} \u2192 (t \u2261 t\u2032) \u2192 \u2200 {u u\u2032} \u2192 (u \u2261 u\u2032) \u2192\n      \u2200 {f f\u2032} \u2192 (\u2200 {t\u2264u t\u2032\u2264u\u2032} \u2192 \u22a4 \u2192 \u211c (f t\u2264u) (f\u2032 t\u2032\u2264u\u2032)) \u2192\n        \u2200 {g g\u2032} \u2192 (\u2200 {u<t u\u2032<t\u2032} \u2192 \u22a4 \u2192 \u211c (g u<t) (g\u2032 u\u2032<t\u2032)) \u2192\n          \u211c (cond (u < t) f g) (cond (u\u2032 < t\u2032) f\u2032 g\u2032)\n  lemma \u211c {t} refl {u} refl f\u211cf\u2032 g\u211cg\u2032 = \n    cond\u00b2 \u211c (u < t) (\u03bb t\u2264u \u2192 f\u211cf\u2032 {t\u2264u} {t\u2264u} tt) (\u03bb u<t \u2192 g\u211cg\u2032 {u<t} {u<t} tt)\n  \n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n  tapp : \u2200 {K \u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} \u2192 Exp \u0393 (\u03a0 K T) \u2192 \u2200 U \u2192 Exp \u0393 (substn T U)\n  eq : \u2200 {\u03b1 T U} \u2192 (set \u03b1 \u220b T \u2263 U) \u2192 (Exp \u0393 T) \u2192 (Exp \u0393 U)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\nM\u27e6 tapp {T = T} M U \u27e7 As as = \n  cast (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (M\u27e6 M \u27e7 As as (T\u27e6 U \u27e7 As))\nM\u27e6 eq T\u2263U M \u27e7 As as = cast (\u2263\u27e6 T\u2263U \u27e7 As) (M\u27e6 M \u27e7 As as)\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\nM\u27e6 tapp {T = T} M U \u27e7\u00b2 \u211cs as\u211cbs = \n  struct-cast (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _)\n    (cast\u00b2 (substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 U \u27e7\u00b2 \u211cs)))\nM\u27e6 eq {\u03b1} {T} {U} T\u2263U M \u27e7\u00b2 {As} {Bs} \u211cs as\u211cbs = \n  struct-cast (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (cast\u00b2 (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs))\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time \n  (app (world {time \u2237 rset \u03b1 \u2237 \u03a3}) tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080))\n\nsignal : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1 \u21d2 rset \u03b1)\nsignal {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (\u03a0 time ((tvar\u2081 \u227c tvar\u2080) \u22b8 \n  ((world {time \u2237 time \u2237 rset \u03b1 \u2237 \u03a3} \u27e8 tvar\u2081 , tvar\u2080 ]) \u22b8 app tvar\u2082 tvar\u2080))))\n\n\u25a7 : \u2200 {\u03a3 \u03b1} \u2192 Typ+ \u03a3 (rset \u03b1) \u2192 Typ+ \u03a3 (rset \u03b1)\n\u25a7 {\u03a3} = app (signal {\u03a3})\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app (taut {[]}) \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = \u25a7 {[]} \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsig : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsig T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsig\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsig\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsig-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (sig T s (sig\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsig-iso T s \u03c3 = refl\n\nsig-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (sig\u207b\u00b9 T s (sig T s \u03c3) \u2261 \u03c3)\nsig-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = sig U s (f (sig\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = sig\u207b\u00b9 U s (f (sig T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = (\u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _))\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 True (s \u2264 u) \u2192 (T at s \u22a8 \u03c3 s _ \u2248[ u ] \u03c4 s _)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (set \u03b1)) (U : STyp (set \u03b2)) \u2192 T\u27ea T \u21a6 U \u27eb \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U f = \u2200 u \u03c3 \u03c4 \u2192 \n  (T \u22a8 \u03c3 \u2248[ u ] \u03c4) \u2192 (U \u22a8 f \u03c3 \u2248[ u ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s u \u2192 True (s \u2264 u) \u2192 \u2200 a b \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u s\u2264u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u s\u2264u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u s\u2264u a c a\u211cc , \u211c-impl-\u2248 U at s u s\u2264u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u s\u2264u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u s\u2264u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u s\u2264u a b a\u2248b))\n  \u211c-impl-\u2248_at (\u25a1 T) s u s\u2264u \u03c3 \u03c4 \u03c3\u211c\u03c4 = \u03bb t s\u2264t t\u2264u \u2192 \n    \u211c-impl-\u2248 T at t u t\u2264u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n      (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s u \u2192 True (s \u2264 u) \u2192 \u2200 a b \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u s\u2264u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u s\u2264u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u s\u2264u a c a\u2248c , \u2248-impl-\u211c U at s u s\u2264u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u s\u2264u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u s\u2264u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u s\u2264u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u s\u2264u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt k\u211ck\u2032 with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt k\u211ck\u2032 | refl\n      = \u2248-impl-\u211c T at t u t\u2264u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) (\u03c3\u2248\u03c4 t s\u2264t t\u2264u) where\n        t\u2264u : True (t \u2264 u)\n        t\u2264u with switch (s < t)\n        t\u2264u | (true , s<t)  = k\u211ck\u2032 {t} refl {s<t} {s<t} tt {\u2264-refl t} {\u2264-refl t} tt \n        t\u2264u | (false , t\u2264s) = \u2264-trans t s u t\u2264s s\u2264u\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s s\u2264u \u2192 \u211c-impl-\u2248 T at s u s\u2264u (\u03c3 s _) (\u03c4 s _) (\u03c3\u211c\u03c4 refl s\u2264u)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl s\u2264u = \u2248-impl-\u211c T at s u s\u2264u (\u03c3 s _) (\u03c4 s _) (\u03c3\u2248\u03c4 s s\u2264u)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (set \u03b1)) (U : STyp (set \u03b2)) (M : Exp [] \u27ea T \u21a6 U \u27eb) \u2192 \n  Causal T U (M\u27e6 M \u27e7 (World , tt) tt)\ncausality T U M u\n  = \u211c-impl-\u2248 (T \u21a6 U) u\n      (M\u27e6 M \u27e7 (World , tt) tt) \n      (M\u27e6 M \u27e7 (World , tt) tt) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ u ] , _) tt)\n","old_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; epoch ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.Delay using ( Delay ; _ms )\nopen import FRP.JS.Bool using ( true ; false ; not )\nopen import FRP.JS.Nat using () renaming ( _<_ to _<n_ )\nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\nsym : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 (a \u2261 b) \u2192 (b \u2261 a)\nsym refl = refl\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\nstruct-sym : \u2200 K {A B C D \u2111 \u211c} \u2192 (A\u2261B : A \u2261 B) \u2192 (C\u2261D : C \u2261 D) \u2192\n  (\u2111 \u2261 struct K A\u2261B \u211c C\u2261D) \u2192 \n    (\u211c \u2261 struct K (sym A\u2261B) \u2111 (sym C\u2261D))\nstruct-sym K refl refl refl = refl\n\nstruct-trans : \u2200 K {A B C D E F}\n  (A\u2261B : A \u2261 B) (B\u2261C : B \u2261 C) (\u211c : K \u220b C \u2194 F) (E\u2261F : E \u2261 F) (D\u2261E : D \u2261 E) \u2192\n    struct K A\u2261B (struct K B\u2261C \u211c E\u2261F) D\u2261E \u2261\n      struct K (trans A\u2261B B\u2261C) \u211c (trans D\u2261E E\u2261F)\nstruct-trans K refl refl \u211c refl refl = refl\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Concatenation of type contexts\n\n_++_ : Kinds \u2192 Kinds \u2192 Kinds\n[]      ++ \u03a5 = \u03a5\n(K \u2237 \u03a3) ++ \u03a5 = K \u2237 (\u03a3 ++ \u03a5)\n\n_\u220b_++_\u220b_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u2200 \u03a5 \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 ++ \u03a5 \u27e7\n[]      \u220b tt       ++ \u03a5 \u220b Bs = Bs\n(K \u2237 \u03a3) \u220b (A , As) ++ \u03a5 \u220b Bs = (A , (\u03a3 \u220b As ++ \u03a5 \u220b Bs))\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03a3 {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 \u2200 \u03a5 {Cs Ds} \u2192 (\u03a5 \u220b Cs \u2194* Ds) \u2192 \n  ((\u03a3 ++ \u03a5) \u220b (\u03a3 \u220b As ++ \u03a5 \u220b Cs) \u2194* (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds))\n[]      \u220b tt       ++\u00b2 \u03a5 \u220b \u2111s = \u2111s\n(K \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s = (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq lt : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 lt \u27e7     = \u03bb t u \u2192 True (t < u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 lt \u27e7\u00b2     = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n_\u227a_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227a_ = capp\u2082 lt\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time (app tvar\u2081 tvar\u2080))\n\ninterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\ninterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227c tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\nconstreq : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nconstreq {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (\u03a0 time \n  ((tvar\u2081 \u227c tvar\u2080) \u22b8 (app (app (app interval tvar\u2083) tvar\u2081) tvar\u2080 \u22b8 app tvar\u2082 tvar\u2080)))))\n\n_\u22b5_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b5_ = app\u2082 constreq\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Susbtitution on type variables under a context\n\n\u03c4substn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 TVar K (\u03a3 ++ (L \u2237 \u03a5)) \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\n\u03c4substn+ []      zero    U = U\n\u03c4substn+ []      (suc \u03c4) U = var \u03c4\n\u03c4substn+ (K \u2237 \u03a3) zero    U = var zero\n\u03c4substn+ (K \u2237 \u03a3) (suc \u03c4) U = weaken (skip K id) (\u03c4substn+ \u03a3 \u03c4 U)\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7 (A , As) Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Bs = trans \n  (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs) \n  (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip K id) (A , (\u03a3 \u220b As ++ _ \u220b Bs)))\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ (J \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s = refl\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 (J \u2237 \u03a3) {\u03a5} {L} {K} (suc \u03c4) U {A , As} {B , Bs} {Cs} {Ds} (\u211c , \u211cs) \u2111s = \n  begin\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) X (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (skip J id) (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n      (struct K\n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs))) \n        (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))) \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 struct-trans K \n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs)))\n       (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)))\n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))\n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 \u03c4substn+ (J \u2237 \u03a3) (suc \u03c4) U \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) )\n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (B , Bs) Ds)    \n  \u220e\n\n-- Substitution on types under a context\n\nsubstn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 Typ (\u03a3 ++ (L \u2237 \u03a5)) K \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\nsubstn+ \u03a3 (const C) U = const C\nsubstn+ \u03a3 (abs K T) U = abs K (substn+ (K \u2237 \u03a3) T U)\nsubstn+ \u03a3 (app S T) U = app (substn+ \u03a3 S U) (substn+ \u03a3 T U)\nsubstn+ \u03a3 (var \u03c4)   U = \u03c4substn+ \u03a3 \u03c4 U\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    T\u27e6 T \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 substn+ \u03a3 T U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7 As Bs = refl\nsubstn+ \u03a3 \u27e6 abs K T \u27e7\u27e6 U \u27e7 As Bs = ext (\u03bb A \u2192 substn+ K \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Bs)\nsubstn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Bs = apply (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Bs) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Bs)\nsubstn+ \u03a3 \u27e6 var \u03c4   \u27e7\u27e6 U \u27e7 As Bs = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = refl\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (abs J {K} T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs J T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u211c\n  \u2261\u27e8 substn+ (J \u2237 \u03a3) \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s \u27e9\n    struct K \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds)\n  \u2261\u27e8 struct-ext J K \n       (\u03bb A \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n       (\u03bb \u211c \u2192 T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b \u211c , \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (\u03bb B \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds) \u211c \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 (abs J T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 Bs Ds) \u211c\n  \u220e)))\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (app {J} {K} S T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  begin\n    T\u27e6 app S T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 cong (T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct J \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 struct-apply J K \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n       (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 substn+ \u03a3 (app S T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u220e\nsubstn+ \u03a3 \u27e6 var \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s\n\n-- Substitution on types\n\nsubstn : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 \u03a3) K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 K\nsubstn = substn+ []\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) (As : \u03a3\u27e6 \u03a3 \u27e7)\u2192\n  T\u27e6 T \u27e7 (T\u27e6 U \u27e7 As , As) \u2261 T\u27e6 substn T U \u27e7 As\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7 tt\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192\n  T\u27e6 T \u27e7\u00b2 (T\u27e6 U \u27e7\u00b2 \u211cs , \u211cs) \u2261 \n    struct K (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 Bs)\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 tt\n\n-- Eta-beta equivalence on types\n\ndata _\u220b_\u2263_ {\u03a3} : \u2200 K \u2192 Typ \u03a3 K \u2192 Typ \u03a3 K \u2192 Set where\n  abs : \u2200 K {L T U} \u2192 (L \u220b T \u2263 U) \u2192 ((K \u21d2 L) \u220b abs K T \u2263 abs K U)\n  app : \u2200 {K L F G T U} \u2192 ((K \u21d2 L) \u220b F \u2263 G) \u2192 (K \u220b T \u2263 U) \u2192 (L \u220b app F T \u2263 app G U)\n  beta : \u2200 {K L} T U \u2192 (L \u220b app (abs K T) U \u2263 substn T U)\n  eta : \u2200 {K L} T \u2192 ((K \u21d2 L) \u220b T \u2263 abs K (app (weaken (skip K id) T) tvar\u2080))\n  \u2263-refl : \u2200 {K T} \u2192 (K \u220b T \u2263 T)\n  \u2263-sym : \u2200 {K T U} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 T)\n  \u2263-trans : \u2200 {K T U V} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 V) \u2192 (K \u220b T \u2263 V)\n\n\u2263\u27e6_\u27e7 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} \u2192 (K \u220b T \u2263 U) \u2192 \u2200 As \u2192 T\u27e6 T \u27e7 As \u2261 T\u27e6 U \u27e7 As\n\u2263\u27e6 abs K T\u2263U \u27e7       As = ext (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As))\n\u2263\u27e6 app F\u2263G T\u2263U \u27e7     As = apply (\u2263\u27e6 F\u2263G \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 beta T U \u27e7        As = substn\u27e6 T \u27e7\u27e6 U \u27e7 As\n\u2263\u27e6 eta {K} T \u27e7       As = ext (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n\u2263\u27e6 \u2263-refl \u27e7          As = refl\n\u2263\u27e6 \u2263-sym T\u2263U \u27e7       As = sym (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As = trans (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As)\n\n\u2263\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} (T\u2263U : K \u220b T \u2263 U) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 \u211cs \u2261 struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n\u2263\u27e6 abs K {L} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 (\u211c , \u211cs) \u27e9\n    struct L (\u2263\u27e6 T\u2263U \u27e7 (A , As)) (T\u27e6 U \u27e7\u00b2 (\u211c , \u211cs)) (\u2263\u27e6 T\u2263U \u27e7 (B , Bs))\n  \u2261\u27e8 struct-ext K L (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As)) (\u03bb \u211c' \u2192 T\u27e6 U \u27e7\u00b2 (\u211c' , \u211cs)) (\u03bb B \u2192 \u2263\u27e6 T\u2263U \u27e7 (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 abs K T\u2263U \u27e7 As) (T\u27e6 abs K U \u27e7\u00b2 \u211cs) (\u2263\u27e6 abs K T\u2263U \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 app {K} {L} {F} {G} {T} {U} F\u2263G T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app F T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 cong (T\u27e6 F \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) \u27e9\n    T\u27e6 F \u27e7\u00b2 \u211cs (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)))\n       (\u2263\u27e6 F\u2263G \u27e7\u00b2 \u211cs) \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n      (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 struct-apply K L\n       (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n       (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct L (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 As) (T\u27e6 app G U \u27e7\u00b2 \u211cs) (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 Bs)\n  \u220e\n\u2263\u27e6 beta T U \u27e7\u00b2 \u211cs = substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs\n\u2263\u27e6 eta {K} {L} T \u27e7\u00b2 {As} {Bs} \u211cs = iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 \n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs \u211c\n  \u2261\u27e8 cong (\u03bb X \u2192 X \u211c) (weaken\u27e6 T \u27e7\u00b2 (skip K id) (\u211c , \u211cs)) \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 T \u27e7 (skip K id) (A , As))\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs))\n      (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) \u211c\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 (skip K id) (A , As)) \n       (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl \u211c refl \u27e9\n    struct L \n      (apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c)\n      (apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl)\n  \u2261\u27e8 struct-ext K L\n       (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c) \n       (\u03bb B \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl) \u211c \u27e9\n    struct (K \u21d2 L) \n      (\u2263\u27e6 eta T \u27e7 As)\n      (T\u27e6 abs K (app (weaken (skip K id) T) (var zero)) \u27e7\u00b2 \u211cs)\n      (\u2263\u27e6 eta T \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 \u2263-refl \u27e7\u00b2 \u211cs = refl\n\u2263\u27e6 \u2263-sym {K} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  struct-sym K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs)\n\u2263\u27e6 \u2263-trans {K} {T} {U} {V} T\u2263U U\u2263V \u27e7\u00b2 {As} {Bs} \u211cs =\n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u2263\u27e6 T\u2263U \u27e7 As) X (\u2263\u27e6 T\u2263U \u27e7 Bs)) (\u2263\u27e6 U\u2263V \u27e7\u00b2 \u211cs) \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (struct K (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs)) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 struct-trans K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct K (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 Bs)\n  \u220e\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  \u227c-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  \u227c-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n  \u227c-absurd : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time ((tvar\u2081 \u227a tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 tvar\u2082)))))\n  \u227c-case : \u2200 {\u03b1} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 time (\u03a0 time (((tvar\u2081 \u227c tvar\u2080) \u22b8 tvar\u2082) \u22b8 (((tvar\u2080 \u227a tvar\u2081) \u22b8 tvar\u2082) \u22b8 tvar\u2082)))))\n\nabsurd : \u2200 {\u03b1} {A : Set \u03b1} b \u2192 True b \u2192 True (not b) \u2192 A\nabsurd true  tt ()\nabsurd false () tt\n\ncase : \u2200 {\u03b1} {A : Set \u03b1} b \u2192 (True (not b) \u2192 A) \u2192 (True b \u2192 A) \u2192 A\ncase true  f g = g tt\ncase false f g = f tt\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7     As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7      As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7      As = \u03bb A B \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7   As = \u2264-refl\nc\u27e6 \u227c-trans \u27e7  As = \u2264-trans\nc\u27e6 \u227c-absurd \u27e7 As = \u03bb A t u \u2192 absurd (t < u)\nc\u27e6 \u227c-case \u27e7   As = \u03bb A t u \u2192 case (u < t)\n\ncase\u00b2 : \u2200 {\u03b1} {A A\u2032 : Set \u03b1} (\u211c : A \u2192 A\u2032 \u2192 Set \u03b1) b \u2192\n  \u2200 {f f\u2032} \u2192 (\u2200 b\u00d7 \u2192 \u211c (f b\u00d7) (f\u2032 b\u00d7)) \u2192\n    \u2200 {g g\u2032} \u2192 (\u2200 b\u2713 \u2192 \u211c (g b\u2713) (g\u2032 b\u2713)) \u2192\n      \u211c (case b f g) (case b f\u2032 g\u2032)\ncase\u00b2 \u211c true  f\u211cf\u2032 g\u211cg\u2032 = g\u211cg\u2032 tt\ncase\u00b2 \u211c false f\u211cf\u2032 g\u211cg\u2032 = f\u211cf\u2032 tt\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2        \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2        \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 \u227c-refl \u27e7\u00b2     \u211cs = _\nc\u27e6 \u227c-trans \u27e7\u00b2    \u211cs = _\nc\u27e6 \u227c-absurd \u27e7\u00b2   \u211cs = \u03bb \u211c {t} t\u2261t {u} u\u2261u {t<u} tt {u\u2264t} tt \u2192 absurd (t < u) t<u u\u2264t\nc\u27e6 \u227c-case {\u03b1} \u27e7\u00b2 \u211cs = lemma where\n  lemma : \u2200 {A A\u2032 : Set \u03b1} (\u211c : A \u2192 A\u2032 \u2192 Set \u03b1) \u2192\n    \u2200 {t t\u2032} \u2192 (t \u2261 t\u2032) \u2192 \u2200 {u u\u2032} \u2192 (u \u2261 u\u2032) \u2192\n      \u2200 {f f\u2032} \u2192 (\u2200 {t\u2264u t\u2032\u2264u\u2032} \u2192 \u22a4 \u2192 \u211c (f t\u2264u) (f\u2032 t\u2032\u2264u\u2032)) \u2192\n        \u2200 {g g\u2032} \u2192 (\u2200 {u<t u\u2032<t\u2032} \u2192 \u22a4 \u2192 \u211c (g u<t) (g\u2032 u\u2032<t\u2032)) \u2192\n          \u211c (case (u < t) f g) (case (u\u2032 < t\u2032) f\u2032 g\u2032)\n  lemma \u211c {t} refl {u} refl f\u211cf\u2032 g\u211cg\u2032 = \n    case\u00b2 \u211c (u < t) (\u03bb t\u2264u \u2192 f\u211cf\u2032 {t\u2264u} {t\u2264u} tt) (\u03bb u<t \u2192 g\u211cg\u2032 {u<t} {u<t} tt)\n  \n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n  tapp : \u2200 {K \u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} \u2192 Exp \u0393 (\u03a0 K T) \u2192 \u2200 U \u2192 Exp \u0393 (substn T U)\n  eq : \u2200 {\u03b1 T U} \u2192 (set \u03b1 \u220b T \u2263 U) \u2192 (Exp \u0393 T) \u2192 (Exp \u0393 U)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\nM\u27e6 tapp {T = T} M U \u27e7 As as = \n  cast (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (M\u27e6 M \u27e7 As as (T\u27e6 U \u27e7 As))\nM\u27e6 eq T\u2263U M \u27e7 As as = cast (\u2263\u27e6 T\u2263U \u27e7 As) (M\u27e6 M \u27e7 As as)\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\nM\u27e6 tapp {T = T} M U \u27e7\u00b2 \u211cs as\u211cbs = \n  struct-cast (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _)\n    (cast\u00b2 (substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 U \u27e7\u00b2 \u211cs)))\nM\u27e6 eq {\u03b1} {T} {U} T\u2263U M \u27e7\u00b2 {As} {Bs} \u211cs as\u211cbs = \n  struct-cast (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (cast\u00b2 (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs))\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app taut \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar\u2080 \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsignal : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsignal T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsignal\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsignal\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsignal-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (signal T s (signal\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsignal-iso T s \u03c3 = refl\n\nsignal-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (signal\u207b\u00b9 T s (signal T s \u03c3) \u2261 \u03c3)\nsignal-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = signal U s (f (signal\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = signal\u207b\u00b9 U s (f (signal T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _)\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 (T at s \u22a8 \u03c3 s \u2248[ u ] \u03c4 s)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192 T\u27ea T \u21d2 U \u27eb s \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U s f = \u2200 t \u03c3 \u03c4 \u2192 \n  (T at s \u22a8 \u03c3 \u2248[ t ] \u03c4) \u2192 (U at s \u22a8 f \u03c3 \u2248[ t ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u a c a\u211cc , \u211c-impl-\u2248 U at s u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u a b a\u2248b))\n  \u211c-impl-\u2248 (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb t s\u2264t t\u2264u \u2192 \u211c-impl-\u2248 T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u a c a\u2248c , \u2248-impl-\u211c U at s u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt t\u2264u with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt t\u2264u | refl = \n      \u2248-impl-\u211c T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u2248\u03c4 t s\u2264t (t\u2264u refl {s\u2264t} {s\u2264t} tt {\u2264-refl t} {\u2264-refl t} tt))\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s \u2192 \u211c-impl-\u2248 T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u211c\u03c4 refl)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl = \u2248-impl-\u211c T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u2248\u03c4 s)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) (M : Exp [] \u27ea [ T \u21d2 U ] \u27eb) s \u2192 \n  Causal T U s (M\u27e6 M \u27e7 (World , tt) tt s)\ncausality T U M s t \n  = \u211c-impl-\u2248 T \u21d2 U at s t \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ t ] , _) tt refl)\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c20d7aa371f03ee7d8b499a517bf99b217c86e2c","subject":"q","message":"q\n\nIgnore-this: b6763cb318b52da897750917a1299480\n\ndarcs-hash:20110205042757-3bd4e-8a21b864a44e2178b929463cf2d0aefc496195cc.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download agda2atp tool (described in above paper) using darcs\n-- with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains interactive proofs\n-- that are used by the combined proofs.\n\n------------------------------------------------------------------------------\n-- Distributive laws on a binary operation\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- Group theory\n\n-- We formalize the theory of groups using Agda postulates for\n-- the group axioms.\n\n-- Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Logic\n\n-- Propositional logic\nopen import Logic.Propositional.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n-- Predicate logic\nopen import Logic.Predicate.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n------------------------------------------------------------------------------\n-- LTC\n\n-- Formalization of (a version of) Azcel's Logical Theory of constructions.\n\n-- LTC base\nopen import LTC.Base.Properties\nopen import LTC.Base.PropertiesATP\nopen import LTC.Base.PropertiesI\n\n-- Booleans\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Bool.PropertiesI\n\n-- Lists\nopen import LTC.Data.List.PropertiesATP\nopen import LTC.Data.List.PropertiesI\n\n-- Naturals numbers: Common properties\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.PropertiesI\n\nopen import LTC.Data.Nat.PropertiesByInductionATP\nopen import LTC.Data.Nat.PropertiesByInductionI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC.Data.Nat.Divisibility.PropertiesATP\nopen import LTC.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC.Data.Nat.Induction.LexicographicATP\nopen import LTC.Data.Nat.Induction.LexicographicI\nopen import LTC.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: List\nopen import LTC.Data.Nat.List.PropertiesATP\nopen import LTC.Data.Nat.List.PropertiesI\n\n-- The GCD algorithm\nopen import LTC.Program.GCD.ProofSpecificationATP\nopen import LTC.Program.GCD.ProofSpecificationI\n\n-- Burstall's sort list algorithm\nopen import LTC.Program.SortList.ProofSpecificationATP\nopen import LTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- LTC-PCF\n\n-- Formalization of a version of Azcel's Logical Theory of constructions.\n\n-- Naturals numbers: Common properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC-PCF.Data.Nat.Induction.LexicographicATP\nopen import LTC-PCF.Data.Nat.Induction.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedATP\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- The division algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- The GCD algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- Axiomatic PA\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- Inductive PA\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, so see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Automatic and Interactive\n-- Proof in First Order Theories of Combinators\" by Ana Bove,\n-- Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez.\n\n------------------------------------------------------------------------------\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n------------------------------------------------------------------------------\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites\n------------------------------------------------------------------------------\n\n-- You can download agda2atp tool (described in above paper) using darcs\n-- with the following command:\n\n-- $ darcs get http:\/\/patch-tag.com\/r\/asr\/agda2atp\n\n-- The agda2atp tool and the FOT code require a modified version of\n-- Agda. See agda2atp\/README.txt for the related instructions.\n\n------------------------------------------------------------------------------\n-- Use\n------------------------------------------------------------------------------\n\n-- Let's suppose you want to use the Metis ATP to prove the group\n-- theory properties stated in the module\n-- GroupTheory.PropertiesATP. First, you should type-check the module using\n-- Agda\n\n-- $ agda -isrc src\/GroupTheory\/PropertiesATP.agda\n\n-- Second, you call the agda2tool using the Metis ATP\n\n-- $ agda2atp -isrc --atp=metis src\/GroupTheory\/PropertiesATP.agda\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains interactive proofs\n-- that are used by the combined proofs.\n\n------------------------------------------------------------------------------\n-- Distributive laws on a binary operation\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- Group theory\n\n-- We formalize the theory of groups using Agda postulates for\n-- the group axioms.\n\n-- Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Logic\n\n-- Propositional logic\nopen import Logic.Propositional.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n-- Predicate logic\nopen import Logic.Predicate.PropertiesATP\nopen import Logic.Predicate.PropertiesI\n\n------------------------------------------------------------------------------\n-- LTC\n\n-- Formalization of (a version of) Azcel's Logical Theory of constructions.\n\n-- LTC base\nopen import LTC.Base.Properties\nopen import LTC.Base.PropertiesATP\nopen import LTC.Base.PropertiesI\n\n-- Booleans\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Bool.PropertiesI\n\n-- Lists\nopen import LTC.Data.List.PropertiesATP\nopen import LTC.Data.List.PropertiesI\n\n-- Naturals numbers: Common properties\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.PropertiesI\n\nopen import LTC.Data.Nat.PropertiesByInductionATP\nopen import LTC.Data.Nat.PropertiesByInductionI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC.Data.Nat.Divisibility.PropertiesATP\nopen import LTC.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC.Data.Nat.Induction.LexicographicATP\nopen import LTC.Data.Nat.Induction.LexicographicI\nopen import LTC.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: List\nopen import LTC.Data.Nat.List.PropertiesATP\nopen import LTC.Data.Nat.List.PropertiesI\n\n-- The GCD algorithm\nopen import LTC.Program.GCD.ProofSpecificationATP\nopen import LTC.Program.GCD.ProofSpecificationI\n\n-- Burstall's sort list algorithm\nopen import LTC.Program.SortList.ProofSpecificationATP\nopen import LTC.Program.SortList.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- LTC-PCF\n\n-- Formalization of a version of Azcel's Logical Theory of constructions.\n\n-- Naturals numbers: Common properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- Naturals numbers: Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.PropertiesI\n\n-- Naturals numbers: Induction\nopen import LTC-PCF.Data.Nat.Induction.LexicographicATP\nopen import LTC-PCF.Data.Nat.Induction.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedATP\nopen import LTC-PCF.Data.Nat.Induction.WellFoundedI\n\n-- Naturals numbers: Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- Naturals numbers: The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- The division algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- The GCD algorithm\nopen import LTC-PCF.Program.GCD.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- Axiomatic PA\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- Inductive PA\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import LTC-PCF.Program.GCD.EquationsATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, so see src\/Agsy.README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cb4557c626443a10a28c55fd7c0881306a8231d1","subject":"Algebra: ^\u207a-\u2238","message":"Algebra: ^\u207a-\u2238\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP using (flip; \u03a0; \u03a0\u2071; Op\u2081; Op\u2082; nest)\nopen import Data.Nat.Base using (\u2115; _\u2264_; _\u2238_; z\u2264n; s\u2264s)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.NP\nopen import Algebra.Raw\nopen \u2261-Reasoning\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\nmodule Implicits where\n  open Algebra.FunctionProperties.NP \u03a0\u2071 {a}{a}{A} _\u2261_ public\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n\n  module From-Magma (magma : Magma A) where\n\n    open Magma magma\n\n    module From-Comm\n             (comm  : Commutative _\u2219_)\n             {x y x' y' : A}\n             (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n           where\n      comm= : x \u2219 y \u2261 x' \u2219 y'\n      comm= = comm \u2666 e \u2666 comm\n\n    module From-Assoc\n             (assoc : Associative _\u2219_)\n           where\n\n      assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n      assocs = assoc , ! assoc\n\n      module _ {c x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n        assoc= = ! assoc \u2666 \u2219= e idp \u2666 assoc\n\n        !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n        !assoc= = assoc \u2666 \u2219= idp e \u2666 ! assoc\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n        inner= = assoc= (!assoc= e)\n\n    module From-Assoc-Comm\n             (assoc : Associative _\u2219_)\n             (comm  : Commutative _\u2219_)\n           where\n      open From-Assoc assoc public\n      open From-Comm  comm  public\n\n      module _ {x y z} where\n        assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n        assoc-comm = assoc= comm\n\n        !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n        !assoc-comm = !assoc= comm\n\n      interchange : Interchange _\u2219_ _\u2219_\n      interchange = assoc= (!assoc= comm)\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n        outer= = \u2219= comm comm \u2666 assoc= (!assoc= e) \u2666 \u2219= comm comm\n\n  module From-Op\u2082 (op : Op\u2082 A) = From-Magma \u27e8 op \u27e9\n\n  module From-Monoid-Ops (mon-ops : Monoid-Ops A) where\n    open Monoid-Ops mon-ops\n    open From-Op\u2082 _\u2219_ public\n\n    module From-LeftIdentity (idl : LeftIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n        is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 idl\n\n    module From-RightIdentity (idr : RightIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n        is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 idr\n\n      module _ {b} where\n        ^\u207a-^\u00b9\u207a : \u2200 n \u2192 b ^\u207a (1+ n) \u2261 b ^\u00b9\u207a n\n        ^\u207a-^\u00b9\u207a 0 = idr\n        ^\u207a-^\u00b9\u207a (1+ n) = \u2219= idp (^\u207a-^\u00b9\u207a n)\n\n        ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n        ^\u207a0-\u03b5 = idp\n\n        ^\u207a1-id : b ^\u207a 1 \u2261 b\n        ^\u207a1-id = idr\n\n        ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n        ^\u207a2-\u2219 = ^\u207a-^\u00b9\u207a 1\n\n        ^\u207a3-\u2219 : b ^\u207a 3 \u2261 b \u2219 (b \u2219 b)\n        ^\u207a3-\u2219 = ^\u207a-^\u00b9\u207a 2\n\n        ^\u207a4-\u2219 : b ^\u207a 4 \u2261 b \u2219 (b \u2219 (b \u2219 b))\n        ^\u207a4-\u2219 = ^\u207a-^\u00b9\u207a 3\n\n    module From-Identities (identities : Identity \u03b5 _\u2219_) where\n      comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n      comm-\u03b5 = snd identities \u2666 ! fst identities\n\n      open From-LeftIdentity  (fst identities) public\n      open From-RightIdentity (snd identities) public\n\n    module From-Assoc-RightIdentity\n             (assoc : Associative _\u2219_)\n             (idr   : RightIdentity \u03b5 _\u2219_)\n             where\n      open From-RightIdentity idr public\n\n      module _ {c x y}(e : x \u2219 y \u2261 \u03b5) where\n        elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n        elim-assoc= = assoc \u2666 \u2219= idp e \u2666 idr\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-inner= = ! assoc \u2666 \u2219= (elim-assoc= e) idp\n\n    module From-Assoc-LeftIdentity\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             where\n      open From-LeftIdentity idl public\n\n      -- Together with ^\u207a0-\u03b5, ^\u207a-+ shows that (b ^\u207a_) is a\n      -- monoid homomorphism from (\u2115,+,0) to (M,\u2219,\u03b5)\n      -- therefore also a group homomorphism\n      ^\u207a-+ : \u2200 m {n b} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n      ^\u207a-+ 0      = ! idl\n      ^\u207a-+ (1+ m) = ap (_\u2219_ _) (^\u207a-+ m) \u2666 ! assoc\n\n      -- Derived from ^\u207a-+ in Algebra.Group\n      ^\u207a-* : \u2200 m {n b} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n      ^\u207a-* 0 = idp\n      ^\u207a-* (1+ m) {n} {b}\n        = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n          b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n          b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n      module _ {c x y} (e : x \u2219 y \u2261 \u03b5) where\n        elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n        elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 idl\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-!inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-!inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module From-Group-Ops (grp-ops : Group-Ops A) where\n    open Group-Ops grp-ops\n    open From-Monoid-Ops mon-ops public\n\n    module From-Assoc-LeftIdentity-LeftInverse\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             (inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-LeftIdentity assoc idl\n\n      \u207b\u00b9-inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      \u207b\u00b9-inverse\u02b3 {x}\n         = x \u2219 x \u207b\u00b9                      \u2261\u27e8 ! idl \u27e9\n           \u03b5 \u2219 (x \u2219 x \u207b\u00b9)                \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9) \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9)              \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5                             \u220e\n\n      cancels-\u2219-left : LeftCancel _\u2219_\n      cancels-\u2219-left {c} {x} {y} e\n        = x              \u2261\u27e8 ! idl \u27e9\n          \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inverse\u02e1 idp \u27e9\n          \u03b5 \u2219 y          \u2261\u27e8 idl \u27e9\n          y              \u220e\n\n      module _ {x y} where\n        unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n        unique-\u03b5-right eq\n          = y               \u2261\u27e8 ! is-\u03b5-left inverse\u02e1 \u27e9\n            x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n            x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n            x \u207b\u00b9 \u2219 x        \u2261\u27e8 inverse\u02e1 \u27e9\n            \u03b5               \u220e\n\n      module _ {x y} where\n        -- _\u207b\u00b9 is a group homomorphism, see Algebra.Group._\u1d52\u1d56\n        \u207b\u00b9-hom\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n        \u207b\u00b9-hom\u2032 = cancels-\u2219-left {x \u2219 y}\n           ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 \u207b\u00b9-inverse\u02b3 \u27e9\n            \u03b5                       \u2261\u27e8 ! \u207b\u00b9-inverse\u02b3 \u27e9\n            x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! idl) \u27e9\n            x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! \u207b\u00b9-inverse\u02b3) idp) \u27e9\n            x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n            x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n            (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n      \u207b\u00b9-hom\u2032= : \u2200 {x y z t} \u2192 x \u2219 y \u2261 z \u2219 t \u2192 y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261 t \u207b\u00b9 \u2219 z \u207b\u00b9\n      \u207b\u00b9-hom\u2032= e = ! \u207b\u00b9-hom\u2032 \u2666 ap _\u207b\u00b9 e \u2666 \u207b\u00b9-hom\u2032\n\n      elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n      elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n        = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n          x \u207b\u00b9 \u2219 y               \u220e\n\n      elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n      elim-\u2219-right-\/ {c} {x} {y}\n        = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n          x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-!inner= \u207b\u00b9-inverse\u02b3 \u27e9\n          x \/ y \u220e\n\n    module From-Assoc-RightIdentity-RightInverse\n             (assoc : Associative _\u2219_)\n             (idr : RightIdentity \u03b5 _\u2219_)\n             (inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-RightIdentity assoc idr\n\n      cancels-\u2219-right : RightCancel _\u2219_\n      cancels-\u2219-right {c} {x} {y} e\n        = x              \u2261\u27e8 ! idr \u27e9\n          x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n          y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inverse\u02b3 \u27e9\n          y \u2219 \u03b5          \u2261\u27e8 idr \u27e9\n          y              \u220e\n\n      inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02e1 {x} = x \u207b\u00b9 \u2219 x                      \u2261\u27e8 ! idr \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 \u03b5                \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-inner= inverse\u02b3 \u27e9\n                  (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9)              \u2261\u27e8 inverse\u02b3 \u27e9\n                  \u03b5 \u220e\n\n      \u207b\u00b9-involutive : Involutive _\u207b\u00b9\n      \u207b\u00b9-involutive {x}\n        = cancels-\u2219-right\n          (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5              \u2261\u27e8 ! inverse\u02b3 \u27e9\n           x \u2219 x \u207b\u00b9 \u220e)\n\n      module _ {x y} where\n        \/-\u2219 : x \u2261 (x \/ y) \u2219 y\n        \/-\u2219 = x               \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02e1) \u27e9\n              x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n              (x \/ y) \u2219 y     \u220e\n\n        \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n        \u2219-\/ = x            \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02b3) \u27e9\n              x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n              (x \u2219 y) \/ y  \u220e\n\n      module _ {x y} where\n        unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n        unique-\u03b5-left eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            y \/ y        \u2261\u27e8 inverse\u02b3 \u27e9\n            \u03b5            \u220e\n\n        x\/y\u2261\u03b5\u2192x\u2261y : x \/ y \u2261 \u03b5 \u2192 x \u2261 y\n        x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5 = cancels-\u2219-right (x\/y\u2261\u03b5 \u2666 ! inverse\u02b3)\n\n        x\/y\u2262\u03b5 : x \u2262 y \u2192 x \/ y \u2262 \u03b5\n        x\/y\u2262\u03b5 x\u2262y x\/y\u2261\u03b5 = x\u2262y (x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5)\n\n      \u03b5\u207b\u00b9\u2261\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n      \u03b5\u207b\u00b9\u2261\u03b5 = unique-\u03b5-left inverse\u02e1\n\n    module From-Assoc-Identities-RightInverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      private\n        idl = fst identities\n        idr = snd identities\n\n      inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02b3 = inv-r\n\n      open From-Assoc assoc\n      open From-Identities identities using (comm-\u03b5)\n      open From-Assoc-RightIdentity assoc idr\n      open From-Assoc-LeftIdentity  assoc idl\n      open From-Assoc-RightIdentity-RightInverse assoc idr inverse\u02b3 public\n      open From-Assoc-LeftIdentity-LeftInverse assoc idl inverse\u02e1 public\n        hiding (\u207b\u00b9-inverse\u02b3)\n\n      module _ {x y} where\n        unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n        unique-\u207b\u00b9 eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            \u03b5 \/ y        \u2261\u27e8 idl \u27e9\n            y \u207b\u00b9         \u220e\n\n      \u207b\u00b9-inj : \u2200 {x y} \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9 \u2192 x \u2261 y\n      \u207b\u00b9-inj {x} {y} e\n         = ! (y              \u2261\u27e8 ! idr \u27e9\n              y \u2219 \u03b5          \u2261\u27e8 \u2219= idp (\u03b5        \u2261\u27e8 ! inverse\u02e1  \u27e9\n                                        x \u207b\u00b9 \u2219 x \u2261\u27e8 \u2219= e idp \u27e9\n                                        y \u207b\u00b9 \u2219 x \u220e) \u27e9\n              y \u2219 (y \u207b\u00b9 \u2219 x) \u2261\u27e8 elim-!assoc= inverse\u02b3 \u27e9\n              x \u220e)\n\n      module _ {b} where\n        ^\u207a-1+ : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n        ^\u207a-1+ 0 = comm-\u03b5\n        ^\u207a-1+ (1+ n) = ap (_\u2219_ b) (^\u207a-1+ n) \u2666 ! assoc\n\n        ^\u207a-\u2238 : \u2200 {m n} \u2192 n \u2264 m \u2192 b ^\u207a(m \u2238 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207a-\u2238 z\u2264n = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207a-\u2238 (s\u2264s {n} {m} n\u2264m) =\n           b ^\u207a(m \u2238 n)                     \u2261\u27e8 ^\u207a-\u2238 n\u2264m \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296 : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^-\u2296 m      0      = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296 0      (1+ n) = ! idl\n        ^-\u2296 (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296 m n \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296' : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^-\u2296' m      0      = ! is-\u03b5-left \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296' 0      (1+ n) = ! idr\n        ^-\u2296' (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296' m n \u27e9\n           (b ^\u207b n \u2219 b ^\u207a m)               \u2261\u27e8 ! elim-inner= inverse\u02e1 \u27e9\n           (b ^\u207b n \u2219 b \u207b\u00b9) \u2219 (b \u2219 b ^\u207a m)  \u2261\u27e8 \u2219= (! \u207b\u00b9-hom\u2032) idp \u27e9\n           (b \u2219 b ^\u207a n)\u207b\u00b9 \u2219 (b \u2219 b ^\u207a m)   \u220e\n\n        ^\u207a-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b ^\u207a m\n        ^\u207a-comm 0      n = ! comm-\u03b5\n        ^\u207a-comm (1+ m) n =\n          b \u2219 b\u1d50 \u2219 b\u207f   \u2261\u27e8 !assoc= (^\u207a-comm m n) \u27e9\n          b \u2219 b\u207f \u2219 b\u1d50   \u2261\u27e8 \u2219= (^\u207a-1+ n) idp \u27e9\n          b\u207f \u2219 b \u2219 b\u1d50   \u2261\u27e8 assoc \u27e9\n          b\u207f \u2219 (b \u2219 b\u1d50) \u220e\n          where\n            b\u207f = b ^\u207a n\n            b\u1d50 = b ^\u207a m\n\n        ^\u207a-^\u207b-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^\u207a-^\u207b-comm m n = ! ^-\u2296 m n \u2666 ^-\u2296' m n\n\n        ^\u207b-^\u207a-comm : \u2200 m n \u2192 b ^\u207b n \u2219 b ^\u207a m \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207b-^\u207a-comm m n = ! ^-\u2296' m n \u2666 ^-\u2296 m n\n\n        ^\u207b-comm : \u2200 m n \u2192 b ^\u207b m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207b m\n        ^\u207b-comm m n = \u207b\u00b9-hom\u2032= (^\u207a-comm n m)\n\n        ^\u207b-1+ : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n        ^\u207b-1+ n = ap _\u207b\u00b9 (^\u207a-1+ n) \u2666 \u207b\u00b9-hom\u2032\n\n        ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n        ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                        \u2666 ! \u207b\u00b9-hom\u2032\n                        \u2666 ap _\u207b\u00b9 (! ^\u207a-1+ n)\n\n        ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n        ^\u207b\u20321-id = idr\n\n        ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n        ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n        ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n        ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\n        -- ^-+ is a group homomorphism defined in Algebra.Group\n        -- Some properties can be derived from it.\n        ^-+ : \u2200 i j \u2192 b ^(i +\u2124 j) \u2261 b ^ i \u2219 b ^ j\n        ^-+ -[1+ m ] -[1+ n ] = ap _\u207b\u00b9\n                                 (ap (\u03bb z \u2192 b ^\u207a(1+ z)) (\u2115\u00b0.+-comm (1+ m) n)\n                              \u2666 ^\u207a-+ (1+ n) {1+ m}) \u2666 \u207b\u00b9-hom\u2032\n        ^-+ -[1+ m ]   (+ n)  = ^-\u2296' n (1+ m)\n        ^-+   (+ m)  -[1+ n ] = ^-\u2296 m (1+ n)\n        ^-+   (+ m)    (+ n)  = ^\u207a-+ m\n\n        -- GroupHomomorphism.f-pres-inv\n        ^-- : \u2200 i \u2192 b ^(- i) \u2261 (b ^ i)\u207b\u00b9\n        ^-- -[1+ n ] = ! \u207b\u00b9-involutive\n        ^-- (+ 0)    = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^-- (+ 1+ n) = idp\n\n        -- GroupHomomorphism.f-\u2212-\/\n        ^-\u2212 : \u2200 i j \u2192 b ^(i \u2212\u2124 j) \u2261 b ^ i \/ b ^ j\n        ^-\u2212 i j = ^-+ i (- j) \u2666 \u2219= idp (^-- j)\n\n        ^-suc : \u2200 i \u2192 b ^(suc\u2124 i) \u2261 b \u2219 b ^ i\n        ^-suc i = ^-+ (+ 1) i \u2666 \u2219= idr idp\n\n        ^-pred : \u2200 i \u2192 b ^(pred\u2124 i) \u2261 b \u207b\u00b9 \u2219 b ^ i\n        ^-pred i = ^-+ (- + 1) i \u2666 ap (\u03bb z \u2192 z \u207b\u00b9 \u2219 b ^ i) idr\n\n        ^-comm : \u2200 i j \u2192 b ^ i \u2219 b ^ j \u2261 b ^ j \u2219 b ^ i\n        ^-comm -[1+ m ] -[1+ n ] = ^\u207b-comm (1+ m) (1+ n)\n        ^-comm -[1+ m ]   (+ n)  = ! ^\u207a-^\u207b-comm n (1+ m)\n        ^-comm   (+ m)  -[1+ n ] = ^\u207a-^\u207b-comm m (1+ n)\n        ^-comm   (+ m)    (+ n)  = ^\u207a-comm m n\n\n      {-\n      ^-* : \u2200 i j \u2192 b ^(i *\u2124 j) \u2261 (b ^ j)^ i\n      ^-* i j = {!!}\n      -}\n\n    module From-Assoc-Identities-Inverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv : Inverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc-Identities-RightInverse assoc identities (snd inv) public\n\n  module From-Ring-Ops (rng-ops : Ring-Ops A) where\n    open Ring-Ops rng-ops\n\n    module DistributesOver\u02b3\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_)\n             where\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = *-comm \u2666 *-+-distr\u02e1 \u2666 += *-comm *-comm\n\n    module DistributesOver\u02e1\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = *-comm \u2666 *-+-distr\u02b3 \u2666 += *-comm *-comm\n\n    module From-+Group-*Identity-DistributesOver\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*1-identity : RightIdentity 1# _*_)\n             (*-+-distrs : _*_ DistributesOver _+_)\n             where\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( \u207b\u00b9-involutive to 0\u2212-involutive\n                     ; cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n                     ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n                     )\n\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = fst *-+-distrs\n\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = snd *-+-distrs\n\n      -0\u22610 : -0# \u2261 0#\n      -0\u22610 = 0\u22120\u22610\n\n      0*-zero : LeftZero 0# _*_\n      0*-zero {x} = cancels-+-left\n        (x + 0# * x      \u2261\u27e8 += (! 1*-identity) idp \u27e9\n         1# * x + 0# * x \u2261\u27e8 ! *-+-distr\u02b3 \u27e9\n         (1# + 0#) * x   \u2261\u27e8 *= +0-identity idp \u27e9\n         1# * x          \u2261\u27e8 1*-identity \u27e9\n         x               \u2261\u27e8 ! +0-identity \u27e9\n         x + 0#          \u220e)\n\n      *0-zero : RightZero 0# _*_\n      *0-zero {x} = cancels-+-right\n        (x * 0# + x      \u2261\u27e8 += idp (! *1-identity) \u27e9\n         x * 0# + x * 1# \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n         x * (0# + 1#)   \u2261\u27e8 *= idp 0+-identity \u27e9\n         x * 1#          \u2261\u27e8 *1-identity \u27e9\n         x               \u2261\u27e8 ! 0+-identity \u27e9\n         0# + x          \u220e)\n\n      2*-spec : \u2200 {n} \u2192 2* n \u2261 2# * n\n      2*-spec = ! += 1*-identity 1*-identity \u2666 ! *-+-distr\u02b3\n\n      0\u2212c+c+x : \u2200 {c x} \u2192 0\u2212 c + c + x \u2261 x\n      0\u2212c+c+x = += 0\u2212-inverse\u02e1 idp \u2666 0+-identity\n\n      0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n      0\u2212-*-distr = cancels-+-right (0\u2212-inverse\u02e1 \u2666 ! 0*-zero \u2666 *= (! 0\u2212-inverse\u02e1) idp \u2666 *-+-distr\u02b3)\n\n      0\u2212-*-distr\u02b3 : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n      0\u2212-*-distr\u02b3 = cancels-+-right (0\u2212-inverse\u02e1 \u2666 (! *0-zero \u2666 *= idp (! 0\u2212-inverse\u02e1)) \u2666 *-+-distr\u02e1)\n\n      -1*-neg : \u2200 {x} \u2192 -1# * x \u2261 0\u2212 x\n      -1*-neg = ! 0\u2212-*-distr \u2666 0\u2212= 1*-identity\n\n      *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n      *-\u2212-distr = *-+-distr\u02e1 \u2666 += idp (! 0\u2212-*-distr\u02b3)\n\n      \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n      \u00b2-0\u2212-distr = ! 0\u2212-*-distr \u2666 0\u2212=(! 0\u2212-*-distr\u02b3) \u2666 0\u2212-involutive\n\n      2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n      2*-*-distr = ! *-+-distr\u02b3\n\n      +-comm : Commutative _+_\n      +-comm {a} {b} =\n          cancels-+-right\n            (cancels-+-left\n              (a + (a + b + b)   \u2261\u27e8 += idp +-assoc \u2666 ! +-assoc \u27e9\n               (a + a) + (b + b) \u2261\u27e8 += 2*-spec 2*-spec \u27e9\n               2# * a  + 2# * b  \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n               2# * (a + b)      \u2261\u27e8 ! 2*-spec \u27e9\n               (a + b) + (a + b) \u2261\u27e8 +-assoc \u2666 += idp (! +-assoc) \u27e9\n               a + (b + a + b)   \u220e))\n\n      0\u2212-+-distr\u2032 : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 y \u2212 x\n      0\u2212-+-distr\u2032 = 0\u2212-hom\u2032\n\n      0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n      0\u2212-+-distr = 0\u2212= +-comm \u2666 0\u2212-+-distr\u2032\n\n  module From-Field-Ops (fld-ops : Field-Ops A) where\n    open Field-Ops fld-ops\n\nmodule Explicits where\n  open Algebra.FunctionProperties.NP \u03a0  {a}{a}{A} _\u2261_ public\n\n  -- REPEATED from above but with explicit arguments\n  module FromOp\u2082\n           (_\u2219_ : Op\u2082 A){x x' y y'}(p : x \u2261 x')(q : y \u2261 y')\n         where\n    op= : x \u2219 y \u2261 x' \u2219 y'\n    op= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  module FromComm\n           (_\u2219_   : Op\u2082 A)\n           (comm  : Commutative _\u2219_)\n           (x y x' y' : A)\n           (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n         where\n    open FromOp\u2082 _\u2219_\n\n    comm= : x \u2219 y \u2261 x' \u2219 y'\n    comm= = comm _ _ \u2666 e \u2666 comm _ _\n\n  module FromAssoc\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n\n         where\n    open FromOp\u2082 _\u2219_\n\n    assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    assocs = assoc , (\u03bb _ _ _ \u2192 ! assoc _ _ _)\n\n    module _ {c x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n      assoc= = ! assoc _ _ _ \u2666 op= e idp \u2666 assoc _ _ _\n\n      !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n      !assoc= = assoc _ _ _ \u2666 op= idp e \u2666 ! assoc _ _ _\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n      inner= = assoc= (!assoc= e)\n\n  module FromAssocComm\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n           (comm  : Commutative _\u2219_)\n         where\n    open FromOp\u2082   _\u2219_       renaming (op= to \u2219=)\n    open FromAssoc _\u2219_ assoc public\n    open FromComm  _\u2219_ comm  public\n\n    module _ x y z where\n      assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n      assoc-comm = assoc= (comm _ _)\n\n      !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n      !assoc-comm = !assoc= (comm _ _)\n\n    interchange : Interchange _\u2219_ _\u2219_\n    interchange _ _ _ _ = assoc= (!assoc= (comm _ _))\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n      outer= = \u2219= (comm _ _) (comm _ _) \u2666 assoc= (!assoc= e) \u2666 \u2219= (comm _ _) (comm _ _)\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj _ _ fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP using (flip; \u03a0; \u03a0\u2071; Op\u2081; Op\u2082; nest)\nopen import Data.Nat.Base using (\u2115)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.NP\nopen import Algebra.Raw\nopen \u2261-Reasoning\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\nmodule Implicits where\n  open Algebra.FunctionProperties.NP \u03a0\u2071 {a}{a}{A} _\u2261_ public\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n\n  module From-Magma (magma : Magma A) where\n\n    open Magma magma\n\n    module From-Comm\n             (comm  : Commutative _\u2219_)\n             {x y x' y' : A}\n             (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n           where\n      comm= : x \u2219 y \u2261 x' \u2219 y'\n      comm= = comm \u2666 e \u2666 comm\n\n    module From-Assoc\n             (assoc : Associative _\u2219_)\n           where\n\n      assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n      assocs = assoc , ! assoc\n\n      module _ {c x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n        assoc= = ! assoc \u2666 \u2219= e idp \u2666 assoc\n\n        !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n        !assoc= = assoc \u2666 \u2219= idp e \u2666 ! assoc\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n        inner= = assoc= (!assoc= e)\n\n    module From-Assoc-Comm\n             (assoc : Associative _\u2219_)\n             (comm  : Commutative _\u2219_)\n           where\n      open From-Assoc assoc public\n      open From-Comm  comm  public\n\n      module _ {x y z} where\n        assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n        assoc-comm = assoc= comm\n\n        !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n        !assoc-comm = !assoc= comm\n\n      interchange : Interchange _\u2219_ _\u2219_\n      interchange = assoc= (!assoc= comm)\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n        outer= = \u2219= comm comm \u2666 assoc= (!assoc= e) \u2666 \u2219= comm comm\n\n  module From-Op\u2082 (op : Op\u2082 A) = From-Magma \u27e8 op \u27e9\n\n  module From-Monoid-Ops (mon-ops : Monoid-Ops A) where\n    open Monoid-Ops mon-ops\n    open From-Op\u2082 _\u2219_ public\n\n    module From-LeftIdentity (idl : LeftIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n        is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 idl\n\n    module From-RightIdentity (idr : RightIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n        is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 idr\n\n      module _ {b} where\n        ^\u207a-^\u00b9\u207a : \u2200 n \u2192 b ^\u207a (1+ n) \u2261 b ^\u00b9\u207a n\n        ^\u207a-^\u00b9\u207a 0 = idr\n        ^\u207a-^\u00b9\u207a (1+ n) = \u2219= idp (^\u207a-^\u00b9\u207a n)\n\n        ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n        ^\u207a0-\u03b5 = idp\n\n        ^\u207a1-id : b ^\u207a 1 \u2261 b\n        ^\u207a1-id = idr\n\n        ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n        ^\u207a2-\u2219 = ^\u207a-^\u00b9\u207a 1\n\n        ^\u207a3-\u2219 : b ^\u207a 3 \u2261 b \u2219 (b \u2219 b)\n        ^\u207a3-\u2219 = ^\u207a-^\u00b9\u207a 2\n\n        ^\u207a4-\u2219 : b ^\u207a 4 \u2261 b \u2219 (b \u2219 (b \u2219 b))\n        ^\u207a4-\u2219 = ^\u207a-^\u00b9\u207a 3\n\n    module From-Identities (identities : Identity \u03b5 _\u2219_) where\n      comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n      comm-\u03b5 = snd identities \u2666 ! fst identities\n\n      open From-LeftIdentity  (fst identities) public\n      open From-RightIdentity (snd identities) public\n\n    module From-Assoc-RightIdentity\n             (assoc : Associative _\u2219_)\n             (idr   : RightIdentity \u03b5 _\u2219_)\n             where\n      open From-RightIdentity idr public\n\n      module _ {c x y}(e : x \u2219 y \u2261 \u03b5) where\n        elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n        elim-assoc= = assoc \u2666 \u2219= idp e \u2666 idr\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-inner= = ! assoc \u2666 \u2219= (elim-assoc= e) idp\n\n    module From-Assoc-LeftIdentity\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             where\n      open From-LeftIdentity idl public\n\n      -- Together with ^\u207a0-\u03b5, ^\u207a-+ shows that (b ^\u207a_) is a\n      -- monoid homomorphism from (\u2115,+,0) to (M,\u2219,\u03b5)\n      -- therefore also a group homomorphism\n      ^\u207a-+ : \u2200 m {n b} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n      ^\u207a-+ 0      = ! idl\n      ^\u207a-+ (1+ m) = ap (_\u2219_ _) (^\u207a-+ m) \u2666 ! assoc\n\n      -- Derived from ^\u207a-+ in Algebra.Group\n      ^\u207a-* : \u2200 m {n b} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n      ^\u207a-* 0 = idp\n      ^\u207a-* (1+ m) {n} {b}\n        = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n          b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n          b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n      module _ {c x y} (e : x \u2219 y \u2261 \u03b5) where\n        elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n        elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 idl\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-!inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-!inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module From-Group-Ops (grp-ops : Group-Ops A) where\n    open Group-Ops grp-ops\n    open From-Monoid-Ops mon-ops public\n\n    module From-Assoc-LeftIdentity-LeftInverse\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             (inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-LeftIdentity assoc idl\n\n      \u207b\u00b9-inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      \u207b\u00b9-inverse\u02b3 {x}\n         = x \u2219 x \u207b\u00b9                      \u2261\u27e8 ! idl \u27e9\n           \u03b5 \u2219 (x \u2219 x \u207b\u00b9)                \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9) \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9)              \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5                             \u220e\n\n      cancels-\u2219-left : LeftCancel _\u2219_\n      cancels-\u2219-left {c} {x} {y} e\n        = x              \u2261\u27e8 ! idl \u27e9\n          \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inverse\u02e1 idp \u27e9\n          \u03b5 \u2219 y          \u2261\u27e8 idl \u27e9\n          y              \u220e\n\n      module _ {x y} where\n        unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n        unique-\u03b5-right eq\n          = y               \u2261\u27e8 ! is-\u03b5-left inverse\u02e1 \u27e9\n            x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n            x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n            x \u207b\u00b9 \u2219 x        \u2261\u27e8 inverse\u02e1 \u27e9\n            \u03b5               \u220e\n\n      module _ {x y} where\n        -- _\u207b\u00b9 is a group homomorphism, see Algebra.Group._\u1d52\u1d56\n        \u207b\u00b9-hom\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n        \u207b\u00b9-hom\u2032 = cancels-\u2219-left {x \u2219 y}\n           ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 \u207b\u00b9-inverse\u02b3 \u27e9\n            \u03b5                       \u2261\u27e8 ! \u207b\u00b9-inverse\u02b3 \u27e9\n            x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! idl) \u27e9\n            x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! \u207b\u00b9-inverse\u02b3) idp) \u27e9\n            x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n            x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n            (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n      \u207b\u00b9-hom\u2032= : \u2200 {x y z t} \u2192 x \u2219 y \u2261 z \u2219 t \u2192 y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261 t \u207b\u00b9 \u2219 z \u207b\u00b9\n      \u207b\u00b9-hom\u2032= e = ! \u207b\u00b9-hom\u2032 \u2666 ap _\u207b\u00b9 e \u2666 \u207b\u00b9-hom\u2032\n\n      elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n      elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n        = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n          x \u207b\u00b9 \u2219 y               \u220e\n\n      elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n      elim-\u2219-right-\/ {c} {x} {y}\n        = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n          x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-!inner= \u207b\u00b9-inverse\u02b3 \u27e9\n          x \/ y \u220e\n\n    module From-Assoc-RightIdentity-RightInverse\n             (assoc : Associative _\u2219_)\n             (idr : RightIdentity \u03b5 _\u2219_)\n             (inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-RightIdentity assoc idr\n\n      cancels-\u2219-right : RightCancel _\u2219_\n      cancels-\u2219-right {c} {x} {y} e\n        = x              \u2261\u27e8 ! idr \u27e9\n          x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n          y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inverse\u02b3 \u27e9\n          y \u2219 \u03b5          \u2261\u27e8 idr \u27e9\n          y              \u220e\n\n      inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02e1 {x} = x \u207b\u00b9 \u2219 x                      \u2261\u27e8 ! idr \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 \u03b5                \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-inner= inverse\u02b3 \u27e9\n                  (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9)              \u2261\u27e8 inverse\u02b3 \u27e9\n                  \u03b5 \u220e\n\n      \u207b\u00b9-involutive : Involutive _\u207b\u00b9\n      \u207b\u00b9-involutive {x}\n        = cancels-\u2219-right\n          (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5              \u2261\u27e8 ! inverse\u02b3 \u27e9\n           x \u2219 x \u207b\u00b9 \u220e)\n\n      module _ {x y} where\n        \/-\u2219 : x \u2261 (x \/ y) \u2219 y\n        \/-\u2219 = x               \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02e1) \u27e9\n              x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n              (x \/ y) \u2219 y     \u220e\n\n        \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n        \u2219-\/ = x            \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02b3) \u27e9\n              x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n              (x \u2219 y) \/ y  \u220e\n\n      module _ {x y} where\n        unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n        unique-\u03b5-left eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            y \/ y        \u2261\u27e8 inverse\u02b3 \u27e9\n            \u03b5            \u220e\n\n        x\/y\u2261\u03b5\u2192x\u2261y : x \/ y \u2261 \u03b5 \u2192 x \u2261 y\n        x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5 = cancels-\u2219-right (x\/y\u2261\u03b5 \u2666 ! inverse\u02b3)\n\n        x\/y\u2262\u03b5 : x \u2262 y \u2192 x \/ y \u2262 \u03b5\n        x\/y\u2262\u03b5 x\u2262y x\/y\u2261\u03b5 = x\u2262y (x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5)\n\n      \u03b5\u207b\u00b9\u2261\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n      \u03b5\u207b\u00b9\u2261\u03b5 = unique-\u03b5-left inverse\u02e1\n\n    module From-Assoc-Identities-RightInverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      private\n        idl = fst identities\n        idr = snd identities\n\n      inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02b3 = inv-r\n\n      open From-Assoc assoc\n      open From-Identities identities using (comm-\u03b5)\n      open From-Assoc-RightIdentity assoc idr\n      open From-Assoc-LeftIdentity  assoc idl\n      open From-Assoc-RightIdentity-RightInverse assoc idr inverse\u02b3 public\n      open From-Assoc-LeftIdentity-LeftInverse assoc idl inverse\u02e1 public\n        hiding (\u207b\u00b9-inverse\u02b3)\n\n      module _ {x y} where\n        unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n        unique-\u207b\u00b9 eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            \u03b5 \/ y        \u2261\u27e8 idl \u27e9\n            y \u207b\u00b9         \u220e\n\n      \u207b\u00b9-inj : \u2200 {x y} \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9 \u2192 x \u2261 y\n      \u207b\u00b9-inj {x} {y} e\n         = ! (y              \u2261\u27e8 ! idr \u27e9\n              y \u2219 \u03b5          \u2261\u27e8 \u2219= idp (\u03b5        \u2261\u27e8 ! inverse\u02e1  \u27e9\n                                        x \u207b\u00b9 \u2219 x \u2261\u27e8 \u2219= e idp \u27e9\n                                        y \u207b\u00b9 \u2219 x \u220e) \u27e9\n              y \u2219 (y \u207b\u00b9 \u2219 x) \u2261\u27e8 elim-!assoc= inverse\u02b3 \u27e9\n              x \u220e)\n\n      module _ {b} where\n        ^\u207a-1+ : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n        ^\u207a-1+ 0 = comm-\u03b5\n        ^\u207a-1+ (1+ n) = ap (_\u2219_ b) (^\u207a-1+ n) \u2666 ! assoc\n\n        -- ^-\u2238 : \u2200 m n \u2192 m > n \u2192 b ^(m \u2238 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n\n        ^-\u2296 : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^-\u2296 m      0      = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296 0      (1+ n) = ! idl\n        ^-\u2296 (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296 m n \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296' : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^-\u2296' m      0      = ! is-\u03b5-left \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296' 0      (1+ n) = ! idr\n        ^-\u2296' (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296' m n \u27e9\n           (b ^\u207b n \u2219 b ^\u207a m)               \u2261\u27e8 ! elim-inner= inverse\u02e1 \u27e9\n           (b ^\u207b n \u2219 b \u207b\u00b9) \u2219 (b \u2219 b ^\u207a m)  \u2261\u27e8 \u2219= (! \u207b\u00b9-hom\u2032) idp \u27e9\n           (b \u2219 b ^\u207a n)\u207b\u00b9 \u2219 (b \u2219 b ^\u207a m)   \u220e\n\n        ^\u207a-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b ^\u207a m\n        ^\u207a-comm 0      n = ! comm-\u03b5\n        ^\u207a-comm (1+ m) n =\n          b \u2219 b\u1d50 \u2219 b\u207f   \u2261\u27e8 !assoc= (^\u207a-comm m n) \u27e9\n          b \u2219 b\u207f \u2219 b\u1d50   \u2261\u27e8 \u2219= (^\u207a-1+ n) idp \u27e9\n          b\u207f \u2219 b \u2219 b\u1d50   \u2261\u27e8 assoc \u27e9\n          b\u207f \u2219 (b \u2219 b\u1d50) \u220e\n          where\n            b\u207f = b ^\u207a n\n            b\u1d50 = b ^\u207a m\n\n        ^\u207a-^\u207b-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^\u207a-^\u207b-comm m n = ! ^-\u2296 m n \u2666 ^-\u2296' m n\n\n        ^\u207b-^\u207a-comm : \u2200 m n \u2192 b ^\u207b n \u2219 b ^\u207a m \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207b-^\u207a-comm m n = ! ^-\u2296' m n \u2666 ^-\u2296 m n\n\n        ^\u207b-comm : \u2200 m n \u2192 b ^\u207b m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207b m\n        ^\u207b-comm m n = \u207b\u00b9-hom\u2032= (^\u207a-comm n m)\n\n        ^\u207b-1+ : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n        ^\u207b-1+ n = ap _\u207b\u00b9 (^\u207a-1+ n) \u2666 \u207b\u00b9-hom\u2032\n\n        ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n        ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                        \u2666 ! \u207b\u00b9-hom\u2032\n                        \u2666 ap _\u207b\u00b9 (! ^\u207a-1+ n)\n\n        ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n        ^\u207b\u20321-id = idr\n\n        ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n        ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n        ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n        ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\n        -- ^-+ is a group homomorphism defined in Algebra.Group\n        -- Some properties can be derived from it.\n        ^-+ : \u2200 i j \u2192 b ^(i +\u2124 j) \u2261 b ^ i \u2219 b ^ j\n        ^-+ -[1+ m ] -[1+ n ] = ap _\u207b\u00b9\n                                 (ap (\u03bb z \u2192 b ^\u207a(1+ z)) (\u2115\u00b0.+-comm (1+ m) n)\n                              \u2666 ^\u207a-+ (1+ n) {1+ m}) \u2666 \u207b\u00b9-hom\u2032\n        ^-+ -[1+ m ]   (+ n)  = ^-\u2296' n (1+ m)\n        ^-+   (+ m)  -[1+ n ] = ^-\u2296 m (1+ n)\n        ^-+   (+ m)    (+ n)  = ^\u207a-+ m\n\n        -- GroupHomomorphism.f-pres-inv\n        ^-- : \u2200 i \u2192 b ^(- i) \u2261 (b ^ i)\u207b\u00b9\n        ^-- -[1+ n ] = ! \u207b\u00b9-involutive\n        ^-- (+ 0)    = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^-- (+ 1+ n) = idp\n\n        -- GroupHomomorphism.f-\u2212-\/\n        ^-\u2212 : \u2200 i j \u2192 b ^(i \u2212\u2124 j) \u2261 b ^ i \/ b ^ j\n        ^-\u2212 i j = ^-+ i (- j) \u2666 \u2219= idp (^-- j)\n\n        ^-suc : \u2200 i \u2192 b ^(suc\u2124 i) \u2261 b \u2219 b ^ i\n        ^-suc i = ^-+ (+ 1) i \u2666 \u2219= idr idp\n\n        ^-pred : \u2200 i \u2192 b ^(pred\u2124 i) \u2261 b \u207b\u00b9 \u2219 b ^ i\n        ^-pred i = ^-+ (- + 1) i \u2666 ap (\u03bb z \u2192 z \u207b\u00b9 \u2219 b ^ i) idr\n\n        ^-comm : \u2200 i j \u2192 b ^ i \u2219 b ^ j \u2261 b ^ j \u2219 b ^ i\n        ^-comm -[1+ m ] -[1+ n ] = ^\u207b-comm (1+ m) (1+ n)\n        ^-comm -[1+ m ]   (+ n)  = ! ^\u207a-^\u207b-comm n (1+ m)\n        ^-comm   (+ m)  -[1+ n ] = ^\u207a-^\u207b-comm m (1+ n)\n        ^-comm   (+ m)    (+ n)  = ^\u207a-comm m n\n\n      {-\n      ^-* : \u2200 i j \u2192 b ^(i *\u2124 j) \u2261 (b ^ j)^ i\n      ^-* i j = {!!}\n      -}\n\n    module From-Assoc-Identities-Inverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv : Inverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc-Identities-RightInverse assoc identities (snd inv) public\n\n  module From-Ring-Ops (rng-ops : Ring-Ops A) where\n    open Ring-Ops rng-ops\n\n    module DistributesOver\u02b3\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_)\n             where\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = *-comm \u2666 *-+-distr\u02e1 \u2666 += *-comm *-comm\n\n    module DistributesOver\u02e1\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = *-comm \u2666 *-+-distr\u02b3 \u2666 += *-comm *-comm\n\n    module From-+Group-*Identity-DistributesOver\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*1-identity : RightIdentity 1# _*_)\n             (*-+-distrs : _*_ DistributesOver _+_)\n             where\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( \u207b\u00b9-involutive to 0\u2212-involutive\n                     ; cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n                     ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n                     )\n\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = fst *-+-distrs\n\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = snd *-+-distrs\n\n      -0\u22610 : -0# \u2261 0#\n      -0\u22610 = 0\u22120\u22610\n\n      0*-zero : LeftZero 0# _*_\n      0*-zero {x} = cancels-+-left\n        (x + 0# * x      \u2261\u27e8 += (! 1*-identity) idp \u27e9\n         1# * x + 0# * x \u2261\u27e8 ! *-+-distr\u02b3 \u27e9\n         (1# + 0#) * x   \u2261\u27e8 *= +0-identity idp \u27e9\n         1# * x          \u2261\u27e8 1*-identity \u27e9\n         x               \u2261\u27e8 ! +0-identity \u27e9\n         x + 0#          \u220e)\n\n      *0-zero : RightZero 0# _*_\n      *0-zero {x} = cancels-+-right\n        (x * 0# + x      \u2261\u27e8 += idp (! *1-identity) \u27e9\n         x * 0# + x * 1# \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n         x * (0# + 1#)   \u2261\u27e8 *= idp 0+-identity \u27e9\n         x * 1#          \u2261\u27e8 *1-identity \u27e9\n         x               \u2261\u27e8 ! 0+-identity \u27e9\n         0# + x          \u220e)\n\n      2*-spec : \u2200 {n} \u2192 2* n \u2261 2# * n\n      2*-spec = ! += 1*-identity 1*-identity \u2666 ! *-+-distr\u02b3\n\n      0\u2212c+c+x : \u2200 {c x} \u2192 0\u2212 c + c + x \u2261 x\n      0\u2212c+c+x = += 0\u2212-inverse\u02e1 idp \u2666 0+-identity\n\n      0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n      0\u2212-*-distr = cancels-+-right (0\u2212-inverse\u02e1 \u2666 ! 0*-zero \u2666 *= (! 0\u2212-inverse\u02e1) idp \u2666 *-+-distr\u02b3)\n\n      0\u2212-*-distr\u02b3 : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n      0\u2212-*-distr\u02b3 = cancels-+-right (0\u2212-inverse\u02e1 \u2666 (! *0-zero \u2666 *= idp (! 0\u2212-inverse\u02e1)) \u2666 *-+-distr\u02e1)\n\n      -1*-neg : \u2200 {x} \u2192 -1# * x \u2261 0\u2212 x\n      -1*-neg = ! 0\u2212-*-distr \u2666 0\u2212= 1*-identity\n\n      *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n      *-\u2212-distr = *-+-distr\u02e1 \u2666 += idp (! 0\u2212-*-distr\u02b3)\n\n      \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n      \u00b2-0\u2212-distr = ! 0\u2212-*-distr \u2666 0\u2212=(! 0\u2212-*-distr\u02b3) \u2666 0\u2212-involutive\n\n      2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n      2*-*-distr = ! *-+-distr\u02b3\n\n      +-comm : Commutative _+_\n      +-comm {a} {b} =\n          cancels-+-right\n            (cancels-+-left\n              (a + (a + b + b)   \u2261\u27e8 += idp +-assoc \u2666 ! +-assoc \u27e9\n               (a + a) + (b + b) \u2261\u27e8 += 2*-spec 2*-spec \u27e9\n               2# * a  + 2# * b  \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n               2# * (a + b)      \u2261\u27e8 ! 2*-spec \u27e9\n               (a + b) + (a + b) \u2261\u27e8 +-assoc \u2666 += idp (! +-assoc) \u27e9\n               a + (b + a + b)   \u220e))\n\n      0\u2212-+-distr\u2032 : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 y \u2212 x\n      0\u2212-+-distr\u2032 = 0\u2212-hom\u2032\n\n      0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n      0\u2212-+-distr = 0\u2212= +-comm \u2666 0\u2212-+-distr\u2032\n\n  module From-Field-Ops (fld-ops : Field-Ops A) where\n    open Field-Ops fld-ops\n\nmodule Explicits where\n  open Algebra.FunctionProperties.NP \u03a0  {a}{a}{A} _\u2261_ public\n\n  -- REPEATED from above but with explicit arguments\n  module FromOp\u2082\n           (_\u2219_ : Op\u2082 A){x x' y y'}(p : x \u2261 x')(q : y \u2261 y')\n         where\n    op= : x \u2219 y \u2261 x' \u2219 y'\n    op= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  module FromComm\n           (_\u2219_   : Op\u2082 A)\n           (comm  : Commutative _\u2219_)\n           (x y x' y' : A)\n           (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n         where\n    open FromOp\u2082 _\u2219_\n\n    comm= : x \u2219 y \u2261 x' \u2219 y'\n    comm= = comm _ _ \u2666 e \u2666 comm _ _\n\n  module FromAssoc\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n\n         where\n    open FromOp\u2082 _\u2219_\n\n    assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    assocs = assoc , (\u03bb _ _ _ \u2192 ! assoc _ _ _)\n\n    module _ {c x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n      assoc= = ! assoc _ _ _ \u2666 op= e idp \u2666 assoc _ _ _\n\n      !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n      !assoc= = assoc _ _ _ \u2666 op= idp e \u2666 ! assoc _ _ _\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n      inner= = assoc= (!assoc= e)\n\n  module FromAssocComm\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n           (comm  : Commutative _\u2219_)\n         where\n    open FromOp\u2082   _\u2219_       renaming (op= to \u2219=)\n    open FromAssoc _\u2219_ assoc public\n    open FromComm  _\u2219_ comm  public\n\n    module _ x y z where\n      assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n      assoc-comm = assoc= (comm _ _)\n\n      !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n      !assoc-comm = !assoc= (comm _ _)\n\n    interchange : Interchange _\u2219_ _\u2219_\n    interchange _ _ _ _ = assoc= (!assoc= (comm _ _))\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n      outer= = \u2219= (comm _ _) (comm _ _) \u2666 assoc= (!assoc= e) \u2666 \u2219= (comm _ _) (comm _ _)\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj _ _ fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a46763a6919ff3ff8682f52edc906ea64703f1cc","subject":"Simplify","message":"Simplify\n","repos":"inc-lc\/ilc-agda","old_file":"Structure\/Bag\/Nehemiah.agda","new_file":"Structure\/Bag\/Nehemiah.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Bags of integers, for Nehemiah plugin.\n--\n-- This module imports postulates about bags of integers\n-- with negative multiplicities as a group under additive union.\n------------------------------------------------------------------------\n\nmodule Structure.Bag.Nehemiah where\n\nopen import Postulate.Bag-Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra using (CommutativeRing)\nopen import Algebra.Structures\nopen import Data.Integer\nopen import Data.Integer.Properties\n  using ()\n  renaming (commutativeRing to \u2124-is-commutativeRing)\nopen import Data.Product\n\ninfixl 9 _\\\\_ -- same as Data.Map.(\\\\)\n_\\\\_ : Bag \u2192 Bag \u2192 Bag\nd \\\\ b = d ++ (negateBag b)\n\n-- Useful properties of abelian groups\ncommutative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (m n : A) \u2192 f m n \u2261 f n m\ncommutative = IsAbelianGroup.comm\n\nassociative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (k m n : A) \u2192 f (f k m) n \u2261 f k (f m n)\nassociative abelian = IsAbelianGroup.assoc abelian\n\nleft-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f (neg n) n \u2261 z\nleft-inverse abelian = proj\u2081 (IsAbelianGroup.inverse abelian)\nright-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n (neg n) \u2261 z\nright-inverse abelian = proj\u2082 (IsAbelianGroup.inverse abelian)\n\nleft-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f z n \u2261 n\nleft-identity abelian = proj\u2081 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\nright-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n z \u2261 n\nright-identity abelian = proj\u2082 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\n\ninstance\n  abelian-int : IsAbelianGroup _\u2261_ _+_ (+ 0) (-_)\n  abelian-int =\n    IsRing.+-isAbelianGroup\n    (IsCommutativeRing.isRing\n    (CommutativeRing.isCommutativeRing\n    \u2124-is-commutativeRing))\ncommutative-int : (m n : \u2124) \u2192 m + n \u2261 n + m\ncommutative-int = commutative abelian-int\nassociative-int : (k m n : \u2124) \u2192 (k + m) + n \u2261 k + (m + n)\nassociative-int = associative abelian-int\nright-inv-int : (n : \u2124) \u2192 n - n \u2261 + 0\nright-inv-int = right-inverse abelian-int\nleft-id-int : (n : \u2124) \u2192 (+ 0) + n \u2261 n\nleft-id-int = left-identity abelian-int\nright-id-int : (n : \u2124) \u2192 n + (+ 0) \u2261 n\nright-id-int = right-identity abelian-int\n\ncommutative-bag : (a b : Bag) \u2192 a ++ b \u2261 b ++ a\ncommutative-bag = commutative abelian-bag\nassociative-bag : (a b c : Bag) \u2192 (a ++ b) ++ c \u2261 a ++ (b ++ c)\nassociative-bag = associative abelian-bag\nright-inv-bag : (b : Bag) \u2192 b \\\\ b \u2261 emptyBag\nright-inv-bag = right-inverse abelian-bag\nleft-id-bag : (b : Bag) \u2192 emptyBag ++ b \u2261 b\nleft-id-bag = left-identity abelian-bag\nright-id-bag : (b : Bag) \u2192 b ++ emptyBag \u2261 b\nright-id-bag = right-identity abelian-bag\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Bags of integers, for Nehemiah plugin.\n--\n-- This module imports postulates about bags of integers\n-- with negative multiplicities as a group under additive union.\n------------------------------------------------------------------------\n\nmodule Structure.Bag.Nehemiah where\n\nopen import Postulate.Bag-Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra using (CommutativeRing)\nopen import Algebra.Structures\nopen import Data.Integer\nimport Data.Integer.Properties\n  renaming (commutativeRing to \u2124-is-commutativeRing)\nopen Data.Integer.Properties using (\u2124-is-commutativeRing)\nopen import Data.Product\n\ninfixl 9 _\\\\_ -- same as Data.Map.(\\\\)\n_\\\\_ : Bag \u2192 Bag \u2192 Bag\nd \\\\ b = d ++ (negateBag b)\n\n-- Useful properties of abelian groups\ncommutative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (m n : A) \u2192 f m n \u2261 f n m\ncommutative = IsAbelianGroup.comm\n\nassociative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (k m n : A) \u2192 f (f k m) n \u2261 f k (f m n)\nassociative abelian = IsSemigroup.assoc (IsMonoid.isSemigroup\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\n\nleft-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f (neg n) n \u2261 z\nleft-inverse abelian = proj\u2081\n  (IsGroup.inverse (IsAbelianGroup.isGroup abelian))\nright-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n (neg n) \u2261 z\nright-inverse abelian = proj\u2082\n  (IsGroup.inverse (IsAbelianGroup.isGroup abelian))\n\nleft-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f z n \u2261 n\nleft-identity abelian = proj\u2081 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\nright-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n z \u2261 n\nright-identity abelian = proj\u2082 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\n\ninstance\n  abelian-int : IsAbelianGroup _\u2261_ _+_ (+ 0) (-_)\n  abelian-int =\n    IsRing.+-isAbelianGroup\n    (IsCommutativeRing.isRing\n    (CommutativeRing.isCommutativeRing\n    \u2124-is-commutativeRing))\ncommutative-int : (m n : \u2124) \u2192 m + n \u2261 n + m\ncommutative-int = commutative abelian-int\nassociative-int : (k m n : \u2124) \u2192 (k + m) + n \u2261 k + (m + n)\nassociative-int = associative abelian-int\nright-inv-int : (n : \u2124) \u2192 n - n \u2261 + 0\nright-inv-int = right-inverse abelian-int\nleft-id-int : (n : \u2124) \u2192 (+ 0) + n \u2261 n\nleft-id-int = left-identity abelian-int\nright-id-int : (n : \u2124) \u2192 n + (+ 0) \u2261 n\nright-id-int = right-identity abelian-int\n\ncommutative-bag : (a b : Bag) \u2192 a ++ b \u2261 b ++ a\ncommutative-bag = commutative abelian-bag\nassociative-bag : (a b c : Bag) \u2192 (a ++ b) ++ c \u2261 a ++ (b ++ c)\nassociative-bag = associative abelian-bag\nright-inv-bag : (b : Bag) \u2192 b \\\\ b \u2261 emptyBag\nright-inv-bag = right-inverse abelian-bag\nleft-id-bag : (b : Bag) \u2192 emptyBag ++ b \u2261 b\nleft-id-bag = left-identity abelian-bag\nright-id-bag : (b : Bag) \u2192 b ++ emptyBag \u2261 b\nright-id-bag = right-identity abelian-bag\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c4ea6ed65b7b21e33e219c3d918df18fa826e7fa","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import htype-decidable\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (pres-lem eps D1 D3 D4 D5) D2\n  pres-lem (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (pres-lem eps D2 D3 D4 D5)\n  pres-lem (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (pres-lem eps D1 D2 D3 D4) x\u2081\n  pres-lem (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (pres-lem eps D1 D2 D3 D4) x\n  pres-lem (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (pres-lem x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = pres-lem2 x y\n\n  -- this is the literal contents of the hole in lem3; it might not go\n  -- through exactly like this.\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst TAConst D2 = TAConst\n  lem-subst {\u0393 = \u0393} {x = x'} (TAVar {x = x} x\u2082) D2\n    with \u0393 x\n  lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | Some x\u2081\n    with natEQ x' x\n  lem-subst (TAVar xin) D2 | Some x\u2083 | Inl refl = {!!}\n  lem-subst (TAVar refl) D2 | Some x\u2083 | Inr x\u2082 = {!!}\n  lem-subst {x = x} (TAVar {x = x'} x\u2082) D2 | None with natEQ x' x\n  lem-subst {x = x} (TAVar x\u2083) D2 | None | Inl refl with natEQ x x\n  lem-subst (TAVar refl) D2 | None | Inl refl | Inl refl = D2\n  lem-subst (TAVar x\u2083) D2 | None | Inl refl | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | None | Inr x\u2082 with natEQ x x'\n  lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inl x\u2082 = abort ((flip x\u2083) x\u2082)\n  lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inr x\u2082 = abort (somenotnone (! x\u2084))\n  lem-subst {\u0393 = \u0393} {x = x} (TALam {x = x'} D1) D2 = {!!}\n  lem-subst (TAAp D1 D2) D3 = TAAp (lem-subst D1 D3) (lem-subst D2 D3)\n  lem-subst (TAEHole x\u2081 x\u2082) D2 = TAEHole x\u2081 {!!}\n  lem-subst (TANEHole x\u2081 D1 x\u2082) D2 = TANEHole x\u2081 (lem-subst D1 D2) {!!}\n  lem-subst (TACast D1 x\u2081) D2 = TACast (lem-subst D1 D2) x\u2081\n  lem-subst (TAFailedCast D1 x\u2081 x\u2082 x\u2083) D2 = TAFailedCast (lem-subst D1 D2) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import htype-decidable\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (pres-lem eps D1 D3 D4 D5) D2\n  pres-lem (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (pres-lem eps D2 D3 D4 D5)\n  pres-lem (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (pres-lem eps D1 D2 D3 D4) x\u2081\n  pres-lem (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (pres-lem eps D1 D2 D3 D4) x\n  pres-lem (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (pres-lem x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = pres-lem2 x y\n\n  -- this is the literal contents of the hole in lem3; it might not go\n  -- through exactly like this.\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst TAConst D2 = TAConst\n  lem-subst {\u0393 = \u0393} {x = x'} (TAVar {x = x} x\u2082) D2\n    with \u0393 x\n  lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | Some x\u2081\n    with natEQ x' x\n  lem-subst (TAVar xin) D2 | Some x\u2083 | Inl refl = {!!}\n  lem-subst (TAVar refl) D2 | Some x\u2083 | Inr x\u2082 = {!!}\n  lem-subst {x = x} (TAVar {x = x'} x\u2082) D2 | None with natEQ x' x\n  lem-subst {x = x} (TAVar x\u2083) D2 | None | Inl refl with natEQ x x\n  lem-subst (TAVar refl) D2 | None | Inl refl | Inl refl = D2\n  lem-subst (TAVar x\u2083) D2 | None | Inl refl | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | None | Inr x\u2082 with natEQ x x'\n  lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inl x\u2082 = abort ((flip x\u2083) x\u2082)\n  lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inr x\u2082 = abort (somenotnone (! x\u2084))\n  lem-subst (TALam D1) D2 = {!!}\n  lem-subst (TAAp D1 D2) D3 = TAAp (lem-subst D1 D3) (lem-subst D2 D3)\n  lem-subst (TAEHole x\u2081 x\u2082) D2 = TAEHole x\u2081 {!!}\n  lem-subst (TANEHole x\u2081 D1 x\u2082) D2 = TANEHole x\u2081 (lem-subst D1 D2) {!!}\n  lem-subst (TACast D1 x\u2081) D2 = TACast (lem-subst D1 D2) x\u2081\n  lem-subst (TAFailedCast D1 x\u2081 x\u2082 x\u2083) D2 = TAFailedCast (lem-subst D1 D2) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"50fd394a723e3e813b89d10bfaa72167a1206f06","subject":"Continue with functional semantics","message":"Continue with functional semantics\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) zero n1 ()\neval-const-mono (lit v) (suc n0) .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) v1 v2 \u2192\n  relV \u03c3 v1 v2 k \u2192\n  relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n  (v1 : Val \u03c4) \u2192\n  \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n  (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV  x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- relT (app s t) (app s t)\n\nrelV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\nrelV-apply = {!!}\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (const (lit v)) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , suc zero , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , n , n , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | Done tv1 n2 | [ t1eq ] with fundamental s _ \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s {!!}) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq) | fundamental t _ \u03c11 \u03c12 \u03c1\u03c1 tv1 (suc n2) {!!} {!eval-mono t \u03c11 tv1 n1 n2 t1eq!}\n... | closure st2 s\u03c12 , sn3 , sn4 , s2eq , svv | tv2 , tn3 , tn4 , t2eq , tvv = {!!}\n--\n-- {!eval s \u03c12 !}\n-- fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) ? ?\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a (suc n) = eval t (a \u2022 \u03c1) n\napply (closure t \u03c1) a zero = TimeOut\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 = eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3d30a4f4dd594d70f1d1e9a8cd983d40b16d53bf","subject":"Added Colist-coind-stronger-ho.","message":"Added Colist-coind-stronger-ho.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GreatestFixedPoints\/Colist.agda","new_file":"notes\/fixed-points\/GreatestFixedPoints\/Colist.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Colist where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Colist is a greatest fixed-point of a functor\n\n-- The functor.\nColistF : (D \u2192 Set) \u2192 D \u2192 Set\nColistF A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n\n-- Colist is the greatest fixed-point of ColistF.\npostulate\n  Colist : D \u2192 Set\n\n  -- Colist is a post-fixed point of ColistF, i.e.\n  --\n  -- Colist \u2264 ColistF Colist.\n  Colist-out-ho : \u2200 {n} \u2192 Colist n \u2192 ColistF Colist n\n\n  -- Colist is the greatest post-fixed point of ColistF, i.e.\n  --\n  -- \u2200 A. A \u2264 ColistF A \u21d2 A \u2264 Colist.\n  Colist-coind-ho :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ColistF.\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n    -- Colist is greater than A.\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n------------------------------------------------------------------------------\n-- First-order versions\n\nColist-out : \u2200 {xs} \u2192\n             Colist xs \u2192\n             xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\nColist-out = Colist-out-ho\n\nColist-coind :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind = Colist-coind-ho\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Colist predicate is also a pre-fixed point of the functional\n-- ColistF, i.e.\n--\n-- ColistF Colist \u2264 Colist.\nColist-in-ho : \u2200 {xs} \u2192 ColistF Colist xs \u2192 Colist xs\nColist-in-ho h = Colist-coind-ho A h' h\n  where\n  A : D \u2192 Set\n  A xs = ColistF Colist xs\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 ColistF A xs\n  h' (inj\u2081 xs\u22610) = inj\u2081 xs\u22610\n  h' (inj\u2082 (x' , xs' , prf , CLxs' )) =\n    inj\u2082 (x' , xs' , prf , Colist-out CLxs')\n\n-- The first-order version.\nColist-in : \u2200 {xs} \u2192\n            xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs') \u2192\n            Colist xs\nColist-in = Colist-in-ho\n\n------------------------------------------------------------------------------\n-- A stronger co-induction principle\n--\n-- From (Paulson, 1997. p. 16).\n\npostulate\n  Colist-coind-stronger-ho :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs \u2228 Colist xs) \u2192\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\nColist-coind-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-ho' A h Axs =\n  Colist-coind-stronger-ho A (\u03bb Ays \u2192 inj\u2081 (h Ays)) Axs\n\n-- The first-order version.\nColist-coind-stronger :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192\n    (xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'))\n    \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger = Colist-coind-stronger-ho\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. In: Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","old_contents":"------------------------------------------------------------------------------\n-- Co-lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Colist where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Colist is a greatest fixed-point of a functor\n\n-- The functor.\nColistF : (D \u2192 Set) \u2192 D \u2192 Set\nColistF A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n\n-- Colist is the greatest fixed-point of ColistF.\npostulate\n  Colist : D \u2192 Set\n\n  -- Colist is a post-fixed point of ColistF, i.e.\n  --\n  -- Colist \u2264 ColistF Colist.\n  Colist-out : \u2200 {xs} \u2192\n               Colist xs \u2192\n               xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\n\n  -- The higher-order version.\n  Colist-out-ho : \u2200 {n} \u2192 Colist n \u2192 ColistF Colist n\n\n  -- Colist is the greatest post-fixed point of ColistF, i.e.\n  --\n  -- \u2200 A. A \u2264 ColistF A \u21d2 A \u2264 Colist.\n  Colist-coind :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ColistF.\n    (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n    -- Colist is greater than A.\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n  -- The higher-order version.\n  Colist-coind-ho :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n------------------------------------------------------------------------------\n-- Colist-out and Colist-out-ho are equivalents\n\nColist-out-fo : \u2200 {xs} \u2192\n                Colist xs \u2192\n                xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\nColist-out-fo = Colist-out-ho\n\nColist-out-ho' : \u2200 {xs} \u2192 Colist xs \u2192 ColistF Colist xs\nColist-out-ho' = Colist-out\n\n------------------------------------------------------------------------------\n-- Colist-coind and Colist-coind-ho are equivalents\n\nColist-coind-fo :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-fo = Colist-coind\n\nColist-coind-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-ho' = Colist-coind\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Colist predicate is also a pre-fixed point of the functional ColistF,\n-- i.e.\n--\n-- ColistF Colist \u2264 Colist.\nColist-in : \u2200 {xs} \u2192\n            xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs') \u2192\n            Colist xs\nColist-in h = Colist-coind A h' h\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n  h' (inj\u2081 xs\u22610) = inj\u2081 xs\u22610\n  h' (inj\u2082 (x' , xs' , prf , CLxs' )) =\n    inj\u2082 (x' , xs' , prf , Colist-out CLxs')\n\n-- The higher-order version.\nColist-in-ho : \u2200 {xs} \u2192 ColistF Colist xs \u2192 Colist xs\nColist-in-ho = Colist-in\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"66d040d178c66458dbc506d2dbd14c35dd762658","subject":"Add 3 more primitives to get a more precise account on space cost","message":"Add 3 more primitives to get a more precise account on space cost\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_)\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Level as L\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\nopen \u2261 using (_\u2261_)\n\nopen import Data.Bits using (Bit; Bits; _\u2192\u1d47_; 0b; 1b)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _\u2218_\n  infixr 1 _\u204f_\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- Products\n    dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n    <tt,> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  _\u204f_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f \u204f g = g \u2218 f\n\n  _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f >>> g = f \u204f g\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = dup \u204f < f \u00d7 g >\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  <_\u00d7_>-on-top-of-<,> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g >-on-top-of-<,> = < fst \u204f f , snd \u204f g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  swap-on-top-<,> : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap-on-top-<,> = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = < f \u00d7 id >\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = < id \u00d7 f >\n\n  fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork f g = second < f , g > \u204f cond\n\n  -- In case you wonder... one can also derive cond with fork\n  cond-on-top-of-fork : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond-on-top-of-fork = fork fst snd\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\u22a4\n  <,tt> = <tt,> \u204f swap\n\n  assoc-default : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc-default = < fst \u204f fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd \u204f snd >\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > \u204f cons\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = < f \u2237 \u229b fs >\n\n  \u229b\u2032 : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 A `\u2192 `Vec B n\n  \u229b\u2032 []       = nil\n  \u229b\u2032 (f \u2237 fs) = < f \u2237\u2032 \u229b\u2032 fs >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 _A `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f xs = \u229b\u2032 (V.map f xs)\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = \u229b (V.replicate f)\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  <,nil> : \u2200 {A B} \u2192 A `\u2192 A `\u00d7 `Vec B 0\n  <,nil> = <,tt> \u204f second nil\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = <,nil> \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id F._\u2218\u2032_\n             (F.const 0b) (F.const 1b) (\u03bb { (b , x , y) \u2192 if b then x else y })\n             (\u03bb x \u2192 x , x) proj\u2081 proj\u2082 (\u03bb f g \u2192 \u00d7.map f g)\n             \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) _ (\u03bb x \u2192 _ , x) (F.const []) (uncurry _\u2237_) V.uncons\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _\u2218_\n                 (const [ 0b ]) (const [ 1b ]) cond\u1d47\n                 dup\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 {x}) <_\u00d7_>\u1d47\n                 (\u03bb {x} \u2192 swap\u1d47 {x}) (\u03bb {x} \u2192 assoc\u1d47 {x}) (const []) id (const []) id id\n  where\n  open BitsFunTypes\n  open FunOps\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = V.take A\n  snd\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 {A} = V.drop A\n  dup\u1d47 : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n  dup\u1d47 xs = xs ++ xs\n  <_\u00d7_>\u1d47 : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  <_\u00d7_>\u1d47 {A} f g x = f (take A x) ++ g (drop A x)\n  cond\u1d47 : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond\u1d47 {A} (b \u2237 xs) = if b then take A xs else drop A xs\n  swap\u1d47 : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap\u1d47 {A} xs = drop A xs ++ take A xs\n  assoc\u1d47 : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc\u1d47 {A} {B} xs = take A (take (A + B) xs) ++ (drop A (take (A + B) xs) ++ drop (A + B) xs) -- < fst\u1d47 \u204f fst\u1d47 , first snd\u1d47 >\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond)\n       (S.dup , T.dup) (S.fst , T.fst) (S.snd , T.snd) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.swap , T.swap) (S.assoc , T.assoc)\n       (S.tt , T.tt) (S.<tt,> , T.<tt,>)\n       (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FunOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond)\n       (S.dup , T.dup) (S.fst , T.fst) (S.snd , T.snd) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.swap , T.swap) (S.assoc , T.assoc)\n       (S.tt , T.tt) (S.<tt,> , T.<tt,>)\n       (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FunOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1;\n            dup = 0; fst = 0; snd = 0; <_\u00d7_> = _\u2294_; swap = 0; assoc = 0;\n            tt = 0; <tt,> = 0;\n            nil = 0; cons = 0; uncons = 0 }\n\nmodule TimeOps = FlatFunsOps timeOps\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 0; <1b> = 0; cond = 1;\n             dup = 1; fst = 0; snd = 0; <_\u00d7_> = _+_; swap = 0; assoc = 0;\n             tt = 0; <tt,> = 0;\n             nil = 0; cons = 0; uncons = 0 }\n\nmodule SpaceOps where\n  Space = \u2115\n  open FlatFunsOps spaceOps public\n\n  singleton\u22610 : singleton \u2261 0\n  singleton\u22610 = \u2261.refl\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_)\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Level as L\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bits using (Bit; Bits; _\u2192\u1d47_; 0b; 1b)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _\u2218_\n  infixr 1 _\u204f_\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Bit\n    <0b> <1b> : \u2200 {_A} \u2192 _A `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- Products\n    dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Vectors\n    nil    : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons   : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  _\u204f_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f \u204f g = g \u2218 f\n\n  _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f >>> g = f \u204f g\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = dup \u204f < f \u00d7 g >\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  <_\u00d7_>-on-top-of-<,> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g >-on-top-of-<,> = < fst \u204f f , snd \u204f g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = f *** id\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = id *** f\n\n  fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork f g = second < f , g > \u204f cond\n\n  -- In case you wonder... one can also derive cond with fork\n  cond-on-top-of-fork : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond-on-top-of-fork = fork fst snd\n\n  constBit : \u2200 {_A} \u2192 Bit \u2192 _A `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc = < fst \u204f fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd \u204f snd >\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u00d7 g > \u204f cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > \u204f cons\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = < f \u2237 \u229b fs >\n\n  \u229b\u2032 : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 A `\u2192 `Vec B n\n  \u229b\u2032 []       = nil\n  \u229b\u2032 (f \u2237 fs) = < f \u2237\u2032 \u229b\u2032 fs >\n\n  constVec : \u2200 {n b _A} {B : Set b} {C} \u2192 (B \u2192 _A `\u2192 C) \u2192 Vec B n \u2192 _A `\u2192 `Vec C n\n  constVec f xs = \u229b\u2032 (V.map f xs)\n\n  constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n  constBits = constVec constBit\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map {zero}  f = nil\n  map {suc n} f = < f \u2237 map f >\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id , nil > \u204f cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd \u204f singleton\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > \u204f cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz \u204f f , tabulate (fs \u204f f) > \u204f cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id F._\u2218\u2032_\n             (F.const 0b) (F.const 1b) (\u03bb { (b , x , y) \u2192 if b then x else y })\n             (\u03bb x \u2192 x , x) proj\u2081 proj\u2082 (\u03bb f g \u2192 \u00d7.map f g)\n             _ (F.const []) (uncurry _\u2237_) V.uncons\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _\u2218_\n                 (const [ 0b ]) (const [ 1b ]) cond\u1d47\n                 dup\u1d47 (\u03bb {A} \u2192 V.take A) (\u03bb {A} \u2192 V.drop A) <_\u00d7_>\u1d47\n                 (const []) (const []) id id\n  where\n  open BitsFunTypes\n  open FunOps\n  dup\u1d47 : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n  dup\u1d47 xs = xs ++ xs\n  <_\u00d7_>\u1d47 : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  <_\u00d7_>\u1d47 {A} f g x = f (take A x) ++ g (drop A x)\n  cond\u1d47 : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n  cond\u1d47 {A} (b \u2237 xs) = if b then take A xs else drop A xs\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond)\n       (S.dup , T.dup) (S.fst , T.fst) (S.snd , T.snd) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FunOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._\u2218_ T._\u2218_)\n       (S.<0b> , T.<0b>) (S.<1b> , T.<1b>) (S.cond , T.cond)\n       (S.dup , T.dup) (S.fst , T.fst) (S.snd , T.snd) (\u00d7.zip S.<_\u00d7_> T.<_\u00d7_>)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FunOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = record {\n            id = 0; _\u2218_ = _+_;\n            <0b> = 0; <1b> = 0; cond = 1;\n            dup = 0; fst = 0; snd = 0; <_\u00d7_> = _\u2294_;\n            tt = 0;\n            nil = 0; cons = 0; uncons = 0 }\n\nmodule TimeOps = FlatFunsOps timeOps\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = record {\n             id = 0; _\u2218_ = _+_;\n             <0b> = 0; <1b> = 0; cond = 1;\n             dup = 1; fst = 0; snd = 0; <_\u00d7_> = _+_;\n             tt = 0;\n             nil = 0; cons = 0; uncons = 0 }\n\nmodule SpaceOps = FlatFunsOps spaceOps\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FlatFunsOps time\u00d7spaceOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e03e05b0d9ad469cb29f032687a431dcfe115212","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 9679e5e5b7e04c672101f1626610d4a6\n\ndarcs-hash:20110406210038-3bd4e-9bc556a6e9785ce5d81eaff93aefd4284f80c4b9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.Properties.AuxiliaryATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 10-N \u2238-N #-}\n{-# ATP prove x<mc91x+11>100 +-N \u2238-N x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 #-}\n\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove mc91<mc91+11 mc91-res-100 102\u226191+11 100<102 #-}\n\n---- Case n \u2264 100\n\npostulate\n  -- Here we only reduce the definition of mc91\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n\npostulate\n  mc91x-res\u2264100 : \u2200 m n \u2192 LE m one-hundred \u2192\n                  mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                  mc91 m \u2261 ninety-one\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","old_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.Properties.AuxiliaryATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 10-N \u2238-N #-}\n{-# ATP prove x<mc91x+11>100 +-N \u2238-N x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 #-}\n\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven) \n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove mc91<mc91+11 mc91-res-100 102\u226191+11 100<102 #-}\n\n---- Case n \u2264 100\n\npostulate\n  -- Here we only reduce the definition of mc91\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n\npostulate\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d5b6a24bbe65ed220da02435d605203270c15075","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: ad938e7eafbcc12667861d22d0600f60\n\ndarcs-hash:20120216155813-3bd4e-d0de42d4bd483dac1f842f071dace36dc7211426.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Existential.agda","new_file":"notes\/thesis\/logical-framework\/Existential.agda","new_contents":"-- Tested with FOT on 16 February 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Existential where\n\nmodule LF where\n  postulate\n    D       : Set\n    \u2203       : (P : D \u2192 Set) \u2192 Set\n    _,_     : {P : D \u2192 Set}(x : D) \u2192 P x \u2192 \u2203 P\n    \u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n    \u2203-proj\u2082 : {P : D \u2192 Set}(p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  postulate P : D \u2192 D \u2192 Set\n\n  \u2203\u2200 : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n  \u2203\u2200 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\nmodule Inductive where\n\n  open import Common.Universe\n  open import Common.Data.Product\n\n  postulate P : D \u2192 D \u2192 Set\n\n  \u2203\u2200-el : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n  \u2203\u2200-el h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n  \u2203\u2200 : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n  \u2203\u2200 (x , Px) y = x , Px y\n","old_contents":"-- Tested with the development version of Agda on 07 February 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Existential where\n\nmodule LF where\n  postulate\n    D       : Set\n    \u2203       : (P : D \u2192 Set) \u2192 Set\n    _,_     : {P : D \u2192 Set}(x : D) \u2192 P x \u2192 \u2203 P\n    \u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n    \u2203-proj\u2082 : {P : D \u2192 Set}(p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  postulate P : D \u2192 D \u2192 Set\n\n  \u2203\u2200 : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n  \u2203\u2200 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\nmodule Inductive where\n\n  open import Common.Universe\n  open import Common.Data.Product\n\n  postulate P : D \u2192 D \u2192 Set\n\n  \u2203\u2200-el : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n  \u2203\u2200-el h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n  \u2203\u2200 : \u2203[ x ](\u2200 y \u2192 P x y) \u2192 \u2200 y \u2192 \u2203[ x ] P x y\n  \u2203\u2200 (x , Px) y = x , Px y\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b796f46c9070bb8ca14a238aaf66de660c0f53cc","subject":"Bool: +\u2227\u21d2\u2228, +to\u2115, +to\u2115\u22641","message":"Bool: +\u2227\u21d2\u2228, +to\u2115, +to\u2115\u22641\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ce8def0c5d4f551bb474ae0b1a721a630fb31baa","subject":"Added ++-Stream.","message":"Added ++-Stream.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream\n\n-----------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functional\n-- StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream (see FOTC.Data.Stream.Type).\nStream-pre-fixed : \u2200 {xs} \u2192\n                   (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs') \u2192\n                   Stream xs\nStream-pre-fixed {xs} h = Stream-coind (\u03bb ys \u2192 ys \u2261 ys) h' refl\n  where\n  h' : xs \u2261 xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 xs' \u2261 xs'\n  h' _ with h\n  ... | x' , xs' , prf , _ = x' , xs' , prf , refl\n\n++-Stream : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 Stream (xs ++ ys)\n++-Stream {xs} {ys} Sxs Sys with Stream-unf Sxs\n... | x' , xs' , prf , Sxs' = subst Stream prf\u2081 prf\u2082\n  where\n  prf\u2081 : x' \u2237 (xs' ++ ys) \u2261 xs ++ ys\n  prf\u2081 = trans (sym (++-\u2237 x' xs' ys)) (++-leftCong (sym prf))\n\n  -- TODO (15 December 2013): Why the termination checker accepts the\n  -- recursive called ++-Stream_Sxs'_Sys?\n  prf\u2082 : Stream (x' \u2237 xs' ++ ys)\n  prf\u2082 = Stream-pre-fixed (x' , xs' ++ ys , refl , ++-Stream Sxs' Sys)\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS h with Stream-unf h\n... | x' , xs' , prf , Sxs' =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\nstreamLength : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength {xs} Sxs = \u2248N-coind (\u03bb m _ \u2192 m \u2261 m) h refl\n  where\n\n  h : length xs \u2261 length xs \u2192 length xs \u2261 zero \u2227 \u221e \u2261 zero\n        \u2228 (\u2203[ m ] \u2203[ n ] length xs \u2261 succ\u2081 m \u2227 \u221e \u2261 succ\u2081 n \u2227 m \u2261 m)\n  h _ with Stream-unf Sxs\n  ... | x' , xs' , xs\u2261x'\u2237xs' , _ = inj\u2082 (length xs' , \u221e , prf , \u221e-eq , refl)\n    where\n    prf : length xs \u2261 succ\u2081 (length xs')\n    prf = trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs')\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream\n\n-----------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functional\n-- StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream (see FOTC.Data.Stream.Type).\nStream-pre-fixed : \u2200 {xs} \u2192\n                   (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs') \u2192\n                   Stream xs\nStream-pre-fixed {xs} h = Stream-coind (\u03bb ys \u2192 ys \u2261 ys) h' refl\n  where\n  h' : xs \u2261 xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 xs' \u2261 xs'\n  h' _ with h\n  ... | x' , xs' , prf , _ = x' , xs' , prf , refl\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS h with Stream-unf h\n... | x' , xs' , prf , Sxs' =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\nstreamLength : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength {xs} Sxs = \u2248N-coind (\u03bb m _ \u2192 m \u2261 m) h refl\n  where\n\n  h : length xs \u2261 length xs \u2192 length xs \u2261 zero \u2227 \u221e \u2261 zero\n        \u2228 (\u2203[ m ] \u2203[ n ] length xs \u2261 succ\u2081 m \u2227 \u221e \u2261 succ\u2081 n \u2227 m \u2261 m)\n  h _ with Stream-unf Sxs\n  ... | x' , xs' , xs\u2261x'\u2237xs' , _ = inj\u2082 (length xs' , \u221e , prf , \u221e-eq , refl)\n    where\n    prf : length xs \u2261 succ\u2081 (length xs')\n    prf = trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7a9286f38f38e747e7ba0c4711b3af9aa7b0d9ac","subject":"flipbased: waste and weaken","message":"flipbased: waste and weaken\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"open import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule flipbased\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n where\n\nCoins = \u2115\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\n-- An experiment\nEXP : \u2115 \u2192 Set\nEXP n = \u21ba n Bit\n\n-- A guessing game\n\u2141? : \u2200 c \u2192 Set\n\u2141? c = Bit \u2192 EXP c\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = return\u21ba\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ x f = join\u21ba (map\u21ba f x)\n\n_=<<_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       (A \u2192 \u21ba n\u2081 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_=<<_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2081 n\u2082 = flip _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\n_>=>_ : \u2200 {n\u2081 n\u2082 a b c} {A : Set a} {B : Set b} {C : Set c}\n        \u2192 (A \u2192 \u21ba n\u2081 B) \u2192 (B \u2192 \u21ba n\u2082 C) \u2192 A \u2192 \u21ba (n\u2081 + n\u2082) C\n(f >=> g) x = f x >>= g\n\nwaste : \u2200 n \u2192 \u21ba n \u22a4\nwaste n = return\u21ba {n} _\n\nweaken : \u2200 m {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken m x = waste m >> x\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 x = x >>= return\u21ba\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 pf x with \u2264\u21d2\u2203 pf\n... | k , \u2261.refl = weaken\u2032 x\n\npure\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure\u21ba = return\u21ba\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = toss >>= return\u21ba\n\n_\u25b9\u21ba_ : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 \u21ba n A \u2192 (A \u2192 B) \u2192 \u21ba n B\nx \u25b9\u21ba f = map\u21ba f x\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\u21ba\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map\u21ba f x\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 \u27ea f \u00b7 mx \u27eb \n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map\u21ba f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\nzip\u21ba : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} \u2192 \u21ba c\u2080 A \u2192 \u21ba c\u2081 B \u2192 \u21ba (c\u2080 + c\u2081) (A \u00d7 B)\nzip\u21ba x y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\n_\u27e8,\u27e9_ = zip\u21ba\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2237\u27e9_ : \u2200 {n\u2081 n\u2082 m a} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 (Vec A m) \u2192 \u21ba (n\u2081 + n\u2082) (Vec A (suc m))\nx \u27e8\u2237\u27e9 xs = \u27ea _\u2237_ \u00b7 x \u00b7 xs \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\nT\u21ba : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\nT\u21ba p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = x \u27e8\u2237\u27e9 replicate\u21ba x\n\nnot\u21ba : \u2200 {n} \u2192 EXP n \u2192 EXP n\nnot\u21ba = map\u21ba not\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2^ m * n) (Vec (Bits n) (2^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFunFromTbl : \u2200 m n \u2192 \u21ba (2^ m * n) (Bits m \u2192 Bits n)\nrandomFunFromTbl m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb\n  where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\nrandomFun : \u2200 m n \u2192 \u21ba (2^\u27e8 m \u27e9* n) (Bits m \u2192 Bits n)\nrandomFun zero    _ = \u27ea const \u00b7 random \u27eb\nrandomFun (suc m) _ = randomFunExt (randomFun m _)\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 return\u1d30 =<< x \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map\u21ba swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"open import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule flipbased\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n where\n\nCoins = \u2115\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\n-- An experiment\nEXP : \u2115 \u2192 Set\nEXP n = \u21ba n Bit\n\n-- A guessing game\n\u2141? : \u2200 c \u2192 Set\n\u2141? c = Bit \u2192 EXP c\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = return\u21ba\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ x f = join\u21ba (map\u21ba f x)\n\n_=<<_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       (A \u2192 \u21ba n\u2081 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_=<<_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2081 n\u2082 = flip _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\n_>=>_ : \u2200 {n\u2081 n\u2082 a b c} {A : Set a} {B : Set b} {C : Set c}\n        \u2192 (A \u2192 \u21ba n\u2081 B) \u2192 (B \u2192 \u21ba n\u2082 C) \u2192 A \u2192 \u21ba (n\u2081 + n\u2082) C\n(f >=> g) x = f x >>= g\n\nweaken : \u2200 m {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken m x = return\u21ba {m} 0 >> x\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 x = x >>= return\u21ba\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 pf x with \u2264\u21d2\u2203 pf\n... | k , \u2261.refl = weaken\u2032 x\n\npure\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure\u21ba = return\u21ba\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = toss >>= return\u21ba\n\n_\u25b9\u21ba_ : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 \u21ba n A \u2192 (A \u2192 B) \u2192 \u21ba n B\nx \u25b9\u21ba f = map\u21ba f x\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\u21ba\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map\u21ba f x\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 \u27ea f \u00b7 mx \u27eb \n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map\u21ba f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\nzip\u21ba : \u2200 {c\u2080 c\u2081 a b} {A : Set a} {B : Set b} \u2192 \u21ba c\u2080 A \u2192 \u21ba c\u2081 B \u2192 \u21ba (c\u2080 + c\u2081) (A \u00d7 B)\nzip\u21ba x y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\n_\u27e8,\u27e9_ = zip\u21ba\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2237\u27e9_ : \u2200 {n\u2081 n\u2082 m a} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 (Vec A m) \u2192 \u21ba (n\u2081 + n\u2082) (Vec A (suc m))\nx \u27e8\u2237\u27e9 xs = \u27ea _\u2237_ \u00b7 x \u00b7 xs \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\nT\u21ba : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\nT\u21ba p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = x \u27e8\u2237\u27e9 replicate\u21ba x\n\nnot\u21ba : \u2200 {n} \u2192 EXP n \u2192 EXP n\nnot\u21ba = map\u21ba not\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2^ m * n) (Vec (Bits n) (2^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFunFromTbl : \u2200 m n \u2192 \u21ba (2^ m * n) (Bits m \u2192 Bits n)\nrandomFunFromTbl m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb\n  where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\nrandomFun : \u2200 m n \u2192 \u21ba (2^\u27e8 m \u27e9* n) (Bits m \u2192 Bits n)\nrandomFun zero    _ = \u27ea const \u00b7 random \u27eb\nrandomFun (suc m) _ = randomFunExt (randomFun m _)\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 return\u1d30 =<< x \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map\u21ba swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"df41c005d15604256ca7d340975689955ae6729f","subject":"update protocols.agda","message":"update protocols.agda\n","repos":"crypto-agda\/protocols","old_file":"protocols.agda","new_file":"protocols.agda","new_contents":"module protocols where\nimport Control.Protocol\nimport Control.Protocol.Additive\nimport Control.Protocol.CLL\nimport Control.Protocol.ClientServer\nimport Control.Protocol.Examples\nimport Control.Protocol.Extend\nimport Control.Protocol.Lift\nimport Control.Protocol.MultiParty\nimport Control.Protocol.Multiplicative\nimport Control.Protocol.Relation\nimport Control.Protocol.Sequence\n","old_contents":"module protocols where\nimport Control.Protocol\nimport Control.Protocol.Additive\nimport Control.Protocol.CLL\nimport Control.Protocol.ClientServer\nimport Control.Protocol.Examples\nimport Control.Protocol.MultiParty\nimport Control.Protocol.Multiplicative\nimport Control.Protocol.Relation\nimport Control.Protocol.Sequence\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c331f69b265880f6b98ff0ecc55dbbd4b5d2d695","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: 364ef2c60263dc8dc66151b308396671\n\ndarcs-hash:20120119215014-3bd4e-c2fdcf5ffb49d2aa51fe222d53b7ffd098ac17d1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/papers\/FoSSaCS-2012\/BottomBottom.agda","new_file":"notes\/papers\/FoSSaCS-2012\/BottomBottom.agda","new_contents":"-- Tested on 08 December 2011.\n\nmodule BottomBottom where\n\nopen import Common.Universe\nopen import Common.Data.Empty\n\npostulate bot\u2081 bot\u2082 : \u22a5\n{-# ATP prove bot\u2081 bot\u2082 #-}\n{-# ATP prove bot\u2082 bot\u2081 #-}\n\n-- $ agda2atp -i src -i notes\/papers\/FoSSaCS-2012\/ notes\/papers\/FoSSaCS-2012\/BottomBottom.agda\n-- Proving the conjecture in \/tmp\/BottomBottom.bot1_8.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot1_8.tptp\n-- Proving the conjecture in \/tmp\/BottomBottom.bot2_8.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot2_8.tptp\n","old_contents":"-- Tested on 08 December 2011.\n\nmodule BottomBottom where\n\nopen import Common.Universe\nopen import Common.Data.Empty\n\npostulate bot\u2081 bot\u2082 : \u22a5\n{-# ATP prove bot\u2081 bot\u2082 #-}\n{-# ATP prove bot\u2082 bot\u2081 #-}\n\n-- $ agda2atp -i src -i notes\/papers\/FoSSaCS-2012\/ notes\/papers\/FoSSaCS-2012\/BottomBottom.agda\n-- Proving the conjecture in \/tmp\/BottomBottom.bot1_8.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot1_8.tptp\n-- Proving the conjecture in \/tmp\/BottomBottom.bot2_8.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot2_8.tptp\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ff7cc70477ddaedcd1fcec0dbd330bbe0d8e9848","subject":"generic-zero-knowledge-interactive: Finishing DLog","message":"generic-zero-knowledge-interactive: Finishing DLog\n","repos":"crypto-agda\/crypto-agda","old_file":"generic-zero-knowledge-interactive.agda","new_file":"generic-zero-knowledge-interactive.agda","new_contents":"open import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\nprivate\n  \u2605 : Set\u2081\n  \u2605 = Set\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (sum\u03c0        : Sum Permutation)\n         (\u03bc\u03c0          : SumProp sum\u03c0)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (\u03bcR\u209a-xtra    : SumProp sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sum\u03c0 \u00d7Sum sumR\u209a-xtra\n\n    \u03bcR\u209a : SumProp sumR\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    sumR : Sum R\n    sumR = sumBit \u00d7Sum sumR\u209a\n\n    \u03bcR : SumProp sumR\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl  : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-cong  : \u2200 {\u03b1 \u03b2 b} (f : G \u2192 G) \u2192 \u03b1 == \u03b2 \u2261 b \u2192 f \u03b1 == f \u03b2 \u2261 b)\n            (==-true  : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (sum\u2124q      : Sum \u2124q)\n            (\u03bc\u2124q        : SumProp sum\u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (sumR\u209a-xtra : Sum R\u209a-xtra)\n            (\u03bcR\u209a-xtra   : SumProp sumR\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s = ==-cong (_\u2219_ (g^ \u03c0)) check-p-s\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 sum\u2124q \u03bc\u2124q R\u209a-xtra sumR\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\nprivate\n  \u2605 : Set\u2081\n  \u2605 = Set\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (sum\u03c0        : Sum Permutation)\n         (\u03bc\u03c0          : SumProp sum\u03c0)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (\u03bcR\u209a-xtra    : SumProp sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sum\u03c0 \u00d7Sum sumR\u209a-xtra\n\n    \u03bcR\u209a : SumProp sumR\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    sumR : Sum R\n    sumR = sumBit \u00d7Sum sumR\u209a\n\n    \u03bcR : SumProp sumR\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl  : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-true  : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (==-false : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 false \u2192 \u03b1 \u2262 \u03b2)\n            (sum\u2124q      : Sum \u2124q)\n            (\u03bc\u2124q        : SumProp sum\u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (sumR\u209a-xtra : Sum R\u209a-xtra)\n            (\u03bcR\u209a-xtra   : SumProp sumR\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s\n      with ==-false check-p-s ?\n   ... | ()\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 sum\u2124q \u03bc\u2124q R\u209a-xtra sumR\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a3a6c0d7b42940b151354af420f8d4469955d5c8","subject":"Simplify Atlas-diff and Atlas-apply.","message":"Simplify Atlas-diff and Atlas-apply.\n\nOld-commit-hash: ada22ebb7416f8c4fb19882d4080166628b1b292\n","repos":"inc-lc\/ilc-agda","old_file":"Atlas\/Change\/Term.agda","new_file":"Atlas\/Change\/Term.agda","new_contents":"module Atlas.Change.Term where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\n\n-- nil-changes\n\nnil-const : \u2200 {\u03b9 : Base} \u2192 Const  \u2205 (base (\u0394Base \u03b9))\nnil-const {\u03b9} = neutral {\u0394Base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9))\nnil-term {Bool}   = curriedConst (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = curriedConst (nil-const {Map \u03ba \u03b9})\n\n-- diff-term and apply-term\n\nopen import Parametric.Change.Term Const \u0394Base\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))\nAtlas-diff {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\nAtlas-diff {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs Atlas-diff) m\u2081 m\u2082)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\nAtlas-apply {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs Atlas-apply) m\u2081 m\u2082)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (\u0394Type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n","old_contents":"module Atlas.Change.Term where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\n\n-- nil-changes\n\nnil-const : \u2200 {\u03b9 : Base} \u2192 Const  \u2205 (base (\u0394Base \u03b9))\nnil-const {\u03b9} = neutral {\u0394Base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9))\nnil-term {Bool}   = curriedConst (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = curriedConst (nil-const {Map \u03ba \u03b9})\n\n-- diff-term and apply-term\n\nopen import Parametric.Change.Term Const \u0394Base\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))\nAtlas-diff {Bool} = abs (abs (curriedConst xor (var (that this)) (var this)))\nAtlas-diff {Map \u03ba \u03b9} = abs (abs (curriedConst zip (abs Atlas-diff) (var (that this)) (var this)))\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = abs (abs (curriedConst xor (var (that this)) (var this)))\nAtlas-apply {Map \u03ba \u03b9} = abs (abs (curriedConst zip (abs Atlas-apply) (var (that this)) (var this)))\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (\u0394Type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d41e35b9fb769cc937f3fa2787fe655ac719f13a","subject":"Atlas: add primitives `lookup`, `fold` and various improvements","message":"Atlas: add primitives `lookup`, `fold` and various improvements\n\nImprovements:\n\n- speed up derivative of zip\n\n- less verbose term constructors for constants\n\n- remove need to support conjunction and union directly\n\nFuture work:\n\n- elimination forms: if-then-else, if-nonempty\n\nOld-commit-hash: 9aafd0c2835ff027b57e44ed2930f4f57147e0de\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\nAtlas-lookup (lookup {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9\n\n-- Model of zip = Haskell Data.List.zipWith\n--\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n--\n-- Behavioral difference: all key-value pairs present\n-- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n-- be iterated over. Neutral element of type `a` or `b`\n-- will be supplied if the key is missing in the\n-- corresponding map.\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\n-- Model of fold = Haskell Data.Map.foldWithKey\n--\n-- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n--\nAtlas-lookup (fold {\u03ba} {a} {b}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n  base b \u21d2 base (Map \u03ba a) \u21d2 base b\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t (const empty)\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app\u2082 (const xor) \u0394x \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n--\n-- TODO: write this and call it Syntax.Term.Plotkin.lift-term\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\n\nzip! f m\u2081 m\u2082 = app (app (app (const zip) f) m\u2081) m\u2082\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\n\nlookup! = {!!} -- TODO: add constant `lookup`\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff s t = app (app (lift-diff Atlas-diff Atlas-apply) s) t\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply s t = app (app (lift-apply Atlas-diff Atlas-apply) s) t\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\n\nupdate! k v m = app (app (app (const update) k) v) m\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- union s t should be:\n-- 1. equal to t if s is neutral\n-- 2. equal to s if t is neutral\n-- 3. equal to s if s == t\n-- 4. whatever it wants to be otherwise\n--\n-- On Booleans, only conjunction can be union.\n--\n-- TODO (later): support conjunction, probably by Boolean\n-- elimination form if-then-else\n\npostulate\n  and! : \u2200 {\u0393} \u2192\n    Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n    Atlas-term \u0393 (base Bool)\n\nunion : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base \u03b9)\nunion {Bool} s t = and! s t\nunion {Map \u03ba \u03b9} s t =\n  let\n    union-term = abs (abs (union (var (that this)) (var this)))\n  in\n    zip! (abs union-term) s t\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    v\u2081 = var (that this)\n    v\u2082 = var this\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    k\u2081\u2082 = var (that (that this))\n    k\u2083\u2084 = var (that (that this))\n    f\u2081\u2082 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2081\u2082) v\u2081) v\u2082)\n      (lookup! k\u2081\u2082 (weaken\u2083 m\u2083))) (lookup! k\u2081\u2082 (weaken\u2083 m\u2084)))))\n    f\u2083\u2084 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2083\u2084)\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2081)))\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2082))) v\u2083) v\u2084)))\n  in\n    -- A correct but inefficient implementation.\n    -- May want to speed it up after constants are finalized.\n    union (zip! f\u2081\u2082 m\u2081 m\u2082) (zip! f\u2083\u2084 m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app (app (weaken\u2081 \u0394f) (var this)) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fb83ee7872bb6b8aba0c02639123149953ef0d90","subject":"s\/Mu\/\u03bc and s\/Nu\/\u03bd.","message":"s\/Mu\/\u03bc and s\/Nu\/\u03bd.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Functors.agda","new_file":"notes\/fixed-points\/Functors.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Functors where\n\n-- The carrier of the initial algebra is (up to isomorphism) a\n-- fixed-point of the functor (Vene 2000, p).\n\n------------------------------------------------------------------------------\n\ndata Bool : Set where\n  false true : Bool\n\ndata _\u228e_  (A B : Set) : Set where\n  inl : A \u2192 A \u228e B\n  inr : B \u2192 A \u228e B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata One : Set where\n   one : One\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = One \u228e X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = One \u228e (A \u00d7 X)\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl one)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl one)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conaturals type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl one)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- TODO: The conat destructor.\npred : Conat \u2192 One \u228e Conat\npred cn with out cn\n... | inl _ = inl one\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl one)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Functors where\n\n-- The carrier of the initial algebra is (up to isomorphism) a\n-- fixed-point of the functor (Vene 2000, p).\n\n------------------------------------------------------------------------------\n\ndata Bool : Set where\n  false true : Bool\n\ndata _\u228e_  (A B : Set) : Set where\n  inl : A \u2192 A \u228e B\n  inr : B \u2192 A \u228e B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata One : Set where\n   one : One\n\npostulate\n  -- The least fixed-point.\n  -- The names came from the Haskell definitions\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  Mu   : (Set \u2192 Set) \u2192 Set\n  In   : {f : Set \u2192 Set} \u2192 f (Mu f) \u2192 Mu f\n  unIn : {f : Set \u2192 Set} \u2192 Mu f \u2192 f (Mu f)\n\npostulate\n  -- The greatest fixed-point.\n  -- The names came from the Haskell definitions\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  Nu   : (Set \u2192 Set) \u2192 Set\n  Wrap : {f : Set \u2192 Set} \u2192 f (Nu f) \u2192 Nu f\n  out  : {f : Set \u2192 Set} \u2192 Nu f \u2192 f (Nu f)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = One \u228e X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = One \u228e (A \u00d7 X)\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = Mu IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = Mu NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl one)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = Mu (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl one)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = Nu IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conaturals type is a greatest fixed-point.\nConat : Set\nConat = Nu NatF\n\nzeroC : Conat\nzeroC = Wrap (inl one)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- TODO: The conat destructor.\npred : Conat \u2192 One \u228e Conat\npred cn with out cn\n... | inl _ = inl one\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = Nu (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl one)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = Nu (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"233ce4867a6fc1f8c29c642b2516cc11c7edd483","subject":"Implement apply-term and diff-term directly.","message":"Implement apply-term and diff-term directly.\n\nBefore this commit: Mutual recursion of apply-term and diff-term encoded\nexplicitly via a product type. After this commit: Mutual recursion\nexpressed with Agda mutual recursion.\n\nOld-commit-hash: 9a7571f8ef5cf6765629a001c46870e002f54d7e\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    (let\n       diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n       apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))\n    ,\n    (let\n       diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n       apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  diff-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  apply-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  apply-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4e5025e3160f775b4805a504187a9e95c764c9b1","subject":"Type.Identities: some more identities","message":"Type.Identities: some more identities\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two using (\ud835\udfda ; 0\u2082 ; 1\u2082 ; [0:_1:_]; twist)\nopen import Data.Fin as Fin using (Fin ; suc ; zero)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; Reveal_is_ ; [_]; tr) renaming (refl to idp; cong\u2082 to ap\u2082; _\u2257_ to _\u223c_)\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n-- for use with ap\u2082 etc.\n_\u27f6_ : \u2200 {a b} \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (b \u2294 a)\nA \u27f6 B = A \u2192 B\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u228e B \u2192 \u2605} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : (x : A) \u2192 B x \u2192 \u2605} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {A B : \u2605} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {A B C : \u2605} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}(A : \ud835\udfd9 \u2192 \u2605) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}(A : \u2605) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}(F G : \ud835\udfd8 \u2192 \u2605) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {A : \ud835\udfd9 \u2192 \u2605} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    \u228e\ud835\udfd8-inl : {{_ : UA}} \u2192 A \u2261 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n    \ud835\udfd9\u00d7-snd : {{_ : UA}} \u2192 (\ud835\udfd9 \u00d7 A) \u2261 A\n    \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n    \u00d7\ud835\udfd9-fst : {{_ : UA}} \u2192 (A \u00d7 \ud835\udfd9) \u2261 A\n    \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n    -- old names\n    A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n    \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {A : \ud835\udfda \u2192 \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u228e-\u03a3 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u00d7-\u03a0 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : \ud835\udfda \u00d7 A \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n  \ud835\udfda\u2192A\u2194A\u00d7A\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2261\ud835\udfd9\u228e : \u2200 {{_ : UA}}\u2192 Maybe A \u2261 (\ud835\udfd9 \u228e A)\n  Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua (equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2)\n      where \u03b2 : (_ : Fin 1) \u2192 _\n            \u03b2 zero = idp\n            \u03b2 (suc ())\n\nmodule _ where\n  isZero? : \u2200 {n}{A : Fin (suc n) \u2192 Set} \u2192 ((i : Fin n) \u2192 A (suc i)) \u2192 A zero\n    \u2192 (i : Fin (suc n)) \u2192 A i\n  isZero? f x zero = x\n  isZero? f x (suc i) = f i\n\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv (isZero? inr (inl _)) [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n    [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n    (isZero? (\u03bb _ \u2192 idp) idp)\n\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero} = ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero} = ap\u2082 _\u00d7_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = ap\u2082 _\u00d7_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero} = ap\u2082 _\u27f6_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = ap\u2082 _\u27f6_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  ap\u2082 _\u228e_ (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 ap\u2082 _\u228e_ (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nmodule _ {{_ : UA}} where\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua (equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp }))\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n{-\nTODO ?\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two using (\ud835\udfda ; 0\u2082 ; 1\u2082 ; [0:_1:_])\nopen import Data.Fin as Fin using (Fin ; suc ; zero)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; Reveal_is_ ; [_]) renaming (subst to tr; refl to idp; cong\u2082 to ap\u2082; _\u2257_ to _\u223c_)\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n-- for use with ap\u2082 etc.\n_\u27f6_ : \u2200 {a b} \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (b \u2294 a)\nA \u27f6 B = A \u2192 B\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u228e B \u2192 \u2605} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : (x : A) \u2192 B x \u2192 \u2605} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {A B : \u2605} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {A B C : \u2605} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}(A : \ud835\udfd9 \u2192 \u2605) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}(A : \u2605) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : FunExt}}(F G : \ud835\udfd8 \u2192 \u2605) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {A : \ud835\udfd9 \u2192 \u2605} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    \u228e\ud835\udfd8-inl : {{_ : UA}} \u2192 A \u2261 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n    \ud835\udfd9\u00d7-snd : {{_ : UA}} \u2192 (\ud835\udfd9 \u00d7 A) \u2261 A\n    \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n    \ud835\udfd9\u00d7-fst : {{_ : UA}} \u2192 (A \u00d7 \ud835\udfd9) \u2261 A\n    \ud835\udfd9\u00d7-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {A : \ud835\udfda \u2192 \u2605} where\n  \u03a3\ud835\udfda-\u228e : {{_ : UA}}{{_ : FunExt}} \u2192 \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u228e-\u03a3 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : {{_ : UA}}{{_ : FunExt}} \u2192 \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-first \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 \u2219 dist-\u00d7-\u03a0 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2261\ud835\udfd9\u228e : \u2200 {{_ : UA}}\u2192 Maybe A \u2261 (\ud835\udfd9 \u228e A)\n  Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\nFin0\u2261\ud835\udfd8 : {{_ : UA}} \u2192 Fin 0 \u2261 \ud835\udfd8\nFin0\u2261\ud835\udfd8 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\nmodule _ where\n  isZero? : \u2200 {n}{A : Fin (suc n) \u2192 Set} \u2192 ((i : Fin n) \u2192 A (suc i)) \u2192 A zero\n    \u2192 (i : Fin (suc n)) \u2192 A i\n  isZero? f x zero = x\n  isZero? f x (suc i) = f i\n\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv (isZero? inr (inl _)) [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n    [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n    (isZero? (\u03bb _ \u2192 idp) idp)\n\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\n  Fin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero} = ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero} = ap\u2082 _\u00d7_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = ap\u2082 _\u00d7_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 ap\u2082 _\u228e_ \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero} = ap\u2082 _\u27f6_ Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = ap\u2082 _\u27f6_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 ap\u2082 _\u00d7_ (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  ap\u2082 _\u228e_ (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (ap\u2082 _\u228e_ Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 ap\u2082 _\u228e_ (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 ap\u2082 _\u228e_ Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6371c2dd29365338ea26008a34552df2e4ee1d90","subject":"Permutations and search","message":"Permutations and search\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    toPerm : Op \u2192 Perm\n    toPerm `id     = `id\n    toPerm `0\u21941    = `0\u21941\n    toPerm `not    = `id -- Important\n    toPerm (`tl f) = `tl (toPerm f)\n    toPerm (f `\u204f g) = toPerm f `\u204f toPerm g\n\n    infixr 9 _\u2219\u2032_\n    _\u2219\u2032_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    f \u2219\u2032 xs = toPerm f P.\u2219 xs\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    almost-\u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) f xs \u2192 f \u2219\u2032 pad \u2295 f \u2219 xs \u2261 f \u2219 (pad \u2295 xs)\n    almost-\u2295-dist-\u2219 pad           `id     xs = refl\n    almost-\u2295-dist-\u2219 pad           `0\u21941    xs = \u2295-dist-0\u21941 pad xs\n    almost-\u2295-dist-\u2219 []            `not    [] = refl\n    almost-\u2295-dist-\u2219 (true  \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 (false \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 []            (`tl f) [] = refl\n    almost-\u2295-dist-\u2219 (p \u2237 pad)     (`tl f) (x \u2237 xs) rewrite almost-\u2295-dist-\u2219 pad f xs = refl\n    almost-\u2295-dist-\u2219 pad           (f `\u204f g) xs rewrite almost-\u2295-dist-\u2219 (f \u2219\u2032 pad) g (f \u2219 xs)\n                                                   | almost-\u2295-dist-\u2219 pad f xs = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-+ to sum-+;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u00b7 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u00b7-comm : \u2200 {m} (x y : A m) \u2192 x \u00b7 y \u2261 y \u00b7 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u00b7-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule SimpleSearch {a} {A : Set a} (_\u00b7_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u00b7_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u00b7\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module Interchange (\u00b7-interchange : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (x \u00b7 z) \u00b7 (y \u00b7 t)) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u00b7 f\u2081 x) \u2261 search f\u2080 \u00b7 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u00b7-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module Interchange; search-const\u03b5\u2261\u03b5)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open Interchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-+ to sum-+;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b51a3256a3c1252ef33def3ad7127f444c50c44e","subject":"Fix \u0394-Context.","message":"Fix \u0394-Context.\n\nOld-commit-hash: ce2c7324266f9f7511dd78f63accb2c3ae8c54fb\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","old_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70bdc97fcd407dacc27f9e32beabb2dadfaa82ba","subject":"otp-sem-sec.agda","message":"otp-sem-sec.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec\nopen import Data.Product\nopen import circuit\nopen import Relation.Binary.PropositionalEquality\n\n-- dup from Game\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nCoins = \u2115\nPorts = \u2115\n\nmodule F'\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n\n  where\n\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n  open Power public\n\n  M = Bits |M|\n  C = Bits |C|\n\n  \u21ba : Coins \u2192 Set \u2192 Set\n  \u21ba c A = Bits c \u2192 A\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c)\n    field\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n      beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n  runSemSec E {mk c} A b R\n      = case SemSecAdv.beh A R\u2080 of \u03bb\n          { (m0m1 , kont) \u2192 b == kont (EXP b E m0m1) R\u2081 }\n    where R\u2080 = take (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n          R\u2081 = drop (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n\n  -- runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n\n  negligible-advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  negligible-advantage EXP = negligible (dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611])\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible-advantage (runSemSec E A)\n\n  |K| = |M|\n  K = Bits |K|\n  -- OTP : K \u2192 M \u2192 C\n  -- OTP = _\u2295_\n\n  -- OTP-SemSec : SemSec (OTP \n\n  Enc = M \u2192 C\n\n  Tr = Enc \u2192 Enc\n\n  -- In general the power might change\n  SemSecTr : Tr \u2192 Set\n  SemSecTr tr = \u2200 {E p} \u2192 SemSec (tr E) p \u2192 SemSec E p\n\n  neg-pres-sem-sec : SemSecTr (_\u2218_ vnot)\n  neg-pres-sem-sec {E} {p} E'-sec A = A-breaks-E\n     where E' : Enc\n           E' = vnot \u2218 E\n           open SemSecAdv A using (c\u2261c\u2080+c\u2081 ; c\u2080 ; c\u2081) renaming (beh to A-beh)\n           A'-beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n           A'-beh = {!!}\n           A' : SemSecAdv p\n           A' = {!!}\n           A'-breaks-E' : negligible-advantage (runSemSec E' A')\n           A'-breaks-E' = E'-sec A'\n           A-breaks-E : negligible-advantage (runSemSec E A)\n           A-breaks-E = {!!}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecTr (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec mask {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = (_\u2295_ mask) \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n-}\n\nmodule F\n  (Size Time : Set)\n  (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  (Cp : Ports \u2192 Ports \u2192 Set)\n  (builder : CircuitBuilder Cp)\n  (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n  (runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o)\n\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n\n  where\n\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n      size  : Size\n      time  : Time\n  open Power public\n\n  M = Bits |M|\n  C = Bits |C|\n\n  \u21ba : Coins \u2192 Set \u2192 Set\n  \u21ba c A = Bits c \u2192 A\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n\n  runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n  runSemSec E {mk c s t} A b R\n      = case SemSecAdv.beh A R\u2080 of \u03bb\n          { (m0m1 , kont) \u2192 {- b == -} kont (EXP b E m0m1) R\u2081 }\n    where R\u2080 = take (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n          R\u2081 = drop (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n\n  -- runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n\n  negligible-advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  negligible-advantage EXP = negligible (dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611])\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible-advantage (runSemSec E A)\n\n  |K| = |M|\n  K = Bits |K|\n  -- OTP : K \u2192 M \u2192 C\n  -- OTP = _\u2295_\n\n  -- OTP-SemSec : SemSec (OTP \n\n  Enc = M \u2192 C\n\n  Tr = Enc \u2192 Enc\n\n  -- In general the power might change\n  SemSecTr : Tr \u2192 Set\n  SemSecTr tr = \u2200 {E p} \u2192 SemSec (tr E) p \u2192 SemSec E p\n\n  neg-pres-sem-sec : SemSecTr (_\u2218_ vnot)\n  neg-pres-sem-sec {E} {p} E'-sec A = A-breaks-E\n     where E' : Enc\n           E' = vnot \u2218 E\n           open SemSecAdv A using (c\u2261c\u2080+c\u2081 ; c\u2080 ; c\u2081) renaming (beh to A-beh)\n           A'-beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n           A'-beh = {!!}\n           A' : SemSecAdv p\n           A' = {!!}\n           A'-breaks-E' : negligible-advantage (runSemSec E' A')\n           A'-breaks-E' = E'-sec A'\n           A-breaks-E : negligible-advantage (runSemSec E A)\n           A-breaks-E = {!!}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecTr (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec mask {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = (_\u2295_ mask) \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n-}\n","old_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec\nopen import Data.Product\nopen import circuit\nopen import Relation.Binary.PropositionalEquality\n\n-- dup from Game\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nCoins = \u2115\nPorts = \u2115\n\nmodule F\n  (Size Time : Set)\n  (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  (Cp : Ports \u2192 Ports \u2192 Set)\n  (builder : CircuitBuilder Cp)\n  (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n  (runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o)\n\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n\n  where\n\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n      size  : Size\n      time  : Time\n  open Power public\n\n  M = Bits |M|\n  C = Bits |C|\n\n  \u21ba : Coins \u2192 Set \u2192 Set\n  \u21ba c A = Bits c \u2192 A\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n\n  runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n  runSemSec E {mk c s t} A b R\n      = case SemSecAdv.beh A R\u2080 of \u03bb\n          { (m0m1 , kont) \u2192 {- b == -} kont (EXP b E m0m1) R\u2081 }\n    where R\u2080 = take (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n          R\u2081 = drop (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n\n  -- runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n\n  negligible-advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  negligible-advantage EXP = negligible (dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611])\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible-advantage (runSemSec E A)\n\n  |K| = |M|\n  K = Bits |K|\n  -- OTP : K \u2192 M \u2192 C\n  -- OTP = _\u2295_\n\n  -- OTP-SemSec : SemSec (OTP \n\n  Enc = M \u2192 C\n\n  Tr = Enc \u2192 Enc\n\n  -- In general the power might change\n  SemSecTr : Tr \u2192 Set\n  SemSecTr tr = \u2200 {E p} \u2192 SemSec (tr E) p \u2192 SemSec E p\n\n  neg-pres-sem-sec : SemSecTr (_\u2218_ vnot)\n  neg-pres-sem-sec {E} {p} E'-sec A = A-breaks-E\n     where E' : Enc\n           E' = vnot \u2218 E\n           open SemSecAdv A using (c\u2261c\u2080+c\u2081 ; c\u2080 ; c\u2081) renaming (beh to A-beh)\n           A'-beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n           A'-beh = {!!}\n           A' : SemSecAdv p\n           A' = {!!}\n           A'-breaks-E' : negligible-advantage (runSemSec E' A')\n           A'-breaks-E' = E'-sec A'\n           A-breaks-E : negligible-advantage (runSemSec E A)\n           A-breaks-E = ?\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecTr (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec mask {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = (_\u2295_ mask) \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b3368bf58769715c5c6847366026d4abb50940fc","subject":"Nat: + <=.isTotalOrder","message":"Nat: + <=.isTotalOrder\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c1ff9f6bce16be7d52bc424c1c012656dec37587","subject":"Implement apply-env and diff-env.","message":"Implement apply-env and diff-env.\n\nOld-commit-hash: 0a4d0485631257cbfb9ee522e8ac6facb88f0721\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : (\u03b9 : Base) \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 (v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 (u v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : (\u03c4 : Type) \u2192 \u27e6 \u03c4 \u27e7 \u2192 Set\n\n  nil-change : \u2200 \u03c4 v \u2192 Change \u03c4 v\n  apply-change : \u2200 \u03c4 \u2192 (v : \u27e6 \u03c4 \u27e7) (dv : Change \u03c4 v) \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) v = Change-base \u03b9 v\n  Change (\u03c3 \u21d2 \u03c4) f = Pair\n    (\u2200 v \u2192 Change \u03c3 v \u2192 Change \u03c4 (f v))\n    (\u03bb \u0394f \u2192 \u2200 v dv \u2192\n      f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e \u0394f (v \u229e\u208d \u03c3 \u208e dv) (nil-change \u03c3 (v \u229e\u208d \u03c3 \u208e dv)) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v dv)\n\n  open Pair public using () renaming\n    ( cdr to is-valid\n    ; car to call-change\n    )\n\n  nil-change \u03c4 v = diff-change \u03c4 v v\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  apply-change (base \u03b9) n \u0394n = apply-change-base \u03b9 n \u0394n\n  apply-change (\u03c3 \u21d2 \u03c4) f \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e call-change \u0394f v (nil-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  diff-change (base \u03b9) m n = diff-change-base \u03b9 m n\n  diff-change (\u03c3 \u21d2 \u03c4) g f = cons (\u03bb v dv \u2192 g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n    (\u03bb v dv \u2192\n      begin\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g ((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g \u25a1 \u229f\u208d \u03c4 \u208e (f (v \u229e\u208d \u03c3 \u208e dv))))\n              (v+[u-v]=u {\u03c3} {v \u229e\u208d \u03c3 \u208e dv} {v \u229e\u208d \u03c3 \u208e dv}) \u27e9\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f (v \u229e\u208d \u03c3 \u208e dv)} \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8  sym (v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f v} )  \u27e9\n        f v \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n      \u220e) where open \u2261-Reasoning\n\n  -- call this lemma \"replace\"?\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (apply-change (\u03c3 \u21d2 \u03c4) v (diff-change (\u03c3 \u21d2 \u03c4) u v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- abbrevitations\n  before : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  before {\u03c4} {v} _ = v\n\n  after : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  after {\u03c4} {v} dv = v \u229e\u208d \u03c4 \u208e dv\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  data \u0394Env : \u2200 (\u0393 : Context) \u2192 \u27e6 \u0393 \u27e7 \u2192 Set where\n    \u2205 : \u0394Env \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1} \u2192\n      (dv : Change \u03c4 v) \u2192\n      (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n      \u0394Env (\u03c4 \u2022 \u0393) (v \u2022 \u03c1)\n\n  ignore : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  ignore {\u0393} {\u03c1} _ = \u03c1\n\n  update : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  update \u2205 = \u2205\n  update {\u03c4 \u2022 \u0393} (dv \u2022 d\u03c1) = after {\u03c4} dv \u2022 update d\u03c1\n\n  apply-env : \u2200 \u0393 \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  apply-env \u0393 \u03c1 d\u03c1 = update {\u0393} d\u03c1\n\n  diff-env : \u2200 \u0393 \u2192 (\u03c0 \u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394Env \u0393 \u03c1\n  diff-env \u2205 \u2205 \u2205 = \u2205\n  diff-env (\u03c4 \u2022 \u0393) (u \u2022 \u03c1) (v \u2022 \u03c0) = diff-change \u03c4 u v \u2022 diff-env \u0393 \u03c1 \u03c0\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : (\u03b9 : Base) \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 (v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 (u v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : (\u03c4 : Type) \u2192 \u27e6 \u03c4 \u27e7 \u2192 Set\n\n  nil-change : \u2200 \u03c4 v \u2192 Change \u03c4 v\n  apply-change : \u2200 \u03c4 \u2192 (v : \u27e6 \u03c4 \u27e7) (dv : Change \u03c4 v) \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) v = Change-base \u03b9 v\n  Change (\u03c3 \u21d2 \u03c4) f = Pair\n    (\u2200 v \u2192 Change \u03c3 v \u2192 Change \u03c4 (f v))\n    (\u03bb \u0394f \u2192 \u2200 v dv \u2192\n      f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e \u0394f (v \u229e\u208d \u03c3 \u208e dv) (nil-change \u03c3 (v \u229e\u208d \u03c3 \u208e dv)) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v dv)\n\n  open Pair public using () renaming\n    ( cdr to is-valid\n    ; car to call-change\n    )\n\n  nil-change \u03c4 v = diff-change \u03c4 v v\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  apply-change (base \u03b9) n \u0394n = apply-change-base \u03b9 n \u0394n\n  apply-change (\u03c3 \u21d2 \u03c4) f \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e call-change \u0394f v (nil-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  diff-change (base \u03b9) m n = diff-change-base \u03b9 m n\n  diff-change (\u03c3 \u21d2 \u03c4) g f = cons (\u03bb v dv \u2192 g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n    (\u03bb v dv \u2192\n      begin\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g ((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g \u25a1 \u229f\u208d \u03c4 \u208e (f (v \u229e\u208d \u03c3 \u208e dv))))\n              (v+[u-v]=u {\u03c3} {v \u229e\u208d \u03c3 \u208e dv} {v \u229e\u208d \u03c3 \u208e dv}) \u27e9\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f (v \u229e\u208d \u03c3 \u208e dv)} \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8  sym (v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f v} )  \u27e9\n        f v \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n      \u220e) where open \u2261-Reasoning\n\n  -- call this lemma \"replace\"?\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (apply-change (\u03c3 \u21d2 \u03c4) v (diff-change (\u03c3 \u21d2 \u03c4) u v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- abbrevitations\n  before : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  before {\u03c4} {v} _ = v\n\n  after : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  after {\u03c4} {v} dv = v \u229e\u208d \u03c4 \u208e dv\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  data \u0394Env : \u2200 (\u0393 : Context) \u2192 \u27e6 \u0393 \u27e7 \u2192 Set where\n    \u2205 : \u0394Env \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1} \u2192\n      (dv : Change \u03c4 v) \u2192\n      (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n      \u0394Env (\u03c4 \u2022 \u0393) (v \u2022 \u03c1)\n\n  ignore : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  ignore {\u0393} {\u03c1} _ = \u03c1\n\n  update : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  update \u2205 = \u2205\n  update {\u03c4 \u2022 \u0393} (dv \u2022 d\u03c1) = after {\u03c4} dv \u2022 update d\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c6af6a479f8ea7cd6a2571bf4479cd29f94d9946","subject":"Stratified Desc model: document.","message":"Stratified Desc model: document.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : A -> Set m}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\n-- In the paper, we have presented Desc as the grammar of inductive\n-- types. Hence, the codes in the paper closely follow this\n-- grammar:\n\ndata DescPaper : Set1 where\n  oneP : DescPaper\n  sigmaP : (S : Set) -> (S -> DescPaper) -> DescPaper\n  indx : DescPaper -> DescPaper\n  hindx : Set -> DescPaper -> DescPaper\n\n-- We take advantage of this model to give you an alternative\n-- presentation. This alternative model is the one implemented in\n-- Epigram. It is also the one which inspired the code for indexed\n-- descriptions.\n\n-- With sigma, we are actually \"quoting\" a standard type-former,\n-- namely:\n--     |Sigma : (S : Set) -> (S -> Set) -> Set|\n-- With:\n--     |sigma : (S : Set) -> (S -> Desc) -> Desc|\n\n-- In the alternative presentation, we go further and present all our\n-- codes as quotations of standard type-formers:\n\ndata Desc {l : Level} : Set (suc l) where\n  id : Desc\n  const : Set l -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set l) -> (S -> Desc) -> Desc\n  pi : (S : Set l) -> (S -> Desc) -> Desc\n\n-- Note that we replace |oneP| by a more general |const| code. Whereas\n-- |oneP| was interpreted as the unit set, |const K| is\n-- interpreted as |K|, for any |K : Set|. Extensionally,\n-- |const K| and |sigma K (\\_ -> Unit)| are equivalent. However,\n-- |const| is *first-order*, unlike its equivalent encoding. From a\n-- definitional perspective, we are giving more opportunities to the\n-- type-system, hence reducing the burden on the programmer. For the same\n-- reason, we introduce |prod| that overlaps with |pi|.\n\n-- This reorganisation is strictly equivalent to the |DescPaper|. For\n-- instance, we can encode |indx| and |hindx| using the following\n-- code:\n\nindx2 : {l : Level} -> Desc {l = l} -> Desc {l = l}\nindx2 D = prod id D\n\nhindx2 : Set -> Desc -> Desc\nhindx2 H D = prod (pi H (\\_ -> id)) D\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : {l : Level} -> Desc -> Set l -> Set l\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu {l : Level}(D : Desc {l = l}) : Set l where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {l : Level}(D : Desc)(X : Set)(P : X -> Set l) -> [| D |] X -> Set l\nAll id          X P x        = P x\nAll (const Z)   X P x        = Unit\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\n\nall : {l : Level}(D : Desc)(X : Set)(P : X -> Set l)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = Void\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\n-- This one is bonus: one could rightfully expect our so-called\n-- functors to have a morphism part! Here it is.\n\nmap : {l : Level}(D : Desc)(X Y : Set l)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n-- Together with the proof that they respect the functor laws:\n\n-- map id = id\nproof-map-id : {l : Level}(D : Desc)(X : Set l)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n-- map (f . g) = map f . map g\nproof-map-compos : {l : Level}(D : Desc)(X Y Z : Set l)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nind : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\nind D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> ind D P ms x) xs)\n-}\n\n-- But the termination checker is unhappy.\n-- So we write the following:\n\nmodule Elim {l : Level}\n            (D : Desc)\n            (P : Mu D -> Set l)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        ind : (x : Mu D) -> P x\n        ind (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = ind x\n        hyps (const Z) z = Void\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \nind : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\nind D P ms x = Elim.ind D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst  natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsu : Nat -> Nat\nsu n = con (Su , n)\n\n-- Now we can get addition for example:\n\nplusCase : (xs : [| NatD |] Nat) ->\n           All NatD Nat (\\_ -> Nat -> Nat) xs -> Nat -> Nat\nplusCase ( Ze , Void ) hs y = y\nplusCase ( Su , n ) hs y = su (hs y)\n\nplus : Nat -> Nat -> Nat\nplus x = ind NatD (\\ _ -> (Nat -> Nat)) plusCase x\n\n-- Do this thing in Epigram, you will see that this is *not*\n-- hieroglyphic with a bit of elaboration.\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\n-- If we weren't such big fans of levitating things, we would\n-- implement finite sets with:\n\n{-\n\ndata En : Set where\n  nE : En\n  cE : En -> En\n\nspi : (e : En)(P : EnumT e -> Set) -> Set\nspi nE P = Unit\nspi (cE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : En)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nE P b ()\nswitch (cE e) P b EZe = fst b\nswitch (cE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n-- But no, we make it fly in Desc:\n\n--****************\n-- En\n--****************\n\n-- As we have no tags here, we use Nat instead of List. \n\nEnD : Desc\nEnD = NatD\n\nEn : Set\nEn = Nat\n\nnE : En\nnE = ze\n\ncE : En -> En\ncE e = su e \n\n--****************\n-- EnumT\n--****************\n\n-- Because I don't want to fall back on wacky unicode symbols, I will\n-- write EnumT for #, EZe for 0, and ESu for 1+. Sorry about that\n\ndata EnumT : (e : En) -> Set where\n  EZe : {e : En} -> EnumT (cE e)\n  ESu : {e : En} -> EnumT e -> EnumT (cE e)\n\n--****************\n-- Small Pi\n--****************\n\n-- This corresponds to the small pi |\\pi|.\n\ncasesSpi : {l : Level}(xs : [| EnD |] En) -> \n           All EnD En (\\e -> (EnumT e -> Set l) -> Set l) xs -> \n           (EnumT (con xs) -> Set l) -> Set l\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : {l : Level}(e : En)(P : EnumT e -> Set l) -> Set l\nspi {x} e P =  ind EnD (\\E -> (EnumT E -> Set x) -> Set x) casesSpi e P\n\n--****************\n-- Switch\n--****************\n\ncasesSwitch : {l : Level}\n              (xs : [| EnD |] En) ->\n              All EnD En (\\e -> (P' : EnumT e -> Set l)\n                                (b' : spi e P')\n                                (x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set l)\n              (b' : spi (con xs) P')\n              (x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Su , n) hs P' b' EZe = fst b'\ncasesSwitch (Su , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\n\nswitch : {l : Level}\n         (e : En)\n         (P : EnumT e -> Set l)\n         (b : spi e P)\n         (x : EnumT e) -> P x\nswitch {x} e P b xs =  ind EnD\n                           (\\e -> (P : EnumT e -> Set x)\n                                  (b : spi e P)\n                                  (xs : EnumT e) -> P xs) \n                           casesSwitch e P b xs \n\n--****************\n-- Desc\n--****************\n\n-- In the following, we implement Desc in itself. As usual, we have a\n-- finite set of constructors -- the name of the codes. Note that we\n-- could really define these as a finite set built above. However, in\n-- Agda, it's horribly verbose. For the sake of clarity, we won't do\n-- that here. \n\ndata DescDef : Set1 where\n  DescId : DescDef\n  DescConst : DescDef\n  DescProd : DescDef\n  DescSigma : DescDef\n  DescPi : DescDef\n\n-- We slightly diverge here from the presentation of the paper: note\n-- the presence of terminating \"const Unit\". Recall our Lisp-ish\n-- notation for nested tuples:\n--    |[a b c]| \n-- Corresponds to \n--    |[a , [ b , [c , []]]]|\n-- So, if we want to write constructors using our Lisp-ish notation, the interpretation \n-- [| DescD |] (Mu DescD) have to evaluates to [ constructor , [ arg1 , [ arg2 , []]]]\n\n-- Hence, we define Desc's code as follow:\n\ndescCases : DescDef -> Desc\ndescCases DescId = const Unit\ndescCases DescConst = sigma Set (\\_ -> const Unit)\ndescCases DescProd = prod id (prod id (const Unit))\ndescCases DescSigma = sigma Set (\\S -> prod (pi (lift S) (\\_ -> id)) (const Unit))\ndescCases DescPi = sigma Set (\\S -> prod (pi (lift S) (\\_ -> id)) (const Unit))\n\nDescD : Desc\nDescD = sigma DescDef descCases\n\nDescIn : Set1\nDescIn = Mu DescD\n\n-- So that the constructors are:\n-- (Note the annoying |pair|s to set the implicit levels. I could not\n--  get rid of the yellow otherwise)\n\nidIn : DescIn\nidIn = con (pair {i = suc zero} {j =  suc zero} DescId Void)\nconstIn : Set -> DescIn\nconstIn K = con (pair {i = suc zero} {j = suc zero} DescConst (K , Void))\nprodIn : (D D' : DescIn) -> DescIn\nprodIn D D' = con (pair {i = suc zero} {j = suc zero} DescProd (D , ( D' , Void )))\nsigmaIn : (S : Set)(D : S -> DescIn) -> DescIn\nsigmaIn S D = con (pair {i = suc zero} {j = suc zero} DescSigma (S , ((\\s -> D (unlift s)) , Void )))\npiIn : (S : Set)(D : S -> DescIn) -> DescIn\npiIn S D = con (pair {i = suc zero} {j = suc zero} DescPi (S , ((\\s -> D (unlift s)) , Void )))\n\n-- At this stage, we could prove the isomorphism between |DescIn| and\n-- |Desc|. While not technically difficult, it is long and\n-- laborious. We have carried this proof on the more complex and\n-- interesting |IDesc| universe, in IDesc.agda.\n\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : {l : Level} -> Set (suc l)\nTagDesc = Sigma En (\\e -> spi e (\\_ -> Desc))\n\nde : TagDesc -> Desc\nde (B , F) = sigma (EnumT B) (\\E -> switch B (\\_ -> Desc) F E)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = ind D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = cE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (de (D ** Y))) ->\n        [| de (D ** X) |] (Mu (de (D ** Y))) ->\n        Mu (de (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (de (D ** X)) ->\n        (X -> Mu (de (D ** Y))) ->\n        Mu (de (D ** Y))\nsubst D X Y x sig = cata (de (D ** X)) (Mu (de (D ** Y))) (apply D X Y sig) x\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : A -> Set m}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc {l : Level} : Set (suc l) where\n  id : Desc\n  const : Set l -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set l) -> (S -> Desc) -> Desc\n  pi : (S : Set l) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : {l : Level} -> Desc -> Set l -> Set l\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu {l : Level}(D : Desc {l = l}) : Set l where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {l : Level}(D : Desc)(X : Set)(P : X -> Set l) -> [| D |] X -> Set l\nAll id          X P x        = P x\nAll (const Z)   X P x        = Unit\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\n\nall : {l : Level}(D : Desc)(X : Set)(P : X -> Set l)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = Void\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : {l : Level}(D : Desc)(X Y : Set l)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : {l : Level}(D : Desc)(X : Set l)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\nproof-map-compos : {l : Level}(D : Desc)(X Y Z : Set l)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\n-- But the termination checker is unhappy.\n-- So we write the following:\n\nmodule Elim {l : Level}\n            (D : Desc)\n            (P : Mu D -> Set l)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = Void\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst  natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsu : Nat -> Nat\nsu n = con (Su , n)\n\nplusCase : (xs : [| NatD |] Nat) ->\n           All NatD Nat (\\_ -> Nat -> Nat) xs -> Nat -> Nat\nplusCase ( Ze , Void ) hs y = y\nplusCase ( Su , n ) hs y = su (hs y)\n\nplus : Nat -> Nat -> Nat\nplus x = induction NatD (\\ _ -> (Nat -> Nat)) plusCase x\n\n-- Do this thing in Epigram, you will see that this is *not*\n-- hieroglyphic with a bit of elaboration.\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\n-- If we weren't such big fans of levitating things, we would\n-- implement finite sets with:\n\n{-\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n-- But no, we make it fly in Desc:\n\n--****************\n-- EnumU\n--****************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = su e \n\n--****************\n-- EnumT\n--****************\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n--****************\n-- Small Pi\n--****************\n\ncasesSpi : {l : Level}(xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (EnumT e -> Set l) -> Set l) xs -> \n           (EnumT (con xs) -> Set l) -> Set l\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : {l : Level}(e : EnumU)(P : EnumT e -> Set l) -> Set l\nspi {x} e P =  induction NatD (\\E -> (EnumT E -> Set x) -> Set x) casesSpi e P\n\n--****************\n-- Switch\n--****************\n\ncasesSwitch : {l : Level}\n              (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set l)\n                                  (b' : spi e P')\n                                  (x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set l)\n              (b' : spi (con xs) P')\n              (x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Su , n) hs P' b' EZe = fst b'\ncasesSwitch (Su , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\n\nswitch : {l : Level}\n         (e : EnumU)\n         (P : EnumT e -> Set l)\n         (b : spi e P)\n         (x : EnumT e) -> P x\nswitch {x} e P b xs =  induction NatD\n                                 (\\e -> (P : EnumT e -> Set x)\n                                        (b : spi e P)\n                                        (xs : EnumT e) -> P xs) \n                                 casesSwitch e P b xs \n\n--****************\n-- Desc\n--****************\n\n-- TODO: explain that if it weren't so verbose\n--       we could use finite sets instead of DescDef\n\ndata DescDef : Set1 where\n  DescId : DescDef\n  DescConst : DescDef\n  DescProd : DescDef\n  DescSigma : DescDef\n  DescPi : DescDef\n\n-- TODO: explain the Units\n\ndescCases : DescDef -> Desc\ndescCases DescId = const Unit\ndescCases DescConst = sigma Set (\\_ -> const Unit)\ndescCases DescProd = prod id (prod id (const Unit))\ndescCases DescSigma = sigma Set (\\S -> prod (pi (lift S) (\\_ -> id)) (const Unit))\ndescCases DescPi = sigma Set (\\S -> prod (pi (lift S) (\\_ -> id)) (const Unit))\n\nDescD : Desc\nDescD = sigma DescDef descCases\n\nDescIn : Set1\nDescIn = Mu DescD\n\n-- TODO: Explain that we can prove iso between Desc and DescIn. \n--       Report to IDesc.\n\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : {l : Level} -> Set (suc l)\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\E -> switch B (\\_ -> Desc) F E)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d3a7e936c967d5c8f0ec2392f3f73ce9c739b925","subject":"agda : completed conor Ex1","message":"agda : completed conor Ex1\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/course\/2017-conor_mcbride_cs410\/Ex1-HC.agda","new_file":"agda\/course\/2017-conor_mcbride_cs410\/Ex1-HC.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n------------------------------------------------------------------------------\n------------------------------------------------------------------------------\n-- CS410 2017\/18 Exercise 1  VECTORS AND FRIENDS (worth 25%)\n------------------------------------------------------------------------------\n------------------------------------------------------------------------------\n\n-- NOTE (19\/9\/17) This file is currently incomplete: more will arrive on\n-- GitHub.\n\n-- MARK SCHEME (transcribed from paper): the (m) numbers add up to slightly\n-- more than 25, so should be taken as the maximum number of marks losable on\n-- the exercise. In fact, I did mark it negatively, but mostly because it was\n-- done so well (with Agda's help) that it was easier to find the errors.\n\n\n------------------------------------------------------------------------------\n-- Dependencies\n------------------------------------------------------------------------------\n\nopen import CS410-Prelude\n\ncong : \u2200 {A B : Set} {x y : A}\n     \u2192 (f : A \u2192 B)\n     \u2192   x ==   y\n       ----------\n     \u2192 f x == f y\ncong f (refl x) =  refl (f x)\n\n------------------------------------------------------------------------------\n-- Vectors\n------------------------------------------------------------------------------\n\ndata Vec (X : Set) : Nat -> Set where  -- like lists, but length-indexed\n  []   :                              Vec X zero\n  _,-_ : {n : Nat} -> X -> Vec X n -> Vec X (suc n)\ninfixr 4 _,-_   -- the \"cons\" operator associates to the right\n\n-- I like to use the asymmetric ,- to remind myself that the element is to\n-- the left and the rest of the list is to the right.\n\n-- Vectors are useful when there are important length-related safety\n-- properties.\n\n------------------------------------------------------------------------------\n-- Heads and Tails\n------------------------------------------------------------------------------\n\n-- We can rule out nasty head and tail errors by insisting on nonemptiness!\n\n--??--1.1-(2)-----------------------------------------------------------------\n\nvHead : {X : Set}{n : Nat} -> Vec X (suc n) -> X\nvHead (x ,- _) = x\n\nvTail : {X : Set}{n : Nat} -> Vec X (suc n) -> Vec X n\nvTail (_ ,- xs) = xs\n\nvHeadTailFact :  {X : Set}{n : Nat}(xs : Vec X (suc n)) ->\n                 (vHead xs ,- vTail xs) == xs\nvHeadTailFact (x ,- xs) = refl (x ,- xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Concatenation and its Inverse\n------------------------------------------------------------------------------\n\n--??--1.2-(2)-----------------------------------------------------------------\n\n_+V_ : {X : Set}{m n : Nat} -> Vec X m -> Vec X n -> Vec X (m +N n)\n[]        +V ys = ys\n(x ,- xs) +V ys = x ,- xs +V ys\ninfixr 4 _+V_\n\nvChop : {X : Set}(m : Nat){n : Nat} -> Vec X (m +N n) -> Vec X m * Vec X n\nvChop  zero         xs = [] , xs\nvChop (suc m) (x ,- xs)\n  with vChop m xs\n... | vm , vn\n  = (x ,- vm) , vn\n\nvChopAppendFact : {X : Set}{m n : Nat}(xs : Vec X m)(ys : Vec X n) ->\n                  vChop m (xs +V ys) == (xs , ys)\nvChopAppendFact       []  ys = refl ([] , ys)\nvChopAppendFact (x ,- xs) ys rewrite vChopAppendFact xs ys = refl ((x ,- xs) , ys)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Map, take I\n------------------------------------------------------------------------------\n\n-- Implement the higher-order function that takes an operation on\n-- elements and does it to each element of a vector. Use recursion\n-- on the vector.\n-- Note that the type tells you the size remains the same.\n\n-- Show that if the elementwise function \"does nothing\", neither does\n-- its vMap. \"map of identity is identity\"\n\n-- Show that two vMaps in a row can be collapsed to just one, or\n-- \"composition of maps is map of compositions\"\n\n--??--1.3-(2)-----------------------------------------------------------------\n\nvMap : {X Y : Set} -> (X -> Y) -> {n : Nat} -> Vec X n -> Vec Y n\nvMap f       []  = []\nvMap f (x ,- xs) = f x ,- vMap f xs\n\nvMapIdFact : {X : Set}{f : X -> X}(feq : (x : X) -> f x == x) ->\n             {n : Nat}(xs : Vec X n) -> vMap f xs == xs\nvMapIdFact feq       []  = refl []\nvMapIdFact feq (x ,- xs)\n  rewrite vMapIdFact feq xs\n  |       feq x\n  = refl (x ,- xs)\n\nvMapCpFact : {X Y Z : Set}{f : Y -> Z}{g : X -> Y}{h : X -> Z}\n               (heq : (x : X) -> f (g x) == h x) ->\n             {n : Nat}(xs : Vec X n) ->\n               vMap f (vMap g xs) == vMap h xs\nvMapCpFact heq       []  = refl []\nvMapCpFact {_}{_}{_}{f} {g} {h} heq (x ,- xs)\n  rewrite heq x\n  |       vMapCpFact {_}{_}{_}{f} {g} {h} heq xs\n  = refl (h x ,- vMap h xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- vMap and +V\n------------------------------------------------------------------------------\n\n-- Show that if you've got two vectors of Xs and a function from X to Y,\n-- and you want to concatenate and map, it doesn't matter which you do\n-- first.\n\n--??--1.4-(1)-----------------------------------------------------------------\n\nvMap+VFact : {X Y : Set}(f : X -> Y) ->\n             {m n : Nat}(xs : Vec X m)(xs' : Vec X n) ->\n             vMap f (xs +V xs') == (vMap f xs +V vMap f xs')\nvMap+VFact f       []  xs' = refl (vMap f xs')\nvMap+VFact f (x ,- xs) xs'\n  rewrite vMap+VFact f xs xs'\n  = refl (f x ,- vMap f xs +V vMap f xs')\n\n--??--------------------------------------------------------------------------\n\n-- Think about what you could prove, relating vMap with vHead, vTail, vChop...\n-- Now google \"Philip Wadler\" \"Theorems for Free\"\n\n-- TODO\n\n------------------------------------------------------------------------------\n-- Applicative Structure (giving mapping and zipping cheaply)\n------------------------------------------------------------------------------\n\n--??--1.5-(2)-----------------------------------------------------------------\n\n-- HINT: you will need to override the default invisibility of n to do this.\n-- HC : replicate\nvPure : {X : Set} -> X -> {n : Nat} -> Vec X n\nvPure x {zero}  = []\nvPure x {suc n} = x ,- vPure x {n}\n\n_$V_ : {X Y : Set}{n : Nat} -> Vec (X -> Y) n -> Vec X n -> Vec Y n\n[]      $V      [] = []\nf ,- fs $V x ,- xs = f x ,- (fs $V xs)\ninfixl 3 _$V_  -- \"Application associates to the left,\n               --  rather as we all did in the sixties.\" (Roger Hindley)\n\n-- Pattern matching and recursion are forbidden for the next two tasks.\n\n-- implement vMap again, but as a one-liner\nvec : {X Y : Set} -> (X -> Y) -> {n : Nat} -> Vec X n -> Vec Y n\nvec f xs = vPure f $V xs\n\n-- implement the operation which pairs up corresponding elements\nvZip : {X Y : Set}{n : Nat} -> Vec X n -> Vec Y n -> Vec (X * Y) n\nvZip xs ys = vec (_,_) xs $V ys\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Applicative Laws\n------------------------------------------------------------------------------\n\n-- According to \"Applicative programming with effects\" by\n--   Conor McBride and Ross Paterson\n-- some laws should hold for applicative functors.\n-- Check that this is the case.\n\n--??--1.6-(2)-----------------------------------------------------------------\n\nvIdentity : {X : Set}{f : X -> X}(feq : (x : X) -> f x == x) ->\n            {n : Nat}(xs : Vec X n) -> (vPure f $V xs) == xs\nvIdentity feq        [] = refl []\nvIdentity feq (x ,- xs) rewrite vIdentity feq xs | feq x = refl (x ,- xs)\n\nvHomomorphism : {X Y : Set}(f : X -> Y)(x : X) ->\n                {n : Nat} -> (vPure f $V vPure x) == vPure (f x) {n}\nvHomomorphism f x {zero}  = refl []\nvHomomorphism f x {suc n} rewrite vHomomorphism f x {n} = refl (f x ,- vPure (f x))\n\nvInterchange : {X Y : Set}{n : Nat}(fs : Vec (X -> Y) n)(x : X) ->\n               (fs $V vPure x) == (vPure (_$ x) $V fs)\nvInterchange []        x = refl []\nvInterchange (f ,- fs) x rewrite vInterchange fs x = refl (f x ,- (vPure (\u03bb x\u2192y \u2192 x\u2192y x) $V fs))\n\nvComposition : {X Y Z : Set}{n : Nat}\n               (fs : Vec (Y -> Z) n)(gs : Vec (X -> Y) n)(xs : Vec X n) ->\n               (vPure _<<_ $V fs $V gs $V xs) == (fs $V (gs $V xs))\nvComposition       []        []        []  = refl []\nvComposition (f ,- fs) (g ,- gs) (x ,- xs)\n  rewrite vComposition fs gs xs\n  = refl (f (g x) ,- (fs $V (gs $V xs)))\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Order-Preserving Embeddings (also known in the business as \"thinnings\")\n------------------------------------------------------------------------------\n\n-- What have these to do with Pascal's Triangle?\n\n-- how to choose N things from M things\ndata _<=_ : Nat -> Nat -> Set where\n  oz :                          zero  <= zero  -- stop\n  os : {n m : Nat} -> n <= m -> suc n <= suc m -- take this one and keep going\n  o' : {n m : Nat} -> n <= m ->     n <= suc m -- skip this one and keep going\n\nrefl-<= : (n : Nat) -> n <= n\nrefl-<= zero    = oz\nrefl-<= (suc n) = os (refl-<= n)\n\ntrans-<= : {n m p : Nat} -> n <= m -> m <= p -> n <= p\ntrans-<=  oz           zero<=p  = zero<=p\ntrans-<= (os n<=m) (os sucm<=p) = os (trans-<=     n<=m  sucm<=p)\ntrans-<= (os n<=m) (o' sucm<=p) = os (trans-<= (o' n<=m) sucm<=p)\ntrans-<= (o' n<=m) (os    m<=p) = o' (trans-<=     n<=m     m<=p)\ntrans-<= (o' n<=m) (o'    m<=p) = o' (trans-<= (o' n<=m)    m<=p)\n\n<=-suc : (x : Nat) -> x <= suc x\n<=-suc  zero   = o' oz\n<=-suc (suc x) = os (<=-suc x)\n\nn<=m\u2192sucn<=sucm : {n m : Nat} \u2192 n <= m -> suc n <= suc m\nn<=m\u2192sucn<=sucm  oz       = os oz\nn<=m\u2192sucn<=sucm (os n<=m) = os (n<=m\u2192sucn<=sucm n<=m)\nn<=m\u2192sucn<=sucm (o' n<=m) = o' (n<=m\u2192sucn<=sucm n<=m)\n\nsucsucn<=m\u2192sucn<=m : {n m : Nat} -> suc (suc n) <= m -> suc n <= m\nsucsucn<=m\u2192sucn<=m (os sucn<=m)  = o' sucn<=m\nsucsucn<=m\u2192sucn<=m (o' (os xxx)) = o' (o' xxx)\nsucsucn<=m\u2192sucn<=m (o' (o' xxx)) = o' (o' (sucsucn<=m\u2192sucn<=m xxx))\n\nsucn<=m\u2192sucn<=sucm : {n m : Nat} -> suc n <= m -> suc n <= suc m\nsucn<=m\u2192sucn<=sucm (os p) = os (o' p)\nsucn<=m\u2192sucn<=sucm (o' p) = o' (sucn<=m\u2192sucn<=sucm p)\n\nzero<=m : {m : Nat} -> zero <= m\nzero<=m  {zero} = oz\nzero<=m {suc m} = o' zero<=m\n\nsucn<=sucm\u2192n<=m : {n m : Nat} -> suc n <= suc m -> n <= m\nsucn<=sucm\u2192n<=m {zero}  {zero}       p  = oz\nsucn<=sucm\u2192n<=m {zero}  {suc m} (os  p) = o' (zero<=m {m})\nsucn<=sucm\u2192n<=m {zero}  {suc m} (o'  p) = o' (zero<=m {m})\nsucn<=sucm\u2192n<=m {suc n} {zero}  (os ())\nsucn<=sucm\u2192n<=m {suc n} {zero}  (o' ())\nsucn<=sucm\u2192n<=m {suc n} {suc m} (os  p) = p\nsucn<=sucm\u2192n<=m {suc n} {suc m} (o'  p) = o' (sucn<=sucm\u2192n<=m p)\n\n-- Find all the values in each of the following <= types.\n-- This is a good opportunity to learn to use C-c C-a with the -l option\n--   (a.k.a. \"google the type\" without \"I feel lucky\")\n-- The -s n option also helps.\n\n--??--1.7-(1)-----------------------------------------------------------------\n\nall0<=4 : Vec (0 <= 4) 1\nall0<=4 = o' (o' (o' (o' oz))) ,- []\n\nall1<=4 : Vec (1 <= 4) 1\nall1<=4 = os (o' (o' (o' oz))) ,- []\n\nall2<=4 : Vec (2 <= 4) 1\nall2<=4 = os (os (o' (o' oz))) ,- []\n\nall3<=4 : Vec (3 <= 4) 1\nall3<=4 = os (os (os (o' oz))) ,- []\n\nall4<=4 : Vec (4 <= 4) 1\nall4<=4 = os (os (os (os oz))) ,- []\n\n-- Prove the following. A massive case analysis \"rant\" is fine.\n\nno5<=4 : 5 <= 4 -> Zero\nno5<=4 (os (os (os (os ()))))\nno5<=4 (os (os (os (o' ()))))\nno5<=4 (os (os (o' (os ()))))\nno5<=4 (os (os (o' (o' ()))))\nno5<=4 (os (o' (os (os ()))))\nno5<=4 (os (o' (os (o' ()))))\nno5<=4 (os (o' (o' (os ()))))\nno5<=4 (os (o' (o' (o' ()))))\nno5<=4 (o' (os (os (os ()))))\nno5<=4 (o' (os (os (o' ()))))\nno5<=4 (o' (os (o' (os ()))))\nno5<=4 (o' (os (o' (o' ()))))\nno5<=4 (o' (o' (os (os ()))))\nno5<=4 (o' (o' (os (o' ()))))\nno5<=4 (o' (o' (o' (os ()))))\nno5<=4 (o' (o' (o' (o' ()))))\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Order-Preserving Embeddings Select From Vectors\n------------------------------------------------------------------------------\n\n-- Use n <= m to encode the choice of n elements from an m-Vector.\n-- The os constructor tells you to take the next element of the vector;\n-- the o' constructor tells you to omit the next element of the vector.\n\n--??--1.8-(2)-----------------------------------------------------------------\n\n_<?=_ : {X : Set}{n m : Nat} -> n <= m -> Vec X m\n                     -> Vec X n\noz    <?=        _  = []\nos th <?= (x ,- xs) = x ,- th <?= xs\no' th <?= (_ ,- xs) =      th <?= xs\n\nvn4 : Vec Nat 4\nvn4 = os (os (os (os oz))) <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ :                     vn4 == (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ =                       refl (1 ,- 2 ,- 3 ,- 4 ,- [])\n\nvn3 : Vec Nat 3\nvn3 = o' (os (o' (os (o' (os oz))))) <?= (1 ,- 2 ,- 3 ,- 4 ,- 5 ,- 6 ,- [])\n_ :                               vn3 ==      (2      ,- 4      ,- 6 ,- [])\n_ =                                      refl (2      ,- 4      ,- 6 ,- [])\n\nvn2 : Vec Nat 2\nvn2 = os (o' (os (o' oz))) <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ :                     vn2 == (1      ,- 3      ,- [])\n_ =                       refl (1      ,- 3      ,- [])\n\nvn2' : Vec Nat 2\nvn2' = o' (os (os (o' oz))) <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ :                          vn2' == (2 ,- 3      ,- [])\n_ =                             refl (2 ,- 3      ,- [])\n\n-- it shouldn't matter whether you map then select or select then map\n\nvMap<?=Fact : {X Y : Set}(f : X -> Y)\n              {n m : Nat}(th : n <= m)(xs : Vec X m) ->\n              vMap f (th <?= xs) == (th <?= vMap f xs)\nvMap<?=Fact f  oz     [] = refl []\nvMap<?=Fact f (os th) (x ,- xs) rewrite vMap<?=Fact f th xs = refl (f x ,- (th <?= vMap f xs))\nvMap<?=Fact f (o' th) (x ,- xs) rewrite vMap<?=Fact f th xs = refl         (th <?= vMap f xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Our Favourite Thinnings\n------------------------------------------------------------------------------\n\n-- Construct the identity thinning and the empty thinning.\n\n--??--1.9-(1)-----------------------------------------------------------------\n\noi : {n : Nat} -> n <= n\noi  {zero} = oz\noi {suc n} = os oi\n\noe : {n : Nat} -> 0 <= n\noe  {zero} = oz\noe {suc n} = o' oe\n\nvnoi : Vec Nat 4\nvnoi = oi  <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_    : vnoi == (1 ,- 2 ,- 3 ,- 4 ,- [])\n_    =    refl (1 ,- 2 ,- 3 ,- 4 ,- [])\n\nvnoe : Vec Nat 0\nvnoe = oe <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_    :                     vnoe == []\n_    =                        refl []\n\n\n--??--------------------------------------------------------------------------\n\n\n-- Show that all empty thinnings are equal to yours.\n\n--??--1.10-(1)----------------------------------------------------------------\n\noeUnique : {n : Nat}(th : 0 <= n) -> th == oe\noeUnique  oz                       = refl  oz\noeUnique (o' i) rewrite oeUnique i = refl (o' oe)\n\n--??--------------------------------------------------------------------------\n\n\n-- Show that there are no thinnings of form big <= small  (TRICKY)\n-- Then show that all the identity thinnings are equal to yours.\n-- Note that you can try the second even if you haven't finished the first.\n-- HINT: you WILL need to expose the invisible numbers.\n-- HINT: check CS410-Prelude for a reminder of >=\n\n--??--1.11-(3)----------------------------------------------------------------\n\noTooBig : {n m : Nat} -> n >= m -> suc n <= m -> Zero\noTooBig  {zero}  {zero} n>=m ()\noTooBig  {zero} {suc m} n>=m (os th) = n>=m\noTooBig  {zero} {suc m} n>=m (o' th) = n>=m\noTooBig {suc n} {suc m} n>=m (os th) = oTooBig n>=m th\noTooBig {suc n} {suc m} n>=m (o' th) = oTooBig {n} {m} n>=m (sucsucn<=m\u2192sucn<=m th)\n\noiUnique : {n : Nat}(n<=n : n <= n) -> n<=n == oi\noiUnique          oz                             = refl oz\noiUnique         (os n<=n) rewrite oiUnique n<=n = refl (os oi)\noiUnique {suc n} (o' sucn<=n)\n  with oTooBig (refl->= n) sucn<=n\n... | ()\n\n--??--------------------------------------------------------------------------\n\n\n-- Show that the identity thinning selects the whole vector\n\n--??--1.12-(1)----------------------------------------------------------------\n\nid-<?= : {X : Set}{n : Nat}(xs : Vec X n) -> (oi <?= xs) == xs\nid-<?=       []  = refl []\nid-<?= (x ,- xs) = cong (x ,-_) (id-<?= xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Composition of Thinnings\n------------------------------------------------------------------------------\n\n-- Define the composition of thinnings and show that selecting by a\n-- composite thinning is like selecting then selecting again.\n-- A small bonus applies to minimizing the length of the proof.\n-- To collect the bonus, you will need to think carefully about\n-- how to make the composition as *lazy* as possible.\n\n--??--1.13-(3)----------------------------------------------------------------\n\n-- my 1st attempt - but everything gets stuff because pattern matching on right instead of left\n_o>>1_ : {p n m : Nat} -> p <= n -> n <= m -> p <= m\np<=zero o>>1 oz      = p<=zero\np<=sucn o>>1 os n<=m = trans-<= p<=sucn (n<=m\u2192sucn<=sucm n<=m)\np<=n    o>>1 o' n<=m = trans-<= p<=n                 (o' n<=m)\n\n-- 2nd attempt - but things further down get stuck\n_o>>2_ : {p n m : Nat} -> p <= n -> n <= m -> p <= m\noz       o>>2 oz          = oz\nos n<=m\u2081 o>>2 os    m\u2081<=m = os     (n<=m\u2081 o>>2    m\u2081<=m)\nos n<=m\u2081 o>>2 o' sucm\u2081<=m = os (o'  n<=m\u2081 o>>2 sucm\u2081<=m)\no' p<=m\u2081 o>>2 os    m\u2081<=m =     o' (p<=m\u2081 o>>2    m\u2081<=m)\no' p<=m\u2081 o>>2 o' sucm\u2081<=m = o' (o'  p<=m\u2081 o>>2 sucm\u2081<=m)\noz       o>>2 o'  zero<=m = o'        (oz o>>2  zero<=m)\n\n-- https:\/\/github.com\/laMudri\/thinning\/blob\/master\/src\/Data\/Thinning.agda\n_o>>_ : \u2200 {m n o} \u2192 m <= n \u2192 n <= o \u2192 m <= o\nm<=z o>> oz   = m<=z\nos \u03b8 o>> os \u03c6 = os (\u03b8 o>> \u03c6)\no' \u03b8 o>> os \u03c6 = o' (\u03b8 o>> \u03c6)\n\u03b8    o>> o' \u03c6 = o' (\u03b8 o>> \u03c6)\n\n\n-- empty thinning returns an empty vector\noe-<?= : {X : Set}{n : Nat}(xs : Vec X n) -> (oe <?= xs) == []\noe-<?=       []  = refl []\noe-<?= (x ,- xs) = oe-<?= xs\n\ncp-<?= : {p n m : Nat}(p<=n : p <= n)(n<=m : n <= m) ->\n         {X : Set}(xs : Vec X m) ->\n         ((p<=n o>> n<=m) <?= xs) == (p<=n <?= (n<=m <?= xs))\ncp-<?=     p<=n   oz             []  = refl (p<=n <?= [])\ncp-<?=     p<=n  (o' n<=m) (_ ,- xs) = cp-<?= p<=n n<=m xs\ncp-<?= (o' p<=n) (os n<=m) (_ ,- xs) = cp-<?= p<=n n<=m xs\ncp-<?= (os p<=n) (os n<=m) (x ,- xs) = cong (x ,-_) (cp-<?= p<=n n<=m xs)\n\n\nzero<=4 : zero <= 4\nzero<=4 = o' (o' (o' (o' oz)))\n\n1<=4    : 1    <= 4\n1<=4    = os (o' (o' (o' oz)))\n\nv04     : Vec Nat zero\nv04     = zero<=4 <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_       :                           v04 == []\n_       =                             refl []\n\nv14     : Vec Nat 1\nv14     =    1<=4 <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_       :      v14 == (1                ,- [])\n_       =        refl (1                ,- [])\n\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Thinning Dominoes\n------------------------------------------------------------------------------\n\n--??--1.14-(3)----------------------------------------------------------------\n\nidThen-o>> : {n m : Nat}(n<=m : n <= m) -> (oi o>> n<=m) == n<=m\nidThen-o>>  oz                               = refl oz\nidThen-o>> (os n<=m) = cong os (idThen-o>> n<=m) -- rewrite idThen-o>> n<=m = refl (os n<=m)\nidThen-o>> (o' n<=m) = cong o' (idThen-o>> n<=m)\n\nidAfter-o>> : {n m : Nat}(n<=m : n <= m) -> (n<=m o>> oi) == n<=m\nidAfter-o>>  oz       = refl oz\nidAfter-o>> (os n<=m) = cong os (idAfter-o>> n<=m) -- rewrite idAfter-o>> n<=m = refl (os n<=m)\nidAfter-o>> (o' n<=m) = cong o' (idAfter-o>> n<=m) -- rewrite idAfter-o>> n<=m = refl (o' n<=m)\n\nassoc-o>> : {q p n m : Nat}(q<=p : q <= p)(p<=n : p <= n)(n<=m : n <= m) ->\n            ((q<=p o>> p<=n) o>> n<=m) == (q<=p o>> (p<=n o>> n<=m))\nassoc-o>>     q<=p      p<=n   oz       = refl (q<=p o>> p<=n)\nassoc-o>>     q<=p      p<=n  (o' n<=m) = cong o' (assoc-o>> q<=p p<=n n<=m)\nassoc-o>>     q<=p  (o' p<=n) (os n<=m) = cong o' (assoc-o>> q<=p p<=n n<=m)\nassoc-o>> (o' q<=p) (os p<=n) (os n<=m) = cong o' (assoc-o>> q<=p p<=n n<=m)\nassoc-o>> (os q<=p) (os p<=n) (os n<=m) = cong os (assoc-o>> q<=p p<=n n<=m)\n\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Vectors as Arrays\n------------------------------------------------------------------------------\n\n-- We can use 1 <= n as the type of bounded indices into a vector and do\n-- a kind of \"array projection\". First we select a 1-element vector from\n-- the n-element vector, then we take its head to get the element out.\n\nvProject : {n : Nat}{X : Set} -> Vec X n -> 1 <= n -> X\nvProject xs i = vHead (i <?= xs)\n\nvp3 : Nat\nvp3 = vProject (1 ,- 2 ,- 3 ,- 4 ,- []) (o' (o' (os (o' oz))))\n_ :                vp3 == 3\n_ =                  refl 3\n\n\n-- Your (TRICKY) mission is to reverse the process, tabulating a function\n-- from indices as a vector. Then show that these operations are inverses.\n\n--??--1.15-(3)----------------------------------------------------------------\n\n-- HINT: composition of functions\nvTabulate : {n : Nat}{X : Set} -> (1 <= n -> X) -> Vec X n\nvTabulate  {zero} _ = []\nvTabulate {suc n} f = f (os (zero<=m {n})) ,- (vTabulate (\u03bb 1<=n \u2192 f (o' 1<=n)))\n\nvt : Vec Nat 4\nvt = vTabulate f\n where\n  f : {n : Nat} -> (1 <= n) \u2192 Nat\n  f (os _) = 1\n  f (o' x) = 1 +N f x\n_ : vt == (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ =  refl (1 ,- 2 ,- 3 ,- 4 ,- [])\n\n-- This should be easy if vTabulate is correct.\nvTabulateProjections : {n : Nat}{X : Set}(xs : Vec X n) ->\n                       vTabulate (vProject xs) == xs\nvTabulateProjections [] = refl []\nvTabulateProjections (x ,- xs) = cong (x ,-_) (vTabulateProjections xs)\n\n-- HINT: oeUnique\nvProjectFromTable : {n : Nat}{X : Set}(f : 1 <= n -> X)(i : 1 <= n) ->\n                    vProject (vTabulate f) i == f i\nvProjectFromTable {suc n} f (os 0<=n)\n  rewrite oeUnique 0<=n | oeUnique (zero<=m {n})\n  = refl (f (os oe))\nvProjectFromTable {suc n} f (o' 1<=n)\n  = vProjectFromTable (\u03bb 1<=n \u2192 f (o' 1<=n)) 1<=n\n\n--??--------------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n------------------------------------------------------------------------------\n------------------------------------------------------------------------------\n-- CS410 2017\/18 Exercise 1  VECTORS AND FRIENDS (worth 25%)\n------------------------------------------------------------------------------\n------------------------------------------------------------------------------\n\n-- NOTE (19\/9\/17) This file is currently incomplete: more will arrive on\n-- GitHub.\n\n-- MARK SCHEME (transcribed from paper): the (m) numbers add up to slightly\n-- more than 25, so should be taken as the maximum number of marks losable on\n-- the exercise. In fact, I did mark it negatively, but mostly because it was\n-- done so well (with Agda's help) that it was easier to find the errors.\n\n\n------------------------------------------------------------------------------\n-- Dependencies\n------------------------------------------------------------------------------\n\nopen import CS410-Prelude\n\ncong : \u2200 {A B : Set} {x y : A}\n     \u2192 (f : A \u2192 B)\n     \u2192   x ==   y\n       ----------\n     \u2192 f x == f y\ncong f (refl x) =  refl (f x)\n\n------------------------------------------------------------------------------\n-- Vectors\n------------------------------------------------------------------------------\n\ndata Vec (X : Set) : Nat -> Set where  -- like lists, but length-indexed\n  []   :                              Vec X zero\n  _,-_ : {n : Nat} -> X -> Vec X n -> Vec X (suc n)\ninfixr 4 _,-_   -- the \"cons\" operator associates to the right\n\n-- I like to use the asymmetric ,- to remind myself that the element is to\n-- the left and the rest of the list is to the right.\n\n-- Vectors are useful when there are important length-related safety\n-- properties.\n\n------------------------------------------------------------------------------\n-- Heads and Tails\n------------------------------------------------------------------------------\n\n-- We can rule out nasty head and tail errors by insisting on nonemptiness!\n\n--??--1.1-(2)-----------------------------------------------------------------\n\nvHead : {X : Set}{n : Nat} -> Vec X (suc n) -> X\nvHead (x ,- _) = x\n\nvTail : {X : Set}{n : Nat} -> Vec X (suc n) -> Vec X n\nvTail (_ ,- xs) = xs\n\nvHeadTailFact :  {X : Set}{n : Nat}(xs : Vec X (suc n)) ->\n                 (vHead xs ,- vTail xs) == xs\nvHeadTailFact (x ,- xs) = refl (x ,- xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Concatenation and its Inverse\n------------------------------------------------------------------------------\n\n--??--1.2-(2)-----------------------------------------------------------------\n\n_+V_ : {X : Set}{m n : Nat} -> Vec X m -> Vec X n -> Vec X (m +N n)\n[]        +V ys = ys\n(x ,- xs) +V ys = x ,- xs +V ys\ninfixr 4 _+V_\n\nvChop : {X : Set}(m : Nat){n : Nat} -> Vec X (m +N n) -> Vec X m * Vec X n\nvChop  zero         xs = [] , xs\nvChop (suc m) (x ,- xs)\n  with vChop m xs\n... | vm , vn\n  = (x ,- vm) , vn\n\nvChopAppendFact : {X : Set}{m n : Nat}(xs : Vec X m)(ys : Vec X n) ->\n                  vChop m (xs +V ys) == (xs , ys)\nvChopAppendFact       []  ys = refl ([] , ys)\nvChopAppendFact (x ,- xs) ys rewrite vChopAppendFact xs ys = refl ((x ,- xs) , ys)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Map, take I\n------------------------------------------------------------------------------\n\n-- Implement the higher-order function that takes an operation on\n-- elements and does it to each element of a vector. Use recursion\n-- on the vector.\n-- Note that the type tells you the size remains the same.\n\n-- Show that if the elementwise function \"does nothing\", neither does\n-- its vMap. \"map of identity is identity\"\n\n-- Show that two vMaps in a row can be collapsed to just one, or\n-- \"composition of maps is map of compositions\"\n\n--??--1.3-(2)-----------------------------------------------------------------\n\nvMap : {X Y : Set} -> (X -> Y) -> {n : Nat} -> Vec X n -> Vec Y n\nvMap f       []  = []\nvMap f (x ,- xs) = f x ,- vMap f xs\n\nvMapIdFact : {X : Set}{f : X -> X}(feq : (x : X) -> f x == x) ->\n             {n : Nat}(xs : Vec X n) -> vMap f xs == xs\nvMapIdFact feq       []  = refl []\nvMapIdFact feq (x ,- xs)\n  rewrite vMapIdFact feq xs\n  |       feq x\n  = refl (x ,- xs)\n\nvMapCpFact : {X Y Z : Set}{f : Y -> Z}{g : X -> Y}{h : X -> Z}\n               (heq : (x : X) -> f (g x) == h x) ->\n             {n : Nat}(xs : Vec X n) ->\n               vMap f (vMap g xs) == vMap h xs\nvMapCpFact heq       []  = refl []\nvMapCpFact {_}{_}{_}{f} {g} {h} heq (x ,- xs)\n  rewrite heq x\n  |       vMapCpFact {_}{_}{_}{f} {g} {h} heq xs\n  = refl (h x ,- vMap h xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- vMap and +V\n------------------------------------------------------------------------------\n\n-- Show that if you've got two vectors of Xs and a function from X to Y,\n-- and you want to concatenate and map, it doesn't matter which you do\n-- first.\n\n--??--1.4-(1)-----------------------------------------------------------------\n\nvMap+VFact : {X Y : Set}(f : X -> Y) ->\n             {m n : Nat}(xs : Vec X m)(xs' : Vec X n) ->\n             vMap f (xs +V xs') == (vMap f xs +V vMap f xs')\nvMap+VFact f       []  xs' = refl (vMap f xs')\nvMap+VFact f (x ,- xs) xs'\n  rewrite vMap+VFact f xs xs'\n  = refl (f x ,- vMap f xs +V vMap f xs')\n\n--??--------------------------------------------------------------------------\n\n-- Think about what you could prove, relating vMap with vHead, vTail, vChop...\n-- Now google \"Philip Wadler\" \"Theorems for Free\"\n\n-- TODO\n\n------------------------------------------------------------------------------\n-- Applicative Structure (giving mapping and zipping cheaply)\n------------------------------------------------------------------------------\n\n--??--1.5-(2)-----------------------------------------------------------------\n\n-- HINT: you will need to override the default invisibility of n to do this.\n-- HC : replicate\nvPure : {X : Set} -> X -> {n : Nat} -> Vec X n\nvPure x {zero}  = []\nvPure x {suc n} = x ,- vPure x {n}\n\n_$V_ : {X Y : Set}{n : Nat} -> Vec (X -> Y) n -> Vec X n -> Vec Y n\n[]      $V      [] = []\nf ,- fs $V x ,- xs = f x ,- (fs $V xs)\ninfixl 3 _$V_  -- \"Application associates to the left,\n               --  rather as we all did in the sixties.\" (Roger Hindley)\n\n-- Pattern matching and recursion are forbidden for the next two tasks.\n\n-- implement vMap again, but as a one-liner\nvec : {X Y : Set} -> (X -> Y) -> {n : Nat} -> Vec X n -> Vec Y n\nvec f xs = vPure f $V xs\n\n-- implement the operation which pairs up corresponding elements\nvZip : {X Y : Set}{n : Nat} -> Vec X n -> Vec Y n -> Vec (X * Y) n\nvZip xs ys = vec (_,_) xs $V ys\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Applicative Laws\n------------------------------------------------------------------------------\n\n-- According to \"Applicative programming with effects\" by\n--   Conor McBride and Ross Paterson\n-- some laws should hold for applicative functors.\n-- Check that this is the case.\n\n--??--1.6-(2)-----------------------------------------------------------------\n\nvIdentity : {X : Set}{f : X -> X}(feq : (x : X) -> f x == x) ->\n            {n : Nat}(xs : Vec X n) -> (vPure f $V xs) == xs\nvIdentity feq        [] = refl []\nvIdentity feq (x ,- xs) rewrite vIdentity feq xs | feq x = refl (x ,- xs)\n\nvHomomorphism : {X Y : Set}(f : X -> Y)(x : X) ->\n                {n : Nat} -> (vPure f $V vPure x) == vPure (f x) {n}\nvHomomorphism f x {zero}  = refl []\nvHomomorphism f x {suc n} rewrite vHomomorphism f x {n} = refl (f x ,- vPure (f x))\n\nvInterchange : {X Y : Set}{n : Nat}(fs : Vec (X -> Y) n)(x : X) ->\n               (fs $V vPure x) == (vPure (_$ x) $V fs)\nvInterchange []        x = refl []\nvInterchange (f ,- fs) x rewrite vInterchange fs x = refl (f x ,- (vPure (\u03bb x\u2192y \u2192 x\u2192y x) $V fs))\n\nvComposition : {X Y Z : Set}{n : Nat}\n               (fs : Vec (Y -> Z) n)(gs : Vec (X -> Y) n)(xs : Vec X n) ->\n               (vPure _<<_ $V fs $V gs $V xs) == (fs $V (gs $V xs))\nvComposition       []        []        []  = refl []\nvComposition (f ,- fs) (g ,- gs) (x ,- xs)\n  rewrite vComposition fs gs xs\n  = refl (f (g x) ,- (fs $V (gs $V xs)))\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Order-Preserving Embeddings (also known in the business as \"thinnings\")\n------------------------------------------------------------------------------\n\n-- What have these to do with Pascal's Triangle?\n\n-- how to choose N things from M things\ndata _<=_ : Nat -> Nat -> Set where\n  oz :                          zero  <= zero  -- stop\n  os : {n m : Nat} -> n <= m -> suc n <= suc m -- take this one and keep going\n  o' : {n m : Nat} -> n <= m ->     n <= suc m -- skip this one and keep going\n\nrefl-<= : (n : Nat) -> n <= n\nrefl-<= zero    = oz\nrefl-<= (suc n) = os (refl-<= n)\n\ntrans-<= : {n m p : Nat} -> n <= m -> m <= p -> n <= p\ntrans-<=  oz           zero<=p  = zero<=p\ntrans-<= (os n<=m) (os sucm<=p) = os (trans-<=     n<=m  sucm<=p)\ntrans-<= (os n<=m) (o' sucm<=p) = os (trans-<= (o' n<=m) sucm<=p)\ntrans-<= (o' n<=m) (os    m<=p) = o' (trans-<=     n<=m     m<=p)\ntrans-<= (o' n<=m) (o'    m<=p) = o' (trans-<= (o' n<=m)    m<=p)\n\n<=-suc : (x : Nat) -> x <= suc x\n<=-suc  zero   = o' oz\n<=-suc (suc x) = os (<=-suc x)\n\nn<=m\u2192sucn<=sucm : {n m : Nat} \u2192 n <= m -> suc n <= suc m\nn<=m\u2192sucn<=sucm  oz       = os oz\nn<=m\u2192sucn<=sucm (os n<=m) = os (n<=m\u2192sucn<=sucm n<=m)\nn<=m\u2192sucn<=sucm (o' n<=m) = o' (n<=m\u2192sucn<=sucm n<=m)\n\nsucsucn<=m\u2192sucn<=m : {n m : Nat} -> suc (suc n) <= m -> suc n <= m\nsucsucn<=m\u2192sucn<=m (os sucn<=m)  = o' sucn<=m\nsucsucn<=m\u2192sucn<=m (o' (os xxx)) = o' (o' xxx)\nsucsucn<=m\u2192sucn<=m (o' (o' xxx)) = o' (o' (sucsucn<=m\u2192sucn<=m xxx))\n\nsucn<=m\u2192sucn<=sucm : {n m : Nat} -> suc n <= m -> suc n <= suc m\nsucn<=m\u2192sucn<=sucm (os p) = os (o' p)\nsucn<=m\u2192sucn<=sucm (o' p) = o' (sucn<=m\u2192sucn<=sucm p)\n\nzero<=m : {m : Nat} -> zero <= m\nzero<=m  {zero} = oz\nzero<=m {suc m} = o' zero<=m\n\nsucn<=sucm\u2192n<=m : {n m : Nat} -> suc n <= suc m -> n <= m\nsucn<=sucm\u2192n<=m {zero}  {zero}       p  = oz\nsucn<=sucm\u2192n<=m {zero}  {suc m} (os  p) = o' (zero<=m {m})\nsucn<=sucm\u2192n<=m {zero}  {suc m} (o'  p) = o' (zero<=m {m})\nsucn<=sucm\u2192n<=m {suc n} {zero}  (os ())\nsucn<=sucm\u2192n<=m {suc n} {zero}  (o' ())\nsucn<=sucm\u2192n<=m {suc n} {suc m} (os  p) = p\nsucn<=sucm\u2192n<=m {suc n} {suc m} (o'  p) = o' (sucn<=sucm\u2192n<=m p)\n\n-- Find all the values in each of the following <= types.\n-- This is a good opportunity to learn to use C-c C-a with the -l option\n--   (a.k.a. \"google the type\" without \"I feel lucky\")\n-- The -s n option also helps.\n\n--??--1.7-(1)-----------------------------------------------------------------\n\nall0<=4 : Vec (0 <= 4) 1\nall0<=4 = o' (o' (o' (o' oz))) ,- []\n\nall1<=4 : Vec (1 <= 4) 1\nall1<=4 = os (o' (o' (o' oz))) ,- []\n\nall2<=4 : Vec (2 <= 4) 1\nall2<=4 = os (os (o' (o' oz))) ,- []\n\nall3<=4 : Vec (3 <= 4) 1\nall3<=4 = os (os (os (o' oz))) ,- []\n\nall4<=4 : Vec (4 <= 4) 1\nall4<=4 = os (os (os (os oz))) ,- []\n\n-- Prove the following. A massive case analysis \"rant\" is fine.\n\nno5<=4 : 5 <= 4 -> Zero\nno5<=4 (os (os (os (os ()))))\nno5<=4 (os (os (os (o' ()))))\nno5<=4 (os (os (o' (os ()))))\nno5<=4 (os (os (o' (o' ()))))\nno5<=4 (os (o' (os (os ()))))\nno5<=4 (os (o' (os (o' ()))))\nno5<=4 (os (o' (o' (os ()))))\nno5<=4 (os (o' (o' (o' ()))))\nno5<=4 (o' (os (os (os ()))))\nno5<=4 (o' (os (os (o' ()))))\nno5<=4 (o' (os (o' (os ()))))\nno5<=4 (o' (os (o' (o' ()))))\nno5<=4 (o' (o' (os (os ()))))\nno5<=4 (o' (o' (os (o' ()))))\nno5<=4 (o' (o' (o' (os ()))))\nno5<=4 (o' (o' (o' (o' ()))))\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Order-Preserving Embeddings Select From Vectors\n------------------------------------------------------------------------------\n\n-- Use n <= m to encode the choice of n elements from an m-Vector.\n-- The os constructor tells you to take the next element of the vector;\n-- the o' constructor tells you to omit the next element of the vector.\n\n--??--1.8-(2)-----------------------------------------------------------------\n\n_<?=_ : {X : Set}{n m : Nat} -> n <= m -> Vec X m\n                     -> Vec X n\noz    <?=        _  = []\nos th <?= (x ,- xs) = x ,- th <?= xs\no' th <?= (_ ,- xs) =      th <?= xs\n\nvn4 : Vec Nat 4\nvn4 = os (os (os (os oz))) <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ :                     vn4 == (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ =                       refl (1 ,- 2 ,- 3 ,- 4 ,- [])\n\nvn3 : Vec Nat 3\nvn3 = o' (os (o' (os (o' (os oz))))) <?= (1 ,- 2 ,- 3 ,- 4 ,- 5 ,- 6 ,- [])\n_ :                               vn3 ==      (2      ,- 4      ,- 6 ,- [])\n_ =                                      refl (2      ,- 4      ,- 6 ,- [])\n\nvn2 : Vec Nat 2\nvn2 = os (o' (os (o' oz))) <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ :                     vn2 == (1      ,- 3      ,- [])\n_ =                       refl (1      ,- 3      ,- [])\n\nvn2' : Vec Nat 2\nvn2' = o' (os (os (o' oz))) <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ :                          vn2' == (2 ,- 3      ,- [])\n_ =                             refl (2 ,- 3      ,- [])\n\n-- it shouldn't matter whether you map then select or select then map\n\nvMap<?=Fact : {X Y : Set}(f : X -> Y)\n              {n m : Nat}(th : n <= m)(xs : Vec X m) ->\n              vMap f (th <?= xs) == (th <?= vMap f xs)\nvMap<?=Fact f  oz     [] = refl []\nvMap<?=Fact f (os th) (x ,- xs) rewrite vMap<?=Fact f th xs = refl (f x ,- (th <?= vMap f xs))\nvMap<?=Fact f (o' th) (x ,- xs) rewrite vMap<?=Fact f th xs = refl         (th <?= vMap f xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Our Favourite Thinnings\n------------------------------------------------------------------------------\n\n-- Construct the identity thinning and the empty thinning.\n\n--??--1.9-(1)-----------------------------------------------------------------\n\noi : {n : Nat} -> n <= n\noi  {zero} = oz\noi {suc n} = os oi\n\noe : {n : Nat} -> 0 <= n\noe  {zero} = oz\noe {suc n} = o' oe\n\nvnoi : Vec Nat 4\nvnoi = oi  <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_    : vnoi == (1 ,- 2 ,- 3 ,- 4 ,- [])\n_    =    refl (1 ,- 2 ,- 3 ,- 4 ,- [])\n\nvnoe : Vec Nat 0\nvnoe = oe <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_    :                     vnoe == []\n_    =                        refl []\n\n\n--??--------------------------------------------------------------------------\n\n\n-- Show that all empty thinnings are equal to yours.\n\n--??--1.10-(1)----------------------------------------------------------------\n\noeUnique : {n : Nat}(th : 0 <= n) -> th == oe\noeUnique  oz                       = refl  oz\noeUnique (o' i) rewrite oeUnique i = refl (o' oe)\n\n--??--------------------------------------------------------------------------\n\n\n-- Show that there are no thinnings of form big <= small  (TRICKY)\n-- Then show that all the identity thinnings are equal to yours.\n-- Note that you can try the second even if you haven't finished the first.\n-- HINT: you WILL need to expose the invisible numbers.\n-- HINT: check CS410-Prelude for a reminder of >=\n\n--??--1.11-(3)----------------------------------------------------------------\n\noTooBig : {n m : Nat} -> n >= m -> suc n <= m -> Zero\noTooBig  {zero}  {zero} n>=m ()\noTooBig  {zero} {suc m} n>=m (os th) = n>=m\noTooBig  {zero} {suc m} n>=m (o' th) = n>=m\noTooBig {suc n} {suc m} n>=m (os th) = oTooBig n>=m th\noTooBig {suc n} {suc m} n>=m (o' th) = oTooBig {n} {m} n>=m (sucsucn<=m\u2192sucn<=m th)\n\noiUnique : {n : Nat}(n<=n : n <= n) -> n<=n == oi\noiUnique          oz                             = refl oz\noiUnique         (os n<=n) rewrite oiUnique n<=n = refl (os oi)\noiUnique {suc n} (o' sucn<=n)\n  with oTooBig (refl->= n) sucn<=n\n... | ()\n\n--??--------------------------------------------------------------------------\n\n\n-- Show that the identity thinning selects the whole vector\n\n--??--1.12-(1)----------------------------------------------------------------\n\nid-<?= : {X : Set}{n : Nat}(xs : Vec X n) -> (oi <?= xs) == xs\nid-<?=       []  = refl []\nid-<?= (x ,- xs) = cong (x ,-_) (id-<?= xs)\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Composition of Thinnings\n------------------------------------------------------------------------------\n\n-- Define the composition of thinnings and show that selecting by a\n-- composite thinning is like selecting then selecting again.\n-- A small bonus applies to minimizing the length of the proof.\n-- To collect the bonus, you will need to think carefully about\n-- how to make the composition as *lazy* as possible.\n\n--??--1.13-(3)----------------------------------------------------------------\n\n-- my 1st attempt - but everything gets stuff because pattern matching on right instead of left\n_o>>1_ : {p n m : Nat} -> p <= n -> n <= m -> p <= m\np<=zero o>>1 oz      = p<=zero\np<=sucn o>>1 os n<=m = trans-<= p<=sucn (n<=m\u2192sucn<=sucm n<=m)\np<=n    o>>1 o' n<=m = trans-<= p<=n                 (o' n<=m)\n\n-- 2nd attempt - but things further down get stuck\n_o>>2_ : {p n m : Nat} -> p <= n -> n <= m -> p <= m\noz       o>>2 oz          = oz\nos n<=m\u2081 o>>2 os    m\u2081<=m = os     (n<=m\u2081 o>>2    m\u2081<=m)\nos n<=m\u2081 o>>2 o' sucm\u2081<=m = os (o'  n<=m\u2081 o>>2 sucm\u2081<=m)\no' p<=m\u2081 o>>2 os    m\u2081<=m =     o' (p<=m\u2081 o>>2    m\u2081<=m)\no' p<=m\u2081 o>>2 o' sucm\u2081<=m = o' (o'  p<=m\u2081 o>>2 sucm\u2081<=m)\noz       o>>2 o'  zero<=m = o'        (oz o>>2  zero<=m)\n\n-- https:\/\/github.com\/laMudri\/thinning\/blob\/master\/src\/Data\/Thinning.agda\n_o>>_ : \u2200 {m n o} \u2192 m <= n \u2192 n <= o \u2192 m <= o\nm<=z o>> oz   = m<=z\nos \u03b8 o>> os \u03c6 = os (\u03b8 o>> \u03c6)\no' \u03b8 o>> os \u03c6 = o' (\u03b8 o>> \u03c6)\n\u03b8    o>> o' \u03c6 = o' (\u03b8 o>> \u03c6)\n\n\n-- empty thinning returns an empty vector\noe-<?= : {X : Set}{n : Nat}(xs : Vec X n) -> (oe <?= xs) == []\noe-<?=       []  = refl []\noe-<?= (x ,- xs) = oe-<?= xs\n\ncp-<?= : {p n m : Nat}(p<=n : p <= n)(n<=m : n <= m) ->\n         {X : Set}(xs : Vec X m) ->\n         ((p<=n o>> n<=m) <?= xs) == (p<=n <?= (n<=m <?= xs))\ncp-<?=     p<=n   oz             []  = refl (p<=n <?= [])\ncp-<?=     p<=n  (o' n<=m) (_ ,- xs) = cp-<?= p<=n n<=m xs\ncp-<?= (o' p<=n) (os n<=m) (_ ,- xs) = cp-<?= p<=n n<=m xs\ncp-<?= (os p<=n) (os n<=m) (x ,- xs) = cong (x ,-_) (cp-<?= p<=n n<=m xs)\n\n\nzero<=4 : zero <= 4\nzero<=4 = o' (o' (o' (o' oz)))\n\n1<=4    : 1    <= 4\n1<=4    = os (o' (o' (o' oz)))\n\nv04     : Vec Nat zero\nv04     = zero<=4 <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_       :                           v04 == []\n_       =                             refl []\n\nv14     : Vec Nat 1\nv14     =    1<=4 <?= (1 ,- 2 ,- 3 ,- 4 ,- [])\n_       :      v14 == (1                ,- [])\n_       =        refl (1                ,- [])\n\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Thinning Dominoes\n------------------------------------------------------------------------------\n\n--??--1.14-(3)----------------------------------------------------------------\n\nidThen-o>> : {n m : Nat}(n<=m : n <= m) -> (oi o>> n<=m) == n<=m\nidThen-o>>  oz                               = refl oz\nidThen-o>> (os n<=m) = cong os (idThen-o>> n<=m) -- rewrite idThen-o>> n<=m = refl (os n<=m)\nidThen-o>> (o' n<=m) = cong o' (idThen-o>> n<=m)\n\nidAfter-o>> : {n m : Nat}(n<=m : n <= m) -> (n<=m o>> oi) == n<=m\nidAfter-o>>  oz       = refl oz\nidAfter-o>> (os n<=m) = cong os (idAfter-o>> n<=m) -- rewrite idAfter-o>> n<=m = refl (os n<=m)\nidAfter-o>> (o' n<=m) = cong o' (idAfter-o>> n<=m) -- rewrite idAfter-o>> n<=m = refl (o' n<=m)\n\nassoc-o>> : {q p n m : Nat}(q<=p : q <= p)(p<=n : p <= n)(n<=m : n <= m) ->\n            ((q<=p o>> p<=n) o>> n<=m) == (q<=p o>> (p<=n o>> n<=m))\nassoc-o>>     q<=p      p<=n   oz       = refl (q<=p o>> p<=n)\nassoc-o>>     q<=p      p<=n  (o' n<=m) = cong o' (assoc-o>> q<=p p<=n n<=m)\nassoc-o>>     q<=p  (o' p<=n) (os n<=m) = cong o' (assoc-o>> q<=p p<=n n<=m)\nassoc-o>> (o' q<=p) (os p<=n) (os n<=m) = cong o' (assoc-o>> q<=p p<=n n<=m)\nassoc-o>> (os q<=p) (os p<=n) (os n<=m) = cong os (assoc-o>> q<=p p<=n n<=m)\n\n\n--??--------------------------------------------------------------------------\n\n\n------------------------------------------------------------------------------\n-- Vectors as Arrays\n------------------------------------------------------------------------------\n\n-- We can use 1 <= n as the type of bounded indices into a vector and do\n-- a kind of \"array projection\". First we select a 1-element vector from\n-- the n-element vector, then we take its head to get the element out.\n\nvProject : {n : Nat}{X : Set} -> Vec X n -> 1 <= n -> X\nvProject xs i = vHead (i <?= xs)\n\nvp3 : Nat\nvp3 = vProject (1 ,- 2 ,- 3 ,- 4 ,- []) (o' (o' (os (o' oz))))\n_ :                vp3 == 3\n_ =                  refl 3\n\n\n-- Your (TRICKY) mission is to reverse the process, tabulating a function\n-- from indices as a vector. Then show that these operations are inverses.\n\n--??--1.15-(3)----------------------------------------------------------------\n\n-- HINT: composition of functions\nvTabulate : {n : Nat}{X : Set} -> (1 <= n -> X) -> Vec X n\nvTabulate  {zero} _ = []\nvTabulate {suc n} f = f (os (zero<=m {n})) ,- (vTabulate (\u03bb 1<=n \u2192 f (o' 1<=n)))\n\nvt : Vec Nat 4\nvt = vTabulate f\n where\n  f : (1 <= 4) \u2192 Nat\n  f             (os _)    = 1\n  f         (o' (os _))   = 2\n  f     (o' (o' (os _)))  = 3\n  f (o' (o' (o' (os _)))) = 4\n_ : vt == (1 ,- 2 ,- 3 ,- 4 ,- [])\n_ =  refl (1 ,- 2 ,- 3 ,- 4 ,- [])\n\n-- This should be easy if vTabulate is correct.\nvTabulateProjections : {n : Nat}{X : Set}(xs : Vec X n) ->\n                       vTabulate (vProject xs) == xs\nvTabulateProjections [] = refl []\nvTabulateProjections (x ,- xs) = cong (x ,-_) (vTabulateProjections xs)\n\n-- HINT: oeUnique\nvProjectFromTable : {n : Nat}{X : Set}(f : 1 <= n -> X)(i : 1 <= n) ->\n                    vProject (vTabulate f) i == f i\nvProjectFromTable f i = {!!}\n\n--??--------------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"00d88d736e355f710a6f1506eb8535eeb024b409","subject":"updated the elgamal example","message":"updated the elgamal example\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Unit\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; Fin\u25b9\u2115) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_; _==_)\nopen import Data.Bit\nopen import Data.Two\nopen import Data.Bits hiding (_==_)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\nimport Game.DDH\nimport Game.IND-CPA\nimport Cipher.ElGamal.Generic\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Explorable.Sum renaming (\u03bcBit to \u03bc\ud835\udfda)\nopen import Explore.Explorable.Product\nopen import Explore.Explorable.Fin\nopen import Relation.Binary.NP\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : Explorable X \u2192 \u2200 u \u2192 Explorable (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7-\u03bc \u03bcU \u03bcX u\u2081\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = count (\u03bcU \u03bc\u2124q u) (run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sym (\u229e-stable x (Bit\u25b9\u2115 \u2218 Adv))\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc]; sucmod-inj to [suc]-inj)\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q-1 : \u2115) ([0]' [1]' : Fin (suc q-1)) where\n  -- open Sum\n  q : \u2115\n  q = suc q-1\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  \u03bc\u2124q : Explorable \u2124q\n  \u03bc\u2124q = \u03bcFinSuc q-1\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q = sum \u03bc\u2124q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  [suc]-stable : SumStableUnder (sum \u03bc\u2124q) [suc]\n  [suc]-stable = \u03bcFinSUI [suc] [suc]-inj\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = [suc]-inj (\u2115\u229e-inj n eq)\n\n  \u2115\u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u2115\u229e_ m)\n  \u2115\u229e-stable m = \u03bcFinSUI (_\u2115\u229e_ m) (\u2115\u229e-inj m)\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin\u25b9\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin\u25b9\u2115 m)\n\n  \u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ m)\n  \u229e-stable m = \u03bcFinSUI (_\u229e_ m) (\u229e-inj m)\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin\u25b9\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin\u25b9\u2115 n)\n\nmodule G-implem (p-1 q-1 : \u2115) (g' 0[p] 1[p] : Fin (suc p-1)) (0[q] 1[q] : Fin (suc q-1)) where\n  open \u2124q-implem q-1 0[q] 1[q] public\n  open \u2124q-implem p-1 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin\u25b9\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule El-Gamal-Generic\n  (\u2124q       : \u2605)\n  (_\u22a0_      : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G        : \u2605)\n  (g        : G)\n  (_^_      : G \u2192 \u2124q \u2192 G)\n  (Message  : \u2605)\n  (_\u2219_      : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_      : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219      : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q      : Explorable \u2124q)\n  (R\u2090       : \u2605)\n  (\u03bcR\u2090      : Explorable R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    R\u2093 : \u2605\n    R\u2093 = \u2124q\n\n    open Cipher.ElGamal.Generic Message \u2124q G g _^_ _\u2219_ _\/_\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\n    open IND-CPA renaming (R to R')\n\n    \u03bcR' : Explorable R'\n    \u03bcR' = \u03bcR\u2090 \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q\n\n    sumR' = sum \u03bcR'\n    \n    R = \ud835\udfda \u00d7 R'\n    \u03bcR = \u03bc\ud835\udfda \u00d7-\u03bc \u03bcR'\n    sumR = sum \u03bcR\n    \n    sumExtR\u2090 = sum-ext \u03bcR\u2090\n    sumExt\u2124q = sum-ext \u03bc\u2124q\n    sumHomR' = sum-hom \u03bcR'\n    sumExtR' = sum-ext \u03bcR'\n    \n    IND-CPA-\u2141 : IND-CPA.Adv \u2192 R \u2192 \ud835\udfda\n    IND-CPA-\u2141 A (b , r\u2090 , r\u2096 , r\u2091 , r\u2093) = IND-CPA.\u2141 b A (r\u2090 , r\u2096 , r\u2091 , r\u2093)\n    \n    module DDH = Game.DDH \u2124q G g _^_ (\ud835\udfda \u00d7 R\u2090)\n\n    DDH-\u2141\u2080 DDH-\u2141\u2081 : DDH.Adv \u2192 R \u2192 \ud835\udfda\n    DDH-\u2141\u2080 A (b , r\u2090 , g\u02e3 , g\u02b8 , g\u1dbb) = DDH.\u2141\u2080 A ((b , r\u2090) , g\u02e3 , g\u02b8 , g\u1dbb)\n    DDH-\u2141\u2081 A (b , r\u2090 , g\u02e3 , g\u02b8 , g\u1dbb) = DDH.\u2141\u2081 A ((b , r\u2090) , g\u02e3 , g\u02b8 , g\u1dbb)\n  \n    transformAdv : IND-CPA.Adv \u2192 DDH.Adv\n    transformAdv (m , d) (b , r\u2090) g\u02e3 g\u02b8 g\u1dbb = b == b\u2032\n                 where mb  = m r\u2090 g\u02e3 b\n                       c   = (g\u02b8 , g\u1dbb \u2219 mb)\n                       b\u2032  = d r\u2090 g\u02e3 c\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 \u2115) \u2192 \u2605\n    f \u2248q g = sum \u03bc\u2124q f \u2261 sum \u03bc\u2124q g\n\n    postulate\n      A : IND-CPA.Adv\n\n    A\u2032 = transformAdv A\n\n    1\/2 : R \u2192 \ud835\udfda\n    1\/2 (b , _) = b\n \n\n    module _ {S} where \n      _\u2248\u1d3f_ : (f g : R \u2192 S) \u2192 \u2605\n      f \u2248\u1d3f g = \u2200 (X : S \u2192 \u2115) \u2192 sumR (X \u2218 f) \u2261 sumR (X \u2218 g) \n\n    OTP-\u2141 : IND-CPA.Adv \u2192 R \u2192 \ud835\udfda\n    OTP-\u2141 (m , d) (b , r\u2090 , x , y , z) = b == d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g ^ x\n            g\u02b8 = g ^ y\n            g\u1dbb = g ^ z\n\n    postulate\n      otp-lem : \u2200 (A : Message \u2192 \u2115) m\u2080 m\u2081 \u2192\n        (\u03bb x \u2192 A((g ^ x) \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A((g ^ x) \u2219 m\u2081))\n \n    goal4 : 1\/2 \u2248\u1d3f OTP-\u2141 A\n    goal4 X = sumR' (\u03bb _ \u2192 X 0b) + sumR' (\u03bb _ \u2192 X 1b)\n            \u2261\u27e8 sym (sumHomR' _  _) \u27e9\n              sumR' (\u03bb _ \u2192 X 0b + X 1b)\n            \u2261\u27e8 sumExtR' (lemma \u2218 B 0b) \u27e9\n              sumR' (Y 0b 0b +\u00b0 Y 1b 0b)\n            \u2261\u27e8 sumHomR' _ _ \u27e9\n              sumR' (Y 0b 0b) + sumR' (Y 1b 0b)\n            \u2261\u27e8 cong (_+_ (sumR' (Y 0b 0b))) lemma2 \u27e9\n              sumR' (Y 0b 0b) + sumR' (Y 1b 1b)\n            \u220e\n      where\n        open \u2261-Reasoning\n        \n        B : \ud835\udfda \u2192 R' \u2192 \ud835\udfda\n        B b (r\u2090 , x , y , z) = proj\u2082 A r\u2090 (g ^ x) (g ^ y , (g ^ z) \u2219 proj\u2081 A r\u2090 (g ^ x) b)\n\n        Y = \u03bb bb bbb r \u2192 X (bb == B bbb r)\n\n        lemma : \u2200 b \u2192 X 0b + X 1b \u2261  X (0b == b) + X (1b == b)\n        lemma 1b = refl\n        lemma 0b = \u2115\u00b0.+-comm (X 0b) _\n         \n        lemma2 : sumR' (Y 1b 0b) \u2261 sumR' (Y 1b 1b)\n        lemma2 = sumExtR\u2090 \u03bb r\u2090 \u2192\n                 sumExt\u2124q \u03bb x \u2192\n                 sumExt\u2124q \u03bb y \u2192\n                   otp-lem (\u03bb m \u2192 X (proj\u2082 A r\u2090 (g ^ x) (g ^ y , m))) (proj\u2081 A r\u2090 (g ^ x) 0') (proj\u2081 A r\u2090 (g ^ x) 1')\n\n                 {-\n                  otp-lem (\u03bb m \u2192 proj\u2082 A r\u2090 (g ^ x) (g ^ y , m))\n                          (proj\u2081 A r\u2090 (g ^ x) 1b)\n                          (proj\u2081 A r\u2090 (g ^ x) 0b)\n                          (\u03bb c \u2192 X (1b == c))\n-}\n\n    module absDist {DIST : \u2605}(Dist : (f g : R \u2192 \ud835\udfda) \u2192 DIST)\n      (dist-cong : \u2200 {f g h i} \u2192 f \u2248\u1d3f g \u2192 h \u2248\u1d3f i \u2192 Dist f h \u2261 Dist g i) where\n      goal : Dist (IND-CPA-\u2141 A) 1\/2 \u2261 Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n      goal = Dist (IND-CPA-\u2141 A) 1\/2\n           \u2261\u27e8 refl \u27e9\n             Dist (DDH-\u2141\u2080 A\u2032) 1\/2\n           \u2261\u27e8 dist-cong (\u03bb _ \u2192 refl) goal4 \u27e9\n             Dist (DDH-\u2141\u2080 A\u2032) (OTP-\u2141 A)\n           \u2261\u27e8 refl \u27e9\n             Dist (DDH-\u2141\u2080 A\u2032) (DDH-\u2141\u2081 A\u2032)\n           \u220e\n        where open \u2261-Reasoning\n\n    Dist : (f g : R \u2192 \ud835\udfda) \u2192 \u2115\n    Dist f g = dist (count \u03bcR f) (count \u03bcR g)\n\n    dist-cong : \u2200  {f g h i} \u2192 f \u2248\u1d3f g \u2192 h \u2248\u1d3f i \u2192 Dist f h \u2261 Dist g i\n    dist-cong {f}{g}{h}{i} f\u2248g h\u2248i = cong\u2082 dist (f\u2248g \ud835\udfda\u25b9\u2115) (h\u2248i \ud835\udfda\u25b9\u2115)\n\n    open absDist Dist (\u03bb {f}{g}{h}{i} \u2192 dist-cong {f}{g}{h}{i})\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    {-\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n    -}\n        \n{-\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ m \u03b4 = \u210b \u03b4 \u2295 m\n{-\n\n    \/-\u2219 : \u2200 x y \u2192 \u210b\u27e8 x \u27e9\u2295 y \/ x \u2261 y\n    \/-\u2219 x y = {!!}\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n\n  _\/_ : G \u2192 G \u2192 G\n  x \/ y = x \u2219 (y \u207b\u00b9)\n\n  postulate\n    \/-\u2022 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n    \u22a0-comm : \u2200 x y \u2192 x \u22a0 y \u2261 y \u22a0 x\n\n  comm-^ : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n  comm-^ \u03b1 x y = (\u03b1 ^ x)^ y\n               \u2261\u27e8 sym (dist-^-\u22a0 \u03b1 x y) \u27e9\n                  \u03b1 ^ (x \u22a0 y)\n               \u2261\u27e8 cong (_^_ \u03b1) (\u22a0-comm x y) \u27e9\n                  \u03b1 ^ (y \u22a0 x)\n               \u2261\u27e8 dist-^-\u22a0 \u03b1 y x \u27e9\n                  (\u03b1 ^ y)^ x\n               \u220e\n    where open \u2261-Reasoning\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u03bcR\u2090 : Explorable (El `R\u2090)\n  \u03bcR\u2090 = \u03bcU \u03bc\u2124q `R\u2090\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum \u03bcR\u2090\n  sumR\u2090-ext = sum-ext \u03bcR\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\/_ \/-\u2022 comm-^ dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090\n  open EB hiding (g^_)\n\n  otp-base-lem : \u2200 (A : G \u2192 Bit) m \u2192 (A \u2218 g^_) \u2248q (A \u2218 g^_ \u2218 _\u229e_ m)\n  otp-base-lem A m = \u229e-stable m (Bit\u25b9\u2115 \u2218 A \u2218 g^_)\n\n  {-\n  postulate\n    -- ddh-hyp : (A : DDH.Adv) \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n  -}\n\n  -- open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 IND-CPA.\u2141 A 0b \u2248\u1d3f IND-CPA.\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Unit\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; Fin\u25b9\u2115) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_; _==_)\nopen import Data.Bit\nopen import Data.Bits hiding (_==_)\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\nimport Game.DDH\nimport Game.IND-CPA\nimport Cipher.ElGamal.Generic\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to Explorable)\nopen import Search.Searchable.Product\nopen import Search.Searchable.Fin\nopen import Relation.Binary.NP\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : Explorable X \u2192 \u2200 u \u2192 Explorable (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7-\u03bc \u03bcU \u03bcX u\u2081\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = count (\u03bcU \u03bc\u2124q u) (run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sym (\u229e-stable x (Bit\u25b9\u2115 \u2218 Adv))\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc]; sucmod-inj to [suc]-inj)\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q-1 : \u2115) ([0]' [1]' : Fin (suc q-1)) where\n  -- open Sum\n  q : \u2115\n  q = suc q-1\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  \u03bc\u2124q : Explorable \u2124q\n  \u03bc\u2124q = \u03bcFinSuc q-1\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q = sum \u03bc\u2124q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  [suc]-stable : SumStableUnder (sum \u03bc\u2124q) [suc]\n  [suc]-stable = \u03bcFinSUI [suc] [suc]-inj\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = [suc]-inj (\u2115\u229e-inj n eq)\n\n  \u2115\u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u2115\u229e_ m)\n  \u2115\u229e-stable m = \u03bcFinSUI (_\u2115\u229e_ m) (\u2115\u229e-inj m)\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin\u25b9\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin\u25b9\u2115 m)\n\n  \u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ m)\n  \u229e-stable m = \u03bcFinSUI (_\u229e_ m) (\u229e-inj m)\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin\u25b9\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin\u25b9\u2115 n)\n\nmodule G-implem (p-1 q-1 : \u2115) (g' 0[p] 1[p] : Fin (suc p-1)) (0[q] 1[q] : Fin (suc q-1)) where\n  open \u2124q-implem q-1 0[q] 1[q] public\n  open \u2124q-implem p-1 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin\u25b9\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : Explorable \u2124q)\n  (\u229e-stable : \u2200 x \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ x))\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule El-Gamal-Generic\n  (\u2124q       : \u2605)\n  (_\u22a0_      : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G        : \u2605)\n  (g        : G)\n  (_^_      : G \u2192 \u2124q \u2192 G)\n  (Message  : \u2605)\n  (_\u2219_      : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_      : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219      : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q      : Explorable \u2124q)\n  (R\u2090       : \u2605)\n  (\u03bcR\u2090      : Explorable R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    R\u2093 : \u2605\n    R\u2093 = \u2124q\n\n    open Cipher.ElGamal.Generic Message \u2124q G g _^_ _\u2219_ _\/_\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\n    open IND-CPA using (R)\n\n    UnusedGame : (i : Bit) \u2192 IND-CPA.Adv \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    UnusedGame i (m , d) (b , r\u2090 , x , y , z) = b == d r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    module DDH = Game.DDH \u2124q G g _^_ R\u2090\n\n    OTP\u2141 : (R\u2090 \u2192 G \u2192 Message) \u2192 (R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit) \u2192 R \u2192 Bit\n    OTP\u2141 M d (r , x , y , z) = d r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : Bit \u2192 IND-CPA.Adv \u2192 DDH.Adv\n    TrA b (m , d) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = d r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : IND-CPA.Adv \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : IND-CPA.Adv \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , d) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    module Unused where\n        like-\u2141 : Bit \u2192 IND-CPA.Game\n        like-\u2141 b (m , d) (r\u2090 , x , y , _z) =\n          d r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n          where g\u02e3 = g^ x\n                g\u02b8 = g^ y\n\n        IND-CPA-\u2141\u2261like-\u2141 : IND-CPA.\u2141 \u2261 like-\u2141\n        IND-CPA-\u2141\u2261like-\u2141 = refl\n\n    -- R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : Explorable R\n    \u03bcR = \u03bcR\u2090 \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q \u00d7-\u03bc \u03bc\u2124q\n\n    #\u1d3f_ : Count R\n    #\u1d3f_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    Re = (f g : R \u2192 Bit) \u2192 \u2605\n    record Tra (_\u2248\u2080_ _\u2248\u2081_ : Re) (f g : R \u2192 Bit) : \u2605 where\n      field\n        h : R \u2192 Bit\n        f\u2248\u2080h : f \u2248\u2080 h\n        h\u2248\u2081g : h \u2248\u2081 g\n\n    record _\u2248\u1d3f_ (f g : R \u2192 Bit) : \u2605 where\n      constructor mk\n      field\n        un-\u2248\u1d3f : #\u1d3f f \u2261 #\u1d3f g\n    open _\u2248\u1d3f_ public\n\n    \u2248\u1d3f-trans : Transitive _\u2248\u1d3f_\n    \u2248\u1d3f-trans (mk p) (mk q) = mk (\u2261.trans p q)\n\n    module \u2248\u1d3f-Reasoning where\n      open Trans-Reasoning _\u2248\u1d3f_ \u2248\u1d3f-trans public using () renaming (_\u2248\u27e8_\u27e9_ to _\u2248\u1d3f\u27e8_\u27e9_)\n      infix  2 _\u220e\n\n      _\u220e : \u2200 x \u2192 x \u2248\u1d3f x\n      _ \u220e = mk refl\n\n    module Proof\n        (ddh-hyp : \u2200 A \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A)\n        (otp-lem : \u2200 A m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : IND-CPA.Adv) (b : Bit)\n      where\n        OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n        OTP\u2141-lem d M\u2080 M\u2081 = mk (\n                           sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y))))\n          where\n          pf : \u2200 r x y \u2192 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2080 d (r , x , y , z))\n                       \u2261 count \u03bc\u2124q (\u03bb z \u2192 OTP\u2141 M\u2081 d (r , x , y , z))\n          pf r x y rewrite otp-lem (d r (g^ x) (g^ y)) (M\u2080 r (g^ x)) (M\u2081 r (g^ x)) = refl\n\n        -- moving this definition above OTP\u2141-lem breaks type-checking: ???\n        \u00acb : Bit\n        \u00acb = not b\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA \u00acb A\n\n        pf0,5 : IND-CPA.\u2141 b A \u2257 DDH.\u2141\u2080 A\u1d47\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf2,5 : DDH.\u2141\u2081 A\u1d47 \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf4,5 : IND-CPA.\u2141 \u00acb A \u2257 DDH.\u2141\u2080 A\u00ac\u1d47\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        open \u2248\u1d3f-Reasoning\n\n        final : IND-CPA.\u2141 b A \u2248\u1d3f IND-CPA.\u2141 \u00acb A\n        final = IND-CPA.\u2141 b A\n              \u2248\u1d3f\u27e8 mk (sum-ext \u03bcR (cong Bit\u25b9\u2115 \u2218 pf0,5)) \u27e9\n                DDH.\u2141\u2080 A\u1d47\n              \u2248\u1d3f\u27e8 ddh-hyp A\u1d47 \u27e9\n                DDH.\u2141\u2081 A\u1d47\n              \u2248\u1d3f\u27e8 OTP\u2141-lem (projD A) (projM A b) (projM A \u00acb) \u27e9\n                DDH.\u2141\u2081 A\u00ac\u1d47\n              \u2248\u1d3f\u27e8 mk (\u2261.sym (un-\u2248\u1d3f (ddh-hyp A\u00ac\u1d47))) \u27e9\n                DDH.\u2141\u2080 A\u00ac\u1d47\n              \u2248\u1d3f\u27e8 mk (\u2261.sym (sum-ext \u03bcR (cong Bit\u25b9\u2115 \u2218 pf4,5))) \u27e9\n                IND-CPA.\u2141 \u00acb A\n              \u220e\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : Explorable \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : Explorable R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ m \u03b4 = \u210b \u03b4 \u2295 m\n{-\n\n    \/-\u2219 : \u2200 x y \u2192 \u210b\u27e8 x \u27e9\u2295 y \/ x \u2261 y\n    \/-\u2219 x y = {!!}\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248\u1d3f OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n\n  _\/_ : G \u2192 G \u2192 G\n  x \/ y = x \u2219 (y \u207b\u00b9)\n\n  postulate\n    \/-\u2022 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n    \u22a0-comm : \u2200 x y \u2192 x \u22a0 y \u2261 y \u22a0 x\n\n  comm-^ : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x\n  comm-^ \u03b1 x y = (\u03b1 ^ x)^ y\n               \u2261\u27e8 sym (dist-^-\u22a0 \u03b1 x y) \u27e9\n                  \u03b1 ^ (x \u22a0 y)\n               \u2261\u27e8 cong (_^_ \u03b1) (\u22a0-comm x y) \u27e9\n                  \u03b1 ^ (y \u22a0 x)\n               \u2261\u27e8 dist-^-\u22a0 \u03b1 y x \u27e9\n                  (\u03b1 ^ y)^ x\n               \u220e\n    where open \u2261-Reasoning\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q \u229e-stable\n\n  \u03bcR\u2090 : Explorable (El `R\u2090)\n  \u03bcR\u2090 = \u03bcU \u03bc\u2124q `R\u2090\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum \u03bcR\u2090\n  sumR\u2090-ext = sum-ext \u03bcR\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\/_ \/-\u2022 comm-^ dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090\n  open EB hiding (g^_)\n\n  otp-base-lem : \u2200 (A : G \u2192 Bit) m \u2192 (A \u2218 g^_) \u2248q (A \u2218 g^_ \u2218 _\u229e_ m)\n  otp-base-lem A m = \u229e-stable m (Bit\u25b9\u2115 \u2218 A \u2218 g^_)\n\n  postulate\n    ddh-hyp : (A : DDH.Adv) \u2192 DDH.\u2141\u2080 A \u2248\u1d3f DDH.\u2141\u2081 A\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 IND-CPA.\u2141 A 0b \u2248\u1d3f IND-CPA.\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5c3cfcb7ee57a3ab3309e20e640f6d43c366be08","subject":"Unlock more in sum-setoid","message":"Unlock more in sum-setoid\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-setoid.agda","new_file":"sum-setoid.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_ ; \u2261-setoid)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nopen import Relation.Binary.Sum\nopen import Relation.Binary.Product.Pointwise\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum-setoid where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 (P  : Search A \u2192 \u2605)\n                       (P\u2219 : \u2200 {s\u2080 s\u2081 : Search A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb _\u2219_ f \u2192 s\u2080 _\u2219_ f \u2219 s\u2081 _\u2219_ f))\n                       (Pf : \u2200 x \u2192 P (\u03bb _ f \u2192 f x))\n                     \u2192  P srch\n\nSumInd : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2081\nSumInd {A} sum = \u2200 (P  : Sum A \u2192 \u2605)\n                   (P+ : \u2200 {s\u2080 s\u2081 : Sum A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f))\n                   (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n                \u2192  P sum\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSearchSgExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSgExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                              f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt {A} s\u1d2c = \u2200 {B} op {f g : A \u2192 B} \u2192 f \u2257 g \u2192 s\u1d2c op f \u2261 s\u1d2c op g\n\nSearchExtFun : \u2200 {A B} \u2192 Search (A \u2192 B) \u2192 \u2605\u2081\nSearchExtFun {A}{B} s\u1d2c\u1d2e = \u2200 {M} op {f g : (A \u2192 B) \u2192 M} \u2192 (\u2200 {\u03c6 \u03c8} \u2192 \u03c6 \u2257 \u03c8 \u2192 f \u03c6 \u2261 g \u03c8) \u2192 s\u1d2c\u1d2e op f \u2261 s\u1d2c\u1d2e op g\n\nSearchExtFun-\u00faber : \u2200 {A B} \u2192 (SF : Search (A \u2192 B)) \u2192 SearchInd SF \u2192 SearchExtFun SF\nSearchExtFun-\u00faber sf sf-ind op {f = f}{g} f\u2248g = sf-ind (\u03bb s \u2192 s op f \u2261 s op g) (\u2261.cong\u2082 op) (\u03bb x \u2192 f\u2248g (\u03bb _ \u2192 \u2261.refl))\n\nSearchExtoid : \u2200 {A : Setoid L.zero L.zero} \u2192 Search (Setoid.Carrier A) \u2192 \u2605\u2081\nSearchExtoid {A} s\u1d2c = \u2200 {M} op {f g : Setoid.Carrier A \u2192 M} \u2192 (\u2200 {x y} \u2192 Setoid._\u2248_ A x y \u2192 f x \u2261 g y) \u2192 s\u1d2c op f \u2261 s\u1d2c op g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192\n                    let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumPropoid (As : Setoid L.zero L.zero) : \u2605\u2081 where\n  constructor _,_\n  private\n    A = Setoid.Carrier As\n  field\n    search     : Search A\n    search-ind : SearchInd search\n\n  search-sg-ext : SearchSgExt search\n  search-sg-ext sg {f} {g} f\u2248\u00b0g = search-ind (\u03bb s \u2192 s _ f \u2248 s _ g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  search-ext : SearchExt search\n  search-ext op = search-ind (\u03bb s \u2192 s _ _ \u2261 s _ _) (\u2261.cong\u2082 op)\n\n  search-mono : SearchMono search\n  search-mono _\u2286_ _\u2219-mono_ {f} {g} f\u2286\u00b0g = search-ind (\u03bb s \u2192 s _ f \u2286 s _ g) _\u2219-mono_ f\u2286\u00b0g\n\n  search-swap : SearchSwap search\n  search-swap sg f {s\u1d2e} pf = search-ind (\u03bb s \u2192 s _ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s _ \u2218 flip f)) (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _))) (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\n  searchMon : SearchMon A\n  searchMon m = search _\u2219_\n    where open Mon m\n\n  search-\u03b5 : Search\u03b5 searchMon\n  search-\u03b5 m = search-ind (\u03bb s \u2192 s _ (const \u03b5) \u2248 \u03b5) (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n    where open Mon m\n\n  search-hom : SearchMonHom searchMon\n  search-hom cm f g = search-ind (\u03bb s \u2192 s _ (f \u2219\u00b0 g) \u2248 s _ f \u2219 s _ g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  search-hom\u2032 :\n      \u2200 {S T}\n        (_+_ : Op\u2082 S)\n        (_*_ : Op\u2082 T)\n        (f   : S \u2192 T)\n        (g   : A \u2192 S)\n        (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n        \u2192 f (search _+_ g) \u2261 search _*_ (f \u2218 g)\n  search-hom\u2032 _+_ _*_ f g hom = search-ind (\u03bb s \u2192 f (s _+_ g) \u2261 s _*_ (f \u2218 g))\n                                           (\u03bb p q \u2192 \u2261.trans (hom _ _) (\u2261.cong\u2082 _*_ p q))\n                                           (\u03bb _ \u2192 \u2261.refl)\n\n  StableUnder : (A \u2192 A) \u2192 \u2605\u2081\n  StableUnder p = \u2200 {B} (op : Op\u2082 B) f \u2192 search op f \u2261 search op (f \u2218 p)\n\n  sum : Sum A\n  sum = search _+_\n\n  sum-ind : SumInd sum\n  sum-ind P P+ Pf = search-ind (\u03bb s \u2192 P (s _+_)) P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = search-ext _+_\n\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  SumStableUnder : (A \u2192 A) \u2192 \u2605\n  SumStableUnder p = \u2200 f \u2192 (\u2200 {x y} \u2192 Setoid._\u2248_ As x y \u2192 f x \u2261.\u2261 f y) \u2192 sum f \u2261 sum (f \u2218 p)\n\n  sumStableUnder : \u2200 {p} \u2192 StableUnder p \u2192 SumStableUnder p\n  sumStableUnder SU-p f _ = SU-p _+_ f\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\n  CountStableUnder : (A \u2192 A) \u2192 \u2605\n  CountStableUnder p = \u2200 f \u2192 (\u2200 {x y} \u2192 Setoid._\u2248_ As x y \u2192 f x \u2261.\u2261 f y) \u2192 count f \u2261 count (f \u2218 p)\n\n  countStableUnder : \u2200 {p} \u2192 SumStableUnder p \u2192 CountStableUnder p\n  countStableUnder sumSU-p f f-pres = sumSU-p (Bool.to\u2115 \u2218 f) (\u2261.cong Bool.to\u2115 \u2218 f-pres)\n\n  search-extoid : SearchExtoid {As} search\n  search-extoid op {f = f}{g} f\u2248g = search-ind (\u03bb s\u2081 \u2192 s\u2081 op f \u2261 s\u2081 op g) (\u2261.cong\u2082 op) (\u03bb x \u2192 f\u2248g (Setoid.refl As))\n\nSumProp : \u2605 \u2192 \u2605\u2081\nSumProp A = SumPropoid (\u2261.setoid A)\n\nopen SumPropoid public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumPropoid A) (\u03bcB : SumPropoid B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumPropoid A) (\u03bcB : SumPropoid B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search \u22a4\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumPropoid A \u2192 SumPropoid B \u2192 SumPropoid (A \u228e-setoid B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) op f = search\u1d2c op (\u03bb x \u2192 search\u1d2e op (curry f x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n               \u2192 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd Ps\u1d2e) P P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 \u2218 curry Pf)\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumPropoid A \u2192 SumPropoid B \u2192 SumPropoid (A \u00d7-setoid B)\n\u03bcA \u00d7\u03bc \u03bcB = _ , search-ind \u03bcA \u00d7SearchInd search-ind \u03bcB\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nvsgsum : \u2200 sg {n} \u2192 let open Sgrp sg in\n                    Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n{-\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumPropoid A \u2192 SumPropoid (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n-}\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n{-\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bc\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf = \n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) \n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum-setoid where\n\nSEToid = Setoid L.zero L.zero\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 (P  : Search A \u2192 \u2605)\n                       (P\u2219 : \u2200 {s\u2080 s\u2081 : Search A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb _\u2219_ f \u2192 s\u2080 _\u2219_ f \u2219 s\u2081 _\u2219_ f))\n                       (Pf : \u2200 x \u2192 P (\u03bb _ f \u2192 f x))\n                     \u2192  P srch\n\nSumInd : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2081\nSumInd {A} sum = \u2200 (P  : Sum A \u2192 \u2605)\n                   (P+ : \u2200 {s\u2080 s\u2081 : Sum A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f))\n                   (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n                \u2192  P sum\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSearchSgExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSgExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                              f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt {A} s\u1d2c = \u2200 {B} op {f g : A \u2192 B} \u2192 f \u2257 g \u2192 s\u1d2c op f \u2261 s\u1d2c op g\n\nSearchExtFun : \u2200 {A B} \u2192 Search (A \u2192 B) \u2192 \u2605\u2081\nSearchExtFun {A}{B} s\u1d2c\u1d2e = \u2200 {M} op {f g : (A \u2192 B) \u2192 M} \u2192 (\u2200 {\u03c6 \u03c8} \u2192 \u03c6 \u2257 \u03c8 \u2192 f \u03c6 \u2261 g \u03c8) \u2192 s\u1d2c\u1d2e op f \u2261 s\u1d2c\u1d2e op g\n\nSearchExtFun-\u00faber : \u2200 {A B} \u2192 (SF : Search (A \u2192 B)) \u2192 SearchInd SF \u2192 SearchExtFun SF\nSearchExtFun-\u00faber sf sf-ind op {f = f}{g} f\u2248g = sf-ind (\u03bb s \u2192 s op f \u2261 s op g) (\u2261.cong\u2082 op) (\u03bb x \u2192 f\u2248g (\u03bb _ \u2192 \u2261.refl))\n\nSearchExtoid : \u2200 {A : Setoid L.zero L.zero} \u2192 Search (Setoid.Carrier A) \u2192 \u2605\u2081\nSearchExtoid {A} s\u1d2c = \u2200 {M} op {f g : Setoid.Carrier A \u2192 M} \u2192 (\u2200 {x y} \u2192 Setoid._\u2248_ A x y \u2192 f x \u2261 g y) \u2192 s\u1d2c op f \u2261 s\u1d2c op g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192\n                    let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumPropoid (As : Setoid L.zero L.zero) : \u2605\u2081 where\n  constructor _,_\n  private\n    A = Setoid.Carrier As\n  field\n    search     : Search A\n    search-ind : SearchInd search\n\n  search-sg-ext : SearchSgExt search\n  search-sg-ext sg {f} {g} f\u2248\u00b0g = search-ind (\u03bb s \u2192 s _ f \u2248 s _ g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  search-ext : SearchExt search\n  search-ext op = search-ind (\u03bb s \u2192 s _ _ \u2261 s _ _) (\u2261.cong\u2082 op)\n\n  search-mono : SearchMono search\n  search-mono _\u2286_ _\u2219-mono_ {f} {g} f\u2286\u00b0g = search-ind (\u03bb s \u2192 s _ f \u2286 s _ g) _\u2219-mono_ f\u2286\u00b0g\n\n  search-swap : SearchSwap search\n  search-swap sg f {s\u1d2e} pf = search-ind (\u03bb s \u2192 s _ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s _ \u2218 flip f)) (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _))) (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\n  searchMon : SearchMon A\n  searchMon m = search _\u2219_\n    where open Mon m\n\n  search-\u03b5 : Search\u03b5 searchMon\n  search-\u03b5 m = search-ind (\u03bb s \u2192 s _ (const \u03b5) \u2248 \u03b5) (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n    where open Mon m\n\n  search-hom : SearchMonHom searchMon\n  search-hom cm f g = search-ind (\u03bb s \u2192 s _ (f \u2219\u00b0 g) \u2248 s _ f \u2219 s _ g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  search-hom\u2032 :\n      \u2200 {S T}\n        (_+_ : Op\u2082 S)\n        (_*_ : Op\u2082 T)\n        (f   : S \u2192 T)\n        (g   : A \u2192 S)\n        (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n        \u2192 f (search _+_ g) \u2261 search _*_ (f \u2218 g)\n  search-hom\u2032 _+_ _*_ f g hom = search-ind (\u03bb s \u2192 f (s _+_ g) \u2261 s _*_ (f \u2218 g))\n                                           (\u03bb p q \u2192 \u2261.trans (hom _ _) (\u2261.cong\u2082 _*_ p q))\n                                           (\u03bb _ \u2192 \u2261.refl)\n\n  StableUnder : (A \u2192 A) \u2192 \u2605\u2081\n  StableUnder p = \u2200 {B} (op : Op\u2082 B) f \u2192 search op f \u2261 search op (f \u2218 p)\n\n  sum : Sum A\n  sum = search _+_\n\n  sum-ind : SumInd sum\n  sum-ind P P+ Pf = search-ind (\u03bb s \u2192 P (s _+_)) P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = search-ext _+_\n\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  SumStableUnder : (A \u2192 A) \u2192 \u2605\n  SumStableUnder p = \u2200 f \u2192 (\u2200 {x y} \u2192 Setoid._\u2248_ As x y \u2192 f x \u2261.\u2261 f y) \u2192 sum f \u2261 sum (f \u2218 p)\n\n  sumStableUnder : \u2200 {p} \u2192 StableUnder p \u2192 SumStableUnder p\n  sumStableUnder SU-p f _ = SU-p _+_ f\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\n  CountStableUnder : (A \u2192 A) \u2192 \u2605\n  CountStableUnder p = \u2200 f \u2192 (\u2200 {x y} \u2192 Setoid._\u2248_ As x y \u2192 f x \u2261.\u2261 f y) \u2192 count f \u2261 count (f \u2218 p)\n\n  countStableUnder : \u2200 {p} \u2192 SumStableUnder p \u2192 CountStableUnder p\n  countStableUnder sumSU-p f f-pres = sumSU-p (Bool.to\u2115 \u2218 f) (\u2261.cong Bool.to\u2115 \u2218 f-pres)\n\n  search-extoid : SearchExtoid {As} search\n  search-extoid op {f = f}{g} f\u2248g = search-ind (\u03bb s\u2081 \u2192 s\u2081 op f \u2261 s\u2081 op g) (\u2261.cong\u2082 op) (\u03bb x \u2192 f\u2248g (Setoid.refl As))\n\nopen SumPropoid public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumPropoid A) (\u03bcB : SumPropoid B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumPropoid A) (\u03bcB : SumPropoid B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n{-\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search \u22a4\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) op f = search\u1d2c op (\u03bb x \u2192 search\u1d2e op (curry f x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n               \u2192 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd Ps\u1d2e) P P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 \u2218 curry Pf)\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u00d7 B)\n\u03bcA \u00d7\u03bc \u03bcB = _ , search-ind \u03bcA \u00d7SearchInd search-ind \u03bcB\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nvsgsum : \u2200 sg {n} \u2192 let open Sgrp sg in\n                    Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bc\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf = \n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) \n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"89fd5fa1fbb32e7dfbfd8995589cef232308cac3","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: 1176ac103b139489f79a9d236b7f707d\n\ndarcs-hash:20100719141536-3bd4e-7a766f5ad4169bb90a7d945092ee748da7180ba2.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/OnlyAxioms\/GeneralHints.agda","new_file":"Test\/Succeed\/OnlyAxioms\/GeneralHints.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the use of general hints\n------------------------------------------------------------------------------\n\nmodule Test.Succeed.OnlyAxioms.GeneralHints where\n\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : (n : D) \u2192 N n \u2192 N (succ n)\n-- The data constructors are general hints. They are translate as axioms.\n{-# ATP hint zN #-}\n{-# ATP hint sN #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the use of general hints\n------------------------------------------------------------------------------\n\nmodule Test.Succeed.OnlyAxioms.GeneralHints where\n\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : (n : D) \u2192 N n \u2192 N (succ n)\n\n-- A general hints. They are translate as axioms.\n{-# ATP hint zN #-}\n{-# ATP hint sN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e64316878d3bef9df458888d59a4cf01835a76c4","subject":"less asinine names #14","message":"less asinine names #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values and errors are disjoint\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = boxedval-not-indet x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole () ind) (ITCastID x\u2081)\n  lem2 (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  lem2 (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  lem2 (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  lem2 (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  lem2 (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors and expressions that step are disjoint\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail caserr-not-step\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!}\n\n    -- congruence caserr-not-step\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter er) = ve bv (ce-castarr er)\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong FHOuter er) = ve bv (ce-castth er)\n  ve (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVVal x) (CECong FHOuter er) = ve (BVVal x) er\n  ve (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  ve (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  ve (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  ve (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  vs (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  vs (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (ce-nehole err)\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (ce-castarr err)\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (ce-castth err)\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (ce-castht err x)\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole () ind) (ITCastID x\u2081)\n  lem2 (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  lem2 (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  lem2 (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  lem2 (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  lem2 (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail cases\n  es (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  es (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  es (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  es (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!}\n\n    -- congruence cases\n  es (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = es ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  es (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = ve (BVVal VLam) ce\n  es (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  es (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  es (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  es (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  es (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = es d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  es (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = es (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  es (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = fs x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  es (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = fe x\u2084 d1err\n  ... | Inr d2err = es d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  es (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  es (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  es (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = es (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  es (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = es (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  es (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- es {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  es (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = es (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c496c41d4f713230d125f5f19f7223eea9aaa2de","subject":"More one-time-pad related lemmas: \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047","message":"More one-time-pad related lemmas: \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047\n","repos":"crypto-agda\/crypto-agda","old_file":"single-bit-one-time-pad.agda","new_file":"single-bit-one-time-pad.agda","new_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP\nopen import Data.Product renaming (map to <_\u00d7_>)\nopen import Data.Nat.NP\nimport Data.Vec.NP as V\nopen V using (Vec; take; drop; drop\u2032; take\u2032; _++_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) where\n    open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n    kont\u2080-not : \u2200 b k \u2192 kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n    kont\u2080-not b k rewrite xor-not-not b k = \u2261.refl\n\n    open \u2261-Reasoning\n\n    lem\u2082 : \u2200 b \u2192 count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem\u2082 b = count\u21ba (runA b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n          \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not b 0b) (kont\u2080-not b 1b) \u27e9\n            count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n          \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n            count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n          \u2261\u27e8 \u2261.refl \u27e9\n            count\u21ba (runA (not b)) \u220e\n\n    lem\u2083 : Safe\u2141? runA\n    lem\u2083 = lem\u2082 0b\n\n    -- A specialized version of lem\u2082 (\u2248lem\u2083)\n    lem\u2084 : Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\n    lem\u2084    = count\u21ba (runA 0b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n            \u2261\u27e8 \u2261.cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b 0b) (kont\u2080-not 0b 1b) \u27e9\n              count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n            \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n              count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n            \u2261\u27e8 \u2261.refl \u27e9\n              count\u21ba (runA 1b) \u220e\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = \u2261.refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\n\u2047 : \u2200 {n} \u2192 \u21ba n (Bits n)\n\u2047 = random\n\n\nlem'' : \u2200 {k} (f : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 2* #\u27e8 f \u27e9\nlem'' f = \u2261.refl\n\nlem' : \u2200 {k} (f g : Bits k \u2192 Bit) \u2192 #\u27e8 f \u2218 tail \u27e9 \u2261 #\u27e8 g \u2218 tail \u27e9 \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\nlem' f g pf = 2*-inj (\u2261.trans (lem'' f) (\u2261.trans pf (\u2261.sym (lem'' g))))\n\ndrop-tail : \u2200 k {n a} {A : Set a} \u2192 drop (suc k) {n} \u2257 drop k \u2218 tail {A = A}\ndrop-tail k (x \u2237 xs) = V.drop-\u2237 k x xs\n\nlemdrop\u2032 : \u2200 {k n} (f : Bits n \u2192 Bit) \u2192 #\u27e8 f \u2218 drop\u2032 k \u27e9 \u2261 \u27e82^ k * #\u27e8 f \u27e9 \u27e9\nlemdrop\u2032 {zero} f = \u2261.refl\nlemdrop\u2032 {suc k} f = #\u27e8 f \u2218 drop\u2032 k \u2218 tail \u27e9\n                   \u2261\u27e8 lem'' (f \u2218 drop\u2032 k) \u27e9\n                     2* #\u27e8 f \u2218 drop\u2032 k \u27e9\n                   \u2261\u27e8 \u2261.cong 2*_ (lemdrop\u2032 {k} f) \u27e9\n                     2* \u27e82^ k * #\u27e8 f \u27e9 \u27e9 \u220e\n                    where open \u2261-Reasoning\n\nvswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nvswap m xs = drop\u2032 m xs ++ take\u2032 m xs\n\n\n\n\n\n\n\u2248\u1d2c\u2032-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c\u2032 toss\n\u2248\u1d2c\u2032-toss true Adv = \u2115\u00b0.+-comm (count\u21ba (Adv true)) _\n\u2248\u1d2c\u2032-toss false Adv = \u2261.refl\n\n\u2248\u1d2c-toss : \u2200 b \u2192 \u27ea b \u27eb\u1d30 \u27e8xor\u27e9 toss \u2248\u1d2c toss\n\u2248\u1d2c-toss b Adv = \u2248\u1d2c\u2032-toss b (return\u1d30 \u2218 Adv)\n\n-- should be equivalent to #-comm if \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 x were convertible to \u27ea _\u2295_ m \u00b7 x \u27eb\n\u2248\u1d2c-\u2047 : \u2200 {k} (m : Bits k) \u2192 \u27ea m \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u2047\n\u2248\u1d2c-\u2047 {zero}  _       _ = \u2261.refl\n\u2248\u1d2c-\u2047 {suc k} (h \u2237 m) Adv\n  rewrite \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 0b))\n        | \u2248\u1d2c-\u2047 m (Adv \u2218 _\u2237_ (h xor 1b))\n        = \u2248\u1d2c\u2032-toss h (\u03bb x \u2192 \u27ea Adv \u2218 _\u2237_ x \u00b7 \u2047 \u27eb)\n\n\u2248\u1d2c-\u2047\u2082 : \u2200 {k} (m\u2080 m\u2081 : Bits k) \u2192 \u27ea m\u2080 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m\u2081 \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2082 {k} m\u2080 m\u2081 = \u2248\u1d2c.trans {k} (\u2248\u1d2c-\u2047 m\u2080) (\u2248\u1d2c.sym {k} (\u2248\u1d2c-\u2047 m\u2081))\n\n\u2248\u1d2c-\u2047\u2083 : \u2200 {k} (m : Bit \u2192 Bits k) (b : Bit) \u2192 \u27ea m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2083 m b = \u2248\u1d2c-\u2047\u2082 (m b) (m (not b))\n\n\u2248\u1d2c-\u2047\u2084 : \u2200 {k} (m : Bits k \u00d7 Bits k) (b : Bit) \u2192 \u27ea proj m b \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047 \u2248\u1d2c \u27ea proj m (not b) \u27eb\u1d30 \u27e8\u2295\u27e9 \u2047\n\u2248\u1d2c-\u2047\u2084 = \u2248\u1d2c-\u2047\u2083 \u2218 proj\n","old_contents":"module single-bit-one-time-pad where\n\nopen import Function\nopen import Data.Bool.NP\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Data.Bits\nopen import flipbased-implem\nopen import program-distance\n\nK = Bit\nM = Bit\nC = Bit\n\nrecord Adv (S\u2080 S\u2081 S\u2082 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    step\u2081 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n    step\u2082 : C \u00d7 S\u2081 \u2192 S\u2082\n    step\u2083 : S\u2082 \u2192 Bit\n\nrecord Adv\u2081 (S\u2080 S\u2081 : Set) ca : Set where\n  constructor mk\n  field\n    step\u2080 : \u21ba ca S\u2080\n    -- step\u2081 s\u2080 = id , s\u2080\n    step\u2082 : C \u00d7 S\u2080 \u2192 S\u2081\n    step\u2083 : S\u2081 \u2192 Bit\n\nAdv\u2082 : \u2200 ca \u2192 Set\nAdv\u2082 ca = \u21ba ca (C \u2192 Bit)\n\nmodule Run\u2141\u2032 {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open Adv A\n  E : M \u2192 \u21ba 1 C\n  E m = toss >>= \u03bb k \u2192 return\u21ba (m xor k)\n  run\u2141\u2032 : \u2141? (ca + 1)\n  run\u2141\u2032 b = step\u2080 >>= \u03bb s\u2080 \u2192\n            case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n            E (m b) >>= \u03bb c \u2192\n            return\u21ba (step\u2083 (step\u2082 (c , s\u2081)))}\n\nmodule Run\u2141 {S\u2080 S\u2081 S\u2082 ca} (E : K \u2192 M \u2192 C) (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n  open Adv A\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n    step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n    case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n    let c = E k (m b) in\n    step\u2083 (step\u2082 (c , s\u2081))}\n\n  run\u2141 : EXP (1 + ca)\n  run\u2141 = toss >>= kont\u2080\n\n  {- looks wrong\nmodule Run\u2141-Properties {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b k : Bit) where\n  open Run\u2141 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = {!refl!}\n  -}\n\nmodule SymAdv (homPrgDist : HomPrgDist) {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  step\u2081\u2032 : S\u2080 \u2192 (Bit \u2192 M) \u00d7 S\u2081\n  step\u2081\u2032 s\u2080 = case step\u2081 s\u2080 of \u03bb { (m , s\u2081) \u2192 (m \u2218 not , s\u2081) }\n  symA : Adv S\u2080 S\u2081 S\u2082 ca\n  symA = mk step\u2080 step\u2081\u2032 step\u2082 (not \u2218 step\u2083)\n\n  symA\u2032 : Adv S\u2080 S\u2081 S\u2082 ca\n  symA\u2032 = mk step\u2080 step\u2081\u2032 step\u2082 step\u2083\n\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  open Run\u2141 _xor_ symA renaming (run\u2141 to runSymA)\n  open Run\u2141 _xor_ symA\u2032 renaming (run\u2141 to runSymA\u2032)\n  not\u21ba : \u2200 {n} \u2192 EXP n \u2192 EXP n\n  not\u21ba = map\u21ba not\n  {-\n  helper : \u2200 {n} (g\u2080 g\u2081 : EXP n) \u2192 g\u2080 ]-[ g\u2081 \u2192 not\u21ba g\u2080 ]-[ not\u21ba g\u2081\n  helper = {!!}\n  lem : runA \u21d3 runSymA\n  lem A-breaks-E = {!helper (uunSymA\u2032 0b) (runSymA\u2032 1)!}\n     where pf : breaks runSymA\n           pf = {!!}\n  -}\n\nmodule Run\u2141\u2082 {ca} (A : Adv\u2082 ca) (b : Bit) where\n  E : Bit \u2192 M \u2192 C\n  E k m = m xor k\n\n  m : Bit \u2192 Bit\n  m = id\n\n  kont\u2080 : \u2141? _\n  kont\u2080 k =\n      A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n\n      {-\n    kont\u2080\u2032 : \u2141? _\n    kont\u2080\u2032 k =\n      conv-Adv A \u25b9\u21ba \u03bb f \u2192\n      f (E k (m b))\n      -}\n\n  run\u2141\u2082 : EXP (1 + ca)\n  run\u2141\u2082 = toss >>= kont\u2080\n\nmodule Run\u2141\u2082-Properties {ca} (A : Adv\u2082 ca) (b k : Bit) where\n  open Run\u2141\u2082 A\n  kont\u2080-not : kont\u2080 b k \u2261 kont\u2080 (not b) (not k)\n  kont\u2080-not rewrite xor-not-not b k = refl\n\nconv-Adv : \u2200 {ca S\u2080 S\u2081 S\u2082} \u2192 Adv S\u2080 S\u2081 S\u2082 ca \u2192 Adv\u2082 ca\nconv-Adv A = step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n             case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n             \u03bb c \u2192 m (step\u2083 (step\u2082 (c , s\u2081)))}\n  where open Adv A\n\nopen \u2261-Reasoning\n\nmodule Conv-Adv-Props (homPrgDist : HomPrgDist) {ca S\u2080 S\u2081 S\u2082} (A : Adv S\u2080 S\u2081 S\u2082 ca) where\n  open HomPrgDist homPrgDist\n  open Adv A\n  open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n  -- open Run\u2141-Properties A\n\n  A\u2032 : Adv\u2082 ca\n  A\u2032 = conv-Adv A\n  open Run\u2141\u2082 A\u2032 using () renaming (kont\u2080 to kont\u2080\u2032; run\u2141\u2082 to runA\u2032)\n\n  kont\u2080\u2032\u2032 : \u2200 b k \u2192 EXP ca\n  kont\u2080\u2032\u2032 b k =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 ((m b) xor k , s\u2081)))}\n\n  kont\u2080\u2032\u2032\u2032 : \u2200 b\u2032 \u2192 EXP ca\n  kont\u2080\u2032\u2032\u2032 b\u2032 =\n      step\u2080 \u25b9\u21ba \u03bb s\u2080 \u2192\n      case step\u2081 s\u2080 of \u03bb {(m , s\u2081) \u2192\n      m (step\u2083 (step\u2082 (b\u2032 , s\u2081)))}\n\n  {-\n  kont\u2032\u2032-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032\u2032 b k)\n  kont\u2032\u2032-lem true true = {!!}\n  kont\u2032\u2032-lem false true = {!!}\n  kont\u2032\u2032-lem true false = {!!}\n  kont\u2032\u2032-lem false false = {!!}\n\n  kont-lem : \u2200 b k \u2192 count\u21ba (kont\u2080 b k) \u2261 count\u21ba (kont\u2080\u2032 b k)\n  kont-lem b true = {!!}\n  kont-lem b false = {!!}\n\n  conv-Adv-lem : runA \u2248\u2141? runA\u2032\n  conv-Adv-lem b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 _+_ (kont-lem b 0b) (kont-lem b 1b) \u27e9\n          count\u21ba (kont\u2080\u2032 b 0b) + count\u21ba (kont\u2080\u2032 b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA\u2032 b) \u220e\n\n  conv-Adv-sound : runA \u21d3 runA\u2032\n  conv-Adv-sound = ]-[-cong-\u2248\u21ba (conv-Adv-lem 0b) (conv-Adv-lem 1b)\n  -}\n-- Cute fact: this is true by computation!\ncount\u21ba-toss->>= : \u2200 {c} (f : \u2141? c) \u2192 count\u21ba (toss >>= f) \u2261 count\u21ba (f 0b) + count\u21ba (f 1b)\ncount\u21ba-toss->>= f = refl\n\n{-\nmodule Run\u2141-Properties' {S\u2080 S\u2081 S\u2082 ca} (A : Adv S\u2080 S\u2081 S\u2082 ca) (b : Bit) where\n    open Run\u2141 _xor_ A renaming (run\u2141 to runA)\n    lem : count\u21ba (runA b) \u2261 count\u21ba (runA (not b))\n    lem = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) {x = kont\u2080 b 0b} {kont\u2080 (not b) 1b} {kont\u2080 b 1b} {kont\u2080 (not b) 0b} {!!} {!!} \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n-}\nopen import program-distance\nopen import Relation.Nullary\n\nlem\u2082 : \u2200 {ca} (A : Adv\u2082 ca) b \u2192 count\u21ba (Run\u2141\u2082.run\u2141\u2082 A b) \u2261 count\u21ba (Run\u2141\u2082.run\u2141\u2082 A (not b))\nlem\u2082 A b = count\u21ba (runA b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 b 0b) + count\u21ba (kont\u2080 b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b) (kont\u2080-not 1b) \u27e9\n          count\u21ba (kont\u2080 (not b) 1b) + count\u21ba (kont\u2080 (not b) 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 (not b) 1b)) _ \u27e9\n          count\u21ba (kont\u2080 (not b) 0b) + count\u21ba (kont\u2080 (not b) 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA (not b)) \u220e\n  where open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n        open Run\u2141\u2082-Properties A b\n\nlem\u2083 : \u2200 {ca} (A : Adv\u2082 ca) \u2192 Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\nlem\u2083 A = lem\u2082 A 0b\n\n-- A specialized version of lem\u2082\nlem\u2084 : \u2200 {ca} (A : Adv\u2082 ca) \u2192 Safe\u2141? (Run\u2141\u2082.run\u2141\u2082 A)\nlem\u2084 A  = count\u21ba (runA 0b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (kont\u2080 0b 0b) + count\u21ba (kont\u2080 0b 1b)\n        \u2261\u27e8 cong\u2082 (_+_ on count\u21ba) (kont\u2080-not 0b) (kont\u2080-not 1b) \u27e9\n          count\u21ba (kont\u2080 1b 1b) + count\u21ba (kont\u2080 1b 0b)\n        \u2261\u27e8 \u2115\u00b0.+-comm (count\u21ba (kont\u2080 1b 1b)) _ \u27e9\n          count\u21ba (kont\u2080 1b 0b) + count\u21ba (kont\u2080 1b 1b)\n        \u2261\u27e8 refl \u27e9\n          count\u21ba (runA 1b) \u220e\n  where open Run\u2141\u2082 A renaming (run\u2141\u2082 to runA)\n        open Run\u2141\u2082-Properties A 0b\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cbb81bff855e175c0ad11d2c2c0312873abe8085","subject":"Agda: sketch proof that foldGroup is indeed an homomorphism","message":"Agda: sketch proof that foldGroup is indeed an homomorphism\n\nI should fill in too many lemmas to make this go through.\n\nOld-commit-hash: 5c3e9a678d7df2110bcc23ec8b1c432ec461e97e\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/FoldableBag.agda","new_file":"experimental\/FoldableBag.agda","new_contents":"module FoldableBag where\n\nopen import FoldableBagParametric\nimport Level as L\n\nopen import Algebra\nopen import Function\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nimport Data.Nat as N\nimport Data.Integer as Z\n  \nopen import Data.Bool \nopen import Data.Maybe\nopen import Data.List as List using (List)\n\nopen import Data.Product\n\n-- Note that hom is what I call foldGroup elsewhere!\n\n-- This is the mathematical definition of group homomorphism. This is mostly\n-- annoying to prove because one can't write the blackboard definition directly\n-- (since bags are not freely generated).\nhomIsAnHom : \u2200 {T} {{oT : Ord T}} G f (b\u2081 b\u2082 : Bag T) \u2192 let open AbelianGroup G in\n  hom G f (union b\u2081 b\u2082) \u2261 hom G f b\u2081 \u2219 hom G f b\u2082\nhomIsAnHom {{oT}} G f (b\u207a\u2081 , b\u207b\u2081) (b\u207a\u2082 , b\u207b\u2082) =\n  begin\n    hom G f (union (b\u207a\u2081 , b\u207b\u2081) (b\u207a\u2082 , b\u207b\u2082))\n  \u2261\u27e8\u27e9\n    hom G f (uunion b\u207a\u2081 b\u207a\u2082 , uunion b\u207b\u2081 b\u207b\u2082)\n  -- TODO: Lots of straightforward group manipulation\n  \u2261\u27e8 {!!} \u27e9\n     hom G f (b\u207a\u2081 , b\u207b\u2081) \u2219 hom G f (b\u207a\u2082 , b\u207b\u2082)\n  \u220e\n   where\n     open AbelianGroup G\n     open \u2261-Reasoning\n     open import Data.List.Properties\n     open import Sorting\n\n     open Sort ord hiding (insert; toList)\n\n-- Simplest possible definition of the derivative of hom (in the semantic domain).\n\nhomDelta : \u2200 {T} {{oT : Ord T}} \u2192 (G : AbelianGroup L.zero L.zero) \u2192 let U = AbelianGroup.Carrier G in (T \u2192 U) \u2192 Bag T \u2192 \u0394Bag T \u2192 U\nhomDelta G f b db = hom G f db\n\n-- This states that the derivative\n\nhomDeltaCorrect : \u2200 G f b db \u2192 let open AbelianGroup G in\n  hom G f (union b db) \u2261 hom G f b \u2219 homDelta G f b db\nhomDeltaCorrect G f b db =\n  begin\n    hom G f (union b db)\n  \u2261\u27e8  homIsAnHom G f b db  \u27e9\n    hom G f b \u2219 hom G f db\n  \u2261\u27e8\u27e9\n    hom G f b \u2219 homDelta G f b db\n  \u220e\n  where\n    open AbelianGroup G\n    open \u2261-Reasoning\n    open import Data.List.Properties\n\n--postulate BagGroup :  \u2200 {T} (oT : Ord T) \u2192 AbelianGroup L.zero L.zero\n\nmap\u2081 : \u2200 {A B} {oA : Ord A} {oB : Ord B} \u2192 (A \u2192 B) \u2192 Bag A \u2192 Bag B\nmap\u2081 f = hom BagGroup (singleton \u2218 f)\n\nflatMap : \u2200 {A B} {oA : Ord A} {oB : Ord B} \u2192 (A \u2192 Bag B) \u2192 Bag A \u2192 Bag B\nflatMap f = hom BagGroup f\n\n-- Use instance arguments for Ord.\n\nfilter : \u2200 {A} {oA : Ord A} \u2192 (A \u2192 Bool) \u2192 Bag A \u2192 Bag A\nfilter p b = hom BagGroup (\u03bb el \u2192 if (p el) then (singleton el) else empty) b\n\n{-\nTODOs:\n\n * Express basic functions using hom:\n   remove \n   ...\n\n * State and prove the main theorems. Define changes on bags (as bags first?),\n   then the derivative of hom, then write the correctness theorem.\n\n-}\n\n{-\npostulate Bag : Set \u2192 Set\n\nhom : let B =  AbelianGroup.Carrier G in \u2200 {A} \u2192 (A \u2192 B) \u2192 Bag A \u2192 B\nhom = {!!}\n-}\n\n\n-- Whoops\u00b2, that's hard to implement. Let's just postulate it.\n--map\u2080 = {!T.map!}\n\n-- map : ({k : Key} \u2192 Value k \u2192 Value k) \u2192 Tree \u2192 Tree\n-- map f (tree t) = tree $ Indexed.map f t\n\n-- _\u2208?_ : Key \u2192 Tree \u2192 Bool\n-- k \u2208? t = maybeToBool (lookup k t)\n\n------------------------------------------------\n-- Elimination forms for trees\n------------------------------------------------\n\n-- -- Naive implementations of union.\n\n-- unionWith : (\u2200 {k} \u2192 Value k \u2192 Value k \u2192 Value k) \u2192\n--             -- Left \u2192 right \u2192 result.\n--             Tree \u2192 Tree \u2192 Tree\n\n-- union : Tree \u2192 Tree \u2192 Tree\n-- union = unionWith const\n\n-- unionsWith : (\u2200 {k} \u2192 Value k \u2192 Value k \u2192 Value k) \u2192 List Tree \u2192 Tree\n\n-- -- Left-biased.\n\n-- unions : List Tree \u2192 Tree\n-- unions = unionsWith const\n\n","old_contents":"module FoldableBag where\n\nopen import FoldableBagParametric\nimport Level as L\n\nopen import Algebra\nopen import Function\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nimport Data.Nat as N\nimport Data.Integer as Z\n  \nopen import Data.Bool \nopen import Data.Maybe\nopen import Data.List as List using (List)\n\nopen import Data.Product\n\n-- Simplest possible definition of the derivative of hom (in the semantic domain).\n\nhomDelta : \u2200 {T} {{oT : Ord T}} \u2192 (G : AbelianGroup L.zero L.zero) \u2192 let U = AbelianGroup.Carrier G in (T \u2192 U) \u2192 Bag T \u2192 \u0394Bag T \u2192 U\nhomDelta G f b db = hom G f db\n\n-- This states that the derivative\n\nhomDeltaCorrect : \u2200 G f b db \u2192 let open AbelianGroup G in\n  hom G f (union b db) \u2261 hom G f b \u2219 homDelta G f b db\nhomDeltaCorrect G f b db =\n  begin\n    hom G f (union b db)\n  \u2261\u27e8 {!!} \u27e9\n    hom G f b \u2219 hom G f db\n  \u2261\u27e8\u27e9\n    hom G f b \u2219 homDelta G f b db\n  \u220e\n  where\n    open AbelianGroup G\n    open \u2261-Reasoning\n    open import Data.List.Properties\n\n--postulate BagGroup :  \u2200 {T} (oT : Ord T) \u2192 AbelianGroup L.zero L.zero\n\nmap\u2081 : \u2200 {A B} {oA : Ord A} {oB : Ord B} \u2192 (A \u2192 B) \u2192 Bag A \u2192 Bag B\nmap\u2081 f = hom BagGroup (singleton \u2218 f)\n\nflatMap : \u2200 {A B} {oA : Ord A} {oB : Ord B} \u2192 (A \u2192 Bag B) \u2192 Bag A \u2192 Bag B\nflatMap f = hom BagGroup f\n\n-- Use instance arguments for Ord.\n\nfilter : \u2200 {A} {oA : Ord A} \u2192 (A \u2192 Bool) \u2192 Bag A \u2192 Bag A\nfilter p b = hom BagGroup (\u03bb el \u2192 if (p el) then (singleton el) else empty) b\n\n{-\nTODOs:\n\n * Express basic functions using hom:\n   remove \n   ...\n\n * State and prove the main theorems. Define changes on bags (as bags first?),\n   then the derivative of hom, then write the correctness theorem.\n\n-}\n\n{-\npostulate Bag : Set \u2192 Set\n\nhom : let B =  AbelianGroup.Carrier G in \u2200 {A} \u2192 (A \u2192 B) \u2192 Bag A \u2192 B\nhom = {!!}\n-}\n\n\n-- Whoops\u00b2, that's hard to implement. Let's just postulate it.\n--map\u2080 = {!T.map!}\n\n-- map : ({k : Key} \u2192 Value k \u2192 Value k) \u2192 Tree \u2192 Tree\n-- map f (tree t) = tree $ Indexed.map f t\n\n-- _\u2208?_ : Key \u2192 Tree \u2192 Bool\n-- k \u2208? t = maybeToBool (lookup k t)\n\n------------------------------------------------\n-- Elimination forms for trees\n------------------------------------------------\n\n-- -- Naive implementations of union.\n\n-- unionWith : (\u2200 {k} \u2192 Value k \u2192 Value k \u2192 Value k) \u2192\n--             -- Left \u2192 right \u2192 result.\n--             Tree \u2192 Tree \u2192 Tree\n\n-- union : Tree \u2192 Tree \u2192 Tree\n-- union = unionWith const\n\n-- unionsWith : (\u2200 {k} \u2192 Value k \u2192 Value k \u2192 Value k) \u2192 List Tree \u2192 Tree\n\n-- -- Left-biased.\n\n-- unions : List Tree \u2192 Tree\n-- unions = unionsWith const\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25f0137572c2e43bdea68c2f240867df19ffa108","subject":"improved proof for leq-trans","message":"improved proof for leq-trans\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n\n\ndata   False : Set where\nrecord True  : Set where\n\n\ntrivial : True\ntrivial = _\n\n\nisTrue : Bool -> Set\nisTrue true  = True\nisTrue false = False\n\n\n_<_ : \u2115 -> \u2115 -> Bool\n_   < O   = false\nO   < S n = true\nS m < S n = m < n\n\nlength : {A : Set} -> List A -> \u2115\nlength [] = O\nlength (x :: xs) = S (length xs)\n\nlookup : {A : Set}(xs : List A)(n : \u2115) -> isTrue (n < length xs) -> A\nlookup (x :: xs) O b = x\nlookup (x :: xs) (S n) b = lookup xs n b\nlookup [] a ()\n\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\n\ndata _\u2264_ : \u2115 -> \u2115 -> Set where\n  \u2264O : {m n : \u2115} -> m == n -> m \u2264 n\n  \u2264I : {m n : \u2115} -> m \u2264  n -> m \u2264 S n\n\n\nleq-trans : {l m n : \u2115} -> l \u2264 m -> m \u2264 n -> l \u2264 n\nleq-trans a (\u2264O refl) = a\nleq-trans a (\u2264I b)    = \u2264I (leq-trans a b)\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n\n\ndata   False : Set where\nrecord True  : Set where\n\n\ntrivial : True\ntrivial = _\n\n\nisTrue : Bool -> Set\nisTrue true  = True\nisTrue false = False\n\n\n_<_ : \u2115 -> \u2115 -> Bool\n_   < O   = false\nO   < S n = true\nS m < S n = m < n\n\nlength : {A : Set} -> List A -> \u2115\nlength [] = O\nlength (x :: xs) = S (length xs)\n\nlookup : {A : Set}(xs : List A)(n : \u2115) -> isTrue (n < length xs) -> A\nlookup (x :: xs) O b = x\nlookup (x :: xs) (S n) b = lookup xs n b\nlookup [] a ()\n\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\n\ndata _\u2264_ : \u2115 -> \u2115 -> Set where\n  \u2264O : {m n : \u2115} -> m == n -> m \u2264 n\n  \u2264I : {m n : \u2115} -> m \u2264  n -> m \u2264 S n\n\n\nleq-trans : {l m n : \u2115} -> l \u2264 m -> m \u2264 n -> l \u2264 n\nleq-trans (\u2264O refl) b = b\nleq-trans a (\u2264O refl) = a\nleq-trans (\u2264I a) (\u2264I b) = \u2264I (leq-trans (\u2264I a) b)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1f9a0adb2fdaddbdd525b0cda3df4325249bfeb4","subject":"Fixed a misplaced importation.","message":"Fixed a misplaced importation.\n\nIgnore-this: f44340563188fa12e9b79aed8492deab\n\ndarcs-hash:20120228152056-3bd4e-143ebb2d8b0da74339da12c68888ca6a759c6afe.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/FOL.agda","new_file":"src\/FOL\/FOL.agda","new_contents":"------------------------------------------------------------------------------\n-- FOL (without equality)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module exported all the logical constants and the\n-- propositional equality. This module is re-exported by the \"base\"\n-- modules whose theories are defined on FOL (without equality)\n\nmodule FOL.FOL where\n\n------------------------------------------------------------------------------\n-- The propositional logical connectives\n\n-- The logical connectives are hard-coded in our translation,\n-- i.e. the symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192, and \u2194 must be used.\n-- N.B. For the implication we use the Agda function type.\nopen import FOL.Data.Empty public using ( \u22a5 ; \u22a5-elim )\nopen import FOL.Data.Product public using ( _\u2227_ ; _,_ ; \u2227-proj\u2081 ; \u2227-proj\u2082 )\nopen import FOL.Data.Sum public using ( _\u2228_ ; [_,_] ; inj\u2081 ; inj\u2082 )\nopen import FOL.Data.Unit public using ( \u22a4 )\nopen import FOL.Relation.Nullary public using ( \u00ac_ )\n\ninfixr 2 _\u2194_\n\n_\u2194_ : Set \u2192 Set \u2192 Set\nP \u2194 Q = (P \u2192 Q) \u2227 (Q \u2192 P)\n\n------------------------------------------------------------------------------\n-- The quantifiers\n\n-- The existential quantifier\n-- The existential quantifier is hard-coded in our translation,\n-- i.e. the symbol \u2203 must be used.\n\nopen import FOL.Data.Product public using ( _,,_ ; \u2203 ; \u2203-elim )\n\n-- The universal quantifier\n-- N.B. For the universal quantifier we use the Agda (dependent)\n-- function type.\n","old_contents":"------------------------------------------------------------------------------\n-- FOL (without equality)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module exported all the logical constants and the\n-- propositional equality. This module is re-exported by the \"base\"\n-- modules whose theories are defined on FOL (without equality)\n\nmodule FOL.FOL where\n\n------------------------------------------------------------------------------\n-- The propositional logical connectives\n\n-- The logical connectives are hard-coded in our translation,\n-- i.e. the symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192, and \u2194 must be used.\n-- N.B. For the implication we use the Agda function type.\nopen import FOL.Data.Empty public using ( \u22a5 ; \u22a5-elim )\nopen import FOL.Data.Product public using ( _\u2227_ ; _,_ ; _,,_ ; \u2227-proj\u2081 ; \u2227-proj\u2082 )\nopen import FOL.Data.Sum public using ( _\u2228_ ; [_,_] ; inj\u2081 ; inj\u2082 )\nopen import FOL.Data.Unit public using ( \u22a4 )\nopen import FOL.Relation.Nullary public using ( \u00ac_ )\n\ninfixr 2 _\u2194_\n\n_\u2194_ : Set \u2192 Set \u2192 Set\nP \u2194 Q = (P \u2192 Q) \u2227 (Q \u2192 P)\n\n------------------------------------------------------------------------------\n-- The quantifiers\n\n-- The existential quantifier\n-- The existential quantifier is hard-coded in our translation,\n-- i.e. the symbol \u2203 must be used.\n\nopen import FOL.Data.Product public using ( _,_ ; \u2203 ; \u2203-elim )\n\n-- The universal quantifier\n-- N.B. For the universal quantifier we use the Agda (dependent)\n-- function type.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"73758b64f31209c83ec68f4efd4b6f450f4edebb","subject":"Example 3 works now\u0302\u00b9. \u0302\u00b9 Other than that the `Bag` type is an abstract postulate.","message":"Example 3 works now\u0302\u00b9.\n\u0302\u00b9 Other than that the `Bag` type is an abstract postulate.\n\nA presentation of example 3 starts from line 297.\n\nThe optimized derivation `derive2` considers the derivative of\na term to be the nil change only if it is closed. Should we have\n\u03b2-equivalence test or some inherited description of whether the\nargument to an abs-term may change, we could handle more examples.\n\nOld-commit-hash: 6488c110b93dc4c760ddc61feef198b55ab4940d\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import TaggedDeltaTypes\n\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nopen import Relation.Binary.Core using (Decidable)\nopen import Relation.Nullary.Core using (yes ; no)\n\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n  where ext = extensionality\n\nproj-H : \u2200 {\u0393 : Context} {\u03c1 : \u0394Env \u0393} {us vs} \u2192\n  Honest \u03c1 (FV-union us vs) \u2192 Honest \u03c1 us \u00d7 Honest \u03c1 vs\nproj-H {\u2205} {us = \u2205} {vs = \u2205} clearly = clearly , clearly\nproj-H {us = alter us} {alter vs} (alter H) =\n  let uss , vss = proj-H H in alter uss , alter vss\nproj-H {us = alter us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  alter uss , abide eq vss\nproj-H {us = abide us} {alter vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , alter vss\nproj-H {us = abide us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , abide eq vss\n\n-- Equivalence proofs are unique\nequivalence-unique : \u2200 {A : Set} {a b : A} \u2192 \u2200 {p q : a \u2261 b} \u2192 p \u2261 q\nequivalence-unique {p = refl} {refl} = refl\n\n-- Product of singletons are singletons (for example)\nproduct-unique : \u2200 {A : Set} {B : A \u2192 Set} \u2192\n  (\u2200 {a b : A} \u2192 a \u2261 b) \u2192\n  (\u2200 {a : A} {c d : B a} \u2192 c \u2261 d) \u2192\n  (\u2200 {p q : \u03a3 A B} \u2192 p \u2261 q)\nproduct-unique {A} {B} lhs rhs {a , c} {b , d}\n  rewrite lhs {a} {b} = cong (_,_ b) rhs\n\n-- Validity proofs are (extensionally) unique (as functions)\nvalidity-unique : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} {dv : \u0394Val \u03c4} \u2192\n  \u2200 {p q : valid v dv} \u2192 p \u2261 q\nvalidity-unique {nats} = equivalence-unique\nvalidity-unique {bags} = refl\nvalidity-unique {\u03c3 \u21d2 \u03c4} =  ext\u00b3 (\u03bb v dv R[v,dv] \u2192\n  product-unique validity-unique equivalence-unique)\n\nstabilityVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (select-just x)) \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\n\nstabilityVar {x = this} (abide proof _) = proof\nstabilityVar {x = that y} (alter H) = stabilityVar {x = y} H\n\nstability : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV t)) \u2192\n    \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Boilerplate begins\nstabilityAbs : \u2200 {\u03c3 \u03c4 \u0393} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV (abs t))) \u2192\n  (v : \u27e6 \u03c3 \u27e7) \u2192\n    \u27e6 t \u27e7 (v \u2022 ignore \u03c1) \u2295\n    \u27e6 derive t \u27e7\u0394 (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (v \u2022 ignore \u03c1)\nstabilityAbs {t = t} {\u03c1} H v with FV t | inspect FV t\n... | abide vars | [ case0 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case0 = abide v\u2295[u\u229dv]=u H\n... | alter vars | [ case1 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case1 = alter H\n-- Boilerplate ends\n\nstability {t = nat n} H = refl\nstability {t = bag b} H = b++\u2205=b\nstability {t = var x} H = stabilityVar H\nstability {t = abs t} {\u03c1} H = extensionality (stabilityAbs {t = t} H)\nstability {t = app s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n  in\n    begin\n      f v \u2295 df v dv (validity {t = t})\n    \u2261\u27e8 sym (corollary s t) \u27e9\n      (f \u2295 df) (v \u2295 dv)\n    \u2261\u27e8 stability {t = s} Hs \u27e8$\u27e9 stability {t = t} Ht \u27e9\n      f v\n    \u220e where open \u2261-Reasoning\nstability {t = add s t} H =\n  let Hs , Ht = proj-H H\n  in cong\u2082 _+_ (stability {t = s} Hs) (stability {t = t} Ht)\nstability {t = map s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n    map = mapBag\n  in\n  begin\n    map f b \u2295 (map (f \u2295 df) (b \u2295 db) \u229d map f b)\n  \u2261\u27e8 b++[d\\\\b]=d \u27e9\n    map (f \u2295 df) (b \u2295 db)\n  \u2261\u27e8 cong\u2082 map (stability {t = s} Hs) (stability {t = t} Ht) \u27e9\n    map f b\n  \u220e where open \u2261-Reasoning\n\neq-map : \u2200 {\u0393}\n  (s    : Term \u0393 (nats \u21d2 nats))\n  (t    : Term \u0393 bags)\n  (\u03c1    : \u0394Env \u0393)\n  (H    : Honest \u03c1 (FV s)) \u2192\n    \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n  \u2261 \u27e6 \u0394map\u2080 s (derive s) t (derive t) \u27e7\u0394 \u03c1 (unrestricted (map s t))\n\neq-map s t \u03c1 H =\n  let\n    ds = derive s\n    dt = derive t\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 (unrestricted t)\n\n    eq1 : \u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (unrestricted (map s t))\n        \u2261 mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n    eq1 = refl\n\n    eq2 : mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n        \u2261 mapBag f dv\n    eq2 = trans\n      (cong (\u03bb hole \u2192 hole \u229d mapBag f v) (trans\n        (cong (\u03bb hole \u2192 mapBag hole (v \u2295 dv)) (stability {t = s} H))\n        map-over-++))\n      [b++d]\\\\b=d\n\n    eq3 : mapBag f dv\n        \u2261 \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n    eq3 = refl\n\n  in sym (trans eq1 (trans eq2 eq3))\n\n-- Vars test\nnone-selected? : \u2200 {\u0393} \u2192 (vs : Vars \u0393) \u2192 (vs \u2261 select-none) \u228e \u22a4\nnone-selected? \u2205 = inj\u2081 refl\nnone-selected? (abide vs) = inj\u2082 tt\nnone-selected? (alter vs) with none-selected? vs\n... | inj\u2081 vs=\u2205 rewrite vs=\u2205 = inj\u2081 refl\n... | inj\u2082 _ = inj\u2082 tt\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 select-none) \u228e \u22a4\nclosed? t = none-selected? (FV t)\n\nvacuous-honesty : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} \u2192 Honest \u03c1 select-none\nvacuous-honesty {\u2205} {\u2205} = clearly\nvacuous-honesty {\u03c4 \u2022 \u0393} {cons _ _ _ \u03c1} = alter (vacuous-honesty {\u03c1 = \u03c1})\n\n-- Immunity of closed terms to dishonest environments\nimmune : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  (FV t \u2261 select-none) \u2192 \u2200 {\u03c1} \u2192 Honest \u03c1 (FV t)\nimmune {t = t} eq rewrite eq = vacuous-honesty\n\n-- Ineffectual first optimization step\nderive1 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive1 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive t)\n... | inj\u2082 tt = derive (map s t)\nderive1 others = derive others\n\nvalid1 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive1 t is-valid-for \u03c1\nvalid1 (nat n) = tt\nvalid1 (bag b) = tt\nvalid1 (var x) = tt\nvalid1 (abs t) = \u03bb _ _ _ \u2192 unrestricted t\nvalid1 (app s t) =\n  cons (unrestricted s) (unrestricted t) (validity {t = t}) tt\nvalid1 (add s t) =\n  cons (unrestricted s) (unrestricted t) tt tt\nvalid1 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (unrestricted s) (unrestricted t) tt tt\n... | inj\u2081 if-closed =\n  cons (unrestricted t) (immune {t = s} if-closed) tt tt\n\ncorrect1 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive1 t \u27e7\u0394 \u03c1 (valid1 t) \u2261 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrect1 {t = nat n} = refl\ncorrect1 {t = bag b} = refl\ncorrect1 {t = var x} = refl\ncorrect1 {t = abs t} = refl\ncorrect1 {t = app s t} = refl\ncorrect1 {t = add s t} = refl\ncorrect1 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt = refl\n... | inj\u2081 if-closed = eq-map s t \u03c1 (immune {t = s} if-closed)\n\n-- derive2 = derive1 + congruence\nderive2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive2 (abs t) = \u0394abs (derive2 t)\nderive2 (app s t) = \u0394app (derive2 s) t (derive2 t)\nderive2 (add s t) = \u0394add (derive2 s) (derive2 t)\nderive2 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive2 t)\n... | inj\u2082 tt = \u0394map\u2080 s (derive2 s) t (derive2 t)\nderive2 others = derive others\n\nvalid2 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive2 t is-valid-for \u03c1\ncorrect2 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t) \u2261 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\nvalid2 (nat n) = tt\nvalid2 (bag b) = tt\nvalid2 (var x) = tt\nvalid2 (abs t) = \u03bb _ _ _ \u2192 valid2 t\nvalid2 (app s t) {\u03c1} = cons (valid2 s) (valid2 t) V tt\n  where\n  V : valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t))\n  V rewrite correct2 {t = t} {\u03c1} = validity {t = t} {\u03c1}\nvalid2 (add s t) = cons (valid2 s) (valid2 t) tt tt\nvalid2 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (valid2 s) (valid2 t) tt tt\n... | inj\u2081 if-closed =\n  cons (valid2 t) (immune {t = s} if-closed) tt tt\n\ncorrect2 {t = nat n} = refl\ncorrect2 {t = bag b} = refl\ncorrect2 {t = var x} = refl\ncorrect2 {t = abs t} = ext\u00b3 (\u03bb _ _ _ \u2192 correct2 {t = t})\ncorrect2 {t = app {\u03c3} {\u03c4} s t} {\u03c1} = \u2245-to-\u2261 eq-all where\n  open import Relation.Binary.HeterogeneousEquality hiding (cong\u2082)\n  import Relation.Binary.HeterogeneousEquality as HET\n  df0 : \u0394Val (\u03c3 \u21d2 \u03c4)\n  df0 = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n  df2 : \u0394Val (\u03c3 \u21d2 \u03c4)\n  df2 = \u27e6 derive2 s \u27e7\u0394 \u03c1 (valid2 s)\n  dv0 : \u0394Val \u03c3\n  dv0 = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  dv2 : \u0394Val \u03c3\n  dv2 = \u27e6 derive2 t \u27e7\u0394 \u03c1 (valid2 t)\n  v   : \u27e6 \u03c3 \u27e7\n  v   = \u27e6 t \u27e7 (ignore \u03c1)\n  eq-df : df2 \u2261 df0\n  eq-df = correct2 {t = s}\n  eq-dv : dv2 \u2261 dv0\n  eq-dv = correct2 {t = t}\n  eq-all : df2 v dv2 (caddr (valid2 (app s t) {\u03c1}))\n         \u2245 df0 v dv0 (validity {t = t} {\u03c1})\n  eq-all rewrite eq-dv = HET.cong\n    (\u03bb hole \u2192 hole v dv0 (validity {t = t} {\u03c1})) (\u2261-to-\u2245 eq-df)\n    -- found without intuition by trial-and-error\ncorrect2 {t = add s t} {\u03c1} =\n  let\n    cs = correct2 {t = s} {\u03c1}\n    ct = correct2 {t = t} {\u03c1}\n  in cong\u2082 _,_ (cong\u2082 _+_ (cong proj\u2081 cs) (cong proj\u2081 ct))\n               (cong\u2082 _+_ (cong proj\u2082 cs) (cong proj\u2082 ct))\ncorrect2 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt = \n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n  in\n    cong\u2082 (\u03bb h1 h2 \u2192 mapBag (f \u2295 h1) (b \u2295 h2) \\\\ mapBag f b)\n    (correct2 {t = s}) (correct2 {t = t})\n... | inj\u2081 if-closed = trans\n  (cong (mapBag (\u27e6 s \u27e7 (ignore \u03c1))) (correct2 {t = t}))\n  (eq-map s t \u03c1 (immune {t = s} if-closed))\n\n---------------\n-- Example 3 --\n---------------\n\n-- [+1] = \u03bb x \u2192 x + 1\n[+1] : \u2200 {\u0393} \u2192 Term \u0393 (nats \u21d2 nats)\n[+1] = (abs (add (var this) (nat 1)))\n\n-- inc = \u03bb bag \u2192 map [+1] bag\ninc : Term \u2205 (bags \u21d2 bags)\ninc = abs (map [+1] (var this))\n\n-- Program transformation in example 3\n-- Sadly, `inc` is not executable, for `Bag` is an abstract type\n-- whose existence we postulated.\n--\n-- derive2 inc = \u03bb bag dbag \u2192 map [+1] dbag\nexample-3 : derive2 inc \u2261 \u0394abs (\u0394map\u2081 [+1] (\u0394var this))\nexample-3 = refl\n\n-- Correctness of optimized derivation on `inc`\n--\n-- \u27e6 derive2 inc \u27e7 <embedded-proof> = \u27e6 derive inc \u27e7 <another-proof>\nexample-3-correct :\n  \u27e6 derive2 inc \u27e7\u0394 \u2205 (valid2 inc) \u2261 \u27e6 derive inc \u27e7\u0394 \u2205 (unrestricted inc)\nexample-3-correct = correct2 {t = inc} {\u2205}\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import TaggedDeltaTypes\n\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Sum hiding (map)\nopen import Data.Product hiding (map)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nopen import Relation.Binary.Core using (Decidable)\nopen import Relation.Nullary.Core using (yes ; no)\n\next\u00b3 : \u2200\n  {A : Set}\n  {B : A \u2192 Set}\n  {C : (a : A) \u2192 B a \u2192 Set }\n  {D : (a : A) \u2192 (b : B a) \u2192 C a b \u2192 Set}\n  {f g : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 D a b c} \u2192\n  ((a : A) (b : B a) (c : C a b) \u2192 f a b c \u2261 g a b c) \u2192 f \u2261 g\n\next\u00b3 fabc=gabc = ext (\u03bb a \u2192 ext (\u03bb b \u2192 ext (\u03bb c \u2192 fabc=gabc a b c)))\n  where ext = extensionality\n\nproj-H : \u2200 {\u0393 : Context} {\u03c1 : \u0394Env \u0393} {us vs} \u2192\n  Honest \u03c1 (FV-union us vs) \u2192 Honest \u03c1 us \u00d7 Honest \u03c1 vs\nproj-H {\u2205} {us = \u2205} {vs = \u2205} clearly = clearly , clearly\nproj-H {us = alter us} {alter vs} (alter H) =\n  let uss , vss = proj-H H in alter uss , alter vss\nproj-H {us = alter us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  alter uss , abide eq vss\nproj-H {us = abide us} {alter vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , alter vss\nproj-H {us = abide us} {abide vs} (abide eq H) =\n  let uss , vss = proj-H H in\n  abide eq uss , abide eq vss\n\nstabilityVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (select-just x)) \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\n\nstabilityVar {x = this} (abide proof _) = proof\nstabilityVar {x = that y} (alter H) = stabilityVar {x = y} H\n\nstability : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV t)) \u2192\n    \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Boilerplate begins\nstabilityAbs : \u2200 {\u03c3 \u03c4 \u0393} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} (H : Honest \u03c1 (FV (abs t))) \u2192\n  (v : \u27e6 \u03c3 \u27e7) \u2192\n    \u27e6 t \u27e7 (v \u2022 ignore \u03c1) \u2295\n    \u27e6 derive t \u27e7\u0394 (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (v \u2022 ignore \u03c1)\nstabilityAbs {t = t} {\u03c1} H v with FV t | inspect FV t\n... | abide vars | [ case0 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case0 = abide v\u2295[u\u229dv]=u H\n... | alter vars | [ case1 ] = stability {t = t} Ht\n  where\n  Ht : Honest (cons v (v \u229d v) R[v,u\u229dv] \u03c1) (FV t)\n  Ht rewrite case1 = alter H\n-- Boilerplate ends\n\nstability {t = nat n} H = refl\nstability {t = bag b} H = b++\u2205=b\nstability {t = var x} H = stabilityVar H\nstability {t = abs t} {\u03c1} H = extensionality (stabilityAbs {t = t} H)\nstability {t = app s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n  in\n    begin\n      f v \u2295 df v dv (validity {t = t})\n    \u2261\u27e8 sym (corollary s t) \u27e9\n      (f \u2295 df) (v \u2295 dv)\n    \u2261\u27e8 stability {t = s} Hs \u27e8$\u27e9 stability {t = t} Ht \u27e9\n      f v\n    \u220e where open \u2261-Reasoning\nstability {t = add s t} H =\n  let Hs , Ht = proj-H H\n  in cong\u2082 _+_ (stability {t = s} Hs) (stability {t = t} Ht)\nstability {t = map s t} {\u03c1} H =\n  let\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n    Hs , Ht = proj-H H\n    map = mapBag\n  in\n  begin\n    map f b \u2295 (map (f \u2295 df) (b \u2295 db) \u229d map f b)\n  \u2261\u27e8 b++[d\\\\b]=d \u27e9\n    map (f \u2295 df) (b \u2295 db)\n  \u2261\u27e8 cong\u2082 map (stability {t = s} Hs) (stability {t = t} Ht) \u27e9\n    map f b\n  \u220e where open \u2261-Reasoning\n\neq-map : \u2200 {\u0393}\n  (s    : Term \u0393 (nats \u21d2 nats))\n  (t    : Term \u0393 bags)\n  (\u03c1    : \u0394Env \u0393)\n  (H    : Honest \u03c1 (FV s)) \u2192\n    \u27e6 \u0394map\u2080 s (derive s) t (derive t) \u27e7\u0394 \u03c1 (unrestricted (map s t))\n  \u2261 \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n\neq-map s t \u03c1 H =\n  let\n    ds = derive s\n    dt = derive t\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 (unrestricted s)\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 (unrestricted t)\n\n    eq1 : \u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (unrestricted (map s t))\n        \u2261 mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n    eq1 = refl\n\n    eq2 : mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n        \u2261 mapBag f dv\n    eq2 = trans\n      (cong (\u03bb hole \u2192 hole \u229d mapBag f v) (trans\n        (cong (\u03bb hole \u2192 mapBag hole (v \u2295 dv)) (stability {t = s} H))\n        map-over-++))\n      [b++d]\\\\b=d\n\n    eq3 : mapBag f dv\n        \u2261 \u27e6 \u0394map\u2081 s (derive t) \u27e7\u0394 \u03c1 (cons (unrestricted t) H tt tt)\n    eq3 = refl\n\n  in trans eq1 (trans eq2 eq3)\n\n-- Vars test\nnone-selected? : \u2200 {\u0393} \u2192 (vs : Vars \u0393) \u2192 (vs \u2261 select-none) \u228e \u22a4\nnone-selected? \u2205 = inj\u2081 refl\nnone-selected? (abide vs) = inj\u2082 tt\nnone-selected? (alter vs) with none-selected? vs\n... | inj\u2081 vs=\u2205 rewrite vs=\u2205 = inj\u2081 refl\n... | inj\u2082 _ = inj\u2082 tt\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 select-none) \u228e \u22a4\nclosed? t = none-selected? (FV t)\n\nvacuous-honesty : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} \u2192 Honest \u03c1 select-none\nvacuous-honesty {\u2205} {\u2205} = clearly\nvacuous-honesty {\u03c4 \u2022 \u0393} {cons _ _ _ \u03c1} = alter (vacuous-honesty {\u03c1 = \u03c1})\n\n-- Immunity of closed terms to dishonest environments\nimmune : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  (FV t \u2261 select-none) \u2192 \u2200 {\u03c1} \u2192 Honest \u03c1 (FV t)\nimmune {t = t} eq rewrite eq = vacuous-honesty\n\n-- Ineffectual first optimization step\nderive1 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\nderive1 (map s t) with closed? s\n... | inj\u2081 is-closed = \u0394map\u2081 s (derive t)\n... | inj\u2082 tt = derive (map s t)\nderive1 others = derive others\n\nvalid1 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192 derive1 t is-valid-for \u03c1\nvalid1 (nat n) = tt\nvalid1 (bag b) = tt\nvalid1 (var x) = tt\nvalid1 (abs t) = \u03bb _ _ _ \u2192 unrestricted t\nvalid1 (app s t) =\n  cons (unrestricted s) (unrestricted t) (validity {t = t}) tt\nvalid1 (add s t) =\n  cons (unrestricted s) (unrestricted t) tt tt\nvalid1 (map s t) {\u03c1} with closed? s\n... | inj\u2082 tt = cons (unrestricted s) (unrestricted t) tt tt\n... | inj\u2081 if-closed =\n  cons (unrestricted t) (immune {t = s} if-closed) tt tt\n\ncorrect1 : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t) \u2261 \u27e6 derive1 t \u27e7\u0394 \u03c1 (valid1 t)\n\ncorrect1 {t = nat n} = refl\ncorrect1 {t = bag b} = refl\ncorrect1 {t = var x} = refl\ncorrect1 {t = abs t} = refl\ncorrect1 {t = app s t} = refl\ncorrect1 {t = add s t} = refl\ncorrect1 {t = map s t} {\u03c1} with closed? s\n... | inj\u2082 tt = refl\n... | inj\u2081 if-closed = eq-map s t \u03c1 (immune {t = s} if-closed)\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a69042e1203e1b0da55918e3c8130ee69193e159","subject":"Boring.","message":"Boring.\n\nIgnore-this: c710f736c882ebdb2b00863199903eef\n\ndarcs-hash:20100412130736-3bd4e-4a089904a683c9be05ed341034af47ec1e08c49c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/Names.agda","new_file":"Test\/Succeed\/Names.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the translation of function, predicates and variables names\n------------------------------------------------------------------------------\n\n-- From the technical manual of TPTP\n-- (http:\/\/www.cs.miami.edu\/~tptp\/TPTP\/TR\/TPTPTR.shtml)\n\n-- ... variables start with upper case letters, ... predicates and\n-- functors either start with lower case and contain alphanumerics and\n-- underscore ...\n\nmodule Test.Succeed.Names where\n\ninfix 4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n-- A funny function name.\npostulate\n  FUN! : D \u2192 D\n\npostulate\n  -- Using a funny function and variable name\n  a\u2081 : (nx\u220e : D) \u2192 FUN! nx\u220e \u2261 nx\u220e\n{-# ATP axiom a\u2081 #-}\n\npostulate\n  foo : (n : D) \u2192 FUN! n \u2261 n\n{-# ATP prove foo #-}\n\n-- A funny predicate name\ndata PRED! : D \u2192 Set where\n\npostulate\n  a\u2082 : (n : D) \u2192 PRED! n\n{-# ATP axiom a\u2082 #-}\n\npostulate\n  bar : (n : D) \u2192 PRED! n\n{-# ATP prove bar #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the translation of function, predicates and variables names\n------------------------------------------------------------------------------\n\n-- From the technical manual of TPTP\n-- http:\/\/www.cs.miami.edu\/~tptp\/TPTP\/TR\/TPTPTR.shtml\n\n-- variables start with upper case letters, ... predicates and\n-- functors either start with lower case and contain alphanumerics and\n-- underscore ...\n\nmodule Test.Succeed.Names where\n\ninfix 4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n-- A funny function name.\npostulate\n  FUN! : D \u2192 D\n\npostulate\n  -- Using a funny function and variable name\n  a\u2081 : (nx\u220e : D) \u2192 FUN! nx\u220e \u2261 nx\u220e\n{-# ATP axiom a\u2081 #-}\n\npostulate\n  foo : (n : D) \u2192 FUN! n \u2261 n\n{-# ATP prove foo #-}\n\n-- A funny predicate name\ndata PRED! : D \u2192 Set where\n\npostulate\n  a\u2082 : (n : D) \u2192 PRED! n\n{-# ATP axiom a\u2082 #-}\n\npostulate\n  bar : (n : D) \u2192 PRED! n\n{-# ATP prove bar #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"eb9b76a2590365332ba7634aba748ff2018711d2","subject":"Boring.","message":"Boring.\n\nIgnore-this: afd837a76a16677be82e439fbd3f3052\n\ndarcs-hash:20100812142917-3bd4e-3afcc41d73968067a6626fddc35217ec3888a7a8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Minimal.agda","new_file":"LTC\/Minimal.agda","new_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\nmodule LTC.Minimal where\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain.\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n------------------------------------------------------------------------------\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties.\n\n-- The substitution is defined in LTC.MinimalER\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Empty public\nopen import LTC.Data.Product public\nopen import LTC.Data.Sum public\nopen import LTC.Data.Unit public\nopen import LTC.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom cB\u2081 #-}\n{-# ATP axiom cB\u2082 #-}\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom cZ\u2081 #-}\n{-# ATP axiom cZ\u2082 #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom cBeta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom cFix #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\nmodule LTC.Minimal where\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain.\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n------------------------------------------------------------------------------\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties.\n\n-- The substitution is defined in LTC.MinimalER\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Empty public\nopen import LTC.Data.Product public\nopen import LTC.Data.Sum public\nopen import LTC.Data.Unit public\nopen import LTC.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom cB\u2081 #-}\n{-# ATP axiom cB\u2082 #-}\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom cZ\u2081 #-}\n{-# ATP axiom cZ\u2082 #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom cBeta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom cFix #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8c57726522691c1d75ef56448841051067ea72e4","subject":"Added missing --without-K option","message":"Added missing --without-K option\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOL\/ImplicitArgumentSubst.agda","new_file":"notes\/FOT\/FOL\/ImplicitArgumentSubst.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing subst using an implicit arguments for the propositional function.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas              #-}\n{-# OPTIONS --no-sized-types                    #-}\n{-# OPTIONS --no-universe-polymorphism          #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K                         #-}\n\nmodule FOT.FOL.ImplicitArgumentSubst where\n\ninfix 7 _\u2261_\n\npostulate\n  D              : Set\n  _\u00b7_            : D \u2192 D \u2192 D\n  zero succ pred : D\n\nsucc\u2081 : D \u2192 D\nsucc\u2081 n = succ \u00b7 n\n\npred\u2081 : D \u2192 D\npred\u2081 n = pred \u00b7 n\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- The propositional function is not an implicit argument.\nsubst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\nsubst A refl Ax = Ax\n\n-- The propositional formula is an implicit argument.\nsubst' : {A : D \u2192 Set} \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\nsubst' refl Ax = Ax\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n-- Conversion rules.\npostulate\n  pred-0 : pred\u2081 zero            \u2261 zero\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n\n-- The FOTC natural numbers type.\ndata N : D \u2192 Set where\n  nzero :               N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Works using subst.\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N nzero          = subst N (sym pred-0) nzero\npred-N (nsucc {n} Nn) = subst N (sym (pred-S n)) Nn\n\n-- Fails using subst'.\npred-N' : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N' nzero          = subst' (sym pred-0) nzero\npred-N' (nsucc {n} Nn) = subst' (sym (pred-S n)) Nn\n","old_contents":"------------------------------------------------------------------------------\n-- Testing subst using an implicit arguments for the propositional function.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas              #-}\n{-# OPTIONS --no-sized-types                    #-}\n{-# OPTIONS --no-universe-polymorphism          #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n\nmodule FOT.FOL.ImplicitArgumentSubst where\n\ninfix 7 _\u2261_\n\npostulate\n  D              : Set\n  _\u00b7_            : D \u2192 D \u2192 D\n  zero succ pred : D\n\nsucc\u2081 : D \u2192 D\nsucc\u2081 n = succ \u00b7 n\n\npred\u2081 : D \u2192 D\npred\u2081 n = pred \u00b7 n\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- The propositional function is not an implicit argument.\nsubst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\nsubst A refl Ax = Ax\n\n-- The propositional formula is an implicit argument.\nsubst' : {A : D \u2192 Set} \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\nsubst' refl Ax = Ax\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n-- Conversion rules.\npostulate\n  pred-0 : pred\u2081 zero            \u2261 zero\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n\n-- The FOTC natural numbers type.\ndata N : D \u2192 Set where\n  nzero :               N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Works using subst.\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N nzero          = subst N (sym pred-0) nzero\npred-N (nsucc {n} Nn) = subst N (sym (pred-S n)) Nn\n\n-- Fails using subst'.\npred-N' : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N' nzero          = subst' (sym pred-0) nzero\npred-N' (nsucc {n} Nn) = subst' (sym (pred-S n)) Nn\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f0ce38074cd7a2b8fbc37eacf555d188fe888e27","subject":"some stuff now works...possibly subject expansion","message":"some stuff now works...possibly subject expansion\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping.agda","new_file":"Agda\/ITyping.agda","new_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\n-- open import Data.Maybe\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\n\ndata IType : Set where\n  o : IType\n  _~>_ : IType -> IType -> IType\n  \u2229 : List IType -> IType\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\no \u225fTI (\u2229 _) = no (\u03bb ())\n\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(_ ~> _) \u225fTI (\u2229 _) = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n(\u2229 _) \u225fTI o = no (\u03bb ())\n(\u2229 _) \u225fTI (_ ~> _) = no (\u03bb ())\n\u2229 [] \u225fTI \u2229 [] = yes refl\n\u2229 [] \u225fTI \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 IType)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\n\ndata _\u2237'_ : IType -> Type -> Set where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237' A -> \u03c4 \u2237' B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n  \u2229-nil : \u2200 {A} ->  \u03c9 \u2237' A\n  \u2229-cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237' A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237' A\n\n\ndata _\u2264\u2229_ : IType -> IType -> Set where\n  base : o \u2264\u2229 o\n  arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n  \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n  \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n  -- \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 : \u2200 {\u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u2229 \u03c4\u2c7c \u2264\u2229 \u2229 \u03c4\u1d62\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {[]} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-nil\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {x \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-cons (\u2229-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl))) (\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2081} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)))\n\n\u2264\u2229-refl : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c4\n\u2264\u2229-refl {o} = base\n\u2264\u2229-refl {\u03c4 ~> \u03c4\u2081} = arr \u2264\u2229-refl \u2264\u2229-refl\n\u2264\u2229-refl {\u2229 []} = \u2229-nil\n\u2264\u2229-refl {\u2229 (x \u2237 x\u2081)} = \u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2082} z \u2192 z)\n\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\n-- _\u2208?_ : \u2200 {a} {A : Set a} -> Atom -> List (Atom \u00d7 A) -> Maybe A\n-- a \u2208? [] = nothing\n-- a \u2208? (l \u2237 ist) with a \u225f proj\u2081 l\n-- ... | yes _ = just (proj\u2082 l)\n-- a \u2208? (l \u2237 ist) | no _ = a \u2208? ist\n\n-- \u22a2->\u039b : \u2200 {\u0393 m t} -> (List (Atom \u00d7 \u2115)) -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- \u22a2->\u039b {m = bv i} _ ()\n-- \u22a2->\u039b {m = fv x} bound \u0393\u22a2m\u2236t with x \u2208? bound\n-- \u22a2->\u039b {m = fv x} {t} bound \u0393\u22a2m\u2236t | just i = bv {t} i\n-- \u22a2->\u039b {m = fv x} bound \u0393\u22a2m\u2236t | nothing = fv x\n-- \u22a2->\u039b {m = lam m} bound (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) = lam \u03c4\u2081 (\u22a2->\u039b ((x , 0) \u2237 bound') (cf (\u2209-cons-l _ _ x\u2209)))\n--   where\n--   x = \u2203fresh (L ++ FV m)\n--   x\u2209 : x \u2209 (L ++ FV m)\n--   x\u2209 = \u2203fresh-spec (L ++ FV m)\n--\n--   bound' : List (Atom \u00d7 \u2115)\n--   bound' = Data.List.map (\u03bb a,i \u2192 (proj\u2081 a,i) , suc (proj\u2082 a,i)) bound\n--\n-- \u22a2->\u039b {m = app t1 t} bound (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b bound \u0393\u22a2s) (\u22a2->\u039b bound \u0393\u22a2t)\n-- \u22a2->\u039b {m = Y \u03c4} bound (Y x) = Y \u03c4\n--\n-- \u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- \u22a2->\u039b = \u22a2->\u039b []\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app t1 t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y x) = Y \u03c4\n\n\u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n\u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n-- ers : \u2200 {t} -> \u039b t -> PTerm\n-- ers (bv i) = bv i\n-- ers (fv x) = fv x\n-- ers (lam A \u039bt) = lam (ers \u039bt)\n-- ers (app \u039bs \u039bt) = app (ers \u039bs) (ers \u039bt)\n-- ers (Y t) = Y t\n\n\n-- data IType\u209b : IType -> Set where\n--   o : IType\u209b o\n--   arr : \u2200 {\u03c4 \u03c4'} -> IType\u209b \u03c4 -> IType\u209b \u03c4' -> IType\u209b (\u03c4 ~> \u03c4')\n--\n-- data IType\u209b\u209b : IType -> Set where\n--   o : IType\u209b\u209b o\n--   arr :  \u2200 {\u03c4 \u03c4'} -> IType\u209b\u209b \u03c4 -> IType\u209b\u209b \u03c4' -> IType\u209b\u209b (\u03c4 ~> \u03c4')\n--   \u2229-nil : IType\u209b\u209b \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4\u1d62} -> IType\u209b \u03c4 ->  IType\u209b\u209b (\u2229 \u03c4\u1d62) -> IType\u209b\u209b (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n--\n-- \u03c4\u209b->\u03c4\u209b\u209b : \u2200 {\u03c4} -> IType\u209b \u03c4 -> IType\u209b\u209b \u03c4\n-- \u03c4\u209b->\u03c4\u209b\u209b o = o\n-- \u03c4\u209b->\u03c4\u209b\u209b (arr \u03c4\u209b \u03c4\u209b\u2081) = arr (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b) (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b\u2081)\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata Y-shape : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  intro\u2081 : \u2200 {A m} -> Y-shape (app (Y A) m)\n  intro\u2082 : \u2200 {A m} -> Y-shape (app m (app (Y A) m))\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} -> (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u2229 \u03c4\u1d62)) \u2208 \u0393) -> (\u03c4\u1d62\u2264\u2229\u03c4 : \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4) ->\n    \u03c4 \u2237' A -> \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4\u2081 \u03c4\u2082} -> \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9 t \u2236 \u03c4\u2081 -> (\u03c4\u2081 ~> \u03c4\u2082) \u2237' (A \u27f6 B) -> \u03c4\u2081 \u2237' A ->\n    \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\u2082\n  \u2229-nil : \u2200 {A \u0393} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) -> \u0393 \u22a9 m\u2005 \u2236 \u03c9\n  \u2229-cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 \u03c4\u1d62) -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n  abs : \u2200 {A B \u0393 \u03c4\u1d62 \u03c4} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4 ) -> \u2229 \u03c4\u1d62 \u2237' A -> \u03c4 \u2237' B ->\n    \u0393 \u22a9 lam A t \u2236 (\u2229 \u03c4\u1d62 ~> \u03c4)\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} -> Wf-ICtxt \u0393 -> \u03c4 \u2237' A -> \u03c4\u2081 \u2237' A -> \u03c4\u2082 \u2237' A ->\n    \u0393 \u22a9 Y A \u2236 ((\u03c4 ~> \u03c4\u2081) ~> \u03c4\u2082)\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n-- \u22a2->\u039b-\u039bTerm : \u2200 {\u0393 t \u03c4} -> {\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4} -> (\u039bTerm (\u22a2->\u039b \u0393\u22a2t))\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = var x\u2081 x\u2082} = var\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = app \u0393\u22a2t \u0393\u22a2t\u2081} = app \u22a2->\u039b-\u039bTerm \u22a2->\u039b-\u039bTerm\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = abs L cf} = {!   !}\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = Y x} = Y\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n\n\u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n\u22a9->\u03b2 (app {s = m} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A) \u0393\u22a9m x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n  app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n\u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n\u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n\u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236\u03c4'~>\u03c4 (app (Y {\u03c4 = \u03c4'} wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 \u03c4\u2082\u2237) \u0393\u22a9m\u2236\u03c4'~>\u03c4')) (Y trm-m) =\n--     app (Y wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 {!   !}) \u0393\u22a9m\u2236\u03c4'~>\u03c4'\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393 trm-m)) (Y x) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082)) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393 trm-m) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2082)\n--\n--\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u03b2m'') \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-n\u2082)) (redL trm-n\u2081 m->\u03b2m'') =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2-redL (\u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 m->\u03b2m'')) wf-\u0393 (app (app (->\u03b2-Term-l m->\u03b2m'') trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redR x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (redR x\u2081 m->\u03b2m'')) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-e\u2082) trm-n\u2082)) (redR trm-m\u2081 n\u2081->e\u2082) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (redR trm-m\u2081 n\u2081->e\u2082))) wf-\u0393 (app (app trm-m\u2081 (->\u03b2-Term-l n\u2081->e\u2082)) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (abs L cf) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (abs L cf)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app trm-lam-m'' trm-n\u2081)) (abs L cf) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (abs L cf))) wf-\u0393 (app (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (cf x\u2209L))) trm-n\u2081)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (beta trm-lam-m\u2081 trm-n\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (beta trm-lam-m\u2081 trm-n\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 wf-\u0393 (app trm-m\u2081^n\u2081 trm-n\u2082)) (beta trm-lam-m\u2081 trm-n\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (beta trm-lam-m\u2081 trm-n\u2081))) wf-\u0393 (app (app trm-lam-m\u2081 trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (Y trm-m\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (Y trm-m\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-Ym\u2081) trm-n\u2081)) (Y trm-m\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (Y trm-m\u2081))) wf-\u0393 (app trm-Ym\u2081 trm-n\u2081)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (redR trm-n m->\u03b2m') = app \u0393\u22a9m'\u2236\u03c4 (\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4\u2081 m->\u03b2m')\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mn'\u2236\u03c4\u1d62 wf-\u0393 trm-mn') (redR trm-m n->\u03b2n') =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mn'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redR trm-m n->\u03b2n')) wf-\u0393 (app trm-m (->\u03b2-Term-l n->\u03b2n'))\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9lam-m'\u2236\u03c4\u1d62 wf-\u0393 trm-lam-x) (abs L x\u2082) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (abs L x\u2082)) wf-\u0393 (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (x\u2082 x\u2209L)))\n-- \u22a9->\u03b2 (abs L cf) (abs L\u2081 x) = abs (L ++ L\u2081) (\u03bb x\u2209L \u2192 \u22a9->\u03b2 (cf (\u2209-cons-l _ _ x\u2209L)) (x (\u2209-cons-r L _ x\u2209L)))\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (app \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'')) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = []} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 {\u0393} (app {s = m} \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = x \u2237 \u03c4\u1d62} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n--   where\n--   \u0393\u22a9m\u2236x~>x : \u0393 \u22a9 m \u2236 (x ~> x)\n--   \u0393\u22a9m\u2236x~>x = {!   !}\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mYm\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-Ym)) (Y trm-m) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mYm\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (Y trm-m)) wf-\u0393 trm-Ym\n","old_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\n-- open import Data.Maybe\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\n\ndata IType : Set where\n  o : IType\n  _~>_ : IType -> IType -> IType\n  \u2229 : List IType -> IType\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\no \u225fTI (\u2229 _) = no (\u03bb ())\n\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(_ ~> _) \u225fTI (\u2229 _) = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n(\u2229 _) \u225fTI o = no (\u03bb ())\n(\u2229 _) \u225fTI (_ ~> _) = no (\u03bb ())\n\u2229 [] \u225fTI \u2229 [] = yes refl\n\u2229 [] \u225fTI \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 IType)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\n\ndata _\u2237'_ : IType -> Type -> Set where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237' A -> \u03c4 \u2237' B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n  \u2229-nil : \u2200 {A} ->  \u03c9 \u2237' A\n  \u2229-cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237' A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237' A\n\n\ndata _\u2264\u2229_ : IType -> IType -> Set where\n  base : o \u2264\u2229 o\n  arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n  \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n  \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n  -- \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 : \u2200 {\u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u2229 \u03c4\u2c7c \u2264\u2229 \u2229 \u03c4\u1d62\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {[]} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-nil\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {x \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-cons (\u2229-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl))) (\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2081} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)))\n\n\u2264\u2229-refl : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c4\n\u2264\u2229-refl {o} = base\n\u2264\u2229-refl {\u03c4 ~> \u03c4\u2081} = arr \u2264\u2229-refl \u2264\u2229-refl\n\u2264\u2229-refl {\u2229 []} = \u2229-nil\n\u2264\u2229-refl {\u2229 (x \u2237 x\u2081)} = \u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2082} z \u2192 z)\n\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\n-- _\u2208?_ : \u2200 {a} {A : Set a} -> Atom -> List (Atom \u00d7 A) -> Maybe A\n-- a \u2208? [] = nothing\n-- a \u2208? (l \u2237 ist) with a \u225f proj\u2081 l\n-- ... | yes _ = just (proj\u2082 l)\n-- a \u2208? (l \u2237 ist) | no _ = a \u2208? ist\n\n-- PTerm->\u039b : \u2200 {\u0393 m t} -> (List (Atom \u00d7 \u2115)) -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- PTerm->\u039b {m = bv i} _ ()\n-- PTerm->\u039b {m = fv x} bound \u0393\u22a2m\u2236t with x \u2208? bound\n-- PTerm->\u039b {m = fv x} {t} bound \u0393\u22a2m\u2236t | just i = bv {t} i\n-- PTerm->\u039b {m = fv x} bound \u0393\u22a2m\u2236t | nothing = fv x\n-- PTerm->\u039b {m = lam m} bound (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) = lam \u03c4\u2081 (PTerm->\u039b ((x , 0) \u2237 bound') (cf (\u2209-cons-l _ _ x\u2209)))\n--   where\n--   x = \u2203fresh (L ++ FV m)\n--   x\u2209 : x \u2209 (L ++ FV m)\n--   x\u2209 = \u2203fresh-spec (L ++ FV m)\n--\n--   bound' : List (Atom \u00d7 \u2115)\n--   bound' = Data.List.map (\u03bb a,i \u2192 (proj\u2081 a,i) , suc (proj\u2082 a,i)) bound\n--\n-- PTerm->\u039b {m = app t1 t} bound (app \u0393\u22a2s \u0393\u22a2t) = app (PTerm->\u039b bound \u0393\u22a2s) (PTerm->\u039b bound \u0393\u22a2t)\n-- PTerm->\u039b {m = Y \u03c4} bound (Y x) = Y \u03c4\n--\n-- PTerm->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- PTerm->\u039b = PTerm->\u039b []\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\nPTerm->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\nPTerm->\u039b {m = bv i} ()\nPTerm->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\nPTerm->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ]\n    (PTerm->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\nPTerm->\u039b {m = app t1 t} (app \u0393\u22a2s \u0393\u22a2t) = app (PTerm->\u039b \u0393\u22a2s) (PTerm->\u039b \u0393\u22a2t)\nPTerm->\u039b {m = Y \u03c4} (Y x) = Y \u03c4\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\n\u039b*^-*^-swap : \u2200 {t : Type} k n x y m -> \u00ac(k \u2261 n) -> \u00ac(x \u2261 y) -> \u039b[_<<_] {t} k x (\u039b[ n << y ] m) \u2261 \u039b[ n << y ] (\u039b[ k << x ] m)\n\u039b*^-*^-swap k n x y (bv i) k\u2260n x\u2260y = refl\n\u039b*^-*^-swap k n x y (fv z) k\u2260n x\u2260y = {!   !}\n\u039b*^-*^-swap k n x y (lam A m) k\u2260n x\u2260y =\n  cong (lam A) (\u039b*^-*^-swap (suc k) (suc n) x y m (\u03bb x\u2081 \u2192 k\u2260n (\u2261-suc x\u2081)) x\u2260y)\n\u039b*^-*^-swap k n x y (app m m') k\u2260n x\u2260y rewrite\n  \u039b*^-*^-swap k n x y m k\u2260n x\u2260y | \u039b*^-*^-swap k n x y m' k\u2260n x\u2260y = refl\n\u039b*^-*^-swap k n x y (Y t) k\u2260n x\u2260y = refl\n\n\nfv-^-\u039b*^-refl : \u2200 x t {k \u0393 \u03c4} -> x \u2209 FV t -> (\u0393\u22a2t^x : \u0393 \u22a2 [ k >> fv x ] t \u2236 \u03c4) -> (\u039b[ k << x ] (PTerm->\u039b \u0393\u22a2t^x) ) ~ t\nfv-^-\u039b*^-refl x (bv n) x\u2209FVt ()\nfv-^-\u039b*^-refl x (fv y) x\u2209FVt \u0393\u22a2t^x with x \u225f y\nfv-^-\u039b*^-refl x (fv .x) x\u2209FVt \u0393\u22a2t^x | yes refl = \u22a5-elim (x\u2209FVt (here refl))\nfv-^-\u039b*^-refl x (fv y) x\u2209FVt \u0393\u22a2t^x | no x\u2260y = fv\nfv-^-\u039b*^-refl x (lam t) {k} {\u0393} x\u2209FVt (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam {!   !}\n  --\n  -- where\n  -- x' = \u2203fresh ( L ++ FV ([ suc k >> fv x ] t) )\n  -- x'\u2209 = \u2203fresh-spec ( L ++ FV ([ suc k >> fv x ] t) )\n  --\n  -- x'\u0393\u22a2[suc-k>>x]t^'x' : ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 ([ suc k >> fv x ] t) ^' x' \u2236 \u03c4\u2082\n  -- x'\u0393\u22a2[suc-k>>x]t^'x' = cf (\u2209-cons-l L (FV ([ suc k >> fv x ] t)) (\u2203fresh-spec (L ++ FV ([ suc k >> fv x ] t))))\n  --\n  --\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 : ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 [ suc k >> fv x ] (t ^' x') \u2236 \u03c4\u2082 \u2261\n  --   ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 ([ suc k >> fv x ] t) ^' x' \u2236 \u03c4\u2082\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 rewrite ^-^-swap (suc k) 0 x x' t (\u03bb ()) (\u03bb x\u2261x' \u2192 {!   !}) = refl\n  --\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'' : ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 [ suc k >> fv x ] (t ^' x') \u2236 \u03c4\u2082\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'' rewrite x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 = x'\u0393\u22a2[suc-k>>x]t^'x'\n  --\n  -- ih'' : \u039b[ (suc k) << x ] (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x'') ~ (t ^' x')\n  -- ih'' = fv-^-\u039b*^-refl x ([ 0 >> fv x' ] t) {!   !} x'\u0393\u22a2[suc-k>>x]t^'x''\n  --\n  -- -- ih' : \u039b[ 0 << x' ] ( \u039b[ (suc k) << x ] (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x') ) ~ t\n  -- -- ih' rewrite x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 = {!   !}\n  --\n  -- ih : \u039b[ (suc k) << x ] ( \u039b[ 0 << x' ] (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x') ) ~ t\n  -- ih rewrite\n  --   \u039b*^-*^-swap {\u03c4\u2082} (suc k) 0 x x' (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x') {!   !} {!   !} |\n  --   x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 = {!   !}\n\nfv-^-\u039b*^-refl x (app s t) x\u2209FVt (app \u0393\u22a2s^x \u0393\u22a2t^x) = app\n  (fv-^-\u039b*^-refl x s (\u2209-cons-l _ _ x\u2209FVt) \u0393\u22a2s^x)\n  (fv-^-\u039b*^-refl x t (\u2209-cons-r (FV s) _ x\u2209FVt) \u0393\u22a2t^x)\nfv-^-\u039b*^-refl x (Y \u03c4) x\u2209FVt (Y x\u2081) = Y\n\n\n\nPTerm->\u039b~ : \u2200 {\u0393 t \u03c4} -> {\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4} -> (PTerm->\u039b \u0393\u22a2t) ~ t\nPTerm->\u039b~ {t = bv i} = \u03bb {\u03c4} \u2192 \u03bb {}\nPTerm->\u039b~ {t = fv x} = \u03bb {\u03c4} {\u0393\u22a2t} \u2192 fv\nPTerm->\u039b~ {t = lam t} {\u0393\u22a2t = abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf} =\n  lam (fv-^-\u039b*^-refl (\u2203fresh (L ++ FV t)) t (\u2209-cons-r L _ (\u2203fresh-spec (L ++ FV t))) (cf (\u2209-cons-l L (FV t) (\u2203fresh-spec (L ++ FV t)))))\n  -- where\n  -- x' = \u2203fresh (L ++ FV t)\n  -- x'\u2237\u0393\u22a2t^'x' = cf (\u2209-cons-l L (FV t) (\u2203fresh-spec (L ++ FV t)))\n  --\n  -- ih' : \u039b[ 0 << x' ] (PTerm->\u039b x'\u2237\u0393\u22a2t^'x') ~ (* x' ^ (t ^' x'))\n  -- ih' = {!   !}\n  --\n  -- sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  -- sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n  --\n  -- ih : \u039b[ 0 << x' ] (PTerm->\u039b x'\u2237\u0393\u22a2t^'x') ~ t\n  -- ih rewrite sub {_} {x'} {\u039b[ 0 << x' ] (PTerm->\u039b x'\u2237\u0393\u22a2t^'x')} {!   !} = ih'\nPTerm->\u039b~ {t = app t t\u2081} {\u0393\u22a2t = app \u0393\u22a2t \u0393\u22a2t\u2081} = app PTerm->\u039b~ PTerm->\u039b~\nPTerm->\u039b~ {t = Y t\u2081} {\u0393\u22a2t = Y x} = Y\n\n\n\u039b->PTerm : \u2200 {t} -> \u039b t -> PTerm\n\u039b->PTerm (bv i) = bv i\n\u039b->PTerm (fv x) = fv x\n\u039b->PTerm (lam A \u039bt) = lam (\u039b->PTerm \u039bt)\n\u039b->PTerm (app \u039bs \u039bt) = app (\u039b->PTerm \u039bs) (\u039b->PTerm \u039bt)\n\u039b->PTerm (Y t) = Y t\n\n\n\n\n\n-- data IType\u209b : IType -> Set where\n--   o : IType\u209b o\n--   arr : \u2200 {\u03c4 \u03c4'} -> IType\u209b \u03c4 -> IType\u209b \u03c4' -> IType\u209b (\u03c4 ~> \u03c4')\n--\n-- data IType\u209b\u209b : IType -> Set where\n--   o : IType\u209b\u209b o\n--   arr :  \u2200 {\u03c4 \u03c4'} -> IType\u209b\u209b \u03c4 -> IType\u209b\u209b \u03c4' -> IType\u209b\u209b (\u03c4 ~> \u03c4')\n--   \u2229-nil : IType\u209b\u209b \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4\u1d62} -> IType\u209b \u03c4 ->  IType\u209b\u209b (\u2229 \u03c4\u1d62) -> IType\u209b\u209b (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n--\n-- \u03c4\u209b->\u03c4\u209b\u209b : \u2200 {\u03c4} -> IType\u209b \u03c4 -> IType\u209b\u209b \u03c4\n-- \u03c4\u209b->\u03c4\u209b\u209b o = o\n-- \u03c4\u209b->\u03c4\u209b\u209b (arr \u03c4\u209b \u03c4\u209b\u2081) = arr (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b) (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b\u2081)\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2) -- app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata Y-shape : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  intro\u2081 : \u2200 {A m} -> Y-shape (app (Y A) m)\n  intro\u2082 : \u2200 {A m} -> Y-shape (app m (app (Y A) m))\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} -> (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u2229 \u03c4\u1d62)) \u2208 \u0393) -> (\u03c4\u1d62\u2264\u2229\u03c4 : \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4) -> \u03c4 \u2237' A ->\n    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4\u2081 \u03c4\u2082} -> \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9 t \u2236 \u03c4\u2081 -> (\u03c4\u2081 ~> \u03c4\u2082) \u2237' (A \u27f6 B) -> \u03c4\u2081 \u2237' B ->\n    \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\u2082\n  \u2229-nil : \u2200 {A \u0393} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) -> \u0393 \u22a9 m\u2005 \u2236 \u03c9\n  \u2229-cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 \u03c4\u1d62) -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n  abs : \u2200 {A B \u0393 \u03c4\u1d62 \u03c4} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4 ) -> \u2229 \u03c4\u1d62 \u2237' A -> \u03c4 \u2237' B -> \u0393 \u22a9 lam A t \u2236 (\u2229 \u03c4\u1d62 ~> \u03c4)\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} -> Wf-ICtxt \u0393 -> \u03c4 \u2237' A -> \u03c4\u2081 \u2237' A -> \u03c4\u2082 \u2237' A ->\n    \u0393 \u22a9 Y A \u2236 ((\u03c4 ~> \u03c4\u2081) ~> \u03c4\u2082)\n\n\n-- \u22a9->\u03b2 : \u2200 {\u0393 m m' \u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236\u03c4'~>\u03c4 (app (Y {\u03c4 = \u03c4'} wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 \u03c4\u2082\u2237) \u0393\u22a9m\u2236\u03c4'~>\u03c4')) (Y trm-m) =\n--     app (Y wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 {!   !}) \u0393\u22a9m\u2236\u03c4'~>\u03c4'\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393 trm-m)) (Y x) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082)) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393 trm-m) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2082)\n--\n--\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u03b2m'') \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-n\u2082)) (redL trm-n\u2081 m->\u03b2m'') =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2-redL (\u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 m->\u03b2m'')) wf-\u0393 (app (app (->\u03b2-Term-l m->\u03b2m'') trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redR x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (redR x\u2081 m->\u03b2m'')) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-e\u2082) trm-n\u2082)) (redR trm-m\u2081 n\u2081->e\u2082) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (redR trm-m\u2081 n\u2081->e\u2082))) wf-\u0393 (app (app trm-m\u2081 (->\u03b2-Term-l n\u2081->e\u2082)) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (abs L cf) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (abs L cf)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app trm-lam-m'' trm-n\u2081)) (abs L cf) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (abs L cf))) wf-\u0393 (app (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (cf x\u2209L))) trm-n\u2081)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (beta trm-lam-m\u2081 trm-n\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (beta trm-lam-m\u2081 trm-n\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 wf-\u0393 (app trm-m\u2081^n\u2081 trm-n\u2082)) (beta trm-lam-m\u2081 trm-n\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (beta trm-lam-m\u2081 trm-n\u2081))) wf-\u0393 (app (app trm-lam-m\u2081 trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (Y trm-m\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (Y trm-m\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-Ym\u2081) trm-n\u2081)) (Y trm-m\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (Y trm-m\u2081))) wf-\u0393 (app trm-Ym\u2081 trm-n\u2081)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (redR trm-n m->\u03b2m') = app \u0393\u22a9m'\u2236\u03c4 (\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4\u2081 m->\u03b2m')\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mn'\u2236\u03c4\u1d62 wf-\u0393 trm-mn') (redR trm-m n->\u03b2n') =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mn'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redR trm-m n->\u03b2n')) wf-\u0393 (app trm-m (->\u03b2-Term-l n->\u03b2n'))\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9lam-m'\u2236\u03c4\u1d62 wf-\u0393 trm-lam-x) (abs L x\u2082) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (abs L x\u2082)) wf-\u0393 (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (x\u2082 x\u2209L)))\n-- \u22a9->\u03b2 (abs L cf) (abs L\u2081 x) = abs (L ++ L\u2081) (\u03bb x\u2209L \u2192 \u22a9->\u03b2 (cf (\u2209-cons-l _ _ x\u2209L)) (x (\u2209-cons-r L _ x\u2209L)))\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (app \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'')) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = []} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 {\u0393} (app {s = m} \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = x \u2237 \u03c4\u1d62} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n--   where\n--   \u0393\u22a9m\u2236x~>x : \u0393 \u22a9 m \u2236 (x ~> x)\n--   \u0393\u22a9m\u2236x~>x = {!   !}\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mYm\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-Ym)) (Y trm-m) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mYm\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (Y trm-m)) wf-\u0393 trm-Ym\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"90a15e4a99a04068b454c8a345f15491d28aef67","subject":"Some changes to a note on co-inductive natural numbers.","message":"Some changes to a note on co-inductive natural numbers.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.ConatSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')\nConat-unf cozero          = inj\u2081 refl\nConat-unf (cosucc {n} Cn) = inj\u2082 (n , \u266d Cn , refl)\n\nConat-pre-fixed : \u2200 {n} \u2192\n                  (n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                  Conat n\nConat-pre-fixed (inj\u2081 h) = subst Conat (sym h) cozero\nConat-pre-fixed (inj\u2082 (n , Cn , h)) = subst Conat (sym h) (cosucc (\u266f Cn))\n\nConat-coind : \u2200 (A : D \u2192 Set) {n} \u2192\n              (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')) \u2192\n              A n \u2192 Conat n\nConat-coind A h An = {!!}\n\npostulate\n  inf    : D\n  inf-eq : inf \u2261 succ\u2081 inf\n{-# ATP axiom inf-eq #-}\n\n{-# NO_TERMINATION_CHECK #-}\ninf-Conat : Conat inf\ninf-Conat = subst Conat (sym inf-eq) (cosucc (\u266f inf-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.ConatSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')\nConat-unf cozero = inj\u2081 refl\nConat-unf (cosucc {n} Cn) = inj\u2082 (n , \u266d Cn , refl)\n\nConat-pre-fixed : \u2200 {n} \u2192\n                  (n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                  Conat n\nConat-pre-fixed (inj\u2081 h) = subst Conat (sym h) cozero\nConat-pre-fixed (inj\u2082 (n , Cn , h)) = subst Conat (sym h) (cosucc (\u266f Cn))\n\nConat-coind : \u2200 (A : D \u2192 Set) {n} \u2192\n              (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')) \u2192\n              A n \u2192 Conat n\nConat-coind A h An = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d555f1b4a408249eaf68dbbeebfcdd95ac7832d1","subject":"improved proof that double is even","message":"improved proof that double is even\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"code\/GoaTest.agda","new_file":"code\/GoaTest.agda","new_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where -- finite type\n  true : Bool\n  false : Bool\n\nidBool : Bool \u2192 Bool -- lambda\nidBool x = x\n\nalwaysTrue : Bool \u2192 Bool\nalwaysTrue x = true\n\nnot : Bool \u2192 Bool -- case defn\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool -- lambda\nnotnot x = not(not(x))\n\n_&_ : Bool \u2192 Bool \u2192 Bool --curried function\ntrue & x = x\nfalse & _ = false\n\ndata \u2115 : Type where -- infinite type\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool -- recursive definition\neven zero = true\neven (succ x) = not (even x)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115 \nzero + y = y\nsucc x + y = x + succ y\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata \u2115List : Type where --list type \n  [] : \u2115List -- empty list \n  _::_ : \u2115 \u2192 \u2115List \u2192 \u2115List -- add number to head of list\n\nmylist : \u2115List \nmylist = 3 :: (4 :: (2 :: [])) -- the list [3, 4, 2]\n\ndata Vector : \u2115 \u2192 Type where -- type family\n  [] : Vector 0\n  _::_ : {n : \u2115} \u2192 \u2115 \u2192 Vector n \u2192 Vector (succ n) \n\nsum : {n : \u2115} \u2192 Vector n \u2192 \u2115\nsum [] = 0\nsum (x :: l) = x + sum l\n\n\ncountdown : (n : \u2115) \u2192 Vector n -- dependent function\ncountdown 0 = []\ncountdown (succ n) = (succ n) :: (countdown n)\n\nsumToN : \u2115 \u2192 \u2115 -- calculation\nsumToN n = sum(countdown n)\n\ndata isEven : \u2115 \u2192 Type where\n  0even : isEven 0\n  +2even : (n : \u2115) \u2192 isEven n \u2192 isEven (succ(succ(n)))\n\n4even : isEven 4\n4even = +2even _ (+2even _ 0even)\n\ndata False : Type where\n\n1odd : isEven 1 \u2192 False\n1odd ()\n\n3odd : isEven 3 \u2192 False\n3odd (+2even .1 ())\n\nhalf : (n : \u2115) \u2192 isEven n \u2192 \u2115\nhalf .0 0even = 0\nhalf .(succ (succ n)) (+2even n pf) = half n pf\n\ndouble : (n : \u2115) \u2192 \u2115\ndouble 0 = 0\ndouble (succ n) = succ(succ(double(n)))\n\nthm : (n  : \u2115) \u2192 isEven (double n)\nthm zero = 0even\nthm (succ n) = +2even _ (thm n)\n\nhalfOfDouble : \u2115 \u2192 \u2115\nhalfOfDouble n = half (double n) (thm n)\n","old_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where -- finite type\n  true : Bool\n  false : Bool\n\nidBool : Bool \u2192 Bool -- lambda\nidBool x = x\n\nalwaysTrue : Bool \u2192 Bool\nalwaysTrue x = true\n\nnot : Bool \u2192 Bool -- case defn\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool -- lambda\nnotnot x = not(not(x))\n\n_&_ : Bool \u2192 Bool \u2192 Bool --curried function\ntrue & x = x\nfalse & _ = false\n\ndata \u2115 : Type where -- infinite type\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool -- recursive definition\neven zero = true\neven (succ x) = not (even x)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115 \nzero + y = y\nsucc x + y = x + succ y\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata \u2115List : Type where --list type \n  [] : \u2115List -- empty list \n  _::_ : \u2115 \u2192 \u2115List \u2192 \u2115List -- add number to head of list\n\nmylist : \u2115List \nmylist = 3 :: (4 :: (2 :: [])) -- the list [3, 4, 2]\n\ndata Vector : \u2115 \u2192 Type where -- type family\n  [] : Vector 0\n  _::_ : {n : \u2115} \u2192 \u2115 \u2192 Vector n \u2192 Vector (succ n) \n\nsum : {n : \u2115} \u2192 Vector n \u2192 \u2115\nsum [] = 0\nsum (x :: l) = x + sum l\n\n\ncountdown : (n : \u2115) \u2192 Vector n -- dependent function\ncountdown 0 = []\ncountdown (succ n) = (succ n) :: (countdown n)\n\nsumToN : \u2115 \u2192 \u2115 -- calculation\nsumToN n = sum(countdown n)\n\ndata isEven : \u2115 \u2192 Type where\n  0even : isEven 0\n  +2even : (n : \u2115) \u2192 isEven n \u2192 isEven (succ(succ(n)))\n\n4even : isEven 4\n4even = +2even _ (+2even _ 0even)\n\ndata False : Type where\n\n1odd : isEven 1 \u2192 False\n1odd ()\n\n3odd : isEven 3 \u2192 False\n3odd (+2even .1 ())\n\nhalf : (n : \u2115) \u2192 isEven n \u2192 \u2115\nhalf .0 0even = 0\nhalf .(succ (succ n)) (+2even n pf) = half n pf\n\ndouble : (n : \u2115) \u2192 \u2115\ndouble 0 = 0\ndouble (succ n) = succ(succ(double(n)))\n\nstep : (n : \u2115) \u2192 (isEven (double n)) \u2192 isEven (double(succ(n)))\nstep n pf = +2even _ pf\n\nthm : (n  : \u2115) \u2192 isEven (double n)\nthm zero = 0even\nthm (succ n) = step _ (thm n)\n\nhalfOfDouble : \u2115 \u2192 \u2115\nhalfOfDouble n = half (double n) (thm n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dd7fc1faaff45990fdfb94040dc68dcdfadf86f0","subject":"Added doc (the bisimilarity relation _\u2248_ is on unbounded lists).","message":"Added doc (the bisimilarity relation _\u2248_ is on unbounded lists).\n\nIgnore-this: dbbdf0d52a1b6a25c1ae4a742876f1b0\n\ndarcs-hash:20120322232946-3bd4e-379500496257d6c261ce7494f21a26034b65f876.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Relation\/Binary\/Bisimilarity.agda","new_file":"src\/FOTC\/Relation\/Binary\/Bisimilarity.agda","new_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation on unbounded lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The bisimilarity relation _\u2248_ on unbounded lists is the greatest\n-- fixed point (by \u2248-gfp\u2081 and \u2248-gfp\u2082) of the bisimulation functional\n-- (see below).\n\n-- The bisimilarity relation on unbounded lists.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation _\u2248_ on unbounded lists is a post-fixed\n-- point of the bisimulation functional (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation _\u2248_ on unbounded lists is the greatest\n-- post-fixed point of the bisimulation functional (see below).\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed-point, the\n-- bisimilarity relation _\u2248_ on unbounded lists is also a pre-fixed\n-- point of the bisimulation functional (see below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional on unbounded\n  -- lists. This module is only for illustrative purposes.\n\n  -- References:\n  --\n  -- \u2022 Peter Dybjer and Herbert Sander. A functional programming\n  --   approach to the specification and verification of concurrent\n  --   systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n  --\n  -- \u2022 Bart Jacobs and Jan Rutten. (Co)algebras and\n  --   (co)induction. EATCS Bulletin, 62:222\u2013259, 1997.\n\n  ----------------------------------------------------------------------------\n  -- The bisimilarity relation _\u2248_ on unbounded lists is the greatest\n  -- post-fixed point of Bisimulation (by post-fp and gpfp).\n\n  -- The bisimulation functional on unbounded lists (adapted from\n  -- Dybjer and Sander 1989, p. 310, and Jacobs and Rutten 1997,\n  -- p. 30).\n  BisimulationF : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BisimulationF _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The bisimilarity relation _\u2248_ on unbounded lists is a post-fixed\n  -- point of Bisimulation, i.e,\n  --\n  -- _\u2248_ \u2264 Bisimulation _\u2248_.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 BisimulationF _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  -- The bisimilarity relation _\u2248_ on unbounded lists is the greatest\n  -- post-fixed point of Bisimulation, i.e\n  --\n  -- \u2200 R. R \u2264 Bisimulation R \u21d2 R \u2264 _\u2248_.\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 BisimulationF _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- Because a greatest post-fixed point is a fixed-point, the\n  -- bisimilarity relation _\u2248_ on unbounded lists is also a pre-fixed\n  -- point of Bisimulation, i.e.\n  --\n  -- Bisimulation _\u2248_ \u2264 _\u2248_.\n  pre-fp : \u2200 {xs ys} \u2192 BisimulationF _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","old_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The bisimilarity relation _\u2248_ is the greatest fixed point (by\n-- \u2248-gfp\u2081 and \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation _\u2248_ is a post-fixed point of the\n-- bisimulation functional (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n-- the bisimulation functional (see below).\n--\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed-point, the\n-- bisimilarity relation _\u2248_ is also a pre-fixed point of the\n-- bisimulation functional (see below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- References:\n  --\n  -- \u2022 Peter Dybjer and Herbert Sander. A functional programming\n  --   approach to the specification and verification of concurrent\n  --   systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n  --\n  -- \u2022 Bart Jacobs and Jan Rutten. (Co)algebras and\n  --   (co)induction. EATCS Bulletin, 62:222\u2013259, 1997.\n\n  ----------------------------------------------------------------------------\n  -- Adapted from (Dybjer and Sander 1989, p. 310). In this paper, the\n  -- authors use the name\n\n  -- as (R :: R') bs'\n\n  -- for the bisimulation functional.\n\n  -- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The bisimulation functional (Jacobs and Rutten 1997, p. 30).\n  BisimulationF : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BisimulationF _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The bisimilarity relation _\u2248_ is a post-fixed point of\n  -- Bisimulation, i.e,\n  --\n  -- _\u2248_ \u2264 Bisimulation _\u2248_.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 BisimulationF _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  -- The bisimilarity relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation, i.e\n  --\n  -- \u2200 R. R \u2264 Bisimulation R \u21d2 R \u2264 _\u2248_.\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 BisimulationF _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- Because a greatest post-fixed point is a fixed-point, the\n  -- bisimilarity relation _\u2248_ is also a pre-fixed point of\n  -- Bisimulation, i.e.\n  --\n  -- Bisimulation _\u2248_ \u2264 _\u2248_.\n  pre-fp : \u2200 {xs ys} \u2192 BisimulationF _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f5c0e7addd33faf63ba9346cb7a85c7384176846","subject":"Added two automatic proofs.","message":"Added two automatic proofs.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Conat\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Data\/Conat\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Conat properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- \u2022 Herbert P. Sander. A logic of functional programs with an\n--   application to concurrency. PhD thesis, Chalmers University of\n--   Technology and University of Gothenburg, Department of Computer\n--   Sciences, 1992.\n\nmodule FOTC.Data.Conat.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n0-Conat : Conat zero\n0-Conat = Conat-coind P h refl\n  where\n  P : D \u2192 Set\n  P n = n \u2261 zero\n  {-# ATP definition P #-}\n\n  postulate h : \u2200 {n} \u2192 P n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] P n' \u2227 n \u2261 succ\u2081 n')\n  {-# ATP prove h #-}\n\n-- Adapted from (Sander 1992, p. 57).\n\u221e-Conat : Conat \u221e\n\u221e-Conat = Conat-coind P h refl\n  where\n  P : D \u2192 Set\n  P n = n \u2261 \u221e\n  {-# ATP definition P #-}\n\n  postulate h : \u2200 {n} \u2192 P n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] P n' \u2227 n \u2261 succ\u2081 n')\n  {-# ATP prove h #-}\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat Nn = Conat-coind N h Nn\n  where\n  h : \u2200 {m} \u2192 N m \u2192 m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 N m' \u2227 m \u2261 succ\u2081 m')\n  h nzero = prf\n    where postulate prf : zero \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 N m' \u2227 zero \u2261 succ\u2081 m')\n          {-# ATP prove prf #-}\n  h (nsucc {m} Nm) = prf\n    where postulate prf : succ\u2081 m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 N m' \u2227 succ\u2081 m \u2261 succ\u2081 m')\n          {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Conat properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- \u2022 Herbert P. Sander. A logic of functional programs with an\n--   application to concurrency. PhD thesis, Chalmers University of\n--   Technology and University of Gothenburg, Department of Computer\n--   Sciences, 1992.\n\nmodule FOTC.Data.Conat.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n0-Conat : Conat zero\n0-Conat = Conat-coind P h refl\n  where\n  P : D \u2192 Set\n  P n = n \u2261 zero\n  {-# ATP definition P #-}\n\n  postulate h : \u2200 {n} \u2192 P n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] P n' \u2227 n \u2261 succ\u2081 n')\n  {-# ATP prove h #-}\n\n-- Adapted from (Sander 1992, p. 57).\n\u221e-Conat : Conat \u221e\n\u221e-Conat = Conat-coind P h refl\n  where\n  P : D \u2192 Set\n  P n = n \u2261 \u221e\n  {-# ATP definition P #-}\n\n  postulate h : \u2200 {n} \u2192 P n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] P n' \u2227 n \u2261 succ\u2081 n')\n  {-# ATP prove h #-}\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat Nn = Conat-coind N h Nn\n  where\n  h : \u2200 {m} \u2192 N m \u2192 m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 N m' \u2227 m \u2261 succ\u2081 m')\n  h nzero          = inj\u2081 refl\n  h (nsucc {m} Nm) = inj\u2082 (m , Nm , refl)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a3df42670b0f46a7af869da9ff572d148b28bb7d","subject":"Fix type inference problem that arose","message":"Fix type inference problem that arose\n\nOnly noticed when tried removing --allow-unsolved-metas. That seems a habit for\nAgda.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3[ \u2261\u0393 \u2208 \u03931 \u2261 \u03932 ]\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT\n        t1\n        (subst (\u03bb \u0393 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4) (sym \u2261\u0393) t2)\n        (v1 \u2022 \u03c11)\n        (subst (\u03bb \u0393 \u2192 \u27e6 \u03c3 \u2022 \u0393 \u27e7Context) (sym \u2261\u0393) (v2 \u2022 \u03c12))\n        k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 {!dt!} t2 (v1 \u2022 \u03c11) d\u03c1 (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3[ \u2261\u0393 \u2208 \u03931 \u2261 \u03932 ]\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT\n        t1\n        (subst (\u03bb \u0393 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4) (sym \u2261\u0393) t2)\n        (v1 \u2022 \u03c11)\n        (subst (\u03bb \u0393 \u2192 \u27e6 \u03c3 \u2022 \u0393 \u27e7Context) (sym \u2261\u0393) (v2 \u2022 \u03c12))\n        k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 {!dt!} t2 (v1 \u2022 \u03c11) d\u03c1 (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a10c358ed179bc75187c8a826b8e779593a32cf2","subject":"clarified question","message":"clarified question\n\nOld-commit-hash: 6c108888ef225e1f4e8b0978db5f3783ef6b834a\n","repos":"inc-lc\/ilc-agda","old_file":"derivation.agda","new_file":"derivation.agda","new_contents":"module derivation where\n\nopen import lambda\n\n-- KO: Is it correct that this file is supposed to contain the syntactic \n<<<<<<< HEAD\n-- variants of the operations in Syntactic.Changes.agda?\n-- If so, why does it have a different (and rather strange) \u0394-Type definition?\n=======\n-- variants of the operations in Changes.agda? What is its relation to\n-- the definitions in Syntactic.Changes?\n-- Why does it have a different (and rather strange) \u0394-Type definition?\n>>>>>>> clarified question\n-- Why are the base cases (bool) all missing? The inductive cases look\n-- boring since they don't actually do anything (which explains why compose and apply are the same)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","old_contents":"module derivation where\n\nopen import lambda\n\n-- KO: Is it correct that this file is supposed to contain the syntactic \n-- variants of the operations in Syntactic.Changes.agda?\n-- If so, why does it have a different (and rather strange) \u0394-Type definition?\n-- Why are the base cases (bool) all missing? The inductive cases look\n-- boring since they don't actually do anything (which explains why compose and apply are the same)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"248a51f14bba78dd9c66b0378c4ee935b70b5a79","subject":"Minor change.","message":"Minor change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationATP.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationATP.agda","new_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.ABP.ProofSpecificationATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1ATP\nopen import FOTC.Program.ABP.Lemma2ATP\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 transfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  postulate prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n                   \u2203[ i' ] \u2203[ is' ] \u2203[ js' ]\n                   is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  {-# ATP prove prf\u2081 lemma\u2081 lemma\u2082 not-Bool #-}\n\n  prf\u2082 : B is (transfer b os\u2080 os\u2081 is)\n  prf\u2082 = b , os\u2080 , os\u2081 , as , bs , cs , ds , helper\n    where\n    as bs cs ds : D\n    as = has (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    bs = hbs (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    cs = hcs (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    ds = hds (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    {-# ATP definition as bs cs ds #-}\n\n    postulate helper : Stream is \u2227 Bit b \u2227 Fair os\u2080 \u2227 Fair os\u2081\n                       \u2227 ABP b is os\u2080 os\u2081 as bs cs ds (transfer b os\u2080 os\u2081 is)\n    {-# ATP prove helper #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.ABP.ProofSpecificationATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1ATP\nopen import FOTC.Program.ABP.Lemma2ATP\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 transfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  postulate prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n                   \u2203[ i' ] \u2203[ is' ] \u2203[ js' ]\n                   is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  {-# ATP prove prf\u2081 lemma\u2081 lemma\u2082 not-Bool #-}\n\n  prf\u2082 : B is (transfer b os\u2080 os\u2081 is)\n  prf\u2082 = b , os\u2080 , os\u2081 , as , bs , cs , ds , helper\n    where\n    as bs cs ds : D\n    as = has (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    bs = hbs (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    cs = hcs (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n    ds = hds (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n\n    {-# ATP definition as bs cs ds #-}\n\n    postulate helper : Stream is \u2227 Bit b \u2227 Fair os\u2080 \u2227 Fair os\u2081\n                       \u2227 ABP b is os\u2080 os\u2081 as bs cs ds (transfer b os\u2080 os\u2081 is)\n    {-# ATP prove helper #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c349b7d6ea0a18743c706db2f6f14e8e3b95f6de","subject":"Logical: avoid a misleading notation","message":"Logical: avoid a misleading notation\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Logical.agda","new_file":"lib\/Relation\/Binary\/Logical.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.Logical where\n\nopen import Level\n\n\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : Set a\u2081) (A\u2082 : Set a\u2082) \u2192 Set _\n\u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63\n\n\u27e6Set\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : Set) \u2192 Set\u2081\n\u27e6Set\u2080\u27e7 = \u27e6Set\u27e7 zero\n\n\u27e6Set\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : Set\u2081) \u2192 Set\u2082\n\u27e6Set\u2081\u27e7 = \u27e6Set\u27e7 (suc zero)\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 Set _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 Set _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\nopen import Data.Product\n\n\nopen import Data.Unit\n\nrecord \u27e6\u22a4\u27e7 (x\u2081 x\u2082 : \u22a4) : Set where\n  constructor \u27e6tt\u27e7\n\nopen import Data.Empty\n\ndata \u27e6\u22a5\u27e7 (x\u2081 x\u2082 : \u22a5) : Set where\n\nopen import Relation.Nullary\n\ninfix 3 \u27e6\u00ac\u27e7_\n\n--\u27e6\u00ac\u27e7_ : (\u27e6Set\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u27e7) \u00ac_ \u00ac_\n\u27e6\u00ac\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63) \u2192 \u00ac A\u2081 \u2192 \u00ac A\u2082 \u2192 Set _\n\u27e6\u00ac\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u22a5\u27e7\n\n{-\n\u27e6\u2203\u27e7 : {A\u2081 A\u2082 : World \u2192 Set} (A\u1d63 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 A\u2081 \u03b1\u2081 \u2192 A\u2082 \u03b1\u2082 \u2192 Set)\n       (p\u2081 : \u2203 A\u2081) (f\u2082 : \u2203 A\u2082) \u2192 Set\n\u27e6\u2203\u27e7 A\u1d63 = \u03bb p\u2081 p\u2082 \u2192 \u03a3[ \u03b1\u1d63 \u2236 \u27e6World\u27e7 (proj\u2081 p\u2081) (proj\u2081 p\u2082) ]\n                       (A\u1d63 \u03b1\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n\nsyntax \u27e6\u2203\u27e7 (\u03bb \u03b1\u1d63 \u2192 f) = \u27e6\u2203\u27e7[ \u03b1\u1d63 ] f\n-}\n\nPred : \u2200 \u2113 {a} (A : Set a) \u2192 Set (a \u2294 suc \u2113)\nPred \u2113 A = A \u2192 Set \u2113\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n--\u27e6Pred\u27e7 {p\u2081} {p\u2082} p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 (p\u2081 \u2294 p\u2082 \u2294 p\u1d63)\n\u27e6Pred\u27e7 {p\u2081} {p\u2082} p\u1d63 A\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 f\u2081 x\u2081 \u2192 f\u2082 x\u2082 \u2192 Set (p\u2081 \u2294 p\u2082 \u2294 p\u1d63)\n\nopen import Relation.Binary\n\nprivate\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 Set a \u2192 Set b \u2192 Set (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 Set \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082} \u2113\u1d63 \u2192\n          (\u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) (REL\u2032 \u2113\u2081) (REL\u2032 \u2113\u2082)\n  \u27e6REL\u27e7\u2032 \u2113\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6Set\u27e7 \u2113\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          {\u2113\u2081 \u2113\u2082} \u2113\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 \u2113\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 \u2113\u2082) \u2192 Set _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 \u2113\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6Set\u27e7 \u2113\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {\u2113\u2081 \u2113\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 \u2113\u2081) (\u223c\u2082 : Rel A\u2082 \u2113\u2082) \u2192 Set _\n\u27e6Rel\u27e7 A\u1d63 \u2113\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 \u2113\u1d63\n\n-- data \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : Set \u2113\u2081} {P\u2082 : Set \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 Set \u2113\u1d63) : Dec P\u2081 \u2192 Dec P\u2082 \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) where\n-- data \u27e6Dec\u27e7 {\u2113\u1d63} : \u27e6Pred\u27e7 (\u27e6Set\u27e7 \u2113\u1d63) \u2113\u1d63 Dec Dec where\ndata \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : Set \u2113\u2081} {P\u2082 : Set \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 Set \u2113\u1d63) : \u27e6Set\u27e7 (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) (Dec P\u2081) (Dec P\u2082) where\n  yes : {-\u2200 {P\u2081 P\u2082 P\u1d63}-} {p\u2081 : P\u2081} {p\u2082 : P\u2082} (p\u1d63 : P\u1d63 p\u2081 p\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (yes p\u2081) (yes p\u2082)\n  no  : {-\u2200 {P\u2081 P\u2082 P\u1d63}-} {\u00acp\u2081 : \u00ac P\u2081} {\u00acp\u2082 : \u00ac P\u2082} (\u00acp\u1d63 : (\u27e6\u00ac\u27e7 P\u1d63) \u00acp\u2081 \u00acp\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (no \u00acp\u2081) (no \u00acp\u2082)\n\n--\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {b\u2081 b\u2082} b\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6REL\u27e7 A\u1d63 B\u1d63 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) Decidable Decidable\n\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n                 {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n                 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {\u223c\u2081 : REL A\u2081 B\u2081 \u2113\u2081} {\u223c\u2082 : REL A\u2082 B\u2082 \u2113\u2082} (\u223c\u1d63 : \u27e6REL\u27e7 A\u1d63 B\u1d63 \u2113\u1d63 \u223c\u2081 \u223c\u2082)\n              \u2192 Decidable \u223c\u2081 \u2192 Decidable \u223c\u2082 \u2192 Set _\n\u27e6Decidable\u27e7 A\u1d63 B\u1d63 _\u223c\u1d63_ = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e8 y\u1d63 \u2236 B\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Dec\u27e7 (x\u1d63 \u223c\u1d63 y\u1d63)\n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.Logical where\n\nopen import Level\n\n\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : Set a\u2081) (A\u2082 : Set a\u2082) \u2192 Set _\n\u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63\n\n\u27e6Set\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : Set) \u2192 Set\u2081\n\u27e6Set\u2080\u27e7 = \u27e6Set\u27e7 zero\n\n\u27e6Set\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : Set\u2081) \u2192 Set\u2082\n\u27e6Set\u2081\u27e7 = \u27e6Set\u27e7 (suc zero)\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2200 A\u2081 \u2192 Set b\u2081} {B\u2082 : \u2200 A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 Set _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 Set _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\nopen import Data.Product\n\n\nopen import Data.Unit\n\nrecord \u27e6\u22a4\u27e7 (x\u2081 x\u2082 : \u22a4) : Set where\n  constructor \u27e6tt\u27e7\n\nopen import Data.Empty\n\ndata \u27e6\u22a5\u27e7 (x\u2081 x\u2082 : \u22a5) : Set where\n\nopen import Relation.Nullary\n\ninfix 3 \u27e6\u00ac\u27e7_\n\n--\u27e6\u00ac\u27e7_ : (\u27e6Set\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u27e7) \u00ac_ \u00ac_\n\u27e6\u00ac\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63) \u2192 \u00ac A\u2081 \u2192 \u00ac A\u2082 \u2192 Set _\n\u27e6\u00ac\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u22a5\u27e7\n\n{-\n\u27e6\u2203\u27e7 : {A\u2081 A\u2082 : World \u2192 Set} (A\u1d63 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 A\u2081 \u03b1\u2081 \u2192 A\u2082 \u03b1\u2082 \u2192 Set)\n       (p\u2081 : \u2203 A\u2081) (f\u2082 : \u2203 A\u2082) \u2192 Set\n\u27e6\u2203\u27e7 A\u1d63 = \u03bb p\u2081 p\u2082 \u2192 \u03a3[ \u03b1\u1d63 \u2236 \u27e6World\u27e7 (proj\u2081 p\u2081) (proj\u2081 p\u2082) ]\n                       (A\u1d63 \u03b1\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n\nsyntax \u27e6\u2203\u27e7 (\u03bb \u03b1\u1d63 \u2192 f) = \u27e6\u2203\u27e7[ \u03b1\u1d63 ] f\n-}\n\nPred : \u2200 \u2113 {a} (A : Set a) \u2192 Set (a \u2294 suc \u2113)\nPred \u2113 A = A \u2192 Set \u2113\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n--\u27e6Pred\u27e7 {p\u2081} {p\u2082} p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 (p\u2081 \u2294 p\u2082 \u2294 p\u1d63)\n\u27e6Pred\u27e7 {p\u2081} {p\u2082} p\u1d63 A\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 f\u2081 x\u2081 \u2192 f\u2082 x\u2082 \u2192 Set (p\u2081 \u2294 p\u2082 \u2294 p\u1d63)\n\nopen import Relation.Binary\n\nprivate\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 Set a \u2192 Set b \u2192 Set (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 Set \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082} \u2113\u1d63 \u2192\n          (\u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) (REL\u2032 \u2113\u2081) (REL\u2032 \u2113\u2082)\n  \u27e6REL\u27e7\u2032 \u2113\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6Set\u27e7 \u2113\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          {\u2113\u2081 \u2113\u2082} \u2113\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 \u2113\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 \u2113\u2082) \u2192 Set _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 \u2113\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6Set\u27e7 \u2113\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {\u2113\u2081 \u2113\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 \u2113\u2081) (\u223c\u2082 : Rel A\u2082 \u2113\u2082) \u2192 Set _\n\u27e6Rel\u27e7 A\u1d63 \u2113\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 \u2113\u1d63\n\n-- data \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : Set \u2113\u2081} {P\u2082 : Set \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 Set \u2113\u1d63) : Dec P\u2081 \u2192 Dec P\u2082 \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) where\n-- data \u27e6Dec\u27e7 {\u2113\u1d63} : \u27e6Pred\u27e7 (\u27e6Set\u27e7 \u2113\u1d63) \u2113\u1d63 Dec Dec where\ndata \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : Set \u2113\u2081} {P\u2082 : Set \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 Set \u2113\u1d63) : \u27e6Set\u27e7 (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) (Dec P\u2081) (Dec P\u2082) where\n  yes : {-\u2200 {P\u2081 P\u2082 P\u1d63}-} {p\u2081 : P\u2081} {p\u2082 : P\u2082} (p\u1d63 : P\u1d63 p\u2081 p\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (yes p\u2081) (yes p\u2082)\n  no  : {-\u2200 {P\u2081 P\u2082 P\u1d63}-} {\u00acp\u2081 : \u00ac P\u2081} {\u00acp\u2082 : \u00ac P\u2082} (\u00acp\u1d63 : (\u27e6\u00ac\u27e7 P\u1d63) \u00acp\u2081 \u00acp\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (no \u00acp\u2081) (no \u00acp\u2082)\n\n--\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {b\u2081 b\u2082} b\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6REL\u27e7 A\u1d63 B\u1d63 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) Decidable Decidable\n\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n                 {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n                 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {\u223c\u2081 : REL A\u2081 B\u2081 \u2113\u2081} {\u223c\u2082 : REL A\u2082 B\u2082 \u2113\u2082} (\u223c\u1d63 : \u27e6REL\u27e7 A\u1d63 B\u1d63 \u2113\u1d63 \u223c\u2081 \u223c\u2082)\n              \u2192 Decidable \u223c\u2081 \u2192 Decidable \u223c\u2082 \u2192 Set _\n\u27e6Decidable\u27e7 A\u1d63 B\u1d63 _\u223c\u1d63_ = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e8 y\u1d63 \u2236 B\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Dec\u27e7 (x\u1d63 \u223c\u1d63 y\u1d63)\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ddde3b69f6588b06031bd633774e1f4669ef00f6","subject":"Cost: adapt to Two","message":"Cost: adapt to Two\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Cost.agda","new_file":"FunUniverse\/Cost.agda","new_contents":"-- NOTE with-K\nmodule FunUniverse.Cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nopen import Data.One\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Level.NP hiding (_\u2294_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_; _\u2192\u1d47_)\n\nopen import FunUniverse.Core\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Const\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\n{-\nseqTimeOpsD : FunOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0\u2082> = 0#; <1\u2082> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            <[]> = 0#; <\u2237> = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FunOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1\u2082-} _ = \u2261.refl\n  constBit\u22610 false {-0\u2082-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n-}\n\nTime = \u2115\nTimeCost = constFuns Time\nSpace = \u2115\nSpaceCost = constFuns Space\n\nseqTimeCat : Category {\u2080} {\u2080} {\ud835\udfd9} (\u03bb _ _ \u2192 Time)\nseqTimeCat = 0 , _+_\n\nseqTimeLin : LinRewiring TimeCost\nseqTimeLin =\n  record {\n    cat = seqTimeCat;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nseqTimeRewiring : Rewiring TimeCost\nseqTimeRewiring =\n  record {\n    linRewiring = seqTimeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _+_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\nseqTimeFork : HasFork TimeCost\nseqTimeFork = record { cond = 1; fork = 1+_+_ }\n\nseqTimeOps : FunOps TimeCost\nseqTimeOps = record { rewiring = seqTimeRewiring; hasFork = seqTimeFork;\n                      <0\u2082> = 0; <1\u2082> = 0 }\n\nseqTimeBij : Bijective TimeCost\nseqTimeBij = FunOps.bijective seqTimeOps\n\ntimeCat : Category (\u03bb _ _ \u2192 Time)\ntimeCat = seqTimeCat\n\ntimeLin : LinRewiring TimeCost\ntimeLin =\n  record {\n    cat = timeCat;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _\u2294_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\ntimeRewiring : Rewiring TimeCost\ntimeRewiring =\n  record {\n    linRewiring = timeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _\u2294_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\ntimeFork : HasFork TimeCost\ntimeFork = record { cond = 1; fork = 1+_\u2294_ }\n\ntimeOps : FunOps TimeCost\ntimeOps = record { rewiring = timeRewiring; hasFork = timeFork;\n                   <0\u2082> = 0; <1\u2082> = 0 }\n\n{-\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                rewiring = record seqTimeRewiring {\n                                             linRewiring = record seqTimeLin { <_\u00d7_> = _\u2294_ };\n                                             <_,_> = _\u2294_};\n                                fork = 1+_\u2294_ }\n                                {-;\n                                <\u2237> = 0; uncons = 0 } -- Without <\u2237> = 0... this definition makes\n                                                       -- the FunOps record def yellow\n                                                       -}\ntimeOps\u2261seqTimeOps = \u2261.refl\n-}\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FunOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0\u2082> = con 0; <1\u2082> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            <[]> = con 0; <\u2237> = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FunOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  open FunOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u22a4\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec\u22a4 {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u22a4\u2261maximum f [] = \u2261.refl\n  constVec\u22a4\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec\u22a4 f bs) 0\n                                              | constVec\u22a4\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\n                                {-\nspaceLin : LinRewiring SpaceCost\nspaceLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nspaceLin\u2261seqTimeLin : spaceLin \u2261 seqTimeLin\nspaceLin\u2261seqTimeLin = \u2261.refl\n\nspaceRewiring : Rewiring TimeCost\nspaceRewiring =\n  record {\n    linRewiring = spaceLin;\n    tt = 0;\n    dup = 1;\n    <[]> = 0;\n    <_,_> = 1+_+_;\n    fst = 0;\n    snd = 0;\n    rewire = \u03bb {_} {o} _ \u2192 o;\n    rewireTbl = \u03bb {_} {o} _ \u2192 o }\n\nspaceFork : HasFork SpaceCost\nspaceFork = record { cond = 1; fork = 1+_+_ }\n\nspaceOps : FunOps SpaceCost\nspaceOps = record { rewiring = spaceRewiring; hasFork = spaceFork;\n                    <0\u2082> = 1; <1\u2082> = 1 }\n\n             {-\n-- So far the space cost model is like the sequential time cost model but makes <0\u2082>,<1\u2082>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0\u2082> = 1; <1\u2082> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; <\u2237> = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n-}\n\nmodule SpaceOps where\n  open FunOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u22a4\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec\u22a4 {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u22a4\u2261sum f [] = \u2261.refl\n  constVec\u22a4\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec\u22a4 f bs) 0\n                                          | constVec\u22a4\u2261sum f bs = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f xs rewrite constVec\u22a4\u2261sum f xs | \u2115\u00b0.+-comm (V.sum (V.map f xs)) 0 = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\n{-\ntime\u00d7spaceOps : FunOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-Ops timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FunOps time\u00d7spaceOps\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- NOTE with-K\nmodule FunUniverse.Cost where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2*_; 2^_; _^_; _\u2294_; module \u2115\u00b0; module \u2294\u00b0; 2*\u2032_)\nopen import Data.Bool using (true; false)\nopen import Data.One\nimport Data.DifferenceNat\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nimport Relation.Binary.PropositionalEquality as \u2261\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_)\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Level.NP hiding (_\u2294_)\n\nopen import Data.Bits using (Bits; 0\u2237_; 1\u2237_; _\u2192\u1d47_)\n\nopen import FunUniverse.Core\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Const\n\nmodule D where\n  open Data.DifferenceNat public renaming (suc to suc#; _+_ to _+#_)\n  _*#_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *# d = 0#\n  suc n *# d = (n *# d) +# d\n  _*#\u2032_ : \u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  zero  *#\u2032 d = 0#\n  suc n *#\u2032 d = d +# (n *#\u2032 d)\n  2*#_ : Diff\u2115 \u2192 Diff\u2115\n  2*# n = n +# n\n  2^#_ : \u2115 \u2192 Diff\u2115\n  2^# zero = 1#\n  2^# suc n = 2*# (2^# n)\n  1+_+_D : Diff\u2115 \u2192 Diff\u2115 \u2192 Diff\u2115\n  1+ x + y D = 1# \u2218\u2032 (x \u2218\u2032 y)\nopen D using (Diff\u2115)\n\nprivate\n  1+_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x + y = 1 + (x + y)\n  1+_\u2294_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  1+ x \u2294 y = 1 + (x \u2294 y)\n  i\u2294i\u2261i : \u2200 i \u2192 i \u2294 i \u2261 i\n  i\u2294i\u2261i zero = \u2261.refl\n  i\u2294i\u2261i (suc i) = \u2261.cong suc (i\u2294i\u2261i i)\n\n{-\nseqTimeOpsD : FunOps (constFuns Diff\u2115)\nseqTimeOpsD = record {\n            id = 0#; _\u2218_ = _\u2218\u2032_;\n            <0b> = 0#; <1b> = 0#; cond = 1#; fork = 1+_+_D; tt = 0#;\n            <_,_> = _\u2218\u2032_; fst = 0#; snd = 0#;\n            dup = 0#; first = F.id; swap = 0#; assoc = 0#;\n            <tt,id> = 0#; snd<tt,> = 0#;\n            <_\u00d7_> = _\u2218\u2032_; second = F.id;\n            <[]> = 0#; <\u2237> = 0#; uncons = 0# }\n            where open D\n\nmodule SeqTimeOpsD where\n  Time = Diff\u2115\n  open D\n  open FunOps seqTimeOpsD public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2257 0#\n  snoc\u22610 zero x = \u2261.refl\n  snoc\u22610 (suc n) x rewrite snoc\u22610 n x = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2257 0#\n  reverse\u22610 zero x = \u2261.refl\n  reverse\u22610 (suc n) x rewrite reverse\u22610 n x | snoc\u22610 n x = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2257 0#\n  replicate\u22610 zero = \u03bb _ \u2192 \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2257 0#\n  constBit\u22610 true  {-1b-} _ = \u2261.refl\n  constBit\u22610 false {-0b-} _ = \u2261.refl\n\n  sum\u2032 : \u2200 {n} \u2192 Vec Diff\u2115 n \u2192 Diff\u2115\n  sum\u2032 = V.foldr (const Diff\u2115) _\u2218\u2032_ id\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec {n} f xs \u2257 sum\u2032 (V.map f xs)\n  constVec\u2261sum f [] x = \u2261.refl\n  constVec\u2261sum {suc n} f (b \u2237 bs) x rewrite \u2115\u00b0.+-comm (f b x \u2294 constVec f bs x) 0\n                                          | constVec\u2261sum f bs x = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2257 0#\n  constBits\u22610 [] x = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) x rewrite constBits\u22610 bs x\n                                       | constBit\u22610 b x = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2257 m *# n\n  foldl\u2261* zero    n x = \u2261.refl\n  foldl\u2261* (suc m) n x rewrite foldl\u2261* m n (n x) = \u2261.refl\n\n  foldr\u2261* : \u2200 m n \u2192 foldr {m} n \u2257 m *#\u2032 n\n  foldr\u2261* zero    n x = \u2261.refl\n  foldr\u2261* (suc m) n x rewrite foldr\u2261* m n x = \u2261.refl\n\n  #nodes : \u2200 i \u2192 Diff\u2115\n  #nodes zero    = 0#\n  #nodes (suc i) = 1# +# #nodes i +# #nodes i\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2257 #nodes i\n  fromBitsFun\u2261 {zero} f x = constBits\u22610 (f []) x\n  fromBitsFun\u2261 {suc i} f x rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_) x\n                                 | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_) (#nodes i x) = \u2261.refl\n-}\n\nTime = \u2115\nTimeCost = constFuns Time\nSpace = \u2115\nSpaceCost = constFuns Space\n\nseqTimeCat : Category {\u2080} {\u2080} {\ud835\udfd9} (\u03bb _ _ \u2192 Time)\nseqTimeCat = 0 , _+_\n\nseqTimeLin : LinRewiring TimeCost\nseqTimeLin =\n  record {\n    cat = seqTimeCat;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nseqTimeRewiring : Rewiring TimeCost\nseqTimeRewiring =\n  record {\n    linRewiring = seqTimeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _+_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\nseqTimeFork : HasFork TimeCost\nseqTimeFork = record { cond = 1; fork = 1+_+_ }\n\nseqTimeOps : FunOps TimeCost\nseqTimeOps = record { rewiring = seqTimeRewiring; hasFork = seqTimeFork;\n                      <0b> = 0; <1b> = 0 }\n\nseqTimeBij : Bijective TimeCost\nseqTimeBij = FunOps.bijective seqTimeOps\n\ntimeCat : Category (\u03bb _ _ \u2192 Time)\ntimeCat = seqTimeCat\n\ntimeLin : LinRewiring TimeCost\ntimeLin =\n  record {\n    cat = timeCat;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _\u2294_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\ntimeRewiring : Rewiring TimeCost\ntimeRewiring =\n  record {\n    linRewiring = timeLin;\n    tt = 0;\n    dup = 0;\n    <[]> = 0;\n    <_,_> = _\u2294_;\n    fst = 0;\n    snd = 0;\n    rewire = const 0;\n    rewireTbl = const 0 }\n\ntimeFork : HasFork TimeCost\ntimeFork = record { cond = 1; fork = 1+_\u2294_ }\n\ntimeOps : FunOps TimeCost\ntimeOps = record { rewiring = timeRewiring; hasFork = timeFork;\n                   <0b> = 0; <1b> = 0 }\n\n{-\ntimeOps\u2261seqTimeOps : timeOps \u2261 record seqTimeOps {\n                                rewiring = record seqTimeRewiring {\n                                             linRewiring = record seqTimeLin { <_\u00d7_> = _\u2294_ };\n                                             <_,_> = _\u2294_};\n                                fork = 1+_\u2294_ }\n                                {-;\n                                <\u2237> = 0; uncons = 0 } -- Without <\u2237> = 0... this definition makes\n                                                       -- the FunOps record def yellow\n                                                       -}\ntimeOps\u2261seqTimeOps = \u2261.refl\n-}\n\n{-\nopen import maxsemiring\nopen \u2294-+-SemiringSolver\ntimeOps' : \u2200 n \u2192 FunOps (constFuns (Polynomial n))\ntimeOps' n = record {\n            id = con 0; _\u2218_ = _:*_;\n            <0b> = con 0; <1b> = con 0; cond = con 1; fork = 1+_+_; tt = con 0;\n            <_,_> = _:*_; fst = con 0; snd = con 0;\n            dup = con 0; <_\u00d7_> = _:+_; swap = con 0; assoc = con 0;\n            <tt,id> = con 0; snd<tt,> = con 0;\n            <[]> = con 0; <\u2237> = con 0; uncons = con 0 }\n   where 1+_+_ : Polynomial n \u2192 Polynomial n \u2192 Polynomial n\n         1+ x + y = con 1 :* (x :* y)\n\nModule TimeOps' where\n  Time = \u2115\n  open FunOps (timeOps' 1) public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} := con 0\n  snoc\u22610 zero = ?\n  snoc\u22610 (suc n) = ?\n-}\n\nmodule TimeOps where\n  open FunOps timeOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2294\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u22610 : \u2200 n \u2192 replicate {n} \u2261 0\n  replicate\u22610 zero = \u2261.refl\n  replicate\u22610 (suc n) = replicate\u22610 n\n\n  constBit\u22610 : \u2200 b \u2192 constBit b \u2261 0\n  constBit\u22610 true  = \u2261.refl\n  constBit\u22610 false = \u2261.refl\n\n  maximum : \u2200 {n} \u2192 Vec \u2115 n \u2192 \u2115\n  maximum = V.foldr (const \u2115) (\u03bb {_} x y \u2192 x \u2294 y) 0\n\n  constVec\u22a4\u2261maximum : \u2200 {n b} {B : Set b} (f : B \u2192 Time) xs \u2192 constVec\u22a4 {n} f xs \u2261 maximum (V.map f xs)\n  constVec\u22a4\u2261maximum f [] = \u2261.refl\n  constVec\u22a4\u2261maximum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b \u2294 constVec\u22a4 f bs) 0\n                                              | constVec\u22a4\u2261maximum f bs = \u2261.refl\n\n  constBits\u22610 : \u2200 {n} xs \u2192 constBits {n} xs \u2261 0\n  constBits\u22610 [] = \u2261.refl\n  constBits\u22610 {suc n} (b \u2237 bs) rewrite constBit\u22610 b\n                                     | constBits\u22610 bs = \u2261.refl\n\n  fromBitsFun\u2261i : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 i\n  fromBitsFun\u2261i {zero}  f rewrite constBits\u22610 (f []) = \u2261.refl\n  fromBitsFun\u2261i {suc i} f rewrite fromBitsFun\u2261i {i} (f \u2218\u2032 1\u2237_)\n                                | fromBitsFun\u2261i {i} (f \u2218\u2032 0\u2237_)\n                                | \u2115\u00b0.+-comm (i \u2294 i) 0\n                                = \u2261.cong suc (i\u2294i\u2261i i)\n\n                                {-\nspaceLin : LinRewiring SpaceCost\nspaceLin =\n  record {\n    id = 0;\n    _\u2218_ = _+_;\n    first = F.id;\n    swap = 0;\n    assoc = 0;\n    <tt,id> = 0;\n    snd<tt,> = 0;\n    <_\u00d7_> = _+_;\n    second = F.id;\n    tt\u2192[] = 0;\n    []\u2192tt = 0;\n    <\u2237> = 0;\n    uncons = 0 }\n\nspaceLin\u2261seqTimeLin : spaceLin \u2261 seqTimeLin\nspaceLin\u2261seqTimeLin = \u2261.refl\n\nspaceRewiring : Rewiring TimeCost\nspaceRewiring =\n  record {\n    linRewiring = spaceLin;\n    tt = 0;\n    dup = 1;\n    <[]> = 0;\n    <_,_> = 1+_+_;\n    fst = 0;\n    snd = 0;\n    rewire = \u03bb {_} {o} _ \u2192 o;\n    rewireTbl = \u03bb {_} {o} _ \u2192 o }\n\nspaceFork : HasFork SpaceCost\nspaceFork = record { cond = 1; fork = 1+_+_ }\n\nspaceOps : FunOps SpaceCost\nspaceOps = record { rewiring = spaceRewiring; hasFork = spaceFork;\n                    <0b> = 1; <1b> = 1 }\n\n             {-\n-- So far the space cost model is like the sequential time cost model but makes <0b>,<1b>,dup\n-- cost one unit of space.\nspaceOps\u2261seqTimeOps : spaceOps \u2261 record seqTimeOps { <0b> = 1; <1b> = 1; dup = 1; <_,_> = 1+_+_\n                                                    ; <\u2237> = 0; uncons = 0 } -- same bug here\nspaceOps\u2261seqTimeOps = \u2261.refl\n-}\n\nmodule SpaceOps where\n  open FunOps spaceOps public\n\n  snoc\u22610 : \u2200 n \u2192 snoc {n} \u2261 0\n  snoc\u22610 zero = \u2261.refl\n  snoc\u22610 (suc n) rewrite snoc\u22610 n = \u2261.refl\n\n  reverse\u22610 : \u2200 n \u2192 reverse {n} \u2261 0\n  reverse\u22610 zero = \u2261.refl\n  reverse\u22610 (suc n) rewrite snoc\u22610 n | reverse\u22610 n = \u2261.refl\n\n  foldl\u2261* : \u2200 m n \u2192 foldl {m} n \u2261 m * n\n  foldl\u2261* zero n = \u2261.refl\n  foldl\u2261* (suc m) n rewrite foldl\u2261* m n\n                          | \u2115\u00b0.+-comm n 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n + n) 0\n                          | \u2115\u00b0.+-comm (m * n) n\n                          = \u2261.refl\n\n  replicate\u2261n : \u2200 n \u2192 replicate {n} \u2261 n\n  replicate\u2261n zero = \u2261.refl\n  replicate\u2261n (suc n) rewrite replicate\u2261n n = \u2261.refl\n\n  constBit\u22611 : \u2200 b \u2192 constBit b \u2261 1\n  constBit\u22611 true  = \u2261.refl\n  constBit\u22611 false = \u2261.refl\n\n  constVec\u22a4\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec\u22a4 {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u22a4\u2261sum f [] = \u2261.refl\n  constVec\u22a4\u2261sum {suc n} f (b \u2237 bs) rewrite \u2115\u00b0.+-comm (f b + constVec\u22a4 f bs) 0\n                                          | constVec\u22a4\u2261sum f bs = \u2261.refl\n\n  constVec\u2261sum : \u2200 {n b} {B : Set b} (f : B \u2192 Space) xs \u2192 constVec {n} f xs \u2261 V.sum (V.map f xs)\n  constVec\u2261sum f xs rewrite constVec\u22a4\u2261sum f xs | \u2115\u00b0.+-comm (V.sum (V.map f xs)) 0 = \u2261.refl\n\n  constBits\u2261n : \u2200 {n} xs \u2192 constBits {n} xs \u2261 n\n  constBits\u2261n [] = \u2261.refl\n  constBits\u2261n {suc n} (b \u2237 bs) rewrite constBit\u22611 b\n                                     | constBits\u2261n bs\n                                     | \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n  fromBitsFun-cost : \u2200 (i o : \u2115) \u2192 \u2115\n  fromBitsFun-cost zero    o = o\n  fromBitsFun-cost (suc i) o = 1 + 2*(fromBitsFun-cost i o)\n\n  fromBitsFun\u2261 : \u2200 {i o} (f : i \u2192\u1d47 o) \u2192 fromBitsFun f \u2261 fromBitsFun-cost i o\n  fromBitsFun\u2261 {zero}  {o} f rewrite constBits\u2261n (f []) | \u2115\u00b0.+-comm o 0 = \u2261.refl\n  fromBitsFun\u2261 {suc i} {o} f rewrite fromBitsFun\u2261 {i} (f \u2218\u2032 1\u2237_)\n                                   | fromBitsFun\u2261 {i} (f \u2218\u2032 0\u2237_)\n                                   | \u2115\u00b0.+-comm (2* (fromBitsFun-cost i o)) 0\n                                   = \u2261.refl\n\n{-\ntime\u00d7spaceOps : FunOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-Ops timeOps spaceOps\n\nmodule Time\u00d7SpaceOps = FunOps time\u00d7spaceOps\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2cd47f65437e3b825c70b2ef65e9b9187895f050","subject":"rel to #45, turns out we do actually already prove the more general preservation statement","message":"rel to #45, turns out we do actually already prove the more general preservation statement\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nopen import structural-assumptions\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans TAConst ()\n  preserve-trans (TAVar x\u2081) ()\n  preserve-trans (TALam _ ta) ()\n  preserve-trans (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt ta ta\u2081\n  preserve-trans (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans (TAEHole x x\u2081) ()\n  preserve-trans (TANEHole x ta x\u2081) ()\n  preserve-trans (TACast ta x) (ITCastID) = ta\n  preserve-trans (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans (TAFailedCast x y z q) ()\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nopen import structural-assumptions\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans TAConst ()\n  preserve-trans (TAVar x\u2081) ()\n  preserve-trans (TALam _ ta) ()\n  preserve-trans (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt ta ta\u2081\n  preserve-trans (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans (TAEHole x x\u2081) ()\n  preserve-trans (TANEHole x ta x\u2081) ()\n  preserve-trans (TACast ta x) (ITCastID) = ta\n  preserve-trans (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans (TAFailedCast x y z q) ()\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d04b0965999db6ed3cee137b6ea8966697cc5bdf","subject":"Removed unnecessary local hypothesis (by Ana Bove).","message":"Removed unnecessary local hypothesis (by Ana Bove).\n\nIgnore-this: c5e32af0a034aa3e43d885358595e928\n\ndarcs-hash:20110406205609-3bd4e-8dbd2fef9ac0b9955b0b19782da6b97a5ec0dc81.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.Properties.AuxiliaryATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 10-N \u2238-N #-}\n{-# ATP prove x<mc91x+11>100 +-N \u2238-N x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 #-}\n\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven) \n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove mc91<mc91+11 mc91-res-100 102\u226191+11 100<102 #-}\n\n---- Case n \u2264 100\n\npostulate\n  -- Here we only reduce the definition of mc91\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n\npostulate\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","old_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.Properties.AuxiliaryATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\n\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 10-N \u2238-N #-}\n{-# ATP prove x<mc91x+11>100 +-N \u2238-N 10-N 11-N x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11\n                             +-comm\n#-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n                        Nmc91>100 111-N 101-N\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 91-N #-}\n{-# ATP prove mc91<mc91+11 mc91-res-100 102\u226191+11 100<102 #-}\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3f147784e99d6ca24756127bd00a60a477993741","subject":"Desc stratified model: descDChoice implicit.","message":"Desc stratified model: descDChoice implicit.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : (l : Level) -> Set l -> DescDConst -> IDesc Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"79eb1bff6fe6f19057332ff8737b10e85457ff5f","subject":"Drop dead comment","message":"Drop dead comment\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n\n  \u27e6_\u27e7TermCacheCBV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 fromCBVCtx \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 cbvToCompType \u03c4 \u27e7CompTypeHidCache\n  \u27e6 t \u27e7TermCacheCBV = \u27e6 fromCBV t \u27e7CompTermCache\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\n\nimport Parametric.Syntax.MType as MType\nimport Parametric.Syntax.MTerm as MTerm\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Denotation.MValue as MValue\nimport Parametric.Denotation.CachingMValue as CachingMValue\nimport Parametric.Denotation.MEvaluation as MEvaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (ValConst : MTerm.ValConstStructure Const)\n    (CompConst : MTerm.CompConstStructure Const)\n    -- I should really switch to records - can it get sillier than this?\n    (\u27e6_\u27e7ValBase : MEvaluation.ValStructure Const \u27e6_\u27e7Base ValConst CompConst)\n    (\u27e6_\u27e7CompBase : MEvaluation.CompStructure Const \u27e6_\u27e7Base ValConst CompConst)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\n\nopen MType.Structure Base\nopen MTerm.Structure Const ValConst CompConst\n\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen MValue.Structure Base \u27e6_\u27e7Base\nopen CachingMValue.Structure Base \u27e6_\u27e7Base\nopen MEvaluation.Structure Const \u27e6_\u27e7Base ValConst CompConst \u27e6_\u27e7ValBase \u27e6_\u27e7CompBase\n\nopen import Base.Denotation.Notation\n\n-- Extension Point: Evaluation of fully applied constants.\nValStructure : Set\nValStructure = \u2200 {\u03a3 \u03c4} \u2192 ValConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\nCompStructure : Set\nCompStructure = \u2200 {\u03a3 \u03c4} \u2192 CompConst \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\nmodule Structure\n       (\u27e6_\u27e7ValBaseTermCache  : ValStructure)\n       (\u27e6_\u27e7CompBaseTermCache : CompStructure)\n    where\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n  \u27e6_\u27e7ValsTermCache : \u2200 {\u0393 \u03a3} \u2192 Vals \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03a3 \u27e7ValCtxHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVarHidCache)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- Copy of \u27e6_\u27e7Vals\n  \u27e6 \u2205 \u27e7ValsTermCache \u03c1 = \u2205\n  \u27e6 vt \u2022 valtms \u27e7ValsTermCache \u03c1 = \u27e6 vt \u27e7ValTermCache \u03c1 \u2022 \u27e6 valtms \u27e7ValsTermCache \u03c1\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVarHidCache \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n  -- No caching, because the arguments are values, so evaluating them does not\n  -- produce intermediate results.\n  \u27e6 vConst c args \u27e7ValTermCache \u03c1 = \u27e6 c \u27e7ValBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- The only caching is done by the interpretation of the constant (because the\n  -- arguments are values so need no caching).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = \u27e6 c \u27e7CompBaseTermCache (\u27e6 args \u27e7ValsTermCache \u03c1)\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 = \u03bb x \u2192 \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n\n  \u27e6_\u27e7TermCacheCBV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 fromCBVCtx \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 cbvToCompType \u03c4 \u27e7CompTypeHidCache\n  \u27e6 t \u27e7TermCacheCBV = \u27e6 fromCBV t \u27e7CompTermCache\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70904b0a4e8a3b7e691d0d859a518ee5341f7a93","subject":"Cleaning.","message":"Cleaning.\n\nIgnore-this: a0d11238690f72ea668e358ccc188abd\n\ndarcs-hash:20110405150812-3bd4e-afdff1a0cdbd4dc4b3edfee982506c8248779ca5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 dom0 =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) (sN Nn) prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn (sN Nn) (nest-\u2264 h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 dom0 =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) (sN Nn) prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn (sN Nn) (nest-\u2264 h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b74c37657f37688ea8fbed425653bb36650135e0","subject":"working on #2, found bug in syntactic sugar for failed casts","message":"working on #2, found bug in syntactic sugar for failed casts\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (pres-lem eps D1 D3 D4 D5) D2\n  pres-lem (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (pres-lem eps D2 D3 D4 D5)\n  pres-lem (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (pres-lem eps D1 D2 D3 D4) x\u2081\n  pres-lem (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (pres-lem eps D1 D2 D3 D4) x\n  pres-lem (FHFailedCast FHOuter) (TAFailedCast D1 x x\u2081 x\u2082) (TACast D2 x\u2083) D3 (FHFailedCast FHOuter) = {!!}\n  pres-lem (FHFailedCast (FHCast eps)) (TAFailedCast D1 x x\u2081 x\u2082) D2 D3 (FHFailedCast (FHCast D4)) = TAFailedCast (pres-lem eps D1 D2 D3 D4) x x\u2081 x\u2082\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x y z w) (FHFailedCast FHOuter) = _ , TACast x TCHole1\n  pres-lem2 (TAFailedCast x y z w) (FHFailedCast (FHCast eps)) = pres-lem2 x eps\n\n  -- this is the literal contents of the hole in lem3; it might not go\n  -- through exactly like this.\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst D1 D2 = {!!}\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = TAAp {!!} {!!}\n  pres-lem (FHAp2 eps) D1 D2 D3 (FHAp2 D4) = TAAp {!!} {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = TANEHole {!!} {!!} {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n  pres-lem (FHFailedCast eps) D1 D2 D3 (FHFailedCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x y z w) (FHFailedCast FHOuter) = _ , TACast x TCHole1\n  pres-lem2 (TAFailedCast x y z w) (FHFailedCast (FHCast eps)) = pres-lem2 x eps\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = {!!} -- todo: this is a lemma\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c0147d23185a96901fb63acfb8b5d86e4e892500","subject":"FunUniverse.BinTree: without-K","message":"FunUniverse.BinTree: without-K\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/BinTree.agda","new_file":"FunUniverse\/BinTree.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.BinTree where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; _\u2264_; s\u2264s; _+_; module \u2115\u2264; module \u2115\u00b0)\nopen import Data.Nat.Properties\nopen import Data.Bool\nopen import Data.Vec using (_++_)\nopen import Data.Bits\n--import Data.Bits.Search as Search\n--open Search.SimpleSearch\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import Composition.Horizontal\nopen import Composition.Vertical\nopen import Composition.Forkable\n\ndata Tree {a} (A : \u2605 a) : \u2115 \u2192 \u2605 a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero}  f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : \u2605 a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : \u2605 a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (false {-0b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (true  {-1b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nfold : \u2200 {n a} {A : \u2605 a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\n{-\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : \u2605 a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n-}\n\nleaf\u207f : \u2200 {n a} {A : \u2605 a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : \u2605 a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : \u2605 a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x = cong\u2082 fork (fromConst\u2261leaf\u207f x) (fromConst\u2261leaf\u207f x)\n\nfromFun\u2218toFun : \u2200 {n a} {A : \u2605 a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x)     = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nweaken\u2264 : \u2200 {m n a} {A : \u2605 a} \u2192 m \u2264 n \u2192 Tree A m \u2192 Tree A n\nweaken\u2264 _       (leaf x)          = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 n {m a} {A : \u2605 a} \u2192 Tree A m \u2192 Tree A (n + m)\nweaken+ n = weaken\u2264 (m\u2264n+m _ n)\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : \u2605 a} {B : \u2605 b}\n        \u2192 Tree A n\u2081 \u2192 (A \u2192 Tree B n\u2082) \u2192 Tree B (n\u2081 + n\u2082)\nleaf {n} x >>= f = weaken+ n (f x)\nfork \u2113 r   >>= f = fork (\u2113 >>= f) (r >>= f)\n\njoin : \u2200 {c\u2081 c\u2082 a} {A : \u2605 a} \u2192 Tree (Tree A c\u2082) c\u2081 \u2192 Tree A (c\u2081 + c\u2082)\njoin t = t >>= id\n\n_\u2192\u1d57_ : (i o : \u2115) \u2192 \u2605\u2080\ni \u2192\u1d57 o = Tree (Bits o) i\n\n_>>>_ : \u2200 {m n a} {A : \u2605 a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\nf >>> g = map (flip lookup g) f\n\n_***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2192\u1d57 o\u2080 \u2192 i\u2081 \u2192\u1d57 o\u2081 \u2192 (i\u2080 + i\u2081) \u2192\u1d57 (o\u2080 + o\u2081)\n(f *** g) = join (map (\u03bb xs \u2192 map (_++_ xs) g) f)\n\nhcomposable : HComposable _\u2192\u1d57_\nhcomposable = mk _>>>_\n\nvcomposable : VComposable _+_ _\u2192\u1d57_\nvcomposable = mk _***_\n\nforkable : Forkable suc _\u2192\u1d57_\nforkable = mk fork\n\n-- This is probably useless by now\ndata Rot {a} {A : \u2605 a} : \u2200 {n} (left right : Tree A n) \u2192 \u2605 a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : \u2605 a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : \u2605 a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\n{- with-K\nRot-trans : \u2200 {n a} {A : \u2605 a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n-- -}\n","old_contents":"-- NOTE with-K\nmodule FunUniverse.BinTree where\n\nopen import Type hiding (\u2605)\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; _\u2264_; s\u2264s; _+_; module \u2115\u2264; module \u2115\u00b0)\nopen import Data.Nat.Properties\nopen import Data.Bool\nopen import Data.Vec using (_++_)\nopen import Data.Bits\n--import Data.Bits.Search as Search\n--open Search.SimpleSearch\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import Composition.Horizontal\nopen import Composition.Vertical\nopen import Composition.Forkable\n\ndata Tree {a} (A : \u2605 a) : \u2115 \u2192 \u2605 a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero}  f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : \u2605 a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : \u2605 a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (false {-0b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (true  {-1b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nfold : \u2200 {n a} {A : \u2605 a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\n{-\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : \u2605 a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n-}\n\nleaf\u207f : \u2200 {n a} {A : \u2605 a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : \u2605 a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : \u2605 a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x rewrite fromConst\u2261leaf\u207f {n} x = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : \u2605 a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x) = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nweaken\u2264 : \u2200 {m n a} {A : \u2605 a} \u2192 m \u2264 n \u2192 Tree A m \u2192 Tree A n\nweaken\u2264 _       (leaf x)          = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 n {m a} {A : \u2605 a} \u2192 Tree A m \u2192 Tree A (n + m)\nweaken+ n = weaken\u2264 (m\u2264n+m _ n)\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : \u2605 a} {B : \u2605 b}\n        \u2192 Tree A n\u2081 \u2192 (A \u2192 Tree B n\u2082) \u2192 Tree B (n\u2081 + n\u2082)\nleaf {n} x >>= f = weaken+ n (f x)\nfork \u2113 r   >>= f = fork (\u2113 >>= f) (r >>= f)\n\njoin : \u2200 {c\u2081 c\u2082 a} {A : \u2605 a} \u2192 Tree (Tree A c\u2082) c\u2081 \u2192 Tree A (c\u2081 + c\u2082)\njoin t = t >>= id\n\n_\u2192\u1d57_ : (i o : \u2115) \u2192 \u2605\u2080\ni \u2192\u1d57 o = Tree (Bits o) i\n\n_>>>_ : \u2200 {m n a} {A : \u2605 a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\nf >>> g = map (flip lookup g) f\n\n_***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2192\u1d57 o\u2080 \u2192 i\u2081 \u2192\u1d57 o\u2081 \u2192 (i\u2080 + i\u2081) \u2192\u1d57 (o\u2080 + o\u2081)\n(f *** g) = join (map (\u03bb xs \u2192 map (_++_ xs) g) f)\n\nhcomposable : HComposable _\u2192\u1d57_\nhcomposable = mk _>>>_\n\nvcomposable : VComposable _+_ _\u2192\u1d57_\nvcomposable = mk _***_\n\nforkable : Forkable suc _\u2192\u1d57_\nforkable = mk fork\n\n-- This is probably useless by now\ndata Rot {a} {A : \u2605 a} : \u2200 {n} (left right : Tree A n) \u2192 \u2605 a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : \u2605 a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : \u2605 a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : \u2605 a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1d642ac95ff88e3ac92e6a4700c3fec8bccc42ed","subject":"nothing works...","message":"nothing works...\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping.agda","new_file":"Agda\/ITyping.agda","new_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\n-- open import Data.Maybe\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\n\ndata IType : Set where\n  o : IType\n  _~>_ : IType -> IType -> IType\n  \u2229 : List IType -> IType\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\no \u225fTI (\u2229 _) = no (\u03bb ())\n\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(_ ~> _) \u225fTI (\u2229 _) = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n(\u2229 _) \u225fTI o = no (\u03bb ())\n(\u2229 _) \u225fTI (_ ~> _) = no (\u03bb ())\n\u2229 [] \u225fTI \u2229 [] = yes refl\n\u2229 [] \u225fTI \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 IType)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\n\ndata _\u2237'_ : IType -> Type -> Set where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237' A -> \u03c4 \u2237' B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n  \u2229-nil : \u2200 {A} ->  \u03c9 \u2237' A\n  \u2229-cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237' A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237' A\n\n\ndata _\u2264\u2229_ : IType -> IType -> Set where\n  base : o \u2264\u2229 o\n  arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n  \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n  \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n  -- \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 : \u2200 {\u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u2229 \u03c4\u2c7c \u2264\u2229 \u2229 \u03c4\u1d62\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {[]} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-nil\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {x \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-cons (\u2229-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl))) (\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2081} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)))\n\n\u2264\u2229-refl : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c4\n\u2264\u2229-refl {o} = base\n\u2264\u2229-refl {\u03c4 ~> \u03c4\u2081} = arr \u2264\u2229-refl \u2264\u2229-refl\n\u2264\u2229-refl {\u2229 []} = \u2229-nil\n\u2264\u2229-refl {\u2229 (x \u2237 x\u2081)} = \u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2082} z \u2192 z)\n\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\n-- _\u2208?_ : \u2200 {a} {A : Set a} -> Atom -> List (Atom \u00d7 A) -> Maybe A\n-- a \u2208? [] = nothing\n-- a \u2208? (l \u2237 ist) with a \u225f proj\u2081 l\n-- ... | yes _ = just (proj\u2082 l)\n-- a \u2208? (l \u2237 ist) | no _ = a \u2208? ist\n\n-- PTerm->\u039b : \u2200 {\u0393 m t} -> (List (Atom \u00d7 \u2115)) -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- PTerm->\u039b {m = bv i} _ ()\n-- PTerm->\u039b {m = fv x} bound \u0393\u22a2m\u2236t with x \u2208? bound\n-- PTerm->\u039b {m = fv x} {t} bound \u0393\u22a2m\u2236t | just i = bv {t} i\n-- PTerm->\u039b {m = fv x} bound \u0393\u22a2m\u2236t | nothing = fv x\n-- PTerm->\u039b {m = lam m} bound (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) = lam \u03c4\u2081 (PTerm->\u039b ((x , 0) \u2237 bound') (cf (\u2209-cons-l _ _ x\u2209)))\n--   where\n--   x = \u2203fresh (L ++ FV m)\n--   x\u2209 : x \u2209 (L ++ FV m)\n--   x\u2209 = \u2203fresh-spec (L ++ FV m)\n--\n--   bound' : List (Atom \u00d7 \u2115)\n--   bound' = Data.List.map (\u03bb a,i \u2192 (proj\u2081 a,i) , suc (proj\u2082 a,i)) bound\n--\n-- PTerm->\u039b {m = app t1 t} bound (app \u0393\u22a2s \u0393\u22a2t) = app (PTerm->\u039b bound \u0393\u22a2s) (PTerm->\u039b bound \u0393\u22a2t)\n-- PTerm->\u039b {m = Y \u03c4} bound (Y x) = Y \u03c4\n--\n-- PTerm->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- PTerm->\u039b = PTerm->\u039b []\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\nPTerm->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\nPTerm->\u039b {m = bv i} ()\nPTerm->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\nPTerm->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ]\n    (PTerm->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\nPTerm->\u039b {m = app t1 t} (app \u0393\u22a2s \u0393\u22a2t) = app (PTerm->\u039b \u0393\u22a2s) (PTerm->\u039b \u0393\u22a2t)\nPTerm->\u039b {m = Y \u03c4} (Y x) = Y \u03c4\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\n\u039b*^-*^-swap : \u2200 {t : Type} k n x y m -> \u00ac(k \u2261 n) -> \u00ac(x \u2261 y) -> \u039b[_<<_] {t} k x (\u039b[ n << y ] m) \u2261 \u039b[ n << y ] (\u039b[ k << x ] m)\n\u039b*^-*^-swap k n x y (bv i) k\u2260n x\u2260y = refl\n\u039b*^-*^-swap k n x y (fv z) k\u2260n x\u2260y = {!   !}\n\u039b*^-*^-swap k n x y (lam A m) k\u2260n x\u2260y =\n  cong (lam A) (\u039b*^-*^-swap (suc k) (suc n) x y m (\u03bb x\u2081 \u2192 k\u2260n (\u2261-suc x\u2081)) x\u2260y)\n\u039b*^-*^-swap k n x y (app m m') k\u2260n x\u2260y rewrite\n  \u039b*^-*^-swap k n x y m k\u2260n x\u2260y | \u039b*^-*^-swap k n x y m' k\u2260n x\u2260y = refl\n\u039b*^-*^-swap k n x y (Y t) k\u2260n x\u2260y = refl\n\n\nfv-^-\u039b*^-refl : \u2200 x t {k \u0393 \u03c4} -> x \u2209 FV t -> (\u0393\u22a2t^x : \u0393 \u22a2 [ k >> fv x ] t \u2236 \u03c4) -> (\u039b[ k << x ] (PTerm->\u039b \u0393\u22a2t^x) ) ~ t\nfv-^-\u039b*^-refl x (bv n) x\u2209FVt ()\nfv-^-\u039b*^-refl x (fv y) x\u2209FVt \u0393\u22a2t^x with x \u225f y\nfv-^-\u039b*^-refl x (fv .x) x\u2209FVt \u0393\u22a2t^x | yes refl = \u22a5-elim (x\u2209FVt (here refl))\nfv-^-\u039b*^-refl x (fv y) x\u2209FVt \u0393\u22a2t^x | no x\u2260y = fv\nfv-^-\u039b*^-refl x (lam t) {k} {\u0393} x\u2209FVt (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam {!   !}\n  --\n  -- where\n  -- x' = \u2203fresh ( L ++ FV ([ suc k >> fv x ] t) )\n  -- x'\u2209 = \u2203fresh-spec ( L ++ FV ([ suc k >> fv x ] t) )\n  --\n  -- x'\u0393\u22a2[suc-k>>x]t^'x' : ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 ([ suc k >> fv x ] t) ^' x' \u2236 \u03c4\u2082\n  -- x'\u0393\u22a2[suc-k>>x]t^'x' = cf (\u2209-cons-l L (FV ([ suc k >> fv x ] t)) (\u2203fresh-spec (L ++ FV ([ suc k >> fv x ] t))))\n  --\n  --\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 : ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 [ suc k >> fv x ] (t ^' x') \u2236 \u03c4\u2082 \u2261\n  --   ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 ([ suc k >> fv x ] t) ^' x' \u2236 \u03c4\u2082\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 rewrite ^-^-swap (suc k) 0 x x' t (\u03bb ()) (\u03bb x\u2261x' \u2192 {!   !}) = refl\n  --\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'' : ((x' , \u03c4\u2081) \u2237 \u0393) \u22a2 [ suc k >> fv x ] (t ^' x') \u2236 \u03c4\u2082\n  -- x'\u0393\u22a2[suc-k>>x]t^'x'' rewrite x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 = x'\u0393\u22a2[suc-k>>x]t^'x'\n  --\n  -- ih'' : \u039b[ (suc k) << x ] (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x'') ~ (t ^' x')\n  -- ih'' = fv-^-\u039b*^-refl x ([ 0 >> fv x' ] t) {!   !} x'\u0393\u22a2[suc-k>>x]t^'x''\n  --\n  -- -- ih' : \u039b[ 0 << x' ] ( \u039b[ (suc k) << x ] (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x') ) ~ t\n  -- -- ih' rewrite x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 = {!   !}\n  --\n  -- ih : \u039b[ (suc k) << x ] ( \u039b[ 0 << x' ] (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x') ) ~ t\n  -- ih rewrite\n  --   \u039b*^-*^-swap {\u03c4\u2082} (suc k) 0 x x' (PTerm->\u039b x'\u0393\u22a2[suc-k>>x]t^'x') {!   !} {!   !} |\n  --   x'\u0393\u22a2[suc-k>>x]t^'x'\u2261 = {!   !}\n\nfv-^-\u039b*^-refl x (app s t) x\u2209FVt (app \u0393\u22a2s^x \u0393\u22a2t^x) = app\n  (fv-^-\u039b*^-refl x s (\u2209-cons-l _ _ x\u2209FVt) \u0393\u22a2s^x)\n  (fv-^-\u039b*^-refl x t (\u2209-cons-r (FV s) _ x\u2209FVt) \u0393\u22a2t^x)\nfv-^-\u039b*^-refl x (Y \u03c4) x\u2209FVt (Y x\u2081) = Y\n\n\n\nPTerm->\u039b~ : \u2200 {\u0393 t \u03c4} -> {\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4} -> (PTerm->\u039b \u0393\u22a2t) ~ t\nPTerm->\u039b~ {t = bv i} = \u03bb {\u03c4} \u2192 \u03bb {}\nPTerm->\u039b~ {t = fv x} = \u03bb {\u03c4} {\u0393\u22a2t} \u2192 fv\nPTerm->\u039b~ {t = lam t} {\u0393\u22a2t = abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf} =\n  lam (fv-^-\u039b*^-refl (\u2203fresh (L ++ FV t)) t (\u2209-cons-r L _ (\u2203fresh-spec (L ++ FV t))) (cf (\u2209-cons-l L (FV t) (\u2203fresh-spec (L ++ FV t)))))\n  -- where\n  -- x' = \u2203fresh (L ++ FV t)\n  -- x'\u2237\u0393\u22a2t^'x' = cf (\u2209-cons-l L (FV t) (\u2203fresh-spec (L ++ FV t)))\n  --\n  -- ih' : \u039b[ 0 << x' ] (PTerm->\u039b x'\u2237\u0393\u22a2t^'x') ~ (* x' ^ (t ^' x'))\n  -- ih' = {!   !}\n  --\n  -- sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  -- sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n  --\n  -- ih : \u039b[ 0 << x' ] (PTerm->\u039b x'\u2237\u0393\u22a2t^'x') ~ t\n  -- ih rewrite sub {_} {x'} {\u039b[ 0 << x' ] (PTerm->\u039b x'\u2237\u0393\u22a2t^'x')} {!   !} = ih'\nPTerm->\u039b~ {t = app t t\u2081} {\u0393\u22a2t = app \u0393\u22a2t \u0393\u22a2t\u2081} = app PTerm->\u039b~ PTerm->\u039b~\nPTerm->\u039b~ {t = Y t\u2081} {\u0393\u22a2t = Y x} = Y\n\n\n\u039b->PTerm : \u2200 {t} -> \u039b t -> PTerm\n\u039b->PTerm (bv i) = bv i\n\u039b->PTerm (fv x) = fv x\n\u039b->PTerm (lam A \u039bt) = lam (\u039b->PTerm \u039bt)\n\u039b->PTerm (app \u039bs \u039bt) = app (\u039b->PTerm \u039bs) (\u039b->PTerm \u039bt)\n\u039b->PTerm (Y t) = Y t\n\n\n\n\n\n-- data IType\u209b : IType -> Set where\n--   o : IType\u209b o\n--   arr : \u2200 {\u03c4 \u03c4'} -> IType\u209b \u03c4 -> IType\u209b \u03c4' -> IType\u209b (\u03c4 ~> \u03c4')\n--\n-- data IType\u209b\u209b : IType -> Set where\n--   o : IType\u209b\u209b o\n--   arr :  \u2200 {\u03c4 \u03c4'} -> IType\u209b\u209b \u03c4 -> IType\u209b\u209b \u03c4' -> IType\u209b\u209b (\u03c4 ~> \u03c4')\n--   \u2229-nil : IType\u209b\u209b \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4\u1d62} -> IType\u209b \u03c4 ->  IType\u209b\u209b (\u2229 \u03c4\u1d62) -> IType\u209b\u209b (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n--\n-- \u03c4\u209b->\u03c4\u209b\u209b : \u2200 {\u03c4} -> IType\u209b \u03c4 -> IType\u209b\u209b \u03c4\n-- \u03c4\u209b->\u03c4\u209b\u209b o = o\n-- \u03c4\u209b->\u03c4\u209b\u209b (arr \u03c4\u209b \u03c4\u209b\u2081) = arr (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b) (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b\u2081)\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2) -- app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata Y-shape : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  intro\u2081 : \u2200 {A m} -> Y-shape (app (Y A) m)\n  intro\u2082 : \u2200 {A m} -> Y-shape (app m (app (Y A) m))\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} -> (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u2229 \u03c4\u1d62)) \u2208 \u0393) -> (\u03c4\u1d62\u2264\u2229\u03c4 : \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4) -> \u03c4 \u2237' A ->\n    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4\u2081 \u03c4\u2082} -> \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9 t \u2236 \u03c4\u2081 -> (\u03c4\u2081 ~> \u03c4\u2082) \u2237' (A \u27f6 B) -> \u03c4\u2081 \u2237' B ->\n    \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\u2082\n  \u2229-nil : \u2200 {A \u0393} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) -> \u0393 \u22a9 m\u2005 \u2236 \u03c9\n  \u2229-cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 \u03c4\u1d62) -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n  abs : \u2200 {A B \u0393 \u03c4\u1d62 \u03c4} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4 ) -> \u2229 \u03c4\u1d62 \u2237' A -> \u03c4 \u2237' B -> \u0393 \u22a9 lam A t \u2236 (\u2229 \u03c4\u1d62 ~> \u03c4)\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} -> Wf-ICtxt \u0393 -> \u03c4 \u2237' A -> \u03c4\u2081 \u2237' A -> \u03c4\u2082 \u2237' A ->\n    \u0393 \u22a9 Y A \u2236 ((\u03c4 ~> \u03c4\u2081) ~> \u03c4\u2082)\n\n\n-- \u22a9->\u03b2 : \u2200 {\u0393 m m' \u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236\u03c4'~>\u03c4 (app (Y {\u03c4 = \u03c4'} wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 \u03c4\u2082\u2237) \u0393\u22a9m\u2236\u03c4'~>\u03c4')) (Y trm-m) =\n--     app (Y wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 {!   !}) \u0393\u22a9m\u2236\u03c4'~>\u03c4'\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393 trm-m)) (Y x) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082)) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393 trm-m) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2082)\n--\n--\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u03b2m'') \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-n\u2082)) (redL trm-n\u2081 m->\u03b2m'') =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2-redL (\u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 m->\u03b2m'')) wf-\u0393 (app (app (->\u03b2-Term-l m->\u03b2m'') trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redR x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (redR x\u2081 m->\u03b2m'')) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-e\u2082) trm-n\u2082)) (redR trm-m\u2081 n\u2081->e\u2082) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (redR trm-m\u2081 n\u2081->e\u2082))) wf-\u0393 (app (app trm-m\u2081 (->\u03b2-Term-l n\u2081->e\u2082)) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (abs L cf) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (abs L cf)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app trm-lam-m'' trm-n\u2081)) (abs L cf) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (abs L cf))) wf-\u0393 (app (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (cf x\u2209L))) trm-n\u2081)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (beta trm-lam-m\u2081 trm-n\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (beta trm-lam-m\u2081 trm-n\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 wf-\u0393 (app trm-m\u2081^n\u2081 trm-n\u2082)) (beta trm-lam-m\u2081 trm-n\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (beta trm-lam-m\u2081 trm-n\u2081))) wf-\u0393 (app (app trm-lam-m\u2081 trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (Y trm-m\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (Y trm-m\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-Ym\u2081) trm-n\u2081)) (Y trm-m\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (Y trm-m\u2081))) wf-\u0393 (app trm-Ym\u2081 trm-n\u2081)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (redR trm-n m->\u03b2m') = app \u0393\u22a9m'\u2236\u03c4 (\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4\u2081 m->\u03b2m')\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mn'\u2236\u03c4\u1d62 wf-\u0393 trm-mn') (redR trm-m n->\u03b2n') =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mn'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redR trm-m n->\u03b2n')) wf-\u0393 (app trm-m (->\u03b2-Term-l n->\u03b2n'))\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9lam-m'\u2236\u03c4\u1d62 wf-\u0393 trm-lam-x) (abs L x\u2082) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (abs L x\u2082)) wf-\u0393 (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (x\u2082 x\u2209L)))\n-- \u22a9->\u03b2 (abs L cf) (abs L\u2081 x) = abs (L ++ L\u2081) (\u03bb x\u2209L \u2192 \u22a9->\u03b2 (cf (\u2209-cons-l _ _ x\u2209L)) (x (\u2209-cons-r L _ x\u2209L)))\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (app \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'')) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = []} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 {\u0393} (app {s = m} \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = x \u2237 \u03c4\u1d62} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n--   where\n--   \u0393\u22a9m\u2236x~>x : \u0393 \u22a9 m \u2236 (x ~> x)\n--   \u0393\u22a9m\u2236x~>x = {!   !}\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mYm\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-Ym)) (Y trm-m) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mYm\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (Y trm-m)) wf-\u0393 trm-Ym\n","old_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\n\n\ndata IType : Set\ndata IType\u209b : Set\n\n\ndata IType\u209b where\n  o : IType\u209b\n  _~>_ : (s : IType) -> (t : IType\u209b) -> IType\u209b\n\ndata IType where\n   \u2229 : List IType\u209b -> IType\n\n\u03c9 = \u2229 []\n\n\u2229' : IType\u209b -> IType\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u209b_ : Decidable {A = IType\u209b} _\u2261_\n\n\u2229 [] \u225fTI \u2229 [] = yes refl\n\u2229 [] \u225fTI \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI\u209b y | (\u2229 \u03c4\u1d62) \u225fTI (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\no \u225fTI\u209b o = yes refl\no \u225fTI\u209b (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI\u209b o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI\u209b (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u209b \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI\u209b (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI\u209b (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI\u209b (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\nICtxt = List (Atom \u00d7 IType)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u209b_ : IType\u209b -> Type -> Set\n\ndata _\u2237'\u209b_ where\n  base : o \u2237'\u209b \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237' A -> \u03c4 \u2237'\u209b B -> (\u03b4 ~> \u03c4) \u2237'\u209b (A \u27f6 B)\n\ndata _\u2237'_ where\n  int : \u2200{\u03c4\u1d62 A} -> (c : \u2200 {\u03c4} -> (\u03c4\u2208\u03c4\u1d62 : \u03c4 \u2208 \u03c4\u1d62) -> \u03c4 \u2237'\u209b A) -> \u2229 \u03c4\u1d62 \u2237' A\n\n\ndata _\u22a9_\u2236_ : ICtxt -> PTerm -> IType\u209b -> Set where\n  var : \u2200 {\u0393 x \u03c4} {\u03c4\u1d62 : List IType\u209b} -> (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u2229 \u03c4\u1d62)) \u2208 \u0393) -> (\u03c4\u2208\u03c4\u1d62 : \u03c4 \u2208 \u03c4\u1d62) ->\n    \u0393 \u22a9 fv x \u2236 \u03c4\n  app : \u2200 {\u0393 s t \u03c4\u1d62 \u03c4} -> \u0393 \u22a9 s \u2236 ((\u2229 \u03c4\u1d62) ~> \u03c4) -> Term t -> (\u0393\u22a9t\u2236\u03c4\u1d62 : \u2200 {\u03c4'} -> (\u03c4'\u2208\u03c4\u1d62 : \u03c4' \u2208 \u03c4\u1d62) -> \u0393 \u22a9 t \u2236 \u03c4') ->\n    \u0393 \u22a9 (app s t) \u2236 \u03c4\n  abs : \u2200 {\u0393 \u03b4 \u03c4} (L : FVars) -> \u2200 {t} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u03b4) \u2237 \u0393) \u22a9 t ^' x \u2236 \u03c4 ) -> \u0393 \u22a9 lam t \u2236 (\u03b4 ~> \u03c4)\n  Y : \u2200 {\u0393 A \u03c4\u1d62 \u03c4} -> Wf-ICtxt \u0393 -> (\u03c4\u2208\u03c4\u1d62 : \u03c4 \u2208 \u03c4\u1d62) -> (\u03c4\u1d62\u2237 : \u2200 {\u03c4} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4 \u2237'\u209b A) ->\n    \u0393 \u22a9 Y A \u2236 (\u2229 (Data.List.map (\u03bb \u03c4\u2096 -> (\u2229 \u03c4\u1d62 ~> \u03c4\u2096)) \u03c4\u1d62) ~> \u03c4)\n\n\u22a9-term : \u2200 {\u0393 m \u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> Term m\n\u22a9-term (var _ _ _) = var\n\u22a9-term (app \u0393\u22a9m\u2236\u03c4 trm-t c) = app (\u22a9-term \u0393\u22a9m\u2236\u03c4) trm-t\n\u22a9-term (abs L cf) = lam L (\u03bb {x} x\u2209L \u2192 \u22a9-term (cf x\u2209L))\n\u22a9-term (Y x \u03c4\u2208\u03c4\u1d62 x\u2081) = Y\n\n-- cex : [] \u22a9 app (lam (lam (bv 0))) (bv 4) \u2236 (\u2229' o ~> o)\n\n\nwfI-cons : \u2200 {\u0393 x \u03c4} -> Wf-ICtxt ((x , \u03c4) \u2237 \u0393) -> Wf-ICtxt \u0393\nwfI-cons (cons x\u2209 wf-\u0393) = wf-\u0393\n\n\n\u22a9-imp-wf\u0393 : \u2200 {\u0393 m \u03c4} -> \u0393 \u22a9 m \u2005\u2236 \u03c4 -> Wf-ICtxt \u0393\n\u22a9-imp-wf\u0393 (var x\u2081 x\u2082 x\u2083) = x\u2081\n\u22a9-imp-wf\u0393 (app \u0393\u22a2m:\u03c4 trm-t c) = \u22a9-imp-wf\u0393 \u0393\u22a2m:\u03c4\n\u22a9-imp-wf\u0393 (abs L cf) = wfI-cons (\u22a9-imp-wf\u0393 (cf x\u2209))\n  where\n  x = \u2203fresh L\n\n  x\u2209 : x \u2209 L\n  x\u2209 = \u2203fresh-spec L\n\n\u22a9-imp-wf\u0393 (Y x \u03c4\u2208\u03c4\u1d62 x\u2081) = x\n\n\nvar-\u22a9-\u2208 : \u2200 {x y \u03c4\u1d62 \u03c4 \u0393} -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 fv y \u2236 \u03c4 -> x \u2261 y -> \u03c4 \u2208 \u03c4\u1d62\nvar-\u22a9-\u2208 {x} {_} {\u03c4\u1d62} {\u03c4} (var {x = .x} {\u03c4\u1d62 = \u03c4\u2c7c} (cons x\u2209 wf-\u0393) \u2229\u03c4\u2c7c\u2208x\u2237\u0393 \u03c4\u2208\u03c4\u2c7c) refl with (\u2229 \u03c4\u2c7c) \u225fTI (\u2229 \u03c4\u1d62)\nvar-\u22a9-\u2208 (var (cons x\u2209 wf-\u0393) \u2229\u03c4\u2c7c\u2208x\u2237\u0393 \u03c4\u2208\u03c4\u2c7c) refl | yes refl = \u03c4\u2208\u03c4\u2c7c\nvar-\u22a9-\u2208 (var (cons x\u2209 wf-\u0393) \u2229\u03c4\u2c7c\u2208x\u2237\u0393 \u03c4\u2208\u03c4\u2c7c) refl | no \u2229\u03c4\u2c7c\u2260\u2229\u03c4\u1d62 = \u22a5-elim (\u2209-\u2237 _ _ (\u03bb x \u2192 \u2229\u03c4\u2c7c\u2260\u2229\u03c4\u1d62 (\u00d7-inj-r x)) (\u2209-dom x\u2209) \u2229\u03c4\u2c7c\u2208x\u2237\u0393)\n\n\n-- substitution lemma.\n\nweakening-\u22a9 : \u2200 {\u0393 \u0393' m \u03c4} -> \u0393 \u2286 \u0393' -> Wf-ICtxt \u0393' -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393' \u22a9 m \u2236 \u03c4\nweakening-\u22a9 \u0393\u2286\u0393' wf-\u0393' (var wf-\u0393 \u2229\u03c4\u1d62\u2208\u0393 \u03c4\u2208\u03c4\u1d62) = var wf-\u0393' (\u0393\u2286\u0393' \u2229\u03c4\u1d62\u2208\u0393) \u03c4\u2208\u03c4\u1d62\nweakening-\u22a9 \u0393\u2286\u0393' wf-\u0393' (app \u0393\u22a9m\u2236\u03c4 x c) = app (weakening-\u22a9 \u0393\u2286\u0393' wf-\u0393' \u0393\u22a9m\u2236\u03c4) x\n  (\u03bb {\u03c4'} \u03c4'\u2208\u03c4\u1d62 \u2192 weakening-\u22a9 \u0393\u2286\u0393' wf-\u0393' (c \u03c4'\u2208\u03c4\u1d62))\nweakening-\u22a9 {\u0393} {\u0393'} \u0393\u2286\u0393' wf-\u0393' (abs {_} {\u03c4\u1d62} {\u03c4} L {m} cf) =\n  abs (L ++ dom \u0393') (\u03bb {x} x\u2209L++\u0393' \u2192\n    weakening-\u22a9 {(x , \u03c4\u1d62) \u2237 \u0393} {(x , \u03c4\u1d62) \u2237 \u0393'} {m ^' x} {\u03c4}\n      (cons-\u2286 \u0393\u2286\u0393')\n      (cons (\u2209-cons-r L _ x\u2209L++\u0393') wf-\u0393')\n      (cf (\u2209-cons-l _ _ x\u2209L++\u0393')))\nweakening-\u22a9 \u0393\u2286\u0393' wf-\u0393' (Y x \u03c4\u2208\u03c4\u1d62 x\u2081) = Y wf-\u0393' \u03c4\u2208\u03c4\u1d62 x\u2081\n\n\nwfI-\u0393-exchange : \u2200 {\u0393 x y \u03c4\u2081 \u03c4\u2082} -> Wf-ICtxt ((x , \u03c4\u2081) \u2237 (y , \u03c4\u2082) \u2237 \u0393) -> Wf-ICtxt ((y , \u03c4\u2082) \u2237 (x , \u03c4\u2081) \u2237 \u0393)\nwfI-\u0393-exchange (cons x\u2209y\u2237\u0393 (cons y\u2209\u0393 wf\u2237\u0393)) =\n  cons (\u2209-\u2237 _ _ (\u03bb y\u2261x \u2192 fv-x\u2260y _ _ x\u2209y\u2237\u0393 (\u2261-sym y\u2261x)) y\u2209\u0393) (cons (\u2209-\u2237-elim _ x\u2209y\u2237\u0393) wf\u2237\u0393)\n\n\nexchange-\u22a9 : \u2200 {\u0393 m x y \u03c4\u2081 \u03c4\u2082 \u03b4} -> ((x , \u03c4\u2081) \u2237 (y , \u03c4\u2082) \u2237 \u0393) \u22a9 m \u2236 \u03b4 -> ((y , \u03c4\u2082) \u2237 (x , \u03c4\u2081) \u2237 \u0393) \u22a9 m \u2236 \u03b4\nexchange-\u22a9 {\u0393} {m} {x} {y} {\u03c4\u2081} {\u03c4\u2082} x\u2237y\u2237\u0393\u22a2m\u2236\u03b4 =\n  weakening-\u22a9 {(x , \u03c4\u2081) \u2237 (y , \u03c4\u2082) \u2237 \u0393} {(y , \u03c4\u2082) \u2237 (x , \u03c4\u2081) \u2237 \u0393}\n    sub (wfI-\u0393-exchange (\u22a9-imp-wf\u0393 x\u2237y\u2237\u0393\u22a2m\u2236\u03b4)) x\u2237y\u2237\u0393\u22a2m\u2236\u03b4\n\n    where\n    sub : (x , \u03c4\u2081) \u2237 (y , \u03c4\u2082) \u2237 \u0393 \u2286 (y , \u03c4\u2082) \u2237 (x , \u03c4\u2081) \u2237 \u0393\n    sub (here px) = there (here px)\n    sub (there (here px)) = here px\n    sub (there (there \u2208)) = there (there \u2208)\n\n\nsubst-\u22a9 : \u2200 {\u0393 m n \u03c4\u1d62 \u03c4\u2082 x} -> Term m -> Term n -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 m \u2236 \u03c4\u2082 -> (\u2200 {\u03c4} -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 n \u2236 \u03c4) ->\n  \u0393 \u22a9 (m [ x ::= n ]) \u2236 \u03c4\u2082\nsubst-\u22a9 {x = x} var trm-n (var {x = y} wf-\u0393 \u2229\u03c4\u1d62\u2208\u0393 \u03c4\u2208\u03c4\u1d62) \u0393\u22a9n\u2236\u03c4\u1d62 with x \u225f y\nsubst-\u22a9 var trm-n (var wf-\u0393 \u2229\u03c4\u1d62\u2208\u0393 \u03c4\u2208\u03c4\u1d62) \u0393\u22a9n\u2236\u03c4\u1d62 | yes refl =\n  \u0393\u22a9n\u2236\u03c4\u1d62 (var-\u22a9-\u2208 (var wf-\u0393 \u2229\u03c4\u1d62\u2208\u0393 \u03c4\u2208\u03c4\u1d62) refl)\nsubst-\u22a9 {\u0393} {x = x} var trm-n (var {x = y} wf-\u0393 \u2229\u03c4\u1d62\u2208\u0393 \u03c4\u2208\u03c4\u1d62) \u0393\u22a9n\u2236\u03c4\u1d62 | no x\u2260y =\n  var {\u0393} {y} (wfI-cons wf-\u0393) (\u2208-\u2237-elim _ _ (\u03bb x\u2229\u03c4\u1d62\u2261y\u2229\u03c4\u2c7c \u2192 x\u2260y (\u00d7-inj-l x\u2229\u03c4\u1d62\u2261y\u2229\u03c4\u2c7c)) \u2229\u03c4\u1d62\u2208\u0393) \u03c4\u2208\u03c4\u1d62\nsubst-\u22a9 {\u0393} {_} {n} {\u03c4\u1d62} {_} {x} (lam L cf) trm-n (abs {_} {\u03c4\u1d62'} {\u03c4\u2082} L' {m} cf') \u0393\u22a9n\u2236\u03c4\u1d62 = abs (x \u2237 L ++ L' ++ dom \u0393) body\n  where\n  body : \u2200 {x' : \u2115} \u2192 x' \u2209 x \u2237 L ++ L' ++ dom \u0393 \u2192 ((x' , \u03c4\u1d62') \u2237 \u0393) \u22a9 (m [ x ::= n ]) ^' x' \u2236 \u03c4\u2082\n  body {x'} x'\u2209 rewrite\n    subst-open-var x' x n m (fv-x\u2260y _ _ x'\u2209) trm-n =\n      subst-\u22a9 {(x' , \u03c4\u1d62') \u2237 \u0393} {m ^' x'} {n} {\u03c4\u1d62} {\u03c4\u2082}\n        (cf (\u2209-\u2237-elim _ (\u2209-cons-l _ _ x'\u2209)))\n        trm-n\n        (exchange-\u22a9 (cf' (\u2209-cons-l _ _ (\u2209-cons-r L _ (\u2209-\u2237-elim _ x'\u2209)))))\n        (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 weakening-\u22a9 there (cons (\u2209-cons-r L' _ (\u2209-cons-r L _ (\u2209-\u2237-elim _ x'\u2209))) (\u22a9-imp-wf\u0393 (\u0393\u22a9n\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62))) (\u0393\u22a9n\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62))\n\nsubst-\u22a9 (app trm-m trm-m\u2081) trm-n (app x\u2237\u0393\u22a9m\u2236\u03c4\u2082 trm-t c) \u0393\u22a9n\u2236\u03c4\u1d62 =\n  app (subst-\u22a9 trm-m trm-n x\u2237\u0393\u22a9m\u2236\u03c4\u2082 \u0393\u22a9n\u2236\u03c4\u1d62) (subst-Term trm-t trm-n) (\u03bb {\u03c4'} \u03c4'\u2208\u03c4\u1d62 \u2192 subst-\u22a9 trm-m\u2081 trm-n (c \u03c4'\u2208\u03c4\u1d62) \u0393\u22a9n\u2236\u03c4\u1d62)\nsubst-\u22a9 Y trm-n (Y (cons x\u2209 wf-\u0393) \u03c4\u2208\u03c4\u1d62 x\u2082) \u0393\u22a9n\u2236\u03c4\u1d62 = Y wf-\u0393 \u03c4\u2208\u03c4\u1d62 x\u2082\n\n\n-- subject Reduction\n^-\u22a9 : \u2200 {\u0393 L m n \u03c4\u1d62 \u03c4\u2082} -> Term n -> ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 m ^' x \u2236 \u03c4\u2082 ) ->\n  (\u2200 {\u03c4} -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 n \u2236 \u03c4) -> \u0393 \u22a9 m ^ n \u2236 \u03c4\u2082\n^-\u22a9 {\u0393} {L} {m} {n} {\u03c4\u1d62} {\u03c4} trm-n cf \u0393\u22a9n\u2236\u03c4\u1d62 = body\n  where\n  x = \u2203fresh (L ++ FV m)\n\n  x\u2209 : x \u2209 (L ++ FV m)\n  x\u2209 = \u2203fresh-spec (L ++ FV m)\n\n  body : \u0393 \u22a9 m ^ n \u2236 \u03c4\n  body rewrite\n    subst-intro x n m (\u2209-cons-r L _ x\u2209) trm-n =\n    subst-\u22a9 (\u22a9-term (cf (\u2209-cons-l _ _ x\u2209))) trm-n (cf (\u2209-cons-l _ _ x\u2209)) \u0393\u22a9n\u2236\u03c4\u1d62\n\n\n->\u03b2-\u22a9 : \u2200 {\u0393 m m' \u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 m' \u2236 \u03c4\n->\u03b2-\u22a9 (app \u0393\u22a9m\u2236\u03c4 x c) (redL x\u2081 m->\u03b2m') = app (->\u03b2-\u22a9 \u0393\u22a9m\u2236\u03c4 m->\u03b2m') x\u2081 c\n->\u03b2-\u22a9 (app \u0393\u22a9m\u2236\u03c4 trm-t c) (redR x\u2081 m->\u03b2m') = app \u0393\u22a9m\u2236\u03c4 (->\u03b2-Term-r m->\u03b2m') (\u03bb \u03c4'\u2208\u03c4\u1d62 \u2192 ->\u03b2-\u22a9 (c \u03c4'\u2208\u03c4\u1d62) m->\u03b2m')\n->\u03b2-\u22a9 (abs L cf) (abs L\u2081 cf\u2081) = abs (L ++ L\u2081) (\u03bb {x} x\u2209L \u2192 ->\u03b2-\u22a9 (cf (\u2209-cons-l _ _ x\u2209L)) (cf\u2081 (\u2209-cons-r L _ x\u2209L)))\n->\u03b2-\u22a9 (app (abs L {m} cf) x \u0393\u22a2n\u2236\u03c4\u1d62) (beta x\u2081 trm-n) = ^-\u22a9 {m = m} trm-n cf \u0393\u22a2n\u2236\u03c4\u1d62\n->\u03b2-\u22a9 (app (Y {\u03c4 = \u03c4} wf-\u0393 \u03c4\u2208\u03c4\u1d62 \u03c4\u2237'\u209b\u03c3) _ \u0393\u22a9m\u2236\u03c4\u1d62) (Y trm-m) =\n  app\n    (\u0393\u22a9m\u2236\u03c4\u1d62 (map-\u2208 (\u03c4 , \u03c4\u2208\u03c4\u1d62 , refl)))\n    (app Y trm-m)\n    (\u03bb \u03c4'\u2208\u03c4\u1d62 \u2192 app (Y wf-\u0393 \u03c4'\u2208\u03c4\u1d62 (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u03c4\u2237'\u209b\u03c3 \u03c4\u2208\u03c4\u1d62)) trm-m (\u03bb \u03c4'\u2208\u03c4\u1d62\u2081 \u2192 \u0393\u22a9m\u2236\u03c4\u1d62 \u03c4'\u2208\u03c4\u1d62\u2081))\n\n-- subject expansion\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5826fd422a35e593bc840ca76dea1d6b1e32dc6d","subject":"updating all","message":"updating all\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import focus-formation\nopen import finality\n\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\nopen import correspondence\nopen import expandability\nopen import expansion-unicity\nopen import type-assignment-unicity\n-- open import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\n-- open import progress-checks\n-- open import progress\n-- open import preservation\n\nopen import complete-preservation\n-- open import complete-progress\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import focus-formation\nopen import finality\n\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\nopen import correspondence\nopen import expandability\nopen import expansion-unicity\nopen import type-assignment-unicity\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\n-- open import canonical-indeterminate-forms\n\n-- open import progress-checks\n\n-- open import preservation\n-- open import progress\nopen import complete-preservation\n-- open import complete-progress\n\n-- open import typed-expansion\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"14dcf28eeffdcf7d1dbadd450170fddf0972e563","subject":"Category.Monad","message":"Category.Monad\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Monad\/NP.agda","new_file":"lib\/Category\/Monad\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Category.Functor.NP\nopen import Type\nopen import Function\nopen import Level.NP\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Vec using (Vec; []; _\u2237_)\n\nmodule Category.Monad.NP {\u2113} where\n\nopen import Category.Monad public\n\nrecord IsMonadic {M : Set \u2113 \u2192 Set \u2113} (rawMonad : RawMonad M) : \u2605_ (\u209b \u2113) where\n  open RawMonad rawMonad\n\n  field\n    return->>= : \u2200 {A B} (f : A \u2192 M B) x \u2192 (return x >>= f) \u2261 f x\n    >>=-return : \u2200 {A} (m : M A) \u2192 (m >>= return) \u2261 m\n    >>=-assoc  : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (k : B \u2192 M C)\n                         \u2192 (mx >>= \u03bb x \u2192 my x >>= k) \u2261 (mx >>= my >>= k)\n\n  >>-assoc  : \u2200 {A B C} (mx : M A) (my : M B) (mz : M C)\n                       \u2192 (mx >> (my >> mz)) \u2261 ((mx >> my) >> mz)\n  >>-assoc mx my mz = >>=-assoc mx (const my) (const mz)\n\n  liftM : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  liftM f x = x >>= return \u2218 f\n\n  <$>-liftM : \u2200 {A B} (f : A \u2192 B) x \u2192 f <$> x \u2261 liftM f x\n  <$>-liftM f x = return->>= (\u03bb g \u2192 liftM g x) f\n\n  liftM-id : \u2200 {A} (mx : M A) \u2192 liftM id mx \u2261 mx\n  liftM-id = >>=-return\n\n  <$>-id : \u2200 {A} (mx : M A) \u2192 id <$> mx \u2261 mx\n  <$>-id mx = trans (<$>-liftM id mx) (liftM-id mx)\n\n  {- requires function extensionality\n  cong-return-\u2218 : \u2200 {A B : \u2605_ \u2113} {f g : A \u2192 B} \u2192 f \u2261 g \u2192 return \u2218 f \u2261 return \u2218 g\n  cong-return-\u2218 = {!!}\n\n  liftM-\u2218 : \u2200 {A B C} (f : B \u2192 C) (g : A \u2192 B) x \u2192 liftM (f \u2218 g) x \u2261 liftM f (liftM g x)\n  liftM-\u2218 f g x = trans (cong (_>>=_ x) {!cong-return-\u2218 {!return->>= (g ?)!}!}) (>>=-assoc x (return \u2218 g) (return \u2218 f))\n\n  functor : Functor rawFunctor\n  functor = record { <$>-id = <$>-id; <$>-\u2218 = {!!} }\n  -}\n\n  vmapM : \u2200 {n a}{A : Set a} {B : Set \u2113} \u2192 (A \u2192 M B) \u2192 Vec A n \u2192 M (Vec B n)\n  vmapM f []       = return []\n  vmapM f (x \u2237 xs) = _\u2237_ <$> f x \u229b vmapM f xs\n\nrecord Monad (M : Set \u2113 \u2192 Set \u2113) : \u2605_ (\u209b \u2113) where\n  constructor _,_\n  field\n    rawMonad  : RawMonad M\n    isMonadic : IsMonadic rawMonad\n\n  open RawMonad  rawMonad  public\n  open IsMonadic isMonadic public\n","old_contents":"","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"93d1536fcf0ff29c501257842452ef0bb16955c2","subject":"renamed variables","message":"renamed variables\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  exb0 : Val 0 \u21d3 0\n  exb0 = \u21d3Base\n\n  exb3 : Val 3 \u21d3 3\n  exb3 = \u21d3Base\n\n  exs331 : Div (Val 3) (Val 3) \u21d3 1\n  exs331 = \u21d3Step \u21d3Base \u21d3Base\n\n  exs931 : Div (Val 9) (Val 3) \u21d3 3\n  exs931 = \u21d3Step \u21d3Base \u21d3Base\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  relate the two definitions:\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, define this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder)) with \u27e6 el \u27e7 | \u27e6 er \u27e7 | correct el sdel | correct er sder\n  correct (Div el er) (ernz , (sdel , sder)) | Pure v1      | Pure Zero         | el\u21d3v1 | er\u21d3Z   = magic (ernz er\u21d3Z)\n  correct (Div el er) (ernz , (sdel , sder)) | Pure v1      | Pure (Succ v2)    | el\u21d3v1 | er\u21d3Sv2 = \u21d3Step el\u21d3v1 er\u21d3Sv2\n  correct (Div el er) (ernz , (sdel , sder)) | Pure v1      | Step Abort \u22a5\u2192FCRN | el\u21d3v1 | ()\n  correct (Div el er) (ernz , (sdel , sder)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7 | sound el | sound er\n  sound (Div el er) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div el er) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div el er) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div el er) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7\n  inDom (Div el er) () | Pure v1      | Pure Zero\n  inDom (Div el er) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div el er) () | Pure _       | Step Abort _\n  inDom (Div el er) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ el\n  wpPartial1 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 {el} {er} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {el} {er} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux el x eq\n  wpPartial1 {el} {er} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {el} {er} () | Step Abort x | eq         | ver\n\n  wpPartial2 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ er\n  wpPartial2 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n  wpPartial2 {el} {er} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux er y eqy\n  wpPartial2 {el} {er} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | ser           | eq2\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n      | complete el (wpPartial1 {el} {er} h)\n      | complete er (wpPartial2 {el} {er} h)\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div el er) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div el er) h | Step Abort x | eq1         | ser           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er n1 n2}\n         ->     el    \u21d3  n1\n         ->        er \u21d3         (Succ n2) -- divisor is non-zero\n         -> Div el er \u21d3 (n1 div (Succ n2))\n\n  exb0 : Val 0 \u21d3 0\n  exb0 = \u21d3Base\n\n  exb3 : Val 3 \u21d3 3\n  exb3 = \u21d3Base\n\n  exs331 : Div (Val 3) (Val 3) \u21d3 1\n  exs331 = \u21d3Step \u21d3Base \u21d3Base\n\n  exs931 : Div (Val 9) (Val 3) \u21d3 3\n  exs931 = \u21d3Step \u21d3Base \u21d3Base\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  relate the two definitions:\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, define this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"1dd4f0f861df9af4f01a33af2bb834b72be708cc","subject":"Implement DependentList on top of Data.List.All.","message":"Implement DependentList on top of Data.List.All.\n\nOld-commit-hash: 32c7c7edadf9940f3c013ca3d4ebfdc8b3cbc4f4\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Data\/DependentList.agda","new_file":"Base\/Data\/DependentList.agda","new_contents":"module Base.Data.DependentList where\n\nopen import Data.List.All public\n  renaming\n    ( All to DependentList\n    ; _\u2237_ to _\u2022_\n    ; [] to \u2205\n    )\n","old_contents":"module Base.Data.DependentList where\n\nopen import Level\n\nopen import Data.Unit\nopen import Data.Product using (\u03a3 ; _\u00d7_ ; _,_)\n\nimport Data.List as List\nopen List using (List ; [] ; _\u2237_)\n\ndata DependentList {a b} {A : Set a}\n  (F : A \u2192 Set b) : (type-args : List A) \u2192 Set (a \u2294 b)\n  where\n  \u2205 : DependentList F []\n  _\u2022_ : \u2200 {type-arg} {other-type-args}\n    (head : F type-arg)\n    (tail : DependentList F other-type-args) \u2192\n    DependentList F (type-arg \u2237 other-type-args)\n\ninfixr 5 _\u2022_\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"816cffb53d3bae1af15f0f477ac1a68ccfa88b05","subject":"add more composision with >>","message":"add more composision with >>\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Sequence.agda","new_file":"Control\/Protocol\/Sequence.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_; first) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2219_; refl; ap; coe; coe!; !_ ; tr)-- renaming (subst to tr)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\nopen import Type.Identities\n\nopen import Control.Protocol.Core\n\nmodule Control.Protocol.Sequence where\n\ninfixl 1 _>>=_ _>>_\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\n>>=-fst : \u2200 P {Q} \u2192 \u27e6 P >>= Q \u27e7 \u2192 \u27e6 P \u27e7\n>>=-fst end q = end\n>>=-fst (recv P) pq = \u03bb m \u2192 >>=-fst (P m) (pq m)\n>>=-fst (send P) (m , pq) = m , >>=-fst (P m) pq\n\n>>=-snd : \u2200 P {Q}(pq : \u27e6 P >>= Q \u27e7)(p : \u27e6 P \u22a5\u27e7) \u2192 \u27e6 Q (telecom P (>>=-fst P pq) p) \u27e7\n>>=-snd end      q        end     = q\n>>=-snd (recv P) pq       (m , p) = >>=-snd (P m) (pq m) p\n>>=-snd (send P) (m , pq) p       = >>=-snd (P m) pq (p m)\n\n>>-snd : \u2200 P {Q} (pq : \u27e6 P >> Q \u27e7)(p : \u27e6 P \u22a5\u27e7) \u2192 \u27e6 Q \u27e7\n>>-snd P = >>=-snd P\n\n[_]_>>>=_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 ((log : Log P) \u2192 \u27e6 Q log \u27e7) \u2192 \u27e6 P >>= Q \u27e7\n[ end    ] p       >>>= q = q _\n[ recv P ] p       >>>= q = \u03bb m \u2192 [ P m ] p m >>>= \u03bb log \u2192 q (m , log)\n[ send P ] (m , p) >>>= q = m , [ P m ] p >>>= \u03bb log \u2192 q (m , log)\n\n[_]_>>>\u1d38_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 (Log P \u2192 \u27e6 Q \u27e7) \u2192 \u27e6 P >> Q \u27e7\n[ P ] p >>>\u1d38 q = [ P ] p >>>= q\n\n[_]_>>>_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P >> Q \u27e7\n[ P ] p >>> q = [ P ] p >>>\u1d38 (const q)\n\nmodule _ {{_ : FunExt}} where\n    >>=-fst-inv : \u2200 P {Q}(p : \u27e6 P \u27e7)(q : ((log : Log P) \u2192 \u27e6 Q log \u27e7)) \u2192 >>=-fst P {Q} ([ P ] p >>>= q) \u2261 p\n    >>=-fst-inv end      end     q = refl\n    >>=-fst-inv (recv P) p       q = \u03bb= \u03bb m \u2192 >>=-fst-inv (P m) (p m) \u03bb log \u2192 q (m , log)\n    >>=-fst-inv (send P) (m , p) q = snd= (>>=-fst-inv (P m) p \u03bb log \u2192 q (m , log))\n\n    >>=-snd-inv : \u2200 P {Q}(p : \u27e6 P \u27e7)(q : ((log : Log P) \u2192 \u27e6 Q log \u27e7))(p' : \u27e6 P \u22a5\u27e7)\n                  \u2192 tr (\u03bb x \u2192 \u27e6 Q (telecom P x p') \u27e7) (>>=-fst-inv P p q)\n                       (>>=-snd P {Q} ([ P ] p >>>= q) p') \u2261 q (telecom P p p')\n    >>=-snd-inv end          end     q p' = refl\n    >>=-snd-inv (recv P) {Q} p       q (m , p') =\n      ap (flip _$_ (>>=-snd (P m) ([ P m ] p m >>>= (\u03bb log \u2192 q (m , log))) p'))\n                                                     (tr-\u03bb= (\u03bb z \u2192 \u27e6 Q (m , telecom (P m) z p') \u27e7)\n                                                               (\u03bb m \u2192 >>=-fst-inv (P m) {Q \u2218 _,_ m} (p m) (q \u2218 _,_ m)))\n                                                \u2219 >>=-snd-inv (P m) {Q \u2218 _,_ m} (p m) (q \u2218 _,_ m) p'\n    >>=-snd-inv (send P) {Q} (m , p) q p' = tr-snd= (\u03bb { (m , p) \u2192 \u27e6 Q (m , telecom (P m) p (p' m)) \u27e7 })\n                                                    (>>=-fst-inv (P m) p (q \u2218 _,_ m))\n                                                    (>>=-snd (P m) {Q \u2218 _,_ m} ([ P m ] p >>>= (q \u2218 _,_ m)) (p' m))\n                                          \u2219 >>=-snd-inv (P m) {Q \u2218 _,_ m} p (\u03bb log \u2192 q (m , log)) (p' m)\n\n    {- hmmm...\n    >>=-uniq : \u2200 P {Q} (pq : \u27e6 P >>= Q \u27e7)(p' : \u27e6 P \u22a5\u27e7) \u2192 pq \u2261 [ P ] (>>=-fst P {Q} pq) >>>= (\u03bb log \u2192 {!>>=-snd P {Q} pq p'!})\n    >>=-uniq = {!!}\n    -}\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\nreplicate-proc : \u2200 n P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 replicate\u1d3e n P \u27e7\nreplicate-proc zero    P p = end\nreplicate-proc (suc n) P p = [ P ] p >>> replicate-proc n P p\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  Log->>=-\u03a3 : \u2200 (P : Proto){Q} \u2192 Log (P >>= Q) \u2261 \u03a3 (Log P) (Log \u2218 Q)\n  Log->>=-\u03a3 end       = ! (\u00d7= End-uniq refl \u2219 \ud835\udfd9\u00d7-snd)\n  Log->>=-\u03a3 (com _ P) = \u03a3=\u2032 _ (\u03bb m \u2192 Log->>=-\u03a3 (P m)) \u2219 \u03a3-assoc\n\n  Log->>-\u00d7 : \u2200 (P : Proto){Q} \u2192 Log (P >> Q) \u2261 (Log P \u00d7 Log Q)\n  Log->>-\u00d7 P = Log->>=-\u03a3 P\n\n  ++Log' : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n  ++Log' P p q = coe! (Log->>=-\u03a3 P) (p , q)\n\n-- Same as ++Log' but computes better\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  ++Log\u2261++Log' : \u2200 P {Q} xs ys  \u2192 ++Log P {Q} xs ys \u2261 ++Log' P xs ys\n  ++Log\u2261++Log' end xs ys\n    = ! coe-\u03b2 _ _\n    \u2219 ap (coe \ud835\udfd9\u00d7-snd) (! coe\u00d7= End-uniq refl)\n    \u2219 coe\u2218coe \ud835\udfd9\u00d7-snd (\u00d7= End-uniq refl) (xs , ys)\n    \u2219 coe-same (! (!-inv (\u00d7= End-uniq refl \u2219 \ud835\udfd9\u00d7-snd))) (xs , ys)\n  ++Log\u2261++Log' (com io P) (m , xs) ys\n    = ap (_,_ m) (++Log\u2261++Log' (P m) xs ys)\n    \u2219 ! coe!\u03a3=\u2032 _ (\u03bb m \u2192 Log->>=-\u03a3 (P m))\n    \u2219 ap (coe (! \u03a3=\u2032 _ (\u03bb m\u2081 \u2192 Log->>=-\u03a3 (P m\u2081)))) (! coe!-\u03b2 _ _)\n    \u2219 coe\u2218coe (! \u03a3=\u2032 _ (\u03bb m \u2192 Log->>=-\u03a3 (P m))) (! \u03a3-assoc) ((m , xs) , ys)\n    \u2219 coe-same (! (hom-!-\u2219 (\u03a3=\u2032 _ (\u03bb m \u2192 Log->>=-\u03a3 (P m))) \u03a3-assoc)) ((m , xs) , ys)\n\nmodule _ {{_ : FunExt}} where\n    >>-right-unit : \u2200 P \u2192 (P >> end) \u2261 P\n    >>-right-unit end       = refl\n    >>-right-unit (com q P) = com= refl refl \u03bb m \u2192 >>-right-unit (P m)\n\n    >>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n                \u2192 (P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y)))) \u2261 ((P >>= Q) >>= R)\n    >>=-assoc end       Q R = refl\n    >>=-assoc (com q P) Q R = com= refl refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\n    dual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261 (dual P >> dual Q)\n    dual->> end        Q = refl\n    dual->> (com io P) Q = com= refl refl \u03bb m \u2192 dual->> (P m) Q\n\n    {- My coe-ap-fu is lacking...\n    dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261 dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n    dual->>= end Q = refl\n    dual->>= (com io P) Q = com= refl refl \u03bb m \u2192 dual->>= (P m) (Q \u2218 _,_ m) \u2219 ap (_>>=_ (dual (P m))) (\u03bb= \u03bb ms \u2192 ap (\u03bb x \u2192 dual (Q x)) (pair= {!!} {!!}))\n    -}\n\n    dual-replicate\u1d3e : \u2200 n P \u2192 dual (replicate\u1d3e n P) \u2261 replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    P = refl\n    dual-replicate\u1d3e (suc n) P = dual->> P (replicate\u1d3e n P) \u2219 ap (_>>_ (dual P)) (dual-replicate\u1d3e n P)\n\n{- An incremental telecom function which makes processes communicate\n   during a matching initial protocol. -}\n>>=-telecom : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n            \u2192 \u27e6 P >>= Q  \u27e7\n            \u2192 \u27e6 P >>= R \u22a5\u27e7\n            \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-telecom end      p0       p1       = _ , p0 , p1\n>>=-telecom (send P) (m , p0) p1       = first (_,_ m) (>>=-telecom (P m) p0 (p1 m))\n>>=-telecom (recv P) p0       (m , p1) = first (_,_ m) (>>=-telecom (P m) (p0 m) p1)\n\n>>-telecom : (P : Proto){Q R : Proto}\n           \u2192 \u27e6 P >> Q  \u27e7\n           \u2192 \u27e6 P >> R \u22a5\u27e7\n           \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-telecom P p q = >>=-telecom P p q\n\n>>-compose : (P Q : Proto){R : Proto}\n  \u2192 \u27e6 P >> Q \u27e7\n  \u2192 \u27e6 dual Q >> R \u27e7\n  \u2192 \u27e6 P >> R \u27e7\n>>-compose end end p>>q q>>r = q>>r\n>>-compose end (send P) (m , p>>q) q>>r = >>-compose end (P m) p>>q (q>>r m)\n>>-compose end (recv P) p>>q (m , q>>r) = >>-compose end (P m) (p>>q m) q>>r\n>>-compose (send P) Q (m , p>>q) q>>r = m , >>-compose (P m) Q p>>q q>>r\n>>-compose (recv P) Q p>>q q>>r m = >>-compose (P m) Q (p>>q m) q>>r\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_; first) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2219_; refl; ap; coe; coe!; !_ ; tr)-- renaming (subst to tr)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\nopen import Type.Identities\n\nopen import Control.Protocol.Core\n\nmodule Control.Protocol.Sequence where\n\ninfixl 1 _>>=_ _>>_\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\n>>=-fst : \u2200 P {Q} \u2192 \u27e6 P >>= Q \u27e7 \u2192 \u27e6 P \u27e7\n>>=-fst end q = end\n>>=-fst (recv P) pq = \u03bb m \u2192 >>=-fst (P m) (pq m)\n>>=-fst (send P) (m , pq) = m , >>=-fst (P m) pq\n\n>>=-snd : \u2200 P {Q}(pq : \u27e6 P >>= Q \u27e7)(p : \u27e6 P \u22a5\u27e7) \u2192 \u27e6 Q (telecom P (>>=-fst P pq) p) \u27e7\n>>=-snd end      q        end     = q\n>>=-snd (recv P) pq       (m , p) = >>=-snd (P m) (pq m) p\n>>=-snd (send P) (m , pq) p       = >>=-snd (P m) pq (p m)\n\n[_]_>>>=_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 ((log : Log P) \u2192 \u27e6 Q log \u27e7) \u2192 \u27e6 P >>= Q \u27e7\n[ end    ] p       >>>= q = q _\n[ recv P ] p       >>>= q = \u03bb m \u2192 [ P m ] p m >>>= \u03bb log \u2192 q (m , log)\n[ send P ] (m , p) >>>= q = m , [ P m ] p >>>= \u03bb log \u2192 q (m , log)\n\n[_]_>>>_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P >> Q \u27e7\n[ P ] p >>> q = [ P ] p >>>= \u03bb _ \u2192 q\n\nmodule _ {{_ : FunExt}} where\n    >>=-fst-inv : \u2200 P {Q}(p : \u27e6 P \u27e7)(q : ((log : Log P) \u2192 \u27e6 Q log \u27e7)) \u2192 >>=-fst P {Q} ([ P ] p >>>= q) \u2261 p\n    >>=-fst-inv end      end     q = refl\n    >>=-fst-inv (recv P) p       q = \u03bb= \u03bb m \u2192 >>=-fst-inv (P m) (p m) \u03bb log \u2192 q (m , log)\n    >>=-fst-inv (send P) (m , p) q = snd= (>>=-fst-inv (P m) p \u03bb log \u2192 q (m , log))\n\n    >>=-snd-inv : \u2200 P {Q}(p : \u27e6 P \u27e7)(q : ((log : Log P) \u2192 \u27e6 Q log \u27e7))(p' : \u27e6 P \u22a5\u27e7)\n                  \u2192 tr (\u03bb x \u2192 \u27e6 Q (telecom P x p') \u27e7) (>>=-fst-inv P p q)\n                       (>>=-snd P {Q} ([ P ] p >>>= q) p') \u2261 q (telecom P p p')\n    >>=-snd-inv end          end     q p' = refl\n    >>=-snd-inv (recv P) {Q} p       q (m , p') =\n      ap (flip _$_ (>>=-snd (P m) ([ P m ] p m >>>= (\u03bb log \u2192 q (m , log))) p'))\n                                                     (tr-\u03bb= (\u03bb z \u2192 \u27e6 Q (m , telecom (P m) z p') \u27e7)\n                                                               (\u03bb m \u2192 >>=-fst-inv (P m) {Q \u2218 _,_ m} (p m) (q \u2218 _,_ m)))\n                                                \u2219 >>=-snd-inv (P m) {Q \u2218 _,_ m} (p m) (q \u2218 _,_ m) p'\n    >>=-snd-inv (send P) {Q} (m , p) q p' = tr-snd= (\u03bb { (m , p) \u2192 \u27e6 Q (m , telecom (P m) p (p' m)) \u27e7 })\n                                                    (>>=-fst-inv (P m) p (q \u2218 _,_ m))\n                                                    (>>=-snd (P m) {Q \u2218 _,_ m} ([ P m ] p >>>= (q \u2218 _,_ m)) (p' m))\n                                          \u2219 >>=-snd-inv (P m) {Q \u2218 _,_ m} p (\u03bb log \u2192 q (m , log)) (p' m)\n\n    {- hmmm...\n    >>=-uniq : \u2200 P {Q} (pq : \u27e6 P >>= Q \u27e7)(p' : \u27e6 P \u22a5\u27e7) \u2192 pq \u2261 [ P ] (>>=-fst P {Q} pq) >>>= (\u03bb log \u2192 {!>>=-snd P {Q} pq p'!})\n    >>=-uniq = {!!}\n    -}\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\nreplicate-proc : \u2200 n P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 replicate\u1d3e n P \u27e7\nreplicate-proc zero    P p = end\nreplicate-proc (suc n) P p = [ P ] p >>> replicate-proc n P p\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  Log->>=-\u03a3 : \u2200 (P : Proto){Q} \u2192 Log (P >>= Q) \u2261 \u03a3 (Log P) (Log \u2218 Q)\n  Log->>=-\u03a3 end       = ! (\u00d7= End-uniq refl \u2219 \ud835\udfd9\u00d7-snd)\n  Log->>=-\u03a3 (com _ P) = \u03a3=\u2032 _ (\u03bb m \u2192 Log->>=-\u03a3 (P m)) \u2219 \u03a3-assoc\n\n  Log->>-\u00d7 : \u2200 (P : Proto){Q} \u2192 Log (P >> Q) \u2261 (Log P \u00d7 Log Q)\n  Log->>-\u00d7 P = Log->>=-\u03a3 P\n\n  ++Log' : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n  ++Log' P p q = coe! (Log->>=-\u03a3 P) (p , q)\n\n-- Same as ++Log' but computes better\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\nmodule _ {{_ : FunExt}} where\n    >>-right-unit : \u2200 P \u2192 (P >> end) \u2261 P\n    >>-right-unit end       = refl\n    >>-right-unit (com q P) = com= refl refl \u03bb m \u2192 >>-right-unit (P m)\n\n    >>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n                \u2192 (P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y)))) \u2261 ((P >>= Q) >>= R)\n    >>=-assoc end       Q R = refl\n    >>=-assoc (com q P) Q R = com= refl refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\n    dual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261 (dual P >> dual Q)\n    dual->> end        Q = refl\n    dual->> (com io P) Q = com= refl refl \u03bb m \u2192 dual->> (P m) Q\n\n    {- My coe-ap-fu is lacking...\n    dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261 dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n    dual->>= end Q = refl\n    dual->>= (com io P) Q = com= refl refl \u03bb m \u2192 dual->>= (P m) (Q \u2218 _,_ m) \u2219 ap (_>>=_ (dual (P m))) (\u03bb= \u03bb ms \u2192 ap (\u03bb x \u2192 dual (Q x)) (pair= {!!} {!!}))\n    -}\n\n    dual-replicate\u1d3e : \u2200 n P \u2192 dual (replicate\u1d3e n P) \u2261 replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    P = refl\n    dual-replicate\u1d3e (suc n) P = dual->> P (replicate\u1d3e n P) \u2219 ap (_>>_ (dual P)) (dual-replicate\u1d3e n P)\n\n{- An incremental telecom function which makes processes communicate\n   during a matching initial protocol. -}\n>>=-telecom : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n            \u2192 \u27e6 P >>= Q  \u27e7\n            \u2192 \u27e6 P >>= R \u22a5\u27e7\n            \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-telecom end      p0       p1       = _ , p0 , p1\n>>=-telecom (send P) (m , p0) p1       = first (_,_ m) (>>=-telecom (P m) p0 (p1 m))\n>>=-telecom (recv P) p0       (m , p1) = first (_,_ m) (>>=-telecom (P m) (p0 m) p1)\n\n>>-telecom : (P : Proto){Q R : Proto}\n           \u2192 \u27e6 P >> Q  \u27e7\n           \u2192 \u27e6 P >> R \u22a5\u27e7\n           \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-telecom P p q = >>=-telecom P p q\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d0326164e3270519a7229d44abdf050a074562b9","subject":"NaPa: Add precedence annotations","message":"NaPa: Add precedence annotations\n","repos":"np\/NomPa","old_file":"lib\/NaPa.agda","new_file":"lib\/NaPa.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n-- vim: iskeyword=a-z,A-Z,48-57,-,+,',#,\/,~,],[,=,_,`,\u2286\nmodule NaPa where\n\nimport Level as L\n-- open import Irrelevance\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2257-Reasoning; module \u2261-Reasoning)\nopen import Data.Nat.NP as Nat using (\u2115; zero; suc; s\u2264s; z\u2264n; \u2264-pred; pred; _<=_; module <=)\n                            renaming (_+_ to _+\u2115_ ; _\u2238_ to _\u2238\u2115_ ; _==_ to _==\u2115_; \u00ac\u2264 to \u00ac\u2264\u2115;\n                                      _<_ to _<\u2115_ ; _\u2264_ to _\u2264\u2115_; _\u2265_ to _\u2265\u2115_; _\u2264?_ to _\u2264?\u2115_)\nopen import Data.Nat.Logical using (\u27e6\u2115\u27e7; zero; suc; \u27e6\u2115\u27e7-setoid; \u27e6\u2115\u27e7-equality; \u27e6\u2115\u27e7-cong)\n                          renaming (_\u225f_ to _\u225f\u27e6\u2115\u27e7_)\nimport Data.Nat.Properties as Nat\nopen import Function\nopen import Function.Equality as \u27f6\u2261 using (_\u27f6_; _\u27e8$\u27e9_)\nopen import Data.Sum.NP as Sum using (_\u228e_; inl; inr; [_,_]\u2032) renaming (map to \u228e-map)\nopen import Data.Bool using (Bool; true; false; if_then_else_; T; not)\nopen import Data.Two renaming (\u2713-not-\u00ac to T'not'\u00ac)\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product.NP using (_,_)\nopen import Data.Maybe.NP using (Maybe; nothing; just; maybe; _\u2192?_)\nopen import Relation.Binary.NP as Bin\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.On as On\nopen import NomPa.Worlds\nopen import NomPa.Worlds.Syntax as WorldSyntax\nopen import NomPa.Subtyping\nopen ListBoolWorlds public hiding (_\u2286_) renaming (_\u2286\u2032_ to _\u2286_)\nimport Category.Monad as Cat\nopen Cat.RawMonad (Data.Maybe.NP.monad {L.zero})\nopen Data.Maybe.NP.FunctorLemmas\n\nprivate\n  module \u2115s = Setoid \u27e6\u2115\u27e7-setoid\n  module \u2115e = Equality \u27e6\u2115\u27e7-equality\n\ninfixr 4 _,_\nrecord Name \u03b1 : Set where\n  constructor _,_\n  field\n    name    : \u2115\n    -- .poofname\u2208\u03b1 : name \u2208 \u03b1\n    name\u2208\u03b1 : name \u2208 \u03b1\n\nopen Name public\n{-\ndata Name \u03b1 : Set where\n  _,_ : (x : \u2115) \u2192 .(x \u2208 \u03b1) \u2192 Name \u03b1\nname : \u2200 {\u03b1} (x : Name \u03b1) \u2192 \u2115\nname (x , _) = x\n\n.name\u2208\u03b1 : \u2200 {\u03b1} (x : Name \u03b1) \u2192 name x \u2208 \u03b1\nname\u2208\u03b1 (_ , pf) = pf\n-}\n\n--mkName : \u2200 {\u03b1} (x : \u2115) \u2192 Irr (x \u2208 \u03b1) \u2192 Name \u03b1\n--mkName x (irr pf) = x , pf\nmkName : \u2200 {\u03b1} (x : \u2115) \u2192 x \u2208 \u03b1 \u2192 Name \u03b1\nmkName x pf = x , pf\n\n--name\u2208\u03b1 : \u2200 {\u03b1} (x : Name \u03b1) \u2192 Irr (name x \u2208 \u03b1)\n--name\u2208\u03b1 (_ , pf) = irr pf\n\ninfix 4 _\u2261\u1d3a_ _\u225f\u1d3a_ _==\u1d3a_\n\n-- prim\n_\u2261\u1d3a_ : \u2200 {\u03b1} (x y : Name \u03b1) \u2192 Set\n_\u2261\u1d3a_ = \u27e6\u2115\u27e7 on name\n\n-- prim\n_\u225f\u1d3a_ : \u2200 {\u03b1} \u2192 Decidable (_\u2261\u1d3a_ {\u03b1})\nx \u225f\u1d3a y = name x \u225f\u27e6\u2115\u27e7 name y\n\n-- prim\n_==\u1d3a_ : \u2200 {\u03b1} (x y : Name \u03b1) \u2192 Bool\n_==\u1d3a_ x y = name x ==\u2115 name y\n\nname-injective : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 name x \u2261 name y \u2192 x \u2261 y\nname-injective {\u03b1} {_ , p\u2081} {a , p\u2082} eq rewrite eq\n--= \u2261.refl {_} {_} {a , p\u2082}\n  = \u2261.cong (_,_ a) (\u2208-uniq \u03b1 p\u2081 p\u2082)\n\nNm : World \u2192 Setoid L.zero L.zero\nNm \u03b1 = On.setoid \u27e6\u2115\u27e7-setoid (name {\u03b1})\n\n\u2261\u1d3a-equality : \u2200 {\u03b1} \u2192 Equality (_\u2261\u1d3a_ {\u03b1})\n\u2261\u1d3a-equality {\u03b1}\n   = record { isEquivalence = Setoid.isEquivalence (Nm \u03b1)\n              -- I would prefered not going to PropEq.\u2261\n            ; subst = \u03bb P \u2192 \u2261.tr P \u2218 name-injective \u2218 Equality.to-propositional \u27e6\u2115\u27e7-equality }\n\nmodule NmEq {\u03b1} = Equality (\u2261\u1d3a-equality {\u03b1})\n\n\u2261\u1d3a\u21d2\u2261 : \u2200 {\u03b1} \u2192 _\u2261\u1d3a_ {\u03b1} \u21d2 _\u2261_\n\u2261\u1d3a\u21d2\u2261 = NmEq.to-propositional\n\n\u2261\u1d3a-cong : \u2200 {\u03b1 \u03b2} (f : Name \u03b1 \u2192 Name \u03b2) \u2192 _\u2261\u1d3a_ =[ f ]\u21d2 _\u2261\u1d3a_\n\u2261\u1d3a-cong f = NmEq.reflexive \u2218 \u2261.cong f \u2218 \u2261\u1d3a\u21d2\u2261\n\n\u2261\u1d3a-\u27e6\u2115\u27e7-cong : \u2200 {\u03b1} (f : Name \u03b1 \u2192 \u2115) \u2192 _\u2261\u1d3a_ =[ f ]\u21d2 \u27e6\u2115\u27e7\n\u2261\u1d3a-\u27e6\u2115\u27e7-cong f = \u2115e.reflexive \u2218 \u2261.cong f \u2218 \u2261\u1d3a\u21d2\u2261\n\n\u2261-\u27e6\u2115\u27e7-cong : \u2200 {a} {A : Set a} (f : A \u2192 \u2115) \u2192 _\u2261_ =[ f ]\u21d2 \u27e6\u2115\u27e7\n\u2261-\u27e6\u2115\u27e7-cong f = \u2115e.reflexive \u2218 \u2261.cong f\n\n\u2261\u1d3a-\u2261-cong : \u2200 {a \u03b1} {A : Set a} (f : Name \u03b1 \u2192 A) \u2192 _\u2261\u1d3a_ =[ f ]\u21d2 _\u2261_\n\u2261\u1d3a-\u2261-cong f = \u2261.cong f \u2218 \u2261\u1d3a\u21d2\u2261\n\n\u27e6\u2115\u27e7-\u2261\u1d3a-cong : \u2200 {\u03b1} (f : \u2115 \u2192 Name \u03b1) \u2192 \u27e6\u2115\u27e7 =[ f ]\u21d2 _\u2261\u1d3a_\n\u27e6\u2115\u27e7-\u2261\u1d3a-cong f = NmEq.reflexive \u2218 \u2261.cong f \u2218 \u2115e.to-propositional\n\n\u27e6\u2115\u27e7-\u2261-cong : \u2200 {a} {A : Set a} (f : \u2115 \u2192 A) \u2192 \u27e6\u2115\u27e7 =[ f ]\u21d2 _\u2261_\n\u27e6\u2115\u27e7-\u2261-cong f = \u2261.cong f \u2218 \u2115e.to-propositional\n\nmodule \u2261\u1d3a-Reasoning {\u03b1} = Bin.Setoid-Reasoning (Nm \u03b1)\n\nopen WorldSymantics listBoolWorlds public\nopen WorldOps listBoolWorlds public\nopen WorldOps listBoolWorlds using ()\nopen NomPa.Subtyping.\u2286-Pack \u2286\u2032\u1d47-pack public hiding (_\u2286_; _\u2288_)\nopen NomPa.Subtyping.SyntacticOnBools \u2286\u2032\u1d47-pack public hiding (minimalSymantics)\n\n-- Here is a first set of core primitives easy to define\n\n-- prim\nzero\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 \u21911)\nzero\u1d3a = 0 , _\n\n-- prim\n\u00acName\u00f8 : \u00ac(Name \u00f8)\n\u00acName\u00f8 (_ , ())\n\n-- prim\nadd\u1d3a : \u2200 {\u03b1} k (x : Name \u03b1) \u2192 Name (\u03b1 +\u1d42 k)\nadd\u1d3a {\u03b1} k x = k +\u2115 name x , proof k where\n  proof : \u2200 k \u2192 k +\u2115 name x \u2208 \u03b1 +\u1d42 k\n  proof zero    = name\u2208\u03b1 x\n  proof (suc k) = proof k\n\ninfixl 6 add\u1d3a\nsyntax add\u1d3a k x = x +\u1d3a k\n\n-- prim\nsubtract\u1d3a : \u2200 {\u03b1} k (x : Name (\u03b1 +\u1d42 k)) \u2192 Name \u03b1\nsubtract\u1d3a k x = name x \u2238\u2115 k , proof (name x) k (name\u2208\u03b1 x) where\n  proof : \u2200 {\u03b1} x k \u2192 (x \u2208 \u03b1 +\u1d42 k) \u2192 (x \u2238\u2115 k \u2208 \u03b1)\n  proof x       zero = id\n  proof zero    (suc k) = \u03bb()\n  proof (suc x) (suc k) = proof x k\n\ninfixl 4 subtract\u1d3a\nsyntax subtract\u1d3a k x = x \u2238\u1d3a k\n\n-- prim\ncoerce\u1d3a : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 \u2286 \u03b2 \u2192 Name \u03b1 \u2192 Name \u03b2\ncoerce\u1d3a \u03b1\u2286\u03b2 x = name x , coe \u03b1\u2286\u03b2 (name x) (name\u2208\u03b1 x)\n\n-- Then we define some convenience functions on top of them.\n\ninfix 0 _\u27e8-because_-\u27e9\n_\u27e8-because_-\u27e9 : \u2200 {\u03b1 \u03b2} \u2192 Name \u03b1 \u2192 \u03b1 \u2286 \u03b2 \u2192 Name \u03b2\n_\u27e8-because_-\u27e9 n pf = coerce\u1d3a pf n\n\nadd\u1d3a\u2191 : \u2200 {\u03b1} \u2113 \u2192 Name \u03b1 \u2192 Name (\u03b1 \u2191 \u2113)\nadd\u1d3a\u2191 \u2113 x = add\u1d3a \u2113 x  \u27e8-because \u2286-+-\u2191 \u2113 -\u27e9\n\nsuc\u1d3a : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name (\u03b1 +1)\nsuc\u1d3a = add\u1d3a 1\n\nsuc\u1d3a\u2191 : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name (\u03b1 \u21911)\nsuc\u1d3a\u2191 = add\u1d3a\u2191 1\n\n-- Then comes <\u1d3a and the necessary intermediate definitions\n\nprivate\n  lem-<\u2113 : \u2200 {\u03b1 \u03b2 x \u2113} \u2192 x \u2264\u2115 \u2113 \u2192 x \u2208 (\u03b1 \u2191 suc \u2113) \u2192 x \u2208 (\u03b2 \u2191 suc \u2113)\n  lem-<\u2113 z\u2264n       = _\n  lem-<\u2113 (s\u2264s m\u2264n) = lem-<\u2113 m\u2264n\n\n  lem-\u2265\u2113 : \u2200 {\u03b1 x \u2113} \u2192 x \u2265\u2115 \u2113 \u2192 x \u2208 \u03b1 \u2191 \u2113 \u2192 x \u2208 \u03b1 +\u1d42 \u2113\n  lem-\u2265\u2113 z\u2264n        = id\n  lem-\u2265\u2113 (s\u2264s m\u2264n)  = lem-\u2265\u2113 m\u2264n\n\n  lem-<\u2113? : \u2200 {\u03b1 \u03b2 n} \u2113 \u2192 n \u2208 \u03b2 \u2191 \u2113 \u2192 n \u2208 (if suc n <= \u2113 then \u03b1 \u2191 \u2113 else \u03b2 +\u1d42 \u2113)\n  lem-<\u2113? {n = n} \u2113 with suc n <= \u2113 | <=.sound (suc n) \u2113 | <=.complete {suc n} {\u2113}\n  ... | true  | n<\u2113 | _    = lem-<\u2113 (n<\u2113 _)\n  ... | false | _   | \u00acn<\u2113 = lem-\u2265\u2113 (\u00ac\u2264\u2115 \u00acn<\u2113)\n\n-- prim\ncmp\u1d3a-bool : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Bool\ncmp\u1d3a-bool \u2113 x = suc (name x) <= \u2113\n\nsyntax cmp\u1d3a-bool k x = x <\u1d3a-bool k\n\n-- prim\ncmp\u1d3a-name : \u2200 {\u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113)) \u2192 Name (if cmp\u1d3a-bool \u2113 x then \u00f8 \u2191 \u2113 else \u03b1 +\u1d42 \u2113)\ncmp\u1d3a-name \u2113 x = name x , lem-<\u2113? \u2113 (name\u2208\u03b1 x)\n\nsyntax cmp\u1d3a-name \u2113 x = x <\u1d3a-name \u2113\n\n-- prim\ncmp\u1d3a-name\u2208\u03b1 : \u2200 {\u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113)) \u2192 name x \u2208 (if cmp\u1d3a-bool \u2113 x then \u00f8 \u2191 \u2113 else \u03b1 +\u1d42 \u2113)\ncmp\u1d3a-name\u2208\u03b1 \u2113 x = lem-<\u2113? \u2113 (name\u2208\u03b1 x)\n\nsyntax cmp\u1d3a-name\u2208\u03b1 \u2113 x = x <\u1d3a-name\u2208\u03b1 \u2113\n\n-- prim\ncmp\u1d3a : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u00f8 \u2191 \u2113) \u228e Name (\u03b1 +\u1d42 \u2113)\ncmp\u1d3a \u2113 x with cmp\u1d3a-bool \u2113 x | cmp\u1d3a-name\u2208\u03b1 \u2113 x\n... | true  | pf = inl (name x , pf)\n... | false | pf = inr (name x , pf)\n\ninfix 4 cmp\u1d3a\nsyntax cmp\u1d3a k x = x <\u1d3a k\n\ncmp\u1d3a-ind : \u2200 {a \u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113))\n            (P : (Name (\u00f8 \u2191 \u2113) \u228e Name (\u03b1 +\u1d42 \u2113)) \u2192 Set a)\n            (Pinl : T (cmp\u1d3a-bool \u2113 x) \u2192 \u2200 pf \u2192 P (inl (name x , pf)))\n            (Pinr : T (not (cmp\u1d3a-bool \u2113 x)) \u2192 \u2200 pf \u2192 P (inr (name x , pf)))\n          \u2192 P (cmp\u1d3a \u2113 x)\ncmp\u1d3a-ind \u2113 x P p\u2081 p\u2082\n with cmp\u1d3a-bool \u2113 x | cmp\u1d3a-name\u2208\u03b1 \u2113 x\n... | true         | pf = p\u2081 _ pf\n... | false        | pf = p\u2082 _ pf\n\neasy-cmp\u1d3a : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u00f8 \u2191 \u2113) \u228e Name (\u03b1 +\u1d42 \u2113)\neasy-cmp\u1d3a zero x               = inr x\neasy-cmp\u1d3a (suc _) (zero , _)   = inl (zero , _)\neasy-cmp\u1d3a (suc \u2113) (suc x , pf) = \u228e-map suc\u1d3a\u2191 suc\u1d3a (easy-cmp\u1d3a \u2113 (x , pf))\n\nsyntax easy-cmp\u1d3a \u2113 x = x <\u1d3a-easy \u2113\n\neasy-cmp\u1d3a\u2257cmp\u1d3a : \u2200 {\u03b1} \u2113 \u2192 easy-cmp\u1d3a {\u03b1} \u2113 \u2257 cmp\u1d3a \u2113\neasy-cmp\u1d3a\u2257cmp\u1d3a zero x = \u2261.cong inr (name-injective \u2261.refl)\neasy-cmp\u1d3a\u2257cmp\u1d3a (suc n) (zero , _) = \u2261.refl\neasy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1} (suc n) (suc x , pf) rewrite easy-cmp\u1d3a\u2257cmp\u1d3a n (x , pf) = helper n x pf\n  where\n    helper : \u2200 \u2113 x pf \u2192 \u228e-map suc\u1d3a\u2191 suc\u1d3a (cmp\u1d3a {\u03b1} \u2113 (x , pf)) \u2261 cmp\u1d3a (suc \u2113) (suc x , pf)\n    helper \u2113 x pf = cmp\u1d3a-ind \u2113 (x , pf) (\u03bb r \u2192 \u228e-map suc\u1d3a\u2191 suc\u1d3a r \u2261 cmp\u1d3a (suc \u2113) (suc x , pf))\n                      (\u03bb h pf' \u2192 cmp\u1d3a-ind (suc \u2113) (suc x , pf) (\u03bb r \u2192 inl (suc x , pf') \u2261 r)\n                                         (\u03bb _ _ \u2192 \u2261.cong inl (name-injective \u2261.refl))\n                                         (\u03bb h' \u2192 \u22a5-elim (T'not'\u00ac h' h)))\n                      (\u03bb h pf' \u2192 cmp\u1d3a-ind (suc \u2113) (suc x , pf) (\u03bb r \u2192 inr (suc x , pf') \u2261 r)\n                                         (\u03bb h' \u2192 \u22a5-elim (T'not'\u00ac h h'))\n                                         (\u03bb _ _ \u2192 \u2261.cong inr (name-injective \u2261.refl)))\n\n-- prim (could be defined with (cmp\u1d3a 1)\nprotect\u21911 : \u2200 {\u03b1 \u03b2} \u2192 (Name \u03b1 \u2192 Name \u03b2) \u2192 Name (\u03b1 \u21911) \u2192 Name (\u03b2 \u21911)\nprotect\u21911 f x with name x | {-irr-} (name\u2208\u03b1 x)\n... | zero  | _  = zero , _\n... | suc m | pf = suc\u1d3a\u2191 (f (m , {-cert-} pf))\n\n-- mkName; _,_; name; name\u2208\u03b1 are forbidden from now on (except in proofs).\n\nzero\u1d3a\u21911+ : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 (1 +\u2115 \u2113))\nzero\u1d3a\u21911+ _ = zero\u1d3a\n\ninfix 10 _\u1d3a\n_\u1d3a : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 suc \u2113)\n_\u1d3a {\u03b1} \u2113 = zero\u1d3a +\u1d3a \u2113 \u27e8-because (\u03b1 \u21911 +\u1d42 \u2113) \u2286\u27e8 \u2286-+-\u2191 \u2113 \u27e9\n                                \u03b1 \u21911 \u2191  \u2113   \u2286\u27e8 \u2286-exch-\u2191-\u2191 \u2286-refl 1 \u2113 \u27e9\n                                \u03b1 \u2191 suc \u2113 \u220e -\u27e9\n   where open \u2286-Reasoning\n\n-- Handy name eliminator\nsubtractWith\u1d3a : \u2200 {a \u03b1} {A : Set a} \u2113 \u2192 A \u2192 (Name \u03b1 \u2192 A) \u2192 Name (\u03b1 \u2191 \u2113) \u2192 A\nsubtractWith\u1d3a \u2113 v f x = [ const v , f \u2218\u2032 subtract\u1d3a \u2113 ]\u2032 (x <\u1d3a \u2113)\n\nsubtract\u1d3a? : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192? Name \u03b1\nsubtract\u1d3a? \u2113 = subtractWith\u1d3a \u2113 nothing just\n\ninfixl 4 subtract\u1d3a?\nsyntax subtract\u1d3a? k x = x \u2238\u1d3a? k\n\nprivate\n module CoeExamples where\n  Name\u03b1+\u2192Name\u03b1\u2191 : \u2200 {\u03b1} k \u2192 Name (\u03b1 +\u1d42 k) \u2192 Name (\u03b1 \u2191 k)\n  Name\u03b1+\u2192Name\u03b1\u2191 k = coerce\u1d3a (\u2286-ctx-+\u2191 \u2286-refl k)\n module Unused where\n  open import Data.List using ([]; _\u2237_; replicate; _++_)\n  n\u2208true\u207f\u207a\u00b9 : \u2200 {\u03b1} n \u2192 n \u2208 replicate (suc n) true ++ \u03b1\n  n\u2208true\u207f\u207a\u00b9 zero    = _\n  n\u2208true\u207f\u207a\u00b9 (suc n) = n\u2208true\u207f\u207a\u00b9 n\n\npred\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 +1) \u2192 Name \u03b1\npred\u1d3a = subtract\u1d3a 1\n\npred\u1d3a? : \u2200 {\u03b1} \u2192 Name (\u03b1 \u21911) \u2192? Name \u03b1\npred\u1d3a? = subtract\u1d3a? 1\n\nName\u2192-to-Nm\u27f6 : \u2200 {b\u2081 b\u2082 \u03b1} {B : Setoid b\u2081 b\u2082} \u2192\n                 (Name \u03b1 \u2192 Setoid.Carrier B) \u2192 Nm \u03b1 \u27f6 B\nName\u2192-to-Nm\u27f6 {\u03b1 = \u03b1} {B} f = record { _\u27e8$\u27e9_ = f; cong = cong\u2032 }\n  where\n  open Setoid B renaming (_\u2248_ to _\u2248B_)\n\n  cong\u2261 : \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2248B f y\n  cong\u2261 \u2261.refl = Setoid.refl B\n\n  cong\u2032 : \u2200 {x y} \u2192 x \u2261\u1d3a y \u2192 f x \u2248B f y\n  cong\u2032 = cong\u2261 \u2218 Equality.to-propositional \u2261\u1d3a-equality\n  -- cong\u2032 = Setoid.reflexive \u2218 Equality.to-propositional \u2261\u1d3a-equality\n\n\u00acName : \u2200 {\u03b1} \u2192 \u03b1 \u2286 \u00f8 \u2192 \u00ac(Name \u03b1)\n\u00acName \u03b1\u2286\u00f8 = \u00acName\u00f8 \u2218 coerce\u1d3a \u03b1\u2286\u00f8\n\nName-elim : \u2200 {a} {A : Set a} {\u03b1} (\u03b1\u2286\u00f8 : \u03b1 \u2286 \u00f8) (x : Name \u03b1) \u2192 A\nName-elim x y with \u00acName x y\n... | ()\n\nName\u00f8-elim : \u2200 {a} {A : Set a} \u2192 Name \u00f8 \u2192 A\nName\u00f8-elim = Name-elim \u2286-refl\n\n\u00acName\u00f8+ : \u2200 k \u2192 \u00ac(Name (\u00f8 +\u1d42 k))\n\u00acName\u00f8+ = \u00acName \u2218 \u2286-\u00f8+\n\nName\u00f8+-elim : \u2200 {a} {A : Set a} k \u2192 Name (\u00f8 +\u1d42 k) \u2192 A\nName\u00f8+-elim = Name-elim \u2218 \u2286-\u00f8+\n\nName\u00f8\u2191\u2192Name\u03b1\u2191 : \u2200 {\u2113} \u2192 Name (\u00f8 \u2191 \u2113) \u2192 \u2200 {\u03b1} \u2192 Name (\u03b1 \u2191 \u2113)\nName\u00f8\u2191\u2192Name\u03b1\u2191 {\u2113} x = x \u27e8-because \u2286-cong-\u2191 \u2286-\u00f8 \u2113 -\u27e9\n\nis0? : \u2200 {\u03b1} \u2192 Name (\u03b1 \u21911) \u2192 Bool\nis0? = cmp\u1d3a-bool 1\n-- is0? = [ const true , const false ]\u2032 (x <\u1d3a 1)\n\nshift\u2115 : \u2200 (\u2113 k n : \u2115) \u2192 \u2115\nshift\u2115 \u2113 k n = if suc n <= \u2113 then n else n +\u2115 k\n\nshift\u1d3a : \u2200 {\u03b1 \u03b2} \u2113 k \u2192 \u03b1 +\u1d42 k \u2286 \u03b2 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u03b2 \u2191 \u2113)\nshift\u1d3a \u2113 k pf n\n  with n <\u1d3a \u2113\n...  | inl n\u2032 = n\u2032       \u27e8-because \u2286-cong-\u2191 \u2286-\u00f8 \u2113 -\u27e9\n...  | inr n\u2032 = n\u2032 +\u1d3a k  \u27e8-because \u2286-trans (\u2286-exch-+-+ \u2286-refl \u2113 k) (\u2286-ctx-+\u2191 pf \u2113) -\u27e9\n\nmodule Singletons where\n\n  SWorld : \u2115 \u2192 World\n  SWorld n = \u00f8 \u21911 +\u1d42 n\n\n  Name\u02e2 : \u2115 \u2192 Set\n  Name\u02e2 = Name \u2218 SWorld\n\n  _\u02e2 : \u2200 n \u2192 Name\u02e2 n\n  _\u02e2 n = zero\u1d3a +\u1d3a n\n\n  add\u02e2 : \u2200 {n} k \u2192 Name\u02e2 n \u2192 Name\u02e2 (k +\u2115 n)\n  add\u02e2 {n} k x = add\u1d3a k x  \u27e8-because \u2286-assoc-+ \u2286-refl n k -\u27e9\n\n  subtract\u02e2 : \u2200 {n} k \u2192 Name\u02e2 (k +\u2115 n) \u2192 Name\u02e2 n\n  subtract\u02e2 {n} k x = subtract\u1d3a k (x \u27e8-because \u2286-assoc-+\u2032 \u2286-refl n k -\u27e9)\n\n{-\nshift\u1d3a-1-1+-\n  shift\u1d3a 1 (suc k) pf n \u2261 shift\u1d3a 1 k ? (shift\u1d3a 1 1 ? n)\n-}\n\n-- Properties\n\ncoerce\u1d3a-names : \u2200 {\u03b1 \u03b2} (\u03b1\u2286\u03b2 : \u03b1 \u2286 \u03b2) \u2192 name \u2218 coerce\u1d3a \u03b1\u2286\u03b2 \u2257 name\ncoerce\u1d3a-names _ _ = \u2261.refl\n\n{-\n<\u1d3a-coerce : \u2200 {\u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113)) \u2192 [ coerce\u1d3a (\u2286-cong-\u2191 \u2286-\u00f8 \u2113) , coerce\u1d3a (\u2286-ctx-+\u2191 \u2286-refl \u2113) ]\u2032 (x <\u1d3a \u2113) \u2261 x\n<\u1d3a-coerce \u2113 x with x <\u1d3a-bool \u2113 | x <\u1d3a-name\u2208\u03b1 \u2113\n... | true  | _ = name-injective \u2261.refl\n... | false | _ = name-injective \u2261.refl\n-}\n\n<\u1d3a-names : \u2200 {\u03b1 \u2113} {x : Name (\u03b1 \u2191 \u2113)} \u2192 [ name , name ]\u2032 (x <\u1d3a \u2113) \u2261 name x\n<\u1d3a-names {\u03b1} {\u2113} {x} with x <\u1d3a-bool \u2113 | x <\u1d3a-name\u2208\u03b1 \u2113\n... | true  | _ = \u2261.refl\n... | false | _ = \u2261.refl\n\nDec-\u2261\u1d3a\u2192\u2261 : \u2200 {\u03b1} \u2192 Decidable (_\u2261\u1d3a_ {\u03b1}) \u2192 Decidable _\u2261_\nDec-\u2261\u1d3a\u2192\u2261 _\u225f_ x y with x \u225f y\n... | yes x\u2261y = yes (\u2261\u1d3a\u21d2\u2261 x\u2261y)\n... | no  x\u2262y = no  (x\u2262y \u2218\u2032 \u2115s.reflexive \u2218\u2032 \u2261.cong name)\n\nshift\u1d3a-\u2113-0\u2261coerce\u1d3a : \u2200 {\u03b1 \u03b2} \u2113 (pf : \u03b1 \u2286 \u03b2) n \u2192 name (shift\u1d3a \u2113 0 pf n) \u2261 name n\nshift\u1d3a-\u2113-0\u2261coerce\u1d3a \u2113 pf x with x <\u1d3a-bool \u2113 | x <\u1d3a-name\u2208\u03b1 \u2113\n... | true  | _ = \u2261.refl\n... | false | _ = \u2261.refl\n\npred-shift\u2115-comm : \u2200 k x \u2192 pred (shift\u2115 1 k (suc x)) \u2261 x +\u2115 k\npred-shift\u2115-comm k x = \u2261.refl\n\n.shift11-pred\u1d3a?-comm :\n  \u2200 {\u03b1 \u03b2} (pf : \u03b1 +1 \u2286 \u03b2) (x : Name (\u03b1 \u21911)) \u2192\n  pred\u1d3a? (shift\u1d3a 1 1 pf x) \u2261 (coerce\u1d3a pf \u2218 suc\u1d3a <$> pred\u1d3a? x)\nshift11-pred\u1d3a?-comm {\u03b1} pf x\n   = <$>-injective\u2081 name-injective (\u2261.trans (helper (name x) {name\u2208\u03b1 x} (suc (name x) \u2264?\u2115 1)) (<$>-assoc (pred\u1d3a? x)))\n  where\n    -- getting the name unpacked (and the Dec), works around an Agda bug in 2.2.8.\n    .helper : \u2200 n {pn : n \u2208 \u03b1 \u21911} \u2192 Dec (n <\u2115 1) \u2192 let x = n , pn in\n             (name <$> (pred\u1d3a? (shift\u1d3a 1 1 pf x))) \u2261 (name \u2218 (coerce\u1d3a pf \u2218 suc\u1d3a) <$> pred\u1d3a? x)\n    helper zero    (no \u00acp)         = \u22a5-elim (\u00acp (s\u2264s z\u2264n))\n    helper .0      (yes (s\u2264s z\u2264n)) = \u2261.refl\n    helper (suc _) (no _)          = \u2261.refl\n\n1+ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name (\u03b1 +1)\n1+ x = x +\u1d3a 1\n\nmodule Name\u2191\u21d4Fin where\n  open import Data.Fin as Fin\n\n  \u21d2 : \u2200 n \u2192 Name (\u00f8 \u2191 n) \u2192 Fin n\n  \u21d2 zero    = Name\u00f8-elim\n  \u21d2 (suc n) = maybe (suc \u2218\u2032 \u21d2 n) zero \u2218\u2032 pred\u1d3a?\n\n  \u21d0 : \u2200 {\u03b1 n} \u2192 Fin n \u2192 Name (\u03b1 \u2191 n)\n  \u21d0 (zero {n})  = zero\u1d3a\u21911+ n\n  \u21d0 (suc i)     = suc\u1d3a\u2191 (\u21d0 i)\n\n{-\n  open Singletons\n  open \u2286-Reasoning\n  f : \u2200 {\u03b1 n} \u2192 Fin n \u2192 Name (\u03b1 \u2191 n)\n  f {\u03b1} {zero} ()\n  f {\u03b1} {suc n} x = coerce\u1d3a c (sName x\u2032) where\n    x\u2032 = Fin.to\u2115 x\n    c\u2032 = \u00f8 \u21911 +[ n \u2238\u2115 x\u2032 ]   \u2286\u27e8 {!!} \u27e9\n         (\u00f8 +[ n \u2238\u2115 x\u2032 ]) \u21911 \u2286\u27e8 \u2286-cong-\u21911 (\u03b1+\u2286\u00f8\u2192\u03b1\u2286\u00f8 _ \u2286-refl) \u27e9\n         \u00f8 \u21911                 \u2286\u27e8 \u2286-cong-\u21911 \u2286-\u00f8 \u27e9\n         \u03b1 \u21911 \u220e\n    c : SWorld x\u2032 \u2286 \u03b1 \u2191 (suc n)\n    c = SWorld x\u2032                \u2286\u27e8 refl \u27e9\n        \u00f8 \u21911 +[ x\u2032 ]             \u2286\u27e8 {!!} \u27e9\n        \u00f8 \u21911 +[ n \u2238\u2115 x\u2032 ] +[ n ] \u2286\u27e8 \u2286-ctx-+\u2191 c\u2032 n \u27e9\n        \u03b1 \u21911 \u2191 n                 \u2286\u27e8 \u2286-exch-\u2191-\u2191 \u2286-refl 1 n \u27e9\n        \u03b1 \u2191 suc n \u220e\n-}\n\n  -- \u21d4 : \u2200 n \u2192 Name (\u00f8 \u2191 n) \u21d4 Fin n\n  -- \u21d4 n = ?\n\n{-\nmodule Bug where\n  record Fin\u2032 (n : \u2115) : Set where\n    constructor mkFin\n    field\n      x    : \u2115\n      .x<n : x <\u2115 n\n  raise : \u2200 {n m} \u2192 Fin\u2032 n \u2192 n \u2264\u2115 m \u2192 Fin\u2032 m\n  raise (mkFin x x<n) n\u2264m = mkFin x (\u2115\u2264.trans x<n n\u2264m)\n  raise-assoc : \u2200 {n m o} p (q : m \u2264\u2115 o) (x : Fin\u2032 n) \u2192 (raise (raise x p) q) \u2261 (raise x (\u2115\u2264.trans p q))\n  raise-assoc p q (mkFin x x<n) with raise-assoc p q (mkFin x x<n)\n  ... | refl = ?\n-}\n\n{-\nmodule Fin\u2032 where\n  record Fin\u2032 (n : \u2115) : Set where\n    constructor mkFin\n    field\n      x    : \u2115\n      .x<n : x <\u2115 n\n  zeroF : \u2200 {n} \u2192 Fin\u2032 (suc n)\n  zeroF = mkFin 0 (s\u2264s z\u2264n)\n  sucF : \u2200 {n} \u2192 Fin\u2032 n \u2192 Fin\u2032 (suc n)\n  sucF (mkFin x x<n) = mkFin (suc x) (s\u2264s x<n)\n\n  raise : \u2200 {n} (x : Fin\u2032 n) k \u2192 Fin\u2032 (n +\u2115 k)\n  raise (mkFin x x<n) k = mkFin x (m\u2264n\u2192m\u2264n+k x<n)\n    where m\u2264n\u2192m\u2264n+k : \u2200 {m n k} \u2192 m \u2264\u2115 n \u2192 m \u2264\u2115 n +\u2115 k\n          m\u2264n\u2192m\u2264n+k z\u2264n        = z\u2264n\n          m\u2264n\u2192m\u2264n+k (s\u2264s m\u2264n)  = s\u2264s (m\u2264n\u2192m\u2264n+k m\u2264n)\n-}\n\n{-\n(x : \u03b1 +\u1d42 n) \u2192 x \u2265 n\n\n(x : \u03b1 +\u1d42 suc n) \u2192 x \u2262 zero\u1d3a\n-}\n\nmodule \u2115\u21d2Name`\u03b1` where\n  open WorldSyntax\n  open WorldSyntax.El listBoolWorlds\n  \u21d2 : \u2200 `\u03b1` x \u2192 x \u2208 El `\u03b1` \u2192 Name (El `\u03b1`)\n  \u21d2 (`\u03b1` `\u21911`) zero    _     = zero , _\n  \u21d2 (`\u03b1` `\u21911`) (suc i) pf    = suc\u1d3a\u2191 (\u21d2 `\u03b1` i pf)\n  \u21d2 (`\u03b1` `+1`) zero    ()\n  \u21d2 (`\u03b1` `+1`) (suc i) pf    = suc\u1d3a (\u21d2 `\u03b1` i pf)\n  \u21d2 `\u00f8`        _       ()\n\nmodule Name`\u03b1`\u21d4Fin where\n  open import Data.Fin as F renaming (_+_ to _+F_)\n  open WorldSyntax\n  open WorldSyntax.El listBoolWorlds\n  open WorldSyntax.Properties listBoolWorlds\n  mutual\n\n    \u21d2+1 : \u2200 `\u03b1` \u2192 Name (El `\u03b1` +1) \u2192 Fin (suc (bnd `\u03b1`))\n    \u21d2+1 `\u03b1` = F.raise 1 \u2218 \u21d2 `\u03b1` \u2218 pred\u1d3a\n\n    \u21d2 : \u2200 `\u03b1` \u2192 Name (El `\u03b1`) \u2192 Fin (bnd `\u03b1`)\n    \u21d2 (`\u03b1` `\u21911`)  = [ inject+ _ \u2218 Name\u2191\u21d4Fin.\u21d2 1 , \u21d2+1 `\u03b1` ]\u2032 \u2218 cmp\u1d3a 1\n    \u21d2 (`\u03b1` `+1`)  = \u21d2+1 `\u03b1`\n    \u21d2 `\u00f8`         = Name\u00f8-elim\n\n  \u21d0\u2081 : \u2200 {n} `\u03b1` \u2192 (x : Fin n) \u2192 to\u2115 x \u2208 El `\u03b1` \u2192 Name (El `\u03b1`)\n  \u21d0\u2081 `\u03b1` = \u2115\u21d2Name`\u03b1`.\u21d2 `\u03b1` \u2218 to\u2115\n\n  x\u2208\u00f8\u2191x : \u2200 {n} (x : Fin n) \u2192 to\u2115 x \u2208 (\u00f8 \u2191 n)\n  x\u2208\u00f8\u2191x zero    = _\n  x\u2208\u00f8\u2191x (suc n) = x\u2208\u00f8\u2191x n\n\n  \u21d0\u2082 : \u2200 {n} \u2192 (x : Fin n) \u2192 Name (\u00f8 \u2191 n)\n  \u21d0\u2082 {n} x with x\u2208\u00f8\u2191x x\n  ...         | y       rewrite comm-El-\u2191 `\u00f8` n = \u21d0\u2081 (`\u00f8` `\u2191` n) x y\n\n  {- I would have prefered a local `where` here\n  \u21d0\u2082 : \u2200 {n} \u2192 (x : Fin n) \u2192 Name (\u00f8 \u2191 n)\n  \u21d0\u2082 {n} x where y = x\u2208\u00f8\u2191x x rewrite comm-El-\u2191 `\u00f8` n = \u21d0\u2081 (`\u00f8` `\u2191` n) x y\n  -}\n\nmodule WellFormedWorld where\n  data WF : World \u2192 Set where\n    `\u00f8 : WF \u00f8\n    _`+_ : \u2200 {\u03b1} (`\u03b1 : WF \u03b1) k \u2192 WF (\u03b1 +\u1d42 k)\n    _`\u2191_ : \u2200 {\u03b1} (`\u03b1 : WF \u03b1) k \u2192 WF (\u03b1 \u2191 k)\n\n_+\u2191\u1d3a_ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \u2200 \u2113 \u2192 Name (\u03b1 \u2191 \u2113)\n_+\u2191\u1d3a_ x \u2113 = coerce\u1d3a (\u2286-ctx-+\u2191 \u2286-refl \u2113) (x +\u1d3a \u2113)\n\nsubtract\u1d3a\u2191 : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u00f8 \u2191 \u2113) \u228e Name \u03b1\nsubtract\u1d3a\u2191 \u2113 x with x <\u1d3a \u2113\n... | inl x\u2032 = inl x\u2032\n... | inr x\u2032 = inr (x\u2032 \u2238\u1d3a \u2113)\n\ninfixl 4 subtract\u1d3a\u2191\nsyntax subtract\u1d3a\u2191 \u2113 x = x -\u2191\u1d3a \u2113\n\ndata WithLim (R : (\u03b1 \u03b2 : World) \u2192 Set) : (\u03b1 \u03b2 : World) \u2192 Set where\n  mkWithLim : \u2200 {\u03b1\u2080 \u03b2\u2080} \u2113 (R\u03b1\u2080\u03b2\u2080 : R \u03b1\u2080 \u03b2\u2080) \u2192 WithLim R (\u03b1\u2080 \u2191 \u2113) (\u03b2\u2080 \u2191 \u2113)\n\nwithLim : \u2200 {R \u03b1 \u03b2} \u2192 R \u03b1 \u03b2 \u2192 WithLim R \u03b1 \u03b2\nwithLim = mkWithLim 0\n\n-- functions known to be ``bad'' and that we don't want to export\nprivate\n module Bad where\n  Name2\u2115 : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \u2115\n  Name2\u2115 = name\n  isZero? : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Bool\n  isZero? (zero  , _) = true\n  isZero? (suc _ , _) = false\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n-- vim: iskeyword=a-z,A-Z,48-57,-,+,',#,\/,~,],[,=,_,`,\u2286\nmodule NaPa where\n\nimport Level as L\n-- open import Irrelevance\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2257-Reasoning; module \u2261-Reasoning)\nopen import Data.Nat.NP as Nat using (\u2115; zero; suc; s\u2264s; z\u2264n; \u2264-pred; pred; _<=_; module <=)\n                            renaming (_+_ to _+\u2115_ ; _\u2238_ to _\u2238\u2115_ ; _==_ to _==\u2115_; \u00ac\u2264 to \u00ac\u2264\u2115;\n                                      _<_ to _<\u2115_ ; _\u2264_ to _\u2264\u2115_; _\u2265_ to _\u2265\u2115_; _\u2264?_ to _\u2264?\u2115_)\nopen import Data.Nat.Logical using (\u27e6\u2115\u27e7; zero; suc; \u27e6\u2115\u27e7-setoid; \u27e6\u2115\u27e7-equality; \u27e6\u2115\u27e7-cong)\n                          renaming (_\u225f_ to _\u225f\u27e6\u2115\u27e7_)\nimport Data.Nat.Properties as Nat\nopen import Function\nopen import Function.Equality as \u27f6\u2261 using (_\u27f6_; _\u27e8$\u27e9_)\nopen import Data.Sum.NP as Sum using (_\u228e_; inl; inr; [_,_]\u2032) renaming (map to \u228e-map)\nopen import Data.Bool using (Bool; true; false; if_then_else_; T; not)\nopen import Data.Two renaming (\u2713-not-\u00ac to T'not'\u00ac)\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product.NP using (_,_)\nopen import Data.Maybe.NP using (Maybe; nothing; just; maybe; _\u2192?_)\nopen import Relation.Binary.NP as Bin\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.On as On\nopen import NomPa.Worlds\nopen import NomPa.Worlds.Syntax as WorldSyntax\nopen import NomPa.Subtyping\nopen ListBoolWorlds public hiding (_\u2286_) renaming (_\u2286\u2032_ to _\u2286_)\nimport Category.Monad as Cat\nopen Cat.RawMonad (Data.Maybe.NP.monad {L.zero})\nopen Data.Maybe.NP.FunctorLemmas\n\nprivate\n  module \u2115s = Setoid \u27e6\u2115\u27e7-setoid\n  module \u2115e = Equality \u27e6\u2115\u27e7-equality\n\ninfixr 4 _,_\nrecord Name \u03b1 : Set where\n  constructor _,_\n  field\n    name    : \u2115\n    -- .poofname\u2208\u03b1 : name \u2208 \u03b1\n    name\u2208\u03b1 : name \u2208 \u03b1\n\nopen Name public\n{-\ndata Name \u03b1 : Set where\n  _,_ : (x : \u2115) \u2192 .(x \u2208 \u03b1) \u2192 Name \u03b1\nname : \u2200 {\u03b1} (x : Name \u03b1) \u2192 \u2115\nname (x , _) = x\n\n.name\u2208\u03b1 : \u2200 {\u03b1} (x : Name \u03b1) \u2192 name x \u2208 \u03b1\nname\u2208\u03b1 (_ , pf) = pf\n-}\n\n--mkName : \u2200 {\u03b1} (x : \u2115) \u2192 Irr (x \u2208 \u03b1) \u2192 Name \u03b1\n--mkName x (irr pf) = x , pf\nmkName : \u2200 {\u03b1} (x : \u2115) \u2192 x \u2208 \u03b1 \u2192 Name \u03b1\nmkName x pf = x , pf\n\n--name\u2208\u03b1 : \u2200 {\u03b1} (x : Name \u03b1) \u2192 Irr (name x \u2208 \u03b1)\n--name\u2208\u03b1 (_ , pf) = irr pf\n\ninfix 4 _\u2261\u1d3a_ _\u225f\u1d3a_ _==\u1d3a_\n\n-- prim\n_\u2261\u1d3a_ : \u2200 {\u03b1} (x y : Name \u03b1) \u2192 Set\n_\u2261\u1d3a_ = \u27e6\u2115\u27e7 on name\n\n-- prim\n_\u225f\u1d3a_ : \u2200 {\u03b1} \u2192 Decidable (_\u2261\u1d3a_ {\u03b1})\nx \u225f\u1d3a y = name x \u225f\u27e6\u2115\u27e7 name y\n\n-- prim\n_==\u1d3a_ : \u2200 {\u03b1} (x y : Name \u03b1) \u2192 Bool\n_==\u1d3a_ x y = name x ==\u2115 name y\n\nname-injective : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 name x \u2261 name y \u2192 x \u2261 y\nname-injective {\u03b1} {_ , p\u2081} {a , p\u2082} eq rewrite eq\n--= \u2261.refl {_} {_} {a , p\u2082}\n  = \u2261.cong (_,_ a) (\u2208-uniq \u03b1 p\u2081 p\u2082)\n\nNm : World \u2192 Setoid L.zero L.zero\nNm \u03b1 = On.setoid \u27e6\u2115\u27e7-setoid (name {\u03b1})\n\n\u2261\u1d3a-equality : \u2200 {\u03b1} \u2192 Equality (_\u2261\u1d3a_ {\u03b1})\n\u2261\u1d3a-equality {\u03b1}\n   = record { isEquivalence = Setoid.isEquivalence (Nm \u03b1)\n              -- I would prefered not going to PropEq.\u2261\n            ; subst = \u03bb P \u2192 \u2261.tr P \u2218 name-injective \u2218 Equality.to-propositional \u27e6\u2115\u27e7-equality }\n\nmodule NmEq {\u03b1} = Equality (\u2261\u1d3a-equality {\u03b1})\n\n\u2261\u1d3a\u21d2\u2261 : \u2200 {\u03b1} \u2192 _\u2261\u1d3a_ {\u03b1} \u21d2 _\u2261_\n\u2261\u1d3a\u21d2\u2261 = NmEq.to-propositional\n\n\u2261\u1d3a-cong : \u2200 {\u03b1 \u03b2} (f : Name \u03b1 \u2192 Name \u03b2) \u2192 _\u2261\u1d3a_ =[ f ]\u21d2 _\u2261\u1d3a_\n\u2261\u1d3a-cong f = NmEq.reflexive \u2218 \u2261.cong f \u2218 \u2261\u1d3a\u21d2\u2261\n\n\u2261\u1d3a-\u27e6\u2115\u27e7-cong : \u2200 {\u03b1} (f : Name \u03b1 \u2192 \u2115) \u2192 _\u2261\u1d3a_ =[ f ]\u21d2 \u27e6\u2115\u27e7\n\u2261\u1d3a-\u27e6\u2115\u27e7-cong f = \u2115e.reflexive \u2218 \u2261.cong f \u2218 \u2261\u1d3a\u21d2\u2261\n\n\u2261-\u27e6\u2115\u27e7-cong : \u2200 {a} {A : Set a} (f : A \u2192 \u2115) \u2192 _\u2261_ =[ f ]\u21d2 \u27e6\u2115\u27e7\n\u2261-\u27e6\u2115\u27e7-cong f = \u2115e.reflexive \u2218 \u2261.cong f\n\n\u2261\u1d3a-\u2261-cong : \u2200 {a \u03b1} {A : Set a} (f : Name \u03b1 \u2192 A) \u2192 _\u2261\u1d3a_ =[ f ]\u21d2 _\u2261_\n\u2261\u1d3a-\u2261-cong f = \u2261.cong f \u2218 \u2261\u1d3a\u21d2\u2261\n\n\u27e6\u2115\u27e7-\u2261\u1d3a-cong : \u2200 {\u03b1} (f : \u2115 \u2192 Name \u03b1) \u2192 \u27e6\u2115\u27e7 =[ f ]\u21d2 _\u2261\u1d3a_\n\u27e6\u2115\u27e7-\u2261\u1d3a-cong f = NmEq.reflexive \u2218 \u2261.cong f \u2218 \u2115e.to-propositional\n\n\u27e6\u2115\u27e7-\u2261-cong : \u2200 {a} {A : Set a} (f : \u2115 \u2192 A) \u2192 \u27e6\u2115\u27e7 =[ f ]\u21d2 _\u2261_\n\u27e6\u2115\u27e7-\u2261-cong f = \u2261.cong f \u2218 \u2115e.to-propositional\n\nmodule \u2261\u1d3a-Reasoning {\u03b1} = Bin.Setoid-Reasoning (Nm \u03b1)\n\nopen WorldSymantics listBoolWorlds public\nopen WorldOps listBoolWorlds public\nopen WorldOps listBoolWorlds using ()\nopen NomPa.Subtyping.\u2286-Pack \u2286\u2032\u1d47-pack public hiding (_\u2286_; _\u2288_)\nopen NomPa.Subtyping.SyntacticOnBools \u2286\u2032\u1d47-pack public hiding (minimalSymantics)\n\n-- Here is a first set of core primitives easy to define\n\n-- prim\nzero\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 \u21911)\nzero\u1d3a = 0 , _\n\n-- prim\n\u00acName\u00f8 : \u00ac(Name \u00f8)\n\u00acName\u00f8 (_ , ())\n\n-- prim\nadd\u1d3a : \u2200 {\u03b1} k (x : Name \u03b1) \u2192 Name (\u03b1 +\u1d42 k)\nadd\u1d3a {\u03b1} k x = k +\u2115 name x , proof k where\n  proof : \u2200 k \u2192 k +\u2115 name x \u2208 \u03b1 +\u1d42 k\n  proof zero    = name\u2208\u03b1 x\n  proof (suc k) = proof k\n\ninfixl 6 add\u1d3a\nsyntax add\u1d3a k x = x +\u1d3a k\n\n-- prim\nsubtract\u1d3a : \u2200 {\u03b1} k (x : Name (\u03b1 +\u1d42 k)) \u2192 Name \u03b1\nsubtract\u1d3a k x = name x \u2238\u2115 k , proof (name x) k (name\u2208\u03b1 x) where\n  proof : \u2200 {\u03b1} x k \u2192 (x \u2208 \u03b1 +\u1d42 k) \u2192 (x \u2238\u2115 k \u2208 \u03b1)\n  proof x       zero = id\n  proof zero    (suc k) = \u03bb()\n  proof (suc x) (suc k) = proof x k\n\ninfixl 4 subtract\u1d3a\nsyntax subtract\u1d3a k x = x \u2238\u1d3a k\n\n-- prim\ncoerce\u1d3a : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 \u2286 \u03b2 \u2192 Name \u03b1 \u2192 Name \u03b2\ncoerce\u1d3a \u03b1\u2286\u03b2 x = name x , coe \u03b1\u2286\u03b2 (name x) (name\u2208\u03b1 x)\n\n-- Then we define some convenience functions on top of them.\n\ninfix 0 _\u27e8-because_-\u27e9\n_\u27e8-because_-\u27e9 : \u2200 {\u03b1 \u03b2} \u2192 Name \u03b1 \u2192 \u03b1 \u2286 \u03b2 \u2192 Name \u03b2\n_\u27e8-because_-\u27e9 n pf = coerce\u1d3a pf n\n\nadd\u1d3a\u2191 : \u2200 {\u03b1} \u2113 \u2192 Name \u03b1 \u2192 Name (\u03b1 \u2191 \u2113)\nadd\u1d3a\u2191 \u2113 x = add\u1d3a \u2113 x  \u27e8-because \u2286-+-\u2191 \u2113 -\u27e9\n\nsuc\u1d3a : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name (\u03b1 +1)\nsuc\u1d3a = add\u1d3a 1\n\nsuc\u1d3a\u2191 : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name (\u03b1 \u21911)\nsuc\u1d3a\u2191 = add\u1d3a\u2191 1\n\n-- Then comes <\u1d3a and the necessary intermediate definitions\n\nprivate\n  lem-<\u2113 : \u2200 {\u03b1 \u03b2 x \u2113} \u2192 x \u2264\u2115 \u2113 \u2192 x \u2208 (\u03b1 \u2191 suc \u2113) \u2192 x \u2208 (\u03b2 \u2191 suc \u2113)\n  lem-<\u2113 z\u2264n       = _\n  lem-<\u2113 (s\u2264s m\u2264n) = lem-<\u2113 m\u2264n\n\n  lem-\u2265\u2113 : \u2200 {\u03b1 x \u2113} \u2192 x \u2265\u2115 \u2113 \u2192 x \u2208 \u03b1 \u2191 \u2113 \u2192 x \u2208 \u03b1 +\u1d42 \u2113\n  lem-\u2265\u2113 z\u2264n        = id\n  lem-\u2265\u2113 (s\u2264s m\u2264n)  = lem-\u2265\u2113 m\u2264n\n\n  lem-<\u2113? : \u2200 {\u03b1 \u03b2 n} \u2113 \u2192 n \u2208 \u03b2 \u2191 \u2113 \u2192 n \u2208 (if suc n <= \u2113 then \u03b1 \u2191 \u2113 else \u03b2 +\u1d42 \u2113)\n  lem-<\u2113? {n = n} \u2113 with suc n <= \u2113 | <=.sound (suc n) \u2113 | <=.complete {suc n} {\u2113}\n  ... | true  | n<\u2113 | _    = lem-<\u2113 (n<\u2113 _)\n  ... | false | _   | \u00acn<\u2113 = lem-\u2265\u2113 (\u00ac\u2264\u2115 \u00acn<\u2113)\n\n-- prim\ncmp\u1d3a-bool : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Bool\ncmp\u1d3a-bool \u2113 x = suc (name x) <= \u2113\n\nsyntax cmp\u1d3a-bool k x = x <\u1d3a-bool k\n\n-- prim\ncmp\u1d3a-name : \u2200 {\u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113)) \u2192 Name (if cmp\u1d3a-bool \u2113 x then \u00f8 \u2191 \u2113 else \u03b1 +\u1d42 \u2113)\ncmp\u1d3a-name \u2113 x = name x , lem-<\u2113? \u2113 (name\u2208\u03b1 x)\n\nsyntax cmp\u1d3a-name \u2113 x = x <\u1d3a-name \u2113\n\n-- prim\ncmp\u1d3a-name\u2208\u03b1 : \u2200 {\u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113)) \u2192 name x \u2208 (if cmp\u1d3a-bool \u2113 x then \u00f8 \u2191 \u2113 else \u03b1 +\u1d42 \u2113)\ncmp\u1d3a-name\u2208\u03b1 \u2113 x = lem-<\u2113? \u2113 (name\u2208\u03b1 x)\n\nsyntax cmp\u1d3a-name\u2208\u03b1 \u2113 x = x <\u1d3a-name\u2208\u03b1 \u2113\n\n-- prim\ncmp\u1d3a : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u00f8 \u2191 \u2113) \u228e Name (\u03b1 +\u1d42 \u2113)\ncmp\u1d3a \u2113 x with cmp\u1d3a-bool \u2113 x | cmp\u1d3a-name\u2208\u03b1 \u2113 x\n... | true  | pf = inl (name x , pf)\n... | false | pf = inr (name x , pf)\n\ninfix 4 cmp\u1d3a\nsyntax cmp\u1d3a k x = x <\u1d3a k\n\ncmp\u1d3a-ind : \u2200 {a \u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113))\n            (P : (Name (\u00f8 \u2191 \u2113) \u228e Name (\u03b1 +\u1d42 \u2113)) \u2192 Set a)\n            (Pinl : T (cmp\u1d3a-bool \u2113 x) \u2192 \u2200 pf \u2192 P (inl (name x , pf)))\n            (Pinr : T (not (cmp\u1d3a-bool \u2113 x)) \u2192 \u2200 pf \u2192 P (inr (name x , pf)))\n          \u2192 P (cmp\u1d3a \u2113 x)\ncmp\u1d3a-ind \u2113 x P p\u2081 p\u2082\n with cmp\u1d3a-bool \u2113 x | cmp\u1d3a-name\u2208\u03b1 \u2113 x\n... | true         | pf = p\u2081 _ pf\n... | false        | pf = p\u2082 _ pf\n\neasy-cmp\u1d3a : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u00f8 \u2191 \u2113) \u228e Name (\u03b1 +\u1d42 \u2113)\neasy-cmp\u1d3a zero x               = inr x\neasy-cmp\u1d3a (suc _) (zero , _)   = inl (zero , _)\neasy-cmp\u1d3a (suc \u2113) (suc x , pf) = \u228e-map suc\u1d3a\u2191 suc\u1d3a (easy-cmp\u1d3a \u2113 (x , pf))\n\nsyntax easy-cmp\u1d3a \u2113 x = x <\u1d3a-easy \u2113\n\neasy-cmp\u1d3a\u2257cmp\u1d3a : \u2200 {\u03b1} \u2113 \u2192 easy-cmp\u1d3a {\u03b1} \u2113 \u2257 cmp\u1d3a \u2113\neasy-cmp\u1d3a\u2257cmp\u1d3a zero x = \u2261.cong inr (name-injective \u2261.refl)\neasy-cmp\u1d3a\u2257cmp\u1d3a (suc n) (zero , _) = \u2261.refl\neasy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1} (suc n) (suc x , pf) rewrite easy-cmp\u1d3a\u2257cmp\u1d3a n (x , pf) = helper n x pf\n  where\n    helper : \u2200 \u2113 x pf \u2192 \u228e-map suc\u1d3a\u2191 suc\u1d3a (cmp\u1d3a {\u03b1} \u2113 (x , pf)) \u2261 cmp\u1d3a (suc \u2113) (suc x , pf)\n    helper \u2113 x pf = cmp\u1d3a-ind \u2113 (x , pf) (\u03bb r \u2192 \u228e-map suc\u1d3a\u2191 suc\u1d3a r \u2261 cmp\u1d3a (suc \u2113) (suc x , pf))\n                      (\u03bb h pf' \u2192 cmp\u1d3a-ind (suc \u2113) (suc x , pf) (\u03bb r \u2192 inl (suc x , pf') \u2261 r)\n                                         (\u03bb _ _ \u2192 \u2261.cong inl (name-injective \u2261.refl))\n                                         (\u03bb h' \u2192 \u22a5-elim (T'not'\u00ac h' h)))\n                      (\u03bb h pf' \u2192 cmp\u1d3a-ind (suc \u2113) (suc x , pf) (\u03bb r \u2192 inr (suc x , pf') \u2261 r)\n                                         (\u03bb h' \u2192 \u22a5-elim (T'not'\u00ac h h'))\n                                         (\u03bb _ _ \u2192 \u2261.cong inr (name-injective \u2261.refl)))\n\n-- prim (could be defined with (cmp\u1d3a 1)\nprotect\u21911 : \u2200 {\u03b1 \u03b2} \u2192 (Name \u03b1 \u2192 Name \u03b2) \u2192 Name (\u03b1 \u21911) \u2192 Name (\u03b2 \u21911)\nprotect\u21911 f x with name x | {-irr-} (name\u2208\u03b1 x)\n... | zero  | _  = zero , _\n... | suc m | pf = suc\u1d3a\u2191 (f (m , {-cert-} pf))\n\n-- mkName; _,_; name; name\u2208\u03b1 are forbidden from now on (except in proofs).\n\nzero\u1d3a\u21911+ : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 (1 +\u2115 \u2113))\nzero\u1d3a\u21911+ _ = zero\u1d3a\n\ninfix 10 _\u1d3a\n_\u1d3a : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 suc \u2113)\n_\u1d3a {\u03b1} \u2113 = zero\u1d3a +\u1d3a \u2113 \u27e8-because (\u03b1 \u21911 +\u1d42 \u2113) \u2286\u27e8 \u2286-+-\u2191 \u2113 \u27e9\n                                \u03b1 \u21911 \u2191  \u2113   \u2286\u27e8 \u2286-exch-\u2191-\u2191 \u2286-refl 1 \u2113 \u27e9\n                                \u03b1 \u2191 suc \u2113 \u220e -\u27e9\n   where open \u2286-Reasoning\n\n-- Handy name eliminator\nsubtractWith\u1d3a : \u2200 {a \u03b1} {A : Set a} \u2113 \u2192 A \u2192 (Name \u03b1 \u2192 A) \u2192 Name (\u03b1 \u2191 \u2113) \u2192 A\nsubtractWith\u1d3a \u2113 v f x = [ const v , f \u2218\u2032 subtract\u1d3a \u2113 ]\u2032 (x <\u1d3a \u2113)\n\nsubtract\u1d3a? : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192? Name \u03b1\nsubtract\u1d3a? \u2113 = subtractWith\u1d3a \u2113 nothing just\n\ninfixl 4 subtract\u1d3a?\nsyntax subtract\u1d3a? k x = x \u2238\u1d3a? k\n\nprivate\n module CoeExamples where\n  Name\u03b1+\u2192Name\u03b1\u2191 : \u2200 {\u03b1} k \u2192 Name (\u03b1 +\u1d42 k) \u2192 Name (\u03b1 \u2191 k)\n  Name\u03b1+\u2192Name\u03b1\u2191 k = coerce\u1d3a (\u2286-ctx-+\u2191 \u2286-refl k)\n module Unused where\n  open import Data.List using ([]; _\u2237_; replicate; _++_)\n  n\u2208true\u207f\u207a\u00b9 : \u2200 {\u03b1} n \u2192 n \u2208 replicate (suc n) true ++ \u03b1\n  n\u2208true\u207f\u207a\u00b9 zero    = _\n  n\u2208true\u207f\u207a\u00b9 (suc n) = n\u2208true\u207f\u207a\u00b9 n\n\npred\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 +1) \u2192 Name \u03b1\npred\u1d3a = subtract\u1d3a 1\n\npred\u1d3a? : \u2200 {\u03b1} \u2192 Name (\u03b1 \u21911) \u2192? Name \u03b1\npred\u1d3a? = subtract\u1d3a? 1\n\nName\u2192-to-Nm\u27f6 : \u2200 {b\u2081 b\u2082 \u03b1} {B : Setoid b\u2081 b\u2082} \u2192\n                 (Name \u03b1 \u2192 Setoid.Carrier B) \u2192 Nm \u03b1 \u27f6 B\nName\u2192-to-Nm\u27f6 {\u03b1 = \u03b1} {B} f = record { _\u27e8$\u27e9_ = f; cong = cong\u2032 }\n  where\n  open Setoid B renaming (_\u2248_ to _\u2248B_)\n\n  cong\u2261 : \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2248B f y\n  cong\u2261 \u2261.refl = Setoid.refl B\n\n  cong\u2032 : \u2200 {x y} \u2192 x \u2261\u1d3a y \u2192 f x \u2248B f y\n  cong\u2032 = cong\u2261 \u2218 Equality.to-propositional \u2261\u1d3a-equality\n  -- cong\u2032 = Setoid.reflexive \u2218 Equality.to-propositional \u2261\u1d3a-equality\n\n\u00acName : \u2200 {\u03b1} \u2192 \u03b1 \u2286 \u00f8 \u2192 \u00ac(Name \u03b1)\n\u00acName \u03b1\u2286\u00f8 = \u00acName\u00f8 \u2218 coerce\u1d3a \u03b1\u2286\u00f8\n\nName-elim : \u2200 {a} {A : Set a} {\u03b1} (\u03b1\u2286\u00f8 : \u03b1 \u2286 \u00f8) (x : Name \u03b1) \u2192 A\nName-elim x y with \u00acName x y\n... | ()\n\nName\u00f8-elim : \u2200 {a} {A : Set a} \u2192 Name \u00f8 \u2192 A\nName\u00f8-elim = Name-elim \u2286-refl\n\n\u00acName\u00f8+ : \u2200 k \u2192 \u00ac(Name (\u00f8 +\u1d42 k))\n\u00acName\u00f8+ = \u00acName \u2218 \u2286-\u00f8+\n\nName\u00f8+-elim : \u2200 {a} {A : Set a} k \u2192 Name (\u00f8 +\u1d42 k) \u2192 A\nName\u00f8+-elim = Name-elim \u2218 \u2286-\u00f8+\n\nName\u00f8\u2191\u2192Name\u03b1\u2191 : \u2200 {\u2113} \u2192 Name (\u00f8 \u2191 \u2113) \u2192 \u2200 {\u03b1} \u2192 Name (\u03b1 \u2191 \u2113)\nName\u00f8\u2191\u2192Name\u03b1\u2191 {\u2113} x = x \u27e8-because \u2286-cong-\u2191 \u2286-\u00f8 \u2113 -\u27e9\n\nis0? : \u2200 {\u03b1} \u2192 Name (\u03b1 \u21911) \u2192 Bool\nis0? = cmp\u1d3a-bool 1\n-- is0? = [ const true , const false ]\u2032 (x <\u1d3a 1)\n\nshift\u2115 : \u2200 (\u2113 k n : \u2115) \u2192 \u2115\nshift\u2115 \u2113 k n = if suc n <= \u2113 then n else n +\u2115 k\n\nshift\u1d3a : \u2200 {\u03b1 \u03b2} \u2113 k \u2192 \u03b1 +\u1d42 k \u2286 \u03b2 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u03b2 \u2191 \u2113)\nshift\u1d3a \u2113 k pf n\n  with n <\u1d3a \u2113\n...  | inl n\u2032 = n\u2032       \u27e8-because \u2286-cong-\u2191 \u2286-\u00f8 \u2113 -\u27e9\n...  | inr n\u2032 = n\u2032 +\u1d3a k  \u27e8-because \u2286-trans (\u2286-exch-+-+ \u2286-refl \u2113 k) (\u2286-ctx-+\u2191 pf \u2113) -\u27e9\n\nmodule Singletons where\n\n  SWorld : \u2115 \u2192 World\n  SWorld n = \u00f8 \u21911 +\u1d42 n\n\n  Name\u02e2 : \u2115 \u2192 Set\n  Name\u02e2 = Name \u2218 SWorld\n\n  _\u02e2 : \u2200 n \u2192 Name\u02e2 n\n  _\u02e2 n = zero\u1d3a +\u1d3a n\n\n  add\u02e2 : \u2200 {n} k \u2192 Name\u02e2 n \u2192 Name\u02e2 (k +\u2115 n)\n  add\u02e2 {n} k x = add\u1d3a k x  \u27e8-because \u2286-assoc-+ \u2286-refl n k -\u27e9\n\n  subtract\u02e2 : \u2200 {n} k \u2192 Name\u02e2 (k +\u2115 n) \u2192 Name\u02e2 n\n  subtract\u02e2 {n} k x = subtract\u1d3a k (x \u27e8-because \u2286-assoc-+\u2032 \u2286-refl n k -\u27e9)\n\n{-\nshift\u1d3a-1-1+-\n  shift\u1d3a 1 (suc k) pf n \u2261 shift\u1d3a 1 k ? (shift\u1d3a 1 1 ? n)\n-}\n\n-- Properties\n\ncoerce\u1d3a-names : \u2200 {\u03b1 \u03b2} (\u03b1\u2286\u03b2 : \u03b1 \u2286 \u03b2) \u2192 name \u2218 coerce\u1d3a \u03b1\u2286\u03b2 \u2257 name\ncoerce\u1d3a-names _ _ = \u2261.refl\n\n{-\n<\u1d3a-coerce : \u2200 {\u03b1} \u2113 (x : Name (\u03b1 \u2191 \u2113)) \u2192 [ coerce\u1d3a (\u2286-cong-\u2191 \u2286-\u00f8 \u2113) , coerce\u1d3a (\u2286-ctx-+\u2191 \u2286-refl \u2113) ]\u2032 (x <\u1d3a \u2113) \u2261 x\n<\u1d3a-coerce \u2113 x with x <\u1d3a-bool \u2113 | x <\u1d3a-name\u2208\u03b1 \u2113\n... | true  | _ = name-injective \u2261.refl\n... | false | _ = name-injective \u2261.refl\n-}\n\n<\u1d3a-names : \u2200 {\u03b1 \u2113} {x : Name (\u03b1 \u2191 \u2113)} \u2192 [ name , name ]\u2032 (x <\u1d3a \u2113) \u2261 name x\n<\u1d3a-names {\u03b1} {\u2113} {x} with x <\u1d3a-bool \u2113 | x <\u1d3a-name\u2208\u03b1 \u2113\n... | true  | _ = \u2261.refl\n... | false | _ = \u2261.refl\n\nDec-\u2261\u1d3a\u2192\u2261 : \u2200 {\u03b1} \u2192 Decidable (_\u2261\u1d3a_ {\u03b1}) \u2192 Decidable _\u2261_\nDec-\u2261\u1d3a\u2192\u2261 _\u225f_ x y with x \u225f y\n... | yes x\u2261y = yes (\u2261\u1d3a\u21d2\u2261 x\u2261y)\n... | no  x\u2262y = no  (x\u2262y \u2218\u2032 \u2115s.reflexive \u2218\u2032 \u2261.cong name)\n\nshift\u1d3a-\u2113-0\u2261coerce\u1d3a : \u2200 {\u03b1 \u03b2} \u2113 (pf : \u03b1 \u2286 \u03b2) n \u2192 name (shift\u1d3a \u2113 0 pf n) \u2261 name n\nshift\u1d3a-\u2113-0\u2261coerce\u1d3a \u2113 pf x with x <\u1d3a-bool \u2113 | x <\u1d3a-name\u2208\u03b1 \u2113\n... | true  | _ = \u2261.refl\n... | false | _ = \u2261.refl\n\npred-shift\u2115-comm : \u2200 k x \u2192 pred (shift\u2115 1 k (suc x)) \u2261 x +\u2115 k\npred-shift\u2115-comm k x = \u2261.refl\n\n.shift11-pred\u1d3a?-comm :\n  \u2200 {\u03b1 \u03b2} (pf : \u03b1 +1 \u2286 \u03b2) (x : Name (\u03b1 \u21911)) \u2192\n  pred\u1d3a? (shift\u1d3a 1 1 pf x) \u2261 coerce\u1d3a pf \u2218 suc\u1d3a <$> pred\u1d3a? x\nshift11-pred\u1d3a?-comm {\u03b1} pf x\n   = <$>-injective\u2081 name-injective (\u2261.trans (helper (name x) {name\u2208\u03b1 x} (suc (name x) \u2264?\u2115 1)) (<$>-assoc (pred\u1d3a? x)))\n  where\n    -- getting the name unpacked (and the Dec), works around an Agda bug in 2.2.8.\n    .helper : \u2200 n {pn : n \u2208 \u03b1 \u21911} \u2192 Dec (n <\u2115 1) \u2192 let x = n , pn in\n             name <$> (pred\u1d3a? (shift\u1d3a 1 1 pf x)) \u2261 name \u2218 (coerce\u1d3a pf \u2218 suc\u1d3a) <$> pred\u1d3a? x\n    helper zero    (no \u00acp)         = \u22a5-elim (\u00acp (s\u2264s z\u2264n))\n    helper .0      (yes (s\u2264s z\u2264n)) = \u2261.refl\n    helper (suc _) (no _)          = \u2261.refl\n\n1+ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name (\u03b1 +1)\n1+ x = x +\u1d3a 1\n\nmodule Name\u2191\u21d4Fin where\n  open import Data.Fin as Fin\n\n  \u21d2 : \u2200 n \u2192 Name (\u00f8 \u2191 n) \u2192 Fin n\n  \u21d2 zero    = Name\u00f8-elim\n  \u21d2 (suc n) = maybe (suc \u2218\u2032 \u21d2 n) zero \u2218\u2032 pred\u1d3a?\n\n  \u21d0 : \u2200 {\u03b1 n} \u2192 Fin n \u2192 Name (\u03b1 \u2191 n)\n  \u21d0 (zero {n})  = zero\u1d3a\u21911+ n\n  \u21d0 (suc i)     = suc\u1d3a\u2191 (\u21d0 i)\n\n{-\n  open Singletons\n  open \u2286-Reasoning\n  f : \u2200 {\u03b1 n} \u2192 Fin n \u2192 Name (\u03b1 \u2191 n)\n  f {\u03b1} {zero} ()\n  f {\u03b1} {suc n} x = coerce\u1d3a c (sName x\u2032) where\n    x\u2032 = Fin.to\u2115 x\n    c\u2032 = \u00f8 \u21911 +[ n \u2238\u2115 x\u2032 ]   \u2286\u27e8 {!!} \u27e9\n         (\u00f8 +[ n \u2238\u2115 x\u2032 ]) \u21911 \u2286\u27e8 \u2286-cong-\u21911 (\u03b1+\u2286\u00f8\u2192\u03b1\u2286\u00f8 _ \u2286-refl) \u27e9\n         \u00f8 \u21911                 \u2286\u27e8 \u2286-cong-\u21911 \u2286-\u00f8 \u27e9\n         \u03b1 \u21911 \u220e\n    c : SWorld x\u2032 \u2286 \u03b1 \u2191 (suc n)\n    c = SWorld x\u2032                \u2286\u27e8 refl \u27e9\n        \u00f8 \u21911 +[ x\u2032 ]             \u2286\u27e8 {!!} \u27e9\n        \u00f8 \u21911 +[ n \u2238\u2115 x\u2032 ] +[ n ] \u2286\u27e8 \u2286-ctx-+\u2191 c\u2032 n \u27e9\n        \u03b1 \u21911 \u2191 n                 \u2286\u27e8 \u2286-exch-\u2191-\u2191 \u2286-refl 1 n \u27e9\n        \u03b1 \u2191 suc n \u220e\n-}\n\n  -- \u21d4 : \u2200 n \u2192 Name (\u00f8 \u2191 n) \u21d4 Fin n\n  -- \u21d4 n = ?\n\n{-\nmodule Bug where\n  record Fin\u2032 (n : \u2115) : Set where\n    constructor mkFin\n    field\n      x    : \u2115\n      .x<n : x <\u2115 n\n  raise : \u2200 {n m} \u2192 Fin\u2032 n \u2192 n \u2264\u2115 m \u2192 Fin\u2032 m\n  raise (mkFin x x<n) n\u2264m = mkFin x (\u2115\u2264.trans x<n n\u2264m)\n  raise-assoc : \u2200 {n m o} p (q : m \u2264\u2115 o) (x : Fin\u2032 n) \u2192 (raise (raise x p) q) \u2261 (raise x (\u2115\u2264.trans p q))\n  raise-assoc p q (mkFin x x<n) with raise-assoc p q (mkFin x x<n)\n  ... | refl = ?\n-}\n\n{-\nmodule Fin\u2032 where\n  record Fin\u2032 (n : \u2115) : Set where\n    constructor mkFin\n    field\n      x    : \u2115\n      .x<n : x <\u2115 n\n  zeroF : \u2200 {n} \u2192 Fin\u2032 (suc n)\n  zeroF = mkFin 0 (s\u2264s z\u2264n)\n  sucF : \u2200 {n} \u2192 Fin\u2032 n \u2192 Fin\u2032 (suc n)\n  sucF (mkFin x x<n) = mkFin (suc x) (s\u2264s x<n)\n\n  raise : \u2200 {n} (x : Fin\u2032 n) k \u2192 Fin\u2032 (n +\u2115 k)\n  raise (mkFin x x<n) k = mkFin x (m\u2264n\u2192m\u2264n+k x<n)\n    where m\u2264n\u2192m\u2264n+k : \u2200 {m n k} \u2192 m \u2264\u2115 n \u2192 m \u2264\u2115 n +\u2115 k\n          m\u2264n\u2192m\u2264n+k z\u2264n        = z\u2264n\n          m\u2264n\u2192m\u2264n+k (s\u2264s m\u2264n)  = s\u2264s (m\u2264n\u2192m\u2264n+k m\u2264n)\n-}\n\n{-\n(x : \u03b1 +\u1d42 n) \u2192 x \u2265 n\n\n(x : \u03b1 +\u1d42 suc n) \u2192 x \u2262 zero\u1d3a\n-}\n\nmodule \u2115\u21d2Name`\u03b1` where\n  open WorldSyntax\n  open WorldSyntax.El listBoolWorlds\n  \u21d2 : \u2200 `\u03b1` x \u2192 x \u2208 El `\u03b1` \u2192 Name (El `\u03b1`)\n  \u21d2 (`\u03b1` `\u21911`) zero    _     = zero , _\n  \u21d2 (`\u03b1` `\u21911`) (suc i) pf    = suc\u1d3a\u2191 (\u21d2 `\u03b1` i pf)\n  \u21d2 (`\u03b1` `+1`) zero    ()\n  \u21d2 (`\u03b1` `+1`) (suc i) pf    = suc\u1d3a (\u21d2 `\u03b1` i pf)\n  \u21d2 `\u00f8`        _       ()\n\nmodule Name`\u03b1`\u21d4Fin where\n  open import Data.Fin as F renaming (_+_ to _+F_)\n  open WorldSyntax\n  open WorldSyntax.El listBoolWorlds\n  open WorldSyntax.Properties listBoolWorlds\n  mutual\n\n    \u21d2+1 : \u2200 `\u03b1` \u2192 Name (El `\u03b1` +1) \u2192 Fin (suc (bnd `\u03b1`))\n    \u21d2+1 `\u03b1` = F.raise 1 \u2218 \u21d2 `\u03b1` \u2218 pred\u1d3a\n\n    \u21d2 : \u2200 `\u03b1` \u2192 Name (El `\u03b1`) \u2192 Fin (bnd `\u03b1`)\n    \u21d2 (`\u03b1` `\u21911`)  = [ inject+ _ \u2218 Name\u2191\u21d4Fin.\u21d2 1 , \u21d2+1 `\u03b1` ]\u2032 \u2218 cmp\u1d3a 1\n    \u21d2 (`\u03b1` `+1`)  = \u21d2+1 `\u03b1`\n    \u21d2 `\u00f8`         = Name\u00f8-elim\n\n  \u21d0\u2081 : \u2200 {n} `\u03b1` \u2192 (x : Fin n) \u2192 to\u2115 x \u2208 El `\u03b1` \u2192 Name (El `\u03b1`)\n  \u21d0\u2081 `\u03b1` = \u2115\u21d2Name`\u03b1`.\u21d2 `\u03b1` \u2218 to\u2115\n\n  x\u2208\u00f8\u2191x : \u2200 {n} (x : Fin n) \u2192 to\u2115 x \u2208 (\u00f8 \u2191 n)\n  x\u2208\u00f8\u2191x zero    = _\n  x\u2208\u00f8\u2191x (suc n) = x\u2208\u00f8\u2191x n\n\n  \u21d0\u2082 : \u2200 {n} \u2192 (x : Fin n) \u2192 Name (\u00f8 \u2191 n)\n  \u21d0\u2082 {n} x with x\u2208\u00f8\u2191x x\n  ...         | y       rewrite comm-El-\u2191 `\u00f8` n = \u21d0\u2081 (`\u00f8` `\u2191` n) x y\n\n  {- I would have prefered a local `where` here\n  \u21d0\u2082 : \u2200 {n} \u2192 (x : Fin n) \u2192 Name (\u00f8 \u2191 n)\n  \u21d0\u2082 {n} x where y = x\u2208\u00f8\u2191x x rewrite comm-El-\u2191 `\u00f8` n = \u21d0\u2081 (`\u00f8` `\u2191` n) x y\n  -}\n\nmodule WellFormedWorld where\n  data WF : World \u2192 Set where\n    `\u00f8 : WF \u00f8\n    _`+_ : \u2200 {\u03b1} (`\u03b1 : WF \u03b1) k \u2192 WF (\u03b1 +\u1d42 k)\n    _`\u2191_ : \u2200 {\u03b1} (`\u03b1 : WF \u03b1) k \u2192 WF (\u03b1 \u2191 k)\n\n_+\u2191\u1d3a_ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \u2200 \u2113 \u2192 Name (\u03b1 \u2191 \u2113)\n_+\u2191\u1d3a_ x \u2113 = coerce\u1d3a (\u2286-ctx-+\u2191 \u2286-refl \u2113) (x +\u1d3a \u2113)\n\nsubtract\u1d3a\u2191 : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 Name (\u00f8 \u2191 \u2113) \u228e Name \u03b1\nsubtract\u1d3a\u2191 \u2113 x with x <\u1d3a \u2113\n... | inl x\u2032 = inl x\u2032\n... | inr x\u2032 = inr (x\u2032 \u2238\u1d3a \u2113)\n\ninfixl 4 subtract\u1d3a\u2191\nsyntax subtract\u1d3a\u2191 \u2113 x = x -\u2191\u1d3a \u2113\n\ndata WithLim (R : (\u03b1 \u03b2 : World) \u2192 Set) : (\u03b1 \u03b2 : World) \u2192 Set where\n  mkWithLim : \u2200 {\u03b1\u2080 \u03b2\u2080} \u2113 (R\u03b1\u2080\u03b2\u2080 : R \u03b1\u2080 \u03b2\u2080) \u2192 WithLim R (\u03b1\u2080 \u2191 \u2113) (\u03b2\u2080 \u2191 \u2113)\n\nwithLim : \u2200 {R \u03b1 \u03b2} \u2192 R \u03b1 \u03b2 \u2192 WithLim R \u03b1 \u03b2\nwithLim = mkWithLim 0\n\n-- functions known to be ``bad'' and that we don't want to export\nprivate\n module Bad where\n  Name2\u2115 : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \u2115\n  Name2\u2115 = name\n  isZero? : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Bool\n  isZero? (zero  , _) = true\n  isZero? (suc _ , _) = false\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9eb10b2925043281b535da3adacd2c3755d73595","subject":"IDesc: enum (hard-coded)","message":"IDesc: enum (hard-coded)\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0c1221f0b13060174c0999dbc0235f071a12e530","subject":"Fixed doc","message":"Fixed doc\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/report\/FOTC\/MutualInductivePredicates.agda","new_file":"notes\/thesis\/report\/FOTC\/MutualInductivePredicates.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of mutual inductive predicates\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n\nmodule FOTC.MutualInductivePredicates where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Using mutual inductive predicates\n\ndata Even : D \u2192 Set\ndata Odd  : D \u2192 Set\n\ndata Even where\n  ezero : Even zero\n  esucc : \u2200 {n} \u2192 Odd n \u2192 Even (succ\u2081 n)\n\ndata Odd where\n  osucc : \u2200 {n} \u2192 Even n \u2192 Odd (succ\u2081 n)\n\n-- Non-mutual induction principles\n\nEven-ind : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 Odd n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 Even n \u2192 A n\nEven-ind A A0 h ezero      = A0\nEven-ind A A0 h (esucc On) = h On\n\nOdd-ind : (A : D \u2192 Set) \u2192\n          (\u2200 {n} \u2192 Even n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 Odd n \u2192 A n\nOdd-ind A h (osucc En) = h En\n\n-- Mutual induction principles (from Coq)\nEven-mutual-ind :\n  (A B : D \u2192 Set) \u2192\n  A zero \u2192\n  (\u2200 {n} \u2192 Odd n \u2192 B n \u2192 A (succ\u2081 n)) \u2192\n  (\u2200 {n} \u2192 Even n \u2192 A n \u2192 B (succ\u2081 n)) \u2192\n  \u2200 {n} \u2192 Even n \u2192 A n\n\nOdd-mutual-ind :\n  (A B : D \u2192 Set) \u2192\n  A zero \u2192\n  (\u2200 {n} \u2192 Odd n \u2192 B n \u2192 A (succ\u2081 n)) \u2192\n  (\u2200 {n} \u2192 Even n \u2192 A n \u2192 B (succ\u2081 n)) \u2192\n  \u2200 {n} \u2192 Odd n \u2192 B n\n\nEven-mutual-ind A B A0 h\u2081 h\u2082 ezero      = A0\nEven-mutual-ind A B A0 h\u2081 h\u2082 (esucc On) = h\u2081 On (Odd-mutual-ind A B A0 h\u2081 h\u2082 On)\n\nOdd-mutual-ind A B A0 h\u2081 h\u2082 (osucc En) = h\u2082 En (Even-mutual-ind A B A0 h\u2081 h\u2082 En)\n\nmodule DisjointSum where\n  -- Using a single inductive predicate on D \u00d7 D (see\n  -- Blanchette\u00a0(2013)).\n\n  _+_ : Set \u2192 Set \u2192 Set\n  _+_ = _\u2228_\n\n  data EvenOdd : D + D \u2192 Set where\n    eozero :                            EvenOdd (inj\u2081 zero)\n    eoodd  : {n : D} \u2192 EvenOdd (inj\u2081 n) \u2192 EvenOdd (inj\u2082 (succ\u2081 n))\n    eoeven : {n : D} \u2192 EvenOdd (inj\u2082 n) \u2192 EvenOdd (inj\u2081 (succ\u2081 n))\n\n  -- Induction principle\n  EvenOdd-ind : (A : D + D \u2192 Set) \u2192\n                A (inj\u2081 zero) \u2192\n                ({n : D} \u2192 A (inj\u2081 n) \u2192 A (inj\u2082 (succ\u2081 n))) \u2192\n                ({n : D} \u2192 A (inj\u2082 n) \u2192 A (inj\u2081 (succ\u2081 n))) \u2192\n                {n : D + D} \u2192 EvenOdd n \u2192 A n\n  EvenOdd-ind A A0 h\u2081 h\u2082 eozero       = A0\n  EvenOdd-ind A A0 h\u2081 h\u2082 (eoodd EOn)  = h\u2081 (EvenOdd-ind A A0 h\u2081 h\u2082 EOn)\n  EvenOdd-ind A A0 h\u2081 h\u2082 (eoeven EOn) = h\u2082 (EvenOdd-ind A A0 h\u2081 h\u2082 EOn)\n\n  -------------------------------------------------------------------------\n  -- From the single inductive predicate to the mutual inductive predicates\n\n  -- Even and Odd from EvenOdd.\n  Even' : D \u2192 Set\n  Even' n = EvenOdd (inj\u2081 n)\n\n  Odd' : D \u2192 Set\n  Odd' n = EvenOdd (inj\u2082 n)\n\n  -- From Even\/Odd to Even'\/Odd'\n  Even\u2192Even' : \u2200 {n} \u2192 Even n \u2192 Even' n\n  Odd\u2192Odd'   : \u2200 {n} \u2192 Odd n \u2192 Odd' n\n\n  Even\u2192Even' ezero = eozero\n  Even\u2192Even' (esucc On) = eoeven (Odd\u2192Odd' On)\n\n  Odd\u2192Odd' (osucc En) = eoodd (Even\u2192Even' En)\n\n  -- From Even'\/Odd' to Even\/Odd\n  Even'\u2192Even : \u2200 {n} \u2192 Even' n \u2192 Even n\n  Odd'\u2192Odd : \u2200 {n} \u2192 Odd' n \u2192 Odd n\n\n  -- Requires K.\n  Even'\u2192Even eozero     = ezero\n  Even'\u2192Even (eoeven h) = esucc (Odd'\u2192Odd h)\n\n  Odd'\u2192Odd (eoodd h) = osucc (Even'\u2192Even h)\n\n  -- TODO (03 December 2012). From EvenOdd-ind to Even-mutual-ind and\n  -- Odd-mutual-ind.\n\nmodule FunctionSpace where\n-- Using a single inductive predicate on D \u2192 D\n\n  data EvenOdd : D \u2192 D \u2192 Set where\n    eozero :                         EvenOdd zero (succ\u2081 zero)\n    eoodd  : \u2200 {m n} \u2192 EvenOdd m n \u2192 EvenOdd m (succ\u2081 m)\n    eoeven : \u2200 {m n} \u2192 EvenOdd m n \u2192 EvenOdd (succ\u2081 n) n\n\n  -- Even and Odd from EvenOdd.\n  -- Even' : D \u2192 Set\n  -- Even' n = EvenOdd zero (succ\u2081 zero) \u2228 EvenOdd (succ\u2081 n) n\n\n  -- Odd' : D \u2192 Set\n  -- Odd' n = EvenOdd n (succ\u2081 n)\n\n  -- -- From Even\/Odd to Even'\/Odd'\n  -- Even\u2192Even' : \u2200 {n} \u2192 Even n \u2192 Even' n\n  -- Odd\u2192Odd'   : \u2200 {n} \u2192 Odd n \u2192 Odd' n\n\n  -- Even\u2192Even' ezero      = inj\u2081 eozero\n  -- Even\u2192Even' (esucc On) = inj\u2082 (eoeven (Odd\u2192Odd' On))\n\n  -- Odd\u2192Odd' (osucc En) = eoodd {!!}\n\n  -- -- From Even'\/Odd' to Even\/Odd\n  -- Even'\u2192Even : \u2200 {n} \u2192 Even' n \u2192 Even n\n  -- Odd'\u2192Odd : \u2200 {n} \u2192 Odd' n \u2192 Odd n\n\n  -- Even'\u2192Even (inj\u2081 x) = {!!}\n  -- Even'\u2192Even (inj\u2082 x) = {!!}\n\n  -- Odd'\u2192Odd h = {!!}\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Blanchette, Jasmin Christian (2013). Relational analysis of\n-- (co)inductive predicates, (co)algebraic datatypes, and\n-- (co)recursive functions. Software Quality Journal 21.1,\n-- pp. 101\u2013126.\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of mutual inductive predicates\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n\nmodule FOTC.MutualInductivePredicates where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Using mutual inductive predicates\n\ndata Even : D \u2192 Set\ndata Odd  : D \u2192 Set\n\ndata Even where\n  ezero : Even zero\n  esucc : \u2200 {n} \u2192 Odd n \u2192 Even (succ\u2081 n)\n\ndata Odd where\n  osucc : \u2200 {n} \u2192 Even n \u2192 Odd (succ\u2081 n)\n\n-- Non-mutual induction principles\n\nEven-ind : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 Odd n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 Even n \u2192 A n\nEven-ind A A0 h ezero      = A0\nEven-ind A A0 h (esucc On) = h On\n\nOdd-ind : (A : D \u2192 Set) \u2192\n          (\u2200 {n} \u2192 Even n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 Odd n \u2192 A n\nOdd-ind A h (osucc En) = h En\n\n-- Mutual induction principles (from Coq)\nEven-mutual-ind :\n  (A B : D \u2192 Set) \u2192\n  A zero \u2192\n  (\u2200 {n} \u2192 Odd n \u2192 B n \u2192 A (succ\u2081 n)) \u2192\n  (\u2200 {n} \u2192 Even n \u2192 A n \u2192 B (succ\u2081 n)) \u2192\n  \u2200 {n} \u2192 Even n \u2192 A n\n\nOdd-mutual-ind :\n  (A B : D \u2192 Set) \u2192\n  A zero \u2192\n  (\u2200 {n} \u2192 Odd n \u2192 B n \u2192 A (succ\u2081 n)) \u2192\n  (\u2200 {n} \u2192 Even n \u2192 A n \u2192 B (succ\u2081 n)) \u2192\n  \u2200 {n} \u2192 Odd n \u2192 B n\n\nEven-mutual-ind A B A0 h\u2081 h\u2082 ezero      = A0\nEven-mutual-ind A B A0 h\u2081 h\u2082 (esucc On) = h\u2081 On (Odd-mutual-ind A B A0 h\u2081 h\u2082 On)\n\nOdd-mutual-ind A B A0 h\u2081 h\u2082 (osucc En) = h\u2082 En (Even-mutual-ind A B A0 h\u2081 h\u2082 En)\n\nmodule DisjointSum where\n  -- Using a single inductive predicate on D \u00d7 D (see Blanchette).\n\n  _+_ : Set \u2192 Set \u2192 Set\n  _+_ = _\u2228_\n\n  data EvenOdd : D + D \u2192 Set where\n    eozero :                            EvenOdd (inj\u2081 zero)\n    eoodd  : {n : D} \u2192 EvenOdd (inj\u2081 n) \u2192 EvenOdd (inj\u2082 (succ\u2081 n))\n    eoeven : {n : D} \u2192 EvenOdd (inj\u2082 n) \u2192 EvenOdd (inj\u2081 (succ\u2081 n))\n\n  -- Induction principle\n  EvenOdd-ind : (A : D + D \u2192 Set) \u2192\n                A (inj\u2081 zero) \u2192\n                ({n : D} \u2192 A (inj\u2081 n) \u2192 A (inj\u2082 (succ\u2081 n))) \u2192\n                ({n : D} \u2192 A (inj\u2082 n) \u2192 A (inj\u2081 (succ\u2081 n))) \u2192\n                {n : D + D} \u2192 EvenOdd n \u2192 A n\n  EvenOdd-ind A A0 h\u2081 h\u2082 eozero       = A0\n  EvenOdd-ind A A0 h\u2081 h\u2082 (eoodd EOn)  = h\u2081 (EvenOdd-ind A A0 h\u2081 h\u2082 EOn)\n  EvenOdd-ind A A0 h\u2081 h\u2082 (eoeven EOn) = h\u2082 (EvenOdd-ind A A0 h\u2081 h\u2082 EOn)\n\n  -------------------------------------------------------------------------\n  -- From the single inductive predicate to the mutual inductive predicates\n\n  -- Even and Odd from EvenOdd.\n  Even' : D \u2192 Set\n  Even' n = EvenOdd (inj\u2081 n)\n\n  Odd' : D \u2192 Set\n  Odd' n = EvenOdd (inj\u2082 n)\n\n  -- From Even\/Odd to Even'\/Odd'\n  Even\u2192Even' : \u2200 {n} \u2192 Even n \u2192 Even' n\n  Odd\u2192Odd'   : \u2200 {n} \u2192 Odd n \u2192 Odd' n\n\n  Even\u2192Even' ezero = eozero\n  Even\u2192Even' (esucc On) = eoeven (Odd\u2192Odd' On)\n\n  Odd\u2192Odd' (osucc En) = eoodd (Even\u2192Even' En)\n\n  -- From Even'\/Odd' to Even\/Odd\n  Even'\u2192Even : \u2200 {n} \u2192 Even' n \u2192 Even n\n  Odd'\u2192Odd : \u2200 {n} \u2192 Odd' n \u2192 Odd n\n\n  -- Requires K.\n  Even'\u2192Even eozero     = ezero\n  Even'\u2192Even (eoeven h) = esucc (Odd'\u2192Odd h)\n\n  Odd'\u2192Odd (eoodd h) = osucc (Even'\u2192Even h)\n\n  -- TODO (03 December 2012). From EvenOdd-ind to Even-mutual-ind and\n  -- Odd-mutual-ind.\n\nmodule FunctionSpace where\n-- Using a single inductive predicate on D \u2192 D\n\n  data EvenOdd : D \u2192 D \u2192 Set where\n    eozero :                         EvenOdd zero (succ\u2081 zero)\n    eoodd  : \u2200 {m n} \u2192 EvenOdd m n \u2192 EvenOdd m (succ\u2081 m)\n    eoeven : \u2200 {m n} \u2192 EvenOdd m n \u2192 EvenOdd (succ\u2081 n) n\n\n  -- Even and Odd from EvenOdd.\n  -- Even' : D \u2192 Set\n  -- Even' n = EvenOdd zero (succ\u2081 zero) \u2228 EvenOdd (succ\u2081 n) n\n\n  -- Odd' : D \u2192 Set\n  -- Odd' n = EvenOdd n (succ\u2081 n)\n\n  -- -- From Even\/Odd to Even'\/Odd'\n  -- Even\u2192Even' : \u2200 {n} \u2192 Even n \u2192 Even' n\n  -- Odd\u2192Odd'   : \u2200 {n} \u2192 Odd n \u2192 Odd' n\n\n  -- Even\u2192Even' ezero      = inj\u2081 eozero\n  -- Even\u2192Even' (esucc On) = inj\u2082 (eoeven (Odd\u2192Odd' On))\n\n  -- Odd\u2192Odd' (osucc En) = eoodd {!!}\n\n  -- -- From Even'\/Odd' to Even\/Odd\n  -- Even'\u2192Even : \u2200 {n} \u2192 Even' n \u2192 Even n\n  -- Odd'\u2192Odd : \u2200 {n} \u2192 Odd' n \u2192 Odd n\n\n  -- Even'\u2192Even (inj\u2081 x) = {!!}\n  -- Even'\u2192Even (inj\u2082 x) = {!!}\n\n  -- Odd'\u2192Odd h = {!!}\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Blanchette, Jasmin Christian (2013). Relational analysis of\n-- (co)inductive predicates, (co)algebraic datatypes, and\n-- (co)recursive functions. Software Quality Journal 21.1,\n-- pp. 101\u2013126.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9a938fbd117bc5580125c2c4d5f666d96a746546","subject":"IDesc: our Hutton's razor","message":"IDesc: our Hutton's razor\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep nat (EZe , t) = t\n              evalOneStep nat ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep nat ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool (EZe , t ) = t\n              evalOneStep bool ((ESu EZe) , (true , (x , _))) = x\n              evalOneStep bool ((ESu EZe) , (false , (_ , y))) = y\n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\nassgnmt : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\nassgnmt ty vars context (l value) = con ( EZe , value )\nassgnmt ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4d9de23b0ab1cd4455e5e2bf819016d170d2ba2c","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V VConst\n\n  -- variables\n  progress (TAVar x) = abort (somenotnone (! x))\n\n  -- lambdas\n  progress (TALam D) = V VLam\n\n  -- applications\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n    -- left applicand value\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n    -- errors propagate\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = E (EAp2 (FVal x) x\u2081)\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = E (EAp2 (FIndet x) x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = E (EAp1 x) -- NB: could have picked the other one here, too; this is a source of non-determinism\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = S {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n    -- indeterminates\n  progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n    -- either applicand steps\n  progress (TAAp D1 x\u2083 D2) | S (d1' , Step x x\u2081 x\u2082)  | _ = S (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082))\n  progress (TAAp D1 x\u2084 D2) | _ | S (d2' , Step x\u2081 x\u2082 x\u2083) = S (_ , Step (FHAp1 {!!} x\u2081) x\u2082 (FHAp1 {!!} x\u2083))\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- non-empty holes\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (_ , Step x x\u2081 x\u2082) = S (_ , Step (FHNEHole x) x\u2081 (FHNEHole x\u2082))\n\n  -- casts\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  progress (TACast D m)  | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V VConst\n\n  -- variables\n  progress (TAVar x) = abort (somenotnone (! x))\n\n  -- lambdas\n  progress (TALam D) = V VLam\n\n  -- applications\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n    -- left applicand value\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n    -- errors propagate\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = E (EAp2 (FVal x) x\u2081)\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = E (EAp2 (FIndet x) x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = E (EAp1 x) -- NB: could have picked the other one here, too; this is a source of non-determinism\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = S {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n    -- indeterminates\n  progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n    -- either applicand steps\n  progress (TAAp D1 x\u2083 D2) | S (d1' , Step x x\u2081 x\u2082)  | _ = S (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082))\n  progress (TAAp D1 x\u2084 D2) | _ | S (d2' , Step x\u2081 x\u2082 x\u2083) = S (_ , Step (FHAp1 {!!} x\u2081) x\u2082 (FHAp1 {!!} x\u2083))\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- non-empty holes\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (_ , Step x x\u2081 x\u2082) = S (_ , Step (FHNEHole x) x\u2081 (FHNEHole x\u2082))\n\n  -- casts\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  progress (TACast D m) | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"da1a35ac8ce78d7836e287a81a5022ed4b6f0a01","subject":"named sums -- im pretty sure its correct and only needs to be indexed by d not also Delta","message":"named sums -- im pretty sure its correct and only needs to be indexed by d not also Delta\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) \u2192 Set where\n    V : \u2200{d} \u2192 d val \u2192 ok d\n    I : \u2200{d} \u2192 d indet \u2192 ok d\n    E : \u2200{d \u0394} \u2192 d err[ \u0394 ] \u2192 ok d\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | V x\u2081\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp D1 x\u2082 D2) | V x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | I x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | E x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | S x = {!!}\n  progress (TACast D x)\n    with progress D\n  progress (TACast D x\u2081) | V x = {!!}\n  progress (TACast D x\u2081) | I x = {!!}\n  progress (TACast D x\u2081) | E x = {!!}\n  progress (TACast D x\u2081) | S x = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') -- todo : make this a record so this is readable\n  progress TAConst = Inl VConst\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n  progress (TALam D) = Inl VLam\n  progress (TAAp D x D\u2081) with progress D | progress D\u2081\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inl x\u2081 = Inr (Inr (Inr (_ , (Step {!!} {!!} {!!}))))\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inr (Inr (Inl x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inr (Inr (Inr x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inl x) | Inl x\u2081 = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inl x\u2081 = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inl x\u2081 = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inl x) | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2081 D\u2081) | Inr (Inl x) | Inr (Inr ih2) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inr (Inr (Inl x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inr (Inr (Inr x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inr (Inr (Inl x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inr (Inr (Inr x\u2081)) = {!!}\n  progress (TAEHole x x\u2081) = Inr (Inl IEHole)\n  progress (TANEHole x D x\u2081) with progress D\n  progress (TANEHole x\u2081 D x\u2082) | Inl x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | Inr (Inl x) = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | Inr (Inr (Inl x)) = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | Inr (Inr (Inr x)) = {!!}\n  progress (TACast D x) with progress D\n  progress (TACast D x\u2081) | Inl x = Inr (Inr (Inr (_ , Step FRefl (ITCast (FVal x) D x\u2081) FRefl)))\n  progress (TACast D x\u2081) | Inr (Inl x) = Inr (Inl (ICast x))\n  progress (TACast D x\u2081) | Inr (Inr (Inl x)) = Inr (Inr (Inl {!!} ))\n  progress (TACast D x\u2081) | Inr (Inr (Inr x)) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"55f82f5473699a95c46c45e4af552047a924fbd3","subject":"more working on lemmas","message":"more working on lemmas\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192 dom-eq \u03941 H1 \u2192 dom-eq \u03942 H2 \u2192 dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union (\u03c01 , \u03c02) (\u03c03 , \u03c04) = (\u03bb n x \u2192 {!!}) ,\n                                  (\u03bb n x \u2192 {!!})\n\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) with holes-delta-ana h x\u2084 | holes-delta-ana h\u2081 x\u2085\n    ... | ih1 | ih2 = dom-union ih1 ih2\n\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  lem-dom-apt : {A : Set} {G : A ctx} {x y : Nat} \u2192 x # G \u2192 dom G y \u2192 x \u2260 y\n  lem-dom-apt {x = x} {y = y} apt dom with natEQ x y\n  lem-dom-apt apt dom | Inl refl = abort (somenotnone (! (\u03c02 dom) \u00b7 apt))\n  lem-dom-apt apt dom | Inr x\u2081 = x\u2081\n\n  lem-apart-disjoint : {A : Set} {H : A ctx} {u : Nat} {x : A} \u2192 u # H \u2192 (\u25a0 (u , x)) ## H\n  lem-apart-disjoint {H = H} apt = (\u03bb n x \u2192 tr (\u03bb qq \u2192 qq # H) (singleton-eq (\u03c02 x)) apt) ,\n                                   (\u03bb n x \u2192 apart-singleton (flip (lem-dom-apt apt x)))\n\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-disjoint (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-disjoint (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  mutual\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint = {!!}\n\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192 dom-eq \u03941 H1 \u2192 dom-eq \u03942 H2 \u2192 dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union (\u03c01 , \u03c02) (\u03c03 , \u03c04) = (\u03bb n x \u2192 {!!}) ,\n                                  (\u03bb n x \u2192 {!!})\n\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) with holes-delta-ana h x\u2084 | holes-delta-ana h\u2081 x\u2085\n    ... | ih1 | ih2 = dom-union ih1 ih2\n\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  lem-apart-disjoint : {A : Set} {H : A ctx} {u : Nat} {x : A} \u2192 u # H \u2192 (\u25a0 (u , x)) ## H\n  lem-apart-disjoint {H = H} apt = (\u03bb n x \u2192 tr (\u03bb qq \u2192 qq # H) (singleton-eq (\u03c02 x)) apt) ,\n                                   (\u03bb n x \u2192 {!!})\n\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-disjoint (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-disjoint (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  mutual\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint = {!!}\n\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ed675a211c38e6ec4c7a16e5e9f640db94a45ff7","subject":"Oh yeah!","message":"Oh yeah!\n","repos":"crypto-agda\/crypto-agda","old_file":"isos-examples.agda","new_file":"isos-examples.agda","new_contents":"module isos-examples where\n\nopen import Function\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nimport Function.Related as FR\nopen import Type hiding (\u2605)\nopen import Data.Product.NP\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality as \u2261\n\n_\u2248\u2082_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 Bit) \u2192 \u2605 _\n_\u2248\u2082_ {A = A} f g = \u03a3 A (T \u2218 f) \u2194 \u03a3 A (T \u2218 g)\n\nmodule _ {a r} {A : \u2605 a} {R : \u2605 r} where\n  _\u2248_ : (f g : A \u2192 R) \u2192 \u2605 _\n  f \u2248 g = \u2200 (O : R \u2192 \u2605 r) \u2192 \u03a3 A (O \u2218 f) \u2194 \u03a3 A (O \u2218 g)\n\n  \u2248-refl : Reflexive {A = A \u2192 R} _\u2248_\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : Transitive {A = A \u2192 R} _\u2248_\n  \u2248-trans p q O = q O FI.\u2218 p O\n\n  \u2248-sym : Symmetric {A = A \u2192 R} _\u2248_\n  \u2248-sym p O = FI.sym (p O)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = FI.Inverse.left-inverse-of f\n    right-f = FI.Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    coe' : (xp : \u03a3 A C) \u2192 C (from f (to f (proj\u2081 xp)))\n    coe' (x , p) = coe x p\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C \u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C \u2218 from f) \u2192 \u03a3 A C\n    \u21d0 (x , p) = from f x , p\n    left : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    left (x , p) rewrite left-f x = refl\n    mk\u03a3\u2261 : \u2200 {a b} {A : \u2605 a} {x y : A} (B : A \u2192 \u2605 b) {p : B x} {q : B y} (xy : x \u2261 y) \u2192 subst B xy p \u2261 q \u2192 (x , p) \u2261 (y , q)\n    mk\u03a3\u2261 _ xy h rewrite xy | h = refl\n    right : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    right p = mk\u03a3\u2261 (C \u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 subst (C \u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | refl | refl = refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C \u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) left right\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n  _\u224b_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b g = (f \u2218 proj\u2081) \u2248 (g \u2218 proj\u2082)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C \u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n\n  _\u224b\u2032_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b\u2032 g = \u2200 (O : R \u2192 \u2605 r) \u2192\n            (B \u00d7 \u03a3 A (O \u2218 f)) \u2194 (A \u00d7 \u03a3 B (O \u2218 g))\n\n  module _ {f : A \u2192 R} {g : B \u2192 R} where\n    open FR.EquationalReasoning\n\n    \u224b\u2032\u2192\u224b : f \u224b\u2032 g \u2192 f \u224b g\n    \u224b\u2032\u2192\u224b h O = \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 h O \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u220e\n\n    \u224b\u2192\u224b\u2032 : f \u224b g \u2192 f \u224b\u2032 g\n    \u224b\u2192\u224b\u2032 h O = (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 h O \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u220e\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n","old_contents":"module isos-examples where\n\nopen import Function\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nimport Function.Related as FR\nopen import Type hiding (\u2605)\nopen import Data.Product.NP\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality as \u2261\n\n_\u2248\u2082_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 Bit) \u2192 \u2605 _\n_\u2248\u2082_ {A = A} f g = \u03a3 A (T \u2218 f) \u2194 \u03a3 A (T \u2218 g)\n\nmodule _ {a r} {A : \u2605 a} {R : \u2605 r} where\n  _\u2248_ : (f g : A \u2192 R) \u2192 \u2605 _\n  f \u2248 g = \u2200 (O : R \u2192 \u2605 r) \u2192 \u03a3 A (O \u2218 f) \u2194 \u03a3 A (O \u2218 g)\n\n  \u2248-refl : Reflexive {A = A \u2192 R} _\u2248_\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : Transitive {A = A \u2192 R} _\u2248_\n  \u2248-trans p q O = q O FI.\u2218 p O\n\n  \u2248-sym : Symmetric {A = A \u2192 R} _\u2248_\n  \u2248-sym p O = FI.sym (p O)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = FI.Inverse.left-inverse-of f\n    right-f = FI.Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    coe' : (xp : \u03a3 A C) \u2192 C (from f (to f (proj\u2081 xp)))\n    coe' (x , p) = coe x p\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C \u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C \u2218 from f) \u2192 \u03a3 A C\n    \u21d0 (x , p) = from f x , p\n    left : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    left (x , p) rewrite left-f x = refl\n    mk\u03a3\u2261 : \u2200 {a b} {A : \u2605 a} {x y : A} (B : A \u2192 \u2605 b) {p : B x} {q : B y} (xy : x \u2261 y) \u2192 subst B xy p \u2261 q \u2192 (x , p) \u2261 (y , q)\n    mk\u03a3\u2261 _ xy h rewrite xy | h = refl\n    right : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    right p = mk\u03a3\u2261 (C \u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 subst (C \u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with left-f (from f (proj\u2081 p)) | right-f (proj\u2081 p)\n                ... | q | r = {!!}\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C \u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) left right\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n  _\u224b_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b g = (f \u2218 proj\u2081) \u2248 (g \u2218 proj\u2082)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C \u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n\n  _\u224b\u2032_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b\u2032 g = \u2200 (O : R \u2192 \u2605 r) \u2192\n            (B \u00d7 \u03a3 A (O \u2218 f)) \u2194 (A \u00d7 \u03a3 B (O \u2218 g))\n\n  module _ {f : A \u2192 R} {g : B \u2192 R} where\n    open FR.EquationalReasoning\n\n    \u224b\u2032\u2192\u224b : f \u224b\u2032 g \u2192 f \u224b g\n    \u224b\u2032\u2192\u224b h O = \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 h O \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u220e\n\n    \u224b\u2192\u224b\u2032 : f \u224b g \u2192 f \u224b\u2032 g\n    \u224b\u2192\u224b\u2032 h O = (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 h O \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u220e\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"acfe61c34821988354d7e11d17d312c10a7db793","subject":"patching correspondence after rewrite","message":"patching correspondence after rewrite\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"correspondence.agda","new_file":"correspondence.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule correspondence where\n  mutual\n    correspondence-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0393 \u22a2 e => \u03c4\n    correspondence-synth ESConst = SConst\n    correspondence-synth (ESVar x\u2081) = SVar x\u2081\n    correspondence-synth (ESLam apt ex) with correspondence-synth ex\n    ... | ih = SLam apt ih\n    correspondence-synth (ESAp x\u2081 x\u2082 x\u2083) = SAp (correspondence-synth x\u2081) x\u2082 (correspondence-ana x\u2083)\n    correspondence-synth ESEHole = SEHole\n    correspondence-synth (ESNEHole ex) = SNEHole (correspondence-synth ex)\n    correspondence-synth (ESAsc x) = SAsc (correspondence-ana x)\n\n    correspondence-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          \u0393 \u22a2 e <= \u03c4\n    correspondence-ana (EALam apt ex) = ALam apt MAArr (correspondence-ana ex)\n    correspondence-ana (EALamHole apt D) = ALam apt MAHole (correspondence-ana D)\n    correspondence-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ASubsume (correspondence-synth x\u2082) x\u2083\n    correspondence-ana EAEHole = ASubsume SEHole TCHole1\n    correspondence-ana (EANEHole x) = ASubsume (SNEHole (correspondence-synth x)) TCHole1\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule correspondence where\n  mutual\n    correspondence-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0393 \u22a2 e => \u03c4\n    correspondence-synth ESConst = SConst\n    correspondence-synth (ESVar x\u2081) = SVar x\u2081\n    correspondence-synth (ESLam apt ex) with correspondence-synth ex\n    ... | ih = SLam apt ih\n    correspondence-synth (ESAp1 x\u2081 x\u2082 x\u2083) = SAp x\u2081 MAHole (correspondence-ana x\u2083)\n    correspondence-synth (ESAp2 ex x\u2081 x\u2082) = SAp (correspondence-synth ex) MAArr (correspondence-ana x\u2081)\n    correspondence-synth (ESAp3 ex x\u2081) = SAp (correspondence-synth ex) MAArr (correspondence-ana x\u2081)\n    correspondence-synth ESEHole = SEHole\n    correspondence-synth (ESNEHole ex) = SNEHole (correspondence-synth ex)\n    correspondence-synth (ESAsc1 x _) = SAsc (correspondence-ana x)\n    correspondence-synth (ESAsc2 x) = SAsc (correspondence-ana x)\n\n    correspondence-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          \u0393 \u22a2 e <= \u03c4\n    correspondence-ana (EALam apt ex) = ALam apt MAArr (correspondence-ana ex)\n    correspondence-ana (EALamHole apt D) = ALam apt MAHole (correspondence-ana D)\n    correspondence-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ASubsume (correspondence-synth x\u2082) x\u2083\n    correspondence-ana EAEHole = ASubsume SEHole TCHole1\n    correspondence-ana (EANEHole x) = ASubsume (SNEHole (correspondence-synth x)) TCHole1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"df6f1da421f3c352c1dc518bcd10f8c588c10760","subject":"Bits: +[0\u2192_,1\u2192_] +case_0\u2192_1\u2192_","message":"Bits: +[0\u2192_,1\u2192_] +case_0\u2192_1\u2192_\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n|not| : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n|not| f = not \u2218 f\n\n|\u2227|-comm : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2227| g \u2257 g |\u2227| f\n|\u2227|-comm f g x with f x | g x\n... | 0b | 0b = refl\n... | 0b | 1b = refl\n... | 1b | 0b = refl\n... | 1b | 1b = refl\n\n\n-- open Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\n\nview\u2237 : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n|not| : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n|not| f = not \u2218 f\n\n|\u2227|-comm : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2227| g \u2257 g |\u2227| f\n|\u2227|-comm f g x with f x | g x\n... | 0b | 0b = refl\n... | 0b | 1b = refl\n... | 1b | 0b = refl\n... | 1b | 1b = refl\n\n\n-- open Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\n\nview\u2237 : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0441c2de83a6d06d54d33561c18e934e2920dfb9","subject":"CachingEvaluation: drop design prototypes","message":"CachingEvaluation: drop design prototypes\n\nIt's interesting to read the comments and see how hard it was to come up\nwith the correct semantics --- we should mention this in the paper (on\nit). This drops 3 (incomplete) versions of the semantics.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bc226d27f7d9d5a70140ecdb8c7cc5e36a5c3871","subject":"Desc model: induction implicit.","message":"Desc model: induction implicit.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d58b089f9039ef40a27115d6806fba756b2c3b8d","subject":"Function.Equalitiy: \u2208 and \u2208\u2032 and equivalence","message":"Function.Equalitiy: \u2208 and \u2208\u2032 and equivalence\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Inverse\/NP.agda","new_file":"lib\/Function\/Inverse\/NP.agda","new_contents":"module Function.Inverse.NP where\n\nopen import Function.Inverse public\nimport Function as F\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Relation.Binary\nopen import Data.Product\nopen import Type hiding (\u2605)\nopen import Relation.Binary.PropositionalEquality using (\u2192-to-\u27f6)\nimport Function.Equality as FE\nopen import Function.LeftInverse using (_LeftInverseOf_; _RightInverseOf_)\n\nisEquivalence : \u2200 {f\u2081 f\u2082} \u2192 IsEquivalence (Inverse {f\u2081} {f\u2082})\nisEquivalence = record\n   { refl  = id\n   ; sym   = sym\n   ; trans = F.flip _\u2218_ }\n\nsetoid : \u2200 {c \u2113} \u2192 Setoid _ _\nsetoid {c} {\u2113} = record\n  { Carrier       = Setoid c \u2113\n  ; _\u2248_           = Inverse\n  ; isEquivalence = isEquivalence }\n\ninverses : \u2200 {f t}\n             {From : \u2605 f} {To : \u2605 t}\n             (to   : From \u2192 To) \u2192\n             (from : To \u2192 From) \u2192\n             (\u2192-to-\u27f6 from) LeftInverseOf (\u2192-to-\u27f6 to) \u2192\n             (\u2192-to-\u27f6 from) RightInverseOf (\u2192-to-\u27f6 to) \u2192\n             From \u2194 To\ninverses to from left right =\n  record { inverse-of = inv }\n  where inv : \u2192-to-\u27f6 from InverseOf \u2192-to-\u27f6 to\n        inv = record { left-inverse-of = left ; right-inverse-of = right }\n\n_$\u2081_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n         {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n       \u2192 Inverse From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\niso $\u2081 x = Inverse.to iso \u27e8$\u27e9 x\n\n_$\u2082_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n         {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n       \u2192 Inverse From To \u2192 Setoid.Carrier To \u2192 Setoid.Carrier From\niso $\u2082 x = Inverse.from iso \u27e8$\u27e9 x\n\nmodule _ {a b ra rb}\n         {A  : Setoid a ra}\n         {B  : Setoid b rb}\n  where\n  module A\u02e2 = Setoid A\n  module B\u02e2 = Setoid B\n  open A\u02e2 renaming (_\u2248_ to _\u2248\u1d2c_)\n  open B\u02e2 renaming (_\u2248_ to _\u2248\u1d2e_)\n\n  module _ (x\u1d62 : Setoid.Carrier A \u00d7 Setoid.Carrier B)\n           (f  : Inverse A B)\n           where\n    x\u2081 = proj\u2081 x\u1d62\n    x\u2082 = proj\u2082 x\u1d62\n\n    _\u2208_  : \u2605 rb\n    _\u2208_  = f $\u2081 x\u2081 \u2248\u1d2e x\u2082\n\n    _\u2208\u2032_ : \u2605 ra\n    _\u2208\u2032_ = x\u2081 \u2248\u1d2c f $\u2082 x\u2082\n\n  \u2208\u2192\u2208\u2032 : \u2200 f x\u1d62 \u2192 x\u1d62 \u2208 f \u2192 x\u1d62 \u2208\u2032 f\n  \u2208\u2192\u2208\u2032 f (x\u2081 , x\u2082) p = A\u02e2.trans (A\u02e2.sym (Inverse.left-inverse-of f x\u2081)) (FE.cong (Inverse.from f) p)\n\n  \u2208\u2032\u2192\u2208 : \u2200 f x\u1d62 \u2192 x\u1d62 \u2208\u2032 f \u2192 x\u1d62 \u2208 f\n  \u2208\u2032\u2192\u2208 f (x\u2081 , x\u2082) p = B\u02e2.trans (FE.cong (Inverse.to f) p) (Inverse.right-inverse-of f x\u2082)\n","old_contents":"module Function.Inverse.NP where\n\nopen import Function.Inverse public\nimport Function as F\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Relation.Binary\nopen import Data.Product\nopen import Type hiding (\u2605)\nopen import Relation.Binary.PropositionalEquality using (\u2192-to-\u27f6)\nopen import Function.LeftInverse using (_LeftInverseOf_; _RightInverseOf_)\n\nisEquivalence : \u2200 {f\u2081 f\u2082} \u2192 IsEquivalence (Inverse {f\u2081} {f\u2082})\nisEquivalence = record\n   { refl  = id\n   ; sym   = sym\n   ; trans = F.flip _\u2218_ }\n\nsetoid : \u2200 {c \u2113} \u2192 Setoid _ _\nsetoid {c} {\u2113} = record\n  { Carrier       = Setoid c \u2113\n  ; _\u2248_           = Inverse\n  ; isEquivalence = isEquivalence }\n\ninverses : \u2200 {f t}\n             {From : \u2605 f} {To : \u2605 t}\n             (to   : From \u2192 To) \u2192\n             (from : To \u2192 From) \u2192\n             (\u2192-to-\u27f6 from) LeftInverseOf (\u2192-to-\u27f6 to) \u2192\n             (\u2192-to-\u27f6 from) RightInverseOf (\u2192-to-\u27f6 to) \u2192\n             From \u2194 To\ninverses to from left right =\n  record { inverse-of = inv }\n  where inv : \u2192-to-\u27f6 from InverseOf \u2192-to-\u27f6 to\n        inv = record { left-inverse-of = left ; right-inverse-of = right }\n\n_$\u2081_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n         {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n       \u2192 Inverse From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\niso $\u2081 x = Inverse.to iso \u27e8$\u27e9 x\n\n_$\u2082_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n         {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n       \u2192 Inverse From To \u2192 Setoid.Carrier To \u2192 Setoid.Carrier From\niso $\u2082 x = Inverse.from iso \u27e8$\u27e9 x\n\n_\u2208_ : \u2200 {a\u2081 a\u2082}\n        {A\u2081 A\u2082  : Setoid a\u2081 a\u2082}\n        (x\u1d62     : Setoid.Carrier A\u2081 \u00d7 Setoid.Carrier A\u2082)\n        (bij    : Inverse A\u2081 A\u2082)\n      \u2192 \u2605 a\u2082\n_\u2208_ {A\u2082 = A\u2082} (x\u2081 , x\u2082) iso = iso $\u2081 x\u2081 \u2248 x\u2082\n  where open Setoid A\u2082\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cc23c6f94c8d824a0d666b561e586235c8601e18","subject":"sum.agda","message":"sum.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n  \nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x \n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = ?\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n(\u03bcA +\u03bc \u03bcB) = mk srch ext mono swp eps hom\n  where\n    s\u1d2c = search \u03bcA\n    s\u1d2e = search \u03bcB\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e \n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB)\n   = mk srch ext mono swp eps hom\n   where\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k \nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSumZer : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZer sum\u1d43 = sum\u1d43 (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMon : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMon sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nrecord SumLinear {A} sum\u1d2c : \u2605 where\n  constructor mk\n  field\n    sum-lin : SumLin {A} sum\u1d2c\n    sum-hom : SumHom sum\u1d2c\n\nSumSwap : \u2200 {A} \u2192 Sum A \u2192 \u2605\u2081\nSumSwap {A} sum\u1d2c = \u2200 {B} f {sum\u1d2e} \u2192 SumLinear {B} sum\u1d2e \u2192 sum\u1d2c (sum\u1d2e \u2218 f) \u2261 sum\u1d2e (sum\u1d2c \u2218 flip f)\n\nmkSumLin : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumHom sum \u2192 SumZer sum \u2192 SumLin sum\nmkSumLin sum-hom sum-zer f zero    = sum-zer\nmkSumLin {sum = sum} sum-hom sum-zer f (suc k) rewrite\n    sum-hom f (\u03bb x \u2192 k * f x)\n  | mkSumLin {sum = sum} sum-hom sum-zer f k = refl\n\nmkSumZer : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZer sum\nmkSumZer sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMon sum \u2192 SumExt sum\nmkSumExt sum-mon f\u2257g = \u2115\u2264.antisym (sum-mon (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mon (\u2115\u2264.reflexive \u2218 sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMon sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    sum : Sum A\n    is-linear : SumLinear sum\n    sum-ext : SumExt sum\n    sum-mon : SumMon sum\n    sum-swap : SumSwap sum\n\n  -- open SumLinear is-linear public\n\n  sum-lin = SumLinear.sum-lin is-linear\n  sum-hom = SumLinear.sum-hom is-linear\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\nsum\u22a4 : Sum \u22a4\nsum\u22a4 f = f _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk sum\u22a4 (mk sum\u22a4-lin sum\u22a4-hom) sum\u22a4-ext sum\u22a4-mon sum\u22a4-swap\n  where\n    sum\u22a4-ext : SumExt sum\u22a4\n    sum\u22a4-ext f\u2257g = f\u2257g _\n\n    sum\u22a4-zer : SumZer sum\u22a4\n    sum\u22a4-zer = refl\n\n    sum\u22a4-lin : SumLin sum\u22a4\n    sum\u22a4-lin f k = refl\n\n    sum\u22a4-hom : SumHom sum\u22a4\n    sum\u22a4-hom f g = refl\n\n    sum\u22a4-mon : SumMon sum\u22a4\n    sum\u22a4-mon f\u2264\u00b0g = f\u2264\u00b0g _\n\n    sum\u22a4-swap : SumSwap sum\u22a4\n    sum\u22a4-swap f sum\u1d2c-linear = refl\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk sumBit (mk sumBit-lin sumBit-hom) sumBit-ext sumBit-mon sumBit-swap\n  where\n    sumBit-ext : SumExt sumBit\n    sumBit-ext f\u2257g rewrite f\u2257g 0b | f\u2257g 1b = refl\n\n    sumBit-zero : sumBit (const 0) \u2261 0\n    sumBit-zero = refl\n\n    sumBit-hom : SumHom sumBit\n    sumBit-hom f g = +-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n\n    sumBit-lin : SumLin sumBit\n    sumBit-lin = mkSumLin sumBit-hom sumBit-zero\n\n    sumBit-mon : SumMon sumBit\n    sumBit-mon f\u2264\u00b0g = f\u2264\u00b0g 0b +-mono f\u2264\u00b0g 1b\n\n    sumBit-swap : SumSwap sumBit\n    sumBit-swap f sum\u1d2c-linear rewrite SumLinear.sum-hom sum\u1d2c-linear (f 0b) (f 1b) = refl\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n(\u03bcA +\u03bc \u03bcB) = mk +-\u03bc (mk +\u03bc-lin +\u03bc-hom) +\u03bc-ext +\u03bc-mon +\u03bc-swap\n  where\n    +-\u03bc = sum \u03bcA +Sum sum \u03bcB\n    +\u03bc-ext : SumExt +-\u03bc \n    +\u03bc-ext f\u2257g rewrite sum-ext \u03bcA (f\u2257g \u2218 inj\u2081) | sum-ext \u03bcB (f\u2257g \u2218 inj\u2082) = refl\n\n    +\u03bc-zero : +-\u03bc (\u03bb _ \u2192 zero) \u2261 zero\n    +\u03bc-zero rewrite sum-lin \u03bcA (\u03bb _ \u2192 0) 0 | sum-lin \u03bcB (\u03bb _ \u2192 0) 0 = refl\n\n    +\u03bc-lin : SumLin +-\u03bc\n    +\u03bc-lin f k rewrite sum-lin \u03bcA (f \u2218 inj\u2081) k | sum-lin \u03bcB (f \u2218 inj\u2082) k \n          = sym (proj\u2081 \u2115\u00b0.distrib k (sum \u03bcA (f \u2218 inj\u2081)) (sum \u03bcB (f \u2218 inj\u2082)))\n\n    +\u03bc-hom : SumHom +-\u03bc\n    +\u03bc-hom f g rewrite sum-hom \u03bcA (f \u2218 inj\u2081) (g \u2218 inj\u2081) | sum-hom \u03bcB (f \u2218 inj\u2082) (g \u2218 inj\u2082)\n          = +-interchange (sum \u03bcA (f \u2218 inj\u2081)) (sum \u03bcA (g \u2218 inj\u2081))\n              (sum \u03bcB (f \u2218 inj\u2082)) (sum \u03bcB (g \u2218 inj\u2082))\n          \n    +\u03bc-mon : SumMon +-\u03bc\n    +\u03bc-mon f\u2264\u00b0g = sum-mon \u03bcA (f\u2264\u00b0g \u2218 inj\u2081) +-mono sum-mon \u03bcB (f\u2264\u00b0g \u2218 inj\u2082)\n\n    +\u03bc-swap : SumSwap +-\u03bc\n    +\u03bc-swap f {sum\u1d2c} sum\u1d2c-linear rewrite SumLinear.sum-hom sum\u1d2c-linear (\u03bb x \u2192 sum \u03bcA (\u03bb y \u2192 f (inj\u2081 y) x)) (\u03bb x \u2192 sum \u03bcB (\u03bb y \u2192 f (inj\u2082 y) x))\n      | sum-swap \u03bcA (f \u2218 inj\u2081) sum\u1d2c-linear\n      | sum-swap \u03bcB (f \u2218 inj\u2082) sum\u1d2c-linear\n      = refl\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n_\u00d7Sum-ext_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} \u2192\n              SumExt sum\u1d2c \u2192 SumExt sum\u1d2e \u2192 SumExt (sum\u1d2c \u00d7Sum sum\u1d2e)\n(sumA-ext \u00d7Sum-ext sumB-ext) f\u2257g = sumA-ext (sumB-ext \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB)\n   = mk \u03bc-\u00d7 (mk lin hom) (sum-ext \u03bcA \u00d7Sum-ext sum-ext \u03bcB) mon \u00d7\u03bc-swap\n   where\n     \u03bc-\u00d7 = sum \u03bcA \u00d7Sum sum \u03bcB\n\n     zero' : \u03bc-\u00d7 (\u03bb _ \u2192 0) \u2261 0\n     zero' rewrite sum-lin \u03bcB (\u03bb _ \u2192 0) 0 = sum-lin \u03bcA (\u03bb _ \u2192 0) 0\n\n     lin : SumLin \u03bc-\u00d7\n     lin f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-lin \u03bcB (\u03bb y \u2192 f (x , y)) k) = sum-lin \u03bcA (sum \u03bcB \u2218 curry f) k\n\n     hom : SumHom \u03bc-\u00d7\n     hom f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-hom \u03bcB (\u03bb y \u2192 f (x , y)) (\u03bb y \u2192 g (x , y))) \n         = sum-hom \u03bcA (sum \u03bcB \u2218 curry f) (sum \u03bcB \u2218 curry g)\n\n     mon : SumMon \u03bc-\u00d7\n     mon f\u2264\u00b0g = sum-mon \u03bcA (sum-mon \u03bcB \u2218 curry f\u2264\u00b0g)\n\n     \u00d7\u03bc-swap : SumSwap \u03bc-\u00d7\n     \u00d7\u03bc-swap f {sum\u02e3} sum\u02e3-linear rewrite sum-ext \u03bcA  (\u03bb x \u2192 sum-swap \u03bcB (curry f x) sum\u02e3-linear)\n       = sum-swap \u03bcA (\u03bb z x \u2192 sum \u03bcB (\u03bb y \u2192 f (z , y) x)) sum\u02e3-linear\n\nsum-0 : \u2200 {A} (\u03bcA : SumProp A) \u2192 sum \u03bcA (const 0) \u2261 0\nsum-0 \u03bcA = sum-lin \u03bcA (\u03bb _ \u2192 0) 0\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k \nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum)\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin n)\n\u03bcFin n = mk (sumFin n) (mk sumFin-lin sumFin-hom) sumFin-ext sumFin-mon sumFin-swap\n  module SumFin where\n    sumFin-ext : SumExt (sumFin n)\n    sumFin-ext f\u2257g rewrite map-ext f\u2257g (allFin n) = refl\n\n    sumFin-lin : SumLin (sumFin n)\n    sumFin-lin f x = sum-distrib\u02e1 f x (allFin n)\n\n    sumFin-zero : SumZer (sumFin n)\n    sumFin-zero = mkSumZer sumFin-lin\n    \n    sumFin-hom : SumHom (sumFin n)\n    sumFin-hom f g = sum-linear f g (allFin n)\n\n    sumFin-mon : SumMon (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = sum\u1d2c (\u03bb x \u2192 sumVec n sum\u1d2c (f \u2218 _\u2237_ x))\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec {A} \u03bcA n = mk (sV n)  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMon (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mon \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n\nsumVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m)\u2192 \u2115) \n           \u2192 sum (\u03bcVec \u03bcA (n + m)) (\u03bb xs \u2192 f xs)\n           \u2261 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\nsumVec-++ zero    \u03bcA f = refl\nsumVec-++ (suc n) \u03bcA f = sum-ext \u03bcA (\u03bb x \u2192 sumVec-++ n \u03bcA (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bb xs ys \u2192 f (xs ++ ys))\n                           (is-linear (\u03bcVec \u03bcA m))\n\n\n\u03bc-view : \u2200 {A B : Set} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk sumB (mk \u03bcB-lin \u03bcB-hom) \u03bcB-ext \u03bcB-mon \u03bcB-swap\n  where\n    sumB : Sum B\n    sumB f = sum \u03bcA (f \u2218 A\u2192B)\n\n    \u03bcB-ext : SumExt sumB\n    \u03bcB-ext f\u2257g = sum-ext \u03bcA (f\u2257g \u2218 A\u2192B)\n\n    \u03bcB-lin : SumLin sumB\n    \u03bcB-lin f k = sum-lin \u03bcA (f \u2218 A\u2192B) k\n\n    \u03bcB-hom : SumHom sumB\n    \u03bcB-hom f g = sum-hom \u03bcA (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    \u03bcB-mon : SumMon sumB\n    \u03bcB-mon f\u2264\u00b0g = sum-mon \u03bcA (f\u2264\u00b0g \u2218 A\u2192B)\n\n    \u03bcB-swap : SumSwap sumB\n    \u03bcB-swap f {sum\u02e3} sum\u02e3-linear = sum-swap \u03bcA (f \u2218 A\u2192B) sum\u02e3-linear\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (cong f \u2218 A\u2248B)\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 refl) f \u03bcA\u00d7B {- or refl -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"11dd97138b62b839e6eff70a2897426660d4fd10","subject":"Rename (ignore|update)-valid-change to (before|after).","message":"Rename (ignore|update)-valid-change to (before|after).\n\nOld-commit-hash: 16288a40704567fce8f03420af8050326b9b9b34\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- Change : Type \u2192 Set\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : Change \u03c3) \u2192 valid v dv \u2192 Change \u03c4\n\n  before : ValidChange \u2286 \u27e6_\u27e7\n  before (cons v _ _) = v\n\n  after : ValidChange \u2286 \u27e6_\u27e7\n  after {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : Change \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 before {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 after {\u03c4})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- Change : Type \u2192 Set\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : Change \u03c3) \u2192 valid v dv \u2192 Change \u03c4\n\n  ignore-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  ignore-valid-change (cons v _ _) = v\n\n  update-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  update-valid-change {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : Change \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-valid-change {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-valid-change {\u03c4})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a2af2da2688a19ee49290626b8c34c74068663b9","subject":"0\u2026 and 1\u2026 => 0\u207f and 1\u207f","message":"0\u2026 and 1\u2026 => 0\u207f and 1\u207f\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u207f) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u207f (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u2026 : \u2200 {n} \u2192 Bits n\n0\u2026 = replicate 0b\n\n-- Warning: 0\u2026 {0} \u2261 1\u2026 {0}\n1\u2026 : \u2200 {n} \u2192 Bits n\n1\u2026 = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u2026\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u2026 \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u2026 x) \u27e9\n         x \u2295 (x \u2295 1\u2026) \u2261\u27e8 sym (\u2295-assoc x x 1\u2026) \u27e9\n         (x \u2295 x) \u2295 1\u2026 \u2261\u27e8 cong (flip _\u2295_ 1\u2026) (\u2295-\u2261 x) \u27e9\n         0\u2026 \u2295 1\u2026       \u2261\u27e8 \u2295-left-identity 1\u2026 \u27e9\n         1\u2026 \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u2026) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u2026 (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u2026 sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ee2399f5b4ae1a7b1bc02f3fb4c5ac4f1d922676","subject":"Rolled back an wrong modification.","message":"Rolled back an wrong modification.\n\nIgnore-this: b9dc25330c7cce1bf705bab719df0d28\n\ndarcs-hash:20110303195314-3bd4e-9809087b6f4b1b979800cd53d676ced8880dffbf.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/Tree\/ClosuresI.agda","new_file":"Draft\/Mirror\/Tree\/ClosuresI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with the closures of the tree type\n------------------------------------------------------------------------------\n\n-- {-# OPTIONS --no-termination-check #-}\n{-# OPTIONS --injective-type-constructors #-}\n\nmodule Draft.Mirror.Tree.ClosuresI where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\nopen import FOTC.Program.Mirror.Mirror\nopen import FOTC.Program.Mirror.ListTree.Closures\nopen import FOTC.Program.Mirror.ListTree.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nnode-ListTree : \u2200 {d ts} \u2192 Tree (node d ts) \u2192 ListTree ts\nnode-ListTree {d} (treeT .d LTts) = LTts\n\npostulate\n  reverse-\u2237' : \u2200 x ys \u2192 reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\n\nmirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\nmirror-Tree (treeT d nilLT) =\n  subst Tree (sym (mirror-eq d [])) (treeT d helper\u2082)\n    where\n      helper\u2081 : rev (map mirror []) [] \u2261 []\n      helper\u2081 =\n        begin\n          rev (map mirror []) []\n          \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n                   (map-[] mirror)\n                   refl\n          \u27e9\n          rev [] []\n            \u2261\u27e8 rev-[] [] \u27e9\n          []\n        \u220e\n\n      helper\u2082 : ListTree (rev (map mirror []) [])\n      helper\u2082 = subst ListTree (sym helper\u2081) nilLT\n\nmirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n  subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d helper)\n\n  where\n     h\u2081 : Tree (node d (reverse (map mirror ts)))\n     h\u2081 = subst Tree (mirror-eq d ts) (mirror-Tree (treeT d LTts))\n\n     h\u2082 : ListTree (reverse (map mirror ts))\n     h\u2082 = node-ListTree h\u2081\n\n     h\u2083 : ListTree ((reverse (map mirror ts)) ++ (mirror \u00b7 t \u2237 []))\n     h\u2083 = ++-ListTree h\u2082 (consLT (mirror-Tree Tt) nilLT)\n\n     h\u2084 : ListTree (reverse (mirror \u00b7 t \u2237 map mirror ts))\n     h\u2084 = subst ListTree ( sym (reverse-\u2237' (mirror \u00b7 t) (map mirror ts))) h\u2083\n\n     helper : ListTree (reverse (map mirror (t \u2237 ts)))\n     helper = subst ListTree\n              (subst (\u03bb x \u2192 reverse x \u2261 reverse (map mirror (t \u2237 ts)))\n                     (map-\u2237 mirror t ts)\n                     refl)\n              h\u2084\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with the closures of the tree type\n------------------------------------------------------------------------------\n\n-- {-# OPTIONS --no-termination-check #-}\n{-# OPTIONS --injective-type-constructors #-}\n\nmodule Draft.Mirror.Tree.ClosuresI where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\nopen import FOTC.Program.Mirror.Mirror\nopen import FOTC.Program.Mirror.ListTree.Closures\nopen import FOTC.Program.Mirror.ListTree.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nnode-ListTree : \u2200 {d ts} \u2192 Tree (node d ts) \u2192 ListTree ts\nnode-ListTree {d} (treeT .d LTts) = LTts\n\npostulate\n  reverse-\u2237' : \u2200 x ys \u2192 reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\n\nmirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\nmirror-Tree (treeT d nilLT) =\n  subst Tree (sym (mirror-eq d [])) (treeT d helper\u2082)\n    where\n      helper\u2081 : rev (map mirror []) [] \u2261 []\n      helper\u2081 =\n        begin\n          rev (map mirror []) []\n          \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n                   (map-[] mirror)\n                   refl\n          \u27e9\n          rev [] []\n            \u2261\u27e8 rev-[] [] \u27e9\n          []\n        \u220e\n\n      helper\u2082 : ListTree (rev (map mirror []) [])\n      helper\u2082 = subst ListTree (sym helper\u2081) nilLT\n\nmirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n  prf (mirror-Tree Tt) (mirror-Tree (treeT d LTts))\n  where\n    postulate prf : Tree (mirror \u00b7 t) \u2192\n                    Tree (mirror \u00b7 (node d ts)) \u2192\n                    Tree (mirror \u00b7 node d (t \u2237 ts))\n\n\n  -- let h\u2081 : Tree (node d (reverse (map mirror ts)))\n  --     h\u2081 = subst Tree (mirror-eq d ts) (mirror-Tree (treeT d LTts))\n\n  --     h\u2082 : ListTree (reverse (map mirror ts))\n  --     h\u2082 = node-ListTree h\u2081\n\n  --     h\u2083 : ListTree ((reverse (map mirror ts)) ++ (mirror \u00b7 t \u2237 []))\n  --     h\u2083 = ++-ListTree h\u2082 (consLT (mirror-Tree Tt) nilLT)\n\n  --     h\u2084 : ListTree (reverse (mirror \u00b7 t \u2237 map mirror ts))\n  --     h\u2084 = subst ListTree ( sym (reverse-\u2237' (mirror \u00b7 t) (map mirror ts))) h\u2083\n\n  --     helper : ListTree (reverse (map mirror (t \u2237 ts)))\n  --     helper = subst ListTree\n  --              (subst (\u03bb x \u2192 reverse x \u2261 reverse (map mirror (t \u2237 ts)))\n  --                     (map-\u2237 mirror t ts)\n  --                     refl)\n  --              h\u2084\n\n\n   -- in subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d helper)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4f3993e47b68e7f72d1a8e5af649750d6d296a3c","subject":"Agda cosmetic fixup","message":"Agda cosmetic fixup\n\nOld-commit-hash: 57dc81427e7e0c57d6c7721c83c7ee900a54b583\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} dfxdx\u2259dgxdx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      open import Postulate.Extensionality\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = dfxdx\u2259dgxdx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : FC.FunctionChange f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} dfxdx\u2259dgxdx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      open import Postulate.Extensionality\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = dfxdx\u2259dgxdx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c9417b5d660274f8df1d9f82827339727e574ac","subject":"Reindent, add line breaks","message":"Reindent, add line breaks\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/MTerm.agda","new_file":"Parametric\/Syntax\/MTerm.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Syntax.MType as MType\n\nmodule Parametric.Syntax.MTerm\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n  where\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Term.Structure Base Const\n\n-- Our extension points are sets of primitives, indexed by the\n-- types of their arguments and their return type.\n\n-- We want different types of constants; some produce values, some produce\n-- computations. In all cases, arguments are assumed to be values (though\n-- individual primitives might require thunks).\n\nValConstStructure : Set\u2081\nValConstStructure = ValContext \u2192 ValType \u2192 Set\n\nCompConstStructure : Set\u2081\nCompConstStructure = ValContext \u2192 CompType \u2192 Set\n\nmodule Structure (ValConst : ValConstStructure) (CompConst : CompConstStructure) where\n  mutual\n    -- Analogues of Terms\n    Vals : ValContext \u2192 ValContext \u2192 Set\n    Vals \u0393 = DependentList (Val \u0393)\n\n    Comps : ValContext \u2192 List CompType \u2192 Set\n    Comps \u0393 = DependentList (Comp \u0393)\n\n    data Val \u0393 : (\u03c4 : ValType) \u2192 Set where\n      vVar : \u2200 {\u03c4} (x : ValVar \u0393 \u03c4) \u2192 Val \u0393 \u03c4\n      -- XXX Do we need thunks? The draft in the paper doesn't have them.\n      -- However, they will start being useful if we deal with CBN source\n      -- languages.\n      vThunk : \u2200 {\u03c4} \u2192 Comp \u0393 \u03c4 \u2192 Val \u0393 (U \u03c4)\n      vConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : ValConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Val \u0393 \u03c4\n\n    data Comp \u0393 : (\u03c4 : CompType) \u2192 Set where\n      cConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : CompConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Comp \u0393 \u03c4\n\n      cForce : \u2200 {\u03c4} \u2192 Val \u0393 (U \u03c4) \u2192 Comp \u0393 \u03c4\n      cReturn : \u2200 {\u03c4} (v : Val \u0393 \u03c4) \u2192 Comp \u0393 (F \u03c4)\n      {-\n      -- Originally, M to x in N. But here we have no names!\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 \u03c4\n      -}\n      -- The following constructor is the main difference between CBPV and this\n      -- monadic calculus. This is better for the caching transformation.\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) (F \u03c4)) \u2192\n        Comp \u0393 (F \u03c4)\n      cAbs : \u2200 {\u03c3 \u03c4} \u2192\n        (t : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 (\u03c3 \u21db \u03c4)\n      cApp : \u2200 {\u03c3 \u03c4} \u2192\n        (s : Comp \u0393 (\u03c3 \u21db \u03c4)) \u2192\n        (t : Val \u0393 \u03c3) \u2192\n        Comp \u0393 \u03c4\n\n  weaken-val : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Val \u0393\u2081 \u03c4 \u2192\n    Val \u0393\u2082 \u03c4\n\n  weaken-comp : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Comp \u0393\u2081 \u03c4 \u2192\n    Comp \u0393\u2082 \u03c4\n\n  weaken-vals : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Vals \u0393\u2081 \u03a3 \u2192\n    Vals \u0393\u2082 \u03a3\n\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vVar x) = vVar (weaken-val-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vThunk x) = vThunk (weaken-comp \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vConst c args) = vConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cConst c args) = cConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cForce x) = cForce (weaken-val \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cReturn v) = cReturn (weaken-val \u0393\u2081\u227c\u0393\u2082 v)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (_into_ {\u03c3} c c\u2081) = (weaken-comp \u0393\u2081\u227c\u0393\u2082 c) into (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c\u2081)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cAbs {\u03c3} c) = cAbs (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cApp s t) = cApp (weaken-comp \u0393\u2081\u227c\u0393\u2082 s) (weaken-val \u0393\u2081\u227c\u0393\u2082 t)\n\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 (px \u2022 ts) = (weaken-val \u0393\u2081\u227c\u0393\u2082 px) \u2022 (weaken-vals \u0393\u2081\u227c\u0393\u2082 ts)\n\n  fromCBN : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBNCtx \u0393) (cbnToCompType \u03c4)\n\n  fromCBNTerms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Vals (fromCBNCtx \u0393) (fromCBNCtx \u03a3)\n  fromCBNTerms \u2205 = \u2205\n  fromCBNTerms (px \u2022 ts) = vThunk (fromCBN px) \u2022 fromCBNTerms ts\n\n  open import UNDEFINED\n  -- This is really supposed to be part of the plugin interface.\n  cbnToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBNCtx \u03a3) (cbnToCompType \u03c4)\n  cbnToCompConst = reveal UNDEFINED\n\n  fromCBN (const c args) = cConst (cbnToCompConst c) (fromCBNTerms args)\n  fromCBN (var x) = cForce (vVar (fromVar cbnToValType x))\n  fromCBN (app s t) = cApp (fromCBN s) (vThunk (fromCBN t))\n  fromCBN (abs t) = cAbs (fromCBN t)\n\n  -- To satisfy termination checking, we'd need to inline fromCBV and weaken: fromCBV needs to produce a term in a bigger context.\n  -- But let's ignore that.\n  {-# NO_TERMINATION_CHECK #-}\n  fromCBV : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n\n  -- This is really supposed to be part of the plugin interface.\n  cbvToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBVCtx \u03a3) (cbvToCompType \u03c4)\n  cbvToCompConst = reveal UNDEFINED\n\n  cbvTermsToComps : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Comps (fromCBVCtx \u0393) (fromCBVToCompList \u03a3)\n  cbvTermsToComps \u2205 = \u2205\n  cbvTermsToComps (px \u2022 ts) = fromCBV px \u2022 cbvTermsToComps ts\n\n\n  dequeValContexts : ValContext \u2192 ValContext \u2192 ValContext\n  dequeValContexts \u2205 \u0393 = \u0393\n  dequeValContexts (x \u2022 \u03a3) \u0393 = dequeValContexts \u03a3 (x \u2022 \u0393)\n\n  dequeValContexts\u227c\u227c : \u2200 \u03a3 \u0393 \u2192 \u0393 \u227c\u227c dequeValContexts \u03a3 \u0393\n  dequeValContexts\u227c\u227c \u2205 \u0393 = \u227c\u227c-refl\n  dequeValContexts\u227c\u227c (x \u2022 \u03a3) \u0393 = \u227c\u227c-trans (drop_\u2022\u2022_ x \u227c\u227c-refl) (dequeValContexts\u227c\u227c \u03a3 (x \u2022 \u0393))\n\n  dequeContexts : Context \u2192 Context \u2192 ValContext\n  dequeContexts \u03a3 \u0393 = dequeValContexts (fromCBVCtx \u03a3) (fromCBVCtx \u0393)\n\n  fromCBVArg :\n    \u2200 {\u03c3 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192\n      (Val (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToValType \u03c4) \u2192\n        Comp (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToCompType \u03c3)) \u2192\n      Comp (fromCBVCtx \u0393) (cbvToCompType \u03c3)\n  fromCBVArg t k =\n    (fromCBV t) into\n      k (vVar vThis)\n\n  fromCBVArgs :\n    \u2200 {\u03a3 \u0393 \u03c4} \u2192 Terms \u0393 \u03a3 \u2192\n      (Vals (dequeContexts \u03a3 \u0393) (fromCBVCtx \u03a3) \u2192\n        Comp (dequeContexts \u03a3 \u0393) (cbvToCompType \u03c4)) \u2192\n      Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n  fromCBVArgs \u2205 k = k \u2205\n  fromCBVArgs {\u03c3 \u2022 \u03a3} {\u0393} (t \u2022 ts) k =\n    fromCBVArg t (\u03bb v \u2192\n      fromCBVArgs\n        (weaken-terms (drop_\u2022_ _ \u227c-refl) ts) (\u03bb vs \u2192\n          k (weaken-val (dequeValContexts\u227c\u227c (fromCBVCtx \u03a3) _) v \u2022 vs)))\n\n  -- Transform a constant application into\n  --\n  --   arg1 to x1 in (arg2 to x2 in ... (argn to xn in (cConst (cbvToCompConst c) (x1 :: x2 :: ... xn)))).\n\n  fromCBVConstCPSRoot : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 Terms \u0393 \u03a3 \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n  fromCBVConstCPSRoot c ts = fromCBVArgs ts (cConst (cbvToCompConst c))\n\n  fromCBV (const c args) = fromCBVConstCPSRoot c args\n  fromCBV (app s t) =\n    (fromCBV s) into\n      (fromCBV (weaken (drop _ \u2022 \u227c-refl) t) into\n        cApp (cForce (vVar (vThat vThis))) (vVar vThis))\n  -- Values\n  fromCBV (var x) = cReturn (vVar (fromVar cbvToValType x))\n  fromCBV (abs t) = cReturn (vThunk (cAbs (fromCBV t)))\n\n  -- This reflects the CBV conversion of values in TLCA '99, but we can't write\n  -- this because it's partial on the Term type. Instead, we duplicate thunking\n  -- at the call site.\n  {-\n  fromCBVToVal : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Val (fromCBVCtx \u0393) (cbvToValType \u03c4)\n  fromCBVToVal (var x) = vVar (fromVar cbvToValType x)\n  fromCBVToVal (abs t) = vThunk (cAbs (fromCBV t))\n\n  fromCBVToVal (const c args) = {!!}\n  fromCBVToVal (app t t\u2081) = {!!} -- Not a value\n  -}\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Syntax.MType as MType\n\nmodule Parametric.Syntax.MTerm\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n  where\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Term.Structure Base Const\n\n-- Our extension points are sets of primitives, indexed by the\n-- types of their arguments and their return type.\n\n-- We want different types of constants; some produce values, some produce\n-- computations. In all cases, arguments are assumed to be values (though\n-- individual primitives might require thunks).\n\nValConstStructure : Set\u2081\nValConstStructure = ValContext \u2192 ValType \u2192 Set\n\nCompConstStructure : Set\u2081\nCompConstStructure = ValContext \u2192 CompType \u2192 Set\n\nmodule Structure (ValConst : ValConstStructure) (CompConst : CompConstStructure) where\n  mutual\n    -- Analogues of Terms\n    Vals : ValContext \u2192 ValContext \u2192 Set\n    Vals \u0393 = DependentList (Val \u0393)\n\n    Comps : ValContext \u2192 List CompType \u2192 Set\n    Comps \u0393 = DependentList (Comp \u0393)\n\n    data Val \u0393 : (\u03c4 : ValType) \u2192 Set where\n      vVar : \u2200 {\u03c4} (x : ValVar \u0393 \u03c4) \u2192 Val \u0393 \u03c4\n      -- XXX Do we need thunks? The draft in the paper doesn't have them.\n      -- However, they will start being useful if we deal with CBN source\n      -- languages.\n      vThunk : \u2200 {\u03c4} \u2192 Comp \u0393 \u03c4 \u2192 Val \u0393 (U \u03c4)\n      vConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : ValConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Val \u0393 \u03c4\n\n    data Comp \u0393 : (\u03c4 : CompType) \u2192 Set where\n      cConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : CompConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Comp \u0393 \u03c4\n\n      cForce : \u2200 {\u03c4} \u2192 Val \u0393 (U \u03c4) \u2192 Comp \u0393 \u03c4\n      cReturn : \u2200 {\u03c4} (v : Val \u0393 \u03c4) \u2192 Comp \u0393 (F \u03c4)\n      {-\n      -- Originally, M to x in N. But here we have no names!\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 \u03c4\n      -}\n      -- The following constructor is the main difference between CBPV and this\n      -- monadic calculus. This is better for the caching transformation.\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) (F \u03c4)) \u2192\n        Comp \u0393 (F \u03c4)\n      cAbs : \u2200 {\u03c3 \u03c4} \u2192\n        (t : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 (\u03c3 \u21db \u03c4)\n      cApp : \u2200 {\u03c3 \u03c4} \u2192\n        (s : Comp \u0393 (\u03c3 \u21db \u03c4)) \u2192\n        (t : Val \u0393 \u03c3) \u2192\n        Comp \u0393 \u03c4\n\n  weaken-val : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Val \u0393\u2081 \u03c4 \u2192\n    Val \u0393\u2082 \u03c4\n\n  weaken-comp : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Comp \u0393\u2081 \u03c4 \u2192\n    Comp \u0393\u2082 \u03c4\n\n  weaken-vals : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Vals \u0393\u2081 \u03a3 \u2192\n    Vals \u0393\u2082 \u03a3\n\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vVar x) = vVar (weaken-val-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vThunk x) = vThunk (weaken-comp \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vConst c args) = vConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cConst c args) = cConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cForce x) = cForce (weaken-val \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cReturn v) = cReturn (weaken-val \u0393\u2081\u227c\u0393\u2082 v)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (_into_ {\u03c3} c c\u2081) = (weaken-comp \u0393\u2081\u227c\u0393\u2082 c) into (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c\u2081)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cAbs {\u03c3} c) = cAbs (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cApp s t) = cApp (weaken-comp \u0393\u2081\u227c\u0393\u2082 s) (weaken-val \u0393\u2081\u227c\u0393\u2082 t)\n\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 (px \u2022 ts) = (weaken-val \u0393\u2081\u227c\u0393\u2082 px) \u2022 (weaken-vals \u0393\u2081\u227c\u0393\u2082 ts)\n\n  fromCBN : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBNCtx \u0393) (cbnToCompType \u03c4)\n\n  fromCBNTerms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Vals (fromCBNCtx \u0393) (fromCBNCtx \u03a3)\n  fromCBNTerms \u2205 = \u2205\n  fromCBNTerms (px \u2022 ts) = vThunk (fromCBN px) \u2022 fromCBNTerms ts\n\n  open import UNDEFINED\n  -- This is really supposed to be part of the plugin interface.\n  cbnToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBNCtx \u03a3) (cbnToCompType \u03c4)\n  cbnToCompConst = reveal UNDEFINED\n\n  fromCBN (const c args) = cConst (cbnToCompConst c) (fromCBNTerms args)\n  fromCBN (var x) = cForce (vVar (fromVar cbnToValType x))\n  fromCBN (app s t) = cApp (fromCBN s) (vThunk (fromCBN t))\n  fromCBN (abs t) = cAbs (fromCBN t)\n\n  -- To satisfy termination checking, we'd need to inline fromCBV and weaken: fromCBV needs to produce a term in a bigger context.\n  -- But let's ignore that.\n  {-# NO_TERMINATION_CHECK #-}\n  fromCBV : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n\n  -- This is really supposed to be part of the plugin interface.\n  cbvToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBVCtx \u03a3) (cbvToCompType \u03c4)\n  cbvToCompConst = reveal UNDEFINED\n\n  cbvTermsToComps : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Comps (fromCBVCtx \u0393) (fromCBVToCompList \u03a3)\n  cbvTermsToComps \u2205 = \u2205\n  cbvTermsToComps (px \u2022 ts) = fromCBV px \u2022 cbvTermsToComps ts\n\n  module _ where\n\n    dequeValContexts : ValContext \u2192 ValContext \u2192 ValContext\n    dequeValContexts \u2205 \u0393 = \u0393\n    dequeValContexts (x \u2022 \u03a3) \u0393 = dequeValContexts \u03a3 (x \u2022 \u0393)\n\n    dequeValContexts\u227c\u227c : \u2200 \u03a3 \u0393 \u2192 \u0393 \u227c\u227c dequeValContexts \u03a3 \u0393\n    dequeValContexts\u227c\u227c \u2205 \u0393 = \u227c\u227c-refl\n    dequeValContexts\u227c\u227c (x \u2022 \u03a3) \u0393 = \u227c\u227c-trans (drop_\u2022\u2022_ x \u227c\u227c-refl) (dequeValContexts\u227c\u227c \u03a3 (x \u2022 \u0393))\n\n    dequeContexts : Context \u2192 Context \u2192 ValContext\n    dequeContexts \u03a3 \u0393 = dequeValContexts (fromCBVCtx \u03a3) (fromCBVCtx \u0393)\n\n    fromCBVArg : \u2200 {\u03c3 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 (Val (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToValType \u03c4) \u2192 Comp (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToCompType \u03c3)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c3)\n    fromCBVArg t k = (fromCBV t) into k (vVar vThis)\n\n    fromCBVArgs : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Terms \u0393 \u03a3 \u2192 (Vals (dequeContexts \u03a3 \u0393) (fromCBVCtx \u03a3) \u2192 Comp (dequeContexts \u03a3 \u0393) (cbvToCompType \u03c4)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    fromCBVArgs \u2205 k = k \u2205\n    fromCBVArgs {\u03c3 \u2022 \u03a3} {\u0393} (t \u2022 ts) k = fromCBVArg t (\u03bb v \u2192 fromCBVArgs (weaken-terms (drop_\u2022_ _ \u227c-refl) ts) (\u03bb vs \u2192 k (weaken-val (dequeValContexts\u227c\u227c (fromCBVCtx \u03a3) _) v \u2022 vs)))\n\n    -- Transform a constant application into\n    --\n    --   arg1 to x1 in (arg2 to x2 in ... (argn to xn in (cConst (cbvToCompConst c) (x1 :: x2 :: ... xn)))).\n\n    fromCBVConstCPSRoot : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 Terms \u0393 \u03a3 \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    -- pass that as a closure to compose with (_\u2022_ x) for each new variable.\n    fromCBVConstCPSRoot c ts = fromCBVArgs ts (\u03bb vs \u2192 cConst (cbvToCompConst c) vs)\n\n  fromCBV (const c args) = fromCBVConstCPSRoot c args\n  fromCBV (app s t) =\n    (fromCBV s) into\n      (fromCBV (weaken (drop _ \u2022 \u227c-refl) t) into\n        cApp (cForce (vVar (vThat vThis))) (vVar vThis))\n  -- Values\n  fromCBV (var x) = cReturn (vVar (fromVar cbvToValType x))\n  fromCBV (abs t) = cReturn (vThunk (cAbs (fromCBV t)))\n\n  -- This reflects the CBV conversion of values in TLCA '99, but we can't write\n  -- this because it's partial on the Term type. Instead, we duplicate thunking\n  -- at the call site.\n  {-\n  fromCBVToVal : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Val (fromCBVCtx \u0393) (cbvToValType \u03c4)\n  fromCBVToVal (var x) = vVar (fromVar cbvToValType x)\n  fromCBVToVal (abs t) = vThunk (cAbs (fromCBV t))\n\n  fromCBVToVal (const c args) = {!!}\n  fromCBVToVal (app t t\u2081) = {!!} -- Not a value\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c5d04c57ebfc9f226305ea61a923331b9742ee7d","subject":"Only reordering.","message":"Only reordering.\n\nIgnore-this: c41f338dcf084cc236557506f6503040\n\ndarcs-hash:20100909125858-3bd4e-185c01385b3db0af31b9ba1e1e2fe88df9c9ee41.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List\/Properties.agda","new_file":"LTC\/Data\/List\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat.Type using ( N ; sN ; zN )\nopen import LTC.Data.List using\n  ( [] ; _\u2237_\n  ; _++_ ; ++-[]\n  ; List ; consL ; nilL\n  ; length\n  ; map\n  ; replicate\n  ; reverse\n  )\n\n------------------------------------------------------------------------------\n-- Closure properties\n\n++-List : {xs ys : D} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL ysL = prf\n  where\n    postulate prf : List ([] ++ ys)\n    {-# ATP prove prf #-}\n\n++-List {ys = ys} (consL x {xs} xsL) ysL = prf (++-List xsL ysL)\n  where\n    postulate prf : List (xs ++ ys) \u2192\n                    List ((x \u2237 xs) ++ ys)\n    {-# ATP prove prf consL #-}\n\nreverse-List : {xs : D} \u2192 List xs \u2192 List (reverse xs)\nreverse-List nilL = prf\n  where\n    postulate prf : List (reverse [])\n    {-# ATP prove prf nilL #-}\n\nreverse-List (consL x {xs} xsL) = prf (reverse-List xsL)\n  where\n    postulate prf : List (reverse xs) \u2192\n                    List (reverse (x \u2237 xs))\n    {-# ATP prove prf consL nilL ++-List #-}\n\n++-leftIdentity : {xs : D} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : {xs : D} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL = prf\n  where\n    postulate prf : [] ++ [] \u2261 []\n    {-# ATP prove prf #-}\n\n++-rightIdentity (consL x {xs} xsL) = prf (++-rightIdentity xsL)\n  where\n    postulate prf : xs ++ [] \u2261 xs \u2192\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n    {-# ATP prove prf #-}\n\n++-assoc : {as bs cs : D} \u2192 List as \u2192 List bs \u2192 List cs \u2192\n           (as ++ bs) ++ cs \u2261 as ++ (bs ++ cs)\n++-assoc .{[]} {bs} {cs} nilL bsL csL = prf\n  where\n    postulate prf : ([] ++ bs) ++ cs \u2261 [] ++ bs ++ cs\n    {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} {bs} {cs} (consL x {xs} xsL) bsL csL =\n  prf (++-assoc xsL bsL csL)\n  where\n    postulate prf : (xs ++ bs) ++ cs \u2261 xs ++ bs ++ cs \u2192\n                    ((x \u2237 xs) ++ bs) ++ cs \u2261 (x \u2237 xs) ++ bs ++ cs\n    {-# ATP prove prf #-}\n\npostulate\n  reverse-[x]\u2261[x] : (x : D) \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nreverse-++ : {xs ys : D} \u2192 List xs \u2192 List ys \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ {ys = ys} nilL ysL = prf\n  where\n    postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n    {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++ {ys = ys} (consL x {xs} xsL) ysL = prf (reverse-++ xsL ysL)\n  where\n    postulate prf : reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs \u2192\n                    reverse ((x \u2237 xs) ++ ys) \u2261 reverse ys ++ reverse (x \u2237 xs)\n    -- E 1.2 cannot prove this postulate with --time=180.\n    {-# ATP prove prf ++-assoc nilL consL reverse-List ++-List #-}\n\nreverse\u00b2 : {xs : D} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse\u00b2 nilL = prf\n  where\n    postulate prf : reverse (reverse []) \u2261 []\n    {-# ATP prove prf #-}\n\nreverse\u00b2 (consL x {xs} xsL) = prf (reverse\u00b2 xsL)\n  where\n    postulate prf : reverse (reverse xs) \u2261 xs \u2192\n                    reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n    {-# ATP prove prf reverse-++ consL nilL reverse-List ++-List #-}\n\nmap-++ : (f : D){xs ys : D} \u2192 List xs \u2192 List ys \u2192\n         map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++ f {ys = ys} nilL ysL = prf\n  where\n    postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n    {-# ATP prove prf #-}\nmap-++ f {ys = ys} (consL x {xs} xsL) ysL = prf (map-++ f xsL ysL)\n  where\n    postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n                    map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n    -- E 1.2 cannot prove this postulate with --time=180.\n    {-# ATP prove prf #-}\n\nlength-replicate : {n : D} \u2192 N n \u2192 (d : D) \u2192 length (replicate n d) \u2261 n\nlength-replicate zN d = prf\n  where\n    postulate prf : length (replicate zero d) \u2261 zero\n    {-# ATP prove prf #-}\nlength-replicate (sN {n} Nn) d = prf (length-replicate Nn d)\n  where\n    postulate prf : length (replicate n d) \u2261 n \u2192\n                    length (replicate (succ n) d) \u2261 succ n\n    {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat.Type using ( N ; sN ; zN )\nopen import LTC.Data.List using\n  ( [] ; _\u2237_\n  ; _++_ ; ++-[]\n  ; List ; consL ; nilL\n  ; length\n  ; map\n  ; replicate\n  ; reverse\n  )\n\n------------------------------------------------------------------------------\n\n++-List : {xs ys : D} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL ysL = prf\n  where\n    postulate prf : List ([] ++ ys)\n    {-# ATP prove prf #-}\n\n++-List {ys = ys} (consL x {xs} xsL) ysL = prf (++-List xsL ysL)\n  where\n    postulate prf : List (xs ++ ys) \u2192\n                    List ((x \u2237 xs) ++ ys)\n    {-# ATP prove prf consL #-}\n\n++-leftIdentity : {xs : D} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : {xs : D} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL = prf\n  where\n    postulate prf : [] ++ [] \u2261 []\n    {-# ATP prove prf #-}\n\n++-rightIdentity (consL x {xs} xsL) = prf (++-rightIdentity xsL)\n  where\n    postulate prf : xs ++ [] \u2261 xs \u2192\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n    {-# ATP prove prf #-}\n\n++-assoc : {as bs cs : D} \u2192 List as \u2192 List bs \u2192 List cs \u2192\n           (as ++ bs) ++ cs \u2261 as ++ (bs ++ cs)\n++-assoc .{[]} {bs} {cs} nilL bsL csL = prf\n  where\n    postulate prf : ([] ++ bs) ++ cs \u2261 [] ++ bs ++ cs\n    {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} {bs} {cs} (consL x {xs} xsL) bsL csL =\n  prf (++-assoc xsL bsL csL)\n  where\n    postulate prf : (xs ++ bs) ++ cs \u2261 xs ++ bs ++ cs \u2192\n                    ((x \u2237 xs) ++ bs) ++ cs \u2261 (x \u2237 xs) ++ bs ++ cs\n    {-# ATP prove prf #-}\n\nreverse-List : {xs : D} \u2192 List xs \u2192 List (reverse xs)\nreverse-List nilL = prf\n  where\n    postulate prf : List (reverse [])\n    {-# ATP prove prf nilL #-}\n\nreverse-List (consL x {xs} xsL) = prf (reverse-List xsL)\n  where\n    postulate prf : List (reverse xs) \u2192\n                    List (reverse (x \u2237 xs))\n    {-# ATP prove prf consL nilL ++-List #-}\n\npostulate\n  reverse-[x]\u2261[x] : (x : D) \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nreverse-++ : {xs ys : D} \u2192 List xs \u2192 List ys \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ {ys = ys} nilL ysL = prf\n  where\n    postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n    {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++ {ys = ys} (consL x {xs} xsL) ysL = prf (reverse-++ xsL ysL)\n  where\n    postulate prf : reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs \u2192\n                    reverse ((x \u2237 xs) ++ ys) \u2261 reverse ys ++ reverse (x \u2237 xs)\n    -- E 1.2 cannot prove this postulate with --time=180.\n    {-# ATP prove prf ++-assoc nilL consL reverse-List ++-List #-}\n\nreverse\u00b2 : {xs : D} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse\u00b2 nilL = prf\n  where\n    postulate prf : reverse (reverse []) \u2261 []\n    {-# ATP prove prf #-}\n\nreverse\u00b2 (consL x {xs} xsL) = prf (reverse\u00b2 xsL)\n  where\n    postulate prf : reverse (reverse xs) \u2261 xs \u2192\n                    reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n    {-# ATP prove prf reverse-++ consL nilL reverse-List ++-List #-}\n\nmap-++ : (f : D){xs ys : D} \u2192 List xs \u2192 List ys \u2192\n         map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++ f {ys = ys} nilL ysL = prf\n  where\n    postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n    {-# ATP prove prf #-}\nmap-++ f {ys = ys} (consL x {xs} xsL) ysL = prf (map-++ f xsL ysL)\n  where\n    postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n                    map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n    -- E 1.2 cannot prove this postulate with --time=180.\n    {-# ATP prove prf #-}\n\nlength-replicate : {n : D} \u2192 N n \u2192 (d : D) \u2192 length (replicate n d) \u2261 n\nlength-replicate zN d = prf\n  where\n    postulate prf : length (replicate zero d) \u2261 zero\n    {-# ATP prove prf #-}\nlength-replicate (sN {n} Nn) d = prf (length-replicate Nn d)\n  where\n    postulate prf : length (replicate n d) \u2261 n \u2192\n                    length (replicate (succ n) d) \u2261 succ n\n    {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"90362087df216c3833f3c0cf381b38211cb62c69","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 4aa7b9214eebfe20641b79ed2a83c775\n\ndarcs-hash:20110810144352-3bd4e-90cdb5bbab3c2ec06f4164ad5483106f752402f3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Relation\/Binary\/Bisimilarity.agda","new_file":"src\/FOTC\/Relation\/Binary\/Bisimilarity.agda","new_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation on FOTC\n------------------------------------------------------------------------------\n\nmodule FOTC.Relation.Binary.Bisimilarity where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation is a post-fixed point of the functor\n-- BisimilarityF (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n           xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation is the greatest post-fixed point of the\n-- functor BisimilarityF (see below).\n\n-- N.B. This is a second-order axiom. In the automatic proofs, we\n-- *must* use an instance. Therefore, we do not add this postulate as\n-- an ATP axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of BisimilarityF.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed point, the\n-- bisimilarity relation is also a pre-fixed point of the functor\n-- BisimilarityF (see below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 R helper h\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n            xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 R xs ys \u2192\n           \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n           R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (x' , xs' , ys' , xs'\u2248ys' , prf) =\n    x' , xs' , ys' , (\u2248-gfp\u2081 xs'\u2248ys') , prf\n\nmodule BisimilarityF where\n  -- In FOTC we won't use the bisimilarity functor. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n  -- 'as (R :: R') bs' (p. 310) for the functor BisimilarityF.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimilarity functor.\n  BisimilarityF : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BisimilarityF _R_ xs ys =\n    \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n      xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The bisimilarity relation is a post-fixed point of BisimilarityF.\n  pfp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 BisimilarityF _\u2248_ xs ys\n  pfp = \u2248-gfp\u2081\n\n  -- The bisimilarity relation is the greatest post-fixed point of\n  -- BisimilarityF.\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of BisimilarityF.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 BisimilarityF _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n","old_contents":"------------------------------------------------------------------------------\n-- Bisimilarity relation on FOTC\n------------------------------------------------------------------------------\n\nmodule FOTC.Relation.Binary.Bisimilarity where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The bisimilarity relation.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The bisimilarity relation is a post-fixed point of the functor\n-- BisimilarityF (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192 xs' \u2248 ys' \u2227\n                                        xs \u2261 x' \u2237 xs' \u2227\n                                        ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- The bisimilarity relation is the greatest post-fixed point of the\n-- functor BisimilarityF (see below).\n\n-- N.B. This is a second-order axiom. In the automatic proofs, we\n-- *must* use an instance. Therefore, we do not add this postulate as\n-- an ATP axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of BisimilarityF.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192 xs' R ys' \u2227\n                                         xs \u2261 x' \u2237 xs' \u2227\n                                         ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed point, the\n-- bisimilarity relation is also a pre-fixed point of the functor\n-- BisimilarityF (see below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192 xs' \u2248 ys' \u2227\n                                       xs \u2261 x' \u2237 xs' \u2227\n                                       ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 R helper h\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192 xs' \u2248 ys' \u2227\n                                         xs \u2261 x' \u2237 xs' \u2227\n                                         ys \u2261 x' \u2237 ys'\n\n  helper : {xs ys : D} \u2192 R xs ys \u2192\n           \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192\n             R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (x' , xs' , ys' , xs'\u2248ys' , prf) =\n    x' , xs' , ys' , (\u2248-gfp\u2081 xs'\u2248ys') , prf\n\nmodule BisimilarityF where\n  -- In FOTC we won't use the bisimilarity functor. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n  -- 'as (R :: R') bs' (p. 310) for the functor BisimilarityF.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimilarity functor.\n  BisimilarityF : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  BisimilarityF _R_ xs ys =\n    \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 \u2203 \u03bb ys' \u2192 xs' R ys' \u2227\n                                 xs \u2261 x' \u2237 xs' \u2227\n                                 ys \u2261 x' \u2237 ys'\n\n  -- The bisimilarity relation is a post-fixed point of BisimilarityF.\n  pfp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 BisimilarityF _\u2248_ xs ys\n  pfp = \u2248-gfp\u2081\n\n  -- The bisimilarity relation is the greatest post-fixed point of\n  -- BisimilarityF.\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of BisimilarityF.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 BisimilarityF _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d68a7b442eeb939be20d860f9cda3f341d04c094","subject":"Added draft version of Stream-gfp\u2082 using the Agda coinductive stuff.","message":"Added draft version of Stream-gfp\u2082 using the Agda coinductive stuff.\n\nIgnore-this: 6c2676f3926657febcc44f83992f3352\n\ndarcs-hash:20110926155004-3bd4e-b325b914bf8d3d38ab3084ce1c2bc672b442c586.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/StreamSL.agda","new_file":"Draft\/FOTC\/Data\/Stream\/StreamSL.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"75429bb69a022fe800c050d4e53fdc2a9ccc9261","subject":"Update PLDI14-List-of-Theorems.agda (for #275).","message":"Update PLDI14-List-of-Theorems.agda (for #275).\n\nThe list included plugin requirements, but we left them out of the\npaper, so the theorem numbers were off.\n\nOld-commit-hash: d4ea643c7474f10c798fcb0d494db9f941e3deb5\n","repos":"inc-lc\/ilc-agda","old_file":"PLDI14-List-of-Theorems.agda","new_file":"PLDI14-List-of-Theorems.agda","new_contents":"module PLDI14-List-of-Theorems where\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important names. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- Definition 3.1 (Domains)\nimport Parametric.Denotation.Value\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Definition 3.2 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.3 (Evaluation)\nimport Parametric.Denotation.Evaluation\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Definition 3.4 (Changes)\n-- Definition 3.5 (Change environments)\nimport Parametric.Change.Validity\nopen Parametric.Change.Validity.Structure using (change-algebra)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Definition 3.6 (Change semantics)\n-- Lemma 3.7 (Change semantics is the derivative of semantics)\nimport Parametric.Change.Specification\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.8 (Erasure)\n-- Lemma 3.9 (The erased version of a change is almost the same)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.10 (\u27e6 t \u27e7\u0394 erases to Derive(t))\n-- Theorem 3.11 (Correctness of differentiation)\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","old_contents":"module PLDI14-List-of-Theorems where\n\nopen import Function\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important names. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- For each plugin requirement, we include its definition and\n-- a concrete instantiation called \"Nehemiah\" with integers and\n-- bags of integers as base types.\n\n-- Plugin Requirement 3.1 (Domains of base types)\nopen import Parametric.Denotation.Value using (Structure)\nopen import     Nehemiah.Denotation.Value using (\u27e6_\u27e7Base)\n\n-- Definition 3.2 (Domains)\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Plugin Requirement 3.3 (Evaluation of constants)\nopen import Parametric.Denotation.Evaluation using (Structure)\nopen import     Nehemiah.Denotation.Evaluation using (\u27e6_\u27e7Const)\n\n-- Definition 3.4 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.5 (Evaluation)\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Plugin Requirement 3.6 (Changes on base types)\nopen import Parametric.Change.Validity using (Structure)\nopen import     Nehemiah.Change.Validity using (change-algebra-base-family)\n\n-- Definition 3.7 (Changes)\nopen Parametric.Change.Validity.Structure using (change-algebra)\n\n-- Definition 3.8 (Change environments)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Plugin Requirement 3.9 (Change semantics for constants)\nopen import Parametric.Change.Specification using (Structure)\nopen import     Nehemiah.Change.Specification using (specification-structure)\n\n-- Definition 3.10 (Change semantics)\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\n\n-- Lemma 3.11 (Change semantics is the derivative of semantics)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.12 (Erasure)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen import Nehemiah.Change.Implementation using (implements-base)\n\n-- Lemma 3.13 (The erased version of a change is almost the same)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.14 (\u27e6 t \u27e7\u0394 erases to Derive(t))\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\n\n-- Theorem 3.15 (Correctness of differentiation)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"716b50ffc397bb978f0dbc40c4b6ddaac717ed83","subject":"bintree: more Sorted,Monotone,evalTree...","message":"bintree: more Sorted,Monotone,evalTree...\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nfromFun-\u2257 : \u2200 {n a} {A : Set a} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\nfromFun-\u2257 {zero}  f\u2257g\n  rewrite f\u2257g [] = \u2261.refl\nfromFun-\u2257 {suc n} f\u2257g\n  rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n        | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n        = \u2261.refl\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\nopen Dummy public\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x\u2080 , x\u2081) = (x\u2080 \u2293\u1d2c x\u2081 , x\u2080 \u2294\u1d2c x\u2081)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule EvalTree where\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {a} {A : Set a} \u2192 Bij \u2192 \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    evalTree `id         = id\n    evalTree (op\u2080 `\u204f op\u2081) = evalTree op\u2081 \u2218 evalTree op\u2080\n    evalTree op {zero} = id\n    evalTree (`id   `\u2237 g) {suc n} = map-outer (evalTree (g 0b)) (evalTree (g 1b))\n    evalTree (`not\u1d2e `\u2237 g) {suc n} = map-outer (evalTree (g 1b)) (evalTree (g 0b)) \u2218 swap\n    evalTree `0\u21941 {suc zero} = id\n    evalTree `0\u21941 {suc (suc n)} = interchange\n\n    evalTree-eval : \u2200 {a} {A : Set a} (f : Bij) {n} (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id t xs = \u2261.refl\n    evalTree-eval `0\u21941 t [] = \u2261.refl\n    evalTree-eval `0\u21941 t (x \u2237 []) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval (f `\u204f f\u2081) {n} t xs = \u2261.trans (evalTree-eval f t xs) (evalTree-eval f\u2081 (evalTree f t) (eval f xs))\n    evalTree-eval (_ `\u2237 x\u2081) t [] = \u2261.refl\n    evalTree-eval (`id `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {a} {A : Set a} (f : Bij) {n} (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule PermTreeProof where\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Bij\n        proof : (t : Tree A n) \u2192 t \u2261 evalTree (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {f = f}{g} (mk `f pf) (mk `g pg)\n      = mk (\u03bb t \u2192 `f (g t) `\u204f `g t)\n           (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (evalTree (`g t)) (pf (g t))))\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = mk (\u03bb _ \u2192 `not) \u03b7fork\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof `f `g = mk (\u03bb t \u2192 `map-outer (perm `f (lft t)) (perm `g (rght t)))\n                               (\u03bb { (fork t u) \u2192 \u2261.cong\u2082 fork (proof `f t) (proof `g u) })\n       where open Perm\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof {A} {f = f} `f = mk map-inner-perm helper\n       where open Perm\n             map-inner-perm = `map-inner \u2218 perm `f \u2218 inner\n             helper : \u2200 t \u2192 t \u2261 evalTree (map-inner-perm t) (map-inner f t)\n             helper (fork (fork a b) (fork c d)) with f (fork b c) | \u2261.sym (proof `f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\nmodule Sorting-Perm-Properties {OT : Set} (_<=\u1d2c_ : OT \u2192 OT \u2192 Bool)\n    (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    _\u2293\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n\n    _\u2294\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open PermTreeProof\n\n    `sort\u2081 : Tree OT 1 \u2192 Bij\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    -- \u2200 x  T (y <=\u1d2c x) \u2192 fork (leaf x) (leaf y) \u2261 fork (leaf (x \u2293 y)) (leaf (x \u2294 y))\n    sort\u2081-proof : Perm {OT} 1 sort\u2081\n    sort\u2081-proof = mk `sort\u2081 helper\n      where helper : \u2200 t \u2192 t \u2261 evalTree (`sort\u2081 t) (sort\u2081 t)\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = sort\u2081-proof\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    module SPP = Sorting-Perm-Properties _<=_ isTotalOrder\n    open EvalTree public using (evalTree)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\nopen Dummy public\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x\u2080 , x\u2081) = (x\u2080 \u2293\u1d2c x\u2081 , x\u2080 \u2294\u1d2c x\u2081)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule Sorting-Perm-Properties {OT : Set} (_<=\u1d2c_ : OT \u2192 OT \u2192 Bool)\n    (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    _\u2293\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n\n    _\u2294\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open import Function.Bijection.SyntaxKit\n\n    evalTree : \u2200 {a} {A : Set a} \u2192 Bij \u2192 \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    evalTree `id         = id\n    evalTree (op\u2080 `\u204f op\u2081) = evalTree op\u2081 \u2218 evalTree op\u2080\n    evalTree op {zero} = id\n    evalTree (`id   `\u2237 g) {suc n} = map-outer (evalTree (g 0b)) (evalTree (g 1b))\n    evalTree (`not\u1d2e `\u2237 g) {suc n} = map-outer (evalTree (g 0b)) (evalTree (g 1b)) \u2218 swap\n    evalTree `0\u21941 {suc zero} = id\n    evalTree `0\u21941 {suc (suc n)} = interchange\n\n    {-\n    evalTree-eval : \u2200 {a} {A : Set a} (f : Bij) {n} (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval (f \u207b\u00b9)\n    evalTree-eval `id t xs = \u2261.refl\n    evalTree-eval `0\u21941 t [] = \u2261.refl\n    evalTree-eval `0\u21941 t (x \u2237 []) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval (f `\u204f f\u2081) {n} t xs = {!evalTree-eval!}\n    evalTree-eval (_ `\u2237 x\u2081) t [] = \u2261.refl\n    evalTree-eval (`id `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 0b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 1b) t xs\n\n    evalTree-eval : \u2200 {a} {A : Set a} (f : Bij) {n} (t : Tree A n) \u2192 toFun t \u2218 eval f \u2257 toFun (evalTree f t)\n    evalTree-eval `id t xs = \u2261.refl\n    evalTree-eval `0\u21941 t [] = \u2261.refl\n    evalTree-eval `0\u21941 t (x \u2237 []) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval (f `\u204f f\u2081) {n} t xs = {!evalTree-eval f\u2081 (evalTree f t) xs!}\n    evalTree-eval (_ `\u2237 x\u2081) t [] = \u2261.refl\n    evalTree-eval (`id `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) u xs\n    -}\n\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Bij\n        proof : (t : Tree A n) \u2192 t \u2261 evalTree (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {f = f}{g} (mk `f pf) (mk `g pg)\n      = mk (\u03bb t \u2192 `f (g t) `\u204f `g t)\n           (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (evalTree (`g t)) (pf (g t))))\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = mk (\u03bb _ \u2192 `not) \u03b7fork\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof `f `g = mk (\u03bb t \u2192 `map-outer (perm `f (lft t)) (perm `g (rght t)))\n                               (\u03bb { (fork t u) \u2192 \u2261.cong\u2082 fork (proof `f t) (proof `g u) })\n       where open Perm\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof {A} {f = f} `f = mk map-inner-perm helper\n       where open Perm\n             map-inner-perm = `map-inner \u2218 perm `f \u2218 inner\n             helper : \u2200 t \u2192 t \u2261 evalTree (map-inner-perm t) (map-inner f t)\n             helper (fork (fork a b) (fork c d)) with f (fork b c) | \u2261.sym (proof `f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\n    `sort\u2081 : Tree OT 1 \u2192 Bij\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    -- \u2200 x  T (y <=\u1d2c x) \u2192 fork (leaf x) (leaf y) \u2261 fork (leaf (x \u2293 y)) (leaf (x \u2294 y))\n    sort\u2081-proof : Perm {OT} 1 sort\u2081\n    sort\u2081-proof = mk `sort\u2081 helper\n      where helper : \u2200 t \u2192 t \u2261 evalTree (`sort\u2081 t) (sort\u2081 t)\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = sort\u2081-proof\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y)\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_\n    module SPP = Sorting-Perm-Properties _<=_ isTotalOrder\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open OperationSyntax\n    mkBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij\n    mkBij f = SPP.Perm.perm SPP.sort-proof (fromFun f)\n\n    {-\n    thm : \u2200 {n} (f : Endo (Bits n)) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n            eval (mkBij f) \u2257 f\n    thm f f-inj = \u03bb xs \u2192 {!\u2261.cong toFun (SPP.Perm.proof SPP.sort-proof (fromFun f))!}\n    -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f1af99798e77e407f6fd94056023ddd19492df6a","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: b34d8c096cb8602a06b0197ff7910750\n\ndarcs-hash:20110527130738-3bd4e-34ed1d919b19c7f09065246ef4e4b509ca05441c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/Nat\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Totality properties\n\n-- We removed the equation pred zero \u2261 zero, so we cannot prove the\n-- totality of the function pred.\n-- pred-N : \u2200 {n} \u2192 N n \u2192 N (pred n)\n-- pred-N zN  = prf\n--   where\n--     postulate prf : N (pred zero)\n--     {-# ATP prove prf #-}\n\n-- pred-N (sN {n} Nn) = prf\n--   where\n--     postulate prf : N (pred (succ n))\n--     {-# ATP prove prf #-}\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} _ zN = prf\n  where\n    postulate prf : N (m \u2238 zero)\n    {-# ATP prove prf #-}\n\n\u2238-N zN (sN {n} _) = prf\n  where\n    postulate prf : N (zero \u2238 succ n)\n    {-# ATP prove prf #-}\n\n\u2238-N (sN {m} Nm) (sN {n} Nn) = prf $ \u2238-N Nm Nn\n  where\n    postulate prf : N (m \u2238 n) \u2192  -- IH.\n                    N (succ m \u2238 succ n)\n    {-# ATP prove prf #-}\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN _ = prf\n  where\n    postulate prf : N (zero + n)\n    {-# ATP prove prf #-}\n+-N {n = n} (sN {m} Nm) Nn = prf $ +-N Nm Nn\n  where\n    postulate prf : N (m + n) \u2192  -- IH.\n                    N (succ m + n)\n    {-# ATP prove prf #-}\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN _ = prf\n  where\n    postulate prf : N (zero * n)\n    {-# ATP prove prf #-}\n*-N {n = n} (sN {m} Nm) Nn = prf $ *-N Nm Nn\n  where\n    postulate prf : N (m * n) \u2192  -- IH.\n                    N (succ m * n)\n    {-# ATP prove prf +-N #-}\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\n+-leftIdentity : \u2200 {n} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) = prf $ +-rightIdentity Nn\n   where\n     postulate prf : n + zero \u2261 n \u2192  -- IH.\n                     succ n + zero \u2261 succ n\n     {-# ATP prove prf #-}\n\n+-assoc : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o} zN _ _ = prf\n  where\n    postulate prf : zero + n + o \u2261 zero + (n + o)\n    {-# ATP prove prf #-}\n+-assoc {n = n} {o} (sN {m} Nm) Nn No = prf $ +-assoc Nm Nn No\n  where\n    postulate prf : m + n + o \u2261 m + (n + o) \u2192  -- IH.\n                    succ m + n + o \u2261 succ m + (n + o)\n    {-# ATP prove prf #-}\n\nx+Sy\u2261S[x+y] : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] {n = n} zN _ = prf\n  where\n    postulate prf : zero + succ n \u2261 succ (zero + n)\n    {-# ATP prove prf #-}\nx+Sy\u2261S[x+y] {n = n} (sN {m} Nm) Nn = prf $ x+Sy\u2261S[x+y] Nm Nn\n  where\n    postulate prf : m + succ n \u2261 succ (m + n) \u2192  -- IH.\n                    succ m + succ n \u2261 succ (succ m + n)\n    {-# ATP prove prf #-}\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN _ = prf\n  where\n    postulate prf : zero + n \u2261 n + zero\n    {-# ATP prove prf +-rightIdentity #-}\n+-comm {n = n} (sN {m} Nm) Nn = prf $ +-comm Nm Nn\n  where\n    postulate prf : m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\nx+S0\u2261Sx : \u2200 {n} \u2192 N n \u2192 n + succ zero \u2261 succ n\nx+S0\u2261Sx zN          = +-0x (succ zero)\nx+S0\u2261Sx (sN {n} Nn) = prf (x+S0\u2261Sx Nn)\n  where\n    postulate prf : n + succ zero \u2261 succ n \u2192  -- IH.\n                    succ n + succ zero \u2261 succ (succ n)\n    {-# ATP prove prf #-}\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN         = \u2238-x0 zero\n\u2238-0x (sN {n} _) = \u2238-0S n\n\nx\u2238x\u22610 : \u2200 {n} \u2192 N n \u2192 n \u2238 n \u2261 zero\nx\u2238x\u22610 zN          = \u2238-x0 zero\nx\u2238x\u22610 (sN {n} Nn) = trans (\u2238-SS n n) (x\u2238x\u22610 Nn)\n\nSx\u2238x\u2261S0 : \u2200 {n} \u2192 N n \u2192 succ n \u2238 n \u2261 succ zero\nSx\u2238x\u2261S0 zN          = \u2238-x0 (succ zero)\nSx\u2238x\u2261S0 (sN {n} Nn) = trans (\u2238-SS (succ n) n) (Sx\u2238x\u2261S0 Nn)\n\n[x+Sy]\u2238y\u2261Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 (m + succ n) \u2238 n \u2261 succ m\n[x+Sy]\u2238y\u2261Sx {n = n} zN Nn = prf\n  where\n    postulate prf : zero + succ n \u2238 n \u2261 succ zero\n    {-# ATP prove prf Sx\u2238x\u2261S0 #-}\n[x+Sy]\u2238y\u2261Sx (sN {m} Nm) zN = prf\n  where\n    postulate prf : succ m + succ zero \u2238 zero \u2261 succ (succ m)\n    {-# ATP prove prf x+S0\u2261Sx #-}\n\n[x+Sy]\u2238y\u2261Sx (sN {m} Nm) (sN {n} Nn) = prf ([x+Sy]\u2238y\u2261Sx (sN Nm) Nn)\n  where\n    postulate prf : succ m + succ n \u2238 n \u2261 succ (succ m) \u2192  -- IH.\n                    succ m + succ (succ n) \u2238 succ n \u2261 succ (succ m)\n    {-# ATP prove prf +-comm #-}\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ = prf\n  where\n    postulate prf : (zero + n) \u2238 (zero + o) \u2261 n \u2238 o\n    {-# ATP prove prf #-}\n\n-- Nice proof by the ATP.\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  prf $ [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No\n  where\n    postulate prf : (m + n) \u2238 (m + o) \u2261 n \u2238 o \u2192  -- IH.\n                    (succ m + n) \u2238 (succ m + o) \u2261 n \u2238 o\n    {-# ATP prove prf #-}\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) = prf $ *-rightZero Nn\n  where\n    postulate prf : n * zero \u2261 zero \u2192  -- IH.\n                    succ n * zero \u2261 zero\n    {-# ATP prove prf #-}\n\npostulate *-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ zero * n \u2261 n\n{-# ATP prove *-leftIdentity +-rightIdentity #-}\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = prf\n  where\n    postulate prf : zero * succ n \u2261 zero + zero * n\n    {-# ATP prove prf #-}\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn = prf (x*Sy\u2261x+xy Nm Nn)\n                                       (+-assoc Nn Nm (*-N Nm Nn))\n                                       (+-assoc Nm Nn (*-N Nm Nn))\n  where\n    -- N.B. We had to feed the ATP with the instances of the associate law\n    postulate prf :  m * succ n \u2261 m + m * n \u2192  -- IH\n                     (n + m) + (m * n) \u2261 n + (m + (m * n)) \u2192  -- Associative law\n                     (m + n) + (m * n) \u2261 m + (n + (m * n)) \u2192  -- Associateve law\n                     succ m * succ n \u2261 succ m + succ m * n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf +-comm #-}\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN _ = prf\n  where\n    postulate prf : zero * n \u2261 n * zero\n    {-# ATP prove prf *-rightZero #-}\n*-comm {n = n} (sN {m} Nm) Nn = prf $ *-comm Nm Nn\n  where\n    postulate prf : m * n \u2261 n * m \u2192  -- IH.\n                    succ m * n \u2261 n * succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x*Sy\u2261x+xy #-}\n\n*-rightIdentity : \u2200 {n} \u2192 N n \u2192 n * succ zero \u2261 n\n*-rightIdentity {n} Nn = trans (*-comm Nn (sN zN)) (*-leftIdentity Nn)\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ = prf\n  where\n    postulate prf : (m \u2238 zero) * o \u2261 m * o \u2238 zero * o\n    {-# ATP prove prf #-}\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No = prf $ \u2238-0x (*-N (sN Nn) No)\n  where\n    postulate prf : zero \u2238 succ n * o \u2261 zero \u2192\n                    (zero \u2238 succ n) * o \u2261 zero * o \u2238 succ n * o\n    {-# ATP prove prf #-}\n\n*\u2238-leftDistributive (sN {m} _) (sN {n} _) zN = prf\n  where\n    postulate prf : (succ m \u2238 succ n) * zero \u2261 succ m * zero \u2238 succ n * zero\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf \u2238-N *-comm #-}\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  prf $ *\u2238-leftDistributive Nm Nn (sN No)\n  where\n    postulate prf : (m \u2238 n) * succ o \u2261 m * succ o \u2238 n * succ o \u2192  -- IH\n                    (succ m \u2238 succ n) * succ o \u2261\n                    succ m * succ o \u2238 succ n * succ o\n    -- E 1.2: CPU time limit exceeded (180 sec).\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf *-N [x+y]\u2238[x+z]\u2261y\u2238z #-}\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} _ _ zN = prf\n  where\n    postulate prf : (m + n) * zero \u2261 m * zero + n * zero\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf *-comm +-rightIdentity *-N +-N #-}\n\n*+-leftDistributive {n = n} zN _ (sN {o} _) = prf\n  where\n    postulate prf :  (zero + n) * succ o \u2261 zero * succ o + n * succ o\n    {-# ATP prove prf #-}\n\n*+-leftDistributive (sN {m} _) zN (sN {o} _) = prf\n  where\n    postulate prf : (succ m + zero) * succ o \u2261 succ m * succ o + zero * succ o\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-rightIdentity *-leftZero *-N #-}\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  prf $ *+-leftDistributive Nm (sN Nn) (sN No)\n    where\n      postulate\n        prf : (m + succ n) * succ o \u2261 m * succ o + succ n * succ o \u2192  -- IH.\n              (succ m + succ n) * succ o \u2261 succ m * succ o + succ n * succ o\n      -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n      -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n      {-# ATP prove prf +-assoc *-N #-}\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u2228 n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 zN      _  _                   = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN Nm) zN _                   = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN {m} Nm) (sN {n} Nn) SmSn\u22610 = \u22a5-elim (0\u2260S prf)\n  where\n    postulate prf : zero \u2261 succ (n + m * succ n)\n    {-# ATP prove prf #-}\n\nxy\u22611\u2192x\u22611\u2228y\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 one \u2192 m \u2261 one \u2228 n \u2261 one\nxy\u22611\u2192x\u22611\u2228y\u22611 {n = n} zN Nn h = \u22a5-elim (0\u2260S (trans (sym (*-leftZero n)) h))\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN {m} Nm) zN h =\n  \u22a5-elim (0\u2260S (trans (sym (*-rightZero (sN Nm))) h))\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN zN) (sN Nn) h = inj\u2081 refl\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN (sN Nm)) (sN zN) h = inj\u2082 refl\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN (sN {m} Nm)) (sN (sN {n} Nn)) h = prf\n  where\n    postulate prf : succ (succ m) \u2261 one \u2228 succ (succ n) \u2261 one\n    {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Totality properties\n\n-- We removed the equation pred zero \u2261 zero, so we cannot prove the\n-- totality of the function pred.\n-- pred-N : \u2200 {n} \u2192 N n \u2192 N (pred n)\n-- pred-N zN  = prf\n--   where\n--     postulate prf : N (pred zero)\n--     {-# ATP prove prf #-}\n\n-- pred-N (sN {n} Nn) = prf\n--   where\n--     postulate prf : N (pred (succ n))\n--     {-# ATP prove prf #-}\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} _ zN = prf\n  where\n    postulate prf : N (m \u2238 zero)\n    {-# ATP prove prf #-}\n\n\u2238-N zN (sN {n} _) = prf\n  where\n    postulate prf : N (zero \u2238 succ n)\n    {-# ATP prove prf #-}\n\n\u2238-N (sN {m} Nm) (sN {n} Nn) = prf $ \u2238-N Nm Nn\n  where\n    postulate prf : N (m \u2238 n) \u2192  -- IH.\n                    N (succ m \u2238 succ n)\n    {-# ATP prove prf #-}\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN _ = prf\n  where\n    postulate prf : N (zero + n)\n    {-# ATP prove prf #-}\n+-N {n = n} (sN {m} Nm) Nn = prf $ +-N Nm Nn\n  where\n    postulate prf : N (m + n) \u2192  -- IH.\n                    N (succ m + n)\n    {-# ATP prove prf #-}\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN _ = prf\n  where\n    postulate prf : N (zero * n)\n    {-# ATP prove prf #-}\n*-N {n = n} (sN {m} Nm) Nn = prf $ *-N Nm Nn\n  where\n    postulate prf : N (m * n) \u2192  -- IH.\n                    N (succ m * n)\n    {-# ATP prove prf +-N #-}\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\n+-leftIdentity : \u2200 {n} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} _ = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zN\n+-rightIdentity (sN {n} Nn) = prf $ +-rightIdentity Nn\n   where\n     postulate prf : n + zero \u2261 n \u2192  -- IH.\n                     succ n + zero \u2261 succ n\n     {-# ATP prove prf #-}\n\n+-assoc : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o} zN _ _ = prf\n  where\n    postulate prf : zero + n + o \u2261 zero + (n + o)\n    {-# ATP prove prf #-}\n+-assoc {n = n} {o} (sN {m} Nm) Nn No = prf $ +-assoc Nm Nn No\n  where\n    postulate prf : m + n + o \u2261 m + (n + o) \u2192  -- IH.\n                    succ m + n + o \u2261 succ m + (n + o)\n    {-# ATP prove prf #-}\n\nx+Sy\u2261S[x+y] : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] {n = n} zN _ = prf\n  where\n    postulate prf : zero + succ n \u2261 succ (zero + n)\n    {-# ATP prove prf #-}\nx+Sy\u2261S[x+y] {n = n} (sN {m} Nm) Nn = prf $ x+Sy\u2261S[x+y] Nm Nn\n  where\n    postulate prf : m + succ n \u2261 succ (m + n) \u2192  -- IH.\n                    succ m + succ n \u2261 succ (succ m + n)\n    {-# ATP prove prf #-}\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN _ = prf\n  where\n    postulate prf : zero + n \u2261 n + zero\n    {-# ATP prove prf +-rightIdentity #-}\n+-comm {n = n} (sN {m} Nm) Nn = prf $ +-comm Nm Nn\n  where\n    postulate prf : m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\nx+S0\u2261Sx : \u2200 {n} \u2192 N n \u2192 n + succ zero \u2261 succ n\nx+S0\u2261Sx zN          = +-0x (succ zero)\nx+S0\u2261Sx (sN {n} Nn) = prf (x+S0\u2261Sx Nn)\n  where\n    postulate prf : n + succ zero \u2261 succ n \u2192  -- IH.\n                    succ n + succ zero \u2261 succ (succ n)\n    {-# ATP prove prf #-}\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN         = \u2238-x0 zero\n\u2238-0x (sN {n} _) = \u2238-0S n\n\nx\u2238x\u22610 : \u2200 {n} \u2192 N n \u2192 n \u2238 n \u2261 zero\nx\u2238x\u22610 zN          = \u2238-x0 zero\nx\u2238x\u22610 (sN {n} Nn) = trans (\u2238-SS n n) (x\u2238x\u22610 Nn)\n\nSx\u2238x\u2261S0 : \u2200 {n} \u2192 N n \u2192 succ n \u2238 n \u2261 succ zero\nSx\u2238x\u2261S0 zN          = \u2238-x0 (succ zero)\nSx\u2238x\u2261S0 (sN {n} Nn) = trans (\u2238-SS (succ n) n) (Sx\u2238x\u2261S0 Nn)\n\n[x+Sy]\u2238y\u2261Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 (m + succ n) \u2238 n \u2261 succ m\n[x+Sy]\u2238y\u2261Sx {n = n} zN Nn = prf\n  where\n    postulate prf : zero + succ n \u2238 n \u2261 succ zero\n    {-# ATP prove prf Sx\u2238x\u2261S0 #-}\n[x+Sy]\u2238y\u2261Sx (sN {m} Nm) zN = prf\n  where\n    postulate prf : succ m + succ zero \u2238 zero \u2261 succ (succ m)\n    {-# ATP prove prf x+S0\u2261Sx #-}\n\n[x+Sy]\u2238y\u2261Sx (sN {m} Nm) (sN {n} Nn) = prf ([x+Sy]\u2238y\u2261Sx (sN Nm) Nn)\n  where\n    postulate prf : succ m + succ n \u2238 n \u2261 succ (succ m) \u2192  -- IH.\n                    succ m + succ (succ n) \u2238 succ n \u2261 succ (succ m)\n    {-# ATP prove prf +-comm #-}\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ = prf\n  where\n    postulate prf : (zero + n) \u2238 (zero + o) \u2261 n \u2238 o\n    {-# ATP prove prf #-}\n\n-- Nice proof by the ATP.\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  prf $ [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No\n  where\n    postulate prf : (m + n) \u2238 (m + o) \u2261 n \u2238 o \u2192  -- IH.\n                    (succ m + n) \u2238 (succ m + o) \u2261 n \u2238 o\n    {-# ATP prove prf #-}\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) = prf $ *-rightZero Nn\n  where\n    postulate prf : n * zero \u2261 zero \u2192  -- IH.\n                    succ n * zero \u2261 zero\n    {-# ATP prove prf #-}\n\npostulate *-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ zero * n \u2261 n\n{-# ATP prove *-leftIdentity +-rightIdentity #-}\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = prf\n  where\n    postulate prf : zero * succ n \u2261 zero + zero * n\n    {-# ATP prove prf #-}\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn = prf (x*Sy\u2261x+xy Nm Nn)\n                                       (+-assoc Nn Nm (*-N Nm Nn))\n                                       (+-assoc Nm Nn (*-N Nm Nn))\n  where\n    -- N.B. We had to feed the ATP with the instances of the associate law\n    postulate prf :  m * succ n \u2261 m + m * n \u2192  -- IH\n                     (n + m) + (m * n) \u2261 n + (m + (m * n)) \u2192  -- Associative law\n                     (m + n) + (m * n) \u2261 m + (n + (m * n)) \u2192  -- Associateve law\n                     succ m * succ n \u2261 succ m + succ m * n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf +-comm #-}\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN _ = prf\n  where\n    postulate prf : zero * n \u2261 n * zero\n    {-# ATP prove prf *-rightZero #-}\n*-comm {n = n} (sN {m} Nm) Nn = prf $ *-comm Nm Nn\n  where\n    postulate prf : m * n \u2261 n * m \u2192  -- IH.\n                    succ m * n \u2261 n * succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x*Sy\u2261x+xy #-}\n\n*-rightIdentity : \u2200 {n} \u2192 N n \u2192 n * succ zero \u2261 n\n*-rightIdentity {n} Nn = trans (*-comm Nn (sN zN)) (*-leftIdentity Nn)\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ = prf\n  where\n    postulate prf : (m \u2238 zero) * o \u2261 m * o \u2238 zero * o\n    {-# ATP prove prf #-}\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No = prf $ \u2238-0x (*-N (sN Nn) No)\n  where\n    postulate prf : zero \u2238 succ n * o \u2261 zero \u2192\n                    (zero \u2238 succ n) * o \u2261 zero * o \u2238 succ n * o\n    {-# ATP prove prf #-}\n\n*\u2238-leftDistributive (sN {m} _) (sN {n} _) zN = prf\n  where\n    postulate prf : (succ m \u2238 succ n) * zero \u2261 succ m * zero \u2238 succ n * zero\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf \u2238-N *-comm #-}\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  prf $ *\u2238-leftDistributive Nm Nn (sN No)\n  where\n    postulate prf : (m \u2238 n) * succ o \u2261 m * succ o \u2238 n * succ o \u2192  -- IH\n                    (succ m \u2238 succ n) * succ o \u2261\n                    succ m * succ o \u2238 succ n * succ o\n    -- E 1.2: CPU time limit exceeded (180 sec).\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf *-N [x+y]\u2238[x+z]\u2261y\u2238z #-}\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} _ _ zN = prf\n  where\n    postulate prf : (m + n) * zero \u2261 m * zero + n * zero\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf *-comm +-rightIdentity *-N +-N #-}\n\n*+-leftDistributive {n = n} zN _ (sN {o} _) = prf\n  where\n    postulate prf :  (zero + n) * succ o \u2261 zero * succ o + n * succ o\n    {-# ATP prove prf #-}\n\n*+-leftDistributive (sN {m} _) zN (sN {o} _) = prf\n  where\n    postulate prf : (succ m + zero) * succ o \u2261 succ m * succ o + zero * succ o\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n      -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-rightIdentity *-leftZero *-N #-}\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  prf $ *+-leftDistributive Nm (sN Nn) (sN No)\n    where\n      postulate\n        prf : (m + succ n) * succ o \u2261 m * succ o + succ n * succ o \u2192  -- IH.\n              (succ m + succ n) * succ o \u2261 succ m * succ o + succ n * succ o\n      -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n      -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n      {-# ATP prove prf +-assoc *-N #-}\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u2228 n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 zN      _  _                   = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN Nm) zN _                   = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN {m} Nm) (sN {n} Nn) SmSn\u22610 = \u22a5-elim (0\u2260S prf)\n  where\n    postulate prf : zero \u2261 succ (n + m * succ n)\n    {-# ATP prove prf #-}\n\nxy\u22611\u2192x\u22611\u2228y\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 one \u2192 m \u2261 one \u2228 n \u2261 one\nxy\u22611\u2192x\u22611\u2228y\u22611 {n = n} zN Nn h = \u22a5-elim (0\u2260S (trans (sym (*-leftZero n)) h))\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN {m} Nm) zN h =\n  \u22a5-elim (0\u2260S (trans (sym (*-rightZero (sN Nm))) h))\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN zN) (sN Nn) h = inj\u2081 refl\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN (sN Nm)) (sN zN) h = inj\u2082 refl\nxy\u22611\u2192x\u22611\u2228y\u22611 (sN (sN {m} Nm)) (sN (sN {n} Nn)) h = prf\n  where\n    postulate prf : succ (succ m) \u2261 one \u2228 succ (succ n) \u2261 one\n    {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0e4f305f9ade206c6c11a01af8ceccbcde55ba30","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: da46b58650678e69c9a80286f1cc0110\n\ndarcs-hash:20110314164406-3bd4e-ac71ffe48514cfbaad3f2a6707d8086060d9ad1b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/LogicalConstants.agda","new_file":"src\/Common\/LogicalConstants.agda","new_contents":"------------------------------------------------------------------------------\n-- Logical constants\n------------------------------------------------------------------------------\n\nmodule Common.LogicalConstants where\n\n-- This module just exported all the logical constants.\n\n------------------------------------------------------------------------------\n-- The propositional logical connectives\n\n-- The logical connectives are hard-coded in our translation,\n-- i.e. the symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192, and \u2194 must be used.\n-- N.B. For the implication we use the Agda function type.\nopen import Common.Data.Empty public using ( \u22a5 ; \u22a5-elim )\nopen import Common.Data.Product public using ( _\u2227_ ; _,_ ; \u2227-proj\u2081 ; \u2227-proj\u2082 )\nopen import Common.Data.Sum public using ( _\u2228_ ; [_,_] ; inj\u2081 ; inj\u2082 )\nopen import Common.Data.Unit public using ( \u22a4 )\nopen import Common.Relation.Nullary public using ( \u00ac_ )\n\ninfixr 2 _\u2194_\n\n_\u2194_ : Set \u2192 Set \u2192 Set\nP \u2194 Q = (P \u2192 Q) \u2227 (Q \u2192 P)\n\n------------------------------------------------------------------------------\n-- The quantifiers\n\n-- The existential quantifier\n-- The existential quantifier is hard-coded in our translation,\n-- i.e. the symbol \u2203 must be used.\n\nopen import Common.Data.Product public using ( _,_ ; \u2203 ; \u2203-proj\u2081 ; \u2203-proj\u2082 )\n\n-- The universal quantifier\n-- N.B. For the universal quantifier we use the Agda (dependent)\n-- function type.\n","old_contents":"------------------------------------------------------------------------------\n-- Logical constants\n------------------------------------------------------------------------------\n\nmodule Common.LogicalConstants where\n\n-- This module just exported all the logical constants.\n\n------------------------------------------------------------------------------\n-- The logical connectives\n\n-- The logical connectives are hard-coded in our translation,\n-- i.e. the symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192, and \u2194 must be used.\n-- For the implication we use the Agda function type.\nopen import Common.Data.Empty public using ( \u22a5 ; \u22a5-elim )\nopen import Common.Data.Product public using ( _\u2227_ ; _,_ ; \u2227-proj\u2081 ; \u2227-proj\u2082 )\nopen import Common.Data.Sum public using ( _\u2228_ ; [_,_] ; inj\u2081 ; inj\u2082 )\nopen import Common.Data.Unit public using ( \u22a4 )\nopen import Common.Relation.Nullary public using ( \u00ac_ )\n\ninfixr 2 _\u2194_\n\n_\u2194_ : Set \u2192 Set \u2192 Set\nP \u2194 Q = (P \u2192 Q) \u2227 (Q \u2192 P)\n\n------------------------------------------------------------------------------\n-- The quantifiers\n\n-- The existential quantifier\n-- The existential quantifier is hard-coded in our translation,\n-- i.e. the symbol \u2203 must be used.\n\nopen import Common.Data.Product public using ( _,_ ; \u2203 ; \u2203-proj\u2081 ; \u2203-proj\u2082 )\n\n-- The universal quantifier\n-- For the universal quantifier we use the Agda (dependent) function type.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ef3f46045d85fef7b038b4703fc7d2fee8d29a8a","subject":"Generalize term equivalence to arbitrary calculi.","message":"Generalize term equivalence to arbitrary calculi.\n\nOld-commit-hash: 239a4be82a8af884669ca62db624782bee407ca1\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Equivalence.agda","new_file":"Denotational\/Equivalence.agda","new_contents":"module Denotational.Equivalence where\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Relation.Binary.PropositionalEquality\n\nopen import Denotational.Notation\n\n-- TERMS\n\n-- Term Equivalence\n--\n-- This module is parametric in the syntax and semantics of types\n-- and terms to make it work for different calculi.\n\nmodule _ where\n  open import Syntactic.Contexts\n  open import Denotational.Environments\n\n  module MakeEquivalence\n      (Type : Set)\n      (\u27e6_\u27e7Type : Type \u2192 Set)\n      (Term : Context Type \u2192 Type \u2192 Set)\n      (\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6_\u27e7Context Type \u27e6_\u27e7Type \u0393 \u2192 \u27e6 \u03c4 \u27e7Type) where\n    module _ {\u0393} {\u03c4} where\n      data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n        ext-t :\n          (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7Term \u03c1 \u2261 \u27e6 t\u2082 \u27e7Term \u03c1) \u2192\n          t\u2081 \u2248 t\u2082\n\n      \u2248-refl : Reflexive _\u2248_\n      \u2248-refl = ext-t (\u03bb \u03c1 \u2192 refl)\n\n      \u2248-sym : Symmetric _\u2248_\n      \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 sym (\u2248 \u03c1))\n\n      \u2248-trans : Transitive _\u2248_\n      \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n      \u2248-isEquivalence : IsEquivalence _\u2248_\n      \u2248-isEquivalence = record\n        { refl  = \u2248-refl\n        ; sym   = \u2248-sym\n        ; trans = \u2248-trans\n        }\n\n    \u2248-setoid : Context Type \u2192 Type \u2192 Setoid _ _\n    \u2248-setoid \u0393 \u03c4 = record\n      { Carrier       = Term \u0393 \u03c4\n      ; _\u2248_           = _\u2248_\n      ; isEquivalence = \u2248-isEquivalence\n      }\n\n    module \u2248-Reasoning where\n      module _ {\u0393 : Context Type} {\u03c4 : Type} where\n        open import Relation.Binary.EqReasoning (\u2248-setoid \u0393 \u03c4) public\n          hiding (_\u2261\u27e8_\u27e9_)\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\n\nopen import Changes\nopen import ChangeContexts\n\n-- Export a version of the equivalence for terms with total\n-- derivatives\n\nopen MakeEquivalence Type \u27e6_\u27e7 Term \u27e6_\u27e7 public\n\n-- Specialized congruence rules\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\n-- Consistency\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n","old_contents":"module Denotational.Equivalence where\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext-t :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open import Relation.Binary.EqReasoning (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f445eb29cb49f5df21f384217e34e41cbb4955d9","subject":"Added fixity to \u2227.","message":"Added fixity to \u2227.\n\nIgnore-this: c367b405b7f5f7c7031020541aca93f\n\ndarcs-hash:20100513032756-3bd4e-29d298c9e1f649bd87c5a74b424423d02d7703b4.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"MyStdLib\/Data\/Product.agda","new_file":"MyStdLib\/Data\/Product.agda","new_contents":"-----------------------------------------------------------------------------\n-- The product data type\n-----------------------------------------------------------------------------\n\nmodule MyStdLib.Data.Product where\n\ninfixr 4 _,_\ninfixr 2 _\u00d7_\n\n-- data \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n--   _,_ : (x : A) (y : B x) \u2192 \u03a3 A B\n\n-- \u2203 : {A : Set} \u2192 (A \u2192 Set) \u2192 Set\n-- \u2203 = \u03a3 _\n\n-- _\u00d7_ : (A B : Set) \u2192 Set\n-- A \u00d7 B = \u03a3 A (\u03bb _ \u2192 B)\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n\u00d7-proj\u2081 : {A B : Set} \u2192 A \u00d7 B \u2192 A\n\u00d7-proj\u2081 (x , y) = x\n\n\u00d7-proj\u2082 : {A B : Set} \u2192 A \u00d7 B \u2192 B\n\u00d7-proj\u2082 (x , y) = y\n\n\n-- N.B. It is not necessary to add the constructor _,_ , nor the fields\n-- proj\u2081, proj\u2082 as hints, because the ATP implements them.\n\n-- _\u2227_ : (A B : Set) \u2192 Set\n-- A \u2227 B = A \u00d7 B\n\ninfixr 2 _\u2227_\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n","old_contents":"-----------------------------------------------------------------------------\n-- The product data type\n-----------------------------------------------------------------------------\n\nmodule MyStdLib.Data.Product where\n\ninfixr 4 _,_\ninfixr 2 _\u00d7_\n\n-- data \u03a3 (A : Set) (B : A \u2192 Set) : Set where\n--   _,_ : (x : A) (y : B x) \u2192 \u03a3 A B\n\n-- \u2203 : {A : Set} \u2192 (A \u2192 Set) \u2192 Set\n-- \u2203 = \u03a3 _\n\n-- _\u00d7_ : (A B : Set) \u2192 Set\n-- A \u00d7 B = \u03a3 A (\u03bb _ \u2192 B)\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n\u00d7-proj\u2081 : {A B : Set} \u2192 A \u00d7 B \u2192 A\n\u00d7-proj\u2081 (x , y) = x\n\n\u00d7-proj\u2082 : {A B : Set} \u2192 A \u00d7 B \u2192 B\n\u00d7-proj\u2082 (x , y) = y\n\n\n-- N.B. It is not necessary to add the constructor _,_ , nor the fields\n-- proj\u2081, proj\u2082 as hints, because the ATP implements them.\n\n-- _\u2227_ : (A B : Set) \u2192 Set\n-- A \u2227 B = A \u00d7 B\n\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"806a51d402bfd1e88a08453decc08aa1cdd332ce","subject":"Only white spaces.","message":"Only white spaces.\n\nIgnore-this: f24a6175e207d3d7a282cdf79c095f9d\n\ndarcs-hash:20100703024037-3bd4e-29afaf143070ef3aad407f2f6566756f8e485c41.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Bool.agda","new_file":"LTC\/Data\/Bool.agda","new_contents":"------------------------------------------------------------------------------\n-- The booleans\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Bool where\n\nopen import LTC.Minimal\n\ninfixr 6 _&&_\n\n------------------------------------------------------------------------------\n-- The LTC booleans type.\ndata Bool : D \u2192 Set where\n  tB : Bool true\n  fB : Bool false\n\n-- The rule of proof by case analysis on 'B'.\nindBool : (P : D \u2192 Set) \u2192 P true \u2192 P false \u2192 {b : D} \u2192 Bool b \u2192 P b\nindBool P pt pf tB = pt\nindBool P pt pf fB = pf\n\n------------------------------------------------------------------------------\n-- Basic functions.\n\n-- The conjunction.\npostulate\n  _&&_  : D \u2192 D \u2192 D\n  &&-tt : true  && true  \u2261 true\n  &&-tf : true  && false \u2261 false\n  &&-ft : false && true  \u2261 false\n  &&-ff : false && false \u2261 false\n{-# ATP axiom &&-tt #-}\n{-# ATP axiom &&-tf #-}\n{-# ATP axiom &&-ft #-}\n{-# ATP axiom &&-ff #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The booleans\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Bool where\n\nopen import LTC.Minimal\n\ninfixr 6 _&&_\n\n------------------------------------------------------------------------------\n-- The LTC booleans type.\ndata Bool : D \u2192 Set where\n  tB : Bool true\n  fB : Bool false\n\n-- The rule of proof by case analysis on 'B'.\nindBool : (P : D \u2192 Set) \u2192 P true \u2192 P false \u2192 {b : D} \u2192 Bool b \u2192 P b\nindBool P pt pf tB = pt\nindBool P pt pf fB = pf\n\n------------------------------------------------------------------------------\n-- Basic functions.\n\n-- The conjunction.\npostulate\n  _&&_  : D \u2192 D \u2192 D\n  &&-tt : true  && true  \u2261 true\n  &&-tf : true  && false \u2261 false\n  &&-ft : false && true  \u2261 false\n  &&-ff : false && false \u2261 false\n{-# ATP axiom &&-tt #-}\n{-# ATP axiom &&-tf #-}\n{-# ATP axiom &&-ft #-}\n{-# ATP axiom &&-ff #-}\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"13f6840654cc9d02a520f0e6cc1fadc57b3c030e","subject":"TDDwI","message":"TDDwI\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"idris\/book\/2017-Type_Driven_Development_with_Idris\/src\/P236-257-predicates-hangman.agda","new_file":"idris\/book\/2017-Type_Driven_Development_with_Idris\/src\/P236-257-predicates-hangman.agda","new_contents":"module P236-257-predicates-hangman where\n\nopen import Agda.Builtin.Equality\nopen import Agda.Builtin.Unit                     using (\u22a4)\nopen import Data.Bool\nopen import Data.Char                             as C hiding (_==_) renaming (_\u225f_ to _\u225fC_)\nopen import Data.Empty\nopen import Data.Nat                              renaming (_\u225f_ to _\u225fN_)\nopen import Data.List                             as L\nopen import Data.Product\nopen import Data.Vec                              as V hiding (_>>=_)\nopen import Data.String                           as S hiding (_==_)\nopen import Data.Sum\nopen import Data.Vec.Membership.Propositional\nopen import Data.Vec.Relation.Unary.Any\nopen import Function.Base\nopen import hcio                                  as HCIO\nopen import IO                                    as IO hiding (_>>=_; _>>_)\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl; cong; cong-app; sym; subst)\nopen import Relation.Nullary\n\nNat = \u2115\n_==_ = _\u2261_\n\n-- ------------------------------------------------------------------------------\n-- -- 'Elem' : guaranteeing a value is in a vector\n\nremoveElem : {a : Set} {n : Nat}\n          -> (value : a) -> (xs : Vec a (suc n)) -> (prf : value \u2208 xs)\n          -> Vec a n\nremoveElem             value (_ \u2237 ys) (here  _)     = ys\nremoveElem {_} {suc k} value (y \u2237 ys) (there later) = y \u2237 removeElem value ys later\n\nexV : Vec Nat 3\nexV = 1 \u2237 2 \u2237 3 \u2237 []\n\ntwoInVector : 2 \u2208 exV\ntwoInVector = there (here refl)\n\nrmTwo : Vec Nat 2\nrmTwo = removeElem 2 exV twoInVector\n\n-- -- auto-implicit args - automatically constructs proofs -- TODO : NOT IN AGDA\n\nremoveElemAuto : {a : Set} {n : Nat}\n              -> (value : a) -> (xs : Vec a (suc n)) -> {prf : value \u2208 xs}\n              -> Vec a n\nremoveElemAuto             value  (_ \u2237 ys) {here      _} = ys\nremoveElemAuto {_} {suc k} value  (y \u2237 ys) {there later} = y \u2237 removeElemAuto value ys {later}\n\nrmTwo' : Vec Nat 2\nrmTwo' = removeElemAuto 2 exV {twoInVector}\n\n-- ------------------------------------------------------------------------------\n-- -- decidable predicates\n\nnotInNil : {A : Set} -> (value : A \u2208 []) -> \u22a5\nnotInNil ()\n\nnotInTail : {A : Set} {n : Nat} {x value : A} {xs : Vec A n}\n         -> (notThere : value \u2208 xs   -> \u22a5)\n         -> (notHere  : (value == x) -> \u22a5)\n         -> value \u2208 (x \u2237 xs)\n         -> \u22a5\nnotInTail notThere notHere (here  v==x)  = notHere  v==x\nnotInTail notThere notHere (there later) = notThere later\n\n-- TODO       v - make it a Set (Generalize)\nisElem : {n : Nat} (value : Nat) -> (xs : Vec Nat n) -> Dec (value \u2208 xs)\nisElem value      [] = no \u03bb ()\nisElem value (x \u2237 xs)\n  with value \u225fN x\n...| yes refl = yes (here refl)\n...| no  notHere\n  with isElem value xs\n...| yes prf      = yes (there prf)\n...| no  notThere = no  (notInTail notThere notHere)\n\nelem : {n : Nat} -> Nat -> (xs : Vec Nat n) -> Bool\nelem value      []  = false\nelem value (x \u2237 xs)\n  with value \u225fN x\n...| no  _ = elem value xs\n...| yes _ = true\n\nexIsElem : Dec    (3 \u2208 1 \u2237 2 \u2237 3 \u2237 [])\nexIsElem = isElem 3   (1 \u2237 2 \u2237 3 \u2237 [])\n\n-- ------------------------------------------------------------------------------\n-- -- exercises\n\ndata Last {A : Set} : (List A) -> A -> Set where\n  LastOne  : {a   : A}                                    -> Last (a \u2237 []) a\n  LastCons : {a x : A} {xs : List A} -> (prf : Last xs a) -> Last (x \u2237 xs) a\n\nexLast123 : Last (1 \u2237 2 \u2237 3 \u2237 []) 3\nexLast123 = LastCons (LastCons LastOne)\n\nexLastTTTF : Last (true \u2237 true \u2237 true \u2237 false \u2237 []) false\nexLastTTTF = LastCons (LastCons (LastCons LastOne))\n\n-- notLastInNil : Last [] value -> Void\n-- notLastInNil  LastOne     impossible\n-- notLastInNil (LastCons _) impossible\n\n-- lastOneImpliesEq : Last [x] a -> x = a\n-- lastOneImpliesEq  LastOne       = Refl\n-- lastOneImpliesEq (LastCons prf) impossible\n\n-- tailStillLast : Last (x :: y :: xs) a -> Last (y :: xs) a\n-- tailStillLast (LastCons yxs) = yxs\n\n-- isLast : DecEq a => (xs : List a) -> (value : a) -> Dec (Last xs value)\n-- isLast            []  a = No notLastInNil\n-- isLast      (x :: []) a =\n--   case decEq x a of\n--     No  xEQaToVoid => No (\\LastPxPa => xEQaToVoid (lastOneImpliesEq LastPxPa))\n--     Yes xEQa       => doYes xEQa\n--  where\n--   doYes : (x = a) -> Dec (Last [x] a)\n--   doYes prf = rewrite prf in (Yes LastOne)\n-- isLast (x :: y :: xs) a =\n--   case isLast (y :: xs) a of\n--     No  lastYXSaToVoid => No  (\\LastXYXSa => void (lastYXSaToVoid (tailStillLast LastXYXSa)))\n--     Yes why            => Yes (LastCons why)\n\n-- exIsLast : Dec (Last [1, 2, 3] 3)\n-- exIsLast = isLast [1,2,3] 3\n\n-- ------------------------------------------------------------------------------\n-- -- hangman\n\ndata WordState : (guesses_remaining : Nat) -> (letters : Nat) -> Set where\n  MkWordState  : (guesses_remaining : Nat)\n              -> {letters : Nat}\n              -> (word    : String)\n              -> (missing : Vec Char letters)\n              -> WordState guesses_remaining letters\n\ndata Finished : Set where\n  Lost : {letters : Nat} (game : WordState 0 (suc letters)) -> Finished\n  Won  : {guesses : Nat} (game : WordState (suc guesses) 0) -> Finished\n\ndata ValidInput : List Char -> Set where\n  Letter : (c : Char) -> ValidInput (c \u2237 [])\n\nisValidNil : ValidInput [] -> \u22a5\nisValidNil ()\n\nisValidTwo : {x y : Char} {xs : List Char} -> ValidInput (x \u2237 y \u2237 xs) -> \u22a5\nisValidTwo ()\n\nisValidInput : (cs : List Char) -> Dec (ValidInput cs)\nisValidInput           []   = no  isValidNil\nisValidInput (x      \u2237 [])  = yes (Letter x)\nisValidInput (x \u2237 (y \u2237 xs)) = no  isValidTwo\n\nisValidString : (s : String) -> Dec (ValidInput (S.toList s))\nisValidString s = isValidInput (S.toList s)\n\n{-# TERMINATING #-}\nreadGuess : IO (\u03a3 (List Char) (\u03bb lc -> ValidInput lc))\nreadGuess = do\n  putStr \"Guess:\"\n  x <- HCIO.getLine\n  case isValidString (S.fromList (L.map C.toUpper (S.toList x))) of \u03bb where\n    (yes prf)    -> return (_ , prf)\n    (no  contra) -> do putStrLn \"Invalid guess\"\n                       readGuess\n\nEither = _\u228e_\n\nisElemV : {n : Nat} (value : Char) -> (xs : Vec Char n) -> Dec (value \u2208 xs)\nisElemV value      [] = no \u03bb ()\nisElemV value (x \u2237 xs)\n  with value \u225fC x\n...| yes refl = yes (here refl)\n...| no  notHere\n  with isElemV value xs\n...| yes prf      = yes (there prf)\n...| no  notThere = no  (notInTail notThere notHere)\n\nprocessGuess : {guesses letters : Nat}\n            -> (letter : Char)\n            ->         WordState (suc guesses) (suc letters)\n            -> Either (WordState      guesses  (suc letters))\n                      (WordState (suc guesses)      letters)\nprocessGuess letter (MkWordState guesses_remaining word missing) =\n  case isElemV letter missing of \u03bb where\n    (yes prf)    -> inj\u2082 (MkWordState  guesses_remaining      word (removeElem letter missing prf))\n    (no  contra) -> inj\u2081 (MkWordState (guesses_remaining \u2238 1) word missing)\n\n-- game : {guesses letters : Nat} -> WordState (suc guesses) (suc letters) -> IO Finished\n-- game {guesses} {letters} st = do\n--   (_ , Letter letter) <- readGuess\n--   case processGuess letter st of \u03bb where\n--     (inj\u2081 l) -> do\n--       putStrLn \"Wrong!\"\n--       case guesses of \u03bb where\n--         zero    -> return (Lost {!!})\n--         (suc k) -> game {!!}\n--     (inj\u2082 r) -> do\n--       putStrLn \"Right!\"\n--       case letters of \u03bb where\n--         zero    -> return (Won {!!})\n--         (suc k) -> game {!!}\n\n-- main : IO ()\n-- main = do\n--   result <- game {guesses=2} (MkWordState \"Test\" ['T', 'E', 'S'])\n--   case result of\n--     Lost (MkWordState word missing) => putStrLn (\"You lose. The word was \" ++ word)\n--     Won game                        => putStrLn \"You win!\"\n\n-- {-\n-- :exec main\n-- -}\n","old_contents":"module P236-257-predicates-hangman where\n\nopen import Agda.Builtin.Equality\nopen import Agda.Builtin.Unit using (\u22a4)\nopen import Data.Bool\nopen import Data.Empty\nopen import Data.Nat renaming (_\u225f_ to _\u225fN_)\nopen import Data.List as L\nopen import Data.Vec as V\nopen import Data.Vec.Membership.Propositional\nopen import Data.Vec.Relation.Unary.Any\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl; cong; cong-app; sym; subst)\nopen import Relation.Nullary\n\nNat = \u2115\n_==_ = _\u2261_\n\n-- ------------------------------------------------------------------------------\n-- -- 'Elem' : guaranteeing a value is in a vector\n\nremoveElem : {a : Set} {n : Nat}\n          -> (value : a) -> (xs : Vec a (suc n)) -> (prf : value \u2208 xs)\n          -> Vec a n\nremoveElem             value (_ \u2237 ys) (here  _)     = ys\nremoveElem {_} {suc k} value (y \u2237 ys) (there later) = y \u2237 removeElem value ys later\n\nexV : Vec Nat 3\nexV = 1 \u2237 2 \u2237 3 \u2237 []\n\ntwoInVector : 2 \u2208 exV\ntwoInVector = there (here refl)\n\nrmTwo : Vec Nat 2\nrmTwo = removeElem 2 exV twoInVector\n\n-- -- auto-implicit args - automatically constructs proofs -- TODO : NOT IN AGDA\n\nremoveElemAuto : {a : Set} {n : Nat}\n              -> (value : a) -> (xs : Vec a (suc n)) -> {prf : value \u2208 xs}\n              -> Vec a n\nremoveElemAuto             value  (_ \u2237 ys) {here      _} = ys\nremoveElemAuto {_} {suc k} value  (y \u2237 ys) {there later} = y \u2237 removeElemAuto value ys {later}\n\nrmTwo' : Vec Nat 2\nrmTwo' = removeElemAuto 2 exV {twoInVector}\n\n-- ------------------------------------------------------------------------------\n-- -- decidable predicates\n\nnotInNil : {A : Set} -> (value : A \u2208 []) -> \u22a5\nnotInNil ()\n\nnotInTail : {A : Set} {n : Nat} {x value : A} {xs : Vec A n}\n         -> (notThere : value \u2208 xs   -> \u22a5)\n         -> (notHere  : (value == x) -> \u22a5)\n         -> value \u2208 (x \u2237 xs)\n         -> \u22a5\nnotInTail notThere notHere (here  v==x)  = notHere  v==x\nnotInTail notThere notHere (there later) = notThere later\n\n-- TODO       v - make it a Set (Generalize)\nisElem : {n : Nat} (value : Nat) -> (xs : Vec Nat n) -> Dec (value \u2208 xs)\nisElem value      [] = no \u03bb ()\nisElem value (x \u2237 xs)\n  with value \u225fN x\n...| yes refl = yes (here refl)\n...| no  notHere\n  with isElem value xs\n...| yes prf      = yes (there prf)\n...| no  notThere = no  (notInTail notThere notHere)\n\nelem : {n : Nat} -> Nat -> (xs : Vec Nat n) -> Bool\nelem value      []  = false\nelem value (x \u2237 xs)\n  with value \u225fN x\n...| no  _ = elem value xs\n...| yes _ = true\n\nexIsElem : Dec    (3 \u2208 1 \u2237 2 \u2237 3 \u2237 [])\nexIsElem = isElem 3   (1 \u2237 2 \u2237 3 \u2237 [])\n\n-- ------------------------------------------------------------------------------\n-- -- exercises\n\n-- data ElemL : a -> List a -> Type where\n--   Here  : ElemL x (x :: xs)\n--   There : (later : ElemL x xs) -> ElemL x (y :: xs)\n\n-- notInNilL : ElemL value [] -> Void\n-- notInNilL  Here     impossible\n-- notInNilL (There _) impossible\n\n-- notInTailL : (notThere : ElemL value xs -> Void) -> (notHere : (value = x) -> Void)\n--           -> ElemL value (x :: xs) -> Void\n-- notInTailL notThere notHere  Here         = notHere  Refl\n-- notInTailL notThere notHere (There later) = notThere later\n\n-- isElemL : DecEq a => (value : a) -> (xs : List a) -> Dec (ElemL value xs)\n-- isElemL value       []  = No notInNilL\n-- isElemL value (x :: xs) =\n--   case decEq value x of\n--     Yes Refl   => Yes Here\n--     No notHere =>\n--       case isElemL value xs of\n--         Yes prf     => Yes (There prf)\n--         No notThere => No (notInTailL notThere notHere)\n\n-- elemL : Eq ty => (value : ty) -> (xs : List ty) -> Bool\n-- elemL value       []  = False\n-- elemL value (x :: xs) =\n--   case value == x of\n--     False => elemL value xs\n--     True  => True\n\n-- exIsElemL : Dec (ElemL 3 [1,2,3])\n-- exIsElemL = isElemL 3 [1,2,3]\n\n-- -----\n\n-- data Last : List a -> a -> Type where\n--   LastOne  : Last [value] value\n--   LastCons : (prf : Last xs value) -> Last (x :: xs) value\n\n-- exLast123 : Last [1,2,3] 3\n-- exLast123 = LastCons (LastCons LastOne)\n\n-- notLastInNil : Last [] value -> Void\n-- notLastInNil  LastOne     impossible\n-- notLastInNil (LastCons _) impossible\n\n-- lastOneImpliesEq : Last [x] a -> x = a\n-- lastOneImpliesEq  LastOne       = Refl\n-- lastOneImpliesEq (LastCons prf) impossible\n\n-- tailStillLast : Last (x :: y :: xs) a -> Last (y :: xs) a\n-- tailStillLast (LastCons yxs) = yxs\n\n-- isLast : DecEq a => (xs : List a) -> (value : a) -> Dec (Last xs value)\n-- isLast            []  a = No notLastInNil\n-- isLast      (x :: []) a =\n--   case decEq x a of\n--     No  xEQaToVoid => No (\\LastPxPa => xEQaToVoid (lastOneImpliesEq LastPxPa))\n--     Yes xEQa       => doYes xEQa\n--  where\n--   doYes : (x = a) -> Dec (Last [x] a)\n--   doYes prf = rewrite prf in (Yes LastOne)\n-- isLast (x :: y :: xs) a =\n--   case isLast (y :: xs) a of\n--     No  lastYXSaToVoid => No  (\\LastXYXSa => void (lastYXSaToVoid (tailStillLast LastXYXSa)))\n--     Yes why            => Yes (LastCons why)\n\n-- exIsLast : Dec (Last [1, 2, 3] 3)\n-- exIsLast = isLast [1,2,3] 3\n\n-- ------------------------------------------------------------------------------\n-- -- hangman\n\n-- data WordState : (guesses_remaining : Nat) -> (letters : Nat) -> Type where\n--   MkWordState : (word : String) -> (missing : Vect letters Char)\n--              -> WordState guesses_remaining letters\n\n-- data Finished : Type where\n--   Lost : (game : WordState 0 (S letters)) -> Finished\n--   Won  : (game : WordState (S guesses) 0) -> Finished\n\n-- data ValidInput : List Char -> Type where\n--   Letter : (c : Char) -> ValidInput [c]\n\n-- isValidNil : ValidInput [] -> Void\n-- isValidNil (Letter _) impossible\n\n-- isValidTwo : ValidInput (x :: y :: xs) -> Void\n-- isValidTwo (Letter _) impossible\n\n-- isValidInput : (cs : List Char) -> Dec (ValidInput cs)\n-- isValidInput             []   = No  isValidNil\n-- isValidInput (x       :: [])  = Yes (Letter x)\n-- isValidInput (x :: (y :: xs)) = No  isValidTwo\n\n-- isValidString : (s : String) -> Dec (ValidInput (unpack s))\n-- isValidString s = isValidInput (unpack s)\n\n-- readGuess : IO (x ** ValidInput x)\n-- readGuess = do\n--   putStr \"Guess:\"\n--   x <- getLine\n--   case isValidString (toUpper x) of\n--     Yes prf    => pure (_ ** prf)\n--     No  contra => do putStrLn \"Invalid guess\"\n--                      readGuess\n-- {-\n-- :exec readGuess >>= \\_ => putStrLn \"YES\"\n-- -}\n\n-- processGuess : (letter : Char)\n--             ->         WordState (S guesses) (S letters)\n--             -> Either (WordState    guesses  (S letters))\n--                       (WordState (S guesses)    letters)\n-- processGuess letter (MkWordState word missing) =\n--   case V.isElem letter missing of\n--     Yes prf    => Right (MkWordState word (removeElemAuto letter missing))\n--     No  contra => Left  (MkWordState word missing)\n\n-- game : WordState (S guesses) (S letters) -> IO Finished\n-- game {guesses} {letters} st = do\n--    (_ ** Letter letter) <- readGuess\n--    case processGuess letter st of\n--      Left l => do\n--        putStrLn \"Wrong!\"\n--        case guesses of\n--          Z   => pure (Lost l)\n--          S k => game l\n--      Right r => do\n--        putStrLn \"Right!\"\n--        case letters of\n--          Z   => pure (Won r)\n--          S k => game r\n\n-- main : IO ()\n-- main = do\n--   result <- game {guesses=2} (MkWordState \"Test\" ['T', 'E', 'S'])\n--   case result of\n--     Lost (MkWordState word missing) => putStrLn (\"You lose. The word was \" ++ word)\n--     Won game                        => putStrLn \"You win!\"\n\n-- {-\n-- :exec main\n-- -}\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"facbec4fc36be9a8232d2a4c95ac44184120ba7c","subject":"Data.Nat.NP: add 2*-spec, 2*\u2032_, 2*\u2032-spec, 2^-spec","message":"Data.Nat.NP: add 2*-spec, 2*\u2032_, 2*\u2032-spec, 2^-spec\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\n{-\npostulate\n  dist-refl  : \u2200 x \u2192 dist x x \u2261 0\n  dist-sym   : \u2200 x y \u2192 dist x y \u2261 dist y x\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-x+    : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n  dist-x*  : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n\n                      | \u2115\u00b0.+-comm 1 (n + n)\n                      | \u2115\u00b0.+-assoc n n 1\n                      | \u2115\u00b0.+-comm n 1\n                      = \u2261.refl\n\n2^_ : \u2115 \u2192 \u2115\n2^ zero  = 1\n2^ suc n = 2* 2^ n\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\n{-\npostulate\n  dist-refl  : \u2200 x \u2192 dist x x \u2261 0\n  dist-sym   : \u2200 x y \u2192 dist x y \u2261 dist y x\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-x+    : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n  dist-x*  : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ zero  = 1\n2^ suc n = 2* 2^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0943bda3d970a1ec75f5dfe1fa182d290d84cf31","subject":"Bits: restore the previous proof","message":"Bits: restore the previous proof\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"90a1126b654ec2e7a15528e4bbd4a65b0ab4f0b9","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Even.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Even.agda","new_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Even : D \u2192 Set where\n  zE :                  Even zero\n  nE : \u2200 {d} \u2192 Even d \u2192 Even (succ\u2081 (succ\u2081 d))\n\nEven-ind : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {d} \u2192 A d \u2192 A (succ\u2081 (succ\u2081 d))) \u2192\n           \u2200 {d} \u2192 Even d \u2192 A d\nEven-ind A A0 h zE      = A0\nEven-ind A A0 h (nE Ed) = h (Even-ind A A0 h Ed)\n","old_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Even : D \u2192 Set where\n  zE :                  Even zero\n  nE : \u2200 {d} \u2192 Even d \u2192 Even (succ\u2081 (succ\u2081 d))\n\nEven-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {d} \u2192 A d \u2192 A (succ\u2081 (succ\u2081 d))) \u2192\n          \u2200 {d} \u2192 Even d \u2192 A d\nEven-ind A A0 h zE      = A0\nEven-ind A A0 h (nE Ed) = h (Even-ind A A0 h Ed)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"90e62144b22dcc8d6181e110e5f313f406aff57f","subject":"change wording","message":"change wording\n","repos":"spire\/spire","old_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\nAn alternative to the Arith codes is to create a *view* of the\nnatural numbers along with a collection of functions over them.\n-}\n\ndata ArithView : \u2115 \u2192 Set where\n  `nat : (n : \u2115) \u2192 ArithView n\n  _`+_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 + n\u2082)\n  _`*_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith with Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nHere is the ArithView specialized with the Fin type.\n-}\n\ndata Fin\u2083 : \u2115 \u2192 Set where\n  fin : (n : \u2115) (i : Fin n) \u2192 Fin\u2083 n\n  fin+ : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 + n\u2082)) \u2192 Fin\u2083 (n\u2081 + n\u2082)\n  fin* : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 * n\u2082)) \u2192 Fin\u2083 (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `ArithView `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `Fin\u2083 : (n : \u2115) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `ArithView n \u27e7 = ArithView n\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `Fin\u2083 n \u27e7 = Fin\u2083 n\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin\u2082 : Tactic\nrm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin\u2082 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin\u2082 : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin\u2082 a (fin i) = proj\u2082\n  (rm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic that simplifies an ArithView index.\nTo target Fin, ideally you want to match on something like:\n`\u03a3 `\u2115 (\u03bb n \u2192 `ArithView n `\u00d7 Fin n)\nBut this requires matching on a \u03bb, and a non-linear occurrence\nof n.\n\nThe example after this one accomplishes the same thing via the\nspecialized Fin\u2083 type instead.\n-}\n\nrm-plus0-ArithView : Tactic\nrm-plus0-ArithView (`ArithView .(n + 0) , (n `+ 0)) =\n  `ArithView n , `nat n \nrm-plus0-ArithView x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify to simplify the \"view\"-based Fin type Fin\u2083.\nThis is most similar to rm-plus0-Fin\u2082, except Fin\u2083 is view-based\nwhile Fin\u2082 is universe-based (where the universe is the codes of\npossible operations).\n\nOn one hand this Fin\u2083 approach is simpler than Fin\u2082, because you do\nnot need to write an eval function or tell it to reduce definitional\nequalities explicitly (like you would to get Arith to compute).\n\nOn the other hand, the Fin\u2082 approach is more powerful because it\nallows you to control when things reduce. This is more like Coq\ntactics, where you might not want something to reduce and you use\n\"simpl\" explicitly when you do. The \"simpl\" tactic is defined below\nvia the eval\u2082 function.\n-}\n\nrm-plus0-Fin\u2083 : Tactic\nrm-plus0-Fin\u2083 (`Fin\u2083 .(n + 0) , fin+ n 0 i) =\n  `Fin\u2083 n\n  ,\n  fin n (subst Fin (sym (plusrident n)) i)\nrm-plus0-Fin\u2083 x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\nsimp-step-n-plus0-Fin : \u2115 \u2192 Tactic\nsimp-step-n-plus0-Fin zero x = x\nsimp-step-n-plus0-Fin (suc n) x =\n  simp-step-n-plus0-Fin n (simp-step-plus0-Fin x)\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","old_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\nAn alternative to the Arith codes is to create a *view* of the\nnatural numbers along with a collection of functions over them.\n-}\n\ndata ArithView : \u2115 \u2192 Set where\n  `nat : (n : \u2115) \u2192 ArithView n\n  _`+_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 + n\u2082)\n  _`*_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nHere is the ArithView specialized into the Fin type.\n-}\n\ndata Fin\u2083 : \u2115 \u2192 Set where\n  fin : (n : \u2115) (i : Fin n) \u2192 Fin\u2083 n\n  fin+ : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 + n\u2082)) \u2192 Fin\u2083 (n\u2081 + n\u2082)\n  fin* : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 * n\u2082)) \u2192 Fin\u2083 (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `ArithView `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `Fin\u2083 : (n : \u2115) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `ArithView n \u27e7 = ArithView n\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `Fin\u2083 n \u27e7 = Fin\u2083 n\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin\u2082 : Tactic\nrm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin\u2082 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin\u2082 : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin\u2082 a (fin i) = proj\u2082\n  (rm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic that simplifies an ArithView index.\nTo target Fin, ideally you want to match on something like:\n`\u03a3 `\u2115 (\u03bb n \u2192 `ArithView n `\u00d7 Fin n)\nBut this requires matching on a \u03bb, and a non-linear occurrence\nof n.\n\nThe example after this one accomplishes the same thing via the\nspecialized Fin\u2083 type instead.\n-}\n\nrm-plus0-ArithView : Tactic\nrm-plus0-ArithView (`ArithView .(n + 0) , (n `+ 0)) =\n  `ArithView n , `nat n \nrm-plus0-ArithView x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify to simplify the \"view\"-based Fin type Fin\u2083.\nThis is most similar to rm-plus0-Fin\u2082, except Fin\u2083 is view-based\nwhile Fin\u2082 is universe-based (where the universe is the codes of\npossible operations).\n\nOn one hand this Fin\u2083 approach is simpler than Fin\u2082, because you do\nnot need to write an eval function or tell it to reduce definitional\nequalities explicitly (like you would to get Arith to compute).\n\nOn the other hand, the Fin\u2082 approach is more powerful because it\nallows you to control when things reduce. This is more like Coq\ntactics, where you might not want something to reduce and you use\n\"simpl\" explicitly when you do. The \"simpl\" tactic is defined below\nvia the eval\u2082 function.\n-}\n\nrm-plus0-Fin\u2083 : Tactic\nrm-plus0-Fin\u2083 (`Fin\u2083 .(n + 0) , fin+ n 0 i) =\n  `Fin\u2083 n\n  ,\n  fin n (subst Fin (sym (plusrident n)) i)\nrm-plus0-Fin\u2083 x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\nsimp-step-n-plus0-Fin : \u2115 \u2192 Tactic\nsimp-step-n-plus0-Fin zero x = x\nsimp-step-n-plus0-Fin (suc n) x =\n  simp-step-n-plus0-Fin n (simp-step-plus0-Fin x)\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"224bef483393cb10b22b2f51c63865cc5ddf471f","subject":"Fix comment (fix #20).","message":"Fix comment (fix #20).\n\nOld-commit-hash: d5a7ef44c7c38bf82aa00466ee0bac803dd860d3\n","repos":"inc-lc\/ilc-agda","old_file":"partial.agda","new_file":"partial.agda","new_contents":"module partial where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with (one-variable) partial derivatives\n--\n-- Features:\n--   * Changes and derivatives are different.\n--   * \u0394 x dx e  describes how e changes if x changes by dx.\n--   * a full denotational semantics is given.\n--   * the domain of change types `\u27e6\u0394 \u03c4\u27e7` is uniformly defined as `\u27e6\u03c4\u27e7 \u2192 \u27e6\u03c4\u27e7`.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\n  -- `\u0394 \u03c4` is the type of changes to `\u03c4`\n  \u0394 : (\u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 \u0394 \u03c4 \u27e7Type = \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 x dx t` maps changes `dx` in `x` to changes in `t`\n  \u0394 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (x : Var \u0393 \u03c4\u2081) (dx : Var \u0393 (\u0394 \u03c4\u2081)) \u2192 (t : Term \u0393 \u03c4\u2082) \u2192 Term \u0393 (\u0394 \u03c4\u2082)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 x dx t \u27e7Term \u03c1 = \u03bb _ \u2192 \u27e6 t \u27e7Term (update x (\u27e6 dx \u27e7 \u03c1) \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 x dx t) = \u0394 (lift {\u0393\u2081} {\u0393\u2082} x) (lift {\u0393\u2081} {\u0393\u2082} dx) (weaken {\u0393\u2081} {\u0393\u2082} t)\n","old_contents":"module partial where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with (one-variable) partial derivatives\n--\n-- Features:\n--   * Changes and derivatives are different.\n--   * \u0394 x e  maps changes of x to changes of e.\n--   * a full denotational semantics is given.\n--   * the domain of change types `\u27e6\u0394 \u03c4\u27e7` is uniformly defined as `\u27e6\u03c4\u27e7 \u2192 \u27e6\u03c4\u27e7`.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\n  -- `\u0394 \u03c4` is the type of changes to `\u03c4`\n  \u0394 : (\u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 \u0394 \u03c4 \u27e7Type = \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 x dx t` maps changes `dx` in `x` to changes in `t`\n  \u0394 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (x : Var \u0393 \u03c4\u2081) (dx : Var \u0393 (\u0394 \u03c4\u2081)) \u2192 (t : Term \u0393 \u03c4\u2082) \u2192 Term \u0393 (\u0394 \u03c4\u2082)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 x dx t \u27e7Term \u03c1 = \u03bb _ \u2192 \u27e6 t \u27e7Term (update x (\u27e6 dx \u27e7 \u03c1) \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 x dx t) = \u0394 (lift {\u0393\u2081} {\u0393\u2082} x) (lift {\u0393\u2081} {\u0393\u2082} dx) (weaken {\u0393\u2081} {\u0393\u2082} t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6a0262f170cea36ade044674b08a0631f7368219","subject":"Formalize function changes.","message":"Formalize function changes.\n\nOld-commit-hash: b5f06c519796a50f29ede0a97ed1db731e73252b\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n\n-- Function changes\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n    changeAlgebra = record\n      { Change = FunctionChange\n      ; update = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n      ; diff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb g f \u2192 ext (\u03bb a \u2192\n          begin\n            f a \u229e (g (a \u229e nil a) \u229f f a)\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n            f a \u229e (g a \u229f f a)\n          \u2261\u27e8 update-diff (g a) (f a) \u27e9\n            g a\n          \u220e)\n        }\n      }\n","old_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Abelian groups induce change algebras\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"554ca676d7b2257485f0305e2a5ed3f5e4701120","subject":"Finalized a version of the mirror example using the standard library.","message":"Finalized a version of the mirror example using the standard library.\n\nIgnore-this: 4aa6c329d508c0a7472f3884d5b85557\n\ndarcs-hash:20110211131217-3bd4e-de3f5e6d21d012ceac60d15886b0a527a423abc8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/MirrorStdLib.agda","new_file":"Draft\/Mirror\/MirrorStdLib.agda","new_contents":"module MirrorStdLib where\n\nopen import Algebra\nopen import Data.List as List hiding ( reverse )\nopen import Data.List.Properties\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\nmodule LM {A : Set} = Monoid (List.monoid A)\n\n------------------------------------------------------------------------------\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n++-rightIdentity : {A : Set}(xs : List A) \u2192 xs ++ [] \u2261 xs\n++-rightIdentity []       = refl\n++-rightIdentity (x \u2237 xs) = cong (_\u2237_ x) (++-rightIdentity xs)\n\nreverse-++ : {A : Set}(xs ys : List A) \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ [] ys       = sym (++-rightIdentity (reverse ys))\nreverse-++ (x \u2237 xs) [] = cong (\u03bb x' \u2192 reverse x' ++ x \u2237 []) (++-rightIdentity xs)\nreverse-++ (x \u2237 xs) (y \u2237 ys) =\n  begin\n    reverse (xs ++ y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 cong (\u03bb x' \u2192 x' ++ x \u2237 []) (reverse-++ xs (y \u2237 ys)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 LM.assoc (reverse (y \u2237 ys)) (reverse xs) (x \u2237 []) \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\ndata Tree (A : Set) : Set where\n  treeT : A \u2192 List (Tree A) \u2192 Tree A\n\nmirror : {A : Set} \u2192 Tree A \u2192 Tree A\nmirror (treeT a ts) = treeT a (reverse (map mirror ts))\n\nmutual\n  mirror\u00b2 : {A : Set} \u2192 (t : Tree A) \u2192 mirror (mirror t) \u2261 t\n  mirror\u00b2 (treeT a [])       = refl\n  mirror\u00b2 (treeT a (t \u2237 ts)) =\n    begin\n      treeT a (reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts) ++\n                                                      mirror t \u2237 []))) \u2261\n                        treeT a x)\n           (aux (t \u2237 ts))\n           refl\n        \u27e9\n      treeT a (t \u2237 ts)\n    \u220e\n\n  aux : {A : Set} \u2192 (ts : List (Tree A)) \u2192\n        reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  aux []       = refl\n  aux (t \u2237 ts) =\n    begin\n      reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 []))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror (reverse (map mirror ts)) (mirror t \u2237 []))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n                          (map mirror (mirror t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++ (map mirror (reverse (map mirror ts)))\n                             (map mirror (mirror t \u2237 [])))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror t \u2237 [])) ++\n              reverse (map mirror (reverse (map mirror ts)))\n              \u2261\u27e8 refl \u27e9\n      mirror (mirror t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror (mirror t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 t)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (aux ts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n","old_contents":"module MirrorStdLib where\n\nopen import Algebra\nopen import Data.List as List hiding ( reverse )\nopen import Data.List.Properties\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\nmodule LM {A : Set} = Monoid (List.monoid A)\n\n------------------------------------------------------------------------------\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n++-rightIdentity : {A : Set}(xs : List A) \u2192 xs ++ [] \u2261 xs\n++-rightIdentity []       = refl\n++-rightIdentity (x \u2237 xs) = cong (_\u2237_ x) (++-rightIdentity xs)\n\nreverse-++ : {A : Set}(xs ys : List A) \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ [] ys       = sym (++-rightIdentity (reverse ys))\nreverse-++ (x \u2237 xs) [] = cong (\u03bb x' \u2192 reverse x' ++ x \u2237 []) (++-rightIdentity xs)\nreverse-++ (x \u2237 xs) (y \u2237 ys) =\n  begin\n    reverse (xs ++ y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 cong (\u03bb x' \u2192 x' ++ x \u2237 []) (reverse-++ xs (y \u2237 ys)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n      \u2261\u27e8 LM.assoc (reverse (y \u2237 ys)) (reverse xs) (x \u2237 []) \u27e9\n    reverse (y \u2237 ys) ++ reverse (x \u2237 xs)\n  \u220e\n\ndata Tree (A : Set) : Set where\n  treeT : A \u2192 List (Tree A) \u2192 Tree A\n\nmirror : {A : Set} \u2192 Tree A \u2192 Tree A\nmirror (treeT a ts) = treeT a (reverse (map mirror ts))\n\nmirror\u00b2 : {A : Set} \u2192 (t : Tree A) \u2192 mirror (mirror t) \u2261 t\nmirror\u00b2 (treeT a [])       = refl\nmirror\u00b2 (treeT a (t \u2237 ts)) =\n  begin\n    treeT a (reverse (map mirror (reverse (map mirror ts) ++ mirror t \u2237 [])))\n      \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts) ++\n                                                    mirror t \u2237 []))) \u2261\n                      treeT a (reverse x))\n         (map-++-commute mirror (reverse (map mirror ts)) (mirror t \u2237 []))\n         refl\n      \u27e9\n    treeT a (reverse (map mirror (reverse (map mirror ts)) ++\n                     (map mirror (mirror t \u2237 []))))\n      \u2261\u27e8 subst (\u03bb x \u2192 treeT a (reverse (map mirror (reverse (map mirror ts)) ++\n                                       (map mirror (mirror t \u2237 [])))) \u2261\n                      treeT a x)\n               (reverse-++ (map mirror (reverse (map mirror ts)))\n                           (map mirror (mirror t \u2237 [])))\n               refl\n      \u27e9\n    treeT a (reverse (map mirror (mirror t \u2237 [])) ++\n             reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 refl \u27e9\n    treeT a (mirror (mirror t) \u2237 reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 subst (\u03bb x \u2192 treeT a (mirror (mirror t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                      treeT a (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n               (mirror\u00b2 t)  -- IH.\n               refl\n      \u27e9\n    treeT a (t \u2237 reverse (map mirror (reverse (map mirror ts))))\n      \u2261\u27e8 {!!} \u27e9\n    treeT a (t \u2237 ts)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c87576d2241944da63e98945fa5620f8df4eb3d1","subject":"Build fix","message":"Build fix\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.BigStep where\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_)\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (derive-var x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ unit ]\u03c4 dv from v1 to v2 = \u22a4\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 derive-var x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!fromtoDerive _ (suc n) s d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!LIE!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!LIE!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.BigStep where\n\nopen import Data.Empty\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_)\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (derive-var x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 derive-var x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!fromtoDerive _ (suc n) s d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!LIE!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!LIE!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"311106e13349d10cdfa60f233bf460ab290ec25f","subject":"README","message":"README\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import Explore.README\n\nopen import Game.DDH\nopen import Game.IND-CPA\nopen import Game.EntropySmoothing\nopen import Game.EntropySmoothing.WithKey\nopen import Game.Transformation.InternalExternal\n\nopen import Crypto.Schemes\nopen import Cipher.ElGamal.Generic\n\nopen import Solver.Linear\nopen import Solver.AddMax\n\nopen import FunUniverse.README\n\nopen import bijection-syntax.README\n\nopen import alea.cpo\n\nopen import circuits.circuit\n\nopen import Composition.Horizontal\nopen import Composition.Vertical\nopen import Composition.Forkable\n","old_contents":"module README where\n\nopen import Game.DDH\nopen import Game.IND-CPA\nopen import Game.EntropySmoothing\nopen import Game.EntropySmoothing.WithKey\nopen import Game.Transformation.InternalExternal\n\nopen import Crypto.Schemes\nopen import Cipher.ElGamal.Generic\n\nopen import Solver.Linear\nopen import Solver.AddMax\n\nopen import FunUniverse.README\n\nopen import bijection-syntax.README\n\n--open import alea.cpo\n\nopen import circuits.circuit\n\nopen import Composition.Horizontal\nopen import Composition.Vertical\nopen import Composition.Forkable\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1023fbb79c34ad706eb34cb8b92af92beb59d44e","subject":"Bijection.NP","message":"Bijection.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Bijection\/NP.agda","new_file":"lib\/Function\/Bijection\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Bijection.NP where\n\nopen import Function.Bijection public\nopen import Function.Surjection\nopen import Function.Equality hiding (flip)\nopen import Relation.Binary\nopen import Data.Product\nopen import Level\n\n_\u27e8$\u27e9\u2081_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Bijection From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\n_\u27e8$\u27e9\u2081_ bij x = Bijection.to bij \u27e8$\u27e9 x\n\n_\u27e8$\u27e9\u2082_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Bijection From To \u2192 Setoid.Carrier To \u2192 Setoid.Carrier From\n_\u27e8$\u27e9\u2082_ bij x = Bijection.from bij \u27e8$\u27e9 x\n\n_\u2208_ : \u2200  {a\u2081 a\u2082}\n         {A\u2081 A\u2082  : Setoid a\u2081 a\u2082}\n         (x\u1d62     : Setoid.Carrier A\u2081 \u00d7 Setoid.Carrier A\u2082)\n         (bij    : Bijection A\u2081 A\u2082)\n      \u2192  Set a\u2082\n_\u2208_ {A\u2082 = A\u2082} (x\u2081 , x\u2082) bij = bij \u27e8$\u27e9\u2081 x\u2081 \u2248 x\u2082\n  where open Setoid A\u2082\n\n-- Bijection is symmetric\nflip : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082} {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n       \u2192 Bijection From To \u2192 Bijection To From\nflip bij =\n  record { to = from\n         ; bijective =\n             record { injective = Surjection.injective surjection\n                    ; surjective =\n                        record { from = to\n                               ; right-inverse-of = left-inverse-of } } }\n  where open Bijection bij\n","old_contents":"{-# OPTIONS --without-K #-}\n{-# OPTIONS --universe-polymorphism #-}\nmodule Function.Bijection.NP where\n\nopen import Function.Bijection public\nopen import Function.Surjection\nopen import Function.Equality\nopen import Relation.Binary\nopen import Data.Product\nopen import Level\n\n_\u27e8$\u27e9\u2081_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Bijection From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\n_\u27e8$\u27e9\u2081_ bij x = Bijection.to bij \u27e8$\u27e9 x\n\n_\u27e8$\u27e9\u2082_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Bijection From To \u2192 Setoid.Carrier To \u2192 Setoid.Carrier From\n_\u27e8$\u27e9\u2082_ bij x = Bijection.from bij \u27e8$\u27e9 x\n\n_\u2208_ : \u2200  {a\u2081 a\u2082}\n         {A\u2081 A\u2082  : Setoid a\u2081 a\u2082}\n         (x\u1d62     : Setoid.Carrier A\u2081 \u00d7 Setoid.Carrier A\u2082)\n         (bij    : Bijection A\u2081 A\u2082)\n      \u2192  Set a\u2082\n_\u2208_ {A\u2082 = A\u2082} (x\u2081 , x\u2082) bij = bij \u27e8$\u27e9\u2081 x\u2081 \u2248 x\u2082\n  where open Setoid A\u2082\n\n-- Bijection is symmetric\nflip : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082} {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n       \u2192 Bijection From To \u2192 Bijection To From\nflip bij =\n  record { to = from\n         ; bijective =\n             record { injective = Surjection.injective surjection\n                    ; surjective =\n                        record { from = to\n                               ; right-inverse-of = left-inverse-of } } }\n  where open Bijection bij\n\n{-\nwip : \u2200  {a\u2081 a\u2082}\n         {A\u2081 A\u2082  : Setoid a\u2081 a\u2082}\n         (x\u1d62     : Setoid.Carrier A\u2081 \u00d7 Setoid.Carrier A\u2082)\n         (bij    : Bijection A\u2081 A\u2082)\n      \u2192 x\u1d62 \u2208 bij \u2192 swap x\u1d62 \u2208 flip bij\nwip (x\u2081 , x\u2082) bij x = ?\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ea2e84b530096fb3b82a0a94709f2fea00131cd6","subject":"Before changing the nextComs to comply with the new definitions of reverseTran, indRevNoComs functions","message":"Before changing the nextComs to comply with the new definitions of reverseTran, indRevNoComs functions\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/IndexLFCo.agda","new_file":"agda\/IndexLFCo.agda","new_contents":"{-# OPTIONS --exact-split #-}\nmodule IndexLFCo where\n\nopen import Common\nopen import LinLogic\nopen import LinLogicProp\nopen import LinFun\nopen import IndexLLProp\nopen import Data.Maybe\n\nmodule _ where\n\n  open import Relation.Binary.PropositionalEquality using (sym)\n  open  Relation.Binary.PropositionalEquality.Deprecated-inspect\n\n\n  -- reverseTran returns nothing if during the reversion, it finds a com.\n  -- or if the cll is transformed.\n  \n\n\n  mutual\n    data ReverseTranT {i u} : \u2200{ll cll rll} \u2192 LFun ll rll \u2192 IndexLL {i} {u} cll rll \u2192 Set u where\n      cr1 : \u2200{cll rll} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 ReverseTranT I iind\n      cr2 : \u2200{ll cll rll ell pll ind lf\u2081 lf x\u2081} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 (rvT\u2081 : ReverseTranT lf\u2081 iind) \u2192 let x = reverseTran lf\u2081 iind rvT\u2081 in (just x\u2081 \u2261 x -\u2098\u1d62 (updInd ell ind)) \u2192 (rvT\u2082 : ReverseTranT lf x\u2081) \u2192 ReverseTranT (_\u2282_ {pll = pll} {ll = ll} {ell = ell} {rll = rll} {ind = ind} lf lf\u2081) iind\n      cr3 : \u2200{ll cll rll ell pll ind lf\u2081 lf} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 (rvT\u2081 : ReverseTranT lf\u2081 iind) \u2192 let x = reverseTran lf\u2081 iind rvT\u2081 in (eq\u2081 : (x -\u2098\u1d62 (updInd ell ind) \u2261 nothing)) \u2192 (eq\u2082 :((updInd ell ind) -\u2098\u1d62 x \u2261 nothing)) \u2192 ReverseTranT (_\u2282_ {pll = pll} {ll = ll} {ell = ell} {rll = rll} {ind = ind} lf lf\u2081) iind\n      cr4 : \u2200{ll cll orll rll lf x} \u2192 {ltr : LLTr orll ll} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 (rvT\u2081 : ReverseTranT lf iind) \u2192 (just x \u2261 tran (reverseTran lf iind rvT\u2081) (revTr ltr)) \u2192 ReverseTranT (tr ltr lf) iind\n\n\n\n    -- reverseTran returns nothing if during the reversion, it finds a com.\n    -- or if the cll is transformed.\n  \n    reverseTran : \u2200{i u ll cll rll} \u2192 (lf : LFun ll rll) \u2192 (iind : IndexLL {i} {u} cll rll) \u2192 ReverseTranT lf iind \u2192 IndexLL cll ll\n    reverseTran .I iind cr1 = iind\n    reverseTran (_\u2282_ {ind = ind} lf lf\u2081) iind (cr2 {x\u2081 = x\u2081} pr x pr\u2081) =  ind +\u1d62 (reverseTran lf x\u2081 pr\u2081)\n    reverseTran (_\u2282_ {ind = ind} _ lf\u2081) iind (cr3 pr eq\u2081 eq\u2082) = revUpdInd ind (reverseTran lf\u2081 iind pr) eq\u2081 eq\u2082\n    reverseTran (tr ltr lf) iind (cr4 {x = x} pr eq) = x\n\n\n  getReverseTranT : \u2200{i u ll cll rll} \u2192 (lf : LFun ll rll) \u2192 (iind : IndexLL {i} {u} cll rll) \u2192 Maybe $ ReverseTranT lf iind\n  getReverseTranT I iind = just cr1\n  getReverseTranT (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) iind with (getReverseTranT lf\u2081 iind)\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | just x with (inspect ((reverseTran lf\u2081 iind x) -\u2098\u1d62 (updInd ell ind)))\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | just x | ((just x\u2081) with-\u2261 eq) with (getReverseTranT lf x\u2081)\n  getReverseTranT (_\u2282_ {pll} {ll} {ell} {rll} {ind} lf lf\u2081) iind | just x | (just x\u2081 with-\u2261 eq) | (just x\u2082) = just (cr2 x (sym eq) x\u2082)\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | just x | (just x\u2081 with-\u2261 eq) | nothing = nothing\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | just x | (nothing with-\u2261 eq) with (inspect ((updInd ell ind) -\u2098\u1d62 (reverseTran lf\u2081 iind x)))\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | just x | (nothing with-\u2261 eq) | (just x\u2081 with-\u2261 eq\u2081) = nothing -- We do not accept transformations that change the cll. The cll definitely changes here. (unless lf only has iind).\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | just x | (nothing with-\u2261 eq) | (nothing with-\u2261 eq\u2081) = just (cr3 x eq eq\u2081)\n  getReverseTranT (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) iind | nothing = nothing\n  getReverseTranT (tr ltr lf) iind with (getReverseTranT lf iind)\n  getReverseTranT (tr ltr lf) iind | just x with (inspect (tran (reverseTran lf iind x) (revTr ltr)))\n  getReverseTranT (tr ltr lf) iind | just x | (just x\u2081 with-\u2261 eq) = just (cr4 x (sym eq))\n  getReverseTranT (tr ltr lf) iind | just x | (nothing with-\u2261 eq) = nothing\n  getReverseTranT (tr ltr lf) iind | nothing = nothing\n  getReverseTranT (com df lf) iind = nothing\n  getReverseTranT (call x) iind = nothing\n\n  data IndRevNoComsT {i u ll pll ell cll} {ind : IndexLL {i} {u} pll ll} {lind : IndexLL cll (replLL ll ind ell)} {lf : LFun pll ell} : Set u where\n    c1 : \u2200{x} \u2192 (just x \u2261 lind -\u2098\u1d62 (updInd ell ind)) \u2192 (ReverseTranT lf x) \u2192 IndRevNoComsT\n    c2 : (lind -\u2098\u1d62 (updInd ell ind) \u2261 nothing) \u2192 ((updInd ell ind) -\u2098\u1d62 lind \u2261 nothing) \u2192 IndRevNoComsT\n\n\n  indRevNoComs : \u2200{i u ll pll ell cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (lf : LFun pll ell) \u2192 IndRevNoComsT {ind = ind} {lind = lind} {lf = lf} \u2192 IndexLL cll ll\n  indRevNoComs ind lind lf (c1 {x = x} b pr) = ind +\u1d62 (reverseTran lf x pr)\n  indRevNoComs ind lind lf (c2 eq\u2081 eq\u2082) = revUpdInd ind lind eq\u2081 eq\u2082\n\n\n-- This is almost the same code as above but it is required in IndexLFCo.\n  getIndRevNoComsT : \u2200{i u ll pll ell cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (lf : LFun pll ell) \u2192 Maybe $ IndRevNoComsT {ind = ind} {lind = lind} {lf = lf}\n  getIndRevNoComsT {ell = ell} ind lind lf with (inspect (lind -\u2098\u1d62 (updInd ell ind)))\n  getIndRevNoComsT {_} {_} {_} {_} {ell} ind lind lf | just x with-\u2261 eq with (getReverseTranT lf x)\n  getIndRevNoComsT {_} {_} {_} {_} {ell} ind lind lf | just x with-\u2261 eq | (just x\u2081) = just (c1 (sym eq) x\u2081)\n  getIndRevNoComsT {_} {_} {_} {_} {ell} ind lind lf | just x with-\u2261 eq | nothing = nothing\n  getIndRevNoComsT {_} {_} {_} {_} {ell} ind lind lf | nothing with-\u2261 eq with (inspect ((updInd ell ind) -\u2098\u1d62 lind))\n  getIndRevNoComsT {_} {_} {_} {_} {ell} ind lind lf | nothing with-\u2261 eq | (just x with-\u2261 eq\u2081) = nothing\n  getIndRevNoComsT {_} {_} {_} {_} {ell} ind lind lf | nothing with-\u2261 eq | (nothing with-\u2261 eq\u2081) = just (c2 eq eq\u2081)\n\n  \ndata IndexLFCo {i u cll} (frll : LinLogic i {u}) : \u2200{ll rll} \u2192 IndexLL cll ll \u2192 LFun {i} {u} ll rll \u2192 Set (u) where\n  _\u2190\u2282 : \u2200{rll pll ell ll ind elf lf lind}\n         \u2192 IndexLFCo frll lind elf\n         \u2192 IndexLFCo frll (ind +\u1d62 lind) (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  \u2282\u2192_ : \u2200{rll pll ell ll ind elf lf lind rs}\n         \u2192 IndexLFCo frll lind lf\n         \u2192 (irnc : IndRevNoComsT {ind = ind} {lind = lind} {lf = elf})\n         \u2192 {prf : rs \u2261 indRevNoComs ind lind elf irnc}\n         \u2192 IndexLFCo frll rs (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  tr   : \u2200{ll orll rll lind rs} \u2192 {ltr : LLTr orll ll} \u2192 {lf : LFun {i} {u} orll rll}\n         \u2192 IndexLFCo frll lind lf\n         \u2192 {prf : just rs \u2261 tran lind (revTr ltr) }\n         \u2192 IndexLFCo frll rs (tr ltr lf) \n  \u2193  : \u2200{rll prfi prfo df lf}\n         \u2192 IndexLFCo  frll \u2193 (com {i} {u} {rll} {cll} {frll} {{prfi}} {{prfo}} df lf)\n\n\n\ndata SetLFCoRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2193    : \u2200{ll cll frll} \u2192 {ind : IndexLL {i} {u} cll oll} \u2192 IndexLFCo frll ind olf \u2192 SetLFCoRem olf ll\n  _\u2190\u2227   : \u2200{rs ls} \u2192 SetLFCoRem olf ls                   \u2192 SetLFCoRem olf (ls \u2227 rs)\n  \u2227\u2192_   : \u2200{rs ls} \u2192 SetLFCoRem olf rs                   \u2192 SetLFCoRem olf (ls \u2227 rs)\n  _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLFCoRem olf ls \u2192 SetLFCoRem olf rs \u2192 SetLFCoRem olf (ls \u2227 rs)\n  _\u2190\u2228   : \u2200{rs ls} \u2192 SetLFCoRem olf ls                   \u2192 SetLFCoRem olf (ls \u2228 rs)\n  \u2228\u2192_   : \u2200{rs ls} \u2192 SetLFCoRem olf rs                   \u2192 SetLFCoRem olf (ls \u2228 rs)\n  _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLFCoRem olf ls \u2192 SetLFCoRem olf rs \u2192 SetLFCoRem olf (ls \u2228 rs)\n  _\u2190\u2202   : \u2200{rs ls} \u2192 SetLFCoRem olf ls                   \u2192 SetLFCoRem olf (ls \u2202 rs)\n  \u2202\u2192_   : \u2200{rs ls} \u2192 SetLFCoRem olf rs                   \u2192 SetLFCoRem olf (ls \u2202 rs)\n  _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLFCoRem olf ls \u2192 SetLFCoRem olf rs \u2192 SetLFCoRem olf (ls \u2202 rs)\n\ndata MSetLFCoRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2205   : \u2200{ll}            \u2192 MSetLFCoRem olf ll\n  \u00ac\u2205  : \u2200{ll} \u2192 SetLFCoRem olf ll \u2192 MSetLFCoRem olf ll\n\n\u2205-addLFCoRem : \u2200{i u ll pll oll orll frll cll} \u2192 {iind : IndexLL cll oll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLFCo frll iind olf\n        \u2192 SetLFCoRem olf ll\n\u2205-addLFCoRem \u2193 m = \u2193 m\n\u2205-addLFCoRem (ind \u2190\u2227) m = (\u2205-addLFCoRem ind m) \u2190\u2227\n\u2205-addLFCoRem (\u2227\u2192 ind) m = \u2227\u2192 (\u2205-addLFCoRem ind m)\n\u2205-addLFCoRem (ind \u2190\u2228) m = (\u2205-addLFCoRem ind m) \u2190\u2228\n\u2205-addLFCoRem (\u2228\u2192 ind) m = \u2228\u2192 (\u2205-addLFCoRem ind m)\n\u2205-addLFCoRem (ind \u2190\u2202) m = (\u2205-addLFCoRem ind m) \u2190\u2202\n\u2205-addLFCoRem (\u2202\u2192 ind) m = \u2202\u2192 (\u2205-addLFCoRem ind m)\n\naddLFCoRem : \u2200{i u ll pll oll orll frll cll} \u2192 {iind : IndexLL cll oll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCoRem olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLFCo frll iind olf\n        \u2192 SetLFCoRem olf ll\naddLFCoRem (\u2193 rm) ind m          = \u2193 m\naddLFCoRem (x \u2190\u2227) \u2193 m            = \u2193 m\naddLFCoRem (\u2227\u2192 x) \u2193 m            = \u2193 m --TODO Here we lose the information that is at lower levels.\naddLFCoRem (x \u2190\u2227\u2192 x\u2081) \u2193 m        = \u2193 m\naddLFCoRem (x \u2190\u2228) \u2193 m            = \u2193 m\naddLFCoRem (\u2228\u2192 x) \u2193 m            = \u2193 m\naddLFCoRem (x \u2190\u2228\u2192 x\u2081) \u2193 m        = \u2193 m\naddLFCoRem (x \u2190\u2202) \u2193 m            = \u2193 m\naddLFCoRem (\u2202\u2192 x) \u2193 m            = \u2193 m\naddLFCoRem (x \u2190\u2202\u2192 x\u2081) \u2193 m        = \u2193 m\naddLFCoRem (s \u2190\u2227) (ind \u2190\u2227) m     = (addLFCoRem s ind m) \u2190\u2227\naddLFCoRem (s \u2190\u2227) (\u2227\u2192 ind) m     = s \u2190\u2227\u2192 (\u2205-addLFCoRem ind m)\naddLFCoRem (\u2227\u2192 s) (ind \u2190\u2227) m     = (\u2205-addLFCoRem ind m) \u2190\u2227\u2192 s\naddLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) m     = \u2227\u2192 addLFCoRem s ind m\naddLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) m = (addLFCoRem s ind m) \u2190\u2227\u2192 s\u2081\naddLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) m = s \u2190\u2227\u2192 (addLFCoRem s\u2081 ind m)\naddLFCoRem (s \u2190\u2228) (ind \u2190\u2228) m     = (addLFCoRem s ind m) \u2190\u2228\naddLFCoRem (s \u2190\u2228) (\u2228\u2192 ind) m     = s \u2190\u2228\u2192 (\u2205-addLFCoRem ind m)\naddLFCoRem (\u2228\u2192 s) (ind \u2190\u2228) m     = (\u2205-addLFCoRem ind m) \u2190\u2228\u2192 s\naddLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) m     = \u2228\u2192 addLFCoRem s ind m\naddLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) m = (addLFCoRem s ind m) \u2190\u2228\u2192 s\u2081\naddLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) m = s \u2190\u2228\u2192 (addLFCoRem s\u2081 ind m)\naddLFCoRem (s \u2190\u2202) (ind \u2190\u2202) m     = (addLFCoRem s ind m) \u2190\u2202\naddLFCoRem (s \u2190\u2202) (\u2202\u2192 ind) m     = s \u2190\u2202\u2192 (\u2205-addLFCoRem ind m)\naddLFCoRem (\u2202\u2192 s) (ind \u2190\u2202) m     = (\u2205-addLFCoRem ind m) \u2190\u2202\u2192 s\naddLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) m     = \u2202\u2192 addLFCoRem s ind m\naddLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) m = (addLFCoRem s ind m) \u2190\u2202\u2192 s\u2081\naddLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) m = s \u2190\u2202\u2192 (addLFCoRem s\u2081 ind m)\n\nmaddLFCoRem : \u2200{i u ll pll oll orll frll cll} \u2192 {iind : IndexLL cll oll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCoRem olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLFCo frll iind olf\n      \u2192 MSetLFCoRem olf ll\nmaddLFCoRem \u2205 ind m = \u00ac\u2205 (\u2205-addLFCoRem ind m)\nmaddLFCoRem (\u00ac\u2205 x) ind m = \u00ac\u2205 (addLFCoRem x ind m)\n\n\ntruncSetLFCoRem : \u2200{i} \u2192 \u2200{u ll oll orll q} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCoRem {i} {u} olf ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 MSetLFCoRem {i} olf q\ntruncSetLFCoRem \u2205 ind = \u2205\ntruncSetLFCoRem (\u00ac\u2205 x) \u2193 = \u00ac\u2205 x\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (ind \u2190\u2227) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227)) (ind \u2190\u2227) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2227\u2192 x)) (ind \u2190\u2227) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (ind \u2190\u2227) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (\u2227\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227)) (\u2227\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (\u2227\u2192 x)) (\u2227\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (\u2227\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (ind \u2190\u2228) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228)) (ind \u2190\u2228) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2228\u2192 x)) (ind \u2190\u2228) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (ind \u2190\u2228) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (\u2228\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228)) (\u2228\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (\u2228\u2192 x)) (\u2228\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (\u2228\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (ind \u2190\u2202) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202)) (ind \u2190\u2202) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2202\u2192 x)) (ind \u2190\u2202) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (ind \u2190\u2202) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (\u2202\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202)) (\u2202\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (\u2202\u2192 x)) (\u2202\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (\u2202\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x\u2081) ind\n\ndelLFCoRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCoRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n      \u2192 MSetLFCoRem {i} olf (replLL ll ind rll)\ndelLFCoRem s \u2193 rll = \u2205\ndelLFCoRem (\u2193 x) (ind \u2190\u2227) rll = \u2205 -- We loose Information.\ndelLFCoRem (s \u2190\u2227) (ind \u2190\u2227) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u2205 = \u2205\ndelLFCoRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227)\ndelLFCoRem (\u2227\u2192 s) (ind \u2190\u2227) rll = \u00ac\u2205 (\u2227\u2192 (s))\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u2205 = \u00ac\u2205 (\u2227\u2192 (s\u2081))\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227\u2192 (s\u2081))\ndelLFCoRem (\u2193 x) (\u2227\u2192 ind) rll = \u2205 --\ndelLFCoRem (s \u2190\u2227) (\u2227\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) rll with (delLFCoRem s ind rll)\ndelLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u2205 = \u2205\ndelLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2227\u2192 x)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll with (delLFCoRem s\u2081 ind rll)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2227\u2192 x)\ndelLFCoRem (\u2193 x) (ind \u2190\u2228) rll = \u2205\ndelLFCoRem (s \u2190\u2228) (ind \u2190\u2228) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u2205 = \u2205\ndelLFCoRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228)\ndelLFCoRem (\u2228\u2192 s) (ind \u2190\u2228) rll = \u00ac\u2205 (\u2228\u2192 (s))\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u2205 = \u00ac\u2205 (\u2228\u2192 (s\u2081))\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228\u2192 (s\u2081))\ndelLFCoRem (\u2193 x) (\u2228\u2192 ind) rll = \u2205\ndelLFCoRem (s \u2190\u2228) (\u2228\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) rll with (delLFCoRem s ind rll)\ndelLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u2205 = \u2205\ndelLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2228\u2192 x)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll with (delLFCoRem s\u2081 ind rll)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2228\u2192 x)\ndelLFCoRem (\u2193 x) (ind \u2190\u2202) rll = \u2205\ndelLFCoRem (s \u2190\u2202) (ind \u2190\u2202) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u2205 = \u2205\ndelLFCoRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202)\ndelLFCoRem (\u2202\u2192 s) (ind \u2190\u2202) rll = \u00ac\u2205 (\u2202\u2192 (s))\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u2205 = \u00ac\u2205 (\u2202\u2192 (s\u2081))\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202\u2192 (s\u2081))\ndelLFCoRem (\u2193 x) (\u2202\u2192 ind) rll = \u2205\ndelLFCoRem (s \u2190\u2202) (\u2202\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) rll with (delLFCoRem s ind rll)\ndelLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u2205 = \u2205\ndelLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2202\u2192 x)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll with (delLFCoRem s\u2081 ind rll)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2202\u2192 x)\n\nmdelLFCoRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCoRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n             \u2192 MSetLFCoRem {i} olf (replLL ll ind rll)\nmdelLFCoRem \u2205 ind rll = \u2205\nmdelLFCoRem (\u00ac\u2205 x) ind rll = delLFCoRem x ind rll\n\ntranLFCoRem : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCoRem {i} olf ll \u2192 (tr : LLTr {i} {u} rll ll)\n       \u2192 SetLFCoRem olf rll\ntranLFCoRem s I                           = s\ntranLFCoRem (s \u2190\u2202) (\u2202c ltr)                = tranLFCoRem (\u2202\u2192 s) ltr\ntranLFCoRem (\u2193 x) (\u2202c ltr)                = \u2193 x\ntranLFCoRem (\u2202\u2192 s) (\u2202c ltr)                = tranLFCoRem (s \u2190\u2202) ltr\ntranLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202c ltr)            = tranLFCoRem (s\u2081 \u2190\u2202\u2192 s) ltr\ntranLFCoRem (s \u2190\u2228) (\u2228c ltr)                = tranLFCoRem (\u2228\u2192 s) ltr\ntranLFCoRem (\u2193 x) (\u2228c ltr)                = \u2193 x\ntranLFCoRem (\u2228\u2192 s) (\u2228c ltr)                = tranLFCoRem (s \u2190\u2228) ltr\ntranLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228c ltr)            = tranLFCoRem (s\u2081 \u2190\u2228\u2192 s) ltr\ntranLFCoRem (s \u2190\u2227) (\u2227c ltr)                = tranLFCoRem (\u2227\u2192 s) ltr\ntranLFCoRem (\u2193 x) (\u2227c ltr)                = \u2193 x\ntranLFCoRem (\u2227\u2192 s) (\u2227c ltr)                = tranLFCoRem (s \u2190\u2227) ltr\ntranLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227c ltr)            = tranLFCoRem (s\u2081 \u2190\u2227\u2192 s) ltr\ntranLFCoRem ((s \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCoRem (s \u2190\u2227) ltr\ntranLFCoRem (\u2193 x) (\u2227\u2227d ltr)          = \u2193 x\ntranLFCoRem ((\u2193 x) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCoRem ((\u2193 x) \u2190\u2227\u2192 ((\u2193 x) \u2190\u2227)) ltr\ntranLFCoRem ((\u2227\u2192 s) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCoRem (\u2227\u2192 (s \u2190\u2227)) ltr\ntranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227) (\u2227\u2227d ltr)      = tranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227)) ltr\ntranLFCoRem (\u2227\u2192 s) (\u2227\u2227d ltr)               = tranLFCoRem (\u2227\u2192 (\u2227\u2192 s)) ltr\ntranLFCoRem ((\u2193 x) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCoRem ((\u2193 x) \u2190\u2227\u2192 ((\u2193 x) \u2190\u2227\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2227) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCoRem (s \u2190\u2227\u2192 (\u2227\u2192 s\u2081)) ltr\ntranLFCoRem ((\u2227\u2192 s) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCoRem (\u2227\u2192 (s \u2190\u2227\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227\u2192 s\u2082) (\u2227\u2227d ltr)  = tranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227\u2192 s\u2082)) ltr\ntranLFCoRem (s \u2190\u2227) (\u00ac\u2227\u2227d ltr)              = tranLFCoRem ((s \u2190\u2227) \u2190\u2227) ltr\ntranLFCoRem (\u2193 x) (\u00ac\u2227\u2227d ltr)              = \u2193 x\ntranLFCoRem (\u2227\u2192 (\u2193 x)) (\u00ac\u2227\u2227d ltr)         = tranLFCoRem ((\u2227\u2192 (\u2193 x)) \u2190\u2227\u2192 (\u2193 x)) ltr\ntranLFCoRem (\u2227\u2192 (s \u2190\u2227)) (\u00ac\u2227\u2227d ltr)         = tranLFCoRem ((\u2227\u2192 s) \u2190\u2227) ltr\ntranLFCoRem (\u2227\u2192 (\u2227\u2192 s)) (\u00ac\u2227\u2227d ltr)         = tranLFCoRem (\u2227\u2192 s) ltr\ntranLFCoRem (\u2227\u2192 (s \u2190\u2227\u2192 s\u2081)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((\u2227\u2192 s) \u2190\u2227\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (\u2193 x)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((s \u2190\u2227\u2192 (\u2193 x)) \u2190\u2227\u2192 (\u2193 x)) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (\u2227\u2192 s\u2081)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((s \u2190\u2227) \u2190\u2227\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227\u2192 s\u2082)) (\u00ac\u2227\u2227d ltr) = tranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227\u2192 s\u2082) ltr\ntranLFCoRem ((s \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCoRem (s \u2190\u2228) ltr\ntranLFCoRem (\u2193 x) (\u2228\u2228d ltr)          = \u2193 x\ntranLFCoRem ((\u2193 x) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCoRem ((\u2193 x) \u2190\u2228\u2192 ((\u2193 x) \u2190\u2228)) ltr\ntranLFCoRem ((\u2228\u2192 s) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCoRem (\u2228\u2192 (s \u2190\u2228)) ltr\ntranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228) (\u2228\u2228d ltr)      = tranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228)) ltr\ntranLFCoRem (\u2228\u2192 s) (\u2228\u2228d ltr)               = tranLFCoRem (\u2228\u2192 (\u2228\u2192 s)) ltr\ntranLFCoRem ((\u2193 x) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCoRem ((\u2193 x) \u2190\u2228\u2192 ((\u2193 x) \u2190\u2228\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2228) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCoRem (s \u2190\u2228\u2192 (\u2228\u2192 s\u2081)) ltr\ntranLFCoRem ((\u2228\u2192 s) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCoRem (\u2228\u2192 (s \u2190\u2228\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228\u2192 s\u2082) (\u2228\u2228d ltr)  = tranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228\u2192 s\u2082)) ltr\ntranLFCoRem (s \u2190\u2228) (\u00ac\u2228\u2228d ltr)              = tranLFCoRem ((s \u2190\u2228) \u2190\u2228) ltr\ntranLFCoRem (\u2193 x) (\u00ac\u2228\u2228d ltr)              = \u2193 x\ntranLFCoRem (\u2228\u2192 (\u2193 x)) (\u00ac\u2228\u2228d ltr)         = tranLFCoRem ((\u2228\u2192 (\u2193 x)) \u2190\u2228\u2192 (\u2193 x)) ltr\ntranLFCoRem (\u2228\u2192 (s \u2190\u2228)) (\u00ac\u2228\u2228d ltr)         = tranLFCoRem ((\u2228\u2192 s) \u2190\u2228) ltr\ntranLFCoRem (\u2228\u2192 (\u2228\u2192 s)) (\u00ac\u2228\u2228d ltr)         = tranLFCoRem (\u2228\u2192 s) ltr\ntranLFCoRem (\u2228\u2192 (s \u2190\u2228\u2192 s\u2081)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((\u2228\u2192 s) \u2190\u2228\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (\u2193 x)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((s \u2190\u2228\u2192 (\u2193 x)) \u2190\u2228\u2192 (\u2193 x)) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (\u2228\u2192 s\u2081)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((s \u2190\u2228) \u2190\u2228\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228\u2192 s\u2082)) (\u00ac\u2228\u2228d ltr) = tranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228\u2192 s\u2082) ltr\ntranLFCoRem ((s \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCoRem (s \u2190\u2202) ltr\ntranLFCoRem (\u2193 x) (\u2202\u2202d ltr)          = \u2193 x\ntranLFCoRem ((\u2193 x) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCoRem ((\u2193 x) \u2190\u2202\u2192 ((\u2193 x) \u2190\u2202)) ltr\ntranLFCoRem ((\u2202\u2192 s) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCoRem (\u2202\u2192 (s \u2190\u2202)) ltr\ntranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202) (\u2202\u2202d ltr)      = tranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202)) ltr\ntranLFCoRem (\u2202\u2192 s) (\u2202\u2202d ltr)               = tranLFCoRem (\u2202\u2192 (\u2202\u2192 s)) ltr\ntranLFCoRem ((\u2193 x) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCoRem ((\u2193 x) \u2190\u2202\u2192 ((\u2193 x) \u2190\u2202\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2202) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCoRem (s \u2190\u2202\u2192 (\u2202\u2192 s\u2081)) ltr\ntranLFCoRem ((\u2202\u2192 s) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCoRem (\u2202\u2192 (s \u2190\u2202\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202\u2192 s\u2082) (\u2202\u2202d ltr)  = tranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202\u2192 s\u2082)) ltr\ntranLFCoRem (s \u2190\u2202) (\u00ac\u2202\u2202d ltr)              = tranLFCoRem ((s \u2190\u2202) \u2190\u2202) ltr\ntranLFCoRem (\u2202\u2192 (s \u2190\u2202)) (\u00ac\u2202\u2202d ltr)         = tranLFCoRem ((\u2202\u2192 s) \u2190\u2202) ltr\ntranLFCoRem (\u2193 x) (\u00ac\u2202\u2202d ltr)         = \u2193 x\ntranLFCoRem (\u2202\u2192 (\u2193 x)) (\u00ac\u2202\u2202d ltr)         = tranLFCoRem ((\u2202\u2192 (\u2193 x)) \u2190\u2202\u2192 (\u2193 x)) ltr\ntranLFCoRem (\u2202\u2192 (\u2202\u2192 s)) (\u00ac\u2202\u2202d ltr)         = tranLFCoRem (\u2202\u2192 s) ltr\ntranLFCoRem (\u2202\u2192 (s \u2190\u2202\u2192 s\u2081)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((\u2202\u2192 s) \u2190\u2202\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (\u2193 x)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((s \u2190\u2202\u2192 (\u2193 x)) \u2190\u2202\u2192 (\u2193 x)) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (\u2202\u2192 s\u2081)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((s \u2190\u2202) \u2190\u2202\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202\u2192 s\u2082)) (\u00ac\u2202\u2202d ltr) = tranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202\u2192 s\u2082) ltr\n\n\nextendLFCoRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 IndexLL {i} {u} pll ll \u2192 SetLFCoRem {i} olf pll \u2192 SetLFCoRem olf ll\nextendLFCoRem \u2193 sr = sr\nextendLFCoRem (ind \u2190\u2227) sr = (extendLFCoRem ind sr) \u2190\u2227\nextendLFCoRem (\u2227\u2192 ind) sr = \u2227\u2192 (extendLFCoRem ind sr)\nextendLFCoRem (ind \u2190\u2228) sr = (extendLFCoRem ind sr) \u2190\u2228\nextendLFCoRem (\u2228\u2192 ind) sr = \u2228\u2192 (extendLFCoRem ind sr)\nextendLFCoRem (ind \u2190\u2202) sr = (extendLFCoRem ind sr) \u2190\u2202\nextendLFCoRem (\u2202\u2192 ind) sr = \u2202\u2192 (extendLFCoRem ind sr)\n\nreplaceLFCoRem : \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 SetLFCoRem {i} olf rll \u2192 SetLFCoRem olf ll \u2192 SetLFCoRem olf (replLL ll ind rll)\nreplaceLFCoRem \u2193 esr sr = esr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (\u2193 x) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2227\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (sr \u2190\u2227) = replaceLFCoRem ind esr sr \u2190\u2227\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (\u2227\u2192 sr) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2227\u2192 sr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (sr \u2190\u2227\u2192 sr\u2081) = (replaceLFCoRem ind esr sr) \u2190\u2227\u2192 sr\u2081\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (\u2193 x) = \u2227\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (sr \u2190\u2227) = sr \u2190\u2227\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (\u2227\u2192 sr) = \u2227\u2192 replaceLFCoRem ind esr sr\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (sr \u2190\u2227\u2192 sr\u2081) = sr \u2190\u2227\u2192 replaceLFCoRem ind esr sr\u2081\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (\u2193 x) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2228\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (sr \u2190\u2228) = replaceLFCoRem ind esr sr \u2190\u2228\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (\u2228\u2192 sr) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2228\u2192 sr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (sr \u2190\u2228\u2192 sr\u2081) = (replaceLFCoRem ind esr sr) \u2190\u2228\u2192 sr\u2081\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (\u2193 x) = \u2228\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (sr \u2190\u2228) = sr \u2190\u2228\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (\u2228\u2192 sr) = \u2228\u2192 replaceLFCoRem ind esr sr\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (sr \u2190\u2228\u2192 sr\u2081) = sr \u2190\u2228\u2192 replaceLFCoRem ind esr sr\u2081\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (\u2193 x) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2202\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (sr \u2190\u2202) = replaceLFCoRem ind esr sr \u2190\u2202\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (\u2202\u2192 sr) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2202\u2192 sr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (sr \u2190\u2202\u2192 sr\u2081) = (replaceLFCoRem ind esr sr) \u2190\u2202\u2192 sr\u2081\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (\u2193 x) = \u2202\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (sr \u2190\u2202) = sr \u2190\u2202\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (\u2202\u2192 sr) = \u2202\u2192 replaceLFCoRem ind esr sr\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (sr \u2190\u2202\u2192 sr\u2081) = sr \u2190\u2202\u2192 replaceLFCoRem ind esr sr\u2081\n\n\nmreplaceLFCoRem :  \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 MSetLFCoRem {i} olf rll \u2192 MSetLFCoRem olf ll \u2192 MSetLFCoRem olf (replLL ll ind rll)\nmreplaceLFCoRem ind \u2205 \u2205 = \u2205\nmreplaceLFCoRem {rll = rll} ind \u2205 (\u00ac\u2205 x) = delLFCoRem x ind rll\nmreplaceLFCoRem {rll = rll} ind (\u00ac\u2205 x) \u2205 = \u00ac\u2205 (extendLFCoRem (updInd rll ind) x)\nmreplaceLFCoRem ind (\u00ac\u2205 x) (\u00ac\u2205 x\u2081) = \u00ac\u2205 (replaceLFCoRem ind x x\u2081)\n","old_contents":"{-# OPTIONS --exact-split #-}\nmodule IndexLFCo where\n\nopen import Common\nopen import LinLogic\nopen import LinLogicProp\nopen import LinFun\nopen import IndexLLProp\nopen import Data.Maybe\n\nmodule _ where\n\n  import Relation.Binary.PropositionalEquality\n  open  Relation.Binary.PropositionalEquality.Deprecated-inspect\n\n\n  -- reverseTran returns nothing if during the reversion, it finds a com.\n  -- or if the cll is transformed.\n  \n  reverseTran : \u2200{i u ll cll rll} \u2192 LFun ll rll \u2192 IndexLL {i} {u} cll rll \u2192 Maybe $ IndexLL cll ll\n  reverseTran I i = just i\n  reverseTran (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) i with (reverseTran lf\u2081 i)\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i | just x with (inspect (x -\u2098\u1d62 (updInd ell ind)))\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i | just x | ((just x\u2081) with-\u2261 eq\u2081) with (reverseTran lf x\u2081)\n  reverseTran (_\u2282_ {pll} {ll} {ell} {_} {ind} lf lf\u2081) i | just x | ((just x\u2081) with-\u2261 eq\u2081) | (just x\u2082) = just $ ind +\u1d62 x\u2082\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i | just x | ((just x\u2081) with-\u2261 eq\u2081) | nothing = nothing\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i | just x | (nothing with-\u2261 eq\u2081) with (inspect ((updInd ell ind) -\u2098\u1d62 x))\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i\u2081 | just x | (nothing with-\u2261 eq\u2081) | ((just x\u2081) with-\u2261 eq\u2082) = nothing -- We do not accept transformations that change the cll. The cll definitely changes here. (unless lf only has I).\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i\u2081 | just x | (nothing with-\u2261 eq\u2081) | (nothing with-\u2261 eq\u2082) = just $ revUpdInd ind x eq\u2081 eq\u2082\n  reverseTran (_\u2282_ {_} {_} {ell} {_} {ind} lf lf\u2081) i | nothing = nothing\n  reverseTran (tr ltr lf) i with (reverseTran lf i)\n  reverseTran (tr ltr lf) i\u2081 | just x = tran x $ revTr ltr\n  reverseTran (tr ltr lf) i\u2081 | nothing = nothing\n  reverseTran (com df lf) i = nothing\n  reverseTran (call x) i = nothing \n\n  data ReverseTranT {i u} : \u2200{ll cll rll} \u2192 LFun ll rll \u2192 IndexLL {i} {u} cll rll \u2192 Set u where\n    cr1 : \u2200{cll rll} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 ReverseTranT I iind\n    cr2 : \u2200{ll cll rll ell pll ind lf\u2081 lf x x\u2081 x\u2082} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 (just x \u2261 reverseTran lf\u2081 iind) \u2192 (just x\u2081 \u2261 x -\u2098\u1d62 (updInd ell ind)) \u2192 (just x\u2082 \u2261 reverseTran lf x\u2081) \u2192 ReverseTranT (_\u2282_ {pll = pll} {ll = ll} {ell = ell} {rll = rll} {ind = ind} lf lf\u2081) iind\n    cr3 : \u2200{ll cll rll ell pll ind lf\u2081 lf x } \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 (just x \u2261 reverseTran lf\u2081 iind) \u2192 (eq\u2081 : (nothing \u2261 x -\u2098\u1d62 (updInd ell ind))) \u2192 (eq\u2082 :((updInd ell ind) -\u2098\u1d62 x \u2261 nothing)) \u2192 ReverseTranT (_\u2282_ {pll = pll} {ll = ll} {ell = ell} {rll = rll} {ind = ind} lf lf\u2081) iind\n    cr4 : \u2200{ll cll orll rll lf x} \u2192 {ltr : LLTr orll ll} \u2192 {iind : IndexLL {i} {u} cll rll} \u2192 (just x \u2261 reverseTran lf iind) \u2192 ReverseTranT (tr ltr lf) iind\n\n  -- This is almost the same code as above but it is required in IndexLFCo.\n  indRevNoComs : \u2200{i u ll pll ell cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLL cll (replLL ll ind ell) \u2192 LFun pll ell \u2192 Maybe $ IndexLL cll ll\n  indRevNoComs {ell = ell} ind lind lf with (inspect (lind -\u2098\u1d62 (updInd ell ind)))\n  indRevNoComs {_} {_} {_} {_} {ell} ind lind lf | just x with-\u2261 eq with (reverseTran lf x)\n  indRevNoComs {_} {_} {_} {_} {ell} ind lind lf | just x with-\u2261 eq | (just x\u2081) = just $ ind +\u1d62 x\u2081\n  indRevNoComs {_} {_} {_} {_} {ell} ind lind lf | just x with-\u2261 eq | nothing = nothing\n  indRevNoComs {_} {_} {_} {_} {ell} ind lind lf | nothing with-\u2261 eq with (inspect ((updInd ell ind) -\u2098\u1d62 lind))\n  indRevNoComs {_} {_} {_} {_} {ell} ind lind lf | nothing with-\u2261 eq | (just x with-\u2261 eq\u2081) = nothing\n  indRevNoComs {_} {_} {_} {_} {ell} ind lind lf | nothing with-\u2261 eq | (nothing with-\u2261 eq\u2081) = just $ revUpdInd ind lind eq eq\u2081\n   \n  data IndRevNoComsT {i u ll pll ell cll} {ind : IndexLL {i} {u} pll ll} {lind : IndexLL cll (replLL ll ind ell)} {lf : LFun pll ell} : Set u where\n    c1 : \u2200{x x\u2081} \u2192 (just x \u2261 lind -\u2098\u1d62 (updInd ell ind)) \u2192 (x\u2081 \u2261 reverseTran lf x) \u2192 IndRevNoComsT\n    c2 : (nothing \u2261 lind -\u2098\u1d62 (updInd ell ind)) \u2192 (nothing \u2261 (updInd ell ind) -\u2098\u1d62 lind) \u2192 IndRevNoComsT\n\ndata IndexLFCo {i u cll} (frll : LinLogic i {u}) : \u2200{ll rll} \u2192 IndexLL cll ll \u2192 LFun {i} {u} ll rll \u2192 Set (u) where\n  _\u2190\u2282 : \u2200{rll pll ell ll ind elf lf lind}\n         \u2192 IndexLFCo frll lind elf\n         \u2192 IndexLFCo frll (ind +\u1d62 lind) (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  \u2282\u2192_ : \u2200{rll pll ell ll ind elf lf lind rs}\n         \u2192 IndexLFCo frll lind lf\n         \u2192 {prf : just rs \u2261 indRevNoComs ind lind elf}\n         \u2192 IndexLFCo frll rs (_\u2282_ {i} {u} {pll} {ll} {ell} {rll} {ind} elf lf)\n  tr   : \u2200{ll orll rll lind rs} \u2192 {ltr : LLTr orll ll} \u2192 {lf : LFun {i} {u} orll rll}\n         \u2192 IndexLFCo frll lind lf\n         \u2192 {prf : just rs \u2261 tran lind (revTr ltr) }\n         \u2192 IndexLFCo frll rs (tr ltr lf) \n  \u2193  : \u2200{rll prfi prfo df lf}\n         \u2192 IndexLFCo  frll \u2193 (com {i} {u} {rll} {cll} {frll} {{prfi}} {{prfo}} df lf)\n\n\n\ndata SetLFCoRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2193    : \u2200{ll cll frll} \u2192 {ind : IndexLL {i} {u} cll oll} \u2192 IndexLFCo frll ind olf \u2192 SetLFCoRem olf ll\n  _\u2190\u2227   : \u2200{rs ls} \u2192 SetLFCoRem olf ls                   \u2192 SetLFCoRem olf (ls \u2227 rs)\n  \u2227\u2192_   : \u2200{rs ls} \u2192 SetLFCoRem olf rs                   \u2192 SetLFCoRem olf (ls \u2227 rs)\n  _\u2190\u2227\u2192_ : \u2200{rs ls} \u2192 SetLFCoRem olf ls \u2192 SetLFCoRem olf rs \u2192 SetLFCoRem olf (ls \u2227 rs)\n  _\u2190\u2228   : \u2200{rs ls} \u2192 SetLFCoRem olf ls                   \u2192 SetLFCoRem olf (ls \u2228 rs)\n  \u2228\u2192_   : \u2200{rs ls} \u2192 SetLFCoRem olf rs                   \u2192 SetLFCoRem olf (ls \u2228 rs)\n  _\u2190\u2228\u2192_ : \u2200{rs ls} \u2192 SetLFCoRem olf ls \u2192 SetLFCoRem olf rs \u2192 SetLFCoRem olf (ls \u2228 rs)\n  _\u2190\u2202   : \u2200{rs ls} \u2192 SetLFCoRem olf ls                   \u2192 SetLFCoRem olf (ls \u2202 rs)\n  \u2202\u2192_   : \u2200{rs ls} \u2192 SetLFCoRem olf rs                   \u2192 SetLFCoRem olf (ls \u2202 rs)\n  _\u2190\u2202\u2192_ : \u2200{rs ls} \u2192 SetLFCoRem olf ls \u2192 SetLFCoRem olf rs \u2192 SetLFCoRem olf (ls \u2202 rs)\n\ndata MSetLFCoRem {i u oll orll} (olf : LFun {i} {u} oll orll) : LinLogic i {u} \u2192 Set (lsuc u) where\n  \u2205   : \u2200{ll}            \u2192 MSetLFCoRem olf ll\n  \u00ac\u2205  : \u2200{ll} \u2192 SetLFCoRem olf ll \u2192 MSetLFCoRem olf ll\n\n\u2205-addLFCoRem : \u2200{i u ll pll oll orll frll cll} \u2192 {iind : IndexLL cll oll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLFCo frll iind olf\n        \u2192 SetLFCoRem olf ll\n\u2205-addLFCoRem \u2193 m = \u2193 m\n\u2205-addLFCoRem (ind \u2190\u2227) m = (\u2205-addLFCoRem ind m) \u2190\u2227\n\u2205-addLFCoRem (\u2227\u2192 ind) m = \u2227\u2192 (\u2205-addLFCoRem ind m)\n\u2205-addLFCoRem (ind \u2190\u2228) m = (\u2205-addLFCoRem ind m) \u2190\u2228\n\u2205-addLFCoRem (\u2228\u2192 ind) m = \u2228\u2192 (\u2205-addLFCoRem ind m)\n\u2205-addLFCoRem (ind \u2190\u2202) m = (\u2205-addLFCoRem ind m) \u2190\u2202\n\u2205-addLFCoRem (\u2202\u2192 ind) m = \u2202\u2192 (\u2205-addLFCoRem ind m)\n\naddLFCoRem : \u2200{i u ll pll oll orll frll cll} \u2192 {iind : IndexLL cll oll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCoRem olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLFCo frll iind olf\n        \u2192 SetLFCoRem olf ll\naddLFCoRem (\u2193 rm) ind m          = \u2193 m\naddLFCoRem (x \u2190\u2227) \u2193 m            = \u2193 m\naddLFCoRem (\u2227\u2192 x) \u2193 m            = \u2193 m --TODO Here we lose the information that is at lower levels.\naddLFCoRem (x \u2190\u2227\u2192 x\u2081) \u2193 m        = \u2193 m\naddLFCoRem (x \u2190\u2228) \u2193 m            = \u2193 m\naddLFCoRem (\u2228\u2192 x) \u2193 m            = \u2193 m\naddLFCoRem (x \u2190\u2228\u2192 x\u2081) \u2193 m        = \u2193 m\naddLFCoRem (x \u2190\u2202) \u2193 m            = \u2193 m\naddLFCoRem (\u2202\u2192 x) \u2193 m            = \u2193 m\naddLFCoRem (x \u2190\u2202\u2192 x\u2081) \u2193 m        = \u2193 m\naddLFCoRem (s \u2190\u2227) (ind \u2190\u2227) m     = (addLFCoRem s ind m) \u2190\u2227\naddLFCoRem (s \u2190\u2227) (\u2227\u2192 ind) m     = s \u2190\u2227\u2192 (\u2205-addLFCoRem ind m)\naddLFCoRem (\u2227\u2192 s) (ind \u2190\u2227) m     = (\u2205-addLFCoRem ind m) \u2190\u2227\u2192 s\naddLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) m     = \u2227\u2192 addLFCoRem s ind m\naddLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) m = (addLFCoRem s ind m) \u2190\u2227\u2192 s\u2081\naddLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) m = s \u2190\u2227\u2192 (addLFCoRem s\u2081 ind m)\naddLFCoRem (s \u2190\u2228) (ind \u2190\u2228) m     = (addLFCoRem s ind m) \u2190\u2228\naddLFCoRem (s \u2190\u2228) (\u2228\u2192 ind) m     = s \u2190\u2228\u2192 (\u2205-addLFCoRem ind m)\naddLFCoRem (\u2228\u2192 s) (ind \u2190\u2228) m     = (\u2205-addLFCoRem ind m) \u2190\u2228\u2192 s\naddLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) m     = \u2228\u2192 addLFCoRem s ind m\naddLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) m = (addLFCoRem s ind m) \u2190\u2228\u2192 s\u2081\naddLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) m = s \u2190\u2228\u2192 (addLFCoRem s\u2081 ind m)\naddLFCoRem (s \u2190\u2202) (ind \u2190\u2202) m     = (addLFCoRem s ind m) \u2190\u2202\naddLFCoRem (s \u2190\u2202) (\u2202\u2192 ind) m     = s \u2190\u2202\u2192 (\u2205-addLFCoRem ind m)\naddLFCoRem (\u2202\u2192 s) (ind \u2190\u2202) m     = (\u2205-addLFCoRem ind m) \u2190\u2202\u2192 s\naddLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) m     = \u2202\u2192 addLFCoRem s ind m\naddLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) m = (addLFCoRem s ind m) \u2190\u2202\u2192 s\u2081\naddLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) m = s \u2190\u2202\u2192 (addLFCoRem s\u2081 ind m)\n\nmaddLFCoRem : \u2200{i u ll pll oll orll frll cll} \u2192 {iind : IndexLL cll oll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCoRem olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 IndexLFCo frll iind olf\n      \u2192 MSetLFCoRem olf ll\nmaddLFCoRem \u2205 ind m = \u00ac\u2205 (\u2205-addLFCoRem ind m)\nmaddLFCoRem (\u00ac\u2205 x) ind m = \u00ac\u2205 (addLFCoRem x ind m)\n\n\ntruncSetLFCoRem : \u2200{i} \u2192 \u2200{u ll oll orll q} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCoRem {i} {u} olf ll \u2192 (ind : IndexLL {i} {u} q ll) \u2192 MSetLFCoRem {i} olf q\ntruncSetLFCoRem \u2205 ind = \u2205\ntruncSetLFCoRem (\u00ac\u2205 x) \u2193 = \u00ac\u2205 x\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (ind \u2190\u2227) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227)) (ind \u2190\u2227) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2227\u2192 x)) (ind \u2190\u2227) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (ind \u2190\u2227) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (\u2227\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227)) (\u2227\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (\u2227\u2192 x)) (\u2227\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2227\u2192 x\u2081)) (\u2227\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (ind \u2190\u2228) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228)) (ind \u2190\u2228) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2228\u2192 x)) (ind \u2190\u2228) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (ind \u2190\u2228) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (\u2228\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228)) (\u2228\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (\u2228\u2192 x)) (\u2228\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2228\u2192 x\u2081)) (\u2228\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x\u2081) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (ind \u2190\u2202) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202)) (ind \u2190\u2202) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2202\u2192 x)) (ind \u2190\u2202) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (ind \u2190\u2202) = truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (\u2193 x)) (\u2202\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202)) (\u2202\u2192 ind) = \u2205\ntruncSetLFCoRem (\u00ac\u2205 (\u2202\u2192 x)) (\u2202\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x) ind\ntruncSetLFCoRem (\u00ac\u2205 (x \u2190\u2202\u2192 x\u2081)) (\u2202\u2192 ind) =  truncSetLFCoRem (\u00ac\u2205 x\u2081) ind\n\ndelLFCoRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCoRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n      \u2192 MSetLFCoRem {i} olf (replLL ll ind rll)\ndelLFCoRem s \u2193 rll = \u2205\ndelLFCoRem (\u2193 x) (ind \u2190\u2227) rll = \u2205 -- We loose Information.\ndelLFCoRem (s \u2190\u2227) (ind \u2190\u2227) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u2205 = \u2205\ndelLFCoRem (s \u2190\u2227) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227)\ndelLFCoRem (\u2227\u2192 s) (ind \u2190\u2227) rll = \u00ac\u2205 (\u2227\u2192 (s))\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u2205 = \u00ac\u2205 (\u2227\u2192 (s\u2081))\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (ind \u2190\u2227) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2227\u2192 (s\u2081))\ndelLFCoRem (\u2193 x) (\u2227\u2192 ind) rll = \u2205 --\ndelLFCoRem (s \u2190\u2227) (\u2227\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) rll with (delLFCoRem s ind rll)\ndelLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u2205 = \u2205\ndelLFCoRem (\u2227\u2192 s) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2227\u2192 x)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll with (delLFCoRem s\u2081 ind rll)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2227)\ndelLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2227\u2192 x)\ndelLFCoRem (\u2193 x) (ind \u2190\u2228) rll = \u2205\ndelLFCoRem (s \u2190\u2228) (ind \u2190\u2228) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u2205 = \u2205\ndelLFCoRem (s \u2190\u2228) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228)\ndelLFCoRem (\u2228\u2192 s) (ind \u2190\u2228) rll = \u00ac\u2205 (\u2228\u2192 (s))\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u2205 = \u00ac\u2205 (\u2228\u2192 (s\u2081))\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (ind \u2190\u2228) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2228\u2192 (s\u2081))\ndelLFCoRem (\u2193 x) (\u2228\u2192 ind) rll = \u2205\ndelLFCoRem (s \u2190\u2228) (\u2228\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) rll with (delLFCoRem s ind rll)\ndelLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u2205 = \u2205\ndelLFCoRem (\u2228\u2192 s) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2228\u2192 x)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll with (delLFCoRem s\u2081 ind rll)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2228)\ndelLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2228\u2192 x)\ndelLFCoRem (\u2193 x) (ind \u2190\u2202) rll = \u2205\ndelLFCoRem (s \u2190\u2202) (ind \u2190\u2202) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u2205 = \u2205\ndelLFCoRem (s \u2190\u2202) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202)\ndelLFCoRem (\u2202\u2192 s) (ind \u2190\u2202) rll = \u00ac\u2205 (\u2202\u2192 (s))\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll with (delLFCoRem s ind rll)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u2205 = \u00ac\u2205 (\u2202\u2192 (s\u2081))\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (ind \u2190\u2202) rll | \u00ac\u2205 x = \u00ac\u2205 (x \u2190\u2202\u2192 (s\u2081))\ndelLFCoRem (\u2193 x) (\u2202\u2192 ind) rll = \u2205\ndelLFCoRem (s \u2190\u2202) (\u2202\u2192 ind) rll = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) rll with (delLFCoRem s ind rll)\ndelLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u2205 = \u2205\ndelLFCoRem (\u2202\u2192 s) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 (\u2202\u2192 x)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll with (delLFCoRem s\u2081 ind rll)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u2205 = \u00ac\u2205 ((s) \u2190\u2202)\ndelLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202\u2192 ind) rll | \u00ac\u2205 x = \u00ac\u2205 ((s) \u2190\u2202\u2192 x)\n\nmdelLFCoRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 MSetLFCoRem {i} olf ll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (rll : LinLogic i)\n             \u2192 MSetLFCoRem {i} olf (replLL ll ind rll)\nmdelLFCoRem \u2205 ind rll = \u2205\nmdelLFCoRem (\u00ac\u2205 x) ind rll = delLFCoRem x ind rll\n\ntranLFCoRem : \u2200{i u oll orll ll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 SetLFCoRem {i} olf ll \u2192 (tr : LLTr {i} {u} rll ll)\n       \u2192 SetLFCoRem olf rll\ntranLFCoRem s I                           = s\ntranLFCoRem (s \u2190\u2202) (\u2202c ltr)                = tranLFCoRem (\u2202\u2192 s) ltr\ntranLFCoRem (\u2193 x) (\u2202c ltr)                = \u2193 x\ntranLFCoRem (\u2202\u2192 s) (\u2202c ltr)                = tranLFCoRem (s \u2190\u2202) ltr\ntranLFCoRem (s \u2190\u2202\u2192 s\u2081) (\u2202c ltr)            = tranLFCoRem (s\u2081 \u2190\u2202\u2192 s) ltr\ntranLFCoRem (s \u2190\u2228) (\u2228c ltr)                = tranLFCoRem (\u2228\u2192 s) ltr\ntranLFCoRem (\u2193 x) (\u2228c ltr)                = \u2193 x\ntranLFCoRem (\u2228\u2192 s) (\u2228c ltr)                = tranLFCoRem (s \u2190\u2228) ltr\ntranLFCoRem (s \u2190\u2228\u2192 s\u2081) (\u2228c ltr)            = tranLFCoRem (s\u2081 \u2190\u2228\u2192 s) ltr\ntranLFCoRem (s \u2190\u2227) (\u2227c ltr)                = tranLFCoRem (\u2227\u2192 s) ltr\ntranLFCoRem (\u2193 x) (\u2227c ltr)                = \u2193 x\ntranLFCoRem (\u2227\u2192 s) (\u2227c ltr)                = tranLFCoRem (s \u2190\u2227) ltr\ntranLFCoRem (s \u2190\u2227\u2192 s\u2081) (\u2227c ltr)            = tranLFCoRem (s\u2081 \u2190\u2227\u2192 s) ltr\ntranLFCoRem ((s \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCoRem (s \u2190\u2227) ltr\ntranLFCoRem (\u2193 x) (\u2227\u2227d ltr)          = \u2193 x\ntranLFCoRem ((\u2193 x) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCoRem ((\u2193 x) \u2190\u2227\u2192 ((\u2193 x) \u2190\u2227)) ltr\ntranLFCoRem ((\u2227\u2192 s) \u2190\u2227) (\u2227\u2227d ltr)          = tranLFCoRem (\u2227\u2192 (s \u2190\u2227)) ltr\ntranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227) (\u2227\u2227d ltr)      = tranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227)) ltr\ntranLFCoRem (\u2227\u2192 s) (\u2227\u2227d ltr)               = tranLFCoRem (\u2227\u2192 (\u2227\u2192 s)) ltr\ntranLFCoRem ((\u2193 x) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCoRem ((\u2193 x) \u2190\u2227\u2192 ((\u2193 x) \u2190\u2227\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2227) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCoRem (s \u2190\u2227\u2192 (\u2227\u2192 s\u2081)) ltr\ntranLFCoRem ((\u2227\u2192 s) \u2190\u2227\u2192 s\u2081) (\u2227\u2227d ltr)      = tranLFCoRem (\u2227\u2192 (s \u2190\u2227\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227\u2192 s\u2082) (\u2227\u2227d ltr)  = tranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227\u2192 s\u2082)) ltr\ntranLFCoRem (s \u2190\u2227) (\u00ac\u2227\u2227d ltr)              = tranLFCoRem ((s \u2190\u2227) \u2190\u2227) ltr\ntranLFCoRem (\u2193 x) (\u00ac\u2227\u2227d ltr)              = \u2193 x\ntranLFCoRem (\u2227\u2192 (\u2193 x)) (\u00ac\u2227\u2227d ltr)         = tranLFCoRem ((\u2227\u2192 (\u2193 x)) \u2190\u2227\u2192 (\u2193 x)) ltr\ntranLFCoRem (\u2227\u2192 (s \u2190\u2227)) (\u00ac\u2227\u2227d ltr)         = tranLFCoRem ((\u2227\u2192 s) \u2190\u2227) ltr\ntranLFCoRem (\u2227\u2192 (\u2227\u2192 s)) (\u00ac\u2227\u2227d ltr)         = tranLFCoRem (\u2227\u2192 s) ltr\ntranLFCoRem (\u2227\u2192 (s \u2190\u2227\u2192 s\u2081)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((\u2227\u2192 s) \u2190\u2227\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (\u2193 x)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((s \u2190\u2227\u2192 (\u2193 x)) \u2190\u2227\u2192 (\u2193 x)) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (\u2227\u2192 s\u2081)) (\u00ac\u2227\u2227d ltr)     = tranLFCoRem ((s \u2190\u2227) \u2190\u2227\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2227\u2192 (s\u2081 \u2190\u2227\u2192 s\u2082)) (\u00ac\u2227\u2227d ltr) = tranLFCoRem ((s \u2190\u2227\u2192 s\u2081) \u2190\u2227\u2192 s\u2082) ltr\ntranLFCoRem ((s \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCoRem (s \u2190\u2228) ltr\ntranLFCoRem (\u2193 x) (\u2228\u2228d ltr)          = \u2193 x\ntranLFCoRem ((\u2193 x) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCoRem ((\u2193 x) \u2190\u2228\u2192 ((\u2193 x) \u2190\u2228)) ltr\ntranLFCoRem ((\u2228\u2192 s) \u2190\u2228) (\u2228\u2228d ltr)          = tranLFCoRem (\u2228\u2192 (s \u2190\u2228)) ltr\ntranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228) (\u2228\u2228d ltr)      = tranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228)) ltr\ntranLFCoRem (\u2228\u2192 s) (\u2228\u2228d ltr)               = tranLFCoRem (\u2228\u2192 (\u2228\u2192 s)) ltr\ntranLFCoRem ((\u2193 x) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCoRem ((\u2193 x) \u2190\u2228\u2192 ((\u2193 x) \u2190\u2228\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2228) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCoRem (s \u2190\u2228\u2192 (\u2228\u2192 s\u2081)) ltr\ntranLFCoRem ((\u2228\u2192 s) \u2190\u2228\u2192 s\u2081) (\u2228\u2228d ltr)      = tranLFCoRem (\u2228\u2192 (s \u2190\u2228\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228\u2192 s\u2082) (\u2228\u2228d ltr)  = tranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228\u2192 s\u2082)) ltr\ntranLFCoRem (s \u2190\u2228) (\u00ac\u2228\u2228d ltr)              = tranLFCoRem ((s \u2190\u2228) \u2190\u2228) ltr\ntranLFCoRem (\u2193 x) (\u00ac\u2228\u2228d ltr)              = \u2193 x\ntranLFCoRem (\u2228\u2192 (\u2193 x)) (\u00ac\u2228\u2228d ltr)         = tranLFCoRem ((\u2228\u2192 (\u2193 x)) \u2190\u2228\u2192 (\u2193 x)) ltr\ntranLFCoRem (\u2228\u2192 (s \u2190\u2228)) (\u00ac\u2228\u2228d ltr)         = tranLFCoRem ((\u2228\u2192 s) \u2190\u2228) ltr\ntranLFCoRem (\u2228\u2192 (\u2228\u2192 s)) (\u00ac\u2228\u2228d ltr)         = tranLFCoRem (\u2228\u2192 s) ltr\ntranLFCoRem (\u2228\u2192 (s \u2190\u2228\u2192 s\u2081)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((\u2228\u2192 s) \u2190\u2228\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (\u2193 x)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((s \u2190\u2228\u2192 (\u2193 x)) \u2190\u2228\u2192 (\u2193 x)) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (\u2228\u2192 s\u2081)) (\u00ac\u2228\u2228d ltr)     = tranLFCoRem ((s \u2190\u2228) \u2190\u2228\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2228\u2192 (s\u2081 \u2190\u2228\u2192 s\u2082)) (\u00ac\u2228\u2228d ltr) = tranLFCoRem ((s \u2190\u2228\u2192 s\u2081) \u2190\u2228\u2192 s\u2082) ltr\ntranLFCoRem ((s \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCoRem (s \u2190\u2202) ltr\ntranLFCoRem (\u2193 x) (\u2202\u2202d ltr)          = \u2193 x\ntranLFCoRem ((\u2193 x) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCoRem ((\u2193 x) \u2190\u2202\u2192 ((\u2193 x) \u2190\u2202)) ltr\ntranLFCoRem ((\u2202\u2192 s) \u2190\u2202) (\u2202\u2202d ltr)          = tranLFCoRem (\u2202\u2192 (s \u2190\u2202)) ltr\ntranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202) (\u2202\u2202d ltr)      = tranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202)) ltr\ntranLFCoRem (\u2202\u2192 s) (\u2202\u2202d ltr)               = tranLFCoRem (\u2202\u2192 (\u2202\u2192 s)) ltr\ntranLFCoRem ((\u2193 x) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCoRem ((\u2193 x) \u2190\u2202\u2192 ((\u2193 x) \u2190\u2202\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2202) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCoRem (s \u2190\u2202\u2192 (\u2202\u2192 s\u2081)) ltr\ntranLFCoRem ((\u2202\u2192 s) \u2190\u2202\u2192 s\u2081) (\u2202\u2202d ltr)      = tranLFCoRem (\u2202\u2192 (s \u2190\u2202\u2192 s\u2081)) ltr\ntranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202\u2192 s\u2082) (\u2202\u2202d ltr)  = tranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202\u2192 s\u2082)) ltr\ntranLFCoRem (s \u2190\u2202) (\u00ac\u2202\u2202d ltr)              = tranLFCoRem ((s \u2190\u2202) \u2190\u2202) ltr\ntranLFCoRem (\u2202\u2192 (s \u2190\u2202)) (\u00ac\u2202\u2202d ltr)         = tranLFCoRem ((\u2202\u2192 s) \u2190\u2202) ltr\ntranLFCoRem (\u2193 x) (\u00ac\u2202\u2202d ltr)         = \u2193 x\ntranLFCoRem (\u2202\u2192 (\u2193 x)) (\u00ac\u2202\u2202d ltr)         = tranLFCoRem ((\u2202\u2192 (\u2193 x)) \u2190\u2202\u2192 (\u2193 x)) ltr\ntranLFCoRem (\u2202\u2192 (\u2202\u2192 s)) (\u00ac\u2202\u2202d ltr)         = tranLFCoRem (\u2202\u2192 s) ltr\ntranLFCoRem (\u2202\u2192 (s \u2190\u2202\u2192 s\u2081)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((\u2202\u2192 s) \u2190\u2202\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (\u2193 x)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((s \u2190\u2202\u2192 (\u2193 x)) \u2190\u2202\u2192 (\u2193 x)) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (\u2202\u2192 s\u2081)) (\u00ac\u2202\u2202d ltr)     = tranLFCoRem ((s \u2190\u2202) \u2190\u2202\u2192 s\u2081) ltr\ntranLFCoRem (s \u2190\u2202\u2192 (s\u2081 \u2190\u2202\u2192 s\u2082)) (\u00ac\u2202\u2202d ltr) = tranLFCoRem ((s \u2190\u2202\u2192 s\u2081) \u2190\u2202\u2192 s\u2082) ltr\n\n\nextendLFCoRem : \u2200{i u oll orll ll pll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 IndexLL {i} {u} pll ll \u2192 SetLFCoRem {i} olf pll \u2192 SetLFCoRem olf ll\nextendLFCoRem \u2193 sr = sr\nextendLFCoRem (ind \u2190\u2227) sr = (extendLFCoRem ind sr) \u2190\u2227\nextendLFCoRem (\u2227\u2192 ind) sr = \u2227\u2192 (extendLFCoRem ind sr)\nextendLFCoRem (ind \u2190\u2228) sr = (extendLFCoRem ind sr) \u2190\u2228\nextendLFCoRem (\u2228\u2192 ind) sr = \u2228\u2192 (extendLFCoRem ind sr)\nextendLFCoRem (ind \u2190\u2202) sr = (extendLFCoRem ind sr) \u2190\u2202\nextendLFCoRem (\u2202\u2192 ind) sr = \u2202\u2192 (extendLFCoRem ind sr)\n\nreplaceLFCoRem : \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 SetLFCoRem {i} olf rll \u2192 SetLFCoRem olf ll \u2192 SetLFCoRem olf (replLL ll ind rll)\nreplaceLFCoRem \u2193 esr sr = esr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (\u2193 x) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2227\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (sr \u2190\u2227) = replaceLFCoRem ind esr sr \u2190\u2227\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (\u2227\u2192 sr) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2227\u2192 sr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2227) esr (sr \u2190\u2227\u2192 sr\u2081) = (replaceLFCoRem ind esr sr) \u2190\u2227\u2192 sr\u2081\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (\u2193 x) = \u2227\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (sr \u2190\u2227) = sr \u2190\u2227\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (\u2227\u2192 sr) = \u2227\u2192 replaceLFCoRem ind esr sr\nreplaceLFCoRem {rll = rll} (\u2227\u2192 ind) esr (sr \u2190\u2227\u2192 sr\u2081) = sr \u2190\u2227\u2192 replaceLFCoRem ind esr sr\u2081\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (\u2193 x) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2228\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (sr \u2190\u2228) = replaceLFCoRem ind esr sr \u2190\u2228\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (\u2228\u2192 sr) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2228\u2192 sr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2228) esr (sr \u2190\u2228\u2192 sr\u2081) = (replaceLFCoRem ind esr sr) \u2190\u2228\u2192 sr\u2081\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (\u2193 x) = \u2228\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (sr \u2190\u2228) = sr \u2190\u2228\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (\u2228\u2192 sr) = \u2228\u2192 replaceLFCoRem ind esr sr\nreplaceLFCoRem {rll = rll} (\u2228\u2192 ind) esr (sr \u2190\u2228\u2192 sr\u2081) = sr \u2190\u2228\u2192 replaceLFCoRem ind esr sr\u2081\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (\u2193 x) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2202\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (sr \u2190\u2202) = replaceLFCoRem ind esr sr \u2190\u2202\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (\u2202\u2192 sr) = (extendLFCoRem (updInd rll ind) esr) \u2190\u2202\u2192 sr\nreplaceLFCoRem {rll = rll} (ind \u2190\u2202) esr (sr \u2190\u2202\u2192 sr\u2081) = (replaceLFCoRem ind esr sr) \u2190\u2202\u2192 sr\u2081\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (\u2193 x) = \u2202\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (sr \u2190\u2202) = sr \u2190\u2202\u2192 (extendLFCoRem (updInd rll ind) esr)\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (\u2202\u2192 sr) = \u2202\u2192 replaceLFCoRem ind esr sr\nreplaceLFCoRem {rll = rll} (\u2202\u2192 ind) esr (sr \u2190\u2202\u2192 sr\u2081) = sr \u2190\u2202\u2192 replaceLFCoRem ind esr sr\u2081\n\n\nmreplaceLFCoRem :  \u2200{i u oll orll ll pll rll} \u2192 {olf : LFun {i} {u} oll orll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 MSetLFCoRem {i} olf rll \u2192 MSetLFCoRem olf ll \u2192 MSetLFCoRem olf (replLL ll ind rll)\nmreplaceLFCoRem ind \u2205 \u2205 = \u2205\nmreplaceLFCoRem {rll = rll} ind \u2205 (\u00ac\u2205 x) = delLFCoRem x ind rll\nmreplaceLFCoRem {rll = rll} ind (\u00ac\u2205 x) \u2205 = \u00ac\u2205 (extendLFCoRem (updInd rll ind) x)\nmreplaceLFCoRem ind (\u00ac\u2205 x) (\u00ac\u2205 x\u2081) = \u00ac\u2205 (replaceLFCoRem ind x x\u2081)\n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"66f8975ebba009ee39f01f368adf5ca9c91f44c0","subject":"cyrus says this rule is OK","message":"cyrus says this rule is OK\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4 -- todo: made a change here\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942) -- todo: should that be \u03c42'?\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4 -- todo: made a change here\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4ed8a98ebf76e60d0812fd41621ee003015d07fc","subject":"Data.Nat.NP: add 2^ and 2*","message":"Data.Nat.NP: add 2^ and 2*\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ zero  = 1\n2^ suc n = 2* 2^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1ba47258fdeeebc176947c926217f52ed2b63a1c","subject":"Tmp definition of nest","message":"Tmp definition of nest\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/NP.agda","new_file":"lib\/Function\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary.PropositionalEquality.NP\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : Set) (n : \u2115) (B : Set) \u2192 Set\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : Set) (n : \u2115) (B : Set\u2081) \u2192 Set\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = refl\n  nest-+ (suc m) n = cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!} \n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very few code rely on it.\n-- \u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n-- \u2026 \u2983 x \u2984 = x\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary.PropositionalEquality.NP\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : Set) (n : \u2115) (B : Set) \u2192 Set\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : Set) (n : \u2115) (B : Set\u2081) \u2192 Set\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\nnest n f x = fold x f n\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = refl\n  nest-+ (suc m) n = cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!} \n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very few code rely on it.\n-- \u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n-- \u2026 \u2983 x \u2984 = x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"873ab6f3148ef9e242cf51188ae07997d12944a9","subject":"whitespace","message":"whitespace\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"structural.agda","new_file":"structural.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule structural where\n  -- todo: there's a freshness concern here; i'm not sure if just\n  -- being apart from \u0393 is good enough. in POPL work it wasn't, we\n  -- wanted really aggressive freshness because of barendrecht's. we\n  -- might be able to generate that freshness from the first two\n  -- premises.\n  postulate\n    weaken-ana-expand : \u2200{ \u0393 e \u03c4 e' \u03c4' \u0394 x \u03c4* } \u2192 x # \u0393\n                                                \u2192 \u0393 \u22a2 e \u21d0 \u03c4 ~> e' :: \u03c4' \u22a3 \u0394\n                                                \u2192 (\u0393 ,, (x , \u03c4*)) \u22a2 e \u21d0 \u03c4 ~> e' :: \u03c4' \u22a3 \u0394\n\n  postulate\n    lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 ## \u03942 \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule structural where\n  -- todo: there's a freshness concern here; i'm not sure if just\n  -- being apart from \u0393 is good enough. in POPL work it wasn't, we\n  -- wanted really aggressive freshness because of barendrecht's. we\n  -- might be able to generate that freshness from the first two\n  -- premises.\n  postulate\n    weaken-ana-expand : \u2200{ \u0393 e \u03c4 e' \u03c4' \u0394 x \u03c4* } \u2192 x # \u0393 \u2192 \u0393 \u22a2 e \u21d0 \u03c4 ~> e' :: \u03c4' \u22a3 \u0394 \u2192 (\u0393 ,, (x , \u03c4*)) \u22a2 e \u21d0 \u03c4 ~> e' :: \u03c4' \u22a3 \u0394\n\n  postulate\n    lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 ## \u03942 \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"007579ee74b2e0742b6e40fda6df9902a49e0c6e","subject":"error in indeterminate -- was matching on any check implicitly","message":"error in indeterminate -- was matching on any check implicitly\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error -- todo\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    -- EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err -- todo: is it an error to ever cast the constant?\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error -- todo\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    -- EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err -- todo: is it an error to ever cast the constant?\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"34e3329d924e7ab6cf08867c60e0b74c3825ca80","subject":"Added TODO.","message":"Added TODO.\n\nIgnore-this: da06cb652662731c3e1126c7ae17d9f3\n\ndarcs-hash:20120228202357-3bd4e-fc56819069220d0ce3ab19886c8084e7ada6ec37.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/MainATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/Properties\/MainATP.agda","new_contents":"-- TODO: Move to the main directory.\n\n------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.Properties.MainATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.LT2MCR-ATP\nopen import FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Program.McCarthy91.Properties.AuxiliaryATP\n\n------------------------------------------------------------------------------\n\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 N (mc91 n) \u2227 LT n (mc91 n + eleven)\nmc91-N-ineq = wfInd-MCR P mc91-N-ineq-aux\n  where\n  P : D \u2192 Set\n  P d = N (mc91 d) \u2227 LT d (mc91 d + eleven)\n\n  mc91-N-ineq-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-N-ineq-aux {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = ( Nmc91>100 Nm m>100 , x<mc91x+11>100 Nm m>100 )\n  ... | inj\u2082 m\u226f100 =\n    let Nm+11 : N (m + eleven)\n        Nm+11 = x+11-N Nm\n\n        ih\u2081 : P (m + eleven)\n        ih\u2081 = f Nm+11 (LT2MCR (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        Nih\u2081 : N (mc91 (m + eleven))\n        Nih\u2081 = \u2227-proj\u2081 ih\u2081\n\n        LTih\u2081 : LT (m + eleven) (mc91 (m + eleven) + eleven)\n        LTih\u2081 = \u2227-proj\u2082 ih\u2081\n\n        m<mc91m+11 : LT m (mc91 (m + eleven))\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm Nih\u2081 11-N LTih\u2081\n\n        ih\u2081 : P (mc91 (m + eleven))\n        ih\u2081 = f Nih\u2081 (LT2MCR Nih\u2081 Nm m\u226f100 m<mc91m+11)\n\n        Nmc91\u2264100 : N (mc91 m)\n        Nmc91\u2264100 = Nmc91\u226f100 m m\u226f100 (\u2227-proj\u2081 ih\u2081)\n\n    in ( Nmc91\u2264100 ,\n         <-trans Nm Nih\u2081 (x+11-N Nmc91\u2264100)\n                 m<mc91m+11\n                 (mc91x+11<mc91x+11 m m\u226f100 (\u2227-proj\u2082 ih\u2081)))\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (GT n one-hundred \u2227 mc91 n \u2261 n \u2238 ten) \u2228\n                         (NGT n one-hundred \u2227 mc91 n \u2261 ninety-one)\nmc91-res = wfInd-MCR P mc91-res-aux\n  where\n  P : D \u2192 Set\n  P d = (GT d one-hundred \u2227 mc91 d \u2261 d \u2238 ten) \u2228\n        (NGT d one-hundred \u2227 mc91 d \u2261 ninety-one)\n\n  mc91-res-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-res-aux {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq-aux m m>100 )\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u226f100 , mc91-res-100' m\u2261100 )\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = inj\u2082 ( m\u226f100 , mc91-res-99' m\u226199 )\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = inj\u2082 ( m\u226f100 , mc91-res-98' m\u226198 )\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = inj\u2082 ( m\u226f100 , mc91-res-97' m\u226197 )\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = inj\u2082 ( m\u226f100 , mc91-res-96' m\u226196 )\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = inj\u2082 ( m\u226f100 , mc91-res-95' m\u226195 )\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = inj\u2082 ( m\u226f100 , mc91-res-94' m\u226194 )\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = inj\u2082 ( m\u226f100 , mc91-res-93' m\u226193 )\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = inj\u2082 ( m\u226f100 , mc91-res-92' m\u226192 )\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = inj\u2082 ( m\u226f100 , mc91-res-91' m\u226191 )\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = inj\u2082 ( m\u226f100 , mc91-res-90' m\u226190 )\n  ... | inj\u2081 m\u226f89 = inj\u2082 ( m\u226f100 , mc91-res-m\u226f89 )\n    where\n    m\u226489 : LE m eighty-nine\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : mc91 (m + eleven) \u2261 ninety-one\n    mc91-res-m+11 with f (x+11-N Nm) (LT2MCR (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n    ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm m\u226489 m+11>100)\n    ... | inj\u2082 ( _ , res ) = res\n\n    mc91-res-m\u226f89 : mc91 m \u2261 ninety-one\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m ninety-one m\u226f100 mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn = \u2227-proj\u2081 (mc91-N-ineq Nn)\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 LT n (mc91 n + eleven)\nmc91-ineq Nn = \u2227-proj\u2082 (mc91-N-ineq Nn)\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\nmc91-res>100 Nn n>100 with mc91-res Nn\n... | inj\u2081 ( _     , res ) = res\n... | inj\u2082 ( n\u226f100 , _ )   = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 NGT n one-hundred \u2192 mc91 n \u2261 ninety-one\nmc91-res\u226f100 Nn n\u226f100 with mc91-res Nn\n... | inj\u2081 ( n>100 , _   ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n... | inj\u2082 ( _     , res ) = res\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.Properties.MainATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.LT2MCR-ATP\nopen import FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Program.McCarthy91.Properties.AuxiliaryATP\n\n------------------------------------------------------------------------------\n\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 N (mc91 n) \u2227 LT n (mc91 n + eleven)\nmc91-N-ineq = wfInd-MCR P mc91-N-ineq-aux\n  where\n  P : D \u2192 Set\n  P d = N (mc91 d) \u2227 LT d (mc91 d + eleven)\n\n  mc91-N-ineq-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-N-ineq-aux {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = ( Nmc91>100 Nm m>100 , x<mc91x+11>100 Nm m>100 )\n  ... | inj\u2082 m\u226f100 =\n    let Nm+11 : N (m + eleven)\n        Nm+11 = x+11-N Nm\n\n        ih\u2081 : P (m + eleven)\n        ih\u2081 = f Nm+11 (LT2MCR (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        Nih\u2081 : N (mc91 (m + eleven))\n        Nih\u2081 = \u2227-proj\u2081 ih\u2081\n\n        LTih\u2081 : LT (m + eleven) (mc91 (m + eleven) + eleven)\n        LTih\u2081 = \u2227-proj\u2082 ih\u2081\n\n        m<mc91m+11 : LT m (mc91 (m + eleven))\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm Nih\u2081 11-N LTih\u2081\n\n        ih\u2081 : P (mc91 (m + eleven))\n        ih\u2081 = f Nih\u2081 (LT2MCR Nih\u2081 Nm m\u226f100 m<mc91m+11)\n\n        Nmc91\u2264100 : N (mc91 m)\n        Nmc91\u2264100 = Nmc91\u226f100 m m\u226f100 (\u2227-proj\u2081 ih\u2081)\n\n    in ( Nmc91\u2264100 ,\n         <-trans Nm Nih\u2081 (x+11-N Nmc91\u2264100)\n                 m<mc91m+11\n                 (mc91x+11<mc91x+11 m m\u226f100 (\u2227-proj\u2082 ih\u2081)))\n\nmc91-res : \u2200 {n} \u2192 N n \u2192 (GT n one-hundred \u2227 mc91 n \u2261 n \u2238 ten) \u2228\n                         (NGT n one-hundred \u2227 mc91 n \u2261 ninety-one)\nmc91-res = wfInd-MCR P mc91-res-aux\n  where\n  P : D \u2192 Set\n  P d = (GT d one-hundred \u2227 mc91 d \u2261 d \u2238 ten) \u2228\n        (NGT d one-hundred \u2227 mc91 d \u2261 ninety-one)\n\n  mc91-res-aux : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 MCR k m \u2192 P k) \u2192 P m\n  mc91-res-aux {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = inj\u2081 ( m>100 , mc91-eq-aux m m>100 )\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = inj\u2082 ( m\u226f100 , mc91-res-100' m\u2261100 )\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = inj\u2082 ( m\u226f100 , mc91-res-99' m\u226199 )\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = inj\u2082 ( m\u226f100 , mc91-res-98' m\u226198 )\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = inj\u2082 ( m\u226f100 , mc91-res-97' m\u226197 )\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = inj\u2082 ( m\u226f100 , mc91-res-96' m\u226196 )\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = inj\u2082 ( m\u226f100 , mc91-res-95' m\u226195 )\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = inj\u2082 ( m\u226f100 , mc91-res-94' m\u226194 )\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = inj\u2082 ( m\u226f100 , mc91-res-93' m\u226193 )\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = inj\u2082 ( m\u226f100 , mc91-res-92' m\u226192 )\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = inj\u2082 ( m\u226f100 , mc91-res-91' m\u226191 )\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = inj\u2082 ( m\u226f100 , mc91-res-90' m\u226190 )\n  ... | inj\u2081 m\u226f89 = inj\u2082 ( m\u226f100 , mc91-res-m\u226f89 )\n    where\n    m\u226489 : LE m eighty-nine\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : mc91 (m + eleven) \u2261 ninety-one\n    mc91-res-m+11 with f (x+11-N Nm) (LT2MCR (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n    ... | inj\u2081 ( m+11>100 , _ ) = \u22a5-elim (x\u226489\u2192x+11>100\u2192\u22a5 Nm m\u226489 m+11>100)\n    ... | inj\u2082 ( _ , res ) = res\n\n    mc91-res-m\u226f89 : mc91 m \u2261 ninety-one\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m ninety-one m\u226f100 mc91-res-m+11 mc91-res-91\n\n------------------------------------------------------------------------------\n-- Main properties\n\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N Nn = \u2227-proj\u2081 (mc91-N-ineq Nn)\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 LT n (mc91 n + eleven)\nmc91-ineq Nn = \u2227-proj\u2082 (mc91-N-ineq Nn)\n\n-- For all n > 100, then mc91 n = n - 10.\nmc91-res>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\nmc91-res>100 Nn n>100 with mc91-res Nn\n... | inj\u2081 ( _     , res ) = res\n... | inj\u2082 ( n\u226f100 , _ )   = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 NGT n one-hundred \u2192 mc91 n \u2261 ninety-one\nmc91-res\u226f100 Nn n\u226f100 with mc91-res Nn\n... | inj\u2081 ( n>100 , _   ) = \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nn 100-N n>100\n                                               (x\u226fy\u2192x\u2264y Nn 100-N n\u226f100))\n... | inj\u2082 ( _     , res ) = res\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7b872fad17fc7f25184c171975bcd5ef11722bc4","subject":"Implicit arguments are eagerly introduced again","message":"Implicit arguments are eagerly introduced again\n","repos":"np\/agda-parametricity","old_file":"lib\/Function\/Param\/Binary.agda","new_file":"lib\/Function\/Param\/Binary.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Type.Param.Binary\nopen import Algebra.FunctionProperties\n\nmodule Function.Param.Binary where\n\nprivate\n  \u2605_ = \u03bb \u2113 \u2192 Set \u2113\n  \u2605\u2080 = Set\u2080\n  \u2605\u2081 = Set\u2081\n  \u2605\u2082 = Set\u2082\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 1 _\u2192\u27e7_\n_\u2192\u27e7_ : \u2200 {a b\u2081 b\u2082 b\u1d63} (A : \u2605 a) {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082) \u2192 \u27e6\u2605\u27e7 _ (A \u2192 B\u2081) (A \u2192 B\u2082)\n(A \u2192\u27e7 B\u1d63) f\u2081 f\u2082 = \u2200 (x : A) \u2192 B\u1d63 (f\u2081 x) (f\u2082 x)\n\n\u03a0\u27e7 : \u2200 {a b\u2081 b\u2082 b\u1d63} (A : \u2605 a) {B\u2081 : A \u2192 \u2605 b\u2081} {B\u2082 : A \u2192 \u2605 b\u2082} (B\u1d63 : (A \u2192\u27e7 \u27e6\u2605\u27e7 b\u1d63) B\u2081 B\u2082) \u2192 \u27e6\u2605\u27e7 _ ((x : A) \u2192 B\u2081 x) ((x : A) \u2192 B\u2082 x)\n\u03a0\u27e7 A B\u1d63 f\u2081 f\u2082 = \u2200 (x : A) \u2192 B\u1d63 x (f\u2081 x) (f\u2082 x)\n\ninfixr 0 \u03a0\u27e7\nsyntax \u03a0\u27e7 A (\u03bb x \u2192 f) = \u27e8 x \u2236 A \u27e9\u2192\u27e7 f\n\n\u2200\u27e7 : \u2200 {a} (A : \u2605 a)\n       {b\u2081 b\u2082 b\u1d63} {B\u2081 : A \u2192 \u2605 b\u2081} {B\u2082 : A \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 x \u2192 B\u2081 x \u2192 B\u2082 x \u2192 \u2605 b\u1d63)\n       (f\u2081 : {x : A} \u2192 B\u2081 x) (f\u2082 : {x : A} \u2192 B\u2082 x) \u2192 \u2605 _\n\u2200\u27e7 A B\u1d63 f\u2081 f\u2082 = {x : A} \u2192 B\u1d63 x (f\u2081 {x}) (f\u2082 {x})\n\ninfixr 0 \u2200\u27e7\nsyntax \u2200\u27e7 A (\u03bb x \u2192 f) = \u2200\u27e8 x \u2236 A \u27e9\u2192\u27e7 f\n\n-- Non universe polymorphic versions\n\ninfixr 1 _\u27e6\u2080\u2192\u2080\u27e7_ _\u27e6\u2080\u2192\u2081\u27e7_ _\u27e6\u2081\u2192\u2082\u27e7_\n\n_\u27e6\u2080\u2192\u2080\u27e7_ : \u2200 {A\u2081 A\u2082 : \u2605\u2080} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605\u2080)\n          {B\u2081 B\u2082 : \u2605\u2080} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605\u2080)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605\u2080\n_\u27e6\u2080\u2192\u2080\u27e7_ {A\u2081} {A\u2082} A\u1d63 {B\u2081} {B\u2082} B\u1d63 f\u2081 f\u2082 =\n          \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n_\u27e6\u2080\u2192\u2081\u27e7_ : \u2200 {A\u2081 A\u2082 : \u2605\u2080} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605\u2080)\n          {B\u2081 B\u2082 : \u2605\u2081} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605\u2081)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605\u2081\n_\u27e6\u2080\u2192\u2081\u27e7_ {A\u2081} {A\u2082} A\u1d63 {B\u2081} {B\u2082} B\u1d63 f\u2081 f\u2082 =\n          \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n_\u27e6\u2081\u2192\u2082\u27e7_ : \u2200 {A\u2081 A\u2082 : \u2605\u2081} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605\u2081)\n          {B\u2081 B\u2082 : \u2605\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605\u2082)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605\u2082\n_\u27e6\u2081\u2192\u2082\u27e7_ {A\u2081} {A\u2082} A\u1d63 {B\u2081} {B\u2082} B\u1d63 f\u2081 f\u2082 =\n          \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n-- open import Relation.Unary.NP using (Pred)\nprivate\n  -- Flipped version of Relation.Unary.Pred which scale better\n  -- to logical relation [Pred] and \u27e6Pred\u27e7\n  Pred : \u2200 \u2113 {a} (A : \u2605 a) \u2192 \u2605 (a \u2294 suc \u2113)\n  Pred \u2113 A = A \u2192 \u2605 \u2113\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n\u27e6Pred\u27e7 p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 p\u1d63\n\nprivate\n  -- from Relation.Binary\n  REL : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 (\u2113 : Level) \u2192 Set (a \u2294 b \u2294 suc \u2113)\n  REL A B \u2113 = A \u2192 B \u2192 Set \u2113\n\n  -- from Relation.Binary\n  Rel : \u2200 {a} \u2192 Set a \u2192 (\u2113 : Level) \u2192 Set (a \u2294 suc \u2113)\n  Rel A \u2113 = REL A A \u2113\n\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 \u2605 \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 r\u2081 r\u2082} r\u1d63 \u2192\n          (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (REL\u2032 r\u2081) (REL\u2032 r\u2082)\n  \u27e6REL\u27e7\u2032 r\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          {r\u2081 r\u2082} r\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 r\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 r\u2082) \u2192 \u2605 _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 r\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {r\u2081 r\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 r\u2081) (\u223c\u2082 : Rel A\u2082 r\u2082) \u2192 \u2605 _\n\u27e6Rel\u27e7 A\u1d63 r\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 r\u1d63\n\n\u27e6Op\u2081\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} (a\u2081 \u2294 a\u2082 \u2294 a\u1d63) \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Op\u2081 Op\u2081\n\u27e6Op\u2081\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\n\u27e6Op\u2082\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} (a\u2081 \u2294 a\u2082 \u2294 a\u1d63) \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Op\u2082 Op\u2082\n\u27e6Op\u2082\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Type.Param.Binary\nopen import Algebra.FunctionProperties\n\nmodule Function.Param.Binary where\n\nprivate\n  \u2605_ = \u03bb \u2113 \u2192 Set \u2113\n  \u2605\u2080 = Set\u2080\n  \u2605\u2081 = Set\u2081\n  \u2605\u2082 = Set\u2082\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 1 _\u2192\u27e7_\n_\u2192\u27e7_ : \u2200 {a b\u2081 b\u2082 b\u1d63} (A : \u2605 a) {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082) \u2192 \u27e6\u2605\u27e7 _ (A \u2192 B\u2081) (A \u2192 B\u2082)\n(A \u2192\u27e7 B\u1d63) f\u2081 f\u2082 = \u2200 (x : A) \u2192 B\u1d63 (f\u2081 x) (f\u2082 x)\n\n\u03a0\u27e7 : \u2200 {a b\u2081 b\u2082 b\u1d63} (A : \u2605 a) {B\u2081 : A \u2192 \u2605 b\u2081} {B\u2082 : A \u2192 \u2605 b\u2082} (B\u1d63 : (A \u2192\u27e7 \u27e6\u2605\u27e7 b\u1d63) B\u2081 B\u2082) \u2192 \u27e6\u2605\u27e7 _ ((x : A) \u2192 B\u2081 x) ((x : A) \u2192 B\u2082 x)\n\u03a0\u27e7 A B\u1d63 f\u2081 f\u2082 = \u2200 (x : A) \u2192 B\u1d63 x (f\u2081 x) (f\u2082 x)\n\ninfixr 0 \u03a0\u27e7\nsyntax \u03a0\u27e7 A (\u03bb x \u2192 f) = \u27e8 x \u2236 A \u27e9\u2192\u27e7 f\n\n\u2200\u27e7 : \u2200 {a} (A : \u2605 a)\n       {b\u2081 b\u2082 b\u1d63} {B\u2081 : A \u2192 \u2605 b\u2081} {B\u2082 : A \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 x \u2192 B\u2081 x \u2192 B\u2082 x \u2192 \u2605 b\u1d63)\n       (f\u2081 : {x : A} \u2192 B\u2081 x) (f\u2082 : {x : A} \u2192 B\u2082 x) \u2192 \u2605 _\n\u2200\u27e7 A B\u1d63 f\u2081 f\u2082 = {x : A} \u2192 B\u1d63 x (f\u2081 {x}) (f\u2082 {x})\n\ninfixr 0 \u2200\u27e7\nsyntax \u2200\u27e7 A (\u03bb x \u2192 f) = \u2200\u27e8 x \u2236 A \u27e9\u2192\u27e7 f\n\n-- Non universe polymorphic versions\n\ninfixr 1 _\u27e6\u2080\u2192\u2080\u27e7_ _\u27e6\u2080\u2192\u2081\u27e7_ _\u27e6\u2081\u2192\u2082\u27e7_\n\n_\u27e6\u2080\u2192\u2080\u27e7_ : \u2200 {A\u2081 A\u2082 : \u2605\u2080} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605\u2080)\n          {B\u2081 B\u2082 : \u2605\u2080} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605\u2080)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605\u2080\n_\u27e6\u2080\u2192\u2080\u27e7_ = \u03bb {A\u2081} {A\u2082} A\u1d63 {B\u2081} {B\u2082} B\u1d63 f\u2081 f\u2082 \u2192\n          \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n_\u27e6\u2080\u2192\u2081\u27e7_ : \u2200 {A\u2081 A\u2082 : \u2605\u2080} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605\u2080)\n          {B\u2081 B\u2082 : \u2605\u2081} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605\u2081)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605\u2081\n_\u27e6\u2080\u2192\u2081\u27e7_ = \u03bb {A\u2081} {A\u2082} A\u1d63 {B\u2081} {B\u2082} B\u1d63 f\u2081 f\u2082 \u2192\n          \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n_\u27e6\u2081\u2192\u2082\u27e7_ : \u2200 {A\u2081 A\u2082 : \u2605\u2081} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605\u2081)\n          {B\u2081 B\u2082 : \u2605\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605\u2082)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605\u2082\n_\u27e6\u2081\u2192\u2082\u27e7_ = \u03bb {A\u2081} {A\u2082} A\u1d63 {B\u2081} {B\u2082} B\u1d63 f\u2081 f\u2082 \u2192\n          \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n-- open import Relation.Unary.NP using (Pred)\nprivate\n  -- Flipped version of Relation.Unary.Pred which scale better\n  -- to logical relation [Pred] and \u27e6Pred\u27e7\n  Pred : \u2200 \u2113 {a} (A : \u2605 a) \u2192 \u2605 (a \u2294 suc \u2113)\n  Pred \u2113 A = A \u2192 \u2605 \u2113\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n\u27e6Pred\u27e7 p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 p\u1d63\n\nprivate\n  -- from Relation.Binary\n  REL : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 (\u2113 : Level) \u2192 Set (a \u2294 b \u2294 suc \u2113)\n  REL A B \u2113 = A \u2192 B \u2192 Set \u2113\n\n  -- from Relation.Binary\n  Rel : \u2200 {a} \u2192 Set a \u2192 (\u2113 : Level) \u2192 Set (a \u2294 suc \u2113)\n  Rel A \u2113 = REL A A \u2113\n\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 \u2605 \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 r\u2081 r\u2082} r\u1d63 \u2192\n          (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (REL\u2032 r\u2081) (REL\u2032 r\u2082)\n  \u27e6REL\u27e7\u2032 r\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          {r\u2081 r\u2082} r\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 r\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 r\u2082) \u2192 \u2605 _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 r\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {r\u2081 r\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 r\u2081) (\u223c\u2082 : Rel A\u2082 r\u2082) \u2192 \u2605 _\n\u27e6Rel\u27e7 A\u1d63 r\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 r\u1d63\n\n\u27e6Op\u2081\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} (a\u2081 \u2294 a\u2082 \u2294 a\u1d63) \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Op\u2081 Op\u2081\n\u27e6Op\u2081\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\n\u27e6Op\u2082\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} (a\u2081 \u2294 a\u2082 \u2294 a\u1d63) \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Op\u2082 Op\u2082\n\u27e6Op\u2082\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5f16ec4ef9eb37cb1dd65ad7a81215fb739810a0","subject":"New\/Changes.intCA: define composition explicitly","message":"New\/Changes.intCA: define composition explicitly\n\nApparently the definition was *inferred* by unification O_O.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da (ada : valid a da) \u2192 WellDefinedFunChangePoint f df a da\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  instance\n    sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 da1 + da2\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","old_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da (ada : valid a da) \u2192 WellDefinedFunChangePoint f df a da\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  instance\n    sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"42e4d3740b2b7f1d0506a9f1946f2c75c8f1c1af","subject":"Messing around, not entirely productively, with ornamental agda.","message":"Messing around, not entirely productively, with ornamental agda.\n","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/Ornament.agda","new_file":"models\/Ornament.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Ornament where\n\nopen import Data.Nat using (\u2115)\nopen import Data.Fin \nopen import Data.Bool\nopen import Data.Product\nopen import Data.Unit\nopen import Data.List\n\ninfix 4 _\u2261_\n\ndata _\u2261_ {a} {A : Set a} (x : A) : A \u2192 Set a where\n  refl : x \u2261 x\n\n--\n\n-- Modified from Conor's original HOAS presentation\n\ndata IndexParam {n} (I : Set) (N : Fin n) : Set\u2081 where\n  -- read an indexed subnote; continue\n  rec : I -> IndexParam I N -> IndexParam I N\n\n  -- stop and return the node's index\n  ret : I -> IndexParam I N\n\nsemantics : \u2200{I} -> \u2200{n : \u2115} -> IndexParam I (from\u2115 n) -> (I -> Set) -> I -> Set\nsemantics = {!!}\n  \nrecord EmbedDataConstr : Set\u2081 where\n  constructor embedDataConstr\n  field arityTyping : {!!}\n\nrecord EmbedDataTy (I A : Set) : Set\u2081 where\n  constructor embedDataTy\n  field indexes : List I\n  field parameters : List A\n  field constrs : List EmbedDataConstr\n\ndata Desc (I : Set) : Set\u2081 where\n  -- read field in A; continue, given its value\n  -- I don't think this is really what we want -- want parametricity\n  arg : (a : Set) -> (a -> Desc I) -> Desc I\n\n  -- read an indexed subnode; continue regardless\n  rec : I -> Desc I -> Desc I\n\n  -- stop reading and return the node\u2019s index\n  ret : I -> Desc I\n\n  -- choice between n different constructors\n  choice : (n : \u2115) -> (Fin n -> Desc I) -> Desc I\n\n-- The semantics (\"extension\") of an indexed container\n\u27e6_\u27e7 : \u2200{I} -> Desc I -> (I -> Set) -> I -> Set\n\u27e6 arg A D \u27e7 R i = \u03a3 A \u03bb a \u2192 \u27e6 D a \u27e7 R i\n\u27e6 rec h D \u27e7 R i = R h \u00d7 \u27e6 D \u27e7 R i\n\u27e6 ret o \u27e7 R i = o \u2261 i\n\u27e6 choice n D \u27e7 R i =  \u03a3 (Fin n) \u03bb #n -> \u27e6 D #n \u27e7 R i\n\n\n-- This is really the least fixpoint operator\ndata Data {I}(D : Desc I)(i : I) : Set where\n  \u27e8_\u27e9 : \u27e6 D \u27e7 (Data D) i -> Data D i\n\n-- TODO think about codata \/ greatest fixed point\n-- data CoData {I}(D : Desc I)(i : I) : Set where\n--   \u27e9_\u27e8 : \n\n--\n\n\nrecord DataConstr : Set where\n  constructor dataConstr\n  field arityTyping : {!!}\n\nrecord DataTy : Set where\n  constructor dataTy\n  field indexes : List \u2115\n  field parameters : List \u2115\n  field constrs : List {!!}\n\n--\n\nNatChoiceDesc : Desc \u22a4\nNatChoiceDesc = choice 2 \u03bb\n  { zero -> ret tt\n  ; (suc n) -> rec tt (ret tt)\n  }\n\nNatChoice : Set\nNatChoice = Data NatChoiceDesc tt\n\nzeChoice : NatChoice\nzeChoice = \u27e8 zero , refl \u27e9\n\nsuChoice : NatChoice -> NatChoice\nsuChoice n = \u27e8 suc zero , n , refl \u27e9\n\n--\n\nNatDesc : Desc \u22a4\nNatDesc = arg Bool \u03bb\n  { true -> ret tt\n  ; false -> rec tt (ret tt)\n  }\n\nNat : Set\nNat = Data NatDesc tt\n\nze : Nat\nze = \u27e8 true , refl \u27e9\n\nsu : Nat -> Nat\nsu n = \u27e8 false , n , refl \u27e9\n\n--\n\nVecDesc : Set -> Desc Nat\nVecDesc X = arg Bool \u03bb\n  { false -> ret ze\n  ; true -> arg X \u03bb _ -> arg Nat \u03bb n -> rec n (ret (su n))\n  }\n\nVec : Set -> Nat -> Set\nVec X n = Data (VecDesc X) n\n\nnil : \u2200{X} -> Vec X ze\nnil = \u27e8 false , refl \u27e9\n\ncons : \u2200{X} -> (n : Nat) -> X -> Vec X n -> Vec X (su n)\ncons n x xs = \u27e8 true , x , n , xs , refl \u27e9\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Ornament where\n\nopen import Data.Nat using (\u2115)\nopen import Data.Fin \nopen import Data.Bool\nopen import Data.Product\nopen import Data.Unit\n\ninfix 4 _\u2261_\n\ndata _\u2261_ {a} {A : Set a} (x : A) : A \u2192 Set a where\n  refl : x \u2261 x\n\n--\n\n-- Modified from Conor's original HOAS presentation\n\ndata Desc (I : Set) : Set\u2081 where\n  -- read field in A; continue, given its value\n  -- I don't think this is really what we want -- want parametricity\n  arg : (a : Set) -> (a -> Desc I) -> Desc I\n\n  -- read an indexed subnode; continue regardless\n  rec : I -> Desc I -> Desc I\n\n  -- stop reading and return the node\u2019s index\n  ret : I -> Desc I\n\n  -- choice between n different constructors\n  choice : (n : \u2115) -> (Fin n -> Desc I) -> Desc I\n\n-- The semantics (\"extension\") of an indexed container\n\u27e6_\u27e7 : \u2200{I} -> Desc I -> (I -> Set) -> I -> Set\n\u27e6 arg A D \u27e7 R i = \u03a3 A \u03bb a \u2192 \u27e6 D a \u27e7 R i\n\u27e6 rec h D \u27e7 R i = R h \u00d7 \u27e6 D \u27e7 R i\n\u27e6 ret o \u27e7 R i = o \u2261 i\n\u27e6 choice n D \u27e7 R i =  \u03a3 (Fin n) \u03bb #n -> \u27e6 D #n \u27e7 R i\n\n\n-- This is really the least fixpoint operator\ndata Data {I}(D : Desc I)(i : I) : Set where\n  \u27e8_\u27e9 : \u27e6 D \u27e7 (Data D) i -> Data D i\n\n-- TODO think about codata \/ greatest fixed point\n-- data CoData {I}(D : Desc I)(i : I) : Set where\n--   \u27e9_\u27e8 : \n\n--\n\nNatChoiceDesc : Desc \u22a4\nNatChoiceDesc = choice 2 \u03bb\n  { zero -> ret tt\n  ; (suc n) -> rec tt (ret tt)\n  }\n\nNatChoice : Set\nNatChoice = Data NatChoiceDesc tt\n\nzeChoice : NatChoice\nzeChoice = \u27e8 zero , refl \u27e9\n\nsuChoice : NatChoice -> NatChoice\nsuChoice n = \u27e8 suc zero , n , refl \u27e9\n\n--\n\nNatDesc : Desc \u22a4\nNatDesc = arg Bool \u03bb\n  { true -> ret tt\n  ; false -> rec tt (ret tt)\n  }\n\nNat : Set\nNat = Data NatDesc tt\n\nze : Nat\nze = \u27e8 true , refl \u27e9\n\nsu : Nat -> Nat\nsu n = \u27e8 false , n , refl \u27e9\n\n--\n\nVecDesc : Set -> Desc Nat\nVecDesc X = arg Bool \u03bb\n  { false -> ret ze\n  ; true -> arg X \u03bb _ -> arg Nat \u03bb n -> rec n (ret (su n))\n  }\n\nVec : Set -> Nat -> Set\nVec X n = Data (VecDesc X) n\n\nnil : \u2200{X} -> Vec X ze\nnil = \u27e8 false , refl \u27e9\n\ncons : \u2200{X} -> (n : Nat) -> X -> Vec X n -> Vec X (su n)\ncons n x xs = \u27e8 true , x , n , xs , refl \u27e9\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a48c9114f79c1e6b68ea775581b1599a7c4a9256","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Predicates.agda","new_file":"notes\/fixed-points\/Predicates.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC types without use data, i.e. using Agda as a logical framework\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Predicates where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- The existential proyections.\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using postulates.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N     : D \u2192 Set\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = nsucc nzero\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    lnil  : List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    lnnil  : ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = lncons nzero (lncons oneN lnnil)\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = nsucc nzero\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    lnil  :                      List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    lnnil  :                             ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = lncons nzero (lncons oneN lnnil)\n\n------------------------------------------------------------------------------\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined types.\n    --\n    -- N.B. We cannot write LFP in first-order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n-- The greatest fixed-point operator.\n\n-- N.B. At the moment, the definitions of LFP and GFP are the same.\n\nmodule GFP where\n\n  postulate\n    -- Greatest fixed-points correspond to coinductively defined types.\n\n    GFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    GFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 GFP f d \u2192 f (GFP f) d\n    GFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (GFP f) d \u2192 GFP f d\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero      = \u22a4\n  -- NatF X (succ\u2081 n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  nsucc : {n : D} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc {n} Nn = LFP\u2082 NatF (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = nsucc nzero\n\n------------------------------------------------------------------------------\n-- The FOTC list type as the least fixed-point of a functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  lnil : List []\n  lnil = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  lcons x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  lnnil : ListN []\n  lnnil = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  lncons {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = lncons nzero (lncons oneN lnnil)\n\n------------------------------------------------------------------------------\n-- The FOTC Colist type as the greatest fixed-point of a functor.\n\nmodule CoList where\n\n  open GFP\n\n  -- Functor for the FOTC Colists type (the same functor that for the\n  -- List type).\n  ColistF : (D \u2192 Set) \u2192 D \u2192 Set\n  ColistF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC Colist type using GFP.\n  Colist : D \u2192 Set\n  Colist = GFP ColistF\n\n  -- The data constructors of Colist.\n  nilCL : Colist []\n  nilCL = GFP\u2082 ColistF [] (inj\u2081 refl)\n\n  consCL : \u2200 x xs \u2192 Colist xs \u2192 Colist (x \u2237 xs)\n  consCL x xs CLxs = GFP\u2082 ColistF (x \u2237 xs) (inj\u2082 (x , xs , refl , CLxs))\n\n  -- Example (finite colist).\n  l : Colist (zero \u2237 true \u2237 [])\n  l = consCL zero (true \u2237 []) (consCL true [] nilCL)\n\n  -- TODO: Example (infinite colist).\n  -- zerosCL : Colist {!!}\n  -- zerosCL = consCL zero {!!} zerosCL\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2081 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 X ds\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- Using StreamF we cannot define this data constructor\n  -- consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  -- consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) {!!}\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2082 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} S = GFP\u2082 StreamF xs (x , x \u2237 xs , S)\n\n  -- The functor StreamF does not link together the parts of the\n  -- stream, so I can get a stream from any stream.\n  bad : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys\n  bad {xs} {ys} S = GFP\u2082 StreamF ys (ys , xs , S)\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2083 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , refl , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} Sx\u2237xs = Sxs\n    where\n    unfoldS : StreamF (GFP StreamF) (x \u2237 xs)\n    unfoldS = GFP\u2081 StreamF (x \u2237 xs) Sx\u2237xs\n\n    e : D\n    e = \u2203-proj\u2081 unfoldS\n\n    Pe : \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pe = \u2203-proj\u2082 unfoldS\n\n    es : D\n    es = \u2203-proj\u2081 Pe\n\n    Pes : x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pes = \u2203-proj\u2082 Pe\n\n    xs\u2261es : xs \u2261 es\n    xs\u2261es = \u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2081 Pes))\n\n    Sxs : GFP StreamF xs\n    Sxs = subst (GFP StreamF) (sym xs\u2261es) (\u2227-proj\u2082 Pes)\n","old_contents":"------------------------------------------------------------------------------\n-- The FOTC types without use data, i.e. using Agda as a logical framework\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Predicates where\n\nopen import FOTC.Base\n\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- The existential proyections.\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using postulates.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N     : D \u2192 Set\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = nsucc nzero\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    lnil  : List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    lnnil  : ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = lncons nzero (lncons oneN lnnil)\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = nsucc nzero\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    lnil  :                      List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    lnnil  :                             ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = lncons nzero (lncons oneN lnnil)\n\n------------------------------------------------------------------------------\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined types.\n    --\n    -- N.B. We cannot write LFP in first-order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n-- The greatest fixed-point operator.\n\n-- N.B. At the moment, the definitions of LFP and GFP are the same.\n\nmodule GFP where\n\n  postulate\n    -- Greatest fixed-points correspond to coinductively defined types.\n\n    GFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    GFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 GFP f d \u2192 f (GFP f) d\n    GFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (GFP f) d \u2192 GFP f d\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero     = \u22a4\n  -- NatF X (succ\u2081 n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  nsucc : {n : D} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc {n} Nn = LFP\u2082 NatF (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  one : D\n  one = succ\u2081 zero\n\n  oneN : N one\n  oneN = nsucc nzero\n\n------------------------------------------------------------------------------\n-- The FOTC list type as the least fixed-point of a functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  lnil : List []\n  lnil = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  lcons x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  lnnil : ListN []\n  lnnil = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  lncons {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = lncons nzero (lncons oneN lnnil)\n\n------------------------------------------------------------------------------\n-- The FOTC Colist type as the greatest fixed-point of a functor.\n\nmodule CoList where\n\n  open GFP\n\n  -- Functor for the FOTC Colists type (the same functor that for the\n  -- List type).\n  ColistF : (D \u2192 Set) \u2192 D \u2192 Set\n  ColistF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC Colist type using GFP.\n  Colist : D \u2192 Set\n  Colist = GFP ColistF\n\n  -- The data constructors of Colist.\n  nilCL : Colist []\n  nilCL = GFP\u2082 ColistF [] (inj\u2081 refl)\n\n  consCL : \u2200 x xs \u2192 Colist xs \u2192 Colist (x \u2237 xs)\n  consCL x xs CLxs = GFP\u2082 ColistF (x \u2237 xs) (inj\u2082 (x , xs , refl , CLxs))\n\n  -- Example (finite colist).\n  l : Colist (zero \u2237 true \u2237 [])\n  l = consCL zero (true \u2237 []) (consCL true [] nilCL)\n\n  -- TODO: Example (infinite colist).\n  -- zerosCL : Colist {!!}\n  -- zerosCL = consCL zero {!!} zerosCL\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2081 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 X ds\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- Using StreamF we cannot define this data constructor\n  -- consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  -- consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) {!!}\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2082 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} S = GFP\u2082 StreamF xs (x , x \u2237 xs , S)\n\n  -- The functor StreamF does not link together the parts of the\n  -- stream, so I can get a stream from any stream.\n  bad : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys\n  bad {xs} {ys} S = GFP\u2082 StreamF ys (ys , xs , S)\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2083 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , refl , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} Sx\u2237xs = Sxs\n    where\n    unfoldS : StreamF (GFP StreamF) (x \u2237 xs)\n    unfoldS = GFP\u2081 StreamF (x \u2237 xs) Sx\u2237xs\n\n    e : D\n    e = \u2203-proj\u2081 unfoldS\n\n    Pe : \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pe = \u2203-proj\u2082 unfoldS\n\n    es : D\n    es = \u2203-proj\u2081 Pe\n\n    Pes : x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pes = \u2203-proj\u2082 Pe\n\n    xs\u2261es : xs \u2261 es\n    xs\u2261es = \u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2081 Pes))\n\n    Sxs : GFP StreamF xs\n    Sxs = subst (GFP StreamF) (sym xs\u2261es) (\u2227-proj\u2082 Pes)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"173e4a9626ae597164c134e37887fd538d81e574","subject":"Updated note on Leibniz's equality.","message":"Updated note on Leibniz's equality.\n\nIgnore-this: fcd2ac13bb245948e504e266cb17d380\n\ndarcs-hash:20120514131953-3bd4e-ea5dcb248c4efd0a6381518bc66d8209a3014571.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/LeibnizEquality.agda","new_file":"notes\/setoids\/LeibnizEquality.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 14 May 2012.\n\nmodule LeibnizEquality where\n\n-- References:\n--\n-- \u2022 Martin Hofmman. Extensional concepts in intensional type\n--   theory. PhD thesis, University of Edinburgh, 1995.\n\n-- \u2022 Zhaohui Luo. Computation and Reasoning. A Type Theory for\n--   Computer Science. Oxford University Press, 1994.\n\n------------------------------------------------------------------------------\n-- The identity type.\ndata _\u2261_ {A : Set}(x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\n-- Leibniz equality (see [Luo 1994, sec. 5.1.3])\n\n-- (From Agda\/examples\/lib\/Logic\/Leibniz.agda)\n_\u2250_ : {A : Set} \u2192 A \u2192 A \u2192 Set\u2081\nx \u2250 y = (P : _ \u2192 Set) \u2192 P x \u2192 P y\n\n-- Properties\n\u2250-refl : {A : Set}{x : A} \u2192 x \u2250 x\n\u2250-refl P Px = Px\n\n\u2250-sym : {A : Set}{x y : A} \u2192 x \u2250 y \u2192 y \u2250 x\n\u2250-sym {x = x} {y} h P Py = h (\u03bb z \u2192 P z \u2192 P x) (\u03bb Px \u2192 Px) Py\n\n\u2250-trans : {A : Set}{x y z : A} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n\u2250-trans h\u2081 h\u2082 P Px = h\u2082 P (h\u2081 P Px)\n\n\u2250-subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2250 y \u2192 P x \u2192 P y\n\u2250-subst P h = h P\n\n------------------------------------------------------------------------------\n-- Leibniz's equality and the identity type\n\n-- \"In the presence of a type of propositions \"Prop\" one can also\n-- define propositional equality by Leibniz's principle.\" [Hofmman\n-- 1995, p. 4]\n\n\u2250\u2192\u2261 : {A : Set}{x y : A} \u2192 x \u2250 y \u2192 x \u2261 y\n\u2250\u2192\u2261 {x = x} h = h (\u03bb z \u2192 x \u2261 z) refl\n\n\u2261\u2192\u2250 : {A : Set}{x y : A} \u2192 x \u2261 y \u2192 x \u2250 y\n\u2261\u2192\u2250 refl P Px = Px\n","old_contents":"-- Tested with Agda 2.2.11 on 18 November 2011.\n\nmodule LeibnizEquality where\n\n------------------------------------------------------------------------------\n-- The identity type\ndata _\u2261_ {A : Set}(x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\n-- Leibniz equality (see [1, sec. 5.1.3])\n\n-- [1] Zhaohui Luo. Computation and Reasoning. A Type Theory for\n--     Computer Science. Oxford University Press, 1994.\n\n-- (From Agda\/examples\/lib\/Logic\/Leibniz.agda)\n_\u2250_ : {A : Set} \u2192 A \u2192 A \u2192 Set\u2081\nx \u2250 y = (P : _ \u2192 Set) \u2192 P x \u2192 P y\n\n-- Properties\n\u2250-refl : {A : Set}{x : A} \u2192 x \u2250 x\n\u2250-refl P Px = Px\n\n\u2250-sym : {A : Set}(x y : A) \u2192 x \u2250 y \u2192 y \u2250 x\n\u2250-sym x y x\u2250y P Py = x\u2250y (\u03bb z \u2192 P z \u2192 P x) (\u03bb Px \u2192 Px) Py\n\n\u2250-trans : {A : Set}{x y z : A} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n\u2250-trans x\u2250y y\u2250z P Px = y\u2250z P (x\u2250y P Px)\n\n\u2250-subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2250 y \u2192 P x \u2192 P y\n\u2250-subst P x\u2250y = x\u2250y P\n\n------------------------------------------------------------------------------\n-- Leibniz's equality and the identity type\n\n-- \"In the presence of a type of propositions \"Prop\" one can also\n-- define propositional equality by Leibniz's principle.\" [2, p. 4]\n\n-- [2] Martin Hofmman. Extensional concepts in intensional type\n--     theory. PhD thesis, University of Edinburgh, 1995\n\n\u2250\u2192\u2261 : {A : Set}{x y : A} \u2192 x \u2250 y \u2192 x \u2261 y\n\u2250\u2192\u2261 {x = x} x\u2250y = x\u2250y (\u03bb z \u2192 x \u2261 z) refl\n\n\u2261\u2192\u2250 : {A : Set}{x y : A} \u2192 x \u2261 y \u2192 x \u2250 y\n\u2261\u2192\u2250 refl P Px = Px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c8e906509576c3b43daa91ec7a1a908973e036e2","subject":"Updated issue #11.","message":"Updated issue #11.\n","repos":"asr\/apia,asr\/apia","old_file":"issues\/Issue11.agda","new_file":"issues\/Issue11.agda","new_contents":"------------------------------------------------------------------------------\n-- All parameters are required in an ATP definition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issue11 where\n\npostulate\n  D          : Set\n  false true : D\n\ndata Bool : D \u2192 Set where\n  btrue  : Bool true\n  bfalse : Bool false\n\nOkBit : D \u2192 Set\nOkBit b = Bool b\n{-# ATP definition OkBit #-}\n\npostulate foo : \u2200 b \u2192 OkBit b \u2192 OkBit b\n{-# ATP prove foo #-}\n\nWrongBit : D \u2192 Set\nWrongBit = Bool\n{-# ATP definition WrongBit #-}\n\npostulate bar : \u2200 b \u2192 WrongBit b \u2192 WrongBit b\n{-# ATP prove bar #-}\n\n-- $ apia --check Issue11.agda\n-- apia: tptp4X found an error in the file \/tmp\/Issue11\/29-bar.tptp\n","old_contents":"------------------------------------------------------------------------------\n-- All parameters are required in an ATP definition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issue11 where\n\npostulate\n  D          : Set\n  false true : D\n\ndata Bool : D \u2192 Set where\n  btrue  : Bool true\n  bfalse : Bool false\n\nOkBit : D \u2192 Set\nOkBit b = Bool b\n{-# ATP definition OkBit #-}\n\npostulate foo : \u2200 b \u2192 OkBit b \u2192 OkBit b\n{-# ATP prove foo #-}\n\nWrongBit : D \u2192 Set\nWrongBit = Bool\n{-# ATP definition WrongBit #-}\n\npostulate bar : \u2200 b \u2192 WrongBit b \u2192 WrongBit b\n{-# ATP prove bar #-}\n\n-- $ apia Issue11.agda\n-- apia: tptp4X found an error in the file \/tmp\/Issue11\/29-bar.tptp\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bc1831c964dc87b1039049f5e54443b7b761ce0e","subject":"GroupHomomorphism","message":"GroupHomomorphism\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/GroupHomomorphism.agda","new_file":"lib\/Explore\/GroupHomomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.GroupHomomorphism where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP hiding (_\u2219_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc    : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)\n         (f : A \u2192 B)\n         (exploreA : Explore zero A)(f-homo : GroupHomomorphism GA GB f)\n         ([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n         (explore-ext : ExploreExt exploreA)\n         where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n  -}\n\n  {-\n   I had some problems with using the standard library definiton of Groups\n   so I rolled my own, therefor I need some boring proofs first\n\n  -}\n\n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n  module _ {X}(z : X)(op : X \u2192 X \u2192 X)\n           (sui : \u2200 k \u2192 StableUnder' exploreA z op (flip (Group._\u2219_ GA) k))\n    where\n  -- this proof isn't actually any hard..\n    thm : \u2200 (O : B \u2192 X) m\u2080 m\u2081 \u2192 exploreA z op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 exploreA z op (\u03bb x \u2192 O (f x * m\u2081))\n    thm O m\u2080 m\u2081 = explore (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma1 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) (O \u2218 f) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma2 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n     where\n      open \u2261-Reasoning\n      explore = exploreA z op\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Explore.GroupHomomorphism where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP hiding (_\u2219_)\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc    : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)(f : A \u2192 B)(exploreA : Explore zero A)(f-homo : GroupHomomorphism GA GB f)([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n              (sui : \u2200 k \u2192 StableUnder exploreA (flip (Group._\u2219_ GA) k))(explore-ext : ExploreExt exploreA) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n  -}\n\n  {-\n   I had some problems with using the standard library definiton of Groups\n   so I rolled my own, therefor I need some boring proofs first\n\n  -}\n\n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n  -- this proof isn't actually any hard..\n  thm : \u2200 {X}(z : X)(op : X \u2192 X \u2192 X)(O : B \u2192 X) m\u2080 m\u2081 \u2192 exploreA z op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 exploreA z op (\u03bb x \u2192 O (f x * m\u2081))\n  thm z op O m\u2080 m\u2081 = explore (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) z op (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma1 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) z op (O \u2218 f) \u27e9\n                   explore (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 explore-ext z op (\u03bb x \u2192 cong O (lemma2 x)) \u27e9\n                   explore (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n    where\n      open \u2261-Reasoning\n      explore = exploreA z op\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fc6bafb81826aa9f71cf7dc2cd0ef72422b499db","subject":"ASL made Induction.* more universe-polymorphic.","message":"ASL made Induction.* more universe-polymorphic.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/QuickSort\/QuickSortBC.agda","new_file":"notes\/FOT\/FOTC\/Program\/QuickSort\/QuickSortBC.agda","new_contents":"------------------------------------------------------------------------------\n-- Quicksort using the Bove-Capretta method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}  -- No accepted!\n\nmodule FOT.FOTC.Program.QuickSort.QuickSortBC where\n\nopen import Data.Bool\nopen import Data.List\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Sum\n\nopen import Induction\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nmodule InvImg =\n  Induction.WellFounded.Inverse-image {A = List \u2115} {\u2115} {_<\u2032_} length\n\n------------------------------------------------------------------------------\n\n_\u2264\u2032?_ : Decidable _\u2264\u2032_\nm \u2264\u2032? n with m \u2264? n\n... | yes p = yes (\u2264\u21d2\u2264\u2032 p)\n... | no \u00acp = no (\u03bb m\u2264\u2032n \u2192 \u00acp (\u2264\u2032\u21d2\u2264 m\u2264\u2032n))\n\n-- Non-terminating quicksort.\n\n{-# NO_TERMINATION_CHECK #-}\nqsNT : List \u2115 \u2192 List \u2115\nqsNT []       = []\nqsNT (x \u2237 xs) = qsNT (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) ++\n                x \u2237 qsNT (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)\n\n-- Domain predicate for quicksort.\ndata QSDom : List \u2115 \u2192 Set where\n  qsDom-[] : QSDom []\n  qsDom-\u2237  : \u2200 {x xs} \u2192\n             (QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             (QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             QSDom (x \u2237 xs)\n\n-- Induction principle associated to the domain predicate of quicksort.\n-- (It was not necessary).\nindQSDom : (P : List \u2115 \u2192 Set) \u2192\n           P [] \u2192\n           (\u2200 {x xs} \u2192 QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (x \u2237 xs)) \u2192\n           (\u2200 {xs} \u2192 QSDom xs \u2192 P xs)\nindQSDom P P[] ih qsDom-[]        = P[]\nindQSDom P P[] ih (qsDom-\u2237 h\u2081 h\u2082) =\n  ih h\u2081 (indQSDom P P[] ih h\u2081) h\u2082 (indQSDom P P[] ih h\u2082)\n\n-- Well-founded relation on lists.\n_\u27ea\u2032_ : {A : Set} \u2192 List A \u2192 List A \u2192 Set\nxs \u27ea\u2032 ys = length xs <\u2032 length ys\n\nwf-\u27ea\u2032 : Well-founded _\u27ea\u2032_\nwf-\u27ea\u2032 = InvImg.well-founded <-well-founded\n\n-- The well-founded induction principle on _\u27ea\u2032_.\n-- postulate wfi-\u27ea\u2032 : (P : List \u2115 \u2192 Set) \u2192\n--                    (\u2200 xs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 xs \u2192 P ys) \u2192 P xs) \u2192\n--                    \u2200 xs \u2192 P xs\n\n-- The quicksort algorithm is total.\n\nfilter-length : {A : Set}(P : A \u2192 Bool) \u2192 \u2200 xs \u2192\n                length (filter P xs) \u2264\u2032 length xs\nfilter-length P []       = \u2264\u2032-refl\nfilter-length P (x \u2237 xs) with P x\n... | true  = \u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length P xs)))\n... | false = \u2264\u2032-step (filter-length P xs)\n\nmodule AllWF = Induction.WellFounded.All wf-\u27ea\u2032\n\nallQSDom : \u2200 xs \u2192 QSDom xs\n-- If we use wfi-\u27ea\u2032 then allQSDom =  wfi-\u27ea\u2032 P ih\nallQSDom = build (AllWF.wfRec-builder _) P ih\n  where\n  P : List \u2115 \u2192 Set\n  P = QSDom\n\n  -- If we use wfi-\u27ea\u2032 then\n  -- ih : \u2200 zs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 zs \u2192 P ys) \u2192 P zs\n  ih :  \u2200 zs \u2192 WfRec _\u27ea\u2032_ P zs \u2192 P zs\n  ih []       h = qsDom-[]\n  ih (z \u2237 zs) h = qsDom-\u2237 prf\u2081 prf\u2082\n    where\n    c\u2081 : \u2115 \u2192 Bool\n    c\u2081 = \u03bb y \u2192 \u230a y \u2264\u2032? z \u230b\n\n    c\u2082 : \u2115 \u2192 Bool\n    c\u2082 = \u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b\n\n    f\u2081 : List \u2115\n    f\u2081 = filter c\u2081 zs\n\n    f\u2082 : List \u2115\n    f\u2082 = filter c\u2082 zs\n\n    prf\u2081 : QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2081 = h f\u2081 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2081 zs))))\n\n    prf\u2082 : QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2082 = h f\u2082 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2082 zs))))\n\n-- Quicksort algorithm by structural recursion on the domain predicate.\nqsDom : \u2200 xs \u2192 QSDom xs \u2192 List \u2115\nqsDom .[]      qsDom-[]        = []\nqsDom (x \u2237 xs) (qsDom-\u2237 h\u2081 h\u2082) =\n  qsDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) h\u2081 ++\n  x \u2237 qsDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) h\u2082\n\n-- The quicksort algorithm.\nqs : List \u2115 \u2192 List \u2115\nqs xs = qsDom xs (allQSDom xs)\n\n-- Testing.\nl\u2081 : List \u2115\nl\u2081 = []\nl\u2082 = 1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 []\nl\u2083 = 5 \u2237 4 \u2237 3 \u2237 2 \u2237 1 \u2237 []\nl\u2084 = 4 \u2237 1 \u2237 3 \u2237 5 \u2237 2 \u2237 []\n\nt\u2081 = qs l\u2081\nt\u2082 = qs l\u2082\nt\u2083 = qs l\u2083\nt\u2084 = qs l\u2084\n","old_contents":"------------------------------------------------------------------------------\n-- Quicksort using the Bove-Capretta method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}  -- No accepted!\n\nmodule FOT.FOTC.Program.QuickSort.QuickSortBC where\n\nopen import Data.Bool\nopen import Data.List\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Sum\n\nopen import Induction\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nmodule InvImg =\n  Induction.WellFounded.Inverse-image {A = List \u2115} {\u2115} {_<\u2032_} length\n\n------------------------------------------------------------------------------\n\n_\u2264\u2032?_ : Decidable _\u2264\u2032_\nm \u2264\u2032? n with m \u2264? n\n... | yes p = yes (\u2264\u21d2\u2264\u2032 p)\n... | no \u00acp = no (\u03bb m\u2264\u2032n \u2192 \u00acp (\u2264\u2032\u21d2\u2264 m\u2264\u2032n))\n\n-- Non-terminating quicksort.\n\n{-# NO_TERMINATION_CHECK #-}\nqsNT : List \u2115 \u2192 List \u2115\nqsNT []       = []\nqsNT (x \u2237 xs) = qsNT (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) ++\n                x \u2237 qsNT (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)\n\n-- Domain predicate for quicksort.\ndata QSDom : List \u2115 \u2192 Set where\n  qsDom-[] : QSDom []\n  qsDom-\u2237  : \u2200 {x xs} \u2192\n             (QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             (QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             QSDom (x \u2237 xs)\n\n-- Induction principle associated to the domain predicate of quicksort.\n-- (It was not necessary).\nindQSDom : (P : List \u2115 \u2192 Set) \u2192\n           P [] \u2192\n           (\u2200 {x xs} \u2192 QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (x \u2237 xs)) \u2192\n           (\u2200 {xs} \u2192 QSDom xs \u2192 P xs)\nindQSDom P P[] ih qsDom-[]        = P[]\nindQSDom P P[] ih (qsDom-\u2237 h\u2081 h\u2082) =\n  ih h\u2081 (indQSDom P P[] ih h\u2081) h\u2082 (indQSDom P P[] ih h\u2082)\n\n-- Well-founded relation on lists.\n_\u27ea\u2032_ : {A : Set} \u2192 List A \u2192 List A \u2192 Set\nxs \u27ea\u2032 ys = length xs <\u2032 length ys\n\nwf-\u27ea\u2032 : Well-founded _\u27ea\u2032_\nwf-\u27ea\u2032 = InvImg.well-founded <-well-founded\n\n-- The well-founded induction principle on _\u27ea\u2032_.\n-- postulate wfi-\u27ea\u2032 : (P : List \u2115 \u2192 Set) \u2192\n--                    (\u2200 xs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 xs \u2192 P ys) \u2192 P xs) \u2192\n--                    \u2200 xs \u2192 P xs\n\n-- The quicksort algorithm is total.\n\nfilter-length : {A : Set}(P : A \u2192 Bool) \u2192 \u2200 xs \u2192\n                length (filter P xs) \u2264\u2032 length xs\nfilter-length P []       = \u2264\u2032-refl\nfilter-length P (x \u2237 xs) with P x\n... | true  = \u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length P xs)))\n... | false = \u2264\u2032-step (filter-length P xs)\n\nmodule AllWF = Induction.WellFounded.All wf-\u27ea\u2032\n\nallQSDom : \u2200 xs \u2192 QSDom xs\n-- If we use wfi-\u27ea\u2032 then allQSDom =  wfi-\u27ea\u2032 P ih\nallQSDom = build AllWF.wfRec-builder P ih\n  where\n  P : List \u2115 \u2192 Set\n  P = QSDom\n\n  -- If we use wfi-\u27ea\u2032 then\n  -- ih : \u2200 zs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 zs \u2192 P ys) \u2192 P zs\n  ih :  \u2200 zs \u2192 WfRec _\u27ea\u2032_ P zs \u2192 P zs\n  ih []       h = qsDom-[]\n  ih (z \u2237 zs) h = qsDom-\u2237 prf\u2081 prf\u2082\n    where\n    c\u2081 : \u2115 \u2192 Bool\n    c\u2081 = \u03bb y \u2192 \u230a y \u2264\u2032? z \u230b\n\n    c\u2082 : \u2115 \u2192 Bool\n    c\u2082 = \u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b\n\n    f\u2081 : List \u2115\n    f\u2081 = filter c\u2081 zs\n\n    f\u2082 : List \u2115\n    f\u2082 = filter c\u2082 zs\n\n    prf\u2081 : QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2081 = h f\u2081 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2081 zs))))\n\n    prf\u2082 : QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2082 = h f\u2082 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2082 zs))))\n\n-- Quicksort algorithm by structural recursion on the domain predicate.\nqsDom : \u2200 xs \u2192 QSDom xs \u2192 List \u2115\nqsDom .[]      qsDom-[]        = []\nqsDom (x \u2237 xs) (qsDom-\u2237 h\u2081 h\u2082) =\n  qsDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) h\u2081 ++\n  x \u2237 qsDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) h\u2082\n\n-- The quicksort algorithm.\nqs : List \u2115 \u2192 List \u2115\nqs xs = qsDom xs (allQSDom xs)\n\n-- Testing.\nl\u2081 : List \u2115\nl\u2081 = []\nl\u2082 = 1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 []\nl\u2083 = 5 \u2237 4 \u2237 3 \u2237 2 \u2237 1 \u2237 []\nl\u2084 = 4 \u2237 1 \u2237 3 \u2237 5 \u2237 2 \u2237 []\n\nt\u2081 = qs l\u2081\nt\u2082 = qs l\u2082\nt\u2083 = qs l\u2083\nt\u2084 = qs l\u2084\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24faa2c5edd91d1db64977d5124ef40e01de433f","subject":"Show how badly reasoning for \u2259 and \u2261 combine","message":"Show how badly reasoning for \u2259 and \u2261 combine\n\n\u2261\u27e8 ... \u27e9 is also available inside \u2259-Reasoning, but it allows using\npropositional equalities on the changes; while that's certainly correct,\nit's not very useful. I'd like to be able to use equalities on base\nvalues.\n\nHowever, improving on might require invasive changes to EqReasoning.\n\nOld-commit-hash: a64324daefc3c0f33c6d7eec0da95f5d16ba9ff7\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    trans\n      (proof (let open \u2259-Reasoning in\n        begin\n          dx\n        \u2259\u27e8 dx\u2259nil-x \u27e9\n          nil x\n        \u220e))\n      (let open \u2261-Reasoning in\n        begin\n          x \u229e (nil x)\n        \u2261\u27e8 update-nil x \u27e9\n          x\n        \u220e)\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4715eafa925674207d26c157c261b83f9d558eee","subject":"+cancel-*-left","message":"+cancel-*-left\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"64c7aa44ab0d86c66f744241c812cc27ab7fb2c6","subject":"Add TODO","message":"Add TODO\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Sums.agda","new_file":"Base\/Change\/Sums.agda","new_contents":"module Base.Change.Sums where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\nopen import Base.Change.Equivalence\nopen import Postulate.Extensionality\nopen import Data.Sum\n\nmodule SumChanges \u2113 (X Y : Set \u2113) {{CX : ChangeAlgebra \u2113 X}} {{CY : ChangeAlgebra \u2113 Y}} where\n  open \u2261-Reasoning\n\n  -- This is an indexed datatype, so it has two constructors per argument. But\n  -- erasure would probably not be smart enough to notice.\n\n  -- Should we rewrite this as two separate datatypes?\n  data SumChange : X \u228e Y \u2192 Set \u2113 where\n    ch\u2081 : \u2200 {x} \u2192 (dx : \u0394 x) \u2192 SumChange (inj\u2081 x)\n    rp\u2081\u2082 : \u2200 {x} \u2192 (y : Y) \u2192 SumChange (inj\u2081 x)\n    ch\u2082 : \u2200 {y} \u2192 (dy : \u0394 y) \u2192 SumChange (inj\u2082 y)\n    rp\u2082\u2081 : \u2200 {y} \u2192 (x : X) \u2192 SumChange (inj\u2082 y)\n\n  _\u2295_ : (v : X \u228e Y) \u2192 SumChange v \u2192 X \u228e Y\n  inj\u2081 x \u2295 ch\u2081 dx = inj\u2081 (x \u229e dx)\n  inj\u2082 y \u2295 ch\u2082 dy = inj\u2082 (y \u229e dy)\n  inj\u2081 x \u2295 rp\u2081\u2082 y = inj\u2082 y\n  inj\u2082 y \u2295 rp\u2082\u2081 x = inj\u2081 x\n\n  _\u229d_ : \u2200 (v\u2082 v\u2081 : X \u228e Y) \u2192 SumChange v\u2081\n  inj\u2081 x\u2082 \u229d inj\u2081 x\u2081 = ch\u2081 (x\u2082 \u229f x\u2081)\n  inj\u2082 y\u2082 \u229d inj\u2082 y\u2081 = ch\u2082 (y\u2082 \u229f y\u2081)\n  inj\u2082 y\u2082 \u229d inj\u2081 x\u2081 = rp\u2081\u2082 y\u2082\n  inj\u2081 x\u2082 \u229d inj\u2082 y\u2081 = rp\u2082\u2081 x\u2082\n\n  s-nil : (v : X \u228e Y) \u2192 SumChange v\n  s-nil (inj\u2081 x) = ch\u2081 (nil x)\n  s-nil (inj\u2082 y) = ch\u2082 (nil y)\n\n  s-update-diff : \u2200 (u v : X \u228e Y) \u2192 v \u2295 (u \u229d v) \u2261 u\n  s-update-diff (inj\u2081 x\u2082) (inj\u2081 x\u2081) = cong inj\u2081 (update-diff x\u2082 x\u2081)\n  s-update-diff (inj\u2082 y\u2082) (inj\u2082 y\u2081) = cong inj\u2082 (update-diff y\u2082 y\u2081)\n  s-update-diff (inj\u2081 x\u2082) (inj\u2082 y\u2081) = refl\n  s-update-diff (inj\u2082 y\u2082) (inj\u2081 x\u2081) = refl\n\n  s-update-nil : \u2200 v \u2192 v \u2295 (s-nil v) \u2261 v\n  s-update-nil (inj\u2081 x) = cong inj\u2081 (update-nil x)\n  s-update-nil (inj\u2082 y) = cong inj\u2082 (update-nil y)\n\n  instance\n    changeAlgebra : ChangeAlgebra \u2113 (X \u228e Y)\n    changeAlgebra = record\n      { Change = SumChange\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = s-nil\n      ; isChangeAlgebra = record\n        { update-diff = s-update-diff\n        ; update-nil = s-update-nil\n        }\n      }\n\n  -- Encode infix ascription.\n  as' : \u2200 {\u2113} (A : Set \u2113) (a : A) \u2192 A\n  as' _ a = a\n  syntax as' A a = a as A\n\n  inj\u2081\u2032 : RawChange (inj\u2081 as (X \u2192 (X \u228e Y)))\n  inj\u2081\u2032 x dx = ch\u2081 dx\n\n  inj\u2081\u2032Derivative : IsDerivative (inj\u2081 as (X \u2192 (X \u228e Y))) inj\u2081\u2032\n  inj\u2081\u2032Derivative x dx = refl\n\n  inj\u2082\u2032 : RawChange (inj\u2082 as (Y \u2192 (X \u228e Y)))\n  inj\u2082\u2032 y dy = ch\u2082 dy\n\n  inj\u2082\u2032Derivative : IsDerivative (inj\u2082 as (Y \u2192 (X \u228e Y))) inj\u2082\u2032\n  inj\u2082\u2032Derivative y dy = refl\n\n  -- Elimination form for sums. This is a less dependently-typed version of\n  -- [_,_].\n  match : \u2200 {Z : Set \u2113} \u2192 (X \u2192 Z) \u2192 (Y \u2192 Z) \u2192 X \u228e Y \u2192 Z\n  match f g (inj\u2081 x) = f x\n  match f g (inj\u2082 y) = g y\n\n  module _ {Z : Set \u2113} {{CZ : ChangeAlgebra \u2113 Z}} where\n    instance\n      X\u2192Z = FunctionChanges.changeAlgebra X Z {{CX}} {{CZ}}\n      --module \u0394X\u2192Z = FunctionChanges X Z {{CX}} {{CZ}}\n      Y\u2192Z = FunctionChanges.changeAlgebra Y Z {{CY}} {{CZ}}\n      --module \u0394Y\u2192Z = FunctionChanges Y Z {{CY}} {{CZ}}\n      X\u228eY\u2192Z = FunctionChanges.changeAlgebra (X \u228e Y) Z {{changeAlgebra}} {{CZ}}\n      Y\u2192Z\u2192X\u228eY\u2192Z = FunctionChanges.changeAlgebra (Y \u2192 Z) (X \u228e Y \u2192 Z) {{Y\u2192Z}} {{X\u228eY\u2192Z}}\n\n    open FunctionChanges using (apply; correct)\n\n    match\u2032\u2080-realizer : (f : X \u2192 Z) \u2192 \u0394 f \u2192 (g : Y \u2192 Z) \u2192 \u0394 g \u2192 (s : X \u228e Y) \u2192 \u0394 s \u2192 \u0394 (match f g s)\n    match\u2032\u2080-realizer f df g dg (inj\u2081 x) (ch\u2081 dx) = apply df x dx\n    match\u2032\u2080-realizer f df g dg (inj\u2081 x) (rp\u2081\u2082 y) = ((g \u229e dg) y) \u229f (f x)\n    match\u2032\u2080-realizer f df g dg (inj\u2082 y) (rp\u2082\u2081 x) = ((f \u229e df) x) \u229f (g y)\n    match\u2032\u2080-realizer f df g dg (inj\u2082 y) (ch\u2082 dy) = apply dg y dy\n\n    match\u2032\u2080-realizer-correct : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192 (s : X \u228e Y) \u2192 (ds : \u0394 s) \u2192 match f g (s \u2295 ds) \u229e match\u2032\u2080-realizer f df g dg (s \u2295 ds) (nil (s \u2295 ds)) \u2261 match f g s \u229e match\u2032\u2080-realizer f df g dg s ds\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2081 x) (ch\u2081 dx) = correct df x dx\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2082 y) (ch\u2082 dy) = correct dg y dy\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2081 x) (rp\u2081\u2082 y) rewrite update-diff ((g \u229e dg) y) (f x) = refl\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2082 y) (rp\u2082\u2081 x) rewrite update-diff ((f \u229e df) x) (g y) = refl\n\n    match\u2032\u2080 : (f : X \u2192 Z) \u2192 \u0394 f \u2192 (g : Y \u2192 Z) \u2192 \u0394 g \u2192 \u0394 (match f g)\n    match\u2032\u2080 f df g dg = record { apply = match\u2032\u2080-realizer f df g dg ; correct = match\u2032\u2080-realizer-correct f df g dg }\n\n    match\u2032-realizer-correct-body : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192 (s : X \u228e Y) \u2192\n      (match f (g \u229e dg) \u229e match\u2032\u2080 f df (g \u229e dg) (nil (g \u229e dg))) s \u2261 (match f g \u229e match\u2032\u2080 f df g dg) s\n\n    match\u2032-realizer-correct-body f df g dg (inj\u2081 x) = refl\n    -- refl doesn't work here. That seems a *huge* bad smell. However, that's simply because we're only updating g, not f\n    match\u2032-realizer-correct-body f df g dg (inj\u2082 y) rewrite update-nil y = update-diff (g y \u229e apply dg y (nil y)) (g y \u229e apply dg y (nil y))\n\n    match\u2032-realizer-correct : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192\n      (match f (g \u229e dg)) \u229e match\u2032\u2080 f df (g \u229e dg) (nil (g \u229e dg)) \u2261 match f g \u229e match\u2032\u2080 f df g dg\n    match\u2032-realizer-correct f df g dg = ext (match\u2032-realizer-correct-body f df g dg)\n    match\u2032 : (f : X \u2192 Z) \u2192 \u0394 f \u2192 \u0394 (match f)\n    match\u2032 f df = record { apply = \u03bb g dg \u2192 match\u2032\u2080 f df g dg ; correct = match\u2032-realizer-correct f df }\n","old_contents":"module Base.Change.Sums where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\nopen import Base.Change.Equivalence\nopen import Postulate.Extensionality\nopen import Data.Sum\n\nmodule SumChanges \u2113 (X Y : Set \u2113) {{CX : ChangeAlgebra \u2113 X}} {{CY : ChangeAlgebra \u2113 Y}} where\n  open \u2261-Reasoning\n\n  data SumChange : X \u228e Y \u2192 Set \u2113 where\n    ch\u2081 : \u2200 {x} \u2192 (dx : \u0394 x) \u2192 SumChange (inj\u2081 x)\n    rp\u2081\u2082 : \u2200 {x} \u2192 (y : Y) \u2192 SumChange (inj\u2081 x)\n    ch\u2082 : \u2200 {y} \u2192 (dy : \u0394 y) \u2192 SumChange (inj\u2082 y)\n    rp\u2082\u2081 : \u2200 {y} \u2192 (x : X) \u2192 SumChange (inj\u2082 y)\n\n  _\u2295_ : (v : X \u228e Y) \u2192 SumChange v \u2192 X \u228e Y\n  inj\u2081 x \u2295 ch\u2081 dx = inj\u2081 (x \u229e dx)\n  inj\u2082 y \u2295 ch\u2082 dy = inj\u2082 (y \u229e dy)\n  inj\u2081 x \u2295 rp\u2081\u2082 y = inj\u2082 y\n  inj\u2082 y \u2295 rp\u2082\u2081 x = inj\u2081 x\n\n  _\u229d_ : \u2200 (v\u2082 v\u2081 : X \u228e Y) \u2192 SumChange v\u2081\n  inj\u2081 x\u2082 \u229d inj\u2081 x\u2081 = ch\u2081 (x\u2082 \u229f x\u2081)\n  inj\u2082 y\u2082 \u229d inj\u2082 y\u2081 = ch\u2082 (y\u2082 \u229f y\u2081)\n  inj\u2082 y\u2082 \u229d inj\u2081 x\u2081 = rp\u2081\u2082 y\u2082\n  inj\u2081 x\u2082 \u229d inj\u2082 y\u2081 = rp\u2082\u2081 x\u2082\n\n  s-nil : (v : X \u228e Y) \u2192 SumChange v\n  s-nil (inj\u2081 x) = ch\u2081 (nil x)\n  s-nil (inj\u2082 y) = ch\u2082 (nil y)\n\n  s-update-diff : \u2200 (u v : X \u228e Y) \u2192 v \u2295 (u \u229d v) \u2261 u\n  s-update-diff (inj\u2081 x\u2082) (inj\u2081 x\u2081) = cong inj\u2081 (update-diff x\u2082 x\u2081)\n  s-update-diff (inj\u2082 y\u2082) (inj\u2082 y\u2081) = cong inj\u2082 (update-diff y\u2082 y\u2081)\n  s-update-diff (inj\u2081 x\u2082) (inj\u2082 y\u2081) = refl\n  s-update-diff (inj\u2082 y\u2082) (inj\u2081 x\u2081) = refl\n\n  s-update-nil : \u2200 v \u2192 v \u2295 (s-nil v) \u2261 v\n  s-update-nil (inj\u2081 x) = cong inj\u2081 (update-nil x)\n  s-update-nil (inj\u2082 y) = cong inj\u2082 (update-nil y)\n\n  instance\n    changeAlgebra : ChangeAlgebra \u2113 (X \u228e Y)\n    changeAlgebra = record\n      { Change = SumChange\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = s-nil\n      ; isChangeAlgebra = record\n        { update-diff = s-update-diff\n        ; update-nil = s-update-nil\n        }\n      }\n\n  -- Encode infix ascription.\n  as' : \u2200 {\u2113} (A : Set \u2113) (a : A) \u2192 A\n  as' _ a = a\n  syntax as' A a = a as A\n\n  inj\u2081\u2032 : RawChange (inj\u2081 as (X \u2192 (X \u228e Y)))\n  inj\u2081\u2032 x dx = ch\u2081 dx\n\n  inj\u2081\u2032Derivative : IsDerivative (inj\u2081 as (X \u2192 (X \u228e Y))) inj\u2081\u2032\n  inj\u2081\u2032Derivative x dx = refl\n\n  inj\u2082\u2032 : RawChange (inj\u2082 as (Y \u2192 (X \u228e Y)))\n  inj\u2082\u2032 y dy = ch\u2082 dy\n\n  inj\u2082\u2032Derivative : IsDerivative (inj\u2082 as (Y \u2192 (X \u228e Y))) inj\u2082\u2032\n  inj\u2082\u2032Derivative y dy = refl\n\n  -- Elimination form for sums. This is a less dependently-typed version of\n  -- [_,_].\n  match : \u2200 {Z : Set \u2113} \u2192 (X \u2192 Z) \u2192 (Y \u2192 Z) \u2192 X \u228e Y \u2192 Z\n  match f g (inj\u2081 x) = f x\n  match f g (inj\u2082 y) = g y\n\n  module _ {Z : Set \u2113} {{CZ : ChangeAlgebra \u2113 Z}} where\n    instance\n      X\u2192Z = FunctionChanges.changeAlgebra X Z {{CX}} {{CZ}}\n      --module \u0394X\u2192Z = FunctionChanges X Z {{CX}} {{CZ}}\n      Y\u2192Z = FunctionChanges.changeAlgebra Y Z {{CY}} {{CZ}}\n      --module \u0394Y\u2192Z = FunctionChanges Y Z {{CY}} {{CZ}}\n      X\u228eY\u2192Z = FunctionChanges.changeAlgebra (X \u228e Y) Z {{changeAlgebra}} {{CZ}}\n      Y\u2192Z\u2192X\u228eY\u2192Z = FunctionChanges.changeAlgebra (Y \u2192 Z) (X \u228e Y \u2192 Z) {{Y\u2192Z}} {{X\u228eY\u2192Z}}\n\n    open FunctionChanges using (apply; correct)\n\n    match\u2032\u2080-realizer : (f : X \u2192 Z) \u2192 \u0394 f \u2192 (g : Y \u2192 Z) \u2192 \u0394 g \u2192 (s : X \u228e Y) \u2192 \u0394 s \u2192 \u0394 (match f g s)\n    match\u2032\u2080-realizer f df g dg (inj\u2081 x) (ch\u2081 dx) = apply df x dx\n    match\u2032\u2080-realizer f df g dg (inj\u2081 x) (rp\u2081\u2082 y) = ((g \u229e dg) y) \u229f (f x)\n    match\u2032\u2080-realizer f df g dg (inj\u2082 y) (rp\u2082\u2081 x) = ((f \u229e df) x) \u229f (g y)\n    match\u2032\u2080-realizer f df g dg (inj\u2082 y) (ch\u2082 dy) = apply dg y dy\n\n    match\u2032\u2080-realizer-correct : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192 (s : X \u228e Y) \u2192 (ds : \u0394 s) \u2192 match f g (s \u2295 ds) \u229e match\u2032\u2080-realizer f df g dg (s \u2295 ds) (nil (s \u2295 ds)) \u2261 match f g s \u229e match\u2032\u2080-realizer f df g dg s ds\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2081 x) (ch\u2081 dx) = correct df x dx\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2082 y) (ch\u2082 dy) = correct dg y dy\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2081 x) (rp\u2081\u2082 y) rewrite update-diff ((g \u229e dg) y) (f x) = refl\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2082 y) (rp\u2082\u2081 x) rewrite update-diff ((f \u229e df) x) (g y) = refl\n\n    match\u2032\u2080 : (f : X \u2192 Z) \u2192 \u0394 f \u2192 (g : Y \u2192 Z) \u2192 \u0394 g \u2192 \u0394 (match f g)\n    match\u2032\u2080 f df g dg = record { apply = match\u2032\u2080-realizer f df g dg ; correct = match\u2032\u2080-realizer-correct f df g dg }\n\n    match\u2032-realizer-correct-body : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192 (s : X \u228e Y) \u2192\n      (match f (g \u229e dg) \u229e match\u2032\u2080 f df (g \u229e dg) (nil (g \u229e dg))) s \u2261 (match f g \u229e match\u2032\u2080 f df g dg) s\n\n    match\u2032-realizer-correct-body f df g dg (inj\u2081 x) = refl\n    -- refl doesn't work here. That seems a *huge* bad smell. However, that's simply because we're only updating g, not f\n    match\u2032-realizer-correct-body f df g dg (inj\u2082 y) rewrite update-nil y = update-diff (g y \u229e apply dg y (nil y)) (g y \u229e apply dg y (nil y))\n\n    match\u2032-realizer-correct : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192\n      (match f (g \u229e dg)) \u229e match\u2032\u2080 f df (g \u229e dg) (nil (g \u229e dg)) \u2261 match f g \u229e match\u2032\u2080 f df g dg\n    match\u2032-realizer-correct f df g dg = ext (match\u2032-realizer-correct-body f df g dg)\n    match\u2032 : (f : X \u2192 Z) \u2192 \u0394 f \u2192 \u0394 (match f)\n    match\u2032 f df = record { apply = \u03bb g dg \u2192 match\u2032\u2080 f df g dg ; correct = match\u2032-realizer-correct f df }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"42f74a6f081d7a079dda2440b4889254c7fdd72d","subject":"Better name for some postulates.","message":"Better name for some postulates.\n\nIgnore-this: 559baa82891f607ac7f829552643731e\n\ndarcs-hash:20100518203545-3bd4e-904f353c6d18e69fdb2f57e345fd5a11a3d9a707.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sd\u22640 = \u22a5-elim prf\n  where\n    -- The proof uses the axiom true\u2260false\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n    -- trans (lt-SS (succ n) zero) (lt-S0 n)\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sd\u22640 = \u22a5-elim prf\n  where\n    -- The proof uses the axiom true\u2260false\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n    -- trans (lt-SS (succ n) zero) (lt-S0 n)\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = SSm\u2264Sn (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate SSm\u2264Sn : LE (succ m) n \u2192\n                       LE (succ (succ m)) (succ n)\n    {-# ATP prove SSm\u2264Sn #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e48eb5797665b1a59f94d0a0f57334e4a279f6cf","subject":"otp-sem-sec: extract out one-time-semantic-security","message":"otp-sem-sec: extract out one-time-semantic-security\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\nopen import program-distance\nopen import one-time-semantic-security\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess' = Guess prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess' = Guess prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6f8fc3f0dd5684daa2717cf2a6aabe315afc114a","subject":"Added iterate.","message":"Added iterate.\n\nIgnore-this: 5813b16ec543343dd4806321dc36a17\n\ndarcs-hash:20101020225528-3bd4e-c19649c2be9498c8fb6843cc75628926cf757117.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List.agda","new_file":"LTC\/Data\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\ninfixr 5 _++_\n\n------------------------------------------------------------------------------\n-- The LTC list type.\nopen import LTC.Data.List.Type public\n\n------------------------------------------------------------------------------\n-- Basic functions\n\npostulate\n  length    : D \u2192 D\n  length-[] :              length []       \u2261 zero\n  length-\u2237  : (d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192      []       ++ ds \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D) \u2192      map f []       \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  -- Behavior: rev xs ys = reverse xs ++ ys\n  rev    : D \u2192 D \u2192 D\n  rev-[] : (es : D) \u2192      rev []       es \u2261 es\n  rev-\u2237  : (d ds es : D) \u2192 rev (d \u2237 ds) es \u2261 rev ds (d \u2237 es)\n{-# ATP axiom rev-[] #-}\n{-# ATP axiom rev-\u2237 #-}\n\nreverse : D \u2192 D\nreverse ds = rev ds []\n{-# ATP definition reverse #-}\n\npostulate\n  replicate   : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192   replicate zero     d \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n-- Reducing lists\n\npostulate\n  foldr    : D \u2192 D \u2192 D \u2192 D\n  foldr-[] : (f n : D) \u2192      foldr f n []       \u2261 n\n  foldr-\u2237  : (f n d ds : D) \u2192 foldr f n (d \u2237 ds) \u2261 f \u2219 d \u2219 (foldr f n ds)\n{-# ATP axiom foldr-[] #-}\n{-# ATP axiom foldr-\u2237 #-}\n\n-- Building lists\n\npostulate\n  iterate    : D \u2192 D \u2192 D\n  iterate-eq : (f x : D) \u2192 iterate f x \u2261 x \u2237 iterate f (f \u2219 x)\n{-# ATP axiom iterate-eq #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\ninfixr 5 _++_\n\n------------------------------------------------------------------------------\n-- The LTC list type.\nopen import LTC.Data.List.Type public\n\n------------------------------------------------------------------------------\n-- Basic functions\n\npostulate\n  length    : D \u2192 D\n  length-[] :              length []       \u2261 zero\n  length-\u2237  : (d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192      []       ++ ds \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D) \u2192      map f []       \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  -- Behavior: rev xs ys = reverse xs ++ ys\n  rev    : D \u2192 D \u2192 D\n  rev-[] : (es : D) \u2192      rev []       es \u2261 es\n  rev-\u2237  : (d ds es : D) \u2192 rev (d \u2237 ds) es \u2261 rev ds (d \u2237 es)\n{-# ATP axiom rev-[] #-}\n{-# ATP axiom rev-\u2237 #-}\n\nreverse : D \u2192 D\nreverse ds = rev ds []\n{-# ATP definition reverse #-}\n\npostulate\n  replicate   : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192   replicate zero     d \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n-- Reducing lists\n\npostulate\n  foldr    : D \u2192 D \u2192 D \u2192 D\n  foldr-[] : (f n : D) \u2192      foldr f n []       \u2261 n\n  foldr-\u2237  : (f n d ds : D) \u2192 foldr f n (d \u2237 ds) \u2261 f \u2219 d \u2219 (foldr f n ds)\n{-# ATP axiom foldr-[] #-}\n{-# ATP axiom foldr-\u2237 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e6783fa18dc50982ee95f12ff34a3d1bd5923a46","subject":"absurd patterns","message":"absurd patterns\n\nSigned-off-by: Shou Ya <8a35a2db84f314be65c58cf56a320d595342f8d7@gmail.com>\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c4b2cbea561a7dcfa3d502581e319465f168d99f","subject":"universe: flip _*_ is better","message":"universe: flip _*_ is better\n","repos":"crypto-agda\/crypto-agda","old_file":"universe.agda","new_file":"universe.agda","new_contents":"module universe where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (\u03a3; _\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ (flip _*_)\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U t\nfold-U u\u2080 {T} uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U P = mk (`\u22a4 , {!!}) {!!} {!!} {!!}\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","old_contents":"module universe where\n\nimport Level as L\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ _*_\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U t\nfold-U u\u2080 {T} uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U = ?\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1df3b6faf282052bf39073d5836d0281bf26e7f9","subject":"Vec: add lemmas about \u203c,group,concat,map,\u229b","message":"Vec: add lemmas about \u203c,group,concat,map,\u229b\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (Type)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; _\u2219_; !_) renaming (refl to idp)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : Type a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 Type a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : Type a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Type a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Type a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id= : \u2200 {a n} {A : Type a}{f : A \u2192 A}\n             \u2192 f \u2257 id \u2192 map f \u2257 id {A = Vec A n}\n    map-id= f= []       = idp\n    map-id= f= (x \u2237 xs) = ap-\u2237 (f= x) (map-id= f= xs)\n\n    map-id : \u2200 {a n} {A : Type a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id = map-id= (\u03bb _ \u2192 idp)\n\n    map-\u2218= : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n               {f : B \u2192 C}{g : A \u2192 B}{h : A \u2192 C} \u2192\n               f \u2218 g \u2257 h \u2192\n               _\u2257_ {A = Vec A n} (map f \u2218 map g) (map h)\n    map-\u2218= fg= []       = idp\n    map-\u2218= fg= (x \u2237 xs) = ap-\u2237 (fg= x) (map-\u2218= fg= xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n              (f : B \u2192 C) (g : A \u2192 B) \u2192\n              _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g v = ! map-\u2218= (\u03bb _ \u2192 idp) v\n\n    map-ext : \u2200 {a b} {A : Type a} {B : Type b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : Type a}\n    (_\u2248_ : A \u2192 A \u2192 Type \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Type \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 Type _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : Type a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= idp idp = idp\n\nmodule With\u2261 {a}{A : Type a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                           ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Type a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Type a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Type a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\ncount\u1da0 : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : Type a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : Type a} where\n  count-++ : \u2200 {m n} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m b} {B : Type b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , idp = idp\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\n  vec= : \u2200 {n}{xs ys : Vec A n} \u2192 (\u2200 i \u2192 (xs \u203c i) \u2261 (ys \u203c i)) \u2192 xs \u2261 ys\n  vec= {xs = []}     {[]}     f= = idp\n  vec= {xs = x \u2237 xs} {y \u2237 ys} f= = ap-\u2237 (f= zero) (vec= (f= \u2218 suc))\n\n  concat-group : \u2200 m n (xs : Vec A (m * n)) \u2192 concat (group m n xs) \u2261 xs\n  concat-group zero    n [] = idp\n  concat-group (suc m) n xs rewrite concat-group m n (drop n xs) = take-drop-lem n xs\n\n  group-concat : \u2200 {m n}(xss : Vec (Vec A n) m) \u2192 group m n (concat xss) \u2261 xss\n  group-concat [] = idp\n  group-concat (xs \u2237 xss)\n    rewrite take-++ _ xs (concat xss)\n          | drop-++ _ xs (concat xss)\n          | group-concat xss\n          = idp\n\n  -- These are also in the stdlib as Data.Vec.Properties.lookup-morphism\n  \u203c-replicate : \u2200 {n}(i : Fin n)(x : A) \u2192 (replicate x \u203c i) \u2261 x\n  \u203c-replicate zero    = \u03bb _ \u2192 idp\n  \u203c-replicate (suc i) = \u203c-replicate i\n\nmodule _ {a b}{A : Type a}{B : Type b} where\n  \u203c-\u229b= : \u2200 {n} i {fs : Vec (A \u2192 B) n}{xs : Vec A n}{f x}\n           (fs= : (fs \u203c i) \u2261 f)\n           (xs= : (xs \u203c i) \u2261 x)\n         \u2192 (fs \u229b xs \u203c i) \u2261 f x\n  \u203c-\u229b= zero    {f \u2237 fs} {x \u2237 xs} f= xs= = ap\u2082 _$_ f= xs=\n  \u203c-\u229b= (suc i) {f \u2237 fs} {x \u2237 xs} f= xs= = \u203c-\u229b= i f= xs=\n\n  \u203c-\u229b : \u2200 {n} i (fs : Vec (A \u2192 B) n) (xs : Vec A n) \u2192\n          (fs \u229b xs \u203c i) \u2261 (fs \u203c i) (xs \u203c i)\n  \u203c-\u229b i fs xs = \u203c-\u229b= i idp idp\n\n  \u203c-map= : \u2200 {n} i (f : A \u2192 B) {xs : Vec A n}{x : A}\n             (xs= : (xs \u203c i) \u2261 x)\n           \u2192 (map f xs \u203c i) \u2261 f x\n  \u203c-map= i f = \u203c-\u229b= i (\u203c-replicate i f)\n\n  \u203c-map : \u2200 {n} i (f : A \u2192 B) (xs : Vec A n) \u2192\n            (map f xs \u203c i) \u2261 f (xs \u203c i)\n  \u203c-map i f xs = \u203c-map= i f idp\n\n  vuncurry : \u2200 {n}(f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\n  vuncurry f (x \u2237 xs) = f x xs\n\n  count-\u2218 : \u2200 {n}(f : A \u2192 B)(pred : B \u2192 Bool) \u2192\n              count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\n  count-\u2218 f pred [] = idp\n  count-\u2218 f pred (x \u2237 xs) with pred (f x)\n  ... | 1b rewrite count-\u2218 f pred xs = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                 | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                 | count-\u2218 f pred xs = idp\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (Type)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : Type a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 Type a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : Type a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Type a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Type a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : Type a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : Type a} {B : Type b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : Type a}\n    (_\u2248_ : A \u2192 A \u2192 Type \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Type \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 Type _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : Type a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= \u2261.refl \u2261.refl = \u2261.refl\n\nmodule With\u2261 {a}{A : Type a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Type a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Type a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Type a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Type a} {B : Type b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | 1b rewrite count-\u2218 f pred xs = idp\n... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : Type a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : Type a} where\n  count-++ : \u2200 {m n} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m b} {B : Type b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"21d4e01e40469d7582aa2dbbb48115b766ff89b9","subject":"universe.agda","message":"universe.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"universe.agda","new_file":"universe.agda","new_contents":"module universe where\n\nimport Level as L\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ _*_\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U t\nfold-U u\u2080 {T} uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U = ?\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","old_contents":"module universe where\n\nimport Level as L\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ _*_\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U\nfold-U {_} {T} u\u2080 uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U = ?\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8181add67909ab3852accd12e8b44969d05c3891","subject":"agda: start implementing object-level change manipulation (see #25)","message":"agda: start implementing object-level change manipulation (see #25)\n\nOld-commit-hash: 766e45dd4d6ce96f539297ae89703c6628adbdf2\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"762783160bf94700b8c5e5a53cff5271250234f8","subject":"Strategy: runT","message":"Strategy: runT\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Strategy.agda","new_file":"Control\/Strategy.agda","new_contents":"module Control.Strategy where\n\nopen import Function.NP\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f)\n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n\n  private\n    Transcript = List (\u03a3 Q R)\n  module TranscriptRun (Oracle : Transcript \u2192 \u03a0 Q R){A} where\n    runT : M A \u2192 Transcript \u2192 A \u00d7 Transcript\n    runT (ask q? cont) t = let r = Oracle t q? in runT (cont r) ((q? , r) \u2237 t)\n    runT (done x)      t = x , t\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Control.Strategy where\n\nopen import Function\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f)\n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e0b2a2036942b7b9010a4d1e45e94ed00a26742d","subject":"Rename Atlas-diff to diff-base.","message":"Rename Atlas-diff to diff-base.\n\nOld-commit-hash: 8940b78431520f22d625a114d12ce0faee05542c\n","repos":"inc-lc\/ilc-agda","old_file":"Atlas\/Change\/Term.agda","new_file":"Atlas\/Change\/Term.agda","new_contents":"module Atlas.Change.Term where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\n\n-- nil-changes\n\nnil-const : \u2200 {\u03b9 : Base} \u2192 Const  \u2205 (base (\u0394Base \u03b9))\nnil-const {\u03b9} = neutral {\u0394Base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9))\nnil-term {Bool}   = curriedConst (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = curriedConst (nil-const {Map \u03ba \u03b9})\n\n-- diff-term and apply-term\n\nopen import Parametric.Change.Term Const \u0394Base\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\ndiff-base : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))\ndiff-base {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\ndiff-base {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs diff-base) m\u2081 m\u2082)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\nAtlas-apply {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs Atlas-apply) m\u2081 m\u2082)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (\u0394Type \u03c4)\ndiff = app\u2082 (lift-diff diff-base Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply diff-base Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n","old_contents":"module Atlas.Change.Term where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\n\n-- nil-changes\n\nnil-const : \u2200 {\u03b9 : Base} \u2192 Const  \u2205 (base (\u0394Base \u03b9))\nnil-const {\u03b9} = neutral {\u0394Base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base (\u0394Base \u03b9))\nnil-term {Bool}   = curriedConst (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = curriedConst (nil-const {Map \u03ba \u03b9})\n\n-- diff-term and apply-term\n\nopen import Parametric.Change.Term Const \u0394Base\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))\nAtlas-diff {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\nAtlas-diff {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs Atlas-diff) m\u2081 m\u2082)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = abs\u2082 (\u03bb b\u2081 b\u2082 \u2192 xor! b\u2081 b\u2082)\nAtlas-apply {Map \u03ba \u03b9} = abs\u2082 (\u03bb m\u2081 m\u2082 \u2192 zip! (abs Atlas-apply) m\u2081 m\u2082)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 (\u0394Type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n  Term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Term \u0393 (base \u03ba) \u2192 Term \u0393 (base \u03b9) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9))) \u2192\n  Term \u0393 (\u0394Type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"eeb25567dcd19ceed0923b89314986eff10eca5f","subject":"Added more files to the readme","message":"Added more files to the readme\n","repos":"crypto-agda\/explore","old_file":"lib\/EXPLORE-README.agda","new_file":"lib\/EXPLORE-README.agda","new_contents":"module EXPLORE-README where\n\nopen import Explore.Type\n\nopen import Explore.Derived\nopen import Explore.Explorable\nopen import Explore.Summable\n\n-- Examples\nopen import Explore.BinTree\nopen import Explore.Product\nopen import Explore.Subset\nopen import Explore.Sum\n\n\n-- Other\nopen import Explore.Monad\nopen import Explore.Syntax\n","old_contents":"module EXPLORE-README where\n\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Sum\n\n-- Examples\nopen import Explore.Sum\nopen import Explore.Product\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dc25180d2799ce525f6411a9250a2ed48d40d199","subject":"game.agda","message":"game.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"game.agda","new_file":"game.agda","new_contents":"module game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Bool -- hiding (_\u225f_)\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R\n  ask : {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R) \u2192 Strategy \u2610_ R\n\nrun : \u2200 {R} \u2192 Strategy id R \u2192 R\nrun (say r)   = r\nrun (ask a f) = run (f a)\n\nopen import Data.Bits\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nmodule CryptoGames where\n\n  module CipherGames (|M| |C| : \u2115) where\n    M : Set\n    M = Bits |M|\n    C : Set\n    C = Bits |C|\n\n    SemSecAdv : Set\n    SemSecAdv = C \u2192 Bit\n\n    CPAAdv : Set\u2081\n    CPAAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit\n\n    data CPAWinner (enc : M \u2192 C) b : Strategy id Bit \u2192 Set\u2081 where\n      win : CPAWinner enc b (say b)\n      ask : \u2200 {m f} \u2192 CPAWinner enc b (f (enc m)) \u2192 CPAWinner enc b (ask (enc m) f)\n\n    CPA2Sem : SemSecAdv \u2192 CPAAdv\n    CPA2Sem semAdv enc = ask (enc 0\u2026) (\u03bb c \u2192 say (semAdv c))\n\n    CPA2Sem-correct : SemSecWinner semAdv \u2192 CPAWinner enc b (CPA2Sem semAdv)\n    CPA2Sem-correct\n\n  module MACGames (M T : Set) where\n    MacAdv : Set\u2081\n    MacAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 T) \u2192 Strategy \u2610_ (M \u00d7 T)\n\n    data MACWinner (mac : M \u2192 T) : Strategy id (M \u00d7 T) \u2192 Set\u2081 where\n      win : \u2200 {m} \u2192 MACWinner mac (say (m , mac m))\n      ask : \u2200 {m f} \u2192 MACWinner mac (f (mac m)) \u2192 MACWinner mac (ask (mac m) f)\n","old_contents":"module game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Bool -- hiding (_\u225f_)\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R\n  ask : {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R) \u2192 Strategy \u2610_ R\n\nrun : \u2200 {R} \u2192 Strategy id R \u2192 R\nrun (say r)   = r\nrun (ask a f) = run (f a)\n\nopen import Data.Bits\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nmodule CryptoGames where\n\n  module CipherGames (M C : Set) where\n    SemSecAdv : Set\n    SemSecAdv = C \u2192 Bit\n\n    CPAAdv : Set\u2081\n    CPAAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit\n\n    data CPAWinner (enc : M \u2192 C) b : Strategy id Bit \u2192 Set\u2081 where\n      win : CPAWinner enc b (say b)\n      ask : \u2200 {m f} \u2192 CPAWinner enc b (f (enc m)) \u2192 CPAWinner enc b (ask (enc m) f)\n\n  module MACGames (M T : Set) where\n    MacAdv : Set\u2081\n    MacAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 T) \u2192 Strategy \u2610_ (M \u00d7 T)\n\n    data MACWinner (mac : M \u2192 T) : Strategy id (M \u00d7 T) \u2192 Set\u2081 where\n      win : \u2200 {m} \u2192 MACWinner mac (say (m , mac m))\n      ask : \u2200 {m f} \u2192 MACWinner mac (f (mac m)) \u2192 MACWinner mac (ask (mac m) f)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a8faac396617459c5743f607567dc09a91193922","subject":"Relate further new-style to old-style validity","message":"Relate further new-style to old-style validity\n\nExtract, out of valid function changes in the \"new style\" from my PhD\nthesis, something similar (but not equivalent) to function changes in\nthe \"old style\" of PLDI'14.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/RelateToValidity.agda","new_file":"Thesis\/RelateToValidity.agda","new_contents":"module Thesis.RelateToValidity where\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\n\nopen import Thesis.Changes\nopen import Thesis.Lang\n\nmodule _ {A : Set} {{CA : ChangeStructure A}} where\n  fromto\u2192valid fromto\u2192valid-2 : \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192 valid a1 da\n  fromto\u2192valid da a1 a2 daa rewrite fromto\u2192\u2295 da a1 _ daa = daa\n  fromto\u2192valid-2 da a1 a2 daa = subst (ch da from a1 to_) (sym (fromto\u2192\u2295 da a1 a2 daa)) daa\n\n  -- The \"inverse\" is so trivial to not be worth calling, hence commented out:\n  -- valid\u2192fromto : \u2200 da (a : A) \u2192 (valid a da) \u2192 ch da from a to (a \u2295 da)\n  -- valid\u2192fromto _ _ daa = daa\n\nmodule _\n  {A : Set} {B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 \u2200 a da \u2192 Set\n  WellDefinedFunChangePoint f df a da = (f \u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangeFromTo\u2032 : \u2200 (f1 : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 Set\n  WellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 valid a da \u2192 WellDefinedFunChangePoint f1 df a da\n\n  open \u2261-Reasoning\n  open import Function\n\n  fromto-incrementalization : \u2200 {f1 f2 : A \u2192 B} {df} \u2192 ch df from f1 to f2 \u2192\n    \u2200 {da a} \u2192 valid a da \u2192\n    f1 a \u2295 df a da \u2261 f2 (a \u2295 da)\n  fromto-incrementalization {f1 = f1} {f2} {df} dff {da} {a} daa = fromto\u2192\u2295 _ _ _ (dff da a _ daa)\n\n  fromto\u2192valid-fun : \u2200 {f1 f2 : A \u2192 B} {df : Ch (A \u2192 B)} \u2192 ch df from f1 to f2 \u2192 \u2200 {a da} (daa : valid a da) \u2192 valid (f1 a) (df a da)\n  fromto\u2192valid-fun {f1} {f2} {df} dff {a} {da} daa = fromto\u2192valid (df a da) (f1 a) _ (dff da a _ daa)\n\n  fromto\u2192WellDefined\u2032 : \u2200 {f1 f2 df} \u2192 ch df from f1 to f2 \u2192\n    WellDefinedFunChangeFromTo\u2032 f1 df\n  fromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n    begin\n      (f1 \u2295 df) (a \u2295 da)\n    \u2261\u27e8 cong (_$ (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n      f2 (a \u2295 da)\n    \u2261\u27e8 sym (fromto-incrementalization dff daa) \u27e9\n      f1 a \u2295 df a da\n    \u220e\n\n  -- \u0394 f is similar to function changes in PLDI'14, but PLDI'14 function changes\n  -- need not be defined on invalid changes; instead, if (df, dff) : \u0394 f, then\n  -- df is a function change that is also defined on invalid changes.\n  --\n  -- Next we define f\u0394. It is closer to the PLDI'14 definition of function\n  -- changes; but it is not defined recursively, so it still produces new-style\n  -- function changes.\n  --\n  -- Hence this is sort of bigger than \u0394 f, though not really.\n  f\u0394 : (A \u2192 B) \u2192 Set\n  f\u0394 f = (a : A) \u2192 \u0394 a \u2192 \u0394 (f a)\n\n  -- We can indeed map \u0394 f into f\u0394 f, via valid-functions-map-\u0394. If this mapping\n  -- were an injection, we could say that f\u0394 f is bigger than \u0394 f. But we'd need\n  -- first to quotient \u0394 f to turn valid-functions-map-\u0394 into an injection:\n  --\n  -- 1. We need to quotient \u0394 f by extensional equivalence of functions.\n  --    Luckily, we can just postulate it.\n  --\n  -- 2. We need to quotient \u0394 f by identifying functions that only differ by\n  --    behavior on invalid changes; such functions can't be distinguished after\n  --    being injected into f\u0394 f.\n  valid-functions-map-\u0394 : \u2200 (f : A \u2192 B) (df : \u0394 f) \u2192 f\u0394 f\n  valid-functions-map-\u0394 f (df , dff) a (da , daa) = df a da , valid-res\n    where\n      valid-res : ch df a da from f a to (f a \u2295 df a da)\n      valid-res rewrite sym (fromto\u2192WellDefined\u2032 dff da a daa) = dff da a (a \u2295 da) daa\n\n  -- Two alternative proofs\n  fromto-functions-map-\u0394-1 fromto-functions-map-\u0394-2 : \u2200 (f1 f2 : A \u2192 B) (df : Ch (A \u2192 B)) \u2192 ch df from f1 to f2 \u2192 f\u0394 f1\n  fromto-functions-map-\u0394-1 f1 f2 df dff a (da , daa) = valid-functions-map-\u0394 f1 (df , fromto\u2192valid df f1 f2 dff) a (da , daa)\n\n  fromto-functions-map-\u0394-2 f1 f2 df dff a (da , daa) = df a da , fromto\u2192valid-fun dff daa\n\nopen import Thesis.LangChanges\n\nfromto\u2192WellDefined\u2032Lang : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032Lang {f1 = f1} {f2} {df} dff da a daa =\n  fromto\u2192WellDefined\u2032 dff da a daa\n","old_contents":"module Thesis.RelateToValidity where\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\n\nopen import Thesis.Changes\nopen import Thesis.Lang\n\nmodule _\n  {A : Set} {B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 \u2200 a da \u2192 Set\n  WellDefinedFunChangePoint f df a da = (f \u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangeFromTo\u2032 : \u2200 (f1 : A \u2192 B) \u2192 (df : Ch (A \u2192 B)) \u2192 Set\n  WellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 ch da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\n  open \u2261-Reasoning\n\n  fromto\u2192WellDefined\u2032 : \u2200 {f1 f2 df} \u2192 ch df from f1 to f2 \u2192\n    WellDefinedFunChangeFromTo\u2032 f1 df\n  fromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n    begin\n      (f1 \u2295 df) (a \u2295 da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n      f2 (a \u2295 da)\n    \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n      f1 a \u2295 df a da\n    \u220e\n\n  -- This is similar (not equivalent) to the old definition of function changes.\n  -- However, such a function need not be defined on invalid changes, unlike the\n  -- functions.\n  --\n  -- Hence this is sort of bigger than \u0394 f, though not really.\n  f\u0394 : (A \u2192 B) \u2192 Set\n  f\u0394 f = (a : A) \u2192 \u0394 a \u2192 \u0394 (f a)\n\n  -- We can indeed map \u0394 f into f\u0394 f, via valid-functions-map-\u0394. If this mapping\n  -- were an injection, we could say that f\u0394 f is bigger than \u0394 f. But we'd need\n  -- first to quotient \u0394 f to turn valid-functions-map-\u0394 into an injection:\n  --\n  -- 1. We need to quotient \u0394 f by extensional equivalence of functions.\n  --    Luckily, we can just postulate it.\n  --\n  -- 2. We need to quotient \u0394 f by identifying functions that only differ by\n  --    behavior on invalid changes; such functions can't be distinguished after\n  --    being injected into f\u0394 f.\n  valid-functions-map-\u0394 : \u2200 (f : A \u2192 B) (df : \u0394 f) \u2192 f\u0394 f\n  valid-functions-map-\u0394 f (df , dff) a (da , daa) = df a da , valid-res\n    where\n      valid-res : ch df a da from f a to (f a \u2295 df a da)\n      valid-res rewrite sym (fromto\u2192WellDefined\u2032 dff da a daa) = dff da a (a \u2295 da) daa\n\n  fromto-functions-map-\u0394 : \u2200 (f1 f2 : A \u2192 B) (df : Ch (A \u2192 B)) \u2192 ch df from f1 to f2 \u2192 f\u0394 f1\n  fromto-functions-map-\u0394 f1 f2 df dff a (da , daa) = valid-functions-map-\u0394 f1 (df , dff\u2032) a (da , daa)\n    where\n      dff\u2032 : ch df from f1 to (f1 \u2295 df)\n      dff\u2032 = subst (ch df from f1 to_) (sym (fromto\u2192\u2295 df f1 f2 dff)) dff\n\nopen import Thesis.LangChanges\n\nfromto\u2192WellDefined\u2032Lang : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032Lang {f1 = f1} {f2} {df} dff da a daa =\n  fromto\u2192WellDefined\u2032 dff da a daa\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5a499d733fe3968065e0ae2b47d1393cf5980209","subject":"Data.Two: add _\u00b2","message":"Data.Two: add _\u00b2\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Two.agda","new_file":"lib\/Data\/Two.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two where\n\nopen import Data.Bool public hiding (if_then_else_) renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\nopen import Data.Bool.Properties\n  public\n  using (isCommutativeSemiring-\u2228-\u2227\n        ;commutativeSemiring-\u2228-\u2227\n        ;module RingSolver\n        ;isCommutativeSemiring-\u2227-\u2228\n        ;commutativeSemiring-\u2227-\u2228\n        ;isBooleanAlgebra\n        ;booleanAlgebra\n        ;commutativeRing-xor-\u2227\n        ;module XorRingSolver\n        ;not-involutive\n        ;not-\u00ac\n        ;\u00ac-not\n        ;\u21d4\u2192\u2261\n        ;proof-irrelevance)\n  renaming\n        (T-\u2261 to \u2713-\u2261\n        ;T-\u2227 to \u2713-\u2227\n        ;T-\u2228 to \u2713-\u2228)\n\nopen import Type using (\u2605_)\n\nopen import Algebra                    using (module CommutativeRing; module CommutativeSemiring)\nopen import Algebra.FunctionProperties using (Op\u2081; Op\u2082)\n\nopen import Data.Nat     using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\nopen import Data.Zero    using (\ud835\udfd8-elim)\nopen import Data.One     using (\ud835\udfd9)\nopen import Data.Product using (proj\u2081; proj\u2082; uncurry; _\u00d7_; _,_)\nopen import Data.Sum     using (_\u228e_; inj\u2081; inj\u2082)\n\nopen import Function.Equivalence using (module Equivalence)\nopen import Function.Equality    using (_\u27e8$\u27e9_)\nopen import Function.NP          using (id; _\u2218_; _\u27e8_\u27e9\u00b0_)\n\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; refl; idp; _\u2219_; !_)\nopen import Relation.Nullary                         using (\u00ac_; Dec; yes; no)\n\nopen Equivalence using (to; from)\n\nmodule Xor\u00b0 = CommutativeRing     commutativeRing-xor-\u2227\nmodule \ud835\udfda\u00b0   = CommutativeSemiring commutativeSemiring-\u2227-\u2228\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: proj\u2081 1: proj\u2082 ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 p q = _\u27e8$\u27e9_ (from \u2713-\u2227) (p , q)\n\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to \u2713-\u2227)\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (\u2713-\u2227 {b\u2081}))\n\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {b\u2081} = _\u27e8$\u27e9_ (to (\u2713-\u2228 {b\u2081}))\n\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 = _\u27e8$\u27e9_ (from \u2713-\u2228) \u2218 inj\u2081\n\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (\u2713-\u2228 {b\u2081})) \u2218 inj\u2082\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not 0\u2082 y = not-involutive y\nxor-not-not 1\u2082 _ = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 (x xor y) \u2261 (x xor z) \u2192 y \u2261 z\nxor-inj\u2081 0\u2082 = id\nxor-inj\u2081 1\u2082 = not-inj\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 (y xor x) \u2261 (z xor x) \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} p = xor-inj\u2081 x (Xor\u00b0.+-comm x y \u2219 p \u2219 Xor\u00b0.+-comm z x)\n\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \ud835\udfd9\ncheck = _\n\nmodule Indexed {a} {A : \u2605 a} where\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 \ud835\udfda)\n    not\u00b0 f = not \u2218 f\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two where\n\nopen import Data.Bool public hiding (if_then_else_) renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\nopen import Data.Bool.Properties\n  public\n  using (isCommutativeSemiring-\u2228-\u2227\n        ;commutativeSemiring-\u2228-\u2227\n        ;module RingSolver\n        ;isCommutativeSemiring-\u2227-\u2228\n        ;commutativeSemiring-\u2227-\u2228\n        ;isBooleanAlgebra\n        ;booleanAlgebra\n        ;commutativeRing-xor-\u2227\n        ;module XorRingSolver\n        ;not-involutive\n        ;not-\u00ac\n        ;\u00ac-not\n        ;\u21d4\u2192\u2261\n        ;proof-irrelevance)\n  renaming\n        (T-\u2261 to \u2713-\u2261\n        ;T-\u2227 to \u2713-\u2227\n        ;T-\u2228 to \u2713-\u2228)\n\nopen import Type using (\u2605_)\n\nopen import Algebra                    using (module CommutativeRing; module CommutativeSemiring)\nopen import Algebra.FunctionProperties using (Op\u2081; Op\u2082)\n\nopen import Data.Nat     using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\nopen import Data.Zero    using (\ud835\udfd8-elim)\nopen import Data.One     using (\ud835\udfd9)\nopen import Data.Product using (proj\u2081; proj\u2082; uncurry; _\u00d7_; _,_)\nopen import Data.Sum     using (_\u228e_; inj\u2081; inj\u2082)\n\nopen import Function.Equivalence using (module Equivalence)\nopen import Function.Equality    using (_\u27e8$\u27e9_)\nopen import Function.NP          using (id; _\u2218_; _\u27e8_\u27e9\u00b0_)\n\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; refl; idp; _\u2219_; !_)\nopen import Relation.Nullary                         using (\u00ac_; Dec; yes; no)\n\nopen Equivalence using (to; from)\n\nmodule Xor\u00b0 = CommutativeRing     commutativeRing-xor-\u2227\nmodule \ud835\udfda\u00b0   = CommutativeSemiring commutativeSemiring-\u2227-\u2228\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 \ud835\udfda \u2192 A\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 \ud835\udfda \u2192 A\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: proj\u2081 1: proj\u2082 ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 p q = _\u27e8$\u27e9_ (from \u2713-\u2227) (p , q)\n\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to \u2713-\u2227)\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (\u2713-\u2227 {b\u2081}))\n\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {b\u2081} = _\u27e8$\u27e9_ (to (\u2713-\u2228 {b\u2081}))\n\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 = _\u27e8$\u27e9_ (from \u2713-\u2228) \u2218 inj\u2081\n\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (\u2713-\u2228 {b\u2081})) \u2218 inj\u2082\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not 0\u2082 y = not-involutive y\nxor-not-not 1\u2082 _ = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 (x xor y) \u2261 (x xor z) \u2192 y \u2261 z\nxor-inj\u2081 0\u2082 = id\nxor-inj\u2081 1\u2082 = not-inj\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 (y xor x) \u2261 (z xor x) \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} p = xor-inj\u2081 x (Xor\u00b0.+-comm x y \u2219 p \u2219 Xor\u00b0.+-comm z x)\n\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \ud835\udfd9\ncheck = _\n\nmodule Indexed {a} {A : \u2605 a} where\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 \ud835\udfda)\n    not\u00b0 f = not \u2218 f\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"203b43b48ea601c9effa30be535350505824ae16","subject":"Added missing pragma.","message":"Added missing pragma.\n\nIgnore-this: a111320f910e8ea0ffcbbb0b4ca06e51\n\ndarcs-hash:20110212092556-3bd4e-964489bd7303cd483e60cd32d1c4175a8fa8ca39.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Unary\/IsN-ATP.agda","new_file":"src\/LTC\/Data\/Nat\/Unary\/IsN-ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- The unary numbers are LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Unary.IsN-ATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.Nat.Unary.Numbers\n\n------------------------------------------------------------------------------\n\npostulate\n  N0  : N zero\n  N1  : N one\n  N10 : N ten\n{-# ATP prove N0 #-}\n{-# ATP prove N1 #-}\n{-# ATP prove N10 #-}\n\npostulate\n  N11 : N eleven\n{-# ATP prove N11 #-}\n\npostulate\n  N89 : N eighty-nine\n  N90 : N ninety\n{-# ATP prove N89 #-}\n{-# ATP prove N90 #-}\n\npostulate\n  N91  : N ninety-one\n  N100 : N one-hundred\n{-# ATP prove N91 #-}\n{-# ATP prove N100 #-}\n\npostulate\n  N101 : N hundred-one\n  N111 : N hundred-eleven\n{-# ATP prove N101 #-}\n{-# ATP prove N111 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The unary numbers are LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Unary.IsN-ATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.Nat.Unary.Numbers\n\n------------------------------------------------------------------------------\n\npostulate\n  N0 : N zero\n  N1  : N one\n  N10 : N ten\n{-# ATP prove N1 #-}\n{-# ATP prove N10 #-}\n\npostulate\n  N11 : N eleven\n{-# ATP prove N11 #-}\n\npostulate\n  N89 : N eighty-nine\n  N90 : N ninety\n{-# ATP prove N89 #-}\n{-# ATP prove N90 #-}\n\npostulate\n  N91  : N ninety-one\n  N100 : N one-hundred\n{-# ATP prove N91 #-}\n{-# ATP prove N100 #-}\n\npostulate\n  N101 : N hundred-one\n  N111 : N hundred-eleven\n{-# ATP prove N101 #-}\n{-# ATP prove N111 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8c8b44eddb11a94a556956c2960a8833cc4caa9f","subject":"Desc stratified model: stratify phi and psi.","message":"Desc stratified model: stratify phi and psi.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dd61a8d1db025c8ea586d233c151e7598bdac4a3","subject":"bit-guessing-game: cleanup imports","message":"bit-guessing-game: cleanup imports\n","repos":"crypto-agda\/crypto-agda","old_file":"bit-guessing-game.agda","new_file":"bit-guessing-game.agda","new_contents":"open import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Data.Bits using (Bit)\n\nopen import flipbased-implem using (Coins; \u21ba)\nopen import program-distance using (PrgDist; module PrgDist)\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n","old_contents":"open import program-distance\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Product\nopen import Relation.Nullary\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6ad96baab8d36671b146b356a5a446ae10e1b832","subject":"Remove apply-env and diff-env.","message":"Remove apply-env and diff-env.\n\nOld-commit-hash: 0901643e49a0e4ff70db8f16278c25bd28348d3a\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Base.Change.Algebra as CA using\n  ( ChangeAlgebra\n  ; ChangeAlgebraFamily\n  ; change-algebra\u208d_\u208e\n  ; family\n  )\nopen import Level\n\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  ----------------\n  -- Parameters --\n  ----------------\n\n  Change-base : (\u03b9 : Base) \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Set\n  Change-base = CA.\u0394\u208d_\u208e\n\n  diff-change-base : \u2200 \u03b9 \u2192 (u v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v\n  diff-change-base = CA.diff\u2032\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  open CA public\n    using\n      (\n      )\n    renaming\n      ( \u0394\u208d_\u208e to Change\n      ; update\u2032 to apply-change\n      ; diff\u2032 to diff-change\n      ; nil\u208d_\u208e to nil-change\n      )\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n  v+[u-v]=u {\u03c4} {u} {v} = CA.update-diff\u208d \u03c4 \u208e u v\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open CA.FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      using\n        (\n        )\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  open CA public using\n    ( before\n    ; after\n    ; before\u208d_\u208e\n    ; after\u208d_\u208e\n    )\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open CA.FunctionChanges public using (cons)\n  open CA public using () renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  open CA.ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  \u0394Env : \u2200 (\u0393 : Context) \u2192 \u27e6 \u0393 \u27e7 \u2192 Set\n  \u0394Env \u0393 \u03c1 = CA.\u0394\u208d \u0393 \u208e \u03c1\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Base.Change.Algebra as CA using\n  ( ChangeAlgebra\n  ; ChangeAlgebraFamily\n  ; change-algebra\u208d_\u208e\n  ; family\n  )\nopen import Level\n\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  ----------------\n  -- Parameters --\n  ----------------\n\n  Change-base : (\u03b9 : Base) \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Set\n  Change-base = CA.\u0394\u208d_\u208e\n\n  diff-change-base : \u2200 \u03b9 \u2192 (u v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v\n  diff-change-base = CA.diff\u2032\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  open CA public\n    using\n      (\n      )\n    renaming\n      ( \u0394\u208d_\u208e to Change\n      ; update\u2032 to apply-change\n      ; diff\u2032 to diff-change\n      ; nil\u208d_\u208e to nil-change\n      )\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n  v+[u-v]=u {\u03c4} {u} {v} = CA.update-diff\u208d \u03c4 \u208e u v\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open CA.FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      using\n        (\n        )\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  open CA public using\n    ( before\n    ; after\n    ; before\u208d_\u208e\n    ; after\u208d_\u208e\n    )\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open CA.FunctionChanges public using (cons)\n  open CA public using () renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  open CA.ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  \u0394Env : \u2200 (\u0393 : Context) \u2192 \u27e6 \u0393 \u27e7 \u2192 Set\n  \u0394Env \u0393 \u03c1 = CA.\u0394\u208d \u0393 \u208e \u03c1\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n\n  apply-env : \u2200 \u0393 \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  apply-env = CA.update\u2032\n\n  diff-env : \u2200 \u0393 \u2192 (\u03c0 \u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394Env \u0393 \u03c1\n  diff-env = CA.diff\u2032\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8ca845b807ca4a79231ed0722e0c33b25c1c0239","subject":"Nat: merge","message":"Nat: merge\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k \u27e9* m \u2261 \u27e82^ k \u27e9* n \u2192 m \u2261 n\n2^-inj zero    eq = eq\n2^-inj (suc k) eq = 2^-inj k (2*-inj eq)\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9db9bce60cd9ca9db04a9a764a7e721ac21ee2f8","subject":"flat-funs: fixities","message":"flat-funs: fixities\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat\nopen import Data.Bits using (Bits)\nimport Level as L\nimport Function as F\nopen import composable\nopen import vcomp\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n\n    _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    constBits : \u2200 {n} \u2192 Bits n \u2192 `\u22a4 `\u2192 `Bits n\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Product renaming (zip to \u00d7-zip; map to \u00d7-map)\nopen import Function\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id (\u03bb f g x \u2192 g (f x)) (\u03bb f g \u2192 \u00d7-map f g) _&&&_ proj\u2081 proj\u2082 const\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id (\u03bb f g \u2192 g \u2218 f) (VComposable._***_ bitsFunVComp) _&&&_ (\u03bb {A} \u2192 take A)\n                    (\u03bb {A} \u2192 drop A) (\u03bb xs _ \u2192 xs ++ [] {- ugly? -})\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe\n                           (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                        (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                        (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4 = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                         (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                         (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ _\u2294_ _\u2294_ 0 0 (const 0)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ _+_ _+_ 0 0 (\u03bb {n} _ \u2192 n)\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nopen import Data.Bits using (Bits)\nimport Level as L\nimport Function as F\nopen import composable\nopen import vcomp\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n\n    _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    constBits : \u2200 {n} \u2192 Bits n \u2192 `\u22a4 `\u2192 `Bits n\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Product renaming (zip to \u00d7-zip; map to \u00d7-map)\nopen import Function\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id (\u03bb f g x \u2192 g (f x)) (\u03bb f g \u2192 \u00d7-map f g) _&&&_ proj\u2081 proj\u2082 const\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id (\u03bb f g \u2192 g \u2218 f) (VComposable._***_ bitsFunVComp) _&&&_ (\u03bb {A} \u2192 take A)\n                    (\u03bb {A} \u2192 drop A) (\u03bb xs _ \u2192 xs ++ [] {- ugly? -})\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe\n                           (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                        (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                        (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4 = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                         (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                         (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ _\u2294_ _\u2294_ 0 0 (const 0)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ _+_ _+_ 0 0 (\u03bb {n} _ \u2192 n)\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c2768a3a06f4ab9a152d35ce0244cffd0435da2f","subject":"IDesc model: cool down on universe polymorphism on Desc","message":"IDesc model: cool down on universe polymorphism on Desc","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c4db16427326278f3e09137cc22a5560b3e4f6b2","subject":"subst proven and with fewer assumptions","message":"subst proven and with fewer assumptions\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\nopen import binders-disjoint-checks\n\nmodule lemmas-subst-ta where\n  mutual\n    binders-envfresh : \u2200{\u0394 \u0393 \u0393' y \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 y # \u0393 \u2192 unbound-in-\u03c3 y \u03c3 \u2192 binders-unique-\u03c3 \u03c3 \u2192 envfresh y \u03c3\n    binders-envfresh {\u0393' = \u0393'} {y = y} (STAId x) apt unbound unique with ctxindirect \u0393' y\n    binders-envfresh {\u0393' = \u0393'} {y = y} (STAId x\u2081) apt unbound unique | Inl x = abort (somenotnone (! (x\u2081 y (\u03c01 x) (\u03c02 x)) \u00b7 apt))\n    binders-envfresh (STAId x\u2081) apt unbound unique | Inr x = EFId x\n    binders-envfresh {\u0393 = \u0393} {y = y} (STASubst  {y = z} subst x\u2081) apt (UB\u03c3Subst x\u2082 unbound neq) (BU\u03c3Subst zz x\u2083 x\u2084) =\n                                                                                    EFSubst (binders-fresh {y = y} x\u2081 zz x\u2082 apt)\n                                                                                            (binders-envfresh subst (apart-extend1 \u0393 neq apt) unbound x\u2083)\n                                                                                            neq\n\n    binders-fresh : \u2200{ \u0394 \u0393 d2 \u03c4 y} \u2192 \u0394 , \u0393 \u22a2 d2 :: \u03c4\n                                      \u2192 binders-unique d2\n                                      \u2192 unbound-in y d2\n                                      \u2192 \u0393 y == None\n                                      \u2192 fresh y d2\n    binders-fresh TAConst BUHole UBConst apt = FConst\n    binders-fresh {y = y} (TAVar {x = x} x\u2081)  BUVar UBVar apt with natEQ y x\n    binders-fresh (TAVar x\u2082) BUVar UBVar apt | Inl refl = abort (somenotnone (! x\u2082 \u00b7 apt))\n    binders-fresh (TAVar x\u2082) BUVar UBVar apt | Inr x\u2081 = FVar x\u2081\n    binders-fresh {y = y} (TALam {x = x} x\u2081 wt) bu2 ub apt  with natEQ y x\n    binders-fresh (TALam x\u2082 wt) bu2 (UBLam2 x\u2081 ub) apt | Inl refl = abort (x\u2081 refl)\n    binders-fresh {\u0393 = \u0393} (TALam {x = x} x\u2082 wt) (BULam bu2 x\u2083) (UBLam2 x\u2084 ub) apt | Inr x\u2081 =  FLam x\u2081 (binders-fresh wt bu2 ub (apart-extend1 \u0393 x\u2084 apt))\n    binders-fresh (TAAp wt wt\u2081)  (BUAp bu2 bu3 x) (UBAp ub ub\u2081) apt = FAp (binders-fresh wt bu2 ub apt) (binders-fresh wt\u2081 bu3 ub\u2081 apt)\n    binders-fresh (TAEHole x\u2081 x\u2082) (BUEHole x) (UBHole x\u2083) apt = FHole (binders-envfresh x\u2082 apt x\u2083 x )\n    binders-fresh (TANEHole x\u2081 wt x\u2082) (BUNEHole bu2 x) (UBNEHole x\u2083 ub) apt = FNEHole (binders-envfresh x\u2082 apt x\u2083 x) (binders-fresh wt bu2  ub apt)\n    binders-fresh (TACast wt x\u2081) (BUCast bu2) (UBCast ub) apt = FCast (binders-fresh wt  bu2  ub apt)\n    binders-fresh (TAFailedCast wt x x\u2081 x\u2082) (BUFailedCast bu2) (UBFailedCast ub) apt = FFailedCast (binders-fresh wt  bu2  ub apt)\n\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  binders-disjoint d1 d2 \u2192\n                  binders-unique d1 \u2192 -- todo: do i need both of these?\n                  binders-unique d2 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt bd bu1 bu2  TAConst wt2 = TAConst\n  lem-subst {x = x} apt bd bu1 bu2  (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2 (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (BDLam bd bd') (BULam bu1' ub) bu2 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) bd bu1' bu2 (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1)\n                                         (weaken-ta (binders-fresh wt2  bu2 bd' y#\u0393) wt2))\n  lem-subst apt (BDAp bd bd\u2081) (BUAp bu1 bu2 x\u2081) bu3 (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt bd bu1 bu3 wt1 wt3) (lem-subst apt bd\u2081 bu2 bu3 wt2 wt3)\n  lem-subst apt bd bu1 bu2 (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (BDNEHole x\u2081 bd) (BUNEHole bu1 x\u2082) bu2 (TANEHole x\u2083 wt1 x\u2084) wt2 = TANEHole x\u2083 (lem-subst apt bd bu1 bu2 wt1 wt2) (STASubst x\u2084 wt2)\n  lem-subst apt (BDCast bd) (BUCast bu1) bu2 (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt bd bu1 bu2 wt1 wt2) x\u2081\n  lem-subst apt (BDFailedCast bd) (BUFailedCast bu1) bu2 (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt bd bu1 bu2 wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\nopen import binders-disjoint-checks\n\nmodule lemmas-subst-ta where\n  mutual\n    binders-envfresh : \u2200{\u0394 \u0393 \u0393' y \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 y # \u0393 \u2192 unbound-in-\u03c3 y \u03c3 \u2192 binders-unique-\u03c3 \u03c3 \u2192 envfresh y \u03c3\n    binders-envfresh {\u0393' = \u0393'} {y = y} (STAId x) apt unbound unique with ctxindirect \u0393' y\n    binders-envfresh {\u0393' = \u0393'} {y = y} (STAId x\u2081) apt unbound unique | Inl x = abort (somenotnone (! (x\u2081 y (\u03c01 x) (\u03c02 x)) \u00b7 apt))\n    binders-envfresh (STAId x\u2081) apt unbound unique | Inr x = EFId x\n    binders-envfresh {\u0393 = \u0393} {y = y} (STASubst  {y = z} subst x\u2081) apt (UB\u03c3Subst x\u2082 unbound neq) (BU\u03c3Subst zz x\u2083 x\u2084) =\n                                                                                    EFSubst {!binders-fresh!}\n                                                                                      -- (binders-fresh x\u2081 {!!} zz {!!} x\u2082 apt)\n                                                                                            (binders-envfresh subst (apart-extend1 \u0393 neq apt) unbound x\u2083)\n                                                                                            neq\n\n    binders-fresh : \u2200{ \u0394 \u0393 d1 d2 \u03c4 y} \u2192 \u0394 , \u0393 \u22a2 d2 :: \u03c4\n                                      \u2192 binders-unique d1 -- todo: ditch?\n                                      \u2192 binders-unique d2\n                                      \u2192 binders-disjoint d1 d2\n                                      \u2192 unbound-in y d2\n                                      \u2192 \u0393 y == None\n                                      \u2192 fresh y d2\n    binders-fresh TAConst bu1 BUHole bd UBConst apt = FConst\n    binders-fresh {y = y}  (TAVar {x = x} x\u2081) bu1 BUVar bd UBVar apt with natEQ y x\n    binders-fresh (TAVar x\u2082) bu1 BUVar bd UBVar apt | Inl refl = abort (somenotnone (! x\u2082 \u00b7 apt))\n    binders-fresh (TAVar x\u2082) bu1 BUVar bd UBVar apt | Inr x\u2081 = FVar x\u2081\n    binders-fresh {y = y} (TALam {x = x} x\u2081 wt) bu1 bu2 bd ub apt  with natEQ y x\n    binders-fresh (TALam x\u2082 wt) bu1 bu2 bd (UBLam2 x\u2081 ub) apt | Inl refl = abort (x\u2081 refl)\n    binders-fresh {\u0393 = \u0393} (TALam {x = x} x\u2082 wt) bu1 (BULam bu2 x\u2083) bd (UBLam2 x\u2084 ub) apt | Inr x\u2081 =  FLam x\u2081 (binders-fresh wt bu1 bu2 (lem-bd-lam bd) ub (apart-extend1 \u0393 x\u2084 apt))\n    binders-fresh (TAAp wt wt\u2081) bu1 (BUAp bu2 bu3 x) bd (UBAp ub ub\u2081) apt = FAp (binders-fresh wt BUHole bu2 BDConst ub apt)\n                                                                                (binders-fresh wt\u2081 bu2 bu3 x ub\u2081 apt)\n    binders-fresh (TAEHole x\u2081 x\u2082) bu1 (BUEHole x) bd (UBHole x\u2083) apt = FHole {!!} -- FHole {!binders-envfresh x\u2082 apt x\u2083 x !}\n    binders-fresh (TANEHole x\u2081 wt x\u2082) bu1 (BUNEHole bu2 x) bd (UBNEHole x\u2083 ub) apt = FNEHole {!binders-envfresh x\u2082 apt x\u2083!} (binders-fresh wt bu1 bu2 (lem-bd-hole bd) ub apt)\n    binders-fresh (TACast wt x\u2081) bu1 (BUCast bu2) bd (UBCast ub) apt = FCast (binders-fresh wt bu1 bu2 (lem-bd-cast bd) ub apt)\n    binders-fresh (TAFailedCast wt x x\u2081 x\u2082) bu1 (BUFailedCast bu2) bd (UBFailedCast ub) apt = FFailedCast (binders-fresh wt bu1 bu2 (lem-bd-failedcast bd) ub apt)\n\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  binders-disjoint d1 d2 \u2192\n                  binders-unique d1 \u2192 -- todo: do i need both of these?\n                  binders-unique d2 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt bd bu1 bu2  TAConst wt2 = TAConst\n  lem-subst {x = x} apt bd bu1 bu2  (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2 (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt bd bu1 bu2  (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (BDLam bd bd') (BULam bu1' ub) bu2 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) bd bu1' bu2 (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1)\n                                         (weaken-ta (binders-fresh wt2 (BULam {\u03c4 = \u03c42} bu1' ub) bu2 (BDLam bd bd') bd' y#\u0393) wt2))\n  lem-subst apt (BDAp bd bd\u2081) (BUAp bu1 bu2 x\u2081) bu3 (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt bd bu1 bu3 wt1 wt3) (lem-subst apt bd\u2081 bu2 bu3 wt2 wt3)\n  lem-subst apt bd bu1 bu2 (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (BDNEHole x\u2081 bd) (BUNEHole bu1 x\u2082) bu2 (TANEHole x\u2083 wt1 x\u2084) wt2 = TANEHole x\u2083 (lem-subst apt bd bu1 bu2 wt1 wt2) (STASubst x\u2084 wt2)\n  lem-subst apt (BDCast bd) (BUCast bu1) bu2 (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt bd bu1 bu2 wt1 wt2) x\u2081\n  lem-subst apt (BDFailedCast bd) (BUFailedCast bu1) bu2 (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt bd bu1 bu2 wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7477ac89b344df0b761310be5b5968a5e2aee9bc","subject":"temporary but correct full-adder","message":"temporary but correct full-adder\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Core.agda","new_file":"FunUniverse\/Core.agda","new_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nopen import FunUniverse.Types public\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\nopen import FunUniverse.Rewiring.Linear\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    -- bijFork f\u2080 f\u2081 (0' , x) = 0' , f\u2080 x\n    -- bijFork f\u2080 f\u2081 (1' , x) = 1' , f\u2081 x\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  -- bijFork\u2032 f\u2080 f\u2081 (0' , x) = 0' , f\u2080 0' x\n  -- bijFork\u2032 f\u2080 f\u2081 (1' , x) = 1' , f\u2081 1' x\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- cond (0' , x\u2080 , x\u2081) = x\u2080\n    -- cond (1' , x\u2080 , x\u2081) = x\u2081\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- fork f\u2080 f\u2081 (0' , x) = f\u2080 x\n    -- fork f\u2080 f\u2081 (1' , x) = f\u2081 x\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  -- Bad idea\u2122\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <1b> <0b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  -- full-adder (C\u1d62\u2099 , A , B) = (C\u2092\u1d64\u209c , S)\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < cout , s >\n    where a    = snd \u204f fst\n          b    = snd \u204f snd\n          cin  = fst\n          s    = second <xor> \u204f <xor>\n          cout = <   < a , b > \u204f <and>\n                 , < < a , b > \u204f <xor> , cin > \u204f <and> >\n               \u204f <or> -- this one can also be a <xor>\n\n  {-\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n  -}\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","old_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nopen import FunUniverse.Types public\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\nopen import FunUniverse.Rewiring.Linear\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    -- bijFork f\u2080 f\u2081 (0' , x) = 0' , f\u2080 x\n    -- bijFork f\u2080 f\u2081 (1' , x) = 1' , f\u2081 x\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  -- bijFork\u2032 f\u2080 f\u2081 (0' , x) = 0' , f\u2080 0' x\n  -- bijFork\u2032 f\u2080 f\u2081 (1' , x) = 1' , f\u2081 1' x\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- cond (0' , x\u2080 , x\u2081) = x\u2080\n    -- cond (1' , x\u2080 , x\u2081) = x\u2081\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- fork f\u2080 f\u2081 (0' , x) = f\u2080 x\n    -- fork f\u2080 f\u2081 (1' , x) = f\u2081 x\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  -- Bad idea\u2122\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <1b> <0b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f4d118ec81bf154d77c7bf0606995bf0bc9ba296","subject":"Nat: +0\u2238_\u22610, +2*-\u2238, +n+k\u2238m, +factor-+-\u2238","message":"Nat: +0\u2238_\u22610, +2*-\u2238, +n+k\u2238m, +factor-+-\u2238\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"32d7275c4b089d42d17adc6bb784438aac4015f2","subject":"Vec: \u229b-replicate, \u229b-take-drop-lem, take2*-++","message":"Vec: \u229b-replicate, \u229b-take-drop-lem, take2*-++\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (Type)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd; map to \u00d7-map)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; _\u2219_; !_) renaming (refl to idp)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : Type a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 Type a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : Type a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Type a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Type a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id= : \u2200 {a n} {A : Type a}{f : A \u2192 A}\n             \u2192 f \u2257 id \u2192 map f \u2257 id {A = Vec A n}\n    map-id= f= []       = idp\n    map-id= f= (x \u2237 xs) = ap-\u2237 (f= x) (map-id= f= xs)\n\n    map-id : \u2200 {a n} {A : Type a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id = map-id= (\u03bb _ \u2192 idp)\n\n    map-\u2218= : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n               {f : B \u2192 C}{g : A \u2192 B}{h : A \u2192 C} \u2192\n               f \u2218 g \u2257 h \u2192\n               _\u2257_ {A = Vec A n} (map f \u2218 map g) (map h)\n    map-\u2218= fg= []       = idp\n    map-\u2218= fg= (x \u2237 xs) = ap-\u2237 (fg= x) (map-\u2218= fg= xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n              (f : B \u2192 C) (g : A \u2192 B) \u2192\n              _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g v = ! map-\u2218= (\u03bb _ \u2192 idp) v\n\n    map-ext : \u2200 {a b} {A : Type a} {B : Type b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : Type a}\n    (_\u2248_ : A \u2192 A \u2192 Type \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Type \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 Type _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : Type a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= idp idp = idp\n\nmodule With\u2261 {a}{A : Type a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                           ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Type a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Type a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Type a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\ncount\u1da0 : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : Type a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : Type a} where\n  count-++ : \u2200 {m n} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m b} {B : Type b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , idp = idp\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , idp = idp\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  map2*-++ : \u2200 {m n}(f : Vec A m \u2192 Vec A n) \u2192 map2* m n f \u2218 uncurry _++_ \u2257 uncurry _++_ \u2218 \u00d7-map f f\n  map2*-++ {m} f (x , y) rewrite take-++ m x y | drop-++ m x y = idp\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\n  vec= : \u2200 {n}{xs ys : Vec A n} \u2192 (\u2200 i \u2192 (xs \u203c i) \u2261 (ys \u203c i)) \u2192 xs \u2261 ys\n  vec= {xs = []}     {[]}     f= = idp\n  vec= {xs = x \u2237 xs} {y \u2237 ys} f= = ap-\u2237 (f= zero) (vec= (f= \u2218 suc))\n\n  concat-group : \u2200 m n (xs : Vec A (m * n)) \u2192 concat (group m n xs) \u2261 xs\n  concat-group zero    n [] = idp\n  concat-group (suc m) n xs rewrite concat-group m n (drop n xs) = take-drop-lem n xs\n\n  group-concat : \u2200 {m n}(xss : Vec (Vec A n) m) \u2192 group m n (concat xss) \u2261 xss\n  group-concat [] = idp\n  group-concat (xs \u2237 xss)\n    rewrite take-++ _ xs (concat xss)\n          | drop-++ _ xs (concat xss)\n          | group-concat xss\n          = idp\n\n  -- These are also in the stdlib as Data.Vec.Properties.lookup-morphism\n  \u203c-replicate : \u2200 {n}(i : Fin n)(x : A) \u2192 (replicate x \u203c i) \u2261 x\n  \u203c-replicate zero    = \u03bb _ \u2192 idp\n  \u203c-replicate (suc i) = \u203c-replicate i\n\nmodule _ {a b}{A : Type a}{B : Type b} where\n  \u203c-\u229b= : \u2200 {n} i {fs : Vec (A \u2192 B) n}{xs : Vec A n}{f x}\n           (fs= : (fs \u203c i) \u2261 f)\n           (xs= : (xs \u203c i) \u2261 x)\n         \u2192 (fs \u229b xs \u203c i) \u2261 f x\n  \u203c-\u229b= zero    {f \u2237 fs} {x \u2237 xs} f= xs= = ap\u2082 _$_ f= xs=\n  \u203c-\u229b= (suc i) {f \u2237 fs} {x \u2237 xs} f= xs= = \u203c-\u229b= i f= xs=\n\n  \u203c-\u229b : \u2200 {n} i (fs : Vec (A \u2192 B) n) (xs : Vec A n) \u2192\n          (fs \u229b xs \u203c i) \u2261 (fs \u203c i) (xs \u203c i)\n  \u203c-\u229b i fs xs = \u203c-\u229b= i idp idp\n\n  \u203c-map= : \u2200 {n} i (f : A \u2192 B) {xs : Vec A n}{x : A}\n             (xs= : (xs \u203c i) \u2261 x)\n           \u2192 (map f xs \u203c i) \u2261 f x\n  \u203c-map= i f = \u203c-\u229b= i (\u203c-replicate i f)\n\n  \u203c-map : \u2200 {n} i (f : A \u2192 B) (xs : Vec A n) \u2192\n            (map f xs \u203c i) \u2261 f (xs \u203c i)\n  \u203c-map i f xs = \u203c-map= i f idp\n\n  \u229b-replicate : \u2200 {n}(fs : Vec (A \u2192 B) n)(x : A) \u2192 fs \u229b replicate x \u2261 map (\u03bb f \u2192 f x) fs\n  \u229b-replicate []       x = idp\n  \u229b-replicate (f \u2237 fs) x = ap (_\u2237_ (f x)) (\u229b-replicate fs x)\n\n  vuncurry : \u2200 {n}(f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\n  vuncurry f (x \u2237 xs) = f x xs\n\n  count-\u2218 : \u2200 {n}(f : A \u2192 B)(pred : B \u2192 Bool) \u2192\n              count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\n  count-\u2218 f pred [] = idp\n  count-\u2218 f pred (x \u2237 xs) with pred (f x)\n  ... | 1b rewrite count-\u2218 f pred xs = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                 | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                 | count-\u2218 f pred xs = idp\n\nmodule _ {a}{A : Type a} where\n  \u229b-take-drop-lem : \u2200 m {n o} (xs : Vec (Vec A (m + n)) o)\n       \u2192 map _++_ (map (take m) xs) \u229b (map (drop m) xs) \u2261 xs\n  \u229b-take-drop-lem m xs = vec= \u03bb i \u2192\n    \u203c-\u229b= i (\u203c-map= i _ (\u203c-map i _ _)) (\u203c-map i _ _) \u2219\n    take-drop-lem m _\n\n  take2*-++ : \u2200 n b (xs ys : Vec A n)\n              \u2192 take2* n b (xs ++ ys) \u2261 proj\u2032 (xs , ys) b\n  take2*-++ n 0b xs ys = take-++ n xs ys\n  take2*-++ n 1b xs ys = drop-++ n xs ys\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (Type)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd; map to \u00d7-map)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; _\u2219_; !_) renaming (refl to idp)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : Type a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 Type a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : Type a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Type a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Type a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id= : \u2200 {a n} {A : Type a}{f : A \u2192 A}\n             \u2192 f \u2257 id \u2192 map f \u2257 id {A = Vec A n}\n    map-id= f= []       = idp\n    map-id= f= (x \u2237 xs) = ap-\u2237 (f= x) (map-id= f= xs)\n\n    map-id : \u2200 {a n} {A : Type a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id = map-id= (\u03bb _ \u2192 idp)\n\n    map-\u2218= : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n               {f : B \u2192 C}{g : A \u2192 B}{h : A \u2192 C} \u2192\n               f \u2218 g \u2257 h \u2192\n               _\u2257_ {A = Vec A n} (map f \u2218 map g) (map h)\n    map-\u2218= fg= []       = idp\n    map-\u2218= fg= (x \u2237 xs) = ap-\u2237 (fg= x) (map-\u2218= fg= xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n              (f : B \u2192 C) (g : A \u2192 B) \u2192\n              _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g v = ! map-\u2218= (\u03bb _ \u2192 idp) v\n\n    map-ext : \u2200 {a b} {A : Type a} {B : Type b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : Type a}\n    (_\u2248_ : A \u2192 A \u2192 Type \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Type \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 Type _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : Type a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= idp idp = idp\n\nmodule With\u2261 {a}{A : Type a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                           ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Type a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Type a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Type a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\ncount\u1da0 : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : Type a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : Type a} where\n  count-++ : \u2200 {m n} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m b} {B : Type b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , idp = idp\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , idp = idp\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  map2*-++ : \u2200 {m n}(f : Vec A m \u2192 Vec A n) \u2192 map2* m n f \u2218 uncurry _++_ \u2257 uncurry _++_ \u2218 \u00d7-map f f\n  map2*-++ {m} f (x , y) rewrite take-++ m x y | drop-++ m x y = idp\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\n  vec= : \u2200 {n}{xs ys : Vec A n} \u2192 (\u2200 i \u2192 (xs \u203c i) \u2261 (ys \u203c i)) \u2192 xs \u2261 ys\n  vec= {xs = []}     {[]}     f= = idp\n  vec= {xs = x \u2237 xs} {y \u2237 ys} f= = ap-\u2237 (f= zero) (vec= (f= \u2218 suc))\n\n  concat-group : \u2200 m n (xs : Vec A (m * n)) \u2192 concat (group m n xs) \u2261 xs\n  concat-group zero    n [] = idp\n  concat-group (suc m) n xs rewrite concat-group m n (drop n xs) = take-drop-lem n xs\n\n  group-concat : \u2200 {m n}(xss : Vec (Vec A n) m) \u2192 group m n (concat xss) \u2261 xss\n  group-concat [] = idp\n  group-concat (xs \u2237 xss)\n    rewrite take-++ _ xs (concat xss)\n          | drop-++ _ xs (concat xss)\n          | group-concat xss\n          = idp\n\n  -- These are also in the stdlib as Data.Vec.Properties.lookup-morphism\n  \u203c-replicate : \u2200 {n}(i : Fin n)(x : A) \u2192 (replicate x \u203c i) \u2261 x\n  \u203c-replicate zero    = \u03bb _ \u2192 idp\n  \u203c-replicate (suc i) = \u203c-replicate i\n\nmodule _ {a b}{A : Type a}{B : Type b} where\n  \u203c-\u229b= : \u2200 {n} i {fs : Vec (A \u2192 B) n}{xs : Vec A n}{f x}\n           (fs= : (fs \u203c i) \u2261 f)\n           (xs= : (xs \u203c i) \u2261 x)\n         \u2192 (fs \u229b xs \u203c i) \u2261 f x\n  \u203c-\u229b= zero    {f \u2237 fs} {x \u2237 xs} f= xs= = ap\u2082 _$_ f= xs=\n  \u203c-\u229b= (suc i) {f \u2237 fs} {x \u2237 xs} f= xs= = \u203c-\u229b= i f= xs=\n\n  \u203c-\u229b : \u2200 {n} i (fs : Vec (A \u2192 B) n) (xs : Vec A n) \u2192\n          (fs \u229b xs \u203c i) \u2261 (fs \u203c i) (xs \u203c i)\n  \u203c-\u229b i fs xs = \u203c-\u229b= i idp idp\n\n  \u203c-map= : \u2200 {n} i (f : A \u2192 B) {xs : Vec A n}{x : A}\n             (xs= : (xs \u203c i) \u2261 x)\n           \u2192 (map f xs \u203c i) \u2261 f x\n  \u203c-map= i f = \u203c-\u229b= i (\u203c-replicate i f)\n\n  \u203c-map : \u2200 {n} i (f : A \u2192 B) (xs : Vec A n) \u2192\n            (map f xs \u203c i) \u2261 f (xs \u203c i)\n  \u203c-map i f xs = \u203c-map= i f idp\n\n  vuncurry : \u2200 {n}(f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\n  vuncurry f (x \u2237 xs) = f x xs\n\n  count-\u2218 : \u2200 {n}(f : A \u2192 B)(pred : B \u2192 Bool) \u2192\n              count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\n  count-\u2218 f pred [] = idp\n  count-\u2218 f pred (x \u2237 xs) with pred (f x)\n  ... | 1b rewrite count-\u2218 f pred xs = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                 | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                 | count-\u2218 f pred xs = idp\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1e082c5262a59293057a9eeb0e687b584640b728","subject":"adding new files","message":"adding new files\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import core\nopen import lemmas-consistency\nopen import lemmas-contexts\nopen import lemmas-matching\nopen import synth-unicity\n\n-- open import correspondence\n-- open import expandability\n-- open import expansion-unicity\n-- open import preservation\n-- open import progress\n-- open import type-assignment-unicity\n-- open import typed-expansion\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import core\n-- open import correspondence\n-- open import expandability\n-- open import expansion-unicity\n-- open import preservation\n-- open import progress\n-- open import type-assignment-unicity\n-- open import typed-expansion\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1e80c580a94d19a680ecf353f02381f2f4328cc2","subject":"proved that boxedvalues arent errors #14","message":"proved that boxedvalues arent errors #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    -- todo: also being used below\n    lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    --- todo: okay this is now getting reused here and in the case of ie\n    --- below, so this is apparently more of an interesting property than\n    --- i'd thought\n    lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n    lem (CECastFail x\u2081 x\u2082 () x\u2084)\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVVal x) (CECong FHOuter er) = ve (BVVal x) er\n  ve (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  ve (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  ve (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  ve (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c14eb81dfab82a646b29c89bdd39e3fa6ea64b9b","subject":"almost got #6","message":"almost got #6\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-progress.agda","new_file":"complete-progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import progress\nopen import htype-decidable\n\nmodule complete-progress where\n\n  data okc : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 okc d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 okc d \u0394\n\n  lem-ind-comp : \u2200{d} \u2192 d dcomplete \u2192 d indet \u2192 \u22a5\n  lem-ind-comp DCVar ()\n  lem-ind-comp DCConst ()\n  lem-ind-comp (DCLam comp x\u2081) ()\n  lem-ind-comp (DCAp comp comp\u2081) (IAp x ind x\u2081) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastArr x\u2082 ind) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastGroundHole x\u2082 ind) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastHoleGround x\u2082 ind x\u2083) = lem-ind-comp comp ind\n  lem-ind-comp (DCFailedCast comp x x\u2081) (IFailedCast x\u2082 GBase GBase x\u2085) = x\u2085 refl\n  lem-ind-comp (DCFailedCast comp x (TCArr () x\u2082)) (IFailedCast x\u2083 GBase GHole x\u2085)\n  lem-ind-comp (DCFailedCast comp (TCArr x ()) x\u2082) (IFailedCast x\u2083 GHole GBase x\u2085)\n  lem-ind-comp (DCFailedCast comp x x\u2081) (IFailedCast x\u2082 GHole GHole x\u2085) = x\u2085 refl\n\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n                       \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                       d dcomplete \u2192\n                       okc d \u0394\n  complete-progress wt comp with progress wt\n  complete-progress wt comp | S x = S x\n  complete-progress wt comp | BV (BVVal x) = V x\n\n  complete-progress wt comp | I x = abort (lem-ind-comp comp x)\n  complete-progress (TACast wt x) (DCCast comp x\u2083 ()) | BV (BVHoleCast x\u2081 x\u2082)\n  complete-progress (TACast wt con) (DCCast comp (TCArr x\u2083 x\u2084) (TCArr x\u2085 x\u2086)) | BV (BVArrCast x\u2081 x\u2082) = {!!}\n  --   with complete-progress wt comp\n  -- complete-progress (TACast wt con) (DCCast comp (TCArr x\u2083 x\u2084) (TCArr x\u2085 x\u2086)) | BV (BVArrCast x\u2081 x\u2082) | V x = {!!}\n  -- complete-progress (TACast wt con) (DCCast comp (TCArr x\u2083 x\u2084) (TCArr x\u2085 x\u2086)) | BV (BVArrCast x\u2081 x\u2082) | S x = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-progress where\n\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n\n  -- nb: don't need (\u0394 , \u2205 \u22a2 d :: \u03c4) because of typed-expansion\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} {e : hexp}\u2192\n                       ecomplete e \u2192\n                       \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                       ok d \u0394\n  complete-progress = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c6c4af59fa8878c0fbe55afd2a6af2fbf5e0a857","subject":"IDesc model: polymorphize the Prelude","message":"IDesc model: polymorphize the Prelude","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i : Level}(A B : Set i) -> Set i\nA * B = Sigma A \\_ -> B\n\nfst : {i : Level}{A : Set i}{B : A -> Set i} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i : Level}{A : Set i}{B : A -> Set i} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i : Level}(A B : Set i) : Set i where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"73ab7604d37cca78023efdb6c5e6ba02ec1f9e26","subject":"Import dependencies more explicitly.","message":"Import dependencies more explicitly.\n\nThis change will allow to instantiate whole modules and thereby\nimprove type inference.\n\nOld-commit-hash: 87c06945274f6c3847e53ebab7cc2a3bdb49fc51\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\nopen import Syntax.Type.Plotkin\n\ndata Atlas-const : Type Atlas-type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nopen import Syntax.Context\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nopen import Syntax.Term.Plotkin\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n  Atlas-term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nopen import Syntax.Language.Calculus\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Type Atlas-type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n  Atlas-term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5ef62706bb4aa429b2c982c7b4d3091c3f768229","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: f350ec11dff554ac99feb79c78632ccf\n\ndarcs-hash:20110409035915-3bd4e-c1257bc5a5fb7f271708a77c4f19511e36b6f251.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/RenderToHTML.agda","new_file":"Draft\/RenderToHTML.agda","new_contents":"------------------------------------------------------------------------------\n-- Draft modules render to HTML\n------------------------------------------------------------------------------\n\nmodule Draft.RenderToHTML where\n\nopen import Draft.FOTC.Program.McCarthy91.Properties.MainATP\n","old_contents":"------------------------------------------------------------------------------\n-- Draft modules render to HTML\n------------------------------------------------------------------------------\n\nmodule Draft.RenderToHTML where\n\nopen import Draft.FOTC.Program.McCarthy91.Properties.MainATP\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b261fb8c61da4cc6364f1db5d6a293829b5300b5","subject":"flipping the order of ~ in the conclusion of the analytic case saves a little bit of a hassle in the synthetic case and makes the proof easier to read; kind of fussy details but seems OK.","message":"flipping the order of ~ in the conclusion of the analytic case saves a little bit of a hassle in the synthetic case and makes the proof easier to read; kind of fussy details but seems OK.\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\nopen import lemmas-consistency\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub = {!!}\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D D\u2081) = TAAp (lem-weaken\u03941 D) (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a1 \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a1 \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4} D = tr (\u03bb q \u2192 q , \u0393 \u22a2 d :: \u03c4) (\u222acomm \u03942 \u03941) (lem-weaken\u03941 D)\n\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp {\u03941 = \u03941} x\u2081 x\u2082 x\u2083 x\u2084)\n      with typed-expansion-ana x\u2083 | typed-expansion-ana x\u2084\n    ... | con1 , ih1 | con2 , ih2  = TAAp (TACast (lem-weaken\u03941 ih1) con1) (TACast (lem-weaken\u03942 {\u03941 = \u03941 }ih2) con2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-synth (ESAsc x)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ~sym x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole1 , TALam ih\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\nopen import lemmas-consistency\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub = {!!}\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D D\u2081) = TAAp (lem-weaken\u03941 D) (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a1 \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a1 \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4} D = tr (\u03bb q \u2192 q , \u0393 \u22a2 d :: \u03c4) (\u222acomm \u03942 \u03941) (lem-weaken\u03941 D)\n\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp {\u03941 = \u03941} x\u2081 x\u2082 x\u2083 x\u2084)\n      with typed-expansion-ana x\u2083 | typed-expansion-ana x\u2084\n    ... | con1 , ih1 | con2 , ih2  = TAAp (TACast (lem-weaken\u03941 ih1) (~sym con1)) (TACast (lem-weaken\u03942 {\u03941 = \u03941 }ih2) (~sym con2)) --todo: is there a notational inconsistency that means we need this ~sym here, or is it actually important?\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-synth (ESAsc x)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih (~sym con) --todo: ditto above\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e7ba95c444768b5dbf31005525e9135ffb935ed5","subject":"fixing a typo in my transcription of a rule, removing the todo","message":"fixing a typo in my transcription of a rule, removing the todo\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192 -- todo: why does \u0394 get thrown away here?\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u2205\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d82e4ba3970fcbb3b576a088f5e032b76234177c","subject":"Improved an example.","message":"Improved an example.\n\nIgnore-this: c5fd2be8b129aa1c7b9543c815ab3a5e\n\ndarcs-hash:20100716163019-3bd4e-8d454cb9bbbb9d152ac0f174f4294ccb6a3d680a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  -- N.B. It is not necessary to define neither the data constructor\n  -- _,_, nor the projections -- because the ATPs implement it.\n  data _\u2227_ (A B : Set) : Set where\n\n  -- Testing the data constructor and the projections.\n  postulate\n    A B     : Set\n    d e     : D\n    _,_     : A \u2192 B \u2192 A \u2227 B\n    \u2227-proj\u2081 : A \u2227 B \u2192 A\n    \u2227-proj\u2082 : A \u2227 B \u2192 B\n  {-# ATP prove _,_ #-}\n  {-# ATP prove \u2227-proj\u2081 #-}\n  {-# ATP prove \u2227-proj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to define neither the data constructors\n  -- inj\u2081 and inj\u2082 nor the disyunction elimination because the ATP\n  -- implements them.\n  data _\u2228_ (A B : Set) : Set where\n\n  -- Testing the data constructors and the elimination.\n  postulate\n    A B C : Set\n    inj\u2081  : A \u2192 A \u2228 B\n    inj\u2082  : B \u2192 A \u2228 B\n    [_,_] : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  {-# ATP prove inj\u2081 #-}\n  {-# ATP prove inj\u2082 #-}\n  {-# ATP prove [_,_] #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\nmodule ExistentialQuantifier where\n\n  data \u2203D (P : D \u2192 Set) : Set where\n\n  postulate\n    test : (d : D) \u2192 \u2203D (\u03bb e \u2192 e \u2261 d)\n  {-# ATP prove test #-}\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  infixr 4 _,_\n  infixr 2 _\u00d7_\n\n  -- We want to use the product type to define the conjunction type\n  record _\u00d7_ (A B : Set) : Set where\n    constructor _,_\n    field\n      proj\u2081 : A\n      proj\u2082 : B\n\n  -- N.B. It is not necessary to add the constructor _,_, nor the fields\n  -- proj\u2081, proj\u2082 as hints, because the ATP implements it.\n\n  _\u2227_ : (A B : Set) \u2192 Set\n  A \u2227 B = A \u00d7 B\n\n  postulate\n    P   : D \u2192 Set\n    d e : D\n\n  -- Testing the conjunction data constructor\n  postulate\n    testAndDataConstructor : P d \u2192 P e \u2192 P d \u2227 P e\n  {-# ATP prove testAndDataConstructor #-}\n\n  -- Testing the first projection\n  postulate\n    testProj\u2081 : P d \u2227 P e \u2192 P d\n  {-# ATP prove testProj\u2081 #-}\n\n  -- Testing the second projection\n  postulate\n    testProj\u2082 : P d \u2227 P e \u2192 P e\n  {-# ATP prove testProj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true  : D\n    false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to add the constructors inj\u2081 and inj\u2082\n  -- nor the elimantion [_,_] as hints, because the ATP implements it.\n\n  data _\u2228_ (A B : Set) : Set where\n    inj\u2081 : (x : A) \u2192 A \u2228 B\n    inj\u2082 : (y : B) \u2192 A \u2228 B\n\n  [_,_] : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  [ f , g ] (inj\u2081 x) = f x\n  [ f , g ] (inj\u2082 y) = g y\n\n  postulate\n    P   : D \u2192 Set\n    d e f : D\n\n  -- Testing the disjunction data constructors\n  postulate\n    inj\u2081Or : P d \u2192 P d \u2228 P e\n    inj\u2082Or : P e \u2192 P d \u2228 P e\n  {-# ATP prove inj\u2081Or #-}\n  {-# ATP prove inj\u2082Or #-}\n\n  -- Testing the disjunction elimination\n  postulate\n    A : Set\n    B : Set\n    orElim :  (P d \u2192 P f) \u2192 (P e \u2192 P f) \u2192 P d \u2228 P e \u2192 P f\n  {-# ATP prove orElim #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\n  module ExistentialQuantifier where\n\n    data \u2203D (P : D \u2192 Set) : Set where\n\n    postulate\n      test1 : (d : D) \u2192 \u2203D (\u03bb e \u2192 e \u2261 d)\n    {-# ATP prove test1 #-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"aa20c4a4543201d892fbb307db07fb7e51d4dc27","subject":"Require related closures to share typing context","message":"Require related closures to share typing context\n\nThis makes the logical relation more restrictive, but is necessary if we want to\nbe able to update a function with a valid function change by combining their\nenvironments.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3[ \u2261\u0393 \u2208 \u03931 \u2261 \u03932 ]\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT\n        t1\n        (subst (\u03bb \u0393 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4) (sym \u2261\u0393) t2)\n        (v1 \u2022 \u03c11)\n        (subst (\u03bb \u0393 \u2192 \u27e6 \u03c3 \u2022 \u0393 \u27e7Context) (sym \u2261\u0393) (v2 \u2022 \u03c12))\n        k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 {!dt!} t2 (v1 \u2022 \u03c11) d\u03c1 (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 {!dt!} t2 (v1 \u2022 \u03c11) d\u03c1 (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n zero k\u2264n v1 v2 x = tt\n  example1 n (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl = intV 1 , 0 , refl , (I.+ 1 , refl)\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n zero k\u2264n v1 v2 x = tt\n  example2 n (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl = intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a3d0049fb45bc75086f915b54b406e2a10843eaf","subject":"Remove cruft","message":"Remove cruft\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n zero k\u2264n v1 v2 x = tt\n  example1 n (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl = intV 1 , 0 , refl , (I.+ 1 , refl)\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n zero k\u2264n v1 v2 x = tt\n  example2 n (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl = intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n zero k\u2264n v1 v2 x = tt\n  example1 n (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl = intV 1 , 0 , refl , (I.+ 1 , refl)\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n zero k\u2264n v1 v2 x = tt\n  example2 n (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl = intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24517a1c7f11ed6d26507675d66237b0677383d0","subject":"Removing two postulates from a draft about the existential quantifier.","message":"Removing two postulates from a draft about the existential quantifier.\n\nIgnore-this: b196e786ad005b27f0400ae93aaa6b85\n\ndarcs-hash:20120423222959-3bd4e-66111d69ea59d3a2fdd45cd6c8367349049dfd69.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/PA\/Inductive\/Witness.agda","new_file":"Draft\/PA\/Inductive\/Witness.agda","new_contents":"-- Tested with FOT on 23 April 2012.\n\nmodule Draft.PA.Inductive.Witness where\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n\nopen import Common.FOL.FOL using ( \u22a5-elim ; _\u2228_ ; [_,_] )\n\nopen import PA.Inductive.Base\nopen import PA.Inductive.Properties\n\n------------------------------------------------------------------------------\n-- The principle of the excluded middle.\npostulate pem : \u2200 {A} \u2192 A \u2228 \u00ac A\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on M\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\ndata \u2203 (A : M \u2192 Set) : Set where\n  _,_ : (x : M) \u2192 A x \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 M\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Non-intuitionistic logic theorems\n\n-- The principle of indirect proof (proof by contradiction).\n\u00ac-elim : \u2200 {A} \u2192 (\u00ac A \u2192 \u22a5) \u2192 A\n\u00ac-elim h = [ (\u03bb a \u2192 a) , (\u03bb \u00aca \u2192 \u22a5-elim (h \u00aca)) ] pem\n\n-- \u2203 in terms of \u2200 and \u00ac.\n\u00ac\u2203\u00ac\u2192\u2200 : {A : M \u2192 Set} \u2192 \u00ac (\u2203[ x ] \u00ac A x) \u2192 \u2200 {x} \u2192 A x\n\u00ac\u2203\u00ac\u2192\u2200 h {x} = \u00ac-elim (\u03bb h\u2081 \u2192 h (x , h\u2081))\n\n------------------------------------------------------------------------------\n\n-- Constructive proof.\nproof\u2081 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2081 = succ zero , x\u2262Sx\n\n-- Non-constructive proof.\nproof\u2082 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2082 = \u00ac-elim (\u03bb h \u2192 x\u2262Sx {succ zero} (\u00ac\u2203\u00ac\u2192\u2200 h))\n\n-- We can extract an existential witness from a constructive proof.\nwitness\u2081 : \u2203-proj\u2081 proof\u2081 \u2261 succ zero\nwitness\u2081 = refl\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n-- witness\u2082 : \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n-- witness\u2082 = {!refl!}\n\n-- Agda error:\n--\n-- \u2203-proj\u2081 (\u00ac-elim (\u03bb h \u2192 x\u2262Sx (\u00ac\u2203\u00ac\u2192\u2200 h))) != succ zero of type M\n-- when checking that the expression refl has type\n-- \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n","old_contents":"-- Tested with FOT on 23 April 2012.\n\nmodule Draft.PA.Inductive.Witness where\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n\nopen import PA.Inductive.Base\nopen import PA.Inductive.Properties\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on M\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\ndata \u2203 (A : M \u2192 Set) : Set where\n  _,_ : (x : M) \u2192 A x \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 M\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Non-intuitionistic logic theorems\n\npostulate\n  -- The principle of indirect proof (proof by contradiction).\n  \u00ac-elim : \u2200 {A} \u2192 (\u00ac A \u2192 \u22a5) \u2192 A\n\n  -- \u2203 in terms of \u2200 and \u00ac.\n  \u00ac\u2203\u00ac\u2192\u2200 : {A : M \u2192 Set} \u2192 \u00ac (\u2203[ x ] \u00ac A x) \u2192 \u2200 {x} \u2192 A x\n\n------------------------------------------------------------------------------\n\n-- Constructive proof.\nproof\u2081 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2081 = succ zero , x\u2262Sx\n\n-- Non-constructive proof.\nproof\u2082 : \u2203[ x ] \u00ac (x \u2261 succ x)\nproof\u2082 = \u00ac-elim (\u03bb h \u2192 x\u2262Sx {succ zero} (\u00ac\u2203\u00ac\u2192\u2200 h))\n\n-- We can extract an existential witness from a constructive proof.\nwitness\u2081 : \u2203-proj\u2081 proof\u2081 \u2261 succ zero\nwitness\u2081 = refl\n\n-- We cannot extract an existential witness from a non-constructive\n-- proof.\n-- witness\u2082 : \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n-- witness\u2082 = {!refl!}\n\n-- Agda error:\n--\n-- \u2203-proj\u2081 (\u00ac-elim (\u03bb h \u2192 x\u2262Sx (\u00ac\u2203\u00ac\u2192\u2200 h))) != succ zero of type M\n-- when checking that the expression refl has type\n-- \u2203-proj\u2081 proof\u2082 \u2261 succ zero\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"58f4fa214575cabd73290c7ae6a92b08bfdbbecb","subject":"prg: some notes about a reduction from xoring with a PRG to breaking the PRG","message":"prg: some notes about a reduction from xoring with a PRG to breaking the PRG\n","repos":"crypto-agda\/crypto-agda","old_file":"prg.agda","new_file":"prg.agda","new_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search _\u2228_ {k} ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search _\u2227_ {k} (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = search-hom _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n\nmodule PRG\u2141\u2082 {k n |R|\u1d43 : \u2115} (PRG : PRGFun k n) where\n\n    module AbsAdv {t} {T : Set t} (funU : FunUniverse T) where\n        open FunUniverse funU\n\n        M   =  Bits n\n        K   =  Bits k\n        `M  = `Bits n\n        `K  = `Bits k\n        R\u1d43  =  Bits |R|\u1d43\n        `R\u1d43 = `Bits |R|\u1d43\n\n        Adv : Set\n        Adv = `R\u1d43 `\u00d7 `M `\u2192 `Bit\n\n        {-\n    red : SemSecAdv |R|\u1d43 \u2192 PRGAdv |R|\u1d43\n    red (mk A\u2080 A\u2081 A\u2082 A\u2083) = \u03bb (r\u1d43 , p) \u2192 let (m , s\u2081) = A\u2081 (A\u2080 r\u1d43) in\n                                       let (b , _) = A\u2082 (p \u2295 m 0b , s\u2081) in b\n                                       -- if b \u2261 0b it means the adversary won\n                                       -- so it's probably the case that p was \"PRG \u2047\"\n\n  -- the semsec game, here adv A is playing against the Encrypt\u2218fst or Encrypt\u2218snd\n  E\u21c4 A : \u2141?\n  E\u21c4 A b = A(E(m b)) -- m comes from A as well\n\n  -- the PRG game, here adv A is playing against \u2047 or PRG\n  \u2047\u21c4 A : \u2141?\n  \u2047\u21c4 A 0b = A PRG\u2047\n  \u2047\u21c4 A 1b = A \u2047\n\n  E' p m = p \u2295 m\n  redA p = \u27e8E' p\u27e9\u21c4 A 0b\n\n  _\u2261#_ -- same count\n  f \u2261# g = #f \u2261 #g\n\n  -- The one time pad cipher\n  OTP m = \u2047 \u2295 m\n\n  -- OTP has the same count than \u2047\n  OTP m \u2261# ??\n  OTP\u21c4 A b \u2261# \u2047\n  OTP\u21c4 A 0b \u2261# OTP\u21c4 A 1b\n\n  -- adv A breaking encryption E implies adv redA breaking PRG\n  E\u21c4 A \u21d3 \u2047\u21c4 redA\n\n  breaks (E\u21c4 A) \u2261 E\u21c4 A 0b ]-[ E\u21c4 A 1b\n  breaks (\u2047\u21c4 redA) \u2261 \u2047\u21c4 redA 0b ]-[ \u2047\u21c4 redA 1b\n                     \u2261 redA PRG\u2047   ]-[ redA \u2047\n\n  \u2047\u21c4 redA 0b \u2261  redA PRG\u2047\n               \u2261  \u27e8E' PRG\u2047\u27e9\u21c4 A 0b\n               \u2261  E\u21c4 A 0b\n              ]-[ E\u21c4 A 1b\n\n  \u2047\u21c4 redA 1b \u2261  redA \u2047\n               \u2261  \u27e8E' \u2047\u27e9\u21c4 A 0b\n               \u2261  OTP\u21c4 A 0b\n               \u2261# OTP\u21c4 A 1b\n               \u2261  \u27e8E' \u2047\u27e9\u21c4 A 1b\n\n  Goal:\n    breaks (\u2047\u21c4 redA) \u21d4 \u2047\u21c4 redA 0b ]-[ \u2047\u21c4 redA 1b\n    -}\n","old_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search _\u2228_ {k} ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search _\u2227_ {k} (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = search-hom _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0bbd2ffd4c6ec84dbfec05471761c2530bb1bd5e","subject":"Only indentation.","message":"Only indentation.\n\nIgnore-this: a51f44620f5cd7506e54250951451c44\n\ndarcs-hash:20100524030049-3bd4e-f2020aed700f1c5e485835df0ca3f6280c49fdca.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Function.Arithmetic.PropertiesER\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0           = true\u2260false (trans (sym 0<0) (lt-00))\n\u00acx<0 (sN {n} Nn) Sn<0 = true\u2260false (trans (sym Sn<0) (lt-S0 n))\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sd\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sd\u22640\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = trans (lt-SS n (succ n)) (x<Sx Nn)\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn) _              = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  begin\n    lt (succ n) (succ m) \u2261\u27e8 lt-SS n m \u27e9\n    lt n m \u2261\u27e8 x<y\u2192x\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn) \u27e9\n    false\n  \u220e\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN               m<0       = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192Sx\u2264y zN (sN zN)          _         = trans (lt-SS zero zero) lt-00\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _         = trans (lt-SS (succ n) zero) (lt-S0 n)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS n (succ m)) (x<y\u2192Sx\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  trans (lt-SS m n) (Sx\u2264y\u2192x<y Nm Nn (trans (sym (lt-SS n (succ m))) SSm\u2264Sn))\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  begin\n    lt (succ m) (succ o) \u2261\u27e8 lt-SS m o \u27e9\n    lt m o \u2261\u27e8 <-trans Nm Nn No\n                       (trans (sym (lt-SS m n)) Sm<Sn)\n                       (trans (sym (lt-SS n o)) Sn<So) \u27e9\n    true\n  \u220e\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  trans (lt-SS o m)\n        (\u2264-trans Nm Nn No\n                 (trans (sym (lt-SS n m)) Sm\u2264Sn)\n                 (trans (sym (lt-SS o n)) Sn\u2264So))\n\nSx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS n m)) Sm\u2264Sn\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m + n) (succ m)   \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m + n) (succ m) \u2261\n                                               lt t (succ m))\n                                        (+-Sx m n)\n                                        refl\n                               \u27e9\n    lt (succ (m + n)) (succ m) \u2261\u27e8 lt-SS (m + n) m \u27e9\n    lt (m + n) m               \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    false\n  \u220e\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN =\n  begin\n    lt (m - zero) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (m - zero) (succ m) \u2261\n                                           lt t (succ m))\n                                    (minus-x0 m)\n                                    refl\n                           \u27e9\n    lt m (succ m)          \u2261\u27e8 x<Sx Nm \u27e9\n    true\n  \u220e\n\nx-y<Sx zN (sN {n} Nn) =\n  begin\n    lt (zero - succ n) (succ zero)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (zero - succ n) (succ zero) \u2261 lt t (succ zero))\n               (minus-0S n)\n               refl\n      \u27e9\n    lt zero (succ zero) \u2261\u27e8 lt-0S zero \u27e9\n    true\n  \u220e\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) =\n  begin\n    lt (succ m - succ n) (succ (succ m))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ (succ m)) \u2261\n                      lt t (succ (succ m)))\n               (minus-SS m n)\n               refl\n      \u27e9\n    lt (m - n) (succ (succ m))\n      \u2261\u27e8 <-trans (minus-N Nm Nn) (sN Nm) (sN (sN Nm))\n                 (x-y<Sx Nm Nn) (x<Sx (sN Nm))\n      \u27e9\n    true\n  \u220e\n\nSx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\nSx-Sy<Sx {m} {n} Nm Nn =\n  begin\n    lt (succ m - succ n) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n)\n                                                     (succ m) \u2261\n                                                  lt t (succ m))\n                                           (minus-SS m n)\n                                           refl\n                                  \u27e9\n    lt (m - n) (succ m)           \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n    \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS m n)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN      Nn 0\u2264n  = trans (+-rightIdentity (minus-N Nn zN))\n                                            (minus-x0 n)\nx\u2264y\u2192y-x+x\u2261y         (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  begin\n    (succ n - succ m) + succ m \u2261\u27e8 subst (\u03bb t \u2192 (succ n - succ m) + succ m \u2261\n                                               t + succ m)\n                                        (minus-SS n m)\n                                        refl\n                               \u27e9\n    (n - m) + succ m           \u2261\u27e8 +-comm (minus-N Nn Nm) (sN Nm) \u27e9\n    succ m + (n - m)           \u2261\u27e8 +-Sx m (n - m) \u27e9\n    succ (m + (n - m))         \u2261\u27e8 subst (\u03bb t \u2192 succ (m + (n - m)) \u2261 succ t )\n                                        (+-comm Nm (minus-N Nn Nm))\n                                        refl\n                               \u27e9\n    succ ((n - m) + m)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((n - m) + m) \u2261 succ t )\n                                        (x\u2264y\u2192y-x+x\u2261y Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm\u2264Sn) )\n                                        refl\n                               \u27e9\n    succ n\n  \u220e\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\n\u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n\u00acxy<00 Nm Nn mn<00 =\n  [ (\u03bb m<0     \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n  ]\n  mn<00\n\n\u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n\u00ac0Sx<00 Nm 0Sm<00 =\n  [ (\u03bb 0<0      \u2192 \u22a5-elim (\u00acx<0 zN 0<0))\n  , (\u03bb 0\u22610\u2227Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) (\u2227-proj\u2082 0\u22610\u2227Sm<0)))\n  ]\n  0Sm<00\n\n\u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n\u00acSxy\u2081<0y\u2082 Nm Nn\u2081 Nn\u2082 Smn\u2081<0n\u2082 =\n  [ (\u03bb Sm<0       \u2192 \u22a5-elim (\u00acx<0 (sN Nm) Sm<0))\n  , (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2260S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n  ]\n  Smn\u2081<0n\u2082\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = inj\u2081 (Sx-Sy<Sx Nm Nn)\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = inj\u2082 (refl , Sx-Sy<Sx Nn Nm)\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Function.Arithmetic.PropertiesER\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0           = true\u2260false (trans (sym 0<0) (lt-00))\n\u00acx<0 (sN {n} Nn) Sn<0 = true\u2260false (trans (sym Sn<0) (lt-S0 n))\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sd\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sd\u22640\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = trans (lt-SS n (succ n)) (x<Sx Nn)\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn) _              = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  begin\n    lt (succ n) (succ m) \u2261\u27e8 lt-SS n m \u27e9\n    lt n m \u2261\u27e8 x<y\u2192x\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn) \u27e9\n    false\n  \u220e\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN               m<0       = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192Sx\u2264y zN (sN zN)          _         = trans (lt-SS zero zero) lt-00\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _         = trans (lt-SS (succ n) zero) (lt-S0 n)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS n (succ m)) (x<y\u2192Sx\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  trans (lt-SS m n) (Sx\u2264y\u2192x<y Nm Nn (trans (sym (lt-SS n (succ m))) SSm\u2264Sn))\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  begin\n    lt (succ m) (succ o) \u2261\u27e8 lt-SS m o \u27e9\n    lt m o \u2261\u27e8 <-trans Nm Nn No\n                       (trans (sym (lt-SS m n)) Sm<Sn)\n                       (trans (sym (lt-SS n o)) Sn<So) \u27e9\n    true\n  \u220e\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  trans (lt-SS o m)\n        (\u2264-trans Nm Nn No\n                 (trans (sym (lt-SS n m)) Sm\u2264Sn)\n                 (trans (sym (lt-SS o n)) Sn\u2264So))\n\nSx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS n m)) Sm\u2264Sn\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m + n) (succ m)   \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m + n) (succ m) \u2261\n                                               lt t (succ m))\n                                        (+-Sx m n)\n                                        refl\n                               \u27e9\n    lt (succ (m + n)) (succ m) \u2261\u27e8 lt-SS (m + n) m \u27e9\n    lt (m + n) m               \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    false\n  \u220e\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN =\n  begin\n    lt (m - zero) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (m - zero) (succ m) \u2261\n                                           lt t (succ m))\n                                    (minus-x0 m)\n                                    refl\n                           \u27e9\n    lt m (succ m)          \u2261\u27e8 x<Sx Nm \u27e9\n    true\n  \u220e\n\nx-y<Sx zN (sN {n} Nn) =\n  begin\n    lt (zero - succ n) (succ zero)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (zero - succ n) (succ zero) \u2261 lt t (succ zero))\n               (minus-0S n)\n               refl\n      \u27e9\n    lt zero (succ zero) \u2261\u27e8 lt-0S zero \u27e9\n    true\n  \u220e\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) =\n  begin\n    lt (succ m - succ n) (succ (succ m))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ (succ m)) \u2261\n                      lt t (succ (succ m)))\n               (minus-SS m n)\n               refl\n      \u27e9\n    lt (m - n) (succ (succ m))\n      \u2261\u27e8 <-trans (minus-N Nm Nn) (sN Nm) (sN (sN Nm))\n                 (x-y<Sx Nm Nn) (x<Sx (sN Nm))\n      \u27e9\n    true\n  \u220e\n\nSx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\nSx-Sy<Sx {m} {n} Nm Nn =\n  begin\n    lt (succ m - succ n) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n)\n                                                     (succ m) \u2261\n                                                  lt t (succ m))\n                                           (minus-SS m n)\n                                           refl\n                                  \u27e9\n    lt (m - n) (succ m)           \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n    \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS m n)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN      Nn 0\u2264n  = trans (+-rightIdentity (minus-N Nn zN))\n                                            (minus-x0 n)\nx\u2264y\u2192y-x+x\u2261y         (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  begin\n    (succ n - succ m) + succ m \u2261\u27e8 subst (\u03bb t \u2192 (succ n - succ m) + succ m \u2261\n                                               t + succ m)\n                                        (minus-SS n m)\n                                        refl\n                               \u27e9\n    (n - m) + succ m           \u2261\u27e8 +-comm (minus-N Nn Nm) (sN Nm) \u27e9\n    succ m + (n - m)           \u2261\u27e8 +-Sx m (n - m) \u27e9\n    succ (m + (n - m))         \u2261\u27e8 subst (\u03bb t \u2192 succ (m + (n - m)) \u2261 succ t )\n                                        (+-comm Nm (minus-N Nn Nm))\n                                        refl\n                               \u27e9\n    succ ((n - m) + m)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((n - m) + m) \u2261 succ t )\n                                        (x\u2264y\u2192y-x+x\u2261y Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm\u2264Sn) )\n                                        refl\n                               \u27e9\n    succ n\n  \u220e\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\n\u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n\u00acxy<00 Nm Nn mn<00 =\n  [ (\u03bb m<0 \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n  ]\n  mn<00\n\n\u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n\u00ac0Sx<00 Nm 0Sm<00 =\n  [ (\u03bb 0<0 \u2192 \u22a5-elim (\u00acx<0 zN 0<0))\n  , (\u03bb 0\u22610\u2227Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) (\u2227-proj\u2082 0\u22610\u2227Sm<0)))\n  ]\n  0Sm<00\n\n\u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n\u00acSxy\u2081<0y\u2082 Nm Nn\u2081 Nn\u2082 Smn\u2081<0n\u2082 =\n  [ (\u03bb Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) Sm<0))\n  , (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2260S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n  ]\n  Smn\u2081<0n\u2082\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = inj\u2081 (Sx-Sy<Sx Nm Nn)\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = inj\u2082 (refl , Sx-Sy<Sx Nn Nm)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9b11adeb704c2ed55a177a1cf21da6a20c39ec2c","subject":"bit-guessing-game: moves stuff to nplib","message":"bit-guessing-game: moves stuff to nplib\n","repos":"crypto-agda\/crypto-agda","old_file":"bit-guessing-game.agda","new_file":"bit-guessing-game.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\nopen import Function\nopen import Data.Bool.NP\n\nopen import Data.Bits\n\nopen import flipbased-implem using (Coins; \u21ba; \u2141) -- ; #\u27e8_\u27e9) renaming (count\u21ba to #\u27e8_\u27e9)\nopen import program-distance using (PrgDist; module PrgDist)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n\nopen PrgDist prgDist\n\nGuessAdv : Coins \u2192 Set\nGuessAdv = \u2141\n\nrunGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u2141 ca\nrunGuess\u2141 A _ = A\n\n-- An oracle: an adversary who can break the guessing game.\nOracle : Coins \u2192 Set\nOracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n-- Any adversary cannot do better than a random guess.\nGuessSec : Coins \u2192 Set\nGuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\nopen import Function\nopen import Data.Bool.NP\n\nopen import Data.Bits\n\nopen import flipbased-implem using (Coins; \u21ba; \u2141) -- ; #\u27e8_\u27e9) renaming (count\u21ba to #\u27e8_\u27e9)\nopen import program-distance using (PrgDist; module PrgDist)\nopen import Relation.Binary.PropositionalEquality\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n\nopen PrgDist prgDist\n\nGuessAdv : Coins \u2192 Set\nGuessAdv = \u2141\n\nrunGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u2141 ca\nrunGuess\u2141 A _ = A\n\n-- An oracle: an adversary who can break the guessing game.\nOracle : Coins \u2192 Set\nOracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n-- Any adversary cannot do better than a random guess.\nGuessSec : Coins \u2192 Set\nGuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\n#\u1d47 : Bool \u2192 \u2115\n#\u1d47 b = if b then 1 else 0\n\n#\u1d47\u22641 : \u2200 b \u2192 #\u1d47 b \u2264 1\n#\u1d47\u22641 true = s\u2264s z\u2264n\n#\u1d47\u22641 false = z\u2264n\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = #\u1d47\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot zero f = refl\n#-\u2218vnot (suc c) f\n  rewrite #-\u2218vnot c (f \u2218 0\u2237_)\n        | #-\u2218vnot c (f \u2218 1\u2237_) = \u2115\u00b0.+-comm #\u27e8 f \u2218 0\u2237_ \u2218 vnot \u27e9 _\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true = refl\n... | false = refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = helper c #\u27e8 not \u2218 f \u2218 0\u2237_ \u27e9 _ (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n  where helper : \u2200 c x y \u2192 x \u2264 2^ c \u2192 y \u2264 2^ c \u2192 (2^ c \u2238 x) + (2^ c \u2238 y) \u2261 2^(1 + c) \u2238 (x + y)\n        helper zero x y x\u2264 y\u2264 = {!!}\n        helper (suc c\u2081) x y x\u2264 y\u2264 = {!helper c\u2081 x y ? ?!}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f8385608bd9eb6f1043f6fd2979b3f64fd2c21d0","subject":"Bool: T and \u2261","message":"Bool: T and \u2261\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\n\u2261\u2192T : \u2200 {b} \u2192 b \u2261 true \u2192 T b\n\u2261\u2192T \u2261.refl = _\n\n\u2261\u2192Tnot : \u2200 {b} \u2192 b \u2261 false \u2192 T (not b)\n\u2261\u2192Tnot \u2261.refl = _\n\nT\u2192\u2261 : \u2200 {b} \u2192 T b \u2192 b \u2261 true\nT\u2192\u2261 {true}  _  = \u2261.refl\nT\u2192\u2261 {false} ()\n\nTnot\u2192\u2261 : \u2200 {b} \u2192 T (not b) \u2192 b \u2261 false\nTnot\u2192\u2261 {false} _  = \u2261.refl\nTnot\u2192\u2261 {true}  ()\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s)\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fbc86b8148cb9d9d9a3add7b4f0b2fd5346bc6e2","subject":"Desc model: iso1 implicit.","message":"Desc model: iso1 implicit.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : {I : Set} -> IDescl I -> IDesc I\niso1 {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"973a194f1e408a138980b8d9e4156302e223bc75","subject":"Define a correct nil on values and environments","message":"Define a correct nil on values and environments\n\nCompose is trickier because I must match on two validity proofs, need an\ninversion lemma for that.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/DeriveCorrect.agda","new_file":"Thesis\/SIRelBigStep\/DeriveCorrect.agda","new_contents":"--\n-- In this module we conclude our proof: we have shown that t, derive-dterm t\n-- and produce related values (in the sense of the fundamental property,\n-- rfundamental3), and for related values v1, dv and v2 we have v1 \u2295 dv \u2261\n-- v2 (because \u2295 agrees with validity, rrelV3\u2192\u2295).\n--\n-- We now put these facts together via derive-correct-si and derive-correct.\n-- This is immediate, even though I spend so much code on it:\n-- Indeed, all theorems are immediate corollaries of what we established (as\n-- explained above); only the statements are longer, especially because I bother\n-- expanding them.\n\nmodule Thesis.SIRelBigStep.DeriveCorrect where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\nopen import Thesis.SIRelBigStep.FundamentalProperty\n\n-- Theorem statement. This theorem still mentions step-indexes explicitly.\nderive-correct-si-type =\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4) \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3-skeleton (\u03bb v1 dv v2 _ \u2192 v1 \u2295 dv \u2261 v2) t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\n-- A verified expansion of the theorem statement.\nderive-correct-si-type-means :\n  derive-correct-si-type \u2261\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j (j<k : j < k) \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct-si-type-means = refl\n\nderive-correct-si : derive-correct-si-type\nderive-correct-si t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rrelT3\u2192\u2295 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 (rfundamental3 _ t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1)\n\n-- Infinitely related environments:\nrrel\u03c13-inf : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12 = \u2200 k \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k\n\n-- A theorem statement for infinitely-related environments. On paper, if we informally\n-- omit the arbitrary derivation step-index as usual, we get a statement without\n-- step-indexes.\nderive-correct :\n  \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j = derive-correct-si t \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 (suc j)) v1 v2 j \u2264-refl\n\nopen import Data.Unit\n\nnil\u03c1 : \u2200 {\u0393 n} (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c1 d\u03c1 \u03c1 n\nnilV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\nnilV (closure t \u03c1) = let d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dclosure (derive-dterm t) \u03c1 d\u03c1 , rfundamental3svv _ (abs t) \u03c1 d\u03c1 \u03c1 \u03c1\u03c1\nnilV (natV n\u2081) = dnatV zero , refl\nnilV (pairV a b) = let 0a , aa = nilV a; 0b , bb = nilV b in dpairV 0a 0b , aa , bb\n\nnil\u03c1 \u2205 = \u2205 , tt\nnil\u03c1 (v \u2022 \u03c1) = let dv , vv = nilV v ; d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n","old_contents":"--\n-- In this module we conclude our proof: we have shown that t, derive-dterm t\n-- and produce related values (in the sense of the fundamental property,\n-- rfundamental3), and for related values v1, dv and v2 we have v1 \u2295 dv \u2261\n-- v2 (because \u2295 agrees with validity, rrelV3\u2192\u2295).\n--\n-- We now put these facts together via derive-correct-si and derive-correct.\n-- This is immediate, even though I spend so much code on it:\n-- Indeed, all theorems are immediate corollaries of what we established (as\n-- explained above); only the statements are longer, especially because I bother\n-- expanding them.\n\nmodule Thesis.SIRelBigStep.DeriveCorrect where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\nopen import Thesis.SIRelBigStep.FundamentalProperty\n\n-- Theorem statement. This theorem still mentions step-indexes explicitly.\nderive-correct-si-type =\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4) \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3-skeleton (\u03bb v1 dv v2 _ \u2192 v1 \u2295 dv \u2261 v2) t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\n-- A verified expansion of the theorem statement.\nderive-correct-si-type-means :\n  derive-correct-si-type \u2261\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j (j<k : j < k) \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct-si-type-means = refl\n\nderive-correct-si : derive-correct-si-type\nderive-correct-si t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rrelT3\u2192\u2295 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 (rfundamental3 _ t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1)\n\n-- Infinitely related environments:\nrrel\u03c13-inf : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12 = \u2200 k \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k\n\n-- A theorem statement for infinitely-related environments. On paper, if we informally\n-- omit the arbitrary derivation step-index as usual, we get a statement without\n-- step-indexes.\nderive-correct :\n  \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j = derive-correct-si t \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 (suc j)) v1 v2 j \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1da0c206672cf5880079e99b71f1940b6df8932e","subject":"Implement NbE for lambda calculus with conditionals (see #50).","message":"Implement NbE for lambda calculus with conditionals (see #50).\n\nThis version is simpler than the code in NormalizationByEvaluation\nin the following ways:\n\n - Instead of supporting arbitrary base types and constants,\n   bool, true, false, and if are hard-coded.\n\n - Instead of supporting arbitrary Kripke structures,\n   metacircular evaluation and symbolic evaluation are hard-coded.\n\nOld-commit-hash: d25aa0871f3efecd4932029a007775f4c22568fb\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2083 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n","old_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"69c50906015b70f129002c9a47abdb7b1c33357e","subject":"reworking canonical indet forms","message":"reworking canonical indet forms\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"canonical-indeterminate-forms.agda","new_file":"canonical-indeterminate-forms.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n\n  -- this type gives somewhat nicer syntax for the output of the canonical\n  -- forms lemma for indeterminates at base type\n  data cif-base : (\u0394 : hctx) (d : ihexp) \u2192 Set where\n    CIFBEHole : \u2200 {\u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393 \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         ((u ::[ \u0393 ] b) \u2208 \u0394) \u00d7\n         (\u0394 , \u2205 \u22a2 \u03c3 :s: \u0393)\n        )\n       \u2192 cif-base \u0394 d\n    CIFBNEHole : \u2200 {\u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393 \u2208 tctx ] \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n         (d' final) \u00d7\n         ((u ::[ \u0393 ] b) \u2208 \u0394) \u00d7\n         (\u0394 , \u2205 \u22a2 \u03c3 :s: \u0393)\n        )\n        \u2192 cif-base \u0394 d\n    CIFBAp : \u2200 {\u0394 d} \u2192\n      \u03a3[ d1 \u2208 ihexp ] \u03a3[ d2 \u2208 ihexp ] \u03a3[ \u03c42 \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n         (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n         (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n         (d1 indet) \u00d7\n         (d2 final) \u00d7\n         ((\u03c43 \u03c44 \u03c43' \u03c44' : htyp) (d1' : ihexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c43 ==> \u03c44 \u21d2 \u03c43' ==> \u03c44' \u27e9))\n        )\n        \u2192 cif-base \u0394 d\n    CIFBCast : \u2200 {\u0394 d} \u2192\n      \u03a3[ d' \u2208 ihexp ]\n        ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u2987\u2988) \u00d7\n         (d' indet) \u00d7\n         ((d'' : ihexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))\n        )\n        \u2192 cif-base \u0394 d\n    CIFBFailedCast : \u2200 {\u0394 d} \u2192\n      \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == d' \u27e8 \u03c4' \u21d2\u2987\u2988\u21d2\u0338 b \u27e9) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n         (\u03c4' ground) \u00d7\n         (\u03c4' \u2260 b)\n        )\n       \u2192 cif-base \u0394 d\n\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       cif-base \u0394 d\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFBAp (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = CIFBEHole (_ , _ , _ , refl , x , x\u2081)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFBNEHole (_ , _ , _ , _ , _ , refl , wt , x\u2082 , x , x\u2081)\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = CIFBCast (_ , refl , wt , ind , x\u2081)\n  canonical-indeterminate-forms-base (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = CIFBFailedCast (_ , _ , refl , x , x\u2085 , x\u2087)\n\n  -- this type gives somewhat nicer syntax for the output of the canonical\n  -- forms lemma for indeterminates at arrow type\n  data cif-arr : (\u0394 : hctx) (d : ihexp) (\u03c41 \u03c42 : htyp) \u2192 Set where\n    CIFAEHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393 \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         ((u ::[ \u0393 ] (\u03c41 ==> \u03c42)) \u2208 \u0394) \u00d7\n         (\u0394 , \u2205 \u22a2 \u03c3 :s: \u0393)\n        )\n      \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFANEHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393 \u2208 tctx ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n         (d' final) \u00d7\n         ((u ::[ \u0393 ] (\u03c41 ==> \u03c42)) \u2208 \u0394) \u00d7\n         (\u0394 , \u2205 \u22a2 \u03c3 :s: \u0393)\n        )\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFAAp : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      \u03a3[ d1 \u2208 ihexp ] \u03a3[ d2 \u2208 ihexp ] \u03a3[ \u03c42' \u2208 htyp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n         (\u0394 , \u2205 \u22a2 d1 :: \u03c42' ==> (\u03c41 ==> \u03c42)) \u00d7\n         (\u0394 , \u2205 \u22a2 d2 :: \u03c42') \u00d7\n         (d1 indet) \u00d7\n         (d2 final) \u00d7\n         ((\u03c43 \u03c44 \u03c43' \u03c44' : htyp) (d1' : ihexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c43 ==> \u03c44 \u21d2 \u03c43' ==> \u03c44' \u27e9))\n        )\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFACast : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n        ((d == d' \u27e8 (\u03c41' ==> \u03c42') \u21d2 (\u03c41 ==> \u03c42) \u27e9) \u00d7\n          (\u0394 , \u2205 \u22a2 d' :: \u03c41' ==> \u03c42') \u00d7\n          (d' indet) \u00d7\n          ((\u03c41' ==> \u03c42') \u2260 (\u03c41 ==> \u03c42))\n        )\n       \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFACastHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      \u03a3[ d' \u2208 ihexp ]\n        ((d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n         (\u03c41 == \u2987\u2988) \u00d7\n         (\u03c42 == \u2987\u2988) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u2987\u2988) \u00d7\n         (d' indet) \u00d7\n         ((d'' : ihexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))\n        )\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFAFailedCast : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n          ((d == (d' \u27e8 \u03c4' \u21d2\u2987\u2988\u21d2\u0338 \u2987\u2988 ==> \u2987\u2988 \u27e9) ) \u00d7\n           (\u03c41 == \u2987\u2988) \u00d7\n           (\u03c42 == \u2987\u2988) \u00d7\n           (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n           (\u03c4' ground) \u00d7\n           (\u03c4' \u2260 (\u2987\u2988 ==> \u2987\u2988))\n           )\n          \u2192 cif-arr \u0394 d \u03c41 \u03c42\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       cif-arr \u0394 d \u03c41 \u03c42\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam _ wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFAAp (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = CIFAEHole (_ , _ , _ , refl , x , x\u2081)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFANEHole (_ , _ , _ , _ , _ , refl , wt , x\u2082 , x , x\u2081)\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = CIFACast (_ , _ , _ , _ , _ , refl , wt , ind , x\u2081)\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = CIFACastHole (_ , refl , refl , refl , wt , ind , x\u2081)\n  canonical-indeterminate-forms-arr (TAFailedCast x x\u2081 GHole x\u2083) (IFailedCast x\u2084 x\u2085 GHole x\u2087) = CIFAFailedCast (_ , _ , refl , refl , refl , x , x\u2085 , x\u2087)\n\n\n  -- this type gives somewhat nicer syntax for the output of the canonical\n  -- forms lemma for indeterminates at hole type\n  data cif-hole : (\u0394 : hctx) (d : ihexp) \u2192 Set where\n    CIFHEHole : \u2200 {\u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393 \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         ((u ::[ \u0393 ] \u2987\u2988) \u2208 \u0394) \u00d7\n         (\u0394 , \u2205 \u22a2 \u03c3 :s: \u0393)\n        )\n      \u2192 cif-hole \u0394 d\n    CIFHNEHole : \u2200 {\u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393 \u2208 tctx ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n         (d' final) \u00d7\n         ((u ::[ \u0393 ] \u2987\u2988) \u2208 \u0394) \u00d7\n         (\u0394 , \u2205 \u22a2 \u03c3 :s: \u0393)\n        )\n      \u2192 cif-hole \u0394 d\n    CIFHAp : \u2200 {\u0394 d} \u2192\n      \u03a3[ d1 \u2208 ihexp ] \u03a3[ d2 \u2208 ihexp ] \u03a3[ \u03c42 \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n         (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n         (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n         (d1 indet) \u00d7\n         (d2 final) \u00d7\n         ((\u03c43 \u03c44 \u03c43' \u03c44' : htyp) (d1' : ihexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c43 ==> \u03c44 \u21d2 \u03c43' ==> \u03c44' \u27e9))\n        )\n      \u2192 cif-hole \u0394 d\n    CIFHCast : \u2200 {\u0394 d} \u2192\n      \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n         (\u03c4' ground) \u00d7\n         (d' indet)\n        )\n      \u2192 cif-hole \u0394 d\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       cif-hole \u0394 d\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFHAp (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = CIFHEHole (_ , _ , _ , refl , x , x\u2081)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFHNEHole (_ , _ , _ , _ , _ , refl , wt , x\u2082 , x , x\u2081)\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = CIFHCast (_ , _ , refl , wt , x\u2081 , ind)\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind ())\n  canonical-indeterminate-forms-hole (TAFailedCast x x\u2081 () x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087)\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam _ wt) () nb na nh\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAEHole x x\u2081) IEHole nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAEHole x x\u2081) IEHole nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAEHole x x\u2081) IEHole nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb z _ _\u2081 \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ _\u2081 z \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ z _\u2081 \u2192 z \u03c4 \u03c4\u2081 refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\nopen import type-assignment-unicity\n\nmodule canonical-indeterminate-forms where\n\n  -- this type gives somewhat nicer syntax for the output of the canonical\n  -- forms lemma for indeterminates at base type\n  data cif-base : (\u0394 : hctx) (d : ihexp) \u2192 Set where\n    CIFBEHole : \u2200 {\u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        ((u ::[ \u0393' ] b) \u2208 \u0394))\n       \u2192 cif-base \u0394 d\n    CIFBNEHole : \u2200 {\u0394 d} \u2192\n      \u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393' \u2208 tctx ] \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        (d' final) \u00d7\n        (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n        ((u ::[ \u0393' ] b) \u2208 \u0394))\n        \u2192 cif-base \u0394 d\n    CIFBAp : \u2200 {\u0394 d} \u2192\n      \u03a3[ d1 \u2208 ihexp ] \u03a3[ d2 \u2208 ihexp ] \u03a3[ \u03c42 \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n        (\u0394 , \u2205 \u22a2 d1 :: \u03c42 ==> b) \u00d7\n        (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n        (d1 indet) \u00d7\n        (d2 final) \u00d7\n        ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : ihexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9)))\n        \u2192 cif-base \u0394 d\n    CIFBCast : \u2200 {\u0394 d} \u2192\n      \u03a3[ d' \u2208 ihexp ]\n        ((d == d' \u27e8 \u2987\u2988 \u21d2 b \u27e9) \u00d7\n        (d' indet) \u00d7\n        ((d'' : ihexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)))\n        \u2192 cif-base \u0394 d\n    CIFBFailedCast : \u2200 {\u0394 d} \u2192\n      \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == d' \u27e8 \u03c4' \u21d2\u2987\u2988\u21d2\u0338 b \u27e9) \u00d7\n        (\u03c4' ground) \u00d7\n        (\u03c4' \u2260 b) \u00d7\n        (\u0394 , \u2205 \u22a2 d' :: \u03c4'))\n       \u2192 cif-base \u0394 d\n\n  canonical-indeterminate-forms-base : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: b \u2192\n                                       d indet \u2192\n                                       cif-base \u0394 d\n  canonical-indeterminate-forms-base TAConst ()\n  canonical-indeterminate-forms-base (TAVar x\u2081) ()\n  canonical-indeterminate-forms-base (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFBAp (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-base (TAEHole x x\u2081) IEHole = CIFBEHole (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-base (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFBNEHole (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x)\n  canonical-indeterminate-forms-base (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) = CIFBCast (_ , refl , ind , x\u2081)\n  canonical-indeterminate-forms-base (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = CIFBFailedCast (_ , _ , refl , x\u2085 , x\u2087 , x)\n\n  -- this type gives somewhat nicer syntax for the output of the canonical\n  -- forms lemma for indeterminates at arrow type\n  data cif-arr : (\u0394 : hctx) (d : ihexp) (\u03c41 \u03c42 : htyp) \u2192 Set where\n    CIFAEHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394)))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFANEHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n        (d' final) \u00d7\n        (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n        ((u ::[ \u0393' ] (\u03c41 ==> \u03c42)) \u2208 \u0394)))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFAAp : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d1 \u2208 ihexp ] \u03a3[ d2 \u2208 ihexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c41' \u2208 htyp ] \u03a3[ \u03c42' \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n        (\u0394 , \u2205 \u22a2 d1 :: \u03c4 ==> (\u03c41' ==> \u03c42')) \u00d7\n        (\u0394 , \u2205 \u22a2 d2 :: \u03c4) \u00d7\n        (d1 indet) \u00d7\n        (d2 final) \u00d7\n        ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : ihexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFACast : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] \u03a3[ \u03c43 \u2208 htyp ] \u03a3[ \u03c44 \u2208 htyp ]\n        ((d == d' \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9) \u00d7\n          (d' indet) \u00d7\n          ((\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44))))\n       \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFACastHole : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d' \u2208 ihexp ]\n        ((\u03c41 == \u2987\u2988) \u00d7\n          (\u03c42 == \u2987\u2988) \u00d7\n          (d == (d' \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u27e9)) \u00d7\n          (d' indet) \u00d7\n          ((d'' : ihexp) (\u03c4' : htyp) \u2192 d' \u2260 (d'' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9))))\n        \u2192 cif-arr \u0394 d \u03c41 \u03c42\n    CIFAFailedCast : \u2200{d \u0394 \u03c41 \u03c42} \u2192\n      (\u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4 \u2208 htyp ]\n          ((d == (d' \u27e8 \u03c4 \u21d2\u2987\u2988\u21d2\u0338 \u2987\u2988 ==> \u2987\u2988 \u27e9) ) \u00d7\n            (\u03c4 ground) \u00d7\n            (\u03c4 \u2260 (\u2987\u2988 ==> \u2987\u2988)) \u00d7\n            (\u0394 , \u2205 \u22a2 d' :: \u03c4)))\n          \u2192 cif-arr \u0394 d \u03c41 \u03c42\n\n  canonical-indeterminate-forms-arr : \u2200{\u0394 d \u03c41 \u03c42 } \u2192\n                                       \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                                       d indet \u2192\n                                       cif-arr \u0394 d \u03c41 \u03c42\n  canonical-indeterminate-forms-arr (TAVar x\u2081) ()\n  canonical-indeterminate-forms-arr (TALam _ wt) ()\n  canonical-indeterminate-forms-arr (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFAAp (_ , _ , _ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-arr (TAEHole x x\u2081) IEHole = CIFAEHole (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-arr (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFANEHole (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x)\n  canonical-indeterminate-forms-arr (TACast wt x) (ICastArr x\u2081 ind) = CIFACast (_ , _ , _ , _ , _ , refl , ind , x\u2081)\n  canonical-indeterminate-forms-arr (TACast wt TCHole2) (ICastHoleGround x\u2081 ind GHole) = CIFACastHole (_ , refl , refl , refl , ind , x\u2081)\n  canonical-indeterminate-forms-arr (TAFailedCast x x\u2081 GHole x\u2083) (IFailedCast x\u2084 x\u2085 GHole x\u2087) = CIFAFailedCast (_ , _ , refl , x\u2085 , x\u2087 , x)\n\n\n  -- this type gives somewhat nicer syntax for the output of the canonical\n  -- forms lemma for indeterminates at hole type\n  data cif-hole : (\u0394 : hctx) (d : ihexp) \u2192 Set where\n    CIFHEHole : \u2200 {\u0394 d} \u2192\n      (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987\u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394)))\n      \u2192 cif-hole \u0394 d\n    CIFHNEHole : \u2200 {\u0394 d} \u2192\n      (\u03a3[ u \u2208 Nat ] \u03a3[ \u03c3 \u2208 env ] \u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0393' \u2208 tctx ]\n        ((d == \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9) \u00d7\n         (d' final) \u00d7\n         (\u0394 , \u2205 \u22a2 d' :: \u03c4') \u00d7\n         ((u ::[ \u0393' ] \u2987\u2988) \u2208 \u0394)))\n      \u2192 cif-hole \u0394 d\n    CIFHAp : \u2200 {\u0394 d} \u2192\n      (\u03a3[ d1 \u2208 ihexp ] \u03a3[ d2 \u2208 ihexp ] \u03a3[ \u03c42 \u2208 htyp ]\n        ((d == d1 \u2218 d2) \u00d7\n         (\u0394 , \u2205 \u22a2 d1 :: (\u03c42 ==> \u2987\u2988)) \u00d7\n         (\u0394 , \u2205 \u22a2 d2 :: \u03c42) \u00d7\n         (d1 indet) \u00d7\n         (d2 final) \u00d7\n         ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : ihexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))))\n      \u2192 cif-hole \u0394 d\n    CIFHCast : \u2200 {\u0394 d} \u2192\n      (\u03a3[ d' \u2208 ihexp ] \u03a3[ \u03c4' \u2208 htyp ]\n        ((d == d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u00d7\n         (\u03c4' ground) \u00d7\n         (d' indet)))\n      \u2192 cif-hole \u0394 d\n\n  canonical-indeterminate-forms-hole : \u2200{\u0394 d} \u2192\n                                       \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                       d indet \u2192\n                                       cif-hole \u0394 d\n  canonical-indeterminate-forms-hole (TAVar x\u2081) ()\n  canonical-indeterminate-forms-hole (TAAp wt wt\u2081) (IAp x ind x\u2081) = CIFHAp (_ , _ , _ , refl , wt , wt\u2081 , ind , x\u2081 , x)\n  canonical-indeterminate-forms-hole (TAEHole x x\u2081) IEHole = CIFHEHole (_ , _ , _ , refl , x)\n  canonical-indeterminate-forms-hole (TANEHole x wt x\u2081) (INEHole x\u2082) = CIFHNEHole (_ , _ , _ , _ , _ , refl , x\u2082 , wt , x )\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastGroundHole x\u2081 ind) = CIFHCast (_ , _ , refl , x\u2081 , ind)\n  canonical-indeterminate-forms-hole (TACast wt x) (ICastHoleGround x\u2081 ind ())\n  canonical-indeterminate-forms-hole (TAFailedCast x x\u2081 () x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087)\n\n  canonical-indeterminate-forms-coverage : \u2200{\u0394 d \u03c4} \u2192\n                                           \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                           d indet \u2192\n                                           \u03c4 \u2260 b \u2192\n                                           ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                           \u03c4 \u2260 \u2987\u2988 \u2192\n                                           \u22a5\n  canonical-indeterminate-forms-coverage TAConst () nb na nh\n  canonical-indeterminate-forms-coverage (TAVar x\u2081) () nb na nh\n  canonical-indeterminate-forms-coverage (TALam _ wt) () nb na nh\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAAp wt wt\u2081) (IAp x ind x\u2081) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAEHole x x\u2081) IEHole nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAEHole x x\u2081) IEHole nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAEHole x x\u2081) IEHole nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TANEHole x wt x\u2081) (INEHole x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastArr x\u2081 ind) nb na nh = na _ _ refl\n  canonical-indeterminate-forms-coverage (TACast wt x) (ICastGroundHole x\u2081 ind) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nb refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = nh refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TACast wt x) (ICastHoleGround x\u2081 ind x\u2082) nb na nh = na \u03c4 \u03c4\u2081 refl\n  canonical-indeterminate-forms-coverage {\u03c4 = b} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb z _ _\u2081 \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u2987\u2988} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ _\u2081 z \u2192 z refl\n  canonical-indeterminate-forms-coverage {\u03c4 = \u03c4 ==> \u03c4\u2081} (TAFailedCast x x\u2081 x\u2082 x\u2083) (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087) = \u03bb _ z _\u2081 \u2192 z \u03c4 \u03c4\u2081 refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"49331a931713b91ed0906229650bb542976cf6ee","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 312a2569acbe9040993995e24d0d024\n\ndarcs-hash:20110307134622-3bd4e-88a8649159e08522998f93f454b6cf35d5aebf26.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/Mirror\/Type.agda","new_file":"src\/FOTC\/Program\/Mirror\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- The types used by the mirror function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.Mirror.Type where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\n------------------------------------------------------------------------------\n-- Tree terms.\npostulate\n  node : D \u2192 D \u2192 D\n\nmutual\n  -- The list of rose trees (called forest).\n  data Forest : D \u2192 Set where\n    nilF  : Forest []\n    consF : \u2200 {t ts} \u2192 Tree t \u2192 Forest ts \u2192 Forest (t \u2237 ts)\n\n  -- The rose tree type.\n  data Tree : D \u2192 Set where\n    treeT : \u2200 d {ts} \u2192 Forest ts \u2192 Tree (node d ts)\n\n{-# ATP hint nilF #-}\n{-# ATP hint consF #-}\n{-# ATP hint treeT #-}\n\n------------------------------------------------------------------------------\n-- From Coq'Art: These induction principles *not cover* the mutual\n-- structure of the types Tree and Rose (p. 401).\n\n-- Induction principle for Tree.\nindTree : (P : D \u2192 Set) \u2192\n          (\u2200 d {ts} \u2192 Forest ts \u2192 P (node d ts)) \u2192\n          \u2200 {t} \u2192 Tree t \u2192 P t\nindTree P h (treeT d Fts) = h d Fts\n\n-- Induction principle for Forest.\nindForest : (P : D \u2192 Set) \u2192\n            P [] \u2192\n            (\u2200 {t ts} \u2192 Tree t \u2192 Forest ts \u2192 P ts \u2192 P (t \u2237 ts)) \u2192\n            \u2200 {ts} \u2192 Forest ts \u2192 P ts\nindForest P P[] h nilF            = P[]\nindForest P P[] h (consF Tt Fts) = h Tt Fts (indForest P P[] h Fts)\n\n------------------------------------------------------------------------------\n-- Mutual induction for Tree and Forest\n\n-- Adapted from the induction principles generate from the Coq command\n-- Scheme Tree_mutual_ind :=\n--   Minimality for Tree Sort Prop\n-- with Forest_mutual_ind :=\n--   Minimality for Forest Sort Prop.\n\nmutual\n  mutualIndTree :\n    {P Q : D \u2192 Set} \u2192\n    (\u2200 d {ts} \u2192 Forest ts \u2192 Q ts \u2192 P (node d ts)) \u2192\n    Q [] \u2192\n    (\u2200 {t ts} \u2192 Tree t \u2192 P t \u2192 Forest ts \u2192 Q ts \u2192 Q (t \u2237 ts)) \u2192\n    \u2200 {t} \u2192 Tree t \u2192 P t\n  mutualIndTree ihP Q[] _   (treeT d nilF)           = ihP d nilF Q[]\n  mutualIndTree ihP Q[] ihQ (treeT d (consF Tt Fts)) =\n    ihP d (consF Tt Fts) (ihQ Tt (mutualIndTree ihP Q[] ihQ Tt)\n                              Fts (mutualIndForest ihP Q[] ihQ Fts))\n\n  mutualIndForest :\n     {P Q : D \u2192 Set} \u2192\n     (\u2200 d {ts} \u2192 Forest ts \u2192 Q ts \u2192 P (node d ts)) \u2192\n     Q [] \u2192\n     (\u2200 {t ts} \u2192 Tree t \u2192 P t \u2192 Forest ts \u2192 Q ts \u2192 Q (t \u2237 ts)) \u2192\n     \u2200 {ts} \u2192 Forest ts \u2192 Q ts\n  mutualIndForest _   Q[] _   nilF           = Q[]\n  mutualIndForest ihP Q[] ihQ (consF Tt Fts) =\n    ihQ Tt (mutualIndTree ihP Q[] ihQ Tt) Fts\n        (mutualIndForest ihP Q[] ihQ Fts)\n","old_contents":"------------------------------------------------------------------------------\n-- The types used by the mirror function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.Mirror.Type where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\n------------------------------------------------------------------------------\n-- Tree terms.\npostulate\n  node : D \u2192 D \u2192 D\n\nmutual\n  -- The list of rose trees (called forest).\n  data Forest : D \u2192 Set where\n    nilF  : Forest []\n    consF : \u2200 {t ts} \u2192 Tree t \u2192 Forest ts \u2192 Forest (t \u2237 ts)\n\n  -- The rose tree type.\n  data Tree : D \u2192 Set where\n    treeT : \u2200 d {ts} \u2192 Forest ts \u2192 Tree (node d ts)\n\n{-# ATP hint nilF #-}\n{-# ATP hint consF #-}\n{-# ATP hint treeT #-}\n\n-- From Coq'Art: These induction principles *not cover* the mutual\n-- structure of the types Tree and Rose (p. 401).\n\n-- Induction principle for Tree.\nindTree : (P : D \u2192 Set) \u2192\n          (\u2200 d {ts} \u2192 Forest ts \u2192 P (node d ts)) \u2192\n          \u2200 {t} \u2192 Tree t \u2192 P t\nindTree P h (treeT d Fts) = h d Fts\n\n-- Induction principle for Forest.\nindForest : (P : D \u2192 Set) \u2192\n            P [] \u2192\n            (\u2200 {t ts} \u2192 Tree t \u2192 Forest ts \u2192 P ts \u2192 P (t \u2237 ts)) \u2192\n            \u2200 {ts} \u2192 Forest ts \u2192 P ts\nindForest P P[] h nilF            = P[]\nindForest P P[] h (consF Tt Fts) = h Tt Fts (indForest P P[] h Fts)\n\n-- Mutual induction for Tree and Forest\n-- From the Coq command\n-- Scheme Tree_mutual_ind :=\n--   Minimality for Tree Sort Prop\n-- with Forest_mutual_ind :=\n--   Minimality for Forest Sort Prop.\n\nmutual\n  mutualIndTree :\n    {P Q : D \u2192 Set} \u2192\n    (\u2200 d {ts} \u2192 Forest ts \u2192 Q ts \u2192 P (node d ts)) \u2192\n    Q [] \u2192\n    (\u2200 {t ts} \u2192 Tree t \u2192 P t \u2192 Forest ts \u2192 Q ts \u2192 Q (t \u2237 ts)) \u2192\n    \u2200 {t} \u2192 Tree t \u2192 P t\n  mutualIndTree ihP Q[] _   (treeT d nilF)           = ihP d nilF Q[]\n  mutualIndTree ihP Q[] ihQ (treeT d (consF Tt Fts)) =\n    ihP d (consF Tt Fts) (ihQ Tt (mutualIndTree ihP Q[] ihQ Tt)\n                         Fts (mutualIndForest ihP Q[] ihQ Fts))\n\n  mutualIndForest :\n     {P Q : D \u2192 Set} \u2192\n     (\u2200 d {ts} \u2192 Forest ts \u2192 Q ts \u2192 P (node d ts)) \u2192\n     Q [] \u2192\n     (\u2200 {t ts} \u2192 Tree t \u2192 P t \u2192 Forest ts \u2192 Q ts \u2192 Q (t \u2237 ts)) \u2192\n     \u2200 {ts} \u2192 Forest ts \u2192 Q ts\n  mutualIndForest _   Q[] _   nilF           = Q[]\n  mutualIndForest ihP Q[] ihQ (consF Tt Fts) =\n    ihQ Tt (mutualIndTree ihP Q[] ihQ Tt) Fts\n        (mutualIndForest ihP Q[] ihQ Fts)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2bc86b16750cda8c1189ff286ba5405a47076a56","subject":"Update Agda version in README.agda","message":"Update Agda version in README.agda\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda (2.4.0), the Agda standard library (version 0.8), generate\n-- Everything.agda with the attached Haskell helper, and finally run Agda on\n-- this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n--\n-- The formalization has four parts:\n--\n--   1. A formalization of change structures. This lives in\n--   Base.Change.Algebra. (What we call \"change structure\" in the\n--   paper, we call \"change algebra\" in the Agda code. We changed\n--   the name when writing the paper, and never got around to\n--   updating the name in the Agda code).\n--\n--   2. Incrementalization framework for first-class functions,\n--   with extension points for plugging in data types and their\n--   incrementalization. This lives in the Parametric.*\n--   hierarchy.\n--\n--   3. An example plugin that provides integers and bags\n--   with negative multiplicity. This lives in the Nehemiah.*\n--   hierarchy. (For some reason, we choose to call this\n--   particular incarnation of the plugin Nehemiah).\n--\n--   4. Other material that is unrelated to the framework\/plugin\n--   distinction. This is all other files.\n\n\n\n-- FORMALIZATION OF CHANGE STRUCTURES\n-- ==================================\n--\n-- Section 2 of the paper, and Base.Change.Algebra in Agda.\n\nimport Base.Change.Algebra\n\n\n\n-- INCREMENTALIZATION FRAMEWORK\n-- ============================\n--\n-- Section 3 of the paper, and Parametric.* hierarchy in Agda.\n--\n-- The extension points are implemented as module parameters. See\n-- detailed explanation in Parametric.Syntax.Type and\n-- Parametric.Syntax.Term. Some extension points are for types,\n-- some for values, and some for proof fragments. In Agda, these\n-- three kinds of entities are unified anyway, so we can encode\n-- all of them as module parameters.\n--\n-- Every module in the Parametric.* hierarchy adds at least one\n-- extension point, so the module hierarchy of a plugin will\n-- typically mirror the Parametric.* hierarchy, defining the\n-- values for these extension points.\n--\n-- Firstly, we have the syntax of types and terms, the erased\n-- change structure for function types and incrementalization as\n-- a term-to-term transformation. The contents of these modules\n-- formalizes Sec. 3.1 and 3.2 of the paper, except for the\n-- second and third line of Figure 3, which is formalized further\n-- below.\n\nimport Parametric.Syntax.Type   -- syntax of types\nimport Parametric.Syntax.Term   -- syntax of terms\nimport Parametric.Change.Type   -- simply-typed changes\nimport Parametric.Change.Derive -- incrementalization\n\n-- Secondly, we define the usual denotational semantics of the\n-- simply-typed lambda calculus in terms of total functions, and\n-- the change semantics in terms of a change structure on total\n-- functions. The contents of these modules formalize Sec. 3.3,\n-- 3.4 and 3.5 of the paper.\n\nimport Parametric.Denotation.Value      -- standard values\nimport Parametric.Denotation.Evaluation -- standard evaluation\nimport Parametric.Change.Validity       -- dependently-typed changes\nimport Parametric.Change.Specification  -- change evaluation\n\n-- Thirdly, we define terms that operate on simply-typed changes,\n-- and connect them to their values. The concents of these\n-- modules formalize the second and third line of Figure 3, as\n-- well as the semantics of these lines.\n\nimport Parametric.Change.Term        -- terms that operate on simply-typed changes\nimport Parametric.Change.Value       -- the values of these terms\nimport Parametric.Change.Evaluation  -- connecting the terms and their values\n\n-- Finally, we prove correctness by connecting the (syntactic)\n-- incrementalization to the (semantic) change evaluation by a\n-- logical relation, and a proof that the values of terms\n-- produced by the incrementalization transformation are related\n-- to the change values of the original terms. The contents of\n-- these modules formalize Sec. 3.6.\n\nimport Parametric.Change.Implementation -- logical relation\nimport Parametric.Change.Correctness    -- main correctness proof\n\n\n\n-- EXAMPLE PLUGIN\n-- ==============\n--\n-- Sec. 3.7 in the paper, and the Nehemiah.* hierarchy in Agda.\n--\n-- The module structure of the plugin follows the structure of\n-- the Parametric.* hierarchy.  For example, the extension point\n-- defined in Parametric.Syntax.Term is instantiated in\n-- Nehemiah.Syntax.Term.\n--\n-- As discussed in Sec. 3.7 of the paper, the point of this\n-- plugin is not to speed up any real programs, but to show \"that\n-- the interface for proof plugins can be implemented\". As a\n-- first step towards proving correctness of the more complicated\n-- plugin with integers, bags and finite maps we implement in\n-- Scala, we choose to define plugin with integers and bags in\n-- Agda. Instead of implementing bags (with negative\n-- multiplicities, like in the paper) in Agda, though, we\n-- postulate that a group of such bags exist. Note that integer\n-- bags with integer multiplicities are actually the free group\n-- given a singleton operation `Integer -> Bag`, so this should\n-- be easy to formalize in principle.\n\n-- Before we start with the plugin, we postulate an abstract data\n-- type for integer bags.\nimport Postulate.Bag-Nehemiah\n\n-- Firstly, we extend the syntax of types and terms, the erased\n-- change structure for function types, and incrementalization as\n-- a term-to-term transformation to account for the data types of\n-- the Nehemiah language. The contents of these modules\n-- instantiate the extension points in Sec. 3.1 and 3.2 of the\n-- paper, except for the second and third line of Figure 3, which\n-- is instantiated further below.\n\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\nimport Nehemiah.Change.Type\nimport Nehemiah.Change.Derive\n\n-- Secondly, we extend the usual denotational semantics and the\n-- change semantics to account for the data types of the Nehemiah\n-- language. The contents of these modules instantiate the\n-- extension points in Sec. 3.3, 3.4 and 3.5 of the paper.\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\n\n-- Thirdly, we extend the terms that operate on simply-typed\n-- changes, and the connection to their values to account for the\n-- data types of the Nehemiah language. The concents of these\n-- modules instantiate the extension points in the second and\n-- third line of Figure 3.\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- Finally, we extend the logical relation and the main\n-- correctness proof to account for the data types in the\n-- Nehemiah language. The contents of these modules instantiate\n-- the extension points defined in Sec. 3.6.\n\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n\n\n-- OTHER MATERIAL\n-- ==============\n--\n-- We postulate extensionality of Agda functions. This postulate\n-- is well known to be compatible with Agda's type theory.\n\nimport Postulate.Extensionality\n\n-- For historical reasons, we reexport Data.List.All from the\n-- standard library under the name DependentList.\n\nimport Base.Data.DependentList\n\n-- This module supports overloading the \u27e6_\u27e7 notation based on\n-- Agda's instance arguments.\n\nimport Base.Denotation.Notation\n\n-- These modules implement contexts including typed de Bruijn\n-- indices to represent bound variables, sets of bound variables,\n-- and environments. These modules are parametric in the set of\n-- types (that are stored in contexts) and the set of values\n-- (that are stored in environments). So these modules are even\n-- more parametric than the Parametric.* hierarchy.\n\nimport Base.Syntax.Context\nimport Base.Syntax.Vars\nimport Base.Denotation.Environment\n\n-- This module contains some helper definitions to merge a\n-- context of values and a context of changes.\nimport Base.Change.Context\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n--\n-- The formalization has four parts:\n--\n--   1. A formalization of change structures. This lives in\n--   Base.Change.Algebra. (What we call \"change structure\" in the\n--   paper, we call \"change algebra\" in the Agda code. We changed\n--   the name when writing the paper, and never got around to\n--   updating the name in the Agda code).\n--\n--   2. Incrementalization framework for first-class functions,\n--   with extension points for plugging in data types and their\n--   incrementalization. This lives in the Parametric.*\n--   hierarchy.\n--\n--   3. An example plugin that provides integers and bags\n--   with negative multiplicity. This lives in the Nehemiah.*\n--   hierarchy. (For some reason, we choose to call this\n--   particular incarnation of the plugin Nehemiah).\n--\n--   4. Other material that is unrelated to the framework\/plugin\n--   distinction. This is all other files.\n\n\n\n-- FORMALIZATION OF CHANGE STRUCTURES\n-- ==================================\n--\n-- Section 2 of the paper, and Base.Change.Algebra in Agda.\n\nimport Base.Change.Algebra\n\n\n\n-- INCREMENTALIZATION FRAMEWORK\n-- ============================\n--\n-- Section 3 of the paper, and Parametric.* hierarchy in Agda.\n--\n-- The extension points are implemented as module parameters. See\n-- detailed explanation in Parametric.Syntax.Type and\n-- Parametric.Syntax.Term. Some extension points are for types,\n-- some for values, and some for proof fragments. In Agda, these\n-- three kinds of entities are unified anyway, so we can encode\n-- all of them as module parameters.\n--\n-- Every module in the Parametric.* hierarchy adds at least one\n-- extension point, so the module hierarchy of a plugin will\n-- typically mirror the Parametric.* hierarchy, defining the\n-- values for these extension points.\n--\n-- Firstly, we have the syntax of types and terms, the erased\n-- change structure for function types and incrementalization as\n-- a term-to-term transformation. The contents of these modules\n-- formalizes Sec. 3.1 and 3.2 of the paper, except for the\n-- second and third line of Figure 3, which is formalized further\n-- below.\n\nimport Parametric.Syntax.Type   -- syntax of types\nimport Parametric.Syntax.Term   -- syntax of terms\nimport Parametric.Change.Type   -- simply-typed changes\nimport Parametric.Change.Derive -- incrementalization\n\n-- Secondly, we define the usual denotational semantics of the\n-- simply-typed lambda calculus in terms of total functions, and\n-- the change semantics in terms of a change structure on total\n-- functions. The contents of these modules formalize Sec. 3.3,\n-- 3.4 and 3.5 of the paper.\n\nimport Parametric.Denotation.Value      -- standard values\nimport Parametric.Denotation.Evaluation -- standard evaluation\nimport Parametric.Change.Validity       -- dependently-typed changes\nimport Parametric.Change.Specification  -- change evaluation\n\n-- Thirdly, we define terms that operate on simply-typed changes,\n-- and connect them to their values. The concents of these\n-- modules formalize the second and third line of Figure 3, as\n-- well as the semantics of these lines.\n\nimport Parametric.Change.Term        -- terms that operate on simply-typed changes\nimport Parametric.Change.Value       -- the values of these terms\nimport Parametric.Change.Evaluation  -- connecting the terms and their values\n\n-- Finally, we prove correctness by connecting the (syntactic)\n-- incrementalization to the (semantic) change evaluation by a\n-- logical relation, and a proof that the values of terms\n-- produced by the incrementalization transformation are related\n-- to the change values of the original terms. The contents of\n-- these modules formalize Sec. 3.6.\n\nimport Parametric.Change.Implementation -- logical relation\nimport Parametric.Change.Correctness    -- main correctness proof\n\n\n\n-- EXAMPLE PLUGIN\n-- ==============\n--\n-- Sec. 3.7 in the paper, and the Nehemiah.* hierarchy in Agda.\n--\n-- The module structure of the plugin follows the structure of\n-- the Parametric.* hierarchy.  For example, the extension point\n-- defined in Parametric.Syntax.Term is instantiated in\n-- Nehemiah.Syntax.Term.\n--\n-- As discussed in Sec. 3.7 of the paper, the point of this\n-- plugin is not to speed up any real programs, but to show \"that\n-- the interface for proof plugins can be implemented\". As a\n-- first step towards proving correctness of the more complicated\n-- plugin with integers, bags and finite maps we implement in\n-- Scala, we choose to define plugin with integers and bags in\n-- Agda. Instead of implementing bags (with negative\n-- multiplicities, like in the paper) in Agda, though, we\n-- postulate that a group of such bags exist. Note that integer\n-- bags with integer multiplicities are actually the free group\n-- given a singleton operation `Integer -> Bag`, so this should\n-- be easy to formalize in principle.\n\n-- Before we start with the plugin, we postulate an abstract data\n-- type for integer bags.\nimport Postulate.Bag-Nehemiah\n\n-- Firstly, we extend the syntax of types and terms, the erased\n-- change structure for function types, and incrementalization as\n-- a term-to-term transformation to account for the data types of\n-- the Nehemiah language. The contents of these modules\n-- instantiate the extension points in Sec. 3.1 and 3.2 of the\n-- paper, except for the second and third line of Figure 3, which\n-- is instantiated further below.\n\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\nimport Nehemiah.Change.Type\nimport Nehemiah.Change.Derive\n\n-- Secondly, we extend the usual denotational semantics and the\n-- change semantics to account for the data types of the Nehemiah\n-- language. The contents of these modules instantiate the\n-- extension points in Sec. 3.3, 3.4 and 3.5 of the paper.\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\n\n-- Thirdly, we extend the terms that operate on simply-typed\n-- changes, and the connection to their values to account for the\n-- data types of the Nehemiah language. The concents of these\n-- modules instantiate the extension points in the second and\n-- third line of Figure 3.\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- Finally, we extend the logical relation and the main\n-- correctness proof to account for the data types in the\n-- Nehemiah language. The contents of these modules instantiate\n-- the extension points defined in Sec. 3.6.\n\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n\n\n-- OTHER MATERIAL\n-- ==============\n--\n-- We postulate extensionality of Agda functions. This postulate\n-- is well known to be compatible with Agda's type theory.\n\nimport Postulate.Extensionality\n\n-- For historical reasons, we reexport Data.List.All from the\n-- standard library under the name DependentList.\n\nimport Base.Data.DependentList\n\n-- This module supports overloading the \u27e6_\u27e7 notation based on\n-- Agda's instance arguments.\n\nimport Base.Denotation.Notation\n\n-- These modules implement contexts including typed de Bruijn\n-- indices to represent bound variables, sets of bound variables,\n-- and environments. These modules are parametric in the set of\n-- types (that are stored in contexts) and the set of values\n-- (that are stored in environments). So these modules are even\n-- more parametric than the Parametric.* hierarchy.\n\nimport Base.Syntax.Context\nimport Base.Syntax.Vars\nimport Base.Denotation.Environment\n\n-- This module contains some helper definitions to merge a\n-- context of values and a context of changes.\nimport Base.Change.Context\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e53498636618bd357e831ed6b7ebeceea19275ad","subject":"Minor change.","message":"Minor change.\n\nIgnore-this: aaa074c4f8803e0021f89a28f22151b4\n\ndarcs-hash:20120606211737-3bd4e-8f2e9efe7f2fae6509e09887226a955fd3215ca4.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC-PCF\/Program\/GCD\/Total\/GCD.agda","new_file":"src\/LTC-PCF\/Program\/GCD\/Total\/GCD.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of the gcd of two natural numbers using the Euclid's algorithm\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Program.GCD.Total.GCD where\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Loop\n\n------------------------------------------------------------------------------\n-- In GHC \u2265 7.2.1 the gcd is a total function, i.e. gcd 0 0 = 0.\n\n-- Instead of defining gcdh : ((D \u2192 D \u2192 D) \u2192 (D \u2192 D \u2192 D)) \u2192 D \u2192 D \u2192 D,\n-- we use the LTC-PCF \u03bb-abstraction and application to avoid use a\n-- polymorphic fixed-point operator.\n\ngcdh : D \u2192 D\ngcdh g = lam (\u03bb m \u2192 lam (\u03bb n \u2192\n           if (iszero\u2081 n)\n              then m\n              else (if (iszero\u2081 m)\n                       then n\n                       else (if (m > n)\n                                then g \u00b7 (m \u2238 n) \u00b7 n\n                                else g \u00b7 m \u00b7 (n \u2238 m)))))\n\ngcd : D \u2192 D \u2192 D\ngcd m n = fix gcdh \u00b7 m \u00b7 n\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of the gcd of two natural numbers using the Euclid's algorithm\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Program.GCD.Total.GCD where\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Loop\n\n------------------------------------------------------------------------------\n-- In GHC \u2265 7.2.1 the gcd is a total function, i.e. gcd 0 0 = 0.\n\n-- Instead of defining gcdh : ((D \u2192 D \u2192 D) \u2192 (D \u2192 D \u2192 D)) \u2192 D \u2192 D \u2192 D,\n-- we use the LTC-PCF \u03bb-abstraction and application to avoid use a\n-- polymorphic fixed-point operator.\n\ngcdh : D \u2192 D\ngcdh g = lam (\u03bb m \u2192 lam (\u03bb n \u2192\n           if (iszero\u2081 n)\n              then m\n              else (if (iszero\u2081 m)\n                       then n\n                       else (if (m > n)\n                                then g \u00b7 (m \u2238 n) \u00b7 n\n                                else g \u00b7 m \u00b7 (n \u2238 m)))))\n\ngcd : D \u2192 D \u2192 D\ngcd m n = fix gcdh \u00b7 m \u00b7 n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1fd2e308e6cbee61ce0c527c7a1c8fa86ae11694","subject":"Remove more dead code","message":"Remove more dead code\n\nOld-commit-hash: 20e82287ead4c99b0bce90e095ed1f7711119d2a\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record.\n  record _\u2259_ dx dy : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u229e dx \u2261 x \u229e dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u229e dx) \u2261 f (x \u229e dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n      -- Unused, but just to test that inference works.\n      lemma : nil f \u2259 dg\n      lemma = \u2259-sym (derivative-is-nil dg fdg)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8f2332ac52b7e8312b57d1d5a4531437d3d874b4","subject":"Unindent fragment","message":"Unindent fragment\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Lang.agda","new_file":"New\/Lang.agda","new_contents":"module New.Lang where\n\nopen import New.Contexts public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n   plus : Const (int \u21d2 int \u21d2 int)\n   minus : Const (int \u21d2 int \u21d2 int)\n   cons : \u2200 {t1 t2} \u2192 Const (t1 \u21d2 t2 \u21d2 pair t1 t2)\n   fst : \u2200 {t1 t2} \u2192 Const (pair t1 t2 \u21d2 t1)\n   snd : \u2200 {t1 t2} \u2192 Const (pair t1 t2 \u21d2 t2)\n   linj : \u2200 {t1 t2} \u2192 Const (t1 \u21d2 sum t1 t2)\n   rinj : \u2200 {t1 t2} \u2192 Const (t2 \u21d2 sum t1 t2)\n   match : \u2200 {t1 t2 t3} \u2192 Const (sum t1 t2 \u21d2 (t1 \u21d2 t3) \u21d2 (t2 \u21d2 t3) \u21d2 t3)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indicies, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n  Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n  Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n  Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n  Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n  Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n  Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n  Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n  Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n  Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n  Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n  Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n  Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n  Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Const cons = \u03bb v1 v2 \u2192 v1 , v2\n\u27e6 plus \u27e7Const = _+_\n\u27e6 minus \u27e7Const = _-_\n\u27e6 fst \u27e7Const (v1 , v2) = v1\n\u27e6 snd \u27e7Const (v1 , v2) = v2\n\u27e6 linj \u27e7Const v1 = inj\u2081 v1\n\u27e6 rinj \u27e7Const v2 = inj\u2082 v2\n\u27e6 match \u27e7Const (inj\u2081 x) f g = f x\n\u27e6 match \u27e7Const (inj\u2082 y) f g = g y\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1)\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7Context) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7Term \u03c1 \u2261 \u27e6 t \u27e7Term (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7\u227c \u03c1)\nweaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\nweaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (const c) \u03c1 = refl\n","old_contents":"module New.Lang where\n\nopen import New.Contexts public\n\nmodule _ where\n  data Const : (\u03c4 : Type) \u2192 Set where\n     plus : Const (int \u21d2 int \u21d2 int)\n     minus : Const (int \u21d2 int \u21d2 int)\n     cons : \u2200 {t1 t2} \u2192 Const (t1 \u21d2 t2 \u21d2 pair t1 t2)\n     fst : \u2200 {t1 t2} \u2192 Const (pair t1 t2 \u21d2 t1)\n     snd : \u2200 {t1 t2} \u2192 Const (pair t1 t2 \u21d2 t2)\n     linj : \u2200 {t1 t2} \u2192 Const (t1 \u21d2 sum t1 t2)\n     rinj : \u2200 {t1 t2} \u2192 Const (t2 \u21d2 sum t1 t2)\n     match : \u2200 {t1 t2 t3} \u2192 Const (sum t1 t2 \u21d2 (t1 \u21d2 t3) \u21d2 (t2 \u21d2 t3) \u21d2 t3)\n\n  data Term (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set where\n    -- constants aka. primitives\n    const : \u2200 {\u03c4} \u2192\n      (c : Const \u03c4) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Const cons = \u03bb v1 v2 \u2192 v1 , v2\n\u27e6 plus \u27e7Const = _+_\n\u27e6 minus \u27e7Const = _-_\n\u27e6 fst \u27e7Const (v1 , v2) = v1\n\u27e6 snd \u27e7Const (v1 , v2) = v2\n\u27e6 linj \u27e7Const v1 = inj\u2081 v1\n\u27e6 rinj \u27e7Const v2 = inj\u2082 v2\n\u27e6 match \u27e7Const (inj\u2081 x) f g = f x\n\u27e6 match \u27e7Const (inj\u2082 y) f g = g y\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1)\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7Context) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7Term \u03c1 \u2261 \u27e6 t \u27e7Term (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7\u227c \u03c1)\nweaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\nweaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (const c) \u03c1 = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e263f70617e85a2289d746acd2ba8ad844ccfa12","subject":"removing unused dependency","message":"removing unused dependency\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     envfresh x \u03c3 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 {\u0393 = \u0393} (EFId x\u2081) (STAId x\u2082) = STAId (\u03bb x \u03c4 x\u2083 \u2192 x\u2208\u222al \u0393 _ x \u03c4 (x\u2082 x \u03c4 x\u2083) )\n    weaken-subst-\u0393 {x = x} {\u0393 = \u0393} (EFSubst x\u2081 efrsh x\u2082) (STASubst {y = y} {\u03c4 = \u03c4'} subst x\u2083) =\n      STASubst (exchange-subst-\u0393 {\u0393 = \u0393} (flip x\u2082) (weaken-subst-\u0393 {\u0393 = \u0393 ,, (y , \u03c4')} efrsh subst))\n               (weaken-ta x\u2081 x\u2083)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192  -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081)))\n                                                                                    {!weaken-ta {\u0393 = \u0393 ,, (y , ?)} x\u2083 wt!}\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 -- = {!weaken-ta ? (TALam x\u2082 wt1)!}\n    with natEQ y x\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = abort ((\u03c01(lem-union-none {\u0393 = \u0393} x\u2083)) refl)\n  lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n    with lem-union-none {\u0393 = \u0393} x\u2083 -- | lem-subst apt wt1  -- probably not the straight IH; need to weaken\n  ... | neq , r =  {!!} --  TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     envfresh x \u03c3 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 {\u0393 = \u0393} (EFId x\u2081) (STAId x\u2082) = STAId (\u03bb x \u03c4 x\u2083 \u2192 x\u2208\u222al \u0393 _ x \u03c4 (x\u2082 x \u03c4 x\u2083) )\n    weaken-subst-\u0393 {x = x} {\u0393 = \u0393} (EFSubst x\u2081 efrsh x\u2082) (STASubst {y = y} {\u03c4 = \u03c4'} subst x\u2083) =\n      STASubst (exchange-subst-\u0393 {\u0393 = \u0393} (flip x\u2082) (weaken-subst-\u0393 {\u0393 = \u0393 ,, (y , \u03c4')} efrsh subst))\n               (weaken-ta x\u2081 x\u2083)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192  -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081)))\n                                                                                    {!weaken-ta {\u0393 = \u0393 ,, (y , ?)} x\u2083 wt!}\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 x\u2081 x\u2083)\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 -- = {!weaken-ta ? (TALam x\u2082 wt1)!}\n    with natEQ y x\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = abort ((\u03c01(lem-union-none {\u0393 = \u0393} x\u2083)) refl)\n  lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n    with lem-union-none {\u0393 = \u0393} x\u2083 -- | lem-subst apt wt1  -- probably not the straight IH; need to weaken\n  ... | neq , r =  {!!} --  TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"51cc48e2708f05e3fbeeabc086b1cf3ac82f6256","subject":"More field","message":"More field\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\n\nrecord RawField {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  -0\u1da0 -1\u1da0 : A\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\nrecord IsField {\u2113} (A : Set \u2113) : Set \u2113 where\n  open FP {\u2113} {A}\n\n  field\n    rawField : RawField A\n  open RawField rawField\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  open RawField rawField public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : IsField A) where\n  open IsField F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : RawField B)(default : Decode-Term B) where\n    module G = RawField G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.Nat using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_ -- _*\u2124_\ndata Tm (A : Set) : Set where\n  -- 0' 1' -1' : Tm A\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : IsField F) where\n    open IsField F-field renaming (0-_ to -_)\n\n    suc\u1da0 : F \u2192 F\n    suc\u1da0 = _+_ 1\u1da0\n\n    nat : \u2115 \u2192 F\n    nat = fold 0\u1da0 suc\u1da0\n\n    eval\u2124 : \u2124 \u2192 F\n    eval\u2124 (\u2124.+ x)    = nat x\n    eval\u2124 \u2124.-[1+ x ] = -(nat (suc x))\n\n    _^\u2115_ : F \u2192 \u2115 \u2192 F\n    b ^\u2115 zero  = 1\u1da0\n    b ^\u2115 suc x = b * b ^\u2115 x\n\n    _^_ : F \u2192 \u2124 \u2192 F\n    b ^ (\u2124.+ x)    = b ^\u2115 x\n    b ^ \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = eval\u2124 x\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\n\nrecord IsField {\u2113} (A : Set \u2113) : Set \u2113 where\n  open FP {\u2113} {A}\n\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0\u1da0 1\u1da0        : A\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0-_         : A \u2192 A\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    _\u207b\u00b9         : A \u2192 A\n    \u207b\u00b9-inverse  : LeftInverse 1\u1da0 _\u207b\u00b9 _*_\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : IsField A) where\n  open IsField F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-Field {b}{B : Set b}(G : IsField B)(default : Decode-Term B) where\n    module G = IsField G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6c9b6ef868dcdf5810941fbfa7ea9897a1316e78","subject":"Proved the two induction principles (with\/without the N n) in a version of N via fixed points stuff.","message":"Proved the two induction principles (with\/without the N n) in a version of N via fixed points stuff.\n\nIgnore-this: 8858ebc76c0720b30ff6eaee77147200\n\ndarcs-hash:20111103160726-3bd4e-be20f7db64d8899a4285f373604d1e04b54c4c28.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_contents":"-- Tested with Agda 2.2.11 on 03 November 2011.\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is a fixed\n-- point of f (TODO: source).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least pre-fixed point of a functor\n\n-- Instead defining the least pre-fixed via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n-- The natural numbers are the least pre-fixed point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is pre-fixed point of NatF.\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081    : \u2200 n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m) \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082    : \u2200 (P : D \u2192 Set) {n} \u2192\n              (n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n) \u2192\n              N n \u2192 P n\n  -- N-lfp\u2082 : \u2200 (P : D \u2192 Set) {n} \u2192  -- Higher-order version\n  --          (NatF P n \u2192 P n) \u2192\n  --          N n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 zero (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN {n} Nn = N-lfp\u2081 (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m)\nN-lfp\u2083 {n} Nn = N-lfp\u2082 P prf Nn\n  where\n  P : D \u2192 Set\n  P x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ\u2081 m \u2227 N m\n\n  prf : n \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf h = [ inj\u2081 , (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) ] h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m\n    prf\u2081 (m , n=Sm , h\u2082) = m , n=Sm , prf\u2082 h\u2082\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = [ (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) , prf\u2083 ] h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n        prf\u2083 (m' , m\u2261Sm' , Nm') = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nindN\u2081 : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 P n \u2192 P (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN\u2081 P P0 is {n} Nn = N-lfp\u2082 P [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 P n\n  prf\u2081 n\u22610 = subst P (sym n\u22610) P0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf\u2082 (m , n\u2261Sm , Pm) = subst P (sym n\u2261Sm) (is Pm)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\nindN\u2082 : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN\u2082 P P0 is {n} Nn = N-lfp\u2082 P [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 P n\n  prf\u2081 n\u22610 = subst P (sym n\u22610) P0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf\u2082 (m , n\u2261Sm , Pm) = subst P (sym n\u2261Sm) (is helper Pm)\n    where\n    helper : N m\n    helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 Nn)\n      where\n      prf\u2083 : n \u2261 zero \u2192 N m\n      prf\u2083 n\u22610 = \u22a5-elim (0\u2260S (trans (sym n\u22610) n\u2261Sm))\n\n      prf\u2084 : \u2203 (\u03bb m' \u2192 n \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2084 (m' , n\u2261Sm' , Nm') =\n        subst N (succInjective (trans (sym n\u2261Sm') n\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-lfp\u2082 P prf Nm\n\n  where\n  P : D \u2192 Set\n  P i = N (i + n)\n\n  prf : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 P m') \u2192 P m\n  prf h = [ prf\u2081 , prf\u2082 ] h\n    where\n    P0 : P zero\n    P0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : m \u2261 zero \u2192 P m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) P0\n\n    is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n    is {i} Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n    prf\u2082 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 P m') \u2192 P m\n    prf\u2082 (m' , m\u2261Sm' , Pm') = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Pm')\n","old_contents":"-- Tested with Agda 2.2.11 on 11 October 2011.\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d               -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e  -- d is the least prefixed-point of f\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d               -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e  -- d is the least prefixed-point of f\n\n-- Thm: If d is the least pre-fixed point of f, then d is a fixed\n-- point of f (TODO: source).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least pre-fixed point of a functor\n\n-- Instead defining the least pre-fixed via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\n-- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n-- NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 X m)\n\n-- The natural numbers are the least pre-fixed point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is pre-fixed point of NatF.\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081 : \u2200 n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m) \u2192 N n\n\n  -- N is the least prefixed-point of NatF.\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082 : \u2200 (P : D \u2192 Set) {n} \u2192\n           (n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n) \u2192\n           N n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 zero (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN {n} Nn = N-lfp\u2081 (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m)\nN-lfp\u2083 {n} Nn = N-lfp\u2082 P prf Nn\n  where\n  P : D \u2192 Set\n  P x = x \u2261 zero \u2228 \u2203 \u03bb m \u2192 x \u2261 succ\u2081 m \u2227 N m\n\n  prf : n \u2261 zero \u2228 \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf h = [ inj\u2081 , (\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081)) ] h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 (m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m'))) \u2192\n           \u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 N m\n    prf\u2081 (m , n=Sm , h\u2082) = m , n=Sm , prf\u2082 h\u2082\n      where\n      prf\u2082 : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n      prf\u2082 h\u2082 = [ (\u03bb h\u2083 \u2192 subst N (sym h\u2083) zN) , prf\u2083 ] h\u2082\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n        prf\u2083 (m' , m\u2261Sm' , Nm') = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N.\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 P n \u2192 P (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 is {n} Nn = N-lfp\u2082 P [ prf\u2081 , prf\u2082 ] Nn\n  where\n  prf\u2081 : n \u2261 zero \u2192 P n\n  prf\u2081 n\u22610 = subst P (sym n\u22610) P0\n\n  prf\u2082 : \u2203 (\u03bb m \u2192 n \u2261 succ\u2081 m \u2227 P m) \u2192 P n\n  prf\u2082 (m , n\u2261Sm , Pm) = subst P (sym n\u2261Sm) (is Pm)\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-lfp\u2082 P prf Nm\n\n  where\n  P : D \u2192 Set\n  P i = N (i + n)\n\n  prf : m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 P m') \u2192 P m\n  prf h = [ prf\u2081 , prf\u2082 ] h\n    where\n    P0 : P zero\n    P0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : m \u2261 zero \u2192 P m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) P0\n\n    is : \u2200 {i} \u2192 P i \u2192 P (succ\u2081 i)\n    is {i} Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n    prf\u2082 : \u2203 (\u03bb m' \u2192 m \u2261 succ\u2081 m' \u2227 P m') \u2192 P m\n    prf\u2082 (m' , m\u2261Sm' , Pm') = subst N (cong (flip _+_ n) (sym m\u2261Sm')) (is Pm')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5812217497a6496aa243479790deaa3e44213e62","subject":"Typo.","message":"Typo.\n\nIgnore-this: a2288d34b15c2ed4a92559dcc5e67eeb\n\ndarcs-hash:20110121144705-3bd4e-015f775f7417633e4ddf04660e981412a1e02742.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Inductive2Axiomatic.agda","new_file":"src\/PA\/Inductive2Axiomatic.agda","new_contents":"------------------------------------------------------------------------------\n-- From inductive PA to axiomatic PA\n------------------------------------------------------------------------------\n\n-- From the definition of PA using Agda data types and primitive\n-- recursive functions for addition and multiplication, we can prove the\n-- Mendelson's axioms [1]\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\n-- [1]. Elliott Mendelson. Introduction to mathematical logic. Chapman\n-- & Hall, 4th edition, 1997, p. 155.\n\nmodule PA.Inductive2Axiomatic where\n\nopen import PA.Inductive.Base\n\nopen import Common.Relation.Nullary\n\n------------------------------------------------------------------------------\n\nS\u2081 :  \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2261 o \u2192 n \u2261 o\nS\u2081 refl refl = refl\n\nS\u2082 : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nS\u2082 refl = refl\n\nS\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\nS\u2083 ()\n\nS\u2084 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\nS\u2084 refl = refl\n\nS\u2085 : \u2200 n \u2192 zero + n \u2261 n\nS\u2085 n = refl\n\nS\u2086 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\nS\u2086 m n = refl\n\nS\u2087 : \u2200 n \u2192 zero * n \u2261 zero\nS\u2087 n = refl\n\nS\u2088 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\nS\u2088 m n = refl\n\nS\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\nS\u2089 P P0 h zero     = P0\nS\u2089 P P0 h (succ n) = h n (S\u2089 P P0 h n)\n","old_contents":"------------------------------------------------------------------------------\n-- From inductive PA to axiomatic PA\n------------------------------------------------------------------------------\n\n-- From the definition of PA using Agda data types and primitive\n-- recursive functions for addition and multiplication, we can prove the\n-- Mendelson's axioms [1]\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\n-- [1]. Elliott Mendelson. Introduction to mathematical logic. Chapman\n-- & Hall, 4th edition, 1997, p. 155)\n\nmodule PA.Inductive2Axiomatic where\n\nopen import PA.Inductive.Base\n\nopen import Common.Relation.Nullary\n\n------------------------------------------------------------------------------\n\nS\u2081 :  \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2261 o \u2192 n \u2261 o\nS\u2081 refl refl = refl\n\nS\u2082 : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nS\u2082 refl = refl\n\nS\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\nS\u2083 ()\n\nS\u2084 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\nS\u2084 refl = refl\n\nS\u2085 : \u2200 n \u2192 zero + n \u2261 n\nS\u2085 n = refl\n\nS\u2086 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\nS\u2086 m n = refl\n\nS\u2087 : \u2200 n \u2192 zero * n \u2261 zero\nS\u2087 n = refl\n\nS\u2088 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\nS\u2088 m n = refl\n\nS\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\nS\u2089 P P0 h zero     = P0\nS\u2089 P P0 h (succ n) = h n (S\u2089 P P0 h n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70f2a709ca8937d59833f7386d7b747be794de06","subject":"parameterised the suit type in Run","message":"parameterised the suit type in Run\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/rummy.agda","new_file":"agda\/rummy.agda","new_contents":"module rummy where\n\nopen import Data.Nat\n\n-- The suit type\ndata Suit : Set where\n  \u2663 : Suit\n  \u2662 : Suit\n  \u2665 : Suit\n  \u2660 : Suit\n\n-- A card consists of a value and a suite.\ndata Card : \u2115 \u2192 Suit \u2192 Set where\n  _of_ : (n : \u2115) (s : Suit) \u2192 Card n s\n\n-- Common names for cards\n\nA : \u2115; A = 1\nK : \u2115; K = 13\nQ : \u2115; Q = 12\nJ : \u2115; J = 11\n\n\n--\n-- The clubs suit\n--\n\nA\u2663  : Card A  \u2663; A\u2663  = A  of \u2663\n2\u2663  : Card 2  \u2663; 2\u2663  = 2  of \u2663\n3\u2663  : Card 3  \u2663; 3\u2663  = 3  of \u2663\n4\u2663  : Card 4  \u2663; 4\u2663  = 4  of \u2663\n5\u2663  : Card 5  \u2663; 5\u2663  = 5  of \u2663\n6\u2663  : Card 6  \u2663; 6\u2663  = 6  of \u2663\n7\u2663  : Card 7  \u2663; 7\u2663  = 7  of \u2663\n8\u2663  : Card 8  \u2663; 8\u2663  = 8  of \u2663\n9\u2663  : Card 9  \u2663; 9\u2663  = 9  of \u2663\n10\u2663 : Card 10 \u2663; 10\u2663 = 10 of \u2663\nJ\u2663  : Card J  \u2663; J\u2663  = J  of \u2663\nQ\u2663  : Card Q  \u2663; Q\u2663  = Q  of \u2663\nK\u2663  : Card K  \u2663; K\u2663  = K  of \u2663\n\n--\n-- The diamond suit\n--\n\nA\u2662  : Card A  \u2662; A\u2662  = A  of \u2662\n2\u2662  : Card 2  \u2662; 2\u2662  = 2  of \u2662\n3\u2662  : Card 3  \u2662; 3\u2662  = 3  of \u2662\n4\u2662  : Card 4  \u2662; 4\u2662  = 4  of \u2662\n5\u2662  : Card 5  \u2662; 5\u2662  = 5  of \u2662\n6\u2662  : Card 6  \u2662; 6\u2662  = 6  of \u2662\n7\u2662  : Card 7  \u2662; 7\u2662  = 7  of \u2662\n8\u2662  : Card 8  \u2662; 8\u2662  = 8  of \u2662\n9\u2662  : Card 9  \u2662; 9\u2662  = 9  of \u2662\n10\u2662 : Card 10 \u2662; 10\u2662 = 10 of \u2662\nJ\u2662  : Card J  \u2662; J\u2662  = J  of \u2662\nQ\u2662  : Card Q  \u2662; Q\u2662  = Q  of \u2662\nK\u2662  : Card K  \u2662; K\u2662  = K  of \u2662\n\n--\n-- The heart suit\n--\n\nA\u2665  : Card A  \u2665; A\u2665  = A  of \u2665\n2\u2665  : Card 2  \u2665; 2\u2665  = 2  of \u2665\n3\u2665  : Card 3  \u2665; 3\u2665  = 3  of \u2665\n4\u2665  : Card 4  \u2665; 4\u2665  = 4  of \u2665\n5\u2665  : Card 5  \u2665; 5\u2665  = 5  of \u2665\n6\u2665  : Card 6  \u2665; 6\u2665  = 6  of \u2665\n7\u2665  : Card 7  \u2665; 7\u2665  = 7  of \u2665\n8\u2665  : Card 8  \u2665; 8\u2665  = 8  of \u2665\n9\u2665  : Card 9  \u2665; 9\u2665  = 9  of \u2665\n10\u2665 : Card 10 \u2665; 10\u2665 = 10 of \u2665\nJ\u2665  : Card J  \u2665; J\u2665  = J  of \u2665\nQ\u2665  : Card Q  \u2665; Q\u2665  = Q  of \u2665\nK\u2665  : Card K  \u2665; K\u2665  = K  of \u2665\n\n--\n-- The spade suit\n--\n\nA\u2660  : Card A  \u2660; A\u2660  = A  of \u2660\n2\u2660  : Card 2  \u2660; 2\u2660  = 2  of \u2660\n3\u2660  : Card 3  \u2660; 3\u2660  = 3  of \u2660\n4\u2660  : Card 4  \u2660; 4\u2660  = 4  of \u2660\n5\u2660  : Card 5  \u2660; 5\u2660  = 5  of \u2660\n6\u2660  : Card 6  \u2660; 6\u2660  = 6  of \u2660\n7\u2660  : Card 7  \u2660; 7\u2660  = 7  of \u2660\n8\u2660  : Card 8  \u2660; 8\u2660  = 8  of \u2660\n9\u2660  : Card 9  \u2660; 9\u2660  = 9  of \u2660\n10\u2660 : Card 10 \u2660; 10\u2660 = 10 of \u2660\nJ\u2660  : Card J  \u2660; J\u2660  = J  of \u2660\nQ\u2660  : Card Q  \u2660; Q\u2660  = Q  of \u2660\nK\u2660  : Card K  \u2660; K\u2660  = K  of \u2660\n\n-- A run of length \u2113.\ndata Run  (suit : Suit) : (start \u2113 : \u2115) \u2192 Set where\n  [] : {start : \u2115} \u2192 Run  suit  start 0\n\n  _,_ : {n \u2113 : \u2115}\n       \u2192 Card n suit\n       \u2192 Run    suit (1 + n)     \u2113\n       \u2192 Run    suit    n     (1 + \u2113)\n\n\n-- A group of length \u2113.\ndata Group : (value \u2113 : \u2115) \u2192 Set where\n  [] : {value : \u2115} \u2192 Group value 0\n\n  _,_ : {suit : Suit} {value \u2113 : \u2115}\n       \u2192 Card   value  suit\n       \u2192 Group  value    \u2113\n       \u2192 Group  value    (1 + \u2113)\n\n-- Some pretty functions for creating runs. You can create a run as\n-- follows:\n--\n-- myrun = \u27e8 A\u2663 , 2\u2663 , 3\u2663 \u27e9\n--\n\n\u27e8_ : {suit : Suit} {n \u2113 : \u2115} \u2192 Run suit n \u2113 \u2192 Run suit n \u2113\n\u27e8 r = r\n\n_\u27e9 : {suit : Suit} {n : \u2115} \u2192 Card n suit \u2192 Run suit n 1\nc \u27e9 = c , []\n\n\n-- Some pretty functions for creating groups. You can create a group\n-- group as follows:\n--\n-- mygroup = \u27e6 A\u2663 , A\u2665 , A\u2662 \u27e7\n--\n\n\u27e6_ : {value \u2113 : \u2115} \u2192 Group value \u2113 \u2192 Group value \u2113\n\u27e6 g = g\n\n\n_\u27e7 : {suit : Suit} {n : \u2115} \u2192 Card n suit \u2192 Group n 1\nc \u27e7 = c , []\n\ninfixr 2 _\u27e9 _\u27e7 _,_\ninfixl 1 \u27e8_ \u27e6_\n\n\nmygroup = \u27e6 A\u2663 , A\u2663 , A\u2660 \u27e7\nmyrun   = \u27e8 A\u2663 , 2\u2663 , 3\u2663 \u27e9\n","old_contents":"module rummy where\n\nopen import Data.Nat\n\n-- The suit type\ndata Suit : Set where\n  \u2663 : Suit\n  \u2662 : Suit\n  \u2665 : Suit\n  \u2660 : Suit\n\n-- A card consists of a value and a suite.\ndata Card : \u2115 \u2192 Suit \u2192 Set where\n  _of_ : (n : \u2115) (s : Suit) \u2192 Card n s\n\n-- Common names for cards\n\nA : \u2115; A = 1\nK : \u2115; K = 13\nQ : \u2115; Q = 12\nJ : \u2115; J = 11\n\n\n--\n-- The clubs suit\n--\n\nA\u2663  : Card A  \u2663; A\u2663  = A  of \u2663\n2\u2663  : Card 2  \u2663; 2\u2663  = 2  of \u2663\n3\u2663  : Card 3  \u2663; 3\u2663  = 3  of \u2663\n4\u2663  : Card 4  \u2663; 4\u2663  = 4  of \u2663\n5\u2663  : Card 5  \u2663; 5\u2663  = 5  of \u2663\n6\u2663  : Card 6  \u2663; 6\u2663  = 6  of \u2663\n7\u2663  : Card 7  \u2663; 7\u2663  = 7  of \u2663\n8\u2663  : Card 8  \u2663; 8\u2663  = 8  of \u2663\n9\u2663  : Card 9  \u2663; 9\u2663  = 9  of \u2663\n10\u2663 : Card 10 \u2663; 10\u2663 = 10 of \u2663\nJ\u2663  : Card J  \u2663; J\u2663  = J  of \u2663\nQ\u2663  : Card Q  \u2663; Q\u2663  = Q  of \u2663\nK\u2663  : Card K  \u2663; K\u2663  = K  of \u2663\n\n--\n-- The diamond suit\n--\n\nA\u2662  : Card A  \u2662; A\u2662  = A  of \u2662\n2\u2662  : Card 2  \u2662; 2\u2662  = 2  of \u2662\n3\u2662  : Card 3  \u2662; 3\u2662  = 3  of \u2662\n4\u2662  : Card 4  \u2662; 4\u2662  = 4  of \u2662\n5\u2662  : Card 5  \u2662; 5\u2662  = 5  of \u2662\n6\u2662  : Card 6  \u2662; 6\u2662  = 6  of \u2662\n7\u2662  : Card 7  \u2662; 7\u2662  = 7  of \u2662\n8\u2662  : Card 8  \u2662; 8\u2662  = 8  of \u2662\n9\u2662  : Card 9  \u2662; 9\u2662  = 9  of \u2662\n10\u2662 : Card 10 \u2662; 10\u2662 = 10 of \u2662\nJ\u2662  : Card J  \u2662; J\u2662  = J  of \u2662\nQ\u2662  : Card Q  \u2662; Q\u2662  = Q  of \u2662\nK\u2662  : Card K  \u2662; K\u2662  = K  of \u2662\n\n--\n-- The heart suit\n--\n\nA\u2665  : Card A  \u2665; A\u2665  = A  of \u2665\n2\u2665  : Card 2  \u2665; 2\u2665  = 2  of \u2665\n3\u2665  : Card 3  \u2665; 3\u2665  = 3  of \u2665\n4\u2665  : Card 4  \u2665; 4\u2665  = 4  of \u2665\n5\u2665  : Card 5  \u2665; 5\u2665  = 5  of \u2665\n6\u2665  : Card 6  \u2665; 6\u2665  = 6  of \u2665\n7\u2665  : Card 7  \u2665; 7\u2665  = 7  of \u2665\n8\u2665  : Card 8  \u2665; 8\u2665  = 8  of \u2665\n9\u2665  : Card 9  \u2665; 9\u2665  = 9  of \u2665\n10\u2665 : Card 10 \u2665; 10\u2665 = 10 of \u2665\nJ\u2665  : Card J  \u2665; J\u2665  = J  of \u2665\nQ\u2665  : Card Q  \u2665; Q\u2665  = Q  of \u2665\nK\u2665  : Card K  \u2665; K\u2665  = K  of \u2665\n\n--\n-- The spade suit\n--\n\nA\u2660  : Card A  \u2660; A\u2660  = A  of \u2660\n2\u2660  : Card 2  \u2660; 2\u2660  = 2  of \u2660\n3\u2660  : Card 3  \u2660; 3\u2660  = 3  of \u2660\n4\u2660  : Card 4  \u2660; 4\u2660  = 4  of \u2660\n5\u2660  : Card 5  \u2660; 5\u2660  = 5  of \u2660\n6\u2660  : Card 6  \u2660; 6\u2660  = 6  of \u2660\n7\u2660  : Card 7  \u2660; 7\u2660  = 7  of \u2660\n8\u2660  : Card 8  \u2660; 8\u2660  = 8  of \u2660\n9\u2660  : Card 9  \u2660; 9\u2660  = 9  of \u2660\n10\u2660 : Card 10 \u2660; 10\u2660 = 10 of \u2660\nJ\u2660  : Card J  \u2660; J\u2660  = J  of \u2660\nQ\u2660  : Card Q  \u2660; Q\u2660  = Q  of \u2660\nK\u2660  : Card K  \u2660; K\u2660  = K  of \u2660\n\n\n-- A run of length \u2113.\ndata Run : Suit \u2192 (start \u2113 : \u2115) \u2192 Set where\n  [] : {suit : Suit} {start : \u2115} \u2192 Run  suit  start 0\n\n  _,_ : {suit : Suit} {n \u2113 : \u2115}\n       \u2192 Card n suit\n       \u2192 Run    suit (1 + n)     \u2113\n       \u2192 Run    suit    n     (1 + \u2113)\n\n\n-- A group of length \u2113.\ndata Group : (value \u2113 : \u2115) \u2192 Set where\n  [] : {value : \u2115} \u2192 Group value 0\n\n  _,_ : {suit : Suit} {value \u2113 : \u2115}\n       \u2192 Card   value  suit\n       \u2192 Group  value    \u2113\n       \u2192 Group  value    (1 + \u2113)\n\n-- Some pretty functions for creating runs. You can create a run as\n-- follows:\n--\n-- myrun = \u27e8 A\u2663 , 2\u2663 , 3\u2663 \u27e9\n--\n\n\u27e8_ : {suit : Suit} {n \u2113 : \u2115} \u2192 Run suit n \u2113 \u2192 Run suit n \u2113\n\u27e8 r = r\n\n_\u27e9 : {suit : Suit} {n : \u2115} \u2192 Card n suit \u2192 Run suit n 1\nc \u27e9 = c , []\n\n\n-- Some pretty functions for creating groups. You can create a group\n-- group as follows:\n--\n-- mygroup = \u27e6 A\u2663 , A\u2665 , A\u2662 \u27e7\n--\n\n\u27e6_ : {value \u2113 : \u2115} \u2192 Group value \u2113 \u2192 Group value \u2113\n\u27e6 g = g\n\n\n_\u27e7 : {suit : Suit} {n : \u2115} \u2192 Card n suit \u2192 Group n 1\nc \u27e7 = c , []\n\ninfixr 2 _\u27e9 _\u27e7 _,_\ninfixl 1 \u27e8_ \u27e6_\n\n\nmygroup = \u27e6 A\u2663 , A\u2663 , A\u2660 \u27e7\nmyrun   = \u27e8 A\u2663 , 2\u2663 , 3\u2663 \u27e9\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"1b459cc9f732afffe31d3c5025ab259aa09dc4ea","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/List\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/List\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationPropertiesI\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n++-leftCong : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 xs ++ zs \u2261 ys ++ zs\n++-leftCong refl = refl\n\n++-rightCong : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 xs ++ ys \u2261 xs ++ zs\n++-rightCong refl = refl\n\nmapCong\u2082 : \u2200 {f xs ys} \u2192 xs \u2261 ys \u2192 map f xs \u2261 map f ys\nmapCong\u2082 refl = refl\n\nrevCong\u2081 : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 rev xs zs \u2261 rev ys zs\nrevCong\u2081 refl = refl\n\nrevCong\u2082 : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 rev xs ys \u2261 rev xs zs\nrevCong\u2082 refl = refl\n\nreverseCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse xs \u2261 reverse ys\nreverseCong refl = refl\n\nreverse'Cong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse' xs \u2261 reverse' ys\nreverse'Cong refl = refl\n\nlengthCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 length xs \u2261 length ys\nlengthCong refl = refl\n\n------------------------------------------------------------------------------\n-- Totality properties\n\nlengthList-N : \u2200 {xs} \u2192 List xs \u2192 N (length xs)\nlengthList-N lnil               = subst N (sym length-[]) nzero\nlengthList-N (lcons x {xs} Lxs) =\n  subst N (sym (length-\u2237 x xs)) (nsucc (lengthList-N Lxs))\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} lnil               Lys = subst List (sym (++-[] ys)) Lys\n++-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (++-\u2237 x xs ys)) (lcons x (++-List Lxs Lys))\n\nmap-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List f lnil               = subst List (sym (map-[] f)) lnil\nmap-List f (lcons x {xs} Lxs) =\n  subst List (sym (map-\u2237 f x xs)) (lcons (f \u00b7 x) (map-List f Lxs))\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} lnil               Lys = subst List (sym (rev-[] ys)) Lys\nrev-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (rev-\u2237 x xs ys)) (rev-List Lxs (lcons x Lys))\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs lnil\n\nreverse'-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse' xs)\nreverse'-List lnil = subst List (sym reverse'-[]) lnil\nreverse'-List (lcons x {xs} Lxs) =\n  subst List (sym (reverse'-\u2237 x xs)) (++-List (reverse'-List Lxs) (lcons x lnil))\n\n-- Length properties\n\nlg-x<lg-x\u2237xs : \u2200 x {xs} \u2192 List xs \u2192 length xs < length (x \u2237 xs)\nlg-x<lg-x\u2237xs x lnil =\n  lt (length []) (length (x \u2237 []))\n    \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt (length []) (length (x \u2237 [])) \u2261 lt t t')\n              length-[]\n              (length-\u2237 x [])\n              refl\n    \u27e9\n  lt zero (succ\u2081 (length []))\n    \u2261\u27e8 lt-0S (length []) \u27e9\n  true \u220e\n\nlg-x<lg-x\u2237xs x (lcons y {xs} Lxs) =\n  lt (length (y \u2237 xs)) (length (x \u2237 y \u2237 xs))\n    \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt (length (y \u2237 xs)) (length (x \u2237 y \u2237 xs)) \u2261 lt t t')\n              (length-\u2237 y xs)\n              (length-\u2237 x (y \u2237 xs))\n              refl\n    \u27e9\n  lt (succ\u2081 (length xs)) (succ\u2081 (length (y \u2237 xs)))\n    \u2261\u27e8 lt-SS (length xs) (length (y \u2237 xs)) \u27e9\n  lt (length xs) (length (y \u2237 xs))\n    \u2261\u27e8 lg-x<lg-x\u2237xs y Lxs \u27e9\n  true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 : \u2200 {xs} \u2192 List xs \u2192 \u00ac (length xs < length [])\nlg-xs<lg-[]\u2192\u22a5 lnil lg-[]<lg-[] = \u22a5-elim (0<0\u2192\u22a5 helper)\n  where\n  helper : zero < zero\n  helper =\n    lt zero zero\n      \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt zero zero \u2261 lt t t')\n                (sym length-[])\n                (sym length-[])\n                refl\n      \u27e9\n    lt (length []) (length [])\n      \u2261\u27e8 lg-[]<lg-[] \u27e9\n    true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 (lcons x {xs} Lxs) lg-x\u2237xs<lg-[] = \u22a5-elim (S<0\u2192\u22a5 helper)\n  where\n  helper : succ\u2081 (length xs) < zero\n  helper =\n    lt (succ\u2081 (length xs)) zero\n      \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt (succ\u2081 (length xs)) zero \u2261 lt t t')\n                (sym (length-\u2237 x xs))\n                (sym length-[])\n                refl\n      \u27e9\n    lt (length (x \u2237 xs)) (length [])\n      \u2261\u27e8 lg-x\u2237xs<lg-[] \u27e9\n    true \u220e\n\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e : \u2200 x xs \u2192 length xs \u2261 \u221e \u2192 length (x \u2237 xs) \u2261 \u221e\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e x xs h =\n  length (x \u2237 xs)   \u2261\u27e8 length-\u2237 x xs \u27e9\n  succ\u2081 (length xs) \u2261\u27e8 succCong h \u27e9\n  succ\u2081 \u221e           \u2261\u27e8 sym \u221e-eq \u27e9\n  \u221e                 \u220e\n\n-- Append properties\n\n++-leftIdentity : \u2200 xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity = ++-[]\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity lnil               = ++-leftIdentity []\n++-rightIdentity (lcons x {xs} Lxs) =\n  (x \u2237 xs) ++ [] \u2261\u27e8 ++-\u2237 x xs [] \u27e9\n  x \u2237 (xs ++ []) \u2261\u27e8 \u2237-rightCong (++-rightIdentity Lxs) \u27e9\n  x \u2237 xs         \u220e\n\n++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc lnil ys zs =\n  ([] ++ ys) ++ zs \u2261\u27e8 ++-leftCong (++-leftIdentity ys) \u27e9\n  ys ++ zs         \u2261\u27e8 sym (++-leftIdentity (ys ++ zs)) \u27e9\n  [] ++ ys ++ zs   \u220e\n\n++-assoc (lcons x {xs} Lxs) ys zs =\n  ((x \u2237 xs) ++ ys) ++ zs \u2261\u27e8 ++-leftCong (++-\u2237 x xs ys) \u27e9\n  (x \u2237 (xs ++ ys)) ++ zs \u2261\u27e8 ++-\u2237 x (xs ++ ys) zs \u27e9\n  x \u2237 ((xs ++ ys) ++ zs) \u2261\u27e8 \u2237-rightCong (++-assoc Lxs ys zs) \u27e9\n  x \u2237 (xs ++ ys ++ zs)   \u2261\u27e8 sym (++-\u2237 x xs (ys ++ zs)) \u27e9\n  (x \u2237 xs) ++ ys ++ zs   \u220e\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f lnil ys =\n  map f ([] ++ ys)     \u2261\u27e8 mapCong\u2082 (++-leftIdentity ys) \u27e9\n  map f ys             \u2261\u27e8 sym (++-leftIdentity (map f ys)) \u27e9\n  [] ++ map f ys       \u2261\u27e8 ++-leftCong (sym (map-[] f)) \u27e9\n  map f [] ++ map f ys \u220e\n\nmap-++-commute f (lcons x {xs} Lxs) ys =\n  map f ((x \u2237 xs) ++ ys)         \u2261\u27e8 mapCong\u2082 (++-\u2237 x xs ys) \u27e9\n  map f (x \u2237 xs ++ ys)           \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n  f \u00b7 x \u2237 map f (xs ++ ys)       \u2261\u27e8 \u2237-rightCong (map-++-commute f Lxs ys) \u27e9\n  f \u00b7 x \u2237 (map f xs ++ map f ys) \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n  (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261\u27e8 ++-leftCong (sym (map-\u2237 f x xs)) \u27e9\n  map f (x \u2237 xs) ++ map f ys     \u220e\n\nmap\u2261[] : \u2200 {f xs} \u2192 List xs \u2192 map f xs \u2261 [] \u2192 xs \u2261 []\nmap\u2261[] lnil                   h = refl\nmap\u2261[] {f} (lcons x {xs} Lxs) h = \u22a5-elim ([]\u2262cons (trans (sym h) (map-\u2237 f x xs)))\n\n-- Reverse properties\n\nreverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\nreverse-[x]\u2261[x] x =\n  rev (x \u2237 []) [] \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 []) \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []          \u220e\n\nrev-++-commute : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute lnil ys =\n  rev [] ys       \u2261\u27e8 rev-[] ys \u27e9\n  ys              \u2261\u27e8 sym (++-leftIdentity ys) \u27e9\n  [] ++ ys        \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  rev [] [] ++ ys \u220e\n\nrev-++-commute (lcons x {xs} Lxs) ys =\n  rev (x \u2237 xs) ys\n    \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n  rev xs (x \u2237 ys)\n    \u2261\u27e8 rev-++-commute Lxs (x \u2237 ys) \u27e9\n  rev xs [] ++ x \u2237 ys\n    \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n             (sym (\n               (x \u2237 []) ++ ys  \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                x \u2237 ([] ++ ys) \u2261\u27e8 \u2237-rightCong (++-leftIdentity ys) \u27e9\n                x \u2237 ys         \u220e\n               )\n             )\n             refl\n    \u27e9\n  rev xs [] ++ (x \u2237 []) ++ ys\n    \u2261\u27e8 sym (++-assoc (rev-List Lxs lnil) (x \u2237 []) ys) \u27e9\n  (rev xs [] ++ (x \u2237 [])) ++ ys\n    \u2261\u27e8 ++-leftCong (sym (rev-++-commute Lxs (x \u2237 []))) \u27e9\n  rev xs (x \u2237 []) ++ ys\n    \u2261\u27e8 ++-leftCong (sym (rev-\u2237 x xs [])) \u27e9\n  rev (x \u2237 xs) [] ++ ys \u220e\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} lnil Lys =\n  reverse ([] ++ ys)       \u2261\u27e8 reverseCong (++-leftIdentity ys) \u27e9\n  reverse ys               \u2261\u27e8 sym (++-rightIdentity (reverse-List Lys)) \u27e9\n  reverse ys ++ []         \u2261\u27e8 ++-rightCong (sym (rev-[] [])) \u27e9\n  reverse ys ++ reverse [] \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) lnil =\n  reverse ((x \u2237 xs) ++ [])\n    \u2261\u27e8 reverseCong (++-rightIdentity (lcons x Lxs)) \u27e9\n  reverse (x \u2237 xs)\n    \u2261\u27e8 sym (++-leftIdentity (reverse (x \u2237 xs))) \u27e9\n  [] ++ reverse (x \u2237 xs)\n     \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  reverse [] ++ reverse (x \u2237 xs) \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) (lcons y {ys} Lys) =\n  reverse ((x \u2237 xs) ++ y \u2237 ys)\n    \u2261\u27e8 refl \u27e9\n  rev ((x \u2237 xs) ++ y \u2237 ys) []\n    \u2261\u27e8 revCong\u2081 (++-\u2237 x xs (y \u2237 ys)) \u27e9\n  rev (x \u2237 (xs ++ y \u2237 ys)) []\n    \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n  rev (xs ++ y \u2237 ys) (x \u2237 [])\n    \u2261\u27e8 rev-++-commute (++-List Lxs (lcons y Lys)) (x \u2237 []) \u27e9\n  rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n    \u2261\u27e8 refl \u27e9\n  reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n    \u2261\u27e8 ++-leftCong (reverse-++-commute Lxs (lcons y Lys)) \u27e9\n  (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n    \u2261\u27e8 ++-assoc (reverse-List (lcons y Lys)) (reverse xs) (x \u2237 []) \u27e9\n  reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n    \u2261\u27e8 refl \u27e9\n  reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n    \u2261\u27e8 ++-rightCong (sym (rev-++-commute Lxs (x \u2237 []))) \u27e9\n  reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n    \u2261\u27e8 ++-rightCong (sym (rev-\u2237 x xs [])) \u27e9\n  reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n    \u2261\u27e8 refl \u27e9\n  reverse (y \u2237 ys) ++ reverse (x \u2237 xs) \u220e\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x lnil =\n  rev (x \u2237 []) []     \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 [])     \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []              \u2261\u27e8 sym (++-leftIdentity (x \u2237 [])) \u27e9\n  [] ++ x \u2237 []        \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  rev [] [] ++ x \u2237 [] \u220e\n\nreverse-\u2237 x (lcons y {ys} Lys) = sym prf\n  where\n  prf : reverse (y \u2237 ys) ++ x \u2237 [] \u2261 reverse (x \u2237 y \u2237 ys)\n  prf =\n    reverse (y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 ++-rightCong (sym (reverse-[x]\u2261[x] x)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse (x \u2237 []))\n      \u2261\u27e8 sym (reverse-++-commute (lcons x lnil) (lcons y Lys)) \u27e9\n    reverse ((x \u2237 []) ++ (y \u2237 ys))\n      \u2261\u27e8 reverseCong (++-\u2237 x [] (y \u2237 ys)) \u27e9\n    reverse (x \u2237 ([] ++ (y \u2237 ys)))\n      \u2261\u27e8 reverseCong (\u2237-rightCong (++-leftIdentity (y \u2237 ys))) \u27e9\n    reverse (x \u2237 y \u2237 ys) \u220e\n\nreverse'-involutive-helper : \u2200 x {ys} \u2192 List ys \u2192\n                             reverse' (ys ++ x \u2237 []) \u2261 x \u2237 reverse' ys\nreverse'-involutive-helper x lnil =\n  reverse' ([] ++ x \u2237 []) \u2261\u27e8 reverse'Cong (++-[] (x \u2237 [])) \u27e9\n  reverse' (x \u2237 [])       \u2261\u27e8 reverse'-\u2237 x [] \u27e9\n  reverse' [] ++ x \u2237 []   \u2261\u27e8 ++-leftCong reverse'-[] \u27e9\n  [] ++ x \u2237 []            \u2261\u27e8 ++-[] (x \u2237 []) \u27e9\n  x \u2237 []                  \u2261\u27e8 \u2237-rightCong (sym reverse'-[]) \u27e9\n  x \u2237 reverse' []         \u220e\n\nreverse'-involutive-helper x (lcons y {ys} Lys) =\n  reverse' ((y \u2237 ys) ++ x \u2237 [])\n    \u2261\u27e8 reverse'Cong (++-\u2237 y ys (x \u2237 [])) \u27e9\n  reverse' (y \u2237 ys ++ x \u2237 [])\n    \u2261\u27e8 reverse'-\u2237 y (ys ++ x \u2237 []) \u27e9\n  reverse' (ys ++ x \u2237 []) ++ y \u2237 []\n    \u2261\u27e8 ++-leftCong (reverse'-involutive-helper x Lys) \u27e9\n  (x \u2237 reverse' ys) ++ (y \u2237 [])\n    \u2261\u27e8 ++-\u2237 x (reverse' ys) (y \u2237 []) \u27e9\n  x \u2237 (reverse' ys ++ y \u2237 [])\n    \u2261\u27e8 \u2237-rightCong (sym (reverse'-\u2237 y ys)) \u27e9\n  x \u2237 reverse' (y \u2237 ys) \u220e\n\n-- Adapted from (Bird and Wadler, 1988 \u00a75.4.2).\nreverse'-involutive : \u2200 {xs} \u2192 List xs \u2192 reverse' (reverse' xs) \u2261 xs\nreverse'-involutive lnil =\n  reverse' (reverse' []) \u2261\u27e8 reverse'Cong reverse'-[] \u27e9\n  reverse' []            \u2261\u27e8 reverse'-[] \u27e9\n  []                     \u220e\n\nreverse'-involutive (lcons x {xs} Lxs) =\n  reverse' (reverse' (x \u2237 xs))\n    \u2261\u27e8 reverse'Cong (reverse'-\u2237 x xs) \u27e9\n  reverse' (reverse' xs ++ (x \u2237 []))\n    \u2261\u27e8 reverse'-involutive-helper x (reverse'-List Lxs) \u27e9\n  x \u2237 reverse' (reverse' xs)\n    \u2261\u27e8 \u2237-rightCong (reverse'-involutive Lxs) \u27e9\n  x \u2237 xs \u220e\n\n------------------------------------------------------------------------------\n-- References\n\n-- \u2022 Bird, R. and Wadler, P. (1988). Introduction to Functional\n--   Programming. Prentice Hall International.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationPropertiesI\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n++-leftCong : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 xs ++ zs \u2261 ys ++ zs\n++-leftCong refl = refl\n\n++-rightCong : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 xs ++ ys \u2261 xs ++ zs\n++-rightCong refl = refl\n\nmapCong\u2082 : \u2200 {f xs ys} \u2192 xs \u2261 ys \u2192 map f xs \u2261 map f ys\nmapCong\u2082 refl = refl\n\nrevCong\u2081 : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 rev xs zs \u2261 rev ys zs\nrevCong\u2081 refl = refl\n\nrevCong\u2082 : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 rev xs ys \u2261 rev xs zs\nrevCong\u2082 refl = refl\n\nreverseCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse xs \u2261 reverse ys\nreverseCong refl = refl\n\nreverse'Cong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse' xs \u2261 reverse' ys\nreverse'Cong refl = refl\n\nlengthCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 length xs \u2261 length ys\nlengthCong refl = refl\n\n------------------------------------------------------------------------------\n-- Totality properties\n\nlengthList-N : \u2200 {xs} \u2192 List xs \u2192 N (length xs)\nlengthList-N lnil               = subst N (sym length-[]) nzero\nlengthList-N (lcons x {xs} Lxs) =\n  subst N (sym (length-\u2237 x xs)) (nsucc (lengthList-N Lxs))\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} lnil               Lys = subst List (sym (++-[] ys)) Lys\n++-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (++-\u2237 x xs ys)) (lcons x (++-List Lxs Lys))\n\nmap-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List f lnil               = subst List (sym (map-[] f)) lnil\nmap-List f (lcons x {xs} Lxs) =\n  subst List (sym (map-\u2237 f x xs)) (lcons (f \u00b7 x) (map-List f Lxs))\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} lnil               Lys = subst List (sym (rev-[] ys)) Lys\nrev-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (rev-\u2237 x xs ys)) (rev-List Lxs (lcons x Lys))\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs lnil\n\nreverse'-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse' xs)\nreverse'-List lnil = subst List (sym reverse'-[]) lnil\nreverse'-List (lcons x {xs} Lxs) =\n  subst List (sym (reverse'-\u2237 x xs)) (++-List (reverse'-List Lxs) (lcons x lnil))\n\n-- Length properties\n\nlg-x<lg-x\u2237xs : \u2200 x {xs} \u2192 List xs \u2192 length xs < length (x \u2237 xs)\nlg-x<lg-x\u2237xs x lnil =\n  lt (length []) (length (x \u2237 []))\n    \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt (length []) (length (x \u2237 [])) \u2261 lt t t')\n              length-[]\n              (length-\u2237 x [])\n              refl\n    \u27e9\n  lt zero (succ\u2081 (length []))\n    \u2261\u27e8 lt-0S (length []) \u27e9\n  true \u220e\n\nlg-x<lg-x\u2237xs x (lcons y {xs} Lxs) =\n  lt (length (y \u2237 xs)) (length (x \u2237 y \u2237 xs))\n    \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt (length (y \u2237 xs)) (length (x \u2237 y \u2237 xs)) \u2261 lt t t')\n              (length-\u2237 y xs)\n              (length-\u2237 x (y \u2237 xs))\n              refl\n    \u27e9\n  lt (succ\u2081 (length xs)) (succ\u2081 (length (y \u2237 xs)))\n    \u2261\u27e8 lt-SS (length xs) (length (y \u2237 xs)) \u27e9\n  lt (length xs) (length (y \u2237 xs))\n    \u2261\u27e8 lg-x<lg-x\u2237xs y Lxs \u27e9\n  true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 : \u2200 {xs} \u2192 List xs \u2192 \u00ac (length xs < length [])\nlg-xs<lg-[]\u2192\u22a5 lnil lg-[]<lg-[] = \u22a5-elim (0<0\u2192\u22a5 helper)\n  where\n  helper : zero < zero\n  helper =\n    lt zero zero\n      \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt zero zero \u2261 lt t t')\n                (sym length-[])\n                (sym length-[])\n                refl\n      \u27e9\n    lt (length []) (length [])\n      \u2261\u27e8 lg-[]<lg-[] \u27e9\n    true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 (lcons x {xs} Lxs) lg-x\u2237xs<lg-[] = \u22a5-elim (S<0\u2192\u22a5 helper)\n  where\n  helper : succ\u2081 (length xs) < zero\n  helper =\n    lt (succ\u2081 (length xs)) zero\n      \u2261\u27e8 subst\u2082 (\u03bb t t' \u2192 lt (succ\u2081 (length xs)) zero \u2261 lt t t')\n                (sym (length-\u2237 x xs))\n                (sym length-[])\n                refl\n      \u27e9\n    lt (length (x \u2237 xs)) (length [])\n      \u2261\u27e8 lg-x\u2237xs<lg-[] \u27e9\n    true \u220e\n\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e : \u2200 x xs \u2192 length xs \u2261 \u221e \u2192 length (x \u2237 xs) \u2261 \u221e\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e x xs h =\n  length (x \u2237 xs)   \u2261\u27e8 length-\u2237 x xs \u27e9\n  succ\u2081 (length xs) \u2261\u27e8 succCong h \u27e9\n  succ\u2081 \u221e           \u2261\u27e8 sym \u221e-eq \u27e9\n  \u221e                 \u220e\n\n-- Append properties\n\n++-leftIdentity : \u2200 xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity = ++-[]\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity lnil               = ++-leftIdentity []\n++-rightIdentity (lcons x {xs} Lxs) =\n  (x \u2237 xs) ++ [] \u2261\u27e8 ++-\u2237 x xs [] \u27e9\n  x \u2237 (xs ++ []) \u2261\u27e8 \u2237-rightCong (++-rightIdentity Lxs) \u27e9\n  x \u2237 xs         \u220e\n\n++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc lnil ys zs =\n  ([] ++ ys) ++ zs \u2261\u27e8 ++-leftCong (++-leftIdentity ys) \u27e9\n  ys ++ zs         \u2261\u27e8 sym (++-leftIdentity (ys ++ zs)) \u27e9\n  [] ++ ys ++ zs   \u220e\n\n++-assoc (lcons x {xs} Lxs) ys zs =\n  ((x \u2237 xs) ++ ys) ++ zs \u2261\u27e8 ++-leftCong (++-\u2237 x xs ys) \u27e9\n  (x \u2237 (xs ++ ys)) ++ zs \u2261\u27e8 ++-\u2237 x (xs ++ ys) zs \u27e9\n  x \u2237 ((xs ++ ys) ++ zs) \u2261\u27e8 \u2237-rightCong (++-assoc Lxs ys zs) \u27e9\n  x \u2237 (xs ++ ys ++ zs)   \u2261\u27e8 sym (++-\u2237 x xs (ys ++ zs)) \u27e9\n  (x \u2237 xs) ++ ys ++ zs   \u220e\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f lnil ys =\n  map f ([] ++ ys)     \u2261\u27e8 mapCong\u2082 (++-leftIdentity ys) \u27e9\n  map f ys             \u2261\u27e8 sym (++-leftIdentity (map f ys)) \u27e9\n  [] ++ map f ys       \u2261\u27e8 ++-leftCong (sym (map-[] f)) \u27e9\n  map f [] ++ map f ys \u220e\n\nmap-++-commute f (lcons x {xs} Lxs) ys =\n  map f ((x \u2237 xs) ++ ys)         \u2261\u27e8 mapCong\u2082 (++-\u2237 x xs ys) \u27e9\n  map f (x \u2237 xs ++ ys)           \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n  f \u00b7 x \u2237 map f (xs ++ ys)       \u2261\u27e8 \u2237-rightCong (map-++-commute f Lxs ys) \u27e9\n  f \u00b7 x \u2237 (map f xs ++ map f ys) \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n  (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261\u27e8 ++-leftCong (sym (map-\u2237 f x xs)) \u27e9\n  map f (x \u2237 xs) ++ map f ys     \u220e\n\nmap\u2261[] : \u2200 {f xs} \u2192 List xs \u2192 map f xs \u2261 [] \u2192 xs \u2261 []\nmap\u2261[] lnil                   h = refl\nmap\u2261[] {f} (lcons x {xs} Lxs) h = \u22a5-elim ([]\u2262cons (trans (sym h) (map-\u2237 f x xs)))\n\n-- Reverse properties\n\nreverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\nreverse-[x]\u2261[x] x =\n  rev (x \u2237 []) [] \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 []) \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []          \u220e\n\nrev-++-commute : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute lnil ys =\n  rev [] ys       \u2261\u27e8 rev-[] ys \u27e9\n  ys              \u2261\u27e8 sym (++-leftIdentity ys) \u27e9\n  [] ++ ys        \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  rev [] [] ++ ys \u220e\n\nrev-++-commute (lcons x {xs} Lxs) ys =\n  rev (x \u2237 xs) ys\n    \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n  rev xs (x \u2237 ys)\n    \u2261\u27e8 rev-++-commute Lxs (x \u2237 ys) \u27e9\n  rev xs [] ++ x \u2237 ys\n    \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n             (sym (\n               (x \u2237 []) ++ ys  \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                x \u2237 ([] ++ ys) \u2261\u27e8 \u2237-rightCong (++-leftIdentity ys) \u27e9\n                x \u2237 ys         \u220e\n               )\n             )\n             refl\n    \u27e9\n  rev xs [] ++ (x \u2237 []) ++ ys\n    \u2261\u27e8 sym (++-assoc (rev-List Lxs lnil) (x \u2237 []) ys) \u27e9\n  (rev xs [] ++ (x \u2237 [])) ++ ys\n    \u2261\u27e8 ++-leftCong (sym (rev-++-commute Lxs (x \u2237 []))) \u27e9\n  rev xs (x \u2237 []) ++ ys\n    \u2261\u27e8 ++-leftCong (sym (rev-\u2237 x xs [])) \u27e9\n  rev (x \u2237 xs) [] ++ ys \u220e\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} lnil Lys =\n  reverse ([] ++ ys)       \u2261\u27e8 reverseCong (++-leftIdentity ys) \u27e9\n  reverse ys               \u2261\u27e8 sym (++-rightIdentity (reverse-List Lys)) \u27e9\n  reverse ys ++ []         \u2261\u27e8 ++-rightCong (sym (rev-[] [])) \u27e9\n  reverse ys ++ reverse [] \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) lnil =\n  reverse ((x \u2237 xs) ++ [])\n    \u2261\u27e8 reverseCong (++-rightIdentity (lcons x Lxs)) \u27e9\n  reverse (x \u2237 xs)\n    \u2261\u27e8 sym (++-leftIdentity (reverse (x \u2237 xs))) \u27e9\n  [] ++ reverse (x \u2237 xs)\n     \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  reverse [] ++ reverse (x \u2237 xs) \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) (lcons y {ys} Lys) =\n  reverse ((x \u2237 xs) ++ y \u2237 ys)\n    \u2261\u27e8 refl \u27e9\n  rev ((x \u2237 xs) ++ y \u2237 ys) []\n    \u2261\u27e8 revCong\u2081 (++-\u2237 x xs (y \u2237 ys)) \u27e9\n  rev (x \u2237 (xs ++ y \u2237 ys)) []\n    \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n  rev (xs ++ y \u2237 ys) (x \u2237 [])\n    \u2261\u27e8 rev-++-commute (++-List Lxs (lcons y Lys)) (x \u2237 []) \u27e9\n  rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n    \u2261\u27e8 refl \u27e9\n  reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n    \u2261\u27e8 ++-leftCong (reverse-++-commute Lxs (lcons y Lys)) \u27e9\n  (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n    \u2261\u27e8 ++-assoc (reverse-List (lcons y Lys)) (reverse xs) (x \u2237 []) \u27e9\n  reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n    \u2261\u27e8 refl \u27e9\n  reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n    \u2261\u27e8 ++-rightCong (sym (rev-++-commute Lxs (x \u2237 []))) \u27e9\n  reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n    \u2261\u27e8 ++-rightCong (sym (rev-\u2237 x xs [])) \u27e9\n  reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n    \u2261\u27e8 refl \u27e9\n  reverse (y \u2237 ys) ++ reverse (x \u2237 xs) \u220e\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x lnil =\n  rev (x \u2237 []) []     \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 [])     \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []              \u2261\u27e8 sym (++-leftIdentity (x \u2237 [])) \u27e9\n  [] ++ x \u2237 []        \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  rev [] [] ++ x \u2237 [] \u220e\n\nreverse-\u2237 x (lcons y {ys} Lys) = sym prf\n  where\n  prf : reverse (y \u2237 ys) ++ x \u2237 [] \u2261 reverse (x \u2237 y \u2237 ys)\n  prf =\n    reverse (y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 ++-rightCong (sym (reverse-[x]\u2261[x] x)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse (x \u2237 []))\n      \u2261\u27e8 sym (reverse-++-commute (lcons x lnil) (lcons y Lys)) \u27e9\n    reverse ((x \u2237 []) ++ (y \u2237 ys))\n      \u2261\u27e8 reverseCong (++-\u2237 x [] (y \u2237 ys)) \u27e9\n    reverse (x \u2237 ([] ++ (y \u2237 ys)))\n      \u2261\u27e8 reverseCong (\u2237-rightCong (++-leftIdentity (y \u2237 ys))) \u27e9\n    reverse (x \u2237 y \u2237 ys) \u220e\n\nreverse'-involutive-helper : \u2200 x {ys} \u2192 List ys \u2192\n                             reverse' (ys ++ x \u2237 []) \u2261 x \u2237 reverse' ys\nreverse'-involutive-helper x lnil =\n  reverse' ([] ++ x \u2237 []) \u2261\u27e8 reverse'Cong (++-[] (x \u2237 [])) \u27e9\n  reverse' (x \u2237 [])       \u2261\u27e8 reverse'-\u2237 x [] \u27e9\n  reverse' [] ++ x \u2237 []   \u2261\u27e8 ++-leftCong reverse'-[] \u27e9\n  [] ++ x \u2237 []            \u2261\u27e8 ++-[] (x \u2237 []) \u27e9\n  x \u2237 []                  \u2261\u27e8 \u2237-rightCong (sym reverse'-[]) \u27e9\n  x \u2237 reverse' []         \u220e\n\nreverse'-involutive-helper x (lcons y {ys} Lys) =\n  reverse' ((y \u2237 ys) ++ x \u2237 [])\n    \u2261\u27e8 reverse'Cong (++-\u2237 y ys (x \u2237 [])) \u27e9\n  reverse' (y \u2237 ys ++ x \u2237 [])\n    \u2261\u27e8 reverse'-\u2237 y (ys ++ x \u2237 []) \u27e9\n  reverse' (ys ++ x \u2237 []) ++ y \u2237 []\n    \u2261\u27e8 ++-leftCong (reverse'-involutive-helper x Lys) \u27e9\n  (x \u2237 reverse' ys) ++ (y \u2237 [])\n    \u2261\u27e8 ++-\u2237 x (reverse' ys) (y \u2237 []) \u27e9\n  x \u2237 (reverse' ys ++ y \u2237 [])\n    \u2261\u27e8 \u2237-rightCong (sym (reverse'-\u2237 y ys)) \u27e9\n  x \u2237 reverse' (y \u2237 ys) \u220e\n\n-- Adapted from (Bird and Wadler, 1988 \u00a75.4.2).\nreverse'-involutive : \u2200 {xs} \u2192 List xs \u2192 reverse' (reverse' xs) \u2261 xs\nreverse'-involutive lnil =\n  reverse' (reverse' []) \u2261\u27e8 reverse'Cong reverse'-[] \u27e9\n  reverse' []            \u2261\u27e8 reverse'-[] \u27e9\n  []                     \u220e\n\nreverse'-involutive (lcons x {xs} Lxs) =\n  reverse' (reverse' (x \u2237 xs))\n    \u2261\u27e8 reverse'Cong (reverse'-\u2237 x xs) \u27e9\n  reverse' (reverse' xs ++ (x \u2237 []))\n    \u2261\u27e8 reverse'-involutive-helper x (reverse'-List Lxs) \u27e9\n  x \u2237 reverse' (reverse' xs) \u2261\u27e8 \u2237-rightCong (reverse'-involutive Lxs) \u27e9\n  x \u2237 xs \u220e\n\n------------------------------------------------------------------------------\n-- References\n\n-- \u2022 Bird, R. and Wadler, P. (1988). Introduction to Functional\n--   Programming. Prentice Hall International.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a2e492e71275027c52c9ce53e09896a0b941fa82","subject":"Cleaning.","message":"Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1NoHelperATP.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1NoHelperATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1NoHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- 30 November 2013. If we don't have the following definitions\n-- outside, the ATPs cannot prove the theorems.\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\nlemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n         Bit b \u2192\n         Fair os\u2081 \u2192\n         Fair os\u2082 \u2192\n         S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n         \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n           Fair os\u2081'\n           \u2227 Fair os\u2082'\n           \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n           \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n            Fair os\u2081'\n            \u2227 Fair os\u2082'\n            \u2227 (as' \u2261 await b i' is' ds'\n              \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n              \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n              \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n              \u2227 js' \u2261 out (not b) \u00b7 bs')\n            \u2227 js \u2261 i' \u2237 js'\n  {-# ATP prove prf #-}\n\n... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n  lemma\u2081 b i' is'\n         (ft\u2081^ ++ os\u2081')\n         (tail\u2081 os\u2082)\n         (await b i' is' ds)\n         (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n         (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n         (corrupt (tail\u2081 os\u2082) \u00b7\n           (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n         js Bb (Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')) (tail-Fair Fos\u2082) ihS\n     where\n     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n     {-# ATP prove os\u2081-eq-helper #-}\n\n     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n     {-# ATP prove as-eq #-}\n\n     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove cs-eq bs-eq #-}\n\n     postulate\n       ds-eq-helper\u2081 :\n         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n     postulate\n       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2082)\n\n     postulate\n       as^-eq-helper\u2081 :\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n     postulate\n       as^-eq-helper\u2082 :\n         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2082 #-}\n\n     as^-eq : as^ b i' is' ds \u2261\n              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove js-eq bs-eq #-}\n\n     ihS : S b\n             (i' \u2237 is')\n             (os\u2081^ os\u2081' ft\u2081^)\n             (os\u2082^ os\u2082)\n             (as^ b i' is' ds)\n             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n             js\n     ihS = as^-eq , refl , refl , refl , js-eq\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1NoHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- 30 November 2013. If we don't have the following definitions\n-- outside, the ATPs cannot prove the theorems.\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n-- Helper function for the ABP lemma 1\nmodule Helper where\n  helper : \u2200 {b i' is' os\u2081 os\u2082 as bs cs ds js} \u2192\n           Bit b \u2192\n           Fair os\u2082 \u2192\n           S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n           \u2200 ft\u2081 os\u2081' \u2192 F*T ft\u2081 \u2192 Fair os\u2081' \u2192 os\u2081 \u2261 ft\u2081 ++ os\u2081' \u2192\n           \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n             Fair os\u2081'\n             \u2227 Fair os\u2082'\n             \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n             \u2227 js \u2261 i' \u2237 js'\n  helper {b} {i'} {is'} {js = js}\n         Bb Fos\u2082 s .(T \u2237 []) os\u2081' f*tnil Fos\u2081' os\u2081-eq = prf\n    where\n    postulate\n      prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n              Fair os\u2081'\n              \u2227 Fair os\u2082'\n              \u2227 (as' \u2261 await b i' is' ds'\n                 \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n                 \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n                 \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n                 \u2227 js' \u2261 out (not b) \u00b7 bs')\n              \u2227 js \u2261 i' \u2237 js'\n    {-# ATP prove prf #-}\n\n  helper {b} {i'} {is'} {os\u2081} {os\u2082} {as} {bs} {cs} {ds} {js} Bb Fos\u2082 s\n         .(F \u2237 ft\u2081^) os\u2081' (f*tcons {ft\u2081^} FTft\u2081^) Fos\u2081' os\u2081-eq =\n         helper Bb (tail-Fair Fos\u2082) ihS ft\u2081^ os\u2081' FTft\u2081^ Fos\u2081' refl\n    where\n    postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n    {-# ATP prove os\u2081-eq-helper #-}\n\n    postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n    {-# ATP prove as-eq #-}\n\n    postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n    {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n    postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n    {-# ATP prove cs-eq bs-eq #-}\n\n    postulate\n      ds-eq-helper\u2081 :\n        os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n        ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n    postulate\n      ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                      ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n    ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n            \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                 (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                 (head-tail-Fair Fos\u2082)\n\n    postulate\n      as^-eq-helper\u2081 :\n        ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n        as^ b i' is' ds \u2261\n          send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n    postulate\n      as^-eq-helper\u2082 :\n        ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n        as^ b i' is' ds \u2261\n          send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    {-# ATP prove as^-eq-helper\u2082 #-}\n\n    as^-eq : as^ b i' is' ds \u2261\n             send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n    as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n    postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n    {-# ATP prove js-eq bs-eq #-}\n\n    ihS : S b\n            (i' \u2237 is')\n            (os\u2081^ os\u2081' ft\u2081^)\n            (os\u2082^ os\u2082)\n            (as^ b i' is' ds)\n            (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n            (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n            (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n            js\n    ihS = as^-eq , refl , refl , refl , js-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\n-- open Helper\n-- lemma\u2081 : \u2200 {b i' is' os\u2081 os\u2082 as bs cs ds js} \u2192\n--          Bit b \u2192\n--          Fair os\u2081 \u2192\n--          Fair os\u2082 \u2192\n--          S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n--          \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--            Fair os\u2081'\n--            \u2227 Fair os\u2082'\n--            \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n--            \u2227 js \u2261 i' \u2237 js'\n-- lemma\u2081 Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n-- ... | ft\u2081 , os\u2081' , FTft\u2081 , prf , Fos\u2081' =\n--   helper Bb Fos\u2082 s ft\u2081 os\u2081' FTft\u2081 Fos\u2081' prf\n\nlemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n         Bit b \u2192\n         Fair os\u2081 \u2192\n         Fair os\u2082 \u2192\n         S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n         \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n           Fair os\u2081'\n           \u2227 Fair os\u2082'\n           \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n           \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n            Fair os\u2081'\n            \u2227 Fair os\u2082'\n            \u2227 (as' \u2261 await b i' is' ds'\n              \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n              \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n              \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n              \u2227 js' \u2261 out (not b) \u00b7 bs')\n            \u2227 js \u2261 i' \u2237 js'\n  {-# ATP prove prf #-}\n\n... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n  lemma\u2081 b i' is'\n         (ft\u2081^ ++ os\u2081')\n         (tail\u2081 os\u2082)\n         (await b i' is' ds)\n         (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n         (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n         (corrupt (tail\u2081 os\u2082) \u00b7\n           (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n         js Bb (Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')) (tail-Fair Fos\u2082) ihS\n     where\n     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n     {-# ATP prove os\u2081-eq-helper #-}\n\n     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n     {-# ATP prove as-eq #-}\n\n     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove cs-eq bs-eq #-}\n\n     postulate\n       ds-eq-helper\u2081 :\n         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n     postulate\n       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2082)\n\n     postulate\n       as^-eq-helper\u2081 :\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n     postulate\n       as^-eq-helper\u2082 :\n         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2082 #-}\n\n     as^-eq : as^ b i' is' ds \u2261\n              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove js-eq bs-eq #-}\n\n     ihS : S b\n             (i' \u2237 is')\n             (os\u2081^ os\u2081' ft\u2081^)\n             (os\u2082^ os\u2082)\n             (as^ b i' is' ds)\n             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n             js\n     ihS = as^-eq , refl , refl , refl , js-eq\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e87a9fc0946ded53de5e1775b46c12ba9a8ba6dc","subject":"ZK.GroupHom.Types: use Eq?","message":"ZK.GroupHom.Types: use Eq?\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom\/Types.agda","new_file":"ZK\/GroupHom\/Types.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Type.Eq\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n\nmodule ZK.GroupHom.Types where\n\n-- P is the \"valid witness\" predicate.\nrecord ZK-hom (G+ G* : Type)(P : G+ \u2192 Type) : Type where\n  field\n    \n    \ud835\udd3e+ : Group G+\n    \ud835\udd3e* : Group G*\n\n    {{G*-eq?}} : Eq? G*\n\n  open Eq? G*-eq?\n  open Additive-Group       \ud835\udd3e+ hiding (_\u2297_) public\n  open Multiplicative-Group \ud835\udd3e* hiding (_^_) public\n\n  field\n    \u03c6 : G+ \u2192 G*\n    \u03c6-hom : GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* \u03c6\n\n    y : G*\n    \u03c6\u21d2P : \u2200 x \u2192 \u03c6 x \u2261 y \u2192 P x\n    P\u21d2\u03c6 : \u2200 x \u2192 P x \u2192 \u03c6 x \u2261 y\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n\nmodule ZK.GroupHom.Types where\n\nrecord GrpHom (G+ G* : Type)(P : G+ \u2192 Type) : Type where\n  field\n    \n    \ud835\udd3e+ : Group G+\n    \ud835\udd3e* : Group G*\n\n    _==_ : G* \u2192 G* \u2192 Bool\n    \u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n    ==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y\n\n  open Additive-Group       \ud835\udd3e+ hiding (_\u2297_) public\n  open Multiplicative-Group \ud835\udd3e* hiding (_^_) public\n\n  field\n    \u03c6 : G+ \u2192 G*\n    \u03c6-hom : GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* \u03c6\n\n    y : G*\n    \u03c6\u21d2P : \u2200 x \u2192 \u03c6 x \u2261 y \u2192 P x\n    P\u21d2\u03c6 : \u2200 x \u2192 P x \u2192 \u03c6 x \u2261 y\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"090d277f6b6c4ef8d7183a825c8257894f59d450","subject":"Cosmetic fixup","message":"Cosmetic fixup\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Value.agda","new_file":"Parametric\/Denotation\/Value.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for standard evaluation (Def. 3.1 and 3.2, Fig. 4c and 4f).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\n\nmodule Parametric.Denotation.Value\n    (Base : Type.Structure)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\n\n-- Extension point: Values for base types.\nStructure : Set\u2081\nStructure = Base \u2192 Set\n\nmodule Structure (\u27e6_\u27e7Base : Structure) where\n  -- We provide: Values for arbitrary types.\n  \u27e6_\u27e7Type : Type \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type = \u27e6 \u03b9 \u27e7Base\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\n  -- This means: Overload \u27e6_\u27e7 to mean \u27e6_\u27e7Type.\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n  -- We also provide: Environments of such values.\n  open import Base.Denotation.Environment Type \u27e6_\u27e7Type public\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for standard evaluation (Def. 3.1 and 3.2, Fig. 4c and 4f).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\n\nmodule Parametric.Denotation.Value\n    (Base : Type.Structure)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\n\n-- Extension point: Values for base types.\nStructure : Set\u2081\nStructure = Base \u2192 Set\n\nmodule Structure (\u27e6_\u27e7Base : Structure) where\n  -- We provide: Values for arbitrary types.\n  \u27e6_\u27e7Type : Type -> Set\n  \u27e6 base \u03b9 \u27e7Type = \u27e6 \u03b9 \u27e7Base\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\n  -- This means: Overload \u27e6_\u27e7 to mean \u27e6_\u27e7Type.\n  meaningOfType : Meaning Type\n  meaningOfType = meaning \u27e6_\u27e7Type\n\n  -- We also provide: Environments of such values.\n  open import Base.Denotation.Environment Type \u27e6_\u27e7Type public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1f22c1a971c7f3e0f08ff77e1da3a894e4bbc666","subject":"Update to the Is-nothing\/Is-just changes...","message":"Update to the Is-nothing\/Is-just changes...\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"-- {-# OPTIONS --without-K #-}\n-- NOTE with-K\nmodule Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Relation.Unary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Product\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\ninfixr 0 _\u2192?_\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\ninfixl 1 _>>=?_\n\n-- More universe-polymorphic than M?._>>=_\n_>>=?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 Maybe A \u2192 (A \u2192 Maybe B) \u2192 Maybe B\nmx >>=? f = maybe f nothing mx\n\ninfixr 1 _=<<?_\n\n-- More universe-polymorphic than M?._=<<_\n_=<<?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Maybe B) \u2192 Maybe A \u2192 Maybe B\nf =<<? mx = mx >>=? f\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f mx = mx >>=? (just \u2218 f)\n\n_<$?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2192 Maybe B \u2192 Maybe A\n_<$?_ x = map? (const x)\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = _=<<?_ id\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \ud835\udfd8\njust? (just _) = \ud835\udfd9\n\njust?\u2192Is-just : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 Is-just x\njust?\u2192Is-just {x = just _}  p = just _\njust?\u2192Is-just {x = nothing} ()\n\nAny\u2192just? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata [Maybe] {a p} {A : \u2605 a} (A\u209a : A \u2192 \u2605 p) : Maybe A \u2192 \u2605 (a \u2294 p) where\n  nothing : [Maybe] A\u209a nothing\n  just    : (A\u209a [\u2192] [Maybe] A\u209a) just\n\n[maybe] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] (\u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] ((A\u209a [\u2192] B\u209a) [\u2192] (B\u209a [\u2192] ([Maybe] A\u209a [\u2192] B\u209a)))))\n                     (maybe {a} {b})\n[maybe] _ _ just\u209a nothing\u209a (just x\u209a) = just\u209a x\u209a\n[maybe] _ _ just\u209a nothing\u209a nothing   = nothing\u209a\n\n[map?] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] \u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] (A\u209a [\u2192] B\u209a) [\u2192] [Maybe] A\u209a [\u2192] [Maybe] B\u209a) (map? {a} {b})\n[map?] _ _ f\u209a (just x\u209a) = just (f\u209a x\u209a)\n[map?] _ _ f\u209a nothing   = nothing\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  just    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  nothing : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (just x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 nothing   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (just x\u1d63) = just (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 nothing   = nothing\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (just x\u1d63) = just x\u1d63\n\u27e6map?-id\u27e7 _ nothing   = nothing\n\nAny-join? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192Is-just : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P x \u2192 Is-just x\nAny\u2192Is-just (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192Is-just\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\n(f \u2218? g) x = g x >>=? f\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 Is-just (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_[\u2192?]_ : \u2200 {a b pa pb} \u2192 ([\u2605] {a} pa [\u2192] [\u2605] {b} pb [\u2192] [\u2605] _) _\u2192?_\nA\u209a [\u2192?] B\u209a = A\u209a [\u2192] [Maybe] B\u209a\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = just (refl-A x)\n  refl refl-A nothing  = nothing\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (just x\u223c\u2081y) = just (sym-A x\u223c\u2081y)\n  sym sym-A nothing     = nothing\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (just x\u223cy) (just y\u223cz) = just (trans' x\u223cy y\u223cz)\n  trans trans' nothing    nothing    = nothing\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (just x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f nothing    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIs-nothing-\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 Is-nothing x \u2192 x \u2261 nothing\nIs-nothing-\u2261nothing nothing = \u2261.refl\nIs-nothing-\u2261nothing (just ())\n\n\u2261nothing-Is-nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 Is-nothing x\n\u2261nothing-Is-nothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  {-\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {f = f} {x}  {y} f-inj =\n    maybe {B = {!\u03bb x \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y!} {!!} x\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n  -}\n  \n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \ud835\udfd8-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\n-- NOTE with-K\nmodule Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Relation.Unary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Product\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\ninfixr 0 _\u2192?_\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\ninfixl 1 _>>=?_\n\n-- More universe-polymorphic than M?._>>=_\n_>>=?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 Maybe A \u2192 (A \u2192 Maybe B) \u2192 Maybe B\nmx >>=? f = maybe f nothing mx\n\ninfixr 1 _=<<?_\n\n-- More universe-polymorphic than M?._=<<_\n_=<<?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Maybe B) \u2192 Maybe A \u2192 Maybe B\nf =<<? mx = mx >>=? f\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f mx = mx >>=? (just \u2218 f)\n\n_<$?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2192 Maybe B \u2192 Maybe A\n_<$?_ x = map? (const x)\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = _=<<?_ id\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \ud835\udfd8\njust? (just _) = \ud835\udfd9\n\njust?\u2192IsJust : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata [Maybe] {a p} {A : \u2605 a} (A\u209a : A \u2192 \u2605 p) : Maybe A \u2192 \u2605 (a \u2294 p) where\n  nothing : [Maybe] A\u209a nothing\n  just    : (A\u209a [\u2192] [Maybe] A\u209a) just\n\n[maybe] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] (\u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] ((A\u209a [\u2192] B\u209a) [\u2192] (B\u209a [\u2192] ([Maybe] A\u209a [\u2192] B\u209a)))))\n                     (maybe {a} {b})\n[maybe] _ _ just\u209a nothing\u209a (just x\u209a) = just\u209a x\u209a\n[maybe] _ _ just\u209a nothing\u209a nothing   = nothing\u209a\n\n[map?] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] \u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] (A\u209a [\u2192] B\u209a) [\u2192] [Maybe] A\u209a [\u2192] [Maybe] B\u209a) (map? {a} {b})\n[map?] _ _ f\u209a (just x\u209a) = just (f\u209a x\u209a)\n[map?] _ _ f\u209a nothing   = nothing\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  just    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  nothing : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (just x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 nothing   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (just x\u1d63) = just (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 nothing   = nothing\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (just x\u1d63) = just x\u1d63\n\u27e6map?-id\u27e7 _ nothing   = nothing\n\nAny-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\n(f \u2218? g) x = g x >>=? f\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_[\u2192?]_ : \u2200 {a b pa pb} \u2192 ([\u2605] {a} pa [\u2192] [\u2605] {b} pb [\u2192] [\u2605] _) _\u2192?_\nA\u209a [\u2192?] B\u209a = A\u209a [\u2192] [Maybe] B\u209a\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = just (refl-A x)\n  refl refl-A nothing  = nothing\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (just x\u223c\u2081y) = just (sym-A x\u223c\u2081y)\n  sym sym-A nothing     = nothing\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (just x\u223cy) (just y\u223cz) = just (trans' x\u223cy y\u223cz)\n  trans trans' nothing    nothing    = nothing\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (just x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f nothing    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing-\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing-\u2261nothing nothing = \u2261.refl\nIsNothing-\u2261nothing (just (lift ()))\n\n\u2261nothing-IsNothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing-IsNothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  {-\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {f = f} {x}  {y} f-inj =\n    maybe {B = {!\u03bb x \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y!} {!!} x\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n  -}\n  \n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \ud835\udfd8-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b8f6e977049a76f280dbc743d24e7c27a134d2ad","subject":"Minor changes.","message":"Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\nN-ind-instance : \u2200 n \u2192\n                 N (zero + n) \u2192\n                 (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n                 \u2200 {m} \u2192 N m \u2192 N (m + n)\nN-ind-instance n = N-ind (\u03bb i \u2192 N (i + n))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 N-ind-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 N-ind #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 ih Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    ih {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\nN-ind-instance : \u2200 n \u2192\n                N (zero + n) \u2192\n                (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n                \u2200 {m} \u2192 N m \u2192 N (m + n)\nN-ind-instance n = N-ind (\u03bb i \u2192 N (i + n))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 N-ind-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 N-ind #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cdf0cf1b60dbea35b07e189a05453c23d29ef38a","subject":"close #20","message":"close #20\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-progress.agda","new_file":"complete-progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-progress where\n\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n\n  -- nb: don't need (\u0394 , \u2205 \u22a2 d :: \u03c4) because of typed-expansion\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} {e : hexp}\u2192\n                       ecomplete e \u2192\n                       \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                       ok d \u0394\n  complete-progress = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-progress where\n\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  -- nb: don't need (\u0394 , \u2205 \u22a2 d :: \u03c4) because of typed-expansion\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} {e : hexp}\u2192\n                       ecomplete e \u2192\n                       \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                       ok d \u0394\n  complete-progress = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"37474cb3311f0482faf34c1cab7fe887fa2cce56","subject":"Added doc: Agda doesn't accept the definition of Stream-build.","message":"Added doc: Agda doesn't accept the definition of Stream-build.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Functors.agda","new_file":"notes\/fixed-points\/Functors.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conat type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- The pred function is the conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n-- From (Leclerc and Paulin-Mohring 1994, p. 195).\n--\n-- TODO (07 January 2014): Agda doesn't accept the definition of\n-- Stream-build.\n{-# NO_TERMINATION_CHECK #-}\nStream-build :\n  {A X : Set} \u2192\n  (X \u2192 StreamF A X) \u2192\n  X \u2192 Stream A\nStream-build f x with f x\n... | a , x' = Wrap (a , Stream-build f x')\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Leclerc, F. and Paulin-Mohring, C. (1994). Programming with Streams\n-- in Coq. A case study : the Sieve of Eratosthenes. In: Types for\n-- Proofs and Programs (TYPES \u201993). Ed. by Barendregt, H. and Nipkow,\n-- T. Vol. 806. LNCS. Springer, pp. 191\u2013212.\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Based on (Vene, 2000).\n\nmodule Functors where\n\ninfixr 1 _+_\ninfixr 2 _\u00d7_\n\ndata Bool : Set where\n  false true : Bool\n\ndata _+_  (A B : Set) : Set where\n  inl : A \u2192 A + B\n  inr : B \u2192 A + B\n\ndata _\u00d7_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u00d7 B\n\n-- The terminal object.\ndata \u22a4 : Set where\n   <> : \u22a4\n\npostulate\n  -- The least fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Mu f = In (f (Mu f))\n\n  -- unIn :: Mu f \u2192 f (Mu f)\n  -- unIn (In x) = x\n\n  \u03bc    : (Set \u2192 Set) \u2192 Set\n  In   : {F : Set \u2192 Set} \u2192 F (\u03bc F) \u2192 \u03bc F\n  unIn : {F : Set \u2192 Set} \u2192 \u03bc F \u2192 F (\u03bc F)\n\npostulate\n  -- The greatest fixed-point.\n\n  -- Haskell definitions:\n\n  -- data Nu f = Wrap (f (Nu f))\n\n  -- out :: Nu f \u2192 (f (Nu f))\n  -- out (Wrap x) = x\n\n  \u03bd    : (Set \u2192 Set) \u2192 Set\n  Wrap : {F : Set \u2192 Set} \u2192 F (\u03bd F) \u2192 \u03bd F\n  out  : {F : Set \u2192 Set} \u2192 \u03bd F \u2192 F (\u03bd F)\n\n------------------------------------------------------------------------------\n-- Functors\n\n--  The identity functor (the functor for the empty and unit types).\nIdF : Set \u2192 Set\nIdF X = X\n\n-- The (co)natural numbers functor.\nNatF : Set \u2192 Set\nNatF X = \u22a4 + X\n\n-- The (co)list functor.\nListF : Set \u2192 Set \u2192 Set\nListF A X = \u22a4 + A \u00d7 X\n\n-- The stream functor.\nStreamF : Set \u2192 Set \u2192 Set\nStreamF A X = A \u00d7 X\n\n------------------------------------------------------------------------------\n-- Types as least fixed-points\n\n-- The empty type is a least fixed-point.\n\u22a5 : Set\n\u22a5 = \u03bc IdF\n\n-- The natural numbers type is a least fixed-point.\nN : Set\nN = \u03bc NatF\n\n-- The data constructors for the natural numbers.\nzero : N\nzero = In (inl <>)\n\nsucc : N \u2192 N\nsucc n = In (inr n)\n\n-- The list type is a least fixed-point.\nList : Set \u2192 Set\nList A = \u03bc (ListF A)\n\n-- The data constructors for List.\nnil : {A : Set} \u2192 List A\nnil = In (inl <>)\n\ncons : {A : Set} \u2192 A \u2192 List A \u2192 List A\ncons x xs = In (inr (x , xs))\n\n------------------------------------------------------------------------------\n-- Types as greatest fixed-points\n\n-- The unit type is a greatest fixed-point.\nUnit : Set\nUnit = \u03bd IdF\n\n-- Non-structural recursion\n-- unit : Nu IdF\n-- unit = Wrap IdF {!unit!}\n\n-- The conat type is a greatest fixed-point.\nConat : Set\nConat = \u03bd NatF\n\nzeroC : Conat\nzeroC = Wrap (inl <>)\n\nsuccC : Conat \u2192 Conat\nsuccC cn = Wrap (inr cn)\n\n-- Non-structural recursion for the definition of \u221eC.\n-- \u221eC : Conat\n-- \u221eC = succC {!\u221eC!}\n\n-- The pred function is the conat destructor.\npred : Conat \u2192 \u22a4 + Conat\npred cn with out cn\n... | inl _ = inl <>\n... | inr x = inr x\n\n-- The colist type is a greatest fixed-point.\nColist : Set \u2192 Set\nColist A = \u03bd (ListF A)\n\n-- The colist data constructors.\nnilCL : {A : Set} \u2192 Colist A\nnilCL = Wrap (inl <>)\n\nconsCL : {A : Set} \u2192 A \u2192 Colist A \u2192 Colist A\nconsCL x xs = Wrap (inr (x , xs))\n\n-- The colist destructors.\nnullCL : {A : Set} \u2192 Colist A \u2192 Bool\nnullCL xs with out xs\n... | inl _ = true\n... | inr _ = false\n\n-- headCL : {A : Set} \u2192 Colist A \u2192 A\n-- headCL {A} xs with out (ListF A) xs\n-- ... | inl t       =  -- Impossible\n-- ... | inr (x , _) = x\n\n-- tailCL : {A : Set} \u2192 Colist A \u2192 Colist A\n-- tailCL {A} xs with out (ListF A) xs\n-- ... | inl t         =  -- Impossible\n-- ... | inr (_ , xs') = xs'\n\n-- The stream type is a greatest fixed-point.\nStream : Set \u2192 Set\nStream A = \u03bd (StreamF A)\n\n-- The stream data constructor.\nconsS : {A : Set} \u2192 A \u2192 Stream A \u2192 Stream A\nconsS x xs = Wrap (x , xs)\n\n-- The stream destructors.\nheadS : {A : Set} \u2192 Stream A \u2192 A\nheadS xs with out xs\n... | x , _ = x\n\ntailS : {A : Set} \u2192 Stream A \u2192 Stream A\ntailS xs with out xs\n... | _ , xs' = xs'\n\n-- From (Leclerc and Paulin-Mohring 1994, p. 195).\n{-# NO_TERMINATION_CHECK #-}\nStream-build :\n  {A X : Set} \u2192\n  (X \u2192 StreamF A X) \u2192\n  X \u2192 Stream A\nStream-build f x with f x\n... | a , x' = Wrap (a , Stream-build f x')\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Leclerc, F. and Paulin-Mohring, C. (1994). Programming with Streams\n-- in Coq. A case study : the Sieve of Eratosthenes. In: Types for\n-- Proofs and Programs (TYPES \u201993). Ed. by Barendregt, H. and Nipkow,\n-- T. Vol. 806. LNCS. Springer, pp. 191\u2013212.\n--\n-- Vene, Varmo (2000). Categorical programming with inductive and\n-- coinductive types. PhD thesis. Faculty of Mathematics: University\n-- of Tartu.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39f10f6cc776bc8a3f06f83f7221e2ec0379280e","subject":"Prove that derivatives are unique","message":"Prove that derivatives are unique\n\nAlso prove needed reusable lemmas, which invert some of the properties\nwe already have.\n\nFixes #332, #333.\n\nOld-commit-hash: 23de9d1fa8c0fdcae447e177ff15873976b90d95\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      f \u229e df\n    \u2261\u27e8 derivative-is-nil df fdf \u27e9\n      f \u229e nil f\n    \u2261\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      f \u229e dg\n    \u220e\n    where\n      open \u2261-Reasoning\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n  --\n  -- Small TODO: It could be even more elegant to define a setoid for \u2259 and\n  -- use equational reasoning on it directly.\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} dfxdx\u2259dgxdx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      open import Postulate.Extensionality\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = dfxdx\u2259dgxdx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"979ab046243c5fc37711031baa48b878c74daf61","subject":"Desc stratified model: from host to embedding","message":"Desc stratified model: from host to embedding","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"185a82d3476a1ace8d82bbd3cb7f288ce84d66bd","subject":"Vec: sum, rot, properties","message":"Vec: sum, rot, properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; module \u2115\u00b0)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : Set a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = refl\n    map-id (x \u2237 xs) = \u2261.cong (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = refl\n    map-\u2218 f g (x \u2237 xs) = \u2261.cong (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x [] = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) rewrite sum-\u2237\u02b3 x xs | \u2115\u00b0.+-assoc x\u2081 (sum xs) x = refl\n\nrot\u2081 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : Set a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 [] = refl\nsum-rot\u2081 (x \u2237 xs) rewrite sum-\u2237\u02b3 x xs = \u2115\u00b0.+-comm x _\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x [] = refl\nmap-\u2237\u02b3 f x (x\u2081 \u2237 xs) rewrite map-\u2237\u02b3 f x xs = refl\n\nsum-map-rot\u2081 : \u2200 {n a} {A : Set a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = refl\nsum-map-rot\u2081 f (x \u2237 xs) rewrite \u2115\u00b0.+-comm (f x) (sum (map f xs))\n                              | \u2261.sym (sum-\u2237\u02b3 (f x) (map f xs))\n                              | \u2261.sym (map-\u2237\u02b3 f x xs)\n                              = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"49156ef8121da70923efd88829faf71b1f88c596","subject":"Explain file contents.","message":"Explain file contents.\n\nOld-commit-hash: ee63b553e1dfecc4fc7d70052659a3a2909b86e6\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with (one-variable) partial derivatives\n--\n-- Features:\n--   * Changes and derivatives are different.\n--   * \u0394 x e  maps changes of x to changes of e.\n--   * a full denotational semantics is given.\n--   * the domain of change types `\u27e6\u0394 \u03c4\u27e7` is uniformly defined as `\u27e6\u03c4\u27e7 \u2192 \u27e6\u03c4\u27e7`.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\n  -- `\u0394 \u03c4` is the type of changes to `\u03c4`\n  \u0394 : (\u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 \u0394 \u03c4 \u27e7Type = \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 x dx t` maps changes `dx` in `x` to changes in `t`\n  \u0394 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (x : Var \u0393 \u03c4\u2081) (dx : Var \u0393 (\u0394 \u03c4\u2081)) \u2192 (t : Term \u0393 \u03c4\u2082) \u2192 Term \u0393 (\u0394 \u03c4\u2082)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 x dx t \u27e7Term \u03c1 = \u03bb _ \u2192 \u27e6 t \u27e7Term (update x (\u27e6 dx \u27e7 \u03c1) \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 x dx t) = \u0394 (lift {\u0393\u2081} {\u0393\u2082} x) (lift {\u0393\u2081} {\u0393\u2082} dx) (weaken {\u0393\u2081} {\u0393\u2082} t)\n","old_contents":"module incremental where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\n  -- `\u0394 \u03c4` is the type of changes to `\u03c4`\n  \u0394 : (\u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 \u0394 \u03c4 \u27e7Type = \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 x dx t` maps changes `dx` in `x` to changes in `t`\n  \u0394 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (x : Var \u0393 \u03c4\u2081) (dx : Var \u0393 (\u0394 \u03c4\u2081)) \u2192 (t : Term \u0393 \u03c4\u2082) \u2192 Term \u0393 (\u0394 \u03c4\u2082)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 x dx t \u27e7Term \u03c1 = \u03bb _ \u2192 \u27e6 t \u27e7Term (update x (\u27e6 dx \u27e7 \u03c1) \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 x dx t) = \u0394 (lift {\u0393\u2081} {\u0393\u2082} x) (lift {\u0393\u2081} {\u0393\u2082} dx) (weaken {\u0393\u2081} {\u0393\u2082} t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c636f62c550d3eb56e72bbc1c976763fc4129c60","subject":"added an actual implementation of ex-fresh + proof","message":"added an actual implementation of ex-fresh + proof\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/Core.agda","new_file":"Agda\/Core.agda","new_contents":"module Core where\n\nopen import Data.Empty\nopen import Data.Nat\nopen import Data.List\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Data.Sum\nopen import Data.Nat.Properties\n\nAtom = \u2115\n\ndata Type : Set where\n  \u03c3 : Type\n  _\u27f6_ : (t\u2081 : Type) -> (t\u2082 : Type) -> Type\n\ndata PTerm : Set where\n  bv : (i : \u2115) -> PTerm\n  fv : (x : Atom) -> PTerm\n  lam : (e : PTerm) -> PTerm\n  app : (e\u2081 : PTerm) -> (e\u2082 : PTerm) -> PTerm\n  Y : (t\u2081 : Type) -> PTerm\n\n[_>>_] : \u2115 -> PTerm -> PTerm -> PTerm\n[ k >> u ] (bv i) with k \u225f i\n... | yes _ = u\n... | no _ = bv i\n[ k >> u ] (fv x) = fv x\n[ k >> u ] (lam t) = lam ([ (suc k) >> u ] t)\n[ k >> u ] (app t1 t2) = app ([ k >> u ] t1) ([ k >> u ] t2)\n[ k >> u ] (Y t) = Y t\n\n_^_ : PTerm -> PTerm -> PTerm\nt ^ u = [ 0 >> u ] t\n\n_^'_ : PTerm -> Atom -> PTerm\nt ^' x = t ^ (fv x)\n\n[_<<_] : \u2115 -> Atom -> PTerm -> PTerm\n[ k << x ] (bv i) = bv i\n[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n[ k << x ] (lam t) = lam ([ (suc k) << x ] t)\n[ k << x ] (app t1 t2) = app ([ k << x ] t1) ([ k << x ] t2)\n[ k << x ] (Y t) = Y t\n\n*_^_ : Atom -> PTerm -> PTerm\n* x ^ t = [ 0 << x ] t\n\n_[_::=_] : PTerm -> Atom -> PTerm -> PTerm\nbv i [ x ::= u ] = bv i\nfv y [ x ::= u ] with x \u225f y\n... | yes _ = u\n... | no _ = fv y\nlam t [ x ::= u ] = lam (t [ x ::= u ])\napp t1 t2 [ x ::= u ] = app (t1 [ x ::= u ]) (t2 [ x ::= u ])\nY t\u2081 [ x ::= u ] = Y t\u2081\n\nFVars = List Atom\n\nFV : PTerm -> FVars\nFV (bv i) = []\nFV (fv x) = x \u2237 []\nFV (lam e) = FV e\nFV (app e\u2081 e\u2082) = FV e\u2081 ++ FV e\u2082\nFV (Y t) = []\n\ndata Term : PTerm -> Set where\n  var : \u2200 {x} -> Term (fv x)\n  lam : (L : FVars) -> \u2200 {e} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> Term (e ^' x)) -> Term (lam e)\n  app : \u2200 {e\u2081 e\u2082} -> Term e\u2081 -> Term e\u2082 -> Term (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> Term (Y t)\n\n\npostulate \u2203fresh : FVars -> Atom\npostulate \u2203fresh-spec : \u2200 L -> \u2203fresh L \u2209 L\n\n\u222a : FVars -> Atom\n\u222a [] = 0\n\u222a (x \u2237 xs) = x \u2294 (\u222a xs)\n\n\u2203fresh-impl : FVars -> Atom\n\u2203fresh-impl list = suc (\u222a list)\n\n_\u2228_ = _\u228e_\n\n\u2115-meet-dist : \u2200 {x y z : \u2115} -> (x \u2264 y) \u2228 (x \u2264 z) -> x \u2264 y \u2294 z\n\u2115-meet-dist {zero} x\u2264y\u228ex\u2264z = z\u2264n\n\u2115-meet-dist {suc x} {zero} {zero} (inj\u2081 ())\n\u2115-meet-dist {suc x} {zero} {zero} (inj\u2082 ())\n\u2115-meet-dist {suc x} {zero} {suc z} (inj\u2081 ())\n\u2115-meet-dist {suc x} {zero} {suc z} (inj\u2082 sx\u2264sz) = sx\u2264sz\n\u2115-meet-dist {suc x} {suc y} {zero} (inj\u2081 sx\u2264sy) = sx\u2264sy\n\u2115-meet-dist {suc x} {suc y} {zero} (inj\u2082 ())\n\u2115-meet-dist {suc x} {suc y} {suc z} (inj\u2081 (s\u2264s x\u2264y)) = s\u2264s (\u2115-meet-dist (inj\u2081 x\u2264y))\n\u2115-meet-dist {suc x} {suc y} {suc z} (inj\u2082 (s\u2264s y\u2264z)) = s\u2264s (\u2115-meet-dist {_} {y} {z} (inj\u2082 y\u2264z))\n\n\u2264-refl : \u2200 {x : \u2115} -> x \u2264 x\n\u2264-refl {zero} = z\u2264n\n\u2264-refl {suc x} = s\u2264s \u2264-refl\n\n\u2208-cons : \u2200 {L} {x y : \u2115} -> x \u2208 (y \u2237 L) -> \u00ac(x \u2261 y) -> x \u2208 L\n\u2208-cons {[]} (here refl) \u00acx\u2261y = \u22a5-elim (\u00acx\u2261y refl)\n\u2208-cons {[]} (there ()) \u00acx\u2261y\n\u2208-cons {y \u2237 L} {x} (here refl) \u00acx\u2261y = \u22a5-elim (\u00acx\u2261y refl)\n\u2208-cons {y \u2237 L} {x} (there x\u2208L) \u00acx\u2261y = x\u2208L\n\n\u2203fresh-impl-spec' : \u2200 x L -> x \u2208 L -> x \u2264 \u222a L\n\u2203fresh-impl-spec' x [] ()\n\u2203fresh-impl-spec' x (y \u2237 L) x\u2208y\u2237L with x \u225f y\n\u2203fresh-impl-spec' x (.x \u2237 L) x\u2208y\u2237L | yes refl = \u2115-meet-dist {x} {x} (inj\u2081 \u2264-refl)\n\u2203fresh-impl-spec' x (y \u2237 L) x\u2208y\u2237L | no \u00acx\u2261y = \u2115-meet-dist {x} {y} (inj\u2082 (\u2203fresh-impl-spec' x L (\u2208-cons x\u2208y\u2237L \u00acx\u2261y)))\n\n\u2203fresh-impl-spec'' : \u2200 x L -> x \u2208 L -> \u00ac (x \u2261 \u2203fresh-impl L)\n\u2203fresh-impl-spec'' .(suc (\u222a L)) L x\u2208L refl = 1+n\u2270n {\u222a L} (\u2203fresh-impl-spec' (suc (\u222a L)) L x\u2208L)\n\n\u2203fresh-impl-spec : \u2200 L -> \u2203fresh-impl L \u2209 L\n\u2203fresh-impl-spec L \u2203fresh-implL\u2208L = \u2203fresh-impl-spec'' (\u2203fresh-impl L) L \u2203fresh-implL\u2208L refl\n","old_contents":"module Core where\n\nopen import Data.Empty\nopen import Data.Nat\nopen import Data.List\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nAtom = \u2115\n\ndata Type : Set where\n  \u03c3 : Type\n  _\u27f6_ : (t\u2081 : Type) -> (t\u2082 : Type) -> Type\n\ndata PTerm : Set where\n  bv : (i : \u2115) -> PTerm\n  fv : (x : Atom) -> PTerm\n  lam : (e : PTerm) -> PTerm\n  app : (e\u2081 : PTerm) -> (e\u2082 : PTerm) -> PTerm\n  Y : (t\u2081 : Type) -> PTerm\n\n[_>>_] : \u2115 -> PTerm -> PTerm -> PTerm\n[ k >> u ] (bv i) with k \u225f i\n... | yes _ = u\n... | no _ = bv i\n[ k >> u ] (fv x) = fv x\n[ k >> u ] (lam t) = lam ([ (suc k) >> u ] t)\n[ k >> u ] (app t1 t2) = app ([ k >> u ] t1) ([ k >> u ] t2)\n[ k >> u ] (Y t) = Y t\n\n_^_ : PTerm -> PTerm -> PTerm\nt ^ u = [ 0 >> u ] t\n\n_^'_ : PTerm -> Atom -> PTerm\nt ^' x = t ^ (fv x)\n\n[_<<_] : \u2115 -> Atom -> PTerm -> PTerm\n[ k << x ] (bv i) = bv i\n[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n[ k << x ] (lam t) = lam ([ (suc k) << x ] t)\n[ k << x ] (app t1 t2) = app ([ k << x ] t1) ([ k << x ] t2)\n[ k << x ] (Y t) = Y t\n\ncls : Atom -> PTerm -> PTerm\ncls x = [ 0 << x ]\n\n_[_::=_] : PTerm -> Atom -> PTerm -> PTerm\nbv i [ x ::= u ] = bv i\nfv y [ x ::= u ] with x \u225f y\n... | yes _ = u\n... | no _ = fv y\nlam t [ x ::= u ] = lam (t [ x ::= u ])\napp t1 t2 [ x ::= u ] = app (t1 [ x ::= u ]) (t2 [ x ::= u ])\nY t\u2081 [ x ::= u ] = Y t\u2081\n\nFVars = List Atom\n\nFV : PTerm -> FVars\nFV (bv i) = []\nFV (fv x) = x \u2237 []\nFV (lam e) = FV e\nFV (app e\u2081 e\u2082) = FV e\u2081 ++ FV e\u2082\nFV (Y t) = []\n\ndata Term : PTerm -> Set where\n  var : \u2200 {x} -> Term (fv x)\n  lam : (L : FVars) -> \u2200 {e} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> Term (e ^' x)) -> Term (lam e)\n  app : \u2200 {e\u2081 e\u2082} -> Term e\u2081 -> Term e\u2082 -> Term (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> Term (Y t)\n\n\npostulate \u2203fresh : FVars -> Atom\npostulate \u2203fresh-spec : \u2200 L -> \u2203fresh L \u2209 L\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9edbf6d8d5fc3aa0140a80c8a798755dcc6eb030","subject":"cases of progress; collapsing a couple of redundant ones into catch all pattern matches","message":"cases of progress; collapsing a couple of redundant ones into catch all pattern matches\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!})\n  progress (TAAp D1 x\u2082 D2) | _ | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _ = E (EAp1 x)\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V VConst = I (INEHole (FVal VConst))\n  progress (TANEHole x\u2081 (TALam D) x\u2082) | V VLam = I (INEHole (FVal VLam))\n  progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = {!!}\n  progress (TACast TAConst TCHole2) | V VConst = {!!}\n  progress (TACast D x\u2081) | V VLam = {!!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d err[ \u0394 ] \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | V x\u2081\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp TAConst () D2) | V VConst | I x\u2081\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!})\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V VConst = I (INEHole (FVal VConst))\n  progress (TANEHole x\u2081 (TALam D) x\u2082) | V VLam = I (INEHole (FVal VLam))\n  progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E {!!}\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = E {!!}\n  progress (TACast TAConst TCHole2) | V VConst = E {!!}\n  progress (TACast D x\u2081) | V VLam = E {!!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E {!!}\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8a318caa1b0dc32e9bb96871534b81ae66b15ee5","subject":"flipping the cast syntax to match the paper (and make sense)","message":"flipping the cast syntax to match the paper (and make sense)\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 \u00eb\n    \u2987_\u2988[_]   : \u00eb \u2192 (Nat \u00d7 subst) \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< t2 > e1') \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< t2 > e2') \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u27e6 u ::[ \u0393 ] \u2987\u2988 \u27e7\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< t > e') \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x t1 t2 e e' t2' \u0394 } \u2192\n              (\u0393 ,, (x , t1)) \u22a2 e \u21d0 t2 ~> e' :: t2' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' :: (t1 ==> t2') \u22a3 \u0394\n      EASubsume : \u2200{e u m \u0393 t' e' \u0394 t} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 m \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                  t ~ t' \u2192\n                  \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u t  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 t ~> \u2987\u2988[ u , id \u0393 ] :: t \u22a3 \u27e6 u ::[ \u0393 ] t \u27e7\n      EANEHole : \u2200{ \u0393 e u t e' t' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 t ~> \u2987 e' \u2988[ u , id \u0393 ] :: t \u22a3 (\u0394 ,, u ::[ \u0393 ] t)\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n    TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n    TAVar : \u2200{\u0394 \u0393 x t} \u2192 \u0394 , (\u0393 ,, (x , t)) \u22a2 X x :: t\n    TALam : \u2200{ \u0394 \u0393 x t1 e t2} \u2192\n            \u0394 , (\u0393 ,, (x , t1)) \u22a2 e :: t2 \u2192\n            \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e :: (t1 ==> t2)\n    TAAp : \u2200{ \u0394 \u0393 e1 e2 t1 t2 t} \u2192\n           \u0394 , \u0393 \u22a2 e1 :: t1 \u2192\n           t1 \u25b8arr (t2 ==> t) \u2192\n           \u0394 , \u0393 \u22a2 e2 :: t2 \u2192\n           \u0394 , \u0393 \u22a2 e1 \u2218 e2 :: t\n    -- TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' } \u2192 -- todo: overloaded form in the paper here\n    -- TANEHole : \u2200 { \u0394 \u0393 e t' \u0393' u \u03c3 t }\n    TACast : \u2200{ \u0394 \u0393 e t t'} \u2192\n           \u0394 , \u0393 \u22a2 e :: t' \u2192\n           t ~ t' \u2192\n           \u0394 , \u0393 \u22a2 < t > e :: t\n\n  -- todo: substition goes here\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< t > e) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n    STHole : \u2200{ e e' u \u03c3 } \u2192\n             e \u21a6 e' \u2192\n             \u2987 e \u2988[ u , \u03c3 ] \u21a6 \u2987 e' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ e1 e2 e1' } \u2192\n            e1 \u21a6 e1' \u2192\n            (e1 \u2218 e2) \u21a6 (e1' \u2218 e2)\n    STAp2 : \u2200{ e1 e2 e2' } \u2192\n            e1 final \u2192\n            e2 \u21a6 e2' \u2192\n            (e1 \u2218 e2) \u21a6 (e1 \u2218 e2')\n    STAp\u03b2 : \u2200{ e1 e2 t x } \u2192\n            e2 final \u2192\n            ((\u00b7\u03bb x [ t ] e1) \u2218 e2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] e2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 \u00eb\n    \u2987_\u2988[_]   : \u00eb \u2192 (Nat \u00d7 subst) \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< e2' > t2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u27e6 u ::[ \u0393 ] \u2987\u2988 \u27e7\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< e' > t) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x t1 t2 e e' t2' \u0394 } \u2192\n              (\u0393 ,, (x , t1)) \u22a2 e \u21d0 t2 ~> e' :: t2' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' :: (t1 ==> t2') \u22a3 \u0394\n      EASubsume : \u2200{e u m \u0393 t' e' \u0394 t} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 m \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                  t ~ t' \u2192\n                  \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u t  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 t ~> \u2987\u2988[ u , id \u0393 ] :: t \u22a3 \u27e6 u ::[ \u0393 ] t \u27e7\n      EANEHole : \u2200{ \u0393 e u t e' t' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 t ~> \u2987 e' \u2988[ u , id \u0393 ] :: t \u22a3 (\u0394 ,, u ::[ \u0393 ] t)\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n    TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n    TAVar : \u2200{\u0394 \u0393 x t} \u2192 \u0394 , (\u0393 ,, (x , t)) \u22a2 X x :: t\n    TALam : \u2200{ \u0394 \u0393 x t1 e t2} \u2192\n            \u0394 , (\u0393 ,, (x , t1)) \u22a2 e :: t2 \u2192\n            \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e :: (t1 ==> t2)\n    TAAp : \u2200{ \u0394 \u0393 e1 e2 t1 t2 t} \u2192\n           \u0394 , \u0393 \u22a2 e1 :: t1 \u2192\n           t1 \u25b8arr (t2 ==> t) \u2192\n           \u0394 , \u0393 \u22a2 e2 :: t2 \u2192\n           \u0394 , \u0393 \u22a2 e1 \u2218 e2 :: t\n    -- TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' } \u2192 -- todo: overloaded form in the paper here\n    -- TANEHole : \u2200 { \u0394 \u0393 e t' \u0393' u \u03c3 t }\n    TACast : \u2200{ \u0394 \u0393 e t t'} \u2192\n           \u0394 , \u0393 \u22a2 e :: t' \u2192\n           t ~ t' \u2192\n           \u0394 , \u0393 \u22a2 < e > t :: t\n\n  -- todo: substition goes here\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n    STHole : \u2200{ e e' u \u03c3 } \u2192\n             e \u21a6 e' \u2192\n             \u2987 e \u2988[ u , \u03c3 ] \u21a6 \u2987 e' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ e1 e2 e1' } \u2192\n            e1 \u21a6 e1' \u2192\n            (e1 \u2218 e2) \u21a6 (e1' \u2218 e2)\n    STAp2 : \u2200{ e1 e2 e2' } \u2192\n            e1 final \u2192\n            e2 \u21a6 e2' \u2192\n            (e1 \u2218 e2) \u21a6 (e1 \u2218 e2')\n    STAp\u03b2 : \u2200{ e1 e2 t x } \u2192\n            e2 final \u2192\n            ((\u00b7\u03bb x [ t ] e1) \u2218 e2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] e2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"80726e324aaf92949e38e84303659fd950b4d263","subject":"knocked out a bunch of easy preservation cases #2","message":"knocked out a bunch of easy preservation cases #2\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = TAAp {!!} {!!}\n  pres-lem (FHAp2 eps) D1 D2 D3 (FHAp2 D4) = TAAp {!!} {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = TANEHole {!!} {!!} {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n  pres-lem (FHFailedCast eps) D1 D2 D3 (FHFailedCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x y z w) (FHFailedCast FHOuter) = _ , TACast x TCHole1\n  pres-lem2 (TAFailedCast x y z w) (FHFailedCast (FHCast eps)) = pres-lem2 x eps\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = {!!} -- todo: this is a lemma\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = {!!}\n  pres-lem (FHAp2 eps) D1 D2 D3 (FHAp2 D4) = {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n  pres-lem (FHFailedCast eps) D1 D2 D3 (FHFailedCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) eps = {!!}\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp ta ta\u2081) (ITLam) = {!!}\n  pres-lem3 (TAAp ta ta\u2081) (ITApCast) = {!!}\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) () -- todo: this is a little surprising; nehole doesn't take a transition but it defieately can still step\n  pres-lem3 (TACast ta x) (ITCastID) = {!!}\n  pres-lem3 (TACast ta x) (ITCastSucceed x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITGround y) = {!!}\n  pres-lem3 (TACast ta x) (ITExpand x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITCastFail w y z) = {!!}\n  pres-lem3 (TAFailedCast x y z q) xx  = {!!}\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ca3e055bd85678679c9742aa612a363169593297","subject":"Fixed wrong importation.","message":"Fixed wrong importation.\n\nIgnore-this: b91d2c992b8642e6c3b42facc2c7ffb\n\ndarcs-hash:20100716151338-3bd4e-120a3d65a2b01e7dfe3f436a5ceb23ec962fa2b9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/DivisionPCF\/EquationsPCF.agda","new_file":"Examples\/DivisionPCF\/EquationsPCF.agda","new_contents":"------------------------------------------------------------------------------\n-- Equations for the division\n------------------------------------------------------------------------------\n\nmodule Examples.DivisionPCF.EquationsPCF where\n\nopen import LTC.Minimal\n\nopen import Examples.DivisionPCF.DivisionPCF\nopen import LTC.Data.NatPCF\nopen import LTC.Data.NatPCF.InequalitiesPCF\nopen import LTC.Data.NatPCF.InequalitiesPCF.PropertiesPCF using ( x\u2265y\u2192x\u226ey )\n\n----------------------------------------------------------------------\n-- The division result when the dividend is minor than the\n-- the divisor\n\npostulate\n  div-x<y : {i j : D} \u2192 LT i j \u2192 div i j \u2261 zero\n{-# ATP prove div-x<y #-}\n\n-- The division result when the dividend is greater or equal than the\n-- the divisor\n\npostulate\n  div-x\u2265y : {i j : D} \u2192 N i \u2192 N j \u2192 GE i j \u2192 div i j \u2261 succ (div (i - j) j)\n{-# ATP prove div-x\u2265y x\u2265y\u2192x\u226ey #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Equations for the division\n------------------------------------------------------------------------------\n\nmodule Examples.DivisionPCF.EquationsPCF where\n\nopen import LTC.Minimal\n\nopen import Examples.DivisionPCF.DivisionPCF\nopen import LTC.Data.NatPCF\nopen import LTC.Data.NatPCF.InequalitiesPCF\nopen import LTC.Data.NatPCF.InequalitiesPCF.PropertiesPCF-ER using ( x\u2265y\u2192x\u226ey )\n\n----------------------------------------------------------------------\n-- The division result when the dividend is minor than the\n-- the divisor\n\npostulate\n  div-x<y : {i j : D} \u2192 LT i j \u2192 div i j \u2261 zero\n{-# ATP prove div-x<y #-}\n\n-- The division result when the dividend is greater or equal than the\n-- the divisor\n\npostulate\n  div-x\u2265y : {i j : D} \u2192 N i \u2192 N j \u2192 GE i j \u2192 div i j \u2261 succ (div (i - j) j)\n{-# ATP prove div-x\u2265y x\u2265y\u2192x\u226ey #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c0fac480f365f96dffb1fcea4e72c4a1d00cb23c","subject":"Removed unnecessary hypothesis.","message":"Removed unnecessary hypothesis.\n\nIgnore-this: 25c9053c3019cb1adfc7df8c66c7c691\n\ndarcs-hash:20110405150506-3bd4e-991423ec03c050d8695db68e7e3bcceecf07d500.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 dom0 =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) (sN Nn) prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn (sN Nn) (nest-\u2264 h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 N n \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 Nz dom0           =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 NSn (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) NSn prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 (dom-N (nest n) h\u2082) h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn NSn (nest-\u2264 Nn h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 Nx dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ccb6d881829a82120858ae65dc823a63ad24858d","subject":"We could not prove the subtraction conversion rules without using totality hypotheses.","message":"We could not prove the subtraction conversion rules without using totality hypotheses.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/LTC-PCF\/Data\/Nat\/SubtractionRecCombinator.agda","new_file":"notes\/FOT\/LTC-PCF\/Data\/Nat\/SubtractionRecCombinator.agda","new_contents":"------------------------------------------------------------------------------\n-- Subtraction using the rec combinator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.LTC-PCF.Data.Nat.SubtractionRecCombinator where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat.Properties hiding ( \u2238-x0 ; \u2238-0x ; \u2238-0S ; \u2238-SS )\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.Equations\nopen import LTC-PCF.Data.Nat.Type\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u2238_\n\n------------------------------------------------------------------------------\n\n-- Subtraction\n_\u2238_ : D \u2192 D \u2192 D\nm \u2238 n = rec n m (lam (\u03bb _ \u2192 lam pred\u2081))\n\n-- Conversion rules (from the Agda standard library)\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S nzero =\n  rec (succ\u2081 zero) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) zero) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 predCong (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x nzero      = \u2238-x0 zero\n\u2238-0x (nsucc Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ nzero =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) zero) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS nzero (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb _ \u2192 lam pred\u2081))  \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS nzero Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (nsucc {m} Nm) (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS (nsucc Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb _ \u2192 lam pred\u2081) n)) \u27e9\n  (lam (\u03bb _ \u2192 lam pred\u2081)) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n------------------------------------------------------------------------------\n-- Conversion rules from the Agda standard library without totality\n-- hypotheses.\n\n-- We could not prove this property.\n-- \u2238-0S\u2081 : \u2200 n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n-- \u2238-0S\u2081 n =\n--   rec (succ\u2081 n) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n--     \u2261\u27e8 rec-S n zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 n \u00b7 (zero \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (zero \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (zero \u2238 n) \u27e9\n--   pred\u2081 (zero \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   zero \u220e\n\n-- We could not prove this property.\n-- \u2238-SS\u2081 : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n-- \u2238-SS\u2081 m n =\n--   rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n--   \u2261\u27e8 rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n--   pred\u2081 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   m \u2238 n \u220e\n\n------------------------------------------------------------------------------\n-- Coq conversion rules\n\n-- Fixpoint minus (n m:nat) : nat :=\n--   match n, m with\n--   | O, _ => n\n--   | S k, O => S k\n--   | S k, S l => k - l\n--   end\n\n\u2238-x0-coq : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0-coq n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n             n            \u220e\n\n\u2238-S0-coq : \u2200 n \u2192 succ\u2081 n \u2238 zero \u2261 succ\u2081 n\n\u2238-S0-coq n =\n  rec zero (succ\u2081 n) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-0 (succ\u2081 n) \u27e9\n  succ\u2081 n \u220e\n\n-- We could not prove this property.\n-- \u2238-SS-coq : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n-- \u2238-SS-coq m n =\n--   rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n--   \u2261\u27e8 rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n--   pred\u2081 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   m \u2238 n \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Subtraction using the rec combinator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.LTC-PCF.Data.Nat.SubtractionRecCombinator where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat.Properties hiding ( \u2238-x0 ; \u2238-0x ; \u2238-0S ; \u2238-SS )\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.Equations\nopen import LTC-PCF.Data.Nat.Type\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u2238_\n\n------------------------------------------------------------------------------\n\n-- Subtraction\n_\u2238_ : D \u2192 D \u2192 D\nm \u2238 n = rec n m (lam (\u03bb _ \u2192 lam (\u03bb x \u2192 pred\u2081 x)))\n\n-- Conversion rules\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S nzero =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 predCong (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x nzero      = \u2238-x0 zero\n\u2238-0x (nsucc Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ nzero =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS nzero (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS nzero Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (nsucc {m} Nm) (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS (nsucc Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n)) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8093f75eef288c7c04d8b9f1c1f35189d2cf3cfa","subject":"more goofing around","message":"more goofing around\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule preservation where\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             \u0394 \u22a2 d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation TAConst (Step _ _ _) = TAConst\n  preservation (TAVar x\u2081) step = abort (somenotnone (! x\u2081))\n  preservation (TALam wt) (Step x\u2081 x\u2082 x\u2083) = {!!}\n  preservation (TAAp wt x wt\u2081) step = {!!}\n  preservation (TAEHole x x\u2081) (Step x\u2082 x\u2083 x\u2084) = {!!}\n  preservation (TANEHole x wt x\u2081) step = {!!}\n  preservation (TACast wt x) step = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule preservation where\n  preservation : (\u0394 : hctx) (d d' : dhexp) (\u03c4 : htyp) \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d12151cac3a0c06a91309d0abf569a9d540d2ed1","subject":"Added nest-x\u22610 and nest-N.","message":"Added nest-x\u22610 and nest-N.\n\nIgnore-this: c3690576d9951142c70d174776b87d3\n\ndarcs-hash:20110405145834-3bd4e-84427e8786b525a2afd0693da15c7b1484b8f47f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 N n \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 Nz dom0           =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 NSn (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) NSn prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 (dom-N (nest n) h\u2082) h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn NSn (nest-\u2264 Nn h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 Nx dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-N : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-N dom0           = subst N (sym nest-0) zN\nnest-N (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-N h\u2082)\n\nnest-\u2264 : \u2200 {n} \u2192 N n \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 Nz dom0           =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 NSn (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (domS n h\u2081 h\u2082)) (nest-N h\u2081) NSn prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 (dom-N (nest n) h\u2082) h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N h\u2081) Nn NSn (nest-\u2264 Nn h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N dn-x) (x\u2264y\u2192x<Sy (nest-N dn-x) Nx (nest-\u2264 Nx dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"78075c4789c38d328bc21172fe14c4988e1eef4d","subject":"Rel.Bin.ToNat: fix","message":"Rel.Bin.ToNat: fix\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/ToNat.agda","new_file":"lib\/Relation\/Binary\/ToNat.agda","new_contents":"import Data.Nat.NP as \u2115\nopen \u2115 using (\u2115; zero; suc; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Function\nopen import Relation.Binary\nopen import Algebra.FunctionProperties\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Relation.Binary.ToNat {a} {A : Set a} (f : A \u2192 \u2115) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) where\n\n_<=_ : A \u2192 A \u2192 Bool\n_<=_ = \u2115._<=_ on f \n\n_\u2264_ : A \u2192 A \u2192 Set\nx \u2264 y = T (x <= y)\n\n_\u2293_ : A \u2192 A \u2192 A\nx \u2293 y = if x <= y then x else y\n\n_\u2294_ : A \u2192 A \u2192 A\nx \u2294 y = if x <= y then y else x\n\nisPreorder : IsPreorder _\u2261_ _\u2264_\nisPreorder = record { isEquivalence = \u2261.isEquivalence\n                    ; reflexive = reflexive\n                    ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n  where open \u2115.<= using (sound; complete)\n        reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 i \u2264 j\n        reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {f i})\n        trans : Transitive _\u2264_\n        trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound (f x) (f y) p) (sound (f y) (f z) q))\n\nisPartialOrder : IsPartialOrder _\u2261_ _\u2264_\nisPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n  where open \u2115.<= using (sound)\n        antisym : Antisymmetric _\u2261_ _\u2264_\n        antisym {x} {y} p q = f-inj (\u2115\u2264.antisym (sound (f x) (f y) p) (sound (f y) (f x) q))\n\nisTotalOrder : IsTotalOrder _\u2261_ _\u2264_\nisTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total (f x) (f y)) }\n  where open \u2115.<= using (complete)\n\nopen IsTotalOrder isTotalOrder\n\n\u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n\u2294-spec {x} {y} p with x <= y\n\u2294-spec _    | true = \u2261.refl\n\u2294-spec ()   | false\n\n\u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n\u2293-spec {x} {y} p with x <= y\n\u2293-spec _    | true = \u2261.refl\n\u2293-spec ()   | false\n\n\u2293-comm : Commutative _\u2261_ _\u2293_\n\u2293-comm x y with x <= y | y <= x | antisym {x} {y} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2294-comm : Commutative _\u2261_ _\u2294_\n\u2294-comm x y with x <= y | y <= x | antisym {y} {x} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n\u2293-\u2264 x y with total x y\n\u2293-\u2264 x y | inj\u2081 p rewrite \u2293-spec p = p\n\u2293-\u2264 x y | inj\u2082 p rewrite \u2293-comm x y | \u2293-spec p = refl\n\n\u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n\u2264-\u2294 x y with total x y\n\u2264-\u2294 x y | inj\u2081 p rewrite \u2294-comm y x | \u2294-spec p = p\n\u2264-\u2294 x y | inj\u2082 p rewrite \u2294-spec p = refl\n\n\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n\u2264-<_,_> {x} {y} {z} x\u2264y y\u2264z with total y z\n... | inj\u2081 p rewrite \u2293-spec p = x\u2264y\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = y\u2264z\n\n\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n\u2264-[_,_] {x} {y} {z} x\u2264z y\u2264z with total x y\n... | inj\u2081 p rewrite \u2294-spec p = y\u2264z\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = x\u2264z\n\n\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n\u2264-\u2293\u2080 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = id\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = flip trans p\n\n\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n\u2264-\u2293\u2081 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = flip trans p\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = id\n\n\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n\u2264-\u2294\u2080 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = trans p\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = id\n\n\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\u2264-\u2294\u2081 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = id\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = trans p\n","old_contents":"import Data.Nat.NP as \u2115\nopen \u2115 using (\u2115; zero; suc; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Function\nopen import Relation.Binary\nopen import Algebra.FunctionProperties\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Relation.Binary.ToNat {a} {A : Set a} (f : A \u2192 \u2115) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) where\n\n_<=_ : A \u2192 A \u2192 Bool\n_<=_ = \u2115._<=_ on f \n\n_\u2264_ : A \u2192 A \u2192 Set\nx \u2264 y = T (x <= y)\n\n_\u2293_ : A \u2192 A \u2192 A\nx \u2293 y = if x <= y then x else y\n\n_\u2294_ : A \u2192 A \u2192 A\nx \u2294 y = if x <= y then y else x\n\nisPreorder : IsPreorder _\u2261_ _\u2264_\nisPreorder = record { isEquivalence = \u2261.isEquivalence\n                    ; reflexive = reflexive\n                    ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n  where open \u2115.<=\n        reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 i \u2264 j\n        reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {f i})\n        trans : Transitive _\u2264_\n        trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound (f x) (f y) p) (sound (f y) (f z) q))\n\nisPartialOrder : IsPartialOrder _\u2261_ _\u2264_\nisPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n  where open \u2115.<= using (sound)\n        antisym : Antisymmetric _\u2261_ _\u2264_\n        antisym {x} {y} p q = f-inj (\u2115\u2264.antisym (sound (f x) (f y) p) (sound (f y) (f x) q))\n\nisTotalOrder : IsTotalOrder _\u2261_ _\u2264_\nisTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total (f x) (f y)) }\n  where open \u2115.<= using (complete)\n\nopen IsTotalOrder isTotalOrder\n\n\u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n\u2294-spec {x} {y} p with x <= y\n\u2294-spec _    | true = \u2261.refl\n\u2294-spec ()   | false\n\n\u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n\u2293-spec {x} {y} p with x <= y\n\u2293-spec _    | true = \u2261.refl\n\u2293-spec ()   | false\n\n\u2293-comm : Commutative _\u2261_ _\u2293_\n\u2293-comm x y with x <= y | y <= x | antisym {x} {y} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2294-comm : Commutative _\u2261_ _\u2294_\n\u2294-comm x y with x <= y | y <= x | antisym {y} {x} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n\u2293-\u2264 x y with total x y\n\u2293-\u2264 x y | inj\u2081 p rewrite \u2293-spec p = p\n\u2293-\u2264 x y | inj\u2082 p rewrite \u2293-comm x y | \u2293-spec p = refl\n\n\u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n\u2264-\u2294 x y with total x y\n\u2264-\u2294 x y | inj\u2081 p rewrite \u2294-comm y x | \u2294-spec p = p\n\u2264-\u2294 x y | inj\u2082 p rewrite \u2294-spec p = refl\n\n\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n\u2264-<_,_> {x} {y} {z} x\u2264y y\u2264z with total y z\n... | inj\u2081 p rewrite \u2293-spec p = x\u2264y\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = y\u2264z\n\n\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n\u2264-[_,_] {x} {y} {z} x\u2264z y\u2264z with total x y\n... | inj\u2081 p rewrite \u2294-spec p = y\u2264z\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = x\u2264z\n\n\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n\u2264-\u2293\u2080 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = id\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = flip trans p\n\n\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n\u2264-\u2293\u2081 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = flip trans p\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = id\n\n\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n\u2264-\u2294\u2080 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = trans p\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = id\n\n\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\u2264-\u2294\u2081 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = id\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = trans p\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b5fdf3d17a49654bf58307b5cda58c3229c0f5a","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: c85750b448aff287e79a31ee0a9493a1\n\ndarcs-hash:20120305232404-3bd4e-72b4f4dfe122fd57996eacaac807c149e85500d7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/Equality.agda","new_file":"src\/FOTC\/Data\/Stream\/Equality.agda","new_contents":"------------------------------------------------------------------------------\n-- Equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality where\n\n-- 2012-02-05. Although this module does not use the predicate Stream,\n-- it is defined in the Stream directory due to the theorem\n--\n-- \u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n--\n-- in the module FOTC.Data.Stream.PropertiesI\/ATP.\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The equality on streams.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The relation _\u2248_ is the greatest post-fixed point (by \u2248-gfp\u2081 and\n-- \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The relation _\u2248_ is a post-fixed point of the bisimulation functional\n-- (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed point, the relation\n-- _\u2248_ is also a pre-fixed point of the bisimulation functional (see\n-- below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- References:\n  --\n  -- \u2022 Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- \u2022 Bart Jacobs and Jan Rutten. (Co)algebras and\n  -- (co)induction. EATCS Bulletin, 62:222\u2013259, 1997.\n\n  ----------------------------------------------------------------------------\n  -- Adapted from (Dybjer and Sander, 1989, p. 310). In this paper, the\n  -- authors use the name\n\n  -- as (R :: R') bs'\n\n  -- for the bisimulation functional.\n\n  -- The bisimulation functional (Jacobs and Rutten, 1997, p. 30).\n  Bisimulation : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  Bisimulation _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The relation _\u2248_ is a post-fixed point of Bisimulation.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Bisimulation _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 Bisimulation _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- The relation _\u2248_ is a pre-fixed point of Bisimulation.\n  pre-fp : \u2200 {xs ys} \u2192 Bisimulation _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","old_contents":"------------------------------------------------------------------------------\n-- Equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2248_\n\n------------------------------------------------------------------------------\n-- The equality on streams.\npostulate\n  _\u2248_ : D \u2192 D \u2192 Set\n\n-- The relation _\u2248_ is the greatest post-fixed point (by \u2248-gfp\u2081 and\n-- \u2248-gfp\u2082) of the bisimulation functional (see below).\n\n-- The relation _\u2248_ is a post-fixed point of the bisimulation functional\n-- (see below).\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n-- N.B. This is an axiom schema. Because in the automatic proofs we\n-- *must* use an instance, we do not add this postulate as an ATP\n-- axiom.\npostulate\n  \u2248-gfp\u2082 : (_R_ : D \u2192 D \u2192 Set) \u2192\n           -- R is a post-fixed point of the bisimulation functional.\n           (\u2200 {xs ys} \u2192 xs R ys \u2192\n            \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n            xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n           -- _\u2248_ is greater than R.\n           \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n\n-- Because a greatest post-fixed point is a fixed point, the relation\n-- _\u2248_ is also a pre-fixed point of the bisimulation functional (see\n-- below).\n\u2248-gfp\u2083 : \u2200 {xs ys} \u2192\n         (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n          xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys') \u2192\n         xs \u2248 ys\n\u2248-gfp\u2083 h = \u2248-gfp\u2082 _R_ helper h\n  where\n  _R_ : D \u2192 D \u2192 Set\n  _R_ xs ys = \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  helper : \u2200 {xs ys} \u2192 xs R ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  helper (_ , _ , _ , xs'\u2248ys' , prf) = _ , _ , _ , \u2248-gfp\u2081 xs'\u2248ys' , prf\n\nprivate\n  module Bisimulation where\n  -- In FOTC we won't use the bisimulation functional. This module is\n  -- only for illustrative purposes.\n\n  -- Adapted from [1]. In this paper the authors use the name\n\n  -- as (R :: R') bs' (p. 310)\n\n  -- for the bisimulation functional.\n\n  -- [1] Peter Dybjer and Herbert Sander. A functional programming\n  -- approach to the specification and verification of concurrent\n  -- systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\n  -- The bisimulation functional (Bart Jacobs and Jan\n  -- Rutten. (Co)algebras and (co)induction. EATCS Bulletin,\n  -- 62:222\u2013259, 1997).\n  Bisimulation : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\n  Bisimulation _R_ xs ys =\n    \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' R ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n\n  -- The relation _\u2248_ is the greatest post-fixed point of\n  -- Bisimulation (by post-fp and gpfp).\n\n  -- The relation _\u2248_ is a post-fixed point of Bisimulation.\n  post-fp : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Bisimulation _\u2248_ xs ys\n  post-fp = \u2248-gfp\u2081\n\n  gpfp : (_R_ : D \u2192 D \u2192 Set) \u2192\n         -- R is a post-fixed point of Bisimulation.\n         (\u2200 {xs ys} \u2192 xs R ys \u2192 Bisimulation _R_ xs ys) \u2192\n         -- _\u2248_ is greater than R.\n         \u2200 {xs ys} \u2192 xs R ys \u2192 xs \u2248 ys\n  gpfp = \u2248-gfp\u2082\n\n  -- The relation _\u2248_ is a pre-fixed point of Bisimulation.\n  pre-fp : \u2200 {xs ys} \u2192 Bisimulation _\u2248_ xs ys \u2192 xs \u2248 ys\n  pre-fp = \u2248-gfp\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"81973895f5d78d8b5a38c25bbd33feacf5ac9374","subject":"Typo.","message":"Typo.\n\nIgnore-this: 5d39e5a0d391478bcb2a2a2d62bf3207\n\ndarcs-hash:20110930182240-3bd4e-f32121202fa2cd4391c83100189d0ea648064e4e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/DataPostulate.agda","new_file":"Draft\/DataPostulate.agda","new_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulated one: Using the induction\n-- principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulated one: Using pattern\n-- matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulated predicate to the data one.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {n} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {n} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulate predicate: Using the\n-- induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulate predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulate predicate to the data predicate\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {n} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {n} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"61adc205859b99f216096bf10da4b6ffe56adfc7","subject":"Simplify _\u22ce_","message":"Simplify _\u22ce_\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Syntax\/Context.agda","new_file":"Base\/Syntax\/Context.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- Typing Contexts\n-- ===============\n\nimport Data.List as List\nopen List public\n  using ()\n  renaming\n    ( [] to \u2205 ; _\u2237_ to _\u2022_\n    ; map to mapContext\n    )\n\nContext : Set\nContext = List.List Type\n\n-- Variables\n-- =========\n--\n-- Here it is clear that we are using de Bruijn indices,\n-- encoded as natural numbers, more or less.\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- Weakening\n-- =========\n--\n-- We provide two approaches for weakening: context prefixes and\n-- and subcontext relationship. In this formalization, we mostly\n-- (maybe exclusively?) use the latter.\n\n-- Context Prefixes\n-- ----------------\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  _\u22ce_ = List._++_\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Subcontexts\n-- -----------\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\n-- This handling of contexts is recommended by [this\n-- email](https:\/\/lists.chalmers.se\/pipermail\/agda\/2011\/003423.html) and\n-- attributed to Conor McBride.\n--\n-- The associated thread discusses a few alternatives solutions, including one\n-- where beta-reduction can handle associativity of ++.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- Typing Contexts\n-- ===============\n\nimport Data.List as List\nopen List public\n  using ()\n  renaming\n    ( [] to \u2205 ; _\u2237_ to _\u2022_\n    ; map to mapContext\n    )\n\nContext : Set\nContext = List.List Type\n\n-- Variables\n-- =========\n--\n-- Here it is clear that we are using de Bruijn indices,\n-- encoded as natural numbers, more or less.\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- Weakening\n-- =========\n--\n-- We provide two approaches for weakening: context prefixes and\n-- and subcontext relationship. In this formalization, we mostly\n-- (maybe exclusively?) use the latter.\n\n-- Context Prefixes\n-- ----------------\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Subcontexts\n-- -----------\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\n-- This handling of contexts is recommended by [this\n-- email](https:\/\/lists.chalmers.se\/pipermail\/agda\/2011\/003423.html) and\n-- attributed to Conor McBride.\n--\n-- The associated thread discusses a few alternatives solutions, including one\n-- where beta-reduction can handle associativity of ++.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"930e08a87ab9d0101427cf092ff607b105b2384b","subject":"tidying","message":"tidying\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"structural.agda","new_file":"structural.agda","new_contents":"","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule structural where\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"20a12b1042a885bee52efcc4c5159ce219d1c3e9","subject":"s\/prf\u2081\/A0 and s\/prf\u2082\/is.","message":"s\/prf\u2081\/A0 and s\/prf\u2082\/is.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Interactive proof using an instance of the induction principle.\n+-N-ind : \u2200 {n} \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = +-N-ind A0 is Nm\n  where\n  A0 : N (zero + n)\n  A0 = subst N (sym (+-0x n)) Nn\n\n  is : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n  is {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2082 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- N-ind\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- Because the ATPs don't handle induction, them cannot prove this\n-- postulate.\n-- {-# ATP prove +-N\u2083 N-ind #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Interactive proof using an instance of the induction principle.\n+-N-ind : \u2200 {n} \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = +-N-ind prf\u2081 prf\u2082 Nm\n  where\n  prf\u2081 : N (zero + n)\n  prf\u2081 = subst N (sym (+-0x n)) Nn\n\n  prf\u2082 : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n  prf\u2082 {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2082 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- N-ind\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- Because the ATPs don't handle induction, them cannot prove this\n-- postulate.\n-- {-# ATP prove +-N\u2083 N-ind #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b30747d2eeac44109ff2baf57c4dd3f3de4a5f02","subject":"Data.Bits: better sucBCarry","message":"Data.Bits: better sucBCarry\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u207f) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u207f (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"434a0a8ff0d6c6d85bdf642063a382c4e724cfdd","subject":"Fixed typo.","message":"Fixed typo.\n\nIgnore-this: 1dff44d99535d1bb494b667bcb944163\n\ndarcs-hash:20101208184847-3bd4e-45d95675ace25a7aee0c929d97204fa6fa3dc534.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/PropertiesER.agda","new_file":"src\/GroupTheory\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Group theory properties (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule GroupTheory.PropertiesER where\n\nopen import GroupTheory.Base\n\nopen import Common.Relation.Binary.EqReasoning using ( _\u2261\u27e8_\u27e9_ ; _\u220e ; begin_ )\nopen import Common.Relation.Binary.PropositionalEquality.PropertiesER\n  using ( subst )\n\n------------------------------------------------------------------------------\n\nrightIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 x \u2219 u \u2261 x) \u2227\n                               (\u2200 u' \u2192 (\u2200 x \u2192 x \u2219 u' \u2261 x) \u2192 u \u2261 u')\nrightIdentityUnique =\n-- Paper proof:\n-- 1.  We know that \u03b5 is a right identity.\n-- 2.  Let suppose there is other right identity u', i.e. \u2200 x \u2192 xu' \u2261 x, then\n-- 2.1 \u03b5   = \u03b5u'  (Hypothesis)\n-- 2.2 \u03b5u' = u    (Left identity)\n-- 2.3 \u03b5   = u    (Transitivity)\n  \u03b5 , rightIdentity , \u03bb u' hyp \u2192 trans (sym (hyp \u03b5)) (leftIdentity u')\n\nleftIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 u \u2219 x \u2261 x) \u2227\n                              (\u2200 u' \u2192 (\u2200 x \u2192 u' \u2219 x \u2261 x) \u2192 u \u2261 u')\nleftIdentityUnique =\n-- Paper proof:\n-- 1.  We know that \u03b5 is a left identity.\n-- 2.  Let's suppose there is other left identity u', i.e. \u2200 x \u2192 u'x \u2261 x, then\n-- 2.1 \u03b5   = u'\u03b5  (Hypothesis)\n-- 2.2 u'\u03b5 = u    (Right identity)\n-- 2.3 \u03b5   = u    (Transitivity)\n  \u03b5 , leftIdentity , \u03bb u' hyp \u2192 trans (sym (hyp \u03b5)) (rightIdentity u')\n\nrightCancellation : \u2200 {x y z} \u2192 y \u2219 x \u2261 z \u2219 x \u2192 y \u2261 z\nrightCancellation {x} {y} {z} yx\u2261zx =\n-- Paper proof:\n-- 1. (yx)x\u207b\u00b9  = (zx)x\u207b\u00b9  (Hypothesis xy = xz).\n-- 2. (y)xx\u207b\u00b9  = (z)xx\u207b\u00b9  (Associative).\n-- 3. y\u03b5       = z\u03b5       (Right inverse).\n-- 4. y        = z        (Right identity).\n  begin\n    y              \u2261\u27e8 sym (rightIdentity y) \u27e9\n    y \u2219 \u03b5          \u2261\u27e8 subst (\u03bb t \u2192 y \u2219 \u03b5 \u2261 y \u2219 t)\n                            (sym (rightInverse x))\n                            refl\n                   \u27e9\n    y \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 sym (assoc y x (x \u207b\u00b9)) \u27e9\n    y \u2219 x \u2219 x \u207b\u00b9   \u2261\u27e8 subst (\u03bb t \u2192 y \u2219 x \u2219 x \u207b\u00b9 \u2261 t \u2219 x \u207b\u00b9) yx\u2261zx refl \u27e9\n    z \u2219 x \u2219 x \u207b\u00b9   \u2261\u27e8 assoc z x (x \u207b\u00b9) \u27e9\n    z \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 subst (\u03bb t \u2192 z \u2219 (x \u2219 x \u207b\u00b9) \u2261 z \u2219 t)\n                            (rightInverse x)\n                            refl\n                   \u27e9\n    z \u2219 \u03b5          \u2261\u27e8 rightIdentity z \u27e9\n    z\n  \u220e\n\nleftCancellation : \u2200 {x y z} \u2192 x \u2219 y \u2261 x \u2219 z \u2192 y \u2261 z\nleftCancellation {x} {y} {z} xy\u2261xz =\n-- Paper proof:\n-- 1. x\u207b\u00b9(xy)  = x\u207b\u00b9(xz)  (Hypothesis xy = xz).\n-- 2. x\u207b\u00b9x(y)  = x\u207b\u00b9x(z)  (Associative).\n-- 3. \u03b5y       = \u03b5z       (Left inverse).\n-- 4. y        = z        (Left identity).\n  begin\n    y              \u2261\u27e8 sym (leftIdentity y) \u27e9\n    \u03b5 \u2219 y          \u2261\u27e8 subst (\u03bb t \u2192 \u03b5 \u2219 y \u2261 t \u2219 y) (sym (leftInverse x)) refl \u27e9\n    x \u207b\u00b9 \u2219 x \u2219 y   \u2261\u27e8 assoc (x \u207b\u00b9) x y \u27e9\n    x \u207b\u00b9 \u2219 (x \u2219 y) \u2261\u27e8 subst (\u03bb t \u2192 x \u207b\u00b9 \u2219 (x \u2219 y) \u2261 x \u207b\u00b9 \u2219 t) xy\u2261xz refl \u27e9\n    x \u207b\u00b9 \u2219 (x \u2219 z) \u2261\u27e8 sym (assoc (x \u207b\u00b9) x z) \u27e9\n    x \u207b\u00b9 \u2219 x \u2219 z   \u2261\u27e8 subst (\u03bb t \u2192 x \u207b\u00b9 \u2219 x \u2219 z \u2261 t \u2219 z) (leftInverse x) refl \u27e9\n    \u03b5 \u2219 z          \u2261\u27e8 leftIdentity z \u27e9\n    z\n  \u220e\n\nrightInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb r \u2192 (x \u2219 r \u2261 \u03b5) \u2227\n                                      (\u2200 r' \u2192 x \u2219 r' \u2261 \u03b5 \u2192 r \u2261 r')\nrightInverseUnique {x} =\n-- Paper proof:\n-- 1.   We know that (x \u207b\u00b9) is a right inverse for x.\n-- 2.   Let's suppose there is other right inverse r for x, i.e. xr \u2261 \u03b5, then\n-- 2.1. xx\u207b\u00b9 = \u03b5  (Right inverse).\n-- 2.2. xr   = \u03b5  (Hypothesis).\n-- 2.3. xx\u207b\u00b9 = xr (Transitivity).\n-- 2.4  x\u207b\u00b9  = r  (Left cancellation).\n  (x \u207b\u00b9) , rightInverse x , \u03bb r' hyp \u2192 leftCancellation\n    ( begin\n        x \u2219 x \u207b\u00b9 \u2261\u27e8 rightInverse x \u27e9\n        \u03b5        \u2261\u27e8 sym hyp \u27e9\n        x \u2219 r'\n      \u220e\n    )\n\n-- A more appropiate version to be used in the proofs.\nrightInverseUnique' : \u2200 {x r} \u2192 x \u2219 r \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 r\nrightInverseUnique' {x} {r} = \u03bb hyp \u2192 leftCancellation\n  ( begin\n      x \u2219 x \u207b\u00b9 \u2261\u27e8 rightInverse x \u27e9\n      \u03b5        \u2261\u27e8 sym hyp \u27e9\n      x \u2219 r\n    \u220e\n  )\n\nleftInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb l \u2192 (l \u2219 x \u2261 \u03b5) \u2227\n                                     (\u2200 l' \u2192 l' \u2219 x \u2261 \u03b5 \u2192 l \u2261 l')\nleftInverseUnique {x} =\n-- Paper proof:\n-- 1.   We know that (x \u207b\u00b9) is a left inverse for x.\n-- 2.   Let's suppose there is other right inverse l for x, i.e. lx \u2261 \u03b5, then\n-- 2.1. x\u207b\u00b9x = \u03b5  (Left inverse).\n-- 2.2. lx   = \u03b5  (Hypothesis).\n-- 2.3. x\u207b\u00b9x = lx (Transitivity).\n-- 2.4  x\u207b\u00b9  = l  (Right cancellation).\n  (x \u207b\u00b9) , leftInverse x , \u03bb l' hyp \u2192 rightCancellation\n    ( begin\n        x \u207b\u00b9 \u2219 x \u2261\u27e8 leftInverse x \u27e9\n        \u03b5 \u2261\u27e8 sym hyp \u27e9\n        l' \u2219 x\n      \u220e\n    )\n\n-- A more appropiate version to be used in the proofs.\nleftInverseUnique' : \u2200 {x l} \u2192 l \u2219 x \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 l\nleftInverseUnique' {x} {l} = \u03bb hyp \u2192 rightCancellation\n  ( begin\n      x \u207b\u00b9 \u2219 x \u2261\u27e8 leftInverse x \u27e9\n      \u03b5        \u2261\u27e8 sym hyp \u27e9\n      l \u2219 x\n    \u220e\n  )\n\n\u207b\u00b9-involutive : \u2200 x \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n-- Paper proof:\n-- 1. x\u207b\u00b9x = \u03b5  (Left inverse).\n-- 2. The previous equation states that x is the unique right\n-- inverse (x\u207b\u00b9)\u207b\u00b9 of x\u207b\u00b9.\n\u207b\u00b9-involutive x = rightInverseUnique' (leftInverse x)\n\nidentityInverse : \u03b5 \u207b\u00b9 \u2261 \u03b5\n-- Paper proof:\n-- 1. \u03b5\u03b5 = \u03b5  (Left\/Right identity).\n-- 2. The previous equation states that \u03b5 is the unique left\/right\n-- inverse \u03b5\u207b\u00b9 of \u03b5.\nidentityInverse = rightInverseUnique' (leftIdentity \u03b5)\n\ninverseDistribution : \u2200 x y \u2192 (x \u2219 y) \u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n-- Paper proof:\n-- (y\u207b\u00b9x\u207b\u00b9)(xy) = y\u207b\u00b9(x \u207b\u00b9(xy))  (Associative).\n--              = y\u207b\u00b9(x \u207b\u00b9x)y    (Associative).\n--              = y\u207b\u00b9(\u03b5y)        (Left inverse).\n--              = y\u207b\u00b9y           (Left identity).\n--              = \u03b5              (Left inverse)\n-- Therefore, y\u207b\u00b9x\u207b\u00b9 is the unique left inverse of xy.\ninverseDistribution x y = leftInverseUnique' y\u207b\u00b9x\u207b\u00b9[xy]\u2261\u03b5\n  where\n    y\u207b\u00b9x\u207b\u00b9[xy]\u2261\u03b5 : y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2219 (x \u2219 y) \u2261 \u03b5\n    y\u207b\u00b9x\u207b\u00b9[xy]\u2261\u03b5 =\n        begin\n          y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2219 (x \u2219 y)   \u2261\u27e8 assoc (y \u207b\u00b9) (x \u207b\u00b9) (x \u2219 y) \u27e9\n          y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 (x \u2219 y)) \u2261\u27e8 subst (\u03bb t \u2192 y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 (x \u2219 y)) \u2261\n                                                  y \u207b\u00b9 \u2219 t)\n                                           (sym (assoc (x \u207b\u00b9) x y))\n                                           refl\n                                  \u27e9\n          y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 x \u2219 y)   \u2261\u27e8 subst (\u03bb t \u2192 y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 x \u2219 y) \u2261\n                                                  y \u207b\u00b9 \u2219 (t \u2219 y))\n                                           (leftInverse x)\n                                           refl\n                                  \u27e9\n          y \u207b\u00b9 \u2219 (\u03b5 \u2219 y)          \u2261\u27e8 subst (\u03bb t \u2192 y \u207b\u00b9 \u2219 (\u03b5 \u2219 y) \u2261 y \u207b\u00b9 \u2219 t)\n                                           (leftIdentity y)\n                                           refl\n                                  \u27e9\n          y \u207b\u00b9 \u2219 y                \u2261\u27e8 leftInverse y \u27e9\n          \u03b5\n        \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory properties (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule GroupTheory.PropertiesER where\n\nopen import GroupTheory.Base\n\nopen import Common.Relation.Binary.EqReasoning using ( _\u2261\u27e8_\u27e9_ ; _\u220e ; begin_ )\nopen import Common.Relation.Binary.PropositionalEquality.PropertiesER\n  using ( subst )\n\n------------------------------------------------------------------------------\n\nrightIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 x \u2219 u \u2261 x) \u2227\n                               (\u2200 u' \u2192 (\u2200 x \u2192 x \u2219 u' \u2261 x) \u2192 u \u2261 u')\nrightIdentityUnique =\n-- Paper proof:\n-- 1.  We know that \u03b5 is a right identity.\n-- 2.  Let suppose there is other right identity u', i.e. \u2200 x \u2192 xu' \u2261 x, then\n-- 2.1 \u03b5   = \u03b5u'  (Hypothesis)\n-- 2.2 \u03b5u' = u    (Left identity)\n-- 2.3 \u03b5   = u    (Transitivity)\n  \u03b5 , rightIdentity , \u03bb u' hyp \u2192 trans (sym (hyp \u03b5)) (leftIdentity u')\n\nleftIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 u \u2219 x \u2261 x) \u2227\n                              (\u2200 u' \u2192 (\u2200 x \u2192 u' \u2219 x \u2261 x) \u2192 u \u2261 u')\nleftIdentityUnique =\n-- Paper proof:\n-- 1.  We know that \u03b5 is a left identity.\n-- 2.  Let's suppose there is other left identity u', i.e. \u2200 x \u2192 u'x \u2261 x, then\n-- 2.1 \u03b5   = u'\u03b5  (Hypothesis)\n-- 2.2 u'\u03b5 = u    (Right identity)\n-- 2.3 \u03b5   = u    (Transitivity)\n  \u03b5 , leftIdentity , \u03bb u' hyp \u2192 trans (sym (hyp \u03b5)) (rightIdentity u')\n\nrightCancellation : \u2200 {x y z} \u2192 y \u2219 x \u2261 z \u2219 x \u2192 y \u2261 z\nrightCancellation {x} {y} {z} yx\u2261zx =\n-- Paper proof:\n-- 1. (yx)x\u207b\u00b9  = (zx)x\u207b\u00b9  (Hypothesis xy = xz).\n-- 2. (y)xx\u207b\u00b9  = (z)xx\u207b\u00b9  (Associative).\n-- 3. y\u03b5       = z\u03b5       (Right inverse).\n-- 4. y        = z        (Right identity).\n  begin\n    y              \u2261\u27e8 sym (rightIdentity y) \u27e9\n    y \u2219 \u03b5          \u2261\u27e8 subst (\u03bb t \u2192 y \u2219 \u03b5 \u2261 y \u2219 t)\n                            (sym (rightInverse x))\n                            refl\n                   \u27e9\n    y \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 sym (assoc y x (x \u207b\u00b9)) \u27e9\n    y \u2219 x \u2219 x \u207b\u00b9   \u2261\u27e8 subst (\u03bb t \u2192 y \u2219 x \u2219 x \u207b\u00b9 \u2261 t \u2219 x \u207b\u00b9) yx\u2261zx refl \u27e9\n    z \u2219 x \u2219 x \u207b\u00b9   \u2261\u27e8 assoc z x (x \u207b\u00b9) \u27e9\n    z \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 subst (\u03bb t \u2192 z \u2219 (x \u2219 x \u207b\u00b9) \u2261 z \u2219 t)\n                            (rightInverse x)\n                            refl\n                   \u27e9\n    z \u2219 \u03b5          \u2261\u27e8 rightIdentity z \u27e9\n    z\n  \u220e\n\nleftCancellation : \u2200 {x y z} \u2192 x \u2219 y \u2261 x \u2219 z \u2192 y \u2261 z\nleftCancellation {x} {y} {z} xy\u2261xz =\n-- Paper proof:\n-- 1. x\u207b\u00b9 (xy) = x\u207b\u00b9(xz)  (Hypothesis xy = xz).\n-- 2. x\u207b\u00b9x(y)  = x\u207b\u00b9x(z)  (Associative)-\n-- 3. \u03b5y       = \u03b5z       (Left inverse).\n-- 4. y        = z        (Left identity).\n  begin\n    y              \u2261\u27e8 sym (leftIdentity y) \u27e9\n    \u03b5 \u2219 y          \u2261\u27e8 subst (\u03bb t \u2192 \u03b5 \u2219 y \u2261 t \u2219 y) (sym (leftInverse x)) refl \u27e9\n    x \u207b\u00b9 \u2219 x \u2219 y   \u2261\u27e8 assoc (x \u207b\u00b9) x y \u27e9\n    x \u207b\u00b9 \u2219 (x \u2219 y) \u2261\u27e8 subst (\u03bb t \u2192 x \u207b\u00b9 \u2219 (x \u2219 y) \u2261 x \u207b\u00b9 \u2219 t) xy\u2261xz refl \u27e9\n    x \u207b\u00b9 \u2219 (x \u2219 z) \u2261\u27e8 sym (assoc (x \u207b\u00b9) x z) \u27e9\n    x \u207b\u00b9 \u2219 x \u2219 z   \u2261\u27e8 subst (\u03bb t \u2192 x \u207b\u00b9 \u2219 x \u2219 z \u2261 t \u2219 z) (leftInverse x) refl \u27e9\n    \u03b5 \u2219 z          \u2261\u27e8 leftIdentity z \u27e9\n    z\n  \u220e\n\nrightInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb r \u2192 (x \u2219 r \u2261 \u03b5) \u2227\n                                      (\u2200 r' \u2192 x \u2219 r' \u2261 \u03b5 \u2192 r \u2261 r')\nrightInverseUnique {x} =\n-- Paper proof:\n-- 1.   We know that (x \u207b\u00b9) is a right inverse for x.\n-- 2.   Let's suppose there is other right inverse r for x, i.e. xr \u2261 \u03b5, then\n-- 2.1. xx\u207b\u00b9 = \u03b5  (Right inverse).\n-- 2.2. xr   = \u03b5  (Hypothesis).\n-- 2.3. xx\u207b\u00b9 = xr (Transitivity).\n-- 2.4  x\u207b\u00b9  = r  (Left cancellation).\n  (x \u207b\u00b9) , rightInverse x , \u03bb r' hyp \u2192 leftCancellation\n    ( begin\n        x \u2219 x \u207b\u00b9 \u2261\u27e8 rightInverse x \u27e9\n        \u03b5        \u2261\u27e8 sym hyp \u27e9\n        x \u2219 r'\n      \u220e\n    )\n\n-- A more appropiate version to be used in the proofs.\nrightInverseUnique' : \u2200 {x r} \u2192 x \u2219 r \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 r\nrightInverseUnique' {x} {r} = \u03bb hyp \u2192 leftCancellation\n  ( begin\n      x \u2219 x \u207b\u00b9 \u2261\u27e8 rightInverse x \u27e9\n      \u03b5        \u2261\u27e8 sym hyp \u27e9\n      x \u2219 r\n    \u220e\n  )\n\nleftInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb l \u2192 (l \u2219 x \u2261 \u03b5) \u2227\n                                     (\u2200 l' \u2192 l' \u2219 x \u2261 \u03b5 \u2192 l \u2261 l')\nleftInverseUnique {x} =\n-- Paper proof:\n-- 1.   We know that (x \u207b\u00b9) is a left inverse for x.\n-- 2.   Let's suppose there is other right inverse l for x, i.e. lx \u2261 \u03b5, then\n-- 2.1. x\u207b\u00b9x = \u03b5  (Left inverse).\n-- 2.2. lx   = \u03b5  (Hypothesis).\n-- 2.3. x\u207b\u00b9x = lx (Transitivity).\n-- 2.4  x\u207b\u00b9  = l  (Right cancellation).\n  (x \u207b\u00b9) , leftInverse x , \u03bb l' hyp \u2192 rightCancellation\n    ( begin\n        x \u207b\u00b9 \u2219 x \u2261\u27e8 leftInverse x \u27e9\n        \u03b5 \u2261\u27e8 sym hyp \u27e9\n        l' \u2219 x\n      \u220e\n    )\n\n-- A more appropiate version to be used in the proofs.\nleftInverseUnique' : \u2200 {x l} \u2192 l \u2219 x \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 l\nleftInverseUnique' {x} {l} = \u03bb hyp \u2192 rightCancellation\n  ( begin\n      x \u207b\u00b9 \u2219 x \u2261\u27e8 leftInverse x \u27e9\n      \u03b5        \u2261\u27e8 sym hyp \u27e9\n      l \u2219 x\n    \u220e\n  )\n\n\u207b\u00b9-involutive : \u2200 x \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n-- Paper proof:\n-- 1. x\u207b\u00b9x = \u03b5  (Left inverse).\n-- 2. The previous equation states that x is the unique right\n-- inverse (x\u207b\u00b9)\u207b\u00b9 of x\u207b\u00b9.\n\u207b\u00b9-involutive x = rightInverseUnique' (leftInverse x)\n\nidentityInverse : \u03b5 \u207b\u00b9 \u2261 \u03b5\n-- Paper proof:\n-- 1. \u03b5\u03b5 = \u03b5  (Left\/Right identity).\n-- 2. The previous equation states that \u03b5 is the unique left\/right\n-- inverse \u03b5\u207b\u00b9 of \u03b5.\nidentityInverse = rightInverseUnique' (leftIdentity \u03b5)\n\ninverseDistribution : \u2200 x y \u2192 (x \u2219 y) \u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n-- Paper proof:\n-- (y\u207b\u00b9x\u207b\u00b9)(xy) = y\u207b\u00b9(x \u207b\u00b9(xy))  (Associative).\n--              = y\u207b\u00b9(x \u207b\u00b9x)y    (Associative).\n--              = y\u207b\u00b9(\u03b5y)        (Left inverse).\n--              = y\u207b\u00b9y           (Left identity).\n--              = \u03b5              (Left inverse)\n-- Therefore, y\u207b\u00b9x\u207b\u00b9 is the unique left inverse of xy.\ninverseDistribution x y = leftInverseUnique' y\u207b\u00b9x\u207b\u00b9[xy]\u2261\u03b5\n  where\n    y\u207b\u00b9x\u207b\u00b9[xy]\u2261\u03b5 : y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2219 (x \u2219 y) \u2261 \u03b5\n    y\u207b\u00b9x\u207b\u00b9[xy]\u2261\u03b5 =\n        begin\n          y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2219 (x \u2219 y)   \u2261\u27e8 assoc (y \u207b\u00b9) (x \u207b\u00b9) (x \u2219 y) \u27e9\n          y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 (x \u2219 y)) \u2261\u27e8 subst (\u03bb t \u2192 y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 (x \u2219 y)) \u2261\n                                                  y \u207b\u00b9 \u2219 t)\n                                           (sym (assoc (x \u207b\u00b9) x y))\n                                           refl\n                                  \u27e9\n          y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 x \u2219 y)   \u2261\u27e8 subst (\u03bb t \u2192 y \u207b\u00b9 \u2219 (x \u207b\u00b9 \u2219 x \u2219 y) \u2261\n                                                  y \u207b\u00b9 \u2219 (t \u2219 y))\n                                           (leftInverse x)\n                                           refl\n                                  \u27e9\n          y \u207b\u00b9 \u2219 (\u03b5 \u2219 y)          \u2261\u27e8 subst (\u03bb t \u2192 y \u207b\u00b9 \u2219 (\u03b5 \u2219 y) \u2261 y \u207b\u00b9 \u2219 t)\n                                           (leftIdentity y)\n                                           refl\n                                  \u27e9\n          y \u207b\u00b9 \u2219 y                \u2261\u27e8 leftInverse y \u27e9\n          \u03b5\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fc330dfe86fa90e6124f63b458c15688d7e815fa","subject":"flailing at context junk","message":"flailing at context junk\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"contexts.agda","new_file":"contexts.agda","new_contents":"open import Prelude\nopen import List\nopen import Nat\n\nmodule contexts where\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  _ctx : Set \u2192 Set\n  A ctx = Nat \u2192 Maybe A\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : {A : Set} \u2192 A ctx\n  \u2205 _ = None\n\n  infixr 100 \u25a0_\n\n  -- membership, or presence, in a context\n  _\u2208_ : {A : Set} (p : Nat \u00d7 A) \u2192 (\u0393 : A ctx) \u2192 Set\n  (x , y) \u2208 \u0393 = (\u0393 x) == Some y\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : {A : Set} (n : Nat) \u2192 (\u0393 : A ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- disjointness for contexts\n  _##_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 Set\n  _##_ {A} \u0393 \u0393'  = ((n : Nat) (a : A) \u2192 \u0393 n  == Some a \u2192 n # \u0393') \u00d7\n                   ((n : Nat) (a : A) \u2192 \u0393' n == Some a \u2192 n # \u0393 )\n\n  -- without: remove a variable from a context\n  _\/\/_ : {A : Set} \u2192 A ctx \u2192 Nat \u2192 A ctx\n  (\u0393 \/\/ x) y with natEQ x y\n  (\u0393 \/\/ x) .x | Inl refl = None\n  (\u0393 \/\/ x) y  | Inr neq  = \u0393 y\n\n  -- a variable is apart from any context from which it is removed\n  aar : {A : Set} \u2192 (\u0393 : A ctx) (x : Nat) \u2192 x # (\u0393 \/\/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {A : Set} \u2192 {\u0393 : A ctx} {n : Nat} {t t' : A} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- warning: this is union but it assumes WITHOUT CHECKING that the\n  -- domains are disjoint.\n  _\u222a_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 A ctx\n  (C1 \u222a C2) x with C1 x\n  (C1 \u222a C2) x | Some x\u2081 = Some x\u2081\n  (C1 \u222a C2) x | None = C2 x\n\n  -- the singleton context\n  \u25a0_ : {A : Set} \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u25a0 (x , a)) y with natEQ x y\n  (\u25a0 (x , a)) .x | Inl refl = Some a\n  ... | Inr _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : {A : Set} \u2192 A ctx \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u0393 ,, (x , t)) = \u0393 \u222a (\u25a0 (x , t))\n\n  infixl 10 _,,_\n\n  x\u2208\u222al : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  x\u2208\u222al \u0393 \u0393' n x xin with \u0393 n\n  x\u2208\u222al \u0393 \u0393' n x\u2081 xin | Some x = xin\n  x\u2208\u222al \u0393 \u0393' n x ()   | None\n\n  lem-stop : {A : Set} (\u0393 : A ctx) (x : Nat) \u2192 (\u03a3[ a \u2208 A ] ((x , a) \u2208 \u0393)) + (x # \u0393)\n  lem-stop \u0393 x with \u0393 x\n  lem-stop \u0393 x | Some x\u2081 = Inl (x\u2081 , refl)\n  lem-stop \u0393 x | None = Inr refl\n\n\n  -- x\u2208\u222ar \u0393 \u0393' n x xin d with lem-stop \u0393 n\n  -- x\u2208\u222ar \u0393 \u0393' n x\u2081 xin d | Inl (a , ain) = {!ain!}\n  -- x\u2208\u222ar \u0393 \u0393' n x\u2081 xin d | Inr x = {!!}\n\n\n  x\u2208\u25a0 : {A : Set} (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u25a0 (n , a))\n  x\u2208\u25a0 n a with natEQ n n\n  x\u2208\u25a0 n a | Inl refl = refl\n  x\u2208\u25a0 n a | Inr x = abort (x refl)\n\n\n  postulate -- TODO\n    \u222acomm : {A : Set} \u2192 (C1 C2 : A ctx) \u2192 (C1 \u222a C2) == (C2 \u222a C1)\n    x\u2208sing : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\n    x\u2208\u222ar : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 \u0393 ## \u0393' \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2081 xin | None   | _ = abort (somenotnone (! xin))\n\n  -- x\u2208\u222a\u25a0 : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 \u222a (\u25a0 (n , a)))\n  -- x\u2208\u222a\u25a0 \u0393 n a with natEQ n n\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inl refl = {!!} -- it might be in \u0393 because i don't know that they're disjoint. this is where that premise gets you\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inr x = {!!}\n\n  -- x\u2208sing \u0393 n a  = {!!} -- x\u2208\u222a2 \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a)\n","old_contents":"open import Prelude\nopen import List\nopen import Nat\n\nmodule contexts where\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  _ctx : Set \u2192 Set\n  A ctx = Nat \u2192 Maybe A\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : {A : Set} \u2192 A ctx\n  \u2205 _ = None\n\n  infixr 100 \u25a0_\n\n  -- membership, or presence, in a context\n  _\u2208_ : {A : Set} (p : Nat \u00d7 A) \u2192 (\u0393 : A ctx) \u2192 Set\n  (x , y) \u2208 \u0393 = (\u0393 x) == Some y\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : {A : Set} (n : Nat) \u2192 (\u0393 : A ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- disjointness for contexts\n  _##_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 Set\n  _##_ {A} \u0393 \u0393'  = ((n : Nat) (a : A) \u2192 \u0393 n == Some a \u2192 \u0393' n == None) \u00d7\n                   ((n : Nat) (a : A) \u2192 \u0393' n == Some a \u2192 \u0393 n == None)\n\n  -- without: remove a variable from a context\n  _\/\/_ : {A : Set} \u2192 A ctx \u2192 Nat \u2192 A ctx\n  (\u0393 \/\/ x) y with natEQ x y\n  (\u0393 \/\/ x) .x | Inl refl = None\n  (\u0393 \/\/ x) y  | Inr neq  = \u0393 y\n\n  -- a variable is apart from any context from which it is removed\n  aar : {A : Set} \u2192 (\u0393 : A ctx) (x : Nat) \u2192 x # (\u0393 \/\/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {A : Set} \u2192 {\u0393 : A ctx} {n : Nat} {t t' : A} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- warning: this is union but it assumes WITHOUT CHECKING that the\n  -- domains are disjoint.\n  _\u222a_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 A ctx\n  (C1 \u222a C2) x with C1 x\n  (C1 \u222a C2) x | Some x\u2081 = Some x\u2081\n  (C1 \u222a C2) x | None = C2 x\n\n  -- the singleton context\n  \u25a0_ : {A : Set} \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u25a0 (x , a)) y with natEQ x y\n  (\u25a0 (x , a)) .x | Inl refl = Some a\n  ... | Inr _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : {A : Set} \u2192 A ctx \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u0393 ,, (x , t)) = \u0393 \u222a (\u25a0 (x , t))\n\n  infixl 10 _,,_\n\n  x\u2208\u222a1 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  x\u2208\u222a1 \u0393 \u0393' n x xin with \u0393 n\n  x\u2208\u222a1 \u0393 \u0393' n x\u2081 xin | Some x = xin\n  x\u2208\u222a1 \u0393 \u0393' n x ()   | None\n\n\n  x\u2208\u25a0 : {A : Set} (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u25a0 (n , a))\n  x\u2208\u25a0 n a with natEQ n n\n  x\u2208\u25a0 n a | Inl refl = refl\n  x\u2208\u25a0 n a | Inr x = abort (x refl)\n\n  postulate -- TODO\n    x\u2208sing : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\n    \u222acomm : {A : Set} \u2192 (C1 C2 : A ctx) \u2192 (C1 \u222a C2) == (C2 \u222a C1)\n\n-- x\u2208sing \u0393 n x with \u0393 n\n  -- x\u2208sing \u0393 n x  | Some y with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inl refl = {!!}\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inr x = abort (x refl)\n  -- x\u2208sing \u0393 n x  | None with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | None | Inl refl = refl\n  -- x\u2208sing \u0393 n x\u2081 | None | Inr x = abort (x refl)\n\n  -- x\u2208\u222a2 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  -- x\u2208\u222a2 \u0393 \u0393' n x xin with \u0393' n | \u0393 n\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2082 xin | Some x | Some x\u2081 = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x refl | Some .x | None = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2081 xin | None   | _ = abort (somenotnone (! xin))\n\n-- x\u2208\u222a\u25a0 : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 \u222a (\u25a0 (n , a)))\n  -- x\u2208\u222a\u25a0 \u0393 n a with natEQ n n\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inl refl = {!!} -- it might be in \u0393 because i don't know that they're disjoint. this is where that premise gets you\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inr x = {!!}\n\n  -- x\u2208sing \u0393 n a  = {!!} -- x\u2208\u222a2 \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b93733dc8edf5e1dc3e416dfa0bca1405b5ae331","subject":"Break lines.","message":"Break lines.\n\nOld-commit-hash: c7c69f51c35066147017459aa226bebe26d68df2\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {T : Type} \u2192 \u27e6 T \u27e7 \u2192 \u27e6 \u0394-Type T \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {T : Type} \u2192 \u27e6 T \u27e7 \u2192 \u27e6 \u0394-Type T \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df = \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192 (valid-\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c2afb2fbaded7b9318925ecaa1719c3921172f34","subject":"[ power function] Added missing hypothesis.","message":"[ power function] Added missing hypothesis.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Pow\/PropertiesATP.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Pow\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Some proofs related to the power function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.Nat.Pow.PropertiesATP where\n\nopen import FOT.FOTC.Data.Nat.Pow\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\npostulate 0^0\u22611 : 0' ^ 0' \u2261 1'\n{-# ATP prove 0^0\u22611 #-}\n\n0^Sx\u22610 : \u2200 {n} \u2192 N n \u2192 0' ^ succ\u2081 n \u2261 0'\n0^Sx\u22610 {.zero} nzero  = prf\n  where postulate prf : 0' ^ succ\u2081 zero \u2261 0'\n        {-# ATP prove prf #-}\n\n0^Sx\u22610 (nsucc {n} Nn) = prf\n  where postulate prf : 0' ^ succ\u2081 (succ\u2081 n) \u2261 0'\n        {-# ATP prove prf #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ 5' \u2264 5' ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn) h\n  where\n  postulate prf : (5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n) \u2192\n                  5' \u2264 succ\u2081 n \u2192\n                  succ\u2081 n ^ 5' \u2264 5' ^ succ\u2081 n\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261 2' ^ (n + 1') \u2238 1'\nthm\u2082 nzero = prf\n  where\n  postulate prf : ((2' ^ zero) \u2238 1') + 1' + ((2' ^ zero) \u2238 1') \u2261\n                  2' ^ (zero + 1') \u2238 1'\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261\n                  2' ^ (n + 1') \u2238 1' \u2192\n                  ((2' ^ succ\u2081 n) \u2238 1') + 1' + ((2' ^ succ\u2081 n) \u2238 1') \u2261\n                  2' ^ (succ\u2081 n + 1') \u2238 1'\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some proofs related to the power function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.Nat.Pow.PropertiesATP where\n\nopen import FOT.FOTC.Data.Nat.Pow\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\npostulate 0^0\u22611 : 0' ^ 0' \u2261 1'\n{-# ATP prove 0^0\u22611 #-}\n\n0^Sx\u22610 : \u2200 {n} \u2192 N n \u2192 0' ^ succ\u2081 n \u2261 0'\n0^Sx\u22610 {.zero} nzero  = prf\n  where postulate prf : 0' ^ succ\u2081 zero \u2261 0'\n        {-# ATP prove prf #-}\n\n0^Sx\u22610 (nsucc {n} Nn) = prf\n  where postulate prf : 0' ^ succ\u2081 (succ\u2081 n) \u2261 0'\n        {-# ATP prove prf #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ 5' \u2264 5' ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn)\n  where\n  postulate prf : (5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n) \u2192 (succ\u2081 n) ^ 5' \u2264 5' ^ (succ\u2081 n)\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261 2' ^ (n + 1') \u2238 1'\nthm\u2082 nzero = prf\n  where\n  postulate prf : ((2' ^ zero) \u2238 1') + 1' + ((2' ^ zero) \u2238 1') \u2261\n                  2' ^ (zero + 1') \u2238 1'\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261\n                  2' ^ (n + 1') \u2238 1' \u2192\n                  ((2' ^ succ\u2081 n) \u2238 1') + 1' + ((2' ^ succ\u2081 n) \u2238 1') \u2261\n                  2' ^ (succ\u2081 n + 1') \u2238 1'\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0a61745088c8125531b7135fa301b6a2a74b26ba","subject":"\u27e6\u2115\u27e7-Reasoning = Setoid-Reasoning \u27e6\u2115\u27e7-setoid","message":"\u27e6\u2115\u27e7-Reasoning = Setoid-Reasoning \u27e6\u2115\u27e7-setoid\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Logical.agda","new_file":"lib\/Data\/Nat\/Logical.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.Logical where\n\nimport Level as L\nopen import Data.Nat using (\u2115; zero; suc; pred; _\u2264_; z\u2264n; s\u2264s; \u2264-pred; _\u2264?_; _+_)\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nimport      Relation.Binary.PropositionalEquality as PropEq\n\ndata \u27e6\u2115\u27e7 : \u27e6Set\u2080\u27e7 \u2115 \u2115 where\n  zero : \u27e6\u2115\u27e7 zero zero\n  suc  : \u2200 {n\u2081 n\u2082} (n\u1d63 : \u27e6\u2115\u27e7 n\u2081 n\u2082) \u2192 \u27e6\u2115\u27e7 (suc n\u2081) (suc n\u2082)\n\n_\u27e6+\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) _+_ _+_\nzero  \u27e6+\u27e7 n = n\nsuc m \u27e6+\u27e7 n = suc (m \u27e6+\u27e7 n)\n\n\u27e6pred\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred pred\n\u27e6pred\u27e7 (suc n\u1d63) = n\u1d63\n\u27e6pred\u27e7 zero     = zero\n\n_\u225f_ : Decidable \u27e6\u2115\u27e7\nzero  \u225f zero   = yes zero\nsuc m \u225f suc n  with m \u225f n\n...            | yes p = yes (suc p)\n...            | no \u00acp = no (\u00acp \u2218 \u27e6pred\u27e7)\nzero  \u225f suc n  = no \u03bb()\nsuc m \u225f zero   = no \u03bb()\n\nprivate\n  refl : Reflexive \u27e6\u2115\u27e7\n  refl {zero}  = zero\n  refl {suc n} = suc (refl {n})\n  sym : Symmetric \u27e6\u2115\u27e7\n  sym zero = zero\n  sym (suc n\u1d63) = suc (sym n\u1d63)\n  trans : Transitive \u27e6\u2115\u27e7\n  trans zero    zero     = zero\n  trans (suc m\u1d63) (suc n\u1d63) = suc (trans m\u1d63 n\u1d63)\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6\u2115\u27e7 \u2113\n  subst _ zero    = id\n  subst P (suc n\u1d63) = subst (P \u2218 suc) n\u1d63\n\n\u27e6\u2115\u27e7-isEquivalence : IsEquivalence \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n\u27e6\u2115\u27e7-isDecEquivalence : IsDecEquivalence \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-isDecEquivalence = record { isEquivalence = \u27e6\u2115\u27e7-isEquivalence; _\u225f_ = _\u225f_ }\n\n\u27e6\u2115\u27e7-equality : Equality \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-equality = record { isEquivalence = \u27e6\u2115\u27e7-isEquivalence; subst = subst }\n\n\u27e6\u2115\u27e7\u21d2\u2261 : \u27e6\u2115\u27e7 \u21d2 PropEq._\u2261_\n\u27e6\u2115\u27e7\u21d2\u2261 = Equality.to-reflexive \u27e6\u2115\u27e7-equality {PropEq._\u2261_} PropEq.refl\n\n\u27e6\u2115\u27e7-setoid : Setoid _ _\n\u27e6\u2115\u27e7-setoid = record { Carrier = \u2115; _\u2248_ = \u27e6\u2115\u27e7; isEquivalence = \u27e6\u2115\u27e7-isEquivalence }\n\n\u27e6\u2115\u27e7-decSetoid : DecSetoid _ _\n\u27e6\u2115\u27e7-decSetoid = record { Carrier = \u2115; _\u2248_ = \u27e6\u2115\u27e7; isDecEquivalence = \u27e6\u2115\u27e7-isDecEquivalence }\n\n\u27e6\u2115\u27e7-cong : \u2200 f \u2192 \u27e6\u2115\u27e7 =[ f ]\u21d2 \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-cong _ zero     = refl\n\u27e6\u2115\u27e7-cong f (suc n\u1d63) = \u27e6\u2115\u27e7-cong (f \u2218 suc) n\u1d63\n\nmodule \u27e6\u2115\u27e7-Reasoning = Setoid-Reasoning \u27e6\u2115\u27e7-setoid\n\ndata \u27e6\u2264\u27e7 : \u27e6REL\u27e7 \u27e6\u2115\u27e7 \u27e6\u2115\u27e7 L.zero _\u2264_ _\u2264_ where\n  z\u2264n : \u2200 {m\u2081 m\u2082} {m\u1d63 : \u27e6\u2115\u27e7 m\u2081 m\u2082} \u2192 \u27e6\u2264\u27e7 zero m\u1d63 z\u2264n z\u2264n\n  s\u2264s : \u2200 {m\u2081 m\u2082 n\u2081 n\u2082 m\u1d63 n\u1d63} {m\u2264n\u2081 : m\u2081 \u2264 n\u2081} {m\u2264n\u2082 : m\u2082 \u2264 n\u2082} (m\u2264n\u1d63 : \u27e6\u2264\u27e7 m\u1d63 n\u1d63 m\u2264n\u2081 m\u2264n\u2082)\n        \u2192 \u27e6\u2264\u27e7 (suc m\u1d63) (suc n\u1d63) (s\u2264s m\u2264n\u2081) (s\u2264s m\u2264n\u2082)\n\n\u27e6\u2264-pred\u27e7 : \u2200 {m\u2081 m\u2082 n\u2081 n\u2082 m\u1d63 n\u1d63} {m\u2264n\u2081 : suc m\u2081 \u2264 suc n\u2081} {m\u2264n\u2082 : suc m\u2082 \u2264 suc n\u2082}\n          \u2192 \u27e6\u2264\u27e7 (suc m\u1d63) (suc n\u1d63) m\u2264n\u2081 m\u2264n\u2082 \u2192 \u27e6\u2264\u27e7 m\u1d63 n\u1d63 (\u2264-pred m\u2264n\u2081) (\u2264-pred m\u2264n\u2082)\n\u27e6\u2264-pred\u27e7 (s\u2264s m\u2264n) = m\u2264n\n\n_\u27e6\u2264?\u27e7_ : \u27e6Decidable\u27e7 \u27e6\u2115\u27e7 \u27e6\u2115\u27e7 \u27e6\u2264\u27e7 _\u2264?_ _\u2264?_\nzero             \u27e6\u2264?\u27e7 _      = yes z\u2264n\nsuc n\u1d63           \u27e6\u2264?\u27e7 zero   = no (\u03bb())\nsuc {m\u2081} {m\u2082} m\u1d63 \u27e6\u2264?\u27e7 suc {n\u2081} {n\u2082} n\u1d63\n with m\u2081 \u2264? n\u2081 | m\u2082 \u2264? n\u2082 | m\u1d63 \u27e6\u2264?\u27e7 n\u1d63\n... | yes _     | yes _      | yes m\u2264n\u1d63 = yes (s\u2264s m\u2264n\u1d63)\n... | no _      | no _       | no \u00acm\u2264n\u1d63 = no (\u00acm\u2264n\u1d63 \u2218 \u27e6\u2264-pred\u27e7)\n... | yes _     | no _       | ()\n... | no _      | yes _      | ()\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.Logical where\n\nimport Level as L\nopen import Data.Nat using (\u2115; zero; suc; pred; _\u2264_; z\u2264n; s\u2264s; \u2264-pred; _\u2264?_; _+_)\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nimport      Relation.Binary.PropositionalEquality as PropEq\n\ndata \u27e6\u2115\u27e7 : \u27e6Set\u2080\u27e7 \u2115 \u2115 where\n  zero : \u27e6\u2115\u27e7 zero zero\n  suc  : \u2200 {n\u2081 n\u2082} (n\u1d63 : \u27e6\u2115\u27e7 n\u2081 n\u2082) \u2192 \u27e6\u2115\u27e7 (suc n\u2081) (suc n\u2082)\n\n_\u27e6+\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) _+_ _+_\nzero  \u27e6+\u27e7 n = n\nsuc m \u27e6+\u27e7 n = suc (m \u27e6+\u27e7 n)\n\n\u27e6pred\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred pred\n\u27e6pred\u27e7 (suc n\u1d63) = n\u1d63\n\u27e6pred\u27e7 zero     = zero\n\n_\u225f_ : Decidable \u27e6\u2115\u27e7\nzero  \u225f zero   = yes zero\nsuc m \u225f suc n  with m \u225f n\n...            | yes p = yes (suc p)\n...            | no \u00acp = no (\u00acp \u2218 \u27e6pred\u27e7)\nzero  \u225f suc n  = no \u03bb()\nsuc m \u225f zero   = no \u03bb()\n\nprivate\n  refl : Reflexive \u27e6\u2115\u27e7\n  refl {zero}  = zero\n  refl {suc n} = suc (refl {n})\n  sym : Symmetric \u27e6\u2115\u27e7\n  sym zero = zero\n  sym (suc n\u1d63) = suc (sym n\u1d63)\n  trans : Transitive \u27e6\u2115\u27e7\n  trans zero    zero     = zero\n  trans (suc m\u1d63) (suc n\u1d63) = suc (trans m\u1d63 n\u1d63)\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6\u2115\u27e7 \u2113\n  subst _ zero    = id\n  subst P (suc n\u1d63) = subst (P \u2218 suc) n\u1d63\n\n\u27e6\u2115\u27e7-isEquivalence : IsEquivalence \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n\u27e6\u2115\u27e7-isDecEquivalence : IsDecEquivalence \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-isDecEquivalence = record { isEquivalence = \u27e6\u2115\u27e7-isEquivalence; _\u225f_ = _\u225f_ }\n\n\u27e6\u2115\u27e7-equality : Equality \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-equality = record { isEquivalence = \u27e6\u2115\u27e7-isEquivalence; subst = subst }\n\n\u27e6\u2115\u27e7\u21d2\u2261 : \u27e6\u2115\u27e7 \u21d2 PropEq._\u2261_\n\u27e6\u2115\u27e7\u21d2\u2261 = Equality.to-reflexive \u27e6\u2115\u27e7-equality {PropEq._\u2261_} PropEq.refl\n\n\u27e6\u2115\u27e7-setoid : Setoid _ _\n\u27e6\u2115\u27e7-setoid = record { Carrier = \u2115; _\u2248_ = \u27e6\u2115\u27e7; isEquivalence = \u27e6\u2115\u27e7-isEquivalence }\n\n\u27e6\u2115\u27e7-decSetoid : DecSetoid _ _\n\u27e6\u2115\u27e7-decSetoid = record { Carrier = \u2115; _\u2248_ = \u27e6\u2115\u27e7; isDecEquivalence = \u27e6\u2115\u27e7-isDecEquivalence }\n\n\u27e6\u2115\u27e7-cong : \u2200 f \u2192 \u27e6\u2115\u27e7 =[ f ]\u21d2 \u27e6\u2115\u27e7\n\u27e6\u2115\u27e7-cong _ zero     = refl\n\u27e6\u2115\u27e7-cong f (suc n\u1d63) = \u27e6\u2115\u27e7-cong (f \u2218 suc) n\u1d63\n\nmodule \u27e6\u2115\u27e7-Reasoning = Trans-Reasoning \u27e6\u2115\u27e7 trans\n\ndata \u27e6\u2264\u27e7 : \u27e6REL\u27e7 \u27e6\u2115\u27e7 \u27e6\u2115\u27e7 L.zero _\u2264_ _\u2264_ where\n  z\u2264n : \u2200 {m\u2081 m\u2082} {m\u1d63 : \u27e6\u2115\u27e7 m\u2081 m\u2082} \u2192 \u27e6\u2264\u27e7 zero m\u1d63 z\u2264n z\u2264n\n  s\u2264s : \u2200 {m\u2081 m\u2082 n\u2081 n\u2082 m\u1d63 n\u1d63} {m\u2264n\u2081 : m\u2081 \u2264 n\u2081} {m\u2264n\u2082 : m\u2082 \u2264 n\u2082} (m\u2264n\u1d63 : \u27e6\u2264\u27e7 m\u1d63 n\u1d63 m\u2264n\u2081 m\u2264n\u2082)\n        \u2192 \u27e6\u2264\u27e7 (suc m\u1d63) (suc n\u1d63) (s\u2264s m\u2264n\u2081) (s\u2264s m\u2264n\u2082)\n\n\u27e6\u2264-pred\u27e7 : \u2200 {m\u2081 m\u2082 n\u2081 n\u2082 m\u1d63 n\u1d63} {m\u2264n\u2081 : suc m\u2081 \u2264 suc n\u2081} {m\u2264n\u2082 : suc m\u2082 \u2264 suc n\u2082}\n          \u2192 \u27e6\u2264\u27e7 (suc m\u1d63) (suc n\u1d63) m\u2264n\u2081 m\u2264n\u2082 \u2192 \u27e6\u2264\u27e7 m\u1d63 n\u1d63 (\u2264-pred m\u2264n\u2081) (\u2264-pred m\u2264n\u2082)\n\u27e6\u2264-pred\u27e7 (s\u2264s m\u2264n) = m\u2264n\n\n_\u27e6\u2264?\u27e7_ : \u27e6Decidable\u27e7 \u27e6\u2115\u27e7 \u27e6\u2115\u27e7 \u27e6\u2264\u27e7 _\u2264?_ _\u2264?_\nzero             \u27e6\u2264?\u27e7 _      = yes z\u2264n\nsuc n\u1d63           \u27e6\u2264?\u27e7 zero   = no (\u03bb())\nsuc {m\u2081} {m\u2082} m\u1d63 \u27e6\u2264?\u27e7 suc {n\u2081} {n\u2082} n\u1d63\n with m\u2081 \u2264? n\u2081 | m\u2082 \u2264? n\u2082 | m\u1d63 \u27e6\u2264?\u27e7 n\u1d63\n... | yes _     | yes _      | yes m\u2264n\u1d63 = yes (s\u2264s m\u2264n\u1d63)\n... | no _      | no _       | no \u00acm\u2264n\u1d63 = no (\u00acm\u2264n\u1d63 \u2218 \u27e6\u2264-pred\u27e7)\n... | yes _     | no _       | ()\n... | no _      | yes _      | ()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"97eb0c6225db27f9c7102dbf268c916391db6175","subject":"refactor","message":"refactor\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-disjointness.agda","new_file":"lemmas-disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule lemmas-disjointness where\n  -- disjointness is commutative\n  ##-comm : {A : Set} {\u03941 \u03942 : A ctx} \u2192 \u03941 ## \u03942 \u2192 \u03942 ## \u03941\n  ##-comm (\u03c01 , \u03c02) = \u03c02 , \u03c01\n\n  -- the empty context is disjoint from any context\n  empty-disj : {A : Set} (\u0393 : A ctx) \u2192 \u2205 ## \u0393\n  empty-disj \u0393 = ed1 , ed2\n   where\n    ed1 : {A : Set} (n : Nat) \u2192 dom {A} \u2205 n \u2192 n # \u0393\n    ed1 n (\u03c01 , ())\n\n    ed2 : {A : Set} (n : Nat) \u2192 dom \u0393 n \u2192  _#_ {A} n \u2205\n    ed2 _ _ = refl\n\n  -- two singleton contexts with different indices are disjoint\n  disjoint-singles : {A : Set} {x y : A} {u1 u2 : Nat} \u2192\n                          u1 \u2260 u2 \u2192\n                          (\u25a0 (u1 , x)) ## (\u25a0 (u2 , y))\n  disjoint-singles {_} {x} {y} {u1} {u2} neq = ds1 , ds2\n    where\n      ds1 : (n : Nat) \u2192 dom (\u25a0 (u1 , x)) n \u2192 n # (\u25a0 (u2 , y))\n      ds1 n d with lem-dom-eq _ _ _ d\n      ds1 .u1 d | refl with natEQ u2 u1\n      ds1 .u1 d | refl | Inl xx = abort (neq (! xx))\n      ds1 .u1 d | refl | Inr x\u2081 = refl\n\n      ds2 : (n : Nat) \u2192 dom (\u25a0 (u2 , y)) n \u2192 n # (\u25a0 (u1 , x))\n      ds2 n d with lem-dom-eq _ _ _ d\n      ds2 .u2 d | refl with natEQ u1 u2\n      ds2 .u2 d | refl | Inl x\u2081 = abort (neq x\u2081)\n      ds2 .u2 d | refl | Inr x\u2081 = refl\n\n  -- the only index of a singleton context is in its domain\n  lem-domsingle : {A : Set} (p : Nat) (q : A) \u2192 dom (\u25a0 (p , q)) p\n  lem-domsingle p q with natEQ p p\n  lem-domsingle p q | Inl refl = q , refl\n  lem-domsingle p q | Inr x\u2081 = abort (x\u2081 refl)\n\n  apart-noteq : {A : Set} (p r : Nat) (q : A) \u2192 p # (\u25a0 (r , q)) \u2192 p \u2260 r\n  apart-noteq p r q apt with natEQ r p\n  apart-noteq p .p q apt | Inl refl = abort (somenotnone apt)\n  apart-noteq p r q apt | Inr x\u2081 = flip x\u2081\n\n  -- if singleton contexts are disjoint, their indices must be disequal\n  singles-notequal : {A : Set} {x y : A} {u1 u2 : Nat} \u2192\n                          (\u25a0 (u1 , x)) ## (\u25a0 (u2 , y)) \u2192\n                          u1 \u2260 u2\n  singles-notequal {A} {x} {y} {u1} {u2} (d1 , d2) = apart-noteq u1 u2 y (d1 u1 (lem-domsingle u1 x))\n\n  -- dual of lem2 above; if two indices are disequal, then either is apart\n  -- from the singleton formed with the other\n  apart-singleton : {A : Set} \u2192 \u2200{x y} \u2192 {\u03c4 : A} \u2192\n                          x \u2260 y \u2192\n                          x # (\u25a0 (y , \u03c4))\n  apart-singleton {A} {x} {y} {\u03c4} neq with natEQ y x\n  apart-singleton neq | Inl x\u2081 = abort ((flip neq) x\u2081)\n  apart-singleton neq | Inr x\u2081 = refl\n\n  -- if an index is apart from two contexts, it's apart from their union as\n  -- well. used below and in other files, so it's outside the local scope.\n  apart-parts : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 n # \u03931 \u2192 n # \u03932 \u2192 n # (\u03931 \u222a \u03932)\n  apart-parts \u03931 \u03932 n apt1 apt2 with \u03931 n\n  apart-parts _ _ n refl apt2 | .None = apt2\n\n  -- this is just for convenience; it shows up a lot.\n  apart-extend1 : {A : Set} \u2192 \u2200{ x y \u03c4} \u2192 (\u0393 : A ctx)  \u2192 x \u2260 y \u2192 x # \u0393 \u2192 x # (\u0393 ,, (y , \u03c4))\n  apart-extend1 {A} {x} {y} {\u03c4} \u0393 neq apt = apart-parts \u0393 (\u25a0 (y , \u03c4)) x apt (apart-singleton neq)\n\n  -- if both parts of a union are disjoint with a target, so is the union\n  disjoint-parts : {A : Set} {\u03931 \u03932 \u03933 : A ctx} \u2192 \u03931 ## \u03933 \u2192 \u03932 ## \u03933 \u2192 (\u03931 \u222a \u03932) ## \u03933\n  disjoint-parts {_} {\u03931} {\u03932} {\u03933} D13 D23 = d31 , d32\n    where\n      dom-split : {A : Set} \u2192 (\u03931 \u03932 : A ctx) (n : Nat) \u2192 dom (\u03931 \u222a \u03932) n \u2192 dom \u03931 n + dom \u03932 n\n      dom-split \u03934 \u03935 n (\u03c01 , \u03c02) with \u03934 n\n      dom-split \u03934 \u03935 n (\u03c01 , \u03c02) | Some x = Inl (x , refl)\n      dom-split \u03934 \u03935 n (\u03c01 , \u03c02) | None = Inr (\u03c01 , \u03c02)\n\n      d31 : (n : Nat) \u2192 dom (\u03931 \u222a \u03932) n \u2192 n # \u03933\n      d31 n D with dom-split \u03931 \u03932 n D\n      d31 n D | Inl x = \u03c01 D13 n x\n      d31 n D | Inr x = \u03c01 D23 n x\n\n      d32 : (n : Nat) \u2192 dom \u03933 n \u2192 n # (\u03931 \u222a \u03932)\n      d32 n D = apart-parts \u03931 \u03932 n (\u03c02 D13 n D) (\u03c02 D23 n D)\n\n  apart-union1 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 n # (\u03931 \u222a \u03932) \u2192 n # \u03931\n  apart-union1 \u03931 \u03932 n aprt with \u03931 n\n  apart-union1 \u03931 \u03932 n () | Some x\n  apart-union1 \u03931 \u03932 n aprt | None = refl\n\n  apart-union2 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 n # (\u03931 \u222a \u03932) \u2192 n # \u03932\n  apart-union2 \u03931 \u03932 n aprt with \u03931 n\n  apart-union2 \u03933 \u03934 n () | Some x\n  apart-union2 \u03933 \u03934 n aprt | None = aprt\n\n  -- if a union is disjoint with a target, so is the left unand\n  disjoint-union1 : {A : Set} {\u03931 \u03932 \u0394 : A ctx} \u2192 (\u03931 \u222a \u03932) ## \u0394 \u2192 \u03931 ## \u0394\n  disjoint-union1 {\u03931 = \u03931} {\u03932 = \u03932} {\u0394 = \u0394} (ud1 , ud2) = du11 , du12\n    where\n      dom-union1 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 dom \u03931 n \u2192 dom (\u03931 \u222a \u03932) n\n      dom-union1 \u03931 \u03932 n (\u03c01 , \u03c02) with \u03931 n\n      dom-union1 \u03931 \u03932 n (\u03c01 , \u03c02) | Some x = x , refl\n      dom-union1 \u03931 \u03932 n (\u03c01 , ()) | None\n\n      du11 : (n : Nat) \u2192 dom \u03931 n \u2192 n # \u0394\n      du11 n dom = ud1 n (dom-union1 \u03931 \u03932 n dom)\n\n      du12 : (n : Nat) \u2192 dom \u0394 n \u2192 n # \u03931\n      du12 n dom = apart-union1 \u03931 \u03932 n (ud2 n dom)\n\n  -- if a union is disjoint with a target, so is the right unand\n  disjoint-union2 : {A : Set} {\u03931 \u03932 \u0394 : A ctx} \u2192 (\u03931 \u222a \u03932) ## \u0394 \u2192 \u03932 ## \u0394\n  disjoint-union2 {\u03931 = \u03931} {\u03932 = \u03932} {\u0394 = \u0394} (ud1 , ud2) = du21 , du22\n    where\n      dom-union2 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 dom \u03932 n \u2192 dom (\u03931 \u222a \u03932) n\n      dom-union2 \u03931 \u03932 n (\u03c01 , \u03c02) with \u03931 n\n      dom-union2 \u03933 \u03934 n (\u03c01 , \u03c02) | Some x = x , refl\n      dom-union2 \u03933 \u03934 n (\u03c01 , \u03c02) | None = \u03c01 , \u03c02\n\n      du21 : (n : Nat) \u2192 dom \u03932 n \u2192 n # \u0394\n      du21 n dom = ud1 n (dom-union2 \u03931  \u03932 n dom)\n\n      du22 : (n : Nat) \u2192 dom \u0394 n \u2192 n # \u03932\n      du22 n dom = apart-union2 \u03931 \u03932 n (ud2 n dom)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule lemmas-disjointness where\n  -- disjointness is commutative\n  ##-comm : {A : Set} {\u03941 \u03942 : A ctx} \u2192 \u03941 ## \u03942 \u2192 \u03942 ## \u03941\n  ##-comm (\u03c01 , \u03c02) = \u03c02 , \u03c01\n\n  -- the empty context is disjoint from any context\n  empty-disj : {A : Set} (\u0393 : A ctx) \u2192 \u2205 ## \u0393\n  empty-disj \u0393 = ed1 , ed2\n   where\n    ed1 : {A : Set} (n : Nat) \u2192 dom {A} \u2205 n \u2192 n # \u0393\n    ed1 n (\u03c01 , ())\n\n    ed2 : {A : Set} (n : Nat) \u2192 dom \u0393 n \u2192  _#_ {A} n \u2205\n    ed2 _ _ = refl\n\n  -- two singleton contexts with different indices are disjoint\n  disjoint-singles : {A : Set} {x y : A} {u1 u2 : Nat} \u2192 u1 \u2260 u2 \u2192 (\u25a0 (u1 , x)) ## (\u25a0 (u2 , y))\n  disjoint-singles {_} {x} {y} {u1} {u2} neq = ds1 , ds2\n    where\n      ds1 : (n : Nat) \u2192 dom (\u25a0 (u1 , x)) n \u2192 n # (\u25a0 (u2 , y))\n      ds1 n d with lem-dom-eq _ _ _ d\n      ds1 .u1 d | refl with natEQ u2 u1\n      ds1 .u1 d | refl | Inl xx = abort (neq (! xx))\n      ds1 .u1 d | refl | Inr x\u2081 = refl\n\n      ds2 : (n : Nat) \u2192 dom (\u25a0 (u2 , y)) n \u2192 n # (\u25a0 (u1 , x))\n      ds2 n d with lem-dom-eq _ _ _ d\n      ds2 .u2 d | refl with natEQ u1 u2\n      ds2 .u2 d | refl | Inl x\u2081 = abort (neq x\u2081)\n      ds2 .u2 d | refl | Inr x\u2081 = refl\n\n  -- the only index of a singleton context is in its domain\n  lem-domsingle : {A : Set} (p : Nat) (q : A) \u2192 dom (\u25a0 (p , q)) p\n  lem-domsingle p q with natEQ p p\n  lem-domsingle p q | Inl refl = q , refl\n  lem-domsingle p q | Inr x\u2081 = abort (x\u2081 refl)\n\n  -- if singleton contexts are disjoint, their indices must be disequal\n  singles-notequal : {A : Set} {x y : A} {u1 u2 : Nat} \u2192\n                          (\u25a0 (u1 , x)) ## (\u25a0 (u2 , y)) \u2192\n                          u1 \u2260 u2\n  singles-notequal {A} {x} {y} {u1} {u2} (d1 , d2) = lem2 u1 u2 y (d1 u1 (lem-domsingle u1 x))\n    where\n      lem2 : {A : Set} (p r : Nat) (q : A) \u2192 p # (\u25a0 (r , q)) \u2192 p \u2260 r\n      lem2 p r q apt with natEQ r p\n      lem2 p .p q apt | Inl refl = abort (somenotnone apt)\n      lem2 p r q apt | Inr x\u2081 = flip x\u2081\n\n  -- dual of lem2 above; if two indices are disequal, then either is apart\n  -- from the singleton formed with the other\n  apart-singleton : {A : Set} \u2192 \u2200{x y} \u2192 {\u03c4 : A} \u2192\n                          x \u2260 y \u2192\n                          x # (\u25a0 (y , \u03c4))\n  apart-singleton {A} {x} {y} {\u03c4} neq with natEQ y x\n  apart-singleton neq | Inl x\u2081 = abort ((flip neq) x\u2081)\n  apart-singleton neq | Inr x\u2081 = refl\n\n  -- if an index is apart from two contexts, it's apart from their union as\n  -- well. used below and in other files, so it's outside the local scope.\n  apart-parts : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 n # \u03931 \u2192 n # \u03932 \u2192 n # (\u03931 \u222a \u03932)\n  apart-parts \u03931 \u03932 n apt1 apt2 with \u03931 n\n  apart-parts _ _ n refl apt2 | .None = apt2\n\n  -- this is just for convenience; it shows up a lot.\n  apart-extend1 : {A : Set} \u2192 \u2200{ x y \u03c4} \u2192 (\u0393 : A ctx)  \u2192 x \u2260 y \u2192 x # \u0393 \u2192 x # (\u0393 ,, (y , \u03c4))\n  apart-extend1 {A} {x} {y} {\u03c4} \u0393 neq apt = apart-parts \u0393 (\u25a0 (y , \u03c4)) x apt (apart-singleton neq)\n\n  -- if both parts of a union are disjoint with a target, so is the union\n  disjoint-parts : {A : Set} {\u03931 \u03932 \u03933 : A ctx} \u2192 \u03931 ## \u03933 \u2192 \u03932 ## \u03933 \u2192 (\u03931 \u222a \u03932) ## \u03933\n  disjoint-parts {_} {\u03931} {\u03932} {\u03933} D13 D23 = d31 , d32\n    where\n      dom-split : {A : Set} \u2192 (\u03931 \u03932 : A ctx) (n : Nat) \u2192 dom (\u03931 \u222a \u03932) n \u2192 dom \u03931 n + dom \u03932 n\n      dom-split \u03934 \u03935 n (\u03c01 , \u03c02) with \u03934 n\n      dom-split \u03934 \u03935 n (\u03c01 , \u03c02) | Some x = Inl (x , refl)\n      dom-split \u03934 \u03935 n (\u03c01 , \u03c02) | None = Inr (\u03c01 , \u03c02)\n\n      d31 : (n : Nat) \u2192 dom (\u03931 \u222a \u03932) n \u2192 n # \u03933\n      d31 n D with dom-split \u03931 \u03932 n D\n      d31 n D | Inl x = \u03c01 D13 n x\n      d31 n D | Inr x = \u03c01 D23 n x\n\n      d32 : (n : Nat) \u2192 dom \u03933 n \u2192 n # (\u03931 \u222a \u03932)\n      d32 n D = apart-parts \u03931 \u03932 n (\u03c02 D13 n D) (\u03c02 D23 n D)\n\n  -- if a union is disjoint with a target, so is the left unand\n  disjoint-union1 : {A : Set} {\u03931 \u03932 \u0394 : A ctx} \u2192 (\u03931 \u222a \u03932) ## \u0394 \u2192 \u03931 ## \u0394\n  disjoint-union1 {\u03931 = \u03931} {\u03932 = \u03932} {\u0394 = \u0394} (ud1 , ud2) = du11 , du12\n    where\n      dom-union1 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 dom \u03931 n \u2192 dom (\u03931 \u222a \u03932) n\n      dom-union1 \u03931 \u03932 n (\u03c01 , \u03c02) with \u03931 n\n      dom-union1 \u03931 \u03932 n (\u03c01 , \u03c02) | Some x = x , refl\n      dom-union1 \u03931 \u03932 n (\u03c01 , ()) | None\n\n      apart-union1 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 n # (\u03931 \u222a \u03932) \u2192 n # \u03931\n      apart-union1 \u03931 \u03932 n aprt with \u03931 n\n      apart-union1 \u03931 \u03932 n () | Some x\n      apart-union1 \u03931 \u03932 n aprt | None = refl\n\n      du11 : (n : Nat) \u2192 dom \u03931 n \u2192 n # \u0394\n      du11 n dom = ud1 n (dom-union1 \u03931 \u03932 n dom)\n\n      du12 : (n : Nat) \u2192 dom \u0394 n \u2192 n # \u03931\n      du12 n dom = apart-union1 \u03931 \u03932 n (ud2 n dom)\n\n  -- if a union is disjoint with a target, so is the right unand\n  disjoint-union2 : {A : Set} {\u03931 \u03932 \u0394 : A ctx} \u2192 (\u03931 \u222a \u03932) ## \u0394 \u2192 \u03932 ## \u0394\n  disjoint-union2 {\u03931 = \u03931} {\u03932 = \u03932} {\u0394 = \u0394} (ud1 , ud2) = du21 , du22\n    where\n      dom-union2 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 dom \u03932 n \u2192 dom (\u03931 \u222a \u03932) n\n      dom-union2 \u03931 \u03932 n (\u03c01 , \u03c02) with \u03931 n\n      dom-union2 \u03933 \u03934 n (\u03c01 , \u03c02) | Some x = x , refl\n      dom-union2 \u03933 \u03934 n (\u03c01 , \u03c02) | None = \u03c01 , \u03c02\n\n      apart-union2 : {A : Set} (\u03931 \u03932 : A ctx) (n : Nat) \u2192 n # (\u03931 \u222a \u03932) \u2192 n # \u03932\n      apart-union2 \u03931 \u03932 n aprt with \u03931 n\n      apart-union2 \u03933 \u03934 n () | Some x\n      apart-union2 \u03933 \u03934 n aprt | None = aprt\n\n      du21 : (n : Nat) \u2192 dom \u03932 n \u2192 n # \u0394\n      du21 n dom = ud1 n (dom-union2 \u03931  \u03932 n dom)\n\n      du22 : (n : Nat) \u2192 dom \u0394 n \u2192 n # \u03932\n      du22 n dom = apart-union2 \u03931 \u03932 n (ud2 n dom)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70c4f27307b20d9efe82deb76b0db0f11a345723","subject":"update HoTT , remove postulate for hae","message":"update HoTT , remove postulate for hae\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/HoTT.agda","new_file":"lib\/HoTT.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\nopen \u2261.\u2261-Reasoning\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  -- could be derived in any groupoid\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  module _ {x y : A} where\n\n    \u2219-assoc : (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n    \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n                 (\u03bb x q r \u2192 idp)\n\n    module _ (p : x \u2261 y) where\n      -- ! is a left-inverse for _\u2219_\n      !-\u2219 : ! p \u2219 p \u2261 idp_ y\n      !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp) p\n\n      -- ! is a right-inverse for _\u2219_\n      \u2219-! : p \u2219 ! p \u2261 idp_ x\n      \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp) p\n\n      -- ! is involutive\n      !-involutive : ! (! p) \u2261 p\n      !-involutive = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp) p\n\n      !p\u2219p = !-\u2219\n      p\u2219!p = \u2219-!\n\n      ==-refl-\u2219 : {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n      ==-refl-\u2219 = ap (flip _\u2219_ p)\n\n      \u2219-==-refl : {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n      \u2219-==-refl qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  module _ {x y : A} where\n    module _ (p : x \u2261 x)(q : x \u2261 y)(e : p \u2219 q \u2261 q) where\n      unique-idp-left : p \u2261 idp\n      unique-idp-left\n        = p              \u2261\u27e8 ! \u2219-refl p \u27e9\n          p \u2219 idp        \u2261\u27e8 ap (_\u2219_ p) (! \u2219-! q) \u27e9\n          p \u2219 (q \u2219 ! q)  \u2261\u27e8 \u2219-assoc p q (! q) \u27e9\n          (p \u2219 q) \u2219 ! q  \u2261\u27e8 ap (flip _\u2219_ (! q)) e \u27e9\n          q \u2219 ! q        \u2261\u27e8 \u2219-! q \u27e9\n          idp            \u220e\n\n  module _ {x y z : A}{p\u2080 p\u2081 : x \u2261 y}{q\u2080 q\u2081 : y \u2261 z}(p : p\u2080 \u2261 p\u2081)(q : q\u2080 \u2261 q\u2081) where\n      \u2219= : p\u2080 \u2219 q\u2080 \u2261 p\u2081 \u2219 q\u2081\n      \u2219= = ap (flip _\u2219_ q\u2080) p \u2219 ap (_\u2219_ p\u2081) q\n\n  module _ {x y z : A} where\n    module _ (p : x \u2261 y)(q : y \u2261 z) where\n      \u2219-\u2219-==-refl : {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n      \u2219-\u2219-==-refl rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n      !p\u2219p\u2219q : ! p \u2219 p \u2219 q \u2261 q\n      !p\u2219p\u2219q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n    p\u2219!p\u2219q : (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n    p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n    p\u2219!q\u2219q : (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n    p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n    p\u2219q\u2219!q : (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n    p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n    \u2219-cancel : {p q : x \u2261 y}(r : y \u2261 z) \u2192 p \u2219 r \u2261 q \u2219 r \u2192 p \u2261 q\n    \u2219-cancel {p = p} {q} r e\n      = p               \u2261\u27e8 ! p\u2219q\u2219!q p r \u27e9\n         p \u2219 r  \u2219 ! r   \u2261\u27e8 \u2219-assoc p r (! r) \u27e9\n        (p \u2219 r) \u2219 ! r   \u2261\u27e8 \u2219= e idp \u27e9\n        (q \u2219 r) \u2219 ! r   \u2261\u27e8 ! \u2219-assoc q r (! r) \u27e9\n        q \u2219 (r  \u2219 ! r)  \u2261\u27e8 p\u2219q\u2219!q q r \u27e9\n        q               \u220e\n\n    \u2219-cancel\u2032 : (p : x \u2261 y){q r : y \u2261 z} \u2192 p \u2219 q \u2261 p \u2219 r \u2192 q \u2261 r\n    \u2219-cancel\u2032 p {q} {r} e\n      = q             \u2261\u27e8 ! !p\u2219p\u2219q p q \u27e9\n        ! p \u2219 p \u2219 q   \u2261\u27e8 ap (_\u2219_ (! p)) e  \u27e9\n        ! p \u2219 p \u2219 r   \u2261\u27e8 !p\u2219p\u2219q p r \u27e9\n        r             \u220e\n\n!-ap : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B){x y}(p : x \u2261 y)\n       \u2192 ! (ap f p) \u2261 ap f (! p)\n!-ap f idp = idp\n\nap-id : \u2200 {a}{A : Set a}{x y : A}(p : x \u2261 y) \u2192 ap id p \u2261 p\nap-id idp = idp\n\nap-\u2218 : \u2200 {a b c}{A : Set a}{B : Set b}{C : Set c}(f : B \u2192 C)(g : A \u2192 B){x y}(p : x \u2261 y)\n       \u2192 ap (f \u2218 g) p \u2261 ap f (ap g p)\nap-\u2218 f g idp = idp\n\nmodule _ {a b}{A : Set a}{B : Set b}{f g : A \u2192 B}(H : \u2200 x \u2192 f x \u2261 g x) where\n  ap-nat : \u2200 {x y}(q : x \u2261 y) \u2192 ap f q \u2219 H _ \u2261 H _ \u2219 ap g q\n  ap-nat idp = ! \u2219-refl _\n\nmodule _ {a}{A : Set a}{f : A \u2192 A}(H : \u2200 x \u2192 f x \u2261 x) where\n  ap-nat-id : \u2200 x \u2192 ap f (H x) \u2261 H (f x)\n  ap-nat-id x = \u2219-cancel (H x) (ap-nat H (H x) \u2219 ap (_\u2219_ (H (f x))) (ap-id (H x)))\n\ntr-\u2218 : \u2200 {a b p}{A : Set a}{B : Set b}(P : B \u2192 Set p)(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 tr (P \u2218 f) p \u2261 tr P (ap f p)\ntr-\u2218 P f idp = idp\n\nmodule _ {a}{A : \u2605_ a} where\n    tr-\u2219\u2032 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) \u2192\n             tr P (p \u2219 q) \u223c tr P q \u2218 tr P p\n    tr-\u2219\u2032 P idp _ _ = idp\n\n    tr-\u2219 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) (pq : x \u2261 z) \u2192\n             pq \u2261 p \u2219 q \u2192\n             tr P pq \u223c tr P q \u2218 tr P p\n    tr-\u2219 P p q ._ idp = tr-\u2219\u2032 P p q\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb _ p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) idp p\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605_(a \u2294 b)\n    HAE {f} f-qinv = \u2200 x \u2192 ap f (F.inv-is-linv x) \u2261 F.inv-is-rinv (f x)\n      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      f-g' : (x : B) \u2192 f (g x) \u2261 x\n      f-g' x = ! ap (f \u2218 g) (f-g x) \u2219 ap f (g-f (g x)) \u2219 f-g x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n\n      hae : HAE (qinv g f-g' g-f)\n      hae a = \u2219-cancel\u2032 (ap (f \u2218 g) (f-g (f a)))\n        (ap (f \u2218 g) (f-g (f a)) \u2219 ap f (g-f a) \u2261\u27e8 \u2219= (ap-nat-id f-g (f a)) (! ap-id (ap f (g-f a))) \u27e9\n         f-g (f (g (f a))) \u2219 ap id (ap f (g-f a))  \u2261\u27e8 ! ap-nat f-g {f (g (f a))} {f a} (ap f (g-f a)) \u27e9\n         ap (f \u2218 g) (ap f (g-f a)) \u2219 f-g (f a)    \u2261\u27e8 ap (flip _\u2219_ (f-g (f a))) (! ap-\u2218 (f \u2218 g) f (g-f a)\n                                                                               \u2219 ap-\u2218 f (g \u2218 f) (g-f a))  \u27e9\n         ap f (ap (\u03bb z \u2192 g (f z)) (g-f a)) \u2219 f-g (f a) \u2261\u27e8 ap (flip _\u2219_ (f-g (f a))) (ap (ap f) (ap-nat-id g-f a)) \u27e9\n         ap f (g-f (g (f a))) \u2219 f-g (f a)      \u2261\u27e8 ! p\u2219!p\u2219q (ap (f \u2218 g) (f-g (f a))) (ap f (g-f (g (f a))) \u2219 f-g (f a)) \u27e9\n         ap (f \u2218 g) (f-g (f a)) \u2219 ! ap (f \u2218 g) (f-g (f a)) \u2219 ap f (g-f (g (f a))) \u2219 f-g (f a)    \u220e\n        )\n\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g' g-f\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module Equiv {a b}{A : \u2605_ a}{B : \u2605_ b}(e : A \u2243 B) where\n    \u00b7\u2192 : A \u2192 B\n    \u00b7\u2192 = fst e\n\n    open Is-equiv (snd e) public\n      renaming (linv to \u00b7\u2190; rinv to \u00b7\u2190\u2032; is-linv to \u00b7\u2190-inv-l; is-rinv to \u00b7\u2190-inv-r)\n\n    -- Equivalences are \"injective\"\n    equiv-inj : {x y : A} \u2192 (\u00b7\u2192 x \u2261 \u00b7\u2192 y \u2192 x \u2261 y)\n    equiv-inj p = ! \u00b7\u2190-inv-l _ \u2219 ap \u00b7\u2190 p \u2219 \u00b7\u2190-inv-l _\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n    <\u2013' = \u00b7\u2190\u2032\n\n    <\u2013-inv-l = \u00b7\u2190-inv-l\n    <\u2013-inv-r = \u00b7\u2190-inv-r\n\n  open Equiv public\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n      prop-has-all-paths : is-prop A \u2192 has-all-paths A\n      prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n      all-paths-is-prop : has-all-paths A \u2192 is-prop A\n      all-paths-is-prop c x y = c x y , canon-path\n        where\n        lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n        lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n        canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n        canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                        (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n      Is-contr\u2192is-prop : Is-contr A \u2192 is-prop A\n      Is-contr\u2192is-prop (x , p) y z\n         = ! p y \u2219 p z\n         , J' (\u03bb y\u2081 z\u2081 q \u2192 ! p y\u2081 \u2219 p z\u2081 \u2261 q) (!-\u2219 \u2218 p)\n\n      {-\n      has-level-up : \u2200 {n} \u2192 has-level n A \u2192 has-level \u27e8S n \u27e9 A\n      has-level-up {\u27e8-2\u27e9} = Is-contr\u2192is-prop\n      has-level-up {\u27e8S n \u27e9} \u03c0 x y p q = {!!}\n\n      Is-contr-is-prop : is-prop (Is-contr A)\n      Is-contr-is-prop (x , p) (y , q) = ?\n\n      module _ {{_ : FunExt}} where\n        is-prop-is-prop : has-all-paths (has-all-paths A)\n        is-prop-is-prop h0 h1 = \u03bb= \u03bb x \u2192 \u03bb= \u03bb y \u2192 {!!}\n      -}\n\n\ud835\udfd8-is-prop : is-prop \ud835\udfd8\n\ud835\udfd8-is-prop () _\n\n\ud835\udfd8-has-all-paths : has-all-paths \ud835\udfd8\n\ud835\udfd8-has-all-paths () _\n\n\ud835\udfd9-is-contr : Is-contr \ud835\udfd9\n\ud835\udfd9-is-contr = _ , \u03bb _ \u2192 idp\n\n\ud835\udfd9-is-prop : is-prop \ud835\udfd9\n\ud835\udfd9-is-prop = Is-contr\u2192is-prop \ud835\udfd9-is-contr\n\n\ud835\udfd9-has-all-paths : has-all-paths \ud835\udfd9\n\ud835\udfd9-has-all-paths _ _ = idp\n\nmodule _ {{_ : FunExt}}{a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n  \u03a0-has-all-paths : (\u2200 x \u2192 has-all-paths (B x)) \u2192 has-all-paths (\u03a0 A B)\n  \u03a0-has-all-paths B-has-all-paths f g\n    = \u03bb= \u03bb _ \u2192 B-has-all-paths _ _ _\n\n  \u03a0-is-prop : (\u2200 x \u2192 is-prop (B x)) \u2192 is-prop (\u03a0 A B)\n  \u03a0-is-prop B-is-prop = all-paths-is-prop (\u03a0-has-all-paths (prop-has-all-paths \u2218 B-is-prop))\n\nmodule _ {{_ : FunExt}}{a}{A : \u2605_ a} where\n    \u00ac-has-all-paths : has-all-paths (\u00ac A)\n    \u00ac-has-all-paths = \u03a0-has-all-paths (\u03bb _ \u2192 \ud835\udfd8-has-all-paths)\n\n    \u00ac-is-prop : is-prop (\u00ac A)\n    \u00ac-is-prop = \u03a0-is-prop (\u03bb _ \u2192 \ud835\udfd8-is-prop)\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same p x = ap (\u03bb X \u2192 coe X x) p\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u00b7\u2192 e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u00b7\u2192 e a) (coe-equiv-\u03b2 e)\n\n  postulate\n    coe!-\u03b2 : (e : A \u2243 B) (b : B) \u2192 coe! (ua e) b \u2261 \u00b7\u2190 e b\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u00b7\u2192-paths-equiv : (x \u2261 y) \u2261 (\u00b7\u2192 e x \u2261 \u00b7\u2192 e y)\n    \u00b7\u2192-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n    \u2013>-paths-equiv = \u00b7\u2192-paths-equiv\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\nopen \u2261.\u2261-Reasoning\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  -- could be derived in any groupoid\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  module _ {x y : A} where\n\n    \u2219-assoc : (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n    \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n                 (\u03bb x q r \u2192 idp)\n\n    module _ (p : x \u2261 y) where\n      -- ! is a left-inverse for _\u2219_\n      !-\u2219 : ! p \u2219 p \u2261 idp_ y\n      !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp) p\n\n      -- ! is a right-inverse for _\u2219_\n      \u2219-! : p \u2219 ! p \u2261 idp_ x\n      \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp) p\n\n      -- ! is involutive\n      !-involutive : ! (! p) \u2261 p\n      !-involutive = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp) p\n\n      !p\u2219p = !-\u2219\n      p\u2219!p = \u2219-!\n\n      ==-refl-\u2219 : {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n      ==-refl-\u2219 = ap (flip _\u2219_ p)\n\n      \u2219-==-refl : {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n      \u2219-==-refl qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  module _ {x y : A} where\n    module _ (p : x \u2261 x)(q : x \u2261 y)(e : p \u2219 q \u2261 q) where\n      unique-idp-left : p \u2261 idp\n      unique-idp-left\n        = p              \u2261\u27e8 ! \u2219-refl p \u27e9\n          p \u2219 idp        \u2261\u27e8 ap (_\u2219_ p) (! \u2219-! q) \u27e9\n          p \u2219 (q \u2219 ! q)  \u2261\u27e8 \u2219-assoc p q (! q) \u27e9\n          (p \u2219 q) \u2219 ! q  \u2261\u27e8 ap (flip _\u2219_ (! q)) e \u27e9\n          q \u2219 ! q        \u2261\u27e8 \u2219-! q \u27e9\n          idp            \u220e\n\n  module _ {x y z : A}{p\u2080 p\u2081 : x \u2261 y}{q\u2080 q\u2081 : y \u2261 z}(p : p\u2080 \u2261 p\u2081)(q : q\u2080 \u2261 q\u2081) where\n      \u2219= : p\u2080 \u2219 q\u2080 \u2261 p\u2081 \u2219 q\u2081\n      \u2219= = ap (flip _\u2219_ q\u2080) p \u2219 ap (_\u2219_ p\u2081) q\n\n  module _ {x y z : A} where\n    module _ (p : x \u2261 y)(q : y \u2261 z) where\n      \u2219-\u2219-==-refl : {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n      \u2219-\u2219-==-refl rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n      !p\u2219p\u2219q : ! p \u2219 p \u2219 q \u2261 q\n      !p\u2219p\u2219q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n    p\u2219!p\u2219q : (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n    p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n    p\u2219!q\u2219q : (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n    p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n    p\u2219q\u2219!q : (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n    p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n    \u2219-cancel : {p q : x \u2261 y}(r : y \u2261 z) \u2192 p \u2219 r \u2261 q \u2219 r \u2192 p \u2261 q\n    \u2219-cancel {p = p} {q} r e\n      = p               \u2261\u27e8 ! p\u2219q\u2219!q p r \u27e9\n         p \u2219 r  \u2219 ! r   \u2261\u27e8 \u2219-assoc p r (! r) \u27e9\n        (p \u2219 r) \u2219 ! r   \u2261\u27e8 \u2219= e idp \u27e9\n        (q \u2219 r) \u2219 ! r   \u2261\u27e8 ! \u2219-assoc q r (! r) \u27e9\n        q \u2219 (r  \u2219 ! r)  \u2261\u27e8 p\u2219q\u2219!q q r \u27e9\n        q               \u220e\n\n!-ap : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B){x y}(p : x \u2261 y)\n       \u2192 ! (ap f p) \u2261 ap f (! p)\n!-ap f idp = idp\n\nap-id : \u2200 {a}{A : Set a}{x y : A}(p : x \u2261 y) \u2192 ap id p \u2261 p\nap-id idp = idp\n\nmodule _ {a b}{A : Set a}{B : Set b}{f g : A \u2192 B}(H : \u2200 x \u2192 f x \u2261 g x) where\n  ap-nat : \u2200 {x y}(q : x \u2261 y) \u2192 ap f q \u2219 H _ \u2261 H _ \u2219 ap g q\n  ap-nat idp = ! \u2219-refl _\n\nmodule _ {a}{A : Set a}{f : A \u2192 A}(H : \u2200 x \u2192 f x \u2261 x) where\n  ap-nat-id : \u2200 x \u2192 ap f (H x) \u2261 H (f x)\n  ap-nat-id x = \u2219-cancel (H x) (ap-nat H (H x) \u2219 ap (_\u2219_ (H (f x))) (ap-id (H x)))\n\ntr-\u2218 : \u2200 {a b p}{A : Set a}{B : Set b}(P : B \u2192 Set p)(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 tr (P \u2218 f) p \u2261 tr P (ap f p)\ntr-\u2218 P f idp = idp\n\nmodule _ {a}{A : \u2605_ a} where\n    tr-\u2219\u2032 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) \u2192\n             tr P (p \u2219 q) \u223c tr P q \u2218 tr P p\n    tr-\u2219\u2032 P idp _ _ = idp\n\n    tr-\u2219 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) (pq : x \u2261 z) \u2192\n             pq \u2261 p \u2219 q \u2192\n             tr P pq \u223c tr P q \u2218 tr P p\n    tr-\u2219 P p q ._ idp = tr-\u2219\u2032 P p q\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb _ p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) idp p\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605_(a \u2294 b)\n    HAE {f} f-qinv = \u2200 x \u2192 ap f (F.inv-is-linv x) \u2261 F.inv-is-rinv (f x)\n      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      f-g' : (x : B) \u2192 f (g x) \u2261 x\n      f-g' x = ! ap (f \u2218 g) (f-g x) \u2219 ap f (g-f (g x)) \u2219 f-g x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n\n      postulate hae : HAE (qinv g f-g' g-f)\n\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g' g-f\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module Equiv {a b}{A : \u2605_ a}{B : \u2605_ b}(e : A \u2243 B) where\n    \u00b7\u2192 : A \u2192 B\n    \u00b7\u2192 = fst e\n\n    open Is-equiv (snd e) public\n      renaming (linv to \u00b7\u2190; rinv to \u00b7\u2190\u2032; is-linv to \u00b7\u2190-inv-l; is-rinv to \u00b7\u2190-inv-r)\n\n    -- Equivalences are \"injective\"\n    equiv-inj : {x y : A} \u2192 (\u00b7\u2192 x \u2261 \u00b7\u2192 y \u2192 x \u2261 y)\n    equiv-inj p = ! \u00b7\u2190-inv-l _ \u2219 ap \u00b7\u2190 p \u2219 \u00b7\u2190-inv-l _\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n    <\u2013' = \u00b7\u2190\u2032\n\n    <\u2013-inv-l = \u00b7\u2190-inv-l\n    <\u2013-inv-r = \u00b7\u2190-inv-r\n\n  open Equiv public\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n      prop-has-all-paths : is-prop A \u2192 has-all-paths A\n      prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n      all-paths-is-prop : has-all-paths A \u2192 is-prop A\n      all-paths-is-prop c x y = c x y , canon-path\n        where\n        lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n        lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n        canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n        canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                        (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n      Is-contr\u2192is-prop : Is-contr A \u2192 is-prop A\n      Is-contr\u2192is-prop (x , p) y z\n         = ! p y \u2219 p z\n         , J' (\u03bb y\u2081 z\u2081 q \u2192 ! p y\u2081 \u2219 p z\u2081 \u2261 q) (!-\u2219 \u2218 p)\n\n      {-\n      has-level-up : \u2200 {n} \u2192 has-level n A \u2192 has-level \u27e8S n \u27e9 A\n      has-level-up {\u27e8-2\u27e9} = Is-contr\u2192is-prop\n      has-level-up {\u27e8S n \u27e9} \u03c0 x y p q = {!!}\n\n      Is-contr-is-prop : is-prop (Is-contr A)\n      Is-contr-is-prop (x , p) (y , q) = ?\n\n      module _ {{_ : FunExt}} where\n        is-prop-is-prop : has-all-paths (has-all-paths A)\n        is-prop-is-prop h0 h1 = \u03bb= \u03bb x \u2192 \u03bb= \u03bb y \u2192 {!!}\n      -}\n\n\ud835\udfd8-is-prop : is-prop \ud835\udfd8\n\ud835\udfd8-is-prop () _\n\n\ud835\udfd8-has-all-paths : has-all-paths \ud835\udfd8\n\ud835\udfd8-has-all-paths () _\n\n\ud835\udfd9-is-contr : Is-contr \ud835\udfd9\n\ud835\udfd9-is-contr = _ , \u03bb _ \u2192 idp\n\n\ud835\udfd9-is-prop : is-prop \ud835\udfd9\n\ud835\udfd9-is-prop = Is-contr\u2192is-prop \ud835\udfd9-is-contr\n\n\ud835\udfd9-has-all-paths : has-all-paths \ud835\udfd9\n\ud835\udfd9-has-all-paths _ _ = idp\n\nmodule _ {{_ : FunExt}}{a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n  \u03a0-has-all-paths : (\u2200 x \u2192 has-all-paths (B x)) \u2192 has-all-paths (\u03a0 A B)\n  \u03a0-has-all-paths B-has-all-paths f g\n    = \u03bb= \u03bb _ \u2192 B-has-all-paths _ _ _\n\n  \u03a0-is-prop : (\u2200 x \u2192 is-prop (B x)) \u2192 is-prop (\u03a0 A B)\n  \u03a0-is-prop B-is-prop = all-paths-is-prop (\u03a0-has-all-paths (prop-has-all-paths \u2218 B-is-prop))\n\nmodule _ {{_ : FunExt}}{a}{A : \u2605_ a} where\n    \u00ac-has-all-paths : has-all-paths (\u00ac A)\n    \u00ac-has-all-paths = \u03a0-has-all-paths (\u03bb _ \u2192 \ud835\udfd8-has-all-paths)\n\n    \u00ac-is-prop : is-prop (\u00ac A)\n    \u00ac-is-prop = \u03a0-is-prop (\u03bb _ \u2192 \ud835\udfd8-is-prop)\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same p x = ap (\u03bb X \u2192 coe X x) p\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u00b7\u2192 e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u00b7\u2192 e a) (coe-equiv-\u03b2 e)\n\n  postulate\n    coe!-\u03b2 : (e : A \u2243 B) (b : B) \u2192 coe! (ua e) b \u2261 \u00b7\u2190 e b\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u00b7\u2192-paths-equiv : (x \u2261 y) \u2261 (\u00b7\u2192 e x \u2261 \u00b7\u2192 e y)\n    \u00b7\u2192-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n    \u2013>-paths-equiv = \u00b7\u2192-paths-equiv\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"230c9fd542aa231ef41ff245f7f17b98aa7e86d5","subject":"True\/False and lt\/le","message":"True\/False and lt\/le\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n\n\ndata   False : Set where\nrecord True  : Set where\n\n\ntrivial : True\ntrivial = _\n\n\nisTrue : Bool -> Set\nisTrue true  = True\nisTrue false = False\n\n\n_<_ : \u2115 -> \u2115 -> Bool\n_   < O   = false\nO   < S n = true\nS m < S n = m < n\n\nlength : {A : Set} -> List A -> \u2115\nlength [] = O\nlength (x :: xs) = S (length xs)\n\nlookup : {A : Set}(xs : List A)(n : \u2115) -> isTrue (n < length xs) -> A\nlookup (x :: xs) O b = x\nlookup (x :: xs) (S n) b = lookup xs n b\nlookup [] a ()\n\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\n\ndata _\u2264_ : \u2115 -> \u2115 -> Set where\n  \u2264O : {m n : \u2115} -> m == n -> m \u2264 n\n  \u2264I : {m n : \u2115} -> m \u2264  n -> m \u2264 S n\n\n\nleq-trans : {l m n : \u2115} -> l \u2264 m -> m \u2264 n -> l \u2264 n\nleq-trans (\u2264O refl) b = b\nleq-trans a (\u2264O refl) = a\nleq-trans (\u2264I a) (\u2264I b) = \u2264I (leq-trans (\u2264I a) b)\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n\n\ndata Fin : \u2115 -> Set where\n  fzero : {n : \u2115} -> Fin (S n)\n  fsuc  : {n : \u2115} -> Fin n -> Fin (S n)\n\n_!_ : {n : \u2115}{A : Set} -> Vec A n -> Fin n -> A\nnil ! ()\ncons n x a ! fzero = x\ncons n x a ! fsuc b = a ! b\n\n\n\ntabulate : {n : \u2115}{A : Set} -> (Fin n -> A) -> Vec A n\ntabulate {O} f = nil\ntabulate {S n} f = cons n (f fzero) (tabulate (f \u2218 fsuc))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fdb5d8f0195c9849f6e208db228bb650fe5eba97","subject":"Rename \u0394base to \u0394Base.","message":"Rename \u0394base to \u0394Base.\n\nOld-commit-hash: e9efdc6e4dd5d76003e901e1ef557ec049d5e59c\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff apply {base \u03b9} = diff , apply\n  lift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n  lift-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394base\n\nopen import Data.Product\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff apply {base \u03b9} = diff , apply\n  lift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n  lift-apply :\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n    (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a2f45eebca69d96dd28c2960f4edd6853d285500","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: d55be4ca0f940a641e282a1d090c1131\n\ndarcs-hash:20110401164324-3bd4e-1a750e87a1161fdbc62774ee881f3bad9df90b27.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(n,x) \u2192 app\u2081(n,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","old_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 P. app\u2081(P,zero) \u2192\n--      (\u2200 x. app1(N,x) \u2192 app\u2081(P,x) \u2192 app\u2081(P,succ x)) \u2192   -- indN\n--      (\u2200 x. app\u2081(N,x) \u2192 app\u2081(P,x))\n------------------------------------------------------------------\n-- \u2200 x y. app\u2081(N,x) \u2192 app\u2081(N,y) \u2192 app\u2081(N, x + y)          -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5736b7a0ce4a179ba60c22dc910773f8ef479967","subject":"Updated doc.","message":"Updated doc.\n\nIgnore-this: f1c847e6477e6855e3eee9457be98c6e\n\ndarcs-hash:20110814055030-3bd4e-c4503592e0b36a3ab621331a0f9330790448d922.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Divisibility\/By0.agda","new_file":"src\/FOTC\/Data\/Nat\/Divisibility\/By0.agda","new_contents":"------------------------------------------------------------------------------\n-- The relation of divisibility on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Divisibility.By0 where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2223_\n\n------------------------------------------------------------------------------\n-- The relation of divisibility.\n-- The symbol is '\\mid' not '|'.\n-- It seems there is not agreement about if 0\u22230, e.g.\n-- * Agda standard library, version 0.5: 0|0\n-- * Coq 8.3pl2: 0\u22230\n-- * Hardy and Wright. An introduction to the theory of numbers. 1975. 4ed: 0\u22240\n-- * Isabelle, version Isabelle2011: 0\u22230\n--\n-- In our definition 0\u22230, which is used to prove properties of the gcd\n-- as it is in GHC 7.2.1, where gcd 0 0 = 0 (see\n-- http:\/\/hackage.haskell.org\/trac\/ghc\/ticket\/3304).\n_\u2223_ : D \u2192 D \u2192 Set\nd \u2223 e = \u2203 \u03bb k \u2192 N k \u2227 e \u2261 k * d\n{-# ATP definition _\u2223_ #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The relation of divisibility on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Divisibility.By0 where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2223_\n\n------------------------------------------------------------------------------\n-- The relation of divisibility.\n-- The symbol is '\\mid' not '|'.\n-- It seems there is not agreement about if 0\u22230, e.g.\n-- * Agda standard library, version 0.5: 0|0\n-- * Coq 8.3pl2: 0\u22230\n-- * Hardy and Wright. An introduction to the theory of numbers. 1975. 4ed: 0\u22240\n-- * Isabelle, version Isabelle2011: 0\u22230\n--\n-- In our definition 0\u22230, which is used to prove properties of the gcd as\n-- it is in GHC HEAD (7.1.20110615), where gcd 0 0 = 0 (see\n-- http:\/\/hackage.haskell.org\/trac\/ghc\/ticket\/3304).\n_\u2223_ : D \u2192 D \u2192 Set\nd \u2223 e = \u2203 \u03bb k \u2192 N k \u2227 e \u2261 k * d\n{-# ATP definition _\u2223_ #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c70cc5026e4cf367700fbc2aec2b7a8e59f156ad","subject":"More Data.Bits operations and proofs","message":"More Data.Bits operations and proofs\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u2026 : \u2200 {n} \u2192 Bits n\n0\u2026 = replicate 0b\n\n-- Warning: 0\u2026 {0} \u2261 1\u2026 {0}\n1\u2026 : \u2200 {n} \u2192 Bits n\n1\u2026 = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u2026\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u2026 \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u2026 x) \u27e9\n         x \u2295 (x \u2295 1\u2026) \u2261\u27e8 sym (\u2295-assoc x x 1\u2026) \u27e9\n         (x \u2295 x) \u2295 1\u2026 \u2261\u27e8 cong (flip _\u2295_ 1\u2026) (\u2295-\u2261 x) \u27e9\n         0\u2026 \u2295 1\u2026       \u2261\u27e8 \u2295-left-identity 1\u2026 \u27e9\n         1\u2026 \u220e where open \u2261-Reasoning\n\nmsb : \u2200 {n} k \u2192 Bits (k + n) \u2192 Bits k\nmsb zero    bs       = []\nmsb (suc k) (b \u2237 bs) = b \u2237 msb k bs\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u2026) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u2026 (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u2026 sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Bool\nopen import Data.Fin using (Fin; zero; suc; to\u2115; #_; inject\u2081)\nopen import Data.Vec hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n{-\ndata Bit : Set where\n  0b 1b : Bit\n-}\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u2026 : \u2200 {n} \u2192 Bits n\n0\u2026 = replicate 0b\n\n-- Warning: 0\u2026 {0} \u2261 1\u2026 {0}\n1\u2026 : \u2200 {n} \u2192 Bits n\n1\u2026 = replicate 1b\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u2026\n\npostulate\n  \u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n  \u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n  \u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n  \u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n  \u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\n  \u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n  -- x \u2295 vnot x \u2261 x \u2295 (1\u2026 \u2295 x) \u2261 (x \u2295 x) \u2295 1\u2026 \u2261 0\u2026 \u2295 1\u2026 \u2261 1\u2026\n\nmsb : \u2200 {n} k \u2192 Bits (k + n) \u2192 Bits k\nmsb zero    bs       = []\nmsb (suc k) (b \u2237 bs) = b \u2237 msb k bs\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n\n11\u1d47 : Bits 2\n11\u1d47 = 1b \u2237 1b \u2237 []\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nbits\u2074 : Vec (Bits 4) 16\nbits\u2074 = fromList $ replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\n0000b = bits\u2074 ! # 0\n0001b = bits\u2074 ! # 1\n0010b = bits\u2074 ! # 2\n0011b = bits\u2074 ! # 3\n0100b = bits\u2074 ! # 4\n0101b = bits\u2074 ! # 5\n0110b = bits\u2074 ! # 6\n0111b = bits\u2074 ! # 7\n1000b = bits\u2074 ! # 8\n1001b = bits\u2074 ! # 9\n1010b = bits\u2074 ! # 10\n1011b = bits\u2074 ! # 11\n1100b = bits\u2074 ! # 12\n1101b = bits\u2074 ! # 13\n1110b = bits\u2074 ! # 14\n1111b = bits\u2074 ! # 15\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n{- alternative\n\u03b7 {zero} = \u03bb _ \u2192 []\n\u03b7 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e9bd1c466f113186ad8e1fa93ea7e5232066eb96","subject":"Desc model: proof-psi-phi implicit and slightly refactored","message":"Desc model: proof-psi-phi implicit and slightly refactored","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-casesW\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (xs : Sigma DescDConst \n                                                  (\u03bb s \u2192 desc (descDChoice I s)\n                                                  (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                            (hs : desc \n                                                  (box (sigma DescDConst (descDChoice I))\n                                                  (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                  P)\n                                            \u2192 P (Void , con xs)\n                      proof-psi-phi-cases (lvar , i) hs = refl\n                      proof-psi-phi-cases (lconst , x) hs = refl\n                      proof-psi-phi-cases (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (i : Unit)\n                                             (xs : Sigma DescDConst\n                                                   (\u03bb s \u2192 desc (descDChoice I s)\n                                                   (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                             (hs : desc \n                                                   (box (sigma DescDConst (descDChoice I))\n                                                   (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                   P)\n                                              \u2192  P (i , con xs)\n                      proof-psi-phi-casesW Void = proof-psi-phi-cases\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7ed997a9f08476266fb59fb488eac08ad40048de","subject":"Disable reflection for now","message":"Disable reflection for now\n","repos":"np\/agda-parametricity","old_file":"parametricity.agda","new_file":"parametricity.agda","new_contents":"module parametricity where\nimport Data.Bool.Param.NatBool\nimport Data.Dec.Param.Binary\nimport Data.Dec.Param.Unary\nimport Data.Fin.Param.Binary\nimport Data.Maybe.Param.Binary\nimport Data.Maybe.Param.Unary\nimport Data.Nat.Param.Binary\nimport Data.One.Param.Binary\nimport Data.One.Param.Unary\nimport Data.Product.Param.Binary\nimport Data.Product.Param.Unary\nimport Data.Sum.Param.Binary\nimport Data.Sum.Param.Unary\nimport Data.Two.Param.Binary\nimport Data.Two.Param.Unary\nimport Data.Zero.Param.Binary\nimport Data.Zero.Param.Unary\nimport Function.Param.Binary\nimport Function.Param.Unary\n{-\nimport Reflection.NP\nimport Reflection.Param\nimport Reflection.Param.Env\nimport Reflection.Param.Tests\n-}\nimport Type.Param.Binary\nimport Type.Param.Unary\n","old_contents":"module parametricity where\nimport Data.Bool.Param.NatBool\nimport Data.Dec.Param.Binary\nimport Data.Dec.Param.Unary\nimport Data.Fin.Param.Binary\nimport Data.Maybe.Param.Binary\nimport Data.Maybe.Param.Unary\nimport Data.Nat.Param.Binary\nimport Data.One.Param.Binary\nimport Data.One.Param.Unary\nimport Data.Product.Param.Binary\nimport Data.Product.Param.Unary\nimport Data.Sum.Param.Binary\nimport Data.Sum.Param.Unary\nimport Data.Two.Param.Binary\nimport Data.Two.Param.Unary\nimport Data.Zero.Param.Binary\nimport Data.Zero.Param.Unary\nimport Function.Param.Binary\nimport Function.Param.Unary\nimport Reflection.NP\nimport Reflection.Param\nimport Reflection.Param.Env\nimport Reflection.Param.Tests\nimport Type.Param.Binary\nimport Type.Param.Unary\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ab777f510ffa03f184a684d07c9b80f46c896dfe","subject":"From\/to N as a least fixed-point to\/from N as data type.","message":"From\/to N as a least fixed-point to\/from N as data type.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_file":"notes\/fixed-points\/LeastFixedPoints.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\n\ninfixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- Instead defining the least fixed-point via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF P n = n \u2261 zero \u2228 (\u2203[ m ] P m \u2227 n \u2261 succ\u2081 m)\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] N n' \u2227 n \u2261 succ\u2081 n') \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- N is a the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082 : (A : D \u2192 Set) \u2192\n           (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n') \u2192 A n) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  -- N-lfp\u2082 : (A : D \u2192 Set) \u2192  -- Higher-order version\n  --          (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192\n  --          \u2200 {n} \u2192 N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nnzero : N zero\nnzero = N-lfp\u2081 (inj\u2081 refl)\n\nnsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nnsucc Nn = N-lfp\u2081 (inj\u2082 (_ , (Nn , refl)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 N m \u2227 n \u2261 succ\u2081 m)\nN-lfp\u2083 Nn = N-lfp\u2082 A prf Nn\n  where\n  A : D \u2192 Set\n  A x = x \u2261 zero \u2228 (\u2203 \u03bb n' \u2192 N n' \u2227 x \u2261 succ\u2081 n')\n\n  prf : \u2200 {n''} \u2192 n'' \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n'' \u2261 succ\u2081 n') \u2192 A n''\n  prf {n''} h = case inj\u2081 ((\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081))) h -- case inj\u2081 prf\u2081 h\n    where\n    prf\u2081 : \u2203 (\u03bb n' \u2192 A n' \u2227 n'' \u2261 succ\u2081 n') \u2192 \u2203 (\u03bb n' \u2192 N n' \u2227 n'' \u2261 succ\u2081 n')\n    prf\u2081 (n' , An' , n''=Sn') = n' , prf\u2082 An' , n''=Sn'\n      where\n      prf\u2082 : A n' \u2192 N n'\n      prf\u2082 An' = case (\u03bb ah \u2192 subst N (sym ah) nzero) prf\u2083 An'\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 N m' \u2227 n' \u2261 succ\u2081 m') \u2192 N n'\n        prf\u2083 (_ , Nm' , m\u2261Sm' ) = subst N (sym m\u2261Sm') (nsucc Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nN-ind : (A : D \u2192 Set) \u2192\n        A zero \u2192\n        (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n        \u2200 {n} \u2192 N n \u2192 A n\nN-ind A A0 h Nn = N-lfp\u2082 A (case prf\u2081 prf\u2082) Nn\n  where\n  prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n  prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n  prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 A m \u2227 n' \u2261 succ\u2081 m) \u2192 A n'\n  prf\u2082 (_ , Am , n'\u2261Sm) = subst A (sym n'\u2261Sm) (h Am)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\n--\n-- 2012-03-06. We cannot proof this principle because N-lfp\u2082 does not\n-- have the hypothesis N n.\n--\n-- indN\u2082 : (A : D \u2192 Set) \u2192\n--        A zero \u2192\n--        (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n--        \u2200 {n} \u2192 N n \u2192 A n\n-- indN\u2082 A A0 is Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n--   where\n--   prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n--   prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n--   prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n--   prf\u2082 {n'} (m , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is helper Am)\n--     where\n--     helper : N m\n--     helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 {!!})\n--       where\n--       prf\u2083 : n' \u2261 zero \u2192 N m\n--       prf\u2083 n'\u22610 = \u22a5-elim (0\u2262S (trans (sym n'\u22610) n'\u2261Sm))\n\n--       prf\u2084 : \u2203 (\u03bb m' \u2192 n' \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n--       prf\u2084 (_ , n'\u2261Sm' , Nm') =\n--         subst N (succInjective (trans (sym n'\u2261Sm') n'\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-lfp\u2082 A prf Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  prf : \u2200 {m'} \u2192 m' \u2261 zero \u2228 \u2203 (\u03bb m'' \u2192 A m'' \u2227 m' \u2261 succ\u2081 m'') \u2192 A m'\n  prf h = case prf\u2081 prf\u2082 h\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : \u2200 {m} \u2192 m \u2261 zero \u2192 A m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) A0\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n    prf\u2082 : \u2200 {m} \u2192 \u2203 (\u03bb m'' \u2192 A m'' \u2227 m \u2261 succ\u2081 m'') \u2192 A m\n    prf\u2082 (_ ,  Am'' , m\u2261Sm'') =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm'')) (is Am'')\n\n------------------------------------------------------------------------------\n-- From\/to N as a least fixed-point to\/from N as data type.\n\nopen import FOTC.Data.Nat.Type renaming\n  ( N to N'\n  ; N-ind to N-ind'\n  ; nsucc to nsucc'\n  ; nzero to nzero'\n  )\n\nthm\u2081 : \u2200 {n} \u2192 N' n \u2192 N n\nthm\u2081 nzero' = nzero\nthm\u2081 (nsucc' Nn) = nsucc (thm\u2081 Nn)\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192 N' n\nthm\u2082 Nn = N-ind N' nzero' nsucc' Nn\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n-- infixl 9 _+_\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\n------------------------------------------------------------------------------\n-- N is a least fixed-point of a functor\n\n-- Instead defining the least fixed-point via a (higher-order)\n-- operator, we will define it using an instance of that operator.\n\n-- The functor\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF P n = n \u2261 zero \u2228 (\u2203[ m ] P m \u2227 n \u2261 succ\u2081 m)\n\n-- The natural numbers are the least fixed-point of NatF.\npostulate\n  N : D \u2192 Set\n\n  -- N is a pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  N-lfp\u2081    : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] N m \u2227 n \u2261 succ\u2081 m) \u2192 N n\n  -- N-lfp\u2081 : \u2200 n \u2192 NatF N n \u2192 N n  -- Higher-order version\n\n  -- N is a the least pre-fixed point of NatF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  N-lfp\u2082    : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ m ] A m \u2227 n \u2261 succ\u2081 m) \u2192 A n) \u2192\n              \u2200 {n} \u2192 N n \u2192 A n\n  -- N-lfp\u2082 : (A : D \u2192 Set) \u2192  -- Higher-order version\n  --          (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192\n  --          \u2200 {n} \u2192 N n \u2192 A n\n\n------------------------------------------------------------------------------\n-- The data constructors of N.\nzN : N zero\nzN = N-lfp\u2081 (inj\u2081 refl)\n\nsN : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\nsN Nn = N-lfp\u2081 (inj\u2082 (_ , (Nn , refl)))\n\n------------------------------------------------------------------------------\n-- Because N is the least pre-fixed point of NatF (i.e. N-lfp\u2081 and\n-- N-lfp\u2082), we can proof that N is also a post-fixed point of NatF.\n\n-- N is a post-fixed point of NatF.\nN-lfp\u2083 : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 N m \u2227 n \u2261 succ\u2081 m)\nN-lfp\u2083 Nn = N-lfp\u2082 A prf Nn\n  where\n  A : D \u2192 Set\n  A x = x \u2261 zero \u2228 (\u2203 \u03bb m \u2192 N m \u2227 x \u2261 succ\u2081 m)\n\n  prf : \u2200 {n'} \u2192 n' \u2261 zero \u2228 (\u2203[ m ] A m \u2227 n' \u2261 succ\u2081 m) \u2192 A n'\n  prf {n'} h = case inj\u2081 ((\u03bb h\u2081 \u2192 inj\u2082 (prf\u2081 h\u2081))) h -- case inj\u2081 prf\u2081 h\n    where\n    prf\u2081 : \u2203 (\u03bb m \u2192 A m \u2227 n' \u2261 succ\u2081 m) \u2192 \u2203 (\u03bb m \u2192 N m \u2227 n' \u2261 succ\u2081 m)\n    prf\u2081 (m , Am , n'=Sm) = m , prf\u2082 Am , n'=Sm\n      where\n      prf\u2082 : A m \u2192 N m\n      prf\u2082 Am = case (\u03bb ah \u2192 subst N (sym ah) zN) prf\u2083 Am\n        where\n        prf\u2083 : \u2203 (\u03bb m' \u2192 N m' \u2227 m \u2261 succ\u2081 m') \u2192 N m\n        prf\u2083 (_ , Nm' , m\u2261Sm' ) = subst N (sym m\u2261Sm') (sN Nm')\n\n------------------------------------------------------------------------------\n-- The induction principle for N *without* the hypothesis N n in the\n-- induction step.\nindN\u2081 : (A : D \u2192 Set) \u2192\n       A zero \u2192\n       (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 A n\nindN\u2081 A A0 is Nn = N-lfp\u2082 A (case prf\u2081 prf\u2082) Nn\n  where\n  prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n  prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n  prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 A m \u2227 n' \u2261 succ\u2081 m) \u2192 A n'\n  prf\u2082 (_ , Am , n'\u2261Sm) = subst A (sym n'\u2261Sm) (is Am)\n\n-- The induction principle for N *with* the hypothesis N n in the\n-- induction step.\n--\n-- 2012-03-06. We cannot proof this principle because N-lfp\u2082 does not\n-- have the hypothesis N n.\n--\n-- indN\u2082 : (A : D \u2192 Set) \u2192\n--        A zero \u2192\n--        (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n--        \u2200 {n} \u2192 N n \u2192 A n\n-- indN\u2082 A A0 is Nn = N-lfp\u2082 A [ prf\u2081 , prf\u2082 ] Nn\n--   where\n--   prf\u2081 : \u2200 {n'} \u2192 n' \u2261 zero \u2192 A n'\n--   prf\u2081 n'\u22610 = subst A (sym n'\u22610) A0\n\n--   prf\u2082 : \u2200 {n'} \u2192 \u2203 (\u03bb m \u2192 n' \u2261 succ\u2081 m \u2227 A m) \u2192 A n'\n--   prf\u2082 {n'} (m , n'\u2261Sm , Am) = subst A (sym n'\u2261Sm) (is helper Am)\n--     where\n--     helper : N m\n--     helper = [ prf\u2083 , prf\u2084 ] (N-lfp\u2083 {!!})\n--       where\n--       prf\u2083 : n' \u2261 zero \u2192 N m\n--       prf\u2083 n'\u22610 = \u22a5-elim (0\u2262S (trans (sym n'\u22610) n'\u2261Sm))\n\n--       prf\u2084 : \u2203 (\u03bb m' \u2192 n' \u2261 succ\u2081 m' \u2227 N m') \u2192 N m\n--       prf\u2084 (_ , n'\u2261Sm' , Nm') =\n--         subst N (succInjective (trans (sym n'\u2261Sm') n'\u2261Sm)) Nm'\n\n------------------------------------------------------------------------------\n-- Example: We will use N-lfp\u2082 as the induction principle on N.\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero    + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-lfp\u2082 A prf Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  prf : \u2200 {m'} \u2192 m' \u2261 zero \u2228 \u2203 (\u03bb m'' \u2192 A m'' \u2227 m' \u2261 succ\u2081 m'') \u2192 A m'\n  prf h = case prf\u2081 prf\u2082 h\n    where\n    A0 : A zero\n    A0 = subst N (sym (+-leftIdentity n)) Nn\n\n    prf\u2081 : \u2200 {m} \u2192 m \u2261 zero \u2192 A m\n    prf\u2081 h\u2081 = subst N (cong (flip _+_ n) (sym h\u2081)) A0\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n    prf\u2082 : \u2200 {m} \u2192 \u2203 (\u03bb m'' \u2192 A m'' \u2227 m \u2261 succ\u2081 m'') \u2192 A m\n    prf\u2082 (_ ,  Am'' , m\u2261Sm'') =\n      subst N (cong (flip _+_ n) (sym m\u2261Sm'')) (is Am'')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39443ea9275f5a1efcd6091bc6e78b7b1a88703e","subject":"[ test-suite ] Fixed withespace issues.","message":"[ test-suite ] Fixed withespace issues.\n","repos":"asr\/apia,asr\/apia","old_file":"test\/succeed\/fol-theorems\/LogicalConstants.agda","new_file":"test\/succeed\/fol-theorems\/LogicalConstants.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the translation of the logical constants\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule LogicalConstants where\n\ninfix  5 \u00ac_\ninfixr 4 _\u2227_\ninfixr 3 _\u2228_\ninfixr 2 _\u21d2_\ninfixr 1 _\u2194_ _\u21d4_\n\n------------------------------------------------------------------------------\n-- Propositional logic\n\n-- The logical constants\n\n-- The logical constants are hard-coded in our implementation,\n-- i.e. the following symbols must be used (it is possible to use Agda\n-- non-dependent function space \u2192 instead of \u21d2).\npostulate\n  \u22a5 \u22a4         : Set -- N.B. the name of the tautology symbol is \"\\top\" not T.\n  \u00ac_          : Set \u2192 Set -- N.B. the right hole.\n  _\u2227_ _\u2228_     : Set \u2192 Set \u2192 Set\n  _\u21d2_ _\u2194_ _\u21d4_ : Set \u2192 Set \u2192 Set\n\n-- We postulate some formulae (which are translated as 0-ary\n-- predicates).\npostulate A B C : Set\n\n-- Testing the conditional using the non-dependent function type.\npostulate A\u2192A : A \u2192 A\n{-# ATP prove A\u2192A #-}\n\n-- The introduction and elimination rules for the propositional\n-- connectives are theorems.\npostulate\n  \u2192I  : (A \u2192 B) \u2192 A \u21d2 B\n  \u2192E  : (A \u21d2 B) \u2192 A \u2192 B\n  \u2227I  : A \u2192 B \u2192 A \u2227 B\n  \u2227E\u2081 : A \u2227 B \u2192 A\n  \u2227E\u2082 : A \u2227 B \u2192 B\n  \u2228I\u2081 : A \u2192 A \u2228 B\n  \u2228I\u2082 : B \u2192 A \u2228 B\n  \u2228E  : (A \u21d2 C) \u2192 (B \u21d2 C) \u2192 A \u2228 B \u2192 C\n  \u22a5E  : \u22a5 \u2192 A\n  \u00acE : (\u00ac A \u2192 \u22a5) \u2192 A\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u00acE #-}\n\n-- Testing other logical constants.\npostulate\n  thm\u2081 : A \u2227 \u22a4 \u2192 A\n  thm\u2082 : A \u2228 \u22a5 \u2192 A\n  thm\u2083 : A \u2194 A\n  thm\u2084 : A \u21d4 A\n{-# ATP prove thm\u2081 #-}\n{-# ATP prove thm\u2082 #-}\n{-# ATP prove thm\u2083 #-}\n{-# ATP prove thm\u2084 #-}\n\n------------------------------------------------------------------------------\n-- Predicate logic\n\n--- The universe of discourse.\npostulate D : Set\n\n-- The propositional equality is hard-coded in our implementation, i.e. the\n-- following symbol must be used.\npostulate _\u2261_ : D \u2192 D \u2192 Set\n\n-- Testing propositional equality.\npostulate refl : \u2200 {x} \u2192 x \u2261 x\n{-# ATP prove refl #-}\n\n-- The quantifiers are hard-coded in our implementation, i.e. the\n-- following symbols must be used (it is possible to use Agda\n-- dependent function space \u2200 x \u2192 A instead of \u22c0).\npostulate\n  \u2203 : (A : D \u2192 Set) \u2192 Set\n  \u22c0 : (A : D \u2192 Set) \u2192 Set\n\n-- We postulate some formulae and propositional functions.\npostulate\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are\n-- theorems.\npostulate\n  \u2200I\u2081 : ((x : D) \u2192 A\u00b9 x) \u2192 \u22c0 A\u00b9\n  \u2200I\u2082 : ((x : D) \u2192 A\u00b9 x) \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200E\u2081 : (t : D) \u2192 \u22c0 A\u00b9 \u2192 A\u00b9 t\n  \u2200E\u2082 : (t : D) \u2192 (\u2200 x \u2192 A\u00b9 x) \u2192 A\u00b9 t\n{-# ATP prove \u2200I\u2081 #-}\n{-# ATP prove \u2200I\u2082 #-}\n{-# ATP prove \u2200E\u2081 #-}\n{-# ATP prove \u2200E\u2082 #-}\n\npostulate \u2203I : (t : D) \u2192 A\u00b9 t \u2192 \u2203 A\u00b9\n{-# ATP prove \u2203I #-}\n\npostulate \u2203E : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n{-# ATP prove \u2203E #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the translation of the logical constants\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule LogicalConstants where\n\ninfix  5 \u00ac_\ninfixr 4 _\u2227_\ninfixr 3 _\u2228_\ninfixr 2 _\u21d2_\ninfixr 1 _\u2194_ _\u21d4_\n\n------------------------------------------------------------------------------\n-- Propositional logic\n\n-- The logical constants\n\n-- The logical constants are hard-coded in our implementation,\n-- i.e. the following symbols must be used (it is possible to use Agda\n-- non-dependent function space \u2192 instead of \u21d2).\npostulate\n  \u22a5 \u22a4         : Set -- N.B. the name of the tautology symbol is \"\\top\" not T.\n  \u00ac_          : Set \u2192 Set -- N.B. the right hole.\n  _\u2227_ _\u2228_     : Set \u2192 Set \u2192 Set\n  _\u21d2_ _\u2194_ _\u21d4_ : Set \u2192 Set \u2192 Set\n\n-- We postulate some formulae (which are translated as 0-ary\n-- predicates).\npostulate A B C : Set\n\n-- Testing the conditional using the non-dependent function type.\npostulate A\u2192A : A \u2192 A\n{-# ATP prove A\u2192A #-}\n\n-- The introduction and elimination rules for the propositional\n-- connectives are theorems.\npostulate\n  \u2192I  : (A \u2192 B) \u2192 A \u21d2 B\n  \u2192E  : (A \u21d2 B) \u2192 A \u2192 B\n  \u2227I  : A \u2192 B \u2192 A \u2227 B\n  \u2227E\u2081 : A \u2227 B \u2192 A\n  \u2227E\u2082 : A \u2227 B \u2192 B\n  \u2228I\u2081 : A \u2192 A \u2228 B\n  \u2228I\u2082 : B \u2192 A \u2228 B\n  \u2228E  : (A \u21d2 C) \u2192 (B \u21d2 C) \u2192 A \u2228 B \u2192 C\n  \u22a5E  : \u22a5 \u2192 A\n  \u00acE : (\u00ac A \u2192 \u22a5 ) \u2192 A\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u00acE #-}\n\n-- Testing other logical constants.\npostulate\n  thm\u2081 : A \u2227 \u22a4 \u2192 A\n  thm\u2082 : A \u2228 \u22a5 \u2192 A\n  thm\u2083 : A \u2194 A\n  thm\u2084 : A \u21d4 A\n{-# ATP prove thm\u2081 #-}\n{-# ATP prove thm\u2082 #-}\n{-# ATP prove thm\u2083 #-}\n{-# ATP prove thm\u2084 #-}\n\n------------------------------------------------------------------------------\n-- Predicate logic\n\n--- The universe of discourse.\npostulate D : Set\n\n-- The propositional equality is hard-coded in our implementation, i.e. the\n-- following symbol must be used.\npostulate _\u2261_ : D \u2192 D \u2192 Set\n\n-- Testing propositional equality.\npostulate refl : \u2200 {x} \u2192 x \u2261 x\n{-# ATP prove refl #-}\n\n-- The quantifiers are hard-coded in our implementation, i.e. the\n-- following symbols must be used (it is possible to use Agda\n-- dependent function space \u2200 x \u2192 A instead of \u22c0).\npostulate\n  \u2203 : (A : D \u2192 Set) \u2192 Set\n  \u22c0 : (A : D \u2192 Set) \u2192 Set\n\n-- We postulate some formulae and propositional functions.\npostulate\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n\npostulate\n  \u2200I\u2081 : ((x : D) \u2192 A\u00b9 x) \u2192 \u22c0 A\u00b9\n  \u2200I\u2082 : ((x : D) \u2192 A\u00b9 x) \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200E\u2081 : (t : D) \u2192 \u22c0 A\u00b9 \u2192 A\u00b9 t\n  \u2200E\u2082 : (t : D) \u2192 (\u2200 x \u2192 A\u00b9 x) \u2192 A\u00b9 t\n{-# ATP prove \u2200I\u2081 #-}\n{-# ATP prove \u2200I\u2082 #-}\n{-# ATP prove \u2200E\u2081 #-}\n{-# ATP prove \u2200E\u2082 #-}\n\npostulate \u2203I : (t : D) \u2192 A\u00b9 t \u2192 \u2203 A\u00b9\n{-# ATP prove \u2203I #-}\n\npostulate \u2203E : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n{-# ATP prove \u2203E #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93039aa0a9ea6ac40d59eda3ad696a0df21829ec","subject":"Better information in the types of some postulates.","message":"Better information in the types of some postulates.\n\nIgnore-this: 8be0603121976d97c85d4ff3f9d0d846\n\ndarcs-hash:20100518143526-3bd4e-b3ea6d1272f6f65ce69c749ab34c168758fa6526.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sd\u22640 = \u22a5-elim prf\n  where\n    -- The proof uses the axiom true\u2260false\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\ntrans-LT : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\ntrans-LT zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\ntrans-LT zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\ntrans-LT zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\ntrans-LT (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\ntrans-LT (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\ntrans-LT (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (trans-LT Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf trans-LT minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sd\u22640 = \u22a5-elim prf\n  where\n    -- The proof uses the axiom true\u2260false\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\ntrans-LT : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\ntrans-LT zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\ntrans-LT zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\ntrans-LT zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\ntrans-LT (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\ntrans-LT (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\ntrans-LT (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (trans-LT Nm Nn No m<n n<o)\n\n  where\n    postulate prf : lt m o \u2261 true \u2192\n                    lt (succ m) (succ o) \u2261 true\n    {-# ATP prove prf #-}\n\n    postulate m<n : lt m n \u2261 true\n    {-# ATP prove m<n #-}\n\n    postulate n<o : lt n o \u2261 true\n    {-# ATP prove n<o #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : lt (m + n) m \u2261 false \u2192\n                    lt (succ m + n) (succ m) \u2261 false\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : lt (m - zero) (succ m) \u2261 true\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : lt (zero - succ n) (succ zero) \u2261 true\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : lt (m - n) (succ m) \u2261 true \u2192\n                    lt (succ m - succ n) (succ (succ m)) \u2261 true\n    {-# ATP prove prf trans-LT minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"280ed4b368d1f518431b29ef44401530f450bd77","subject":"\u22a4-U is uniq","message":"\u22a4-U is uniq\n","repos":"crypto-agda\/crypto-agda","old_file":"universe.agda","new_file":"universe.agda","new_contents":"module universe where\n\nimport Level as L\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen import Function\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (\u03a3; _\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ (flip _*_)\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = _ -- Agda figures out that there is only one such universe\n\n\u22a4-U-uniq : {U\u2080 U\u2081 : Universe \u22a4} \u2192 U\u2080 \u2261 U\u2081\n\u22a4-U-uniq = \u2261.refl\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U t\nfold-U u\u2080 {T} uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U P = mk (`\u22a4 , {!!}) {!!} {!!} {!!}\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","old_contents":"module universe where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (\u03a3; _\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\n-- An interface for small, finite & discrete universes of types.\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\n-- In Set-U, types are simply represented by Agda types (Set).\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\n-- In Bits-U, a type is represented by a natural number\n-- representing the width of the type in a binary representation.\n-- A natural embedding in Set is the Bits type (aka Vec Bool).\nBits-U : Universe \u2115\nBits-U = mk 0 1 _+_ (flip _*_)\n\n-- In Fin-U, a type is represented by a natural number\n-- representing the cardinality of the type.\n-- A natural embedding in Set is the Fin type.\nFin-U : Universe \u2115\nFin-U = mk 1 2 _*_ _^_\n\n-- In \u22a4-U, there is only one possible type.\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n-- Take the product of two universes. All types have two components, one from\n-- each of the forming universes.\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\n-- Sym-U is a \u201csymantic\u201d (a mix of syntax and semantics) representation\n-- for types. Symantic types are those defined only in term of the\n-- Universe interface. [See the \u201cFinally Tagless\u201d approach by Oleg Kiselyov.]\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T \u2192 T\n\n-- Abstract syntax tree from types.\ndata Ty : Set where\n  \u22a4\u2032 Bit\u2032 : Ty\n  _\u00d7\u2032_    : Ty \u2192 Ty \u2192 Ty\n  _^\u2032_    : Ty \u2192 \u2115 \u2192 Ty\n\n-- Ty-U is the universe of the syntactic represented types.\nTy-U : Universe Ty\nTy-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\n-- Turn a syntactic type into a symantic one.\n-- Alternatively:\n--   * a syntactic type is turned into a type of any $given universe.\n--   * the catamorphism for syntactic types.\nfold-U : \u2200 {t} \u2192 Ty \u2192 Sym-U t\nfold-U u\u2080 {T} uni = go u\u2080\n  where open Universe uni\n        go : Ty \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\n{-\n\u03a3-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 (P : T \u2192 Set) \u2192 Universe (\u03a3 T P)\n\u03a3-U T-U P = mk (`\u22a4 , {!!}) {!!} {!!} {!!}\n  where open Universe T-U\n-}\n\n-- The type of universe unary operators or universe transformers.\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\n-- The type of universe binary operators.\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2bfad1b0d566319d587cb13e3e068b7a6db39fa9","subject":"typo in nu","message":"typo in nu\n","repos":"crypto-agda\/protocols","old_file":"partensor\/Shallow\/Term.agda","new_file":"partensor\/Shallow\/Term.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product hiding (zip)\n                         renaming (_,_ to \u27e8_,_\u27e9; proj\u2081 to fst; proj\u2082 to snd;\n                                   map to \u00d7map)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum\nopen import Data.Nat\n{-\nopen import Data.Vec\nopen import Data.Fin\n-}\n-- open import Data.List\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP hiding ([_]; J)\nopen import partensor.Shallow.Dom\nimport partensor.Shallow.Session as Session\nimport partensor.Shallow.Map as Map\nimport partensor.Shallow.Env as Env\nimport partensor.Shallow.Proto as Proto\nopen Session hiding (Ended)\nopen Env     hiding (_\/\u2080_; _\/\u2081_; _\/_; Ended)\nopen Proto   hiding ()\nopen import partensor.Shallow.Equiv\n\nmodule partensor.Shallow.Term where\n\ndata \u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n  end :\n    Ended I\n    \u2192 \u27e8 I \u27e9\n\n  \u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I )\n      (P : \u2200 c\u2080 c\u2081 \u2192 \u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n  \u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 \u27e8 I [\/\u2026] l ,[ E\/ l , c\u2080 \u21a6 S\u2080 , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n  split :\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : \u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : \u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n  nu :\n    \u2200 {S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (P : \u2200 c\u2080 c\u2081 \u2192 \u27e8 I ,[ \u03b5 , c\u2080 \u21a6 S\u2080 , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n\n{-\n  module _ {\u03b4F}(F : Env \u03b4F) where\n    \u2297-split : \u2200 {\u03b4 \u03b4s}{I : Proto \u03b4s}{E : Env \u03b4}\n                 (l : Doms'.[ \u03b4 ]\u2208 \u03b4s)\n                 -- (l : [ E ]\u2208 I)\n                 -- {c S}(l' : c \u21a6 S \u2208 E)\n                 (P : \u27e8 I \u27e9)\n                 (\u03ba : \u2200 {c S}\n                        -- E \n                        (l' : c \u21a6 S \u2208 E)\n                        {J : Proto \u03b4s}\n                        (\u03c3s : Selections \u03b4s)\n                        (A0 : AtMost 0 \u03c3s)\n                        (P\u2080 : \u27e8 J \/\u2080 \u03c3s ,[ (E Env.\/ l' , c \u21a6 end) ] ,[ F ] \u27e9)\n                        (P\u2081 : \u27e8 J \/\u2081 \u03c3s ,[ (E Env.\/*   , c \u21a6   S) ] ,[ F ] \u27e9)\n                      \u2192 T\u27e8 J {-\/ l-} ,[ F ] \u27e9)\n              \u2192 T\u27e8 I \/ l ,[ F ] \u27e9\n    \u2297-split l (end x) \u03ba = {!!} -- elim\n    \u2297-split l (\u214b-inp l\u2081 P) \u03ba = T-\u214b-inp {!l\u2081!} {!!}\n    \u2297-split l (\u2297-out l\u2081 l\u2081' P) \u03ba = {!!}\n    \u2297-split l (split Z A1 P P\u2081) \u03ba = {!!}\n    \u2297-split l (nu D P) \u03ba = {!!}\n-}\n\ndata T\u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n T-\u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n T-\u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n T-end : \u2200 (E : Ended I) \u2192 T\u27e8 I \u27e9\n\n T-cut :\n    \u2200 {S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (\u03c3s : Selections \u03b4I)\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/\u2080 \u03c3s ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/\u2081 \u03c3s ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n T-split :\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : T\u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : T\u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\ndata TC\u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n TC-\u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 TC\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 TC\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TC\u27e8 I \u27e9\n\n TC-\u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 TC\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TC\u27e8 I \u27e9\n\n TC-end : \u2200 (E : Ended I) \u2192 TC\u27e8 I \u27e9\n\n TC-split :\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : TC\u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : TC\u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 TC\u27e8 I \u27e9\n\n\n{-\ncut : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 T\u27e8 I \u27e9 \u2192 TC\u27e8 I \u27e9\ncut (T-\u2297-out l \u03c3s \u03c3E A0 P\u2080 P\u2081) = TC-\u2297-out l \u03c3s \u03c3E A0 (\u03bb c\u2080 \u2192 cut (P\u2080 c\u2080)) (\u03bb c\u2081 \u2192 cut (P\u2081 c\u2081))\ncut (T-\u214b-inp l P) = TC-\u214b-inp l (\u03bb c\u2080 c\u2081 \u2192 cut (P c\u2080 c\u2081))\ncut (T-end E) = TC-end E\ncut (T-cut D \u03c3s A0 P\u2080 P\u2081) = {! TC-cut D \u03c3s A0 {!\u03bb c\u2080 \u2192 ? !} {!!} !}\ncut (T-split \u03c3s A1 P P\u2081) = {!!}\n\n{-\n-- no split version,\ndata TNS\u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n T-\u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n\u2190      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 TNS\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 TNS\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 S\u2081 ] \u27e9)\n   \u2192 TNS\u27e8 I \u27e9\n\n T-\u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 TNS\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TNS\u27e8 I \u27e9\n\n T-end : \u2200 (E : Ended I) \u2192 TNS\u27e8 I \u27e9\n\n T-cut :\n    \u2200 {S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (\u03c3s : Selections \u03b4I)\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 TNS\u27e8 I \/\u2080 \u03c3s ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 TNS\u27e8 I \/\u2081 \u03c3s ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TNS\u27e8 I \u27e9\n-}\n\n\n{-\ndata T\u27e8_\u27e9 {\u03b4s}(I : Proto \u03b4s) : Set\u2081 where\n  \u2297 :\n    \u2200 {c S\u2080 S\u2081}\n      (l  : [ c \u21a6 S\u2080 \u2297 S\u2081 ]\u2208 I)\n      (\u03c3s : Selections \u03b4s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/ l \/\u2080 \u03c3 ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/ l \/\u2081 \u03c3 ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n  \u214b :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 I \/ l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n  end :\n    Ended I\n    \u2192 T\u27e8 I \u27e9\n\n  cut :\n    \u2200 {S\u2080 S\u2081}\n      (\u03c3s : Selections \u03b4s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/\u2080 \u03c3 ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/\u2081 \u03c3 ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n      (D : Dual S\u2080 S\u2081)\n    \u2192 T\u27e8 I \u27e9\n\n{-\ndata T\u27e8_\u27e9 {\u03b4} (E : Env \u03b4) : Set\u2081 where\n  \u2297 :\n    \u2200 {c S\u2080 S\u2081}\n      (l  : c \u21a6 (S\u2080 \u2297 S\u2081) \u2208 E)\n      (\u03c3  : Selection \u03b4)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 (E Env.\/ l) \/\u2080 \u03c3 , c\u2080 \u21a6 S\u2080 \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 (E Env.\/ l) \/\u2081 \u03c3 , c\u2081 \u21a6 S\u2081 \u27e9)\n    \u2192 T\u27e8 E \u27e9\n\n  \u214b :\n    \u2200 {c S\u2080 S\u2081}\n      (l : c \u21a6 (S\u2080 \u214b S\u2081) \u2208 E)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 (E \/ l) , c\u2080 \u21a6 S\u2080 , c\u2081 \u21a6 S\u2081 \u27e9)\n    \u2192 T\u27e8 E \u27e9\n\n  end :\n    Env.Ended E\n    \u2192 T\u27e8 E \u27e9\n\n  cut :\n    \u2200 {S\u2080 S\u2081}\n      (\u03c3  : Selection \u03b4)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 E \/\u2080 \u03c3 , c\u2080 \u21a6 S\u2080 \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 E \/\u2081 \u03c3 , c\u2081 \u21a6 S\u2081 \u27e9)\n      (D : Dual S\u2080 S\u2081)\n    \u2192 T\u27e8 E \u27e9\n\n{-\ndata Env : Dom \u2192 Set\u2081 where\n  \u03b5     : Env \u03b5\n  _,_\u21a6_ : \u2200 {\u03b4} (E : Env \u03b4) c (S : Session) \u2192 Env (\u03b4 , c)\n-}\n\n{-\n\n{-\n\n{-\nDom = \u2115\nEnv : Dom \u2192 Set\u2081\nEnv n = Vec Session n\n\nLayout : Dom \u2192 Set\nLayout n = Vec \ud835\udfda n\n\nprojLayout : \u2200 {n} (f : Session \u2192 \ud835\udfda \u2192 Session) (\u0394 : Env n) \u2192 Layout n \u2192 Env n\nprojLayout f = zipWith f\n\n_\/\u2080_ : \u2200 {n} (\u0394 : Env n) \u2192 Layout n \u2192 Env n\n_\/\u2080_ = projLayout (\u03bb S \u2192 [0: S 1: end ])\n\n_\/\u2081_ : \u2200 {n} (\u0394 : Env n) \u2192 Layout n \u2192 Env n\n_\/\u2081_ = projLayout (\u03bb S \u2192 [0: end 1: S ])\n\ndata \u27e8_\u27e9 {n} (\u0394 : Env n) : Set\u2081 where\n  \u2297 :\n    \u2200 {c S\u2080 S\u2081}\n      (l  : \u0394 [ c ]= S\u2080 \u2297 S\u2081)\n      (L  : Layout n)\n      (P\u2080 : \u27e8 \u0394 \/\u2080 L [ c ]\u2254 S\u2080 \u27e9)\n      (P\u2081 : \u27e8 \u0394 \/\u2081 L [ c ]\u2254 S\u2081 \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n  \u214b :\n    \u2200 {c S\u2080 S\u2081}\n      (l : \u0394 [ c ]= S\u2080 \u214b S\u2081)\n      (P : \u0394 [ c ]=* (S\u2080 \u2237 S\u2081 \u2237 []))\n    \u2192 \u27e8 \u0394 \u27e9\n    -}\n\n{-\ndata Layout : Env \u2192 Set where\n  \u03b5 : Layout \u03b5\n  _,_\u21a6_ : \u2200 {\u03c3} (L : Layout \u03c3) c {S} (b : \ud835\udfda) \u2192 Layout (\u03c3 , c \u21a6 S)\n\nprojLayout : (f : \ud835\udfda \u2192 Session \u2192 Session) (\u0394 : Env) \u2192 Layout \u0394 \u2192 Env\nprojLayout f \u03b5 \u03b5 = \u03b5\nprojLayout f (\u0394 , .c \u21a6 S) (L , c \u21a6 b) = projLayout f \u0394 L , c \u21a6 f b S\n\n_\/\u2080_ : (\u0394 : Env) \u2192 Layout \u0394 \u2192 Env\n_\/\u2080_ = projLayout (\u03bb b S \u2192 [0: S 1: end ] b)\n\n_\/\u2081_ : (\u0394 : Env) \u2192 Layout \u0394 \u2192 Env\n_\/\u2081_ = projLayout (\u03bb b S \u2192 [0: end 1: S ] b)\n\ndata \u27e8_\u27e9 (\u0394 : Env) : Set\u2081 where\n{-\n  \u214b-in :\n    \u2200 {c S T}\n      (l : [ c \u21a6 (\u214b S T) ]\u2208 \u0394 )\n      (P : \u2200 d e \u2192 \u27e8 \u0394 [ l \u2254* \u00b7 ,[ d \u21a6 S ] ,[ e \u21a6 T ] ] \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n-}\n  \u2297-out :\n    \u2200 {c S T}\n      (l : c \u21a6 \u2297 S T \u2208 \u0394)\n      (L : Layout \n      (P : \u2200 d \u2192 \u27e8 \u0394 [ l \u2254 S ] \/\u2080 L \u27e9)\n      (Q : \u2200 e \u2192 \u27e8 \u0394 [ l \u2254 T ] \/\u2081 L \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n\n{-\n  split :\n    \u2200 {\u0394\u2080 \u0394\u2081}\n    \u2192 ZipP 1 \u0394 \u0394\u2080 \u0394\u2081 \u2192 \u27e8 \u0394\u2080 \u27e9 \u2192 \u27e8 \u0394\u2081 \u27e9 \u2192 \u27e8 \u0394 \u27e9\n\n  end :\n    EndedProto \u0394\n    \u2192 \u27e8 \u0394 \u27e9\n\n  nu : \u2200 {\u0394' S T}\n       \u2192 (l : []\u2208 \u0394')\n       \u2192 (\u2200 c d \u2192 \u27e8 \u0394' [ l \u2254* (\u00b7 ,[ c \u21a6 S ] ,[ d \u21a6 T ]) ]' \u27e9)\n       \u2192 Dual S T\n       \u2192 \u0394 \u2261 \u0394' [ l \u2254* \u00b7 ]'\n       \u2192 \u27e8 \u0394 \u27e9\n\n{-\ndata \u27e8_\u27e9 (\u0394 : Proto) : Set\u2081 where\n  \u214b-in :\n    \u2200 {c S T}\n      (l : [ c \u21a6 (\u214b S T) ]\u2208 \u0394 )\n      (P : \u2200 d e \u2192 \u27e8 \u0394 [ l \u2254* \u00b7 ,[ d \u21a6 S ] ,[ e \u21a6 T ] ] \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n\n  \u2297-out :\n    \u2200 {c S T \u03b7}\n      (l : [ \u03b7 ]\u2208 \u0394)\n      (l' : c \u21a6 \u2297 S T \u2208 \u03b7)\n      (P : \u2200 d e \u2192 \u27e8 \u0394 [ l \u2254* \u00b7 ,[ \u03b7 [ l' += \u03b5 , d \u21a6 S , e \u21a6 T ]\u03b7 ] ] \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n\n  split :\n    \u2200 {\u0394\u2080 \u0394\u2081}\n    \u2192 ZipP 1 \u0394 \u0394\u2080 \u0394\u2081 \u2192 \u27e8 \u0394\u2080 \u27e9 \u2192 \u27e8 \u0394\u2081 \u27e9 \u2192 \u27e8 \u0394 \u27e9\n\n  end :\n    EndedProto \u0394\n    \u2192 \u27e8 \u0394 \u27e9\n\n  nu : \u2200 {\u0394' S T}\n       \u2192 (l : []\u2208 \u0394')\n       \u2192 (\u2200 c d \u2192 \u27e8 \u0394' [ l \u2254* (\u00b7 ,[ c \u21a6 S ] ,[ d \u21a6 T ]) ]' \u27e9)\n       \u2192 Dual S T\n       \u2192 \u0394 \u2261 \u0394' [ l \u2254* \u00b7 ]'\n       \u2192 \u27e8 \u0394 \u27e9\n\n\n{-\n-- cut\u2081 : \u2200 {\u0394 \u0394' S}(l : S \u2208 \u0394)(l' : dual S \u2208 \u0394') \u2192 \u27e8 \u0394 \u27e9 \u2192 \u27e8 \u0394' \u27e9 \u2192 \u27e8 \u0394 [ l \u2254 end ] \u214b \u0394' [ l' \u2254 end ] \u27e9\ncut\u2081 : \u2200 {\u0394 \u0394' S}(l : S \u2208 \u0394)(l' : dual S \u2208 \u0394')\n      \u2192 \u27e8 \u0394 \u27e9 \u2192 \u27e8 \u0394' \u27e9 \u2192 \u27e8 \u0394 [ l \u2254 end ] \u214b \u0394' [ l' \u2254 end ] \u27e9\ncut\u2081\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product hiding (zip)\n                         renaming (_,_ to \u27e8_,_\u27e9; proj\u2081 to fst; proj\u2082 to snd;\n                                   map to \u00d7map)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum\nopen import Data.Nat\n{-\nopen import Data.Vec\nopen import Data.Fin\n-}\n-- open import Data.List\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP hiding ([_]; J)\nopen import partensor.Shallow.Dom\nimport partensor.Shallow.Session as Session\nimport partensor.Shallow.Map as Map\nimport partensor.Shallow.Env as Env\nimport partensor.Shallow.Proto as Proto\nopen Session hiding (Ended)\nopen Env     hiding (_\/\u2080_; _\/\u2081_; _\/_; Ended)\nopen Proto   hiding ()\nopen import partensor.Shallow.Equiv\n\nmodule partensor.Shallow.Term where\n\ndata \u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n  end :\n    Ended I\n    \u2192 \u27e8 I \u27e9\n\n  \u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I )\n      (P : \u2200 c\u2080 c\u2081 \u2192 \u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n  \u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 \u27e8 I [\/\u2026] l ,[ E\/ l , c\u2080 \u21a6 S\u2080 , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n  split :\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : \u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : \u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n  nu :\n    \u2200 {S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (P : \u2200 c\u2080 c\u2081 \u2192 \u27e8 I ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 \u27e8 I \u27e9\n\n\n{-\n  module _ {\u03b4F}(F : Env \u03b4F) where\n    \u2297-split : \u2200 {\u03b4 \u03b4s}{I : Proto \u03b4s}{E : Env \u03b4}\n                 (l : Doms'.[ \u03b4 ]\u2208 \u03b4s)\n                 -- (l : [ E ]\u2208 I)\n                 -- {c S}(l' : c \u21a6 S \u2208 E)\n                 (P : \u27e8 I \u27e9)\n                 (\u03ba : \u2200 {c S}\n                        -- E \n                        (l' : c \u21a6 S \u2208 E)\n                        {J : Proto \u03b4s}\n                        (\u03c3s : Selections \u03b4s)\n                        (A0 : AtMost 0 \u03c3s)\n                        (P\u2080 : \u27e8 J \/\u2080 \u03c3s ,[ (E Env.\/ l' , c \u21a6 end) ] ,[ F ] \u27e9)\n                        (P\u2081 : \u27e8 J \/\u2081 \u03c3s ,[ (E Env.\/*   , c \u21a6   S) ] ,[ F ] \u27e9)\n                      \u2192 T\u27e8 J {-\/ l-} ,[ F ] \u27e9)\n              \u2192 T\u27e8 I \/ l ,[ F ] \u27e9\n    \u2297-split l (end x) \u03ba = {!!} -- elim\n    \u2297-split l (\u214b-inp l\u2081 P) \u03ba = T-\u214b-inp {!l\u2081!} {!!}\n    \u2297-split l (\u2297-out l\u2081 l\u2081' P) \u03ba = {!!}\n    \u2297-split l (split Z A1 P P\u2081) \u03ba = {!!}\n    \u2297-split l (nu D P) \u03ba = {!!}\n-}\n\ndata T\u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n T-\u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n T-\u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n T-end : \u2200 (E : Ended I) \u2192 T\u27e8 I \u27e9\n\n T-cut :\n    \u2200 {S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (\u03c3s : Selections \u03b4I)\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/\u2080 \u03c3s ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/\u2081 \u03c3s ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n T-split :\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : T\u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : T\u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\ndata TC\u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n TC-\u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 TC\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 TC\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TC\u27e8 I \u27e9\n\n TC-\u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 TC\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TC\u27e8 I \u27e9\n\n TC-end : \u2200 (E : Ended I) \u2192 TC\u27e8 I \u27e9\n\n TC-split :\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : TC\u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : TC\u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 TC\u27e8 I \u27e9\n\n\n{-\ncut : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 T\u27e8 I \u27e9 \u2192 TC\u27e8 I \u27e9\ncut (T-\u2297-out l \u03c3s \u03c3E A0 P\u2080 P\u2081) = TC-\u2297-out l \u03c3s \u03c3E A0 (\u03bb c\u2080 \u2192 cut (P\u2080 c\u2080)) (\u03bb c\u2081 \u2192 cut (P\u2081 c\u2081))\ncut (T-\u214b-inp l P) = TC-\u214b-inp l (\u03bb c\u2080 c\u2081 \u2192 cut (P c\u2080 c\u2081))\ncut (T-end E) = TC-end E\ncut (T-cut D \u03c3s A0 P\u2080 P\u2081) = {! TC-cut D \u03c3s A0 {!\u03bb c\u2080 \u2192 ? !} {!!} !}\ncut (T-split \u03c3s A1 P P\u2081) = {!!}\n\n{-\n-- no split version,\ndata TNS\u27e8_\u27e9 {\u03b4I}(I : Proto \u03b4I) : Set\u2081 where\n T-\u2297-out :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n\u2190      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 TNS\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 TNS\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 S\u2081 ] \u27e9)\n   \u2192 TNS\u27e8 I \u27e9\n\n T-\u214b-inp :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 TNS\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TNS\u27e8 I \u27e9\n\n T-end : \u2200 (E : Ended I) \u2192 TNS\u27e8 I \u27e9\n\n T-cut :\n    \u2200 {S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (\u03c3s : Selections \u03b4I)\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 TNS\u27e8 I \/\u2080 \u03c3s ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 TNS\u27e8 I \/\u2081 \u03c3s ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 TNS\u27e8 I \u27e9\n-}\n\n\n{-\ndata T\u27e8_\u27e9 {\u03b4s}(I : Proto \u03b4s) : Set\u2081 where\n  \u2297 :\n    \u2200 {c S\u2080 S\u2081}\n      (l  : [ c \u21a6 S\u2080 \u2297 S\u2081 ]\u2208 I)\n      (\u03c3s : Selections \u03b4s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/ l \/\u2080 \u03c3 ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/ l \/\u2081 \u03c3 ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n  \u214b :\n    \u2200 {c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 I \/ l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9\n\n  end :\n    Ended I\n    \u2192 T\u27e8 I \u27e9\n\n  cut :\n    \u2200 {S\u2080 S\u2081}\n      (\u03c3s : Selections \u03b4s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/\u2080 \u03c3 ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/\u2081 \u03c3 ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n      (D : Dual S\u2080 S\u2081)\n    \u2192 T\u27e8 I \u27e9\n\n{-\ndata T\u27e8_\u27e9 {\u03b4} (E : Env \u03b4) : Set\u2081 where\n  \u2297 :\n    \u2200 {c S\u2080 S\u2081}\n      (l  : c \u21a6 (S\u2080 \u2297 S\u2081) \u2208 E)\n      (\u03c3  : Selection \u03b4)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 (E Env.\/ l) \/\u2080 \u03c3 , c\u2080 \u21a6 S\u2080 \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 (E Env.\/ l) \/\u2081 \u03c3 , c\u2081 \u21a6 S\u2081 \u27e9)\n    \u2192 T\u27e8 E \u27e9\n\n  \u214b :\n    \u2200 {c S\u2080 S\u2081}\n      (l : c \u21a6 (S\u2080 \u214b S\u2081) \u2208 E)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 (E \/ l) , c\u2080 \u21a6 S\u2080 , c\u2081 \u21a6 S\u2081 \u27e9)\n    \u2192 T\u27e8 E \u27e9\n\n  end :\n    Env.Ended E\n    \u2192 T\u27e8 E \u27e9\n\n  cut :\n    \u2200 {S\u2080 S\u2081}\n      (\u03c3  : Selection \u03b4)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 E \/\u2080 \u03c3 , c\u2080 \u21a6 S\u2080 \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 E \/\u2081 \u03c3 , c\u2081 \u21a6 S\u2081 \u27e9)\n      (D : Dual S\u2080 S\u2081)\n    \u2192 T\u27e8 E \u27e9\n\n{-\ndata Env : Dom \u2192 Set\u2081 where\n  \u03b5     : Env \u03b5\n  _,_\u21a6_ : \u2200 {\u03b4} (E : Env \u03b4) c (S : Session) \u2192 Env (\u03b4 , c)\n-}\n\n{-\n\n{-\n\n{-\nDom = \u2115\nEnv : Dom \u2192 Set\u2081\nEnv n = Vec Session n\n\nLayout : Dom \u2192 Set\nLayout n = Vec \ud835\udfda n\n\nprojLayout : \u2200 {n} (f : Session \u2192 \ud835\udfda \u2192 Session) (\u0394 : Env n) \u2192 Layout n \u2192 Env n\nprojLayout f = zipWith f\n\n_\/\u2080_ : \u2200 {n} (\u0394 : Env n) \u2192 Layout n \u2192 Env n\n_\/\u2080_ = projLayout (\u03bb S \u2192 [0: S 1: end ])\n\n_\/\u2081_ : \u2200 {n} (\u0394 : Env n) \u2192 Layout n \u2192 Env n\n_\/\u2081_ = projLayout (\u03bb S \u2192 [0: end 1: S ])\n\ndata \u27e8_\u27e9 {n} (\u0394 : Env n) : Set\u2081 where\n  \u2297 :\n    \u2200 {c S\u2080 S\u2081}\n      (l  : \u0394 [ c ]= S\u2080 \u2297 S\u2081)\n      (L  : Layout n)\n      (P\u2080 : \u27e8 \u0394 \/\u2080 L [ c ]\u2254 S\u2080 \u27e9)\n      (P\u2081 : \u27e8 \u0394 \/\u2081 L [ c ]\u2254 S\u2081 \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n  \u214b :\n    \u2200 {c S\u2080 S\u2081}\n      (l : \u0394 [ c ]= S\u2080 \u214b S\u2081)\n      (P : \u0394 [ c ]=* (S\u2080 \u2237 S\u2081 \u2237 []))\n    \u2192 \u27e8 \u0394 \u27e9\n    -}\n\n{-\ndata Layout : Env \u2192 Set where\n  \u03b5 : Layout \u03b5\n  _,_\u21a6_ : \u2200 {\u03c3} (L : Layout \u03c3) c {S} (b : \ud835\udfda) \u2192 Layout (\u03c3 , c \u21a6 S)\n\nprojLayout : (f : \ud835\udfda \u2192 Session \u2192 Session) (\u0394 : Env) \u2192 Layout \u0394 \u2192 Env\nprojLayout f \u03b5 \u03b5 = \u03b5\nprojLayout f (\u0394 , .c \u21a6 S) (L , c \u21a6 b) = projLayout f \u0394 L , c \u21a6 f b S\n\n_\/\u2080_ : (\u0394 : Env) \u2192 Layout \u0394 \u2192 Env\n_\/\u2080_ = projLayout (\u03bb b S \u2192 [0: S 1: end ] b)\n\n_\/\u2081_ : (\u0394 : Env) \u2192 Layout \u0394 \u2192 Env\n_\/\u2081_ = projLayout (\u03bb b S \u2192 [0: end 1: S ] b)\n\ndata \u27e8_\u27e9 (\u0394 : Env) : Set\u2081 where\n{-\n  \u214b-in :\n    \u2200 {c S T}\n      (l : [ c \u21a6 (\u214b S T) ]\u2208 \u0394 )\n      (P : \u2200 d e \u2192 \u27e8 \u0394 [ l \u2254* \u00b7 ,[ d \u21a6 S ] ,[ e \u21a6 T ] ] \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n-}\n  \u2297-out :\n    \u2200 {c S T}\n      (l : c \u21a6 \u2297 S T \u2208 \u0394)\n      (L : Layout \n      (P : \u2200 d \u2192 \u27e8 \u0394 [ l \u2254 S ] \/\u2080 L \u27e9)\n      (Q : \u2200 e \u2192 \u27e8 \u0394 [ l \u2254 T ] \/\u2081 L \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n\n{-\n  split :\n    \u2200 {\u0394\u2080 \u0394\u2081}\n    \u2192 ZipP 1 \u0394 \u0394\u2080 \u0394\u2081 \u2192 \u27e8 \u0394\u2080 \u27e9 \u2192 \u27e8 \u0394\u2081 \u27e9 \u2192 \u27e8 \u0394 \u27e9\n\n  end :\n    EndedProto \u0394\n    \u2192 \u27e8 \u0394 \u27e9\n\n  nu : \u2200 {\u0394' S T}\n       \u2192 (l : []\u2208 \u0394')\n       \u2192 (\u2200 c d \u2192 \u27e8 \u0394' [ l \u2254* (\u00b7 ,[ c \u21a6 S ] ,[ d \u21a6 T ]) ]' \u27e9)\n       \u2192 Dual S T\n       \u2192 \u0394 \u2261 \u0394' [ l \u2254* \u00b7 ]'\n       \u2192 \u27e8 \u0394 \u27e9\n\n{-\ndata \u27e8_\u27e9 (\u0394 : Proto) : Set\u2081 where\n  \u214b-in :\n    \u2200 {c S T}\n      (l : [ c \u21a6 (\u214b S T) ]\u2208 \u0394 )\n      (P : \u2200 d e \u2192 \u27e8 \u0394 [ l \u2254* \u00b7 ,[ d \u21a6 S ] ,[ e \u21a6 T ] ] \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n\n  \u2297-out :\n    \u2200 {c S T \u03b7}\n      (l : [ \u03b7 ]\u2208 \u0394)\n      (l' : c \u21a6 \u2297 S T \u2208 \u03b7)\n      (P : \u2200 d e \u2192 \u27e8 \u0394 [ l \u2254* \u00b7 ,[ \u03b7 [ l' += \u03b5 , d \u21a6 S , e \u21a6 T ]\u03b7 ] ] \u27e9)\n    \u2192 \u27e8 \u0394 \u27e9\n\n  split :\n    \u2200 {\u0394\u2080 \u0394\u2081}\n    \u2192 ZipP 1 \u0394 \u0394\u2080 \u0394\u2081 \u2192 \u27e8 \u0394\u2080 \u27e9 \u2192 \u27e8 \u0394\u2081 \u27e9 \u2192 \u27e8 \u0394 \u27e9\n\n  end :\n    EndedProto \u0394\n    \u2192 \u27e8 \u0394 \u27e9\n\n  nu : \u2200 {\u0394' S T}\n       \u2192 (l : []\u2208 \u0394')\n       \u2192 (\u2200 c d \u2192 \u27e8 \u0394' [ l \u2254* (\u00b7 ,[ c \u21a6 S ] ,[ d \u21a6 T ]) ]' \u27e9)\n       \u2192 Dual S T\n       \u2192 \u0394 \u2261 \u0394' [ l \u2254* \u00b7 ]'\n       \u2192 \u27e8 \u0394 \u27e9\n\n\n{-\n-- cut\u2081 : \u2200 {\u0394 \u0394' S}(l : S \u2208 \u0394)(l' : dual S \u2208 \u0394') \u2192 \u27e8 \u0394 \u27e9 \u2192 \u27e8 \u0394' \u27e9 \u2192 \u27e8 \u0394 [ l \u2254 end ] \u214b \u0394' [ l' \u2254 end ] \u27e9\ncut\u2081 : \u2200 {\u0394 \u0394' S}(l : S \u2208 \u0394)(l' : dual S \u2208 \u0394')\n      \u2192 \u27e8 \u0394 \u27e9 \u2192 \u27e8 \u0394' \u27e9 \u2192 \u27e8 \u0394 [ l \u2254 end ] \u214b \u0394' [ l' \u2254 end ] \u27e9\ncut\u2081\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c95e1205e145e5df19df5cd214f280b1cb28dee4","subject":"Drop more dead code","message":"Drop more dead code\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Failed variants, the step-indexes are wrong:\n-- dapply-equiv-2 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc n)) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-2 (suc n) t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 (suc zero) (const c) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (var x) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (app t t\u2081) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (abs t) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc (suc n)) t d\u03c1 \u03c11 da a1 = refl\n\n-- Worse:\n-- dapply-equiv-0 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-0 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc zero) t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc (suc n)) t d\u03c1 \u03c11 da a1 = {!!}\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c0c2b55b3414f4e873b25c2539eddd1223ce50c4","subject":"Changed doc.","message":"Changed doc.\n\nIgnore-this: 366baccf864db612a0e4c1ad19b9e50f\n\ndarcs-hash:20101102155907-3bd4e-75efa3d253c4d85ec08a05b10b7573ff46f349fe.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Lib\/Relation\/Binary\/EqReasoning.agda","new_file":"src\/Lib\/Relation\/Binary\/EqReasoning.agda","new_contents":"-----------------------------------------------------------------------------\n-- Equality reasoning\n-----------------------------------------------------------------------------\n\nmodule Lib.Relation.Binary.EqReasoning\n  {A     : Set}\n  ( _\u2261_  : A \u2192 A \u2192 Set)\n  (refl  : {x : A} \u2192 x \u2261 x)\n  (trans : {x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n  where\n\ninfix 4 _\u2243_\n\nprivate\n  data _\u2243_ (x y : A) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\n------------------------------------------------------------------------------\n-- From the Ulf's thesis.\n\nmodule Original where\n\n  infix  2 chain>_\n  infixl 2 _===_by_\n  infix  1 _qed\n\n  chain>_ : (x : A) \u2192 x \u2243 x\n  chain> x = prf (refl {x})\n\n  _===_by_ : {x y : A} \u2192 x \u2243 y \u2192 (z : A) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\n------------------------------------------------------------------------------\n-- From the standard library (Relation.Binary.PreorderReasoning).\n\nmodule StdLib where\n\n  infix  1 begin_\n  infixr 2 _\u2261\u27e8_\u27e9_\n  infix  2 _\u220e\n\n  begin_ : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : A){y z : A} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : A) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n","old_contents":"-----------------------------------------------------------------------------\n-- Equality reasoning\n-----------------------------------------------------------------------------\n\nmodule Lib.Relation.Binary.EqReasoning\n  {A     : Set}\n  ( _\u2261_  : A \u2192 A \u2192 Set)\n  (refl  : {x : A} \u2192 x \u2261 x)\n  (trans : {x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n  where\n\ninfix 4 _\u2243_\n\nprivate\n  data _\u2243_ (x y : A) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\n------------------------------------------------------------------------------\n-- A version from Ulf's thesis.\nmodule Original where\n\n  infix  2 chain>_\n  infixl 2 _===_by_\n  infix  1 _qed\n\n  chain>_ : (x : A) \u2192 x \u2243 x\n  chain> x = prf (refl {x})\n\n  _===_by_ : {x y : A} \u2192 x \u2243 y \u2192 (z : A) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\n------------------------------------------------------------------------------\n-- A version from the standard library (Relation.Binary.PreorderReasoning).\nmodule StdLib where\n\n  infix  1 begin_\n  infixr 2 _\u2261\u27e8_\u27e9_\n  infix  2 _\u220e\n\n  begin_ : {x y : A} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : A){y z : A} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : A) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3fe4f896e2d6561d531630dd8f4a46f8c362b9ab","subject":"add trace to Choreography","message":"add trace to Choreography\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5bd1cb31b6ff25934b1212da8538587845a850b9","subject":"\u2261: Hide subst, expose tr...","message":"\u2261: Hide subst, expose tr...\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Type hiding (\u2605)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product using (\u03a3; _,_)\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning; subst)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\nprivate\n  module Dummy {a} {A : \u2605 a} where\n    open IsEquivalence (isEquivalence {a} {A}) public hiding (refl; sym; trans)\nopen Dummy public\n\nPathOver : \u2200 {i j} {A : \u2605 i} (B : A \u2192 \u2605 j)\n  {x y : A} (p : x \u2261 y) (u : B x) (v : B y) \u2192 \u2605 j\nPathOver B refl u v = (u \u2261 v)\n\nsyntax PathOver B p u v =\n  u \u2261 v [ B \u2193 p ]\n\n{- one day this will work -}\n-- pattern idp = refl\n\n-- Some definitions with the names from Agda-HoTT\n-- this is only a temporary state of affairs...\nmodule _ {a} {A : \u2605 a} where\n\n    idp : {x : A} \u2192 x \u2261 x\n    idp = refl\n\n    infixr 8 _\u2219_ _\u2219'_\n\n    _\u2219_ _\u2219'_ : {x y z : A} \u2192 (x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n\n    refl \u2219 q  = q\n    q \u2219' refl = q\n\n    !_ : {x y : A} \u2192 (x \u2261 y \u2192 y \u2261 x)\n    !_ refl = refl\n\nap\u2193 : \u2200 {i j k} {A : \u2605 i} {B : A \u2192 \u2605 j} {C : A \u2192 \u2605 k}\n  (g : {a : A} \u2192 B a \u2192 C a) {x y : A} {p : x \u2261 y}\n  {u : B x} {v : B y}\n  \u2192 (u \u2261 v [ B \u2193 p ] \u2192 g u \u2261 g v [ C \u2193 p ])\nap\u2193 g {p = refl} refl = refl\n\nap : \u2200 {i j} {A : \u2605 i} {B : \u2605 j} (f : A \u2192 B) {x y : A}\n  \u2192 (x \u2261 y \u2192 f x \u2261 f y)\nap f p = ap\u2193 {A = \ud835\udfd9} f {p = refl} p\n\napd : \u2200 {i j} {A : \u2605 i} {B : A \u2192 \u2605 j} (f : (a : A) \u2192 B a) {x y : A}\n  \u2192 (p : x \u2261 y) \u2192 f x \u2261 f y [ B \u2193 p ]\napd f refl = refl\n\ncoe : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 A \u2192 B\ncoe refl x = x\n\ncoe! : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 B \u2192 A\ncoe! p = coe (! p)\n\nmodule _ {\u2113 \u2113p}\n         {A : Set \u2113}\n         (P : A \u2192 Set \u2113p)\n         {x y : A}\n         (p : x \u2261 y)\n         where\n    tr : P x \u2192 P y\n    tr = coe (ap P p)\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2080 a\u2081 b\u2080 b\u2081 c\u2080 c\u2081}\n        \u2192 a\u2080 \u2261 a\u2081 \u2192 b\u2080 \u2261 b\u2081 \u2192 c\u2080 \u2261 c\u2081 \u2192 f a\u2080 b\u2080 c\u2080 \u2261 f a\u2081 b\u2081 c\u2081\ncong\u2083 f refl refl refl = refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} (f g : A \u2192 B \u2192 C) \u2192 \u2605 _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\nmodule _ {a} {A : \u2605 a} where\n  J-orig : \u2200 {b} (P : (x y : A) \u2192 x \u2261 y \u2192 \u2605_ b) \u2192 (\u2200 x \u2192 P x x refl) \u2192 \u2200 {x y} (p : x \u2261 y) \u2192 P x y p\n  J-orig P p refl = p _\n\n  -- This version is better suited to our identity type which has the first argument as a parameter.\n  -- (due to Paulin-Mohring)\n  J : \u2200 {b} {x : A} (P : (y : A) \u2192 x \u2261 y \u2192 \u2605_ b) \u2192 P x refl \u2192 \u2200 {y} (p : x \u2261 y) \u2192 P y p\n  J P p refl = p\n\n  injective : InjectiveRel A _\u2261_\n  injective p q = p \u2219 ! q\n\n  surjective : SurjectiveRel A _\u2261_\n  surjective p q = ! p \u2219 q\n\n  bijective : BijectiveRel A _\u2261_\n  bijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\n  Equalizer : \u2200 {b} {B : A \u2192 \u2605 b} (f g : (x : A) \u2192 B x) \u2192 \u2605 _\n  Equalizer f g = \u03a3 A (\u03bb x \u2192 f x \u2261 g x)\n  {- In a category theory context this type would the object 'E'\n     and 'proj\u2081' would be the morphism 'eq : E \u2192 A' such that\n     given any object O, and morphism 'm : O \u2192 A', there exists\n     a unique morphism 'u : O \u2192 E' such that 'eq \u2218 u \u2261 m'.\n  -}\n\n  Pullback : \u2200 {b c} {B : \u2605 b} {C : \u2605 c} (f : A \u2192 C) (g : B \u2192 C) \u2192 \u2605 _\n  Pullback {B = B} f g = \u03a3 A (\u03bb x \u2192 \u03a3 B (\u03bb y \u2192 f x \u2261 g y))\n\nmodule \u2261-Reasoning {a} {A : \u2605 a} where\n  open Setoid-Reasoning (setoid A) public\n    renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_;\n                    _\u2248\u27e8by-definition\u27e9_ to _\u2261\u27e8by-definition\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : \u2605 a} {B : \u2605 b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public\n    renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_;\n                    _\u2248\u27e8by-definition\u27e9_ to _\u2257\u27e8by-definition\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2080 a\u2081 a\u1d63}\n         {A\u2080 A\u2081} (A\u1d63 : \u27e6\u2605\u27e7 {a\u2080} {a\u2081} a\u1d63 A\u2080 A\u2081)\n         {x\u2080 x\u2081} (x\u1d63 : A\u1d63 x\u2080 x\u2081)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a\u1d63) (_\u2261_ x\u2080) (_\u2261_ x\u2081) where\n    -- : \u2200 {y\u2080 y\u2081} (y\u1d63 : A\u1d63 y\u2080 y\u2081) \u2192 x\u2080 \u2261 y\u2080 \u2192 x\u2081 \u2261 y\u2081 \u2192 \u2605\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","old_contents":"{-# OPTIONS --without-K #-}\n-- move this to propeq\nmodule Relation.Binary.PropositionalEquality.NP where\n\nopen import Type hiding (\u2605)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product using (\u03a3; _,_)\nopen import Relation.Binary.PropositionalEquality public hiding (module \u2261-Reasoning)\nopen import Relation.Binary.NP\nopen import Relation.Binary.Bijection\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\n\nprivate\n  module Dummy {a} {A : \u2605 a} where\n    open IsEquivalence (isEquivalence {a} {A}) public hiding (refl; sym; trans)\nopen Dummy public\n\nPathOver : \u2200 {i j} {A : \u2605 i} (B : A \u2192 \u2605 j)\n  {x y : A} (p : x \u2261 y) (u : B x) (v : B y) \u2192 \u2605 j\nPathOver B refl u v = (u \u2261 v)\n\nsyntax PathOver B p u v =\n  u \u2261 v [ B \u2193 p ]\n\n{- one day this will work -}\n-- pattern idp = refl\n\n-- Some definitions with the names from Agda-HoTT\n-- this is only a temporary state of affairs...\nmodule _ {a} {A : \u2605 a} where\n\n    idp : {x : A} \u2192 x \u2261 x\n    idp = refl\n\n    infixr 8 _\u2219_ _\u2219'_\n\n    _\u2219_ _\u2219'_ : {x y z : A} \u2192 (x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z)\n\n    refl \u2219 q  = q\n    q \u2219' refl = q\n\n    !_ : {x y : A} \u2192 (x \u2261 y \u2192 y \u2261 x)\n    !_ refl = refl\n\nap\u2193 : \u2200 {i j k} {A : \u2605 i} {B : A \u2192 \u2605 j} {C : A \u2192 \u2605 k}\n  (g : {a : A} \u2192 B a \u2192 C a) {x y : A} {p : x \u2261 y}\n  {u : B x} {v : B y}\n  \u2192 (u \u2261 v [ B \u2193 p ] \u2192 g u \u2261 g v [ C \u2193 p ])\nap\u2193 g {p = refl} refl = refl\n\nap : \u2200 {i j} {A : \u2605 i} {B : \u2605 j} (f : A \u2192 B) {x y : A}\n  \u2192 (x \u2261 y \u2192 f x \u2261 f y)\nap f p = ap\u2193 {A = \ud835\udfd9} f {p = refl} p\n\napd : \u2200 {i j} {A : \u2605 i} {B : A \u2192 \u2605 j} (f : (a : A) \u2192 B a) {x y : A}\n  \u2192 (p : x \u2261 y) \u2192 f x \u2261 f y [ B \u2193 p ]\napd f refl = refl\n\ncoe : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 A \u2192 B\ncoe refl x = x\n\ncoe! : \u2200 {i} {A B : \u2605 i} (p : A \u2261 B) \u2192 B \u2192 A\ncoe! p x = coe (! p) x\n\ncong\u2083 : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n          (f : A \u2192 B \u2192 C \u2192 D) {a\u2081 a\u2082 b\u2081 b\u2082 c\u2081 c\u2082}\n        \u2192 a\u2081 \u2261 a\u2082 \u2192 b\u2081 \u2261 b\u2082 \u2192 c\u2081 \u2261 c\u2082 \u2192 f a\u2081 b\u2081 c\u2081 \u2261 f a\u2082 b\u2082 c\u2082\ncong\u2083 f refl refl refl = refl\n\n_\u2257\u2082_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} (f g : A \u2192 B \u2192 C) \u2192 \u2605 _\nf \u2257\u2082 g = \u2200 x y \u2192 f x y \u2261 g x y\n\nmodule _ {a} {A : \u2605 a} where\n  J-orig : \u2200 {b} (P : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ b) \u2192 (\u2200 {x} \u2192 P {x} {x} refl) \u2192 \u2200 {x y} (p : x \u2261 y) \u2192 P p\n  J-orig P p refl = p\n\n  -- This version is better suited to our identity type which has the first argument as a parameter.\n  -- (due to Paulin-Mohring)\n  J : \u2200 {b} {x : A} (P : {y : A} \u2192 x \u2261 y \u2192 \u2605_ b) \u2192 P {x} refl \u2192 \u2200 {y} (p : x \u2261 y) \u2192 P p\n  J P p refl = p\n\n  injective : InjectiveRel A _\u2261_\n  injective p q = p \u2219 ! q\n\n  surjective : SurjectiveRel A _\u2261_\n  surjective p q = ! p \u2219 q\n\n  bijective : BijectiveRel A _\u2261_\n  bijective = record { injectiveREL = injective; surjectiveREL = surjective }\n\n  Equalizer : \u2200 {b} {B : A \u2192 \u2605 b} (f g : (x : A) \u2192 B x) \u2192 \u2605 _\n  Equalizer f g = \u03a3 A (\u03bb x \u2192 f x \u2261 g x)\n  {- In a category theory context this type would the object 'E'\n     and 'proj\u2081' would be the morphism 'eq : E \u2192 A' such that\n     given any object O, and morphism 'm : O \u2192 A', there exists\n     a unique morphism 'u : O \u2192 E' such that 'eq \u2218 u \u2261 m'.\n  -}\n\n  Pullback : \u2200 {b c} {B : \u2605 b} {C : \u2605 c} (f : A \u2192 C) (g : B \u2192 C) \u2192 \u2605 _\n  Pullback {B = B} f g = \u03a3 A (\u03bb x \u2192 \u03a3 B (\u03bb y \u2192 f x \u2261 g y))\n\nmodule \u2261-Reasoning {a} {A : \u2605 a} where\n  open Setoid-Reasoning (setoid A) public\n    renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_;\n                    _\u2248\u27e8-by-definition-\u27e9_ to _\u2261\u27e8-by-definition-\u27e9_)\n\nmodule \u2257-Reasoning {a b} {A : \u2605 a} {B : \u2605 b} where\n  open Setoid-Reasoning (A \u2192-setoid B) public\n    renaming (_\u2248\u27e8_\u27e9_ to _\u2257\u27e8_\u27e9_;\n                    _\u2248\u27e8-by-definition-\u27e9_ to _\u2257\u27e8-by-definition-\u27e9_)\n\ndata \u27e6\u2261\u27e7 {a\u2081 a\u2082 a\u1d63}\n         {A\u2081 A\u2082} (A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082)\n         {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       : (A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a\u1d63) (_\u2261_ x\u2081) (_\u2261_ x\u2082) where\n    -- : \u2200 {y\u2081 y\u2082} (y\u1d63 : A\u1d63 y\u2081 y\u2082) \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 \u2605\n  \u27e6refl\u27e7 : \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63 refl refl\n\n-- Double checking level 0 with a direct \u27e6_\u27e7 encoding\nprivate\n  \u27e6\u2261\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) _\u2261_ _\u2261_\n  \u27e6\u2261\u27e7\u2032 = \u27e6\u2261\u27e7\n\n  \u27e6refl\u27e7\u2032 : (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 x\u1d63) refl refl\n  \u27e6refl\u27e7\u2032 _ _ = \u27e6refl\u27e7\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"17e1714a982e54a859db29053557e0922ad03e60","subject":"[ quicksort ] Improved test.","message":"[ quicksort ] Improved test.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/QuickSort\/QuickSortBC.agda","new_file":"notes\/FOT\/FOTC\/Program\/QuickSort\/QuickSortBC.agda","new_contents":"------------------------------------------------------------------------------\n-- Quicksort using the Bove-Capretta method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Program.QuickSort.QuickSortBC where\n\nopen import Data.Bool.Base\nopen import Data.List\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Sum\n\nopen import Induction\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nmodule InvImg =\n  Induction.WellFounded.Inverse-image {A = List \u2115} {\u2115} {_<\u2032_} length\n\n------------------------------------------------------------------------------\n\n_\u2264\u2032?_ : Decidable _\u2264\u2032_\nm \u2264\u2032? n with m \u2264? n\n... | yes p = yes (\u2264\u21d2\u2264\u2032 p)\n... | no \u00acp = no (\u03bb m\u2264\u2032n \u2192 \u00acp (\u2264\u2032\u21d2\u2264 m\u2264\u2032n))\n\n-- Non-terminating quicksort.\n\n{-# TERMINATING #-}\nqsNT : List \u2115 \u2192 List \u2115\nqsNT []       = []\nqsNT (x \u2237 xs) = qsNT (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) ++\n                x \u2237 qsNT (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)\n\n-- Domain predicate for quicksort.\ndata QSDom : List \u2115 \u2192 Set where\n  qsDom-[] : QSDom []\n  qsDom-\u2237  : \u2200 {x xs} \u2192\n             (QSDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             (QSDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             QSDom (x \u2237 xs)\n\n-- Induction principle associated to the domain predicate of quicksort.\n-- (It was not necessary).\nQSDom-ind : (P : List \u2115 \u2192 Set) \u2192\n            P [] \u2192\n            (\u2200 {x xs} \u2192 QSDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        P (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        QSDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        P (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        P (x \u2237 xs)) \u2192\n            (\u2200 {xs} \u2192 QSDom xs \u2192 P xs)\nQSDom-ind P P[] ih qsDom-[]        = P[]\nQSDom-ind P P[] ih (qsDom-\u2237 h\u2081 h\u2082) =\n  ih h\u2081 (QSDom-ind P P[] ih h\u2081) h\u2082 (QSDom-ind P P[] ih h\u2082)\n\n-- Well-founded relation on lists.\n_\u27ea\u2032_ : {A : Set} \u2192 List A \u2192 List A \u2192 Set\nxs \u27ea\u2032 ys = length xs <\u2032 length ys\n\nwf-\u27ea\u2032 : Well-founded _\u27ea\u2032_\nwf-\u27ea\u2032 = InvImg.well-founded <\u2032-well-founded\n\n-- The well-founded induction principle on _\u27ea\u2032_.\n-- postulate wfi-\u27ea\u2032 : (P : List \u2115 \u2192 Set) \u2192\n--                    (\u2200 xs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 xs \u2192 P ys) \u2192 P xs) \u2192\n--                    \u2200 xs \u2192 P xs\n\n-- The quicksort algorithm is total.\n\nfilter-length : {A : Set}(P : A \u2192 Bool) \u2192 \u2200 xs \u2192\n                length (boolFilter P xs) \u2264\u2032 length xs\nfilter-length P []       = \u2264\u2032-refl\nfilter-length P (x \u2237 xs) with P x\n... | true  = \u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length P xs)))\n... | false = \u2264\u2032-step (filter-length P xs)\n\nmodule AllWF = Induction.WellFounded.All wf-\u27ea\u2032\n\nallQSDom : \u2200 xs \u2192 QSDom xs\n-- If we use wfi-\u27ea\u2032 then allQSDom =  wfi-\u27ea\u2032 P ih\nallQSDom = build (AllWF.wfRec-builder _) P ih\n  where\n  P : List \u2115 \u2192 Set\n  P = QSDom\n\n  -- If we use wfi-\u27ea\u2032 then\n  -- ih : \u2200 zs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 zs \u2192 P ys) \u2192 P zs\n  ih :  \u2200 zs \u2192 WfRec _\u27ea\u2032_ P zs \u2192 P zs\n  ih []       h = qsDom-[]\n  ih (z \u2237 zs) h = qsDom-\u2237 prf\u2081 prf\u2082\n    where\n    c\u2081 : \u2115 \u2192 Bool\n    c\u2081 = \u03bb y \u2192 \u230a y \u2264\u2032? z \u230b\n\n    c\u2082 : \u2115 \u2192 Bool\n    c\u2082 = \u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b\n\n    f\u2081 : List \u2115\n    f\u2081 = boolFilter c\u2081 zs\n\n    f\u2082 : List \u2115\n    f\u2082 = boolFilter c\u2082 zs\n\n    prf\u2081 : QSDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2081 = h f\u2081 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2081 zs))))\n\n    prf\u2082 : QSDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2082 = h f\u2082 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2082 zs))))\n\n-- Quicksort algorithm by structural recursion on the domain predicate.\nqsDom : \u2200 xs \u2192 QSDom xs \u2192 List \u2115\nqsDom .[]      qsDom-[]        = []\nqsDom (x \u2237 xs) (qsDom-\u2237 h\u2081 h\u2082) =\n  qsDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) h\u2081 ++\n  x \u2237 qsDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) h\u2082\n\n-- The quicksort algorithm.\nqs : List \u2115 \u2192 List \u2115\nqs xs = qsDom xs (allQSDom xs)\n\n-- Testing.\nl\u2081 : List \u2115\nl\u2081 = []\nl\u2082 = 1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 []\nl\u2083 = 5 \u2237 4 \u2237 3 \u2237 2 \u2237 1 \u2237 []\nl\u2084 = 4 \u2237 1 \u2237 3 \u2237 5 \u2237 2 \u2237 []\n\nt\u2081 : qs l\u2081 \u2261 l\u2081\nt\u2081 = refl\n\nt\u2082 : qs l\u2082 \u2261 l\u2082\nt\u2082 = refl\n\nt\u2083 : qs l\u2083 \u2261 l\u2082\nt\u2083 = refl\n\nt\u2084 : qs l\u2084 \u2261 l\u2082\nt\u2084 = refl\n","old_contents":"------------------------------------------------------------------------------\n-- Quicksort using the Bove-Capretta method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Program.QuickSort.QuickSortBC where\n\nopen import Data.Bool.Base\nopen import Data.List\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Sum\n\nopen import Induction\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nmodule InvImg =\n  Induction.WellFounded.Inverse-image {A = List \u2115} {\u2115} {_<\u2032_} length\n\n------------------------------------------------------------------------------\n\n_\u2264\u2032?_ : Decidable _\u2264\u2032_\nm \u2264\u2032? n with m \u2264? n\n... | yes p = yes (\u2264\u21d2\u2264\u2032 p)\n... | no \u00acp = no (\u03bb m\u2264\u2032n \u2192 \u00acp (\u2264\u2032\u21d2\u2264 m\u2264\u2032n))\n\n-- Non-terminating quicksort.\n\n{-# TERMINATING #-}\nqsNT : List \u2115 \u2192 List \u2115\nqsNT []       = []\nqsNT (x \u2237 xs) = qsNT (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) ++\n                x \u2237 qsNT (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)\n\n-- Domain predicate for quicksort.\ndata QSDom : List \u2115 \u2192 Set where\n  qsDom-[] : QSDom []\n  qsDom-\u2237  : \u2200 {x xs} \u2192\n             (QSDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             (QSDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             QSDom (x \u2237 xs)\n\n-- Induction principle associated to the domain predicate of quicksort.\n-- (It was not necessary).\nQSDom-ind : (P : List \u2115 \u2192 Set) \u2192\n            P [] \u2192\n            (\u2200 {x xs} \u2192 QSDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        P (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        QSDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        P (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                        P (x \u2237 xs)) \u2192\n            (\u2200 {xs} \u2192 QSDom xs \u2192 P xs)\nQSDom-ind P P[] ih qsDom-[]        = P[]\nQSDom-ind P P[] ih (qsDom-\u2237 h\u2081 h\u2082) =\n  ih h\u2081 (QSDom-ind P P[] ih h\u2081) h\u2082 (QSDom-ind P P[] ih h\u2082)\n\n-- Well-founded relation on lists.\n_\u27ea\u2032_ : {A : Set} \u2192 List A \u2192 List A \u2192 Set\nxs \u27ea\u2032 ys = length xs <\u2032 length ys\n\nwf-\u27ea\u2032 : Well-founded _\u27ea\u2032_\nwf-\u27ea\u2032 = InvImg.well-founded <\u2032-well-founded\n\n-- The well-founded induction principle on _\u27ea\u2032_.\n-- postulate wfi-\u27ea\u2032 : (P : List \u2115 \u2192 Set) \u2192\n--                    (\u2200 xs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 xs \u2192 P ys) \u2192 P xs) \u2192\n--                    \u2200 xs \u2192 P xs\n\n-- The quicksort algorithm is total.\n\nfilter-length : {A : Set}(P : A \u2192 Bool) \u2192 \u2200 xs \u2192\n                length (boolFilter P xs) \u2264\u2032 length xs\nfilter-length P []       = \u2264\u2032-refl\nfilter-length P (x \u2237 xs) with P x\n... | true  = \u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length P xs)))\n... | false = \u2264\u2032-step (filter-length P xs)\n\nmodule AllWF = Induction.WellFounded.All wf-\u27ea\u2032\n\nallQSDom : \u2200 xs \u2192 QSDom xs\n-- If we use wfi-\u27ea\u2032 then allQSDom =  wfi-\u27ea\u2032 P ih\nallQSDom = build (AllWF.wfRec-builder _) P ih\n  where\n  P : List \u2115 \u2192 Set\n  P = QSDom\n\n  -- If we use wfi-\u27ea\u2032 then\n  -- ih : \u2200 zs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 zs \u2192 P ys) \u2192 P zs\n  ih :  \u2200 zs \u2192 WfRec _\u27ea\u2032_ P zs \u2192 P zs\n  ih []       h = qsDom-[]\n  ih (z \u2237 zs) h = qsDom-\u2237 prf\u2081 prf\u2082\n    where\n    c\u2081 : \u2115 \u2192 Bool\n    c\u2081 = \u03bb y \u2192 \u230a y \u2264\u2032? z \u230b\n\n    c\u2082 : \u2115 \u2192 Bool\n    c\u2082 = \u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b\n\n    f\u2081 : List \u2115\n    f\u2081 = boolFilter c\u2081 zs\n\n    f\u2082 : List \u2115\n    f\u2082 = boolFilter c\u2082 zs\n\n    prf\u2081 : QSDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2081 = h f\u2081 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2081 zs))))\n\n    prf\u2082 : QSDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b) zs)\n    prf\u2082 = h f\u2082 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2082 zs))))\n\n-- Quicksort algorithm by structural recursion on the domain predicate.\nqsDom : \u2200 xs \u2192 QSDom xs \u2192 List \u2115\nqsDom .[]      qsDom-[]        = []\nqsDom (x \u2237 xs) (qsDom-\u2237 h\u2081 h\u2082) =\n  qsDom (boolFilter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) h\u2081 ++\n  x \u2237 qsDom (boolFilter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) h\u2082\n\n-- The quicksort algorithm.\nqs : List \u2115 \u2192 List \u2115\nqs xs = qsDom xs (allQSDom xs)\n\n-- Testing.\nl\u2081 : List \u2115\nl\u2081 = []\nl\u2082 = 1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 []\nl\u2083 = 5 \u2237 4 \u2237 3 \u2237 2 \u2237 1 \u2237 []\nl\u2084 = 4 \u2237 1 \u2237 3 \u2237 5 \u2237 2 \u2237 []\n\nt\u2081 = qs l\u2081\nt\u2082 = qs l\u2082\nt\u2083 = qs l\u2083\nt\u2084 = qs l\u2084\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24e521dd9caa347bcbbd77ef5a71ec996a4947ba","subject":"modular-exp [still called ecc]: better use of the Alegbra modules","message":"modular-exp [still called ecc]: better use of the Alegbra modules\n","repos":"crypto-agda\/crypto-agda","old_file":"ECC\/ecc.agda","new_file":"ECC\/ecc.agda","new_contents":"open import Relation.Binary.PropositionalEquality.NP\nopen import Data.Two.Base\nopen import Data.List\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen Implicits\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Ring\n\nmodule ecc\n  {\ud835\udd3d : Set} (\ud835\udd3d-ring : Ring \ud835\udd3d)\n  {\u2119 : Set} (\u2119-monoid : Commutative-Monoid \u2119)\n  where\n\nopen module \ud835\udd3d = Ring \ud835\udd3d-ring\n\nopen module \u2295 = Commutative-Monoid \u2119-monoid\n  renaming\n    ( _\u2219_ to _\u2295_\n    ; \u2219= to \u2295=\n    ; \u03b5\u2219-identity to \u03b5\u2295-identity\n    ; \u2219\u03b5-identity to \u2295\u03b5-identity\n    ; _\u00b2 to 2\u00b7_\n    )\n\n2\u00b7-\u2295-distr : \u2200 {P Q} \u2192 2\u00b7 (P \u2295 Q) \u2261 2\u00b7 P \u2295 2\u00b7 Q\n2\u00b7-\u2295-distr = \u2295.interchange\n\n2\u00b7-\u2295 : \u2200 {P Q R} \u2192 2\u00b7 P \u2295 (Q \u2295 R) \u2261 (P \u2295 Q) \u2295 (P \u2295 R)\n2\u00b7-\u2295 = \u2295.interchange\n\n-- NOT used yet\nmultiply-bin : List \ud835\udfda \u2192 \u2119 \u2192 \u2119\nmultiply-bin scalar P = go scalar\n  where\n    go : List \ud835\udfda \u2192 \u2119\n    go []       = P\n    go (b \u2237 bs) = [0: x\u2080 1: x\u2081 ] b\n      where x\u2080 = 2\u00b7 go bs\n            x\u2081 = P \u2295 x\u2080\n\n{-\nmultiply : \ud835\udd3d \u2192 \u2119 \u2192 \u2119\nmultiply scalar P =\n  -- if scalar == 0 or scalar >= N: raise Exception(\"Invalid Scalar\/Private Key\")\n    multiply-bin (bin scalar) P\n\n_\u00b7_ = multiply\ninfixr 8 _\u00b7_\n-}\n\ninfixl 7 1+2*_\n1+2*_ = \u03bb x \u2192 1+ 2* x\n\ndata Parity-View : \ud835\udd3d \u2192 Set where\n  zero\u27e8_\u27e9    : \u2200 {n} \u2192 n \u2261 0# \u2192 Parity-View n\n  even_by\u27e8_\u27e9 : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 2* m    \u2192 Parity-View n\n  odd_by\u27e8_\u27e9  : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 1+ 2* m \u2192 Parity-View n\n\ncast_by\u27e8_\u27e9 : \u2200 {x y} \u2192 Parity-View x \u2192 y \u2261 x \u2192 Parity-View y\ncast zero\u27e8 x\u2091 \u27e9       by\u27e8 y\u2091 \u27e9 = zero\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast even x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = even x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast odd  x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = odd  x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\n\ninfixr 8 _\u00b7\u209a_\n_\u00b7\u209a_ : \u2200 {n} (p : Parity-View n) \u2192 \u2119 \u2192 \u2119\nzero\u27e8 e \u27e9      \u00b7\u209a P = \u03b5\neven p by\u27e8 e \u27e9 \u00b7\u209a P = 2\u00b7 (p \u00b7\u209a P)\nodd  p by\u27e8 e \u27e9 \u00b7\u209a P = P \u2295 (2\u00b7 (p \u00b7\u209a P))\n\n_+2*_ : \ud835\udfda \u2192 \ud835\udd3d \u2192 \ud835\udd3d\n0\u2082 +2* m =   2* m\n1\u2082 +2* m = 1+2* m\n\n{-\npostulate\n    bin-2* : \u2200 {n} \u2192 bin (2* n) \u2261 0\u2082 \u2237 bin n\n    bin-1+2* : \u2200 {n} \u2192 bin (1+2* n) \u2261 1\u2082 \u2237 bin n\n\nbin-+2* : (b : \ud835\udfda)(n : \ud835\udd3d) \u2192 bin (b +2* n) \u2261 b \u2237 bin n\nbin-+2* 1\u2082 n = bin-1+2*\nbin-+2* 0\u2082 n = bin-2*\n-}\n\n-- (msb) most significant bit first\nbin\u209a : \u2200 {n} \u2192 Parity-View n \u2192 List \ud835\udfda\nbin\u209a zero\u27e8 e \u27e9      = []\nbin\u209a even p by\u27e8 e \u27e9 = 0\u2082 \u2237 bin\u209a p\nbin\u209a odd  p by\u27e8 e \u27e9 = 1\u2082 \u2237 bin\u209a p\n\nhalf : \u2200 {n} \u2192 Parity-View n \u2192 \ud835\udd3d\nhalf zero\u27e8 _ \u27e9            = 0#\nhalf (even_by\u27e8_\u27e9 {m} _ _) = m\nhalf (odd_by\u27e8_\u27e9  {m} _ _) = m\n\n{-\nbin-parity : \u2200 {n} (p : ParityView n) \u2192 bin n \u2261 parity p \u2237 bin (half p)\nbin-parity (even n) = bin-2*\nbin-parity (odd n)  = bin-1+2*\n-}\n\ninfixl 6 1+\u209a_ _+\u209a_\n1+\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (1+ x)\n1+\u209a zero\u27e8 e \u27e9      = odd zero\u27e8 refl \u27e9 by\u27e8 ap 1+_ (e \u2219 ! 0+-identity) \u27e9\n1+\u209a even p by\u27e8 e \u27e9 = odd p      by\u27e8 ap 1+_ e \u27e9\n1+\u209a odd  p by\u27e8 e \u27e9 = even 1+\u209a p by\u27e8 ap 1+_ e \u2219 ! +-assoc \u2219 +-interchange \u27e9\n\n_+\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x + y)\nzero\u27e8 x\u2091 \u27e9       +\u209a y\u209a        = cast y\u209a by\u27e8 ap (\u03bb z \u2192 z + _) x\u2091 \u2219 0+-identity \u27e9\nx\u209a              +\u209a zero\u27e8 y\u2091 \u27e9 = cast x\u209a by\u27e8 ap (_+_ _) y\u2091 \u2219 +-comm \u2219 0+-identity \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even     x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-interchange \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-comm \u2219 +-assoc \u2219 ap 1+_ (+-comm \u2219 +-interchange) \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-assoc \u2219 ap 1+_ +-interchange \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = even 1+\u209a (x\u209a +\u209a y\u209a) by\u27e8 += x\u2091 y\u2091 \u2219 +-on-sides refl +-interchange \u27e9\n\ninfixl 7 2*\u209a_\n2*\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (2* x)\n2*\u209a x\u209a = x\u209a +\u209a x\u209a\n\nopen \u2261-Reasoning\n\nmodule _ {P Q} where\n    \u00b7\u209a-\u2295-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a (P \u2295 Q) \u2261 x\u209a \u00b7\u209a P \u2295 x\u209a \u00b7\u209a Q\n    \u00b7\u209a-\u2295-distr zero\u27e8 x\u2091 \u27e9       = ! \u03b5\u2295-identity\n    \u00b7\u209a-\u2295-distr even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u2295-distr x\u209a) \u2219 2\u00b7-\u2295-distr\n    \u00b7\u209a-\u2295-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap (\u03bb z \u2192 P \u2295 Q \u2295 2\u00b7 z) (\u00b7\u209a-\u2295-distr x\u209a)\n                               \u2219 ap (\u03bb z \u2192 P \u2295 Q \u2295 z) (! 2\u00b7-\u2295)\n                               \u2219 \u2295.interchange\n\nmodule _ {P} where\n    cast-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u2091 : y \u2261 x) \u2192 cast x\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P\n    cast-\u00b7\u209a-distr zero\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n\n    1+\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (1+\u209a x\u209a) \u00b7\u209a P \u2261 P \u2295 x\u209a \u00b7\u209a P\n    1+\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9       = ap (_\u2295_ P) \u03b5\u2295-identity\n    1+\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 = refl\n    1+\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (1+\u209a-\u00b7\u209a-distr x\u209a) \u2219 \u2295.interchange \u2219 \u2295.assoc\n\n    +\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y)\n                 \u2192 (x\u209a +\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P\n    +\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = cast-\u00b7\u209a-distr y\u209a _ \u2219 ! \u03b5\u2295-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 \u2295.assoc-comm\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 ! \u2295.assoc\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9\n       = (odd x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd y\u209a by\u27e8 y\u2091 \u27e9) \u00b7\u209a P\n       \u2261\u27e8by-definition\u27e9\n         2\u00b7((1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P)\n       \u2261\u27e8 ap 2\u00b7_ helper \u27e9\n         2\u00b7(P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 ap (_\u2295_ (2\u00b7 P)) 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P) \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295 \u27e9\n         P \u2295 2\u00b7(x\u209a \u00b7\u209a P) \u2295 (P \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8by-definition\u27e9\n         odd x\u209a by\u27e8 x\u2091 \u27e9 \u00b7\u209a P \u2295 odd y\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P\n       \u220e\n        where helper = (1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P\n                     \u2261\u27e8 1+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) \u27e9\n                       P \u2295 ((x\u209a +\u209a y\u209a) \u00b7\u209a P)\n                     \u2261\u27e8 ap (_\u2295_ P) (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u27e9\n                       P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P)\n                     \u220e\n\n*-1+-distr : \u2200 {x y} \u2192 x * (1+ y) \u2261 x + x * y\n*-1+-distr = *-+-distr\u02e1 \u2219 += *1-identity refl\n\n1+-*-distr : \u2200 {x y} \u2192 (1+ x) * y \u2261 y + x * y\n1+-*-distr = *-+-distr\u02b3 \u2219 += 1*-identity refl\n\ninfixl 7 _*\u209a_\n_*\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x * y)\nzero\u27e8 x\u2091 \u27e9       *\u209a y\u209a              = zero\u27e8 *= x\u2091 refl \u2219 0*-zero \u27e9\nx\u209a              *\u209a zero\u27e8 y\u2091 \u27e9       = zero\u27e8 *= refl y\u2091 \u2219 *0-zero \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 *\u209a y\u209a              = even (x\u209a *\u209a y\u209a) by\u27e8 *= x\u2091 refl \u2219 *-+-distr\u02b3 \u27e9\nx\u209a              *\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even (x\u209a *\u209a y\u209a) by\u27e8 *= refl y\u2091 \u2219 *-+-distr\u02e1 \u27e9\nodd x\u209a by\u27e8 x\u2091 \u27e9  *\u209a odd y\u209a by\u27e8 y\u2091 \u27e9  = odd  (x\u209a +\u209a y\u209a +\u209a 2*\u209a (x\u209a *\u209a y\u209a)) by\u27e8 *= x\u2091 y\u2091 \u2219 helper \u27e9\n   where\n     x = _\n     y = _\n     helper = (1+2* x)*(1+2* y)\n            \u2261\u27e8 1+-*-distr \u27e9\n              1+2* y + 2* x * 1+2* y\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2* y + z)\n                 (2* x * 1+2* y\n                 \u2261\u27e8 *-1+-distr \u27e9\n                 (2* x + 2* x * 2* y)\n                 \u2261\u27e8 += refl *-+-distr\u02e1 \u2219 +-interchange \u27e9\n                 (2* (x + 2* x * y))\n                 \u220e) \u27e9\n              1+2* y + 2* (x + 2* x * y)\n            \u2261\u27e8 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n              1+2*(y + (x + 2* x * y))\n            \u2261\u27e8 ap 1+2*_ (! +-assoc \u2219 += +-comm refl) \u27e9\n              1+2*(x + y + 2* x * y)\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2*(x + y + z)) *-+-distr\u02b3 \u27e9\n              1+2*(x + y + 2* (x * y))\n            \u220e\n\nmodule _ {P} where\n    2\u00b7-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 2\u00b7(x\u209a \u00b7\u209a P) \u2261 x\u209a \u00b7\u209a 2\u00b7 P\n    2\u00b7-\u00b7\u209a-distr x\u209a = ! \u00b7\u209a-\u2295-distr x\u209a\n\n    2*\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (2*\u209a x\u209a) \u00b7\u209a P \u2261 2\u00b7(x\u209a \u00b7\u209a P)\n    2*\u209a-\u00b7\u209a-distr x\u209a = +\u209a-\u00b7\u209a-distr x\u209a x\u209a\n\n\u00b7\u209a-\u03b5 : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a \u03b5 \u2261 \u03b5\n\u00b7\u209a-\u03b5 zero\u27e8 x\u2091 \u27e9       = refl\n\u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\u00b7\u209a-\u03b5 odd x\u209a by\u27e8 x\u2091 \u27e9  = \u03b5\u2295-identity \u2219 ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\n*\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y) {P} \u2192 (x\u209a *\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a y\u209a \u00b7\u209a P\n*\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = refl\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 odd  x\u209a by\u27e8 x\u2091 \u27e9\n\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a even y\u209a by\u27e8 y\u2091 \u27e9)\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9)\n\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2091 \u27e9 y\u209a) \u2219 2\u00b7-\u2295-distr \u2219 ap (\u03bb z \u2192 2\u00b7 (y\u209a \u00b7\u209a P) \u2295 2\u00b7 z) (2\u00b7-\u00b7\u209a-distr x\u209a)\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap (_\u2295_ P)\n     (ap 2\u00b7_\n        (+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) (2*\u209a (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) (2*\u209a-\u00b7\u209a-distr (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= \u2295.comm (ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u00b7\u209a-distr x\u209a) \u2219 \u2295.assoc\n        \u2219 \u2295= refl (! \u00b7\u209a-\u2295-distr x\u209a) ) \u2219 2\u00b7-\u2295-distr)\n     \u2219 \u2295.!assoc\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"--open import prelude renaming (Bool to \ud835\udfda; true to 1\u2082; false to 0\u2082)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Data.Two.Base\nopen import Data.List\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen Implicits\nopen import Algebra.Raw\nopen import Algebra.Field\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Group\n\nmodule ecc\n  (\u2119 : Set)\n  (\u2119-monoid : Commutative-Monoid \u2119)\n  (Number : Set)\n  (+-comm-mon : Commutative-Monoid Number)\n  (*-mon : Monoid Number)\n  (open Additive-Commutative-Monoid +-comm-mon)\n  (open Multiplicative-Monoid *-mon)\n  (+-*-distr : \u2200 {x y z} \u2192 (x + y) * z \u2261 x * z + y * z)\n  (*-+-distr : \u2200 {x y z} \u2192 x * (y + z) \u2261 x * y + x * z)\n  (0*-zero : \u2200 {x} \u2192 0# * x \u2261 0#)\n  (*0-zero : \u2200 {x} \u2192 x * 0# \u2261 0#)\n   --(modinv-*-distr : \u2200 {x y} \u2192 modinv (x * y) \u2261 modinv x * modinv y)\n   --(modinv-modinv : \u2200 {x} \u2192 modinv (modinv x) \u2261 x)\n   --(*-assoc : \u2200 {x y z} \u2192 (x * y) * z \u2261 x * (y * z))\n   --(*-comm : \u2200 {x y} \u2192 x * y \u2261 y * x)\n   --(modinv-cancel : \u2200 {x y} \u2192 x * modinv x * y \u2261 y)\n   --(2*1\u2099 : 2* 1# \u2261 2\u2099)\n   --(2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u2099 * n)\n  where\n\nmodule \u2295 = Commutative-Monoid \u2119-monoid\nopen \u2295\n  renaming\n    ( _\u2219_ to _\u2295_\n    ; \u2219= to \u2295=\n    ; \u03b5\u2219-identity to \u03b5\u2295-identity\n    ; \u2219\u03b5-identity to \u2295\u03b5-identity\n    )\n\n2\u00b7_ : \u2119 \u2192 \u2119\n2\u00b7 P = P \u2295 P\n\n2\u00b7-\u2295-distr : \u2200 {P Q} \u2192 2\u00b7 (P \u2295 Q) \u2261 2\u00b7 P \u2295 2\u00b7 Q\n2\u00b7-\u2295-distr = \u2295.interchange\n\n2\u00b7-\u2295 : \u2200 {P Q R} \u2192 2\u00b7 P \u2295 (Q \u2295 R) \u2261 (P \u2295 Q) \u2295 (P \u2295 R)\n2\u00b7-\u2295 = \u2295.interchange\n\n{-\nec-multiply-bin : List \ud835\udfda \u2192 \u2119 \u2192 \u2119\nec-multiply-bin scalar P = go scalar\n  where\n    go : List \ud835\udfda \u2192 \u2119\n    go []       = P\n    go (b \u2237 bs) = [0: x\u2080 1: x\u2081 ] b\n      where x\u2080 = 2\u00b7 go bs\n            x\u2081 = P \u2295 x\u2080\n\nec-multiply : Number \u2192 \u2119 \u2192 \u2119\nec-multiply scalar P =\n  -- if scalar == 0 or scalar >= N: raise Exception(\"Invalid Scalar\/Private Key\")\n    ec-multiply-bin (bin scalar) P\n\n_\u00b7_ = ec-multiply\ninfixr 8 _\u00b7_\n-}\n\n\nopen From-Op\u2082.From-Assoc-Comm _+_ +-assoc +-comm\n  renaming ( on-sides to +-on-sides)\n\ninfixl 6 1+_\ninfixl 7 2*_ 1+2*_\n1+_ = \u03bb x \u2192 1# + x\n2*_ = \u03bb x \u2192 x + x\n1+2*_ = \u03bb x \u2192 1+ 2* x\n\ndata Parity-View : Number \u2192 Set where\n  zero\u27e8_\u27e9    : \u2200 {n} \u2192 n \u2261 0# \u2192 Parity-View n\n  even_by\u27e8_\u27e9 : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 2* m    \u2192 Parity-View n\n  odd_by\u27e8_\u27e9  : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 1+ 2* m \u2192 Parity-View n\n\ncast_by\u27e8_\u27e9 : \u2200 {x y} \u2192 Parity-View x \u2192 y \u2261 x \u2192 Parity-View y\ncast zero\u27e8 x\u2091 \u27e9       by\u27e8 y\u2091 \u27e9 = zero\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast even x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = even x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast odd  x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = odd  x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\n\ninfixr 8 _\u00b7\u209a_\n_\u00b7\u209a_ : \u2200 {n} (p : Parity-View n) \u2192 \u2119 \u2192 \u2119\nzero\u27e8 e \u27e9      \u00b7\u209a P = \u03b5\neven p by\u27e8 e \u27e9 \u00b7\u209a P = 2\u00b7 (p \u00b7\u209a P)\nodd  p by\u27e8 e \u27e9 \u00b7\u209a P = P \u2295 (2\u00b7 (p \u00b7\u209a P))\n\n_+2*_ : \ud835\udfda \u2192 Number \u2192 Number\n0\u2082 +2* m =   2* m\n1\u2082 +2* m = 1+2* m\n\n{-\npostulate\n    bin-2* : \u2200 {n} \u2192 bin (2* n) \u2261 0\u2082 \u2237 bin n\n    bin-1+2* : \u2200 {n} \u2192 bin (1+2* n) \u2261 1\u2082 \u2237 bin n\n\nbin-+2* : (b : \ud835\udfda)(n : Number) \u2192 bin (b +2* n) \u2261 b \u2237 bin n\nbin-+2* 1\u2082 n = bin-1+2*\nbin-+2* 0\u2082 n = bin-2*\n-}\n\n-- (msb) most significant bit first\nbin\u209a : \u2200 {n} \u2192 Parity-View n \u2192 List \ud835\udfda\nbin\u209a zero\u27e8 e \u27e9      = []\nbin\u209a even p by\u27e8 e \u27e9 = 0\u2082 \u2237 bin\u209a p\nbin\u209a odd  p by\u27e8 e \u27e9 = 1\u2082 \u2237 bin\u209a p\n\nhalf : \u2200 {n} \u2192 Parity-View n \u2192 Number\nhalf zero\u27e8 _ \u27e9            = 0#\nhalf (even_by\u27e8_\u27e9 {m} _ _) = m\nhalf (odd_by\u27e8_\u27e9  {m} _ _) = m\n\n{-\nbin-parity : \u2200 {n} (p : ParityView n) \u2192 bin n \u2261 parity p \u2237 bin (half p)\nbin-parity (even n) = bin-2*\nbin-parity (odd n)  = bin-1+2*\n-}\n\ninfixl 6 1+\u209a_ _+\u209a_\n1+\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (1+ x)\n1+\u209a zero\u27e8 e \u27e9      = odd zero\u27e8 refl \u27e9 by\u27e8 ap 1+_ (e \u2219 ! 0+-identity) \u27e9\n1+\u209a even p by\u27e8 e \u27e9 = odd p      by\u27e8 ap 1+_ e \u27e9\n1+\u209a odd  p by\u27e8 e \u27e9 = even 1+\u209a p by\u27e8 ap 1+_ e \u2219 ! +-assoc \u2219 +-interchange \u27e9\n\n_+\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x + y)\nzero\u27e8 x\u2091 \u27e9       +\u209a y\u209a        = cast y\u209a by\u27e8 ap (\u03bb z \u2192 z + _) x\u2091 \u2219 0+-identity \u27e9\nx\u209a              +\u209a zero\u27e8 y\u2091 \u27e9 = cast x\u209a by\u27e8 ap (_+_ _) y\u2091 \u2219 +-comm \u2219 0+-identity \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even     x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-interchange \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-comm \u2219 +-assoc \u2219 ap 1+_ (+-comm \u2219 +-interchange) \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-assoc \u2219 ap 1+_ +-interchange \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = even 1+\u209a (x\u209a +\u209a y\u209a) by\u27e8 += x\u2091 y\u2091 \u2219 +-on-sides refl +-interchange \u27e9\n\ninfixl 7 2*\u209a_\n2*\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (2* x)\n2*\u209a x\u209a = x\u209a +\u209a x\u209a\n\nopen \u2261-Reasoning\n\nmodule _ {P Q} where\n    \u00b7\u209a-\u2295-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a (P \u2295 Q) \u2261 x\u209a \u00b7\u209a P \u2295 x\u209a \u00b7\u209a Q\n    \u00b7\u209a-\u2295-distr zero\u27e8 x\u2091 \u27e9       = ! \u03b5\u2295-identity\n    \u00b7\u209a-\u2295-distr even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u2295-distr x\u209a) \u2219 2\u00b7-\u2295-distr\n    \u00b7\u209a-\u2295-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap (\u03bb z \u2192 P \u2295 Q \u2295 2\u00b7 z) (\u00b7\u209a-\u2295-distr x\u209a)\n                               \u2219 ap (\u03bb z \u2192 P \u2295 Q \u2295 z) (! 2\u00b7-\u2295)\n                               \u2219 \u2295.interchange\n\nmodule _ {P} where\n    cast-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u2091 : y \u2261 x) \u2192 cast x\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P\n    cast-\u00b7\u209a-distr zero\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n\n    1+\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (1+\u209a x\u209a) \u00b7\u209a P \u2261 P \u2295 x\u209a \u00b7\u209a P\n    1+\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9       = ap (_\u2295_ P) \u03b5\u2295-identity\n    1+\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 = refl\n    1+\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (1+\u209a-\u00b7\u209a-distr x\u209a) \u2219 \u2295.interchange \u2219 \u2295.assoc\n\n    +\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y)\n                 \u2192 (x\u209a +\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P\n    +\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = cast-\u00b7\u209a-distr y\u209a _ \u2219 ! \u03b5\u2295-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 \u2295.assoc-comm\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 ! \u2295.assoc\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9\n       = (odd x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd y\u209a by\u27e8 y\u2091 \u27e9) \u00b7\u209a P\n       \u2261\u27e8by-definition\u27e9\n         2\u00b7((1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P)\n       \u2261\u27e8 ap 2\u00b7_ helper \u27e9\n         2\u00b7(P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 ap (_\u2295_ (2\u00b7 P)) 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P) \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295 \u27e9\n         P \u2295 2\u00b7(x\u209a \u00b7\u209a P) \u2295 (P \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8by-definition\u27e9\n         odd x\u209a by\u27e8 x\u2091 \u27e9 \u00b7\u209a P \u2295 odd y\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P\n       \u220e\n        where helper = (1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P\n                     \u2261\u27e8 1+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) \u27e9\n                       P \u2295 ((x\u209a +\u209a y\u209a) \u00b7\u209a P)\n                     \u2261\u27e8 ap (_\u2295_ P) (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u27e9\n                       P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P)\n                     \u220e\n\n*-1+-distr : \u2200 {x y} \u2192 x * (1+ y) \u2261 x + x * y\n*-1+-distr = *-+-distr \u2219 += *1-identity refl\n\n1+-*-distr : \u2200 {x y} \u2192 (1+ x) * y \u2261 y + x * y\n1+-*-distr = +-*-distr \u2219 += 1*-identity refl\n\ninfixl 7 _*\u209a_\n_*\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x * y)\nzero\u27e8 x\u2091 \u27e9       *\u209a y\u209a              = zero\u27e8 *= x\u2091 refl \u2219 0*-zero \u27e9\nx\u209a              *\u209a zero\u27e8 y\u2091 \u27e9       = zero\u27e8 *= refl y\u2091 \u2219 *0-zero \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 *\u209a y\u209a              = even (x\u209a *\u209a y\u209a) by\u27e8 *= x\u2091 refl \u2219 +-*-distr \u27e9\nx\u209a              *\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even (x\u209a *\u209a y\u209a) by\u27e8 *= refl y\u2091 \u2219 *-+-distr \u27e9\nodd x\u209a by\u27e8 x\u2091 \u27e9  *\u209a odd y\u209a by\u27e8 y\u2091 \u27e9  = odd  (x\u209a +\u209a y\u209a +\u209a 2*\u209a (x\u209a *\u209a y\u209a)) by\u27e8 *= x\u2091 y\u2091 \u2219 helper \u27e9\n   where\n     x = _\n     y = _\n     helper = (1+2* x)*(1+2* y)\n            \u2261\u27e8 1+-*-distr \u27e9\n              1+2* y + 2* x * 1+2* y\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2* y + z)\n                 (2* x * 1+2* y\n                 \u2261\u27e8 *-1+-distr \u27e9\n                 (2* x + 2* x * 2* y)\n                 \u2261\u27e8 += refl *-+-distr \u2219 +-interchange \u27e9\n                 (2* (x + 2* x * y))\n                 \u220e) \u27e9\n              1+2* y + 2* (x + 2* x * y)\n            \u2261\u27e8 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n              1+2*(y + (x + 2* x * y))\n            \u2261\u27e8 ap 1+2*_ (! +-assoc \u2219 += +-comm refl) \u27e9\n              1+2*(x + y + 2* x * y)\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2*(x + y + z)) +-*-distr \u27e9\n              1+2*(x + y + 2* (x * y))\n            \u220e\n\nmodule _ {P} where\n    2\u00b7-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 2\u00b7(x\u209a \u00b7\u209a P) \u2261 x\u209a \u00b7\u209a 2\u00b7 P\n    2\u00b7-\u00b7\u209a-distr x\u209a = ! \u00b7\u209a-\u2295-distr x\u209a\n\n    2*\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (2*\u209a x\u209a) \u00b7\u209a P \u2261 2\u00b7(x\u209a \u00b7\u209a P)\n    2*\u209a-\u00b7\u209a-distr x\u209a = +\u209a-\u00b7\u209a-distr x\u209a x\u209a\n\n\u00b7\u209a-\u03b5 : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a \u03b5 \u2261 \u03b5\n\u00b7\u209a-\u03b5 zero\u27e8 x\u2091 \u27e9       = refl\n\u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\u00b7\u209a-\u03b5 odd x\u209a by\u27e8 x\u2091 \u27e9  = \u03b5\u2295-identity \u2219 ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\n*\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y) {P} \u2192 (x\u209a *\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a y\u209a \u00b7\u209a P\n*\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = refl\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 odd  x\u209a by\u27e8 x\u2091 \u27e9\n\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a even y\u209a by\u27e8 y\u2091 \u27e9)\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9)\n\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2091 \u27e9 y\u209a) \u2219 2\u00b7-\u2295-distr \u2219 ap (\u03bb z \u2192 2\u00b7 (y\u209a \u00b7\u209a P) \u2295 2\u00b7 z) (2\u00b7-\u00b7\u209a-distr x\u209a)\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap (_\u2295_ P)\n     (ap 2\u00b7_\n        (+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) (2*\u209a (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) (2*\u209a-\u00b7\u209a-distr (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= \u2295.comm (ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u00b7\u209a-distr x\u209a) \u2219 \u2295.assoc\n        \u2219 \u2295= refl (! \u00b7\u209a-\u2295-distr x\u209a) ) \u2219 2\u00b7-\u2295-distr)\n     \u2219 ! \u2295.assoc\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"98ad94b5c101b6c25846cd7ea172de203c4fbfe7","subject":"From discussion; fix Extract so that it now depends on \u03a3-protocol","message":"From discussion; fix Extract so that it now depends on \u03a3-protocol\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Strong-Fiat-Shamir.agda","new_file":"ZK\/Strong-Fiat-Shamir.agda","new_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; done; ask)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  v \u21c4 p = \u03a3-Protocol.\u03a3-game (v , p)\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct p = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Protocol p\n    in \u03a3-game r Y c \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 \u03a3-proto Y : \u2605 where\n    constructor mk\n    open \u03a3-Protocol \u03a3-proto using (\u03a3-verifier)\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n      -- The \u03a3-transcript verify\n      verify\u2080 : \u03a3-verifier Y (mk get-A get-c\u2080 get-f\u2080) \u2261 1\u2082\n      verify\u2081 : \u03a3-verifier Y (mk get-A get-c\u2081 get-f\u2081) \u2261 1\u2082\n\n  record Special-Soundness \u03a3-proto : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 \u03a3-proto Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 \u03a3-proto Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              -- For the transformation we technically only need Simulate, no proofs...\n              -- but we take the proofs as well\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 FS-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness \u03a3-proto)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        K0 : Prfs \u2192 Transcript \u2192 ExtractorServerPart\n        K0 prfs init-transcript past-history on-going-transcript q = {!!}\n\n        K1 : Prfs \u2192 Transcript \u2192 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n        K1 [] init-transcript = done {!!}\n        K1 ((Y , \u03c0) \u2237 prfs) init-transcript = ask _ (\u03bb pt \u2192 {!!})\n\n        K : Extractor\n        K prfs init-transcript = K0 prfs init-transcript , K1 prfs init-transcript\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; done; ask)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {RP R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  (v \u21c4 p) r Y c = \u03a3-Protocol.\u03a3-game (v , p) r Y c\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct (v , p) = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (mk (get-A r Y) c (get-f r Y c)) \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 Y : \u2605 where\n    constructor mk\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n  record Special-Soundness : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 Challenge \u00d7 \u03a3-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        K0 : Prfs \u2192 Transcript \u2192 ExtractorServerPart\n        K0 prfs init-transcript past-history on-going-transcript q = {!!}\n\n        K1 : Prfs \u2192 Transcript \u2192 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n        K1 [] init-transcript = done {!!}\n        K1 ((Y , \u03c0) \u2237 prfs) init-transcript = ask _ (\u03bb pt \u2192 {!!})\n\n        K : Extractor\n        K prfs init-transcript = K0 prfs init-transcript , K1 prfs init-transcript\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"46be4d1b485e1175b4bd6a02fc26e7aca4e230a6","subject":"#31 missed one file","message":"#31 missed one file\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = {!!}\n  pres-lem (FHAp2 eps) D1 D2 D3 (FHAp2 D4) = {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp ta ta\u2081) (ITLam) = {!!}\n  pres-lem3 (TAAp ta ta\u2081) (ITApCast) = {!!}\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) () -- todo: this is a little surprising; nehole doesn't take a transition but it defieately can still step\n  pres-lem3 (TACast ta x) (ITCastID) = {!!}\n  pres-lem3 (TACast ta x) (ITCastSucceed x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITGround y) = {!!}\n  pres-lem3 (TACast ta x) (ITExpand x\u2082) = {!!}\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = {!!}\n  pres-lem (FHAp2 x eps) D1 D2 D3 (FHAp2 x\u2081 D4) = {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 x eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp ta ta\u2081) (ITLam x\u2081) = {!!}\n  pres-lem3 (TAAp ta ta\u2081) (ITApCast x x\u2081) = {!!}\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) () -- todo: this is a little surprising; nehole doesn't take a transition but it defieately can still step\n  pres-lem3 (TACast ta x) (ITCastID x\u2081) = {!!}\n  pres-lem3 (TACast ta x) (ITCastSucceed x\u2081 x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITGround x\u2081 y) = {!!}\n  pres-lem3 (TACast ta x) (ITExpand x\u2081 x\u2082) = {!!}\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a4ccc6de21d71408222178ac8ab7f25f9ac6fb50","subject":"Simplified the development.","message":"Simplified the development.\n\nEasier to read.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DCBSets.agda","new_file":"dialectica-cats\/DCBSets.agda","new_contents":"module DCBSets where\n\nopen import prelude\nopen import relations\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (\u03a3[ \u03b1 \u2208 (U \u2192 X \u2192 Set) ]( \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x) ))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , \u03b1 , _) (V , Y , _ , \u03b2 , _) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , n\u2081 , \u03b1 , _)} {(V , Y , n\u2082 , \u03b2 , _)} {(W , Z , n\u2083 , \u03b3 , _)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , n , \u03b1 , _)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , \u03b1 , _)}{(V , Y , _ , \u03b2 , _)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , \u03b1 , _}\n         {V , Y , _ , \u03b2 , _}\n         {W , Z , _ , \u03b3 , _}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{W , Z , _ , \u03b3 , _}{S , T , _ , \u03b9 , _}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n\u2297d : \u2200{U V X Y}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2  : V \u2192 Y \u2192 Set}{d\u2081 : (u : U) (x : X) \u2192 Dec (\u03b1 u x)}{d\u2082 : (u : V) (x : Y) \u2192 Dec (\u03b2 u x)} \u2192 (u : \u03a3 U (\u03bb x \u2192 V)) (x : \u03a3 X (\u03bb x\u2081 \u2192 Y)) \u2192 Dec ((\u03b1 \u2297\u1d63 \u03b2) u x)\n\u2297d {U}{V}{X}{Y}{\u03b1}{\u03b2}{d\u2081}{d\u2082} (u , v) (x , y) with d\u2081 u x | d\u2082 v y\n... | yes z | yes z' = yes (z , z')\n... | yes z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n... | no z | yes z' = no (\u03bb z'' \u2192 z (fst z''))\n... | no z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2 , \u2297d {\u03b1 = \u03b1} {\u03b2}{d\u2081}{d\u2082})\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , \u03b1 , _)}{(V , Y , _ , \u03b2 , _)}{(W , Z , _ , \u03b3 , _)}{(S , T , _ , \u03b4 , _)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {A B : Obj} \u2192 Hom (A \u2297\u2092 B) A\n\u03c0\u2081 {U , X , n\u2081 , \u03b1 , d\u2081}{V , Y , n\u2082 , \u03b2 , d\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {A B : Obj} \u2192 Hom (A \u2297\u2092 B) B\n\u03c0\u2082 {U , X , n\u2081 , \u03b1 , d\u2081}{V , Y , n\u2082 , \u03b2 , d\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n\ncart-ar : {A B C : Obj}\n  \u2192 (f : Hom C A)\n  \u2192 (g : Hom C B)\n  \u2192 Hom C (A \u2297\u2092 B)\ncart-ar {U , X , x , \u03b1 , d\u2081}{V , Y , y , \u03b2 , d\u2082}{W , Z , z , \u03b3 , d\u2083} (f , F , p\u2081) (g , G , p\u2082) = trans-\u00d7 f g ,  cart-ar-d , cond\n where\n   cart-ar-d : W \u2192 \u03a3 X (\u03bb x \u2192 Y) \u2192 Z\n   cart-ar-d w (x , y) with d\u2082 (g w) y\n   ... | yes t = F w x\n   ... | no t = G w y\n\n   cond : {u : W} {y : \u03a3 X (\u03bb x \u2192 Y)} \u2192 \u03b3 u (cart-ar-d u y) \u2192 (\u03b1 \u2297\u1d63 \u03b2) (f u , g u) y\n   cond {w}{x , y} p\u2083 with d\u2082 (g w) y\n   ... | yes t = p\u2081 p\u2083 , t\n   ... | no t = \u22a5-elim (t (p\u2082 p\u2083)) , p\u2082  p\u2083\n\ncart-diag\u2081 : \u2200{A B C : Obj}\n  \u2192 {f : Hom C A}\n  \u2192 {g : Hom C B}\n  \u2192 (cart-ar f g) \u25cb \u03c0\u2081 \u2261h f\ncart-diag\u2081 = {!!}\n\ncart-diag\u2082 : \u2200{A B C : Obj}\n  \u2192 {f : Hom C A}\n  \u2192 {g : Hom C B}\n  \u2192 (cart-ar f g) \u25cb \u03c0\u2082 \u2261h g\ncart-diag\u2082 = {!!}\n","old_contents":"module DCBSets where\n\nopen import prelude\nopen import relations\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (\u03a3[ \u03b1 \u2208 (U \u2192 X \u2192 Set) ]( \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x) ))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , \u03b1 , _) (V , Y , _ , \u03b2 , _) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , n\u2081 , \u03b1 , _)} {(V , Y , n\u2082 , \u03b2 , _)} {(W , Z , n\u2083 , \u03b3 , _)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , n , \u03b1 , _)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , \u03b1 , _)}{(V , Y , _ , \u03b2 , _)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , \u03b1 , _}\n         {V , Y , _ , \u03b2 , _}\n         {W , Z , _ , \u03b3 , _}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{W , Z , _ , \u03b3 , _}{S , T , _ , \u03b9 , _}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n\u2297d : \u2200{U V X Y}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2  : V \u2192 Y \u2192 Set}{d\u2081 : (u : U) (x : X) \u2192 Dec (\u03b1 u x)}{d\u2082 : (u : V) (x : Y) \u2192 Dec (\u03b2 u x)} \u2192 (u : \u03a3 U (\u03bb x \u2192 V)) (x : \u03a3 X (\u03bb x\u2081 \u2192 Y)) \u2192 Dec ((\u03b1 \u2297\u1d63 \u03b2) u x)\n\u2297d {U}{V}{X}{Y}{\u03b1}{\u03b2}{d\u2081}{d\u2082} (u , v) (x , y) with d\u2081 u x | d\u2082 v y\n... | yes z | yes z' = yes (z , z')\n... | yes z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n... | no z | yes z' = no (\u03bb z'' \u2192 z (fst z''))\n... | no z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2 , \u2297d {\u03b1 = \u03b1} {\u03b2}{d\u2081}{d\u2082})\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , \u03b1 , _)}{(V , Y , _ , \u03b2 , _)}{(W , Z , _ , \u03b3 , _)}{(S , T , _ , \u03b4 , _)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}\n    \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n    \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}\n    \u2192 Hom ((U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082)) (U , X , n\u2081 , \u03b1 , d\u2081)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}\n    \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n    \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}    \n    \u2192 Hom ((U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082)) (V , Y , n\u2082 , \u03b2 , d\u2082)\n\u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n\ncart-ar : {U X V Y W Z : Set}\n  \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n  \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n  \u2192 {n\u2081 : \u22a4 \u2192 X}\n  \u2192 {n\u2082 : \u22a4 \u2192 Y}\n  \u2192 {n\u2083 : \u22a4 \u2192 Z}\n  \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n  \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}\n  \u2192 {d\u2083 : \u2200(w : W)(z : Z) \u2192 Dec (\u03b3 w z)}  \n  \u2192 Hom (W , Z , n\u2083 , \u03b3 , d\u2083) (U , X , n\u2081 , \u03b1 , d\u2081)\n  \u2192 Hom (W , Z , n\u2083 , \u03b3 , d\u2083) (V , Y , n\u2082 , \u03b2 , d\u2082)\n  \u2192 Hom (W , Z , n\u2083 , \u03b3 , d\u2083) ((U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082))\ncart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3}{n\u2081}{n\u2082}{n\u2083}{d\u2081}{d\u2082}{d\u2083} (f , F , p\u2081) (g , G , p\u2082) = trans-\u00d7 f g ,  cart-ar-d , cond\n where\n   cart-ar-d : W \u2192 \u03a3 X (\u03bb x \u2192 Y) \u2192 Z\n   cart-ar-d w (x , y) with d\u2082 (g w) y\n   ... | yes t = F w x\n   ... | no t = G w y\n\n   cond : {u : W} {y : \u03a3 X (\u03bb x \u2192 Y)} \u2192 \u03b3 u (cart-ar-d u y) \u2192 (\u03b1 \u2297\u1d63 \u03b2) (f u , g u) y\n   cond {w}{x , y} p\u2083 with d\u2082 (g w) y\n   ... | yes t = p\u2081 p\u2083 , t\n   ... | no t = \u22a5-elim (t (p\u2082 p\u2083)) , p\u2082  p\u2083\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64f718df73e48b29bce2de2b6ecc8bbe59b45056","subject":"Use instance arguments for vowel predicate.","message":"Use instance arguments for vowel predicate.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  add-acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  add-grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  add-circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\n\u03b1-acute : \u03b1\u2032 accent\n\u03b1-acute = add-acute\n\n\u03b1-grave : \u03b1\u2032 accent\n\u03b1-grave = add-grave\n\n\u03b1-circumflex : \u03b1\u2032 accent\n\u03b1-circumflex = add-circumflex \u03b1-long-vowel\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth add-smooth-lower-vowel\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough add-rough-vowel\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\n\u03b5-acute : \u03b5\u2032 accent\n\u03b5-acute = add-acute\n\n\u03b5-grave : \u03b5\u2032 accent\n\u03b5-grave = add-grave\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth add-smooth-lower-vowel\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough add-rough-vowel\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\n\u03b7-acute : \u03b7\u2032 accent\n\u03b7-acute = add-acute\n\n\u03b7-grave : \u03b7\u2032 accent\n\u03b7-grave = add-grave\n\n\u03b7-circumflex : \u03b7\u2032 accent\n\u03b7-circumflex = add-circumflex \u03b7-long-vowel\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth add-smooth-lower-vowel\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough add-rough-vowel\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\n\u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\n\u03b9-acute : \u03b9\u2032 accent\n\u03b9-acute = add-acute\n\n\u03b9-grave : \u03b9\u2032 accent\n\u03b9-grave = add-grave\n\n\u03b9-circumflex : \u03b9\u2032 accent\n\u03b9-circumflex = add-circumflex \u03b9-long-vowel\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth add-smooth-lower-vowel\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing \u03b1-grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing \u03b1-grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing \u03b1-acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing \u03b1-acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing \u03b1-circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing \u03b1-circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing \u03b1-grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing \u03b1-grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing \u03b1-acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing \u03b1-acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing \u03b1-circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing \u03b1-circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent \u03b1-grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent \u03b1-acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota \u03b1-grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota \u03b1-grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota \u03b1-acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota \u03b1-acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota \u03b1-circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota \u03b1-circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota \u03b1-grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota \u03b1-grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota \u03b1-acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota \u03b1-acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota \u03b1-circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota \u03b1-circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota \u03b1-grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota \u03b1-acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent \u03b1-circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota \u03b1-circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent \u03b1-grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent \u03b1-acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent add-acute = acute-mark\nletter-to-accent add-grave = grave-mark\nletter-to-accent (add-circumflex x) = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 v vowel \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 v vowel \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 v vowel \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 v vowel \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 v vowel \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 v vowel \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 v vowel \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  add-acute : \u2200 {\u2113} \u2192 \u2113 vowel \u2192 \u2113 accent\n  add-grave : \u2200 {\u2113} \u2192 \u2113 vowel \u2192 \u2113 accent\n  add-circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\n\u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short \u03b1-vowel (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u03b1-vowel \u00ac\u03b1-always-short\n\n\u03b1-acute : \u03b1\u2032 accent\n\u03b1-acute = add-acute \u03b1-vowel\n\n\u03b1-grave : \u03b1\u2032 accent\n\u03b1-grave = add-grave \u03b1-vowel\n\n\u03b1-circumflex : \u03b1\u2032 accent\n\u03b1-circumflex = add-circumflex \u03b1-long-vowel\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth (add-smooth-lower-vowel \u03b1-vowel)\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b1-vowel (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough (add-rough-vowel \u03b1-vowel)\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough (add-rough-vowel \u03b1-vowel)\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript \u03b1-vowel here\n\n-- \u0395 \u03b5\n\u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short \u03b5-vowel here\n\n\u03b5-acute : \u03b5\u2032 accent\n\u03b5-acute = add-acute \u03b5-vowel\n\n\u03b5-grave : \u03b5\u2032 accent\n\u03b5-grave = add-grave \u03b5-vowel\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth (add-smooth-lower-vowel \u03b5-vowel)\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b5-vowel (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough (add-rough-vowel \u03b5-vowel)\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough (add-rough-vowel \u03b5-vowel)\n\n-- \u0397 \u03b7\n\u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short \u03b7-vowel (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u03b7-vowel \u00ac\u03b7-always-short\n\n\u03b7-acute : \u03b7\u2032 accent\n\u03b7-acute = add-acute \u03b7-vowel\n\n\u03b7-grave : \u03b7\u2032 accent\n\u03b7-grave = add-grave \u03b7-vowel\n\n\u03b7-circumflex : \u03b7\u2032 accent\n\u03b7-circumflex = add-circumflex \u03b7-long-vowel\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth (add-smooth-lower-vowel \u03b7-vowel)\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b7-vowel (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough (add-rough-vowel \u03b7-vowel)\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough (add-rough-vowel \u03b7-vowel)\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript \u03b7-vowel (there here)\n\n-- \u0399 \u03b9\n\ninstance\n  \u03b9-vowel : \u03b9\u2032 vowel\n  \u03b9-vowel = is-vowel (there (there (there here)))\n\n  \u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n  \u00ac\u03b9-always-short (is-always-short \u03b9-vowel (there (there ())))\n\n  \u03b9-long-vowel : \u03b9\u2032 long-vowel\n  \u03b9-long-vowel = make-long-vowel \u03b9-vowel \u00ac\u03b9-always-short\n\n  \u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n  \u03b9-not-\u03c5 ()\n\n\u03b9-acute : \u03b9\u2032 accent\n\u03b9-acute = add-acute \u03b9-vowel\n\n\u03b9-grave : \u03b9\u2032 accent\n\u03b9-grave = add-grave \u03b9-vowel\n\n\u03b9-circumflex : \u03b9\u2032 accent\n\u03b9-circumflex = add-circumflex \u03b9-long-vowel\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth (add-smooth-lower-vowel \u03b9-vowel)\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b9-vowel (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough (add-rough-vowel \u03b9-vowel)\n\n-- \u039f \u03bf\nf : \u2200 {\u2113} \u2192 {{ p : \u2113 vowel }} \u2192 \u2113 accent\nf {{ p }} = add-acute p\n\n\u03b9-acutes : Token \u03b9\u2032 lower\n\u03b9-acutes = with-accent f\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing \u03b1-grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing \u03b1-grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing \u03b1-acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing \u03b1-acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing \u03b1-circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing \u03b1-circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing \u03b1-grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing \u03b1-grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing \u03b1-acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing \u03b1-acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing \u03b1-circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing \u03b1-circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent \u03b1-grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent \u03b1-acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota \u03b1-grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota \u03b1-grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota \u03b1-acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota \u03b1-acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota \u03b1-circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota \u03b1-circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota \u03b1-grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota \u03b1-grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota \u03b1-acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota \u03b1-acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota \u03b1-circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota \u03b1-circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota \u03b1-grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota \u03b1-acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent \u03b1-circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota \u03b1-circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent \u03b1-grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent \u03b1-acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent (add-acute x) = acute-mark\nletter-to-accent (add-grave x) = grave-mark\nletter-to-accent (add-circumflex x) = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"69f8f4c538d8b29e2c36026bdecd77c83dc74714","subject":"Cosmetic refactoring","message":"Cosmetic refactoring\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7f06d89624fee2acacb5b09b999896749ec95635","subject":"Added issue related to the erasure of proof terms.","message":"Added issue related to the erasure of proof terms.\n\nIgnore-this: 20fecaf1f2a20f09bc0d4b0e6c34f040\n\ndarcs-hash:20110820025417-3bd4e-bef58805884c49a9f400122415aa8975444763d7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/ProofTerm1.agda","new_file":"Issues\/ProofTerm1.agda","new_contents":"module Issues.ProofTerm1 where\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  P   : D \u2192 Set\n  Q   : D \u2192 D \u2192 Set\n\nfoo : \u2200 {a b} \u2192 P b \u2192 Q b b \u2192 D\nfoo {a} {b} Pb Qbb = a\n  where\n  c : D\n  c = a\n  {-# ATP definition c #-}\n\n  postulate bar : c \u2261 a\n  {-# ATP prove bar #-}\n","old_contents":"module Issues.ProofTerm1 where\n\npostulate\n  D          : Set\n  true false : D\n  not        : D \u2192 D\n\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\ndata Bool : D \u2192 Set where\n  tB : Bool true\n  fB : Bool false\n\nfoo : \u2200 {b0 b1 b2} \u2192\n      Bool b1 \u2192\n      Bool b2 \u2192\n      b0 \u2261 not b1 \u2192\n      b2 \u2261 b2\nfoo      tB Bb2 b0-eq = refl\nfoo {b0} fB Bb2 b0-eq = refl\n  where\n  as : D\n  as = b0\n\n  bs : D\n  bs = not as\n  {-# ATP definition bs #-}\n\n  postulate bar : bs \u2261 bs\n  {-# ATP prove bar #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"424733cef6ca27a96e796c9bd22e39ee6c97d27b","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/ABP.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/ABP.agda","new_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module define the ABP following the presentation in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ABP where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- ABP equations\n\npostulate\n  send ack out corrupt : D \u2192 D\n  await                : D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate send-eq : \u2200 b i is ds \u2192\n                    send b \u00b7 (i \u2237 is) \u00b7 ds \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom send-eq #-}\n\npostulate\n  await-ok\u2261 : \u2200 b b' i is ds \u2192\n              b \u2261 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 send (not b) \u00b7 is \u00b7 ds\n\n  await-ok\u2262 : \u2200 b b' i is ds \u2192\n              b \u2262 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n\n  await-error : \u2200 b i is ds \u2192\n                await b i is (error \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom await-ok\u2261 await-ok\u2262 await-error #-}\n\npostulate\n  ack-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192\n            ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 b \u2237 ack (not b) \u00b7 bs\n\n  ack-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192\n            ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n\n  ack-error : \u2200 b bs \u2192 ack b \u00b7 (error \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n{-# ATP axiom ack-ok\u2261 ack-ok\u2262 ack-error #-}\n\npostulate\n  out-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192\n            out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 i \u2237 out (not b) \u00b7 bs\n\n  out-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192\n            out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 out b \u00b7 bs\n\n  out-error : \u2200 b bs \u2192 out b \u00b7 (error \u2237 bs) \u2261 out b \u00b7 bs\n{-# ATP axiom out-ok\u2261 out-ok\u2262 out-error #-}\n\npostulate\n  corrupt-T : \u2200 os x xs \u2192\n              corrupt (T \u2237 os) \u00b7 (x \u2237 xs) \u2261 ok x \u2237 corrupt os \u00b7 xs\n  corrupt-F : \u2200 os x xs \u2192\n              corrupt (F \u2237 os) \u00b7 (x \u2237 xs) \u2261 error \u2237 corrupt os \u00b7 xs\n{-# ATP axiom corrupt-T corrupt-F #-}\n\npostulate has hbs hcs hds : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate has-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2081 \u00b7 is \u00b7 (hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom has-eq #-}\n\npostulate hbs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2081 \u00b7 (has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hbs-eq #-}\n\npostulate hcs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2082 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hcs-eq #-}\n\npostulate hds-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2082 \u00b7 (hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hds-eq #-}\n\npostulate\n  transfer    : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  transfer-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                transfer f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2083 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom transfer-eq #-}\n\npostulate\n  abpTransfer : D \u2192 D \u2192 D \u2192 D \u2192 D\n  abpTransfer-eq :\n    \u2200 b os\u2081 os\u2082 is \u2192\n      abpTransfer b os\u2081 os\u2082 is \u2261\n      transfer (send b) (ack b) (out b) (corrupt os\u2081) (corrupt os\u2082) is\n{-# ATP axiom abpTransfer-eq #-}\n\n------------------------------------------------------------------------------\n-- ABP relations\n\n-- Abbreviation for the recursive equations of the alternating bit\n-- protocol.\nABP : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nABP b is os\u2081 os\u2082 as bs cs ds js =\n  as \u2261 send b \u00b7 is \u00b7 ds\n  \u2227 bs \u2261 corrupt os\u2081 \u00b7 as\n  \u2227 cs \u2261 ack b \u00b7 bs\n  \u2227 ds \u2261 corrupt os\u2082 \u00b7 cs\n  \u2227 js \u2261 out b \u00b7 bs\n{-# ATP definition ABP #-}\n\nABP' : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nABP' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' =\n  ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n  \u2227 as' \u2261 await b i' is' ds'  -- Typo in ds'.\n  \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n  \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n  \u2227 js' \u2261 out (not b) \u00b7 bs'\n{-# ATP definition ABP' #-}\n\n-- Auxiliary bisimulation.\nB : D \u2192 D \u2192 Set\nB is js = \u2203[ b ] \u2203[ os\u2081 ] \u2203[ os\u2082 ] \u2203[ as ] \u2203[ bs ] \u2203[ cs ] \u2203[ ds ]\n            Stream is\n            \u2227 Bit b\n            \u2227 Fair os\u2081\n            \u2227 Fair os\u2082\n            \u2227 ABP b is os\u2081 os\u2082 as bs cs ds js\n{-# ATP definition B #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module define the ABP following the presentation in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ABP where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- ABP equations\n\npostulate\n  send ack out corrupt : D \u2192 D\n  await                : D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate send-eq : \u2200 b i is ds \u2192\n                    send b \u00b7 (i \u2237 is) \u00b7 ds \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom send-eq #-}\n\npostulate\n  await-ok\u2261 : \u2200 b b' i is ds \u2192\n              b \u2261 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 send (not b) \u00b7 is \u00b7 ds\n\n  await-ok\u2262 : \u2200 b b' i is ds \u2192\n              b \u2262 b' \u2192\n              await b i is (ok b' \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n\n  await-error : \u2200 b i is ds \u2192\n                await b i is (error \u2237 ds) \u2261 < i , b > \u2237 await b i is ds\n{-# ATP axiom await-ok\u2261 await-ok\u2262 await-error #-}\n\npostulate\n  ack-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192 ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 b \u2237 ack (not b) \u00b7 bs\n\n  ack-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192 ack b \u00b7 (ok < i , b' > \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n\n  ack-error : \u2200 b bs \u2192 ack b \u00b7 (error \u2237 bs) \u2261 not b \u2237 ack b \u00b7 bs\n{-# ATP axiom ack-ok\u2261 ack-ok\u2262 ack-error #-}\n\npostulate\n  out-ok\u2261 : \u2200 b b' i bs \u2192\n            b \u2261 b' \u2192 out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 i \u2237 out (not b) \u00b7 bs\n\n  out-ok\u2262 : \u2200 b b' i bs \u2192\n            b \u2262 b' \u2192 out b \u00b7 (ok < i , b' > \u2237 bs) \u2261 out b \u00b7 bs\n\n  out-error : \u2200 b bs \u2192 out b \u00b7 (error \u2237 bs) \u2261 out b \u00b7 bs\n{-# ATP axiom out-ok\u2261 out-ok\u2262 out-error #-}\n\npostulate\n  corrupt-T : \u2200 os x xs \u2192\n              corrupt (T \u2237 os) \u00b7 (x \u2237 xs) \u2261 ok x \u2237 corrupt os \u00b7 xs\n  corrupt-F : \u2200 os x xs \u2192\n              corrupt (F \u2237 os) \u00b7 (x \u2237 xs) \u2261 error \u2237 corrupt os \u00b7 xs\n{-# ATP axiom corrupt-T corrupt-F #-}\n\npostulate has hbs hcs hds : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n\npostulate has-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2081 \u00b7 is \u00b7 (hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom has-eq #-}\n\npostulate hbs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2081 \u00b7 (has f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hbs-eq #-}\n\npostulate hcs-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2082 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hcs-eq #-}\n\npostulate hds-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                   hds f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 g\u2082 \u00b7 (hcs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom hds-eq #-}\n\npostulate\n  transfer    : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  transfer-eq : \u2200 f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2192\n                transfer f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is \u2261 f\u2083 \u00b7 (hbs f\u2081 f\u2082 f\u2083 g\u2081 g\u2082 is)\n{-# ATP axiom transfer-eq #-}\n\npostulate\n  abpTransfer    : D \u2192 D \u2192 D \u2192 D \u2192 D\n  abpTransfer-eq : \u2200 b os\u2081 os\u2082 is \u2192\n                   abpTransfer b os\u2081 os\u2082 is \u2261\n                     transfer (send b) (ack b) (out b) (corrupt os\u2081)\n                       (corrupt os\u2082) is\n{-# ATP axiom abpTransfer-eq #-}\n\n------------------------------------------------------------------------------\n-- ABP relations\n\n-- Abbreviation for the recursive equations of the alternating bit\n-- protocol.\nABP : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nABP b is os\u2081 os\u2082 as bs cs ds js =\n  as \u2261 send b \u00b7 is \u00b7 ds\n  \u2227 bs \u2261 corrupt os\u2081 \u00b7 as\n  \u2227 cs \u2261 ack b \u00b7 bs\n  \u2227 ds \u2261 corrupt os\u2082 \u00b7 cs\n  \u2227 js \u2261 out b \u00b7 bs\n{-# ATP definition ABP #-}\n\nABP' : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 Set\nABP' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' =\n  ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n  \u2227 as' \u2261 await b i' is' ds'  -- Typo in ds'.\n  \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n  \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n  \u2227 js' \u2261 out (not b) \u00b7 bs'\n{-# ATP definition ABP' #-}\n\n-- Auxiliary bisimulation.\nB : D \u2192 D \u2192 Set\nB is js = \u2203[ b ] \u2203[ os\u2081 ] \u2203[ os\u2082 ] \u2203[ as ] \u2203[ bs ] \u2203[ cs ] \u2203[ ds ]\n            Stream is\n            \u2227 Bit b\n            \u2227 Fair os\u2081\n            \u2227 Fair os\u2082\n            \u2227 ABP b is os\u2081 os\u2082 as bs cs ds js\n{-# ATP definition B #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8562df227abd884dae0eee9c954cd6cf57baf080","subject":"Add more shorthands for fully applied constants.","message":"Add more shorthands for fully applied constants.\n\nOld-commit-hash: 8f2cea4a25c293f39c27128e0f9eaedf726c41b5\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Type Atlas-type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\ntrue! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\ntrue! = const true\n\nfalse! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool)\nfalse! = const false\n\nxor! : \u2200 {\u0393} \u2192\n  Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n  Atlas-term \u0393 (base Bool)\nxor! = app\u2082 (const xor)\n\nempty! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nempty! = const empty\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t empty!\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (xor! \u0394x \u0394y))))\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Type Atlas-type \u2192 Set where\n  true  : Atlas-const\n    (base Bool)\n\n  false : Atlas-const\n    (base Bool)\n\n  xor   : Atlas-const\n    (base Bool \u21d2 base Bool \u21d2 base Bool)\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base (Map \u03ba \u03b9))\n\n  -- `update key val my-map` would\n  -- - insert if `key` is not present in `my-map`\n  -- - delete if `val` is the neutral element\n  -- - make an update otherwise\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9))\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n    (base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9)\n\n  -- Model of zip = Haskell Data.List.zipWith\n  --\n  -- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n  --\n  -- Behavioral difference: all key-value pairs present\n  -- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n  -- be iterated over. Neutral element of type `a` or `b`\n  -- will be supplied if the key is missing in the\n  -- corresponding map.\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n    ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n     base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c))\n\n  -- Model of fold = Haskell Data.Map.foldWithKey\n  --\n  -- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n   ((base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n    base b \u21d2 base (Map \u03ba a) \u21d2 base b)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base \u03b9)\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const (base (Atlas-\u0394base \u03b9))\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t (const empty)\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393 \u03c4} \u2192 (c : Atlas-const \u03c4) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app\u2082 (const xor) \u0394x \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a972b1df4e68988e6b14aa4fabb1e1ec3114fc5f","subject":"import inside","message":"import inside\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/Evens.agda","new_file":"livecode\/Evens.agda","new_contents":"open import Base\n\nopen import Nat\n\nopen import Logic\n\nmodule Evens  where\n\ndata Even : \u2115 \u2192 Type where \n  0even : Even 0\n  _+2even : {n : \u2115} \u2192 Even n \u2192 Even (succ (succ n))\n\n4even = (0even +2even) +2even \n\n1odd : Even 1 \u2192 False\n1odd ()\n\nmodule n|n+1even where\n  P : \u2115 \u2192 Type\n  P n = (Even n) \u2295 (Even (succ n))\n\n  step : {n : \u2115} \u2192 P n \u2192 P (succ n)\n  step (\u03b9\u2081 p) = \u03b9\u2082(p +2even)\n  step (\u03b9\u2082 p) = \u03b9\u2081 p\n\n  thm : (n : \u2115) \u2192 P n\n  thm 0 = \u03b9\u2081 0even\n  thm (succ n) = step (thm n) \n\nvalue : {n : \u2115} \u2192 Even n \u2192 \u2115\nvalue 0even = 0\nvalue (p +2even) = succ (succ (value p))\n\nhalf : (n : \u2115) \u2192 Even n \u2192 \u2115\nhalf .0 0even = 0\nhalf (succ (succ n)) (p +2even) = succ (half n p)\n\ndouble : (n : \u2115) \u2192 \u03a3 \u2115 Even\ndouble 0 = [ 0 , 0even ]\ndouble (succ n) = [ (succ (succ (proj\u2081 (double n)))) , (proj\u2082 (double n)) +2even ]\n\nhalfdouble : \u2115 \u2192 \u2115\nhalfdouble n = half (proj\u2081 (double n)) (proj\u2082 (double n))\n\n\n\nmodule Collatz where\n  open n|n+1even\n\n  frompf : (n : \u2115) \u2192 P n \u2192 \u2115\n  frompf n (\u03b9\u2081 p) = half n p\n  frompf n (\u03b9\u2082 p) = (3 * n) + 1\n\n  fn = \u03bb n \u2192 frompf n (thm n) \n","old_contents":"open import Base\n\nopen import Nat\n\nopen import Logic\n\nmodule Evens  where\n\ndata Even : \u2115 \u2192 Type where \n  0even : Even 0\n  _+2even : {n : \u2115} \u2192 Even n \u2192 Even (succ (succ n))\n\n4even = (0even +2even) +2even \n\n1odd : Even 1 \u2192 False\n1odd ()\n\nmodule n|n+1even where\n  P : \u2115 \u2192 Type\n  P n = (Even n) \u2295 (Even (succ n))\n\n  step : {n : \u2115} \u2192 P n \u2192 P (succ n)\n  step (\u03b9\u2081 p) = \u03b9\u2082(p +2even)\n  step (\u03b9\u2082 p) = \u03b9\u2081 p\n\n  thm : (n : \u2115) \u2192 P n\n  thm 0 = \u03b9\u2081 0even\n  thm (succ n) = step (thm n) \n\nvalue : {n : \u2115} \u2192 Even n \u2192 \u2115\nvalue 0even = 0\nvalue (p +2even) = succ (succ (value p))\n\nhalf : (n : \u2115) \u2192 Even n \u2192 \u2115\nhalf .0 0even = 0\nhalf (succ (succ n)) (p +2even) = succ (half n p)\n\ndouble : (n : \u2115) \u2192 \u03a3 \u2115 Even\ndouble 0 = [ 0 , 0even ]\ndouble (succ n) = [ (succ (succ (proj\u2081 (double n)))) , (proj\u2082 (double n)) +2even ]\n\nhalfdouble : \u2115 \u2192 \u2115\nhalfdouble n = half (proj\u2081 (double n)) (proj\u2082 (double n))\n\n\n\nmodule Collatz where\n  frompf : (n : \u2115) \u2192 n|n+1even.P n \u2192 \u2115\n  frompf n (\u03b9\u2081 p) = half n p\n  frompf n (\u03b9\u2082 p) = (3 * n) + 1\n\n  fn = \u03bb n \u2192 frompf n (n|n+1even.thm n) \n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b7ed92cc07ec0994e63f4a661631f95efcdfc0c3","subject":"IDesc: our Hutton's razor","message":"IDesc: our Hutton's razor","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep nat (EZe , t) = t\n              evalOneStep nat ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep nat ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool (EZe , t ) = t\n              evalOneStep bool ((ESu EZe) , (true , (x , _))) = x\n              evalOneStep bool ((ESu EZe) , (false , (_ , y))) = y\n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\nassgnmt : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\nassgnmt ty vars context (l value) = con ( EZe , value )\nassgnmt ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e61274b2f5a2819b62a095df9b8fcf0938a8a4ed","subject":"Added x\u2264y\u2192x\u226fy.","message":"Added x\u2264y\u2192x\u226fy.\n\nIgnore-this: fdd2e8d0bfdad9b528d0155a9bfcaa68\n\ndarcs-hash:20110509170827-3bd4e-3172a8930e333ca14ecbe49c93e2b5762f295b25.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function using ( _$_ )\n\nopen import FOTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The FOTC natural numbers type\n        )\nopen import FOTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import FOTC.Data.Nat.Inequalities.Properties public\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 GE n n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\nx<Sy\u2192x\u2264y zN Nn 0<Sn       = 0\u2264x Nn\nx<Sy\u2192x\u2264y (sN Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m (succ n)\nx\u2264y\u2192x<Sy {n = n} zN      Nn 0\u2264n  = <-0S n\nx\u2264y\u2192x<Sy         (sN Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 NGT m n\nx\u2264y\u2192x\u226fy zN          Nn          0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (sN Nm)     zN          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (<-SS m (succ n))) Sm\u2264Sn))\n  where\n    postulate prf : NGT m n \u2192  -- IH.\n                    NGT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<y\u2238x Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (cong succ m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function using ( _$_ )\n\nopen import FOTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The FOTC natural numbers type\n        )\nopen import FOTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import FOTC.Data.Nat.Inequalities.Properties public\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 GE n n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\nx<Sy\u2192x\u2264y zN Nn 0<Sn       = 0\u2264x Nn\nx<Sy\u2192x\u2264y (sN Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m (succ n)\nx\u2264y\u2192x<Sy {n = n} zN      Nn 0\u2264n  = <-0S n\nx\u2264y\u2192x<Sy         (sN Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<y\u2238x Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (cong succ m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0acc9e27ea5875caf158f7560d37506ab5684008","subject":"Helios: better account to the \"query hacking\"","message":"Helios: better account to the \"query hacking\"\n","repos":"crypto-agda\/crypto-agda","old_file":"Helios.agda","new_file":"Helios.agda","new_contents":"open import Type\nopen import Function\nopen import Data.List\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Fin using (Fin; zero; suc) renaming (to\u2115 to Fin\u25b9\u2115)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) hiding (zip; map)\n{-\nopen import Data.Zero\nopen import Data.One\n-}\nopen import Data.Two\nopen import Data.Nat.NP hiding (_==_; _<_)\n--open import Control.Strategy\n\nmodule Helios\n  (VoterId : \u2605)\n  (PublicKey : \u2605)\n  (SecretKey : \u2605)\n  (NIZKSecretKey : PublicKey \u2192 \u2605)\n  (Vote   : \u2605)\n  (Ballot : \u2605)\n  (Decryption-share : \u2605)\n  (PublicInfo : \u2605)\n  (R-setup : \u2605)\n  (R-vote : \u2605)\n  (vote : PublicKey \u2192 R-vote \u2192 VoterId \u2192 Vote \u2192 Ballot)\n  {-\n  (let Ballots = List (VoterId \u00d7 Ballot)\n       Decryption-shares = List Decryption-share\n  )\n  -}\n  (validate-ballots :\n     let Ballots = List (VoterId \u00d7 Ballot)\n         Decryption-shares = List Decryption-share in\n     PublicKey \u2192 Ballots \u2192 \ud835\udfda)\n  (Tally : \u2605) -- could be a valid tally or a special symbol \u22a5\n  (fake-decrytion-share :\n     let Ballots = List (VoterId \u00d7 Ballot)\n         Decryption-shares = List Decryption-share in\n     Tally \u2192 Ballots \u2192 Decryption-share)\n  (R-adversary : \u2605)\n  where\n\nBallots = List (VoterId \u00d7 Ballot)\nDecryption-shares = List Decryption-share\n\nmodule _ {A : \u2605} where\n  map-n\u1d57\u02b0 : \u2115 \u2192 (A \u2192 A) \u2192 List A \u2192 List A\n  map-n\u1d57\u02b0 n       f []       = []\n  map-n\u1d57\u02b0 zero    f (x \u2237 xs) = f x \u2237 xs\n  map-n\u1d57\u02b0 (suc n) f (x \u2237 xs) = x \u2237 map-n\u1d57\u02b0 n f xs\n\n_<_ : \u2115 \u2192 \u2115 \u2192 \ud835\udfda\nx < y = suc x <= y\n\nmodule _ {k : \u2115} {- sequences -} where\n     Seq : \u2605 \u2192 \u2605\n     Seq A = Fin k \u2192 A\n\n     {-\n     replicateSeq : {A : \u2605} \u2192 Fin k \u2192 A \u2192 A \u2192 Seq A\n     replicateSeq n x y i = case {!(i < n)!} 0: x 1: y\n     -}\n\nrecord KeyPair : \u2605 where\n  constructor mk\n  field\n    pk : PublicKey\n    sk : SecretKey\n    zk : NIZKSecretKey pk\n\n\nrecord BB : \u2605 where\n  constructor <_,,_,,_,,_>\n  field\n    Y-pk              : PublicKey\n    \u03c0-pk              : NIZKSecretKey Y-pk\n    ballots           : Ballots\n    decryption-shares : Decryption-shares\nopen BB public\n\nMake-decryption-shares = KeyPair \u2192 Ballots \u2192 Decryption-shares\nSimulate-decryption-shares = (bs\u2080 : Ballots) \u2192 Make-decryption-shares\n\nvalidate : PublicKey \u2192 Ballots \u2192 VoterId \u00d7 Ballot \u2192 \ud835\udfda\nvalidate pk bs vb = validate-ballots pk (vb \u2237 bs)\n\ndata Q : \u2605 where\n  ask-vote   : (v_ : Vote \u00b2) \u2192 Q\n  ask-ballot : (b  : Ballot) \u2192 Q\n\nmap-vote\u00b2-Q : (Vote \u00b2 \u2192 Vote \u00b2) \u2192 Q \u2192 Q\nmap-vote\u00b2-Q f (ask-vote v_) = ask-vote (f v_)\nmap-vote\u00b2-Q f (ask-ballot b) = ask-ballot b\n\nVotingPhase\u2080 : \u2605\nVotingPhase\u2080 = Ballots \u2192 VoterId \u00d7 Q\n\nVotingPhase : \u2605\nVotingPhase = List VotingPhase\u2080\n\nmodule Tallying\n  (decryption-shares : Make-decryption-shares)\n  (result : BB \u2192 Tally)\n  where\n\n  tally-bb : KeyPair \u2192 Ballots \u2192 BB\n  tally-bb (mk Y sk zk) bs = < Y ,, zk ,, bs ,, decryption-shares (mk Y sk zk) bs >\n\n  tally : KeyPair \u2192 Ballots \u2192 Tally\n  tally kp bs = result (tally-bb kp bs)\n\nmodule Run-common (pk : PublicKey) where\n    run-vote-query\u00b9 : R-vote \u2192 VoterId \u2192 Vote \u2192 Ballots \u2192 Ballots\n    run-vote-query\u00b9 r-vote vid v bs = case validate pk bs (vid , b)\n                                        0: bs\n                                        1: ((vid , b) \u2237 bs)\n      where\n        b : Ballot\n        b = vote pk r-vote vid v\n\n    run-vote-query : R-vote \u00b2 \u2192 VoterId \u2192 Vote \u00b2 \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-vote-query r-vote vid v_ bs_ i = run-vote-query\u00b9 (r-vote i) vid (v_ i) (bs_ i)\n\n    run-ballot-query : (\u03b2 : \ud835\udfda) \u2192 VoterId \u2192 Ballot \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-ballot-query \u03b2 vid b bs_ i =\n        case validate pk (bs_ \u03b2) (vid , b)\n          0: bs_ i\n          1: case validate pk (bs_ (not \u03b2)) (vid , b)\n               0: case i\n                    0: ((vid , b) \u2237 bs_ 0\u2082)\n                    1: bs_ 1\u2082\n               1: ((vid , b) \u2237 bs_ i)\n\nmodule Run (\u03b2 : \ud835\udfda) (pk : PublicKey) where\n    open Run-common pk\n\n    run-query : R-vote \u00b2 \u2192 VoterId \u2192 Q \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-query r-vote vid (ask-vote v_)  = run-vote-query r-vote vid v_\n    run-query r-vote vid (ask-ballot b) = run-ballot-query \u03b2 vid b\n\n    run-voting-phase : List (VotingPhase\u2080 \u00d7 R-vote \u00b2) \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-voting-phase []                  bs_ = bs_\n    run-voting-phase ((q , r-vote) \u2237 qs) bs_ =\n       run-voting-phase qs (uncurry (run-query r-vote) (q (bs_ \u03b2)) bs_)\n\nHack-run-vote-query : \u2115 \u2192 \u2605\nHack-run-vote-query #q = Fin #q \u2192 \ud835\udfda \u2192 \ud835\udfda\n\nmodule _ {-(max-q-1 : \u2115)-} where\n  -- max-q = \u2115.suc max-q-1\n  Adversary = R-adversary \u2192\n              PublicKey \u2192\n              -- \u03a3 _ \u03bb (#q : Fin max-q) \u2192\n              VotingPhase \u00d7 -- (Fin\u25b9\u2115 #q) \u00d7\n              (BB \u2192\n              \ud835\udfda)\n\n  map-vote\u00b2-in-voting-phase : (Vote \u00b2 \u2192 Vote \u00b2) \u2192 VotingPhase\u2080 \u2192 VotingPhase\u2080\n  map-vote\u00b2-in-voting-phase f = _\u2218_ (second (map-vote\u00b2-Q f))\n\n  map-Advesary : (VotingPhase \u2192 VotingPhase) \u2192 Adversary \u2192 Adversary\n  map-Advesary f A r\u2090 pk = first f (A r\u2090 pk)\n\n  module CommonChallenger\n              (setup : R-setup \u2192 KeyPair)\n              (r-setup : R-setup)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              where\n    kp = setup r-setup\n    open KeyPair kp public\n\n    A-setup = A r-adversary pk\n\n    A-voting-phase : VotingPhase\n    A-voting-phase = fst A-setup\n\n  -- The real challenger (has access to \u03b2)\n  module Game (setup : R-setup \u2192 KeyPair)\n              (decryption-shares : Make-decryption-shares)\n\n              (r-setup : R-setup)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              -- (hack-run-vote-queries : VotingPhase \u2192 VotingPhase)\n              -- (hack-run-vote-query : (x : Fin max-q) \u2192 Hack-run-vote-query (Fin\u25b9\u2115 x))\n              (r-votes : List (R-vote \u00b2))\n              (simulate-decryption-shares : Simulate-decryption-shares)\n              (\u03b2 : \ud835\udfda)\n              where\n\n    open CommonChallenger setup r-setup r-adversary A public\n\n    open Run \u03b2 pk\n\n    -- zip : \u2200 {n m} {o : Fin n} (xs : Vec A (Fin\u25b9\u2115 o)) (ys : Vec A\n\n    voting-phase : List (VotingPhase\u2080 \u00d7 R-vote \u00b2)\n    voting-phase = zip A-voting-phase r-votes\n\n    bs_ : Ballots \u00b2\n    bs_ = run-voting-phase voting-phase (const [])\n\n    decryption-shares-\u03b2 =\n      (case \u03b2\n         0: decryption-shares\n         1: simulate-decryption-shares (bs_ 0\u2082))\n      kp (bs_ \u03b2)\n\n    bb : BB\n    bb = record { Y-pk = pk ; \u03c0-pk = zk ; ballots = bs_ \u03b2 ; decryption-shares = decryption-shares-\u03b2 }\n\n    exp : \ud835\udfda\n    exp = snd A-setup bb\n\n    game = exp == \u03b2\n\n  module Unused-simulate-decryption-shares\n              (decryption-shares : Make-decryption-shares)\n              (result : BB \u2192 Tally)\n              (kp  : KeyPair)\n              (bs_ : Ballots \u00b2)\n           where\n    open KeyPair kp\n    open Tallying decryption-shares result\n    simulated-decryption-shares : Decryption-shares\n    simulated-decryption-shares = fake-decrytion-share (tally kp (bs_ 0\u2082)) (bs_ 1\u2082) \u2237 []\n\n    {-\n  module Simulated-Challenger\n              (r-setup : R-setup)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n            where\n    open CommonChallenger r-setup r-adversary A public\n-}\n  module Proof (setup : R-setup \u2192 KeyPair)\n               (result : BB \u2192 Tally)\n               (decryption-shares  decryption-shares\u2090 : Make-decryption-shares)\n               (r-setup : R-setup)\n               (r-adversary : R-adversary)\n               (A : Adversary)\n               -- (r-votes : (x : Fin max-q) (y : Fin (Fin\u25b9\u2115 x)) \u2192 R-vote \u00b2)\n               (r-votes : List (R-vote \u00b2))\n               (simulate-decryption-shares : Simulate-decryption-shares)\n               (simulate-zksecret : (pk : PublicKey) \u2192 NIZKSecretKey pk)\n               where\n     hack-voting-phase\u2080 = map-vote\u00b2-in-voting-phase (\u03bb v i \u2192 v (not i))\n\n     hack-all-queries : Adversary \u2192 Adversary\n     hack-all-queries = map-Advesary (map hack-voting-phase\u2080)\n\n     hack-query : \u2115 \u2192 Adversary \u2192 Adversary\n     hack-query n = map-Advesary (map-n\u1d57\u02b0 n hack-voting-phase\u2080)\n\n     hack-upto : \u2115 \u2192 Adversary \u2192 Adversary\n     hack-upto zero    = id\n     hack-upto (suc n) = hack-upto n \u2218 hack-query n\n\n     module EXP = Game setup decryption-shares r-setup r-adversary A r-votes simulate-decryption-shares\n     G\u2080 = EXP.exp 0\u2082\n     G\u2081 = EXP.exp 1\u2082\n\n     setup\u2090 : R-setup \u2192 KeyPair\n     setup\u2090 r-setup = record (setup r-setup) { zk = simulate-zksecret _ }\n\n     {-\n     from G\u2080\n       change setup             to setup\u2090\n       change decryption-shares to decryption-shares\u2090\n\n       because \u03b2 \u2261 0\u2082 simulate-decryption-shares is not used\n     -}\n     module EXP\u2090 = Game setup\u2090 decryption-shares\u2090 r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     G\u2090 = EXP\u2090.exp\n\n     -- TODO: G\u2080 \u2248 G\u2090\n\n     simulate-decryption-shares-B : Make-decryption-shares\n     simulate-decryption-shares-B kp bs = fake-decrytion-share t bs \u2237 []\n       where\n         open Tallying decryption-shares\u2090 result\n         t  = tally kp bs\n\n     {-\n     from G\u2090\n       change decryption-shares\u2090 to simulate-decryption-shares-B\n     -}\n     module EXP-B = Game setup\u2090 simulate-decryption-shares-B r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n     G-B = EXP-B.exp\n\n     {-\n     G-C-last 0 is EXP-B\n     -}\n     module EXP-C-n\u1d57\u02b0 (n : \u2115) = Game setup\u2090 simulate-decryption-shares-B r-setup r-adversary (hack-query n A) r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n     module EXP-C-upto (n : \u2115) = Game setup\u2090 simulate-decryption-shares-B r-setup r-adversary (hack-upto n A) r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     {-\n     like G-B but all vote queries are \"hacked\", namely moved from left(0\u2082) to right(1\u2082)\n     -}\n     module EXP-C = Game setup\u2090 simulate-decryption-shares-B r-setup r-adversary (hack-all-queries A) r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n     G-C = EXP-C.exp\n\n     {- (revert G\u2090\u2192G-B)\n     from G-C\n       change simulate-decryption-shares-B to decryption-shares\u2090\n     -}\n     module EXP-D = Game setup\u2090 decryption-shares\u2090 r-setup r-adversary (hack-all-queries A) r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n     G-D = EXP-D.exp\n\n     {- (revert G\u2080\u2192G\u2090)\n     from G-D\n       change setup\u2090             to setup\n       change decryption-shares\u2090 to decryption-shares\n     -}\n     module EXP-E = Game setup decryption-shares r-setup r-adversary (hack-all-queries A) r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n     G-E = EXP-E.exp\n\n     -- G-E \u2248 G\u2081\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function\nopen import Data.List\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\n{-\nopen import Data.Zero\nopen import Data.One\n-}\nopen import Data.Two\nopen import Data.Nat.NP hiding (_==_; _<_)\nopen import Control.Strategy\n\nmodule Helios\n  (VoterId : \u2605)\n  (PublicKey : \u2605)\n  (SecretKey : \u2605)\n  (NIZKSecretKey : PublicKey \u2192 \u2605)\n  (Vote   : \u2605)\n  (Ballot : \u2605)\n  (Decryption-share : \u2605)\n  (PublicInfo : \u2605)\n  (R-setup : \u2605)\n  (R-vote : \u2605)\n  (vote : PublicKey \u2192 R-vote \u2192 VoterId \u2192 Vote \u2192 Ballot)\n  (let Ballots = List (VoterId \u00d7 Ballot)\n       Decryption-shares = List Decryption-share\n  )\n  (validate-ballots : PublicKey \u2192 Ballots \u2192 \ud835\udfda)\n  (Tally : \u2605) -- could be a valid tally or a special symbol \u22a5\n  (fake-decrytion-share : Tally \u2192 Ballots \u2192 Decryption-share)\n  (R-adversary : \u2605)\n  where\n\nrecord KeyPair : \u2605 where\n  constructor mk\n  field\n    pk : PublicKey\n    sk : SecretKey\n    zk : NIZKSecretKey pk\n\n\nrecord BB : \u2605 where\n  constructor <_,,_,,_,,_>\n  field\n    Y-pk              : PublicKey\n    \u03c0-pk              : NIZKSecretKey Y-pk\n    ballots           : Ballots\n    decryption-shares : Decryption-shares\nopen BB public\n\nMake-decryption-shares = KeyPair \u2192 Ballots \u2192 Decryption-shares\nSimulate-decryption-shares = (bs\u2080 : Ballots) \u2192 Make-decryption-shares\n\nvalidate : PublicKey \u2192 Ballots \u2192 VoterId \u00d7 Ballot \u2192 \ud835\udfda\nvalidate pk bs vb = validate-ballots pk (vb \u2237 bs)\n\ndata Q : \u2605 where\n  ask-vote   : (v_ : Vote \u00b2) \u2192 Q\n  ask-ballot : (b  : Ballot) \u2192 Q\n\nVotingPhase = List (Ballots \u2192 VoterId \u00d7 Q)\n\nmodule Tallying\n  (decryption-shares : Make-decryption-shares)\n  (result : BB \u2192 Tally)\n  where\n\n  tally-bb : KeyPair \u2192 Ballots \u2192 BB\n  tally-bb (mk Y sk zk) bs = < Y ,, zk ,, bs ,, decryption-shares (mk Y sk zk) bs >\n\n  tally : KeyPair \u2192 Ballots \u2192 Tally\n  tally kp bs = result (tally-bb kp bs)\n\nmodule Run-common\n           (pk : PublicKey)\n           (r-votes : VoterId \u2192 R-vote \u00b2) where\n    run-vote-query\u00b9 : R-vote \u2192 VoterId \u2192 Vote \u2192 Ballots \u2192 Ballots\n    run-vote-query\u00b9 r-vote vid v bs = case validate pk bs (vid , b)\n                                        0: bs\n                                        1: ((vid , b) \u2237 bs)\n      where\n        b : Ballot\n        b = vote pk r-vote vid v\n\n    run-vote-query : (hack-run-vote-query : \ud835\udfda \u2192 \ud835\udfda) \u2192 VoterId \u2192 Vote \u00b2 \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-vote-query hack-run-vote-query vid v_ bs_ i = run-vote-query\u00b9 (r-votes vid i) vid (v_ (hack-run-vote-query i)) (bs_ i)\n\n    run-ballot-query : (\u03b2 : \ud835\udfda) \u2192 VoterId \u2192 Ballot \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-ballot-query \u03b2 vid b bs_ i =\n        case validate pk (bs_ \u03b2) (vid , b)\n          0: bs_ i\n          1: case validate pk (bs_ (not \u03b2)) (vid , b)\n               0: case i\n                    0: ((vid , b) \u2237 bs_ 0\u2082)\n                    1: bs_ 1\u2082\n               1: ((vid , b) \u2237 bs_ i)\n\nHack-run-vote-query = \u2115 \u2192 \ud835\udfda \u2192 \ud835\udfda\n\nmodule Run (\u03b2 : \ud835\udfda)\n           (hack-run-vote-query : Hack-run-vote-query)\n           (pk : PublicKey)\n           (r-votes : VoterId \u2192 R-vote \u00b2) where\n    open Run-common pk r-votes\n\n    run-query : (hack-run-vote-query : \ud835\udfda \u2192 \ud835\udfda) \u2192 VoterId \u2192 Q \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-query hack-run-vote-query vid (ask-vote v_)  = run-vote-query hack-run-vote-query vid v_\n    run-query hack-run-vote-query vid (ask-ballot b) = run-ballot-query \u03b2 vid b\n\n    run-voting-phase : VotingPhase \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-voting-phase []       bs_ = bs_\n    run-voting-phase (q \u2237 qs) bs_ = run-voting-phase qs (uncurry (run-query (hack-run-vote-query (length qs))) (q (bs_ \u03b2)) bs_)\n\nmodule _ where\n  Adversary = R-adversary \u2192\n              PublicKey \u2192\n              VotingPhase \u00d7\n              (BB \u2192\n              \ud835\udfda)\n\n  module CommonChallenger\n              (setup : R-setup \u2192 KeyPair)\n              (r-setup : R-setup)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              where\n    kp = setup r-setup\n    open KeyPair kp public\n\n    A-setup = A r-adversary pk\n\n    A-voting-phase : VotingPhase\n    A-voting-phase = fst A-setup\n\n  -- The real challenger (has access to \u03b2)\n  module Game (setup : R-setup \u2192 KeyPair)\n              (decryption-shares : Make-decryption-shares)\n              (hack-run-vote-query : Hack-run-vote-query)\n\n              (r-setup : R-setup)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (simulate-decryption-shares : Simulate-decryption-shares)\n              (\u03b2 : \ud835\udfda)\n              where\n    open CommonChallenger setup r-setup r-adversary A public\n\n    open Run \u03b2 hack-run-vote-query pk r-votes\n\n    bs_ : Ballots \u00b2\n    bs_ = run-voting-phase A-voting-phase (const [])\n\n    decryption-shares-\u03b2 =\n      (case \u03b2\n         0: decryption-shares\n         1: simulate-decryption-shares (bs_ 0\u2082))\n      kp (bs_ \u03b2)\n\n    bb : BB\n    bb = record { Y-pk = pk ; \u03c0-pk = zk ; ballots = bs_ \u03b2 ; decryption-shares = decryption-shares-\u03b2 }\n\n    exp : \ud835\udfda\n    exp = snd A-setup bb\n\n    game = exp == \u03b2\n\n  module Unused-simulate-decryption-shares\n              (decryption-shares : Make-decryption-shares)\n              (result : BB \u2192 Tally)\n              (kp  : KeyPair)\n              (bs_ : Ballots \u00b2)\n           where\n    open KeyPair kp\n    open Tallying decryption-shares result\n    simulated-decryption-shares : Decryption-shares\n    simulated-decryption-shares = fake-decrytion-share (tally kp (bs_ 0\u2082)) (bs_ 1\u2082) \u2237 []\n\n    {-\n  module Simulated-Challenger\n              (r-setup : R-setup)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n            where\n    open CommonChallenger r-setup r-adversary A public\n-}\n  module Proof (setup : R-setup \u2192 KeyPair)\n               (result : BB \u2192 Tally)\n               (decryption-shares  decryption-shares\u2090 : Make-decryption-shares)\n               (r-setup : R-setup)\n               (r-adversary : R-adversary)\n               (A : Adversary)\n               (r-votes : VoterId \u2192 R-vote \u00b2)\n               (simulate-decryption-shares : Simulate-decryption-shares)\n               (simulate-zksecret : (pk : PublicKey) \u2192 NIZKSecretKey pk)\n               where\n     hack-no-queries : Hack-run-vote-query\n     hack-no-queries = const id\n     module EXP = Game setup decryption-shares hack-no-queries r-setup r-adversary A r-votes simulate-decryption-shares\n     G\u2080 = EXP.exp 0\u2082\n     G\u2081 = EXP.exp 1\u2082\n\n     setup\u2090 : R-setup \u2192 KeyPair\n     setup\u2090 r-setup = record (setup r-setup) { zk = simulate-zksecret _ }\n\n     module EXP\u2090 = Game setup\u2090 decryption-shares\u2090 hack-no-queries r-setup r-adversary A r-votes simulate-decryption-shares 0\u2082\n\n     G\u2090 = EXP\u2090.exp\n\n     simulate-decryption-shares-B : Make-decryption-shares\n     simulate-decryption-shares-B kp bs = fake-decrytion-share t bs \u2237 []\n       where\n         open Tallying decryption-shares\u2090 result\n         t  = tally kp bs\n\n     module EXP-B = Game setup\u2090 simulate-decryption-shares-B hack-no-queries r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     G-B = EXP-B.exp\n\n     hack-all-queries : Hack-run-vote-query\n     hack-all-queries = const not\n\n     Seq : \u2605 \u2192 \u2605\n     Seq A = \u2115 \u2192 A\n\n     _<_ : \u2115 \u2192 \u2115 \u2192 \ud835\udfda\n     x < y = suc x <= y\n\n     replicateSeq : {A : \u2605} \u2192 \u2115 \u2192 A \u2192 A \u2192 Seq A\n     replicateSeq n x y i = case (i < n) 0: x 1: y\n\n     hack-queries-up-to : \u2115 \u2192 Hack-run-vote-query\n     hack-queries-up-to n i = replicateSeq (n id not i\n\n     module EXP-C-up-to (n : \u2115) = Game setup\u2090 simulate-decryption-shares-B (hack-queries-up-to n) r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     module EXP-C = Game setup\u2090 simulate-decryption-shares-B hack-all-queries r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     G-C = EXP-C.exp\n\n     {- EXP-C-up-to zero -}\n\n     -- G\u2080\u223cG\u2090\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0f705a1c4095ccf40c3d074017f09490f3550b02","subject":"Simplified a proof.","message":"Simplified a proof.\n\nIgnore-this: 9a3f841d137bca087a14ef28ac6d7416\n\ndarcs-hash:20110217160257-3bd4e-34a47035e8fcc6e1f03998b1ad83625b9bd8c0ca.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/WellFoundedInductionATP.agda","new_file":"Draft\/McCarthy91\/WellFoundedInductionATP.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n----------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check #-}\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule Draft.McCarthy91.WellFoundedInductionATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\nopen import Draft.McCarthy91.RelationATP\n\nopen import LTC.Data.Nat\n\n----------------------------------------------------------------------------\n\n-- Adapted from LTC.Data.Nat.Induction.WellFoundedI.wfInd-LT\u2081.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 MCR n m \u2192 P n) \u2192 P m) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P accH Nn = accH Nn (helper Nn)\n  where\n    helper : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR n m \u2192 P n\n    helper Nm zN 0\u00abm = \u22a5-elim (0\u00abx\u2192\u22a5 Nm 0\u00abm)\n\n    -- This equation does not pass the termination check.\n    helper zN Nn n\u00ab0 = accH Nn\n      (\u03bb {n'} Nn' n'\u00abn \u2192\n        let n'\u00ab0 : MCR n' zero\n            n'\u00ab0 = \u00ab-trans Nn' Nn zN n'\u00abn n\u00ab0\n\n        in helper zN Nn' n'\u00ab0\n      )\n\n    helper (sN {m} Nm) (sN {n} Nn) Sn\u00abSm = helper Nm (sN Nn) Sn\u00abm\n      where\n        Sn\u00abm : MCR (succ n) m\n        Sn\u00abm = x\u00abSy\u2192x\u00aby (sN Nn) Nm Sn\u00abSm\n","old_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n----------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check #-}\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule Draft.McCarthy91.WellFoundedInductionATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\nopen import Draft.McCarthy91.RelationATP\n\nopen import LTC.Data.Nat\n\n----------------------------------------------------------------------------\n\n-- Adapted from LTC.Data.Nat.Induction.WellFoundedI.wfInd-LT\u2081.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 MCR n m \u2192 P n) \u2192 P m) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P accH Nn = accH Nn (helper Nn)\n  where\n    helper : \u2200 {m n} \u2192 N m \u2192 N n \u2192 MCR n m \u2192 P n\n    helper Nm zN 0\u00abm = \u22a5-elim (0\u00abx\u2192\u22a5 Nm 0\u00abm)\n\n    -- This equation does not pass the termination check.\n    helper zN (sN Nn) Sn\u00ab0 = accH (sN Nn)\n      (\u03bb {n'} Nn' n'\u00abSn \u2192\n        let n'\u00ab0 : MCR n' zero\n            n'\u00ab0 = \u00ab-trans Nn' (sN Nn) zN n'\u00abSn Sn\u00ab0\n\n        in helper zN Nn' n'\u00ab0\n      )\n\n    helper (sN {m} Nm) (sN {n} Nn) Sn\u00abSm = accH (sN Nn)\n      (\u03bb {n'} Nn' n'\u00abSn \u2192\n        let n\u00abm : MCR n m\n            n\u00abm = Sx\u00abSy\u2192x\u00aby Nn Nm Sn\u00abSm\n\n            n'\u00abn : MCR n' n\n            n'\u00abn = x\u00abSy\u2192x\u00aby Nn' Nn n'\u00abSn\n\n            n'\u00abm : MCR n' m\n            n'\u00abm = \u00ab-trans Nn' Nn Nm n'\u00abn n\u00abm\n\n        in  helper Nm Nn' n'\u00abm\n      )\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"61132f19dfcf62a56249c0213a52227a09d7d368","subject":"working out a few more cases for removing casts; its not quite right but more cases might help me know why","message":"working out a few more cases for removing casts; its not quite right but more cases might help me know why\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"cast-inert.agda","new_file":"cast-inert.agda","new_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import typed-expansion\nopen import lemmas-gcomplete\nopen import lemmas-complete\nopen import progress-checks\nopen import finality\nopen import preservation\n\nmodule cast-inert where\n  -- if a term is compelete and well typed, then the casts inside are all\n  -- identity casts and there are no failed casts\n  cast-inert : \u2200{\u0394 \u0393 d \u03c4} \u2192\n                  \u0393 gcomplete \u2192\n                  d dcomplete \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                  cast-id d -- all casts are id, and there are no failed casts\n  cast-inert gc dc TAConst = CIConst\n  cast-inert gc dc (TAVar x\u2081) = CIVar\n  cast-inert gc (DCLam dc x\u2081) (TALam x\u2082 wt) = CILam (cast-inert (gcomp-extend gc x\u2081 x\u2082) dc wt)\n  cast-inert gc (DCAp dc dc\u2081) (TAAp wt wt\u2081) = CIAp (cast-inert gc dc wt)\n                                                   (cast-inert gc dc\u2081 wt\u2081)\n  cast-inert gc () (TAEHole x x\u2081)\n  cast-inert gc () (TANEHole x wt x\u2081)\n  cast-inert gc (DCCast dc x x\u2081) (TACast wt x\u2082)\n    with eq-complete-consist x x\u2081 x\u2082\n  ... | refl = CICast (cast-inert gc dc wt)\n  cast-inert gc () (TAFailedCast wt x x\u2081 x\u2082)\n\n  -- relates expressions to the same thing with all identity casts removed\n  data no-id-casts : dhexp \u2192 dhexp \u2192 Set where\n    NICConst  : no-id-casts c c\n    NICVar    : \u2200{x} \u2192 no-id-casts (X x) (X x)\n    NICLam    : \u2200{x \u03c4 d d'} \u2192 no-id-casts d d' \u2192 no-id-casts (\u00b7\u03bb x [ \u03c4 ] d) (\u00b7\u03bb x [ \u03c4 ] d')\n    NICHole   : \u2200{u} \u2192 no-id-casts (\u2987\u2988\u27e8 u \u27e9) (\u2987\u2988\u27e8 u \u27e9)\n    NICNEHole : \u2200{d d' u} \u2192 no-id-casts d d' \u2192 no-id-casts (\u2987 d \u2988\u27e8 u \u27e9) (\u2987 d' \u2988\u27e8 u \u27e9)\n    NICAp     : \u2200{d1 d2 d1' d2'} \u2192 no-id-casts d1 d1' \u2192 no-id-casts d2 d2' \u2192 no-id-casts (d1 \u2218 d2) (d1' \u2218 d2')\n    NICCast   : \u2200{d d' \u03c4} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) d'\n    NICFailed : \u2200{d d' \u03c41 \u03c42} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- removing identity casts doesn't change the type\n  no-id-casts-type : \u2200{\u0393 \u0394 d \u03c4 d' } \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                                      no-id-casts d d' \u2192\n                                      \u0394 , \u0393 \u22a2 d' :: \u03c4\n  no-id-casts-type TAConst NICConst = TAConst\n  no-id-casts-type (TAVar x\u2081) NICVar = TAVar x\u2081\n  no-id-casts-type (TALam x\u2081 wt) (NICLam nic) = TALam x\u2081 (no-id-casts-type wt nic)\n  no-id-casts-type (TAAp wt wt\u2081) (NICAp nic nic\u2081) = TAAp (no-id-casts-type wt nic) (no-id-casts-type wt\u2081 nic\u2081)\n  no-id-casts-type (TAEHole x x\u2081) NICHole = TAEHole x x\u2081\n  no-id-casts-type (TANEHole x wt x\u2081) (NICNEHole nic) = TANEHole x (no-id-casts-type wt nic) x\u2081\n  no-id-casts-type (TACast wt x) (NICCast nic) = no-id-casts-type wt nic\n  no-id-casts-type (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) = TAFailedCast (no-id-casts-type wt nic) x x\u2081 x\u2082\n\n  -- simple lemma used below; if an application is final so is each\n  -- applicand. saves some redundant pattern matching.\n  ap-final : \u2200{d1 d2} \u2192 (d1 \u2218 d2) final \u2192 d1 final \u00d7 d2 final\n  ap-final (FBoxed (BVVal ()))\n  ap-final (FIndet (IAp x x\u2081 x\u2082)) = FIndet x\u2081 , x\u2082\n\n  -- removing identity casts doesn't change the ultimate answer produced\n  remove-casts-same : \u2200{\u0394 d1 d2 d1' d2' \u0393 \u03c4 } \u2192\n                           \u0394 , \u0393 \u22a2 d1 :: \u03c4 \u2192\n                           no-id-casts d1 d2 \u2192\n                           d1 \u21a6* d1' \u2192\n                           d2 \u21a6* d2' \u2192\n                           d1' final \u2192\n                           d2' final \u2192\n                           d1' == d2'\n  remove-casts-same TAConst NICConst step1 step2 f1 f2 = ! (finality* (FBoxed (BVVal VConst)) step1) \u00b7 finality* (FBoxed (BVVal VConst)) step2\n\n  remove-casts-same (TAVar x\u2081) NICVar MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TAVar x\u2081) NICVar MSRefl step2 (FIndet ()) f2\n  remove-casts-same (TAVar x\u2081) NICVar (MSStep (Step FHOuter () FHOuter) step1) step2 f1 f2\n\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl MSRefl (FBoxed (BVVal VLam)) (FBoxed (BVVal VLam)) = {!!} -- ap1 (\u03bb qq \u2192 \u00b7\u03bb _ [ _ ] qq) {!!}\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl MSRefl _ (FIndet ())\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl MSRefl (FIndet ()) _\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) _ (MSStep x\u2082 step2) f1 f2 = abort (boxedval-not-step (BVVal VLam) ( _ , x\u2082))\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) (MSStep x\u2082 step1) _ f1 f2 = abort (boxedval-not-step (BVVal VLam) ( _ , x\u2082))\n\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) MSRefl MSRefl (FIndet (IAp x x\u2081 x\u2082)) f2 with remove-casts-same wt nic MSRefl MSRefl (FIndet x\u2081) (\u03c01 (ap-final f2))\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) MSRefl MSRefl (FIndet (IAp x x\u2081 x\u2082)) f2 | refl with remove-casts-same wt\u2081 nic\u2081 MSRefl MSRefl x\u2082 (\u03c02 (ap-final f2))\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) MSRefl MSRefl (FIndet (IAp x x\u2081 x\u2082)) f2 | refl | refl = refl\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) _ (MSStep x step2) (FIndet (IAp x\u2081 x\u2082 x\u2083)) f2 = {!!}\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) (MSStep x step1) step2 f1 f2 = {!!}\n\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl MSRefl (FIndet IEHole) (FBoxed (BVVal ()))\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl MSRefl (FIndet IEHole) (FIndet IEHole) = refl\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl (MSStep (Step FHOuter () FHOuter) step2) (FIndet IEHole) f2\n  remove-casts-same (TAEHole x x\u2081) NICHole (MSStep (Step FHOuter () FHOuter) step1) step2 f1 f2\n\n  remove-casts-same (TANEHole x wt x\u2081) (NICNEHole nic) step1 step2 f1 f2 = {!!}\n\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TACast wt TCRefl) (NICCast nic) MSRefl step2 (FBoxed (BVArrCast x\u2081 x\u2082)) f2 = abort (x\u2081 refl)\n  remove-casts-same (TACast wt (TCArr x x\u2081)) (NICCast nic) MSRefl step2 (FBoxed (BVArrCast x\u2082 x\u2083)) f2 = abort (x\u2082 refl)\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FBoxed (BVHoleCast () x\u2082)) f2\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FIndet (ICastArr x\u2081 x\u2082)) f2 = abort (x\u2081 refl)\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FIndet (ICastGroundHole () x\u2082)) f2\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FIndet (ICastHoleGround x\u2081 x\u2082 ())) f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep (Step FHOuter ITCastID FHOuter) step1) step2 f1 f2 = remove-casts-same wt nic step1 step2 f1 f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep (Step FHOuter (ITCastSucceed x\u2081) FHOuter) step1) step2 f1 f2 = remove-casts-same wt nic\n                                                                                                                         (MSStep (Step FHOuter ITCastID FHOuter) step1) step2 f1 f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep (Step FHOuter (ITCastFail x\u2081 () x\u2083) FHOuter) step1) step2 f1 f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep (Step FHOuter (ITGround ()) FHOuter) step1) step2 f1 f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep (Step FHOuter (ITExpand ()) FHOuter) step1) step2 f1 f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep (Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) step1) step2 f1 f2 = {!!}\n\n    -- remove-casts-same wt nic (MSStep (Step x\u2081 x\u2082 x\u2083) {!!}) step2 f1 f2\n\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) MSRefl MSRefl (FBoxed (BVVal ())) f2\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) MSRefl MSRefl (FIndet (IFailedCast x\u2083 x\u2084 x\u2085 x\u2086)) (FBoxed (BVVal ()))\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) MSRefl MSRefl (FIndet (IFailedCast x\u2083 x\u2084 x\u2085 x\u2086)) (FIndet (IFailedCast x\u2087 x\u2088 x\u2089 x\u2081\u2080))\n    with remove-casts-same wt nic MSRefl MSRefl x\u2083 x\u2087\n  ... | refl = refl\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) MSRefl (MSStep x\u2083 step2) (FBoxed (BVVal ())) f2\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) MSRefl (MSStep x\u2083 step2) (FIndet (IFailedCast x\u2084 x\u2085 x\u2086 x\u2087)) f2 = {!!}\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) (MSStep x\u2083 step1) MSRefl f1 f2 = {!!}\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) (MSStep x\u2083 step1) (MSStep x\u2084 step2) f1 f2 = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import core\nopen import contexts\nopen import typed-expansion\nopen import lemmas-gcomplete\nopen import lemmas-complete\nopen import progress-checks\nopen import finality\n\nmodule cast-inert where\n  -- if a term is compelete and well typed, then the casts inside are all\n  -- identity casts and there are no failed casts\n  cast-inert : \u2200{\u0394 \u0393 d \u03c4} \u2192\n                  \u0393 gcomplete \u2192\n                  d dcomplete \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                  cast-id d\n  cast-inert gc dc TAConst = CIConst\n  cast-inert gc dc (TAVar x\u2081) = CIVar\n  cast-inert gc (DCLam dc x\u2081) (TALam x\u2082 wt) = CILam (cast-inert (gcomp-extend gc x\u2081 x\u2082) dc wt)\n  cast-inert gc (DCAp dc dc\u2081) (TAAp wt wt\u2081) = CIAp (cast-inert gc dc wt)\n                                                   (cast-inert gc dc\u2081 wt\u2081)\n  cast-inert gc () (TAEHole x x\u2081)\n  cast-inert gc () (TANEHole x wt x\u2081)\n  cast-inert gc (DCCast dc x x\u2081) (TACast wt x\u2082)\n    with eq-complete-consist x x\u2081 x\u2082\n  ... | refl = CICast (cast-inert gc dc wt)\n  cast-inert gc () (TAFailedCast wt x x\u2081 x\u2082)\n\n  -- relates expressions to the same thing with all identity casts removed\n  data no-id-casts : dhexp \u2192 dhexp \u2192 Set where\n    NICConst  : no-id-casts c c\n    NICVar    : \u2200{x} \u2192 no-id-casts (X x) (X x)\n    NICLam    : \u2200{x \u03c4 d d'} \u2192 no-id-casts d d' \u2192 no-id-casts (\u00b7\u03bb x [ \u03c4 ] d) (\u00b7\u03bb x [ \u03c4 ] d')\n    NICHole   : \u2200{u} \u2192 no-id-casts (\u2987\u2988\u27e8 u \u27e9) (\u2987\u2988\u27e8 u \u27e9)\n    NICNEHole : \u2200{d d' u} \u2192 no-id-casts d d' \u2192 no-id-casts (\u2987 d \u2988\u27e8 u \u27e9) (\u2987 d' \u2988\u27e8 u \u27e9)\n    NICAp     : \u2200{d1 d2 d1' d2'} \u2192 no-id-casts d1 d1' \u2192 no-id-casts d2 d2' \u2192 no-id-casts (d1 \u2218 d2) (d1' \u2218 d2')\n    NICCast   : \u2200{d d' \u03c4} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) d'\n    NICFailed : \u2200{d d' \u03c41 \u03c42} \u2192 no-id-casts d d' \u2192 no-id-casts (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- removing identity casts doesn't change the type\n  no-id-casts-type : \u2200{\u0393 \u0394 d \u03c4 d' } \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192 no-id-casts d d' \u2192 \u0394 , \u0393 \u22a2 d' :: \u03c4\n  no-id-casts-type TAConst NICConst = TAConst\n  no-id-casts-type (TAVar x\u2081) NICVar = TAVar x\u2081\n  no-id-casts-type (TALam x\u2081 wt) (NICLam nic) = TALam x\u2081 (no-id-casts-type wt nic)\n  no-id-casts-type (TAAp wt wt\u2081) (NICAp nic nic\u2081) = TAAp (no-id-casts-type wt nic) (no-id-casts-type wt\u2081 nic\u2081)\n  no-id-casts-type (TAEHole x x\u2081) NICHole = TAEHole x x\u2081\n  no-id-casts-type (TANEHole x wt x\u2081) (NICNEHole nic) = TANEHole x (no-id-casts-type wt nic) x\u2081\n  no-id-casts-type (TACast wt x) (NICCast nic) = no-id-casts-type wt nic\n  no-id-casts-type (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) = TAFailedCast (no-id-casts-type wt nic) x x\u2081 x\u2082\n\n\n  -- removing identity casts doesn't change the ultimate answer produced\n  remove-casts-same : \u2200{\u0394 d1 d2 d1' d2' \u0393 \u03c4 } \u2192\n                           \u0394 , \u0393 \u22a2 d1 :: \u03c4 \u2192\n                           no-id-casts d1 d2 \u2192\n                           d1 \u21a6* d1' \u2192\n                           d2 \u21a6* d2' \u2192\n                           d1' final \u2192\n                           d2' final \u2192\n                           d1' == d2'\n  remove-casts-same TAConst NICConst step1 step2 f1 f2 = ! (finality* (FBoxed (BVVal VConst)) step1) \u00b7 finality* (FBoxed (BVVal VConst)) step2\n\n  remove-casts-same (TAVar x\u2081) NICVar MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TAVar x\u2081) NICVar MSRefl step2 (FIndet ()) f2\n  remove-casts-same (TAVar x\u2081) NICVar (MSStep (Step FHOuter () FHOuter) step1) step2 f1 f2\n\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl MSRefl (FBoxed (BVVal VLam)) (FBoxed (BVVal VLam)) = {!!} -- ap1 (\u03bb qq \u2192 \u00b7\u03bb _ [ _ ] qq) {!!}\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl MSRefl _ (FIndet ())\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl MSRefl (FIndet ()) _\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) MSRefl (MSStep x\u2082 step2) f1 f2 = abort (boxedval-not-step (BVVal VLam) ( _ , x\u2082))\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) (MSStep x\u2082 step1) MSRefl f1 f2 = abort (boxedval-not-step (BVVal VLam) ( _ , x\u2082))\n  remove-casts-same (TALam x\u2081 wt) (NICLam nic) (MSStep x\u2082 step1) (MSStep x\u2083 step2) f1 f2 = {!!}\n\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) MSRefl step2 (FIndet (IAp x x\u2081 x\u2082)) f2 = {!!}\n  remove-casts-same (TAAp wt wt\u2081) (NICAp nic nic\u2081) (MSStep x step1) step2 f1 f2 = {!!}\n\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl MSRefl (FIndet IEHole) (FBoxed (BVVal ()))\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl MSRefl (FIndet IEHole) (FIndet IEHole) = refl\n  remove-casts-same (TAEHole x x\u2081) NICHole MSRefl (MSStep (Step FHOuter () FHOuter) step2) (FIndet IEHole) f2\n  remove-casts-same (TAEHole x x\u2081) NICHole (MSStep (Step FHOuter () FHOuter) step1) step2 f1 f2\n\n  remove-casts-same (TANEHole x wt x\u2081) (NICNEHole nic) step1 step2 f1 f2 = {!!}\n\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FBoxed (BVVal ())) f2\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FBoxed (BVArrCast x\u2081 x\u2082)) f2 = {!!}\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FBoxed (BVHoleCast () x\u2082)) f2\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FIndet (ICastArr x\u2081 x\u2082)) f2 = {!!}\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FIndet (ICastGroundHole () x\u2082)) f2\n  remove-casts-same (TACast wt x) (NICCast nic) MSRefl step2 (FIndet (ICastHoleGround x\u2081 x\u2082 ())) f2\n  remove-casts-same (TACast wt x) (NICCast nic) (MSStep x\u2081 step1) step2 f1 f2 = {!!}\n\n  remove-casts-same (TAFailedCast wt x x\u2081 x\u2082) (NICFailed nic) step1 step2 f1 f2 = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"caaa7157a5a86db63d515710f87d0c806b77e3e3","subject":"Removed unnecessary import.","message":"Removed unnecessary import.\n\nIgnore-this: 8e1e69ee36df5076ff21e131d2de95f6\n\ndarcs-hash:20111013131446-3bd4e-e8671806671297c7debf1f8543409ea2935a3218.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/Even.agda","new_file":"Draft\/FOTC\/Data\/Nat\/Even.agda","new_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\n\ndata Even : D \u2192 Set where\n  zeroeven : Even zero\n  nexteven : \u2200 {d} \u2192 Even d \u2192 Even (succ (succ d))\n\nindEven : (P : D \u2192 Set) \u2192\n          P zero \u2192\n          (\u2200 {d} \u2192 P d \u2192 P (succ (succ d))) \u2192\n          \u2200 {d} \u2192 Even d \u2192 P d\nindEven P P0 h zeroeven      = P0\nindEven P P0 h (nexteven Ed) = h (indEven P P0 h Ed)\n","old_contents":"module Draft.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\n\ndata Even : D \u2192 Set where\n  zeroeven : Even zero\n  nexteven : \u2200 {d} \u2192 Even d \u2192 Even (succ (succ d))\n\nindEven : (P : D \u2192 Set) \u2192\n          P zero \u2192\n          (\u2200 {d} \u2192 P d \u2192 P (succ (succ d))) \u2192\n          \u2200 {d} \u2192 Even d \u2192 P d\nindEven P P0 h zeroeven      = P0\nindEven P P0 h (nexteven Ed) = h (indEven P P0 h Ed)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0dd0444807737b1fb543e964f7e4558b35fd2654","subject":"Possible agda interactive bug.","message":"Possible agda interactive bug.\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/LinFunContructw.agda","new_file":"agda\/LinFunContructw.agda","new_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n  \nmodule _ where\n\n  open IndexLLProp\n\n  boo : \u2200{ i u ll pll ell cs} \u2192 \u2200 s \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL ell ll) \u2192 \u2200 \u00achob \u00achoh\n        \u2192 let bind = \u00acho-shr-morph s eq ind \u00achob in\n          let hind = \u00acho-shr-morph s eq lind \u00achoh in\n          Ordered\u1d62 bind hind \u2192 Ordered\u1d62 ind lind\n  boo \u2193 () ind lind \u00achob \u00achoh ord\n\n  boo (s \u2190\u2227) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2193)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2193)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (a\u2264\u1d62b y)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 z)) (b\u2264\u1d62a y)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x\u2081)\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2193) \n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (s \u2190\u2227) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq lind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s lind)\n    \u00acnho x = \u00achoh (hitsAtLeastOnce\u2190\u2227\u2190\u2227 x)\n  boo (s \u2190\u2227) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (s \u2190\u2227) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo {ll = lll \u2227 rll} (s \u2190\u2227) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s \n  boo {ll = lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 with shrink rll (fillAllLower rll) | shr-fAL-id rll\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b x) | \u2205 | .rll | refl = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a x) | \u2205 | .rll | refl = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)\n  boo {ll = lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u00ac\u2205 x with shrink rll (fillAllLower rll) | shr-fAL-id rll\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | .rll | refl = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)\n  boo {u = _} {lll \u2227 rll} (s \u2190\u2227) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | .rll | refl = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)\n  boo (\u2227\u2192 s) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2227\u2192\u2193)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2227\u2192\u2193)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (a\u2264\u1d62b y)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] with r where\n   r = boo s ieq ind lind (\u03bb z \u2192 \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (\u03bb z \u2192 \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 z)) (b\u2264\u1d62a y)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | a\u2264\u1d62b x\u2081 = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 y)) | \u00ac\u2205 x | [ ieq ] | b\u2264\u1d62a x\u2081 = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x\u2081)\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) refl (\u2227\u2192 ind) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2227\u2192\u2193) \n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s\n  boo (\u2227\u2192 s) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord | \u2205 | [ ieq ] = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq lind))  where\n    \u00acnho : \u00ac (hitsAtLeastOnce s lind)\n    \u00acnho x = \u00achoh (hitsAtLeastOnce\u2227\u2192\u2227\u2192 x)\n  boo (\u2227\u2192 s) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (a\u2264\u1d62b ()) | \u00ac\u2205 x | [ ieq ]\n  boo (\u2227\u2192 s) refl (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh (b\u2264\u1d62a ()) | \u00ac\u2205 x | [ ieq ]\n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s \n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 with shrink lll (fillAllLower lll) | shr-fAL-id lll\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b x) | \u2205 | .lll | refl = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a x) | \u2205 | .lll | refl = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)\n  boo {ll = lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x with shrink lll (fillAllLower lll) | shr-fAL-id lll\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | .lll | refl = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 y)\n  boo {u = _} {lll \u2227 rll} (\u2227\u2192 s) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)) | \u00ac\u2205 x | .lll | refl = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 y)\n\n\n\n\n  boo (s \u2190\u2227\u2192 s\u2081) eq \u2193 lind \u00achob \u00achoh ord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) \u2193 \u00achob \u00achoh ord = \u22a5-elim (\u00achoh hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord with complL\u209b s | inspect complL\u209b s | complL\u209b s\u2081\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u2205 | [ ieq ] | e = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s ieq ind)) where\n    \u00acnho : \u00ac (hitsAtLeastOnce s ind)\n    \u00acnho x = \u00achob (hitsAtLeastOnce\u2190\u2227\u2192\u2190\u2227 x)\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u2205 with boo s ieq ind lind {!!} {!!} ord\n  ... | g = {!!}\n  boo (s \u2190\u2227\u2192 s\u2081) refl (ind \u2190\u2227) (lind \u2190\u2227) \u00achob \u00achoh ord | \u00ac\u2205 x | [ ieq ] | \u00ac\u2205 x\u2081 = {!!}\n  boo (s \u2190\u2227\u2192 s\u2081) eq (ind \u2190\u2227) (\u2227\u2192 lind) \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2227\u2192 s\u2081) eq (\u2227\u2192 ind) lind \u00achob \u00achoh ord = {!!}\n\n  boo (s \u2190\u2228) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (\u2228\u2192 s) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2228\u2192 s\u2081) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2202) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (\u2202\u2192 s) eq ind lind \u00achob \u00achoh ord = {!!}\n  boo (s \u2190\u2202\u2192 s\u2081) eq ind lind \u00achob \u00achoh ord = {!!}\n\n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n    hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n          \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n          \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n          \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n    hf2 \u2205 mnho s lf eqs = \u2205\n    hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (\u03bb z \u2192 tranLFMIndexLL lf (\u00ac\u2205 z)) x\n    hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n    hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n\n\n  data MLFun {i u ll rll n dt df} (mind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s mind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    \u2205 : \u2200{ts} \u2192 (\u2205 \u2261 mind) \u2192 (complL\u209b s \u2261 \u2205)\n        \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun mind s m\u00acho lf\n    \u00ac\u2205\u2205 : \u2200 {ind cs} \u2192 (\u00ac\u2205 ind \u2261 mind) \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n          \u2192 (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n          \u2192 (cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) rll\n          \u2192 MLFun mind s m\u00acho lf \n    \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf mind in\n           (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 mind s m\u00acho lf eqs) cll))\n           \u2192 MLFun mind s m\u00acho lf \n           --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n           -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n    -- s here does contain the ind.\n  -- TODO IMPORTANT After test , we need to remove \u2282 that contain I in the first place.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test ind s mnho lf with complL\u209b s | inspect complL\u209b s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u2205 | [ r ] = IMPOSSIBLE\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] with complL\u209b x | inspect complL\u209b x\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u2205 | [ t ] = \u2205 refl e (sym r) t\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf | \u2205 | [ e ] = \u22a5-elim (\u00acho (compl\u2261\u2205\u21d2ho s e x)) \n\n  test \u2205 s mnho I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n  test (\u00ac\u2205 x\u2081) s (\u00ac\u2205 \u00acho) I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  ... | \u2205 indeq ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n  ... | \u00ac\u2205\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00ac\u2205\u00ac\u2205 eq tseq ceqo ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n  ... | \u2205 indeq ceq eqs ceqo with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  ... | \u00ac\u2205 mx | r with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u2205 = IMPOSSIBLE -- IMPOSSIBLE and not provable with the current information we have.\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x with compl\u2261\u00ac\u2205\u21d2replace-compl\u2261\u00ac\u2205 s lind eq teq ceq x\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , proj\u2084 with complL\u209b (replacePartOf s to x at ind)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 refl eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , () | .\u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi1 eqs1 cll1 t1\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 {ts = ts} indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 {cs} ceqi1 eqs1 ceqo1 t1\n    = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 ((\u03bb cll \u2192 subst (\u03bb z \u2192 LFun z (shrink rll _)) (shrink-repl\u2261\u2205 s lind eq teq ceq _ ceqo t) (t1 cll))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    t : complL\u209b (replacePartOf s to ts at lind) \u2261 \u00ac\u2205 cs\n    t with trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    ... | g with replacePartOf s to ts at lind\n    t | refl | e = ceqi1\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq refl ceqo | \u2205 | [ () ] | .(\u00ac\u2205 _)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind\u2081} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t\n       with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi () ceqo t | \u2205 | [ meq ] | \u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ () ] | \u00ac\u2205 _\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u2205 indeq ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl ceqo t | \u00ac\u2205 .(replacePartOf s to _ at lind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | .(\u00ac\u2205 _) with complL\u209b ts | ct where\n    m = replacePartOf s to ts at lind\n    mind = subst (\u03bb z \u2192 IndexLL z (replLL ll lind ell)) (replLL-\u2193 lind) ((a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))\n    ct = compl\u2261\u2205\u21d2compltr\u2261\u2205 m ceq1 mind (sym (tr-repl\u21d2id s lind ts))\n  test {rll = rll} {ll} {n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl () t | \u00ac\u2205 .(replacePartOf s to ts at ind) | [ refl ] | \u2205 indeq ceq1 eqs1 ceqo1 | .(\u00ac\u2205 ts) | .\u2205 | refl\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi\u2081 eqs\u2081 cll1 x\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 ceqi1 eqs1 ceqo1 t1\n      = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 (\u03bb cll \u2192 _\u2282_ {ind = hind} (t cll) (subst (\u03bb z \u2192 LFun z _) (sym (ho-shrink-repl-comm s lind eq teq ceqi ceqo w ceqi1)) (t1 cll))) where\n    w = trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    hind = ho-shr-morph s eq lind (sym teq) ceqi\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with isLTi lind ind\n  ... | no \u00acp with test {n = n} {dt} {df} nind mx (\u00ac\u2205 n\u00acho) lf\u2081 where\n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    nind = \u00ac\u2205 (\u00acord-morph ind lind ell (flipNotOrd\u1d62 nord))\n    n\u00acho = \u00acord&\u00acho-del\u21d2\u00acho ind s \u00acho lind nord (sym meq)\n  ... | \u2205 () ceq eqs ceqo\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u2205 {ind = tind} indeq ceqi eqs cll t\n    = \u00ac\u2205\u2205 refl eq tseq cll (_\u2282_ {ind = sind} lf {!best!}) where \n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    hind = \u00acho-shr-morph s eq lind ((trunc\u2261\u2205\u21d2\u00acho s lind teq))\n    bind = \u00acho-shr-morph s eq ind \u00acho\n    sind = \u00acord-morph hind bind cll {!!}\n    best = replLL-\u00acordab\u2261ba bind cll hind ell {!!}\n\n    find = \u00acord-morph ind lind ell (flipNotOrd\u1d62 nord)\n    nind = \u00acord-morph lind ind cll nord\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t = {!!}\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | yes p = {!!}\n  test (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] = {!!}\n  test ind s mnho (tr ltr lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (com df\u2081 lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (call x\u2081) | \u00ac\u2205 x | eq = {!!}\n\n\n\n\n\n--  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | \u00acI\u00ac\u2205 ceqi () ceqo x | .(\u00ac\u2205 _)\n--\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n--\n--\n---- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n----   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n----     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n---- --    is = test \u2205 {!!} \u2205 lf\u2081\n----   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n----   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n----   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n----   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n----     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n----     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n----   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n--  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n--  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n--  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n--  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n--  \n--  \n  \n  \n  \n  \n  \n","old_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n  \nmodule _ where\n\n  open IndexLLProp \n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n    hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n          \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n          \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n          \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n    hf2 \u2205 mnho s lf eqs = \u2205\n    hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (\u03bb z \u2192 tranLFMIndexLL lf (\u00ac\u2205 z)) x\n    hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n    hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n\n\n  data MLFun {i u ll rll n dt df} (mind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s mind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    \u2205 : \u2200{ts} \u2192 (\u2205 \u2261 mind) \u2192 (complL\u209b s \u2261 \u2205)\n        \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun mind s m\u00acho lf\n    \u00ac\u2205\u2205 : \u2200 {ind cs} \u2192 (\u00ac\u2205 ind \u2261 mind) \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n          \u2192 (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n          \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) rll)\n          \u2192 MLFun mind s m\u00acho lf \n    \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf mind in\n           (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 mind s m\u00acho lf eqs) cll))\n           \u2192 MLFun mind s m\u00acho lf \n           --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n           -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n    -- s here does contain the ind.\n  -- TODO IMPORTANT After test , we need to remove \u2282 that contain I in the first place.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test ind s mnho lf with complL\u209b s | inspect complL\u209b s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u2205 | [ r ] = IMPOSSIBLE\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] with complL\u209b x | inspect complL\u209b x\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u2205 | [ t ] = \u2205 {!!} e (sym r) t\n  test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf | \u2205 | [ e ] = \u22a5-elim (\u00acho (compl\u2261\u2205\u21d2ho s e x)) \n\n  test \u2205 s mnho I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n  test (\u00ac\u2205 x\u2081) s (\u00ac\u2205 \u00acho) I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  ... | \u2205 indeq ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi eqs t\n  ... | \u00ac\u2205\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00ac\u2205\u00ac\u2205 eq tseq ceqo ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n    g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n  ... | \u2205 indeq ceq eqs ceqo with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  ... | \u00ac\u2205 mx | r with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u2205 = IMPOSSIBLE -- IMPOSSIBLE and not provable with the current information we have.\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x with compl\u2261\u00ac\u2205\u21d2replace-compl\u2261\u00ac\u2205 s lind eq teq ceq x\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , proj\u2084 with complL\u209b (replacePartOf s to x at ind)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 refl eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , () | .\u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi1 eqs1 t1\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 {ts = ts} indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 {cs} ceqi1 eqs1 ceqo1 t1\n    = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 ((\u03bb cll \u2192 subst (\u03bb z \u2192 LFun z (shrink rll _)) (shrink-repl\u2261\u2205 s lind eq teq ceq _ ceqo t) (t1 cll))) where\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    t : complL\u209b (replacePartOf s to ts at lind) \u2261 \u00ac\u2205 cs\n    t with trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    ... | g with replacePartOf s to ts at lind\n    t | refl | e = ceqi1\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq refl ceqo | \u2205 | [ () ] | .(\u00ac\u2205 _)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind\u2081} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u2205 () ceqi eqs t\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t\n       with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi () ceqo t | \u2205 | [ meq ] | \u2205\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ () ] | \u00ac\u2205 _\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u2205 indeq ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl ceqo t | \u00ac\u2205 .(replacePartOf s to _ at lind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | .(\u00ac\u2205 _) with complL\u209b ts | ct where\n    m = replacePartOf s to ts at lind\n    mind = subst (\u03bb z \u2192 IndexLL z (replLL ll lind ell)) (replLL-\u2193 lind) ((a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))\n    ct = compl\u2261\u2205\u21d2compltr\u2261\u2205 m ceq1 mind (sym (tr-repl\u21d2id s lind ts))\n  test {rll = rll} {ll} {n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl () t | \u00ac\u2205 .(replacePartOf s to ts at ind) | [ refl ] | \u2205 indeq ceq1 eqs1 ceqo1 | .(\u00ac\u2205 ts) | .\u2205 | refl\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi\u2081 eqs\u2081 x\u2081\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 ceqi1 eqs1 ceqo1 t1\n      = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 (\u03bb cll \u2192 _\u2282_ {ind = hind} (t cll) (subst (\u03bb z \u2192 LFun z _) (sym (ho-shrink-repl-comm s lind eq teq ceqi ceqo w ceqi1)) (t1 cll))) where\n    w = trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n    hind = ho-shr-morph s eq lind (sym teq) ceqi\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with isLTi lind ind\n  ... | no \u00acp with test {n = n} {dt} {df} nind mx (\u00ac\u2205 n\u00acho) lf\u2081 where\n    nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n    nind = \u00ac\u2205 (\u00acord-morph ind lind ell (flipNotOrd\u1d62 nord))\n    n\u00acho = \u00acord&\u00acho-del\u21d2\u00acho ind s \u00acho lind nord (sym meq)\n  ... | \u2205 () ceq eqs ceqo\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u2205 refl ceqi eqs t = {!!}\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t = {!!}\n  test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | yes p = {!!}\n  test (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] = {!!}\n  test ind s mnho (tr ltr lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (com df\u2081 lf) | \u00ac\u2205 x | eq = {!!}\n  test ind s mnho (call x\u2081) | \u00ac\u2205 x | eq = {!!}\n\n\n\n\n\n--  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | \u00acI\u00ac\u2205 ceqi () ceqo x | .(\u00ac\u2205 _)\n--\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n--  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n--\n--\n---- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n----   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n----     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n---- --    is = test \u2205 {!!} \u2205 lf\u2081\n----   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n----   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n----   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n----   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n----     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n----     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n----   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n--  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n--  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n--  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n--  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n--  \n--  \n  \n  \n  \n  \n  \n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"ff27c0a3f9695d2063af723c6c66408aef6ed3be","subject":"More regular names for validity for type","message":"More regular names for validity for type\n\nMatch validity for contexts.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ]\u03c4 da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ]\u03c4 db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ]\u03c4 \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = sym (right-id-int n)\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ]\u03c4 da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ]\u03c4 da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ]\u03c4 da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ]\u03c4 dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ]\u03c4 df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ] da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ] db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n[ sum \u03c3 \u03c4 ] dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = sym (right-id-int n)\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db9309775433335735edcd2ca4c5943b4a19e2aa","subject":"Validity of derivatives using proof-embedded \u0394-terms","message":"Validity of derivatives using proof-embedded \u0394-terms\n\nOld-commit-hash: 21985b6314506084e5b5171cd083140a95f42d05\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\ndata \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms\n-- (because it embeds validity proofs),\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                          valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n  \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394 \u03c1\n    dh = \u27e6 df \u27e7\u0394 \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\n\u0394-equiv : \u2200 {\u03c4 : Type} \u2192 (du : \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4) (dv : \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4) \u2192 Set\n\u0394-equiv {\u03c4} du dv =\n  \u2200 {v : \u27e6 \u03c4 \u27e7} (R[v,du] : valid v du) (R[v,dv] : valid v dv) \u2192\n    v \u2295 du \u2261 v \u2295 dv\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u0394-equiv (\u27e6 derive t \u27e7 \u03c1) (\u27e6 t \u27e7 (update \u03c1) \u229d \u27e6 t \u27e7 (ignore \u03c1))\n\ncorollary : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\ncorollary {\u03c4} {\u0393} {t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    v\u2032 = \u27e6 t \u27e7 (update \u03c1)\n  in\n    begin\n      v \u2295 \u27e6 derive t \u27e7 \u03c1\n    \u2261\u27e8 correctness {\u03c4} {\u0393} {t} {\u03c1} {v} validity R[v,u\u229dv] \u27e9\n      v \u2295 (v\u2032 \u229d v)\n    \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n      v\u2032\n    \u220e where open \u2261-Reasoning\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag b\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {nats} {\u0393} {nat n} = refl\nvalidity {bags} {\u0393} {bag b} = tt\nvalidity {\u03c4} {\u0393} {var x} = validity-var x\nvalidity {nats} {\u0393} {add s t} = cong\u2082 _+_ R[s,ds] R[t,dt]\n  where R[s,ds] = validity {nats} {\u0393} {s}\n        R[t,dt] = validity {nats} {\u0393} {t}\nvalidity {bags} {\u0393} {map f b} = tt\n\nvalidity {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 corollary {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (corollary {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\nvalidity {\u03c4} {\u0393} {app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\ncorrectness = {!!}\n\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\ndata \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms,\n-- (because it embeds validity proofs)\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192(ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                                 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {_} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {_} {u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car    : A\n    cadr   : B car\n    caddr  : C car cadr\n    cadddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n  \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394 \u03c1\n    dh = \u27e6 df \u27e7\u0394 \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\n\u0394-equiv : \u2200 {\u03c4 : Type} \u2192 (du : \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4) (dv : \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4) \u2192 Set\n\u0394-equiv {\u03c4} du dv =\n  \u2200 {v : \u27e6 \u03c4 \u27e7} (R[v,du] : valid v du) (R[v,dv] : valid v dv) \u2192\n    v \u2295 du \u2261 v \u2295 dv\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u0394-equiv (\u27e6 derive t \u27e7 \u03c1) (\u27e6 t \u27e7 (update \u03c1) \u229d \u27e6 t \u27e7 (ignore \u03c1))\n\ncorollary : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\ncorollary {\u03c4} {\u0393} {t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    v\u2032 = \u27e6 t \u27e7 (update \u03c1)\n  in\n    begin\n      v \u2295 \u27e6 derive t \u27e7 \u03c1\n    \u2261\u27e8 correctness {\u03c4} {\u0393} {t} {\u03c1} {v} validity R[v,u\u229dv] \u27e9\n      v \u2295 (v\u2032 \u229d v)\n    \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n      v\u2032\n    \u220e where open \u2261-Reasoning\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag b\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {nats} {\u0393} {nat n} = refl\nvalidity {bags} {\u0393} {bag b} = tt\nvalidity {\u03c4} {\u0393} {var x} = validity-var x\nvalidity {nats} {\u0393} {add s t} = cong\u2082 _+_ R[s,ds] R[t,dt]\n  where R[s,ds] = validity {nats} {\u0393} {s}\n        R[t,dt] = validity {nats} {\u0393} {t}\nvalidity {bags} {\u0393} {map f b} = tt\n\nvalidity {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v  dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {_} {_} {t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 corollary {_} {_} {t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 {!!} \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\n\nvalidity {\u03c4} {\u0393} {app s t} = {!!}\n\ncorrectness = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3c3d82d54d0b54ea4c89be26771242fae2b6faf0","subject":"Banana :D","message":"Banana :D\n","repos":"banacorn\/agda-mode,banacorn\/agda-mode,banacorn\/agda-mode","old_file":"test\/Banana.agda","new_file":"test\/Banana.agda","new_contents":"module Banana where\n\ndata Banana : Set where\n    raw : Banana\n    ripe : Banana\n    rotten : Banana\n\na : Banana\na = {!!}\n\nb : Banana\nb = {!!}\n","old_contents":"module Banana where\n\ndata Banana : Set where\n    peeled : Banana\n    unpeeled : Banana\n\ndata Nat : Set where\n  Z : Nat\n  S : Nat -> Nat\n\na : Banana\na = {!!}\n\nb : Banana\nb = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4a1bf78bc1454bf93894609ece1b3125d9e2ce64","subject":"Typos.","message":"Typos.\n\nIgnore-this: b93037029d2add29e31efadb8744c5b3\n\ndarcs-hash:20120204164223-3bd4e-e7bbfb7655b2839a5c2aa5890b9eb16cc3354155.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Axiomatic\/Base.agda","new_file":"src\/PA\/Axiomatic\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\ninfix  7  _\u2263_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the program agda2atp as the ATPs equality.\npostulate _\u2263_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2263 n \u2192 m \u2263 o \u2192 n \u2263 o\n  S\u2082 : \u2200 {m n} \u2192 m \u2263 n \u2192 succ m \u2263 succ n\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2263 succ n)\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2263 succ n \u2192 m \u2263 n\n  S\u2085 : \u2200 n \u2192 zero + n \u2263 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2263 succ (m + n)\n  S\u2087 : \u2200 n \u2192 zero * n \u2263 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2263 n + m * n\n{-# ATP axiom S\u2081 S\u2082 S\u2083 S\u2084 S\u2085 S\u2086 S\u2087 S\u2088 #-}\n\n-- The axiom S\u2089 is a higher-order one, therefore we do not translate\n-- it as an ATP axiom.\npostulate S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfix  7  _\u2263_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the program agda2atp as the ATPs equality.\npostulate _\u2263_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2263 n \u2192 m \u2263 o \u2192 n \u2263 o\n  S\u2082 : \u2200 {m n} \u2192 m \u2263 n \u2192 succ m \u2263 succ n\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2263 succ n)\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2263 succ n \u2192 m \u2263 n\n  S\u2085 : \u2200 n \u2192 zero + n \u2263 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2263 succ (m + n)\n  S\u2087 : \u2200 n \u2192 zero   * n \u2263 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2263 n + m * n\n{-# ATP axiom S\u2081 S\u2082 S\u2083 S\u2084 S\u2085 S\u2086 S\u2087 S\u2088 #-}\n\n-- The axiom S\u2089 is a higher-order one, therefore we do not translate\n-- it as an ATP axiom.\npostulate S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"690f2df38e24ddcd80d24056254c61364cc071fd","subject":"Add \u2259-cong\u2082","message":"Add \u2259-cong\u2082\n\nUseful for next steps.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a} {A : Set a} {{ca : ChangeAlgebra A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 x) \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 (nil x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\n  -- \\begin{lemma}[Equivalence cancellation]\n  --   |v2 `ominus` v1 `doe` dv| holds if and only if |v2 = v1 `oplus`\n  --   dv|, for any |v1, v2 `elem` V| and |dv `elem` Dt ^ v1|.\n  -- \\end{lemma}\n\n  equiv-cancel-1 : \u2200 x' dx \u2192 (x' \u229f x) \u2259 dx \u2192 x' \u2261 x \u229e dx\n  equiv-cancel-1 x' dx (doe x'\u229fx\u2259dx) = trans (sym (update-diff x' x)) x'\u229fx\u2259dx\n  equiv-cancel-2 : \u2200 x' dx \u2192 x' \u2261 x \u229e dx \u2192 x' \u229f x \u2259 dx\n  equiv-cancel-2 _ dx refl = diff-update\n\nmodule _ {a} {b} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} where\n\n  \u2259-cong\u2082 : \u2200 {c} {C : Set c} {x y}\n       (f : A \u2192 B \u2192 C) {dx1 dx2 dy1 dy2} \u2192 dx1 \u2259 dx2 \u2192 dy1 \u2259 dy2 \u2192 f (x \u229e dx1) (y \u229e dy1) \u2261 f (x \u229e dx2) (y \u229e dy2)\n  \u2259-cong\u2082 f dx1\u2259dx2 dy1\u2259dy2 = cong\u2082 f (proof dx1\u2259dx2) (proof dy1\u2259dy2)\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    df\u2081 \u2259\u208d f \u208e df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259\u208d f \u208e df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    df a da \u2259 f (a \u229e da) \u229f f a\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 df x dx\u2081 \u2259 df x dx\u2082\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 df x dx\u2081 \u2259 df x dx\u2082\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x (dx\u2081 dx\u2082 : \u0394 x) \u2192 dx\u2081 \u2259\u208d x \u208e dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (apply (nil f))\n  nil-is-derivative f a da =\n    begin\n      f a \u229e apply (nil f) a da\n    \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n      (f \u229e nil f) (a \u229e da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da)) (update-nil f) \u27e9\n      f (a \u229e da)\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- If we have two derivatives, they're both nil, hence they're equivalent.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  equiv-derivative-is-derivative : \u2200 {f : A \u2192 B} df\u2081 df\u2082 \u2192 IsDerivative f (apply df\u2081) \u2192 df\u2081 \u2259\u208d f \u208e df\u2082 \u2192 IsDerivative f (apply df\u2082)\n  equiv-derivative-is-derivative {f} df\u2081 df\u2082 IsDerivative-f-df\u2081 df\u2081\u2259df\u2082 a da =\n    begin\n      f a \u229e apply df\u2082 a da\n    \u2261\u27e8 sym (incrementalization f df\u2082 a da) \u27e9\n      (f \u229e df\u2082) (a \u229e da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da)) lemma \u27e9\n      f (a \u229e da)\n    \u220e\n    where\n      open \u2261-Reasoning\n      lemma : f \u229e df\u2082 \u2261 f\n      lemma =\n        begin\n          f \u229e df\u2082\n        \u2261\u27e8 proof (\u2259-sym df\u2081\u2259df\u2082) \u27e9\n          f \u229e df\u2081\n        \u2261\u27e8 derivative-is-\u229e-unit df\u2081 IsDerivative-f-df\u2081 \u27e9\n          f\n        \u220e\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil v) \u2259 nil (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil v)\n        \u2261\u27e8 proof v (nil v) \u27e9\n          f (v \u229e (nil v))\n        \u2261\u27e8 cong f (update-nil v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil (f v)) \u27e9\n          f v \u229e nil (f v)\n        \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a} {A : Set a} {{ca : ChangeAlgebra A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 x) \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 (nil x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\n  -- \\begin{lemma}[Equivalence cancellation]\n  --   |v2 `ominus` v1 `doe` dv| holds if and only if |v2 = v1 `oplus`\n  --   dv|, for any |v1, v2 `elem` V| and |dv `elem` Dt ^ v1|.\n  -- \\end{lemma}\n\n  equiv-cancel-1 : \u2200 x' dx \u2192 (x' \u229f x) \u2259 dx \u2192 x' \u2261 x \u229e dx\n  equiv-cancel-1 x' dx (doe x'\u229fx\u2259dx) = trans (sym (update-diff x' x)) x'\u229fx\u2259dx\n  equiv-cancel-2 : \u2200 x' dx \u2192 x' \u2261 x \u229e dx \u2192 x' \u229f x \u2259 dx\n  equiv-cancel-2 _ dx refl = diff-update\n\nmodule _ {a} {b} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} where\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    df\u2081 \u2259\u208d f \u208e df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259\u208d f \u208e df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    df a da \u2259 f (a \u229e da) \u229f f a\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 df x dx\u2081 \u2259 df x dx\u2082\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 df x dx\u2081 \u2259 df x dx\u2082\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x (dx\u2081 dx\u2082 : \u0394 x) \u2192 dx\u2081 \u2259\u208d x \u208e dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (apply (nil f))\n  nil-is-derivative f a da =\n    begin\n      f a \u229e apply (nil f) a da\n    \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n      (f \u229e nil f) (a \u229e da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da)) (update-nil f) \u27e9\n      f (a \u229e da)\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- If we have two derivatives, they're both nil, hence they're equivalent.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  equiv-derivative-is-derivative : \u2200 {f : A \u2192 B} df\u2081 df\u2082 \u2192 IsDerivative f (apply df\u2081) \u2192 df\u2081 \u2259\u208d f \u208e df\u2082 \u2192 IsDerivative f (apply df\u2082)\n  equiv-derivative-is-derivative {f} df\u2081 df\u2082 IsDerivative-f-df\u2081 df\u2081\u2259df\u2082 a da =\n    begin\n      f a \u229e apply df\u2082 a da\n    \u2261\u27e8 sym (incrementalization f df\u2082 a da) \u27e9\n      (f \u229e df\u2082) (a \u229e da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da)) lemma \u27e9\n      f (a \u229e da)\n    \u220e\n    where\n      open \u2261-Reasoning\n      lemma : f \u229e df\u2082 \u2261 f\n      lemma =\n        begin\n          f \u229e df\u2082\n        \u2261\u27e8 proof (\u2259-sym df\u2081\u2259df\u2082) \u27e9\n          f \u229e df\u2081\n        \u2261\u27e8 derivative-is-\u229e-unit df\u2081 IsDerivative-f-df\u2081 \u27e9\n          f\n        \u220e\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil v) \u2259 nil (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil v)\n        \u2261\u27e8 proof v (nil v) \u27e9\n          f (v \u229e (nil v))\n        \u2261\u27e8 cong f (update-nil v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil (f v)) \u27e9\n          f v \u229e nil (f v)\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c2bfe50b41af438f040e8a9bb60f83ffe00e3dbe","subject":"Adding witness, statement and valid witnesses to SigmaProtocol","message":"Adding witness, statement and valid witnesses to SigmaProtocol\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/SigmaProtocol.agda","new_file":"ZK\/SigmaProtocol.agda","new_contents":"open import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\n\nmodule ZK.SigmaProtocol\n          (Commitment : Type) -- Prover commitments\n          (Challenge  : Type) -- Verifier challenges\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Witness    : Type) -- Prover's witness\n          (Statement  : Type) -- Set of possible statements of the zk-proof\n          (_\u2208_ : Witness \u2192 Statement \u2192 Type) -- Valid witness for a given statement\n       where\n\n-- The prover is made of two steps.\n-- * First, send the commitment.\n-- * Second, receive a challenge and send back the response.\n--\n-- Therefor one represents a prover as a pair (a record) made\n-- of a commitment (get-A) and a function (get-f) from challenges\n-- to responses.\n--\n-- Note that one might wish the function get-f to be partial.\n--\n-- This covers any kind of prover, either honest or dishonest\n-- Notice as well that such should depend on some randomness.\n-- So such a prover as type: SomeRandomness \u2192 Prover\nrecord Prover-Interaction : Type where\n  constructor _,_\n  field\n    get-A : Commitment\n    get-f : Challenge \u2192 Response\n\nProver : Type\nProver = (w : Witness)(Y : Statement) \u2192 Prover-Interaction\n\n-- A transcript of the interaction between the prover and the\n-- verifier.\n--\n-- Note that in the case of an interactive zero-knowledge\n-- proof the transcript alone does not prove anything. It can usually\n-- be simulated if one knows the challenge before the commitment.\n--\n-- The only way to be convinced by an interactive prover is to be\/trust the verifier,\n-- namely to receive the commitment first and send a randomly picked challenge\n-- thereafter. So the proof is not transferable.\n--\n-- The other way is by making the zero-knowledge proof non-interactive, which forces the\n-- challenge to be a cryptographic hash of the commitment and the statement.\n-- See the StrongFiatShamir module and the corresponding paper for more details.\nrecord Transcript : Type where\n  constructor mk\n  field\n    get-A : Commitment\n    get-c : Challenge\n    get-f : Response\n\n-- The actual behavior of an interactive verifier is as follows:\n-- * Given a statement\n-- * Receive a commitment from the prover\n-- * Send a randomly chosen challenge to the prover\n-- * Receive a response to that challenge\n-- Since the first two points are common to all \u03a3-protocols we describe\n-- the verifier behavior as a computable test on the transcript.\nVerifier : Type\nVerifier = (Y : Statement)(t : Transcript) \u2192 Bool\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 {w Y} c \u2192 w \u2208 Y \u2192 \u2713 (verifier Y (run (prover w Y) c))\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (Y : Statement) (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 Y c s \u2192 let A = simulator Y c s in\n              \u2713(verifier Y (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator         : Simulator\n    correct-simulator : Correct-simulator verifier simulator\n\n-- A pair of \"Transcript\"s such that the commitment is shared\n-- and the challenges are different.\nrecord Transcript\u00b2 (verifier : Verifier) (Y : Statement) : Type where\n  constructor mk\n  field\n    -- The commitment is shared\n    get-A         : Commitment\n\n    -- The challenges...\n    get-c\u2080 get-c\u2081 : Challenge\n\n    -- ...are different\n    c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n    -- The responses\/proofs are arbitrary\n    get-f\u2080 get-f\u2081 : Response\n\n  -- The two transcripts\n  t\u2080 : Transcript\n  t\u2080 = mk get-A get-c\u2080 get-f\u2080\n  t\u2081 : Transcript\n  t\u2081 = mk get-A get-c\u2081 get-f\u2081\n\n  field\n    -- The transcripts verify\n    verify\u2080 : \u2713 (verifier Y t\u2080)\n    verify\u2081 : \u2713 (verifier Y t\u2081)\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = \u2200 Y \u2192 Transcript\u00b2 verifier Y \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 Y t\u00b2 \u2192 extractor Y t\u00b2 \u2208 Y\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor             : Extractor verifier\n    extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n","old_contents":"open import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\n\nmodule ZK.SigmaProtocol\n          (Commitment : Type) -- Prover commitments\n          (Challenge  : Type) -- Verifier challenges\n          (Response   : Type) -- Prover responses\/proofs to challenges\n       where\n\n-- The prover is made of two steps.\n-- * First, send the commitment.\n-- * Second, receive a challenge and send back the response.\n--\n-- Therefor one represents a prover as a pair (a record) made\n-- of a commitment (get-A) and a function (get-f) from challenges\n-- to responses.\n--\n-- Note that one might wish the function get-f to be partial.\n--\n-- Notice as well that such should depend on some randomness.\n-- So such a prover as type: SomeRandomness \u2192 Prover\nrecord Prover : Type where\n  constructor _,_\n  field\n    get-A : Commitment\n    get-f : Challenge \u2192 Response\n\n-- A transcript of the interaction between the prover and the\n-- verifier.\n--\n-- Note that in the case of an interactive zero-knowledge\n-- proof the transcript alone does not prove anything. It can usually\n-- be simulated if one knows the challenge before the commitment.\n--\n-- The only to be convinced by an interactive prover is to be\/trust the verifier,\n-- namely to receive the commitment first and send a randomly picked challenge\n-- thereafter.\n--\n-- The other way is the non-interactive zero-knowledge proof which forces the\n-- challenge to be a cryptographic hash of the commitment and the statement.\n-- See the StrongFiatShamir module and the corresponding paper for more details.\nrecord Transcript : Type where\n  constructor mk\n  field\n    get-A : Commitment\n    get-c : Challenge\n    get-f : Response\n\n-- The actual behavior of an interactive verifier is as follows:\n-- * Receive a commitment from the prover\n-- * Send a randomly chosen challenge to the prover\n-- * Receive a response to that challenge\n-- Since the first two points are common to all \u03a3-protocols we describe\n-- the verifier behavior as a computable test on the transcript.\nVerifier : Type\nVerifier = Transcript \u2192 Bool\n\n-- To run the interaction, one only need the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n-- Correctness: a \u03a3-protocol is said to be correct if for any challenge,\n-- the (honest) verifier accepts what the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 c \u2192 \u2713 (verifier (run prover c))\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713(verifier (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is coverd by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if c and s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the prover and the verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator         : Simulator\n    correct-simulator : Correct-simulator verifier simulator\n\n-- Two provers being equivalent, namely pointwise equal.\nrecord EqProver (p\u2080 p\u2081 : Prover) : Type where\n  constructor _,_\n  module P\u2080 = Prover p\u2080\n  module P\u2081 = Prover p\u2081\n  field\n    eq-A : P\u2080.get-A \u2261 P\u2081.get-A\n    eq-f : \u2200 c \u2192 P\u2080.get-f c \u2261 P\u2081.get-f c\n\n-- A pair of \"Transcript\"s such that the commitment is shared\n-- and the challenges are different.\nrecord Transcript\u00b2 (verifier : Verifier) : Type where\n  constructor mk\n  field\n    -- The commitment is shared\n    get-A         : Commitment\n\n    -- The challenges...\n    get-c\u2080 get-c\u2081 : Challenge\n\n    -- ...are different\n    c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n    -- The responses\/proofs are arbitrary\n    get-f\u2080 get-f\u2081 : Response\n\n  -- The two transcripts\n  t\u2080 : Transcript\n  t\u2080 = mk get-A get-c\u2080 get-f\u2080\n  t\u2081 : Transcript\n  t\u2081 = mk get-A get-c\u2081 get-f\u2081\n\n  field\n    -- The transcripts verify\n    verify\u2080 : \u2713 (verifier t\u2080)\n    verify\u2081 : \u2713 (verifier t\u2081)\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = Transcript\u00b2 verifier \u2192 Prover\n\nExtractor-Exact : (prover : Prover) (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtractor-Exact prover verifier extractor = \u2200 t\u00b2 \u2192 EqProver (extractor t\u00b2) prover\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor       : Extractor verifier\n    extractor-exact : Extractor-Exact prover verifier extractor\n\n  {- In the module StrongFiatShamir the extractor is as follows:\n\n    Extract    : (t : \u03a3-Transcript\u00b2 verifier) \u2192 W\n    Extract-ok : (t : \u03a3-Transcript\u00b2 verifier) \u2192 L (Extract t) Y\n\n    This version does not include the language of statements (\u039b : Type),\n    the statements (Y : \u0394), the witnesses (W : Type), nor the witnesses\n    relation (L : W \u2192 \u039b \u2192 Type).\n    Therefor the extraction is described as extracting a honest prover\n    from the pair of transcripts.\n  -}\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e375143fb586adf77b0d2fc59b043bec77b61a14","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Base.agda","new_file":"src\/fot\/FOTC\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\n-- Common definitions.\nopen import Common.DefinitionsATP public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | loop\n\npostulate\n  _\u00b7_                    : D \u2192 D \u2192 D  -- FOTC application.\n  true false if          : D          -- FOTC partial Booleans.\n  zero succ pred iszero  : D          -- FOTC partial natural numbers.\n  loop                   : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 n = succ \u00b7 n\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 n = pred \u00b7 n\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 n = iszero \u00b7 n\n  -- {-# ATP definition iszero\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as non-logical axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for Booleans.\n-- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n-- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\n-- N.B. We don't need this equation.\n-- pred-0 :       pred \u00b7 zero     \u2261 zero\n-- pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\npostulate\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\n-- iszero-0 :       iszero \u00b7 zero       \u2261 true\n-- iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\npostulate\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 n \u2192 iszero\u2081 (succ\u2081 n) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rule for loop.\n--\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a first-order logic axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\n--  0\u2262S : \u2200 {n} \u2192 zero \u2262 succ \u00b7 n\npostulate\n  true\u2262false : true \u2262 false\n  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ\u2081 n\n{-# ATP axiom true\u2262false 0\u2262S #-}\n\n------------------------------------------------------------------------------\n-- FOTC combinators for lists, colists, streams, etc.\n\nmodule BList where\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 8 _\u2237_\n\n  -- List constants.\n  postulate\n    [] cons head tail null : D  -- FOTC lists.\n\n  -- Definitions\n  abstract\n    _\u2237_  : D \u2192 D \u2192 D\n    x \u2237 xs = cons \u00b7 x \u00b7 xs\n    -- {-# ATP definition _\u2237_ #-}\n\n    head\u2081 : D \u2192 D\n    head\u2081 xs = head \u00b7 xs\n    -- {-# ATP definition head\u2081 #-}\n\n    tail\u2081 : D \u2192 D\n    tail\u2081 xs = tail \u00b7 xs\n    -- {-# ATP definition tail\u2081 #-}\n\n    null\u2081 : D \u2192 D\n    null\u2081 xs = null \u00b7 xs\n    -- {-# ATP definition null\u2081 #-}\n\n  -- Conversion rules for null.\n  -- null-[] :          null \u00b7 []              \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  postulate\n    null-[] :          null\u2081 []       \u2261 true\n    null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n  -- Conversion rule for head.\n  -- head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  postulate head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n  {-# ATP axiom head-\u2237 #-}\n\n  -- Conversion rule for tail.\n  -- tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  postulate tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n  {-# ATP axiom tail-\u2237 #-}\n\n  -- Discrimination rules\n  -- postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 cons \u00b7 x \u00b7 xs\n  postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 x \u2237 xs\n","old_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\n-- Common definitions.\nopen import Common.DefinitionsATP public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | loop\n\npostulate\n  _\u00b7_                    : D \u2192 D \u2192 D  -- FOTC application.\n  true false if          : D          -- FOTC partial Booleans.\n  zero succ pred iszero  : D          -- FOTC partial natural numbers.\n  loop                   : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 n = succ \u00b7 n\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 n = pred \u00b7 n\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 n = iszero \u00b7 n\n  -- {-# ATP definition iszero\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as non-logical axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for Booleans.\n-- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n-- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\n-- N.B. We don't need this equation.\n-- pred-0 :       pred \u00b7 zero     \u2261 zero\n-- pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\npostulate\n  pred-S : \u2200 n \u2192 pred\u2081 (succ\u2081 n) \u2261 n\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\n-- iszero-0 :       iszero \u00b7 zero       \u2261 true\n-- iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\npostulate\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 n \u2192 iszero\u2081 (succ\u2081 n) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rule for loop.\n--\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a first-order logic axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\n--  0\u2262S : \u2200 {n} \u2192 zero \u2262 succ \u00b7 n\npostulate\n  true\u2262false : true \u2262 false\n  0\u2262S        : \u2200 {n} \u2192 zero \u2262 succ\u2081 n\n{-# ATP axiom true\u2262false 0\u2262S #-}\n\n------------------------------------------------------------------------------\n-- FOTC combinators for lists, colists, streams, etc.\n\nmodule BList where\n\n  -- We add 3 to the fixities of the standard library.\n  infixr 8 _\u2237_\n\n  -- List constants.\n  postulate\n    [] cons head tail null : D  -- FOTC lists.\n\n  -- Definitions\n  abstract\n    _\u2237_  : D \u2192 D \u2192 D\n    x \u2237 xs = cons \u00b7 x \u00b7 xs\n    -- {-# ATP definition _\u2237_ #-}\n\n    head\u2081 : D \u2192 D\n    head\u2081 xs = head \u00b7 xs\n    -- {-# ATP definition head\u2081 #-}\n\n    tail\u2081 : D \u2192 D\n    tail\u2081 xs = tail \u00b7 xs\n    -- {-# ATP definition tail\u2081 #-}\n\n    null\u2081 : D \u2192 D\n    null\u2081 xs = null \u00b7 xs\n    -- {-# ATP definition null\u2081 #-}\n\n  -- Conversion rules for null.\n  -- null-[] :          null \u00b7 nil             \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  postulate\n    null-[] :          null\u2081 []       \u2261 true\n    null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n  -- Conversion rule for head.\n  -- head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  postulate head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n  {-# ATP axiom head-\u2237 #-}\n\n  -- Conversion rule for tail.\n  -- tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  postulate tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n  {-# ATP axiom tail-\u2237 #-}\n\n  -- Discrimination rules\n  -- postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 cons \u00b7 x \u00b7 xs\n  postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 x \u2237 xs\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a67acbd79b2cd0069cb05abbbe315e1dce006f39","subject":"easy subst cases","message":"easy subst cases\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  -- todo: i'm worried this may actually be false without knowing that x \u2260\n  -- y, but that's kind of what we usually mean on paper anyway\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   x \u2260 y \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} x\u2260y xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') (funext swap) xy\n    where\n      swap : (z : Nat) \u2192 (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) z == (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) z\n      swap = {!!}\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y -- todo: maybe?\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- this is false\n  --\n  -- apart-fresh : \u2200{ x \u0393 \u0394 d \u03c4} \u2192\n  --               x # \u0393 \u2192\n  --               \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n  --               fresh x d\n  -- apart-fresh aprt TAConst = FConst\n  -- apart-fresh {x = x} aprt (TAVar {x = y} x\u2082) with natEQ x y\n  -- apart-fresh aprt (TAVar x\u2083) | Inl refl = abort (somenotnone (! x\u2083 \u00b7 aprt))\n  -- apart-fresh aprt (TAVar x\u2083) | Inr x\u2082 = FVar x\u2082\n  -- apart-fresh {x = x} aprt (TALam {x = y} x\u2082 wt) with natEQ x y\n  -- apart-fresh aprt (TALam x\u2083 wt) | Inl refl = {!!}\n  -- apart-fresh aprt (TALam x\u2083 wt) | Inr x\u2082 = FLam x\u2082\n  -- apart-fresh aprt (TAAp wt wt\u2081) = FAp (apart-fresh aprt wt) (apart-fresh aprt wt\u2081)\n  -- apart-fresh aprt (TAEHole x\u2081 x\u2082) = FHole {!!}\n  -- apart-fresh aprt (TANEHole x\u2081 wt x\u2082) = FNEHole {!!} (apart-fresh aprt wt)\n  -- apart-fresh aprt (TACast wt x\u2081) = FCast (apart-fresh aprt wt)\n  -- apart-fresh aprt (TAFailedCast wt x\u2081 x\u2082 x\u2083) = FFailedCast (apart-fresh aprt wt)\n\n\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     x # \u0393 \u2192 -- todo should this be envfresh x \u03c3\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 = {!!}\n    -- weaken-subst-\u0393 {x = x} {\u0393 = \u0393} {\u03c4 = \u03c41} apt (STAId x\u2081) = STAId (\u03bb y \u03c42 x\u2082 \u2192 x\u2208\u222al \u0393 (\u25a0 (x , \u03c41)) _ _ (x\u2081 y \u03c42 x\u2082))\n    -- weaken-subst-\u0393 {x = x} {\u0393 = \u0393} {\u03c4 = \u03c41} apt (STASubst {y = y} {\u03c4 = \u03c42} s x\u2081) with natEQ x y\n    -- weaken-subst-\u0393 {x = x} {\u0393 = \u0393} {\u03c4 = \u03c41} apt (STASubst {\u03c4 = \u03c42} s x\u2082) | Inl refl = STASubst {!!} (weaken-ta apt x\u2082)\n    -- weaken-subst-\u0393 apt (STASubst s x\u2082) | Inr x\u2081 = STASubst {!!} (weaken-ta apt x\u2082)\n    -- = STASubst {!!} (weaken-ta apt x\u2081)\n\n      -- STASubst (weaken-subst-\u0393 {x = y} {\u0393 = \u0393 ,, (x , \u03c41)} {\u03c4 = \u03c42} {!!} {!!}) (weaken-ta apt x\u2081)\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192\n                -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta {x} {\u0393} {_} {_} {\u03c4} {\u03c4'} (FVar x\u2082) (TAVar x\u2083) = TAVar (x\u2208\u222al \u0393 (\u25a0 (x , \u03c4')) _ _ x\u2083)\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081))) (weaken-ta {\u0393 = {!\u25a0 (y , ? )!} \u222a \u0393} x\u2083 wt)\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = TAAp (weaken-ta frsh wt) (weaken-ta frsh\u2081 wt\u2081)\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = TAEHole x\u2082 (weaken-subst-\u0393 {!!} x\u2083)\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = TANEHole x\u2082 (weaken-ta frsh wt) (weaken-subst-\u0393 {!!} x\u2083)\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = TACast (weaken-ta frsh wt) x\u2081\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = TAFailedCast (weaken-ta frsh wt) x\u2081 x\u2082 x\u2083\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 = {!(TALam x\u2082 wt1)!}\n  --   with natEQ y x\n  -- lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = {!!}\n  -- lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n  --   with lem-union-none {\u0393 = \u0393} x\u2083\n  -- ... | _ , r = TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n\n\n  -- with lem-union-none {\u0393 = \u0393} x\u2082 | lem-subst apt wt1 {!!}\n  -- ... | l , r | ih = TALam r {!ih!}\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import structural\n\nmodule lemmas-subst-ta where\n  lem-subst-\u03c3 : \u2200{\u0394 x \u0393 \u03c41 \u03c3 \u0393' d } \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 \u03c3 :s: \u0393' \u2192\n                  \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 Subst d x \u03c3 :s: \u0393'\n  lem-subst-\u03c3 s wt = STASubst s wt\n\n  exchange-subst-\u0393 : \u2200{\u0394 \u0393 x y \u03c41 \u03c42 \u03c3 \u0393'} \u2192\n                   \u0394 , (\u0393 ,, (x , \u03c41) ,, (y , \u03c42)) \u22a2 \u03c3 :s: \u0393' \u2192\n                   \u0394 , (\u0393 ,, (y , \u03c42) ,, (x , \u03c41)) \u22a2 \u03c3 :s: \u0393'\n  exchange-subst-\u0393 {\u0394} {\u0393} {x} {y} {\u03c41} {\u03c42} {\u03c3} {\u0393'} xy = tr (\u03bb qq \u2192 \u0394 , qq \u22a2 \u03c3 :s: \u0393') {!!} xy -- todo: i'm worried this may actually be false without knowing that x \u2260 y\n\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y -- todo: maybe?\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- this is false\n  --\n  -- apart-fresh : \u2200{ x \u0393 \u0394 d \u03c4} \u2192\n  --               x # \u0393 \u2192\n  --               \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n  --               fresh x d\n  -- apart-fresh aprt TAConst = FConst\n  -- apart-fresh {x = x} aprt (TAVar {x = y} x\u2082) with natEQ x y\n  -- apart-fresh aprt (TAVar x\u2083) | Inl refl = abort (somenotnone (! x\u2083 \u00b7 aprt))\n  -- apart-fresh aprt (TAVar x\u2083) | Inr x\u2082 = FVar x\u2082\n  -- apart-fresh {x = x} aprt (TALam {x = y} x\u2082 wt) with natEQ x y\n  -- apart-fresh aprt (TALam x\u2083 wt) | Inl refl = {!!}\n  -- apart-fresh aprt (TALam x\u2083 wt) | Inr x\u2082 = FLam x\u2082\n  -- apart-fresh aprt (TAAp wt wt\u2081) = FAp (apart-fresh aprt wt) (apart-fresh aprt wt\u2081)\n  -- apart-fresh aprt (TAEHole x\u2081 x\u2082) = FHole {!!}\n  -- apart-fresh aprt (TANEHole x\u2081 wt x\u2082) = FNEHole {!!} (apart-fresh aprt wt)\n  -- apart-fresh aprt (TACast wt x\u2081) = FCast (apart-fresh aprt wt)\n  -- apart-fresh aprt (TAFailedCast wt x\u2081 x\u2082 x\u2083) = FFailedCast (apart-fresh aprt wt)\n\n\n  mutual\n    weaken-subst-\u0393 : \u2200{ x \u0393 \u0394 \u03c3 \u0393' \u03c4} \u2192\n                     x # \u0393 \u2192\n                     \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                     \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 \u03c3 :s: \u0393'\n    weaken-subst-\u0393 = {!!}\n    -- weaken-subst-\u0393 {x = x} {\u0393 = \u0393} {\u03c4 = \u03c41} apt (STAId x\u2081) = STAId (\u03bb y \u03c42 x\u2082 \u2192 x\u2208\u222al \u0393 (\u25a0 (x , \u03c41)) _ _ (x\u2081 y \u03c42 x\u2082))\n    -- weaken-subst-\u0393 {x = x} {\u0393 = \u0393} {\u03c4 = \u03c41} apt (STASubst {y = y} {\u03c4 = \u03c42} s x\u2081) with natEQ x y\n    -- weaken-subst-\u0393 {x = x} {\u0393 = \u0393} {\u03c4 = \u03c41} apt (STASubst {\u03c4 = \u03c42} s x\u2082) | Inl refl = STASubst {!!} (weaken-ta apt x\u2082)\n    -- weaken-subst-\u0393 apt (STASubst s x\u2082) | Inr x\u2081 = STASubst {!!} (weaken-ta apt x\u2082)\n    -- = STASubst {!!} (weaken-ta apt x\u2081)\n\n      -- STASubst (weaken-subst-\u0393 {x = y} {\u0393 = \u0393 ,, (x , \u03c41)} {\u03c4 = \u03c42} {!!} {!!}) (weaken-ta apt x\u2081)\n\n\n    weaken-ta : \u2200{x \u0393 \u0394 d \u03c4 \u03c4'} \u2192\n                fresh x d \u2192\n                -- x # \u0393 ?\n                \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                \u0394 , \u0393 ,, (x , \u03c4') \u22a2 d :: \u03c4\n    weaken-ta _ TAConst = TAConst\n    weaken-ta (FVar x\u2082) (TAVar x\u2083) = {!!}\n    weaken-ta {x = x} frsh (TALam {x = y} x\u2082 wt) with natEQ x y\n    weaken-ta (FLam x\u2081 x\u2082) (TALam x\u2083 wt) | Inl refl = abort (x\u2081 refl)\n    weaken-ta {\u0393 = \u0393} {\u03c4' = \u03c4'} (FLam x\u2081 x\u2083) (TALam {x = y} x\u2084 wt) | Inr x\u2082 = TALam (apart-parts \u0393 _ _ x\u2084 (apart-singleton (flip x\u2081))) (weaken-ta {\u0393 = {!\u25a0 (y , )!} \u222a \u0393} x\u2083 {!!})\n    weaken-ta (FAp frsh frsh\u2081) (TAAp wt wt\u2081) = {!!}\n    weaken-ta (FHole x\u2081) (TAEHole x\u2082 x\u2083) = {!!}\n    weaken-ta (FNEHole x\u2081 frsh) (TANEHole x\u2082 wt x\u2083) = {!!}\n    weaken-ta (FCast frsh) (TACast wt x\u2081) = {!!}\n    weaken-ta (FFailedCast frsh) (TAFailedCast wt x\u2081 x\u2082 x\u2083) = {!!}\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0393 = \u0393} {x = x} apt (TALam  {x = y} x\u2082 wt1) wt2 = {!(TALam x\u2082 wt1)!}\n  --   with natEQ y x\n  -- lem-subst {\u0393 = \u0393} {x = x} apt (TALam x\u2083 wt1) wt2 | Inl refl = {!!}\n  -- lem-subst {\u0393 = \u0393} apt (TALam x\u2083 wt1) wt2 | Inr x\u2082\n  --   with lem-union-none {\u0393 = \u0393} x\u2083\n  -- ... | _ , r = TALam r (lem-subst {!!} (weaken-ta {!!} {!!}) (weaken-ta r wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (lem-subst-\u03c3 sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (lem-subst-\u03c3  x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n\n\n  -- with lem-union-none {\u0393 = \u0393} x\u2082 | lem-subst apt wt1 {!!}\n  -- ... | l , r | ih = TALam r {!ih!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0993352ebcb9f6c2ff726d127fe8a158a3a2b2bc","subject":"sFS: extract template","message":"sFS: extract template\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Strong-Fiat-Shamir.agda","new_file":"ZK\/Strong-Fiat-Shamir.agda","new_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; done; ask)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {RP R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  (v \u21c4 p) r Y c = \u03a3-Protocol.\u03a3-game (v , p) r Y c\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct (v , p) = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (mk (get-A r Y) c (get-f r Y c)) \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 Y : \u2605 where\n    constructor mk\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n  record Special-Soundness : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 Challenge \u00d7 \u03a3-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        K0 : Prfs \u2192 Transcript \u2192 ExtractorServerPart\n        K0 prfs init-transcript past-history on-going-transcript q = {!!}\n\n        K1 : Prfs \u2192 Transcript \u2192 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n        K1 [] init-transcript = done {!!}\n        K1 ((Y , \u03c0) \u2237 prfs) init-transcript = ask _ (\u03bb pt \u2192 {!!})\n\n        K : Extractor\n        K prfs init-transcript = K0 prfs init-transcript , K1 prfs init-transcript\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; done; ask)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {RP R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  (v \u21c4 p) r Y c = \u03a3-Protocol.\u03a3-game (v , p) r Y c\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct (v , p) = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (mk (get-A r Y) c (get-f r Y c)) \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 Y : \u2605 where\n    constructor mk\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n  record Special-Soundness : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 Challenge \u00d7 \u03a3-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"61e42684557f0558479feff752b0e10624471cdc","subject":"b++[\u2205\\\\b]=\u2205 on singletons","message":"b++[\u2205\\\\b]=\u2205 on singletons\n\nOld-commit-hash: 543e4d821c724adb7f23c22e3b55834703e98c57\n","repos":"inc-lc\/ilc-agda","old_file":"Data\/NatBag\/Properties.agda","new_file":"Data\/NatBag\/Properties.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"6e8742bd2626fba7ae73d8e0aa22f298daa641f1","subject":"Minor changes.","message":"Minor changes.\n\nIgnore-this: 39165e534e6710e8748364b14ae24619\n\ndarcs-hash:20110930182440-3bd4e-a7b80cd649d6858fbfe0b0da1c7ecf804cd0b45f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/DataPostulate.agda","new_file":"Draft\/DataPostulate.agda","new_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulated one: Using the induction\n-- principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulated one: Using pattern\n-- matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulated predicate to the data one.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {_} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {_} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulated one: Using the induction\n-- principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulated one: Using pattern\n-- matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulated predicate to the data one.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {n} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {n} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c68ae3394af059da44a270c4286cea18d3987019","subject":"Document Term and Terms.","message":"Document Term and Terms.\n\nOld-commit-hash: 7f9cbb4cffc6d9343c93f495fd39e0d6b1bcd855\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 TermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 TermConstructor \u0393 \u03a3 \u03c4\u2032\n\n-- helper for lift-\u03b7-const, don't try to understand at home\nlift-\u03b7-const-rec : \u2200 {\u03a3 \u0393 \u03c4} \u2192 (Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4) \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const-rec {\u2205} k = k \u2205\nlift-\u03b7-const-rec {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 lift-\u03b7-const-rec (\u03bb ts \u2192 k (t \u2022 ts))\n\nlift-\u03b7-const : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 TermConstructor \u0393 \u03a3 \u03c4\nlift-\u03b7-const constant = lift-\u03b7-const-rec (const constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4bb07f285806d91a05d50734a3f9f7333114e440","subject":"Vec: +rev-app","message":"Vec: +rev-app\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n      `tl : Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id     \u2219 xs       = xs\n    (f `\u204f g) \u2219 xs       = g \u2219 f \u2219 xs\n    `0\u21941    \u2219 xs       = 0\u21941 xs\n    (`tl f) \u2219 []       = []\n    (`tl f) \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : Perm \u2192 Perm \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) [] = refl\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) [] = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl _)   [] = refl\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 n {A : Set a} (x : A) f \u2192 f \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl _) = refl\n    \u2219-replicate (suc n) x (`tl f) rewrite \u2219-replicate n x f = refl\n    \u2219-replicate n x (f `\u204f g) rewrite \u2219-replicate n x f | \u2219-replicate n x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map {n} f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate n f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : Set (a L.\u2294 b) where\n      `id `0\u21941 : Bij\n      _`\u204f_ : Bij \u2192 Bij \u2192 Bij\n      _`\u2237_ : Bij\u1d2c \u2192 (A \u2192 Bij) \u2192 Bij\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n             \u2192 (B \u2192 A)\n             \u2192 (Bij\u1d2c \u2192 Bij\u1d2e)\n             \u2192 Bij A Bij\u1d2c \u2192 Bij B Bij\u1d2e\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : Endo Bij\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    infixr 9 _\u2219_\n    _\u2219_ : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id \u2219 xs             = xs\n    (f `\u204f g)   \u2219 xs       = g \u2219 f \u2219 xs\n    `0\u21941      \u2219 xs       = 0\u21941 xs\n    (f\u1d2c `\u2237 f) \u2219 []       = []\n    (f\u1d2c `\u2237 f) \u2219 (x \u2237 xs) = eval\u1d2c f\u1d2c x \u2237 f x \u2219 xs\n\n    _\u2257\u2032_ : Bij \u2192 Bij \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk Bij (\u03bb f xs \u2192 _\u2219_ f {n} xs) _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : Endo Bij\n    `tl f = id\u1d2c `\u2237 const f\n\n    module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : Perm \u2192 Bij\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 \u03c0 {n} (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) [] = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n      `tl : Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id     \u2219 xs       = xs\n    (f `\u204f g) \u2219 xs       = g \u2219 f \u2219 xs\n    `0\u21941    \u2219 xs       = 0\u21941 xs\n    (`tl f) \u2219 []       = []\n    (`tl f) \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : Perm \u2192 Perm \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) [] = refl\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) [] = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl _)   [] = refl\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 n {A : Set a} (x : A) f \u2192 f \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl _) = refl\n    \u2219-replicate (suc n) x (`tl f) rewrite \u2219-replicate n x f = refl\n    \u2219-replicate n x (f `\u204f g) rewrite \u2219-replicate n x f | \u2219-replicate n x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map {n} f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate n f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : Set (a L.\u2294 b) where\n      `id `0\u21941 : Bij\n      _`\u204f_ : Bij \u2192 Bij \u2192 Bij\n      _`\u2237_ : Bij\u1d2c \u2192 (A \u2192 Bij) \u2192 Bij\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n             \u2192 (B \u2192 A)\n             \u2192 (Bij\u1d2c \u2192 Bij\u1d2e)\n             \u2192 Bij A Bij\u1d2c \u2192 Bij B Bij\u1d2e\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : Endo Bij\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    infixr 9 _\u2219_\n    _\u2219_ : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id \u2219 xs             = xs\n    (f `\u204f g)   \u2219 xs       = g \u2219 f \u2219 xs\n    `0\u21941      \u2219 xs       = 0\u21941 xs\n    (f\u1d2c `\u2237 f) \u2219 []       = []\n    (f\u1d2c `\u2237 f) \u2219 (x \u2237 xs) = eval\u1d2c f\u1d2c x \u2237 f x \u2219 xs\n\n    _\u2257\u2032_ : Bij \u2192 Bij \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk Bij (\u03bb f xs \u2192 _\u2219_ f {n} xs) _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : Endo Bij\n    `tl f = id\u1d2c `\u2237 const f\n\n    module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : Perm \u2192 Bij\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 \u03c0 {n} (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) [] = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1fdac5f6ef63d7efdf6e98dd8d61464ebdd29e37","subject":"Cosmetic change.","message":"Cosmetic change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/Terms.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/Terms.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP terms\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.ABP.Terms where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\n\n------------------------------------------------------------------------------\n-- N.B. We did not define @Bit = Bool@ due to the issue #11.\nBit : D \u2192 Set\nBit b = Bool b\n{-# ATP definition Bit #-}\n\nF T : D\nF = false\nT = true\n{-# ATP definition F T #-}\n\npostulate\n  <_,_> : D \u2192 D \u2192 D\n  ok    : D \u2192 D\n","old_contents":"------------------------------------------------------------------------------\n-- ABP terms\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.ABP.Terms where\n\nopen import FOTC.Base\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\n\n------------------------------------------------------------------------------\n-- N.B. We did not define @Bit = Bool@ due to the issue #11.\nBit : D \u2192 Set\nBit b = Bool b\n{-# ATP definition Bit #-}\n\nF : D\nF = false\n{-# ATP definition F #-}\n\nT : D\nT = true\n{-# ATP definition T #-}\n\npostulate\n  <_,_> : D \u2192 D \u2192 D\n  ok    : D \u2192 D\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f5b981ea7905d4deb015eced08bb201e6b26490f","subject":"FiniteField: rename as \u2124[_], and improve imports from BigI","message":"FiniteField: rename as \u2124[_], and improve imports from BigI\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/JS\/BigI\/FiniteField.agda","new_file":"Crypto\/JS\/BigI\/FiniteField.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import FFI.JS using (JS[_]; _++_; return; _>>_)\nopen import FFI.JS.Check\n  using    (check!)\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Algebra.Raw\nopen import Algebra.Field\n\n-- TODO carry on a primality proof of q\nmodule Crypto.JS.BigI.FiniteField (q : BigI) where\n\nabstract\n  \u2124[_] : Set\n  \u2124[_] = BigI\n\n  private\n    \u2124q : Set\n    \u2124q = \u2124[_]\n\n    mod-q : BigI \u2192 \u2124q\n    mod-q x = mod x q\n\n  -- There are two ways to go from BigI to \u2124q: BigI\u25b9\u2124[ q ] and mod-q\n  -- Use BigI\u25b9\u2124[ q ] for untrusted input data and mod-q for internal\n  -- computation.\n  BigI\u25b9\u2124[_] : BigI \u2192 JS[ \u2124q ]\n  BigI\u25b9\u2124[_] x =\n  -- Console.log \"BigI\u25b9\u2124[_]\"\n    check! \"below the modulus\" (x <I q)\n           (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++\n                  \" is less than x:\" ++ toString x) >>\n    check! \"positivity\" (x \u2265I 0I)\n           (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") >>\n    return x\n\n  check-non-zero : \u2124q \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \u2124q \u2192 BigI\n  repr x = x\n\n  0# 1# : \u2124q\n  0# = 0I\n  1# = 1I\n\n  -- One could choose here to return a dummy value when 0 is given.\n  -- Instead we throw an exception which in some circumstances could\n  -- be bad if the runtime semantics is too eager.\n  1\/_ : \u2124q \u2192 \u2124q\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : \u2124q \u2192 BigI \u2192 \u2124q\n  x ^ y = modPow x y q\n\n_\u2297_ : \u2124q \u2192 BigI \u2192 \u2124q\nx \u2297 y = mod-q (multiply (repr x) y)\n\n_+_ _\u2212_ _*_ _\/_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = x \u2297 repr y\nx \/ y = x * 1\/ y\n\n0\u2212_ : \u2124q \u2192 \u2124q\n0\u2212 x = mod-q (negate (repr x))\n\nsum prod : List \u2124q \u2192 \u2124q\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\ninstance\n  \u2124[_]-Eq? : Eq? \u2124q\n  \u2124[_]-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124q \u2192 \u2124q \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\n+-mon-ops : Monoid-Ops \u2124q\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \u2124q\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \u2124q\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \u2124q\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \u2124q\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \u2124q\nfld = fld-ops , fld-struct\n\nmodule fld = Field fld\n\nopen fld using (+-grp) public\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import FFI.JS using (Bool; trace-call; _++_)\nopen import FFI.JS.Check\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Algebra.Raw\nopen import Algebra.Field\n\n-- TODO carry on a primality proof of q\nmodule Crypto.JS.BigI.FiniteField (q : BigI) where\n\nabstract\n  -- \u2124q\n  \ud835\udd3d : Set\n  \ud835\udd3d = BigI\n\n  private\n    mod-q : BigI \u2192 \ud835\udd3d\n    mod-q x = mod x q\n\n  -- There are two ways to go from BigI to \ud835\udd3d: BigI\u25b9\ud835\udd3d and mod-q\n  -- Use BigI\u25b9\ud835\udd3d for untrusted input data and mod-q for internal\n  -- computation.\n  BigI\u25b9\ud835\udd3d : BigI \u2192 \ud835\udd3d\n  BigI\u25b9\ud835\udd3d = -- trace-call \"BigI\u25b9\ud835\udd3d \"\n    \u03bb x \u2192\n      (check? (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++ \" is less than x:\" ++ toString x)\n         (check? (x \u2265I 0I)\n            (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") x))\n\n  check-non-zero : \ud835\udd3d \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \ud835\udd3d \u2192 BigI\n  repr x = x\n\n  0# 1# : \ud835\udd3d\n  0# = 0I\n  1# = 1I\n\n  1\/_ : \ud835\udd3d \u2192 \ud835\udd3d\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : \ud835\udd3d \u2192 BigI \u2192 \ud835\udd3d\n  x ^ y = modPow x y q\n\n_\u2297_ : \ud835\udd3d \u2192 BigI \u2192 \ud835\udd3d\nx \u2297 y = mod-q (multiply (repr x) y)\n\n_+_ _\u2212_ _*_ _\/_ : \ud835\udd3d \u2192 \ud835\udd3d \u2192 \ud835\udd3d\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = x \u2297 repr y\nx \/ y = x * 1\/ y\n\n0\u2212_ : \ud835\udd3d \u2192 \ud835\udd3d\n0\u2212 x = mod-q (negate (repr x))\n\nsum prod : List \ud835\udd3d \u2192 \ud835\udd3d\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\ninstance\n  \ud835\udd3d-Eq? : Eq? \ud835\udd3d\n  \ud835\udd3d-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \ud835\udd3d \u2192 \ud835\udd3d \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\n+-mon-ops : Monoid-Ops \ud835\udd3d\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \ud835\udd3d\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \ud835\udd3d\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \ud835\udd3d\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \ud835\udd3d\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \ud835\udd3d\nfld = fld-ops , fld-struct\n\nmodule fld = Field fld\n\nopen fld using (+-grp) public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ccc6cb84ca74473835502698f489fd834009e3d4","subject":"more bounty from added premise #14","message":"more bounty from added premise #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    -- todo: also being used below\n    lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    --- todo: okay this is now getting reused here and in the case of ie\n    --- below, so this is apparently more of an interesting property than\n    --- i'd thought\n    lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n    lem (CECastFail x\u2081 x\u2082 () x\u2084)\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVVal x) (CECong FHOuter er) = ve (BVVal x) er\n  ve (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  ve (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  ve (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  ve (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  vs (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  vs (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole () ind) (ITCastID x\u2081)\n  lem2 (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  lem2 (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  lem2 (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  lem2 (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  lem2 (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    -- todo: also being used below\n    lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    --- todo: okay this is now getting reused here and in the case of ie\n    --- below, so this is apparently more of an interesting property than\n    --- i'd thought\n    lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n    lem (CECastFail x\u2081 x\u2082 () x\u2084)\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVVal x) (CECong FHOuter er) = ve (BVVal x) er\n  ve (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  ve (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  ve (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  ve (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f012817db8b092a4d81050b974438691f73b1586","subject":"Desc model: add the map operator","message":"Desc model: add the map operator\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms )) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms )) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5d3aa98ffc0c22e45257913557e911cc654c4777","subject":"Use \u27e6\u2295\u208d_\u208e\u27e7 and \u27e6\u229d\u208d_\u208e\u27e7 in Popl14.Change.Evaluation.","message":"Use \u27e6\u2295\u208d_\u208e\u27e7 and \u27e6\u229d\u208d_\u208e\u27e7 in Popl14.Change.Evaluation.\n\nOld-commit-hash: a05b08ff42508e49817f899523f4849e9334e057\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Change\/Evaluation.agda","new_file":"Popl14\/Change\/Evaluation.agda","new_contents":"module Popl14.Change.Evaluation where\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Popl14.Change.Type\nopen import Popl14.Change.Term\nopen import Popl14.Change.Value\nopen import Popl14.Denotation.Value\nopen import Popl14.Denotation.Evaluation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\n\nopen import Postulate.Extensionality\n\n-- unique names with unambiguous types\n-- to help type inference figure things out\nprivate\n  module Disambiguation where\n    infixr 9 _\u22c6_\n    _\u22c6_ : Type \u2192 Context \u2192 Context\n    _\u22c6_ = _\u2022_\n\n-- Relating `diff` with `diff-term`, `apply` with `apply-term`\nmeaning-\u2295 : \u2200 {\u03c4 \u0393}\n  {t : Term \u0393 \u03c4} {\u0394t : Term \u0393 (\u0394Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295 \u0394t \u27e7 \u03c1\n\nmeaning-\u229d : \u2200 {\u03c4 \u0393}\n  {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d t \u27e7 \u03c1\n\nmeaning-\u2295 {base base-int} = refl\nmeaning-\u2295 {base base-bag} = refl\nmeaning-\u2295 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {\u0394t} {\u03c1} = ext (\u03bb v \u2192\n  let\n    \u0393\u2032 = \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0394Type (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n    \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7\n    \u03c1\u2032 = v \u2022 (\u27e6 t \u27e7 \u03c1) \u2022 (\u27e6 \u0394t \u27e7 \u03c1) \u2022 \u03c1\n    x  : Term \u0393\u2032 \u03c3\n    x  = var this\n    f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n    f  = var (that this)\n    \u0394f : Term \u0393\u2032 (\u0394Type (\u03c3 \u21d2 \u03c4))\n    \u0394f = var (that (that this))\n    y  = app f x\n    \u0394y = app (app \u0394f x) (x \u229d x)\n  in\n    begin\n      \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v hole)\n         (meaning-\u229d {s = x} {x} {\u03c1\u2032}) \u27e9\n      \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (\u27e6 x \u229d x \u27e7 \u03c1\u2032)\n    \u2261\u27e8 meaning-\u2295 {t = y} {\u0394t = \u0394y} {\u03c1\u2032} \u27e9\n      \u27e6 y \u2295 \u0394y \u27e7 \u03c1\u2032\n    \u220e)\n  where\n    open \u2261-Reasoning\n    open Disambiguation\n\nmeaning-\u229d {int} = refl\nmeaning-\u229d {bag} = refl\nmeaning-\u229d {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} =\n  ext (\u03bb v \u2192 ext (\u03bb \u0394v \u2192\n  let\n    \u0393\u2032 = \u0394Type \u03c3 \u22c6 \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n    \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7Context\n    \u03c1\u2032 = \u0394v \u2022 v \u2022 \u27e6 t \u27e7Term \u03c1 \u2022 \u27e6 s \u27e7Term \u03c1 \u2022 \u03c1\n    \u0394x : Term \u0393\u2032 (\u0394Type \u03c3)\n    \u0394x = var this\n    x  : Term \u0393\u2032 \u03c3\n    x  = var (that this)\n    f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n    f  = var (that (that this))\n    g  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n    g  = var (that (that (that this)))\n    y  = app f x\n    y\u2032 = app g (x \u2295 \u0394x)\n  in\n    begin\n      \u27e6 s \u27e7 \u03c1 (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 s \u27e7 \u03c1 hole \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v)\n         (meaning-\u2295 {t = x} {\u0394t = \u0394x} {\u03c1\u2032}) \u27e9\n      \u27e6 s \u27e7 \u03c1 (\u27e6 x \u2295 \u0394x \u27e7 \u03c1\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n    \u2261\u27e8 meaning-\u229d {s = y\u2032} {y} {\u03c1\u2032} \u27e9\n      \u27e6 y\u2032 \u229d y \u27e7 \u03c1\u2032\n    \u220e))\n  where\n    open \u2261-Reasoning\n    open Disambiguation\n","old_contents":"module Popl14.Change.Evaluation where\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Popl14.Change.Type\nopen import Popl14.Change.Term\nopen import Popl14.Change.Value\nopen import Popl14.Denotation.Value\nopen import Popl14.Denotation.Evaluation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\n\nopen import Postulate.Extensionality\n\n-- unique names with unambiguous types\n-- to help type inference figure things out\nprivate\n  module Disambiguation where\n    infixr 9 _\u22c6_\n    _\u22c6_ : Type \u2192 Context \u2192 Context\n    _\u22c6_ = _\u2022_\n\n-- Relating `diff` with `diff-term`, `apply` with `apply-term`\nmeaning-\u2295 : \u2200 {\u03c4 \u0393}\n  {t : Term \u0393 \u03c4} {\u0394t : Term \u0393 (\u0394Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  let\n    _+_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  in\n    \u27e6 t \u27e7 \u03c1 + \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295 \u0394t \u27e7 \u03c1\n\nmeaning-\u229d : \u2200 {\u03c4 \u0393}\n  {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  let\n    _-_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  in\n    \u27e6 s \u27e7 \u03c1 - \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d t \u27e7 \u03c1\n\nmeaning-\u2295 {base base-int} = refl\nmeaning-\u2295 {base base-bag} = refl\nmeaning-\u2295 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {\u0394t} {\u03c1} = ext (\u03bb v \u2192\n  let\n    \u0393\u2032 = \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0394Type (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n    \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7\n    \u03c1\u2032 = v \u2022 (\u27e6 t \u27e7 \u03c1) \u2022 (\u27e6 \u0394t \u27e7 \u03c1) \u2022 \u03c1\n    _-\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n    _+\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n    x  : Term \u0393\u2032 \u03c3\n    x  = var this\n    f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n    f  = var (that this)\n    \u0394f : Term \u0393\u2032 (\u0394Type (\u03c3 \u21d2 \u03c4))\n    \u0394f = var (that (that this))\n    y  = app f x\n    \u0394y = app (app \u0394f x) (x \u229d x)\n  in\n    begin\n      \u27e6 t \u27e7 \u03c1 v +\u2081 \u27e6 \u0394t \u27e7 \u03c1 v (v -\u2080 v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v +\u2081 \u27e6 \u0394t \u27e7 \u03c1 v hole)\n         (meaning-\u229d {s = x} {x} {\u03c1\u2032}) \u27e9\n      \u27e6 t \u27e7 \u03c1 v +\u2081 \u27e6 \u0394t \u27e7 \u03c1 v (\u27e6 x \u229d x \u27e7 \u03c1\u2032)\n    \u2261\u27e8 meaning-\u2295 {t = y} {\u0394t = \u0394y} {\u03c1\u2032} \u27e9\n      \u27e6 y \u2295 \u0394y \u27e7 \u03c1\u2032\n    \u220e)\n  where\n    open \u2261-Reasoning\n    open Disambiguation\n\nmeaning-\u229d {int} = refl\nmeaning-\u229d {bag} = refl\nmeaning-\u229d {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} =\n  ext (\u03bb v \u2192 ext (\u03bb \u0394v \u2192\n  let\n    \u0393\u2032 = \u0394Type \u03c3 \u22c6 \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n    \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7Context\n    \u03c1\u2032 = \u0394v \u2022 v \u2022 \u27e6 t \u27e7Term \u03c1 \u2022 \u27e6 s \u27e7Term \u03c1 \u2022 \u03c1\n    \u0394x : Term \u0393\u2032 (\u0394Type \u03c3)\n    \u0394x = var this\n    x  : Term \u0393\u2032 \u03c3\n    x  = var (that this)\n    f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n    f  = var (that (that this))\n    g  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n    g  = var (that (that (that this)))\n    y  = app f x\n    y\u2032 = app g (x \u2295 \u0394x)\n    _+\u2080_ = _\u27e6\u2295\u27e7_ {\u03c3}\n    _-\u2081_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  in\n    begin\n      \u27e6 s \u27e7 \u03c1 (v +\u2080 \u0394v) -\u2081 \u27e6 t \u27e7 \u03c1 v\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 s \u27e7 \u03c1 hole -\u2081 \u27e6 t \u27e7 \u03c1 v)\n         (meaning-\u2295 {t = x} {\u0394t = \u0394x} {\u03c1\u2032}) \u27e9\n      \u27e6 s \u27e7 \u03c1 (\u27e6 x \u2295 \u0394x \u27e7 \u03c1\u2032) -\u2081 \u27e6 t \u27e7 \u03c1 v\n    \u2261\u27e8 meaning-\u229d {s = y\u2032} {y} {\u03c1\u2032} \u27e9\n      \u27e6 y\u2032 \u229d y \u27e7 \u03c1\u2032\n    \u220e))\n  where\n    open \u2261-Reasoning\n    open Disambiguation\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"11c958d27e9b6b11cc77f3b177e03eaad4a5f3e7","subject":"Define derivatives for change algebra families.","message":"Define derivatives for change algebra families.\n\nOld-commit-hash: 7734b87b24dad9b86ce93e2c34acb376d1aab35a\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"3999589273f140eb97b3bdf2b0e19bf8272cba13","subject":"removing redundant rules after removing marks; guess at eval context rules","message":"removing redundant rules after removing marks; guess at eval context rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192 -- lol hax: d final?\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           d1 final \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 } \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2081 \u03b5) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2082 d) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3  \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 evalctx \u2192\n             (< \u03c4 > \u03b5 ) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHFinal : \u2200{d} \u2192 d final \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d2' \u27e7\n    FHAp2 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d1' \u27e7\n    FHEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 (u , \u03c3 ) \u27e9 == \u2299 \u27e6 \u2987\u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHNEHoleFinal : \u2200{ d u \u03c3} \u2192\n              d final \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2299 \u27e6 \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c4 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (< \u03c4 > d) == < \u03c4 > \u03b5 \u27e6 d' \u27e7\n    FHCastFinal : \u2200{d \u03c4} \u2192\n                 d final \u2192\n                 (< \u03c4 > d) == \u2299 \u27e6 < \u03c4 > d \u27e7\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           \u0394 \u22a2 d \u21a6 d'\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3  \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3  \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192 -- lol hax: d final?\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           d1 final \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 } \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2081 \u03b5) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2082 d) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3  \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 evalctx \u2192\n             (< \u03c4 > \u03b5 ) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHFinal : \u2200{d} \u2192 d final \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d2' \u27e7\n    FHAp2 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d1' \u27e7\n    FHEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 (u , \u03c3 ) \u27e9 == \u2299 \u27e6 \u2987\u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e7\n    FHNEHoleEvaled : \u2200{ d u \u03c3} \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2299 \u27e6 \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e7\n    FHNEHoleInside : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHNEHoleFinal : \u2200{ d u \u03c3} \u2192\n              d final \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2299 \u27e6 \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c4 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (< \u03c4 > d) == < \u03c4 > \u03b5 \u27e6 d' \u27e7\n    FHCastFinal : \u2200{d \u03c4} \u2192\n                 d final \u2192\n                 (< \u03c4 > d) == \u2299 \u27e6 < \u03c4 > d \u27e7\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3  \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3  \u27e9\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3  \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3  \u27e9\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           \u0394 \u22a2 d \u21a6 d'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"69fe0b1567d5b5439ff7565a79deffd679e3a82b","subject":"Make P.C.Type conform to plugin pattern.","message":"Make P.C.Type conform to plugin pattern.\n\nOld-commit-hash: 8dc76b168a0b8d9f859a69834d52d91e2f34b235\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Type.agda","new_file":"Parametric\/Change\/Type.agda","new_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Change.Type\n  (Base : Type.Structure)\n  where\n\nopen Type.Structure Base\n\nStructure : Set\nStructure = Base \u2192 Base\n\nmodule Structure (\u0394Base : Structure) where\n  \u0394Type : Type \u2192 Type\n  \u0394Type (base \u03b9) = base (\u0394Base \u03b9)\n  \u0394Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394Type \u03c3 \u21d2 \u0394Type \u03c4\n\n  open import Base.Change.Context \u0394Type public\n","old_contents":"module Parametric.Change.Type\n  (Base : Set)\n  where\n\nimport Parametric.Syntax.Type as Type\nopen Type.Structure Base\n\nStructure : Set\nStructure = Base \u2192 Base\n\nmodule Structure (\u0394Base : Structure) where\n  \u0394Type : Type \u2192 Type\n  \u0394Type (base \u03b9) = base (\u0394Base \u03b9)\n  \u0394Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394Type \u03c3 \u21d2 \u0394Type \u03c4\n\n  open import Base.Change.Context \u0394Type public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2ffcd7ac6160c8248eab42d51b48950e1e1b0529","subject":"Cleanup","message":"Cleanup\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Terms that operate on changes (Fig. 3).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\n-- Extension point 1: A term for \u229d on base types.\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\n-- Extension point 2: A term for \u2295 on base types.\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (apply-base : ApplyStructure)\n    (diff-base  : DiffStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  -- We provide: terms for \u2295 and \u229d on arbitrary types.\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       absV 3 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       absV 4 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x)\n\n  apply : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply _ = app\u2082 apply-term\n\n  diff : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff _ = app\u2082 diff-term\n\n  infixl 6 apply diff\n\n  syntax apply \u03c4 x \u0394x = \u0394x \u2295\u208d \u03c4 \u208e x\n  syntax diff \u03c4 x y = x \u229d\u208d \u03c4 \u208e y\n\n  infixl 6 _\u2295_ _\u229d_\n\n  _\u2295_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  _\u2295_ {\u03c4} = apply \u03c4\n\n  _\u229d_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  _\u229d_ {\u03c4} = diff \u03c4\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Terms that operate on changes (Fig. 3).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Const : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\n-- Extension point 1: A term for \u229d on base types.\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\n-- Extension point 2: A term for \u2295 on base types.\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (apply-base : ApplyStructure)\n    (diff-base  : DiffStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  -- We provide: terms for \u2295 and \u229d on arbitrary types.\n  apply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n  diff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  apply-term {base \u03b9} = apply-base\n  apply-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c3} {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c4} {\u0393}) \u0394t t\n     in\n       absV 3 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff-term {base \u03b9} = diff-base\n  diff-term {\u03c3 \u21d2 \u03c4} =\n    (let\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff-term {\u03c4} {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply-term {\u03c3} {\u0393}) \u0394t t\n     in\n       absV 4 (\u03bb g f x \u0394x \u2192 app g (x \u2295\u03c3 \u0394x) \u229d\u03c4 app f x))\n\n  apply : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply _ = app\u2082 apply-term\n\n  diff : \u2200 \u03c4 {\u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff _ = app\u2082 diff-term\n\n  infixl 6 apply diff\n\n  syntax apply \u03c4 x \u0394x = \u0394x \u2295\u208d \u03c4 \u208e x\n  syntax diff \u03c4 x y = x \u229d\u208d \u03c4 \u208e y\n\n  infixl 6 _\u2295_ _\u229d_\n\n  _\u2295_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  _\u2295_ {\u03c4} = apply \u03c4\n\n  _\u229d_ : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  _\u229d_ {\u03c4} = diff \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fa694cbb245f34881704373fc3bac9cc23fbe0c5","subject":"game: more","message":"game: more\n","repos":"crypto-agda\/crypto-agda","old_file":"game.agda","new_file":"game.agda","new_contents":"module game where\n\nopen import Level using (Lift) renaming (_\u2294_ to _L\u2294_)\nopen import Function.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.List.Any\nopen import Data.Bool\nopen import Relation.Binary.PropositionalEquality\nopen Membership-\u2261\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\n{-\n- cost\n- non-det\n- queries\n- partial (like decryption with a mac)\n-}\n\nopen import Data.Bits\nopen import flipbased\n\nBits\u27e8_\u27e9\u2192_ : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a\nBits\u27e8 n \u27e9\u2192 A = Bits n \u2192 A\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : \u2200 k \u2192 Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R 0\n  ask : \u2200 {k} {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R k) \u2192 Strategy \u2610_ R k\n  rnd : \u2200 {n k} {A : Set} \u2192 \u21ba n A \u2192 (A \u2192 Strategy \u2610_ R k) \u2192 Strategy \u2610_ R (n + k)\n\nrun : \u2200 {k R} \u2192 Strategy id R k \u2192 \u21ba k R\nrun (say r)    = \u27ea r \u27eb\nrun (ask a f)  = run (f a)\nrun (rnd nd f) = nd >>= (\u03bb xs \u2192 run (f xs))\n\npostulate\n      Proba : Set\n      Pr[_\u22611] : \u2200 {k} (EXP : \u21ba k Bit) \u2192 Proba\n      _\u22611\/2^_ : Proba \u2192 \u2115 \u2192 Set\n      Pr[toss\u22611] : Pr[ toss \u22611] \u22611\/2^ 1\n      Pr[random\u2261_] : \u2200 {n} (x : Bits n) \u2192 Pr[ \u27ea _==_ x \u00b7 random \u27eb \u22611] \u22611\/2^ n\n      negligible-proba : Proba\n      negligible-advantage-proba : Proba \u2192 Proba \u2192 Set\n\n_is-negligible : Proba \u2192 Set\np is-negligible = negligible-advantage-proba p negligible-proba\n\nnegligible-advantage : \u2200 {k} (EXP : Bit \u2192 \u21ba k Bit) \u2192 Set\nnegligible-advantage EXP = negligible-advantage-proba Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\npostulate\n  negligible-advantage-xor : \u2200 {k} (EXP\u2081 EXP\u2082 : \u21ba k Bit) \u2192 EXP\u2081 \u2261 \u27ea _xor_ 1b \u00b7 EXP\u2082 \u27eb \u2192 negligible-advantage-proba Pr[ EXP\u2081 \u22611] Pr[ EXP\u2082 \u22611]\n\n  -- whatch out\n  negligible-advantage-trans : \u2200 {p\u2081 p\u2082 p\u2083} \u2192 negligible-advantage-proba p\u2081 p\u2082 \u2192 negligible-advantage-proba p\u2082 p\u2083 \u2192 negligible-advantage-proba p\u2081 p\u2083\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nmodule CryptoGames where\n\n  module CipherGames (|K| |M| |C| : \u2115) where\n    K = Bits |K|\n    M = Bits |M|\n    C = Bits |C|\n\n    record SemSecAdv k : Set where\n      constructor mk\n      field\n        {k\u2080 k\u2081}   : \u2115\n        k\u2261k\u2080+k\u2081  : k \u2261 k\u2080 + k\u2081\n        gen-m\u2080m\u2081  : \u21ba k\u2080 (M \u00d7 M)\n        stat-test : C \u2192 \u21ba k\u2081 Bit\n\n    mk\u2032 : \u2200 {k\u2080 k\u2081} \u2192 \u21ba k\u2080 (M \u00d7 M) \u2192 (C \u2192 \u21ba k\u2081 Bit) \u2192 SemSecAdv (k\u2080 + k\u2081)\n    mk\u2032 = mk refl\n\n    module DeterministicSemanticSecurity where\n      runDetSemSec : \u2200 {k} (enc : M \u2192 C) \u2192 SemSecAdv k \u2192 Bit \u2192 \u21ba k Bit\n      runDetSemSec enc (mk refl gen-m\u2080m\u2081 f) b = gen-m\u2080m\u2081 >>= \u03bb m\u2080m\u2081 \u2192 f (EXP b enc m\u2080m\u2081)\n\n      DetSemSecWinner : \u2200 (enc : M \u2192 C) b {k} (Adv : SemSecAdv k) \u2192 \u21ba k Set\n      DetSemSecWinner enc b Adv = \u21baT (runDetSemSec enc Adv b)\n\n      DetSemSec : \u2200 k (enc : M \u2192 C) \u2192 Set\n      DetSemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runDetSemSec enc Adv)\n\n      luckySemSec : SemSecAdv 1\n      luckySemSec = mk\u2032 \u27ea 0\u207f , 0\u207f \u27eb\u1d30 (\u03bb _ \u2192 toss)\n\n      -- see OTPDetSemSec\n\n    module DeterministicCPASecurity where\n      DetCPAAdv : \u2200 k \u2192 Set\u2081\n      DetCPAAdv k = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit k\n\n      data DetCPAWinner (enc : M \u00d7 M \u2192 C) b (ms : List (M \u00d7 M)) : \u2200 {k} (q : \u2115) \u2192 Strategy id Bit k \u2192 Set\u2081 where\n        win : \u2200 {q} \u2192 DetCPAWinner enc b ms q (say b)\n        ask : \u2200 {k q m\u2080m\u2081 f} \u2192 m\u2080m\u2081 \u2209 ms \u2192 DetCPAWinner enc b (m\u2080m\u2081 \u2237 ms) q (f (enc m\u2080m\u2081)) \u2192\n              DetCPAWinner enc b ms {k} (suc q) (ask (enc m\u2080m\u2081) f)\n        rnd : \u2200 {A : Set} {n k q bs} {r : \u21ba n A} {f : A \u2192 _}\n              \u2192 DetCPAWinner enc b ms {k} q (f bs) \u2192 DetCPAWinner enc b ms q (rnd r f)\n\n      runDetCPA : \u2200 {k} (enc : M \u2192 C) (Adv : DetCPAAdv k) \u2192 Bit \u2192 \u21ba k Bit\n      runDetCPA enc Adv b = run (Adv (EXP b enc))\n\n      -- no use of CPAWinner\n      DetCPASec : \u2200 k (enc : M \u2192 C) \u2192 Set\u2081\n      DetCPASec k enc = \u2200 (Adv : DetCPAAdv k) \u2192 negligible-advantage (runDetCPA enc Adv)\n\n    CPAAdv : \u2200 k \u2192 Set\u2081\n    CPAAdv k = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 (\u21ba |K| C)) \u2192 Strategy \u2610_ Bit k\n\n    sem2CPAAdv : \u2200 {k} \u2192 SemSecAdv k \u2192 CPAAdv (|K| + k)\n    sem2CPAAdv (mk {k\u2080} {k\u2081} refl gen-m\u2080m\u2081 f) {\u2610_} enc =\n      subst (Strategy \u2610_ Bit) helper\n        (rnd gen-m\u2080m\u2081 (\u03bb m\u2080m\u2081 \u2192\n          ask (enc m\u2080m\u2081) (\u03bb c \u2192\n            rnd c (\u03bb c' \u2192 rnd (f c') say))))\n      where open import Data.Nat.Properties\n            open SemiringSolver\n            helper : k\u2080 + (|K| + (k\u2081 + 0)) \u2261 |K| + (k\u2080 + k\u2081)\n            helper = solve 3 (\u03bb k\u2080 |K| k\u2081 \u2192 k\u2080 :+ (|K| :+ (k\u2081 :+ con 0)) := |K| :+ (k\u2080 :+ k\u2081))\n                             refl k\u2080 |K| k\u2081\n\n    data CPAWinner (enc : M \u00d7 M \u2192 \u21ba |K| C) b : \u2200 {k} q \u2192 Strategy id Bit k \u2192 Set\u2081 where\n      win : \u2200 {q} \u2192 CPAWinner enc b q (say b)\n      ask : \u2200 {k q m\u2080m\u2081 f} \u2192 CPAWinner enc b {k} q (f (enc m\u2080m\u2081)) \u2192\n            CPAWinner enc b (suc q) (ask (enc m\u2080m\u2081) f)\n      rnd : \u2200 {A : Set} {n k q bs} {r : \u21ba n A} {f : A \u2192 _}\n            \u2192 CPAWinner enc b {k} q (f bs) \u2192 CPAWinner enc b q (rnd r f)\n\n    det : \u2200 {k a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 A \u2192 \u21ba k B\n    det f x = \u27ea f x \u27eb\n\n    runCPA : \u2200 {k} (enc : M \u2192 \u21ba |K| C) (Adv : CPAAdv k) \u2192 Bit \u2192 \u21ba k Bit\n    runCPA enc Adv b = run (Adv (EXP b enc))\n\n    -- no use of CPAWinner\n    CPASec : \u2200 k (enc : M \u2192 \u21ba |K| C) \u2192 Set\u2081\n    CPASec k enc = \u2200 (Adv : CPAAdv k) \u2192 negligible-advantage (runCPA enc Adv)\n\n    SemSecWinner : \u2200 (enc : M \u00d7 M \u2192 \u21ba |K| C) b k \u2192 SemSecAdv k \u2192 Set\u2081\n    SemSecWinner enc b k Adv = CPAWinner enc b 1 (sem2CPAAdv Adv enc)\n\n    runSemSec : \u2200 {k} (enc : M \u2192 \u21ba |K| C) \u2192 SemSecAdv k \u2192 Bit \u2192 \u21ba (|K| + k) Bit\n    runSemSec enc Adv = runCPA enc (sem2CPAAdv Adv)\n\n    -- no use of SemSecWinner\n    SemSec : \u2200 k (enc : M \u2192 \u21ba |K| C) \u2192 Set\n    SemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runSemSec enc Adv)\n\n    weakenWinner : \u2200 {enc b k q\u2080 q\u2081} {s : Strategy id Bit k} \u2192 CPAWinner enc b q\u2080 s \u2192 CPAWinner enc b (q\u2080 + q\u2081) s\n    weakenWinner win     = win\n    weakenWinner (ask w) = ask (weakenWinner w)\n    weakenWinner (rnd {n = n} {k} {bs = bs} w) = rnd {n = n} {k} {bs = bs} (weakenWinner w)\n\n    CPA2Sem-correct : \u2200 {enc b k q} {semAdv : SemSecAdv k}\n                      \u2192 SemSecWinner enc b k semAdv \u2192 CPAWinner enc b (suc q) (sem2CPAAdv semAdv enc)\n    CPA2Sem-correct = weakenWinner\n\n  record Forgery {a} {A : Set a} (P : A \u2192 Set) (queries : List A) : Set a where\n    field\n      forgery         : A\n      fresh-forgery   : forgery \u2209 queries\n      correct-forgery : P forgery\n  open Forgery public\n\n  module MACGames (|M| |Tag| : \u2115) where\n    M = Bits |M|\n    Tag = Bits |Tag|\n\n    -- One might also want that the adversary sends the trials to the challenger.\n    -- The challenger might then either end the game or maybe return the info to\n    -- the adversary.\n    MACAdv : Set\u2081\n    MACAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 Tag) \u2192 Strategy \u2610_ (M \u00d7 Tag) 0\n\n    -- or\n    -- MACAdv = \u2200 {\u2610_ Done} (mac : M \u2192 \u2610 Tag) (submit : M \u00d7 Tag \u2192 \u2610 Done) \u2192 Strategy \u2610_ Done\n\n    -- or\n    -- MACAdv = \u2200 {\u2610_ Done} (mac : M \u2192 \u2610 Tag)\n    --                       (try : M \u00d7 Tag \u2192 \u2610 Bool)\n    --                       (submit : M \u00d7 Tag \u2192 \u2610 Done) \u2192 Strategy \u2610_ Done\n\n    module MACGames' (mac : M \u2192 Tag) where\n      CorrectMAC : M \u00d7 Tag \u2192 Set\n      CorrectMAC (m , t) = mac m \u2261 t\n\n      correctMAC? : M \u00d7 Tag \u2192 Bool\n      correctMAC? (m , t) = mac m == t\n\n      data MACWinner (mts : List (M \u00d7 Tag)) : (q : \u2115) \u2192 Strategy id (M \u00d7 Tag) 0 \u2192 Set\u2081 where\n        win : \u2200 {q} (f : Forgery CorrectMAC mts) \u2192 MACWinner mts q (say (forgery f))\n        ask : \u2200 {m f q} \u2192 MACWinner ((m , mac m) \u2237 mts) q (f (mac m))\n                        \u2192 MACWinner mts (suc q) (ask (mac m) f)\n\n      SecMAC : Set\u2081\n      SecMAC = \u2200 (A : MACAdv) \u2192 Pr[ \u27ea correctMAC? \u00b7 run (A mac) \u27eb \u22611] \u22611\/2^ |Tag| -- check that bound\n\n  module CiphertextIntegrity (M C : Set) where\n    CIAdv : Set\u2081\n    CIAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ C 0\n\n    module CiphertextIntegrity' (enc : M \u2192 C) (dec : C \u2192? M) where\n      CorrectCiphertext : C \u2192 Set\n      CorrectCiphertext c = IsJust (dec c)\n\n      data CIWinner (cs : List C) : (q : \u2115) \u2192 Strategy id C 0 \u2192 Set\u2081 where\n        win : \u2200 {q} (f : Forgery CorrectCiphertext cs) \u2192 CIWinner cs q (say (forgery f))\n        ask : \u2200 {m f q} \u2192 CIWinner (enc m \u2237 cs) q (f (enc m))\n                        \u2192 CIWinner cs (suc q) (ask (enc m) f)\n\n  module AuthenticatedEncryption (|K| |M| |C| : \u2115) where\n    open CipherGames |K| |M| |C|\n    open CiphertextIntegrity M C\n\n    data AERes : Set where\n      distinguish   : Bit \u2192 AERes\n      found-forgery : C \u2192 AERes\n\n    AEAdv : Set\u2081\n    AEAdv = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) \u2192 Strategy \u2610_ AERes 0\n\n    CCAAdv : Set\u2081\n    CCAAdv = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) (dec : C \u2192 \u2610 (Maybe M)) \u2192 Strategy \u2610_ Bit 0\n\n    module AE' (enc : M \u00d7 M \u2192 C) (dec : C \u2192? M) where\n      open CiphertextIntegrity' (\u03bb m \u2192 enc (m , m)) dec\n\n      data AEWinner (cs : List C) b : (q : \u2115) \u2192 Strategy id AERes 0 \u2192 Set\u2081 where\n        win-distinguish : \u2200 {q} \u2192 AEWinner cs b q (say (distinguish b))\n        win-forgery : \u2200 {q} (f : Forgery CorrectCiphertext cs)\n                       \u2192 AEWinner cs b q (say (found-forgery (forgery f)))\n        ask : \u2200 {m\u2080m\u2081 f q} \u2192 AEWinner cs b q (f (enc m\u2080m\u2081)) \u2192\n              AEWinner cs b (suc q) (ask (enc m\u2080m\u2081) f)\n\n      data CCAWinner b : (q : \u2115) \u2192 Strategy id Bit 0 \u2192 Set\u2081 where\n        win : \u2200 {q} \u2192 CCAWinner b q (say b)\n        ask-enc : \u2200 {m\u2080m\u2081 f q} \u2192 CCAWinner b q (f (enc m\u2080m\u2081)) \u2192\n                    CCAWinner b (suc q) (ask (enc m\u2080m\u2081) f)\n        ask-dec : \u2200 {c f q} \u2192 CCAWinner b q (f (dec c)) \u2192\n                    CCAWinner b (suc q) (ask (dec c) f)\n\n  -- CBC with rand IV is not CCA secure\n\n      -- AE \u21d2 CCA Secure\n\n  module OTPDetSemSec {n} (key : Bits n) where\n    open CipherGames n n n\n    open DeterministicSemanticSecurity\n\n    postulate\n      OTPDetSemSec : \u2200 {k} \u2192 DetSemSec k (OTP key)\n{-\n    OTPDetSemSec (mk refl gen-m\u2080m\u2081 stat-test) = {!!}\n    -- DetSemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runDetSemSec enc Adv)\n    -- runDetSemSec enc (mk refl gen-m\u2080m\u2081 f) b = gen-m\u2080m\u2081 >>= \u03bb m\u2080m\u2081 \u2192 f (EXP b enc m\u2080m\u2081)\n\nPr[ runDetSemSec enc Adv b \u2261 b ]\n-}\n\n  Enc : \u2200 |K| |M| |C| \u2192 Set\n  Enc |K| |M| |C| = Bits |K| \u2192 Bits |M| \u2192 Bits |C|\n\n  module DerivedBlockCiphers\n    {|K| |M| |C|} {E : Enc |K| |M| |C|}\n    (E-sec : \u2200 {k} key \u2192 CipherGames.DeterministicSemanticSecurity.DetSemSec |K| |M| |C| k (E key)) where\n\n    open module CipherGames' {|M| |C|} = CipherGames |K| |M| |C|\n    open DeterministicSemanticSecurity\n\n    E\u2080 : Enc |K| |M| |C|\n    E\u2080 k m = 1\u207f \u2295 E k m\n\n    postulate\n      lem' : \u2200 {k} (Adv : SemSecAdv k) b (f : Bits k \u2192 Bits |C|) \u2192 runDetSemSec {k} (_\u2295_ 1\u207f \u2218 f) Adv b \u2261 \u27ea _xor_ 1b \u00b7 runDetSemSec f Adv b \u27eb\n    -- lem' Adv b f = {!!}\n\n      lem'' : \u2200 {k} (Adv : SemSecAdv k) b (f : Bits k \u2192 Bits |C|) \u2192 \u27ea _xor_ 1b \u00b7 runDetSemSec {k} (_\u2295_ 1\u207f \u2218 f) Adv b \u27eb \u2261 runDetSemSec f Adv b\n    -- lem'' Adv b f = {!!}\n\n    -- postulate\n    E\u2080-sec : \u2200 key \u2192 DetSemSec |M| (E\u2080 key)\n    E\u2080-sec key Adv = negligible-advantage-trans (negligible-advantage-xor (runDetSemSec (E\u2080 key) Adv 0b) (runDetSemSec (E key) Adv 0b) (lem' Adv 0b (E key))) (negligible-advantage-trans (E-sec key Adv) (negligible-advantage-xor (runDetSemSec (E key) Adv 1b) (runDetSemSec (E\u2080 key) Adv 1b) (sym (lem'' Adv 1b (E key)))))\n\n    E\u2081 : Enc |K| |M| (1 + |C|)\n    E\u2081 k m = 0\u2237 E k m\n\n    -- E\u2081-adv : \u2200 {k} key \u2192 SemSecAdv k (E key) \u2192 SemSecAdv k (E\u2081 key)\n    -- E\u2081-adv key Adv = {!E-sec!}\n\n    postulate\n      E\u2081-sec : \u2200 {k} key \u2192 DetSemSec k (E\u2081 key)\n    -- E\u2081-sec key Adv = {!E-sec key ?!}\n\n    -- runDetSemSec enc adv b \u2248 runDetSemSec enc' adv b\n","old_contents":"module game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Bool -- hiding (_\u225f_)\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R\n  ask : {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R) \u2192 Strategy \u2610_ R\n\nrun : \u2200 {R} \u2192 Strategy id R \u2192 R\nrun (say r)   = r\nrun (ask a f) = run (f a)\n\nopen import Data.Bits\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nmodule CryptoGames where\n\n  module CipherGames (|M| |C| : \u2115) where\n    M : Set\n    M = Bits |M|\n    C : Set\n    C = Bits |C|\n\n    SemSecAdv : Set\n    SemSecAdv = C \u2192 Bit\n\n    CPAAdv : Set\u2081\n    CPAAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit\n\n    data CPAWinner (enc : M \u2192 C) b : Strategy id Bit \u2192 Set\u2081 where\n      win : CPAWinner enc b (say b)\n      ask : \u2200 {m f} \u2192 CPAWinner enc b (f (enc m)) \u2192 CPAWinner enc b (ask (enc m) f)\n\n    CPA2Sem : SemSecAdv \u2192 CPAAdv\n    CPA2Sem semAdv enc = ask (enc 0\u2026) (\u03bb c \u2192 say (semAdv c))\n\n    CPA2Sem-correct : SemSecWinner semAdv \u2192 CPAWinner enc b (CPA2Sem semAdv)\n    CPA2Sem-correct\n\n  module MACGames (M T : Set) where\n    MacAdv : Set\u2081\n    MacAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 T) \u2192 Strategy \u2610_ (M \u00d7 T)\n\n    data MACWinner (mac : M \u2192 T) : Strategy id (M \u00d7 T) \u2192 Set\u2081 where\n      win : \u2200 {m} \u2192 MACWinner mac (say (m , mac m))\n      ask : \u2200 {m f} \u2192 MACWinner mac (f (mac m)) \u2192 MACWinner mac (ask (mac m) f)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"164fe8668797943a96ba4cc480aec8464a647e26","subject":"Group change structures need no abelian group","message":"Group change structures need no abelian group\n\nThe previous proof was unnecessarily restrictive for no clear reason.\nOTOH, aggregation can take advantage of abelian groups.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set c)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c}\n    (Carrier : Set c) : Set (suc c) where\n  field\n    Change : Carrier \u2192 Set c\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} {A : Set a} (P : A \u2192 Set p): Set (suc p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 b)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (p \u2294 q)\nRawChange\u208d x , y \u208e f = \u2200 px (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isGroup : IsGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsGroup isGroup\n      hiding\n        ( refl\n        ; sym\n        )\n      renaming\n        ( \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    _\u229d_ : A \u2192 A \u2192 A\n    v \u229d u = (u \u207b\u00b9) \u2295 v\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8\u27e9\n               v \u2295 ((v \u207b\u00b9) \u2295 u)\n            \u2261\u27e8  sym (assoc _ _ _) \u27e9\n               (v \u2295 (v \u207b\u00b9)) \u2295 u\n            \u2261\u27e8 proj\u2082 inverse v \u27e8\u2295\u27e9 refl \u27e9\n               \u03b5 \u2295 u\n            \u2261\u27e8 proj\u2081 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\nmodule AbelianGroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} (A : Set a) (B : Set b) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g (a \u229e da \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n\n    funNil : (f : A \u2192 B) \u2192 FunctionChange f\n    funNil = \u03bb f \u2192 funDiff f f\n    private\n      -- Realizer for funNil\n      funNil-realizer : (f : A \u2192 B) \u2192 RawChange f\n      funNil-realizer f = \u03bb a da \u2192 f (a \u229e da) \u229f f a\n      -- Note that it cannot use nil (f a), that would not be a correct function change!\n      funNil-correct : \u2200 f a da \u2192 funNil-realizer f a da \u2261 apply (funNil f) a da\n      funNil-correct f a da = refl\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebraFun : ChangeAlgebra (A \u2192 B)\n\n      changeAlgebraFun = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n-- Reexport a few members with A and B marked as implicit parameters. This\n-- matters especially for changeAlgebra, since otherwise it can't be used for\n-- instance search.\nmodule _\n    {a} {b} {A : Set a} {B : Set b} {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    open FunctionChanges A B {{CA}} {{CB}} public\n      using\n        ( changeAlgebraFun\n        ; apply\n        ; correct\n        ; incrementalization\n        ; DerivativeAsChange\n        ; FunctionChange\n        ; nil-is-derivative\n        )\n\nmodule BinaryFunctionChanges\n    {a} {b} {c} (A : Set a) (B : Set b) (C : Set c) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} {{CC : ChangeAlgebra C}} where\n    incrementalization-binary : \u2200 (f : A \u2192 B \u2192 C) df a da b db \u2192\n      (f \u229e df) (a \u229e da) (b \u229e db) \u2261 f a b \u229e apply (apply df a da) b db\n    incrementalization-binary f df x dx y dy\n      rewrite cong (\u03bb \u25a1 \u2192 \u25a1 (y \u229e dy)) (incrementalization f df x dx)\n      = incrementalization (f x) (apply df x dx) y dy\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 \u0394 pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 \u0394 pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebraListChanges : ChangeAlgebraFamily (All P)\n\n    changeAlgebraListChanges = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set c)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c}\n    (Carrier : Set c) : Set (suc c) where\n  field\n    Change : Carrier \u2192 Set c\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} {A : Set a} (P : A \u2192 Set p): Set (suc p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nRawChange : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  Set (a \u2294 b)\nRawChange f = \u2200 a (da : \u0394 a) \u2192 \u0394 (f a)\n\nIsDerivative : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra A}} \u2192\n  {{CB : ChangeAlgebra B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : RawChange f) \u2192\n  Set (a \u2294 b)\nIsDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of IsDerivative for change algebra families.\nRawChange\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192 Set (p \u2294 q)\nRawChange\u208d x , y \u208e f = \u2200 px (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)\n\nIsDerivative\u208d_,_\u208e : \u2200 {a b p q} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily P}} \u2192\n  {{CQ : ChangeAlgebraFamily Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : RawChange\u208d_,_\u208e x y f) \u2192\n  Set (p \u2294 q)\nIsDerivative\u208d_,_\u208e {P = P} {{CP}} {{CQ}} x y f df = IsDerivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d_\u208e {{CP}} x\n  CQy = change-algebra\u208d_\u208e {{CQ}} y\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebraGroup : ChangeAlgebra A\n    changeAlgebraGroup = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} (A : Set a) (B : Set b) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b) where\n      field\n        -- Definition 2.6a\n        apply : RawChange f\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g (a \u229e da \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n\n    funNil : (f : A \u2192 B) \u2192 FunctionChange f\n    funNil = \u03bb f \u2192 funDiff f f\n    private\n      -- Realizer for funNil\n      funNil-realizer : (f : A \u2192 B) \u2192 RawChange f\n      funNil-realizer f = \u03bb a da \u2192 f (a \u229e da) \u229f f a\n      -- Note that it cannot use nil (f a), that would not be a correct function change!\n      funNil-correct : \u2200 f a da \u2192 funNil-realizer f a da \u2261 apply (funNil f) a da\n      funNil-correct f a da = refl\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebraFun : ChangeAlgebra (A \u2192 B)\n\n      changeAlgebraFun = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e (nil a)) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper. However, the derivative of f is just\n    -- the apply component of `nil f`, not the full `nil f`, which also includes\n    -- a proof. This is not an issue in the paper, which is formulated in a\n    -- proof-irrelevant metalanguage.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      IsDerivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n    -- Show that any derivative is a valid function change.\n\n    -- In the paper, this is never actually stated. We just prove that nil\n    -- changes are derivatives; the paper keeps talking about \"the derivative\",\n    -- suggesting derivatives are unique. If derivatives were unique, we could\n    -- say that the nil change is *the* derivative, hence the derivative is the\n    -- nil change (hence also a change).\n    --\n    -- In fact, derivatives are only unique up to change equivalence and\n    -- extensional equality; this is proven in Base.Change.Equivalence.derivative-unique.\n    --\n    Derivative-is-valid : \u2200 {f : A \u2192 B} df (IsDerivative-f-df : IsDerivative f df) a da \u2192\n      f (a \u229e da) \u229e (df (a \u229e da) (nil (a \u229e da))) \u2261 f a \u229e df a da\n    Derivative-is-valid {f} df IsDerivative-f-df a da rewrite IsDerivative-f-df (a \u229e da) (nil (a \u229e da)) | update-nil (a \u229e da) = sym (IsDerivative-f-df a da)\n\n    DerivativeAsChange : \u2200 {f : A \u2192 B} {df} (IsDerivative-f-df : IsDerivative f df) \u2192 \u0394 f\n    DerivativeAsChange {df = df} IsDerivative-f-df = record { apply = df ; correct = Derivative-is-valid df IsDerivative-f-df }\n    -- In Equivalence.agda, derivative-is-nil-alternative then proves that a derivative is also a nil change.\n\n-- Reexport a few members with A and B marked as implicit parameters. This\n-- matters especially for changeAlgebra, since otherwise it can't be used for\n-- instance search.\nmodule _\n    {a} {b} {A : Set a} {B : Set b} {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}}\n  where\n    open FunctionChanges A B {{CA}} {{CB}} public\n      using\n        ( changeAlgebraFun\n        ; apply\n        ; correct\n        ; incrementalization\n        ; DerivativeAsChange\n        ; FunctionChange\n        ; nil-is-derivative\n        )\n\nmodule BinaryFunctionChanges\n    {a} {b} {c} (A : Set a) (B : Set b) (C : Set c) {{CA : ChangeAlgebra A}} {{CB : ChangeAlgebra B}} {{CC : ChangeAlgebra C}} where\n    incrementalization-binary : \u2200 (f : A \u2192 B \u2192 C) df a da b db \u2192\n      (f \u229e df) (a \u229e da) (b \u229e db) \u2261 f a b \u229e apply (apply df a da) b db\n    incrementalization-binary f df x dx y dy\n      rewrite cong (\u03bb \u25a1 \u2192 \u25a1 (y \u229e dy)) (incrementalization f df x dx)\n      = incrementalization (f x) (apply df x dx) y dy\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 \u0394 pxs  \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 \u0394 pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebraListChanges : ChangeAlgebraFamily (All P)\n\n    changeAlgebraListChanges = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39c291ae589f2cde15f722e7c410688983e59bae","subject":"Typo.","message":"Typo.\n\nIgnore-this: 7d863544a50ca658d5e4ceb2894bb038\n\ndarcs-hash:20120222194729-3bd4e-7d4675c37d9d3d6ee2053bec0f9827728462a3ce.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Axiomatic\/Standard\/Base.agda","new_file":"src\/PA\/Axiomatic\/Standard\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Standard.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe.\n-- We chose the symbol M because there are non-standard models of\n-- Peano Arithmetic, where the domain is not the set of natural\n-- numbers.\nopen import FOL.Universe public renaming ( D to M )\n\n-- FOL with equality.\nopen import FOL.FOL-Eq public\n\n-- Non-logical constants\npostulate\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- Proper axioms (see [p. 263, 1], [p. 28, 2])\n--\n-- [1] Mosh\u00e9 Machover. Set theory, logic and their\n--     limitations. Cambridge University Press, 1996.\n--\n-- [2] Petr H\u00e1jeck and Pavel Pudl\u00e1k. Metamathematics of First-Order\n--     Arithmetic. Springer, 1998. 2nd printing.\n\n-- A\u2081. 0 \u2260 succ n\n-- A\u2082. succ m = succ n \u2192 m = n\n-- A\u2083. 0 + n = n\n-- A\u2084. succ m + n = succ (m + n)\n-- A\u2085. 0 * n = 0\n-- A\u2086. succ m * n = (m * n) + n\n-- A\u2087. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  A\u2081 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\n  A\u2082 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n  A\u2083 : \u2200 n \u2192 zero + n \u2261 n\n  A\u2084 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  A\u2085 : \u2200 n \u2192 zero * n \u2261 zero\n  A\u2086 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n{-# ATP axiom A\u2081 A\u2082 A\u2083 A\u2084 A\u2085 A\u2086 #-}\n\n-- A\u2087 is an axiom schema, therefore we do not translate it to TPTP.\npostulate A\u2087 : (P : M \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Standard.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe.\n-- We chose the symbol M because there are non-standard models of\n-- Peano Arithmetic, where the domain is not the set of natural\n-- numbers.\nopen import FOL.Universe public renaming ( D to M )\n\n-- FOL with equality.\nopen import FOL.FOL-Eq public\n\n-- Non-logical constants\npostulate\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- Proper axioms (see [p. 263, 1], [p. 28, 2])\n--\n-- [1] Mosh\u00e9 Machover. Set theory, logic and their\n--     limitations. Cambridge University Press, 1996.\n--\n-- [2] Petr H\u00e1jeck and Pavel Pudl\u00e1k. Metamathematics of First-Order\n--     Arithmetic. Springer, 1998. 2nd printing.\n\n-- A\u2081. 0 \u2260 succ n\n-- A\u2082. succ m = succ n \u2192 m = n\n-- A\u2083. 0 + n = n\n-- A\u2084. succ m + n = succ (m + n)\n-- A\u2085. 0 * n = 0\n-- A\u2086. succ m * n = (m * n) + m\n-- A\u2087. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  A\u2081 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\n  A\u2082 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n  A\u2083 : \u2200 n \u2192 zero + n \u2261 n\n  A\u2084 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  A\u2085 : \u2200 n \u2192 zero * n \u2261 zero\n  A\u2086 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n{-# ATP axiom A\u2081 A\u2082 A\u2083 A\u2084 A\u2085 A\u2086 #-}\n\n-- A\u2087 is an axiom schema, therefore we do not translate it to TPTP.\npostulate A\u2087 : (P : M \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"644151df0e927fd182a6cd6115bd0f84128b983e","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: fad85ace1c4b968f92501f9ce9408d8d\n\ndarcs-hash:20111129153543-3bd4e-23c2aed9e8a9fab6f56b60bda9286fb19f32cd9d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Inequalities.agda","new_file":"src\/FOTC\/Data\/Nat\/Inequalities.agda","new_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  _<_  : D \u2192 D \u2192 D\n  <-00 :         zero    < zero    \u2261 false\n  <-0S : \u2200 d \u2192   zero    < succ\u2081 d \u2261 true\n  <-S0 : \u2200 d \u2192   succ\u2081 d < zero    \u2261 false\n  <-SS : \u2200 d e \u2192 succ\u2081 d < succ\u2081 e \u2261 d < e\n{-# ATP axiom <-00 <-0S <-S0 <-SS #-}\n\n_\u2264_ : D \u2192 D \u2192 D\nd \u2264 e = d < succ\u2081 e\n{-# ATP definition _\u2264_ #-}\n\n_>_ : D \u2192 D \u2192 D\nd > e = e < d\n{-# ATP definition _>_ #-}\n\n_\u2265_ : D \u2192 D \u2192 D\n_\u2265_ d e = e \u2264 d\n{-# ATP definition _\u2265_ #-}\n\n------------------------------------------------------------------------\n-- The data types\n\nGT : D \u2192 D \u2192 Set\nGT d e = d > e \u2261 true\n{-# ATP definition GT #-}\n\nNGT : D \u2192 D \u2192 Set\nNGT d e = d > e \u2261 false\n{-# ATP definition NGT #-}\n\nLT : D \u2192 D \u2192 Set\nLT d e = d < e \u2261 true\n{-# ATP definition LT #-}\n\nNLT : D \u2192 D \u2192 Set\nNLT d e = d < e \u2261 false\n{-# ATP definition NLT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = d \u2264 e \u2261 true\n{-# ATP definition LE #-}\n\nNLE : D \u2192 D \u2192 Set\nNLE d e = d \u2264 e \u2261 false\n{-# ATP definition NLE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = d \u2265 e \u2261 true\n{-# ATP definition GE #-}\n\nNGE : D \u2192 D \u2192 Set\nNGE d e = d \u2265 e \u2261 false\n{-# ATP definition NGE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order.\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  _<_  : D \u2192 D \u2192 D\n  <-00 :         zero   < zero     \u2261 false\n  <-0S : \u2200 d \u2192   zero   < succ\u2081 d  \u2261 true\n  <-S0 : \u2200 d \u2192   succ\u2081 d < zero    \u2261 false\n  <-SS : \u2200 d e \u2192 succ\u2081 d < succ\u2081 e \u2261 d < e\n{-# ATP axiom <-00 <-0S <-S0 <-SS #-}\n\n_\u2264_ : D \u2192 D \u2192 D\nd \u2264 e = d < succ\u2081 e\n{-# ATP definition _\u2264_ #-}\n\n_>_ : D \u2192 D \u2192 D\nd > e = e < d\n{-# ATP definition _>_ #-}\n\n_\u2265_ : D \u2192 D \u2192 D\n_\u2265_ d e = e \u2264 d\n{-# ATP definition _\u2265_ #-}\n\n------------------------------------------------------------------------\n-- The data types\n\nGT : D \u2192 D \u2192 Set\nGT d e = d > e \u2261 true\n{-# ATP definition GT #-}\n\nNGT : D \u2192 D \u2192 Set\nNGT d e = d > e \u2261 false\n{-# ATP definition NGT #-}\n\nLT : D \u2192 D \u2192 Set\nLT d e = d < e \u2261 true\n{-# ATP definition LT #-}\n\nNLT : D \u2192 D \u2192 Set\nNLT d e = d < e \u2261 false\n{-# ATP definition NLT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = d \u2264 e \u2261 true\n{-# ATP definition LE #-}\n\nNLE : D \u2192 D \u2192 Set\nNLE d e = d \u2264 e \u2261 false\n{-# ATP definition NLE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = d \u2265 e \u2261 true\n{-# ATP definition GE #-}\n\nNGE : D \u2192 D \u2192 Set\nNGE d e = d \u2265 e \u2261 false\n{-# ATP definition NGE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order.\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a019fa8cb011fee51c4966ba7da21015a5e3cff6","subject":"Fixed a thesis' note.","message":"Fixed a thesis' note.\n\nIgnore-this: 51b11ae0e26ec1d6b9ae9a97ae82b7ad\n\ndarcs-hash:20120507171011-3bd4e-02a0be94501e902f0ff93fd66e88947a1c567c46.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Disjunction.agda","new_file":"notes\/thesis\/logical-framework\/Disjunction.agda","new_contents":"-- Tested with the development version of Agda on 07 May 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Disjunction where\n\nmodule LF where\n  postulate\n    _\u2228_   : Set \u2192 Set \u2192 Set\n    inj\u2081  : {A B : Set} \u2192 A \u2192 A \u2228 B\n    inj\u2082  : {A B : Set} \u2192 B \u2192 A \u2228 B\n    [_,_] : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n\n  \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 B \u2228 A\n  \u2228-comm = [ inj\u2082 , inj\u2081 ]\n\nmodule Inductive where\n\n  open import Common.FOL.FOL\n\n  \u2228-comm-el : {A B : Set} \u2192 A \u2228 B \u2192 B \u2228 A\n  \u2228-comm-el = [ inj\u2082 , inj\u2081 ]\n\n  \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 B \u2228 A\n  \u2228-comm (inj\u2081 a) = inj\u2082 a\n  \u2228-comm (inj\u2082 b) = inj\u2081 b\n","old_contents":"-- Tested with the development version of Agda on 07 February 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Disjunction where\n\nmodule LF where\n  postulate\n    _\u2228_   : Set \u2192 Set \u2192 Set\n    inj\u2081  : {A B : Set} \u2192 A \u2192 A \u2228 B\n    inj\u2082  : {A B : Set} \u2192 B \u2192 A \u2228 B\n    [_,_] : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n\n  \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 B \u2228 A\n  \u2228-comm = [ inj\u2082 , inj\u2081 ]\n\nmodule Inductive where\n\n  open import Common.Data.Sum\n\n  \u2228-comm-el : {A B : Set} \u2192 A \u2228 B \u2192 B \u2228 A\n  \u2228-comm-el = [ inj\u2082 , inj\u2081 ]\n\n  \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 B \u2228 A\n  \u2228-comm (inj\u2081 a) = inj\u2082 a\n  \u2228-comm (inj\u2082 b) = inj\u2081 b\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9544171825909548817d2be2274c9f24d65e4fb2","subject":"Reported an issue related to syntax declarations.","message":"Reported an issue related to syntax declarations.\n\nIgnore-this: a5aef065a36f032ae2da76bef1734318\n\ndarcs-hash:20111201152937-3bd4e-0ac00063e610409016194cc8d8698622e0116549.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/ExistentialSyntax.agda","new_file":"notes\/ExistentialSyntax.agda","new_contents":"-- Tested on 01 December 2011.\n\nmodule ExistentialSyntax where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 {d} \u2192 d \u2261 d\n  d    : D\n\nmodule \u2203\u2081 where\n  infixr 4 _,_\n\n  data \u2203 (P : D \u2192 Set) : Set where\n    _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n  syntax \u2203 (\u03bb x \u2192 e)  = \u2203[ x ] e\n\n  t\u2081 : \u2203 \u03bb x \u2192 x \u2261 x\n  t\u2081 = d , refl\n\n  t\u2082 : \u2203[ x ] x \u2261 x\n  t\u2082 = d , refl\n\n  t\u2083 : \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 x \u2261 y\n  t\u2083 = d , d , refl\n\n  t\u2084 : \u2203[ x ] \u2203[ y ] x \u2261 y\n  t\u2084 = d , d ,  refl\n\nmodule \u2203\u2082 where\n  infixr 4 _,_,_\n\n  data \u2203\u2082 (P : D \u2192 D \u2192 Set) : Set where\n    _,_,_ : (x y : D) \u2192 P x y \u2192 \u2203\u2082 P\n\n  -- Agda issue: 536\n  -- syntax \u2203\u2082 (\u03bb x y \u2192 e) = \u2203\u2082[ x , y ] e\n\n  t\u2081 : \u2203\u2082 \u03bb x y \u2192 x \u2261 y\n  t\u2081 = d , d , refl\n","old_contents":"-- Tested on 01 December 2011.\n\nmodule ExistentialSyntax where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 {d} \u2192 d \u2261 d\n  d    : D\n\nmodule \u2203\u2081 where\n  infixr 4 _,_\n\n  data \u2203 (P : D \u2192 Set) : Set where\n    _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n  syntax \u2203 (\u03bb x \u2192 e)  = \u2203[ x ] e\n\n  t\u2081 : \u2203 \u03bb x \u2192 x \u2261 x\n  t\u2081 = d , refl\n\n  t\u2082 : \u2203[ x ] x \u2261 x\n  t\u2082 = d , refl\n\n  t\u2083 : \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 x \u2261 y\n  t\u2083 = d , d , refl\n\n  t\u2084 : \u2203[ x ] \u2203[ y ] x \u2261 y\n  t\u2084 = d , d ,  refl\n\nmodule \u2203\u2082 where\n  infixr 4 _,_,_\n\n  data \u2203\u2082 (P : D \u2192 D \u2192 Set) : Set where\n    _,_,_ : (x y : D) \u2192 P x y \u2192 \u2203\u2082 P\n\n  -- Parse error\n  -- syntax \u2203\u2082 (\u03bb x y \u2192 e) = \u2203\u2082[ x y ] e\n\n  t\u2081 : \u2203\u2082 \u03bb x y \u2192 x \u2261 y\n  t\u2081 = d , d , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"46d74d7fe8c8e1bbca68e63370beb849eebfa191","subject":"Defined a mutual block.","message":"Defined a mutual block.\n\nIgnore-this: b506af7202e0a2460bd133e2025c5f58\n\ndarcs-hash:20100713235715-3bd4e-63461410827016b602ad261a238b453d759cf5f6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/SortList\/Closures\/ListOrd.agda","new_file":"Examples\/SortList\/Closures\/ListOrd.agda","new_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to ListOrd\n------------------------------------------------------------------------------\n\nmodule Examples.SortList.Closures.ListOrd where\n\nopen import LTC.Minimal\n\nopen import Examples.SortList.Closures.Bool\n  using ( \u2264-ItemList-Bool ; \u2264-Lists-Bool ; isListOrd-Bool)\nopen import Examples.SortList.Closures.List using ( flatten-List )\nopen import Examples.SortList.Closures.TreeOrd\n  using ( leftSubTree-TreeOrd ; rightSubTree-TreeOrd )\nopen import Examples.SortList.Properties using ( subList-ListOrd ; xs\u2264[] )\nopen import Examples.SortList.SortList\n\nopen import LTC.Data.Bool.Properties\n  using ( \u2264-Bool ; x&&y\u2261true\u2192x\u2261true ; x&&y\u2261true\u2192y\u2261true )\nopen import LTC.Data.Nat.List.Type\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.List\n\nopen import Postulates using ( ++-ListOrd-aux\u2081 )\n\n------------------------------------------------------------------------------\n-- Auxiliar functions\n\n++-ListOrd-aux\u2082 : {item is js : D} \u2192 N item \u2192 ListN is \u2192 ListN js \u2192\n                  LE-ItemList item is \u2192\n                  LE-Lists is js \u2192\n                  LE-ItemList item (is ++ js)\n++-ListOrd-aux\u2082 {item} {js = js} Nitem nilLN LNjs item\u2264niL niL\u2264js = prf\n  where\n    postulate prf : LE-ItemList item ([] ++ js)\n    {-# ATP prove prf ++-ListOrd-aux\u2081 #-}\n\n++-ListOrd-aux\u2082 {item} {js = js} Nitem\n               (consLN {i} {is} Ni LNis) LNjs item\u2264i\u2237is i\u2237is\u2264js =\n  prf (++-ListOrd-aux\u2082 Nitem LNis LNjs\n        (x&&y\u2261true\u2192y\u2261true (\u2264-Bool Nitem Ni)\n                          (\u2264-ItemList-Bool Nitem LNis)\n                          (trans (sym (\u2264-ItemList-\u2237 item i is)) item\u2264i\u2237is))\n        (x&&y\u2261true\u2192y\u2261true (\u2264-ItemList-Bool Ni LNis)\n                          (\u2264-Lists-Bool LNis LNjs)\n                          (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : LE-ItemList item (is ++ js) \u2192 -- IH.\n                    LE-ItemList item ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-Bool \u2264-ItemList-Bool x&&y\u2261true\u2192x\u2261true #-}\n\n------------------------------------------------------------------------------\n-- Append preserves the order.\n++-ListOrd : {is js : D} \u2192 ListN is \u2192 ListN js \u2192 ListOrd is \u2192 ListOrd js \u2192\n             LE-Lists is js \u2192\n             ListOrd (is ++ js)\n\n++-ListOrd {js = js} nilLN LNjs LOis LOjs is\u2264js = prf\n  where\n    postulate prf : ListOrd ([] ++ js)\n    {-# ATP prove prf #-}\n\n++-ListOrd {js = js} (consLN {i} {is} Ni LNis) LNjs LOi\u2237is LOjs i\u2237is\u2264js =\n  prf (++-ListOrd LNis LNjs (subList-ListOrd Ni LNis LOi\u2237is) LOjs\n                  (x&&y\u2261true\u2192y\u2261true\n                    (\u2264-ItemList-Bool Ni LNis)\n                    (\u2264-Lists-Bool LNis LNjs)\n                    (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : ListOrd (is ++ js) \u2192 -- IH.\n                    ListOrd ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-ItemList-Bool \u2264-Lists-Bool\n                      x&&y\u2261true\u2192x\u2261true x&&y\u2261true\u2192y\u2261true\n                      ++-ListOrd-aux\u2082\n    #-}\n\n------------------------------------------------------------------------------\nmutual\n  -- If t is ordered then (flatten t) is ordered\n  flatten-ListOrd : {t : D} \u2192 Tree t \u2192 TreeOrd t \u2192 ListOrd (flatten t)\n  flatten-ListOrd nilT TOnilT = prf\n    where\n      postulate prf : ListOrd (flatten nilTree)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd (tipT {i} Ni ) TOtipT = prf\n    where\n      postulate prf : ListOrd (flatten (tip i))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082 ) TOnodeT =\n    prf (flatten-ListOrd Tt\u2081 (leftSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n        (flatten-ListOrd Tt\u2082 (rightSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n        (flatten-ListOrd-aux Tt\u2081 Ni Tt\u2082 TOnodeT)\n    where\n      postulate prf : ListOrd (flatten t\u2081) \u2192 -- IH.\n                      ListOrd (flatten t\u2082) \u2192 -- IH.\n                      LE-Lists (flatten t\u2081) (flatten t\u2082) \u2192\n                      ListOrd (flatten (node t\u2081 i t\u2082))\n      {-# ATP prove prf ++-ListOrd flatten-List\n                  leftSubTree-TreeOrd rightSubTree-TreeOrd\n      #-}\n\n  flatten-ListOrd-aux : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192\n                        Tree t\u2082 \u2192 TreeOrd (node t\u2081 i t\u2082) \u2192\n                        LE-Lists (flatten t\u2081) (flatten t\u2082)\n  flatten-ListOrd-aux {t\u2082 = t\u2082} nilT Niaux Tt\u2082 _ = prf\n    where\n      postulate prf : LE-Lists (flatten nilTree) (flatten t\u2082)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni nilT TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten nilTree)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni (tipT {k} Nk) TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten (tip k))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni (nodeT {ta} {k} {tb} Tta Nk Ttb )\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten (node ta k tb))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb) Ni nilT TOnode =\n    prf (flatten-List (nodeT Tta Nj Ttb))\n        (flatten-ListOrd (nodeT Tta Nj Ttb)\n        (leftSubTree-TreeOrd (nodeT Tta Nj Ttb) Ni nilT TOnode))\n    where\n      postulate prf : ListN (flatten (node ta j tb)) \u2192\n                      ListOrd (flatten (node ta j tb)) \u2192 -- mutual IH.\n                      LE-Lists (flatten (node ta j tb)) (flatten nilTree)\n      {-# ATP prove prf xs\u2264[] #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb)\n                      Ni\n                      (tipT {k} Nk)\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (node ta j tb)) (flatten (tip k))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb)\n                      Ni\n                      (nodeT {tc} {k} {td} Ttc Nk Ttd)\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (node ta j tb))\n                                (flatten (node tc k td))\n","old_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to ListOrd\n------------------------------------------------------------------------------\n\nmodule Examples.SortList.Closures.ListOrd where\n\nopen import LTC.Minimal\n\nopen import Examples.SortList.Closures.Bool\n  using ( \u2264-ItemList-Bool ; \u2264-Lists-Bool ; isListOrd-Bool)\nopen import Examples.SortList.Closures.List using ( flatten-List )\nopen import Examples.SortList.Closures.TreeOrd\n  using ( leftSubTree-TreeOrd ; rightSubTree-TreeOrd )\nopen import Examples.SortList.Properties using ( subList-ListOrd ; xs\u2264[] )\nopen import Examples.SortList.SortList\n\nopen import LTC.Data.Bool.Properties\n  using ( \u2264-Bool ; x&&y\u2261true\u2192x\u2261true ; x&&y\u2261true\u2192y\u2261true )\nopen import LTC.Data.Nat.List.Type\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.List\n\nopen import Postulates using ( ++-ListOrd-aux\u2081 )\n\n------------------------------------------------------------------------------\n-- Auxiliar functions\n\n++-ListOrd-aux\u2082 : {item is js : D} \u2192 N item \u2192 ListN is \u2192 ListN js \u2192\n                  LE-ItemList item is \u2192\n                  LE-Lists is js \u2192\n                  LE-ItemList item (is ++ js)\n++-ListOrd-aux\u2082 {item} {js = js} Nitem nilLN LNjs item\u2264niL niL\u2264js = prf\n  where\n    postulate prf : LE-ItemList item ([] ++ js)\n    {-# ATP prove prf ++-ListOrd-aux\u2081 #-}\n\n++-ListOrd-aux\u2082 {item} {js = js} Nitem\n               (consLN {i} {is} Ni LNis) LNjs item\u2264i\u2237is i\u2237is\u2264js =\n  prf (++-ListOrd-aux\u2082 Nitem LNis LNjs\n        (x&&y\u2261true\u2192y\u2261true (\u2264-Bool Nitem Ni)\n                          (\u2264-ItemList-Bool Nitem LNis)\n                          (trans (sym (\u2264-ItemList-\u2237 item i is)) item\u2264i\u2237is))\n        (x&&y\u2261true\u2192y\u2261true (\u2264-ItemList-Bool Ni LNis)\n                          (\u2264-Lists-Bool LNis LNjs)\n                          (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : LE-ItemList item (is ++ js) \u2192 -- IH.\n                    LE-ItemList item ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-Bool \u2264-ItemList-Bool x&&y\u2261true\u2192x\u2261true #-}\n\n------------------------------------------------------------------------------\n-- Append preserves the order.\n++-ListOrd : {is js : D} \u2192 ListN is \u2192 ListN js \u2192 ListOrd is \u2192 ListOrd js \u2192\n             LE-Lists is js \u2192\n             ListOrd (is ++ js)\n\n++-ListOrd {js = js} nilLN LNjs LOis LOjs is\u2264js = prf\n  where\n    postulate prf : ListOrd ([] ++ js)\n    {-# ATP prove prf #-}\n\n++-ListOrd {js = js} (consLN {i} {is} Ni LNis) LNjs LOi\u2237is LOjs i\u2237is\u2264js =\n  prf (++-ListOrd LNis LNjs (subList-ListOrd Ni LNis LOi\u2237is) LOjs\n                  (x&&y\u2261true\u2192y\u2261true\n                    (\u2264-ItemList-Bool Ni LNis)\n                    (\u2264-Lists-Bool LNis LNjs)\n                    (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : ListOrd (is ++ js) \u2192 -- IH.\n                    ListOrd ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-ItemList-Bool \u2264-Lists-Bool\n                      x&&y\u2261true\u2192x\u2261true x&&y\u2261true\u2192y\u2261true\n                      ++-ListOrd-aux\u2082\n    #-}\n\n------------------------------------------------------------------------------\n-- If t is ordered then (flatten t) is ordered\nflatten-ListOrd : {t : D} \u2192 Tree t \u2192 TreeOrd t \u2192 ListOrd (flatten t)\nflatten-ListOrd nilT TOnilT = prf\n  where\n    postulate prf : ListOrd (flatten nilTree)\n    {-# ATP prove prf #-}\n\nflatten-ListOrd (tipT {i} Ni ) TOtipT = prf\n  where\n    postulate prf : ListOrd (flatten (tip i))\n    {-# ATP prove prf #-}\n\nflatten-ListOrd (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082 ) TOnodeT =\n  prf (flatten-ListOrd Tt\u2081 (leftSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n      (flatten-ListOrd Tt\u2082 (rightSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n      (flatten-ListOrd-aux Tt\u2081 Ni Tt\u2082 TOnodeT)\n  where\n\n    postulate prf : ListOrd (flatten t\u2081) \u2192 -- IH.\n                    ListOrd (flatten t\u2082) \u2192 -- IH.\n                    LE-Lists (flatten t\u2081) (flatten t\u2082) \u2192\n                    ListOrd (flatten (node t\u2081 i t\u2082))\n    {-# ATP prove prf ++-ListOrd flatten-List\n                  leftSubTree-TreeOrd rightSubTree-TreeOrd\n    #-}\n\n    flatten-ListOrd-aux : {taux\u2081 iaux taux\u2082 : D} \u2192 Tree taux\u2081 \u2192 N iaux \u2192\n                          Tree taux\u2082 \u2192 TreeOrd (node taux\u2081 iaux taux\u2082) \u2192\n                          LE-Lists (flatten taux\u2081) (flatten taux\u2082)\n    flatten-ListOrd-aux {taux\u2082 = taux\u2082} nilT Niaux Ttaux\u2082 _ = prf-aux\n      where\n        postulate prf-aux : LE-Lists (flatten nilTree) (flatten taux\u2082)\n        {-# ATP prove prf-aux #-}\n\n    flatten-ListOrd-aux (tipT {j} Nj) Niaux nilT TOaux = prf-aux\n      where\n        postulate prf-aux : LE-Lists (flatten (tip j)) (flatten nilTree)\n        {-# ATP prove prf-aux #-}\n\n    flatten-ListOrd-aux (tipT {j} Nj) Niaux (tipT {k} Nk) TOaux = prf-aux\n      where\n        postulate prf-aux : LE-Lists (flatten (tip j)) (flatten (tip k))\n        {-# ATP prove prf-aux #-}\n\n    flatten-ListOrd-aux (tipT {j} Nj) Niaux (nodeT {ta} {k} {tb} Tta Nk Ttb )\n                        TOaux = prf-aux\n      where\n        postulate prf-aux : LE-Lists (flatten (tip j)) (flatten (node ta k tb))\n        {-# ATP prove prf-aux #-}\n\n    flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb) Niaux nilT TOaux =\n      prf-aux (flatten-List (nodeT Tta Nj Ttb))\n              (flatten-ListOrd (nodeT Tta Nj Ttb)\n              (leftSubTree-TreeOrd (nodeT Tta Nj Ttb) Niaux nilT TOaux))\n      where\n        postulate prf-aux : ListN (flatten (node ta j tb)) \u2192\n                            ListOrd (flatten (node ta j tb)) \u2192 --IH.\n                            LE-Lists (flatten (node ta j tb)) (flatten nilTree)\n        {-# ATP prove prf-aux xs\u2264[] #-}\n\n    flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb) Niaux (tipT {k} Nk)\n                        TOaux = prf-aux\n      where\n        postulate prf-aux : LE-Lists (flatten (node ta j tb)) (flatten (tip k))\n        {-# ATP prove prf-aux #-}\n\n    flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb) Niaux\n                        (nodeT {tc} {k} {td} Ttc Nk Ttd) TOaux = prf-aux\n                        where\n                        postulate prf-aux : LE-Lists (flatten (node ta j tb))\n                                                     (flatten (node tc k td))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a85af9b943432b57c0d896d7efb1dfbc86002ce2","subject":"added addition and multiplication of natural numnbers","message":"added addition and multiplication of natural numnbers\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat.agda","new_file":"agda\/Nat.agda","new_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n","old_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"6ea9a3346907120a4fb8624d5f83b9f25bde8837","subject":"lift-op\u2082 is in nplib now","message":"lift-op\u2082 is in nplib now\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Explorable.agda","new_file":"lib\/Explore\/Explorable.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base\nopen import Data.Indexed\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin) renaming (zero to fzero)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product.NP renaming (map to \u00d7-map) hiding (first)\nopen import Data.Sum.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary.NP\nopen import Relation.Binary\nopen import Relation.Binary.Sum using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise using (_\u00d7-cong_)\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen import HoTT\nopen Equivalences\nopen \u2261 using (_\u2261_)\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule _ {m a} {A : \u2605 a} where\n  open EM {a} m\n  gfilter-explore : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore m A \u2192 Explore m B\n  gfilter-explore f e\u1d2c = e\u1d2c >>= \u03bb x \u2192 maybe (\u03bb \u03b7 \u2192 point-explore \u03b7) empty-explore (f x)\n\n  filter-explore : (A \u2192 \ud835\udfda) \u2192 Explore m A \u2192 Explore m A\n  filter-explore p = gfilter-explore \u03bb x \u2192 [0: nothing 1: just x ] (p x)\n\n  -- monoidal exploration: explore A with a monoid M\n  explore-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-monoid e\u1d2c M = e\u1d2c \u03b5 _\u00b7_ where open Mon M renaming (_\u2219_ to _\u00b7_)\n\n  explore-endo : Explore m A \u2192 Explore m A\n  explore-endo e\u1d2c \u03b5 op f = e\u1d2c id _\u2218\u2032_ (op \u2218 f) \u03b5\n\n  explore-endo-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-endo-monoid = explore-monoid \u2218 explore-endo\n\n  explore-backward : Explore m A \u2192 Explore m A\n  explore-backward e\u1d2c \u03b5 _\u2219_ f = e\u1d2c \u03b5 (flip _\u2219_) f\n\n  -- explore-backward \u2218 explore-backward = id\n  -- (m : a comm monoid) \u2192 explore-backward m = explore m\n\nprivate\n  module FindForward {a} {A : \u2605 a} (explore : Explore a A) where\n    find? : Find? A\n    find? = explore nothing (M?._\u2223_ _)\n\n    first : Maybe A\n    first = find? just\n\n    findKey : FindKey A\n    findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule ExplorePlug {\u2113 a} {A : \u2605 a} where\n  record ExploreIndKit p (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (a \u2294 \u209b \u2113 \u2294 p) where\n    constructor mk\n    field\n      P\u03b5 : P empty-explore\n      P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081)\n      Pf : \u2200 x \u2192 P (point-explore x)\n\n  _$kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p} {e : Explore \u2113 A}\n           \u2192 ExploreInd p e \u2192 ExploreIndKit p P \u2192 P e\n  _$kit_ {P = P} ind (mk P\u03b5 P\u2219 Pf) = ind P P\u03b5 P\u2219 Pf\n\n  _,-kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p}{Q : Explore \u2113 A \u2192 \u2605 p}\n            \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\n  Pk ,-kit Qk = mk (P\u03b5 Pk , P\u03b5 Qk)\n                   (\u03bb x y \u2192 P\u2219 Pk (fst x) (fst y) , P\u2219 Qk (snd x) (snd y))\n                   (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n               where open ExploreIndKit\n\n  ExploreInd-Extra : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 _\n  ExploreInd-Extra p exp =\n    \u2200 (Q     : Explore \u2113 A \u2192 \u2605 p)\n      (Q-kit : ExploreIndKit p Q)\n      (P     : Explore \u2113 A \u2192 \u2605 p)\n      (P\u03b5    : P empty-explore)\n      (P\u2219    : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 Q e\u2080 \u2192 Q e\u2081 \u2192 P e\u2080 \u2192 P e\u2081\n               \u2192 P (merge-explore e\u2080 e\u2081))\n      (Pf    : \u2200 x \u2192 P (point-explore x))\n    \u2192 P exp\n\n  to-extra : \u2200 {p} {e : Explore \u2113 A} \u2192 ExploreInd p e \u2192 ExploreInd-Extra p e\n  to-extra e-ind Q Q-kit P P\u03b5 P\u2219 Pf =\n   snd (e-ind (Q \u00d7\u00b0 P)\n           (Q\u03b5 , P\u03b5)\n           (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n           (\u03bb x \u2192 Qf x , Pf x))\n   where open ExploreIndKit Q-kit renaming (P\u03b5 to Q\u03b5; P\u2219 to Q\u2219; Pf to Qf)\n\n  ExplorePlug : \u2200 {m} (M : Monoid \u2113 m) (e : Explore _ A) \u2192 \u2605 _\n  ExplorePlug M e = \u2200 f x \u2192 e\u2218 \u03b5 _\u2219_ f \u2219 x \u2248 e\u2218 x _\u2219_ f\n     where open Mon M\n           e\u2218 = explore-endo e\n\n  plugKit : \u2200 {m} (M : Monoid \u2113 m) \u2192 ExploreIndKit _ (ExplorePlug M)\n  plugKit M = mk (\u03bb _ \u2192 fst identity)\n                 (\u03bb Ps Ps' f x \u2192\n                    trans (\u2219-cong (! Ps _ _) refl)\n                          (trans (assoc _ _ _)\n                                 (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n                 (\u03bb x f _ \u2192 \u2219-cong (snd identity (f x)) refl)\n       where open Mon M\n\nmodule FromExplore\n    {a} {A : \u2605 a}\n    (explore : \u2200 {\u2113} \u2192 Explore \u2113 A) where\n\n  module _ {\u2113} where\n    with-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-monoid = explore-monoid explore\n\n    with\u2218 : Explore \u2113 A\n    with\u2218 = explore-endo explore\n\n    with-endo-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-endo-monoid = explore-endo-monoid explore\n\n    backward : Explore \u2113 A\n    backward = explore-backward explore\n\n    gfilter : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore \u2113 B\n    gfilter f = gfilter-explore f explore\n\n    filter : (A \u2192 \ud835\udfda) \u2192 Explore \u2113 A\n    filter p = filter-explore p explore\n\n  sum : Sum A\n  sum = explore 0 _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore 1 _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore 1\u2082 _\u2227_\n  and   = big-\u2227\n  all   = big-\u2227\n\n  big-\u2228 = explore 0\u2082 _\u2228_\n  or    = big-\u2228\n  any   = big-\u2228\n\n  big-xor = explore 0\u2082 _xor_\n\n  bin-tree : BinTree A\n  bin-tree = explore empty fork leaf\n\n  list : List A\n  list = explore List.[] _++_ List.[_]\n\n  module FindBackward = FindForward backward\n\n  findLast? : Find? A\n  findLast? = FindBackward.find?\n\n  last : Maybe A\n  last = FindBackward.first\n\n  findLastKey : FindKey A\n  findLastKey = FindBackward.findKey\n\n  open FindForward explore public\n\nmodule FromLookup\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (lookup : \u2200 {\u2113} \u2192 Lookup {\u2113} explore)\n    where\n\n  module CheckDec! {\u2113}{P : A \u2192 \u2605 \u2113}(decP : \u2200 x \u2192 Dec (P x)) where\n    CheckDec! : \u2605 _\n    CheckDec! = explore (Lift \ud835\udfd9) _\u00d7_ \u03bb x \u2192 \u2713 \u230a decP x \u230b\n\n    checkDec! : {p\u2713 : CheckDec!} \u2192 \u2200 x \u2192 P x\n    checkDec! {p\u2713} x = toWitness (lookup p\u2713 x)\n\nmodule FromExploreInd\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (explore-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p explore)\n    where\n\n  open FromExplore explore public\n\n  module _ {\u2113 p} where\n    explore-mon-ext : ExploreMonExt {\u2113} p explore\n    explore-mon-ext m {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ _ f \u2248 s _ _ g) refl \u2219-cong f\u2248\u00b0g\n      where open Mon m\n\n    explore-mono : ExploreMono {\u2113} p explore\n    explore-mono _\u2286_ z\u2286 _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n      explore-ind (\u03bb e \u2192 e _ _ f \u2286 e _ _ g) z\u2286 _\u2219-mono_ f\u2286\u00b0g\n\n    open ExplorePlug {\u2113} {a} {A}\n\n    explore\u2218-plug : (M : Monoid \u2113 \u2113) \u2192 ExplorePlug M explore\n    explore\u2218-plug M = explore-ind $kit plugKit M\n\n    module _ (M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = explore-ind (\u03bb e \u2192 \u2200 z \u2192 e \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo e z _\u2219_ f)\n                                                (fst identity) (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _))) (\u03bb _ _ \u2192 refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n    explore\u2218-ind : \u2200 (M : Monoid \u2113 \u2113) \u2192 BigOpMonInd \u2113 M (with-endo-monoid M)\n    explore\u2218-ind M P P\u03b5 P\u2219 Pf P\u2248 =\n      snd (explore-ind (\u03bb e \u2192 ExplorePlug M e \u00d7 P (\u03bb f \u2192 e id _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n                 (const (fst identity) , P\u03b5)\n                 (\u03bb {e} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {e} {s'} (fst Ps) (fst Ps')\n                                    , P\u2248 (\u03bb f \u2192 fst Ps f _) (P\u2219 (snd Ps) (snd Ps')))\n                 (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                      , P\u2248 (\u03bb f \u2192 ! snd identity _) (Pf x)))\n      where open Mon M\n\n    explore-swap : \u2200 {b} \u2192 ExploreSwap {\u2113} p explore {b}\n    explore-swap mon {e\u1d2e} e\u1d2e-\u03b5 pf f =\n      explore-ind (\u03bb e \u2192 e _ _ (e\u1d2e \u2218 f) \u2248 e\u1d2e (e _ _ \u2218 flip f))\n                  (! e\u1d2e-\u03b5)\n                  (\u03bb p q \u2192 trans (\u2219-cong p q) (! pf _ _))\n                  (\u03bb _ \u2192 refl)\n      where open Mon mon\n\n    explore-\u03b5 : Explore\u03b5 {\u2113} p explore\n    explore-\u03b5 M = explore-ind (\u03bb e \u2192 e \u03b5 _ (const \u03b5) \u2248 \u03b5)\n                              refl\n                              (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (fst identity \u03b5))\n                              (\u03bb _ \u2192 refl)\n      where open Mon M\n\n    explore-hom : ExploreHom {\u2113} p explore\n    explore-hom cm f g = explore-ind (\u03bb e \u2192 e _ _ (f \u2219\u00b0 g) \u2248 e _ _ f \u2219 e _ _ g)\n                                     (! fst identity \u03b5)\n                                     (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _))\n                                     (\u03bb _ \u2192 refl)\n      where open CMon cm\n\n    explore-lin\u02e1 : ExploreLin\u02e1 {\u2113} p explore\n    explore-lin\u02e1 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e \u03b5 _\u2219_ f) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    explore-lin\u02b3 : ExploreLin\u02b3 {\u2113} p explore\n    explore-lin\u02b3 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e \u03b5 _\u2219_ f \u25ce k) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    module ProductMonoid\n        {M : \u2605\u2080} (\u03b5\u2098 : M) (_\u2295\u2098_ : Op\u2082 M)\n        {N : \u2605\u2080} (\u03b5\u2099 : N) (_\u2295\u2099_ : Op\u2082 N)\n        where\n        \u03b5 = (\u03b5\u2098 , \u03b5\u2099)\n        _\u2295_ : Op\u2082 (M \u00d7 N)\n        (x\u2098 , x\u2099) \u2295 (y\u2098 , y\u2099) = (x\u2098 \u2295\u2098 y\u2098 , x\u2099 \u2295\u2099 y\u2099)\n\n        explore-product-monoid :\n          \u2200 f\u2098 f\u2099 \u2192 explore \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (explore \u03b5\u2098 _\u2295\u2098_ f\u2098 , explore \u03b5\u2099 _\u2295\u2099_ f\u2099)\n        explore-product-monoid f\u2098 f\u2099 =\n          explore-ind (\u03bb e \u2192 e \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (e \u03b5\u2098 _\u2295\u2098_ f\u2098 , e \u03b5\u2099 _\u2295\u2099_ f\u2099)) \u2261.refl (\u2261.ap\u2082 _\u2295_) (\u03bb _ \u2192 \u2261.refl)\n  {-\n  empty-explore:\n    \u03b5 \u2261 (\u03b5\u2098 , \u03b5\u2099) \u2713\n  point-explore (x , y):\n    < f\u2098 , f\u2099 > (x , y) \u2261 (f\u2098 x , f\u2099 y) \u2713\n  merge-explore e\u2080 e\u2081:\n    e\u2080 \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2295 e\u2081 \u03b5 _\u2295_ < f\u2098 , f\u2099 >\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099) \u2295 (e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 \u2295 e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099 \u2295 e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n  -}\n\n  module _ {\u2113} where\n    reify : Reify {\u2113} explore\n    reify = explore-ind (\u03bb e\u1d2c \u2192 \u03a0\u1d49 e\u1d2c _) _ _,_\n\n    unfocus : Unfocus {\u2113} explore\n    unfocus = explore-ind Unfocus (\u03bb{ (lift ()) }) (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n    module _ {\u2113\u1d63 a\u1d63} {A\u1d63 : A \u2192 A \u2192 \u2605 a\u1d63}\n             (A\u1d63-refl : Reflexive A\u1d63) where\n      \u27e6explore\u27e7 : \u27e6Explore\u27e7 \u2113\u1d63 A\u1d63 (explore {\u2113}) (explore {\u2113})\n      \u27e6explore\u27e7 M\u1d63 z\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb e \u2192 M\u1d63 (e _ _ _) (e _ _ _)) z\u1d63 (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n    explore-ext : ExploreExt {\u2113} explore\n    explore-ext \u03b5 op = explore-ind (\u03bb e \u2192 e _ _ _ \u2261 e _ _ _) \u2261.refl (\u2261.ap\u2082 op)\n\n  module LiftHom\n       {m p}\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 p)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (zero : S)\n       (_+_  : Op\u2082 S)\n       (one  : T)\n       (_*_  : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom-0-1 : f zero \u2248 one)\n       (hom-+-* : \u2200 {x y} \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore zero _+_ g) \u2248 explore one _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb e \u2192 f (e zero _+_ g) \u2248 e one _*_ (f \u2218 g))\n                               hom-0-1\n                               (\u03bb p q \u2192 \u2248-trans hom-+-* (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  module _ {\u2113} {P : A \u2192 \u2605_ \u2113} where\n    open LiftHom {S = \u2605_ \u2113} {\u2605_ \u2113} (\u03bb A B \u2192 B \u2192 A) id _\u2218\u2032_\n                 (Lift \ud835\udfd8) _\u228e_ (Lift \ud835\udfd9) _\u00d7_\n                 (\u03bb f g \u2192 \u00d7-map f g) Dec P (const (no (\u03bb{ (lift ()) })))\n                 (uncurry Dec-\u228e)\n                 public renaming (lift-hom to lift-Dec)\n\n  module FromFocus {p} (focus : Focus {p} explore) where\n    Dec-\u03a3 : \u2200 {P} \u2192 \u03a0 A (Dec \u2218 P) \u2192 Dec (\u03a3 A P)\n    Dec-\u03a3 = map-Dec unfocus focus \u2218 lift-Dec \u2218 reify\n\n  lift-hom-\u2261 :\n      \u2200 {m} {S T : \u2605 m}\n        (zero : S)\n        (_+_  : Op\u2082 S)\n        (one  : T)\n        (_*_  : Op\u2082 T)\n        (f    : S \u2192 T)\n        (g    : A \u2192 S)\n        (hom-0-1 : f zero \u2261 one)\n        (hom-+-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore zero _+_ g) \u2261 explore one _*_ (f \u2218 g)\n  lift-hom-\u2261 z _+_ o _*_ = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans z _+_ o _*_ (\u2261.ap\u2082 _*_)\n\n  sum-ind : SumInd sum\n  sum-ind P P0 P+ Pf = explore-ind (\u03bb e \u2192 P (e 0 _+_)) P0 P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext 0 _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ z\u2264n _+-mono_\n\n  sum-swap' : SumSwap sum\n  sum-swap' {sum\u1d2e = s\u1d2e} s\u1d2e-0 hom f =\n    sum-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2261 s\u1d2e (s \u2218 flip f))\n            (! s\u1d2e-0)\n            (\u03bb p q \u2192 (ap\u2082 _+_ p q) \u2219 (! hom _ _)) (\u03bb _ \u2192 refl)\n    where open \u2261\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.ap\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sum-const : SumConst sum\n  sum-const x = sum-ext (\u03bb _ \u2192 ! snd \u2115\u00b0.*-identity x) \u2219 sum-lin (const 1) x \u2219 \u2115\u00b0.*-comm x Card\n    where open \u2261\n\n  exploreStableUnder\u2192sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  exploreStableUnder\u2192sumStableUnder SU-p = SU-p 0 _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sumStableUnder\u2192countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  sumStableUnder\u2192countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  diff-list = with-endo-monoid (List.monoid A) List.[_]\n\n  {-\n  list\u2261diff-list : list \u2261 diff-list\n  list\u2261diff-list = {!explore-endo-monoid-spec (List.monoid A) List.[_]!}\n  -}\n\n  lift-sum : \u2200 \u2113 \u2192 Sum A\n  lift-sum \u2113 f = lower {\u2080} {\u2113} (explore (lift 0) (lift-op\u2082 _+_) (lift \u2218 f))\n\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin : \u2200 {{_ : UA}}(f : A \u2192 \u2115) \u2192 Fin (lift-sum _ f) \u2261 \u03a3\u1d49 explore (Fin \u2218 f)\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin f = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans (lift 0) (lift-op\u2082 _+_) (Lift \ud835\udfd8) _\u228e_ \u228e= (Fin \u2218 lower) (lift \u2218 f) (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id) (! Fin-\u228e-+)\n    where open \u2261\n\nmodule FromTwoExploreInd\n    {a} {A : \u2605 a}\n    {e\u1d2c : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (e\u1d2c-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2c)\n    {b} {B : \u2605 b}\n    {e\u1d2e : \u2200 {\u2113} \u2192 Explore \u2113 B}\n    (e\u1d2e-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2e)\n    where\n\n    module A = FromExploreInd e\u1d2c-ind\n    module B = FromExploreInd e\u1d2e-ind\n\n    module _ {c \u2113}(cm : CommutativeMonoid c \u2113) where\n        open CMon cm\n\n        op\u1d2c = e\u1d2c \u03b5 _\u2219_\n        op\u1d2e = e\u1d2e \u03b5 _\u2219_\n\n        -- TODO use lift-hom\n        explore-swap' : \u2200 f \u2192 op\u1d2c (op\u1d2e \u2218 f) \u2248 op\u1d2e (op\u1d2c \u2218 flip f)\n        explore-swap' = A.explore-swap m (B.explore-\u03b5 m) (B.explore-hom cm)\n\n    sum-swap : \u2200 f \u2192 A.sum (B.sum \u2218 f) \u2261 B.sum (A.sum \u2218 flip f)\n    sum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n\nmodule FromTwoAdequate-sum\n  {{_ : UA}}{{_ : FunExt}}\n  {A}{B}\n  {sum\u1d2c : Sum A}{sum\u1d2e : Sum B}\n  (open Adequacy _\u2261_)\n  (sum\u1d2c-adq : Adequate-sum sum\u1d2c)\n  (sum\u1d2e-adq : Adequate-sum sum\u1d2e) where\n\n  open \u2261\n  sumStableUnder : (p : A \u2243 B)(f : B \u2192 \u2115)\n                 \u2192 sum\u1d2c (f \u2218 \u00b7\u2192 p) \u2261 sum\u1d2e f\n  sumStableUnder p f = Fin-injective (sum\u1d2c-adq (f \u2218 \u00b7\u2192 p)\n                                      \u2219 \u03a3-fst\u2243 p _\n                                      \u2219 ! sum\u1d2e-adq f)\n\n  sumStableUnder\u2032 : (p : A \u2243 B)(f : A \u2192 \u2115)\n                 \u2192 sum\u1d2c f \u2261 sum\u1d2e (f \u2218 <\u2013 p)\n  sumStableUnder\u2032 p f = Fin-injective (sum\u1d2c-adq f\n                                      \u2219 \u03a3-fst\u2243\u2032 p _\n                                      \u2219 ! sum\u1d2e-adq (f \u2218 <\u2013 p))\n\nmodule FromAdequate-sum\n  {A}\n  {sum : Sum A}\n  (open Adequacy _\u2261_)\n  (sum-adq : Adequate-sum sum)\n  {{_ : UA}}{{_ : FunExt}}\n  where\n\n  open FromTwoAdequate-sum sum-adq sum-adq public\n  open \u2261\n\n  sum-ext : SumExt sum\n  sum-ext = ap sum \u2218 \u03bb=\n\n  private\n    count : Count A\n    count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  private\n    module M {p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) where\n      private\n\n        P = \u03bb x \u2192 p x \u2261 1\u2082\n        Q = \u03bb x \u2192 q x \u2261 1\u2082\n        \u00acP = \u03bb x \u2192 p x \u2261 0\u2082\n        \u00acQ = \u03bb x \u2192 q x \u2261 0\u2082\n\n        \u03c0 : \u03a3 A P \u2261 \u03a3 A Q\n        \u03c0 = ! \u03a3=\u2032 _ (count-\u2261 p)\n            \u2219 ! (sum-adq (\ud835\udfda\u25b9\u2115 \u2218 p))\n            \u2219 ap Fin same-count\n            \u2219 sum-adq (\ud835\udfda\u25b9\u2115 \u2218 q)\n            \u2219 \u03a3=\u2032 _ (count-\u2261 q)\n\n        lem1 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 qx \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 (not px) * \ud835\udfda\u25b9\u2115 qx\n        lem1 1\u2082 1\u2082 = \u2261.refl\n        lem1 1\u2082 0\u2082 = \u2261.refl\n        lem1 0\u2082 1\u2082 = \u2261.refl\n        lem1 0\u2082 0\u2082 = \u2261.refl\n\n        lem2 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 px \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 px * \ud835\udfda\u25b9\u2115 (not qx)\n        lem2 1\u2082 1\u2082 = \u2261.refl\n        lem2 1\u2082 0\u2082 = \u2261.refl\n        lem2 0\u2082 1\u2082 = \u2261.refl\n        lem2 0\u2082 0\u2082 = \u2261.refl\n\n        lemma1 : \u2200 px qx \u2192 (qx \u2261 1\u2082) \u2261 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 0\u2082 \u00d7 qx \u2261 1\u2082))\n        lemma1 px qx = ! Fin-\u2261-\u22611\u2082 qx\n                     \u2219 ap Fin (lem1 px qx)\n                     \u2219 ! Fin-\u228e-+\n                     \u2219 \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22610\u2082 px) (Fin-\u2261-\u22611\u2082 qx))\n\n        lemma2 : \u2200 px qx \u2192 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 1\u2082 \u00d7 qx \u2261 0\u2082)) \u2261 (px \u2261 1\u2082)\n        lemma2 px qx = ! \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22611\u2082 px) (Fin-\u2261-\u22610\u2082 qx)) \u2219 Fin-\u228e-+ \u2219 ap Fin (! lem2 px qx) \u2219 Fin-\u2261-\u22611\u2082 px\n\n        \u03c0' : (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192  P x \u00d7 \u00acQ x))\n           \u2261 (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192 \u00acP x \u00d7  Q x))\n        \u03c0' = \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n           \u2219 ! \u03a3\u228e-split\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma2 (p x) (q x))\n           \u2219 \u03c0\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma1 (p x) (q x))\n           \u2219 \u03a3\u228e-split\n           \u2219 ! \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n\n        \u03c0'' : \u03a3 A (P \u00d7\u00b0 \u00acQ) \u2261 \u03a3 A (\u00acP \u00d7\u00b0 Q)\n        \u03c0'' = Fin\u228e-injective (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u03c0'\n\n      open EquivalentSubsets \u03c0'' public\n\n  same-count\u2192iso : \u2200{p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) \u2192 p \u2261 q \u2218 M.\u03c0 {p} {q} same-count\n  same-count\u2192iso {p} {q} sc = M.prop {p} {q} sc\n\nmodule From\u27e6Explore\u27e7\n    {-a-} {A : \u2605\u2080 {- a-}}\n    {explore   : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (\u27e6explore\u27e7 : \u2200 {\u2113\u2080 \u2113\u2081} \u2113\u1d63 \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 _\u2261_ explore explore)\n    {{_ : UA}}\n    where\n  open FromExplore explore\n\n  module AlsoInFromExploreInd\n          {\u2113}(M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = \u27e6explore\u27e7 \u2080 {C} {C \u2192 C}\n                                              (\u03bb r s \u2192 \u2200 z \u2192 r \u2219 z \u2248 s z)\n                                              (fst identity)\n                                              (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _)))\n                                              (\u03bb x\u1d63 _ \u2192 \u2219-cong (reflexive (\u2261.ap f x\u1d63)) refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n  open \u2261\n  module _ (f : A \u2192 \u2115) where\n    sum\u21d2\u03a3\u1d49 : Fin (explore 0 _+_ f) \u2261 explore (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 f)\n    sum\u21d2\u03a3\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb n X \u2192 Fin n \u2261 X)\n                (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id)\n                (\u03bb p q \u2192 ! Fin-\u228e-+ \u2219 \u228e= p q)\n                (ap (Fin \u2218 f))\n\n    product\u21d2\u03a0\u1d49 : Fin (explore 1 _*_ f) \u2261 explore (Lift \ud835\udfd9) _\u00d7_ (Fin \u2218 f)\n    product\u21d2\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                    (\u03bb n X \u2192 Fin n \u2261 X)\n                    (Fin1\u2261\ud835\udfd9 \u2219 ! Lift\u2261id)\n                    (\u03bb p q \u2192 ! Fin-\u00d7-* \u2219 \u00d7= p q)\n                    (ap (Fin \u2218 f))\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    \u2713all-\u03a0\u1d49 : \u2713 (all f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    \u2713all-\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb b X \u2192 \u2713 b \u2261 X)\n                (! Lift\u2261id)\n                (\u03bb p q \u2192 \u2713-\u2227-\u00d7 _ _ \u2219 \u00d7= p q)\n                (ap (\u2713 \u2218 f))\n\n    \u2713any\u2192\u03a3\u1d49 : \u2713 (any f) \u2192 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    \u2713any\u2192\u03a3\u1d49 p = \u27e6explore\u27e7 {\u2080} {\u209b \u2080} \u2081\n                         (\u03bb b (X : \u2605\u2080) \u2192 Lift (\u2713 b) \u2192 X)\n                         (\u03bb x \u2192 lift (lower x))\n                         (\u03bb { {0\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inr (y\u1d63 z\u1d63)\n                            ; {1\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inl (x\u1d63 _) })\n                         (\u03bb x\u1d63 x \u2192 tr (\u2713 \u2218 f) x\u1d63 (lower x)) (lift p)\n\n  module FromAdequate-\u03a3\u1d49\n           (adequate-\u03a3\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-sum : Adequate-sum _\u2261_ sum\n    adequate-sum f = sum\u21d2\u03a3\u1d49 f \u2219 adequate-\u03a3\u1d49 (Fin \u2218 f)\n\n    open FromAdequate-sum adequate-sum public\n\n    adequate-any : Adequate-any -\u2192- any\n    adequate-any f e = coe (adequate-\u03a3\u1d49 (\u2713 \u2218 f)) (\u2713any\u2192\u03a3\u1d49 f e)\n\n  module FromAdequate-\u03a0\u1d49\n           (adequate-\u03a0\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-product : Adequate-product _\u2261_ product\n    adequate-product f = product\u21d2\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 (Fin \u2218 f)\n\n    adequate-all : Adequate-all _\u2261_ all\n    adequate-all f = \u2713all-\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 _\n\n    check! : (f : A \u2192 \ud835\udfda) {pf : \u2713 (all f)} \u2192 (\u2200 x \u2192 \u2713 (f x))\n    check! f {pf} = coe (adequate-all f) pf\n\n{-\nmodule ExplorableRecord where\n    record Explorable A : \u2605\u2081 where\n      constructor mk\n      field\n        explore     : Explore\u2080 A\n        explore-ind : ExploreInd\u2080 explore\n\n      open FromExploreInd explore-ind\n      field\n        adequate-sum     : Adequate-sum sum\n    --  adequate-product : AdequateProduct product\n\n      open FromExploreInd explore-ind public\n\n    open Explorable public\n\n    ExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\n    ExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\n    record Funable A : \u2605\u2082 where\n      constructor _,_\n      field\n        explorable : Explorable A\n        negative   : ExploreForFun A\n\n    module DistFun {A} (\u03bcA : Explorable A)\n                       (\u03bcA\u2192 : ExploreForFun A)\n                       {B} (\u03bcB : Explorable B){X}\n                       (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                       (0\u2032  : X)\n                       (_+_ : X \u2192 X \u2192 X)\n                       (_*_ : X \u2192 X \u2192 X) where\n\n      \u03a3\u1d2e = explore \u03bcB 0\u2032 _+_\n      \u03a0' = explore \u03bcA 0\u2032 _*_\n      \u03a3' = explore (\u03bcA\u2192 \u03bcB) 0\u2032 _+_\n\n      DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\n    DistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\n    DistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                       \u2200 _*_ \u2192 Zero _\u2248_ \u03b5 _*_ \u2192 _DistributesOver_ _\u2248_ _*_ _\u2219_ \u2192 _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                       \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ \u03b5 _\u2219_ _*_\n\n    DistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\n    DistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03bc-iso : \u2200 {A B} \u2192 (A \u2243 B) \u2192 Explorable A \u2192 Explorable B\n        \u03bc-iso {A}{B} A\u2243B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n          where\n            open \u2261\n            A\u2192B = \u2013> A\u2243B\n            ade = \u03bb f \u2192 adequate-sum \u03bcA (f \u2218 A\u2192B) \u2219 \u03a3-fst\u2243 A\u2243B _\n\n        -- I guess this could be more general\n        \u03bc-iso-preserve : \u2200 {A B} (A\u2243B : A \u2243 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2243B \u03bcA) (f \u2218 <\u2013 A\u2243B)\n        \u03bc-iso-preserve A\u2243B f \u03bcA = sum-ext \u03bcA (\u03bb x \u2192 ap f (! (<\u2013-inv-l A\u2243B x)))\n          where open \u2261\n\n          {-\n        \u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n        \u03bcLift = \u03bc-iso {!(! Lift\u2194id)!}\n          where open \u2261\n          -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base\nopen import Data.Indexed\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin) renaming (zero to fzero)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product.NP renaming (map to \u00d7-map) hiding (first)\nopen import Data.Sum.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary.NP\nopen import Relation.Binary\nopen import Relation.Binary.Sum using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise using (_\u00d7-cong_)\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen import HoTT\nopen Equivalences\nopen \u2261 using (_\u2261_)\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule _ {m a} {A : \u2605 a} where\n  open EM {a} m\n  gfilter-explore : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore m A \u2192 Explore m B\n  gfilter-explore f e\u1d2c = e\u1d2c >>= \u03bb x \u2192 maybe (\u03bb \u03b7 \u2192 point-explore \u03b7) empty-explore (f x)\n\n  filter-explore : (A \u2192 \ud835\udfda) \u2192 Explore m A \u2192 Explore m A\n  filter-explore p = gfilter-explore \u03bb x \u2192 [0: nothing 1: just x ] (p x)\n\n  -- monoidal exploration: explore A with a monoid M\n  explore-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-monoid e\u1d2c M = e\u1d2c \u03b5 _\u00b7_ where open Mon M renaming (_\u2219_ to _\u00b7_)\n\n  explore-endo : Explore m A \u2192 Explore m A\n  explore-endo e\u1d2c \u03b5 op f = e\u1d2c id _\u2218\u2032_ (op \u2218 f) \u03b5\n\n  explore-endo-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-endo-monoid = explore-monoid \u2218 explore-endo\n\n  explore-backward : Explore m A \u2192 Explore m A\n  explore-backward e\u1d2c \u03b5 _\u2219_ f = e\u1d2c \u03b5 (flip _\u2219_) f\n\n  -- explore-backward \u2218 explore-backward = id\n  -- (m : a comm monoid) \u2192 explore-backward m = explore m\n\nprivate\n  module FindForward {a} {A : \u2605 a} (explore : Explore a A) where\n    find? : Find? A\n    find? = explore nothing (M?._\u2223_ _)\n\n    first : Maybe A\n    first = find? just\n\n    findKey : FindKey A\n    findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule ExplorePlug {\u2113 a} {A : \u2605 a} where\n  record ExploreIndKit p (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (a \u2294 \u209b \u2113 \u2294 p) where\n    constructor mk\n    field\n      P\u03b5 : P empty-explore\n      P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081)\n      Pf : \u2200 x \u2192 P (point-explore x)\n\n  _$kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p} {e : Explore \u2113 A}\n           \u2192 ExploreInd p e \u2192 ExploreIndKit p P \u2192 P e\n  _$kit_ {P = P} ind (mk P\u03b5 P\u2219 Pf) = ind P P\u03b5 P\u2219 Pf\n\n  _,-kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p}{Q : Explore \u2113 A \u2192 \u2605 p}\n            \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\n  Pk ,-kit Qk = mk (P\u03b5 Pk , P\u03b5 Qk)\n                   (\u03bb x y \u2192 P\u2219 Pk (fst x) (fst y) , P\u2219 Qk (snd x) (snd y))\n                   (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n               where open ExploreIndKit\n\n  ExploreInd-Extra : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 _\n  ExploreInd-Extra p exp =\n    \u2200 (Q     : Explore \u2113 A \u2192 \u2605 p)\n      (Q-kit : ExploreIndKit p Q)\n      (P     : Explore \u2113 A \u2192 \u2605 p)\n      (P\u03b5    : P empty-explore)\n      (P\u2219    : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 Q e\u2080 \u2192 Q e\u2081 \u2192 P e\u2080 \u2192 P e\u2081\n               \u2192 P (merge-explore e\u2080 e\u2081))\n      (Pf    : \u2200 x \u2192 P (point-explore x))\n    \u2192 P exp\n\n  to-extra : \u2200 {p} {e : Explore \u2113 A} \u2192 ExploreInd p e \u2192 ExploreInd-Extra p e\n  to-extra e-ind Q Q-kit P P\u03b5 P\u2219 Pf =\n   snd (e-ind (Q \u00d7\u00b0 P)\n           (Q\u03b5 , P\u03b5)\n           (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n           (\u03bb x \u2192 Qf x , Pf x))\n   where open ExploreIndKit Q-kit renaming (P\u03b5 to Q\u03b5; P\u2219 to Q\u2219; Pf to Qf)\n\n  ExplorePlug : \u2200 {m} (M : Monoid \u2113 m) (e : Explore _ A) \u2192 \u2605 _\n  ExplorePlug M e = \u2200 f x \u2192 e\u2218 \u03b5 _\u2219_ f \u2219 x \u2248 e\u2218 x _\u2219_ f\n     where open Mon M\n           e\u2218 = explore-endo e\n\n  plugKit : \u2200 {m} (M : Monoid \u2113 m) \u2192 ExploreIndKit _ (ExplorePlug M)\n  plugKit M = mk (\u03bb _ \u2192 fst identity)\n                 (\u03bb Ps Ps' f x \u2192\n                    trans (\u2219-cong (! Ps _ _) refl)\n                          (trans (assoc _ _ _)\n                                 (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n                 (\u03bb x f _ \u2192 \u2219-cong (snd identity (f x)) refl)\n       where open Mon M\n\nmodule FromExplore\n    {a} {A : \u2605 a}\n    (explore : \u2200 {\u2113} \u2192 Explore \u2113 A) where\n\n  module _ {\u2113} where\n    with-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-monoid = explore-monoid explore\n\n    with\u2218 : Explore \u2113 A\n    with\u2218 = explore-endo explore\n\n    with-endo-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-endo-monoid = explore-endo-monoid explore\n\n    backward : Explore \u2113 A\n    backward = explore-backward explore\n\n    gfilter : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore \u2113 B\n    gfilter f = gfilter-explore f explore\n\n    filter : (A \u2192 \ud835\udfda) \u2192 Explore \u2113 A\n    filter p = filter-explore p explore\n\n  sum : Sum A\n  sum = explore 0 _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore 1 _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore 1\u2082 _\u2227_\n  and   = big-\u2227\n  all   = big-\u2227\n\n  big-\u2228 = explore 0\u2082 _\u2228_\n  or    = big-\u2228\n  any   = big-\u2228\n\n  big-xor = explore 0\u2082 _xor_\n\n  bin-tree : BinTree A\n  bin-tree = explore empty fork leaf\n\n  list : List A\n  list = explore List.[] _++_ List.[_]\n\n  module FindBackward = FindForward backward\n\n  findLast? : Find? A\n  findLast? = FindBackward.find?\n\n  last : Maybe A\n  last = FindBackward.first\n\n  findLastKey : FindKey A\n  findLastKey = FindBackward.findKey\n\n  open FindForward explore public\n\nmodule FromLookup\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (lookup : \u2200 {\u2113} \u2192 Lookup {\u2113} explore)\n    where\n\n  module CheckDec! {\u2113}{P : A \u2192 \u2605 \u2113}(decP : \u2200 x \u2192 Dec (P x)) where\n    CheckDec! : \u2605 _\n    CheckDec! = explore (Lift \ud835\udfd9) _\u00d7_ \u03bb x \u2192 \u2713 \u230a decP x \u230b\n\n    checkDec! : {p\u2713 : CheckDec!} \u2192 \u2200 x \u2192 P x\n    checkDec! {p\u2713} x = toWitness (lookup p\u2713 x)\n\nmodule FromExploreInd\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (explore-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p explore)\n    where\n\n  open FromExplore explore public\n\n  module _ {\u2113 p} where\n    explore-mon-ext : ExploreMonExt {\u2113} p explore\n    explore-mon-ext m {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ _ f \u2248 s _ _ g) refl \u2219-cong f\u2248\u00b0g\n      where open Mon m\n\n    explore-mono : ExploreMono {\u2113} p explore\n    explore-mono _\u2286_ z\u2286 _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n      explore-ind (\u03bb e \u2192 e _ _ f \u2286 e _ _ g) z\u2286 _\u2219-mono_ f\u2286\u00b0g\n\n    open ExplorePlug {\u2113} {a} {A}\n\n    explore\u2218-plug : (M : Monoid \u2113 \u2113) \u2192 ExplorePlug M explore\n    explore\u2218-plug M = explore-ind $kit plugKit M\n\n    module _ (M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = explore-ind (\u03bb e \u2192 \u2200 z \u2192 e \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo e z _\u2219_ f)\n                                                (fst identity) (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _))) (\u03bb _ _ \u2192 refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n    explore\u2218-ind : \u2200 (M : Monoid \u2113 \u2113) \u2192 BigOpMonInd \u2113 M (with-endo-monoid M)\n    explore\u2218-ind M P P\u03b5 P\u2219 Pf P\u2248 =\n      snd (explore-ind (\u03bb e \u2192 ExplorePlug M e \u00d7 P (\u03bb f \u2192 e id _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n                 (const (fst identity) , P\u03b5)\n                 (\u03bb {e} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {e} {s'} (fst Ps) (fst Ps')\n                                    , P\u2248 (\u03bb f \u2192 fst Ps f _) (P\u2219 (snd Ps) (snd Ps')))\n                 (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                      , P\u2248 (\u03bb f \u2192 ! snd identity _) (Pf x)))\n      where open Mon M\n\n    explore-swap : \u2200 {b} \u2192 ExploreSwap {\u2113} p explore {b}\n    explore-swap mon {e\u1d2e} e\u1d2e-\u03b5 pf f =\n      explore-ind (\u03bb e \u2192 e _ _ (e\u1d2e \u2218 f) \u2248 e\u1d2e (e _ _ \u2218 flip f))\n                  (! e\u1d2e-\u03b5)\n                  (\u03bb p q \u2192 trans (\u2219-cong p q) (! pf _ _))\n                  (\u03bb _ \u2192 refl)\n      where open Mon mon\n\n    explore-\u03b5 : Explore\u03b5 {\u2113} p explore\n    explore-\u03b5 M = explore-ind (\u03bb e \u2192 e \u03b5 _ (const \u03b5) \u2248 \u03b5)\n                              refl\n                              (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (fst identity \u03b5))\n                              (\u03bb _ \u2192 refl)\n      where open Mon M\n\n    explore-hom : ExploreHom {\u2113} p explore\n    explore-hom cm f g = explore-ind (\u03bb e \u2192 e _ _ (f \u2219\u00b0 g) \u2248 e _ _ f \u2219 e _ _ g)\n                                     (! fst identity \u03b5)\n                                     (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _))\n                                     (\u03bb _ \u2192 refl)\n      where open CMon cm\n\n    explore-lin\u02e1 : ExploreLin\u02e1 {\u2113} p explore\n    explore-lin\u02e1 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e \u03b5 _\u2219_ f) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    explore-lin\u02b3 : ExploreLin\u02b3 {\u2113} p explore\n    explore-lin\u02b3 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e \u03b5 _\u2219_ f \u25ce k) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    module ProductMonoid\n        {M : \u2605\u2080} (\u03b5\u2098 : M) (_\u2295\u2098_ : Op\u2082 M)\n        {N : \u2605\u2080} (\u03b5\u2099 : N) (_\u2295\u2099_ : Op\u2082 N)\n        where\n        \u03b5 = (\u03b5\u2098 , \u03b5\u2099)\n        _\u2295_ : Op\u2082 (M \u00d7 N)\n        (x\u2098 , x\u2099) \u2295 (y\u2098 , y\u2099) = (x\u2098 \u2295\u2098 y\u2098 , x\u2099 \u2295\u2099 y\u2099)\n\n        explore-product-monoid :\n          \u2200 f\u2098 f\u2099 \u2192 explore \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (explore \u03b5\u2098 _\u2295\u2098_ f\u2098 , explore \u03b5\u2099 _\u2295\u2099_ f\u2099)\n        explore-product-monoid f\u2098 f\u2099 =\n          explore-ind (\u03bb e \u2192 e \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (e \u03b5\u2098 _\u2295\u2098_ f\u2098 , e \u03b5\u2099 _\u2295\u2099_ f\u2099)) \u2261.refl (\u2261.ap\u2082 _\u2295_) (\u03bb _ \u2192 \u2261.refl)\n  {-\n  empty-explore:\n    \u03b5 \u2261 (\u03b5\u2098 , \u03b5\u2099) \u2713\n  point-explore (x , y):\n    < f\u2098 , f\u2099 > (x , y) \u2261 (f\u2098 x , f\u2099 y) \u2713\n  merge-explore e\u2080 e\u2081:\n    e\u2080 \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2295 e\u2081 \u03b5 _\u2295_ < f\u2098 , f\u2099 >\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099) \u2295 (e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 \u2295 e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099 \u2295 e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n  -}\n\n  module _ {\u2113} where\n    reify : Reify {\u2113} explore\n    reify = explore-ind (\u03bb e\u1d2c \u2192 \u03a0\u1d49 e\u1d2c _) _ _,_\n\n    unfocus : Unfocus {\u2113} explore\n    unfocus = explore-ind Unfocus (\u03bb{ (lift ()) }) (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n    module _ {\u2113\u1d63 a\u1d63} {A\u1d63 : A \u2192 A \u2192 \u2605 a\u1d63}\n             (A\u1d63-refl : Reflexive A\u1d63) where\n      \u27e6explore\u27e7 : \u27e6Explore\u27e7 \u2113\u1d63 A\u1d63 (explore {\u2113}) (explore {\u2113})\n      \u27e6explore\u27e7 M\u1d63 z\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb e \u2192 M\u1d63 (e _ _ _) (e _ _ _)) z\u1d63 (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n    explore-ext : ExploreExt {\u2113} explore\n    explore-ext \u03b5 op = explore-ind (\u03bb e \u2192 e _ _ _ \u2261 e _ _ _) \u2261.refl (\u2261.ap\u2082 op)\n\n  module LiftHom\n       {m p}\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 p)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (zero : S)\n       (_+_  : Op\u2082 S)\n       (one  : T)\n       (_*_  : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom-0-1 : f zero \u2248 one)\n       (hom-+-* : \u2200 {x y} \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore zero _+_ g) \u2248 explore one _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb e \u2192 f (e zero _+_ g) \u2248 e one _*_ (f \u2218 g))\n                               hom-0-1\n                               (\u03bb p q \u2192 \u2248-trans hom-+-* (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  module _ {\u2113} {P : A \u2192 \u2605_ \u2113} where\n    open LiftHom {S = \u2605_ \u2113} {\u2605_ \u2113} (\u03bb A B \u2192 B \u2192 A) id _\u2218\u2032_\n                 (Lift \ud835\udfd8) _\u228e_ (Lift \ud835\udfd9) _\u00d7_\n                 (\u03bb f g \u2192 \u00d7-map f g) Dec P (const (no (\u03bb{ (lift ()) })))\n                 (uncurry Dec-\u228e)\n                 public renaming (lift-hom to lift-Dec)\n\n  module FromFocus {p} (focus : Focus {p} explore) where\n    Dec-\u03a3 : \u2200 {P} \u2192 \u03a0 A (Dec \u2218 P) \u2192 Dec (\u03a3 A P)\n    Dec-\u03a3 = map-Dec unfocus focus \u2218 lift-Dec \u2218 reify\n\n  lift-hom-\u2261 :\n      \u2200 {m} {S T : \u2605 m}\n        (zero : S)\n        (_+_  : Op\u2082 S)\n        (one  : T)\n        (_*_  : Op\u2082 T)\n        (f    : S \u2192 T)\n        (g    : A \u2192 S)\n        (hom-0-1 : f zero \u2261 one)\n        (hom-+-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore zero _+_ g) \u2261 explore one _*_ (f \u2218 g)\n  lift-hom-\u2261 z _+_ o _*_ = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans z _+_ o _*_ (\u2261.ap\u2082 _*_)\n\n  sum-ind : SumInd sum\n  sum-ind P P0 P+ Pf = explore-ind (\u03bb e \u2192 P (e 0 _+_)) P0 P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext 0 _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ z\u2264n _+-mono_\n\n  sum-swap' : SumSwap sum\n  sum-swap' {sum\u1d2e = s\u1d2e} s\u1d2e-0 hom f =\n    sum-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2261 s\u1d2e (s \u2218 flip f))\n            (! s\u1d2e-0)\n            (\u03bb p q \u2192 (ap\u2082 _+_ p q) \u2219 (! hom _ _)) (\u03bb _ \u2192 refl)\n    where open \u2261\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.ap\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sum-const : SumConst sum\n  sum-const x = sum-ext (\u03bb _ \u2192 ! snd \u2115\u00b0.*-identity x) \u2219 sum-lin (const 1) x \u2219 \u2115\u00b0.*-comm x Card\n    where open \u2261\n\n  exploreStableUnder\u2192sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  exploreStableUnder\u2192sumStableUnder SU-p = SU-p 0 _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sumStableUnder\u2192countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  sumStableUnder\u2192countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  diff-list = with-endo-monoid (List.monoid A) List.[_]\n\n  {-\n  list\u2261diff-list : list \u2261 diff-list\n  list\u2261diff-list = {!explore-endo-monoid-spec (List.monoid A) List.[_]!}\n  -}\n\n  lift-op\u2082 : \u2200 {a}{A : \u2605_ a}(op : Op\u2082 A){\u2113} \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A\n  lift-op\u2082 op (lift x) (lift y) = lift (op x y)\n\n  lift-sum : \u2200 \u2113 \u2192 Sum A\n  lift-sum \u2113 f = lower {\u2080} {\u2113} (explore (lift 0) (lift-op\u2082 _+_) (lift \u2218 f))\n\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin : \u2200 {{_ : UA}}(f : A \u2192 \u2115) \u2192 Fin (lift-sum _ f) \u2261 \u03a3\u1d49 explore (Fin \u2218 f)\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin f = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans (lift 0) (lift-op\u2082 _+_) (Lift \ud835\udfd8) _\u228e_ \u228e= (Fin \u2218 lower) (lift \u2218 f) (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id) (! Fin-\u228e-+)\n    where open \u2261\n\nmodule FromTwoExploreInd\n    {a} {A : \u2605 a}\n    {e\u1d2c : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (e\u1d2c-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2c)\n    {b} {B : \u2605 b}\n    {e\u1d2e : \u2200 {\u2113} \u2192 Explore \u2113 B}\n    (e\u1d2e-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2e)\n    where\n\n    module A = FromExploreInd e\u1d2c-ind\n    module B = FromExploreInd e\u1d2e-ind\n\n    module _ {c \u2113}(cm : CommutativeMonoid c \u2113) where\n        open CMon cm\n\n        op\u1d2c = e\u1d2c \u03b5 _\u2219_\n        op\u1d2e = e\u1d2e \u03b5 _\u2219_\n\n        -- TODO use lift-hom\n        explore-swap' : \u2200 f \u2192 op\u1d2c (op\u1d2e \u2218 f) \u2248 op\u1d2e (op\u1d2c \u2218 flip f)\n        explore-swap' = A.explore-swap m (B.explore-\u03b5 m) (B.explore-hom cm)\n\n    sum-swap : \u2200 f \u2192 A.sum (B.sum \u2218 f) \u2261 B.sum (A.sum \u2218 flip f)\n    sum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n\nmodule FromTwoAdequate-sum\n  {{_ : UA}}{{_ : FunExt}}\n  {A}{B}\n  {sum\u1d2c : Sum A}{sum\u1d2e : Sum B}\n  (open Adequacy _\u2261_)\n  (sum\u1d2c-adq : Adequate-sum sum\u1d2c)\n  (sum\u1d2e-adq : Adequate-sum sum\u1d2e) where\n\n  open \u2261\n  sumStableUnder : (p : A \u2243 B)(f : B \u2192 \u2115)\n                 \u2192 sum\u1d2c (f \u2218 \u00b7\u2192 p) \u2261 sum\u1d2e f\n  sumStableUnder p f = Fin-injective (sum\u1d2c-adq (f \u2218 \u00b7\u2192 p)\n                                      \u2219 \u03a3-fst\u2243 p _\n                                      \u2219 ! sum\u1d2e-adq f)\n\n  sumStableUnder\u2032 : (p : A \u2243 B)(f : A \u2192 \u2115)\n                 \u2192 sum\u1d2c f \u2261 sum\u1d2e (f \u2218 <\u2013 p)\n  sumStableUnder\u2032 p f = Fin-injective (sum\u1d2c-adq f\n                                      \u2219 \u03a3-fst\u2243\u2032 p _\n                                      \u2219 ! sum\u1d2e-adq (f \u2218 <\u2013 p))\n\nmodule FromAdequate-sum\n  {A}\n  {sum : Sum A}\n  (open Adequacy _\u2261_)\n  (sum-adq : Adequate-sum sum)\n  {{_ : UA}}{{_ : FunExt}}\n  where\n\n  open FromTwoAdequate-sum sum-adq sum-adq public\n  open \u2261\n\n  sum-ext : SumExt sum\n  sum-ext = ap sum \u2218 \u03bb=\n\n  private\n    count : Count A\n    count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  private\n    module M {p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) where\n      private\n\n        P = \u03bb x \u2192 p x \u2261 1\u2082\n        Q = \u03bb x \u2192 q x \u2261 1\u2082\n        \u00acP = \u03bb x \u2192 p x \u2261 0\u2082\n        \u00acQ = \u03bb x \u2192 q x \u2261 0\u2082\n\n        \u03c0 : \u03a3 A P \u2261 \u03a3 A Q\n        \u03c0 = ! \u03a3=\u2032 _ (count-\u2261 p)\n            \u2219 ! (sum-adq (\ud835\udfda\u25b9\u2115 \u2218 p))\n            \u2219 ap Fin same-count\n            \u2219 sum-adq (\ud835\udfda\u25b9\u2115 \u2218 q)\n            \u2219 \u03a3=\u2032 _ (count-\u2261 q)\n\n        lem1 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 qx \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 (not px) * \ud835\udfda\u25b9\u2115 qx\n        lem1 1\u2082 1\u2082 = \u2261.refl\n        lem1 1\u2082 0\u2082 = \u2261.refl\n        lem1 0\u2082 1\u2082 = \u2261.refl\n        lem1 0\u2082 0\u2082 = \u2261.refl\n\n        lem2 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 px \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 px * \ud835\udfda\u25b9\u2115 (not qx)\n        lem2 1\u2082 1\u2082 = \u2261.refl\n        lem2 1\u2082 0\u2082 = \u2261.refl\n        lem2 0\u2082 1\u2082 = \u2261.refl\n        lem2 0\u2082 0\u2082 = \u2261.refl\n\n        lemma1 : \u2200 px qx \u2192 (qx \u2261 1\u2082) \u2261 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 0\u2082 \u00d7 qx \u2261 1\u2082))\n        lemma1 px qx = ! Fin-\u2261-\u22611\u2082 qx\n                     \u2219 ap Fin (lem1 px qx)\n                     \u2219 ! Fin-\u228e-+\n                     \u2219 \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22610\u2082 px) (Fin-\u2261-\u22611\u2082 qx))\n\n        lemma2 : \u2200 px qx \u2192 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 1\u2082 \u00d7 qx \u2261 0\u2082)) \u2261 (px \u2261 1\u2082)\n        lemma2 px qx = ! \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22611\u2082 px) (Fin-\u2261-\u22610\u2082 qx)) \u2219 Fin-\u228e-+ \u2219 ap Fin (! lem2 px qx) \u2219 Fin-\u2261-\u22611\u2082 px\n\n        \u03c0' : (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192  P x \u00d7 \u00acQ x))\n           \u2261 (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192 \u00acP x \u00d7  Q x))\n        \u03c0' = \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n           \u2219 ! \u03a3\u228e-split\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma2 (p x) (q x))\n           \u2219 \u03c0\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma1 (p x) (q x))\n           \u2219 \u03a3\u228e-split\n           \u2219 ! \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n\n        \u03c0'' : \u03a3 A (P \u00d7\u00b0 \u00acQ) \u2261 \u03a3 A (\u00acP \u00d7\u00b0 Q)\n        \u03c0'' = Fin\u228e-injective (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u03c0'\n\n      open EquivalentSubsets \u03c0'' public\n\n  same-count\u2192iso : \u2200{p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) \u2192 p \u2261 q \u2218 M.\u03c0 {p} {q} same-count\n  same-count\u2192iso {p} {q} sc = M.prop {p} {q} sc\n\nmodule From\u27e6Explore\u27e7\n    {-a-} {A : \u2605\u2080 {- a-}}\n    {explore   : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (\u27e6explore\u27e7 : \u2200 {\u2113\u2080 \u2113\u2081} \u2113\u1d63 \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 _\u2261_ explore explore)\n    {{_ : UA}}\n    where\n  open FromExplore explore\n\n  module AlsoInFromExploreInd\n          {\u2113}(M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = \u27e6explore\u27e7 \u2080 {C} {C \u2192 C}\n                                              (\u03bb r s \u2192 \u2200 z \u2192 r \u2219 z \u2248 s z)\n                                              (fst identity)\n                                              (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _)))\n                                              (\u03bb x\u1d63 _ \u2192 \u2219-cong (reflexive (\u2261.ap f x\u1d63)) refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n  open \u2261\n  module _ (f : A \u2192 \u2115) where\n    sum\u21d2\u03a3\u1d49 : Fin (explore 0 _+_ f) \u2261 explore (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 f)\n    sum\u21d2\u03a3\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb n X \u2192 Fin n \u2261 X)\n                (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id)\n                (\u03bb p q \u2192 ! Fin-\u228e-+ \u2219 \u228e= p q)\n                (ap (Fin \u2218 f))\n\n    product\u21d2\u03a0\u1d49 : Fin (explore 1 _*_ f) \u2261 explore (Lift \ud835\udfd9) _\u00d7_ (Fin \u2218 f)\n    product\u21d2\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                    (\u03bb n X \u2192 Fin n \u2261 X)\n                    (Fin1\u2261\ud835\udfd9 \u2219 ! Lift\u2261id)\n                    (\u03bb p q \u2192 ! Fin-\u00d7-* \u2219 \u00d7= p q)\n                    (ap (Fin \u2218 f))\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    \u2713all-\u03a0\u1d49 : \u2713 (all f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    \u2713all-\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb b X \u2192 \u2713 b \u2261 X)\n                (! Lift\u2261id)\n                (\u03bb p q \u2192 \u2713-\u2227-\u00d7 _ _ \u2219 \u00d7= p q)\n                (ap (\u2713 \u2218 f))\n\n    \u2713any\u2192\u03a3\u1d49 : \u2713 (any f) \u2192 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    \u2713any\u2192\u03a3\u1d49 p = \u27e6explore\u27e7 {\u2080} {\u209b \u2080} \u2081\n                         (\u03bb b (X : \u2605\u2080) \u2192 Lift (\u2713 b) \u2192 X)\n                         (\u03bb x \u2192 lift (lower x))\n                         (\u03bb { {0\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inr (y\u1d63 z\u1d63)\n                            ; {1\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inl (x\u1d63 _) })\n                         (\u03bb x\u1d63 x \u2192 tr (\u2713 \u2218 f) x\u1d63 (lower x)) (lift p)\n\n  module FromAdequate-\u03a3\u1d49\n           (adequate-\u03a3\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-sum : Adequate-sum _\u2261_ sum\n    adequate-sum f = sum\u21d2\u03a3\u1d49 f \u2219 adequate-\u03a3\u1d49 (Fin \u2218 f)\n\n    open FromAdequate-sum adequate-sum public\n\n    adequate-any : Adequate-any -\u2192- any\n    adequate-any f e = coe (adequate-\u03a3\u1d49 (\u2713 \u2218 f)) (\u2713any\u2192\u03a3\u1d49 f e)\n\n  module FromAdequate-\u03a0\u1d49\n           (adequate-\u03a0\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-product : Adequate-product _\u2261_ product\n    adequate-product f = product\u21d2\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 (Fin \u2218 f)\n\n    adequate-all : Adequate-all _\u2261_ all\n    adequate-all f = \u2713all-\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 _\n\n    check! : (f : A \u2192 \ud835\udfda) {pf : \u2713 (all f)} \u2192 (\u2200 x \u2192 \u2713 (f x))\n    check! f {pf} = coe (adequate-all f) pf\n\n{-\nmodule ExplorableRecord where\n    record Explorable A : \u2605\u2081 where\n      constructor mk\n      field\n        explore     : Explore\u2080 A\n        explore-ind : ExploreInd\u2080 explore\n\n      open FromExploreInd explore-ind\n      field\n        adequate-sum     : Adequate-sum sum\n    --  adequate-product : AdequateProduct product\n\n      open FromExploreInd explore-ind public\n\n    open Explorable public\n\n    ExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\n    ExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\n    record Funable A : \u2605\u2082 where\n      constructor _,_\n      field\n        explorable : Explorable A\n        negative   : ExploreForFun A\n\n    module DistFun {A} (\u03bcA : Explorable A)\n                       (\u03bcA\u2192 : ExploreForFun A)\n                       {B} (\u03bcB : Explorable B){X}\n                       (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                       (0\u2032  : X)\n                       (_+_ : X \u2192 X \u2192 X)\n                       (_*_ : X \u2192 X \u2192 X) where\n\n      \u03a3\u1d2e = explore \u03bcB 0\u2032 _+_\n      \u03a0' = explore \u03bcA 0\u2032 _*_\n      \u03a3' = explore (\u03bcA\u2192 \u03bcB) 0\u2032 _+_\n\n      DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\n    DistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\n    DistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                       \u2200 _*_ \u2192 Zero _\u2248_ \u03b5 _*_ \u2192 _DistributesOver_ _\u2248_ _*_ _\u2219_ \u2192 _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                       \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ \u03b5 _\u2219_ _*_\n\n    DistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\n    DistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03bc-iso : \u2200 {A B} \u2192 (A \u2243 B) \u2192 Explorable A \u2192 Explorable B\n        \u03bc-iso {A}{B} A\u2243B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n          where\n            open \u2261\n            A\u2192B = \u2013> A\u2243B\n            ade = \u03bb f \u2192 adequate-sum \u03bcA (f \u2218 A\u2192B) \u2219 \u03a3-fst\u2243 A\u2243B _\n\n        -- I guess this could be more general\n        \u03bc-iso-preserve : \u2200 {A B} (A\u2243B : A \u2243 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2243B \u03bcA) (f \u2218 <\u2013 A\u2243B)\n        \u03bc-iso-preserve A\u2243B f \u03bcA = sum-ext \u03bcA (\u03bb x \u2192 ap f (! (<\u2013-inv-l A\u2243B x)))\n          where open \u2261\n\n          {-\n        \u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n        \u03bcLift = \u03bc-iso {!(! Lift\u2194id)!}\n          where open \u2261\n          -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c02ff8189b7a1b202c050483ac966ea765126ec6","subject":"Fixed UpTran","message":"Fixed UpTran\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/IndexLLProp.agda","new_file":"agda\/IndexLLProp.agda","new_contents":"{-# OPTIONS --exact-split #-}\n-- {-# OPTIONS --show-implicit #-}\n\nmodule IndexLLProp where\n\nopen import Common\nopen import LinLogic\nopen import Data.Maybe\nopen import Data.Product\nimport Relation.Binary.HeterogeneousEquality\nopen import Relation.Binary.PropositionalEquality using (subst ; cong ; subst\u2082)\n\ndata _\u2245\u1d62_ {i u gll} : \u2200{fll ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set where\n  \u2245\u1d62\u2193 :  \u2193 \u2245\u1d62 \u2193\n  \u2245\u1d62\u2190\u2227 : \u2200{fll li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2227 ri} (sind \u2190\u2227) (bind \u2190\u2227)\n  \u2245\u1d62\u2227\u2192 : \u2200{fll li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2227 ri} (\u2227\u2192 sind) (\u2227\u2192 bind)\n  \u2245\u1d62\u2190\u2228 : \u2200{fll li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2228 ri} (sind \u2190\u2228) (bind \u2190\u2228)\n  \u2245\u1d62\u2228\u2192 : \u2200{fll li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2228 ri} (\u2228\u2192 sind) (\u2228\u2192 bind)\n  \u2245\u1d62\u2190\u2202 : \u2200{fll li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2202 ri} (sind \u2190\u2202) (bind \u2190\u2202)\n  \u2245\u1d62\u2202\u2192 : \u2200{fll li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2202 ri} (\u2202\u2192 sind) (\u2202\u2192 bind)\n\n\ndata _\u2264\u1d62_ {i u gll fll} : \u2200{ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set where\n  \u2264\u1d62\u2193 : {ind : IndexLL fll gll} \u2192 \u2193 \u2264\u1d62 ind\n  \u2264\u1d62\u2190\u2227 : \u2200{li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2227 ri} (sind \u2190\u2227) (bind \u2190\u2227)\n  \u2264\u1d62\u2227\u2192 : \u2200{li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2227 ri} (\u2227\u2192 sind) (\u2227\u2192 bind)\n  \u2264\u1d62\u2190\u2228 : \u2200{li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2228 ri} (sind \u2190\u2228) (bind \u2190\u2228)\n  \u2264\u1d62\u2228\u2192 : \u2200{li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2228 ri} (\u2228\u2192 sind) (\u2228\u2192 bind)\n  \u2264\u1d62\u2190\u2202 : \u2200{li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2202 ri} (sind \u2190\u2202) (bind \u2190\u2202)\n  \u2264\u1d62\u2202\u2192 : \u2200{li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2202 ri} (\u2202\u2192 sind) (\u2202\u2192 bind)\n\n\u2264\u1d62-reflexive : \u2200{i u gll ll} \u2192 (ind : IndexLL {i} {u} gll ll) \u2192 ind \u2264\u1d62 ind\n\u2264\u1d62-reflexive \u2193 = \u2264\u1d62\u2193\n\u2264\u1d62-reflexive (ind \u2190\u2227) = \u2264\u1d62\u2190\u2227 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (\u2227\u2192 ind) = \u2264\u1d62\u2227\u2192 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (ind \u2190\u2228) = \u2264\u1d62\u2190\u2228 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (\u2228\u2192 ind) = \u2264\u1d62\u2228\u2192 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (ind \u2190\u2202) = \u2264\u1d62\u2190\u2202 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (\u2202\u2192 ind) = \u2264\u1d62\u2202\u2192 (\u2264\u1d62-reflexive ind)\n\n\u2264\u1d62-transitive : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (b \u2264\u1d62 c) \u2192 (a \u2264\u1d62 c)\n\u2264\u1d62-transitive \u2264\u1d62\u2193 ltbc = \u2264\u1d62\u2193\n\u2264\u1d62-transitive (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltbc) = \u2264\u1d62\u2190\u2227 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltbc) = \u2264\u1d62\u2227\u2192 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltbc) = \u2264\u1d62\u2190\u2228 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltbc) = \u2264\u1d62\u2228\u2192 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltbc) = \u2264\u1d62\u2190\u2202 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltbc) = \u2264\u1d62\u2202\u2192 (\u2264\u1d62-transitive ltab ltbc)\n\n\nisLTi : \u2200{i u gll ll fll} \u2192 (s : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL fll ll) \u2192 Dec (s \u2264\u1d62 b)\nisLTi \u2193 b = yes \u2264\u1d62\u2193\nisLTi (s \u2190\u2227) \u2193 = no (\u03bb ())\nisLTi (s \u2190\u2227) (b \u2190\u2227) with (isLTi s b)\nisLTi (s \u2190\u2227) (b \u2190\u2227) | yes p = yes $ \u2264\u1d62\u2190\u2227 p\nisLTi (s \u2190\u2227) (b \u2190\u2227) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2190\u2227 p) \u2192 \u00acp p }) \nisLTi (s \u2190\u2227) (\u2227\u2192 b) = no (\u03bb ())\nisLTi (\u2227\u2192 s) \u2193 = no (\u03bb ())\nisLTi (\u2227\u2192 s) (b \u2190\u2227) = no (\u03bb ())\nisLTi (\u2227\u2192 s) (\u2227\u2192 b) with (isLTi s b)\nisLTi (\u2227\u2192 s) (\u2227\u2192 b) | yes p = yes (\u2264\u1d62\u2227\u2192 p)\nisLTi (\u2227\u2192 s) (\u2227\u2192 b) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2227\u2192 p) \u2192 \u00acp p })\nisLTi (s \u2190\u2228) \u2193 = no (\u03bb ())\nisLTi (s \u2190\u2228) (b \u2190\u2228) with (isLTi s b)\nisLTi (s \u2190\u2228) (b \u2190\u2228) | yes p = yes $ \u2264\u1d62\u2190\u2228 p\nisLTi (s \u2190\u2228) (b \u2190\u2228) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2190\u2228 p) \u2192 \u00acp p }) \nisLTi (s \u2190\u2228) (\u2228\u2192 b) = no (\u03bb ())\nisLTi (\u2228\u2192 s) \u2193 = no (\u03bb ())\nisLTi (\u2228\u2192 s) (b \u2190\u2228) = no (\u03bb ())\nisLTi (\u2228\u2192 s) (\u2228\u2192 b) with (isLTi s b)\nisLTi (\u2228\u2192 s) (\u2228\u2192 b) | yes p = yes (\u2264\u1d62\u2228\u2192 p)\nisLTi (\u2228\u2192 s) (\u2228\u2192 b) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2228\u2192 p) \u2192 \u00acp p })\nisLTi (s \u2190\u2202) \u2193 = no (\u03bb ())\nisLTi (s \u2190\u2202) (b \u2190\u2202) with (isLTi s b)\nisLTi (s \u2190\u2202) (b \u2190\u2202) | yes p = yes $ \u2264\u1d62\u2190\u2202 p\nisLTi (s \u2190\u2202) (b \u2190\u2202) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2190\u2202 p) \u2192 \u00acp p }) \nisLTi (s \u2190\u2202) (\u2202\u2192 b) = no (\u03bb ())\nisLTi (\u2202\u2192 s) \u2193 = no (\u03bb ())\nisLTi (\u2202\u2192 s) (b \u2190\u2202) = no (\u03bb ())\nisLTi (\u2202\u2192 s) (\u2202\u2192 b) with (isLTi s b)\nisLTi (\u2202\u2192 s) (\u2202\u2192 b) | yes p = yes (\u2264\u1d62\u2202\u2192 p)\nisLTi (\u2202\u2192 s) (\u2202\u2192 b) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2202\u2192 p) \u2192 \u00acp p })\n\nisEq\u1d62 : \u2200{u i ll rll} \u2192 (a : IndexLL {i} {u} rll ll) \u2192 (b : IndexLL rll ll) \u2192 Dec (a \u2261 b)\nisEq\u1d62 \u2193 \u2193 = yes refl\nisEq\u1d62 \u2193 (b \u2190\u2227) = no (\u03bb ())\nisEq\u1d62 \u2193 (\u2227\u2192 b) = no (\u03bb ())\nisEq\u1d62 \u2193 (b \u2190\u2228) = no (\u03bb ())\nisEq\u1d62 \u2193 (\u2228\u2192 b) = no (\u03bb ())\nisEq\u1d62 \u2193 (b \u2190\u2202) = no (\u03bb ())\nisEq\u1d62 \u2193 (\u2202\u2192 b) = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2227) \u2193 = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2227) (b \u2190\u2227) with isEq\u1d62 a b\nisEq\u1d62 (a \u2190\u2227) (.a \u2190\u2227) | yes refl = yes refl\nisEq\u1d62 (a \u2190\u2227) (b \u2190\u2227) | no \u00acp = no hf where\n  hf : \u00ac ((a \u2190\u2227) \u2261 (b \u2190\u2227))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2227) (\u2227\u2192 b) = no (\u03bb ())\nisEq\u1d62 (\u2227\u2192 a) \u2193 = no (\u03bb ())\nisEq\u1d62 (\u2227\u2192 a) (b \u2190\u2227) = no (\u03bb ())\nisEq\u1d62 (\u2227\u2192 a) (\u2227\u2192 b)  with isEq\u1d62 a b\nisEq\u1d62 (\u2227\u2192 a) (\u2227\u2192 .a) | yes refl = yes refl\nisEq\u1d62 (\u2227\u2192 a) (\u2227\u2192 b) | no \u00acp = no hf where\n  hf : \u00ac ((\u2227\u2192 a) \u2261 (\u2227\u2192 b))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2228) \u2193 = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2228) (b \u2190\u2228) with isEq\u1d62 a b\nisEq\u1d62 (a \u2190\u2228) (.a \u2190\u2228) | yes refl = yes refl\nisEq\u1d62 (a \u2190\u2228) (b \u2190\u2228) | no \u00acp = no hf where\n  hf : \u00ac ((a \u2190\u2228) \u2261 (b \u2190\u2228))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2228) (\u2228\u2192 b) = no (\u03bb ())\nisEq\u1d62 (\u2228\u2192 a) \u2193 = no (\u03bb ())\nisEq\u1d62 (\u2228\u2192 a) (b \u2190\u2228) = no (\u03bb ())\nisEq\u1d62 (\u2228\u2192 a) (\u2228\u2192 b)  with isEq\u1d62 a b\nisEq\u1d62 (\u2228\u2192 a) (\u2228\u2192 .a) | yes refl = yes refl\nisEq\u1d62 (\u2228\u2192 a) (\u2228\u2192 b) | no \u00acp = no hf where\n  hf : \u00ac ((\u2228\u2192 a) \u2261 (\u2228\u2192 b))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2202) \u2193 = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2202) (b \u2190\u2202) with isEq\u1d62 a b\nisEq\u1d62 (a \u2190\u2202) (.a \u2190\u2202) | yes refl = yes refl\nisEq\u1d62 (a \u2190\u2202) (b \u2190\u2202) | no \u00acp = no hf where\n  hf : \u00ac ((a \u2190\u2202) \u2261 (b \u2190\u2202))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2202) (\u2202\u2192 b) = no (\u03bb ())\nisEq\u1d62 (\u2202\u2192 a) \u2193 = no (\u03bb ())\nisEq\u1d62 (\u2202\u2192 a) (b \u2190\u2202) = no (\u03bb ())\nisEq\u1d62 (\u2202\u2192 a) (\u2202\u2192 b)  with isEq\u1d62 a b\nisEq\u1d62 (\u2202\u2192 a) (\u2202\u2192 .a) | yes refl = yes refl\nisEq\u1d62 (\u2202\u2192 a) (\u2202\u2192 b) | no \u00acp = no hf where\n  hf : \u00ac ((\u2202\u2192 a) \u2261 (\u2202\u2192 b))\n  hf refl = \u00acp refl\n\n\n\n\n\nmodule _ where\n\n  open import Data.Vec\n\n\n  ind\u03c4\u21d2\u00acle : \u2200{i u rll ll n dt df} \u2192 (ind : IndexLL {i} {u} (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (ind2 : IndexLL rll ll) \u2192 \u00ac (ind2 \u2245\u1d62 ind) \u2192 \u00ac (ind \u2264\u1d62 ind2)\n  ind\u03c4\u21d2\u00acle \u2193 \u2193 neq = \u03bb _ \u2192 neq \u2245\u1d62\u2193\n  ind\u03c4\u21d2\u00acle (x \u2190\u2227) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (x \u2190\u2227) (y \u2190\u2227) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2190\u2227 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2190\u2227 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2227) (\u2227\u2192 y) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2227\u2192 x) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2227\u2192 x) (y \u2190\u2227) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2227\u2192 x) (\u2227\u2192 y) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2227\u2192 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2227\u2192 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2228) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (x \u2190\u2228) (y \u2190\u2228) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2190\u2228 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2190\u2228 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2228) (\u2228\u2192 y) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2228\u2192 x) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2228\u2192 x) (y \u2190\u2228) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2228\u2192 x) (\u2228\u2192 y) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2228\u2192 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2228\u2192 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2202) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (x \u2190\u2202) (y \u2190\u2202) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2190\u2202 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2190\u2202 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2202) (\u2202\u2192 y) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2202\u2192 x) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2202\u2192 x) (y \u2190\u2202) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2202\u2192 x) (\u2202\u2192 y) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2202\u2192 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2202\u2192 x) \u2192 r x}\n  \n\n\n \n\ndata Ordered\u1d62 {i u gll fll ll} (a : IndexLL {i} {u} gll ll) (b : IndexLL {i} {u} fll ll) : Set where\n  a\u2264\u1d62b : a \u2264\u1d62 b \u2192 Ordered\u1d62 a b\n  b\u2264\u1d62a : b \u2264\u1d62 a \u2192 Ordered\u1d62 a b\n\nisOrd\u1d62 : \u2200{i u gll fll ll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL {i} {u} fll ll)\n         \u2192 Dec (Ordered\u1d62 a b)\nisOrd\u1d62 a b with (isLTi a b) | (isLTi b a)\nisOrd\u1d62 a b | yes p | r = yes (a\u2264\u1d62b p)\nisOrd\u1d62 a b | no \u00acp | (yes p) = yes (b\u2264\u1d62a p)\nisOrd\u1d62 a b | no \u00acp | (no \u00acp\u2081) = no ((\u03bb { (a\u2264\u1d62b p) \u2192 \u00acp p\n                                       ; (b\u2264\u1d62a p) \u2192 \u00acp\u2081 p }))\n\n\n\nflipOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n           \u2192 Ordered\u1d62 a b \u2192 Ordered\u1d62 b a\nflipOrd\u1d62 (a\u2264\u1d62b x) = b\u2264\u1d62a x\nflipOrd\u1d62 (b\u2264\u1d62a x) = a\u2264\u1d62b x\n\nflipNotOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac Ordered\u1d62 a b \u2192 \u00ac Ordered\u1d62 b a\nflipNotOrd\u1d62 nord = \u03bb x \u2192 nord (flipOrd\u1d62 x) \n\n\n\u00aclt\u00acgt\u21d2\u00acOrd : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac (a \u2264\u1d62 b) \u2192 \u00ac (b \u2264\u1d62 a) \u2192 \u00ac(Ordered\u1d62 a b)\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (a\u2264\u1d62b x) = nlt x\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (b\u2264\u1d62a x) = ngt x\n\n\na,c\u2264\u1d62b\u21d2ordac : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (c \u2264\u1d62 b) \u2192 Ordered\u1d62 a c\na,c\u2264\u1d62b\u21d2ordac ltab \u2264\u1d62\u2193 = b\u2264\u1d62a \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2190\u2227 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2227\u2192 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2190\u2228 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2228\u2192 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2190\u2202 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2202\u2192 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)\n\n\n\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 \u00ac Ordered\u1d62 a c \u2192 \u00ac Ordered\u1d62 b c\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (a\u2264\u1d62b x) = \u22a5-elim (nord (a\u2264\u1d62b (\u2264\u1d62-transitive lt x)))\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (b\u2264\u1d62a x) = \u22a5-elim (nord (a,c\u2264\u1d62b\u21d2ordac lt x))\n\n\n_-\u1d62_ : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll) \u2192 (sind \u2264\u1d62 bind)\n       \u2192 IndexLL cll pll\n(bind -\u1d62 .\u2193) \u2264\u1d62\u2193 = bind\n((bind \u2190\u2227) -\u1d62 (sind \u2190\u2227)) (\u2264\u1d62\u2190\u2227 eq) = (bind -\u1d62 sind) eq\n((\u2227\u2192 bind) -\u1d62 (\u2227\u2192 sind)) (\u2264\u1d62\u2227\u2192 eq) = (bind -\u1d62 sind) eq\n((bind \u2190\u2228) -\u1d62 (sind \u2190\u2228)) (\u2264\u1d62\u2190\u2228 eq) = (bind -\u1d62 sind) eq\n((\u2228\u2192 bind) -\u1d62 (\u2228\u2192 sind)) (\u2264\u1d62\u2228\u2192 eq) = (bind -\u1d62 sind) eq\n((bind \u2190\u2202) -\u1d62 (sind \u2190\u2202)) (\u2264\u1d62\u2190\u2202 eq) = (bind -\u1d62 sind) eq\n((\u2202\u2192 bind) -\u1d62 (\u2202\u2192 sind)) (\u2264\u1d62\u2202\u2192 eq) = (bind -\u1d62 sind) eq\n\n\nreplLL-\u2193 : \u2200{i u pll ell ll} \u2192 \u2200(ind : IndexLL {i} {u} pll ll)\n           \u2192 replLL pll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell \u2261 ell\nreplLL-\u2193 \u2193 = refl\nreplLL-\u2193 (ind \u2190\u2227) = replLL-\u2193 ind\nreplLL-\u2193 (\u2227\u2192 ind) = replLL-\u2193 ind\nreplLL-\u2193 (ind \u2190\u2228) = replLL-\u2193 ind\nreplLL-\u2193 (\u2228\u2192 ind) = replLL-\u2193 ind\nreplLL-\u2193 (ind \u2190\u2202) = replLL-\u2193 ind\nreplLL-\u2193 (\u2202\u2192 ind) = replLL-\u2193 ind\n\n\nind-rpl\u2193 : \u2200{i u ll pll cll ell} \u2192 (ind : IndexLL {i} {u} pll ll)\n        \u2192 IndexLL cll (replLL pll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell) \u2192 IndexLL cll ell\nind-rpl\u2193 {_} {_} {_} {pll} {cll} {ell} ind x\n  =  subst (\u03bb x \u2192 x) (cong (\u03bb x \u2192 IndexLL cll x) (replLL-\u2193 {ell = ell} ind)) x \n\n\n\na\u2264\u1d62b-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n             \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll \u2192 (lt : emi \u2264\u1d62 ind)\n             \u2192 IndexLL (replLL fll ((ind -\u1d62 emi) lt) frll) (replLL ll ind frll) \na\u2264\u1d62b-morph \u2193 ind frll \u2264\u1d62\u2193 = \u2193\na\u2264\u1d62b-morph (emi \u2190\u2227) (ind \u2190\u2227) frll (\u2264\u1d62\u2190\u2227 lt) = a\u2264\u1d62b-morph emi ind frll lt \u2190\u2227\na\u2264\u1d62b-morph (\u2227\u2192 emi) (\u2227\u2192 ind) frll (\u2264\u1d62\u2227\u2192 lt) = \u2227\u2192 a\u2264\u1d62b-morph emi ind frll lt\na\u2264\u1d62b-morph (emi \u2190\u2228) (ind \u2190\u2228) frll (\u2264\u1d62\u2190\u2228 lt) = a\u2264\u1d62b-morph emi ind frll lt \u2190\u2228\na\u2264\u1d62b-morph (\u2228\u2192 emi) (\u2228\u2192 ind) frll (\u2264\u1d62\u2228\u2192 lt) = \u2228\u2192 a\u2264\u1d62b-morph emi ind frll lt\na\u2264\u1d62b-morph (emi \u2190\u2202) (ind \u2190\u2202) frll (\u2264\u1d62\u2190\u2202 lt) = a\u2264\u1d62b-morph emi ind frll lt \u2190\u2202\na\u2264\u1d62b-morph (\u2202\u2192 emi) (\u2202\u2192 ind) frll (\u2264\u1d62\u2202\u2192 lt) = \u2202\u2192 a\u2264\u1d62b-morph emi ind frll lt\n\n\n\na\u2264\u1d62b-morph-id : \u2200{i u ll rll}\n             \u2192 (ind : IndexLL {i} {u} rll ll)\n             \u2192 subst\u2082 (\u03bb x y \u2192 IndexLL x y) (replLL-\u2193 {ell = rll} ind) (replLL-id ll ind rll refl) (a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind)) \u2261 ind\na\u2264\u1d62b-morph-id {i} {u} {ll} {.ll} \u2193 = refl\na\u2264\u1d62b-morph-id {i} {u} {(li \u2227 _)} {rll} (ind \u2190\u2227) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL li ind rll | replLL-id li ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {li \u2227 _} {.h} (ind \u2190\u2227) | h | refl | .li | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(_ \u2227 ri)} {rll} (\u2227\u2192 ind) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ri ind rll | replLL-id ri ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {_ \u2227 ri} {.h} (\u2227\u2192 ind) | h | refl | .ri | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(li \u2228 _)} {rll} (ind \u2190\u2228) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL li ind rll | replLL-id li ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {li \u2228 _} {.h} (ind \u2190\u2228) | h | refl | .li | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(_ \u2228 ri)} {rll} (\u2228\u2192 ind) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ri ind rll | replLL-id ri ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {_ \u2228 ri} {.h} (\u2228\u2192 ind) | h | refl | .ri | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(li \u2202 _)} {rll} (ind \u2190\u2202) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL li ind rll | replLL-id li ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {li \u2202 _} {.h} (ind \u2190\u2202) | h | refl | .li | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(_ \u2202 ri)} {rll} (\u2202\u2192 ind) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ri ind rll | replLL-id ri ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {_ \u2202 ri} {.h} (\u2202\u2192 ind) | h | refl | .ri | refl | .ind | refl = refl\n\n\nmodule _ where\n\n  open Relation.Binary.HeterogeneousEquality\n\n  a\u2264\u1d62b-morph-hid : \u2200{i u ll rll}\n               \u2192 (ind : IndexLL {i} {u} rll ll)\n               \u2192 (a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind)) \u2245 ind\n  a\u2264\u1d62b-morph-hid {ll = ll} {rll = rll} ind with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ll ind rll | replLL-id ll ind rll refl | (a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind)) | a\u2264\u1d62b-morph-id ind\n  a\u2264\u1d62b-morph-hid {_} {_} {.r} {.q} .y | q | refl | r | refl | y | refl = refl\n\n\n\nreplLL-a\u2264b\u2261a : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 gll\n               \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll \u2192 (lt : emi \u2264\u1d62 ind)\n               \u2192 replLL (replLL ll ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll \u2261 replLL ll emi gll\nreplLL-a\u2264b\u2261a \u2193 gll ind frll \u2264\u1d62\u2193 = refl\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (emi \u2190\u2227) gll (ind \u2190\u2227) frll (\u2264\u1d62\u2190\u2227 lt)\n  with (replLL (replLL li ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (emi \u2190\u2227) gll (ind \u2190\u2227) frll (\u2264\u1d62\u2190\u2227 lt) | .(replLL li emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (\u2227\u2192 emi) gll (\u2227\u2192 ind) frll (\u2264\u1d62\u2227\u2192 lt)\n  with (replLL (replLL ri ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (\u2227\u2192 emi) gll (\u2227\u2192 ind) frll (\u2264\u1d62\u2227\u2192 lt) | .(replLL ri emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (emi \u2190\u2228) gll (ind \u2190\u2228) frll (\u2264\u1d62\u2190\u2228 lt)\n  with (replLL (replLL li ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (emi \u2190\u2228) gll (ind \u2190\u2228) frll (\u2264\u1d62\u2190\u2228 lt) | .(replLL li emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (\u2228\u2192 emi) gll (\u2228\u2192 ind) frll (\u2264\u1d62\u2228\u2192 lt)\n  with (replLL (replLL ri ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (\u2228\u2192 emi) gll (\u2228\u2192 ind) frll (\u2264\u1d62\u2228\u2192 lt) | .(replLL ri emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (emi \u2190\u2202) gll (ind \u2190\u2202) frll (\u2264\u1d62\u2190\u2202 lt)\n  with (replLL (replLL li ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (emi \u2190\u2202) gll (ind \u2190\u2202) frll (\u2264\u1d62\u2190\u2202 lt) | .(replLL li emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (\u2202\u2192 emi) gll (\u2202\u2192 ind) frll (\u2264\u1d62\u2202\u2192 lt)\n  with (replLL (replLL ri ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (\u2202\u2192 emi) gll (\u2202\u2192 ind) frll (\u2264\u1d62\u2202\u2192 lt) | .(replLL ri emi gll) | refl = refl\n\n\n\u00acord-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n             \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll \u2192 .(nord : \u00ac Ordered\u1d62 ind emi)\n             \u2192 IndexLL fll (replLL ll ind frll)\n\u00acord-morph \u2193 ind frll nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n\u00acord-morph (emi \u2190\u2227) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193)) \n\u00acord-morph (emi \u2190\u2227) (ind \u2190\u2227) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 lt))\n                }))\n... | r = r \u2190\u2227\n\u00acord-morph (emi \u2190\u2227) (\u2227\u2192 ind) frll nord = emi \u2190\u2227\n\u00acord-morph (\u2227\u2192 emi) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph (\u2227\u2192 emi) (ind \u2190\u2227) frll nord = \u2227\u2192 emi\n\u00acord-morph (\u2227\u2192 emi) (\u2227\u2192 ind) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 lt))\n                }))\n... | r = \u2227\u2192 r\n\u00acord-morph (emi \u2190\u2228) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193)) \n\u00acord-morph (emi \u2190\u2228) (ind \u2190\u2228) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 lt))\n                }))\n... | r = r \u2190\u2228\n\u00acord-morph (emi \u2190\u2228) (\u2228\u2192 ind) frll nord = emi \u2190\u2228\n\u00acord-morph (\u2228\u2192 emi) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph (\u2228\u2192 emi) (ind \u2190\u2228) frll nord = \u2228\u2192 emi\n\u00acord-morph (\u2228\u2192 emi) (\u2228\u2192 ind) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 lt))\n                }))\n... | r = \u2228\u2192 r\n\u00acord-morph (emi \u2190\u2202) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193) )\n\u00acord-morph (emi \u2190\u2202) (ind \u2190\u2202) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 lt))\n                }))\n... | r = r \u2190\u2202\n\u00acord-morph (emi \u2190\u2202) (\u2202\u2192 ind) frll nord = emi \u2190\u2202\n\u00acord-morph (\u2202\u2192 emi) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph (\u2202\u2192 emi) (ind \u2190\u2202) frll nord = \u2202\u2192 emi\n\u00acord-morph (\u2202\u2192 emi) (\u2202\u2192 ind) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 lt))\n                }))\n... | r = \u2202\u2192 r\n\n\nreplLL-\u00acordab\u2261ba : \u2200{i u rll ll fll}\n  \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 gll\n  \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll\n  \u2192 .(nord : \u00ac Ordered\u1d62 ind emi)\n  \u2192 replLL (replLL ll ind frll) (\u00acord-morph emi ind frll nord) gll \u2261 replLL (replLL ll emi gll) (\u00acord-morph ind emi gll (flipNotOrd\u1d62 nord)) frll\nreplLL-\u00acordab\u2261ba \u2193 gll ind frll nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba (emi \u2190\u2227) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2227 ri} (emi \u2190\u2227) gll (ind \u2190\u2227) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x))})\n... | r = cong (\u03bb x \u2192 x \u2227 ri) r\nreplLL-\u00acordab\u2261ba (emi \u2190\u2227) gll (\u2227\u2192 ind) frll nord = refl\nreplLL-\u00acordab\u2261ba (\u2227\u2192 emi) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2227 ri} (\u2227\u2192 emi) gll (\u2227\u2192 ind) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x))})\n... | r = cong (\u03bb x \u2192 li \u2227 x) r\nreplLL-\u00acordab\u2261ba (\u2227\u2192 emi) gll (ind \u2190\u2227) frll nord = refl\nreplLL-\u00acordab\u2261ba (emi \u2190\u2228) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2228 ri} (emi \u2190\u2228) gll (ind \u2190\u2228) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x))})\n... | r = cong (\u03bb x \u2192 x \u2228 ri) r\nreplLL-\u00acordab\u2261ba (emi \u2190\u2228) gll (\u2228\u2192 ind) frll nord = refl\nreplLL-\u00acordab\u2261ba (\u2228\u2192 emi) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2228 ri} (\u2228\u2192 emi) gll (\u2228\u2192 ind) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x))})\n... | r = cong (\u03bb x \u2192 li \u2228 x) r\nreplLL-\u00acordab\u2261ba (\u2228\u2192 emi) gll (ind \u2190\u2228) frll nord = refl\nreplLL-\u00acordab\u2261ba (emi \u2190\u2202) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2202 ri} (emi \u2190\u2202) gll (ind \u2190\u2202) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x))})\n... | r = cong (\u03bb x \u2192 x \u2202 ri) r\nreplLL-\u00acordab\u2261ba (emi \u2190\u2202) gll (\u2202\u2192 ind) frll nord = refl\nreplLL-\u00acordab\u2261ba (\u2202\u2192 emi) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2202 ri} (\u2202\u2192 emi) gll (\u2202\u2192 ind) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x))})\n... | r = cong (\u03bb x \u2192 li \u2202 x) r\nreplLL-\u00acordab\u2261ba (\u2202\u2192 emi) gll (ind \u2190\u2202) frll nord = refl\n \n\nlemma\u2081-\u00acord-a\u2264\u1d62b : \u2200{i u ll pll rll fll}\n      \u2192 (emi : IndexLL {i} {u} fll ll)\n      \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n      \u2192 (omi : IndexLL pll (replLL ll ind ell))\n      \u2192 .(nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) omi)\n      \u2192 IndexLL pll ll\nlemma\u2081-\u00acord-a\u2264\u1d62b \u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (omi \u2190\u2227) nord\n     = (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n                                  ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)) })) \u2190\u2227\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 omi) nord = \u2227\u2192 omi\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (omi \u2190\u2227) nord = omi \u2190\u2227\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 omi) nord\n     = \u2227\u2192 (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x))\n                                     ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)) })) \nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (omi \u2190\u2228) nord\n     = (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x))\n                                  ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)) })) \u2190\u2228\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 omi) nord = \u2228\u2192 omi\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (omi \u2190\u2228) nord = omi \u2190\u2228\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 omi) nord\n     = \u2228\u2192 (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x))\n                                     ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)) })) \nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (omi \u2190\u2202) nord\n     = (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x))\n                                  ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)) })) \u2190\u2202\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 omi) nord = \u2202\u2192 omi\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (omi \u2190\u2202) nord = omi \u2190\u2202\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 omi) nord\n     = \u2202\u2192 (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x))\n                                     ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)) })) \n\n\n\n\nrlemma\u2081\u21d2\u00acord : \u2200{i u ll pll rll fll}\n      \u2192 (emi : IndexLL {i} {u} fll ll)\n      \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n      \u2192 (omi : IndexLL pll (replLL ll ind ell))\n      \u2192 .(nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) omi)\n      \u2192 \u00ac Ordered\u1d62 emi (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi nord)\nrlemma\u2081\u21d2\u00acord \u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (omi \u2190\u2227) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 omi) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll li)\n       \u2192 (omi : IndexLL pll ri)\n       \u2192 \u00ac Ordered\u1d62 (emi \u2190\u2227) (\u2227\u2192 omi)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 omi) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (omi \u2190\u2227) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll ri)\n       \u2192 (omi : IndexLL pll li)\n       \u2192 \u00ac Ordered\u1d62 (\u2227\u2192 emi) (omi \u2190\u2227)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (omi \u2190\u2228) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 omi) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll li)\n       \u2192 (omi : IndexLL pll ri)\n       \u2192 \u00ac Ordered\u1d62 (emi \u2190\u2228) (\u2228\u2192 omi)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 omi) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (omi \u2190\u2228) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll ri)\n       \u2192 (omi : IndexLL pll li)\n       \u2192 \u00ac Ordered\u1d62 (\u2228\u2192 emi) (omi \u2190\u2228)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (omi \u2190\u2202) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 omi) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll li)\n       \u2192 (omi : IndexLL pll ri)\n       \u2192 \u00ac Ordered\u1d62 (emi \u2190\u2202) (\u2202\u2192 omi)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 omi) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (omi \u2190\u2202) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll ri)\n       \u2192 (omi : IndexLL pll li)\n       \u2192 \u00ac Ordered\u1d62 (\u2202\u2192 emi) (omi \u2190\u2202)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\n\n\n\u00acord-morph$lemma\u2081\u2261I : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 lind (a\u2264\u1d62b-morph emi ind ell lt))\n       \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind (flipNotOrd\u1d62 nord)) ind ell (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind (flipNotOrd\u1d62 nord))) \u2261 lind)\n\u00acord-morph$lemma\u2081\u2261I ell \u2193 ind \u2264\u1d62\u2193 lind nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (lind \u2190\u2227) nord = cong (\u03bb x \u2192 x \u2190\u2227) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 lind) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (lind \u2190\u2227) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 lind) nord = cong (\u03bb x \u2192 \u2227\u2192 x) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (lind \u2190\u2228) nord = cong (\u03bb x \u2192 x \u2190\u2228) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 lind) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (lind \u2190\u2228) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 lind) nord = cong (\u03bb x \u2192 \u2228\u2192 x) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (lind \u2190\u2202) nord = cong (\u03bb x \u2192 x \u2190\u2202) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 lind) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (lind \u2190\u2202) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 lind) nord = cong (\u03bb x \u2192 \u2202\u2192 x) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 lt)) }))\n\n\n\n_+\u1d62_ : \u2200{i u pll cll ll} \u2192 IndexLL {i} {u} pll ll \u2192 IndexLL cll pll \u2192 IndexLL cll ll\n_+\u1d62_ \u2193 is = is\n_+\u1d62_ (if \u2190\u2227) is = (_+\u1d62_ if is) \u2190\u2227\n_+\u1d62_ (\u2227\u2192 if) is = \u2227\u2192 (_+\u1d62_ if is)\n_+\u1d62_ (if \u2190\u2228) is = (_+\u1d62_ if is) \u2190\u2228\n_+\u1d62_ (\u2228\u2192 if) is = \u2228\u2192 (_+\u1d62_ if is)\n_+\u1d62_ (if \u2190\u2202) is = (_+\u1d62_ if is) \u2190\u2202\n_+\u1d62_ (\u2202\u2192 if) is = \u2202\u2192 (_+\u1d62_ if is)\n\n\n\n\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 : \u2200{i u pll cll ll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n           \u2192 ind \u2264\u1d62 (ind +\u1d62 lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 \u2193 lind = \u2264\u1d62\u2193\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2227) lind = \u2264\u1d62\u2190\u2227 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2227\u2192 ind) lind = \u2264\u1d62\u2227\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2228) lind = \u2264\u1d62\u2190\u2228 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2228\u2192 ind) lind = \u2264\u1d62\u2228\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2202) lind = \u2264\u1d62\u2190\u2202 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2202\u2192 ind) lind = \u2264\u1d62\u2202\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n\n\na+\u2193\u2261a : \u2200{i u pll ll} \u2192 (a : IndexLL {i} {u} pll ll) \u2192 a +\u1d62 \u2193 \u2261 a\na+\u2193\u2261a \u2193 = refl\na+\u2193\u2261a (a \u2190\u2227) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (a \u2190\u2227) | .a | refl = refl\na+\u2193\u2261a (\u2227\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (\u2227\u2192 a) | .a | refl = refl\na+\u2193\u2261a (a \u2190\u2228) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (a \u2190\u2228) | .a | refl = refl\na+\u2193\u2261a (\u2228\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (\u2228\u2192 a) | .a | refl = refl\na+\u2193\u2261a (a \u2190\u2202) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (a \u2190\u2202) | .a | refl = refl\na+\u2193\u2261a (\u2202\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (\u2202\u2192 a) | .a | refl = refl\n\n\n[a+b]-a=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n          \u2192 (b : IndexLL rll fll)\n          \u2192 ((a +\u1d62 b) -\u1d62 a) (+\u1d62\u21d2l\u2264\u1d62+\u1d62 a b) \u2261 b\n[a+b]-a=b \u2193 b = refl\n[a+b]-a=b (a \u2190\u2227) b = [a+b]-a=b a b\n[a+b]-a=b (\u2227\u2192 a) b = [a+b]-a=b a b\n[a+b]-a=b (a \u2190\u2228) b = [a+b]-a=b a b\n[a+b]-a=b (\u2228\u2192 a) b = [a+b]-a=b a b\n[a+b]-a=b (a \u2190\u2202) b = [a+b]-a=b a b\n[a+b]-a=b (\u2202\u2192 a) b = [a+b]-a=b a b\n\na+[b-a]=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n            \u2192 (b : IndexLL rll ll)\n            \u2192 (lt : a \u2264\u1d62 b)\n            \u2192 (a +\u1d62 (b -\u1d62 a) lt) \u2261 b\na+[b-a]=b \u2193 b \u2264\u1d62\u2193 = refl\na+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) | .b | refl = refl\na+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) | .b | refl = refl\na+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) | .b | refl = refl\na+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt)with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt) | .b | refl = refl\na+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) | .b | refl = refl\na+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) | .b | refl = refl\n\n\n\n-- A predicate that is true if pll is not transformed by ltr.\n\ndata UpTran {i u} : \u2200 {ll pll rll} \u2192 IndexLL pll ll \u2192 LLTr {i} {u} rll ll \u2192 Set u where\n  indI : \u2200{pll ll} \u2192 {ind : IndexLL pll ll} \u2192 UpTran ind I\n  \u2190\u2202\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (\u2202\u2192 ind) ltr\n         \u2192 UpTran (ind \u2190\u2202) (\u2202c ltr)\n  \u2202\u2192\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (ind \u2190\u2202) ltr\n         \u2192 UpTran (\u2202\u2192 ind) (\u2202c ltr)\n  \u2190\u2228\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (\u2228\u2192 ind) ltr\n         \u2192 UpTran (ind \u2190\u2228) (\u2228c ltr)\n  \u2228\u2192\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (ind \u2190\u2228) ltr\n         \u2192 UpTran (\u2228\u2192 ind) (\u2228c ltr)\n  \u2190\u2227\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (\u2227\u2192 ind) ltr\n         \u2192 UpTran (ind \u2190\u2227) (\u2227c ltr)\n  \u2227\u2192\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (ind \u2190\u2227) ltr\n         \u2192 UpTran (\u2227\u2192 ind) (\u2227c ltr)\n  \u2190\u2227]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n            \u2192 UpTran {rll = rll} (ind \u2190\u2227) ltr \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)\n  \u2227\u2192]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (ind \u2190\u2227)) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr)\n  \u2227\u2192\u2227\u2227d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 ind)) ltr\n            \u2192 UpTran {ll = ((lli \u2227 lri) \u2227 ri)} (\u2227\u2192 ind) (\u2227\u2227d ltr)\n  \u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n            \u2192 UpTran {rll = rll} ((ind \u2190\u2227) \u2190\u2227) ltr\n            \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr)\n  \u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} ((\u2227\u2192 ind) \u2190\u2227) ltr\n            \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr)\n  \u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 ind) ltr\n            \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr)\n  \u2190\u2228]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n            \u2192 UpTran {rll = rll} (ind \u2190\u2228) ltr \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)\n  \u2228\u2192]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (ind \u2190\u2228)) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr)\n  \u2228\u2192\u2228\u2228d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 ind)) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} (\u2228\u2192 ind) (\u2228\u2228d ltr)\n  \u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n            \u2192 UpTran {rll = rll} ((ind \u2190\u2228) \u2190\u2228) ltr\n            \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr)\n  \u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} ((\u2228\u2192 ind) \u2190\u2228) ltr\n            \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr)\n  \u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 ind) ltr\n            \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr)\n  \u2190\u2202]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n            \u2192 UpTran {rll = rll} (ind \u2190\u2202) ltr \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)\n  \u2202\u2192]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (ind \u2190\u2202)) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr)\n  \u2202\u2192\u2202\u2202d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 ind)) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} (\u2202\u2192 ind) (\u2202\u2202d ltr)\n  \u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n            \u2192 UpTran {rll = rll} ((ind \u2190\u2202) \u2190\u2202) ltr\n            \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr)\n  \u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} ((\u2202\u2192 ind) \u2190\u2202) ltr\n            \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr)\n  \u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 ind) ltr\n            \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr)\n\nisUpTran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 Dec (UpTran ind ltr)\nisUpTran ind I = yes indI\nisUpTran \u2193 (\u2202c ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2202) (\u2202c ltr) with (isUpTran (\u2202\u2192 ind) ltr)\nisUpTran (ind \u2190\u2202) (\u2202c ltr) | yes ut = yes (\u2190\u2202\u2202c ut)\nisUpTran (ind \u2190\u2202) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2190\u2202\u2202c ut) \u2192 \u00acut ut})\nisUpTran (\u2202\u2192 ind) (\u2202c ltr)  with (isUpTran (ind \u2190\u2202) ltr)\nisUpTran (\u2202\u2192 ind) (\u2202c ltr) | yes ut = yes (\u2202\u2192\u2202c ut)\nisUpTran (\u2202\u2192 ind) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2202\u2192\u2202c ut) \u2192 \u00acut ut})\nisUpTran \u2193 (\u2228c ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2228) (\u2228c ltr) with (isUpTran (\u2228\u2192 ind) ltr)\nisUpTran (ind \u2190\u2228) (\u2228c ltr) | yes p = yes (\u2190\u2228\u2228c p)\nisUpTran (ind \u2190\u2228) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u2228c p) \u2192 \u00acp p}) \nisUpTran (\u2228\u2192 ind) (\u2228c ltr) with (isUpTran (ind \u2190\u2228) ltr)\nisUpTran (\u2228\u2192 ind) (\u2228c ltr) | yes p = yes (\u2228\u2192\u2228c p)\nisUpTran (\u2228\u2192 ind) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228c ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2227c ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2227) (\u2227c ltr) with (isUpTran (\u2227\u2192 ind) ltr)\nisUpTran (ind \u2190\u2227) (\u2227c ltr) | yes p = yes (\u2190\u2227\u2227c p)\nisUpTran (ind \u2190\u2227) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u2227c ut) \u2192 \u00acp ut}) \nisUpTran (\u2227\u2192 ind) (\u2227c ltr) with (isUpTran (ind \u2190\u2227) ltr)\nisUpTran (\u2227\u2192 ind) (\u2227c ltr) | yes p = yes (\u2227\u2192\u2227c p)\nisUpTran (\u2227\u2192 ind) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227c ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (ind \u2190\u2227) ltr)\nisUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2190\u2227]\u2190\u2227\u2227\u2227d p)\nisUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \nisUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (ind \u2190\u2227)) ltr)\nisUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192]\u2190\u2227\u2227\u2227d p)\nisUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \nisUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 ind)) ltr)\nisUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192\u2227\u2227d p)\nisUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) with (isUpTran ((ind \u2190\u2227) \u2190\u2227) ltr)\nisUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2190\u2227\u00ac\u2227\u2227d p)\nisUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) with (isUpTran ((\u2227\u2192 ind) \u2190\u2227) ltr)\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d p)\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 ind) ltr)\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d p)\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (ind \u2190\u2228) ltr)\nisUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2190\u2228]\u2190\u2228\u2228\u2228d p)\nisUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \nisUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (ind \u2190\u2228)) ltr)\nisUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192]\u2190\u2228\u2228\u2228d p)\nisUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \nisUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 ind)) ltr)\nisUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192\u2228\u2228d p)\nisUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) with (isUpTran ((ind \u2190\u2228) \u2190\u2228) ltr)\nisUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2190\u2228\u00ac\u2228\u2228d p)\nisUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) with (isUpTran ((\u2228\u2192 ind) \u2190\u2228) ltr)\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d p)\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 ind) ltr)\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d p)\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (ind \u2190\u2202) ltr)\nisUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2190\u2202]\u2190\u2202\u2202\u2202d p)\nisUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (ind \u2190\u2202)) ltr)\nisUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192]\u2190\u2202\u2202\u2202d p)\nisUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 ind)) ltr)\nisUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192\u2202\u2202d p)\nisUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) with (isUpTran ((ind \u2190\u2202) \u2190\u2202) ltr)\nisUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2190\u2202\u00ac\u2202\u2202d p)\nisUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) with (isUpTran ((\u2202\u2192 ind) \u2190\u2202) ltr)\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d p)\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 ind) ltr)\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d p)\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n\n\n\ntran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 UpTran ind ltr\n       \u2192 IndexLL pll rll\ntran ind I indI = ind \ntran \u2193 (\u2202c ltr) () \ntran (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c ut) = tran (\u2202\u2192 ind) ltr ut\ntran (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c ut) =  tran (ind \u2190\u2202) ltr ut\ntran \u2193 (\u2228c ltr) () \ntran (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c ut) = tran (\u2228\u2192 ind) ltr ut\ntran (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c ut) = tran (ind \u2190\u2228) ltr ut\ntran \u2193 (\u2227c ltr) () \ntran (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c ut) = tran (\u2227\u2192 ind) ltr ut\ntran (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c ut) = tran (ind \u2190\u2227) ltr ut\ntran \u2193 (\u2227\u2227d ltr) () \ntran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) () \ntran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d ut) = tran (ind \u2190\u2227) ltr ut\ntran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d ut) = tran (\u2227\u2192 (ind \u2190\u2227)) ltr ut\ntran (\u2227\u2192 ind) (\u2227\u2227d ltr) (\u2227\u2192\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 ind)) ltr ut\ntran \u2193 (\u00ac\u2227\u2227d ltr) () \ntran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((ind \u2190\u2227) \u2190\u2227) ltr ut\ntran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) () \ntran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((\u2227\u2192 ind) \u2190\u2227) ltr ut\ntran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) = tran (\u2227\u2192 ind) ltr ut\ntran \u2193 (\u2228\u2228d ltr) () \ntran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) () \ntran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d ut) = tran (ind \u2190\u2228) ltr ut\ntran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d ut) = tran (\u2228\u2192 (ind \u2190\u2228)) ltr ut\ntran (\u2228\u2192 ind) (\u2228\u2228d ltr) (\u2228\u2192\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 ind)) ltr ut\ntran \u2193 (\u00ac\u2228\u2228d ltr) () \ntran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((ind \u2190\u2228) \u2190\u2228) ltr ut\ntran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) () \ntran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((\u2228\u2192 ind) \u2190\u2228) ltr ut\ntran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) = tran (\u2228\u2192 ind) ltr ut\ntran \u2193 (\u2202\u2202d ltr) () \ntran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) () \ntran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d ut) = tran (ind \u2190\u2202) ltr ut\ntran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d ut) = tran (\u2202\u2192 (ind \u2190\u2202)) ltr ut\ntran (\u2202\u2192 ind) (\u2202\u2202d ltr) (\u2202\u2192\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 ind)) ltr ut\ntran \u2193 (\u00ac\u2202\u2202d ltr) () \ntran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((ind \u2190\u2202) \u2190\u2202) ltr ut\ntran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) () \ntran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((\u2202\u2192 ind) \u2190\u2202) ltr ut\ntran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) = tran (\u2202\u2192 ind) ltr ut\n\n\ntr-ltr-morph : \u2200 {i u ll pll orll} \u2192 \u2200 frll \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} orll ll) \u2192 (rvT : UpTran ind ltr) \u2192 LLTr (replLL orll (tran ind ltr rvT) frll) (replLL ll ind frll)\ntr-ltr-morph frll \u2193 .I indI = I\ntr-ltr-morph frll (ind \u2190\u2227) I indI = I\ntr-ltr-morph frll (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c rvT) with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n... | r = \u2227c r\ntr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) ltr rvT\n... | r = \u00ac\u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 ind) I indI = I\ntr-ltr-morph frll (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n... | r = \u2227c r\ntr-ltr-morph frll (\u2227\u2192 ind) (\u2227\u2227d ltr) (\u2227\u2192\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) ltr rvT\n... | r = \u00ac\u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d rvT)  with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n... | r = \u00ac\u2227\u2227d r\ntr-ltr-morph frll (ind \u2190\u2228) I indI = I\ntr-ltr-morph frll (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c rvT) with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n... | r = \u2228c r\ntr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) ltr rvT\n... | r = \u00ac\u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 ind) I indI = I\ntr-ltr-morph frll (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n... | r = \u2228c r\ntr-ltr-morph frll (\u2228\u2192 ind) (\u2228\u2228d ltr) (\u2228\u2192\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) ltr rvT\n... | r = \u00ac\u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d rvT)  with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n... | r = \u00ac\u2228\u2228d r\ntr-ltr-morph frll (ind \u2190\u2202) I indI = I\ntr-ltr-morph frll (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c rvT) with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n... | r = \u2202c r\ntr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) ltr rvT\n... | r = \u00ac\u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 ind) I indI = I\ntr-ltr-morph frll (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n... | r = \u2202c r\ntr-ltr-morph frll (\u2202\u2192 ind) (\u2202\u2202d ltr) (\u2202\u2192\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) ltr rvT\n... | r = \u00ac\u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d rvT)  with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n... | r = \u00ac\u2202\u2202d r\n\n\n\ndrepl=>repl+ : \u2200{i u pll ll cll frll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n               \u2192 (replLL ll ind (replLL pll lind frll)) \u2261 replLL ll (ind +\u1d62 lind) frll\ndrepl=>repl+ \u2193 lind = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (ind \u2190\u2227) lind\n             with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (ind \u2190\u2227) lind\n             | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (\u2227\u2192 ind) lind\n             with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (\u2227\u2192 ind) lind\n             | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (ind \u2190\u2228) lind\n             with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (ind \u2190\u2228) lind\n             | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (\u2228\u2192 ind) lind\n             with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (\u2228\u2192 ind) lind\n             | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (ind \u2190\u2202) lind\n             with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (ind \u2190\u2202) lind\n             | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (\u2202\u2192 ind) lind\n             with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (\u2202\u2192 ind) lind\n             | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n\n\n\nrepl-r=>l : \u2200{i u pll ll cll frll} \u2192 \u2200 ell \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (x : IndexLL cll (replLL ll ind ell)) \u2192 let uind = a\u2264\u1d62b-morph ind ind ell (\u2264\u1d62-reflexive ind)\n       in (ltuindx : uind \u2264\u1d62 x)\n       \u2192 (replLL ll ind (replLL ell (ind-rpl\u2193 ind ((x -\u1d62 uind) ltuindx)) frll) \u2261 replLL (replLL ll ind ell) x frll)\nrepl-r=>l ell \u2193 x \u2264\u1d62\u2193 = refl\nrepl-r=>l {ll = li \u2227 ri} ell (ind \u2190\u2227) (x \u2190\u2227) (\u2264\u1d62\u2190\u2227 ltuindx) = cong (\u03bb x \u2192 x \u2227 ri) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2227 ri} ell (\u2227\u2192 ind) (\u2227\u2192 x) (\u2264\u1d62\u2227\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2227 x) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2228 ri} ell (ind \u2190\u2228) (x \u2190\u2228) (\u2264\u1d62\u2190\u2228 ltuindx) = cong (\u03bb x \u2192 x \u2228 ri) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2228 ri} ell (\u2228\u2192 ind) (\u2228\u2192 x) (\u2264\u1d62\u2228\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2228 x) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2202 ri} ell (ind \u2190\u2202) (x \u2190\u2202) (\u2264\u1d62\u2190\u2202 ltuindx) = cong (\u03bb x \u2192 x \u2202 ri) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2202 ri} ell (\u2202\u2192 ind) (\u2202\u2192 x) (\u2264\u1d62\u2202\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2202 x) (repl-r=>l ell ind x ltuindx)\n\n\n\n-- Deprecated\nupdInd : \u2200{i u rll ll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} rll ll)\n         \u2192 IndexLL {i} {u} nrll (replLL ll ind nrll)\nupdInd nrll \u2193 = \u2193\nupdInd nrll (ind \u2190\u2227) = (updInd nrll ind) \u2190\u2227\nupdInd nrll (\u2227\u2192 ind) = \u2227\u2192 (updInd nrll ind)\nupdInd nrll (ind \u2190\u2228) = (updInd nrll ind) \u2190\u2228\nupdInd nrll (\u2228\u2192 ind) = \u2228\u2192 (updInd nrll ind)\nupdInd nrll (ind \u2190\u2202) = (updInd nrll ind) \u2190\u2202\nupdInd nrll (\u2202\u2192 ind) = \u2202\u2192 (updInd nrll ind)\n\n-- Deprecated\n-- Maybe instead of this function use a\u2264\u1d62b-morph\nupdIndGen : \u2200{i u pll ll cll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n            \u2192 IndexLL {i} {u} (replLL pll lind nrll) (replLL ll (ind +\u1d62 lind) nrll)\nupdIndGen nrll \u2193 lind = \u2193\nupdIndGen nrll (ind \u2190\u2227) lind = (updIndGen nrll ind lind) \u2190\u2227\nupdIndGen nrll (\u2227\u2192 ind) lind = \u2227\u2192 (updIndGen nrll ind lind)\nupdIndGen nrll (ind \u2190\u2228) lind = (updIndGen nrll ind lind) \u2190\u2228\nupdIndGen nrll (\u2228\u2192 ind) lind = \u2228\u2192 (updIndGen nrll ind lind)\nupdIndGen nrll (ind \u2190\u2202) lind = (updIndGen nrll ind lind) \u2190\u2202\nupdIndGen nrll (\u2202\u2192 ind) lind = \u2202\u2192 (updIndGen nrll ind lind)\n\n\n\n-- TODO This negation was writen so as to return nothing if \u00ac (b \u2264\u1d62 a)\n_-\u2098\u1d62_ : \u2200 {i u pll cll ll} \u2192 IndexLL {i} {u} cll ll \u2192 IndexLL pll ll \u2192 Maybe $ IndexLL cll pll\na -\u2098\u1d62 \u2193 = just a\n\u2193 -\u2098\u1d62 (b \u2190\u2227) = nothing\n(a \u2190\u2227) -\u2098\u1d62 (b \u2190\u2227) = a -\u2098\u1d62 b\n(\u2227\u2192 a) -\u2098\u1d62 (b \u2190\u2227) = nothing\n\u2193 -\u2098\u1d62 (\u2227\u2192 b) = nothing\n(a \u2190\u2227) -\u2098\u1d62 (\u2227\u2192 b) = nothing\n(\u2227\u2192 a) -\u2098\u1d62 (\u2227\u2192 b) = a -\u2098\u1d62 b\n\u2193 -\u2098\u1d62 (b \u2190\u2228) = nothing\n(a \u2190\u2228) -\u2098\u1d62 (b \u2190\u2228) = a -\u2098\u1d62 b\n(\u2228\u2192 a) -\u2098\u1d62 (b \u2190\u2228) = nothing\n\u2193 -\u2098\u1d62 (\u2228\u2192 b) = nothing\n(a \u2190\u2228) -\u2098\u1d62 (\u2228\u2192 b) = nothing\n(\u2228\u2192 a) -\u2098\u1d62 (\u2228\u2192 b) = a -\u2098\u1d62 b\n\u2193 -\u2098\u1d62 (b \u2190\u2202) = nothing\n(a \u2190\u2202) -\u2098\u1d62 (b \u2190\u2202) = a -\u2098\u1d62 b\n(\u2202\u2192 a) -\u2098\u1d62 (b \u2190\u2202) = nothing\n\u2193 -\u2098\u1d62 (\u2202\u2192 b) = nothing\n(a \u2190\u2202) -\u2098\u1d62 (\u2202\u2192 b) = nothing\n(\u2202\u2192 a) -\u2098\u1d62 (\u2202\u2192 b) = a -\u2098\u1d62 b\n\n-- Deprecated\nb-s\u2261j\u21d2s\u2264\u1d62b : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll)\n             \u2192  \u03a3 (IndexLL {i} {u} cll pll) (\u03bb x \u2192 (bind -\u2098\u1d62 sind) \u2261 just x) \u2192 sind \u2264\u1d62 bind\nb-s\u2261j\u21d2s\u2264\u1d62b bind \u2193 (rs , x) = \u2264\u1d62\u2193\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2227) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (sind \u2190\u2227) (rs , x) = \u2264\u1d62\u2190\u2227 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (sind \u2190\u2227) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2227\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (\u2227\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (\u2227\u2192 sind) (rs , x) = \u2264\u1d62\u2227\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2228) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (sind \u2190\u2228) (rs , x) = \u2264\u1d62\u2190\u2228 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (sind \u2190\u2228) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2228\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (\u2228\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (\u2228\u2192 sind) (rs , x) =  \u2264\u1d62\u2228\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2202) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (sind \u2190\u2202) (rs , x) = \u2264\u1d62\u2190\u2202 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (sind \u2190\u2202) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2202\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (\u2202\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (\u2202\u2192 sind) (rs , x) = \u2264\u1d62\u2202\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n\n-- Deprecated\nrevUpdInd : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll (replLL ll b ell))\n            \u2192 a -\u2098\u1d62 (updInd ell b) \u2261 nothing \u2192 (updInd ell b) -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll ll\nrevUpdInd \u2193 a () b-a\nrevUpdInd (b \u2190\u2227) \u2193 refl ()\nrevUpdInd (b \u2190\u2227) (a \u2190\u2227) a-b b-a = revUpdInd b a a-b b-a \u2190\u2227\nrevUpdInd (b \u2190\u2227) (\u2227\u2192 a) a-b b-a = \u2227\u2192 a\nrevUpdInd (\u2227\u2192 b) \u2193 refl ()\nrevUpdInd (\u2227\u2192 b) (a \u2190\u2227) a-b b-a = a \u2190\u2227\nrevUpdInd (\u2227\u2192 b) (\u2227\u2192 a) a-b b-a = \u2227\u2192 revUpdInd b a a-b b-a\nrevUpdInd (b \u2190\u2228) \u2193 refl ()\nrevUpdInd (b \u2190\u2228) (a \u2190\u2228) a-b b-a = revUpdInd b a a-b b-a \u2190\u2228\nrevUpdInd (b \u2190\u2228) (\u2228\u2192 a) a-b b-a = \u2228\u2192 a\nrevUpdInd (\u2228\u2192 b) \u2193 refl ()\nrevUpdInd (\u2228\u2192 b) (a \u2190\u2228) a-b b-a = a \u2190\u2228\nrevUpdInd (\u2228\u2192 b) (\u2228\u2192 a) a-b b-a = \u2228\u2192 revUpdInd b a a-b b-a\nrevUpdInd (b \u2190\u2202) \u2193 refl ()\nrevUpdInd (b \u2190\u2202) (a \u2190\u2202) a-b b-a = revUpdInd b a a-b b-a \u2190\u2202\nrevUpdInd (b \u2190\u2202) (\u2202\u2192 a) a-b b-a = \u2202\u2192 a\nrevUpdInd (\u2202\u2192 b) \u2193 refl ()\nrevUpdInd (\u2202\u2192 b) (a \u2190\u2202) a-b b-a = a \u2190\u2202\nrevUpdInd (\u2202\u2192 b) (\u2202\u2192 a) a-b b-a = \u2202\u2192 revUpdInd b a a-b b-a\n\n\n-- Deprecated\nupdIndPart : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll ll)\n             \u2192 a -\u2098\u1d62 b \u2261 nothing \u2192 b -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll (replLL ll b ell)\nupdIndPart \u2193 a () eq2\nupdIndPart (b \u2190\u2227) \u2193 refl ()\nupdIndPart (b \u2190\u2227) (a \u2190\u2227) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2227\nupdIndPart (b \u2190\u2227) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 a\nupdIndPart (\u2227\u2192 b) \u2193 refl ()\nupdIndPart (\u2227\u2192 b) (a \u2190\u2227) eq1 eq2 = a \u2190\u2227\nupdIndPart (\u2227\u2192 b) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 updIndPart b a eq1 eq2\nupdIndPart (b \u2190\u2228) \u2193 refl ()\nupdIndPart (b \u2190\u2228) (a \u2190\u2228) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2228\nupdIndPart (b \u2190\u2228) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 a\nupdIndPart (\u2228\u2192 b) \u2193 refl ()\nupdIndPart (\u2228\u2192 b) (a \u2190\u2228) eq1 eq2 = a \u2190\u2228\nupdIndPart (\u2228\u2192 b) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 updIndPart b a eq1 eq2\nupdIndPart (b \u2190\u2202) \u2193 refl ()\nupdIndPart (b \u2190\u2202) (a \u2190\u2202) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2202\nupdIndPart (b \u2190\u2202) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 a\nupdIndPart (\u2202\u2192 b) \u2193 refl ()\nupdIndPart (\u2202\u2192 b) (a \u2190\u2202) eq1 eq2 = a \u2190\u2202\nupdIndPart (\u2202\u2192 b) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 updIndPart b a eq1 eq2\n\n-- Deprecated\nrev\u21d2rs-i\u2261n : \u2200{i u ll cll ell pll} \u2192 (ind : IndexLL pll ll)\n             \u2192 (lind : IndexLL {i} {u} cll (replLL ll ind ell))\n             \u2192 (eq\u2081 : (lind -\u2098\u1d62 (updInd ell ind) \u2261 nothing))\n             \u2192 (eq\u2082 : ((updInd ell ind) -\u2098\u1d62 lind \u2261 nothing))\n             \u2192 let rs = revUpdInd ind lind eq\u2081 eq\u2082 in\n                 ((rs -\u2098\u1d62 ind) \u2261 nothing) \u00d7 ((ind -\u2098\u1d62 rs) \u2261 nothing)\nrev\u21d2rs-i\u2261n \u2193 lind () eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2227) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (ind \u2190\u2227) (lind \u2190\u2227) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2227) (\u2227\u2192 lind) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2227\u2192 ind) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (\u2227\u2192 ind) (lind \u2190\u2227) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2227\u2192 ind) (\u2227\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2228) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (ind \u2190\u2228) (lind \u2190\u2228) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2228) (\u2228\u2192 lind) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2228\u2192 ind) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (\u2228\u2192 ind) (lind \u2190\u2228) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2228\u2192 ind) (\u2228\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2202) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (ind \u2190\u2202) (lind \u2190\u2202) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2202) (\u2202\u2192 lind) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2202\u2192 ind) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (\u2202\u2192 ind) (lind \u2190\u2202) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2202\u2192 ind) (\u2202\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n\n\n\n\n-- Deprecated\nremRepl : \u2200{i u ll frll ell pll cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n          \u2192 replLL (replLL ll (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell \u2261 replLL ll ind ell\nremRepl \u2193 lind = refl\nremRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (ind \u2190\u2227) lind\n        with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2227 ri} {frll} {ell} (ind \u2190\u2227) lind | .(replLL li ind ell) | refl = refl\nremRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (\u2227\u2192 ind) lind\n        with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2227 ri} {frll} {ell} (\u2227\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\nremRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (ind \u2190\u2228) lind\n        with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2228 ri} {frll} {ell} (ind \u2190\u2228) lind | .(replLL li ind ell) | refl = refl\nremRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (\u2228\u2192 ind) lind\n        with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2228 ri} {frll} {ell} (\u2228\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\nremRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (ind \u2190\u2202) lind\n        with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2202 ri} {frll} {ell} (ind \u2190\u2202) lind | .(replLL li ind ell) | refl = refl\nremRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (\u2202\u2192 ind) lind\n        with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2202 ri} {frll} {ell} (\u2202\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n\n-- Deprecated\nreplLL-inv : \u2200{i u ll ell pll} \u2192 (ind : IndexLL {i} {u} pll ll)\n             \u2192 replLL (replLL ll ind ell) (updInd ell ind) pll \u2261 ll\nreplLL-inv \u2193 = refl\nreplLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (ind \u2190\u2227)\n           with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (ind \u2190\u2227) | .li | refl = refl\nreplLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (\u2227\u2192 ind)\n           with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (\u2227\u2192 ind) | .ri | refl = refl\nreplLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (ind \u2190\u2228)\n           with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (ind \u2190\u2228) | .li | refl = refl\nreplLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (\u2228\u2192 ind)\n           with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (\u2228\u2192 ind) | .ri | refl = refl\nreplLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (ind \u2190\u2202)\n           with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (ind \u2190\u2202) | .li | refl = refl\nreplLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (\u2202\u2192 ind)\n           with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (\u2202\u2192 ind) | .ri | refl = refl\n\n\n","old_contents":"{-# OPTIONS --exact-split #-}\n-- {-# OPTIONS --show-implicit #-}\n\nmodule IndexLLProp where\n\nopen import Common\nopen import LinLogic\nopen import Data.Maybe\nopen import Data.Product\nimport Relation.Binary.HeterogeneousEquality\nopen import Relation.Binary.PropositionalEquality using (subst ; cong ; subst\u2082)\n\ndata _\u2245\u1d62_ {i u gll} : \u2200{fll ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set where\n  \u2245\u1d62\u2193 :  \u2193 \u2245\u1d62 \u2193\n  \u2245\u1d62\u2190\u2227 : \u2200{fll li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2227 ri} (sind \u2190\u2227) (bind \u2190\u2227)\n  \u2245\u1d62\u2227\u2192 : \u2200{fll li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2227 ri} (\u2227\u2192 sind) (\u2227\u2192 bind)\n  \u2245\u1d62\u2190\u2228 : \u2200{fll li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2228 ri} (sind \u2190\u2228) (bind \u2190\u2228)\n  \u2245\u1d62\u2228\u2192 : \u2200{fll li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2228 ri} (\u2228\u2192 sind) (\u2228\u2192 bind)\n  \u2245\u1d62\u2190\u2202 : \u2200{fll li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2202 ri} (sind \u2190\u2202) (bind \u2190\u2202)\n  \u2245\u1d62\u2202\u2192 : \u2200{fll li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = li \u2202 ri} (\u2202\u2192 sind) (\u2202\u2192 bind)\n\n\ndata _\u2264\u1d62_ {i u gll fll} : \u2200{ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set where\n  \u2264\u1d62\u2193 : {ind : IndexLL fll gll} \u2192 \u2193 \u2264\u1d62 ind\n  \u2264\u1d62\u2190\u2227 : \u2200{li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2227 ri} (sind \u2190\u2227) (bind \u2190\u2227)\n  \u2264\u1d62\u2227\u2192 : \u2200{li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2227 ri} (\u2227\u2192 sind) (\u2227\u2192 bind)\n  \u2264\u1d62\u2190\u2228 : \u2200{li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2228 ri} (sind \u2190\u2228) (bind \u2190\u2228)\n  \u2264\u1d62\u2228\u2192 : \u2200{li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2228 ri} (\u2228\u2192 sind) (\u2228\u2192 bind)\n  \u2264\u1d62\u2190\u2202 : \u2200{li ri} \u2192 {sind : IndexLL gll li} \u2192 {bind : IndexLL fll li} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2202 ri} (sind \u2190\u2202) (bind \u2190\u2202)\n  \u2264\u1d62\u2202\u2192 : \u2200{li ri} \u2192 {sind : IndexLL gll ri} \u2192 {bind : IndexLL fll ri} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = li \u2202 ri} (\u2202\u2192 sind) (\u2202\u2192 bind)\n\n\u2264\u1d62-reflexive : \u2200{i u gll ll} \u2192 (ind : IndexLL {i} {u} gll ll) \u2192 ind \u2264\u1d62 ind\n\u2264\u1d62-reflexive \u2193 = \u2264\u1d62\u2193\n\u2264\u1d62-reflexive (ind \u2190\u2227) = \u2264\u1d62\u2190\u2227 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (\u2227\u2192 ind) = \u2264\u1d62\u2227\u2192 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (ind \u2190\u2228) = \u2264\u1d62\u2190\u2228 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (\u2228\u2192 ind) = \u2264\u1d62\u2228\u2192 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (ind \u2190\u2202) = \u2264\u1d62\u2190\u2202 (\u2264\u1d62-reflexive ind)\n\u2264\u1d62-reflexive (\u2202\u2192 ind) = \u2264\u1d62\u2202\u2192 (\u2264\u1d62-reflexive ind)\n\n\u2264\u1d62-transitive : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (b \u2264\u1d62 c) \u2192 (a \u2264\u1d62 c)\n\u2264\u1d62-transitive \u2264\u1d62\u2193 ltbc = \u2264\u1d62\u2193\n\u2264\u1d62-transitive (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltbc) = \u2264\u1d62\u2190\u2227 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltbc) = \u2264\u1d62\u2227\u2192 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltbc) = \u2264\u1d62\u2190\u2228 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltbc) = \u2264\u1d62\u2228\u2192 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltbc) = \u2264\u1d62\u2190\u2202 (\u2264\u1d62-transitive ltab ltbc)\n\u2264\u1d62-transitive (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltbc) = \u2264\u1d62\u2202\u2192 (\u2264\u1d62-transitive ltab ltbc)\n\n\nisLTi : \u2200{i u gll ll fll} \u2192 (s : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL fll ll) \u2192 Dec (s \u2264\u1d62 b)\nisLTi \u2193 b = yes \u2264\u1d62\u2193\nisLTi (s \u2190\u2227) \u2193 = no (\u03bb ())\nisLTi (s \u2190\u2227) (b \u2190\u2227) with (isLTi s b)\nisLTi (s \u2190\u2227) (b \u2190\u2227) | yes p = yes $ \u2264\u1d62\u2190\u2227 p\nisLTi (s \u2190\u2227) (b \u2190\u2227) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2190\u2227 p) \u2192 \u00acp p }) \nisLTi (s \u2190\u2227) (\u2227\u2192 b) = no (\u03bb ())\nisLTi (\u2227\u2192 s) \u2193 = no (\u03bb ())\nisLTi (\u2227\u2192 s) (b \u2190\u2227) = no (\u03bb ())\nisLTi (\u2227\u2192 s) (\u2227\u2192 b) with (isLTi s b)\nisLTi (\u2227\u2192 s) (\u2227\u2192 b) | yes p = yes (\u2264\u1d62\u2227\u2192 p)\nisLTi (\u2227\u2192 s) (\u2227\u2192 b) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2227\u2192 p) \u2192 \u00acp p })\nisLTi (s \u2190\u2228) \u2193 = no (\u03bb ())\nisLTi (s \u2190\u2228) (b \u2190\u2228) with (isLTi s b)\nisLTi (s \u2190\u2228) (b \u2190\u2228) | yes p = yes $ \u2264\u1d62\u2190\u2228 p\nisLTi (s \u2190\u2228) (b \u2190\u2228) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2190\u2228 p) \u2192 \u00acp p }) \nisLTi (s \u2190\u2228) (\u2228\u2192 b) = no (\u03bb ())\nisLTi (\u2228\u2192 s) \u2193 = no (\u03bb ())\nisLTi (\u2228\u2192 s) (b \u2190\u2228) = no (\u03bb ())\nisLTi (\u2228\u2192 s) (\u2228\u2192 b) with (isLTi s b)\nisLTi (\u2228\u2192 s) (\u2228\u2192 b) | yes p = yes (\u2264\u1d62\u2228\u2192 p)\nisLTi (\u2228\u2192 s) (\u2228\u2192 b) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2228\u2192 p) \u2192 \u00acp p })\nisLTi (s \u2190\u2202) \u2193 = no (\u03bb ())\nisLTi (s \u2190\u2202) (b \u2190\u2202) with (isLTi s b)\nisLTi (s \u2190\u2202) (b \u2190\u2202) | yes p = yes $ \u2264\u1d62\u2190\u2202 p\nisLTi (s \u2190\u2202) (b \u2190\u2202) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2190\u2202 p) \u2192 \u00acp p }) \nisLTi (s \u2190\u2202) (\u2202\u2192 b) = no (\u03bb ())\nisLTi (\u2202\u2192 s) \u2193 = no (\u03bb ())\nisLTi (\u2202\u2192 s) (b \u2190\u2202) = no (\u03bb ())\nisLTi (\u2202\u2192 s) (\u2202\u2192 b) with (isLTi s b)\nisLTi (\u2202\u2192 s) (\u2202\u2192 b) | yes p = yes (\u2264\u1d62\u2202\u2192 p)\nisLTi (\u2202\u2192 s) (\u2202\u2192 b) | no \u00acp = no (\u03bb {(\u2264\u1d62\u2202\u2192 p) \u2192 \u00acp p })\n\nisEq\u1d62 : \u2200{u i ll rll} \u2192 (a : IndexLL {i} {u} rll ll) \u2192 (b : IndexLL rll ll) \u2192 Dec (a \u2261 b)\nisEq\u1d62 \u2193 \u2193 = yes refl\nisEq\u1d62 \u2193 (b \u2190\u2227) = no (\u03bb ())\nisEq\u1d62 \u2193 (\u2227\u2192 b) = no (\u03bb ())\nisEq\u1d62 \u2193 (b \u2190\u2228) = no (\u03bb ())\nisEq\u1d62 \u2193 (\u2228\u2192 b) = no (\u03bb ())\nisEq\u1d62 \u2193 (b \u2190\u2202) = no (\u03bb ())\nisEq\u1d62 \u2193 (\u2202\u2192 b) = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2227) \u2193 = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2227) (b \u2190\u2227) with isEq\u1d62 a b\nisEq\u1d62 (a \u2190\u2227) (.a \u2190\u2227) | yes refl = yes refl\nisEq\u1d62 (a \u2190\u2227) (b \u2190\u2227) | no \u00acp = no hf where\n  hf : \u00ac ((a \u2190\u2227) \u2261 (b \u2190\u2227))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2227) (\u2227\u2192 b) = no (\u03bb ())\nisEq\u1d62 (\u2227\u2192 a) \u2193 = no (\u03bb ())\nisEq\u1d62 (\u2227\u2192 a) (b \u2190\u2227) = no (\u03bb ())\nisEq\u1d62 (\u2227\u2192 a) (\u2227\u2192 b)  with isEq\u1d62 a b\nisEq\u1d62 (\u2227\u2192 a) (\u2227\u2192 .a) | yes refl = yes refl\nisEq\u1d62 (\u2227\u2192 a) (\u2227\u2192 b) | no \u00acp = no hf where\n  hf : \u00ac ((\u2227\u2192 a) \u2261 (\u2227\u2192 b))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2228) \u2193 = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2228) (b \u2190\u2228) with isEq\u1d62 a b\nisEq\u1d62 (a \u2190\u2228) (.a \u2190\u2228) | yes refl = yes refl\nisEq\u1d62 (a \u2190\u2228) (b \u2190\u2228) | no \u00acp = no hf where\n  hf : \u00ac ((a \u2190\u2228) \u2261 (b \u2190\u2228))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2228) (\u2228\u2192 b) = no (\u03bb ())\nisEq\u1d62 (\u2228\u2192 a) \u2193 = no (\u03bb ())\nisEq\u1d62 (\u2228\u2192 a) (b \u2190\u2228) = no (\u03bb ())\nisEq\u1d62 (\u2228\u2192 a) (\u2228\u2192 b)  with isEq\u1d62 a b\nisEq\u1d62 (\u2228\u2192 a) (\u2228\u2192 .a) | yes refl = yes refl\nisEq\u1d62 (\u2228\u2192 a) (\u2228\u2192 b) | no \u00acp = no hf where\n  hf : \u00ac ((\u2228\u2192 a) \u2261 (\u2228\u2192 b))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2202) \u2193 = no (\u03bb ())\nisEq\u1d62 (a \u2190\u2202) (b \u2190\u2202) with isEq\u1d62 a b\nisEq\u1d62 (a \u2190\u2202) (.a \u2190\u2202) | yes refl = yes refl\nisEq\u1d62 (a \u2190\u2202) (b \u2190\u2202) | no \u00acp = no hf where\n  hf : \u00ac ((a \u2190\u2202) \u2261 (b \u2190\u2202))\n  hf refl = \u00acp refl\nisEq\u1d62 (a \u2190\u2202) (\u2202\u2192 b) = no (\u03bb ())\nisEq\u1d62 (\u2202\u2192 a) \u2193 = no (\u03bb ())\nisEq\u1d62 (\u2202\u2192 a) (b \u2190\u2202) = no (\u03bb ())\nisEq\u1d62 (\u2202\u2192 a) (\u2202\u2192 b)  with isEq\u1d62 a b\nisEq\u1d62 (\u2202\u2192 a) (\u2202\u2192 .a) | yes refl = yes refl\nisEq\u1d62 (\u2202\u2192 a) (\u2202\u2192 b) | no \u00acp = no hf where\n  hf : \u00ac ((\u2202\u2192 a) \u2261 (\u2202\u2192 b))\n  hf refl = \u00acp refl\n\n\n\n\n\nmodule _ where\n\n  open import Data.Vec\n\n\n  ind\u03c4\u21d2\u00acle : \u2200{i u rll ll n dt df} \u2192 (ind : IndexLL {i} {u} (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (ind2 : IndexLL rll ll) \u2192 \u00ac (ind2 \u2245\u1d62 ind) \u2192 \u00ac (ind \u2264\u1d62 ind2)\n  ind\u03c4\u21d2\u00acle \u2193 \u2193 neq = \u03bb _ \u2192 neq \u2245\u1d62\u2193\n  ind\u03c4\u21d2\u00acle (x \u2190\u2227) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (x \u2190\u2227) (y \u2190\u2227) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2190\u2227 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2190\u2227 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2227) (\u2227\u2192 y) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2227\u2192 x) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2227\u2192 x) (y \u2190\u2227) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2227\u2192 x) (\u2227\u2192 y) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2227\u2192 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2227\u2192 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2228) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (x \u2190\u2228) (y \u2190\u2228) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2190\u2228 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2190\u2228 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2228) (\u2228\u2192 y) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2228\u2192 x) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2228\u2192 x) (y \u2190\u2228) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2228\u2192 x) (\u2228\u2192 y) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2228\u2192 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2228\u2192 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2202) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (x \u2190\u2202) (y \u2190\u2202) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2190\u2202 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2190\u2202 x) \u2192 r x}\n  ind\u03c4\u21d2\u00acle (x \u2190\u2202) (\u2202\u2192 y) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2202\u2192 x) \u2193 neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2202\u2192 x) (y \u2190\u2202) neq = \u03bb ()\n  ind\u03c4\u21d2\u00acle (\u2202\u2192 x) (\u2202\u2192 y) neq with ind\u03c4\u21d2\u00acle x y ineq where\n    ineq : \u00ac (y \u2245\u1d62 x)\n    ineq z = neq (\u2245\u1d62\u2202\u2192 z)\n  ... | r = \u03bb { (\u2264\u1d62\u2202\u2192 x) \u2192 r x}\n  \n\n\n \n\ndata Ordered\u1d62 {i u gll fll ll} (a : IndexLL {i} {u} gll ll) (b : IndexLL {i} {u} fll ll) : Set where\n  a\u2264\u1d62b : a \u2264\u1d62 b \u2192 Ordered\u1d62 a b\n  b\u2264\u1d62a : b \u2264\u1d62 a \u2192 Ordered\u1d62 a b\n\nisOrd\u1d62 : \u2200{i u gll fll ll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL {i} {u} fll ll)\n         \u2192 Dec (Ordered\u1d62 a b)\nisOrd\u1d62 a b with (isLTi a b) | (isLTi b a)\nisOrd\u1d62 a b | yes p | r = yes (a\u2264\u1d62b p)\nisOrd\u1d62 a b | no \u00acp | (yes p) = yes (b\u2264\u1d62a p)\nisOrd\u1d62 a b | no \u00acp | (no \u00acp\u2081) = no ((\u03bb { (a\u2264\u1d62b p) \u2192 \u00acp p\n                                       ; (b\u2264\u1d62a p) \u2192 \u00acp\u2081 p }))\n\n\n\nflipOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n           \u2192 Ordered\u1d62 a b \u2192 Ordered\u1d62 b a\nflipOrd\u1d62 (a\u2264\u1d62b x) = b\u2264\u1d62a x\nflipOrd\u1d62 (b\u2264\u1d62a x) = a\u2264\u1d62b x\n\nflipNotOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac Ordered\u1d62 a b \u2192 \u00ac Ordered\u1d62 b a\nflipNotOrd\u1d62 nord = \u03bb x \u2192 nord (flipOrd\u1d62 x) \n\n\n\u00aclt\u00acgt\u21d2\u00acOrd : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac (a \u2264\u1d62 b) \u2192 \u00ac (b \u2264\u1d62 a) \u2192 \u00ac(Ordered\u1d62 a b)\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (a\u2264\u1d62b x) = nlt x\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (b\u2264\u1d62a x) = ngt x\n\n\na,c\u2264\u1d62b\u21d2ordac : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (c \u2264\u1d62 b) \u2192 Ordered\u1d62 a c\na,c\u2264\u1d62b\u21d2ordac ltab \u2264\u1d62\u2193 = b\u2264\u1d62a \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2190\u2227 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2227 ltab) (\u2264\u1d62\u2190\u2227 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2227\u2192 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2227\u2192 ltab) (\u2264\u1d62\u2227\u2192 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2190\u2228 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2228 ltab) (\u2264\u1d62\u2190\u2228 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2228\u2192 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2228\u2192 ltab) (\u2264\u1d62\u2228\u2192 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2190\u2202 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2190\u2202 ltab) (\u2264\u1d62\u2190\u2202 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 (\u2264\u1d62\u2202\u2192 ltac) = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltac) with (a,c\u2264\u1d62b\u21d2ordac ltab ltac)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltac) | a\u2264\u1d62b x = a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x)\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62\u2202\u2192 ltab) (\u2264\u1d62\u2202\u2192 ltac) | b\u2264\u1d62a x = b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)\n\n\n\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 \u00ac Ordered\u1d62 a c \u2192 \u00ac Ordered\u1d62 b c\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (a\u2264\u1d62b x) = \u22a5-elim (nord (a\u2264\u1d62b (\u2264\u1d62-transitive lt x)))\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (b\u2264\u1d62a x) = \u22a5-elim (nord (a,c\u2264\u1d62b\u21d2ordac lt x))\n\n\n_-\u1d62_ : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll) \u2192 (sind \u2264\u1d62 bind)\n       \u2192 IndexLL cll pll\n(bind -\u1d62 .\u2193) \u2264\u1d62\u2193 = bind\n((bind \u2190\u2227) -\u1d62 (sind \u2190\u2227)) (\u2264\u1d62\u2190\u2227 eq) = (bind -\u1d62 sind) eq\n((\u2227\u2192 bind) -\u1d62 (\u2227\u2192 sind)) (\u2264\u1d62\u2227\u2192 eq) = (bind -\u1d62 sind) eq\n((bind \u2190\u2228) -\u1d62 (sind \u2190\u2228)) (\u2264\u1d62\u2190\u2228 eq) = (bind -\u1d62 sind) eq\n((\u2228\u2192 bind) -\u1d62 (\u2228\u2192 sind)) (\u2264\u1d62\u2228\u2192 eq) = (bind -\u1d62 sind) eq\n((bind \u2190\u2202) -\u1d62 (sind \u2190\u2202)) (\u2264\u1d62\u2190\u2202 eq) = (bind -\u1d62 sind) eq\n((\u2202\u2192 bind) -\u1d62 (\u2202\u2192 sind)) (\u2264\u1d62\u2202\u2192 eq) = (bind -\u1d62 sind) eq\n\n\nreplLL-\u2193 : \u2200{i u pll ell ll} \u2192 \u2200(ind : IndexLL {i} {u} pll ll)\n           \u2192 replLL pll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell \u2261 ell\nreplLL-\u2193 \u2193 = refl\nreplLL-\u2193 (ind \u2190\u2227) = replLL-\u2193 ind\nreplLL-\u2193 (\u2227\u2192 ind) = replLL-\u2193 ind\nreplLL-\u2193 (ind \u2190\u2228) = replLL-\u2193 ind\nreplLL-\u2193 (\u2228\u2192 ind) = replLL-\u2193 ind\nreplLL-\u2193 (ind \u2190\u2202) = replLL-\u2193 ind\nreplLL-\u2193 (\u2202\u2192 ind) = replLL-\u2193 ind\n\n\nind-rpl\u2193 : \u2200{i u ll pll cll ell} \u2192 (ind : IndexLL {i} {u} pll ll)\n        \u2192 IndexLL cll (replLL pll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell) \u2192 IndexLL cll ell\nind-rpl\u2193 {_} {_} {_} {pll} {cll} {ell} ind x\n  =  subst (\u03bb x \u2192 x) (cong (\u03bb x \u2192 IndexLL cll x) (replLL-\u2193 {ell = ell} ind)) x \n\n\n\na\u2264\u1d62b-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n             \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll \u2192 (lt : emi \u2264\u1d62 ind)\n             \u2192 IndexLL (replLL fll ((ind -\u1d62 emi) lt) frll) (replLL ll ind frll) \na\u2264\u1d62b-morph \u2193 ind frll \u2264\u1d62\u2193 = \u2193\na\u2264\u1d62b-morph (emi \u2190\u2227) (ind \u2190\u2227) frll (\u2264\u1d62\u2190\u2227 lt) = a\u2264\u1d62b-morph emi ind frll lt \u2190\u2227\na\u2264\u1d62b-morph (\u2227\u2192 emi) (\u2227\u2192 ind) frll (\u2264\u1d62\u2227\u2192 lt) = \u2227\u2192 a\u2264\u1d62b-morph emi ind frll lt\na\u2264\u1d62b-morph (emi \u2190\u2228) (ind \u2190\u2228) frll (\u2264\u1d62\u2190\u2228 lt) = a\u2264\u1d62b-morph emi ind frll lt \u2190\u2228\na\u2264\u1d62b-morph (\u2228\u2192 emi) (\u2228\u2192 ind) frll (\u2264\u1d62\u2228\u2192 lt) = \u2228\u2192 a\u2264\u1d62b-morph emi ind frll lt\na\u2264\u1d62b-morph (emi \u2190\u2202) (ind \u2190\u2202) frll (\u2264\u1d62\u2190\u2202 lt) = a\u2264\u1d62b-morph emi ind frll lt \u2190\u2202\na\u2264\u1d62b-morph (\u2202\u2192 emi) (\u2202\u2192 ind) frll (\u2264\u1d62\u2202\u2192 lt) = \u2202\u2192 a\u2264\u1d62b-morph emi ind frll lt\n\n\n\na\u2264\u1d62b-morph-id : \u2200{i u ll rll}\n             \u2192 (ind : IndexLL {i} {u} rll ll)\n             \u2192 subst\u2082 (\u03bb x y \u2192 IndexLL x y) (replLL-\u2193 {ell = rll} ind) (replLL-id ll ind rll refl) (a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind)) \u2261 ind\na\u2264\u1d62b-morph-id {i} {u} {ll} {.ll} \u2193 = refl\na\u2264\u1d62b-morph-id {i} {u} {(li \u2227 _)} {rll} (ind \u2190\u2227) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL li ind rll | replLL-id li ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {li \u2227 _} {.h} (ind \u2190\u2227) | h | refl | .li | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(_ \u2227 ri)} {rll} (\u2227\u2192 ind) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ri ind rll | replLL-id ri ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {_ \u2227 ri} {.h} (\u2227\u2192 ind) | h | refl | .ri | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(li \u2228 _)} {rll} (ind \u2190\u2228) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL li ind rll | replLL-id li ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {li \u2228 _} {.h} (ind \u2190\u2228) | h | refl | .li | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(_ \u2228 ri)} {rll} (\u2228\u2192 ind) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ri ind rll | replLL-id ri ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {_ \u2228 ri} {.h} (\u2228\u2192 ind) | h | refl | .ri | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(li \u2202 _)} {rll} (ind \u2190\u2202) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL li ind rll | replLL-id li ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {li \u2202 _} {.h} (ind \u2190\u2202) | h | refl | .li | refl | .ind | refl = refl\na\u2264\u1d62b-morph-id {i} {u} {(_ \u2202 ri)} {rll} (\u2202\u2192 ind) with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ri ind rll | replLL-id ri ind rll refl | a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind) | a\u2264\u1d62b-morph-id ind\na\u2264\u1d62b-morph-id {i} {u} {_ \u2202 ri} {.h} (\u2202\u2192 ind) | h | refl | .ri | refl | .ind | refl = refl\n\n\nmodule _ where\n\n  open Relation.Binary.HeterogeneousEquality\n\n  a\u2264\u1d62b-morph-hid : \u2200{i u ll rll}\n               \u2192 (ind : IndexLL {i} {u} rll ll)\n               \u2192 (a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind)) \u2245 ind\n  a\u2264\u1d62b-morph-hid {ll = ll} {rll = rll} ind with replLL rll ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll | replLL-\u2193 {ell = rll} ind | replLL ll ind rll | replLL-id ll ind rll refl | (a\u2264\u1d62b-morph ind ind rll (\u2264\u1d62-reflexive ind)) | a\u2264\u1d62b-morph-id ind\n  a\u2264\u1d62b-morph-hid {_} {_} {.r} {.q} .y | q | refl | r | refl | y | refl = refl\n\n\n\nreplLL-a\u2264b\u2261a : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 gll\n               \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll \u2192 (lt : emi \u2264\u1d62 ind)\n               \u2192 replLL (replLL ll ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll \u2261 replLL ll emi gll\nreplLL-a\u2264b\u2261a \u2193 gll ind frll \u2264\u1d62\u2193 = refl\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (emi \u2190\u2227) gll (ind \u2190\u2227) frll (\u2264\u1d62\u2190\u2227 lt)\n  with (replLL (replLL li ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (emi \u2190\u2227) gll (ind \u2190\u2227) frll (\u2264\u1d62\u2190\u2227 lt) | .(replLL li emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (\u2227\u2192 emi) gll (\u2227\u2192 ind) frll (\u2264\u1d62\u2227\u2192 lt)\n  with (replLL (replLL ri ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2227 ri} (\u2227\u2192 emi) gll (\u2227\u2192 ind) frll (\u2264\u1d62\u2227\u2192 lt) | .(replLL ri emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (emi \u2190\u2228) gll (ind \u2190\u2228) frll (\u2264\u1d62\u2190\u2228 lt)\n  with (replLL (replLL li ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (emi \u2190\u2228) gll (ind \u2190\u2228) frll (\u2264\u1d62\u2190\u2228 lt) | .(replLL li emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (\u2228\u2192 emi) gll (\u2228\u2192 ind) frll (\u2264\u1d62\u2228\u2192 lt)\n  with (replLL (replLL ri ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2228 ri} (\u2228\u2192 emi) gll (\u2228\u2192 ind) frll (\u2264\u1d62\u2228\u2192 lt) | .(replLL ri emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (emi \u2190\u2202) gll (ind \u2190\u2202) frll (\u2264\u1d62\u2190\u2202 lt)\n  with (replLL (replLL li ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (emi \u2190\u2202) gll (ind \u2190\u2202) frll (\u2264\u1d62\u2190\u2202 lt) | .(replLL li emi gll) | refl = refl\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (\u2202\u2192 emi) gll (\u2202\u2192 ind) frll (\u2264\u1d62\u2202\u2192 lt)\n  with (replLL (replLL ri ind frll) (a\u2264\u1d62b-morph emi ind frll lt) gll)\n       | (replLL-a\u2264b\u2261a emi gll ind frll lt)\nreplLL-a\u2264b\u2261a {ll = li \u2202 ri} (\u2202\u2192 emi) gll (\u2202\u2192 ind) frll (\u2264\u1d62\u2202\u2192 lt) | .(replLL ri emi gll) | refl = refl\n\n\n\u00acord-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n             \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll \u2192 .(nord : \u00ac Ordered\u1d62 ind emi)\n             \u2192 IndexLL fll (replLL ll ind frll)\n\u00acord-morph \u2193 ind frll nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n\u00acord-morph (emi \u2190\u2227) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193)) \n\u00acord-morph (emi \u2190\u2227) (ind \u2190\u2227) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 lt))\n                }))\n... | r = r \u2190\u2227\n\u00acord-morph (emi \u2190\u2227) (\u2227\u2192 ind) frll nord = emi \u2190\u2227\n\u00acord-morph (\u2227\u2192 emi) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph (\u2227\u2192 emi) (ind \u2190\u2227) frll nord = \u2227\u2192 emi\n\u00acord-morph (\u2227\u2192 emi) (\u2227\u2192 ind) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 lt))\n                }))\n... | r = \u2227\u2192 r\n\u00acord-morph (emi \u2190\u2228) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193)) \n\u00acord-morph (emi \u2190\u2228) (ind \u2190\u2228) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 lt))\n                }))\n... | r = r \u2190\u2228\n\u00acord-morph (emi \u2190\u2228) (\u2228\u2192 ind) frll nord = emi \u2190\u2228\n\u00acord-morph (\u2228\u2192 emi) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph (\u2228\u2192 emi) (ind \u2190\u2228) frll nord = \u2228\u2192 emi\n\u00acord-morph (\u2228\u2192 emi) (\u2228\u2192 ind) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 lt))\n                }))\n... | r = \u2228\u2192 r\n\u00acord-morph (emi \u2190\u2202) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193) )\n\u00acord-morph (emi \u2190\u2202) (ind \u2190\u2202) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 lt))\n                }))\n... | r = r \u2190\u2202\n\u00acord-morph (emi \u2190\u2202) (\u2202\u2192 ind) frll nord = emi \u2190\u2202\n\u00acord-morph (\u2202\u2192 emi) \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph (\u2202\u2192 emi) (ind \u2190\u2202) frll nord = \u2202\u2192 emi\n\u00acord-morph (\u2202\u2192 emi) (\u2202\u2192 ind) frll nord\n           with (\u00acord-morph emi ind frll\n             (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 lt))\n                ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 lt))\n                }))\n... | r = \u2202\u2192 r\n\n\nreplLL-\u00acordab\u2261ba : \u2200{i u rll ll fll}\n  \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 gll\n  \u2192 (ind : IndexLL rll ll) \u2192 \u2200 frll\n  \u2192 .(nord : \u00ac Ordered\u1d62 ind emi)\n  \u2192 replLL (replLL ll ind frll) (\u00acord-morph emi ind frll nord) gll \u2261 replLL (replLL ll emi gll) (\u00acord-morph ind emi gll (flipNotOrd\u1d62 nord)) frll\nreplLL-\u00acordab\u2261ba \u2193 gll ind frll nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba (emi \u2190\u2227) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2227 ri} (emi \u2190\u2227) gll (ind \u2190\u2227) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x))})\n... | r = cong (\u03bb x \u2192 x \u2227 ri) r\nreplLL-\u00acordab\u2261ba (emi \u2190\u2227) gll (\u2227\u2192 ind) frll nord = refl\nreplLL-\u00acordab\u2261ba (\u2227\u2192 emi) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2227 ri} (\u2227\u2192 emi) gll (\u2227\u2192 ind) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x))})\n... | r = cong (\u03bb x \u2192 li \u2227 x) r\nreplLL-\u00acordab\u2261ba (\u2227\u2192 emi) gll (ind \u2190\u2227) frll nord = refl\nreplLL-\u00acordab\u2261ba (emi \u2190\u2228) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2228 ri} (emi \u2190\u2228) gll (ind \u2190\u2228) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x))})\n... | r = cong (\u03bb x \u2192 x \u2228 ri) r\nreplLL-\u00acordab\u2261ba (emi \u2190\u2228) gll (\u2228\u2192 ind) frll nord = refl\nreplLL-\u00acordab\u2261ba (\u2228\u2192 emi) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2228 ri} (\u2228\u2192 emi) gll (\u2228\u2192 ind) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x))})\n... | r = cong (\u03bb x \u2192 li \u2228 x) r\nreplLL-\u00acordab\u2261ba (\u2228\u2192 emi) gll (ind \u2190\u2228) frll nord = refl\nreplLL-\u00acordab\u2261ba (emi \u2190\u2202) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2202 ri} (emi \u2190\u2202) gll (ind \u2190\u2202) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x))})\n... | r = cong (\u03bb x \u2192 x \u2202 ri) r\nreplLL-\u00acordab\u2261ba (emi \u2190\u2202) gll (\u2202\u2192 ind) frll nord = refl\nreplLL-\u00acordab\u2261ba (\u2202\u2192 emi) gll \u2193 frll nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nreplLL-\u00acordab\u2261ba {ll = li \u2202 ri} (\u2202\u2192 emi) gll (\u2202\u2192 ind) frll nord\n  with replLL-\u00acordab\u2261ba emi gll ind frll hf where\n    .hf : (\u00ac Ordered\u1d62 ind emi)\n    hf = (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x))\n            ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x))})\n... | r = cong (\u03bb x \u2192 li \u2202 x) r\nreplLL-\u00acordab\u2261ba (\u2202\u2192 emi) gll (ind \u2190\u2202) frll nord = refl\n \n\nlemma\u2081-\u00acord-a\u2264\u1d62b : \u2200{i u ll pll rll fll}\n      \u2192 (emi : IndexLL {i} {u} fll ll)\n      \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n      \u2192 (omi : IndexLL pll (replLL ll ind ell))\n      \u2192 .(nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) omi)\n      \u2192 IndexLL pll ll\nlemma\u2081-\u00acord-a\u2264\u1d62b \u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (omi \u2190\u2227) nord\n     = (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n                                  ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)) })) \u2190\u2227\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 omi) nord = \u2227\u2192 omi\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (omi \u2190\u2227) nord = omi \u2190\u2227\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 omi) nord\n     = \u2227\u2192 (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x))\n                                     ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)) })) \nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (omi \u2190\u2228) nord\n     = (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x))\n                                  ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)) })) \u2190\u2228\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 omi) nord = \u2228\u2192 omi\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (omi \u2190\u2228) nord = omi \u2190\u2228\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 omi) nord\n     = \u2228\u2192 (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x))\n                                     ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)) })) \nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (omi \u2190\u2202) nord\n     = (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x))\n                                  ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)) })) \u2190\u2202\nlemma\u2081-\u00acord-a\u2264\u1d62b (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 omi) nord = \u2202\u2192 omi\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (omi \u2190\u2202) nord = omi \u2190\u2202\nlemma\u2081-\u00acord-a\u2264\u1d62b (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 omi) nord\n     = \u2202\u2192 (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x))\n                                     ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)) })) \n\n\n\n\nrlemma\u2081\u21d2\u00acord : \u2200{i u ll pll rll fll}\n      \u2192 (emi : IndexLL {i} {u} fll ll)\n      \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n      \u2192 (omi : IndexLL pll (replLL ll ind ell))\n      \u2192 .(nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) omi)\n      \u2192 \u00ac Ordered\u1d62 emi (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi nord)\nrlemma\u2081\u21d2\u00acord \u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (omi \u2190\u2227) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (emi \u2190\u2227) (ind \u2190\u2227) ell (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 omi) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll li)\n       \u2192 (omi : IndexLL pll ri)\n       \u2192 \u00ac Ordered\u1d62 (emi \u2190\u2227) (\u2227\u2192 omi)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 omi) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (\u2227\u2192 emi) (\u2227\u2192 ind) ell (\u2264\u1d62\u2227\u2192 lt) (omi \u2190\u2227) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll ri)\n       \u2192 (omi : IndexLL pll li)\n       \u2192 \u00ac Ordered\u1d62 (\u2227\u2192 emi) (omi \u2190\u2227)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (omi \u2190\u2228) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (emi \u2190\u2228) (ind \u2190\u2228) ell (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 omi) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll li)\n       \u2192 (omi : IndexLL pll ri)\n       \u2192 \u00ac Ordered\u1d62 (emi \u2190\u2228) (\u2228\u2192 omi)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 omi) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (\u2228\u2192 emi) (\u2228\u2192 ind) ell (\u2264\u1d62\u2228\u2192 lt) (omi \u2190\u2228) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll ri)\n       \u2192 (omi : IndexLL pll li)\n       \u2192 \u00ac Ordered\u1d62 (\u2228\u2192 emi) (omi \u2190\u2228)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (omi \u2190\u2202) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (emi \u2190\u2202) (ind \u2190\u2202) ell (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 omi) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll li)\n       \u2192 (omi : IndexLL pll ri)\n       \u2192 \u00ac Ordered\u1d62 (emi \u2190\u2202) (\u2202\u2192 omi)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\nrlemma\u2081\u21d2\u00acord (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\nrlemma\u2081\u21d2\u00acord (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 omi) nord\n  = \u03bb { (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x)) \u2192 n (a\u2264\u1d62b x)\n      ; (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)) \u2192 n (b\u2264\u1d62a x) } where\n  n = (rlemma\u2081\u21d2\u00acord emi ind ell lt omi (\u03bb { (a\u2264\u1d62b x) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 x))\n                                          ; (b\u2264\u1d62a x) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 x)) }))\nrlemma\u2081\u21d2\u00acord {i} {u} {pll = pll} {fll = fll} (\u2202\u2192 emi) (\u2202\u2192 ind) ell (\u2264\u1d62\u2202\u2192 lt) (omi \u2190\u2202) nord = hf emi omi  where\n  hf : \u2200{li ri} \u2192 (emi : IndexLL {i} {u} fll ri)\n       \u2192 (omi : IndexLL pll li)\n       \u2192 \u00ac Ordered\u1d62 (\u2202\u2192 emi) (omi \u2190\u2202)\n  hf emi omi (a\u2264\u1d62b ())\n  hf emi omi (b\u2264\u1d62a ())\n\n\n\u00acord-morph$lemma\u2081\u2261I : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 lind (a\u2264\u1d62b-morph emi ind ell lt))\n       \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind (flipNotOrd\u1d62 nord)) ind ell (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind (flipNotOrd\u1d62 nord))) \u2261 lind)\n\u00acord-morph$lemma\u2081\u2261I ell \u2193 ind \u2264\u1d62\u2193 lind nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (lind \u2190\u2227) nord = cong (\u03bb x \u2192 x \u2190\u2227) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2227 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2227 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 lind) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (lind \u2190\u2227) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 lind) nord = cong (\u03bb x \u2192 \u2227\u2192 x) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2227\u2192 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2227\u2192 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (lind \u2190\u2228) nord = cong (\u03bb x \u2192 x \u2190\u2228) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2228 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2228 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 lind) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (lind \u2190\u2228) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 lind) nord = cong (\u03bb x \u2192 \u2228\u2192 x) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2228\u2192 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2228\u2192 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (lind \u2190\u2202) nord = cong (\u03bb x \u2192 x \u2190\u2202) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2190\u2202 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2190\u2202 lt)) }))\n\u00acord-morph$lemma\u2081\u2261I ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 lind) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n\u00acord-morph$lemma\u2081\u2261I ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (lind \u2190\u2202) nord = refl\n\u00acord-morph$lemma\u2081\u2261I ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 lind) nord = cong (\u03bb x \u2192 \u2202\u2192 x) r where\n  r = (\u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind (\u03bb { (a\u2264\u1d62b lt) \u2192 nord (a\u2264\u1d62b (\u2264\u1d62\u2202\u2192 lt))\n                                   ; (b\u2264\u1d62a lt) \u2192 nord (b\u2264\u1d62a (\u2264\u1d62\u2202\u2192 lt)) }))\n\n\n\n_+\u1d62_ : \u2200{i u pll cll ll} \u2192 IndexLL {i} {u} pll ll \u2192 IndexLL cll pll \u2192 IndexLL cll ll\n_+\u1d62_ \u2193 is = is\n_+\u1d62_ (if \u2190\u2227) is = (_+\u1d62_ if is) \u2190\u2227\n_+\u1d62_ (\u2227\u2192 if) is = \u2227\u2192 (_+\u1d62_ if is)\n_+\u1d62_ (if \u2190\u2228) is = (_+\u1d62_ if is) \u2190\u2228\n_+\u1d62_ (\u2228\u2192 if) is = \u2228\u2192 (_+\u1d62_ if is)\n_+\u1d62_ (if \u2190\u2202) is = (_+\u1d62_ if is) \u2190\u2202\n_+\u1d62_ (\u2202\u2192 if) is = \u2202\u2192 (_+\u1d62_ if is)\n\n\n\n\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 : \u2200{i u pll cll ll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n           \u2192 ind \u2264\u1d62 (ind +\u1d62 lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 \u2193 lind = \u2264\u1d62\u2193\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2227) lind = \u2264\u1d62\u2190\u2227 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2227\u2192 ind) lind = \u2264\u1d62\u2227\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2228) lind = \u2264\u1d62\u2190\u2228 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2228\u2192 ind) lind = \u2264\u1d62\u2228\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2202) lind = \u2264\u1d62\u2190\u2202 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n+\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2202\u2192 ind) lind = \u2264\u1d62\u2202\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n\n\na+\u2193\u2261a : \u2200{i u pll ll} \u2192 (a : IndexLL {i} {u} pll ll) \u2192 a +\u1d62 \u2193 \u2261 a\na+\u2193\u2261a \u2193 = refl\na+\u2193\u2261a (a \u2190\u2227) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (a \u2190\u2227) | .a | refl = refl\na+\u2193\u2261a (\u2227\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (\u2227\u2192 a) | .a | refl = refl\na+\u2193\u2261a (a \u2190\u2228) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (a \u2190\u2228) | .a | refl = refl\na+\u2193\u2261a (\u2228\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (\u2228\u2192 a) | .a | refl = refl\na+\u2193\u2261a (a \u2190\u2202) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (a \u2190\u2202) | .a | refl = refl\na+\u2193\u2261a (\u2202\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\na+\u2193\u2261a (\u2202\u2192 a) | .a | refl = refl\n\n\n[a+b]-a=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n          \u2192 (b : IndexLL rll fll)\n          \u2192 ((a +\u1d62 b) -\u1d62 a) (+\u1d62\u21d2l\u2264\u1d62+\u1d62 a b) \u2261 b\n[a+b]-a=b \u2193 b = refl\n[a+b]-a=b (a \u2190\u2227) b = [a+b]-a=b a b\n[a+b]-a=b (\u2227\u2192 a) b = [a+b]-a=b a b\n[a+b]-a=b (a \u2190\u2228) b = [a+b]-a=b a b\n[a+b]-a=b (\u2228\u2192 a) b = [a+b]-a=b a b\n[a+b]-a=b (a \u2190\u2202) b = [a+b]-a=b a b\n[a+b]-a=b (\u2202\u2192 a) b = [a+b]-a=b a b\n\na+[b-a]=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n            \u2192 (b : IndexLL rll ll)\n            \u2192 (lt : a \u2264\u1d62 b)\n            \u2192 (a +\u1d62 (b -\u1d62 a) lt) \u2261 b\na+[b-a]=b \u2193 b \u2264\u1d62\u2193 = refl\na+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) | .b | refl = refl\na+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) | .b | refl = refl\na+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) | .b | refl = refl\na+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt)with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt) | .b | refl = refl\na+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) | .b | refl = refl\na+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\na+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) | .b | refl = refl\n\n\n\n-- A predicate that is true if pll is not transformed by ltr.\n\ndata UpTran {i u} : \u2200 {ll pll rll} \u2192 IndexLL pll ll \u2192 LLTr {i} {u} rll ll \u2192 Set u where\n  indI : \u2200{pll ll} \u2192 {ind : IndexLL pll ll} \u2192 UpTran ind I\n  \u2190\u2202\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (\u2202\u2192 ind) ltr\n         \u2192 UpTran (ind \u2190\u2202) (\u2202c ltr)\n  \u2202\u2192\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (ind \u2190\u2202) ltr\n         \u2192 UpTran (\u2202\u2192 ind) (\u2202c ltr)\n  \u2190\u2228\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (\u2228\u2192 ind) ltr\n         \u2192 UpTran (ind \u2190\u2228) (\u2228c ltr)\n  \u2228\u2192\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (ind \u2190\u2228) ltr\n         \u2192 UpTran (\u2228\u2192 ind) (\u2228c ltr)\n  \u2190\u2227\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (\u2227\u2192 ind) ltr\n         \u2192 UpTran (ind \u2190\u2227) (\u2227c ltr)\n  \u2227\u2192\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (ind \u2190\u2227) ltr\n         \u2192 UpTran (\u2227\u2192 ind) (\u2227c ltr)\n  \u2190\u2227]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n            \u2192 UpTran {rll = rll} (ind \u2190\u2227) ltr \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)\n  \u2227\u2192]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (ind \u2190\u2227)) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr)\n  \u2227\u2192[\u2190\u2227\u2227\u2227d : \u2200{pll lli lri rri rli rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2227))) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 (rli \u2227 rri)} (\u2227\u2192 (ind \u2190\u2227)) (\u2227\u2227d ltr)\n  \u2227\u2192[\u2227\u2192\u2227\u2227d : \u2200{pll lli lri rri rli rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 (\u2227\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 (rli \u2227 rri)} (\u2227\u2192 (\u2227\u2192 ind)) (\u2227\u2227d ltr)\n  \u2227\u2192[\u2190\u2228\u2227\u2227d : \u2200{pll lli lri rri rli rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2228))) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 (rli \u2228 rri)} (\u2227\u2192 (ind \u2190\u2228)) (\u2227\u2227d ltr)\n  \u2227\u2192[\u2228\u2192\u2227\u2227d : \u2200{pll lli lri rri rli rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 (\u2228\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 (rli \u2228 rri)} (\u2227\u2192 (\u2228\u2192 ind)) (\u2227\u2227d ltr)\n  \u2227\u2192[\u2190\u2202\u2227\u2227d : \u2200{pll lli lri rri rli rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2202))) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 (rli \u2202 rri)} (\u2227\u2192 (ind \u2190\u2202)) (\u2227\u2227d ltr)\n  \u2227\u2192[\u2202\u2192\u2227\u2227d : \u2200{pll lli lri rri rli rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 (\u2202\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2227 lri) \u2227 (rli \u2202 rri)} (\u2227\u2192 (\u2202\u2192 ind)) (\u2227\u2227d ltr)\n  \u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n            \u2192 UpTran {rll = rll} ((ind \u2190\u2227) \u2190\u2227) ltr\n            \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr)\n  \u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} ((\u2227\u2192 ind) \u2190\u2227) ltr\n            \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr)\n  \u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2227\u2192 ind) ltr\n            \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr)\n  \u2190\u2228]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n            \u2192 UpTran {rll = rll} (ind \u2190\u2228) ltr \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)\n  \u2228\u2192]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (ind \u2190\u2228)) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr)\n  \u2228\u2192[\u2190\u2227\u2228\u2228d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2227))) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 (rli \u2227 rri)} (\u2228\u2192 (ind \u2190\u2227)) (\u2228\u2228d ltr)\n  \u2228\u2192[\u2227\u2192\u2228\u2228d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 (\u2227\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 (rli \u2227 rri)} (\u2228\u2192 (\u2227\u2192 ind)) (\u2228\u2228d ltr)\n  \u2228\u2192[\u2190\u2228\u2228\u2228d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2228))) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 (rli \u2228 rri)} (\u2228\u2192 (ind \u2190\u2228)) (\u2228\u2228d ltr)\n  \u2228\u2192[\u2228\u2192\u2228\u2228d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 (\u2228\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 (rli \u2228 rri)} (\u2228\u2192 (\u2228\u2192 ind)) (\u2228\u2228d ltr)\n  \u2228\u2192[\u2190\u2202\u2228\u2228d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2202))) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 (rli \u2202 rri)} (\u2228\u2192 (ind \u2190\u2202)) (\u2228\u2228d ltr)\n  \u2228\u2192[\u2202\u2192\u2228\u2228d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 (\u2202\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2228 lri) \u2228 (rli \u2202 rri)} (\u2228\u2192 (\u2202\u2192 ind)) (\u2228\u2228d ltr)\n  \u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n            \u2192 UpTran {rll = rll} ((ind \u2190\u2228) \u2190\u2228) ltr\n            \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr)\n  \u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} ((\u2228\u2192 ind) \u2190\u2228) ltr\n            \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr)\n  \u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2228\u2192 ind) ltr\n            \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr)\n  \u2190\u2202]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n            \u2192 UpTran {rll = rll} (ind \u2190\u2202) ltr \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)\n  \u2202\u2192]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (ind \u2190\u2202)) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr)\n  \u2202\u2192[\u2190\u2227\u2202\u2202d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2227))) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 (rli \u2227 rri)} (\u2202\u2192 (ind \u2190\u2227)) (\u2202\u2202d ltr)\n  \u2202\u2192[\u2227\u2192\u2202\u2202d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 (\u2227\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 (rli \u2227 rri)} (\u2202\u2192 (\u2227\u2192 ind)) (\u2202\u2202d ltr)\n  \u2202\u2192[\u2190\u2228\u2202\u2202d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2228))) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 (rli \u2228 rri)} (\u2202\u2192 (ind \u2190\u2228)) (\u2202\u2202d ltr)\n  \u2202\u2192[\u2228\u2192\u2202\u2202d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 (\u2228\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 (rli \u2228 rri)} (\u2202\u2192 (\u2228\u2192 ind)) (\u2202\u2202d ltr)\n  \u2202\u2192[\u2190\u2202\u2202\u2202d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2202))) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 (rli \u2202 rri)} (\u2202\u2192 (ind \u2190\u2202)) (\u2202\u2202d ltr)\n  \u2202\u2192[\u2202\u2192\u2202\u2202d : \u2200{pll lli lri rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 (\u2202\u2192 ind))) ltr\n            \u2192 UpTran {ll = (lli \u2202 lri) \u2202 (rli \u2202 rri)} (\u2202\u2192 (\u2202\u2192 ind)) (\u2202\u2202d ltr)\n  \u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n            \u2192 UpTran {rll = rll} ((ind \u2190\u2202) \u2190\u2202) ltr\n            \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr)\n  \u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n            \u2192 UpTran {rll = rll} ((\u2202\u2192 ind) \u2190\u2202) ltr\n            \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr)\n  \u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n            \u2192 UpTran {rll = rll} (\u2202\u2192 ind) ltr\n            \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr)\n\nisUpTran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 Dec (UpTran ind ltr)\nisUpTran ind I = yes indI\nisUpTran \u2193 (\u2202c ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2202) (\u2202c ltr) with (isUpTran (\u2202\u2192 ind) ltr)\nisUpTran (ind \u2190\u2202) (\u2202c ltr) | yes ut = yes (\u2190\u2202\u2202c ut)\nisUpTran (ind \u2190\u2202) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2190\u2202\u2202c ut) \u2192 \u00acut ut})\nisUpTran (\u2202\u2192 ind) (\u2202c ltr)  with (isUpTran (ind \u2190\u2202) ltr)\nisUpTran (\u2202\u2192 ind) (\u2202c ltr) | yes ut = yes (\u2202\u2192\u2202c ut)\nisUpTran (\u2202\u2192 ind) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2202\u2192\u2202c ut) \u2192 \u00acut ut})\nisUpTran \u2193 (\u2228c ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2228) (\u2228c ltr) with (isUpTran (\u2228\u2192 ind) ltr)\nisUpTran (ind \u2190\u2228) (\u2228c ltr) | yes p = yes (\u2190\u2228\u2228c p)\nisUpTran (ind \u2190\u2228) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u2228c p) \u2192 \u00acp p}) \nisUpTran (\u2228\u2192 ind) (\u2228c ltr) with (isUpTran (ind \u2190\u2228) ltr)\nisUpTran (\u2228\u2192 ind) (\u2228c ltr) | yes p = yes (\u2228\u2192\u2228c p)\nisUpTran (\u2228\u2192 ind) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228c ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2227c ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2227) (\u2227c ltr) with (isUpTran (\u2227\u2192 ind) ltr)\nisUpTran (ind \u2190\u2227) (\u2227c ltr) | yes p = yes (\u2190\u2227\u2227c p)\nisUpTran (ind \u2190\u2227) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u2227c ut) \u2192 \u00acp ut}) \nisUpTran (\u2227\u2192 ind) (\u2227c ltr) with (isUpTran (ind \u2190\u2227) ltr)\nisUpTran (\u2227\u2192 ind) (\u2227c ltr) | yes p = yes (\u2227\u2192\u2227c p)\nisUpTran (\u2227\u2192 ind) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227c ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (ind \u2190\u2227) ltr)\nisUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2190\u2227]\u2190\u2227\u2227\u2227d p)\nisUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \nisUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (ind \u2190\u2227)) ltr)\nisUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192]\u2190\u2227\u2227\u2227d p)\nisUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \nisUpTran (\u2227\u2192 \u2193) (\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2227))) ltr)\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2227\u2227\u2227d p)\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u2227\u2227d ltr) | no \u00acp =  no (\u03bb {(\u2227\u2192[\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 (\u2227\u2192 ind))) ltr)\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2227\u2192\u2227\u2227d p)\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u2227\u2227d ltr) | no \u00acp =  no (\u03bb {(\u2227\u2192[\u2227\u2192\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (ind \u2190\u2228)) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2228))) ltr)\nisUpTran (\u2227\u2192 (ind \u2190\u2228)) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2228\u2227\u2227d p)\nisUpTran (\u2227\u2192 (ind \u2190\u2228)) (\u2227\u2227d ltr) | no \u00acp =  no (\u03bb {(\u2227\u2192[\u2190\u2228\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (\u2228\u2192 ind)) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 (\u2228\u2192 ind))) ltr)\nisUpTran (\u2227\u2192 (\u2228\u2192 ind)) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2228\u2192\u2227\u2227d p)\nisUpTran (\u2227\u2192 (\u2228\u2192 ind)) (\u2227\u2227d ltr) | no \u00acp =  no (\u03bb {(\u2227\u2192[\u2228\u2192\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (ind \u2190\u2202)) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2202))) ltr)\nisUpTran (\u2227\u2192 (ind \u2190\u2202)) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2202\u2227\u2227d p)\nisUpTran (\u2227\u2192 (ind \u2190\u2202)) (\u2227\u2227d ltr) | no \u00acp =  no (\u03bb {(\u2227\u2192[\u2190\u2202\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (\u2202\u2192 ind)) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 (\u2202\u2192 ind))) ltr)\nisUpTran (\u2227\u2192 (\u2202\u2192 ind)) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2202\u2192\u2227\u2227d p)\nisUpTran (\u2227\u2192 (\u2202\u2192 ind)) (\u2227\u2227d ltr) | no \u00acp =  no (\u03bb {(\u2227\u2192[\u2202\u2192\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) with (isUpTran ((ind \u2190\u2227) \u2190\u2227) ltr)\nisUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2190\u2227\u00ac\u2227\u2227d p)\nisUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) with (isUpTran ((\u2227\u2192 ind) \u2190\u2227) ltr)\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d p)\nisUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 ind) ltr)\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d p)\nisUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (ind \u2190\u2228) ltr)\nisUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2190\u2228]\u2190\u2228\u2228\u2228d p)\nisUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \nisUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (ind \u2190\u2228)) ltr)\nisUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192]\u2190\u2228\u2228\u2228d p)\nisUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \nisUpTran (\u2228\u2192 \u2193) (\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (\u2228\u2192 (ind \u2190\u2227)) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2227))) ltr)\nisUpTran (\u2228\u2192 (ind \u2190\u2227)) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2227\u2228\u2228d p)\nisUpTran (\u2228\u2192 (ind \u2190\u2227)) (\u2228\u2228d ltr) | no \u00acp =  no (\u03bb {(\u2228\u2192[\u2190\u2227\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (\u2227\u2192 ind)) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 (\u2227\u2192 ind))) ltr)\nisUpTran (\u2228\u2192 (\u2227\u2192 ind)) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2227\u2192\u2228\u2228d p)\nisUpTran (\u2228\u2192 (\u2227\u2192 ind)) (\u2228\u2228d ltr) | no \u00acp =  no (\u03bb {(\u2228\u2192[\u2227\u2192\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2228))) ltr)\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2228\u2228\u2228d p)\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u2228\u2228d ltr) | no \u00acp =  no (\u03bb {(\u2228\u2192[\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 (\u2228\u2192 ind))) ltr)\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2228\u2192\u2228\u2228d p)\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u2228\u2228d ltr) | no \u00acp =  no (\u03bb {(\u2228\u2192[\u2228\u2192\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (ind \u2190\u2202)) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2202))) ltr)\nisUpTran (\u2228\u2192 (ind \u2190\u2202)) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2202\u2228\u2228d p)\nisUpTran (\u2228\u2192 (ind \u2190\u2202)) (\u2228\u2228d ltr) | no \u00acp =  no (\u03bb {(\u2228\u2192[\u2190\u2202\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (\u2202\u2192 ind)) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 (\u2202\u2192 ind))) ltr)\nisUpTran (\u2228\u2192 (\u2202\u2192 ind)) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2202\u2192\u2228\u2228d p)\nisUpTran (\u2228\u2192 (\u2202\u2192 ind)) (\u2228\u2228d ltr) | no \u00acp =  no (\u03bb {(\u2228\u2192[\u2202\u2192\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) with (isUpTran ((ind \u2190\u2228) \u2190\u2228) ltr)\nisUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2190\u2228\u00ac\u2228\u2228d p)\nisUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) with (isUpTran ((\u2228\u2192 ind) \u2190\u2228) ltr)\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d p)\nisUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 ind) ltr)\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d p)\nisUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (ind \u2190\u2202) ltr)\nisUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2190\u2202]\u2190\u2202\u2202\u2202d p)\nisUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (ind \u2190\u2202)) ltr)\nisUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192]\u2190\u2202\u2202\u2202d p)\nisUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 \u2193) (\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (\u2202\u2192 (ind \u2190\u2227)) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2227))) ltr)\nisUpTran (\u2202\u2192 (ind \u2190\u2227)) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2227\u2202\u2202d p)\nisUpTran (\u2202\u2192 (ind \u2190\u2227)) (\u2202\u2202d ltr) | no \u00acp =  no (\u03bb {(\u2202\u2192[\u2190\u2227\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (\u2227\u2192 ind)) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 (\u2227\u2192 ind))) ltr)\nisUpTran (\u2202\u2192 (\u2227\u2192 ind)) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2227\u2192\u2202\u2202d p)\nisUpTran (\u2202\u2192 (\u2227\u2192 ind)) (\u2202\u2202d ltr) | no \u00acp =  no (\u03bb {(\u2202\u2192[\u2227\u2192\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (ind \u2190\u2228)) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2228))) ltr)\nisUpTran (\u2202\u2192 (ind \u2190\u2228)) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2228\u2202\u2202d p)\nisUpTran (\u2202\u2192 (ind \u2190\u2228)) (\u2202\u2202d ltr) | no \u00acp =  no (\u03bb {(\u2202\u2192[\u2190\u2228\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (\u2228\u2192 ind)) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 (\u2228\u2192 ind))) ltr)\nisUpTran (\u2202\u2192 (\u2228\u2192 ind)) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2228\u2192\u2202\u2202d p)\nisUpTran (\u2202\u2192 (\u2228\u2192 ind)) (\u2202\u2202d ltr) | no \u00acp =  no (\u03bb {(\u2202\u2192[\u2228\u2192\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2202))) ltr)\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2202\u2202\u2202d p)\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u2202\u2202d ltr) | no \u00acp =  no (\u03bb {(\u2202\u2192[\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 (\u2202\u2192 ind))) ltr)\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2202\u2192\u2202\u2202d p)\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u2202\u2202d ltr) | no \u00acp =  no (\u03bb {(\u2202\u2192[\u2202\u2192\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran \u2193 (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) with (isUpTran ((ind \u2190\u2202) \u2190\u2202) ltr)\nisUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2190\u2202\u00ac\u2202\u2202d p)\nisUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) with (isUpTran ((\u2202\u2192 ind) \u2190\u2202) ltr)\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d p)\nisUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 ind) ltr)\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d p)\nisUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n\n\n\ntran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 UpTran ind ltr\n       \u2192 IndexLL pll rll\ntran ind I indI = ind \ntran \u2193 (\u2202c ltr) () \ntran (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c ut) = tran (\u2202\u2192 ind) ltr ut\ntran (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c ut) =  tran (ind \u2190\u2202) ltr ut\ntran \u2193 (\u2228c ltr) () \ntran (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c ut) = tran (\u2228\u2192 ind) ltr ut\ntran (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c ut) = tran (ind \u2190\u2228) ltr ut\ntran \u2193 (\u2227c ltr) () \ntran (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c ut) = tran (\u2227\u2192 ind) ltr ut\ntran (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c ut) = tran (ind \u2190\u2227) ltr ut\ntran \u2193 (\u2227\u2227d ltr) () \ntran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) () \ntran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d ut) = tran (ind \u2190\u2227) ltr ut\ntran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d ut) = tran (\u2227\u2192 (ind \u2190\u2227)) ltr ut\ntran (\u2227\u2192 \u2193) (\u2227\u2227d ltr) ()\ntran (\u2227\u2192 (ind \u2190\u2227)) (\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2227))) ltr ut\ntran (\u2227\u2192 (\u2227\u2192 ind)) (\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 (\u2227\u2192 ind))) ltr ut\ntran (\u2227\u2192 (ind \u2190\u2228)) (\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2228\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2228))) ltr ut\ntran (\u2227\u2192 (\u2228\u2192 ind)) (\u2227\u2227d ltr) (\u2227\u2192[\u2228\u2192\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 (\u2228\u2192 ind))) ltr ut\ntran (\u2227\u2192 (ind \u2190\u2202)) (\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2202\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2202))) ltr ut\ntran (\u2227\u2192 (\u2202\u2192 ind)) (\u2227\u2227d ltr) (\u2227\u2192[\u2202\u2192\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 (\u2202\u2192 ind))) ltr ut\ntran \u2193 (\u00ac\u2227\u2227d ltr) () \ntran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((ind \u2190\u2227) \u2190\u2227) ltr ut\ntran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) () \ntran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((\u2227\u2192 ind) \u2190\u2227) ltr ut\ntran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) = tran (\u2227\u2192 ind) ltr ut\ntran \u2193 (\u2228\u2228d ltr) () \ntran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) () \ntran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d ut) = tran (ind \u2190\u2228) ltr ut\ntran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d ut) = tran (\u2228\u2192 (ind \u2190\u2228)) ltr ut\ntran (\u2228\u2192 \u2193) (\u2228\u2228d ltr) ()\ntran (\u2228\u2192 (ind \u2190\u2227)) (\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2227\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2227))) ltr ut\ntran (\u2228\u2192 (\u2227\u2192 ind)) (\u2228\u2228d ltr) (\u2228\u2192[\u2227\u2192\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 (\u2227\u2192 ind))) ltr ut\ntran (\u2228\u2192 (ind \u2190\u2228)) (\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2228))) ltr ut\ntran (\u2228\u2192 (\u2228\u2192 ind)) (\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 (\u2228\u2192 ind))) ltr ut\ntran (\u2228\u2192 (ind \u2190\u2202)) (\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2202\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2202))) ltr ut\ntran (\u2228\u2192 (\u2202\u2192 ind)) (\u2228\u2228d ltr) (\u2228\u2192[\u2202\u2192\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 (\u2202\u2192 ind))) ltr ut\ntran \u2193 (\u00ac\u2228\u2228d ltr) () \ntran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((ind \u2190\u2228) \u2190\u2228) ltr ut\ntran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) () \ntran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((\u2228\u2192 ind) \u2190\u2228) ltr ut\ntran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) = tran (\u2228\u2192 ind) ltr ut\ntran \u2193 (\u2202\u2202d ltr) () \ntran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) () \ntran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d ut) = tran (ind \u2190\u2202) ltr ut\ntran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d ut) = tran (\u2202\u2192 (ind \u2190\u2202)) ltr ut\ntran (\u2202\u2192 \u2193) (\u2202\u2202d ltr) ()\ntran (\u2202\u2192 (ind \u2190\u2227)) (\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2227\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2227))) ltr ut\ntran (\u2202\u2192 (\u2227\u2192 ind)) (\u2202\u2202d ltr) (\u2202\u2192[\u2227\u2192\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 (\u2227\u2192 ind))) ltr ut\ntran (\u2202\u2192 (ind \u2190\u2228)) (\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2228\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2228))) ltr ut\ntran (\u2202\u2192 (\u2228\u2192 ind)) (\u2202\u2202d ltr) (\u2202\u2192[\u2228\u2192\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 (\u2228\u2192 ind))) ltr ut\ntran (\u2202\u2192 (ind \u2190\u2202)) (\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2202))) ltr ut\ntran (\u2202\u2192 (\u2202\u2192 ind)) (\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 (\u2202\u2192 ind))) ltr ut\ntran \u2193 (\u00ac\u2202\u2202d ltr) () \ntran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((ind \u2190\u2202) \u2190\u2202) ltr ut\ntran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) () \ntran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((\u2202\u2192 ind) \u2190\u2202) ltr ut\ntran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) = tran (\u2202\u2192 ind) ltr ut\n\n\ntr-ltr-morph : \u2200 {i u ll pll orll} \u2192 \u2200 frll \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} orll ll) \u2192 (rvT : UpTran ind ltr) \u2192 LLTr (replLL orll (tran ind ltr rvT) frll) (replLL ll ind frll)\ntr-ltr-morph frll \u2193 .I indI = I\ntr-ltr-morph frll (ind \u2190\u2227) I indI = I\ntr-ltr-morph frll (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c rvT) with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n... | r = \u2227c r\ntr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) ltr rvT\n... | r = \u00ac\u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 ind) I indI = I\ntr-ltr-morph frll (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n... | r = \u2227c r\ntr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) (\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2227))) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) (\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 (\u2227\u2192 ind))) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2228)) (\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2228\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2228))) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (\u2228\u2192 ind)) (\u2227\u2227d ltr) (\u2227\u2192[\u2228\u2192\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 (\u2228\u2192 ind))) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2202)) (\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2202\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 (ind \u2190\u2202))) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (\u2202\u2192 ind)) (\u2227\u2227d ltr) (\u2227\u2192[\u2202\u2192\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 (\u2202\u2192 ind))) ltr rvT\n... | r = \u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) ltr rvT\n... | r = \u00ac\u2227\u2227d r\ntr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d rvT)  with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n... | r = \u00ac\u2227\u2227d r\ntr-ltr-morph frll (ind \u2190\u2228) I indI = I\ntr-ltr-morph frll (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c rvT) with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n... | r = \u2228c r\ntr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) ltr rvT\n... | r = \u00ac\u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 ind) I indI = I\ntr-ltr-morph frll (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n... | r = \u2228c r\ntr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) (\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2228))) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) (\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 (\u2228\u2192 ind))) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2227)) (\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2227\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2227))) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (\u2227\u2192 ind)) (\u2228\u2228d ltr) (\u2228\u2192[\u2227\u2192\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 (\u2227\u2192 ind))) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2202)) (\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2202\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 (ind \u2190\u2202))) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (\u2202\u2192 ind)) (\u2228\u2228d ltr) (\u2228\u2192[\u2202\u2192\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 (\u2202\u2192 ind))) ltr rvT\n... | r = \u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) ltr rvT\n... | r = \u00ac\u2228\u2228d r\ntr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d rvT)  with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n... | r = \u00ac\u2228\u2228d r\ntr-ltr-morph frll (ind \u2190\u2202) I indI = I\ntr-ltr-morph frll (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c rvT) with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n... | r = \u2202c r\ntr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) ltr rvT\n... | r = \u00ac\u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 ind) I indI = I\ntr-ltr-morph frll (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n... | r = \u2202c r\ntr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) (\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2202))) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) (\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 (\u2202\u2192 ind))) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2228)) (\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2228\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2228))) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (\u2228\u2192 ind)) (\u2202\u2202d ltr) (\u2202\u2192[\u2228\u2192\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 (\u2228\u2192 ind))) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2227)) (\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2227\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 (ind \u2190\u2227))) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (\u2227\u2192 ind)) (\u2202\u2202d ltr) (\u2202\u2192[\u2227\u2192\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 (\u2227\u2192 ind))) ltr rvT\n... | r = \u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) ltr rvT\n... | r = \u00ac\u2202\u2202d r\ntr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d rvT)  with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n... | r = \u00ac\u2202\u2202d r\n\n\n\ndrepl=>repl+ : \u2200{i u pll ll cll frll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n               \u2192 (replLL ll ind (replLL pll lind frll)) \u2261 replLL ll (ind +\u1d62 lind) frll\ndrepl=>repl+ \u2193 lind = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (ind \u2190\u2227) lind\n             with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (ind \u2190\u2227) lind\n             | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (\u2227\u2192 ind) lind\n             with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (\u2227\u2192 ind) lind\n             | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (ind \u2190\u2228) lind\n             with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (ind \u2190\u2228) lind\n             | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (\u2228\u2192 ind) lind\n             with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (\u2228\u2192 ind) lind\n             | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (ind \u2190\u2202) lind\n             with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (ind \u2190\u2202) lind\n             | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\ndrepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (\u2202\u2192 ind) lind\n             with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\ndrepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (\u2202\u2192 ind) lind\n             | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n\n\n\nrepl-r=>l : \u2200{i u pll ll cll frll} \u2192 \u2200 ell \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (x : IndexLL cll (replLL ll ind ell)) \u2192 let uind = a\u2264\u1d62b-morph ind ind ell (\u2264\u1d62-reflexive ind)\n       in (ltuindx : uind \u2264\u1d62 x)\n       \u2192 (replLL ll ind (replLL ell (ind-rpl\u2193 ind ((x -\u1d62 uind) ltuindx)) frll) \u2261 replLL (replLL ll ind ell) x frll)\nrepl-r=>l ell \u2193 x \u2264\u1d62\u2193 = refl\nrepl-r=>l {ll = li \u2227 ri} ell (ind \u2190\u2227) (x \u2190\u2227) (\u2264\u1d62\u2190\u2227 ltuindx) = cong (\u03bb x \u2192 x \u2227 ri) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2227 ri} ell (\u2227\u2192 ind) (\u2227\u2192 x) (\u2264\u1d62\u2227\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2227 x) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2228 ri} ell (ind \u2190\u2228) (x \u2190\u2228) (\u2264\u1d62\u2190\u2228 ltuindx) = cong (\u03bb x \u2192 x \u2228 ri) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2228 ri} ell (\u2228\u2192 ind) (\u2228\u2192 x) (\u2264\u1d62\u2228\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2228 x) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2202 ri} ell (ind \u2190\u2202) (x \u2190\u2202) (\u2264\u1d62\u2190\u2202 ltuindx) = cong (\u03bb x \u2192 x \u2202 ri) (repl-r=>l ell ind x ltuindx)\nrepl-r=>l {ll = li \u2202 ri} ell (\u2202\u2192 ind) (\u2202\u2192 x) (\u2264\u1d62\u2202\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2202 x) (repl-r=>l ell ind x ltuindx)\n\n\n\n-- Deprecated\nupdInd : \u2200{i u rll ll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} rll ll)\n         \u2192 IndexLL {i} {u} nrll (replLL ll ind nrll)\nupdInd nrll \u2193 = \u2193\nupdInd nrll (ind \u2190\u2227) = (updInd nrll ind) \u2190\u2227\nupdInd nrll (\u2227\u2192 ind) = \u2227\u2192 (updInd nrll ind)\nupdInd nrll (ind \u2190\u2228) = (updInd nrll ind) \u2190\u2228\nupdInd nrll (\u2228\u2192 ind) = \u2228\u2192 (updInd nrll ind)\nupdInd nrll (ind \u2190\u2202) = (updInd nrll ind) \u2190\u2202\nupdInd nrll (\u2202\u2192 ind) = \u2202\u2192 (updInd nrll ind)\n\n-- Deprecated\n-- Maybe instead of this function use a\u2264\u1d62b-morph\nupdIndGen : \u2200{i u pll ll cll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n            \u2192 IndexLL {i} {u} (replLL pll lind nrll) (replLL ll (ind +\u1d62 lind) nrll)\nupdIndGen nrll \u2193 lind = \u2193\nupdIndGen nrll (ind \u2190\u2227) lind = (updIndGen nrll ind lind) \u2190\u2227\nupdIndGen nrll (\u2227\u2192 ind) lind = \u2227\u2192 (updIndGen nrll ind lind)\nupdIndGen nrll (ind \u2190\u2228) lind = (updIndGen nrll ind lind) \u2190\u2228\nupdIndGen nrll (\u2228\u2192 ind) lind = \u2228\u2192 (updIndGen nrll ind lind)\nupdIndGen nrll (ind \u2190\u2202) lind = (updIndGen nrll ind lind) \u2190\u2202\nupdIndGen nrll (\u2202\u2192 ind) lind = \u2202\u2192 (updIndGen nrll ind lind)\n\n\n\n-- TODO This negation was writen so as to return nothing if \u00ac (b \u2264\u1d62 a)\n_-\u2098\u1d62_ : \u2200 {i u pll cll ll} \u2192 IndexLL {i} {u} cll ll \u2192 IndexLL pll ll \u2192 Maybe $ IndexLL cll pll\na -\u2098\u1d62 \u2193 = just a\n\u2193 -\u2098\u1d62 (b \u2190\u2227) = nothing\n(a \u2190\u2227) -\u2098\u1d62 (b \u2190\u2227) = a -\u2098\u1d62 b\n(\u2227\u2192 a) -\u2098\u1d62 (b \u2190\u2227) = nothing\n\u2193 -\u2098\u1d62 (\u2227\u2192 b) = nothing\n(a \u2190\u2227) -\u2098\u1d62 (\u2227\u2192 b) = nothing\n(\u2227\u2192 a) -\u2098\u1d62 (\u2227\u2192 b) = a -\u2098\u1d62 b\n\u2193 -\u2098\u1d62 (b \u2190\u2228) = nothing\n(a \u2190\u2228) -\u2098\u1d62 (b \u2190\u2228) = a -\u2098\u1d62 b\n(\u2228\u2192 a) -\u2098\u1d62 (b \u2190\u2228) = nothing\n\u2193 -\u2098\u1d62 (\u2228\u2192 b) = nothing\n(a \u2190\u2228) -\u2098\u1d62 (\u2228\u2192 b) = nothing\n(\u2228\u2192 a) -\u2098\u1d62 (\u2228\u2192 b) = a -\u2098\u1d62 b\n\u2193 -\u2098\u1d62 (b \u2190\u2202) = nothing\n(a \u2190\u2202) -\u2098\u1d62 (b \u2190\u2202) = a -\u2098\u1d62 b\n(\u2202\u2192 a) -\u2098\u1d62 (b \u2190\u2202) = nothing\n\u2193 -\u2098\u1d62 (\u2202\u2192 b) = nothing\n(a \u2190\u2202) -\u2098\u1d62 (\u2202\u2192 b) = nothing\n(\u2202\u2192 a) -\u2098\u1d62 (\u2202\u2192 b) = a -\u2098\u1d62 b\n\n-- Deprecated\nb-s\u2261j\u21d2s\u2264\u1d62b : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll)\n             \u2192  \u03a3 (IndexLL {i} {u} cll pll) (\u03bb x \u2192 (bind -\u2098\u1d62 sind) \u2261 just x) \u2192 sind \u2264\u1d62 bind\nb-s\u2261j\u21d2s\u2264\u1d62b bind \u2193 (rs , x) = \u2264\u1d62\u2193\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2227) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (sind \u2190\u2227) (rs , x) = \u2264\u1d62\u2190\u2227 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (sind \u2190\u2227) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2227\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (\u2227\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (\u2227\u2192 sind) (rs , x) = \u2264\u1d62\u2227\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2228) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (sind \u2190\u2228) (rs , x) = \u2264\u1d62\u2190\u2228 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (sind \u2190\u2228) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2228\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (\u2228\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (\u2228\u2192 sind) (rs , x) =  \u2264\u1d62\u2228\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2202) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (sind \u2190\u2202) (rs , x) = \u2264\u1d62\u2190\u2202 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (sind \u2190\u2202) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2202\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (\u2202\u2192 sind) (rs , ())\nb-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (\u2202\u2192 sind) (rs , x) = \u2264\u1d62\u2202\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n\n-- Deprecated\nrevUpdInd : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll (replLL ll b ell))\n            \u2192 a -\u2098\u1d62 (updInd ell b) \u2261 nothing \u2192 (updInd ell b) -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll ll\nrevUpdInd \u2193 a () b-a\nrevUpdInd (b \u2190\u2227) \u2193 refl ()\nrevUpdInd (b \u2190\u2227) (a \u2190\u2227) a-b b-a = revUpdInd b a a-b b-a \u2190\u2227\nrevUpdInd (b \u2190\u2227) (\u2227\u2192 a) a-b b-a = \u2227\u2192 a\nrevUpdInd (\u2227\u2192 b) \u2193 refl ()\nrevUpdInd (\u2227\u2192 b) (a \u2190\u2227) a-b b-a = a \u2190\u2227\nrevUpdInd (\u2227\u2192 b) (\u2227\u2192 a) a-b b-a = \u2227\u2192 revUpdInd b a a-b b-a\nrevUpdInd (b \u2190\u2228) \u2193 refl ()\nrevUpdInd (b \u2190\u2228) (a \u2190\u2228) a-b b-a = revUpdInd b a a-b b-a \u2190\u2228\nrevUpdInd (b \u2190\u2228) (\u2228\u2192 a) a-b b-a = \u2228\u2192 a\nrevUpdInd (\u2228\u2192 b) \u2193 refl ()\nrevUpdInd (\u2228\u2192 b) (a \u2190\u2228) a-b b-a = a \u2190\u2228\nrevUpdInd (\u2228\u2192 b) (\u2228\u2192 a) a-b b-a = \u2228\u2192 revUpdInd b a a-b b-a\nrevUpdInd (b \u2190\u2202) \u2193 refl ()\nrevUpdInd (b \u2190\u2202) (a \u2190\u2202) a-b b-a = revUpdInd b a a-b b-a \u2190\u2202\nrevUpdInd (b \u2190\u2202) (\u2202\u2192 a) a-b b-a = \u2202\u2192 a\nrevUpdInd (\u2202\u2192 b) \u2193 refl ()\nrevUpdInd (\u2202\u2192 b) (a \u2190\u2202) a-b b-a = a \u2190\u2202\nrevUpdInd (\u2202\u2192 b) (\u2202\u2192 a) a-b b-a = \u2202\u2192 revUpdInd b a a-b b-a\n\n\n-- Deprecated\nupdIndPart : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll ll)\n             \u2192 a -\u2098\u1d62 b \u2261 nothing \u2192 b -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll (replLL ll b ell)\nupdIndPart \u2193 a () eq2\nupdIndPart (b \u2190\u2227) \u2193 refl ()\nupdIndPart (b \u2190\u2227) (a \u2190\u2227) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2227\nupdIndPart (b \u2190\u2227) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 a\nupdIndPart (\u2227\u2192 b) \u2193 refl ()\nupdIndPart (\u2227\u2192 b) (a \u2190\u2227) eq1 eq2 = a \u2190\u2227\nupdIndPart (\u2227\u2192 b) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 updIndPart b a eq1 eq2\nupdIndPart (b \u2190\u2228) \u2193 refl ()\nupdIndPart (b \u2190\u2228) (a \u2190\u2228) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2228\nupdIndPart (b \u2190\u2228) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 a\nupdIndPart (\u2228\u2192 b) \u2193 refl ()\nupdIndPart (\u2228\u2192 b) (a \u2190\u2228) eq1 eq2 = a \u2190\u2228\nupdIndPart (\u2228\u2192 b) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 updIndPart b a eq1 eq2\nupdIndPart (b \u2190\u2202) \u2193 refl ()\nupdIndPart (b \u2190\u2202) (a \u2190\u2202) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2202\nupdIndPart (b \u2190\u2202) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 a\nupdIndPart (\u2202\u2192 b) \u2193 refl ()\nupdIndPart (\u2202\u2192 b) (a \u2190\u2202) eq1 eq2 = a \u2190\u2202\nupdIndPart (\u2202\u2192 b) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 updIndPart b a eq1 eq2\n\n-- Deprecated\nrev\u21d2rs-i\u2261n : \u2200{i u ll cll ell pll} \u2192 (ind : IndexLL pll ll)\n             \u2192 (lind : IndexLL {i} {u} cll (replLL ll ind ell))\n             \u2192 (eq\u2081 : (lind -\u2098\u1d62 (updInd ell ind) \u2261 nothing))\n             \u2192 (eq\u2082 : ((updInd ell ind) -\u2098\u1d62 lind \u2261 nothing))\n             \u2192 let rs = revUpdInd ind lind eq\u2081 eq\u2082 in\n                 ((rs -\u2098\u1d62 ind) \u2261 nothing) \u00d7 ((ind -\u2098\u1d62 rs) \u2261 nothing)\nrev\u21d2rs-i\u2261n \u2193 lind () eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2227) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (ind \u2190\u2227) (lind \u2190\u2227) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2227) (\u2227\u2192 lind) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2227\u2192 ind) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (\u2227\u2192 ind) (lind \u2190\u2227) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2227\u2192 ind) (\u2227\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2228) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (ind \u2190\u2228) (lind \u2190\u2228) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2228) (\u2228\u2192 lind) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2228\u2192 ind) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (\u2228\u2192 ind) (lind \u2190\u2228) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2228\u2192 ind) (\u2228\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2202) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (ind \u2190\u2202) (lind \u2190\u2202) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\nrev\u21d2rs-i\u2261n (ind \u2190\u2202) (\u2202\u2192 lind) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2202\u2192 ind) \u2193 eq1 ()\nrev\u21d2rs-i\u2261n (\u2202\u2192 ind) (lind \u2190\u2202) eq1 eq2 = refl , refl\nrev\u21d2rs-i\u2261n (\u2202\u2192 ind) (\u2202\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n\n\n\n\n-- Deprecated\nremRepl : \u2200{i u ll frll ell pll cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n          \u2192 replLL (replLL ll (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell \u2261 replLL ll ind ell\nremRepl \u2193 lind = refl\nremRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (ind \u2190\u2227) lind\n        with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2227 ri} {frll} {ell} (ind \u2190\u2227) lind | .(replLL li ind ell) | refl = refl\nremRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (\u2227\u2192 ind) lind\n        with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2227 ri} {frll} {ell} (\u2227\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\nremRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (ind \u2190\u2228) lind\n        with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2228 ri} {frll} {ell} (ind \u2190\u2228) lind | .(replLL li ind ell) | refl = refl\nremRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (\u2228\u2192 ind) lind\n        with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2228 ri} {frll} {ell} (\u2228\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\nremRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (ind \u2190\u2202) lind\n        with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2202 ri} {frll} {ell} (ind \u2190\u2202) lind | .(replLL li ind ell) | refl = refl\nremRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (\u2202\u2192 ind) lind\n        with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n        | (remRepl {frll = frll} {ell = ell} ind lind)\nremRepl {_} {_} {li \u2202 ri} {frll} {ell} (\u2202\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n\n-- Deprecated\nreplLL-inv : \u2200{i u ll ell pll} \u2192 (ind : IndexLL {i} {u} pll ll)\n             \u2192 replLL (replLL ll ind ell) (updInd ell ind) pll \u2261 ll\nreplLL-inv \u2193 = refl\nreplLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (ind \u2190\u2227)\n           with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (ind \u2190\u2227) | .li | refl = refl\nreplLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (\u2227\u2192 ind)\n           with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (\u2227\u2192 ind) | .ri | refl = refl\nreplLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (ind \u2190\u2228)\n           with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (ind \u2190\u2228) | .li | refl = refl\nreplLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (\u2228\u2192 ind)\n           with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (\u2228\u2192 ind) | .ri | refl = refl\nreplLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (ind \u2190\u2202)\n           with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (ind \u2190\u2202) | .li | refl = refl\nreplLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (\u2202\u2192 ind)\n           with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\nreplLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (\u2202\u2192 ind) | .ri | refl = refl\n\n\n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"fc7f5ef989cb7469151bd5269df43071bd52c380","subject":"Removed a telescope.","message":"Removed a telescope.\n\nIgnore-this: e6070a68e84421df13a46dbd756e488f\n\ndarcs-hash:20100503221942-3bd4e-16ab13126f16d7a1d4ea963465f1cb1df295f59a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Minimal.agda","new_file":"LTC\/Minimal.agda","new_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\n------------------------------------------------------------------------------\n\nmodule LTC.Minimal where\n\n-- Standard library imports\n-- open import Data.Fin using ( Fin )\n-- open import Data.Nat using ( \u2115 )\n-- open import Data.String\n-- open import Relation.Binary.PropositionalEquality\n-- open \u2261-Reasoning\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain\n------------------------------------------------------------------------------\n\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language\n------------------------------------------------------------------------------\n\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  fixFO  : D\n\n------------------------------------------------------------------------------\n-- Equality: identity type\n------------------------------------------------------------------------------\n\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n------------------------------------------------------------------------------\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Product public\nopen import MyStdLib.Data.Empty public\nopen import MyStdLib.Data.Product public\nopen import MyStdLib.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n------------------------------------------------------------------------------\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n------------------------------------------------------------------------------\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac ( zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\n------------------------------------------------------------------------------\n\nmodule LTC.Minimal where\n\n-- Standard library imports\n-- open import Data.Fin using ( Fin )\n-- open import Data.Nat using ( \u2115 )\n-- open import Data.String\n-- open import Relation.Binary.PropositionalEquality\n-- open \u2261-Reasoning\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain\n------------------------------------------------------------------------------\n\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language\n------------------------------------------------------------------------------\n\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  fixFO  : D\n\n------------------------------------------------------------------------------\n-- Equality: identity type\n------------------------------------------------------------------------------\n\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n------------------------------------------------------------------------------\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Product public\nopen import MyStdLib.Data.Empty public\nopen import MyStdLib.Data.Product public\nopen import MyStdLib.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n------------------------------------------------------------------------------\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D)(a : D) \u2192 (lam f) \u2219 a \u2261 f a\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n------------------------------------------------------------------------------\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac ( zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"799ecf7906eb3e8eafef355dc3f96dfd5a71014e","subject":"one-time-semantic-security: Improve change-adv and add \u2257A\u2192\u21d3","message":"one-time-semantic-security: Improve change-adv and add \u2257A\u2192\u21d3\n","repos":"crypto-agda\/crypto-agda","old_file":"one-time-semantic-security.agda","new_file":"one-time-semantic-security.agda","new_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Product using (\u2203; module \u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nopen import Data.Vec using (splitAt; take; drop)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import Data.Bits\n\nopen import flat-funs\nopen import program-distance\nopen import flipbased-implem\n\n-- Capture the various size parameters required\n-- for a cipher.\n--\n-- M  is the message space (|M| is the size of messages)\n-- C  is the cipher text space\n-- R\u1d49 is the randomness used by the cipher.\nrecord EncParams : Set where\n  constructor mk\n  field\n    |M| |C| |R|\u1d49 : \u2115\n\n  M  = Bits |M|\n  C  = Bits |C|\n  R\u1d49 = Bits |R|\u1d49\n\n  -- The type of the encryption function\n  Enc : Set\n  Enc = M \u2192 \u21ba |R|\u1d49 C\n\nmodule EncParams\u00b2 (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams ep\u2080 public using ()\n    renaming (|M| to |M|\u2080; |C| to |C|\u2080; |R|\u1d49 to |R|\u1d49\u2080; M to M\u2080;\n              C to C\u2080; R\u1d49 to R\u1d49\u2080; Enc to Enc\u2080)\n  open EncParams ep\u2081 public using ()\n    renaming (|M| to |M|\u2081; |C| to |C|\u2081; |R|\u1d49 to |R|\u1d49\u2081; M to M\u2081;\n              C to C\u2081; R\u1d49 to R\u1d49\u2081; Enc to Enc\u2081)\n  Tr   = Enc\u2080 \u2192 Enc\u2081\n  Tr\u207b\u00b9 = Enc\u2081 \u2192 Enc\u2080\n\nmodule EncParams\u00b2-same-|R|\u1d49 (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  ep\u2081 = EncParams.mk |M|\u2081 |C|\u2081 (EncParams.|R|\u1d49 ep\u2080)\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n\nmodule \u266dEncParams {t} {T : Set t}\n                  (\u266dFuns : FlatFuns T)\n                  (ep : EncParams) where\n  open FlatFuns \u266dFuns\n  open EncParams ep public\n\n  `M  = `Bits |M|\n  `C  = `Bits |C|\n  `R\u1d49 = `Bits |R|\u1d49\n\nmodule \u266dEncParams\u00b2 {t} {T : Set t}\n                   (\u266dFuns : FlatFuns T)\n                   (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n  open \u266dEncParams \u266dFuns ep\u2080 public using () renaming (`M to `M\u2080; `C to `C\u2080; `R\u1d49 to `R\u1d49\u2080)\n  open \u266dEncParams \u266dFuns ep\u2081 public using () renaming (`M to `M\u2081; `C to `C\u2081; `R\u1d49 to `R\u1d49\u2081)\n\nmodule AbsSemSec {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  record SemSecAdv (ep : EncParams) |R|\u1d43 : Set where\n    constructor mk\n\n    open \u266dEncParams \u266dFuns ep\n\n    field\n      {|S|} : \u2115\n\n    `S  = `Bits |S|\n    S   =  Bits |S|\n\n    -- Randomness adversary\n    `R\u1d43 = `Bits |R|\u1d43\n    R\u1d43  =  Bits |R|\u1d43\n\n    field\n      step\u2080 : `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `S\n      step\u2081 : `C `\u00d7 `S `\u2192 `Bit\n\n    M\u00b2  = Bit \u2192 M\n    `M\u00b2 = `Bit `\u2192 `M\n\n    module Ops (\u266dops : FlatFunsOps \u266dFuns) where\n      open FlatFunsOps \u266dops\n      observe : `C `\u00d7 `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `Bit\n      observe = idO *** step\u2080 >>> \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 >>> idO *** step\u2081\n        where \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 = < snd >>> fst , < fst , snd >>> snd > >\n\n    open \u266dEncParams \u266dFuns ep public\n\n  SemSecReduction : \u2200 ep\u2080 ep\u2081 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction ep\u2080 ep\u2081 f = \u2200 {c} \u2192 SemSecAdv ep\u2080 c \u2192 SemSecAdv ep\u2081 (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) where\n  open PrgDist prgDist\n  open AbsSemSec fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  module FunSemSecAdv {ep : EncParams} {|R|\u1d43} (A : SemSecAdv ep |R|\u1d43) where\n    open SemSecAdv A public\n    open Ops fun\u266dOps public\n\n    step\u2080F : R\u1d43 \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R|\u1d43 (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  module RunSemSec (ep : EncParams) where\n    open EncParams ep using (M; C; |R|\u1d49; Enc)\n\n    -- Returing 0 means Chal wins, Adv looses\n    --          1 means Adv  wins, Chal looses\n    runSemSec : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    runSemSec E A b\n      = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n      where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n    _\u21c4_ : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    _\u21c4_ = runSemSec\n\n    _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : SemSecAdv ep p) \u2192 Set\n    A\u2080 \u2257A A\u2081 = observe A\u2080 \u2257 observe A\u2081\n      where open FunSemSecAdv\n\n    change-adv : \u2200 {ca} (A\u2080 A\u2081 : SemSecAdv ep ca) E \u2192 A\u2080 \u2257A A\u2081 \u2192 (E \u21c4 A\u2080) \u2257\u2141 (E \u21c4 A\u2081)\n    change-adv {ca} _ _ _ pf b R with splitAt ca R\n    change-adv A\u2080 A\u2081 E pf b ._ | pre \u03a3., post , \u2261.refl = \u2261.trans (\u2261.cong proj\u2082 (helper\u2080 A\u2080)) helper\u2082\n       where open FunSemSecAdv\n             helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post , pre)\n             helper\u2082 = \u2261.cong (\u03bb m \u2192 step\u2081 A\u2081 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2081 pre)))\n                              (helper\u2080 A\u2081)\n\n    _\u2257E_ : \u2200 (E\u2080 E\u2081 : Enc) \u2192 Set\n    E\u2080 \u2257E E\u2081 = \u2200 m \u2192 E\u2080 m \u2257\u21ba E\u2081 m\n\n    \u2257E\u2192\u2257\u2141 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c)\n               \u2192 (E\u2080 \u21c4 A) \u2257\u2141 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 {c} A b R\n      rewrite E\u2080\u2257E\u2081 (proj (proj\u2081 (SemSecAdv.step\u2080 A (take c R))) b) (drop c R) = \u2261.refl\n\n    \u2257A\u2192\u21d3 : \u2200 {c} (A\u2080 A\u2081 : SemSecAdv ep c) \u2192 A\u2080 \u2257A A\u2081 \u2192 \u2200 E \u2192 (E \u21c4 A\u2080) \u21d3 (E \u21c4 A\u2081)\n    \u2257A\u2192\u21d3 A\u2080 A\u2081 A\u2080\u2257A\u2081 E = extensional-reduction (change-adv A\u2080 A\u2081 E A\u2080\u2257A\u2081)\n\n    \u2257E\u2192\u21d3 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c) \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u21d3 E\u2080\u2257E\u2081 A = extensional-reduction (\u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 A)\n\n  module SemSecReductions (ep\u2080 ep\u2081 : EncParams) (f : Coins \u2192 Coins) where\n    open EncParams\u00b2 ep\u2080 ep\u2081\n    open RunSemSec ep\u2080 public using () renaming (_\u21c4_ to _\u21c4\u2080_; _\u2257E_ to _\u2257E\u2080_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2080)\n    open RunSemSec ep\u2081 public using () renaming (_\u21c4_ to _\u21c4\u2081_; _\u2257E_ to _\u2257E\u2081_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2081)\n\n    Reduction : Set\n    Reduction = SemSecReduction ep\u2081 ep\u2080 f\n\n    SemSecTr : Tr \u2192 Set\n    SemSecTr tr =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u207b\u00b9 : Tr\u207b\u00b9 \u2192 Set\n    SemSecTr\u207b\u00b9 tr\u207b\u00b9 =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u2192SemSecTr\u207b\u00b9 : \u2200 tr tr\u207b\u00b9 (tr-right-inv : \u2200 E \u2192 tr (tr\u207b\u00b9 E) \u2257E\u2081 E)\n                          \u2192 SemSecTr tr \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9\n    SemSecTr\u2192SemSecTr\u207b\u00b9 _ tr\u207b\u00b9 tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n            helper E A A-breaks-E = pf (tr\u207b\u00b9 E) A A-breaks-tr-tr\u207b\u00b9-E\n              where A-breaks-tr-tr\u207b\u00b9-E = \u2257E\u2192\u21d3\u2081 (\u03bb m R \u2192 \u2261.sym (tr-inv E m R)) A A-breaks-E\n\n    SemSecTr\u207b\u00b9\u2192SemSecTr : \u2200 tr tr\u207b\u00b9 (tr-left-inv : \u2200 E \u2192 tr\u207b\u00b9 (tr E) \u2257E\u2080 E)\n                          \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9 \u2192 SemSecTr tr\n    SemSecTr\u207b\u00b9\u2192SemSecTr tr _ tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n            helper E {c} A A-breaks-E = \u2257E\u2192\u21d3\u2080 (tr-inv E) (red A) (pf (tr E) A A-breaks-E)\n\n","old_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Product using (\u2203; module \u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nopen import Data.Vec using (splitAt; take; drop)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import Data.Bits\n\nopen import flat-funs\nopen import program-distance\nopen import flipbased-implem\n\n-- Capture the various size parameters required\n-- for a cipher.\n--\n-- M  is the message space (|M| is the size of messages)\n-- C  is the cipher text space\n-- R\u1d49 is the randomness used by the cipher.\nrecord EncParams : Set where\n  constructor mk\n  field\n    |M| |C| |R|\u1d49 : \u2115\n\n  M  = Bits |M|\n  C  = Bits |C|\n  R\u1d49 = Bits |R|\u1d49\n\n  -- The type of the encryption function\n  Enc : Set\n  Enc = M \u2192 \u21ba |R|\u1d49 C\n\nmodule EncParams\u00b2 (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams ep\u2080 public using ()\n    renaming (|M| to |M|\u2080; |C| to |C|\u2080; |R|\u1d49 to |R|\u1d49\u2080; M to M\u2080;\n              C to C\u2080; R\u1d49 to R\u1d49\u2080; Enc to Enc\u2080)\n  open EncParams ep\u2081 public using ()\n    renaming (|M| to |M|\u2081; |C| to |C|\u2081; |R|\u1d49 to |R|\u1d49\u2081; M to M\u2081;\n              C to C\u2081; R\u1d49 to R\u1d49\u2081; Enc to Enc\u2081)\n  Tr   = Enc\u2080 \u2192 Enc\u2081\n  Tr\u207b\u00b9 = Enc\u2081 \u2192 Enc\u2080\n\nmodule EncParams\u00b2-same-|R|\u1d49 (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  ep\u2081 = EncParams.mk |M|\u2081 |C|\u2081 (EncParams.|R|\u1d49 ep\u2080)\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n\nmodule \u266dEncParams {t} {T : Set t}\n                  (\u266dFuns : FlatFuns T)\n                  (ep : EncParams) where\n  open FlatFuns \u266dFuns\n  open EncParams ep public\n\n  `M  = `Bits |M|\n  `C  = `Bits |C|\n  `R\u1d49 = `Bits |R|\u1d49\n\nmodule \u266dEncParams\u00b2 {t} {T : Set t}\n                   (\u266dFuns : FlatFuns T)\n                   (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n  open \u266dEncParams \u266dFuns ep\u2080 public using () renaming (`M to `M\u2080; `C to `C\u2080; `R\u1d49 to `R\u1d49\u2080)\n  open \u266dEncParams \u266dFuns ep\u2081 public using () renaming (`M to `M\u2081; `C to `C\u2081; `R\u1d49 to `R\u1d49\u2081)\n\nmodule AbsSemSec {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  record SemSecAdv (ep : EncParams) |R|\u1d43 : Set where\n    constructor mk\n\n    open \u266dEncParams \u266dFuns ep\n\n    field\n      {|S|} : \u2115\n\n    `S  = `Bits |S|\n    S   =  Bits |S|\n\n    -- Randomness adversary\n    `R\u1d43 = `Bits |R|\u1d43\n    R\u1d43  =  Bits |R|\u1d43\n\n    field\n      step\u2080 : `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `S\n      step\u2081 : `C `\u00d7 `S `\u2192 `Bit\n\n    M\u00b2  = Bit \u2192 M\n    `M\u00b2 = `Bit `\u2192 `M\n\n    module Ops (\u266dops : FlatFunsOps \u266dFuns) where\n      open FlatFunsOps \u266dops\n      observe : `C `\u00d7 `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `Bit\n      observe = idO *** step\u2080 >>> \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 >>> idO *** step\u2081\n        where \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 = < snd >>> fst , < fst , snd >>> snd > >\n\n    open \u266dEncParams \u266dFuns ep public\n\n  SemSecReduction : \u2200 ep\u2080 ep\u2081 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction ep\u2080 ep\u2081 f = \u2200 {c} \u2192 SemSecAdv ep\u2080 c \u2192 SemSecAdv ep\u2081 (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) where\n  open PrgDist prgDist\n  open AbsSemSec fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  module FunSemSecAdv {ep : EncParams} {|R|\u1d43} (A : SemSecAdv ep |R|\u1d43) where\n    open SemSecAdv A public\n    open Ops fun\u266dOps public\n\n    step\u2080F : R\u1d43 \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R|\u1d43 (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  module RunSemSec (ep : EncParams) where\n    open EncParams ep using (M; C; |R|\u1d49; Enc)\n\n    -- Returing 0 means Chal wins, Adv looses\n    --          1 means Adv  wins, Chal looses\n    runSemSec : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    runSemSec E A b\n      = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n      where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n    _\u21c4_ : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    _\u21c4_ = runSemSec\n\n    _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : SemSecAdv ep p) \u2192 Set\n    A\u2080 \u2257A A\u2081 = observe A\u2080 \u2257 observe A\u2081\n      where open FunSemSecAdv\n\n    change-adv : \u2200 {ca E} {A\u2081 A\u2082 : SemSecAdv ep ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n    change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n    change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , \u2261.refl = \u2261.trans (\u2261.cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n       where open FunSemSecAdv\n             helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post , pre)\n             helper\u2082 = \u2261.cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                              (helper\u2080 A\u2082)\n\n    _\u2257E_ : \u2200 (E\u2080 E\u2081 : Enc) \u2192 Set\n    E\u2080 \u2257E E\u2081 = \u2200 m \u2192 E\u2080 m \u2257\u21ba E\u2081 m\n\n    \u2257E\u2192\u2257\u2141 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c)\n               \u2192 (E\u2080 \u21c4 A) \u2257\u2141 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 {c} A b R\n      rewrite E\u2080\u2257E\u2081 (proj (proj\u2081 (SemSecAdv.step\u2080 A (take c R))) b) (drop c R) = \u2261.refl\n\n    \u2257E\u2192\u21d3 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c) \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u21d3 E\u2080\u2257E\u2081 A = extensional-reduction (\u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 A)\n\n  module SemSecReductions (ep\u2080 ep\u2081 : EncParams) (f : Coins \u2192 Coins) where\n    open EncParams\u00b2 ep\u2080 ep\u2081\n    open RunSemSec ep\u2080 public using () renaming (_\u21c4_ to _\u21c4\u2080_; _\u2257E_ to _\u2257E\u2080_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2080)\n    open RunSemSec ep\u2081 public using () renaming (_\u21c4_ to _\u21c4\u2081_; _\u2257E_ to _\u2257E\u2081_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2081)\n\n    Reduction : Set\n    Reduction = SemSecReduction ep\u2081 ep\u2080 f\n\n    SemSecTr : Tr \u2192 Set\n    SemSecTr tr =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u207b\u00b9 : Tr\u207b\u00b9 \u2192 Set\n    SemSecTr\u207b\u00b9 tr\u207b\u00b9 =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u2192SemSecTr\u207b\u00b9 : \u2200 tr tr\u207b\u00b9 (tr-right-inv : \u2200 E \u2192 tr (tr\u207b\u00b9 E) \u2257E\u2081 E)\n                          \u2192 SemSecTr tr \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9\n    SemSecTr\u2192SemSecTr\u207b\u00b9 _ tr\u207b\u00b9 tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n            helper E A A-breaks-E = pf (tr\u207b\u00b9 E) A A-breaks-tr-tr\u207b\u00b9-E\n              where A-breaks-tr-tr\u207b\u00b9-E = \u2257E\u2192\u21d3\u2081 (\u03bb m R \u2192 \u2261.sym (tr-inv E m R)) A A-breaks-E\n\n    SemSecTr\u207b\u00b9\u2192SemSecTr : \u2200 tr tr\u207b\u00b9 (tr-left-inv : \u2200 E \u2192 tr\u207b\u00b9 (tr E) \u2257E\u2080 E)\n                          \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9 \u2192 SemSecTr tr\n    SemSecTr\u207b\u00b9\u2192SemSecTr tr _ tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n            helper E {c} A A-breaks-E = \u2257E\u2192\u21d3\u2080 (tr-inv E) (red A) (pf (tr E) A A-breaks-E)\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"916b02851cc08a915427025cc0100d845c71a694","subject":"Removed unnecessary ATP hypothesis.","message":"Removed unnecessary ATP hypothesis.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddComm.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddComm.agda","new_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition of total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Auxiliary properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 1: Interactive proof using pattern matching\n\n+-comm\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2081 {n = n} nzero Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm\u2081 {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm\u2081 Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 2: Interactive proof using the induction principle\n\n+-comm\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2082 {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 3: Combined proof using pattern matching\n\n+-comm\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2083 {n = n} nzero Nn = prf\n  where postulate prf : zero + n \u2261 n + zero\n        {-# ATP prove prf +-rightIdentity #-}\n+-comm\u2083 {n = n} (nsucc {m} Nm) Nn = prf (+-comm\u2083 Nm Nn)\n  where postulate prf : m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m\n        {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 4: Combined proof using the induction principle\n\n+-comm\u2084 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2084 {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-rightIdentity #-}\n\n  postulate is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 5: Combined proof using an instance of the induction\n-- principle\n\n+-comm-ind-instance :\n  \u2200 n \u2192\n  zero + n \u2261 n + zero \u2192\n  (\u2200 {m} \u2192 m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m) \u2192\n  \u2200 {m} \u2192 N m \u2192 m + n \u2261 n + m\n+-comm-ind-instance n = N-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n-- TODO (25 October 2012): Why is it not necessary the hypothesis\n-- +-rightIdentity?\npostulate +-comm\u2085 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n{-# ATP prove +-comm\u2085 +-comm-ind-instance x+Sy\u2261S[x+y] #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition of total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Auxiliary properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 1: Interactive proof using pattern matching\n\n+-comm\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2081 {n = n} nzero Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm\u2081 {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm\u2081 Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 2: Interactive proof using the induction principle\n\n+-comm\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2082 {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n\n                \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n)\n                \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i)\n                \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i \u220e\n\n------------------------------------------------------------------------------\n-- Approach 3: Combined proof using pattern matching\n\n+-comm\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2083 {n = n} nzero Nn = prf\n  where postulate prf : zero + n \u2261 n + zero\n        {-# ATP prove prf +-rightIdentity #-}\n+-comm\u2083 {n = n} (nsucc {m} Nm) Nn = prf (+-comm\u2083 Nm Nn)\n  where postulate prf : m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m\n        {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 4: Combined proof using the induction principle\n\n+-comm\u2084 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2084 {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-rightIdentity #-}\n\n  postulate is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 5: Combined proof using instance of the induction\n-- principle\n\n+-comm-ind : \u2200 n \u2192\n            (zero + n \u2261 n + zero) \u2192\n            (\u2200 {m} \u2192 m + n \u2261 n + m  \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m) \u2192\n            \u2200 {m} \u2192 N m \u2192 m + n \u2261 n + m\n+-comm-ind n = N-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n-- TODO (25 October 2012) Why is it not necessary the hypothesis\n-- +-rightIdentity ?\npostulate +-comm\u2085 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n{-# ATP prove +-comm\u2085 +-comm-ind +-rightIdentity x+Sy\u2261S[x+y] #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0eea2d6f9a5d402f97258811d86a6df70db52b33","subject":"Algebra: a few more lemmas","message":"Algebra: a few more lemmas\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b7c23b5a867bd278a0c46d5353000555bc031a6","subject":"Added x\u2264y\u2192x-y\u22610 (by Ana Bove).","message":"Added x\u2264y\u2192x-y\u22610 (by Ana Bove).\n\nIgnore-this: fa2d051e8f718179b58746fc80b2a0dd\n\ndarcs-hash:20110210164711-3bd4e-262b0518a2141dda47047d0d64406b023ca59c1f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ab1099f8573c544e98531d58aac709002e109268","subject":"Type: add \u2605_ for the universal type of type","message":"Type: add \u2605_ for the universal type of type\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type.agda","new_file":"lib\/Type.agda","new_contents":"module Type where\n\nopen import Level\n\n\u2605\u2082 : Set\u2083\n\u2605\u2082 = Set\u2082\n\n\u2605\u2081 : \u2605\u2082\n\u2605\u2081 = Set\u2081\n\n\u2605\u2080 : \u2605\u2081\n\u2605\u2080 = Set\n\n\u2605 : \u2605\u2081\n\u2605 = \u2605\u2080\n\n\u2605_ : \u2200 \u2113 \u2192 Set (suc \u2113)\n\u2605_ \u2113 = Set \u2113\n","old_contents":"module Type where\n\n\u2605\u2082 : Set\u2083\n\u2605\u2082 = Set\u2082\n\n\u2605\u2081 : \u2605\u2082\n\u2605\u2081 = Set\u2081\n\n\u2605 : \u2605\u2081\n\u2605 = Set\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"69bae70b3a6a842a5dc502d4fbe96699c5d9c3d8","subject":"Minor change.","message":"Minor change.\n\nIgnore-this: f9c4671f0b508c9d6efe9b29ab6a3821\n\ndarcs-hash:20111219052427-3bd4e-9f7ca75d0028398c1652641b0c60dc788faa62e8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs\" by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4c385c13c66f0d0c9a21a2f6d2703c0cead5b754","subject":"Updated doc.","message":"Updated doc.\n\nIgnore-this: ea90bf60f58408e68cf3a15d6617c5fc\n\ndarcs-hash:20111219052141-3bd4e-f5d3cfb1252093bcb2fb811b494239f0956342d3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs\" by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Reasoning about Functional Programs\" by Ana Bove, Peter Dybjer, and\n-- Andr\u00e9s Sicard-Ram\u00edrez.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first-order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first-order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fbe7e0e1f8ed8086c50b3b454f06eb3aa8ffd061","subject":"Fix proof of fundamental theorem","message":"Fix proof of fundamental theorem\n\nI used an assertion on indexes that wasn't true.\nOnce I fixed the assertion, I made the mistake of actually proving them.\nOn ANF, this would be simpler.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStepSILR2.agda","new_file":"Thesis\/BigStepSILR2.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"96bd98b071931690e68c52eeb8e376ff57c724b6","subject":"Added Everything.agda","message":"Added Everything.agda\n\nIgnore-this: 5b827a13f800b7eb7dfb69c1bb56e07b\n\ndarcs-hash:20110219141613-3bd4e-e0ea1583b15f4e8809516d5ba59954b5d0a31c86.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/PropertiesATP.agda","new_file":"Draft\/Mirror\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check  #-}\n\nmodule Draft.Mirror.PropertiesATP where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesATP\nopen import Draft.Mirror.ListTree.Closures\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesATP using ( reverse-[x]\u2261[x] )\nopen import Draft.Mirror.ListTree.PropertiesATP\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\nmirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\nmirror\u00b2 (treeT d nilLT) = prf\n  where\n    postulate prf : mirror \u00b7 (mirror \u00b7 node d []) \u2261 node d []\n    {-# ATP prove prf #-}\n\nmirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n  prf (mirror\u00b2 Tt) (mirror\u00b2 (treeT d LTts))\n  where\n    postulate prf : mirror \u00b7 (mirror \u00b7 t) \u2261 t \u2192 -- IH.\n                    mirror \u00b7 (mirror \u00b7 node d ts) \u2261 node d ts \u2192 -- IH.\n                    mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 node d (t \u2237 ts)\n    {-# ATP prove prf reverse-\u2237 map-ListTree mirror-Tree map-++-commute\n                      reverse-++-commute rev-ListTree reverse-[x]\u2261[x]\n                      ++-leftIdentity\n    #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-termination-check  #-}\n\nmodule Draft.Mirror.PropertiesATP where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesATP\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesATP using ( reverse-[x]\u2261[x] )\nopen import Draft.Mirror.ListTree.PropertiesATP\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\nmirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\nmirror\u00b2 (treeT d nilLT) = prf\n  where\n    postulate prf : mirror \u00b7 (mirror \u00b7 node d []) \u2261 node d []\n    {-# ATP prove prf #-}\n\nmirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n  prf (mirror\u00b2 Tt) (mirror\u00b2 (treeT d LTts))\n  where\n    postulate prf : mirror \u00b7 (mirror \u00b7 t) \u2261 t \u2192 -- IH.\n                    mirror \u00b7 (mirror \u00b7 node d ts) \u2261 node d ts \u2192 -- IH.\n                    mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 node d (t \u2237 ts)\n    {-# ATP prove prf reverse-\u2237 map-ListTree mirror-Tree map-++ reverse-++ rev-ListTree reverse-[x]\u2261[x] ++-leftIdentity #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1f0dfe5d9a7e7f52ba8dafc944cd1cfaf5aa8a18","subject":"Vec: various permutations","message":"Vec: various permutations\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b048020c005aaee3b1005342f88eee8a828a1e4a","subject":"Category\/Profunctor","message":"Category\/Profunctor\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Profunctor.agda","new_file":"lib\/Category\/Profunctor.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Based on profunctors 3.1 haskell package\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\nopen import Data.Product\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n  dimap : \u2200 {A B C D} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 B \u219d C \u2192 A \u219d D\n  dimap f g = lmap f \u2218 rmap g\n\nrecord Lenticular {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor mk\n  field\n    profunctor : Profunctor _\u219d_\n    lenticular : \u2200 {A B} \u2192 A \u219d B \u2192 A \u219d (A \u00d7 B)\n  open Profunctor profunctor public\n\n\u2192Profunctor : Profunctor {\u2113} -\u2192-\n\u2192Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\n\u2192Lenticular : Lenticular {\u2113} -\u2192-\n\u2192Lenticular = mk \u2192Profunctor (\u03bb f x \u2192 x , f x)\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nupStarRawFunctor : \u2200 {F A} \u2192 RawFunctor F \u2192 RawFunctor (UpStar F A)\nupStarRawFunctor fun = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n  where open RawFunctor fun\n\nupStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (UpStar F)\nupStarProfunctor fun = flip _\u2218\u2032_\n                     , RawFunctor._<$>_ (upStarRawFunctor fun)\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStarProfunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\ndownStarRawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStarRawFunctor = record { _<$>_ = _\u2218\u2032_ }\n","old_contents":"import Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n\u2192Profunctor : Profunctor {\u2113} -\u2192-\n\u2192Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nupStarRawFunctor : \u2200 {F A} \u2192 RawFunctor F \u2192 RawFunctor (UpStar F A)\nupStarRawFunctor fun = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n  where open RawFunctor fun\n\nupStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (UpStar F)\nupStarProfunctor fun = flip _\u2218\u2032_\n                     , RawFunctor._<$>_ (upStarRawFunctor fun)\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStarProfunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\ndownStarRawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStarRawFunctor = record { _<$>_ = _\u2218\u2032_ }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"31c5947ec2ff1027636628ab15004c15d304b422","subject":"Agda standard library changed the type of _\u2248_. (The first argument is now an equality).","message":"Agda standard library changed the type of _\u2248_.\n(The first argument is now an equality).\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/MapIterate\/MapIterateSL.agda","new_file":"notes\/FOT\/FOTC\/Program\/MapIterate\/MapIterateSL.agda","new_contents":"------------------------------------------------------------------------------\n-- The map-iterate property using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: Nils Anders Danielsson and Thorsten Altenkirch. Mixing\n-- Induction and Coinduction. Draft 2009.\n\nmodule FOT.FOTC.Program.MapIterate.MapIterateSL where\n\nopen import Coinduction\n\nopen import Data.Stream\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\nmap-iterate : \u2200 {A} (f : A \u2192 A) (x : A) \u2192\n              map f (iterate f x) \u2248 iterate f (f x)\nmap-iterate f x = refl \u2237 \u266f map-iterate f (f x)\n","old_contents":"------------------------------------------------------------------------------\n-- The map-iterate property using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: Nils Anders Danielsson and Thorsten Altenkirch. Mixing\n-- Induction and Coinduction. Draft 2009.\n\nmodule FOT.FOTC.Program.MapIterate.MapIterateSL where\n\nopen import Coinduction\n\nopen import Data.Stream\n\n------------------------------------------------------------------------------\n\nmap-iterate : \u2200 {A} (f : A \u2192 A) (x : A) \u2192\n              map f (iterate f x) \u2248 iterate f (f x)\nmap-iterate f x = f x \u2237 \u266f map-iterate f (f x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"78bcef35a43173b501f51932b84fa2cdfc150ca9","subject":"IDesc model: paranoid alpha-renaming.","message":"IDesc model: paranoid alpha-renaming.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set (suc l)) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set (suc l)} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set (suc l)}(R : I -> IDesc {l = l} I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set (suc l)}(D : IDesc I)(X : I -> Set l) -> desc D X -> IDesc (Sigma I X)\nbox (var i)     X x        = var (i , x)\nbox (const _)   X x        = const Unit\nbox (prod D D') X (d , d') = prod (box D X d) (box D' X d')\nbox (sigma S T) X (a , b)  = box (T a) X b\nbox (pi S T)    X f        = pi S (\\s -> box (T s) X (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set (suc l) -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\nIDescD : {l : Level}(I : Set (suc l)) -> IDesc {l = suc l} Unit\nIDescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : {l : Level}(I : Set (suc l)) -> Unit -> Set (suc l)\nIDescl0 {x} I = IMu {l = suc x} (\\_ -> IDescD {l = x} I)\n\nIDescl : {l : Level}(I : Set (suc l)) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set (suc l)}(i : I) -> IDescl I\nvarl {x} i = con (lvar {l = suc x} , i) \n\nconstl : {l : Level}{I : Set (suc l)}(X : Set l) -> IDescl I\nconstl {x} X = con (lconst {l = suc x} , X)\n\nprodl : {l : Level}{I : Set (suc l)}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lprod {l = suc x} , (D , D'))\n\n\npil : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lpi {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n\nsigmal : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (lsigma {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n           \n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set (suc l)}\n        (xs : desc (IDescD I) (IMu (\u03bb _ -> IDescD I)))\n        (hs : desc (box (IDescD I) (IMu (\u03bb _ -> IDescD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s) )\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set (suc l)} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> IDescD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set (suc l)} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set (suc l)} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-psi-phi : {l : Level}(I : Set (suc l)) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> IDescD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n    where P : Sigma Unit (IMu (\\ x -> IDescD I)) -> Set (suc x)\n          P ( Void , D ) = psi (phi D) == D\n          proof-psi-phi-cases : (i : Unit)\n                                (xs : desc (IDescD I) (IDescl0 I))\n                                (hs : desc (box (IDescD I) (IDescl0 I) xs) P)\n                                -> P (i , con xs)\n          proof-psi-phi-cases Void (lvar , i) hs = refl\n          proof-psi-phi-cases Void (lconst , x) hs = refl\n          proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n          proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                               (trans (reflFun (\\ s -> psi (phi (T (lifter (unlift s)))))\n                                                                               (\\ s -> psi (phi (T (s))))\n                                                                               (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                           (proof-lift-unlift-eq s)))\n                                                                       (reflFun (\\s -> psi (phi (T s))) \n                                                                                T \n                                                                                hs)) \n          proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                  (trans (reflFun (\\ s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                  (\\ s \u2192 psi (phi (T (s))))\n                                                                                  (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                              (proof-lift-unlift-eq s)))\n                                                                         (reflFun (\\s -> psi (phi (T s))) \n                                                                                  T \n                                                                                  hs))","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set (suc l)) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set (suc l)} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set (suc l)}(R : I -> IDesc {l = l} I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set (suc l)}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const Unit\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set (suc l)}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set (suc l) -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\nIDescD : {l : Level}(I : Set (suc l)) -> IDesc {l = suc l} Unit\nIDescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : {l : Level}(I : Set (suc l)) -> Unit -> Set (suc l)\nIDescl0 {x} I = IMu {l = suc x} (\\_ -> IDescD {l = x} I)\n\nIDescl : {l : Level}(I : Set (suc l)) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set (suc l)}(i : I) -> IDescl I\nvarl {x} i = con (lvar {l = suc x} , i) \n\nconstl : {l : Level}{I : Set (suc l)}(X : Set l) -> IDescl I\nconstl {x} X = con (lconst {l = suc x} , X)\n\nprodl : {l : Level}{I : Set (suc l)}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lprod {l = suc x} , (D , D'))\n\n\npil : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lpi {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n\nsigmal : {l : Level}{I : Set (suc l)}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (lsigma {l = suc x} , pair {i = suc x}{j = suc x} S (\\s -> T (unlift s)))\n           \n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set (suc l)}\n        (xs : desc (IDescD I) (IMu (\u03bb _ -> IDescD I)))\n        (hs : desc (box (IDescD I) (IMu (\u03bb _ -> IDescD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s) )\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set (suc l)} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> IDescD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set (suc l)} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set (suc l)} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-psi-phi : {l : Level}(I : Set (suc l)) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> IDescD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n    where P : Sigma Unit (IMu (\\ x -> IDescD I)) -> Set (suc x)\n          P ( Void , D ) = psi (phi D) == D\n          proof-psi-phi-cases : (i : Unit)\n                                (xs : desc (IDescD I) (IDescl0 I))\n                                (hs : desc (box (IDescD I) (IDescl0 I) xs) P)\n                                -> P (i , con xs)\n          proof-psi-phi-cases Void (lvar , i) hs = refl\n          proof-psi-phi-cases Void (lconst , x) hs = refl\n          proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n          proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                               (trans (reflFun (\\ s -> psi (phi (T (lifter (unlift s)))))\n                                                                               (\\ s -> psi (phi (T (s))))\n                                                                               (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                           (proof-lift-unlift-eq s)))\n                                                                       (reflFun (\\s -> psi (phi (T s))) \n                                                                                T \n                                                                                hs)) \n          proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                  (trans (reflFun (\\ s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                  (\\ s \u2192 psi (phi (T (s))))\n                                                                                  (\\s -> cong (\\ s -> psi (phi (T (s))))\n                                                                                              (proof-lift-unlift-eq s)))\n                                                                         (reflFun (\\s -> psi (phi (T s))) \n                                                                                  T \n                                                                                  hs))","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1981f3a1302ba721654cb2ea01ccca86987e55af","subject":"IDescTT model: clean-up.","message":"IDescTT model: clean-up.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const X\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> (xs : desc D X) -> desc (All D X xs) R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = k\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (All D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n{-\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = induction R P m i xs\n-}\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (All (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (All (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (All D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a9241a950cd1412a4a93e2beeca1d0d51823855c","subject":"restructuring and making headway on #5","message":"restructuring and making headway on #5\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i (FVal x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = vs x (_ , Step p q r)\n  is (IAp i (FIndet x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n  -- final expressions are not errors (not one of the 6 cases for progress)\n  fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  fe (FVal x) er = ve x er\n  fe (FIndet x) er = ie x er\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 er) (d' , Step (FHAp2 x) x\u2081 x\u2082) = {!!}\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole er) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i (FVal x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = vs x (_ , Step p q r)\n  is (IAp i (FIndet x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n--  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n\n  --es (ENEHole e) x = {!e!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step FHNEHoleEvaled x\u2082 x\u2083) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleInside x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  -- es (ENEHole e) (d' , Step (FHFinal (FVal ())) x\u2081 x\u2082)\n  -- es (ENEHole e) (d' , Step (FHFinal (FIndet (INEHole x))) () x\u2082)\n  -- es (ENEHole e) (d' , Step FHNEHoleEvaled () x\u2082)\n  -- es (ENEHole e) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FVal ())))\n  -- es (ENEHole (ECastError x x\u2081)) (_ , Step (FHNEHoleFinal x\u2082) (ITNEHole x\u2083) (FHFinal (FIndet (INEHole x\u2084)))) = {!x\u2081!}\n  -- es (ENEHole (EAp1 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (EAp2 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ENEHole e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ECastProp e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) FHNEHoleEvaled) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8ab606ba2d4c372e9471c6a84124d3711414edc4","subject":"worked out a few cases with @cyrus- , also removed a premise from one of the subsume rules that we decided we didnt need","message":"worked out a few cases with @cyrus- , also removed a premise from one of the subsume rules that we decided we didnt need\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , {!!} -- ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc x\u2081) x\u2082) = {!!}\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083) = _ , _ , _ , EASubsume {!!} {!!} (ESLam x\u2082 {!!}) x\u2083\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAHole x\u2081) x\u2082) = {!!}\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D'  = {!!}\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , {!!} -- ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana (ASubsume wt x\u2081) with expandability-synth wt\n    ... | d' , \u0394' , D' = d' , \u0394' , _ , EASubsume (\u03bb x \u2192 {!!}) (\u03bb x \u2192 {!!}) D' x\u2081 -- what are u and e'?\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D'  = {!!}\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"27d658e7fbaa6a2f19e0521364ede55ae18489ca","subject":"FunExt: cleanup","message":"FunExt: cleanup\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Extensionality.agda","new_file":"lib\/Function\/Extensionality.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\n-- See Function.Extensionality.Implicit for the extensionality of {x : A} \u2192 B x\n\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\npostulate\n    FunExt : Set\n\nmodule _ {a b}{A : Set a}{B : A \u2192 Set b} where\n  infix 4 _\u223c_\n\n  _\u223c_   : (f\u2080 f\u2081 : (x : A) \u2192 B x) \u2192 Set _\n  f\u2080 \u223c f\u2081 = (x : A) \u2192 f\u2080 x \u2261 f\u2081 x\n\n  module _ {f\u2080 f\u2081 : (x : A) \u2192 B x} where\n    happly : f\u2080 \u2261 f\u2081 \u2192 f\u2080 \u223c f\u2081\n    happly p x = ap (\u03bb f\u2080 \u2192 f\u2080 x) p\n\n    module _ {{fe : FunExt}} where\n      postulate\n        \u03bb= : (f= : f\u2080 \u223c f\u2081) \u2192 f\u2080 \u2261 f\u2081\n\n        happly-\u03bb= : (f= : f\u2080 \u223c f\u2081) \u2192 happly (\u03bb= f=) \u2261 f=\n\n        \u03bb=-happly : (\u03b1 : f\u2080 \u2261 f\u2081) \u2192 \u03bb= (happly \u03b1) \u2261 \u03b1\n\n        -- This should be derivable if I had a proper proof of \u03bb=\n        tr-\u03bb= : \u2200 {p x}(P : B x \u2192 Set p)(f= : f\u2080 \u223c f\u2081)\n                \u2192 tr (\u03bb f\u2080 \u2192 P (f\u2080 x)) (\u03bb= f=) \u2261 tr P (f= x)\n\n      \u03bb=\u2071 : (f= : \u2200 {x} \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n      \u03bb=\u2071 f= = \u03bb= \u03bb x \u2192 f= {x}\n\n  !-\u03b1-\u03bb= : \u2200 {f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n             (\u03b1 : f\u2080 \u2261 f\u2081) \u2192 ! \u03b1 \u2261 \u03bb= (!_ \u2218 happly \u03b1)\n  !-\u03b1-\u03bb= refl = ! \u03bb=-happly refl\n\nmodule _ {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}} where\n  !-\u03bb= : (f= : f\u2080 \u223c f\u2081) \u2192 ! (\u03bb= f=) \u2261 \u03bb= (!_ \u2218 f=)\n  !-\u03bb= f= = !-\u03b1-\u03bb= (\u03bb= f=) \u2219 ap \u03bb= (\u03bb= (\u03bb x \u2192 ap !_ (happly (happly-\u03bb= f=) x)))\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP  -- renaming (subst to tr)\n\nhapply : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}\n  \u2192 f \u2261 g \u2192 (x : A) \u2192 f x \u2261 g x\nhapply p x = ap (\u03bb f \u2192 f x) p\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n           (f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n\n    happly-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 happly (\u03bb= fg) \u2261 fg\n\n    \u03bb=-happly : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (\u03b1 : f \u2261 g) \u2192 \u03bb= (happly \u03b1) \u2261 \u03b1\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}{{fe : FunExt}}(fg : (x : A) \u2192 f x \u2261 g x)\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n\n\n!-\u03b1-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (\u03b1 : f \u2261 g) \u2192 ! \u03b1 \u2261 \u03bb= (!_ \u2218 happly \u03b1)\n!-\u03b1-\u03bb= refl = ! \u03bb=-happly refl\n\n!-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 ! (\u03bb= fg) \u2261 \u03bb= (!_ \u2218 fg)\n!-\u03bb= fg = !-\u03b1-\u03bb= (\u03bb= fg) \u2219 ap \u03bb= (\u03bb= (\u03bb x \u2192 ap !_ (happly (happly-\u03bb= fg) x)))\n\nmodule _ {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}} where\n  \u03bb=\u2071 : (f= : \u2200 {x} \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n  \u03bb=\u2071 f= = \u03bb= \u03bb x \u2192 f= {x}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c7cbfe14459b4e5607ccf51d8bc3c0f7af8f303b","subject":"Move FFI.JS.BigI.FiniteField back to crypto-agda","message":"Move FFI.JS.BigI.FiniteField back to crypto-agda\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS\/BigI\/FiniteField.agda","new_file":"lib\/FFI\/JS\/BigI\/FiniteField.agda","new_contents":"","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import FFI.JS using (\ud835\udfd9; Number; Bool; true; false; String; warn-check; check; trace-call; _++_)\nopen import FFI.JS.BigI\nopen import Data.List.Base hiding (sum; _++_)\n{-\nopen import Algebra.Raw\nopen import Algebra.Field\n-}\n\nmodule FFI.JS.BigI.FiniteField (q : BigI) where\n\nabstract\n  \ud835\udd3d : Set\n  \ud835\udd3d = BigI\n\n  private\n    mod-q : BigI \u2192 \ud835\udd3d\n    mod-q x = mod x q\n\n    check' : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\n    -- check' = warn-check\n    check' = check\n\n  -- There is two ways to go from BigI to \u2124q: check and mod-q\n  -- Use check for untrusted input data and mod-q for internal\n  -- computation.\n  fromBigI : BigI \u2192 \ud835\udd3d\n  fromBigI = -- trace-call \"BigI\u25b9\u2124q \"\n    \u03bb x \u2192\n      (check' (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++ \" is less than x:\" ++ toString x)\n         (check' (x \u2265I 0I)\n            (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") x))\n\n  check-non-zero : \ud835\udd3d \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \ud835\udd3d \u2192 BigI\n  repr x = x\n\n  0# 1# : \ud835\udd3d\n  0# = 0I\n  1# = 1I\n\n  1\/_ : Op\u2081 \ud835\udd3d\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : Op\u2082 \ud835\udd3d\n  x ^ y = modPow (repr x) (repr y) q\n\n_+_ _\u2212_ _*_ _\/_ : Op\u2082 \ud835\udd3d\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = mod-q (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\n0\u2212_ : Op\u2081 \ud835\udd3d\n0\u2212 x = mod-q (negate (repr x))\n\n_==_ : (x y : \ud835\udd3d) \u2192 Bool\nx == y = equals (repr x) (repr y)\n\nsum prod : List \ud835\udd3d \u2192 \ud835\udd3d\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\n{-\n+-mon-ops : Monoid-Ops \ud835\udd3d\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \ud835\udd3d\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \ud835\udd3d\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \ud835\udd3d\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \ud835\udd3d\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \ud835\udd3d\nfld = fld-ops , fld-struct\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0d80106e1e3a5cd18ab85358a3986b8f394ac646","subject":"Update: Field.Solver","message":"Update: Field.Solver\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field\/Solver.agda","new_file":"lib\/Algebra\/Field\/Solver.agda","new_contents":"import Algebra.Field as Field\nimport Algebra.RingSolver as RS\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Level\nopen import Data.Nat as Nat using (\u2115)\nopen import Data.Fin\nopen import Data.One\nopen import Data.Product\n\nopen import Function\n\nmodule Algebra.Field.Solver where\n\ndata R-Op : Set where\n  [+] [*] [-] : R-Op\n\ndata F-Op : Set where\n  [+] [*] [-] [\/] : F-Op\n\ndata Tm (F Op Var : Set) : Set where\n  op  : (o : Op) (t\u2081 t\u2082 : Tm F Op Var) \u2192 Tm F Op Var\n  con : F \u2192 Tm F Op Var\n  var : Var \u2192 Tm F Op Var\n\ninfixl 6 _:+_ _:-_\ninfixl 7 _:*_ _:\/_\npattern _:+_ x y = op [+] x y\npattern _:*_ x y = op [*] x y\npattern _:-_ x y = op [-] x y\npattern _:\/_ x y = op [\/] x y\n\nR-include : R-Op \u2192 F-Op\nR-include [+] = [+]\nR-include [*] = [*]\nR-include [-] = [-]\n\nmodule _ {F Op Var : Set}(evalOp : Op \u2192 F \u2192 F \u2192 F)(\u03c1 : Var \u2192 F) where\n  eval : Tm F Op Var \u2192 F\n  eval (op o t u) = evalOp o (eval t) (eval u)\n  eval (con x) = x\n  eval (var x) = \u03c1 x\n\nmodule _\n  {Var : Set}\n  {F : Set}\n  (\ud835\udd3d : Field.Field-Ops F)\n  where\n\n  module \ud835\udd3d = Field.Field-Ops \ud835\udd3d\n\n  F-Op-eval : F-Op \u2192 F \u2192 F \u2192 F\n  F-Op-eval [+] x y = x \ud835\udd3d.+ y\n  F-Op-eval [*] x y = x \ud835\udd3d.* y\n  F-Op-eval [-] x y = x \ud835\udd3d.\u2212 y\n  F-Op-eval [\/] x y = x \ud835\udd3d.\/ y\n\n  R-Op-eval : R-Op \u2192 F \u2192 F \u2192 F\n  R-Op-eval o = F-Op-eval (R-include o)\n\n  evalF : (\u03c1 : Var \u2192 F) \u2192 Tm F F-Op Var \u2192 F\n  evalR : (\u03c1 : Var \u2192 F) \u2192 Tm F R-Op Var \u2192 F\n  evalF = eval F-Op-eval\n  evalR = eval R-Op-eval\n\n  infix 2 _\/_:[_,_]\n  record Div \u03c1 (t : Tm F F-Op Var) : Set where\n    constructor _\/_:[_,_]\n    field\n      N : Tm F R-Op Var\n      D : Tm F R-Op Var\n      .non-zero-D : evalR \u03c1 D \u2262 \ud835\udd3d.`0\n      .is-correct : evalF \u03c1 t \u2261 evalR \u03c1 N \ud835\udd3d.\/ evalR \u03c1 D\n\n  module _ (FS : Field.Field-Struct \ud835\udd3d)(\u03c1 : Var \u2192 F) where\n    open Field.Field-Struct FS\n\n    private\n      TmF = Tm F F-Op Var\n      TmR = Tm F R-Op Var\n\n      Div' = Div \u03c1\n      ev = evalF \u03c1\n\n    _+Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :+ u)\n    (N \/ D :[ nD , ok ]) +Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* D\u2081 :+ N\u2081 :* D \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , += ok ok\u2081 \u2219 +-quotient nD nD\u2081 ]\n\n    _*Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :* u)\n    (N \/ D :[ nD , ok ]) *Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* N\u2081 \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , *= ok ok\u2081 \u2219 *-quotient nD nD\u2081 ]\n\n    _\u2212Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :- u)\n    (N \/ D :[ nD , ok ]) \u2212Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* D\u2081 :- N\u2081 :* D \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , \u2212= ok ok\u2081 \u2219 \u2212-quotient nD nD\u2081 ]\n\n    _\/Div_:[_] : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 ev u \u2262 `0 \u2192 Div' (t :\/ u)\n    (N \/ D :[ nD , ok ]) \/Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ]) :[ u\u22620 ]\n      = (N :* D\u2081) \/ (D :* N\u2081)\n      :[ no-zero-divisor nD nN\u2081 , \/= ok ok\u2081 \u2219 \/-quotient nN\u2081 nD nD\u2081 ]\n      where .nN\u2081 : evalR \u03c1 N\u2081 \u2262 `0\n            nN\u2081 p = u\u22620 (ok\u2081 \u2219 *= p idp \u2219 0*-zero)\n\n    -- con-Div and var-Div are basically the same but not so convenient to unify\n    con-Div : \u2200 x \u2192 Div' (con x)\n    var-Div : \u2200 x \u2192 Div' (var x)\n\n    con-Div x = con x \/ con \ud835\udd3d.`1 :[ 1\u22620 , ! \/1-id ]\n    var-Div x = var x \/ con \ud835\udd3d.`1 :[ 1\u22620 , ! \/1-id ]\n\n    Assumptions : TmF \u2192 Set\n    Assumptions (op [+] t u) = Assumptions t \u00d7 Assumptions u\n    Assumptions (op [*] t u) = Assumptions t \u00d7 Assumptions u\n    Assumptions (op [-] t u) = Assumptions t \u00d7 Assumptions u\n    Assumptions (op [\/] t u) = Assumptions t \u00d7 Assumptions u \u00d7 ev u \u2262 `0\n    Assumptions (con x) = Lift \ud835\udfd9\n    Assumptions (var x) = Lift \ud835\udfd9\n\n    to-ring : \u2200 (t : TmF) \u2192 Assumptions t \u2192 Div' t\n    to-ring (op [+] t u) (at , au)       = to-ring t at +Div to-ring u au\n    to-ring (op [*] t u) (at , au)       = to-ring t at *Div to-ring u au\n    to-ring (op [-] t u) (at , au)       = to-ring t at \u2212Div to-ring u au\n    to-ring (op [\/] t u) (at , au , u\u22620) = to-ring t at \/Div to-ring u au :[ u\u22620 ]\n    to-ring (con x) _                    = con-Div x\n    to-ring (var x) _                    = var-Div x\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Algebra.Field as Field renaming (Field-Ops to RawField ; module Field-Ops to RawField)\nimport Algebra.RingSolver as RS\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Level\nopen import Data.Nat as Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin\nopen import Data.One\nopen import Data.Product\n\nopen import Function\n\nmodule Algebra.Field.Solver\n  {f}{F : Set f}\n  (\ud835\udd3d : Field.RawField F) where\n\nmodule \ud835\udd3d = Field.RawField \ud835\udd3d\n\n\ndata R-Op : Set where\n  [+] [*] [-] : R-Op\n\ndata F-Op : Set where\n  [+] [*] [-] [\/] : F-Op\n\nF-Op-eval : F-Op \u2192 F \u2192 F \u2192 F\nF-Op-eval [+] x y = x \ud835\udd3d.+ y\nF-Op-eval [*] x y = x \ud835\udd3d.* y\nF-Op-eval [-] x y = x \ud835\udd3d.\u2212 y\nF-Op-eval [\/] x y = x \ud835\udd3d.\/ y\n\nR-include : R-Op \u2192 F-Op\nR-include [+] = [+]\nR-include [*] = [*]\nR-include [-] = [-]\n\nR-Op-eval : R-Op \u2192 F \u2192 F \u2192 F\nR-Op-eval op = F-Op-eval (R-include op)\n\n\ndata Tm (Op : Set)(m : \u2115) : Set f where\n  op : (o : Op) (t\u2081 t\u2082 : Tm Op m) \u2192 Tm Op m\n  con : F \u2192 Tm Op m\n  var : Fin m \u2192 Tm Op m\n\ninfixl 6 _:+_ _:-_\ninfixl 7 _:*_ _:\/_\npattern _:+_ x y = op [+] x y\npattern _:*_ x y = op [*] x y\npattern _:-_ x y = op [-] x y\npattern _:\/_ x y = op [\/] x y\n\nmodule _ {m}{Op : Set}(evalOp : Op \u2192 F \u2192 F \u2192 F)(\u03c1 : Fin m \u2192 F) where\n  eval : Tm Op m \u2192 F\n  eval (op o tm tm\u2081) = evalOp o (eval tm) (eval tm\u2081)\n  eval (con x) = x\n  eval (var x) = \u03c1 x\n\ninfix 2 _\/_:[_,_]\nrecord Div {m} \u03c1 (tm : Tm F-Op m) : Set f where\n  constructor _\/_:[_,_]\n  field\n    N : Tm R-Op m\n    D : Tm R-Op m\n    .nonZero-D : eval R-Op-eval \u03c1 D \u2262 \ud835\udd3d.0\u1da0\n    .is-Correct : eval F-Op-eval \u03c1 tm \u2261 eval R-Op-eval \u03c1 N \ud835\udd3d.\/ eval R-Op-eval \u03c1 D\n\nmodule _ (FS : Field.Field-Struct \ud835\udd3d) where\n  open Field.Field-Struct FS -- {{...}}\n\n  {-\n  toRingOp : \u2200 {m} \u2192 F-Op \u2192 Div {m} \u2192 Div {m} \u2192 Div {m}\n  toRingOp [+] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* D' :+ N' :* D  \/ D :* D'\n                                                :[ (\u03bb \u03c1 \u2192 noZeroDivisor (P \u03c1) (P' \u03c1) ) ]\n  toRingOp [*] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* N' \/ D :* D' :[ (\u03bb \u03c1 \u2192 noZeroDivisor (P \u03c1) (P' \u03c1)) ]\n  toRingOp [-] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* D' :- N' :* D \/ D :* D'\n                                                :[ (\u03bb \u03c1 \u2192 noZeroDivisor (P \u03c1) (P' \u03c1)) ]\n  toRingOp [\/] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* D' \/ N' :* D :[ {!!} ]\n  -}\n\n\n  module _ {m} (\u03c1 : Fin m \u2192 F) where\n    Assumptions : Tm F-Op m \u2192 Set f\n    Assumptions (op [+] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081\n    Assumptions (op [*] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081\n    Assumptions (op [-] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081\n    Assumptions (op [\/] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081 \u00d7 eval F-Op-eval \u03c1 tm\u2081 \u2262 0\u1da0\n    Assumptions (con x) = Lift \ud835\udfd9\n    Assumptions (var x) = Lift \ud835\udfd9\n\n    toRing : \u2200 (tm : Tm F-Op m) \u2192 Assumptions tm \u2192 Div {m} \u03c1 tm\n    toRing (op [+] tm tm\u2081) (atm , atm\u2081) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = N :* D\u2081 :+ N\u2081 :* D \/ D :* D\u2081\n                                               :[ noZeroDivisor nD nD\u2081 , += isC isC\u2081 \u2219 +-quotient nD nD\u2081 ]\n    toRing (op [*] tm tm\u2081) (atm , atm\u2081) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = N :* N\u2081 \/ D :* D\u2081\n                                               :[ noZeroDivisor nD nD\u2081 , *= isC isC\u2081 \u2219 *-quotient nD nD\u2081 ]\n    toRing (op [-] tm tm\u2081) (atm , atm\u2081) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = N :* D\u2081 :- N\u2081 :* D \/ D :* D\u2081\n                                               :[ noZeroDivisor nD nD\u2081 , \u2212= isC isC\u2081 \u2219 \u2212-quotient nD nD\u2081 ]\n    toRing (op [\/] tm tm\u2081) (atm , atm\u2081 , tm\u2081\/=0) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = (N :* D\u2081) \/ (D :* N\u2081)\n                                               :[ noZeroDivisor nD nN\u2081 , \/= isC isC\u2081 \u2219 \/-quotient nN\u2081 nD nD\u2081 ]\n      where .nN\u2081 : eval R-Op-eval \u03c1 N\u2081 \u2262 0\u1da0\n            nN\u2081 p = tm\u2081\/=0 (isC\u2081 \u2219 *= p refl \u2219 0*-zero)\n    toRing (con x) _ = con x \/ con \ud835\udd3d.1\u1da0 :[ 0\u22621 \u2218 !_ , ! (*= refl 1\u207b\u00b9\u22611 \u2219 *1-identity) ]\n    toRing (var x) _ = var x \/ con \ud835\udd3d.1\u1da0 :[ 0\u22621 \u2218 !_ , ! (*= refl 1\u207b\u00b9\u22611 \u2219 *1-identity) ]\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"baa82d46ce4e47ef731df82054bb6b93b0f34ec1","subject":"removed IType_i, because it was useless...simply using List IType now","message":"removed IType_i, because it was useless...simply using List IType now\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping.agda","new_file":"Agda\/ITyping.agda","new_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List.Properties\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\ndata IType : Set where\n  o : IType\n  _~>_ : List IType -> List IType -> IType\n\n\n\u03c9 : List IType\n\u03c9 = []\n\n\u2229' : IType -> List IType\n\u2229' x = (x \u2237 [])\n\n_\u2229_ : IType -> IType -> List IType\nA \u2229 B = A \u2237 B \u2237 []\n\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n-- list-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> (x \u2237 \u03c4\u1d62) \u2261 (y \u2237 \u03c4\u2c7c) -> \u03c4\u1d62 \u2261 \u03c4\u2c7c\n-- list-inj-cons refl = refl\n--\n-- list-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> (x \u2237 \u03c4\u1d62) \u2261 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n-- \u2229-inj refl = refl\n--\n-- list-\u2237-\u2261 : \u2200 {x y : IType} {xs ys} -> x \u2261 y -> xs \u2261 ys -> (x \u2237 xs) \u2261 (y \u2237 ys)\n-- list-\u2237-\u2261 refl refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u2097_ : Decidable {A = List IType} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u2097 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u2097 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n[] \u225fTI\u2097 [] = yes refl\n[] \u225fTI\u2097 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 [] = no (\u03bb ())\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | \u03c4\u1d62 \u225fTI\u2097 \u03c4\u2c7c\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (proj\u2082 (\u2237-injective \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c)))\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (proj\u2081 (\u2237-injective \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c)))\n\n\n\nICtxt = List (Atom \u00d7 ((List IType) \u00d7 Type))\n\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u2097_ : List IType -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u2097 A -> \u03c4 \u2237'\u2097 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\ndata _\u2237'\u2097_ where\n  nil : \u2200 {A} -> \u03c9 \u2237'\u2097 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} -> \u03c4 \u2237' A -> \u03c4\u1d62 \u2237'\u2097 A -> (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u2097 A\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {A \u0393 x \u03c4} ->\n    (x\u2209 : x \u2209 dom \u0393) -> \u03c4 \u2237'\u2097 A -> Wf-ICtxt \u0393 ->\n    --------------------------------------------\n            Wf-ICtxt ((x , (\u03c4 , A)) \u2237 \u0393)\n\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u2097[_]_ : List IType -> Type -> List IType -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u2097[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u2097[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u2097[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u2097 A ->\n    -----------\n    \u03c9 \u2286\u2097[ A ] \u03c4\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> \u03c4'\u1d62 \u2286\u2097[ A ] \u03c4\u1d62 ->\n    -------------------------------------------------------\n                    (\u03c4' \u2237 \u03c4'\u1d62) \u2286\u2097[ A ] \u03c4\u1d62\n\n\n\u2237'\u2097-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2237'\u2097 A -> \u03c4 \u2237' A\n\u2237'\u2097-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u2097-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u2097-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u2097-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u2097-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u2097-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u2097 A -> \u03c4 \u2286\u2097[ A ] \u03c4\n\u2286\u2097-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u2097 A -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u2097-refl \u03c4\u2237\u1d62A) (\u2286\u2097-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u2097-refl {\u03c4 = []} nil = nil nil\n\u2286\u2097-refl {A} {\u03c4 \u2237 \u03c4\u1d62} \u03c4\u03c4\u1d62\u2237A = \u2286\u2097-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u2097-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u2097-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u2097-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u2097-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n\n\u2286\u2097-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u03c4\u2081 \u2286\u2097[ A ] \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n\u2286\u2097-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n\u2286\u2097-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u2097-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n\n\n\u2286\u2097-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u03c4 \u2237 \u03c4\u1d62) \u2286\u2097[ A ] \u03c9 -> \u22a5\n\u2286\u2097-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286-\u2237'-l : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4 \u2237' A\n\u2286-\u2237'-l base = base\n\u2286-\u2237'-l (arr _ _ x _) = x\n\n\u2286\u2097-\u2237'\u2097-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u2097 A\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\u2286\u2097-\u2237'\u2097-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u1d62 \u2237'\u2097 A\n\u2286\u2097-\u2237'\u2097-l (nil x) = nil\n\u2286\u2097-\u2237'\u2097-l (cons (_ , _ , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) = cons (\u2286-\u2237'-l \u03c4'\u2286\u03c4) (\u2286\u2097-\u2237'\u2097-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\n\n\u2286\u2097-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n  \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u2097[ A ] \u03c4\u2096 ->\n  ---------------------------------\n            \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2096\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\n\u2286\u2097-trans {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u2097-\u2237'\u2097-r \u03c4\u2c7c\u2286\u03c4\u2096)\n\u2286\u2097-trans {\u03c4\u1d62 = \u03c4' \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u2097-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u2097-trans {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} {\u03c4\u2c7c} {\u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n  cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n    where\n    \u03c4'' = proj\u2081 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n    \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n    \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u2097-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u2097-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n\n\n\u2286->\u2286\u2097 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u2097[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u2097 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app s t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y _) = Y \u03c4\n\n-- \u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n-- \u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u2097_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> List IType -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u03c4\u1d62 , A)) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u2097 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u2097[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u2097 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , (\u03c4 , A)) \u2237 \u0393) \u22a9\u2097 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> (\u03c4 ~> \u03c4') \u2237' (A \u27f6 B) ->\n    --------------------------------------------------------------------------------------------------------------\n                                                \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2081) \u00d7 ((\u03c4 ~> \u03c4) \u2286[ A \u27f6 A ] \u03c4')) -> \u03c4\u2081 \u2237'\u2097 (A \u27f6 A) -> \u03c4\u2082 \u2286\u2097[ A ] \u03c4 -> --  \u03c4\u2081 \u2286[ (A \u27f6 A) \u27f6 A ]((\u2229' (\u03c4 ~> \u03c4)) ~> \u03c4)\n    -----------------------------------------------------------------------------------------\n                                    \u0393 \u22a9 Y A \u2236 (\u03c4\u2081 ~> \u03c4\u2082)\n\ndata _\u22a9\u2097_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u2097 m\u2005 \u2236 \u03c9\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u2097 m\u2005 \u2236 \u03c4\u1d62 ->\n    --------------------------------\n          \u0393 \u22a9\u2097 m\u2005 \u2236 (\u03c4 \u2237 \u03c4\u1d62)\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n\ndata _\u2286\u0393_ : ICtxt -> ICtxt -> Set where\n  nil : \u2200 {\u0393} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          [] \u2286\u0393 \u0393\n  cons : \u2200 {A x \u03c4' \u0393 \u0393'} ->\n    \u2203(\u03bb \u03c4 -> ((x , (\u03c4 , A)) \u2208 \u0393') \u00d7 (\u03c4' \u2286\u2097[ A ] \u03c4)) -> \u0393 \u2286\u0393 \u0393' ->\n    ------------------------------------------------------------\n                      ((x , (\u03c4' , A)) \u2237 \u0393) \u2286\u0393 \u0393'\n\n\n\u22a9\u2097-wf-\u0393 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> Wf-ICtxt \u0393\n\u22a9\u2097-wf-\u0393 (nil wf-\u0393) = wf-\u0393\n\u22a9\u2097-wf-\u0393 (cons _ \u0393\u22a9\u2097m\u2236\u03c4) = \u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097m\u2236\u03c4\n\n\n\u2286\u0393-wf\u0393' : \u2200 {\u0393 \u0393'} -> \u0393 \u2286\u0393 \u0393' -> Wf-ICtxt \u0393'\n\u2286\u0393-wf\u0393' (nil wf-\u0393') = wf-\u0393'\n\u2286\u0393-wf\u0393' (cons _ \u0393\u2286\u0393') = \u2286\u0393-wf\u0393' \u0393\u2286\u0393'\n\nwf-\u0393-\u2237'\u2097 : \u2200 {A x \u03c4 \u0393} -> (x , (\u03c4 , A)) \u2208 \u0393 -> Wf-ICtxt \u0393 -> \u03c4 \u2237'\u2097 A\nwf-\u0393-\u2237'\u2097 (here refl) (cons _ x\u2081 _) = x\u2081\nwf-\u0393-\u2237'\u2097 (there x\u03c4A\u2208\u0393) (cons _ _ wf-\u0393) = wf-\u0393-\u2237'\u2097 x\u03c4A\u2208\u0393 wf-\u0393\n\n\n\u2286\u0393-\u2286 : \u2200 {\u0393 \u0393'} -> Wf-ICtxt \u0393' -> \u0393 \u2286 \u0393' -> \u0393 \u2286\u0393 \u0393'\n\u2286\u0393-\u2286 {[]} wf-\u0393' \u0393\u2286\u0393' = nil wf-\u0393'\n\u2286\u0393-\u2286 {(x , \u03c4 , A) \u2237 \u0393} wf-\u0393' \u0393\u2286\u0393' = cons\n  (\u03c4 , ((\u0393\u2286\u0393' (here refl)) , \u2286\u2097-refl (wf-\u0393-\u2237'\u2097 (\u0393\u2286\u0393' (here refl)) wf-\u0393')))\n  (\u2286\u0393-\u2286 wf-\u0393' (\u03bb {x\u2081} z \u2192 \u0393\u2286\u0393' (there z)))\n\n-- \u2286\u0393-refl {[]} _ = nil nil\n-- \u2286\u0393-refl {(x , \u03c4 , A) \u2237 \u0393} (cons _ \u03c4\u2237A wf-\u0393) = cons (\u03c4 , (here refl , \u2286\u2097-refl \u03c4\u2237A)) {!   !}\n\n\n\u22a9-\u2237' : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4 \u2237' A\n\u22a9\u2097-\u2237'\u2097 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4 \u2237'\u2097 A\n\n\u22a9-\u2237' (var _ _ \u03c4\u2286\u03c4\u1d62) = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (app _ _ \u03c4\u2286\u03c4\u1d62 _) = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (abs _ _ x) = x\n\u22a9-\u2237' (Y _ _ \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) = arr \u03c4\u2081\u2237A\u27f6A (\u2286\u2097-\u2237'\u2097-l \u03c4\u2082\u2286\u03c4)\n\n\u22a9\u2097-\u2237'\u2097 (nil _) = nil\n\u22a9\u2097-\u2237'\u2097 (cons \u0393\u22a9m\u2236\u03c4 \u0393\u22a9\u2097m\u2236\u03c4\u1d62) = cons (\u22a9-\u2237' \u0393\u22a9m\u2236\u03c4) (\u22a9\u2097-\u2237'\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u1d62)\n\n\n\u22a9\u2097-\u2208-\u22a9 : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4\u1d62} -> \u0393 \u22a9\u2097 m \u2236 \u03c4\u1d62 -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9\u2097-\u2208-\u22a9 (nil _) ()\n\u22a9\u2097-\u2208-\u22a9 (cons \u0393\u22a9m\u2236\u03c4 x) (here refl) = \u0393\u22a9m\u2236\u03c4\n\u22a9\u2097-\u2208-\u22a9 (cons _ \u0393\u22a9\u2097m\u2236\u03c4\u1d62) (there \u03c4\u2208\u03c4\u1d62) = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n\u2208-\u2286\u0393-trans : \u2200 {A x \u03c4\u1d62} {\u0393 \u0393'} -> (x , (\u03c4\u1d62 , A)) \u2208 \u0393 -> \u0393 \u2286\u0393 \u0393' -> \u2203(\u03bb \u03c4\u1d62' -> ((x , (\u03c4\u1d62' , A)) \u2208 \u0393') \u00d7 \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62')\n\u2208-\u2286\u0393-trans (here refl) (cons x _) = x\n\u2208-\u2286\u0393-trans (there x\u2208L) (cons _ L\u2286L') = \u2208-\u2286\u0393-trans x\u2208L L\u2286L'\n\n\n\u2286\u0393-\u2237 : \u2200 {A x \u03c4\u1d62 \u0393 \u0393'} -> x \u2209 dom \u0393' -> \u03c4\u1d62 \u2237'\u2097 A -> \u0393 \u2286\u0393 \u0393' -> \u0393 \u2286\u0393 ((x , \u03c4\u1d62 , A) \u2237 \u0393')\n\u2286\u0393-\u2237 {\u0393 = []} x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393' = nil (cons x\u2209\u0393' \u03c4\u1d62\u2237A (\u2286\u0393-wf\u0393' \u0393\u2286\u0393'))\n\u2286\u0393-\u2237 {\u0393 = (x , \u03c4\u1d62 , A) \u2237 \u0393} x\u2209\u0393' \u03c4\u1d62\u2237A (cons (proj\u2081 , proj\u2082 , proj\u2083) \u0393\u2286\u0393') =\n  cons (proj\u2081 , ((there proj\u2082) , proj\u2083)) (\u2286\u0393-\u2237 x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393')\n\n\nsub\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4\nsub\u1d62\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u2097 m \u2236 \u03c4\n\nsub\u0393 (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u0393\u2286\u0393' = var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u2097-trans \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub\u0393 (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u0393\u2286\u0393' = app (sub\u0393 \u0393\u22a9m\u2236\u03c4 \u0393\u2286\u0393') (sub\u1d62\u0393 x \u0393\u2286\u0393') x\u2081 x\u2082\nsub\u0393 {\u0393' = \u0393'} (abs {\u03c4 = \u03c4} L cf (arr \u03c4\u2237A \u03c4'\u2237B)) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\u0393\n    (cf (\u2209-cons-l _ _ x\u2209))\n    (cons\n      (\u03c4 , (here refl) , (\u2286\u2097-refl \u03c4\u2237A))\n      (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2237A \u03c4'\u2237B)\nsub\u0393 (Y x x\u2081 x\u2082 x\u2083) \u0393\u2286\u0393' = Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') x\u2081 x\u2082 x\u2083\n\nsub\u1d62\u0393 (nil wf-\u0393) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62\u0393 (cons x \u0393\u22a9\u2097m\u2236\u03c4) \u0393\u2286\u0393' = cons (sub\u0393 x \u0393\u2286\u0393') (sub\u1d62\u0393 \u0393\u22a9\u2097m\u2236\u03c4 \u0393\u2286\u0393')\n\n\n\u2208-\u2286\u2097-trans : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2c7c) \u00d7 \u03c4 \u2286[ A ] \u03c4')\n\u2208-\u2286\u2097-trans (here refl) (cons x _) = x\n\u2208-\u2286\u2097-trans (there \u03c4\u2208\u03c4\u1d62) (cons _ \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2208-\u2286\u2097-trans \u03c4\u2208\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\nsub : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4'\nsub\u1d62 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4' \u2286\u2097[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u2097 m \u2236 \u03c4'\n\nsub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u2097-trans (\u2286\u2097-trans (\u2286->\u2286\u2097 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62) \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub (app \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u22a9\u2097t\u2236\u03c4\u2081 \u03c4\u2286\u03c4\u2082 \u03c4\u2237A) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = app\n  (sub\u0393 \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u2286\u0393')\n  (sub\u1d62\u0393 \u0393\u22a9\u2097t\u2236\u03c4\u2081 \u0393\u2286\u0393')\n  (\u2286\u2097-trans (\u2286->\u2286\u2097 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u2082)\n  \u03c4\u2237A\nsub {_} {\u0393} {\u0393'} (abs {\u03c4 = \u03c4} {\u03c4'} L {t} cf \u03c4~>\u03c4'\u2237A\u27f6B) (arr {A} {\u03c4\u2081\u2081 = \u03c4\u2081\u2081} \u03c4\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4' (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B) x\u2083) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\n    (cf (\u2209-cons-l _ _ x\u2209))\n    \u03c4\u2081\u2082\u2286\u03c4'\n    (cons (\u03c4\u2081\u2081 , (here refl) , \u03c4\u2286\u03c4\u2081\u2081) (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2081\u2081\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B)\nsub (Y wf-\u0393 (\u03c4' , \u03c4'\u2208\u03c4\u2081 , \u03c4~>\u03c4\u2286\u03c4') \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) (arr {\u03c4\u2081\u2081 = \u03c4\u2081'} \u03c4\u2081\u2286\u03c4\u2081' \u03c4\u2082'\u2286\u03c4\u2082 (arr \u2229\u03c4\u2081'\u2237A\u27f6A \u03c4\u2082'\u2237A) x\u2084) \u0393\u2286\u0393' =\n  Y ((\u2286\u0393-wf\u0393' \u0393\u2286\u0393')) (\u03c4'' , (\u03c4''\u2208\u03c4\u2081' , \u2286-trans \u03c4~>\u03c4\u2286\u03c4' \u03c4'\u2286\u03c4'')) \u2229\u03c4\u2081'\u2237A\u27f6A (\u2286\u2097-trans \u03c4\u2082'\u2286\u03c4\u2082 \u03c4\u2082\u2286\u03c4)\n  where\n  \u03c4'' = proj\u2081 (\u2208-\u2286\u2097-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081')\n  \u03c4''\u2208\u03c4\u2081' = proj\u2081 (proj\u2082 (\u2208-\u2286\u2097-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n  \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2208-\u2286\u2097-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n\nsub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4 (nil x) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4'\u1d62\u2286\u03c4\u1d62) \u0393\u2286\u0393' with \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n... | \u0393\u22a9m\u2236\u03c4 = cons (sub \u0393\u22a9m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393') (sub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4'\u1d62\u2286\u03c4\u1d62 \u0393\u2286\u0393')\n\n\n\n\n\n\n~>\u2229 : \u2200 {X A B C} -> (A ~> B) \u2237' X -> (A ~> C) \u2237' X ->\n  \u2229' (A ~> (B ++ C)) \u2286\u2097[ X ] ((A ~> B) \u2237 (A ~> C) \u2237 [])\n~>\u2229 A~>B\u2237X A~>C\u2237X = cons ({!   !} , ({!   !} , {!   !})) (nil (cons A~>B\u2237X (cons A~>C\u2237X nil)))\n\n\n\u0393\u22a9Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u0393 \u22a9 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4')\n\u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app {\u03c4\u2081 = \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62} {\u03c4'}\n  (Y {\u03c4 = \u03c4''} wf-\u0393 (_ , \u03c4\u2081~>\u03c4\u2082\u2208\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 , arr {\u03c4\u2082\u2081 = \u03c4\u2081} {\u03c4\u2082} \u03c4\u2081\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082 \u03c4''~>\u03c4''\u2237A\u27f6A (arr \u03c4\u2081\u2237A \u03c4\u2082\u2237A)) \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A \u03c4'\u2286\u03c4'')\n  \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n  \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A\u2082) =\n  \u03c4\u2082 , \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 , \u03c4\u2286\u03c4\u2082\n\n  where\n  \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z)\n  \u03c4\u2286\u03c4'' = \u2286\u2097-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''\n  \u03c4\u2286\u03c4\u2082 = \u2286\u2097-trans \u03c4\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082\n\n  \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u03c4\u2081 ~> \u03c4\u2082)\n  \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 \u03c4\u2081~>\u03c4\u2082\u2208\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n\n  \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u03c4\u2082 ~> \u03c4\u2082)\n  \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 = sub \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 (arr (\u2286\u2097-trans \u03c4\u2081\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082) (\u2286\u2097-refl \u03c4\u2082\u2237A) (arr \u03c4\u2082\u2237A \u03c4\u2082\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2082\u2237A)) \u0393\u2286\u0393-refl\n\n\n\n\u0393\u22a9\u2097Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u00ac(\u03c4 \u2261 \u03c9) -> \u0393 \u22a9\u2097 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 \u03c4 \u2286\u2097[ A ] \u03c4')\n\u0393\u22a9\u2097Ym-max \u03c4\u2260\u03c9 (nil wf-\u0393) = \u22a5-elim (\u03c4\u2260\u03c9 refl)\n\u0393\u22a9\u2097Ym-max \u03c4\u2260\u03c9 (cons x (nil wf-\u0393)) = \u0393\u22a9Ym-max x\n\u0393\u22a9\u2097Ym-max {A} {\u0393} {m} \u03c4\u2260\u03c9 (cons {\u03c4 = \u03c4} x (cons {\u03c4 = \u03c4\u2082} {\u03c4\u1d62} x\u2081 \u0393\u22a9\u2097Ym\u2236\u03c4)) =\n  {!   !}\n  where\n  ih : \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 (\u03c4\u2082 \u2237 \u03c4\u1d62) \u2286\u2097[ A ] \u03c4')\n  ih = \u0393\u22a9\u2097Ym-max {!   !} (cons {\u03c4 = \u03c4\u2082} {\u03c4\u1d62} x\u2081 \u0393\u22a9\u2097Ym\u2236\u03c4)\n\n  \u03c4\u1d62' = proj\u2081 ih\n  \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62' = proj\u2081 (proj\u2082 ih)\n  \u03c4\u2082\u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 ih)\n\n  \u03c4' = proj\u2081 (\u0393\u22a9Ym-max x)\n  \u0393\u22a9m\u2236\u03c4'~>\u03c4' = proj\u2081 (proj\u2082 (\u0393\u22a9Ym-max x))\n  \u03c4\u2286\u03c4' = proj\u2082 (proj\u2082 (\u0393\u22a9Ym-max x))\n\n\n\n\u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n\u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (nil wf-\u0393) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y _) = app\n  (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n    wf-\u0393\n    ((\u03c9 ~> \u2229' \u03c4) , here refl , arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr nil \u03c4\u2237A))\n    (cons (arr []\u2237A \u03c4\u2237A) nil)\n    (\u2286\u2097-refl \u03c4\u2237A))\n  (cons (sub \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (arr (nil []\u2237A) \u03c4\u2286\u03c4\u1d62 (arr []\u2237A \u03c4\u2237A) (arr []\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n  (\u2286\u2097-refl \u03c4\u2237A)\n  (cons (arr []\u2237A \u03c4\u2237A) nil)\n\n  where\n  \u03c4\u2237A = \u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62\n\n\u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u03c4' \u2237 \u03c4'\u1d62} {\u03c4\u1d62} \u0393\u22a9m\u2236\u03c4'~>\u03c4\u1d62 \u0393\u22a9\u2097Ym\u2236\u03c4' \u03c4\u2286\u03c4\u1d62 \u03c4'\u2237A) (Y _) = {!   !}\n  where\n  \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 (\u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097Ym\u2236\u03c4') (\u03bb {x} z \u2192 z)\n\n  \u03c4'' = proj\u2081 (\u0393\u22a9\u2097Ym-max (\u03bb x \u2192 {!   !}) \u0393\u22a9\u2097Ym\u2236\u03c4')\n\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4'' : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u03c4'')\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4'' = proj\u2081 (proj\u2082 (\u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4'))\n\n  \u03c4'\u03c4'\u1d62\u2286\u03c4'' : (\u03c4' \u2237 \u03c4'\u1d62) \u2286\u2097[ A ] \u03c4''\n  \u03c4'\u03c4'\u1d62\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4'))\n\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4\u1d62 : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u2229' \u03c4)\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4\u1d62 = sub \u0393\u22a9m\u2236\u03c4'~>\u03c4\u1d62 (arr \u03c4'\u03c4'\u1d62\u2286\u03c4'' \u03c4\u2286\u03c4\u1d62 (arr (\u2286\u2097-\u2237'\u2097-r \u03c4'\u03c4'\u1d62\u2286\u03c4'') (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)) (arr \u03c4'\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) \u0393\u2286\u0393-refl\n\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (nil wf-\u0393) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y _) = app\n--   (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n--     wf-\u0393\n--     ((\u03c9 ~> \u2229' \u03c4) , here refl , arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr nil \u03c4\u2237A))\n--     (cons (arr []\u2237A \u03c4\u2237A) nil)\n--     (\u2286\u2097-refl \u03c4\u2237A))\n--   (cons (sub \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (arr (nil []\u2237A) \u03c4\u2286\u03c4\u1d62 (arr []\u2237A \u03c4\u2237A) (arr []\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n--   (\u2286\u2097-refl \u03c4\u2237A)\n--   (cons (arr []\u2237A \u03c4\u2237A) nil)\n--\n--\n-- -- \u22a9->\u03b2 (app {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} \u0393\u22a9m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082 (cons (app (Y wf-\u0393 (proj\u2081 , () , proj\u2083) x\u2081 x\u2082) (nil wf-\u0393\u2081) \u03c4\u2081\u2286\u03c4\u2081\u2082 x\u2085) \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62) \u03c4\u2286\u03c4\u2082 \u03c4\u2081\u2237A) (Y x)\n-- -- \u22a9->\u03b2 {\u03c4 = \u03c4} (app {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} {\u03c4\u2082} \u0393\u22a9m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082 (cons (app {\u03c4\u2081 = \u2229 (\u03c4\u2081\u2081 \u2237 \u03c4\u2081\u2081\u1d62)} {\u03c4\u2081\u2082} (Y {\u03c4 = \u03c4'} wf-\u0393 (\u03c4'' , \u03c4''\u2208 \u03c4\u2081\u2081, proj\u2083) x\u2081 x\u2082) (cons \u0393\u22a9m\u2236\u03c4\u2081\u2081 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081\u1d62) \u03c4\u2081\u2286\u03c4\u2081\u2082 x\u2085) \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62) \u03c4\u2286\u03c4\u2082 \u03c4\u2081\u2237A) (Y _) =\n-- \u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} {\u03c4\u2082}\n--   \u0393\u22a9m\u2236\u03c4\u2081\u1d62\u03c4\u2081\u1d62~>\u03c4\u2082\n--   (cons\n--     (app {\u03c4\u2081 = \u2229 (\u03c4\u2081\u2081)} {\u03c4\u2081\u2082}\n--       (Y {\u03c4 = \u03c4'} wf-\u0393 (_ , \u03c4''\u2208\u03c4\u2081\u2081 , arr {\u03c4\u2082\u2081 = \u03c4''\u2081} {\u03c4''\u2082} \u03c4''\u2081\u2286\u03c4' \u03c4'\u2286\u03c4''\u2082 \u03c4'~>\u03c4'\u2237A\u27f6A (arr \u03c4''\u2081\u2237A \u03c4''\u2082\u2237A)) \u03c4\u2081\u2081\u2237A\u27f6A \u03c4\u2081\u2082\u2286\u03c4')\n--       \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081 \u03c4\u2081\u2286\u03c4\u2081\u2082 \u03c4\u2081\u2081\u2237A\u27f6A\u2082)\n--     \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62)\n--   \u03c4\u2286\u03c4\u2082\n--   \u03c4\u2081\u2237A) (Y _) =\n--   {!   !}\n--\n--   where\n--   \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z)\n--\n--   \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 : \u0393 \u22a9 m \u2236 (\u03c4''\u2081 ~> \u03c4''\u2082)\n--   \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081 \u03c4''\u2208\u03c4\u2081\u2081\n--\n--   \u0393\u22a9m\u2236\u03c4''\u2082~>\u03c4''\u2082 : \u0393 \u22a9 m \u2236 (\u03c4''\u2082 ~> \u03c4''\u2082)\n--   \u0393\u22a9m\u2236\u03c4''\u2082~>\u03c4''\u2082 = sub \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 (arr (\u2286\u2097-trans \u03c4''\u2081\u2286\u03c4' \u03c4'\u2286\u03c4''\u2082) (\u2286\u2097-refl \u03c4''\u2082\u2237A) (arr \u03c4''\u2082\u2237A \u03c4''\u2082\u2237A) (arr \u03c4''\u2081\u2237A \u03c4''\u2082\u2237A)) \u0393\u2286\u0393-refl\n--\n--   \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u2229' \u03c4\u2081 ~> \u03c4\u2082)\n--   \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 = sub \u0393\u22a9m\u2236\u03c4\u2081\u1d62\u03c4\u2081\u1d62~>\u03c4\u2082 (arr {!   !} {!   !} {!   !} {!   !}) \u0393\u2286\u0393-refl\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u2097Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","old_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\ndata IType : Set\ndata IType\u1d62 : Set\n\ndata IType where\n  o : IType\n  _~>_ : IType\u1d62 -> IType\u1d62 -> IType\n\ndata IType\u1d62 where\n  \u2229 : List IType -> IType\u1d62\n\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\u1d62\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\u2229-\u2237-\u2261 : \u2200 {x y : IType} {xs ys} -> x \u2261 y -> \u2229 xs \u2261 \u2229 ys -> \u2229 (x \u2237 xs) \u2261 \u2229 (y \u2237 ys)\n\u2229-\u2237-\u2261 refl refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u1d62_ : Decidable {A = IType\u1d62} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u1d62 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u1d62 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n\u2229 [] \u225fTI\u1d62 \u2229 [] = yes refl\n\u2229 [] \u225fTI\u1d62 \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI\u1d62 (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 (IType\u1d62 \u00d7 Type))\n\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u1d62_ : IType\u1d62 -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u1d62 A -> \u03c4 \u2237'\u1d62 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {A \u0393 x \u03c4} ->\n    (x\u2209 : x \u2209 dom \u0393) -> \u03c4 \u2237'\u1d62 A -> Wf-ICtxt \u0393 ->\n    --------------------------------------------\n              Wf-ICtxt ((x , (\u03c4 , A)) \u2237 \u0393)\n\ndata _\u2237'\u1d62_ where\n  nil : \u2200 {A} ->  \u03c9 \u2237'\u1d62 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237'\u1d62 A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u1d62 A\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u1d62[_]_ : IType\u1d62 -> Type -> IType\u1d62 -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u1d62[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u1d62[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u1d62[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u1d62 A ->\n    -----------\n    \u03c9 \u2286\u1d62[ A ] \u03c4\n  -- \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229' \u03c4) \u2286[ A ] (\u2229 \u03c4\u1d62)\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> (\u2229 \u03c4'\u1d62) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62) ->\n    -------------------------------------------------------------\n                  (\u2229 (\u03c4' \u2237 \u03c4'\u1d62)) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62)\n\n-- data _\u2264\u2229_ : IType -> IType -> Set where\n--   base : o \u2264\u2229 o\n--   arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n--   \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n--   \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n--   \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\u2237'\u1d62-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A -> \u03c4 \u2237' A\n\u2237'\u1d62-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u1d62-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u1d62-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u1d62-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u1d62-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u1d62-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u1d62 A -> \u03c4 \u2286\u1d62[ A ] \u03c4\n\u2286\u1d62-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> \u2229 \u03c4\u1d62 \u2286\u1d62[ A ] \u2229 \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u1d62-refl \u03c4\u2237\u1d62A) (\u2286\u1d62-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u1d62-refl {\u03c4 = \u2229 []} nil = nil nil\n\u2286\u1d62-refl {A} {\u2229 (\u03c4 \u2237 \u03c4\u1d62)} \u03c4\u03c4\u1d62\u2237A = \u2286\u1d62-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u1d62-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u1d62-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n\n\u2286\u1d62-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u2229 \u03c4\u2081 \u2286\u1d62[ A ] \u2229 \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n\u2286\u1d62-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n\u2286\u1d62-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u1d62-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n\n\n\u2286\u1d62-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u2229 (\u03c4 \u2237 \u03c4\u1d62)) \u2286\u1d62[ A ] \u03c9 -> \u22a5\n\u2286\u1d62-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286-\u2237'-l : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4 \u2237' A\n\u2286-\u2237'-l base = base\n\u2286-\u2237'-l (arr _ _ x _) = x\n\n\u2286\u1d62-\u2237'\u1d62-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u1d62-\u2237'\u1d62-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\u2286\u1d62-\u2237'\u1d62-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u1d62 \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-l (nil x) = nil\n\u2286\u1d62-\u2237'\u1d62-l (cons (_ , _ , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) = cons (\u2286-\u2237'-l \u03c4'\u2286\u03c4) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\n\n\u2286\u1d62-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n  \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u1d62[ A ] \u03c4\u2096 ->\n  ---------------------------------\n            \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2096\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2c7c\u2286\u03c4\u2096)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u1d62-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} {\u2229 \u03c4\u2c7c} {\u2229 \u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n  cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u1d62-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n    where\n    \u03c4'' = proj\u2081 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n    \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n    \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} {\u2229 (_ \u2237 \u03c4\u2c7c)} {\u2229 \u03c4\u2096} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) (cons x \u03c4\u2c7c\u2286\u03c4\u2096) = cons (\u03c4 , ({!   !} , {!   !})) {!   !}\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 = {!   !}\n\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u1d62-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u1d62-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n\n\n\u2286->\u2286\u1d62 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u1d62[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u1d62 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app s t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y _) = Y \u03c4\n\n-- \u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n-- \u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u1d62_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType\u1d62 -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : IType\u1d62} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u03c4\u1d62 , A)) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u1d62[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u1d62 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u1d62[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u1d62 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , (\u03c4 , A)) \u2237 \u0393) \u22a9\u1d62 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> (\u03c4 ~> \u03c4') \u2237' (A \u27f6 B) ->\n    --------------------------------------------------------------------------------------------------------------\n                                                \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2081) \u00d7 ((\u03c4 ~> \u03c4) \u2286[ A \u27f6 A ] \u03c4')) -> (\u2229 \u03c4\u2081) \u2237'\u1d62 (A \u27f6 A) -> \u03c4\u2082 \u2286\u1d62[ A ] \u03c4 -> --  \u03c4\u2081 \u2286[ (A \u27f6 A) \u27f6 A ]((\u2229' (\u03c4 ~> \u03c4)) ~> \u03c4)\n    -----------------------------------------------------------------------------------------\n                                    \u0393 \u22a9 Y A \u2236 ((\u2229 \u03c4\u2081) ~> \u03c4\u2082)\n\ndata _\u22a9\u1d62_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 \u03c9\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 \u03c4\u1d62) ->\n    --------------------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n\ndata _\u2286\u0393_ : ICtxt -> ICtxt -> Set where\n  nil : \u2200 {\u0393} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          [] \u2286\u0393 \u0393\n  cons : \u2200 {A x \u03c4' \u0393 \u0393'} ->\n    \u2203(\u03bb \u03c4 -> ((x , (\u03c4 , A)) \u2208 \u0393') \u00d7 (\u03c4' \u2286\u1d62[ A ] \u03c4)) -> \u0393 \u2286\u0393 \u0393' ->\n    ------------------------------------------------------------\n                      ((x , (\u03c4' , A)) \u2237 \u0393) \u2286\u0393 \u0393'\n\n\n\u2286\u0393-wf\u0393' : \u2200 {\u0393 \u0393'} -> \u0393 \u2286\u0393 \u0393' -> Wf-ICtxt \u0393'\n\u2286\u0393-wf\u0393' (nil wf-\u0393') = wf-\u0393'\n\u2286\u0393-wf\u0393' (cons _ \u0393\u2286\u0393') = \u2286\u0393-wf\u0393' \u0393\u2286\u0393'\n\n\n\n\u22a9-\u2237' : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4 \u2237' A\n\u22a9\u1d62-\u2237'\u1d62 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u03c4 \u2237'\u1d62 A\n\n\u22a9-\u2237' (var _ _ \u03c4\u2286\u03c4\u1d62) = \u2237'\u1d62-\u2208 (here refl) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (app _ _ \u03c4\u2286\u03c4\u1d62 _) = \u2237'\u1d62-\u2208 (here refl) (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (abs _ _ x) = x\n\u22a9-\u2237' (Y _ _ \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) = arr \u03c4\u2081\u2237A\u27f6A (\u2286\u1d62-\u2237'\u1d62-l \u03c4\u2082\u2286\u03c4)\n\n\u22a9\u1d62-\u2237'\u1d62 (nil _) = nil\n\u22a9\u1d62-\u2237'\u1d62 (cons \u0393\u22a9m\u2236\u03c4 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62) = cons (\u22a9-\u2237' \u0393\u22a9m\u2236\u03c4) (\u22a9\u1d62-\u2237'\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62)\n\n\n\u22a9\u1d62-\u2208-\u22a9 : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4\u1d62} -> \u0393 \u22a9\u1d62 m \u2236 \u2229 \u03c4\u1d62 -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9\u1d62-\u2208-\u22a9 (nil _) ()\n\u22a9\u1d62-\u2208-\u22a9 (cons \u0393\u22a9m\u2236\u03c4 x) (here refl) = \u0393\u22a9m\u2236\u03c4\n\u22a9\u1d62-\u2208-\u22a9 (cons _ \u0393\u22a9\u1d62m\u2236\u03c4\u1d62) (there \u03c4\u2208\u03c4\u1d62) = \u22a9\u1d62-\u2208-\u22a9 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n\u2208-\u2286\u0393-trans : \u2200 {A x \u03c4\u1d62} {\u0393 \u0393'} -> (x , (\u03c4\u1d62 , A)) \u2208 \u0393 -> \u0393 \u2286\u0393 \u0393' -> \u2203(\u03bb \u03c4\u1d62' -> ((x , (\u03c4\u1d62' , A)) \u2208 \u0393') \u00d7 \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u1d62')\n\u2208-\u2286\u0393-trans (here refl) (cons x _) = x\n\u2208-\u2286\u0393-trans (there x\u2208L) (cons _ L\u2286L') = \u2208-\u2286\u0393-trans x\u2208L L\u2286L'\n\n\n\u2286\u0393-\u2237 : \u2200 {A x \u03c4\u1d62 \u0393 \u0393'} -> x \u2209 dom \u0393' -> \u03c4\u1d62 \u2237'\u1d62 A -> \u0393 \u2286\u0393 \u0393' -> \u0393 \u2286\u0393 ((x , \u03c4\u1d62 , A) \u2237 \u0393')\n\u2286\u0393-\u2237 {\u0393 = []} x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393' = nil (cons x\u2209\u0393' \u03c4\u1d62\u2237A (\u2286\u0393-wf\u0393' \u0393\u2286\u0393'))\n\u2286\u0393-\u2237 {\u0393 = (x , \u03c4\u1d62 , A) \u2237 \u0393} x\u2209\u0393' \u03c4\u1d62\u2237A (cons (proj\u2081 , proj\u2082 , proj\u2083) \u0393\u2286\u0393') =\n  cons (proj\u2081 , ((there proj\u2082) , proj\u2083)) (\u2286\u0393-\u2237 x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393')\n\n\nsub\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4\nsub\u1d62\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u1d62 m \u2236 \u03c4\n\nsub\u0393 (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u0393\u2286\u0393' = var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u1d62-trans \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub\u0393 (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u0393\u2286\u0393' = app (sub\u0393 \u0393\u22a9m\u2236\u03c4 \u0393\u2286\u0393') (sub\u1d62\u0393 x \u0393\u2286\u0393') x\u2081 x\u2082\nsub\u0393 {\u0393' = \u0393'} (abs {\u03c4 = \u03c4} L cf (arr \u03c4\u2237A \u03c4'\u2237B)) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\u0393\n    (cf (\u2209-cons-l _ _ x\u2209))\n    (cons\n      (\u03c4 , (here refl) , (\u2286\u1d62-refl \u03c4\u2237A))\n      (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2237A \u03c4'\u2237B)\nsub\u0393 (Y x x\u2081 x\u2082 x\u2083) \u0393\u2286\u0393' = Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') x\u2081 x\u2082 x\u2083\n\nsub\u1d62\u0393 (nil wf-\u0393) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62\u0393 (cons x \u0393\u22a9\u1d62m\u2236\u03c4) \u0393\u2286\u0393' = cons (sub\u0393 x \u0393\u2286\u0393') (sub\u1d62\u0393 \u0393\u22a9\u1d62m\u2236\u03c4 \u0393\u2286\u0393')\n\n\n\u2208-\u2286\u1d62-trans : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229 \u03c4\u1d62) \u2286\u1d62[ A ] (\u2229 \u03c4\u2c7c) -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2c7c) \u00d7 \u03c4 \u2286[ A ] \u03c4')\n\u2208-\u2286\u1d62-trans (here refl) (cons x _) = x\n\u2208-\u2286\u1d62-trans (there \u03c4\u2208\u03c4\u1d62) (cons _ \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2208-\u2286\u1d62-trans \u03c4\u2208\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\nsub : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4'\nsub\u1d62 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9\u1d62 m \u2236 \u03c4 -> \u03c4' \u2286\u1d62[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u1d62 m \u2236 \u03c4'\n\nsub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u1d62-trans (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62) \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub (app \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u22a9\u1d62t\u2236\u03c4\u2081 \u03c4\u2286\u03c4\u2082 \u03c4\u2237A) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = app\n  (sub\u0393 \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u2286\u0393')\n  (sub\u1d62\u0393 \u0393\u22a9\u1d62t\u2236\u03c4\u2081 \u0393\u2286\u0393')\n  (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u2082)\n  \u03c4\u2237A\nsub {_} {\u0393} {\u0393'} (abs {\u03c4 = \u03c4} {\u03c4'} L {t} cf \u03c4~>\u03c4'\u2237A\u27f6B) (arr {A} {\u03c4\u2081\u2081 = \u03c4\u2081\u2081} \u03c4\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4' (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B) x\u2083) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\n    (cf (\u2209-cons-l _ _ x\u2209))\n    \u03c4\u2081\u2082\u2286\u03c4'\n    (cons (\u03c4\u2081\u2081 , (here refl) , \u03c4\u2286\u03c4\u2081\u2081) (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2081\u2081\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B)\nsub (Y wf-\u0393 (\u03c4' , \u03c4'\u2208\u03c4\u2081 , \u03c4~>\u03c4\u2286\u03c4') \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) (arr {\u03c4\u2081\u2081 = \u2229 \u03c4\u2081'} \u03c4\u2081\u2286\u03c4\u2081' \u03c4\u2082'\u2286\u03c4\u2082 (arr \u2229\u03c4\u2081'\u2237A\u27f6A \u03c4\u2082'\u2237A) x\u2084) \u0393\u2286\u0393' =\n  Y ((\u2286\u0393-wf\u0393' \u0393\u2286\u0393')) (\u03c4'' , (\u03c4''\u2208\u03c4\u2081' , \u2286-trans \u03c4~>\u03c4\u2286\u03c4' \u03c4'\u2286\u03c4'')) \u2229\u03c4\u2081'\u2237A\u27f6A (\u2286\u1d62-trans \u03c4\u2082'\u2286\u03c4\u2082 \u03c4\u2082\u2286\u03c4)\n  where\n  \u03c4'' = proj\u2081 (\u2208-\u2286\u1d62-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081')\n  \u03c4''\u2208\u03c4\u2081' = proj\u2081 (proj\u2082 (\u2208-\u2286\u1d62-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n  \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2208-\u2286\u1d62-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n\nsub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4 (nil x) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4'\u1d62\u2286\u03c4\u1d62) \u0393\u2286\u0393' with \u22a9\u1d62-\u2208-\u22a9 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n... | \u0393\u22a9m\u2236\u03c4 = cons (sub \u0393\u22a9m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393') (sub\u1d62 \u0393\u22a9\u1d62m\u2236\u03c4\u1d62 \u03c4'\u1d62\u2286\u03c4\u1d62 \u0393\u2286\u0393')\n\n\n\n\n-- \u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u1d62Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9e33de6b050c2a37f154b0e2d8e4beed3e79cc94","subject":"Small cleanups","message":"Small cleanups\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\nchangeSemVar : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSemVar t \u03c1 d\u03c1 = \u27e6 deriveVar t \u27e7Var (alternate \u03c1 d\u03c1)\n\nchangeSem : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSem t \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1))\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (v : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 v \u27e7Var (changeSemVar v)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1))\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term (changeSem t)\n\ncorrectDerive (const ()) \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 changeSem t \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (changeSem t \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 changeSem t \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295\n       \u27e6 derive t \u27e7Term (alternate \u03c11 d\u03c11)\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term (a \u2022 \u03c1) \u2295 \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const ()) \u03c1 d\u03c1 \u03c1d\u03c1\n","old_contents":"module New.Correctness where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\nchangeSemVar : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSemVar t \u03c1 d\u03c1 = \u27e6 deriveVar t \u27e7Var (alternate \u03c1 d\u03c1)\n\nchangeSem : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\nchangeSem t \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1))\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (v : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 v \u27e7Var (changeSemVar v)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1))\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term (changeSem t)\n\ncorrectDerive (const ()) \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive {\u0393} (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 changeSem t \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (changeSem s \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (changeSem t \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive {\u0393} (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 changeSem t \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive {\u0393 = \u0393} (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295\n       \u27e6 derive t \u27e7Term (alternate \u03c11 d\u03c11)\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term (a \u2022 \u03c1) \u2295 \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n      \u220e\n      )\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const ()) \u03c1 d\u03c1 \u03c1d\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"778f7ad4c660413c31589f449514267c50aefe9a","subject":"Restore the type params for ZK-chaum-pedersen-pok-elgamal-rnd","message":"Restore the type params for ZK-chaum-pedersen-pok-elgamal-rnd\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker.agda","new_file":"ZK\/JSChecker.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.ZqZp as ZqZp\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\nrecord ZK-chaum-pedersen-pok-elgamal-rnd (\u2124q \u2124p\u2605 : Set) : Set where\n  field\n    m c s : \u2124q\n    g p q y \u03b1 \u03b2 A B : \u2124p\u2605\n\nzk-check-chaum-pedersen-pok-elgamal-rnd! : ZK-chaum-pedersen-pok-elgamal-rnd BigI BigI \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd! pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks!\n      >> check! \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check! \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    params = record\n      { primality-test-probability-bound = primality-test-probability-bound\n      ; min-bits-q = min-bits-q\n      ; min-bits-p = min-bits-p\n      ; qI = I.q\n      ; pI = I.p\n      ; gI = I.g\n      }\n    open module [\u2124q]\u2124p\u2605 = ZqZp params\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n    check! \"type of statement\" (typ === fromString cpt)\n                               (\u03bb _ \u2192 \"Expected type of statement: \" ++ cpt ++ \" not \" ++ toString typ)\n >> zk-check-chaum-pedersen-pok-elgamal-rnd! pok\n module Zk-check where\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.ZqZp as ZqZp\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- TODO bug (undefined)!\nrecord ZK-chaum-pedersen-pok-elgamal-rnd {--(\u2124q \u2124p\u2605 : Set)--} : Set where\n  field\n    m c s : BigI {--\u2124q--}\n    g p q y \u03b1 \u03b2 A B : BigI --\u2124p\u2605\n\nzk-check-chaum-pedersen-pok-elgamal-rnd! : ZK-chaum-pedersen-pok-elgamal-rnd {-BigI BigI-} \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd! pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks!\n      >> check! \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check! \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module ZK-check-chaum-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    params = record\n      { primality-test-probability-bound = primality-test-probability-bound\n      ; min-bits-q = min-bits-q\n      ; min-bits-p = min-bits-p\n      ; qI = I.q\n      ; pI = I.p\n      ; gI = I.g\n      }\n    open module [\u2124q]\u2124p\u2605 = ZqZp params\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n    check! \"type of statement\" (typ === fromString cpt)\n                               (\u03bb _ \u2192 \"Expected type of statement: \" ++ cpt ++ \" not \" ++ toString typ)\n >> zk-check-chaum-pedersen-pok-elgamal-rnd! pok\n module Zk-check where\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f44b292352a83a6d7005c590597734364cc640d4","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 4508bb8c9230c5e134fb68a3f18e1b46\n\ndarcs-hash:20120223034355-3bd4e-d6cf87bc79b145b0863c84b642a1ac9363302e23.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites, tested versions of the ATPS, and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fossacs-2012\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import PredicateLogic.Base\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemata\nopen import PredicateLogic.SchemataATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import FOL.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import FOL.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Standard.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.Standard.PropertiesATP\nopen import PA.Axiomatic.Standard.PropertiesI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.EliminationProperties\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fossacs-2012\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import PredicateLogic.Base\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemata\nopen import PredicateLogic.SchemataATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import FOL.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import FOL.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Standard.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.Standard.PropertiesATP\nopen import PA.Axiomatic.Standard.PropertiesI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.EliminationProperties\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0572405d5e21384cb228e8c6aca51fb63ee7b696","subject":"CachingEvaluation: switch to M{Type,Term}","message":"CachingEvaluation: switch to M{Type,Term}\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\u2081\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u03c4\u2081 )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\u2081\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values (where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things)!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.CBPVType as CBPVType\n  open CBPVType.Structure Base\n  open import Parametric.Syntax.CBPVTerm as CBPVTerm\n  open CBPVTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n  \u27e6 \u03c4\u2081 \u03a0 \u03c4\u2082 \u27e7CompType = \u27e6 \u03c4\u2081 \u27e7CompType \u00d7 \u27e6 \u03c4\u2082 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.>\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\u2081\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u03c4\u2081 )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6 \u03c4\u2081 \u03a0 \u03c4\u2082 \u27e7CompTypeHidCache = \u27e6 \u03c4\u2081 \u27e7CompTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\u2081\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values (where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things)!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cProduce v) \u03c1 = , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n\n  -- But if we alter _into_ as described above, composing the caches works!\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into2 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cProduce time, we just need the nested into2 to do their job, because as I\n  -- said intermediate results are a a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using into2 for the same reasons - which makes\n  -- sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into2 case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cProduce. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"244c3b4285d455271121fa1cc4982d804ffe9437","subject":"Fix a note","message":"Fix a note\n","repos":"crypto-agda\/crypto-agda","old_file":"Solver\/Linear.agda","new_file":"Solver\/Linear.agda","new_contents":"-- NOTO with-K\nmodule Solver.Linear where\n\nopen import FunUniverse.Types\nopen import FunUniverse.Rewiring.Linear\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat as \u2115 using (\u2115)\nimport Solver.Linear.Syntax as LinSyn\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl; _\u2262_)\nopen import Relation.Nullary\nimport Data.String as String\nmodule String\u2264 = StrictTotalOrder String.strictTotalOrder\nopen import Data.Fin using (Fin)\nopen import Data.Fin.Props using (strictTotalOrder) renaming (_\u225f_ to _\u225f\u1da0_)\nopen import Data.Vec using (Vec; lookup)\nopen import Data.Vec.N-ary using (N-ary ; _$\u207f_)\nopen import Data.Vec using (allFin) renaming (map to vmap)\n\nmodule Syntax\n  {Var : Set}\n  (_\u2264V\u2082_ : Var \u2192 Var \u2192 \ud835\udfda)\n  (_\u225fV_ : Decidable (_\u2261_ {A = Var}))\n  {a} {A : Set a}\n  {funU : FunUniverse A}\n  (linRewiring : LinRewiring funU)\n  (\u0393 : Var \u2192 A)\n where\n\n  open FunUniverse funU\n  open LinRewiring linRewiring\n  open LinSyn Var\n\n  eval : Syn \u2192 A\n  eval (var x) = \u0393 x\n  eval tt = `\u22a4\n  eval (s , s\u2081) = eval s `\u00d7 eval s\u2081\n\n  EvalEq : Eq \u2192 Set\n  EvalEq e = eval (LHS e) `\u2192 eval (RHS e)\n\n  data R : Syn \u2192 Syn \u2192 Set a where\n    _``\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    `<id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R (A , tt) A\n    `<tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R (tt , A) A\n    `<tt,id> : \u2200 {A} \u2192 R A (tt , A)\n    `<id,tt> : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_``\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    `assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    `assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    `id : \u2200 {A} \u2192 R A A\n    `swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  `\u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  `\u27e8 `id \u00d7 `id \u27e9 = `id\n  `\u27e8 r\u2081  \u00d7  r\u2082 \u27e9 = \u27e8 r\u2081 ``\u00d7 r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  `id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b `id = r\u2081\n  `<tt,id>\u207b\u00b9 \u223b `<tt,id> = `id\n  `<id,tt>\u207b\u00b9 \u223b `<id,tt> = `id\n  `<tt,id> \u223b `<tt,id>\u207b\u00b9 = `id\n  `<id,tt> \u223b `<id,tt>\u207b\u00b9 = `id\n  `swap \u223b `<id,tt>\u207b\u00b9 = `<tt,id>\u207b\u00b9\n  `swap \u223b `<tt,id>\u207b\u00b9 = `<id,tt>\u207b\u00b9\n  `<id,tt> \u223b `swap = `<tt,id>\n  `<tt,id> \u223b `swap = `<id,tt>\n  `assoc \u223b `assoc\u207b\u00b9 = `id\n  `assoc \u223b (`assoc\u207b\u00b9 ``\u223b r) = r\n  `assoc\u207b\u00b9 \u223b `assoc = `id\n  `assoc\u207b\u00b9 \u223b (`assoc ``\u223b r) = r\n  `swap \u223b `swap = `id\n  `swap \u223b (`swap ``\u223b r) = r\n  (r\u2081 ``\u223b r\u2082) \u223b r\u2083 = r\u2081 ``\u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 ``\u00d7 r\u2082 \u27e9 \u223b \u27e8 r\u2083 ``\u00d7 r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 ``\u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 ``\u00d7 r\u2082 \u27e9 \u223b ( \u27e8 r\u2083 ``\u00d7 r\u2084 \u27e9 ``\u223b r\u2085) with \u27e8 r\u2081 \u223b r\u2083 ``\u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0`id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 ``\u223b r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 ``\u223b r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r ``\u223b r\u2081) = sym r\u2081 \u223b sym r\n  sym `<id,tt> = `<id,tt>\u207b\u00b9\n  sym `<tt,id> = `<tt,id>\u207b\u00b9\n  sym `<id,tt>\u207b\u00b9 = `<id,tt>\n  sym `<tt,id>\u207b\u00b9 = `<tt,id>\n  sym \u27e8 r ``\u00d7 r\u2081 \u27e9 = \u27e8 sym r ``\u00d7 sym r\u2081 \u27e9\n  sym `assoc = `assoc\u207b\u00b9\n  sym `assoc\u207b\u00b9 = `assoc\n  sym `id = `id\n  sym `swap = `swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 eval S `\u2192 eval S'\n  proof\u2081 (r ``\u223b r\u2081) = proof\u2081 r \u204f proof\u2081 r\u2081\n  proof\u2081 `<id,tt> = <id,tt>\n  proof\u2081 `<tt,id> = <tt,id>\n  proof\u2081 `<id,tt>\u207b\u00b9 = fst<,tt>\n  proof\u2081 `<tt,id>\u207b\u00b9 = snd<tt,>\n  proof\u2081 \u27e8 `id ``\u00d7 r \u27e9 = second (proof\u2081 r)\n  proof\u2081 \u27e8 r ``\u00d7 `id \u27e9 = first  (proof\u2081 r)\n  proof\u2081 \u27e8 r ``\u00d7 r\u2081 \u27e9  = < proof\u2081 r \u00d7 proof\u2081 r\u2081 >\n  proof\u2081 `assoc = assoc\u2032\n  proof\u2081 `assoc\u207b\u00b9 = assoc\n  proof\u2081 `id = id\n  proof\u2081 `swap = swap\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 eval S' `\u2192 eval S\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set a where\n    tt : NF tt\n    var : \u2200 x \u2192 NF (var x)\n    var_::_ : \u2200 {S} i \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set a where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 `<tt,id>\n  merge (var i) n2 = (var i :: n2) \u22a2 `id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 `id ``\u00d7 p \u27e9 \u223b `assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 `id\n  norm tt = tt \u22a2 `id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 ``\u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} x \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 `<id,tt>\n  insert y (var i) with y \u2264V\u2082 i\n  ... | 1\u2082 = (var y :: var i) \u22a2 `id\n  ... | 0\u2082 = (var i :: var y) \u22a2 `swap\n  insert y (var i :: n1) with y \u2264V\u2082 i\n  ... | 1\u2082 = (var y :: (var i :: n1)) \u22a2 `id\n  ... | 0\u2082 with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 `id ``\u00d7 p \u27e9 \u223b (`assoc \u223b (\u27e8 `swap ``\u00d7 `id \u27e9 \u223b `assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 `id\n  sort (var i) = var i \u22a2 `id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 `id ``\u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = `id\n\n  var-inj : \u2200 {i j} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x \u225fV x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  data equation-not-ok : Set where\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = equation-not-ok\n\n  EqOk? : Eq \u2192 Set\n  EqOk? e = CHECK (NFP.S' (normal (LHS e))) (NFP.S' (normal (RHS e)))\n\n  rewire : (e : Eq) {prf : EqOk? e} \u2192 EvalEq e\n  rewire (s\u2081 \u21a6 s\u2082) {eq} with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire (_ \u21a6 _) {()} | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\nmodule Syntax\u1da0 {a} {A : Set a} {funU : FunUniverse A} (linRewiring : LinRewiring funU) where\n  module _ {n} where\n    open LinSyn (Fin n) public\n\n  module _ {n} (\u0393 : Vec A n)\n           (f : N-ary n (LinSyn.Syn (Fin n)) (LinSyn.Eq (Fin n)))\n           where\n    open Syntax {Fin n} ((\u03bb x y \u2192 \u230a StrictTotalOrder._<?_ (strictTotalOrder n) x y \u230b))\n                _\u225f\u1da0_ {a} {A} {funU} linRewiring (\u03bb i \u2192 lookup i \u0393) public\n\n    private\n      e = f $\u207f vmap Syn.var (allFin n)\n\n    rewire\u1da0 : {e-ok : EqOk? e} \u2192 EvalEq e\n    rewire\u1da0 {e-ok} = rewire e {e-ok}\n\nmodule Syntax\u02e2 {a} {A} {funU} linRewiring \u0393 where\n  open Syntax (\u03bb x y \u2192 \u230a String\u2264._<?_ x y \u230b) String._\u225f_ {a} {A} {funU} linRewiring \u0393 public\n  open import Solver.Linear.Parser\n\n  rewire\u02e2 : \u2200 s {s-ok} \u2192\n            let e = parseEq\u02e2 s {s-ok} in\n            {e-ok : EqOk? e} \u2192 EvalEq e\n  rewire\u02e2 s {s-ok} {e-ok} = rewire (parseEq\u02e2 s {s-ok}) {e-ok}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- NOTE with-K\nmodule Solver.Linear where\n\nopen import FunUniverse.Types\nopen import FunUniverse.Rewiring.Linear\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat as \u2115 using (\u2115)\nimport Solver.Linear.Syntax as LinSyn\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl; _\u2262_)\nopen import Relation.Nullary\nimport Data.String as String\nmodule String\u2264 = StrictTotalOrder String.strictTotalOrder\nopen import Data.Fin using (Fin)\nopen import Data.Fin.Props using (strictTotalOrder) renaming (_\u225f_ to _\u225f\u1da0_)\nopen import Data.Vec using (Vec; lookup)\nopen import Data.Vec.N-ary using (N-ary ; _$\u207f_)\nopen import Data.Vec using (allFin) renaming (map to vmap)\n\nmodule Syntax\n  {Var : Set}\n  (_\u2264V\u2082_ : Var \u2192 Var \u2192 \ud835\udfda)\n  (_\u225fV_ : Decidable (_\u2261_ {A = Var}))\n  {a} {A : Set a}\n  {funU : FunUniverse A}\n  (linRewiring : LinRewiring funU)\n  (\u0393 : Var \u2192 A)\n where\n\n  open FunUniverse funU\n  open LinRewiring linRewiring\n  open LinSyn Var\n\n  eval : Syn \u2192 A\n  eval (var x) = \u0393 x\n  eval tt = `\u22a4\n  eval (s , s\u2081) = eval s `\u00d7 eval s\u2081\n\n  EvalEq : Eq \u2192 Set\n  EvalEq e = eval (LHS e) `\u2192 eval (RHS e)\n\n  data R : Syn \u2192 Syn \u2192 Set a where\n    _``\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    `<id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R (A , tt) A\n    `<tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R (tt , A) A\n    `<tt,id> : \u2200 {A} \u2192 R A (tt , A)\n    `<id,tt> : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_``\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    `assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    `assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    `id : \u2200 {A} \u2192 R A A\n    `swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  `\u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  `\u27e8 `id \u00d7 `id \u27e9 = `id\n  `\u27e8 r\u2081  \u00d7  r\u2082 \u27e9 = \u27e8 r\u2081 ``\u00d7 r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  `id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b `id = r\u2081\n  `<tt,id>\u207b\u00b9 \u223b `<tt,id> = `id\n  `<id,tt>\u207b\u00b9 \u223b `<id,tt> = `id\n  `<tt,id> \u223b `<tt,id>\u207b\u00b9 = `id\n  `<id,tt> \u223b `<id,tt>\u207b\u00b9 = `id\n  `swap \u223b `<id,tt>\u207b\u00b9 = `<tt,id>\u207b\u00b9\n  `swap \u223b `<tt,id>\u207b\u00b9 = `<id,tt>\u207b\u00b9\n  `<id,tt> \u223b `swap = `<tt,id>\n  `<tt,id> \u223b `swap = `<id,tt>\n  `assoc \u223b `assoc\u207b\u00b9 = `id\n  `assoc \u223b (`assoc\u207b\u00b9 ``\u223b r) = r\n  `assoc\u207b\u00b9 \u223b `assoc = `id\n  `assoc\u207b\u00b9 \u223b (`assoc ``\u223b r) = r\n  `swap \u223b `swap = `id\n  `swap \u223b (`swap ``\u223b r) = r\n  (r\u2081 ``\u223b r\u2082) \u223b r\u2083 = r\u2081 ``\u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 ``\u00d7 r\u2082 \u27e9 \u223b \u27e8 r\u2083 ``\u00d7 r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 ``\u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 ``\u00d7 r\u2082 \u27e9 \u223b ( \u27e8 r\u2083 ``\u00d7 r\u2084 \u27e9 ``\u223b r\u2085) with \u27e8 r\u2081 \u223b r\u2083 ``\u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0`id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 ``\u223b r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 ``\u223b r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r ``\u223b r\u2081) = sym r\u2081 \u223b sym r\n  sym `<id,tt> = `<id,tt>\u207b\u00b9\n  sym `<tt,id> = `<tt,id>\u207b\u00b9\n  sym `<id,tt>\u207b\u00b9 = `<id,tt>\n  sym `<tt,id>\u207b\u00b9 = `<tt,id>\n  sym \u27e8 r ``\u00d7 r\u2081 \u27e9 = \u27e8 sym r ``\u00d7 sym r\u2081 \u27e9\n  sym `assoc = `assoc\u207b\u00b9\n  sym `assoc\u207b\u00b9 = `assoc\n  sym `id = `id\n  sym `swap = `swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 eval S `\u2192 eval S'\n  proof\u2081 (r ``\u223b r\u2081) = proof\u2081 r \u204f proof\u2081 r\u2081\n  proof\u2081 `<id,tt> = <id,tt>\n  proof\u2081 `<tt,id> = <tt,id>\n  proof\u2081 `<id,tt>\u207b\u00b9 = fst<,tt>\n  proof\u2081 `<tt,id>\u207b\u00b9 = snd<tt,>\n  proof\u2081 \u27e8 `id ``\u00d7 r \u27e9 = second (proof\u2081 r)\n  proof\u2081 \u27e8 r ``\u00d7 `id \u27e9 = first  (proof\u2081 r)\n  proof\u2081 \u27e8 r ``\u00d7 r\u2081 \u27e9  = < proof\u2081 r \u00d7 proof\u2081 r\u2081 >\n  proof\u2081 `assoc = assoc\u2032\n  proof\u2081 `assoc\u207b\u00b9 = assoc\n  proof\u2081 `id = id\n  proof\u2081 `swap = swap\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 eval S' `\u2192 eval S\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set a where\n    tt : NF tt\n    var : \u2200 x \u2192 NF (var x)\n    var_::_ : \u2200 {S} i \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set a where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 `<tt,id>\n  merge (var i) n2 = (var i :: n2) \u22a2 `id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 `id ``\u00d7 p \u27e9 \u223b `assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 `id\n  norm tt = tt \u22a2 `id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 ``\u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} x \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 `<id,tt>\n  insert y (var i) with y \u2264V\u2082 i\n  ... | 1\u2082 = (var y :: var i) \u22a2 `id\n  ... | 0\u2082 = (var i :: var y) \u22a2 `swap\n  insert y (var i :: n1) with y \u2264V\u2082 i\n  ... | 1\u2082 = (var y :: (var i :: n1)) \u22a2 `id\n  ... | 0\u2082 with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 `id ``\u00d7 p \u27e9 \u223b (`assoc \u223b (\u27e8 `swap ``\u00d7 `id \u27e9 \u223b `assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 `id\n  sort (var i) = var i \u22a2 `id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 `id ``\u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = `id\n\n  var-inj : \u2200 {i j} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x \u225fV x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  data equation-not-ok : Set where\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = equation-not-ok\n\n  EqOk? : Eq \u2192 Set\n  EqOk? e = CHECK (NFP.S' (normal (LHS e))) (NFP.S' (normal (RHS e)))\n\n  rewire : (e : Eq) {prf : EqOk? e} \u2192 EvalEq e\n  rewire (s\u2081 \u21a6 s\u2082) {eq} with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire (_ \u21a6 _) {()} | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\nmodule Syntax\u1da0 {a} {A : Set a} {funU : FunUniverse A} (linRewiring : LinRewiring funU) where\n  module _ {n} where\n    open LinSyn (Fin n) public\n\n  module _ {n} (\u0393 : Vec A n)\n           (f : N-ary n (LinSyn.Syn (Fin n)) (LinSyn.Eq (Fin n)))\n           where\n    open Syntax {Fin n} ((\u03bb x y \u2192 \u230a StrictTotalOrder._<?_ (strictTotalOrder n) x y \u230b))\n                _\u225f\u1da0_ {a} {A} {funU} linRewiring (\u03bb i \u2192 lookup i \u0393) public\n\n    private\n      e = f $\u207f vmap Syn.var (allFin n)\n\n    rewire\u1da0 : {e-ok : EqOk? e} \u2192 EvalEq e\n    rewire\u1da0 {e-ok} = rewire e {e-ok}\n\nmodule Syntax\u02e2 {a} {A} {funU} linRewiring \u0393 where\n  open Syntax (\u03bb x y \u2192 \u230a String\u2264._<?_ x y \u230b) String._\u225f_ {a} {A} {funU} linRewiring \u0393 public\n  open import Solver.Linear.Parser\n\n  rewire\u02e2 : \u2200 s {s-ok} \u2192\n            let e = parseEq\u02e2 s {s-ok} in\n            {e-ok : EqOk? e} \u2192 EvalEq e\n  rewire\u02e2 s {s-ok} {e-ok} = rewire (parseEq\u02e2 s {s-ok}) {e-ok}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"31b0809974b692e148c09162379a05141e38419f","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 24dc33dfbb72399aa465f5826dda43ee\n\ndarcs-hash:20110511135414-3bd4e-b7772086e62e149171f7da14329c07df8eb500c3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/Collatz\/Data\/Nat.agda","new_file":"src\/FOTC\/Program\/Collatz\/Data\/Nat.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC natural numbers (added for the Collatz function example)\n\n-- (We don't want populate the FOTC library with more FOL axioms)\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.Collatz.Data.Nat where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\n\ninfixr 12 _^_\ninfixl 10 _\/_\n\n------------------------------------------------------------------------------\n\npostulate\n  _\/_   : D \u2192 D \u2192 D\n  \/-x<y : \u2200 {m n} \u2192 LT m n \u2192 m \/ n \u2261 zero\n  \/-x\u2265y : \u2200 {m n} \u2192 GE m n \u2192 m \/ n \u2261 succ ((m \u2238 n) \/ n)\n{-# ATP axiom \/-x<y #-}\n{-# ATP axiom \/-x\u2265y #-}\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192   n ^ zero   \u2261 one\n  ^-S : \u2200 m n \u2192 m ^ succ n \u2261 m * m ^ n\n{-# ATP axiom ^-0 #-}\n{-# ATP axiom ^-S #-}\n\n-- Some predicates\n\nEven : D \u2192 Set\nEven n = \u2203 \u03bb k \u2192 n \u2261 two * k\n\nOdd : D \u2192 Set\nOdd n = \u2203 \u03bb k \u2192 n \u2261 two * k + one\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC natural numbers (added for the Collatz function example)\n\n-- (We don't want populate the FOTC library with more FOL axioms)\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.Collatz.Data.Nat where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\n\ninfixr 12 _^_\ninfixl 10 _\/_\n\n------------------------------------------------------------------------------\n\npostulate\n  _\/_   : D \u2192 D \u2192 D\n  \/-x<y : \u2200 {m n} \u2192 LT m n \u2192 m \/ n \u2261 zero\n  \/-x\u2265y : \u2200 {m n} \u2192 GE m n \u2192 m \/ n \u2261 succ ((m \u2238 n) \/ n)\n{-# ATP axiom \/-x<y #-}\n{-# ATP axiom \/-x\u2265y #-}\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192   n ^ zero   \u2261 one\n  ^-S : \u2200 m n \u2192 m ^ succ n \u2261 m * m ^ n\n{-# ATP axiom ^-0 #-}\n{-# ATP axiom ^-S #-}\n\n-- Some predicates\nEven : D \u2192 Set\nEven n = \u2203 (\u03bb k \u2192 n \u2261 two * k)\n\nOdd : D \u2192 Set\nOdd n = \u2203 (\u03bb k \u2192 n \u2261 two * k + one)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"611ff4923bf4d5eb58f58db07f5da7d855582edf","subject":"Field: add \/1-id","message":"Field: add \/1-id\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Data.Nat.NP using (\u2115; zero; fold) renaming (suc to 1+)\nopen import Data.Integer using (\u2124; +_; -[1+_])\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Group\nopen import Algebra.Group.Abelian\nopen import HoTT\n\nmodule Algebra.Field where\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 2+_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-grp-ops : Group-Ops A\n\n  open Additive-Group-Ops       +-grp-ops public\n    renaming (2\u2297_ to 2*_)\n  open Multiplicative-Group-Ops *-grp-ops public\n\n  suc : A \u2192 A\n  suc = _+_ `1\n\n  pred : A \u2192 A\n  pred x = x \u2212 `1\n\n  `2 `3 `-0 `-1 : A\n  `2  = suc `1\n  `3  = suc `2\n  `-0 = 0\u2212 `0\n  `-1 = 0\u2212 `1\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0    ] = `0\n  \u2115[ 1+ n ] = fold `1 suc n\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ + x      ] = \u2115[ x ]\n  \u2124[ -[1+ x ] ] = 0\u2212 \u2115[ 1+ x ]\n\n  -- TODO use _^\u207a_ and _^_ from group\/monoid\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0      = `1\n  b ^\u2115 (1+ e) = fold b (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (+ x)    = b ^\u2115 x\n  b ^\u2124 -[1+ x ] = (b ^\u2115 (1+ x))\u207b\u00b9\n\n  2+_ : A \u2192 A\n  2+ x = `2 + x\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open Field-Ops field-ops public\n  open \u2261-Reasoning\n\n  field\n    0\u22621                  : `0 \u2262 `1\n    +-abelian-grp-struct : Abelian-Group-Struct +-grp-ops\n\n    *-comm-mon-struct    : Commutative-Monoid-Struct *-mon-ops\n    \u207b\u00b9-inverse           : InverseNonZero `0 `1 _\u207b\u00b9 _*_\n    *-+-distrs           : _*_ DistributesOver _+_\n\n  open Additive-Abelian-Group-Struct +-abelian-grp-struct public\n  -- 0\u2212-involutive   : Involutive 0\u2212_\n  -- cancels-+-left  : LeftCancel _+_\n  -- cancels-+-right : RightCancel _+_\n  -- ...\n\n  open Multiplicative-Commutative-Monoid-Struct *-comm-mon-struct public\n\n  +-grp-struct : Group-Struct +-grp-ops\n  +-grp-struct = Abelian-Group-Struct.grp-struct +-abelian-grp-struct\n\n  +-abelian-grp : Abelian-Group A\n  +-abelian-grp = +-grp-ops , +-abelian-grp-struct\n\n  +-grp : Group A\n  +-grp = +-grp-ops , +-grp-struct\n\n  -- Since the Abelian-Group module includes all what Group and\n  -- Monoid provide, we can expose a single module.\n  module +-Grp = Abelian-Group +-abelian-grp\n\n  *-mon-struct : Monoid-Struct *-mon-ops\n  *-mon-struct = Commutative-Monoid-Struct.mon-struct *-comm-mon-struct\n\n  *-mon : Monoid A\n  *-mon = *-mon-ops , *-mon-struct\n\n  *-comm-mon : Commutative-Monoid A\n  *-comm-mon = *-mon-ops , *-comm-mon-struct\n\n  module *-Mon = Commutative-Monoid *-comm-mon\n\n  1\u22620 : `1 \u2262 `0\n  1\u22620 e = 0\u22621 (! e)\n\n  NonZero : A \u2192 Set \u2113\n  NonZero x = x \u2262 `0\n\n  A\/0 = \u03a3 A NonZero\n\n  \u207b\u00b9-left-inverse : LeftInverseNonZero `0 `1 _\u207b\u00b9 _*_\n  \u207b\u00b9-left-inverse = fst \u207b\u00b9-inverse\n\n  \u207b\u00b9-right-inverse : RightInverseNonZero `0 `1 _\u207b\u00b9 _*_\n  \u207b\u00b9-right-inverse = snd \u207b\u00b9-inverse\n\n  *-+-distr : _*_ DistributesOver\u02e1 _+_\n  *-+-distr = fst *-+-distrs\n\n  *0-zero : RightZero `0 _*_\n  *0-zero = cancels-+-right  (+= idp (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= idp 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero `0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  c\u207b\u00b9*c*x : \u2200 {c x} \u2192 c \u2262 `0 \u2192 c \u207b\u00b9 * c * x \u2261 x\n  c\u207b\u00b9*c*x c\u22620 = *= (\u207b\u00b9-left-inverse c\u22620) idp \u2219 1*-identity\n\n  cancels-*-left : LeftCancelNonZero `0 _*_\n  cancels-*-left c\u22620 p = ! c\u207b\u00b9*c*x c\u22620 \u2219 *-!assoc= p \u2219 c\u207b\u00b9*c*x c\u22620\n\n  cancels-*-right : RightCancelNonZero `0 _*_\n  cancels-*-right c\u22620 p = cancels-*-left c\u22620 (*-comm= p)\n\n  unique-0 : \u2200 {x y} \u2192 x \u2262 `0 \u2192 x * y \u2261 `0 \u2192 y \u2261 `0\n  unique-0 nx xy\u22610 = cancels-*-left nx (xy\u22610 \u2219 ! *0-zero)\n\n  no-zero-divisor : \u2200 {x y} \u2192 x \u2262 `0 \u2192 y \u2262 `0 \u2192 x * y \u2262 `0\n  no-zero-divisor nx ny x*y\u22610 = ny (unique-0 nx x*y\u22610)\n\n  _*\/0_ : A\/0 \u2192 A\/0 \u2192 A\/0\n  (x , x\u22620) *\/0 (y , y\u22620) = x * y , no-zero-divisor x\u22620 y\u22620\n\n  ``1 : A\/0\n  ``1 = `1 , 1\u22620\n\n  *-mon-ops\/0 : Monoid-Ops A\/0\n  *-mon-ops\/0 = _*\/0_ , ``1\n\n  \u207b\u00b9-non-zero : \u2200 {x} \u2192 x \u2262 `0 \u2192 x \u207b\u00b9 \u2262 `0\n  \u207b\u00b9-non-zero x\u22620 = \u03bb p \u2192 0\u22621 (! 0*-zero \u2219 *= (! p) idp \u2219 \u207b\u00b9-left-inverse x\u22620)\n\n  _\u207b\u00b9\/0 : A\/0 \u2192 A\/0\n  (x , x\u22620) \u207b\u00b9\/0 = x \u207b\u00b9 , \u207b\u00b9-non-zero x\u22620\n\n  *-grp-ops\/0 : Group-Ops A\/0\n  *-grp-ops\/0 = *-mon-ops\/0 , _\u207b\u00b9\/0\n\n  module _ {{_ : FunExt}} where\n    A\/0= : {x y : A\/0}(p : fst x \u2261 fst y) \u2192 x \u2261 y\n    A\/0= p = pair= p (\u00ac-has-all-paths _ _)\n\n    *-mon-struct\/0 : Monoid-Struct *-mon-ops\/0\n    *-mon-struct\/0 = (A\/0= *-assoc , A\/0= *-!assoc)\n                   , A\/0= 1*-identity , A\/0= *1-identity\n\n    \u207b\u00b9-left-inverse\/0 : LeftInverse ``1 _\u207b\u00b9\/0 _*\/0_\n    \u207b\u00b9-left-inverse\/0 {x , x\u22620} = A\/0= (\u207b\u00b9-left-inverse x\u22620)\n\n    *-grp-struct\/0 : Group-Struct *-grp-ops\/0\n    *-grp-struct\/0 = *-mon-struct\/0\n                   , (\u03bb {x} \u2192 \u207b\u00b9-left-inverse\/0 {x}) ,\n                     (\u03bb {x} \u2192 A\/0= (\u207b\u00b9-right-inverse (snd x)))\n\n    *-grp\/0 : Group A\/0\n    *-grp\/0 = *-grp-ops\/0 , *-grp-struct\/0\n\n    *-abelian-grp-struct\/0 : Abelian-Group-Struct *-grp-ops\/0\n    *-abelian-grp-struct\/0 = *-grp-struct\/0 , A\/0= *-comm\n\n    *-abelian-grp\/0 : Abelian-Group A\/0\n    *-abelian-grp\/0 = *-grp-ops\/0 , *-abelian-grp-struct\/0\n    -- module *-Grp\/0 = Abelian-Group *-abelian-grp\/0\n\n  0\u2212-left-inverse : LeftInverse `0 0\u2212_ _+_\n  0\u2212-left-inverse = fst 0\u2212-inverse\n\n  0\u2212-right-inverse : RightInverse `0 0\u2212_ _+_\n  0\u2212-right-inverse = snd 0\u2212-inverse\n\n  0\u2212= : \u2200 {x y} \u2192 x \u2261 y \u2192 0\u2212 x \u2261 0\u2212 y\n  0\u2212= = ap 0\u2212_\n\n  \u207b\u00b9= : \u2200 {x y} \u2192 x \u2261 y \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9\n  \u207b\u00b9= = ap _\u207b\u00b9\n\n  *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n  *-+-distr\u02b3 = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  \/-+-distr\u02b3 : _\/_ DistributesOver\u02b3 _+_\n  \/-+-distr\u02b3 = *-+-distr\u02b3\n\n  0\u2212c+c+x : \u2200 {c x} \u2192 0\u2212 c + c + x \u2261 x\n  0\u2212c+c+x = += 0\u2212-left-inverse idp \u2219 0+-identity\n\n  -0\u22610 : `-0 \u2261 `0\n  -0\u22610 = 0\u22120\u22610\n\n  0\u2262-1 : `0 \u2262 `-1\n  0\u2262-1 0\u2261-1 = 0\u22621 (! 0\u2212-left-inverse \u2219 +-comm \u2219 ! ap suc 0\u2261-1 \u2219 +0-identity)\n\n  0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n  0\u2212-*-distr = cancels-+-right (0\u2212-left-inverse \u2219 ! 0*-zero \u2219 *= (! 0\u2212-left-inverse) idp \u2219 *-+-distr\u02b3)\n\n  -1*-neg : \u2200 {x} \u2192 `-1 * x \u2261 0\u2212 x\n  -1*-neg = ! 0\u2212-*-distr \u2219 0\u2212= 1*-identity\n\n  1\u207b\u00b9\u22611 : `1 \u207b\u00b9 \u2261 `1\n  1\u207b\u00b9\u22611 = ! *1-identity \u2219 \u207b\u00b9-left-inverse (\u03bb p \u2192 0\u22621 (! p))\n\n  \/1-id : \u2200 {x} \u2192 x \/ `1 \u2261 x\n  \/1-id = *= idp 1\u207b\u00b9\u22611 \u2219 *1-identity\n\n  2*-spec : \u2200 {n} \u2192 2* n \u2261 `2 * n\n  2*-spec = ! += 1*-identity 1*-identity \u2219 ! *-+-distr\u02b3\n\n  *-unique-inverse : \u2200 {x y} \u2192 x \u2262 `0 \u2192 x * y \u2261 `1 \u2192 y \u2261 x \u207b\u00b9\n  *-unique-inverse x\u22620 xy=1 = ! 1*-identity \u2219 *= (! \u207b\u00b9-left-inverse x\u22620) idp \u2219 *-assoc \u2219 *= idp xy=1 \u2219 *1-identity\n\n  \u207b\u00b9-involutive : \u2200 {x} \u2192 x \u2262 `0 \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n  \u207b\u00b9-involutive x\u22620 = ! *-unique-inverse (\u207b\u00b9-non-zero x\u22620) (\u207b\u00b9-left-inverse x\u22620)\n\n  \u207b\u00b9*-distr : \u2200 {x y} \u2192 x \u2262 `0 \u2192 y \u2262 `0 \u2192 (x * y)\u207b\u00b9 \u2261 x \u207b\u00b9  \/ y\n  \u207b\u00b9*-distr x\u22620 y\u22620 =\n    ! (*-unique-inverse (no-zero-divisor x\u22620 y\u22620) (*= idp *-comm \u2219 *-assoc \u2219 *= idp (! *-assoc \u2219 *= (\u207b\u00b9-right-inverse y\u22620) idp \u2219 1*-identity) \u2219 \u207b\u00b9-right-inverse x\u22620))\n\n  +-quotient : \u2200 {a b a' b'} \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n    \u2192 (a \/ b) + (a' \/ b') \u2261 (a * b' + a' * b) \/ (b * b')\n  +-quotient b b'\n    = += (*= (! *1-identity \u2219 *= idp (! \u207b\u00b9-left-inverse b') \u2219 *= idp *-comm \u2219 ! *-assoc) idp \u2219\n          *-assoc \u2219 *= idp (! \u207b\u00b9*-distr b' b \u2219 \u207b\u00b9= *-comm))\n         (*= (! *1-identity \u2219 *= idp (! \u207b\u00b9-left-inverse b ) \u2219 *= idp *-comm) idp \u2219\n          *-!assoc= *-assoc \u2219 *= idp (! \u207b\u00b9*-distr b b'))\n    \u2219 ! \/-+-distr\u02b3\n\n  \u2212-quotient : \u2200 {a b a' b'} \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n    \u2192 (a \/ b) \u2212 (a' \/ b') \u2261 (a * b' \u2212 a' * b) \/ (b * b')\n  \u2212-quotient b b' = += idp 0\u2212-*-distr \u2219 +-quotient b b' \u2219 \/= (+= idp (! 0\u2212-*-distr)) idp\n\n  *-quotient : \u2200 {a b a' b'} \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n    \u2192 (a \/ b) * (a' \/ b') \u2261 (a * a') \/ (b * b')\n  *-quotient b\u22620 b'\u22620 = *-interchange \u2219 *= idp (! \u207b\u00b9*-distr b\u22620 b'\u22620)\n\n  \/-quotient : \u2200 {a b a' b'} \u2192 a' \u2262 `0 \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n     \u2192 (a \/ b) \/ (a' \/ b') \u2261 (a * b') \/ (b * a')\n  \/-quotient a' b b' = *= idp (\u207b\u00b9*-distr a' (\u207b\u00b9-non-zero b') \u2219 *= idp (\u207b\u00b9-involutive b') \u2219 *-comm)\n    \u2219 *-interchange \u2219 *= idp (! \u207b\u00b9*-distr b a')\n\n  0\u2212-*-distr' : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n  0\u2212-*-distr' = 0\u2212= *-comm \u2219 0\u2212-*-distr \u2219 *-comm\n\n  *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n  *-\u2212-distr = *-+-distr \u2219 += idp (! 0\u2212-*-distr')\n\n  0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n  0\u2212-+-distr = 0\u2212-hom\n\n  \u00b2-*-distr : \u2200 {x y} \u2192 (x * y)\u00b2 \u2261 x \u00b2 * y \u00b2\n  \u00b2-*-distr = *-interchange\n\n  \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n  \u00b2-0\u2212-distr {x} = ! 0\u2212-*-distr \u2219 0\u2212= (! 0\u2212-*-distr') \u2219 0\u2212-involutive\n\n  2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n  2*-*-distr = ! *-+-distr\u02b3\n\n  \u00b2-+-distr : \u2200 {x y} \u2192 (x + y)\u00b2 \u2261 x \u00b2 + y \u00b2 + 2* x * y\n  \u00b2-+-distr {x} {y} = (x + y)\u00b2\n                    \u2261\u27e8 *-+-distr \u27e9\n                      (x + y) * x + (x + y) * y\n                    \u2261\u27e8 += *-+-distr\u02b3 *-+-distr\u02b3 \u27e9\n                      x \u00b2 + y * x + (x * y + y \u00b2)\n                    \u2261\u27e8 += (+= idp *-comm) +-comm \u2219 +-interchange \u27e9\n                      x \u00b2 + y \u00b2 + 2*(x * y)\n                    \u2261\u27e8 += idp 2*-*-distr \u27e9\n                       x \u00b2 + y \u00b2 + 2* x * y\n                    \u220e\n\n  \u00b2-\u2212-distr : \u2200 {x y} \u2192 (x \u2212 y)\u00b2 \u2261 x \u00b2 + y \u00b2 \u2212 2* x * y\n  \u00b2-\u2212-distr = \u00b2-+-distr \u2219 += (+= idp \u00b2-0\u2212-distr) (*-comm \u2219 ! 0\u2212-*-distr \u2219 0\u2212= *-comm)\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Data.Nat.NP using (\u2115; zero; fold) renaming (suc to 1+)\nopen import Data.Integer using (\u2124; +_; -[1+_])\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Group\nopen import Algebra.Group.Abelian\nopen import HoTT\n\nmodule Algebra.Field where\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 2+_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-grp-ops : Group-Ops A\n\n  open Additive-Group-Ops       +-grp-ops public\n    renaming (2\u2297_ to 2*_)\n  open Multiplicative-Group-Ops *-grp-ops public\n\n  suc : A \u2192 A\n  suc = _+_ `1\n\n  pred : A \u2192 A\n  pred x = x \u2212 `1\n\n  `2 `3 `-0 `-1 : A\n  `2  = suc `1\n  `3  = suc `2\n  `-0 = 0\u2212 `0\n  `-1 = 0\u2212 `1\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0    ] = `0\n  \u2115[ 1+ n ] = fold `1 suc n\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ + x      ] = \u2115[ x ]\n  \u2124[ -[1+ x ] ] = 0\u2212 \u2115[ 1+ x ]\n\n  -- TODO use _^\u207a_ and _^_ from group\/monoid\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0      = `1\n  b ^\u2115 (1+ e) = fold b (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (+ x)    = b ^\u2115 x\n  b ^\u2124 -[1+ x ] = (b ^\u2115 (1+ x))\u207b\u00b9\n\n  2+_ : A \u2192 A\n  2+ x = `2 + x\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open Field-Ops field-ops public\n  open \u2261-Reasoning\n\n  field\n    0\u22621                  : `0 \u2262 `1\n    +-abelian-grp-struct : Abelian-Group-Struct +-grp-ops\n\n    *-comm-mon-struct    : Commutative-Monoid-Struct *-mon-ops\n    \u207b\u00b9-inverse           : InverseNonZero `0 `1 _\u207b\u00b9 _*_\n    *-+-distrs           : _*_ DistributesOver _+_\n\n  open Additive-Abelian-Group-Struct +-abelian-grp-struct public\n  -- 0\u2212-involutive   : Involutive 0\u2212_\n  -- cancels-+-left  : LeftCancel _+_\n  -- cancels-+-right : RightCancel _+_\n  -- ...\n\n  open Multiplicative-Commutative-Monoid-Struct *-comm-mon-struct public\n\n  +-grp-struct : Group-Struct +-grp-ops\n  +-grp-struct = Abelian-Group-Struct.grp-struct +-abelian-grp-struct\n\n  +-abelian-grp : Abelian-Group A\n  +-abelian-grp = +-grp-ops , +-abelian-grp-struct\n\n  +-grp : Group A\n  +-grp = +-grp-ops , +-grp-struct\n\n  -- Since the Abelian-Group module includes all what Group and\n  -- Monoid provide, we can expose a single module.\n  module +-Grp = Abelian-Group +-abelian-grp\n\n  *-mon-struct : Monoid-Struct *-mon-ops\n  *-mon-struct = Commutative-Monoid-Struct.mon-struct *-comm-mon-struct\n\n  *-mon : Monoid A\n  *-mon = *-mon-ops , *-mon-struct\n\n  *-comm-mon : Commutative-Monoid A\n  *-comm-mon = *-mon-ops , *-comm-mon-struct\n\n  module *-Mon = Commutative-Monoid *-comm-mon\n\n  1\u22620 : `1 \u2262 `0\n  1\u22620 e = 0\u22621 (! e)\n\n  NonZero : A \u2192 Set \u2113\n  NonZero x = x \u2262 `0\n\n  A\/0 = \u03a3 A NonZero\n\n  \u207b\u00b9-left-inverse : LeftInverseNonZero `0 `1 _\u207b\u00b9 _*_\n  \u207b\u00b9-left-inverse = fst \u207b\u00b9-inverse\n\n  \u207b\u00b9-right-inverse : RightInverseNonZero `0 `1 _\u207b\u00b9 _*_\n  \u207b\u00b9-right-inverse = snd \u207b\u00b9-inverse\n\n  *-+-distr : _*_ DistributesOver\u02e1 _+_\n  *-+-distr = fst *-+-distrs\n\n  *0-zero : RightZero `0 _*_\n  *0-zero = cancels-+-right  (+= idp (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= idp 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero `0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  c\u207b\u00b9*c*x : \u2200 {c x} \u2192 c \u2262 `0 \u2192 c \u207b\u00b9 * c * x \u2261 x\n  c\u207b\u00b9*c*x c\u22620 = *= (\u207b\u00b9-left-inverse c\u22620) idp \u2219 1*-identity\n\n  cancels-*-left : LeftCancelNonZero `0 _*_\n  cancels-*-left c\u22620 p = ! c\u207b\u00b9*c*x c\u22620 \u2219 *-!assoc= p \u2219 c\u207b\u00b9*c*x c\u22620\n\n  cancels-*-right : RightCancelNonZero `0 _*_\n  cancels-*-right c\u22620 p = cancels-*-left c\u22620 (*-comm= p)\n\n  unique-0 : \u2200 {x y} \u2192 x \u2262 `0 \u2192 x * y \u2261 `0 \u2192 y \u2261 `0\n  unique-0 nx xy\u22610 = cancels-*-left nx (xy\u22610 \u2219 ! *0-zero)\n\n  no-zero-divisor : \u2200 {x y} \u2192 x \u2262 `0 \u2192 y \u2262 `0 \u2192 x * y \u2262 `0\n  no-zero-divisor nx ny x*y\u22610 = ny (unique-0 nx x*y\u22610)\n\n  _*\/0_ : A\/0 \u2192 A\/0 \u2192 A\/0\n  (x , x\u22620) *\/0 (y , y\u22620) = x * y , no-zero-divisor x\u22620 y\u22620\n\n  ``1 : A\/0\n  ``1 = `1 , 1\u22620\n\n  *-mon-ops\/0 : Monoid-Ops A\/0\n  *-mon-ops\/0 = _*\/0_ , ``1\n\n  \u207b\u00b9-non-zero : \u2200 {x} \u2192 x \u2262 `0 \u2192 x \u207b\u00b9 \u2262 `0\n  \u207b\u00b9-non-zero x\u22620 = \u03bb p \u2192 0\u22621 (! 0*-zero \u2219 *= (! p) idp \u2219 \u207b\u00b9-left-inverse x\u22620)\n\n  _\u207b\u00b9\/0 : A\/0 \u2192 A\/0\n  (x , x\u22620) \u207b\u00b9\/0 = x \u207b\u00b9 , \u207b\u00b9-non-zero x\u22620\n\n  *-grp-ops\/0 : Group-Ops A\/0\n  *-grp-ops\/0 = *-mon-ops\/0 , _\u207b\u00b9\/0\n\n  module _ {{_ : FunExt}} where\n    A\/0= : {x y : A\/0}(p : fst x \u2261 fst y) \u2192 x \u2261 y\n    A\/0= p = pair= p (\u00ac-has-all-paths _ _)\n\n    *-mon-struct\/0 : Monoid-Struct *-mon-ops\/0\n    *-mon-struct\/0 = (A\/0= *-assoc , A\/0= *-!assoc)\n                   , A\/0= 1*-identity , A\/0= *1-identity\n\n    \u207b\u00b9-left-inverse\/0 : LeftInverse ``1 _\u207b\u00b9\/0 _*\/0_\n    \u207b\u00b9-left-inverse\/0 {x , x\u22620} = A\/0= (\u207b\u00b9-left-inverse x\u22620)\n\n    *-grp-struct\/0 : Group-Struct *-grp-ops\/0\n    *-grp-struct\/0 = *-mon-struct\/0\n                   , (\u03bb {x} \u2192 \u207b\u00b9-left-inverse\/0 {x}) ,\n                     (\u03bb {x} \u2192 A\/0= (\u207b\u00b9-right-inverse (snd x)))\n\n    *-grp\/0 : Group A\/0\n    *-grp\/0 = *-grp-ops\/0 , *-grp-struct\/0\n\n    *-abelian-grp-struct\/0 : Abelian-Group-Struct *-grp-ops\/0\n    *-abelian-grp-struct\/0 = *-grp-struct\/0 , A\/0= *-comm\n\n    *-abelian-grp\/0 : Abelian-Group A\/0\n    *-abelian-grp\/0 = *-grp-ops\/0 , *-abelian-grp-struct\/0\n    -- module *-Grp\/0 = Abelian-Group *-abelian-grp\/0\n\n  0\u2212-left-inverse : LeftInverse `0 0\u2212_ _+_\n  0\u2212-left-inverse = fst 0\u2212-inverse\n\n  0\u2212-right-inverse : RightInverse `0 0\u2212_ _+_\n  0\u2212-right-inverse = snd 0\u2212-inverse\n\n  0\u2212= : \u2200 {x y} \u2192 x \u2261 y \u2192 0\u2212 x \u2261 0\u2212 y\n  0\u2212= = ap 0\u2212_\n\n  \u207b\u00b9= : \u2200 {x y} \u2192 x \u2261 y \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9\n  \u207b\u00b9= = ap _\u207b\u00b9\n\n  *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n  *-+-distr\u02b3 = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  \/-+-distr\u02b3 : _\/_ DistributesOver\u02b3 _+_\n  \/-+-distr\u02b3 = *-+-distr\u02b3\n\n  0\u2212c+c+x : \u2200 {c x} \u2192 0\u2212 c + c + x \u2261 x\n  0\u2212c+c+x = += 0\u2212-left-inverse idp \u2219 0+-identity\n\n  -0\u22610 : `-0 \u2261 `0\n  -0\u22610 = 0\u22120\u22610\n\n  0\u2262-1 : `0 \u2262 `-1\n  0\u2262-1 0\u2261-1 = 0\u22621 (! 0\u2212-left-inverse \u2219 +-comm \u2219 ! ap suc 0\u2261-1 \u2219 +0-identity)\n\n  0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n  0\u2212-*-distr = cancels-+-right (0\u2212-left-inverse \u2219 ! 0*-zero \u2219 *= (! 0\u2212-left-inverse) idp \u2219 *-+-distr\u02b3)\n\n  -1*-neg : \u2200 {x} \u2192 `-1 * x \u2261 0\u2212 x\n  -1*-neg = ! 0\u2212-*-distr \u2219 0\u2212= 1*-identity\n\n  1\u207b\u00b9\u22611 : `1 \u207b\u00b9 \u2261 `1\n  1\u207b\u00b9\u22611 = ! *1-identity \u2219 \u207b\u00b9-left-inverse (\u03bb p \u2192 0\u22621 (! p))\n\n  2*-spec : \u2200 {n} \u2192 2* n \u2261 `2 * n\n  2*-spec = ! += 1*-identity 1*-identity \u2219 ! *-+-distr\u02b3\n\n  *-unique-inverse : \u2200 {x y} \u2192 x \u2262 `0 \u2192 x * y \u2261 `1 \u2192 y \u2261 x \u207b\u00b9\n  *-unique-inverse x\u22620 xy=1 = ! 1*-identity \u2219 *= (! \u207b\u00b9-left-inverse x\u22620) idp \u2219 *-assoc \u2219 *= idp xy=1 \u2219 *1-identity\n\n  \u207b\u00b9-involutive : \u2200 {x} \u2192 x \u2262 `0 \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n  \u207b\u00b9-involutive x\u22620 = ! *-unique-inverse (\u207b\u00b9-non-zero x\u22620) (\u207b\u00b9-left-inverse x\u22620)\n\n  \u207b\u00b9*-distr : \u2200 {x y} \u2192 x \u2262 `0 \u2192 y \u2262 `0 \u2192 (x * y)\u207b\u00b9 \u2261 x \u207b\u00b9  \/ y\n  \u207b\u00b9*-distr x\u22620 y\u22620 =\n    ! (*-unique-inverse (no-zero-divisor x\u22620 y\u22620) (*= idp *-comm \u2219 *-assoc \u2219 *= idp (! *-assoc \u2219 *= (\u207b\u00b9-right-inverse y\u22620) idp \u2219 1*-identity) \u2219 \u207b\u00b9-right-inverse x\u22620))\n\n  +-quotient : \u2200 {a b a' b'} \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n    \u2192 (a \/ b) + (a' \/ b') \u2261 (a * b' + a' * b) \/ (b * b')\n  +-quotient b b'\n    = += (*= (! *1-identity \u2219 *= idp (! \u207b\u00b9-left-inverse b') \u2219 *= idp *-comm \u2219 ! *-assoc) idp \u2219\n          *-assoc \u2219 *= idp (! \u207b\u00b9*-distr b' b \u2219 \u207b\u00b9= *-comm))\n         (*= (! *1-identity \u2219 *= idp (! \u207b\u00b9-left-inverse b ) \u2219 *= idp *-comm) idp \u2219\n          *-!assoc= *-assoc \u2219 *= idp (! \u207b\u00b9*-distr b b'))\n    \u2219 ! \/-+-distr\u02b3\n\n  \u2212-quotient : \u2200 {a b a' b'} \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n    \u2192 (a \/ b) \u2212 (a' \/ b') \u2261 (a * b' \u2212 a' * b) \/ (b * b')\n  \u2212-quotient b b' = += idp 0\u2212-*-distr \u2219 +-quotient b b' \u2219 \/= (+= idp (! 0\u2212-*-distr)) idp\n\n  *-quotient : \u2200 {a b a' b'} \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n    \u2192 (a \/ b) * (a' \/ b') \u2261 (a * a') \/ (b * b')\n  *-quotient b\u22620 b'\u22620 = *-interchange \u2219 *= idp (! \u207b\u00b9*-distr b\u22620 b'\u22620)\n\n  \/-quotient : \u2200 {a b a' b'} \u2192 a' \u2262 `0 \u2192 b \u2262 `0 \u2192 b' \u2262 `0\n     \u2192 (a \/ b) \/ (a' \/ b') \u2261 (a * b') \/ (b * a')\n  \/-quotient a' b b' = *= idp (\u207b\u00b9*-distr a' (\u207b\u00b9-non-zero b') \u2219 *= idp (\u207b\u00b9-involutive b') \u2219 *-comm)\n    \u2219 *-interchange \u2219 *= idp (! \u207b\u00b9*-distr b a')\n\n  0\u2212-*-distr' : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n  0\u2212-*-distr' = 0\u2212= *-comm \u2219 0\u2212-*-distr \u2219 *-comm\n\n  *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n  *-\u2212-distr = *-+-distr \u2219 += idp (! 0\u2212-*-distr')\n\n  0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n  0\u2212-+-distr = 0\u2212-hom\n\n  \u00b2-*-distr : \u2200 {x y} \u2192 (x * y)\u00b2 \u2261 x \u00b2 * y \u00b2\n  \u00b2-*-distr = *-interchange\n\n  \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n  \u00b2-0\u2212-distr {x} = ! 0\u2212-*-distr \u2219 0\u2212= (! 0\u2212-*-distr') \u2219 0\u2212-involutive\n\n  2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n  2*-*-distr = ! *-+-distr\u02b3\n\n  \u00b2-+-distr : \u2200 {x y} \u2192 (x + y)\u00b2 \u2261 x \u00b2 + y \u00b2 + 2* x * y\n  \u00b2-+-distr {x} {y} = (x + y)\u00b2\n                    \u2261\u27e8 *-+-distr \u27e9\n                      (x + y) * x + (x + y) * y\n                    \u2261\u27e8 += *-+-distr\u02b3 *-+-distr\u02b3 \u27e9\n                      x \u00b2 + y * x + (x * y + y \u00b2)\n                    \u2261\u27e8 += (+= idp *-comm) +-comm \u2219 +-interchange \u27e9\n                      x \u00b2 + y \u00b2 + 2*(x * y)\n                    \u2261\u27e8 += idp 2*-*-distr \u27e9\n                       x \u00b2 + y \u00b2 + 2* x * y\n                    \u220e\n\n  \u00b2-\u2212-distr : \u2200 {x y} \u2192 (x \u2212 y)\u00b2 \u2261 x \u00b2 + y \u00b2 \u2212 2* x * y\n  \u00b2-\u2212-distr = \u00b2-+-distr \u2219 += (+= idp \u00b2-0\u2212-distr) (*-comm \u2219 ! 0\u2212-*-distr \u2219 0\u2212= *-comm)\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dd0edff3a115d46e15a08cc72e26a8b23e339aa2","subject":"a couple of cases of progress","message":"a couple of cases of progress\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d err[ \u0394 ] \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | V x\u2081\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp TAConst () D2) | V VConst | I x\u2081\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!})\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V VConst = I (INEHole (FVal VConst))\n  progress (TANEHole x\u2081 (TALam D) x\u2082) | V VLam = I (INEHole (FVal VLam))\n  progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E {!!}\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = E {!!}\n  progress (TACast TAConst TCHole2) | V VConst = E {!!}\n  progress (TACast D x\u2081) | V VLam = E {!!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E {!!}\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d err[ \u0394 ] \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | V x\u2081\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp D1 x\u2082 D2) | V x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | E x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | I x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | E x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | S x = {!!}\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = E {!!}\n  progress (TACast TAConst TCHole2) | V VConst = E {!!}\n  progress (TACast D x\u2081) | V VLam = {!!}\n  progress (TACast D x\u2081) | I x = {!!}\n  progress (TACast D x\u2081) | E x = {!!}\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5d5974ad4a0cc6d7d273f5095bdc17620902f26b","subject":"Removed unnecessary local hint.","message":"Removed unnecessary local hint.\n\nIgnore-this: 20861fa0526b90dd5a4befd9057767da\n\ndarcs-hash:20110215033029-3bd4e-49b903264c9a8debc8c99919549787d57ab29f20.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\n-- TODO: To move to LTC.Data.Nat.Inequalities.Properties.\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x-y\u21920<Sx-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x-y\u21920<Sx-y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x-y\u21920<Sx-y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x-y\u21920<Sx-y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x-y\u21920<Sx-y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\n-- TODO: To move to LTC.Data.Nat.Inequalities.Properties.\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x-y\u21920<Sx-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x-y\u21920<Sx-y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x-y\u21920<Sx-y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x-y\u21920<Sx-y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x-y\u21920<Sx-y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx \u2238-SS x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5a188fcb33ba607e1af5024d482118e96b9ee5fd","subject":"Update tests","message":"Update tests\n","repos":"np\/agda-parametricity","old_file":"lib\/Reflection\/Param\/Tests.agda","new_file":"lib\/Reflection\/Param\/Tests.agda","new_contents":"{-# OPTIONS -vtc.unquote.decl:20 -vtc.unquote.def:20 #-}\n{-# OPTIONS --without-K #-}\nopen import Level hiding (zero; suc)\nopen import Data.Unit renaming (\u22a4 to \ud835\udfd9; tt to 0\u2081)\nopen import Data.Bool\n  using    (not)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082)\nopen import Data.String.Base using (String)\nopen import Data.Float       using (Float)\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; refl) renaming (_\u2257_ to _~_)\n\nopen import Function.Param.Unary\nopen import Function.Param.Binary\nopen import Type.Param.Unary\nopen import Type.Param.Binary\nopen import Data.Two.Param.Binary\nopen import Data.Nat.Param.Binary\nopen import Reflection.NP\nopen import Reflection.Param\nopen import Reflection.Param.Env\n\nmodule Reflection.Param.Tests where\n\nimport Reflection.Printer as Pr\nopen Pr using (var;con;def;lam;pi;sort;unknown;showTerm;showType;showDef;showFunDef;showClauses)\n\n-- Local \"imports\" to avoid depending on nplib\nprivate\n  postulate\n    opaque : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2192 B \u2192 B\n    -- opaque-rule : \u2200 {x} y \u2192 opaque x y \u2261 y\n\n  \u2605\u2080 = Set\u2080\n  \u2605\u2081 = Set\u2081\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\n{-\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n-}\n\n_\u2261-no-hints_ : Term \u2192 Term \u2192 Set\nt \u2261-no-hints u = noHintsTerm t \u2261 noHintsTerm u\n\n_\u2261-def-no-hints_ : Definition \u2192 Definition \u2192 Set\nt \u2261-def-no-hints u = noHintsDefinition t \u2261 noHintsDefinition u\n\np[Set\u2080]-type = param-type-by-name (\u03b5 1) (quote [Set\u2080])\np[Set\u2080] = param-clauses-by-name (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl p[Set\u2080]-type)\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261-no-hints Get-term.from-clauses p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nu-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\nunquoteDef u-p[Set\u2080] = p[Set\u2080]\n\nFalse : Set\u2081\nFalse = (A : Set) \u2192 A\n\nparam1-False-type = param-type-by-name (\u03b5 1) (quote False)\nparam1-False-term = param-term-by-name (\u03b5 1) (quote False)\n\nparam1-False-type-check : [Set\u2081] False \u2261 unquote (unEl param1-False-type)\nparam1-False-type-check = refl\n\n[False] : unquote (unEl param1-False-type)\n[False] = unquote param1-False-term\n\n[Level] : [Set\u2080] Level\n[Level] _ = \ud835\udfd9\n\n[String] : [Set\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [Set\u2080] Float\n[Float] _ = \ud835\udfd9\n\n-- Levels are parametric, hence the relation is total\n\u27e6Level\u27e7 : \u27e6Set\u2080\u27e7 Level Level\n\u27e6Level\u27e7 _ _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6Set\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6Set\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda)      = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115)      = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float)  = quote [Float]\ndefDefEnv1 (quote \u2605\u2080)     = quote [Set\u2080]\ndefDefEnv1 (quote \u2605\u2081)     = quote [Set\u2081]\ndefDefEnv1 (quote False)  = quote [False]\ndefDefEnv1 (quote Level)  = quote [Level]\ndefDefEnv1 n              = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082)         = quote [0\u2082]\ndefConEnv1 (quote 1\u2082)         = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero)     = quote [zero]\ndefConEnv1 (quote \u2115.suc)      = quote [suc]\ndefConEnv1 (quote Level.zero) = quote 0\u2081\ndefConEnv1 (quote Level.suc)  = quote 0\u2081\ndefConEnv1 n                  = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda)      = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115)      = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote \u2605\u2080)     = quote \u27e6Set\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081)     = quote \u27e6Set\u2081\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float)  = quote \u27e6Float\u27e7\ndefDefEnv2 (quote Level)  = quote \u27e6Level\u27e7\ndefDefEnv2 n              = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082)         = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082)         = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero)     = quote \u27e6\u2115\u27e7.\u27e6zero\u27e7\ndefConEnv2 (quote \u2115.suc)      = quote \u27e6\u2115\u27e7.\u27e6suc\u27e7\ndefConEnv2 (quote Level.zero) = quote 0\u2081\ndefConEnv2 (quote Level.suc)  = quote 0\u2081\ndefConEnv2 n                  = opaque \"defConEnv2\" n\n\ndefEnv0 : Env' 0\ndefEnv0 = record (\u03b5 0)\n                 { pConT = con\n                 ; pConP = con\n                 ; pDef  = id }\n\ndefEnv1 : Env' 1\ndefEnv1 = record (\u03b5 1)\n  { pConP = con \u2218\u2032 defConEnv1\n  ; pConT = con \u2218\u2032 defConEnv1\n  ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record (\u03b5 2)\n  { pConP = con \u2218\u2032 defConEnv2\n  ; pConT = con \u2218\u2032 defConEnv2\n  ; pDef = defDefEnv2 }\n\nparam1-[False]-type = param-type-by-name defEnv1 (quote [False])\nparam1-[False]-term = param-term-by-name defEnv1 (quote [False])\n\n{-\nmodule Const where\n  postulate\n    A  : Set\u2080\n    A\u1d63 : A \u2192 A \u2192 Set\u2080\n  data Wrapper : Set where\n    wrap : A \u2192 Wrapper\n\n  idWrapper : Wrapper \u2192 Wrapper\n  idWrapper (wrap x) = wrap x\n\n  data \u27e6Wrapper\u27e7 : Wrapper \u2192 Wrapper \u2192 Set\u2080 where\n    \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7) wrap wrap\n\n  wrapperEnv = record (\u03b5 2)\n   { pDef = [ quote Wrapper       \u2254 quote \u27e6Wrapper\u27e7  ] id\n   ; pConP = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   ; pConT = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   }\n\n  unquoteDecl \u27e6idWrapper\u27e7 = param-rec-def-by-name wrapperEnv (quote idWrapper) \u27e6idWrapper\u27e7\n-}\n\ndata Wrapper (A : Set\u2080) : Set\u2080 where\n  wrap : A \u2192 Wrapper A\n\nidWrapper : \u2200 {A} \u2192 Wrapper A \u2192 Wrapper A\nidWrapper (wrap x) = wrap x\n\nmodule Param where\n\n  data [Wrapper] {A : Set} (A\u209a : A \u2192 Set\u2080)\n     : Wrapper A \u2192 Set\u2080 where\n    [wrap] : (A\u209a [\u2192] [Wrapper] A\u209a) wrap\n\n  {-\n  [Wrapper]-env = record (\u03b5 1)\n    { pDef = [ quote Wrapper \u2254 quote [Wrapper] ] id\n    ; pConP = [ quote wrap \u2254 con (quote [wrap])  ] con\n    ; pConT = [ quote wrap \u2254 conSkip' (quote [wrap]) ] con\n    }\n\n  unquoteDecl [idWrapper] =\n    param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]\n  -}\n  {-\n  [idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n  -- [idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n  -}\n\n{-\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6Wrapper\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) Wrapper Wrapper\n\nprivate\n  \u27e6Wrapper\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6Wrapper\u27e7 where\n  \u27e6wrap\u27e7 : unquote (\u27e6Wrapper\u27e7-ctor (quote Wrapper.wrap))\n-}\n\n{-\ndata \u27e6Wrapper\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Wrapper A\u2080 \u2192 Wrapper A\u2081 \u2192 Set\u2080 where\n  \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A\u1d63) wrap wrap\n\n\u27e6Wrapper\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] id\n  ; pConP = [ quote wrap \u2254 con (quote \u27e6wrap\u27e7)  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 3 (quote \u27e6wrap\u27e7) ] con\n  }\n\n\u27e6idWrapper\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\n\u27e6idWrapper\u27e71 {x0} {x1} (x2) {._} {._} (\u27e6wrap\u27e7 {x3} {x4} x5)\n  = \u27e6wrap\u27e7 {_} {_} {_} {x3} {x4} x5\n\n\u27e6idWrapper\u27e7-clauses =\n  clause\n  (arg (arg-info hidden  relevant) (var \"A0\") \u2237\n   arg (arg-info hidden  relevant) (var \"A1\") \u2237\n   arg (arg-info visible relevant) (var \"Ar\") \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info visible relevant)\n   (con (quote \u27e6wrap\u27e7)\n    (arg (arg-info hidden  relevant) (var \"x0\") \u2237\n     arg (arg-info hidden  relevant) (var \"x1\") \u2237\n     arg (arg-info visible relevant) (var \"xr\") \u2237 []))\n   \u2237 [])\n  (con (quote \u27e6wrap\u27e7)\n   (arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant)  (var 2 []) \u2237\n    arg (arg-info hidden relevant)  (var 1 []) \u2237\n    arg (arg-info visible relevant) (var 0 []) \u2237 []))\n  \u2237 []\n\n\u27e6idWrapper\u27e72 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\nunquoteDef \u27e6idWrapper\u27e72 = \u27e6idWrapper\u27e7-clauses\n\nunquoteDecl \u27e6idWrapper\u27e7 =\n  param-rec-def-by-name \u27e6Wrapper\u27e7-env (quote idWrapper) \u27e6idWrapper\u27e7\n-}\n\ndata Bot (A : Set\u2080) : Set\u2080 where\n  bot : Bot A \u2192 Bot A\n\ngobot : \u2200 {A} \u2192 Bot A \u2192 A\ngobot (bot x) = gobot x\n\ndata [Bot] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Bot A \u2192 Set\u2080 where\n  [bot] : ([Bot] A\u209a [\u2192] [Bot] A\u209a) bot\n\n[Bot]-env = record (\u03b5 1)\n  { pDef = [ quote Bot \u2254 quote [Bot] ] id\n  ; pConP = [ quote bot \u2254 con (quote [bot])  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 2 (quote [bot]) ] con\n  }\n\n[gobot]' : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n[gobot]' {x0} (x1) {._} ([bot] {x2} x3)\n  = [gobot]' {x0} x1 {x2} x3\n\n-- [gobot]' = showClauses \"[gobot]'\" (param-rec-clauses-by-name [Bot]-env (quote gobot) (quote [gobot]'))\n\n[gobot]2 : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n\n[gobot]2-clauses =\n  clause\n    (arg (arg-info hidden  relevant) (var \"A\u1d620\") \u2237\n     arg (arg-info visible relevant) (var \"A\u1d63\") \u2237\n     arg (arg-info hidden  relevant) dot \u2237\n     arg (arg-info visible relevant)\n     (con (quote [bot])\n      (arg (arg-info hidden  relevant) (var \"x\u1d620\") \u2237\n       arg (arg-info visible relevant) (var \"x\u1d63\") \u2237 []))\n     \u2237 [])\n    (def (quote [gobot]2)\n     (arg (arg-info hidden  relevant) (var 3 []) \u2237\n      arg (arg-info visible relevant) (var 2 []) \u2237\n      arg (arg-info hidden  relevant) (var 1 []) \u2237\n      arg (arg-info visible relevant) (var 0 []) \u2237 []))\n    \u2237 []\n\nunquoteDef [gobot]2 = [gobot]2-clauses\n\nunquoteDecl [gobot] =\n  param-rec-def-by-name [Bot]-env (quote gobot) [gobot]\n\ndata \u27e6Bot\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Bot A\u2080 \u2192 Bot A\u2081 \u2192 Set\u2080 where\n  \u27e6bot\u27e7 : (\u27e6Bot\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Bot\u27e7 A\u1d63) bot bot\n\n\u27e6Bot\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Bot \u2254 quote \u27e6Bot\u27e7 ] id\n  ; pConP = [ quote bot \u2254 con (quote \u27e6bot\u27e7)  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 3 (quote \u27e6bot\u27e7) ] con\n  }\n\n\u27e6gobot\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Bot\u27e7 A \u27e6\u2192\u27e7 A) gobot gobot\n\u27e6gobot\u27e71 {x0} {x1} x2 {._} {._} (\u27e6bot\u27e7 {x3} {x4} x5)\n  = \u27e6gobot\u27e71 {x0} {x1} x2 {x3} {x4} x5\n\nunquoteDecl \u27e6gobot\u27e7 =\n  param-rec-def-by-name \u27e6Bot\u27e7-env (quote gobot) \u27e6gobot\u27e7\n\nid\u2080 : {A : Set\u2080} \u2192 A \u2192 A\nid\u2080 x = x\n\n\u27e6id\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A \u27e6\u2192\u27e7 A) id\u2080 id\u2080\n\u27e6id\u2080\u27e7 = \u03bb {x\u2081} {x\u2082} x\u1d63 {x\u2083} {x\u2084} x\u1d63\u2081 \u2192 x\u1d63\u2081\n\ndata List\u2080 (A : Set) : Set where\n  []  : List\u2080 A\n  _\u2237_ : A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} (f : A \u2192 B) (xs : List\u2080 A) \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\n-- idList\u2080 : List\u2080 \u2115 \u2192 List\u2080 \u2115\nidList\u2080 []       = []\nidList\u2080 {A} (x \u2237 xs) = idList\u2080 {A} xs\n\ndata \u27e6List\u2080\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080) : List\u2080 A\u2080 \u2192 List\u2080 A\u2081 \u2192 Set\u2080 where\n  \u27e6[]\u27e7  : \u27e6List\u2080\u27e7 A\u1d63 [] []\n  _\u27e6\u2237\u27e7_ : (A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63) _\u2237_ _\u2237_\n\ncon\u27e6List\u2080\u27e7 = conSkip' 3\n\u27e6List\u2080\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] id\n  ; pConP = [ quote List\u2080.[]  \u2254 con (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  ; pConT = [ quote List\u2080.[]  \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  }\n\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\n-- \u27e6idList\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A) idList\u2080 idList\u2080\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n\n{-\n\u27e6map\u2080\u27e7 : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : List\u2080 (x0)} \u2192 {x10 : List\u2080 (x1)} \u2192 (x11 : \u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 \u27e6List\u2080\u27e7 {x3} {x4} (x5) (map\u2080 {x0} {x3} (x6) (x9)) (map\u2080 {x1} {x4} (x7) (x10))\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (\u27e6[]\u27e7 )  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (_\u27e6\u2237\u27e7_ {x11} {x12} (x13) {x14} {x15} (x16) )  = _\u27e6\u2237\u27e7_ {x6 (x11)} {x7 (x12)} (x8 {x11} {x12} (x13)) {map\u2080 {x0} {x3} (x6) (x14)} {map\u2080 {x1} {x4} (x7) (x15)} (\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {x14} {x15} (x16))\n-}\n\nunquoteDecl \u27e6map\u2080\u27e7\n = param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) \u27e6map\u2080\u27e7\n\n{-\nmap-nat : \u2200 (f : \u2200 {X} \u2192 List\u2080 X \u2192 List\u2080 X)\n            {A B : Set} (g : A \u2192 B)\n          \u2192 f \u2218 map\u2080 g ~ map\u2080 g \u2218 f\nmap-nat f g x = {!\u27e6map\u2080\u27e7 _\u2261_ _\u2261_ {g}!}\n\n  -- The generated type is bigger since it is a familly for no reason.\n  data \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\n  private\n    \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\n  data \u27e6List\u2080\u27e7 where\n    \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n    _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n  \u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                        [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n             (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n-}\n\ndata Maybe' (A : Set) : Set\u2081 where\n  nothing : Maybe' A\n  just    : A \u2192 Maybe' A\n\n{-\n-- Set\u2081 is here because \u27e6List\u2080\u27e7 is not using parameters, hence gets bigger.\n-- This only happens without-K given the new rules for data types.\ndata List\u2080 : (A : Set) \u2192 Set\u2081 where\n  []  : \u2200 {A} \u2192 List\u2080 A\n  _\u2237_ : \u2200 {A} \u2192 A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} \u2192 (A \u2192 B) \u2192 List\u2080 A \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\nidList\u2080 []       = []\nidList\u2080 (x \u2237 xs) = x \u2237 idList\u2080 xs\n\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\nprivate\n  \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6List\u2080\u27e7 where\n  \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n  _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n\u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                      [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n           (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n\n-- test = \u27e6[]\u27e7 {{!showType (type (quote List\u2080.[]))!}} {{!!}} {!!}\n-}\n\n{-\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n-}\n\n{-\n\u27e6map\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote map\u2080)))\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {[]} {[]} \u27e6[]\u27e7\n  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {._ \u2237 ._}\n  {._ \u2237 ._}\n  (_\u27e6\u2237\u27e7_ {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085 {x\u2081\u2086} {x\u2081\u2087} x\u2081\u2088)\n  = _\u27e6\u2237\u27e7_ {x\u2081\u2080 x\u2081\u2083} {x\u2081\u2081 x\u2081\u2084} (x\u2081\u2082 {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085)\n    {map\u2080 {x\u2084} {x\u2087} x\u2081\u2080 x\u2081\u2086} {map\u2080 {x\u2085} {x\u2088} x\u2081\u2081 x\u2081\u2087}\n    (\u27e6map\u2080\u27e7 {x\u2084} {x\u2085} {x\u2086} {x\u2087} {x\u2088} {x\u2089} {x\u2081\u2080} {x\u2081\u2081} x\u2081\u2082 {x\u2081\u2086} {x\u2081\u2087}\n     x\u2081\u2088)\n-}\n\n{-\nunquoteDef \u27e6map\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote \u27e6map\u2080\u27e7)\n-}\n\n\n{-\nfoo : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : Reflection.Param.List\u2080 (x0)} \u2192 {x10 : Reflection.Param.List\u2080 (x1)} \u2192 (x11 : Reflection.Param.\u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 Reflection.Param.\u27e6List\u2080\u27e7 {x3} {x4} (x5) (Reflection.Param.map\u2080 {x0} {x3} (x6) (x9)) (Reflection.Param.map\u2080 {x1} {x4} (x7) (x10))\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7 )  = Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x} {x} (x) {xs} {xs} (xs) )  = Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x0 (x0)} {x0 (x0)} (x0 {x0} {x0} (x0)) {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} (Reflection.Param.test' {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0))\n-}\n-}\n\n-- test' = {! showFunDef \"foo\" (param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote test'))!}\n\nopen import Function.Param.Unary\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\n\nrevealed-[\u2192]' : \u2200 (A : Set\u2080) (A\u209a : A \u2192 Set\u2080)\n                  (B : Set\u2080) (B\u209a : B \u2192 Set\u2080)\n                  (f : A \u2192 B) \u2192 Set\u2080\nunquoteDef revealed-[\u2192]' = Get-clauses.from-def revealed-[\u2192]\n\nrevelator-[\u2192] : ({A : Set} (A\u209a : A \u2192 Set) {B : Set} (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set)\n              \u2192  (A : Set) (A\u209a : A \u2192 Set) (B : Set) (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set\nunquoteDef revelator-[\u2192] = Revelator.clauses (type (quote _[\u2080\u2192\u2080]_))\n\np-[\u2192]-type = param-type-by-name    (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]      = param-clauses-by-name (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\n\np-[\u2192]' = \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n           {A\u209a : A \u2192 Set\u2080}  (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 A\u209a x \u2192 Set\u2080)\n           {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n           {B\u209a : B \u2192 Set\u2080}  (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 B\u209a x \u2192 Set\u2080)\n           {f : A \u2192 B}      (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n         \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f\n         \u2192 Set\n\np-[\u2192]'-test : p-[\u2192]' \u2261 unquote (unEl p-[\u2192]-type)\np-[\u2192]'-test = refl\n\n[[\u2192]] : unquote (unEl p-[\u2192]-type)\nunquoteDef [[\u2192]] = p-[\u2192]\n\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : [[\u2192]] [\u2115] [[\u2115]] [\u2115] [[\u2115]] [suc] [suc]\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7          \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\npred' : \u2115 \u2192 \u2115\npred' = \u03bb { zero    \u2192 zero\n          ; (suc m) \u2192 m }\n\n\u27e6pred'\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred' pred'\nunquoteDef \u27e6pred'\u27e7 = param-clauses-by-name defEnv2 (quote pred')\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7-type = unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7)))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : \u27e6\u27e6Set\u2080\u27e7\u27e7-type \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = param-clauses-by-name defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261-no-hints Get-term.from-clauses p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7'' : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\nunquoteDef \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\n\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' : _\u2261_ {A = \u27e6\u27e6Set\u2080\u27e7\u27e7-type} \u27e6\u27e6Set\u2080\u27e7\u27e7'' \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' = refl\n\ntest-p0-\u27e6Set\u2080\u27e7 : pTerm defEnv0 (quoteTerm \u27e6Set\u2080\u27e7) \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-p0-\u27e6Set\u2080\u27e7 = refl\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\np1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n[\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\ntest-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\ntest-p1\u2115\u2192\u2115 = refl\n\np2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\ntest-p2\u2115\u2192\u2115 = refl\n\np\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261-no-hints quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 = refl\nZERO : Set\u2081\nZERO = (A : Set\u2080) \u2192 A\n\u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n\u27e6ZERO\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\npZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\nq\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\ntest-pZERO : pZERO \u2261-no-hints q\u27e6ZERO\u27e7\ntest-pZERO = refl\nID : Set\u2081\nID = (A : Set\u2080) \u2192 A \u2192 A\n\u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n\u27e6ID\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n  \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\npID = pTerm (\u03b5 2) (quoteTerm ID)\nq\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\ntest-ID : q\u27e6ID\u27e7 \u2261-no-hints pID\ntest-ID = refl\n\n\u27e6not\u27e7' : (\u27e6\ud835\udfda\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) not not\nunquoteDef \u27e6not\u27e7' = param-clauses-by-name defEnv2 (quote not)\ntest-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 \u27e6not\u27e7' x\u1d63\ntest-not \u27e60\u2082\u27e7 = refl\ntest-not \u27e61\u2082\u27e7 = refl\n\n[pred]' : ([\u2115] [\u2192] [\u2115]) pred\nunquoteDef [pred]' = param-clauses-by-name defEnv1 (quote pred)\n\ntest-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\ntest-p1-pred [zero]     = refl\ntest-p1-pred ([suc] n\u209a) = refl\n\n\u27e6pred\u27e7' : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred pred\nunquoteDef \u27e6pred\u27e7' = param-clauses-by-name defEnv2 (quote pred)\n\ntest-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\ntest-p2-pred \u27e6zero\u27e7     = refl\ntest-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\np\/2 = param-rec-def-by-name defEnv2 (quote _\/2)\nq\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\nunquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\ntest-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261-def-no-hints q\u27e6\/2\u27e7\ntest-\/2 = refl\ntest-\/2' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\ntest-\/2' \u27e6zero\u27e7 = refl\ntest-\/2' (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\ntest-\/2' (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) rewrite test-\/2' n\u1d63 = refl\n\np+ = param-rec-def-by-name defEnv2 (quote _+\u2115_)\nq\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\nunquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\ntest-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261-def-no-hints q\u27e6+\u27e7\ntest-+ = refl\ntest-+' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\ntest-+' \u27e6zero\u27e7    n'\u1d63 = refl\ntest-+' (\u27e6suc\u27e7 n\u1d63) n'\u1d63 rewrite test-+' n\u1d63 n'\u1d63 = refl\n\n{-\nis-good : String \u2192 \ud835\udfda\nis-good \"good\" = 1\u2082\nis-good _      = 0\u2082\n\n\u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n\u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n\u27e6is-good\u27e7 {_}      refl = {!!}\n\nmy-good = unquote (lit (string \"good\"))\nmy-good-test : my-good \u2261 \"good\"\nmy-good-test = refl\n-}\n\n{-\n\u27e6is-good\u27e7' : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\nunquoteDef \u27e6is-good\u27e7' = param-clauses-by-name defEnv2 (quote is-good)\ntest-is-good = {!!}\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS -vtc.unquote.decl:20 -vtc.unquote.def:20 #-}\n{-# OPTIONS --without-K #-}\nopen import Level hiding (zero; suc)\nopen import Data.Unit renaming (\u22a4 to \ud835\udfd9; tt to 0\u2081)\nopen import Data.Bool\n  using    (not)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082)\nopen import Data.String.Core using (String)\nopen import Data.Float       using (Float)\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; refl) renaming (_\u2257_ to _~_)\n\nopen import Function.Param.Unary\nopen import Function.Param.Binary\nopen import Type.Param.Unary\nopen import Type.Param.Binary\nopen import Data.Two.Param.Binary\nopen import Data.Nat.Param.Binary\nopen import Reflection.NP\nopen import Reflection.Param\nopen import Reflection.Param.Env\n\nmodule Reflection.Param.Tests where\n\nimport Reflection.Printer as Pr\nopen Pr using (var;con;def;lam;pi;sort;unknown;showTerm;showType;showDef;showFunDef;showClauses)\n\n-- Local \"imports\" to avoid depending on nplib\nprivate\n  postulate\n    opaque : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2192 B \u2192 B\n    -- opaque-rule : \u2200 {x} y \u2192 opaque x y \u2261 y\n\n  \u2605\u2080 = Set\u2080\n  \u2605\u2081 = Set\u2081\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\n{-\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n-}\n\n_\u2261-no-hints_ : Term \u2192 Term \u2192 Set\nt \u2261-no-hints u = noHintsTerm t \u2261 noHintsTerm u\n\n_\u2261-def-no-hints_ : Definition \u2192 Definition \u2192 Set\nt \u2261-def-no-hints u = noHintsDefinition t \u2261 noHintsDefinition u\n\np[Set\u2080]-type = param-type-by-name (\u03b5 1) (quote [Set\u2080])\np[Set\u2080] = param-clauses-by-name (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl p[Set\u2080]-type)\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261-no-hints Get-term.from-clauses p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nu-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\nunquoteDef u-p[Set\u2080] = p[Set\u2080]\n\nFalse : Set\u2081\nFalse = (A : Set) \u2192 A\n\nparam1-False-type = param-type-by-name (\u03b5 1) (quote False)\nparam1-False-term = param-term-by-name (\u03b5 1) (quote False)\n\nparam1-False-type-check : [Set\u2081] False \u2261 unquote (unEl param1-False-type)\nparam1-False-type-check = refl\n\n[False] : unquote (unEl param1-False-type)\n[False] = unquote param1-False-term\n\n[Level] : [Set\u2080] Level\n[Level] _ = \ud835\udfd9\n\n[String] : [Set\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [Set\u2080] Float\n[Float] _ = \ud835\udfd9\n\n-- Levels are parametric, hence the relation is total\n\u27e6Level\u27e7 : \u27e6Set\u2080\u27e7 Level Level\n\u27e6Level\u27e7 _ _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6Set\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6Set\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda)      = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115)      = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float)  = quote [Float]\ndefDefEnv1 (quote \u2605\u2080)     = quote [Set\u2080]\ndefDefEnv1 (quote \u2605\u2081)     = quote [Set\u2081]\ndefDefEnv1 (quote False)  = quote [False]\ndefDefEnv1 (quote Level)  = quote [Level]\ndefDefEnv1 n              = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082)         = quote [0\u2082]\ndefConEnv1 (quote 1\u2082)         = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero)     = quote [zero]\ndefConEnv1 (quote \u2115.suc)      = quote [suc]\ndefConEnv1 (quote Level.zero) = quote 0\u2081\ndefConEnv1 (quote Level.suc)  = quote 0\u2081\ndefConEnv1 n                  = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda)      = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115)      = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote \u2605\u2080)     = quote \u27e6Set\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081)     = quote \u27e6Set\u2081\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float)  = quote \u27e6Float\u27e7\ndefDefEnv2 (quote Level)  = quote \u27e6Level\u27e7\ndefDefEnv2 n              = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082)         = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082)         = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero)     = quote \u27e6\u2115\u27e7.\u27e6zero\u27e7\ndefConEnv2 (quote \u2115.suc)      = quote \u27e6\u2115\u27e7.\u27e6suc\u27e7\ndefConEnv2 (quote Level.zero) = quote 0\u2081\ndefConEnv2 (quote Level.suc)  = quote 0\u2081\ndefConEnv2 n                  = opaque \"defConEnv2\" n\n\ndefEnv0 : Env' 0\ndefEnv0 = record (\u03b5 0)\n                 { pConT = con\n                 ; pConP = con\n                 ; pDef  = id }\n\ndefEnv1 : Env' 1\ndefEnv1 = record (\u03b5 1)\n  { pConP = con \u2218\u2032 defConEnv1\n  ; pConT = con \u2218\u2032 defConEnv1\n  ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record (\u03b5 2)\n  { pConP = con \u2218\u2032 defConEnv2\n  ; pConT = con \u2218\u2032 defConEnv2\n  ; pDef = defDefEnv2 }\n\nparam1-[False]-type = param-type-by-name defEnv1 (quote [False])\nparam1-[False]-term = param-term-by-name defEnv1 (quote [False])\n\n{-\nmodule Const where\n  postulate\n    A  : Set\u2080\n    A\u1d63 : A \u2192 A \u2192 Set\u2080\n  data Wrapper : Set where\n    wrap : A \u2192 Wrapper\n\n  idWrapper : Wrapper \u2192 Wrapper\n  idWrapper (wrap x) = wrap x\n\n  data \u27e6Wrapper\u27e7 : Wrapper \u2192 Wrapper \u2192 Set\u2080 where\n    \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7) wrap wrap\n\n  wrapperEnv = record (\u03b5 2)\n   { pDef = [ quote Wrapper       \u2254 quote \u27e6Wrapper\u27e7  ] id\n   ; pConP = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   ; pConT = [ quote Wrapper.wrap \u2254 con (quote \u27e6Wrapper\u27e7.\u27e6wrap\u27e7) ] con\n   }\n\n  unquoteDecl \u27e6idWrapper\u27e7 = param-rec-def-by-name wrapperEnv (quote idWrapper) \u27e6idWrapper\u27e7\n-}\n\ndata Wrapper (A : Set\u2080) : Set\u2080 where\n  wrap : A \u2192 Wrapper A\n\nidWrapper : \u2200 {A} \u2192 Wrapper A \u2192 Wrapper A\nidWrapper (wrap x) = wrap x\n\ndata [Wrapper] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Wrapper A \u2192 Set\u2080 where\n  [wrap] : (A\u209a [\u2192] [Wrapper] A\u209a) wrap\n\n[Wrapper]-env = record (\u03b5 1)\n  { pDef = [ quote Wrapper \u2254 quote [Wrapper] ] id\n  ; pConP = [ quote wrap \u2254 con (quote [wrap])  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 2 (quote [wrap]) ] con\n  }\n\nunquoteDecl [idWrapper] =\n  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]\n\n  {-\n[idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n-- [idWrapper] = {!  param-rec-def-by-name [Wrapper]-env (quote idWrapper) [idWrapper]!}\n-}\n\n{-\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6Wrapper\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) Wrapper Wrapper\n\nprivate\n  \u27e6Wrapper\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6Wrapper\u27e7 where\n  \u27e6wrap\u27e7 : unquote (\u27e6Wrapper\u27e7-ctor (quote Wrapper.wrap))\n-}\ndata \u27e6Wrapper\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Wrapper A\u2080 \u2192 Wrapper A\u2081 \u2192 Set\u2080 where\n  \u27e6wrap\u27e7 : (A\u1d63 \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A\u1d63) wrap wrap\n\n\u27e6Wrapper\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Wrapper \u2254 quote \u27e6Wrapper\u27e7 ] id\n  ; pConP = [ quote wrap \u2254 con (quote \u27e6wrap\u27e7)  ] con\n  ; pConT = [ quote wrap \u2254 conSkip' 3 (quote \u27e6wrap\u27e7) ] con\n  }\n\n\u27e6idWrapper\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\n\u27e6idWrapper\u27e71 {x0} {x1} (x2) {._} {._} (\u27e6wrap\u27e7 {x3} {x4} x5)\n  = \u27e6wrap\u27e7 {_} {_} {_} {x3} {x4} x5\n\n\u27e6idWrapper\u27e7-clauses =\n  clause\n  (arg (arg-info hidden  relevant) (var \"A0\") \u2237\n   arg (arg-info hidden  relevant) (var \"A1\") \u2237\n   arg (arg-info visible relevant) (var \"Ar\") \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info hidden  relevant) dot \u2237\n   arg (arg-info visible relevant)\n   (con (quote \u27e6wrap\u27e7)\n    (arg (arg-info hidden  relevant) (var \"x0\") \u2237\n     arg (arg-info hidden  relevant) (var \"x1\") \u2237\n     arg (arg-info visible relevant) (var \"xr\") \u2237 []))\n   \u2237 [])\n  (con (quote \u27e6wrap\u27e7)\n   (arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant) unknown \u2237\n    arg (arg-info hidden relevant)  (var 2 []) \u2237\n    arg (arg-info hidden relevant)  (var 1 []) \u2237\n    arg (arg-info visible relevant) (var 0 []) \u2237 []))\n  \u2237 []\n\n\u27e6idWrapper\u27e72 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A \u27e6\u2192\u27e7 \u27e6Wrapper\u27e7 A) idWrapper idWrapper\nunquoteDef \u27e6idWrapper\u27e72 = \u27e6idWrapper\u27e7-clauses\n\nunquoteDecl \u27e6idWrapper\u27e7 =\n  param-rec-def-by-name \u27e6Wrapper\u27e7-env (quote idWrapper) \u27e6idWrapper\u27e7\n\ndata Bot (A : Set\u2080) : Set\u2080 where\n  bot : Bot A \u2192 Bot A\n\ngobot : \u2200 {A} \u2192 Bot A \u2192 A\ngobot (bot x) = gobot x\n\ndata [Bot] {A : Set} (A\u209a : A \u2192 Set\u2080)\n   : Bot A \u2192 Set\u2080 where\n  [bot] : ([Bot] A\u209a [\u2192] [Bot] A\u209a) bot\n\n[Bot]-env = record (\u03b5 1)\n  { pDef = [ quote Bot \u2254 quote [Bot] ] id\n  ; pConP = [ quote bot \u2254 con (quote [bot])  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 2 (quote [bot]) ] con\n  }\n\n[gobot]' : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n[gobot]' {x0} (x1) {._} ([bot] {x2} x3)\n  = [gobot]' {x0} x1 {x2} x3\n\n-- [gobot]' = {!showClauses \"[gobot]'\" (param-rec-clauses-by-name [Bot]-env (quote gobot) (quote [gobot]'))!}\n\n[gobot]2 : (\u2200\u27e8 A \u2236 [Set\u2080] \u27e9[\u2192] [Bot] A [\u2192] A) gobot\n\n[gobot]2-clauses =\n  clause\n    (arg (arg-info hidden  relevant) (var \"A\u1d620\") \u2237\n     arg (arg-info visible relevant) (var \"A\u1d63\") \u2237\n     arg (arg-info hidden  relevant) dot \u2237\n     arg (arg-info visible relevant)\n     (con (quote [bot])\n      (arg (arg-info hidden  relevant) (var \"x\u1d620\") \u2237\n       arg (arg-info visible relevant) (var \"x\u1d63\") \u2237 []))\n     \u2237 [])\n    (def (quote [gobot]2)\n     (arg (arg-info hidden  relevant) (var 4 []) \u2237\n      arg (arg-info visible relevant) (var 3 []) \u2237\n      arg (arg-info hidden  relevant) (var 1 []) \u2237\n      arg (arg-info visible relevant) (var 0 []) \u2237 []))\n    \u2237 []\n\nunquoteDef [gobot]2 = [gobot]2-clauses\n\nunquoteDecl [gobot] =\n  param-rec-def-by-name [Bot]-env (quote gobot) [gobot]\n\ndata \u27e6Bot\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n   : Bot A\u2080 \u2192 Bot A\u2081 \u2192 Set\u2080 where\n  \u27e6bot\u27e7 : (\u27e6Bot\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Bot\u27e7 A\u1d63) bot bot\n\n\u27e6Bot\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote Bot \u2254 quote \u27e6Bot\u27e7 ] id\n  ; pConP = [ quote bot \u2254 con (quote \u27e6bot\u27e7)  ] con\n  ; pConT = [ quote bot \u2254 conSkip' 3 (quote \u27e6bot\u27e7) ] con\n  }\n\n\u27e6gobot\u27e71 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Bot\u27e7 A \u27e6\u2192\u27e7 A) gobot gobot\n\u27e6gobot\u27e71 {x0} {x1} x2 {._} {._} (\u27e6bot\u27e7 {x3} {x4} x5)\n  = \u27e6gobot\u27e71 {x0} {x1} x2 {x3} {x4} x5\n\nunquoteDecl \u27e6gobot\u27e7 =\n  param-rec-def-by-name \u27e6Bot\u27e7-env (quote gobot) \u27e6gobot\u27e7\n\nid\u2080 : {A : Set\u2080} \u2192 A \u2192 A\nid\u2080 x = x\n\n\u27e6id\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 A \u27e6\u2192\u27e7 A) id\u2080 id\u2080\n\u27e6id\u2080\u27e7 = \u03bb {x\u2081} {x\u2082} x\u1d63 {x\u2083} {x\u2084} x\u1d63\u2081 \u2192 x\u1d63\u2081\n\ndata List\u2080 (A : Set) : Set where\n  []  : List\u2080 A\n  _\u2237_ : A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} (f : A \u2192 B) (xs : List\u2080 A) \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\n-- idList\u2080 : List\u2080 \u2115 \u2192 List\u2080 \u2115\nidList\u2080 []       = []\nidList\u2080 {A} (x \u2237 xs) = idList\u2080 {A} xs\n\ndata \u27e6List\u2080\u27e7 {A\u2080 A\u2081 : Set} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080) : List\u2080 A\u2080 \u2192 List\u2080 A\u2081 \u2192 Set\u2080 where\n  \u27e6[]\u27e7  : \u27e6List\u2080\u27e7 A\u1d63 [] []\n  _\u27e6\u2237\u27e7_ : (A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A\u1d63) _\u2237_ _\u2237_\n\ncon\u27e6List\u2080\u27e7 = conSkip' 3\n\u27e6List\u2080\u27e7-env = record (\u03b5 2)\n  { pDef = [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] id\n  ; pConP = [ quote List\u2080.[]  \u2254 con (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  ; pConT = [ quote List\u2080.[]  \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7)  ]\n           ([ quote List\u2080._\u2237_ \u2254 con\u27e6List\u2080\u27e7 (quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_) ]\n            con)\n  }\n\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\n-- \u27e6idList\u2080\u27e7 : (\u2200\u27e8 A \u2236 \u27e6Set\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A \u27e6\u2192\u27e7 \u27e6List\u2080\u27e7 A) idList\u2080 idList\u2080\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n\n{-\n\u27e6map\u2080\u27e7 : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : List\u2080 (x0)} \u2192 {x10 : List\u2080 (x1)} \u2192 (x11 : \u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 \u27e6List\u2080\u27e7 {x3} {x4} (x5) (map\u2080 {x0} {x3} (x6) (x9)) (map\u2080 {x1} {x4} (x7) (x10))\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (\u27e6[]\u27e7 )  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {._} {._} (_\u27e6\u2237\u27e7_ {x11} {x12} (x13) {x14} {x15} (x16) )  = _\u27e6\u2237\u27e7_ {x6 (x11)} {x7 (x12)} (x8 {x11} {x12} (x13)) {map\u2080 {x0} {x3} (x6) (x14)} {map\u2080 {x1} {x4} (x7) (x15)} (\u27e6map\u2080\u27e7 {x0} {x1} (x2) {x3} {x4} (x5) {x6} {x7} (x8) {x14} {x15} (x16))\n-}\n\nunquoteDecl \u27e6map\u2080\u27e7\n = param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) \u27e6map\u2080\u27e7\n\n{-\nmap-nat : \u2200 (f : \u2200 {X} \u2192 List\u2080 X \u2192 List\u2080 X)\n            {A B : Set} (g : A \u2192 B)\n          \u2192 f \u2218 map\u2080 g ~ map\u2080 g \u2218 f\nmap-nat f g x = {!\u27e6map\u2080\u27e7 _\u2261_ _\u2261_ {g}!}\n\n  -- The generated type is bigger since it is a familly for no reason.\n  data \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\n  private\n    \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\n  data \u27e6List\u2080\u27e7 where\n    \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n    _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n  \u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                        [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n             (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n-}\n\ndata Maybe' (A : Set) : Set\u2081 where\n  nothing : Maybe' A\n  just    : A \u2192 Maybe' A\n\n{-\n-- Set\u2081 is here because \u27e6List\u2080\u27e7 is not using parameters, hence gets bigger.\n-- This only happens without-K given the new rules for data types.\ndata List\u2080 : (A : Set) \u2192 Set\u2081 where\n  []  : \u2200 {A} \u2192 List\u2080 A\n  _\u2237_ : \u2200 {A} \u2192 A \u2192 List\u2080 A \u2192 List\u2080 A\n\nmap\u2080 : \u2200 {A B} \u2192 (A \u2192 B) \u2192 List\u2080 A \u2192 List\u2080 B\nmap\u2080 f []       = []\nmap\u2080 f (x \u2237 xs) = f x \u2237 map\u2080 f xs\n\nidList\u2080 : \u2200 {A} \u2192 List\u2080 A \u2192 List\u2080 A\nidList\u2080 []       = []\nidList\u2080 (x \u2237 xs) = x \u2237 idList\u2080 xs\n\n-- The generated type bigger since it is a familly for no reason.\ndata \u27e6List\u2080\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2081\u27e7) List\u2080 List\u2080\n\nprivate\n  \u27e6List\u2080\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2)) c)\n\ndata \u27e6List\u2080\u27e7 where\n  \u27e6[]\u27e7  : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080.[]))\n  _\u27e6\u2237\u27e7_ : unquote (\u27e6List\u2080\u27e7-ctor (quote List\u2080._\u2237_))\n\n\u27e6List\u2080\u27e7-env = extConEnv ([ quote List\u2080.[]  \u2254 quote \u27e6List\u2080\u27e7.\u27e6[]\u27e7  ] \u2218\n                      [ quote List\u2080._\u2237_ \u2254 quote \u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ ])\n           (extDefEnv [ quote List\u2080 \u2254 quote \u27e6List\u2080\u27e7 ] (\u03b5 2))\n\n-- test = \u27e6[]\u27e7 {{!showType (type (quote List\u2080.[]))!}} {{!!}} {!!}\n-}\n\n{-\n\u27e6idList\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080)))\nunquoteDef \u27e6idList\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote idList\u2080) (quote \u27e6idList\u2080\u27e7)\n-}\n\n{-\n\u27e6map\u2080\u27e7 : unquote (unEl (param-type-by-name \u27e6List\u2080\u27e7-env (quote map\u2080)))\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {[]} {[]} \u27e6[]\u27e7\n  = \u27e6[]\u27e7\n\u27e6map\u2080\u27e7 {x} {x\u2081} {x\u2082} {x\u2083} {x\u2084} {x\u2085} {x\u2086} {x\u2087} x\u2088 {._ \u2237 ._}\n  {._ \u2237 ._}\n  (_\u27e6\u2237\u27e7_ {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085 {x\u2081\u2086} {x\u2081\u2087} x\u2081\u2088)\n  = _\u27e6\u2237\u27e7_ {x\u2081\u2080 x\u2081\u2083} {x\u2081\u2081 x\u2081\u2084} (x\u2081\u2082 {x\u2081\u2083} {x\u2081\u2084} x\u2081\u2085)\n    {map\u2080 {x\u2084} {x\u2087} x\u2081\u2080 x\u2081\u2086} {map\u2080 {x\u2085} {x\u2088} x\u2081\u2081 x\u2081\u2087}\n    (\u27e6map\u2080\u27e7 {x\u2084} {x\u2085} {x\u2086} {x\u2087} {x\u2088} {x\u2089} {x\u2081\u2080} {x\u2081\u2081} x\u2081\u2082 {x\u2081\u2086} {x\u2081\u2087}\n     x\u2081\u2088)\n-}\n\n{-\nunquoteDef \u27e6map\u2080\u27e7 = param-rec-clauses-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote \u27e6map\u2080\u27e7)\n-}\n\n\n{-\nfoo : {x0 : Set0} \u2192 {x1 : Set0} \u2192 (x2 : (x2 : x0) \u2192 (x3 : x1) \u2192 Set0) \u2192 {x3 : Set0} \u2192 {x4 : Set0} \u2192 (x5 : (x5 : x3) \u2192 (x6 : x4) \u2192 Set0) \u2192 {x6 : (x6 : x0) \u2192 x3} \u2192 {x7 : (x7 : x1) \u2192 x4} \u2192 (x8 : {x8 : x0} \u2192 {x9 : x1} \u2192 (x10 : x2 (x8) (x9)) \u2192 x5 (x6 (x8)) (x7 (x9))) \u2192 {x9 : Reflection.Param.List\u2080 (x0)} \u2192 {x10 : Reflection.Param.List\u2080 (x1)} \u2192 (x11 : Reflection.Param.\u27e6List\u2080\u27e7 {x0} {x1} (x2) (x9) (x10)) \u2192 Reflection.Param.\u27e6List\u2080\u27e7 {x3} {x4} (x5) (Reflection.Param.map\u2080 {x0} {x3} (x6) (x9)) (Reflection.Param.map\u2080 {x1} {x4} (x7) (x10))\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7 )  = Reflection.Param.\u27e6List\u2080\u27e7.\u27e6[]\u27e7\nfoo {A} {A} (A) {B} {B} (B) {f} {f} (f) {._} {._} (Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x} {x} (x) {xs} {xs} (xs) )  = Reflection.Param.\u27e6List\u2080\u27e7._\u27e6\u2237\u27e7_ {x0 (x0)} {x0 (x0)} (x0 {x0} {x0} (x0)) {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} {Reflection.Param.map\u2080 {x0} {x0} (x0) (x0)} (Reflection.Param.test' {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0) {x0} {x0} (x0))\n-}\n\n-- test' = {! showFunDef \"foo\" (param-rec-def-by-name \u27e6List\u2080\u27e7-env (quote map\u2080) (quote test'))!}\n\nopen import Function.Param.Unary\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\n\nrevealed-[\u2192]' : \u2200 (A : Set\u2080) (A\u209a : A \u2192 Set\u2080)\n                  (B : Set\u2080) (B\u209a : B \u2192 Set\u2080)\n                  (f : A \u2192 B) \u2192 Set\u2080\nunquoteDef revealed-[\u2192]' = Get-clauses.from-def revealed-[\u2192]\n\nrevelator-[\u2192] : ({A : Set} (A\u209a : A \u2192 Set) {B : Set} (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set)\n              \u2192  (A : Set) (A\u209a : A \u2192 Set) (B : Set) (B\u209a : B \u2192 Set) (f : A \u2192 B) \u2192 Set\nunquoteDef revelator-[\u2192] = Revelator.clauses (type (quote _[\u2080\u2192\u2080]_))\n\np-[\u2192]-type = param-type-by-name    (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]      = param-clauses-by-name (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\n\np-[\u2192]' = \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n           {A\u209a : A \u2192 Set\u2080}  (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 A\u209a x \u2192 Set\u2080)\n           {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n           {B\u209a : B \u2192 Set\u2080}  (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 B\u209a x \u2192 Set\u2080)\n           {f : A \u2192 B}      (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n         \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f\n         \u2192 Set\n\np-[\u2192]'-test : p-[\u2192]' \u2261 unquote (unEl p-[\u2192]-type)\np-[\u2192]'-test = refl\n\n[[\u2192]] : unquote (unEl p-[\u2192]-type)\nunquoteDef [[\u2192]] = p-[\u2192]\n\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : [[\u2192]] [\u2115] [[\u2115]] [\u2115] [[\u2115]] [suc] [suc]\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7          \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\npred' : \u2115 \u2192 \u2115\npred' = \u03bb { zero    \u2192 zero\n          ; (suc m) \u2192 m }\n\n\u27e6pred'\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred' pred'\nunquoteDef \u27e6pred'\u27e7 = param-clauses-by-name defEnv2 (quote pred')\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7-type = unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7)))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : \u27e6\u27e6Set\u2080\u27e7\u27e7-type \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = param-clauses-by-name defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261-no-hints quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261-no-hints Get-term.from-clauses p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7'' : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\nunquoteDef \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\n\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' : _\u2261_ {A = \u27e6\u27e6Set\u2080\u27e7\u27e7-type} \u27e6\u27e6Set\u2080\u27e7\u27e7'' \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' = refl\n\ntest-p0-\u27e6Set\u2080\u27e7 : pTerm defEnv0 (quoteTerm \u27e6Set\u2080\u27e7) \u2261-no-hints quoteTerm \u27e6Set\u2080\u27e7\ntest-p0-\u27e6Set\u2080\u27e7 = refl\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\np1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n[\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\ntest-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\ntest-p1\u2115\u2192\u2115 = refl\n\np2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\ntest-p2\u2115\u2192\u2115 = refl\n\np\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n\u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261-no-hints quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\ntest-p\u2115\u2192\u2115\u2192\u2115 = refl\nZERO : Set\u2081\nZERO = (A : Set\u2080) \u2192 A\n\u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n\u27e6ZERO\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\npZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\nq\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\ntest-pZERO : pZERO \u2261-no-hints q\u27e6ZERO\u27e7\ntest-pZERO = refl\nID : Set\u2081\nID = (A : Set\u2080) \u2192 A \u2192 A\n\u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n\u27e6ID\u27e7 f\u2080 f\u2081 =\n  {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n  {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n  \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\npID = pTerm (\u03b5 2) (quoteTerm ID)\nq\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\ntest-ID : q\u27e6ID\u27e7 \u2261-no-hints pID\ntest-ID = refl\n\n\u27e6not\u27e7' : (\u27e6\ud835\udfda\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) not not\nunquoteDef \u27e6not\u27e7' = param-clauses-by-name defEnv2 (quote not)\ntest-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 \u27e6not\u27e7' x\u1d63\ntest-not \u27e60\u2082\u27e7 = refl\ntest-not \u27e61\u2082\u27e7 = refl\n\n[pred]' : ([\u2115] [\u2192] [\u2115]) pred\nunquoteDef [pred]' = param-clauses-by-name defEnv1 (quote pred)\n\ntest-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\ntest-p1-pred [zero]     = refl\ntest-p1-pred ([suc] n\u209a) = refl\n\n\u27e6pred\u27e7' : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) pred pred\nunquoteDef \u27e6pred\u27e7' = param-clauses-by-name defEnv2 (quote pred)\n\ntest-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\ntest-p2-pred \u27e6zero\u27e7     = refl\ntest-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\np\/2 = param-rec-def-by-name defEnv2 (quote _\/2)\nq\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\nunquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\ntest-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261-def-no-hints q\u27e6\/2\u27e7\ntest-\/2 = refl\ntest-\/2' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\ntest-\/2' \u27e6zero\u27e7 = refl\ntest-\/2' (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\ntest-\/2' (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) rewrite test-\/2' n\u1d63 = refl\n\np+ = param-rec-def-by-name defEnv2 (quote _+\u2115_)\nq\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\nunquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\ntest-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261-def-no-hints q\u27e6+\u27e7\ntest-+ = refl\ntest-+' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\ntest-+' \u27e6zero\u27e7    n'\u1d63 = refl\ntest-+' (\u27e6suc\u27e7 n\u1d63) n'\u1d63 rewrite test-+' n\u1d63 n'\u1d63 = refl\n\n{-\nis-good : String \u2192 \ud835\udfda\nis-good \"good\" = 1\u2082\nis-good _      = 0\u2082\n\n\u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n\u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n\u27e6is-good\u27e7 {_}      refl = {!!}\n\nmy-good = unquote (lit (string \"good\"))\nmy-good-test : my-good \u2261 \"good\"\nmy-good-test = refl\n-}\n\n{-\n\u27e6is-good\u27e7' : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\nunquoteDef \u27e6is-good\u27e7' = param-clauses-by-name defEnv2 (quote is-good)\ntest-is-good = {!!}\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"228737ab3f938c852b193616934118f6dadf2a13","subject":"Added \u2203D-proj\u2081 and \u2203D-proj\u2082.","message":"Added \u2203D-proj\u2081 and \u2203D-proj\u2082.\n\nIgnore-this: f21fd3f648ab025a8a8d0d43d126b3e0\n\ndarcs-hash:20100603223545-3bd4e-c1dc41f639f19f458ef39d81f2fe308882d7df23.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Product.agda","new_file":"LTC\/Data\/Product.agda","new_contents":"------------------------------------------------------------------------------\n-- The existential quantifier type on D\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Product where\n\nopen import LTC.Minimal.Core\n-- open import MyStdLib.Data.Product\n\n------------------------------------------------------------------------------\ninfixr 4 _,_\n\n-- \u2203D : (D \u2192 Set) \u2192 Set\n-- \u2203D P = \u2203 P\n\ndata \u2203D (P : D \u2192 Set) : Set where\n  _,_ : (d : D) (Pd : P d) \u2192 \u2203D P\n\n\u2203D-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203D P \u2192 D\n\u2203D-proj\u2081 (x , _ ) = x\n\n\u2203D-proj\u2082 : {P : D \u2192 Set} \u2192 (x-px : \u2203D P) \u2192 P (\u2203D-proj\u2081 x-px)\n\u2203D-proj\u2082 (_ , px) = px\n","old_contents":"------------------------------------------------------------------------------\n-- The existential quantifier type on D\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Product where\n\nopen import LTC.Minimal.Core\n-- open import MyStdLib.Data.Product\n\n------------------------------------------------------------------------------\ninfixr 4 _,_\n\n-- \u2203D : (D \u2192 Set) \u2192 Set\n-- \u2203D P = \u2203 P\n\ndata \u2203D (P : D \u2192 Set) : Set where\n  _,_ : (d : D) (Pd : P d) \u2192 \u2203D P\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"11a231bdddafa77b575a6e41f74818355516023b","subject":"Before performing radical changes , abastractions to simplify the proofs.","message":"Before performing radical changes , abastractions to simplify the proofs.\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/IndexLLProp.agda","new_file":"agda\/IndexLLProp.agda","new_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule IndexLLProp where\n\nopen import Common\nopen import LinLogic\nopen import Data.Maybe\nimport Relation.Binary.HeterogeneousEquality\nopen import Relation.Binary.PropositionalEquality using (trans ; sym ; subst ; cong ; subst\u2082)\n\ndata _\u2245\u1d62_ {i u gll} : \u2200{fll ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set (lsuc u) where\n  \u2245\u1d62\u2193 :  \u2193 \u2245\u1d62 \u2193\n  \u2245\u1d62ic : \u2200{fll ict ll tll il eq} \u2192 {sind : IndexLL gll ll} \u2192 {bind : IndexLL fll ll} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n\n\n-- TODO Is this needed ?\n\u2245\u1d62-spec : \u2200{i u ll ict tll il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll}\n          \u2192 {bind : IndexLL fll ll} \u2192 _\u2245\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n          \u2192 (sind \u2245\u1d62 bind)\n\u2245\u1d62-spec (\u2245\u1d62ic a) = a\n\n\u00ac\u2245\u1d62-spec : \u2200{i u ll ict tll il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll}\n          \u2192 {bind : IndexLL fll ll}\n          \u2192 \u00ac (_\u2245\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}}))\n           \u2192 \u00ac (sind \u2245\u1d62 bind)\n\u00ac\u2245\u1d62-spec rl = \u03bb z \u2192 rl (\u2245\u1d62ic z)\n\n\u2245\u1d62-reflexive : \u2200{i u rll ll} \u2192 (a : IndexLL {i} {u} rll ll) \u2192 a \u2245\u1d62 a\n\u2245\u1d62-reflexive \u2193 = \u2245\u1d62\u2193\n\u2245\u1d62-reflexive (ic x) = \u2245\u1d62ic (\u2245\u1d62-reflexive x)\n\n\n\n\u2261-to-\u2245\u1d62 : \u2200{i u rll ll} \u2192 (a b : IndexLL {i} {u} rll ll) \u2192 a \u2261 b \u2192 a \u2245\u1d62 b\n\u2261-to-\u2245\u1d62 a .a refl = \u2245\u1d62-reflexive a\n\n\n\ndata _\u2264\u1d62_ {i u gll fll} : \u2200{ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set (lsuc u) where\n  \u2264\u1d62\u2193 : {ind : IndexLL fll gll} \u2192 \u2193 \u2264\u1d62 ind\n  \u2264\u1d62ic : \u2200{ll ict tll il eq} \u2192 {sind : IndexLL gll ll} \u2192 {bind : IndexLL fll ll} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n\nmodule _ where\n\n  open import Relation.Binary.PropositionalEquality\n\n  \u2264\u1d62-spec : \u2200{i u ll ict tll il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll}\n            \u2192 {bind : IndexLL fll ll} \u2192 _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n            \u2192 (sind \u2264\u1d62 bind)\n  \u2264\u1d62-spec (\u2264\u1d62ic rl) = rl\n\n\n\u00ac\u2264\u1d62-ext : \u2200{i u ll tll ict il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll} \u2192 {bind : IndexLL fll ll} \u2192 \u00ac (sind \u2264\u1d62 bind) \u2192 \u00ac ( _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}}))\n\u00ac\u2264\u1d62-ext rl = \u03bb z \u2192 rl (\u2264\u1d62-spec z)\n\n\u2264\u1d62-extT : \u2200{i u ll tll ict il eq gll fll} \u2192 (sind : IndexLL {i} {u} gll ll) \u2192 (bind : IndexLL fll ll) \u2192 Set (lsuc u)\n\u2264\u1d62-extT {i} {u} {ll} {tll} {ict} {il} {eq} {gll} {fll} sind bind = _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n\n\n\u2264\u1d62-ext : \u2200{i u ll tll ict all eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll} \u2192 {bind : IndexLL fll ll} \u2192 (sind \u2264\u1d62 bind) \u2192 \u2264\u1d62-extT {tll = tll} {ict} {all} {eq} sind bind\n\u2264\u1d62-ext {i} {u} {ll} {tll} {ict} {all} {eq} {gll} {fll} {sind} {bind} rl = \u2264\u1d62ic {ll = ll} {ict} {tll} {all} {eq} {sind = sind} {bind} rl\n\n\n\u2264\u1d62-reflexive : \u2200{i u gll ll} \u2192 (ind : IndexLL {i} {u} gll ll) \u2192 ind \u2264\u1d62 ind\n\u2264\u1d62-reflexive \u2193 = \u2264\u1d62\u2193\n\u2264\u1d62-reflexive (ic {ll = ll} {tll} {ict} x {{eq}}) = \u2264\u1d62ic {ll = ll} {ict} {tll} {_} {eq} (\u2264\u1d62-reflexive x)\n\n\n\u2264\u1d62-transitive : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (b \u2264\u1d62 c) \u2192 (a \u2264\u1d62 c)\n\u2264\u1d62-transitive \u2264\u1d62\u2193 b = \u2264\u1d62\u2193\n\u2264\u1d62-transitive (\u2264\u1d62ic {ict = ict} {tll} {_} {eq} x) (\u2264\u1d62ic y) = \u2264\u1d62ic {ict = ict} {tll} {_} {eq} (\u2264\u1d62-transitive x y)\n\n\nmodule _ where\n\n  mutual\n  \n    isLTi-abs1 : \u2200{u i ll tll ict il eq rll gll ica icb} \u2192 \u2200 a b\n                 \u2192 ica \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} a {{eq}}\n                 \u2192 icb \u2261 ic {i} {u} {gll} {ll = ll} {tll} {ict} {il} b {{eq}}\n                 \u2192 Dec (a \u2264\u1d62 b)\n                 \u2192 Dec (ica \u2264\u1d62 icb)\n    isLTi-abs1 a b refl refl (yes p) = yes (\u2264\u1d62-ext p)\n    isLTi-abs1 a b refl refl (no \u00acp) = no (\u03bb p \u2192 \u00acp (\u2264\u1d62-spec p))\n\n\n    isLTi-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb rll gll ica icb} \u2192 \u2200 a b\n                \u2192 ica \u2261 ic {i} {u} {rll} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n                \u2192 icb \u2261 ic {i} {u} {gll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n               \u2192 IndU icta ictb eqa eqb\n                \u2192 Dec (ica \u2264\u1d62 icb)\n    isLTi-abs a b iceqa iceqb (ictEq refl refl refl refl) = isLTi-abs1 a b iceqa iceqb (isLTi a b)\n    isLTi-abs a b refl refl (ict\u00acEq \u00acicteq reqa reqb) = no \u03bb { (\u2264\u1d62ic x) \u2192 \u00acicteq refl}\n    \n    isLTi : \u2200{i u gll ll fll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL fll ll) \u2192 Dec (a \u2264\u1d62 b)\n    isLTi \u2193 b = yes \u2264\u1d62\u2193\n    isLTi (ic a) \u2193 = no (\u03bb ())\n    isLTi (ic a) (ic b) = isLTi-abs a b refl refl compIndU\n\n\n\n\nmodule _ where\n  mutual \n    isEq\u1d62-abs1 : \u2200{u i ll tll ict il eq rll} \u2192 {a : IndexLL {i} {u} rll ll} \u2192 {b : IndexLL rll ll} \u2192 Dec (a \u2261 b) \u2192 Dec (ic {ll = ll} {tll} {ict} {il} a {{eq}} \u2261 ic {ll = ll} {tll} {ict} b {{eq}})\n    isEq\u1d62-abs1 (yes refl) = yes refl\n    isEq\u1d62-abs1 (no \u00acp) = no \u03bb { refl \u2192 \u00acp refl}\n\n    isEq\u1d62-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb rll ica icb} \u2192 \u2200 a b\n            \u2192 ica \u2261 ic {i} {u} {rll} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n            \u2192 icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n            \u2192 IndU icta ictb eqa eqb\n            \u2192 Dec (ica \u2261 icb)\n    isEq\u1d62-abs a b refl refl (ictEq refl refl refl refl) = isEq\u1d62-abs1 (isEq\u1d62 a b)\n    isEq\u1d62-abs a b refl refl (ict\u00acEq \u00acicteq reqa reqb) = no \u03bb { refl \u2192 \u00acicteq refl}\n    \n    isEq\u1d62 : \u2200{u i ll rll} \u2192 (a : IndexLL {i} {u} rll ll) \u2192 (b : IndexLL rll ll) \u2192 Dec (a \u2261 b)\n    isEq\u1d62 \u2193 \u2193 = yes refl\n    isEq\u1d62 \u2193 (ic b) = no \u03bb ()\n    isEq\u1d62 (ic a) \u2193 = no \u03bb ()\n    isEq\u1d62 ica@(ic a) icb@(ic b) = isEq\u1d62-abs a b refl refl compIndU\n   \n\n\nmodule _ where\n\n  open import Data.Vec\n\n  mutual\n\n    ind\u03c4\u21d2\u00acle-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb rll n dt df ica icb} \u2192 \u2200 a b\n            \u2192 ica \u2261 ic {i} {u} {\u03c4 {n = n} {dt} df} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n            \u2192 icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n            \u2192 \u00ac icb \u2245\u1d62 ica\n            \u2192 IndU icta ictb eqa eqb\n            \u2192 \u00ac ica \u2264\u1d62 icb\n    ind\u03c4\u21d2\u00acle-abs a b refl refl neq (ictEq refl refl refl refl) = \u00ac\u2264\u1d62-ext (ind\u03c4\u21d2\u00acle a b (\u00ac\u2245\u1d62-spec neq))\n    ind\u03c4\u21d2\u00acle-abs a b refl refl neq (ict\u00acEq \u00acicteq reqa reqb) = \u03bb { (\u2264\u1d62ic x) \u2192 \u00acicteq refl}\n\n-- Is there a better way to express this?\n    ind\u03c4\u21d2\u00acle : \u2200{i u rll ll n dt df} \u2192 (ind : IndexLL {i} {u} (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (ind2 : IndexLL rll ll) \u2192 \u00ac (ind2 \u2245\u1d62 ind) \u2192 \u00ac (ind \u2264\u1d62 ind2)\n    ind\u03c4\u21d2\u00acle \u2193 \u2193 neq = \u03bb _ \u2192 neq \u2245\u1d62\u2193\n    ind\u03c4\u21d2\u00acle (ic ind1) \u2193 neq = \u03bb ()\n    ind\u03c4\u21d2\u00acle (ic ind1) (ic ind2 ) neq = ind\u03c4\u21d2\u00acle-abs ind1 ind2 refl refl neq compIndU\n\n\nmodule _ where\n\n\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 : \u2200{i u rll ll n dt df}\n                \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll) (lind : IndexLL rll ll)\n                \u2192 \u00ac (lind \u2264\u1d62 ind) \u2192 \u00ac (lind \u2245\u1d62 ind)\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 \u2193 \u2193 neq = \u03bb _ \u2192 neq \u2264\u1d62\u2193\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 (ic ind) \u2193 neq = \u03bb ()\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 (ic ind) (ic lind) neq = \u03bb { (\u2245\u1d62ic x) \u2192 ind\u03c4&\u00acge\u21d2\u00ac\u2245 ind lind (\u03bb z \u2192 neq (\u2264\u1d62ic z)) x}\n\n \n\ndata Ordered\u1d62 {i u gll fll ll} (a : IndexLL {i} {u} gll ll) (b : IndexLL {i} {u} fll ll) : Set (lsuc u) where\n  a\u2264\u1d62b : a \u2264\u1d62 b \u2192 Ordered\u1d62 a b\n  b\u2264\u1d62a : b \u2264\u1d62 a \u2192 Ordered\u1d62 a b\n\n\n\nord-spec : \u2200{i u rll ll ict tll il eq fll} \u2192 {emi : IndexLL {i} {u} fll ll}\n           \u2192 {ind : IndexLL rll ll} \u2192 Ordered\u1d62 (ic {tll = tll} {ict} {il} ind {{eq}}) (ic {tll = tll} {ict} {il} emi {{eq}}) \u2192 Ordered\u1d62 ind emi\nord-spec (a\u2264\u1d62b x) = a\u2264\u1d62b (\u2264\u1d62-spec x)\nord-spec (b\u2264\u1d62a x) = b\u2264\u1d62a (\u2264\u1d62-spec x)\n\nord-ext : \u2200{i u rll ll ict tll il eq fll} \u2192 {emi : IndexLL {i} {u} fll ll}\n           \u2192 {ind : IndexLL rll ll} \u2192 Ordered\u1d62 ind emi \u2192 Ordered\u1d62 (ic {tll = tll} {ict} {il} ind {{eq}}) (ic {tll = tll} {ict} {il} emi {{eq}})\nord-ext (a\u2264\u1d62b x) = a\u2264\u1d62b (\u2264\u1d62-ext x)\nord-ext (b\u2264\u1d62a x) = b\u2264\u1d62a (\u2264\u1d62-ext x)\n\n\nord-spec\u2218ord-ext\u2261id : \u2200{i u ll ict tll il eq fll rll} \u2192 {ind : IndexLL {i} {u} fll ll} \u2192 {lind : IndexLL rll ll} \u2192 (ord : Ordered\u1d62 ind lind) \u2192 ord-spec {ict = ict} {tll = tll} {il} {eq} (ord-ext {ict = ict} {tll = tll} ord) \u2261 ord\nord-spec\u2218ord-ext\u2261id (a\u2264\u1d62b x) = refl\nord-spec\u2218ord-ext\u2261id (b\u2264\u1d62a x) = refl\n\n\n\nmodule _ where\n\n  isOrd\u1d62-abs : \u2200{i u gll fll ll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL {i} {u} fll ll)\n               \u2192 Dec (a \u2264\u1d62 b) \u2192 Dec (b \u2264\u1d62 a)\n               \u2192 Dec (Ordered\u1d62 a b)\n  isOrd\u1d62-abs a b (yes p) r = yes (a\u2264\u1d62b p)\n  isOrd\u1d62-abs a b (no \u00acp) (yes p) = yes (b\u2264\u1d62a p)\n  isOrd\u1d62-abs a b (no \u00acp) (no \u00acp\u2081) = no (\u03bb { (a\u2264\u1d62b x) \u2192 \u00acp x ; (b\u2264\u1d62a x) \u2192 \u00acp\u2081 x})\n  \n  isOrd\u1d62 : \u2200{i u gll fll ll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL {i} {u} fll ll)\n           \u2192 Dec (Ordered\u1d62 a b)\n  isOrd\u1d62 a b = isOrd\u1d62-abs a b (isLTi a b) (isLTi b a) \n\n\n\n\nflipOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n           \u2192 Ordered\u1d62 a b \u2192 Ordered\u1d62 b a\nflipOrd\u1d62 (a\u2264\u1d62b x) = b\u2264\u1d62a x\nflipOrd\u1d62 (b\u2264\u1d62a x) = a\u2264\u1d62b x\n\nflipNotOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac Ordered\u1d62 a b \u2192 \u00ac Ordered\u1d62 b a\nflipNotOrd\u1d62 nord = \u03bb x \u2192 nord (flipOrd\u1d62 x) \n\n\n\u00aclt\u00acgt\u21d2\u00acOrd : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac (a \u2264\u1d62 b) \u2192 \u00ac (b \u2264\u1d62 a) \u2192 \u00ac(Ordered\u1d62 a b)\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (a\u2264\u1d62b x) = nlt x\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (b\u2264\u1d62a x) = ngt x\n\n\u00acord-spec : \u2200{i u rll ll ict tll il eq fll} \u2192 {emi : IndexLL {i} {u} fll ll}\n            \u2192 {ind : IndexLL rll ll} \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} {il} ind {{eq}}) (ic {tll = tll} {ict} {il} emi {{eq}})) \u2192 \u00ac Ordered\u1d62 ind emi\n\u00acord-spec nord ord = nord (ord-ext ord)\n\n\n\n\u00acict\u21d2\u00acord : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll ica icb} \u2192 \u2200 a b\n       \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n       \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n       \u2192 \u00ac icta \u2261 ictb\n       \u2192 \u00ac Ordered\u1d62 icb ica\n\u00acict\u21d2\u00acord a b refl refl \u00acp (a\u2264\u1d62b (\u2264\u1d62ic x)) = \u00acp refl\n\u00acict\u21d2\u00acord a b refl refl \u00acp (b\u2264\u1d62a (\u2264\u1d62ic x)) = \u00acp refl\n\n\nind\u03c4&\u00acge\u21d2\u00acOrd : \u2200{i u rll ll n dt df} \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n                          (lind : IndexLL rll ll) \u2192 \u00ac (lind \u2264\u1d62 ind) \u2192 \u00ac Ordered\u1d62 ind lind\nind\u03c4&\u00acge\u21d2\u00acOrd ind lind neq (a\u2264\u1d62b x) = ind\u03c4\u21d2\u00acle ind lind (ind\u03c4&\u00acge\u21d2\u00ac\u2245 ind lind neq) x\nind\u03c4&\u00acge\u21d2\u00acOrd ind lind neq (b\u2264\u1d62a x) = neq x                        \n\n\n\na,c\u2264\u1d62b\u21d2ordac : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (c \u2264\u1d62 b) \u2192 Ordered\u1d62 a c\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 ltbc = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62ic ltab) \u2264\u1d62\u2193 = b\u2264\u1d62a \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62ic ltab) (\u2264\u1d62ic ltcb) = ord-ext (a,c\u2264\u1d62b\u21d2ordac ltab ltcb)\n\n\n\n\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 \u00ac Ordered\u1d62 a c \u2192 \u00ac Ordered\u1d62 b c\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (a\u2264\u1d62b x) = \u22a5-elim (nord (a\u2264\u1d62b (\u2264\u1d62-transitive lt x)))\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (b\u2264\u1d62a x) = \u22a5-elim (nord (a,c\u2264\u1d62b\u21d2ordac lt x))\n\n\n_-\u1d62_ : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll) \u2192 (sind \u2264\u1d62 bind)\n       \u2192 IndexLL cll pll\n(bind -\u1d62 .\u2193) \u2264\u1d62\u2193 = bind\n((ic bind) -\u1d62 (ic sind)) (\u2264\u1d62ic lt) = (bind -\u1d62 sind) lt\n\n\n\n\nreplLL-\u2193 : \u2200{i u pll ell ll} \u2192 \u2200(ind : IndexLL {i} {u} pll ll)\n           \u2192 replLL ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell \u2261 ell\nreplLL-\u2193 \u2193 = refl\nreplLL-\u2193 (ic ind) = replLL-\u2193 ind\n\n\n\nind-rpl\u2193 : \u2200{i u ll pll cll ell} \u2192 (ind : IndexLL {i} {u} pll ll)\n        \u2192 IndexLL cll (replLL ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell) \u2192 IndexLL cll ell\nind-rpl\u2193 {_} {_} {_} {pll} {cll} {ell} ind x\n  =  subst (\u03bb x \u2192 x) (cong (\u03bb x \u2192 IndexLL cll x) (replLL-\u2193 {ell = ell} ind)) x \n\n\n\na\u2264\u1d62b-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n             \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (lt : emi \u2264\u1d62 ind)\n             \u2192 IndexLL (replLL ((ind -\u1d62 emi) lt) frll) (replLL ind frll) \na\u2264\u1d62b-morph .\u2193 ind \u2264\u1d62\u2193 = \u2193\na\u2264\u1d62b-morph (ic {tll = tll} {ict = ict} emi) (ic ind) (\u2264\u1d62ic lt) = ic {tll = tll} {ict = ict} (a\u2264\u1d62b-morph emi ind lt)\n\n\n\n\nmodule _ where\n\n  a\u2264\u1d62b-morph-id-abs : \u2200{i u ll tll ict rll}\n               \u2192 {ind : IndexLL {i} {u} rll ll}\n               \u2192 \u2200 {w\u2081T} \u2192 (w\u2081 : w\u2081T \u2261 rll)  -- w\u2081T : replLL ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll\n               \u2192 \u2200 {w\u2082T} \u2192 (w\u2082 : w\u2082T \u2261 ll) -- w\u2082T : replLL li ind rl\n               \u2192 (w\u2083 : IndexLL w\u2081T w\u2082T) -- w\u2083 : (a\u2264\u1d62b-morph ind ind (\u2264\u1d62-reflexive ind))\n               \u2192 (w\u2084 : subst\u2082 (\u03bb x y \u2192 IndexLL x y) w\u2081 w\u2082 w\u2083 \u2261 ind) -- recursive step\n               \u2192 subst\u2082 IndexLL w\u2081 \n                   (cong (\u03bb x \u2192 il[ expLLT x ict tll ]) w\u2082)\n                     (ic {tll = tll} {ict} w\u2083) \u2261 ic {tll = tll} {ict} ind\n  a\u2264\u1d62b-morph-id-abs refl refl _ refl = refl\n\n\n  a\u2264\u1d62b-morph-id : \u2200{i u ll rll}\n               \u2192 (ind : IndexLL {i} {u} rll ll)\n               \u2192 subst\u2082 (\u03bb x y \u2192 IndexLL x y) (replLL-\u2193 {ell = rll} ind) (replLL-id ind rll refl) (a\u2264\u1d62b-morph ind ind (\u2264\u1d62-reflexive ind)) \u2261 ind\n  a\u2264\u1d62b-morph-id \u2193 = refl\n  a\u2264\u1d62b-morph-id {rll = rll} (ic ind {{refl}}) = a\u2264\u1d62b-morph-id-abs (replLL-\u2193 {ell = rll} ind) (replLL-id ind rll refl) (a\u2264\u1d62b-morph ind ind (\u2264\u1d62-reflexive ind)) (a\u2264\u1d62b-morph-id ind)\n\n\n\nreplLL-a\u2264b\u2261a : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 {gll}\n               \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (lt : emi \u2264\u1d62 ind)\n               \u2192 replLL (a\u2264\u1d62b-morph emi ind {frll = frll} lt) gll \u2261 replLL emi gll\nreplLL-a\u2264b\u2261a .\u2193 ind \u2264\u1d62\u2193 = refl\nreplLL-a\u2264b\u2261a (ic {tll = tll} {ict} emi) (ic ind) (\u2264\u1d62ic lt) = cong (\u03bb x \u2192 il[ expLLT x ict tll ]) (replLL-a\u2264b\u2261a emi ind lt)\n\nmodule _ where\n\n  mutual\n  \n    \u00acord-morph-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll frll ica icb} \u2192 \u2200 a b\n              \u2192 ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n              \u2192 icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n              \u2192 \u00ac Ordered\u1d62 icb ica\n              \u2192 IndU icta ictb eqa eqb\n              \u2192 IndexLL fll il[ expLLT (replLL b frll) ictb tllb ]\n    \u00acord-morph-abs {tllb = tllb} {ictb} {frll = frll} a b refl refl nord (ictEq _ _ _ refl)\n        = ic {ll = _} {tllb} {ictb} (\u00acord-morph a b {frll = frll} (\u00acord-spec nord))\n    \u00acord-morph-abs {icta = icta} {eqa = eqa} {eqb = eqb} {frll = frll} a b refl refl nord (ict\u00acEq \u00acicteq refl refl) = ic {tll\n        = replLL b frll} {ict = icta} a {{rexpLLT\u21d2req \u00acicteq eqa eqb}}\n\n\n    \u00acord-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n                 \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (nord : \u00ac Ordered\u1d62 ind emi)\n                 \u2192 IndexLL fll (replLL ind frll)\n    \u00acord-morph emi \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    \u00acord-morph \u2193 (ic ind) nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    \u00acord-morph (ic {ll = lle} {tlle} {icte} {il} emi {{eqe}}) (ic {ll = lli} {tlli} {icti} ind {{eqi}}) {frll} nord\n        = \u00acord-morph-abs emi ind refl refl nord compIndU\n\n\nmodule _ where\n\nmutual\n\n    \u00acord-morph-\u00acord-ir-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll ica icb frll} \u2192 \u2200 a b\n      \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n      \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n      \u2192 (nord1 nord2 : \u00ac Ordered\u1d62 icb ica)\n      \u2192 (w : IndU icta ictb eqa eqb)\n      \u2192  \u00acord-morph-abs {frll = frll} a b iceqa iceqb nord1 w \u2261 \u00acord-morph-abs a b iceqa iceqb nord2 w\n    \u00acord-morph-\u00acord-ir-abs {tllb = tllb} {ictb} {frll = frll} a b refl refl nord1 nord2 (ictEq refl refl refl refl)\n        = cong (\u03bb z \u2192 ic {ll = _} {tllb} {ictb} z) (\u00acord-morph-\u00acord-ir a b {frll} (\u00acord-spec nord1) (\u00acord-spec nord2))\n    \u00acord-morph-\u00acord-ir-abs ica icb refl refl nord1 nord2 (ict\u00acEq \u00acicteq refl refl) = refl\n\n\n    \u00acord-morph-\u00acord-ir : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n                         \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (nord nord2 : \u00ac Ordered\u1d62 ind emi)\n                         \u2192 \u00acord-morph emi ind {frll} nord \u2261 \u00acord-morph emi ind {frll} nord2\n    \u00acord-morph-\u00acord-ir \u2193 ind {frll} nord nord2 = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    \u00acord-morph-\u00acord-ir (ic emi) \u2193 {frll} nord nord2 = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    \u00acord-morph-\u00acord-ir ice@(ic emi) ici@(ic ind) {frll} nord nord2 = \u00acord-morph-\u00acord-ir-abs emi ind refl refl nord nord2 compIndU \n\n\n\nmodule _ where\n\n  mutual\n\n    replLL-\u00acordab\u2261ba-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll ica icb gll frll} \u2192 \u2200 a b\n      \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n      \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n      \u2192 (nord : \u00ac Ordered\u1d62 icb ica)\n      \u2192 (fnord : \u00ac Ordered\u1d62 ica icb)\n      \u2192 \u2200 w1 w2\n      \u2192 replLL\n          (\u00acord-morph-abs {frll = frll} a b iceqa iceqb nord w1) gll\n        \u2261\n        replLL\n          (\u00acord-morph-abs {frll = gll} b a iceqb iceqa fnord w2) frll\n    replLL-\u00acordab\u2261ba-abs {tllb = tllb} {ictb} {eqb} {gll = gll} {frll} a b refl refl nord fnord (ictEq _ _ _ refl) (ictEq _ _ _ refl)\n      =  cong (\u03bb z \u2192 il[ expLLT z ictb tllb ])\n              (replLL-\u00acordab\u2261ba a b (\u00acord-spec nord) (\u00acord-spec fnord))\n    replLL-\u00acordab\u2261ba-abs a b iceqa iceqb nord fnord (ictEq icteq lleq tlleq eqeq) (ict\u00acEq \u00acicteq reqa reqb) = \u22a5-elim (\u00acicteq (sym icteq))\n    replLL-\u00acordab\u2261ba-abs a b iceqa iceqb nord fnord (ict\u00acEq \u00acicteq reqa reqb) (ictEq icteq lleq tlleq eqeq) = \u22a5-elim (\u00acicteq (sym icteq))\n    replLL-\u00acordab\u2261ba-abs {eqa = eqa} {eqb = eqb} a b refl refl nord fnord (ict\u00acEq _ refl refl) (ict\u00acEq \u00acicteq refl refl) = cong il[_] (rexpLLT\u21d2req \u00acicteq eqb eqa)\n\n\n    replLL-\u00acordab\u2261ba : \u2200{i u rll ll fll}\n      \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 {gll}\n      \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll}\n      \u2192 (nord : \u00ac Ordered\u1d62 ind emi)\n      \u2192 (fnord : \u00ac Ordered\u1d62 emi ind)\n      \u2192 replLL (\u00acord-morph emi ind {frll} nord) gll \u2261 replLL (\u00acord-morph ind emi {gll} fnord) frll\n    replLL-\u00acordab\u2261ba \u2193 ind nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    replLL-\u00acordab\u2261ba (ic emi) \u2193 nord fnord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    replLL-\u00acordab\u2261ba ice@(ic emi) ici@(ic ind) nord fnord = replLL-\u00acordab\u2261ba-abs emi ind refl refl nord fnord compIndU compIndU \n\n\n\n\nmodule _ where\n\n  mutual\n\n    lemma\u2081-\u00acord-a\u2264\u1d62b-abs : \u2200{u i ll tll ict il eq llc tllc ictc fll rll pll ica icb ell} \u2192 \u2200 a b c\n               \u2192 \u2200 {icc eqc}\n               \u2192 ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}}\n               \u2192 icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}}\n               \u2192 icc \u2261 ic {i} {u} {pll} {ll = llc} {tllc} {ictc} {expLLT (replLL b ell) ict tll} c {{eqc}}\n               \u2192 \u2200 lt\n               \u2192 \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) icc\n               \u2192 IndU ict ictc refl eqc\n               \u2192 IndexLL pll il[ il ]\n    lemma\u2081-\u00acord-a\u2264\u1d62b-abs {tll = tll} {ict} {eq = eq} {ell = ell} a b c iceqa iceqb refl lt nord (ictEq _ _ _ refl) = ic {tll = tll} {ict = ict} (lemma\u2081-\u00acord-a\u2264\u1d62b a b ell lt c (\u00acord-spec nord)) {{eq}}\n    lemma\u2081-\u00acord-a\u2264\u1d62b-abs {ll = ll} {eq = eq} {ictc = ictc} a b c {eqc = eqc} iceqa iceqb iceqc lt nord (ict\u00acEq \u00acicteq refl refl) = ic {tll = ll} {ictc} c {{trans eq (sym (rexpLLT\u21d2req \u00acicteq refl eqc))}}\n\n\n    lemma\u2081-\u00acord-a\u2264\u1d62b : \u2200{i u ll pll rll fll}\n          \u2192 (emi : IndexLL {i} {u} fll ll)\n          \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n          \u2192 (omi : IndexLL pll (replLL ind ell))\n          \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind {ell} lt) omi)\n          \u2192 IndexLL pll ll\n    lemma\u2081-\u00acord-a\u2264\u1d62b .\u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    lemma\u2081-\u00acord-a\u2264\u1d62b .(ic _) .(ic _) ell (\u2264\u1d62ic lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    lemma\u2081-\u00acord-a\u2264\u1d62b ica@(ic a) icb@(ic b {{eq}}) ell (\u2264\u1d62ic lt) icc@(ic c) nord = lemma\u2081-\u00acord-a\u2264\u1d62b-abs {eq = eq} a b c refl refl refl lt nord compIndU\n\n\n\nmodule _ where\n\nmutual\n\n    \u00acord-morph\u21d2\u00acord-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll frll ica icb} \u2192 \u2200 a b\n           \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n           \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n           \u2192 (nord : \u00ac Ordered\u1d62 icb ica)\n           \u2192 (w : IndU icta ictb eqa eqb)\n           \u2192  \u00ac Ordered\u1d62 (ic {tll = tllb} {ict = ictb} (a\u2264\u1d62b-morph b b {frll = frll} (\u2264\u1d62-reflexive b)))\n                         (\u00acord-morph-abs {frll = frll} a b iceqa iceqb nord w)\n    \u00acord-morph\u21d2\u00acord-abs {frll = frll} a b refl refl nord (ictEq icteq lleq tlleq refl) = \u03bb x \u2192 r (ord-spec x) where\n      r = \u00acord-morph\u21d2\u00acord a b {frll} (\u00acord-spec nord)\n    \u00acord-morph\u21d2\u00acord-abs a b refl refl nord (ict\u00acEq \u00acicteq refl refl) = \u00acict\u21d2\u00acord a (a\u2264\u1d62b-morph b b (\u2264\u1d62-reflexive b)) refl refl \u00acicteq\n  \n  \n    \u00acord-morph\u21d2\u00acord : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n          \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (nord : \u00ac Ordered\u1d62 ind emi)\n          \u2192 \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph ind ind {frll} (\u2264\u1d62-reflexive ind)) (\u00acord-morph emi ind nord)\n    \u00acord-morph\u21d2\u00acord \u2193 ind nord = \u03bb _ \u2192 nord (b\u2264\u1d62a \u2264\u1d62\u2193)\n    \u00acord-morph\u21d2\u00acord (ic emi) \u2193 nord = \u03bb _ \u2192 nord (a\u2264\u1d62b \u2264\u1d62\u2193)\n    \u00acord-morph\u21d2\u00acord (ic emi) (ic ind) nord = \u00acord-morph\u21d2\u00acord-abs emi ind refl refl nord compIndU\n\n\n\nmodule _ where\n\n  mutual\n\n    rlemma\u2081\u21d2\u00acord-abs : \u2200{u i ll tll ict il eq llc tllc ictc fll rll pll ica icb ell} \u2192 \u2200 a b c\n           \u2192 \u2200{eqc icc}\n           \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}})\n           \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}})\n           \u2192 (iceqc : icc \u2261 ic {i} {u} {pll} {ll = llc} {tllc} {ictc} {expLLT (replLL b ell) ict tll} c {{eqc}})\n           \u2192 \u2200 lt\n           \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) icc)\n           \u2192 (w : IndU ict ictc refl eqc)\n           \u2192 \u00ac Ordered\u1d62 ica\n                        (lemma\u2081-\u00acord-a\u2264\u1d62b-abs a b c iceqa iceqb iceqc lt nord w)\n    rlemma\u2081\u21d2\u00acord-abs {ell = ell} a b c refl refl refl lt nord (ictEq icteq lleq tlleq refl) = \u03bb x \u2192 r (ord-spec x) where\n      r = rlemma\u2081\u21d2\u00acord a b ell lt c (\u00acord-spec nord)\n    rlemma\u2081\u21d2\u00acord-abs a b c refl refl refl lt nord (ict\u00acEq \u00acicteq refl refl) = \u00acict\u21d2\u00acord c a refl refl (\u03bb x \u2192 \u00acicteq (sym x))\n\n    rlemma\u2081\u21d2\u00acord : \u2200{i u ll pll rll fll}\n       \u2192 (emi : IndexLL {i} {u} fll ll)\n       \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n       \u2192 (omi : IndexLL pll (replLL ind ell))\n       \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind lt) omi)\n       \u2192 \u00ac Ordered\u1d62 emi (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi nord)\n    rlemma\u2081\u21d2\u00acord .\u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    rlemma\u2081\u21d2\u00acord .(ic _) .(ic _) ell (\u2264\u1d62ic lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    rlemma\u2081\u21d2\u00acord (ic emi) (ic ind) ell (\u2264\u1d62ic lt) (ic omi) nord = rlemma\u2081\u21d2\u00acord-abs emi ind omi refl refl refl lt nord compIndU\n\n\n\nmodule _ where\n\n  mutual \n\n    \u00acord-morph$lemma\u2081\u2261I-abs1 : \u2200{u i ll tll ict il eq fll rll pll ica icb ell} \u2192 \u2200 a b c\n               \u2192 \u2200 {icc}\n               \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}})\n               \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}})\n               \u2192 (iceqc : icc \u2261 ic {i} {u} {pll} {ll = replLL b ell} {tll} {ict} {expLLT (replLL b ell) ict tll} c {{refl}})\n               \u2192 \u2200 lt\n               \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) (ic {tll = tll} {ict} c {{refl}}))\n               \u2192 (w : IndU ict ict eq eq)\n               \u2192 \u00acord-morph-abs\n                   (lemma\u2081-\u00acord-a\u2264\u1d62b a b ell lt c (\u03bb ord \u2192 nord (ord-ext ord))) b refl refl\n                     (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc (\u2264\u1d62ic lt)\n                       (\u03bb x \u2192 rlemma\u2081\u21d2\u00acord a b ell lt c (\u03bb ord \u2192 nord (ord-ext ord)) (ord-spec x))) w\n                 \u2261 ic {tll = tll} {ict} c {{refl}}\n    \u00acord-morph$lemma\u2081\u2261I-abs1 {ell = ell} a b c refl refl refl lt nord (ictEq icteq lleq tlleq refl) = {!r!} where\n      r = \u00acord-morph$lemma\u2081\u2261I ell a b lt c (\u00acord-spec nord)\n    \u00acord-morph$lemma\u2081\u2261I-abs1 a b c refl refl refl lt nord (ict\u00acEq \u00acicteq reqa reqb) = {!!}          \n\n\n    \u00acord-morph$lemma\u2081\u2261I-abs : \u2200{u i ll tll ict il eq llc tllc ictc fll rll pll ica icb ell} \u2192 \u2200 a b c\n               \u2192 \u2200 {icc eqc}\n               \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}})\n               \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}})\n               \u2192 (iceqc : icc \u2261 ic {i} {u} {pll} {ll = llc} {tllc} {ictc} {expLLT (replLL b ell) ict tll} c {{eqc}})\n               \u2192 \u2200 lt\n               \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) (ic c))\n               \u2192 (w : IndU ict ictc refl eqc)\n               \u2192 \u00acord-morph\n                   (lemma\u2081-\u00acord-a\u2264\u1d62b-abs {eq = eq} {ell = ell} a b c refl refl refl lt nord w) (ic b) {frll = ell} (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc (\u2264\u1d62ic lt) (rlemma\u2081\u21d2\u00acord-abs a b c refl refl refl lt nord w)) \u2261 icc\n    \u00acord-morph$lemma\u2081\u2261I-abs a b c refl refl refl lt nord (ictEq icteq lleq tlleq refl) = {!!} -- \u00acord-morph$lemma\u2081\u2261I-abs1 a b c refl refl refl lt nord compIndU where\n--      r = \u00acord-morph$lemma\u2081\u2261I ell a b lt c (\u00acord-spec nord)\n    \u00acord-morph$lemma\u2081\u2261I-abs a b c refl refl refl lt nord (ict\u00acEq \u00acicteq refl refl) = {!!} \n\n\n\n    \u00acord-morph$lemma\u2081\u2261I : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind {ell} lt) lind)\n          \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind nord)) \u2261 lind)\n    \u00acord-morph$lemma\u2081\u2261I ell .\u2193 ind \u2264\u1d62\u2193 lind nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    \u00acord-morph$lemma\u2081\u2261I ell .(ic _) .(ic _) (\u2264\u1d62ic lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    \u00acord-morph$lemma\u2081\u2261I ell ice@(ic emi) ici@(ic ind) clt@(\u2264\u1d62ic lt) icl@(ic lind) nord = {!!} -- \u00acord-morph$lemma\u2081\u2261I-abs emi ind lind refl refl refl lt nord  compIndU compIndU (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc clt (rlemma\u2081\u21d2\u00acord ice ici ell clt icl nord))\n\n\n    \u00acord-morph$lemma\u2081\u2261I' : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind lt) lind) \u2192 (lnord : \u00ac Ordered\u1d62 ind (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord))\n        \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind lnord \u2261 lind)\n    \u00acord-morph$lemma\u2081\u2261I' = {!!}\n\n\n-- module _ where\n\n--   \u00acord-morph$lemma\u2081\u2261I' : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) lind) \u2192 (lnord : \u00ac Ordered\u1d62 ind (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord))\n--          \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind ell lnord \u2261 lind)\n--   \u00acord-morph$lemma\u2081\u2261I' ell \u2193 ind \u2264\u1d62\u2193 lind nord _ = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (lind \u2190\u2227) nord lnord = cong (\u03bb x \u2192 x \u2190\u2227) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 lind) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (lind \u2190\u2227) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 lind) nord lnord = cong (\u03bb x \u2192 \u2227\u2192 x) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (lind \u2190\u2228) nord lnord = cong (\u03bb x \u2192 x \u2190\u2228) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 lind) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (lind \u2190\u2228) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 lind) nord lnord = cong (\u03bb x \u2192 \u2228\u2192 x) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (lind \u2190\u2202) nord lnord = cong (\u03bb x \u2192 x \u2190\u2202) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 lind) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (lind \u2190\u2202) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 lind) nord lnord = cong (\u03bb x \u2192 \u2202\u2192 x) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n    \n--   \u00acord-morph$lemma\u2081\u2261I : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) lind)\n--        \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind ell (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind nord)) \u2261 lind)\n--   \u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind nord = \u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind nord (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind nord))\n\n\n\n-- _+\u1d62_ : \u2200{i u pll cll ll} \u2192 IndexLL {i} {u} pll ll \u2192 IndexLL cll pll \u2192 IndexLL cll ll\n-- _+\u1d62_ \u2193 is = is\n-- _+\u1d62_ (if \u2190\u2227) is = (_+\u1d62_ if is) \u2190\u2227\n-- _+\u1d62_ (\u2227\u2192 if) is = \u2227\u2192 (_+\u1d62_ if is)\n-- _+\u1d62_ (if \u2190\u2228) is = (_+\u1d62_ if is) \u2190\u2228\n-- _+\u1d62_ (\u2228\u2192 if) is = \u2228\u2192 (_+\u1d62_ if is)\n-- _+\u1d62_ (if \u2190\u2202) is = (_+\u1d62_ if is) \u2190\u2202\n-- _+\u1d62_ (\u2202\u2192 if) is = \u2202\u2192 (_+\u1d62_ if is)\n\n\n\n\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 : \u2200{i u pll cll ll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--            \u2192 ind \u2264\u1d62 (ind +\u1d62 lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 \u2193 lind = \u2264\u1d62\u2193\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2227) lind = \u2264\u1d62\u2190\u2227 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2227\u2192 ind) lind = \u2264\u1d62\u2227\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2228) lind = \u2264\u1d62\u2190\u2228 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2228\u2192 ind) lind = \u2264\u1d62\u2228\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2202) lind = \u2264\u1d62\u2190\u2202 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2202\u2192 ind) lind = \u2264\u1d62\u2202\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n\n\n-- a+\u2193\u2261a : \u2200{i u pll ll} \u2192 (a : IndexLL {i} {u} pll ll) \u2192 a +\u1d62 \u2193 \u2261 a\n-- a+\u2193\u2261a \u2193 = refl\n-- a+\u2193\u2261a (a \u2190\u2227) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (a \u2190\u2227) | .a | refl = refl\n-- a+\u2193\u2261a (\u2227\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (\u2227\u2192 a) | .a | refl = refl\n-- a+\u2193\u2261a (a \u2190\u2228) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (a \u2190\u2228) | .a | refl = refl\n-- a+\u2193\u2261a (\u2228\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (\u2228\u2192 a) | .a | refl = refl\n-- a+\u2193\u2261a (a \u2190\u2202) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (a \u2190\u2202) | .a | refl = refl\n-- a+\u2193\u2261a (\u2202\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (\u2202\u2192 a) | .a | refl = refl\n\n\n-- [a+b]-a=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n--           \u2192 (b : IndexLL rll fll)\n--           \u2192 ((a +\u1d62 b) -\u1d62 a) (+\u1d62\u21d2l\u2264\u1d62+\u1d62 a b) \u2261 b\n-- [a+b]-a=b \u2193 b = refl\n-- [a+b]-a=b (a \u2190\u2227) b = [a+b]-a=b a b\n-- [a+b]-a=b (\u2227\u2192 a) b = [a+b]-a=b a b\n-- [a+b]-a=b (a \u2190\u2228) b = [a+b]-a=b a b\n-- [a+b]-a=b (\u2228\u2192 a) b = [a+b]-a=b a b\n-- [a+b]-a=b (a \u2190\u2202) b = [a+b]-a=b a b\n-- [a+b]-a=b (\u2202\u2192 a) b = [a+b]-a=b a b\n\n-- a+[b-a]=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n--             \u2192 (b : IndexLL rll ll)\n--             \u2192 (lt : a \u2264\u1d62 b)\n--             \u2192 (a +\u1d62 (b -\u1d62 a) lt) \u2261 b\n-- a+[b-a]=b \u2193 b \u2264\u1d62\u2193 = refl\n-- a+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) | .b | refl = refl\n-- a+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) | .b | refl = refl\n-- a+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) | .b | refl = refl\n-- a+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt)with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt) | .b | refl = refl\n-- a+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) | .b | refl = refl\n-- a+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) | .b | refl = refl\n\n\n\n-- -- A predicate that is true if pll is not transformed by ltr.\n\n-- data UpTran {i u} : \u2200 {ll pll rll} \u2192 IndexLL pll ll \u2192 LLTr {i} {u} rll ll \u2192 Set (lsuc u) where\n--   indI : \u2200{pll ll} \u2192 {ind : IndexLL pll ll} \u2192 UpTran ind I\n--   \u2190\u2202\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (\u2202\u2192 ind) ltr\n--          \u2192 UpTran (ind \u2190\u2202) (\u2202c ltr)\n--   \u2202\u2192\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (ind \u2190\u2202) ltr\n--          \u2192 UpTran (\u2202\u2192 ind) (\u2202c ltr)\n--   \u2190\u2228\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (\u2228\u2192 ind) ltr\n--          \u2192 UpTran (ind \u2190\u2228) (\u2228c ltr)\n--   \u2228\u2192\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (ind \u2190\u2228) ltr\n--          \u2192 UpTran (\u2228\u2192 ind) (\u2228c ltr)\n--   \u2190\u2227\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (\u2227\u2192 ind) ltr\n--          \u2192 UpTran (ind \u2190\u2227) (\u2227c ltr)\n--   \u2227\u2192\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (ind \u2190\u2227) ltr\n--          \u2192 UpTran (\u2227\u2192 ind) (\u2227c ltr)\n--   \u2190\u2227]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n--             \u2192 UpTran {rll = rll} (ind \u2190\u2227) ltr \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)\n--   \u2227\u2192]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n--             \u2192 UpTran {rll = rll} (\u2227\u2192 (ind \u2190\u2227)) ltr\n--             \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr)\n--   \u2227\u2192\u2227\u2227d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n--             \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 ind)) ltr\n--             \u2192 UpTran {ll = ((lli \u2227 lri) \u2227 ri)} (\u2227\u2192 ind) (\u2227\u2227d ltr)\n--   \u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n--             \u2192 UpTran {rll = rll} ((ind \u2190\u2227) \u2190\u2227) ltr\n--             \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr)\n--   \u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n--             \u2192 UpTran {rll = rll} ((\u2227\u2192 ind) \u2190\u2227) ltr\n--             \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr)\n--   \u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n--             \u2192 UpTran {rll = rll} (\u2227\u2192 ind) ltr\n--             \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr)\n--   \u2190\u2228]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n--             \u2192 UpTran {rll = rll} (ind \u2190\u2228) ltr \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)\n--   \u2228\u2192]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n--             \u2192 UpTran {rll = rll} (\u2228\u2192 (ind \u2190\u2228)) ltr\n--             \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr)\n--   \u2228\u2192\u2228\u2228d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n--             \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 ind)) ltr\n--             \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} (\u2228\u2192 ind) (\u2228\u2228d ltr)\n--   \u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n--             \u2192 UpTran {rll = rll} ((ind \u2190\u2228) \u2190\u2228) ltr\n--             \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr)\n--   \u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n--             \u2192 UpTran {rll = rll} ((\u2228\u2192 ind) \u2190\u2228) ltr\n--             \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr)\n--   \u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n--             \u2192 UpTran {rll = rll} (\u2228\u2192 ind) ltr\n--             \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr)\n--   \u2190\u2202]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n--             \u2192 UpTran {rll = rll} (ind \u2190\u2202) ltr \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)\n--   \u2202\u2192]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n--             \u2192 UpTran {rll = rll} (\u2202\u2192 (ind \u2190\u2202)) ltr\n--             \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr)\n--   \u2202\u2192\u2202\u2202d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n--             \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 ind)) ltr\n--             \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} (\u2202\u2192 ind) (\u2202\u2202d ltr)\n--   \u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n--             \u2192 UpTran {rll = rll} ((ind \u2190\u2202) \u2190\u2202) ltr\n--             \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr)\n--   \u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n--             \u2192 UpTran {rll = rll} ((\u2202\u2192 ind) \u2190\u2202) ltr\n--             \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr)\n--   \u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n--             \u2192 UpTran {rll = rll} (\u2202\u2192 ind) ltr\n--             \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr)\n\n-- isUpTran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 Dec (UpTran ind ltr)\n-- isUpTran ind I = yes indI\n-- isUpTran \u2193 (\u2202c ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2202) (\u2202c ltr) with (isUpTran (\u2202\u2192 ind) ltr)\n-- isUpTran (ind \u2190\u2202) (\u2202c ltr) | yes ut = yes (\u2190\u2202\u2202c ut)\n-- isUpTran (ind \u2190\u2202) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2190\u2202\u2202c ut) \u2192 \u00acut ut})\n-- isUpTran (\u2202\u2192 ind) (\u2202c ltr)  with (isUpTran (ind \u2190\u2202) ltr)\n-- isUpTran (\u2202\u2192 ind) (\u2202c ltr) | yes ut = yes (\u2202\u2192\u2202c ut)\n-- isUpTran (\u2202\u2192 ind) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2202\u2192\u2202c ut) \u2192 \u00acut ut})\n-- isUpTran \u2193 (\u2228c ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2228) (\u2228c ltr) with (isUpTran (\u2228\u2192 ind) ltr)\n-- isUpTran (ind \u2190\u2228) (\u2228c ltr) | yes p = yes (\u2190\u2228\u2228c p)\n-- isUpTran (ind \u2190\u2228) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u2228c p) \u2192 \u00acp p}) \n-- isUpTran (\u2228\u2192 ind) (\u2228c ltr) with (isUpTran (ind \u2190\u2228) ltr)\n-- isUpTran (\u2228\u2192 ind) (\u2228c ltr) | yes p = yes (\u2228\u2192\u2228c p)\n-- isUpTran (\u2228\u2192 ind) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228c ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2227c ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2227) (\u2227c ltr) with (isUpTran (\u2227\u2192 ind) ltr)\n-- isUpTran (ind \u2190\u2227) (\u2227c ltr) | yes p = yes (\u2190\u2227\u2227c p)\n-- isUpTran (ind \u2190\u2227) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u2227c ut) \u2192 \u00acp ut}) \n-- isUpTran (\u2227\u2192 ind) (\u2227c ltr) with (isUpTran (ind \u2190\u2227) ltr)\n-- isUpTran (\u2227\u2192 ind) (\u2227c ltr) | yes p = yes (\u2227\u2192\u2227c p)\n-- isUpTran (\u2227\u2192 ind) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227c ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (ind \u2190\u2227) ltr)\n-- isUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2190\u2227]\u2190\u2227\u2227\u2227d p)\n-- isUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \n-- isUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (ind \u2190\u2227)) ltr)\n-- isUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192]\u2190\u2227\u2227\u2227d p)\n-- isUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \n-- isUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 ind)) ltr)\n-- isUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192\u2227\u2227d p)\n-- isUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) with (isUpTran ((ind \u2190\u2227) \u2190\u2227) ltr)\n-- isUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2190\u2227\u00ac\u2227\u2227d p)\n-- isUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) with (isUpTran ((\u2227\u2192 ind) \u2190\u2227) ltr)\n-- isUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d p)\n-- isUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 ind) ltr)\n-- isUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d p)\n-- isUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (ind \u2190\u2228) ltr)\n-- isUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2190\u2228]\u2190\u2228\u2228\u2228d p)\n-- isUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \n-- isUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (ind \u2190\u2228)) ltr)\n-- isUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192]\u2190\u2228\u2228\u2228d p)\n-- isUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \n-- isUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 ind)) ltr)\n-- isUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192\u2228\u2228d p)\n-- isUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) with (isUpTran ((ind \u2190\u2228) \u2190\u2228) ltr)\n-- isUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2190\u2228\u00ac\u2228\u2228d p)\n-- isUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) with (isUpTran ((\u2228\u2192 ind) \u2190\u2228) ltr)\n-- isUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d p)\n-- isUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 ind) ltr)\n-- isUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d p)\n-- isUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (ind \u2190\u2202) ltr)\n-- isUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2190\u2202]\u2190\u2202\u2202\u2202d p)\n-- isUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (ind \u2190\u2202)) ltr)\n-- isUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192]\u2190\u2202\u2202\u2202d p)\n-- isUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 ind)) ltr)\n-- isUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192\u2202\u2202d p)\n-- isUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) with (isUpTran ((ind \u2190\u2202) \u2190\u2202) ltr)\n-- isUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2190\u2202\u00ac\u2202\u2202d p)\n-- isUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) with (isUpTran ((\u2202\u2192 ind) \u2190\u2202) ltr)\n-- isUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d p)\n-- isUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 ind) ltr)\n-- isUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d p)\n-- isUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n\n\n-- indLow\u21d2UpTran : \u2200 {i u rll ll n dt df} \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n--                 \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 UpTran ind ltr\n-- indLow\u21d2UpTran \u2193 I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2227) I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2227) (\u2227c ltr) = \u2190\u2227\u2227c r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 ind) ltr\n-- indLow\u21d2UpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) = \u2190\u2227]\u2190\u2227\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2227) ltr\n-- indLow\u21d2UpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) = \u2227\u2192]\u2190\u2227\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 (ind \u2190\u2227)) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) = \u2190\u2227\u00ac\u2227\u2227d r where\n--   r = indLow\u21d2UpTran ((ind \u2190\u2227) \u2190\u2227) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 ind) I = indI\n-- indLow\u21d2UpTran (\u2227\u2192 ind) (\u2227c ltr) = \u2227\u2192\u2227c r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2227) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) = \u2227\u2192\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 (\u2227\u2192 ind)) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) = \u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d r where\n--   r = indLow\u21d2UpTran ((\u2227\u2192 ind) \u2190\u2227) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) = \u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 ind) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2228) I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2228) (\u2228c ltr) = \u2190\u2228\u2228c r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 ind) ltr\n-- indLow\u21d2UpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) = \u2190\u2228]\u2190\u2228\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2228) ltr\n-- indLow\u21d2UpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) = \u2228\u2192]\u2190\u2228\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 (ind \u2190\u2228)) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) = \u2190\u2228\u00ac\u2228\u2228d r where\n--   r = indLow\u21d2UpTran ((ind \u2190\u2228) \u2190\u2228) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 ind) I = indI\n-- indLow\u21d2UpTran (\u2228\u2192 ind) (\u2228c ltr) = \u2228\u2192\u2228c r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2228) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) = \u2228\u2192\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 (\u2228\u2192 ind)) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) = \u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d r where\n--   r = indLow\u21d2UpTran ((\u2228\u2192 ind) \u2190\u2228) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) = \u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 ind) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2202) I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2202) (\u2202c ltr) = \u2190\u2202\u2202c r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 ind) ltr\n-- indLow\u21d2UpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) = \u2190\u2202]\u2190\u2202\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2202) ltr\n-- indLow\u21d2UpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) = \u2202\u2192]\u2190\u2202\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 (ind \u2190\u2202)) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) = \u2190\u2202\u00ac\u2202\u2202d r where\n--   r = indLow\u21d2UpTran ((ind \u2190\u2202) \u2190\u2202) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 ind) I = indI\n-- indLow\u21d2UpTran (\u2202\u2192 ind) (\u2202c ltr) = \u2202\u2192\u2202c r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2202) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) = \u2202\u2192\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 (\u2202\u2192 ind)) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) = \u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d r where\n--   r = indLow\u21d2UpTran ((\u2202\u2192 ind) \u2190\u2202) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) = \u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 ind) ltr\n\n\n\n-- tran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 UpTran ind ltr\n--        \u2192 IndexLL pll rll\n-- tran ind I indI = ind \n-- tran \u2193 (\u2202c ltr) () \n-- tran (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c ut) = tran (\u2202\u2192 ind) ltr ut\n-- tran (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c ut) =  tran (ind \u2190\u2202) ltr ut\n-- tran \u2193 (\u2228c ltr) () \n-- tran (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c ut) = tran (\u2228\u2192 ind) ltr ut\n-- tran (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c ut) = tran (ind \u2190\u2228) ltr ut\n-- tran \u2193 (\u2227c ltr) () \n-- tran (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c ut) = tran (\u2227\u2192 ind) ltr ut\n-- tran (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c ut) = tran (ind \u2190\u2227) ltr ut\n-- tran \u2193 (\u2227\u2227d ltr) () \n-- tran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) () \n-- tran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d ut) = tran (ind \u2190\u2227) ltr ut\n-- tran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d ut) = tran (\u2227\u2192 (ind \u2190\u2227)) ltr ut\n-- tran (\u2227\u2192 ind) (\u2227\u2227d ltr) (\u2227\u2192\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 ind)) ltr ut\n-- tran \u2193 (\u00ac\u2227\u2227d ltr) () \n-- tran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((ind \u2190\u2227) \u2190\u2227) ltr ut\n-- tran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) () \n-- tran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((\u2227\u2192 ind) \u2190\u2227) ltr ut\n-- tran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) = tran (\u2227\u2192 ind) ltr ut\n-- tran \u2193 (\u2228\u2228d ltr) () \n-- tran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) () \n-- tran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d ut) = tran (ind \u2190\u2228) ltr ut\n-- tran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d ut) = tran (\u2228\u2192 (ind \u2190\u2228)) ltr ut\n-- tran (\u2228\u2192 ind) (\u2228\u2228d ltr) (\u2228\u2192\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 ind)) ltr ut\n-- tran \u2193 (\u00ac\u2228\u2228d ltr) () \n-- tran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((ind \u2190\u2228) \u2190\u2228) ltr ut\n-- tran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) () \n-- tran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((\u2228\u2192 ind) \u2190\u2228) ltr ut\n-- tran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) = tran (\u2228\u2192 ind) ltr ut\n-- tran \u2193 (\u2202\u2202d ltr) () \n-- tran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) () \n-- tran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d ut) = tran (ind \u2190\u2202) ltr ut\n-- tran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d ut) = tran (\u2202\u2192 (ind \u2190\u2202)) ltr ut\n-- tran (\u2202\u2192 ind) (\u2202\u2202d ltr) (\u2202\u2192\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 ind)) ltr ut\n-- tran \u2193 (\u00ac\u2202\u2202d ltr) () \n-- tran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((ind \u2190\u2202) \u2190\u2202) ltr ut\n-- tran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) () \n-- tran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((\u2202\u2192 ind) \u2190\u2202) ltr ut\n-- tran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) = tran (\u2202\u2192 ind) ltr ut\n\n\n-- tr-ltr-morph : \u2200 {i u ll pll orll} \u2192 \u2200 frll \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} orll ll) \u2192 (rvT : UpTran ind ltr) \u2192 LLTr (replLL orll (tran ind ltr rvT) frll) (replLL ll ind frll)\n-- tr-ltr-morph frll \u2193 .I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2227) I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c rvT) with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n-- ... | r = \u2227c r\n-- tr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n-- ... | r = \u2227\u2227d r\n-- tr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) ltr rvT\n-- ... | r = \u2227\u2227d r\n-- tr-ltr-morph frll (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) ltr rvT\n-- ... | r = \u00ac\u2227\u2227d r\n-- tr-ltr-morph frll (\u2227\u2192 ind) I indI = I\n-- tr-ltr-morph frll (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n-- ... | r = \u2227c r\n-- tr-ltr-morph frll (\u2227\u2192 ind) (\u2227\u2227d ltr) (\u2227\u2192\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) ltr rvT\n-- ... | r = \u2227\u2227d r\n-- tr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) ltr rvT\n-- ... | r = \u00ac\u2227\u2227d r\n-- tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d rvT)  with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n-- ... | r = \u00ac\u2227\u2227d r\n-- tr-ltr-morph frll (ind \u2190\u2228) I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c rvT) with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n-- ... | r = \u2228c r\n-- tr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n-- ... | r = \u2228\u2228d r\n-- tr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) ltr rvT\n-- ... | r = \u2228\u2228d r\n-- tr-ltr-morph frll (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) ltr rvT\n-- ... | r = \u00ac\u2228\u2228d r\n-- tr-ltr-morph frll (\u2228\u2192 ind) I indI = I\n-- tr-ltr-morph frll (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n-- ... | r = \u2228c r\n-- tr-ltr-morph frll (\u2228\u2192 ind) (\u2228\u2228d ltr) (\u2228\u2192\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) ltr rvT\n-- ... | r = \u2228\u2228d r\n-- tr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) ltr rvT\n-- ... | r = \u00ac\u2228\u2228d r\n-- tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d rvT)  with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n-- ... | r = \u00ac\u2228\u2228d r\n-- tr-ltr-morph frll (ind \u2190\u2202) I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c rvT) with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n-- ... | r = \u2202c r\n-- tr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n-- ... | r = \u2202\u2202d r\n-- tr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) ltr rvT\n-- ... | r = \u2202\u2202d r\n-- tr-ltr-morph frll (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) ltr rvT\n-- ... | r = \u00ac\u2202\u2202d r\n-- tr-ltr-morph frll (\u2202\u2192 ind) I indI = I\n-- tr-ltr-morph frll (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n-- ... | r = \u2202c r\n-- tr-ltr-morph frll (\u2202\u2192 ind) (\u2202\u2202d ltr) (\u2202\u2192\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) ltr rvT\n-- ... | r = \u2202\u2202d r\n-- tr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) ltr rvT\n-- ... | r = \u00ac\u2202\u2202d r\n-- tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d rvT)  with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n-- ... | r = \u00ac\u2202\u2202d r\n\n\n\n-- drepl=>repl+ : \u2200{i u pll ll cll frll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--                \u2192 (replLL ll ind (replLL pll lind frll)) \u2261 replLL ll (ind +\u1d62 lind) frll\n-- drepl=>repl+ \u2193 lind = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (ind \u2190\u2227) lind\n--              with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (ind \u2190\u2227) lind\n--              | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (\u2227\u2192 ind) lind\n--              with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (\u2227\u2192 ind) lind\n--              | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (ind \u2190\u2228) lind\n--              with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (ind \u2190\u2228) lind\n--              | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (\u2228\u2192 ind) lind\n--              with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (\u2228\u2192 ind) lind\n--              | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (ind \u2190\u2202) lind\n--              with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (ind \u2190\u2202) lind\n--              | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (\u2202\u2192 ind) lind\n--              with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (\u2202\u2192 ind) lind\n--              | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n\n\n\n-- repl-r=>l : \u2200{i u pll ll cll frll} \u2192 \u2200 ell \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (x : IndexLL cll (replLL ll ind ell)) \u2192 let uind = a\u2264\u1d62b-morph ind ind ell (\u2264\u1d62-reflexive ind)\n--        in (ltuindx : uind \u2264\u1d62 x)\n--        \u2192 (replLL ll ind (replLL ell (ind-rpl\u2193 ind ((x -\u1d62 uind) ltuindx)) frll) \u2261 replLL (replLL ll ind ell) x frll)\n-- repl-r=>l ell \u2193 x \u2264\u1d62\u2193 = refl\n-- repl-r=>l {ll = li \u2227 ri} ell (ind \u2190\u2227) (x \u2190\u2227) (\u2264\u1d62\u2190\u2227 ltuindx) = cong (\u03bb x \u2192 x \u2227 ri) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2227 ri} ell (\u2227\u2192 ind) (\u2227\u2192 x) (\u2264\u1d62\u2227\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2227 x) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2228 ri} ell (ind \u2190\u2228) (x \u2190\u2228) (\u2264\u1d62\u2190\u2228 ltuindx) = cong (\u03bb x \u2192 x \u2228 ri) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2228 ri} ell (\u2228\u2192 ind) (\u2228\u2192 x) (\u2264\u1d62\u2228\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2228 x) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2202 ri} ell (ind \u2190\u2202) (x \u2190\u2202) (\u2264\u1d62\u2190\u2202 ltuindx) = cong (\u03bb x \u2192 x \u2202 ri) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2202 ri} ell (\u2202\u2192 ind) (\u2202\u2192 x) (\u2264\u1d62\u2202\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2202 x) (repl-r=>l ell ind x ltuindx)\n\n\n\n-- -- Deprecated\n-- updInd : \u2200{i u rll ll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} rll ll)\n--          \u2192 IndexLL {i} {u} nrll (replLL ll ind nrll)\n-- updInd nrll \u2193 = \u2193\n-- updInd nrll (ind \u2190\u2227) = (updInd nrll ind) \u2190\u2227\n-- updInd nrll (\u2227\u2192 ind) = \u2227\u2192 (updInd nrll ind)\n-- updInd nrll (ind \u2190\u2228) = (updInd nrll ind) \u2190\u2228\n-- updInd nrll (\u2228\u2192 ind) = \u2228\u2192 (updInd nrll ind)\n-- updInd nrll (ind \u2190\u2202) = (updInd nrll ind) \u2190\u2202\n-- updInd nrll (\u2202\u2192 ind) = \u2202\u2192 (updInd nrll ind)\n\n-- -- Deprecated\n-- -- Maybe instead of this function use a\u2264\u1d62b-morph\n-- updIndGen : \u2200{i u pll ll cll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--             \u2192 IndexLL {i} {u} (replLL pll lind nrll) (replLL ll (ind +\u1d62 lind) nrll)\n-- updIndGen nrll \u2193 lind = \u2193\n-- updIndGen nrll (ind \u2190\u2227) lind = (updIndGen nrll ind lind) \u2190\u2227\n-- updIndGen nrll (\u2227\u2192 ind) lind = \u2227\u2192 (updIndGen nrll ind lind)\n-- updIndGen nrll (ind \u2190\u2228) lind = (updIndGen nrll ind lind) \u2190\u2228\n-- updIndGen nrll (\u2228\u2192 ind) lind = \u2228\u2192 (updIndGen nrll ind lind)\n-- updIndGen nrll (ind \u2190\u2202) lind = (updIndGen nrll ind lind) \u2190\u2202\n-- updIndGen nrll (\u2202\u2192 ind) lind = \u2202\u2192 (updIndGen nrll ind lind)\n\n\n\n-- -- TODO This negation was writen so as to return nothing if \u00ac (b \u2264\u1d62 a)\n-- _-\u2098\u1d62_ : \u2200 {i u pll cll ll} \u2192 IndexLL {i} {u} cll ll \u2192 IndexLL pll ll \u2192 Maybe $ IndexLL cll pll\n-- a -\u2098\u1d62 \u2193 = just a\n-- \u2193 -\u2098\u1d62 (b \u2190\u2227) = nothing\n-- (a \u2190\u2227) -\u2098\u1d62 (b \u2190\u2227) = a -\u2098\u1d62 b\n-- (\u2227\u2192 a) -\u2098\u1d62 (b \u2190\u2227) = nothing\n-- \u2193 -\u2098\u1d62 (\u2227\u2192 b) = nothing\n-- (a \u2190\u2227) -\u2098\u1d62 (\u2227\u2192 b) = nothing\n-- (\u2227\u2192 a) -\u2098\u1d62 (\u2227\u2192 b) = a -\u2098\u1d62 b\n-- \u2193 -\u2098\u1d62 (b \u2190\u2228) = nothing\n-- (a \u2190\u2228) -\u2098\u1d62 (b \u2190\u2228) = a -\u2098\u1d62 b\n-- (\u2228\u2192 a) -\u2098\u1d62 (b \u2190\u2228) = nothing\n-- \u2193 -\u2098\u1d62 (\u2228\u2192 b) = nothing\n-- (a \u2190\u2228) -\u2098\u1d62 (\u2228\u2192 b) = nothing\n-- (\u2228\u2192 a) -\u2098\u1d62 (\u2228\u2192 b) = a -\u2098\u1d62 b\n-- \u2193 -\u2098\u1d62 (b \u2190\u2202) = nothing\n-- (a \u2190\u2202) -\u2098\u1d62 (b \u2190\u2202) = a -\u2098\u1d62 b\n-- (\u2202\u2192 a) -\u2098\u1d62 (b \u2190\u2202) = nothing\n-- \u2193 -\u2098\u1d62 (\u2202\u2192 b) = nothing\n-- (a \u2190\u2202) -\u2098\u1d62 (\u2202\u2192 b) = nothing\n-- (\u2202\u2192 a) -\u2098\u1d62 (\u2202\u2192 b) = a -\u2098\u1d62 b\n\n-- -- Deprecated\n-- b-s\u2261j\u21d2s\u2264\u1d62b : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll)\n--              \u2192  \u03a3 (IndexLL {i} {u} cll pll) (\u03bb x \u2192 (bind -\u2098\u1d62 sind) \u2261 just x) \u2192 sind \u2264\u1d62 bind\n-- b-s\u2261j\u21d2s\u2264\u1d62b bind \u2193 (rs , x) = \u2264\u1d62\u2193\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2227) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (sind \u2190\u2227) (rs , x) = \u2264\u1d62\u2190\u2227 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (sind \u2190\u2227) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2227\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (\u2227\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (\u2227\u2192 sind) (rs , x) = \u2264\u1d62\u2227\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2228) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (sind \u2190\u2228) (rs , x) = \u2264\u1d62\u2190\u2228 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (sind \u2190\u2228) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2228\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (\u2228\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (\u2228\u2192 sind) (rs , x) =  \u2264\u1d62\u2228\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2202) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (sind \u2190\u2202) (rs , x) = \u2264\u1d62\u2190\u2202 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (sind \u2190\u2202) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2202\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (\u2202\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (\u2202\u2192 sind) (rs , x) = \u2264\u1d62\u2202\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n\n-- -- Deprecated\n-- revUpdInd : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll (replLL ll b ell))\n--             \u2192 a -\u2098\u1d62 (updInd ell b) \u2261 nothing \u2192 (updInd ell b) -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll ll\n-- revUpdInd \u2193 a () b-a\n-- revUpdInd (b \u2190\u2227) \u2193 refl ()\n-- revUpdInd (b \u2190\u2227) (a \u2190\u2227) a-b b-a = revUpdInd b a a-b b-a \u2190\u2227\n-- revUpdInd (b \u2190\u2227) (\u2227\u2192 a) a-b b-a = \u2227\u2192 a\n-- revUpdInd (\u2227\u2192 b) \u2193 refl ()\n-- revUpdInd (\u2227\u2192 b) (a \u2190\u2227) a-b b-a = a \u2190\u2227\n-- revUpdInd (\u2227\u2192 b) (\u2227\u2192 a) a-b b-a = \u2227\u2192 revUpdInd b a a-b b-a\n-- revUpdInd (b \u2190\u2228) \u2193 refl ()\n-- revUpdInd (b \u2190\u2228) (a \u2190\u2228) a-b b-a = revUpdInd b a a-b b-a \u2190\u2228\n-- revUpdInd (b \u2190\u2228) (\u2228\u2192 a) a-b b-a = \u2228\u2192 a\n-- revUpdInd (\u2228\u2192 b) \u2193 refl ()\n-- revUpdInd (\u2228\u2192 b) (a \u2190\u2228) a-b b-a = a \u2190\u2228\n-- revUpdInd (\u2228\u2192 b) (\u2228\u2192 a) a-b b-a = \u2228\u2192 revUpdInd b a a-b b-a\n-- revUpdInd (b \u2190\u2202) \u2193 refl ()\n-- revUpdInd (b \u2190\u2202) (a \u2190\u2202) a-b b-a = revUpdInd b a a-b b-a \u2190\u2202\n-- revUpdInd (b \u2190\u2202) (\u2202\u2192 a) a-b b-a = \u2202\u2192 a\n-- revUpdInd (\u2202\u2192 b) \u2193 refl ()\n-- revUpdInd (\u2202\u2192 b) (a \u2190\u2202) a-b b-a = a \u2190\u2202\n-- revUpdInd (\u2202\u2192 b) (\u2202\u2192 a) a-b b-a = \u2202\u2192 revUpdInd b a a-b b-a\n\n\n-- -- Deprecated\n-- updIndPart : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll ll)\n--              \u2192 a -\u2098\u1d62 b \u2261 nothing \u2192 b -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll (replLL ll b ell)\n-- updIndPart \u2193 a () eq2\n-- updIndPart (b \u2190\u2227) \u2193 refl ()\n-- updIndPart (b \u2190\u2227) (a \u2190\u2227) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2227\n-- updIndPart (b \u2190\u2227) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 a\n-- updIndPart (\u2227\u2192 b) \u2193 refl ()\n-- updIndPart (\u2227\u2192 b) (a \u2190\u2227) eq1 eq2 = a \u2190\u2227\n-- updIndPart (\u2227\u2192 b) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 updIndPart b a eq1 eq2\n-- updIndPart (b \u2190\u2228) \u2193 refl ()\n-- updIndPart (b \u2190\u2228) (a \u2190\u2228) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2228\n-- updIndPart (b \u2190\u2228) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 a\n-- updIndPart (\u2228\u2192 b) \u2193 refl ()\n-- updIndPart (\u2228\u2192 b) (a \u2190\u2228) eq1 eq2 = a \u2190\u2228\n-- updIndPart (\u2228\u2192 b) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 updIndPart b a eq1 eq2\n-- updIndPart (b \u2190\u2202) \u2193 refl ()\n-- updIndPart (b \u2190\u2202) (a \u2190\u2202) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2202\n-- updIndPart (b \u2190\u2202) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 a\n-- updIndPart (\u2202\u2192 b) \u2193 refl ()\n-- updIndPart (\u2202\u2192 b) (a \u2190\u2202) eq1 eq2 = a \u2190\u2202\n-- updIndPart (\u2202\u2192 b) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 updIndPart b a eq1 eq2\n\n-- -- Deprecated\n-- rev\u21d2rs-i\u2261n : \u2200{i u ll cll ell pll} \u2192 (ind : IndexLL pll ll)\n--              \u2192 (lind : IndexLL {i} {u} cll (replLL ll ind ell))\n--              \u2192 (eq\u2081 : (lind -\u2098\u1d62 (updInd ell ind) \u2261 nothing))\n--              \u2192 (eq\u2082 : ((updInd ell ind) -\u2098\u1d62 lind \u2261 nothing))\n--              \u2192 let rs = revUpdInd ind lind eq\u2081 eq\u2082 in\n--                  ((rs -\u2098\u1d62 ind) \u2261 nothing) \u00d7 ((ind -\u2098\u1d62 rs) \u2261 nothing)\n-- rev\u21d2rs-i\u2261n \u2193 lind () eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2227) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2227) (lind \u2190\u2227) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2227) (\u2227\u2192 lind) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2227\u2192 ind) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (\u2227\u2192 ind) (lind \u2190\u2227) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2227\u2192 ind) (\u2227\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2228) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2228) (lind \u2190\u2228) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2228) (\u2228\u2192 lind) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2228\u2192 ind) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (\u2228\u2192 ind) (lind \u2190\u2228) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2228\u2192 ind) (\u2228\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2202) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2202) (lind \u2190\u2202) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2202) (\u2202\u2192 lind) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2202\u2192 ind) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (\u2202\u2192 ind) (lind \u2190\u2202) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2202\u2192 ind) (\u2202\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n\n\n\n\n-- -- Deprecated\n-- remRepl : \u2200{i u ll frll ell pll cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--           \u2192 replLL (replLL ll (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell \u2261 replLL ll ind ell\n-- remRepl \u2193 lind = refl\n-- remRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (ind \u2190\u2227) lind\n--         with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2227 ri} {frll} {ell} (ind \u2190\u2227) lind | .(replLL li ind ell) | refl = refl\n-- remRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (\u2227\u2192 ind) lind\n--         with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2227 ri} {frll} {ell} (\u2227\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n-- remRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (ind \u2190\u2228) lind\n--         with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2228 ri} {frll} {ell} (ind \u2190\u2228) lind | .(replLL li ind ell) | refl = refl\n-- remRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (\u2228\u2192 ind) lind\n--         with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2228 ri} {frll} {ell} (\u2228\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n-- remRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (ind \u2190\u2202) lind\n--         with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2202 ri} {frll} {ell} (ind \u2190\u2202) lind | .(replLL li ind ell) | refl = refl\n-- remRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (\u2202\u2192 ind) lind\n--         with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2202 ri} {frll} {ell} (\u2202\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n\n-- -- Deprecated\n-- replLL-inv : \u2200{i u ll ell pll} \u2192 (ind : IndexLL {i} {u} pll ll)\n--              \u2192 replLL (replLL ll ind ell) (updInd ell ind) pll \u2261 ll\n-- replLL-inv \u2193 = refl\n-- replLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (ind \u2190\u2227)\n--            with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (ind \u2190\u2227) | .li | refl = refl\n-- replLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (\u2227\u2192 ind)\n--            with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (\u2227\u2192 ind) | .ri | refl = refl\n-- replLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (ind \u2190\u2228)\n--            with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (ind \u2190\u2228) | .li | refl = refl\n-- replLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (\u2228\u2192 ind)\n--            with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (\u2228\u2192 ind) | .ri | refl = refl\n-- replLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (ind \u2190\u2202)\n--            with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (ind \u2190\u2202) | .li | refl = refl\n-- replLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (\u2202\u2192 ind)\n--            with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (\u2202\u2192 ind) | .ri | refl = refl\n\n\n","old_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule IndexLLProp where\n\nopen import Common\nopen import LinLogic\nopen import Data.Maybe\nimport Relation.Binary.HeterogeneousEquality\nopen import Relation.Binary.PropositionalEquality using (trans ; sym ; subst ; cong ; subst\u2082)\n\ndata _\u2245\u1d62_ {i u gll} : \u2200{fll ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set (lsuc u) where\n  \u2245\u1d62\u2193 :  \u2193 \u2245\u1d62 \u2193\n  \u2245\u1d62ic : \u2200{fll ict ll tll il eq} \u2192 {sind : IndexLL gll ll} \u2192 {bind : IndexLL fll ll} \u2192 (sind \u2245\u1d62 bind)\n         \u2192 _\u2245\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n\n\n-- TODO Is this needed ?\n\u2245\u1d62-spec : \u2200{i u ll ict tll il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll}\n          \u2192 {bind : IndexLL fll ll} \u2192 _\u2245\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n          \u2192 (sind \u2245\u1d62 bind)\n\u2245\u1d62-spec (\u2245\u1d62ic a) = a\n\n\u00ac\u2245\u1d62-spec : \u2200{i u ll ict tll il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll}\n          \u2192 {bind : IndexLL fll ll}\n          \u2192 \u00ac (_\u2245\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}}))\n           \u2192 \u00ac (sind \u2245\u1d62 bind)\n\u00ac\u2245\u1d62-spec rl = \u03bb z \u2192 rl (\u2245\u1d62ic z)\n\n\u2245\u1d62-reflexive : \u2200{i u rll ll} \u2192 (a : IndexLL {i} {u} rll ll) \u2192 a \u2245\u1d62 a\n\u2245\u1d62-reflexive \u2193 = \u2245\u1d62\u2193\n\u2245\u1d62-reflexive (ic x) = \u2245\u1d62ic (\u2245\u1d62-reflexive x)\n\n\n\n\u2261-to-\u2245\u1d62 : \u2200{i u rll ll} \u2192 (a b : IndexLL {i} {u} rll ll) \u2192 a \u2261 b \u2192 a \u2245\u1d62 b\n\u2261-to-\u2245\u1d62 a .a refl = \u2245\u1d62-reflexive a\n\n\n\ndata _\u2264\u1d62_ {i u gll fll} : \u2200{ll} \u2192 IndexLL {i} {u} gll ll \u2192 IndexLL {i} {u} fll ll \u2192 Set (lsuc u) where\n  \u2264\u1d62\u2193 : {ind : IndexLL fll gll} \u2192 \u2193 \u2264\u1d62 ind\n  \u2264\u1d62ic : \u2200{ll ict tll il eq} \u2192 {sind : IndexLL gll ll} \u2192 {bind : IndexLL fll ll} \u2192 (sind \u2264\u1d62 bind)\n         \u2192 _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n\nmodule _ where\n\n  open import Relation.Binary.PropositionalEquality\n\n  \u2264\u1d62-spec : \u2200{i u ll ict tll il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll}\n            \u2192 {bind : IndexLL fll ll} \u2192 _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n            \u2192 (sind \u2264\u1d62 bind)\n  \u2264\u1d62-spec (\u2264\u1d62ic rl) = rl\n\n\n\u00ac\u2264\u1d62-ext : \u2200{i u ll tll ict il eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll} \u2192 {bind : IndexLL fll ll} \u2192 \u00ac (sind \u2264\u1d62 bind) \u2192 \u00ac ( _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}}))\n\u00ac\u2264\u1d62-ext rl = \u03bb z \u2192 rl (\u2264\u1d62-spec z)\n\n\u2264\u1d62-extT : \u2200{i u ll tll ict il eq gll fll} \u2192 (sind : IndexLL {i} {u} gll ll) \u2192 (bind : IndexLL fll ll) \u2192 Set (lsuc u)\n\u2264\u1d62-extT {i} {u} {ll} {tll} {ict} {il} {eq} {gll} {fll} sind bind = _\u2264\u1d62_ {ll = il[ il ]} (ic {ll = ll} {tll} {ict} sind {{eq}}) (ic {ll = ll} {tll} {ict} bind {{eq}})\n\n\n\u2264\u1d62-ext : \u2200{i u ll tll ict all eq gll fll} \u2192 {sind : IndexLL {i} {u} gll ll} \u2192 {bind : IndexLL fll ll} \u2192 (sind \u2264\u1d62 bind) \u2192 \u2264\u1d62-extT {tll = tll} {ict} {all} {eq} sind bind\n\u2264\u1d62-ext {i} {u} {ll} {tll} {ict} {all} {eq} {gll} {fll} {sind} {bind} rl = \u2264\u1d62ic {ll = ll} {ict} {tll} {all} {eq} {sind = sind} {bind} rl\n\n\n\u2264\u1d62-reflexive : \u2200{i u gll ll} \u2192 (ind : IndexLL {i} {u} gll ll) \u2192 ind \u2264\u1d62 ind\n\u2264\u1d62-reflexive \u2193 = \u2264\u1d62\u2193\n\u2264\u1d62-reflexive (ic {ll = ll} {tll} {ict} x {{eq}}) = \u2264\u1d62ic {ll = ll} {ict} {tll} {_} {eq} (\u2264\u1d62-reflexive x)\n\n\n\u2264\u1d62-transitive : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (b \u2264\u1d62 c) \u2192 (a \u2264\u1d62 c)\n\u2264\u1d62-transitive \u2264\u1d62\u2193 b = \u2264\u1d62\u2193\n\u2264\u1d62-transitive (\u2264\u1d62ic {ict = ict} {tll} {_} {eq} x) (\u2264\u1d62ic y) = \u2264\u1d62ic {ict = ict} {tll} {_} {eq} (\u2264\u1d62-transitive x y)\n\n\nmodule _ where\n\n  mutual\n  \n    isLTi-abs1 : \u2200{u i ll tll ict il eq rll gll ica icb} \u2192 \u2200 a b\n                 \u2192 ica \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} a {{eq}}\n                 \u2192 icb \u2261 ic {i} {u} {gll} {ll = ll} {tll} {ict} {il} b {{eq}}\n                 \u2192 Dec (a \u2264\u1d62 b)\n                 \u2192 Dec (ica \u2264\u1d62 icb)\n    isLTi-abs1 a b refl refl (yes p) = yes (\u2264\u1d62-ext p)\n    isLTi-abs1 a b refl refl (no \u00acp) = no (\u03bb p \u2192 \u00acp (\u2264\u1d62-spec p))\n\n\n    isLTi-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb rll gll ica icb} \u2192 \u2200 a b\n                \u2192 ica \u2261 ic {i} {u} {rll} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n                \u2192 icb \u2261 ic {i} {u} {gll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n               \u2192 IndU icta ictb eqa eqb\n                \u2192 Dec (ica \u2264\u1d62 icb)\n    isLTi-abs a b iceqa iceqb (ictEq refl refl refl refl) = isLTi-abs1 a b iceqa iceqb (isLTi a b)\n    isLTi-abs a b refl refl (ict\u00acEq \u00acicteq reqa reqb) = no \u03bb { (\u2264\u1d62ic x) \u2192 \u00acicteq refl}\n    \n    isLTi : \u2200{i u gll ll fll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL fll ll) \u2192 Dec (a \u2264\u1d62 b)\n    isLTi \u2193 b = yes \u2264\u1d62\u2193\n    isLTi (ic a) \u2193 = no (\u03bb ())\n    isLTi (ic a) (ic b) = isLTi-abs a b refl refl compIndU\n\n\n\n\nmodule _ where\n  mutual \n    isEq\u1d62-abs1 : \u2200{u i ll tll ict il eq rll} \u2192 {a : IndexLL {i} {u} rll ll} \u2192 {b : IndexLL rll ll} \u2192 Dec (a \u2261 b) \u2192 Dec (ic {ll = ll} {tll} {ict} {il} a {{eq}} \u2261 ic {ll = ll} {tll} {ict} b {{eq}})\n    isEq\u1d62-abs1 (yes refl) = yes refl\n    isEq\u1d62-abs1 (no \u00acp) = no \u03bb { refl \u2192 \u00acp refl}\n\n    isEq\u1d62-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb rll ica icb} \u2192 \u2200 a b\n            \u2192 ica \u2261 ic {i} {u} {rll} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n            \u2192 icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n            \u2192 IndU icta ictb eqa eqb\n            \u2192 Dec (ica \u2261 icb)\n    isEq\u1d62-abs a b refl refl (ictEq refl refl refl refl) = isEq\u1d62-abs1 (isEq\u1d62 a b)\n    isEq\u1d62-abs a b refl refl (ict\u00acEq \u00acicteq reqa reqb) = no \u03bb { refl \u2192 \u00acicteq refl}\n    \n    isEq\u1d62 : \u2200{u i ll rll} \u2192 (a : IndexLL {i} {u} rll ll) \u2192 (b : IndexLL rll ll) \u2192 Dec (a \u2261 b)\n    isEq\u1d62 \u2193 \u2193 = yes refl\n    isEq\u1d62 \u2193 (ic b) = no \u03bb ()\n    isEq\u1d62 (ic a) \u2193 = no \u03bb ()\n    isEq\u1d62 ica@(ic a) icb@(ic b) = isEq\u1d62-abs a b refl refl compIndU\n   \n\n\nmodule _ where\n\n  open import Data.Vec\n\n  mutual\n\n    ind\u03c4\u21d2\u00acle-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb rll n dt df ica icb} \u2192 \u2200 a b\n            \u2192 ica \u2261 ic {i} {u} {\u03c4 {n = n} {dt} df} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n            \u2192 icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n            \u2192 \u00ac icb \u2245\u1d62 ica\n            \u2192 IndU icta ictb eqa eqb\n            \u2192 \u00ac ica \u2264\u1d62 icb\n    ind\u03c4\u21d2\u00acle-abs a b refl refl neq (ictEq refl refl refl refl) = \u00ac\u2264\u1d62-ext (ind\u03c4\u21d2\u00acle a b (\u00ac\u2245\u1d62-spec neq))\n    ind\u03c4\u21d2\u00acle-abs a b refl refl neq (ict\u00acEq \u00acicteq reqa reqb) = \u03bb { (\u2264\u1d62ic x) \u2192 \u00acicteq refl}\n\n-- Is there a better way to express this?\n    ind\u03c4\u21d2\u00acle : \u2200{i u rll ll n dt df} \u2192 (ind : IndexLL {i} {u} (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (ind2 : IndexLL rll ll) \u2192 \u00ac (ind2 \u2245\u1d62 ind) \u2192 \u00ac (ind \u2264\u1d62 ind2)\n    ind\u03c4\u21d2\u00acle \u2193 \u2193 neq = \u03bb _ \u2192 neq \u2245\u1d62\u2193\n    ind\u03c4\u21d2\u00acle (ic ind1) \u2193 neq = \u03bb ()\n    ind\u03c4\u21d2\u00acle (ic ind1) (ic ind2 ) neq = ind\u03c4\u21d2\u00acle-abs ind1 ind2 refl refl neq compIndU\n\n\nmodule _ where\n\n\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 : \u2200{i u rll ll n dt df}\n                \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll) (lind : IndexLL rll ll)\n                \u2192 \u00ac (lind \u2264\u1d62 ind) \u2192 \u00ac (lind \u2245\u1d62 ind)\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 \u2193 \u2193 neq = \u03bb _ \u2192 neq \u2264\u1d62\u2193\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 (ic ind) \u2193 neq = \u03bb ()\n  ind\u03c4&\u00acge\u21d2\u00ac\u2245 (ic ind) (ic lind) neq = \u03bb { (\u2245\u1d62ic x) \u2192 ind\u03c4&\u00acge\u21d2\u00ac\u2245 ind lind (\u03bb z \u2192 neq (\u2264\u1d62ic z)) x}\n\n \n\ndata Ordered\u1d62 {i u gll fll ll} (a : IndexLL {i} {u} gll ll) (b : IndexLL {i} {u} fll ll) : Set (lsuc u) where\n  a\u2264\u1d62b : a \u2264\u1d62 b \u2192 Ordered\u1d62 a b\n  b\u2264\u1d62a : b \u2264\u1d62 a \u2192 Ordered\u1d62 a b\n\n\n\nord-spec : \u2200{i u rll ll ict tll il eq fll} \u2192 {emi : IndexLL {i} {u} fll ll}\n           \u2192 {ind : IndexLL rll ll} \u2192 Ordered\u1d62 (ic {tll = tll} {ict} {il} ind {{eq}}) (ic {tll = tll} {ict} {il} emi {{eq}}) \u2192 Ordered\u1d62 ind emi\nord-spec (a\u2264\u1d62b x) = a\u2264\u1d62b (\u2264\u1d62-spec x)\nord-spec (b\u2264\u1d62a x) = b\u2264\u1d62a (\u2264\u1d62-spec x)\n\nord-ext : \u2200{i u rll ll ict tll il eq fll} \u2192 {emi : IndexLL {i} {u} fll ll}\n           \u2192 {ind : IndexLL rll ll} \u2192 Ordered\u1d62 ind emi \u2192 Ordered\u1d62 (ic {tll = tll} {ict} {il} ind {{eq}}) (ic {tll = tll} {ict} {il} emi {{eq}})\nord-ext (a\u2264\u1d62b x) = a\u2264\u1d62b (\u2264\u1d62-ext x)\nord-ext (b\u2264\u1d62a x) = b\u2264\u1d62a (\u2264\u1d62-ext x)\n\n\nord-spec\u2218ord-ext\u2261id : \u2200{i u ll ict tll il eq fll rll} \u2192 {ind : IndexLL {i} {u} fll ll} \u2192 {lind : IndexLL rll ll} \u2192 (ord : Ordered\u1d62 ind lind) \u2192 ord-spec {ict = ict} {tll = tll} {il} {eq} (ord-ext {ict = ict} {tll = tll} ord) \u2261 ord\nord-spec\u2218ord-ext\u2261id (a\u2264\u1d62b x) = refl\nord-spec\u2218ord-ext\u2261id (b\u2264\u1d62a x) = refl\n\n\n\nmodule _ where\n\n  isOrd\u1d62-abs : \u2200{i u gll fll ll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL {i} {u} fll ll)\n               \u2192 Dec (a \u2264\u1d62 b) \u2192 Dec (b \u2264\u1d62 a)\n               \u2192 Dec (Ordered\u1d62 a b)\n  isOrd\u1d62-abs a b (yes p) r = yes (a\u2264\u1d62b p)\n  isOrd\u1d62-abs a b (no \u00acp) (yes p) = yes (b\u2264\u1d62a p)\n  isOrd\u1d62-abs a b (no \u00acp) (no \u00acp\u2081) = no (\u03bb { (a\u2264\u1d62b x) \u2192 \u00acp x ; (b\u2264\u1d62a x) \u2192 \u00acp\u2081 x})\n  \n  isOrd\u1d62 : \u2200{i u gll fll ll} \u2192 (a : IndexLL {i} {u} gll ll) \u2192 (b : IndexLL {i} {u} fll ll)\n           \u2192 Dec (Ordered\u1d62 a b)\n  isOrd\u1d62 a b = isOrd\u1d62-abs a b (isLTi a b) (isLTi b a) \n\n\n\n\nflipOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n           \u2192 Ordered\u1d62 a b \u2192 Ordered\u1d62 b a\nflipOrd\u1d62 (a\u2264\u1d62b x) = b\u2264\u1d62a x\nflipOrd\u1d62 (b\u2264\u1d62a x) = a\u2264\u1d62b x\n\nflipNotOrd\u1d62 : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac Ordered\u1d62 a b \u2192 \u00ac Ordered\u1d62 b a\nflipNotOrd\u1d62 nord = \u03bb x \u2192 nord (flipOrd\u1d62 x) \n\n\n\u00aclt\u00acgt\u21d2\u00acOrd : \u2200{i u gll fll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL {i} {u} fll ll}\n              \u2192 \u00ac (a \u2264\u1d62 b) \u2192 \u00ac (b \u2264\u1d62 a) \u2192 \u00ac(Ordered\u1d62 a b)\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (a\u2264\u1d62b x) = nlt x\n\u00aclt\u00acgt\u21d2\u00acOrd nlt ngt (b\u2264\u1d62a x) = ngt x\n\n\u00acord-spec : \u2200{i u rll ll ict tll il eq fll} \u2192 {emi : IndexLL {i} {u} fll ll}\n            \u2192 {ind : IndexLL rll ll} \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} {il} ind {{eq}}) (ic {tll = tll} {ict} {il} emi {{eq}})) \u2192 \u00ac Ordered\u1d62 ind emi\n\u00acord-spec nord ord = nord (ord-ext ord)\n\n\n\n\u00acict\u21d2\u00acord : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll ica icb} \u2192 \u2200 a b\n       \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n       \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n       \u2192 \u00ac icta \u2261 ictb\n       \u2192 \u00ac Ordered\u1d62 icb ica\n\u00acict\u21d2\u00acord a b refl refl \u00acp (a\u2264\u1d62b (\u2264\u1d62ic x)) = \u00acp refl\n\u00acict\u21d2\u00acord a b refl refl \u00acp (b\u2264\u1d62a (\u2264\u1d62ic x)) = \u00acp refl\n\n\nind\u03c4&\u00acge\u21d2\u00acOrd : \u2200{i u rll ll n dt df} \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n                          (lind : IndexLL rll ll) \u2192 \u00ac (lind \u2264\u1d62 ind) \u2192 \u00ac Ordered\u1d62 ind lind\nind\u03c4&\u00acge\u21d2\u00acOrd ind lind neq (a\u2264\u1d62b x) = ind\u03c4\u21d2\u00acle ind lind (ind\u03c4&\u00acge\u21d2\u00ac\u2245 ind lind neq) x\nind\u03c4&\u00acge\u21d2\u00acOrd ind lind neq (b\u2264\u1d62a x) = neq x                        \n\n\n\na,c\u2264\u1d62b\u21d2ordac : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 (c \u2264\u1d62 b) \u2192 Ordered\u1d62 a c\na,c\u2264\u1d62b\u21d2ordac \u2264\u1d62\u2193 ltbc = a\u2264\u1d62b \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62ic ltab) \u2264\u1d62\u2193 = b\u2264\u1d62a \u2264\u1d62\u2193\na,c\u2264\u1d62b\u21d2ordac (\u2264\u1d62ic ltab) (\u2264\u1d62ic ltcb) = ord-ext (a,c\u2264\u1d62b\u21d2ordac ltab ltcb)\n\n\n\n\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc : \u2200{i u gll fll mll ll} \u2192 {a : IndexLL {i} {u} gll ll} \u2192 {b : IndexLL fll ll} \u2192 {c : IndexLL mll ll} \u2192 (a \u2264\u1d62 b) \u2192 \u00ac Ordered\u1d62 a c \u2192 \u00ac Ordered\u1d62 b c\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (a\u2264\u1d62b x) = \u22a5-elim (nord (a\u2264\u1d62b (\u2264\u1d62-transitive lt x)))\na\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt nord (b\u2264\u1d62a x) = \u22a5-elim (nord (a,c\u2264\u1d62b\u21d2ordac lt x))\n\n\n_-\u1d62_ : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll) \u2192 (sind \u2264\u1d62 bind)\n       \u2192 IndexLL cll pll\n(bind -\u1d62 .\u2193) \u2264\u1d62\u2193 = bind\n((ic bind) -\u1d62 (ic sind)) (\u2264\u1d62ic lt) = (bind -\u1d62 sind) lt\n\n\n\n\nreplLL-\u2193 : \u2200{i u pll ell ll} \u2192 \u2200(ind : IndexLL {i} {u} pll ll)\n           \u2192 replLL ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell \u2261 ell\nreplLL-\u2193 \u2193 = refl\nreplLL-\u2193 (ic ind) = replLL-\u2193 ind\n\n\n\nind-rpl\u2193 : \u2200{i u ll pll cll ell} \u2192 (ind : IndexLL {i} {u} pll ll)\n        \u2192 IndexLL cll (replLL ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) ell) \u2192 IndexLL cll ell\nind-rpl\u2193 {_} {_} {_} {pll} {cll} {ell} ind x\n  =  subst (\u03bb x \u2192 x) (cong (\u03bb x \u2192 IndexLL cll x) (replLL-\u2193 {ell = ell} ind)) x \n\n\n\na\u2264\u1d62b-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n             \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (lt : emi \u2264\u1d62 ind)\n             \u2192 IndexLL (replLL ((ind -\u1d62 emi) lt) frll) (replLL ind frll) \na\u2264\u1d62b-morph .\u2193 ind \u2264\u1d62\u2193 = \u2193\na\u2264\u1d62b-morph (ic {tll = tll} {ict = ict} emi) (ic ind) (\u2264\u1d62ic lt) = ic {tll = tll} {ict = ict} (a\u2264\u1d62b-morph emi ind lt)\n\n\n\n\nmodule _ where\n\n  a\u2264\u1d62b-morph-id-abs : \u2200{i u ll tll ict rll}\n               \u2192 {ind : IndexLL {i} {u} rll ll}\n               \u2192 \u2200 {w\u2081T} \u2192 (w\u2081 : w\u2081T \u2261 rll)  -- w\u2081T : replLL ((ind -\u1d62 ind) (\u2264\u1d62-reflexive ind)) rll\n               \u2192 \u2200 {w\u2082T} \u2192 (w\u2082 : w\u2082T \u2261 ll) -- w\u2082T : replLL li ind rl\n               \u2192 (w\u2083 : IndexLL w\u2081T w\u2082T) -- w\u2083 : (a\u2264\u1d62b-morph ind ind (\u2264\u1d62-reflexive ind))\n               \u2192 (w\u2084 : subst\u2082 (\u03bb x y \u2192 IndexLL x y) w\u2081 w\u2082 w\u2083 \u2261 ind) -- recursive step\n               \u2192 subst\u2082 IndexLL w\u2081 \n                   (cong (\u03bb x \u2192 il[ expLLT x ict tll ]) w\u2082)\n                     (ic {tll = tll} {ict} w\u2083) \u2261 ic {tll = tll} {ict} ind\n  a\u2264\u1d62b-morph-id-abs refl refl _ refl = refl\n\n\n  a\u2264\u1d62b-morph-id : \u2200{i u ll rll}\n               \u2192 (ind : IndexLL {i} {u} rll ll)\n               \u2192 subst\u2082 (\u03bb x y \u2192 IndexLL x y) (replLL-\u2193 {ell = rll} ind) (replLL-id ind rll refl) (a\u2264\u1d62b-morph ind ind (\u2264\u1d62-reflexive ind)) \u2261 ind\n  a\u2264\u1d62b-morph-id \u2193 = refl\n  a\u2264\u1d62b-morph-id {rll = rll} (ic ind {{refl}}) = a\u2264\u1d62b-morph-id-abs (replLL-\u2193 {ell = rll} ind) (replLL-id ind rll refl) (a\u2264\u1d62b-morph ind ind (\u2264\u1d62-reflexive ind)) (a\u2264\u1d62b-morph-id ind)\n\n\n\nreplLL-a\u2264b\u2261a : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 {gll}\n               \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (lt : emi \u2264\u1d62 ind)\n               \u2192 replLL (a\u2264\u1d62b-morph emi ind {frll = frll} lt) gll \u2261 replLL emi gll\nreplLL-a\u2264b\u2261a .\u2193 ind \u2264\u1d62\u2193 = refl\nreplLL-a\u2264b\u2261a (ic {tll = tll} {ict} emi) (ic ind) (\u2264\u1d62ic lt) = cong (\u03bb x \u2192 il[ expLLT x ict tll ]) (replLL-a\u2264b\u2261a emi ind lt)\n\nmodule _ where\n\n  mutual\n  \n    \u00acord-morph-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll frll ica icb} \u2192 \u2200 a b\n              \u2192 ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}}\n              \u2192 icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}}\n              \u2192 \u00ac Ordered\u1d62 icb ica\n              \u2192 IndU icta ictb eqa eqb\n              \u2192 IndexLL fll il[ expLLT (replLL b frll) ictb tllb ]\n    \u00acord-morph-abs {tllb = tllb} {ictb} {frll = frll} a b refl refl nord (ictEq _ _ _ refl)\n        = ic {ll = _} {tllb} {ictb} (\u00acord-morph a b {frll = frll} (\u00acord-spec nord))\n    \u00acord-morph-abs {icta = icta} {eqa = eqa} {eqb = eqb} {frll = frll} a b refl refl nord (ict\u00acEq \u00acicteq refl refl) = ic {tll\n        = replLL b frll} {ict = icta} a {{rexpLLT\u21d2req \u00acicteq eqa eqb}}\n\n\n    \u00acord-morph : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n                 \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (nord : \u00ac Ordered\u1d62 ind emi)\n                 \u2192 IndexLL fll (replLL ind frll)\n    \u00acord-morph emi \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    \u00acord-morph \u2193 (ic ind) nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    \u00acord-morph (ic {ll = lle} {tlle} {icte} {il} emi {{eqe}}) (ic {ll = lli} {tlli} {icti} ind {{eqi}}) {frll} nord\n        = \u00acord-morph-abs emi ind refl refl nord compIndU\n\n\nmodule _ where\n\nmutual\n\n    \u00acord-morph-\u00acord-ir-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll ica icb frll} \u2192 \u2200 a b\n      \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n      \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n      \u2192 (nord1 nord2 : \u00ac Ordered\u1d62 icb ica)\n      \u2192 (w : IndU icta ictb eqa eqb)\n      \u2192  \u00acord-morph-abs {frll = frll} a b iceqa iceqb nord1 w \u2261 \u00acord-morph-abs a b iceqa iceqb nord2 w\n    \u00acord-morph-\u00acord-ir-abs {tllb = tllb} {ictb} {frll = frll} a b refl refl nord1 nord2 (ictEq refl refl refl refl)\n        = cong (\u03bb z \u2192 ic {ll = _} {tllb} {ictb} z) (\u00acord-morph-\u00acord-ir a b {frll} (\u00acord-spec nord1) (\u00acord-spec nord2))\n    \u00acord-morph-\u00acord-ir-abs ica icb refl refl nord1 nord2 (ict\u00acEq \u00acicteq refl refl) = refl\n\n\n    \u00acord-morph-\u00acord-ir : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n                         \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (nord nord2 : \u00ac Ordered\u1d62 ind emi)\n                         \u2192 \u00acord-morph emi ind {frll} nord \u2261 \u00acord-morph emi ind {frll} nord2\n    \u00acord-morph-\u00acord-ir \u2193 ind {frll} nord nord2 = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    \u00acord-morph-\u00acord-ir (ic emi) \u2193 {frll} nord nord2 = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    \u00acord-morph-\u00acord-ir ice@(ic emi) ici@(ic ind) {frll} nord nord2 = \u00acord-morph-\u00acord-ir-abs emi ind refl refl nord nord2 compIndU \n\n\n\nmodule _ where\n\n  mutual\n\n    replLL-\u00acordab\u2261ba-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll ica icb gll frll} \u2192 \u2200 a b\n      \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n      \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n      \u2192 (nord : \u00ac Ordered\u1d62 icb ica)\n      \u2192 \u2200 w1 w2\n      \u2192 replLL\n          (\u00acord-morph-abs {frll = frll} a b iceqa iceqb nord w1) gll\n        \u2261\n        replLL\n          (\u00acord-morph-abs {frll = gll} b a iceqb iceqa (\u03bb x \u2192 nord (flipOrd\u1d62 x)) w2) frll\n    replLL-\u00acordab\u2261ba-abs {tllb = tllb} {ictb} {eqb} {gll = gll} {frll} a b refl refl nord (ictEq refl refl refl refl) (ictEq refl refl refl refl)\n      =  cong (\u03bb z \u2192 il[ expLLT z ictb tllb ])\n            (subst\n              (\u03bb z \u2192 replLL (\u00acord-morph a b (\u00acord-spec nord)) gll \u2261 replLL z frll)\n              (\u00acord-morph-\u00acord-ir b a (flipNotOrd\u1d62 (\u00acord-spec nord)) (\u00acord-spec (flipNotOrd\u1d62 nord)))\n              (replLL-\u00acordab\u2261ba a b (\u00acord-spec nord)))\n    replLL-\u00acordab\u2261ba-abs a b iceqa iceqb nord (ictEq icteq lleq tlleq eqeq) (ict\u00acEq \u00acicteq reqa reqb) = \u22a5-elim (\u00acicteq (sym icteq))\n    replLL-\u00acordab\u2261ba-abs a b iceqa iceqb nord (ict\u00acEq \u00acicteq reqa reqb) (ictEq icteq lleq tlleq eqeq) = \u22a5-elim (\u00acicteq (sym icteq))\n    replLL-\u00acordab\u2261ba-abs {eqa = eqa} {eqb = eqb} a b refl refl nord (ict\u00acEq _ refl refl) (ict\u00acEq \u00acicteq refl refl) = cong il[_] (rexpLLT\u21d2req \u00acicteq eqb eqa)\n\n\n    replLL-\u00acordab\u2261ba : \u2200{i u rll ll fll}\n      \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 \u2200 {gll}\n      \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll}\n      \u2192 (nord : \u00ac Ordered\u1d62 ind emi)\n      \u2192 replLL (\u00acord-morph emi ind {frll} nord) gll \u2261 replLL (\u00acord-morph ind emi {gll} (flipNotOrd\u1d62 nord)) frll\n    replLL-\u00acordab\u2261ba \u2193 ind nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    replLL-\u00acordab\u2261ba (ic emi) \u2193 nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    replLL-\u00acordab\u2261ba ice@(ic emi) ici@(ic ind) nord = replLL-\u00acordab\u2261ba-abs emi ind refl refl nord compIndU compIndU \n\n\n\n\nmodule _ where\n\n  mutual\n\n    lemma\u2081-\u00acord-a\u2264\u1d62b-abs : \u2200{u i ll tll ict il eq llc tllc ictc fll rll pll ica icb ell} \u2192 \u2200 a b c\n               \u2192 \u2200 {icc eqc}\n               \u2192 ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}}\n               \u2192 icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}}\n               \u2192 icc \u2261 ic {i} {u} {pll} {ll = llc} {tllc} {ictc} {expLLT (replLL b ell) ict tll} c {{eqc}}\n               \u2192 \u2200 lt\n               \u2192 \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) icc\n               \u2192 IndU ict ictc refl eqc\n               \u2192 IndexLL pll il[ il ]\n    lemma\u2081-\u00acord-a\u2264\u1d62b-abs {tll = tll} {ict} {eq = eq} {ell = ell} a b c iceqa iceqb refl lt nord (ictEq _ _ _ refl) = ic {tll = tll} {ict = ict} (lemma\u2081-\u00acord-a\u2264\u1d62b a b ell lt c (\u00acord-spec nord)) {{eq}}\n    lemma\u2081-\u00acord-a\u2264\u1d62b-abs {ll = ll} {eq = eq} {ictc = ictc} a b c {eqc = eqc} iceqa iceqb iceqc lt nord (ict\u00acEq \u00acicteq refl refl) = ic {tll = ll} {ictc} c {{trans eq (sym (rexpLLT\u21d2req \u00acicteq refl eqc))}}\n\n\n    lemma\u2081-\u00acord-a\u2264\u1d62b : \u2200{i u ll pll rll fll}\n          \u2192 (emi : IndexLL {i} {u} fll ll)\n          \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n          \u2192 (omi : IndexLL pll (replLL ind ell))\n          \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind {ell} lt) omi)\n          \u2192 IndexLL pll ll\n    lemma\u2081-\u00acord-a\u2264\u1d62b .\u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    lemma\u2081-\u00acord-a\u2264\u1d62b .(ic _) .(ic _) ell (\u2264\u1d62ic lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    lemma\u2081-\u00acord-a\u2264\u1d62b ica@(ic a) icb@(ic b {{eq}}) ell (\u2264\u1d62ic lt) icc@(ic c) nord = lemma\u2081-\u00acord-a\u2264\u1d62b-abs {eq = eq} a b c refl refl refl lt nord compIndU\n\n\n\nmodule _ where\n\nmutual\n\n    \u00acord-morph\u21d2\u00acord-abs : \u2200{u i lla tlla icta il eqa llb tllb ictb eqb fll rll frll ica icb} \u2192 \u2200 a b\n           \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = lla} {tlla} {icta} {il} a {{eqa}})\n           \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = llb} {tllb} {ictb} {il} b {{eqb}})\n           \u2192 (nord : \u00ac Ordered\u1d62 icb ica)\n           \u2192 (w : IndU icta ictb eqa eqb)\n           \u2192  \u00ac Ordered\u1d62 (ic {tll = tllb} {ict = ictb} (a\u2264\u1d62b-morph b b {frll = frll} (\u2264\u1d62-reflexive b)))\n                         (\u00acord-morph-abs {frll = frll} a b iceqa iceqb nord w)\n    \u00acord-morph\u21d2\u00acord-abs {frll = frll} a b refl refl nord (ictEq icteq lleq tlleq refl) = \u03bb x \u2192 r (ord-spec x) where\n      r = \u00acord-morph\u21d2\u00acord a b {frll} (\u00acord-spec nord)\n    \u00acord-morph\u21d2\u00acord-abs a b refl refl nord (ict\u00acEq \u00acicteq refl refl) = \u00acict\u21d2\u00acord a (a\u2264\u1d62b-morph b b (\u2264\u1d62-reflexive b)) refl refl \u00acicteq\n  \n  \n    \u00acord-morph\u21d2\u00acord : \u2200{i u rll ll fll} \u2192 (emi : IndexLL {i} {u} fll ll)\n          \u2192 (ind : IndexLL rll ll) \u2192 \u2200 {frll} \u2192 (nord : \u00ac Ordered\u1d62 ind emi)\n          \u2192 \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph ind ind {frll} (\u2264\u1d62-reflexive ind)) (\u00acord-morph emi ind nord)\n    \u00acord-morph\u21d2\u00acord \u2193 ind nord = \u03bb _ \u2192 nord (b\u2264\u1d62a \u2264\u1d62\u2193)\n    \u00acord-morph\u21d2\u00acord (ic emi) \u2193 nord = \u03bb _ \u2192 nord (a\u2264\u1d62b \u2264\u1d62\u2193)\n    \u00acord-morph\u21d2\u00acord (ic emi) (ic ind) nord = \u00acord-morph\u21d2\u00acord-abs emi ind refl refl nord compIndU\n\n\n\nmodule _ where\n\n  mutual\n\n    rlemma\u2081\u21d2\u00acord-abs : \u2200{u i ll tll ict il eq llc tllc ictc fll rll pll ica icb ell} \u2192 \u2200 a b c\n           \u2192 \u2200{eqc icc}\n           \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}})\n           \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}})\n           \u2192 (iceqc : icc \u2261 ic {i} {u} {pll} {ll = llc} {tllc} {ictc} {expLLT (replLL b ell) ict tll} c {{eqc}})\n           \u2192 \u2200 lt\n           \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) icc)\n           \u2192 (w : IndU ict ictc refl eqc)\n           \u2192 \u00ac Ordered\u1d62 ica\n                        (lemma\u2081-\u00acord-a\u2264\u1d62b-abs a b c iceqa iceqb iceqc lt nord w)\n    rlemma\u2081\u21d2\u00acord-abs {ell = ell} a b c refl refl refl lt nord (ictEq icteq lleq tlleq refl) = \u03bb x \u2192 r (ord-spec x) where\n      r = rlemma\u2081\u21d2\u00acord a b ell lt c (\u00acord-spec nord)\n    rlemma\u2081\u21d2\u00acord-abs a b c refl refl refl lt nord (ict\u00acEq \u00acicteq refl refl) = \u00acict\u21d2\u00acord c a refl refl (\u03bb x \u2192 \u00acicteq (sym x))\n\n    rlemma\u2081\u21d2\u00acord : \u2200{i u ll pll rll fll}\n       \u2192 (emi : IndexLL {i} {u} fll ll)\n       \u2192 (ind : IndexLL rll ll) \u2192 \u2200 ell \u2192 (lt : emi \u2264\u1d62 ind)\n       \u2192 (omi : IndexLL pll (replLL ind ell))\n       \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind lt) omi)\n       \u2192 \u00ac Ordered\u1d62 emi (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt omi nord)\n    rlemma\u2081\u21d2\u00acord .\u2193 ind ell \u2264\u1d62\u2193 omi nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    rlemma\u2081\u21d2\u00acord .(ic _) .(ic _) ell (\u2264\u1d62ic lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    rlemma\u2081\u21d2\u00acord (ic emi) (ic ind) ell (\u2264\u1d62ic lt) (ic omi) nord = rlemma\u2081\u21d2\u00acord-abs emi ind omi refl refl refl lt nord compIndU\n\n\n\nmodule _ where\n\n  mutual \n\n    \u00acord-morph$lemma\u2081\u2261I-abs1 : \u2200{u i ll tll ict il eq fll rll pll ica icb ell} \u2192 \u2200 a b c\n               \u2192 \u2200 {icc}\n               \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}})\n               \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}})\n               \u2192 (iceqc : icc \u2261 ic {i} {u} {pll} {ll = replLL b ell} {tll} {ict} {expLLT (replLL b ell) ict tll} c {{refl}})\n               \u2192 \u2200 lt\n               \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) (ic {tll = tll} {ict} c {{refl}}))\n               \u2192 (w : IndU ict ict eq eq)\n               \u2192 \u00acord-morph-abs\n                   (lemma\u2081-\u00acord-a\u2264\u1d62b a b ell lt c (\u03bb ord \u2192 nord (ord-ext ord))) b refl refl\n                     (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc (\u2264\u1d62ic lt)\n                       (\u03bb x \u2192 rlemma\u2081\u21d2\u00acord a b ell lt c (\u03bb ord \u2192 nord (ord-ext ord)) (ord-spec x))) w\n                 \u2261 ic {tll = tll} {ict} c {{refl}}\n    \u00acord-morph$lemma\u2081\u2261I-abs1 {ell = ell} a b c refl refl refl lt nord (ictEq icteq lleq tlleq refl) = {!r!} where\n      r = \u00acord-morph$lemma\u2081\u2261I ell a b lt c (\u00acord-spec nord)\n    \u00acord-morph$lemma\u2081\u2261I-abs1 a b c refl refl refl lt nord (ict\u00acEq \u00acicteq reqa reqb) = {!!}          \n\n\n    \u00acord-morph$lemma\u2081\u2261I-abs : \u2200{u i ll tll ict il eq llc tllc ictc fll rll pll ica icb ell} \u2192 \u2200 a b c\n               \u2192 \u2200 {icc eqc}\n               \u2192 (iceqa : ica \u2261 ic {i} {u} {fll} {ll = ll} {tll} {ict} {il} a {{eq}})\n               \u2192 (iceqb : icb \u2261 ic {i} {u} {rll} {ll = ll} {tll} {ict} {il} b {{eq}})\n               \u2192 (iceqc : icc \u2261 ic {i} {u} {pll} {ll = llc} {tllc} {ictc} {expLLT (replLL b ell) ict tll} c {{eqc}})\n               \u2192 \u2200 lt\n               \u2192 (nord : \u00ac Ordered\u1d62 (ic {tll = tll} {ict} (a\u2264\u1d62b-morph a b {ell} lt)) (ic c))\n               \u2192 (w : IndU ict ictc refl eqc)\n               \u2192 \u00acord-morph\n                   (lemma\u2081-\u00acord-a\u2264\u1d62b-abs {eq = eq} {ell = ell} a b c refl refl refl lt nord w) (ic b) {frll = ell} (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc (\u2264\u1d62ic lt) (rlemma\u2081\u21d2\u00acord-abs a b c refl refl refl lt nord w)) \u2261 icc\n    \u00acord-morph$lemma\u2081\u2261I-abs a b c refl refl refl lt nord (ictEq icteq lleq tlleq refl) = {!!} -- \u00acord-morph$lemma\u2081\u2261I-abs1 a b c refl refl refl lt nord compIndU where\n--      r = \u00acord-morph$lemma\u2081\u2261I ell a b lt c (\u00acord-spec nord)\n    \u00acord-morph$lemma\u2081\u2261I-abs a b c refl refl refl lt nord (ict\u00acEq \u00acicteq refl refl) = {!!} \n\n\n\n    \u00acord-morph$lemma\u2081\u2261I : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind {ell} lt) lind)\n          \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind nord)) \u2261 lind)\n    \u00acord-morph$lemma\u2081\u2261I ell .\u2193 ind \u2264\u1d62\u2193 lind nord = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n    \u00acord-morph$lemma\u2081\u2261I ell .(ic _) .(ic _) (\u2264\u1d62ic lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n    \u00acord-morph$lemma\u2081\u2261I ell ice@(ic emi) ici@(ic ind) clt@(\u2264\u1d62ic lt) icl@(ic lind) nord = {!!} -- \u00acord-morph$lemma\u2081\u2261I-abs emi ind lind refl refl refl lt nord  compIndU compIndU (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc clt (rlemma\u2081\u21d2\u00acord ice ici ell clt icl nord))\n\n\n    \u00acord-morph$lemma\u2081\u2261I' : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind lt) lind) \u2192 (lnord : \u00ac Ordered\u1d62 ind (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord))\n        \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind lnord \u2261 lind)\n    \u00acord-morph$lemma\u2081\u2261I' = {!!}\n\n\n-- module _ where\n\n--   \u00acord-morph$lemma\u2081\u2261I' : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) lind) \u2192 (lnord : \u00ac Ordered\u1d62 ind (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord))\n--          \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind ell lnord \u2261 lind)\n--   \u00acord-morph$lemma\u2081\u2261I' ell \u2193 ind \u2264\u1d62\u2193 lind nord _ = \u22a5-elim (nord (a\u2264\u1d62b \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (lind \u2190\u2227) nord lnord = cong (\u03bb x \u2192 x \u2190\u2227) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2227) (ind \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) (\u2227\u2192 lind) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (lind \u2190\u2227) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2227\u2192 emi) (\u2227\u2192 ind) (\u2264\u1d62\u2227\u2192 lt) (\u2227\u2192 lind) nord lnord = cong (\u03bb x \u2192 \u2227\u2192 x) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (lind \u2190\u2228) nord lnord = cong (\u03bb x \u2192 x \u2190\u2228) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2228) (ind \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) (\u2228\u2192 lind) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) \u2193 nord = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (lind \u2190\u2228) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2228\u2192 emi) (\u2228\u2192 ind) (\u2264\u1d62\u2228\u2192 lt) (\u2228\u2192 lind) nord lnord = cong (\u03bb x \u2192 \u2228\u2192 x) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (lind \u2190\u2202) nord lnord = cong (\u03bb x \u2192 x \u2190\u2202) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n--   \u00acord-morph$lemma\u2081\u2261I' ell (emi \u2190\u2202) (ind \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) (\u2202\u2192 lind) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) \u2193 nord _ = \u22a5-elim (nord (b\u2264\u1d62a \u2264\u1d62\u2193))\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (lind \u2190\u2202) nord _ = refl\n--   \u00acord-morph$lemma\u2081\u2261I' ell (\u2202\u2192 emi) (\u2202\u2192 ind) (\u2264\u1d62\u2202\u2192 lt) (\u2202\u2192 lind) nord lnord = cong (\u03bb x \u2192 \u2202\u2192 x) r where\n--     r = (\u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind (\u00acord-spec nord)) (\u00acord-spec lnord)\n    \n--   \u00acord-morph$lemma\u2081\u2261I : \u2200{i u pll ll cll fll} \u2192 \u2200 ell \u2192 (emi : IndexLL {i} {u} fll ll) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lt : emi \u2264\u1d62 ind) \u2192 (lind : IndexLL cll (replLL ll ind ell)) \u2192 (nord : \u00ac Ordered\u1d62 (a\u2264\u1d62b-morph emi ind ell lt) lind)\n--        \u2192 (\u00acord-morph (lemma\u2081-\u00acord-a\u2264\u1d62b emi ind ell lt lind nord) ind ell (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind nord)) \u2261 lind)\n--   \u00acord-morph$lemma\u2081\u2261I ell emi ind lt lind nord = \u00acord-morph$lemma\u2081\u2261I' ell emi ind lt lind nord (a\u2264\u1d62b&\u00acordac\u21d2\u00acordbc lt (rlemma\u2081\u21d2\u00acord emi ind ell lt lind nord))\n\n\n\n-- _+\u1d62_ : \u2200{i u pll cll ll} \u2192 IndexLL {i} {u} pll ll \u2192 IndexLL cll pll \u2192 IndexLL cll ll\n-- _+\u1d62_ \u2193 is = is\n-- _+\u1d62_ (if \u2190\u2227) is = (_+\u1d62_ if is) \u2190\u2227\n-- _+\u1d62_ (\u2227\u2192 if) is = \u2227\u2192 (_+\u1d62_ if is)\n-- _+\u1d62_ (if \u2190\u2228) is = (_+\u1d62_ if is) \u2190\u2228\n-- _+\u1d62_ (\u2228\u2192 if) is = \u2228\u2192 (_+\u1d62_ if is)\n-- _+\u1d62_ (if \u2190\u2202) is = (_+\u1d62_ if is) \u2190\u2202\n-- _+\u1d62_ (\u2202\u2192 if) is = \u2202\u2192 (_+\u1d62_ if is)\n\n\n\n\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 : \u2200{i u pll cll ll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--            \u2192 ind \u2264\u1d62 (ind +\u1d62 lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 \u2193 lind = \u2264\u1d62\u2193\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2227) lind = \u2264\u1d62\u2190\u2227 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2227\u2192 ind) lind = \u2264\u1d62\u2227\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2228) lind = \u2264\u1d62\u2190\u2228 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2228\u2192 ind) lind = \u2264\u1d62\u2228\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (ind \u2190\u2202) lind = \u2264\u1d62\u2190\u2202 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n-- +\u1d62\u21d2l\u2264\u1d62+\u1d62 (\u2202\u2192 ind) lind = \u2264\u1d62\u2202\u2192 (+\u1d62\u21d2l\u2264\u1d62+\u1d62 ind lind)\n\n\n-- a+\u2193\u2261a : \u2200{i u pll ll} \u2192 (a : IndexLL {i} {u} pll ll) \u2192 a +\u1d62 \u2193 \u2261 a\n-- a+\u2193\u2261a \u2193 = refl\n-- a+\u2193\u2261a (a \u2190\u2227) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (a \u2190\u2227) | .a | refl = refl\n-- a+\u2193\u2261a (\u2227\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (\u2227\u2192 a) | .a | refl = refl\n-- a+\u2193\u2261a (a \u2190\u2228) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (a \u2190\u2228) | .a | refl = refl\n-- a+\u2193\u2261a (\u2228\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (\u2228\u2192 a) | .a | refl = refl\n-- a+\u2193\u2261a (a \u2190\u2202) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (a \u2190\u2202) | .a | refl = refl\n-- a+\u2193\u2261a (\u2202\u2192 a) with (a +\u1d62 \u2193) | (a+\u2193\u2261a a)\n-- a+\u2193\u2261a (\u2202\u2192 a) | .a | refl = refl\n\n\n-- [a+b]-a=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n--           \u2192 (b : IndexLL rll fll)\n--           \u2192 ((a +\u1d62 b) -\u1d62 a) (+\u1d62\u21d2l\u2264\u1d62+\u1d62 a b) \u2261 b\n-- [a+b]-a=b \u2193 b = refl\n-- [a+b]-a=b (a \u2190\u2227) b = [a+b]-a=b a b\n-- [a+b]-a=b (\u2227\u2192 a) b = [a+b]-a=b a b\n-- [a+b]-a=b (a \u2190\u2228) b = [a+b]-a=b a b\n-- [a+b]-a=b (\u2228\u2192 a) b = [a+b]-a=b a b\n-- [a+b]-a=b (a \u2190\u2202) b = [a+b]-a=b a b\n-- [a+b]-a=b (\u2202\u2192 a) b = [a+b]-a=b a b\n\n-- a+[b-a]=b : \u2200{i u rll ll fll} \u2192 (a : IndexLL {i} {u} fll ll)\n--             \u2192 (b : IndexLL rll ll)\n--             \u2192 (lt : a \u2264\u1d62 b)\n--             \u2192 (a +\u1d62 (b -\u1d62 a) lt) \u2261 b\n-- a+[b-a]=b \u2193 b \u2264\u1d62\u2193 = refl\n-- a+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (a \u2190\u2227) (b \u2190\u2227) (\u2264\u1d62\u2190\u2227 lt) | .b | refl = refl\n-- a+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (\u2227\u2192 a) (\u2227\u2192 b) (\u2264\u1d62\u2227\u2192 lt) | .b | refl = refl\n-- a+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (a \u2190\u2228) (b \u2190\u2228) (\u2264\u1d62\u2190\u2228 lt) | .b | refl = refl\n-- a+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt)with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (\u2228\u2192 a) (\u2228\u2192 b) (\u2264\u1d62\u2228\u2192 lt) | .b | refl = refl\n-- a+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (a \u2190\u2202) (b \u2190\u2202) (\u2264\u1d62\u2190\u2202 lt) | .b | refl = refl\n-- a+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) with (a +\u1d62 ((b -\u1d62 a) lt)) | (a+[b-a]=b a b lt)\n-- a+[b-a]=b (\u2202\u2192 a) (\u2202\u2192 b) (\u2264\u1d62\u2202\u2192 lt) | .b | refl = refl\n\n\n\n-- -- A predicate that is true if pll is not transformed by ltr.\n\n-- data UpTran {i u} : \u2200 {ll pll rll} \u2192 IndexLL pll ll \u2192 LLTr {i} {u} rll ll \u2192 Set (lsuc u) where\n--   indI : \u2200{pll ll} \u2192 {ind : IndexLL pll ll} \u2192 UpTran ind I\n--   \u2190\u2202\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (\u2202\u2192 ind) ltr\n--          \u2192 UpTran (ind \u2190\u2202) (\u2202c ltr)\n--   \u2202\u2192\u2202c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2202 ri} {rll = rll} (ind \u2190\u2202) ltr\n--          \u2192 UpTran (\u2202\u2192 ind) (\u2202c ltr)\n--   \u2190\u2228\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (\u2228\u2192 ind) ltr\n--          \u2192 UpTran (ind \u2190\u2228) (\u2228c ltr)\n--   \u2228\u2192\u2228c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2228 ri} {rll = rll} (ind \u2190\u2228) ltr\n--          \u2192 UpTran (\u2228\u2192 ind) (\u2228c ltr)\n--   \u2190\u2227\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll ri} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (\u2227\u2192 ind) ltr\n--          \u2192 UpTran (ind \u2190\u2227) (\u2227c ltr)\n--   \u2227\u2192\u2227c : \u2200{pll li ri rll ltr} \u2192 {ind : IndexLL pll li} \u2192 UpTran {ll = li \u2227 ri} {rll = rll} (ind \u2190\u2227) ltr\n--          \u2192 UpTran (\u2227\u2192 ind) (\u2227c ltr)\n--   \u2190\u2227]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n--             \u2192 UpTran {rll = rll} (ind \u2190\u2227) ltr \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr)\n--   \u2227\u2192]\u2190\u2227\u2227\u2227d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n--             \u2192 UpTran {rll = rll} (\u2227\u2192 (ind \u2190\u2227)) ltr\n--             \u2192 UpTran {ll = (lli \u2227 lri) \u2227 ri} ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr)\n--   \u2227\u2192\u2227\u2227d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n--             \u2192 UpTran {rll = rll} (\u2227\u2192 (\u2227\u2192 ind)) ltr\n--             \u2192 UpTran {ll = ((lli \u2227 lri) \u2227 ri)} (\u2227\u2192 ind) (\u2227\u2227d ltr)\n--   \u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n--             \u2192 UpTran {rll = rll} ((ind \u2190\u2227) \u2190\u2227) ltr\n--             \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr)\n--   \u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n--             \u2192 UpTran {rll = rll} ((\u2227\u2192 ind) \u2190\u2227) ltr\n--             \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr)\n--   \u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n--             \u2192 UpTran {rll = rll} (\u2227\u2192 ind) ltr\n--             \u2192 UpTran {ll = li \u2227 (rli \u2227 rri)} (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr)\n--   \u2190\u2228]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n--             \u2192 UpTran {rll = rll} (ind \u2190\u2228) ltr \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr)\n--   \u2228\u2192]\u2190\u2228\u2228\u2228d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n--             \u2192 UpTran {rll = rll} (\u2228\u2192 (ind \u2190\u2228)) ltr\n--             \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr)\n--   \u2228\u2192\u2228\u2228d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n--             \u2192 UpTran {rll = rll} (\u2228\u2192 (\u2228\u2192 ind)) ltr\n--             \u2192 UpTran {ll = (lli \u2228 lri) \u2228 ri} (\u2228\u2192 ind) (\u2228\u2228d ltr)\n--   \u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n--             \u2192 UpTran {rll = rll} ((ind \u2190\u2228) \u2190\u2228) ltr\n--             \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr)\n--   \u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n--             \u2192 UpTran {rll = rll} ((\u2228\u2192 ind) \u2190\u2228) ltr\n--             \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr)\n--   \u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n--             \u2192 UpTran {rll = rll} (\u2228\u2192 ind) ltr\n--             \u2192 UpTran {ll = li \u2228 (rli \u2228 rri)} (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr)\n--   \u2190\u2202]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lli}\n--             \u2192 UpTran {rll = rll} (ind \u2190\u2202) ltr \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr)\n--   \u2202\u2192]\u2190\u2202\u2202\u2202d : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll lri}\n--             \u2192 UpTran {rll = rll} (\u2202\u2192 (ind \u2190\u2202)) ltr\n--             \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr)\n--   \u2202\u2192\u2202\u2202d    : \u2200{pll lli lri ri rll ltr} \u2192 {ind : IndexLL pll ri}\n--             \u2192 UpTran {rll = rll} (\u2202\u2192 (\u2202\u2192 ind)) ltr\n--             \u2192 UpTran {ll = (lli \u2202 lri) \u2202 ri} (\u2202\u2192 ind) (\u2202\u2202d ltr)\n--   \u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll li}\n--             \u2192 UpTran {rll = rll} ((ind \u2190\u2202) \u2190\u2202) ltr\n--             \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr)\n--   \u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rli}\n--             \u2192 UpTran {rll = rll} ((\u2202\u2192 ind) \u2190\u2202) ltr\n--             \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr)\n--   \u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d : \u2200{pll li rli rri rll ltr} \u2192 {ind : IndexLL pll rri}\n--             \u2192 UpTran {rll = rll} (\u2202\u2192 ind) ltr\n--             \u2192 UpTran {ll = li \u2202 (rli \u2202 rri)} (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr)\n\n-- isUpTran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 Dec (UpTran ind ltr)\n-- isUpTran ind I = yes indI\n-- isUpTran \u2193 (\u2202c ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2202) (\u2202c ltr) with (isUpTran (\u2202\u2192 ind) ltr)\n-- isUpTran (ind \u2190\u2202) (\u2202c ltr) | yes ut = yes (\u2190\u2202\u2202c ut)\n-- isUpTran (ind \u2190\u2202) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2190\u2202\u2202c ut) \u2192 \u00acut ut})\n-- isUpTran (\u2202\u2192 ind) (\u2202c ltr)  with (isUpTran (ind \u2190\u2202) ltr)\n-- isUpTran (\u2202\u2192 ind) (\u2202c ltr) | yes ut = yes (\u2202\u2192\u2202c ut)\n-- isUpTran (\u2202\u2192 ind) (\u2202c ltr) | no \u00acut = no (\u03bb {(\u2202\u2192\u2202c ut) \u2192 \u00acut ut})\n-- isUpTran \u2193 (\u2228c ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2228) (\u2228c ltr) with (isUpTran (\u2228\u2192 ind) ltr)\n-- isUpTran (ind \u2190\u2228) (\u2228c ltr) | yes p = yes (\u2190\u2228\u2228c p)\n-- isUpTran (ind \u2190\u2228) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u2228c p) \u2192 \u00acp p}) \n-- isUpTran (\u2228\u2192 ind) (\u2228c ltr) with (isUpTran (ind \u2190\u2228) ltr)\n-- isUpTran (\u2228\u2192 ind) (\u2228c ltr) | yes p = yes (\u2228\u2192\u2228c p)\n-- isUpTran (\u2228\u2192 ind) (\u2228c ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228c ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2227c ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2227) (\u2227c ltr) with (isUpTran (\u2227\u2192 ind) ltr)\n-- isUpTran (ind \u2190\u2227) (\u2227c ltr) | yes p = yes (\u2190\u2227\u2227c p)\n-- isUpTran (ind \u2190\u2227) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u2227c ut) \u2192 \u00acp ut}) \n-- isUpTran (\u2227\u2192 ind) (\u2227c ltr) with (isUpTran (ind \u2190\u2227) ltr)\n-- isUpTran (\u2227\u2192 ind) (\u2227c ltr) | yes p = yes (\u2227\u2192\u2227c p)\n-- isUpTran (\u2227\u2192 ind) (\u2227c ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227c ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (ind \u2190\u2227) ltr)\n-- isUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2190\u2227]\u2190\u2227\u2227\u2227d p)\n-- isUpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \n-- isUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (ind \u2190\u2227)) ltr)\n-- isUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192]\u2190\u2227\u2227\u2227d p)\n-- isUpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192]\u2190\u2227\u2227\u2227d ut) \u2192 \u00acp ut}) \n-- isUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 (\u2227\u2192 ind)) ltr)\n-- isUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) | yes p = yes (\u2227\u2192\u2227\u2227d p)\n-- isUpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) with (isUpTran ((ind \u2190\u2227) \u2190\u2227) ltr)\n-- isUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2190\u2227\u00ac\u2227\u2227d p)\n-- isUpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) = no (\u03bb ())\n-- isUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) with (isUpTran ((\u2227\u2192 ind) \u2190\u2227) ltr)\n-- isUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d p)\n-- isUpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) with (isUpTran (\u2227\u2192 ind) ltr)\n-- isUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | yes p = yes (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d p)\n-- isUpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) | no \u00acp = no (\u03bb {(\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (ind \u2190\u2228) ltr)\n-- isUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2190\u2228]\u2190\u2228\u2228\u2228d p)\n-- isUpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \n-- isUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (ind \u2190\u2228)) ltr)\n-- isUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192]\u2190\u2228\u2228\u2228d p)\n-- isUpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192]\u2190\u2228\u2228\u2228d ut) \u2192 \u00acp ut}) \n-- isUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 (\u2228\u2192 ind)) ltr)\n-- isUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) | yes p = yes (\u2228\u2192\u2228\u2228d p)\n-- isUpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) with (isUpTran ((ind \u2190\u2228) \u2190\u2228) ltr)\n-- isUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2190\u2228\u00ac\u2228\u2228d p)\n-- isUpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) = no (\u03bb ())\n-- isUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) with (isUpTran ((\u2228\u2192 ind) \u2190\u2228) ltr)\n-- isUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d p)\n-- isUpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) with (isUpTran (\u2228\u2192 ind) ltr)\n-- isUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | yes p = yes (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d p)\n-- isUpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) | no \u00acp = no (\u03bb {(\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (ind \u2190\u2202) ltr)\n-- isUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2190\u2202]\u2190\u2202\u2202\u2202d p)\n-- isUpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (ind \u2190\u2202)) ltr)\n-- isUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192]\u2190\u2202\u2202\u2202d p)\n-- isUpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192]\u2190\u2202\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 (\u2202\u2192 ind)) ltr)\n-- isUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) | yes p = yes (\u2202\u2192\u2202\u2202d p)\n-- isUpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran \u2193 (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) with (isUpTran ((ind \u2190\u2202) \u2190\u2202) ltr)\n-- isUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2190\u2202\u00ac\u2202\u2202d p)\n-- isUpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) = no (\u03bb ())\n-- isUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) with (isUpTran ((\u2202\u2192 ind) \u2190\u2202) ltr)\n-- isUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d p)\n-- isUpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n-- isUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) with (isUpTran (\u2202\u2192 ind) ltr)\n-- isUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | yes p = yes (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d p)\n-- isUpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) | no \u00acp = no (\u03bb {(\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) \u2192 \u00acp ut})\n\n\n-- indLow\u21d2UpTran : \u2200 {i u rll ll n dt df} \u2192 (ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n--                 \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 UpTran ind ltr\n-- indLow\u21d2UpTran \u2193 I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2227) I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2227) (\u2227c ltr) = \u2190\u2227\u2227c r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 ind) ltr\n-- indLow\u21d2UpTran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) = \u2190\u2227]\u2190\u2227\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2227) ltr\n-- indLow\u21d2UpTran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) = \u2227\u2192]\u2190\u2227\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 (ind \u2190\u2227)) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) = \u2190\u2227\u00ac\u2227\u2227d r where\n--   r = indLow\u21d2UpTran ((ind \u2190\u2227) \u2190\u2227) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 ind) I = indI\n-- indLow\u21d2UpTran (\u2227\u2192 ind) (\u2227c ltr) = \u2227\u2192\u2227c r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2227) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 ind) (\u2227\u2227d ltr) = \u2227\u2192\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 (\u2227\u2192 ind)) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) = \u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d r where\n--   r = indLow\u21d2UpTran ((\u2227\u2192 ind) \u2190\u2227) ltr\n-- indLow\u21d2UpTran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) = \u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d r where\n--   r = indLow\u21d2UpTran (\u2227\u2192 ind) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2228) I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2228) (\u2228c ltr) = \u2190\u2228\u2228c r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 ind) ltr\n-- indLow\u21d2UpTran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) = \u2190\u2228]\u2190\u2228\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2228) ltr\n-- indLow\u21d2UpTran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) = \u2228\u2192]\u2190\u2228\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 (ind \u2190\u2228)) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) = \u2190\u2228\u00ac\u2228\u2228d r where\n--   r = indLow\u21d2UpTran ((ind \u2190\u2228) \u2190\u2228) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 ind) I = indI\n-- indLow\u21d2UpTran (\u2228\u2192 ind) (\u2228c ltr) = \u2228\u2192\u2228c r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2228) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 ind) (\u2228\u2228d ltr) = \u2228\u2192\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 (\u2228\u2192 ind)) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) = \u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d r where\n--   r = indLow\u21d2UpTran ((\u2228\u2192 ind) \u2190\u2228) ltr\n-- indLow\u21d2UpTran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) = \u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d r where\n--   r = indLow\u21d2UpTran (\u2228\u2192 ind) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2202) I = indI\n-- indLow\u21d2UpTran (ind \u2190\u2202) (\u2202c ltr) = \u2190\u2202\u2202c r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 ind) ltr\n-- indLow\u21d2UpTran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) = \u2190\u2202]\u2190\u2202\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2202) ltr\n-- indLow\u21d2UpTran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) = \u2202\u2192]\u2190\u2202\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 (ind \u2190\u2202)) ltr\n-- indLow\u21d2UpTran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) = \u2190\u2202\u00ac\u2202\u2202d r where\n--   r = indLow\u21d2UpTran ((ind \u2190\u2202) \u2190\u2202) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 ind) I = indI\n-- indLow\u21d2UpTran (\u2202\u2192 ind) (\u2202c ltr) = \u2202\u2192\u2202c r where\n--   r = indLow\u21d2UpTran (ind \u2190\u2202) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 ind) (\u2202\u2202d ltr) = \u2202\u2192\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 (\u2202\u2192 ind)) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) = \u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d r where\n--   r = indLow\u21d2UpTran ((\u2202\u2192 ind) \u2190\u2202) ltr\n-- indLow\u21d2UpTran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) = \u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d r where\n--   r = indLow\u21d2UpTran (\u2202\u2192 ind) ltr\n\n\n\n-- tran : \u2200 {i u ll pll rll} \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 UpTran ind ltr\n--        \u2192 IndexLL pll rll\n-- tran ind I indI = ind \n-- tran \u2193 (\u2202c ltr) () \n-- tran (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c ut) = tran (\u2202\u2192 ind) ltr ut\n-- tran (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c ut) =  tran (ind \u2190\u2202) ltr ut\n-- tran \u2193 (\u2228c ltr) () \n-- tran (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c ut) = tran (\u2228\u2192 ind) ltr ut\n-- tran (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c ut) = tran (ind \u2190\u2228) ltr ut\n-- tran \u2193 (\u2227c ltr) () \n-- tran (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c ut) = tran (\u2227\u2192 ind) ltr ut\n-- tran (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c ut) = tran (ind \u2190\u2227) ltr ut\n-- tran \u2193 (\u2227\u2227d ltr) () \n-- tran (\u2193 \u2190\u2227) (\u2227\u2227d ltr) () \n-- tran ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d ut) = tran (ind \u2190\u2227) ltr ut\n-- tran ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d ut) = tran (\u2227\u2192 (ind \u2190\u2227)) ltr ut\n-- tran (\u2227\u2192 ind) (\u2227\u2227d ltr) (\u2227\u2192\u2227\u2227d ut) = tran (\u2227\u2192 (\u2227\u2192 ind)) ltr ut\n-- tran \u2193 (\u00ac\u2227\u2227d ltr) () \n-- tran (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((ind \u2190\u2227) \u2190\u2227) ltr ut\n-- tran (\u2227\u2192 \u2193) (\u00ac\u2227\u2227d ltr) () \n-- tran (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d ut) = tran ((\u2227\u2192 ind) \u2190\u2227) ltr ut\n-- tran (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d ut) = tran (\u2227\u2192 ind) ltr ut\n-- tran \u2193 (\u2228\u2228d ltr) () \n-- tran (\u2193 \u2190\u2228) (\u2228\u2228d ltr) () \n-- tran ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d ut) = tran (ind \u2190\u2228) ltr ut\n-- tran ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d ut) = tran (\u2228\u2192 (ind \u2190\u2228)) ltr ut\n-- tran (\u2228\u2192 ind) (\u2228\u2228d ltr) (\u2228\u2192\u2228\u2228d ut) = tran (\u2228\u2192 (\u2228\u2192 ind)) ltr ut\n-- tran \u2193 (\u00ac\u2228\u2228d ltr) () \n-- tran (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((ind \u2190\u2228) \u2190\u2228) ltr ut\n-- tran (\u2228\u2192 \u2193) (\u00ac\u2228\u2228d ltr) () \n-- tran (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d ut) = tran ((\u2228\u2192 ind) \u2190\u2228) ltr ut\n-- tran (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d ut) = tran (\u2228\u2192 ind) ltr ut\n-- tran \u2193 (\u2202\u2202d ltr) () \n-- tran (\u2193 \u2190\u2202) (\u2202\u2202d ltr) () \n-- tran ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d ut) = tran (ind \u2190\u2202) ltr ut\n-- tran ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d ut) = tran (\u2202\u2192 (ind \u2190\u2202)) ltr ut\n-- tran (\u2202\u2192 ind) (\u2202\u2202d ltr) (\u2202\u2192\u2202\u2202d ut) = tran (\u2202\u2192 (\u2202\u2192 ind)) ltr ut\n-- tran \u2193 (\u00ac\u2202\u2202d ltr) () \n-- tran (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((ind \u2190\u2202) \u2190\u2202) ltr ut\n-- tran (\u2202\u2192 \u2193) (\u00ac\u2202\u2202d ltr) () \n-- tran (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d ut) = tran ((\u2202\u2192 ind) \u2190\u2202) ltr ut\n-- tran (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d ut) = tran (\u2202\u2192 ind) ltr ut\n\n\n-- tr-ltr-morph : \u2200 {i u ll pll orll} \u2192 \u2200 frll \u2192 (ind : IndexLL pll ll) \u2192 (ltr : LLTr {i} {u} orll ll) \u2192 (rvT : UpTran ind ltr) \u2192 LLTr (replLL orll (tran ind ltr rvT) frll) (replLL ll ind frll)\n-- tr-ltr-morph frll \u2193 .I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2227) I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2227) (\u2227c ltr) (\u2190\u2227\u2227c rvT) with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n-- ... | r = \u2227c r\n-- tr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) (\u2227\u2227d ltr) (\u2190\u2227]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n-- ... | r = \u2227\u2227d r\n-- tr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) (\u2227\u2227d ltr) (\u2227\u2192]\u2190\u2227\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) ltr rvT\n-- ... | r = \u2227\u2227d r\n-- tr-ltr-morph frll (ind \u2190\u2227) (\u00ac\u2227\u2227d ltr) (\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((ind \u2190\u2227) \u2190\u2227) ltr rvT\n-- ... | r = \u00ac\u2227\u2227d r\n-- tr-ltr-morph frll (\u2227\u2192 ind) I indI = I\n-- tr-ltr-morph frll (\u2227\u2192 ind) (\u2227c ltr) (\u2227\u2192\u2227c rvT) with tr-ltr-morph frll (ind \u2190\u2227) ltr rvT\n-- ... | r = \u2227c r\n-- tr-ltr-morph frll (\u2227\u2192 ind) (\u2227\u2227d ltr) (\u2227\u2192\u2227\u2227d rvT) with tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) ltr rvT\n-- ... | r = \u2227\u2227d r\n-- tr-ltr-morph frll (\u2227\u2192 (ind \u2190\u2227)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2190\u2227\u00ac\u2227\u2227d rvT) with tr-ltr-morph frll ((\u2227\u2192 ind) \u2190\u2227) ltr rvT\n-- ... | r = \u00ac\u2227\u2227d r\n-- tr-ltr-morph frll (\u2227\u2192 (\u2227\u2192 ind)) (\u00ac\u2227\u2227d ltr) (\u2227\u2192[\u2227\u2192\u00ac\u2227\u2227d rvT)  with tr-ltr-morph frll (\u2227\u2192 ind) ltr rvT\n-- ... | r = \u00ac\u2227\u2227d r\n-- tr-ltr-morph frll (ind \u2190\u2228) I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2228) (\u2228c ltr) (\u2190\u2228\u2228c rvT) with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n-- ... | r = \u2228c r\n-- tr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) (\u2228\u2228d ltr) (\u2190\u2228]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n-- ... | r = \u2228\u2228d r\n-- tr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) (\u2228\u2228d ltr) (\u2228\u2192]\u2190\u2228\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) ltr rvT\n-- ... | r = \u2228\u2228d r\n-- tr-ltr-morph frll (ind \u2190\u2228) (\u00ac\u2228\u2228d ltr) (\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((ind \u2190\u2228) \u2190\u2228) ltr rvT\n-- ... | r = \u00ac\u2228\u2228d r\n-- tr-ltr-morph frll (\u2228\u2192 ind) I indI = I\n-- tr-ltr-morph frll (\u2228\u2192 ind) (\u2228c ltr) (\u2228\u2192\u2228c rvT) with tr-ltr-morph frll (ind \u2190\u2228) ltr rvT\n-- ... | r = \u2228c r\n-- tr-ltr-morph frll (\u2228\u2192 ind) (\u2228\u2228d ltr) (\u2228\u2192\u2228\u2228d rvT) with tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) ltr rvT\n-- ... | r = \u2228\u2228d r\n-- tr-ltr-morph frll (\u2228\u2192 (ind \u2190\u2228)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2190\u2228\u00ac\u2228\u2228d rvT) with tr-ltr-morph frll ((\u2228\u2192 ind) \u2190\u2228) ltr rvT\n-- ... | r = \u00ac\u2228\u2228d r\n-- tr-ltr-morph frll (\u2228\u2192 (\u2228\u2192 ind)) (\u00ac\u2228\u2228d ltr) (\u2228\u2192[\u2228\u2192\u00ac\u2228\u2228d rvT)  with tr-ltr-morph frll (\u2228\u2192 ind) ltr rvT\n-- ... | r = \u00ac\u2228\u2228d r\n-- tr-ltr-morph frll (ind \u2190\u2202) I indI = I\n-- tr-ltr-morph frll (ind \u2190\u2202) (\u2202c ltr) (\u2190\u2202\u2202c rvT) with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n-- ... | r = \u2202c r\n-- tr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) (\u2202\u2202d ltr) (\u2190\u2202]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n-- ... | r = \u2202\u2202d r\n-- tr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) (\u2202\u2202d ltr) (\u2202\u2192]\u2190\u2202\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) ltr rvT\n-- ... | r = \u2202\u2202d r\n-- tr-ltr-morph frll (ind \u2190\u2202) (\u00ac\u2202\u2202d ltr) (\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((ind \u2190\u2202) \u2190\u2202) ltr rvT\n-- ... | r = \u00ac\u2202\u2202d r\n-- tr-ltr-morph frll (\u2202\u2192 ind) I indI = I\n-- tr-ltr-morph frll (\u2202\u2192 ind) (\u2202c ltr) (\u2202\u2192\u2202c rvT) with tr-ltr-morph frll (ind \u2190\u2202) ltr rvT\n-- ... | r = \u2202c r\n-- tr-ltr-morph frll (\u2202\u2192 ind) (\u2202\u2202d ltr) (\u2202\u2192\u2202\u2202d rvT) with tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) ltr rvT\n-- ... | r = \u2202\u2202d r\n-- tr-ltr-morph frll (\u2202\u2192 (ind \u2190\u2202)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2190\u2202\u00ac\u2202\u2202d rvT) with tr-ltr-morph frll ((\u2202\u2192 ind) \u2190\u2202) ltr rvT\n-- ... | r = \u00ac\u2202\u2202d r\n-- tr-ltr-morph frll (\u2202\u2192 (\u2202\u2192 ind)) (\u00ac\u2202\u2202d ltr) (\u2202\u2192[\u2202\u2192\u00ac\u2202\u2202d rvT)  with tr-ltr-morph frll (\u2202\u2192 ind) ltr rvT\n-- ... | r = \u00ac\u2202\u2202d r\n\n\n\n-- drepl=>repl+ : \u2200{i u pll ll cll frll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--                \u2192 (replLL ll ind (replLL pll lind frll)) \u2261 replLL ll (ind +\u1d62 lind) frll\n-- drepl=>repl+ \u2193 lind = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (ind \u2190\u2227) lind\n--              with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (ind \u2190\u2227) lind\n--              | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2227 ri} {frll = frll} (\u2227\u2192 ind) lind\n--              with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2227 ri} {_} {frll} (\u2227\u2192 ind) lind\n--              | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (ind \u2190\u2228) lind\n--              with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (ind \u2190\u2228) lind\n--              | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2228 ri} {frll = frll} (\u2228\u2192 ind) lind\n--              with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2228 ri} {_} {frll} (\u2228\u2192 ind) lind\n--              | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (ind \u2190\u2202) lind\n--              with (replLL li ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (ind \u2190\u2202) lind\n--              | .(replLL li (ind +\u1d62 lind) frll) | refl = refl\n-- drepl=>repl+ {pll = pll} {ll = li \u2202 ri} {frll = frll} (\u2202\u2192 ind) lind\n--              with (replLL ri ind (replLL pll lind frll)) | (drepl=>repl+ {frll = frll} ind lind)\n-- drepl=>repl+ {_} {_} {pll} {li \u2202 ri} {_} {frll} (\u2202\u2192 ind) lind\n--              | .(replLL ri (ind +\u1d62 lind) frll) | refl = refl\n\n\n\n-- repl-r=>l : \u2200{i u pll ll cll frll} \u2192 \u2200 ell \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (x : IndexLL cll (replLL ll ind ell)) \u2192 let uind = a\u2264\u1d62b-morph ind ind ell (\u2264\u1d62-reflexive ind)\n--        in (ltuindx : uind \u2264\u1d62 x)\n--        \u2192 (replLL ll ind (replLL ell (ind-rpl\u2193 ind ((x -\u1d62 uind) ltuindx)) frll) \u2261 replLL (replLL ll ind ell) x frll)\n-- repl-r=>l ell \u2193 x \u2264\u1d62\u2193 = refl\n-- repl-r=>l {ll = li \u2227 ri} ell (ind \u2190\u2227) (x \u2190\u2227) (\u2264\u1d62\u2190\u2227 ltuindx) = cong (\u03bb x \u2192 x \u2227 ri) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2227 ri} ell (\u2227\u2192 ind) (\u2227\u2192 x) (\u2264\u1d62\u2227\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2227 x) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2228 ri} ell (ind \u2190\u2228) (x \u2190\u2228) (\u2264\u1d62\u2190\u2228 ltuindx) = cong (\u03bb x \u2192 x \u2228 ri) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2228 ri} ell (\u2228\u2192 ind) (\u2228\u2192 x) (\u2264\u1d62\u2228\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2228 x) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2202 ri} ell (ind \u2190\u2202) (x \u2190\u2202) (\u2264\u1d62\u2190\u2202 ltuindx) = cong (\u03bb x \u2192 x \u2202 ri) (repl-r=>l ell ind x ltuindx)\n-- repl-r=>l {ll = li \u2202 ri} ell (\u2202\u2192 ind) (\u2202\u2192 x) (\u2264\u1d62\u2202\u2192 ltuindx) = cong (\u03bb x \u2192 li \u2202 x) (repl-r=>l ell ind x ltuindx)\n\n\n\n-- -- Deprecated\n-- updInd : \u2200{i u rll ll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} rll ll)\n--          \u2192 IndexLL {i} {u} nrll (replLL ll ind nrll)\n-- updInd nrll \u2193 = \u2193\n-- updInd nrll (ind \u2190\u2227) = (updInd nrll ind) \u2190\u2227\n-- updInd nrll (\u2227\u2192 ind) = \u2227\u2192 (updInd nrll ind)\n-- updInd nrll (ind \u2190\u2228) = (updInd nrll ind) \u2190\u2228\n-- updInd nrll (\u2228\u2192 ind) = \u2228\u2192 (updInd nrll ind)\n-- updInd nrll (ind \u2190\u2202) = (updInd nrll ind) \u2190\u2202\n-- updInd nrll (\u2202\u2192 ind) = \u2202\u2192 (updInd nrll ind)\n\n-- -- Deprecated\n-- -- Maybe instead of this function use a\u2264\u1d62b-morph\n-- updIndGen : \u2200{i u pll ll cll} \u2192 \u2200 nrll \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--             \u2192 IndexLL {i} {u} (replLL pll lind nrll) (replLL ll (ind +\u1d62 lind) nrll)\n-- updIndGen nrll \u2193 lind = \u2193\n-- updIndGen nrll (ind \u2190\u2227) lind = (updIndGen nrll ind lind) \u2190\u2227\n-- updIndGen nrll (\u2227\u2192 ind) lind = \u2227\u2192 (updIndGen nrll ind lind)\n-- updIndGen nrll (ind \u2190\u2228) lind = (updIndGen nrll ind lind) \u2190\u2228\n-- updIndGen nrll (\u2228\u2192 ind) lind = \u2228\u2192 (updIndGen nrll ind lind)\n-- updIndGen nrll (ind \u2190\u2202) lind = (updIndGen nrll ind lind) \u2190\u2202\n-- updIndGen nrll (\u2202\u2192 ind) lind = \u2202\u2192 (updIndGen nrll ind lind)\n\n\n\n-- -- TODO This negation was writen so as to return nothing if \u00ac (b \u2264\u1d62 a)\n-- _-\u2098\u1d62_ : \u2200 {i u pll cll ll} \u2192 IndexLL {i} {u} cll ll \u2192 IndexLL pll ll \u2192 Maybe $ IndexLL cll pll\n-- a -\u2098\u1d62 \u2193 = just a\n-- \u2193 -\u2098\u1d62 (b \u2190\u2227) = nothing\n-- (a \u2190\u2227) -\u2098\u1d62 (b \u2190\u2227) = a -\u2098\u1d62 b\n-- (\u2227\u2192 a) -\u2098\u1d62 (b \u2190\u2227) = nothing\n-- \u2193 -\u2098\u1d62 (\u2227\u2192 b) = nothing\n-- (a \u2190\u2227) -\u2098\u1d62 (\u2227\u2192 b) = nothing\n-- (\u2227\u2192 a) -\u2098\u1d62 (\u2227\u2192 b) = a -\u2098\u1d62 b\n-- \u2193 -\u2098\u1d62 (b \u2190\u2228) = nothing\n-- (a \u2190\u2228) -\u2098\u1d62 (b \u2190\u2228) = a -\u2098\u1d62 b\n-- (\u2228\u2192 a) -\u2098\u1d62 (b \u2190\u2228) = nothing\n-- \u2193 -\u2098\u1d62 (\u2228\u2192 b) = nothing\n-- (a \u2190\u2228) -\u2098\u1d62 (\u2228\u2192 b) = nothing\n-- (\u2228\u2192 a) -\u2098\u1d62 (\u2228\u2192 b) = a -\u2098\u1d62 b\n-- \u2193 -\u2098\u1d62 (b \u2190\u2202) = nothing\n-- (a \u2190\u2202) -\u2098\u1d62 (b \u2190\u2202) = a -\u2098\u1d62 b\n-- (\u2202\u2192 a) -\u2098\u1d62 (b \u2190\u2202) = nothing\n-- \u2193 -\u2098\u1d62 (\u2202\u2192 b) = nothing\n-- (a \u2190\u2202) -\u2098\u1d62 (\u2202\u2192 b) = nothing\n-- (\u2202\u2192 a) -\u2098\u1d62 (\u2202\u2192 b) = a -\u2098\u1d62 b\n\n-- -- Deprecated\n-- b-s\u2261j\u21d2s\u2264\u1d62b : \u2200 {i u pll cll ll} \u2192 (bind : IndexLL {i} {u} cll ll) \u2192 (sind : IndexLL pll ll)\n--              \u2192  \u03a3 (IndexLL {i} {u} cll pll) (\u03bb x \u2192 (bind -\u2098\u1d62 sind) \u2261 just x) \u2192 sind \u2264\u1d62 bind\n-- b-s\u2261j\u21d2s\u2264\u1d62b bind \u2193 (rs , x) = \u2264\u1d62\u2193\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2227) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (sind \u2190\u2227) (rs , x) = \u2264\u1d62\u2190\u2227 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (sind \u2190\u2227) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2227\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2227) (\u2227\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2227\u2192 bind) (\u2227\u2192 sind) (rs , x) = \u2264\u1d62\u2227\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2228) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (sind \u2190\u2228) (rs , x) = \u2264\u1d62\u2190\u2228 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (sind \u2190\u2228) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2228\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2228) (\u2228\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2228\u2192 bind) (\u2228\u2192 sind) (rs , x) =  \u2264\u1d62\u2228\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (sind \u2190\u2202) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (sind \u2190\u2202) (rs , x) = \u2264\u1d62\u2190\u2202 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (sind \u2190\u2202) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b \u2193 (\u2202\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (bind \u2190\u2202) (\u2202\u2192 sind) (rs , ())\n-- b-s\u2261j\u21d2s\u2264\u1d62b (\u2202\u2192 bind) (\u2202\u2192 sind) (rs , x) = \u2264\u1d62\u2202\u2192 $ b-s\u2261j\u21d2s\u2264\u1d62b bind sind (rs , x)\n\n-- -- Deprecated\n-- revUpdInd : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll (replLL ll b ell))\n--             \u2192 a -\u2098\u1d62 (updInd ell b) \u2261 nothing \u2192 (updInd ell b) -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll ll\n-- revUpdInd \u2193 a () b-a\n-- revUpdInd (b \u2190\u2227) \u2193 refl ()\n-- revUpdInd (b \u2190\u2227) (a \u2190\u2227) a-b b-a = revUpdInd b a a-b b-a \u2190\u2227\n-- revUpdInd (b \u2190\u2227) (\u2227\u2192 a) a-b b-a = \u2227\u2192 a\n-- revUpdInd (\u2227\u2192 b) \u2193 refl ()\n-- revUpdInd (\u2227\u2192 b) (a \u2190\u2227) a-b b-a = a \u2190\u2227\n-- revUpdInd (\u2227\u2192 b) (\u2227\u2192 a) a-b b-a = \u2227\u2192 revUpdInd b a a-b b-a\n-- revUpdInd (b \u2190\u2228) \u2193 refl ()\n-- revUpdInd (b \u2190\u2228) (a \u2190\u2228) a-b b-a = revUpdInd b a a-b b-a \u2190\u2228\n-- revUpdInd (b \u2190\u2228) (\u2228\u2192 a) a-b b-a = \u2228\u2192 a\n-- revUpdInd (\u2228\u2192 b) \u2193 refl ()\n-- revUpdInd (\u2228\u2192 b) (a \u2190\u2228) a-b b-a = a \u2190\u2228\n-- revUpdInd (\u2228\u2192 b) (\u2228\u2192 a) a-b b-a = \u2228\u2192 revUpdInd b a a-b b-a\n-- revUpdInd (b \u2190\u2202) \u2193 refl ()\n-- revUpdInd (b \u2190\u2202) (a \u2190\u2202) a-b b-a = revUpdInd b a a-b b-a \u2190\u2202\n-- revUpdInd (b \u2190\u2202) (\u2202\u2192 a) a-b b-a = \u2202\u2192 a\n-- revUpdInd (\u2202\u2192 b) \u2193 refl ()\n-- revUpdInd (\u2202\u2192 b) (a \u2190\u2202) a-b b-a = a \u2190\u2202\n-- revUpdInd (\u2202\u2192 b) (\u2202\u2192 a) a-b b-a = \u2202\u2192 revUpdInd b a a-b b-a\n\n\n-- -- Deprecated\n-- updIndPart : \u2200{i u ll cll ell pll} \u2192 (b : IndexLL pll ll) \u2192 (a : IndexLL {i} {u} cll ll)\n--              \u2192 a -\u2098\u1d62 b \u2261 nothing \u2192 b -\u2098\u1d62 a \u2261 nothing \u2192 IndexLL cll (replLL ll b ell)\n-- updIndPart \u2193 a () eq2\n-- updIndPart (b \u2190\u2227) \u2193 refl ()\n-- updIndPart (b \u2190\u2227) (a \u2190\u2227) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2227\n-- updIndPart (b \u2190\u2227) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 a\n-- updIndPart (\u2227\u2192 b) \u2193 refl ()\n-- updIndPart (\u2227\u2192 b) (a \u2190\u2227) eq1 eq2 = a \u2190\u2227\n-- updIndPart (\u2227\u2192 b) (\u2227\u2192 a) eq1 eq2 = \u2227\u2192 updIndPart b a eq1 eq2\n-- updIndPart (b \u2190\u2228) \u2193 refl ()\n-- updIndPart (b \u2190\u2228) (a \u2190\u2228) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2228\n-- updIndPart (b \u2190\u2228) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 a\n-- updIndPart (\u2228\u2192 b) \u2193 refl ()\n-- updIndPart (\u2228\u2192 b) (a \u2190\u2228) eq1 eq2 = a \u2190\u2228\n-- updIndPart (\u2228\u2192 b) (\u2228\u2192 a) eq1 eq2 = \u2228\u2192 updIndPart b a eq1 eq2\n-- updIndPart (b \u2190\u2202) \u2193 refl ()\n-- updIndPart (b \u2190\u2202) (a \u2190\u2202) eq1 eq2 = updIndPart b a eq1 eq2 \u2190\u2202\n-- updIndPart (b \u2190\u2202) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 a\n-- updIndPart (\u2202\u2192 b) \u2193 refl ()\n-- updIndPart (\u2202\u2192 b) (a \u2190\u2202) eq1 eq2 = a \u2190\u2202\n-- updIndPart (\u2202\u2192 b) (\u2202\u2192 a) eq1 eq2 = \u2202\u2192 updIndPart b a eq1 eq2\n\n-- -- Deprecated\n-- rev\u21d2rs-i\u2261n : \u2200{i u ll cll ell pll} \u2192 (ind : IndexLL pll ll)\n--              \u2192 (lind : IndexLL {i} {u} cll (replLL ll ind ell))\n--              \u2192 (eq\u2081 : (lind -\u2098\u1d62 (updInd ell ind) \u2261 nothing))\n--              \u2192 (eq\u2082 : ((updInd ell ind) -\u2098\u1d62 lind \u2261 nothing))\n--              \u2192 let rs = revUpdInd ind lind eq\u2081 eq\u2082 in\n--                  ((rs -\u2098\u1d62 ind) \u2261 nothing) \u00d7 ((ind -\u2098\u1d62 rs) \u2261 nothing)\n-- rev\u21d2rs-i\u2261n \u2193 lind () eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2227) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2227) (lind \u2190\u2227) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2227) (\u2227\u2192 lind) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2227\u2192 ind) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (\u2227\u2192 ind) (lind \u2190\u2227) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2227\u2192 ind) (\u2227\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2228) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2228) (lind \u2190\u2228) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2228) (\u2228\u2192 lind) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2228\u2192 ind) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (\u2228\u2192 ind) (lind \u2190\u2228) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2228\u2192 ind) (\u2228\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2202) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2202) (lind \u2190\u2202) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n-- rev\u21d2rs-i\u2261n (ind \u2190\u2202) (\u2202\u2192 lind) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2202\u2192 ind) \u2193 eq1 ()\n-- rev\u21d2rs-i\u2261n (\u2202\u2192 ind) (lind \u2190\u2202) eq1 eq2 = refl , refl\n-- rev\u21d2rs-i\u2261n (\u2202\u2192 ind) (\u2202\u2192 lind) eq1 eq2 = rev\u21d2rs-i\u2261n ind lind eq1 eq2\n\n\n\n\n-- -- Deprecated\n-- remRepl : \u2200{i u ll frll ell pll cll} \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL cll pll)\n--           \u2192 replLL (replLL ll (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell \u2261 replLL ll ind ell\n-- remRepl \u2193 lind = refl\n-- remRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (ind \u2190\u2227) lind\n--         with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2227 ri} {frll} {ell} (ind \u2190\u2227) lind | .(replLL li ind ell) | refl = refl\n-- remRepl {ll = li \u2227 ri} {frll = frll} {ell = ell} (\u2227\u2192 ind) lind\n--         with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2227 ri} {frll} {ell} (\u2227\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n-- remRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (ind \u2190\u2228) lind\n--         with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2228 ri} {frll} {ell} (ind \u2190\u2228) lind | .(replLL li ind ell) | refl = refl\n-- remRepl {ll = li \u2228 ri} {frll = frll} {ell = ell} (\u2228\u2192 ind) lind\n--         with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2228 ri} {frll} {ell} (\u2228\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n-- remRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (ind \u2190\u2202) lind\n--         with (replLL (replLL li (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2202 ri} {frll} {ell} (ind \u2190\u2202) lind | .(replLL li ind ell) | refl = refl\n-- remRepl {ll = li \u2202 ri} {frll = frll} {ell = ell} (\u2202\u2192 ind) lind\n--         with (replLL (replLL ri (ind +\u1d62 lind) frll) (updIndGen frll ind lind) ell)\n--         | (remRepl {frll = frll} {ell = ell} ind lind)\n-- remRepl {_} {_} {li \u2202 ri} {frll} {ell} (\u2202\u2192 ind) lind | .(replLL ri ind ell) | refl = refl\n\n-- -- Deprecated\n-- replLL-inv : \u2200{i u ll ell pll} \u2192 (ind : IndexLL {i} {u} pll ll)\n--              \u2192 replLL (replLL ll ind ell) (updInd ell ind) pll \u2261 ll\n-- replLL-inv \u2193 = refl\n-- replLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (ind \u2190\u2227)\n--            with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (ind \u2190\u2227) | .li | refl = refl\n-- replLL-inv {ll = li \u2227 ri} {ell = ell} {pll = pll} (\u2227\u2192 ind)\n--            with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2227 ri} {ell} {pll} (\u2227\u2192 ind) | .ri | refl = refl\n-- replLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (ind \u2190\u2228)\n--            with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (ind \u2190\u2228) | .li | refl = refl\n-- replLL-inv {ll = li \u2228 ri} {ell = ell} {pll = pll} (\u2228\u2192 ind)\n--            with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2228 ri} {ell} {pll} (\u2228\u2192 ind) | .ri | refl = refl\n-- replLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (ind \u2190\u2202)\n--            with (replLL (replLL li ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (ind \u2190\u2202) | .li | refl = refl\n-- replLL-inv {ll = li \u2202 ri} {ell = ell} {pll = pll} (\u2202\u2192 ind)\n--            with (replLL (replLL ri ind ell) (updInd ell ind) pll) | (replLL-inv {ell = ell} ind)\n-- replLL-inv {_} {_} {li \u2202 ri} {ell} {pll} (\u2202\u2192 ind) | .ri | refl = refl\n\n\n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"4794eb75e9ea83b6fa7129d87a8098c535227abb","subject":"bin: sorting WIP","message":"bin: sorting WIP\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_)\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n\n{-\nmodule Sorting {a} {A : Set a} (sort\u1d2c : A \u00d7 A \u2192 A \u00d7 A) where\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    = case sort\u1d2c (x\u2080 , x\u2081) of (\u03bb { (y\u2080 , y\u2081) \u2192 fork (leaf y\u2080) (leaf y\u2081) })\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\nswap-\u00d7 : \u2200 {n} \u2192 Bits n \u00d7 Bits n \u2192 Bits n \u00d7 Bits n\nswap-\u00d7 (x , y) = ?\n\nmodule BitsSorting m where\n\n    module S = Sorting (swap-\u00d7 {m})\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    lem : \u2200 {n} (t : Tree (Bits n) n) \u2192 toFun (sort n t) \u2257 id\n    lem = ?\n-}\n","old_contents":"module prefect-bintree where\n\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"23195232ac9f45758a208ab48715260ca5c37781","subject":"FunUniverse\/Rewiring\/Linear: add mapAccum","message":"FunUniverse\/Rewiring\/Linear: add mapAccum\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Rewiring\/Linear.agda","new_file":"FunUniverse\/Rewiring\/Linear.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Level.NP\nopen import Type\nopen import FunUniverse.Types\nopen import Data.One using (\ud835\udfd9)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport FunUniverse.Category\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\n\nmodule FunUniverse.Rewiring.Linear where\n\nrecord LinRewiring {t} {T : \u2605_ t} (funU : FunUniverse T) : \u2605_ t where\n  constructor mk\n  open FunUniverse funU\n  open FunUniverse.Category _`\u2192_\n\n  field\n    cat : Category\n  open Category cat\n\n  field\n    -- Products (group 2 primitive functions or derived from group 1)\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\ud835\udfd9 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\ud835\udfd9 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    tt\u2192[]  : \u2200 {A} \u2192 `\ud835\udfd9 `\u2192 `Vec A 0\n    []\u2192tt  : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\ud835\udfd9\n    <\u2237>    : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open CompositionNotations _\u2218_ public\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  infixr 3 _***_\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  <id,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\ud835\udfd9\n  <id,tt> = <tt,id> \u204f swap\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\ud835\udfd9 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\ud835\udfd9 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  fst<,tt> : \u2200 {A} \u2192 A `\u00d7 `\ud835\udfd9 `\u2192 A\n  fst<,tt> = swap \u204f snd<tt,>\n\n  fst<,[]> : \u2200 {A B} \u2192 A `\u00d7 `Vec B 0 `\u2192 A\n  fst<,[]> = second []\u2192tt \u204f fst<,tt>\n\n  snd<[],> : \u2200 {A B} \u2192 `Vec A 0 `\u00d7 B `\u2192 B\n  snd<[],> = first []\u2192tt \u204f snd<tt,>\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  swap-top : \u2200 {A B C} \u2192 A `\u00d7 B `\u00d7 C `\u2192 B `\u00d7 A `\u00d7 C\n  swap-top = assoc-first swap\n\n  around : \u2200 {A B C D X} \u2192 (A `\u00d7 B `\u2192 C `\u00d7 D) \u2192 A `\u00d7 X `\u00d7 B `\u2192 C `\u00d7 X `\u00d7 D\n  around f = swap-top \u204f second f \u204f swap-top\n\n  inner : \u2200 {A B C D} \u2192 (A `\u2192 B) \u2192 C `\u00d7 A `\u00d7 D `\u2192 C `\u00d7 B `\u00d7 D\n  inner f = swap-top \u204f first f \u204f swap-top\n\n  {-\n\n  X ------------> X\n        +---+\n  A --->|   |---> C\n        | f |\n  B --->|   |---> D\n        +---+\n  Y ------------> Y\n\n  -}\n  inner2 : \u2200 {A B C D X Y} \u2192 (A `\u00d7 B `\u2192 C `\u00d7 D) \u2192 (X `\u00d7 A) `\u00d7 (B `\u00d7 Y) `\u2192 (X `\u00d7 C) `\u00d7 (D `\u00d7 Y)\n  inner2 f = assoc-first (assoc-second f)\n\n  <_\u00d7\u2081_> : \u2200 {A B C D E F} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 (C `\u2192 F) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 F\n  < f \u00d7\u2081 g > = assoc\u2032 \u204f < f \u00d7 g > \u204f assoc\n\n  <_\u00d7\u2082_> : \u2200 {A B C D E F} \u2192 (A `\u2192 D) \u2192 (B `\u00d7 C `\u2192 E `\u00d7 F) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (D `\u00d7 E) `\u00d7 F\n  < f \u00d7\u2082 g > = assoc \u204f < f \u00d7 g > \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = inner2 swap \u204f < f \u00d7 g >\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f <\u2237>\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\ud835\udfd9 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f <\u2237>\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\ud835\udfd9 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f <\u2237>\n\n  <_\u2237[]> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237[]> = < f \u2237\u2032tt\u204f tt\u2192[] >\n\n  <\u2237[]> : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  <\u2237[]> = < id \u2237[]>\n\n  <[],_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <[], f > = <tt\u204f tt\u2192[] , f >\n\n  <_,[]> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,[]> = < f ,tt\u204f tt\u2192[] >\n\n  head<\u2237> : \u2200 {A} \u2192 `Vec A 1 `\u2192 A\n  head<\u2237> = uncons \u204f fst<,[]>\n\n  constVec\u22a4 : \u2200 {n a} {A : \u2605_ a} {B} \u2192 (A \u2192 `\ud835\udfd9 `\u2192 B) \u2192 Vec A n \u2192 `\ud835\udfd9 `\u2192 `Vec B n\n  constVec\u22a4 f [] = tt\u2192[]\n  constVec\u22a4 f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec\u22a4 f xs >\n\n  []\u2192[] : \u2200 {A B} \u2192 `Vec A 0 `\u2192 `Vec B 0\n  []\u2192[] = []\u2192tt \u204f tt\u2192[]\n\n  <[],[]>\u2192[] : \u2200 {A B C} \u2192 (`Vec A 0 `\u00d7 `Vec B 0) `\u2192 `Vec C 0\n  <[],[]>\u2192[] = fst<,[]> \u204f []\u2192[]\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = []\u2192[]\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst<,[]>\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd<[],>\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  mapAccum : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 S `\u00d7 `Vec B n\n  mapAccum {zero}  F = second []\u2192[]\n  mapAccum {suc n} F = second uncons\n                     \u204f assoc-first F \u204f around (mapAccum F)\n                     \u204f second <\u2237>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = <[],[]>\u2192[]\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f <\u2237>\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd<[],> \u2237[]>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f <\u2237>\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = id\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd<[],>\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f <\u2237>\n\n  <_++_> : \u2200 {m n A} \u2192 (`\ud835\udfd9 `\u2192 `Vec A m) \u2192 (`\ud835\udfd9 `\u2192 `Vec A n) \u2192\n                         `\ud835\udfd9 `\u2192 `Vec A (m + n)\n  < f ++ g > = <tt\u204f f , g > \u204f append\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <[], id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first <\u2237>\n\n  rot-left : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A (n + m)\n  rot-left m = splitAt m \u204f swap \u204f append\n\n  rot-right : \u2200 {m} n {A} \u2192 `Vec A (m + n) `\u2192 `Vec A (n + m)\n  rot-right {m} _ = rot-left m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head<\u2237>\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = []\u2192[]\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = []\u2192[]\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f <\u2237>\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate\u22a4 : \u2200 n \u2192 `\ud835\udfd9 `\u2192 `Vec `\ud835\udfd9 n\n  replicate\u22a4 _ = constVec\u22a4 (\u03bb _ \u2192 id) (V.replicate {A = \ud835\udfd9} _)\n\n  open Category cat public\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Level.NP\nopen import Type\nopen import FunUniverse.Types\nopen import Data.One using (\ud835\udfd9)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport FunUniverse.Category\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\n\nmodule FunUniverse.Rewiring.Linear where\n\nrecord LinRewiring {t} {T : \u2605_ t} (funU : FunUniverse T) : \u2605_ t where\n  constructor mk\n  open FunUniverse funU\n  open FunUniverse.Category _`\u2192_\n\n  field\n    cat : Category\n  open Category cat\n\n  field\n    -- Products (group 2 primitive functions or derived from group 1)\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\ud835\udfd9 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\ud835\udfd9 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    tt\u2192[]  : \u2200 {A} \u2192 `\ud835\udfd9 `\u2192 `Vec A 0\n    []\u2192tt  : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\ud835\udfd9\n    <\u2237>    : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open CompositionNotations _\u2218_ public\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  infixr 3 _***_\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  <id,tt> : \u2200 {A} \u2192 A `\u2192 A `\u00d7 `\ud835\udfd9\n  <id,tt> = <tt,id> \u204f swap\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\ud835\udfd9 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\ud835\udfd9 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  fst<,tt> : \u2200 {A} \u2192 A `\u00d7 `\ud835\udfd9 `\u2192 A\n  fst<,tt> = swap \u204f snd<tt,>\n\n  fst<,[]> : \u2200 {A B} \u2192 A `\u00d7 `Vec B 0 `\u2192 A\n  fst<,[]> = second []\u2192tt \u204f fst<,tt>\n\n  snd<[],> : \u2200 {A B} \u2192 `Vec A 0 `\u00d7 B `\u2192 B\n  snd<[],> = first []\u2192tt \u204f snd<tt,>\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  swap-top : \u2200 {A B C} \u2192 A `\u00d7 B `\u00d7 C `\u2192 B `\u00d7 A `\u00d7 C\n  swap-top = assoc-first swap\n\n  around : \u2200 {A B C D X} \u2192 (A `\u00d7 B `\u2192 C `\u00d7 D) \u2192 A `\u00d7 X `\u00d7 B `\u2192 C `\u00d7 X `\u00d7 D\n  around f = swap-top \u204f second f \u204f swap-top\n\n  inner : \u2200 {A B C D} \u2192 (A `\u2192 B) \u2192 C `\u00d7 A `\u00d7 D `\u2192 C `\u00d7 B `\u00d7 D\n  inner f = swap-top \u204f first f \u204f swap-top\n\n  {-\n\n  X ------------> X\n        +---+\n  A --->|   |---> C\n        | f |\n  B --->|   |---> D\n        +---+\n  Y ------------> Y\n\n  -}\n  inner2 : \u2200 {A B C D X Y} \u2192 (A `\u00d7 B `\u2192 C `\u00d7 D) \u2192 (X `\u00d7 A) `\u00d7 (B `\u00d7 Y) `\u2192 (X `\u00d7 C) `\u00d7 (D `\u00d7 Y)\n  inner2 f = assoc-first (assoc-second f)\n\n  <_\u00d7\u2081_> : \u2200 {A B C D E F} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 (C `\u2192 F) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 F\n  < f \u00d7\u2081 g > = assoc\u2032 \u204f < f \u00d7 g > \u204f assoc\n\n  <_\u00d7\u2082_> : \u2200 {A B C D E F} \u2192 (A `\u2192 D) \u2192 (B `\u00d7 C `\u2192 E `\u00d7 F) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (D `\u00d7 E) `\u00d7 F\n  < f \u00d7\u2082 g > = assoc \u204f < f \u00d7 g > \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = inner2 swap \u204f < f \u00d7 g >\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f <\u2237>\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\ud835\udfd9 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f <\u2237>\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\ud835\udfd9 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f <\u2237>\n\n  <_\u2237[]> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237[]> = < f \u2237\u2032tt\u204f tt\u2192[] >\n\n  <\u2237[]> : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  <\u2237[]> = < id \u2237[]>\n\n  <[],_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <[], f > = <tt\u204f tt\u2192[] , f >\n\n  <_,[]> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,[]> = < f ,tt\u204f tt\u2192[] >\n\n  head<\u2237> : \u2200 {A} \u2192 `Vec A 1 `\u2192 A\n  head<\u2237> = uncons \u204f fst<,[]>\n\n  constVec\u22a4 : \u2200 {n a} {A : \u2605_ a} {B} \u2192 (A \u2192 `\ud835\udfd9 `\u2192 B) \u2192 Vec A n \u2192 `\ud835\udfd9 `\u2192 `Vec B n\n  constVec\u22a4 f [] = tt\u2192[]\n  constVec\u22a4 f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec\u22a4 f xs >\n\n  []\u2192[] : \u2200 {A B} \u2192 `Vec A 0 `\u2192 `Vec B 0\n  []\u2192[] = []\u2192tt \u204f tt\u2192[]\n\n  <[],[]>\u2192[] : \u2200 {A B C} \u2192 (`Vec A 0 `\u00d7 `Vec B 0) `\u2192 `Vec C 0\n  <[],[]>\u2192[] = fst<,[]> \u204f []\u2192[]\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = []\u2192[]\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst<,[]>\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd<[],>\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = <[],[]>\u2192[]\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f <\u2237>\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd<[],> \u2237[]>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f <\u2237>\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = id\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd<[],>\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f <\u2237>\n\n  <_++_> : \u2200 {m n A} \u2192 (`\ud835\udfd9 `\u2192 `Vec A m) \u2192 (`\ud835\udfd9 `\u2192 `Vec A n) \u2192\n                         `\ud835\udfd9 `\u2192 `Vec A (m + n)\n  < f ++ g > = <tt\u204f f , g > \u204f append\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <[], id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first <\u2237>\n\n  rot-left : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A (n + m)\n  rot-left m = splitAt m \u204f swap \u204f append\n\n  rot-right : \u2200 {m} n {A} \u2192 `Vec A (m + n) `\u2192 `Vec A (n + m)\n  rot-right {m} _ = rot-left m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head<\u2237>\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = []\u2192[]\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = []\u2192[]\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f <\u2237>\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate\u22a4 : \u2200 n \u2192 `\ud835\udfd9 `\u2192 `Vec `\ud835\udfd9 n\n  replicate\u22a4 _ = constVec\u22a4 (\u03bb _ \u2192 id) (V.replicate {A = \ud835\udfd9} _)\n\n  open Category cat public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c061026524ddd7377f197b91848cc8b11d9b0e6e","subject":"[ doc ] Fixed.","message":"[ doc ] Fixed.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/report\/FOTC\/EquivalenceInductivePredicateN.agda","new_file":"notes\/thesis\/report\/FOTC\/EquivalenceInductivePredicateN.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalent approaches for implement the inductive predicate N\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.EquivalenceInductivePredicateN where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Using succ : D instead of succ\u2081 : D \u2192 D.\n\nmodule Constant where\n\n  module LFP where\n\n    NatF : (D \u2192 Set) \u2192 D \u2192 Set\n    NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n    postulate\n      N         : D \u2192 Set\n      N-in-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n      N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n    N-in = N-in-ho\n\n    N-ind' : (A : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n             \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' = N-ind'-ho\n\n    --------------------------------------------------------------------------\n    -- The data constructors of N using LFP.\n\n    nzero : N zero\n    nzero = N-in (inj\u2081 refl)\n\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n    nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n    --------------------------------------------------------------------------\n    -- The induction principle of N using LFP.\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h = N-ind' A h'\n      where\n      h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n      h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n      h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  module Data where\n\n    data N : D \u2192 Set where\n      nzero : N zero\n      nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h nzero      = A0\n    N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n    --------------------------------------------------------------------------\n    -- The introduction rule of N using data.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n    N-in {n} h = case prf\u2081 prf\u2082 h\n      where\n      prf\u2081 : n \u2261 zero \u2192 N n\n      prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n      prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n      prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n    --------------------------------------------------------------------------\n    -- The induction principle for N using data.\n    N-ind' :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' A h = N-ind A h\u2081 h\u2082\n      where\n      h\u2081 :  A zero\n      h\u2081 = h (inj\u2081 refl)\n\n      h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n      h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n\n------------------------------------------------------------------------------\n-- Using succ\u2081 : D \u2192 D instead of succ : D.\n\nmodule UnaryFunction where\n\n  module LFP where\n\n    NatF : (D \u2192 Set) \u2192 D \u2192 Set\n    NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n    postulate\n      N         : D \u2192 Set\n      N-in-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n      N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n    N-in = N-in-ho\n\n    N-ind' : (A : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n             \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' = N-ind'-ho\n\n    --------------------------------------------------------------------------\n    -- The data constructors of N using LFP.\n\n    nzero : N zero\n    nzero = N-in (inj\u2081 refl)\n\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n    nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n    --------------------------------------------------------------------------\n    -- The induction principle of N using LFP.\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h = N-ind' A h'\n      where\n      h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n      h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n      h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  module Data where\n\n    data N : D \u2192 Set where\n      nzero : N zero\n      nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h nzero      = A0\n    N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n    --------------------------------------------------------------------------\n    -- The introduction rule of N using data.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n    N-in {n} h = case prf\u2081 prf\u2082 h\n      where\n      prf\u2081 : n \u2261 zero \u2192 N n\n      prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n      prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n      prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n    --------------------------------------------------------------------------\n    -- The induction principle for N using data.\n    N-ind' :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' A h = N-ind A h\u2081 h\u2082\n      where\n      h\u2081 :  A zero\n      h\u2081 = h (inj\u2081 refl)\n\n      h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n      h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalent approaches for implement the inductive predicate N\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.EquivalenceInductivePredicateN where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Using succ : D instead of succ\u2081 : D \u2192 D.\n\nmodule Constant where\n\n  module LFP where\n\n    NatF : (D \u2192 Set) \u2192 D \u2192 Set\n    NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n    postulate\n      N         : D \u2192 Set\n      N-in-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n      N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n    N-in = N-in-ho\n\n    N-ind' : (A : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n             \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' = N-ind'-ho\n\n    --------------------------------------------------------------------------\n    -- The data constructors of N using data.\n\n    nzero : N zero\n    nzero = N-in (inj\u2081 refl)\n\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n    nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n    --------------------------------------------------------------------------\n    -- The induction principle of N using data.\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h = N-ind' A h'\n      where\n      h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n      h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n      h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  module Data where\n\n    data N : D \u2192 Set where\n      nzero : N zero\n      nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h nzero      = A0\n    N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n    --------------------------------------------------------------------------\n    -- The introduction rule of N using LFP.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n    N-in {n} h = case prf\u2081 prf\u2082 h\n      where\n      prf\u2081 : n \u2261 zero \u2192 N n\n      prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n      prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n      prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n    --------------------------------------------------------------------------\n    -- The induction principle for N using LFP.\n    N-ind' :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' A h = N-ind A h\u2081 h\u2082\n      where\n      h\u2081 :  A zero\n      h\u2081 = h (inj\u2081 refl)\n\n      h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n      h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n\n------------------------------------------------------------------------------\n-- Using succ\u2081 : D \u2192 D instead of succ : D.\n\nmodule UnaryFunction where\n\n  module LFP where\n\n    NatF : (D \u2192 Set) \u2192 D \u2192 Set\n    NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n    postulate\n      N         : D \u2192 Set\n      N-in-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n      N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n    N-in = N-in-ho\n\n    N-ind' : (A : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n             \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' = N-ind'-ho\n\n    --------------------------------------------------------------------------\n    -- The data constructors of N using data.\n\n    nzero : N zero\n    nzero = N-in (inj\u2081 refl)\n\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n    nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n    --------------------------------------------------------------------------\n    -- The induction principle of N using data.\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h = N-ind' A h'\n      where\n      h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n      h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n      h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  module Data where\n\n    data N : D \u2192 Set where\n      nzero : N zero\n      nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h nzero      = A0\n    N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n    --------------------------------------------------------------------------\n    -- The introduction rule of N using LFP.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n    N-in {n} h = case prf\u2081 prf\u2082 h\n      where\n      prf\u2081 : n \u2261 zero \u2192 N n\n      prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n      prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n      prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n    --------------------------------------------------------------------------\n    -- The induction principle for N using LFP.\n    N-ind' :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' A h = N-ind A h\u2081 h\u2082\n      where\n      h\u2081 :  A zero\n      h\u2081 = h (inj\u2081 refl)\n\n      h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n      h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"236fac115d705655c7f7e553240eb252cc193380","subject":"+FFI.JS.fromAny","message":"+FFI.JS.fromAny\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS.agda","new_file":"lib\/FFI\/JS.agda","new_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate substring : String \u2192 Number \u2192 Number \u2192 String\n{-# COMPILED_JS substring require(\"libagda\").substring #-}\n\n-- substring with only one argument provided\npostulate substring1 : String \u2192 Number \u2192 String\n{-# COMPILED_JS substring1 require(\"libagda\").substring1 #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate fromAny : {A : Set} \u2192 A \u2192 JSValue\n{-# COMPILED_JS fromAny function(A) { return function(x) { return x; }; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char require(\"libagda\").StringToChar #-}\n\npostulate checkTypeof : (type : String) \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS checkTypeof require(\"libagda\").checkTypeof #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber require(\"libagda\").checkTypeof(\"number\") #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString require(\"libagda\").checkTypeof(\"string\") #-}\n\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray require(\"libagda\").checkTypeof(\"array\") #-}\n\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject require(\"libagda\").checkTypeof(\"object\") #-}\n\npostulate castBool : JSValue \u2192 Bool\n{-# COMPILED_JS castBool require(\"libagda\").checkTypeof(\"bool\") #-}\n\ncastChar : JSValue \u2192 Char\ncastChar = String\u25b9Char \u2218 castString\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate undefinedJS : JSValue\n{-# COMPILED_JS undefinedJS undefined #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\n-- Same type as trace but does not print anything\n-- Usage:\n--   open import FFI.JS renaming (no-trace to trace)\nno-trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\nno-trace _ inp f = f inp\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\npostulate is-null : JSValue \u2192 Bool\n{-# COMPILED_JS is-null function(x) { return (x == null); } #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = toString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nJS[_] : Set \u2192 Set\nJS[ A ] = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\n-- Old name\nCallback1 = JS[_]\n\nJS! : Set\nJS! = JS[ \ud835\udfd9 ]\n\nJS[_,_] : Set \u2192 Set \u2192 Set\nJS[ A , B ] = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback2 = JS[_,_]\n\npostulate assert : Bool \u2192 JS!\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\npostulate return : {A : Set}(x : A) \u2192 JS[ A ]\n{-# COMPILED_JS return function(A) { return function(x) { return function(cb) { return cb(x); }; }; } #-}\n\n{- Note about _>>_, _>>=_ and _>>==_, instead of using the corresponding call0, call1,\n   call2 from libagda. It's preferable to inline their definitions as compiled\n   statements. The reason is that these COMPILED_JS statements uses a call-by-name\n   semantics with strong reduction.\n\n   The worse is for _>>_ which would have a poor run-time semantics,\n   where the second command is needlessly computed.\n   Worse given the use of partial functions such as throw, checkTypeof\n   cast{String,Number...} this can lead to abort the program.\n-}\n\ninfixr 0  _>>_ _>>=_ _>>==_\n\npostulate _>>=_ : {A B : Set}(cmd : JS[ A ])(cb : A \u2192 JS[ B ]) \u2192 JS[ B ]\n{-# COMPILED_JS _>>=_ function(A) { return function(B) { return function(cmd) { return function(k) { return function(cb) { return cmd(function(x) { return k(x)(cb); }); }; }; }; }; } #-}\n\npostulate _>>==_ : {A B C : Set}(cmd : JS[ A , B ])(cb : A \u2192 B \u2192 JS[ C ]) \u2192 JS[ C ]\n{-# COMPILED_JS _>>==_ function(A) { return function(B) { return function(C) { return function(cmd) { return function(k) { return function(cb) { return cmd(function(x, y) { return k(x)(y)(cb); }); }; }; }; }; }; } #-}\n\npostulate _>>_ : {A : Set} \u2192 JS! \u2192 JS[ A ] \u2192 JS[ A ]\n{-# COMPILED_JS _>>_ function(A) { return function(cmd) { return function(k) { return function(cb) { return cmd(function(x) { return k(cb); }); }; }; }; } #-}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate substring : String \u2192 Number \u2192 Number \u2192 String\n{-# COMPILED_JS substring require(\"libagda\").substring #-}\n\n-- substring with only one argument provided\npostulate substring1 : String \u2192 Number \u2192 String\n{-# COMPILED_JS substring1 require(\"libagda\").substring1 #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char require(\"libagda\").StringToChar #-}\n\npostulate checkTypeof : (type : String) \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS checkTypeof require(\"libagda\").checkTypeof #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber require(\"libagda\").checkTypeof(\"number\") #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString require(\"libagda\").checkTypeof(\"string\") #-}\n\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray require(\"libagda\").checkTypeof(\"array\") #-}\n\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject require(\"libagda\").checkTypeof(\"object\") #-}\n\npostulate castBool : JSValue \u2192 Bool\n{-# COMPILED_JS castBool require(\"libagda\").checkTypeof(\"bool\") #-}\n\ncastChar : JSValue \u2192 Char\ncastChar = String\u25b9Char \u2218 castString\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate undefinedJS : JSValue\n{-# COMPILED_JS undefinedJS undefined #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\n-- Same type as trace but does not print anything\n-- Usage:\n--   open import FFI.JS renaming (no-trace to trace)\nno-trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\nno-trace _ inp f = f inp\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\npostulate is-null : JSValue \u2192 Bool\n{-# COMPILED_JS is-null function(x) { return (x == null); } #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = toString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nJS[_] : Set \u2192 Set\nJS[ A ] = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\n-- Old name\nCallback1 = JS[_]\n\nJS! : Set\nJS! = JS[ \ud835\udfd9 ]\n\nJS[_,_] : Set \u2192 Set \u2192 Set\nJS[ A , B ] = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback2 = JS[_,_]\n\npostulate assert : Bool \u2192 JS!\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\npostulate return : {A : Set}(x : A) \u2192 JS[ A ]\n{-# COMPILED_JS return function(A) { return function(x) { return function(cb) { return cb(x); }; }; } #-}\n\n{- Note about _>>_, _>>=_ and _>>==_, instead of using the corresponding call0, call1,\n   call2 from libagda. It's preferable to inline their definitions as compiled\n   statements. The reason is that these COMPILED_JS statements uses a call-by-name\n   semantics with strong reduction.\n\n   The worse is for _>>_ which would have a poor run-time semantics,\n   where the second command is needlessly computed.\n   Worse given the use of partial functions such as throw, checkTypeof\n   cast{String,Number...} this can lead to abort the program.\n-}\n\ninfixr 0  _>>_ _>>=_ _>>==_\n\npostulate _>>=_ : {A B : Set}(cmd : JS[ A ])(cb : A \u2192 JS[ B ]) \u2192 JS[ B ]\n{-# COMPILED_JS _>>=_ function(A) { return function(B) { return function(cmd) { return function(k) { return function(cb) { return cmd(function(x) { return k(x)(cb); }); }; }; }; }; } #-}\n\npostulate _>>==_ : {A B C : Set}(cmd : JS[ A , B ])(cb : A \u2192 B \u2192 JS[ C ]) \u2192 JS[ C ]\n{-# COMPILED_JS _>>==_ function(A) { return function(B) { return function(C) { return function(cmd) { return function(k) { return function(cb) { return cmd(function(x, y) { return k(x)(y)(cb); }); }; }; }; }; }; } #-}\n\npostulate _>>_ : {A : Set} \u2192 JS! \u2192 JS[ A ] \u2192 JS[ A ]\n{-# COMPILED_JS _>>_ function(A) { return function(cmd) { return function(k) { return function(cb) { return cmd(function(x) { return k(cb); }); }; }; }; } #-}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"da398ec544b07c1d537b1836b3ded6a8cbdfd9de","subject":"scraps towards #14","message":"scraps towards #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n      lem (CECong x\u2082 ce) = {!!}\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err -- this feels extremely strange\n    ie (INEHole x) (CECong FHOuter err) = {!!}\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err) = {!!}\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = {!!}\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = {!!}\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = {!!}\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2c661467da93de9a39c40cddc61abeeca5350793","subject":"Make some arguments of validVar explicit.","message":"Make some arguments of validVar explicit.\n\nOld-commit-hash: dc6d178e02b8bdd4f772dddbba1bf337f3b1d74c\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Specification\/Canon-Popl14.agda","new_file":"Denotation\/Specification\/Canon-Popl14.agda","new_contents":"module Denotation.Specification.Canon-Popl14 where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192 Change \u03c4\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to future arguments\nvalidity : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  valid {\u03c4} (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to free variables\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1 \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Change \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 + \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) d\u03c1 = - \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 empty d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 ++ \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) d\u03c1 = \u27e6 x \u27e7\u0394Var d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) d\u03c1 =\n     (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n  (validity {\u03c3} t d\u03c1))\n\u27e6_\u27e7\u0394 (abs t) d\u03c1 = \u03bb v \u2192\n  \u27e6 t \u27e7\u0394 (v \u2022 d\u03c1)\n\nvalidVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : \u0394Env \u0393) \u2192 valid {\u03c4} (\u27e6 x \u27e7 (ignore d\u03c1)) (\u27e6 x \u27e7\u0394Var d\u03c1)\nvalidVar this (cons v \u0394v R[v,\u0394v] \u2022 _) = R[v,\u0394v]\nvalidVar {\u03c4} (that x) (cons _ _ _ \u2022 d\u03c1) = validVar {\u03c4} x d\u03c1\n\nvalidity (intlit n)    d\u03c1 = tt\nvalidity (add s t)     d\u03c1 = tt\nvalidity (minus t)     d\u03c1 = tt\nvalidity (empty)       d\u03c1 = tt\nvalidity (insert s t)  d\u03c1 = tt\nvalidity (union s t)   d\u03c1 = tt\nvalidity (negate t)    d\u03c1 = tt\nvalidity (flatmap s t) d\u03c1 = tt\nvalidity (sum t)       d\u03c1 = tt\n\nvalidity {\u03c4} (var x) d\u03c1 = validVar {\u03c4} x d\u03c1\n\nvalidity (app s t) d\u03c1 =\n  let\n    v = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394v = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    proj\u2081 (validity s d\u03c1 (cons v \u0394v (validity t d\u03c1)))\n\nvalidity {\u03c3 \u21d2 \u03c4} (abs t) d\u03c1 = \u03bb v \u2192\n  let\n    v\u2032 = after {\u03c3} v\n    \u0394v\u2032 = v\u2032 \u229f\u208d \u03c3 \u208e v\u2032\n    d\u03c1\u2081 = v \u2022 d\u03c1\n    d\u03c1\u2082 = nil-valid-change \u03c3 v\u2032 \u2022 d\u03c1\n  in\n    validity t d\u03c1\u2081\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2082) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {d\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {d\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore d\u03c1\u2081) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar {x = this  } {cons v dv R[v,dv] \u2022 d\u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u2022 d\u03c1} = correctVar {x = y} {d\u03c1}\n\ncorrectness {t = intlit n} = right-id-int n\ncorrectness {t = add s t} {d\u03c1} = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _+_ (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = minus t} {d\u03c1} = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong -_ (correctness {t = t}))\n\ncorrectness {t = empty} = right-id-bag emptyBag\ncorrectness {t = insert s t} {d\u03c1} =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness {t = s}))\n         (correctness {t = t}) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness {t = union s t} {d\u03c1} = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _++_ (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = negate t} {d\u03c1} = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong negateBag (correctness {t = t}))\n\ncorrectness {t = flatmap s t} {d\u03c1} =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = sum t} {d\u03c1} =\n  let\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness {t = t}))\n\ncorrectness {\u03c4} {t = var x} = correctVar {\u03c4} {x = x}\ncorrectness {t = app {\u03c3} {\u03c4} s t} {d\u03c1} =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 (ignore d\u03c1)\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e \u0394f (cons u \u0394u (validity {\u03c3} t d\u03c1))\n    \u2261\u27e8 sym (proj\u2082 (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons u \u0394u (validity {\u03c3} t d\u03c1)))) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} {t = s} \u27e8$\u27e9 correctness {\u03c3} {t = t} \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} {abs t} {d\u03c1} = ext (\u03bb v \u2192\n  let\n    d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393)\n    d\u03c1\u2032 = nil-valid-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} {t = t} {d\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {d\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 d\u03c1)\n      (\u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7 (ignore d\u03c1))\n    \u229e\u208d \u03c4 \u208e\n    (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1))\n\ncorollary {\u03c3} {\u03c4} s t {d\u03c1} = proj\u2082\n  (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons (\u27e6 t \u27e7 (ignore d\u03c1))\n     (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1)))\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  (v : ValidChange \u03c3)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} v)\n  \u2261 \u27e6 t \u27e7 \u2205 (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 \u2205 v\n\ncorollary-closed {\u03c3} {\u03c4} {t = t} v =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205\n  in\n    begin\n      f (after {\u03c3} v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} v))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} {t = t})) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} v)\n    \u2261\u27e8 proj\u2082 (validity {\u03c3 \u21d2 \u03c4} t \u2205 v) \u27e9\n      f (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u0394f v\n    \u220e where open \u2261-Reasoning\n","old_contents":"module Denotation.Specification.Canon-Popl14 where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192 Change \u03c4\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to future arguments\nvalidity : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  valid {\u03c4} (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to free variables\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1 \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Change \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 + \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) d\u03c1 = - \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 empty d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 ++ \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) d\u03c1 = \u27e6 x \u27e7\u0394Var d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) d\u03c1 =\n     (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n  (validity {\u03c3} t d\u03c1))\n\u27e6_\u27e7\u0394 (abs t) d\u03c1 = \u03bb v \u2192\n  \u27e6 t \u27e7\u0394 (v \u2022 d\u03c1)\n\nvalidVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 : \u0394Env \u0393} \u2192 valid {\u03c4} (\u27e6 x \u27e7 (ignore d\u03c1)) (\u27e6 x \u27e7\u0394Var d\u03c1)\nvalidVar this {cons v \u0394v R[v,\u0394v] \u2022 _} = R[v,\u0394v]\nvalidVar {\u03c4} (that x) {cons _ _ _ \u2022 d\u03c1} = validVar {\u03c4} x\n\nvalidity (intlit n)    d\u03c1 = tt\nvalidity (add s t)     d\u03c1 = tt\nvalidity (minus t)     d\u03c1 = tt\nvalidity (empty)       d\u03c1 = tt\nvalidity (insert s t)  d\u03c1 = tt\nvalidity (union s t)   d\u03c1 = tt\nvalidity (negate t)    d\u03c1 = tt\nvalidity (flatmap s t) d\u03c1 = tt\nvalidity (sum t)       d\u03c1 = tt\n\nvalidity {\u03c4} (var x) d\u03c1 = validVar {\u03c4} x\n\nvalidity (app s t) d\u03c1 =\n  let\n    v = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394v = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    proj\u2081 (validity s d\u03c1 (cons v \u0394v (validity t d\u03c1)))\n\nvalidity {\u03c3 \u21d2 \u03c4} (abs t) d\u03c1 = \u03bb v \u2192\n  let\n    v\u2032 = after {\u03c3} v\n    \u0394v\u2032 = v\u2032 \u229f\u208d \u03c3 \u208e v\u2032\n    d\u03c1\u2081 = v \u2022 d\u03c1\n    d\u03c1\u2082 = nil-valid-change \u03c3 v\u2032 \u2022 d\u03c1\n  in\n    validity t d\u03c1\u2081\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2082) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {d\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {d\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore d\u03c1\u2081) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar {x = this  } {cons v dv R[v,dv] \u2022 d\u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u2022 d\u03c1} = correctVar {x = y} {d\u03c1}\n\ncorrectness {t = intlit n} = right-id-int n\ncorrectness {t = add s t} {d\u03c1} = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _+_ (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = minus t} {d\u03c1} = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong -_ (correctness {t = t}))\n\ncorrectness {t = empty} = right-id-bag emptyBag\ncorrectness {t = insert s t} {d\u03c1} =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness {t = s}))\n         (correctness {t = t}) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness {t = union s t} {d\u03c1} = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _++_ (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = negate t} {d\u03c1} = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong negateBag (correctness {t = t}))\n\ncorrectness {t = flatmap s t} {d\u03c1} =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = sum t} {d\u03c1} =\n  let\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness {t = t}))\n\ncorrectness {\u03c4} {t = var x} = correctVar {\u03c4} {x = x}\ncorrectness {t = app {\u03c3} {\u03c4} s t} {d\u03c1} =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 (ignore d\u03c1)\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e \u0394f (cons u \u0394u (validity {\u03c3} t d\u03c1))\n    \u2261\u27e8 sym (proj\u2082 (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons u \u0394u (validity {\u03c3} t d\u03c1)))) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} {t = s} \u27e8$\u27e9 correctness {\u03c3} {t = t} \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} {abs t} {d\u03c1} = ext (\u03bb v \u2192\n  let\n    d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393)\n    d\u03c1\u2032 = nil-valid-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} {t = t} {d\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {d\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 d\u03c1)\n      (\u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7 (ignore d\u03c1))\n    \u229e\u208d \u03c4 \u208e\n    (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1))\n\ncorollary {\u03c3} {\u03c4} s t {d\u03c1} = proj\u2082\n  (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons (\u27e6 t \u27e7 (ignore d\u03c1))\n     (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1)))\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  (v : ValidChange \u03c3)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} v)\n  \u2261 \u27e6 t \u27e7 \u2205 (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 \u2205 v\n\ncorollary-closed {\u03c3} {\u03c4} {t = t} v =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205\n  in\n    begin\n      f (after {\u03c3} v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} v))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} {t = t})) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} v)\n    \u2261\u27e8 proj\u2082 (validity {\u03c3 \u21d2 \u03c4} t \u2205 v) \u27e9\n      f (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u0394f v\n    \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ad4a717b9aad6b45b1d211c0e38442ad25612cf3","subject":"Use instance arguments for breathing.","message":"Use instance arguments for breathing.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth add-smooth-lower-vowel\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough add-rough-vowel\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth add-smooth-lower-vowel\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough add-rough-vowel\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth add-smooth-lower-vowel\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough add-rough-vowel\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth add-smooth-lower-vowel\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4cf12c20e250262b7319cb5988e6eafa771d7330","subject":"Data.Indexed","message":"Data.Indexed\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Indexed.agda","new_file":"lib\/Data\/Indexed.agda","new_contents":"open import Type            hiding (\u2605)\nopen import Level           using (_\u2294_; suc; Level)\nopen import Function.NP     using (id; const; _\u2218_; _\u2218\u2032_; _\u02e2_; _\u27e8_\u27e9\u00b0_)\nopen import Data.Sum        using (_\u228e_; inj\u2081; inj\u2082)\nopen import Data.Bool       using (Bool; true; false)\nopen import Data.Unit       using (\u22a4)\nopen import Data.List       using (List; []; _\u2237_)\nopen import Data.Maybe      using (Maybe; nothing; just)\nopen import Data.Empty      using (\u22a5)\nopen import Data.Product.NP using (\u2203; _,_; _\u00d7_)\n\nmodule Data.Indexed {i} {Ix : \u2605 i} where\n\n\u2605\u00b0 : \u2200 \u2113 \u2192 \u2605 _\n\u2605\u00b0 \u2113 = Ix \u2192 \u2605 \u2113\n\n\u2605\u2080\u00b0 : \u2605 _\n\u2605\u2080\u00b0 = Ix \u2192 \u2605\u2080\n\n\u2605\u2081\u00b0 : \u2605 _\n\u2605\u2081\u00b0 = Ix \u2192 \u2605\u2082\n\n\u2200\u00b0 : \u2200 {f} (F : \u2605\u00b0 f) \u2192 \u2605 _\n\u2200\u00b0 F = \u2200 {x} \u2192 F x\n\n\u2203\u00b0 : \u2200 {f} (F : \u2605\u00b0 f) \u2192 \u2605 _\n\u2203\u00b0 F = \u2203[ x ] F x\n\npure\u00b0 : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (Ix \u2192 A)\npure\u00b0 = const\n\n-- an alias for pure\u00b0\nK\u00b0 : \u2200 {a} (A : \u2605 a) \u2192 \u2605\u00b0 _\nK\u00b0 = pure\u00b0\n\nCmp\u00b0 : (F : \u2605\u00b0 _) (i j : Ix) \u2192 \u2605\u2080\nCmp\u00b0 F i j = F i \u2192 F j \u2192 Bool\n\n\u03a0\u00b0 : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2200 {i} \u2192 F i \u2192 \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n\u03a0\u00b0 F G i = (x : F i) \u2192 G x i\n\n-- this version used to work (type-checking its uses) better than that\ninfixr 0 _\u2192\u00b0_\n_\u2192\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 (f \u2294 g)\nF \u2192\u00b0 G = \u03a0\u00b0 F (const G)\n-- expanded: (F \u2192\u00b0 G) i = F i \u2192 G i\n\ninfixr 0 _\u21a6\u00b0_\n_\u21a6\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605 _\n(F \u21a6\u00b0 G) = \u2200\u00b0(F \u2192\u00b0 G)\n\nid\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 F \u21a6\u00b0 F\nid\u00b0 = id\n\n_\u2218\u00b0_ : \u2200 {f g h} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} {H : \u2605\u00b0 h} \u2192 G \u21a6\u00b0 H \u2192 F \u21a6\u00b0 G \u2192 F \u21a6\u00b0 H\nf \u2218\u00b0 g = f \u2218 g\n\ninfixr 0 _$\u00b0_ _$\u00b0\u2032_\n\n_$\u00b0_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2200 {i} \u2192 F i \u2192 \u2605\u00b0 g}\n        \u2192 \u03a0\u00b0 F G \u21a6\u00b0 \u03a0\u00b0 F G\n_$\u00b0_ = id\n\n_$\u00b0\u2032_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g}\n        \u2192 \u2200\u00b0 ((F \u2192\u00b0 G) \u2192\u00b0 F \u2192\u00b0 G)\n_$\u00b0\u2032_ = id\n\ninfixl 4 _\u229b\u00b0_\n\n_\u229b\u00b0_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n         \u2192 (Ix \u2192 A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\n_\u229b\u00b0_ = _\u02e2_\n\nliftA\u00b0 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n          \u2192 (A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\nliftA\u00b0 = _\u2218\u2032_\n-- liftA\u00b0 f x = pure\u00b0 f \u229b\u00b0 x\n\nliftA\u2082\u00b0 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n           \u2192 (A \u2192 B \u2192 C)\n           \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B) \u2192 (Ix \u2192 C))\nliftA\u2082\u00b0 f x y = x \u27e8 f \u27e9\u00b0 y\n-- liftA\u2082\u00b0 f x y = pure\u00b0 f \u229b\u00b0 x \u229b\u00b0 y\n\nList\u00b0 : \u2200 {a} \u2192 \u2605\u00b0 a \u2192 \u2605\u00b0 a\nList\u00b0 = liftA\u00b0 List\n\n[]\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(List\u00b0 F)\n[]\u00b0 = []\n\n_\u2237\u00b0_ : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(F \u2192\u00b0 List\u00b0 F \u2192\u00b0 List\u00b0 F)\n_\u2237\u00b0_ = _\u2237_\n\nMaybe\u00b0 : \u2200 {a} \u2192 \u2605\u00b0 a \u2192 \u2605\u00b0 a\nMaybe\u00b0 = liftA\u00b0 Maybe\n\nnothing\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(Maybe\u00b0 F)\nnothing\u00b0 = nothing\n\njust\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(F \u2192\u00b0 Maybe\u00b0 F)\njust\u00b0 = just\n\ninfixr 2 _\u00d7\u00b0_\n_\u00d7\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n_\u00d7\u00b0_ = liftA\u2082\u00b0 _\u00d7_\n\n_,\u00b0_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 \u2200\u00b0 (F \u2192\u00b0 G \u2192\u00b0 F \u00d7\u00b0 G)\n_,\u00b0_ = _,_\n\npack\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 (x : Ix) \u2192 F x \u2192 \u2203\u00b0 F\npack\u00b0 = _,_\n\ninfixr 1 _\u228e\u00b0_\n\n_\u228e\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n_\u228e\u00b0_ = liftA\u2082\u00b0 _\u228e_\n\ninj\u2081\u00b0 : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 F \u21a6\u00b0 F \u228e\u00b0 G\ninj\u2081\u00b0 = inj\u2081\n\ninj\u2082\u00b0 : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 G \u21a6\u00b0 F \u228e\u00b0 G\ninj\u2082\u00b0 = inj\u2082\n\n\u22a4\u00b0 : \u2605\u00b0 _\n\u22a4\u00b0 = pure\u00b0 \u22a4\n\n\u22a5\u00b0 : \u2605\u00b0 _\n\u22a5\u00b0 = pure\u00b0 \u22a5\n\n\u00ac\u00b0 : \u2200 {\u2113} \u2192 \u2605\u00b0 \u2113 \u2192 \u2605\u00b0 _\n\u00ac\u00b0 F = F \u2192\u00b0 \u22a5\u00b0\n\n-- This is the type of |map| functions, the fmap function on Ix-indexed\n-- functors.\nMap\u00b0 : \u2200 {a a' b b'} (F : \u2605\u00b0 a \u2192 \u2605\u00b0 a')\n                     (G : \u2605\u00b0 b \u2192 \u2605\u00b0 b') \u2192 \u2605 _\nMap\u00b0 F G = \u2200 {A B} \u2192 (A \u21a6\u00b0 B) \u2192 F A \u21a6\u00b0 G B\n\nmap\u00b0-K\u00b0 : \u2200 {a a' b b'}\n            {F : \u2605 a \u2192 \u2605 a'}\n            {G : \u2605 b \u2192 \u2605 b'}\n            (fmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 F A \u2192 G B)\n            {A B} \u2192 (K\u00b0 A \u21a6\u00b0 K\u00b0 B) \u2192 K\u00b0 (F A) \u21a6\u00b0 K\u00b0 (G B)\nmap\u00b0-K\u00b0 fmap f {i} = fmap (f {i})\n","old_contents":"open import Type            hiding (\u2605)\nopen import Level           using (_\u2294_; suc; Level)\nopen import Function.NP     using (id; const; _\u2218_; _\u2218\u2032_; _\u02e2_; _\u27e8_\u27e9\u00b0_)\nopen import Data.Sum        using (_\u228e_; inj\u2081; inj\u2082)\nopen import Data.Bool       using (Bool; true; false)\nopen import Data.Unit       using (\u22a4)\nopen import Data.List       using (List; []; _\u2237_)\nopen import Data.Maybe      using (Maybe; nothing; just)\nopen import Data.Empty      using (\u22a5)\nopen import Data.Product.NP using (\u2203; _,_; _\u00d7_)\n\nmodule Data.Indexed {i} {Ix : \u2605 i} where\n\n\u2605\u00b0 : \u2200 \u2113 \u2192 \u2605 _\n\u2605\u00b0 \u2113 = Ix \u2192 \u2605 \u2113\n\n\u2605\u2080\u00b0 : \u2605 _\n\u2605\u2080\u00b0 = Ix \u2192 \u2605\n\n\u2605\u2081\u00b0 : \u2605 _\n\u2605\u2081\u00b0 = Ix \u2192 \u2605\u2082\n\n\u2200\u00b0 : \u2200 {f} (F : \u2605\u00b0 f) \u2192 \u2605 _\n\u2200\u00b0 F = \u2200 {x} \u2192 F x\n\n\u2203\u00b0 : \u2200 {f} (F : \u2605\u00b0 f) \u2192 \u2605 _\n\u2203\u00b0 F = \u2203[ x ] F x\n\npure\u00b0 : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (Ix \u2192 A)\npure\u00b0 = const\n\n-- an alias for pure\u00b0\nK\u00b0 : \u2200 {a} (A : \u2605 a) \u2192 \u2605\u00b0 _\nK\u00b0 = pure\u00b0\n\nCmp\u00b0 : (F : \u2605\u00b0 _) (i j : Ix) \u2192 \u2605\u2080\nCmp\u00b0 F i j = F i \u2192 F j \u2192 Bool\n\n\u03a0\u00b0 : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2200 {i} \u2192 F i \u2192 \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n\u03a0\u00b0 F G i = (x : F i) \u2192 G x i\n\n-- this version used to work (type-checking its uses) better than that\ninfixr 0 _\u2192\u00b0_\n_\u2192\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 (f \u2294 g)\nF \u2192\u00b0 G = \u03a0\u00b0 F (const G)\n-- expanded: (F \u2192\u00b0 G) i = F i \u2192 G i\n\ninfixr 0 _\u21a6\u00b0_\n_\u21a6\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605 _\n(F \u21a6\u00b0 G) = \u2200\u00b0(F \u2192\u00b0 G)\n\nid\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 F \u21a6\u00b0 F\nid\u00b0 = id\n\n_\u2218\u00b0_ : \u2200 {f g h} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} {H : \u2605\u00b0 h} \u2192 G \u21a6\u00b0 H \u2192 F \u21a6\u00b0 G \u2192 F \u21a6\u00b0 H\nf \u2218\u00b0 g = f \u2218 g\n\ninfixr 0 _$\u00b0_ _$\u00b0\u2032_\n\n_$\u00b0_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2200 {i} \u2192 F i \u2192 \u2605\u00b0 g}\n        \u2192 \u03a0\u00b0 F G \u21a6\u00b0 \u03a0\u00b0 F G\n_$\u00b0_ = id\n\n_$\u00b0\u2032_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g}\n        \u2192 \u2200\u00b0 ((F \u2192\u00b0 G) \u2192\u00b0 F \u2192\u00b0 G)\n_$\u00b0\u2032_ = id\n\ninfixl 4 _\u229b\u00b0_\n\n_\u229b\u00b0_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n         \u2192 (Ix \u2192 A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\n_\u229b\u00b0_ = _\u02e2_\n\nliftA\u00b0 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n          \u2192 (A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\nliftA\u00b0 = _\u2218\u2032_\n-- liftA\u00b0 f x = pure\u00b0 f \u229b\u00b0 x\n\nliftA\u2082\u00b0 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n           \u2192 (A \u2192 B \u2192 C)\n           \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B) \u2192 (Ix \u2192 C))\nliftA\u2082\u00b0 f x y = x \u27e8 f \u27e9\u00b0 y\n-- liftA\u2082\u00b0 f x y = pure\u00b0 f \u229b\u00b0 x \u229b\u00b0 y\n\nList\u00b0 : \u2200 {a} \u2192 \u2605\u00b0 a \u2192 \u2605\u00b0 a\nList\u00b0 = liftA\u00b0 List\n\n[]\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(List\u00b0 F)\n[]\u00b0 = []\n\n_\u2237\u00b0_ : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(F \u2192\u00b0 List\u00b0 F \u2192\u00b0 List\u00b0 F)\n_\u2237\u00b0_ = _\u2237_\n\nMaybe\u00b0 : \u2200 {a} \u2192 \u2605\u00b0 a \u2192 \u2605\u00b0 a\nMaybe\u00b0 = liftA\u00b0 Maybe\n\nnothing\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(Maybe\u00b0 F)\nnothing\u00b0 = nothing\n\njust\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(F \u2192\u00b0 Maybe\u00b0 F)\njust\u00b0 = just\n\ninfixr 2 _\u00d7\u00b0_\n_\u00d7\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n_\u00d7\u00b0_ = liftA\u2082\u00b0 _\u00d7_\n\n_,\u00b0_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 \u2200\u00b0 (F \u2192\u00b0 G \u2192\u00b0 F \u00d7\u00b0 G)\n_,\u00b0_ = _,_\n\npack\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 (x : Ix) \u2192 F x \u2192 \u2203\u00b0 F\npack\u00b0 = _,_\n\ninfixr 1 _\u228e\u00b0_\n\n_\u228e\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n_\u228e\u00b0_ = liftA\u2082\u00b0 _\u228e_\n\ninj\u2081\u00b0 : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 F \u21a6\u00b0 F \u228e\u00b0 G\ninj\u2081\u00b0 = inj\u2081\n\ninj\u2082\u00b0 : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 G \u21a6\u00b0 F \u228e\u00b0 G\ninj\u2082\u00b0 = inj\u2082\n\n\u22a4\u00b0 : \u2605\u00b0 _\n\u22a4\u00b0 = pure\u00b0 \u22a4\n\n\u22a5\u00b0 : \u2605\u00b0 _\n\u22a5\u00b0 = pure\u00b0 \u22a5\n\n\u00ac\u00b0 : \u2200 {\u2113} \u2192 \u2605\u00b0 \u2113 \u2192 \u2605\u00b0 _\n\u00ac\u00b0 F = F \u2192\u00b0 \u22a5\u00b0\n\n-- This is the type of |map| functions, the fmap function on Ix-indexed\n-- functors.\nMap\u00b0 : \u2200 {a a' b b'} (F : \u2605\u00b0 a \u2192 \u2605\u00b0 a')\n                     (G : \u2605\u00b0 b \u2192 \u2605\u00b0 b') \u2192 \u2605 _\nMap\u00b0 F G = \u2200 {A B} \u2192 (A \u21a6\u00b0 B) \u2192 F A \u21a6\u00b0 G B\n\nmap\u00b0-K\u00b0 : \u2200 {a a' b b'}\n            {F : \u2605 a \u2192 \u2605 a'}\n            {G : \u2605 b \u2192 \u2605 b'}\n            (fmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 F A \u2192 G B)\n            {A B} \u2192 (K\u00b0 A \u21a6\u00b0 K\u00b0 B) \u2192 K\u00b0 (F A) \u21a6\u00b0 K\u00b0 (G B)\nmap\u00b0-K\u00b0 fmap f {i} = fmap (f {i})\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b9a42862bd5e427adec1dda3d912bc2e2d93491a","subject":"IDesc model: make that damned fix-point. Stuck with a dead lift to make most constructors.","message":"IDesc model: make that damned fix-point. Stuck with a dead lift to make most constructors.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\n{-\nvarl : (l : Level)(I : Set l) -> IDescl l I\nvarl x I = con (lvar , {!!})\n-}\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lconst , X)\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lprod , (D , D'))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"297a51fcf9284f7368d567dfc8047a920951eb0d","subject":"IDesc model: modelled up to elimination, with no crazy flag but a mutual definition.","message":"IDesc model: modelled up to elimination, with no crazy flag but a mutual definition.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3ee4b4fed60cc477d5e3672148d7f6f90db3368f","subject":"Revert \"Convert to stdlib\"","message":"Revert \"Convert to stdlib\"\n\nThis reverts commit 68af677d56d4f0a4c1aaa95134ae06f0813500c1.\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"open import Agda.Builtin.Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n{-# BUILTIN LIST List #-}\n{-# BUILTIN NIL  \u2218    #-}\n{-# BUILTIN CONS _\u2237_  #-}\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n\n\nmain = (closure proofs (5 \u2237 \u2218))\n","old_contents":"open import Data.Bool\nopen import Data.List\nopen import Data.Nat\n\n\n----------------------------------------\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n----------------------------------------\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 []))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 []\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 [])) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (5 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (6 \u2237 3 \u2237 [])) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (6 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 []) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 []) (3 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 [])) \u2237\n  (model (10 \u2237 9 \u2237 []) (1 \u2237 [])) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 []) (4 \u2237 11 \u2237 8 \u2237 [])) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 []) (11 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 []) (11 \u2237 3 \u2237 [])) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 []) ([])) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 []) (12 \u2237 [])) \u2237 []\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n\n\nmain = (closure proofs (5 \u2237 []))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3d40afcf1e4c32b449544225ef68e0b8bb0556f6","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-ind :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-ind-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ind\/N-ind-ho\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind-ho\n\n  N-ind-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-ind-ho' = N-ind\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-unf : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-unf = N-ind A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  -- Higher-order version.\n  N-unf-ho : \u2200 {n} \u2192 N n \u2192 NatF N n\n  N-unf-ho = N-unf\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-ind.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-ind B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-ind\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-ind A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-ind as the induction principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d        \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ\u2081 d) + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-ind A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-unf.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-unf Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-ind.\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-ind :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-ind-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ind\/N-ind-ho\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind-ho\n\n  N-ind-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-ind-ho' = N-ind\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-unf : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-unf = N-ind A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  -- Higher-order version.\n  N-unf-ho : \u2200 {n} \u2192 N n \u2192 NatF N n\n  N-unf-ho = N-unf\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-ind.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-ind B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-ind\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-ind A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-ind as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d        \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ\u2081 d) + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-ind A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-unf.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-unf Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-ind.\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a43d8badf1978702b8e181a6088db80412b1c798","subject":"Updated thesis' examples.","message":"Updated thesis' examples.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/AgdaIntroduction.agda","new_file":"notes\/thesis\/AgdaIntroduction.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n{-# OPTIONS --no-coverage-check #-}\n\nmodule AgdaIntroduction where\n\ninfixr 5 _\u2237_ _++_\ninfix  4 _\u2261_\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115     #-}\n{-# BUILTIN ZERO    zero  #-}\n{-# BUILTIN SUC     succ  #-}\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A 0\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0      + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin 0 \u2192 A\nmagic ()\n\n-- The with constructor\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven 0        = true\neven (succ n) = odd n\n\nodd 0        = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Normalisation\n\ndata _\u2261_ {A : Set}: A \u2192 A \u2192 Set where\n  refl : {a : A} \u2192 a \u2261 a\n\nlength : {A : Set} \u2192 List A \u2192 \u2115\nlength []       = 0\nlength (x \u2237 xs) = 1 + length xs\n\n_++_ : {A : Set} \u2192 List A \u2192 List A \u2192 List A\n[]       ++ ys = ys\n(x \u2237 xs) ++ ys = x \u2237 xs ++ ys\n\nsuccCong : {m n : \u2115} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsuccCong refl = refl\n\nthm : {A : Set}(xs ys : List A) \u2192 length (xs ++ ys) \u2261 length xs + length ys\nthm [] ys       = refl\nthm (x \u2237 xs) ys = succCong (thm xs ys)\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack 0        n        = succ n\nack (succ m) 0        = ack m 1\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n{-# NO_TERMINATION_CHECK #-}\nf : \u2115 \u2192 \u2115\nf n = f n\n\n{-# NO_TERMINATION_CHECK #-}\nnest : \u2115 \u2192 \u2115\nnest 0        = 0\nnest (succ n) = nest (nest n)\n\n-- Combinators for equational reasoning\n\npostulate\n  sym   : {A : Set}{a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\n  trans : {A : Set}{a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\n  subst : {A : Set}(P : A \u2192 Set){a b : A} \u2192 a \u2261 b \u2192 P a \u2192 P b\n\npostulate\n  _*_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  *-comm          : \u2200 m n \u2192 m * n \u2261 n * m\n  *-rightIdentity : \u2200 n \u2192 n * 1 \u2261 n\n\n*-leftIdentity : \u2200 n \u2192 1 * n \u2261 n\n*-leftIdentity n =\n  trans {\u2115} {1 * n} {n * 1} {n} (*-comm 1 n) (*-rightIdentity n)\n\nmodule ER\n  {A     : Set}\n  (_\u223c_   : A \u2192 A \u2192 Set)\n  (refl  : \u2200 {x} \u2192 x \u223c x)\n  (trans : \u2200 {x y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z)\n  where\n\n  infixr 5 _\u223c\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  _\u223c\u27e8_\u27e9_ : \u2200 x {y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z\n  _ \u223c\u27e8 x\u223cy \u27e9 y\u223cz = trans x\u223cy y\u223cz\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _\u220e _ = refl\n\nopen module \u2261-Reasoning = ER _\u2261_ (refl {\u2115}) (trans {\u2115})\n  renaming ( _\u223c\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_ )\n\n*-leftIdentity' : \u2200 n \u2192 1 * n \u2261 n\n*-leftIdentity' n =\n  1 * n \u2261\u27e8 *-comm 1 n \u27e9\n  n * 1 \u2261\u27e8 *-rightIdentity n \u27e9\n  n     \u220e\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n{-# OPTIONS --no-coverage-check #-}\n\nmodule AgdaIntroduction where\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A zero\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin zero \u2192 A\nmagic ()\n\n-- The with constructor\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven zero     = true\neven (succ n) = odd n\n\nodd zero     = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero     n        = succ n\nack (succ m) zero     = ack m (succ zero)\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n{-# NO_TERMINATION_CHECK #-}\nf : \u2115 \u2192 \u2115\nf n = f n\n\n{-# NO_TERMINATION_CHECK #-}\nnest : \u2115 \u2192 \u2115\nnest zero     = zero\nnest (succ n) = nest (nest n)\n\n-- Combinators for equational reasoning\n\ninfix 4 _\u2261_\n\npostulate\n  _\u2261_   : {A : Set} \u2192 A \u2192 A \u2192 Set\n  refl  : {A : Set}{a : A} \u2192 a \u2261 a\n  sym   : {A : Set}{a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\n  trans : {A : Set}{a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\n  subst : {A : Set}(P : A \u2192 Set){a b : A} \u2192 a \u2261 b \u2192 P a \u2192 P b\n\none : \u2115\none = succ zero\n\npostulate\n  _*_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  *-comm          : \u2200 m n \u2192 m * n \u2261 n * m\n  *-rightIdentity : \u2200 n \u2192 n * one \u2261 n\n\n*-leftIdentity : \u2200 n \u2192 one * n \u2261 n\n*-leftIdentity n =\n  trans {\u2115} {one * n} {n * one} {n} (*-comm one n) (*-rightIdentity n)\n\nmodule ER\n  {A     : Set}\n  (_\u223c_   : A \u2192 A \u2192 Set)\n  (refl  : \u2200 {x} \u2192 x \u223c x)\n  (trans : \u2200 {x y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z)\n  where\n\n  infixr 5 _\u223c\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  _\u223c\u27e8_\u27e9_ : \u2200 x {y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z\n  _ \u223c\u27e8 x\u223cy \u27e9 y\u223cz = trans x\u223cy y\u223cz\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _\u220e _ = refl\n\nopen module \u2261-Reasoning = ER _\u2261_ (refl {\u2115}) (trans {\u2115})\n  renaming ( _\u223c\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_ )\n\n*-leftIdentity' : \u2200 n \u2192 one * n \u2261 n\n*-leftIdentity' n =\n  one * n \u2261\u27e8 *-comm one n \u27e9\n  n * one \u2261\u27e8 *-rightIdentity n \u27e9\n  n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d5474bdaa0fd042a44d5b978da302fb3cd09ac7b","subject":"Modified an test.","message":"Modified an test.\n\nIgnore-this: a8f1b26bb9fecf5294fd2ceb5033df78\n\ndarcs-hash:20100723031640-3bd4e-26324ce43c88b8e65557069122a7183a5a1fd097.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  5 \u00ac_\ninfixr 4 _\u2227_\ninfixr 3 _\u2228_\ninfixr 2 _\u21d2_\ninfixr 1 _\u2194_\n\n------------------------------------------------------------------------------\n-- Propositional logic\n\n-- The logical connectives\n\n-- The logical connectives are hard-coded in our implementation,\n-- i.e. the following symbols must be used.\npostulate\n  \u22a5 \u22a4     : Set -- N.B. the name of the tautology symbol is \"\\top\", not T.\n  \u00ac_      : Set \u2192 Set -- N.B. the right hole.\n  _\u2227_ _\u2228_ : Set \u2192 Set \u2192 Set\n  _\u21d2_ _\u2194_ : Set \u2192 Set \u2192 Set\n\n-- We postulate some propositional variables (which are translated as\n-- 0-ary predicates).\npostulate\n  P Q R : Set\n\n-- The introduction and elimination rules for the propositional\n-- connectives are theorems.\npostulate\n  \u2192I  : (P \u2192 Q) \u2192 P \u21d2 Q\n  \u2192E  : (P \u21d2 Q) \u2192 P \u2192 Q\n  \u2227I  : P \u2192 Q \u2192 P \u2227 Q\n  \u2227E\u2081 : P \u2227 Q \u2192 P\n  \u2227E\u2082 : P \u2227 Q \u2192 Q\n  \u2228I\u2081 : P \u2192 P \u2228 Q\n  \u2228I\u2082 : Q \u2192 P \u2228 Q\n  \u2228E  : (P \u21d2 R) \u2192 (Q \u21d2 R) \u2192 P \u2228 Q \u2192 R\n  \u22a5E  : \u22a5 \u2192 P\n  \u00acE : (\u00ac P \u2192 \u22a5 ) \u2192 P\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u00acE #-}\n\n------------------------------------------------------------------------------\n-- Predicate logic\n\n-- The universe of discourse.\npostulate\n  D : Set\n\n-- The quantifiers are hard-coded in our implementation, i.e. the\n-- following symbols must be used.\npostulate\n  \u2203D : (P : D \u2192 Set) \u2192 Set\n\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 (x : D) \u2192 P\u00b9 x -- TODO\n  \u2200E : ((x : D) \u2192 P\u00b9 x) \u2192 (t : D) \u2192 P\u00b9 t\n  -- This elimination rule cannot prove in Coq because in Coq we can\n  -- have empty domains. We do not have this problem because the ATPs\n  -- assume a non-empty domain.\n  \u2203I : ((x : D) \u2192 P\u00b9 x) \u2192 \u2203D P\u00b9\n--  TODO \u2203E : \u2203D (\u03bb x \u2192 P\u00b9 x) \u2192 ((y : D) \u2192 P\u00b9 y \u2192 Q\u00b9 y) \u2192 (z : D) \u2192 Q\u00b9 z\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203I #-}\n-- {-# ATP prove \u2203E #-}\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  -- N.B. It is not necessary to define neither the data constructor\n  -- _,_, nor the projections -- because the ATPs implement it.\n  data _\u2227_ (A B : Set) : Set where\n\n  -- Testing the data constructor and the projections.\n  postulate\n    A B     : Set\n    _,_     : A \u2192 B \u2192 A \u2227 B\n    \u2227-proj\u2081 : A \u2227 B \u2192 A\n    \u2227-proj\u2082 : A \u2227 B \u2192 B\n  {-# ATP prove _,_ #-}\n  {-# ATP prove \u2227-proj\u2081 #-}\n  {-# ATP prove \u2227-proj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac_\n\n  data \u22a5 : Set where\n\n  \u00ac_ : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to define neither the data constructors\n  -- inj\u2081 and inj\u2082 nor the disyunction elimination because the ATP\n  -- implements them.\n  data _\u2228_ (A B : Set) : Set where\n\n  -- Testing the data constructors and the elimination.\n  postulate\n    A B C : Set\n    inj\u2081  : A \u2192 A \u2228 B\n    inj\u2082  : B \u2192 A \u2228 B\n    [_,_] : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  {-# ATP prove inj\u2081 #-}\n  {-# ATP prove inj\u2082 #-}\n  {-# ATP prove [_,_] #-}\n\n------------------------------------------------------------------------------\n-- The equivalence\n\nmodule Equivalence where\n\n  postulate _\u2194_ : Set \u2192 Set \u2192 Set\n\n  postulate\n    A      : Set\n    ident  : A \u2194 A\n  {-# ATP prove ident #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\nmodule ExistentialQuantifier where\n\n  data \u2203D (P : D \u2192 Set) : Set where\n\n  -- Testing the data constructor.\n  postulate\n    P    : D \u2192 Set\n    _,_  : (d : D) \u2192 P d \u2192 \u2203D (\u03bb e \u2192 P e)\n  {-# ATP prove _,_ #-}\n\n  -- TODO: What about the eliminators?\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e65e839e7dafe9dee2fd14772544c39dc40453f3","subject":"Data.Fin.Param.Binary is broken","message":"Data.Fin.Param.Binary is broken\n","repos":"np\/agda-parametricity","old_file":"lib\/Data\/Fin\/Param\/Binary.agda","new_file":"lib\/Data\/Fin\/Param\/Binary.agda","new_contents":"{-# OPTIONS --with-K #-}\nmodule Data.Fin.Param.Binary where\n\nopen import Data.Fin\nopen import Data.Nat\nopen import Function\n\nopen import Data.Nat.Param.Binary\nopen import Reflection.NP\nopen import Reflection.Param\nopen import Reflection.Param.Env\nopen import Function.Param.Binary\nopen import Type.Param.Binary\n\n{-\n-- See Data.Fin.Logical for an example\n\ndata \u27e6Fin\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) Fin Fin where\n  \u27e6zero\u27e7 : \u2200 {n\u2080 n\u2081}(n : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6Fin\u27e7 (\u27e6suc\u27e7 n) zero zero\n  \u27e6suc\u27e7  : (\u2200\u27e8 n \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Fin\u27e7 n \u27e6\u2192\u27e7 \u27e6Fin\u27e7 (\u27e6suc\u27e7 n)) suc suc\n-}\n\n{- TODO\ndata \u27e6Fin\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) Fin Fin\n\nprivate\n  \u27e6Fin\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote Fin \u2254 quote \u27e6Fin\u27e7 ] \u27e6\u2115\u27e7-env) c)\n\ndata \u27e6Fin\u27e7 where\n  \u27e6zero\u27e7 : unquote (\u27e6Fin\u27e7-ctor (quote Fin.zero))\n  \u27e6suc\u27e7  : unquote (\u27e6Fin\u27e7-ctor (quote Fin.suc ))\n\n{-\ninject\u2081 : \u2200 {m} \u2192 Fin m \u2192 Fin (suc m)\ninject\u2081 zero    = zero\ninject\u2081 (suc i) = suc (inject\u2081 i)\n-}\n\ndefEnv2Fin = extConEnv ([ quote Fin.zero \u2254 quote \u27e6Fin\u27e7.\u27e6zero\u27e7 ] \u2218\n                        [ quote Fin.suc  \u2254 quote \u27e6Fin\u27e7.\u27e6suc\u27e7  ])\n             (extDefEnv [ quote Fin \u2254 quote \u27e6Fin\u27e7 ] \u27e6\u2115\u27e7-env)\n\nopen import Data.Fin using (inject\u2081)\n\nunquoteDecl \u27e6inject\u2081\u27e7 = pFunNameRec defEnv2Fin (quote inject\u2081) \u27e6inject\u2081\u27e7\n-}\n","old_contents":"{-# OPTIONS --with-K #-}\nmodule Data.Fin.Param.Binary where\n\nopen import Data.Fin\nopen import Data.Nat\nopen import Function\n\nopen import Data.Nat.Param.Binary\nopen import Reflection.NP\nopen import Reflection.Param\nopen import Reflection.Param.Env\nopen import Function.Param.Binary\nopen import Type.Param.Binary\n\ndata \u27e6Fin\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) Fin Fin\n\nprivate\n  \u27e6Fin\u27e7-ctor = \u03bb c \u2192 unEl (param-ctor-by-name (extDefEnv [ quote Fin \u2254 quote \u27e6Fin\u27e7 ] \u27e6\u2115\u27e7-env) c)\n\ndata \u27e6Fin\u27e7 where\n  \u27e6zero\u27e7 : unquote (\u27e6Fin\u27e7-ctor (quote Fin.zero))\n  \u27e6suc\u27e7  : unquote (\u27e6Fin\u27e7-ctor (quote Fin.suc ))\n\n{-\ninject\u2081 : \u2200 {m} \u2192 Fin m \u2192 Fin (suc m)\ninject\u2081 zero    = zero\ninject\u2081 (suc i) = suc (inject\u2081 i)\n-}\n\ndefEnv2Fin = extConEnv ([ quote Fin.zero \u2254 quote \u27e6Fin\u27e7.\u27e6zero\u27e7 ] \u2218\n                        [ quote Fin.suc  \u2254 quote \u27e6Fin\u27e7.\u27e6suc\u27e7  ])\n             (extDefEnv [ quote Fin \u2254 quote \u27e6Fin\u27e7 ] \u27e6\u2115\u27e7-env)\n\nopen import Data.Fin using (inject\u2081)\n\nunquoteDecl \u27e6inject\u2081\u27e7 = pFunNameRec defEnv2Fin (quote inject\u2081) \u27e6inject\u2081\u27e7\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"37a180784ff6d77a0e7c83e705073ed948b63373","subject":"IDescTT model: clean-up.","message":"IDescTT model: clean-up.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const X\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> (xs : desc D X) -> desc (All D X xs) R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = k\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (All D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n{-\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = induction R P m i xs\n-}\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (All (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (All (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (All D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9d0cc71c989df6ce77e178afb372046e10dcc3ee","subject":"Adapt one-time-semantic-security","message":"Adapt one-time-semantic-security\n","repos":"crypto-agda\/crypto-agda","old_file":"one-time-semantic-security.agda","new_file":"one-time-semantic-security.agda","new_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Product using (\u2203; module \u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nimport Data.Vec as V\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import Data.Bits\n\nopen import flat-funs\nopen import program-distance\nopen import flipbased-implem\n\n-- Capture the various size parameters required\n-- for a cipher.\n--\n-- M  is the message space (|M| is the size of messages)\n-- C  is the cipher text space\n-- R\u1d49 is the randomness used by the cipher.\nrecord EncParams : Set where\n  constructor mk\n  field\n    |M| |C| |R|\u1d49 : \u2115\n\n  M  = Bits |M|\n  C  = Bits |C|\n  R\u1d49 = Bits |R|\u1d49\n\n  -- The type of the encryption function\n  Enc : Set\n  Enc = M \u2192 \u21ba |R|\u1d49 C\n\nmodule EncParams\u00b2 (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams ep\u2080 public using ()\n    renaming (|M| to |M|\u2080; |C| to |C|\u2080; |R|\u1d49 to |R|\u1d49\u2080; M to M\u2080;\n              C to C\u2080; R\u1d49 to R\u1d49\u2080; Enc to Enc\u2080)\n  open EncParams ep\u2081 public using ()\n    renaming (|M| to |M|\u2081; |C| to |C|\u2081; |R|\u1d49 to |R|\u1d49\u2081; M to M\u2081;\n              C to C\u2081; R\u1d49 to R\u1d49\u2081; Enc to Enc\u2081)\n  Tr   = Enc\u2080 \u2192 Enc\u2081\n  Tr\u207b\u00b9 = Enc\u2081 \u2192 Enc\u2080\n\nmodule EncParams\u00b2-same-|R|\u1d49 (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  ep\u2081 = EncParams.mk |M|\u2081 |C|\u2081 (EncParams.|R|\u1d49 ep\u2080)\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n\nmodule \u266dEncParams {t} {T : Set t}\n                  (\u266dFuns : FlatFuns T)\n                  (ep : EncParams) where\n  open FlatFuns \u266dFuns\n  open EncParams ep public\n\n  `M  = `Bits |M|\n  `C  = `Bits |C|\n  `R\u1d49 = `Bits |R|\u1d49\n\nmodule \u266dEncParams\u00b2 {t} {T : Set t}\n                   (\u266dFuns : FlatFuns T)\n                   (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n  open \u266dEncParams \u266dFuns ep\u2080 public using () renaming (`M to `M\u2080; `C to `C\u2080; `R\u1d49 to `R\u1d49\u2080)\n  open \u266dEncParams \u266dFuns ep\u2081 public using () renaming (`M to `M\u2081; `C to `C\u2081; `R\u1d49 to `R\u1d49\u2081)\n\nmodule AbsSemSec {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  record SemSecAdv (ep : EncParams) |R|\u1d43 : Set where\n    constructor mk\n\n    open \u266dEncParams \u266dFuns ep\n\n    field\n      {|S|} : \u2115\n\n    `S  = `Bits |S|\n    S   =  Bits |S|\n\n    -- Randomness adversary\n    `R\u1d43 = `Bits |R|\u1d43\n    R\u1d43  =  Bits |R|\u1d43\n\n    field\n      step\u2080 : `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `S\n      step\u2081 : `C `\u00d7 `S `\u2192 `Bit\n\n    M\u00b2  = Bit \u2192 M\n    `M\u00b2 = `Bit `\u2192 `M\n\n    module Ops (\u266dops : FlatFunsOps \u266dFuns) where\n      open FlatFunsOps \u266dops\n      observe : `C `\u00d7 `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `Bit\n      observe = idO *** step\u2080 >>> \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 >>> idO *** step\u2081\n        where \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 = < snd >>> fst , < fst , snd >>> snd > >\n\n    open \u266dEncParams \u266dFuns ep public\n\n  SemSecReduction : \u2200 ep\u2080 ep\u2081 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction ep\u2080 ep\u2081 f = \u2200 {c} \u2192 SemSecAdv ep\u2080 c \u2192 SemSecAdv ep\u2081 (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) where\n  open PrgDist prgDist\n  open AbsSemSec fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  module FunSemSecAdv {ep : EncParams} {|R|\u1d43} (A : SemSecAdv ep |R|\u1d43) where\n    open SemSecAdv A public\n    open Ops fun\u266dOps public\n\n    step\u2080F : R\u1d43 \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R|\u1d43 (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  module RunSemSec (ep : EncParams) where\n    open EncParams ep using (M; C; |R|\u1d49; Enc)\n\n    -- Returing 0 means Chal wins, Adv looses\n    --          1 means Adv  wins, Chal looses\n    runSemSec : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    runSemSec E A b\n      = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n      where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n    _\u21c4_ : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    _\u21c4_ = runSemSec\n\n    _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : SemSecAdv ep p) \u2192 Set\n    A\u2080 \u2257A A\u2081 = observe A\u2080 \u2257 observe A\u2081\n      where open FunSemSecAdv\n\n    change-adv : \u2200 {ca} (A\u2080 A\u2081 : SemSecAdv ep ca) E \u2192 A\u2080 \u2257A A\u2081 \u2192 (E \u21c4 A\u2080) \u2257\u2141 (E \u21c4 A\u2081)\n    change-adv {ca} _ _ _ pf b R with V.splitAt ca R\n    change-adv A\u2080 A\u2081 E pf b ._ | pre \u03a3., post , \u2261.refl = \u2261.trans (\u2261.cong proj\u2082 (helper\u2080 A\u2080)) helper\u2082\n       where open FunSemSecAdv\n             helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post , pre)\n             helper\u2082 = \u2261.cong (\u03bb m \u2192 step\u2081 A\u2081 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2081 pre)))\n                              (helper\u2080 A\u2081)\n\n    _\u2257E_ : \u2200 (E\u2080 E\u2081 : Enc) \u2192 Set\n    E\u2080 \u2257E E\u2081 = \u2200 m \u2192 E\u2080 m \u2257\u21ba E\u2081 m\n\n    \u2257E\u2192\u2257\u2141 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c)\n               \u2192 (E\u2080 \u21c4 A) \u2257\u2141 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 {c} A b R\n      rewrite E\u2080\u2257E\u2081 (proj (proj\u2081 (SemSecAdv.step\u2080 A (V.take c R))) b) (V.drop c R) = \u2261.refl\n\n    \u2257A\u2192\u21d3 : \u2200 {c} (A\u2080 A\u2081 : SemSecAdv ep c) \u2192 A\u2080 \u2257A A\u2081 \u2192 \u2200 E \u2192 (E \u21c4 A\u2080) \u21d3 (E \u21c4 A\u2081)\n    \u2257A\u2192\u21d3 A\u2080 A\u2081 A\u2080\u2257A\u2081 E = extensional-reduction (change-adv A\u2080 A\u2081 E A\u2080\u2257A\u2081)\n\n    \u2257E\u2192\u21d3 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c) \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u21d3 E\u2080\u2257E\u2081 A = extensional-reduction (\u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 A)\n\n  module SemSecReductions (ep\u2080 ep\u2081 : EncParams) (f : Coins \u2192 Coins) where\n    open EncParams\u00b2 ep\u2080 ep\u2081\n    open RunSemSec ep\u2080 public using () renaming (_\u21c4_ to _\u21c4\u2080_; _\u2257E_ to _\u2257E\u2080_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2080)\n    open RunSemSec ep\u2081 public using () renaming (_\u21c4_ to _\u21c4\u2081_; _\u2257E_ to _\u2257E\u2081_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2081)\n\n    Reduction : Set\n    Reduction = SemSecReduction ep\u2081 ep\u2080 f\n\n    SemSecTr : Tr \u2192 Set\n    SemSecTr tr =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u207b\u00b9 : Tr\u207b\u00b9 \u2192 Set\n    SemSecTr\u207b\u00b9 tr\u207b\u00b9 =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u2192SemSecTr\u207b\u00b9 : \u2200 tr tr\u207b\u00b9 (tr-right-inv : \u2200 E \u2192 tr (tr\u207b\u00b9 E) \u2257E\u2081 E)\n                          \u2192 SemSecTr tr \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9\n    SemSecTr\u2192SemSecTr\u207b\u00b9 _ tr\u207b\u00b9 tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n            helper E A A-breaks-E = pf (tr\u207b\u00b9 E) A A-breaks-tr-tr\u207b\u00b9-E\n              where A-breaks-tr-tr\u207b\u00b9-E = \u2257E\u2192\u21d3\u2081 (\u03bb m R \u2192 \u2261.sym (tr-inv E m R)) A A-breaks-E\n\n    SemSecTr\u207b\u00b9\u2192SemSecTr : \u2200 tr tr\u207b\u00b9 (tr-left-inv : \u2200 E \u2192 tr\u207b\u00b9 (tr E) \u2257E\u2080 E)\n                          \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9 \u2192 SemSecTr tr\n    SemSecTr\u207b\u00b9\u2192SemSecTr tr _ tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n            helper E {c} A A-breaks-E = \u2257E\u2192\u21d3\u2080 (tr-inv E) (red A) (pf (tr E) A A-breaks-E)\n\n","old_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Product using (\u2203; module \u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nopen import Data.Vec using (splitAt; take; drop)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import Data.Bits\n\nopen import flat-funs\nopen import program-distance\nopen import flipbased-implem\n\n-- Capture the various size parameters required\n-- for a cipher.\n--\n-- M  is the message space (|M| is the size of messages)\n-- C  is the cipher text space\n-- R\u1d49 is the randomness used by the cipher.\nrecord EncParams : Set where\n  constructor mk\n  field\n    |M| |C| |R|\u1d49 : \u2115\n\n  M  = Bits |M|\n  C  = Bits |C|\n  R\u1d49 = Bits |R|\u1d49\n\n  -- The type of the encryption function\n  Enc : Set\n  Enc = M \u2192 \u21ba |R|\u1d49 C\n\nmodule EncParams\u00b2 (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams ep\u2080 public using ()\n    renaming (|M| to |M|\u2080; |C| to |C|\u2080; |R|\u1d49 to |R|\u1d49\u2080; M to M\u2080;\n              C to C\u2080; R\u1d49 to R\u1d49\u2080; Enc to Enc\u2080)\n  open EncParams ep\u2081 public using ()\n    renaming (|M| to |M|\u2081; |C| to |C|\u2081; |R|\u1d49 to |R|\u1d49\u2081; M to M\u2081;\n              C to C\u2081; R\u1d49 to R\u1d49\u2081; Enc to Enc\u2081)\n  Tr   = Enc\u2080 \u2192 Enc\u2081\n  Tr\u207b\u00b9 = Enc\u2081 \u2192 Enc\u2080\n\nmodule EncParams\u00b2-same-|R|\u1d49 (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  ep\u2081 = EncParams.mk |M|\u2081 |C|\u2081 (EncParams.|R|\u1d49 ep\u2080)\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n\nmodule \u266dEncParams {t} {T : Set t}\n                  (\u266dFuns : FlatFuns T)\n                  (ep : EncParams) where\n  open FlatFuns \u266dFuns\n  open EncParams ep public\n\n  `M  = `Bits |M|\n  `C  = `Bits |C|\n  `R\u1d49 = `Bits |R|\u1d49\n\nmodule \u266dEncParams\u00b2 {t} {T : Set t}\n                   (\u266dFuns : FlatFuns T)\n                   (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n  open \u266dEncParams \u266dFuns ep\u2080 public using () renaming (`M to `M\u2080; `C to `C\u2080; `R\u1d49 to `R\u1d49\u2080)\n  open \u266dEncParams \u266dFuns ep\u2081 public using () renaming (`M to `M\u2081; `C to `C\u2081; `R\u1d49 to `R\u1d49\u2081)\n\nmodule AbsSemSec {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  record SemSecAdv (ep : EncParams) |R|\u1d43 : Set where\n    constructor mk\n\n    open \u266dEncParams \u266dFuns ep\n\n    field\n      {|S|} : \u2115\n\n    `S  = `Bits |S|\n    S   =  Bits |S|\n\n    -- Randomness adversary\n    `R\u1d43 = `Bits |R|\u1d43\n    R\u1d43  =  Bits |R|\u1d43\n\n    field\n      step\u2080 : `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `S\n      step\u2081 : `C `\u00d7 `S `\u2192 `Bit\n\n    M\u00b2  = Bit \u2192 M\n    `M\u00b2 = `Bit `\u2192 `M\n\n    module Ops (\u266dops : FlatFunsOps \u266dFuns) where\n      open FlatFunsOps \u266dops\n      observe : `C `\u00d7 `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `Bit\n      observe = idO *** step\u2080 >>> \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 >>> idO *** step\u2081\n        where \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 = < snd >>> fst , < fst , snd >>> snd > >\n\n    open \u266dEncParams \u266dFuns ep public\n\n  SemSecReduction : \u2200 ep\u2080 ep\u2081 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction ep\u2080 ep\u2081 f = \u2200 {c} \u2192 SemSecAdv ep\u2080 c \u2192 SemSecAdv ep\u2081 (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) where\n  open PrgDist prgDist\n  open AbsSemSec fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  module FunSemSecAdv {ep : EncParams} {|R|\u1d43} (A : SemSecAdv ep |R|\u1d43) where\n    open SemSecAdv A public\n    open Ops fun\u266dOps public\n\n    step\u2080F : R\u1d43 \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R|\u1d43 (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  module RunSemSec (ep : EncParams) where\n    open EncParams ep using (M; C; |R|\u1d49; Enc)\n\n    -- Returing 0 means Chal wins, Adv looses\n    --          1 means Adv  wins, Chal looses\n    runSemSec : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    runSemSec E A b\n      = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n      where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n    _\u21c4_ : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    _\u21c4_ = runSemSec\n\n    _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : SemSecAdv ep p) \u2192 Set\n    A\u2080 \u2257A A\u2081 = observe A\u2080 \u2257 observe A\u2081\n      where open FunSemSecAdv\n\n    change-adv : \u2200 {ca} (A\u2080 A\u2081 : SemSecAdv ep ca) E \u2192 A\u2080 \u2257A A\u2081 \u2192 (E \u21c4 A\u2080) \u2257\u2141 (E \u21c4 A\u2081)\n    change-adv {ca} _ _ _ pf b R with splitAt ca R\n    change-adv A\u2080 A\u2081 E pf b ._ | pre \u03a3., post , \u2261.refl = \u2261.trans (\u2261.cong proj\u2082 (helper\u2080 A\u2080)) helper\u2082\n       where open FunSemSecAdv\n             helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post , pre)\n             helper\u2082 = \u2261.cong (\u03bb m \u2192 step\u2081 A\u2081 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2081 pre)))\n                              (helper\u2080 A\u2081)\n\n    _\u2257E_ : \u2200 (E\u2080 E\u2081 : Enc) \u2192 Set\n    E\u2080 \u2257E E\u2081 = \u2200 m \u2192 E\u2080 m \u2257\u21ba E\u2081 m\n\n    \u2257E\u2192\u2257\u2141 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c)\n               \u2192 (E\u2080 \u21c4 A) \u2257\u2141 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 {c} A b R\n      rewrite E\u2080\u2257E\u2081 (proj (proj\u2081 (SemSecAdv.step\u2080 A (take c R))) b) (drop c R) = \u2261.refl\n\n    \u2257A\u2192\u21d3 : \u2200 {c} (A\u2080 A\u2081 : SemSecAdv ep c) \u2192 A\u2080 \u2257A A\u2081 \u2192 \u2200 E \u2192 (E \u21c4 A\u2080) \u21d3 (E \u21c4 A\u2081)\n    \u2257A\u2192\u21d3 A\u2080 A\u2081 A\u2080\u2257A\u2081 E = extensional-reduction (change-adv A\u2080 A\u2081 E A\u2080\u2257A\u2081)\n\n    \u2257E\u2192\u21d3 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c) \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u21d3 E\u2080\u2257E\u2081 A = extensional-reduction (\u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 A)\n\n  module SemSecReductions (ep\u2080 ep\u2081 : EncParams) (f : Coins \u2192 Coins) where\n    open EncParams\u00b2 ep\u2080 ep\u2081\n    open RunSemSec ep\u2080 public using () renaming (_\u21c4_ to _\u21c4\u2080_; _\u2257E_ to _\u2257E\u2080_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2080)\n    open RunSemSec ep\u2081 public using () renaming (_\u21c4_ to _\u21c4\u2081_; _\u2257E_ to _\u2257E\u2081_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2081)\n\n    Reduction : Set\n    Reduction = SemSecReduction ep\u2081 ep\u2080 f\n\n    SemSecTr : Tr \u2192 Set\n    SemSecTr tr =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u207b\u00b9 : Tr\u207b\u00b9 \u2192 Set\n    SemSecTr\u207b\u00b9 tr\u207b\u00b9 =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u2192SemSecTr\u207b\u00b9 : \u2200 tr tr\u207b\u00b9 (tr-right-inv : \u2200 E \u2192 tr (tr\u207b\u00b9 E) \u2257E\u2081 E)\n                          \u2192 SemSecTr tr \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9\n    SemSecTr\u2192SemSecTr\u207b\u00b9 _ tr\u207b\u00b9 tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n            helper E A A-breaks-E = pf (tr\u207b\u00b9 E) A A-breaks-tr-tr\u207b\u00b9-E\n              where A-breaks-tr-tr\u207b\u00b9-E = \u2257E\u2192\u21d3\u2081 (\u03bb m R \u2192 \u2261.sym (tr-inv E m R)) A A-breaks-E\n\n    SemSecTr\u207b\u00b9\u2192SemSecTr : \u2200 tr tr\u207b\u00b9 (tr-left-inv : \u2200 E \u2192 tr\u207b\u00b9 (tr E) \u2257E\u2080 E)\n                          \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9 \u2192 SemSecTr tr\n    SemSecTr\u207b\u00b9\u2192SemSecTr tr _ tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n            helper E {c} A A-breaks-E = \u2257E\u2192\u21d3\u2080 (tr-inv E) (red A) (pf (tr E) A A-breaks-E)\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b16560417a6803cf36fd7e7ebbd7d94261e4af5d","subject":"Desc model: implicit box","message":"Desc model: implicit box\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box I D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box Unit (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ca3585f55fde86ac7d664ac346618396c7308e11","subject":"Implement apply-valid-change.","message":"Implement apply-valid-change.\n\nOld-commit-hash: 64cc4483b462c94a945aa312124cbdf090ffd619\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = ValidChange \u03c3 \u2192 Change \u03c4\n\n  before : ValidChange \u2286 \u27e6_\u27e7\n  before (cons v _ _) = v\n\n  after : ValidChange \u2286 \u27e6_\u27e7\n  after {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  change : ValidChange \u2286 Change\n  change (cons _ dv _) = dv\n\n  apply-valid-change : \u2200 \u03c4 \u2192 (v : \u27e6 \u03c4 \u27e7) (dv : ValidChange \u03c4) \u2192 \u27e6 \u03c4 \u27e7\n  apply-valid-change \u03c4 v dv = apply-change \u03c4 v (change dv)\n\n  diff-valid-change : \u2200 \u03c4 \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 ValidChange \u03c4\n  diff-valid-change \u03c4 u v = cons v (u \u229f\u208d \u03c4 \u208e v) (R[v,u-v] {\u03c4} {u} {v})\n\n  nil-valid-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 ValidChange \u03c4\n  nil-valid-change \u03c4 v = diff-valid-change \u03c4 v v\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f (nil-valid-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u2192 g (after v) \u229f\u208d \u03c4 \u208e f (before v)\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : ValidChange \u03c3)\n    \u2192 valid {\u03c4} (f (before v)) (\u0394f v)\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after v) \u2261 f (before v) \u229e\u208d \u03c4 \u208e \u0394f v\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u2192\n    let\n      w\u2032 = after {\u03c3} w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v (before {\u03c3} w)}) \u27e9\n        v (before {\u03c3} w) \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- applicative structure\n  call-valid-change : \u2200 \u03c3 \u03c4 \u2192\n    ValidChange (\u03c3 \u21d2 \u03c4) \u2192\n    ValidChange \u03c3 \u2192\n    ValidChange \u03c4\n  call-valid-change \u03c3 \u03c4 (cons f \u0394f R[f,\u0394f]) \u0394v =\n    cons (f (before {\u03c3} \u0394v)) (\u0394f \u0394v) (proj\u2081 (R[f,\u0394f] \u0394v))\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 before {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 after {\u03c4})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = ValidChange \u03c3 \u2192 Change \u03c4\n\n  before : ValidChange \u2286 \u27e6_\u27e7\n  before (cons v _ _) = v\n\n  after : ValidChange \u2286 \u27e6_\u27e7\n  after {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  change : ValidChange \u2286 Change\n  change (cons _ dv _) = dv\n\n  diff-valid-change : \u2200 \u03c4 \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 ValidChange \u03c4\n  diff-valid-change \u03c4 u v = cons v (u \u229f\u208d \u03c4 \u208e v) (R[v,u-v] {\u03c4} {u} {v})\n\n  nil-valid-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 ValidChange \u03c4\n  nil-valid-change \u03c4 v = diff-valid-change \u03c4 v v\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f (nil-valid-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u2192 g (after v) \u229f\u208d \u03c4 \u208e f (before v)\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : ValidChange \u03c3)\n    \u2192 valid {\u03c4} (f (before v)) (\u0394f v)\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after v) \u2261 f (before v) \u229e\u208d \u03c4 \u208e \u0394f v\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u2192\n    let\n      w\u2032 = after {\u03c3} w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v (before {\u03c3} w)}) \u27e9\n        v (before {\u03c3} w) \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- applicative structure\n  call-valid-change : \u2200 \u03c3 \u03c4 \u2192\n    ValidChange (\u03c3 \u21d2 \u03c4) \u2192\n    ValidChange \u03c3 \u2192\n    ValidChange \u03c4\n  call-valid-change \u03c3 \u03c4 (cons f \u0394f R[f,\u0394f]) \u0394v =\n    cons (f (before {\u03c3} \u0394v)) (\u0394f \u0394v) (proj\u2081 (R[f,\u0394f] \u0394v))\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 before {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 after {\u03c4})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"baf1ba3db9a1cae8240c23a739c1fa92746c0a64","subject":"NatChal: based on NumChal now","message":"NatChal: based on NumChal now\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom\/NatChal.agda","new_file":"ZK\/GroupHom\/NatChal.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function using (flip)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum.NP\nopen import Data.Zero\nopen import Data.Fin.NP using (Fin\u25b9\u2115)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nopen import Data.Nat.NP hiding (_==_; _^_) renaming (_+_ to _+\u2115_; _*_ to _*\u2115_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Primality\nopen import Data.Nat.DivMod.NP\nimport Data.Nat.ModInv\nimport ZK.SigmaProtocol.KnownStatement\nimport ZK.GroupHom\nopen import ZK.GroupHom.NumChal\n\nmodule ZK.GroupHom.NatChal\n  (G+ G* : Type)\n  (\ud835\udd3e+ : Group G+)\n  (\ud835\udd3e* : Group G*)\n  (_==_ : G* \u2192 G* \u2192 Bool)\n  (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n  (==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y)\n  (\u03c6 : G+ \u2192 G*)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* \u03c6)\n  (Y : G*)\n\n  (q : \u2115)\n  (q-prime : Prime q)\n\n  (open Multiplicative-Group \ud835\udd3e*)\n\n  -- Can be turned into a dynamic test!\n  (G*-order : Y ^\u207a q \u2261 1#)\n\n  (open Data.Nat.ModInv q q-prime)\n  (missing : (x : \u2115) \u2192 x \u2262 0 \u2192 (1\/ x *\u2115 x) mod\u2115 q \u2261 1)\n  where\n\nopen Additive-Group \ud835\udd3e+\nmodule \u03c6 = GroupHomomorphism \u03c6-hom\nmodule \ud835\udd3e* = Group \ud835\udd3e*\n\nhelp! : \u2200 x y \u2192 y < x \u2192 x \u2238 y \u2262 0\nhelp! ._ .0 (s\u2264s z\u2264n) ()\nhelp! ._ ._ (s\u2264s (s\u2264s p)) x = help! _ _ (s\u2264s p) x\n              \n\u2115-package : Package\n\u2115-package = record\n              { Num = \u2115\n              ; Prime = Prime\n              ; _<_ = _<_\n              ; 0\u207f = 0\n              ; 1\u207f = 1\n              ; <-\u2238\u22620 = help!\n              ; _+\u207f_ = _+\u2115_\n              ; _\u2238\u207f_ = _\u2238_\n              ; _*\u207f_ = _*\u2115_\n              ; compare = \u2115cmp.compare\n              ; G+ = G+\n              ; G* = G*\n              ; \ud835\udd3e+ = \ud835\udd3e+\n              ; \ud835\udd3e* = \ud835\udd3e*\n              ; _==_ = _==_\n              ; \u2713-== = \u2713-==\n              ; ==-\u2713 = ==-\u2713\n              ; _\u2297\u207f_ = _\u2297\u207a_\n              ; _^\u207f_ = _^\u207a_\n              ; 1^\u207f = 1^\u207a\n              ; \u03c6 = \u03c6\n              ; \u03c6-hom = \u03c6-hom\n              ; \u03c6-hom-iterated = \u03bb {_}{x} \u2192 \u03c6.hom-iterated\u207a x\n              ; q = q\n              ; q-prime = q-prime\n              ; _div-q = \u03bb x \u2192 x div q\n              ; _mod-q = \u03bb x \u2192 x mod\u2115 q\n              ; div-mod-q-prop\u207f = \u03bb x \u2192 divModProp\u2115 x q\n              ; inv-mod-q = 1\/\n              ; inv-mod-q-prop = missing\n              ; Y = Y\n              ; G*-order = G*-order\n              ; ^\u207f1\u207f+\u207f = idp\n              ; ^\u207f-* = \u03bb {x} {y} \u2192 \ud835\udd3e*.^\u207a-* y {x} {Y}\n              ; ^\u207f-\u2238\u207f = \u03bb {x} {y} y<x \u2192 \ud835\udd3e*.^\u207a-\u2238 {Y} {x} {y} (\u2264-pred (\u2264-step y<x))\n              }\n\nopen FromPackage \u2115-package public\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function using (flip)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum.NP\nopen import Data.Zero\nopen import Data.Fin.NP using (Fin\u25b9\u2115)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nopen import Data.Nat.NP hiding (_==_; _^_) renaming (_+_ to _+\u2115_; _*_ to _*\u2115_)\nopen import Data.Nat.Properties\nimport ZK.SigmaProtocol.KnownStatement\nimport ZK.GroupHom\n\nmodule ZK.GroupHom.NatChal\n  where\npostulate\n  G+ G* : Type\n  \ud835\udd3e+ : Group G+\n  \ud835\udd3e* : Group G*\n  _==_ : G* \u2192 G* \u2192 Bool\n  \u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n  ==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y\n  \u03c6 : G+ \u2192 G*\n  \u03c6-hom : GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* \u03c6\n  Y : G*\n  q : \u2115\n\nopen Additive-Group \ud835\udd3e+\nopen Multiplicative-Group \ud835\udd3e*\n\n-- TODO\npostulate\n  _div_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  _mod_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  div-mod-spec : \u2200 {n q} \u2192 n \u2261 (n mod q) +\u2115 q *\u2115 (n div q)\n  1\/ : \u2115 \u2192 \u2115\n  1\/-prop : \u2200 n \u2192 (1\/ n *\u2115 n) mod q \u2261 1\n\n  -- Can be turned into a dynamic test!\n  G*-order : Y ^\u207a q \u2261 1#\n\nopen \u2261-Reasoning\n\nmodule \u03c6 = GroupHomomorphism \u03c6-hom\nmodule \ud835\udd3e* = Group \ud835\udd3e*\n\nY^-^-1\/-id : \u2200 {x y} \u2192 x > y \u2192 (Y ^\u207a(x \u2238 y))^\u207a(1\/(x \u2238 y)) \u2261 Y\nY^-^-1\/-id {x} {y} x>y\n  = (Y ^\u207a d)^\u207a(1\/ d)     \u2261\u27e8 ! \ud835\udd3e*.^\u207a-* (1\/ d) \u27e9\n    Y ^\u207a(1\/ d *\u2115 d)      \u2261\u27e8 ap (_^\u207a_ Y) div-mod-spec \u27e9\n    Y ^\u207a(r +\u2115 q *\u2115 m)    \u2261\u27e8 ap (\u03bb z \u2192 Y ^\u207a(z +\u2115 q *\u2115 m)) (1\/-prop d) \u27e9\n    Y ^\u207a(1+(q *\u2115 m))     \u2261\u27e8 ap (\u03bb z \u2192 Y ^\u207a(1+ z)) (\u2115\u00b0.*-comm q m) \u27e9\n    Y ^\u207a(1+(m *\u2115 q))     \u2261\u27e8by-definition\u27e9\n    Y * Y ^\u207a(m *\u2115 q)     \u2261\u27e8 *= idp (\ud835\udd3e*.^\u207a-* m) \u27e9\n    Y * (Y ^\u207a q)^\u207a m     \u2261\u27e8 ap (\u03bb z \u2192 Y * z ^\u207a m) G*-order \u27e9\n    Y * 1# ^\u207a m          \u2261\u27e8 *= idp (1^\u207a m) \u27e9\n    Y * 1#               \u2261\u27e8 *1-identity \u27e9\n    Y \u220e\n    where d = x \u2238 y\n          e = 1\/ d *\u2115 d\n          m = e div q\n          r = e mod q\n\nswap? : {c\u2080 c\u2081 : \u2115} \u2192 c\u2080 \u2262 c\u2081 \u2192 (c\u2080 > c\u2081) \u228e (c\u2081 > c\u2080)\nswap? {x} {y} i with \u2115cmp.compare x y\nswap? i | tri< a \u00acb \u00acc = inr a\nswap? i | tri> \u00aca \u00acb c = inl c\nswap? i | tri\u2248 \u00aca b \u00acc = \ud835\udfd8-elim (i b)\n\nopen ZK.GroupHom \ud835\udd3e+ \ud835\udd3e* _==_ \u2713-== ==-\u2713 _>_ swap? _\u2297\u207a_ _^\u207a_ _\u2238_ 1\/\n                 \u03c6 \u03c6-hom (\u03bb {x} {n} \u2192 \u03c6.hom-iterated\u207a n)\n                 Y\n                 (\u03bb {x}{y}i \u2192 \ud835\udd3e*.^\u207a-\u2238 {Y}{x}{y} (>\u2192\u2265 i))\n                 (\u03bb{x}{y}i \u2192 Y^-^-1\/-id{x}{y}i)\n  public\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ce1eeccd327d499bc58a352ee0b57ebaadf449f7","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: ec7283e160acc6e4bc8397a7432be950\n\ndarcs-hash:20110306215407-3bd4e-ce81fdf6dda1dc9bcce86985ca2368fe6fc4791e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Mirror\/TreeR\/Induction\/Acc\/WellFoundedInduction.agda","new_file":"Draft\/FOTC\/Program\/Mirror\/TreeR\/Induction\/Acc\/WellFoundedInduction.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation TreeT\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Program.Mirror.TreeR.Induction.Acc.WellFoundedInduction\n       where\n\nopen import FOTC.Base\n\nopen import Draft.FOTC.Program.Mirror.Induction.Acc.WellFounded\nopen import Draft.FOTC.Program.Mirror.TreeR\nopen import FOTC.Program.Mirror.Type\n\n------------------------------------------------------------------------------\n-- The relation TreeR is well-founded.\npostulate\n  wf-TreeR : WellFounded TreeR\n\n-- Well-founded induction on the relation TreeT.\nwfInd-TreeR :\n  (P : D \u2192 Set) \u2192\n  (\u2200 {t} \u2192 Tree t \u2192 (\u2200 {t'} \u2192 Tree t' \u2192 TreeR t' t \u2192 P t') \u2192 P t) \u2192\n  \u2200 {t} \u2192 Tree t \u2192 P t\nwfInd-TreeR P = WellFoundedInduction wf-TreeR\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation TreeT\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Program.Mirror.TreeR.Induction.Acc.WellFoundedInduction\n       where\n\nopen import FOTC.Base\n\nopen import Draft.FOTC.Program.Mirror.Induction.Acc.WellFounded\nopen import FOTC.Program.Mirror.Type\nopen import Draft.FOTC.Program.Mirror.TreeR\n\n------------------------------------------------------------------------------\n-- The relation TreeR is well-founded.\npostulate\n  wf-TreeR : WellFounded TreeR\n\n-- Well-founded induction on the relation TreeT.\nwfInd-TreeR :\n  (P : D \u2192 Set) \u2192\n  (\u2200 {t} \u2192 Tree t \u2192 (\u2200 {t'} \u2192 Tree t' \u2192 TreeR t' t \u2192 P t') \u2192 P t) \u2192\n  \u2200 {t} \u2192 Tree t \u2192 P t\nwfInd-TreeR P = WellFoundedInduction wf-TreeR\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fbf5d17dabb1058927efc56a1b0c092c4e68e355","subject":"Change \u0394 to expect the change in a free variable.","message":"Change \u0394 to expect the change in a free variable.\n\nProposed by Paolo in a whiteboard discussion today.\n\nOld-commit-hash: bce0357c249c6ae66ad8fe002147ad07623d842f\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\n  -- `\u0394 \u03c4` is the type of changes to `\u03c4`\n  \u0394 : (\u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 \u0394 \u03c4 \u27e7Type = \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 x dx t` maps changes `dx` in `x` to changes in `t`\n  \u0394 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (x : Var \u0393 \u03c4\u2081) (dx : Var \u0393 (\u0394 \u03c4\u2081)) \u2192 (t : Term \u0393 \u03c4\u2082) \u2192 Term \u0393 (\u0394 \u03c4\u2082)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 x dx t \u27e7Term \u03c1 = \u03bb _ \u2192 \u27e6 t \u27e7Term (update x (\u27e6 dx \u27e7 \u03c1) \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 x dx t) = \u0394 (lift {\u0393\u2081} {\u0393\u2082} x) (lift {\u0393\u2081} {\u0393\u2082} dx) (weaken {\u0393\u2081} {\u0393\u2082} t)\n","old_contents":"module incremental where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\n  -- `\u0394 \u03c4` is the type of changes to `\u03c4`\n  \u0394 : (\u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 \u0394 \u03c4 \u27e7Type = \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 x t` maps changes in `x` to changes in `t`\n  \u0394 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (x : Var \u0393 \u03c4\u2081) \u2192 (t : Term \u0393 \u03c4\u2082) \u2192 Term \u0393 (\u0394 \u03c4\u2081 \u21d2 \u0394 \u03c4\u2082)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 x t \u27e7Term \u03c1 = \u03bb \u0394x _ \u2192 \u27e6 t \u27e7Term (update x \u0394x \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 x t) = \u0394 (lift {\u0393\u2081} {\u0393\u2082} x) (weaken {\u0393\u2081} {\u0393\u2082} t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"28e34c3b73b6de3da8a10271fd4286d87e4b4b54","subject":"IDesc model: 0wn3d. No type-in-type but a serious Otis lift. Got up to making Desc constructors in Desc.","message":"IDesc model: 0wn3d. No type-in-type but a serious Otis lift. Got up to making Desc constructors in Desc.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\n{-\nvarl : (l : Level)(I : Set l) -> IDescl l I\nvarl x I = con (lvar , {!!})\n-}\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lconst , X)\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lprod , (D , D'))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1d32ad0cbc8582d30660167abe83ddda4a4cf6d1","subject":"Dec: add Dec-{\ud835\udfd8,\ud835\udfd9,\u00d7,\u2192}","message":"Dec: add Dec-{\ud835\udfd8,\ud835\udfd9,\u00d7,\u2192}\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Nullary\/NP.agda","new_file":"lib\/Relation\/Nullary\/NP.agda","new_contents":"module Relation.Nullary.NP where\n\nopen import Type\nopen import Function\nopen import Data.Zero\nopen import Data.One\nopen import Data.Sum\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary public\n\nDec-\ud835\udfd8 : Dec \ud835\udfd8\nDec-\ud835\udfd8 = no id\n\nDec-\ud835\udfd9 : Dec \ud835\udfd9\nDec-\ud835\udfd9 = yes _\n\nmodule _ {a b} {A : \u2605_ a} {B : \u2605_ b} where\n    Dec-\u228e : Dec A \u2192 Dec B \u2192 Dec (A \u228e B)\n    Dec-\u228e (yes p) _       = yes (inj\u2081 p)\n    Dec-\u228e (no \u00acp) (yes q) = yes (inj\u2082 q)\n    Dec-\u228e (no \u00acp) (no \u00acq) = no [ \u00acp , \u00acq ]\n\n    Dec-\u00d7 : Dec A \u2192 Dec B \u2192 Dec (A \u00d7 B)\n    Dec-\u00d7 (no \u00acp) _       = no  (\u00acp \u2218 fst)\n    Dec-\u00d7 (yes _) (no \u00acq) = no  (\u00acq \u2218 snd)\n    Dec-\u00d7 (yes p) (yes q) = yes (p , q)\n\n    Dec-\u2192 : Dec A \u2192 Dec B \u2192 Dec (A \u2192 B)\n    Dec-\u2192 _       (yes q) = yes (\u03bb _ \u2192 q)\n    Dec-\u2192 (no \u00acp) _       = yes (\ud835\udfd8-elim \u2218 \u00acp)\n    Dec-\u2192 (yes p) (no \u00acq) = no  (\u03bb f \u2192 \u00acq (f p))\n\n    module _ (to : A \u2192 B)(from : B \u2192 A) where\n\n      map-Dec : Dec A \u2192 Dec B\n      map-Dec (yes p) = yes (to p)\n      map-Dec (no \u00acp) = no  (\u00acp \u2218 from)\n","old_contents":"module Relation.Nullary.NP where\n\nopen import Type\nopen import Data.Sum\nopen import Relation.Nullary public\n\nmodule _ {a b} {A : \u2605_ a} {B : \u2605_ b} where\n    Dec-\u228e : Dec A \u2192 Dec B \u2192 Dec (A \u228e B)\n    Dec-\u228e (yes p) _       = yes (inj\u2081 p)\n    Dec-\u228e (no \u00acp) (yes p) = yes (inj\u2082 p)\n    Dec-\u228e (no \u00acp) (no \u00acq) = no [ \u00acp , \u00acq ]\n\n    module _ (to : A \u2192 B)(from : B \u2192 A) where\n\n      map-Dec : Dec A \u2192 Dec B\n      map-Dec (yes p) = yes (to p)\n      map-Dec (no \u00acp) = no (\u03bb z \u2192 \u00acp (from z))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"185b8d2b2e01ea54dc618e4f0b62acdf7f02b75e","subject":"More field work","message":"More field work\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  -0\u1da0 -1\u1da0 : A\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[_] = fold 0\u1da0 suc\u1da0\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 e = fold 1\u1da0 (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : Field A) where\n  open Field F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : Field-Ops B)(default : Decode-Term B) where\n    module G = Field-Ops G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_\ndata Tm (A : Set) : Set where\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\npattern  0' = lit (\u2124.+ 0)\npattern  1' = lit (\u2124.+ 1)\npattern  2' = lit (\u2124.+ 1)\npattern  3' = lit (\u2124.+ 1)\npattern -1' = lit \u2124.-[1+ 0 ]\npattern -2' = lit \u2124.-[1+ 1 ]\npattern -3' = lit \u2124.-[1+ 2 ]\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : Field F) where\n    open Field F-field renaming (0-_ to -_)\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = \u2124[ x ]\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\n\nrecord RawField {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  -0\u1da0 -1\u1da0 : A\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\nrecord IsField {\u2113} (A : Set \u2113) : Set \u2113 where\n  open FP {\u2113} {A}\n\n  field\n    rawField : RawField A\n  open RawField rawField\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  open RawField rawField public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : IsField A) where\n  open IsField F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : RawField B)(default : Decode-Term B) where\n    module G = RawField G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.Nat using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_ -- _*\u2124_\ndata Tm (A : Set) : Set where\n  -- 0' 1' -1' : Tm A\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : IsField F) where\n    open IsField F-field renaming (0-_ to -_)\n\n    suc\u1da0 : F \u2192 F\n    suc\u1da0 = _+_ 1\u1da0\n\n    nat : \u2115 \u2192 F\n    nat = fold 0\u1da0 suc\u1da0\n\n    eval\u2124 : \u2124 \u2192 F\n    eval\u2124 (\u2124.+ x)    = nat x\n    eval\u2124 \u2124.-[1+ x ] = -(nat (suc x))\n\n    _^\u2115_ : F \u2192 \u2115 \u2192 F\n    b ^\u2115 zero  = 1\u1da0\n    b ^\u2115 suc x = b * b ^\u2115 x\n\n    _^_ : F \u2192 \u2124 \u2192 F\n    b ^ (\u2124.+ x)    = b ^\u2115 x\n    b ^ \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = eval\u2124 x\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"12f04daf9839e8532dba6b4f15f4236b779a46f1","subject":"Added noted about equational reasoing combinators.","message":"Added noted about equational reasoing combinators.\n\nIgnore-this: f3ef6989511dd2e241dcd9e0eb58d71f\n\ndarcs-hash:20120131052649-3bd4e-32deeffe9167cfb6d62be7949c641fa26d0d7e1f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/ComparingEqReasoning.agda","new_file":"notes\/ComparingEqReasoning.agda","new_contents":"------------------------------------------------------------------------------\n-- Comparing styles for equational reasoning\n------------------------------------------------------------------------------\n\nmodule ComparingEqReasoning where\n\ninfix  7 _\u2261_\ninfixl 9  _+_\n\ndata _\u2261_ {A : Set}(x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\ntrans : {A : Set}{x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl refl = refl\n\npostulate\n  \u2115               : Set\n  zero            : \u2115\n  _+_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  +-comm          : (m n : \u2115) \u2192 m + n \u2261 n + m\n  +-rightIdentity : (n : \u2115) \u2192 n + zero \u2261 n\n\nmodule Ulf where\n  -- From Ulf's thesis (p. 112).\n\n  infix  7 _\u2243_\n  infix  5 chain>_\n  infixl 5 _===_by_\n  infix  4 _qed\n\n  data _\u2243_ (x y : \u2115) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\n  chain>_ : (x : \u2115) \u2192 x \u2243 x\n  chain> x = prf refl\n\n  _===_by_ : {x y : \u2115} \u2192 x \u2243 y \u2192 (z : \u2115) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : \u2115} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\n  -- Example\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    chain> zero + n\n       === n + zero by +-comm zero n\n       === n        by +-rightIdentity n\n    qed\n\nmodule Nils where\n  -- Adapted from the standard library (Relation.Binary.PreorderReasoning).\n\n  infix  7 _\u2243_\n  infix  4 begin_\n  infixr 5 _\u2261\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  data _\u2243_ (x y : \u2115) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\n  begin_ : {x y : \u2115} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : \u2115){y z : \u2115} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : \u2115) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n\n  -- Example.\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    begin\n      zero + n \u2261\u27e8 +-comm zero n \u27e9\n      n + zero \u2261\u27e8 +-rightIdentity n \u27e9\n      n\n    \u220e\n\nmodule NonWrapper where\n\n  -- A set of combinators without request a wrapper data type.\n\n  -- From: Shin-Cheng Mu, Hsiang-Shang Ko, and Patrick\n  -- Jansson. Algebra of programming in Agda: Dependent types for\n  -- relational program derivation. Journal of Functional Programming,\n  -- 19(5):545\u2013579, 2009\n\n  infixr 5 _\u2261\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  _\u2261\u27e8_\u27e9_ : (x : \u2115){y z : \u2115} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  _ \u2261\u27e8 x\u2261y \u27e9 y\u2261z = trans x\u2261y y\u2261z\n\n  _\u220e : (x : \u2115) \u2192 x \u2261 x\n  _\u220e _ = refl\n\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    zero + n\n      \u2261\u27e8 +-comm zero n \u27e9\n    n + zero\n      \u2261\u27e8 +-rightIdentity n \u27e9\n    n \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Comparing Ulf and Nils styles for equational reasoning\n------------------------------------------------------------------------------\n\nmodule ComparingEqReasoning where\n\ninfix  7 _\u2261_\ninfixl 9  _+_\n\ndata _\u2261_ {A : Set}(x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\ntrans : {A : Set}{x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl refl = refl\n\npostulate\n  \u2115               : Set\n  zero            : \u2115\n  _+_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  +-comm          : (m n : \u2115) \u2192 m + n \u2261 n + m\n  +-rightIdentity : (n : \u2115) \u2192 n + zero \u2261 n\n\ninfix 7 _\u2243_\n\nprivate\n  data _\u2243_ (x y : \u2115) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\nmodule Ulf where\n  -- Adapted from the Ulf's thesis (p. 112).\n\n  infix  5 chain>_\n  infixl 5 _===_by_\n  infix  4 _qed\n\n  chain>_ : (x : \u2115) \u2192 x \u2243 x\n  chain> x = prf refl\n\n  _===_by_ : {x y : \u2115} \u2192 x \u2243 y \u2192 (z : \u2115) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : \u2115} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\n  -- Example\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    chain> zero + n\n       === n + zero by +-comm zero n\n       === n        by +-rightIdentity n\n    qed\n\nmodule Nils where\n  -- Adapted from the standard library (Relation.Binary.PreorderReasoning).\n\n  infix  4 begin_\n  infixr 5 _\u2261\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  begin_ : {x y : \u2115} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : \u2115){y z : \u2115} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : \u2115) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n\n  -- Example.\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    begin\n      zero + n \u2261\u27e8 +-comm zero n \u27e9\n      n + zero \u2261\u27e8 +-rightIdentity n \u27e9\n      n\n    \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0ea795d1835bb4e398a04244310fe6e1eed4388f","subject":"one-time-semantic-security: add \u266dEncParams\u00b2","message":"one-time-semantic-security: add \u266dEncParams\u00b2\n","repos":"crypto-agda\/crypto-agda","old_file":"one-time-semantic-security.agda","new_file":"one-time-semantic-security.agda","new_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Product using (\u2203; module \u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nopen import Data.Vec using (splitAt; take; drop)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import Data.Bits\n\nopen import flat-funs\nopen import program-distance\nopen import flipbased-implem\n\n-- Capture the various size parameters required\n-- for a cipher.\n--\n-- M  is the message space (|M| is the size of messages)\n-- C  is the cipher text space\n-- R\u1d49 is the randomness used by the cipher.\nrecord EncParams : Set where\n  constructor mk\n  field\n    |M| |C| |R|\u1d49 : \u2115\n\n  M  = Bits |M|\n  C  = Bits |C|\n  R\u1d49 = Bits |R|\u1d49\n\n  -- The type of the encryption function\n  Enc : Set\n  Enc = M \u2192 \u21ba |R|\u1d49 C\n\nmodule EncParams\u00b2 (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams ep\u2080 public using ()\n    renaming (|M| to |M|\u2080; |C| to |C|\u2080; |R|\u1d49 to |R|\u1d49\u2080; M to M\u2080;\n              C to C\u2080; R\u1d49 to R\u1d49\u2080; Enc to Enc\u2080)\n  open EncParams ep\u2081 public using ()\n    renaming (|M| to |M|\u2081; |C| to |C|\u2081; |R|\u1d49 to |R|\u1d49\u2081; M to M\u2081;\n              C to C\u2081; R\u1d49 to R\u1d49\u2081; Enc to Enc\u2081)\n  Tr   = Enc\u2080 \u2192 Enc\u2081\n  Tr\u207b\u00b9 = Enc\u2081 \u2192 Enc\u2080\n\nmodule EncParams\u00b2-same-|R|\u1d49 (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  ep\u2081 = EncParams.mk |M|\u2081 |C|\u2081 (EncParams.|R|\u1d49 ep\u2080)\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n\nmodule \u266dEncParams {t} {T : Set t}\n                  (\u266dFuns : FlatFuns T)\n                  (ep : EncParams) where\n  open FlatFuns \u266dFuns\n  open EncParams ep public\n\n  `M  = `Bits |M|\n  `C  = `Bits |C|\n  `R\u1d49 = `Bits |R|\u1d49\n\nmodule \u266dEncParams\u00b2 {t} {T : Set t}\n                   (\u266dFuns : FlatFuns T)\n                   (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n  open \u266dEncParams \u266dFuns ep\u2080 public using () renaming (`M to `M\u2080; `C to `C\u2080; `R\u1d49 to `R\u1d49\u2080)\n  open \u266dEncParams \u266dFuns ep\u2081 public using () renaming (`M to `M\u2081; `C to `C\u2081; `R\u1d49 to `R\u1d49\u2081)\n\nmodule AbsSemSec {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  record SemSecAdv (ep : EncParams) |R|\u1d43 : Set where\n    constructor mk\n\n    open \u266dEncParams \u266dFuns ep\n\n    field\n      {|S|} : \u2115\n\n    `S  = `Bits |S|\n    S   =  Bits |S|\n\n    -- Randomness adversary\n    `R\u1d43 = `Bits |R|\u1d43\n    R\u1d43  =  Bits |R|\u1d43\n\n    field\n      step\u2080 : `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `S\n      step\u2081 : `C `\u00d7 `S `\u2192 `Bit\n\n    M\u00b2  = Bit \u2192 M\n    `M\u00b2 = `Bit `\u2192 `M\n\n    module Ops (\u266dops : FlatFunsOps \u266dFuns) where\n      open FlatFunsOps \u266dops\n      observe : `C `\u00d7 `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `Bit\n      observe = idO *** step\u2080 >>> \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 >>> idO *** step\u2081\n        where \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 = < snd >>> fst , < fst , snd >>> snd > >\n\n    open \u266dEncParams \u266dFuns ep public\n\n  SemSecReduction : \u2200 ep\u2080 ep\u2081 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction ep\u2080 ep\u2081 f = \u2200 {c} \u2192 SemSecAdv ep\u2080 c \u2192 SemSecAdv ep\u2081 (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) where\n  open PrgDist prgDist\n  open AbsSemSec fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  module FunSemSecAdv {ep : EncParams} {|R|\u1d43} (A : SemSecAdv ep |R|\u1d43) where\n    open SemSecAdv A public\n    open Ops fun\u266dOps public\n\n    step\u2080F : R\u1d43 \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R|\u1d43 (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  module RunSemSec (ep : EncParams) where\n    open EncParams ep using (M; C; |R|\u1d49; Enc)\n\n    -- Returing 0 means Chal wins, Adv looses\n    --          1 means Adv  wins, Chal looses\n    runSemSec : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    runSemSec E A b\n      = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n      where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n    _\u21c4_ : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    _\u21c4_ = runSemSec\n\n    _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : SemSecAdv ep p) \u2192 Set\n    A\u2080 \u2257A A\u2081 = observe A\u2080 \u2257 observe A\u2081\n      where open FunSemSecAdv\n\n    change-adv : \u2200 {ca E} {A\u2081 A\u2082 : SemSecAdv ep ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n    change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n    change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , \u2261.refl = \u2261.trans (\u2261.cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n       where open FunSemSecAdv\n             helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post , pre)\n             helper\u2082 = \u2261.cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                              (helper\u2080 A\u2082)\n\n    _\u2257E_ : \u2200 (E\u2080 E\u2081 : Enc) \u2192 Set\n    E\u2080 \u2257E E\u2081 = \u2200 m \u2192 E\u2080 m \u2257\u21ba E\u2081 m\n\n    \u2257E\u2192\u2257\u2141 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c)\n               \u2192 (E\u2080 \u21c4 A) \u2257\u2141 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 {c} A b R\n      rewrite E\u2080\u2257E\u2081 (proj (proj\u2081 (SemSecAdv.step\u2080 A (take c R))) b) (drop c R) = \u2261.refl\n\n    \u2257E\u2192\u21d3 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c) \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u21d3 E\u2080\u2257E\u2081 A = extensional-reduction (\u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 A)\n\n  module SemSecReductions (ep\u2080 ep\u2081 : EncParams) (f : Coins \u2192 Coins) where\n    open EncParams\u00b2 ep\u2080 ep\u2081\n    open RunSemSec ep\u2080 public using () renaming (_\u21c4_ to _\u21c4\u2080_; _\u2257E_ to _\u2257E\u2080_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2080)\n    open RunSemSec ep\u2081 public using () renaming (_\u21c4_ to _\u21c4\u2081_; _\u2257E_ to _\u2257E\u2081_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2081)\n\n    Reduction : Set\n    Reduction = SemSecReduction ep\u2081 ep\u2080 f\n\n    SemSecTr : Tr \u2192 Set\n    SemSecTr tr =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u207b\u00b9 : Tr\u207b\u00b9 \u2192 Set\n    SemSecTr\u207b\u00b9 tr\u207b\u00b9 =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u2192SemSecTr\u207b\u00b9 : \u2200 tr tr\u207b\u00b9 (tr-right-inv : \u2200 E \u2192 tr (tr\u207b\u00b9 E) \u2257E\u2081 E)\n                          \u2192 SemSecTr tr \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9\n    SemSecTr\u2192SemSecTr\u207b\u00b9 _ tr\u207b\u00b9 tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n            helper E A A-breaks-E = pf (tr\u207b\u00b9 E) A A-breaks-tr-tr\u207b\u00b9-E\n              where A-breaks-tr-tr\u207b\u00b9-E = \u2257E\u2192\u21d3\u2081 (\u03bb m R \u2192 \u2261.sym (tr-inv E m R)) A A-breaks-E\n\n    SemSecTr\u207b\u00b9\u2192SemSecTr : \u2200 tr tr\u207b\u00b9 (tr-left-inv : \u2200 E \u2192 tr\u207b\u00b9 (tr E) \u2257E\u2080 E)\n                          \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9 \u2192 SemSecTr tr\n    SemSecTr\u207b\u00b9\u2192SemSecTr tr _ tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n            helper E {c} A A-breaks-E = \u2257E\u2192\u21d3\u2080 (tr-inv E) (red A) (pf (tr E) A A-breaks-E)\n\n","old_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Vec using (splitAt; take; drop)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import Data.Bits\n\nopen import flat-funs\nopen import program-distance\nopen import flipbased-implem\n\n-- Capture the various size parameters required\n-- for a cipher.\n--\n-- M  is the message space (|M| is the size of messages)\n-- C  is the cipher text space\n-- R\u1d49 is the randomness used by the cipher.\nrecord EncParams : Set where\n  constructor mk\n  field\n    |M| |C| |R|\u1d49 : \u2115\n\n  M  = Bits |M|\n  C  = Bits |C|\n  R\u1d49 = Bits |R|\u1d49\n\n  -- The type of the encryption function\n  Enc : Set\n  Enc = M \u2192 \u21ba |R|\u1d49 C\n\nmodule EncParams\u00b2 (ep\u2080 ep\u2081 : EncParams) where\n  open EncParams ep\u2080 public using ()\n    renaming (|M| to |M|\u2080; |C| to |C|\u2080; |R|\u1d49 to |R|\u1d49\u2080; M to M\u2080;\n              C to C\u2080; R\u1d49 to R\u1d49\u2080; Enc to Enc\u2080)\n  open EncParams ep\u2081 public using ()\n    renaming (|M| to |M|\u2081; |C| to |C|\u2081; |R|\u1d49 to |R|\u1d49\u2081; M to M\u2081;\n              C to C\u2081; R\u1d49 to R\u1d49\u2081; Enc to Enc\u2081)\n  Tr   = Enc\u2080 \u2192 Enc\u2081\n  Tr\u207b\u00b9 = Enc\u2081 \u2192 Enc\u2080\n\nmodule EncParams\u00b2-same-|R|\u1d49 (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  ep\u2081 = EncParams.mk |M|\u2081 |C|\u2081 (EncParams.|R|\u1d49 ep\u2080)\n  open EncParams\u00b2 ep\u2080 ep\u2081 public\n\nmodule \u266dEncParams {t} {T : Set t}\n                  (\u266dFuns : FlatFuns T)\n                  (ep : EncParams) where\n  open FlatFuns \u266dFuns\n  open EncParams ep public\n\n  `M  = `Bits |M|\n  `C  = `Bits |C|\n  `R\u1d49 = `Bits |R|\u1d49\n\nmodule AbsSemSec {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  record SemSecAdv (ep : EncParams) |R|\u1d43 : Set where\n    constructor mk\n\n    open \u266dEncParams \u266dFuns ep\n\n    field\n      {|S|} : \u2115\n\n    `S  = `Bits |S|\n    S   =  Bits |S|\n\n    -- Randomness adversary\n    `R\u1d43 = `Bits |R|\u1d43\n    R\u1d43  =  Bits |R|\u1d43\n\n    field\n      step\u2080 : `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `S\n      step\u2081 : `C `\u00d7 `S `\u2192 `Bit\n\n    M\u00b2  = Bit \u2192 M\n    `M\u00b2 = `Bit `\u2192 `M\n\n    module Ops (\u266dops : FlatFunsOps \u266dFuns) where\n      open FlatFunsOps \u266dops\n      observe : `C `\u00d7 `R\u1d43 `\u2192 (`M `\u00d7 `M) `\u00d7 `Bit\n      observe = idO *** step\u2080 >>> \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 >>> idO *** step\u2081\n        where \u27e8C,\u27e8M,S\u27e9\u27e9\u2192\u27e8M\u00b8\u27e8C,S\u27e9\u27e9 = < snd >>> fst , < fst , snd >>> snd > >\n\n    open \u266dEncParams \u266dFuns ep public\n\n  SemSecReduction : \u2200 ep\u2080 ep\u2081 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction ep\u2080 ep\u2081 f = \u2200 {c} \u2192 SemSecAdv ep\u2080 c \u2192 SemSecAdv ep\u2081 (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) where\n  open PrgDist prgDist\n  open AbsSemSec fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  module FunSemSecAdv {ep : EncParams} {|R|\u1d43} (A : SemSecAdv ep |R|\u1d43) where\n    open SemSecAdv A public\n    open Ops fun\u266dOps public\n\n    step\u2080F : R\u1d43 \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R|\u1d43 (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  module RunSemSec (ep : EncParams) where\n    open EncParams ep using (M; C; |R|\u1d49; Enc)\n\n    -- Returing 0 means Chal wins, Adv looses\n    --          1 means Adv  wins, Chal looses\n    runSemSec : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    runSemSec E A b\n      = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n      where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n    _\u21c4_ : \u2200 {|R|\u1d43} (E : Enc) (A : SemSecAdv ep |R|\u1d43) b \u2192 \u21ba (|R|\u1d43 + |R|\u1d49) Bit\n    _\u21c4_ = runSemSec\n\n    _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : SemSecAdv ep p) \u2192 Set\n    A\u2080 \u2257A A\u2081 = observe A\u2080 \u2257 observe A\u2081\n      where open FunSemSecAdv\n\n    change-adv : \u2200 {ca E} {A\u2081 A\u2082 : SemSecAdv ep ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n    change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n    change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , \u2261.refl = \u2261.trans (\u2261.cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n       where open FunSemSecAdv\n             helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post , pre)\n             helper\u2082 = \u2261.cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                              (helper\u2080 A\u2082)\n\n    _\u2257E_ : \u2200 (E\u2080 E\u2081 : Enc) \u2192 Set\n    E\u2080 \u2257E E\u2081 = \u2200 m \u2192 E\u2080 m \u2257\u21ba E\u2081 m\n\n    \u2257E\u2192\u2257\u2141 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c)\n               \u2192 (E\u2080 \u21c4 A) \u2257\u2141 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 {c} A b R\n      rewrite E\u2080\u2257E\u2081 (proj (proj\u2081 (SemSecAdv.step\u2080 A (take c R))) b) (drop c R) = \u2261.refl\n\n    \u2257E\u2192\u21d3 : \u2200 {E\u2080 E\u2081} \u2192 E\u2080 \u2257E E\u2081 \u2192 \u2200 {c} (A : SemSecAdv ep c) \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 A)\n    \u2257E\u2192\u21d3 E\u2080\u2257E\u2081 A = extensional-reduction (\u2257E\u2192\u2257\u2141 E\u2080\u2257E\u2081 A)\n\n  module SemSecReductions (ep\u2080 ep\u2081 : EncParams) (f : Coins \u2192 Coins) where\n    open EncParams\u00b2 ep\u2080 ep\u2081\n    open RunSemSec ep\u2080 public using () renaming (_\u21c4_ to _\u21c4\u2080_; _\u2257E_ to _\u2257E\u2080_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2080)\n    open RunSemSec ep\u2081 public using () renaming (_\u21c4_ to _\u21c4\u2081_; _\u2257E_ to _\u2257E\u2081_; \u2257E\u2192\u21d3 to \u2257E\u2192\u21d3\u2081)\n\n    Reduction : Set\n    Reduction = SemSecReduction ep\u2081 ep\u2080 f\n\n    SemSecTr : Tr \u2192 Set\n    SemSecTr tr =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u207b\u00b9 : Tr\u207b\u00b9 \u2192 Set\n    SemSecTr\u207b\u00b9 tr\u207b\u00b9 =\n      \u2203 \u03bb (red : Reduction) \u2192\n        \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n\n    SemSecTr\u2192SemSecTr\u207b\u00b9 : \u2200 tr tr\u207b\u00b9 (tr-right-inv : \u2200 E \u2192 tr (tr\u207b\u00b9 E) \u2257E\u2081 E)\n                          \u2192 SemSecTr tr \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9\n    SemSecTr\u2192SemSecTr\u207b\u00b9 _ tr\u207b\u00b9 tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (E \u21c4\u2081 A) \u21d3 (tr\u207b\u00b9 E \u21c4\u2080 red {c} A)\n            helper E A A-breaks-E = pf (tr\u207b\u00b9 E) A A-breaks-tr-tr\u207b\u00b9-E\n              where A-breaks-tr-tr\u207b\u00b9-E = \u2257E\u2192\u21d3\u2081 (\u03bb m R \u2192 \u2261.sym (tr-inv E m R)) A A-breaks-E\n\n    SemSecTr\u207b\u00b9\u2192SemSecTr : \u2200 tr tr\u207b\u00b9 (tr-left-inv : \u2200 E \u2192 tr\u207b\u00b9 (tr E) \u2257E\u2080 E)\n                          \u2192 SemSecTr\u207b\u00b9 tr\u207b\u00b9 \u2192 SemSecTr tr\n    SemSecTr\u207b\u00b9\u2192SemSecTr tr _ tr-inv (red , pf) = red , helper\n      where helper : \u2200 E {c} A \u2192 (tr E \u21c4\u2081 A) \u21d3 (E \u21c4\u2080 red {c} A)\n            helper E {c} A A-breaks-E = \u2257E\u2192\u21d3\u2080 (tr-inv E) (red A) (pf (tr E) A A-breaks-E)\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d09373bb2c59280a99b8d697d5fc5dbc0dbf2774","subject":"Renaming P to pmf","message":"Renaming P to pmf\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      P : U \u2192 [0,1]\n      sumP\u22611 : sumP P allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs))\n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"02977387055c6b24c091b1fe7b444d03f9d5ef14","subject":"Cosmetic change.","message":"Cosmetic change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"test\/theorems\/Definition12.agda","new_file":"test\/theorems\/Definition12.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the translation of definitions\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Definition12 where\n\ninfixl 9 _\u00b7_\ninfix  7 _\u2261_\n\npostulate\n  D       : Set\n  _\u2261_     : D \u2192 D \u2192 Set\n  true if : D\n  _\u00b7_     : D \u2192 D \u2192 D\n\npostulate if-true : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n{-# ATP axiom if-true #-}\n\nif_then_else_ : D \u2192 D \u2192 D \u2192 D\nif b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n{-# ATP definition if_then_else_ #-}\n\n-- We test the translation of a definition with holes.\npostulate foo : \u2200 d\u2081 d\u2082 \u2192 if true then d\u2081 else d\u2082 \u2261 d\u2081\n{-# ATP prove foo #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the translation of definitions\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Definition12 where\n\ninfixl 9 _\u00b7_\ninfix  7 _\u2261_\n\npostulate\n  D       : Set\n  _\u2261_     : D \u2192 D \u2192 Set\n  true if : D\n  _\u00b7_     : D \u2192 D \u2192 D\n\npostulate if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n{-# ATP axiom if-true #-}\n\nif_then_else_ : D \u2192 D \u2192 D \u2192 D\nif b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n{-# ATP definition if_then_else_ #-}\n\n-- We test the translation of a definition with holes.\npostulate foo : \u2200 d\u2081 d\u2082 \u2192 if true then d\u2081 else d\u2082 \u2261 d\u2081\n{-# ATP prove foo #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"793bc5fa084352764d8cc9e78b906f652b743537","subject":"Use instance argument in circumflex constructor.","message":"Use instance argument in circumflex constructor.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth add-smooth-lower-vowel\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough add-rough-vowel\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth add-smooth-lower-vowel\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough add-rough-vowel\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth add-smooth-lower-vowel\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough add-rough-vowel\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth add-smooth-lower-vowel\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  add-circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\n\u03b1-circumflex : \u03b1\u2032 accent\n\u03b1-circumflex = add-circumflex \u03b1-long-vowel\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth add-smooth-lower-vowel\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough add-rough-vowel\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough add-rough-vowel\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth add-smooth-lower-vowel\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough add-rough-vowel\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\n\u03b7-circumflex : \u03b7\u2032 accent\n\u03b7-circumflex = add-circumflex \u03b7-long-vowel\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth add-smooth-lower-vowel\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough add-rough-vowel\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough add-rough-vowel\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\n\u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\n\u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n\u03b9-not-\u03c5 ()\n\n\u03b9-circumflex : \u03b9\u2032 accent\n\u03b9-circumflex = add-circumflex \u03b9-long-vowel\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth add-smooth-lower-vowel\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough add-rough-vowel\n\n-- \u039f \u03bf\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing \u03b1-circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing \u03b1-circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing \u03b1-circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing \u03b1-circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota \u03b1-circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota \u03b1-circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota \u03b1-circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota \u03b1-circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent \u03b1-circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota \u03b1-circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent (add-circumflex x) = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"47053f9937d5fb34ba3b9b4ec51504dafc0fd6e9","subject":"updating esap rule","message":"updating esap rule\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4 \u21d2 \u03c4' \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d1' d2 \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)  \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d d' \u03c4 \u03c4' } \u2192\n                        d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           \u0394 \u22a2 d \u21a6 d'\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFinal : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41 \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4 \u21d2 \u03c4' \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d1' d2 \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)  \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d d' \u03c4 \u03c4' } \u2192\n                        d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           \u0394 \u22a2 d \u21a6 d'\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFinal : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b01a7346ed1e6f78cfbc40a59d3d9a3858fbdab3","subject":"working on some comments","message":"working on some comments\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- the type of type contexts, i.e. \u0393s in the judegments below\n  tctx : Set\n  tctx = htyp ctx\n\n  mutual\n    data env : Set where\n      Id : (\u0393 : tctx) \u2192 env\n      Subst : (d : dhexp) \u2192 (y : Nat) \u2192 env \u2192 env\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 env) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 env) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- convenient notation for chaining together two agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- the type of hole contexts, i.e. \u0394s in the judgements\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- notation for a triple to match the CMTT syntax\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- the hole name u does not appear in the term e\n  data hole-name-new : (e : hexp) (u : Nat) \u2192 Set where\n    HNConst : \u2200{u} \u2192 hole-name-new c u\n    HNAsc : \u2200{e \u03c4 u} \u2192\n            hole-name-new e u \u2192\n            hole-name-new (e \u00b7: \u03c4) u\n    HNVar : \u2200{x u} \u2192 hole-name-new (X x) u\n    HNLam1 : \u2200{x e u} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x e) u\n    HNLam2 : \u2200{x e u \u03c4} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x [ \u03c4 ] e) u\n    HNHole : \u2200{u u'} \u2192\n             u' \u2260 u \u2192\n             hole-name-new (\u2987\u2988[ u' ]) u\n    HNNEHole : \u2200{u u' e} \u2192\n               u' \u2260 u \u2192\n               hole-name-new e u \u2192\n               hole-name-new (\u2987 e \u2988[ u' ]) u\n    HNAp : \u2200{ u e1 e2 } \u2192\n           hole-name-new e1 u \u2192\n           hole-name-new e2 u \u2192\n           hole-name-new (e1 \u2218 e2) u\n\n  -- two terms that do not share any hole names\n  data holes-disjoint : (e1 : hexp) \u2192 (e2 : hexp) \u2192 Set where\n    HDConst : \u2200{e} \u2192 holes-disjoint c e\n    HDAsc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (e1 \u00b7: \u03c4) e2\n    HDVar : \u2200{x e} \u2192 holes-disjoint (X x) e\n    HDLam1 : \u2200{x e1 e2} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x e1) e2\n    HDLam2 : \u2200{x e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x [ \u03c4 ] e1) e2\n    HDHole : \u2200{u e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint (\u2987\u2988[ u ]) e2\n    HDNEHole : \u2200{u e1 e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u2987 e1 \u2988[ u ]) e2\n    HDAp :  \u2200{e1 e2 e3} \u2192 holes-disjoint e1 e3 \u2192 holes-disjoint e2 e3 \u2192 holes-disjoint (e1 \u2218 e2) e3\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {x : Nat} \u2192\n                 (x , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X x => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 holes-disjoint e1 e2 \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 hole-name-new e u \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those external expressions without holes\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  -- those internal expressions without holes\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only produce complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n  -- those internal expressions where every cast is the identity cast and\n  -- there are no failed casts\n  data cast-id : dhexp \u2192 Set where\n    CIConst  : cast-id c\n    CIVar    : \u2200{x} \u2192 cast-id (X x)\n    CILam    : \u2200{x \u03c4 d} \u2192 cast-id d \u2192 cast-id (\u00b7\u03bb x [ \u03c4 ] d)\n    CIHole   : \u2200{u} \u2192 cast-id (\u2987\u2988\u27e8 u \u27e9)\n    CINEHole : \u2200{d u} \u2192 cast-id d \u2192 cast-id (\u2987 d \u2988\u27e8 u \u27e9)\n    CIAp     : \u2200{d1 d2} \u2192 cast-id d1 \u2192 cast-id d2 \u2192 cast-id (d1 \u2218 d2)\n    CICast   : \u2200{d \u03c4} \u2192 cast-id d \u2192 cast-id (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9)\n\n  -- expansion\n  mutual\n    -- synthesis\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              holes-disjoint e1 e2 \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , Id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u2987\u2988)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    -- analysis\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u03c4)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground types\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- todo: clean up above, dom?\n    data _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 env \u2192 tctx \u2192 Set where\n      STAId : \u2200{\u0393 \u0393' \u0394} \u2192\n                  ((x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393' \u2192 (x , \u03c4) \u2208 \u0393) \u2192\n                  \u0394 , \u0393 \u22a2 Id \u0393' :s: \u0393'\n      STASubst : \u2200{\u0393 \u0394 \u03c3 y \u0393' d \u03c4 } \u2192\n               \u0394 , \u0393 ,, (y , \u03c4) \u22a2 \u03c3 :s: \u0393' \u2192\n               \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n               \u0394 , \u0393 \u22a2 Subst d y \u03c3 :s: \u0393'\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              x # \u0393 \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution\n  --\n  -- todo: if substitution lemma is hard to prove, maybe get a premise that\n  -- it's final; analagous to \"value substitution\". or define it\n  -- judgementally instead of as a function.\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d')\n    with natEQ x y\n  [ d \/ y ] (\u00b7\u03bb .y [ \u03c4 ] d') | Inl refl = \u00b7\u03bb y [ \u03c4 ] d'\n  [ d \/ y ] (\u00b7\u03bb x [ \u03c4 ] d')  | Inr x\u2081 = \u00b7\u03bb x [ \u03c4 ] ( [ d \/ y ] d')\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- applying an environment to an expression\n  apply-env : env \u2192 dhexp \u2192 dhexp\n  apply-env (Id \u0393) d = d\n  apply-env (Subst d y \u03c3) d' = [ d \/ y ] ( apply-env \u03c3 d')\n\n  -- values\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- boxed values\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate forms\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final expressions\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet    \u2192 d final\n\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 env ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n  -- note: this judgement is redundant: in the absence of the premises in\n  -- the red brackets, all syntactically well formed ectxs are valid. with\n  -- finality premises, that's not true, and that would propagate through\n  -- additions to the calculus. so we leave it here for clarity but note\n  -- that, as written, in any use case its either trival to prove or\n  -- provides no additional information\n\n   --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red brackets\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red brackets\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  -- matched ground types\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red brackets\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1)\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red brackets\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red brackets\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red brackets\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red brackets\n               -- d2 final \u2192 -- red brackets\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  -- single step (in contextual evaluation sense)\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  -- reflexive transitive closure of single steps into multi steps\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- freshness\n  mutual\n    -- ... with respect to a hole context\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    -- ... for inernal expressions\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- ... for external expressions\n  data freshh : Nat \u2192 hexp \u2192 Set where\n    FRHConst : \u2200{x} \u2192 freshh x c\n    FRHAsc   : \u2200{x e \u03c4} \u2192 freshh x e \u2192 freshh x (e \u00b7: \u03c4)\n    FRHVar   : \u2200{x y} \u2192 x \u2260 y \u2192 freshh x (X y)\n    FRHLam1  : \u2200{x y e} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y e)\n    FRHLam2  : \u2200{x \u03c4 e y} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y [ \u03c4 ] e)\n    FRHEHole : \u2200{x u} \u2192 freshh x (\u2987\u2988[ u ])\n    FRHNEHole : \u2200{x u e} \u2192 freshh x e \u2192 freshh x (\u2987 e \u2988[ u ])\n    FRHAp : \u2200{x e1 e2} \u2192 freshh x e1 \u2192 freshh x e2 \u2192 freshh x (e1 \u2218 e2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  tctx : Set\n  tctx = htyp ctx\n\n  mutual\n    data env : Set where\n      Id : (\u0393 : tctx) \u2192 env\n      Subst : (d : dhexp) \u2192 (y : Nat) \u2192 env \u2192 env\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 env) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 env) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- this is just fancy notation for a triple to match the CMTT syntax\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- defines when a name for a hole does not yet appear in a term\n  data hole-name-new : (e : hexp) (u : Nat) \u2192 Set where\n    HNConst : \u2200{u} \u2192 hole-name-new c u\n    HNAsc : \u2200{e \u03c4 u} \u2192\n            hole-name-new e u \u2192\n            hole-name-new (e \u00b7: \u03c4) u\n    HNVar : \u2200{x u} \u2192 hole-name-new (X x) u\n    HNLam1 : \u2200{x e u} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x e) u\n    HNLam2 : \u2200{x e u \u03c4} \u2192\n             hole-name-new e u \u2192\n             hole-name-new (\u00b7\u03bb x [ \u03c4 ] e) u\n    HNHole : \u2200{u u'} \u2192\n             u' \u2260 u \u2192\n             hole-name-new (\u2987\u2988[ u' ]) u\n    HNNEHole : \u2200{u u' e} \u2192\n               u' \u2260 u \u2192\n               hole-name-new e u \u2192\n               hole-name-new (\u2987 e \u2988[ u' ]) u\n    HNAp : \u2200{ u e1 e2 } \u2192\n           hole-name-new e1 u \u2192\n           hole-name-new e2 u \u2192\n           hole-name-new (e1 \u2218 e2) u\n\n  -- describes when the collection of hole names used in two terms do not\n  -- overlap\n  data holes-disjoint : (e1 : hexp) \u2192 (e2 : hexp) \u2192 Set where\n    HDConst : \u2200{e} \u2192 holes-disjoint c e\n    HDAsc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (e1 \u00b7: \u03c4) e2\n    HDVar : \u2200{x e} \u2192 holes-disjoint (X x) e\n    HDLam1 : \u2200{x e1 e2} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x e1) e2\n    HDLam2 : \u2200{x e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x [ \u03c4 ] e1) e2\n    HDHole : \u2200{u e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint (\u2987\u2988[ u ]) e2\n    HDNEHole : \u2200{u e1 e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u2987 e1 \u2988[ u ]) e2\n    HDAp :  \u2200{e1 e2 e3} \u2192 holes-disjoint e1 e3 \u2192 holes-disjoint e2 e3 \u2192 holes-disjoint (e1 \u2218 e2) e3\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {x : Nat} \u2192\n                 (x , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X x => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 holes-disjoint e1 e2 \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 hole-name-new e u \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes anywhere\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those expressions without holes anywhere\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only know about complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n  -- those d for which every cast inside is the identity cast and there are\n  -- no failed casts\n  data cast-id : dhexp \u2192 Set where\n    CIConst  : cast-id c\n    CIVar    : \u2200{x} \u2192 cast-id (X x)\n    CILam    : \u2200{x \u03c4 d} \u2192 cast-id d \u2192 cast-id (\u00b7\u03bb x [ \u03c4 ] d)\n    CIHole   : \u2200{u} \u2192 cast-id (\u2987\u2988\u27e8 u \u27e9)\n    CINEHole : \u2200{d u} \u2192 cast-id d \u2192 cast-id (\u2987 d \u2988\u27e8 u \u27e9)\n    CIAp     : \u2200{d1 d2} \u2192 cast-id d1 \u2192 cast-id d2 \u2192 cast-id (d1 \u2218 d2)\n    CICast   : \u2200{d \u03c4} \u2192 cast-id d \u2192 cast-id (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9)\n\n  -- _extends_ : {A : Set} (\u0393 \u0393' : A ctx) \u2192 Set\n  -- _extends_ {A} \u0393' \u0393  = (x : Nat) \u2192 dom \u0393 x \u2192 \u0393' x == \u0393 x\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              holes-disjoint e1 e2 \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                -- \u0393' extends \u0393 \u2192 -- todo note here about weakening away everything you don't need\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , Id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 --\u0393' extends \u0393 \u2192 -- todo note here about weakening away everything you don't need\n                 \u0394 ## (\u25a0 (u , \u0393 , \u2987\u2988)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0394 ## (\u25a0 (u , \u0393 , \u03c4)) \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , Id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    -- _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 env \u2192 tctx \u2192 Set\n    -- \u0394 , \u0393 \u22a2 Id \u03931 :s: \u03932 = \u03931 == \u03932\n    -- \u0394 , \u0393 \u22a2 Subst d y \u03c3 :s: \u0393' = (\u0394 , \u0393 \u22a2 \u03c3 :s: (\u0393' \\\\ y))\n    --                 \u00d7 \u03a3[ \u03c4 \u2208 htyp ] (\u0393' y == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4) -- todo: dom here\n\n    -- todo: clean up above, dom?\n    data _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 env \u2192 tctx \u2192 Set where\n      STAId : \u2200{\u0393 \u0393' \u0394} \u2192\n                  ((x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393' \u2192 (x , \u03c4) \u2208 \u0393) \u2192\n                  \u0394 , \u0393 \u22a2 Id \u0393' :s: \u0393'\n      STASubst : \u2200{\u0393 \u0394 \u03c3 y \u0393' d \u03c4 } \u2192\n               \u0394 , \u0393 ,, (y , \u03c4) \u22a2 \u03c3 :s: \u0393' \u2192\n               \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n               \u0394 , \u0393 \u22a2 Subst d y \u03c3 :s: \u0393'\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              x # \u0393 \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution\n  --\n  -- todo: if substitution lemma is hard to prove, maybe get a premise that\n  -- it's final; analagous to \"value substitution\". or define it\n  -- judgementally instead of as a function.\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d')\n    with natEQ x y\n  [ d \/ y ] (\u00b7\u03bb .y [ \u03c4 ] d') | Inl refl = \u00b7\u03bb y [ \u03c4 ] d'\n  [ d \/ y ] (\u00b7\u03bb x [ \u03c4 ] d')  | Inr x\u2081 = \u00b7\u03bb x [ \u03c4 ] ( [ d \/ y ] d')\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , Subst d y \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- applying an environment to an expression\n  apply-env : env \u2192 dhexp \u2192 dhexp\n  apply-env (Id \u0393) d = d\n  apply-env (Subst d y \u03c3) d' = [ d \/ y ] ( apply-env \u03c3 d')\n\n  -- values\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- boxed values\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate forms\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final expressions\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet    \u2192 d final\n\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 env ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n  -- note: this judgement is redundant: in the absence of the premises in\n  -- the red brackets, all syntactically well formed ectxs are valid. with\n  -- finality premises, that's not true, and that would propagate through\n  -- additions to the calculus. so we leave it here for clarity but note\n  -- that, as written, in any use case its either trival to prove or\n  -- provides no additional information\n\n   --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red brackets\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red brackets\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  -- matched ground types\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red brackets\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1)\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red brackets\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red brackets\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red brackets\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red brackets\n               -- d2 final \u2192 -- red brackets\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red brackets\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  -- single step (in contextual evaluation sense)\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  -- reflexive transitive closure of single steps into multi steps\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- freshness, used in substitution and disjointness proofs\n  mutual\n    data envfresh : Nat \u2192 env \u2192 Set where\n      EFId : \u2200{x \u0393} \u2192 x # \u0393 \u2192 envfresh x (Id \u0393)\n      EFSubst : \u2200{x d \u03c3 y} \u2192 fresh x d\n                           \u2192 envfresh x \u03c3\n                           \u2192 x \u2260 y\n                           \u2192 envfresh x (Subst d y \u03c3)\n\n    data fresh : Nat \u2192 dhexp \u2192 Set where\n      FConst : \u2200{x} \u2192 fresh x c\n      FVar   : \u2200{x y} \u2192 x \u2260 y \u2192 fresh x (X y)\n      FLam   : \u2200{x y \u03c4 d} \u2192 x \u2260 y \u2192 fresh x d \u2192 fresh x (\u00b7\u03bb y [ \u03c4 ] d)\n      FHole  : \u2200{x u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x (\u2987\u2988\u27e8 u , \u03c3 \u27e9)\n      FNEHole : \u2200{x d u \u03c3} \u2192 envfresh x \u03c3 \u2192 fresh x d \u2192 fresh x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9)\n      FAp     : \u2200{x d1 d2} \u2192 fresh x d1 \u2192 fresh x d2 \u2192 fresh x (d1 \u2218 d2)\n      FCast   : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n      FFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 fresh x d \u2192 fresh x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  data freshh : Nat \u2192 hexp \u2192 Set where\n    FRHConst : \u2200{x} \u2192 freshh x c\n    FRHAsc   : \u2200{x e \u03c4} \u2192 freshh x e \u2192 freshh x (e \u00b7: \u03c4)\n    FRHVar   : \u2200{x y} \u2192 x \u2260 y \u2192 freshh x (X y)\n    FRHLam1  : \u2200{x y e} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y e)\n    FRHLam2  : \u2200{x \u03c4 e y} \u2192 x \u2260 y \u2192 freshh x e \u2192 freshh x (\u00b7\u03bb y [ \u03c4 ] e)\n    FRHEHole : \u2200{x u} \u2192 freshh x (\u2987\u2988[ u ]) --todo: x \u2260 u?\n    FRHNEHole : \u2200{x u e} \u2192 freshh x e \u2192 freshh x (\u2987 e \u2988[ u ])  --todo: x \u2260 u?\n    FRHAp : \u2200{x e1 e2} \u2192 freshh x e1 \u2192 freshh x e2 \u2192 freshh x (e1 \u2218 e2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"29b66fbf69e4dede85829204e2c88990ddd88381","subject":"Product changes: complete primitive derivatives","message":"Product changes: complete primitive derivatives\n\nDerive both _,_ and uncurry.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Products.agda","new_file":"Base\/Change\/Products.agda","new_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  -- An interesting alternative definition allows omitting the nil change of a\n  -- component when that nil change can be computed from the type. For instance, the nil change for integers is always the same.\n\n  -- However, the nil change for function isn't always the same (unless we\n  -- defunctionalize them first), so nil changes for functions can't be omitted.\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-nil : (v : A \u00d7 B) \u2192 PChange v\n  p-nil (a , b) = (nil a , nil b)\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  p-update-nil : (v : A \u00d7 B) \u2192 v \u2295 (p-nil v) \u2261 v\n  p-update-nil (a , b) =\n    let v = (a , b)\n    in\n      begin\n        v \u2295 (p-nil v)\n      \u2261\u27e8\u27e9\n        (a \u229e nil a , b \u229e nil b)\n      \u2261\u27e8 cong\u2082 _,_ (update-nil a) (update-nil b)\u27e9\n         (a , b)\n      \u2261\u27e8\u27e9\n        v\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = p-nil\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      ; update-nil = p-update-nil\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n  -- Morally, the following is a change:\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032-realizer : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032-realizer a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of\n  -- its own validity, and because after application it does not contain a\n  -- proof.\n  --\n  -- However, the above is (morally) a \"realizer\" of the actual change, since it\n  -- only represents its computational behavior, not its proof manipulation.\n\n  -- Hence, we need to do some additional work.\n\n  _,_\u2032-realizer-correct _,_\u2032-realizer-correct-detailed :\n    (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192\n      (a , b \u229e db) \u229e (_,_\u2032-realizer a da (b \u229e db) (nil (b \u229e db))) \u2261 (a , b) \u229e (_,_\u2032-realizer a da b db)\n  _,_\u2032-realizer-correct a da b db rewrite update-nil (b \u229e db) = refl\n\n  _,_\u2032-realizer-correct-detailed a da b db =\n    begin\n      (a , b \u229e db) \u229e (_,_\u2032-realizer a da) (b \u229e db) (nil (b \u229e db))\n    \u2261\u27e8\u27e9\n      a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n      a \u229e da , b \u229e db\n    \u2261\u27e8\u27e9\n      (a , b) \u229e (_,_\u2032-realizer a da) b db\n    \u220e\n\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 \u0394 (_,_ a)\n  _,_\u2032 a da = record { apply = _,_\u2032-realizer a da ; correct = \u03bb b db \u2192 _,_\u2032-realizer-correct a da b db }\n\n  _,_\u2032-Derivative : Derivative _,_ _,_\u2032\n  _,_\u2032-Derivative a da = ext (\u03bb b \u2192 cong (_,_ (a \u229e da)) (update-nil b))\n    where\n      open import Postulate.Extensionality\n\n  -- Define specialized variant of uncurry, and derive it.\n  uncurry\u2080 : \u2200 {C : Set \u2113} \u2192 (A \u2192 B \u2192 C) \u2192 A \u00d7 B \u2192 C\n  uncurry\u2080 f (a , b) = f a b\n\n  module _ {C : Set \u2113} {{CC : ChangeAlgebra \u2113 C}} where\n    B\u2192C : ChangeAlgebra \u2113 (B \u2192 C)\n    B\u2192C = FunctionChanges.changeAlgebra B C\n    A\u2192B\u2192C : ChangeAlgebra \u2113 (A \u2192 B \u2192 C)\n    A\u2192B\u2192C = FunctionChanges.changeAlgebra A (B \u2192 C)\n    A\u00d7B\u2192C : ChangeAlgebra \u2113 (A \u00d7 B \u2192 C)\n    A\u00d7B\u2192C = FunctionChanges.changeAlgebra (A \u00d7 B) C\n    module \u0394B\u2192C = FunctionChanges B C {{CB}} {{CC}}\n    module \u0394A\u2192B\u2192C = FunctionChanges A (B \u2192 C) {{CA}} {{B\u2192C}}\n    module \u0394A\u00d7B\u2192C = FunctionChanges (A \u00d7 B) C {{changeAlgebra}} {{CC}}\n\n    uncurry\u2080\u2032-realizer : (f : A \u2192 B \u2192 C) \u2192 \u0394 f \u2192 (p : A \u00d7 B) \u2192 \u0394 p \u2192 \u0394 (uncurry\u2080 f p)\n    uncurry\u2080\u2032-realizer f df (a , b) (da , db) = \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df a da) b db\n\n    uncurry\u2080\u2032-realizer-correct uncurry\u2080\u2032-realizer-correct-detailed :\n      \u2200 (f : A \u2192 B \u2192 C) (df : \u0394 f) (p : A \u00d7 B) (dp : \u0394 p) \u2192\n        uncurry\u2080 f (p \u2295 dp) \u229e uncurry\u2080\u2032-realizer f df (p \u2295 dp) (nil (p \u229e dp)) \u2261 uncurry\u2080 f p \u229e uncurry\u2080\u2032-realizer f df p dp\n\n    -- Hard to read\n    uncurry\u2080\u2032-realizer-correct f df (a , b) (da , db)\n      rewrite sym (\u0394B\u2192C.incrementalization (f (a \u229e da)) (\u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db)))\n      | update-nil (b \u229e db)\n      | {- cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) -} (sym (\u0394A\u2192B\u2192C.incrementalization f df (a \u229e da) (nil (a \u229e da))))\n      | update-nil (a \u229e da)\n      | cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (\u0394A\u2192B\u2192C.incrementalization f df a da)\n      | \u0394B\u2192C.incrementalization (f a) (\u0394A\u2192B\u2192C.apply df a da) b db\n      = refl\n\n    -- Verbose, but it shows all the intermediate steps.\n    uncurry\u2080\u2032-realizer-correct-detailed f df (a , b) (da , db) =\n      begin\n        uncurry\u2080 f (a \u229e da , b \u229e db) \u229e uncurry\u2080\u2032-realizer f df (a \u229e da , b \u229e db) (nil (a \u229e da , b \u229e db))\n      \u2261\u27e8\u27e9\n        f (a \u229e da) (b \u229e db) \u229e \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8 sym (\u0394B\u2192C.incrementalization (f (a \u229e da)) (\u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db) (nil (b \u229e db))) \u27e9\n        (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) ((b \u229e db) \u229e (nil (b \u229e db)))\n      \u2261\u27e8 cong-lem\u2080 \u27e9\n        (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n      \u2261\u27e8 sym cong-lem\u2082 \u27e9\n        ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n      \u2261\u27e8 cong-lem\u2081 \u27e9\n        (f \u229e df) (a \u229e da) (b \u229e db)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (\u0394A\u2192B\u2192C.incrementalization f df a da) \u27e9\n        (f a \u229e \u0394A\u2192B\u2192C.apply df a da) (b \u229e db)\n      \u2261\u27e8 \u0394B\u2192C.incrementalization (f a) (\u0394A\u2192B\u2192C.apply df a da) b db \u27e9\n        f a b \u229e \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df a da) b db\n      \u2261\u27e8\u27e9\n        uncurry\u2080 f (a , b) \u229e uncurry\u2080\u2032-realizer f df (a , b) (da ,  db)\n      \u220e\n      where\n        cong-lem\u2080 :\n            (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) ((b \u229e db) \u229e (nil (b \u229e db)))\n            \u2261\n            (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n        cong-lem\u2080 rewrite update-nil (b \u229e db) = refl\n\n        cong-lem\u2081 :\n                  ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n                  \u2261\n                  (f \u229e df) (a \u229e da) (b \u229e db)\n        cong-lem\u2081 rewrite update-nil (a \u229e da) = refl\n\n        cong-lem\u2082 :\n                  ((f \u229e df) ((a \u229e da) \u229e (nil (a \u229e da)))) (b \u229e db)\n                  \u2261\n                  (f (a \u229e da) \u229e \u0394A\u2192B\u2192C.apply df (a \u229e da) (nil (a \u229e da))) (b \u229e db)\n        cong-lem\u2082 = cong (\u03bb \u25a1 \u2192 \u25a1 (b \u229e db)) (\u0394A\u2192B\u2192C.incrementalization f df (a \u229e da) (nil (a \u229e da)))\n\n    uncurry\u2080\u2032 : (f : A \u2192 B \u2192 C) \u2192 \u0394 f \u2192 \u0394 (uncurry f)\n    uncurry\u2080\u2032 f df = record\n      { apply = uncurry\u2080\u2032-realizer f df\n      ; correct = uncurry\u2080\u2032-realizer-correct f df }\n\n    -- Now proving that uncurry\u2080\u2032 is a derivative is trivial!\n    uncurry\u2080\u2032Derivative\u2080 : Derivative {{CB = A\u00d7B\u2192C}} uncurry\u2080 uncurry\u2080\u2032\n    uncurry\u2080\u2032Derivative\u2080 f df = refl\n\n    -- If you wonder what's going on, here's the step-by-step proof, going purely by definitional equality.\n    uncurry\u2080\u2032Derivative : Derivative {{CB = A\u00d7B\u2192C}} uncurry\u2080 uncurry\u2080\u2032\n    uncurry\u2080\u2032Derivative f df =\n      begin\n        uncurry\u2080 f \u229e uncurry\u2080\u2032 f df\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 uncurry\u2080 f (a , b) \u229e \u0394A\u00d7B\u2192C.apply (uncurry\u2080\u2032 f df) (a , b) (nil (a , b))})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 f a b \u229e \u0394B\u2192C.apply (\u0394A\u2192B\u2192C.apply df a (nil a)) b (nil b)})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 (f a \u229e \u0394A\u2192B\u2192C.apply df a (nil a)) b})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 (f \u229e df) a b})\n      \u2261\u27e8\u27e9\n        (\u03bb {(a , b) \u2192 uncurry\u2080 (f \u229e df) (a , b)})\n      \u2261\u27e8\u27e9\n        uncurry\u2080 (f \u229e df)\n      \u220e\n","old_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  -- An interesting alternative definition allows omitting the nil change of a\n  -- component when that nil change can be computed from the type. For instance, the nil change for integers is always the same.\n\n  -- However, the nil change for function isn't always the same (unless we\n  -- defunctionalize them first), so nil changes for functions can't be omitted.\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-nil : (v : A \u00d7 B) \u2192 PChange v\n  p-nil (a , b) = (nil a , nil b)\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  p-update-nil : (v : A \u00d7 B) \u2192 v \u2295 (p-nil v) \u2261 v\n  p-update-nil (a , b) =\n    let v = (a , b)\n    in\n      begin\n        v \u2295 (p-nil v)\n      \u2261\u27e8\u27e9\n        (a \u229e nil a , b \u229e nil b)\n      \u2261\u27e8 cong\u2082 _,_ (update-nil a) (update-nil b)\u27e9\n         (a , b)\n      \u2261\u27e8\u27e9\n        v\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = p-nil\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      ; update-nil = p-update-nil\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  -- We should do the same for uncurry instead.\n\n  -- Morally, the following is a change:\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032 a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of its own validity, and because after application it does not contain a proof.\n\n  -- As a consequence, proving that's a derivative requires extra effort.\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n  -- Give a trivial definition of the change of _,_.\n  -- This is simply _,_ \u2296 _,_, together with its generic proof of correctness.\n  _,_\u2032-real : \u0394 _,_\n  _,_\u2032-real = nil _,_\n  _,_\u2032-real-Derivative : Derivative {{CA}} {{B\u2192A\u00d7B}} _,_ (\u0394A\u2192B\u2192A\u00d7B.apply _,_\u2032-real)\n  _,_\u2032-real-Derivative =\n    FunctionChanges.nil-is-derivative A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}} _,_\n\n  -- Formalizing a proof that _,_\u2032 is morally a derivative of _,_ appears\n  -- instead unnecessarily hard, essentially, because this is a curried\n  -- function.\n\n  -- Let's now instead prove by hand that _,_\u2032 a da is a change for _,_ a.\n\n  _,_\u2032\u2032 :  (a : A) \u2192 \u0394 a \u2192\n      \u0394 {{B\u2192A\u00d7B}} (\u03bb b \u2192 (a , b))\n  _,_\u2032\u2032 a da = record\n    { apply = _,_\u2032 a da\n    ; correct = \u03bb b db \u2192\n      begin\n        (a , b \u229e db) \u229e (_,_\u2032 a da) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n        a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9\n        (a , b) \u229e (_,_\u2032 a da) b db\n      \u220e\n    }\n\n  -- We might want to provide a proof schema for curried functions at once,\n  -- starting from more reasonable correctness equation.\n\n{-\n  _,_\u2032\u2032\u2032 : \u0394 {{A\u2192B\u2192A\u00d7B}} _,_\n  _,_\u2032\u2032\u2032 = record\n    { apply = _,_\u2032\u2032\n    ; correct = \u03bb a da \u2192\n      begin\n        update\n        (_,_ (a \u229e da))\n        (_,_\u2032\u2032 (a \u229e da) (nil (a \u229e da)))\n      \u2261\u27e8 {!!} \u27e9\n        update (_,_ a) (_,_\u2032\u2032 a da)\n      \u220e\n    }\n    where\n      -- This is needed to use update above.\n      -- Passing the change structure seems hard with the given operators; maybe I'm just using them wrongly.\n      open ChangeAlgebra B\u2192A\u00d7B hiding (nil)\n{-\n  {!\n      begin\n        (_,_ (a \u229e da)) \u229e _,_\u2032\u2032 (a \u229e da) (nil (a \u229e da))\n      {- \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db)) -}\n      \u2261\u27e8 ? \u27e9\n        {-a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9-}\n       (_,_ a) \u229e (_,_\u2032\u2032 a da)\n      \u220e!}\n    }\n-}\n  open import Postulate.Extensionality\n\n\n  _,_\u2032Derivative :\n    Derivative {{CA}} {{B\u2192A\u00d7B}} _,_  _,_\u2032\u2032\n  _,_\u2032Derivative a da =\n    begin\n      _\u229e_ {{B\u2192A\u00d7B}} (_,_ a) (_,_\u2032\u2032 a da)\n    \u2261\u27e8\u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    --ext (\u03bb b \u2192 cong (\u03bb \u25a1 \u2192  (a , b) \u229e \u25a1) (update-nil {{?}} b))\n    \u2261\u27e8 {!!} \u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    \u2261\u27e8 sym {!\u0394A\u2192B\u2192A\u00d7B.incrementalization _,_ _,_\u2032\u2032\u2032 a da!} \u27e9\n  --FunctionChanges.incrementalization A (B \u2192 A \u00d7 B) {{CA}} {{{!B\u2192A\u00d7B!}}} _,_ {!!} {!!} {!!}\n       _,_ (a \u229e da)\n    \u220e\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5fc5337a591b330872fb00e18af9df6656c26d4f","subject":"Patch logical relation to relate different values","message":"Patch logical relation to relate different values\n\nThis is a minimal example. Still, it's now clear that related elements can be\ndifferent: I even added example where I relate extensionally different functions!\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n zero k\u2264n v1 v2 x = tt\n  example1 n (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl = intV 1 , 0 , refl , (I.+ 1 , refl)\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n zero k\u2264n v1 v2 x = tt\n  example2 n (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl = intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV nat (intV v1) (intV v2) n = v1 \u2261 v2\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Here, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f5e7af9c1bbb5345b1c835a535a9510eb11ecf29","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: d359ee38718a622b194856dab40357b2\n\ndarcs-hash:20100604032132-3bd4e-6d5c26afeb9476c1bf88a2d12a15a8d5d2f876f5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Division\/IsCorrectER.agda","new_file":"Examples\/Division\/IsCorrectER.agda","new_contents":"------------------------------------------------------------------------------\n-- The division result is correct\n------------------------------------------------------------------------------\n\nmodule Examples.Division.IsCorrectER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import Examples.Division.EquationsER\nopen import Examples.Division.Specification\n\nopen import LTC.Data.N\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF-ER\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.InequalitiesPCF\n\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IsNER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\n-- The division result is correct when the dividend is less than\n-- the divisor.\n\ndiv-x<y-aux : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192 i \u2261 j * (div i j) + i\ndiv-x<y-aux {i} {j} Ni Nj i<j = sym\n    ( begin\n        j * (div i j) + i \u2261\u27e8 prf\u2081 \u27e9\n        j * zero + i      \u2261\u27e8 prf\u2082 \u27e9\n        zero + i          \u2261\u27e8 +-leftIdentity Ni \u27e9\n        i\n      \u220e\n    )\n    where\n      prf\u2081 : j * (div i j) + i \u2261 j * zero + i\n      prf\u2081 = subst (\u03bb x \u2192 j * x + i \u2261 j * zero + i )\n                   (sym (div-x<y i<j ))\n                   refl\n\n      prf\u2082 : j * zero + i \u2261 zero + i\n      prf\u2082 = subst (\u03bb x \u2192 x + i \u2261 zero + i)\n                   (sym (*-rightZero Nj))\n                   refl\n\ndiv-x<y-correct : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192\n                  (\u2203D (\u03bb r \u2192 N r \u2227 LT r j \u2227 i \u2261 j * (div i j) + r))\ndiv-x<y-correct {i} Ni Nj i<j =\n  i , (Ni , (i<j , (div-x<y-aux Ni Nj i<j)))\n\n\n-- The division result is correct when the dividend is greater or equal\n-- than the divisor.\n-- Using the inductive hypothesis 'ih' we know that\n-- 'i - j = j * (div (i - j) j) + r'. From\n-- that we get 'i = j * (succ (div (i - j) j)) + r', and\n-- we know 'div i j = succ (div (i - j) j); therefore we\n-- get 'i = j * div i j + r'.\n\npostulate\n  aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n        i - j \u2261 j * (div (i - j) j) + r \u2192\n        i \u2261 j * succ (div (i - j) j) + r\n\ndiv-x\u2265y-aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n              GE i j \u2192\n              i - j \u2261 j * (div (i - j) j) + r \u2192\n              i \u2261 j * (div i j) + r\ndiv-x\u2265y-aux {i} {j} {r} Ni Nj Nr i\u2265j auxH =\n  begin\n    i \u2261\u27e8 aux Ni Nj Nr auxH \u27e9\n    j * succ (div (i - j) j) + r \u2261\u27e8 prf \u27e9\n    j * (div i j) + r\n  \u220e\n\n  where prf : j * succ (div (i - j) j) + r \u2261 j * (div i j) + r\n        prf = subst (\u03bb x \u2192 j * x + r \u2261 j * (div i j) + r )\n                    (div-x\u2265y i\u2265j  )\n                    refl\n\ndiv-x\u2265y-correct : {i j : D} \u2192 N i \u2192 N j \u2192\n                  (ih : DIV (i - j) j (div (i - j) j)) \u2192\n                  GE i j \u2192\n                  \u2203D (\u03bb r \u2192 (N r) \u2227 (LT r j) \u2227 (i \u2261 j * (div i j) + r))\ndiv-x\u2265y-correct {i} {j} Ni Nj ih i\u2265j =\n  r , (Nr , (r<j , (div-x\u2265y-aux Ni Nj Nr i\u2265j auxH)))\n\n    where\n      -- The parts of the inductive hipothesis ih\n      r : D\n      r = \u2203D-proj\u2081 (\u2227-proj\u2082 ih)\n\n      r-correct : N r \u2227 LT r j \u2227 i - j \u2261 j * (div (i - j) j) + r\n      r-correct = \u2203D-proj\u2082 (\u2227-proj\u2082 ih)\n\n      Nr : N r\n      Nr = \u2227-proj\u2081 r-correct\n\n      r<j : LT r j\n      r<j = \u2227-proj\u2081 (\u2227-proj\u2082 r-correct)\n\n      auxH : i - j \u2261 j * (div (i - j) j) + r\n      auxH = \u2227-proj\u2082 (\u2227-proj\u2082 r-correct)\n","old_contents":"------------------------------------------------------------------------------\n-- The division result is correct\n------------------------------------------------------------------------------\n\nmodule Examples.Division.IsCorrectER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import Examples.Division.EquationsER\nopen import Examples.Division.Specification\n\nopen import LTC.Data.N\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF-ER\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.InequalitiesPCF\n\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IsNER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\n-- The division result is correct when the dividend is less than\n-- the divisor.\n\ndiv-x<y-aux : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192 i \u2261 j * (div i j) + i\ndiv-x<y-aux {i} {j} Ni Nj i<j = sym\n    ( begin\n        j * (div i j) + i \u2261\u27e8 prf\u2081 \u27e9\n        j * zero + i      \u2261\u27e8 prf\u2082 \u27e9\n        zero + i          \u2261\u27e8 +-leftIdentity Ni \u27e9\n        i\n      \u220e\n    )\n    where\n      prf\u2081 : j * (div i j) + i \u2261 j * zero + i\n      prf\u2081 = subst (\u03bb x \u2192 j * x + i \u2261 j * zero + i )\n                   (sym (div-x<y i<j ))\n                   refl\n\n      prf\u2082 : j * zero + i \u2261 zero + i\n      prf\u2082 = subst (\u03bb x \u2192 x + i \u2261 zero + i)\n                   (sym (*-rightZero Nj))\n                   refl\n\ndiv-x<y-correct : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192\n                  (\u2203D (\u03bb r \u2192 N r \u2227 LT r j \u2227 i \u2261 j * (div i j) + r))\ndiv-x<y-correct {i} Ni Nj i<j =\n  i , (Ni , (i<j , (div-x<y-aux Ni Nj i<j)))\n\n\n-- The division result is correct when the dividend is greater or equal\n-- than the divisor.\n-- Using the inductive hypothesis 'ih' we know that\n-- 'i - j = j * (div (i - j) j) + r'. From\n-- that we get 'i = j * (succ (div (i - j) j)) + r', and\n-- we know 'div i j = succ (div (i - j) j); therefore we\n-- get 'i = j * div i j + r'.\n\npostulate\n  aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n        i - j \u2261 j * (div (i - j) j) + r \u2192\n        i \u2261 j * succ (div (i - j) j) + r\n\ndiv-x\u2265y-aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n              GE i j \u2192\n              i - j \u2261 j * (div (i - j) j) + r \u2192\n              i \u2261 j * (div i j) + r\ndiv-x\u2265y-aux {i} {j} {r} Ni Nj Nr i\u2265j auxH =\n  begin\n    i \u2261\u27e8 aux Ni Nj Nr auxH \u27e9\n    j * succ (div (i - j) j) + r \u2261\u27e8 prf \u27e9\n    j * (div i j) + r\n  \u220e\n\n  where prf : j * succ (div (i - j) j) + r \u2261 j * (div i j) + r\n        prf = subst (\u03bb x \u2192 j * x + r \u2261 j * (div i j) + r )\n                    (div-x\u2265y i\u2265j  )\n                    refl\n\ndiv-x\u2265y-correct : {i j : D} \u2192 N i \u2192 N j \u2192\n                  (ih : DIV (i - j) j (div (i - j) j)) \u2192\n                  GE i j \u2192\n                  \u2203D (\u03bb r \u2192 (N r) \u2227 (LT r j) \u2227 (i \u2261 j * (div i j) + r))\ndiv-x\u2265y-correct {i} {j} Ni Nj ih i\u2265j =\n  r , (Nr , (r<j , (div-x\u2265y-aux Ni Nj Nr i\u2265j auxH)))\n\n    where\n      -- The parts of the inductive hipothesis 'ih-i-j'\n      r : D\n      r = \u2203D-proj\u2081 (\u2227-proj\u2082 ih)\n\n      r-correct : N r \u2227 LT r j \u2227 i - j \u2261 j * (div (i - j) j) + r\n      r-correct = \u2203D-proj\u2082 (\u2227-proj\u2082 ih)\n\n      Nr : N r\n      Nr = \u2227-proj\u2081 r-correct\n\n      r<j : LT r j\n      r<j = \u2227-proj\u2081 (\u2227-proj\u2082 r-correct)\n\n      auxH : i - j \u2261 j * (div (i - j) j) + r\n      auxH = \u2227-proj\u2082 (\u2227-proj\u2082 r-correct)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fdf9a63d26a644855d906d2de774b44bfc7ba4ae","subject":"-\u2297\u214b-dual +dual-\u214b +dual-\u2297","message":"-\u2297\u214b-dual +dual-\u214b +dual-\u2297\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Multiplicative.agda","new_file":"Control\/Protocol\/Multiplicative.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                              ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                              ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                              ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                              ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  dual-\u214b : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  dual-\u214b end Q = refl\n  dual-\u214b (recv P) Q = com=\u2032 _ \u03bb m \u2192 dual-\u214b (P m) _\n  dual-\u214b (send P) end = refl\n  dual-\u214b (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 dual-\u214b (send P) (Q n)\n  dual-\u214b (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 dual-\u214b (P m) (send Q))\n    ,inr: (\u03bb n \u2192 dual-\u214b (send P) (Q n)) ]\n\n  dual-\u2297 : \u2200 P Q \u2192 dual (P \u2297 Q) \u2261 dual P \u214b dual Q\n  dual-\u2297 end Q = refl\n  dual-\u2297 (send P) Q = com=\u2032 _ \u03bb m \u2192 dual-\u2297 (P m) _\n  dual-\u2297 (recv P) end = refl\n  dual-\u2297 (recv P) (send Q) = com=\u2032 _ \u03bb n \u2192 dual-\u2297 (recv P) (Q n)\n  dual-\u2297 (recv P) (recv Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 dual-\u2297 (P m) (recv Q))\n    ,inr: (\u03bb n \u2192 dual-\u2297 (recv P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  fwd : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  fwd end      = end\n  fwd (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (fwd (P x))\n  fwd (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (fwd (P x))\n\n  \u214b-id = fwd\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\ndata View-\u2218-proto : (P Q R : Proto) \u2192 \u2605\u2081 where\n  sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R\n           \u2192 View-\u2218-proto (send P) (send Q) R\n  recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n           \u2192 View-\u2218-proto (recv P) Q R\n  recvR-sendR : \u2200 {MP MQ MR}ioP(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                \u2192 View-\u2218-proto (com ioP P) (recv Q) (send R)\n  recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n           \u2192 View-\u2218-proto (send P) (recv Q) (recv R)\n  endL : \u2200 Q R \u2192 View-\u2218-proto end Q R\n  sendLM : \u2200 {MP}(P : MP \u2192 Proto)R \u2192 View-\u2218-proto (send P) end R\n  recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) end\n\nview-\u2218-proto : \u2200 P Q R \u2192 View-\u2218-proto P Q R\nview-\u2218-proto end      Q        R        = endL Q R\nview-\u2218-proto (recv P) Q        R        = recvLL P Q R\nview-\u2218-proto (send P) end      R        = sendLM P R\nview-\u2218-proto (send P) (recv Q) end      = recvL-sendR P Q\nview-\u2218-proto (send P) (recv Q) (recv R) = recvRR P Q R\nview-\u2218-proto (send P) (recv Q) (send R) = recvR-sendR Out P Q R\nview-\u2218-proto (send P) (send Q) R        = sendLL P Q R\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    \u214b-\u2218 : \u2200 {P Q R} \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218 {P} {Q} {R} pq qr with view-\u2218-proto P Q R\n    \u214b-\u2218 (inl m , pq) qr | sendLL P Q R = \u214b-sendL R m (\u214b-\u2218 {P m} {send Q} {R} pq qr)\n    \u214b-\u2218 (inr m , pq) qr | sendLL P Q R = \u214b-\u2218 {send P} {Q m} pq (qr m)\n    \u214b-\u2218 pq qr           | recvLL P Q M = \u03bb m \u2192 \u214b-\u2218 {P m} (pq m) qr\n    \u214b-\u2218 pq (inl m , qr) | recvR-sendR ioP P Q R = \u214b-\u2218 {com ioP P} {Q m} {send R} (coe (\u214b-recvR (com ioP P) Q) pq m) qr\n    \u214b-\u2218 pq (inr m , qr) | recvR-sendR ioP P Q R = \u214b-sendR (com ioP P) m (\u214b-\u2218 {com ioP P} {recv Q} {R m} pq qr)\n    \u214b-\u2218 pq qr           | recvRR P Q R = \u03bb m \u2192 \u214b-\u2218 {send P} {recv Q} {R m} pq (qr m)\n    \u214b-\u2218 pq qr           | endL Q R = \u22b8-apply {Q} {R} qr pq\n    \u214b-\u2218 (m , pq) qr     | sendLM P R = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218 pq (m , qr)     | recvL-sendR P Q = \u214b-\u2218 {send P} {Q m} {end} (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                              ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                              ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                              ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                              ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  \u2297\u214b-dual end Q = refl\n  \u2297\u214b-dual (recv P) Q = com=\u2032 _ \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (send P) end = refl\n  \u2297\u214b-dual (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)\n  \u2297\u214b-dual (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (send Q))\n    ,inr: (\u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  fwd : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  fwd end      = end\n  fwd (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (fwd (P x))\n  fwd (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (fwd (P x))\n\n  \u214b-id = fwd\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\ndata View-\u2218-proto : (P Q R : Proto) \u2192 \u2605\u2081 where\n  sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R\n           \u2192 View-\u2218-proto (send P) (send Q) R\n  recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n           \u2192 View-\u2218-proto (recv P) Q R\n  recvR-sendR : \u2200 {MP MQ MR}ioP(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                \u2192 View-\u2218-proto (com ioP P) (recv Q) (send R)\n  recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n           \u2192 View-\u2218-proto (send P) (recv Q) (recv R)\n  endL : \u2200 Q R \u2192 View-\u2218-proto end Q R\n  sendLM : \u2200 {MP}(P : MP \u2192 Proto)R \u2192 View-\u2218-proto (send P) end R\n  recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) end\n\nview-\u2218-proto : \u2200 P Q R \u2192 View-\u2218-proto P Q R\nview-\u2218-proto end      Q        R        = endL Q R\nview-\u2218-proto (recv P) Q        R        = recvLL P Q R\nview-\u2218-proto (send P) end      R        = sendLM P R\nview-\u2218-proto (send P) (recv Q) end      = recvL-sendR P Q\nview-\u2218-proto (send P) (recv Q) (recv R) = recvRR P Q R\nview-\u2218-proto (send P) (recv Q) (send R) = recvR-sendR Out P Q R\nview-\u2218-proto (send P) (send Q) R        = sendLL P Q R\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    \u214b-\u2218 : \u2200 {P Q R} \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218 {P} {Q} {R} pq qr with view-\u2218-proto P Q R\n    \u214b-\u2218 (inl m , pq) qr | sendLL P Q R = \u214b-sendL R m (\u214b-\u2218 {P m} {send Q} {R} pq qr)\n    \u214b-\u2218 (inr m , pq) qr | sendLL P Q R = \u214b-\u2218 {send P} {Q m} pq (qr m)\n    \u214b-\u2218 pq qr           | recvLL P Q M = \u03bb m \u2192 \u214b-\u2218 {P m} (pq m) qr\n    \u214b-\u2218 pq (inl m , qr) | recvR-sendR ioP P Q R = \u214b-\u2218 {com ioP P} {Q m} {send R} (coe (\u214b-recvR (com ioP P) Q) pq m) qr\n    \u214b-\u2218 pq (inr m , qr) | recvR-sendR ioP P Q R = \u214b-sendR (com ioP P) m (\u214b-\u2218 {com ioP P} {recv Q} {R m} pq qr)\n    \u214b-\u2218 pq qr           | recvRR P Q R = \u03bb m \u2192 \u214b-\u2218 {send P} {recv Q} {R m} pq (qr m)\n    \u214b-\u2218 pq qr           | endL Q R = \u22b8-apply {Q} {R} qr pq\n    \u214b-\u2218 (m , pq) qr     | sendLM P R = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218 pq (m , qr)     | recvL-sendR P Q = \u214b-\u2218 {send P} {Q m} {end} (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a089edb05f5612fdfe68ee7885df9911de2a0e69","subject":"Added bag-equality to Tree a la NAD","message":"Added bag-equality to Tree a la NAD\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym swp\u2081 = swp\u2081\nSwp-sym swp\u2082 = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary using (Dec ; yes ; no)\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  open import Function.Inverse\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = Function.Inverse.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 Function.Inverse.sym (t1\u2248s1 x)) (\u03bb x \u2192 Function.Inverse.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = Function.Inverse.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  (x \u2237\u2262 neg) eq = neg (cong (_\u2237_ x) eq)\n\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n{-\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' = ?\n-}\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym swp\u2081 = swp\u2081\nSwp-sym swp\u2082 = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0df4acf5077af5d5a3e048a7350a8537aa505d03","subject":"LamportOneBit: verifkey-is-hash-sig and sign-both-reveals-signkey","message":"LamportOneBit: verifkey-is-hash-sig and sign-both-reveals-signkey\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/LamportOneBit.agda","new_file":"Crypto\/Sig\/LamportOneBit.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Crypto.Sig.LamportOneBit\n          (#secret : \u2115)\n          (#digest : \u2115)\n          (hash    : Bits #secret \u2192 Bits #digest)\n          where\n\n#signkey   = 2* #secret\n#verifkey  = 2* #digest\n#signature = #secret\n\nDigest     = Bits #digest\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nmodule signkey (sk : SignKey) where\n  skL = take #secret sk\n  skH = drop #secret sk\n  vkL = hash skL\n  vkH = hash skH\n  skP = skL , skH\n\n-- Derive the public key by hashing each secret\nverif-key : SignKey \u2192 VerifKey\nverif-key = map2* #secret #digest hash\n\nmodule verifkey (vk : VerifKey) where\n  vkL = take #digest vk\n  vkH = drop #digest vk\n\n-- Key generation\nkey-gen : SignKey \u2192 VerifKey \u00d7 SignKey\nkey-gen sk = vk , sk\n  module key-gen where\n    vk = verif-key sk\n\n-- Sign a single bit message\nsign : SignKey \u2192 Bit \u2192 Signature\nsign sk b = take2* _ b sk\n\n-- Verify the signature of a single bit message\nverify : VerifKey \u2192 Bit \u2192 Signature \u2192 Bit\nverify vk b sig = take2* _ b vk == hash sig\n\nverifkey-is-hash-sig : \u2200 sk b \u2192 take2* _ b (verif-key sk) \u2261 hash (sign sk b)\nverifkey-is-hash-sig sk 0b = take-++ #digest (hash (take #secret sk)) _\nverifkey-is-hash-sig sk 1b = drop-++ #digest (hash (take #secret sk)) _\n\nverify-correct-sig : \u2200 sk b \u2192 verify (verif-key sk) b (sign sk b) \u2261 1b\nverify-correct-sig sk b = ==-reflexive (verifkey-is-hash-sig sk b)\n\nsign-both-reveals-signkey : \u2200 sk \u2192 sign sk 0b ++ sign sk 1b \u2261 sk\nsign-both-reveals-signkey = take-drop-lem #secret\n\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\n\nmodule Assuming-injectivity (hash-inj : Injective hash) where\n\n  -- If one considers the hash function injective, then, so is verif-key\n  verif-key-inj : \u2200 {sk1 sk2} \u2192 verif-key sk1 \u2261 verif-key sk2 \u2192 sk1 \u2261 sk2\n  verif-key-inj {sk1} {sk2} e\n    = take-drop= #secret sk1 sk2\n        (hash-inj (++-inj\u2081 e))\n        (hash-inj (++-inj\u2082 sk1.vkL sk2.vkL e))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n\n  -- Therefor under this assumption, different secret keys means different public keys\n  verif-key-corrolary : \u2200 {sk1 sk2} \u2192 sk1 \u2262 sk2 \u2192 verif-key sk1 \u2262 verif-key sk2\n  verif-key-corrolary {sk1} {sk2} sk\u2262 vk= = sk\u2262 (verif-key-inj vk=)\n\n  -- If one considers the hash function injective then there is\n  -- only one signing key which can sign correctly all (0 and 1)\n  -- the messages\n  signkey-uniqness :\n        \u2200 sk1 sk2 \u2192\n          (\u2200 b \u2192 verify (verif-key sk1) b (sign sk2 b) \u2261 1b) \u2192\n          sk1 \u2261 sk2\n  signkey-uniqness sk1 sk2 e\n    = take-drop= #secret sk1 sk2 (lemmaL (e 0b)) (lemmaH (e 1b))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n      lemmaL : verify (verif-key sk1) 0b (sign sk2 0b) \u2261 1b \u2192\n              sk1.skL \u2261 sk2.skL\n      lemmaL e0\n        rewrite take-++ #digest sk1.vkL sk1.vkH\n              | hash-inj (==\u21d2\u2261 e0)\n              = refl\n\n      lemmaH : verify (verif-key sk1) 1b (sign sk2 1b) \u2261 1b \u2192\n               sk1.skH \u2261 sk2.skH\n      lemmaH e1\n        rewrite drop-++ #digest sk1.vkL sk1.vkH\n              | hash-inj (==\u21d2\u2261 e1)\n              = refl\n\nmodule Assuming-invertibility\n         (unhash : Digest \u2192 Secret)\n         (unhash-hash : \u2200 x \u2192 unhash (hash x) \u2261 x)\n         (hash-unhash : \u2200 x \u2192 hash (unhash x) \u2261 x)\n         where\n\n  -- EXERCISES\n  {-\n  recover-signkey : VerifKey \u2192 SignKey\n  recover-signkey = {!!}\n\n  recover-signkey-correct : \u2200 sk \u2192 recover-signkey (verif-key sk) \u2261 sk\n  recover-signkey-correct = {!!}\n\n  forgesig : VerifKey \u2192 Bit \u2192 Signature\n  forgesig = {!!}\n\n  forgesig-correct : \u2200 vk b \u2192 verify vk b (forgesig vk b) \u2261 1b\n  forgesig-correct = {!!}\n  -}\n\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Crypto.Sig.LamportOneBit\n          (#secret : \u2115)\n          (#digest : \u2115)\n          (hash    : Bits #secret \u2192 Bits #digest)\n          where\n\n#signkey   = 2* #secret\n#verifkey  = 2* #digest\n#signature = #secret\n\nDigest     = Bits #digest\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nmodule signkey (sk : SignKey) where\n  skL = take #secret sk\n  skH = drop #secret sk\n  vkL = hash skL\n  vkH = hash skH\n\n-- Derive the public key by hashing each secret\nverif-key : SignKey \u2192 VerifKey\nverif-key = map2* #secret #digest hash\n\nmodule verifkey (vk : VerifKey) where\n  vkL = take #digest vk\n  vkH = drop #digest vk\n\n-- Key generation\nkey-gen : SignKey \u2192 VerifKey \u00d7 SignKey\nkey-gen sk = vk , sk\n  module key-gen where\n    vk = verif-key sk\n\n-- Sign a single bit message\nsign : SignKey \u2192 Bit \u2192 Signature\nsign sk b = take2* _ b sk\n\n-- Verify the signature of a single bit message\nverify : VerifKey \u2192 Bit \u2192 Signature \u2192 Bit\nverify vk b sig = take2* _ b vk == hash sig\n\nverify-correct-sig : \u2200 sk b \u2192 verify (verif-key sk) b (sign sk b) \u2261 1b\nverify-correct-sig sk 0b = ==-reflexive (take-++ #digest (hash (take #secret sk)) _)\nverify-correct-sig sk 1b = ==-reflexive (drop-++ #digest (hash (take #secret sk)) _)\n\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\n\nmodule Assuming-injectivity (hash-inj : Injective hash) where\n\n  -- If one considers the hash function injective, then, so is verif-key\n  verif-key-inj : \u2200 {sk1 sk2} \u2192 verif-key sk1 \u2261 verif-key sk2 \u2192 sk1 \u2261 sk2\n  verif-key-inj {sk1} {sk2} e\n    = take-drop= #secret sk1 sk2\n        (hash-inj (++-inj\u2081 e))\n        (hash-inj (++-inj\u2082 sk1.vkL sk2.vkL e))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n\n  -- Therefor under this assumption, different secret keys means different public keys\n  verif-key-corrolary : \u2200 {sk1 sk2} \u2192 sk1 \u2262 sk2 \u2192 verif-key sk1 \u2262 verif-key sk2\n  verif-key-corrolary {sk1} {sk2} sk\u2262 vk= = sk\u2262 (verif-key-inj vk=)\n\n  -- If one considers the hash function injective then there is\n  -- only one signing key which can sign correctly all (0 and 1)\n  -- the messages\n  signkey-uniqness :\n        \u2200 sk1 sk2 \u2192\n          (\u2200 b \u2192 verify (verif-key sk1) b (sign sk2 b) \u2261 1b) \u2192\n          sk1 \u2261 sk2\n  signkey-uniqness sk1 sk2 e\n    = take-drop= #secret sk1 sk2 (lemmaL (e 0b)) (lemmaH (e 1b))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n      lemmaL : verify (verif-key sk1) 0b (sign sk2 0b) \u2261 1b \u2192\n              sk1.skL \u2261 sk2.skL\n      lemmaL e0\n        rewrite take-++ #digest sk1.vkL sk1.vkH\n              | hash-inj (==\u21d2\u2261 e0)\n              = refl\n\n      lemmaH : verify (verif-key sk1) 1b (sign sk2 1b) \u2261 1b \u2192\n               sk1.skH \u2261 sk2.skH\n      lemmaH e1\n        rewrite drop-++ #digest sk1.vkL sk1.vkH\n              | hash-inj (==\u21d2\u2261 e1)\n              = refl\n\nmodule Assuming-invertibility\n         (unhash : Digest \u2192 Secret)\n         (unhash-hash : \u2200 x \u2192 unhash (hash x) \u2261 x)\n         (hash-unhash : \u2200 x \u2192 hash (unhash x) \u2261 x)\n         where\n\n  -- EXERCISES\n  {-\n  recover-signkey : VerifKey \u2192 SignKey\n  recover-signkey = {!!}\n\n  recover-signkey-correct : \u2200 sk \u2192 recover-signkey (verif-key sk) \u2261 sk\n  recover-signkey-correct = {!!}\n\n  forgesig : VerifKey \u2192 Bit \u2192 Signature\n  forgesig = {!!}\n\n  forgesig-correct : \u2200 vk b \u2192 verify vk b (forgesig vk b) \u2261 1b\n  forgesig-correct = {!!}\n  -}\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"39a7b981322292b495ec7563c4db2d563d951ae5","subject":"Field.Solver: some progress but far from done...","message":"Field.Solver: some progress but far from done...\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field\/Solver.agda","new_file":"lib\/Algebra\/Field\/Solver.agda","new_contents":"import Algebra.Field as Field\n\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nopen import Level\nopen import Data.Nat as Nat using (\u2115) renaming (suc to 1+_)\nopen import Data.One\nopen import Data.Product\n\nopen import Function\n\nmodule Algebra.Field.Solver where\n\ndata R-Op : Set where\n  [+] [*] [-] : R-Op\n\ndata F-Op : Set where\n  [+] [*] [-] [\/] : F-Op\n\ndata Tm (F Op Var : Set) : Set where\n  op  : (o : Op) (t\u2081 t\u2082 : Tm F Op Var) \u2192 Tm F Op Var\n  con : F \u2192 Tm F Op Var\n  var : Var \u2192 Tm F Op Var\n\ninfixl 6 _:+_ _:-_\ninfixl 7 _:*_ _:\/_\npattern _:+_ x y = op [+] x y\npattern _:*_ x y = op [*] x y\npattern _:-_ x y = op [-] x y\npattern _:\/_ x y = op [\/] x y\n\nR-include : R-Op \u2192 F-Op\nR-include [+] = [+]\nR-include [*] = [*]\nR-include [-] = [-]\n\nmodule _ {F Op Var : Set}(evalOp : Op \u2192 F \u2192 F \u2192 F)(\u03c1 : Var \u2192 F) where\n  eval : Tm F Op Var \u2192 F\n  eval (op o t u) = evalOp o (eval t) (eval u)\n  eval (con x) = x\n  eval (var x) = \u03c1 x\n\nmodule _ {F Var : Set}(evF : Tm F F-Op Var \u2192 Set) where\n  Assumptions : Tm F F-Op Var \u2192 Set\n  Assumptions (op [+] t u) = Assumptions t \u00d7 Assumptions u\n  Assumptions (op [*] t u) = Assumptions t \u00d7 Assumptions u\n  Assumptions (op [-] t u) = Assumptions t \u00d7 Assumptions u\n  Assumptions (op [\/] t u) = Assumptions t \u00d7 Assumptions u \u00d7 evF u\n  Assumptions (con x) = Lift \ud835\udfd9\n  Assumptions (var x) = Lift \ud835\udfd9\n\nmodule _\n  {Var : Set}\n  {F : Set}\n  (\ud835\udd3d : Field.Field-Ops F)\n  where\n\n  open Field.Field-Ops \ud835\udd3d\n\n  F-Op-eval : F-Op \u2192 F \u2192 F \u2192 F\n  F-Op-eval [+] x y = x + y\n  F-Op-eval [*] x y = x * y\n  F-Op-eval [-] x y = x \u2212 y\n  F-Op-eval [\/] x y = x \/ y\n\n  R-Op-eval : R-Op \u2192 F \u2192 F \u2192 F\n  R-Op-eval o = F-Op-eval (R-include o)\n\n  evalF : (\u03c1 : Var \u2192 F) \u2192 Tm F F-Op Var \u2192 F\n  evalR : (\u03c1 : Var \u2192 F) \u2192 Tm F R-Op Var \u2192 F\n  evalF = eval F-Op-eval\n  evalR = eval R-Op-eval\n\n  private\n      TmF = Tm F F-Op Var\n      TmR = Tm F R-Op Var\n\n  infix 2 _\/_:[_,_]\n  record Div (evF : TmF \u2192 F)(evR : TmR \u2192 F)(t : TmF) : Set where\n    constructor _\/_:[_,_]\n    field\n      N : TmR\n      D : TmR\n      .non-zero-D : evR D \u2262 0#\n      .is-correct : evF t \u2261 evR N \/ evR D\n\n  module _ (FS : Field.Field-Struct \ud835\udd3d)(\u03c1 : Var \u2192 F) where\n    open Field.Field-Struct FS\n\n    private\n      Div' = Div (evalF \u03c1) (evalR \u03c1)\n      evF = evalF \u03c1\n\n    NonZeroF : TmF \u2192 Set\n    NonZeroF x = evF x \u2262 0#\n\n    _+Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :+ u)\n    (N \/ D :[ nD , ok ]) +Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* D\u2081 :+ N\u2081 :* D \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , += ok ok\u2081 \u2219 +-quotient nD nD\u2081 ]\n\n    _*Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :* u)\n    (N \/ D :[ nD , ok ]) *Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* N\u2081 \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , *= ok ok\u2081 \u2219 *-quotient nD nD\u2081 ]\n\n    _\u2212Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :- u)\n    (N \/ D :[ nD , ok ]) \u2212Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* D\u2081 :- N\u2081 :* D \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , \u2212= ok ok\u2081 \u2219 \u2212-quotient nD nD\u2081 ]\n\n    _\/Div_:[_] : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 evF u \u2262 0# \u2192 Div' (t :\/ u)\n    (N \/ D :[ nD , ok ]) \/Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ]) :[ u\u22620 ]\n      = (N :* D\u2081) \/ (D :* N\u2081)\n      :[ no-zero-divisor nD nN\u2081 , \/= ok ok\u2081 \u2219 \/-quotient nN\u2081 nD nD\u2081 ]\n      where .nN\u2081 : evalR \u03c1 N\u2081 \u2262 0#\n            nN\u2081 p = u\u22620 (ok\u2081 \u2219 *= p idp \u2219 0*-zero)\n\n    -- con-Div and var-Div are basically the same but not so convenient to unify\n    con-Div : \u2200 x \u2192 Div' (con x)\n    var-Div : \u2200 x \u2192 Div' (var x)\n\n    con-Div x = con x \/ con 1# :[ 1\u22620 , ! \/1-id ]\n    var-Div x = var x \/ con 1# :[ 1\u22620 , ! \/1-id ]\n\n    to-ring : \u2200 (t : TmF) \u2192 Assumptions NonZeroF t \u2192 Div' t\n    to-ring (op [+] t u) (at , au)       = to-ring t at +Div to-ring u au\n    to-ring (op [*] t u) (at , au)       = to-ring t at *Div to-ring u au\n    to-ring (op [-] t u) (at , au)       = to-ring t at \u2212Div to-ring u au\n    to-ring (op [\/] t u) (at , au , u\u22620) = to-ring t at \/Div to-ring u au :[ u\u22620 ]\n    to-ring (con x) _                    = con-Div x\n    to-ring (var x) _                    = var-Div x\n\n{-\n    Eqn : Set _\n    Eqn = TmF \u00d7 TmF\n\n    divR : \u2200 {t} \u2192 Div' t \u2192 TmR\n    divR (N \/ D :[ nD , ok ]) = {!!}\n\n    to-ring' : \u2200 (t : TmF) \u2192 Assumptions t \u2192 TmR\n    to-ring' t a = {!!}\n    -}\n\nopen import Relation.Binary\nmodule _\n  {F : Set}\n  (\ud835\udd3d : Field.Field F)\n  (_\u225fR_ : Decidable {A = F} _\u2261_)\n\n  where\n  open import Algebra using (RawRing)\n  open import Algebra.RingSolver.AlmostCommutativeRing\n\n  private\n    module \ud835\udd3d = Field.Field \ud835\udd3d\n\n  open \ud835\udd3d\n  R : AlmostCommutativeRing _ _\n  R = record\n        { Carrier = F\n        ; _\u2248_ = _\u2261_\n        ; _+_ = _+_\n        ; _*_ = _*_\n        ; -_ = 0\u2212_\n        ; 0# = 0#\n        ; 1# = 1#\n        ; isAlmostCommutativeRing =\n          record\n            { isCommutativeSemiring =\n              record\n              { +-isCommutativeMonoid =\n                record\n                { isSemigroup =\n                  record\n                  { isEquivalence = \u2261.isEquivalence\n                  ; assoc = \u03bb _ _ _ \u2192 +-assoc\n                  ; \u2219-cong = += }\n                ; identity\u02e1 = \u03bb _ \u2192 0+-identity\n                ; comm = \u03bb _ _ \u2192 +-comm }\n              ; *-isCommutativeMonoid =\n                record\n                { isSemigroup =\n                  record\n                  { isEquivalence = \u2261.isEquivalence\n                  ; assoc = \u03bb _ _ _ \u2192 *-assoc\n                  ; \u2219-cong = *= }\n                ; identity\u02e1 = \u03bb _ \u2192 1*-identity\n                ; comm = \u03bb _ _ \u2192 *-comm }\n              ; distrib\u02b3 = \u03bb _ _ _ \u2192 *-+-distr\u02b3\n              ; zero\u02e1 = \u03bb _ \u2192 0*-zero\n              }\n            ; -\u203fcong = 0\u2212=\n            ; -\u203f*-distrib\u02e1 = \u03bb _ _ \u2192 ! 0\u2212-*-distr\n            ; -\u203f+-comm = \u03bb _ _ \u2192 ! 0\u2212-+-distr\n            }\n        }\n  import Algebra.RingSolver\n  import Algebra.RingSolver.Simple\n\n{- I try first with the RingSolver.Simple, otherwise here is what we need\n\n  Coeff : RawRing _\n  Coeff = {!!}\n\n  morphism : Coeff -Raw-AlmostCommutative\u27f6 R\n  morphism = {!!}\n\n  _coeff\u225f_ : Decidable (Induced-equivalence morphism)\n  x coeff\u225f y = {!!}\n\n  module RS = Algebra.RingSolver Coeff R morphism _coeff\u225f_\n-}\n\n  module RS = Algebra.RingSolver.Simple R _\u225fR_\n\n  import Relation.Binary.Reflection as Reflection\n  open Reflection (\u2261.setoid F) RS.var RS.\u27e6_\u27e7 RS.\u27e6_\u27e7\u2193 RS.correct public\n      using (prove; solve; solve\u2081) renaming (_\u229c_ to _:=_)\n\n  test-+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n  test-+-comm m n = solve\u2081 2 (\u03bb x y \u2192 x RS.:+ y := y RS.:+ x) m n refl\n\n  \u2115\u25b9P : \u2115 \u2192 \u2200 {n} \u2192 RS.Polynomial n\n  \u2115\u25b9P 0 = RS.con 0#\n  \u2115\u25b9P 1 = RS.con 1#\n  \u2115\u25b9P (1+ x) = RS.con 1# RS.:+ \u2115\u25b9P x\n\n  {-\n  test-0\u2212-inverse : \u2200 m \u2192 m + 0\u2212 m \u2261 0#\n  test-0\u2212-inverse m = solve\u2081 1 (\u03bb x \u2192 x RS.:+ (RS.:- x) := RS.con 0#) m refl\n\n  test-0\u2212-involutive : \u2200 m \u2192 0\u2212 (0\u2212 m) \u2261 m\n  test-0\u2212-involutive m = solve\u2081 1 (\u03bb x \u2192 RS.:- (RS.:- x) := x) m refl\n\n  test-2* : \u2200 n \u2192 2# * n \u2261 n + n\n  test-2* n = solve\u2081 1 (\u03bb x \u2192 (\u2115\u25b9P 2 RS.:* x) := (x RS.:+ x)) n refl\n  -}\n\n  open import Data.Vec\n  open import Data.Fin using (Fin)\n\n  poly : \u2200 {n} \u2192 Tm F R-Op (Fin n) \u2192 RS.Polynomial n\n  poly (op [+] t u) = poly t RS.:+ poly u\n  poly (op [*] t u) = poly t RS.:* poly u\n  poly (op [-] t u) = poly t RS.:- poly u\n  poly (con x) = RS.con x\n  poly (var x) = RS.var x\n\n  {-\n  FO = \ud835\udd3d.field-ops\n  FS = \ud835\udd3d.field-struct\n\n  Expr : \u2115 \u2192 Set\n  Expr n = \u03a3 (Tm F F-Op (Fin n))\n    -- this is probably wrong...\n    (\u03bb t \u2192 (\u03c1 : Fin n \u2192 F) \u2192 Assumptions (NonZeroF FO FS \u03c1) t)\n\n  Sem : Set\n  Sem = F\n\n  -- this takes only the numerator in account\n  \u27e6_\u27e7 : \u2200 {n} \u2192 Expr n \u2192 Vec F n \u2192 Sem\n  \u27e6 t , a \u27e7 \u03c1 = RS.\u27e6 poly (Div.N (to-ring FO FS (flip lookup \u03c1) t (a (flip lookup \u03c1)))) \u27e7 \u03c1\n\n  -- this takes only the numerator in account\n  \u27e6_\u27e7\u2193 : \u2200 {n} \u2192 Expr n \u2192 Vec F n \u2192 Sem\n  \u27e6 t , a \u27e7\u2193 \u03c1 = RS.\u27e6 poly (Div.N (to-ring FO FS (flip lookup \u03c1) t (a (flip lookup \u03c1)))) \u27e7\u2193 \u03c1\n\n  correct : \u2200 {n} (e : Expr n) \u03c1 \u2192 \u27e6 e \u27e7\u2193 \u03c1 \u2261 \u27e6 e \u27e7 \u03c1\n  correct (op o t t\u2081 , a) \u03c1 = {!!}\n  correct (con x , a) \u03c1 = {!idp!}\n  correct (var x , a) \u03c1 = {!!}\n\n  open Reflection \u2261.setoid var \u27e6_\u27e7 \u27e6_\u27e7\u2193 correct public\n    using (prove; solve) renaming (_\u229c_ to _:=_)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Algebra.Field as Field\nimport Algebra.RingSolver as RS\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Level\nopen import Data.Nat as Nat using (\u2115)\nopen import Data.Fin\nopen import Data.One\nopen import Data.Product\n\nopen import Function\n\nmodule Algebra.Field.Solver where\n\ndata R-Op : Set where\n  [+] [*] [-] : R-Op\n\ndata F-Op : Set where\n  [+] [*] [-] [\/] : F-Op\n\ndata Tm (F Op Var : Set) : Set where\n  op  : (o : Op) (t\u2081 t\u2082 : Tm F Op Var) \u2192 Tm F Op Var\n  con : F \u2192 Tm F Op Var\n  var : Var \u2192 Tm F Op Var\n\ninfixl 6 _:+_ _:-_\ninfixl 7 _:*_ _:\/_\npattern _:+_ x y = op [+] x y\npattern _:*_ x y = op [*] x y\npattern _:-_ x y = op [-] x y\npattern _:\/_ x y = op [\/] x y\n\nR-include : R-Op \u2192 F-Op\nR-include [+] = [+]\nR-include [*] = [*]\nR-include [-] = [-]\n\nmodule _ {F Op Var : Set}(evalOp : Op \u2192 F \u2192 F \u2192 F)(\u03c1 : Var \u2192 F) where\n  eval : Tm F Op Var \u2192 F\n  eval (op o t u) = evalOp o (eval t) (eval u)\n  eval (con x) = x\n  eval (var x) = \u03c1 x\n\nmodule _\n  {Var : Set}\n  {F : Set}\n  (\ud835\udd3d : Field.Field-Ops F)\n  where\n\n  module \ud835\udd3d = Field.Field-Ops \ud835\udd3d\n\n  F-Op-eval : F-Op \u2192 F \u2192 F \u2192 F\n  F-Op-eval [+] x y = x \ud835\udd3d.+ y\n  F-Op-eval [*] x y = x \ud835\udd3d.* y\n  F-Op-eval [-] x y = x \ud835\udd3d.\u2212 y\n  F-Op-eval [\/] x y = x \ud835\udd3d.\/ y\n\n  R-Op-eval : R-Op \u2192 F \u2192 F \u2192 F\n  R-Op-eval o = F-Op-eval (R-include o)\n\n  evalF : (\u03c1 : Var \u2192 F) \u2192 Tm F F-Op Var \u2192 F\n  evalR : (\u03c1 : Var \u2192 F) \u2192 Tm F R-Op Var \u2192 F\n  evalF = eval F-Op-eval\n  evalR = eval R-Op-eval\n\n  infix 2 _\/_:[_,_]\n  record Div \u03c1 (t : Tm F F-Op Var) : Set where\n    constructor _\/_:[_,_]\n    field\n      N : Tm F R-Op Var\n      D : Tm F R-Op Var\n      .non-zero-D : evalR \u03c1 D \u2262 \ud835\udd3d.`0\n      .is-correct : evalF \u03c1 t \u2261 evalR \u03c1 N \ud835\udd3d.\/ evalR \u03c1 D\n\n  module _ (FS : Field.Field-Struct \ud835\udd3d)(\u03c1 : Var \u2192 F) where\n    open Field.Field-Struct FS\n\n    private\n      TmF = Tm F F-Op Var\n      TmR = Tm F R-Op Var\n\n      Div' = Div \u03c1\n      ev = evalF \u03c1\n\n    _+Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :+ u)\n    (N \/ D :[ nD , ok ]) +Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* D\u2081 :+ N\u2081 :* D \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , += ok ok\u2081 \u2219 +-quotient nD nD\u2081 ]\n\n    _*Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :* u)\n    (N \/ D :[ nD , ok ]) *Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* N\u2081 \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , *= ok ok\u2081 \u2219 *-quotient nD nD\u2081 ]\n\n    _\u2212Div_ : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 Div' (t :- u)\n    (N \/ D :[ nD , ok ]) \u2212Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ])\n       = N :* D\u2081 :- N\u2081 :* D \/ D :* D\u2081\n       :[ no-zero-divisor nD nD\u2081 , \u2212= ok ok\u2081 \u2219 \u2212-quotient nD nD\u2081 ]\n\n    _\/Div_:[_] : \u2200 {t u} \u2192 Div' t \u2192 Div' u \u2192 ev u \u2262 `0 \u2192 Div' (t :\/ u)\n    (N \/ D :[ nD , ok ]) \/Div (N\u2081 \/ D\u2081 :[ nD\u2081 , ok\u2081 ]) :[ u\u22620 ]\n      = (N :* D\u2081) \/ (D :* N\u2081)\n      :[ no-zero-divisor nD nN\u2081 , \/= ok ok\u2081 \u2219 \/-quotient nN\u2081 nD nD\u2081 ]\n      where .nN\u2081 : evalR \u03c1 N\u2081 \u2262 `0\n            nN\u2081 p = u\u22620 (ok\u2081 \u2219 *= p idp \u2219 0*-zero)\n\n    -- con-Div and var-Div are basically the same but not so convenient to unify\n    con-Div : \u2200 x \u2192 Div' (con x)\n    var-Div : \u2200 x \u2192 Div' (var x)\n\n    con-Div x = con x \/ con \ud835\udd3d.`1 :[ 1\u22620 , ! \/1-id ]\n    var-Div x = var x \/ con \ud835\udd3d.`1 :[ 1\u22620 , ! \/1-id ]\n\n    Assumptions : TmF \u2192 Set\n    Assumptions (op [+] t u) = Assumptions t \u00d7 Assumptions u\n    Assumptions (op [*] t u) = Assumptions t \u00d7 Assumptions u\n    Assumptions (op [-] t u) = Assumptions t \u00d7 Assumptions u\n    Assumptions (op [\/] t u) = Assumptions t \u00d7 Assumptions u \u00d7 ev u \u2262 `0\n    Assumptions (con x) = Lift \ud835\udfd9\n    Assumptions (var x) = Lift \ud835\udfd9\n\n    to-ring : \u2200 (t : TmF) \u2192 Assumptions t \u2192 Div' t\n    to-ring (op [+] t u) (at , au)       = to-ring t at +Div to-ring u au\n    to-ring (op [*] t u) (at , au)       = to-ring t at *Div to-ring u au\n    to-ring (op [-] t u) (at , au)       = to-ring t at \u2212Div to-ring u au\n    to-ring (op [\/] t u) (at , au , u\u22620) = to-ring t at \/Div to-ring u au :[ u\u22620 ]\n    to-ring (con x) _                    = con-Div x\n    to-ring (var x) _                    = var-Div x\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"89a4ee211b25df9df1f048cc2ebcfb3d84643f88","subject":"Parser: idiom brakets","message":"Parser: idiom brakets\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Text\/Parser.agda","new_file":"lib\/Text\/Parser.agda","new_contents":"open import Data.Maybe.NP as Maybe\nopen import Data.Nat\nopen import Data.Unit\nopen import Data.Char renaming (_==_ to _==\u1d9c_)\nopen import Data.String as String\nopen import Data.List as L using (List; []; _\u2237_; null; filter)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bool\nopen import Function\nimport Level as L\nimport Category.Monad as Cat\nimport Category.Monad.Partiality.NP as Pa\n\nmodule Text.Parser where\n\nopen M? L.zero using () renaming (_=<<_ to _=<<?_; _\u229b_ to _\u229b?_; join to join?)\n\nParser : Set \u2192 Set\nParser A = List Char \u2192? (A \u00d7 List Char)\n\npure : \u2200 {A} \u2192 A \u2192 Parser A\npure a s = just (a , s)\n\ninfixr 3 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : Parser A) \u2192 Parser A\n_\u27e8|\u27e9_ f g s = maybe\u2032 just (g s) $ f s\n\nempty : \u2200 {A} \u2192 Parser A\nempty _ = nothing\n\nsat : (Char \u2192 Bool) \u2192 Parser Char\nsat pred (x \u2237 xs) = (if pred x then pure x else empty) xs\nsat _    []       = nothing\n\neof : Parser \u22a4\neof s = if null s then pure _ [] else empty []\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 Parser A \u2192 List Char \u2192? A\nrunParser p s = map? proj\u2081 (p s)\n\ninfixr 0 _>>=_\n_>>=_ : \u2200 {A B} \u2192 Parser A \u2192 (A \u2192 Parser B) \u2192 Parser B\n_>>=_ pa f = maybe\u2032 (uncurry f) nothing \u2218 pa\n\ninfixl 0 _=<<_\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 Parser B) \u2192 Parser A \u2192 Parser B\n_=<<_ f pa = pa >>= f\n\njoin : \u2200 {A} \u2192 Parser (Parser A) \u2192 Parser A\njoin = _=<<_ id\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\nmap f pa = pa >>= pure \u2218 f\n\ninfixl 4 _\u229b_ _*>_ _<*_\n_\u229b_ : \u2200 {A B} \u2192 Parser (A \u2192 B) \u2192 Parser A \u2192 Parser B\n_\u229b_ pf pa = pf >>= \u03bb f \u2192 pa >>= \u03bb a \u2192 pure (f a)\n\n\u27ea_\u00b7_\u27eb : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192\n            (A \u2192 B \u2192 C) \u2192 Parser A \u2192 Parser B \u2192 Parser C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D} \u2192 (A \u2192 B \u2192 C \u2192 D)\n            \u2192 Parser A \u2192 Parser B \u2192 Parser C \u2192 Parser D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_*>_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser B\np\u2081 *> p\u2082 = p\u2081 >>= const p\u2082\n\n_<*_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser A\np\u2081 <* p\u2082 = p\u2081 >>= \u03bb x \u2192 p\u2082 *> pure x\n\nchoices : \u2200 {A} \u2192 List (Parser A) \u2192 Parser A\nchoices = L.foldr _\u27e8|\u27e9_ empty\n\n{-\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  many p = some p \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  some p = pure _\u2237_ \u229b p \u229b many p\n-}\n\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  many _ 0       = empty\n  many p (suc n) = some p n \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  some p n = pure _\u2237_ \u229b p \u229b many p n\n\nmanySat : (Char \u2192 Bool) \u2192 Parser (List Char)\nmanySat p []       = pure [] []\nmanySat p (x \u2237 xs) = if p x then map? (first (_\u2237_ x)) (manySat p xs) else pure [] (x \u2237 xs)\n\nsomeSat : (Char \u2192 Bool) \u2192 Parser (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nchar : Char \u2192 Parser \u22a4\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nprivate\n  notElem : Char \u2192 List Char \u2192 Bool\n  notElem c = null \u2218 filter (_==\u1d9c_ c)\n\n  syntax notElem c cs = c `notElem` cs\n\n  elem : Char \u2192 List Char \u2192 Bool\n  elem c = not \u2218 notElem c\n\n  syntax elem c cs = c `elem` cs\n\noneOf : List Char \u2192 Parser Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\nnoneOf : List Char \u2192 Parser Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nsomeOneOf : List Char \u2192 Parser (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeNoneOf : List Char \u2192 Parser (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nmanyOneOf : List Char \u2192 Parser (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyNoneOf : List Char \u2192 Parser (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n","old_contents":"open import Data.Maybe.NP as Maybe\nopen import Data.Nat\nopen import Data.Unit\nopen import Data.Char renaming (_==_ to _==\u1d9c_)\nopen import Data.String as String\nopen import Data.List as L using (List; []; _\u2237_; null; filter)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bool\nopen import Function\nimport Level as L\nimport Category.Monad as Cat\nimport Category.Monad.Partiality.NP as Pa\n\nmodule Text.Parser where\n\nopen M? L.zero using () renaming (_=<<_ to _=<<?_; _\u229b_ to _\u229b?_; join to join?)\n\nParser : Set \u2192 Set\nParser A = List Char \u2192? (A \u00d7 List Char)\n\npure : \u2200 {A} \u2192 A \u2192 Parser A\npure a s = just (a , s)\n\ninfixr 3 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : Parser A) \u2192 Parser A\n_\u27e8|\u27e9_ f g s = maybe\u2032 just (g s) $ f s\n\nempty : \u2200 {A} \u2192 Parser A\nempty _ = nothing\n\nsat : (Char \u2192 Bool) \u2192 Parser Char\nsat pred (x \u2237 xs) = (if pred x then pure x else empty) xs\nsat _    []       = nothing\n\neof : Parser \u22a4\neof s = if null s then pure _ [] else empty []\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 Parser A \u2192 List Char \u2192? A\nrunParser p s = map? proj\u2081 (p s)\n\ninfixr 0 _>>=_\n_>>=_ : \u2200 {A B} \u2192 Parser A \u2192 (A \u2192 Parser B) \u2192 Parser B\n_>>=_ pa f = maybe\u2032 (uncurry f) nothing \u2218 pa\n\ninfixl 0 _=<<_\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 Parser B) \u2192 Parser A \u2192 Parser B\n_=<<_ f pa = pa >>= f\n\njoin : \u2200 {A} \u2192 Parser (Parser A) \u2192 Parser A\njoin = _=<<_ id\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\nmap f pa = pa >>= pure \u2218 f\n\ninfixl 4 _\u229b_ _*>_ _<*_\n_\u229b_ : \u2200 {A B} \u2192 Parser (A \u2192 B) \u2192 Parser A \u2192 Parser B\n_\u229b_ pf pa = pf >>= \u03bb f \u2192 pa >>= \u03bb a \u2192 pure (f a)\n\n_*>_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser B\np\u2081 *> p\u2082 = p\u2081 >>= const p\u2082\n\n_<*_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser A\np\u2081 <* p\u2082 = p\u2081 >>= \u03bb x \u2192 p\u2082 *> pure x\n\nchoices : \u2200 {A} \u2192 List (Parser A) \u2192 Parser A\nchoices = L.foldr _\u27e8|\u27e9_ empty\n\n{-\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  many p = some p \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  some p = pure _\u2237_ \u229b p \u229b many p\n-}\n\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  many _ 0       = empty\n  many p (suc n) = some p n \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  some p n = pure _\u2237_ \u229b p \u229b many p n\n\nmanySat : (Char \u2192 Bool) \u2192 Parser (List Char)\nmanySat p []       = pure [] []\nmanySat p (x \u2237 xs) = if p x then map? (first (_\u2237_ x)) (manySat p xs) else pure [] (x \u2237 xs)\n\nsomeSat : (Char \u2192 Bool) \u2192 Parser (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nchar : Char \u2192 Parser \u22a4\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nprivate\n  notElem : Char \u2192 List Char \u2192 Bool\n  notElem c = null \u2218 filter (_==\u1d9c_ c)\n\n  syntax notElem c cs = c `notElem` cs\n\n  elem : Char \u2192 List Char \u2192 Bool\n  elem c = not \u2218 notElem c\n\n  syntax elem c cs = c `elem` cs\n\noneOf : List Char \u2192 Parser Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\nnoneOf : List Char \u2192 Parser Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nsomeOneOf : List Char \u2192 Parser (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeNoneOf : List Char \u2192 Parser (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nmanyOneOf : List Char \u2192 Parser (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyNoneOf : List Char \u2192 Parser (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"00a7767b9e6a38ec98ff6336651129e2786a6a98","subject":"Fix Data.Fin.Logical","message":"Fix Data.Fin.Logical\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/Logical.agda","new_file":"lib\/Data\/Fin\/Logical.agda","new_contents":"{-# OPTIONS --with-K #-}\nmodule Data.Fin.Logical where\n\nopen import Data.Nat.Logical\nopen import Data.Nat.Param.Binary\n-- open import Data.Fin.Param.Binary\n\nopen import Function\nopen import Data.Fin\nopen import Relation.Binary\nopen import Relation.Binary.Logical\n\nprivate\n  module \u27e6\u2115\u27e7s = Setoid \u27e6\u2115\u27e7-setoid\n\ndata \u27e6Fin\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) Fin Fin where\n  \u27e6zero\u27e7 : \u2200 {n\u2081 n\u2082} {n\u1d63 : \u27e6\u2115\u27e7 n\u2081 n\u2082} \u2192 \u27e6Fin\u27e7 (\u27e6suc\u27e7 n\u1d63) zero zero\n  \u27e6suc\u27e7  : \u2200 {n\u2081 n\u2082} {n\u1d63 : \u27e6\u2115\u27e7 n\u2081 n\u2082} {x\u2081 x\u2082} (x\u1d63 : \u27e6Fin\u27e7 n\u1d63 x\u2081 x\u2082) \u2192 \u27e6Fin\u27e7 (\u27e6suc\u27e7 n\u1d63) (suc x\u2081) (suc x\u2082)\n\nprivate\n  refl : \u2200 {n} \u2192 Reflexive (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  refl {x = zero}  = \u27e6zero\u27e7\n  refl {x = suc _} = \u27e6suc\u27e7 refl\n\n  sym : \u2200 {n} \u2192 Symmetric (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  sym \u27e6zero\u27e7 = \u27e6zero\u27e7\n  sym (\u27e6suc\u27e7 x\u1d63) = \u27e6suc\u27e7 (sym x\u1d63)\n\n  trans : \u2200 {n} \u2192 Transitive (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  trans \u27e6zero\u27e7 x\u1d63 = x\u1d63\n  trans (\u27e6suc\u27e7 x\u1d63) (\u27e6suc\u27e7 y\u1d63) = \u27e6suc\u27e7 (trans x\u1d63 y\u1d63)\n\n  subst : \u2200 {\u2113 n} \u2192 Substitutive (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl) \u2113\n  subst _ \u27e6zero\u27e7    = id\n  subst P (\u27e6suc\u27e7 x\u1d63) = subst (P \u2218 suc) x\u1d63\n\n  isEquivalence : \u2200 {n} \u2192 IsEquivalence (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n","old_contents":"-- NOTE with-K\nmodule Data.Fin.Logical where\n\nopen import Data.Nat.Param.Binary\nopen import Data.Fin.Param.Binary\n\nopen import Function\nopen import Data.Fin\nopen import Relation.Binary\nopen import Relation.Binary.Logical\n\nprivate\n  module \u27e6\u2115\u27e7s = Setoid \u27e6\u2115\u27e7-setoid\n\n{-\ndata \u27e6Fin\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u2080\u27e7) Fin Fin where\n  \u27e6zero\u27e7 : \u2200 {n\u2081 n\u2082} {n\u1d63 : \u27e6\u2115\u27e7 n\u2081 n\u2082} \u2192 \u27e6Fin\u27e7 (suc n\u1d63) zero zero\n  \u27e6suc\u27e7  : \u2200 {n\u2081 n\u2082} {n\u1d63 : \u27e6\u2115\u27e7 n\u2081 n\u2082} {x\u2081 x\u2082} (x\u1d63 : \u27e6Fin\u27e7 n\u1d63 x\u2081 x\u2082) \u2192 \u27e6Fin\u27e7 (suc n\u1d63) (suc x\u2081) (suc x\u2082)\n-}\n\nprivate\n  refl : \u2200 {n} \u2192 Reflexive (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  refl {x = zero}  = \u27e6zero\u27e7\n  refl {x = suc _} = \u27e6suc\u27e7 refl\n\n  sym : \u2200 {n} \u2192 Symmetric (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  sym \u27e6zero\u27e7 = \u27e6zero\u27e7\n  sym (\u27e6suc\u27e7 x\u1d63) = \u27e6suc\u27e7 (sym x\u1d63)\n\n  trans : \u2200 {n} \u2192 Transitive (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  trans \u27e6zero\u27e7 x\u1d63 = x\u1d63\n  trans (\u27e6suc\u27e7 x\u1d63) (\u27e6suc\u27e7 y\u1d63) = \u27e6suc\u27e7 (trans x\u1d63 y\u1d63)\n\n  subst : \u2200 {\u2113 n} \u2192 Substitutive (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl) \u2113\n  subst _ \u27e6zero\u27e7    = id\n  subst P (\u27e6suc\u27e7 x\u1d63) = subst (P \u2218 suc) x\u1d63\n\n  isEquivalence : \u2200 {n} \u2192 IsEquivalence (\u27e6Fin\u27e7 {n} \u27e6\u2115\u27e7s.refl)\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"32219bddbd5b5eb5870ca86007023cc8c25d49d9","subject":"Some more isos","message":"Some more isos\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n\n-- requires extensionality\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\n-- requires extensionality\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 : \u2200 {a} (F : \ud835\udfd8 \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfd8 F \u2194 \ud835\udfd8\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 F = inverses proj\u2081 (\u03bb ()) (\u03bb { ((), _) }) (\u03bb ())\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP : Set\n    \u03a3AP = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082)\n\n    \u03a3AQ : Set\n    \u03a3AQ = \u03a3 A (\u03bb x \u2192 q x \u2261 1\u2082)\n\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    {-\n    module M\n      (f : \u03a3AP \u2192 \u03a3AQ)\n      (f-1 : \u03a3AQ \u2192 \u03a3AP)\n      (f-1f : \u2200 x \u2192 f-1 (f x) \u2261 x)\n      (ff-1 : \u2200 x \u2192 f (f-1 x) \u2261 x)\n       where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' (x , (p , nq)) = let y = f (x , p) in proj\u2081 y , {!proj\u2082 y!} , (proj\u2082 y)\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = {!!}\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = {!!}\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = {!!}\n    -}\n\n    module Work-In-Progress\n      (f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ)\n      (f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ)\n      (f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x)\n      (ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x)\n       where\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f x px nqx) in proj\u2081 (f-1 (proj\u2081 (f x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      f-1f x px nqx = \u2261.cong proj\u2081 (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = proj\u2082 (f-1 x px nqx) in proj\u2081 (f (proj\u2081 (f-1 x px nqx)) (proj\u2081 y) (proj\u2082 y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong proj\u2081 (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = proj\u2081 (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = proj\u2081 (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f-1 x z1 z2)) (\u2261.proof-irrelevance ppx px0) (\u2261.proof-irrelevance qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.cong\u2082 (\u03bb z1 z2 \u2192 proj\u2081 (f x z1 z2)) (\u2261.proof-irrelevance ppx px1) (\u2261.proof-irrelevance qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c001 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = proj\u2081 (proj\u2082 fx) in let qfx = proj\u2082 (proj\u2082 fx) in \u2261.trans (\u03c010 (proj\u2081 fx) (p (proj\u2081 fx)) (q (proj\u2081 fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = proj\u2082 (proj\u2082 (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = proj\u2082 (proj\u2082 (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 proj\u2081 (f x)) , (\u03bb x \u2192 proj\u2082 (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n-- {-\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8F\u2194\ud835\udfd9 = inverses _ (const (\u03bb())) (ext\ud835\udfd8 (\u03bb ())) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n{-\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8b8bca8c9d3a602e4d79805a514dc3d6e48720ee","subject":"proof of final confluence modulo postulate close #39","message":"proof of final confluence modulo postulate close #39\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import focus-formation\nopen import ground-decidable\nopen import matched-ground-invariant\nopen import finality\n\nopen import lemmas-ground\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\nopen import correspondence\nopen import expandability\nopen import expansion-unicity\nopen import type-assignment-unicity\n-- open import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import lemmas-progress-checks\nopen import progress-checks\nopen import progress\n-- open import preservation\n\nopen import complete-preservation\nopen import complete-progress\n-- open import complete-expansion\n\n-- open import instantiation\n-- open import commutativity\n-- open import confluence\nopen import final-confluence\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import focus-formation\nopen import ground-decidable\nopen import matched-ground-invariant\nopen import finality\n\nopen import lemmas-ground\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\nopen import correspondence\nopen import expandability\nopen import expansion-unicity\nopen import type-assignment-unicity\n-- open import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import lemmas-progress-checks\nopen import progress-checks\nopen import progress\n-- open import preservation\n\nopen import complete-preservation\nopen import complete-progress\n-- open import complete-expansion\n\n-- open import instantiation\n-- open import commutativity\n-- open import confluence\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25243c5b239f9daaca537e8e584ade88e7ce1f39","subject":"Stream.NP: hide unfinished code","message":"Stream.NP: hide unfinished code\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Stream\/NP.agda","new_file":"lib\/Data\/Stream\/NP.agda","new_contents":"--TODO {-# OPTIONS --without-K #-}\nmodule Data.Stream.NP where\n\nopen import Type\nimport Level as L\nopen import Data.Nat\nopen import Function\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One  using (\ud835\udfd9)\nopen import Data.Two  using (\ud835\udfda; 0\u2082; 1\u2082; not)\nopen import Function.Equality using (_\u27f6_)\nopen import Data.Product using (\u03a3; _,_; _\u00d7_; uncurry; \u2203; proj\u2081; proj\u2082)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2262_)\nopen import Relation.Nullary\nopen import Relation.Binary.NP\n\nnot\u2262id : \u2200 b \u2192 not b \u2262 b\nnot\u2262id 0\u2082 ()\nnot\u2262id 1\u2082 ()\n\n-- TODO use: not-involutive\nnot\u2218not\u2261id : \u2200 b \u2192 b \u2261 not (not b)\nnot\u2218not\u2261id 0\u2082 = \u2261.refl\nnot\u2218not\u2261id 1\u2082 = \u2261.refl\n\nmodule M1 where\n  Stream : \u2605 \u2192 \u2605\n  Stream A = \u2115 \u2192 A\n\n  setoid : Setoid L.zero L.zero \u2192 Setoid _ _\n  setoid s = record\n    { Carrier       = Stream A\n    ; _\u2248_           = \u03bb xs ys \u2192 \u2200 n \u2192 S._\u2248_ (xs n) (ys n)\n    ; isEquivalence = record\n      { refl  = \u03bb n \u2192 S.refl\n      ; sym   = \u03bb p n \u2192 S.sym (p n)\n      ; trans = \u03bb p q n \u2192 S.trans (p n) (q n)\n      }\n    }\n    where module S = Setoid s\n          A = S.Carrier\n\n  map : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 Stream A \u2192 Stream B\n  map f g x = f (g x)\n\n  diagonal : \u2200 {A} \u2192 Stream (Stream A) \u2192 Stream A\n  diagonal f n = f n n\n\n  cantor : Stream (Stream \ud835\udfda) \u2192 Stream \ud835\udfda\n  cantor = map not \u2218 diagonal\n\n  _\u2249_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  xs \u2249 ys = \u2203 \u03bb n \u2192 xs n \u2262 ys n\n\n  cantor-lem : \u2200 xss n \u2192 cantor xss \u2249 xss n\n  cantor-lem xss n = n , not\u2262id _\n\n  _\u2248_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  xs \u2248 ys = \u2200 n \u2192 xs n \u2261 ys n\n\n  \u2249-sound : \u2200 {A} {xs ys : Stream A} \u2192 xs \u2249 ys \u2192 \u00ac(xs \u2248 ys)\n  \u2249-sound (n , p) q = p (q n)\n\n  \u2249\u2192\u2262 : \u2200 {A} {xs ys : Stream A} \u2192 xs \u2249 ys \u2192 xs \u2262 ys\n  \u2249\u2192\u2262 (_ , f) \u2261.refl = f \u2261.refl\n\n  _\u2208_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2208 xss = \u2203 \u03bb n \u2192 xs \u2248 xss n\n\n  _\u2209_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2209 xss = \u00ac(xs \u2208 xss)\n\n  cantor-thm : \u2200 xss \u2192 cantor xss \u2209 xss\n  cantor-thm xss (n , pn) = proj\u2082 (cantor-lem xss n) (pn n)\n\n  -- Meaning that their exists a set that is bigger than \u2115.\n  -- A nice thing with this statement is that it only involves: \u2605,\u2192,\u2200,\u2203,\u2115,\u2262(\u00ac(\ud835\udfd8),\u2261)\n  cantor-thm2 : \u2203 \u03bb (S : \u2605) \u2192 \u2200 (f : \u2115 \u2192 S) \u2192 \u2203 \u03bb (e : S) \u2192 \u2200 n \u2192 e \u2262 f n\n  cantor-thm2 = Stream \ud835\udfda , (\u03bb f \u2192 cantor f , \u2249\u2192\u2262 \u2218 cantor-lem f)\n\n  -- Meaning that their exists a set that is bigger than \u2115.\n  -- A nice thing with this statement is that it only involves: \u2605,\u2192,\u2200,\u2203,\u2115,\u2261,\ud835\udfd8\n  cantor-thm3 : \u2203 \u03bb (S : \u2605) \u2192 \u2200 (f : \u2115 \u2192 S) \u2192 \u2203 \u03bb (e : S) \u2192 \u2200 n \u2192 e \u2261 f n \u2192 \ud835\udfd8\n  cantor-thm3 = cantor-thm2\n\nmodule M2 where\n  Stream : \u2605 \u2192 \u2605\n  Stream A = \u2115 \u2192 A\n\n  head : \u2200 {A} \u2192 Stream A \u2192 A\n  head f = f zero\n\n  tail : \u2200 {A} \u2192 Stream A \u2192 Stream A\n  tail f = f \u2218 suc\n\n  map : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 Stream A \u2192 Stream B\n  map f g x = f (g x)\n\n  diagonal : \u2200 {A} \u2192 Stream (Stream A) \u2192 Stream A\n  -- diagonal xs zero    = head (head xs)\n  -- diagonal xs (suc n) = diagonal (tail (map tail xs)) n\n  diagonal f n = f n n\n\n  cantor : Stream (Stream \ud835\udfda) \u2192 Stream \ud835\udfda\n  cantor = map not \u2218 diagonal\n\n  All : \u2200 {A} (P : A \u2192 \u2605) \u2192 Stream A \u2192 \u2605\n  All P xs = \u2200 n \u2192 P (xs n)\n\n  Any : \u2200 {A} (P : A \u2192 \u2605) \u2192 Stream A \u2192 \u2605\n  Any P xs = \u2203 \u03bb n \u2192 P (xs n)\n\n  zipWith : \u2200 {A B C : \u2605} (f : A \u2192 B \u2192 C) \u2192 Stream A \u2192 Stream B \u2192 Stream C\n  -- zipWith f xs ys zero    = f (head xs) (head ys)\n  -- zipWith f xs ys (suc n) = zipWith f (tail xs) (tail ys) n\n  zipWith f xs ys n = f (xs n) (ys n)\n\n  zip : \u2200 {A B : \u2605} \u2192 Stream A \u2192 Stream B \u2192 Stream (A \u00d7 B)\n  zip = zipWith _,_\n\n  ZipAll : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\n  ZipAll _\u223c_ xs ys = All (uncurry _\u223c_) (zip xs ys)\n\n  ZipAny : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\n  ZipAny _\u223c_ xs ys = Any (uncurry _\u223c_) (zip xs ys)\n\n  module ZipAllProps {A : \u2605} (_\u223c_ : A \u2192 A \u2192 \u2605) where\n    refl : Reflexive _\u223c_ \u2192 Reflexive (ZipAll _\u223c_)\n    refl re _ = re\n\n    trans : Transitive _\u223c_ \u2192 Transitive (ZipAll _\u223c_)\n    trans tr p q n = tr (p n) (q n)\n\n    sym : Symmetric _\u223c_ \u2192 Symmetric (ZipAll _\u223c_)\n    sym sy p = sy \u2218 p\n\n  ZipAll-setoid : Setoid L.zero L.zero \u2192 Setoid _ _\n  ZipAll-setoid s = record\n    { Carrier       = Stream A\n    ; _\u2248_           = ZipAll {A} S._\u2248_\n    ; isEquivalence = record\n      { refl  = Z.refl S.refl\n      ; sym   = Z.sym S.sym\n      ; trans = Z.trans S.trans\n      }\n    }\n    where module S = Setoid s\n          A = S.Carrier\n          module Z = ZipAllProps S._\u2248_\n\n  _\u2249_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  _\u2249_ = ZipAny _\u2262_\n\n  cantor-lem : \u2200 xss \u2192 All (\u03bb xs \u2192 cantor xss \u2249 xs) xss\n  cantor-lem xss n = n , (not\u2262id _)\n\n  _\u2248_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  _\u2248_ = ZipAll _\u2261_\n\n  _\u2208_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2208 xss = Any (_\u2248_ xs) xss\n\n  _\u2209_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2209 xss = \u00ac(xs \u2208 xss)\n\n  cantor-thm : \u2200 xss \u2192 cantor xss \u2209 xss\n  cantor-thm xss (n , pn) = proj\u2082 (cantor-lem xss n) (pn n)\n\n{-\n  cantor-thm xss zero    = zero , not\u2262id _\n  cantor-thm xss (suc n) = suc {!!} , {!proj\u2082 hi!}\n     where hi : CantorArg (tail xss) (tail xss n)\n           hi = cantor-thm (tail xss) n\n           hi' : \u2203 \u03bb m \u2192 {!!} ?\n           hi' = hi\n-}\nimport Data.Stream\nopen import Coinduction\nimport Function.Related as R\n\nmodule All {A : \u2605} (P : A \u2192 \u2605) where\n  open Data.Stream using (Stream; _\u2237_)\n\n  data All : Stream A \u2192 \u2605 where\n    _\u2237_ : \u2200 {x xs} (px : P x) \u2192 \u221e (All (\u266d xs)) \u2192 All (x \u2237 xs)\n\n  head : \u2200 {xs} \u2192 All xs \u2192 P (Data.Stream.head xs)\n  head (px \u2237 _) = px\n\n  tail : \u2200 {xs} \u2192 All xs \u2192 All (Data.Stream.tail xs)\n  tail (_ \u2237 pxs) = \u266d pxs\n\nopen All using (All; _\u2237_)\n\nopen Data.Stream public\n\ndata Any {A : \u2605} (P : A \u2192 \u2605) : Stream A \u2192 \u2605 where\n  here  : \u2200 {x xs} (px : P x) \u2192 Any P (x \u2237 xs)\n  there : \u2200 {x xs} \u2192 Any P (\u266d xs) \u2192 Any P (x \u2237 xs)\n\nzip : \u2200 {A B : \u2605} \u2192 Stream A \u2192 Stream B \u2192 Stream (A \u00d7 B)\nzip = zipWith _,_\n\nZipAll : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\nZipAll _\u223c_ xs ys = All (uncurry _\u223c_) (zip xs ys)\n\nZipAny : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\nZipAny _\u223c_ xs ys = Any (uncurry _\u223c_) (zip xs ys)\n\nmodule ZipAllProps {A : \u2605} (_\u223c_ : A \u2192 A \u2192 \u2605) where\n  data ZipAllD {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) : Stream A \u2192 Stream B \u2192 \u2605 where\n    _\u2237_ : \u2200 {x y xs ys} (x\u223cy : x \u223c y) \u2192 \u221e (ZipAllD _\u223c_ (\u266d xs) (\u266d ys)) \u2192 ZipAllD _\u223c_ (x \u2237 xs) (y \u2237 ys)\n  \u2192All-uncurry : \u2200 {xs ys : Stream A} \u2192 ZipAllD _\u223c_ xs ys \u2192 All (uncurry _\u223c_) (zip xs ys)\n  \u2192All-uncurry (x\u223cy \u2237 p) = x\u223cy \u2237 \u266f \u2192All-uncurry (\u266d p)\n\n  \u2190All-uncurry : \u2200 {xs ys : Stream A} \u2192 All (uncurry _\u223c_) (zip xs ys) \u2192 ZipAllD _\u223c_ xs ys\n  -- \u2190All-uncurry (x \u2237 xs) (y \u2237 ys) (px \u2237 pxs) = px \u2237 \u266f \u2190All-uncurry (\u266d xs) (\u266d ys) (\u266d pxs)\n  \u2190All-uncurry (px \u2237 pxs) = px \u2237 \u266f \u2190All-uncurry (\u266d pxs)\n\n  refl : Reflexive _\u223c_ \u2192 Reflexive (ZipAll _\u223c_)\n  refl re {x \u2237 xs} = re \u2237 \u266f refl re\n\n  trans : Transitive _\u223c_ \u2192 Transitive (ZipAll _\u223c_)\n  trans tr (x\u223cy \u2237 p) (y\u223cz \u2237 q) = tr x\u223cy y\u223cz \u2237 \u266f trans tr (\u266d p) (\u266d q)\n\n  sym : Symmetric _\u223c_ \u2192 Symmetric (ZipAll _\u223c_)\n  sym sy (x\u223cy \u2237 p) = sy x\u223cy \u2237 \u266f sym sy (\u266d p)\n\nZipAll-setoid : Setoid L.zero L.zero \u2192 Setoid _ _\nZipAll-setoid s = record\n  { Carrier       = Stream A\n  ; _\u2248_           = ZipAll {A} S._\u2248_\n  ; isEquivalence = record\n    { refl  = Z.refl S.refl\n    ; sym   = Z.sym S.sym\n    ; trans = Z.trans S.trans\n    }\n  }\n  where module S = Setoid s\n        A = S.Carrier\n        module Z = ZipAllProps S._\u2248_\n\nmodule \u2248-Reasoning {A : \u2605} = Setoid-Reasoning (setoid A)\n\nmodule M {A : \u2605} where\n  open Setoid (setoid A) public using (refl; trans; sym)\nopen M public\n\ndiagonal : \u2200 {A : \u2605} \u2192 Stream (Stream A) \u2192 Stream A\ndiagonal ((x \u2237 xs) \u2237 xss) = x \u2237 \u266f diagonal (map tail (\u266d xss))\n\n-- nats = 0 \u2237 map suc nats\n-- tab f = map f nats\n-- diag = map (head \u2218 head) \u2218 iterate (tail \u2218 tail)\n-- diag xs = tab (\u03bb i \u2192 xs ! i ! i)\n\ncantor : Stream (Stream \ud835\udfda) \u2192 Stream \ud835\udfda\ncantor = map not \u2218 diagonal\n\ninfix 4 _\u2208'_\n\ndata _\u2208'_ {A} : Stream A \u2192 Stream (Stream A) \u2192 \u2605 where\n  here  : \u2200 {x y xs}   (x\u2248y  : x \u2248 y)     \u2192 x \u2208' y \u2237 xs\n  there : \u2200 {x y z xs} (x\u2248y  : x \u2248 y) (x\u2208xs : x \u2208' \u266d xs) \u2192 y \u2208' z \u2237 xs\n\n_\u2209'_ : \u2200 {A} (xs : Stream A) (xss : Stream (Stream A)) \u2192 \u2605\nxs \u2209' xss = \u00ac(xs \u2208' xss)\n\n\u2248-head : \u2200 {A} {x y : A} {xs ys} \u2192 x \u2237 xs \u2248 y \u2237 ys \u2192 x \u2261 y\n\u2248-head (_ \u2237 _) = \u2261.refl\n\nnot\u2237 : \u2200 b bs bs' \u2192 \u00ac(not b \u2237 bs \u2248 b \u2237 bs')\nnot\u2237 0\u2082 _ _ ()\nnot\u2237 1\u2082 _ _ ()\n\n_>>=_ : \u2200 {A B : \u2605} \u2192 Stream A \u2192 (A \u2192 Stream B) \u2192 Stream B\ns >>= f = diagonal (map f s)\n\nap : \u2200 {A B : \u2605} \u2192 Stream (A \u2192 B) \u2192 Stream A \u2192 Stream B\nap fs xs = fs >>= \u03bb f \u2192\n           xs >>= \u03bb x \u2192\n           repeat (f x)\n\nzap : \u2200 {A B : \u2605} \u2192 Stream (A \u2192 B) \u2192 Stream A \u2192 Stream B\nzap = zipWith id\n\ninfix 4 _\u224b_\n\n_\u224b_ : \u2200 {A} \u2192 Stream (Stream A) \u2192 Stream (Stream A) \u2192 \u2605\n_\u224b_ = ZipAll _\u2248_\n\n_\u2249_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n-- xs \u2249 ys = \u00ac(xs \u2248 ys)\n_\u2249_ = ZipAny _\u2262_\n\n{-\ncant-thm : \u2200 xss \u2192 All (_\u2249_ (cantor xss)) xss\ncant-thm xss = here (not\u2262id _) \u2237 \u266f pf (cant-thm (tail (map tail xss)))\n  where\n    open R.EquationalReasoning\n\n    infixr 2 _\u2192\u27e8_\u27e9_\n\n    _\u2192\u27e8_\u27e9_ : \u2200 {x y z} (X : Set x) {Y : Set y} {Z : Set z} \u2192\n              (X \u2192 Y) \u2192 (Y \u2192 Z) \u2192 X \u2192 Z\n    (X \u2192\u27e8 X\u2192Y \u27e9 Y\u2192Z) x = Y\u2192Z (X\u2192Y x)\n\n    pf =   All (_\u2249_ (cantor (tail (map tail xss)))) (tail (map tail xss))\n         \u2192\u27e8 _\u2237_ {!!} \u2218 \u266f_ \u27e9\n           All (_\u2249_ (cantor (tail (map tail xss)))) (map tail xss)\n         \u2192\u27e8 {!All.tail!} \u27e9\n           All (_\u2249_ (cantor xss)) (tail xss) \u220e\n-}\n{-\ncant-thm ((x \u2237 xs) \u2237 xs\u2081) = (not\u2237 _ _ _) \u2237 \u266f {!cant-thm (\u266d xs\u2081)!}\n-}\n\n{-\nmap-cong' : \u2200 {A B} (f : Stream A \u2192 Stream B) {xs ys} \u2192\n           xs \u224b ys \u2192 map f xs \u224b map f ys\nmap-cong' f (x\u2248 \u2237 xs\u2248) = {!!} \u2237 \u266f map-cong' f (\u266d xs\u2248)\n-}\n\ntail-cong : \u2200 {A} {xs ys : Stream A} \u2192 xs \u2248 ys \u2192 tail xs \u2248 tail ys\ntail-cong (_ \u2237 xs\u2248) = \u266d xs\u2248\n\nmap-tail-cong' : \u2200 {A} {xs ys : Stream (Stream A)} \u2192\n           xs \u224b ys \u2192 map tail xs \u224b map tail ys\nmap-tail-cong' (x\u2248 \u2237 xs\u2248) = tail-cong x\u2248 \u2237 \u266f map-tail-cong' (\u266d xs\u2248)\n\ndiagonal-cong : \u2200 {A} {xs ys : Stream (Stream A)} \u2192\n                  xs \u224b ys \u2192 diagonal xs \u2248 diagonal ys\ndiagonal-cong ((x \u2237 xs\u2248) \u2237 xss\u2248) = x \u2237 \u266f diagonal-cong (map-tail-cong' (\u266d xss\u2248))\n\nmap-tail-repeat : \u2200 {A} (xs : Stream A) \u2192 map tail (map repeat xs) \u2248 map repeat xs\nmap-tail-repeat (x \u2237 xs) = _ \u2237 \u266f map-tail-repeat (\u266d xs)\n\n-- map not (diag (map tail xss)) = tail (map not (diag xss))\n\nmodule \u224b {A} = Setoid (ZipAll-setoid (setoid A))\n\n{-\ncantor-tail : \u2200 (xss : Stream (Stream \ud835\udfda)) \u2192 cantor (map tail (tail xss)) \u2248 tail (cantor xss)\ncantor-tail ((x \u2237 xs) \u2237 xs\u2081) = map-cong not (diagonal-cong (map-tail-cong' \u224b.refl))\n-}\n\n\u2248-\u2208' : \u2200 {A : \u2605} {xs ys : Stream A} {xss} \u2192 xs \u2248 ys \u2192 xs \u2208' xss \u2192 ys \u2208' xss\n\u2248-\u2208' p (here x\u2248y) = here (trans (sym p) x\u2248y)\n\u2248-\u2208' p (there x\u2248y q) = there (trans x\u2248y p) q\n\n{-\nmodule MM where\n  cantor-thm : \u2200 (xss : Stream (Stream \ud835\udfda)) \u2192 cantor xss \u2209' xss\n  \u2208'-tail : \u2200 {xs ys} {xss : Stream (Stream \ud835\udfda)} \u2192 xs \u2208' xss \u2192 \u00ac(tail xs \u2248 ys)\n               -- ys \u2208' map tail xss\n\n  cantor-thm ((x \u2237 xs) \u2237 xss) (here x\u2248y) = not\u2237 x _ _ x\u2248y\n  cantor-thm ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss)\n    = \u2208'-tail cs\u2208xss (\u266d ys\u2248zs)\n    -- = cantor-thm (map tail (\u266d xss)) (\u2208'-tail cs\u2208xss (\u266d ys\u2248zs))\n  \u2208'-tail (here (x \u2237 xs\u2248)) q = {!!}\n  \u2208'-tail (there (x \u2237 xs\u2248) p) q = {!!}\n-}\n\n\u2208'-tail : \u2200 {A : \u2605} {xs} {xss : Stream (Stream A)} \u2192 xs \u2208' xss \u2192 tail xs \u2208' map tail xss\n\u2208'-tail (here (x \u2237 xs\u2248)) = here (\u266d xs\u2248)\n\u2208'-tail (there (x \u2237 xs\u2248) p) = there (\u266d xs\u2248) (\u2208'-tail p)\n\n--  there : \u2200 {x y z xs} (x\u2248y  : x \u2248 y) (x\u2208xs : x \u2208' \u266d xs) \u2192 y \u2208' z \u2237 xs\n\n{-\ncantor-thm : \u2200 (xss : Stream (Stream \ud835\udfda)) \u2192 cantor xss \u2209' xss\ncantor-thm ((x \u2237 xs) \u2237 xss) (here x\u2248y) = not\u2237 x _ _ x\u2248y\ncantor-thm ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss)\n  = cantor-thm (map tail (\u266d xss)) (\u2248-\u2208' (\u266d ys\u2248zs) (\u2208'-tail cs\u2208xss))\n\nbu : \u2200 {A} {xs : Stream A} {xss : Stream (Stream A)} \u2192 xs \u2208 xss \u2192 \u2115 \u2192 \u2605\nbu here zero = \ud835\udfd9\nbu here (suc n) = \ud835\udfd8\nbu (there p) zero = \ud835\udfd8\nbu (there p) (suc n) = {!S\u2208' p n!}\n-}\n\nS\u2208' : \u2200 {A} {xs : Stream A} {xss : Stream (Stream A)} \u2192 xs \u2208' xss \u2192 \u2115 \u2192 \u2605\nS\u2208' (here _) zero = \ud835\udfd9\nS\u2208' (here _) (suc n) = \ud835\udfd8\nS\u2208' (there _ p) zero = \ud835\udfd8\nS\u2208' (there _ p) (suc n) = S\u2208' p n\n\n{-\ndata E : \u2115 \u2192 \u2115 \u2192 \u2605 where\n  zero : E zero zero\n  suc  : \u2200 {m n} \u2192 E m n \u2192 E (suc m) (suc n)\n-}\n\nS-tail : \u2200 {xs : Stream \ud835\udfda} {xss} (p : xs \u2208' xss) n \u2192 S\u2208' p n \u2192 S\u2208' (\u2208'-tail p) n\nS-tail (here (_ \u2237 _)) zero _ = _\nS-tail (here _) (suc n) ()\nS-tail (there _ _) zero ()\nS-tail (there (x \u2237 xs\u2248) p) (suc n) q = S-tail p n q\n\n{-\nbar : \u2200 {xs ys : Stream \ud835\udfda} {xss} (pp : xs \u2248 ys) (p : xs \u2208' xss) n \u2192 S\u2208' p n \u2192 S\u2208' (\u2248-\u2208' pp p) n\nbar pp p n q = {!!}\n\ncantor-thm' : \u2200 (xss : Stream (Stream \ud835\udfda)) (p : cantor xss \u2208' xss) n \u2192 S\u2208' p n \u2192 \ud835\udfd8\ncantor-thm' ((x \u2237 xs) \u2237 xss) (here x\u2248y) zero _ = not\u2237 x _ _ x\u2248y\ncantor-thm' ((x \u2237 xs) \u2237 xss) (here x\u2248y) (suc _) ()\ncantor-thm' ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss) (suc n) q\n  = cantor-thm' (map tail (\u266d xss)) (\u2248-\u2208' (\u266d ys\u2248zs) (\u2208'-tail cs\u2208xss)) n (bar (\u266d ys\u2248zs) _ n (S-tail cs\u2208xss n q))\ncantor-thm' ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss) zero ()\n-}\n\n{-\nlem : \u2200 {A B : \u2605} (fs : Stream (A \u2192 B)) xs \u2192 ap fs xs \u2248 zap fs xs\nlem (f \u2237 fs) (x \u2237 xs) -- = trans (f x \u2237 \u266f pf) (f x \u2237 \u266f lem (\u266d fs) (\u266d xs))\n        = ap (f \u2237 fs) (x \u2237 xs)\n       \u2248\u27e8 refl \u27e9\n          ((f \u2237 fs) >>= \u03bb f' \u2192\n          (x \u2237 xs) >>= \u03bb x' \u2192\n          repeat (f' x'))\n       \u2248\u27e8 refl \u27e9\n          ((f \u2237 fs) >>= \u03bb f' \u2192\n          diagonal (map (repeat \u2218 f') (x \u2237 xs)))\n       \u2248\u27e8 f x \u2237 \u266f diagonal-cong (map-tail-cong' {!pf!}) \u27e9\n          ((f \u2237 fs) >>= ff)\n       \u2248\u27e8 refl \u27e9\n          diagonal (map ff (f \u2237 fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          diagonal (ff f \u2237 \u266f map ff (\u266d fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          diagonal (ff' \u2237 \u266f map ff (\u266d fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          diagonal (ff'' \u2237 \u266f map ff (\u266d fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          f x \u2237 \u266f diagonal (map tail (map ff (\u266d fs)))\n       \u2248\u27e8 f x \u2237 \u266f pf \u27e9\n          f x \u2237 \u266f ap (\u266d fs) (\u266d xs)\n       \u2248\u27e8 f x \u2237 \u266f lem (\u266d fs) (\u266d xs) \u27e9\n          f x \u2237 \u266f zap (\u266d fs) (\u266d xs)\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          zap (f \u2237 fs) (x \u2237 xs)\n       \u220e\n  where\n    open \u2248-Reasoning\n    ff = \u03bb f' \u2192 diagonal (repeat (f' x) \u2237 \u266f map (repeat \u2218 f') (\u266d xs))\n    ff' = diagonal ((f x \u2237 \u266f repeat (f x)) \u2237 \u266f map (repeat \u2218 f) (\u266d xs))\n    ff'' = f x \u2237 \u266f diagonal (map tail (map (repeat \u2218 f) (\u266d xs)))\n    pf3 = map tail (map ff (\u266d fs))\n           \u2248\u27e8 {!!} \u27e9\n          map (tail \u2218 ff) (\u266d fs)\n           \u2248\u27e8 {!!} \u27e9\n          map ff (\u266d fs)\n    pf = diagonal (map tail (map ff (\u266d fs)))\n           \u2248\u27e8 {!pf3!} \u27e9\n          diagonal (map ff (\u266d fs))\n           \u2248\u27e8 {!!} \u27e9\n          ap (\u266d fs) (\u266d xs)\n         \u220e\n    pf2 = \u03bb f' \u2192 diagonal (map (repeat \u2218 f') (x \u2237 xs))\n       \u2248\u27e8 f' x \u2237 \u266f refl \u27e9\n          ff f'\n        \u220e\n-}\n","old_contents":"--TODO {-# OPTIONS --without-K #-}\nmodule Data.Stream.NP where\n\nopen import Type\nimport Level as L\nopen import Data.Nat\nopen import Function\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One  using (\ud835\udfd9)\nopen import Data.Two  using (\ud835\udfda; 0\u2082; 1\u2082; not)\nopen import Function.Equality using (_\u27f6_)\nopen import Data.Product using (\u03a3; _,_; _\u00d7_; uncurry; \u2203; proj\u2081; proj\u2082)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2262_)\nopen import Relation.Nullary\nopen import Relation.Binary.NP\n\nnot\u2262id : \u2200 b \u2192 not b \u2262 b\nnot\u2262id 0\u2082 ()\nnot\u2262id 1\u2082 ()\n\n-- TODO use: not-involutive\nnot\u2218not\u2261id : \u2200 b \u2192 b \u2261 not (not b)\nnot\u2218not\u2261id 0\u2082 = \u2261.refl\nnot\u2218not\u2261id 1\u2082 = \u2261.refl\n\nmodule M1 where\n  Stream : \u2605 \u2192 \u2605\n  Stream A = \u2115 \u2192 A\n\n  setoid : Setoid L.zero L.zero \u2192 Setoid _ _\n  setoid s = record\n    { Carrier       = Stream A\n    ; _\u2248_           = \u03bb xs ys \u2192 \u2200 n \u2192 S._\u2248_ (xs n) (ys n)\n    ; isEquivalence = record\n      { refl  = \u03bb n \u2192 S.refl\n      ; sym   = \u03bb p n \u2192 S.sym (p n)\n      ; trans = \u03bb p q n \u2192 S.trans (p n) (q n)\n      }\n    }\n    where module S = Setoid s\n          A = S.Carrier\n\n  map : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 Stream A \u2192 Stream B\n  map f g x = f (g x)\n\n  diagonal : \u2200 {A} \u2192 Stream (Stream A) \u2192 Stream A\n  diagonal f n = f n n\n\n  cantor : Stream (Stream \ud835\udfda) \u2192 Stream \ud835\udfda\n  cantor = map not \u2218 diagonal\n\n  _\u2249_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  xs \u2249 ys = \u2203 \u03bb n \u2192 xs n \u2262 ys n\n\n  cantor-lem : \u2200 xss n \u2192 cantor xss \u2249 xss n\n  cantor-lem xss n = n , not\u2262id _\n\n  _\u2248_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  xs \u2248 ys = \u2200 n \u2192 xs n \u2261 ys n\n\n  \u2249-sound : \u2200 {A} {xs ys : Stream A} \u2192 xs \u2249 ys \u2192 \u00ac(xs \u2248 ys)\n  \u2249-sound (n , p) q = p (q n)\n\n  \u2249\u2192\u2262 : \u2200 {A} {xs ys : Stream A} \u2192 xs \u2249 ys \u2192 xs \u2262 ys\n  \u2249\u2192\u2262 (_ , f) \u2261.refl = f \u2261.refl\n\n  _\u2208_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2208 xss = \u2203 \u03bb n \u2192 xs \u2248 xss n\n\n  _\u2209_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2209 xss = \u00ac(xs \u2208 xss)\n\n  cantor-thm : \u2200 xss \u2192 cantor xss \u2209 xss\n  cantor-thm xss (n , pn) = proj\u2082 (cantor-lem xss n) (pn n)\n\n  -- Meaning that their exists a set that is bigger than \u2115.\n  -- A nice thing with this statement is that it only involves: \u2605,\u2192,\u2200,\u2203,\u2115,\u2262(\u00ac(\ud835\udfd8),\u2261)\n  cantor-thm2 : \u2203 \u03bb (S : \u2605) \u2192 \u2200 (f : \u2115 \u2192 S) \u2192 \u2203 \u03bb (e : S) \u2192 \u2200 n \u2192 e \u2262 f n\n  cantor-thm2 = Stream \ud835\udfda , (\u03bb f \u2192 cantor f , \u2249\u2192\u2262 \u2218 cantor-lem f)\n\n  -- Meaning that their exists a set that is bigger than \u2115.\n  -- A nice thing with this statement is that it only involves: \u2605,\u2192,\u2200,\u2203,\u2115,\u2261,\ud835\udfd8\n  cantor-thm3 : \u2203 \u03bb (S : \u2605) \u2192 \u2200 (f : \u2115 \u2192 S) \u2192 \u2203 \u03bb (e : S) \u2192 \u2200 n \u2192 e \u2261 f n \u2192 \ud835\udfd8\n  cantor-thm3 = cantor-thm2\n\nmodule M2 where\n  Stream : \u2605 \u2192 \u2605\n  Stream A = \u2115 \u2192 A\n\n  head : \u2200 {A} \u2192 Stream A \u2192 A\n  head f = f zero\n\n  tail : \u2200 {A} \u2192 Stream A \u2192 Stream A\n  tail f = f \u2218 suc\n\n  map : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 Stream A \u2192 Stream B\n  map f g x = f (g x)\n\n  diagonal : \u2200 {A} \u2192 Stream (Stream A) \u2192 Stream A\n  -- diagonal xs zero    = head (head xs)\n  -- diagonal xs (suc n) = diagonal (tail (map tail xs)) n\n  diagonal f n = f n n\n\n  cantor : Stream (Stream \ud835\udfda) \u2192 Stream \ud835\udfda\n  cantor = map not \u2218 diagonal\n\n  All : \u2200 {A} (P : A \u2192 \u2605) \u2192 Stream A \u2192 \u2605\n  All P xs = \u2200 n \u2192 P (xs n)\n\n  Any : \u2200 {A} (P : A \u2192 \u2605) \u2192 Stream A \u2192 \u2605\n  Any P xs = \u2203 \u03bb n \u2192 P (xs n)\n\n  zipWith : \u2200 {A B C : \u2605} (f : A \u2192 B \u2192 C) \u2192 Stream A \u2192 Stream B \u2192 Stream C\n  -- zipWith f xs ys zero    = f (head xs) (head ys)\n  -- zipWith f xs ys (suc n) = zipWith f (tail xs) (tail ys) n\n  zipWith f xs ys n = f (xs n) (ys n)\n\n  zip : \u2200 {A B : \u2605} \u2192 Stream A \u2192 Stream B \u2192 Stream (A \u00d7 B)\n  zip = zipWith _,_\n\n  ZipAll : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\n  ZipAll _\u223c_ xs ys = All (uncurry _\u223c_) (zip xs ys)\n\n  ZipAny : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\n  ZipAny _\u223c_ xs ys = Any (uncurry _\u223c_) (zip xs ys)\n\n  module ZipAllProps {A : \u2605} (_\u223c_ : A \u2192 A \u2192 \u2605) where\n    refl : Reflexive _\u223c_ \u2192 Reflexive (ZipAll _\u223c_)\n    refl re _ = re\n\n    trans : Transitive _\u223c_ \u2192 Transitive (ZipAll _\u223c_)\n    trans tr p q n = tr (p n) (q n)\n\n    sym : Symmetric _\u223c_ \u2192 Symmetric (ZipAll _\u223c_)\n    sym sy p = sy \u2218 p\n\n  ZipAll-setoid : Setoid L.zero L.zero \u2192 Setoid _ _\n  ZipAll-setoid s = record\n    { Carrier       = Stream A\n    ; _\u2248_           = ZipAll {A} S._\u2248_\n    ; isEquivalence = record\n      { refl  = Z.refl S.refl\n      ; sym   = Z.sym S.sym\n      ; trans = Z.trans S.trans\n      }\n    }\n    where module S = Setoid s\n          A = S.Carrier\n          module Z = ZipAllProps S._\u2248_\n\n  _\u2249_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  _\u2249_ = ZipAny _\u2262_\n\n  cantor-lem : \u2200 xss \u2192 All (\u03bb xs \u2192 cantor xss \u2249 xs) xss\n  cantor-lem xss n = n , (not\u2262id _)\n\n  _\u2248_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n  _\u2248_ = ZipAll _\u2261_\n\n  _\u2208_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2208 xss = Any (_\u2248_ xs) xss\n\n  _\u2209_ : \u2200 {A} \u2192 Stream A \u2192 Stream (Stream A) \u2192 \u2605\n  xs \u2209 xss = \u00ac(xs \u2208 xss)\n\n  cantor-thm : \u2200 xss \u2192 cantor xss \u2209 xss\n  cantor-thm xss (n , pn) = proj\u2082 (cantor-lem xss n) (pn n)\n\n{-\n  cantor-thm xss zero    = zero , not\u2262id _\n  cantor-thm xss (suc n) = suc {!!} , {!proj\u2082 hi!}\n     where hi : CantorArg (tail xss) (tail xss n)\n           hi = cantor-thm (tail xss) n\n           hi' : \u2203 \u03bb m \u2192 {!!} ?\n           hi' = hi\n-}\nimport Data.Stream\nopen import Coinduction\nimport Function.Related as R\n\nmodule All {A : \u2605} (P : A \u2192 \u2605) where\n  open Data.Stream using (Stream; _\u2237_)\n\n  data All : Stream A \u2192 \u2605 where\n    _\u2237_ : \u2200 {x xs} (px : P x) \u2192 \u221e (All (\u266d xs)) \u2192 All (x \u2237 xs)\n\n  head : \u2200 {xs} \u2192 All xs \u2192 P (Data.Stream.head xs)\n  head (px \u2237 _) = px\n\n  tail : \u2200 {xs} \u2192 All xs \u2192 All (Data.Stream.tail xs)\n  tail (_ \u2237 pxs) = \u266d pxs\n\nopen All using (All; _\u2237_)\n\nopen Data.Stream public\n\ndata Any {A : \u2605} (P : A \u2192 \u2605) : Stream A \u2192 \u2605 where\n  here  : \u2200 {x xs} (px : P x) \u2192 Any P (x \u2237 xs)\n  there : \u2200 {x xs} \u2192 Any P (\u266d xs) \u2192 Any P (x \u2237 xs)\n\nzip : \u2200 {A B : \u2605} \u2192 Stream A \u2192 Stream B \u2192 Stream (A \u00d7 B)\nzip = zipWith _,_\n\nZipAll : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\nZipAll _\u223c_ xs ys = All (uncurry _\u223c_) (zip xs ys)\n\nZipAny : \u2200 {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) \u2192 Stream A \u2192 Stream B \u2192 \u2605\nZipAny _\u223c_ xs ys = Any (uncurry _\u223c_) (zip xs ys)\n\nmodule ZipAllProps {A : \u2605} (_\u223c_ : A \u2192 A \u2192 \u2605) where\n  data ZipAllD {A B : \u2605} (_\u223c_ : A \u2192 B \u2192 \u2605) : Stream A \u2192 Stream B \u2192 \u2605 where\n    _\u2237_ : \u2200 {x y xs ys} (x\u223cy : x \u223c y) \u2192 \u221e (ZipAllD _\u223c_ (\u266d xs) (\u266d ys)) \u2192 ZipAllD _\u223c_ (x \u2237 xs) (y \u2237 ys)\n  \u2192All-uncurry : \u2200 {xs ys : Stream A} \u2192 ZipAllD _\u223c_ xs ys \u2192 All (uncurry _\u223c_) (zip xs ys)\n  \u2192All-uncurry (x\u223cy \u2237 p) = x\u223cy \u2237 \u266f \u2192All-uncurry (\u266d p)\n\n  \u2190All-uncurry : \u2200 {xs ys : Stream A} \u2192 All (uncurry _\u223c_) (zip xs ys) \u2192 ZipAllD _\u223c_ xs ys\n  -- \u2190All-uncurry (x \u2237 xs) (y \u2237 ys) (px \u2237 pxs) = px \u2237 \u266f \u2190All-uncurry (\u266d xs) (\u266d ys) (\u266d pxs)\n  \u2190All-uncurry (px \u2237 pxs) = px \u2237 \u266f \u2190All-uncurry (\u266d pxs)\n\n  refl : Reflexive _\u223c_ \u2192 Reflexive (ZipAll _\u223c_)\n  refl re {x \u2237 xs} = re \u2237 \u266f refl re\n\n  trans : Transitive _\u223c_ \u2192 Transitive (ZipAll _\u223c_)\n  trans tr (x\u223cy \u2237 p) (y\u223cz \u2237 q) = tr x\u223cy y\u223cz \u2237 \u266f trans tr (\u266d p) (\u266d q)\n\n  sym : Symmetric _\u223c_ \u2192 Symmetric (ZipAll _\u223c_)\n  sym sy (x\u223cy \u2237 p) = sy x\u223cy \u2237 \u266f sym sy (\u266d p)\n\nZipAll-setoid : Setoid L.zero L.zero \u2192 Setoid _ _\nZipAll-setoid s = record\n  { Carrier       = Stream A\n  ; _\u2248_           = ZipAll {A} S._\u2248_\n  ; isEquivalence = record\n    { refl  = Z.refl S.refl\n    ; sym   = Z.sym S.sym\n    ; trans = Z.trans S.trans\n    }\n  }\n  where module S = Setoid s\n        A = S.Carrier\n        module Z = ZipAllProps S._\u2248_\n\nmodule \u2248-Reasoning {A : \u2605} = Setoid-Reasoning (setoid A)\n\nmodule M {A : \u2605} where\n  open Setoid (setoid A) public using (refl; trans; sym)\nopen M public\n\ndiagonal : \u2200 {A : \u2605} \u2192 Stream (Stream A) \u2192 Stream A\ndiagonal ((x \u2237 xs) \u2237 xss) = x \u2237 \u266f diagonal (map tail (\u266d xss))\n\n-- nats = 0 \u2237 map suc nats\n-- tab f = map f nats\n-- diag = map (head \u2218 head) \u2218 iterate (tail \u2218 tail)\n-- diag xs = tab (\u03bb i \u2192 xs ! i ! i)\n\ncantor : Stream (Stream \ud835\udfda) \u2192 Stream \ud835\udfda\ncantor = map not \u2218 diagonal\n\ninfix 4 _\u2208'_\n\ndata _\u2208'_ {A} : Stream A \u2192 Stream (Stream A) \u2192 \u2605 where\n  here  : \u2200 {x y xs}   (x\u2248y  : x \u2248 y)     \u2192 x \u2208' y \u2237 xs\n  there : \u2200 {x y z xs} (x\u2248y  : x \u2248 y) (x\u2208xs : x \u2208' \u266d xs) \u2192 y \u2208' z \u2237 xs\n\n_\u2209'_ : \u2200 {A} (xs : Stream A) (xss : Stream (Stream A)) \u2192 \u2605\nxs \u2209' xss = \u00ac(xs \u2208' xss)\n\n\u2248-head : \u2200 {A} {x y : A} {xs ys} \u2192 x \u2237 xs \u2248 y \u2237 ys \u2192 x \u2261 y\n\u2248-head (_ \u2237 _) = \u2261.refl\n\nnot\u2237 : \u2200 b bs bs' \u2192 \u00ac(not b \u2237 bs \u2248 b \u2237 bs')\nnot\u2237 0\u2082 _ _ ()\nnot\u2237 1\u2082 _ _ ()\n\n_>>=_ : \u2200 {A B : \u2605} \u2192 Stream A \u2192 (A \u2192 Stream B) \u2192 Stream B\ns >>= f = diagonal (map f s)\n\nap : \u2200 {A B : \u2605} \u2192 Stream (A \u2192 B) \u2192 Stream A \u2192 Stream B\nap fs xs = fs >>= \u03bb f \u2192\n           xs >>= \u03bb x \u2192\n           repeat (f x)\n\nzap : \u2200 {A B : \u2605} \u2192 Stream (A \u2192 B) \u2192 Stream A \u2192 Stream B\nzap = zipWith id\n\ninfix 4 _\u224b_\n\n_\u224b_ : \u2200 {A} \u2192 Stream (Stream A) \u2192 Stream (Stream A) \u2192 \u2605\n_\u224b_ = ZipAll _\u2248_\n\n_\u2249_ : \u2200 {A} \u2192 Stream A \u2192 Stream A \u2192 \u2605\n-- xs \u2249 ys = \u00ac(xs \u2248 ys)\n_\u2249_ = ZipAny _\u2262_\n\n{-\ncant-thm : \u2200 xss \u2192 All (_\u2249_ (cantor xss)) xss\ncant-thm xss = here (not\u2262id _) \u2237 \u266f pf (cant-thm (tail (map tail xss)))\n  where\n    open R.EquationalReasoning\n\n    infixr 2 _\u2192\u27e8_\u27e9_\n\n    _\u2192\u27e8_\u27e9_ : \u2200 {x y z} (X : Set x) {Y : Set y} {Z : Set z} \u2192\n              (X \u2192 Y) \u2192 (Y \u2192 Z) \u2192 X \u2192 Z\n    (X \u2192\u27e8 X\u2192Y \u27e9 Y\u2192Z) x = Y\u2192Z (X\u2192Y x)\n\n    pf =   All (_\u2249_ (cantor (tail (map tail xss)))) (tail (map tail xss))\n         \u2192\u27e8 _\u2237_ {!!} \u2218 \u266f_ \u27e9\n           All (_\u2249_ (cantor (tail (map tail xss)))) (map tail xss)\n         \u2192\u27e8 {!All.tail!} \u27e9\n           All (_\u2249_ (cantor xss)) (tail xss) \u220e\n-}\n{-\ncant-thm ((x \u2237 xs) \u2237 xs\u2081) = (not\u2237 _ _ _) \u2237 \u266f {!cant-thm (\u266d xs\u2081)!}\n-}\n\n{-\nmap-cong' : \u2200 {A B} (f : Stream A \u2192 Stream B) {xs ys} \u2192\n           xs \u224b ys \u2192 map f xs \u224b map f ys\nmap-cong' f (x\u2248 \u2237 xs\u2248) = {!!} \u2237 \u266f map-cong' f (\u266d xs\u2248)\n-}\n\ntail-cong : \u2200 {A} {xs ys : Stream A} \u2192 xs \u2248 ys \u2192 tail xs \u2248 tail ys\ntail-cong (_ \u2237 xs\u2248) = \u266d xs\u2248\n\nmap-tail-cong' : \u2200 {A} {xs ys : Stream (Stream A)} \u2192\n           xs \u224b ys \u2192 map tail xs \u224b map tail ys\nmap-tail-cong' (x\u2248 \u2237 xs\u2248) = tail-cong x\u2248 \u2237 \u266f map-tail-cong' (\u266d xs\u2248)\n\ndiagonal-cong : \u2200 {A} {xs ys : Stream (Stream A)} \u2192\n                  xs \u224b ys \u2192 diagonal xs \u2248 diagonal ys\ndiagonal-cong ((x \u2237 xs\u2248) \u2237 xss\u2248) = x \u2237 \u266f diagonal-cong (map-tail-cong' (\u266d xss\u2248))\n\nmap-tail-repeat : \u2200 {A} (xs : Stream A) \u2192 map tail (map repeat xs) \u2248 map repeat xs\nmap-tail-repeat (x \u2237 xs) = _ \u2237 \u266f map-tail-repeat (\u266d xs)\n\n-- map not (diag (map tail xss)) = tail (map not (diag xss))\n\nmodule \u224b {A} = Setoid (ZipAll-setoid (setoid A))\n\n{-\ncantor-tail : \u2200 (xss : Stream (Stream \ud835\udfda)) \u2192 cantor (map tail (tail xss)) \u2248 tail (cantor xss)\ncantor-tail ((x \u2237 xs) \u2237 xs\u2081) = map-cong not (diagonal-cong (map-tail-cong' \u224b.refl))\n-}\n\n\u2248-\u2208' : \u2200 {A : \u2605} {xs ys : Stream A} {xss} \u2192 xs \u2248 ys \u2192 xs \u2208' xss \u2192 ys \u2208' xss\n\u2248-\u2208' p (here x\u2248y) = here (trans (sym p) x\u2248y)\n\u2248-\u2208' p (there x\u2248y q) = there (trans x\u2248y p) q\n\nmodule MM where\n  cantor-thm : \u2200 (xss : Stream (Stream \ud835\udfda)) \u2192 cantor xss \u2209' xss\n  \u2208'-tail : \u2200 {xs ys} {xss : Stream (Stream \ud835\udfda)} \u2192 xs \u2208' xss \u2192 \u00ac(tail xs \u2248 ys)\n               -- ys \u2208' map tail xss\n\n  cantor-thm ((x \u2237 xs) \u2237 xss) (here x\u2248y) = not\u2237 x _ _ x\u2248y\n  cantor-thm ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss)\n    = \u2208'-tail cs\u2208xss (\u266d ys\u2248zs)\n    -- = cantor-thm (map tail (\u266d xss)) (\u2208'-tail cs\u2208xss (\u266d ys\u2248zs))\n\n  \u2208'-tail (here (x \u2237 xs\u2248)) q = {!!}\n  \u2208'-tail (there (x \u2237 xs\u2248) p) q = {!!}\n\n\u2208'-tail : \u2200 {A : \u2605} {xs} {xss : Stream (Stream A)} \u2192 xs \u2208' xss \u2192 tail xs \u2208' map tail xss\n\u2208'-tail (here (x \u2237 xs\u2248)) = here (\u266d xs\u2248)\n\u2208'-tail (there (x \u2237 xs\u2248) p) = there (\u266d xs\u2248) (\u2208'-tail p)\n\n--  there : \u2200 {x y z xs} (x\u2248y  : x \u2248 y) (x\u2208xs : x \u2208' \u266d xs) \u2192 y \u2208' z \u2237 xs\n\n{-\ncantor-thm : \u2200 (xss : Stream (Stream \ud835\udfda)) \u2192 cantor xss \u2209' xss\ncantor-thm ((x \u2237 xs) \u2237 xss) (here x\u2248y) = not\u2237 x _ _ x\u2248y\ncantor-thm ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss)\n  = cantor-thm (map tail (\u266d xss)) (\u2248-\u2208' (\u266d ys\u2248zs) (\u2208'-tail cs\u2208xss))\n\nbu : \u2200 {A} {xs : Stream A} {xss : Stream (Stream A)} \u2192 xs \u2208 xss \u2192 \u2115 \u2192 \u2605\nbu here zero = \ud835\udfd9\nbu here (suc n) = \ud835\udfd8\nbu (there p) zero = \ud835\udfd8\nbu (there p) (suc n) = {!S\u2208' p n!}\n-}\n\nS\u2208' : \u2200 {A} {xs : Stream A} {xss : Stream (Stream A)} \u2192 xs \u2208' xss \u2192 \u2115 \u2192 \u2605\nS\u2208' (here _) zero = \ud835\udfd9\nS\u2208' (here _) (suc n) = \ud835\udfd8\nS\u2208' (there _ p) zero = \ud835\udfd8\nS\u2208' (there _ p) (suc n) = S\u2208' p n\n\n{-\ndata E : \u2115 \u2192 \u2115 \u2192 \u2605 where\n  zero : E zero zero\n  suc  : \u2200 {m n} \u2192 E m n \u2192 E (suc m) (suc n)\n-}\n\nS-tail : \u2200 {xs : Stream \ud835\udfda} {xss} (p : xs \u2208' xss) n \u2192 S\u2208' p n \u2192 S\u2208' (\u2208'-tail p) n\nS-tail (here (_ \u2237 _)) zero _ = _\nS-tail (here _) (suc n) ()\nS-tail (there _ _) zero ()\nS-tail (there (x \u2237 xs\u2248) p) (suc n) q = S-tail p n q\n\n{-\nbar : \u2200 {xs ys : Stream \ud835\udfda} {xss} (pp : xs \u2248 ys) (p : xs \u2208' xss) n \u2192 S\u2208' p n \u2192 S\u2208' (\u2248-\u2208' pp p) n\nbar pp p n q = {!!}\n\ncantor-thm' : \u2200 (xss : Stream (Stream \ud835\udfda)) (p : cantor xss \u2208' xss) n \u2192 S\u2208' p n \u2192 \ud835\udfd8\ncantor-thm' ((x \u2237 xs) \u2237 xss) (here x\u2248y) zero _ = not\u2237 x _ _ x\u2248y\ncantor-thm' ((x \u2237 xs) \u2237 xss) (here x\u2248y) (suc _) ()\ncantor-thm' ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss) (suc n) q\n  = cantor-thm' (map tail (\u266d xss)) (\u2248-\u2208' (\u266d ys\u2248zs) (\u2208'-tail cs\u2208xss)) n (bar (\u266d ys\u2248zs) _ n (S-tail cs\u2208xss n q))\ncantor-thm' ((x \u2237 xs) \u2237 xss) (there {._ \u2237 ys} (._ \u2237 ys\u2248zs) cs\u2208xss) zero ()\n-}\n\n{-\nlem : \u2200 {A B : \u2605} (fs : Stream (A \u2192 B)) xs \u2192 ap fs xs \u2248 zap fs xs\nlem (f \u2237 fs) (x \u2237 xs) -- = trans (f x \u2237 \u266f pf) (f x \u2237 \u266f lem (\u266d fs) (\u266d xs))\n        = ap (f \u2237 fs) (x \u2237 xs)\n       \u2248\u27e8 refl \u27e9\n          ((f \u2237 fs) >>= \u03bb f' \u2192\n          (x \u2237 xs) >>= \u03bb x' \u2192\n          repeat (f' x'))\n       \u2248\u27e8 refl \u27e9\n          ((f \u2237 fs) >>= \u03bb f' \u2192\n          diagonal (map (repeat \u2218 f') (x \u2237 xs)))\n       \u2248\u27e8 f x \u2237 \u266f diagonal-cong (map-tail-cong' {!pf!}) \u27e9\n          ((f \u2237 fs) >>= ff)\n       \u2248\u27e8 refl \u27e9\n          diagonal (map ff (f \u2237 fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          diagonal (ff f \u2237 \u266f map ff (\u266d fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          diagonal (ff' \u2237 \u266f map ff (\u266d fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          diagonal (ff'' \u2237 \u266f map ff (\u266d fs))\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          f x \u2237 \u266f diagonal (map tail (map ff (\u266d fs)))\n       \u2248\u27e8 f x \u2237 \u266f pf \u27e9\n          f x \u2237 \u266f ap (\u266d fs) (\u266d xs)\n       \u2248\u27e8 f x \u2237 \u266f lem (\u266d fs) (\u266d xs) \u27e9\n          f x \u2237 \u266f zap (\u266d fs) (\u266d xs)\n       \u2248\u27e8 f x \u2237 \u266f refl \u27e9\n          zap (f \u2237 fs) (x \u2237 xs)\n       \u220e\n  where\n    open \u2248-Reasoning\n    ff = \u03bb f' \u2192 diagonal (repeat (f' x) \u2237 \u266f map (repeat \u2218 f') (\u266d xs))\n    ff' = diagonal ((f x \u2237 \u266f repeat (f x)) \u2237 \u266f map (repeat \u2218 f) (\u266d xs))\n    ff'' = f x \u2237 \u266f diagonal (map tail (map (repeat \u2218 f) (\u266d xs)))\n    pf3 = map tail (map ff (\u266d fs))\n           \u2248\u27e8 {!!} \u27e9\n          map (tail \u2218 ff) (\u266d fs)\n           \u2248\u27e8 {!!} \u27e9\n          map ff (\u266d fs)\n    pf = diagonal (map tail (map ff (\u266d fs)))\n           \u2248\u27e8 {!pf3!} \u27e9\n          diagonal (map ff (\u266d fs))\n           \u2248\u27e8 {!!} \u27e9\n          ap (\u266d fs) (\u266d xs)\n         \u220e\n    pf2 = \u03bb f' \u2192 diagonal (map (repeat \u2218 f') (x \u2237 xs))\n       \u2248\u27e8 f' x \u2237 \u266f refl \u27e9\n          ff f'\n        \u220e\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f27bad04b4c485c526496aad466ae08f20368fd4","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 53a244faf7f8e27a0c931c9733733a5b\n\ndarcs-hash:20110412185226-3bd4e-e45ce69ae53cd21e5e8d2ef0440afc26bfee4acd.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/Induction\/Acc\/WellFoundedInductionATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/Induction\/Acc\/WellFoundedInductionATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n------------------------------------------------------------------------------\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\n\nopen import FOTC.Induction.WellFounded\n\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.PropertiesATP\n\n-- Parametrized modules\n\nopen module InvImg =\n  FOTC.Induction.WellFounded.InverseImage {N} {N} {LT} fnMCR-N\n\n------------------------------------------------------------------------------\n-- The relation MCR is well-founded (using the inverse image combinator).\nwf-MCR : WellFounded MCR\nwf-MCR Nn = wellFounded wf-LT Nn\n\n-- Well-founded induction on the relation MCR.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 MCR m n \u2192 P m) \u2192 P n) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P = WellFoundedInduction wf-MCR\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n------------------------------------------------------------------------------\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionATP\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\n\nopen import FOTC.Induction.WellFounded\n\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.PropertiesATP\n\n-- Parametrized modules\n\nopen module InvImg =\n  FOTC.Induction.WellFounded.InverseImage {N} {N} {LT} fnMCR-N\n\n------------------------------------------------------------------------------\n-- The relation MCR is well-founded.\nwf-MCR : WellFounded MCR\nwf-MCR Nn = wellFounded wf-LT Nn\n\n-- Well-founded induction on the relation MCR.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 MCR m n \u2192 P m) \u2192 P n) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P = WellFoundedInduction wf-MCR\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"360d3d7b24155bb187d2e20671c85eba6715750d","subject":"Minor change.","message":"Minor change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-positivity-check #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\nmodule FOTC where\n\nmodule Aczel-CA where\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    \u2250-refl  : \u2200 {x} \u2192 x \u2250 x\n    \u2250-sym   : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    \u2250-trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    \u2250-cong  : \u2200 {x x' y y'} \u2192 x \u2250 y \u2192 x' \u2250 y' \u2192 x \u00b7 x' \u2250 y \u00b7 y'\n    K-eq    : \u2200 x y \u2192 K \u00b7 x \u00b7 y \u2250 x\n    S-eq    : \u2200 x y z \u2192 S \u00b7 x \u00b7 y \u00b7 z \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate \u2250-subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- we can proof\n\n  \u2250\u2192\u2261 : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  \u2250\u2192\u2261 {x} h = \u2250-subst (\u03bb z \u2192 x \u2261 z) h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule FOTC where\n\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    \u2250-refl   : \u2200 {x} \u2192 x \u2250 x\n    \u2250-sym    : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    \u2250-trans  : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    \u2250-cong   : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 y\u2081 \u2192 x\u2082 \u2250 y\u2082 \u2192 x\u2081 \u00b7 x\u2082 \u2250 y\u2081 \u00b7 y\u2082\n    if-true  : \u2200 t t' \u2192 if \u00b7 true \u00b7 t \u00b7 t' \u2250 t\n    if-false : \u2200 t t' \u2192 if \u00b7 false \u00b7 t \u00b7 t' \u2250 t'\n    pred-0   : pred \u00b7 zero \u2250 zero\n    pred-S   : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2250 n\n    iszero-0 : iszero \u00b7 zero \u2250 true\n    iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2250 false\n    beta     : \u2200 f a \u2192 lam f \u00b7 a \u2250 f a\n    fix-eq   : \u2200 f \u2192 fix f \u2250 f (fix f)\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A h = h A\n\n  -- and the congruency\n\n  cong  : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \u00b7 y \u2261 x' \u00b7 y'\n  cong {x} {x'} {y} {y'} h\u2081 h\u2082 A Axx' =\n    h\u2082 (\u03bb z \u2192 A (x' \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y)) Axx')\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-positivity-check #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\nmodule FOTC where\n\nmodule Aczel-CA where\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    \u2250-refl  : \u2200 {x} \u2192 x \u2250 x\n    \u2250-sym   : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    \u2250-trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    \u2250-cong  : \u2200 {x x' y y'} \u2192 x \u2250 y \u2192 x' \u2250 y' \u2192 x \u00b7 x' \u2250 y \u00b7 y'\n    K-eq    : \u2200 x y \u2192 K \u00b7 x \u00b7 y \u2250 x\n    S-eq    : \u2200 x y z \u2192 S \u00b7 x \u00b7 y \u00b7 z \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate \u2250-subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- we can proof\n\n  \u2250\u2192\u2261 : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  \u2250\u2192\u2261 {x} h = \u2250-subst (\u03bb z \u2192 x \u2261 z) h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule FOTC where\n\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    \u2250-refl   : \u2200 {x} \u2192 x \u2250 x\n    \u2250-sym    : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    \u2250-trans  : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    \u2250-cong   : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 y\u2081 \u2192 x\u2082 \u2250 y\u2082 \u2192 x\u2081 \u00b7 x\u2082 \u2250 y\u2081 \u00b7 y\u2082\n    if-true  : \u2200 t t' \u2192 if \u00b7 true \u00b7 t \u00b7 t' \u2250 t\n    if-false : \u2200 t t' \u2192 if \u00b7 false \u00b7 t \u00b7 t' \u2250 t'\n    pred-0   : pred \u00b7 zero \u2250 zero\n    pred-S   : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2250 n\n    iszero-0 : iszero \u00b7 zero \u2250 true\n    iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2250 false\n    beta     : \u2200 f a \u2192 lam f \u00b7 a \u2250 f a\n    fix-eq   : \u2200 f \u2192 fix f \u2250 f (fix f)\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \u00b7 y \u2261 x' \u00b7 y'\n  cong {x} {x'} {y} {y'} h\u2081 h\u2082 A Axx' =\n    h\u2082 (\u03bb z \u2192 A (x' \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y)) Axx')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"71b4bbc42f967d1637bcccb87d390a1160cb0728","subject":"Added arithmetic properties (by Ana Bove).","message":"Added arithmetic properties (by Ana Bove).\n\nIgnore-this: f8ca4edfb0d79ea91c483d20425cea27\n\ndarcs-hash:20110218152346-3bd4e-da0de11b185f6eb1497944803f490c5e0b5ede28.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111'  : eleven + one-hundred \u2261 hundred-eleven\n  n111   : one-hundred + eleven \u2261 hundred-eleven\n  n101'  : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101   : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'   : hundred-one \u2238 ten \u2261 ninety-one\n  n91    : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102'  : eleven + ninety-one \u2261 hundred-two\n  n102   : ninety-one + eleven \u2261 hundred-two\n  n100'  : eleven + eighty-nine \u2261 one-hundred\n  n100   : eighty-nine + eleven \u2261 one-hundred\n  n110   : ninety-nine + eleven \u2261 hundred-ten\n  n109   : ninety-eight + eleven \u2261 hundred-nine\n  n108   : ninety-seven + eleven \u2261 hundred-eight\n  n107   : ninety-six + eleven \u2261 hundred-seven\n  n106   : ninety-five + eleven \u2261 hundred-six\n  n105   : ninety-four + eleven \u2261 hundred-five\n  n104   : ninety-three + eleven \u2261 hundred-four\n  n103   : ninety-two + eleven \u2261 hundred-three\n  n101'' : ninety + eleven \u2261 hundred-one\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n{-# ATP prove n110 #-}\n{-# ATP prove n109 #-}\n{-# ATP prove n108 #-}\n{-# ATP prove n107 #-}\n{-# ATP prove n106 #-}\n{-# ATP prove n105 #-}\n{-# ATP prove n104 #-}\n{-# ATP prove n103 #-}\n{-# ATP prove n101'' #-}\n\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 n100 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n  n100' : eleven + eighty-nine \u2261 one-hundred\n  n100  : eighty-nine + eleven \u2261 one-hundred\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y N10 1+x-N 11+x-N x\u223810-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N N89 N100 n100 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a94e0f9dc0b53ba1ab8a27b813a9d4e981d8e399","subject":"valuable golfing","message":"valuable golfing\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen L using (Lift)\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to SumProp)\nopen import Search.Searchable.Product\nopen import Search.Derived\n\nmodule sum where\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search _ A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bcLift\u22a4 : SumProp (Lift \u22a4)\n\u03bcLift\u22a4 = _ , ind\n  where\n    srch : Search _ (Lift \u22a4)\n    srch _ f = f _\n\n    ind : SearchInd _ srch\n    ind _ _ Pf = Pf _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search _ \u22a4\n    srch _ f = f _\n\n    ind : SearchInd _ srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search _ Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd _ searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {m A B} \u2192 Search m A \u2192 Search m B \u2192 Search m (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {m p A B} {s\u1d2c : Search m A} {s\u1d2e : Search m B} \u2192\n                 SearchInd p s\u1d2c \u2192 SearchInd p s\u1d2e \u2192 SearchInd p (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nS\u03a0\u03a3\u207b : \u2200 {m A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n       \u2192 Search m ((x : A) (y : B x) \u2192 C (x , y))\n       \u2192 Search m (\u03a0 (\u03a3 A B) C)\nS\u03a0\u03a3\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry)\n\nS\u03a0\u03a3\u207b-ind : \u2200 {m p A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n           \u2192 {s : Search m ((x : A) (y : B x) \u2192 C (x , y))}\n           \u2192 SearchInd p s\n           \u2192 SearchInd p (S\u03a0\u03a3\u207b s)\nS\u03a0\u03a3\u207b-ind ind P P\u2219 Pf = ind (P \u2218 S\u03a0\u03a3\u207b) P\u2219 (Pf \u2218 uncurry)\n\nS\u00d7\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 B \u2192 C) \u2192 Search m (A \u00d7 B \u2192 C)\nS\u00d7\u207b = S\u03a0\u03a3\u207b\n\nS\u00d7\u207b-ind : \u2200 {m p A B C}\n          \u2192 {s : Search m (A \u2192 B \u2192 C)}\n          \u2192 SearchInd p s\n          \u2192 SearchInd p (S\u00d7\u207b s)\nS\u00d7\u207b-ind = S\u03a0\u03a3\u207b-ind\n\nS\u03a0\u228e\u207b : \u2200 {m A B} {C : A \u228e B \u2192 \u2605 _}\n    -- \u2192 Search m (\u03a0 A (C \u2218 inj\u2081)) \u2192 Search m (\u03a0 B (C \u2218 inj\u2082))\n       \u2192 Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n       \u2192 Search m (\u03a0 (A \u228e B) C)\nS\u03a0\u228e\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry [_,_])\n\nS\u03a0\u228e\u207b-ind : \u2200 {m p A B} {C : A \u228e B \u2192 \u2605 _}\n             {- {sA : Search m (\u03a0 A (C \u2218 inj\u2081))}\n             (iA : SearchInd p sA)\n             {sB : Search m (\u03a0 B (C \u2218 inj\u2082))}\n             (iB : SearchInd p sB) -}\n             {s : Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))}\n             (i : SearchInd p s)\n           \u2192 SearchInd p (S\u03a0\u228e\u207b {C = C} s) -- A sB)\nS\u03a0\u228e\u207b-ind i P P\u2219 Pf = i (P \u2218 S\u03a0\u228e\u207b) P\u2219 (Pf \u2218 uncurry [_,_])\n\n{- For each A\u2192C function\n   and each B\u2192C function\n   an A\u228eB\u2192C function is yield\n -}\nS\u228e\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 C) \u2192 Search m (B \u2192 C)\n                  \u2192 Search m (A \u228e B \u2192 C)\nS\u228e\u207b sA sB =  S\u03a0\u228e\u207b (sA \u00d7Search sB)\n\nS\u22a4 : \u2200 {m A} \u2192 Search m A \u2192 Search m (\u22a4 \u2192 A)\nS\u22a4 sA _\u2219_ f = sA _\u2219_ (f \u2218 const)\n\nS\u03a0Bit : \u2200 {m A} \u2192 Search m (A 0b) \u2192 Search m (A 1b)\n                \u2192 Search m (\u03a0 Bit A)\nS\u03a0Bit sA\u2080 sA\u2081 _\u2219_ f = sA\u2080 _\u2219_ \u03bb x \u2192 sA\u2081 _\u2219_ \u03bb y \u2192 f (Cond _ y x)\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search _ B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd _ search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 {c \u2113} (m : Monoid c \u2113) {n} \u2192\n          let open Mon m in\n          Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Mon m\n\nvsgsum : \u2200 {c \u2113} (sg : Semigroup c \u2113) {n} \u2192\n           let open Sgrp sg in\n           Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 {m} n \u2192 Search m (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd _ (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {m A} n \u2192 Search m A \u2192 Search m (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\nmodule _ {A}(\u03bcA : SumProp A)\n  (cm       : CommutativeMonoid L.zero L.zero)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_      : let open CMon cm in C  \u2192 C \u2192 C)\n  (distrib  : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_ : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm\n\n  \u03bcF = \u03bcFun \u03bcA\n\n  -- Sum over A values\n  \u03a3\u1d2c = search \u03bcA _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin (suc I) \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = search \u03bcF _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 search (\u03bcFinSuc I) _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 F')\n    \u2248\u27e8 sym (search-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 F')) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 F'))\n    \u2248\u27e8 search-sg-ext \u03bcA semigroup (\u03bb j \u2192 sym (search-lin\u02e1 \u03bcF monoid _\u25ce_ (\u03a0' I \u2218 F') (F zero j) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I (F' f))\n    \u220e\n    where\n      F' : (Fin (suc I) \u2192 A) \u2192 Fin (suc I) \u2192 C\n      F' = _\u02e2_ (F \u2218 suc)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen L using (Lift)\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to SumProp)\nopen import Search.Searchable.Product\nopen import Search.Derived\n\nmodule sum where\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search _ A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bcLift\u22a4 : SumProp (Lift \u22a4)\n\u03bcLift\u22a4 = _ , ind\n  where\n    srch : Search _ (Lift \u22a4)\n    srch _ f = f _\n\n    ind : SearchInd _ srch\n    ind _ _ Pf = Pf _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search _ \u22a4\n    srch _ f = f _\n\n    ind : SearchInd _ srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search _ Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd _ searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {m A B} \u2192 Search m A \u2192 Search m B \u2192 Search m (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {m p A B} {s\u1d2c : Search m A} {s\u1d2e : Search m B} \u2192\n                 SearchInd p s\u1d2c \u2192 SearchInd p s\u1d2e \u2192 SearchInd p (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nS\u03a0\u03a3\u207b : \u2200 {m A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n       \u2192 Search m ((x : A) (y : B x) \u2192 C (x , y))\n       \u2192 Search m (\u03a0 (\u03a3 A B) C)\nS\u03a0\u03a3\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry)\n\nS\u03a0\u03a3\u207b-ind : \u2200 {m p A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n           \u2192 {s : Search m ((x : A) (y : B x) \u2192 C (x , y))}\n           \u2192 SearchInd p s\n           \u2192 SearchInd p (S\u03a0\u03a3\u207b s)\nS\u03a0\u03a3\u207b-ind ind P P\u2219 Pf = ind (P \u2218 S\u03a0\u03a3\u207b) P\u2219 (Pf \u2218 uncurry)\n\nS\u00d7\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 B \u2192 C) \u2192 Search m (A \u00d7 B \u2192 C)\nS\u00d7\u207b = S\u03a0\u03a3\u207b\n\nS\u00d7\u207b-ind : \u2200 {m p A B C}\n          \u2192 {s : Search m (A \u2192 B \u2192 C)}\n          \u2192 SearchInd p s\n          \u2192 SearchInd p (S\u00d7\u207b s)\nS\u00d7\u207b-ind = S\u03a0\u03a3\u207b-ind\n\nS\u03a0\u228e\u207b : \u2200 {m A B} {C : A \u228e B \u2192 \u2605 _}\n    -- \u2192 Search m (\u03a0 A (C \u2218 inj\u2081)) \u2192 Search m (\u03a0 B (C \u2218 inj\u2082))\n       \u2192 Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n       \u2192 Search m (\u03a0 (A \u228e B) C)\nS\u03a0\u228e\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry [_,_])\n\nS\u03a0\u228e\u207b-ind : \u2200 {m p A B} {C : A \u228e B \u2192 \u2605 _}\n             {- {sA : Search m (\u03a0 A (C \u2218 inj\u2081))}\n             (iA : SearchInd p sA)\n             {sB : Search m (\u03a0 B (C \u2218 inj\u2082))}\n             (iB : SearchInd p sB) -}\n             {s : Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))}\n             (i : SearchInd p s)\n           \u2192 SearchInd p (S\u03a0\u228e\u207b {C = C} s) -- A sB)\nS\u03a0\u228e\u207b-ind i P P\u2219 Pf = i (P \u2218 S\u03a0\u228e\u207b) P\u2219 (Pf \u2218 uncurry [_,_])\n\n{- For each A\u2192C function\n   and each B\u2192C function\n   an A\u228eB\u2192C function is yield\n -}\nS\u228e\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 C) \u2192 Search m (B \u2192 C)\n                  \u2192 Search m (A \u228e B \u2192 C)\nS\u228e\u207b sA sB =  S\u03a0\u228e\u207b (sA \u00d7Search sB)\n\nS\u22a4 : \u2200 {m A} \u2192 Search m A \u2192 Search m (\u22a4 \u2192 A)\nS\u22a4 sA _\u2219_ f = sA _\u2219_ (f \u2218 const)\n\nS\u03a0Bit : \u2200 {m A} \u2192 Search m (A 0b) \u2192 Search m (A 1b)\n                \u2192 Search m (\u03a0 Bit A)\nS\u03a0Bit sA\u2080 sA\u2081 _\u2219_ f = sA\u2080 _\u2219_ \u03bb x \u2192 sA\u2081 _\u2219_ \u03bb y \u2192 f (Cond _ y x)\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search _ B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd _ search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 {c \u2113} (m : Monoid c \u2113) {n} \u2192\n          let open Mon m in\n          Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Mon m\n\nvsgsum : \u2200 {c \u2113} (sg : Semigroup c \u2113) {n} \u2192\n           let open Sgrp sg in\n           Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 {m} n \u2192 Search m (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd _ (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {m A} n \u2192 Search m A \u2192 Search m (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n\nmodule _ {A}(\u03bcA : SumProp A)\n  (cmonoid : CommutativeMonoid L.zero L.zero)\n  (_\u25ce_      : let open CMon cmonoid in C  \u2192 C \u2192 C)\n  (distrib  : let open CMon cmonoid in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_ : let open CMon cmonoid in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cmonoid\n  s\u1d2c = search \u03bcA _\u2219_\n\n  bigDistr : \u2200 I F \u2192 search (\u03bcFinSuc I) _\u25ce_ (search \u03bcA          _\u2219_ \u2218 F)\n                   \u2248 search (\u03bcFun \u03bcA)   _\u2219_ (search (\u03bcFinSuc I) _\u25ce_ \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = s\u1d2c (F zero) \u25ce \u03a0' I (s\u1d2c \u2218 Fj)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I Fj \u27e9\n      (s\u1d2c \u2218 F) zero \u25ce \u03a3F (\u03a0' I \u2218 F')\n    \u2248\u27e8 sym (search-lin\u02e1 \u03bcF monoid _\u25ce_ (\u03a0' I \u2218 F') (s\u1d2c (F zero)) (proj\u2081 distrib)) \u27e9\n      \u03a3F (\u03bb f \u2192 (s\u1d2c \u2218 F) zero \u25ce \u03a0' I (F' f))\n    \u2248\u27e8 search-sg-ext \u03bcF semigroup\n           (\u03bb f \u2192 sym (search-lin\u02b3 \u03bcA  monoid _\u25ce_ (F zero) (\u03a0' I (F' f)) (proj\u2082 distrib))) \u27e9\n      \u03a3F (\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a0' I (F' f)))\n    \u2248\u27e8 search-sg-ext \u03bcF semigroup {(\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a0' I (F' f)))} (\u03bb f \u2192 refl) \u27e9\n      \u03a3F (\u03bb f \u2192 s\u1d2c (\u03bb j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i))))\n    \u2248\u27e8 search-swap \u03bcF semigroup (\u03bb f j \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i)))\n      {s\u1d2e = s\u1d2c} (search-hom \u03bcA cmonoid) \u27e9\n      s\u1d2c (\u03bb j \u2192 \u03a3F (\u03bb f \u2192 \u03a0' (suc I) (\u03bb i \u2192 F i (extend \u03bcA j f i))))\n    \u220e\n    where\n      F' : (Fin (suc I) \u2192 A) \u2192 Fin (suc I) \u2192 C\n      F' = \u03bb f i \u2192 F (suc i) (f i)\n      \u03bcF = \u03bcFun \u03bcA {suc I}\n      \u03a0' = \u03bb i \u2192 search (\u03bcFinSuc i) _\u25ce_\n      \u03a3F = search (\u03bcFun \u03bcA {suc I}) _\u2219_\n      Fj = \u03bb i j \u2192 F (suc i) j\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0e34d3e67b7c1c95ec233b1b5fb3101c535bb088","subject":"Labelled Desc functions using eliminators, which use ind.","message":"Labelled Desc functions using eliminators, which use ind.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/LabelledDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/LabelledDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.LabelledDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\ndata Cases2 (A : Set) : Enum \u2192 Set where -- Vec, Enum is \u2115, Tag is Fin\n  [] : Cases2 A []\n  _\u2237_ : \u2200{l E} \u2192 A \u2192 Cases2 A E \u2192 Cases2 A (l \u2237 E)\n\n-- lookup\ncase2 : (E : Enum) (A : Set) \u2192 Cases2 A E \u2192 Tag E \u2192 A\ncase2 (l \u2237 E) A (c \u2237 cs) here = c\ncase2 (l \u2237 E) A (c \u2237 cs) (there t) = case2 E A cs t\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `End : (i : I) \u2192 Desc I\n  `\u03a3 `\u03a0 : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j)   X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`\u03a3 A B)   X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`\u03a0 A B)   X i = (a : A) \u2192 El I (B a) X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`End j) X P i q = \u22a4\nAll I (`\u03a3 A B) X P i (a , b) = All I (B a) X P i b\nAll I (`\u03a0 A B) X P i f = (a : A) \u2192 All I (B a) X P i (f a)\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : (i : I) (xs : El I D (\u03bc I D) i) \u2192 All I D (\u03bc I D) P i xs \u2192 P i (con xs))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El I D\u2081 (\u03bc I D\u2081) i) \u2192 All I D\u2081 (\u03bc I D\u2081) P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`\u03a3 A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`\u03a0 A B) i f = \u03bb a \u2192 hyps I D P pcon (B a) i (f a)\n\n----------------------------------------------------------------------\n\n\u2115T : Enum\n\u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecT : Enum\nVecT = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115C : Tag \u2115T \u2192 Desc \u22a4\n\u2115C = caseD \u2115T \u22a4\n  ( `End tt\n  , `Rec tt (`End tt)\n  , tt\n  )\n\n\u2115D : Desc \u22a4\n\u2115D = `\u03a3 (Tag \u2115T) \u2115C\n\n\u2115D2 : Desc \u22a4\n\u2115D2 = `\u03a3 (Tag \u2115T) (case2 \u2115T (Desc \u22a4)\n  ( `End tt\n  \u2237 `Rec tt (`End tt)\n  \u2237 []\n  ))\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u22a4 \u2115D\n\nzero : \u2115 tt\nzero = con (here , refl)\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = con (there here , n , refl)\n\nVecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD VecT (\u2115 tt)\n  ( `End zero\n  , `\u03a3 (\u2115 tt) (\u03bb n \u2192 `\u03a3 A \u03bb _ \u2192 `Rec n (`End (suc n)))\n  , tt\n  )\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = `\u03a3 (Tag VecT) (VecC A)\n\nVec : (A : Set) (n : \u2115 tt) \u2192 Set\nVec A n = \u03bc (\u2115 tt) (VecD A) n\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = con (here , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = con (there here , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb u t,c \u2192 case \u2115T\n      (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n             (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n             \u2192 \u2115 u \u2192 \u2115 u\n      )\n      ( (\u03bb q ih n \u2192 n)\n      , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n      (proj\u2081 t,c)\n      (proj\u2082 t,c)\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb u t,c \u2192 case \u2115T\n      (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n             (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n             \u2192 \u2115 u \u2192 \u2115 u\n      )\n      ( (\u03bb q ih n \u2192 zero)\n      , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n      (proj\u2081 t,c)\n      (proj\u2082 t,c)\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb m t,c \u2192 case VecT\n      (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n             (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n             (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n      )\n      ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n      (proj\u2081 t,c)\n      (proj\u2082 t,c)\n    )\n  \n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb n t,c \u2192 case VecT\n      (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n             (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n             \u2192 Vec A (mult n m)\n      )\n      ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n',xs,xss,q ih,tt \u2192\n          let n' = proj\u2081 n',xs,xss,q\n              xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n      (proj\u2081 t,c)\n      (proj\u2082 t,c)\n    )\n\n----------------------------------------------------------------------\n\nmodule Eliminator where\n\n  elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n    (pzero : P zero)\n    (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n    (n : \u2115 tt)\n    \u2192 P n\n  elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n    (\u03bb u t,c \u2192 case \u2115T\n      (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n             (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n             \u2192 P (con (t , c))\n      )\n      ( (\u03bb q ih \u2192\n          elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n            pzero\n            u q\n        )\n      , (\u03bb n,q ih,tt \u2192\n          elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n            (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n            u (proj\u2082 n,q)\n        )\n      , tt\n      )\n      (proj\u2081 t,c)\n      (proj\u2082 t,c)\n    )\n    tt\n  \n  elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n    (pnil : P zero (nil A))\n    (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n    (n : \u2115 tt)\n    (xs : Vec A n)\n    \u2192 P n xs\n  elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n    (\u03bb n t,c \u2192 case VecT\n      (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n             (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n             \u2192 P n (con (t , c))\n      )\n      ( (\u03bb q ih \u2192\n          elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n            pnil\n            n q\n        )\n      , (\u03bb n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              x = proj\u2081 (proj\u2082 n',x,xs,q)\n              xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n            (pcons n' x xs ih )\n            n q\n        )\n      , tt\n      )\n      (proj\u2081 t,c)\n      (proj\u2082 t,c)\n    )\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.LabelledDesc where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\ndata Cases2 (A : Set) : Enum \u2192 Set where -- Vec, Enum is \u2115, Tag is Fin\n  [] : Cases2 A []\n  _\u2237_ : \u2200{l E} \u2192 A \u2192 Cases2 A E \u2192 Cases2 A (l \u2237 E)\n\n-- lookup\ncase2 : (E : Enum) (A : Set) \u2192 Cases2 A E \u2192 Tag E \u2192 A\ncase2 (l \u2237 E) A (c \u2237 cs) here = c\ncase2 (l \u2237 E) A (c \u2237 cs) (there t) = case2 E A cs t\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `End : (i : I) \u2192 Desc I\n  `\u03a3 `\u03a0 : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j)   X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`\u03a3 A B)   X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`\u03a0 A B)   X i = (a : A) \u2192 El I (B a) X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`End j) X P i q = \u22a4\nAll I (`\u03a3 A B) X P i (a , b) = All I (B a) X P i b\nAll I (`\u03a0 A B) X P i f = (a : A) \u2192 All I (B a) X P i (f a)\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : (i : I) (xs : El I D (\u03bc I D) i) \u2192 All I D (\u03bc I D) P i xs \u2192 P i (con xs))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El I D\u2081 (\u03bc I D\u2081) i) \u2192 All I D\u2081 (\u03bc I D\u2081) P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`\u03a3 A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`\u03a0 A B) i f = \u03bb a \u2192 hyps I D P pcon (B a) i (f a)\n\n----------------------------------------------------------------------\n\n\u2115T : Enum\n\u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecT : Enum\nVecT = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115C : Tag \u2115T \u2192 Desc \u22a4\n\u2115C = caseD \u2115T \u22a4\n  ( `End tt\n  , `Rec tt (`End tt)\n  , tt\n  )\n\n\u2115D : Desc \u22a4\n\u2115D = `\u03a3 (Tag \u2115T) \u2115C\n\n\u2115D2 : Desc \u22a4\n\u2115D2 = `\u03a3 (Tag \u2115T) (case2 \u2115T (Desc \u22a4)\n  ( `End tt\n  \u2237 `Rec tt (`End tt)\n  \u2237 []\n  ))\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u22a4 \u2115D\n\n-- zero : \u2115 tt\npattern zero = con (here , refl)\n\n-- suc : \u2115 tt \u2192 \u2115 tt\npattern suc n = con (there here , n , refl)\n\nVecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD VecT (\u2115 tt)\n  ( `End zero\n  , `\u03a3 (\u2115 tt) (\u03bb n \u2192 `\u03a3 A \u03bb _ \u2192 `Rec n (`End (suc n)))\n  , tt\n  )\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = `\u03a3 (Tag VecT) (VecC A)\n\nVec : (A : Set) (n : \u2115 tt) \u2192 Set\nVec A n = \u03bc (\u2115 tt) (VecD A) n\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = con (here , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = con (there here , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\nadd : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nadd = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n  (\u03bb u xs \u2192 case \u2115T\n    (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n           (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n           \u2192 \u2115 u \u2192 \u2115 u\n    )\n    ( (\u03bb q ih n \u2192 n)\n    , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n    , tt\n    )\n    (proj\u2081 xs)\n    (proj\u2082 xs)\n  )\n  tt\n\nmult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nmult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n  (\u03bb u xs \u2192 case \u2115T\n    (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n           (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n           \u2192 \u2115 u \u2192 \u2115 u\n    )\n    ( (\u03bb q ih n \u2192 zero)\n    , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n    , tt\n    )\n    (proj\u2081 xs)\n    (proj\u2082 xs)\n  )\n  tt\n\nappend : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \nappend A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb m xs \u2192 case VecT\n    (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n           (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n           (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    )\n    ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n    , (\u03bb m',x,xs,q ih,tt n ys \u2192\n        let m' = proj\u2081 m',x,xs,q\n            x = proj\u2081 (proj\u2082 m',x,xs,q)\n            q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n            ih = proj\u2081 ih,tt\n        in\n        subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n      )\n    , tt\n    )\n    (proj\u2081 xs)\n    (proj\u2082 xs)\n  )\n\nconcat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n  (\u03bb n xss \u2192 case VecT\n    (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n           (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n           \u2192 Vec A (mult n m)\n    )\n    ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n    , (\u03bb n',xs,xss,q ih,tt \u2192\n        let n' = proj\u2081 n',xs,xss,q\n            xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n            q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n            ih = proj\u2081 ih,tt\n        in\n        subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n      )\n    , tt\n    )\n    (proj\u2081 xss)\n    (proj\u2082 xss)\n  )\n\n-------------------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f4fc96e9711d5a4df23b48c16442921c93f27a42","subject":"more code","message":"more code\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  module _ {A B : \u2605}{f : A \u2192 B}(f\u1d31 : Equiv f) where\n    open Equiv f\u1d31\n    inv : B \u2192 A\n    inv = linv \u2218 f \u2218 rinv\n\n    inv-equiv : Equiv inv\n    inv-equiv = record { linv = f\n                       ; is-linv = \u03bb x \u2192 cong f (is-linv (rinv x)) \u2219 is-rinv x\n                       ; rinv = f\n                       ; is-rinv = \u03bb x \u2192 cong linv (is-rinv (f x)) \u2219 is-linv x }\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u2243-refl : Reflexive _\u2243_\n  \u2243-refl = _ , id\u1d31\n\n  \u2243-sym : Symmetric _\u2243_\n  \u2243-sym (_ , f\u1d31) = _ , inv-equiv f\u1d31\n\n  \u2243-trans : Transitive _\u2243_\n  \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com_ \u2113 : \u2605_(\u209b \u2113) where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605_ \u2113\n        P  : M \u2192 Proto_ \u2113\n\n    data Proto_ \u2113 : \u2605_(\u209b \u2113) where\n      end : Proto_ \u2113\n      com : Com_ \u2113 \u2192 Proto_ \u2113\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\nCom   : \u2605\u2081\nCom   = Com_ \u2080\n\n{-\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n-}\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e M P) (\u03a0\u1d3e M Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c (mk _ M P) = \u03a3\u1d9c M \u03bb m \u2192 source-of (P m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk io M P) = mk (dual\u1d35\u1d3c io) M \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113) (P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7\u2032 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end        \u27e7\u2032 = Lift \ud835\udfd9\n\u27e6 com' q M P \u27e7\u2032 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\u2032\n\n\u27e6_\u27e7 : Proto\u2080 \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl M x) (com refl .M x\u2081) = com refl M (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    open Equivalences funExt\n\n    module _ {A} {B : A \u2192 \u2605} where\n        foo : (\u2200 (x : A) \u2192 B x \u2243 \ud835\udfd9) \u2192 \u03a0 A B \u2243 \ud835\udfd9\n        foo f = {!!}\n          where\n            to : \u03a0 A B \u2192 \ud835\udfd9\n            to = _\n\n            from : \ud835\udfd9 \u2192 \u03a0 A B\n            from _ x = {!f x!}\n\n            to\u1d31 : \u03a0 A B \u2243 \ud835\udfd9\n            to\u1d31 = _ , record { linv = {!!} ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n\n    sink\u2243\ud835\udfd9 : \u2200 P \u2192 \u27e6 sink-of P \u27e7 \u2243 \ud835\udfd9\n    sink\u2243\ud835\udfd9 end = \u2243-refl\n    sink\u2243\ud835\udfd9 (com' _ _ P) = {!sink\u2243\ud835\udfd9 (P ?)!}\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end         _        ys = ys\n++Log (com' q M P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n\nmodule Mobility where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         \u03a0\u1d3e (Lift \u27e6 P  \u27e7)  \u03bb p \u2192\n         \u03a0\u1d3e (Lift \u27e6 P \u22a5\u27e7)  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Lift (Log P)) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\u2032\n  com-proc P (lift p) (lift p\u22a5) = lift (tele-com P p p\u22a5) , _\n\n\n{-\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Log P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = \u03a0\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n    StaticServer  = \u03a3\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end Q = \u2261\u1d3e-refl _\ndual->> (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\ndual->> (\u03a3\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 subst id (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (\u03a0\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (\u03a3\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong funExt (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP\u1d9c >>\u1d9c S = record P\u1d9c { P = \u03bb m \u2192 S (P m) }\n  where open Com_ P\u1d9c\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e M P) (\u03a0\u1d3e M Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 source-of (Com.P c m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Tele : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n{-\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = end\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server = dual \u2218 Client\n\n    Server' : \u2115 \u2192 Proto\n    Server' zero    = end\n    Server' (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server' n\n\n    Server\u2261Server' : \u2200 n \u2192 Server n \u2261\u1d3e Server' n\n    Server\u2261Server' zero    = end\n    Server\u2261Server' (suc n) = com refl Query (\u03bb m \u2192 com refl (Resp m) (\u03bb m' \u2192 Server\u2261Server' n))\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7b8b1db0d9c55c9f6341847c502db86cc1537bd0","subject":"Remove instance arguments.","message":"Remove instance arguments.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script\/Unicode.agda","new_file":"agda\/Text\/Greek\/Script\/Unicode.agda","new_contents":"module Text.Greek.Script.Unicode where\n\nopen import Text.Greek.Script\n\n-- U+0390 - U+03CE\n\u0390 : Token -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = with-accent-diaeresis \u03b9\u2032 lower (acute \u03b9-vowel) \u03b9-diaeresis\n\u0391 : Token -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked \u03b1\u2032 upper\n\u0392 : Token -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked \u03b2\u2032 upper\n\u0393 : Token -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked \u03b3\u2032 upper\n\u0394 : Token -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked \u03b4\u2032 upper\n\u0395 : Token -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked \u03b5\u2032 upper\n\u0396 : Token -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked \u03b6\u2032 upper\n\u0397 : Token -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked \u03b7\u2032 upper\n\u0398 : Token -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked \u03b8\u2032 upper\n\u0399 : Token -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked \u03b9\u2032 upper\n\u039a : Token -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked \u03ba\u2032 upper\n\u039b : Token -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked \u03bb\u2032 upper\n\u039c : Token -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked \u03bc\u2032 upper\n\u039d : Token -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked \u03bd\u2032 upper\n\u039e : Token -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked \u03be\u2032 upper\n\u039f : Token -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked \u03bf\u2032 upper\n\u03a0 : Token -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked \u03c0\u2032 upper\n\u03a1 : Token -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked \u03c1\u2032 upper\n                  -- U+03A2 <reserved>\n\u03a3 : Token -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked \u03c3\u2032 upper\n\u03a4 : Token -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked \u03c4\u2032 upper\n\u03a5 : Token -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked \u03c5\u2032 upper\n\u03a6 : Token -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked \u03c6\u2032 upper\n\u03a7 : Token -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked \u03c7\u2032 upper\n\u03a8 : Token -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked \u03c8\u2032 upper\n\u03a9 : Token -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked \u03c9\u2032 upper\n\u03aa : Token -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = with-diaeresis \u03b9\u2032 upper \u03b9-diaeresis\n\u03ab : Token -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = with-diaeresis \u03c5\u2032 upper \u03c5-diaeresis\n\u03ac : Token -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = with-accent \u03b1\u2032 lower (acute \u03b1-vowel)\n\u03ad : Token -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = with-accent \u03b5\u2032 lower (acute \u03b5-vowel)\n\u03ae : Token -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = with-accent \u03b7\u2032 lower (acute \u03b7-vowel)\n\u03af : Token -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = with-accent \u03b9\u2032 lower (acute \u03b9-vowel)\n\u03b0 : Token -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = with-accent-diaeresis \u03c5\u2032 lower (acute \u03c5-vowel) \u03c5-diaeresis\n\u03b1 : Token -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked \u03b1\u2032 lower\n\u03b2 : Token -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked \u03b2\u2032 lower\n\u03b3 : Token -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked \u03b3\u2032 lower\n\u03b4 : Token -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked \u03b4\u2032 lower\n\u03b5 : Token -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked \u03b5\u2032 lower\n\u03b6 : Token -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked \u03b6\u2032 lower\n\u03b7 : Token -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked \u03b7\u2032 lower\n\u03b8 : Token -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked \u03b8\u2032 lower\n\u03b9 : Token -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked \u03b9\u2032 lower\n\u03ba : Token -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked \u03ba\u2032 lower\n\u2219\u03bb : Token -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked \u03bb\u2032 lower\n\u03bc : Token -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked \u03bc\u2032 lower\n\u03bd : Token -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked \u03bd\u2032 lower\n\u03be : Token -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked \u03be\u2032 lower\n\u03bf : Token -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked \u03bf\u2032 lower\n\u03c0 : Token -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked \u03c0\u2032 lower\n\u03c1 : Token -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked \u03c1\u2032 lower\n\u03c2 : Token -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = final \u03c3\u2032 lower \u03c3-final\n\u03c3 : Token -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked \u03c3\u2032 lower\n\u03c4 : Token -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked \u03c4\u2032 lower\n\u03c5 : Token -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked \u03c5\u2032 lower\n\u03c6 : Token -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked \u03c6\u2032 lower\n\u03c7 : Token -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked \u03c7\u2032 lower\n\u03c8 : Token -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked \u03c8\u2032 lower\n\u03c9 : Token -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked \u03c9\u2032 lower\n\u03ca : Token -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = with-diaeresis \u03b9\u2032 lower \u03b9-diaeresis\n\u03cb : Token -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = with-diaeresis \u03c5\u2032 lower \u03c5-diaeresis\n\u03cc : Token -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = with-accent \u03bf\u2032 lower (acute \u03bf-vowel)\n\u03cd : Token -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = with-accent \u03c5\u2032 lower (acute \u03c5-vowel)\n\u03ce : Token -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = with-accent \u03c9\u2032 lower (acute \u03c9-vowel)\n\n-- U+1F00 - U+1FFF\n\u1f00 : Token -- U+1F00  \u1f00 GREEK SMALL LETTER ALPHA WITH PSILI\n\u1f00 = with-breathing \u03b1\u2032 lower (smooth \u03b1-smooth)\n\u1f01 : Token -- U+1F01  \u1f01 GREEK SMALL LETTER ALPHA WITH DASIA\n\u1f01 = with-breathing \u03b1\u2032 lower (rough \u03b1-rough)\n\u1f02 : Token -- U+1F02  \u1f02 GREEK SMALL LETTER ALPHA WITH PSILI AND VARIA\n\u1f02 = with-accent-breathing \u03b1\u2032 lower (grave \u03b1-vowel) (smooth \u03b1-smooth)\n\u1f03 : Token -- U+1F03  \u1f03 GREEK SMALL LETTER ALPHA WITH DASIA AND VARIA\n\u1f03 = with-accent-breathing \u03b1\u2032 lower (grave \u03b1-vowel) (rough \u03b1-rough)\n\u1f04 : Token -- U+1F04  \u1f04 GREEK SMALL LETTER ALPHA WITH PSILI AND OXIA\n\u1f04 = with-accent-breathing \u03b1\u2032 lower (acute \u03b1-vowel) (smooth \u03b1-smooth)\n\u1f05 : Token -- U+1F05  \u1f05 GREEK SMALL LETTER ALPHA WITH DASIA AND OXIA\n\u1f05 = with-accent-breathing \u03b1\u2032 lower (acute \u03b1-vowel) (rough \u03b1-rough)\n\u1f06 : Token -- U+1F06  \u1f06 GREEK SMALL LETTER ALPHA WITH PSILI AND PERISPOMENI\n\u1f06 = with-accent-breathing \u03b1\u2032 lower (circumflex \u03b1-long-vowel) (smooth \u03b1-smooth)\n\u1f07 : Token -- U+1F07  \u1f07 GREEK SMALL LETTER ALPHA WITH DASIA AND PERISPOMENI\n\u1f07 = with-accent-breathing \u03b1\u2032 lower (circumflex \u03b1-long-vowel) (rough \u03b1-rough)\n\u1f08 : Token -- U+1F08  \u1f08 GREEK CAPITAL LETTER ALPHA WITH PSILI\n\u1f08 = with-breathing \u03b1\u2032 upper (smooth \u0391-smooth)\n\u1f09 : Token -- U+1F09  \u1f09 GREEK CAPITAL LETTER ALPHA WITH DASIA\n\u1f09 = with-breathing \u03b1\u2032 upper (rough \u03b1-rough)\n\u1f0a : Token -- U+1F0A  \u1f0a GREEK CAPITAL LETTER ALPHA WITH PSILI AND VARIA\n\u1f0a = with-accent-breathing \u03b1\u2032 upper (grave \u03b1-vowel) (smooth \u0391-smooth)\n\u1f0b : Token -- U+1F0B  \u1f0b GREEK CAPITAL LETTER ALPHA WITH DASIA AND VARIA\n\u1f0b = with-accent-breathing \u03b1\u2032 upper (grave \u03b1-vowel) (rough \u03b1-rough)\n\u1f0c : Token -- U+1F0C  \u1f0c GREEK CAPITAL LETTER ALPHA WITH PSILI AND OXIA\n\u1f0c = with-accent-breathing \u03b1\u2032 upper (acute \u03b1-vowel) (smooth \u0391-smooth)\n\u1f0d : Token -- U+1F0D  \u1f0d GREEK CAPITAL LETTER ALPHA WITH DASIA AND OXIA\n\u1f0d = with-accent-breathing \u03b1\u2032 upper (acute \u03b1-vowel) (rough \u03b1-rough)\n\u1f0e : Token -- U+1F0E  \u1f0e GREEK CAPITAL LETTER ALPHA WITH PSILI AND PERISPOMENI\n\u1f0e = with-accent-breathing \u03b1\u2032 upper (circumflex \u03b1-long-vowel) (smooth \u0391-smooth)\n\u1f0f : Token -- U+1F0F  \u1f0f GREEK CAPITAL LETTER ALPHA WITH DASIA AND PERISPOMENI\n\u1f0f = with-accent-breathing \u03b1\u2032 upper (circumflex \u03b1-long-vowel) (rough \u03b1-rough)\n\u1f10 : Token -- U+1F10  \u1f10 GREEK SMALL LETTER EPSILON WITH PSILI\n\u1f10 = with-breathing \u03b5\u2032 lower (smooth \u03b5-smooth)\n\u1f11 : Token -- U+1F11  \u1f11 GREEK SMALL LETTER EPSILON WITH DASIA\n\u1f11 = with-breathing \u03b5\u2032 lower (rough \u03b5-rough)\n\u1f12 : Token -- U+1F12  \u1f12 GREEK SMALL LETTER EPSILON WITH PSILI AND VARIA\n\u1f12 = with-accent-breathing \u03b5\u2032 lower (grave \u03b5-vowel) (smooth \u03b5-smooth)\n\u1f13 : Token -- U+1F13  \u1f13 GREEK SMALL LETTER EPSILON WITH DASIA AND VARIA\n\u1f13 = with-accent-breathing \u03b5\u2032 lower (grave \u03b5-vowel) (rough \u03b5-rough)\n\u1f14 : Token -- U+1F14  \u1f14 GREEK SMALL LETTER EPSILON WITH PSILI AND OXIA\n\u1f14 = with-accent-breathing \u03b5\u2032 lower (acute \u03b5-vowel) (smooth \u03b5-smooth)\n\u1f15 : Token -- U+1F15  \u1f15 GREEK SMALL LETTER EPSILON WITH DASIA AND OXIA\n\u1f15 = with-accent-breathing \u03b5\u2032 lower (acute \u03b5-vowel) (rough \u03b5-rough)\n-- U+1F16\n-- U+1F17\n\u1f18 : Token -- U+1F18  \u1f18 GREEK CAPITAL LETTER EPSILON WITH PSILI\n\u1f18 = with-breathing \u03b5\u2032 upper (smooth \u0395-smooth)\n\u1f19 : Token -- U+1F19  \u1f19 GREEK CAPITAL LETTER EPSILON WITH DASIA\n\u1f19 = with-breathing \u03b5\u2032 upper (rough \u03b5-rough)\n\u1f1a : Token -- U+1F1A  \u1f1a GREEK CAPITAL LETTER EPSILON WITH PSILI AND VARIA\n\u1f1a = with-accent-breathing \u03b5\u2032 upper (grave \u03b5-vowel) (smooth \u0395-smooth)\n\u1f1b : Token -- U+1F1B  \u1f1b GREEK CAPITAL LETTER EPSILON WITH DASIA AND VARIA\n\u1f1b = with-accent-breathing \u03b5\u2032 upper (grave \u03b5-vowel) (rough \u03b5-rough)\n\u1f1c : Token -- U+1F1C  \u1f1c GREEK CAPITAL LETTER EPSILON WITH PSILI AND OXIA\n\u1f1c = with-accent-breathing \u03b5\u2032 upper (acute \u03b5-vowel) (smooth \u0395-smooth)\n\u1f1d : Token -- U+1F1D  \u1f1d GREEK CAPITAL LETTER EPSILON WITH DASIA AND OXIA\n\u1f1d = with-accent-breathing \u03b5\u2032 upper (acute \u03b5-vowel) (rough \u03b5-rough)\n-- U+1F1E\n-- U+1F1F\n\u1f20 : Token -- U+1F20  \u1f20 GREEK SMALL LETTER ETA WITH PSILI\n\u1f20 = with-breathing \u03b7\u2032 lower (smooth \u03b7-smooth)\n\u1f21 : Token -- U+1F21  \u1f21 GREEK SMALL LETTER ETA WITH DASIA\n\u1f21 = with-breathing \u03b7\u2032 lower (rough \u03b7-rough)\n\u1f22 : Token -- U+1F22  \u1f22 GREEK SMALL LETTER ETA WITH PSILI AND VARIA\n\u1f22 = with-accent-breathing \u03b7\u2032 lower (grave \u03b7-vowel) (smooth \u03b7-smooth)\n\u1f23 : Token -- U+1F23  \u1f23 GREEK SMALL LETTER ETA WITH DASIA AND VARIA\n\u1f23 = with-accent-breathing \u03b7\u2032 lower (grave \u03b7-vowel) (rough \u03b7-rough)\n\u1f24 : Token -- U+1F24  \u1f24 GREEK SMALL LETTER ETA WITH PSILI AND OXIA\n\u1f24 = with-accent-breathing \u03b7\u2032 lower (acute \u03b7-vowel) (smooth \u03b7-smooth)\n\u1f25 : Token -- U+1F25  \u1f25 GREEK SMALL LETTER ETA WITH DASIA AND OXIA\n\u1f25 = with-accent-breathing \u03b7\u2032 lower (acute \u03b7-vowel) (rough \u03b7-rough)\n\u1f26 : Token -- U+1F26  \u1f26 GREEK SMALL LETTER ETA WITH PSILI AND PERISPOMENI\n\u1f26 = with-accent-breathing \u03b7\u2032 lower (circumflex \u03b7-long-vowel) (smooth \u03b7-smooth)\n\u1f27 : Token -- U+1F27  \u1f27 GREEK SMALL LETTER ETA WITH DASIA AND PERISPOMENI\n\u1f27 = with-accent-breathing \u03b7\u2032 lower (circumflex \u03b7-long-vowel) (rough \u03b7-rough)\n\u1f28 : Token -- U+1F28  \u1f28 GREEK CAPITAL LETTER ETA WITH PSILI\n\u1f28 = with-breathing \u03b7\u2032 upper (smooth \u0397-smooth)\n\u1f29 : Token -- U+1F29  \u1f29 GREEK CAPITAL LETTER ETA WITH DASIA\n\u1f29 = with-breathing \u03b7\u2032 upper (rough \u03b7-rough)\n\u1f2a : Token -- U+1F2A  \u1f2a GREEK CAPITAL LETTER ETA WITH PSILI AND VARIA\n\u1f2a = with-accent-breathing \u03b7\u2032 upper (grave \u03b7-vowel) (smooth \u0397-smooth)\n\u1f2b : Token -- U+1F2B  \u1f2b GREEK CAPITAL LETTER ETA WITH DASIA AND VARIA\n\u1f2b = with-accent-breathing \u03b7\u2032 upper (grave \u03b7-vowel) (rough \u03b7-rough)\n\u1f2c : Token -- U+1F2C  \u1f2c GREEK CAPITAL LETTER ETA WITH PSILI AND OXIA\n\u1f2c = with-accent-breathing \u03b7\u2032 upper (acute \u03b7-vowel) (smooth \u0397-smooth)\n\u1f2d : Token -- U+1F2D  \u1f2d GREEK CAPITAL LETTER ETA WITH DASIA AND OXIA\n\u1f2d = with-accent-breathing \u03b7\u2032 upper (acute \u03b7-vowel) (rough \u03b7-rough)\n\u1f2e : Token -- U+1F2E  \u1f2e GREEK CAPITAL LETTER ETA WITH PSILI AND PERISPOMENI\n\u1f2e = with-accent-breathing \u03b7\u2032 upper (circumflex \u03b7-long-vowel) (smooth \u0397-smooth)\n\u1f2f : Token -- U+1F2F  \u1f2f GREEK CAPITAL LETTER ETA WITH DASIA AND PERISPOMENI\n\u1f2f = with-accent-breathing \u03b7\u2032 upper (circumflex \u03b7-long-vowel) (rough \u03b7-rough)\n\u1f30 : Token -- U+1F30  \u1f30 GREEK SMALL LETTER IOTA WITH PSILI\n\u1f30 = with-breathing \u03b9\u2032 lower (smooth \u03b9-smooth)\n\u1f31 : Token -- U+1F31  \u1f31 GREEK SMALL LETTER IOTA WITH DASIA\n\u1f31 = with-breathing \u03b9\u2032 lower (rough \u03b9-rough)\n\u1f32 : Token -- U+1F32  \u1f32 GREEK SMALL LETTER IOTA WITH PSILI AND VARIA\n\u1f32 = with-accent-breathing \u03b9\u2032 lower (grave \u03b9-vowel) (smooth \u03b9-smooth)\n\u1f33 : Token -- U+1F33  \u1f33 GREEK SMALL LETTER IOTA WITH DASIA AND VARIA\n\u1f33 = with-accent-breathing \u03b9\u2032 lower (grave \u03b9-vowel) (rough \u03b9-rough)\n\u1f34 : Token -- U+1F34  \u1f34 GREEK SMALL LETTER IOTA WITH PSILI AND OXIA\n\u1f34 = with-accent-breathing \u03b9\u2032 lower (acute \u03b9-vowel) (smooth \u03b9-smooth)\n\u1f35 : Token -- U+1F35  \u1f35 GREEK SMALL LETTER IOTA WITH DASIA AND OXIA\n\u1f35 = with-accent-breathing \u03b9\u2032 lower (acute \u03b9-vowel) (rough \u03b9-rough)\n\u1f36 : Token -- U+1F36  \u1f36 GREEK SMALL LETTER IOTA WITH PSILI AND PERISPOMENI\n\u1f36 = with-accent-breathing \u03b9\u2032 lower (circumflex \u03b9-long-vowel) (smooth \u03b9-smooth)\n\u1f37 : Token -- U+1F37  \u1f37 GREEK SMALL LETTER IOTA WITH DASIA AND PERISPOMENI\n\u1f37 = with-accent-breathing \u03b9\u2032 lower (circumflex \u03b9-long-vowel) (rough \u03b9-rough)\n\u1f38 : Token -- U+1F38  \u1f38 GREEK CAPITAL LETTER IOTA WITH PSILI\n\u1f38 = with-breathing \u03b9\u2032 upper (smooth \u0399-smooth)\n\u1f39 : Token -- U+1F39  \u1f39 GREEK CAPITAL LETTER IOTA WITH DASIA\n\u1f39 = with-breathing \u03b9\u2032 upper (rough \u03b9-rough)\n\u1f3a : Token -- U+1F3A  \u1f3a GREEK CAPITAL LETTER IOTA WITH PSILI AND VARIA\n\u1f3a = with-accent-breathing \u03b9\u2032 upper (grave \u03b9-vowel) (smooth \u0399-smooth)\n\u1f3b : Token -- U+1F3B  \u1f3b GREEK CAPITAL LETTER IOTA WITH DASIA AND VARIA\n\u1f3b = with-accent-breathing \u03b9\u2032 upper (grave \u03b9-vowel) (rough \u03b9-rough)\n\u1f3c : Token -- U+1F3C  \u1f3c GREEK CAPITAL LETTER IOTA WITH PSILI AND OXIA\n\u1f3c = with-accent-breathing \u03b9\u2032 upper (acute \u03b9-vowel) (smooth \u0399-smooth)\n\u1f3d : Token -- U+1F3D  \u1f3d GREEK CAPITAL LETTER IOTA WITH DASIA AND OXIA\n\u1f3d = with-accent-breathing \u03b9\u2032 upper (acute \u03b9-vowel) (rough \u03b9-rough)\n\u1f3e : Token -- U+1F3E  \u1f3e GREEK CAPITAL LETTER IOTA WITH PSILI AND PERISPOMENI\n\u1f3e = with-accent-breathing \u03b9\u2032 upper (circumflex \u03b9-long-vowel) (smooth \u0399-smooth)\n\u1f3f : Token -- U+1F3F  \u1f3f GREEK CAPITAL LETTER IOTA WITH DASIA AND PERISPOMENI\n\u1f3f = with-accent-breathing \u03b9\u2032 upper (circumflex \u03b9-long-vowel) (rough \u03b9-rough)\n\u1f40 : Token -- U+1F40  \u1f40 GREEK SMALL LETTER OMICRON WITH PSILI\n\u1f40 = with-breathing \u03bf\u2032 lower (smooth \u03bf-smooth)\n\u1f41 : Token -- U+1F41  \u1f41 GREEK SMALL LETTER OMICRON WITH DASIA\n\u1f41 = with-breathing \u03bf\u2032 lower (rough \u03bf-rough)\n\u1f42 : Token -- U+1F42  \u1f42 GREEK SMALL LETTER OMICRON WITH PSILI AND VARIA\n\u1f42 = with-accent-breathing \u03bf\u2032 lower (grave \u03bf-vowel) (smooth \u03bf-smooth)\n\u1f43 : Token -- U+1F43  \u1f43 GREEK SMALL LETTER OMICRON WITH DASIA AND VARIA\n\u1f43 = with-accent-breathing \u03bf\u2032 lower (grave \u03bf-vowel) (rough \u03bf-rough)\n\u1f44 : Token -- U+1F44  \u1f44 GREEK SMALL LETTER OMICRON WITH PSILI AND OXIA\n\u1f44 = with-accent-breathing \u03bf\u2032 lower (acute \u03bf-vowel) (smooth \u03bf-smooth)\n\u1f45 : Token -- U+1F45  \u1f45 GREEK SMALL LETTER OMICRON WITH DASIA AND OXIA\n\u1f45 = with-accent-breathing \u03bf\u2032 lower (acute \u03bf-vowel) (rough \u03bf-rough)\n-- U+1F46\n-- U+1F47\n\u1f48 : Token -- U+1F48  \u1f48 GREEK CAPITAL LETTER OMICRON WITH PSILI\n\u1f48 = with-breathing \u03bf\u2032 upper (smooth \u039f-smooth)\n\u1f49 : Token -- U+1F49  \u1f49 GREEK CAPITAL LETTER OMICRON WITH DASIA\n\u1f49 = with-breathing \u03bf\u2032 upper (rough \u03bf-rough)\n\u1f4a : Token -- U+1F4A  \u1f4a GREEK CAPITAL LETTER OMICRON WITH PSILI AND VARIA\n\u1f4a = with-accent-breathing \u03bf\u2032 upper (grave \u03bf-vowel) (smooth \u039f-smooth)\n\u1f4b : Token -- U+1F4B  \u1f4b GREEK CAPITAL LETTER OMICRON WITH DASIA AND VARIA\n\u1f4b = with-accent-breathing \u03bf\u2032 upper (grave \u03bf-vowel) (rough \u03bf-rough)\n\u1f4c : Token -- U+1F4C  \u1f4c GREEK CAPITAL LETTER OMICRON WITH PSILI AND OXIA\n\u1f4c = with-accent-breathing \u03bf\u2032 upper (acute \u03bf-vowel) (smooth \u039f-smooth)\n\u1f4d : Token -- U+1F4D  \u1f4d GREEK CAPITAL LETTER OMICRON WITH DASIA AND OXIA\n\u1f4d = with-accent-breathing \u03bf\u2032 upper (acute \u03bf-vowel) (rough \u03bf-rough)\n-- U+1F4E\n-- U+1F4F\n\u1f50 : Token -- U+1F50  \u1f50 GREEK SMALL LETTER UPSILON WITH PSILI\n\u1f50 = with-breathing \u03c5\u2032 lower (smooth \u03c5-smooth)\n\u1f51 : Token -- U+1F51  \u1f51 GREEK SMALL LETTER UPSILON WITH DASIA\n\u1f51 = with-breathing \u03c5\u2032 lower (rough \u03c5-rough)\n\u1f52 : Token -- U+1F52  \u1f52 GREEK SMALL LETTER UPSILON WITH PSILI AND VARIA\n\u1f52 = with-accent-breathing \u03c5\u2032 lower (grave \u03c5-vowel) (smooth \u03c5-smooth)\n\u1f53 : Token -- U+1F53  \u1f53 GREEK SMALL LETTER UPSILON WITH DASIA AND VARIA\n\u1f53 = with-accent-breathing \u03c5\u2032 lower (grave \u03c5-vowel) (rough \u03c5-rough)\n\u1f54 : Token -- U+1F54  \u1f54 GREEK SMALL LETTER UPSILON WITH PSILI AND OXIA\n\u1f54 = with-accent-breathing \u03c5\u2032 lower (acute \u03c5-vowel) (smooth \u03c5-smooth)\n\u1f55 : Token -- U+1F55  \u1f55 GREEK SMALL LETTER UPSILON WITH DASIA AND OXIA\n\u1f55 = with-accent-breathing \u03c5\u2032 lower (acute \u03c5-vowel) (rough \u03c5-rough)\n\u1f56 : Token -- U+1F56  \u1f56 GREEK SMALL LETTER UPSILON WITH PSILI AND PERISPOMENI\n\u1f56 = with-accent-breathing \u03c5\u2032 lower (circumflex \u03c5-long-vowel) (smooth \u03c5-smooth)\n\u1f57 : Token -- U+1F57  \u1f57 GREEK SMALL LETTER UPSILON WITH DASIA AND PERISPOMENI\n\u1f57 = with-accent-breathing \u03c5\u2032 lower (circumflex \u03c5-long-vowel) (rough \u03c5-rough)\n-- U+1F58\n\u1f59 : Token -- U+1F59  \u1f59 GREEK CAPITAL LETTER UPSILON WITH DASIA\n\u1f59 = with-breathing \u03c5\u2032 upper (rough \u03c5-rough)\n-- U+1F5A\n\u1f5b : Token -- U+1F5B  \u1f5b GREEK CAPITAL LETTER UPSILON WITH DASIA AND VARIA\n\u1f5b = with-accent-breathing \u03c5\u2032 upper (grave \u03c5-vowel) (rough \u03c5-rough)\n-- U+1F5C\n\u1f5d : Token -- U+1F5D  \u1f5d GREEK CAPITAL LETTER UPSILON WITH DASIA AND OXIA\n\u1f5d = with-accent-breathing \u03c5\u2032 upper (acute \u03c5-vowel) (rough \u03c5-rough)\n-- U+1F5E\n\u1f5f : Token -- U+1F5F  \u1f5f GREEK CAPITAL LETTER UPSILON WITH DASIA AND PERISPOMENI\n\u1f5f = with-accent-breathing \u03c5\u2032 upper (circumflex \u03c5-long-vowel) (rough \u03c5-rough)\n\u1f60 : Token -- U+1F60  \u1f60 GREEK SMALL LETTER OMEGA WITH PSILI\n\u1f60 = with-breathing \u03c9\u2032 lower (smooth \u03c9-smooth)\n\u1f61 : Token -- U+1F61  \u1f61 GREEK SMALL LETTER OMEGA WITH DASIA\n\u1f61 = with-breathing \u03c9\u2032 lower (rough \u03c9-rough)\n\u1f62 : Token -- U+1F62  \u1f62 GREEK SMALL LETTER OMEGA WITH PSILI AND VARIA\n\u1f62 = with-accent-breathing \u03c9\u2032 lower (grave \u03c9-vowel) (smooth \u03c9-smooth)\n\u1f63 : Token -- U+1F63  \u1f63 GREEK SMALL LETTER OMEGA WITH DASIA AND VARIA\n\u1f63 = with-accent-breathing \u03c9\u2032 lower (grave \u03c9-vowel) (rough \u03c9-rough)\n\u1f64 : Token -- U+1F64  \u1f64 GREEK SMALL LETTER OMEGA WITH PSILI AND OXIA\n\u1f64 = with-accent-breathing \u03c9\u2032 lower (acute \u03c9-vowel) (smooth \u03c9-smooth)\n\u1f65 : Token -- U+1F65  \u1f65 GREEK SMALL LETTER OMEGA WITH DASIA AND OXIA\n\u1f65 = with-accent-breathing \u03c9\u2032 lower (acute \u03c9-vowel) (rough \u03c9-rough)\n\u1f66 : Token -- U+1F66  \u1f66 GREEK SMALL LETTER OMEGA WITH PSILI AND PERISPOMENI\n\u1f66 = with-accent-breathing \u03c9\u2032 lower (circumflex \u03c9-long-vowel) (smooth \u03c9-smooth)\n\u1f67 : Token -- U+1F67  \u1f67 GREEK SMALL LETTER OMEGA WITH DASIA AND PERISPOMENI\n\u1f67 = with-accent-breathing \u03c9\u2032 lower (circumflex \u03c9-long-vowel) (rough \u03c9-rough)\n\u1f68 : Token -- U+1F68  \u1f68 GREEK CAPITAL LETTER OMEGA WITH PSILI\n\u1f68 = with-breathing \u03c9\u2032 upper (smooth \u03a9-smooth)\n\u1f69 : Token -- U+1F69  \u1f69 GREEK CAPITAL LETTER OMEGA WITH DASIA\n\u1f69 = with-breathing \u03c9\u2032 upper (rough \u03c9-rough)\n\u1f6a : Token -- U+1F6A  \u1f6a GREEK CAPITAL LETTER OMEGA WITH PSILI AND VARIA\n\u1f6a = with-accent-breathing \u03c9\u2032 upper (grave \u03c9-vowel) (smooth \u03a9-smooth)\n\u1f6b : Token -- U+1F6B  \u1f6b GREEK CAPITAL LETTER OMEGA WITH DASIA AND VARIA\n\u1f6b = with-accent-breathing \u03c9\u2032 upper (grave \u03c9-vowel) (rough \u03c9-rough)\n\u1f6c : Token -- U+1F6C  \u1f6c GREEK CAPITAL LETTER OMEGA WITH PSILI AND OXIA\n\u1f6c = with-accent-breathing \u03c9\u2032 upper (acute \u03c9-vowel) (smooth \u03a9-smooth)\n\u1f6d : Token -- U+1F6D  \u1f6d GREEK CAPITAL LETTER OMEGA WITH DASIA AND OXIA\n\u1f6d = with-accent-breathing \u03c9\u2032 upper (acute \u03c9-vowel) (rough \u03c9-rough)\n\u1f6e : Token -- U+1F6E  \u1f6e GREEK CAPITAL LETTER OMEGA WITH PSILI AND PERISPOMENI\n\u1f6e = with-accent-breathing \u03c9\u2032 upper (circumflex \u03c9-long-vowel) (smooth \u03a9-smooth)\n\u1f6f : Token -- U+1F6F  \u1f6f GREEK CAPITAL LETTER OMEGA WITH DASIA AND PERISPOMENI\n\u1f6f = with-accent-breathing \u03c9\u2032 upper (circumflex \u03c9-long-vowel) (rough \u03c9-rough)\n\u1f70 : Token -- U+1F70  \u1f70 GREEK SMALL LETTER ALPHA WITH VARIA\n\u1f70 = with-accent \u03b1\u2032 lower (grave \u03b1-vowel)\n\u1f71 : Token -- U+1F71  \u1f71 GREEK SMALL LETTER ALPHA WITH OXIA\n\u1f71 = with-accent \u03b1\u2032 lower (acute \u03b1-vowel)\n\u1f72 : Token -- U+1F72  \u1f72 GREEK SMALL LETTER EPSILON WITH VARIA\n\u1f72 = with-accent \u03b5\u2032 lower (grave \u03b5-vowel)\n\u1f73 : Token -- U+1F73  \u1f73 GREEK SMALL LETTER EPSILON WITH OXIA\n\u1f73 = with-accent \u03b5\u2032 lower (acute \u03b5-vowel)\n\u1f74 : Token -- U+1F74  \u1f74 GREEK SMALL LETTER ETA WITH VARIA\n\u1f74 = with-accent \u03b7\u2032 lower (grave \u03b7-vowel)\n\u1f75 : Token -- U+1F75  \u1f75 GREEK SMALL LETTER ETA WITH OXIA\n\u1f75 = with-accent \u03b7\u2032 lower (acute \u03b7-vowel)\n\u1f76 : Token -- U+1F76  \u1f76 GREEK SMALL LETTER IOTA WITH VARIA\n\u1f76 = with-accent \u03b9\u2032 lower (grave \u03b9-vowel)\n\u1f77 : Token -- U+1F77  \u1f77 GREEK SMALL LETTER IOTA WITH OXIA\n\u1f77 = with-accent \u03b9\u2032 lower (acute \u03b9-vowel)\n\u1f78 : Token -- U+1F78  \u1f78 GREEK SMALL LETTER OMICRON WITH VARIA\n\u1f78 = with-accent \u03bf\u2032 lower (grave \u03bf-vowel)\n\u1f79 : Token -- U+1F79  \u1f79 GREEK SMALL LETTER OMICRON WITH OXIA\n\u1f79 = with-accent \u03bf\u2032 lower (acute \u03bf-vowel)\n\u1f7a : Token -- U+1F7A  \u1f7a GREEK SMALL LETTER UPSILON WITH VARIA\n\u1f7a = with-accent \u03c5\u2032 lower (grave \u03c5-vowel)\n\u1f7b : Token -- U+1F7B  \u1f7b GREEK SMALL LETTER UPSILON WITH OXIA\n\u1f7b = with-accent \u03c5\u2032 lower (acute \u03c5-vowel)\n\u1f7c : Token -- U+1F7C  \u1f7c GREEK SMALL LETTER OMEGA WITH VARIA\n\u1f7c = with-accent \u03c9\u2032 lower (grave \u03c9-vowel)\n\u1f7d : Token -- U+1F7D  \u1f7d GREEK SMALL LETTER OMEGA WITH OXIA\n\u1f7d = with-accent \u03c9\u2032 lower (acute \u03c9-vowel)\n-- U+1F7E\n-- U+1F7F\n\u1f80 : Token -- U+1F80  \u1f80 GREEK SMALL LETTER ALPHA WITH PSILI AND YPOGEGRAMMENI\n\u1f80 = with-breathing-iota \u03b1\u2032 lower (smooth \u03b1-smooth) \u03b1-iota-subscript\n\u1f81 : Token -- U+1F81  \u1f81 GREEK SMALL LETTER ALPHA WITH DASIA AND YPOGEGRAMMENI\n\u1f81 = with-breathing-iota \u03b1\u2032 lower (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f82 : Token -- U+1F82  \u1f82 GREEK SMALL LETTER ALPHA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1f82 = with-accent-breathing-iota \u03b1\u2032 lower (grave \u03b1-vowel) (smooth \u03b1-smooth) \u03b1-iota-subscript\n\u1f83 : Token -- U+1F83  \u1f83 GREEK SMALL LETTER ALPHA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1f83 = with-accent-breathing-iota \u03b1\u2032 lower (grave \u03b1-vowel) (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f84 : Token -- U+1F84  \u1f84 GREEK SMALL LETTER ALPHA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1f84 = with-accent-breathing-iota \u03b1\u2032 lower (acute \u03b1-vowel) (smooth \u03b1-smooth) \u03b1-iota-subscript\n\u1f85 : Token -- U+1F85  \u1f85 GREEK SMALL LETTER ALPHA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1f85 = with-accent-breathing-iota \u03b1\u2032 lower (acute \u03b1-vowel) (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f86 : Token -- U+1F86  \u1f86 GREEK SMALL LETTER ALPHA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f86 = with-accent-breathing-iota \u03b1\u2032 lower (circumflex \u03b1-long-vowel) (smooth \u03b1-smooth) \u03b1-iota-subscript\n\u1f87 : Token -- U+1F87  \u1f87 GREEK SMALL LETTER ALPHA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f87 = with-accent-breathing-iota \u03b1\u2032 lower (circumflex \u03b1-long-vowel) (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f88 : Token -- U+1F88  \u1f88 GREEK CAPITAL LETTER ALPHA WITH PSILI AND PROSGEGRAMMENI\n\u1f88 = with-breathing-iota \u03b1\u2032 upper (smooth \u0391-smooth) \u03b1-iota-subscript\n\u1f89 : Token -- U+1F89  \u1f89 GREEK CAPITAL LETTER ALPHA WITH DASIA AND PROSGEGRAMMENI\n\u1f89 = with-breathing-iota \u03b1\u2032 upper (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f8a : Token -- U+1F8A  \u1f8a GREEK CAPITAL LETTER ALPHA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1f8a = with-accent-breathing-iota \u03b1\u2032 upper (grave \u03b1-vowel) (smooth \u0391-smooth) \u03b1-iota-subscript\n\u1f8b : Token -- U+1F8B  \u1f8b GREEK CAPITAL LETTER ALPHA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1f8b = with-accent-breathing-iota \u03b1\u2032 upper (grave \u03b1-vowel) (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f8c : Token -- U+1F8C  \u1f8c GREEK CAPITAL LETTER ALPHA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1f8c = with-accent-breathing-iota \u03b1\u2032 upper (acute \u03b1-vowel) (smooth \u0391-smooth) \u03b1-iota-subscript\n\u1f8d : Token -- U+1F8D  \u1f8d GREEK CAPITAL LETTER ALPHA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1f8d = with-accent-breathing-iota \u03b1\u2032 upper (acute \u03b1-vowel) (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f8e : Token -- U+1F8E  \u1f8e GREEK CAPITAL LETTER ALPHA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f8e = with-accent-breathing-iota \u03b1\u2032 upper (circumflex \u03b1-long-vowel) (smooth \u0391-smooth) \u03b1-iota-subscript\n\u1f8f : Token -- U+1F8F  \u1f8f GREEK CAPITAL LETTER ALPHA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f8f = with-accent-breathing-iota \u03b1\u2032 upper (circumflex \u03b1-long-vowel) (rough \u03b1-rough) \u03b1-iota-subscript\n\u1f90 : Token -- U+1F90  \u1f90 GREEK SMALL LETTER ETA WITH PSILI AND YPOGEGRAMMENI\n\u1f90 = with-breathing-iota \u03b7\u2032 lower (smooth \u03b7-smooth) \u03b7-iota-subscript\n\u1f91 : Token -- U+1F91  \u1f91 GREEK SMALL LETTER ETA WITH DASIA AND YPOGEGRAMMENI\n\u1f91 = with-breathing-iota \u03b7\u2032 lower (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f92 : Token -- U+1F92  \u1f92 GREEK SMALL LETTER ETA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1f92 = with-accent-breathing-iota \u03b7\u2032 lower (grave \u03b7-vowel) (smooth \u03b7-smooth) \u03b7-iota-subscript\n\u1f93 : Token -- U+1F93  \u1f93 GREEK SMALL LETTER ETA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1f93 = with-accent-breathing-iota \u03b7\u2032 lower (grave \u03b7-vowel) (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f94 : Token -- U+1F94  \u1f94 GREEK SMALL LETTER ETA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1f94 = with-accent-breathing-iota \u03b7\u2032 lower (acute \u03b7-vowel) (smooth \u03b7-smooth) \u03b7-iota-subscript\n\u1f95 : Token -- U+1F95  \u1f95 GREEK SMALL LETTER ETA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1f95 = with-accent-breathing-iota \u03b7\u2032 lower (acute \u03b7-vowel) (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f96 : Token -- U+1F96  \u1f96 GREEK SMALL LETTER ETA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f96 = with-accent-breathing-iota \u03b7\u2032 lower (circumflex \u03b7-long-vowel) (smooth \u03b7-smooth) \u03b7-iota-subscript\n\u1f97 : Token -- U+1F97  \u1f97 GREEK SMALL LETTER ETA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f97 = with-accent-breathing-iota \u03b7\u2032 lower (circumflex \u03b7-long-vowel) (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f98 : Token -- U+1F98  \u1f98 GREEK CAPITAL LETTER ETA WITH PSILI AND PROSGEGRAMMENI\n\u1f98 = with-breathing-iota \u03b7\u2032 upper (smooth \u0397-smooth) \u03b7-iota-subscript\n\u1f99 : Token -- U+1F99  \u1f99 GREEK CAPITAL LETTER ETA WITH DASIA AND PROSGEGRAMMENI\n\u1f99 = with-breathing-iota \u03b7\u2032 upper (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f9a : Token -- U+1F9A  \u1f9a GREEK CAPITAL LETTER ETA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1f9a = with-accent-breathing-iota \u03b7\u2032 upper (grave \u03b7-vowel) (smooth \u0397-smooth) \u03b7-iota-subscript\n\u1f9b : Token -- U+1F9B  \u1f9b GREEK CAPITAL LETTER ETA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1f9b = with-accent-breathing-iota \u03b7\u2032 upper (grave \u03b7-vowel) (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f9c : Token -- U+1F9C  \u1f9c GREEK CAPITAL LETTER ETA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1f9c = with-accent-breathing-iota \u03b7\u2032 upper (acute \u03b7-vowel) (smooth \u0397-smooth) \u03b7-iota-subscript\n\u1f9d : Token -- U+1F9D  \u1f9d GREEK CAPITAL LETTER ETA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1f9d = with-accent-breathing-iota \u03b7\u2032 upper (acute \u03b7-vowel) (rough \u03b7-rough) \u03b7-iota-subscript\n\u1f9e : Token -- U+1F9E  \u1f9e GREEK CAPITAL LETTER ETA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f9e = with-accent-breathing-iota \u03b7\u2032 upper (circumflex \u03b7-long-vowel) (smooth \u0397-smooth) \u03b7-iota-subscript\n\u1f9f : Token -- U+1F9F  \u1f9f GREEK CAPITAL LETTER ETA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f9f = with-accent-breathing-iota \u03b7\u2032 upper (circumflex \u03b7-long-vowel) (rough \u03b7-rough) \u03b7-iota-subscript\n\u1fa0 : Token -- U+1FA0  \u1fa0 GREEK SMALL LETTER OMEGA WITH PSILI AND YPOGEGRAMMENI\n\u1fa0 = with-breathing-iota \u03c9\u2032 lower (smooth \u03c9-smooth) \u03c9-iota-subscript\n\u1fa1 : Token -- U+1FA1  \u1fa1 GREEK SMALL LETTER OMEGA WITH DASIA AND YPOGEGRAMMENI\n\u1fa1 = with-breathing-iota \u03c9\u2032 lower (rough \u03c9-rough) \u03c9-iota-subscript\n\u1fa2 : Token -- U+1FA2  \u1fa2 GREEK SMALL LETTER OMEGA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1fa2 = with-accent-breathing-iota \u03c9\u2032 lower (grave \u03c9-vowel) (smooth \u03c9-smooth) \u03c9-iota-subscript\n\u1fa3 : Token -- U+1FA3  \u1fa3 GREEK SMALL LETTER OMEGA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1fa3 = with-accent-breathing-iota \u03c9\u2032 lower (grave \u03c9-vowel) (rough \u03c9-rough) \u03c9-iota-subscript\n\u1fa4 : Token -- U+1FA4  \u1fa4 GREEK SMALL LETTER OMEGA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1fa4 = with-accent-breathing-iota \u03c9\u2032 lower (acute \u03c9-vowel) (smooth \u03c9-smooth) \u03c9-iota-subscript\n\u1fa5 : Token -- U+1FA5  \u1fa5 GREEK SMALL LETTER OMEGA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1fa5 = with-accent-breathing-iota \u03c9\u2032 lower (acute \u03c9-vowel) (rough \u03c9-rough) \u03c9-iota-subscript\n\u1fa6 : Token -- U+1FA6  \u1fa6 GREEK SMALL LETTER OMEGA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1fa6 = with-accent-breathing-iota \u03c9\u2032 lower (circumflex \u03c9-long-vowel) (smooth \u03c9-smooth) \u03c9-iota-subscript\n\u1fa7 : Token -- U+1FA7  \u1fa7 GREEK SMALL LETTER OMEGA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1fa7 = with-accent-breathing-iota \u03c9\u2032 lower (circumflex \u03c9-long-vowel) (rough \u03c9-rough) \u03c9-iota-subscript\n\u1fa8 : Token -- U+1FA8  \u1fa8 GREEK CAPITAL LETTER OMEGA WITH PSILI AND PROSGEGRAMMENI\n\u1fa8 = with-breathing-iota \u03c9\u2032 upper (smooth \u03a9-smooth) \u03c9-iota-subscript\n\u1fa9 : Token -- U+1FA9  \u1fa9 GREEK CAPITAL LETTER OMEGA WITH DASIA AND PROSGEGRAMMENI\n\u1fa9 = with-breathing-iota \u03c9\u2032 upper (rough \u03c9-rough) \u03c9-iota-subscript\n\u1faa : Token -- U+1FAA  \u1faa GREEK CAPITAL LETTER OMEGA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1faa = with-accent-breathing-iota \u03c9\u2032 upper (grave \u03c9-vowel) (smooth \u03a9-smooth) \u03c9-iota-subscript\n\u1fab : Token -- U+1FAB  \u1fab GREEK CAPITAL LETTER OMEGA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1fab = with-accent-breathing-iota \u03c9\u2032 upper (grave \u03c9-vowel) (rough \u03c9-rough) \u03c9-iota-subscript\n\u1fac : Token -- U+1FAC  \u1fac GREEK CAPITAL LETTER OMEGA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1fac = with-accent-breathing-iota \u03c9\u2032 upper (acute \u03c9-vowel) (smooth \u03a9-smooth) \u03c9-iota-subscript\n\u1fad : Token -- U+1FAD  \u1fad GREEK CAPITAL LETTER OMEGA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1fad = with-accent-breathing-iota \u03c9\u2032 upper (acute \u03c9-vowel) (rough \u03c9-rough) \u03c9-iota-subscript\n\u1fae : Token -- U+1FAE  \u1fae GREEK CAPITAL LETTER OMEGA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1fae = with-accent-breathing-iota \u03c9\u2032 upper (circumflex \u03c9-long-vowel) (smooth \u03a9-smooth) \u03c9-iota-subscript\n\u1faf : Token -- U+1FAF  \u1faf GREEK CAPITAL LETTER OMEGA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1faf = with-accent-breathing-iota \u03c9\u2032 upper (circumflex \u03c9-long-vowel) (rough \u03c9-rough) \u03c9-iota-subscript\n-- U+1FB0 \u1fb0 GREEK SMALL LETTER ALPHA WITH VRACHY\n-- U+1FB1 \u1fb1 GREEK SMALL LETTER ALPHA WITH MACRON\n\u1fb2 : Token -- U+1FB2  \u1fb2 GREEK SMALL LETTER ALPHA WITH VARIA AND YPOGEGRAMMENI\n\u1fb2 = with-accent-iota \u03b1\u2032 lower (grave \u03b1-vowel) \u03b1-iota-subscript\n\u1fb3 : Token -- U+1FB3  \u1fb3 GREEK SMALL LETTER ALPHA WITH YPOGEGRAMMENI\n\u1fb3 = with-iota \u03b1\u2032 lower \u03b1-iota-subscript\n\u1fb4 : Token -- U+1FB4  \u1fb4 GREEK SMALL LETTER ALPHA WITH OXIA AND YPOGEGRAMMENI\n\u1fb4 = with-accent-iota \u03b1\u2032 lower (acute \u03b1-vowel) \u03b1-iota-subscript\n-- U+1FB5\n\u1fb6 : Token -- U+1FB6  \u1fb6 GREEK SMALL LETTER ALPHA WITH PERISPOMENI\n\u1fb6 = with-accent \u03b1\u2032 lower (circumflex \u03b1-long-vowel)\n\u1fb7 : Token -- U+1FB7  \u1fb7 GREEK SMALL LETTER ALPHA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1fb7 = with-accent-iota \u03b1\u2032 lower (circumflex \u03b1-long-vowel) \u03b1-iota-subscript\n-- U+1FB8 \u1fb8 GREEK CAPITAL LETTER ALPHA WITH VRACHY\n-- U+1FB9 \u1fb9 GREEK CAPITAL LETTER ALPHA WITH MACRON\n\u1fba : Token -- U+1FBA  \u1fba GREEK CAPITAL LETTER ALPHA WITH VARIA\n\u1fba = with-accent \u03b1\u2032 upper (grave \u03b1-vowel)\n\u1fbb : Token -- U+1FBB  \u1fbb GREEK CAPITAL LETTER ALPHA WITH OXIA\n\u1fbb = with-accent \u03b1\u2032 upper (acute \u03b1-vowel)\n\u1fbc : Token -- U+1FBC  \u1fbc GREEK CAPITAL LETTER ALPHA WITH PROSGEGRAMMENI\n\u1fbc = with-iota \u03b1\u2032 upper \u03b1-iota-subscript\n-- U+1FBD  \u1fbd  GREEK KORONIS\n-- U+1FBE  \u1fbe  GREEK PROSGEGRAMMENI\n-- U+1FBF  \u1fbf  GREEK PSILI\n-- U+1FC0  \u1fc0  GREEK PERISPOMENI\n-- U+1FC1  \u1fc1  GREEK DIALYTIKA AND PERISPOMENI\n\u1fc2 : Token -- U+1FC2  \u1fc2 GREEK SMALL LETTER ETA WITH VARIA AND YPOGEGRAMMENI accent (circumflex \u03b1-long-vowel)\n\u1fc2 = with-accent-iota \u03b7\u2032 lower (grave \u03b7-vowel) \u03b7-iota-subscript\n\u1fc3 : Token -- U+1FC3  \u1fc3 GREEK SMALL LETTER ETA WITH YPOGEGRAMMENI\n\u1fc3 = with-iota \u03b7\u2032 lower \u03b7-iota-subscript\n\u1fc4 : Token -- U+1FC4  \u1fc4 GREEK SMALL LETTER ETA WITH OXIA AND YPOGEGRAMMENI\n\u1fc4 = with-accent-iota \u03b7\u2032 lower (acute \u03b7-vowel) \u03b7-iota-subscript\n-- U+1FC5\n\u1fc6 : Token -- U+1FC6  \u1fc6 GREEK SMALL LETTER ETA WITH PERISPOMENI\n\u1fc6 = with-accent \u03b7\u2032 lower (circumflex \u03b7-long-vowel)\n\u1fc7 : Token -- U+1FC7  \u1fc7 GREEK SMALL LETTER ETA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1fc7 = with-accent-iota \u03b7\u2032 lower (circumflex \u03b7-long-vowel) \u03b7-iota-subscript\n\u1fc8 : Token -- U+1FC8  \u1fc8 GREEK CAPITAL LETTER EPSILON WITH VARIA\n\u1fc8 = with-accent \u03b5\u2032 upper (grave \u03b5-vowel)\n\u1fc9 : Token -- U+1FC9  \u1fc9 GREEK CAPITAL LETTER EPSILON WITH OXIA\n\u1fc9 = with-accent \u03b5\u2032 upper (acute \u03b5-vowel)\n\u1fca : Token -- U+1FCA  \u1fca GREEK CAPITAL LETTER ETA WITH VARIA\n\u1fca = with-accent \u03b7\u2032 upper (grave \u03b7-vowel)\n\u1fcb : Token -- U+1FCB  \u1fcb GREEK CAPITAL LETTER ETA WITH OXIA\n\u1fcb = with-accent \u03b7\u2032 upper (acute \u03b7-vowel)\n\u1fcc : Token -- U+1FCC  \u1fcc GREEK CAPITAL LETTER ETA WITH PROSGEGRAMMENI\n\u1fcc = with-iota \u03b7\u2032 upper \u03b7-iota-subscript\n-- U+1FCD  \u1fcd GREEK PSILI AND VARIA\n-- U+1FCE  \u1fce GREEK PSILI AND OXIA\n-- U+1FCF  \u1fcf GREEK PSILI AND PERISPOMENI\n-- U+1FD0  \u1fd0 GREEK SMALL LETTER IOTA WITH VRACHY\n-- U+1FD1  \u1fd1 GREEK SMALL LETTER IOTA WITH MACRON\n\u1fd2 : Token -- U+1FD2  \u1fd2 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND VARIA\n\u1fd2 = with-accent-diaeresis \u03b9\u2032 lower (grave \u03b9-vowel) \u03b9-diaeresis\n\u1fd3 : Token -- U+1FD3  \u1fd3 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND OXIA\n\u1fd3 = with-accent-diaeresis \u03b9\u2032 lower (acute \u03b9-vowel) \u03b9-diaeresis\n-- U+1FD4\n-- U+1FD5\n\u1fd6 : Token -- U+1FD6  \u1fd6 GREEK SMALL LETTER IOTA WITH PERISPOMENI\n\u1fd6 = with-accent \u03b9\u2032 lower (circumflex \u03b9-long-vowel)\n\u1fd7 : Token -- U+1FD7  \u1fd7 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND PERISPOMENI\n\u1fd7 = with-accent-diaeresis \u03b9\u2032 lower (circumflex \u03b9-long-vowel) \u03b9-diaeresis\n-- U+1FD8 \u1fd8 GREEK CAPITAL LETTER IOTA WITH VRACHY\n-- U+1FD9 \u1fd9 GREEK CAPITAL LETTER IOTA WITH MACRON\n\u1fda : Token -- U+1FDA  \u1fda GREEK CAPITAL LETTER IOTA WITH VARIA\n\u1fda = with-accent \u03b9\u2032 upper (grave \u03b9-vowel)\n\u1fdb : Token -- U+1FDB  \u1fdb GREEK CAPITAL LETTER IOTA WITH OXIA\n\u1fdb = with-accent \u03b9\u2032 upper (acute \u03b9-vowel)\n-- U+1FDC\n-- U+1FDD  \u1fdd GREEK DASIA AND VARIA\n-- U+1FDE  \u1fde GREEK DASIA AND OXIA\n-- U+1FDF  \u1fdf GREEK DASIA AND PERISPOMENI\n-- U+1FE0 \u1fe0 GREEK SMALL LETTER UPSILON WITH VRACHY\n-- U+1FE1 \u1fe1 GREEK SMALL LETTER UPSILON WITH MACRON\n\u1fe2 : Token -- U+1FE2  \u1fe2 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND VARIA\n\u1fe2 = with-accent-diaeresis \u03c5\u2032 lower (grave \u03c5-vowel) \u03c5-diaeresis\n\u1fe3 : Token -- U+1FE3  \u1fe3 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND OXIA\n\u1fe3 = with-accent-diaeresis \u03c5\u2032 lower (acute \u03c5-vowel) \u03c5-diaeresis\n\u1fe4 : Token -- U+1FE4 \u1fe4 GREEK SMALL LETTER RHO WITH PSILI\n\u1fe4 = with-breathing \u03c1\u2032 lower (smooth \u03c1-smooth)\n\u1fe5 : Token -- U+1FE5 \u1fe5 GREEK SMALL LETTER RHO WITH DASIA\n\u1fe5 = with-breathing \u03c1\u2032 lower (rough \u03c1-rough)\n\u1fe6 : Token -- U+1FE6  \u1fe6 GREEK SMALL LETTER UPSILON WITH PERISPOMENI\n\u1fe6 = with-accent \u03c5\u2032 lower (circumflex \u03c5-long-vowel)\n\u1fe7 : Token -- U+1FE7  \u1fe7 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND PERISPOMENI\n\u1fe7 = with-accent-diaeresis \u03c5\u2032 lower (circumflex \u03c5-long-vowel) \u03c5-diaeresis\n-- U+1FE8 \u1fe8 GREEK CAPITAL LETTER UPSILON WITH VRACHY\n-- U+1FE9 \u1fe9 GREEK CAPITAL LETTER UPSILON WITH MACRON\n\u1fea : Token -- U+1FEA  \u1fea GREEK CAPITAL LETTER UPSILON WITH VARIA\n\u1fea = with-accent \u03c5\u2032 upper (grave \u03c5-vowel)\n\u1feb : Token -- U+1FEB  \u1feb GREEK CAPITAL LETTER UPSILON WITH OXIA\n\u1feb = with-accent \u03c5\u2032 upper (acute \u03c5-vowel)\n\u1fec : Token -- U+1FEC \u1fec GREEK CAPITAL LETTER RHO WITH DASIA\n\u1fec = with-breathing \u03c1\u2032 upper (rough \u03c1-rough)\n-- U+1FED  \u1fed GREEK DIALYTIKA AND VARIA\n-- U+1FEE  \u1fee GREEK DIALYTIKA AND OXIA\n-- U+1FEF  \u1fef GREEK VARIA\n-- U+1FF0\n-- U+1FF1\n\u1ff2 : Token -- U+1FF2  \u1ff2 GREEK SMALL LETTER OMEGA WITH VARIA AND YPOGEGRAMMENI\n\u1ff2 = with-accent-iota \u03c9\u2032 lower (grave \u03c9-vowel) \u03c9-iota-subscript\n\u1ff3 : Token -- U+1FF3  \u1ff3 GREEK SMALL LETTER OMEGA WITH YPOGEGRAMMENI\n\u1ff3 = with-iota \u03c9\u2032 lower \u03c9-iota-subscript\n\u1ff4 : Token -- U+1FF4  \u1ff4 GREEK SMALL LETTER OMEGA WITH OXIA AND YPOGEGRAMMENI\n\u1ff4 = with-accent-iota \u03c9\u2032 lower (acute \u03c9-vowel) \u03c9-iota-subscript\n-- U+1FF5\n\u1ff6 : Token -- U+1FF6  \u1ff6 GREEK SMALL LETTER OMEGA WITH PERISPOMENI\n\u1ff6 = with-accent \u03c9\u2032 lower (circumflex \u03c9-long-vowel)\n\u1ff7 : Token -- U+1FF7  \u1ff7 GREEK SMALL LETTER OMEGA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1ff7 = with-accent-iota \u03c9\u2032 lower (circumflex \u03c9-long-vowel) \u03c9-iota-subscript\n\u1ff8 : Token -- U+1FF8  \u1ff8 GREEK CAPITAL LETTER OMICRON WITH VARIA\n\u1ff8 = with-accent \u03bf\u2032 upper (grave \u03bf-vowel)\n\u1ff9 : Token -- U+1FF9  \u1ff9 GREEK CAPITAL LETTER OMICRON WITH OXIA\n\u1ff9 = with-accent \u03bf\u2032 upper (acute \u03bf-vowel)\n\u1ffa : Token -- U+1FFA  \u1ffa GREEK CAPITAL LETTER OMEGA WITH VARIA\n\u1ffa = with-accent \u03c9\u2032 upper (grave \u03c9-vowel)\n\u1ffb : Token -- U+1FFB  \u1ffb GREEK CAPITAL LETTER OMEGA WITH OXIA\n\u1ffb = with-accent \u03c9\u2032 upper (acute \u03c9-vowel)\n\u1ffc : Token -- U+1FFC  \u1ffc GREEK CAPITAL LETTER OMEGA WITH PROSGEGRAMMENI\n\u1ffc = with-iota \u03c9\u2032 upper \u03c9-iota-subscript\n-- U+1FFD  \u1ffd GREEK OXIA\n-- U+1FFE  \u1ffe GREEK DASIA\n-- U+1FFF\n","old_contents":"module Text.Greek.Script.Unicode where\n\nopen import Text.Greek.Script\n\n-- U+0390 - U+03CE\n\u0390 : Token -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = with-accent-diaeresis \u03b9\u2032 lower acute\n\u0391 : Token -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked \u03b1\u2032 upper\n\u0392 : Token -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked \u03b2\u2032 upper\n\u0393 : Token -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked \u03b3\u2032 upper\n\u0394 : Token -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked \u03b4\u2032 upper\n\u0395 : Token -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked \u03b5\u2032 upper\n\u0396 : Token -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked \u03b6\u2032 upper\n\u0397 : Token -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked \u03b7\u2032 upper\n\u0398 : Token -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked \u03b8\u2032 upper\n\u0399 : Token -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked \u03b9\u2032 upper\n\u039a : Token -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked \u03ba\u2032 upper\n\u039b : Token -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked \u03bb\u2032 upper\n\u039c : Token -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked \u03bc\u2032 upper\n\u039d : Token -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked \u03bd\u2032 upper\n\u039e : Token -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked \u03be\u2032 upper\n\u039f : Token -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked \u03bf\u2032 upper\n\u03a0 : Token -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked \u03c0\u2032 upper\n\u03a1 : Token -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked \u03c1\u2032 upper\n                  -- U+03A2 <reserved>\n\u03a3 : Token -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked \u03c3\u2032 upper\n\u03a4 : Token -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked \u03c4\u2032 upper\n\u03a5 : Token -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked \u03c5\u2032 upper\n\u03a6 : Token -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked \u03c6\u2032 upper\n\u03a7 : Token -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked \u03c7\u2032 upper\n\u03a8 : Token -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked \u03c8\u2032 upper\n\u03a9 : Token -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked \u03c9\u2032 upper\n\u03aa : Token -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = with-diaeresis \u03b9\u2032 upper\n\u03ab : Token -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = with-diaeresis \u03c5\u2032 upper\n\u03ac : Token -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = with-accent \u03b1\u2032 lower acute\n\u03ad : Token -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = with-accent \u03b5\u2032 lower acute\n\u03ae : Token -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = with-accent \u03b7\u2032 lower acute\n\u03af : Token -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = with-accent \u03b9\u2032 lower acute\n\u03b0 : Token -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = with-accent-diaeresis \u03c5\u2032 lower acute\n\u03b1 : Token -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked \u03b1\u2032 lower\n\u03b2 : Token -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked \u03b2\u2032 lower\n\u03b3 : Token -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked \u03b3\u2032 lower\n\u03b4 : Token -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked \u03b4\u2032 lower\n\u03b5 : Token -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked \u03b5\u2032 lower\n\u03b6 : Token -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked \u03b6\u2032 lower\n\u03b7 : Token -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked \u03b7\u2032 lower\n\u03b8 : Token -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked \u03b8\u2032 lower\n\u03b9 : Token -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked \u03b9\u2032 lower\n\u03ba : Token -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked \u03ba\u2032 lower\n\u2219\u03bb : Token -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked \u03bb\u2032 lower\n\u03bc : Token -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked \u03bc\u2032 lower\n\u03bd : Token -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked \u03bd\u2032 lower\n\u03be : Token -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked \u03be\u2032 lower\n\u03bf : Token -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked \u03bf\u2032 lower\n\u03c0 : Token -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked \u03c0\u2032 lower\n\u03c1 : Token -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked \u03c1\u2032 lower\n\u03c2 : Token -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = final \u03c3\u2032 lower\n\u03c3 : Token -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked \u03c3\u2032 lower\n\u03c4 : Token -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked \u03c4\u2032 lower\n\u03c5 : Token -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked \u03c5\u2032 lower\n\u03c6 : Token -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked \u03c6\u2032 lower\n\u03c7 : Token -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked \u03c7\u2032 lower\n\u03c8 : Token -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked \u03c8\u2032 lower\n\u03c9 : Token -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked \u03c9\u2032 lower\n\u03ca : Token -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = with-diaeresis \u03b9\u2032 lower\n\u03cb : Token -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = with-diaeresis \u03c5\u2032 lower\n\u03cc : Token -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = with-accent \u03bf\u2032 lower acute\n\u03cd : Token -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = with-accent \u03c5\u2032 lower acute\n\u03ce : Token -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = with-accent \u03c9\u2032 lower acute\n\n-- U+1F00 - U+1FFF\n\u1f00 : Token -- U+1F00  \u1f00 GREEK SMALL LETTER ALPHA WITH PSILI\n\u1f00 = with-breathing \u03b1\u2032 lower smooth\n\u1f01 : Token -- U+1F01  \u1f01 GREEK SMALL LETTER ALPHA WITH DASIA\n\u1f01 = with-breathing \u03b1\u2032 lower rough\n\u1f02 : Token -- U+1F02  \u1f02 GREEK SMALL LETTER ALPHA WITH PSILI AND VARIA\n\u1f02 = with-accent-breathing \u03b1\u2032 lower grave smooth\n\u1f03 : Token -- U+1F03  \u1f03 GREEK SMALL LETTER ALPHA WITH DASIA AND VARIA\n\u1f03 = with-accent-breathing \u03b1\u2032 lower grave rough\n\u1f04 : Token -- U+1F04  \u1f04 GREEK SMALL LETTER ALPHA WITH PSILI AND OXIA\n\u1f04 = with-accent-breathing \u03b1\u2032 lower acute smooth\n\u1f05 : Token -- U+1F05  \u1f05 GREEK SMALL LETTER ALPHA WITH DASIA AND OXIA\n\u1f05 = with-accent-breathing \u03b1\u2032 lower acute rough\n\u1f06 : Token -- U+1F06  \u1f06 GREEK SMALL LETTER ALPHA WITH PSILI AND PERISPOMENI\n\u1f06 = with-accent-breathing \u03b1\u2032 lower circumflex smooth\n\u1f07 : Token -- U+1F07  \u1f07 GREEK SMALL LETTER ALPHA WITH DASIA AND PERISPOMENI\n\u1f07 = with-accent-breathing \u03b1\u2032 lower circumflex rough\n\u1f08 : Token -- U+1F08  \u1f08 GREEK CAPITAL LETTER ALPHA WITH PSILI\n\u1f08 = with-breathing \u03b1\u2032 upper smooth\n\u1f09 : Token -- U+1F09  \u1f09 GREEK CAPITAL LETTER ALPHA WITH DASIA\n\u1f09 = with-breathing \u03b1\u2032 upper rough\n\u1f0a : Token -- U+1F0A  \u1f0a GREEK CAPITAL LETTER ALPHA WITH PSILI AND VARIA\n\u1f0a = with-accent-breathing \u03b1\u2032 upper grave smooth\n\u1f0b : Token -- U+1F0B  \u1f0b GREEK CAPITAL LETTER ALPHA WITH DASIA AND VARIA\n\u1f0b = with-accent-breathing \u03b1\u2032 upper grave rough\n\u1f0c : Token -- U+1F0C  \u1f0c GREEK CAPITAL LETTER ALPHA WITH PSILI AND OXIA\n\u1f0c = with-accent-breathing \u03b1\u2032 upper acute smooth\n\u1f0d : Token -- U+1F0D  \u1f0d GREEK CAPITAL LETTER ALPHA WITH DASIA AND OXIA\n\u1f0d = with-accent-breathing \u03b1\u2032 upper acute rough\n\u1f0e : Token -- U+1F0E  \u1f0e GREEK CAPITAL LETTER ALPHA WITH PSILI AND PERISPOMENI\n\u1f0e = with-accent-breathing \u03b1\u2032 upper circumflex smooth\n\u1f0f : Token -- U+1F0F  \u1f0f GREEK CAPITAL LETTER ALPHA WITH DASIA AND PERISPOMENI\n\u1f0f = with-accent-breathing \u03b1\u2032 upper circumflex rough\n\u1f10 : Token -- U+1F10  \u1f10 GREEK SMALL LETTER EPSILON WITH PSILI\n\u1f10 = with-breathing \u03b5\u2032 lower smooth\n\u1f11 : Token -- U+1F11  \u1f11 GREEK SMALL LETTER EPSILON WITH DASIA\n\u1f11 = with-breathing \u03b5\u2032 lower rough\n\u1f12 : Token -- U+1F12  \u1f12 GREEK SMALL LETTER EPSILON WITH PSILI AND VARIA\n\u1f12 = with-accent-breathing \u03b5\u2032 lower grave smooth\n\u1f13 : Token -- U+1F13  \u1f13 GREEK SMALL LETTER EPSILON WITH DASIA AND VARIA\n\u1f13 = with-accent-breathing \u03b5\u2032 lower grave rough\n\u1f14 : Token -- U+1F14  \u1f14 GREEK SMALL LETTER EPSILON WITH PSILI AND OXIA\n\u1f14 = with-accent-breathing \u03b5\u2032 lower acute smooth\n\u1f15 : Token -- U+1F15  \u1f15 GREEK SMALL LETTER EPSILON WITH DASIA AND OXIA\n\u1f15 = with-accent-breathing \u03b5\u2032 lower acute rough\n-- U+1F16\n-- U+1F17\n\u1f18 : Token -- U+1F18  \u1f18 GREEK CAPITAL LETTER EPSILON WITH PSILI\n\u1f18 = with-breathing \u03b5\u2032 upper smooth\n\u1f19 : Token -- U+1F19  \u1f19 GREEK CAPITAL LETTER EPSILON WITH DASIA\n\u1f19 = with-breathing \u03b5\u2032 upper rough\n\u1f1a : Token -- U+1F1A  \u1f1a GREEK CAPITAL LETTER EPSILON WITH PSILI AND VARIA\n\u1f1a = with-accent-breathing \u03b5\u2032 upper grave smooth\n\u1f1b : Token -- U+1F1B  \u1f1b GREEK CAPITAL LETTER EPSILON WITH DASIA AND VARIA\n\u1f1b = with-accent-breathing \u03b5\u2032 upper grave rough\n\u1f1c : Token -- U+1F1C  \u1f1c GREEK CAPITAL LETTER EPSILON WITH PSILI AND OXIA\n\u1f1c = with-accent-breathing \u03b5\u2032 upper acute smooth\n\u1f1d : Token -- U+1F1D  \u1f1d GREEK CAPITAL LETTER EPSILON WITH DASIA AND OXIA\n\u1f1d = with-accent-breathing \u03b5\u2032 upper acute rough\n-- U+1F1E\n-- U+1F1F\n\u1f20 : Token -- U+1F20  \u1f20 GREEK SMALL LETTER ETA WITH PSILI\n\u1f20 = with-breathing \u03b7\u2032 lower smooth\n\u1f21 : Token -- U+1F21  \u1f21 GREEK SMALL LETTER ETA WITH DASIA\n\u1f21 = with-breathing \u03b7\u2032 lower rough\n\u1f22 : Token -- U+1F22  \u1f22 GREEK SMALL LETTER ETA WITH PSILI AND VARIA\n\u1f22 = with-accent-breathing \u03b7\u2032 lower grave smooth\n\u1f23 : Token -- U+1F23  \u1f23 GREEK SMALL LETTER ETA WITH DASIA AND VARIA\n\u1f23 = with-accent-breathing \u03b7\u2032 lower grave rough\n\u1f24 : Token -- U+1F24  \u1f24 GREEK SMALL LETTER ETA WITH PSILI AND OXIA\n\u1f24 = with-accent-breathing \u03b7\u2032 lower acute smooth\n\u1f25 : Token -- U+1F25  \u1f25 GREEK SMALL LETTER ETA WITH DASIA AND OXIA\n\u1f25 = with-accent-breathing \u03b7\u2032 lower acute rough\n\u1f26 : Token -- U+1F26  \u1f26 GREEK SMALL LETTER ETA WITH PSILI AND PERISPOMENI\n\u1f26 = with-accent-breathing \u03b7\u2032 lower circumflex smooth\n\u1f27 : Token -- U+1F27  \u1f27 GREEK SMALL LETTER ETA WITH DASIA AND PERISPOMENI\n\u1f27 = with-accent-breathing \u03b7\u2032 lower circumflex rough\n\u1f28 : Token -- U+1F28  \u1f28 GREEK CAPITAL LETTER ETA WITH PSILI\n\u1f28 = with-breathing \u03b7\u2032 upper smooth\n\u1f29 : Token -- U+1F29  \u1f29 GREEK CAPITAL LETTER ETA WITH DASIA\n\u1f29 = with-breathing \u03b7\u2032 upper rough\n\u1f2a : Token -- U+1F2A  \u1f2a GREEK CAPITAL LETTER ETA WITH PSILI AND VARIA\n\u1f2a = with-accent-breathing \u03b7\u2032 upper grave smooth\n\u1f2b : Token -- U+1F2B  \u1f2b GREEK CAPITAL LETTER ETA WITH DASIA AND VARIA\n\u1f2b = with-accent-breathing \u03b7\u2032 upper grave rough\n\u1f2c : Token -- U+1F2C  \u1f2c GREEK CAPITAL LETTER ETA WITH PSILI AND OXIA\n\u1f2c = with-accent-breathing \u03b7\u2032 upper acute smooth\n\u1f2d : Token -- U+1F2D  \u1f2d GREEK CAPITAL LETTER ETA WITH DASIA AND OXIA\n\u1f2d = with-accent-breathing \u03b7\u2032 upper acute rough\n\u1f2e : Token -- U+1F2E  \u1f2e GREEK CAPITAL LETTER ETA WITH PSILI AND PERISPOMENI\n\u1f2e = with-accent-breathing \u03b7\u2032 upper circumflex smooth\n\u1f2f : Token -- U+1F2F  \u1f2f GREEK CAPITAL LETTER ETA WITH DASIA AND PERISPOMENI\n\u1f2f = with-accent-breathing \u03b7\u2032 upper circumflex rough\n\u1f30 : Token -- U+1F30  \u1f30 GREEK SMALL LETTER IOTA WITH PSILI\n\u1f30 = with-breathing \u03b9\u2032 lower smooth\n\u1f31 : Token -- U+1F31  \u1f31 GREEK SMALL LETTER IOTA WITH DASIA\n\u1f31 = with-breathing \u03b9\u2032 lower rough\n\u1f32 : Token -- U+1F32  \u1f32 GREEK SMALL LETTER IOTA WITH PSILI AND VARIA\n\u1f32 = with-accent-breathing \u03b9\u2032 lower grave smooth\n\u1f33 : Token -- U+1F33  \u1f33 GREEK SMALL LETTER IOTA WITH DASIA AND VARIA\n\u1f33 = with-accent-breathing \u03b9\u2032 lower grave rough\n\u1f34 : Token -- U+1F34  \u1f34 GREEK SMALL LETTER IOTA WITH PSILI AND OXIA\n\u1f34 = with-accent-breathing \u03b9\u2032 lower acute smooth\n\u1f35 : Token -- U+1F35  \u1f35 GREEK SMALL LETTER IOTA WITH DASIA AND OXIA\n\u1f35 = with-accent-breathing \u03b9\u2032 lower acute rough\n\u1f36 : Token -- U+1F36  \u1f36 GREEK SMALL LETTER IOTA WITH PSILI AND PERISPOMENI\n\u1f36 = with-accent-breathing \u03b9\u2032 lower circumflex smooth\n\u1f37 : Token -- U+1F37  \u1f37 GREEK SMALL LETTER IOTA WITH DASIA AND PERISPOMENI\n\u1f37 = with-accent-breathing \u03b9\u2032 lower circumflex rough\n\u1f38 : Token -- U+1F38  \u1f38 GREEK CAPITAL LETTER IOTA WITH PSILI\n\u1f38 = with-breathing \u03b9\u2032 upper smooth\n\u1f39 : Token -- U+1F39  \u1f39 GREEK CAPITAL LETTER IOTA WITH DASIA\n\u1f39 = with-breathing \u03b9\u2032 upper rough\n\u1f3a : Token -- U+1F3A  \u1f3a GREEK CAPITAL LETTER IOTA WITH PSILI AND VARIA\n\u1f3a = with-accent-breathing \u03b9\u2032 upper grave smooth\n\u1f3b : Token -- U+1F3B  \u1f3b GREEK CAPITAL LETTER IOTA WITH DASIA AND VARIA\n\u1f3b = with-accent-breathing \u03b9\u2032 upper grave rough\n\u1f3c : Token -- U+1F3C  \u1f3c GREEK CAPITAL LETTER IOTA WITH PSILI AND OXIA\n\u1f3c = with-accent-breathing \u03b9\u2032 upper acute smooth\n\u1f3d : Token -- U+1F3D  \u1f3d GREEK CAPITAL LETTER IOTA WITH DASIA AND OXIA\n\u1f3d = with-accent-breathing \u03b9\u2032 upper acute rough\n\u1f3e : Token -- U+1F3E  \u1f3e GREEK CAPITAL LETTER IOTA WITH PSILI AND PERISPOMENI\n\u1f3e = with-accent-breathing \u03b9\u2032 upper circumflex smooth\n\u1f3f : Token -- U+1F3F  \u1f3f GREEK CAPITAL LETTER IOTA WITH DASIA AND PERISPOMENI\n\u1f3f = with-accent-breathing \u03b9\u2032 upper circumflex rough\n\u1f40 : Token -- U+1F40  \u1f40 GREEK SMALL LETTER OMICRON WITH PSILI\n\u1f40 = with-breathing \u03bf\u2032 lower smooth\n\u1f41 : Token -- U+1F41  \u1f41 GREEK SMALL LETTER OMICRON WITH DASIA\n\u1f41 = with-breathing \u03bf\u2032 lower rough\n\u1f42 : Token -- U+1F42  \u1f42 GREEK SMALL LETTER OMICRON WITH PSILI AND VARIA\n\u1f42 = with-accent-breathing \u03bf\u2032 lower grave smooth\n\u1f43 : Token -- U+1F43  \u1f43 GREEK SMALL LETTER OMICRON WITH DASIA AND VARIA\n\u1f43 = with-accent-breathing \u03bf\u2032 lower grave rough\n\u1f44 : Token -- U+1F44  \u1f44 GREEK SMALL LETTER OMICRON WITH PSILI AND OXIA\n\u1f44 = with-accent-breathing \u03bf\u2032 lower acute smooth\n\u1f45 : Token -- U+1F45  \u1f45 GREEK SMALL LETTER OMICRON WITH DASIA AND OXIA\n\u1f45 = with-accent-breathing \u03bf\u2032 lower acute rough\n-- U+1F46\n-- U+1F47\n\u1f48 : Token -- U+1F48  \u1f48 GREEK CAPITAL LETTER OMICRON WITH PSILI\n\u1f48 = with-breathing \u03bf\u2032 upper smooth\n\u1f49 : Token -- U+1F49  \u1f49 GREEK CAPITAL LETTER OMICRON WITH DASIA\n\u1f49 = with-breathing \u03bf\u2032 upper rough\n\u1f4a : Token -- U+1F4A  \u1f4a GREEK CAPITAL LETTER OMICRON WITH PSILI AND VARIA\n\u1f4a = with-accent-breathing \u03bf\u2032 upper grave smooth\n\u1f4b : Token -- U+1F4B  \u1f4b GREEK CAPITAL LETTER OMICRON WITH DASIA AND VARIA\n\u1f4b = with-accent-breathing \u03bf\u2032 upper grave rough\n\u1f4c : Token -- U+1F4C  \u1f4c GREEK CAPITAL LETTER OMICRON WITH PSILI AND OXIA\n\u1f4c = with-accent-breathing \u03bf\u2032 upper acute smooth\n\u1f4d : Token -- U+1F4D  \u1f4d GREEK CAPITAL LETTER OMICRON WITH DASIA AND OXIA\n\u1f4d = with-accent-breathing \u03bf\u2032 upper acute rough\n-- U+1F4E\n-- U+1F4F\n\u1f50 : Token -- U+1F50  \u1f50 GREEK SMALL LETTER UPSILON WITH PSILI\n\u1f50 = with-breathing \u03c5\u2032 lower smooth\n\u1f51 : Token -- U+1F51  \u1f51 GREEK SMALL LETTER UPSILON WITH DASIA\n\u1f51 = with-breathing \u03c5\u2032 lower rough\n\u1f52 : Token -- U+1F52  \u1f52 GREEK SMALL LETTER UPSILON WITH PSILI AND VARIA\n\u1f52 = with-accent-breathing \u03c5\u2032 lower grave smooth\n\u1f53 : Token -- U+1F53  \u1f53 GREEK SMALL LETTER UPSILON WITH DASIA AND VARIA\n\u1f53 = with-accent-breathing \u03c5\u2032 lower grave rough\n\u1f54 : Token -- U+1F54  \u1f54 GREEK SMALL LETTER UPSILON WITH PSILI AND OXIA\n\u1f54 = with-accent-breathing \u03c5\u2032 lower acute smooth\n\u1f55 : Token -- U+1F55  \u1f55 GREEK SMALL LETTER UPSILON WITH DASIA AND OXIA\n\u1f55 = with-accent-breathing \u03c5\u2032 lower acute rough\n\u1f56 : Token -- U+1F56  \u1f56 GREEK SMALL LETTER UPSILON WITH PSILI AND PERISPOMENI\n\u1f56 = with-accent-breathing \u03c5\u2032 lower circumflex smooth\n\u1f57 : Token -- U+1F57  \u1f57 GREEK SMALL LETTER UPSILON WITH DASIA AND PERISPOMENI\n\u1f57 = with-accent-breathing \u03c5\u2032 lower circumflex rough\n-- U+1F58\n\u1f59 : Token -- U+1F59  \u1f59 GREEK CAPITAL LETTER UPSILON WITH DASIA\n\u1f59 = with-breathing \u03c5\u2032 upper rough\n-- U+1F5A\n\u1f5b : Token -- U+1F5B  \u1f5b GREEK CAPITAL LETTER UPSILON WITH DASIA AND VARIA\n\u1f5b = with-accent-breathing \u03c5\u2032 upper grave rough\n-- U+1F5C\n\u1f5d : Token -- U+1F5D  \u1f5d GREEK CAPITAL LETTER UPSILON WITH DASIA AND OXIA\n\u1f5d = with-accent-breathing \u03c5\u2032 upper acute rough\n-- U+1F5E\n\u1f5f : Token -- U+1F5F  \u1f5f GREEK CAPITAL LETTER UPSILON WITH DASIA AND PERISPOMENI\n\u1f5f = with-accent-breathing \u03c5\u2032 upper circumflex rough\n\u1f60 : Token -- U+1F60  \u1f60 GREEK SMALL LETTER OMEGA WITH PSILI\n\u1f60 = with-breathing \u03c9\u2032 lower smooth\n\u1f61 : Token -- U+1F61  \u1f61 GREEK SMALL LETTER OMEGA WITH DASIA\n\u1f61 = with-breathing \u03c9\u2032 lower rough\n\u1f62 : Token -- U+1F62  \u1f62 GREEK SMALL LETTER OMEGA WITH PSILI AND VARIA\n\u1f62 = with-accent-breathing \u03c9\u2032 lower grave smooth\n\u1f63 : Token -- U+1F63  \u1f63 GREEK SMALL LETTER OMEGA WITH DASIA AND VARIA\n\u1f63 = with-accent-breathing \u03c9\u2032 lower grave rough\n\u1f64 : Token -- U+1F64  \u1f64 GREEK SMALL LETTER OMEGA WITH PSILI AND OXIA\n\u1f64 = with-accent-breathing \u03c9\u2032 lower acute smooth\n\u1f65 : Token -- U+1F65  \u1f65 GREEK SMALL LETTER OMEGA WITH DASIA AND OXIA\n\u1f65 = with-accent-breathing \u03c9\u2032 lower acute rough\n\u1f66 : Token -- U+1F66  \u1f66 GREEK SMALL LETTER OMEGA WITH PSILI AND PERISPOMENI\n\u1f66 = with-accent-breathing \u03c9\u2032 lower circumflex smooth\n\u1f67 : Token -- U+1F67  \u1f67 GREEK SMALL LETTER OMEGA WITH DASIA AND PERISPOMENI\n\u1f67 = with-accent-breathing \u03c9\u2032 lower circumflex rough\n\u1f68 : Token -- U+1F68  \u1f68 GREEK CAPITAL LETTER OMEGA WITH PSILI\n\u1f68 = with-breathing \u03c9\u2032 upper smooth\n\u1f69 : Token -- U+1F69  \u1f69 GREEK CAPITAL LETTER OMEGA WITH DASIA\n\u1f69 = with-breathing \u03c9\u2032 upper rough\n\u1f6a : Token -- U+1F6A  \u1f6a GREEK CAPITAL LETTER OMEGA WITH PSILI AND VARIA\n\u1f6a = with-accent-breathing \u03c9\u2032 upper grave smooth\n\u1f6b : Token -- U+1F6B  \u1f6b GREEK CAPITAL LETTER OMEGA WITH DASIA AND VARIA\n\u1f6b = with-accent-breathing \u03c9\u2032 upper grave rough\n\u1f6c : Token -- U+1F6C  \u1f6c GREEK CAPITAL LETTER OMEGA WITH PSILI AND OXIA\n\u1f6c = with-accent-breathing \u03c9\u2032 upper acute smooth\n\u1f6d : Token -- U+1F6D  \u1f6d GREEK CAPITAL LETTER OMEGA WITH DASIA AND OXIA\n\u1f6d = with-accent-breathing \u03c9\u2032 upper acute rough\n\u1f6e : Token -- U+1F6E  \u1f6e GREEK CAPITAL LETTER OMEGA WITH PSILI AND PERISPOMENI\n\u1f6e = with-accent-breathing \u03c9\u2032 upper circumflex smooth\n\u1f6f : Token -- U+1F6F  \u1f6f GREEK CAPITAL LETTER OMEGA WITH DASIA AND PERISPOMENI\n\u1f6f = with-accent-breathing \u03c9\u2032 upper circumflex rough\n\u1f70 : Token -- U+1F70  \u1f70 GREEK SMALL LETTER ALPHA WITH VARIA\n\u1f70 = with-accent \u03b1\u2032 lower grave\n\u1f71 : Token -- U+1F71  \u1f71 GREEK SMALL LETTER ALPHA WITH OXIA\n\u1f71 = with-accent \u03b1\u2032 lower acute\n\u1f72 : Token -- U+1F72  \u1f72 GREEK SMALL LETTER EPSILON WITH VARIA\n\u1f72 = with-accent \u03b5\u2032 lower grave\n\u1f73 : Token -- U+1F73  \u1f73 GREEK SMALL LETTER EPSILON WITH OXIA\n\u1f73 = with-accent \u03b5\u2032 lower acute\n\u1f74 : Token -- U+1F74  \u1f74 GREEK SMALL LETTER ETA WITH VARIA\n\u1f74 = with-accent \u03b7\u2032 lower grave\n\u1f75 : Token -- U+1F75  \u1f75 GREEK SMALL LETTER ETA WITH OXIA\n\u1f75 = with-accent \u03b7\u2032 lower acute\n\u1f76 : Token -- U+1F76  \u1f76 GREEK SMALL LETTER IOTA WITH VARIA\n\u1f76 = with-accent \u03b9\u2032 lower grave\n\u1f77 : Token -- U+1F77  \u1f77 GREEK SMALL LETTER IOTA WITH OXIA\n\u1f77 = with-accent \u03b9\u2032 lower acute\n\u1f78 : Token -- U+1F78  \u1f78 GREEK SMALL LETTER OMICRON WITH VARIA\n\u1f78 = with-accent \u03bf\u2032 lower grave\n\u1f79 : Token -- U+1F79  \u1f79 GREEK SMALL LETTER OMICRON WITH OXIA\n\u1f79 = with-accent \u03bf\u2032 lower acute\n\u1f7a : Token -- U+1F7A  \u1f7a GREEK SMALL LETTER UPSILON WITH VARIA\n\u1f7a = with-accent \u03c5\u2032 lower grave\n\u1f7b : Token -- U+1F7B  \u1f7b GREEK SMALL LETTER UPSILON WITH OXIA\n\u1f7b = with-accent \u03c5\u2032 lower acute\n\u1f7c : Token -- U+1F7C  \u1f7c GREEK SMALL LETTER OMEGA WITH VARIA\n\u1f7c = with-accent \u03c9\u2032 lower grave\n\u1f7d : Token -- U+1F7D  \u1f7d GREEK SMALL LETTER OMEGA WITH OXIA\n\u1f7d = with-accent \u03c9\u2032 lower acute\n-- U+1F7E\n-- U+1F7F\n\u1f80 : Token -- U+1F80  \u1f80 GREEK SMALL LETTER ALPHA WITH PSILI AND YPOGEGRAMMENI\n\u1f80 = with-breathing-iota \u03b1\u2032 lower smooth\n\u1f81 : Token -- U+1F81  \u1f81 GREEK SMALL LETTER ALPHA WITH DASIA AND YPOGEGRAMMENI\n\u1f81 = with-breathing-iota \u03b1\u2032 lower rough\n\u1f82 : Token -- U+1F82  \u1f82 GREEK SMALL LETTER ALPHA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1f82 = with-accent-breathing-iota \u03b1\u2032 lower grave smooth\n\u1f83 : Token -- U+1F83  \u1f83 GREEK SMALL LETTER ALPHA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1f83 = with-accent-breathing \u03b1\u2032 lower grave rough\n\u1f84 : Token -- U+1F84  \u1f84 GREEK SMALL LETTER ALPHA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1f84 = with-accent-breathing-iota \u03b1\u2032 lower acute smooth\n\u1f85 : Token -- U+1F85  \u1f85 GREEK SMALL LETTER ALPHA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1f85 = with-accent-breathing-iota \u03b1\u2032 lower acute rough\n\u1f86 : Token -- U+1F86  \u1f86 GREEK SMALL LETTER ALPHA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f86 = with-accent-breathing-iota \u03b1\u2032 lower circumflex smooth\n\u1f87 : Token -- U+1F87  \u1f87 GREEK SMALL LETTER ALPHA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f87 = with-accent-breathing-iota \u03b1\u2032 lower circumflex rough\n\u1f88 : Token -- U+1F88  \u1f88 GREEK CAPITAL LETTER ALPHA WITH PSILI AND PROSGEGRAMMENI\n\u1f88 = with-breathing-iota \u03b1\u2032 upper smooth\n\u1f89 : Token -- U+1F89  \u1f89 GREEK CAPITAL LETTER ALPHA WITH DASIA AND PROSGEGRAMMENI\n\u1f89 = with-breathing-iota \u03b1\u2032 upper rough\n\u1f8a : Token -- U+1F8A  \u1f8a GREEK CAPITAL LETTER ALPHA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1f8a = with-accent-breathing-iota \u03b1\u2032 upper grave smooth\n\u1f8b : Token -- U+1F8B  \u1f8b GREEK CAPITAL LETTER ALPHA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1f8b = with-accent-breathing-iota \u03b1\u2032 upper grave rough\n\u1f8c : Token -- U+1F8C  \u1f8c GREEK CAPITAL LETTER ALPHA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1f8c = with-accent-breathing-iota \u03b1\u2032 upper acute smooth\n\u1f8d : Token -- U+1F8D  \u1f8d GREEK CAPITAL LETTER ALPHA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1f8d = with-accent-breathing-iota \u03b1\u2032 upper acute rough\n\u1f8e : Token -- U+1F8E  \u1f8e GREEK CAPITAL LETTER ALPHA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f8e = with-accent-breathing-iota \u03b1\u2032 upper circumflex smooth\n\u1f8f : Token -- U+1F8F  \u1f8f GREEK CAPITAL LETTER ALPHA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f8f = with-accent-breathing-iota \u03b1\u2032 upper circumflex rough\n\u1f90 : Token -- U+1F90  \u1f90 GREEK SMALL LETTER ETA WITH PSILI AND YPOGEGRAMMENI\n\u1f90 = with-breathing-iota \u03b7\u2032 lower smooth\n\u1f91 : Token -- U+1F91  \u1f91 GREEK SMALL LETTER ETA WITH DASIA AND YPOGEGRAMMENI\n\u1f91 = with-breathing-iota \u03b7\u2032 lower rough\n\u1f92 : Token -- U+1F92  \u1f92 GREEK SMALL LETTER ETA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1f92 = with-accent-breathing-iota \u03b7\u2032 lower grave smooth\n\u1f93 : Token -- U+1F93  \u1f93 GREEK SMALL LETTER ETA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1f93 = with-accent-breathing-iota \u03b7\u2032 lower grave rough\n\u1f94 : Token -- U+1F94  \u1f94 GREEK SMALL LETTER ETA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1f94 = with-accent-breathing-iota \u03b7\u2032 lower acute smooth\n\u1f95 : Token -- U+1F95  \u1f95 GREEK SMALL LETTER ETA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1f95 = with-accent-breathing-iota \u03b7\u2032 lower acute rough\n\u1f96 : Token -- U+1F96  \u1f96 GREEK SMALL LETTER ETA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f96 = with-accent-breathing-iota \u03b7\u2032 lower circumflex smooth\n\u1f97 : Token -- U+1F97  \u1f97 GREEK SMALL LETTER ETA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1f97 = with-accent-breathing-iota \u03b7\u2032 lower circumflex rough\n\u1f98 : Token -- U+1F98  \u1f98 GREEK CAPITAL LETTER ETA WITH PSILI AND PROSGEGRAMMENI\n\u1f98 = with-breathing-iota \u03b7\u2032 upper smooth\n\u1f99 : Token -- U+1F99  \u1f99 GREEK CAPITAL LETTER ETA WITH DASIA AND PROSGEGRAMMENI\n\u1f99 = with-breathing-iota \u03b7\u2032 upper rough\n\u1f9a : Token -- U+1F9A  \u1f9a GREEK CAPITAL LETTER ETA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1f9a = with-accent-breathing-iota \u03b7\u2032 upper grave smooth\n\u1f9b : Token -- U+1F9B  \u1f9b GREEK CAPITAL LETTER ETA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1f9b = with-accent-breathing-iota \u03b7\u2032 upper grave rough\n\u1f9c : Token -- U+1F9C  \u1f9c GREEK CAPITAL LETTER ETA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1f9c = with-accent-breathing-iota \u03b7\u2032 upper acute smooth\n\u1f9d : Token -- U+1F9D  \u1f9d GREEK CAPITAL LETTER ETA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1f9d = with-accent-breathing-iota \u03b7\u2032 upper acute rough\n\u1f9e : Token -- U+1F9E  \u1f9e GREEK CAPITAL LETTER ETA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f9e = with-accent-breathing-iota \u03b7\u2032 upper circumflex smooth\n\u1f9f : Token -- U+1F9F  \u1f9f GREEK CAPITAL LETTER ETA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1f9f = with-accent-breathing-iota \u03b7\u2032 upper circumflex rough\n\u1fa0 : Token -- U+1FA0  \u1fa0 GREEK SMALL LETTER OMEGA WITH PSILI AND YPOGEGRAMMENI\n\u1fa0 = with-breathing-iota \u03c9\u2032 lower smooth\n\u1fa1 : Token -- U+1FA1  \u1fa1 GREEK SMALL LETTER OMEGA WITH DASIA AND YPOGEGRAMMENI\n\u1fa1 = with-breathing-iota \u03c9\u2032 lower rough\n\u1fa2 : Token -- U+1FA2  \u1fa2 GREEK SMALL LETTER OMEGA WITH PSILI AND VARIA AND YPOGEGRAMMENI\n\u1fa2 = with-accent-breathing-iota \u03c9\u2032 lower grave smooth\n\u1fa3 : Token -- U+1FA3  \u1fa3 GREEK SMALL LETTER OMEGA WITH DASIA AND VARIA AND YPOGEGRAMMENI\n\u1fa3 = with-accent-breathing-iota \u03c9\u2032 lower grave rough\n\u1fa4 : Token -- U+1FA4  \u1fa4 GREEK SMALL LETTER OMEGA WITH PSILI AND OXIA AND YPOGEGRAMMENI\n\u1fa4 = with-accent-breathing-iota \u03c9\u2032 lower acute smooth\n\u1fa5 : Token -- U+1FA5  \u1fa5 GREEK SMALL LETTER OMEGA WITH DASIA AND OXIA AND YPOGEGRAMMENI\n\u1fa5 = with-accent-breathing-iota \u03c9\u2032 lower acute rough\n\u1fa6 : Token -- U+1FA6  \u1fa6 GREEK SMALL LETTER OMEGA WITH PSILI AND PERISPOMENI AND YPOGEGRAMMENI\n\u1fa6 = with-accent-breathing-iota \u03c9\u2032 lower circumflex smooth\n\u1fa7 : Token -- U+1FA7  \u1fa7 GREEK SMALL LETTER OMEGA WITH DASIA AND PERISPOMENI AND YPOGEGRAMMENI\n\u1fa7 = with-accent-breathing-iota \u03c9\u2032 lower circumflex rough\n\u1fa8 : Token -- U+1FA8  \u1fa8 GREEK CAPITAL LETTER OMEGA WITH PSILI AND PROSGEGRAMMENI\n\u1fa8 = with-breathing-iota \u03c9\u2032 upper smooth\n\u1fa9 : Token -- U+1FA9  \u1fa9 GREEK CAPITAL LETTER OMEGA WITH DASIA AND PROSGEGRAMMENI\n\u1fa9 = with-breathing-iota \u03c9\u2032 upper rough\n\u1faa : Token -- U+1FAA  \u1faa GREEK CAPITAL LETTER OMEGA WITH PSILI AND VARIA AND PROSGEGRAMMENI\n\u1faa = with-accent-breathing-iota \u03c9\u2032 upper grave smooth\n\u1fab : Token -- U+1FAB  \u1fab GREEK CAPITAL LETTER OMEGA WITH DASIA AND VARIA AND PROSGEGRAMMENI\n\u1fab = with-accent-breathing-iota \u03c9\u2032 upper grave rough\n\u1fac : Token -- U+1FAC  \u1fac GREEK CAPITAL LETTER OMEGA WITH PSILI AND OXIA AND PROSGEGRAMMENI\n\u1fac = with-accent-breathing-iota \u03c9\u2032 upper acute smooth\n\u1fad : Token -- U+1FAD  \u1fad GREEK CAPITAL LETTER OMEGA WITH DASIA AND OXIA AND PROSGEGRAMMENI\n\u1fad = with-accent-breathing-iota \u03c9\u2032 upper acute rough\n\u1fae : Token -- U+1FAE  \u1fae GREEK CAPITAL LETTER OMEGA WITH PSILI AND PERISPOMENI AND PROSGEGRAMMENI\n\u1fae = with-accent-breathing-iota \u03c9\u2032 upper circumflex smooth\n\u1faf : Token -- U+1FAF  \u1faf GREEK CAPITAL LETTER OMEGA WITH DASIA AND PERISPOMENI AND PROSGEGRAMMENI\n\u1faf = with-accent-breathing-iota \u03c9\u2032 upper circumflex rough\n-- U+1FB0 \u1fb0 GREEK SMALL LETTER ALPHA WITH VRACHY\n-- U+1FB1 \u1fb1 GREEK SMALL LETTER ALPHA WITH MACRON\n\u1fb2 : Token -- U+1FB2  \u1fb2 GREEK SMALL LETTER ALPHA WITH VARIA AND YPOGEGRAMMENI\n\u1fb2 = with-accent-iota \u03b1\u2032 lower grave\n\u1fb3 : Token -- U+1FB3  \u1fb3 GREEK SMALL LETTER ALPHA WITH YPOGEGRAMMENI\n\u1fb3 = with-iota \u03b1\u2032 lower\n\u1fb4 : Token -- U+1FB4  \u1fb4 GREEK SMALL LETTER ALPHA WITH OXIA AND YPOGEGRAMMENI\n\u1fb4 = with-accent-iota \u03b1\u2032 lower acute\n-- U+1FB5\n\u1fb6 : Token -- U+1FB6  \u1fb6 GREEK SMALL LETTER ALPHA WITH PERISPOMENI\n\u1fb6 = with-accent-iota \u03b1\u2032 lower circumflex\n\u1fb7 : Token -- U+1FB7  \u1fb7 GREEK SMALL LETTER ALPHA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1fb7 = with-accent-iota \u03b1\u2032 lower circumflex\n-- U+1FB8 \u1fb8 GREEK CAPITAL LETTER ALPHA WITH VRACHY\n-- U+1FB9 \u1fb9 GREEK CAPITAL LETTER ALPHA WITH MACRON\n\u1fba : Token -- U+1FBA  \u1fba GREEK CAPITAL LETTER ALPHA WITH VARIA\n\u1fba = with-accent \u03b1\u2032 upper grave\n\u1fbb : Token -- U+1FBB  \u1fbb GREEK CAPITAL LETTER ALPHA WITH OXIA\n\u1fbb = with-accent \u03b1\u2032 upper acute\n\u1fbc : Token -- U+1FBC  \u1fbc GREEK CAPITAL LETTER ALPHA WITH PROSGEGRAMMENI\n\u1fbc = with-iota \u03b1\u2032 upper\n-- U+1FBD  \u1fbd  GREEK KORONIS\n-- U+1FBE  \u1fbe  GREEK PROSGEGRAMMENI\n-- U+1FBF  \u1fbf  GREEK PSILI\n-- U+1FC0  \u1fc0  GREEK PERISPOMENI\n-- U+1FC1  \u1fc1  GREEK DIALYTIKA AND PERISPOMENI\n\u1fc2 : Token -- U+1FC2  \u1fc2 GREEK SMALL LETTER ETA WITH VARIA AND YPOGEGRAMMENI accent circumflex\n\u1fc2 = with-accent-iota \u03b7\u2032 lower grave\n\u1fc3 : Token -- U+1FC3  \u1fc3 GREEK SMALL LETTER ETA WITH YPOGEGRAMMENI\n\u1fc3 = with-iota \u03b7\u2032 lower\n\u1fc4 : Token -- U+1FC4  \u1fc4 GREEK SMALL LETTER ETA WITH OXIA AND YPOGEGRAMMENI\n\u1fc4 = with-accent-iota \u03b7\u2032 lower acute\n-- U+1FC5\n\u1fc6 : Token -- U+1FC6  \u1fc6 GREEK SMALL LETTER ETA WITH PERISPOMENI\n\u1fc6 = with-accent \u03b7\u2032 lower circumflex\n\u1fc7 : Token -- U+1FC7  \u1fc7 GREEK SMALL LETTER ETA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1fc7 = with-accent-iota \u03b7\u2032 lower circumflex\n\u1fc8 : Token -- U+1FC8  \u1fc8 GREEK CAPITAL LETTER EPSILON WITH VARIA\n\u1fc8 = with-accent \u03b5\u2032 upper grave\n\u1fc9 : Token -- U+1FC9  \u1fc9 GREEK CAPITAL LETTER EPSILON WITH OXIA\n\u1fc9 = with-accent \u03b5\u2032 upper acute\n\u1fca : Token -- U+1FCA  \u1fca GREEK CAPITAL LETTER ETA WITH VARIA\n\u1fca = with-accent \u03b7\u2032 upper grave\n\u1fcb : Token -- U+1FCB  \u1fcb GREEK CAPITAL LETTER ETA WITH OXIA\n\u1fcb = with-accent \u03b7\u2032 upper acute\n\u1fcc : Token -- U+1FCC  \u1fcc GREEK CAPITAL LETTER ETA WITH PROSGEGRAMMENI\n\u1fcc = with-iota \u03b7\u2032 upper\n-- U+1FCD  \u1fcd GREEK PSILI AND VARIA\n-- U+1FCE  \u1fce GREEK PSILI AND OXIA\n-- U+1FCF  \u1fcf GREEK PSILI AND PERISPOMENI\n-- U+1FD0  \u1fd0 GREEK SMALL LETTER IOTA WITH VRACHY\n-- U+1FD1  \u1fd1 GREEK SMALL LETTER IOTA WITH MACRON\n\u1fd2 : Token -- U+1FD2  \u1fd2 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND VARIA\n\u1fd2 = with-accent-diaeresis \u03b9\u2032 lower grave\n\u1fd3 : Token -- U+1FD3  \u1fd3 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND OXIA\n\u1fd3 = with-accent-diaeresis \u03b9\u2032 lower acute\n-- U+1FD4\n-- U+1FD5\n\u1fd6 : Token -- U+1FD6  \u1fd6 GREEK SMALL LETTER IOTA WITH PERISPOMENI\n\u1fd6 = with-accent \u03b9\u2032 lower circumflex\n\u1fd7 : Token -- U+1FD7  \u1fd7 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND PERISPOMENI\n\u1fd7 = with-accent-diaeresis \u03b9\u2032 lower circumflex\n-- U+1FD8 \u1fd8 GREEK CAPITAL LETTER IOTA WITH VRACHY\n-- U+1FD9 \u1fd9 GREEK CAPITAL LETTER IOTA WITH MACRON\n\u1fda : Token -- U+1FDA  \u1fda GREEK CAPITAL LETTER IOTA WITH VARIA\n\u1fda = with-accent \u03b9\u2032 upper grave\n\u1fdb : Token -- U+1FDB  \u1fdb GREEK CAPITAL LETTER IOTA WITH OXIA\n\u1fdb = with-accent \u03b9\u2032 upper acute\n-- U+1FDC\n-- U+1FDD  \u1fdd GREEK DASIA AND VARIA\n-- U+1FDE  \u1fde GREEK DASIA AND OXIA\n-- U+1FDF  \u1fdf GREEK DASIA AND PERISPOMENI\n-- U+1FE0 \u1fe0 GREEK SMALL LETTER UPSILON WITH VRACHY\n-- U+1FE1 \u1fe1 GREEK SMALL LETTER UPSILON WITH MACRON\n\u1fe2 : Token -- U+1FE2  \u1fe2 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND VARIA\n\u1fe2 = with-accent-diaeresis \u03c5\u2032 lower grave\n\u1fe3 : Token -- U+1FE3  \u1fe3 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND OXIA\n\u1fe3 = with-accent-diaeresis \u03c5\u2032 lower acute\n\u1fe4 : Token -- U+1FE4 \u1fe4 GREEK SMALL LETTER RHO WITH PSILI\n\u1fe4 = with-breathing \u03c1\u2032 lower smooth\n\u1fe5 : Token -- U+1FE5 \u1fe5 GREEK SMALL LETTER RHO WITH DASIA\n\u1fe5 = with-breathing \u03c1\u2032 lower rough\n\u1fe6 : Token -- U+1FE6  \u1fe6 GREEK SMALL LETTER UPSILON WITH PERISPOMENI\n\u1fe6 = with-accent \u03c5\u2032 lower circumflex\n\u1fe7 : Token -- U+1FE7  \u1fe7 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND PERISPOMENI\n\u1fe7 = with-accent-diaeresis \u03c5\u2032 lower circumflex\n-- U+1FE8 \u1fe8 GREEK CAPITAL LETTER UPSILON WITH VRACHY\n-- U+1FE9 \u1fe9 GREEK CAPITAL LETTER UPSILON WITH MACRON\n\u1fea : Token -- U+1FEA  \u1fea GREEK CAPITAL LETTER UPSILON WITH VARIA\n\u1fea = with-accent \u03c5\u2032 upper grave\n\u1feb : Token -- U+1FEB  \u1feb GREEK CAPITAL LETTER UPSILON WITH OXIA\n\u1feb = with-accent \u03c5\u2032 upper acute\n\u1fec : Token -- U+1FEC \u1fec GREEK CAPITAL LETTER RHO WITH DASIA\n\u1fec = with-breathing \u03c1\u2032 upper rough\n-- U+1FED  \u1fed GREEK DIALYTIKA AND VARIA\n-- U+1FEE  \u1fee GREEK DIALYTIKA AND OXIA\n-- U+1FEF  \u1fef GREEK VARIA\n-- U+1FF0\n-- U+1FF1\n\u1ff2 : Token -- U+1FF2  \u1ff2 GREEK SMALL LETTER OMEGA WITH VARIA AND YPOGEGRAMMENI\n\u1ff2 = with-accent-iota \u03c9\u2032 lower grave\n\u1ff3 : Token -- U+1FF3  \u1ff3 GREEK SMALL LETTER OMEGA WITH YPOGEGRAMMENI\n\u1ff3 = with-iota \u03c9\u2032 lower\n\u1ff4 : Token -- U+1FF4  \u1ff4 GREEK SMALL LETTER OMEGA WITH OXIA AND YPOGEGRAMMENI\n\u1ff4 = with-accent-iota \u03c9\u2032 lower acute\n-- U+1FF5\n\u1ff6 : Token -- U+1FF6  \u1ff6 GREEK SMALL LETTER OMEGA WITH PERISPOMENI\n\u1ff6 = with-accent \u03c9\u2032 lower circumflex\n\u1ff7 : Token -- U+1FF7  \u1ff7 GREEK SMALL LETTER OMEGA WITH PERISPOMENI AND YPOGEGRAMMENI\n\u1ff7 = with-accent-iota \u03c9\u2032 lower circumflex\n\u1ff8 : Token -- U+1FF8  \u1ff8 GREEK CAPITAL LETTER OMICRON WITH VARIA\n\u1ff8 = with-accent \u03bf\u2032 upper grave\n\u1ff9 : Token -- U+1FF9  \u1ff9 GREEK CAPITAL LETTER OMICRON WITH OXIA\n\u1ff9 = with-accent \u03bf\u2032 upper acute\n\u1ffa : Token -- U+1FFA  \u1ffa GREEK CAPITAL LETTER OMEGA WITH VARIA\n\u1ffa = with-accent \u03c9\u2032 upper grave\n\u1ffb : Token -- U+1FFB  \u1ffb GREEK CAPITAL LETTER OMEGA WITH OXIA\n\u1ffb = with-accent \u03c9\u2032 upper acute\n\u1ffc : Token -- U+1FFC  \u1ffc GREEK CAPITAL LETTER OMEGA WITH PROSGEGRAMMENI\n\u1ffc = with-iota \u03c9\u2032 upper\n-- U+1FFD  \u1ffd GREEK OXIA\n-- U+1FFE  \u1ffe GREEK DASIA\n-- U+1FFF\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f2de0a468936233e3d595f33b3cdc3fdcc6f4d4d","subject":"Comment _[mod_]","message":"Comment _[mod_]\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\nopen import Type hiding (\u2605)\n-- cleanup\nimport Level\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : \u2605\u2080\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : \u2605 a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n|not| : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n|not| f = not \u2218 f\n\n|\u2227|-comm : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2227| g \u2257 g |\u2227| f\n|\u2227|-comm f g x with f x | g x\n... | 0b | 0b = refl\n... | 0b | 1b = refl\n... | 1b | 0b = refl\n... | 1b | 1b = refl\n\n\n-- open Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : \u2605 a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : \u2605 a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 \u2605\u2080 where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : \u2605 a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : \u2605 a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\nopen import Type hiding (\u2605)\n-- cleanup\nimport Level\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : \u2605\u2080\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : \u2605 a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n|not| : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n|not| f = not \u2218 f\n\n|\u2227|-comm : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2227| g \u2257 g |\u2227| f\n|\u2227|-comm f g x with f x | g x\n... | 0b | 0b = refl\n... | 0b | 1b = refl\n... | 1b | 0b = refl\n... | 1b | 1b = refl\n\n\n-- open Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : \u2605 a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : \u2605 a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 \u2605\u2080 where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : \u2605 a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : \u2605 a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c181f41ef6789d4aae16e419e24ffafcc82f2930","subject":"agda: finish lift-term  (no proof) - see #22","message":"agda: finish lift-term  (no proof) - see #22\n\nI finally completed lift-term . For now, I don't want to know what it\ntakes to prove it correct.\n\nFor the proofs, it'll be probably better to reduce the number of\nspecial cases in the implementation and use lift-term -weakenOne more\noften.\n\nOld-commit-hash: 64c448b29b44c2d565e8aef74c977ea68d19eb25\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality using (sym)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality using (sym)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d6a14aa7b3c4d8a0f393b2eed14e14c2bdc9a124","subject":"add identity (dual P \u214b P)","message":"add identity (dual P \u214b P)\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 \u2297\u214b-dual (P m) _)\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = 0\u2081\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 \u2297\u214b-dual (P m) _)\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2f85b11aea376a63afbdd8643cc6af8c00769fa1","subject":"Added N92, ..., N99.","message":"Added N92, ..., N99.\n\nIgnore-this: 15470ce8f8307c4d1efbc233b4f22c57\n\ndarcs-hash:20110215220045-3bd4e-69a5d3b054fc1ddbd83411fd8b0c6b6448d70a42.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Unary\/IsN-ATP.agda","new_file":"src\/LTC\/Data\/Nat\/Unary\/IsN-ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- The unary numbers are LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Unary.IsN-ATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.Nat.Unary.Numbers\n\n------------------------------------------------------------------------------\n\npostulate\n  N0  : N zero\n  N1  : N one\n  N10 : N ten\n{-# ATP prove N0 #-}\n{-# ATP prove N1 #-}\n{-# ATP prove N10 #-}\n\npostulate\n  N11 : N eleven\n{-# ATP prove N11 #-}\n\npostulate\n  N89 : N eighty-nine\n  N90 : N ninety\n{-# ATP prove N89 #-}\n{-# ATP prove N90 #-}\n\npostulate\n  N91  : N ninety-one\n  N92  : N ninety-two\n  N93  : N ninety-three\n  N94  : N ninety-four\n  N95  : N ninety-five\n  N96  : N ninety-six\n  N97  : N ninety-seven\n  N98  : N ninety-eight\n  N99  : N ninety-nine\n  N100 : N one-hundred\n{-# ATP prove N91 #-}\n{-# ATP prove N92 #-}\n{-# ATP prove N93 #-}\n{-# ATP prove N94 #-}\n{-# ATP prove N95 #-}\n{-# ATP prove N96 #-}\n{-# ATP prove N97 #-}\n{-# ATP prove N98 #-}\n{-# ATP prove N99 #-}\n{-# ATP prove N100 #-}\n\npostulate\n  N101 : N hundred-one\n  N111 : N hundred-eleven\n{-# ATP prove N101 #-}\n{-# ATP prove N111 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The unary numbers are LTC natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Unary.IsN-ATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.Nat.Unary.Numbers\n\n------------------------------------------------------------------------------\n\npostulate\n  N0  : N zero\n  N1  : N one\n  N10 : N ten\n{-# ATP prove N0 #-}\n{-# ATP prove N1 #-}\n{-# ATP prove N10 #-}\n\npostulate\n  N11 : N eleven\n{-# ATP prove N11 #-}\n\npostulate\n  N89 : N eighty-nine\n  N90 : N ninety\n{-# ATP prove N89 #-}\n{-# ATP prove N90 #-}\n\npostulate\n  N91  : N ninety-one\n  N100 : N one-hundred\n{-# ATP prove N91 #-}\n{-# ATP prove N100 #-}\n\npostulate\n  N101 : N hundred-one\n  N111 : N hundred-eleven\n{-# ATP prove N101 #-}\n{-# ATP prove N111 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"96d0b01de293adbd79b12f62ebeb86111a48758f","subject":"updating to second PDF","message":"updating to second PDF\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41 \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4 \u21d2 \u03c4' \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d1' d2 \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)  \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d d' \u03c4 \u03c4' } \u2192\n                        d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           \u0394 \u22a2 d \u21a6 d'\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFinal : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41 \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4 \u21d2 \u03c4' \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d1' d2 \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)  \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d indet \u2192 d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           \u0394 \u22a2 d \u21a6 d'\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFinal : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"14a054cfa4101df7b7bd2679f22f9f5e4fb6ad59","subject":"Adapt file to Agda 2.5.3 and agda-stdlib 0.14","message":"Adapt file to Agda 2.5.3 and agda-stdlib 0.14\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalUntyped.agda","new_file":"Thesis\/ANormalUntyped.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set\n\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  term : \u2200 {v} n (t : Term \u0393) \u2192\n    \u03c1 \u22a2 t \u2193[ n ] v \u2192\n    \u03c1 F\u22a2 term t \u2193[ n ] v\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {dt : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs dt \u2193 dclosure dt \u03c1 d\u03c1\n  dterm : \u2200 {dv} (dt : DTerm \u0393) \u2192\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 F\u22a2 dterm dt \u2193 dv\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k < n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 =  <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n =  proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n =  proj\u2082 (rrelTV3 n)\n\n  s-rrelV3 :\n    \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n    \u2200 j \u2192 j < k \u2192 rrelV3-Type (k \u2238 j)\n  s-rrelV3 k rec-rrelTV3 rrelV3-k 0 _ = rrelV3-k\n  s-rrelV3 k rec-rrelTV3 rrelV3-k (suc j) sucj<k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt j k sucj<k))\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    -- rrelV3-Type n\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n      (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    \u2200 j (j<k : j < k) n2 \u2192\n    (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 k rec-rrelTV3 rrelV3-k j j<k v1 dv v2\n    -- where\n    --   s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n    --   -- If j = 0, we can't do a recursive call on (k - j) because that's not\n    --   -- well-founded. Luckily, we don't need to, just use relV as defined at\n    --   -- the same level.\n    --   -- Yet, this will be a pain in proofs.\n    --   s-rrelV3 0 _ = rrelV3-k\n    --   s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelV3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 _ _ _ = \u22a5\n\nopen import Postulate.Extensionality\nmutual\n  rrelT3-equiv : \u2200 k {\u0393} {t1 : Fun \u0393} {dt t2 \u03c11 d\u03c1 \u03c12} \u2192\n    rrelT3 k t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2261\n    \u2200 j (j<k : j < k) n2 \u2192\n    \u2200 (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 j v1 dv v2\n  rrelT3-equiv n = cong (\u03bb \u25a1 \u2192 \u2200 j \u2192 \u25a1 j) (ext (\u03bb j \u2192 {!!}))\n\n  rrelV3-equiv : \u2200 n {\u03931 \u03932 \u0394\u0393 t1 dt t2 \u03c11 \u03c1' d\u03c1 \u03c12} \u2192\n    rrelV3 n (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) \u2261 \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        rrelV3 k v1 dv v2 \u2192\n        rrelT3 k\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-equiv n = cong (\u03bb \u25a1 \u2192 \u03a3 (_ \u2261 _ \u00d7 _ \u2261 _) \u25a1) (ext (\u03bb { (eq1 , eq2) \u2192\n    cong\n      (\u03bb \u25a1 \u2192\n         (\u03bb { (refl , refl)\n                \u2192 \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192 \u25a1 k k<n v1 dv v2\n            })\n         eq1 eq2)\n      (ext (\u03bb x \u2192 ext {!!}))}))\n  --\n  -- cong (\u03bb \u25a1 \u2192 \u2200 k \u2192 (k<n : k <\u2032 n) \u2192 \u25a1 k k<n)\n  -- rrelV3-equiv k\n\nrrel\u03c13 : \u2115 \u2192 \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13 n \u2205 \u2205 \u2205 \u2205 = \u22a4\nrrel\u03c13 n (Uni \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) = rrelV3 n v1 dv v2 \u00d7 rrel\u03c13 n \u0393 \u03c11 d\u03c1 \u03c12\n\n--\nrfundamental3 : \u2200 {\u0393} (t : Fun \u0393) \u2192 \u2200 k \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 k \u0393 \u03c11 d\u03c1 \u03c12) \u2192 rrelT3 k t (derive-dfun t) t \u03c11 d\u03c1 \u03c12\nrfundamental3 (term x) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 j j<k n2 v1 v2 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 = {!!}\nrfundamental3 {\u0393} (abs t) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .0 j<k .0 (closure .t .\u03c11) (closure .t .\u03c12) abs abs = dclosure (derive-dfun t) \u03c11 d\u03c1 , 0 , dabs , (refl , refl) , {!body !}\n  where\n    body1 : \u2200 k\u2081 (k\u2081<k : k\u2081 <\u2032 k) \u2192 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 rrelV3 k\u2081 v1 dv v2 \u2192 rrelT3 k\u2081 t (derive-dfun t) t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12)\n    body1 = \u03bb k\u2081 k\u2081<k v1 dv v2 x v3 v4 j n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n    rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n    !}\n  -- \u03bb\n  -- { k\u2081 k<n v1 dv v2 vv v3 v4 0 n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- !}\n  -- -- (Some.wfRec-builder rrelTV3-Type rrelTV3-step k\u2081\n  --                     -- (<-well-founded\u2032 k k\u2081 k<n))\n\n  -- -- All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k k\u2081 k<n : rrelTV3-Type k\u2081\n  -- --\n  -- ; k\u2081 k<n v1 dv v2 vv v3 v4 (suc j) n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- !}\n  -- -- rfundamental3 t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 {!suc j !} n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- }\n\n  --   -- \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n  --   -- \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  --   foo : s-rrelV3 k\n  --     (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k)\n  --     (rrelV3-step k (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k))\n  --     j j<k v1 (dclosure (derive-dfun t) \u03c11 d\u03c1) v2\n  --   foo with j\n  --   ... | s = {!!}\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.ANormalUntyped where\n\nopen import Data.Empty\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Nat\nimport Data.Integer.Base as I\nopen I using (\u2124)\nopen import Data.Integer.Base using (\u2124)\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nrecord DType : Set where\n  constructor DUni\n\nopen import Base.Syntax.Context Type public\nimport Base.Syntax.Context DType as DC\n\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = DUni \u2022 \u0394\u0394 \u0393\n\nderive-dvar : \u2200 {\u0394} \u2192 (x : Var \u0394 Uni) \u2192 DC.Var (\u0394\u0394 \u0394) DUni\nderive-dvar this = DC.this\nderive-dvar (that x) = DC.that (derive-dvar x)\n\ndata DTerm : (\u0394 : Context) \u2192 Set where\n  dvar : \u2200 {\u0394} (x : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    DTerm \u0394\n  dlett : \u2200 {\u0394} \u2192\n    (f : Var \u0394 Uni) \u2192\n    (x : Var \u0394 Uni) \u2192\n    (t : Term (Uni \u2022 \u0394)) \u2192\n    (df : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dx : DC.Var (\u0394\u0394 \u0394) DUni) \u2192\n    (dt : DTerm (Uni \u2022 \u0394)) \u2192\n    DTerm \u0394\n\nderive-dterm : \u2200 {\u0394} \u2192 (t : Term \u0394) \u2192 DTerm \u0394\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- I don't necessarily recommend having a separate syntactic category for\n-- functions, but we should prove a fundamental lemma for them too, somehow.\n-- I'll probably end up with some ANF allowing lambdas to do the semantics.\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\ndata DFun (\u0394 : Context) : Set where\n  dterm : DTerm \u0394 \u2192 DFun \u0394\n  dabs : DFun (Uni \u2022 \u0394) \u2192 DFun \u0394\n\nderive-dfun : \u2200 {\u0394} \u2192 (t : Fun \u0394) \u2192 DFun \u0394\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\ndata DVal : DType \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\nimport Base.Denotation.Environment DType DVal as D\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2115) \u2192 Val Uni\n\ndata DVal where\n  dclosure : \u2200 {\u0393} \u2192 (dt : DFun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) \u2192 DVal DUni\n  dintV : \u2200 (n : \u2124) \u2192 DVal DUni\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\n-- XXX separate syntax is a bit dangerous. Also, do I want to be so accurate relative to the original model?\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Term \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set\n\ndata _F\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : Fun \u0393 \u2192 \u2115 \u2192 Val Uni \u2192 Set where\n  abs : \u2200 {t : Fun (Uni \u2022 \u0393)} \u2192\n    \u03c1 F\u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  term : \u2200 {v} n (t : Term \u0393) \u2192\n    \u03c1 \u22a2 t \u2193[ n ] v \u2192\n    \u03c1 F\u22a2 term t \u2193[ n ] v\n\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) where\n  var : \u2200 (x : Var \u0393 Uni) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lett : \u2200 n1 n2 {\u0393' \u03c1\u2032 v1 v2 v3} {f x t t'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    \u03c1 \u22a2 lett f x t \u2193[ suc (suc (n1 + n2)) ] v3\n  -- lit : \u2200 n \u2192\n  --   \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- data _D_\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : D.\u27e6 \u0394\u0394 \u0393 \u27e7Context) : DTerm \u0393 \u2192 \u2115 \u2192 DVal DUni \u2192 Set where\n--   dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar x \u2193[ 0 ] (D.\u27e6 x \u27e7Var d\u03c1)\n--   dlett : \u2200 n1 n2 n3 n4 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n--     \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n--     \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n--     (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n--     (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n--     -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n--     -- definitions.\n--     \u03c1 D d\u03c1 \u22a2 dvar df \u2193[ 0 ] dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n--     \u03c1 D d\u03c1 \u22a2 dvar dx \u2193[ 0 ] dv1 \u2192\n--     (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193[ n3 ] dv2 \u2192\n--     (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193[ n4 ] dv3 \u2192\n--     \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193[ suc (suc (n1 + n2)) ] dv3\n\n\n-- Do I need to damn count steps here? No.\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DTerm \u0393 \u2192 DVal DUni \u2192 Set\n\ndata _D_F\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : DFun \u0393 \u2192 DVal DUni \u2192 Set where\n  dabs : \u2200 {dt : DFun (Uni \u2022 \u0393)} \u2192\n    \u03c1 D d\u03c1 F\u22a2 dabs dt \u2193 dclosure dt \u03c1 d\u03c1\n  dterm : \u2200 {dv} (dt : DTerm \u0393) \u2192\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 F\u22a2 dterm dt \u2193 dv\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) where\n  dvar : \u2200 (x : DC.Var (\u0394\u0394 \u0393) DUni) \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar x \u2193 (D.\u27e6 x \u27e7Var d\u03c1)\n  dlett : \u2200 n1 n2 {\u0393' \u03c1\u2032 \u03c1'' d\u03c1' v1 v2 v3 dv1 dv2 dv3} {f x t df dx dt t' dt'}  \u2192\n    \u03c1 \u22a2 var f \u2193[ 0 ] closure {\u0393'} t' \u03c1\u2032 \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] v1 \u2192\n    (v1 \u2022 \u03c1\u2032) F\u22a2 t' \u2193[ n1 ] v2 \u2192\n    (v2 \u2022 \u03c1) \u22a2 t \u2193[ n2 ] v3 \u2192\n    -- With a valid input \u03c1' and \u03c1'' coincide? Varies among plausible validity\n    -- definitions.\n    \u03c1 D d\u03c1 \u22a2 dvar df \u2193 dclosure {\u0393'} dt' \u03c1'' d\u03c1' \u2192\n    \u03c1 D d\u03c1 \u22a2 dvar dx \u2193 dv1 \u2192\n    (v1 \u2022 \u03c1'') D (dv1 \u2022 d\u03c1') F\u22a2 dt' \u2193 dv2 \u2192\n    (v2 \u2022 \u03c1) D (dv2 \u2022 d\u03c1) \u22a2 dt \u2193 dv3 \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett f x t df dx dt \u2193 dv3\n\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen import Relation.Binary hiding (_\u21d2_)\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsuc\u2238suc : \u2200 m n \u2192 n < m \u2192 suc (m \u2238 suc n) \u2261 m \u2238 n\nsuc\u2238suc (suc m) zero (s\u2264s n<m) = refl\nsuc\u2238suc (suc m) (suc n) (s\u2264s n<m) = suc\u2238suc m n n<m\n\nlemlt : \u2200 j k \u2192 suc j < k \u2192 k \u2238 suc j < k\nlemlt j (suc k) (s\u2264s j<k) = s\u2264s (\u2238-mono {k} {k} {j} {0} \u2264-refl z\u2264n)\n\nlemlt\u2032 : \u2200 j k \u2192 suc j <\u2032 k \u2192 k \u2238 suc j <\u2032 k\nlemlt\u2032 j k j<\u2032k = \u2264\u21d2\u2264\u2032 (lemlt j k (\u2264\u2032\u21d2\u2264 j<\u2032k))\n\nopen import Induction.WellFounded\nopen import Induction.Nat\nrrelT3-Type : \u2115 \u2192 Set\u2081\nrrelT3-Type n =\n  \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n  (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n\nrrelV3-Type : \u2115 \u2192 Set\u2081\nrrelV3-Type n = \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n\nrrelTV3-Type : \u2115 \u2192 Set\u2081\nrrelTV3-Type n = rrelT3-Type n \u00d7 rrelV3-Type n\n\n-- The type of the function availalbe for recursive calls.\nrrelTV3-recSubCallsT : \u2115 \u2192 Set\u2081\nrrelTV3-recSubCallsT n = (k : \u2115) \u2192 k <\u2032 n \u2192 rrelTV3-Type k\n\nmutual\n  -- Assemble together\n  rrelTV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192 rrelTV3-Type n\n  rrelTV3-step n rec-rrelTV3 =\n    let rrelV3-n = rrelV3-step n rec-rrelTV3\n    in rrelT3-step n rec-rrelTV3 rrelV3-n , rrelV3-n\n  rrelTV3 = <-rec rrelTV3-Type rrelTV3-step\n  rrelT3 : \u2200 n \u2192 rrelT3-Type n\n  rrelT3 n = proj\u2081 (rrelTV3 n)\n  rrelV3 : \u2200 n \u2192 rrelV3-Type n\n  rrelV3 n = proj\u2082 (rrelTV3 n)\n\n  s-rrelV3 :\n    \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n    \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  s-rrelV3 k rec-rrelTV3 rrelV3-k 0 _ = rrelV3-k\n  s-rrelV3 k rec-rrelTV3 rrelV3-k (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n  -- Have the same context for t1, dt and t2? Yeah, good, that's what we want in the end...\n  -- Though we might want more flexibility till when we have replacement\n  -- changes.\n  rrelT3-step : \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192\n    rrelV3-Type k \u2192\n    -- rrelV3-Type n\n    \u2200 {\u0393} \u2192 (t1 : Fun \u0393) (dt : DFun \u0393) (t2 : Fun \u0393)\n      (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n  rrelT3-step k rec-rrelTV3 rrelV3-k =\n    \u03bb t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2192\n    \u2200 j (j<k : j <\u2032 k) n2 \u2192\n    (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    s-rrelV3 k rec-rrelTV3 rrelV3-k j j<k v1 dv v2\n    -- where\n    --   s-rrelV3 : \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n    --   -- If j = 0, we can't do a recursive call on (k - j) because that's not\n    --   -- well-founded. Luckily, we don't need to, just use relV as defined at\n    --   -- the same level.\n    --   -- Yet, this will be a pain in proofs.\n    --   s-rrelV3 0 _ = rrelV3-k\n    --   s-rrelV3 (suc j) sucj<\u2032k = proj\u2082 (rec-rrelTV3 (k \u2238 suc j) (lemlt\u2032 j k sucj<\u2032k))\n\n  rrelV3-step : \u2200 n \u2192 rrelTV3-recSubCallsT n \u2192\n    -- rrelV3-Type n\n    \u2200 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 Set\n  rrelV3-step n rec-rrelTV3 (intV v1) (dintV dv) (intV v2) = dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV3-step n rec-rrelTV3 (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) =\n      -- Require a proof that the two contexts match:\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        proj\u2082 (rec-rrelTV3 k k<n) v1 dv v2 \u2192\n        proj\u2081 (rec-rrelTV3 k k<n)\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-step n rec-rrelTV3 _ _ _ = \u22a5\n\nopen import Postulate.Extensionality\nmutual\n  rrelT3-equiv : \u2200 k {\u0393} {t1 : Fun \u0393} {dt t2 \u03c11 d\u03c1 \u03c12} \u2192\n    rrelT3 k t1 dt t2 \u03c11 d\u03c1 \u03c12 \u2261\n    \u2200 j (j<k : j <\u2032 k) n2 \u2192\n    \u2200 (v1 v2 : Val Uni) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 F\u22a2 t1 \u2193[ j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 F\u22a2 t2 \u2193[ n2 ] v2) \u2192\n    \u03a3[ dv \u2208 DVal DUni ] \u03a3[ dn \u2208 \u2115 ]\n    \u03c11 D d\u03c1 F\u22a2 dt \u2193 dv \u00d7\n    rrelV3 j v1 dv v2\n  rrelT3-equiv n = cong (\u03bb \u25a1 \u2192 \u2200 j \u2192 \u25a1 j) (ext (\u03bb j \u2192 {!!}))\n\n  rrelV3-equiv : \u2200 n {\u03931 \u03932 \u0394\u0393 t1 dt t2 \u03c11 \u03c1' d\u03c1 \u03c12} \u2192\n    rrelV3 n (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1' d\u03c1) (closure {\u03932} t2 \u03c12) \u2261 \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192\n        rrelV3 k v1 dv v2 \u2192\n        rrelT3 k\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n      }\n  rrelV3-equiv n = cong (\u03bb \u25a1 \u2192 \u03a3 (_ \u2261 _ \u00d7 _ \u2261 _) \u25a1) (ext (\u03bb { (eq1 , eq2) \u2192\n    cong\n      (\u03bb \u25a1 \u2192\n         (\u03bb { (refl , refl)\n                \u2192 \u2200 (k : \u2115) (k<n : k <\u2032 n) v1 dv v2 \u2192 \u25a1 k k<n v1 dv v2\n            })\n         eq1 eq2)\n      (ext (\u03bb x \u2192 ext {!!}))}))\n  --\n  -- cong (\u03bb \u25a1 \u2192 \u2200 k \u2192 (k<n : k <\u2032 n) \u2192 \u25a1 k k<n)\n  -- rrelV3-equiv k\n\nrrel\u03c13 : \u2115 \u2192 \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13 n \u2205 \u2205 \u2205 \u2205 = \u22a4\nrrel\u03c13 n (Uni \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) = rrelV3 n v1 dv v2 \u00d7 rrel\u03c13 n \u0393 \u03c11 d\u03c1 \u03c12\n\n--\nrfundamental3 : \u2200 {\u0393} (t : Fun \u0393) \u2192 \u2200 k \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 k \u0393 \u03c11 d\u03c1 \u03c12) \u2192 rrelT3 k t (derive-dfun t) t \u03c11 d\u03c1 \u03c12\nrfundamental3 (term x) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 j j<k n2 v1 v2 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 = {!!}\nrfundamental3 {\u0393} (abs t) k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .0 j<k .0 (closure .t .\u03c11) (closure .t .\u03c12) abs abs = dclosure (derive-dfun t) \u03c11 d\u03c1 , 0 , dabs , (refl , refl) , {!body !}\n  where\n    body1 : \u2200 k\u2081 (k\u2081<k : k\u2081 <\u2032 k) \u2192 (v1 : Val Uni) (dv : DVal DUni) (v2 : Val Uni) \u2192 rrelV3 k\u2081 v1 dv v2 \u2192 rrelT3 k\u2081 t (derive-dfun t) t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12)\n    body1 = \u03bb k\u2081 k\u2081<k v1 dv v2 x v3 v4 j n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n    rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n    !}\n  -- \u03bb\n  -- { k\u2081 k<n v1 dv v2 vv v3 v4 0 n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- rfundamental3 {Uni \u2022 \u0393} t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 0 n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- !}\n  -- -- (Some.wfRec-builder rrelTV3-Type rrelTV3-step k\u2081\n  --                     -- (<-well-founded\u2032 k k\u2081 k<n))\n\n  -- -- All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k k\u2081 k<n : rrelTV3-Type k\u2081\n  -- --\n  -- ; k\u2081 k<n v1 dv v2 vv v3 v4 (suc j) n2 j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192 {!\n  -- !}\n  -- -- rfundamental3 t k\u2081 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) ({!vv!} , {!\u03c1\u03c1!}) v3 v4 {!suc j !} n2 {!j<k\u2081 !} {! \u03c11\u22a2t1\u2193[j]v1 !} \u03c12\u22a2t2\u2193[n2]v2\n  -- }\n\n  --   -- \u2200 k \u2192 rrelTV3-recSubCallsT k \u2192 rrelV3-Type k \u2192\n  --   -- \u2200 j \u2192 j <\u2032 k \u2192 rrelV3-Type (k \u2238 j)\n  --   foo : s-rrelV3 k\n  --     (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k)\n  --     (rrelV3-step k (All.wfRec-builder <-well-founded _ rrelTV3-Type rrelTV3-step k))\n  --     j j<k v1 (dclosure (derive-dfun t) \u03c11 d\u03c1) v2\n  --   foo with j\n  --   ... | s = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9990f1bc8c4e0468ccc3fc89c7d352e19ee94044","subject":"matrix om levels voor te stellen","message":"matrix om levels voor te stellen\n","repos":"toonn\/popartt","old_file":"koopa.agda","new_file":"koopa.agda","new_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin\n  open import Data.List\n  open import Data.Vec renaming (map to vmap; lookup to vlookup)\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa Color : Set where\n    _KT : Color \u2192 KoopaTroopa Color\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n\n  data _follows_ : Position \u2192 Position \u2192 Set where\n    stay  : \u2200 {p} \u2192 p follows p\n    next  : \u2200 {x y mat} \u2192 pos (suc x) y mat follows pos x y mat\n    back  : \u2200 {x y mat} \u2192 pos x y mat follows pos (suc x) y mat\n    -- jump  : \u2200 {x y mat} \u2192 pos x (suc y) mat follows pos x y mat\n    fall  : \u2200 {x y mat} \u2192 pos x y mat follows pos x (suc y) mat\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  example_level : Matrix Position \n  example_level = []\n\n  ex_path : Path (Red KT) (pos 0 0 solid) (pos 0 0 solid)\n  ex_path = pos 0 0 solid \u21a0\u27e8 next \u27e9\n            pos 1 0 solid \u21a0\u27e8 back \u27e9\n            pos 0 0 solid \u21a0\u27e8 stay \u27e9 []\n","old_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\nopen import Data.Nat\n\ndata Color : Set where\n  Green : Color\n  Red : Color\n\ndata KoopaTroopa Color : Set where\n  _KT : Color \u2192 KoopaTroopa Color\n\nrecord Position : Set where\n  constructor pos\n  field\n    x : \u2115\n    y : \u2115\n\ndata _follows_ : Position \u2192 Position \u2192 Set where\n  still : \u2200 {p} \u2192 p follows p\n  next  : \u2200 {x y} \u2192 pos (suc x) y follows pos x y\n  back  : \u2200 {x y} \u2192 pos x y follows pos (suc x) y\n  -- jump  : \u2200 {x y} \u2192 pos x (suc y) follows pos x y\n  fall  : \u2200 {x y} \u2192 pos x y follows pos x (suc y)\n\n\ndata Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n  []  : \u2200 {p} \u2192 Path Koopa p p\n  _\u21a0[_]_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n           \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\nex_path : Path (Red KT) (pos 0 0) (pos 0 0)\nex_path = pos 0 0 \u21a0[ next ] (pos 1 0 \u21a0[ back ] [])\n","returncode":0,"stderr":"","license":"bsd-2-clause","lang":"Agda"}
{"commit":"3247753c272064f435d9e0390ca496ed0d6217d4","subject":"sha1: just display the cost","message":"sha1: just display the cost\n","repos":"crypto-agda\/crypto-agda","old_file":"sha1.agda","new_file":"sha1.agda","new_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1 where\n\nmodule FunSHA1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n    Word : T\n    Word = `Bits 32\n\n    map\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\n    map\u00b2\u02b7 = zipWith\n\n    map\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\n    map\u02b7 = map\n\n    lift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\n    lift f g = g \u204f f\n\n    lift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n    lift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n    `not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    `not = lift (map\u02b7 not)\n\n    infixr 3 _`\u2295_\n    _`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\n    infixr 3 _`\u2227_\n    _`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\n    infixr 2 _`\u2228_\n    _`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\n    open import Solver.Linear\n\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                   \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                   \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    <\u229e> adder : Word `\u00d7 Word `\u2192 Word\n    adder = <tt\u204f <0b> , id > \u204f iter full-adder\n    <\u229e> = adder\n\n    infixl 4 _`\u229e_\n    _`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n    _`\u229e_ = lift\u2082 <\u229e>\n\n    <_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n    < f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n    <_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                                  \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                                  \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n    < f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n    <<<\u2085 : `Endo Word\n    <<<\u2085 = rot-left 5\n\n    <<<\u2083\u2080 : `Endo Word\n    <<<\u2083\u2080 = rot-left 30\n\n    Word\u00b2 = Word `\u00d7 Word\n    Word\u00b3 = Word `\u00d7 Word\u00b2\n    Word\u2074 = Word `\u00d7 Word\u00b3\n    Word\u2075 = Word `\u00d7 Word\u2074\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 =  constBits (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0'\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0'\n            F\u25b9\ud835\udfda (suc _) = 1'\n\n            {-\n    [_-_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m - n mod p ] = {!!}\n\n    [_+_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m + n mod p ] = {!!}\n    -}\n\n    #\u02b7 = bits 32\n\n    <\u229e\u2075> : Word\u2075 `\u00d7 Word\u2075 `\u2192 Word\u2075\n    <\u229e\u2075> = helper \u204f < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 <\u229e> > > > >\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {C} {D} {E} {F} {G} {H} {I} {J} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 C \u2237 D \u2237 E \u2237 F \u2237 G \u2237 H \u2237 I \u2237 J \u2237 [])\n                  (\u03bb a b c d e f g h i j \u2192\n                    ((a , b , c , d , e) , (f , g , h , i , j) \u21a6\n                     ((a , f) , (b , g) , (c , h) , (d , i) , (e , j))))\n\n    iterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\n    iterate\u207f zero    f = id\n    iterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n    _\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n    _\u00b2\u2070 = iterate\u207f 20\n\n    module _ where\n\n      K\u2080 = #\u02b7 0x5A827999\n      K\u2082 = #\u02b7 0x6ED9EBA1\n      K\u2084 = #\u02b7 0x8F1BBCDC\n      K\u2086 = #\u02b7 0xCA62C1D6\n\n      H0 = #\u02b7 0x67452301\n      H1 = #\u02b7 0xEFCDAB89\n      H2 = #\u02b7 0x98BADCFE\n      H3 = #\u02b7 0x10325476\n      H4 = #\u02b7 0xC3D2E1F0\n\n      A B C D E : Word\u2075 `\u2192 Word\n      A = fst\n      B = snd \u204f fst\n      C = snd \u204f snd \u204f fst\n      D = snd \u204f snd \u204f snd \u204f fst\n      E = snd \u204f snd \u204f snd \u204f snd\n\n      F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n      F\u2082 = B `\u2295 C `\u2295 D\n      F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n      F\u2086 = F\u2082\n\n      module _ (F : Word\u2075 `\u2192 Word)\n               (K : `\ud835\udfd9  `\u2192 Word)\n               (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n          where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n      module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n        W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n        W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n        W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n        W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n        W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n        Iteration\u2078\u2070 : `Endo Word\u2075\n        Iteration\u2078\u2070 =\n          (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n          (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n          (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n          (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n        pad0s : \u2115 \u2192 \u2115\n        pad0s zero = 512 \u2238 65\n        pad0s (suc _) = STUCK where postulate STUCK : \u2115\n        -- pad0s n = [ 512 - [ n + 65 mod 512 ] mod 512 ]\n\n        paddedLength : \u2115 \u2192 \u2115\n        paddedLength n = n + (1 + pad0s n + 64)\n\n        padding : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (paddedLength n)\n        padding {n} = < id ,tt\u204f <1\u2237 < <0\u207f> {pad0s n} ++ bits 64 n > > > \u204f append\n\n        ite : Endo (`Endo Word\u2075)\n        ite f = dup \u204f first f \u204f <\u229e\u2075>\n\n        hash-block : `Endo Word\u2075\n        hash-block = ite Iteration\u2078\u2070\n\n      ite' : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word)\n             \u2192 ((Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `Endo Word\u2075) \u2192 `Endo Word\u2075\n      ite' zero    W f = id\n      ite' (suc n) W f = f (W zero) \u204f ite' n (W \u2218 suc) f\n\n      SHA1 : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1 n W = < H0 , H1 , H2 , H3 , H4 >\n               \u204f ite' n W hash-block\n\n      SHA1-on-0s : `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1-on-0s = SHA1 1 (\u03bb _ _ \u2192 <0\u207f>)\n\nmodule AgdaSHA1 where\n  open import FunUniverse.Agda\n  open FunSHA1 agdaFunOps\n  open import Data.Two\n\nopen import IO\nimport IO.Primitive\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Coinduction\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1' = putStr \"1\"\nputBit 0' = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\nput\u00d7 : \u2200 {A B : Set} \u2192 (A \u2192 IO \ud835\udfd9) \u2192 (B \u2192 IO \ud835\udfd9) \u2192 (A \u00d7 B) \u2192 IO \ud835\udfd9\nput\u00d7 pA pB (x , y) = \u266f pA x >> \u266f pB y\n{-\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits\n          putBits))) AgdaSHA1.SHA1-on-0s)\n-}\nfirstBit : \u2200 {A : Set} \u2192 (V.Vec \ud835\udfda 32 \u00d7 A) \u2192 \ud835\udfda\nfirstBit ((b \u2237 _) , _) = b\nimport FunUniverse.Cost as Cost\nopen import Data.Nat.Show\nsha1-cost : \u2115\nsha1-cost = FunSHA1.SHA1-on-0s Cost.timeOps\nmain : IO.Primitive.IO \ud835\udfd9\n--main = IO.run (putBit (firstBit (AgdaSHA1.SHA1-on-0s _)))\nmain = IO.run (putStrLn (show sha1-cost))\n\n-- -}\n","old_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1 where\n\nmodule FunSHA1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n    Word : T\n    Word = `Bits 32\n\n    map\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\n    map\u00b2\u02b7 = zipWith\n\n    map\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\n    map\u02b7 = map\n\n    lift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\n    lift f g = g \u204f f\n\n    lift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n    lift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n    `not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    `not = lift (map\u02b7 not)\n\n    infixr 3 _`\u2295_\n    _`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\n    infixr 3 _`\u2227_\n    _`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\n    infixr 2 _`\u2228_\n    _`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\n    open import Solver.Linear\n\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                   \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                   \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    <\u229e> adder : Word `\u00d7 Word `\u2192 Word\n    adder = <tt\u204f <0b> , id > \u204f iter full-adder\n    <\u229e> = adder\n\n    infixl 4 _`\u229e_\n    _`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n    _`\u229e_ = lift\u2082 <\u229e>\n\n    <_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n    < f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n    <_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                                  \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                                  \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n    < f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n    <<<\u2085 : `Endo Word\n    <<<\u2085 = rot-left 5\n\n    <<<\u2083\u2080 : `Endo Word\n    <<<\u2083\u2080 = rot-left 30\n\n    Word\u00b2 = Word `\u00d7 Word\n    Word\u00b3 = Word `\u00d7 Word\u00b2\n    Word\u2074 = Word `\u00d7 Word\u00b3\n    Word\u2075 = Word `\u00d7 Word\u2074\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 =  constBits (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0'\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0'\n            F\u25b9\ud835\udfda (suc _) = 1'\n\n            {-\n    [_-_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m - n mod p ] = {!!}\n\n    [_+_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m + n mod p ] = {!!}\n    -}\n\n    #\u02b7 = bits 32\n\n    <\u229e\u2075> : Word\u2075 `\u00d7 Word\u2075 `\u2192 Word\u2075\n    <\u229e\u2075> = helper \u204f < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 < <\u229e> \u00d7 <\u229e> > > > >\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {C} {D} {E} {F} {G} {H} {I} {J} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 C \u2237 D \u2237 E \u2237 F \u2237 G \u2237 H \u2237 I \u2237 J \u2237 [])\n                  (\u03bb a b c d e f g h i j \u2192\n                    ((a , b , c , d , e) , (f , g , h , i , j) \u21a6\n                     ((a , f) , (b , g) , (c , h) , (d , i) , (e , j))))\n\n    iterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\n    iterate\u207f zero    f = id\n    iterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n    _\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n    _\u00b2\u2070 = iterate\u207f 20\n\n    module _ where\n\n      K\u2080 = #\u02b7 0x5A827999\n      K\u2082 = #\u02b7 0x6ED9EBA1\n      K\u2084 = #\u02b7 0x8F1BBCDC\n      K\u2086 = #\u02b7 0xCA62C1D6\n\n      H0 = #\u02b7 0x67452301\n      H1 = #\u02b7 0xEFCDAB89\n      H2 = #\u02b7 0x98BADCFE\n      H3 = #\u02b7 0x10325476\n      H4 = #\u02b7 0xC3D2E1F0\n\n      A B C D E : Word\u2075 `\u2192 Word\n      A = fst\n      B = snd \u204f fst\n      C = snd \u204f snd \u204f fst\n      D = snd \u204f snd \u204f snd \u204f fst\n      E = snd \u204f snd \u204f snd \u204f snd\n\n      F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n      F\u2082 = B `\u2295 C `\u2295 D\n      F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n      F\u2086 = F\u2082\n\n      module _ (F : Word\u2075 `\u2192 Word)\n               (K : `\ud835\udfd9  `\u2192 Word)\n               (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n          where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n      module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n        W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n        W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n        W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n        W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n        W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n        Iteration\u2078\u2070 : `Endo Word\u2075\n        Iteration\u2078\u2070 =\n          (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n          (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n          (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n          (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n        pad0s : \u2115 \u2192 \u2115\n        pad0s zero = 512 \u2238 65\n        pad0s (suc _) = STUCK where postulate STUCK : \u2115\n        -- pad0s n = [ 512 - [ n + 65 mod 512 ] mod 512 ]\n\n        paddedLength : \u2115 \u2192 \u2115\n        paddedLength n = n + (1 + pad0s n + 64)\n\n        padding : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (paddedLength n)\n        padding {n} = < id ,tt\u204f <1\u2237 < <0\u207f> {pad0s n} ++ bits 64 n > > > \u204f append\n\n        ite : Endo (`Endo Word\u2075)\n        ite f = dup \u204f first f \u204f <\u229e\u2075>\n\n        hash-block : `Endo Word\u2075\n        hash-block = ite Iteration\u2078\u2070\n\n      ite' : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word)\n             \u2192 ((Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `Endo Word\u2075) \u2192 `Endo Word\u2075\n      ite' zero    W f = id\n      ite' (suc n) W f = f (W zero) \u204f ite' n (W \u2218 suc) f\n\n      SHA1 : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1 n W = < H0 , H1 , H2 , H3 , H4 >\n               \u204f ite' n W hash-block\n\nmodule AgdaSHA1 where\n  open import FunUniverse.Agda\n  open FunSHA1 agdaFunOps\n  open import Data.Two\n\n  SHA1-on-0s : Word\u2075\n  SHA1-on-0s = SHA1 1 (\u03bb _ _ _ \u2192 V.replicate 0') _\n\nopen import IO\nimport IO.Primitive\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Coinduction\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1' = putStr \"1\"\nputBit 0' = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\nput\u00d7 : \u2200 {A B : Set} \u2192 (A \u2192 IO \ud835\udfd9) \u2192 (B \u2192 IO \ud835\udfd9) \u2192 (A \u00d7 B) \u2192 IO \ud835\udfd9\nput\u00d7 pA pB (x , y) = \u266f pA x >> \u266f pB y\n{-\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits\n          putBits))) AgdaSHA1.SHA1-on-0s)\n-}\nfirstBit : \u2200 {A : Set} \u2192 (V.Vec \ud835\udfda 32 \u00d7 A) \u2192 \ud835\udfda\nfirstBit ((b \u2237 _) , _) = b\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (putBit (firstBit AgdaSHA1.SHA1-on-0s))\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f67d5433e55c497b4348966a18f984631b3f4591","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: 4f681c804a67430813f97c1cc20b44f4\n\ndarcs-hash:20120217140437-3bd4e-99fea7a6d09b6da3ebeb66c427a3870abfba4dab.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Data.Stream.Equality\n\n-----------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h\u2081 with (Stream-gfp\u2081 h\u2081)\n... | e , es , Ses , h\u2082 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2082))) Ses\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n            \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  helper\u2081 {ws} (zs , ws\u2248zs) with \u2248-gfp\u2081 ws\u2248zs\n  ... | w' , ws' , zs' , ws'\u2248zs' , ws\u2261w'\u2237ws' , _ =\n    w' , ws' , (zs' , ws'\u2248zs') , ws\u2261w'\u2237ws'\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  helper\u2082  {zs} (ws , ws\u2248zs) with \u2248-gfp\u2081 ws\u2248zs\n  ... | w' , ws' , zs' , ws'\u2248zs' , _ , zs\u2261w'\u2237zs' =\n    w' , zs' , (ws' , ws'\u2248zs') , zs\u2261w'\u2237zs'\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Data.Stream.Equality\n\n-----------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h with (Stream-gfp\u2081 h)\n... | e , es , Ses , x\u2237xs\u2261e\u2237es =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective x\u2237xs\u2261e\u2237es))) Ses\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P\u2081 helper\u2081 (ys , xs\u2248ys)\n                         , Stream-gfp\u2082 P\u2082 helper\u2082 (xs , xs\u2248ys)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  helper\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192\n            \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  helper\u2081 {ws} (zs , ws\u2248zs) with \u2248-gfp\u2081 ws\u2248zs\n  ... | w' , ws' , zs' , ws'\u2248zs' , ws\u2261w'\u2237ws' , _ =\n    w' , ws' , (zs' , ws'\u2248zs') , ws\u2261w'\u2237ws'\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  helper\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  helper\u2082  {zs} (ws , ws\u2248zs) with \u2248-gfp\u2081 ws\u2248zs\n  ... | w' , ws' , zs' , ws'\u2248zs' , _ , zs\u2261w'\u2237zs' =\n    w' , zs' , (ws' , ws'\u2248zs') , zs\u2261w'\u2237zs'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3e20088c8ccfe364dc298f6a83bede2050be2af2","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 981da50b6a4557259e63b1fa4b1d4776\n\ndarcs-hash:20101122135209-3bd4e-172a2e198641c1e2c9f1f1fed9823867e89255ca.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PCF\/Examples\/GCD\/EquationsER.agda","new_file":"src\/PCF\/Examples\/GCD\/EquationsER.agda","new_contents":"------------------------------------------------------------------------------\n-- Equations for the greatest common divisor\n------------------------------------------------------------------------------\n\nmodule PCF.Examples.GCD.EquationsER where\n\nopen import LTC.Base\nopen import LTC.BaseER using ( subst )\n\nimport Lib.Relation.Binary.EqReasoning\nopen module APER = Lib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\nopen import PCF.Examples.GCD.GCD using ( gcd ; gcdh )\n\nopen import PCF.LTC.Data.Nat\n  using ( _-_\n        ; N ; sN -- The LTC natural numbers type.\n        )\nopen import PCF.LTC.Data.Nat.Inequalities using ( _>_ ; GT ; LE )\nopen import PCF.LTC.Data.Nat.Inequalities.PropertiesER using ( x\u2264y\u2192x\u226fy )\n\n------------------------------------------------------------------------------\n\n-- Note: This module was written for the version of gcd using the\n-- lambda abstractions, but we can use it with the version of gcd\n-- using super-combinators.\n\nprivate\n  -- Before to prove some properties for 'gcd i j' it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (gcd-s\u2081,\n  -- gcd-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q\n  -- (e.g proof\u2082\u208b\u2083 : (m n : D) \u2192 gcd-s\u2082 m n \u2192 gcd-s\u2083 m n).\n\n  -- The functions gcd-00, gcd-Sm0, gcd-0Sn, gcd-m>n and\n  -- gcd-m>n show the use of the states gcd-s\u2081, gcd-s\u2082, ..., and the\n  -- proofs associated with the execution steps.\n\n  ----------------------------------------------------------------------------\n  -- The steps\n\n  -- Initially, the conversion rule fix-f is applied.\n  gcd-s\u2081 : D \u2192 D \u2192 D\n  gcd-s\u2081 m n = gcdh (fix gcdh) \u2219 m \u2219 n\n\n  -- First argument application.\n  gcd-s\u2082 : D \u2192 D\n  gcd-s\u2082 m = lam (\u03bb n \u2192\n                    if (isZero n)\n                       then (if (isZero m)\n                                then error\n                                else m)\n                       else (if (isZero m)\n                                then n\n                                else (if (m > n)\n                                         then fix gcdh \u2219 (m - n) \u2219 n\n                                         else fix gcdh \u2219 m \u2219 (n - m))))\n\n  -- Second argument application.\n  gcd-s\u2083 : D \u2192 D \u2192 D\n  gcd-s\u2083 m n = if (isZero n)\n                  then (if (isZero m)\n                           then error\n                           else m)\n                  else (if (isZero m)\n                           then n\n                           else (if (m > n)\n                                    then fix gcdh \u2219 (m - n) \u2219 n\n                                    else fix gcdh \u2219 m \u2219 (n - m)))\n\n  -- Conversion (first if_then_else) 'isZero n = b'.\n  gcd-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  gcd-s\u2084 m n b = if b\n                  then (if (isZero m)\n                           then error\n                           else m)\n                  else (if (isZero m)\n                           then n\n                           else (if (m > n)\n                                    then fix gcdh \u2219 (m - n) \u2219 n\n                                    else fix gcdh \u2219 m \u2219 (n - m)))\n\n  -- Conversion first if_then_else when 'if true ...'.\n  gcd-s\u2085 : D \u2192 D\n  gcd-s\u2085 m  = if (isZero m) then error else m\n\n  -- Conversion first if_then_else when 'if false ...'.\n  gcd-s\u2086 : D \u2192 D \u2192 D\n  gcd-s\u2086 m n = if (isZero m)\n                  then n\n                  else (if (m > n)\n                           then fix gcdh \u2219 (m - n) \u2219 n\n                           else fix gcdh \u2219 m \u2219 (n - m))\n\n  -- Conversion (second if_then_else) 'isZero m = b'.\n  gcd-s\u2087 : D \u2192 D \u2192 D\n  gcd-s\u2087 m b = if b then error else m\n\n  -- Conversion (third if_then_else) 'isZero m = b'.\n  gcd-s\u2088 : D \u2192 D \u2192 D \u2192 D\n  gcd-s\u2088 m n b = if b\n                    then n\n                    else (if (m > n)\n                             then fix gcdh \u2219 (m - n) \u2219 n\n                             else fix gcdh \u2219 m \u2219 (n - m))\n\n  -- Conversion third if_then_else, when 'if false ...'.\n  gcd-s\u2089 : D \u2192 D \u2192 D\n  gcd-s\u2089 m n = if (m > n)\n                   then fix gcdh \u2219 (m - n) \u2219 n\n                   else fix gcdh \u2219 m \u2219 (n - m)\n\n  -- Conversion (fourth if_then_else) 'gt m n = b'.\n  gcd-s\u2081\u2080 : D \u2192 D \u2192 D \u2192 D\n  gcd-s\u2081\u2080 m n b = if b\n                     then fix gcdh \u2219 (m - n) \u2219 n\n                     else fix gcdh \u2219 m \u2219 (n - m)\n\n  ----------------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n  To prove the execution steps\n  (e.g. proof\u2083\u208b\u2084 : (m n : D) \u2192 gcd-s\u2083 m n \u2192 gcd_s\u2084 m n),\n  we usually need to prove that\n\n                         C [m] \u2261 C [n] (1)\n\n  given that\n                             m \u2261 n,    (2)\n\n  where (2) is a conversion rule usually.\n  We prove (1) using\n  'subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y'\n  where\n   P is given by \u03bb m \u2192 C [m ] \u2261 C [n],\n   x \u2261 y is given n \u2261 m (actually, we use sym (m \u2261 n)), and\n   P x is given by C [n] \u2261 C [n] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-f.\n  proof\u2080\u208b\u2081 : (m n : D) \u2192 fix gcdh \u2219 m \u2219 n \u2261 gcd-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u2219 m \u2219 n \u2261 gcdh (fix gcdh) \u2219 m \u2219 n)\n                       (sym (fix-f gcdh))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : (m n : D) \u2192 gcd-s\u2081 m n \u2261 gcd-s\u2082 m \u2219 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u2219 n \u2261 gcd-s\u2082 m \u2219 n)\n                       (sym (beta gcd-s\u2082 m))\n                       refl\n\n  -- Second argument application.\n  proof\u2082\u208b\u2083 : (m n : D) \u2192 gcd-s\u2082 m \u2219 n \u2261 gcd-s\u2083 m n\n  proof\u2082\u208b\u2083 m n  = beta (gcd-s\u2083 m) n\n\n  -- Conversion (first if_then_else) 'isZero n = b' using that proof.\n  proof\u2083\u208b\u2084 : (m n b : D) \u2192 isZero n \u2261 b \u2192 gcd-s\u2083 m n \u2261 gcd-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b prf = subst (\u03bb x \u2192 gcd-s\u2084 m n x \u2261 gcd-s\u2084 m n b)\n                             (sym prf)\n                             refl\n\n  -- Conversion first if_then_else when 'if true ...' using if-true.\n  proof\u2084\u208b\u2085 : (m n : D) \u2192 gcd-s\u2084 m n true \u2261 gcd-s\u2085 m\n  proof\u2084\u208b\u2085 m _ = if-true (gcd-s\u2085 m)\n\n  -- Conversion first if_then_else when 'if false ...' using if-false.\n  proof\u2084\u208b\u2086 : (m n : D) \u2192 gcd-s\u2084 m n false \u2261 gcd-s\u2086 m n\n  proof\u2084\u208b\u2086 m n = if-false (gcd-s\u2086 m n)\n\n  -- -- Conversion (second if_then_else) 'isZero m = b' using that proof.\n  proof\u2085\u208b\u2087 : (m b : D) \u2192 isZero m \u2261 b \u2192 gcd-s\u2085 m \u2261 gcd-s\u2087 m b\n  proof\u2085\u208b\u2087 m b prf = subst (\u03bb x \u2192 gcd-s\u2087 m x \u2261 gcd-s\u2087 m b)\n                           (sym prf)\n                           refl\n\n  -- Conversion (third if_then_else) 'isZero m = b' using that proof.\n  proof\u2086\u208b\u2088 : (m n b : D) \u2192 isZero m \u2261 b \u2192 gcd-s\u2086 m n \u2261 gcd-s\u2088 m n b\n  proof\u2086\u208b\u2088 m n b prf = subst (\u03bb x \u2192 gcd-s\u2088 m n x \u2261 gcd-s\u2088 m n b)\n                             (sym prf)\n                             refl\n\n  -- Conversion second if_then_else when 'if true ...' using if-true.\n  proof\u2087\u208a : (m : D) \u2192 gcd-s\u2087 m true \u2261 error\n  proof\u2087\u208a _ = if-true error\n\n  -- Conversion second if_then_else when 'if false ...' using if-false.\n  proof\u2087\u208b : (m : D) \u2192 gcd-s\u2087 m false \u2261 m\n  proof\u2087\u208b m = if-false m\n\n  -- Conversion third if_then_else when 'if true ...' using if-true.\n  proof\u2088\u208a : (m n : D) \u2192 gcd-s\u2088 m n true \u2261 n\n  proof\u2088\u208a _ n = if-true n\n\n  -- Conversion third if_then_else when 'if false ...' using if-false.\n  proof\u2088\u208b\u2089 : (m n : D) \u2192 gcd-s\u2088 m n false \u2261 gcd-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = if-false (gcd-s\u2089 m n)\n\n  -- Conversion (fourth if_then_else) 'gt m n = b' using that proof.\n  proof\u2089\u208b\u2081\u2080 : (m n b : D) \u2192 m > n \u2261 b \u2192 gcd-s\u2089 m n \u2261 gcd-s\u2081\u2080 m n b\n  proof\u2089\u208b\u2081\u2080 m n b prf = subst (\u03bb x \u2192 gcd-s\u2081\u2080 m n x \u2261 gcd-s\u2081\u2080 m n b)\n                              (sym prf)\n                              refl\n\n  -- Conversion fourth if_then_else when 'if true ...' using if-true.\n  proof\u2081\u2080\u208a : (m n : D) \u2192 gcd-s\u2081\u2080 m n true \u2261 fix gcdh \u2219 (m - n) \u2219 n\n  proof\u2081\u2080\u208a m n = if-true (fix gcdh \u2219 (m - n) \u2219 n)\n\n  -- Conversion fourth if_then_else when 'if was ...' using if-false.\n  proof\u2081\u2080\u208b : (m n : D) \u2192 gcd-s\u2081\u2080 m n false \u2261 fix gcdh \u2219 m \u2219 (n - m)\n  proof\u2081\u2080\u208b m n = if-false (fix gcdh \u2219 m \u2219 (n - m))\n\n------------------------------------------------------------------------------\n-- The five equations for gcd\n\n-- First equation.\n-- We do not use this equation for reasoning about gcd.\ngcd-00 : gcd zero zero \u2261 error\ngcd-00 =\n  begin\n    gcd zero zero         \u2261\u27e8 proof\u2080\u208b\u2081 zero zero \u27e9\n    gcd-s\u2081 zero zero      \u2261\u27e8 proof\u2081\u208b\u2082 zero zero \u27e9\n    gcd-s\u2082 zero \u2219 zero    \u2261\u27e8 proof\u2082\u208b\u2083 zero zero \u27e9\n    gcd-s\u2083 zero zero      \u2261\u27e8 proof\u2083\u208b\u2084 zero zero true isZero-0 \u27e9\n    gcd-s\u2084 zero zero true \u2261\u27e8 proof\u2084\u208b\u2085 zero zero \u27e9\n    gcd-s\u2085 zero           \u2261\u27e8 proof\u2085\u208b\u2087 zero true isZero-0 \u27e9\n    gcd-s\u2087 zero true      \u2261\u27e8 proof\u2087\u208a  zero \u27e9\n    error\n  \u220e\n\n-- Second equation.\ngcd-S0 : (m : D) \u2192 gcd (succ m) zero \u2261 succ m\ngcd-S0 m =\n  begin\n    gcd (succ m) zero         \u2261\u27e8 proof\u2080\u208b\u2081 (succ m) zero \u27e9\n    gcd-s\u2081 (succ m) zero      \u2261\u27e8 proof\u2081\u208b\u2082 (succ m) zero \u27e9\n    gcd-s\u2082 (succ m) \u2219 zero    \u2261\u27e8 proof\u2082\u208b\u2083 (succ m) zero \u27e9\n    gcd-s\u2083 (succ m) zero      \u2261\u27e8 proof\u2083\u208b\u2084 (succ m) zero true isZero-0 \u27e9\n    gcd-s\u2084 (succ m) zero true \u2261\u27e8 proof\u2084\u208b\u2085 (succ m) zero \u27e9\n    gcd-s\u2085 (succ m)           \u2261\u27e8 proof\u2085\u208b\u2087 (succ m) false (isZero-S m) \u27e9\n    gcd-s\u2087 (succ m) false     \u2261\u27e8 proof\u2087\u208b  (succ m) \u27e9\n    succ m\n  \u220e\n\n-- Third equation.\ngcd-0S : (n : D) \u2192 gcd zero (succ n) \u2261 succ n\ngcd-0S n =\n  begin\n    gcd zero (succ n)          \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ n) \u27e9\n    gcd-s\u2081 zero (succ n)       \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ n) \u27e9\n    gcd-s\u2082 zero \u2219 (succ n)     \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ n) \u27e9\n    gcd-s\u2083 zero (succ n)       \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ n) false (isZero-S n) \u27e9\n    gcd-s\u2084 zero (succ n) false \u2261\u27e8 proof\u2084\u208b\u2086 zero (succ n) \u27e9\n    gcd-s\u2086 zero (succ n)       \u2261\u27e8 proof\u2086\u208b\u2088 zero (succ n) true isZero-0 \u27e9\n    gcd-s\u2088 zero (succ n) true  \u2261\u27e8 proof\u2088\u208a  zero (succ n) \u27e9\n    succ n\n  \u220e\n\n-- Fourth equation.\ngcd-S>S : (m n : D) \u2192 GT (succ m) (succ n) \u2192\n          gcd (succ m) (succ n) \u2261 gcd (succ m - succ n) (succ n)\n\ngcd-S>S m n Sm>Sn =\n  begin\n    gcd (succ m) (succ n)          \u2261\u27e8 proof\u2080\u208b\u2081 (succ m) (succ n) \u27e9\n    gcd-s\u2081 (succ m) (succ n)       \u2261\u27e8 proof\u2081\u208b\u2082 (succ m) (succ n) \u27e9\n    gcd-s\u2082 (succ m) \u2219 (succ n)     \u2261\u27e8 proof\u2082\u208b\u2083 (succ m) (succ n) \u27e9\n    gcd-s\u2083 (succ m) (succ n)       \u2261\u27e8 proof\u2083\u208b\u2084 (succ m) (succ n)\n                                               false (isZero-S n)\n                                   \u27e9\n    gcd-s\u2084 (succ m) (succ n) false \u2261\u27e8 proof\u2084\u208b\u2086 (succ m) (succ n) \u27e9\n    gcd-s\u2086 (succ m) (succ n)       \u2261\u27e8 proof\u2086\u208b\u2088 (succ m) (succ n)\n                                               false (isZero-S m)\n                                   \u27e9\n    gcd-s\u2088 (succ m) (succ n) false \u2261\u27e8 proof\u2088\u208b\u2089 (succ m) (succ n) \u27e9\n    gcd-s\u2089 (succ m) (succ n)       \u2261\u27e8 proof\u2089\u208b\u2081\u2080 (succ m) (succ n) true Sm>Sn \u27e9\n    gcd-s\u2081\u2080 (succ m) (succ n) true \u2261\u27e8 proof\u2081\u2080\u208a  (succ m) (succ n) \u27e9\n    fix gcdh \u2219 (succ m - succ n) \u2219 succ n\n  \u220e\n\n-- TODO: This equation requires N m and N n.\n-- Fifth equation.\ngcd-S\u2264S : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) (succ n) \u2192\n          gcd (succ m) (succ n) \u2261 gcd (succ m) (succ n - succ m)\ngcd-S\u2264S {m} {n} Nm Nn Sm\u2264Sn =\n  begin\n    gcd (succ m) (succ n)           \u2261\u27e8 proof\u2080\u208b\u2081 (succ m) (succ n) \u27e9\n    gcd-s\u2081 (succ m) (succ n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ m) (succ n) \u27e9\n    gcd-s\u2082 (succ m) \u2219 (succ n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ m) (succ n) \u27e9\n    gcd-s\u2083 (succ m) (succ n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ m) (succ n)\n                                                false (isZero-S n)\n                                    \u27e9\n    gcd-s\u2084 (succ m) (succ n) false  \u2261\u27e8 proof\u2084\u208b\u2086 (succ m) (succ n) \u27e9\n    gcd-s\u2086 (succ m) (succ n)        \u2261\u27e8 proof\u2086\u208b\u2088 (succ m) (succ n)\n                                                false (isZero-S m)\n                                    \u27e9\n    gcd-s\u2088 (succ m) (succ n) false  \u2261\u27e8 proof\u2088\u208b\u2089 (succ m) (succ n) \u27e9\n    gcd-s\u2089 (succ m) (succ n)        \u2261\u27e8 proof\u2089\u208b\u2081\u2080 (succ m) (succ n)\n                                                 false\n                                                 (x\u2264y\u2192x\u226fy (sN Nm) (sN Nn) Sm\u2264Sn)\n                                    \u27e9\n    gcd-s\u2081\u2080 (succ m) (succ n) false \u2261\u27e8 proof\u2081\u2080\u208b (succ m) (succ n) \u27e9\n    fix gcdh \u2219 succ m \u2219 (succ n - succ m)\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Equations for the greatest common divisor\n------------------------------------------------------------------------------\n\nmodule PCF.Examples.GCD.EquationsER where\n\nopen import LTC.Base\nopen import LTC.BaseER using ( subst )\n\nimport Lib.Relation.Binary.EqReasoning\nopen module APER = Lib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\nopen import PCF.Examples.GCD.GCD using ( gcd ; gcdh )\n\nopen import PCF.LTC.Data.Nat\n  using ( _-_\n        ; N ; sN -- The LTC natural numbers type.\n        )\nopen import PCF.LTC.Data.Nat.Inequalities using ( _>_ ; GT ; LE )\nopen import PCF.LTC.Data.Nat.Inequalities.PropertiesER using ( x\u2264y\u2192x\u226fy )\n\n------------------------------------------------------------------------------\n\n-- Note: This module was written for the version of gcd using the\n-- lambda abstractions, but we can use it with the version of gcd\n-- using super-combinators.\n\nprivate\n  -- Before to prove some properties for 'gcd i j' it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (gcd-s\u2081,\n  -- gcd-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q\n  -- (e.g proof\u2082\u208b\u2083 : (m n : D) \u2192 gcd-s\u2082 m n \u2192 gcd-s\u2083 m n).\n\n  -- The functions gcd-00, gcd-Sm0, gcd-0Sn, gcd-m>n and\n  -- gcd-m>n show the use of the states gcd-s\u2081, gcd-s\u2082, ..., and the\n  -- proofs associated with the execution steps.\n\n  ----------------------------------------------------------------------------\n  -- The steps\n\n  -- Initially, the conversion rule fix-f is applied.\n  gcd-s\u2081 : D \u2192 D \u2192 D\n  gcd-s\u2081 m n = gcdh (fix gcdh) \u2219 m \u2219 n\n\n  -- First argument application.\n  gcd-s\u2082 : D \u2192 D\n  gcd-s\u2082 m = lam (\u03bb n \u2192\n                    if (isZero n)\n                       then (if (isZero m)\n                                then error\n                                else m)\n                       else (if (isZero m)\n                                then n\n                                else (if (m > n)\n                                         then fix gcdh \u2219 (m - n) \u2219 n\n                                         else fix gcdh \u2219 m \u2219 (n - m))))\n\n  -- Second argument application.\n  gcd-s\u2083 : D \u2192 D \u2192 D\n  gcd-s\u2083 m n = if (isZero n)\n                  then (if (isZero m)\n                           then error\n                           else m)\n                  else (if (isZero m)\n                           then n\n                           else (if (m > n)\n                                    then fix gcdh \u2219 (m - n) \u2219 n\n                                    else fix gcdh \u2219 m \u2219 (n - m)))\n\n  -- Conversion (first if_then_else) 'isZero n = b'.\n  gcd-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  gcd-s\u2084 m n b = if b\n                  then (if (isZero m)\n                           then error\n                           else m)\n                  else (if (isZero m)\n                           then n\n                           else (if (m > n)\n                                    then fix gcdh \u2219 (m - n) \u2219 n\n                                    else fix gcdh \u2219 m \u2219 (n - m)))\n\n  -- Conversion first if_then_else when 'if true ...'.\n  gcd-s\u2085 : D \u2192 D\n  gcd-s\u2085 m  = if (isZero m) then error else m\n\n  -- Conversion first if_then_else when 'if false ...'.\n  gcd-s\u2086 : D \u2192 D \u2192 D\n  gcd-s\u2086 m n = if (isZero m)\n                  then n\n                  else (if (m > n)\n                           then fix gcdh \u2219 (m - n) \u2219 n\n                           else fix gcdh \u2219 m \u2219 (n - m))\n\n  -- Conversion (second if_then_else) 'isZero m = b'.\n  gcd-s\u2087 : D \u2192 D \u2192 D\n  gcd-s\u2087 m b = if b then error else m\n\n  -- Conversion (third if_then_else) 'isZero m = b'.\n  gcd-s\u2088 : D \u2192 D \u2192 D \u2192 D\n  gcd-s\u2088 m n b = if b\n                    then n\n                    else (if (m > n)\n                             then fix gcdh \u2219 (m - n) \u2219 n\n                             else fix gcdh \u2219 m \u2219 (n - m))\n\n  -- Conversion third if_then_else, when 'if false ...'.\n  gcd-s\u2089 : D \u2192 D \u2192 D\n  gcd-s\u2089 m n = if (m > n)\n                   then fix gcdh \u2219 (m - n) \u2219 n\n                   else fix gcdh \u2219 m \u2219 (n - m)\n\n  -- Conversion (fourth if_then_else) 'gt m n = b'.\n  gcd-s\u2081\u2080 : D \u2192 D \u2192 D \u2192 D\n  gcd-s\u2081\u2080 m n b = if b\n                     then fix gcdh \u2219 (m - n) \u2219 n\n                     else fix gcdh \u2219 m \u2219 (n - m)\n\n  ----------------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n  To prove the execution steps\n  (e.g. proof\u2083\u208b\u2084 : (m n : D) \u2192 gcd-s\u2083 m n \u2192 gcd_s\u2084 m n),\n  we usually need to prove that\n\n                         C [m] \u2261 C [n] (1)\n\n  given that\n                             m \u2261 n,    (2)\n\n  where (2) is a conversion rule usually.\n  We prove (1) using\n  'subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y'\n  where\n   P is given by \u03bb m \u2192 C [m ] \u2261 C [n],\n   x \u2261 y is given n \u2261 m (actually, we use sym (m \u2261 n)), and\n   P x is given by C [n] \u2261 C [n] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-f.\n  proof\u2080\u208b\u2081 : (m n : D) \u2192 fix gcdh \u2219 m \u2219 n \u2261 gcd-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u2219 m \u2219 n \u2261 gcdh (fix gcdh) \u2219 m \u2219 n)\n                       (sym (fix-f gcdh))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : (m n : D) \u2192 gcd-s\u2081 m n \u2261 gcd-s\u2082 m \u2219 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u2219 n \u2261 gcd-s\u2082 m \u2219 n) (sym (beta gcd-s\u2082 m)) refl\n\n  -- Second argument application.\n  proof\u2082\u208b\u2083 : (m n : D) \u2192 gcd-s\u2082 m \u2219 n \u2261 gcd-s\u2083 m n\n  proof\u2082\u208b\u2083 m n  = beta (gcd-s\u2083 m) n\n\n  -- Conversion (first if_then_else) 'isZero n = b' using that proof.\n  proof\u2083\u208b\u2084 : (m n b : D) \u2192 isZero n \u2261 b \u2192 gcd-s\u2083 m n \u2261 gcd-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b prf = subst (\u03bb x \u2192 gcd-s\u2084 m n x \u2261 gcd-s\u2084 m n b)\n                             (sym prf )\n                             refl\n\n  -- Conversion first if_then_else when 'if true ...' using if-true.\n  proof\u2084\u208b\u2085 : (m n : D) \u2192 gcd-s\u2084 m n true \u2261 gcd-s\u2085 m\n  proof\u2084\u208b\u2085 m _ = if-true (gcd-s\u2085 m)\n\n  -- Conversion first if_then_else when 'if false ...' using if-false.\n  proof\u2084\u208b\u2086 : (m n : D) \u2192 gcd-s\u2084 m n false \u2261 gcd-s\u2086 m n\n  proof\u2084\u208b\u2086 m n = if-false (gcd-s\u2086 m n)\n\n  -- -- Conversion (second if_then_else) 'isZero m = b' using that proof.\n  proof\u2085\u208b\u2087 : (m b : D) \u2192 isZero m \u2261 b \u2192 gcd-s\u2085 m \u2261 gcd-s\u2087 m b\n  proof\u2085\u208b\u2087 m b prf = subst (\u03bb x \u2192 gcd-s\u2087 m x \u2261 gcd-s\u2087 m b)\n                           (sym prf )\n                           refl\n\n  -- Conversion (third if_then_else) 'isZero m = b' using that proof.\n  proof\u2086\u208b\u2088 : (m n b : D) \u2192 isZero m \u2261 b \u2192 gcd-s\u2086 m n \u2261 gcd-s\u2088 m n b\n  proof\u2086\u208b\u2088 m n b prf = subst (\u03bb x \u2192 gcd-s\u2088 m n x \u2261 gcd-s\u2088 m n b)\n                              (sym prf)\n                              refl\n\n  -- Conversion second if_then_else when 'if true ...' using if-true.\n  proof\u2087\u208a : (m : D) \u2192 gcd-s\u2087 m true \u2261 error\n  proof\u2087\u208a _ = if-true error\n\n  -- Conversion second if_then_else when 'if false ...' using if-false.\n  proof\u2087\u208b : (m : D) \u2192 gcd-s\u2087 m false \u2261 m\n  proof\u2087\u208b m = if-false m\n\n  -- Conversion third if_then_else when 'if true ...' using if-true.\n  proof\u2088\u208a : (m n : D) \u2192 gcd-s\u2088 m n true \u2261 n\n  proof\u2088\u208a _ n = if-true n\n\n  -- Conversion third if_then_else when 'if false ...' using if-false.\n  proof\u2088\u208b\u2089 : (m n : D) \u2192 gcd-s\u2088 m n false \u2261 gcd-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = if-false (gcd-s\u2089 m n)\n\n  -- Conversion (fourth if_then_else) 'gt m n = b' using that proof.\n  proof\u2089\u208b\u2081\u2080 : (m n b : D) \u2192 m > n \u2261 b \u2192 gcd-s\u2089 m n \u2261 gcd-s\u2081\u2080 m n b\n  proof\u2089\u208b\u2081\u2080 m n b prf = subst (\u03bb x \u2192 gcd-s\u2081\u2080 m n x \u2261 gcd-s\u2081\u2080 m n b)\n                              (sym prf)\n                              refl\n\n  -- Conversion fourth if_then_else when 'if true ...' using if-true.\n  proof\u2081\u2080\u208a : (m n : D) \u2192 gcd-s\u2081\u2080 m n true \u2261 fix gcdh \u2219 (m - n) \u2219 n\n  proof\u2081\u2080\u208a m n = if-true (fix gcdh \u2219 (m - n) \u2219 n)\n\n  -- Conversion fourth if_then_else when 'if was ...' using if-false.\n  proof\u2081\u2080\u208b : (m n : D) \u2192 gcd-s\u2081\u2080 m n false \u2261 fix gcdh \u2219 m \u2219 (n - m)\n  proof\u2081\u2080\u208b m n = if-false (fix gcdh \u2219 m \u2219 (n - m))\n\n------------------------------------------------------------------------------\n-- The five equations for gcd\n\n-- First equation.\n-- We do not use this equation for reasoning about gcd.\ngcd-00 : gcd zero zero \u2261 error\ngcd-00 =\n  begin\n    gcd zero zero         \u2261\u27e8 proof\u2080\u208b\u2081 zero zero \u27e9\n    gcd-s\u2081 zero zero      \u2261\u27e8 proof\u2081\u208b\u2082 zero zero \u27e9\n    gcd-s\u2082 zero \u2219 zero    \u2261\u27e8 proof\u2082\u208b\u2083 zero zero \u27e9\n    gcd-s\u2083 zero zero      \u2261\u27e8 proof\u2083\u208b\u2084 zero zero true isZero-0 \u27e9\n    gcd-s\u2084 zero zero true \u2261\u27e8 proof\u2084\u208b\u2085 zero zero \u27e9\n    gcd-s\u2085 zero           \u2261\u27e8 proof\u2085\u208b\u2087 zero true isZero-0 \u27e9\n    gcd-s\u2087 zero true      \u2261\u27e8 proof\u2087\u208a  zero \u27e9\n    error\n  \u220e\n\n-- Second equation.\ngcd-S0 : (m : D) \u2192 gcd (succ m) zero \u2261 succ m\ngcd-S0 m =\n  begin\n    gcd (succ m) zero         \u2261\u27e8 proof\u2080\u208b\u2081 (succ m) zero \u27e9\n    gcd-s\u2081 (succ m) zero      \u2261\u27e8 proof\u2081\u208b\u2082 (succ m) zero \u27e9\n    gcd-s\u2082 (succ m) \u2219 zero    \u2261\u27e8 proof\u2082\u208b\u2083 (succ m) zero \u27e9\n    gcd-s\u2083 (succ m) zero      \u2261\u27e8 proof\u2083\u208b\u2084 (succ m) zero true isZero-0 \u27e9\n    gcd-s\u2084 (succ m) zero true \u2261\u27e8 proof\u2084\u208b\u2085 (succ m) zero \u27e9\n    gcd-s\u2085 (succ m)           \u2261\u27e8 proof\u2085\u208b\u2087 (succ m) false (isZero-S m) \u27e9\n    gcd-s\u2087 (succ m) false     \u2261\u27e8 proof\u2087\u208b  (succ m) \u27e9\n    succ m\n  \u220e\n\n-- Third equation.\ngcd-0S : (n : D) \u2192 gcd zero (succ n) \u2261 succ n\ngcd-0S n =\n  begin\n    gcd zero (succ n)          \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ n) \u27e9\n    gcd-s\u2081 zero (succ n)       \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ n) \u27e9\n    gcd-s\u2082 zero \u2219 (succ n)     \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ n) \u27e9\n    gcd-s\u2083 zero (succ n)       \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ n) false (isZero-S n) \u27e9\n    gcd-s\u2084 zero (succ n) false \u2261\u27e8 proof\u2084\u208b\u2086 zero (succ n) \u27e9\n    gcd-s\u2086 zero (succ n)       \u2261\u27e8 proof\u2086\u208b\u2088 zero (succ n) true isZero-0 \u27e9\n    gcd-s\u2088 zero (succ n) true  \u2261\u27e8 proof\u2088\u208a  zero (succ n) \u27e9\n    succ n\n  \u220e\n\n-- Fourth equation.\ngcd-S>S : (m n : D) \u2192 GT (succ m) (succ n) \u2192\n          gcd (succ m) (succ n) \u2261 gcd (succ m - succ n) (succ n)\n\ngcd-S>S m n Sm>Sn =\n  begin\n    gcd (succ m) (succ n)          \u2261\u27e8 proof\u2080\u208b\u2081 (succ m) (succ n) \u27e9\n    gcd-s\u2081 (succ m) (succ n)       \u2261\u27e8 proof\u2081\u208b\u2082 (succ m) (succ n) \u27e9\n    gcd-s\u2082 (succ m) \u2219 (succ n)     \u2261\u27e8 proof\u2082\u208b\u2083 (succ m) (succ n) \u27e9\n    gcd-s\u2083 (succ m) (succ n)       \u2261\u27e8 proof\u2083\u208b\u2084 (succ m) (succ n)\n                                               false (isZero-S n)\n                                   \u27e9\n    gcd-s\u2084 (succ m) (succ n) false \u2261\u27e8 proof\u2084\u208b\u2086 (succ m) (succ n) \u27e9\n    gcd-s\u2086 (succ m) (succ n)       \u2261\u27e8 proof\u2086\u208b\u2088 (succ m) (succ n)\n                                               false (isZero-S m)\n                                   \u27e9\n    gcd-s\u2088 (succ m) (succ n) false \u2261\u27e8 proof\u2088\u208b\u2089 (succ m) (succ n) \u27e9\n    gcd-s\u2089 (succ m) (succ n)       \u2261\u27e8 proof\u2089\u208b\u2081\u2080 (succ m) (succ n) true Sm>Sn \u27e9\n    gcd-s\u2081\u2080 (succ m) (succ n) true \u2261\u27e8 proof\u2081\u2080\u208a  (succ m) (succ n) \u27e9\n    fix gcdh \u2219 (succ m - succ n) \u2219 succ n\n  \u220e\n\n-- TODO: This equation requires N m and N n.\n-- Fifth equation.\ngcd-S\u2264S : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) (succ n) \u2192\n          gcd (succ m) (succ n) \u2261 gcd (succ m) (succ n - succ m)\ngcd-S\u2264S {m} {n} Nm Nn Sm\u2264Sn =\n  begin\n    gcd (succ m) (succ n)           \u2261\u27e8 proof\u2080\u208b\u2081 (succ m) (succ n) \u27e9\n    gcd-s\u2081 (succ m) (succ n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ m) (succ n) \u27e9\n    gcd-s\u2082 (succ m) \u2219 (succ n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ m) (succ n) \u27e9\n    gcd-s\u2083 (succ m) (succ n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ m) (succ n)\n                                                false (isZero-S n)\n                                    \u27e9\n    gcd-s\u2084 (succ m) (succ n) false  \u2261\u27e8 proof\u2084\u208b\u2086 (succ m) (succ n) \u27e9\n    gcd-s\u2086 (succ m) (succ n)        \u2261\u27e8 proof\u2086\u208b\u2088 (succ m) (succ n)\n                                                false (isZero-S m)\n                                    \u27e9\n    gcd-s\u2088 (succ m) (succ n) false  \u2261\u27e8 proof\u2088\u208b\u2089 (succ m) (succ n) \u27e9\n    gcd-s\u2089 (succ m) (succ n)        \u2261\u27e8 proof\u2089\u208b\u2081\u2080 (succ m) (succ n)\n                                                 false\n                                                 (x\u2264y\u2192x\u226fy (sN Nm) (sN Nn) Sm\u2264Sn)\n                                    \u27e9\n    gcd-s\u2081\u2080 (succ m) (succ n) false \u2261\u27e8 proof\u2081\u2080\u208b (succ m) (succ n) \u27e9\n    fix gcdh \u2219 succ m \u2219 (succ n - succ m)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1c61c2c0a02e79cbfeec0aa80af4793d610159ba","subject":"Added missing module.","message":"Added missing module.\n\nIgnore-this: b1d8c2d0e90bb5f4874cf5181001321\n\ndarcs-hash:20110306025334-3bd4e-fde575b74ea4bced9c1a439dec94a46ed58839c0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Everything.agda","new_file":"src\/Common\/Everything.agda","new_contents":"------------------------------------------------------------------------------\n-- All the common modules\n------------------------------------------------------------------------------\n\nmodule Common.Everything where\n\nopen import Common.Data.Empty\nopen import Common.Data.Product\nopen import Common.Data.Sum\nopen import Common.Data.Unit\n\nopen import Common.Function\n\nopen import Common.LogicalConstants\n\nopen import Common.Relation.Binary.PreorderReasoning\nopen import Common.Relation.Binary.PropositionalEquality\nopen import Common.Relation.Nullary\nopen import Common.Relation.Unary\n\nopen import Common.Universe\n","old_contents":"------------------------------------------------------------------------------\n-- All the common modules\n------------------------------------------------------------------------------\n\nmodule Common.Everything where\n\nopen import Common.Data.Empty\nopen import Common.Data.Product\nopen import Common.Data.Sum\nopen import Common.Data.Unit\n\nopen import Common.Function\n\nopen import Common.LogicalConstants\n\nopen import Common.Relation.Binary.PreorderReasoning\nopen import Common.Relation.Binary.PropositionalEquality\nopen import Common.Relation.Unary\n\nopen import Common.Universe\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"434fe3c2998ce1169be647ea2e9ddd3f99734157","subject":"Uber nicification using SearchInd","message":"Uber nicification using SearchInd\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192\n                    let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ind  : SearchInd search\n\n  search-ind' : SearchInd' search\n  search-ind' P P\u2219 {f} Pf = search-ind (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\n  search-ext : SearchExt search\n  search-ext sg {f} {g} f\u2248\u00b0g = search-ind (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  search-mono : SearchMono search\n  search-mono _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = search-ind (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\n  search-swap : SearchSwap search\n  search-swap sg f {s\u1d2e} pf = search-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2248 s\u1d2e (s \u2218 flip f)) (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _))) (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\n  searchMon : SearchMon A\n  searchMon m = search _\u2219_\n    where open Mon m\n\n  search-\u03b5 : Search\u03b5 searchMon\n  search-\u03b5 m = search-ind' Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n    where open Mon m\n\n  search-hom : SearchMonHom searchMon\n  search-hom cm f g = search-ind (\u03bb s \u2192 s (f \u2219\u00b0 g) \u2248 s f \u2219 s g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  sum : Sum A\n  sum = search _+_\n\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ind\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    {- derived from ind\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n    -}\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\n    {- derived from ind\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n    -}\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk _ (search-ind \u03bcA +SearchInd search-ind \u03bcB)\n    {- derived from ind\n  where\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n    -}\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) = mk _ (search-ind \u03bcA \u00d7SearchInd search-ind \u03bcB)\n{- derived from SearchInd\n   where\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n             -}\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n    {- derived from ind\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n    -}\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x))\n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nsearchInd' : \u2200 {A} {s\u1d2c : Search A} \u2192 SearchInd s\u1d2c \u2192 SearchInd' s\u1d2c\nsearchInd' Ps\u1d2c P P\u2219 {f} Pf = Ps\u1d2c (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nsearch-mono' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchMono s\u1d2c\nsearch-mono' s\u1d2c Ps\u1d2c _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nsearch-\u03b5' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 Search\u03b5 (searchMonFromSearch s\u1d2c)\nsearch-\u03b5' s\u1d2c Ps\u1d2c m = searchInd' Ps\u1d2c Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n  where open Mon m\n\nsearch-ext' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchExt s\u1d2c\nsearch-ext' s\u1d2c Ps\u1d2c sg {f} {g} f\u2248\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n  where open Sgrp sg\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n-- _+\u03bc_ {A} {B} \u03bcA \u03bcB = {!!}\n-- {-\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk srch ext mono swp eps hom\n  where\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e {C} P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) -- = {!!}\n-- {-\n   = mk srch ext mono swp eps hom\n   where\n     srch : Search _ -- (A \u00d7 B)\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"76f3d8f43234ad60dff660d754a5b8e92cf3376f","subject":"Syntax.Term.Plotkin: shorthand for multiple applications","message":"Syntax.Term.Plotkin: shorthand for multiple applications\n\nOld-commit-hash: 42ffea36d6e6c6b9f6844d6df1f226eaf41fa5df\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"module Syntax.Term.Plotkin where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen import Syntax.Type.Plotkin\nopen import Syntax.Context\n\ndata Term\n  {B : Set {- of base types -}}\n  {C : Set {- of constants -}}\n  {type-of : C \u2192 Type B}\n  (\u0393 : Context {Type B}) :\n  (\u03c4 : Type B) \u2192 Set\n  where\n  const : (c : C) \u2192 Term \u0393 (type-of c)\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term {B} {C} {type-of} \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term {B} {C} {type-of} \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term {B} {C} {type-of} (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {B C \u22a2 \u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term {B} {C} {\u22a2} \u0393\u2081 \u03c4 \u2192\n  Term {B} {C} {\u22a2} \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {B C \u22a2 \u0393 \u03c3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term {B} {C} {\u22a2} \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","old_contents":"module Syntax.Term.Plotkin where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen import Syntax.Type.Plotkin\nopen import Syntax.Context\n\ndata Term\n  {B : Set {- of base types -}}\n  {C : Set {- of constants -}}\n  {type-of : C \u2192 Type B}\n  (\u0393 : Context {Type B}) :\n  (\u03c4 : Type B) \u2192 Set\n  where\n  const : (c : C) \u2192 Term \u0393 (type-of c)\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term {B} {C} {type-of} \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term {B} {C} {type-of} \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term {B} {C} {type-of} (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {B C type-of} {\u0394base : B \u2192 Type B} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term {B} {C} {type-of}\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {B C \u22a2 \u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term {B} {C} {\u22a2} \u0393\u2081 \u03c4 \u2192\n  Term {B} {C} {\u22a2} \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {B C \u22a2 \u0393 \u03c3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {B C \u22a2 \u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term {B} {C} {\u22a2} \u0393 \u03c4 \u2192 Term {B} {C} {\u22a2} (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25211123a2a5c13c97beb85f352ac6d566ddab20","subject":"Nat: extract some broken stuff","message":"Nat: extract some broken stuff\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b449d13e3646b32805c070f92dda8005db345f82","subject":"agda\/...: weakenMore: fix line breaks","message":"agda\/...: weakenMore: fix line breaks\n\nOld-commit-hash: 4456d93c39d967f4493e69cf3974ef17c22488b6\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore \u0393\u2032 t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    weakenMore : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore {\u0393} \u0393\u2032 t =\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (doWeakenMore (take (\u0394-Context \u0393) \u0393\u2032) (drop (\u0394-Context \u0393) \u0393\u2032)\n          (substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032))\n            (\u0394 t)))\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore \u0393\u2032 t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    weakenMore : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore {\u0393} \u0393\u2032 t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (doWeakenMore (take (\u0394-Context \u0393) \u0393\u2032) (drop (\u0394-Context \u0393) \u0393\u2032) (\n          substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t)))\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"28cb21b371c9f68684c496bb3d47a35bbb634ebb","subject":"[ README ] Added thesis information.","message":"[ README ] Added thesis information.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/README.agda","new_file":"src\/fot\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- Code accompanying the PhD thesis \"Reasoning about Functional\n-- Programs by Combining Interactive and Automatic Proofs\" by Andr\u00e9s\n-- Sicard-Ram\u00edrez and the paper \"Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs\" by Ana\n-- Bove, Peter Dybjer and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Thesis, paper, prerequisites, tested versions of the ATPS and use\n-- see https:\/\/github.com\/asr\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see examples\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Paper, prerequisites, tested versions of the ATPS and use\n\n-- See https:\/\/github.com\/asr\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see examples\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8403c80b80206ebcbe4f1d3acfac2cfce30552ca","subject":"Drop unused defs","message":"Drop unused defs\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  --open import Base.Data.DependentList public -- No\n  --\n  -- Workaround https:\/\/github.com\/agda\/agda\/issues\/1985 by making this import\n  -- private; many clients will need to start importing this now.\n  open import Base.Data.DependentList\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_]; reverse)\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4res = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4res) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4res [] = Term \u0393 \u03c4res\n    --hoasArgType \u0393 \u03c4res (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4res \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4res = foldr _ _\u21d2_ \u03c4res\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4res} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4res \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4res \u03c4s)\n\n    drop-\u227c-l : \u2200 {\u0393 \u0393\u2032 \u03c4} \u2192 (\u03c4 \u2022 \u0393 \u227c \u0393\u2032) \u2192 \u0393 \u227c \u0393\u2032\n    drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081 = \u227c-trans (drop _ \u2022 \u227c-refl) \u0393\u2032\u227c\u0393\u2032\u2081\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN []        f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN (\u03c4\u2081 \u2237 \u03c4s) f =\n      abs (absN \u03c4s\n        (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f\n            {\u0393\u227c\u0393\u2032 = drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081}\n            (weaken \u0393\u2032\u227c\u0393\u2032\u2081 (var this))))\n\n    -- Using a similar trick, we can declare absV (where V stands for variadic),\n    -- which takes the N implicit type arguments individually, collects them and\n    -- passes them on to absN. This is inspired by the example in the Agda 2.4.0\n    -- release notes about support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType zero       \u03c4s = absNType (reverse \u03c4s)\n    absVType (suc n) \u03c4s = {\u03c4\u2081 : Type} \u2192 absVType n (\u03c4\u2081 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN (reverse \u03c4s)\n    -- (Support for varying arity does not work here, so {\u03c4\u2081} must be bound in\n    -- the right-hand side).\n    absVAux \u03c4s (suc n) {\u03c4\u2081} = absVAux (\u03c4\u2081 \u2237 \u03c4s) n --\u03bb {\u03c4\u2081} \u2192 absVAux (\u03c4\u2081 \u2237 \u03c4s) n\n\n    -- \"Initialize\" accumulator to the empty list.\n    absV = absVAux []\n    -- We could maybe define absV directly, without going through absN, by\n    -- making hoasArgType and hoasResType also variadic, but I don't see a clear\n    -- advantage.\n\n  open AbsNHelpers using (absV) public\n\n  {-\n  Example types:\n  absV 1:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4res)\n\n  absV 2:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n       Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4res)\n\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  --open import Base.Data.DependentList public -- No\n  --\n  -- Workaround https:\/\/github.com\/agda\/agda\/issues\/1985 by making this import\n  -- private; many clients will need to start importing this now.\n  open import Base.Data.DependentList\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_]; reverse)\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4res = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4res) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4res [] = Term \u0393 \u03c4res\n    --hoasArgType \u0393 \u03c4res (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4res \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4res = foldr _ _\u21d2_ \u03c4res\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4res} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4res \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4res \u03c4s)\n\n    drop-\u227c-l : \u2200 {\u0393 \u0393\u2032 \u03c4} \u2192 (\u03c4 \u2022 \u0393 \u227c \u0393\u2032) \u2192 \u0393 \u227c \u0393\u2032\n    drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081 = \u227c-trans (drop _ \u2022 \u227c-refl) \u0393\u2032\u227c\u0393\u2032\u2081\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN []        f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN (\u03c4\u2081 \u2237 \u03c4s) f =\n      abs (absN \u03c4s\n        (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f\n            {\u0393\u227c\u0393\u2032 = drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081}\n            (weaken \u0393\u2032\u227c\u0393\u2032\u2081 (var this))))\n\n    -- Using a similar trick, we can declare absV (where V stands for variadic),\n    -- which takes the N implicit type arguments individually, collects them and\n    -- passes them on to absN. This is inspired by the example in the Agda 2.4.0\n    -- release notes about support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType zero       \u03c4s = absNType (reverse \u03c4s)\n    absVType (suc n) \u03c4s = {\u03c4\u2081 : Type} \u2192 absVType n (\u03c4\u2081 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN (reverse \u03c4s)\n    -- (Support for varying arity does not work here, so {\u03c4\u2081} must be bound in\n    -- the right-hand side).\n    absVAux \u03c4s (suc n) {\u03c4\u2081} = absVAux (\u03c4\u2081 \u2237 \u03c4s) n --\u03bb {\u03c4\u2081} \u2192 absVAux (\u03c4\u2081 \u2237 \u03c4s) n\n\n    -- \"Initialize\" accumulator to the empty list.\n    absV = absVAux []\n    -- We could maybe define absV directly, without going through absN, by\n    -- making hoasArgType and hoasResType also variadic, but I don't see a clear\n    -- advantage.\n\n  open AbsNHelpers using (absV) public\n\n  {-\n  Example types:\n  absV 1:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4res)\n\n  absV 2:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n       Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4res)\n\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9674d48c878af55c923334a6c4759a6335bafd5f","subject":"Removed unnecessary hints (by Ana Bove).","message":"Removed unnecessary hints (by Ana Bove).\n\nIgnore-this: 1f1f7d18784c58d2e39bb1509cd23235\n\ndarcs-hash:20110504155017-3bd4e-3dccf53ac404bb617abd1f5c3596128634633e36.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/ArithmeticATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.ArithmeticATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven                 one-hundred\n  111>100  : GT (one-hundred + eleven)         one-hundred\n{-# ATP prove 101>100 101\u2261100+11-10 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven)  one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven)   one-hundred\n  95+11>100 : GT (ninety-five + eleven)  one-hundred\n  94+11>100 : GT (ninety-four + eleven)  one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven)   one-hundred\n  91+11>100 : GT (ninety-one + eleven)   one-hundred\n  90+11>100 : GT (ninety + eleven)       one-hundred\n{-# ATP prove 99+11>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\nx+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\nx+11-N Nn = +-N Nn 11-N\n\nx+11\u223810\u2261Sx : \u2200 {n} \u2192 N n \u2192 (n + eleven) \u2238 ten \u2261 succ n\nx+11\u223810\u2261Sx Nn = [x+Sy]\u2238y\u2261Sx Nn 10-N\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y 10-N 11-N +-N \u2238-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N 89-N 100-N\n                              100\u226189+11\n#-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule FOTC.Program.McCarthy91.ArithmeticATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  91\u2261[100+11\u223810]\u223810 : (one-hundred + eleven \u2238 ten) \u2238 ten \u2261 ninety-one\n{-# ATP prove 91\u2261[100+11\u223810]\u223810 #-}\n\npostulate\n  100\u226189+11     : eighty-nine + eleven         \u2261 one-hundred\n  101\u226190+11\u2261101 : ninety + eleven              \u2261 hundred-one\n  101\u2261100+11-10 : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  102\u226191+11     : ninety-one + eleven          \u2261 hundred-two\n  103\u226192+1      : ninety-two + eleven          \u2261 hundred-three\n  104\u226193+11     : ninety-three + eleven        \u2261 hundred-four\n  105\u226194+11     : ninety-four + eleven         \u2261 hundred-five\n  106\u226195+11     : ninety-five + eleven         \u2261 hundred-six\n  107\u226196+11     : ninety-six + eleven          \u2261 hundred-seven\n  108\u226197+11     : ninety-seven + eleven        \u2261 hundred-eight\n  109\u226199+11     : ninety-eight + eleven        \u2261 hundred-nine\n  110\u226199+11     : ninety-nine + eleven         \u2261 hundred-ten\n  111\u2261100+11    : one-hundred + eleven         \u2261 hundred-eleven\n\n{-# ATP prove 100\u226189+11 #-}\n{-# ATP prove 101\u226190+11\u2261101 #-}\n{-# ATP prove 101\u2261100+11-10 #-}\n{-# ATP prove 102\u226191+11 #-}\n{-# ATP prove 103\u226192+1 #-}\n{-# ATP prove 104\u226193+11 #-}\n{-# ATP prove 105\u226194+11 #-}\n{-# ATP prove 106\u226195+11 #-}\n{-# ATP prove 107\u226196+11 #-}\n{-# ATP prove 108\u226197+11 #-}\n{-# ATP prove 109\u226199+11 #-}\n{-# ATP prove 110\u226199+11 #-}\n{-# ATP prove 111\u2261100+11 #-}\n\npostulate\n  101>100' : GT hundred-one                    one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  102>100  : GT hundred-two                    one-hundred\n  103>100  : GT hundred-three                  one-hundred\n  104>100  : GT hundred-four                   one-hundred\n  105>100  : GT hundred-five                   one-hundred\n  106>100  : GT hundred-six                    one-hundred\n  107>100  : GT hundred-seven                  one-hundred\n  108>100  : GT hundred-eight                  one-hundred\n  109>100  : GT hundred-nine                   one-hundred\n  110>100  : GT hundred-ten                    one-hundred\n  111>100' : GT hundred-eleven                 one-hundred\n  111>100  : GT (one-hundred + eleven)         one-hundred\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' 101\u2261100+11-10 #-}\n{-# ATP prove 102>100 #-}\n{-# ATP prove 103>100 #-}\n{-# ATP prove 104>100 #-}\n{-# ATP prove 105>100 #-}\n{-# ATP prove 106>100 #-}\n{-# ATP prove 107>100 #-}\n{-# ATP prove 108>100 #-}\n{-# ATP prove 109>100 #-}\n{-# ATP prove 110>100 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' 111\u2261100+11 #-}\n\npostulate\n  99+11>100 : GT (ninety-nine + eleven)  one-hundred\n  98+11>100 : GT (ninety-eight + eleven) one-hundred\n  97+11>100 : GT (ninety-seven + eleven) one-hundred\n  96+11>100 : GT (ninety-six + eleven)   one-hundred\n  95+11>100 : GT (ninety-five + eleven)  one-hundred\n  94+11>100 : GT (ninety-four + eleven)  one-hundred\n  93+11>100 : GT (ninety-three + eleven) one-hundred\n  92+11>100 : GT (ninety-two + eleven)   one-hundred\n  91+11>100 : GT (ninety-one + eleven)   one-hundred\n  90+11>100 : GT (ninety + eleven)       one-hundred\n{-# ATP prove 99+11>100 110>100 110\u226199+11 #-}\n{-# ATP prove 98+11>100 109>100 109\u226199+11 #-}\n{-# ATP prove 97+11>100 108>100 108\u226197+11 #-}\n{-# ATP prove 96+11>100 107>100 107\u226196+11 #-}\n{-# ATP prove 95+11>100 106>100 106\u226195+11 #-}\n{-# ATP prove 94+11>100 105>100 105\u226194+11 #-}\n{-# ATP prove 93+11>100 104>100 104\u226193+11 #-}\n{-# ATP prove 92+11>100 103>100 103\u226192+1 #-}\n{-# ATP prove 91+11>100 102>100 102\u226191+11 #-}\n{-# ATP prove 90+11>100 101>100' 101\u226190+11\u2261101 #-}\n\npostulate\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' 102\u226191+11 #-}\n\nx+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\nx+11-N Nn = +-N Nn 11-N\n\nx+11\u223810\u2261Sx : \u2200 {n} \u2192 N n \u2192 (n + eleven) \u2238 ten \u2261 succ n\nx+11\u223810\u2261Sx Nn = [x+Sy]\u2238y\u2261Sx Nn 10-N\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}\n\npostulate x+1\u2264x\u223810+11 : \u2200 {n} \u2192 N n \u2192 LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove x+1\u2264x\u223810+11 x\u2264y+x\u2238y 10-N 11-N +-N \u2238-N +-comm #-}\n\npostulate x\u226489\u2192x+11>100\u2192\u22a5 : \u2200 {n} \u2192 N n \u2192 LE n eighty-nine \u2192\n                            GT (n + eleven) one-hundred \u2192 \u22a5\n{-# ATP prove x\u226489\u2192x+11>100\u2192\u22a5 x>y\u2192x\u2264y\u2192\u22a5 x\u2264y\u2192x+k\u2264y+k x+11-N 89-N 100-N\n                              100\u226189+11\n#-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c74b6edc6717310c5129c01f4ba55bb264922307","subject":"elgamal","message":"elgamal\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\nopen import sum\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : SumProp X \u2192 \u2200 u \u2192 SumProp (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7\u03bc \u03bcU \u03bcX u\u2081\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum (\u03bcU \u03bc\u2124q u) (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  -- open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  \u03bc\u2124q : SumProp \u2124q\n  \u03bc\u2124q = \u03bcFin q\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  {-\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n  -}\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum \u03bc\u2124q (f \u2218 [suc]) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n  {-\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 d (r , x , y , _) = d r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 d (r , x , y , z) = d r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 d b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) d\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 d (b , x , y , z , r) = DDH\u2141 d b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba d) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (d b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2219_ : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_ : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (R\u2090 : \u2605)\n  (\u03bcR\u2090 : SumProp R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2219 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R\u2090 = (R\u2090 \u2192 PubKey \u2192 Bit \u2192 M)\n              \u00d7 (R\u2090 \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , d) b (r\u2090 , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      d r\u2090 pk (Enc pk (m r\u2090 pk b) y)\n\n      -- Unused\n    Game : (i : Bit) \u2192 \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , d) (b , r\u2090 , x , y , z) = b ==\u1d47 d r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R\u2090} \u2192 Game 0b \u2261 Game0 {R\u2090}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M d (r , x , y , z) = d r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R\u2090} \u2192 Bit \u2192 EncAdv R\u2090 \u2192 DDHAdv R\u2090\n    TrA b (m , d) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = d r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , d) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = d r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    like-SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , d) b (r\u2090 , x , y , _z) =\n      d r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    -- open Sum\n\n    R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : SumProp R\n    \u03bcR = \u03bcR\u2090 \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q\n\n    #R_ : Count R\n    #R_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n    f \u2248R g = #R f \u2261 #R g\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module Proof\n        (ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n        (otp-lem : \u2200 (A : Message \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : EncAdv R\u2090) (b : Bit)\n      where\n\n        OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248R OTP\u2141 M\u2081 d\n        OTP\u2141-lem d M\u2080 M\u2081 = sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y)))\n          where\n          f0 = \u03bb M r x y z \u2192 OTP\u2141 M d (r , x , y , z)\n          f1 = \u03bb M r x y \u2192 sum \u03bc\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n          f2 = \u03bb M r x \u2192 sum \u03bc\u2124q (f1 M r x)\n          f3 = \u03bb M r \u2192 sum \u03bc\u2124q (f2 M r)\n          pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n          pf r x y = pf'\n            where g\u02e3 = g^ x\n                  g\u02b8 = g^ y\n                  m\u2080 = M\u2080 r g\u02e3\n                  m\u2081 = M\u2081 r g\u02e3\n                  f5 = \u03bb M z \u2192 d r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                  pf' : f5 m\u2080 \u2248q f5 m\u2081\n                  pf' rewrite otp-lem (d r g\u02e3 g\u02b8) m\u2080 m\u2081 = refl\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA (not b) A\n\n        pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n        pf1 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf0,5)\n\n        pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n        pf2 = ddh-hyp A\u1d47\n\n        pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n        pf3 = OTP\u2141-lem (projD A) (projM A b) (projM A (not b))\n\n        pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n        pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf5 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf4,5)\n\n        final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n        final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    \n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ = {!!}\n\n    {-\n    \/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    \/-\u2219 x y = ?\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 d M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 d \u2248R OTP\u2141 M\u2081 d\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum `R\u2090\n  sumR\u2090-ext = sum-ext `R\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\u207b\u00b9 dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext\n  open EB hiding (g^_)\n\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\nopen import sum\n\nmodule elgamal where\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : SumProp X \u2192 \u2200 u \u2192 SumProp (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7\u03bc \u03bcU \u03bcX u\u2081\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum (\u03bcU \u03bc\u2124q u) (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  -- open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  \u03bc\u2124q : SumProp \u2124q\n  \u03bc\u2124q = \u03bcFin q\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  {-\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n  -}\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum \u03bc\u2124q (f \u2218 [suc]) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n  {-\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 D (r , x , y , _) = D r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 D (r , x , y , z) = D r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba D) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (D b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2219_ : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_ : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (R\u2090 : \u2605)\n  (\u03bcR\u2090 : SumProp R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2219 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R\u2090 = (R\u2090 \u2192 PubKey \u2192 Bit \u2192 M)\n              \u00d7 (R\u2090 \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , D) b (r\u2090 , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      D r\u2090 pk (Enc pk (m r\u2090 pk b) y)\n\n      -- Unused\n    Game : (i : Bit) \u2192 \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , D) (b , r\u2090 , x , y , z) = b ==\u1d47 D r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R\u2090} \u2192 Game 0b \u2261 Game0 {R\u2090}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M D (r , x , y , z) = D r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R\u2090} \u2192 Bit \u2192 EncAdv R\u2090 \u2192 DDHAdv R\u2090\n    TrA b (m , D) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = D r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , D) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = D r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    like-SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , D) b (r\u2090 , x , y , _z) =\n      D r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    -- open Sum\n\n    R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : SumProp R\n    \u03bcR = \u03bcR\u2090 \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q\n\n    #R_ : Count R\n    #R_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n    f \u2248R g = #R f \u2261 #R g\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module Proof\n        (ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n        (otp-lem : \u2200 (A : Message \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : EncAdv R\u2090) (b : Bit)\n      where\n\n        OTP\u2141-lem : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248R OTP\u2141 M\u2081 D\n        OTP\u2141-lem D M\u2080 M\u2081 = sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y)))\n          where\n          f0 = \u03bb M r x y z \u2192 OTP\u2141 M D (r , x , y , z)\n          f1 = \u03bb M r x y \u2192 sum \u03bc\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n          f2 = \u03bb M r x \u2192 sum \u03bc\u2124q (f1 M r x)\n          f3 = \u03bb M r \u2192 sum \u03bc\u2124q (f2 M r)\n          pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n          pf r x y = pf'\n            where g\u02e3 = g^ x\n                  g\u02b8 = g^ y\n                  m\u2080 = M\u2080 r g\u02e3\n                  m\u2081 = M\u2081 r g\u02e3\n                  f5 = \u03bb M z \u2192 D r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                  pf' : f5 m\u2080 \u2248q f5 m\u2081\n                  pf' rewrite otp-lem (D r g\u02e3 g\u02b8) m\u2080 m\u2081 = refl\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA (not b) A\n\n        pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n        pf1 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf0,5)\n\n        pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n        pf2 = ddh-hyp A\u1d47\n\n        pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n        pf3 = OTP\u2141-lem (projD A) (projM A b) (projM A (not b))\n\n        pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n        pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf5 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf4,5)\n\n        final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n        final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    \n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ = {!!}\n\n    {-\n    \/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    \/-\u2219 x y = ?\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248R OTP\u2141 M\u2081 D\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum `R\u2090\n  sumR\u2090-ext = sum-ext `R\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\u207b\u00b9 dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext\n  open EB hiding (g^_)\n\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"95b5472603f17d0c76fb65322be65206a57290ec","subject":"Updated doc.","message":"Updated doc.\n\nIgnore-this: a6aabc86b6b265dca9d848d1c23abbdf\n\ndarcs-hash:20110928103226-3bd4e-4e6e7646a7c4608c48ba6bdadb01a6af973cf629.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/List\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/List\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.List.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL Lys = prf\n  where\n  postulate prf : List ([] ++ ys)\n  {-# ATP prove prf #-}\n\n++-List {ys = ys} (consL x {xs} Lxs) Lys = prf $ ++-List Lxs Lys\n  where\n  postulate prf : List (xs ++ ys) \u2192  -- IH.\n                  List ((x \u2237 xs) ++ ys)\n  {-# ATP prove prf consL #-}\n\nmap-List : \u2200 {xs} f \u2192 List xs \u2192 List (map f xs)\nmap-List f nilL = prf\n  where\n  postulate prf : List (map f [])\n  {-# ATP prove prf nilL #-}\nmap-List f (consL x {xs} Lxs) = prf $ map-List f Lxs\n  where\n  postulate prf : List (map f xs) \u2192  -- IH.\n                  List (map f (x \u2237 xs))\n  {-# ATP prove prf consL #-}\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} nilL Lys = prf\n  where\n  postulate prf : List (rev [] ys)\n  {-# ATP prove prf nilL #-}\nrev-List {ys = ys} (consL x {xs} Lxs) Lys = prf $ rev-List Lxs (consL x Lys)\n  where\n  postulate prf : List (rev xs (x \u2237 ys)) \u2192  -- IH.\n                  List (rev (x \u2237 xs) ys)\n  {-# ATP prove prf #-}\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs nilL\n\n-- Length properties\n\nlength-replicate : \u2200 d {n} \u2192 N n \u2192 length (replicate n d) \u2261 n\nlength-replicate d zN = prf\n  where\n  postulate prf : length (replicate zero d) \u2261 zero\n  {-# ATP prove prf #-}\nlength-replicate d (sN {n} Nn) = prf $ length-replicate d Nn\n  where\n  postulate prf : length (replicate n d) \u2261 n \u2192  -- IH.\n                  length (replicate (succ n) d) \u2261 succ n\n  {-# ATP prove prf #-}\n\n-- Append properties\n\n++-leftIdentity : \u2200 {xs} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL = prf\n  where\n  postulate prf : [] ++ [] \u2261 []\n  {-# ATP prove prf #-}\n\n++-rightIdentity (consL x {xs} Lxs) = prf $ ++-rightIdentity Lxs\n  where\n  postulate prf : xs ++ [] \u2261 xs \u2192  -- IH.\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n  {-# ATP prove prf #-}\n\n++-assoc : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n           (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc .{[]} {ys} {zs} nilL Lys Lzs = prf\n  where\n  postulate prf : ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs\n  {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} {ys} {zs} (consL x {xs} Lxs) Lys Lzs =\n  prf $ ++-assoc Lxs Lys Lzs\n  where\n  postulate prf : (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192  -- IH.\n                  ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs\n  {-# ATP prove prf #-}\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs ys} \u2192 List xs \u2192 List ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f {ys = ys} nilL Lys = prf\n  where\n  postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n  {-# ATP prove prf #-}\n\nmap-++-commute f {ys = ys} (consL x {xs} Lxs) Lys =\n  prf $ map-++-commute f Lxs Lys\n  where\n  postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192  -- IH.\n                  map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove prf #-}\n\n-- Reverse properties\n\n-- N.B. This property does not depend of the type of lists.\npostulate reverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nrev-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute {ys = ys} nilL Lys = prf\n  where\n  postulate prf : rev [] ys \u2261 rev [] [] ++ ys\n  {-# ATP prove prf #-}\nrev-++-commute {ys = ys} (consL x {xs} Lxs) Lys =\n  prf (rev-++-commute Lxs (consL x Lys)) (rev-++-commute Lxs (consL x nilL))\n  where\n  postulate prf : rev xs (x \u2237 ys) \u2261 rev xs [] ++ x \u2237 ys \u2192  -- IH.\n                  rev xs (x \u2237 []) \u2261 rev xs [] ++ x \u2237 [] \u2192  -- IH.\n                  rev (x \u2237 xs) ys \u2261 rev (x \u2237 xs) [] ++ ys\n  -- E 1.2: CPU time limit exceeded (180 sec).\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  -- Vampire 0.6 (revision 903): (Default) memory limit (timeout 180 sec).\n  {-# ATP prove prf consL nilL ++-assoc rev-List ++-List #-}\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} nilL Lys = prf\n  where\n  postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n  {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++-commute (consL x {xs} Lxs) nilL = prf\n  where\n  postulate prf : reverse ((x \u2237 xs) ++ []) \u2261 reverse [] ++ reverse (x \u2237 xs)\n  {-# ATP prove prf ++-rightIdentity #-}\n\nreverse-++-commute (consL x {xs} Lxs) (consL y {ys} Lys) =\n  prf $ reverse-++-commute Lxs (consL y Lys)\n  where\n  postulate prf : reverse (xs ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                           reverse xs \u2192  -- IH.\n                  reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                                 reverse (x \u2237 xs)\n  -- E 1.2: CPU time limit exceeded (180 sec).\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  -- Vampire 0.6 (revision 903): (Default) memory limit (using timeout 180 sec).\n  {-# ATP prove prf consL nilL reverse-List ++-List rev-++-commute ++-assoc #-}\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x nilL = prf\n  where\n  postulate prf : reverse (x \u2237 []) \u2261 reverse [] ++ x \u2237 []\n  {-# ATP prove prf consL nilL ++-leftIdentity #-}\n\nreverse-\u2237 x (consL y {ys} Lys) = prf\n  where\n  postulate prf : reverse (x \u2237 y \u2237 ys) \u2261 reverse (y \u2237 ys) ++ x \u2237 []\n  {-# ATP prove prf consL nilL reverse-[x]\u2261[x] reverse-++-commute\n                    ++-leftIdentity\n  #-}\n\nreverse\u00b2 : \u2200 {xs} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse\u00b2 nilL = prf\n  where\n  postulate prf : reverse (reverse []) \u2261 []\n  {-# ATP prove prf #-}\n\nreverse\u00b2 (consL x {xs} Lxs) = prf $ reverse\u00b2 Lxs\n  where\n  postulate prf : reverse (reverse xs) \u2261 xs \u2192  -- IH.\n                  reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n  -- Equinox 5.0alpha (2010-06-29): TIMEOUT (180 seconds).\n  -- Metis 2.3 (release 20110531): SZS status Unknown (using timeout 180 sec).\n  -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n  {-# ATP prove prf consL nilL rev-List rev-++-commute reverse-++-commute\n                    ++-List ++-rightIdentity\n  #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.List.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL Lys = prf\n  where\n  postulate prf : List ([] ++ ys)\n  {-# ATP prove prf #-}\n\n++-List {ys = ys} (consL x {xs} Lxs) Lys = prf $ ++-List Lxs Lys\n  where\n  postulate prf : List (xs ++ ys) \u2192  -- IH.\n                  List ((x \u2237 xs) ++ ys)\n  {-# ATP prove prf consL #-}\n\nmap-List : \u2200 {xs} f \u2192 List xs \u2192 List (map f xs)\nmap-List f nilL = prf\n  where\n  postulate prf : List (map f [])\n  {-# ATP prove prf nilL #-}\nmap-List f (consL x {xs} Lxs) = prf $ map-List f Lxs\n  where\n  postulate prf : List (map f xs) \u2192  -- IH.\n                  List (map f (x \u2237 xs))\n  {-# ATP prove prf consL #-}\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} nilL Lys = prf\n  where\n  postulate prf : List (rev [] ys)\n  {-# ATP prove prf nilL #-}\nrev-List {ys = ys} (consL x {xs} Lxs) Lys = prf $ rev-List Lxs (consL x Lys)\n  where\n  postulate prf : List (rev xs (x \u2237 ys)) \u2192  -- IH.\n                  List (rev (x \u2237 xs) ys)\n  {-# ATP prove prf #-}\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs nilL\n\n-- Length properties\n\nlength-replicate : \u2200 d {n} \u2192 N n \u2192 length (replicate n d) \u2261 n\nlength-replicate d zN = prf\n  where\n  postulate prf : length (replicate zero d) \u2261 zero\n  {-# ATP prove prf #-}\nlength-replicate d (sN {n} Nn) = prf $ length-replicate d Nn\n  where\n  postulate prf : length (replicate n d) \u2261 n \u2192  -- IH.\n                  length (replicate (succ n) d) \u2261 succ n\n  {-# ATP prove prf #-}\n\n-- Append properties\n\n++-leftIdentity : \u2200 {xs} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL = prf\n  where\n  postulate prf : [] ++ [] \u2261 []\n  {-# ATP prove prf #-}\n\n++-rightIdentity (consL x {xs} Lxs) = prf $ ++-rightIdentity Lxs\n  where\n  postulate prf : xs ++ [] \u2261 xs \u2192  -- IH.\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n  {-# ATP prove prf #-}\n\n++-assoc : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n           (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc .{[]} {ys} {zs} nilL Lys Lzs = prf\n  where\n  postulate prf : ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs\n  {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} {ys} {zs} (consL x {xs} Lxs) Lys Lzs =\n  prf $ ++-assoc Lxs Lys Lzs\n  where\n  postulate prf : (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192  -- IH.\n                  ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs\n  {-# ATP prove prf #-}\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs ys} \u2192 List xs \u2192 List ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f {ys = ys} nilL Lys = prf\n  where\n  postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n  {-# ATP prove prf #-}\n\nmap-++-commute f {ys = ys} (consL x {xs} Lxs) Lys =\n  prf $ map-++-commute f Lxs Lys\n  where\n  postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192  -- IH.\n                  map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove prf #-}\n\n-- Reverse properties\n\n-- N.B. This property does not depend of the type of lists.\npostulate reverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nrev-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute {ys = ys} nilL Lys = prf\n  where\n  postulate prf : rev [] ys \u2261 rev [] [] ++ ys\n  {-# ATP prove prf #-}\nrev-++-commute {ys = ys} (consL x {xs} Lxs) Lys =\n  prf (rev-++-commute Lxs (consL x Lys)) (rev-++-commute Lxs (consL x nilL))\n  where\n  postulate prf : rev xs (x \u2237 ys) \u2261 rev xs [] ++ x \u2237 ys \u2192  -- IH.\n                  rev xs (x \u2237 []) \u2261 rev xs [] ++ x \u2237 [] \u2192  -- IH.\n                  rev (x \u2237 xs) ys \u2261 rev (x \u2237 xs) [] ++ ys\n  -- E 1.2: CPU time limit exceeded (180 sec).\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  -- Vampire 0.6 (revision 903): (Default) memory limit (timeout 180 sec).\n  {-# ATP prove prf consL nilL ++-assoc rev-List ++-List #-}\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} nilL Lys = prf\n  where\n  postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n  {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++-commute (consL x {xs} Lxs) nilL = prf\n  where\n  postulate prf : reverse ((x \u2237 xs) ++ []) \u2261 reverse [] ++ reverse (x \u2237 xs)\n  {-# ATP prove prf ++-rightIdentity #-}\n\nreverse-++-commute (consL x {xs} Lxs) (consL y {ys} Lys) =\n  prf $ reverse-++-commute Lxs (consL y Lys)\n  where\n  postulate prf : reverse (xs ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                           reverse xs \u2192  -- IH.\n                  reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                                 reverse (x \u2237 xs)\n  -- E 1.2: CPU time limit exceeded (180 sec).\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  -- Vampire 0.6 (revision 903): (Default) memory limit (using timeout 180 sec).\n  {-# ATP prove prf consL nilL reverse-List ++-List rev-++-commute ++-assoc #-}\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x nilL = prf\n  where\n  postulate prf : reverse (x \u2237 []) \u2261 reverse [] ++ x \u2237 []\n  {-# ATP prove prf consL nilL ++-leftIdentity #-}\n\nreverse-\u2237 x (consL y {ys} Lys) = prf\n  where\n  postulate prf : reverse (x \u2237 y \u2237 ys) \u2261 reverse (y \u2237 ys) ++ x \u2237 []\n  {-# ATP prove prf consL nilL reverse-[x]\u2261[x] reverse-++-commute\n                    ++-leftIdentity\n  #-}\n\nreverse\u00b2 : \u2200 {xs} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse\u00b2 nilL = prf\n  where\n  postulate prf : reverse (reverse []) \u2261 []\n  {-# ATP prove prf #-}\n\nreverse\u00b2 (consL x {xs} Lxs) = prf $ reverse\u00b2 Lxs\n  where\n  postulate prf : reverse (reverse xs) \u2261 xs \u2192  -- IH.\n                  reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n  -- Equinox 5.0alpha (2010-06-29): TIMEOUT (180 seconds).\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n  {-# ATP prove prf consL nilL rev-List rev-++-commute reverse-++-commute\n                    ++-List ++-rightIdentity\n  #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a99adca7b9200e42269bc0c7a1abc201ce5853be","subject":"Desc model: iso2 implicit.","message":"Desc model: iso2 implicit.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"73e49ad9e7d81fcc6cef8db579209f9cca7bd325","subject":"Typo.","message":"Typo.\n\nIgnore-this: 761dc4d20a1c8a8582057b2a0fee7750\n\ndarcs-hash:20120522120402-3bd4e-936cb06cbe7dcc0acc1ec02e7ae1d1a397eaba42.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/Collatz\/Data\/Nat\/PropertiesATP.agda","new_file":"src\/FOTC\/Program\/Collatz\/Data\/Nat\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (added for the Collatz function example)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.Data.Nat.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesATP\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n\n^-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m ^ n)\n^-N {m} Nm zN          = subst N (sym (^-0 m)) (sN zN)\n^-N {m} Nm (sN {n} Nn) = subst N (sym (^-S m n)) (*-N Nm (^-N Nm Nn))\n\npostulate 2*SSx\u22652 : \u2200 {n} \u2192 N n \u2192 GE (two * succ\u2081 (succ\u2081 n)) two\n{-# ATP prove 2*SSx\u22652 #-}\n\n2x\/2\u2261x : \u2200 {n} \u2192 N n \u2192 two * n \/ two \u2261 n\n2x\/2\u2261x zN = prf\n  where\n  postulate prf : two * zero \/ two \u2261 zero\n  {-# ATP prove prf *-rightZero #-}\n\n2x\/2\u2261x (sN zN) = prf\n  where\n  postulate prf : two * succ\u2081 zero \/ two \u2261 succ\u2081 zero\n  {-# ATP prove prf *-rightIdentity x\u2265x x\u2238x\u22610 #-}\n\n2x\/2\u2261x (sN (sN {n} Nn)) = prf (2x\/2\u2261x (sN Nn))\n  where\n  postulate prf : two * succ\u2081 n \/ two \u2261 succ\u2081 n \u2192  -- IH.\n                  two * succ\u2081 (succ\u2081 n) \/ two \u2261 succ\u2081 (succ\u2081 n)\n  {-# ATP prove prf 2*SSx\u22652 +-rightIdentity +-comm +-N #-}\n\npostulate 2^[x+1]\/2\u22612^x : \u2200 {n} \u2192 N n \u2192 (two ^ (succ\u2081 n)) \/ two \u2261 two ^ n\n{-# ATP prove 2^[x+1]\/2\u22612^x 2x\/2\u2261x ^-N #-}\n\nSx\u22612^0\u2192x\u22610 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2261 two ^ zero \u2192 n \u2261 zero\nSx\u22612^0\u2192x\u22610 zN         _       = refl\nSx\u22612^0\u2192x\u22610(sN {n} Nn) SSn\u22612^0 = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\n+\u22382 : \u2200 {n} \u2192 N n \u2192 \u00ac (n \u2261 zero) \u2192 \u00ac (n \u2261 one) \u2192 n \u2261 succ\u2081 (succ\u2081 (n \u2238 two))\n+\u22382 zN               n\u22620 n\u22621 = \u22a5-elim (n\u22620 refl)\n+\u22382 (sN zN)          n\u22620 n\u22621 = \u22a5-elim (n\u22621 refl)\n+\u22382 (sN (sN {n} Nn)) n\u22620 n\u22621 = prf\n  where\n  -- See the interactive proof.\n  postulate prf : succ\u2081 (succ\u2081 n) \u2261 succ\u2081 (succ\u2081 (succ\u2081 (succ\u2081 n) \u2238 two))\n\n2^x\u22620 : \u2200 {n} \u2192 N n \u2192 \u00ac (two ^ n \u2261 zero)\n2^x\u22620 zN          h = \u22a5-elim (0\u2262S (trans (sym h) (^-0 two)))\n2^x\u22620 (sN {n} Nn) h = prf (2^x\u22620 Nn)\n  where\n  postulate prf : \u00ac (two ^ n \u2261 zero) \u2192  -- IH.\n                  \u22a5\n  {-# ATP prove prf xy\u22610\u2192x\u22610\u2228y\u22610 ^-N #-}\n\n-- TODO.\npostulate 2^[x+1]\u22621 : \u2200 {n} \u2192 N n \u2192 \u00ac (two ^ (succ\u2081 n) \u2261 one)\n\nSx-Even\u2192x-Odd : \u2200 {n} \u2192 N n \u2192 Even (succ\u2081 n) \u2192 Odd n\nSx-Even\u2192x-Odd zN          h = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\nSx-Even\u2192x-Odd (sN {n} Nn) h = trans (sym (even-S (succ\u2081 n))) h\n\nSx-Odd\u2192x-Even : \u2200 {n} \u2192 N n \u2192 Odd (succ\u2081 n) \u2192 Even n\nSx-Odd\u2192x-Even zN          _ = even-0\nSx-Odd\u2192x-Even (sN {n} Nn) h = trans (sym (odd-S (succ\u2081 n))) h\n\nmutual\n  \u2238-Even : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Even m \u2192 Even n \u2192 Even (m \u2238 n)\n  \u2238-Even {m} Nm zN                   h\u2081 _ = subst Even (sym (\u2238-x0 m)) h\u2081\n  \u2238-Even     zN          (sN {n} Nn) h\u2081 _ = subst Even (sym (\u2238-0S n)) h\u2081\n  \u2238-Even     (sN {m} Nm) (sN {n} Nn) h\u2081 h\u2082 = prf\n    where\n    postulate prf : Even (succ\u2081 m \u2238 succ\u2081 n)\n    {-# ATP prove prf \u2238-Odd Sx-Even\u2192x-Odd #-}\n\n  \u2238-Odd : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Odd m \u2192 Odd n \u2192 Even (m \u2238 n)\n  \u2238-Odd zN          Nn          h\u2081 _  = \u22a5-elim (true\u2262false (trans (sym h\u2081) odd-0))\n  \u2238-Odd (sN Nm)     zN          _  h\u2082 = \u22a5-elim (true\u2262false (trans (sym h\u2082) odd-0))\n  \u2238-Odd (sN {m} Nm) (sN {n} Nn) h\u2081 h\u2082 = prf\n    where\n    postulate prf : Even (succ\u2081 m \u2238 succ\u2081 n)\n    {-# ATP prove prf \u2238-Even Sx-Odd\u2192x-Even #-}\n\nx-Even\u2192SSx-Even : \u2200 {n} \u2192 N n \u2192 Even n \u2192 Even (succ\u2081 (succ\u2081 n))\nx-Even\u2192SSx-Even zN h = prf\n  where\n  postulate prf : Even (succ\u2081 (succ\u2081 zero))\n  {-# ATP prove prf #-}\n\nx-Even\u2192SSx-Even (sN {n} Nn) h = prf\n  where\n  postulate prf : Even (succ\u2081 (succ\u2081 (succ\u2081 n)))\n  {-# ATP prove prf #-}\n\nx+x-Even : \u2200 {n} \u2192 N n \u2192 Even (n + n)\nx+x-Even zN          = subst Even (sym (+-rightIdentity zN)) even-0\nx+x-Even (sN {n} Nn) = prf (x+x-Even Nn)\n  where\n  postulate prf : Even (n + n) \u2192  -- IH.\n                  Even (succ\u2081 n + succ\u2081 n)\n  {-# ATP prove prf x-Even\u2192SSx-Even +-N +-comm #-}\n\n2x-Even : \u2200 {n} \u2192 N n \u2192 Even (two * n)\n2x-Even zN          = subst Even (sym (*-rightZero 2-N)) even-0\n2x-Even (sN {n} Nn) = prf\n  where\n  postulate prf : Even (two * succ\u2081 n)\n  {-# ATP prove prf x-Even\u2192SSx-Even x+x-Even +-N +-comm +-rightIdentity #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (added for the Collatz function example)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.Data.Nat.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesATP\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n\n^-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m ^ n)\n^-N {m} Nm zN          = subst N (sym (^-0 m)) (sN zN)\n^-N {m} Nm (sN {n} Nn) = subst N (sym (^-S m n)) (*-N Nm (^-N Nm Nn))\n\npostulate 2*SSx\u22652 : \u2200 {n} \u2192 N n \u2192 GE (two * succ\u2081 (succ\u2081 n)) two\n{-# ATP prove 2*SSx\u22652 #-}\n\n2x\/2\u2261x : \u2200 {n} \u2192 N n \u2192 two * n \/ two \u2261 n\n2x\/2\u2261x zN = prf\n  where\n  postulate prf : two * zero \/ two \u2261 zero\n  {-# ATP prove prf *-rightZero #-}\n\n2x\/2\u2261x (sN zN) = prf\n  where\n  postulate prf : two * succ\u2081 zero \/ two \u2261 succ\u2081 zero\n  {-# ATP prove prf *-rightIdentity x\u2265x x\u2238x\u22610 #-}\n\n2x\/2\u2261x (sN (sN {n} Nn)) = prf (2x\/2\u2261x (sN Nn))\n  where\n  postulate prf : two * succ\u2081 n \/ two \u2261 succ\u2081 n \u2192  -- IH.\n                  two * succ\u2081 (succ\u2081 n) \/ two \u2261 succ\u2081 (succ\u2081 n)\n  {-# ATP prove prf 2*SSx\u22652 +-rightIdentity +-comm +-N #-}\n\npostulate 2^[x+1]\/2\u22612^x : \u2200 {n} \u2192 N n \u2192 (two ^ (succ\u2081 n)) \/ two \u2261 two ^ n\n{-# ATP prove 2^[x+1]\/2\u22612^x 2x\/2\u2261x ^-N #-}\n\nSx\u22612^0\u2192x\u22610 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2261 two ^ zero \u2192 n \u2261 zero\nSx\u22612^0\u2192x\u22610 zN         _       = refl\nSx\u22612^0\u2192x\u22610(sN {n} Nn) SSn\u22612^0 = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\n+\u22382 : \u2200 {n} \u2192 N n \u2192 \u00ac (n \u2261 zero) \u2192 \u00ac (n \u2261 one) \u2192 n \u2261 succ\u2081 (succ\u2081 (n \u2238 two))\n+\u22382 zN               n\u22620 n\u22621 = \u22a5-elim (n\u22620 refl)\n+\u22382 (sN zN)          n\u22620 n\u22621 = \u22a5-elim (n\u22621 refl)\n+\u22382 (sN (sN {n} Nn)) n\u22620 n\u22621 = prf\n  where\n  -- See the interactive proof.\n  postulate prf : succ\u2081 (succ\u2081 n) \u2261 succ\u2081 (succ\u2081 (succ\u2081 (succ\u2081 n) \u2238 two))\n\n2^x\u22620 : \u2200 {n} \u2192 N n \u2192 \u00ac (two ^ n \u2261 zero)\n2^x\u22620 zN          h = \u22a5-elim (0\u2262S (trans (sym h) (^-0 two)))\n2^x\u22620 (sN {n} Nn) h = prf (2^x\u22620 Nn)\n  where\n  postulate prf : \u00ac (two ^ n \u2261 zero) \u2192  -- IH.\n                  \u22a5\n  {-# ATP prove prf xy\u22610\u2192x\u22610\u2228y\u22610 ^-N #-}\n\n-- ToDo.\npostulate 2^[x+1]\u22621 : \u2200 {n} \u2192 N n \u2192 \u00ac (two ^ (succ\u2081 n) \u2261 one)\n\nSx-Even\u2192x-Odd : \u2200 {n} \u2192 N n \u2192 Even (succ\u2081 n) \u2192 Odd n\nSx-Even\u2192x-Odd zN          h = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\nSx-Even\u2192x-Odd (sN {n} Nn) h = trans (sym (even-S (succ\u2081 n))) h\n\nSx-Odd\u2192x-Even : \u2200 {n} \u2192 N n \u2192 Odd (succ\u2081 n) \u2192 Even n\nSx-Odd\u2192x-Even zN          _ = even-0\nSx-Odd\u2192x-Even (sN {n} Nn) h = trans (sym (odd-S (succ\u2081 n))) h\n\nmutual\n  \u2238-Even : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Even m \u2192 Even n \u2192 Even (m \u2238 n)\n  \u2238-Even {m} Nm zN                   h\u2081 _ = subst Even (sym (\u2238-x0 m)) h\u2081\n  \u2238-Even     zN          (sN {n} Nn) h\u2081 _ = subst Even (sym (\u2238-0S n)) h\u2081\n  \u2238-Even     (sN {m} Nm) (sN {n} Nn) h\u2081 h\u2082 = prf\n    where\n    postulate prf : Even (succ\u2081 m \u2238 succ\u2081 n)\n    {-# ATP prove prf \u2238-Odd Sx-Even\u2192x-Odd #-}\n\n  \u2238-Odd : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Odd m \u2192 Odd n \u2192 Even (m \u2238 n)\n  \u2238-Odd zN          Nn          h\u2081 _  = \u22a5-elim (true\u2262false (trans (sym h\u2081) odd-0))\n  \u2238-Odd (sN Nm)     zN          _  h\u2082 = \u22a5-elim (true\u2262false (trans (sym h\u2082) odd-0))\n  \u2238-Odd (sN {m} Nm) (sN {n} Nn) h\u2081 h\u2082 = prf\n    where\n    postulate prf : Even (succ\u2081 m \u2238 succ\u2081 n)\n    {-# ATP prove prf \u2238-Even Sx-Odd\u2192x-Even #-}\n\nx-Even\u2192SSx-Even : \u2200 {n} \u2192 N n \u2192 Even n \u2192 Even (succ\u2081 (succ\u2081 n))\nx-Even\u2192SSx-Even zN h = prf\n  where\n  postulate prf : Even (succ\u2081 (succ\u2081 zero))\n  {-# ATP prove prf #-}\n\nx-Even\u2192SSx-Even (sN {n} Nn) h = prf\n  where\n  postulate prf : Even (succ\u2081 (succ\u2081 (succ\u2081 n)))\n  {-# ATP prove prf #-}\n\nx+x-Even : \u2200 {n} \u2192 N n \u2192 Even (n + n)\nx+x-Even zN          = subst Even (sym (+-rightIdentity zN)) even-0\nx+x-Even (sN {n} Nn) = prf (x+x-Even Nn)\n  where\n  postulate prf : Even (n + n) \u2192  -- IH.\n                  Even (succ\u2081 n + succ\u2081 n)\n  {-# ATP prove prf x-Even\u2192SSx-Even +-N +-comm #-}\n\n2x-Even : \u2200 {n} \u2192 N n \u2192 Even (two * n)\n2x-Even zN          = subst Even (sym (*-rightZero 2-N)) even-0\n2x-Even (sN {n} Nn) = prf\n  where\n  postulate prf : Even (two * succ\u2081 n)\n  {-# ATP prove prf x-Even\u2192SSx-Even x+x-Even +-N +-comm +-rightIdentity #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2f9ab597d2f59a9287d166337384bd7ce1d4ed40","subject":"implicit args to get rid of unresolved unification problems","message":"implicit args to get rid of unresolved unification problems\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub {\u0393 = \u0393} x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!}\n  lem-idsub x d () | None\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 = {!!}\n\n  sing\u222a' : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (x : A) (n' : Nat) \u2192 (\u0393 ,, (n , x)) n' == ((\u0393 \u222a (\u25a0 (n , x))) n')\n  sing\u222a' \u0393 n x n' with natEQ n n'\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl with \u0393 n\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | Some x = {!!}\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | None with natEQ n n\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | None | Inl refl = refl\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | None | Inr x = abort (x refl)\n  sing\u222a' \u0393 n x\u2081 n' | Inr x with \u0393 n'\n  sing\u222a' \u0393 n x\u2082 n' | Inr x\u2081 | Some x = refl\n  sing\u222a' \u0393 n x\u2081 n' | Inr x  | None with natEQ n n'\n  sing\u222a' \u0393 n' x\u2082 .n' | Inr x\u2081 | None | Inl refl = abort (x\u2081 refl)\n  sing\u222a' \u0393 n x\u2082 n' | Inr x\u2081 | None | Inr x = refl\n\n  sing\u222a : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (x : A) \u2192 (\u0393 ,, (n , x)) == \u0393 \u222a (\u25a0 (n , x))\n  sing\u222a \u0393 n x = funext (sing\u222a' \u0393 n x)\n\n  lem-weaken\u0394single : \u2200{\u0394 \u0393 d \u03c4 u \u0393' \u03c4'} \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u0394 ,, u ::[ \u0393' ] \u03c4') , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u0394single = {!!}\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u0394single {u = u} ih1) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u0394single {u = u} ih1) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub {\u0393 = \u0393} x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!}\n  lem-idsub x d () | None\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 = {!!}\n\n  sing\u222a' : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (x : A) (n' : Nat) \u2192 (\u0393 ,, (n , x)) n' == ((\u0393 \u222a (\u25a0 (n , x))) n')\n  sing\u222a' \u0393 n x n' with natEQ n n'\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl with \u0393 n\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | Some x = {!!}\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | None with natEQ n n\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | None | Inl refl = refl\n  sing\u222a' \u0393 n x\u2081 .n | Inl refl | None | Inr x = abort (x refl)\n  sing\u222a' \u0393 n x\u2081 n' | Inr x with \u0393 n'\n  sing\u222a' \u0393 n x\u2082 n' | Inr x\u2081 | Some x = refl\n  sing\u222a' \u0393 n x\u2081 n' | Inr x  | None with natEQ n n'\n  sing\u222a' \u0393 n' x\u2082 .n' | Inr x\u2081 | None | Inl refl = abort (x\u2081 refl)\n  sing\u222a' \u0393 n x\u2082 n' | Inr x\u2081 | None | Inr x = refl\n\n  sing\u222a : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (x : A) \u2192 (\u0393 ,, (n , x)) == \u0393 \u222a (\u25a0 (n , x))\n  sing\u222a \u0393 n x = funext (sing\u222a' \u0393 n x)\n\n  lem-weaken\u0394single : \u2200{\u0394 \u0393 d \u03c4 u \u0393' \u03c4'} \u2192 \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u0394 ,, u ::[ \u0393' ] \u03c4') , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u0394single = {!!}\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u0394single ih1) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u0394single ih1) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7d869fc4e7c75329194b7528e171ab4be57aa354","subject":"Maybe: idiom brakets","message":"Maybe: idiom brakets\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B : Set \u2113} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n  \u27ea f \u00b7 x \u27eb = map? f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C : Set \u2113} \u2192\n              (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map? f x \u229b y\n\n  \u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D : Set \u2113} \u2192 (A \u2192 B \u2192 C \u2192 D)\n              \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n  \u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map? f x \u229b y \u229b z\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {A} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 m n A \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {A} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 m n A \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"423caa9deadb31c21899a501a78f9372a1c83264","subject":"Try alternative proofs for lemmas","message":"Try alternative proofs for lemmas\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import Function hiding (const)\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\nopen import New.FunctionLemmas\nopen import New.Unused\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\n\nmodule _ {t1 t2 t3 : Type}\n  (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (df : Cht (t1 \u21d2 t3))\n  (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (dg : Cht (t2 \u21d2 t3))\n  where\n  private\n    \u0393 = sum t1 t2 \u2022\n      (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n      (t2 \u21d2 t3) \u2022\n      (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n      (t1 \u21d2 t3) \u2022\n      sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n      sum t1 t2 \u2022 \u2205\n  module _ where\n    private\n      \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n    changeMatchSem-lem1 :\n      \u2200 a1 b2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n      \u2261\n        g b2 \u2295 dg b2 (nil b2) \u229d f a1\n    changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n  module _ where\n    private\n      \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n    changeMatchSem-lem2 :\n      \u2200 b1 a2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n      \u2261\n        f a2 \u2295 df a2 (nil a2) \u229d g b1\n    changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\ncorrectDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\ncorrectDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\ncorrectDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n  where\n    lemma : \u2200 s f g \u2192\n      \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n      (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n    lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n    lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\nvalidDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\nvalidDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\nvalidDeriveConst (match {t1} {t2} {t3}) =\n  ternary-valid dmatch-valid dmatch-eq\n  where\n    open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n    dmatch-valid : ternary-valid-preserve-hp\n    dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n    dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n    dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n    dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n    dmatch-eq : ternary-valid-eq-hp\n    dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n      rewrite update-nil (a \u2295 da)\n      | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n    dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n      rewrite update-nil (b \u2295 db)\n      | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n    dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      | update-nil b2\n      | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n      | update-nil (g b2 \u2295 dg b2 (nil b2))\n      = refl\n    dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      | update-nil a2\n      | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n      | update-nil (f a2 \u2295 df a2 (nil a2))\n      = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext $ \u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    -- equal-future-expand-derivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term (correctDerive t)\n    --   \u03c11 d\u03c11 \u03c11d\u03c11\n    --   (a \u2022 (\u03c1 \u2295 d\u03c1))\n    --   (cong (_\u2022 \u03c1 \u2295 d\u03c1) (sym (update-nil a)))\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     equal-future-derivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term (correctDerive t)\n       \u03c11 d\u03c11 \u03c11d\u03c11\n       \u03c12 d\u03c12 \u03c12d\u03c12\n       (cong (\u03bb a\u2032 \u2192 (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)))\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","old_contents":"module New.Correctness where\n\nopen import Function hiding (const)\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\nopen import New.FunctionLemmas\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\n\nmodule _ {t1 t2 t3 : Type}\n  (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (df : Cht (t1 \u21d2 t3))\n  (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (dg : Cht (t2 \u21d2 t3))\n  where\n  private\n    \u0393 = sum t1 t2 \u2022\n      (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n      (t2 \u21d2 t3) \u2022\n      (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n      (t1 \u21d2 t3) \u2022\n      sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n      sum t1 t2 \u2022 \u2205\n  module _ where\n    private\n      \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n    changeMatchSem-lem1 :\n      \u2200 a1 b2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n      \u2261\n        g b2 \u2295 dg b2 (nil b2) \u229d f a1\n    changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n  module _ where\n    private\n      \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n    changeMatchSem-lem2 :\n      \u2200 b1 a2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n      \u2261\n        f a2 \u2295 df a2 (nil a2) \u229d g b1\n    changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\ncorrectDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\ncorrectDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\ncorrectDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n  where\n    lemma : \u2200 s f g \u2192\n      \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n      (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n    lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n    lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\nvalidDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\nvalidDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\nvalidDeriveConst (match {t1} {t2} {t3}) =\n  ternary-valid dmatch-valid dmatch-eq\n  where\n    open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n    dmatch-valid : ternary-valid-preserve-hp\n    dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n    dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n    dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n    dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n    dmatch-eq : ternary-valid-eq-hp\n    dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n      rewrite update-nil (a \u2295 da)\n      | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n    dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n      rewrite update-nil (b \u2295 db)\n      | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n    dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      | update-nil b2\n      | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n      | update-nil (g b2 \u2295 dg b2 (nil b2))\n      = refl\n    dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      | update-nil a2\n      | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n      | update-nil (f a2 \u2295 df a2 (nil a2))\n      = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext $ \u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term \u03c12 \u2295 \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"34d7fbb3638b7289c435d28a592534b642996fad","subject":"Optimize boolean operations (see #50).","message":"Optimize boolean operations (see #50).\n\nI seem to need (all of) these optimiziations in order to get\nreasonable results out of derive. Still not perfect.\n\nOld-commit-hash: a2d52942fd8ef26144cf5af3a57cb8fa565d92dd\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\nsame-var : \u2200 {\u0393 \u03c4} \u2192 (x\u2081 x\u2082 : Var \u0393 \u03c4) \u2192 Bool\nsame-var this this = true\nsame-var this (that x\u2082) = false\nsame-var (that x\u2081) this = false\nsame-var (that x\u2081) (that x\u2082) = same-var x\u2081 x\u2082\n\nsame-term : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 Bool\nsame-term (abs t\u2081) (abs t\u2082) = same-term t\u2081 t\u2082\nsame-term (abs t\u2081) _ = false\nsame-term (app {\u0393} {\u03c4\u2081} t\u2081 t\u2082) (app .{\u0393} {\u03c4\u2082} t\u2083 t\u2084) with \u03c4\u2081 \u225f \u03c4\u2082\nsame-term (app t\u2081 t\u2082) (app t\u2083 t\u2084) | yes refl = same-term t\u2081 t\u2083 \u2227 same-term t\u2082 t\u2084\nsame-term (app t\u2081 t\u2082) (app t\u2083 t\u2084) | no \u00acp = false\nsame-term (app t\u2081 t\u2082) _ = false\nsame-term (var x\u2081) (var x\u2082) = same-var x\u2081 x\u2082\nsame-term (var x\u2081) _ = false\nsame-term true true = true\nsame-term true _ = false\nsame-term false false = true\nsame-term false _ = false\nsame-term (if t\u2081 t\u2082 t\u2083) (if t\u2084 t\u2085 t\u2086) = same-term t\u2081 t\u2084 \u2227 same-term t\u2082 t\u2085 \u2227 same-term t\u2083 t\u2086\nsame-term (if t\u2081 t\u2082 t\u2083) _ = false\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nif\u2032 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nif\u2032 true t\u2082 t\u2083 = t\u2082\nif\u2032 false t\u2082 t\u2083 = t\u2083\nif\u2032 t\u2081 true false = t\u2081\nif\u2032 t\u2081 t\u2082 t\u2083 = if same-term t\u2082 t\u2083 then t\u2082 else if t\u2081 t\u2082 t\u2083\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if\u2032 t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n\n-- SYMBOLIC DERIVATION\n--\n-- compare derive with _\u27ea_\u27eb_Term\n-- and derive-var with \u27e6_\u27e7Var\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = bool\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\nxor\u2083 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nxor\u2083 t\u2081 t\u2082 t\u2083\n  = if\u2032 t\u2081 (if\u2032 t\u2082 t\u2083 (if\u2032 t\u2083 false true)) (if\u2032 t\u2082 (if\u2032 t\u2083 false true) t\u2083)\n\nderive-if : \u2200 {\u03c4 \u0393\u2081} \u2192\n  (t dt : Term \u0393\u2081 bool) \u2192\n  (v\u2081 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2081 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  (v\u2082 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2082 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-if {\u03c4\u2081 \u21d2 \u03c4\u2082} t dt v\u2081 dv\u2081 v\u2082 dv\u2082 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 let \u227c\u2083 = \u227c-trans \u227c\u2081 \u227c\u2082 in\n    derive-if (weaken \u227c\u2083 t) (weaken \u227c\u2083 dt)\n              (v\u2081 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2081 \u227c\u2081 v \u227c\u2082 dv)\n              (v\u2082 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2082 \u227c\u2081 v \u227c\u2082 dv)\nderive-if {bool} t\u2081 dt\u2081 t\u2082 dt\u2082 t\u2083 dt\u2083 =\n  if\u2032 dt\u2081 (if\u2032 t\u2081 (xor\u2083 dt\u2083 t\u2082 t\u2083) (xor\u2083 dt\u2082 t\u2082 t\u2083)) (if\u2032 t\u2081 dt\u2083 dt\u2082)\n\nderive-var : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-var this (dv SymEnv.\u2022 d\u03c1) = dv\nderive-var (that x) (dv SymEnv.\u2022 d\u03c1) = derive-var x d\u03c1\n\nderive : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive (abs t) \u03c1 d\u03c1 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 derive t (liftVal \u227c\u2082 v SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) \u03c1) (dv SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) d\u03c1)\nderive (app t\u2081 t\u2082) \u03c1 d\u03c1 =\n  derive t\u2081 \u03c1 d\u03c1 \u227c-refl (_ \u27ea t\u2082 \u27ebTerm \u03c1) \u227c-refl (derive t\u2082 \u03c1 d\u03c1)\nderive (var x) \u03c1 d\u03c1 = derive-var x d\u03c1\nderive true \u03c1 d\u03c1 = false\nderive false \u03c1 d\u03c1 = false\nderive {\u0393\u2081} (if t\u2081 t\u2082 t\u2083) \u03c1 d\u03c1 =\n  derive-if (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (derive t\u2081 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (derive t\u2082 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2083 \u27ebTerm \u03c1) (derive t\u2083 \u03c1 d\u03c1)\n","old_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n\n-- SYMBOLIC DERIVATION\n--\n-- compare derive with _\u27ea_\u27eb_Term\n-- and derive-var with \u27e6_\u27e7Var\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = bool\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\nxor\u2083 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nxor\u2083 t\u2081 t\u2082 t\u2083\n  = if t\u2081 (if t\u2082 t\u2083 (if t\u2083 false true)) (if t\u2082 (if t\u2083 false true) t\u2083)\n\nderive-if : \u2200 {\u03c4 \u0393\u2081} \u2192\n  (t dt : Term \u0393\u2081 bool) \u2192\n  (v\u2081 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2081 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  (v\u2082 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2082 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-if {\u03c4\u2081 \u21d2 \u03c4\u2082} t dt v\u2081 dv\u2081 v\u2082 dv\u2082 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 let \u227c\u2083 = \u227c-trans \u227c\u2081 \u227c\u2082 in\n    derive-if (weaken \u227c\u2083 t) (weaken \u227c\u2083 dt)\n              (v\u2081 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2081 \u227c\u2081 v \u227c\u2082 dv)\n              (v\u2082 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2082 \u227c\u2081 v \u227c\u2082 dv)\nderive-if {bool} t\u2081 dt\u2081 t\u2082 dt\u2082 t\u2083 dt\u2083 =\n  if dt\u2081 (if t\u2081 (xor\u2083 t\u2082 t\u2083 dt\u2083) (xor\u2083 t\u2082 t\u2083 dt\u2082)) (if t\u2081 dt\u2083 dt\u2082)\n\nderive-var : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-var this (dv SymEnv.\u2022 d\u03c1) = dv\nderive-var (that x) (dv SymEnv.\u2022 d\u03c1) = derive-var x d\u03c1\n\nderive : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive (abs t) \u03c1 d\u03c1 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 derive t (liftVal \u227c\u2082 v SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) \u03c1) (dv SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) d\u03c1)\nderive (app t\u2081 t\u2082) \u03c1 d\u03c1 =\n  derive t\u2081 \u03c1 d\u03c1 \u227c-refl (_ \u27ea t\u2082 \u27ebTerm \u03c1) \u227c-refl (derive t\u2082 \u03c1 d\u03c1)\nderive (var x) \u03c1 d\u03c1 = derive-var x d\u03c1\nderive true \u03c1 d\u03c1 = false\nderive false \u03c1 d\u03c1 = false\nderive {\u0393\u2081} (if t\u2081 t\u2082 t\u2083) \u03c1 d\u03c1 =\n  derive-if (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (derive t\u2081 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (derive t\u2082 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2083 \u27ebTerm \u03c1) (derive t\u2083 \u03c1 d\u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c7ad461731109d522f831464e795886500cbf40d","subject":"Add instructions for users to README.agda","message":"Add instructions for users to README.agda\n\nOld-commit-hash: c0018596eda8908801ba4aa7eeec5aa0ef9452b5\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will help with the\n-- \"generate Everything.agda\" part.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help.\n\n\nimport Popl14.Syntax.Term\nimport Base.Syntax.Context\nimport Base.Change.Context\nimport Popl14.Change.Derive\n{- Language definition of Calc. Atlas -}\nimport Atlas.Syntax.Term\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Parametric.Syntax.Term\nimport Popl14.Change.Term\nimport Atlas.Syntax.Type\nimport Parametric.Syntax.Type\nimport Popl14.Syntax.Type\nimport Base.Syntax.Vars\n\nimport Popl14.Change.Validity\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Popl14.Change.Correctness\nimport Theorem.Irrelevance-Popl14\nimport Base.Denotation.Environment\nimport Popl14.Denotation.Evaluation\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Popl14.Change.Implementation\nimport Base.Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Popl14.Change.Specification\nimport Popl14.Denotation.Value\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport UNDEFINED\n\nimport Everything\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\n\nimport Popl14.Syntax.Term\nimport Base.Syntax.Context\nimport Base.Change.Context\nimport Popl14.Change.Derive\n{- Language definition of Calc. Atlas -}\nimport Atlas.Syntax.Term\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Parametric.Syntax.Term\nimport Popl14.Change.Term\nimport Atlas.Syntax.Type\nimport Parametric.Syntax.Type\nimport Popl14.Syntax.Type\nimport Base.Syntax.Vars\n\nimport Popl14.Change.Validity\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Popl14.Change.Correctness\nimport Theorem.Irrelevance-Popl14\nimport Base.Denotation.Environment\nimport Popl14.Denotation.Evaluation\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Popl14.Change.Implementation\nimport Base.Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Popl14.Change.Specification\nimport Popl14.Denotation.Value\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport UNDEFINED\n\nimport Everything\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e0e680ef43802a44d857ab43a4f4064b804817c8","subject":"Yet more theorems for thesis","message":"Yet more theorems for thesis\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\n  -- \\begin{lemma}[Equivalence cancellation]\n  --   |v2 `ominus` v1 `doe` dv| holds if and only if |v2 = v1 `oplus`\n  --   dv|, for any |v1, v2 `elem` V| and |dv `elem` Dt ^ v1|.\n  -- \\end{lemma}\n\n  equiv-cancel-1 : \u2200 x' dx \u2192 _\u2259_ {{ca}} (x' \u229f x) dx \u2192 x' \u2261 x \u229e dx\n  equiv-cancel-1 x' dx (doe x'\u229fx\u2259dx) = trans (sym (update-diff x' x)) x'\u229fx\u2259dx\n  equiv-cancel-2 : \u2200 x' dx \u2192 x' \u2261 x \u229e dx \u2192 _\u2259_ {{ca}} (x' \u229f x) dx\n  equiv-cancel-2 _ dx refl = diff-update\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CB}} (df a da) (f (a \u229e da) \u229f f a)\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (apply (nil f))\n  nil-is-derivative f a da =\n    begin\n      f a \u229e apply (nil f) a da\n    \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n      (f \u229e nil f) (a \u229e da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (_\u229e_ a da)) (update-nil f) \u27e9\n      f (a \u229e da)\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- If we have two derivatives, they're both nil, hence they're equivalent.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  equiv-derivative-is-derivative : \u2200 {f : A \u2192 B} (df\u2081 df\u2082 : \u0394 f) \u2192 IsDerivative f (apply df\u2081) \u2192  _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 IsDerivative f (apply df\u2082)\n  equiv-derivative-is-derivative {f} df\u2081 df\u2082 IsDerivative-f-df\u2081 df\u2081\u2259df\u2082 a da =\n    begin\n      f a \u229e apply df\u2082 a da\n    \u2261\u27e8 sym (incrementalization f df\u2082 a da) \u27e9\n      (f \u229e df\u2082) (a \u229e da)\n    \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (_\u229e_ a da)) lemma \u27e9\n      f (a \u229e da)\n    \u220e\n    where\n      open \u2261-Reasoning\n      lemma : f \u229e df\u2082 \u2261 f\n      lemma =\n        begin\n          f \u229e df\u2082\n        \u2261\u27e8 proof (\u2259-sym df\u2081\u2259df\u2082) \u27e9\n          f \u229e df\u2081\n        \u2261\u27e8 derivative-is-\u229e-unit df\u2081 IsDerivative-f-df\u2081 \u27e9\n          f\n        \u220e\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 (dx : \u0394 {{ca}} x) \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil {{ca}} x)\n    \u2261\u27e8 update-nil {{ca}} x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 (dx : \u0394 {{ca}} x) \u2192 x \u229e dx \u2261 x \u2192 _\u2259_ {{ca}} dx (nil {{ca}} x)\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil {{ca}} x) \u27e9\n      x \u229e nil {{ca}} x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff {{ca}} x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by\n  --   equiv-fun-changes-respect, and its corollaries fun-change-respects and\n  --   equiv-fun-changes-funs.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx : \u0394 {{ca}} x} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : _\u229e_ {{ca}} x (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff {{ca}} (x \u229e dx) x\n\n  -- \\begin{lemma}[Equivalence cancellation]\n  --   |v2 `ominus` v1 `doe` dv| holds if and only if |v2 = v1 `oplus`\n  --   dv|, for any |v1, v2 `elem` V| and |dv `elem` Dt ^ v1|.\n  -- \\end{lemma}\n\n  equiv-cancel-1 : \u2200 x' dx \u2192 _\u2259_ {{ca}} (x' \u229f x) dx \u2192 x' \u2261 x \u229e dx\n  equiv-cancel-1 x' dx (doe x'\u229fx\u2259dx) = trans (sym (update-diff x' x)) x'\u229fx\u2259dx\n  equiv-cancel-2 : \u2200 x' dx \u2192 x' \u2261 x \u229e dx \u2192 _\u2259_ {{ca}} (x' \u229f x) dx\n  equiv-cancel-2 _ dx refl = diff-update\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization; DerivativeAsChange)\n  open FC.FunctionChange\n\n  equiv-fun-changes-respect : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  equiv-fun-changes-respect {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} (df : \u0394 f) \u2192\n    dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply df x dx\u2082\n  fun-change-respects df dx\u2081\u2259dx\u2082 = equiv-fun-changes-respect (\u2259-refl {dx = df}) dx\u2081\u2259dx\u2082\n\n  -- D.o.e. function changes behave like the same function (up to d.o.e.).\n  equiv-fun-changes-funs : \u2200 {x : A} {dx : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082 : \u0394 f} \u2192\n    _\u2259_ {{changeAlgebra}} df\u2081 df\u2082 \u2192 apply df\u2081 x dx \u2259 apply df\u2082 x dx\n  equiv-fun-changes-funs {dx = dx} df\u2081\u2259df\u2082 = equiv-fun-changes-respect df\u2081\u2259df\u2082 (\u2259-refl {dx = dx})\n\n  derivative-doe-characterization : \u2200 {a : A} {da : \u0394 a}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CB}} (df a da) (f (a \u229e da) \u229f f a)\n  derivative-doe-characterization {a} {da} {f} {df} is-derivative = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f a \u229e df a da \u2261 f a \u229e (f (a \u229e da) \u229f f a)\n      lemma =\n        begin\n          (f a \u229e df a da)\n        \u2261\u27e8 is-derivative a da \u27e9\n          (f (a \u229e da))\n        \u2261\u27e8 sym (update-diff (f (a \u229e da)) (f a)) \u27e9\n          (f a \u229e (f (a \u229e da) \u229f f a))\n        \u220e\n\n  derivative-respects-doe : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    begin\n      df x dx\u2081\n    \u2259\u27e8 derivative-doe-characterization is-derivative \u27e9\n      f (x \u229e dx\u2081) \u229f f x\n    \u2261\u27e8 cong (\u03bb v \u2192 f v  \u229f f x) (proof dx\u2081\u2259dx\u2082) \u27e9\n      f (x \u229e dx\u2082) \u229f f x\n    \u2259\u27e8 \u2259-sym (derivative-doe-characterization is-derivative) \u27e9\n      df x dx\u2082\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is also a corollary of fun-changes-respect\n  derivative-respects-doe-alt : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x}\n    {f : A \u2192 B} {df : RawChange f} (is-derivative : IsDerivative f df) \u2192\n    _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192  _\u2259_ {{CB}} (df x dx\u2081) (df x dx\u2082)\n  derivative-respects-doe-alt {x} {dx\u2081} {dx\u2082} {f} {df} is-derivative dx\u2081\u2259dx\u2082 =\n    fun-change-respects (DerivativeAsChange is-derivative) dx\u2081\u2259dx\u2082\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 _\u2259_ {{CA}} dx\u2081 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that IsDerivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> IsDerivative f (apply df).\n  -- Now, we try to prove that if IsDerivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil {{CA}} x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil {{CA}} x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    IsDerivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  derivative-is-nil-alternative : \u2200 {f : A \u2192 B} df \u2192\n    (IsDerivative-f-df : IsDerivative f df) \u2192 DerivativeAsChange IsDerivative-f-df \u2259 nil f\n  derivative-is-nil-alternative df IsDerivative-f-df = derivative-is-nil (DerivativeAsChange IsDerivative-f-df) IsDerivative-f-df\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 IsDerivative f (apply df) \u2192 IsDerivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    \u2200 (f : A \u2192 B) df \u2192 IsDerivative f df \u2192\n    \u2200 v \u2192 df v (nil {{CA}} v) \u2259 nil {{CB}} (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil {{CB}} (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil {{CA}} v)\n        \u2261\u27e8 proof v (nil {{CA}} v) \u27e9\n          f (v \u229e (nil {{CA}} v))\n        \u2261\u27e8 cong f (update-nil {{CA}} v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil {{CB}} (f v)) \u27e9\n          f v \u229e nil {{CB}} (f v)\n        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"97ac1ebb0cbaecc1af38b1d9916fb946e06656a7","subject":"remove redundant comment","message":"remove redundant comment\n\nOld-commit-hash: 72138e09941c4514d0acef9dc39a91cd83e8077c\n","repos":"inc-lc\/ilc-agda","old_file":"PLDI14-List-of-Theorems.agda","new_file":"PLDI14-List-of-Theorems.agda","new_contents":"module PLDI14-List-of-Theorems where\n\nopen import Function\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important name. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- For each plugin requirement, we include its definition and\n-- a concrete instantiation called \"Popl14\" with integers and\n-- bags of integers as base types.\n\n-- Plugin Requirement 3.1 (Domains of base types)\nopen import Parametric.Denotation.Value using (Structure)\nopen import     Popl14.Denotation.Value using (\u27e6_\u27e7Base)\n\n-- Definition 3.2 (Domains)\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Plugin Requirement 3.3 (Evaluation of constants)\nopen import Parametric.Denotation.Evaluation using (Structure)\nopen import     Popl14.Denotation.Evaluation using (\u27e6_\u27e7Const)\n\n-- Definition 3.4 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.5 (Evaluation)\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Plugin Requirement 3.6 (Changes on base types)\nopen import Parametric.Change.Validity using (Structure)\nopen import     Popl14.Change.Validity using (change-algebra-base-family)\n\n-- Definition 3.7 (Changes)\nopen Parametric.Change.Validity.Structure using (change-algebra)\n\n-- Definition 3.8 (Change environments)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Plugin Requirement 3.9 (Change semantics for constants)\nopen import Parametric.Change.Specification using (Structure)\nopen import     Popl14.Change.Specification using (specification-structure)\n\n-- Definition 3.10 (Change semantics)\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\n\n-- Lemma 3.11 (Change semantics is the derivative of semantics)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.12 (Erasure)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen import Popl14.Change.Implementation using (implements-base)\n\n-- Lemma 3.13 (The erased version of a change is almost the same)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.14 (\u27e6 t \u27e7\u0394 erases to Derive(t))\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\n\n-- Theorem 3.15 (Correctness of differentiation)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","old_contents":"module PLDI14-List-of-Theorems where\n\nopen import Function\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important name. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\n-- (d) ChangeAlgebra.diff\n-- (e) IsChangeAlgebra.update-diff\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- For each plugin requirement, we include its definition and\n-- a concrete instantiation called \"Popl14\" with integers and\n-- bags of integers as base types.\n\n-- Plugin Requirement 3.1 (Domains of base types)\nopen import Parametric.Denotation.Value using (Structure)\nopen import     Popl14.Denotation.Value using (\u27e6_\u27e7Base)\n\n-- Definition 3.2 (Domains)\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Plugin Requirement 3.3 (Evaluation of constants)\nopen import Parametric.Denotation.Evaluation using (Structure)\nopen import     Popl14.Denotation.Evaluation using (\u27e6_\u27e7Const)\n\n-- Definition 3.4 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.5 (Evaluation)\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Plugin Requirement 3.6 (Changes on base types)\nopen import Parametric.Change.Validity using (Structure)\nopen import     Popl14.Change.Validity using (change-algebra-base-family)\n\n-- Definition 3.7 (Changes)\nopen Parametric.Change.Validity.Structure using (change-algebra)\n\n-- Definition 3.8 (Change environments)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Plugin Requirement 3.9 (Change semantics for constants)\nopen import Parametric.Change.Specification using (Structure)\nopen import     Popl14.Change.Specification using (specification-structure)\n\n-- Definition 3.10 (Change semantics)\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\n\n-- Lemma 3.11 (Change semantics is the derivative of semantics)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.12 (Erasure)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen import Popl14.Change.Implementation using (implements-base)\n\n-- Lemma 3.13 (The erased version of a change is almost the same)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.14 (\u27e6 t \u27e7\u0394 erases to Derive(t))\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\n\n-- Theorem 3.15 (Correctness of differentiation)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f49359e19fa0221bed11d6cc608a65246ef31d9a","subject":"Algebra: re-org properties for DistributesOver","message":"Algebra: re-org properties for DistributesOver\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP using (flip; \u03a0; \u03a0\u2071; Op\u2081; Op\u2082; nest)\nopen import Data.Nat.Base using (\u2115; _\u2264_; _\u2238_; z\u2264n; s\u2264s)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.NP\nopen import Algebra.Raw\nopen \u2261-Reasoning\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\nmodule Implicits where\n  open Algebra.FunctionProperties.NP \u03a0\u2071 {a}{a}{A} _\u2261_ public\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n\n  module From-Magma (magma : Magma A) where\n\n    open Magma magma\n\n    module From-Comm\n             (comm  : Commutative _\u2219_)\n             {x y x' y' : A}\n             (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n           where\n      comm= : x \u2219 y \u2261 x' \u2219 y'\n      comm= = comm \u2666 e \u2666 comm\n\n    module From-Assoc\n             (assoc : Associative _\u2219_)\n           where\n\n      assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n      assocs = assoc , ! assoc\n\n      module _ {c x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n        assoc= = ! assoc \u2666 \u2219= e idp \u2666 assoc\n\n        !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n        !assoc= = assoc \u2666 \u2219= idp e \u2666 ! assoc\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n        inner= = assoc= (!assoc= e)\n\n    module From-Assoc-Comm\n             (assoc : Associative _\u2219_)\n             (comm  : Commutative _\u2219_)\n           where\n      open From-Assoc assoc public\n      open From-Comm  comm  public\n\n      module _ {x y z} where\n        assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n        assoc-comm = assoc= comm\n\n        !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n        !assoc-comm = !assoc= comm\n\n      interchange : Interchange _\u2219_ _\u2219_\n      interchange = assoc= (!assoc= comm)\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n        outer= = \u2219= comm comm \u2666 assoc= (!assoc= e) \u2666 \u2219= comm comm\n\n  module From-Op\u2082 (op : Op\u2082 A) = From-Magma \u27e8 op \u27e9\n\n  module From-Monoid-Ops (mon-ops : Monoid-Ops A) where\n    open Monoid-Ops mon-ops\n    open From-Op\u2082 _\u2219_ public\n\n    module From-LeftIdentity (idl : LeftIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n        is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 idl\n\n    module From-RightIdentity (idr : RightIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n        is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 idr\n\n      module _ {b} where\n        ^\u207a-^\u00b9\u207a : \u2200 n \u2192 b ^\u207a (1+ n) \u2261 b ^\u00b9\u207a n\n        ^\u207a-^\u00b9\u207a 0 = idr\n        ^\u207a-^\u00b9\u207a (1+ n) = \u2219= idp (^\u207a-^\u00b9\u207a n)\n\n        ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n        ^\u207a0-\u03b5 = idp\n\n        ^\u207a1-id : b ^\u207a 1 \u2261 b\n        ^\u207a1-id = idr\n\n        ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n        ^\u207a2-\u2219 = ^\u207a-^\u00b9\u207a 1\n\n        ^\u207a3-\u2219 : b ^\u207a 3 \u2261 b \u2219 (b \u2219 b)\n        ^\u207a3-\u2219 = ^\u207a-^\u00b9\u207a 2\n\n        ^\u207a4-\u2219 : b ^\u207a 4 \u2261 b \u2219 (b \u2219 (b \u2219 b))\n        ^\u207a4-\u2219 = ^\u207a-^\u00b9\u207a 3\n\n    module From-Identities (identities : Identity \u03b5 _\u2219_) where\n      comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n      comm-\u03b5 = snd identities \u2666 ! fst identities\n\n      open From-LeftIdentity  (fst identities) public\n      open From-RightIdentity (snd identities) public\n\n    module From-Assoc-RightIdentity\n             (assoc : Associative _\u2219_)\n             (idr   : RightIdentity \u03b5 _\u2219_)\n             where\n      open From-RightIdentity idr public\n\n      module _ {c x y}(e : x \u2219 y \u2261 \u03b5) where\n        elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n        elim-assoc= = assoc \u2666 \u2219= idp e \u2666 idr\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-inner= = ! assoc \u2666 \u2219= (elim-assoc= e) idp\n\n    module From-Assoc-LeftIdentity\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             where\n      open From-LeftIdentity idl public\n\n      -- Together with ^\u207a0-\u03b5, ^\u207a-+ shows that (b ^\u207a_) is a\n      -- monoid homomorphism from (\u2115,+,0) to (M,\u2219,\u03b5)\n      -- therefore also a group homomorphism\n      ^\u207a-+ : \u2200 m {n b} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n      ^\u207a-+ 0      = ! idl\n      ^\u207a-+ (1+ m) = ap (_\u2219_ _) (^\u207a-+ m) \u2666 ! assoc\n\n      -- Derived from ^\u207a-+ in Algebra.Group\n      ^\u207a-* : \u2200 m {n b} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n      ^\u207a-* 0 = idp\n      ^\u207a-* (1+ m) {n} {b}\n        = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n          b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n          b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n      module _ {c x y} (e : x \u2219 y \u2261 \u03b5) where\n        elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n        elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 idl\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-!inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-!inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module From-Group-Ops (grp-ops : Group-Ops A) where\n    open Group-Ops grp-ops\n    open From-Monoid-Ops mon-ops public\n\n    module From-Assoc-LeftIdentity-LeftInverse\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             (inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-LeftIdentity assoc idl\n\n      \u207b\u00b9-inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      \u207b\u00b9-inverse\u02b3 {x}\n         = x \u2219 x \u207b\u00b9                      \u2261\u27e8 ! idl \u27e9\n           \u03b5 \u2219 (x \u2219 x \u207b\u00b9)                \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9) \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9)              \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5                             \u220e\n\n      cancels-\u2219-left : LeftCancel _\u2219_\n      cancels-\u2219-left {c} {x} {y} e\n        = x              \u2261\u27e8 ! idl \u27e9\n          \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inverse\u02e1 idp \u27e9\n          \u03b5 \u2219 y          \u2261\u27e8 idl \u27e9\n          y              \u220e\n\n      module _ {x y} where\n        unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n        unique-\u03b5-right eq\n          = y               \u2261\u27e8 ! is-\u03b5-left inverse\u02e1 \u27e9\n            x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n            x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n            x \u207b\u00b9 \u2219 x        \u2261\u27e8 inverse\u02e1 \u27e9\n            \u03b5               \u220e\n\n      module _ {x y} where\n        -- _\u207b\u00b9 is a group homomorphism, see Algebra.Group._\u1d52\u1d56\n        \u207b\u00b9-hom\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n        \u207b\u00b9-hom\u2032 = cancels-\u2219-left {x \u2219 y}\n           ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 \u207b\u00b9-inverse\u02b3 \u27e9\n            \u03b5                       \u2261\u27e8 ! \u207b\u00b9-inverse\u02b3 \u27e9\n            x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! idl) \u27e9\n            x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! \u207b\u00b9-inverse\u02b3) idp) \u27e9\n            x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n            x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n            (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n      \u207b\u00b9-hom\u2032= : \u2200 {x y z t} \u2192 x \u2219 y \u2261 z \u2219 t \u2192 y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261 t \u207b\u00b9 \u2219 z \u207b\u00b9\n      \u207b\u00b9-hom\u2032= e = ! \u207b\u00b9-hom\u2032 \u2666 ap _\u207b\u00b9 e \u2666 \u207b\u00b9-hom\u2032\n\n      elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n      elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n        = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n          x \u207b\u00b9 \u2219 y               \u220e\n\n      elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n      elim-\u2219-right-\/ {c} {x} {y}\n        = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n          x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-!inner= \u207b\u00b9-inverse\u02b3 \u27e9\n          x \/ y \u220e\n\n    module From-Assoc-RightIdentity-RightInverse\n             (assoc : Associative _\u2219_)\n             (idr : RightIdentity \u03b5 _\u2219_)\n             (inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-RightIdentity assoc idr\n\n      cancels-\u2219-right : RightCancel _\u2219_\n      cancels-\u2219-right {c} {x} {y} e\n        = x              \u2261\u27e8 ! idr \u27e9\n          x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n          y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inverse\u02b3 \u27e9\n          y \u2219 \u03b5          \u2261\u27e8 idr \u27e9\n          y              \u220e\n\n      inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02e1 {x} = x \u207b\u00b9 \u2219 x                      \u2261\u27e8 ! idr \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 \u03b5                \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-inner= inverse\u02b3 \u27e9\n                  (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9)              \u2261\u27e8 inverse\u02b3 \u27e9\n                  \u03b5 \u220e\n\n      \u207b\u00b9-involutive : Involutive _\u207b\u00b9\n      \u207b\u00b9-involutive {x}\n        = cancels-\u2219-right\n          (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5              \u2261\u27e8 ! inverse\u02b3 \u27e9\n           x \u2219 x \u207b\u00b9 \u220e)\n\n      module _ {x y} where\n        \/-\u2219 : x \u2261 (x \/ y) \u2219 y\n        \/-\u2219 = x               \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02e1) \u27e9\n              x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n              (x \/ y) \u2219 y     \u220e\n\n        \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n        \u2219-\/ = x            \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02b3) \u27e9\n              x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n              (x \u2219 y) \/ y  \u220e\n\n      module _ {x y} where\n        unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n        unique-\u03b5-left eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            y \/ y        \u2261\u27e8 inverse\u02b3 \u27e9\n            \u03b5            \u220e\n\n        x\/y\u2261\u03b5\u2192x\u2261y : x \/ y \u2261 \u03b5 \u2192 x \u2261 y\n        x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5 = cancels-\u2219-right (x\/y\u2261\u03b5 \u2666 ! inverse\u02b3)\n\n        x\/y\u2262\u03b5 : x \u2262 y \u2192 x \/ y \u2262 \u03b5\n        x\/y\u2262\u03b5 x\u2262y x\/y\u2261\u03b5 = x\u2262y (x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5)\n\n      \u03b5\u207b\u00b9\u2261\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n      \u03b5\u207b\u00b9\u2261\u03b5 = unique-\u03b5-left inverse\u02e1\n\n    module From-Assoc-Identities-RightInverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      private\n        idl = fst identities\n        idr = snd identities\n\n      inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02b3 = inv-r\n\n      open From-Assoc assoc\n      open From-Identities identities using (comm-\u03b5)\n      open From-Assoc-RightIdentity assoc idr\n      open From-Assoc-LeftIdentity  assoc idl\n      open From-Assoc-RightIdentity-RightInverse assoc idr inverse\u02b3 public\n      open From-Assoc-LeftIdentity-LeftInverse assoc idl inverse\u02e1 public\n        hiding (\u207b\u00b9-inverse\u02b3)\n\n      module _ {x y} where\n        unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n        unique-\u207b\u00b9 eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            \u03b5 \/ y        \u2261\u27e8 idl \u27e9\n            y \u207b\u00b9         \u220e\n\n      \u207b\u00b9-inj : \u2200 {x y} \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9 \u2192 x \u2261 y\n      \u207b\u00b9-inj {x} {y} e\n         = ! (y              \u2261\u27e8 ! idr \u27e9\n              y \u2219 \u03b5          \u2261\u27e8 \u2219= idp (\u03b5        \u2261\u27e8 ! inverse\u02e1  \u27e9\n                                        x \u207b\u00b9 \u2219 x \u2261\u27e8 \u2219= e idp \u27e9\n                                        y \u207b\u00b9 \u2219 x \u220e) \u27e9\n              y \u2219 (y \u207b\u00b9 \u2219 x) \u2261\u27e8 elim-!assoc= inverse\u02b3 \u27e9\n              x \u220e)\n\n      module _ {b} where\n        ^\u207a-1+ : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n        ^\u207a-1+ 0 = comm-\u03b5\n        ^\u207a-1+ (1+ n) = ap (_\u2219_ b) (^\u207a-1+ n) \u2666 ! assoc\n\n        ^\u207a-\u2238 : \u2200 {m n} \u2192 n \u2264 m \u2192 b ^\u207a(m \u2238 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207a-\u2238 z\u2264n = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207a-\u2238 (s\u2264s {n} {m} n\u2264m) =\n           b ^\u207a(m \u2238 n)                     \u2261\u27e8 ^\u207a-\u2238 n\u2264m \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296 : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^-\u2296 m      0      = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296 0      (1+ n) = ! idl\n        ^-\u2296 (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296 m n \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296' : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^-\u2296' m      0      = ! is-\u03b5-left \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296' 0      (1+ n) = ! idr\n        ^-\u2296' (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296' m n \u27e9\n           (b ^\u207b n \u2219 b ^\u207a m)               \u2261\u27e8 ! elim-inner= inverse\u02e1 \u27e9\n           (b ^\u207b n \u2219 b \u207b\u00b9) \u2219 (b \u2219 b ^\u207a m)  \u2261\u27e8 \u2219= (! \u207b\u00b9-hom\u2032) idp \u27e9\n           (b \u2219 b ^\u207a n)\u207b\u00b9 \u2219 (b \u2219 b ^\u207a m)   \u220e\n\n        ^\u207a-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b ^\u207a m\n        ^\u207a-comm 0      n = ! comm-\u03b5\n        ^\u207a-comm (1+ m) n =\n          b \u2219 b\u1d50 \u2219 b\u207f   \u2261\u27e8 !assoc= (^\u207a-comm m n) \u27e9\n          b \u2219 b\u207f \u2219 b\u1d50   \u2261\u27e8 \u2219= (^\u207a-1+ n) idp \u27e9\n          b\u207f \u2219 b \u2219 b\u1d50   \u2261\u27e8 assoc \u27e9\n          b\u207f \u2219 (b \u2219 b\u1d50) \u220e\n          where\n            b\u207f = b ^\u207a n\n            b\u1d50 = b ^\u207a m\n\n        ^\u207a-^\u207b-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^\u207a-^\u207b-comm m n = ! ^-\u2296 m n \u2666 ^-\u2296' m n\n\n        ^\u207b-^\u207a-comm : \u2200 m n \u2192 b ^\u207b n \u2219 b ^\u207a m \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207b-^\u207a-comm m n = ! ^-\u2296' m n \u2666 ^-\u2296 m n\n\n        ^\u207b-comm : \u2200 m n \u2192 b ^\u207b m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207b m\n        ^\u207b-comm m n = \u207b\u00b9-hom\u2032= (^\u207a-comm n m)\n\n        ^\u207b-1+ : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n        ^\u207b-1+ n = ap _\u207b\u00b9 (^\u207a-1+ n) \u2666 \u207b\u00b9-hom\u2032\n\n        ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n        ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                        \u2666 ! \u207b\u00b9-hom\u2032\n                        \u2666 ap _\u207b\u00b9 (! ^\u207a-1+ n)\n\n        ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n        ^\u207b\u20321-id = idr\n\n        ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n        ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n        ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n        ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\n        -- ^-+ is a group homomorphism defined in Algebra.Group\n        -- Some properties can be derived from it.\n        ^-+ : \u2200 i j \u2192 b ^(i +\u2124 j) \u2261 b ^ i \u2219 b ^ j\n        ^-+ -[1+ m ] -[1+ n ] = ap _\u207b\u00b9\n                                 (ap (\u03bb z \u2192 b ^\u207a(1+ z)) (\u2115\u00b0.+-comm (1+ m) n)\n                              \u2666 ^\u207a-+ (1+ n) {1+ m}) \u2666 \u207b\u00b9-hom\u2032\n        ^-+ -[1+ m ]   (+ n)  = ^-\u2296' n (1+ m)\n        ^-+   (+ m)  -[1+ n ] = ^-\u2296 m (1+ n)\n        ^-+   (+ m)    (+ n)  = ^\u207a-+ m\n\n        -- GroupHomomorphism.f-pres-inv\n        ^-- : \u2200 i \u2192 b ^(- i) \u2261 (b ^ i)\u207b\u00b9\n        ^-- -[1+ n ] = ! \u207b\u00b9-involutive\n        ^-- (+ 0)    = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^-- (+ 1+ n) = idp\n\n        -- GroupHomomorphism.f-\u2212-\/\n        ^-\u2212 : \u2200 i j \u2192 b ^(i \u2212\u2124 j) \u2261 b ^ i \/ b ^ j\n        ^-\u2212 i j = ^-+ i (- j) \u2666 \u2219= idp (^-- j)\n\n        ^-suc : \u2200 i \u2192 b ^(suc\u2124 i) \u2261 b \u2219 b ^ i\n        ^-suc i = ^-+ (+ 1) i \u2666 \u2219= idr idp\n\n        ^-pred : \u2200 i \u2192 b ^(pred\u2124 i) \u2261 b \u207b\u00b9 \u2219 b ^ i\n        ^-pred i = ^-+ (- + 1) i \u2666 ap (\u03bb z \u2192 z \u207b\u00b9 \u2219 b ^ i) idr\n\n        ^-comm : \u2200 i j \u2192 b ^ i \u2219 b ^ j \u2261 b ^ j \u2219 b ^ i\n        ^-comm -[1+ m ] -[1+ n ] = ^\u207b-comm (1+ m) (1+ n)\n        ^-comm -[1+ m ]   (+ n)  = ! ^\u207a-^\u207b-comm n (1+ m)\n        ^-comm   (+ m)  -[1+ n ] = ^\u207a-^\u207b-comm m (1+ n)\n        ^-comm   (+ m)    (+ n)  = ^\u207a-comm m n\n\n      {-\n      ^-* : \u2200 i j \u2192 b ^(i *\u2124 j) \u2261 (b ^ j)^ i\n      ^-* i j = {!!}\n      -}\n\n    module From-Assoc-Identities-Inverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv : Inverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc-Identities-RightInverse assoc identities (snd inv) public\n\n  module From-Ring-Ops (rng-ops : Ring-Ops A) where\n    open Ring-Ops rng-ops\n\n    module DistributesOver\u02b3\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_)\n             where\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = *-comm \u2666 *-+-distr\u02e1 \u2666 += *-comm *-comm\n\n    module DistributesOver\u02e1\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = *-comm \u2666 *-+-distr\u02b3 \u2666 += *-comm *-comm\n\n    module From-+Group-1*Identity-DistributesOver\u02b3\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n                     ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n                     )\n\n      -0\u22610 : -0# \u2261 0#\n      -0\u22610 = 0\u22120\u22610\n\n      0*-zero : LeftZero 0# _*_\n      0*-zero {x} = cancels-+-left\n        (x + 0# * x      \u2261\u27e8 += (! 1*-identity) idp \u27e9\n         1# * x + 0# * x \u2261\u27e8 ! *-+-distr\u02b3 \u27e9\n         (1# + 0#) * x   \u2261\u27e8 *= +0-identity idp \u27e9\n         1# * x          \u2261\u27e8 1*-identity \u27e9\n         x               \u2261\u27e8 ! +0-identity \u27e9\n         x + 0#          \u220e)\n\n      2*-spec : \u2200 {n} \u2192 2* n \u2261 2# * n\n      2*-spec = ! += 1*-identity 1*-identity \u2666 ! *-+-distr\u02b3\n\n      0\u2212c+c+x : \u2200 {c x} \u2192 0\u2212 c + c + x \u2261 x\n      0\u2212c+c+x = += 0\u2212-inverse\u02e1 idp \u2666 0+-identity\n\n      0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n      0\u2212-*-distr = cancels-+-right (0\u2212-inverse\u02e1 \u2666 ! 0*-zero \u2666 *= (! 0\u2212-inverse\u02e1) idp \u2666 *-+-distr\u02b3)\n\n      -1*-neg : \u2200 {x} \u2192 -1# * x \u2261 0\u2212 x\n      -1*-neg = ! 0\u2212-*-distr \u2666 0\u2212= 1*-identity\n\n      2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n      2*-*-distr = ! *-+-distr\u02b3\n\n      0\u2212-+-distr\u2032 : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 y \u2212 x\n      0\u2212-+-distr\u2032 = 0\u2212-hom\u2032\n\n    module From-+Group-*Identity-DistributesOver\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*1-identity : RightIdentity 1# _*_)\n             (*-+-distrs : _*_ DistributesOver _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = fst *-+-distrs\n\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = snd *-+-distrs\n\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( \u207b\u00b9-involutive to 0\u2212-involutive\n                     ; cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     )\n\n      open From-+Group-1*Identity-DistributesOver\u02b3\n             +-assoc 0+-identity +0-identity 0\u2212-inverse\u02b3\n             1*-identity *-+-distr\u02b3\n             public\n\n      *0-zero : RightZero 0# _*_\n      *0-zero {x} = cancels-+-right\n        (x * 0# + x      \u2261\u27e8 += idp (! *1-identity) \u27e9\n         x * 0# + x * 1# \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n         x * (0# + 1#)   \u2261\u27e8 *= idp 0+-identity \u27e9\n         x * 1#          \u2261\u27e8 *1-identity \u27e9\n         x               \u2261\u27e8 ! 0+-identity \u27e9\n         0# + x          \u220e)\n\n      0\u2212-*-distr\u02b3 : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n      0\u2212-*-distr\u02b3 = cancels-+-right (0\u2212-inverse\u02e1 \u2666 (! *0-zero \u2666 *= idp (! 0\u2212-inverse\u02e1)) \u2666 *-+-distr\u02e1)\n\n      *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n      *-\u2212-distr = *-+-distr\u02e1 \u2666 += idp (! 0\u2212-*-distr\u02b3)\n\n      \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n      \u00b2-0\u2212-distr = ! 0\u2212-*-distr \u2666 0\u2212=(! 0\u2212-*-distr\u02b3) \u2666 0\u2212-involutive\n\n      +-comm : Commutative _+_\n      +-comm {a} {b} =\n          cancels-+-right\n            (cancels-+-left\n              (a + (a + b + b)   \u2261\u27e8 += idp +-assoc \u2666 ! +-assoc \u27e9\n               (a + a) + (b + b) \u2261\u27e8 += 2*-spec 2*-spec \u27e9\n               2# * a  + 2# * b  \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n               2# * (a + b)      \u2261\u27e8 ! 2*-spec \u27e9\n               (a + b) + (a + b) \u2261\u27e8 +-assoc \u2666 += idp (! +-assoc) \u27e9\n               a + (b + a + b)   \u220e))\n\n      0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n      0\u2212-+-distr = 0\u2212= +-comm \u2666 0\u2212-+-distr\u2032\n\n  module From-Field-Ops (fld-ops : Field-Ops A) where\n    open Field-Ops fld-ops\n\nmodule Explicits where\n  open Algebra.FunctionProperties.NP \u03a0  {a}{a}{A} _\u2261_ public\n\n  -- REPEATED from above but with explicit arguments\n  module FromOp\u2082\n           (_\u2219_ : Op\u2082 A){x x' y y'}(p : x \u2261 x')(q : y \u2261 y')\n         where\n    op= : x \u2219 y \u2261 x' \u2219 y'\n    op= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  module FromComm\n           (_\u2219_   : Op\u2082 A)\n           (comm  : Commutative _\u2219_)\n           (x y x' y' : A)\n           (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n         where\n    open FromOp\u2082 _\u2219_\n\n    comm= : x \u2219 y \u2261 x' \u2219 y'\n    comm= = comm _ _ \u2666 e \u2666 comm _ _\n\n  module FromAssoc\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n\n         where\n    open FromOp\u2082 _\u2219_\n\n    assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    assocs = assoc , (\u03bb _ _ _ \u2192 ! assoc _ _ _)\n\n    module _ {c x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n      assoc= = ! assoc _ _ _ \u2666 op= e idp \u2666 assoc _ _ _\n\n      !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n      !assoc= = assoc _ _ _ \u2666 op= idp e \u2666 ! assoc _ _ _\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n      inner= = assoc= (!assoc= e)\n\n  module FromAssocComm\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n           (comm  : Commutative _\u2219_)\n         where\n    open FromOp\u2082   _\u2219_       renaming (op= to \u2219=)\n    open FromAssoc _\u2219_ assoc public\n    open FromComm  _\u2219_ comm  public\n\n    module _ x y z where\n      assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n      assoc-comm = assoc= (comm _ _)\n\n      !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n      !assoc-comm = !assoc= (comm _ _)\n\n    interchange : Interchange _\u2219_ _\u2219_\n    interchange _ _ _ _ = assoc= (!assoc= (comm _ _))\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n      outer= = \u2219= (comm _ _) (comm _ _) \u2666 assoc= (!assoc= e) \u2666 \u2219= (comm _ _) (comm _ _)\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj _ _ fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP using (flip; \u03a0; \u03a0\u2071; Op\u2081; Op\u2082; nest)\nopen import Data.Nat.Base using (\u2115; _\u2264_; _\u2238_; z\u2264n; s\u2264s)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.NP\nopen import Algebra.Raw\nopen \u2261-Reasoning\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\nmodule Implicits where\n  open Algebra.FunctionProperties.NP \u03a0\u2071 {a}{a}{A} _\u2261_ public\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n\n  module From-Magma (magma : Magma A) where\n\n    open Magma magma\n\n    module From-Comm\n             (comm  : Commutative _\u2219_)\n             {x y x' y' : A}\n             (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n           where\n      comm= : x \u2219 y \u2261 x' \u2219 y'\n      comm= = comm \u2666 e \u2666 comm\n\n    module From-Assoc\n             (assoc : Associative _\u2219_)\n           where\n\n      assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n      assocs = assoc , ! assoc\n\n      module _ {c x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n        assoc= = ! assoc \u2666 \u2219= e idp \u2666 assoc\n\n        !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n        !assoc= = assoc \u2666 \u2219= idp e \u2666 ! assoc\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n        inner= = assoc= (!assoc= e)\n\n    module From-Assoc-Comm\n             (assoc : Associative _\u2219_)\n             (comm  : Commutative _\u2219_)\n           where\n      open From-Assoc assoc public\n      open From-Comm  comm  public\n\n      module _ {x y z} where\n        assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n        assoc-comm = assoc= comm\n\n        !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n        !assoc-comm = !assoc= comm\n\n      interchange : Interchange _\u2219_ _\u2219_\n      interchange = assoc= (!assoc= comm)\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n        outer= = \u2219= comm comm \u2666 assoc= (!assoc= e) \u2666 \u2219= comm comm\n\n  module From-Op\u2082 (op : Op\u2082 A) = From-Magma \u27e8 op \u27e9\n\n  module From-Monoid-Ops (mon-ops : Monoid-Ops A) where\n    open Monoid-Ops mon-ops\n    open From-Op\u2082 _\u2219_ public\n\n    module From-LeftIdentity (idl : LeftIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n        is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 idl\n\n    module From-RightIdentity (idr : RightIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n        is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 idr\n\n      module _ {b} where\n        ^\u207a-^\u00b9\u207a : \u2200 n \u2192 b ^\u207a (1+ n) \u2261 b ^\u00b9\u207a n\n        ^\u207a-^\u00b9\u207a 0 = idr\n        ^\u207a-^\u00b9\u207a (1+ n) = \u2219= idp (^\u207a-^\u00b9\u207a n)\n\n        ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n        ^\u207a0-\u03b5 = idp\n\n        ^\u207a1-id : b ^\u207a 1 \u2261 b\n        ^\u207a1-id = idr\n\n        ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n        ^\u207a2-\u2219 = ^\u207a-^\u00b9\u207a 1\n\n        ^\u207a3-\u2219 : b ^\u207a 3 \u2261 b \u2219 (b \u2219 b)\n        ^\u207a3-\u2219 = ^\u207a-^\u00b9\u207a 2\n\n        ^\u207a4-\u2219 : b ^\u207a 4 \u2261 b \u2219 (b \u2219 (b \u2219 b))\n        ^\u207a4-\u2219 = ^\u207a-^\u00b9\u207a 3\n\n    module From-Identities (identities : Identity \u03b5 _\u2219_) where\n      comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n      comm-\u03b5 = snd identities \u2666 ! fst identities\n\n      open From-LeftIdentity  (fst identities) public\n      open From-RightIdentity (snd identities) public\n\n    module From-Assoc-RightIdentity\n             (assoc : Associative _\u2219_)\n             (idr   : RightIdentity \u03b5 _\u2219_)\n             where\n      open From-RightIdentity idr public\n\n      module _ {c x y}(e : x \u2219 y \u2261 \u03b5) where\n        elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n        elim-assoc= = assoc \u2666 \u2219= idp e \u2666 idr\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-inner= = ! assoc \u2666 \u2219= (elim-assoc= e) idp\n\n    module From-Assoc-LeftIdentity\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             where\n      open From-LeftIdentity idl public\n\n      -- Together with ^\u207a0-\u03b5, ^\u207a-+ shows that (b ^\u207a_) is a\n      -- monoid homomorphism from (\u2115,+,0) to (M,\u2219,\u03b5)\n      -- therefore also a group homomorphism\n      ^\u207a-+ : \u2200 m {n b} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n      ^\u207a-+ 0      = ! idl\n      ^\u207a-+ (1+ m) = ap (_\u2219_ _) (^\u207a-+ m) \u2666 ! assoc\n\n      -- Derived from ^\u207a-+ in Algebra.Group\n      ^\u207a-* : \u2200 m {n b} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n      ^\u207a-* 0 = idp\n      ^\u207a-* (1+ m) {n} {b}\n        = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n          b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n          b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n      module _ {c x y} (e : x \u2219 y \u2261 \u03b5) where\n        elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n        elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 idl\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-!inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-!inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module From-Group-Ops (grp-ops : Group-Ops A) where\n    open Group-Ops grp-ops\n    open From-Monoid-Ops mon-ops public\n\n    module From-Assoc-LeftIdentity-LeftInverse\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             (inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-LeftIdentity assoc idl\n\n      \u207b\u00b9-inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      \u207b\u00b9-inverse\u02b3 {x}\n         = x \u2219 x \u207b\u00b9                      \u2261\u27e8 ! idl \u27e9\n           \u03b5 \u2219 (x \u2219 x \u207b\u00b9)                \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9) \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9)              \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5                             \u220e\n\n      cancels-\u2219-left : LeftCancel _\u2219_\n      cancels-\u2219-left {c} {x} {y} e\n        = x              \u2261\u27e8 ! idl \u27e9\n          \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inverse\u02e1 idp \u27e9\n          \u03b5 \u2219 y          \u2261\u27e8 idl \u27e9\n          y              \u220e\n\n      module _ {x y} where\n        unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n        unique-\u03b5-right eq\n          = y               \u2261\u27e8 ! is-\u03b5-left inverse\u02e1 \u27e9\n            x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n            x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n            x \u207b\u00b9 \u2219 x        \u2261\u27e8 inverse\u02e1 \u27e9\n            \u03b5               \u220e\n\n      module _ {x y} where\n        -- _\u207b\u00b9 is a group homomorphism, see Algebra.Group._\u1d52\u1d56\n        \u207b\u00b9-hom\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n        \u207b\u00b9-hom\u2032 = cancels-\u2219-left {x \u2219 y}\n           ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 \u207b\u00b9-inverse\u02b3 \u27e9\n            \u03b5                       \u2261\u27e8 ! \u207b\u00b9-inverse\u02b3 \u27e9\n            x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! idl) \u27e9\n            x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! \u207b\u00b9-inverse\u02b3) idp) \u27e9\n            x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n            x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n            (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n      \u207b\u00b9-hom\u2032= : \u2200 {x y z t} \u2192 x \u2219 y \u2261 z \u2219 t \u2192 y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261 t \u207b\u00b9 \u2219 z \u207b\u00b9\n      \u207b\u00b9-hom\u2032= e = ! \u207b\u00b9-hom\u2032 \u2666 ap _\u207b\u00b9 e \u2666 \u207b\u00b9-hom\u2032\n\n      elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n      elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n        = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n          x \u207b\u00b9 \u2219 y               \u220e\n\n      elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n      elim-\u2219-right-\/ {c} {x} {y}\n        = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n          x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-!inner= \u207b\u00b9-inverse\u02b3 \u27e9\n          x \/ y \u220e\n\n    module From-Assoc-RightIdentity-RightInverse\n             (assoc : Associative _\u2219_)\n             (idr : RightIdentity \u03b5 _\u2219_)\n             (inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-RightIdentity assoc idr\n\n      cancels-\u2219-right : RightCancel _\u2219_\n      cancels-\u2219-right {c} {x} {y} e\n        = x              \u2261\u27e8 ! idr \u27e9\n          x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n          y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inverse\u02b3 \u27e9\n          y \u2219 \u03b5          \u2261\u27e8 idr \u27e9\n          y              \u220e\n\n      inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02e1 {x} = x \u207b\u00b9 \u2219 x                      \u2261\u27e8 ! idr \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 \u03b5                \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n                  (x \u207b\u00b9 \u2219 x) \u2219 (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-inner= inverse\u02b3 \u27e9\n                  (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9)              \u2261\u27e8 inverse\u02b3 \u27e9\n                  \u03b5 \u220e\n\n      \u207b\u00b9-involutive : Involutive _\u207b\u00b9\n      \u207b\u00b9-involutive {x}\n        = cancels-\u2219-right\n          (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5              \u2261\u27e8 ! inverse\u02b3 \u27e9\n           x \u2219 x \u207b\u00b9 \u220e)\n\n      module _ {x y} where\n        \/-\u2219 : x \u2261 (x \/ y) \u2219 y\n        \/-\u2219 = x               \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02e1) \u27e9\n              x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n              (x \/ y) \u2219 y     \u220e\n\n        \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n        \u2219-\/ = x            \u2261\u27e8 ! idr \u27e9\n              x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inverse\u02b3) \u27e9\n              x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n              (x \u2219 y) \/ y  \u220e\n\n      module _ {x y} where\n        unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n        unique-\u03b5-left eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            y \/ y        \u2261\u27e8 inverse\u02b3 \u27e9\n            \u03b5            \u220e\n\n        x\/y\u2261\u03b5\u2192x\u2261y : x \/ y \u2261 \u03b5 \u2192 x \u2261 y\n        x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5 = cancels-\u2219-right (x\/y\u2261\u03b5 \u2666 ! inverse\u02b3)\n\n        x\/y\u2262\u03b5 : x \u2262 y \u2192 x \/ y \u2262 \u03b5\n        x\/y\u2262\u03b5 x\u2262y x\/y\u2261\u03b5 = x\u2262y (x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5)\n\n      \u03b5\u207b\u00b9\u2261\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n      \u03b5\u207b\u00b9\u2261\u03b5 = unique-\u03b5-left inverse\u02e1\n\n    module From-Assoc-Identities-RightInverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      private\n        idl = fst identities\n        idr = snd identities\n\n      inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02b3 = inv-r\n\n      open From-Assoc assoc\n      open From-Identities identities using (comm-\u03b5)\n      open From-Assoc-RightIdentity assoc idr\n      open From-Assoc-LeftIdentity  assoc idl\n      open From-Assoc-RightIdentity-RightInverse assoc idr inverse\u02b3 public\n      open From-Assoc-LeftIdentity-LeftInverse assoc idl inverse\u02e1 public\n        hiding (\u207b\u00b9-inverse\u02b3)\n\n      module _ {x y} where\n        unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n        unique-\u207b\u00b9 eq\n          = x            \u2261\u27e8 \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            \u03b5 \/ y        \u2261\u27e8 idl \u27e9\n            y \u207b\u00b9         \u220e\n\n      \u207b\u00b9-inj : \u2200 {x y} \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9 \u2192 x \u2261 y\n      \u207b\u00b9-inj {x} {y} e\n         = ! (y              \u2261\u27e8 ! idr \u27e9\n              y \u2219 \u03b5          \u2261\u27e8 \u2219= idp (\u03b5        \u2261\u27e8 ! inverse\u02e1  \u27e9\n                                        x \u207b\u00b9 \u2219 x \u2261\u27e8 \u2219= e idp \u27e9\n                                        y \u207b\u00b9 \u2219 x \u220e) \u27e9\n              y \u2219 (y \u207b\u00b9 \u2219 x) \u2261\u27e8 elim-!assoc= inverse\u02b3 \u27e9\n              x \u220e)\n\n      module _ {b} where\n        ^\u207a-1+ : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n        ^\u207a-1+ 0 = comm-\u03b5\n        ^\u207a-1+ (1+ n) = ap (_\u2219_ b) (^\u207a-1+ n) \u2666 ! assoc\n\n        ^\u207a-\u2238 : \u2200 {m n} \u2192 n \u2264 m \u2192 b ^\u207a(m \u2238 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207a-\u2238 z\u2264n = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207a-\u2238 (s\u2264s {n} {m} n\u2264m) =\n           b ^\u207a(m \u2238 n)                     \u2261\u27e8 ^\u207a-\u2238 n\u2264m \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296 : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^-\u2296 m      0      = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296 0      (1+ n) = ! idl\n        ^-\u2296 (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296 m n \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296' : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^-\u2296' m      0      = ! is-\u03b5-left \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296' 0      (1+ n) = ! idr\n        ^-\u2296' (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296' m n \u27e9\n           (b ^\u207b n \u2219 b ^\u207a m)               \u2261\u27e8 ! elim-inner= inverse\u02e1 \u27e9\n           (b ^\u207b n \u2219 b \u207b\u00b9) \u2219 (b \u2219 b ^\u207a m)  \u2261\u27e8 \u2219= (! \u207b\u00b9-hom\u2032) idp \u27e9\n           (b \u2219 b ^\u207a n)\u207b\u00b9 \u2219 (b \u2219 b ^\u207a m)   \u220e\n\n        ^\u207a-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b ^\u207a m\n        ^\u207a-comm 0      n = ! comm-\u03b5\n        ^\u207a-comm (1+ m) n =\n          b \u2219 b\u1d50 \u2219 b\u207f   \u2261\u27e8 !assoc= (^\u207a-comm m n) \u27e9\n          b \u2219 b\u207f \u2219 b\u1d50   \u2261\u27e8 \u2219= (^\u207a-1+ n) idp \u27e9\n          b\u207f \u2219 b \u2219 b\u1d50   \u2261\u27e8 assoc \u27e9\n          b\u207f \u2219 (b \u2219 b\u1d50) \u220e\n          where\n            b\u207f = b ^\u207a n\n            b\u1d50 = b ^\u207a m\n\n        ^\u207a-^\u207b-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^\u207a-^\u207b-comm m n = ! ^-\u2296 m n \u2666 ^-\u2296' m n\n\n        ^\u207b-^\u207a-comm : \u2200 m n \u2192 b ^\u207b n \u2219 b ^\u207a m \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207b-^\u207a-comm m n = ! ^-\u2296' m n \u2666 ^-\u2296 m n\n\n        ^\u207b-comm : \u2200 m n \u2192 b ^\u207b m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207b m\n        ^\u207b-comm m n = \u207b\u00b9-hom\u2032= (^\u207a-comm n m)\n\n        ^\u207b-1+ : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n        ^\u207b-1+ n = ap _\u207b\u00b9 (^\u207a-1+ n) \u2666 \u207b\u00b9-hom\u2032\n\n        ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n        ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                        \u2666 ! \u207b\u00b9-hom\u2032\n                        \u2666 ap _\u207b\u00b9 (! ^\u207a-1+ n)\n\n        ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n        ^\u207b\u20321-id = idr\n\n        ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n        ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n        ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n        ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\n        -- ^-+ is a group homomorphism defined in Algebra.Group\n        -- Some properties can be derived from it.\n        ^-+ : \u2200 i j \u2192 b ^(i +\u2124 j) \u2261 b ^ i \u2219 b ^ j\n        ^-+ -[1+ m ] -[1+ n ] = ap _\u207b\u00b9\n                                 (ap (\u03bb z \u2192 b ^\u207a(1+ z)) (\u2115\u00b0.+-comm (1+ m) n)\n                              \u2666 ^\u207a-+ (1+ n) {1+ m}) \u2666 \u207b\u00b9-hom\u2032\n        ^-+ -[1+ m ]   (+ n)  = ^-\u2296' n (1+ m)\n        ^-+   (+ m)  -[1+ n ] = ^-\u2296 m (1+ n)\n        ^-+   (+ m)    (+ n)  = ^\u207a-+ m\n\n        -- GroupHomomorphism.f-pres-inv\n        ^-- : \u2200 i \u2192 b ^(- i) \u2261 (b ^ i)\u207b\u00b9\n        ^-- -[1+ n ] = ! \u207b\u00b9-involutive\n        ^-- (+ 0)    = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^-- (+ 1+ n) = idp\n\n        -- GroupHomomorphism.f-\u2212-\/\n        ^-\u2212 : \u2200 i j \u2192 b ^(i \u2212\u2124 j) \u2261 b ^ i \/ b ^ j\n        ^-\u2212 i j = ^-+ i (- j) \u2666 \u2219= idp (^-- j)\n\n        ^-suc : \u2200 i \u2192 b ^(suc\u2124 i) \u2261 b \u2219 b ^ i\n        ^-suc i = ^-+ (+ 1) i \u2666 \u2219= idr idp\n\n        ^-pred : \u2200 i \u2192 b ^(pred\u2124 i) \u2261 b \u207b\u00b9 \u2219 b ^ i\n        ^-pred i = ^-+ (- + 1) i \u2666 ap (\u03bb z \u2192 z \u207b\u00b9 \u2219 b ^ i) idr\n\n        ^-comm : \u2200 i j \u2192 b ^ i \u2219 b ^ j \u2261 b ^ j \u2219 b ^ i\n        ^-comm -[1+ m ] -[1+ n ] = ^\u207b-comm (1+ m) (1+ n)\n        ^-comm -[1+ m ]   (+ n)  = ! ^\u207a-^\u207b-comm n (1+ m)\n        ^-comm   (+ m)  -[1+ n ] = ^\u207a-^\u207b-comm m (1+ n)\n        ^-comm   (+ m)    (+ n)  = ^\u207a-comm m n\n\n      {-\n      ^-* : \u2200 i j \u2192 b ^(i *\u2124 j) \u2261 (b ^ j)^ i\n      ^-* i j = {!!}\n      -}\n\n    module From-Assoc-Identities-Inverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv : Inverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc-Identities-RightInverse assoc identities (snd inv) public\n\n  module From-Ring-Ops (rng-ops : Ring-Ops A) where\n    open Ring-Ops rng-ops\n\n    module DistributesOver\u02b3\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_)\n             where\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = *-comm \u2666 *-+-distr\u02e1 \u2666 += *-comm *-comm\n\n    module DistributesOver\u02e1\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = *-comm \u2666 *-+-distr\u02b3 \u2666 += *-comm *-comm\n\n    module From-+Group-*Identity-DistributesOver\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*1-identity : RightIdentity 1# _*_)\n             (*-+-distrs : _*_ DistributesOver _+_)\n             where\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( \u207b\u00b9-involutive to 0\u2212-involutive\n                     ; cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n                     ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n                     )\n\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = fst *-+-distrs\n\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = snd *-+-distrs\n\n      -0\u22610 : -0# \u2261 0#\n      -0\u22610 = 0\u22120\u22610\n\n      0*-zero : LeftZero 0# _*_\n      0*-zero {x} = cancels-+-left\n        (x + 0# * x      \u2261\u27e8 += (! 1*-identity) idp \u27e9\n         1# * x + 0# * x \u2261\u27e8 ! *-+-distr\u02b3 \u27e9\n         (1# + 0#) * x   \u2261\u27e8 *= +0-identity idp \u27e9\n         1# * x          \u2261\u27e8 1*-identity \u27e9\n         x               \u2261\u27e8 ! +0-identity \u27e9\n         x + 0#          \u220e)\n\n      *0-zero : RightZero 0# _*_\n      *0-zero {x} = cancels-+-right\n        (x * 0# + x      \u2261\u27e8 += idp (! *1-identity) \u27e9\n         x * 0# + x * 1# \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n         x * (0# + 1#)   \u2261\u27e8 *= idp 0+-identity \u27e9\n         x * 1#          \u2261\u27e8 *1-identity \u27e9\n         x               \u2261\u27e8 ! 0+-identity \u27e9\n         0# + x          \u220e)\n\n      2*-spec : \u2200 {n} \u2192 2* n \u2261 2# * n\n      2*-spec = ! += 1*-identity 1*-identity \u2666 ! *-+-distr\u02b3\n\n      0\u2212c+c+x : \u2200 {c x} \u2192 0\u2212 c + c + x \u2261 x\n      0\u2212c+c+x = += 0\u2212-inverse\u02e1 idp \u2666 0+-identity\n\n      0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n      0\u2212-*-distr = cancels-+-right (0\u2212-inverse\u02e1 \u2666 ! 0*-zero \u2666 *= (! 0\u2212-inverse\u02e1) idp \u2666 *-+-distr\u02b3)\n\n      0\u2212-*-distr\u02b3 : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n      0\u2212-*-distr\u02b3 = cancels-+-right (0\u2212-inverse\u02e1 \u2666 (! *0-zero \u2666 *= idp (! 0\u2212-inverse\u02e1)) \u2666 *-+-distr\u02e1)\n\n      -1*-neg : \u2200 {x} \u2192 -1# * x \u2261 0\u2212 x\n      -1*-neg = ! 0\u2212-*-distr \u2666 0\u2212= 1*-identity\n\n      *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n      *-\u2212-distr = *-+-distr\u02e1 \u2666 += idp (! 0\u2212-*-distr\u02b3)\n\n      \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n      \u00b2-0\u2212-distr = ! 0\u2212-*-distr \u2666 0\u2212=(! 0\u2212-*-distr\u02b3) \u2666 0\u2212-involutive\n\n      2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n      2*-*-distr = ! *-+-distr\u02b3\n\n      +-comm : Commutative _+_\n      +-comm {a} {b} =\n          cancels-+-right\n            (cancels-+-left\n              (a + (a + b + b)   \u2261\u27e8 += idp +-assoc \u2666 ! +-assoc \u27e9\n               (a + a) + (b + b) \u2261\u27e8 += 2*-spec 2*-spec \u27e9\n               2# * a  + 2# * b  \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n               2# * (a + b)      \u2261\u27e8 ! 2*-spec \u27e9\n               (a + b) + (a + b) \u2261\u27e8 +-assoc \u2666 += idp (! +-assoc) \u27e9\n               a + (b + a + b)   \u220e))\n\n      0\u2212-+-distr\u2032 : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 y \u2212 x\n      0\u2212-+-distr\u2032 = 0\u2212-hom\u2032\n\n      0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n      0\u2212-+-distr = 0\u2212= +-comm \u2666 0\u2212-+-distr\u2032\n\n  module From-Field-Ops (fld-ops : Field-Ops A) where\n    open Field-Ops fld-ops\n\nmodule Explicits where\n  open Algebra.FunctionProperties.NP \u03a0  {a}{a}{A} _\u2261_ public\n\n  -- REPEATED from above but with explicit arguments\n  module FromOp\u2082\n           (_\u2219_ : Op\u2082 A){x x' y y'}(p : x \u2261 x')(q : y \u2261 y')\n         where\n    op= : x \u2219 y \u2261 x' \u2219 y'\n    op= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  module FromComm\n           (_\u2219_   : Op\u2082 A)\n           (comm  : Commutative _\u2219_)\n           (x y x' y' : A)\n           (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n         where\n    open FromOp\u2082 _\u2219_\n\n    comm= : x \u2219 y \u2261 x' \u2219 y'\n    comm= = comm _ _ \u2666 e \u2666 comm _ _\n\n  module FromAssoc\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n\n         where\n    open FromOp\u2082 _\u2219_\n\n    assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    assocs = assoc , (\u03bb _ _ _ \u2192 ! assoc _ _ _)\n\n    module _ {c x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n      assoc= = ! assoc _ _ _ \u2666 op= e idp \u2666 assoc _ _ _\n\n      !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n      !assoc= = assoc _ _ _ \u2666 op= idp e \u2666 ! assoc _ _ _\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n      inner= = assoc= (!assoc= e)\n\n  module FromAssocComm\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n           (comm  : Commutative _\u2219_)\n         where\n    open FromOp\u2082   _\u2219_       renaming (op= to \u2219=)\n    open FromAssoc _\u2219_ assoc public\n    open FromComm  _\u2219_ comm  public\n\n    module _ x y z where\n      assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n      assoc-comm = assoc= (comm _ _)\n\n      !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n      !assoc-comm = !assoc= (comm _ _)\n\n    interchange : Interchange _\u2219_ _\u2219_\n    interchange _ _ _ _ = assoc= (!assoc= (comm _ _))\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n      outer= = \u2219= (comm _ _) (comm _ _) \u2666 assoc= (!assoc= e) \u2666 \u2219= (comm _ _) (comm _ _)\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj _ _ fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2a89fd42bb8ce1732c096717e76a1a769870efff","subject":"added flip to cpo lib","message":"added flip to cpo lib\n","repos":"crypto-agda\/crypto-agda","old_file":"alea\/cpo.agda","new_file":"alea\/cpo.agda","new_contents":"{-# OPTIONS --copatterns #-}\nmodule alea.cpo where\n\nimport Data.Nat.NP as Nat\n\nimport Relation.Binary.PropositionalEquality as \u2261\n\nrecord Order (A : Set) : Set\u2081 where\n  constructor mk\n  infix 4 _\u2261_ _\u2264_\n  field\n    _\u2264_ : A \u2192 A \u2192 Set\n    _\u2261_ : A \u2192 A \u2192 Set\n    reflexive : \u2200 {x} \u2192 x \u2264 x\n    antisym : \u2200 {x y} \u2192 x \u2264 y \u2192 y \u2264 x \u2192 x \u2261 y\n    transitive : \u2200 {x y z} \u2192 x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z\n\n  \u2261-reflexive : \u2200 {x} \u2192 x \u2261 x\n  \u2261-reflexive = antisym reflexive reflexive\n    \n\u2115 : Order Nat.\u2115\n\u2115 = mk Nat._\u2264_ \u2261._\u2261_ Nat.\u2115\u2264.refl Nat.\u2115\u2264.antisym Nat.\u2115\u2264.trans\n\nfunOrder : \u2200 {A B : Set} \u2192 Order B \u2192 Order (A \u2192 B)\nfunOrder or = mk (\u03bb f g  \u2192 \u2200 x \u2192 f x \u2264 g x) (\u03bb f g \u2192 \u2200 x \u2192 f x \u2261 g x)\n              (\u03bb x\u2081 \u2192 reflexive)\n              (\u03bb f g x \u2192 antisym (f x) (g x)) \n              (\u03bb f g x  \u2192 transitive (f x) (g x))\n  where open Order or\n\n_o\u2192_ : \u2200 A {B} \u2192 Order B \u2192 Order (A \u2192 B)\nA o\u2192 ob = funOrder {A} ob\n  \nrecord _\u2192m_ {A B}(oa : Order A)(ob : Order B) : Set where\n  coinductive\n  constructor mk\n  field\n    _$_ : A \u2192 B\n    .mon : \u2200 {x y} \u2192 Order._\u2264_ oa x y \u2192 Order._\u2264_ ob (_$_ x) (_$_ y)\nopen _\u2192m_ using (_$_ ; mon)\n\n_o\u2192m_ : \u2200 {A B}(oa : Order A)(ob : Order B) \u2192 Order (oa \u2192m ob)\n_o\u2192m_ {A} oa ob = mk (\u03bb f g \u2192 _$_ f \u2264 _$_ g)\n                     (\u03bb f g \u2192 _$_ f \u2261 _$_ g)\n                     reflexive antisym transitive\n  where open Order (A o\u2192 ob)\n\nswapm : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\nswapm f = mk (\u03bb b \u2192 mk (\u03bb a \u2192 (f $ a) $ b) (\u03bb r \u2192 mon f r b )) (\u03bb r x \u2192 mon (f $ x) r)\n\nswapm' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n_$_ (_$_ (swapm' f) b) a = (f $ a) $ b\nmon (_$_ (swapm' f) b) r = mon f r b\nmon (swapm' f) r x = mon (f $ x) r\n\n{-\nswapm'' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n(swapm'' f $ b)    $ a = (f $ a) $ b\nmon (swapm' f $ b) r   = mon f r b\nmon (swapm' f)     r x = mon (f $ x) r\n-}\n\n_\u2218m_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\nf \u2218m g = mk (\u03bb x \u2192 f $ (g $ x)) (\u03bb x \u2192 mon f (mon g x))\n\n_\u2218m'_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\n_$_ (f \u2218m' g) x = f $ (g $ x)\nmon (f \u2218m' g) x = mon f (mon g x)\n\n_$m_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$m_ {oa = oa}{oc} f b = mk (\u03bb a \u2192 (f $ a) b) (\u03bb x\u2264y \u2192 mon f x\u2264y b )\n\n_$m'_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$_ (f $m' b) a   = (f $ a) b\nmon (f $m' b) x\u2264y = mon f x\u2264y b\n\nUB : \u2200 {A} \u2192 Order A \u2192 A \u2192 Set\nUB or a = \u2200 x \u2192 x \u2264 a\n  where open Order or\n\nrecord cpo {D} (o : Order D) : Set where\n  constructor mk\n  open Order o\n\n  field\n    d0 : D\n    lub : (f : \u2115 \u2192m o) \u2192 D\n\n  -- proofs\n  field\n    .Dbot : \u2200 x \u2192 d0 \u2264 x\n    .lubUB : \u2200 (f : \u2115 \u2192m o) n \u2192 (f $ n) \u2264 lub f\n    .lubLUB : \u2200 (f : \u2115 \u2192m o) x \u2192 (\u2200 n \u2192 (f $ n) \u2264 x) \u2192 lub f \u2264 x\n\n  .lub-ext : \u2200 {f g} \u2192 Order._\u2264_ (\u2115 o\u2192m o) f g \u2192 lub f \u2264 lub g\n  lub-ext {f} {g} f\u2264g = lubLUB f (lub g) (\u03bb n \u2192 transitive (f\u2264g n) (lubUB g n))\n\n  lubm : (\u2115 o\u2192m o) \u2192m o\n  lubm = mk lub (\u03bb {a}{b} x\u2081 \u2192 lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081)\n\n  lubm' : (\u2115 o\u2192m o) \u2192m o\n  _$_ lubm' = lub\n  mon lubm' {a}{b} x\u2081 = lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081\n\nmodule _ where\n  .lub-swap : \u2200 {D}{o : Order D}(c : cpo o)\n             \u2192 let open cpo c\n                   open Order o\n               in \u2200 f \u2192 lub (lubm \u2218m f) \u2264 lub (lubm \u2218m swapm f)\n  lub-swap {D}{o} c f = lubLUB (lubm \u2218m f) (lub (lubm \u2218m swapm f))\n                     (\u03bb n \u2192 lubLUB (f $ n) (lub (lubm \u2218m swapm f))\n                     (\u03bb n\u2081 \u2192 transitive (lubUB (swapm f $ n\u2081) n) (lubUB (lubm \u2218m swapm f) n\u2081)))\n    where\n      open cpo c\n      open Order o\n\n  funcpo : \u2200 {A D}{od : Order D} \u2192 cpo od \u2192 cpo (A o\u2192 od)\n  funcpo cpod = mk (\u03bb x \u2192 d0) (\u03bb f x \u2192 lub (f $m x)) (\u03bb x x\u2081 \u2192 Dbot (x x\u2081)) (\u03bb f n x \u2192 lubUB (f $m x) n)\n    (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (f $m x\u2082) (x x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n    where\n      open cpo cpod\n      open Nat\n      open \u2261 using (_\u2261_)\n\n  _c\u2192m_ : \u2200 {A D}(oa : Order A){od : Order D} \u2192 cpo od \u2192 cpo (oa o\u2192m od)\n  _c\u2192m_ {A} oa {od} cpod = mk (mk (\u03bb x \u2192 d0) (\u03bb x\u2081 \u2192 Order.reflexive od))\n                          (\u03bb f \u2192 mk (cpo.lub (funcpo {A} cpod)\n                                       (mk (\u03bb n a \u2192 (f $ n) $ a) (mon f)))\n                                    (\u03bb {x}{y}x\u2264y \u2192 lubLUB _ _ (\u03bb n \u2192 transitive (mon (f $ n) x\u2264y) (lubUB (mk (\u03bb v \u2192 (f $ v) $ y) (\u03bb r \u2192 mon f r y)) n))))\n                          (\u03bb x x\u2081 \u2192 Dbot (x $ x\u2081))\n                          (\u03bb f n x \u2192 lubUB (mk (\u03bb m \u2192 (f $ m) $ x) (\u03bb x\u2082 \u2192 mon f x\u2082 x)) n)\n                          (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (mk (\u03bb n \u2192 _$_ (f $ n)) (mon f) $m x\u2082) (x $ x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n    where\n      open cpo cpod\n      open Order od renaming (_\u2264_ to _\u2264\u1d48_)\n      open Order oa using () renaming (_\u2264_ to _\u2264\u1d43_)\n\nrecord continous {A B}{oa : Order A}{ob : Order B}\n                 (c1 : cpo oa)(c2 : cpo ob)(f : oa \u2192m ob) : Set where\n  constructor mk\n  open cpo c1 renaming (lub to luba ; lubUB to lubUBa ; lubLUB to lubLUBa)\n  open cpo c2\n  open Order ob\n  field\n    .contIntro : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2264 lub (f \u2218m h)\n\n  .contIntro' : \u2200 (h : \u2115 \u2192m oa) \u2192 lub (f \u2218m h) \u2264 f $ luba h\n  contIntro' h = lubLUB (f \u2218m h) (f $ luba h) (\u03bb n \u2192 mon f (lubUBa _ n))\n\n  .contEq : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2261 lub (f \u2218m h)\n  contEq h = antisym (contIntro h) (contIntro' h)\n  \nopen continous\n\nmodule Distr\n  (Ur  : Set) -- the set [0,1]\n  (1\/_+1 : Nat.\u2115 \u2192 Ur)\n  (_+_ : Ur \u2192 Ur \u2192 Ur)\n  (_\u00d7_ : Ur \u2192 Ur \u2192 Ur)\n  (oUr : Order Ur)\n  (\u2264-cong-+ : let open Order oUr in \u2200 {x y z w} \u2192 x \u2264 z \u2192 y \u2264 w \u2192 x + y \u2264 z + w)\n  (\u2264-cong-\u00d7 : let open Order oUr in \u2200 {x y z w} \u2192 x \u2264 z \u2192 y \u2264 w \u2192 x \u00d7 y \u2264 z \u00d7 w)\n  (U   : cpo oUr) where\n\n  record distr A : Set where\n    constructor mk\n    field\n      \u03bc : (A o\u2192 oUr) \u2192m oUr\n      .muContinous : \u2200 (h : \u2115 \u2192m (A o\u2192 oUr)) \u2192 Order._\u2264_ oUr (\u03bc $ cpo.lub (funcpo U) h) (cpo.lub U (\u03bc \u2218m h)) -- continous (funcpo U) U \u03bc\n\n  open distr\n\n  module _ A where\n    module M = cpo ((A o\u2192 oUr) c\u2192m U)\n\n    distrOrder : Order (distr A)\n    distrOrder = mk (\u03bb f g \u2192 \u03bc f \u2264 \u03bc g) (\u03bb f g \u2192 \u03bc f \u2261 \u03bc g)\n      (\u03bb {x} \u2192 reflexive {\u03bc x}) (\u03bb {x} {y} \u2192 antisym {\u03bc x} {\u03bc y}) (\u03bb {x} {y} {z} \u2192 transitive {\u03bc x} {\u03bc y} {\u03bc z})\n      where open Order ((A o\u2192 oUr) o\u2192m oUr)\n\n    \u03bcm : distrOrder \u2192m ((A o\u2192 oUr) o\u2192m oUr)\n    \u03bcm = mk \u03bc (\u03bb x\u2081 x\u2082 \u2192 x\u2081 x\u2082)\n\n    distrcpo : cpo distrOrder\n    distrcpo = mk (mk M.d0 (\u03bb h \u2192 cpo.Dbot U (cpo.lub U (M.d0 \u2218m h))))\n\n                    (\u03bb f \u2192 mk (M.lub (\u03bcm \u2218m f)) (\u03bb h \u2192 transitive\n                                                       (cpo.lub-ext U (\u03bb a \u2192 muContinous (f $ a) h))\n                                                       (lub-swap U (mk (\u03bb z \u2192 mk (\u03bb z\u2081 \u2192 \u03bc (f $ z) $ (h $ z\u2081))\n                                                                                 (\u03bb x \u2192 mon (\u03bc (f $ z)) (mon h x)))\n                                                                       (\u03bb {x}{y} x\u2081 x\u2082 \u2192\n                                                                              mon f x\u2081 (\u03bb x\u2083 \u2192 (h $ x\u2082) x\u2083))))))\n                    (\u03bb x x\u2081 \u2192 cpo.Dbot U (\u03bc x $ x\u2081))\n                    (\u03bb f n x \u2192 cpo.lubUB U _ _ )\n                    (\u03bb f x x\u2081 x\u2082 \u2192 cpo.lubLUB U _ _ (\u03bb n \u2192 x\u2081 n x\u2082))\n       where open Order oUr\n\n  module FIXPOINTS {d} {D : Order d} {c : cpo D} where\n    open cpo c\n    open Order D\n\n    iter : (f : D \u2192m D) \u2192 Nat.\u2115 \u2192 d\n    iter f Nat.zero = d0\n    iter f (Nat.suc n) = f $ iter f n\n\n    iter-mon : (f : D \u2192m D) \u2192 \u2115 \u2192m D\n    iter-mon f = mk (iter f) mono\n      where\n        .mono : \u2200 {x y} \u2192 x Nat.\u2264 y \u2192 iter f x \u2264 iter f y\n        mono Nat.z\u2264n = Dbot _\n        mono (Nat.s\u2264s prf) = mon f (mono prf)\n\n    fix : (f : D \u2192m D) \u2192 d\n    fix f = lub (iter-mon f)\n\n    .fix-le : (F : D \u2192m D) \u2192 fix F \u2264 F $ (fix F)\n    fix-le F = lubLUB _ _ prf\n      where\n        .prf : (n : Nat.\u2115) \u2192 iter-mon F $ n \u2264 F $ fix F\n        prf Nat.zero = Dbot (F $ fix F)\n        prf (Nat.suc n) = mon F (lubUB (iter-mon F) n)\n\n\n\n  module _ where\n\n    open Order oUr\n    open cpo U\n    open import Data.Bool\n\n    postulate\n      EXPLODE : \u2200 {A : Set} \u2192 A\n\n\n    flip : distr Bool\n    _$_ (\u03bc flip) f = (1\/ 1 +1 \u00d7 f true) + (1\/ 1 +1 \u00d7 f false)\n    mon (\u03bc flip) r = \u2264-cong-+ (\u2264-cong-\u00d7 reflexive (r true)) (\u2264-cong-\u00d7 reflexive (r false))\n    muContinous flip h = EXPLODE\n\n    Munit : \u2200 {A} \u2192 A \u2192 distr A\n    Munit x = mk (mk (\u03bb f \u2192 f x) (\u03bb x\u2264y \u2192 x\u2264y x)) (\u03bb h \u2192 reflexive)\n\n    Mlet : \u2200 {A B} \u2192 distr A \u2192 (A \u2192 distr B) \u2192 distr B\n    Mlet m k = mk (mk (\u03bb f \u2192 \u03bc m $ (\u03bb x \u2192 \u03bc (k x) $ f)) (\u03bb x\u2264y \u2192 mon (\u03bc m) (\u03bb x \u2192 mon (\u03bc (k x)) x\u2264y)))\n       (\u03bb h \u2192  transitive (mon (\u03bc m) (\u03bb x \u2192 muContinous (k x) h))\n                   (muContinous m (mk (\u03bb z z\u2081 \u2192 \u03bc (k z\u2081) $ (h $ z)) (\u03bb x\u2081 x\u2082 \u2192 mon (\u03bc (k x\u2082)) (mon h x\u2081)))))\n\n    Munit-simpl : \u2200 {A}(q : A \u2192 Ur) x \u2192 \u03bc (Munit x) $ q \u2261 q x\n    Munit-simpl q x = \u2261-reflexive\n\n    Mlet-simpl : \u2200 {A B}(m : distr A)(M : A \u2192 distr B)(f : B \u2192 Ur)\n               \u2192 \u03bc (Mlet m M) $ f \u2261 \u03bc m $ (\u03bb x \u2192 \u03bc (M x) $ f)\n    Mlet-simpl m M f = \u2261-reflexive\n\n    .Mlet-le-compat : \u2200 {A B} (m1 m2 : distr A)(M1 M2 : A \u2192 distr B)\n      \u2192 Order._\u2264_ (distrOrder A) m1 m2\n      \u2192 Order._\u2264_ (A o\u2192 distrOrder B) M1 M2\n      \u2192 Order._\u2264_ (distrOrder B) (Mlet m1 M1) (Mlet m2 M2)\n    Mlet-le-compat m1 m2 M1 M2 m1\u2264m2 M1\u2264M2 f = transitive (mon (\u03bc m1) (\u03bb x \u2192 M1\u2264M2 x f)) (m1\u2264m2 _)\n\n    -- Monad laws\n\n    Mlet-unit : \u2200 {A B} (x : A)(m : A \u2192 distr B)\n              \u2192 Order._\u2261_ (distrOrder B) (Mlet (Munit x) m) (m x)\n    Mlet-unit x m f = \u2261-reflexive\n\n    Mlet-ext : \u2200 {A}(m : distr A) \u2192 Order._\u2261_ (distrOrder A) (Mlet m Munit) m\n    Mlet-ext m f = \u2261-reflexive\n\n    Mlet-assoc : \u2200 {A B C}(m1 : distr A)(m2 : A \u2192 distr B)(m3 : B \u2192 distr C)\n               \u2192 Order._\u2261_ (distrOrder C)\n                  (Mlet (Mlet m1 m2) m3)\n                  (Mlet m1 (\u03bb x \u2192 Mlet (m2 x) m3))\n    Mlet-assoc m1 m2 m3 f = \u2261-reflexive\n\n","old_contents":"{-# OPTIONS --copatterns #-}\nmodule alea.cpo where\n\nimport Data.Nat.NP as Nat\n\nimport Relation.Binary.PropositionalEquality as \u2261\n\nrecord Order (A : Set) : Set\u2081 where\n  constructor mk\n  infix 4 _\u2261_ _\u2264_\n  field\n    _\u2264_ : A \u2192 A \u2192 Set\n    _\u2261_ : A \u2192 A \u2192 Set\n    reflexive : \u2200 {x} \u2192 x \u2264 x\n    antisym : \u2200 {x y} \u2192 x \u2264 y \u2192 y \u2264 x \u2192 x \u2261 y\n    transitive : \u2200 {x y z} \u2192 x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z\n\n  \u2261-reflexive : \u2200 {x} \u2192 x \u2261 x\n  \u2261-reflexive = antisym reflexive reflexive\n    \n\u2115 : Order Nat.\u2115\n\u2115 = mk Nat._\u2264_ \u2261._\u2261_ Nat.\u2115\u2264.refl Nat.\u2115\u2264.antisym Nat.\u2115\u2264.trans\n\nfunOrder : \u2200 {A B : Set} \u2192 Order B \u2192 Order (A \u2192 B)\nfunOrder or = mk (\u03bb f g  \u2192 \u2200 x \u2192 f x \u2264 g x) (\u03bb f g \u2192 \u2200 x \u2192 f x \u2261 g x)\n              (\u03bb x\u2081 \u2192 reflexive)\n              (\u03bb f g x \u2192 antisym (f x) (g x)) \n              (\u03bb f g x  \u2192 transitive (f x) (g x))\n  where open Order or\n\n_o\u2192_ : \u2200 A {B} \u2192 Order B \u2192 Order (A \u2192 B)\nA o\u2192 ob = funOrder {A} ob\n  \nrecord _\u2192m_ {A B}(oa : Order A)(ob : Order B) : Set where\n  coinductive\n  constructor mk\n  field\n    _$_ : A \u2192 B\n    .mon : \u2200 {x y} \u2192 Order._\u2264_ oa x y \u2192 Order._\u2264_ ob (_$_ x) (_$_ y)\nopen _\u2192m_ using (_$_ ; mon)\n\n_o\u2192m_ : \u2200 {A B}(oa : Order A)(ob : Order B) \u2192 Order (oa \u2192m ob)\n_o\u2192m_ {A} oa ob = mk (\u03bb f g \u2192 _$_ f \u2264 _$_ g)\n                     (\u03bb f g \u2192 _$_ f \u2261 _$_ g)\n                     reflexive antisym transitive\n  where open Order (A o\u2192 ob)\n\nswapm : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\nswapm f = mk (\u03bb b \u2192 mk (\u03bb a \u2192 (f $ a) $ b) (\u03bb r \u2192 mon f r b )) (\u03bb r x \u2192 mon (f $ x) r)\n\nswapm' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n_$_ (_$_ (swapm' f) b) a = (f $ a) $ b\nmon (_$_ (swapm' f) b) r = mon f r b\nmon (swapm' f) r x = mon (f $ x) r\n\n{-\nswapm'' : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n      \u2192 oa \u2192m (ob o\u2192m oc) \u2192 ob \u2192m (oa o\u2192m oc)\n(swapm'' f $ b)    $ a = (f $ a) $ b\nmon (swapm' f $ b) r   = mon f r b\nmon (swapm' f)     r x = mon (f $ x) r\n-}\n\n_\u2218m_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\nf \u2218m g = mk (\u03bb x \u2192 f $ (g $ x)) (\u03bb x \u2192 mon f (mon g x))\n\n_\u2218m'_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\n_$_ (f \u2218m' g) x = f $ (g $ x)\nmon (f \u2218m' g) x = mon f (mon g x)\n\n_$m_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$m_ {oa = oa}{oc} f b = mk (\u03bb a \u2192 (f $ a) b) (\u03bb x\u2264y \u2192 mon f x\u2264y b )\n\n_$m'_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$_ (f $m' b) a   = (f $ a) b\nmon (f $m' b) x\u2264y = mon f x\u2264y b\n\nUB : \u2200 {A} \u2192 Order A \u2192 A \u2192 Set\nUB or a = \u2200 x \u2192 x \u2264 a\n  where open Order or\n\nrecord cpo {D} (o : Order D) : Set where\n  constructor mk\n  open Order o\n\n  field\n    d0 : D\n    lub : (f : \u2115 \u2192m o) \u2192 D\n\n  -- proofs\n  field\n    .Dbot : \u2200 x \u2192 d0 \u2264 x\n    .lubUB : \u2200 (f : \u2115 \u2192m o) n \u2192 (f $ n) \u2264 lub f\n    .lubLUB : \u2200 (f : \u2115 \u2192m o) x \u2192 (\u2200 n \u2192 (f $ n) \u2264 x) \u2192 lub f \u2264 x\n\n  .lub-ext : \u2200 {f g} \u2192 Order._\u2264_ (\u2115 o\u2192m o) f g \u2192 lub f \u2264 lub g\n  lub-ext {f} {g} f\u2264g = lubLUB f (lub g) (\u03bb n \u2192 transitive (f\u2264g n) (lubUB g n))\n\n  lubm : (\u2115 o\u2192m o) \u2192m o\n  lubm = mk lub (\u03bb {a}{b} x\u2081 \u2192 lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081)\n\n  lubm' : (\u2115 o\u2192m o) \u2192m o\n  _$_ lubm' = lub\n  mon lubm' {a}{b} x\u2081 = lub-ext {mk (\u03bb z \u2192 a $ z) (mon a)}{mk (\u03bb z \u2192 b $ z) (mon b)} x\u2081\n\n.lub-swap : \u2200 {D}{o : Order D}(c : cpo o)\n           \u2192 let open cpo c\n                 open Order o\n             in \u2200 f \u2192 lub (lubm \u2218m f) \u2264 lub (lubm \u2218m swapm f)\nlub-swap {D}{o} c f = lubLUB (lubm \u2218m f) (lub (lubm \u2218m swapm f))\n                   (\u03bb n \u2192 lubLUB (f $ n) (lub (lubm \u2218m swapm f))\n                   (\u03bb n\u2081 \u2192 transitive (lubUB (swapm f $ n\u2081) n) (lubUB (lubm \u2218m swapm f) n\u2081)))\n  where\n    open cpo c\n    open Order o\n\nfuncpo : \u2200 {A D}{od : Order D} \u2192 cpo od \u2192 cpo (A o\u2192 od)\nfuncpo cpod = mk (\u03bb x \u2192 d0) (\u03bb f x \u2192 lub (f $m x)) (\u03bb x x\u2081 \u2192 Dbot (x x\u2081)) (\u03bb f n x \u2192 lubUB (f $m x) n)\n  (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (f $m x\u2082) (x x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n  where\n    open cpo cpod\n    open Nat\n    open \u2261 using (_\u2261_)\n\n_c\u2192m_ : \u2200 {A D}(oa : Order A){od : Order D} \u2192 cpo od \u2192 cpo (oa o\u2192m od)\n_c\u2192m_ {A} oa {od} cpod = mk (mk (\u03bb x \u2192 d0) (\u03bb x\u2081 \u2192 Order.reflexive od))\n                        (\u03bb f \u2192 mk (cpo.lub (funcpo {A} cpod)\n                                     (mk (\u03bb n a \u2192 (f $ n) $ a) (mon f)))\n                                  (\u03bb {x}{y}x\u2264y \u2192 lubLUB _ _ (\u03bb n \u2192 transitive (mon (f $ n) x\u2264y) (lubUB (mk (\u03bb v \u2192 (f $ v) $ y) (\u03bb r \u2192 mon f r y)) n))))\n                        (\u03bb x x\u2081 \u2192 Dbot (x $ x\u2081))\n                        (\u03bb f n x \u2192 lubUB (mk (\u03bb m \u2192 (f $ m) $ x) (\u03bb x\u2082 \u2192 mon f x\u2082 x)) n)\n                        (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (mk (\u03bb n \u2192 _$_ (f $ n)) (mon f) $m x\u2082) (x $ x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n  where\n    open cpo cpod\n    open Order od renaming (_\u2264_ to _\u2264\u1d48_)\n    open Order oa using () renaming (_\u2264_ to _\u2264\u1d43_)\n\nrecord continous {A B}{oa : Order A}{ob : Order B}\n                 (c1 : cpo oa)(c2 : cpo ob)(f : oa \u2192m ob) : Set where\n  constructor mk\n  open cpo c1 renaming (lub to luba ; lubUB to lubUBa ; lubLUB to lubLUBa)\n  open cpo c2\n  open Order ob\n  field\n    .contIntro : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2264 lub (f \u2218m h)\n\n  .contIntro' : \u2200 (h : \u2115 \u2192m oa) \u2192 lub (f \u2218m h) \u2264 f $ luba h\n  contIntro' h = lubLUB (f \u2218m h) (f $ luba h) (\u03bb n \u2192 mon f (lubUBa _ n))\n\n  .contEq : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2261 lub (f \u2218m h)\n  contEq h = antisym (contIntro h) (contIntro' h)\n  \nopen continous\n\nmodule Distr\n  (Ur  : Set) -- the set [0,1]\n  (oUr : Order Ur)\n  (U   : cpo oUr) where\n\n  record distr A : Set where\n    constructor mk\n    field\n      \u03bc : (A o\u2192 oUr) \u2192m oUr\n      .muContinous : \u2200 (h : \u2115 \u2192m (A o\u2192 oUr)) \u2192 Order._\u2264_ oUr (\u03bc $ cpo.lub (funcpo U) h) (cpo.lub U (\u03bc \u2218m h)) -- continous (funcpo U) U \u03bc\n\n  open distr\n\n  module _ A where\n    module M = cpo ((A o\u2192 oUr) c\u2192m U)\n\n    distrOrder : Order (distr A)\n    distrOrder = mk (\u03bb f g \u2192 \u03bc f \u2264 \u03bc g) (\u03bb f g \u2192 \u03bc f \u2261 \u03bc g)\n      (\u03bb {x} \u2192 reflexive {\u03bc x}) (\u03bb {x} {y} \u2192 antisym {\u03bc x} {\u03bc y}) (\u03bb {x} {y} {z} \u2192 transitive {\u03bc x} {\u03bc y} {\u03bc z})\n      where open Order ((A o\u2192 oUr) o\u2192m oUr)\n\n    \u03bcm : distrOrder \u2192m ((A o\u2192 oUr) o\u2192m oUr)\n    \u03bcm = mk \u03bc (\u03bb x\u2081 x\u2082 \u2192 x\u2081 x\u2082)\n\n    distrcpo : cpo distrOrder\n    distrcpo = mk (mk M.d0 (\u03bb h \u2192 cpo.Dbot U (cpo.lub U (M.d0 \u2218m h))))\n\n                    (\u03bb f \u2192 mk (M.lub (\u03bcm \u2218m f)) (\u03bb h \u2192 transitive\n                                                       (cpo.lub-ext U (\u03bb a \u2192 muContinous (f $ a) h))\n                                                       (lub-swap U (mk (\u03bb z \u2192 mk (\u03bb z\u2081 \u2192 \u03bc (f $ z) $ (h $ z\u2081))\n                                                                                 (\u03bb x \u2192 mon (\u03bc (f $ z)) (mon h x)))\n                                                                       (\u03bb {x}{y} x\u2081 x\u2082 \u2192\n                                                                              mon f x\u2081 (\u03bb x\u2083 \u2192 (h $ x\u2082) x\u2083))))))\n                    (\u03bb x x\u2081 \u2192 cpo.Dbot U (\u03bc x $ x\u2081))\n                    (\u03bb f n x \u2192 cpo.lubUB U _ _ )\n                    (\u03bb f x x\u2081 x\u2082 \u2192 cpo.lubLUB U _ _ (\u03bb n \u2192 x\u2081 n x\u2082))\n       where open Order oUr\n\n  module FIXPOINTS {d} {D : Order d} {c : cpo D} where\n    open cpo c\n    open Order D\n\n    iter : (f : D \u2192m D) \u2192 Nat.\u2115 \u2192 d\n    iter f Nat.zero = d0\n    iter f (Nat.suc n) = f $ iter f n\n\n    iter-mon : (f : D \u2192m D) \u2192 \u2115 \u2192m D\n    iter-mon f = mk (iter f) mono\n      where\n        .mono : \u2200 {x y} \u2192 x Nat.\u2264 y \u2192 iter f x \u2264 iter f y\n        mono Nat.z\u2264n = Dbot _\n        mono (Nat.s\u2264s prf) = mon f (mono prf)\n\n    fix : (f : D \u2192m D) \u2192 d\n    fix f = lub (iter-mon f)\n\n    .fix-le : (F : D \u2192m D) \u2192 fix F \u2264 F $ (fix F)\n    fix-le F = lubLUB _ _ prf\n      where\n        .prf : (n : Nat.\u2115) \u2192 iter-mon F $ n \u2264 F $ fix F\n        prf Nat.zero = Dbot (F $ fix F)\n        prf (Nat.suc n) = mon F (lubUB (iter-mon F) n)\n\n\n\n  module _ where\n\n    open Order oUr\n    open cpo U\n\n    Munit : \u2200 {A} \u2192 A \u2192 distr A\n    Munit x = mk (mk (\u03bb f \u2192 f x) (\u03bb x\u2264y \u2192 x\u2264y x)) (\u03bb h \u2192 reflexive)\n\n    Mlet : \u2200 {A B} \u2192 distr A \u2192 (A \u2192 distr B) \u2192 distr B\n    Mlet m k = mk (mk (\u03bb f \u2192 \u03bc m $ (\u03bb x \u2192 \u03bc (k x) $ f)) (\u03bb x\u2264y \u2192 mon (\u03bc m) (\u03bb x \u2192 mon (\u03bc (k x)) x\u2264y)))\n       (\u03bb h \u2192  transitive (mon (\u03bc m) (\u03bb x \u2192 muContinous (k x) h))\n                   (muContinous m (mk (\u03bb z z\u2081 \u2192 \u03bc (k z\u2081) $ (h $ z)) (\u03bb x\u2081 x\u2082 \u2192 mon (\u03bc (k x\u2082)) (mon h x\u2081)))))\n\n    Munit-simpl : \u2200 {A}(q : A \u2192 Ur) x \u2192 \u03bc (Munit x) $ q \u2261 q x\n    Munit-simpl q x = \u2261-reflexive\n\n    Mlet-simpl : \u2200 {A B}(m : distr A)(M : A \u2192 distr B)(f : B \u2192 Ur)\n               \u2192 \u03bc (Mlet m M) $ f \u2261 \u03bc m $ (\u03bb x \u2192 \u03bc (M x) $ f)\n    Mlet-simpl m M f = \u2261-reflexive\n\n    .Mlet-le-compat : \u2200 {A B} (m1 m2 : distr A)(M1 M2 : A \u2192 distr B)\n      \u2192 Order._\u2264_ (distrOrder A) m1 m2\n      \u2192 Order._\u2264_ (A o\u2192 distrOrder B) M1 M2\n      \u2192 Order._\u2264_ (distrOrder B) (Mlet m1 M1) (Mlet m2 M2)\n    Mlet-le-compat m1 m2 M1 M2 m1\u2264m2 M1\u2264M2 f = transitive (mon (\u03bc m1) (\u03bb x \u2192 M1\u2264M2 x f)) (m1\u2264m2 _)\n\n    -- Monad laws\n\n    Mlet-unit : \u2200 {A B} (x : A)(m : A \u2192 distr B)\n              \u2192 Order._\u2261_ (distrOrder B) (Mlet (Munit x) m) (m x)\n    Mlet-unit x m f = \u2261-reflexive\n\n    Mlet-ext : \u2200 {A}(m : distr A) \u2192 Order._\u2261_ (distrOrder A) (Mlet m Munit) m\n    Mlet-ext m f = \u2261-reflexive\n\n    Mlet-assoc : \u2200 {A B C}(m1 : distr A)(m2 : A \u2192 distr B)(m3 : B \u2192 distr C)\n               \u2192 Order._\u2261_ (distrOrder C)\n                  (Mlet (Mlet m1 m2) m3)\n                  (Mlet m1 (\u03bb x \u2192 Mlet (m2 x) m3))\n    Mlet-assoc m1 m2 m3 f = \u2261-reflexive\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"63a5b7a9d521ea452ee92bacd134b6149ec7ebf8","subject":"agda: encode validity of changes with a logical relation (#33)","message":"agda: encode validity of changes with a logical relation (#33)\n\nOld-commit-hash: f93419dc1c1cc1cfafe1a9dc82fb0e4f8cabd242\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {T : Type} \u2192 \u27e6 T \u27e7 \u2192 \u27e6 \u0394-Type T \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df = \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192 (valid-\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4}\n  (\u03c1\u2081 : \u27e6 \u0393prefix \u27e7) (\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7)\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\n\nlift-term-doWeakenOne-ignore\u2032 \u0393prefix \u2205 \u03c1\u2081 \u2205 t = refl\n--This step gives the proof idea.\nlift-term-doWeakenOne-ignore\u2032 \u2205 (\u03c4 \u2022 \u0393rest) \u2205 (dv \u2022 v \u2022 \u03c1\u2082) t = lift-term-doWeakenOne-ignore\u2032 (\u03c4 \u2022 \u2205) \u0393rest (v \u2022 \u2205) \u03c1\u2082 t\n\nlift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix) (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} (v\u2080 \u2022 \u03c1\u2081) (dv \u2022 v \u2022 \u03c1\u2082) t = {!!}\n--Look at C-c C-, for the normalized goal.\n-- The solution here should be something like:\n  --lift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest {\u03c4} ((v\u2080 \u2022 \u03c1\u2081) \u222a (v \u2022 \u2205)) \u03c1\u2082  ?\n\n-- accompanied by enough subst to make it typecheck, and maybe by\n-- weakenOne-ignore to account for weakenOne - or maybe not since\n-- weakenOne is a definitional equality. PG\n-- Not planning to finish this, it looks quite horrible.\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4}\n  (\u03c1\u2081 : \u27e6 \u0393prefix \u27e7) (\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7)\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\n\nlift-term-doWeakenOne-ignore\u2032 \u0393prefix \u2205 \u03c1\u2081 \u2205 t = refl\n--This step gives the proof idea.\nlift-term-doWeakenOne-ignore\u2032 \u2205 (\u03c4 \u2022 \u0393rest) \u2205 (dv \u2022 v \u2022 \u03c1\u2082) t = lift-term-doWeakenOne-ignore\u2032 (\u03c4 \u2022 \u2205) \u0393rest (v \u2022 \u2205) \u03c1\u2082 t\n\nlift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix) (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} (v\u2080 \u2022 \u03c1\u2081) (dv \u2022 v \u2022 \u03c1\u2082) t = {!!}\n--Look at C-c C-, for the normalized goal.\n-- The solution here should be something like:\n  --lift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest {\u03c4} ((v\u2080 \u2022 \u03c1\u2081) \u222a (v \u2022 \u2205)) \u03c1\u2082  ?\n\n-- accompanied by enough subst to make it typecheck, and maybe by\n-- weakenOne-ignore to account for weakenOne - or maybe not since\n-- weakenOne is a definitional equality. PG\n-- Not planning to finish this, it looks quite horrible.\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ceb7acd1c67116365b51503065e55f947cda63a5","subject":"agda\/...: simplify weakenMore","message":"agda\/...: simplify weakenMore\n\n* Inline some definitions where appropriate.\n* Remove useless pattern matching on context.\n\nOld-commit-hash: a4c4aee377597f7c17b5fa7d8b1c8a02492041c0\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore \u0393\u2032 t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    weakenMore : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore {\u0393} \u0393\u2032 t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (doWeakenMore (take (\u0394-Context \u0393) \u0393\u2032) (drop (\u0394-Context \u0393) \u0393\u2032) (\n          substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t)))\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore \u0393\u2032 t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore {\u0393} \u0393\u2032 t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"383c5269b477b96439181ea4a35d11ce7a075a93","subject":"Parser: look*","message":"Parser: look*\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Text\/Parser.agda","new_file":"lib\/Text\/Parser.agda","new_contents":"open import Data.Maybe.NP as Maybe\nopen import Data.Nat\nopen import Data.Unit\nopen import Data.Char renaming (_==_ to _==\u1d9c_)\nopen import Data.String as String\nopen import Data.List as L using (List; []; _\u2237_; null; filter)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bool\nopen import Function\nimport Level as L\nimport Category.Monad as Cat\nimport Category.Monad.Partiality.NP as Pa\n\nmodule Text.Parser where\n\nopen M? L.zero using () renaming (_=<<_ to _=<<?_; _\u229b_ to _\u229b?_; join to join?)\n\nParser : Set \u2192 Set\nParser A = List Char \u2192? (A \u00d7 List Char)\n\npure : \u2200 {A} \u2192 A \u2192 Parser A\npure a s = just (a , s)\n\ninfixr 3 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : Parser A) \u2192 Parser A\n_\u27e8|\u27e9_ f g s = maybe\u2032 just (g s) $ f s\n\nempty : \u2200 {A} \u2192 Parser A\nempty _ = nothing\n\nsat : (Char \u2192 Bool) \u2192 Parser Char\nsat pred (x \u2237 xs) = (if pred x then pure x else empty) xs\nsat _    []       = nothing\n\neof : Parser \u22a4\neof s = if null s then pure _ [] else empty []\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 Parser A \u2192 List Char \u2192? A\nrunParser p s = map? proj\u2081 (p s)\n\ninfixr 0 _>>=_\n_>>=_ : \u2200 {A B} \u2192 Parser A \u2192 (A \u2192 Parser B) \u2192 Parser B\n_>>=_ pa f = maybe\u2032 (uncurry f) nothing \u2218 pa\n\ninfixl 0 _=<<_\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 Parser B) \u2192 Parser A \u2192 Parser B\n_=<<_ f pa = pa >>= f\n\njoin : \u2200 {A} \u2192 Parser (Parser A) \u2192 Parser A\njoin = _=<<_ id\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\nmap f pa = pa >>= pure \u2218 f\n\ninfixl 4 _\u229b_ _*>_ _<*_\n_\u229b_ : \u2200 {A B} \u2192 Parser (A \u2192 B) \u2192 Parser A \u2192 Parser B\n_\u229b_ pf pa = pf >>= \u03bb f \u2192 pa >>= \u03bb a \u2192 pure (f a)\n\n\u27ea_\u00b7_\u27eb : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192\n            (A \u2192 B \u2192 C) \u2192 Parser A \u2192 Parser B \u2192 Parser C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D} \u2192 (A \u2192 B \u2192 C \u2192 D)\n            \u2192 Parser A \u2192 Parser B \u2192 Parser C \u2192 Parser D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_*>_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser B\np\u2081 *> p\u2082 = p\u2081 >>= const p\u2082\n\n_<*_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser A\np\u2081 <* p\u2082 = p\u2081 >>= \u03bb x \u2192 p\u2082 *> pure x\n\nchoices : \u2200 {A} \u2192 List (Parser A) \u2192 Parser A\nchoices = L.foldr _\u27e8|\u27e9_ empty\n\n{-\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  many p = some p \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  some p = pure _\u2237_ \u229b p \u229b many p\n-}\n\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  many _ 0       = empty\n  many p (suc n) = some p n \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  some p n = pure _\u2237_ \u229b p \u229b many p n\n\nmanySat : (Char \u2192 Bool) \u2192 Parser (List Char)\nmanySat p []       = pure [] []\nmanySat p (x \u2237 xs) = if p x then map? (first (_\u2237_ x)) (manySat p xs) else pure [] (x \u2237 xs)\n\nsomeSat : (Char \u2192 Bool) \u2192 Parser (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nchar : Char \u2192 Parser \u22a4\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nprivate\n  notElem : Char \u2192 List Char \u2192 Bool\n  notElem c = null \u2218 filter (_==\u1d9c_ c)\n\n  syntax notElem c cs = c `notElem` cs\n\n  elem : Char \u2192 List Char \u2192 Bool\n  elem c = not \u2218 notElem c\n\n  syntax elem c cs = c `elem` cs\n\noneOf : List Char \u2192 Parser Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\nnoneOf : List Char \u2192 Parser Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nsomeOneOf : List Char \u2192 Parser (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeNoneOf : List Char \u2192 Parser (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nmanyOneOf : List Char \u2192 Parser (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyNoneOf : List Char \u2192 Parser (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n\nlookSat : (List Char \u2192 Bool) \u2192 Parser \u22a4\nlookSat p xs = if p xs then just (_ , xs) else nothing\n\nlookSatHead : (Char \u2192 Bool) \u2192 Parser \u22a4\nlookSatHead p\u1d9c = lookSat p\n    where p : List Char \u2192 Bool\n          p (x \u2237 xs) = p\u1d9c x\n          p []       = false\n\nlookHeadNoneOf : List Char \u2192 Parser \u22a4\nlookHeadNoneOf cs = lookSatHead (\u03bb c \u2192 c `notElem` cs)\n\nlookHeadSomeOf : List Char \u2192 Parser \u22a4\nlookHeadSomeOf cs = lookSatHead (\u03bb c \u2192 c `elem` cs)\n","old_contents":"open import Data.Maybe.NP as Maybe\nopen import Data.Nat\nopen import Data.Unit\nopen import Data.Char renaming (_==_ to _==\u1d9c_)\nopen import Data.String as String\nopen import Data.List as L using (List; []; _\u2237_; null; filter)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bool\nopen import Function\nimport Level as L\nimport Category.Monad as Cat\nimport Category.Monad.Partiality.NP as Pa\n\nmodule Text.Parser where\n\nopen M? L.zero using () renaming (_=<<_ to _=<<?_; _\u229b_ to _\u229b?_; join to join?)\n\nParser : Set \u2192 Set\nParser A = List Char \u2192? (A \u00d7 List Char)\n\npure : \u2200 {A} \u2192 A \u2192 Parser A\npure a s = just (a , s)\n\ninfixr 3 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : Parser A) \u2192 Parser A\n_\u27e8|\u27e9_ f g s = maybe\u2032 just (g s) $ f s\n\nempty : \u2200 {A} \u2192 Parser A\nempty _ = nothing\n\nsat : (Char \u2192 Bool) \u2192 Parser Char\nsat pred (x \u2237 xs) = (if pred x then pure x else empty) xs\nsat _    []       = nothing\n\neof : Parser \u22a4\neof s = if null s then pure _ [] else empty []\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 Parser A \u2192 List Char \u2192? A\nrunParser p s = map? proj\u2081 (p s)\n\ninfixr 0 _>>=_\n_>>=_ : \u2200 {A B} \u2192 Parser A \u2192 (A \u2192 Parser B) \u2192 Parser B\n_>>=_ pa f = maybe\u2032 (uncurry f) nothing \u2218 pa\n\ninfixl 0 _=<<_\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 Parser B) \u2192 Parser A \u2192 Parser B\n_=<<_ f pa = pa >>= f\n\njoin : \u2200 {A} \u2192 Parser (Parser A) \u2192 Parser A\njoin = _=<<_ id\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\nmap f pa = pa >>= pure \u2218 f\n\ninfixl 4 _\u229b_ _*>_ _<*_\n_\u229b_ : \u2200 {A B} \u2192 Parser (A \u2192 B) \u2192 Parser A \u2192 Parser B\n_\u229b_ pf pa = pf >>= \u03bb f \u2192 pa >>= \u03bb a \u2192 pure (f a)\n\n\u27ea_\u00b7_\u27eb : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192\n            (A \u2192 B \u2192 C) \u2192 Parser A \u2192 Parser B \u2192 Parser C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D} \u2192 (A \u2192 B \u2192 C \u2192 D)\n            \u2192 Parser A \u2192 Parser B \u2192 Parser C \u2192 Parser D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_*>_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser B\np\u2081 *> p\u2082 = p\u2081 >>= const p\u2082\n\n_<*_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser A\np\u2081 <* p\u2082 = p\u2081 >>= \u03bb x \u2192 p\u2082 *> pure x\n\nchoices : \u2200 {A} \u2192 List (Parser A) \u2192 Parser A\nchoices = L.foldr _\u27e8|\u27e9_ empty\n\n{-\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  many p = some p \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 Parser (List A)\n  some p = pure _\u2237_ \u229b p \u229b many p\n-}\n\nmutual\n  many : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  many _ 0       = empty\n  many p (suc n) = some p n \u27e8|\u27e9 pure []\n\n  some : \u2200 {A} \u2192 Parser A \u2192 \u2115 \u2192 Parser (List A)\n  some p n = pure _\u2237_ \u229b p \u229b many p n\n\nmanySat : (Char \u2192 Bool) \u2192 Parser (List Char)\nmanySat p []       = pure [] []\nmanySat p (x \u2237 xs) = if p x then map? (first (_\u2237_ x)) (manySat p xs) else pure [] (x \u2237 xs)\n\nsomeSat : (Char \u2192 Bool) \u2192 Parser (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nchar : Char \u2192 Parser \u22a4\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nprivate\n  notElem : Char \u2192 List Char \u2192 Bool\n  notElem c = null \u2218 filter (_==\u1d9c_ c)\n\n  syntax notElem c cs = c `notElem` cs\n\n  elem : Char \u2192 List Char \u2192 Bool\n  elem c = not \u2218 notElem c\n\n  syntax elem c cs = c `elem` cs\n\noneOf : List Char \u2192 Parser Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\nnoneOf : List Char \u2192 Parser Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nsomeOneOf : List Char \u2192 Parser (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeNoneOf : List Char \u2192 Parser (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nmanyOneOf : List Char \u2192 Parser (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyNoneOf : List Char \u2192 Parser (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2644e82b6c7f9d3ef8b5fc7b8f2231038f63c3c2","subject":"Do more computation in example","message":"Do more computation in example\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = Decl \u2115E End End\n  (End tt , Rec tt (End tt) , tt)\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = Decl VecE\n  (Arg Set \u03bb _ \u2192 End)\n  (\u03bb _ \u2192 Arg \u2115 \u03bb _ \u2192 End)\n  (\u03bb A \u2192\n    End (zero , tt)\n  , IArg \u2115 (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n  , tt\n  )\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} {n : \u2115} (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} {m n : \u2115} (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} {m} {n} = elim VecR\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb x xs ih ys \u2192 cons x (ih ys))\n  m\n\nconcat : {A : Set} {m n : \u2115} (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} {m} {n} = elim VecR\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb xs xss ih \u2192 append xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : \u2115\none = suc zero\n\ntwo : \u2115\ntwo = suc one\n\nthree : \u2115\nthree = suc two\n\n[1,2] : Vec \u2115 two\n[1,2] = cons one (cons two nil)\n\n[3] : Vec \u2115 one\n[3] = cons three nil\n\n[1,2,3] : Vec \u2115 three\n[1,2,3] = cons one (cons two (cons three nil))\n\ntest-append : [1,2,3] \u2261 append [1,2] [3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n----------------------------------------------------------------------\n\n\u2115R : Data\n\u2115R = Decl \u2115E End End\n  (End tt , Rec tt (End tt) , tt)\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = Decl VecE\n  (Arg Set \u03bb _ \u2192 End)\n  (\u03bb _ \u2192 Arg \u2115 \u03bb _ \u2192 End)\n  (\u03bb A \u2192\n    End (zero , tt)\n  , IArg \u2115 (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n  , tt\n  )\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} {n : \u2115} (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : \u2115 \u2192 \u2115 \u2192 \u2115\nmult = elim \u2115R\n  (\u03bb n \u2192 \u2115 \u2192 \u2115)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : {A : Set} {m n : \u2115} (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend {A} {m} {n} = elim VecR\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb x xs ih ys \u2192 cons x (ih ys))\n  m\n\nconcat : {A : Set} {m n : \u2115} (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat {A} {m} {n} = elim VecR\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  nil\n  (\u03bb xs xss ih \u2192 append xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : \u2115\none = suc zero\n\ntwo : \u2115\ntwo = suc one\n\nthree : \u2115\nthree = suc two\n\n[1] : Vec \u2115 one\n[1] = cons one nil\n\n[2,3] : Vec \u2115 two\n[2,3] = cons two (cons three nil)\n\n[1,2,3] : Vec \u2115 three\n[1,2,3] = cons one (cons two (cons three nil))\n\ntest-append : [1,2,3] \u2261 append [1] [2,3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a737167afb8fb406b75d8e0d52fdfb3fa6c50035","subject":"Rename locals in proof for clarity","message":"Rename locals in proof for clarity\n\nWas re-reading this proof for the thesis.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Specification.agda","new_file":"Parametric\/Change\/Specification.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = record\n    { apply = \u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    ; correct = \u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e\n    } where open \u2261-Reasoning\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = record\n    { apply = \u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    ; correct = \u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e\n    } where open \u2261-Reasoning\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f = \u27e6 s \u27e7 \u03c1\n      g = \u27e6 s \u27e7 (after-env d\u03c1)\n      u = \u27e6 t \u27e7 \u03c1\n      v = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f u \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u \u0394u) \u27e9\n        (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        g v\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"59757767c7fbc1a56deaecbdaf8bb0f7dfa223a0","subject":"Removed unnecessary theorem.","message":"Removed unnecessary theorem.\n\nIgnore-this: 963ee1868f49c41d4bf656d22e342e3b\n\ndarcs-hash:20120203163945-3bd4e-36cf54af3b805e9b96b2e1341e4b10269d06a560.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/PropertiesI.agda","new_file":"src\/GroupTheory\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Group theory properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.PropertiesI where\n\nopen import GroupTheory.Base\nopen import GroupTheory.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- Adapted from the standard library.\ny\u2261x\u207b\u00b9[xy] : \u2200 a b \u2192 b \u2261 a \u207b\u00b9 \u00b7 (a \u00b7 b)\ny\u2261x\u207b\u00b9[xy] a b =\n  b              \u2261\u27e8 sym (leftIdentity b) \u27e9\n  \u03b5 \u00b7 b          \u2261\u27e8 subst (\u03bb t \u2192 \u03b5 \u00b7 b \u2261 t \u00b7 b ) (sym (leftInverse a)) refl \u27e9\n  a \u207b\u00b9 \u00b7 a \u00b7 b   \u2261\u27e8 assoc (a \u207b\u00b9) a b \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 b) \u220e\n\n-- Adapted from the standard library.\nx\u2261[xy]y\u207b\u00b9 : \u2200 a b \u2192 a \u2261 (a \u00b7 b) \u00b7 b \u207b\u00b9\nx\u2261[xy]y\u207b\u00b9 a b =\n  a              \u2261\u27e8 sym (rightIdentity a) \u27e9\n  a \u00b7 \u03b5          \u2261\u27e8 subst (\u03bb t \u2192 a \u00b7 \u03b5 \u2261 a \u00b7 t ) (sym (rightInverse b)) refl \u27e9\n  a \u00b7 (b \u00b7 b \u207b\u00b9) \u2261\u27e8 sym (assoc a b (b \u207b\u00b9)) \u27e9\n  a \u00b7 b \u00b7 b \u207b\u00b9 \u220e\n\nrightIdentityUnique : \u2200 r \u2192 (\u2200 a \u2192 a \u00b7 r \u2261 a) \u2192 r \u2261 \u03b5\n-- Paper proof (Saunders Mac Lane and Garret Birkhoff. Algebra. AMS\n-- Chelsea Publishing, 3rd edition, 1999. p. 48):\n--\n-- 1. r  = \u03b5r (\u03b5 is an identity)\n-- 2. \u03b5r = r  (hypothesis)\n-- 3. r  = \u03b5  (transitivity)\nrightIdentityUnique r h = trans (sym (leftIdentity r)) (h \u03b5)\n\n-- A more appropiate version to be used in the proofs.\n-- Adapted from the standard library.\nrightIdentityUnique' : \u2200 a r \u2192 a \u00b7 r \u2261 a \u2192 r \u2261 \u03b5\nrightIdentityUnique' a r h =\n  r              \u2261\u27e8 y\u2261x\u207b\u00b9[xy] a r \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 r) \u2261\u27e8 cong (_\u00b7_ (a \u207b\u00b9)) h \u27e9\n  a \u207b\u00b9 \u00b7 a       \u2261\u27e8 leftInverse a \u27e9\n  \u03b5 \u220e\n\nleftIdentityUnique : \u2200 l \u2192 (\u2200 a \u2192 l \u00b7 a \u2261 a) \u2192 l \u2261 \u03b5\n-- Paper proof:\n-- 1. l  = le (\u03b5 is an identity)\n-- 2. le = e  (hypothesis)\n-- 3. l  = e  (transitivity)\nleftIdentityUnique l h = trans (sym (rightIdentity l)) (h \u03b5)\n\n-- A more appropiate version to be used in the proofs.\n-- Adapted from the standard library.\nleftIdentityUnique' : \u2200 a l \u2192 l \u00b7 a \u2261 a \u2192 l \u2261 \u03b5\nleftIdentityUnique' a l h =\n  l            \u2261\u27e8 x\u2261[xy]y\u207b\u00b9 l a \u27e9\n  l \u00b7 a \u00b7 a \u207b\u00b9 \u2261\u27e8 subst (\u03bb t \u2192 l \u00b7 a \u00b7 a \u207b\u00b9 \u2261 t \u00b7 a \u207b\u00b9 ) h refl \u27e9\n  a \u00b7 a \u207b\u00b9 \u2261\u27e8 rightInverse a \u27e9\n  \u03b5 \u220e\n\nrightCancellation : \u2200 {a b c} \u2192 b \u00b7 a \u2261 c \u00b7 a \u2192 b \u2261 c\nrightCancellation {a} {b} {c} h =\n-- Paper proof:\n-- 1. (ba)a\u207b\u00b9  = (ca)a\u207b\u00b9  (hypothesis ab = ac)\n-- 2. (b)aa\u207b\u00b9  = (c)aa\u207b\u00b9  (associative axiom)\n-- 3. b\u03b5       = c\u03b5       (right-inverse axiom for a\u207b\u00b9)\n-- 4. b        = c        (right-identity axiom)\n  b              \u2261\u27e8 sym (rightIdentity b) \u27e9\n  b \u00b7 \u03b5          \u2261\u27e8 subst (\u03bb t \u2192 b \u00b7 \u03b5 \u2261 b \u00b7 t) (sym (rightInverse a)) refl \u27e9\n  b \u00b7 (a \u00b7 a \u207b\u00b9) \u2261\u27e8 sym (assoc b a (a \u207b\u00b9)) \u27e9\n  b \u00b7 a \u00b7 a \u207b\u00b9   \u2261\u27e8 subst (\u03bb t \u2192 b \u00b7 a \u00b7 a \u207b\u00b9 \u2261 t \u00b7 a \u207b\u00b9) h refl \u27e9\n  c \u00b7 a \u00b7 a \u207b\u00b9   \u2261\u27e8 assoc c a (a \u207b\u00b9) \u27e9\n  c \u00b7 (a \u00b7 a \u207b\u00b9) \u2261\u27e8 subst (\u03bb t \u2192 c \u00b7 (a \u00b7 a \u207b\u00b9) \u2261 c \u00b7 t) (rightInverse a) refl \u27e9\n  c \u00b7 \u03b5          \u2261\u27e8 rightIdentity c \u27e9\n  c \u220e\n\nleftCancellation : \u2200 {a b c} \u2192 a \u00b7 b \u2261 a \u00b7 c \u2192 b \u2261 c\nleftCancellation {a} {b} {c} h =\n-- Paper proof:\n-- 1. a\u207b\u00b9(ab)  = a\u207b\u00b9(ac)  (hypothesis ab = ac)\n-- 2. a\u207b\u00b9a(b)  = a\u207b\u00b9a(c)  (associative axiom)\n-- 3. \u03b5b       = \u03b5c       (left-inverse axiom for a\u207b\u00b9)\n-- 4. b        = c        (left-identity axiom)\n  b              \u2261\u27e8 sym (leftIdentity b) \u27e9\n  \u03b5 \u00b7 b          \u2261\u27e8 subst (\u03bb t \u2192 \u03b5 \u00b7 b \u2261 t \u00b7 b) (sym (leftInverse a)) refl \u27e9\n  a \u207b\u00b9 \u00b7 a \u00b7 b   \u2261\u27e8 assoc (a \u207b\u00b9) a b \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 b) \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 (a \u00b7 b) \u2261 a \u207b\u00b9 \u00b7 t) h refl \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 c) \u2261\u27e8 sym (assoc (a \u207b\u00b9) a c) \u27e9\n  a \u207b\u00b9 \u00b7 a \u00b7 c   \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 a \u00b7 c \u2261 t \u00b7 c) (leftInverse a) refl \u27e9\n  \u03b5 \u00b7 c          \u2261\u27e8 leftIdentity c \u27e9\n  c \u220e\n\nx\u2261y\u2192xz\u2261yz : \u2200 {a b c} \u2192 a \u2261 b \u2192 a \u00b7 c \u2261 b \u00b7 c\nx\u2261y\u2192xz\u2261yz refl = refl\n\nrightInverseUnique : \u2200 {a} \u2192 \u2203[ r ] (a \u00b7 r \u2261 \u03b5) \u2227\n                                    (\u2200 r' \u2192 a \u00b7 r' \u2261 \u03b5 \u2192 r \u2261 r')\nrightInverseUnique {a} =\n-- Paper proof:\n-- 1.   We know that (a\u207b\u00b9) is a right inverse for a.\n-- 2.   Let's suppose there is other right inverse r for a, i.e. ar \u2261 \u03b5, then\n-- 2.1. aa\u207b\u00b9 = \u03b5  (right-inverse axiom)\n-- 2.2. ar   = \u03b5  (hypothesis)\n-- 2.3. aa\u207b\u00b9 = ar (transitivity)\n-- 2.4  a\u207b\u00b9  = a  (left-cancellation)\n  (a \u207b\u00b9) , rightInverse a , prf\n    where\n    prf : \u2200 r' \u2192 a \u00b7 r' \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 r'\n    prf r' ar'\u2261\u03b5 = leftCancellation aa\u207b\u00b9\u2261ar'\n      where\n      aa\u207b\u00b9\u2261ar' : a \u00b7 a \u207b\u00b9 \u2261 a \u00b7 r'\n      aa\u207b\u00b9\u2261ar' = a \u00b7 a \u207b\u00b9 \u2261\u27e8 rightInverse a \u27e9\n                 \u03b5        \u2261\u27e8 sym ar'\u2261\u03b5 \u27e9\n                 a \u00b7 r' \u220e\n\n-- A more appropiate version to be used in the proofs.\nrightInverseUnique' : \u2200 {a r} \u2192 a \u00b7 r \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 r\nrightInverseUnique' {a} {r} ar\u2261\u03b5 = leftCancellation aa\u207b\u00b9\u2261ar\n  where\n  aa\u207b\u00b9\u2261ar : a \u00b7 a \u207b\u00b9 \u2261 a \u00b7 r\n  aa\u207b\u00b9\u2261ar = a \u00b7 a \u207b\u00b9 \u2261\u27e8 rightInverse a \u27e9\n            \u03b5        \u2261\u27e8 sym ar\u2261\u03b5 \u27e9\n            a \u00b7 r \u220e\n\nleftInverseUnique : \u2200 {a} \u2192 \u2203[ l ] (l \u00b7 a \u2261 \u03b5) \u2227\n                                   (\u2200 l' \u2192 l' \u00b7 a \u2261 \u03b5 \u2192 l \u2261 l')\nleftInverseUnique {a} =\n-- Paper proof:\n-- 1.   We know that (a\u207b\u00b9) is a left inverse for a.\n-- 2.   Let's suppose there is other right inverse l for a, i.e. la \u2261 \u03b5, then\n-- 2.1. a\u207b\u00b9a = \u03b5  (left-inverse axiom)\n-- 2.2. la   = \u03b5  (hypothesis)\n-- 2.3. a\u207b\u00b9a = la (transitivity)\n-- 2.4  a\u207b\u00b9  = l  (right-cancellation)\n  (a \u207b\u00b9) , leftInverse a , prf\n    where\n    prf : \u2200 l' \u2192 l' \u00b7 a \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 l'\n    prf l' l'a\u2261\u03b5 = rightCancellation a\u207b\u00b9a\u2261l'a\n      where\n      a\u207b\u00b9a\u2261l'a : a \u207b\u00b9 \u00b7 a \u2261 l' \u00b7 a\n      a\u207b\u00b9a\u2261l'a = a \u207b\u00b9 \u00b7 a \u2261\u27e8 leftInverse a \u27e9\n                 \u03b5        \u2261\u27e8 sym l'a\u2261\u03b5 \u27e9\n                 l' \u00b7 a \u220e\n\n-- A more appropiate version to be used in the proofs.\nleftInverseUnique' : \u2200 {a l} \u2192 l \u00b7 a \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 l\nleftInverseUnique' {a} {l} la\u2261\u03b5 = rightCancellation a\u207b\u00b9a\u2261la\n  where\n  a\u207b\u00b9a\u2261la : a \u207b\u00b9 \u00b7 a \u2261 l \u00b7 a\n  a\u207b\u00b9a\u2261la = a \u207b\u00b9 \u00b7 a \u2261\u27e8 leftInverse a \u27e9\n            \u03b5        \u2261\u27e8 sym la\u2261\u03b5 \u27e9\n            l \u00b7 a \u220e\n\n\u207b\u00b9-involutive : \u2200 a \u2192 a \u207b\u00b9 \u207b\u00b9 \u2261 a\n-- Paper proof:\n-- 1. a\u207b\u00b9a = \u03b5  (left-inverse axiom)\n-- 2. The previous equation states that a is the unique right\n-- inverse (a\u207b\u00b9)\u207b\u00b9 of a\u207b\u00b9.\n\u207b\u00b9-involutive a = rightInverseUnique' (leftInverse a)\n\nidentityInverse : \u03b5 \u207b\u00b9 \u2261 \u03b5\n-- Paper proof:\n-- 1. \u03b5\u03b5 = \u03b5  (left\/right-identity axiom)\n-- 2. The previous equation states that \u03b5 is the unique left\/right\n-- inverse \u03b5\u207b\u00b9 of \u03b5.\nidentityInverse = rightInverseUnique' (leftIdentity \u03b5)\n\ninverseDistribution : \u2200 a b \u2192 (a \u00b7 b) \u207b\u00b9 \u2261 b \u207b\u00b9 \u00b7 a \u207b\u00b9\n-- Paper proof:\n-- (b\u207b\u00b9a\u207b\u00b9)(ab) = b\u207b\u00b9(a\u207b\u00b9(ab))  (associative axiom)\n--              = b\u207b\u00b9(a\u207b\u00b9a)b    (associative axiom)\n--              = b\u207b\u00b9(\u03b5b)       (left-inverse axiom)\n--              = b\u207b\u00b9b          (left-identity axiom)\n--              = \u03b5             (left-inverse axiom)\n-- Therefore, b\u207b\u00b9a\u207b\u00b9 is the unique left inverse of ab.\ninverseDistribution a b = leftInverseUnique' b\u207b\u00b9a\u207b\u00b9[ab]\u2261\u03b5\n  where\n  b\u207b\u00b9a\u207b\u00b9[ab]\u2261\u03b5 : b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (a \u00b7 b) \u2261 \u03b5\n  b\u207b\u00b9a\u207b\u00b9[ab]\u2261\u03b5 =\n    b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (a \u00b7 b)\n      \u2261\u27e8 assoc (b \u207b\u00b9) (a \u207b\u00b9) (a \u00b7 b) \u27e9\n    b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (a \u00b7 b))\n      \u2261\u27e8 subst (\u03bb t \u2192 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (a \u00b7 b)) \u2261 b \u207b\u00b9 \u00b7 t)\n               (sym (assoc (a \u207b\u00b9) a b))\n               refl\n      \u27e9\n    b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 a \u00b7 b)\n      \u2261\u27e8 subst (\u03bb t \u2192 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 a \u00b7 b) \u2261 b \u207b\u00b9 \u00b7 (t \u00b7 b))\n               (leftInverse a)\n               refl\n      \u27e9\n    b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 b)\n      \u2261\u27e8 subst (\u03bb t \u2192 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 b) \u2261 b \u207b\u00b9 \u00b7 t)\n               (leftIdentity b)\n               refl\n      \u27e9\n    b \u207b\u00b9 \u00b7 b\n      \u2261\u27e8 leftInverse b \u27e9\n    \u03b5 \u220e\n\n-- If the square of every element is the identity, the system is commutative.\n-- From: TPTP v5.3.0. File: Problems\/GRP\/GRP001-2.p\nx\u00b2\u2261\u03b5\u2192comm : (\u2200 a \u2192 a \u00b7 a \u2261 \u03b5) \u2192 \u2200 {b c d} \u2192 b \u00b7 c \u2261 d \u2192 c \u00b7 b \u2261 d\n-- Paper proof:\n-- 1. d(bc)  = dd  (hypothesis bc = d)\n-- 2. d(bc)  = \u03b5   (hypothesis dd = \u03b5)\n-- 3. d(bc)c = c   (by 2)\n-- 4. db(cc) = c   (associativity axiom)\n-- 5. db     = c   (hypothesis cc = \u03b5)\n-- 6. (db)b  = cb  (by 5)\n-- 7. d(bb)  = cb  (associativity axiom)\n-- 6. d      = cb  (hypothesis bb = \u03b5)\nx\u00b2\u2261\u03b5\u2192comm h {b} {c} {d} bc\u2261d = sym d\u2261cb\n  where\n  db\u2261c : d \u00b7 b \u2261 c\n  db\u2261c =\n    d \u00b7 b\n      \u2261\u27e8 sym (rightIdentity (d \u00b7 b)) \u27e9\n    d \u00b7 b \u00b7 \u03b5\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 b \u00b7 \u03b5 \u2261 d \u00b7 b \u00b7 t) (sym (h c)) refl \u27e9\n    d \u00b7 b \u00b7 (c \u00b7 c)\n      \u2261\u27e8 assoc d b (c \u00b7 c) \u27e9\n    d \u00b7 (b \u00b7 (c \u00b7 c))\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 (b \u00b7 (c \u00b7 c)) \u2261 d \u00b7 t) (sym (assoc b c c)) refl \u27e9\n    d \u00b7 ((b \u00b7 c) \u00b7 c)\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 ((b \u00b7 c) \u00b7 c) \u2261 d \u00b7 t)\n               (subst (\u03bb t \u2192 (b \u00b7 c) \u00b7 c \u2261 t \u00b7 c )\n                      bc\u2261d\n                      refl\n               )\n               refl\n      \u27e9\n    d \u00b7 (d \u00b7 c)\n      \u2261\u27e8 sym (assoc d d c) \u27e9\n    d \u00b7 d \u00b7 c\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 d \u00b7 c \u2261 t \u00b7 c ) (h d) refl \u27e9\n    \u03b5 \u00b7 c\n      \u2261\u27e8 leftIdentity c \u27e9\n    c \u220e\n\n  d\u2261cb : d \u2261 c \u00b7 b\n  d\u2261cb =\n    d           \u2261\u27e8 sym (rightIdentity d) \u27e9\n    d \u00b7 \u03b5       \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 \u03b5 \u2261 d \u00b7 t) (sym (h b)) refl \u27e9\n    d \u00b7 (b \u00b7 b) \u2261\u27e8 sym (assoc d b b) \u27e9\n    d \u00b7 b \u00b7 b   \u2261\u27e8 x\u2261y\u2192xz\u2261yz db\u2261c \u27e9\n    c \u00b7 b \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.PropertiesI where\n\nopen import GroupTheory.Base\nopen import GroupTheory.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- Adapted from the standard library.\ny\u2261x\u207b\u00b9[xy] : \u2200 a b \u2192 b \u2261 a \u207b\u00b9 \u00b7 (a \u00b7 b)\ny\u2261x\u207b\u00b9[xy] a b =\n  b              \u2261\u27e8 sym (leftIdentity b) \u27e9\n  \u03b5 \u00b7 b          \u2261\u27e8 subst (\u03bb t \u2192 \u03b5 \u00b7 b \u2261 t \u00b7 b ) (sym (leftInverse a)) refl \u27e9\n  a \u207b\u00b9 \u00b7 a \u00b7 b   \u2261\u27e8 assoc (a \u207b\u00b9) a b \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 b) \u220e\n\n-- Adapted from the standard library.\nx\u2261[xy]y\u207b\u00b9 : \u2200 a b \u2192 a \u2261 (a \u00b7 b) \u00b7 b \u207b\u00b9\nx\u2261[xy]y\u207b\u00b9 a b =\n  a              \u2261\u27e8 sym (rightIdentity a) \u27e9\n  a \u00b7 \u03b5          \u2261\u27e8 subst (\u03bb t \u2192 a \u00b7 \u03b5 \u2261 a \u00b7 t ) (sym (rightInverse b)) refl \u27e9\n  a \u00b7 (b \u00b7 b \u207b\u00b9) \u2261\u27e8 sym (assoc a b (b \u207b\u00b9)) \u27e9\n  a \u00b7 b \u00b7 b \u207b\u00b9 \u220e\n\nrightIdentityUnique : \u2200 r \u2192 (\u2200 a \u2192 a \u00b7 r \u2261 a) \u2192 r \u2261 \u03b5\n-- Paper proof (Saunders Mac Lane and Garret Birkhoff. Algebra. AMS\n-- Chelsea Publishing, 3rd edition, 1999. p. 48):\n--\n-- 1. r  = \u03b5r (\u03b5 is an identity)\n-- 2. \u03b5r = r  (hypothesis)\n-- 3. r  = \u03b5  (transitivity)\nrightIdentityUnique r h = trans (sym (leftIdentity r)) (h \u03b5)\n\n-- A more appropiate version to be used in the proofs.\n-- Adapted from the standard library.\nrightIdentityUnique' : \u2200 a r \u2192 a \u00b7 r \u2261 a \u2192 r \u2261 \u03b5\nrightIdentityUnique' a r h =\n  r              \u2261\u27e8 y\u2261x\u207b\u00b9[xy] a r \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 r) \u2261\u27e8 cong (_\u00b7_ (a \u207b\u00b9)) h \u27e9\n  a \u207b\u00b9 \u00b7 a       \u2261\u27e8 leftInverse a \u27e9\n  \u03b5 \u220e\n\nleftIdentityUnique : \u2200 l \u2192 (\u2200 a \u2192 l \u00b7 a \u2261 a) \u2192 l \u2261 \u03b5\n-- Paper proof:\n-- 1. l  = le (\u03b5 is an identity)\n-- 2. le = e  (hypothesis)\n-- 3. l  = e  (transitivity)\nleftIdentityUnique l h = trans (sym (rightIdentity l)) (h \u03b5)\n\n-- A more appropiate version to be used in the proofs.\n-- Adapted from the standard library.\nleftIdentityUnique' : \u2200 a l \u2192 l \u00b7 a \u2261 a \u2192 l \u2261 \u03b5\nleftIdentityUnique' a l h =\n  l            \u2261\u27e8 x\u2261[xy]y\u207b\u00b9 l a \u27e9\n  l \u00b7 a \u00b7 a \u207b\u00b9 \u2261\u27e8 subst (\u03bb t \u2192 l \u00b7 a \u00b7 a \u207b\u00b9 \u2261 t \u00b7 a \u207b\u00b9 ) h refl \u27e9\n  a \u00b7 a \u207b\u00b9 \u2261\u27e8 rightInverse a \u27e9\n  \u03b5 \u220e\n\nrightCancellation : \u2200 {a b c} \u2192 b \u00b7 a \u2261 c \u00b7 a \u2192 b \u2261 c\nrightCancellation {a} {b} {c} h =\n-- Paper proof:\n-- 1. (ba)a\u207b\u00b9  = (ca)a\u207b\u00b9  (hypothesis ab = ac)\n-- 2. (b)aa\u207b\u00b9  = (c)aa\u207b\u00b9  (associative axiom)\n-- 3. b\u03b5       = c\u03b5       (right-inverse axiom for a\u207b\u00b9)\n-- 4. b        = c        (right-identity axiom)\n  b              \u2261\u27e8 sym (rightIdentity b) \u27e9\n  b \u00b7 \u03b5          \u2261\u27e8 subst (\u03bb t \u2192 b \u00b7 \u03b5 \u2261 b \u00b7 t) (sym (rightInverse a)) refl \u27e9\n  b \u00b7 (a \u00b7 a \u207b\u00b9) \u2261\u27e8 sym (assoc b a (a \u207b\u00b9)) \u27e9\n  b \u00b7 a \u00b7 a \u207b\u00b9   \u2261\u27e8 subst (\u03bb t \u2192 b \u00b7 a \u00b7 a \u207b\u00b9 \u2261 t \u00b7 a \u207b\u00b9) h refl \u27e9\n  c \u00b7 a \u00b7 a \u207b\u00b9   \u2261\u27e8 assoc c a (a \u207b\u00b9) \u27e9\n  c \u00b7 (a \u00b7 a \u207b\u00b9) \u2261\u27e8 subst (\u03bb t \u2192 c \u00b7 (a \u00b7 a \u207b\u00b9) \u2261 c \u00b7 t) (rightInverse a) refl \u27e9\n  c \u00b7 \u03b5          \u2261\u27e8 rightIdentity c \u27e9\n  c \u220e\n\nleftCancellation : \u2200 {a b c} \u2192 a \u00b7 b \u2261 a \u00b7 c \u2192 b \u2261 c\nleftCancellation {a} {b} {c} h =\n-- Paper proof:\n-- 1. a\u207b\u00b9(ab)  = a\u207b\u00b9(ac)  (hypothesis ab = ac)\n-- 2. a\u207b\u00b9a(b)  = a\u207b\u00b9a(c)  (associative axiom)\n-- 3. \u03b5b       = \u03b5c       (left-inverse axiom for a\u207b\u00b9)\n-- 4. b        = c        (left-identity axiom)\n  b              \u2261\u27e8 sym (leftIdentity b) \u27e9\n  \u03b5 \u00b7 b          \u2261\u27e8 subst (\u03bb t \u2192 \u03b5 \u00b7 b \u2261 t \u00b7 b) (sym (leftInverse a)) refl \u27e9\n  a \u207b\u00b9 \u00b7 a \u00b7 b   \u2261\u27e8 assoc (a \u207b\u00b9) a b \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 b) \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 (a \u00b7 b) \u2261 a \u207b\u00b9 \u00b7 t) h refl \u27e9\n  a \u207b\u00b9 \u00b7 (a \u00b7 c) \u2261\u27e8 sym (assoc (a \u207b\u00b9) a c) \u27e9\n  a \u207b\u00b9 \u00b7 a \u00b7 c   \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 a \u00b7 c \u2261 t \u00b7 c) (leftInverse a) refl \u27e9\n  \u03b5 \u00b7 c          \u2261\u27e8 leftIdentity c \u27e9\n  c \u220e\n\nx\u2261y\u2192xz\u2261yz : \u2200 {a b c} \u2192 a \u2261 b \u2192 a \u00b7 c \u2261 b \u00b7 c\nx\u2261y\u2192xz\u2261yz refl = refl\n\nx\u2261y\u2192zx\u2261zy : \u2200 {a b c} \u2192 a \u2261 b \u2192 c \u00b7 a \u2261 c \u00b7 b\nx\u2261y\u2192zx\u2261zy refl = refl\n\nrightInverseUnique : \u2200 {a} \u2192 \u2203[ r ] (a \u00b7 r \u2261 \u03b5) \u2227\n                                    (\u2200 r' \u2192 a \u00b7 r' \u2261 \u03b5 \u2192 r \u2261 r')\nrightInverseUnique {a} =\n-- Paper proof:\n-- 1.   We know that (a\u207b\u00b9) is a right inverse for a.\n-- 2.   Let's suppose there is other right inverse r for a, i.e. ar \u2261 \u03b5, then\n-- 2.1. aa\u207b\u00b9 = \u03b5  (right-inverse axiom)\n-- 2.2. ar   = \u03b5  (hypothesis)\n-- 2.3. aa\u207b\u00b9 = ar (transitivity)\n-- 2.4  a\u207b\u00b9  = a  (left-cancellation)\n  (a \u207b\u00b9) , rightInverse a , prf\n    where\n    prf : \u2200 r' \u2192 a \u00b7 r' \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 r'\n    prf r' ar'\u2261\u03b5 = leftCancellation aa\u207b\u00b9\u2261ar'\n      where\n      aa\u207b\u00b9\u2261ar' : a \u00b7 a \u207b\u00b9 \u2261 a \u00b7 r'\n      aa\u207b\u00b9\u2261ar' = a \u00b7 a \u207b\u00b9 \u2261\u27e8 rightInverse a \u27e9\n                 \u03b5        \u2261\u27e8 sym ar'\u2261\u03b5 \u27e9\n                 a \u00b7 r' \u220e\n\n-- A more appropiate version to be used in the proofs.\nrightInverseUnique' : \u2200 {a r} \u2192 a \u00b7 r \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 r\nrightInverseUnique' {a} {r} ar\u2261\u03b5 = leftCancellation aa\u207b\u00b9\u2261ar\n  where\n  aa\u207b\u00b9\u2261ar : a \u00b7 a \u207b\u00b9 \u2261 a \u00b7 r\n  aa\u207b\u00b9\u2261ar = a \u00b7 a \u207b\u00b9 \u2261\u27e8 rightInverse a \u27e9\n            \u03b5        \u2261\u27e8 sym ar\u2261\u03b5 \u27e9\n            a \u00b7 r \u220e\n\nleftInverseUnique : \u2200 {a} \u2192 \u2203[ l ] (l \u00b7 a \u2261 \u03b5) \u2227\n                                   (\u2200 l' \u2192 l' \u00b7 a \u2261 \u03b5 \u2192 l \u2261 l')\nleftInverseUnique {a} =\n-- Paper proof:\n-- 1.   We know that (a\u207b\u00b9) is a left inverse for a.\n-- 2.   Let's suppose there is other right inverse l for a, i.e. la \u2261 \u03b5, then\n-- 2.1. a\u207b\u00b9a = \u03b5  (left-inverse axiom)\n-- 2.2. la   = \u03b5  (hypothesis)\n-- 2.3. a\u207b\u00b9a = la (transitivity)\n-- 2.4  a\u207b\u00b9  = l  (right-cancellation)\n  (a \u207b\u00b9) , leftInverse a , prf\n    where\n    prf : \u2200 l' \u2192 l' \u00b7 a \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 l'\n    prf l' l'a\u2261\u03b5 = rightCancellation a\u207b\u00b9a\u2261l'a\n      where\n      a\u207b\u00b9a\u2261l'a : a \u207b\u00b9 \u00b7 a \u2261 l' \u00b7 a\n      a\u207b\u00b9a\u2261l'a = a \u207b\u00b9 \u00b7 a \u2261\u27e8 leftInverse a \u27e9\n                 \u03b5        \u2261\u27e8 sym l'a\u2261\u03b5 \u27e9\n                 l' \u00b7 a \u220e\n\n-- A more appropiate version to be used in the proofs.\nleftInverseUnique' : \u2200 {a l} \u2192 l \u00b7 a \u2261 \u03b5 \u2192 a \u207b\u00b9 \u2261 l\nleftInverseUnique' {a} {l} la\u2261\u03b5 = rightCancellation a\u207b\u00b9a\u2261la\n  where\n  a\u207b\u00b9a\u2261la : a \u207b\u00b9 \u00b7 a \u2261 l \u00b7 a\n  a\u207b\u00b9a\u2261la = a \u207b\u00b9 \u00b7 a \u2261\u27e8 leftInverse a \u27e9\n            \u03b5        \u2261\u27e8 sym la\u2261\u03b5 \u27e9\n            l \u00b7 a \u220e\n\n\u207b\u00b9-involutive : \u2200 a \u2192 a \u207b\u00b9 \u207b\u00b9 \u2261 a\n-- Paper proof:\n-- 1. a\u207b\u00b9a = \u03b5  (left-inverse axiom)\n-- 2. The previous equation states that a is the unique right\n-- inverse (a\u207b\u00b9)\u207b\u00b9 of a\u207b\u00b9.\n\u207b\u00b9-involutive a = rightInverseUnique' (leftInverse a)\n\nidentityInverse : \u03b5 \u207b\u00b9 \u2261 \u03b5\n-- Paper proof:\n-- 1. \u03b5\u03b5 = \u03b5  (left\/right-identity axiom)\n-- 2. The previous equation states that \u03b5 is the unique left\/right\n-- inverse \u03b5\u207b\u00b9 of \u03b5.\nidentityInverse = rightInverseUnique' (leftIdentity \u03b5)\n\ninverseDistribution : \u2200 a b \u2192 (a \u00b7 b) \u207b\u00b9 \u2261 b \u207b\u00b9 \u00b7 a \u207b\u00b9\n-- Paper proof:\n-- (b\u207b\u00b9a\u207b\u00b9)(ab) = b\u207b\u00b9(a\u207b\u00b9(ab))  (associative axiom)\n--              = b\u207b\u00b9(a\u207b\u00b9a)b    (associative axiom)\n--              = b\u207b\u00b9(\u03b5b)       (left-inverse axiom)\n--              = b\u207b\u00b9b          (left-identity axiom)\n--              = \u03b5             (left-inverse axiom)\n-- Therefore, b\u207b\u00b9a\u207b\u00b9 is the unique left inverse of ab.\ninverseDistribution a b = leftInverseUnique' b\u207b\u00b9a\u207b\u00b9[ab]\u2261\u03b5\n  where\n  b\u207b\u00b9a\u207b\u00b9[ab]\u2261\u03b5 : b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (a \u00b7 b) \u2261 \u03b5\n  b\u207b\u00b9a\u207b\u00b9[ab]\u2261\u03b5 =\n    b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (a \u00b7 b)\n      \u2261\u27e8 assoc (b \u207b\u00b9) (a \u207b\u00b9) (a \u00b7 b) \u27e9\n    b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (a \u00b7 b))\n      \u2261\u27e8 subst (\u03bb t \u2192 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (a \u00b7 b)) \u2261 b \u207b\u00b9 \u00b7 t)\n               (sym (assoc (a \u207b\u00b9) a b))\n               refl\n      \u27e9\n    b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 a \u00b7 b)\n      \u2261\u27e8 subst (\u03bb t \u2192 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 a \u00b7 b) \u2261 b \u207b\u00b9 \u00b7 (t \u00b7 b))\n               (leftInverse a)\n               refl\n      \u27e9\n    b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 b)\n      \u2261\u27e8 subst (\u03bb t \u2192 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 b) \u2261 b \u207b\u00b9 \u00b7 t)\n               (leftIdentity b)\n               refl\n      \u27e9\n    b \u207b\u00b9 \u00b7 b\n      \u2261\u27e8 leftInverse b \u27e9\n    \u03b5 \u220e\n\n-- If the square of every element is the identity, the system is commutative.\n-- From: TPTP v5.3.0. File: Problems\/GRP\/GRP001-2.p\nx\u00b2\u2261\u03b5\u2192comm : (\u2200 a \u2192 a \u00b7 a \u2261 \u03b5) \u2192 \u2200 {b c d} \u2192 b \u00b7 c \u2261 d \u2192 c \u00b7 b \u2261 d\n-- Paper proof:\n-- 1. d(bc)  = dd  (hypothesis bc = d)\n-- 2. d(bc)  = \u03b5   (hypothesis dd = \u03b5)\n-- 3. d(bc)c = c   (by 2)\n-- 4. db(cc) = c   (associativity axiom)\n-- 5. db     = c   (hypothesis cc = \u03b5)\n-- 6. (db)b  = cb  (by 5)\n-- 7. d(bb)  = cb  (associativity axiom)\n-- 6. d      = cb  (hypothesis bb = \u03b5)\nx\u00b2\u2261\u03b5\u2192comm h {b} {c} {d} bc\u2261d = sym d\u2261cb\n  where\n  db\u2261c : d \u00b7 b \u2261 c\n  db\u2261c =\n    d \u00b7 b\n      \u2261\u27e8 sym (rightIdentity (d \u00b7 b)) \u27e9\n    d \u00b7 b \u00b7 \u03b5\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 b \u00b7 \u03b5 \u2261 d \u00b7 b \u00b7 t) (sym (h c)) refl \u27e9\n    d \u00b7 b \u00b7 (c \u00b7 c)\n      \u2261\u27e8 assoc d b (c \u00b7 c) \u27e9\n    d \u00b7 (b \u00b7 (c \u00b7 c))\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 (b \u00b7 (c \u00b7 c)) \u2261 d \u00b7 t) (sym (assoc b c c)) refl \u27e9\n    d \u00b7 ((b \u00b7 c) \u00b7 c)\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 ((b \u00b7 c) \u00b7 c) \u2261 d \u00b7 t)\n               (subst (\u03bb t \u2192 (b \u00b7 c) \u00b7 c \u2261 t \u00b7 c )\n                      bc\u2261d\n                      refl\n               )\n               refl\n      \u27e9\n    d \u00b7 (d \u00b7 c)\n      \u2261\u27e8 sym (assoc d d c) \u27e9\n    d \u00b7 d \u00b7 c\n      \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 d \u00b7 c \u2261 t \u00b7 c ) (h d) refl \u27e9\n    \u03b5 \u00b7 c\n      \u2261\u27e8 leftIdentity c \u27e9\n    c \u220e\n\n  d\u2261cb : d \u2261 c \u00b7 b\n  d\u2261cb =\n    d           \u2261\u27e8 sym (rightIdentity d) \u27e9\n    d \u00b7 \u03b5       \u2261\u27e8 subst (\u03bb t \u2192 d \u00b7 \u03b5 \u2261 d \u00b7 t) (sym (h b)) refl \u27e9\n    d \u00b7 (b \u00b7 b) \u2261\u27e8 sym (assoc d b b) \u27e9\n    d \u00b7 b \u00b7 b   \u2261\u27e8 x\u2261y\u2192xz\u2261yz db\u2261c \u27e9\n    c \u00b7 b \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"edb843084fa4776541fc471c53ff01b92f978afe","subject":"Explain how to delete *.agdai files.","message":"Explain how to delete *.agdai files.\n\nOld-commit-hash: ef86b1405cd59d268c9156ada86e7e0e6a1e3199\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n----------------------------\n-- INCREMENTAL \u03bb-CALCULUS --\n----------------------------\n--\n--\n-- IMPORTANT FILES\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda formalization.\n--\n-- PLDI14-List-of-Theorems.agda\n--   For each theorem, lemma or definition in the PLDI 2014 submission,\n--   it points to the corresponding Agda object.\n--\n--\n-- LOCATION OF AGDA MODULES\n--\n-- To find the file containing an Agda module, replace the dots in its\n-- full name by directory separators. The result is the file's path relative\n-- to this directory. For example, Parametric.Syntax.Type is defined in\n-- Parametric\/Syntax\/Type.agda.\n--\n--\n-- HOW TO TYPE CHECK EVERYTHING\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","old_contents":"module README where\n\n----------------------------\n-- INCREMENTAL \u03bb-CALCULUS --\n----------------------------\n--\n--\n-- IMPORTANT FILES\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda formalization.\n--\n-- PLDI14-List-of-Theorems.agda\n--   For each theorem, lemma or definition in the PLDI 2014 submission,\n--   it points to the corresponding Agda object.\n--\n--\n-- LOCATION OF AGDA MODULES\n--\n-- To find the file containing an Agda module, replace the dots in its\n-- full name by directory separators. The result is the file's path relative\n-- to this directory. For example, Parametric.Syntax.Type is defined in\n-- Parametric\/Syntax\/Type.agda.\n--\n--\n-- HOW TO TYPE CHECK EVERYTHING\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help.\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c4d3eb2622e33fb8644afd36c5783613e493b957","subject":"this looks random\u2122","message":"this looks random\u2122\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function using (id; _\u2218_; const; _$_)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\nflip\u2032 : M 1 Bit\nflip\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\nflip : \u2200 {n} \u2192 M (1 + n) Bit\nflip = weaken\u2032 flip\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nflips : \u2200 {n} \u2192 M n (Bits n)\n-- flips = coerce ? (replicateM flip) -- specialized version for now to avoid coerce\nflips {zero}  = \u27ea [] \u27eb\nflips {suc _} = \u27ea _\u2237_ \u00b7 flip\u2032 \u00b7 flips \u27eb\n\n-- Another name for flips\nrandom : \u2200 {n} \u2192 M n (Bits n)\nrandom = flips\n\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 flips \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  SecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  SecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 flips \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                         this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : SecPRF prf \n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = map f flips\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 flip\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = flip\u2032 \u27e8,\u27e9 flip\u2032 \n\n  ex\u2083 = \u2200 {n} \u2192 SecPRF {n} _\u2295_\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 SecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function using (id; _\u2218_; const; _$_)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\nflip\u2032 : M 1 Bit\nflip\u2032 = mk (\u03bb { (b \u2237 []) \u2192 b })\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 x = mk (\u03bb _ \u2192 x)\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\nflip : \u2200 {n} \u2192 M (1 + n) Bit\nflip = weaken\u2032 flip\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nflips : \u2200 {n} \u2192 M n (Bits n)\n-- flips = coerce ? (replicateM flip) -- specialized version for now to avoid coerce\nflips {zero}  = \u27ea [] \u27eb\nflips {suc _} = \u27ea _\u2237_ \u00b7 flip\u2032 \u00b7 flips \u27eb\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : Bits k \u2192 Bits n) \u2192 Set\n  SecPRG prg = \u27ea prg \u00b7 flips \u27eb \u224b flips\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  SecPRF : \u2200 {k m n} (prf : Bits k \u2192 Bits m \u2192 Bits n) \u2192 Set\n  SecPRF prf = \u2200 {xs} \u2192 \u27ea prf \u00b7 flips \u00b7 \u27ea xs \u27eb\u2032 \u27eb \u224b flips\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : SecPRF prf \n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = map f flips\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 flip\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = flip\u2032 \u27e8,\u27e9 flip\u2032 \n\n  ex\u2083 = \u2200 {n} {xs : Bits n} \u2192 map (_\u2295_ xs) flips \u2248 flips\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 SecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"53cdcb2ae886c6b7a349cdc64787d67c220ede2f","subject":"Reflection.Decoding","message":"Reflection.Decoding\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Decoding.agda","new_file":"lib\/Reflection\/Decoding.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Reflection.NP\nopen import Data.Maybe\nopen import Data.Nat\nopen import Data.List as L\nopen import Data.Vec hiding (_>>=_;_\u229b_)\nopen import Data.Product\nopen import Data.Bool\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Level.NP\nimport Category.Monad as Cat\n\nmodule Reflection.Decoding where\n\nmodule M? = Cat.RawMonad (Data.Maybe.monad {\u2080})\nopen M? using () renaming (_\u229b_ to _\u229b?_)\n\nD : \u2605 \u2192 \u2605\nD A = Term \u2192 Maybe A\n\npureD : \u2200 {A} \u2192 A \u2192 D A\npureD a _ = just a\n\ninfixr 5 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : D A) \u2192 D A\n_\u27e8|\u27e9_ f g t = maybe\u2032 just (g t) (f t)\n\nemptyD : \u2200 {A} \u2192 D A\nemptyD _ = nothing\n\nmanyD : \u2200 {A} \u2192 List (D A) \u2192 D A\nmanyD = L.foldr _\u27e8|\u27e9_ emptyD\n\nmapD : \u2200 {A B} \u2192 (A \u2192 B) \u2192 D A \u2192 D B\nmapD f da = maybe\u2032 (just \u2218 f) nothing \u2218 da\n\njoinD : \u2200 {A} \u2192 D (D A) \u2192 D A\njoinD dda t = maybe\u2032 (\u03bb f \u2192 f t) nothing (dda t)\n\n_>>=_ : \u2200 {A B} \u2192 D A \u2192 (A \u2192 D B) \u2192 D B\n_>>=_ da f t with da t\n... | just a  = f a t\n... | nothing = nothing\n\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 D B) \u2192 D A \u2192 D B\n_=<<_ f da = da >>= f\n\ninfixl 4 _\u229bD_\n_\u229bD_ : \u2200 {A B} \u2192 D (A \u2192 B) \u2192 D A \u2192 D B\n_\u229bD_ df da = df >>= \u03bb f \u2192 da >>= \u03bb a \u2192 pureD (f a)\n\nArgSpec : \u2605\nArgSpec = Implicit? \u00d7 Relevant?\n\nArgSpecs : \u2605\nArgSpecs = List ArgSpec\n\ndeArg : ArgSpec \u2192 Arg Term \u2192 Maybe Term\ndeArg (im? , r?) (arg im?\u2032 r?\u2032 t) =\n  if \u230a im? \u225f-Implicit? im?\u2032 \u230b \u2227 \u230a r? \u225f-Relevant? r?\u2032 \u230b then just t else nothing\n\ndeArgs : (argSpecs : ArgSpecs) \u2192 Args \u2192 Maybe (Vec Term (length argSpecs))\ndeArgs []       []       = just []\ndeArgs (x \u2237 xs) (y \u2237 ys) = just _\u2237_ \u229b? deArg x y \u229b? deArgs xs ys\ndeArgs []       (_ \u2237 _)  = nothing\ndeArgs (_ \u2237 _)  []       = nothing\n\ndeCon : Name \u2192 (argSpecs : ArgSpecs) \u2192 D (Vec Term (length argSpecs))\ndeCon nm argSpecs (con nm\u2032 args) =\n  if nm == nm\u2032 then deArgs argSpecs args else nothing\ndeCon _ _ _ = nothing\n\ndeDef : Name \u2192 (argSpecs : ArgSpecs) \u2192 D (Vec Term (length argSpecs))\ndeDef nm argSpecs (def nm\u2032 args) =\n  if nm == nm\u2032 then deArgs argSpecs args else nothing\ndeDef _ _ _ = nothing\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Reflection.NP\nopen import Data.Maybe\nopen import Data.Nat\nopen import Data.List as L\nopen import Data.Vec hiding (_>>=_;_\u229b_)\nopen import Data.Product\nopen import Data.Bool\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Level\nimport Category.Monad as Cat\n\nmodule Reflection.Decoding where\n\nmodule M? = Cat.RawMonad (Data.Maybe.monad {zero})\nopen M? using () renaming (_\u229b_ to _\u229b?_)\n\nD : \u2605 \u2192 \u2605\nD A = Term \u2192 Maybe A\n\npureD : \u2200 {A} \u2192 A \u2192 D A\npureD a _ = just a\n\ninfixr 5 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : D A) \u2192 D A\n_\u27e8|\u27e9_ f g t = maybe\u2032 just (g t) (f t)\n\nemptyD : \u2200 {A} \u2192 D A\nemptyD _ = nothing\n\nmanyD : \u2200 {A} \u2192 List (D A) \u2192 D A\nmanyD = L.foldr _\u27e8|\u27e9_ emptyD\n\nmapD : \u2200 {A B} \u2192 (A \u2192 B) \u2192 D A \u2192 D B\nmapD f da = maybe\u2032 (just \u2218 f) nothing \u2218 da\n\njoinD : \u2200 {A} \u2192 D (D A) \u2192 D A\njoinD dda t = maybe\u2032 (\u03bb f \u2192 f t) nothing (dda t)\n\n_>>=_ : \u2200 {A B} \u2192 D A \u2192 (A \u2192 D B) \u2192 D B\n_>>=_ da f t with da t\n... | just a  = f a t\n... | nothing = nothing\n\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 D B) \u2192 D A \u2192 D B\n_=<<_ f da = da >>= f\n\ninfixl 4 _\u229bD_\n_\u229bD_ : \u2200 {A B} \u2192 D (A \u2192 B) \u2192 D A \u2192 D B\n_\u229bD_ df da = df >>= \u03bb f \u2192 da >>= \u03bb a \u2192 pureD (f a)\n\nArgSpec : \u2605\nArgSpec = Implicit? \u00d7 Relevant?\n\nArgSpecs : \u2605\nArgSpecs = List ArgSpec\n\ndeArg : ArgSpec \u2192 Arg Term \u2192 Maybe Term\ndeArg (im? , r?) (arg im?\u2032 r?\u2032 t) =\n  if \u230a im? \u225f-Implicit? im?\u2032 \u230b \u2227 \u230a r? \u225f-Relevant? r?\u2032 \u230b then just t else nothing\n\ndeArgs : (argSpecs : ArgSpecs) \u2192 Args \u2192 Maybe (Vec Term (length argSpecs))\ndeArgs []       []       = just []\ndeArgs (x \u2237 xs) (y \u2237 ys) = just _\u2237_ \u229b? deArg x y \u229b? deArgs xs ys\ndeArgs []       (_ \u2237 _)  = nothing\ndeArgs (_ \u2237 _)  []       = nothing\n\ndeCon : Name \u2192 (argSpecs : ArgSpecs) \u2192 D (Vec Term (length argSpecs))\ndeCon nm argSpecs (con nm\u2032 args) =\n  if nm == nm\u2032 then deArgs argSpecs args else nothing\ndeCon _ _ _ = nothing\n\ndeDef : Name \u2192 (argSpecs : ArgSpecs) \u2192 D (Vec Term (length argSpecs))\ndeDef nm argSpecs (def nm\u2032 args) =\n  if nm == nm\u2032 then deArgs argSpecs args else nothing\ndeDef _ _ _ = nothing\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7afae4cfbca7fae22dafb448533867ac175e0fbd","subject":"Typo.","message":"Typo.\n\nIgnore-this: 3cf7e909b08c6c2a8356b77182e6b711\n\ndarcs-hash:20120217142752-3bd4e-c678949fbbf22fbbd42b682523318682d68f9750.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/Existential.agda","new_file":"Draft\/FOTC\/Data\/Stream\/Existential.agda","new_contents":"------------------------------------------------------------------------------\n-- Existential quantifier: pattern matching vs elimination\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 17 February 2012.\n\nmodule Draft.FOTC.Data.Stream.Existential where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Data.Stream.Equality\n\n------------------------------------------------------------------------------\n\n-- A proof using pattern matching (via the with mechanism).\ntailS\u2081 : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS\u2081 {x} {xs} h\u2081 with (Stream-gfp\u2081 h\u2081)\n... | e , es , Ses , h\u2082 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2082))) Ses\n\n-- A proof using existential elimination.\ntailS\u2082 : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS\u2082 {x} {xs} h =\n  \u2203-elim (Stream-gfp\u2081 h)\n    (\u03bb e prf\u2081 \u2192 \u2203-elim prf\u2081\n       (\u03bb es prf\u2082 \u2192 subst Stream\n                          (sym (\u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2082 prf\u2082))))\n                          (\u2227-proj\u2081 prf\u2082)))\n","old_contents":"------------------------------------------------------------------------------\n-- Existential quantifier: pattern matching vs elimination\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 17 February 2012.\n\nmodule Draft.FOTC.Data.Stream.Existential where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Data.Stream.Equality\n\n------------------------------------------------------------------------------\n\n-- A proof using pattern matching (via the with mechanisms).\ntailS\u2081 : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS\u2081 {x} {xs} h\u2081 with (Stream-gfp\u2081 h\u2081)\n... | e , es , Ses , h\u2082 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2082))) Ses\n\n-- A proof using elimination.\ntailS\u2082 : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS\u2082 {x} {xs} h =\n  \u2203-elim (Stream-gfp\u2081 h)\n    (\u03bb e prf\u2081 \u2192 \u2203-elim prf\u2081\n       (\u03bb es prf\u2082 \u2192 subst Stream\n                          (sym (\u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2082 prf\u2082))))\n                          (\u2227-proj\u2081 prf\u2082)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a06adfefda06e416bc4f111d7085cd3b5ad42568","subject":"Document P.C.Evaluation.","message":"Document P.C.Evaluation.\n\nOld-commit-hash: d56cb423e28dc729478353d4e2d2c4fb422d2f8f\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Evaluation.agda","new_file":"Parametric\/Change\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Connecting Parametric.Change.Term and Parametric.Change.Value.\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\n\nmodule Parametric.Change.Evaluation\n    {Base : Type.Structure}\n    {Const : Term.Structure Base}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\nopen import Postulate.Extensionality\n\n-- Extension point 1: Relating \u229d and its value on base types\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 {\u0393} \u2192\n  {t : Term \u0393 (base \u03b9)} {\u0394t : Term \u0393 (\u0394Type (base \u03b9))} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d base \u03b9 \u208e \u0394t \u27e7 \u03c1\n\n-- Extension point 2: Relating \u2295 and its value on base types\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 {\u0393} \u2192\n  {s : Term \u0393 (base \u03b9)} {t : Term \u0393 (base \u03b9)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d base \u03b9 \u208e t \u27e7 \u03c1\n\nmodule Structure\n    (meaning-\u2295-base : ApplyStructure)\n    (meaning-\u229d-base : DiffStructure)\n  where\n\n  -- unique names with unambiguous types\n  -- to help type inference figure things out\n  private\n    module Disambiguation where\n      infixr 9 _\u22c6_\n      _\u22c6_ : Type \u2192 Context \u2192 Context\n      _\u22c6_ = _\u2022_\n\n  -- We provide: Relating \u2295 and \u229d and their values on arbitrary types.\n  meaning-\u2295 : \u2200 {\u03c4 \u0393}\n    {t : Term \u0393 \u03c4} {\u0394t : Term \u0393 (\u0394Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d \u03c4 \u208e \u0394t \u27e7 \u03c1\n\n  meaning-\u229d : \u2200 {\u03c4 \u0393}\n    {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d \u03c4 \u208e t \u27e7 \u03c1\n\n  meaning-\u2295 {base \u03b9} {\u0393} {\u03c4} {\u0394t} {\u03c1} = meaning-\u2295-base \u03b9 {\u0393} {\u03c4} {\u0394t} {\u03c1}\n  meaning-\u2295 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {\u0394t} {\u03c1} = ext (\u03bb v \u2192\n    let\n      \u0393\u2032 = \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0394Type (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7\n      \u03c1\u2032 = v \u2022 (\u27e6 t \u27e7 \u03c1) \u2022 (\u27e6 \u0394t \u27e7 \u03c1) \u2022 \u03c1\n      x  : Term \u0393\u2032 \u03c3\n      x  = var this\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that this)\n      \u0394f : Term \u0393\u2032 (\u0394Type (\u03c3 \u21d2 \u03c4))\n      \u0394f = var (that (that this))\n      y  = app f x\n      \u0394y = app (app \u0394f x) (x \u229d x)\n    in\n      begin\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v hole)\n           (meaning-\u229d {s = x} {x} {\u03c1\u2032}) \u27e9\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (\u27e6 x \u229d x \u27e7 \u03c1\u2032)\n      \u2261\u27e8 meaning-\u2295 {t = y} {\u0394t = \u0394y} {\u03c1\u2032} \u27e9\n        \u27e6 y \u2295\u208d \u03c4 \u208e \u0394y \u27e7 \u03c1\u2032\n      \u220e)\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n\n  meaning-\u229d {base \u03b9} {\u0393} {s} {t} {\u03c1} = meaning-\u229d-base \u03b9 {\u0393} {s} {t} {\u03c1}\n  meaning-\u229d {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} =\n    ext (\u03bb v \u2192 ext (\u03bb \u0394v \u2192\n    let\n      \u0393\u2032 = \u0394Type \u03c3 \u22c6 \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7Context\n      \u03c1\u2032 = \u0394v \u2022 v \u2022 \u27e6 t \u27e7Term \u03c1 \u2022 \u27e6 s \u27e7Term \u03c1 \u2022 \u03c1\n      \u0394x : Term \u0393\u2032 (\u0394Type \u03c3)\n      \u0394x = var this\n      x  : Term \u0393\u2032 \u03c3\n      x  = var (that this)\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that (that this))\n      g  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      g  = var (that (that (that this)))\n      y  = app f x\n      y\u2032 = app g (x \u2295\u208d \u03c3 \u208e \u0394x)\n    in\n      begin\n        \u27e6 s \u27e7 \u03c1 (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 s \u27e7 \u03c1 hole \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v)\n           (meaning-\u2295 {t = x} {\u0394t = \u0394x} {\u03c1\u2032}) \u27e9\n        \u27e6 s \u27e7 \u03c1 (\u27e6 x \u2295\u208d \u03c3 \u208e \u0394x \u27e7 \u03c1\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 meaning-\u229d {s = y\u2032} {y} {\u03c1\u2032} \u27e9\n        \u27e6 y\u2032 \u229d y \u27e7 \u03c1\u2032\n      \u220e))\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\n\nmodule Parametric.Change.Evaluation\n    {Base : Type.Structure}\n    {Const : Term.Structure Base}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\nopen import Postulate.Extensionality\n\n-- Relating `diff` with `diff-term`, `apply` with `apply-term`\n\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 {\u0393} \u2192\n  {t : Term \u0393 (base \u03b9)} {\u0394t : Term \u0393 (\u0394Type (base \u03b9))} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d base \u03b9 \u208e \u0394t \u27e7 \u03c1\n\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 {\u0393} \u2192\n  {s : Term \u0393 (base \u03b9)} {t : Term \u0393 (base \u03b9)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d base \u03b9 \u208e t \u27e7 \u03c1\n\nmodule Structure\n    (meaning-\u2295-base : ApplyStructure)\n    (meaning-\u229d-base : DiffStructure)\n  where\n\n  -- unique names with unambiguous types\n  -- to help type inference figure things out\n  private\n    module Disambiguation where\n      infixr 9 _\u22c6_\n      _\u22c6_ : Type \u2192 Context \u2192 Context\n      _\u22c6_ = _\u2022_\n\n  meaning-\u2295 : \u2200 {\u03c4 \u0393}\n    {t : Term \u0393 \u03c4} {\u0394t : Term \u0393 (\u0394Type \u03c4)} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 \u2261 \u27e6 t \u2295\u208d \u03c4 \u208e \u0394t \u27e7 \u03c1\n\n  meaning-\u229d : \u2200 {\u03c4 \u0393}\n    {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n    \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 s \u229d\u208d \u03c4 \u208e t \u27e7 \u03c1\n\n  meaning-\u2295 {base \u03b9} {\u0393} {\u03c4} {\u0394t} {\u03c1} = meaning-\u2295-base \u03b9 {\u0393} {\u03c4} {\u0394t} {\u03c1}\n  meaning-\u2295 {\u03c3 \u21d2 \u03c4} {\u0393} {t} {\u0394t} {\u03c1} = ext (\u03bb v \u2192\n    let\n      \u0393\u2032 = \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0394Type (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7\n      \u03c1\u2032 = v \u2022 (\u27e6 t \u27e7 \u03c1) \u2022 (\u27e6 \u0394t \u27e7 \u03c1) \u2022 \u03c1\n      x  : Term \u0393\u2032 \u03c3\n      x  = var this\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that this)\n      \u0394f : Term \u0393\u2032 (\u0394Type (\u03c3 \u21d2 \u03c4))\n      \u0394f = var (that (that this))\n      y  = app f x\n      \u0394y = app (app \u0394f x) (x \u229d x)\n    in\n      begin\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v hole)\n           (meaning-\u229d {s = x} {x} {\u03c1\u2032}) \u27e9\n        \u27e6 t \u27e7 \u03c1 v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u27e6 \u0394t \u27e7 \u03c1 v (\u27e6 x \u229d x \u27e7 \u03c1\u2032)\n      \u2261\u27e8 meaning-\u2295 {t = y} {\u0394t = \u0394y} {\u03c1\u2032} \u27e9\n        \u27e6 y \u2295\u208d \u03c4 \u208e \u0394y \u27e7 \u03c1\u2032\n      \u220e)\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n\n  meaning-\u229d {base \u03b9} {\u0393} {s} {t} {\u03c1} = meaning-\u229d-base \u03b9 {\u0393} {s} {t} {\u03c1}\n  meaning-\u229d {\u03c3 \u21d2 \u03c4} {\u0393} {s} {t} {\u03c1} =\n    ext (\u03bb v \u2192 ext (\u03bb \u0394v \u2192\n    let\n      \u0393\u2032 = \u0394Type \u03c3 \u22c6 \u03c3 \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 (\u03c3 \u21d2 \u03c4) \u22c6 \u0393\n      \u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7Context\n      \u03c1\u2032 = \u0394v \u2022 v \u2022 \u27e6 t \u27e7Term \u03c1 \u2022 \u27e6 s \u27e7Term \u03c1 \u2022 \u03c1\n      \u0394x : Term \u0393\u2032 (\u0394Type \u03c3)\n      \u0394x = var this\n      x  : Term \u0393\u2032 \u03c3\n      x  = var (that this)\n      f  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      f  = var (that (that this))\n      g  : Term \u0393\u2032 (\u03c3 \u21d2 \u03c4)\n      g  = var (that (that (that this)))\n      y  = app f x\n      y\u2032 = app g (x \u2295\u208d \u03c3 \u208e \u0394x)\n    in\n      begin\n        \u27e6 s \u27e7 \u03c1 (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 s \u27e7 \u03c1 hole \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v)\n           (meaning-\u2295 {t = x} {\u0394t = \u0394x} {\u03c1\u2032}) \u27e9\n        \u27e6 s \u27e7 \u03c1 (\u27e6 x \u2295\u208d \u03c3 \u208e \u0394x \u27e7 \u03c1\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 \u27e6 t \u27e7 \u03c1 v\n      \u2261\u27e8 meaning-\u229d {s = y\u2032} {y} {\u03c1\u2032} \u27e9\n        \u27e6 y\u2032 \u229d y \u27e7 \u03c1\u2032\n      \u220e))\n    where\n      open \u2261-Reasoning\n      open Disambiguation\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"159728c45a04b2e0a3562bdd6c354eb5e30eeadf","subject":"Minor changes to the draft of Stream-gfp\u2082.","message":"Minor changes to the draft of Stream-gfp\u2082.\n\nIgnore-this: b56f2ef0794edb40eaf8d0c961cb97f5\n\ndarcs-hash:20110926171750-3bd4e-2472e96be53f2ef43fd7a40099cacefbad9c46df.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/StreamSL.agda","new_file":"Draft\/FOTC\/Data\/Stream\/StreamSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC streams using the Agda coinductive combinators\n------------------------------------------------------------------------------\n\n-- Tested with the development version Agda and the standard library\n-- on 26 September 2011.\n\nmodule StreamSL where\n\nopen import Data.Product renaming ( _\u00d7_ to _\u2227_ )\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata Stream : D \u2192 Set where\n  consS : \u2200 x {xs} \u2192 \u221e (Stream xs) \u2192 Stream (x \u2237 xs)\n\nStream-gfp\u2082 : (P : D \u2192 Set) \u2192\n              -- P is post-fixed point of StreamF.\n              (\u2200 {xs} \u2192 P xs \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n              -- Stream is greater than P.\n              \u2200 {xs} \u2192 P xs \u2192 Stream xs\nStream-gfp\u2082 P h {xs} Pxs = subst Stream (sym xs\u2261x'\u2237xs') prf\n  where\n    x' : D\n    x' = proj\u2081 (h Pxs)\n\n    xs' : D\n    xs' = proj\u2081 (proj\u2082 (h Pxs))\n\n    Pxs' : P xs'\n    Pxs' = proj\u2081 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' = proj\u2082 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    prf : Stream (x' \u2237 xs')\n    prf = consS x' (\u266f (Stream-gfp\u2082 P h Pxs'))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC streams using the Agda coinductive stuff\n------------------------------------------------------------------------------\n\n-- Tested with the development version Agda and the standard library\n-- on 26 September 2011.\n\nmodule Draft.FOTC.Data.Stream.StreamSL where\n\nopen import Data.Product renaming ( _\u00d7_ to _\u2227_ )\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\npostulate\n  D : Set\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata Stream : D \u2192 Set where\n  consS : (x xs : D) \u2192 \u221e (Stream xs) \u2192 Stream (x \u2237 xs)\n\nStream-gfp\u2082 : (P : D \u2192 Set) \u2192\n              -- P is post-fixed point of StreamF.\n              (\u2200 {xs} \u2192 P xs \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n              -- Stream is greater than P.\n              \u2200 {xs} \u2192 P xs \u2192 Stream xs\nStream-gfp\u2082 P h {xs} Pxs = subst Stream (sym xs\u2261x'\u2237xs') prf\n  where\n    x' : D\n    x' = proj\u2081 (h Pxs)\n\n    xs' : D\n    xs' = proj\u2081 (proj\u2082 (h Pxs))\n\n    Pxs' : P xs'\n    Pxs' = proj\u2081 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' = proj\u2082 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    Sxs' : Stream xs'\n    Sxs' = Stream-gfp\u2082 P h Pxs'\n\n    prf : Stream (x' \u2237 xs')\n    prf = consS x' xs' (\u266f Sxs')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"216b6d5cbdfd5e77bcf40eac442f7e843109e3b4","subject":"Vec: Bij is now typed","message":"Vec: Bij is now typed\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 Set where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 {n} {A : Set a} (x : A) (f : Perm n) \u2192 f \u2219 replicate x \u2261 replicate x\n    \u2219-replicate x `id = refl\n    \u2219-replicate x `0\u21941 = refl\n    \u2219-replicate x (`tl f) rewrite \u2219-replicate x f = refl\n    \u2219-replicate x (f `\u204f g) rewrite \u2219-replicate x f | \u2219-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : \u2115 \u2192 Set (a L.\u2294 b) where\n      `id : \u2200 {n} \u2192 Bij n\n      `0\u21941 : \u2200 {n} \u2192 Bij (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij n\n      _`\u2237_ : \u2200 {n} \u2192 Bij\u1d2c \u2192 (A \u2192 Bij n) \u2192 Bij (1 + n)\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n               (f\u1d2e\u1d2c : B \u2192 A)\n               (f   : Bij\u1d2c \u2192 Bij\u1d2e)\n               {n} \u2192 Bij A Bij\u1d2c n \u2192 Bij B Bij\u1d2e n\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Bij n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    eval `id       = id\n    eval (f `\u204f g)   = eval g \u2218 eval f\n    eval `0\u21941      = 0\u21941\n    eval (f\u1d2c `\u2237 f) = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : \u2200 {n} \u2192 Bij n \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Set _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Bij n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Bij n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk (Bij n) eval _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : \u2200 {n} \u2192 Perm n \u2192 Bij n\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 {n} \u03c0 (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n      `tl : Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    eval `id             = id\n    eval (f `\u204f g)        = eval g \u2218 eval f\n    eval `0\u21941            = 0\u21941\n    eval (`tl f) {zero}  = id\n    eval (`tl f) {suc n} = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : Perm \u2192 Perm \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) [] = refl\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) [] = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl _)   [] = refl\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 n {A : Set a} (x : A) f \u2192 f \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl _) = refl\n    \u2219-replicate (suc n) x (`tl f) rewrite \u2219-replicate n x f = refl\n    \u2219-replicate n x (f `\u204f g) rewrite \u2219-replicate n x f | \u2219-replicate n x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map {n} f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate n f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : Set (a L.\u2294 b) where\n      `id `0\u21941 : Bij\n      _`\u204f_ : Bij \u2192 Bij \u2192 Bij\n      _`\u2237_ : Bij\u1d2c \u2192 (A \u2192 Bij) \u2192 Bij\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n             \u2192 (B \u2192 A)\n             \u2192 (Bij\u1d2c \u2192 Bij\u1d2e)\n             \u2192 Bij A Bij\u1d2c \u2192 Bij B Bij\u1d2e\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : Endo Bij\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    eval `id               = id\n    eval (f `\u204f g)          = eval g \u2218 eval f\n    eval `0\u21941              = 0\u21941\n    eval (f\u1d2c `\u2237 f) {zero}  = id\n    eval (f\u1d2c `\u2237 f) {suc n} = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : Bij \u2192 Bij \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk Bij (\u03bb f \u2192 eval f {n}) _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : Endo Bij\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : Perm \u2192 Bij\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 \u03c0 {n} (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) [] = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"61de78a178ea4a23092e726b277f92787528c970","subject":"Make some arguments of correctness explicit.","message":"Make some arguments of correctness explicit.\n\nOld-commit-hash: b8f4ff3d6b2f0ea72c79a94c10ae3ec1b545c2ca\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Specification\/Canon-Popl14.agda","new_file":"Denotation\/Specification\/Canon-Popl14.agda","new_contents":"module Denotation.Specification.Canon-Popl14 where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192 Change \u03c4\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to future arguments\nvalidity : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  valid {\u03c4} (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to free variables\ncorrectness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393)\n  \u2192 \u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1 \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Change \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 + \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) d\u03c1 = - \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 empty d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 ++ \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) d\u03c1 = \u27e6 x \u27e7\u0394Var d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) d\u03c1 =\n     (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n  (validity {\u03c3} t d\u03c1))\n\u27e6_\u27e7\u0394 (abs t) d\u03c1 = \u03bb v \u2192\n  \u27e6 t \u27e7\u0394 (v \u2022 d\u03c1)\n\nvalidVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : \u0394Env \u0393) \u2192 valid {\u03c4} (\u27e6 x \u27e7 (ignore d\u03c1)) (\u27e6 x \u27e7\u0394Var d\u03c1)\nvalidVar this (cons v \u0394v R[v,\u0394v] \u2022 _) = R[v,\u0394v]\nvalidVar {\u03c4} (that x) (cons _ _ _ \u2022 d\u03c1) = validVar {\u03c4} x d\u03c1\n\nvalidity (intlit n)    d\u03c1 = tt\nvalidity (add s t)     d\u03c1 = tt\nvalidity (minus t)     d\u03c1 = tt\nvalidity (empty)       d\u03c1 = tt\nvalidity (insert s t)  d\u03c1 = tt\nvalidity (union s t)   d\u03c1 = tt\nvalidity (negate t)    d\u03c1 = tt\nvalidity (flatmap s t) d\u03c1 = tt\nvalidity (sum t)       d\u03c1 = tt\n\nvalidity {\u03c4} (var x) d\u03c1 = validVar {\u03c4} x d\u03c1\n\nvalidity (app s t) d\u03c1 =\n  let\n    v = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394v = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    proj\u2081 (validity s d\u03c1 (cons v \u0394v (validity t d\u03c1)))\n\nvalidity {\u03c3 \u21d2 \u03c4} (abs t) d\u03c1 = \u03bb v \u2192\n  let\n    v\u2032 = after {\u03c3} v\n    \u0394v\u2032 = v\u2032 \u229f\u208d \u03c3 \u208e v\u2032\n    d\u03c1\u2081 = v \u2022 d\u03c1\n    d\u03c1\u2082 = nil-valid-change \u03c3 v\u2032 \u2022 d\u03c1\n  in\n    validity t d\u03c1\u2081\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2082) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2082\n    \u2261\u27e8 correctness t d\u03c1\u2082 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2081)\n    \u2261\u27e8 sym (correctness t d\u03c1\u2081) \u27e9\n      \u27e6 t \u27e7 (ignore d\u03c1\u2081) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar {x = this  } {cons v dv R[v,dv] \u2022 d\u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u2022 d\u03c1} = correctVar {x = y} {d\u03c1}\n\ncorrectness (intlit n) d\u03c1 = right-id-int n\ncorrectness (add s t) d\u03c1 = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _+_ (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (minus t) d\u03c1 = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong -_ (correctness t d\u03c1))\n\ncorrectness empty d\u03c1 = right-id-bag emptyBag\ncorrectness (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness s d\u03c1))\n         (correctness t d\u03c1) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness (union s t) d\u03c1 = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _++_ (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (negate t) d\u03c1 = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong negateBag (correctness t d\u03c1))\n\ncorrectness (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (sum t) d\u03c1 =\n  let\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness t d\u03c1))\n\ncorrectness {\u03c4} (var x) d\u03c1 = correctVar {\u03c4} {x = x}\ncorrectness (app {\u03c3} {\u03c4} s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 (ignore d\u03c1)\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e \u0394f (cons u \u0394u (validity {\u03c3} t d\u03c1))\n    \u2261\u27e8 sym (proj\u2082 (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons u \u0394u (validity {\u03c3} t d\u03c1)))) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t d\u03c1 \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) d\u03c1 = ext (\u03bb v \u2192\n  let\n    d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393)\n    d\u03c1\u2032 = nil-valid-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} t d\u03c1\u2032 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {d\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 d\u03c1)\n      (\u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7 (ignore d\u03c1))\n    \u229e\u208d \u03c4 \u208e\n    (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1))\n\ncorollary {\u03c3} {\u03c4} s t {d\u03c1} = proj\u2082\n  (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons (\u27e6 t \u27e7 (ignore d\u03c1))\n     (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1)))\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  (v : ValidChange \u03c3)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} v)\n  \u2261 \u27e6 t \u27e7 \u2205 (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 \u2205 v\n\ncorollary-closed {\u03c3} {\u03c4} {t = t} v =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205\n  in\n    begin\n      f (after {\u03c3} v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} v))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205)) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} v)\n    \u2261\u27e8 proj\u2082 (validity {\u03c3 \u21d2 \u03c4} t \u2205 v) \u27e9\n      f (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u0394f v\n    \u220e where open \u2261-Reasoning\n","old_contents":"module Denotation.Specification.Canon-Popl14 where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192 Change \u03c4\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to future arguments\nvalidity : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  valid {\u03c4} (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to free variables\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1 \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Change \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 + \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) d\u03c1 = - \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 empty d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 ++ \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) d\u03c1 = \u27e6 x \u27e7\u0394Var d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) d\u03c1 =\n     (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n  (validity {\u03c3} t d\u03c1))\n\u27e6_\u27e7\u0394 (abs t) d\u03c1 = \u03bb v \u2192\n  \u27e6 t \u27e7\u0394 (v \u2022 d\u03c1)\n\nvalidVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : \u0394Env \u0393) \u2192 valid {\u03c4} (\u27e6 x \u27e7 (ignore d\u03c1)) (\u27e6 x \u27e7\u0394Var d\u03c1)\nvalidVar this (cons v \u0394v R[v,\u0394v] \u2022 _) = R[v,\u0394v]\nvalidVar {\u03c4} (that x) (cons _ _ _ \u2022 d\u03c1) = validVar {\u03c4} x d\u03c1\n\nvalidity (intlit n)    d\u03c1 = tt\nvalidity (add s t)     d\u03c1 = tt\nvalidity (minus t)     d\u03c1 = tt\nvalidity (empty)       d\u03c1 = tt\nvalidity (insert s t)  d\u03c1 = tt\nvalidity (union s t)   d\u03c1 = tt\nvalidity (negate t)    d\u03c1 = tt\nvalidity (flatmap s t) d\u03c1 = tt\nvalidity (sum t)       d\u03c1 = tt\n\nvalidity {\u03c4} (var x) d\u03c1 = validVar {\u03c4} x d\u03c1\n\nvalidity (app s t) d\u03c1 =\n  let\n    v = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394v = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    proj\u2081 (validity s d\u03c1 (cons v \u0394v (validity t d\u03c1)))\n\nvalidity {\u03c3 \u21d2 \u03c4} (abs t) d\u03c1 = \u03bb v \u2192\n  let\n    v\u2032 = after {\u03c3} v\n    \u0394v\u2032 = v\u2032 \u229f\u208d \u03c3 \u208e v\u2032\n    d\u03c1\u2081 = v \u2022 d\u03c1\n    d\u03c1\u2082 = nil-valid-change \u03c3 v\u2032 \u2022 d\u03c1\n  in\n    validity t d\u03c1\u2081\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2082) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {d\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {d\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore d\u03c1\u2081) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar {x = this  } {cons v dv R[v,dv] \u2022 d\u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u2022 d\u03c1} = correctVar {x = y} {d\u03c1}\n\ncorrectness {t = intlit n} = right-id-int n\ncorrectness {t = add s t} {d\u03c1} = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _+_ (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = minus t} {d\u03c1} = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong -_ (correctness {t = t}))\n\ncorrectness {t = empty} = right-id-bag emptyBag\ncorrectness {t = insert s t} {d\u03c1} =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness {t = s}))\n         (correctness {t = t}) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness {t = union s t} {d\u03c1} = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _++_ (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = negate t} {d\u03c1} = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong negateBag (correctness {t = t}))\n\ncorrectness {t = flatmap s t} {d\u03c1} =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness {t = s}) (correctness {t = t}))\ncorrectness {t = sum t} {d\u03c1} =\n  let\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness {t = t}))\n\ncorrectness {\u03c4} {t = var x} = correctVar {\u03c4} {x = x}\ncorrectness {t = app {\u03c3} {\u03c4} s t} {d\u03c1} =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 (ignore d\u03c1)\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e \u0394f (cons u \u0394u (validity {\u03c3} t d\u03c1))\n    \u2261\u27e8 sym (proj\u2082 (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons u \u0394u (validity {\u03c3} t d\u03c1)))) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} {t = s} \u27e8$\u27e9 correctness {\u03c3} {t = t} \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} {abs t} {d\u03c1} = ext (\u03bb v \u2192\n  let\n    d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393)\n    d\u03c1\u2032 = nil-valid-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} {t = t} {d\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {d\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 d\u03c1)\n      (\u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7 (ignore d\u03c1))\n    \u229e\u208d \u03c4 \u208e\n    (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1))\n\ncorollary {\u03c3} {\u03c4} s t {d\u03c1} = proj\u2082\n  (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons (\u27e6 t \u27e7 (ignore d\u03c1))\n     (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1)))\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  (v : ValidChange \u03c3)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} v)\n  \u2261 \u27e6 t \u27e7 \u2205 (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 \u2205 v\n\ncorollary-closed {\u03c3} {\u03c4} {t = t} v =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205\n  in\n    begin\n      f (after {\u03c3} v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} v))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} {t = t})) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} v)\n    \u2261\u27e8 proj\u2082 (validity {\u03c3 \u21d2 \u03c4} t \u2205 v) \u27e9\n      f (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u0394f v\n    \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4a572a589c2922066967371376a6b95d1f26b5de","subject":"Step towards showing sort is a permutation","message":"Step towards showing sort is a permutation\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\nopen Dummy public\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule Sorting-Perm-Properties {OT : Set}\n  (_\u2293\u1d2c_ _\u2294\u1d2c_ : (xs ys : OT) \u2192 OT) where\n    \n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    import Data.Bits as B\n    open B.OperationSyntax renaming (_\u2219_ to _\u2022_) -- \n\n\n    infixr 1 _`\u2218_\n    _`\u2218_ = \u03bb x y \u2192 y `\u204f x\n\n    eval : {A : Set} \u2192 Op \u2192 \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    eval `id t        = t\n    eval `0\u21941 t       = {!!}\n    eval `not t       = {!!}\n    eval (`tl op) t   = {!!}\n    eval (`if0 op) t  = {!!}\n    eval (op `\u204f op\u2081) t = eval op\u2081 (eval op t)\n\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Op\n        proof : (t : Tree A n) \u2192 t \u2261  eval (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {A}{n}{f}{g} (mk \u03c0f pf) (mk \u03c0g pg) = mk (\u03bb t \u2192 \u03c0g t `\u2218 \u03c0f (g t)) \n      (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (eval (\u03c0g t)) (pf (g t))) )\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = {!!}\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof = {!!}\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof = {!!}\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = mk {!!} (\u03bb { (fork (leaf x) (leaf y))  \u2192 {!!} })\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    -- open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\n    {-\nmodule M {n} where\n  postulate\n    _\u2264_ : {-\u2200 {n} \u2192-} Bits n \u2192 Bits n \u2192 Set\n    _\u2294_ : Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : Bits n \u2192 Bits n \u2192 Bits n\n    isPreorder : IsPreorder _\u2261_ _\u2264_\n    \u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n    \u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n    \u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n    \u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n    \u2293-comm : Commutative _\u2261_ _\u2293_\n    \u2294-comm : Commutative _\u2261_ _\u2294_\n    \u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n    \u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n    \u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n    \u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n    \u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n    \u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\nmodule BitsSorting {m} where\n    open M {m}\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2294-spec \u2293-spec \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open SortedData _\u2264_\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\nopen Dummy public\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    -- open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\n    {-\nmodule M {n} where\n  postulate\n    _\u2264_ : {-\u2200 {n} \u2192-} Bits n \u2192 Bits n \u2192 Set\n    _\u2294_ : Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : Bits n \u2192 Bits n \u2192 Bits n\n    isPreorder : IsPreorder _\u2261_ _\u2264_\n    \u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n    \u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n    \u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n    \u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n    \u2293-comm : Commutative _\u2261_ _\u2293_\n    \u2294-comm : Commutative _\u2261_ _\u2294_\n    \u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n    \u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n    \u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n    \u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n    \u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n    \u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\nmodule BitsSorting {m} where\n    open M {m}\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2294-spec \u2293-spec \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open SortedData _\u2264_\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a5785f204960f15cb961e70848b6a8cb07de092e","subject":"Typo.","message":"Typo.\n\nIgnore-this: 819d9faf183e259333bdc5680661df3b\n\ndarcs-hash:20110630160032-3bd4e-3a623d5fd2001f7323dfe7850e28a80f2a733554.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FixedPoints\/Predicates.agda","new_file":"Draft\/FixedPoints\/Predicates.agda","new_contents":"module Draft.FixedPoints.Predicates where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\n-- The FOTC types without use data, i.e. using Agda as a logical framework.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N  : D \u2192 Set\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\n  -- Example.\n  one : D\n  one = succ zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    nilL  : List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    nilLN  : ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\n  -- Example.\n  one : D\n  one = succ zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    nilL  :                      List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    nilLN  :                             ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined predicates.\n    --\n    -- N.B. We cannot write LFP in first order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero     = \u22a4\n  -- NatF X (succ n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  zN : N zero\n  zN = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n  sN {n} Nn = LFP\u2082 NatF (succ n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  one : D\n  one = succ zero\n\n  oneN : N one\n  oneN = sN zN\n\n------------------------------------------------------------------------------\n\n-- The FOTC list type as the least fixed-point of functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  nilL : List []\n  nilL = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  consL x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n------------------------------------------------------------------------------\n\n-- The FOTC list of natural numbers type as the least fixed-point of functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  nilLN : ListN []\n  nilLN = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  consLN {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n","old_contents":"module Draft.FixedPoints.Predicates where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\n-- The FOTC types without use data, i.e. using Agda as a logical framework.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N  : D \u2192 Set\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\n  -- Example.\n  one : D\n  one = succ zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    nilL  : List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    nilLN  : ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    zN : N zero\n    sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\n  -- Example.\n  one : D\n  one = succ zero\n\n  oneN : N one\n  oneN = sN zN\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    nilL  :                      List []\n    consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    nilLN  :                             ListN []\n    consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n\n------------------------------------------------------------------------------\n\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined predicates.\n    --\n    -- N.B. We cannot write LFP in first order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set.\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 (d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero     = \u22a4\n  -- NatF X (succ n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  zN : N zero\n  zN = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n  sN {n} Nn = LFP\u2082 NatF (succ n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  one : D\n  one = succ zero\n\n  oneN : N one\n  oneN = sN zN\n\n------------------------------------------------------------------------------\n\n-- The FOTC list type as the least fixed-point of functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  nilL : List []\n  nilL = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  consL : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  consL x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = consL zero (consL true nilL)\n\n------------------------------------------------------------------------------\n\n-- The FOTC list of natural numbers type as the least fixed-point of functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  nilLN : ListN []\n  nilLN = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  consLN : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  consLN {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 one \u2237 [])\n  ln = consLN zN (consLN oneN nilLN)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"80463c85413df224feab7c464101dbfd5ff7ec77","subject":"Atlas: improve specification of union","message":"Atlas: improve specification of union\n\nOld-commit-hash: cda40f70859c26a48a9077ad39b93d3b9a0af045\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n--\n-- TODO: write this and call it Syntax.Term.Plotkin.lift-term\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\n\nzip! f m\u2081 m\u2082 = app (app (app (const zip) f) m\u2081) m\u2082\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\n\nlookup! = {!!} -- TODO: add constant `lookup`\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff s t = app (app (lift-diff Atlas-diff Atlas-apply) s) t\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply s t = app (app (lift-apply Atlas-diff Atlas-apply) s) t\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\n\nupdate! k v m = app (app (app (const update) k) v) m\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- union s t should be:\n-- 1. equal to t if s is neutral\n-- 2. equal to s if t is neutral\n-- 3. equal to s if s == t\n-- 4. whatever it wants to be otherwise\n--\n-- On Booleans, only conjunction can be union.\n--\n-- TODO (later): support conjunction, probably by Boolean\n-- elimination form if-then-else\n\npostulate\n  and! : \u2200 {\u0393} \u2192\n    Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n    Atlas-term \u0393 (base Bool)\n\nunion : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base \u03b9)\nunion {Bool} s t = and! s t\nunion {Map \u03ba \u03b9} s t =\n  let\n    union-term = abs (abs (union (var (that this)) (var this)))\n  in\n    zip! (abs union-term) s t\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    v\u2081 = var (that this)\n    v\u2082 = var this\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    k\u2081\u2082 = var (that (that this))\n    k\u2083\u2084 = var (that (that this))\n    f\u2081\u2082 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2081\u2082) v\u2081) v\u2082)\n      (lookup! k\u2081\u2082 (weaken\u2083 m\u2083))) (lookup! k\u2081\u2082 (weaken\u2083 m\u2084)))))\n    f\u2083\u2084 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2083\u2084)\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2081)))\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2082))) v\u2083) v\u2084)))\n  in\n    -- A correct but inefficient implementation.\n    -- May want to speed it up after constants are finalized.\n    union (zip! f\u2081\u2082 m\u2081 m\u2082) (zip! f\u2083\u2084 m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app (app (weaken\u2081 \u0394f) (var this)) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n--\n-- TODO: write this and call it Syntax.Term.Plotkin.lift-term\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\n\nzip! f m\u2081 m\u2082 = app (app (app (const zip) f) m\u2081) m\u2082\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\n\nlookup! = {!!} -- TODO: add constant `lookup`\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff s t = app (app (lift-diff Atlas-diff Atlas-apply) s) t\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply s t = app (app (lift-apply Atlas-diff Atlas-apply) s) t\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\n\nupdate! k v m = app (app (app (const update) k) v) m\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- The binary operator such that\n-- union t t \u2261 t\n\n-- To implement union, we need for each non-map base-type\n-- an operator such that `op v v = v` on all values `v`.\n-- for Booleans, conjunction is good enough.\n--\n-- TODO (later): support conjunction, probably by Boolean\n-- elimination form if-then-else\n\npostulate\n  and! : \u2200 {\u0393} \u2192\n    Atlas-term \u0393 (base Bool) \u2192 Atlas-term \u0393 (base Bool) \u2192\n    Atlas-term \u0393 (base Bool)\n\nunion : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base \u03b9)\nunion {Bool} s t = and! s t\nunion {Map \u03ba \u03b9} s t =\n  let\n    union-term = abs (abs (union (var (that this)) (var this)))\n  in\n    zip! (abs union-term) s t\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    v\u2081 = var (that this)\n    v\u2082 = var this\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    k\u2081\u2082 = var (that (that this))\n    k\u2083\u2084 = var (that (that this))\n    f\u2081\u2082 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2081\u2082) v\u2081) v\u2082)\n      (lookup! k\u2081\u2082 (weaken\u2083 m\u2083))) (lookup! k\u2081\u2082 (weaken\u2083 m\u2084)))))\n    f\u2083\u2084 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2083\u2084)\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2081)))\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2082))) v\u2083) v\u2084)))\n  in\n    -- A correct but inefficient implementation.\n    -- May want to speed it up after constants are finalized.\n    union (zip! f\u2081\u2082 m\u2081 m\u2082) (zip! f\u2083\u2084 m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app (app (weaken\u2081 \u0394f) (var this)) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2c54443576bbae481639179d8b3bc48566274aad","subject":"Implement FV more parametrically.","message":"Implement FV more parametrically.\n\nOld-commit-hash: 470f2fd7a3e73bb1ac01fce483f8aa80e4ec51e3\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/FreeVars\/Popl14.agda","new_file":"Syntax\/FreeVars\/Popl14.agda","new_contents":"module Syntax.FreeVars.Popl14 where\n\n-- Free variables of Calculus Popl14\n--\n-- Contents\n-- FV: get the free variables of a term\n-- closed?: test if a term is closed, producing a witness if yes.\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Change.Term\nopen import Base.Syntax.Vars Type public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Sum\n\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\nFV (const \u03b9 ts) = FV-terms ts\nFV (var x) = singleton x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV s \u222a FV t\n\nFV-terms \u2205 = none\nFV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\nclosed? t = empty? (FV t)\n","old_contents":"module Syntax.FreeVars.Popl14 where\n\n-- Free variables of Calculus Popl14\n--\n-- Contents\n-- FV: get the free variables of a term\n-- closed?: test if a term is closed, producing a witness if yes.\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Change.Term\nopen import Base.Syntax.Vars Type public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Sum\n\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV {\u0393 = \u0393} (intlit n) = none\nFV (add s t) = FV s \u222a FV t\nFV (minus t) = FV t\n\nFV {\u0393 = \u0393} empty = none\nFV (insert s t) = FV s \u222a FV t\nFV (union s t) = FV s \u222a FV t\nFV (negate t) = FV t\n\nFV (flatmap s t) = FV s \u222a FV t\nFV (sum t) = FV t\n\nFV (var x) = singleton x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV s \u222a FV t\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\nclosed? t = empty? (FV t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9de1f0244d90f47104d4e71fa150623325bbf77e","subject":"Changed a fixity.","message":"Changed a fixity.\n\nIgnore-this: f005d3be4a2e6600badbfc6a902043f8\n\ndarcs-hash:20110527133036-3bd4e-1d633166d6ce0ca3d8cfd249e39dcf9300bb30d0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Data\/Product.agda","new_file":"src\/Common\/Data\/Product.agda","new_contents":"------------------------------------------------------------------------------\n-- Products\n------------------------------------------------------------------------------\n\nmodule Common.Data.Product where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (witness : D) \u2192 P witness \u2192 \u2203 P\n\n\u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : {P : D \u2192 Set}(\u2203p : \u2203 P) \u2192 P (\u2203-proj\u2081 \u2203p)\n\u2203-proj\u2082 (_ , px) = px\n","old_contents":"------------------------------------------------------------------------------\n-- Products\n------------------------------------------------------------------------------\n\nmodule Common.Data.Product where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfixr 6 _,_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : {A B : Set} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , y) = x\n\n\u2227-proj\u2082 : {A B : Set} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (x , y) = y\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (witness : D) \u2192 P witness \u2192 \u2203 P\n\n\u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : {P : D \u2192 Set}(\u2203p : \u2203 P) \u2192 P (\u2203-proj\u2081 \u2203p)\n\u2203-proj\u2082 (_ , px) = px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5314beec206b3cd4ec31c4d080f0d139b91763c5","subject":"prg: extract stuff to nplib","message":"prg: extract stuff to nplib\n","repos":"crypto-agda\/crypto-agda","old_file":"prg.agda","new_file":"prg.agda","new_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search {k} _\u2228_ ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search {k} _\u2227_ (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = search-comm _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n","old_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search {k} _\u2228_ ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search {k} _\u2227_ (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n\n  always : \u2200 n \u2192 Bits n \u2192 Bit\n  always _ _ = 1b\n  never  : \u2200 n \u2192 Bits n \u2192 Bit\n  never _ _ = 0b\n\n  search-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                   (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n  search-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n    where\n      go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n      go zero = refl\n      go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n  #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n  #never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n  #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n  #always\u22612^_ zero = refl\n  #always\u22612^_ (suc n) = cong\u2082 _+_ pf pf where pf = #always\u22612^ n\n\n  ==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n  ==-comm [] [] = refl\n  ==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n  count\u1d47 : Bit \u2192 \u2115\n  count\u1d47 b = if b then 1 else 0\n\n  #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n  #\u27e8== [] \u27e9 = refl\n  #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n  #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n  \u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n  \u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n  \u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                          (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n  \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n  \u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n  #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n  #-+ f f0 f1 rewrite f0 | f1 = refl\n\n  take-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , refl = refl\n  \n  drop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , refl = refl\n  \n  #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                  \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n  #-take-drop zero n f g with f []\n  ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n  ... | false = #never\u22610 n\n  #-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                    (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                  ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                  (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                         (#-take-drop m n (f \u2218 0\u2237_) g))\n                                    (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                  ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                  (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                         (#-take-drop m n (f \u2218 1\u2237_) g)))\n                             (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n  #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                  \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n  #-drop-take m n f g =\n             #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n           \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n             #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n           \u2261\u27e8 #-take-drop m n g f \u27e9\n             #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n           \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n             #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n           \u220e\n    where open \u2261-Reasoning\n\n  #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n  #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n               \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                 #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n               \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                 2^ n * #\u27e8 f \u27e9\n               \u220e\n    where open \u2261-Reasoning\n\n  #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n  #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n               \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                 #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n               \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                 2^ n * #\u27e8 f \u27e9\n               \u220e\n    where open \u2261-Reasoning\n\n  #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n  #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n  #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n  #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n  ... | true  | true  | _ = s\u2264s z\u2264n\n  ... | true  | false | p = \u22a5-elim (p _)\n  ... | false | _     | _ = z\u2264n\n  #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                  +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n  #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n  #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n  #-\u2227-\u2228\u1d47 false y = refl\n\n  #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n  #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n  #-\u2227-\u2228 {suc n} f g =\n    trans\n      (trans\n         (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                 #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                 #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                 #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n         (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                    (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n      where open SemiringSolver\n            helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n            helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n  #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n  #\u2228' {zero} f g with f []\n  ... | true  = s\u2264s z\u2264n\n  ... | false = \u2115\u2264.refl\n  #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                               #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                      (\u2115\u2264.reflexive\n                        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n      where open SemiringSolver\n            helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n            helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n  #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n  #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n  \u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u200c\u2228 y)\n  \u2227\u21d2\u2228 true y = _\n  \u2227\u21d2\u2228 false y = \u03bb ()\n\n  #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n  #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n  lem' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n  lem' f g x with f x\n  ... | true = refl\n  ... | false = sym (not-involutive _)\n\n  lem : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\n  lem op f g = \u2257-cong-search op {-(f |\u2228| g) (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))-} (lem' f g) \n\n  lem'' :\n    \u2200 {n a b}\n      {A : Set a} {B : Set b}\n      (_+_ : A \u2192 A \u2192 A)\n      (_*_ : B \u2192 B \u2192 B)\n      (f : A \u2192 B)\n      (p : Bits n \u2192 A)\n      (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\n  lem'' {zero} _+_ _*_ f p hom = refl\n  lem'' {suc n} _+_ _*_ f p hom = trans (hom _ _) (cong\u2082 _*_ (lem'' {n} _+_ _*_ f (p \u2218 0\u2237_) hom) (lem'' _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n  [0\u2194_] : \u2200 {n} \u2192 Fin n \u2192 Bits n \u2192 Bits n\n  [0\u2194_] {zero}  i xs = xs\n  [0\u2194_] {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n  [0\u21941] : Bits 2 \u2192 Bits 2\n  [0\u21941] = [0\u2194 suc zero ]\n\n  [0\u21941]-spec : [0\u21941] \u2257 (\u03bb { (x \u2237 y \u2237 []) \u2192 y \u2237 x \u2237 [] })\n  [0\u21941]-spec (x \u2237 y \u2237 []) = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  de-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\n  de-morgan true  _ = refl\n  de-morgan false _ = refl\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = lem'' _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6fb4f53de4b0623f84666134743b31aa442e024e","subject":"making the offending thing in contexts a postulate. need to undo this later","message":"making the offending thing in contexts a postulate. need to undo this later\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"contexts.agda","new_file":"contexts.agda","new_contents":"open import Prelude\nopen import List\nopen import Nat\n\nmodule contexts where\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  _ctx : Set \u2192 Set\n  A ctx = Nat \u2192 Maybe A\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : {A : Set} \u2192 A ctx\n  \u2205 _ = None\n\n  infixr 100 \u25a0_\n\n  -- membership, or presence, in a context\n  _\u2208_ : {A : Set} (p : Nat \u00d7 A) \u2192 (\u0393 : A ctx) \u2192 Set\n  (x , y) \u2208 \u0393 = (\u0393 x) == Some y\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : {A : Set} (n : Nat) \u2192 (\u0393 : A ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- disjointness for contexts\n  _##_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 Set\n  _##_ {A} \u0393 \u0393'  = ((n : Nat) (a : A) \u2192 \u0393 n == Some a \u2192 \u0393' n == None) \u00d7\n                   ((n : Nat) (a : A) \u2192 \u0393' n == Some a \u2192 \u0393 n == None)\n\n  -- without: remove a variable from a context\n  _\/_ : {A : Set} \u2192 A ctx \u2192 Nat \u2192 A ctx\n  (\u0393 \/ x) y with natEQ x y\n  (\u0393 \/ x) .x | Inl refl = None\n  (\u0393 \/ x) y  | Inr neq  = \u0393 y\n\n  -- a variable is apart from any context from which it is removed\n  aar : {A : Set} \u2192 (\u0393 : A ctx) (x : Nat) \u2192 x # (\u0393 \/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {A : Set} \u2192 {\u0393 : A ctx} {n : Nat} {t t' : A} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- warning: this is union but it assumes WITHOUT CHECKING that the\n  -- domains are disjoint.\n  _\u222a_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 A ctx\n  (C1 \u222a C2) x with C1 x\n  (C1 \u222a C2) x | Some x\u2081 = Some x\u2081\n  (C1 \u222a C2) x | None = C2 x\n\n  -- the singleton context\n  \u25a0_ : {A : Set} \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u25a0 (x , a)) y with natEQ x y\n  (\u25a0 (x , a)) .x | Inl refl = Some a\n  ... | Inr _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : {A : Set} \u2192 A ctx \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u0393 ,, (x , t)) = \u0393 \u222a (\u25a0 (x , t))\n\n  infixl 10 _,,_\n\n  -- x\u2208sing \u0393 n x with \u0393 n\n  -- x\u2208sing \u0393 n x  | Some y with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inl refl = {!!}\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inr x = abort (x refl)\n  -- x\u2208sing \u0393 n x  | None with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | None | Inl refl = refl\n  -- x\u2208sing \u0393 n x\u2081 | None | Inr x = abort (x refl)\n\n  x\u2208\u222a1 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  x\u2208\u222a1 \u0393 \u0393' n x xin with \u0393 n\n  x\u2208\u222a1 \u0393 \u0393' n x\u2081 xin | Some x = xin\n  x\u2208\u222a1 \u0393 \u0393' n x ()   | None\n\n  -- x\u2208\u222a2 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  -- x\u2208\u222a2 \u0393 \u0393' n x xin with \u0393' n | \u0393 n\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2082 xin | Some x | Some x\u2081 = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x refl | Some .x | None = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2081 xin | None   | _ = abort (somenotnone (! xin))\n\n  x\u2208\u25a0 : {A : Set} (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u25a0 (n , a))\n  x\u2208\u25a0 n a with natEQ n n\n  x\u2208\u25a0 n a | Inl refl = refl\n  x\u2208\u25a0 n a | Inr x = abort (x refl)\n\n  -- x\u2208\u222a\u25a0 : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 \u222a (\u25a0 (n , a)))\n  -- x\u2208\u222a\u25a0 \u0393 n a with natEQ n n\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inl refl = {!!} -- it might be in \u0393 because i don't know that they're disjoint. this is where that premise gets you\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inr x = {!!}\n\n\n  postulate -- TODO\n    x\u2208sing : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\n  -- x\u2208sing \u0393 n a  = {!!} -- x\u2208\u222a2 \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a)\n","old_contents":"open import Prelude\nopen import List\nopen import Nat\n\nmodule contexts where\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  _ctx : Set \u2192 Set\n  A ctx = Nat \u2192 Maybe A\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : {A : Set} \u2192 A ctx\n  \u2205 _ = None\n\n  infixr 100 \u25a0_\n\n  -- membership, or presence, in a context\n  _\u2208_ : {A : Set} (p : Nat \u00d7 A) \u2192 (\u0393 : A ctx) \u2192 Set\n  (x , y) \u2208 \u0393 = (\u0393 x) == Some y\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : {A : Set} (n : Nat) \u2192 (\u0393 : A ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- disjointness for contexts\n  _##_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 Set\n  _##_ {A} \u0393 \u0393'  = ((n : Nat) (a : A) \u2192 \u0393 n == Some a \u2192 \u0393' n == None) \u00d7\n                   ((n : Nat) (a : A) \u2192 \u0393' n == Some a \u2192 \u0393 n == None)\n\n  -- without: remove a variable from a context\n  _\/_ : {A : Set} \u2192 A ctx \u2192 Nat \u2192 A ctx\n  (\u0393 \/ x) y with natEQ x y\n  (\u0393 \/ x) .x | Inl refl = None\n  (\u0393 \/ x) y  | Inr neq  = \u0393 y\n\n  -- a variable is apart from any context from which it is removed\n  aar : {A : Set} \u2192 (\u0393 : A ctx) (x : Nat) \u2192 x # (\u0393 \/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {A : Set} \u2192 {\u0393 : A ctx} {n : Nat} {t t' : A} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- warning: this is union but it assumes WITHOUT CHECKING that the\n  -- domains are disjoint.\n  _\u222a_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 A ctx\n  (C1 \u222a C2) x with C1 x\n  (C1 \u222a C2) x | Some x\u2081 = Some x\u2081\n  (C1 \u222a C2) x | None = C2 x\n\n  -- the singleton context\n  \u25a0_ : {A : Set} \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u25a0 (x , a)) y with natEQ x y\n  (\u25a0 (x , a)) .x | Inl refl = Some a\n  ... | Inr _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : {A : Set} \u2192 A ctx \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u0393 ,, (x , t)) = \u0393 \u222a (\u25a0 (x , t))\n\n  infixl 10 _,,_\n\n  -- x\u2208sing \u0393 n x with \u0393 n\n  -- x\u2208sing \u0393 n x  | Some y with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inl refl = {!!}\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inr x = abort (x refl)\n  -- x\u2208sing \u0393 n x  | None with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | None | Inl refl = refl\n  -- x\u2208sing \u0393 n x\u2081 | None | Inr x = abort (x refl)\n\n  x\u2208\u222a1 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  x\u2208\u222a1 \u0393 \u0393' n x xin with \u0393 n\n  x\u2208\u222a1 \u0393 \u0393' n x\u2081 xin | Some x = xin\n  x\u2208\u222a1 \u0393 \u0393' n x ()   | None\n\n  -- x\u2208\u222a2 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  -- x\u2208\u222a2 \u0393 \u0393' n x xin with \u0393' n | \u0393 n\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2082 xin | Some x | Some x\u2081 = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x refl | Some .x | None = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2081 xin | None   | _ = abort (somenotnone (! xin))\n\n  x\u2208\u25a0 : {A : Set} (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u25a0 (n , a))\n  x\u2208\u25a0 n a with natEQ n n\n  x\u2208\u25a0 n a | Inl refl = refl\n  x\u2208\u25a0 n a | Inr x = abort (x refl)\n\n  x\u2208\u222a\u25a0 : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 \u222a (\u25a0 (n , a)))\n  x\u2208\u222a\u25a0 \u0393 n a with natEQ n n\n  x\u2208\u222a\u25a0 \u0393 n a | Inl refl = {!!} -- it might be in \u0393 because i don't know that they're disjoint. this is where that premise gets you\n  x\u2208\u222a\u25a0 \u0393 n a | Inr x = {!!}\n\n\n  x\u2208sing : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\n  x\u2208sing \u0393 n a  = {!!} -- x\u2208\u222a2 \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"78f135597c86ef6a04edcf684ee90bfa3d9c9312","subject":"otp-sem-sec: extract PrgDist out","message":"otp-sem-sec: extract PrgDist out\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess' = Guess prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9ee719e758610cf053a5da9fc2b4074cba343c5e","subject":"flipbased: many renamings","message":"flipbased: many renamings\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail) renaming (map to vmap)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\n-- If you are not allowed to toss any coin, then you are deterministic.\nDet : \u2200 {a} \u2192 Set a \u2192 Set a\nDet = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet (mk f) = f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\nreturn\u1d30 = mk \u2218 const\n\npure\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\npure\u1d30 = return\u1d30\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 \u21ba m A \u2192 \u21ba n A\ncoerce \u2261.refl = id\n\n-- Weakened version of toss\ntoss\u1d42 : \u2200 {n} \u2192 \u21ba (1 + n) Bit\ntoss\u1d42 = weaken\u2032 toss\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn = weaken\u2032 \u2218 return\u1d30\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 \u21ba.run (f (\u21ba.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 (A \u2192 \u21ba n\u2082 B) \u2192 \u21ba (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 A \u2192 \u21ba n\u2082 B \u2192 \u21ba (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap f x = mk (f \u2218 \u21ba.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       \u21ba n\u2081 (A \u2192 B) \u2192 \u21ba n\u2082 A \u2192 \u21ba (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        \u21ba m A \u2192 (A \u2192 B \u2192 C) \u2192 \u21ba n B \u2192 \u21ba (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 \u21ba n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n\u27ea_\u27eb\u1d30 = pure\u1d30\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba o C \u2192 \u21ba (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 \u21ba n A \u2192 \u21ba n A \u2192 \u21ba (suc n) A\nchoose x y = toss >>= \u03bb b \u2192 if b then x else y\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 \u21ba m A \u2192 \u21ba n B \u2192 \u21ba (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 \u21ba n\u2081 Bit \u2192 \u21ba n\u2082 Bit \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8==\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 \u21ba n\u2081 (Bits m) \u2192 \u21ba n\u2082 (Bits m) \u2192 \u21ba (n\u2081 + n\u2082) Bit\nx \u27e8==\u27e9 y = \u27ea _==_ \u00b7 x \u00b7 y \u27eb\n\n\u21baT : \u2200 {k} \u2192 \u21ba k Bit \u2192 \u21ba k Set\n\u21baT p = \u27ea T \u00b7 p \u27eb\n\nreplicate\u21ba : \u2200 {n m} {a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (n * m) (Vec A n)\nreplicate\u21ba {zero}  _ = \u27ea [] \u27eb\nreplicate\u21ba {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicate\u21ba x \u27eb\n\nrandom : \u2200 {n} \u2192 \u21ba n (Bits n)\n-- random = coerce ? (replicate\u21ba toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicate\u21ba random\n\nrandomFun : \u2200 m n \u2192 \u21ba (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 \u21ba k (Bits n \u2192 A) \u2192 \u21ba (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 \u21ba (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (\u21ba n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = \u21ba n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = \u21ba n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 \u21ba n\u2081 A \u2192 \u21ba n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u1d30 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 \u21ba k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 \u21ba n B} \u2192 return\u1d30 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : \u21ba n A} \u2192 x >>=\u2032 return\u1d30 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : \u21ba n\u2081 A} {f : A \u2192 \u21ba n\u2082 B} {g : B \u2192 \u21ba n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss \u27e8xor\u27e9 \u27ea x \u27eb\u1d30 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss \u27e8,\u27e9 toss\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 M n A \u2192 M n A \u2192 M (suc n) A\nchoose x y = toss\u2032 >>= \u03bb b \u2192 if b then x else y\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 M k (Bits n \u2192 A) \u2192 M (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 M (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"04d35c79d07fd2ed884e0e0cd8ce4b2469247f0f","subject":"circuits: More specs to be","message":"circuits: More specs to be\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n-}\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6d2ae9598b28ead9859dad39062be73f75c967ab","subject":"IDesc: enum (hard-coded)","message":"IDesc: enum (hard-coded)","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"01f65d8c41d4542bc93acf116c3a4a32c9128352","subject":"Agda: Remove agda\/ModalLogic.agda","message":"Agda: Remove agda\/ModalLogic.agda\n\nThis file doesn't belong to the main branch, and in fact it arguably doesn't\nbelong in this repo at all - it just uses the Agda metatheory library started\nfrom Tillmann.\n\nOld-commit-hash: 31da94b0816ff85ac9bddb0ebcd37be4291760b8\n","repos":"inc-lc\/ilc-agda","old_file":"ModalLogic.agda","new_file":"ModalLogic.agda","new_contents":"","old_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotational.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntactic.Contexts Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n--\n-- Attempt 1 at weakening, subcontext-based. This works, but it is not what we\n-- need later (because of the way substitution is stated in the paper).\n--\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be\n-- fixed and lift introduced by hand.\n\nweaken : \u2200 {\u0393 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393 \u0394\u2081 \u03c4 \u2192 Term \u0393 \u0394\u2082 \u03c4\nweaken \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (app t t\u2081) = app (weaken \u0394\u2032 t) (weaken \u0394\u2032 t\u2081)\nweaken \u0394\u2032 (ovar x) = ovar x\nweaken \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0394\u2032 (box t) = box (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0394\u2032 t in-\n          weaken (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n\nopen import Relation.Binary.PropositionalEquality\n\n-- These two lemmas could be merged.\n\nweaken-\u227c : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u0394 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u0394 \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u0394 \u03c4\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} {\u0393\u2083} (abs {\u03c4\u2081} t) =  abs (weaken-\u227c {\u03c4\u2081 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (app t t\u2081) = app (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t) (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (ovar x) = ovar (Prefixes.lift {\u0393\u2081} {\u0393\u2082} x)\nweaken-\u227c (mvar u) = mvar u\nweaken-\u227c (box t) = box t\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c {\u0393\u2081} {\u0393\u2082} t in- weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081\n\n\nweaken-\u227c-\u0394 : \u2200 {\u0394\u2081 \u0394\u2082 \u0394\u2083 \u0393 \u03c4} \u2192 Term \u0393 (\u0394\u2081 \u22ce \u0394\u2083) \u03c4 \u2192 Term \u0393 (\u0394\u2081 \u22ce \u0394\u2082 \u22ce \u0394\u2083) \u03c4\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (abs t) = abs (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (app t t\u2081) = app (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t) (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t\u2081)\nweaken-\u227c-\u0394 (ovar x) = ovar x\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (mvar u) = mvar (Prefixes.lift {\u0394\u2081} {\u0394\u2082} u)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (box t) = box (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t in- weaken-\u227c-\u0394 {_ \u2022 \u0394\u2081} {\u0394\u2082} t\u2081 \n\nsubstVar : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (v : Var (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstVar {\u2205} t  this = t\nsubstVar {\u2205} t (that v) = ovar v\nsubstVar {\u03c4 \u2022 \u0393} t  this = ovar this\nsubstVar {\u03c4 \u2022 \u0393} t (that v) = weaken-\u227c {\u2205} {\u03c4 \u2022 \u2205} (substVar {\u0393} t v)\n\nsubstTerm : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (t\u2082 : Term (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u0394 \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (abs {\u03c4\u2082} t\u2082) = abs ( substTerm {\u03c4\u2082 \u2022 \u0393} {\u0393\u2032} t\u2081 t\u2082 )\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (app t\u2082 t\u2083) =  app (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082) (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2083) \nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (ovar v) =  substVar {\u0393} {\u0393\u2032} t\u2081 v\nsubstTerm t\u2081 (mvar u) = mvar u\nsubstTerm t\u2081 (box t\u2082) = box t\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (let-box_in-_ {\u03c4\u2083} t\u2082 t\u2083) =\n  let-box substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082 in-\n    substTerm {\u0393} {\u0393\u2032}\n      (weaken-\u227c-\u0394 {\u2205} {\u03c4\u2083 \u2022 \u2205} t\u2081)\n      t\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"173b46708ae3e42dbbde8642f54fd5a1d04760c2","subject":"Monad: more universe-polymorphic","message":"Monad: more universe-polymorphic\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Monad.agda","new_file":"lib\/Explore\/Monad.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\n\nmodule Explore.Monad \u2113 where\n\nM = Explore \u2113\n\nmodule _ {A : Set} where\n    return : A \u2192 M A\n    return x _\u2219_ f = f x\n\n    return-ind : \u2200 {p} x \u2192 ExploreInd p (return x)\n    return-ind x P _P\u2219_ Pf = Pf x\n\nmodule _ {A B : Set} where\n\n    infixl 1 _>>=_ _>>=-ind_\n    _>>=_ : M A \u2192 (A \u2192 M B) \u2192 M B\n    (exp\u1d2c >>= exp\u1d2e) _\u2219_ f = exp\u1d2c _\u2219_ \u03bb x \u2192 exp\u1d2e x _\u2219_ f\n\n    _>>=-ind_ : \u2200 {p}\n                  {exp\u1d2c : M A} \u2192 ExploreInd p exp\u1d2c \u2192\n                  {exp\u1d2e : A \u2192 M B} \u2192 (\u2200 x \u2192 ExploreInd p (exp\u1d2e x))\n               \u2192 ExploreInd p (exp\u1d2c >>= exp\u1d2e)\n    _>>=-ind_ {exp\u1d2c} P\u1d2c {exp\u1d2e} P\u1d2e P _P\u2219_ Pf\n      = P\u1d2c (\u03bb e \u2192 P (e >>= exp\u1d2e)) _P\u2219_ \u03bb x \u2192 P\u1d2e x P _P\u2219_ Pf\n\n    map : (A \u2192 B) \u2192 M A \u2192 M B\n    map f e = e >>= return \u2218 f\n\n    map-ind : \u2200 {p} (f : A \u2192 B) {exp\u1d2c : M A} \u2192 ExploreInd p exp\u1d2c \u2192 ExploreInd p (map f exp\u1d2c)\n    map-ind f P\u1d2c = P\u1d2c >>=-ind return-ind \u2218 f\n\n    private\n        {- This law is for free: map f e = e >>= return \u2218 f -}\n        map' : (A \u2192 B) \u2192 M A \u2192 M B\n        map' f exp _\u2219_ g = exp _\u2219_ (g \u2218 f)\n\n        map'\u2261map : map' \u2261 map\n        map'\u2261map = refl\n\n-- universe issues...\n-- join : M (M A) \u2192 M A\n\nmodule _ {A B C : Set} (m : M A) (f : A \u2192 M B) (g : B \u2192 M C) where\n    infix 0 _\u2261e_\n    _\u2261e_ = \u03bb {A} \u2192 _\u2261_ {A = M A}\n\n    left-identity : f \u2261 (\u03bb x \u2192 return x >>= f)\n    left-identity = refl\n\n    right-identity : m \u2261e m >>= return\n    right-identity = refl\n\n    associativity : m >>= (\u03bb x \u2192 f x >>= g) \u2261e (m >>= f) >>= g\n    associativity = refl\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\n\nmodule Explore.Monad \u2113 where\n\nM = Explore \u2113\n\nmodule _ {A : Set} where\n    return : A \u2192 M A\n    return x _\u2219_ f = f x\n\n    return-ind : \u2200 {p} x \u2192 ExploreInd p (return x)\n    return-ind x P _P\u2219_ Pf = Pf x\n\nmodule _ {A B : Set} where\n\n    infixl 1 _>>=_ _>>=-ind_\n    _>>=_ : M A \u2192 (A \u2192 M B) \u2192 M B\n    (exp\u1d2c >>= exp\u1d2e) _\u2219_ f = exp\u1d2c _\u2219_ \u03bb x \u2192 exp\u1d2e x _\u2219_ f\n\n    _>>=-ind_ : \u2200 {exp\u1d2c : M A} \u2192 ExploreInd\u2080 exp\u1d2c \u2192\n                 {exp\u1d2e : A \u2192 M B} \u2192 (\u2200 x \u2192 ExploreInd\u2080 (exp\u1d2e x))\n               \u2192 ExploreInd\u2080 (exp\u1d2c >>= exp\u1d2e)\n    _>>=-ind_ {exp\u1d2c} P\u1d2c {exp\u1d2e} P\u1d2e P _P\u2219_ Pf\n      = P\u1d2c (\u03bb e \u2192 P (e >>= exp\u1d2e)) _P\u2219_ \u03bb x \u2192 P\u1d2e x P _P\u2219_ Pf\n\n    map : (A \u2192 B) \u2192 M A \u2192 M B\n    map f e = e >>= return \u2218 f\n\n    map-ind : (f : A \u2192 B) {exp\u1d2c : M A} \u2192 ExploreInd\u2080 exp\u1d2c \u2192 ExploreInd\u2080 (map f exp\u1d2c)\n    map-ind f P\u1d2c = P\u1d2c >>=-ind return-ind \u2218 f\n\n    private\n        {- This law is for free: map f e = e >>= return \u2218 f -}\n        map' : (A \u2192 B) \u2192 M A \u2192 M B\n        map' f exp _\u2219_ g = exp _\u2219_ (g \u2218 f)\n\n        map'\u2261map : map' \u2261 map\n        map'\u2261map = refl\n\n-- universe issues...\n-- join : M (M A) \u2192 M A\n\nmodule _ {A B C : Set} (m : M A) (f : A \u2192 M B) (g : B \u2192 M C) where\n    infix 0 _\u2261e_\n    _\u2261e_ = \u03bb {A} \u2192 _\u2261_ {A = M A}\n\n    left-identity : f \u2261 (\u03bb x \u2192 return x >>= f)\n    left-identity = refl\n\n    right-identity : m \u2261e m >>= return\n    right-identity = refl\n\n    associativity : m >>= (\u03bb x \u2192 f x >>= g) \u2261e (m >>= f) >>= g\n    associativity = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"10b4256cb118009f3220457feb1e509fcaf0ef1e","subject":"Fixed doc.","message":"Fixed doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/Common\/FOL\/FOL.agda","new_file":"src\/fot\/Common\/FOL\/FOL.agda","new_contents":"------------------------------------------------------------------------------\n-- First-order logic (without equality)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- This module is re-exported by the \"base\" modules whose theories are\n-- defined on first-order logic (without equality).\n\n-- The logical connectives are hard-coded in our translation, i.e. the\n-- symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192 and \u2194 must be used.\n--\n-- N.B. For the implication we use the Agda function type.\n--\n-- N.B For the universal quantifier we use the Agda (dependent)\n-- function type.\n\nmodule Common.FOL.FOL where\n\ninfixr 4 _,_\ninfix  3 \u00ac_\ninfixr 2 _\u2227_\ninfix  2 \u2203\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\n\n----------------------------------------------------------------------------\n-- The universe of discourse\/universal domain.\npostulate D : Set\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : \u2200 {A B} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (a , _) = a\n\n\u2227-proj\u2082 : \u2200 {A B} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (_ , b) = b\n\n-----------------------------------------------------------------------------\n-- The disjunction data type.\n\n-- It is not necessary to add the data constructors inj\u2081 and inj\u2082 as\n-- axioms because the ATPs implement them.\ndata _\u2228_ (A B : Set) : Set where\n  inj\u2081 : A \u2192 A \u2228 B\n  inj\u2082 : B \u2192 A \u2228 B\n\n-- It is not strictly necessary define the eliminator `case` because\n-- the ATPs implement it. For the same reason, it is not necessary to\n-- add it as a (general\/local) hint.\ncase : \u2200 {A B} \u2192 {C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\ncase f g (inj\u2081 a) = f a\ncase f g (inj\u2082 b) = g b\n\n------------------------------------------------------------------------------\n-- The empty type.\ndata \u22a5 : Set where\n\n\u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n\u22a5-elim ()\n\n------------------------------------------------------------------------------\n-- The unit type.\n-- N.B. The name of this type is \"\\top\", not T.\ndata \u22a4 : Set where\n  tt : \u22a4\n\n------------------------------------------------------------------------------\n-- Negation.\n-- The underscore allows to write for example '\u00ac \u00ac A' instead of '\u00ac (\u00ac A)'.\n\u00ac_ : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n\n------------------------------------------------------------------------------\n-- Biconditional.\n_\u2194_ : Set \u2192 Set \u2192 Set\nA \u2194 B = (A \u2192 B) \u2227 (B \u2192 A)\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n-- 2012-03-05: We avoid to use the existential elimination or the\n-- existential projections because we use pattern matching (and the\n-- Agda's with constructor).\n\n-- The existential elimination.\n--\n-- NB. We do not use the usual type theory elimination with two\n-- projections because we are working in first-order logic where we do\n-- not need extract a witness from an existence proof.\n-- \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n-- \u2203-elim (_ , Ax) h = h Ax\n\n-- The existential proyections.\n-- \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n-- \u2203-proj\u2081 (x , _) = x\n\n-- \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n-- \u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Properties\n\n\u2192-trans : {A B C : Set} \u2192 (A \u2192 B) \u2192 (B \u2192 C) \u2192 A \u2192 C\n\u2192-trans f g a = g (f a)\n","old_contents":"------------------------------------------------------------------------------\n-- First-order logic (without equality)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\n-- This module is re-exported by the \"base\" modules whose theories are\n-- defined on first-order logic (without equality).\n\n-- The logical connectives are hard-coded in our translation, i.e. the\n-- symbols \u22a5, \u22a4, \u00ac, \u2227, \u2228, \u2192 and \u2194 must be used.\n--\n-- N.B. For the implication we use the Agda function type.\n--\n-- N.B For the universal quantifier we use the Agda (dependent)\n-- function type.\n\nmodule Common.FOL.FOL where\n\ninfixr 4 _,_\ninfix  3 \u00ac_\ninfixr 2 _\u2227_\ninfix  2 \u2203\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\n\n----------------------------------------------------------------------------\n-- The universe of discourse\/universal domain.\npostulate D : Set\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : \u2200 {A B} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (a , _) = a\n\n\u2227-proj\u2082 : \u2200 {A B} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (_ , b) = b\n\n-----------------------------------------------------------------------------\n-- The disjunction data type.\n\n-- It is not necessary to add the data constructors inj\u2081 and inj\u2082 as\n-- axioms because the ATPs implement them.\ndata _\u2228_ (A B : Set) : Set where\n  inj\u2081 : A \u2192 A \u2228 B\n  inj\u2082 : B \u2192 A \u2228 B\n\n-- It is not strictly necessary define the eliminator [_,_] because\n-- the ATPs implement it. For the same reason, it is not necessary to\n-- add it as a (general\/local) hint.\ncase : \u2200 {A B} \u2192 {C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\ncase f g (inj\u2081 a) = f a\ncase f g (inj\u2082 b) = g b\n\n------------------------------------------------------------------------------\n-- The empty type.\ndata \u22a5 : Set where\n\n\u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n\u22a5-elim ()\n\n------------------------------------------------------------------------------\n-- The unit type.\n-- N.B. The name of this type is \"\\top\", not T.\ndata \u22a4 : Set where\n  tt : \u22a4\n\n------------------------------------------------------------------------------\n-- Negation.\n-- The underscore allows to write for example '\u00ac \u00ac A' instead of '\u00ac (\u00ac A)'.\n\u00ac_ : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n\n------------------------------------------------------------------------------\n-- Biconditional.\n_\u2194_ : Set \u2192 Set \u2192 Set\nA \u2194 B = (A \u2192 B) \u2227 (B \u2192 A)\n\n------------------------------------------------------------------------------\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\n-- Sugar syntax for the existential quantifier.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n-- 2012-03-05: We avoid to use the existential elimination or the\n-- existential projections because we use pattern matching (and the\n-- Agda's with constructor).\n\n-- The existential elimination.\n--\n-- NB. We do not use the usual type theory elimination with two\n-- projections because we are working in first-order logic where we do\n-- not need extract a witness from an existence proof.\n-- \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n-- \u2203-elim (_ , Ax) h = h Ax\n\n-- The existential proyections.\n-- \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n-- \u2203-proj\u2081 (x , _) = x\n\n-- \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n-- \u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- Properties\n\n\u2192-trans : {A B C : Set} \u2192 (A \u2192 B) \u2192 (B \u2192 C) \u2192 A \u2192 C\n\u2192-trans f g a = g (f a)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"11d31b09dc6225ec2ecf82796b8b0461c9408f24","subject":"Bits: add if0 to the operations which makes them much more powerful","message":"Bits: add if0 to the operations which makes them much more powerful\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    toPerm : Op \u2192 Perm\n    toPerm `id     = `id\n    toPerm `0\u21941    = `0\u21941\n    toPerm `not    = `id -- Important\n    toPerm (`tl f) = `tl (toPerm f)\n    toPerm (f `\u204f g) = toPerm f `\u204f toPerm g\n\n    infixr 9 _\u2219\u2032_\n    _\u2219\u2032_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    f \u2219\u2032 xs = toPerm f P.\u2219 xs\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    almost-\u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) f xs \u2192 f \u2219\u2032 pad \u2295 f \u2219 xs \u2261 f \u2219 (pad \u2295 xs)\n    almost-\u2295-dist-\u2219 pad           `id     xs = refl\n    almost-\u2295-dist-\u2219 pad           `0\u21941    xs = \u2295-dist-0\u21941 pad xs\n    almost-\u2295-dist-\u2219 []            `not    [] = refl\n    almost-\u2295-dist-\u2219 (true  \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 (false \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 []            (`tl f) [] = refl\n    almost-\u2295-dist-\u2219 (p \u2237 pad)     (`tl f) (x \u2237 xs) rewrite almost-\u2295-dist-\u2219 pad f xs = refl\n    almost-\u2295-dist-\u2219 pad           (f `\u204f g) xs rewrite almost-\u2295-dist-\u2219 (f \u2219\u2032 pad) g (f \u2219 xs)\n                                                   | almost-\u2295-dist-\u2219 pad f xs = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-+ to sum-+;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    toPerm : Op \u2192 Perm\n    toPerm `id     = `id\n    toPerm `0\u21941    = `0\u21941\n    toPerm `not    = `id -- Important\n    toPerm (`tl f) = `tl (toPerm f)\n    toPerm (f `\u204f g) = toPerm f `\u204f toPerm g\n\n    infixr 9 _\u2219\u2032_\n    _\u2219\u2032_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    f \u2219\u2032 xs = toPerm f P.\u2219 xs\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    almost-\u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) f xs \u2192 f \u2219\u2032 pad \u2295 f \u2219 xs \u2261 f \u2219 (pad \u2295 xs)\n    almost-\u2295-dist-\u2219 pad           `id     xs = refl\n    almost-\u2295-dist-\u2219 pad           `0\u21941    xs = \u2295-dist-0\u21941 pad xs\n    almost-\u2295-dist-\u2219 []            `not    [] = refl\n    almost-\u2295-dist-\u2219 (true  \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 (false \u2237 pad) `not    (x \u2237 xs) = refl\n    almost-\u2295-dist-\u2219 []            (`tl f) [] = refl\n    almost-\u2295-dist-\u2219 (p \u2237 pad)     (`tl f) (x \u2237 xs) rewrite almost-\u2295-dist-\u2219 pad f xs = refl\n    almost-\u2295-dist-\u2219 pad           (f `\u204f g) xs rewrite almost-\u2295-dist-\u2219 (f \u2219\u2032 pad) g (f \u2219 xs)\n                                                   | almost-\u2295-dist-\u2219 pad f xs = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                     search {m + n} f\n                   \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n        search-+ {zero} f = refl\n        search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                                 | search-+ {m} (f \u2218 1\u2237_) = refl\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-+ to sum-+;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e71e428c1e63d766e7896b94aa406645ef65a52e","subject":"Agda: implement substitution for modal logic","message":"Agda: implement substitution for modal logic\n\nOld-commit-hash: e8580d4ac39e998fdf42c16064112cd2c0326763\n","repos":"inc-lc\/ilc-agda","old_file":"ModalLogic.agda","new_file":"ModalLogic.agda","new_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotational.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntactic.Contexts Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n-- Attempt 1 at weakening, subcontext-based. This works, but it is not what we need later.\n--\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be\n-- fixed and lift introduced by hand.\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393\u2081 \u0394\u2081 \u03c4 \u2192 Term \u0393\u2082 \u0394\u2082 \u03c4\nweaken \u0393\u2032 \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken (keep \u03c4\u2081 \u2022 \u0393\u2032) \u0394\u2032 t)\nweaken \u0393\u2032 \u0394\u2032 (app t t\u2081) = app (weaken \u0393\u2032 \u0394\u2032 t) (weaken \u0393\u2032 \u0394\u2032 t\u2081)\nweaken \u0393\u2032 \u0394\u2032 (ovar x) = ovar (lift \u0393\u2032 x)\nweaken \u0393\u2032 \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0393\u2032 \u0394\u2032 (box t) = box (weaken \u2205 \u0394\u2032 t)\nweaken \u0393\u2032 \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0393\u2032 \u0394\u2032 t in-\n          weaken \u0393\u2032 (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n\n\nopen import Relation.Binary.PropositionalEquality\n\nweaken-\u227c : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u0394 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u0394 \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u0394 \u03c4\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} {\u0393\u2083} (abs {\u03c4\u2081} t) =  abs (weaken-\u227c {\u03c4\u2081 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (app t t\u2081) = app (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t) (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (ovar x) = ovar (Prefixes.lift {\u0393\u2081} {\u0393\u2082} x)\nweaken-\u227c (mvar u) = mvar u\nweaken-\u227c (box t) = box t\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c {\u0393\u2081} {\u0393\u2082} t in- weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081\n\nsubstVar : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (v : Var (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstVar {\u2205} t  this = t\nsubstVar {\u2205} t (that v) = ovar v\nsubstVar {\u03c4 \u2022 \u0393} t  this = ovar this\nsubstVar {\u03c4 \u2022 \u0393} t (that v) = weaken-\u227c {\u2205} {\u03c4 \u2022 \u2205} (substVar {\u0393} t v)\n\nsubstTerm : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (t\u2082 : Term (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u0394 \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (abs {\u03c4\u2082} t\u2082) = abs ( substTerm {\u03c4\u2082 \u2022 \u0393} {\u0393\u2032} t\u2081 t\u2082 )\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (app t\u2082 t\u2083) =  app (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082) (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2083) \nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (ovar v) =  substVar {\u0393} {\u0393\u2032} t\u2081 v\nsubstTerm t\u2081 (mvar u) = mvar u\nsubstTerm t\u2081 (box t\u2082) = box t\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (let-box_in-_ {\u03c4\u2083} t\u2082 t\u2083) =\n  let-box substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082 in-\n    substTerm {\u0393} {\u0393\u2032}\n      (weaken \u227c-refl (drop \u03c4\u2083 \u2022 \u227c-refl) t\u2081)\n      t\u2083\n","old_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotational.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntactic.Contexts Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be fixed.\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393\u2081 \u0394\u2081 \u03c4 \u2192 Term \u0393\u2082 \u0394\u2082 \u03c4\nweaken \u0393\u2032 \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken (keep \u03c4\u2081 \u2022 \u0393\u2032) \u0394\u2032 t)\nweaken \u0393\u2032 \u0394\u2032 (app t t\u2081) = app (weaken \u0393\u2032 \u0394\u2032 t) (weaken \u0393\u2032 \u0394\u2032 t\u2081)\nweaken \u0393\u2032 \u0394\u2032 (ovar x) = ovar (lift \u0393\u2032 x)\nweaken \u0393\u2032 \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0393\u2032 \u0394\u2032 (box t) = box (weaken \u2205 \u0394\u2032 t)\nweaken \u0393\u2032 \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0393\u2032 \u0394\u2032 t in-\n          weaken \u0393\u2032 (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5b7065f4ffbe921589246c1ae5f497f51a441887","subject":"filpbased-implem: upgrade","message":"filpbased-implem: upgrade\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-implem.agda","new_file":"flipbased-implem.agda","new_contents":"module flipbased-implem where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Vec\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Data.Fin as Fin\nopen \u2261 using (_\u2257_; _\u2261_)\nopen Fin using (Fin; suc)\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run\u21ba : Bits n \u2192 A\n\nopen \u21ba public\n\nprivate\n  -- If you are not allowed to toss any coin, then you are deterministic.\n  Det : \u2200 {a} \u2192 Set a \u2192 Set a\n  Det = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet f = run\u21ba f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn\u21ba = mk \u2218 const\n\nmap\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap\u21ba f x = mk (f \u2218 run\u21ba x)\n-- map\u21ba f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin\u21ba {n\u2081} x = mk (\u03bb bs \u2192 run\u21ba (run\u21ba x (take _ bs)) (drop n\u2081 bs))\n-- join\u21ba x = x >>= id\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\n_\u2257\u21ba_ : \u2200 {c a} {A : Set a} (f g : \u21ba c A) \u2192 Set a\nf \u2257\u21ba g = run\u21ba f \u2257 run\u21ba g\n\n\u2141 : \u2115 \u2192 Set\n\u2141 n = \u21ba n Bit\n\n_\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c) \u2192 Set\n\u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n\u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n\u2257\u2141-trans p q b R = \u2261.trans (p b R) (q b R)\n\ncount\u21ba\u1da0 : \u2200 {c} \u2192 \u2141 c \u2192 Fin (suc (2^ c))\ncount\u21ba\u1da0 f = #\u27e8 run\u21ba f \u27e9\u1da0\n\ncount\u21ba : \u2200 {c} \u2192 \u2141 c \u2192 \u2115\ncount\u21ba = Fin.to\u2115 \u2218 count\u21ba\u1da0\n\n_\u223c[_]\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141_ {m} {n} f _\u223c_ g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\n_\u223c[_]\u2141\u2032_ : \u2200 {n} \u2192 \u2141 n \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141\u2032_ {n} f _\u223c_ g = count\u21ba f \u223c count\u21ba g\n\n_\u2248\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\nf \u2248\u2141 g = f \u223c[ _\u2261_ ]\u2141 g\n\n_\u2248\u2141\u2032_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\nf \u2248\u2141\u2032 g = f \u223c[ _\u2261_ ]\u2141\u2032 g\n\n\u2248\u2141-refl : \u2200 {n} {f : \u2141 n} \u2192 f \u2248\u2141 f\n\u2248\u2141-refl = \u2261.refl\n\n\u2248\u2141-sym : \u2200 {n} \u2192 Symmetric {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-sym = \u2261.sym\n\n\u2248\u2141-trans : \u2200 {n} \u2192 Transitive {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-trans = \u2261.trans\n\n\u2257\u21d2\u2248\u2141 : \u2200 {c} {f g : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f \u2248\u2141 g\n\u2257\u21d2\u2248\u2141 pf rewrite ext-# pf = \u2261.refl\n\n\u2248\u2141\u2032\u21d2\u2248\u2141 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141\u2032 g \u2192 f \u2248\u2141 g\n\u2248\u2141\u2032\u21d2\u2248\u2141 eq rewrite eq = \u2261.refl\n\n\u2248\u2141\u21d2\u2248\u2141\u2032 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141 g \u2192 f \u2248\u2141\u2032 g\n\u2248\u2141\u21d2\u2248\u2141\u2032 {n} = 2^-inj n\n\n\u2248\u2141-cong : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141 f' \u2192 g \u2248\u2141 g'\n\u2248\u2141-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n\n\u2248\u2141\u2032-cong : \u2200 {c} {f g f' g' : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141\u2032 f' \u2192 g \u2248\u2141\u2032 g'\n\u2248\u2141\u2032-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n","old_contents":"module flipbased-implem where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Vec\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Data.Fin as Fin\nopen \u2261 using (_\u2257_; _\u2261_)\nopen Fin using (Fin; suc)\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run\u21ba : Bits n \u2192 A\n\nopen \u21ba public\n\nprivate\n  -- If you are not allowed to toss any coin, then you are deterministic.\n  Det : \u2200 {a} \u2192 Set a \u2192 Set a\n  Det = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet f = run\u21ba f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn\u21ba = mk \u2218 const\n\nmap\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap\u21ba f x = mk (f \u2218 run\u21ba x)\n-- map\u21ba f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin\u21ba {n\u2081} x = mk (\u03bb bs \u2192 run\u21ba (run\u21ba x (take _ bs)) (drop n\u2081 bs))\n-- join\u21ba x = x >>= id\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\n_\u2257\u21ba_ : \u2200 {c a} {A : Set a} (f g : \u21ba c A) \u2192 Set a\nf \u2257\u21ba g = run\u21ba f \u2257 run\u21ba g\n\n\u2141 : \u2115 \u2192 Set\n\u2141 n = \u21ba n Bit\n\n_\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c) \u2192 Set\n\u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n\u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n\u2257\u2141-trans p q b R = \u2261.trans (p b R) (q b R)\n\ncount\u21ba\u1da0 : \u2200 {c} \u2192 \u2141 c \u2192 Fin (suc (2^ c))\ncount\u21ba\u1da0 f = #\u27e8 run\u21ba f \u27e9\u1da0\n\ncount\u21ba : \u2200 {c} \u2192 \u2141 c \u2192 \u2115\ncount\u21ba = Fin.to\u2115 \u2218 count\u21ba\u1da0\n\n_\u223c[_]\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141_ {m} {n} f _\u223c_ g = \u27e82^ n \u27e9*( count\u21ba f ) \u223c \u27e82^ m \u27e9*( count\u21ba g )\n\n_\u223c[_]\u2141\u2032_ : \u2200 {n} \u2192 \u2141 n \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141\u2032_ {n} f _\u223c_ g = count\u21ba f \u223c count\u21ba g\n\n_\u2248\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\nf \u2248\u2141 g = f \u223c[ _\u2261_ ]\u2141 g\n\n_\u2248\u2141\u2032_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\nf \u2248\u2141\u2032 g = f \u223c[ _\u2261_ ]\u2141\u2032 g\n\n\u2248\u2141-refl : \u2200 {n} {f : \u2141 n} \u2192 f \u2248\u2141 f\n\u2248\u2141-refl = \u2261.refl\n\n\u2248\u2141-sym : \u2200 {n} \u2192 Symmetric {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-sym = \u2261.sym\n\n\u2248\u2141-trans : \u2200 {n} \u2192 Transitive {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-trans = \u2261.trans\n\n\u2257\u21d2\u2248\u2141 : \u2200 {c} {f g : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f \u2248\u2141 g\n\u2257\u21d2\u2248\u2141 {c} {f} {g} pf rewrite ext-# pf = \u2261.refl\n\n\u2248\u2141\u2032\u21d2\u2248\u2141 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141\u2032 g \u2192 f \u2248\u2141 g\n\u2248\u2141\u2032\u21d2\u2248\u2141 {n} eq rewrite eq = \u2261.refl\n\n\u2248\u2141\u21d2\u2248\u2141\u2032 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141 g \u2192 f \u2248\u2141\u2032 g\n\u2248\u2141\u21d2\u2248\u2141\u2032 {n} = 2^-inj n\n\n\u2248\u2141-cong : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141 f' \u2192 g \u2248\u2141 g'\n\u2248\u2141-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n\n\u2248\u2141\u2032-cong : \u2200 {c} {f g f' g' : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141\u2032 f' \u2192 g \u2248\u2141\u2032 g'\n\u2248\u2141\u2032-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0121df71ae1936654d40af1f95bed309c811c342","subject":"Added Feferman's succesor-onto axiom.","message":"Added Feferman's succesor-onto axiom.\n\nIgnore-this: f8e5dd5fc95cb3511c1f5eceb125d1a\n\ndarcs-hash:20120411040706-3bd4e-fd47fc56ae51618fab68aaa23d6db6c65c21d1ce.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/PropertiesI.agda","new_file":"src\/FOTC\/Data\/Nat\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- M. J. Beeson. Proving programs and programming proofs. In Ruth\n-- Barcan Marcus, George J. W. Dorn, and Paul Weingartner,\n-- editors. Logic, Methodology and Philosophy of Science VII (1983),\n-- volume 114 of Studies in Logic and the Foundations of\n-- Mathematics. North-Holland Publishing Company, 1988, pages 51\u201382.\n\nmodule FOTC.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong h = cong\u2082 _+_ h refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong h = cong\u2082 _+_ refl h\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong h = cong\u2082 _*_ h refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong h = cong\u2082 _*_ refl h\n\n------------------------------------------------------------------------------\n\n-- We removed the equation pred zero \u2261 zero, so we cannot prove the\n-- totality of the function pred.\n-- pred-N : \u2200 {n} \u2192 N n \u2192 N (pred n)\n-- pred-N zN = subst N (sym pred-0) zN\n-- pred-N (sN {n} Nn) = subst N (sym $ pred-S n) Nn\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm          zN          = subst N (sym $ \u2238-x0 m) Nm\n\u2238-N     zN          (sN {n} _)  = subst N (sym $ \u2238-0S n) zN\n\u2238-N     (sN {m} Nm) (sN {n} Nn) = subst N (sym $ \u2238-SS m n) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t)\n               (+-rightIdentity Nn)\n               refl\n        )\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym $ +-leftIdentity n) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym $ +-Sx m n) (sN (+-N Nm Nn))\n\n+-assoc : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o} zN Nn No =\n  zero + n + o \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o        \u2261\u27e8 sym $ +-leftIdentity (n + o) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc {n = n} {o} (sN {m} Nm) Nn No =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm Nn No) refl \u27e9\n  succ\u2081 (m + (n + o)) \u2261\u27e8 sym $ +-Sx m (n + o) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} n \u2192 N m \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] n zN =\n  zero + succ\u2081 n\n    \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym $ +-leftIdentity n) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] n (sN {m} Nm) =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n      \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] n Nm) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym $ +-Sx m n) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym $ +-rightIdentity Nn \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym $ x+Sy\u2261S[x+y] m Nn \u27e9\n  n + succ\u2081 m \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN         = \u2238-x0 zero\n\u2238-0x (sN {n} _) = \u2238-0S n\n\nx\u2238x\u22610 : \u2200 {n} \u2192 N n \u2192 n \u2238 n \u2261 zero\nx\u2238x\u22610 zN          = \u2238-x0 zero\nx\u2238x\u22610 (sN {n} Nn) = trans (\u2238-SS n n) (x\u2238x\u22610 Nn)\n\nSx\u2238x\u2261S0 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2238 n \u2261 succ\u2081 zero\nSx\u2238x\u2261S0 zN          = \u2238-x0 (succ\u2081 zero)\nSx\u2238x\u2261S0 (sN {n} Nn) = trans (\u2238-SS (succ\u2081 n) n) (Sx\u2238x\u2261S0 Nn)\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n  n \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n  n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (m + n) (m + o) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n*-leftZero : \u2200 {n} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N         zN          Nn = subst N (sym $ *-leftZero Nn) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym $ *-Sx m n) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero Nn) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN Nn = sym\n  (\n    zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero Nn) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym $ *-leftZero (sN Nn) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym $ +-assoc Nn Nm (*-N Nm Nn))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm Nn (*-N Nm Nn))\n               refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym $ +-Sx m (n + m * n) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym $ *-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero Nn) (sym $ *-rightZero Nn)\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n   \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m   \u2261\u27e8 sym $ x*Sy\u2261x+xy Nn Nm \u27e9\n  n * succ\u2081 m \u220e\n\n*-rightIdentity : \u2200 {n} \u2192 N n \u2192 n * succ\u2081 zero \u2261 n\n*-rightIdentity {n} Nn = trans (*-comm Nn (sN zN)) (*-leftIdentity Nn)\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o \u2261\u27e8 sym $ \u2238-x0 (m * o) \u27e9\n  m * o \u2238 zero \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym $ *-0x o) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S n) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) No) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o) (sym $ *-0x o) refl \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) zN) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym $ *-0x (succ\u2081 m))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n                    (\u2238-SS m n)\n                    refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n     \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym $ [x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No)) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym $ *-Sx n (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero\n    \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)\n    \u2261\u27e8 *-0x (m + n) \u27e9\n  zero\n    \u2261\u27e8 sym $ *-0x m \u27e9\n  zero * m\n    \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero\n    \u2261\u27e8 sym $ +-rightIdentity (*-N Nm zN) \u27e9\n  m * zero + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n             (trans (sym $ *-0x n) (*-comm zN Nn))\n             refl\n    \u27e9\n  m * zero + n * zero \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym $ +-leftIdentity (n * succ\u2081 o) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym $ *-0x (succ\u2081 o))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym $ +-rightIdentity (*-N (sN Nm) (sN No)) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym $ *-leftZero (sN No))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym $ +-assoc (sN No) (*-N Nm (sN No)) (*-N (sN Nn) (sN No)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u2228 n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 zN      _  _                   = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN Nm) zN _                   = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN {m} Nm) (sN {n} Nn) SmSn\u22610 = \u22a5-elim (0\u2262S prf)\n  where\n  prf : zero \u2261 succ\u2081 (n + m * succ\u2081 n)\n  prf = zero                  \u2261\u27e8 sym SmSn\u22610 \u27e9\n        succ\u2081 m * succ\u2081 n     \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n        succ\u2081 n + m * succ\u2081 n \u2261\u27e8 +-Sx n (m * succ\u2081 n) \u27e9\n        succ\u2081 (n + m * succ\u2081 n) \u220e\n\n-- See the combined proof.\npostulate xy\u22611\u2192x\u22611\u2228y\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 one \u2192 m \u2261 one \u2228 n \u2261 one\n\n-- Feferman's axiom as presented by (Beeson 1986, p. 74).\nsucc-onto : \u2200 {n} \u2192 N n \u2192 \u00ac (n \u2261 zero) \u2192 succ\u2081 (pred\u2081 n) \u2261 n\nsucc-onto zN          h = \u22a5-elim (h refl)\nsucc-onto (sN {n} Nn) h = cong succ\u2081 (pred-S n)\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong h = cong\u2082 _+_ h refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong h = cong\u2082 _+_ refl h\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong h = cong\u2082 _*_ h refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong h = cong\u2082 _*_ refl h\n\n------------------------------------------------------------------------------\n\n-- We removed the equation pred zero \u2261 zero, so we cannot prove the\n-- totality of the function pred.\n-- pred-N : \u2200 {n} \u2192 N n \u2192 N (pred n)\n-- pred-N zN = subst N (sym pred-0) zN\n-- pred-N (sN {n} Nn) = subst N (sym $ pred-S n) Nn\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm          zN          = subst N (sym $ \u2238-x0 m) Nm\n\u2238-N     zN          (sN {n} _)  = subst N (sym $ \u2238-0S n) zN\n\u2238-N     (sN {m} Nm) (sN {n} Nn) = subst N (sym $ \u2238-SS m n) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t)\n               (+-rightIdentity Nn)\n               refl\n        )\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym $ +-leftIdentity n) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym $ +-Sx m n) (sN (+-N Nm Nn))\n\n+-assoc : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc {n = n} {o} zN Nn No =\n  zero + n + o \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o        \u2261\u27e8 sym $ +-leftIdentity (n + o) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc {n = n} {o} (sN {m} Nm) Nn No =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm Nn No) refl \u27e9\n  succ\u2081 (m + (n + o)) \u2261\u27e8 sym $ +-Sx m (n + o) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} n \u2192 N m \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] n zN =\n  zero + succ\u2081 n\n    \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym $ +-leftIdentity n) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] n (sN {m} Nm) =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n      \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] n Nm) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym $ +-Sx m n) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym $ +-rightIdentity Nn \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym $ x+Sy\u2261S[x+y] m Nn \u27e9\n  n + succ\u2081 m \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN         = \u2238-x0 zero\n\u2238-0x (sN {n} _) = \u2238-0S n\n\nx\u2238x\u22610 : \u2200 {n} \u2192 N n \u2192 n \u2238 n \u2261 zero\nx\u2238x\u22610 zN          = \u2238-x0 zero\nx\u2238x\u22610 (sN {n} Nn) = trans (\u2238-SS n n) (x\u2238x\u22610 Nn)\n\nSx\u2238x\u2261S0 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2238 n \u2261 succ\u2081 zero\nSx\u2238x\u2261S0 zN          = \u2238-x0 (succ\u2081 zero)\nSx\u2238x\u2261S0 (sN {n} Nn) = trans (\u2238-SS (succ\u2081 n) n) (Sx\u2238x\u2261S0 Nn)\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n  n \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n  n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (m + n) (m + o) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n*-leftZero : \u2200 {n} \u2192 N n \u2192 zero * n \u2261 zero\n*-leftZero {n} _ = *-0x n\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N         zN          Nn = subst N (sym $ *-leftZero Nn) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym $ *-Sx m n) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zN\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero Nn) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN Nn = sym\n  (\n    zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero Nn) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym $ *-leftZero (sN Nn) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym $ +-assoc Nn Nm (*-N Nm Nn))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm Nn (*-N Nm Nn))\n               refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym $ +-Sx m (n + m * n) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym $ *-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero Nn) (sym $ *-rightZero Nn)\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n   \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m   \u2261\u27e8 sym $ x*Sy\u2261x+xy Nn Nm \u27e9\n  n * succ\u2081 m \u220e\n\n*-rightIdentity : \u2200 {n} \u2192 N n \u2192 n * succ\u2081 zero \u2261 n\n*-rightIdentity {n} Nn = trans (*-comm Nn (sN zN)) (*-leftIdentity Nn)\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o \u2261\u27e8 sym $ \u2238-x0 (m * o) \u27e9\n  m * o \u2238 zero \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym $ *-0x o) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S n) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) No) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o) (sym $ *-0x o) refl \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) zN) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym $ *-0x (succ\u2081 m))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n                    (\u2238-SS m n)\n                    refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n     \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym $ [x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No)) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym $ *-Sx n (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero\n    \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)\n    \u2261\u27e8 *-0x (m + n) \u27e9\n  zero\n    \u2261\u27e8 sym $ *-0x m \u27e9\n  zero * m\n    \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero\n    \u2261\u27e8 sym $ +-rightIdentity (*-N Nm zN) \u27e9\n  m * zero + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n             (trans (sym $ *-0x n) (*-comm zN Nn))\n             refl\n    \u27e9\n  m * zero + n * zero \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym $ +-leftIdentity (n * succ\u2081 o) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym $ *-0x (succ\u2081 o))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym $ +-rightIdentity (*-N (sN Nm) (sN No)) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym $ *-leftZero (sN No))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym $ +-assoc (sN No) (*-N Nm (sN No)) (*-N (sN Nn) (sN No)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u2228 n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 zN      _  _                   = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN Nm) zN _                   = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (sN {m} Nm) (sN {n} Nn) SmSn\u22610 = \u22a5-elim (0\u2262S prf)\n  where\n  prf : zero \u2261 succ\u2081 (n + m * succ\u2081 n)\n  prf = zero                  \u2261\u27e8 sym SmSn\u22610 \u27e9\n        succ\u2081 m * succ\u2081 n     \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n        succ\u2081 n + m * succ\u2081 n \u2261\u27e8 +-Sx n (m * succ\u2081 n) \u27e9\n        succ\u2081 (n + m * succ\u2081 n) \u220e\n\n-- See the combined proof.\npostulate xy\u22611\u2192x\u22611\u2228y\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 one \u2192 m \u2261 one \u2228 n \u2261 one\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d02c0c3ef60a8f1e0b8ace3fdea7c8bb38c61281","subject":"Agda: rename type parameter of `that`","message":"Agda: rename type parameter of `that`\n\nOld-commit-hash: 0b30e5858214cb3ecab7a1661b7a2a79ac6bc871\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Context.agda","new_file":"Syntax\/Context.agda","new_contents":"module Syntax.Context\n    {Type : Set}\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Specialized congruence rules\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 8 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  lift (keep \u03c4 \u2022 \u227c\u2081) this = this\n  lift (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (lift \u227c\u2081 x)\n  lift (drop \u03c4 \u2022 \u227c\u2081) x = that (lift \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","old_contents":"module Syntax.Context\n    {Type : Set}\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Specialized congruence rules\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 8 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  lift (keep \u03c4 \u2022 \u227c\u2081) this = this\n  lift (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (lift \u227c\u2081 x)\n  lift (drop \u03c4 \u2022 \u227c\u2081) x = that (lift \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cd132a2ada356a76c361341940070bcfccc2c571","subject":"add explore function for Lift \ud835\udfd8","message":"add explore function for Lift \ud835\udfd8\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Zero.agda","new_file":"lib\/Explore\/Zero.agda","new_contents":"open import Type\nopen import Type.Identities\nopen import Level.NP\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Explorable\nopen import Data.Zero\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; refl; _\u2219_; !_)\nopen import HoTT\nopen Equivalences\n\nimport Explore.Monad\nopen import Explore.Isomorphism\n\nmodule Explore.Zero where\n\n\nmodule _ {\u2113} where\n    \ud835\udfd8\u1d49 : Explore \u2113 \ud835\udfd8\n    \ud835\udfd8\u1d49 = empty-explore\n    {- or\n    \ud835\udfd8\u1d49 \u03b5 _ _ = \u03b5\n    -}\n\n    \ud835\udfd8\u2071 : \u2200 {p} \u2192 ExploreInd p \ud835\udfd8\u1d49\n    \ud835\udfd8\u2071 = empty-explore-ind\n    {- or\n    \ud835\udfd8\u2071 _ P\u03b5 _ _ = P\u03b5\n    -}\n\nmodule _ {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {R : \ud835\udfd8 \u2192 \ud835\udfd8 \u2192 \u2605\u2080} where\n    \u27e6\ud835\udfd8\u1d49\u27e7 : \u27e6Explore\u27e7\u1d64 \u2113\u2081 \u2113\u2082 \u2113\u1d63 R \ud835\udfd8\u1d49 \ud835\udfd8\u1d49\n    \u27e6\ud835\udfd8\u1d49\u27e7 _ \u03b5\u1d63 _ _ = \u03b5\u1d63\n\nopen Explorable\u2080  \ud835\udfd8\u2071 public using () renaming (sum     to \ud835\udfd8\u02e2; product to \ud835\udfd8\u1d56)\n\nmodule _ {\u2113} where\n    open Explorable\u209b  {\u2113} \ud835\udfd8\u2071 public using () renaming (reify    to \ud835\udfd8\u02b3)\n    open Explorable\u209b\u209b {\u2113} \ud835\udfd8\u2071 public using () renaming (unfocus  to \ud835\udfd8\u1d58)\n\n    \ud835\udfd8\u02e1 : Lookup {\u2113} \ud835\udfd8\u1d49\n    \ud835\udfd8\u02e1 _ ()\n\n    \ud835\udfd8\u1da0 : Focus {\u2113} \ud835\udfd8\u1d49\n    \ud835\udfd8\u1da0 ((), _)\n\n    module _ {{_ : UA}} where\n        \u03a3\u1d49\ud835\udfd8-ok : Adequate-\u03a3\u1d49 {\u2113} \ud835\udfd8\u1d49\n        \u03a3\u1d49\ud835\udfd8-ok _ = ! \u03a3\ud835\udfd8-lift\u2218fst\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03a0\u1d49\ud835\udfd8-ok : Adequate-\u03a0\u1d49 {\u2113} \ud835\udfd8\u1d49\n        \u03a0\u1d49\ud835\udfd8-ok _ = ! \u03a0\ud835\udfd8-uniq _\n\nmodule _ {{_ : UA}} where\n    \ud835\udfd8\u02e2-ok : Adequate-sum \ud835\udfd8\u02e2\n    \ud835\udfd8\u02e2-ok _ = Fin0\u2261\ud835\udfd8 \u2219 ! \u03a3\ud835\udfd8-fst\n\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n    \ud835\udfd8\u1d56-ok : Adequate-product \ud835\udfd8\u1d56\n    \ud835\udfd8\u1d56-ok _ = Fin1\u2261\ud835\udfd9 \u2219 ! (\u03a0\ud835\udfd8-uniq\u2080 _)\n\nexplore\ud835\udfd8          = \ud835\udfd8\u1d49\nexplore\ud835\udfd8-ind      = \ud835\udfd8\u2071\nlookup\ud835\udfd8           = \ud835\udfd8\u02e1\nreify\ud835\udfd8            = \ud835\udfd8\u02b3\nfocus\ud835\udfd8            = \ud835\udfd8\u1da0\nunfocus\ud835\udfd8          = \ud835\udfd8\u1d58\nsum\ud835\udfd8              = \ud835\udfd8\u02e2\nadequate-sum\ud835\udfd8     = \ud835\udfd8\u02e2-ok\nproduct\ud835\udfd8          = \ud835\udfd8\u1d56\nadequate-product\ud835\udfd8 = \ud835\udfd8\u1d56-ok\n\n\nLift\ud835\udfd8\u1d49 : \u2200 {m} \u2192 Explore m (Lift \ud835\udfd8)\nLift\ud835\udfd8\u1d49 = explore-iso (\u2243-sym Lift\u2243id) \ud835\udfd8\u1d49\n\n\u03a3\u1d49Lift\ud835\udfd8-ok : \u2200 {{_ : UA}}{{_ : FunExt}}{m} \u2192 Adequate-\u03a3\u1d49 {m} Lift\ud835\udfd8\u1d49\n\u03a3\u1d49Lift\ud835\udfd8-ok = \u03a3-iso-ok (\u2243-sym Lift\u2243id) {A\u1d49 = \ud835\udfd8\u1d49} \u03a3\u1d49\ud835\udfd8-ok\n\n-- DEPRECATED\nmodule _ {{_ : UA}} where\n    open ExplorableRecord\n    \u03bc\ud835\udfd8 : Explorable \ud835\udfd8\n    \u03bc\ud835\udfd8 = mk _ \ud835\udfd8\u2071 \ud835\udfd8\u02e2-ok\n-- -}\n","old_contents":"open import Type\nopen import Type.Identities\nopen import Level.NP\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Explorable\nopen import Data.Zero\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; refl; _\u2219_; !_)\nopen import HoTT\nopen Equivalences\n\nimport Explore.Monad\n\nmodule Explore.Zero where\n\n\nmodule _ {\u2113} where\n    \ud835\udfd8\u1d49 : Explore \u2113 \ud835\udfd8\n    \ud835\udfd8\u1d49 = empty-explore\n    {- or\n    \ud835\udfd8\u1d49 \u03b5 _ _ = \u03b5\n    -}\n\n    \ud835\udfd8\u2071 : \u2200 {p} \u2192 ExploreInd p \ud835\udfd8\u1d49\n    \ud835\udfd8\u2071 = empty-explore-ind\n    {- or\n    \ud835\udfd8\u2071 _ P\u03b5 _ _ = P\u03b5\n    -}\n\nmodule _ {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {R : \ud835\udfd8 \u2192 \ud835\udfd8 \u2192 \u2605\u2080} where\n    \u27e6\ud835\udfd8\u1d49\u27e7 : \u27e6Explore\u27e7\u1d64 \u2113\u2081 \u2113\u2082 \u2113\u1d63 R \ud835\udfd8\u1d49 \ud835\udfd8\u1d49\n    \u27e6\ud835\udfd8\u1d49\u27e7 _ \u03b5\u1d63 _ _ = \u03b5\u1d63\n\nopen Explorable\u2080  \ud835\udfd8\u2071 public using () renaming (sum     to \ud835\udfd8\u02e2; product to \ud835\udfd8\u1d56)\n\nmodule _ {\u2113} where\n    open Explorable\u209b  {\u2113} \ud835\udfd8\u2071 public using () renaming (reify    to \ud835\udfd8\u02b3)\n    open Explorable\u209b\u209b {\u2113} \ud835\udfd8\u2071 public using () renaming (unfocus  to \ud835\udfd8\u1d58)\n\n    \ud835\udfd8\u02e1 : Lookup {\u2113} \ud835\udfd8\u1d49\n    \ud835\udfd8\u02e1 _ ()\n\n    \ud835\udfd8\u1da0 : Focus {\u2113} \ud835\udfd8\u1d49\n    \ud835\udfd8\u1da0 ((), _)\n\n    module _ {{_ : UA}} where\n        \u03a3\u1d49\ud835\udfd8-ok : Adequate-\u03a3\u1d49 {\u2113} \ud835\udfd8\u1d49\n        \u03a3\u1d49\ud835\udfd8-ok _ = ! \u03a3\ud835\udfd8-lift\u2218fst\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03a0\u1d49\ud835\udfd8-ok : Adequate-\u03a0\u1d49 {\u2113} \ud835\udfd8\u1d49\n        \u03a0\u1d49\ud835\udfd8-ok _ = ! \u03a0\ud835\udfd8-uniq _\n\nmodule _ {{_ : UA}} where\n    \ud835\udfd8\u02e2-ok : Adequate-sum \ud835\udfd8\u02e2\n    \ud835\udfd8\u02e2-ok _ = Fin0\u2261\ud835\udfd8 \u2219 ! \u03a3\ud835\udfd8-fst\n\nmodule _ {{_ : UA}}{{_ : FunExt}} where\n    \ud835\udfd8\u1d56-ok : Adequate-product \ud835\udfd8\u1d56\n    \ud835\udfd8\u1d56-ok _ = Fin1\u2261\ud835\udfd9 \u2219 ! (\u03a0\ud835\udfd8-uniq\u2080 _)\n\nexplore\ud835\udfd8          = \ud835\udfd8\u1d49\nexplore\ud835\udfd8-ind      = \ud835\udfd8\u2071\nlookup\ud835\udfd8           = \ud835\udfd8\u02e1\nreify\ud835\udfd8            = \ud835\udfd8\u02b3\nfocus\ud835\udfd8            = \ud835\udfd8\u1da0\nunfocus\ud835\udfd8          = \ud835\udfd8\u1d58\nsum\ud835\udfd8              = \ud835\udfd8\u02e2\nadequate-sum\ud835\udfd8     = \ud835\udfd8\u02e2-ok\nproduct\ud835\udfd8          = \ud835\udfd8\u1d56\nadequate-product\ud835\udfd8 = \ud835\udfd8\u1d56-ok\n\n-- DEPRECATED\nmodule _ {{_ : UA}} where\n    open ExplorableRecord\n    \u03bc\ud835\udfd8 : Explorable \ud835\udfd8\n    \u03bc\ud835\udfd8 = mk _ \ud835\udfd8\u2071 \ud835\udfd8\u02e2-ok\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8ca7269a8d5fdf6fa1be312947dc48c2a3d6e746","subject":"Maybe is injective","message":"Maybe is injective\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\n\nimport Function as F\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_)\n\nmodule CancelMaybe where\n  private\n    module _ {A : Set} where\n      just-inj : {x y : A} \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\n      just-inj \u2261.refl = \u2261.refl\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 Inverse.to f \u27e8$\u27e9 just x \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 Inverse.from f \u27e8$\u27e9 just x \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n          to = \u03bb x \u2192 Inverse.to f \u27e8$\u27e9 x\n          from = \u03bb x \u2192 Inverse.from f \u27e8$\u27e9 x\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to (just x) | tof-tot x\n          \u21d2 x | just x\u2081 | _ = x\u2081\n          \u21d2 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from (just x) | fof-tot x\n          \u21d0 x | just x\u2081 | p = x\u2081\n          \u21d0 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to (just x)\n                  | tof-tot x\n                  | from (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | Inverse.left-inverse-of f (just x)\n                  | Inverse.right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-inj (\u2261.trans (\u2261.sym (Inverse.left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong from c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \u22a5-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \u22a5-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from (just x)\n                  | fof-tot x\n                  | to (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | Inverse.right-inverse-of f (just x)\n                  | Inverse.left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-inj (\u2261.trans (\u2261.sym (Inverse.right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong to c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \u22a5-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \u22a5-elim (p \u2261.refl)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : Inverse.to f \u27e8$\u27e9 nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 Inverse.to f \u27e8$\u27e9 just x \u2262 nothing\n      tof-tot x eq with Inverse.injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : Inverse.from f \u27e8$\u27e9 nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (\u03bb x \u2192 Inverse.from f \u27e8$\u27e9 x) f-empty)) (Inverse.left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 Inverse.from f \u27e8$\u27e9 just x \u2262 nothing\n      fof-tot x eq with Inverse.injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n          to = \u03bb x \u2192 Inverse.to f \u27e8$\u27e9 x\n          from = \u03bb x \u2192 Inverse.from f \u27e8$\u27e9 x\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to (just x)\n          ... | nothing = to nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from (just x)\n          ... | nothing = from nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to (just x) | Inverse.left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to nothing | Inverse.left-inverse-of f nothing\n          \u21d0\u21d2 (just x\u2081) | nothing | p | just x | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from (just x) | Inverse.right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from nothing | Inverse.right-inverse-of f nothing\n          \u21d2\u21d0 (just x\u2081) | nothing | p | just x | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : Inverse.to g \u27e8$\u27e9 nothing \u2261 nothing\n      g-empty = \u2261.refl\n\n      iso' : A \u2194 B\n      iso' = iso g g-empty\n  CancelMaybe : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\n  CancelMaybe f = iso' f\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = from\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      from : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      from = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nMaybe\u2194Lift\u22a4\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \u22a4 \u228e A)\nMaybe\u2194Lift\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\u22a4 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \u22a4\nVec0\u2194Lift\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\u22a4 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses (\u03bb { (x \u2237 xs) \u2192 x , xs }) (uncurry _\u2237_)\n             (\u03bb { (x \u2237 xs) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\u22a4\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \u22a4) _ _ \u2218 (Maybe\u2194Lift\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9 \n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n","old_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\n\nimport Function as F\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = from\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      from : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      from = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nMaybe\u2194Lift\u22a4\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \u22a4 \u228e A)\nMaybe\u2194Lift\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\u22a4 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \u22a4\nVec0\u2194Lift\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\u22a4 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses (\u03bb { (x \u2237 xs) \u2192 x , xs }) (uncurry _\u2237_)\n             (\u03bb { (x \u2237 xs) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\u22a4\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \u22a4) _ _ \u2218 (Maybe\u2194Lift\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9 \n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4ef9fc1364b49c77c08afc6f02cda83061bb47a9","subject":"IDesc model: substitution ok.","message":"IDesc model: substitution ok.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (domNat domBool : EnumU)\n            (gammaNat : spi domNat (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi domBool (\\_ -> IMu closeTerm' bool))\n            (sigma : (domNat domBool : EnumU)\n                     (gammaNat : spi domNat (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi domBool (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var (chooseDom ty domNat domBool) ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (toIDesc Type (exprFree ** (\\ty -> Val ty + Var (chooseDom ty domNat domBool) ty))) ty ->\n            IMu closeTerm' ty\nsubstExpr domNat domBool gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var (chooseDom ty domNat domBool) ty)\n           Val\n           exprFree\n           (sig domNat domBool gammaNat gammaBool)\n           ty\n           term\n\n\n-- sigmaExpr : (domNat domBool : EnumU)\n--             (gammaNat : spi domNat (\\_ -> IMu closeTerm' nat))\n--             (gammaBool : spi domBool (\\_ -> IMu closeTerm' bool))\n--             (ty : Type) ->\n--             (Val ty + Var (chooseDom ty domNat domBool) ty) ->\n--             IMu closeTerm' ty\n-- sigmaExpr domNat domBool gammaNat gammaBool ty v = \n--     discharge' ty\n--                (chooseDom ty domNat domBool)\n--                (chooseGamma ty (chooseDom ty domNat domBool) gammaNat gammaBool)\n--                v\n\n-- discharge' : (ty : Type)\n--             (vars : EnumU)\n--             (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n--             (Val ty + Var vars ty) -> \n--             IMu closeTerm' ty\n\n-- chooseGamma : (ty : Type)\n--               (dom : EnumU)\n--               (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n--               (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n--               spi dom (\\_ -> IMu closeTerm' ty)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8aefca99408c27e4a214ce4b0721a4e4794eeb87","subject":"Rename Atlas-\u0394const to \u0394Const.","message":"Rename Atlas-\u0394const to \u0394Const.\n\nOld-commit-hash: e7ed19ec0ca363e486fe72ac858facfcd192e057\n","repos":"inc-lc\/ilc-agda","old_file":"Atlas\/Change\/Derivation.agda","new_file":"Atlas\/Change\/Derivation.agda","new_contents":"module Atlas.Change.Derivation where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\nopen import Atlas.Change.Term\n\nopen import Base.Syntax.Context Type\nopen import Base.Change.Context \u0394Type\n\n\u0394Const : \u2200 {\u0393 \u03a3 \u03c4} \u2192 (c : Const \u03a3 \u03c4) \u2192\n  Term \u0393 (internalizeContext (\u0394Context\u2032 \u03a3) (\u0394Type \u03c4))\n\n\u0394Const true  = false!\n\u0394Const false = false!\n\n\u0394Const xor = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 xor! \u0394x \u0394y)\n\n\u0394Const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\n\u0394Const update = abs\u2086 (\u03bb k \u0394k v \u0394v m \u0394m \u2192\n  insert (apply \u0394k k) (apply \u0394v v) (delete k v \u0394m))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\n\u0394Const lookup = abs\u2084 (\u03bb k \u0394k m \u0394m \u2192\n  let\n    k\u2032 = apply \u0394k k\n  in\n    (diff (apply {base _} (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n          (lookup! k m)))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\n\u0394Const zip = abs\u2086 (\u03bb f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 \u2192\n  let\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082)\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\n\u0394Const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n","old_contents":"module Atlas.Change.Derivation where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\nopen import Atlas.Change.Term\n\nopen import Base.Syntax.Context Type\nopen import Base.Change.Context \u0394Type\n\nAtlas-\u0394const : \u2200 {\u0393 \u03a3 \u03c4} \u2192 (c : Const \u03a3 \u03c4) \u2192\n  Term \u0393 (internalizeContext (\u0394Context\u2032 \u03a3) (\u0394Type \u03c4))\n\nAtlas-\u0394const true  = false!\nAtlas-\u0394const false = false!\n\nAtlas-\u0394const xor = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 xor! \u0394x \u0394y)\n\nAtlas-\u0394const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update = abs\u2086 (\u03bb k \u0394k v \u0394v m \u0394m \u2192\n  insert (apply \u0394k k) (apply \u0394v v) (delete k v \u0394m))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup = abs\u2084 (\u03bb k \u0394k m \u0394m \u2192\n  let\n    k\u2032 = apply \u0394k k\n  in\n    (diff (apply {base _} (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n          (lookup! k m)))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip = abs\u2086 (\u03bb f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 \u2192\n  let\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082)\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7eea4a12567c14dd3c2adfed8d3eb19fdbfdd0f4","subject":"lib\/Data\/Bool\/NP.agda","message":"lib\/Data\/Bool\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0a07d340be5502be6364113d66bcbc23f4bd6c0b","subject":"Cosmetics.","message":"Cosmetics.\n","repos":"asr\/apia,asr\/apia","old_file":"test\/succeed\/fol-theorems\/Existential.agda","new_file":"test\/succeed\/fol-theorems\/Existential.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the existential quantifier\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule Existential where\n\npostulate\n  D : Set\n  A : D \u2192 Set\n  \u2203 : (A : D \u2192 Set) \u2192 Set\n\npostulate foo : (\u2203 \u03bb x \u2192 A x) \u2192 (\u2203 \u03bb x \u2192 A x)\n{-# ATP prove foo #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the existential quantifier\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule Existential where\n\npostulate\n  D   : Set\n  A   : D \u2192 Set\n  \u2203   : (A : D \u2192 Set) \u2192 Set\n\npostulate foo : (\u2203 \u03bb x \u2192 A x) \u2192 (\u2203 \u03bb x \u2192 A x)\n{-# ATP prove foo #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5a7916f3a545c7c82adc8b37f392e5158f0514c6","subject":"Replaced the name of the ATP for a TPTP role (axiom).","message":"Replaced the name of the ATP for a TPTP role (axiom).\n\nIgnore-this: b6a44e660727168c0a3363f49bb145de\n\ndarcs-hash:20100316222710-3bd4e-a8d819b778389f6a79ae16a54d60734676b46667.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"test\/Logic.agda","new_file":"test\/Logic.agda","new_contents":"module Logic where\n\ninfix  30 _\u2261_\n\npostulate\n  D    : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\nimport LogicalConstants\nopen module LC = LogicalConstants D\n\npostulate\n f10 : (A : Set)   \u2192 A Implies A\n f20 : (A B : Set) \u2192 A And B Implies B And A\n f30 : (A : Set)   \u2192 Not (A And True) Equiv (Not A Or False)\n f40 : (d : D)     \u2192 d \u2261 d\n f50 : (d1 d2 : D) \u2192 (d1 \u2261 d2) Implies (d2 \u2261 d1)\n\n f60 : ForAll (\u03bb (x : D) \u2192 x \u2261 x)\n f70 : (x : D) \u2192 x \u2261 x\n f80 : ForAll (\u03bb (x : D) \u2192 Exists ( \u03bb (y : D) \u2192 x \u2261 y))\n\n{-# EXTERNAL axiom f10 #-}\n{-# EXTERNAL axiom f20 #-}\n{-# EXTERNAL axiom f30 #-}\n{-# EXTERNAL axiom f40 #-}\n{-# EXTERNAL axiom f50 #-}\n{-# EXTERNAL axiom f60 #-}\n{-# EXTERNAL axiom f70 #-}\n{-# EXTERNAL axiom f80 #-}\n","old_contents":"module Logic where\n\ninfix  30 _\u2261_\n\npostulate\n  D    : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\nimport LogicalConstants\nopen module LC = LogicalConstants D\n\npostulate\n f10 : (A : Set)   \u2192 A Implies A\n f20 : (A B : Set) \u2192 A And B Implies B And A\n f30 : (A : Set)   \u2192 Not (A And True) Equiv (Not A Or False)\n f40 : (d : D)     \u2192 d \u2261 d\n f50 : (d1 d2 : D) \u2192 (d1 \u2261 d2) Implies (d2 \u2261 d1)\n\n f60 : ForAll (\u03bb (x : D) \u2192 x \u2261 x)\n f70 : (x : D) \u2192 x \u2261 x\n f80 : ForAll (\u03bb (x : D) \u2192 Exists ( \u03bb (y : D) \u2192 x \u2261 y))\n\n{-# EXTERNAL Equinox f10 #-}\n{-# EXTERNAL Equinox f20 #-}\n{-# EXTERNAL Equinox f30 #-}\n{-# EXTERNAL Equinox f40 #-}\n{-# EXTERNAL Equinox f50 #-}\n{-# EXTERNAL Equinox f60 #-}\n{-# EXTERNAL Equinox f70 #-}\n{-# EXTERNAL Equinox f80 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"af1d5371ab683fac0cfee48a20cabc40e4da7d41","subject":"document a broken (so far) proof","message":"document a broken (so far) proof\n","repos":"crypto-agda\/crypto-agda","old_file":"program-distance.agda","new_file":"program-distance.agda","new_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) \u2265 2^((m + n) \u2238 k)\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 x y z t \u2192 x + ((y + z) \u2238 t) \u2261 y + ((x + z) \u2238 t)\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n           -- 2\u1d50(dist 2\u1d52#g 2\u207f#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n  ... | q rewrite sym (dist-2^* m (\u27e82^ o \u27e9* (count\u21ba g)) (\u27e82^ n \u27e9* (count\u21ba h)))\n           -- dist 2\u1d502\u1d52#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m o (count\u21ba g)\n           -- dist 2\u1d522\u1d50#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | sym f\u2248g\n           -- dist 2\u1d522\u207f#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm o n (count\u21ba f)\n           -- dist 2\u207f2\u1d52#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m n (count\u21ba h)\n           -- dist 2\u207f2\u1d52#f 2\u207f2\u1d50#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | dist-2^* n (\u27e82^ o \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba h))\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-+ m (n + o \u2238 k) 1\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d50\u207a\u207f\u207a\u1d52\u207b\u1d4f\n                | helper m n o k\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f\u207a\u1d50\u207a\u1d52\u207b\u1d4f\n                | sym (2^-+ n (m + o \u2238 k) 1)\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f2\u1d50\u207a\u1d52\u207b\u1d4f\n                = 2^*-mono\u2032 n q\n           -- dist 2\u1d52#f 2\u1d50#h \u2264 2\u1d50\u207a\u1d52\u207b\u1d4f\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_ (\u03bb {m n f g} x \u2192 ]-[-sym {m} {n} {f} {g} x) (\u03bb {m n o f g h} x y \u2192 ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} x y)\n","old_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) \u2265 2^((m + n) \u2238 k)\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 x y z t \u2192 x + ((y + z) \u2238 t) \u2261 y + ((x + z) \u2238 t)\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n  ... | q rewrite sym (dist-2^* m (\u27e82^ o \u27e9* (count\u21ba g)) (\u27e82^ n \u27e9* (count\u21ba h)))\n                | 2^-comm m o (count\u21ba g)\n                | sym f\u2248g\n                | 2^-comm o n (count\u21ba f)\n                | 2^-comm m n (count\u21ba h)\n                | dist-2^* n (\u27e82^ o \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba h))\n                | 2^-+ m (n + o \u2238 k) 1\n                | helper m n o k\n                | sym (2^-+ n (m + o \u2238 k) 1)\n                = 2^*-mono\u2032 n q\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_ (\u03bb {m n f g} x \u2192 ]-[-sym {m} {n} {f} {g} x) (\u03bb {m n o f g h} x y \u2192 ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} x y)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"17ce6886538cc17544064789b4f2d5bd1e7cfc72","subject":"Fixed bug. We added to the modified version of Agda an error message about a bad ATP pragma.","message":"Fixed bug. We added to the modified version of Agda an error message about a bad ATP pragma.\n\nIgnore-this: 9f9872e7fb0f52a921621da36c6c67e2\n  + (see mgda\/test\/fail\/ATPBadLocalHint.agda)\n\ndarcs-hash:20110322140048-3bd4e-a100e88d2ea22c7971d4bc8a8657e2ae4b4d38d6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bugs\/BadErrorMessage.agda","new_file":"Draft\/Bugs\/BadErrorMessage.agda","new_contents":"","old_contents":"-- Reported by Ana\n\nmodule Draft.Bugs.BadErrorMessage where\n\nopen import LTC.Base\nopen import LTC.Data.Nat\n\nS\u2238N2N : \u2200 {n} \u2192 N n \u2192 (\u2200 m \u2192 N m \u2192 N (n \u2238 m)) \u2192 \u2200 m \u2192 N m \u2192 N (succ n \u2238 m)\nS\u2238N2N {n} Nn h m = indN P' p0 pS\n  where P' : D \u2192 Set\n        P' x = N (succ n \u2238 x)\n        postulate p0 : P' zero\n                  pS : \u2200 {x} \u2192 N x \u2192 P' x \u2192 P' (succ x)\n        {-# ATP prove p0 Nn #-}\n\n-- Agda reports:\n\n-- An internal error has occurred. Please report this as\n-- a bug.  Location of the error:\n-- src\/full\/Agda\/Syntax\/Translation\/ConcreteToAbstract.hs:951\n\n-- The problem is we only accept definitions or data constructors in\n-- the local hints, i.e we cannot use Nn.\n\n-- TODO: To change the error message.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9eb37404d25db3e50aec9a6c0a7d38ed90f90bfc","subject":"Modified a note.","message":"Modified a note.\n\nIgnore-this: e363e2dbdc25ffe336fd77c86ab3cbc1\n\ndarcs-hash:20100810033516-3bd4e-5302ff591024eec2fc3e30a67559c06c534b54e1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/InternalSyntax.agda","new_file":"notes\/InternalSyntax.agda","new_contents":"------------------------------------------------------------------------------\n-- Agda internal syntax\n------------------------------------------------------------------------------\n\nmodule InternalSyntax where\n\n-- Based on Agda.Syntax.Internal\n\npostulate\n  A B C : Set\n\n------------------------------------------------------------------------------\n-- Examples\n\n-- The term is Def\npostulate defTerm : A\n-- El (Type (Lit (LitLevel  0))) (Def InternalSyntax.A [])\n\n-- The term is Fun\npostulate funTerm\u2081 : A \u2192 B\n-- {-# ATP prove funTerm\u2081 #-}\n\n-- El (Type (Lit (LitLevel  0)))\n--    (Fun (El (Type (Lit (LitLevel  0)))\n--             (Def InternalSyntax.A []))\n--         (El (Type (Lit (LitLevel  0)))\n--             (Def InternalSyntax.B [])))\n\npostulate funTerm\u2082 : A \u2192 B \u2192 C\n-- {-# ATP prove funTerm\u2082 #-}\n\n-- El (Type (Lit (LitLevel  0)))\n--    (Fun (El (Type (Lit (LitLevel  0)))\n--             (Def InternalSyntax.A []))\n--         (El (Type (Lit (LitLevel  0)))\n--             (Fun (El (Type (Lit (LitLevel  0)))\n--                      (Def InternalSyntax.B []))\n--                  (El (Type (Lit (LitLevel  0)))\n--                      (Def InternalSyntax.C [])))))\n\n\n-- The term is a (fake) Pi\npostulate piTerm : (a : A) \u2192 B\n-- {-# ATP prove piTerm #-}\n\n-- El (Type (Lit (LitLevel  0)))\n--    (Pi (El (Type (Lit (LitLevel  0)))\n--            (Def InternalSyntax.A []))\n--        (Abs \"a\" El (Type (Lit (LitLevel  0)))\n--                    (Def InternalSyntax.B [])))\n","old_contents":"------------------------------------------------------------------------------\n-- Agda internal syntax\n------------------------------------------------------------------------------\n\nmodule InternalSyntax where\n\n-- Based on Agda.Syntax.Internal\n\npostulate A : Set\n\n------------------------------------------------------------------------------\n-- Examples\n\n-- The term is Def\npostulate termDef : A\n-- El (Type (Lit (LitLevel  0))) (Def InternalSyntax.A [])\n\n-- The term is Fun\npostulate termFun1 : A \u2192 A\n-- El (Type (Lit (LitLevel  0)))\n--    (Fun (El (Type (Lit (LitLevel  0)))\n--             (Def InternalSyntax.A []))\n--         (El (Type (Lit (LitLevel  0)))\n--             (Def InternalSyntax.A [])))\n\npostulate termFun2 : A \u2192 A \u2192 A\n-- El (Type (Lit (LitLevel  0)))\n--    (Fun (El (Type (Lit (LitLevel  0)))\n--             (Def InternalSyntax.A []))\n--         (El (Type (Lit (LitLevel  0)))\n--             (Fun (El (Type (Lit (LitLevel  0)))\n--                      (Def InternalSyntax.A []))\n--                  (El (Type (Lit (LitLevel  0)))\n--                      (Def InternalSyntax.A [])))))\n\n-- The term is a (fake) Pi\npostulate termPi : (a : A) \u2192 A\n-- El (Type (Lit (LitLevel  0)))\n--    (Pi (El (Type (Lit (LitLevel  0)))\n--            (Def InternalSyntax.A []))\n--        (Abs \"a\" El (Type (Lit (LitLevel  0)))\n--                    (Def InternalSyntax.A [])))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f1d81fb8accc71c6f2eadb1dfe17738a90355873","subject":"Desc stratified model: from embedding to host","message":"Desc stratified model: from embedding to host\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"93d37cdce2649c4b0183b57d1b8bbccc43f8551b","subject":"agda: Add missing import (fix #32)","message":"agda: Add missing import (fix #32)\n\nOld-commit-hash: 3f1f4360ac2faf4df8b3ff3175ae5449bfb510be\n","repos":"inc-lc\/ilc-agda","old_file":"TotalNaturalSemantics.agda","new_file":"TotalNaturalSemantics.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"d0df7845258df353bcf850c309c4a494f9869248","subject":"Reorder code","message":"Reorder code\n\nSeparate non-CBPV and CBPV version of the code\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7TermCache \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7TermCache \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c3ae3858a74e759886e35af1f9a827078a446f2b","subject":"Avoided temporal directories names with Unicodes symbols.","message":"Avoided temporal directories names with Unicodes symbols.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddComm.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddComm.agda","new_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition of total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Auxiliary properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm-ind-instance :\n  \u2200 n \u2192\n  zero + n \u2261 n + zero \u2192\n  (\u2200 {m} \u2192 m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m) \u2192\n  \u2200 {m} \u2192 N m \u2192 m + n \u2261 n + m\n+-comm-ind-instance n = N-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n------------------------------------------------------------------------------\n-- Approach 1: Interactive proof using pattern matching\n\nmodule M1 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} nzero Nn =\n    zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n    n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n    n + zero \u220e\n\n  +-comm {n = n} (nsucc {m} Nm) Nn =\n    succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n    succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm Nm Nn) \u27e9\n    succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n    n + succ\u2081 m   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 2: Combined proof using pattern matching\n\nmodule M2 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} nzero Nn = prf\n    where postulate prf : zero + n \u2261 n + zero\n          {-# ATP prove prf +-rightIdentity #-}\n  +-comm {n = n} (nsucc {m} Nm) Nn = prf (+-comm Nm Nn)\n    where postulate prf : m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m\n          {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 3: Interactive proof using the induction principle\n\nmodule M3 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = i + n \u2261 n + i\n\n    A0 : A zero\n    A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n         n + zero \u220e\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n                succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n                succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n                n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 4: Combined proof using the induction principle\n\nmodule M4 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = i + n \u2261 n + i\n    {-# ATP definition A #-}\n\n    postulate A0 : A zero\n    {-# ATP prove A0 +-rightIdentity #-}\n\n    postulate is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 5: Interactive proof using an instance of the induction\n-- principle\n\nmodule M5 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = +-comm-ind-instance n A0 is Nm\n    where\n    A0 : zero + n \u2261 n + zero\n    A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n         n + zero \u220e\n\n    is : \u2200 {i} \u2192 i + n \u2261 n + i \u2192 succ\u2081 i + n \u2261 n + succ\u2081 i\n    is {i} ih = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n                succ\u2081 (i + n) \u2261\u27e8 succCong ih \u27e9\n                succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n                n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 6: Combined proof using an instance of the induction\n-- principle\n\nmodule M6 where\n\n  -- TODO (25 October 2012): Why is it not necessary the hypothesis\n  -- +-rightIdentity?\n  postulate +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  {-# ATP prove +-comm +-comm-ind-instance x+Sy\u2261S[x+y] #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition of total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Auxiliary properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm-ind-instance :\n  \u2200 n \u2192\n  zero + n \u2261 n + zero \u2192\n  (\u2200 {m} \u2192 m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m) \u2192\n  \u2200 {m} \u2192 N m \u2192 m + n \u2261 n + m\n+-comm-ind-instance n = N-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n------------------------------------------------------------------------------\n-- Approach 1: Interactive proof using pattern matching\n\nmodule M\u2081 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} nzero Nn =\n    zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n    n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n    n + zero \u220e\n\n  +-comm {n = n} (nsucc {m} Nm) Nn =\n    succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n    succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm Nm Nn) \u27e9\n    succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n    n + succ\u2081 m   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 2: Combined proof using pattern matching\n\nmodule M\u2082 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} nzero Nn = prf\n    where postulate prf : zero + n \u2261 n + zero\n          {-# ATP prove prf +-rightIdentity #-}\n  +-comm {n = n} (nsucc {m} Nm) Nn = prf (+-comm Nm Nn)\n    where postulate prf : m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m\n          {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 3: Interactive proof using the induction principle\n\nmodule M\u2083 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = i + n \u2261 n + i\n\n    A0 : A zero\n    A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n         n + zero \u220e\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n                succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n                succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n                n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 4: Combined proof using the induction principle\n\nmodule M\u2084 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = i + n \u2261 n + i\n    {-# ATP definition A #-}\n\n    postulate A0 : A zero\n    {-# ATP prove A0 +-rightIdentity #-}\n\n    postulate is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 5: Interactive proof using an instance of the induction\n-- principle\n\nmodule M\u2085 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = +-comm-ind-instance n A0 is Nm\n    where\n    A0 : zero + n \u2261 n + zero\n    A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n         n + zero \u220e\n\n    is : \u2200 {i} \u2192 i + n \u2261 n + i \u2192 succ\u2081 i + n \u2261 n + succ\u2081 i\n    is {i} ih = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n                succ\u2081 (i + n) \u2261\u27e8 succCong ih \u27e9\n                succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n                n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 6: Combined proof using an instance of the induction\n-- principle\n\nmodule M\u2086 where\n\n  -- TODO (25 October 2012): Why is it not necessary the hypothesis\n  -- +-rightIdentity?\n  postulate +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  {-# ATP prove +-comm +-comm-ind-instance x+Sy\u2261S[x+y] #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9654b48257da5d67093ffa9653ed62f10df0bfa4","subject":"Rework composition-sem-sec-reduction.agda","message":"Rework composition-sem-sec-reduction.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"composition-sem-sec-reduction.agda","new_file":"composition-sem-sec-reduction.agda","new_contents":"module composition-sem-sec-reduction where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP using (_,_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\nopen import program-distance\nopen import one-time-semantic-security\nimport bit-guessing-game\n\n-- One focuses on a particular kind of transformation\n-- over a cipher, namely pre and post composing functions\n-- to a given cipher E.\n-- This transformation can change the size of input\/output\n-- of the given cipher.\nmodule Comp (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  open EncParams\u00b2-same-|R|\u1d49 ep\u2080 |M|\u2081 |C|\u2081\n  open FlatFunsOps fun\u266dOps\n\n  comp : (pre-E : M\u2081 \u2192 M\u2080) (post-E : C\u2080 \u2192 C\u2081) \u2192 Tr\n  comp pre-E post-E E = pre-E >>> E >>> map\u21ba post-E\n\n-- This module shows how to adapt an adversary that is supposed to break\n-- one time semantic security of some cipher E enhanced by pre and post\n-- composition to an adversary breaking one time semantic security of the\n-- original cipher E.\n--\n-- The adversary transformation works for any notion of function space\n-- (FlatFuns) and the corresponding operations (FlatFunsOps).\n-- Because of this abstraction the following construction can be safely\n-- instantiated for at least three uses:\n--   * When provided with a concrete notion of function like circuits\n--     or at least functions over bit-vectors, one get a concrete circuit\n--     transformation process.\n--   * When provided with a high-level function space with real products\n--     then proving properties about the code is made easy.\n--   * When provided with a notion of cost such as time or space then\n--     one get a the cost of the resulting adversary given the cost of\n--     the inputs (adversary...).\nmodule CompRed {t} {T : Set t}\n               {\u266dFuns   : FlatFuns T}\n               (\u266dops    : FlatFunsOps \u266dFuns)\n               (ep\u2080 ep\u2081 : EncParams) where\n  open FlatFuns \u266dFuns\n  open FlatFunsOps \u266dops\n  open AbsSemSec \u266dFuns\n  open \u266dEncParams\u00b2 \u266dFuns ep\u2080 ep\u2081 using (`M\u2080; `C\u2080; `M\u2081; `C\u2081)\n\n  comp-red : (pre-E  : `M\u2081 `\u2192 `M\u2080)\n             (post-E : `C\u2080 `\u2192 `C\u2081)\n           \u2192 SemSecReduction ep\u2081 ep\u2080 _\n  comp-red pre-E post-E (mk A\u2080 A\u2081) =\n    mk (A\u2080 >>> (pre-E *** pre-E) *** idO) (post-E *** idO >>> A\u2081)\n\nmodule CompSec (prgDist : PrgDist) (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  open PrgDist prgDist\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist\n  open AbsSemSec fun\u266dFuns\n  open EncParams\u00b2-same-|R|\u1d49 ep\u2080 |M|\u2081 |C|\u2081\n  open SemSecReductions ep\u2080 ep\u2081 id\n  open CompRed fun\u266dOps ep\u2080 ep\u2081\n\n  private\n    module Comp-implicit {x y z} = Comp x y z\n  open Comp-implicit public\n\n  comp-pres-sem-sec : \u2200 pre-E post-E \u2192 SemSecTr (comp pre-E post-E)\n  comp-pres-sem-sec pre-E post-E = comp-red pre-E post-E , \u03bb E A \u2192 id\n\n  comp-pres-sem-sec\u207b\u00b9 : \u2200 pre-E pre-E\u207b\u00b9\n                          (pre-E-right-inv : pre-E \u2218 pre-E\u207b\u00b9 \u2257 id)\n                          post-E post-E\u207b\u00b9\n                          (post-E-left-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                        \u2192 SemSecTr\u207b\u00b9 (comp pre-E post-E)\n  comp-pres-sem-sec\u207b\u00b9 pre-E pre-E\u207b\u00b9 pre-E-inv post-E post-E\u207b\u00b9 post-E-inv =\n    SemSecTr\u2192SemSecTr\u207b\u00b9\n      (comp pre-E\u207b\u00b9 post-E\u207b\u00b9)\n      (comp pre-E post-E)\n      (\u03bb E m R \u2192 trans (post-E-inv _) (cong (\u03bb x \u2192 run\u21ba (E x) R) (pre-E-inv _)))\n      (comp-pres-sem-sec pre-E\u207b\u00b9 post-E\u207b\u00b9)\n\nmodule PostNegSec (prgDist : PrgDist) ep where\n  open EncParams ep\n  open EncParams\u00b2 ep ep using (Tr)\n  open CompSec prgDist ep |M| |C|\n  open FunSemSec prgDist\n  open SemSecReductions ep ep id\n\n  post-neg : Tr\n  post-neg = comp id vnot\n\n  post-neg-pres-sem-sec : SemSecTr post-neg\n  post-neg-pres-sem-sec = comp-pres-sem-sec id vnot\n\n  post-neg-pres-sem-sec\u207b\u00b9 : SemSecTr\u207b\u00b9 post-neg\n  post-neg-pres-sem-sec\u207b\u00b9 = comp-pres-sem-sec\u207b\u00b9 id id (\u03bb _ \u2192 refl) vnot vnot vnot\u2218vnot\u2257id\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess-prgDist     = bit-guessing-game prgDist\n  module FunSemSec-prgDist = FunSemSec prgDist\n  module CompSec-prgDist   = CompSec prgDist\n","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\nopen import program-distance\nopen import one-time-semantic-security\nimport bit-guessing-game\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\n\n\n\n\n\n\nmodule CompRed {t} {T : Set t}\n               {\u266dFuns : FlatFuns T}\n               (\u266dops : FlatFunsOps \u266dFuns)\n               (ep ep' : EncParams) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec \u266dFuns\n\n    open EncParams ep using (|M|; |C|)\n    open EncParams ep' using () renaming (|M| to |M|'; |C| to |C|')\n\n    M = `Bits |M|\n    C = `Bits |C|\n    M' = `Bits |M|'\n    C' = `Bits |C|'\n\n    comp-red : (pre-E : M' `\u2192 M)\n               (post-E : C `\u2192 C') \u2192 SemSecReduction ep' ep _\n    comp-red pre-E post-E (mk A\u2080 A\u2081) = mk (A\u2080 >>> (pre-E *** pre-E) *** idO) (post-E *** idO >>> A\u2081)\n\nmodule Comp (ep : EncParams) (|M|' |C|' : \u2115) where\n  open EncParams ep\n\n  ep' : EncParams\n  ep' = EncParams.mk |M|' |C|' |R|e\n\n  open EncParams ep' using () renaming (Enc to Enc'; M to M'; C to C')\n\n  Tr = Enc \u2192 Enc'\n\n  open FlatFunsOps fun\u266dOps\n\n  comp : (pre-E : M' \u2192 M) (post-E : C \u2192 C') \u2192 Tr\n  comp pre-E post-E E = pre-E >>> E >>> {-weaken\u2264 |R|e\u2264|R|e' \u2218-} map\u21ba post-E\n\nmodule CompSec (prgDist : PrgDist) (ep : EncParams) (|M|' |C|' : \u2115) where\n  open PrgDist prgDist\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist\n  open AbsSemSec fun\u266dFuns\n\n  open EncParams ep\n\n  ep' : EncParams\n  ep' = EncParams.mk |M|' |C|' |R|e\n\n  open EncParams ep' using () renaming (Enc to Enc'; M to M'; C to C')\n\n  module CompSec' (pre-E : M' \u2192 M) (post-E : C \u2192 C') where\n    open SemSecReductions ep ep'\n    open CompRed fun\u266dOps ep ep' hiding (M; C)\n    open Comp ep |M|' |C|'\n\n    comp-pres-sem-sec : SemSecTr id (comp pre-E post-E)\n    comp-pres-sem-sec = comp-red pre-E post-E , \u03bb E A \u2192 id\n\n  open SemSecReductions ep ep'\n  open CompRed fun\u266dOps ep ep' hiding (M; C)\n  open CompSec' public\n  module Comp' {x y z} = Comp x y z\n  open Comp' hiding (Tr)\n  comp-pres-sem-sec\u207b\u00b9 : \u2200 pre-E pre-E\u207b\u00b9\n                          (pre-E-right-inv : pre-E \u2218 pre-E\u207b\u00b9 \u2257 id)\n                          post-E post-E\u207b\u00b9\n                          (post-E-left-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                        \u2192 SemSecTr\u207b\u00b9 id (comp pre-E post-E)\n  comp-pres-sem-sec\u207b\u00b9 pre-E pre-E\u207b\u00b9 pre-E-inv post-E post-E\u207b\u00b9 post-E-inv =\n    SemSecTr\u2192SemSecTr\u207b\u00b9\n      (comp pre-E\u207b\u00b9 post-E\u207b\u00b9)\n      (comp pre-E post-E)\n      (\u03bb E m R \u2192 trans (post-E-inv _) (cong (\u03bb x \u2192 run\u21ba (E x) R) (pre-E-inv _)))\n      (comp-pres-sem-sec pre-E\u207b\u00b9 post-E\u207b\u00b9)\n\nmodule PostNegSec (prgDist : PrgDist) ep where\n  open EncParams ep\n  open Comp ep |M| |C| hiding (Tr)\n  open CompSec prgDist ep |M| |C|\n  open FunSemSec prgDist\n  open SemSecReductions ep ep\n\n  post-neg : Tr\n  post-neg = comp id vnot\n\n  post-neg-pres-sem-sec : SemSecTr id post-neg\n  post-neg-pres-sem-sec = comp-pres-sem-sec id vnot\n\n  post-neg-pres-sem-sec\u207b\u00b9 : SemSecTr\u207b\u00b9 id post-neg\n  post-neg-pres-sem-sec\u207b\u00b9 = comp-pres-sem-sec\u207b\u00b9 id id (\u03bb _ \u2192 refl) vnot vnot vnot\u2218vnot\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess' = bit-guessing-game prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module CompSec' = CompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e164d1dbe315b39e6c0d1d122f378b5518e8db4e","subject":"Added x>y\u2192x\u2264y\u2192\u22a5  (by Ana Bove).","message":"Added x>y\u2192x\u2264y\u2192\u22a5  (by Ana Bove).\n\nIgnore-this: 43a3cf055ae2cf5f599671e7674e06d0\n\ndarcs-hash:20110211221300-3bd4e-39cb32f77164f5222aefc6c1ae1c854a6570f18d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ \u00ac0<0 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ \u00acS<0 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ \u00acS<0 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x-y\u21920<Sx-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x-y\u21920<Sx-y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x-y\u21920<Sx-y zN (sN {n} Nn) h = \u22a5-elim (\u00acx<0 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x-y\u21920<Sx-y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x-y\u21920<Sx-y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ \u00ac0<0 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ \u00acS<0 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ \u00acS<0 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x-y\u21920<Sx-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x-y\u21920<Sx-y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x-y\u21920<Sx-y zN (sN {n} Nn) h = \u22a5-elim (\u00acx<0 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x-y\u21920<Sx-y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x-y\u21920<Sx-y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57e54a4fe45f2b1b2f38b06b249dea34cea8c3e2","subject":"K: without-K","message":"K: without-K\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/K.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/K.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Relation.Binary.PropositionalEquality.K where\n\nopen import Level.NP using (\u209b; _\u2294_)\nopen import Type using (\u2605\u2080; \u2605_)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; J; refl; !_; tr)\nopen import Relation.Binary.NP using (Decidable)\nopen import Relation.Nullary using (yes)\nopen import Function using (flip)\n\nmodule _ {a} {A : \u2605 a} where\n    K-Axiom : \u2605 a\n    K-Axiom = \u2200 {x : A} (p : x \u2261 x) \u2192 p \u2261 refl\n\n    K-Elim : \u2200 {\u2113} \u2192 \u2605(\u209b \u2113 \u2294 a)\n    K-Elim {\u2113} = {x : A}\n                 (P : x \u2261 x \u2192 \u2605 \u2113)\n                 (P-refl : P refl) (p : x \u2261 x) \u2192 P p\n\n    UIP-Axiom : \u2605 a\n    UIP-Axiom = \u2200 {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    K\u2192UIP : K-Axiom \u2192 UIP-Axiom\n    K\u2192UIP K {x} p q = J (\u03bb y q \u2192 (p : x \u2261 y) \u2192 p \u2261 q) K q p\n\n    UIP\u2192K : UIP-Axiom \u2192 K-Axiom\n    UIP\u2192K UIP p = UIP p refl\n\n    K-Elim\u2192K-Axiom : K-Elim \u2192 K-Axiom\n    K-Elim\u2192K-Axiom K-elim = K-elim (flip _\u2261_ refl) refl\n\n    K-Axiom\u2192K-Elim : \u2200 {\u2113} \u2192\u00a0K-Axiom \u2192 K-Elim {\u2113}\n    K-Axiom\u2192K-Elim K P P-refl p = tr P (! K\u2192UIP K p refl) P-refl\n\npostulate\n  KA : \u2605\u2080\n  K  : \u2200 {a} {A : \u2605 a} {{_ : KA}} {x : A} \u2192 (p : x \u2261 x) \u2192 p \u2261 refl\n\nmodule _ {a} {A : \u2605 a} {{_ : KA}} {x : A} where\n  K-elim : \u2200 {\u2113} (P : x \u2261 x \u2192 \u2605 \u2113)\n             (P-refl : P refl) (p : x \u2261 x) \u2192 P p\n  K-elim = K-Axiom\u2192K-Elim K\n\n  proof-irrelevance : {y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n  proof-irrelevance = K\u2192UIP K\n\n  UIP  = proof-irrelevance\n  _\u2261\u2261_ = proof-irrelevance\n\n  _\u225f\u2261_ : \u2200 {y : A} \u2192 Decidable {A = x \u2261 y} _\u2261_\n  _\u225f\u2261_ p q = yes (proof-irrelevance p q)\n\n  K-def : K {x = x} refl \u2261 refl\n  K-def = K (K refl)\n-- -}\n-- -}\n-- -}\n","old_contents":"module Relation.Binary.PropositionalEquality.K where\n\nopen import Type hiding (\u2605)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; J; refl)\nopen import Relation.Binary.NP using (Decidable)\nopen import Relation.Nullary using (yes)\n\npostulate\n  KA : \u2605\u2080\n\nmodule _ {a} {A : \u2605 a} {{_ : KA}} where\n\n  K : {x : A} (p : x \u2261 x) \u2192 p \u2261 refl\n  K refl = refl\n\n  proof-irrelevance : {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n  proof-irrelevance {x} p q = J (\u03bb y q \u2192 (p : x \u2261 y) \u2192 p \u2261 q) K q p\n\n  _\u2261\u2261_ : {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n  _\u2261\u2261_ = proof-irrelevance\n\n  _\u225f\u2261_ : \u2200 {i j : A} \u2192 Decidable {A = i \u2261 j} _\u2261_\n  _\u225f\u2261_ p q = yes (proof-irrelevance p q)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8407e2c488789cec10e51331385e8db8d7d2de00","subject":"Maybe","message":"Maybe\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"module Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = M?.join _\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\njust?\u2192IsJust : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\nAny-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\nf \u2218? g = join? \u2218 map? f \u2218 g\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6\u2605\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing  = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \u22a5-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n","old_contents":"module Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = M?.join _\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\njust?\u2192IsJust : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\nAny-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\nf \u2218? g = join? \u2218 map? f \u2218 g\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6\u2605\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing  = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \u22a5-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7f67ec294571e2bf50ed243bee66018ee9cab79f","subject":"commenting out some cases that no longer work with more precise indeterminate forms -- this reveals some stepping rules that are missing","message":"commenting out some cases that no longer work with more precise indeterminate forms -- this reveals some stepping rules that are missing\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!})\n  progress (TAAp D1 x\u2082 D2) | _ | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _ = E (EAp1 x)\n  -- progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I IEHole | V x\u2081 = I (IAp IEHole (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I (INEHole x) | V x\u2081 = I (IAp (INEHole x) (FVal x\u2081))\n  progress (TAAp D1 x\u2083 D2) | I (IAp x x\u2081) | V x\u2082 = I (IAp (IAp x x\u2081) (FVal x\u2082))\n  progress (TAAp D1 x\u2082 D2) | I (ICast x) | V x\u2081 = I (IAp (ICast x) (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , (Step {!!} (ITLam (FIndet {!!})) {!!}))\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  -- progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {\u03c3 = \u03c3} {u = u} {m = \u2717}  x x\u2081) = S (\u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 , {!!})\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V VConst = {!!} -- I (INEHole (FVal VConst))\n  progress (TANEHole x\u2081 (TALam D) x\u2082) | V VLam = {!!} -- I (INEHole (FVal VLam))\n  progress (TANEHole x\u2081 D x\u2082) | I x = {!!} -- I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = {!!}\n  progress (TACast TAConst TCHole2) | V VConst = {!!}\n  progress (TACast D x\u2081) | V VLam = {!x\u2081!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 ok d \u0394\n  progress TAConst = V VConst\n  progress (TAVar x) = abort (somenotnone (! x))\n  progress (TALam D) = V VLam\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress {\u0394 = \u0394} (TAAp D1 x\u2082 D2) | V VLam | V x\u2081 = S {\u0394 = \u0394} (_ , Step FRefl (ITLam (FVal x\u2081)) FRefl)\n  progress (TAAp (TALam D1) MAArr D2) | V VLam | I x\u2081 = S (_ , Step (FAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!})\n  progress (TAAp D1 x\u2082 D2) | _ | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _ = E (EAp1 x)\n  -- progress (TAAp D1 x\u2082 D2) | V x | S x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | I IEHole | V x\u2081 = I (IAp IEHole (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I (INEHole x) | V x\u2081 = I (IAp (INEHole x) (FVal x\u2081))\n  progress (TAAp D1 x\u2083 D2) | I (IAp x x\u2081) | V x\u2082 = I (IAp (IAp x x\u2081) (FVal x\u2082))\n  progress (TAAp D1 x\u2082 D2) | I (ICast x) | V x\u2081 = I (IAp (ICast x) (FVal x\u2081))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , (Step {!!} (ITLam (FIndet {!!})) {!!}))\n  progress (TAAp D1 x\u2082 D2) | S x | V x\u2081 = {!!}\n  progress (TAAp D1 x\u2082 D2) | S x | I x\u2081 = {!!}\n  -- progress (TAAp D1 x\u2082 D2) | S x | S x\u2081 = {!!}\n  progress (TAEHole x x\u2081) = I IEHole\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole x\u2081 D x\u2082) | V VConst = I (INEHole (FVal VConst))\n  progress (TANEHole x\u2081 (TALam D) x\u2082) | V VLam = I (INEHole (FVal VLam))\n  progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole x\u2083 D x\u2084) | S (d , Step x x\u2081 x\u2082) = S (_ , (Step (FNEHole x) x\u2081 (FNEHole x\u2082)))\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst TCRefl)  | V VConst = {!!}\n  progress (TACast TAConst TCHole2) | V VConst = {!!}\n  progress (TACast D x\u2081) | V VLam = {!x\u2081!}\n  progress (TACast D x\u2081) | I x = I (ICast x)\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S ( _ , Step (FCast x) x\u2081 (FCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"04ab50d04bb94dee97ff51a36ebc5921045b7b9c","subject":"_>>_ is no longer a definition this the CBV semantics was getting in the way","message":"_>>_ is no longer a definition this the CBV semantics was getting in the way\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS.agda","new_file":"lib\/FFI\/JS.agda","new_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate substring : String \u2192 Number \u2192 Number \u2192 String\n{-# COMPILED_JS substring require(\"libagda\").substring #-}\n\n-- substring with only one argument provided\npostulate substring1 : String \u2192 Number \u2192 String\n{-# COMPILED_JS substring1 require(\"libagda\").substring1 #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char require(\"libagda\").StringToChar #-}\n\npostulate checkTypeof : (type : String) \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS checkTypeof require(\"libagda\").checkTypeof #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber require(\"libagda\").checkTypeof(\"number\") #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString require(\"libagda\").checkTypeof(\"string\") #-}\n\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray require(\"libagda\").checkTypeof(\"array\") #-}\n\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject require(\"libagda\").checkTypeof(\"object\") #-}\n\npostulate castBool : JSValue \u2192 Bool\n{-# COMPILED_JS castBool require(\"libagda\").checkTypeof(\"bool\") #-}\n\ncastChar : JSValue \u2192 Char\ncastChar = String\u25b9Char \u2218 castString\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate undefinedJS : JSValue\n{-# COMPILED_JS undefinedJS undefined #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\n-- Same type as trace but does not print anything\n-- Usage:\n--   open import FFI.JS renaming (no-trace to trace)\nno-trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\nno-trace _ inp f = f inp\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\npostulate is-null : JSValue \u2192 Bool\n{-# COMPILED_JS is-null function(x) { return (x == null); } #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = toString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nJS! : Set\nJS! = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 JS!\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\n\npostulate _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n{-# COMPILED_JS _!\u2081_ require(\"libagda\").call1 #-}\n\npostulate _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n{-# COMPILED_JS _!\u2082_ require(\"libagda\").call2 #-}\n\npostulate _>>_ : JS! \u2192 JS! \u2192 JS!\n{-# COMPILED_JS _>>_ function(x) { return function (y) { return require(\"libagda\").call1(\"*\")(x)(function (z) { return y; }); }; } #-}\n-- Unfortunately so far such a definition can have a poor run-time semantics, where the second\n-- is needlessly computed. Worse given the use of partial functions such as assert, throw,\n-- cast{String,Number...} this can lead to abort the program.\n-- cmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate substring : String \u2192 Number \u2192 Number \u2192 String\n{-# COMPILED_JS substring require(\"libagda\").substring #-}\n\n-- substring with only one argument provided\npostulate substring1 : String \u2192 Number \u2192 String\n{-# COMPILED_JS substring1 require(\"libagda\").substring1 #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char require(\"libagda\").StringToChar #-}\n\npostulate checkTypeof : (type : String) \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS checkTypeof require(\"libagda\").checkTypeof #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber require(\"libagda\").checkTypeof(\"number\") #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString require(\"libagda\").checkTypeof(\"string\") #-}\n\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray require(\"libagda\").checkTypeof(\"array\") #-}\n\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject require(\"libagda\").checkTypeof(\"object\") #-}\n\npostulate castBool : JSValue \u2192 Bool\n{-# COMPILED_JS castBool require(\"libagda\").checkTypeof(\"bool\") #-}\n\ncastChar : JSValue \u2192 Char\ncastChar = String\u25b9Char \u2218 castString\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate undefinedJS : JSValue\n{-# COMPILED_JS undefinedJS undefined #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\n-- Same type as trace but does not print anything\n-- Usage:\n--   open import FFI.JS renaming (no-trace to trace)\nno-trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\nno-trace _ inp f = f inp\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\npostulate is-null : JSValue \u2192 Bool\n{-# COMPILED_JS is-null function(x) { return (x == null); } #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = toString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nJS! : Set\nJS! = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 JS!\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\n\npostulate _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n{-# COMPILED_JS _!\u2081_ require(\"libagda\").call1 #-}\n\npostulate _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n{-# COMPILED_JS _!\u2082_ require(\"libagda\").call2 #-}\n\n_>>_ : JS! \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6206561902ba89232563d784b73fa3eecfcc3f03","subject":"Added x<Sy\u2192x<y\u2228x\u2261y.","message":"Added x<Sy\u2192x<y\u2228x\u2261y.\n\nIgnore-this: 7a3604db707115ffe1d99d7a2508bcb1\n\ndarcs-hash:20100525024018-3bd4e-e83f8f49f5bd7c151711a82f9c9536f9203f8174.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Equalities.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\npostulate\n  \u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x x\u22650 #-}\n\npostulate\n  \u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n{-# ATP prove \u00acS\u22640 #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-00\nx\u2264x (sN {m} Nm) = prf (x\u2264x Nm)\n  where\n    postulate prf : LE m m \u2192 LE (succ m) (succ m)\n    {-# ATP prove prf #-}\n\n-- TODO: Why not a dot pattern?\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  prf (\u2264-trans Nm Nn No m\u2264n n\u2264o)\n    where\n      postulate m\u2264n : LE m n\n      {-# ATP prove m\u2264n #-}\n\n      postulate n\u2264o : LE n o\n      {-# ATP prove n\u2264o #-}\n\n      postulate prf : LE m o \u2192 LE (succ m) (succ o)\n      {-# ATP prove prf #-}\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x<Sy Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LT m (succ n) \u2192 LT (succ m) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx<Sy\u2192x<y\u2228x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim (\u00acx<0 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ n))) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 sN #-}\n\npostulate\n  \u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 sN #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\npostulate\n  \u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x x\u22650 #-}\n\npostulate\n  \u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n{-# ATP prove \u00acS\u22640 #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-00\nx\u2264x (sN {m} Nm) = prf (x\u2264x Nm)\n  where\n    postulate prf : LE m m \u2192 LE (succ m) (succ m)\n    {-# ATP prove prf #-}\n\n-- TODO: Why not a dot pattern?\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  prf (\u2264-trans Nm Nn No m\u2264n n\u2264o)\n    where\n      postulate m\u2264n : LE m n\n      {-# ATP prove m\u2264n #-}\n\n      postulate n\u2264o : LE n o\n      {-# ATP prove n\u2264o #-}\n\n      postulate prf : LE m o \u2192 LE (succ m) (succ o)\n      {-# ATP prove prf #-}\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x<Sy Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LT m (succ n) \u2192 LT (succ m) (succ (succ n))\n    {-# ATP prove prf #-}\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 sN #-}\n\npostulate\n  \u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 sN #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bc7d263f196c0ad7095bdd3211d7f045fe8caa1c","subject":"Rename E\u2264E into E<E","message":"Rename E\u2264E into E<E\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E<E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  1-I_ : [0,1] \u2192 [0,1]\n  1-I 0I    = 1I\n  1-I 1I    = 0I\n  1-I (x I) = (1-E x) I\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I  +I x    = x\n  x I +I 0I   = x I\n  1I  +I _    = 1I -- troublesome\n  x I +I 1I   = 1I -- troublesome\n  x I +I x\u2081 I = (x +E x\u2081) I -- faithful\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E<E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E<E pf) = E<E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E<E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E<E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E<E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E<E x\u2081) (E<E x\u2082) = E<E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E<E x\u2081) E\u2261E = E<E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E<E x\u2082) = E<E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E<E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E<E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E<E x\u2082) = E<E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  1-I_ : [0,1] \u2192 [0,1]\n  1-I 0I    = 1I\n  1-I 1I    = 0I\n  1-I (x I) = (1-E x) I\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I  +I x    = x\n  x I +I 0I   = x I\n  1I  +I _    = 1I -- troublesome\n  x I +I 1I   = 1I -- troublesome\n  x I +I x\u2081 I = (x +E x\u2081) I -- faithful\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4d0d904cd9e41814eac61a4cc741d1f33c069bb3","subject":"cleaning up a little bit. i should be able to avoid totally recapitualting the synth argument in ana..","message":"cleaning up a little bit. i should be able to avoid totally recapitualting the synth argument in ana..\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081)\n      with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc wt) x\u2082)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' with htype-dec x \u03c4'\n    expandability-ana {\u0393} {e \u00b7: .x} (ASubsume (SAsc wt) x\u2082) | d' , \u0394' , x , D' | Inl refl = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAsc2 D') x\u2082\n    expandability-ana {\u0393} {e \u00b7: x} (ASubsume (SAsc wt) x\u2082) | d' , \u0394' , \u03c4' , D' | Inr x\u2081 = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAsc1 D' x\u2081) x\u2082\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083)\n      with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EASubsume (\u03bb u \u2192 \u03bb ()) (\u03bb e' u \u2192 \u03bb ()) (ESLam x\u2082 D) x\u2083\n    expandability-ana {e = e1 \u2218 e2} (ASubsume (SAp wt1 MAHole wt2) x\u2082)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2 = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAp1 {!!} wt1 {!!} D2) x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081)\n      with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc wt) x\u2082)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' with htype-dec x \u03c4'\n    expandability-ana {\u0393} {e \u00b7: .x} (ASubsume (SAsc wt) x\u2082) | d' , \u0394' , x , D' | Inl refl = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAsc2 D') x\u2082\n    expandability-ana {\u0393} {e \u00b7: x} (ASubsume (SAsc wt) x\u2082) | d' , \u0394' , \u03c4' , D' | Inr x\u2081 = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAsc1 D' x\u2081) x\u2082\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083)\n      with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EASubsume (\u03bb u \u2192 \u03bb ()) (\u03bb e' u \u2192 \u03bb ()) (ESLam x\u2082 D) x\u2083\n    expandability-ana {e = e1 \u2218 e2} (ASubsume (SAp wt1 MAHole wt2) x\u2082)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2 = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAp1 {!!} wt1 {!!} D2) x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"17e27e3a4a22d33b05b8089a388f6ae0b5a5f50c","subject":"Binary.Logical: cleanup","message":"Binary.Logical: cleanup\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Logical.agda","new_file":"lib\/Relation\/Binary\/Logical.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.Logical where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\nopen import Relation.Nullary\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\n\n\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63\n\n\u27e6\u2605\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2080) \u2192 \u2605\u2081\n\u27e6\u2605\u2080\u27e7 = \u27e6\u2605\u27e7 zero\n\n\u27e6\u2605\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2081) \u2192 \u2605\u2082\n\u27e6\u2605\u2081\u27e7 = \u27e6\u2605\u27e7 (suc zero)\n\n-- old name\n\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : \u2605 a\u2081) (A\u2082 : \u2605 a\u2082) \u2192 \u2605 _\n\u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63\n\n-- old name\n\u27e6Set\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : Set) \u2192 Set\u2081\n\u27e6Set\u2080\u27e7 = \u27e6\u2605\u2080\u27e7\n\n-- old name\n\u27e6Set\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : Set\u2081) \u2192 Set\u2082\n\u27e6Set\u2081\u27e7 = \u27e6\u2605\u2081\u27e7\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 \u2605 _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 \u2605 _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\nrecord \u27e6\u22a4\u27e7 (x\u2081 x\u2082 : \u22a4) : \u2605\u2080 where\n  constructor \u27e6tt\u27e7\n\ndata \u27e6\u22a5\u27e7 (x\u2081 x\u2082 : \u22a5) : \u2605\u2080 where\n\ninfix 3 \u27e6\u00ac\u27e7_\n\n\u27e6\u00ac\u27e7_ : \u2200 {a\u2081 a\u2082 a\u209a} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u209a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) \u00ac_ \u00ac_\n\u27e6\u00ac\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u22a5\u27e7\n\n-- Products \u27e6\u03a3\u27e7, \u27e6\u2203\u27e7, \u27e6\u00d7\u27e7 are in Data.Product.NP\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n\u27e6Pred\u27e7 p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 p\u1d63\n\nprivate\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 \u2605 \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 r\u2081 r\u2082} r\u1d63 \u2192\n          (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (REL\u2032 r\u2081) (REL\u2032 r\u2082)\n  \u27e6REL\u27e7\u2032 r\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          {r\u2081 r\u2082} r\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 r\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 r\u2082) \u2192 \u2605 _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 r\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 r\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {r\u2081 r\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 r\u2081) (\u223c\u2082 : Rel A\u2082 r\u2082) \u2192 \u2605 _\n\u27e6Rel\u27e7 A\u1d63 r\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 r\u1d63\n\ndata \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : \u2605 \u2113\u2081} {P\u2082 : \u2605 \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 \u2605 \u2113\u1d63) : \u27e6\u2605\u27e7 (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) (Dec P\u2081) (Dec P\u2082) where\n  yes : {p\u2081 : P\u2081} {p\u2082 : P\u2082} (p\u1d63 : P\u1d63 p\u2081 p\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (yes p\u2081) (yes p\u2082)\n  no  : {\u00acp\u2081 : \u00ac P\u2081} {\u00acp\u2082 : \u00ac P\u2082} (\u00acp\u1d63 : (\u27e6\u00ac\u27e7 P\u1d63) \u00acp\u2081 \u00acp\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (no \u00acp\u2081) (no \u00acp\u2082)\n\nprivate\n  \u27e6Dec\u27e7' : \u2200 {p\u2081} {p\u2082} {p\u1d63} \u2192 \u27e6Pred\u27e7 {p\u2081} {p\u2082} _ (\u27e6\u2605\u27e7 p\u1d63) Dec Dec\n  \u27e6Dec\u27e7' = \u27e6Dec\u27e7\n\n\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082 \u2113\u1d63}\n              \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7\n                 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e9\u27e6\u2192\u27e7\n                   \u27e6REL\u27e7 A\u1d63 B\u1d63 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Decidable Decidable\n\u27e6Decidable\u27e7 A\u1d63 B\u1d63 _\u223c\u1d63_ = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e8 y\u1d63 \u2236 B\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Dec\u27e7 (x\u1d63 \u223c\u1d63 y\u1d63)\n\n\u27e6Op\u2081\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a) Op\u2081 Op\u2081\n\u27e6Op\u2081\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n\n\u27e6Op\u2082\u27e7 : \u2200 {a} \u2192 (\u27e6\u2605\u27e7 {a} {a} a \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 a) Op\u2082 Op\u2082\n\u27e6Op\u2082\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.Logical where\n\nopen import Type\nopen import Level\n\n\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : Set a\u2081) (A\u2082 : Set a\u2082) \u2192 Set _\n\u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63\n\n\u27e6\u2605\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605) \u2192 \u2605\u2081\n\u27e6\u2605\u2080\u27e7 = \u27e6\u2605\u27e7 zero\n\n\u27e6\u2605\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : \u2605\u2081) \u2192 \u2605\u2082\n\u27e6\u2605\u2081\u27e7 = \u27e6\u2605\u27e7 (suc zero)\n\n-- old name\n\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082} a\u1d63 (A\u2081 : Set a\u2081) (A\u2082 : Set a\u2082) \u2192 Set _\n\u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082 = A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63\n\n-- old name\n\u27e6Set\u2080\u27e7 : \u2200 (A\u2081 A\u2082 : Set) \u2192 Set\u2081\n\u27e6Set\u2080\u27e7 = \u27e6\u2605\u2080\u27e7\n\n-- old name\n\u27e6Set\u2081\u27e7 : \u2200 (A\u2081 A\u2082 : Set\u2081) \u2192 Set\u2082\n\u27e6Set\u2081\u27e7 = \u27e6\u2605\u2081\u27e7\n\n\u27e6\u03a0\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u03a0\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\ninfixr 0 \u27e6\u03a0\u27e7\nsyntax \u27e6\u03a0\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\n\u27e6\u03a0\u27e7e : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : (x : A\u2081) \u2192 B\u2081 x) (f\u2082 : (x : A\u2082) \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u03a0\u27e7e A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 x\u2081 x\u2082 (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 x\u2081) (f\u2082 x\u2082)\n\n\u27e6\u2200\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082} (B\u1d63 : \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n         (f\u2081 : {x : A\u2081} \u2192 B\u2081 x) (f\u2082 : {x : A\u2082} \u2192 B\u2082 x) \u2192 Set _\n\u27e6\u2200\u27e7 A\u1d63 B\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 B\u1d63 x\u1d63 (f\u2081 {x\u2081}) (f\u2082 {x\u2082})\n\ninfixr 0 \u27e6\u2200\u27e7\nsyntax \u27e6\u2200\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 f) = \u2200\u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 f\n\ninfixr 1 _\u27e6\u2192\u27e7_\n_\u27e6\u2192\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 Set _\nA\u1d63 \u27e6\u2192\u27e7 B\u1d63 = \u27e6\u03a0\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\ninfixr 0 _\u27e6\u2192\u27e7e_\n_\u27e6\u2192\u27e7e_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (f\u2081 : A\u2081 \u2192 B\u2081) (f\u2082 : A\u2082 \u2192 B\u2082) \u2192 Set _\n_\u27e6\u2192\u27e7e_ A\u1d63 B\u1d63 = \u27e6\u03a0\u27e7e A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\nopen import Data.Product\n\n\nopen import Data.Unit\n\nrecord \u27e6\u22a4\u27e7 (x\u2081 x\u2082 : \u22a4) : \u2605 where\n  constructor \u27e6tt\u27e7\n\nopen import Data.Empty\n\ndata \u27e6\u22a5\u27e7 (x\u2081 x\u2082 : \u22a5) : \u2605 where\n\nopen import Relation.Nullary\n\ninfix 3 \u27e6\u00ac\u27e7_\n\n--\u27e6\u00ac\u27e7_ : (\u27e6\u2605\u27e7 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7) \u00ac_ \u00ac_\n\u27e6\u00ac\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63) \u2192 \u00ac A\u2081 \u2192 \u00ac A\u2082 \u2192 Set _\n\u27e6\u00ac\u27e7 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u22a5\u27e7\n\n{-\n\u27e6\u2203\u27e7 : {A\u2081 A\u2082 : World \u2192 \u2605} (A\u1d63 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 A\u2081 \u03b1\u2081 \u2192 A\u2082 \u03b1\u2082 \u2192 \u2605)\n       (p\u2081 : \u2203 A\u2081) (f\u2082 : \u2203 A\u2082) \u2192 \u2605\n\u27e6\u2203\u27e7 A\u1d63 = \u03bb p\u2081 p\u2082 \u2192 \u03a3[ \u03b1\u1d63 \u2236 \u27e6World\u27e7 (proj\u2081 p\u2081) (proj\u2081 p\u2082) ]\n                       (A\u1d63 \u03b1\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n\nsyntax \u27e6\u2203\u27e7 (\u03bb \u03b1\u1d63 \u2192 f) = \u27e6\u2203\u27e7[ \u03b1\u1d63 ] f\n-}\n\nPred : \u2200 \u2113 {a} (A : Set a) \u2192 Set (a \u2294 suc \u2113)\nPred \u2113 A = A \u2192 Set \u2113\n\n\u27e6Pred\u27e7 : \u2200 {p\u2081 p\u2082} p\u1d63 {a\u2081 a\u2082 a\u1d63} \u2192 (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (Pred p\u2081) (Pred p\u2082)\n--\u27e6Pred\u27e7 {p\u2081} {p\u2082} p\u1d63 A\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 (p\u2081 \u2294 p\u2082 \u2294 p\u1d63)\n\u27e6Pred\u27e7 {p\u2081} {p\u2082} p\u1d63 A\u1d63 = \u03bb f\u2081 f\u2082 \u2192 \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) \u2192 f\u2081 x\u2081 \u2192 f\u2082 x\u2082 \u2192 Set (p\u2081 \u2294 p\u2082 \u2294 p\u1d63)\n\nopen import Relation.Binary\n\nprivate\n  REL\u2032 : \u2200 \u2113 {a b} \u2192 Set a \u2192 Set b \u2192 Set (a \u2294 b \u2294 suc \u2113)\n  REL\u2032 \u2113 A B = A \u2192 B \u2192 Set \u2113\n\n  \u27e6REL\u27e7\u2032 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082} \u2113\u1d63 \u2192\n          (\u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b\u2081} {b\u2082} b\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) (REL\u2032 \u2113\u2081) (REL\u2032 \u2113\u2082)\n  \u27e6REL\u27e7\u2032 \u2113\u1d63 A\u1d63 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 \u2113\u1d63)\n\n\u27e6REL\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          {\u2113\u2081 \u2113\u2082} \u2113\u1d63 (\u223c\u2081 : REL A\u2081 B\u2081 \u2113\u2081) (\u223c\u2082 : REL A\u2082 B\u2082 \u2113\u2082) \u2192 Set _\n\u27e6REL\u27e7 A\u1d63 B\u1d63 \u2113\u1d63 = A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 (\u27e6\u2605\u27e7 \u2113\u1d63)\n\n\u27e6Rel\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {\u2113\u2081 \u2113\u2082} \u2113\u1d63 (\u223c\u2081 : Rel A\u2081 \u2113\u2081) (\u223c\u2082 : Rel A\u2082 \u2113\u2082) \u2192 Set _\n\u27e6Rel\u27e7 A\u1d63 \u2113\u1d63 = \u27e6REL\u27e7 A\u1d63 A\u1d63 \u2113\u1d63\n\n-- data \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : Set \u2113\u2081} {P\u2082 : Set \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 Set \u2113\u1d63) : Dec P\u2081 \u2192 Dec P\u2082 \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) where\n-- data \u27e6Dec\u27e7 {\u2113\u1d63} : \u27e6Pred\u27e7 (\u27e6\u2605\u27e7 \u2113\u1d63) \u2113\u1d63 Dec Dec where\ndata \u27e6Dec\u27e7 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {P\u2081 : Set \u2113\u2081} {P\u2082 : Set \u2113\u2082} (P\u1d63 : P\u2081 \u2192 P\u2082 \u2192 Set \u2113\u1d63) : \u27e6\u2605\u27e7 (\u2113\u2081 \u2294 \u2113\u2082 \u2294 \u2113\u1d63) (Dec P\u2081) (Dec P\u2082) where\n  yes : {-\u2200 {P\u2081 P\u2082 P\u1d63}-} {p\u2081 : P\u2081} {p\u2082 : P\u2082} (p\u1d63 : P\u1d63 p\u2081 p\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (yes p\u2081) (yes p\u2082)\n  no  : {-\u2200 {P\u2081 P\u2082 P\u1d63}-} {\u00acp\u2081 : \u00ac P\u2081} {\u00acp\u2082 : \u00ac P\u2082} (\u00acp\u1d63 : (\u27e6\u00ac\u27e7 P\u1d63) \u00acp\u2081 \u00acp\u2082) \u2192 \u27e6Dec\u27e7 P\u1d63 (no \u00acp\u2081) (no \u00acp\u2082)\n\n--\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63 b\u2081 b\u2082 b\u1d63 \u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 {b\u2081 b\u2082} b\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6REL\u27e7 A\u1d63 B\u1d63 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) Decidable Decidable\n\u27e6Decidable\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n                 {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n                 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} {\u223c\u2081 : REL A\u2081 B\u2081 \u2113\u2081} {\u223c\u2082 : REL A\u2082 B\u2082 \u2113\u2082} (\u223c\u1d63 : \u27e6REL\u27e7 A\u1d63 B\u1d63 \u2113\u1d63 \u223c\u2081 \u223c\u2082)\n              \u2192 Decidable \u223c\u2081 \u2192 Decidable \u223c\u2082 \u2192 Set _\n\u27e6Decidable\u27e7 A\u1d63 B\u1d63 _\u223c\u1d63_ = \u27e8 x\u1d63 \u2236 A\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e8 y\u1d63 \u2236 B\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Dec\u27e7 (x\u1d63 \u223c\u1d63 y\u1d63)\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8f10f1845130f0f880c090ec1c369b667839e6ec","subject":"Adaptation to minor change in newer Agda","message":"Adaptation to minor change in newer Agda\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  open import Base.Data.DependentList public\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_]; reverse)\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4res = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4res) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4res [] = Term \u0393 \u03c4res\n    --hoasArgType \u0393 \u03c4res (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4res \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4res = foldr _ _\u21d2_ \u03c4res\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4res} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4res \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4res \u03c4s)\n\n    drop-\u227c-l : \u2200 {\u0393 \u0393\u2032 \u03c4} \u2192 (\u03c4 \u2022 \u0393 \u227c \u0393\u2032) \u2192 \u0393 \u227c \u0393\u2032\n    drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081 = \u227c-trans (drop _ \u2022 \u227c-refl) \u0393\u2032\u227c\u0393\u2032\u2081\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN []        f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN (\u03c4\u2081 \u2237 \u03c4s) f =\n      abs (absN \u03c4s\n        (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f\n            {\u0393\u227c\u0393\u2032 = drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081}\n            (weaken \u0393\u2032\u227c\u0393\u2032\u2081 (var this))))\n\n    -- Using a similar trick, we can declare absV (where V stands for variadic),\n    -- which takes the N implicit type arguments individually, collects them and\n    -- passes them on to absN. This is inspired by the example in the Agda 2.4.0\n    -- release notes about support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType zero       \u03c4s = absNType (reverse \u03c4s)\n    absVType (suc n) \u03c4s = {\u03c4\u2081 : Type} \u2192 absVType n (\u03c4\u2081 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN (reverse \u03c4s)\n    -- (Support for varying arity does not work here, so {\u03c4\u2081} must be bound in\n    -- the right-hand side).\n    absVAux \u03c4s (suc n) {\u03c4\u2081} = absVAux (\u03c4\u2081 \u2237 \u03c4s) n --\u03bb {\u03c4\u2081} \u2192 absVAux (\u03c4\u2081 \u2237 \u03c4s) n\n\n    -- \"Initialize\" accumulator to the empty list.\n    absV = absVAux []\n    -- We could maybe define absV directly, without going through absN, by\n    -- making hoasArgType and hoasResType also variadic, but I don't see a clear\n    -- advantage.\n\n  open AbsNHelpers using (absV) public\n\n  {-\n  Example types:\n  absV 1:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4res)\n\n  absV 2:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n       Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4res)\n\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  open import Base.Data.DependentList public\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_]; reverse)\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4res = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4res) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4res [] = Term \u0393 \u03c4res\n    --hoasArgType \u0393 \u03c4res (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4res \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4res = foldr _ _\u21d2_ \u03c4res\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4res} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4res \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4res \u03c4s)\n\n    drop-\u227c-l : \u2200 {\u0393 \u0393\u2032 \u03c4} \u2192 (\u03c4 \u2022 \u0393 \u227c \u0393\u2032) \u2192 \u0393 \u227c \u0393\u2032\n    drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081 = \u227c-trans (drop _ \u2022 \u227c-refl) \u0393\u2032\u227c\u0393\u2032\u2081\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN []        f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN (\u03c4\u2081 \u2237 \u03c4s) f =\n      abs (absN \u03c4s\n        (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f\n            {\u0393\u227c\u0393\u2032 = drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081}\n            (weaken \u0393\u2032\u227c\u0393\u2032\u2081 (var this))))\n\n    -- Using a similar trick, we can declare absV (where V stands for variadic),\n    -- which takes the N implicit type arguments individually, collects them and\n    -- passes them on to absN. This is inspired by the example in the Agda 2.4.0\n    -- release notes about support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType zero       \u03c4s = absNType (reverse \u03c4s)\n    absVType (suc n) \u03c4s = {\u03c4\u2081 : Type} \u2192 absVType n (\u03c4\u2081 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN (reverse \u03c4s)\n    -- (Support for varying arity does not work here, so {\u03c4\u2081} must be bound in\n    -- the right-hand side).\n    absVAux \u03c4s (suc n) = \u03bb {\u03c4\u2081} \u2192 absVAux (\u03c4\u2081 \u2237 \u03c4s) n\n\n    -- \"Initialize\" accumulator to the empty list.\n    absV = absVAux []\n    -- We could maybe define absV directly, without going through absN, by\n    -- making hoasArgType and hoasResType also variadic, but I don't see a clear\n    -- advantage.\n\n  open AbsNHelpers using (absV) public\n\n  {-\n  Example types:\n  absV 1:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4res)\n\n  absV 2:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4res : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n       Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4res) \u2192\n      Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4res)\n\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"48e352d611a13315454cb7afddbef107b207b99a","subject":"Add commented out attempt at eval-strengthen","message":"Add commented out attempt at eval-strengthen\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n zero k\u2264n v1 v2 x = tt\n  example1 n (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl = intV 1 , 0 , refl , (I.+ 1 , refl)\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n zero k\u2264n v1 v2 x = tt\n  example2 n (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl = intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq = {!!}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n    relV \u03c3 v1 v2 k \u2192\n    relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n zero k\u2264n v1 v2 x = tt\n  example1 n (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl = intV 1 , 0 , refl , (I.+ 1 , refl)\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n zero k\u2264n v1 v2 x = tt\n  example2 n (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl = intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) ff k k\u2264m = ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl =  v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 \u0393} (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t v1 \u03c12 n-j\n--   n1 sv1 sv2\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n1)\n--   n2 tv1 tv2\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n2)))\n--   (eq : apply sv1 tv1 n2 \u2261 Done v1 n-j) \u2192\n--   -- (tvv)\n--   -- (eqv1)\n--   \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03a3[ n3 \u2208 \u2115 ] eval (app s t) \u03c12 n2 \u2261 Done v2 n3 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n-- relV-apply s t v1 \u03c1 n-j n1 (closure st \u03c11) (closure st2 \u03c12) svv n2 tv1 tv2 tvv eq = {! !}\n-- -- relV-apply s t v1 \u03c12 n-j _ _ sv2 svv _ tv1 tv2 tvv eq\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , (\u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1))\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono _ _ (s\u2264s (n\u22641+n n2)) _ _ _ tvv ) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cb60f905c70aa857cc07ef5d1d560ff291e255d5","subject":"Sequence: combining and splitting >>= processes","message":"Sequence: combining and splitting >>= processes\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Sequence.agda","new_file":"Control\/Protocol\/Sequence.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_; first) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2219_; refl; ap; coe; coe!; !_) renaming (subst to tr)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol.Core\n\nmodule Control.Protocol.Sequence where\n\ninfixl 1 _>>=_ _>>_\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\n>>=-fst : \u2200 P {Q} \u2192 \u27e6 P >>= Q \u27e7 \u2192 \u27e6 P \u27e7\n>>=-fst end q = end\n>>=-fst (recv P) pq = \u03bb m \u2192 >>=-fst (P m) (pq m)\n>>=-fst (send P) (m , pq) = m , >>=-fst (P m) pq\n\n>>=-snd : \u2200 P {Q}(pq : \u27e6 P >>= Q \u27e7)(p : \u27e6 P \u22a5\u27e7) \u2192 \u27e6 Q (telecom P (>>=-fst P pq) p) \u27e7\n>>=-snd end      q        end     = q\n>>=-snd (recv P) pq       (m , p) = >>=-snd (P m) (pq m) p\n>>=-snd (send P) (m , pq) p       = >>=-snd (P m) pq (p m)\n\n[_]_>>>=_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 ((log : Log P) \u2192 \u27e6 Q log \u27e7) \u2192 \u27e6 P >>= Q \u27e7\n[ end    ] p       >>>= q = q _\n[ recv P ] p       >>>= q = \u03bb m \u2192 [ P m ] p m >>>= \u03bb log \u2192 q (m , log)\n[ send P ] (m , p) >>>= q = m , [ P m ] p >>>= \u03bb log \u2192 q (m , log)\n\n[_]_>>>_ : \u2200 P {Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P >> Q \u27e7\n[ P ] p >>> q = [ P ] p >>>= \u03bb _ \u2192 q\n\nmodule _ {{_ : FunExt}} where\n    >>=-fst-inv : \u2200 P {Q}(p : \u27e6 P \u27e7)(q : ((log : Log P) \u2192 \u27e6 Q log \u27e7)) \u2192 >>=-fst P {Q} ([ P ] p >>>= q) \u2261 p\n    >>=-fst-inv end      end     q = refl\n    >>=-fst-inv (recv P) p       q = \u03bb= \u03bb m \u2192 >>=-fst-inv (P m) (p m) \u03bb log \u2192 q (m , log)\n    >>=-fst-inv (send P) (m , p) q = snd= (>>=-fst-inv (P m) p \u03bb log \u2192 q (m , log))\n\n    >>=-snd-inv : \u2200 P {Q}(p : \u27e6 P \u27e7)(q : ((log : Log P) \u2192 \u27e6 Q log \u27e7))(p' : \u27e6 P \u22a5\u27e7)\n                  \u2192 tr (\u03bb x \u2192 \u27e6 Q (telecom P x p') \u27e7) (>>=-fst-inv P p q)\n                       (>>=-snd P {Q} ([ P ] p >>>= q) p') \u2261 q (telecom P p p')\n    >>=-snd-inv end          end     q p' = refl\n    >>=-snd-inv (recv P) {Q} p       q (m , p') = ap (flip _$_ (>>=-snd (P m) ([ P m ] p m >>>= (\u03bb log \u2192 q (m , log))) p'))\n                                                     (tr-\u03bb= (\u03bb z \u2192 \u27e6 Q (m , telecom (P m) z p') \u27e7)\n                                                               (\u03bb m \u2192 >>=-fst-inv (P m) (p m) (q \u2218 _,_ m)))\n                                                \u2219 >>=-snd-inv (P m) {Q \u2218 _,_ m} (p m) (\u03bb log \u2192 q (m , log)) p'\n    >>=-snd-inv (send P) {Q} (m , p) q p' = tr-snd= (\u03bb { (m , p) \u2192 \u27e6 Q (m , telecom (P m) p (p' m)) \u27e7 })\n                                                    (>>=-fst-inv (P m) p (q \u2218 _,_ m))\n                                                    (>>=-snd (P m) {Q \u2218 _,_ m} ([ P m ] p >>>= (q \u2218 _,_ m)) (p' m))\n                                          \u2219 >>=-snd-inv (P m) {Q \u2218 _,_ m} p (\u03bb log \u2192 q (m , log)) (p' m)\n\n    {- hmmm...\n    >>=-uniq : \u2200 P {Q} (pq : \u27e6 P >>= Q \u27e7)(p' : \u27e6 P \u22a5\u27e7) \u2192 pq \u2261 [ P ] (>>=-fst P {Q} pq) >>>= (\u03bb log \u2192 {!>>=-snd P {Q} pq p'!})\n    >>=-uniq = {!!}\n    -}\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\nreplicate-proc : \u2200 n P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 replicate\u1d3e n P \u27e7\nreplicate-proc zero    P p = end\nreplicate-proc (suc n) P p = [ P ] p >>> replicate-proc n P p\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  Log->>=-\u03a3 : \u2200 (P : Proto){Q} \u2192 Log (P >>= Q) \u2261 \u03a3 (Log P) (Log \u2218 Q)\n  Log->>=-\u03a3 end       = ! (\u00d7= End-uniq refl \u2219 \ud835\udfd9\u00d7-snd)\n  Log->>=-\u03a3 (com _ P) = \u03a3=\u2032 _ (\u03bb m \u2192 Log->>=-\u03a3 (P m)) \u2219 \u03a3-assoc\n\n  Log->>-\u00d7 : \u2200 (P : Proto){Q} \u2192 Log (P >> Q) \u2261 (Log P \u00d7 Log Q)\n  Log->>-\u00d7 P = Log->>=-\u03a3 P\n\n  ++Log' : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n  ++Log' P p q = coe! (Log->>=-\u03a3 P) (p , q)\n\n-- Same as ++Log' but computes better\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\nmodule _ {{_ : FunExt}} where\n    >>-right-unit : \u2200 P \u2192 (P >> end) \u2261 P\n    >>-right-unit end       = refl\n    >>-right-unit (com q P) = com= refl refl \u03bb m \u2192 >>-right-unit (P m)\n\n    >>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n                \u2192 (P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y)))) \u2261 ((P >>= Q) >>= R)\n    >>=-assoc end       Q R = refl\n    >>=-assoc (com q P) Q R = com= refl refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\n    dual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261 (dual P >> dual Q)\n    dual->> end        Q = refl\n    dual->> (com io P) Q = com= refl refl \u03bb m \u2192 dual->> (P m) Q\n\n    {- My coe-ap-fu is lacking...\n    dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261 dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n    dual->>= end Q = refl\n    dual->>= (com io P) Q = com= refl refl \u03bb m \u2192 dual->>= (P m) (Q \u2218 _,_ m) \u2219 ap (_>>=_ (dual (P m))) (\u03bb= \u03bb ms \u2192 ap (\u03bb x \u2192 dual (Q x)) (pair= {!!} {!!}))\n    -}\n\n    dual-replicate\u1d3e : \u2200 n P \u2192 dual (replicate\u1d3e n P) \u2261 replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    P = refl\n    dual-replicate\u1d3e (suc n) P = dual->> P (replicate\u1d3e n P) \u2219 ap (_>>_ (dual P)) (dual-replicate\u1d3e n P)\n\n{- An incremental telecom function which makes processes communicate\n   during a matching initial protocol. -}\n>>=-telecom : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n            \u2192 \u27e6 P >>= Q  \u27e7\n            \u2192 \u27e6 P >>= R \u22a5\u27e7\n            \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-telecom end      p0       p1       = _ , p0 , p1\n>>=-telecom (send P) (m , p0) p1       = first (_,_ m) (>>=-telecom (P m) p0 (p1 m))\n>>=-telecom (recv P) p0       (m , p1) = first (_,_ m) (>>=-telecom (P m) (p0 m) p1)\n\n>>-telecom : (P : Proto){Q R : Proto}\n           \u2192 \u27e6 P >> Q  \u27e7\n           \u2192 \u27e6 P >> R \u22a5\u27e7\n           \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-telecom P p q = >>=-telecom P p q\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_; first)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol.Core\n\nmodule Control.Protocol.Sequence where\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\nmodule _ {{_ : FunExt}} where\n    >>=-right-unit : \u2200 P \u2192 (P >> end) \u2261 P\n    >>=-right-unit end       = refl\n    >>=-right-unit (com q P) = com= refl refl \u03bb m \u2192 >>=-right-unit (P m)\n\n    >>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n                \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261 ((P >>= Q) >>= R)\n    >>=-assoc end       Q R = refl\n    >>=-assoc (com q P) Q R = com= refl refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\n    dual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261 dual P >> dual Q\n    dual->> end        Q = refl\n    dual->> (com io P) Q = com= refl refl \u03bb m \u2192 dual->> (P m) Q\n\n    {- My coe-ap-fu is lacking...\n    dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261 dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n    dual->>= end Q = refl\n    dual->>= (com io P) Q = com= refl refl \u03bb m \u2192 dual->>= (P m) (Q \u2218 _,_ m) \u2219 ap (_>>=_ (dual (P m))) (\u03bb= \u03bb ms \u2192 ap (\u03bb x \u2192 dual (Q x)) (pair= {!!} {!!}))\n    -}\n\n    dual-replicate\u1d3e : \u2200 n P \u2192 dual (replicate\u1d3e n P) \u2261 replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    P = refl\n    dual-replicate\u1d3e (suc n) P = dual->> P (replicate\u1d3e n P) \u2219 ap (_>>_ (dual P)) (dual-replicate\u1d3e n P)\n\n{- An incremental telecom function which makes processes communicate\n   during a matching initial protocol. -}\n>>=-telecom : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n            \u2192 \u27e6 P >>= Q  \u27e7\n            \u2192 \u27e6 P >>= R \u22a5\u27e7\n            \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-telecom end      p0       p1       = _ , p0 , p1\n>>=-telecom (send P) (m , p0) p1       = first (_,_ m) (>>=-telecom (P m) p0 (p1 m))\n>>=-telecom (recv P) p0       (m , p1) = first (_,_ m) (>>=-telecom (P m) (p0 m) p1)\n\n>>-telecom : (P : Proto){Q R : Proto}\n           \u2192 \u27e6 P >> Q  \u27e7\n           \u2192 \u27e6 P >> R \u22a5\u27e7\n           \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-telecom P p q = >>=-telecom P p q\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"93d05fd346dba30aade6b93c7bb8f6fe7cc3d329","subject":"Missing change in the last patch.","message":"Missing change in the last patch.\n\nIgnore-this: 3211beabafe1322efb98fa955570a8c1\n\ndarcs-hash:20110725165551-3bd4e-e1faddbbf8a037a1388aefb01f6ebe11ac35575b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/RenderToHTML.agda","new_file":"Draft\/RenderToHTML.agda","new_contents":"------------------------------------------------------------------------------\n-- Draft modules render to HTML\n------------------------------------------------------------------------------\n\nmodule Draft.RenderToHTML where\n\nopen import Draft.FOTC.Data.Stream.PropertiesATP\n","old_contents":"------------------------------------------------------------------------------\n-- Draft modules render to HTML\n------------------------------------------------------------------------------\n\nmodule Draft.RenderToHTML where\n\nopen import Draft.FixedPoints.LeastFixedPoints\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"49d45eb3a75431d8220026e47e886fe61730d145","subject":"Continue playing around with Smyth in Agda.","message":"Continue playing around with Smyth in Agda.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda2\/Text\/Greek\/Smyth.agda","new_file":"agda2\/Text\/Greek\/Smyth.agda","new_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc\u2032 : OlderLetter\n  \u03d8\u2032 : OlderLetter\n  \u03e1\u2032 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata AlwaysShort : Letter \u2192 Set where\n  \u03b5 : AlwaysShort \u03b5\n  \u03bf : AlwaysShort \u03bf\n\ndata AlwaysLong : Letter \u2192 Set where\n  \u03b7 : AlwaysLong \u03b7\n  \u03c9 : AlwaysLong \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  always-short : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysShort \u2113 \u2192 VowelWithLength v short\n  always-long : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysLong \u2113 \u2192 VowelWithLength v long\n  \u03b1-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b1 vl\n  \u03b9-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b9 vl\n  \u03c5-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03c5 vl\n\nalwaysShort-Vowel : \u2200 {\u2113} \u2192 AlwaysShort \u2113 \u2192 Vowel \u2113\nalwaysShort-Vowel \u03b5 = \u03b5\nalwaysShort-Vowel \u03bf = \u03bf\n\nalwaysLong-Vowel : \u2200 {\u2113} \u2192 AlwaysLong \u2113 \u2192 Vowel \u2113\nalwaysLong-Vowel \u03b7 = \u03b7\nalwaysLong-Vowel \u03c9 = \u03c9\n\n-- Smyth \u00a75\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9 : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5 : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n\n-- Smyth \u00a76\ndata SpuriousDiphthong : \u2200 {v\u2081 v\u2082} \u2192 Diphthong v\u2081 v\u2082 \u2192 Set where\n  \u03b5\u03b9-sp : SpuriousDiphthong \u03b5\u03b9\n  \u03bf\u03c5-sp : SpuriousDiphthong \u03bf\u03c5\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03c5 = closed\ntongue-position-vowel \u03c9 = open-medium\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5 = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata ConsonantSound : Letter \u2192 Set where\n  \u03b2 : ConsonantSound \u03b2\n  \u03b3 : ConsonantSound \u03b3\n  \u03b4 : ConsonantSound \u03b4\n  \u03b6 : ConsonantSound \u03b6\n  \u03b8 : ConsonantSound \u03b8\n  \u03ba : ConsonantSound \u03ba\n  \u03bb\u2032 : ConsonantSound \u03bb\u2032\n  \u03bc : ConsonantSound \u03bc\n  \u03bd : ConsonantSound \u03bd\n  \u03be : ConsonantSound \u03be\n  \u03c0 : ConsonantSound \u03c0\n  \u03c1 : ConsonantSound \u03c1\n  \u03c3 : ConsonantSound \u03c3\n  \u03c4 : ConsonantSound \u03c4\n  \u03c6 : ConsonantSound \u03c6\n  \u03c7 : ConsonantSound \u03c7\n  \u03c8 : ConsonantSound \u03c8\n  \u03b3-nasal : ConsonantSound \u03b3\n  \u03c1\u0314 : ConsonantSound \u03c1\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b3-nasal : Voiced \u03b3-nasal\n  \u03b6 : Voiced \u03b6\n\n-- Smyth \u00a715 a, b\ndata Voiceless : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c1\u0314 : Voiceless \u03c1\u0314\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n\n\n\n-- TODO (needs a unified model)\n\n-- Accent\n-- Smyth \u00a74 All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78 Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n\n","old_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1\u2032 capital = '\u0391'\nform \u03b1\u2032 small = '\u03b1'\nform \u03b2\u2032 capital = '\u0392'\nform \u03b2\u2032 small = '\u03b2'\nform \u03b3\u2032 capital = '\u0393'\nform \u03b3\u2032 small = '\u03b3'\nform \u03b4\u2032 capital = '\u0394'\nform \u03b4\u2032 small = '\u03b4'\nform \u03b5\u2032 capital = '\u0395'\nform \u03b5\u2032 small = '\u03b5'\nform \u03b6\u2032 capital = '\u0396'\nform \u03b6\u2032 small = '\u03b6'\nform \u03b7\u2032 capital = '\u0397'\nform \u03b7\u2032 small = '\u03b7'\nform \u03b8\u2032 capital = '\u0398'\nform \u03b8\u2032 small = '\u03b8'\nform \u03b9\u2032 capital = '\u0399'\nform \u03b9\u2032 small = '\u03b9'\nform \u03ba\u2032 capital = '\u039a'\nform \u03ba\u2032 small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc\u2032 capital = '\u039c'\nform \u03bc\u2032 small = '\u03bc'\nform \u03bd\u2032 capital = '\u039d'\nform \u03bd\u2032 small = '\u03bd'\nform \u03be\u2032 capital = '\u039e'\nform \u03be\u2032 small = '\u03be'\nform \u03bf\u2032 capital = '\u039f'\nform \u03bf\u2032 small = '\u03bf'\nform \u03c0\u2032 capital = '\u03a0'\nform \u03c0\u2032 small = '\u03c0'\nform \u03c1\u2032 capital = '\u03a1'\nform \u03c1\u2032 small = '\u03c1'\nform \u03c3\u2032 capital = '\u03a3'\nform \u03c3\u2032 small = '\u03c3'\nform \u03c4\u2032 capital = '\u03a4'\nform \u03c4\u2032 small = '\u03c4'\nform \u03c5\u2032 capital = '\u03a5'\nform \u03c5\u2032 small = '\u03c5'\nform \u03c6\u2032 capital = '\u03a6'\nform \u03c6\u2032 small = '\u03c6'\nform \u03c7\u2032 capital = '\u03a7'\nform \u03c7\u2032 small = '\u03c7'\nform \u03c8\u2032 capital = '\u03a8'\nform \u03c8\u2032 small = '\u03c8'\nform \u03c9\u2032 capital = '\u03a9'\nform \u03c9\u2032 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  not-end : PositionInWord\n  end-of-word : PositionInWord\n\nform-in-word : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 Char\nform-in-word \u03c3\u2032 small end-of-word = '\u03c2'\nform-in-word \u2113 c _ = form \u2113 c\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc\u2032 : OlderLetter\n  \u03d8\u2032 : OlderLetter\n  \u03e1\u2032 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1-vowel : Vowel \u03b1\u2032\n  \u03b5-vowel : Vowel \u03b5\u2032\n  \u03b7-vowel : Vowel \u03b7\u2032\n  \u03b9-vowel : Vowel \u03b9\u2032\n  \u03bf-vowel : Vowel \u03bf\u2032\n  \u03c5-vowel : Vowel \u03c5\u2032\n  \u03c9-vowel : Vowel \u03c9\u2032\n\ndata AlwaysShort : Letter \u2192 Set where\n  \u03b5-always-short : Vowel \u03b5\u2032 \u2192 AlwaysShort \u03b5\u2032\n  \u03bf-always-short : Vowel \u03bf\u2032 \u2192 AlwaysShort \u03bf\u2032\n\ndata AlwaysLong : Letter \u2192 Set where\n  \u03b7-always-long : Vowel \u03b7\u2032 \u2192 AlwaysLong \u03b7\u2032\n  \u03c9-always-long : Vowel \u03c9\u2032 \u2192 AlwaysLong \u03c9\u2032\n\n-- Smyth \u00a74 All vowels with the circumflex (149) are long\n\n-- Smyth \u00a7149\ndata Accent : Set where\n  acute circumflex grave : Accent\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a64586fb81337cb9798232af765e9d370f34540d","subject":"Nat: +-\u2264-inj sucx\u2270x \u00acn\u2264x<n \u00acn+\u2264y<n","message":"Nat: +-\u2264-inj sucx\u2270x \u00acn\u2264x<n \u00acn+\u2264y<n\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1b008fa84f7f38229355703ba23b5226918bcd80","subject":"Vec: shallow-\u03b7","message":"Vec: shallow-\u03b7\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : Set a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = \u2261.refl\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n      `tl : Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    eval `id             = id\n    eval (f `\u204f g)        = eval g \u2218 eval f\n    eval `0\u21941            = 0\u21941\n    eval (`tl f) {zero}  = id\n    eval (`tl f) {suc n} = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : Perm \u2192 Perm \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) [] = refl\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) [] = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl _)   [] = refl\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 n {A : Set a} (x : A) f \u2192 f \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl _) = refl\n    \u2219-replicate (suc n) x (`tl f) rewrite \u2219-replicate n x f = refl\n    \u2219-replicate n x (f `\u204f g) rewrite \u2219-replicate n x f | \u2219-replicate n x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map {n} f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate n f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : Set (a L.\u2294 b) where\n      `id `0\u21941 : Bij\n      _`\u204f_ : Bij \u2192 Bij \u2192 Bij\n      _`\u2237_ : Bij\u1d2c \u2192 (A \u2192 Bij) \u2192 Bij\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n             \u2192 (B \u2192 A)\n             \u2192 (Bij\u1d2c \u2192 Bij\u1d2e)\n             \u2192 Bij A Bij\u1d2c \u2192 Bij B Bij\u1d2e\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : Endo Bij\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    eval `id               = id\n    eval (f `\u204f g)          = eval g \u2218 eval f\n    eval `0\u21941              = 0\u21941\n    eval (f\u1d2c `\u2237 f) {zero}  = id\n    eval (f\u1d2c `\u2237 f) {suc n} = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : Bij \u2192 Bij \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk Bij (\u03bb f \u2192 eval f {n}) _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : Endo Bij\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : Perm \u2192 Bij\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 \u03c0 {n} (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) [] = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n      `tl : Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    eval : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    eval `id             = id\n    eval (f `\u204f g)        = eval g \u2218 eval f\n    eval `0\u21941            = 0\u21941\n    eval (`tl f) {zero}  = id\n    eval (`tl f) {suc n} = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : Perm \u2192 Perm \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) [] = refl\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) [] = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl _)   [] = refl\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 n {A : Set a} (x : A) f \u2192 f \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl _) = refl\n    \u2219-replicate (suc n) x (`tl f) rewrite \u2219-replicate n x f = refl\n    \u2219-replicate n x (f `\u204f g) rewrite \u2219-replicate n x f | \u2219-replicate n x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map {n} f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate n f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : Set (a L.\u2294 b) where\n      `id `0\u21941 : Bij\n      _`\u204f_ : Bij \u2192 Bij \u2192 Bij\n      _`\u2237_ : Bij\u1d2c \u2192 (A \u2192 Bij) \u2192 Bij\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n             \u2192 (B \u2192 A)\n             \u2192 (Bij\u1d2c \u2192 Bij\u1d2e)\n             \u2192 Bij A Bij\u1d2c \u2192 Bij B Bij\u1d2e\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : Endo Bij\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    eval : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    eval `id               = id\n    eval (f `\u204f g)          = eval g \u2218 eval f\n    eval `0\u21941              = 0\u21941\n    eval (f\u1d2c `\u2237 f) {zero}  = id\n    eval (f\u1d2c `\u2237 f) {suc n} = \u03bb xs \u2192 eval\u1d2c f\u1d2c (head xs) \u2237 eval (f (head xs)) (tail xs)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    _\u2219_ = eval\n\n    _\u2257\u2032_ : Bij \u2192 Bij \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk Bij (\u03bb f \u2192 eval f {n}) _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : Endo Bij\n    `tl f = id\u1d2c `\u2237 const f\n\n    private\n      module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : Perm \u2192 Bij\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 \u03c0 {n} (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) [] = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    private\n      module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f176af328cdd4ce50bb76ddf395f24090dc5bf5d","subject":"Added thesis example.","message":"Added thesis example.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/LogicalFramework\/AdequacyTheorems.agda","new_file":"notes\/thesis\/LogicalFramework\/AdequacyTheorems.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.AdequacyTheorems where\n\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\nmodule Example10 where\n  postulate\n    A B C : Set\n    f\u2081 : A \u2192 C\n    f\u2082 : B \u2192 C\n\n  f : A \u2228 B \u2192 C\n  f (inj\u2081 a) = f\u2081 a\n  f (inj\u2082 b) = f\u2082 b\n\n  f' : A \u2228 B \u2192 C\n  f' = case f\u2081 f\u2082\n\nmodule Example20 where\n\n  f : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f {A} {t} {.t} refl At = d At\n    where\n    postulate d : A t \u2192 A t\n\n  f' : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f' {A} h At = subst A h At\n\nmodule Example30 where\n  postulate\n    A B C E : Set\n    f\u2081 : A \u2192 E\n    f\u2082 : B \u2192 E\n    f\u2083 : C \u2192 E\n\n  f : (A \u2228 B) \u2228 C \u2192 E\n  f (inj\u2081 (inj\u2081 a)) = f\u2081 a\n  f (inj\u2081 (inj\u2082 b)) = f\u2082 b\n  f (inj\u2082 c)        = f\u2083 c\n\n  f' : (A \u2228 B) \u2228 C \u2192 E\n  f' = case (case f\u2081 f\u2082) f\u2083\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.AdequacyTheorems where\n\n-- First-order logic with equality.\nopen import Common.FOL.FOL-Eq public\n\nmodule Example10 where\n  postulate\n    A B C : Set\n    d : A \u2192 C\n    e : B \u2192 C\n\n  f : A \u2228 B \u2192 C\n  f (inj\u2081 a) = d a\n  f (inj\u2082 b) = e b\n\n  f' : A \u2228 B \u2192 C\n  f' = case d e\n\nmodule Example20 where\n\n  f : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f {A} {t} {.t} refl At = d At\n    where\n    postulate d : A t \u2192 A t\n\n  f' : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f' {A} h At = subst A h At\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fa4a76664ec96efcc42536b2b04df27085ddfbe5","subject":"bintree: another sorted predicate","message":"bintree: another sorted predicate\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n  -}\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nmodule Sorted {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : (x : A) \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low\u209c high\u209c low\u1d64 high\u1d64} \u2192\n             Sorted t low\u209c high\u209c \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             high\u209c \u2264\u1d2c low\u1d64 \u2192\n             Sorted (fork t u) low\u209c high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds (leaf ._)       = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n{-\n    data _\u2264\u1d3e_ : \u2200 {x y t} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set where\n      here-here   : here \u2264\u1d3e here\n      left-left   : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 left p \u2264\u1d3e left q\n      right-right : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 right p \u2264\u1d3e right q\n      left-right  : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 left p \u2264\u1d3e right q\n      -}\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = Sorted _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' (leaf ._)       here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)    (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)    (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)    (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted' _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    postulate dec-\u2264 : \u2200 x y \u2192 Dec (x \u2264\u1d2c y)\n    postulate dec-\u2294 : \u2200 x y \u2192 x \u2294\u1d2c y \u2261 x \u228e x \u2294\u1d2c y \u2261 y\n    postulate dec-\u2293 : \u2200 x y \u2192 x \u2293\u1d2c y \u2261 x \u228e x \u2293\u1d2c y \u2261 y\n\n    merge-spec : \u2200 {n} (t u : Tree A n) \u2192\n                 Sorted t \u2192 Sorted u \u2192 Sorted (merge t u)\n    merge-spec (leaf x) (leaf y) sx sy (left here) (left here) pf = \u2264\u1d2c.refl\n    merge-spec (leaf x) (leaf y) sx sy (left here) (right here) (0-1 ._ ._) = \u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)\n    merge-spec (leaf x) (leaf y) sx sy (right p) (left q) ()\n    merge-spec (leaf x) (leaf y) sx sy (right here) (right here) pf = \u2264\u1d2c.refl\n    merge-spec (fork t\u2080 t\u2081) (fork u\u2080 u\u2081) st su p q p\u2264q\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 -- | merge-spec t\u2080 u\u2080 ? ? | merge-spec t\u2081 u\u2081 ? ?\n    ... | fork l m\u2080    | fork m\u2081 h\n      with merge m\u2080 m\u2081 -- | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 -- | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with p | q | p\u2264q\n    ... | left  pp | left  qq | there ._ pp\u2264qq = {!merge-spec t\u2080 u\u2080 !}\n    ... | left  pp | right qq | 0-1 ._ ._ = {!!}\n    ... | right pp | left qq  | ()\n    ... | right pp | right qq | there ._ pp\u2264qq = {!!}\n\n    {-\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high\u209c = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high\u209c = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nmodule Sorted {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : (x : A) \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low\u209c high\u209c low\u1d64 high\u1d64} \u2192\n             Sorted t low\u209c high\u209c \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             high\u209c \u2264\u1d2c low\u1d64 \u2192\n             Sorted (fork t u) low\u209c high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds (leaf ._)       = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high\u209c = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high\u209c = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5d45b1db3c2d72b528f915ac0771a527c10ff2a6","subject":"Using unary function symbols in a note.","message":"Using unary function symbols in a note.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-ind :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-ind-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ind\/N-ind-ho\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind-ho\n\n  N-ind-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-ind-ho' = N-ind\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-unf : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-unf = N-ind A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-ind.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-ind B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-ind\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-ind A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-ind as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d        \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ\u2081 d) + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-ind A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-unf.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-unf Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-ind.\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base hiding ( pred-0 ; pred-S )\nopen import FOTC.Base.PropertiesI hiding ( predCong )\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-ind :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-ind-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ind\/N-ind-ho\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind-ho\n\n  N-ind-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-ind-ho' = N-ind\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-unf : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n')\n  N-unf = N-ind A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-ind.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-ind B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-ind\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-ind A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-ind as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d         \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ \u00b7 d) + e \u2261 succ \u00b7 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-ind A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ \u00b7 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-unf.\n\n  postulate\n    pred-0 : pred \u00b7 zero             \u2261 zero\n    pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n\n  predCong : \u2200 {m n} \u2192 m \u2261 n \u2192 pred \u00b7 m \u2261 pred \u00b7 n\n  predCong refl = refl\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred \u00b7 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-unf Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred \u00b7 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N (pred \u00b7 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ \u00b7 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-ind.\n\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a8b747cdfd660f56f63a228e6107d182214c8d4c","subject":"Do not import Relation.Nullary.Core","message":"Do not import Relation.Nullary.Core\n","repos":"np\/agda-parametricity","old_file":"lib\/Data\/Bool\/Param\/NatBool.agda","new_file":"lib\/Data\/Bool\/Param\/NatBool.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- This is an example of the use of logical relations\nmodule Data.Bool.Param.NatBool where\n\nopen import Data.Nat\nimport Data.Bool\n\nopen import Relation.Nullary\nopen import Type.Param.Binary\nopen import Function.Param.Binary\n\nrecord BOOL : Set\u2081 where\n  constructor mkBOOL\n  field\n    B           : Set\n    true false  : B\n    _\u2228_         : B \u2192 B \u2192 B\n\nmodule Implem where\n  \u2115Bool : Set\n  \u2115Bool = \u2115\n\n  false : \u2115Bool\n  false = 0\n\n  true : \u2115Bool\n  true = 1\n\n  _\u2228_ : \u2115Bool \u2192 \u2115Bool \u2192 \u2115Bool\n  m \u2228 n = m + n\n\n  -- this function breaks the abstraction\/model\n  is4? : \u2115Bool \u2192 \u2115Bool\n  is4? 4 = true\n  is4? _ = false\n\n  bool : BOOL\n  bool = mkBOOL \u2115Bool true false _\u2228_\n\nmodule Model where\n  open Implem\n\n  data \u27e6\u2115Bool\u27e7 : (x y : \u2115Bool) \u2192 Set where\n    \u27e6false\u27e7 : \u27e6\u2115Bool\u27e7 0 0\n    \u27e6true\u27e7  : \u2200 {m n} \u2192 \u27e6\u2115Bool\u27e7 (suc m) (suc n)\n\n  _\u27e6\u2228\u27e7_ : (\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) _\u2228_ _\u2228_\n  \u27e6false\u27e7  \u27e6\u2228\u27e7  x = x\n  \u27e6true\u27e7   \u27e6\u2228\u27e7  _ = \u27e6true\u27e7\n\n  \u00ac\u27e6is4?\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) is4? is4?)\n  \u00ac\u27e6is4?\u27e7 \u27e6is4?\u27e7 with \u27e6is4?\u27e7 {4} {3} \u27e6true\u27e7\n  ...               | ()\n\nreference : BOOL\nreference = mkBOOL Ref.Bool Ref.true Ref.false Ref._\u2228_\n  where module Ref = Data.Bool\n\nrecord BOOL-Sound (bool : BOOL) : Set\u2081 where\n  constructor mk\u27e6BOOL\u27e7\n  open BOOL bool\n  field\n    \u27e6B\u27e7      : \u27e6Set\u2080\u27e7 B B\n    \u27e6true\u27e7   : \u27e6B\u27e7 true true\n    \u27e6false\u27e7  : \u27e6B\u27e7 false false\n    _\u27e6\u2228\u27e7_    : (\u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7) _\u2228_ _\u2228_\n\nreference-sound : BOOL-Sound reference\nreference-sound = mk\u27e6BOOL\u27e7 _\u2261_ refl refl _\u27e6\u2228\u27e7_\n  where module Ref = Data.Bool\n        open import Relation.Binary.PropositionalEquality\n        _\u27e6\u2228\u27e7_ : (_\u2261_ \u27e6\u2192\u27e7 _\u2261_ \u27e6\u2192\u27e7 _\u2261_) Ref._\u2228_ Ref._\u2228_\n        _\u27e6\u2228\u27e7_ refl refl = refl\n\nBOOL-sound : BOOL-Sound Implem.bool\nBOOL-sound = mk\u27e6BOOL\u27e7 \u27e6\u2115Bool\u27e7 \u27e6true\u27e7 \u27e6false\u27e7 _\u27e6\u2228\u27e7_\n  where open Implem\n        open Model\n\nrecord \u27e6BOOL\u27e7 (x y : BOOL) : Set\u2081 where\n  constructor mk\u27e6BOOL\u27e7\n  module X = BOOL x\n  module Y = BOOL y\n  field\n    \u27e6B\u27e7      : \u27e6Set\u2080\u27e7 X.B Y.B\n    \u27e6true\u27e7   : \u27e6B\u27e7 X.true Y.true\n    \u27e6false\u27e7  : \u27e6B\u27e7 X.false Y.false\n    _\u27e6\u2228\u27e7_    : (\u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7) X._\u2228_ Y._\u2228_\n\n\u27e6Implem-bool\u27e7 : \u27e6BOOL\u27e7 Implem.bool Implem.bool\n\u27e6Implem-bool\u27e7 = mk\u27e6BOOL\u27e7 \u27e6\u2115Bool\u27e7 \u27e6true\u27e7 \u27e6false\u27e7 _\u27e6\u2228\u27e7_\n  where open Implem\n        open Model\n\n\u27e6reference\u27e7 : \u27e6BOOL\u27e7 reference reference\n\u27e6reference\u27e7 = mk\u27e6BOOL\u27e7 _\u2261_ refl refl _\u27e6\u2228\u27e7_\n  where module Ref = Data.Bool\n        open import Relation.Binary.PropositionalEquality\n        _\u27e6\u2228\u27e7_ : (_\u2261_ \u27e6\u2192\u27e7 _\u2261_ \u27e6\u2192\u27e7 _\u2261_) Ref._\u2228_ Ref._\u2228_\n        _\u27e6\u2228\u27e7_ refl refl = refl\n\n\u27e6bool-ref\u27e7 : \u27e6BOOL\u27e7 Implem.bool reference\n\u27e6bool-ref\u27e7 = mk\u27e6BOOL\u27e7 \u27e6B\u27e7 \u27e6true\u27e7 \u27e6false\u27e7 _\u27e6\u2228\u27e7_\n  where\n    open Implem    renaming (_\u2228_ to _\u2228\u2081_)\n    open Data.Bool renaming (false to false\u2082; true to true\u2082; _\u2228_ to _\u2228\u2082_)\n\n    data \u27e6B\u27e7 : \u2115Bool \u2192 Bool \u2192 Set where\n      \u27e6false\u27e7 : \u27e6B\u27e7 0 false\u2082\n      \u27e6true\u27e7  : \u2200 {m} \u2192 \u27e6B\u27e7 (suc m) true\u2082\n\n    _\u27e6\u2228\u27e7_ : (\u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7) _\u2228\u2081_ _\u2228\u2082_\n    \u27e6false\u27e7  \u27e6\u2228\u27e7  x = x\n    \u27e6true\u27e7   \u27e6\u2228\u27e7  _ = \u27e6true\u27e7\n\nClient : (A : BOOL \u2192 Set) \u2192 Set\u2081\nClient A = (bool : BOOL) \u2192 A bool\n\n{-\nopen import Data.Zero\nopen import Data.One\n\n\u27e6\u2115Bool\u27e7 : (x y : \u2115Bool) \u2192 Set\n\u27e6\u2115Bool\u27e7 0        0        = \ud835\udfd9\n\u27e6\u2115Bool\u27e7 (suc _)  (suc _)  = \ud835\udfd9\n\u27e6\u2115Bool\u27e7 _        _        = \ud835\udfd8\n\n_\u27e6\u2228\u27e7_ : (\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) _\u2228_ _\u2228_\n_\u27e6\u2228\u27e7_ _ {zero} {suc _} ()\n_\u27e6\u2228\u27e7_ _ {suc _} {zero} ()\n_\u27e6\u2228\u27e7_ {zero} {suc _} () _\n_\u27e6\u2228\u27e7_ {suc _} {zero} () _\n_\u27e6\u2228\u27e7_ {zero} {zero} _ {zero} {zero} _ = _\n_\u27e6\u2228\u27e7_ {zero} {zero} _ {suc _} {suc _} _ = _\n_\u27e6\u2228\u27e7_ {suc _} {suc _} _ {zero} {zero} _ = _\n_\u27e6\u2228\u27e7_ {suc _} {suc _} _ {suc _} {suc _} _ = _\n\n\u00ac\u27e6is4?\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) is4? is4?)\n\u00ac\u27e6is4?\u27e7 \u27e6is4?\u27e7 = \u27e6is4?\u27e7 {4} {2} _\n-}\n{-\n\nimport Data.Bool as B\nopen B using (Bool)\n\ntoBool : \u2115Bool \u2192 Bool\ntoBool 0 = B.false\ntoBool _ = B.true\n\nnot : \u2115Bool \u2192 \u2115Bool\nnot 0 = true\nnot _ = false\n\neven : \u2115Bool \u2192 \u2115Bool\neven zero     = true\neven (suc n)  = not (even n)\n\n\u00ac\u27e6even\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) even even)\n\u00ac\u27e6even\u27e7 \u27e6even\u27e7 with \u27e6even\u27e7 {1} {2} \u27e6true\u27e7\n... | ()\n\nopen import Data.Nat.Logical\n\nto\u2115 : \u2115Bool \u2192 \u2115\nto\u2115 n = n\n\n\u00ac\u27e6to\u2115\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) to\u2115 to\u2115)\n\u00ac\u27e6to\u2115\u27e7 \u27e6to\u2115\u27e7 with \u27e6to\u2115\u27e7 {1} {2} \u27e6true\u27e7\n... | suc ()\n-}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- This is an example of the use of logical relations\nmodule Data.Bool.Param.NatBool where\n\nopen import Data.Nat\nimport Data.Bool\n\nopen import Relation.Nullary.Core\nopen import Type.Param.Binary\nopen import Function.Param.Binary\n\nrecord BOOL : Set\u2081 where\n  constructor mkBOOL\n  field\n    B           : Set\n    true false  : B\n    _\u2228_         : B \u2192 B \u2192 B\n\nmodule Implem where\n  \u2115Bool : Set\n  \u2115Bool = \u2115\n\n  false : \u2115Bool\n  false = 0\n\n  true : \u2115Bool\n  true = 1\n\n  _\u2228_ : \u2115Bool \u2192 \u2115Bool \u2192 \u2115Bool\n  m \u2228 n = m + n\n\n  -- this function breaks the abstraction\/model\n  is4? : \u2115Bool \u2192 \u2115Bool\n  is4? 4 = true\n  is4? _ = false\n\n  bool : BOOL\n  bool = mkBOOL \u2115Bool true false _\u2228_\n\nmodule Model where\n  open Implem\n\n  data \u27e6\u2115Bool\u27e7 : (x y : \u2115Bool) \u2192 Set where\n    \u27e6false\u27e7 : \u27e6\u2115Bool\u27e7 0 0\n    \u27e6true\u27e7  : \u2200 {m n} \u2192 \u27e6\u2115Bool\u27e7 (suc m) (suc n)\n\n  _\u27e6\u2228\u27e7_ : (\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) _\u2228_ _\u2228_\n  \u27e6false\u27e7  \u27e6\u2228\u27e7  x = x\n  \u27e6true\u27e7   \u27e6\u2228\u27e7  _ = \u27e6true\u27e7\n\n  \u00ac\u27e6is4?\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) is4? is4?)\n  \u00ac\u27e6is4?\u27e7 \u27e6is4?\u27e7 with \u27e6is4?\u27e7 {4} {3} \u27e6true\u27e7\n  ...               | ()\n\nreference : BOOL\nreference = mkBOOL Ref.Bool Ref.true Ref.false Ref._\u2228_\n  where module Ref = Data.Bool\n\nrecord BOOL-Sound (bool : BOOL) : Set\u2081 where\n  constructor mk\u27e6BOOL\u27e7\n  open BOOL bool\n  field\n    \u27e6B\u27e7      : \u27e6Set\u2080\u27e7 B B\n    \u27e6true\u27e7   : \u27e6B\u27e7 true true\n    \u27e6false\u27e7  : \u27e6B\u27e7 false false\n    _\u27e6\u2228\u27e7_    : (\u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7) _\u2228_ _\u2228_\n\nreference-sound : BOOL-Sound reference\nreference-sound = mk\u27e6BOOL\u27e7 _\u2261_ refl refl _\u27e6\u2228\u27e7_\n  where module Ref = Data.Bool\n        open import Relation.Binary.PropositionalEquality\n        _\u27e6\u2228\u27e7_ : (_\u2261_ \u27e6\u2192\u27e7 _\u2261_ \u27e6\u2192\u27e7 _\u2261_) Ref._\u2228_ Ref._\u2228_\n        _\u27e6\u2228\u27e7_ refl refl = refl\n\nBOOL-sound : BOOL-Sound Implem.bool\nBOOL-sound = mk\u27e6BOOL\u27e7 \u27e6\u2115Bool\u27e7 \u27e6true\u27e7 \u27e6false\u27e7 _\u27e6\u2228\u27e7_\n  where open Implem\n        open Model\n\nrecord \u27e6BOOL\u27e7 (x y : BOOL) : Set\u2081 where\n  constructor mk\u27e6BOOL\u27e7\n  module X = BOOL x\n  module Y = BOOL y\n  field\n    \u27e6B\u27e7      : \u27e6Set\u2080\u27e7 X.B Y.B\n    \u27e6true\u27e7   : \u27e6B\u27e7 X.true Y.true\n    \u27e6false\u27e7  : \u27e6B\u27e7 X.false Y.false\n    _\u27e6\u2228\u27e7_    : (\u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7) X._\u2228_ Y._\u2228_\n\n\u27e6Implem-bool\u27e7 : \u27e6BOOL\u27e7 Implem.bool Implem.bool\n\u27e6Implem-bool\u27e7 = mk\u27e6BOOL\u27e7 \u27e6\u2115Bool\u27e7 \u27e6true\u27e7 \u27e6false\u27e7 _\u27e6\u2228\u27e7_\n  where open Implem\n        open Model\n\n\u27e6reference\u27e7 : \u27e6BOOL\u27e7 reference reference\n\u27e6reference\u27e7 = mk\u27e6BOOL\u27e7 _\u2261_ refl refl _\u27e6\u2228\u27e7_\n  where module Ref = Data.Bool\n        open import Relation.Binary.PropositionalEquality\n        _\u27e6\u2228\u27e7_ : (_\u2261_ \u27e6\u2192\u27e7 _\u2261_ \u27e6\u2192\u27e7 _\u2261_) Ref._\u2228_ Ref._\u2228_\n        _\u27e6\u2228\u27e7_ refl refl = refl\n\n\u27e6bool-ref\u27e7 : \u27e6BOOL\u27e7 Implem.bool reference\n\u27e6bool-ref\u27e7 = mk\u27e6BOOL\u27e7 \u27e6B\u27e7 \u27e6true\u27e7 \u27e6false\u27e7 _\u27e6\u2228\u27e7_\n  where\n    open Implem    renaming (_\u2228_ to _\u2228\u2081_)\n    open Data.Bool renaming (false to false\u2082; true to true\u2082; _\u2228_ to _\u2228\u2082_)\n\n    data \u27e6B\u27e7 : \u2115Bool \u2192 Bool \u2192 Set where\n      \u27e6false\u27e7 : \u27e6B\u27e7 0 false\u2082\n      \u27e6true\u27e7  : \u2200 {m} \u2192 \u27e6B\u27e7 (suc m) true\u2082\n\n    _\u27e6\u2228\u27e7_ : (\u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7 \u27e6\u2192\u27e7 \u27e6B\u27e7) _\u2228\u2081_ _\u2228\u2082_\n    \u27e6false\u27e7  \u27e6\u2228\u27e7  x = x\n    \u27e6true\u27e7   \u27e6\u2228\u27e7  _ = \u27e6true\u27e7\n\nClient : (A : BOOL \u2192 Set) \u2192 Set\u2081\nClient A = (bool : BOOL) \u2192 A bool\n\n{-\nopen import Data.Zero\nopen import Data.One\n\n\u27e6\u2115Bool\u27e7 : (x y : \u2115Bool) \u2192 Set\n\u27e6\u2115Bool\u27e7 0        0        = \ud835\udfd9\n\u27e6\u2115Bool\u27e7 (suc _)  (suc _)  = \ud835\udfd9\n\u27e6\u2115Bool\u27e7 _        _        = \ud835\udfd8\n\n_\u27e6\u2228\u27e7_ : (\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) _\u2228_ _\u2228_\n_\u27e6\u2228\u27e7_ _ {zero} {suc _} ()\n_\u27e6\u2228\u27e7_ _ {suc _} {zero} ()\n_\u27e6\u2228\u27e7_ {zero} {suc _} () _\n_\u27e6\u2228\u27e7_ {suc _} {zero} () _\n_\u27e6\u2228\u27e7_ {zero} {zero} _ {zero} {zero} _ = _\n_\u27e6\u2228\u27e7_ {zero} {zero} _ {suc _} {suc _} _ = _\n_\u27e6\u2228\u27e7_ {suc _} {suc _} _ {zero} {zero} _ = _\n_\u27e6\u2228\u27e7_ {suc _} {suc _} _ {suc _} {suc _} _ = _\n\n\u00ac\u27e6is4?\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) is4? is4?)\n\u00ac\u27e6is4?\u27e7 \u27e6is4?\u27e7 = \u27e6is4?\u27e7 {4} {2} _\n-}\n{-\n\nimport Data.Bool as B\nopen B using (Bool)\n\ntoBool : \u2115Bool \u2192 Bool\ntoBool 0 = B.false\ntoBool _ = B.true\n\nnot : \u2115Bool \u2192 \u2115Bool\nnot 0 = true\nnot _ = false\n\neven : \u2115Bool \u2192 \u2115Bool\neven zero     = true\neven (suc n)  = not (even n)\n\n\u00ac\u27e6even\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115Bool\u27e7) even even)\n\u00ac\u27e6even\u27e7 \u27e6even\u27e7 with \u27e6even\u27e7 {1} {2} \u27e6true\u27e7\n... | ()\n\nopen import Data.Nat.Logical\n\nto\u2115 : \u2115Bool \u2192 \u2115\nto\u2115 n = n\n\n\u00ac\u27e6to\u2115\u27e7 : \u00ac((\u27e6\u2115Bool\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) to\u2115 to\u2115)\n\u00ac\u27e6to\u2115\u27e7 \u27e6to\u2115\u27e7 with \u27e6to\u2115\u27e7 {1} {2} \u27e6true\u27e7\n... | suc ()\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"aa1582002d81126f197d6bab84ab20fd046902fc","subject":"goofing around with new rules","message":"goofing around with new rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') -- todo : make this a record so this is readable\n  progress TAConst = Inl VConst\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n  progress (TALam D) = Inl VLam\n  progress (TAAp D x D\u2081) with progress D | progress D\u2081\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inl x\u2081 = Inr (Inr (Inr (_ , (Step {!!} {!!} {!!}))))\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inr (Inr (Inl x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inl x | Inr (Inr (Inr x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inl x) | Inl x\u2081 = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inl x\u2081 = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inl x\u2081 = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inl x) | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2081 D\u2081) | Inr (Inl x) | Inr (Inr ih2) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inr (Inl x\u2081) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inr (Inr (Inl x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inl x)) | Inr (Inr (Inr x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inr (Inr (Inl x\u2081)) = {!!}\n  progress (TAAp D x\u2082 D\u2081) | Inr (Inr (Inr x)) | Inr (Inr (Inr x\u2081)) = {!!}\n  progress (TAEHole x x\u2081) = Inr (Inl IEHole)\n  progress (TANEHole x D x\u2081) with progress D\n  progress (TANEHole x\u2081 D x\u2082) | Inl x = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | Inr (Inl x) = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | Inr (Inr (Inl x)) = {!!}\n  progress (TANEHole x\u2081 D x\u2082) | Inr (Inr (Inr x)) = {!!}\n  progress (TACast D x) with progress D\n  progress (TACast D x\u2081) | Inl x = Inr (Inr (Inr (_ , Step FRefl (ITCast (FVal x) D x\u2081) FRefl)))\n  progress (TACast D x\u2081) | Inr (Inl x) = Inr (Inl (ICast x))\n  progress (TACast D x\u2081) | Inr (Inr (Inl x)) = Inr (Inr (Inl {!!} ))\n  progress (TACast D x\u2081) | Inr (Inr (Inr x)) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule progress where\n  progress : (\u0394 : hctx) (d : dhexp) (\u03c4 : htyp) \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  progress = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b061c79b5879df5c6015a46731aa08ce2f4ec0de","subject":"Added Stream-gfp\u2081.","message":"Added Stream-gfp\u2081.\n\nIgnore-this: ecb2ad21ee57f0174263434a111b2869\n\ndarcs-hash:20110926215314-3bd4e-38c884055e8d963b22d33fce58e06b2dacea8594.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/StreamSL.agda","new_file":"Draft\/FOTC\/Data\/Stream\/StreamSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC streams using the Agda coinductive combinators\n------------------------------------------------------------------------------\n\n-- Tested with the development version Agda and the standard library\n-- on 26 September 2011.\n\nmodule StreamSL where\n\nopen import Data.Product renaming ( _\u00d7_ to _\u2227_ )\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata Stream : D \u2192 Set where\n  consS : \u2200 x {xs} \u2192 \u221e (Stream xs) \u2192 Stream (x \u2237 xs)\n\nStream-gfp\u2081 : \u2200 {xs} \u2192 Stream xs \u2192\n              \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 Stream xs' \u2227 xs \u2261 x' \u2237 xs'\nStream-gfp\u2081 (consS x' {xs'} Sxs') = x' , xs' , \u266d Sxs' , refl\n\nStream-gfp\u2082 : (P : D \u2192 Set) \u2192\n              -- P is post-fixed point of StreamF.\n              (\u2200 {xs} \u2192 P xs \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n              -- Stream is greater than P.\n              \u2200 {xs} \u2192 P xs \u2192 Stream xs\nStream-gfp\u2082 P h {xs} Pxs = subst Stream (sym xs\u2261x'\u2237xs') prf\n  where\n    x' : D\n    x' = proj\u2081 (h Pxs)\n\n    xs' : D\n    xs' = proj\u2081 (proj\u2082 (h Pxs))\n\n    Pxs' : P xs'\n    Pxs' = proj\u2081 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' = proj\u2082 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    prf : Stream (x' \u2237 xs')\n    prf = consS x' (\u266f (Stream-gfp\u2082 P h Pxs'))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC streams using the Agda coinductive combinators\n------------------------------------------------------------------------------\n\n-- Tested with the development version Agda and the standard library\n-- on 26 September 2011.\n\nmodule StreamSL where\n\nopen import Data.Product renaming ( _\u00d7_ to _\u2227_ )\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata Stream : D \u2192 Set where\n  consS : \u2200 x {xs} \u2192 \u221e (Stream xs) \u2192 Stream (x \u2237 xs)\n\nStream-gfp\u2082 : (P : D \u2192 Set) \u2192\n              -- P is post-fixed point of StreamF.\n              (\u2200 {xs} \u2192 P xs \u2192 \u2203 \u03bb x' \u2192 \u2203 \u03bb xs' \u2192 P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n              -- Stream is greater than P.\n              \u2200 {xs} \u2192 P xs \u2192 Stream xs\nStream-gfp\u2082 P h {xs} Pxs = subst Stream (sym xs\u2261x'\u2237xs') prf\n  where\n    x' : D\n    x' = proj\u2081 (h Pxs)\n\n    xs' : D\n    xs' = proj\u2081 (proj\u2082 (h Pxs))\n\n    Pxs' : P xs'\n    Pxs' = proj\u2081 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' = proj\u2082 (proj\u2082 (proj\u2082 (h Pxs)))\n\n    prf : Stream (x' \u2237 xs')\n    prf = consS x' (\u266f (Stream-gfp\u2082 P h Pxs'))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"be0274dd27e35d690791eb80a7b4b66635414047","subject":"ANormalDTerm: remove bad experiments","message":"ANormalDTerm: remove bad experiments\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalDTerm.agda","new_file":"Thesis\/ANormalDTerm.agda","new_contents":"module Thesis.ANormalDTerm where\n\nopen import Thesis.ANormal\n\n\n\u0394\u0394 : Context \u2192 Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u0394\u0394 \u0393\n\u0394\u03c4 = \u0394t\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = \u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set\n-- [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205 = \u22a4\n-- [ \u03c4 \u2022 \u0394 ]\u0394 dv \u2022 d\u03c1 from v1 \u2022 \u03c11 to (v2 \u2022 \u03c12) = [ \u03c4 ]\u03c4 dv from v1 to v2 \u00d7 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12\n\ndata [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0394 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0394 ]\u0394 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nfromtoDeriveDVar : \u2200 {\u0394 \u03c4} \u2192 (x : Var \u0394 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12 \u2192\n  [ \u03c4 ]\u03c4 (\u27e6 derive-dvar x \u27e7Var d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveDVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveDVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveDVar x d\u03c1\u03c1\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm : (\u0394 : Context) (\u03c4 : Type) \u2192 Set where\n  dvar : \u2200 {\u0394 \u03c4} (x : Var (\u0394\u0394 \u0394) (\u0394t \u03c4)) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u0394 \u03c4 \u03c3 \u03c4\u2081} \u2192\n    (f : Var \u0394 (\u03c3 \u21d2 \u03c4\u2081)) \u2192\n    (x : Var \u0394 \u03c3) \u2192\n    (t : Term (\u03c4\u2081 \u2022 \u0394) \u03c4) \u2192\n    (df : Var (\u0394\u0394 \u0394) (\u0394\u03c4 (\u03c3 \u21d2 \u03c4\u2081))) \u2192\n    (dx : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)) \u2192\n    (dt : DTerm (\u03c4\u2081 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\ndata DFun (\u0394 : Context) : (\u03c4 : Type) \u2192 Set where\n  dterm : \u2200 {\u03c4} \u2192 DTerm \u0394 \u03c4 \u2192 DFun \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4} \u2192 DFun (\u03c3 \u2022 \u0394) \u03c4 \u2192 DFun \u0394 (\u03c3 \u21d2 \u03c4)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\nderive-dterm (var x) = dvar (derive-dvar x)\nderive-dterm (lett f x t) =\n  dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n\u27e6_\u27e7DTerm : \u2200 {\u0394 \u03c4} \u2192 DTerm \u0394 \u03c4 \u2192 \u27e6 \u0394 \u27e7Context \u2192 \u27e6 \u0394\u0394 \u0394 \u27e7Context \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n\u27e6 dvar x \u27e7DTerm \u03c1 d\u03c1 = \u27e6 x \u27e7Var d\u03c1\n\u27e6 dlett f x t df dx dt \u27e7DTerm \u03c1 d\u03c1 =\n  let v = (\u27e6 x \u27e7Var \u03c1)\n  in\n    \u27e6 dt \u27e7DTerm\n      (\u27e6 f \u27e7Var \u03c1 v \u2022 \u03c1)\n      (\u27e6 df \u27e7Var d\u03c1 v (\u27e6 dx \u27e7Var d\u03c1) \u2022 d\u03c1)\n\nfromtoDeriveDTerm : \u2200 {\u0394 \u03c4} \u2192 (t : Term \u0394 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12 \u2192\n  [ \u03c4 ]\u03c4 (\u27e6 derive-dterm t \u27e7DTerm \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDeriveDTerm (var x) d\u03c1\u03c1 = fromtoDeriveDVar x d\u03c1\u03c1\nfromtoDeriveDTerm (lett f x t) d\u03c1\u03c1 =\n  let fromToF = fromtoDeriveDVar f d\u03c1\u03c1\n      fromToX = fromtoDeriveDVar x d\u03c1\u03c1\n      fromToFX = fromToF _ _ _ fromToX\n  in fromtoDeriveDTerm t (fromToFX v\u2022 d\u03c1\u03c1)\n\nderive-dfun : \u2200 {\u0394 \u03c3} \u2192 (t : Fun \u0394 \u03c3) \u2192 DFun \u0394 \u03c3\nderive-dfun (term t) = dterm (derive-dterm t)\nderive-dfun (abs f) = dabs (derive-dfun f)\n\n\u27e6_\u27e7DFun : \u2200 {\u0394 \u03c4} \u2192 DFun \u0394 \u03c4 \u2192 \u27e6 \u0394 \u27e7Context \u2192 \u27e6 \u0394\u0394 \u0394 \u27e7Context \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n\u27e6 dterm t \u27e7DFun = \u27e6 t \u27e7DTerm\n\u27e6 dabs df \u27e7DFun \u03c1 d\u03c1 = \u03bb v dv \u2192 \u27e6 df \u27e7DFun (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n\nfromtoDeriveDFun : \u2200 {\u0394 \u03c4} \u2192 (f : Fun \u0394 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12 \u2192\n  [ \u03c4 ]\u03c4 (\u27e6 derive-dfun f \u27e7DFun \u03c11 d\u03c1) from (\u27e6 f \u27e7Fun \u03c11) to (\u27e6 f \u27e7Fun \u03c12)\nfromtoDeriveDFun (term t) = fromtoDeriveDTerm t\nfromtoDeriveDFun (abs f) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192 fromtoDeriveDFun f (dvv v\u2022 d\u03c1\u03c1)\n","old_contents":"module Thesis.ANormalDTerm where\n\nopen import Thesis.ANormal\n\n{-\nTry defining an alternative syntax for differential terms. We'll need this for\ncache terms. Maybe?\n\nOptions:\n\n1. use standard untyped lambda calculus as a target language. This requires much\nmore weakening in the syntax, which is a nuisance in proofs. Maybe use de Bruijn\nlevels instead?\n\n2. use a specialized syntax as a target language.\n\n3. use separate namespaces, environments, and so on?\n\nAlso, caching uses quite a bit of sugar on top of lambda calculus, with pairs\nand what not.\n-}\n\n{-\n-- To define an alternative syntax for the derivation output, let's remember that:\nderive (var x) = dvar dx\nderive (let y = f x in t) = let y = f x ; dy = df x dx in derive t\n-}\n\n-- Attempt 1 leads to function calls in datatype indexes. That's very very bad!\nmodule Try1 where\n\n-- data DTerm : (\u0393 : Context) (\u03c4 : Type) \u2192 Set where\n--   dvar : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192\n--     DTerm (\u0394\u0393 \u0393) (\u0394t \u03c4)\n--   dlett : \u2200 {\u0393 \u03c3 \u03c4\u2081 \u03c4} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192 DTerm (\u0394t \u03c4\u2081 \u2022 \u03c4\u2081 \u2022 \u0394\u0393 \u0393) (\u0394t \u03c4) \u2192 DTerm (\u0394\u0393 \u0393) (\u0394t \u03c4)\n\n-- derive-dterm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 DTerm (\u0394\u0393 \u0393) (\u0394t \u03c4)\n-- derive-dterm (var x) = dvar x\n-- derive-dterm (lett f x t) = dlett f x (derive-dterm t)\n\n-- Case split on dt FAILS!\n-- \u27e6_\u27e7DTerm : \u2200 {\u0393 \u03c4} \u2192 DTerm (\u0394\u0393 \u0393) (\u0394t \u03c4) \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n-- \u27e6 dt \u27e7DTerm \u03c1 = {!dt!}\n\n\u0394\u0394 : Context \u2192 Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u0394\u0394 \u0393\n\u0394\u03c4 = \u0394t\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = \u27e6 \u0394\u0394 \u0394 \u27e7Context\n\n-- changeStructure\u0394 : \u2200 \u0394 \u2192 ChangeStructure \u27e6 \u0394 \u27e7Context\n-- changeStructure\u0394 = {!!}\n-- ch_from_to_ {{changeStructure\u0394 \u0394}}\n-- [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set\n-- [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205 = \u22a4\n-- [ \u03c4 \u2022 \u0394 ]\u0394 dv \u2022 d\u03c1 from v1 \u2022 \u03c11 to (v2 \u2022 \u03c12) = [ \u03c4 ]\u03c4 dv from v1 to v2 \u00d7 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12\n\n-- \u0394 is bound to the change type! Not good!\ndata [_]\u0394_from_to_ : \u2200 \u0394 \u2192 Ch\u0394 \u0394 \u2192 (\u03c11 \u03c12 : \u27e6 \u0394 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0394 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0394 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0394 ]\u0394 (dv \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nmodule Try3 where\n  derive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\n  derive-dvar this = this\n  derive-dvar (that x) = that (derive-dvar x)\n\n  fromtoDeriveDVar : \u2200 {\u0394 \u03c4} \u2192 (x : Var \u0394 \u03c4) \u2192\n    \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 derive-dvar x \u27e7Var d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\n  fromtoDeriveDVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\n  fromtoDeriveDVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveDVar x d\u03c1\u03c1\n\n  -- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n  -- a result of type (\u0394t \u03c4).\n  data DTerm : (\u0394 : Context) (\u03c4 : Type) \u2192 Set where\n    dvar : \u2200 {\u0394 \u03c4} (x : Var (\u0394\u0394 \u0394) (\u0394t \u03c4)) \u2192\n      DTerm \u0394 \u03c4\n    dlett : \u2200 {\u0394 \u03c4 \u03c3 \u03c4\u2081} \u2192\n      (f : Var \u0394 (\u03c3 \u21d2 \u03c4\u2081)) \u2192\n      (x : Var \u0394 \u03c3) \u2192\n      (t : Term (\u03c4\u2081 \u2022 \u0394) \u03c4) \u2192\n      (df : Var (\u0394\u0394 \u0394) (\u0394\u03c4 (\u03c3 \u21d2 \u03c4\u2081))) \u2192\n      (dx : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)) \u2192\n      (dt : DTerm (\u03c4\u2081 \u2022 \u0394) \u03c4) \u2192\n      DTerm \u0394 \u03c4\n\n  data DFun (\u0394 : Context) : (\u03c4 : Type) \u2192 Set where\n    dterm : \u2200 {\u03c4} \u2192 DTerm \u0394 \u03c4 \u2192 DFun \u0394 \u03c4\n    dabs : \u2200 {\u03c3 \u03c4} \u2192 DFun (\u03c3 \u2022 \u0394) \u03c4 \u2192 DFun \u0394 (\u03c3 \u21d2 \u03c4)\n\n  derive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n  derive-dterm (var x) = dvar (derive-dvar x)\n  derive-dterm (lett f x t) =\n    dlett f x t (derive-dvar f) (derive-dvar x) (derive-dterm t)\n\n  \u27e6_\u27e7DTerm : \u2200 {\u0394 \u03c4} \u2192 DTerm \u0394 \u03c4 \u2192 \u27e6 \u0394 \u27e7Context \u2192 \u27e6 \u0394\u0394 \u0394 \u27e7Context \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n  \u27e6 dvar x \u27e7DTerm \u03c1 d\u03c1 = \u27e6 x \u27e7Var d\u03c1\n  \u27e6 dlett f x t df dx dt \u27e7DTerm \u03c1 d\u03c1 =\n    let v = (\u27e6 x \u27e7Var \u03c1)\n    in\n      \u27e6 dt \u27e7DTerm\n        (\u27e6 f \u27e7Var \u03c1 v \u2022 \u03c1)\n        (\u27e6 df \u27e7Var d\u03c1 v (\u27e6 dx \u27e7Var d\u03c1) \u2022 d\u03c1)\n\n  fromtoDeriveDTerm : \u2200 {\u0394 \u03c4} \u2192 (t : Term \u0394 \u03c4) \u2192\n    \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 derive-dterm t \u27e7DTerm \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\n  fromtoDeriveDTerm (var x) d\u03c1\u03c1 = fromtoDeriveDVar x d\u03c1\u03c1\n  fromtoDeriveDTerm (lett f x t) d\u03c1\u03c1 =\n    let fromToF = fromtoDeriveDVar f d\u03c1\u03c1\n        fromToX = fromtoDeriveDVar x d\u03c1\u03c1\n        fromToFX = fromToF _ _ _ fromToX\n    in fromtoDeriveDTerm t (fromToFX v\u2022 d\u03c1\u03c1)\n\n  derive-dfun : \u2200 {\u0394 \u03c3} \u2192 (t : Fun \u0394 \u03c3) \u2192 DFun \u0394 \u03c3\n  derive-dfun (term t) = dterm (derive-dterm t)\n  derive-dfun (abs f) = dabs (derive-dfun f)\n\n  \u27e6_\u27e7DFun : \u2200 {\u0394 \u03c4} \u2192 DFun \u0394 \u03c4 \u2192 \u27e6 \u0394 \u27e7Context \u2192 \u27e6 \u0394\u0394 \u0394 \u27e7Context \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n  \u27e6 dterm t \u27e7DFun = \u27e6 t \u27e7DTerm\n  \u27e6 dabs df \u27e7DFun \u03c1 d\u03c1 = \u03bb v dv \u2192 \u27e6 df \u27e7DFun (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n\n  fromtoDeriveDFun : \u2200 {\u0394 \u03c4} \u2192 (f : Fun \u0394 \u03c4) \u2192\n    \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0394 ]\u0394 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 derive-dfun f \u27e7DFun \u03c11 d\u03c1) from (\u27e6 f \u27e7Fun \u03c11) to (\u27e6 f \u27e7Fun \u03c12)\n  fromtoDeriveDFun (term t) = fromtoDeriveDTerm t\n  fromtoDeriveDFun (abs f) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192 fromtoDeriveDFun f (dvv v\u2022 d\u03c1\u03c1)\n\nmodule Try2 where\n  -- This is literally isomorphic to the source type :-| derivation is part of erasure...\n  data DTerm : (\u0393 : Context) (\u03c4 : Type) \u2192 Set where\n    dvar : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192\n      DTerm \u0393 \u03c4\n  --  dlett : \u2200 {\u0393 \u03c3 \u03c4\u2081 \u03c4} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192 DTerm (\u0394t \u03c4\u2081 \u2022 \u03c4\u2081 \u2022 \u0394\u0393 \u0393) (\u0394t \u03c4) \u2192 DTerm (\u0394\u0393 \u0393) (\u0394t \u03c4)\n\n    dlett : \u2200 {\u0393 \u03c4 \u03c3 \u03c4\u2081} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n      DTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 \u2192\n      DTerm \u0393 \u03c4\n\n  derive-dterm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 DTerm \u0393 \u03c4\n  derive-dterm (var x) = dvar x\n  derive-dterm (lett f x t) = dlett f x (derive-dterm t)\n\n  -- Incremental semantics over terms. Same results as base semantics over change\n  -- terms. But less weakening.\n  \u27e6_\u27e7DTerm : \u2200 {\u0393 \u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n  \u27e6 dvar x \u27e7DTerm d\u03c1 = \u27e6 derive-var x \u27e7Var d\u03c1\n  \u27e6 dlett f x dt \u27e7DTerm d\u03c1 =\n    let \u03c1 = \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n    in \u27e6 dt \u27e7DTerm\n      ( \u27e6 derive-var f \u27e7Var d\u03c1 (\u27e6 x \u27e7Var \u03c1) (\u27e6 derive-var x \u27e7Var d\u03c1)\n      \u2022 \u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1)\n      \u2022 d\u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3ef9b9505385dfad155ad9a494a161385271f576","subject":"ZK.GroupHom.ElGamal: add proofs of equality of message (two styles)","message":"ZK.GroupHom.ElGamal: add proofs of equality of message (two styles)\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom\/ElGamal.agda","new_file":"ZK\/GroupHom\/ElGamal.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Type.Eq\nopen import Function.NP using (flip; _\u2218_; it)\nopen import Data.Product.NP\nopen import Data.Two hiding (_\u00b2)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (idp; _\u2261_; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Algebra.Group.Homomorphism\nimport ZK.GroupHom\nopen import ZK.GroupHom.Types\n\nmodule ZK.GroupHom.ElGamal\n  (G+ G* : Type)\n  (\ud835\udd3e+ : Group G+)\n  (\ud835\udd3e* : Group G*)\n  (G*-eq? : Eq? G*)\n  (_^_ : G* \u2192 G+ \u2192 G*)\n  (^-hom : \u2200 b \u2192 GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* (_^_ b))\n  (g : G*)\n  where\n\nmodule \ud835\udd3e* = Group \ud835\udd3e*\nopen Additive-Group \ud835\udd3e+\nopen module MG = Multiplicative-Group \ud835\udd3e* hiding (_^_; _\u00b2)\nmodule ^ b = GroupHomomorphism (^-hom b)\n\n_\u00b2 : Type \u2192 Type\nA \u00b2 = A \u00d7 A\n\n-- TODO: Re-use another module\nmodule ElGamal-encryption where\n  PubKey  = G*\n  PrivKey = G+\n  EncRnd  = G+ {- randomness used for encryption of ct -}\n  Message = G* {- plain text message -}\n  CipherText = G* \u00d7 G*\n\n  module CipherText (ct : CipherText) where\n    \u03b1 \u03b2 : G*\n    \u03b1 = fst ct\n    \u03b2 = snd ct\n\n  pub-of : PrivKey \u2192 PubKey\n  pub-of sk = g ^ sk\n\n  enc : PubKey \u2192 Message \u2192 EncRnd \u2192 CipherText\n  enc y M r = \u03b1 , \u03b2\n    module enc where\n      \u03b1 = g ^ r\n      \u03b2 = (y ^ r) * M\n\n  dec : PrivKey \u2192 CipherText \u2192 Message\n  dec sk (\u03b1 , \u03b2) = (\u03b1 ^ sk)\u207b\u00b9 * \u03b2\n\nopen ElGamal-encryption\n\nopen Product\n\n_\u00b2-grp : {A : Type} \u2192 Group A \u2192 Group (A \u00b2)\n\ud835\udd38 \u00b2-grp = \u00d7-grp \ud835\udd38 \ud835\udd38\n\n\ud835\udd44 : Group Message\n\ud835\udd44 = \ud835\udd3e*\n\n\u2102\ud835\udd4b : Group CipherText\n\u2102\ud835\udd4b = \u00d7-grp \ud835\udd3e* \ud835\udd3e*\n\n-- CP : (g\u2080 g\u2081 u\u2080 u\u2081 : G) (w : \u2124q) \u2192 Type\n-- CP g\u2080 g\u2081 u\u2080 u\u2081 w = (g\u2080 ^ w \u2261 u\u2080) \u00d7 (g\u2081 ^ w \u2261 u\u2081)\n\n-- CP : (g\u2080 g\u2081 u\u2080 u\u2081 : G*) (w : G+) \u2192 Type\n-- CP g\u2080 g\u2081 u\u2080 u\u2081 w = \u2713 (((g\u2080 ^ w) == u\u2080) \u2227 ((g\u2081 ^ w) == u\u2081))\n\nmodule Known-enc-rnd\n         (y  : PubKey)\n         (M  : Message)\n         (ct : CipherText)\n         where\n  module ct = CipherText ct\n\n  Valid-witness : EncRnd \u2192 Type\n  Valid-witness r = enc y M r \u2261 ct\n\n  zk-hom : ZK-hom _ _ Valid-witness\n  zk-hom = record\n    { \u03c6-hom = < ^-hom g , ^-hom y >-hom\n    ; y = ct.\u03b1 , ct.\u03b2 \/ M\n    ; \u03c6\u21d2P = \u03bb _ e \u2192 ap\u2082 (\u03bb p q \u2192 fst p , q) e\n                        (ap (flip _*_ M \u2218 snd) e \u2219 ! \/-*)\n    ; P\u21d2\u03c6 = \u03bb _ e \u2192 ap\u2082 _,_ (ap fst e)\n                            (*-\/ \u2219 ap (flip _\/_ M) (ap snd e))\n    }\n\nopen \u2261-Reasoning\n\n\/\u207b\u00b9-* : \u2200 {x y} \u2192 (x \/ y)\u207b\u00b9 * x \u2261 y\n\/\u207b\u00b9-* {x} {y} =\n  (x \/ y)\u207b\u00b9 * x        \u2261\u27e8 ap (\u03bb z \u2192 z * x) \u207b\u00b9-hom\u2032 \u27e9\n  y \u207b\u00b9 \u207b\u00b9 * x \u207b\u00b9 * x   \u2261\u27e8 *-assoc \u27e9\n  y \u207b\u00b9 \u207b\u00b9 * (x \u207b\u00b9 * x) \u2261\u27e8 ap\u2082 _*_ \u207b\u00b9-involutive (fst \u207b\u00b9-inverse) \u27e9\n  y * 1#               \u2261\u27e8 *1-identity \u27e9\n  y                    \u220e\n\n\/-\u207b\u00b9* : \u2200 {x y} \u2192 x \/ (y \u207b\u00b9 * x) \u2261 y\n\/-\u207b\u00b9* {x} {y} =\n  x * (y \u207b\u00b9 * x) \u207b\u00b9    \u2261\u27e8 ap (_*_ x) \u207b\u00b9-hom\u2032 \u27e9\n  x * (x \u207b\u00b9 * y \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 ! *-assoc \u27e9\n  (x * x \u207b\u00b9) * y \u207b\u00b9 \u207b\u00b9 \u2261\u27e8 *= (snd \u207b\u00b9-inverse) \u207b\u00b9-involutive \u27e9\n  1# * y               \u2261\u27e8 1*-identity \u27e9\n  y                    \u220e\n\nx\u207b\u00b9y\u22611\u2192x\u2261y : \u2200 {x y} \u2192 x \u207b\u00b9 * y \u2261 1# \u2192 x \u2261 y\nx\u207b\u00b9y\u22611\u2192x\u2261y e = cancels-*-left (fst \u207b\u00b9-inverse \u2219 ! e)\n\nmodule Known-dec\n         (y  : PubKey)\n         (M  : Message)\n         (ct : CipherText)\n         where\n  module ct = CipherText ct\n\n  Valid-witness : PrivKey \u2192 Type\n  Valid-witness sk = (g ^ sk \u2261 y) \u00d7 (dec sk ct \u2261 M)\n\n  zk-hom : ZK-hom _ _ Valid-witness\n  zk-hom = record\n    { \u03c6-hom = < ^-hom g , ^-hom ct.\u03b1 >-hom\n    ; y = y , ct.\u03b2 \/ M\n    ; \u03c6\u21d2P = \u03bb x e \u2192 ap fst e , ap (\u03bb z \u2192 z \u207b\u00b9 * ct.\u03b2) (ap snd e) \u2219 \/\u207b\u00b9-*\n    ; P\u21d2\u03c6 = \u03bb x e \u2192 ap\u2082 _,_ (fst e) (! \/-\u207b\u00b9* \u2219 ap (_\/_ ct.\u03b2) (snd e))\n    }\n\n-- Inverse of ciphertexts\n_\u207b\u00b9CT : CipherText \u2192 CipherText\n(\u03b1 , \u03b2)\u207b\u00b9CT = \u03b1 \u207b\u00b9 , \u03b2 \u207b\u00b9\n\n-- Division of ciphertexts\n_\/CT_ : CipherText \u2192 CipherText \u2192 CipherText\n(\u03b1\u2080 , \u03b2\u2080) \/CT (\u03b1\u2081 , \u03b2\u2081) = (\u03b1\u2080 \/ \u03b1\u2081) , (\u03b2\u2080 \/ \u03b2\u2081)\n\n-- TODO: Move this elsewhere (Cipher.ElGamal.Homomorphic)\nimport Algebra.FunctionProperties.Eq as FP\nopen FP.Implicits\nopen import Algebra.Group.Abelian\nmodule From-*-comm\n         (*-comm : Commutative _*_)\n         where\n  private\n    module \ud835\udd3e*-comm = Abelian-Group-Struct (\ud835\udd3e*.grp-struct , *-comm)\n\n  hom-enc : (y : PubKey) \u2192 GroupHomomorphism (\u00d7-grp \ud835\udd44 \ud835\udd3e+) \u2102\ud835\udd4b (uncurry (enc y))\n  hom-enc y = mk \u03bb { {M\u2080 , r\u2080} {M\u2081 , r\u2081} \u2192\n    ap\u2082 _,_ (^.hom _)\n        (enc.\u03b2 y (M\u2080 * M\u2081) (r\u2080 + r\u2081)   \u2261\u27e8by-definition\u27e9\n         y ^ (r\u2080 + r\u2081)     * (M\u2080 * M\u2081) \u2261\u27e8 *= (^.hom y) idp \u27e9\n         (y ^ r\u2080 * y ^ r\u2081) * (M\u2080 * M\u2081) \u2261\u27e8 \ud835\udd3e*-comm.interchange \u27e9\n         enc.\u03b2 y M\u2080 r\u2080 * enc.\u03b2 y M\u2081 r\u2081 \u220e) }\n\n  module hom-enc y = GroupHomomorphism (hom-enc y)\n\n  -- The encryption of the inverse is the inverse of the encryption\n  -- (notice that the randomnesses seems to be negated only because we give an additive notation to our G+ group)\n  enc-\u207b\u00b9 : \u2200 {y M r} \u2192 enc y (M \u207b\u00b9) (0\u2212 r) \u2261 enc y M r \u207b\u00b9CT\n  enc-\u207b\u00b9 = hom-enc.pres-inv _\n\n  -- The encryption of the division is the division of the encryptions\n  -- (notice that the randomnesses seems to be subtracted only because we give an additive notation to our G+ group)\n  enc-\/ : \u2200 {y M\u2080 r\u2080 M\u2081 r\u2081} \u2192 enc y (M\u2080 \/ M\u2081) (r\u2080 \u2212 r\u2081) \u2261 enc y M\u2080 r\u2080 \/CT enc y M\u2081 r\u2081\n  enc-\/ = hom-enc.\u2212-\/ _\n\n  -- Alice wants to prove to the public that she encrypted the\n  -- same message for two (potentially different) recepients\n  -- without revealing the content of the message or the randomness\n  -- used for the encryptions.\n  module Message-equality-enc\n           (y\u2080  y\u2081  : PubKey)\n           (ct\u2080 ct\u2081 : CipherText)\n           where\n    module ct\u2080 = CipherText ct\u2080\n    module ct\u2081 = CipherText ct\u2081\n\n    Witness = Message \u00d7 EncRnd \u00b2\n    Statement = CipherText \u00b2\n\n    Valid-witness : Witness \u2192 Type\n    Valid-witness (M , r\u2080 , r\u2081) = enc y\u2080 M r\u2080 \u2261 ct\u2080 \u00d7 enc y\u2081 M r\u2081 \u2261 ct\u2081\n\n    private\n        \u03b8 : Message \u00d7 EncRnd \u00b2 \u2192 (Message \u00d7 EncRnd)\u00b2\n        \u03b8 (M , r\u2080 , r\u2081) = ((M , r\u2080) , (M , r\u2081))\n\n        \u03b8-hom : GroupHomomorphism (\u00d7-grp \ud835\udd44 (\ud835\udd3e+ \u00b2-grp)) ((\u00d7-grp \ud835\udd44 \ud835\udd3e+)\u00b2-grp) \u03b8\n        \u03b8-hom = mk idp\n\n    \u03c6-hom : GroupHomomorphism _ _ _\n    \u03c6-hom = < hom-enc y\u2080 \u00d7 hom-enc y\u2081 >-hom \u2218-hom \u03b8-hom\n\n    zk-hom : ZK-hom _ _ Valid-witness\n    zk-hom = record\n      { \u03c6-hom = \u03c6-hom\n      ; y = ct\u2080 , ct\u2081\n      ; \u03c6\u21d2P = \u03bb _ e \u2192 ap fst e , ap snd e\n      ; P\u21d2\u03c6 = \u03bb x e \u2192 ap\u2082 _,_ (fst e) (snd e)\n      }\n\n  module From-flip-^-hom\n           (flip-^-hom : \u2200 x \u2192 GroupHomomorphism \ud835\udd3e* \ud835\udd3e* (flip _^_ x))\n         where\n    module flip-^ x = GroupHomomorphism (flip-^-hom x)\n\n    hom-dec : (x : PrivKey) \u2192 GroupHomomorphism \u2102\ud835\udd4b \ud835\udd44 (dec x)\n    hom-dec x = mk \u03bb { {\u03b1\u2080 , \u03b2\u2080} {\u03b1\u2081 , \u03b2\u2081} \u2192\n      dec x (\u03b1\u2080 * \u03b1\u2081 , \u03b2\u2080 * \u03b2\u2081)           \u2261\u27e8by-definition\u27e9\n      ((\u03b1\u2080 * \u03b1\u2081) ^ x)\u207b\u00b9 * (\u03b2\u2080 * \u03b2\u2081)       \u2261\u27e8 ap (\u03bb z \u2192 _*_ (_\u207b\u00b9 z) (_*_ \u03b2\u2080 \u03b2\u2081)) (flip-^.hom x) \u27e9\n      (\u03b1\u2080 ^ x * \u03b1\u2081 ^ x)\u207b\u00b9 * (\u03b2\u2080 * \u03b2\u2081)     \u2261\u27e8 *= \ud835\udd3e*-comm.\u207b\u00b9-hom idp \u27e9\n      (\u03b1\u2080 ^ x)\u207b\u00b9 * (\u03b1\u2081 ^ x)\u207b\u00b9 * (\u03b2\u2080 * \u03b2\u2081) \u2261\u27e8 \ud835\udd3e*-comm.interchange \u27e9\n      dec x (\u03b1\u2080 , \u03b2\u2080) * dec x (\u03b1\u2081 , \u03b2\u2081)   \u220e }\n\n    module hom-dec x = GroupHomomorphism (hom-dec x)\n\n    -- Bob wants to prove to the public that he decrypted two\n    -- ciphertexts which decrypt to the same message,\n    -- without revealing the content of the message or his\n    -- secret key.\n    -- The two ciphertexts are encrypted using the same public key.\n    module Message-equality-dec\n         (y : PubKey)\n         (ct\u2080 ct\u2081 : CipherText)\n         where\n      private\n        module ct\u2080 = CipherText ct\u2080\n        module ct\u2081 = CipherText ct\u2081\n\n        \u03b1\/ = ct\u2080.\u03b1 \/ ct\u2081.\u03b1\n        \u03b2\/ = ct\u2080.\u03b2 \/ ct\u2081.\u03b2\n\n      Valid-witness : PrivKey \u2192 Type\n      Valid-witness sk = pub-of sk \u2261 y \u00d7 dec sk ct\u2080 \u2261 dec sk ct\u2081\n\n      zk-hom : ZK-hom _ _ Valid-witness\n      zk-hom = record\n        { \u03c6-hom = < ^-hom g , ^-hom (ct\u2080.\u03b1 \/ ct\u2081.\u03b1) >-hom\n        ; y = y , ct\u2080.\u03b2 \/ ct\u2081.\u03b2\n        ; \u03c6\u21d2P = \u03bb x e \u2192\n                  ap fst e ,\n                  x\/y\u22611\u2192x\u2261y\n                  (dec x ct\u2080 \/ dec x ct\u2081 \u2261\u27e8 ! hom-dec.\u2212-\/ x \u27e9\n                   dec x (ct\u2080 \/CT ct\u2081) \u2261\u27e8by-definition\u27e9\n                   dec x (\u03b1\/ , \u03b2\/) \u2261\u27e8 ! ap (\u03bb z \u2192 dec x (\u03b1\/ , snd z)) e \u27e9\n                   dec x (\u03b1\/ , (\u03b1\/)^ x) \u2261\u27e8 fst \u207b\u00b9-inverse \u27e9\n                   1# \u220e)\n        ; P\u21d2\u03c6 = \u03bb x e \u2192 ap\u2082 _,_ (fst e)\n                  (x\u207b\u00b9y\u22611\u2192x\u2261y\n                     (dec x (\u03b1\/ , \u03b2\/)       \u2261\u27e8 hom-dec.\u2212-\/ x \u27e9\n                      dec x ct\u2080 \/ dec x ct\u2081 \u2261\u27e8 \/= (snd e) idp \u27e9\n                      dec x ct\u2081 \/ dec x ct\u2081 \u2261\u27e8 snd \u207b\u00b9-inverse \u27e9\n                      1#                    \u220e))\n        }\n\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function using (flip; _\u2218_)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Two\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Algebra.Group.Homomorphism\nimport ZK.SigmaProtocol.KnownStatement\nimport ZK.GroupHom\nopen import ZK.GroupHom.Types\n\nmodule ZK.GroupHom.ElGamal\n  (G+ G* : Type)\n  (\ud835\udd3e+ : Group G+)\n  (\ud835\udd3e* : Group G*)\n  (_==_ : G* \u2192 G* \u2192 \ud835\udfda)\n  (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n  (==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y)\n  (_^_ : G* \u2192 G+ \u2192 G*)\n  (^-hom : \u2200 b \u2192 GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* (_^_ b))\n  (g : G*)\n  where\n\n  open Additive-Group \ud835\udd3e+\n  open Multiplicative-Group \ud835\udd3e* hiding (_^_)\n\n-- TODO: Re-use another module\n  module ElGamal-encryption where\n    record CipherText : Type where\n      constructor _,_\n      field\n        \u03b1 \u03b2 : G*\n\n    PubKey  = G*\n    PrivKey = G+\n    EncRnd  = G+ {- randomness used for encryption of ct -}\n    Message = G* {- plain text message -}\n\n    enc : PubKey \u2192 EncRnd \u2192 Message \u2192 CipherText\n    enc y r M = \u03b1 , \u03b2 where\n      \u03b1 = g ^ r\n      \u03b2 = (y ^ r) * M\n\n    dec : PrivKey \u2192 CipherText \u2192 Message\n    dec x ct = (\u03b1 ^ x)\u207b\u00b9 * \u03b2 where\n      open CipherText ct\n\n  open ElGamal-encryption\n\n  -- CP : (g\u2080 g\u2081 u\u2080 u\u2081 : G) (w : \u2124q) \u2192 Type\n  -- CP g\u2080 g\u2081 u\u2080 u\u2081 w = (g\u2080 ^ w \u2261 u\u2080) \u00d7 (g\u2081 ^ w \u2261 u\u2081)\n\n  -- CP : (g\u2080 g\u2081 u\u2080 u\u2081 : G*) (w : G+) \u2192 Type\n  -- CP g\u2080 g\u2081 u\u2080 u\u2081 w = \u2713 (((g\u2080 ^ w) == u\u2080) \u2227 ((g\u2081 ^ w) == u\u2081))\n\n  module _ (y  : PubKey)\n           (M  : Message)\n           (ct : CipherText)\n           where\n    module CT = CipherText ct\n\n    KnownEncRnd : EncRnd \u2192 Type\n    KnownEncRnd r = enc y r M \u2261 ct\n\n    known-enc-rnd : GrpHom _ _ KnownEncRnd\n    known-enc-rnd = record\n      { \ud835\udd3e+ = \ud835\udd3e+\n      ; \ud835\udd3e* = Product.\u00d7-grp \ud835\udd3e* \ud835\udd3e*\n      ; _==_ = \u03bb x y \u2192 fst x == fst y \u2227 snd x == snd y\n      ; \u2713-== = \u03bb e \u2192 \u2713\u2227 (\u2713-== (ap fst e)) (\u2713-== (ap snd e))\n      ; ==-\u2713 = \u03bb e \u2192 let p = \u2713\u2227\u00d7 e in\n                     ap\u2082 _,_ (==-\u2713 (fst p))\n                             (==-\u2713 (snd p))\n      ; \u03c6 = _\n      ; \u03c6-hom = Pair.pair-hom _ _ _ (_^_ g) (^-hom g)\n                                    (_^_ y) (^-hom y)\n      ; y = CT.\u03b1 , CT.\u03b2 \/ M\n      ; \u03c6\u21d2P = \u03bb _ e \u2192 ap\u2082 (\u03bb p q \u2192 fst p , q) e\n                          (ap (flip _*_ M \u2218 snd) e \u2219 ! \/-*)\n      ; P\u21d2\u03c6 = \u03bb _ e \u2192 ap\u2082 _,_ (ap CipherText.\u03b1 e)\n                              (*-\/ \u2219 ap (flip _\/_ M) (ap CipherText.\u03b2 e))\n      }\n\n    KnownDec : PrivKey \u2192 Type\n    KnownDec x = (g ^ x \u2261 y) \u00d7 (dec x ct \u2261 M)\n\n    open \u2261-Reasoning\n\n    \/\u207b\u00b9-* : \u2200 {x y} \u2192 (x \/ y)\u207b\u00b9 * x \u2261 y\n    \/\u207b\u00b9-* {x} {y} =\n      (x \/ y)\u207b\u00b9 * x        \u2261\u27e8 ap (\u03bb z \u2192 z * x) \u207b\u00b9-hom\u2032 \u27e9\n      y \u207b\u00b9 \u207b\u00b9 * x \u207b\u00b9 * x   \u2261\u27e8 *-assoc \u27e9\n      y \u207b\u00b9 \u207b\u00b9 * (x \u207b\u00b9 * x) \u2261\u27e8 ap\u2082 _*_ \u207b\u00b9-involutive (fst \u207b\u00b9-inverse) \u27e9\n      y * 1#               \u2261\u27e8 *1-identity \u27e9\n      y                    \u220e\n\n    \/-\u207b\u00b9* : \u2200 {x y} \u2192 x \/ (y \u207b\u00b9 * x) \u2261 y\n    \/-\u207b\u00b9* {x} {y} =\n      x * (y \u207b\u00b9 * x) \u207b\u00b9    \u2261\u27e8 ap (_*_ x) \u207b\u00b9-hom\u2032 \u27e9\n      x * (x \u207b\u00b9 * y \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 ! *-assoc \u27e9\n      (x * x \u207b\u00b9) * y \u207b\u00b9 \u207b\u00b9 \u2261\u27e8 *= (snd \u207b\u00b9-inverse) \u207b\u00b9-involutive \u27e9\n      1# * y               \u2261\u27e8 1*-identity \u27e9\n      y                    \u220e\n\n    known-dec : GrpHom _ _ KnownDec\n    known-dec = record\n      { \ud835\udd3e+ = \ud835\udd3e+\n      ; \ud835\udd3e* = Product.\u00d7-grp \ud835\udd3e* \ud835\udd3e*\n      ; _==_ = \u03bb x y \u2192 fst x == fst y \u2227 snd x == snd y\n      ; \u2713-== = \u03bb e \u2192 \u2713\u2227 (\u2713-== (ap fst e)) (\u2713-== (ap snd e))\n      ; ==-\u2713 = \u03bb e \u2192 let p = \u2713\u2227\u00d7 e in\n                     ap\u2082 _,_ (==-\u2713 (fst p))\n                             (==-\u2713 (snd p))\n      ; \u03c6 = _\n      ; \u03c6-hom = Pair.pair-hom _ _ _ (_^_ g)    (^-hom g)\n                                    (_^_ CT.\u03b1) (^-hom CT.\u03b1)\n      ; y = y , CT.\u03b2 \/ M\n      ; \u03c6\u21d2P = \u03bb x e \u2192 ap fst e , ap (\u03bb z \u2192 z \u207b\u00b9 * CT.\u03b2) (ap snd e) \u2219 \/\u207b\u00b9-*\n      ; P\u21d2\u03c6 = \u03bb x e \u2192 ap\u2082 _,_ (fst e) (! \/-\u207b\u00b9* \u2219 ap (_\/_ CT.\u03b2) (snd e))\n      }\n\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"691999f7938d50adf566436b8eea2a85fb1b2674","subject":"Desc model: implicit refl, cong, cong2, reflFun, desc, and IMu","message":"Desc model: implicit refl, cong, cong2, reflFun, desc, and IMu\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box I D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box I (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box Unit (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : (A : Set)(B : Set)(f : A -> B)(x y : A) -> x == y -> f x == f y\ncong A B f x .x refl = refl\n\ncong2 : (A B C : Set)(f : A -> B -> C)(x y : A)(z t : B) -> \n        x == y -> z == t -> f x z == f y t\ncong2 A B C f x .x z .z refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : (A : Set)(B : Set)(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc Unit (descD I) (IMu Unit (\u03bb _ -> descD I)))\n        (hs : desc (Sigma Unit (IMu Unit (\u03bb _ -> descD I)))\n              (box Unit (descD I) (IMu Unit (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 (IDesc I) \n                                                (IDesc I)\n                                                (IDesc I) \n                                                prod \n                                                (iso1 I (iso2 I D))\n                                                D \n                                                (iso1 I (iso2 I D'))\n                                                D' p q\nproof-iso1-iso2 I (pi S T) = cong (S \u2192 IDesc I)\n                                  (IDesc I)\n                                  (pi S) \n                                  (\\x -> iso1 I (iso2 I (T x)))\n                                  T \n                                  (reflFun S (IDesc I)\n                                             (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                             T\n                                             (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (S \u2192 IDesc I)\n                                     (IDesc I)\n                                     (sigma S) \n                                     (\\x -> iso1 I (iso2 I (T x)))\n                                     T \n                                     (reflFun S \n                                              (IDesc I)\n                                              (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- induction : (I : Set)\n--             (R : I -> IDesc I)\n--             (P : Sigma I (IMu I R) -> Set)\n--             (m : (i : I)\n--                  (xs : desc I (R i) (IMu I R))\n--                  (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n--                  P ( i , con xs)) ->\n--             (i : I)(x : IMu I R i) -> P ( i , x )\n\n\nP : (I : Set) -> Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (IDescl I) \n                                          (IDescl I) \n                                          (IDescl I)\n                                          (prodl I) \n                                          (iso2 I (iso1 I D))\n                                          D \n                                          (iso2 I (iso1 I D'))\n                                          D' p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                          (IDescl I)\n                                                                          (pil I S) \n                                                                          (\\x -> iso2 I (iso1 I (T x)))\n                                                                          T \n                                                                          (reflFun S \n                                                                                   (IDescl I)\n                                                                                   (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                             (IDescl I)\n                                                                             (sigmal I S) \n                                                                             (\\x -> iso2 I (iso1 I (T x)))\n                                                                             T \n                                                                             (reflFun S \n                                                                                      (IDescl I)\n                                                                                      (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e4d4b6ad73f4a5ff7c6d8ecdd7d9729374c7670b","subject":"Modified an test.","message":"Modified an test.\n\nIgnore-this: 30e5806c631a8cb97aad3bc3c86b42f2\n\ndarcs-hash:20100803031659-3bd4e-82749625a9fcb9680e9b52cf09533165cb2e04d7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/DefinitionsInsideWhereClauses.agda","new_file":"Test\/Succeed\/DefinitionsInsideWhereClauses.agda","new_contents":"module Test.Succeed.DefinitionsInsideWhereClauses where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n+-rightIdentity : {n : D} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity {n} Nn = indN Q Q0 iStep Nn\n  where\n    Q : D \u2192 Set\n    Q i = i + zero \u2261 i\n    {-# ATP definition Q #-}\n\n    postulate\n      Q0 : Q zero\n    {-# ATP prove Q0 #-}\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192 Q i \u2192 Q (succ i)\n    {-# ATP prove iStep #-}\n","old_contents":"module Test.Succeed.DefinitionsInsideWhereClauses where\n\ninfixl 6 _+_\ninfix  4 _\u2261_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n+-leftIdentity : {n : D} \u2192 N n \u2192 zero + n \u2261 n\n+-leftIdentity {n} Nn = indN P P0 iStep Nn\n  where\n    P : D \u2192 Set\n    P i = zero + i \u2261 i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f410e5bd152cb99e1e74520b117041e013a62c04","subject":"Examples: update","message":"Examples: update\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Examples.agda","new_file":"Control\/Protocol\/Examples.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (_,_)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.Two hiding (_\u225f_)\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; coe)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Examples where\n\nmodule MonomorphicSky (P : Proto\u2080) where\n  Cloud : Proto\u2080\n  Cloud = recv \u03bb (t   : \u27e6 P  \u27e7) \u2192\n          recv \u03bb (u   : \u27e6 P \u22a5\u27e7) \u2192\n          send \u03bb (log : Log P)  \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud t u = telecom P t u , _\n\nmodule PolySky where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P : Proto\u2080) \u2192\n          lift\u1d3e \u2081 (MonomorphicSky.Cloud P)\n  cloud : \u27e6 Cloud \u27e7\n  cloud P = lift-proc \u2081 (MonomorphicSky.Cloud P) (MonomorphicSky.cloud P)\n\nmodule PolySky' where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P   : Proto\u2080) \u2192\n          recv \u03bb (t   : Lift \u27e6 P  \u27e7)  \u2192\n          recv \u03bb (u   : Lift \u27e6 P \u22a5\u27e7)  \u2192\n          send \u03bb (log : Lift (Log P)) \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud P (lift t) (lift u) = lift (telecom P t u) , _\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = send\u2610 \u03bb (s : Secret) \u2192\n             recv  \u03bb (g : Guess)  \u2192\n             send  \u03bb (_ : S\u27e8 s \u27e9) \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g [ m ] = g , _\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = send \u03bb (g\u02b3 : G)  \u2192 -- commitment\n                 recv \u03bb (c  : \u2124q) \u2192 -- challenge\n                 send \u03bb (s  : \u2124q) \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- He is honest but its type does not say it\n        -- The definition for \"prover\" below is derived from\n        -- a well typed version.\n        prover' : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover' x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        verifier : (c : \u2124q) \u2192 \u27e6 Verifier \u27e7\n        verifier c _ = c , _\n\n        module _ ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) where\n            Honest-Prover : Proto\n            Honest-Prover\n              = send\u2610 \u03bb (r  : \u2124q)                \u2192 -- ideal commitment\n                send  \u03bb (g\u02b3 : S\u27e8 g ^ r \u27e9)        \u2192 -- real  commitment\n                recv  \u03bb (c  : \u2124q)                \u2192 -- challenge\n                send  \u03bb (s  : S\u27e8 r + (c * x) \u27e9)  \u2192 -- response\n                recv  \u03bb (_  : Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u2192\n                end\n\n            Honest-Verifier : Proto\n            Honest-Verifier = dual Honest-Prover\n\n            honest-verifier' : (c : \u2124q) \u2192 \u27e6 Honest-Verifier \u27e7\n            honest-verifier' c [ r ] g\u02b3 = c , \u03bb s \u2192 _ \u225f _ , _\n\n            sim-honest-prover : \u27e6 Honest-Prover \u22b8 Prover \u27e7\n            sim-honest-prover [ r ] S[ g\u02b3 \u2225 g\u02b3= ] = inr g\u02b3 , \u03bb c \u2192\n                                                    inl c  , \u03bb { S[ s \u2225 s= ] \u2192\n                                                    inr s  ,\n                                                    _ \u225f _  ,\n                                                    end }\n\n            -- The next version is derived from sim-honest-prover\n            honest-prover\u2192prover' : \u27e6 Honest-Prover \u27e7 \u2192 \u27e6 Prover \u27e7\n            honest-prover\u2192prover' ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n            module _ {{_ : FunExt}}{{_ : UA}} where\n                sim-honest-verifier : \u27e6 Verifier \u22b8 Honest-Verifier \u27e7\n                sim-honest-verifier = coe (\u214b-comm Honest-Verifier _) sim-honest-prover\n\n                honest-prover\u2192prover : \u27e6 Honest-Prover \u27e7 \u2192 \u27e6 Prover \u27e7\n                honest-prover\u2192prover = \u22b8-apply {Honest-Prover} sim-honest-prover\n\n                verifier\u2192honest-verifier : \u27e6 Verifier \u27e7 \u2192 \u27e6 Honest-Verifier \u27e7\n                verifier\u2192honest-verifier = \u22b8-apply {Verifier} {Honest-Verifier} sim-honest-verifier\n\n                honest-verifier : (c : \u2124q) \u2192 \u27e6 Honest-Verifier \u27e7\n                honest-verifier c = verifier\u2192honest-verifier (verifier c)\n\n        module _ (x r : \u2124q) where\n            honest-prover : \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n            honest-prover = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n            -- agsy can do it\n\n            module _ {{_ : FunExt}}{{_ : UA}} where\n                prover : \u27e6 Prover \u27e7\n                prover = honest-prover\u2192prover x S[ g ^ x ] honest-prover\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (_,_)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.Two hiding (_\u225f_)\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; coe)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Lift\n\nmodule Control.Protocol.Examples where\n\nmodule MonomorphicSky (P : Proto\u2080) where\n  Cloud : Proto\u2080\n  Cloud = recv \u03bb (t   : \u27e6 P  \u27e7) \u2192\n          recv \u03bb (u   : \u27e6 P \u22a5\u27e7) \u2192\n          send \u03bb (log : Log P)  \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud t u = telecom P t u , _\n\nmodule PolySky where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P : Proto\u2080) \u2192\n          lift\u1d3e \u2081 (MonomorphicSky.Cloud P)\n  cloud : \u27e6 Cloud \u27e7\n  cloud P = lift-proc \u2081 (MonomorphicSky.Cloud P) (MonomorphicSky.cloud P)\n\nmodule PolySky' where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P   : Proto\u2080) \u2192\n          recv \u03bb (t   : Lift \u27e6 P  \u27e7)  \u2192\n          recv \u03bb (u   : Lift \u27e6 P \u22a5\u27e7)  \u2192\n          send \u03bb (log : Lift (Log P)) \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud P (lift t) (lift u) = lift (telecom P t u) , _\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = send\u2610 \u03bb (s : Secret) \u2192\n             recv  \u03bb (g : Guess)  \u2192\n             send  \u03bb (_ : S\u27e8 s \u27e9) \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g [ m ] = g , _\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = send \u03bb (g\u02b3 : G)  \u2192 -- commitment\n                 recv \u03bb (c  : \u2124q) \u2192 -- challenge\n                 send \u03bb (s  : \u2124q) \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- He is honest but its type does not say it\n        -- The definition for \"prover\" below is derived from\n        -- a well typed version.\n        prover' : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover' x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        verifier : (c : \u2124q) \u2192 \u27e6 Verifier \u27e7\n        verifier c _ = c , _\n\n        module _ ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) where\n            Honest-Prover : Proto\n            Honest-Prover\n              = send\u2610 \u03bb (r  : \u2124q)                \u2192 -- ideal commitment\n                send  \u03bb (g\u02b3 : S\u27e8 g ^ r \u27e9)        \u2192 -- real  commitment\n                recv  \u03bb (c  : \u2124q)                \u2192 -- challenge\n                send  \u03bb (s  : S\u27e8 r + (c * x) \u27e9)  \u2192 -- response\n                recv  \u03bb (_  : Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u2192\n                end\n\n            Honest-Verifier : Proto\n            Honest-Verifier = dual Honest-Prover\n\n            honest-verifier' : (c : \u2124q) \u2192 \u27e6 Honest-Verifier \u27e7\n            honest-verifier' c [ r ] g\u02b3 = c , \u03bb s \u2192 _ \u225f _ , _\n\n            sim-honest-prover : \u27e6 Honest-Prover \u22b8 Prover \u27e7\n            sim-honest-prover [ r ] S[ g\u02b3 \u2225 g\u02b3= ] = inr g\u02b3 , \u03bb c \u2192\n                                                    inl c  , \u03bb { S[ s \u2225 s= ] \u2192\n                                                    inr s  ,\n                                                    _ \u225f _  ,\n                                                    end }\n\n            -- The next version is derived from sim-honest-prover\n            honest-prover\u2192prover' : \u27e6 Honest-Prover \u27e7 \u2192 \u27e6 Prover \u27e7\n            honest-prover\u2192prover' ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n            module _ {{_ : FunExt}}{{_ : UA}} where\n                sim-honest-verifier : \u27e6 Verifier \u22b8 Honest-Verifier \u27e7\n                sim-honest-verifier = coe (\u214b-comm Honest-Verifier _) sim-honest-prover\n\n                honest-prover\u2192prover : \u27e6 Honest-Prover \u27e7 \u2192 \u27e6 Prover \u27e7\n                honest-prover\u2192prover = \u22b8-apply {Honest-Prover} sim-honest-prover\n\n                verifier\u2192honest-verifier : \u27e6 Verifier \u27e7 \u2192 \u27e6 Honest-Verifier \u27e7\n                verifier\u2192honest-verifier = \u22b8-apply {Verifier} {Honest-Verifier} sim-honest-verifier\n\n                honest-verifier : (c : \u2124q) \u2192 \u27e6 Honest-Verifier \u27e7\n                honest-verifier c = verifier\u2192honest-verifier (verifier c)\n\n        module _ (x r : \u2124q) where\n            honest-prover : \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n            honest-prover = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n            -- agsy can do it\n\n            module _ {{_ : FunExt}}{{_ : UA}} where\n                prover : \u27e6 Prover \u27e7\n                prover = honest-prover\u2192prover x S[ g ^ x ] honest-prover\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e21a64aa0cf9c6920125f7ec1f3d5e3376a2b2b1","subject":"Drop dead experiment","message":"Drop dead experiment\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ] da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ] db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n[ sum \u03c3 \u03c4 ] dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n-- module _ {\u2113\u2081} {\u2113\u2082}\n--   {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n\n  -- data SValid3 : A \u228e B \u2192 Ch (A \u228e B) \u2192 Set (\u2113\u2081 Level.\u2294 \u2113\u2082) where\n-- module _ {\u03c3 \u03c4 : Type} where\n--   data SValid3 : Cht (sum \u03c3 \u03c4) \u2192 (s1 s2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set where\n--     sv\u2081 : \u2200 da a1 a2 \u2192 [ \u03c3 ] da from a1 to a2 \u2192 SValid3 (convert\u2081 (ch\u2081 da)) (inj\u2081 a1) (inj\u2081 a2)\n    -- sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid3 (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    -- sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid3 (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    -- svrp\u2081 : \u2200 a1 b2 \u2192 SValid3 (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    -- svrp\u2082 : \u2200 b1 a2 \u2192 SValid3 (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ] da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ] db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n[ sum \u03c3 \u03c4 ] dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n-- Doesn't work, the resulting datatype wouldn't be strictly positive.\n-- [ sum \u03c3 \u03c4 ] dv from v1 to v2 = SValid3 dv v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"74083f502d411a2be738d568c2831185e08720d7","subject":"Assume a less stupid property of == on G","message":"Assume a less stupid property of == on G\n","repos":"crypto-agda\/crypto-agda","old_file":"generic-zero-knowledge-interactive.agda","new_file":"generic-zero-knowledge-interactive.agda","new_contents":"open import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\nprivate\n  \u2605 : Set\u2081\n  \u2605 = Set\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (sum\u03c0        : Sum Permutation)\n         (\u03bc\u03c0          : SumProp sum\u03c0)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (\u03bcR\u209a-xtra    : SumProp sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sum\u03c0 \u00d7Sum sumR\u209a-xtra\n\n    \u03bcR\u209a : SumProp sumR\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    sumR : Sum R\n    sumR = sumBit \u00d7Sum sumR\u209a\n\n    \u03bcR : SumProp sumR\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl    : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-cong-\u2219  : \u2200 {\u03b1 \u03b2 b} \u03b3 \u2192 \u03b1 == \u03b2 \u2261 b \u2192 (\u03b3 \u2219 \u03b1) == (\u03b3 \u2219 \u03b2) \u2261 b)\n            (==-true    : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (sum\u2124q      : Sum \u2124q)\n            (\u03bc\u2124q        : SumProp sum\u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (sumR\u209a-xtra : Sum R\u209a-xtra)\n            (\u03bcR\u209a-xtra   : SumProp sumR\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s = ==-cong-\u2219 (g^ \u03c0) check-p-s\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 sum\u2124q \u03bc\u2124q R\u209a-xtra sumR\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Data.Bool.NP as Bool hiding (check)\nopen import Data.Nat\nopen import Data.Maybe\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import sum\n\nmodule generic-zero-knowledge-interactive where\n\nprivate\n  \u2605 : Set\u2081\n  \u2605 = Set\n\n-- A random argument, this is only a formal notation to\n-- indicate that the argument is supposed to be picked\n-- at random uniformly. (do not confuse with our randomness\n-- monad).\nrecord \u21ba (A : \u2605) : \u2605 where\n  constructor rand\n  field get : A\n\nmodule M (Permutation : \u2605)\n         (_\u207b\u00b9         : Endo Permutation)\n         (sum\u03c0        : Sum Permutation)\n         (\u03bc\u03c0          : SumProp sum\u03c0)\n\n         (R\u209a-xtra     : \u2605) -- extra prover\/adversary randomness\n         (sumR\u209a-xtra  : Sum R\u209a-xtra)\n         (\u03bcR\u209a-xtra    : SumProp sumR\u209a-xtra)\n\n         (Problem     : \u2605)\n         (_==_        : Problem \u2192 Problem \u2192 Bit)\n         (==-refl     : \u2200 {pb} \u2192 (pb == pb) \u2261 true)\n         (_\u2219P_        : Permutation \u2192 Endo Problem)\n         (\u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x)\n\n         (Solution    : \u2605)\n         (_\u2219S_        : Permutation \u2192 Endo Solution)\n         (\u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x)\n\n         (check       : Problem \u2192 Solution \u2192 Bit)\n         (check-\u2219     : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s)\n\n         (easy-pb     : Permutation \u2192 Problem)\n         (easy-sol    : Permutation \u2192 Solution)\n         (check-easy  : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true)\n    where\n\n    -- prover\/adversary randomness\n    R\u209a : \u2605\n    R\u209a = Permutation \u00d7 R\u209a-xtra\n\n    sumR\u209a : Sum R\u209a\n    sumR\u209a = sum\u03c0 \u00d7Sum sumR\u209a-xtra\n\n    \u03bcR\u209a : SumProp sumR\u209a\n    \u03bcR\u209a = \u03bc\u03c0 \u00d7\u03bc \u03bcR\u209a-xtra\n\n    R = Bit \u00d7 R\u209a\n\n    sumR : Sum R\n    sumR = sumBit \u00d7Sum sumR\u209a\n\n    \u03bcR : SumProp sumR\n    \u03bcR = \u03bcBit \u00d7\u03bc \u03bcR\u209a\n\n    check-\u03c0 : Problem \u2192 Solution \u2192 R\u209a \u2192 Bit\n    check-\u03c0 p s (\u03c0 , _) = check (\u03c0 \u2219P p) (\u03c0 \u2219S s)\n\n    otp-\u2219-check : let #_ = count \u03bcR\u209a\n                  in\n                  \u2200 p\u2080 s\u2080 p\u2081 s\u2081 \u2192\n                    check p\u2080 s\u2080 \u2261 check p\u2081 s\u2081 \u2192\n                    #(check-\u03c0 p\u2080 s\u2080) \u2261 #(check-\u03c0 p\u2081 s\u2081)\n    otp-\u2219-check p\u2080 s\u2080 p\u2081 s\u2081 check-pf =\n      count-ext \u03bcR\u209a {f = check-\u03c0 p\u2080 s\u2080} {check-\u03c0 p\u2081 s\u2081} (\u03bb \u03c0,r \u2192\n        check-\u03c0 p\u2080 s\u2080 \u03c0,r \u2261\u27e8 check-\u2219 p\u2080 s\u2080 (proj\u2081 \u03c0,r) \u27e9\n        check p\u2080 s\u2080       \u2261\u27e8 check-pf \u27e9\n        check p\u2081 s\u2081       \u2261\u27e8 sym (check-\u2219 p\u2081 s\u2081 (proj\u2081 \u03c0,r)) \u27e9\n        check-\u03c0 p\u2081 s\u2081 \u03c0,r \u220e)\n        where open \u2261-Reasoning\n\n    #_ : (\u21ba (Bit \u00d7 Permutation \u00d7 R\u209a-xtra) \u2192 Bit) \u2192 \u2115\n    # f = count \u03bcR (f \u2218 rand)\n\n    _\u2261#_ : (f g : \u21ba (Bit \u00d7 R\u209a) \u2192 Bit) \u2192 \u2605\n    f \u2261# g = # f \u2261 # g\n\n{-\n    otp-\u2219 : let otp = \u03bb O pb s \u2192 count \u03bcR\u209a (\u03bb { (\u03c0 , _) \u2192 O (\u03c0 \u2219P pb) (\u03c0 \u2219S s) }) in\n            \u2200 pb\u2080 s\u2080 pb\u2081 s\u2081 \u2192\n              check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081 \u2192\n              (O : _ \u2192 _ \u2192 Bit) \u2192 otp O pb\u2080 s\u2080 \u2261 otp O pb\u2081 s\u2081\n    otp-\u2219 pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf O = {!(\u03bc\u03c0 \u00d7Sum-proj\u2082 \u03bcR\u209a-xtra    ?!}\n-}\n    Answer : Bit \u2192 \u2605\n    Answer false{-0b-} = Permutation\n    Answer true {-1b-} = Solution\n\n    answer : Permutation \u2192 Solution \u2192 \u2200 b \u2192 Answer b\n    answer \u03c0 _ false = \u03c0\n    answer _ s true  = s\n\n    -- The prover is the advesary in the generic terminology,\n    -- and the verifier is the challenger.\n    DepProver : \u2605\n    DepProver = Problem \u2192 \u21ba R\u209a \u2192 (b : Bit) \u2192 Problem \u00d7 Answer b\n\n    Prover\u2080 : \u2605\n    Prover\u2080 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Permutation\n\n    Prover\u2081 : \u2605\n    Prover\u2081 = Problem \u2192 \u21ba R\u209a \u2192 Problem \u00d7 Solution\n\n    Prover : \u2605\n    Prover = Prover\u2080 \u00d7 Prover\u2081\n\n    prover : DepProver \u2192 Prover\n    prover dpr = (\u03bb pb r \u2192 dpr pb r 0b) , (\u03bb pb r \u2192 dpr pb r 1b)\n\n    depProver : Prover \u2192 DepProver\n    depProver (pr\u2080 , pr\u2081) pb r false = pr\u2080 pb r\n    depProver (pr\u2080 , pr\u2081) pb r true  = pr\u2081 pb r\n\n    -- Here we show that the explicit commitment step seems useless given\n    -- the formalization. The verifier can \"trust\" the prover on the fact\n    -- that any choice is going to be govern only by the problem and the\n    -- randomness.\n    module WithCommitment (Commitment : \u2605)\n                          (AnswerWC   : Bit \u2192 \u2605)\n                          (reveal     : \u2200 b \u2192 Commitment \u2192 AnswerWC b \u2192 Problem \u00d7 Answer b) where\n        ProverWC = (Problem \u2192 R\u209a \u2192 Commitment)\n                 \u00d7 (Problem \u2192 R\u209a \u2192 (b : Bit) \u2192 AnswerWC b)\n\n        depProver' : ProverWC \u2192 DepProver\n        depProver' (pr\u2080 , pr\u2081) pb (rand r\u209a) b = reveal b (pr\u2080 pb r\u209a) (pr\u2081 pb r\u209a b)\n\n    Verif : Problem \u2192 \u2200 b \u2192 Problem \u00d7 Answer b \u2192 Bit\n    Verif pb false{-0b-} (\u03c0\u2219pb , \u03c0)   = (\u03c0 \u2219P pb) == \u03c0\u2219pb\n    Verif pb true {-1b-} (\u03c0\u2219pb , \u03c0\u2219s) = check \u03c0\u2219pb \u03c0\u2219s\n\n    _\u21c4\u2032_ : Problem \u2192 DepProver \u2192 Bit \u2192 \u21ba R\u209a \u2192 Bit\n    (pb \u21c4\u2032 pr) b (rand r\u209a) = Verif pb b (pr pb (rand r\u209a) b)\n\n    _\u21c4_ : Problem \u2192 DepProver \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    (pb \u21c4 pr) (rand (b , r\u209a)) = (pb \u21c4\u2032 pr) b (rand r\u209a)\n\n    _\u21c4''_ : Problem \u2192 Prover \u2192 \u21ba (Bit \u00d7 R\u209a) \u2192 Bit\n    pb \u21c4'' pr = pb \u21c4 depProver pr\n\n    honest : (Problem \u2192 Maybe Solution) \u2192 DepProver\n    honest solve pb (rand (\u03c0 , r\u209a)) b = (\u03c0 \u2219P pb , answer \u03c0 sol b)\n      module Honest where\n        sol : Solution\n        sol with solve pb\n        ...    | just sol = \u03c0 \u2219S sol\n        ...    | nothing  = \u03c0 \u2219S easy-sol \u03c0\n\n    module WithCorrectSolver (pb      : Problem)\n                             (s       : Solution)\n                             (check-s : check pb s \u2261 true)\n                             where\n\n        -- When the honest prover has a solution, he gets accepted\n        -- unconditionally by the verifier.\n        honest-accepted : \u2200 r \u2192 (pb \u21c4 honest (const (just s))) r \u2261 1b\n        honest-accepted (rand (true  , \u03c0 , r\u209a)) rewrite check-\u2219 pb s \u03c0 = check-s\n        honest-accepted (rand (false , \u03c0 , r\u209a)) = ==-refl\n\n    honest-\u2141 = \u03bb pb s \u2192 (pb \u21c4 honest (const (just s)))\n\n    module HonestLeakZeroKnowledge (pb\u2080 pb\u2081  : Problem)\n                                   (s\u2080 s\u2081    : Solution)\n                                   (check-pf : check pb\u2080 s\u2080 \u2261 check pb\u2081 s\u2081) where\n\n        helper : \u2200 r\u209a \u2192 Bool.to\u2115 ((pb\u2080 \u21c4\u2032 honest (const (just s\u2080))) 0b (rand r\u209a))\n                      \u2261 Bool.to\u2115 ((pb\u2081 \u21c4\u2032 honest (const (just s\u2081))) 0b (rand r\u209a))\n        helper (\u03c0 , r\u209a) rewrite ==-refl {\u03c0 \u2219P pb\u2080} | ==-refl {\u03c0 \u2219P pb\u2081} = refl\n\n        honest-leak : honest-\u2141 pb\u2080 s\u2080 \u2261# honest-\u2141 pb\u2081 s\u2081\n        honest-leak rewrite otp-\u2219-check pb\u2080 s\u2080 pb\u2081 s\u2081 check-pf | sum-ext \u03bcR\u209a helper = refl\n\n    module HonestLeakZeroKnowledge' (pb       : Problem)\n                                    (s\u2080 s\u2081    : Solution)\n                                    (check-pf : check pb s\u2080 \u2261 check pb s\u2081) where\n\n        honest-leak : honest-\u2141 pb s\u2080 \u2261# honest-\u2141 pb s\u2081\n        honest-leak = HonestLeakZeroKnowledge.honest-leak pb pb s\u2080 s\u2081 check-pf\n\n    -- Predicts b=b\u2032\n    cheater : \u2200 b\u2032 \u2192 DepProver\n    cheater b\u2032 pb (rand (\u03c0 , _)) b = \u03c0 \u2219P (case b\u2032 0\u2192 pb 1\u2192 easy-pb \u03c0)\n                                   , answer \u03c0 (\u03c0 \u2219S easy-sol \u03c0) b\n\n    -- If cheater predicts correctly, verifer accepts him\n    cheater-accepted : \u2200 b pb r\u209a \u2192 (pb \u21c4\u2032 cheater b) b r\u209a \u2261 1b\n    cheater-accepted true  pb (rand (\u03c0 , r\u209a)) = check-easy \u03c0\n    cheater-accepted false pb (rand (\u03c0 , r\u209a)) = ==-refl\n\n    -- If cheater predicts incorrecty, verifier rejects him\n    module CheaterRejected (pb : Problem)\n                           (not-easy-sol : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P pb) (\u03c0 \u2219S easy-sol \u03c0) \u2261 false)\n                           (not-easy-pb  : \u2200 \u03c0 \u2192 ((\u03c0 \u2219P pb) == (\u03c0 \u2219P easy-pb \u03c0)) \u2261 false) where\n\n        cheater-rejected : \u2200 b r\u209a \u2192 (pb \u21c4\u2032 cheater (not b)) b r\u209a \u2261 0b\n        cheater-rejected true  (rand (\u03c0 , r\u209a)) = not-easy-sol \u03c0\n        cheater-rejected false (rand (\u03c0 , r\u209a)) = not-easy-pb \u03c0\n\nmodule DLog (\u2124q  : \u2605)\n            (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n            (\u229f_  : \u2124q \u2192 \u2124q)\n            (G   : \u2605)\n            (g   : G)\n            (_^_ : G \u2192 \u2124q \u2192 G)\n            (_\u2219_ : G \u2192 G \u2192 G)\n            (\u229f-\u229e : \u2200 \u03c0 x \u2192 (\u229f \u03c0) \u229e (\u03c0 \u229e x) \u2261 x)\n            (^\u229f-\u2219 : \u2200 \u03b1 \u03b2 x \u2192 ((\u03b1 ^ (\u229f x)) \u2219 ((\u03b1 ^ x) \u2219 \u03b2)) \u2261 \u03b2)\n            -- (\u2219-assoc : \u2200 \u03b1 \u03b2 \u03b3 \u2192 \u03b1 \u2219 (\u03b2 \u2219 \u03b3) \u2261 (\u03b1 \u2219 \u03b2) \u2219 \u03b3)\n            (dist-^-\u229e : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u229e y) \u2261 (\u03b1 ^ x) \u2219 (\u03b1 ^ y))\n            (_==_ : G \u2192 G \u2192 Bool)\n            (==-refl  : \u2200 {\u03b1} \u2192 (\u03b1 == \u03b1) \u2261 true)\n            (==-cong  : \u2200 {\u03b1 \u03b2 b} (f : G \u2192 G) \u2192 \u03b1 == \u03b2 \u2261 b \u2192 f \u03b1 == f \u03b2 \u2261 b)\n            (==-true  : \u2200 {\u03b1 \u03b2} \u2192 \u03b1 == \u03b2 \u2261 true \u2192 \u03b1 \u2261 \u03b2)\n            (sum\u2124q      : Sum \u2124q)\n            (\u03bc\u2124q        : SumProp sum\u2124q)\n            (R\u209a-xtra    : \u2605) -- extra prover\/adversary randomness\n            (sumR\u209a-xtra : Sum R\u209a-xtra)\n            (\u03bcR\u209a-xtra   : SumProp sumR\u209a-xtra)\n            (some-\u2124q    : \u2124q)\n   where\n\n   Permutation = \u2124q\n   Problem = G\n   Solution = \u2124q\n\n   _\u207b\u00b9 : Endo Permutation\n   \u03c0 \u207b\u00b9 = \u229f \u03c0\n\n   g^_ : \u2124q \u2192 G\n   g^ x = g ^ x\n\n   _\u2219P_ : Permutation \u2192 Endo Problem\n   \u03c0 \u2219P p = g^ \u03c0 \u2219 p\n\n   \u207b\u00b9-inverseP : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219P (\u03c0 \u2219P x) \u2261 x\n   \u207b\u00b9-inverseP \u03c0 x rewrite ^\u229f-\u2219 g x \u03c0 = refl\n\n   _\u2219S_ : Permutation \u2192 Endo Solution\n   \u03c0 \u2219S s = \u03c0 \u229e s\n\n   \u207b\u00b9-inverseS : \u2200 \u03c0 x \u2192 \u03c0 \u207b\u00b9 \u2219S (\u03c0 \u2219S x) \u2261 x\n   \u207b\u00b9-inverseS = \u229f-\u229e\n\n   check : Problem \u2192 Solution \u2192 Bit\n   check p s = p == g^ s\n\n   check-\u2219' : \u2200 p s \u03c0 b \u2192 check p s \u2261 b \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 b\n   check-\u2219' p s \u03c0 true  check-p-s rewrite dist-^-\u229e g \u03c0 s | ==-true check-p-s = ==-refl\n   check-\u2219' p s \u03c0 false check-p-s rewrite dist-^-\u229e g \u03c0 s = ==-cong (_\u2219_ (g^ \u03c0)) check-p-s\n\n   check-\u2219 : \u2200 p s \u03c0 \u2192 check (\u03c0 \u2219P p) (\u03c0 \u2219S s) \u2261 check p s\n   check-\u2219 p s \u03c0 = check-\u2219' p s \u03c0 (check p s) refl\n\n   easy-sol : Permutation \u2192 Solution\n   easy-sol \u03c0 = some-\u2124q\n\n   easy-pb : Permutation \u2192 Problem\n   easy-pb \u03c0 = g^(easy-sol \u03c0)\n\n   check-easy : \u2200 \u03c0 \u2192 check (\u03c0 \u2219P easy-pb \u03c0) (\u03c0 \u2219S easy-sol \u03c0) \u2261 true\n   check-easy \u03c0 rewrite dist-^-\u229e g \u03c0 (easy-sol \u03c0) = ==-refl\n\n   open M Permutation _\u207b\u00b9 sum\u2124q \u03bc\u2124q R\u209a-xtra sumR\u209a-xtra \u03bcR\u209a-xtra\n          Problem _==_ ==-refl _\u2219P_ \u207b\u00b9-inverseP Solution _\u2219S_ \u207b\u00b9-inverseS check check-\u2219 easy-pb easy-sol check-easy\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f08dd49903bfaa812eaf76a759476e548741a747","subject":"-otp-sem-sec","message":"-otp-sem-sec\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\nopen import program-distance\nopen import one-time-semantic-security\nimport bit-guessing-game\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\n\n\n\n\n\n\nmodule CompRed {t} {T : Set t}\n               {\u266dFuns : FlatFuns T}\n               (\u266dops : FlatFunsOps \u266dFuns)\n               (ep ep' : EncParams) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec \u266dFuns\n\n    open EncParams ep using (|M|; |C|)\n    open EncParams ep' using () renaming (|M| to |M|'; |C| to |C|')\n\n    M = `Bits |M|\n    C = `Bits |C|\n    M' = `Bits |M|'\n    C' = `Bits |C|'\n\n    comp-red : (pre-E : M' `\u2192 M)\n               (post-E : C `\u2192 C') \u2192 SemSecReduction ep' ep _\n    comp-red pre-E post-E (mk A\u2080 A\u2081) = mk (A\u2080 >>> (pre-E *** pre-E) *** idO) (post-E *** idO >>> A\u2081)\n\nmodule Comp (ep : EncParams) (|M|' |C|' : \u2115) where\n  open EncParams ep\n\n  ep' : EncParams\n  ep' = EncParams.mk |M|' |C|' |R|e\n\n  open EncParams ep' using () renaming (Enc to Enc'; M to M'; C to C')\n\n  Tr = Enc \u2192 Enc'\n\n  open FlatFunsOps fun\u266dOps\n\n  comp : (pre-E : M' \u2192 M) (post-E : C \u2192 C') \u2192 Tr\n  comp pre-E post-E E = pre-E >>> E >>> {-weaken\u2264 |R|e\u2264|R|e' \u2218-} map\u21ba post-E\n\nmodule CompSec (prgDist : PrgDist) (ep : EncParams) (|M|' |C|' : \u2115) where\n  open PrgDist prgDist\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist\n  open AbsSemSec fun\u266dFuns\n\n  open EncParams ep\n\n  ep' : EncParams\n  ep' = EncParams.mk |M|' |C|' |R|e\n\n  open EncParams ep' using () renaming (Enc to Enc'; M to M'; C to C')\n\n  module CompSec' (pre-E : M' \u2192 M) (post-E : C \u2192 C') where\n    open SemSecReductions ep ep'\n    open CompRed fun\u266dOps ep ep' hiding (M; C)\n    open Comp ep |M|' |C|'\n\n    comp-pres-sem-sec : SemSecTr id (comp pre-E post-E)\n    comp-pres-sem-sec = comp-red pre-E post-E , \u03bb E A \u2192 id\n\n  open SemSecReductions ep ep'\n  open CompRed fun\u266dOps ep ep' hiding (M; C)\n  open CompSec' public\n  module Comp' {x y z} = Comp x y z\n  open Comp' hiding (Tr)\n  comp-pres-sem-sec\u207b\u00b9 : \u2200 pre-E pre-E\u207b\u00b9\n                          (pre-E-right-inv : pre-E \u2218 pre-E\u207b\u00b9 \u2257 id)\n                          post-E post-E\u207b\u00b9\n                          (post-E-left-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                        \u2192 SemSecTr\u207b\u00b9 id (comp pre-E post-E)\n  comp-pres-sem-sec\u207b\u00b9 pre-E pre-E\u207b\u00b9 pre-E-inv post-E post-E\u207b\u00b9 post-E-inv =\n    SemSecTr\u2192SemSecTr\u207b\u00b9\n      (comp pre-E\u207b\u00b9 post-E\u207b\u00b9)\n      (comp pre-E post-E)\n      (\u03bb E m R \u2192 trans (post-E-inv _) (cong (\u03bb x \u2192 run\u21ba (E x) R) (pre-E-inv _)))\n      (comp-pres-sem-sec pre-E\u207b\u00b9 post-E\u207b\u00b9)\n\nmodule PostNegSec (prgDist : PrgDist) ep where\n  open EncParams ep\n  open Comp ep |M| |C| hiding (Tr)\n  open CompSec prgDist ep |M| |C|\n  open FunSemSec prgDist\n  open SemSecReductions ep ep\n\n  post-neg : Tr\n  post-neg = comp id vnot\n\n  post-neg-pres-sem-sec : SemSecTr id post-neg\n  post-neg-pres-sem-sec = comp-pres-sem-sec id vnot\n\n  post-neg-pres-sem-sec\u207b\u00b9 : SemSecTr\u207b\u00b9 id post-neg\n  post-neg-pres-sem-sec\u207b\u00b9 = comp-pres-sem-sec\u207b\u00b9 id id (\u03bb _ \u2192 refl) vnot vnot vnot\u2218vnot\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess' = bit-guessing-game prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module CompSec' = CompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0a71d6d8c1fe3e2b05a48534610115b33b695899","subject":"Possible agda bug","message":"Possible agda bug\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/LinFunContructw.agda","new_file":"agda\/LinFunContructw.agda","new_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n\n\n\n\nmodule _ where\n\n  open IndexLLProp \n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n    hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n          \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n          \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n          \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n    hf2 \u2205 mnho s lf eqs = \u2205\n    hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (tranLFMIndexLL lf) (\u00ac\u2205 x)\n    hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n    hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n      \n  data MLFun {i u ll rll n dt df} (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s ind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    I : \u2200{ts} \u2192 (complL\u209b s \u2261 \u2205) \n        \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun ind s m\u00acho lf\n    \u00acI\u2205 : \u2200 {cs} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n          \u2192 let tind = tranLFMIndexLL lf ind in\n          (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n          \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll ind s ceqi m\u00acho cll) rll)\n          \u2192 MLFun ind s m\u00acho lf \n    \u00acI\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf ind in\n           (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll ind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 ind s m\u00acho lf eqs) cll))\n           \u2192 MLFun ind s m\u00acho lf \n           --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n           -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n    -- s here does contain the ind.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test \u2205 s \u2205 lf with complL\u209b s | inspect complL\u209b s\n  test \u2205 s \u2205 lf | \u2205 | [ eq ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n  test \u2205 s \u2205 lf | \u2205 | [ eq ] | \u2205 | e = IMPOSSIBLE\n  test \u2205 s \u2205 lf | \u2205 | [ eq ] | \u00ac\u2205 x | e with complL\u209b x | inspect complL\u209b x\n  test \u2205 s \u2205 lf | \u2205 | [ eq ] | \u00ac\u2205 x | [ e ] | \u2205 | [ t ] = I eq (sym e) t\n  test \u2205 s \u2205 lf | \u2205 | [ eq ] | \u00ac\u2205 x | e | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n  test \u2205 s \u2205 I | \u00ac\u2205 x | [ eq ] = \u00acI\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with tranLFMSetLL lf\u2081 (\u00ac\u2205 mx) | inspect (\u03bb z \u2192 tranLFMSetLL lf\u2081 (\u00ac\u2205 z)) mx | test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  ... | \u00ac\u2205 ts1 | [ ts1eq ] | I ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  ... | \u00ac\u2205 ts1 | [ ts1eq ] | \u00acI\u2205 ceqi eqs t = \u22a5-elim (r (trans eqs ts1eq)) where\n    r : \u00ac (\u2205 \u2261 ((MSetLL rll) \u220b (\u00ac\u2205 ts1)))\n    r ()\n  ... | \u00ac\u2205 ts1 | [ ts1eq ] | \u00acI\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00acI\u00ac\u2205 eq tseq (subst (\u03bb z \u2192 complL\u209b z \u2261 \u00ac\u2205 cts) (sym r) ceqo) ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n    r : ts1 \u2261 ts\n    r with trans eqs ts1eq\n    r | refl = refl\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) ts1eq)\n    g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n  ... | \u2205 | [ ts1eq ] | I ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  ... | \u2205 | [ ts1eq ] | \u00acI\u2205 ceqi eqs t = \u00acI\u2205 eq tseq (\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (subst (\u03bb a \u2192 LFun a rll) (shrink-repl-comm s lind eq teq meq ceqi) (t z))) where \n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) ts1eq)\n  ... | \u2205 | [ ts1eq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo t with tranLFMSetLL lf\u2081 (\u00ac\u2205 mx)\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u2205 | [ () ] | \u00acI\u00ac\u2205 ceqi refl ceqo t | .(\u00ac\u2205 _)\n\n\n\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ () ] | \u00acI\u00ac\u2205 ceqi refl ceqo x | .(\u00ac\u2205 _)\n\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n\n\n-- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n--     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --    is = test \u2205 {!!} \u2205 lf\u2081\n--   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n--   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n--   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n--   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n--     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n--     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n--   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n--   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n--   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n  \n  \n  \n  \n  \n  \n  \n","old_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n\n\n\n\nmodule _ where\n\n  open IndexLLProp \n\n  private\n    data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n      \u2205 : M\u00acho s \u2205\n      \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n           \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n    hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n         \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n         \u2192 LinLogic i {u}\n    hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n    hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n    hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n          \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n          \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n          \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n    hf2 \u2205 mnho s lf eqs = \u2205\n    hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (tranLFMIndexLL lf) (\u00ac\u2205 x)\n    hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n    hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n      \n  data MLFun {i u ll rll n dt df} (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n             (s : SetLL ll) (m\u00acho : M\u00acho s ind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n    I : (complL\u209b s \u2261 \u2205) \u2192  MLFun ind s m\u00acho lf\n    -- \u2200{ts} \u2192 \n    --  \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205)\n    -- The above cannot be proved because we haven't told the function that s is special, its transformation will eventually go into a specific com except one input.\n    -- thus for the I constructor to be, lf should not contain com or calls.\n    \u00acI\u2205 : \u2200 {cs} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n          \u2192 let tind = tranLFMIndexLL lf ind in\n          (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n          \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll ind s ceqi m\u00acho cll) rll)\n          \u2192 MLFun ind s m\u00acho lf \n    \u00acI\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n           \u2192 let tind = tranLFMIndexLL lf ind in\n           (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n           \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll ind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 ind s m\u00acho lf eqs) cll))\n           \u2192 MLFun ind s m\u00acho lf \n           -- We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n           -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n           -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n  \n    -- s here does contain the ind.\n  test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n         \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n  test \u2205 s \u2205 lf with complL\u209b s | inspect complL\u209b s\n  test \u2205 s \u2205 lf | \u2205 | [ eq ] = I eq\n  test \u2205 s \u2205 I | \u00ac\u2205 x | [ eq ] = \u00acI\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n  test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n  ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n  ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n  ... | \u00ac\u2205 mx | [ meq ] with tranLFMSetLL lf\u2081 (\u00ac\u2205 mx) | inspect (\u03bb z \u2192 tranLFMSetLL lf\u2081 (\u00ac\u2205 z)) mx | test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n  ... | \u00ac\u2205 ts1 | [ ts1eq ] | I ceq = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  ... | \u00ac\u2205 ts1 | [ ts1eq ] | \u00acI\u2205 ceqi eqs t = \u22a5-elim (r (trans eqs ts1eq)) where\n    r : \u00ac (\u2205 \u2261 ((MSetLL rll) \u220b (\u00ac\u2205 ts1)))\n    r ()\n  ... | \u00ac\u2205 ts1 | [ ts1eq ] | \u00acI\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00acI\u00ac\u2205 eq tseq (subst (\u03bb z \u2192 complL\u209b z \u2261 \u00ac\u2205 cts) (sym r) ceqo) ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n    r : ts1 \u2261 ts\n    r with trans eqs ts1eq\n    r | refl = refl\n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) ts1eq)\n    g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n  ... | \u2205 | [ ts1eq ] | I ceq = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n  ... | \u2205 | [ ts1eq ] | \u00acI\u2205 ceqi eqs t = \u00acI\u2205 eq tseq (\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (subst (\u03bb a \u2192 LFun a rll) (shrink-repl-comm s lind eq teq meq ceqi) (t z))) where \n    tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) ts1eq)\n  ... | \u2205 | [ ts1eq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo t with tranLFMSetLL lf\u2081 (\u00ac\u2205 mx)\n  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u2205 | [ () ] | \u00acI\u00ac\u2205 ceqi refl ceqo t | .(\u00ac\u2205 _)\n\n\n\n  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n  ... | \u2205 | tseq = {!!}\n  ... | \u00ac\u2205 ts | tseq = {!!}\n\n\n-- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n--     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --    is = test \u2205 {!!} \u2205 lf\u2081\n--   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n--   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n--   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n--   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n--     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n--     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n--   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n--   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n--   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n  \n  \n  \n  \n  \n  \n  \n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"b8dde2c06e42b85e96e152523d6ff6cdbfb92314","subject":"Function: +\u03a0","message":"Function: +\u03a0\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/NP.agda","new_file":"lib\/Function\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Algebra\nopen import Algebra.Structures\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Product\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : Set a) (B : Set b) \u2192 Set (a L.\u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = \u2261.refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = \u2261.refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = \u2261.refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = \u2261.refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = \u2261.refl\n  nest-+ (suc m) n = \u2261.cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = \u2261.cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = \u2261.refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 \u2261.refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 \u2261.cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 \u2261.refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 \u2261.refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261.\u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\n\u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : Set i} {A : Set a} {B : A \u2192 Set b} {C : (x : A) \u2192 B x \u2192 Set c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : Set a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 Set \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : Set a) = EndoMonoid-\u2248 {A = A} \u2261.isEquivalence (\u2261.cong\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : Set a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2261.\u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 \u2261.trans (p (h x)) (\u2261.cong g (q x)))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nimport Level as L\nopen import Type\nopen import Algebra\nopen import Algebra.Structures\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Product\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : Set a) (B : Set b) \u2192 Set (a L.\u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605) (n : \u2115) (B : \u2605) \u2192 \u2605\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = \u2261.refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = \u2261.refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = \u2261.refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = \u2261.refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = \u2261.refl\n  nest-+ (suc m) n = \u2261.cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = \u2261.cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = \u2261.refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 \u2261.refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 \u2261.cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 \u2261.refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 \u2261.refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261.\u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!} \n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\n\u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : Set i} {A : Set a} {B : A \u2192 Set b} {C : (x : A) \u2192 B x \u2192 Set c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : Set a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 Set \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : Set a) = EndoMonoid-\u2248 {A = A} \u2261.isEquivalence (\u2261.cong\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : Set a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2261.\u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 \u2261.trans (p (h x)) (\u2261.cong g (q x)))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ccb724c55f5701acf132f0a3045f0a1c2d116f96","subject":"agda: Update alternative definition of Valid-\u0394","message":"agda: Update alternative definition of Valid-\u0394\n\nRationale for maintaining this alternative definition: If all proofs\ngo through with the current definition of Valid-\u0394, which can be\nexpressed as a datatype, we'll switch to defining it with a datatype.\n\nOld-commit-hash: a21df87af1c12f4d94d3710d37e6e4eedfcc83b0\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/ValidChanges.agda","new_file":"Denotational\/ValidChanges.agda","new_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid-\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","old_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"884a5f0cedea9599476b8420c36f53c0676e8551","subject":"Add turnstile","message":"Add turnstile\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n--------------------\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n--------------------\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218 = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n--------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\nmodusponens : \u2115 \u2192 Arrow \u2192 Arrow\nmodusponens n (\u21d2 q)   = (\u21d2 q)\nmodusponens n (p \u21d2 q) with (n \u2261 p)\n...                      | true  = modusponens n q\n...                      | false = p \u21d2 (modusponens n q)\n\n\nreduce : \u2115 \u2192 List Arrow \u2192 List Arrow\nreduce n \u2218 = \u2218\nreduce n (arr \u2237 rst) = (modusponens n arr) \u2237 (reduce n rst)\n\n\nsearch : List Arrow \u2192 List \u2115\nsearch \u2218 = \u2218\nsearch ((\u21d2 n) \u2237 rst)   = n \u2237 (search (reduce n rst))\nsearch ((n \u21d2 q) \u2237 rst) with (n \u2208 (search rst))\n...                       | true  = search (q \u2237 rst)\n...                       | false = search rst\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q) = q \u2208 (search cs)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n--------------------\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n--------------------\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218 = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n--------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\nmodusponens : \u2115 \u2192 Arrow \u2192 Arrow\nmodusponens n (\u21d2 q)   = (\u21d2 q)\nmodusponens n (p \u21d2 q) with (n \u2261 p)\n...                      | true  = modusponens n q\n...                      | false = p \u21d2 (modusponens n q)\n\n\nreduce : \u2115 \u2192 List Arrow \u2192 List Arrow\nreduce n \u2218 = \u2218\nreduce n (arr \u2237 rst) = (modusponens n arr) \u2237 (reduce n rst)\n\n\nsearch : List Arrow \u2192 List \u2115\nsearch \u2218 = \u2218\nsearch ((\u21d2 n) \u2237 rst)   = n \u2237 (search (reduce n rst))\nsearch ((n \u21d2 q) \u2237 rst) with (n \u2208 (search rst))\n...                       | true  = search (q \u2237 rst)\n...                       | false = search rst\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1881f087ce62e803d0be7c04de43f96b707b2fe2","subject":"+Fin1\u2194\u22a4","message":"+Fin1\u2194\u22a4\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Bit using (Bit; 0b; 1b; proj)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_)\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \u22a5-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \u22a5-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \u22a5-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \u22a5-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\u22a4 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \u22a4\n\u03a3\u2261\u2194\u22a4 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : S \u2194 T\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\u22a4\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \u22a4 \u228e A)\nMaybe\u2194Lift\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\u22a4 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \u22a4\nVec0\u2194Lift\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\u22a4 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses (\u03bb { (x \u2237 xs) \u2192 x , xs }) (uncurry _\u2237_)\n             (\u03bb { (x \u2237 xs) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin1\u2194\u22a4 : Fin 1 \u2194 \u22a4\nFin1\u2194\u22a4 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\u22a4\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \u22a4) _ _ \u2218 (Maybe\u2194Lift\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nBit\u2194\u22a4\u228e\u22a4 : Bit \u2194 (\u22a4 \u228e \u22a4)\nBit\u2194\u22a4\u228e\u22a4 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0b , F.const 1b ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : Bit) \u2192 _\n  \u21d0\u21d2 1b = \u2261.refl\n  \u21d0\u21d2 0b = \u2261.refl\n  \u21d2\u21d0 : (_ : \u22a4 \u228e \u22a4) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9\n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n-- -}\n","old_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen L using (Lift; lower; lift)\nopen import Type hiding (\u2605)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Bit using (Bit; 0b; 1b; proj)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_)\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \u22a5-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \u22a5-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \u22a5-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \u22a5-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \u22a5-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid L.zero L.zero\n\n\u03a3\u2261\u2194\u22a4 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \u22a4\n\u03a3\u2261\u2194\u22a4 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.subst (C F.\u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with to f (from f (proj\u2081 p)) | right-f (proj\u2081 p) | left-f (from f (proj\u2081 p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : S \u2194 T\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n{- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : S \u2194 T\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) {!\u21d0\u21d2!} \u21d2\u21d0\n-}\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \u22a5;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a5) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \u22a5 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \u22a4;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  left-identity : LeftIdentity (\u2261.setoid \u22a4) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2082\n      ; cong = proj\u2082\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      ; cong  = < proj\u2081 F.\u2218 proj\u2081 , < proj\u2082 F.\u2218 proj\u2081 , proj\u2082 > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      ; cong  = < < proj\u2081 , proj\u2081 F.\u2218 proj\u2082 > , proj\u2082 F.\u2218 proj\u2082 >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \u22a5\n  ; 1# = \u2261.setoid \u22a4\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = L.zero}{f\u2082 = L.zero})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \u22a5) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = proj\u2081\n      ; cong = proj\u2081\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u22a5-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u22a5-elim (proj\u2081 x)\n      ; right-inverse-of = \u03bb x \u2192 \u22a5-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nMaybe\u2194Lift\u22a4\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \u22a4 \u228e A)\nMaybe\u2194Lift\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\u22a4 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \u22a4\nVec0\u2194Lift\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\u22a4 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses (\u03bb { (x \u2237 xs) \u2192 x , xs }) (uncurry _\u2237_)\n             (\u03bb { (x \u2237 xs) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n\u22a4\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a4 \u00d7 A) \u2194 A\n\u22a4\u00d7A\u2194A = proj\u2081 \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\nA\u00d7\u22a4\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \u22a4) \u2194 A\nA\u00d7\u22a4\u2194A = proj\u2082 \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n\u22a5\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u228e A) \u2194 A\n\u22a5\u228eA\u2194A = proj\u2081 \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\nA\u228e\u22a5\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \u22a5) \u2194 A\nA\u228e\u22a5\u2194A = proj\u2082 \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n\u22a5\u00d7A\u2194\u22a5 : \u2200 {A : \u2605\u2080} \u2192 (\u22a5 \u00d7 A) \u2194 \u22a5\n\u22a5\u00d7A\u2194\u22a5 = Lift\u2194id \u2218 proj\u2081 \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = A\u228e\u22a5\u2194A \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\u22a4\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \u22a4) _ _ \u2218 (Maybe\u2194Lift\u22a4\u228e \u228e-cong id)\n\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \u22a5 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \u22a5\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\nBit\u2194\u22a4\u228e\u22a4 : Bit \u2194 (\u22a4 \u228e \u22a4)\nBit\u2194\u22a4\u228e\u22a4 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0b , F.const 1b ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : Bit) \u2192 _\n  \u21d0\u21d2 1b = \u2261.refl\n  \u21d0\u21d2 0b = \u2261.refl\n  \u21d2\u21d0 : (_ : \u22a4 \u228e \u22a4) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\u22a5\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\u22a5\u2194Fin m \u228e-cong Maybe^\u22a5\u2194Fin n)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\u22a5 \u00d7-cong id \u27e9\n                 (\u22a5 \u00d7 Fin n) \u2194\u27e8 \u22a5\u00d7A\u2194\u22a5 \u27e9\n                 \u22a5 \u2194\u27e8 sym Fin0\u2194\u22a5 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\u22a4\u228eFin \u00d7-cong id \u27e9\n                    ((\u22a4 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \u22a4 (Fin m) \u27e9\n                    ((\u22a4 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \u22a4\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"eae398a7bc4f9c6305c05cff7cbddd24c96846f1","subject":"comma and fst for \u2297","message":"comma and fst for \u2297\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 \u2297\u214b-dual (P m) _)\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = 0\u2081\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n  comma\u1d3e P Q p q with view-proc P p | view-proc Q q\n  comma\u1d3e .(com (mk Out M P)) Q .(m , p) q | send M P m p | q' = m , comma\u1d3e (P m) Q p q\n  comma\u1d3e .(com (mk In M P)) .(com (mk Out M\u2081 P\u2081)) p .(m , p\u2081) | recv M P .p | send M\u2081 P\u2081 m p\u2081 = m , comma\u1d3e (\u03a0\u1d3e M P) (P\u2081 m) p p\u2081\n  comma\u1d3e ._ ._ ._ ._ | recv M P p | recv M\u2081 P\u2081 x = [inl: (\u03bb m \u2192 comma\u1d3e (P m) (\u03a0\u1d3e _ P\u2081) (p m) x)\n                                                   ,inr: (\u03bb m' \u2192 comma\u1d3e (\u03a0\u1d3e _ P) (P\u2081 m') p (x m') ) ]\n  comma\u1d3e ._ ._ ._ ._ | recv M P x | end = x\n  comma\u1d3e .end Q .0\u2081 q | end | q' = q\n\n  \u2297\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297\u1d3e-fst end      Q        pq       = _\n  \u2297\u1d3e-fst (\u03a3\u1d3e M P) Q        (m , pq) = m , \u2297\u1d3e-fst (P m) Q pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) end      pq       = pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (_ , pq) = \u2297\u1d3e-fst (\u03a0\u1d3e M P) (Q _) pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-fst (P m) (\u03a0\u1d3e N Q) (pq (inl m))\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 \u2297\u214b-dual (P m) _)\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = 0\u2081\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"334cce029d0399722d5aa02f5101458e2597cdb6","subject":"bintree: re-org","message":"bintree: re-org\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Sum hiding (map)\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen Alternative-Reverse\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nprivate\n  module Dummy {a} {A : Set a} where\n    fromFun : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Tree A n\n    fromFun {zero} f = leaf (f [])\n    fromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\n    toFun : \u2200 {n} \u2192 Tree A n \u2192 Bits n \u2192 A\n    toFun (leaf x) _ = x\n    toFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\n    toFun\u2218fromFun : \u2200 {n} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\n    toFun\u2218fromFun {zero}  f [] = \u2261.refl\n    toFun\u2218fromFun {suc n} f (false \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\n    toFun\u2218fromFun {suc n} f (true \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\n    fromFun\u2218toFun : \u2200 {n} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\n    fromFun\u2218toFun (leaf x) = \u2261.refl\n    fromFun\u2218toFun (fork t\u2080 t\u2081)\n      rewrite fromFun\u2218toFun t\u2080\n            | fromFun\u2218toFun t\u2081 = \u2261.refl\n\n    toFun\u2192fromFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\n    toFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\n    toFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n      rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n            | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\n    fromFun\u2192toFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\n    fromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\n    fromFun-\u2257 : \u2200 {n} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\n    fromFun-\u2257 {zero}  f\u2257g\n      rewrite f\u2257g [] = \u2261.refl\n    fromFun-\u2257 {suc n} f\u2257g\n      rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n            | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n            = \u2261.refl\n\n    lookup : \u2200 {n} \u2192 Bits n \u2192 Tree A n \u2192 A\n    lookup = flip toFun\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\n\n    first : \u2200 {n} \u2192 Tree A n \u2192 A\n    first (leaf x)   = x\n    first (fork t _) = first t\n\n    last : \u2200 {n} \u2192 Tree A n \u2192 A\n    last (leaf x)   = x\n    last (fork _ t) = last t\n\n    -- Returns the flat vector of leaves underlying the perfect binary tree.\n    toVec : \u2200 {n} \u2192 Tree A n \u2192 Vec A (2^ n)\n    toVec (leaf x)     = x \u2237 []\n    toVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\n    lookup' : \u2200 {m n} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\n    lookup' [] t = t\n    lookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n    update' : \u2200 {m n} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\n    update' []        val t = val\n    update' (b \u2237 key) val (fork t u) = if b then fork t (update' key val u)\n                                            else fork (update' key val t) u\n\nopen Dummy public\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n = n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\nmodule Rot where\n    data Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n      leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n      fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n             Rot left\u2080 left\u2081 \u2192\n             Rot right\u2080 right\u2081 \u2192\n             Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n      krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n             Rot left\u2080 right\u2081 \u2192\n             Rot right\u2080 left\u2081 \u2192\n             Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\n    Rot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\n    Rot-refl {x = leaf x} = leaf x\n    Rot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\n    Rot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\n    Rot-sym (leaf x) = leaf x\n    Rot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\n    Rot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\n    Rot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\n    Rot-trans (leaf x) q = q\n    Rot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\n    Rot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\n    Rot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n    Rot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\nmodule SwpOp where\n    data SwpOp : \u2115 \u2192 Set where\n      \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n      _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n      first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n      swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n      swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\n    data Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n      \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n      _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n      first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n             Perm tA tB \u2192\n             Perm (fork tA tC) (fork tB tC)\n\n      swp : \u2200 {n} {tA tB : Tree A n} \u2192\n             Perm (fork tA tB) (fork tB tA)\n\n      swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                     Perm (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tA tD) (fork tC tB))\n\n    data Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n      \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n      swp : \u2200 {n} {tA tB : Tree A n} \u2192\n             Perm0\u2194 (fork tA tB) (fork tB tA)\n\n      first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n             Perm0\u2194 tA tB \u2192\n             Perm0\u2194 (fork tA tC) (fork tB tC)\n\n      firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                     Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                     Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tE tB) (fork tF tD))\n\n      extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                     Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                     Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tE tB) (fork tC tF))\n\n    -- Star Perm0\u2194 can then model any permutation\n\n    infixr 1 _\u204f_\n\n    second-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n               Perm right\u2080 right\u2081 \u2192\n               Perm (fork left right\u2080) (fork left right\u2081)\n    second-perm f = swp \u204f first f \u204f swp\n\n    second-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n    second-swpop f = swp \u204f first f \u204f swp\n\n    <_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n               Perm left\u2080 left\u2081 \u2192\n               Perm right\u2080 right\u2081 \u2192\n               Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n    < f \u00d7 g >-perm = first f \u204f second-perm g\n\n    swp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n              Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n    swp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\n    swp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n             Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\n    swp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\n    swp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                     Perm (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tC tB) (fork tA tD))\n    swp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\n    Swp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\n    Swp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\n    Swp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\n    Swp\u21d2Perm swp\u2081 = swp\n    Swp\u21d2Perm swp\u2082 = swp\u2082-perm\n\n    Swp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\n    Swp\u2605\u21d2Perm \u03b5         = \u03b5\n    Swp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\n    swp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\n    swp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\n    on-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\n    on-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\n    on-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\n    on-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n    0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n    0\u2194 [] = \u03b5\n    0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 []) = swp\n    0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n    0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\n    commSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\n    commSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n    [_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n    [ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n    [_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n    [ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n    _$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n    \u03b5           $swp t = t\n    (f \u204f g)     $swp t = g $swp (f $swp t)\n    (first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\n    swp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\n    swp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\n    swpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\n    swpRel \u03b5           _          = \u03b5\n    swpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\n    swpRel (first f)   (fork _ _) = first (swpRel f _)\n    swpRel swp         (fork _ _) = swp\n    swpRel swp-seconds\n     (fork (fork _ _) (fork _ _)) = swp-seconds\n\n    [0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n    [0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\n    swpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\n    swpOp' \u03b5 = \u03b5\n    swpOp' (first f) = first (swpOp' f)\n    swpOp' swp = swp\n    swpOp' (firsts f) = on-firsts (swpOp' f)\n    swpOp' (extremes f) = on-extremes (swpOp' f)\n\n    swpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\n    swpOp \u03b5 = \u03b5\n    swpOp (f \u204f g) = swpOp f \u204f  swpOp g\n    swpOp (first f) = first (swpOp f)\n    swpOp swp = swp\n    swpOp swp-seconds = swp-seconds\n\n    swpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\n    swpOp-sym \u03b5 = \u03b5\n    swpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\n    swpOp-sym (first f) = first (swpOp-sym f)\n    swpOp-sym swp = swp\n    swpOp-sym swp-seconds = swp-seconds\n\n    swpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\n    swpOp-sym-involutive \u03b5 = \u2261.refl\n    swpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\n    swpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\n    swpOp-sym-involutive swp = \u2261.refl\n    swpOp-sym-involutive swp-seconds = \u2261.refl\n\n    swpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\n    swpOp-sym-sound \u03b5 t = \u2261.refl\n    swpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\n    swpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\n    swpOp-sym-sound swp (fork _ _) = \u2261.refl\n    swpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\n    module \u00acswp-comm where\n      data X : Set where\n        A B C D E F G H : X\n      n : \u2115\n      n = 3\n      t : Tree X n\n      t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n      f : SwpOp n\n      f = swp\n      g : SwpOp n\n      g = first swp\n      pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n      pf ()\n\n    swp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\n    swp-leaf \u03b5 x = \u2261.refl\n    swp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = \u2261.refl\n\n    swpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\n    swpOp-sound \u03b5 = \u2261.refl\n    swpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = \u2261.refl\n    swpOp-sound (first f) rewrite swpOp-sound f = \u2261.refl\n    swpOp-sound swp = \u2261.refl\n    swpOp-sound swp-seconds = \u2261.refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open Rot\n  open SwpOp\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2261.\u2192-to-\u27f6 fun\n    ; from       = \u2261.\u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       fun (left path)  = right path\n       fun (right path) = left path\n\n       inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 fun (fun p) \u2261 p\n       inv (left p)  = \u2261.refl\n       inv (right p) = \u2261.refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2261.\u2192-to-\u27f6 fun\n    ; from       = \u2261.\u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = \u2261.refl\n       inv (left (right p)) = \u2261.refl\n       inv (right (left p)) = \u2261.refl\n       inv (right (right p)) = \u2261.refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2261.\u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2261.\u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg \u2261.refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  {-\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n  -}\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\n  \u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n  \u2208-fromFun f here      = \u2261.refl\n  \u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n  \u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n  \u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n  \u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nmodule fold-Properties {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n  open Rot\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = \u2261.refl\n  fold-rot (fork rot rot\u2081) = \u2261.cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = \u2261.refl\n  fold-swp (right pf) rewrite fold-swp pf = \u2261.refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = \u2261.refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = \u2261.refl\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    open new-approach\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    open new-approach\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x\u2080 , x\u2081) = (x\u2080 \u2293\u1d2c x\u2081 , x\u2080 \u2294\u1d2c x\u2081)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule EvalTree where\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {n a} {A : Set a} \u2192 Bij n \u2192 Endo (Tree A n)\n    evalTree `id          = id\n    evalTree (op\u2080 `\u204f op\u2081) = evalTree op\u2081 \u2218 evalTree op\u2080\n    evalTree (`id   `\u2237 g) = map-outer (evalTree (g 0b)) (evalTree (g 1b))\n    evalTree (`not\u1d2e `\u2237 g) = map-outer (evalTree (g 1b)) (evalTree (g 0b)) \u2218 swap\n    evalTree `0\u21941         = interchange\n\n    evalTree-eval : \u2200 {n a} {A : Set a} (f : Bij n) (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id t xs = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval (f `\u204f f\u2081) t xs = \u2261.trans (evalTree-eval f t xs) (evalTree-eval f\u2081 (evalTree f t) (eval f xs))\n    evalTree-eval (`id `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {n a} {A : Set a} (f : Bij n) (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule PermTreeProof where\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Bij n\n        proof : (t : Tree A n) \u2192 t \u2261 evalTree (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {f = f}{g} (mk `f pf) (mk `g pg)\n      = mk (\u03bb t \u2192 `f (g t) `\u204f `g t)\n           (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (evalTree (`g t)) (pf (g t))))\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = mk (\u03bb _ \u2192 `not) \u03b7fork\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof `f `g = mk (\u03bb t \u2192 `map-outer (perm `f (lft t)) (perm `g (rght t)))\n                               (\u03bb { (fork t u) \u2192 \u2261.cong\u2082 fork (proof `f t) (proof `g u) })\n       where open Perm\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof {A} {f = f} `f = mk map-inner-perm helper\n       where open Perm\n             map-inner-perm = `map-inner \u2218 perm `f \u2218 inner\n             helper : \u2200 t \u2192 t \u2261 evalTree (map-inner-perm t) (map-inner f t)\n             helper (fork (fork a b) (fork c d)) with f (fork b c) | \u2261.sym (proof `f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\nmodule Sorting-Perm-Properties {OT : Set} (_<=\u1d2c_ : OT \u2192 OT \u2192 Bool)\n    (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    _\u2293\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n\n    _\u2294\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open PermTreeProof\n\n    `sort\u2081 : Tree OT 1 \u2192 Bij 1\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    -- \u2200 x  T (y <=\u1d2c x) \u2192 fork (leaf x) (leaf y) \u2261 fork (leaf (x \u2293 y)) (leaf (x \u2294 y))\n    sort\u2081-proof : Perm {OT} 1 sort\u2081\n    sort\u2081-proof = mk `sort\u2081 helper\n      where helper : \u2200 t \u2192 t \u2261 evalTree (`sort\u2081 t) (sort\u2081 t)\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = sort\u2081-proof\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    module SPP = Sorting-Perm-Properties _<=_ isTotalOrder\n    open EvalTree public using (evalTree)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Sum hiding (map)\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen Alternative-Reverse\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nfromFun-\u2257 : \u2200 {n a} {A : Set a} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\nfromFun-\u2257 {zero}  f\u2257g\n  rewrite f\u2257g [] = \u2261.refl\nfromFun-\u2257 {suc n} f\u2257g\n  rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n        | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n        = \u2261.refl\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\nopen Dummy public\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\nmodule Rot where\n    data Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n      leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n      fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n             Rot left\u2080 left\u2081 \u2192\n             Rot right\u2080 right\u2081 \u2192\n             Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n      krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n             Rot left\u2080 right\u2081 \u2192\n             Rot right\u2080 left\u2081 \u2192\n             Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\n    Rot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\n    Rot-refl {x = leaf x} = leaf x\n    Rot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\n    Rot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\n    Rot-sym (leaf x) = leaf x\n    Rot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\n    Rot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\n    Rot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\n    Rot-trans (leaf x) q = q\n    Rot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\n    Rot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\n    Rot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n    Rot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\nmodule SwpOp where\n    data SwpOp : \u2115 \u2192 Set where\n      \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n      _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n      first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n      swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n      swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\n    data Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n      \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n      _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n      first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n             Perm tA tB \u2192\n             Perm (fork tA tC) (fork tB tC)\n\n      swp : \u2200 {n} {tA tB : Tree A n} \u2192\n             Perm (fork tA tB) (fork tB tA)\n\n      swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                     Perm (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tA tD) (fork tC tB))\n\n    data Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n      \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n      swp : \u2200 {n} {tA tB : Tree A n} \u2192\n             Perm0\u2194 (fork tA tB) (fork tB tA)\n\n      first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n             Perm0\u2194 tA tB \u2192\n             Perm0\u2194 (fork tA tC) (fork tB tC)\n\n      firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                     Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                     Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tE tB) (fork tF tD))\n\n      extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                     Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                     Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tE tB) (fork tC tF))\n\n    -- Star Perm0\u2194 can then model any permutation\n\n    infixr 1 _\u204f_\n\n    second-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n               Perm right\u2080 right\u2081 \u2192\n               Perm (fork left right\u2080) (fork left right\u2081)\n    second-perm f = swp \u204f first f \u204f swp\n\n    second-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n    second-swpop f = swp \u204f first f \u204f swp\n\n    <_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n               Perm left\u2080 left\u2081 \u2192\n               Perm right\u2080 right\u2081 \u2192\n               Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n    < f \u00d7 g >-perm = first f \u204f second-perm g\n\n    swp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n              Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n    swp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\n    swp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n             Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\n    swp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\n    swp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                     Perm (fork (fork tA tB) (fork tC tD))\n                              (fork (fork tC tB) (fork tA tD))\n    swp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\n    Swp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\n    Swp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\n    Swp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\n    Swp\u21d2Perm swp\u2081 = swp\n    Swp\u21d2Perm swp\u2082 = swp\u2082-perm\n\n    Swp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\n    Swp\u2605\u21d2Perm \u03b5         = \u03b5\n    Swp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\n    swp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\n    swp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\n    on-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\n    on-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\n    on-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\n    on-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n    0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n    0\u2194 [] = \u03b5\n    0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 []) = swp\n    0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n    0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\n    commSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\n    commSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n    [_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n    [ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n    [_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n    [ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n    _$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n    \u03b5           $swp t = t\n    (f \u204f g)     $swp t = g $swp (f $swp t)\n    (first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\n    swp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\n    swp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\n    swpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\n    swpRel \u03b5           _          = \u03b5\n    swpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\n    swpRel (first f)   (fork _ _) = first (swpRel f _)\n    swpRel swp         (fork _ _) = swp\n    swpRel swp-seconds\n     (fork (fork _ _) (fork _ _)) = swp-seconds\n\n    [0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n    [0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\n    swpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\n    swpOp' \u03b5 = \u03b5\n    swpOp' (first f) = first (swpOp' f)\n    swpOp' swp = swp\n    swpOp' (firsts f) = on-firsts (swpOp' f)\n    swpOp' (extremes f) = on-extremes (swpOp' f)\n\n    swpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\n    swpOp \u03b5 = \u03b5\n    swpOp (f \u204f g) = swpOp f \u204f  swpOp g\n    swpOp (first f) = first (swpOp f)\n    swpOp swp = swp\n    swpOp swp-seconds = swp-seconds\n\n    swpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\n    swpOp-sym \u03b5 = \u03b5\n    swpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\n    swpOp-sym (first f) = first (swpOp-sym f)\n    swpOp-sym swp = swp\n    swpOp-sym swp-seconds = swp-seconds\n\n    swpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\n    swpOp-sym-involutive \u03b5 = \u2261.refl\n    swpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\n    swpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\n    swpOp-sym-involutive swp = \u2261.refl\n    swpOp-sym-involutive swp-seconds = \u2261.refl\n\n    swpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\n    swpOp-sym-sound \u03b5 t = \u2261.refl\n    swpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\n    swpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\n    swpOp-sym-sound swp (fork _ _) = \u2261.refl\n    swpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\n    module \u00acswp-comm where\n      data X : Set where\n        A B C D E F G H : X\n      n : \u2115\n      n = 3\n      t : Tree X n\n      t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n      f : SwpOp n\n      f = swp\n      g : SwpOp n\n      g = first swp\n      pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n      pf ()\n\n    swp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\n    swp-leaf \u03b5 x = \u2261.refl\n    swp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = \u2261.refl\n\n    swpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\n    swpOp-sound \u03b5 = \u2261.refl\n    swpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = \u2261.refl\n    swpOp-sound (first f) rewrite swpOp-sound f = \u2261.refl\n    swpOp-sound swp = \u2261.refl\n    swpOp-sound swp-seconds = \u2261.refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\n\nmodule new-approach where\n\n  open Rot\n  open SwpOp\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2261.\u2192-to-\u27f6 fun\n    ; from       = \u2261.\u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       fun (left path)  = right path\n       fun (right path) = left path\n\n       inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 fun (fun p) \u2261 p\n       inv (left p)  = \u2261.refl\n       inv (right p) = \u2261.refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2261.\u2192-to-\u27f6 fun\n    ; from       = \u2261.\u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = \u2261.refl\n       inv (left (right p)) = \u2261.refl\n       inv (right (left p)) = \u2261.refl\n       inv (right (right p)) = \u2261.refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2261.\u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2261.\u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg \u2261.refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  {-\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n  -}\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\n  \u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n  \u2208-fromFun f here      = \u2261.refl\n  \u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n  \u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n  \u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n  \u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nmodule fold-Properties {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n  open Rot\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = \u2261.refl\n  fold-rot (fork rot rot\u2081) = \u2261.cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = \u2261.refl\n  fold-swp (right pf) rewrite fold-swp pf = \u2261.refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = \u2261.refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = \u2261.refl\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    open new-approach\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    open new-approach\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x\u2080 , x\u2081) = (x\u2080 \u2293\u1d2c x\u2081 , x\u2080 \u2294\u1d2c x\u2081)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule EvalTree where\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {n a} {A : Set a} \u2192 Bij n \u2192 Endo (Tree A n)\n    evalTree `id          = id\n    evalTree (op\u2080 `\u204f op\u2081) = evalTree op\u2081 \u2218 evalTree op\u2080\n    evalTree (`id   `\u2237 g) = map-outer (evalTree (g 0b)) (evalTree (g 1b))\n    evalTree (`not\u1d2e `\u2237 g) = map-outer (evalTree (g 1b)) (evalTree (g 0b)) \u2218 swap\n    evalTree `0\u21941         = interchange\n\n    evalTree-eval : \u2200 {n a} {A : Set a} (f : Bij n) (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id t xs = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 true \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (true \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork a b) (fork c d)) (false \u2237 false \u2237 xs) = \u2261.refl\n    evalTree-eval (f `\u204f f\u2081) t xs = \u2261.trans (evalTree-eval f t xs) (evalTree-eval f\u2081 (evalTree f t) (eval f xs))\n    evalTree-eval (`id `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {n a} {A : Set a} (f : Bij n) (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule PermTreeProof where\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Bij n\n        proof : (t : Tree A n) \u2192 t \u2261 evalTree (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {f = f}{g} (mk `f pf) (mk `g pg)\n      = mk (\u03bb t \u2192 `f (g t) `\u204f `g t)\n           (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (evalTree (`g t)) (pf (g t))))\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = mk (\u03bb _ \u2192 `not) \u03b7fork\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof `f `g = mk (\u03bb t \u2192 `map-outer (perm `f (lft t)) (perm `g (rght t)))\n                               (\u03bb { (fork t u) \u2192 \u2261.cong\u2082 fork (proof `f t) (proof `g u) })\n       where open Perm\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof {A} {f = f} `f = mk map-inner-perm helper\n       where open Perm\n             map-inner-perm = `map-inner \u2218 perm `f \u2218 inner\n             helper : \u2200 t \u2192 t \u2261 evalTree (map-inner-perm t) (map-inner f t)\n             helper (fork (fork a b) (fork c d)) with f (fork b c) | \u2261.sym (proof `f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\nmodule Sorting-Perm-Properties {OT : Set} (_<=\u1d2c_ : OT \u2192 OT \u2192 Bool)\n    (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    _\u2293\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n\n    _\u2294\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open PermTreeProof\n\n    `sort\u2081 : Tree OT 1 \u2192 Bij 1\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    -- \u2200 x  T (y <=\u1d2c x) \u2192 fork (leaf x) (leaf y) \u2261 fork (leaf (x \u2293 y)) (leaf (x \u2294 y))\n    sort\u2081-proof : Perm {OT} 1 sort\u2081\n    sort\u2081-proof = mk `sort\u2081 helper\n      where helper : \u2200 t \u2192 t \u2261 evalTree (`sort\u2081 t) (sort\u2081 t)\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = sort\u2081-proof\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    module SPP = Sorting-Perm-Properties _<=_ isTotalOrder\n    open EvalTree public using (evalTree)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5938d3baec1046be00080400a0f59df41478f811","subject":"Agda: have separate lemmas for weakening contexts","message":"Agda: have separate lemmas for weakening contexts\n\nThis resembles the paper treatment; moreover, the separate lemmas are easier to\ngenerate, because most cases can be auto-generated but only for one of the\nlemmas.\n\nOld-commit-hash: f419fd7acfdb0841d9771834036e2c5f24f10055\n","repos":"inc-lc\/ilc-agda","old_file":"ModalLogic.agda","new_file":"ModalLogic.agda","new_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotational.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntactic.Contexts Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n--\n-- Attempt 1 at weakening, subcontext-based. This works, but it is not what we\n-- need later (because of the way substitution is stated in the paper).\n--\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be\n-- fixed and lift introduced by hand.\n\nweaken : \u2200 {\u0393 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393 \u0394\u2081 \u03c4 \u2192 Term \u0393 \u0394\u2082 \u03c4\nweaken \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (app t t\u2081) = app (weaken \u0394\u2032 t) (weaken \u0394\u2032 t\u2081)\nweaken \u0394\u2032 (ovar x) = ovar x\nweaken \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0394\u2032 (box t) = box (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0394\u2032 t in-\n          weaken (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n\nopen import Relation.Binary.PropositionalEquality\n\n-- These two lemmas could be merged.\n\nweaken-\u227c : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u0394 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u0394 \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u0394 \u03c4\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} {\u0393\u2083} (abs {\u03c4\u2081} t) =  abs (weaken-\u227c {\u03c4\u2081 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (app t t\u2081) = app (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t) (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (ovar x) = ovar (Prefixes.lift {\u0393\u2081} {\u0393\u2082} x)\nweaken-\u227c (mvar u) = mvar u\nweaken-\u227c (box t) = box t\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c {\u0393\u2081} {\u0393\u2082} t in- weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081\n\n\nweaken-\u227c-\u0394 : \u2200 {\u0394\u2081 \u0394\u2082 \u0394\u2083 \u0393 \u03c4} \u2192 Term \u0393 (\u0394\u2081 \u22ce \u0394\u2083) \u03c4 \u2192 Term \u0393 (\u0394\u2081 \u22ce \u0394\u2082 \u22ce \u0394\u2083) \u03c4\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (abs t) = abs (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (app t t\u2081) = app (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t) (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t\u2081)\nweaken-\u227c-\u0394 (ovar x) = ovar x\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (mvar u) = mvar (Prefixes.lift {\u0394\u2081} {\u0394\u2082} u)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (box t) = box (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t in- weaken-\u227c-\u0394 {_ \u2022 \u0394\u2081} {\u0394\u2082} t\u2081 \n\nsubstVar : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (v : Var (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstVar {\u2205} t  this = t\nsubstVar {\u2205} t (that v) = ovar v\nsubstVar {\u03c4 \u2022 \u0393} t  this = ovar this\nsubstVar {\u03c4 \u2022 \u0393} t (that v) = weaken-\u227c {\u2205} {\u03c4 \u2022 \u2205} (substVar {\u0393} t v)\n\nsubstTerm : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (t\u2082 : Term (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u0394 \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (abs {\u03c4\u2082} t\u2082) = abs ( substTerm {\u03c4\u2082 \u2022 \u0393} {\u0393\u2032} t\u2081 t\u2082 )\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (app t\u2082 t\u2083) =  app (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082) (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2083) \nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (ovar v) =  substVar {\u0393} {\u0393\u2032} t\u2081 v\nsubstTerm t\u2081 (mvar u) = mvar u\nsubstTerm t\u2081 (box t\u2082) = box t\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (let-box_in-_ {\u03c4\u2083} t\u2082 t\u2083) =\n  let-box substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082 in-\n    substTerm {\u0393} {\u0393\u2032}\n      (weaken-\u227c-\u0394 {\u2205} {\u03c4\u2083 \u2022 \u2205} t\u2081)\n      t\u2083\n","old_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotational.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntactic.Contexts Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n--\n-- Attempt 1 at weakening, subcontext-based. This works, but it is not what we\n-- need later (because of the way substitution is stated in the paper).\n--\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be\n-- fixed and lift introduced by hand.\n\nweaken : \u2200 {\u0393 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393 \u0394\u2081 \u03c4 \u2192 Term \u0393 \u0394\u2082 \u03c4\nweaken \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (app t t\u2081) = app (weaken \u0394\u2032 t) (weaken \u0394\u2032 t\u2081)\nweaken \u0394\u2032 (ovar x) = ovar x\nweaken \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0394\u2032 (box t) = box (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0394\u2032 t in-\n          weaken (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n\n\nopen import Relation.Binary.PropositionalEquality\n\nweaken-\u227c : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u0394 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u0394 \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u0394 \u03c4\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} {\u0393\u2083} (abs {\u03c4\u2081} t) =  abs (weaken-\u227c {\u03c4\u2081 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (app t t\u2081) = app (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t) (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (ovar x) = ovar (Prefixes.lift {\u0393\u2081} {\u0393\u2082} x)\nweaken-\u227c (mvar u) = mvar u\nweaken-\u227c (box t) = box t\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c {\u0393\u2081} {\u0393\u2082} t in- weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081\n\n{-\nweaken-\u0394-1 : \u2200 {\u0393 \u0394 \u03c4 \u03c4\u2032} \u2192 Term \u0393 \u0394 \u03c4 \u2192 Term \u0393 (\u03c4\u2032 \u2022 \u0394) \u03c4\nweaken-\u0394-1 (abs t) = abs (weaken-\u0394-1 t)\nweaken-\u0394-1 (app t t\u2081) = app (weaken-\u0394-1 t) (weaken-\u0394-1 t\u2081)\nweaken-\u0394-1 (ovar x) = ovar x\nweaken-\u0394-1 (mvar u) = mvar (that u)\nweaken-\u0394-1 (box t) = box (weaken-\u0394-1 t)\n-- This is not enough...\nweaken-\u0394-1 (let-box t in- t\u2081) = let-box weaken-\u0394-1 t in- {!weaken-\u0394-1 t\u2081!}\n-}\n\nsubstVar : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (v : Var (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstVar {\u2205} t  this = t\nsubstVar {\u2205} t (that v) = ovar v\nsubstVar {\u03c4 \u2022 \u0393} t  this = ovar this\nsubstVar {\u03c4 \u2022 \u0393} t (that v) = weaken-\u227c {\u2205} {\u03c4 \u2022 \u2205} (substVar {\u0393} t v)\n\nsubstTerm : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (t\u2082 : Term (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u0394 \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (abs {\u03c4\u2082} t\u2082) = abs ( substTerm {\u03c4\u2082 \u2022 \u0393} {\u0393\u2032} t\u2081 t\u2082 )\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (app t\u2082 t\u2083) =  app (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082) (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2083) \nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (ovar v) =  substVar {\u0393} {\u0393\u2032} t\u2081 v\nsubstTerm t\u2081 (mvar u) = mvar u\nsubstTerm t\u2081 (box t\u2082) = box t\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (let-box_in-_ {\u03c4\u2083} t\u2082 t\u2083) =\n  let-box substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082 in-\n    substTerm {\u0393} {\u0393\u2032}\n      --(weaken-\u0394-1 t\u2081)\n      (weaken (drop \u03c4\u2083 \u2022 \u227c-refl) t\u2081)\n      t\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"17d8ac64334f4d73653a2b77c3198f81fa229fac","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 300e04a63db5641f8b9616ebd205959c\n\ndarcs-hash:20120320125722-3bd4e-6defb55ec3f391d451470ff41f0534191b2908eb.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Base.agda","new_file":"src\/FOTC\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n* Co-inductively defined predicates   * Greatest fixed-points\n-}\n\n-- References:\n--\n-- \u2022 Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n--   theory with one universe. In Miettinen and V\u00e4\u00e4nanen,\n--   editors. Proc. of the Symposium on Mathematical Logic (Oulu,\n--   1974), Report No. 2, Department of Philosopy, University of\n--   Helsinki, Helsinki, 1977, pages 1\u201332.\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfixr 8 _\u2237_\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- FOL with equality.\nopen import Common.FOL.FOL-Eq public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | nil  | cons  | null | head   | tail\n--      | loop\n\npostulate\n  _\u00b7_                     : D \u2192 D \u2192 D  -- FOTC application.\n  true false if           : D          -- FOTC partial booleans.\n  zero succ pred iszero   : D          -- FOTC partial natural numbers.\n  nil cons head tail null : D          -- FOTC lists.\n  loop                    : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing and\n-- looking for an optimization for the ATPs.\n\n-- 2012-03-20. The definitions are inside an abstract block because\n-- the conversion rules (see below) are based on them, so want to\n-- avoid their expansion.\n\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 d = succ \u00b7 d\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 d = pred \u00b7 d\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 d = iszero \u00b7 d\n  -- {-# ATP definition iszero\u2081 #-}\n\n  [] : D\n  [] = nil\n  -- {-# ATP definition [] #-}\n\n  _\u2237_  : D \u2192 D \u2192 D\n  x \u2237 xs = cons \u00b7 x \u00b7 xs\n  -- {-# ATP definition _\u2237_ #-}\n\n  head\u2081 : D \u2192 D\n  head\u2081 xs = head \u00b7 xs\n  -- {-# ATP definition head\u2081 #-}\n\n  tail\u2081 : D \u2192 D\n  tail\u2081 xs = tail \u00b7 xs\n  -- {-# ATP definition tail\u2081 #-}\n\n  null\u2081 : D \u2192 D\n  null\u2081 xs = null \u00b7 xs\n  -- {-# ATP definition null\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977, p. 8)\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as FOL non-logical\n-- axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- term constants.\n\n-- Conversion rules for booleans.\npostulate\n  -- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n  -- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred \u00b7 zero     \u2261 zero\n  -- pred-S : \u2200 d \u2192 pred \u00b7 (succ \u00b7 d) \u2261 d\n  pred-S : \u2200 d \u2192 pred\u2081 (succ\u2081 d) \u2261 d\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\npostulate\n  -- iszero-0 :       iszero \u00b7 zero       \u2261 true\n  -- iszero-S : \u2200 d \u2192 iszero \u00b7 (succ \u00b7 d) \u2261 false\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 d \u2192 iszero\u2081 (succ\u2081 d) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rules for null.\npostulate\n  -- null-[] :          null \u00b7 nil             \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  null-[] :          null\u2081 []       \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\npostulate\n--  head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\npostulate\n--  tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n-- Conversion rule for loop.\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a FOL axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2262false : \u00ac (true \u2261 false)\n--  0\u2262S        : \u2200 {d} \u2192 \u00ac (zero \u2261 succ \u00b7 d)\n  0\u2262S        : \u2200 {d} \u2192 \u00ac (zero \u2261 succ\u2081 d)\n{-# ATP axiom true\u2262false 0\u2262S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The first-order theory of combinators (FOTC) base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\nFOTC                                  The logical framework (Agda)\n\n* Logical constants                   * Curry-Howard isomorphism\n* Equality                            * Identity type\n* Term language and conversion rules  * Postulates\n* Inductively defined predicates      * Inductive families\n-}\n\nmodule FOTC.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfixr 8 _\u2237_\ninfix  8 if_then_else_\n\n------------------------------------------------------------------------------\n-- FOL with equality.\nopen import Common.FOL.FOL-Eq public\n\n------------------------------------------------------------------------------\n-- The term language of FOTC correspond to the PCF terms.\n\n--   t ::= x   | t \u00b7 t |\n--      | true | false | if\n--      | 0    | succ  | pred | iszero\n--      | nil  | cons  | null | head   | tail\n--      | loop\n\n-- NB. We define the PCF terms as constants. After that, we will\n-- define some function symbols based on these constants for\n-- convenience in writing.\n\npostulate\n  _\u00b7_                     : D \u2192 D \u2192 D  -- FOTC application.\n  true false if           : D          -- FOTC partial booleans.\n  zero succ pred iszero   : D          -- FOTC partial natural numbers.\n  nil cons head tail null : D          -- FOTC lists.\n  loop                    : D          -- FOTC looping programs.\n\n------------------------------------------------------------------------------\n-- Definitions\n\n-- We define some function symbols for convenience in writing.\nabstract\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  if b then d\u2081 else d\u2082 = if \u00b7 b \u00b7 d\u2081 \u00b7 d\u2082\n  -- {-# ATP definition if_then_else_ #-}\n\n  succ\u2081 : D \u2192 D\n  succ\u2081 d = succ \u00b7 d\n  -- {-# ATP definition succ\u2081 #-}\n\n  pred\u2081 : D \u2192 D\n  pred\u2081 d = pred \u00b7 d\n  -- {-# ATP definition pred\u2081 #-}\n\n  iszero\u2081 : D \u2192 D\n  iszero\u2081 d = iszero \u00b7 d\n  -- {-# ATP definition iszero\u2081 #-}\n\n  [] : D\n  [] = nil\n  -- {-# ATP definition [] #-}\n\n  _\u2237_  : D \u2192 D \u2192 D\n  x \u2237 xs = cons \u00b7 x \u00b7 xs\n  -- {-# ATP definition _\u2237_ #-}\n\n  head\u2081 : D \u2192 D\n  head\u2081 xs = head \u00b7 xs\n  -- {-# ATP definition head\u2081 #-}\n\n  tail\u2081 : D \u2192 D\n  tail\u2081 xs = tail \u00b7 xs\n  -- {-# ATP definition tail\u2081 #-}\n\n  null\u2081 : D \u2192 D\n  null\u2081 xs = null \u00b7 xs\n  -- {-# ATP definition null\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- The conversion relation _conv_ satifies (Aczel 1977. The strength\n-- of Martin-L\u00f6f's intuitionistic type theory with one universe,\n-- p. 8).\n--\n-- x conv y <=> FOTC \u22a2 x \u2261 y,\n--\n-- therefore, we introduce the conversion rules as FOL non-logical\n-- axioms.\n\n-- N.B. Looking for an optimization for the ATPs, we write the\n-- conversion rules on the defined function symbols instead of on the\n-- PCF constants.\n\n-- Conversion rules for booleans.\npostulate\n  -- if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n  -- if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true if-false #-}\n\n-- Conversion rules for pred.\npostulate\n  -- N.B. We don't need this equation.\n  -- pred-0 :       pred \u00b7 zero     \u2261 zero\n  -- pred-S : \u2200 d \u2192 pred \u00b7 (succ \u00b7 d) \u2261 d\n  pred-S : \u2200 d \u2192 pred\u2081 (succ\u2081 d) \u2261 d\n{-# ATP axiom pred-S #-}\n\n-- Conversion rules for iszero.\npostulate\n  -- iszero-0 :       iszero \u00b7 zero       \u2261 true\n  -- iszero-S : \u2200 d \u2192 iszero \u00b7 (succ \u00b7 d) \u2261 false\n  iszero-0 :       iszero\u2081 zero      \u2261 true\n  iszero-S : \u2200 d \u2192 iszero\u2081 (succ\u2081 d) \u2261 false\n{-# ATP axiom iszero-0 iszero-S #-}\n\n-- Conversion rules for null.\npostulate\n  -- null-[] :          null \u00b7 nil             \u2261 true\n  -- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\n  null-[] :          null\u2081 []       \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\npostulate\n--  head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\n  head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\npostulate\n--  tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\n  tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n-- Conversion rule for loop.\n-- The equation loop-eq adds anything to the logic (because\n-- reflexivity is already an axiom of equality), therefore we won't\n-- add this equation as a FOL axiom.\npostulate loop-eq : loop \u2261 loop\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2262false : \u00ac (true \u2261 false)\n--  0\u2262S        : \u2200 {d} \u2192 \u00ac (zero \u2261 succ \u00b7 d)\n  0\u2262S        : \u2200 {d} \u2192 \u00ac (zero \u2261 succ\u2081 d)\n{-# ATP axiom true\u2262false 0\u2262S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3d48254fa7870a537bb583af2a8d95c83cd8d58b","subject":"Added TODOs.","message":"Added TODOs.\n\nIgnore-this: c0268f8e06c0cb85594dc394a7c5bcec\n\ndarcs-hash:20100719034157-3bd4e-17dade58a748cf108250a61023ee394e0afaf7c5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  -- N.B. It is not necessary to define neither the data constructor\n  -- _,_, nor the projections -- because the ATPs implement it.\n  data _\u2227_ (A B : Set) : Set where\n\n  -- Testing the data constructor and the projections.\n  postulate\n    A B     : Set\n    _,_     : A \u2192 B \u2192 A \u2227 B\n    \u2227-proj\u2081 : A \u2227 B \u2192 A\n    \u2227-proj\u2082 : A \u2227 B \u2192 B\n  {-# ATP prove _,_ #-}\n  {-# ATP prove \u2227-proj\u2081 #-}\n  {-# ATP prove \u2227-proj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to define neither the data constructors\n  -- inj\u2081 and inj\u2082 nor the disyunction elimination because the ATP\n  -- implements them.\n  data _\u2228_ (A B : Set) : Set where\n\n  -- Testing the data constructors and the elimination.\n  postulate\n    A B C : Set\n    inj\u2081  : A \u2192 A \u2228 B\n    inj\u2082  : B \u2192 A \u2228 B\n    [_,_] : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  {-# ATP prove inj\u2081 #-}\n  {-# ATP prove inj\u2082 #-}\n  {-# ATP prove [_,_] #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\nmodule ExistentialQuantifier where\n\n  data \u2203D (P : D \u2192 Set) : Set where\n\n  -- TODO: We cannot prove the data constructor _,_ due to the\n  -- eta-conversion, i.e.  Agda internally translates '\u2203D (\u03bb d \u2192 P d)' to\n  -- '\u2203D P'.\n\n  -- postulate\n  --   P        : D \u2192 Set\n  --   _,_      : (d : D) \u2192 P d \u2192 \u2203D (\u03bb d \u2192 P d)\n  -- {-# ATP prove _,_ #-}\n\n  -- TODO: What about the eliminators?\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  -- N.B. It is not necessary to define neither the data constructor\n  -- _,_, nor the projections -- because the ATPs implement it.\n  data _\u2227_ (A B : Set) : Set where\n\n  -- Testing the data constructor and the projections.\n  postulate\n    A B     : Set\n    _,_     : A \u2192 B \u2192 A \u2227 B\n    \u2227-proj\u2081 : A \u2227 B \u2192 A\n    \u2227-proj\u2082 : A \u2227 B \u2192 B\n  {-# ATP prove _,_ #-}\n  {-# ATP prove \u2227-proj\u2081 #-}\n  {-# ATP prove \u2227-proj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n\n------------------------------------------------------------------------------\n-- The disjunction data type\n\nmodule Disjunction where\n  infixr 1 _\u2228_\n\n  -- N.B. It is not necessary to define neither the data constructors\n  -- inj\u2081 and inj\u2082 nor the disyunction elimination because the ATP\n  -- implements them.\n  data _\u2228_ (A B : Set) : Set where\n\n  -- Testing the data constructors and the elimination.\n  postulate\n    A B C : Set\n    inj\u2081  : A \u2192 A \u2228 B\n    inj\u2082  : B \u2192 A \u2228 B\n    [_,_] : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  {-# ATP prove inj\u2081 #-}\n  {-# ATP prove inj\u2082 #-}\n  {-# ATP prove [_,_] #-}\n\n------------------------------------------------------------------------------\n-- The existential type of D\n\nmodule ExistentialQuantifier where\n\n  data \u2203D (P : D \u2192 Set) : Set where\n\n  postulate\n    test : (d : D) \u2192 \u2203D (\u03bb e \u2192 e \u2261 d)\n  {-# ATP prove test #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f3ffef18b815e9e7d4a28a0a2b691debc4aa0f00","subject":"wokring on a proof that the relation of disjoint hole names is symmetric","message":"wokring on a proof that the relation of disjoint hole names is symmetric\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nopen import structural\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n\n  mutual\n    -- this looks good but may not *quite* work because of the\n    -- weakening calls in the two lambda cases\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint hd (EASubsume x x\u2081 x\u2082 x\u2083) E2 = expand-synth-disjoint hd x\u2082 E2\n    expand-ana-disjoint (HDLam1 hd) (EALam x\u2081 x\u2082 ex1) E2 = expand-ana-disjoint hd ex1 (weaken-ana-expand x\u2081 E2)\n    expand-ana-disjoint (HDHole x) EAEHole E2 = ##-comm (expand-new-disjoint-ana x E2)\n    expand-ana-disjoint (HDNEHole x hd) (EANEHole x\u2081 x\u2082) E2 = disjoint-parts (expand-synth-disjoint hd x\u2082 E2) (##-comm (expand-new-disjoint-ana x E2))\n\n    expand-synth-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d2 \u03c41 ~> e1' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-synth-disjoint HDConst ESConst ana = empty-disj _\n    expand-synth-disjoint (HDAsc hd) (ESAsc x) ana = expand-ana-disjoint hd x ana\n    expand-synth-disjoint HDVar (ESVar x\u2081) ana = empty-disj _\n    expand-synth-disjoint (HDLam1 hd) () ana\n    expand-synth-disjoint (HDLam2 hd) (ESLam x\u2081 synth) ana = expand-synth-disjoint hd synth (weaken-ana-expand x\u2081 ana)\n    expand-synth-disjoint (HDHole x) ESEHole ana = ##-comm (expand-new-disjoint-ana x ana)\n    expand-synth-disjoint (HDNEHole x hd) (ESNEHole x\u2081 synth) ana = disjoint-parts (expand-synth-disjoint hd synth ana) (##-comm (expand-new-disjoint-ana x ana))\n    expand-synth-disjoint (HDAp hd hd\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) ana = disjoint-parts (expand-ana-disjoint hd x\u2084 ana) (expand-ana-disjoint hd\u2081 x\u2085 ana)\n\n\n\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) = HDAsc {!!}\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) = {!!}\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) = {!!}\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) = {!!}\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) = {!!}\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) = {!!}\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) = {!!}\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) = {!!}\n\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since u == u; it's not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (abeit vacuously)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nopen import structural\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n\n  mutual\n    -- this looks good but may not *quite* work because of the\n    -- weakening calls in the two lambda cases\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint hd (EASubsume x x\u2081 x\u2082 x\u2083) E2 = expand-synth-disjoint hd x\u2082 E2\n    expand-ana-disjoint (HDLam1 hd) (EALam x\u2081 x\u2082 ex1) E2 = expand-ana-disjoint hd ex1 (weaken-ana-expand x\u2081 E2)\n    expand-ana-disjoint (HDHole x) EAEHole E2 = ##-comm (expand-new-disjoint-ana x E2)\n    expand-ana-disjoint (HDNEHole x hd) (EANEHole x\u2081 x\u2082) E2 = disjoint-parts (expand-synth-disjoint hd x\u2082 E2) (##-comm (expand-new-disjoint-ana x E2))\n\n    expand-synth-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c42'\u00a0\u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d2 \u03c41 ~> e1' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-synth-disjoint HDConst ESConst ana = empty-disj _\n    expand-synth-disjoint (HDAsc hd) (ESAsc x) ana = expand-ana-disjoint hd x ana\n    expand-synth-disjoint HDVar (ESVar x\u2081) ana = empty-disj _\n    expand-synth-disjoint (HDLam1 hd) () ana\n    expand-synth-disjoint (HDLam2 hd) (ESLam x\u2081 synth) ana = expand-synth-disjoint hd synth (weaken-ana-expand x\u2081 ana)\n    expand-synth-disjoint (HDHole x) ESEHole ana = ##-comm (expand-new-disjoint-ana x ana)\n    expand-synth-disjoint (HDNEHole x hd) (ESNEHole x\u2081 synth) ana = disjoint-parts (expand-synth-disjoint hd synth ana) (##-comm (expand-new-disjoint-ana x ana))\n    expand-synth-disjoint (HDAp hd hd\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) ana = disjoint-parts (expand-ana-disjoint hd x\u2084 ana) (expand-ana-disjoint hd\u2081 x\u2085 ana)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b5aa26aa273178564c9e3df4846793f1e96e2e81","subject":"Disable unused instance","message":"Disable unused instance\n\nAgda will not infer explicit arguments of instance arguments. On the\nother hand, trying to make this into a proper instance (with an implicit\nargument) makes typechecking too much slower or not terminating (more\nprecisely, on some modules typechecking took longer than I wanted to\nwait and I didn't investigate further).\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily \u27e6_\u27e7Base\n\nmodule Structure {{change-algebra-base : Structure}} where\n  -- change algebras\n\n  open CA public renaming\n    -- Constructors for All\u2032\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = changeAlgebraFun {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra-family : ChangeAlgebraFamily \u27e6_\u27e7Type\n    change-algebra-family = family change-algebra\n\n  -- function changes\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChange {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebraListChanges to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily \u27e6_\u27e7Base\n\nmodule Structure {{change-algebra-base : Structure}} where\n  -- change algebras\n\n  open CA public renaming\n    -- Constructors for All\u2032\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = changeAlgebraFun {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n    change-algebra-family : ChangeAlgebraFamily \u27e6_\u27e7Type\n    change-algebra-family = family change-algebra\n\n  -- function changes\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChange {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebraListChanges to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"96370ea10bf79aded7964a371528432082c2248e","subject":"Added thesis's examples.","message":"Added thesis's examples.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/AgdaIntroduction.agda","new_file":"notes\/thesis\/AgdaIntroduction.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n{-# OPTIONS --no-coverage-check #-}\n\nmodule AgdaIntroduction where\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A zero\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin zero \u2192 A\nmagic ()\n\n-- The with constructor\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven zero     = true\neven (succ n) = odd n\n\nodd zero     = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero     n        = succ n\nack (succ m) zero     = ack m (succ zero)\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n{-# NO_TERMINATION_CHECK #-}\nf : \u2115 \u2192 \u2115\nf n = f n\n\n{-# NO_TERMINATION_CHECK #-}\nnest : \u2115 \u2192 \u2115\nnest zero     = zero\nnest (succ n) = nest (nest n)\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule AgdaIntroduction where\n\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A zero\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\nmagic : {A : Set} \u2192 Fin zero \u2192 A\nmagic ()\n\nhead : {A : Set}{n : \u2115} \u2192 Vec A (succ n) \u2192 A\nhead (x \u2237 xs) = x\n\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven zero     = true\neven (succ n) = odd n\n\nodd zero     = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"21f2853bf5c789013e20d1c99c9553c0213085d2","subject":"Restore a renaming...","message":"Restore a renaming...\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP using (flip; \u03a0; \u03a0\u2071; Op\u2081; Op\u2082; nest)\nopen import Data.Nat.Base using (\u2115; _\u2264_; _\u2238_; z\u2264n; s\u2264s)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.NP\nopen import Algebra.Raw\nopen \u2261-Reasoning\nopen import HoTT using (module Equivalences)\nopen Equivalences\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\nmodule Implicits where\n  open Algebra.FunctionProperties.NP \u03a0\u2071 {a}{a}{A} _\u2261_ public\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n\n  module From-Magma (magma : Magma A) where\n\n    open Magma magma\n\n    module From-Comm\n             (comm  : Commutative _\u2219_)\n             {x y x' y' : A}\n             (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n           where\n      comm= : x \u2219 y \u2261 x' \u2219 y'\n      comm= = comm \u2666 e \u2666 comm\n\n    module From-Assoc\n             (assoc : Associative _\u2219_)\n           where\n\n      assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n      assocs = assoc , ! assoc\n\n      module _ {c x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n        assoc= = ! assoc \u2666 \u2219= e idp \u2666 assoc\n\n        !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n        !assoc= = assoc \u2666 \u2219= idp e \u2666 ! assoc\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n        inner= = assoc= (!assoc= e)\n\n    module From-Assoc-Comm\n             (assoc : Associative _\u2219_)\n             (comm  : Commutative _\u2219_)\n           where\n      open From-Assoc assoc public\n      open From-Comm  comm  public\n\n      module _ {x y z} where\n        assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n        assoc-comm = assoc= comm\n\n        !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n        !assoc-comm = !assoc= comm\n\n      interchange : Interchange _\u2219_ _\u2219_\n      interchange = assoc= (!assoc= comm)\n\n      on-sides : \u2200 {x x' y y' z z' t t'}\n                 \u2192 x \u2219 z \u2261 x' \u2219 z'\n                 \u2192 y \u2219 t \u2261 y' \u2219 t'\n                 \u2192 (x \u2219 y) \u2219 (z \u2219 t) \u2261 (x' \u2219 y') \u2219 (z' \u2219 t')\n      on-sides p q = interchange \u2666 \u2219= p q \u2666 interchange\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n        outer= = \u2219= comm comm \u2666 assoc= (!assoc= e) \u2666 \u2219= comm comm\n\n  module From-Op\u2082 (op : Op\u2082 A) = From-Magma \u27e8 op \u27e9\n\n  module From-Monoid-Ops (mon-ops : Monoid-Ops A) where\n    open Monoid-Ops mon-ops\n    open From-Op\u2082 _\u2219_ public\n\n    module From-LeftIdentity (idl : LeftIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n        is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 idl\n\n      \u03b5^\u207a : \u2200 n \u2192 \u03b5 ^\u207a n \u2261 \u03b5\n      \u03b5^\u207a 0      = idp\n      \u03b5^\u207a (1+ n) = idl \u2666 \u03b5^\u207a n\n\n    module From-RightIdentity (idr : RightIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n        is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 idr\n\n      module _ {b} where\n        ^\u207a-^\u00b9\u207a : \u2200 n \u2192 b ^\u207a (1+ n) \u2261 b ^\u00b9\u207a n\n        ^\u207a-^\u00b9\u207a 0 = idr\n        ^\u207a-^\u00b9\u207a (1+ n) = \u2219= idp (^\u207a-^\u00b9\u207a n)\n\n        ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n        ^\u207a0-\u03b5 = idp\n\n        ^\u207a1-id : b ^\u207a 1 \u2261 b\n        ^\u207a1-id = idr\n\n        ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n        ^\u207a2-\u2219 = ^\u207a-^\u00b9\u207a 1\n\n        ^\u207a3-\u2219 : b ^\u207a 3 \u2261 b \u2219 (b \u2219 b)\n        ^\u207a3-\u2219 = ^\u207a-^\u00b9\u207a 2\n\n        ^\u207a4-\u2219 : b ^\u207a 4 \u2261 b \u2219 (b \u2219 (b \u2219 b))\n        ^\u207a4-\u2219 = ^\u207a-^\u00b9\u207a 3\n\n    module From-Identities (identities : Identity \u03b5 _\u2219_) where\n      comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n      comm-\u03b5 = snd identities \u2666 ! fst identities\n\n      open From-LeftIdentity  (fst identities) public\n      open From-RightIdentity (snd identities) public\n\n    module From-Assoc-RightIdentity\n             (assoc : Associative _\u2219_)\n             (idr   : RightIdentity \u03b5 _\u2219_)\n             where\n      open From-RightIdentity idr\n\n      module _ {c x y}(e : x \u2219 y \u2261 \u03b5) where\n        elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n        elim-assoc= = assoc \u2666 \u2219= idp e \u2666 idr\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-inner= = ! assoc \u2666 \u2219= (elim-assoc= e) idp\n\n    module From-Assoc-LeftIdentity\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             where\n      open From-LeftIdentity idl\n\n      -- Together with ^\u207a0-\u03b5, ^\u207a-+ shows that (b ^\u207a_) is a\n      -- monoid homomorphism from (\u2115,+,0) to (M,\u2219,\u03b5)\n      -- therefore also a group homomorphism\n      ^\u207a-+ : \u2200 m {n b} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n      ^\u207a-+ 0      = ! idl\n      ^\u207a-+ (1+ m) = ap (_\u2219_ _) (^\u207a-+ m) \u2666 ! assoc\n\n      -- Derived from ^\u207a-+ in Algebra.Group\n      ^\u207a-* : \u2200 m {n b} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n      ^\u207a-* 0 = idp\n      ^\u207a-* (1+ m) {n} {b}\n        = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n          b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n          b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n      module _ {c x y} (e : x \u2219 y \u2261 \u03b5) where\n        elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n        elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 idl\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-!inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-!inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module From-Group-Ops (grp-ops : Group-Ops A) where\n    open Group-Ops grp-ops\n    open From-Monoid-Ops mon-ops public\n\n    module From-Assoc-LeftIdentity-LeftInverse\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             (inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-LeftIdentity idl\n      open From-Assoc-LeftIdentity assoc idl\n\n      \u207b\u00b9-inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      \u207b\u00b9-inverse\u02b3 {x}\n         = x \u2219 x \u207b\u00b9                      \u2261\u27e8 ! idl \u27e9\n           \u03b5 \u2219 (x \u2219 x \u207b\u00b9)                \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9) \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9)              \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5                             \u220e\n\n      cancels-\u2219-left : LeftCancel _\u2219_\n      cancels-\u2219-left {c} {x} {y} e\n        = x              \u2261\u27e8 ! idl \u27e9\n          \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inverse\u02e1 idp \u27e9\n          \u03b5 \u2219 y          \u2261\u27e8 idl \u27e9\n          y              \u220e\n\n      module _ {x y} where\n        unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n        unique-\u03b5-right eq\n          = y               \u2261\u27e8 ! is-\u03b5-left inverse\u02e1 \u27e9\n            x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n            x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n            x \u207b\u00b9 \u2219 x        \u2261\u27e8 inverse\u02e1 \u27e9\n            \u03b5               \u220e\n\n      \u03b5^\u207b : \u2200 n \u2192 \u03b5 ^\u207b n \u2261 \u03b5\n      \u03b5^\u207b n = \u207b\u00b9= (\u03b5^\u207a n) \u2666 unique-\u03b5-right \u207b\u00b9-inverse\u02b3\n\n      \u03b5^ : \u2200 i \u2192 \u03b5 ^ i \u2261 \u03b5\n      \u03b5^ -[1+ n ] = \u03b5^\u207b(1+ n)\n      \u03b5^ (+ n)    = \u03b5^\u207a n\n\n      module _ {x y} where\n        -- _\u207b\u00b9 is a group homomorphism, see Algebra.Group._\u1d52\u1d56\n        \u207b\u00b9-hom\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n        \u207b\u00b9-hom\u2032 = cancels-\u2219-left {x \u2219 y}\n           ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 \u207b\u00b9-inverse\u02b3 \u27e9\n            \u03b5                       \u2261\u27e8 ! \u207b\u00b9-inverse\u02b3 \u27e9\n            x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! idl) \u27e9\n            x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! \u207b\u00b9-inverse\u02b3) idp) \u27e9\n            x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n            x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n            (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n        \u207b\u00b9-\u2219-distr\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \/ x\n        \u207b\u00b9-\u2219-distr\u2032 = \u207b\u00b9-hom\u2032\n\n      \u207b\u00b9-hom\u2032= : \u2200 {x y z t} \u2192 x \u2219 y \u2261 z \u2219 t \u2192 y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261 t \u207b\u00b9 \u2219 z \u207b\u00b9\n      \u207b\u00b9-hom\u2032= e = ! \u207b\u00b9-hom\u2032 \u2666 ap _\u207b\u00b9 e \u2666 \u207b\u00b9-hom\u2032\n\n      elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n      elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n        = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n          x \u207b\u00b9 \u2219 y               \u220e\n\n      elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n      elim-\u2219-right-\/ {c} {x} {y}\n        = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n          x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-!inner= \u207b\u00b9-inverse\u02b3 \u27e9\n          x \/ y              \u220e\n\n    module From-Assoc-RightIdentity-RightInverse\n             (assoc : Associative _\u2219_)\n             (idr : RightIdentity \u03b5 _\u2219_)\n             (inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-RightIdentity assoc idr\n\n      cancels-\u2219-right : RightCancel _\u2219_\n      cancels-\u2219-right {c} {x} {y} e\n        = x              \u2261\u27e8 ! idr \u27e9\n          x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n          y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inverse\u02b3 \u27e9\n          y \u2219 \u03b5          \u2261\u27e8 idr \u27e9\n          y              \u220e\n\n      inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02e1 {x}\n        = x \u207b\u00b9 \u2219 x                      \u2261\u27e8 ! idr \u27e9\n          (x \u207b\u00b9 \u2219 x) \u2219 \u03b5                \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          (x \u207b\u00b9 \u2219 x) \u2219 (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-inner= inverse\u02b3 \u27e9\n          (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9)              \u2261\u27e8 inverse\u02b3 \u27e9\n          \u03b5                             \u220e\n\n      \u207b\u00b9-involutive : Involutive _\u207b\u00b9\n      \u207b\u00b9-involutive {x}\n        = cancels-\u2219-right\n          (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5              \u2261\u27e8 ! inverse\u02b3 \u27e9\n           x \u2219 x \u207b\u00b9       \u220e)\n\n      module _ {x y} where\n        \/-\u2219 : (x \/ y) \u2219 y \u2261 x\n        \/-\u2219 = elim-assoc= inverse\u02e1\n\n        \u2219-\/ : (x \u2219 y) \/ y \u2261 x\n        \u2219-\/ = elim-assoc= inverse\u02b3\n\n      module _ {x y} where\n        unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n        unique-\u03b5-left eq\n          = x            \u2261\u27e8 ! \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            y \/ y        \u2261\u27e8 inverse\u02b3 \u27e9\n            \u03b5            \u220e\n\n        x\/y\u2261\u03b5\u2192x\u2261y : x \/ y \u2261 \u03b5 \u2192 x \u2261 y\n        x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5 = cancels-\u2219-right (x\/y\u2261\u03b5 \u2666 ! inverse\u02b3)\n\n        x\/y\u2262\u03b5 : x \u2262 y \u2192 x \/ y \u2262 \u03b5\n        x\/y\u2262\u03b5 x\u2262y x\/y\u2261\u03b5 = x\u2262y (x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5)\n\n      \u03b5\u207b\u00b9\u2261\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n      \u03b5\u207b\u00b9\u2261\u03b5 = unique-\u03b5-left inverse\u02e1\n\n    module From-Assoc-Identities-RightInverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      private\n        idl = fst identities\n        idr = snd identities\n\n      inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02b3 = inv-r\n\n      open From-Assoc assoc\n      open From-Identities identities using (comm-\u03b5)\n      open From-LeftIdentity        idl\n      open From-RightIdentity       idr\n      open From-Assoc-LeftIdentity  assoc idl\n      open From-Assoc-RightIdentity assoc idr\n      open From-Assoc-RightIdentity-RightInverse assoc idr inverse\u02b3 public\n      open From-Assoc-LeftIdentity-LeftInverse assoc idl inverse\u02e1 public\n        hiding (\u207b\u00b9-inverse\u02b3)\n\n      module _ {x y} where\n        \u2219-\/\u2032 : y \u2219 (x \/\u2032 y) \u2261 x\n        \u2219-\/\u2032 = elim-!assoc= inverse\u02b3\n\n        \/\u2032-\u2219 : (y \u2219 x) \/\u2032 y \u2261 x\n        \/\u2032-\u2219 = elim-!assoc= inverse\u02e1\n\n        \/-\/\u2032 : x \/ (x \/\u2032 y) \u2261 y\n        \/-\/\u2032 =\n          x \u2219 (y \u207b\u00b9 \u2219 x) \u207b\u00b9    \u2261\u27e8 \u2219= idp \u207b\u00b9-hom\u2032 \u27e9\n          x \u2219 (x \u207b\u00b9 \u2219 y \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-!assoc= inverse\u02b3 \u27e9\n          y \u207b\u00b9 \u207b\u00b9              \u2261\u27e8 \u207b\u00b9-involutive \u27e9\n          y                    \u220e\n\n        \/\u2032-\/ : x \/\u2032 (x \/ y) \u2261 y\n        \/\u2032-\/ =\n          (x \/ y)\u207b\u00b9 \u2219 x        \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          y \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2219 x   \u2261\u27e8 elim-assoc= inverse\u02e1 \u27e9\n          y \u207b\u00b9 \u207b\u00b9              \u2261\u27e8 \u207b\u00b9-involutive \u27e9\n          y                    \u220e\n\n      \u2219-is-equiv : \u2200 {k} \u2192 Is-equiv (_\u2219_ k)\n      \u2219-is-equiv {k} = is-equiv (_\u2219_ (k \u207b\u00b9))\n                                (\u03bb _ \u2192 ! assoc \u2666 \u2219= inverse\u02b3 idp \u2666 idl)\n                                (\u03bb _ \u2192 ! assoc \u2666 \u2219= inverse\u02e1 idp \u2666 idl)\n\n      flip-\u2219-is-equiv : \u2200 {k} \u2192 Is-equiv (flip _\u2219_ k)\n      flip-\u2219-is-equiv {k} = is-equiv (flip _\u2219_ (k \u207b\u00b9))\n                                     (\u03bb _ \u2192 assoc \u2666 \u2219= idp inverse\u02e1 \u2666 idr)\n                                     (\u03bb _ \u2192 assoc \u2666 \u2219= idp inverse\u02b3 \u2666 idr)\n\n      module _ {x y} where\n        unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n        unique-\u207b\u00b9 eq\n          = x            \u2261\u27e8 ! \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            \u03b5 \/ y        \u2261\u27e8 idl \u27e9\n            y \u207b\u00b9         \u220e\n\n      \u207b\u00b9-inj : \u2200 {x y} \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9 \u2192 x \u2261 y\n      \u207b\u00b9-inj {x} {y} e\n         = ! (y              \u2261\u27e8 ! idr \u27e9\n              y \u2219 \u03b5          \u2261\u27e8 \u2219= idp (\u03b5        \u2261\u27e8 ! inverse\u02e1  \u27e9\n                                        x \u207b\u00b9 \u2219 x \u2261\u27e8 \u2219= e idp \u27e9\n                                        y \u207b\u00b9 \u2219 x \u220e) \u27e9\n              y \u2219 (y \u207b\u00b9 \u2219 x) \u2261\u27e8 \u2219-\/\u2032 \u27e9\n              x \u220e)\n\n      module _ {b} where\n        ^\u207a-1+ : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n        ^\u207a-1+ 0 = comm-\u03b5\n        ^\u207a-1+ (1+ n) = ap (_\u2219_ b) (^\u207a-1+ n) \u2666 ! assoc\n\n        ^\u207a-\u2238 : \u2200 {m n} \u2192 n \u2264 m \u2192 b ^\u207a(m \u2238 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207a-\u2238 z\u2264n = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207a-\u2238 (s\u2264s {n} {m} n\u2264m) =\n           b ^\u207a(m \u2238 n)                     \u2261\u27e8 ^\u207a-\u2238 n\u2264m \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296 : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^-\u2296 m      0      = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296 0      (1+ n) = ! idl\n        ^-\u2296 (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296 m n \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296' : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^-\u2296' m      0      = ! is-\u03b5-left \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296' 0      (1+ n) = ! idr\n        ^-\u2296' (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296' m n \u27e9\n           (b ^\u207b n \u2219 b ^\u207a m)               \u2261\u27e8 ! elim-inner= inverse\u02e1 \u27e9\n           (b ^\u207b n \u2219 b \u207b\u00b9) \u2219 (b \u2219 b ^\u207a m)  \u2261\u27e8 \u2219= (! \u207b\u00b9-hom\u2032) idp \u27e9\n           (b \u2219 b ^\u207a n)\u207b\u00b9 \u2219 (b \u2219 b ^\u207a m)   \u220e\n\n        ^\u207a-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b ^\u207a m\n        ^\u207a-comm 0      n = ! comm-\u03b5\n        ^\u207a-comm (1+ m) n =\n          b \u2219 b\u1d50 \u2219 b\u207f   \u2261\u27e8 !assoc= (^\u207a-comm m n) \u27e9\n          b \u2219 b\u207f \u2219 b\u1d50   \u2261\u27e8 \u2219= (^\u207a-1+ n) idp \u27e9\n          b\u207f \u2219 b \u2219 b\u1d50   \u2261\u27e8 assoc \u27e9\n          b\u207f \u2219 (b \u2219 b\u1d50) \u220e\n          where\n            b\u207f = b ^\u207a n\n            b\u1d50 = b ^\u207a m\n\n        ^\u207a-^\u207b-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^\u207a-^\u207b-comm m n = ! ^-\u2296 m n \u2666 ^-\u2296' m n\n\n        ^\u207b-^\u207a-comm : \u2200 m n \u2192 b ^\u207b n \u2219 b ^\u207a m \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207b-^\u207a-comm m n = ! ^-\u2296' m n \u2666 ^-\u2296 m n\n\n        ^\u207b-comm : \u2200 m n \u2192 b ^\u207b m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207b m\n        ^\u207b-comm m n = \u207b\u00b9-hom\u2032= (^\u207a-comm n m)\n\n        ^\u207b-1+ : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n        ^\u207b-1+ n = ap _\u207b\u00b9 (^\u207a-1+ n) \u2666 \u207b\u00b9-hom\u2032\n\n        ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n        ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                        \u2666 ! \u207b\u00b9-hom\u2032\n                        \u2666 ap _\u207b\u00b9 (! ^\u207a-1+ n)\n\n        ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n        ^\u207b\u20321-id = idr\n\n        ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n        ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n        ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n        ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\n        -- ^-+ is a group homomorphism defined in Algebra.Group\n        -- Some properties can be derived from it.\n        ^-+ : \u2200 i j \u2192 b ^(i +\u2124 j) \u2261 b ^ i \u2219 b ^ j\n        ^-+ -[1+ m ] -[1+ n ] = ap _\u207b\u00b9\n                                 (ap (\u03bb z \u2192 b ^\u207a(1+ z)) (\u2115\u00b0.+-comm (1+ m) n)\n                              \u2666 ^\u207a-+ (1+ n) {1+ m}) \u2666 \u207b\u00b9-hom\u2032\n        ^-+ -[1+ m ]   (+ n)  = ^-\u2296' n (1+ m)\n        ^-+   (+ m)  -[1+ n ] = ^-\u2296 m (1+ n)\n        ^-+   (+ m)    (+ n)  = ^\u207a-+ m\n\n        -- GroupHomomorphism.f-pres-inv\n        ^-- : \u2200 i \u2192 b ^(- i) \u2261 (b ^ i)\u207b\u00b9\n        ^-- -[1+ n ] = ! \u207b\u00b9-involutive\n        ^-- (+ 0)    = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^-- (+ 1+ n) = idp\n\n        -- GroupHomomorphism.f-\u2212-\/\n        ^-\u2212 : \u2200 i j \u2192 b ^(i \u2212\u2124 j) \u2261 b ^ i \/ b ^ j\n        ^-\u2212 i j = ^-+ i (- j) \u2666 \u2219= idp (^-- j)\n\n        ^-suc : \u2200 i \u2192 b ^(suc\u2124 i) \u2261 b \u2219 b ^ i\n        ^-suc i = ^-+ (+ 1) i \u2666 \u2219= idr idp\n\n        ^-pred : \u2200 i \u2192 b ^(pred\u2124 i) \u2261 b \u207b\u00b9 \u2219 b ^ i\n        ^-pred i = ^-+ (- + 1) i \u2666 ap (\u03bb z \u2192 z \u207b\u00b9 \u2219 b ^ i) idr\n\n        ^-comm : \u2200 i j \u2192 b ^ i \u2219 b ^ j \u2261 b ^ j \u2219 b ^ i\n        ^-comm -[1+ m ] -[1+ n ] = ^\u207b-comm (1+ m) (1+ n)\n        ^-comm -[1+ m ]   (+ n)  = ! ^\u207a-^\u207b-comm n (1+ m)\n        ^-comm   (+ m)  -[1+ n ] = ^\u207a-^\u207b-comm m (1+ n)\n        ^-comm   (+ m)    (+ n)  = ^\u207a-comm m n\n\n      {-\n      ^-* : \u2200 i j \u2192 b ^(i *\u2124 j) \u2261 (b ^ j)^ i\n      ^-* i j = {!!}\n      -}\n\n    module From-Assoc-Identities-Inverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv : Inverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc-Identities-RightInverse assoc identities (snd inv) public\n\n  module From-Ring-Ops (rng-ops : Ring-Ops A) where\n    open Ring-Ops rng-ops\n\n    module DistributesOver\u02b3\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_)\n             where\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = *-comm \u2666 *-+-distr\u02e1 \u2666 += *-comm *-comm\n\n    module DistributesOver\u02e1\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = *-comm \u2666 *-+-distr\u02b3 \u2666 += *-comm *-comm\n\n    module From-+Group-1*Identity-DistributesOver\u02b3\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n                     ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n                     )\n\n      -0\u22610 : -0# \u2261 0#\n      -0\u22610 = 0\u22120\u22610\n\n      0*-zero : LeftZero 0# _*_\n      0*-zero {x} = cancels-+-left\n        (x + 0# * x      \u2261\u27e8 += (! 1*-identity) idp \u27e9\n         1# * x + 0# * x \u2261\u27e8 ! *-+-distr\u02b3 \u27e9\n         (1# + 0#) * x   \u2261\u27e8 *= +0-identity idp \u27e9\n         1# * x          \u2261\u27e8 1*-identity \u27e9\n         x               \u2261\u27e8 ! +0-identity \u27e9\n         x + 0#          \u220e)\n\n      2*-spec : \u2200 {n} \u2192 2* n \u2261 2# * n\n      2*-spec = ! += 1*-identity 1*-identity \u2666 ! *-+-distr\u02b3\n\n      0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n      0\u2212-*-distr = cancels-+-right (0\u2212-inverse\u02e1 \u2666 ! 0*-zero \u2666 *= (! 0\u2212-inverse\u02e1) idp \u2666 *-+-distr\u02b3)\n\n      -1*-neg : \u2200 {x} \u2192 -1# * x \u2261 0\u2212 x\n      -1*-neg = ! 0\u2212-*-distr \u2666 0\u2212= 1*-identity\n\n      2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n      2*-*-distr = ! *-+-distr\u02b3\n\n    module From-+Group-*Identity-DistributesOver\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*1-identity : RightIdentity 1# _*_)\n             (*-+-distrs : _*_ DistributesOver _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = fst *-+-distrs\n\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = snd *-+-distrs\n\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( \u207b\u00b9-involutive to 0\u2212-involutive\n                     ; cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u207b\u00b9-\u2219-distr\u2032 to 0\u2212-+-distr\u2032\n                     )\n\n      open From-+Group-1*Identity-DistributesOver\u02b3\n             +-assoc 0+-identity +0-identity 0\u2212-inverse\u02b3\n             1*-identity *-+-distr\u02b3\n             public\n\n      *0-zero : RightZero 0# _*_\n      *0-zero {x} = cancels-+-right\n        (x * 0# + x      \u2261\u27e8 += idp (! *1-identity) \u27e9\n         x * 0# + x * 1# \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n         x * (0# + 1#)   \u2261\u27e8 *= idp 0+-identity \u27e9\n         x * 1#          \u2261\u27e8 *1-identity \u27e9\n         x               \u2261\u27e8 ! 0+-identity \u27e9\n         0# + x          \u220e)\n\n      0\u2212-*-distr\u02b3 : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n      0\u2212-*-distr\u02b3 = cancels-+-right (0\u2212-inverse\u02e1 \u2666 (! *0-zero \u2666 *= idp (! 0\u2212-inverse\u02e1)) \u2666 *-+-distr\u02e1)\n\n      *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n      *-\u2212-distr = *-+-distr\u02e1 \u2666 += idp (! 0\u2212-*-distr\u02b3)\n\n      \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n      \u00b2-0\u2212-distr = ! 0\u2212-*-distr \u2666 0\u2212=(! 0\u2212-*-distr\u02b3) \u2666 0\u2212-involutive\n\n      +-comm : Commutative _+_\n      +-comm {a} {b} =\n          cancels-+-right\n            (cancels-+-left\n              (a + (a + b + b)   \u2261\u27e8 += idp +-assoc \u2666 ! +-assoc \u27e9\n               (a + a) + (b + b) \u2261\u27e8 += 2*-spec 2*-spec \u27e9\n               2# * a  + 2# * b  \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n               2# * (a + b)      \u2261\u27e8 ! 2*-spec \u27e9\n               (a + b) + (a + b) \u2261\u27e8 +-assoc \u2666 += idp (! +-assoc) \u27e9\n               a + (b + a + b)   \u220e))\n\n      0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n      0\u2212-+-distr = 0\u2212= +-comm \u2666 0\u2212-+-distr\u2032\n\n  module From-Field-Ops (fld-ops : Field-Ops A) where\n    open Field-Ops fld-ops\n\nmodule Explicits where\n  open Algebra.FunctionProperties.NP \u03a0  {a}{a}{A} _\u2261_ public\n\n  -- REPEATED from above but with explicit arguments\n  module FromOp\u2082\n           (_\u2219_ : Op\u2082 A){x x' y y'}(p : x \u2261 x')(q : y \u2261 y')\n         where\n    op= : x \u2219 y \u2261 x' \u2219 y'\n    op= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  module FromComm\n           (_\u2219_   : Op\u2082 A)\n           (comm  : Commutative _\u2219_)\n           (x y x' y' : A)\n           (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n         where\n    open FromOp\u2082 _\u2219_\n\n    comm= : x \u2219 y \u2261 x' \u2219 y'\n    comm= = comm _ _ \u2666 e \u2666 comm _ _\n\n  module FromAssoc\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n\n         where\n    open FromOp\u2082 _\u2219_\n\n    assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    assocs = assoc , (\u03bb _ _ _ \u2192 ! assoc _ _ _)\n\n    module _ {c x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n      assoc= = ! assoc _ _ _ \u2666 op= e idp \u2666 assoc _ _ _\n\n      !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n      !assoc= = assoc _ _ _ \u2666 op= idp e \u2666 ! assoc _ _ _\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n      inner= = assoc= (!assoc= e)\n\n  module FromAssocComm\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n           (comm  : Commutative _\u2219_)\n         where\n    open FromOp\u2082   _\u2219_       renaming (op= to \u2219=)\n    open FromAssoc _\u2219_ assoc public\n    open FromComm  _\u2219_ comm  public\n\n    module _ x y z where\n      assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n      assoc-comm = assoc= (comm _ _)\n\n      !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n      !assoc-comm = !assoc= (comm _ _)\n\n    interchange : Interchange _\u2219_ _\u2219_\n    interchange _ _ _ _ = assoc= (!assoc= (comm _ _))\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n      outer= = \u2219= (comm _ _) (comm _ _) \u2666 assoc= (!assoc= e) \u2666 \u2219= (comm _ _) (comm _ _)\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj _ _ fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP using (flip; \u03a0; \u03a0\u2071; Op\u2081; Op\u2082; nest)\nopen import Data.Nat.Base using (\u2115; _\u2264_; _\u2238_; z\u2264n; s\u2264s)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.NP\nopen import Algebra.Raw\nopen \u2261-Reasoning\nopen import HoTT using (module Equivalences)\nopen Equivalences\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\nmodule Implicits where\n  open Algebra.FunctionProperties.NP \u03a0\u2071 {a}{a}{A} _\u2261_ public\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n\n  module From-Magma (magma : Magma A) where\n\n    open Magma magma\n\n    module From-Comm\n             (comm  : Commutative _\u2219_)\n             {x y x' y' : A}\n             (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n           where\n      comm= : x \u2219 y \u2261 x' \u2219 y'\n      comm= = comm \u2666 e \u2666 comm\n\n    module From-Assoc\n             (assoc : Associative _\u2219_)\n           where\n\n      assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n      assocs = assoc , ! assoc\n\n      module _ {c x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n        assoc= = ! assoc \u2666 \u2219= e idp \u2666 assoc\n\n        !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n        !assoc= = assoc \u2666 \u2219= idp e \u2666 ! assoc\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n        inner= = assoc= (!assoc= e)\n\n    module From-Assoc-Comm\n             (assoc : Associative _\u2219_)\n             (comm  : Commutative _\u2219_)\n           where\n      open From-Assoc assoc public\n      open From-Comm  comm  public\n\n      module _ {x y z} where\n        assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n        assoc-comm = assoc= comm\n\n        !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n        !assoc-comm = !assoc= comm\n\n      interchange : Interchange _\u2219_ _\u2219_\n      interchange = assoc= (!assoc= comm)\n\n      on-sides : \u2200 {x x' y y' z z' t t'}\n                 \u2192 x \u2219 z \u2261 x' \u2219 z'\n                 \u2192 y \u2219 t \u2261 y' \u2219 t'\n                 \u2192 (x \u2219 y) \u2219 (z \u2219 t) \u2261 (x' \u2219 y') \u2219 (z' \u2219 t')\n      on-sides p q = interchange \u2666 \u2219= p q \u2666 interchange\n\n      module _ {c d x y x' y' : A}\n               (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n        outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n        outer= = \u2219= comm comm \u2666 assoc= (!assoc= e) \u2666 \u2219= comm comm\n\n  module From-Op\u2082 (op : Op\u2082 A) = From-Magma \u27e8 op \u27e9\n\n  module From-Monoid-Ops (mon-ops : Monoid-Ops A) where\n    open Monoid-Ops mon-ops\n    open From-Op\u2082 _\u2219_ public\n\n    module From-LeftIdentity (idl : LeftIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n        is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 idl\n\n      \u03b5^\u207a : \u2200 n \u2192 \u03b5 ^\u207a n \u2261 \u03b5\n      \u03b5^\u207a 0      = idp\n      \u03b5^\u207a (1+ n) = idl \u2666 \u03b5^\u207a n\n\n    module From-RightIdentity (idr : RightIdentity \u03b5 _\u2219_) where\n      module _ {x y} where\n        is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n        is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 idr\n\n      module _ {b} where\n        ^\u207a-^\u00b9\u207a : \u2200 n \u2192 b ^\u207a (1+ n) \u2261 b ^\u00b9\u207a n\n        ^\u207a-^\u00b9\u207a 0 = idr\n        ^\u207a-^\u00b9\u207a (1+ n) = \u2219= idp (^\u207a-^\u00b9\u207a n)\n\n        ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n        ^\u207a0-\u03b5 = idp\n\n        ^\u207a1-id : b ^\u207a 1 \u2261 b\n        ^\u207a1-id = idr\n\n        ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n        ^\u207a2-\u2219 = ^\u207a-^\u00b9\u207a 1\n\n        ^\u207a3-\u2219 : b ^\u207a 3 \u2261 b \u2219 (b \u2219 b)\n        ^\u207a3-\u2219 = ^\u207a-^\u00b9\u207a 2\n\n        ^\u207a4-\u2219 : b ^\u207a 4 \u2261 b \u2219 (b \u2219 (b \u2219 b))\n        ^\u207a4-\u2219 = ^\u207a-^\u00b9\u207a 3\n\n    module From-Identities (identities : Identity \u03b5 _\u2219_) where\n      comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n      comm-\u03b5 = snd identities \u2666 ! fst identities\n\n      open From-LeftIdentity  (fst identities) public\n      open From-RightIdentity (snd identities) public\n\n    module From-Assoc-RightIdentity\n             (assoc : Associative _\u2219_)\n             (idr   : RightIdentity \u03b5 _\u2219_)\n             where\n      open From-RightIdentity idr\n\n      module _ {c x y}(e : x \u2219 y \u2261 \u03b5) where\n        elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n        elim-assoc= = assoc \u2666 \u2219= idp e \u2666 idr\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-inner= = ! assoc \u2666 \u2219= (elim-assoc= e) idp\n\n    module From-Assoc-LeftIdentity\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             where\n      open From-LeftIdentity idl\n\n      -- Together with ^\u207a0-\u03b5, ^\u207a-+ shows that (b ^\u207a_) is a\n      -- monoid homomorphism from (\u2115,+,0) to (M,\u2219,\u03b5)\n      -- therefore also a group homomorphism\n      ^\u207a-+ : \u2200 m {n b} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n      ^\u207a-+ 0      = ! idl\n      ^\u207a-+ (1+ m) = ap (_\u2219_ _) (^\u207a-+ m) \u2666 ! assoc\n\n      -- Derived from ^\u207a-+ in Algebra.Group\n      ^\u207a-* : \u2200 m {n b} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n      ^\u207a-* 0 = idp\n      ^\u207a-* (1+ m) {n} {b}\n        = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n          b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n          b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n      module _ {c x y} (e : x \u2219 y \u2261 \u03b5) where\n        elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n        elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 idl\n\n      module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n        elim-!inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n        elim-!inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module From-Group-Ops (grp-ops : Group-Ops A) where\n    open Group-Ops grp-ops\n    open From-Monoid-Ops mon-ops public\n\n    module From-Assoc-LeftIdentity-LeftInverse\n             (assoc : Associative _\u2219_)\n             (idl : LeftIdentity \u03b5 _\u2219_)\n             (inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-LeftIdentity idl\n      open From-Assoc-LeftIdentity assoc idl\n\n      \u207b\u00b9-inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      \u207b\u00b9-inverse\u02b3 {x}\n         = x \u2219 x \u207b\u00b9                      \u2261\u27e8 ! idl \u27e9\n           \u03b5 \u2219 (x \u2219 x \u207b\u00b9)                \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9) \u2219 (x \u2219 x \u207b\u00b9) \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n           (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9)              \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5                             \u220e\n\n      cancels-\u2219-left : LeftCancel _\u2219_\n      cancels-\u2219-left {c} {x} {y} e\n        = x              \u2261\u27e8 ! idl \u27e9\n          \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inverse\u02e1) idp \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n          c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inverse\u02e1 idp \u27e9\n          \u03b5 \u2219 y          \u2261\u27e8 idl \u27e9\n          y              \u220e\n\n      module _ {x y} where\n        unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n        unique-\u03b5-right eq\n          = y               \u2261\u27e8 ! is-\u03b5-left inverse\u02e1 \u27e9\n            x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n            x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n            x \u207b\u00b9 \u2219 x        \u2261\u27e8 inverse\u02e1 \u27e9\n            \u03b5               \u220e\n\n      \u03b5^\u207b : \u2200 n \u2192 \u03b5 ^\u207b n \u2261 \u03b5\n      \u03b5^\u207b n = \u207b\u00b9= (\u03b5^\u207a n) \u2666 unique-\u03b5-right \u207b\u00b9-inverse\u02b3\n\n      \u03b5^ : \u2200 i \u2192 \u03b5 ^ i \u2261 \u03b5\n      \u03b5^ -[1+ n ] = \u03b5^\u207b(1+ n)\n      \u03b5^ (+ n)    = \u03b5^\u207a n\n\n      module _ {x y} where\n        -- _\u207b\u00b9 is a group homomorphism, see Algebra.Group._\u1d52\u1d56\n        \u207b\u00b9-hom\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n        \u207b\u00b9-hom\u2032 = cancels-\u2219-left {x \u2219 y}\n           ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 \u207b\u00b9-inverse\u02b3 \u27e9\n            \u03b5                       \u2261\u27e8 ! \u207b\u00b9-inverse\u02b3 \u27e9\n            x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! idl) \u27e9\n            x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! \u207b\u00b9-inverse\u02b3) idp) \u27e9\n            x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n            x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n            (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n        \u207b\u00b9-\u2219-distr\u2032 : (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \/ x\n        \u207b\u00b9-\u2219-distr\u2032 = \u207b\u00b9-hom\u2032\n\n      \u207b\u00b9-hom\u2032= : \u2200 {x y z t} \u2192 x \u2219 y \u2261 z \u2219 t \u2192 y \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261 t \u207b\u00b9 \u2219 z \u207b\u00b9\n      \u207b\u00b9-hom\u2032= e = ! \u207b\u00b9-hom\u2032 \u2666 ap _\u207b\u00b9 e \u2666 \u207b\u00b9-hom\u2032\n\n      elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n      elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n        = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-!inner= inverse\u02e1 \u27e9\n          x \u207b\u00b9 \u2219 y               \u220e\n\n      elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n      elim-\u2219-right-\/ {c} {x} {y}\n        = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n          x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-!inner= \u207b\u00b9-inverse\u02b3 \u27e9\n          x \/ y              \u220e\n\n    module From-Assoc-RightIdentity-RightInverse\n             (assoc : Associative _\u2219_)\n             (idr : RightIdentity \u03b5 _\u2219_)\n             (inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc assoc\n      open From-Assoc-RightIdentity assoc idr\n\n      cancels-\u2219-right : RightCancel _\u2219_\n      cancels-\u2219-right {c} {x} {y} e\n        = x              \u2261\u27e8 ! idr \u27e9\n          x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n          y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inverse\u02b3 \u27e9\n          y \u2219 \u03b5          \u2261\u27e8 idr \u27e9\n          y              \u220e\n\n      inverse\u02e1 : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02e1 {x}\n        = x \u207b\u00b9 \u2219 x                      \u2261\u27e8 ! idr \u27e9\n          (x \u207b\u00b9 \u2219 x) \u2219 \u03b5                \u2261\u27e8 \u2219= idp (! inverse\u02b3) \u27e9\n          (x \u207b\u00b9 \u2219 x) \u2219 (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-inner= inverse\u02b3 \u27e9\n          (x \u207b\u00b9 \u2219 x \u207b\u00b9 \u207b\u00b9)              \u2261\u27e8 inverse\u02b3 \u27e9\n          \u03b5                             \u220e\n\n      \u207b\u00b9-involutive : Involutive _\u207b\u00b9\n      \u207b\u00b9-involutive {x}\n        = cancels-\u2219-right\n          (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inverse\u02e1 \u27e9\n           \u03b5              \u2261\u27e8 ! inverse\u02b3 \u27e9\n           x \u2219 x \u207b\u00b9       \u220e)\n\n      module _ {x y} where\n        \/-\u2219 : (x \/ y) \u2219 y \u2261 x\n        \/-\u2219 = elim-assoc= inverse\u02e1\n\n        \u2219-\/ : (x \u2219 y) \/ y \u2261 x\n        \u2219-\/ = elim-assoc= inverse\u02b3\n\n      module _ {x y} where\n        unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n        unique-\u03b5-left eq\n          = x            \u2261\u27e8 ! \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            y \/ y        \u2261\u27e8 inverse\u02b3 \u27e9\n            \u03b5            \u220e\n\n        x\/y\u2261\u03b5\u2192x\u2261y : x \/ y \u2261 \u03b5 \u2192 x \u2261 y\n        x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5 = cancels-\u2219-right (x\/y\u2261\u03b5 \u2666 ! inverse\u02b3)\n\n        x\/y\u2262\u03b5 : x \u2262 y \u2192 x \/ y \u2262 \u03b5\n        x\/y\u2262\u03b5 x\u2262y x\/y\u2261\u03b5 = x\u2262y (x\/y\u2261\u03b5\u2192x\u2261y x\/y\u2261\u03b5)\n\n      \u03b5\u207b\u00b9\u2261\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n      \u03b5\u207b\u00b9\u2261\u03b5 = unique-\u03b5-left inverse\u02e1\n\n    module From-Assoc-Identities-RightInverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      private\n        idl = fst identities\n        idr = snd identities\n\n      inverse\u02b3 : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n      inverse\u02b3 = inv-r\n\n      open From-Assoc assoc\n      open From-Identities identities using (comm-\u03b5)\n      open From-LeftIdentity        idl\n      open From-RightIdentity       idr\n      open From-Assoc-LeftIdentity  assoc idl\n      open From-Assoc-RightIdentity assoc idr\n      open From-Assoc-RightIdentity-RightInverse assoc idr inverse\u02b3 public\n      open From-Assoc-LeftIdentity-LeftInverse assoc idl inverse\u02e1 public\n        hiding (\u207b\u00b9-inverse\u02b3)\n\n      module _ {x y} where\n        \u2219-\/\u2032 : y \u2219 (x \/\u2032 y) \u2261 x\n        \u2219-\/\u2032 = elim-!assoc= inverse\u02b3\n\n        \/\u2032-\u2219 : (y \u2219 x) \/\u2032 y \u2261 x\n        \/\u2032-\u2219 = elim-!assoc= inverse\u02e1\n\n        \/-\/\u2032 : x \/ (x \/\u2032 y) \u2261 y\n        \/-\/\u2032 =\n          x \u2219 (y \u207b\u00b9 \u2219 x) \u207b\u00b9    \u2261\u27e8 \u2219= idp \u207b\u00b9-hom\u2032 \u27e9\n          x \u2219 (x \u207b\u00b9 \u2219 y \u207b\u00b9 \u207b\u00b9) \u2261\u27e8 elim-!assoc= inverse\u02b3 \u27e9\n          y \u207b\u00b9 \u207b\u00b9              \u2261\u27e8 \u207b\u00b9-involutive \u27e9\n          y                    \u220e\n\n        \/\u2032-\/ : x \/\u2032 (x \/ y) \u2261 y\n        \/\u2032-\/ =\n          (x \/ y)\u207b\u00b9 \u2219 x        \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n          y \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2219 x   \u2261\u27e8 elim-assoc= inverse\u02e1 \u27e9\n          y \u207b\u00b9 \u207b\u00b9              \u2261\u27e8 \u207b\u00b9-involutive \u27e9\n          y                    \u220e\n\n      \u2219-is-equiv : \u2200 {k} \u2192 Is-equiv (_\u2219_ k)\n      \u2219-is-equiv {k} = is-equiv (_\u2219_ (k \u207b\u00b9))\n                                (\u03bb _ \u2192 ! assoc \u2666 \u2219= inverse\u02b3 idp \u2666 idl)\n                                (\u03bb _ \u2192 ! assoc \u2666 \u2219= inverse\u02e1 idp \u2666 idl)\n\n      flip-\u2219-is-equiv : \u2200 {k} \u2192 Is-equiv (flip _\u2219_ k)\n      flip-\u2219-is-equiv {k} = is-equiv (flip _\u2219_ (k \u207b\u00b9))\n                                     (\u03bb _ \u2192 assoc \u2666 \u2219= idp inverse\u02e1 \u2666 idr)\n                                     (\u03bb _ \u2192 assoc \u2666 \u2219= idp inverse\u02b3 \u2666 idr)\n\n      module _ {x y} where\n        unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n        unique-\u207b\u00b9 eq\n          = x            \u2261\u27e8 ! \u2219-\/ \u27e9\n            (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n            \u03b5 \/ y        \u2261\u27e8 idl \u27e9\n            y \u207b\u00b9         \u220e\n\n      \u207b\u00b9-inj : \u2200 {x y} \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9 \u2192 x \u2261 y\n      \u207b\u00b9-inj {x} {y} e\n         = ! (y              \u2261\u27e8 ! idr \u27e9\n              y \u2219 \u03b5          \u2261\u27e8 \u2219= idp (\u03b5        \u2261\u27e8 ! inverse\u02e1  \u27e9\n                                        x \u207b\u00b9 \u2219 x \u2261\u27e8 \u2219= e idp \u27e9\n                                        y \u207b\u00b9 \u2219 x \u220e) \u27e9\n              y \u2219 (y \u207b\u00b9 \u2219 x) \u2261\u27e8 \u2219-\/\u2032 \u27e9\n              x \u220e)\n\n      module _ {b} where\n        ^\u207a-1+ : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n        ^\u207a-1+ 0 = comm-\u03b5\n        ^\u207a-1+ (1+ n) = ap (_\u2219_ b) (^\u207a-1+ n) \u2666 ! assoc\n\n        ^\u207a-\u2238 : \u2200 {m n} \u2192 n \u2264 m \u2192 b ^\u207a(m \u2238 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207a-\u2238 z\u2264n = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207a-\u2238 (s\u2264s {n} {m} n\u2264m) =\n           b ^\u207a(m \u2238 n)                     \u2261\u27e8 ^\u207a-\u2238 n\u2264m \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296 : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^-\u2296 m      0      = ! is-\u03b5-right \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296 0      (1+ n) = ! idl\n        ^-\u2296 (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296 m n \u27e9\n           (b ^\u207a m \u2219 b ^\u207b n)               \u2261\u27e8 ! elim-inner= inverse\u02b3 \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b \u207b\u00b9 \u2219 b ^\u207b n)  \u2261\u27e8 \u2219= idp (! \u207b\u00b9-hom\u2032) \u27e9\n           (b ^\u207a m \u2219 b) \u2219 (b ^\u207a n \u2219 b)\u207b\u00b9   \u2261\u27e8 \u2219= (! ^\u207a-1+ m) (ap _\u207b\u00b9 (! ^\u207a-1+ n)) \u27e9\n           (b \u2219 b ^\u207a m) \u2219 (b \u2219 b ^\u207a n)\u207b\u00b9   \u220e\n\n        ^-\u2296' : \u2200 m n \u2192 b ^(m \u2296 n) \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^-\u2296' m      0      = ! is-\u03b5-left \u03b5\u207b\u00b9\u2261\u03b5\n        ^-\u2296' 0      (1+ n) = ! idr\n        ^-\u2296' (1+ m) (1+ n) =\n           b ^(m \u2296 n)                      \u2261\u27e8 ^-\u2296' m n \u27e9\n           (b ^\u207b n \u2219 b ^\u207a m)               \u2261\u27e8 ! elim-inner= inverse\u02e1 \u27e9\n           (b ^\u207b n \u2219 b \u207b\u00b9) \u2219 (b \u2219 b ^\u207a m)  \u2261\u27e8 \u2219= (! \u207b\u00b9-hom\u2032) idp \u27e9\n           (b \u2219 b ^\u207a n)\u207b\u00b9 \u2219 (b \u2219 b ^\u207a m)   \u220e\n\n        ^\u207a-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b ^\u207a m\n        ^\u207a-comm 0      n = ! comm-\u03b5\n        ^\u207a-comm (1+ m) n =\n          b \u2219 b\u1d50 \u2219 b\u207f   \u2261\u27e8 !assoc= (^\u207a-comm m n) \u27e9\n          b \u2219 b\u207f \u2219 b\u1d50   \u2261\u27e8 \u2219= (^\u207a-1+ n) idp \u27e9\n          b\u207f \u2219 b \u2219 b\u1d50   \u2261\u27e8 assoc \u27e9\n          b\u207f \u2219 (b \u2219 b\u1d50) \u220e\n          where\n            b\u207f = b ^\u207a n\n            b\u1d50 = b ^\u207a m\n\n        ^\u207a-^\u207b-comm : \u2200 m n \u2192 b ^\u207a m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207a m\n        ^\u207a-^\u207b-comm m n = ! ^-\u2296 m n \u2666 ^-\u2296' m n\n\n        ^\u207b-^\u207a-comm : \u2200 m n \u2192 b ^\u207b n \u2219 b ^\u207a m \u2261 b ^\u207a m \u2219 b ^\u207b n\n        ^\u207b-^\u207a-comm m n = ! ^-\u2296' m n \u2666 ^-\u2296 m n\n\n        ^\u207b-comm : \u2200 m n \u2192 b ^\u207b m \u2219 b ^\u207b n \u2261 b ^\u207b n \u2219 b ^\u207b m\n        ^\u207b-comm m n = \u207b\u00b9-hom\u2032= (^\u207a-comm n m)\n\n        ^\u207b-1+ : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n        ^\u207b-1+ n = ap _\u207b\u00b9 (^\u207a-1+ n) \u2666 \u207b\u00b9-hom\u2032\n\n        ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n        ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                        \u2666 ! \u207b\u00b9-hom\u2032\n                        \u2666 ap _\u207b\u00b9 (! ^\u207a-1+ n)\n\n        ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n        ^\u207b\u20321-id = idr\n\n        ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n        ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n        ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n        ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n        ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\n        -- ^-+ is a group homomorphism defined in Algebra.Group\n        -- Some properties can be derived from it.\n        ^-+ : \u2200 i j \u2192 b ^(i +\u2124 j) \u2261 b ^ i \u2219 b ^ j\n        ^-+ -[1+ m ] -[1+ n ] = ap _\u207b\u00b9\n                                 (ap (\u03bb z \u2192 b ^\u207a(1+ z)) (\u2115\u00b0.+-comm (1+ m) n)\n                              \u2666 ^\u207a-+ (1+ n) {1+ m}) \u2666 \u207b\u00b9-hom\u2032\n        ^-+ -[1+ m ]   (+ n)  = ^-\u2296' n (1+ m)\n        ^-+   (+ m)  -[1+ n ] = ^-\u2296 m (1+ n)\n        ^-+   (+ m)    (+ n)  = ^\u207a-+ m\n\n        -- GroupHomomorphism.f-pres-inv\n        ^-- : \u2200 i \u2192 b ^(- i) \u2261 (b ^ i)\u207b\u00b9\n        ^-- -[1+ n ] = ! \u207b\u00b9-involutive\n        ^-- (+ 0)    = ! \u03b5\u207b\u00b9\u2261\u03b5\n        ^-- (+ 1+ n) = idp\n\n        -- GroupHomomorphism.f-\u2212-\/\n        ^-\u2212 : \u2200 i j \u2192 b ^(i \u2212\u2124 j) \u2261 b ^ i \/ b ^ j\n        ^-\u2212 i j = ^-+ i (- j) \u2666 \u2219= idp (^-- j)\n\n        ^-suc : \u2200 i \u2192 b ^(suc\u2124 i) \u2261 b \u2219 b ^ i\n        ^-suc i = ^-+ (+ 1) i \u2666 \u2219= idr idp\n\n        ^-pred : \u2200 i \u2192 b ^(pred\u2124 i) \u2261 b \u207b\u00b9 \u2219 b ^ i\n        ^-pred i = ^-+ (- + 1) i \u2666 ap (\u03bb z \u2192 z \u207b\u00b9 \u2219 b ^ i) idr\n\n        ^-comm : \u2200 i j \u2192 b ^ i \u2219 b ^ j \u2261 b ^ j \u2219 b ^ i\n        ^-comm -[1+ m ] -[1+ n ] = ^\u207b-comm (1+ m) (1+ n)\n        ^-comm -[1+ m ]   (+ n)  = ! ^\u207a-^\u207b-comm n (1+ m)\n        ^-comm   (+ m)  -[1+ n ] = ^\u207a-^\u207b-comm m (1+ n)\n        ^-comm   (+ m)    (+ n)  = ^\u207a-comm m n\n\n      {-\n      ^-* : \u2200 i j \u2192 b ^(i *\u2124 j) \u2261 (b ^ j)^ i\n      ^-* i j = {!!}\n      -}\n\n    module From-Assoc-Identities-Inverse\n             (assoc : Associative _\u2219_)\n             (identities : Identity \u03b5 _\u2219_)\n             (inv : Inverse \u03b5 _\u207b\u00b9 _\u2219_)\n             where\n      open From-Assoc-Identities-RightInverse assoc identities (snd inv) public\n\n  module From-Ring-Ops (rng-ops : Ring-Ops A) where\n    open Ring-Ops rng-ops\n\n    module DistributesOver\u02b3\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_)\n             where\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = *-comm \u2666 *-+-distr\u02e1 \u2666 += *-comm *-comm\n\n    module DistributesOver\u02e1\n             (*-comm : Commutative _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = *-comm \u2666 *-+-distr\u02b3 \u2666 += *-comm *-comm\n\n    module From-+Group-1*Identity-DistributesOver\u02b3\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_)\n             where\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     ; \u03b5\u207b\u00b9\u2261\u03b5 to 0\u22120\u22610\n                     ; \u207b\u00b9-hom\u2032 to 0\u2212-hom\u2032\n                     )\n\n      -0\u22610 : -0# \u2261 0#\n      -0\u22610 = 0\u22120\u22610\n\n      0*-zero : LeftZero 0# _*_\n      0*-zero {x} = cancels-+-left\n        (x + 0# * x      \u2261\u27e8 += (! 1*-identity) idp \u27e9\n         1# * x + 0# * x \u2261\u27e8 ! *-+-distr\u02b3 \u27e9\n         (1# + 0#) * x   \u2261\u27e8 *= +0-identity idp \u27e9\n         1# * x          \u2261\u27e8 1*-identity \u27e9\n         x               \u2261\u27e8 ! +0-identity \u27e9\n         x + 0#          \u220e)\n\n      2*-spec : \u2200 {n} \u2192 2* n \u2261 2# * n\n      2*-spec = ! += 1*-identity 1*-identity \u2666 ! *-+-distr\u02b3\n\n      0\u2212-*-distr : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 (0\u2212 x) * y\n      0\u2212-*-distr = cancels-+-right (0\u2212-inverse\u02e1 \u2666 ! 0*-zero \u2666 *= (! 0\u2212-inverse\u02e1) idp \u2666 *-+-distr\u02b3)\n\n      -1*-neg : \u2200 {x} \u2192 -1# * x \u2261 0\u2212 x\n      -1*-neg = ! 0\u2212-*-distr \u2666 0\u2212= 1*-identity\n\n      2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n      2*-*-distr = ! *-+-distr\u02b3\n\n    module From-+Group-*Identity-DistributesOver\n             (+-assoc : Associative _+_)\n             (0+-identity : LeftIdentity 0# _+_)\n             (+0-identity : RightIdentity 0# _+_)\n             (0\u2212-inverse\u02b3 : RightInverse 0# 0\u2212_ _+_)\n             (1*-identity : LeftIdentity 1# _*_)\n             (*1-identity : RightIdentity 1# _*_)\n             (*-+-distrs : _*_ DistributesOver _+_)\n             where\n      *-+-distr\u02e1 : _*_ DistributesOver\u02e1 _+_\n      *-+-distr\u02e1 = fst *-+-distrs\n\n      *-+-distr\u02b3 : _*_ DistributesOver\u02b3 _+_\n      *-+-distr\u02b3 = snd *-+-distrs\n\n      open From-Group-Ops.From-Assoc-Identities-RightInverse\n             +-grp-ops\n             +-assoc (0+-identity , +0-identity) 0\u2212-inverse\u02b3\n            renaming ( \u207b\u00b9-involutive to 0\u2212-involutive\n                     ; cancels-\u2219-left to cancels-+-left\n                     ; cancels-\u2219-right to cancels-+-right\n                     ; inverse\u02e1 to 0\u2212-inverse\u02e1\n                     )\n\n      open From-+Group-1*Identity-DistributesOver\u02b3\n             +-assoc 0+-identity +0-identity 0\u2212-inverse\u02b3\n             1*-identity *-+-distr\u02b3\n             public\n\n      *0-zero : RightZero 0# _*_\n      *0-zero {x} = cancels-+-right\n        (x * 0# + x      \u2261\u27e8 += idp (! *1-identity) \u27e9\n         x * 0# + x * 1# \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n         x * (0# + 1#)   \u2261\u27e8 *= idp 0+-identity \u27e9\n         x * 1#          \u2261\u27e8 *1-identity \u27e9\n         x               \u2261\u27e8 ! 0+-identity \u27e9\n         0# + x          \u220e)\n\n      0\u2212-*-distr\u02b3 : \u2200 {x y} \u2192 0\u2212(x * y) \u2261 x * (0\u2212 y)\n      0\u2212-*-distr\u02b3 = cancels-+-right (0\u2212-inverse\u02e1 \u2666 (! *0-zero \u2666 *= idp (! 0\u2212-inverse\u02e1)) \u2666 *-+-distr\u02e1)\n\n      *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n      *-\u2212-distr = *-+-distr\u02e1 \u2666 += idp (! 0\u2212-*-distr\u02b3)\n\n      \u00b2-0\u2212-distr : \u2200 {x} \u2192 (0\u2212 x)\u00b2 \u2261 x \u00b2\n      \u00b2-0\u2212-distr = ! 0\u2212-*-distr \u2666 0\u2212=(! 0\u2212-*-distr\u02b3) \u2666 0\u2212-involutive\n\n      +-comm : Commutative _+_\n      +-comm {a} {b} =\n          cancels-+-right\n            (cancels-+-left\n              (a + (a + b + b)   \u2261\u27e8 += idp +-assoc \u2666 ! +-assoc \u27e9\n               (a + a) + (b + b) \u2261\u27e8 += 2*-spec 2*-spec \u27e9\n               2# * a  + 2# * b  \u2261\u27e8 ! *-+-distr\u02e1 \u27e9\n               2# * (a + b)      \u2261\u27e8 ! 2*-spec \u27e9\n               (a + b) + (a + b) \u2261\u27e8 +-assoc \u2666 += idp (! +-assoc) \u27e9\n               a + (b + a + b)   \u220e))\n\n      0\u2212-+-distr : \u2200 {x y} \u2192 0\u2212(x + y) \u2261 0\u2212 x \u2212 y\n      0\u2212-+-distr = 0\u2212= +-comm \u2666 0\u2212-+-distr\u2032\n\n  module From-Field-Ops (fld-ops : Field-Ops A) where\n    open Field-Ops fld-ops\n\nmodule Explicits where\n  open Algebra.FunctionProperties.NP \u03a0  {a}{a}{A} _\u2261_ public\n\n  -- REPEATED from above but with explicit arguments\n  module FromOp\u2082\n           (_\u2219_ : Op\u2082 A){x x' y y'}(p : x \u2261 x')(q : y \u2261 y')\n         where\n    op= : x \u2219 y \u2261 x' \u2219 y'\n    op= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  module FromComm\n           (_\u2219_   : Op\u2082 A)\n           (comm  : Commutative _\u2219_)\n           (x y x' y' : A)\n           (e : (y \u2219 x) \u2261 (y' \u2219 x'))\n         where\n    open FromOp\u2082 _\u2219_\n\n    comm= : x \u2219 y \u2261 x' \u2219 y'\n    comm= = comm _ _ \u2666 e \u2666 comm _ _\n\n  module FromAssoc\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n\n         where\n    open FromOp\u2082 _\u2219_\n\n    assocs : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    assocs = assoc , (\u03bb _ _ _ \u2192 ! assoc _ _ _)\n\n    module _ {c x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      assoc= : x \u2219 (y \u2219 c) \u2261 x' \u2219 (y' \u2219 c)\n      assoc= = ! assoc _ _ _ \u2666 op= e idp \u2666 assoc _ _ _\n\n      !assoc= : (c \u2219 x) \u2219 y \u2261 (c \u2219 x') \u2219 y'\n      !assoc= = assoc _ _ _ \u2666 op= idp e \u2666 ! assoc _ _ _\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 (c \u2219 x') \u2219 (y' \u2219 d)\n      inner= = assoc= (!assoc= e)\n\n  module FromAssocComm\n           (_\u2219_   : Op\u2082 A)\n           (assoc : Associative _\u2219_)\n           (comm  : Commutative _\u2219_)\n         where\n    open FromOp\u2082   _\u2219_       renaming (op= to \u2219=)\n    open FromAssoc _\u2219_ assoc public\n    open FromComm  _\u2219_ comm  public\n\n    module _ x y z where\n      assoc-comm : x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n      assoc-comm = assoc= (comm _ _)\n\n      !assoc-comm : (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 y\n      !assoc-comm = !assoc= (comm _ _)\n\n    interchange : Interchange _\u2219_ _\u2219_\n    interchange _ _ _ _ = assoc= (!assoc= (comm _ _))\n\n    module _ {c d x y x' y' : A}\n             (e : (x \u2219 y) \u2261 (x' \u2219 y')) where\n      outer= : (x \u2219 c) \u2219 (d \u2219 y) \u2261 (x' \u2219 c) \u2219 (d \u2219 y')\n      outer= = \u2219= (comm _ _) (comm _ _) \u2666 assoc= (!assoc= e) \u2666 \u2219= (comm _ _) (comm _ _)\n\n  module _ {b}{B : Set b} where\n    open Morphisms {B = B} _\u2261_ public\n\n    module _ {f : A \u2192 B} where\n      Injective-\u00acConflict : Injective f \u2192 \u00ac(Conflict f)\n      Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj _ _ fx\u2261fy)\n\n      Conflict-\u00acInjective : Conflict f \u2192 \u00ac(Injective f)\n      Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e54de835c5c64b833c3cccb76823da8ca3bd58f6","subject":"README: -vcomp,-composable","message":"README: -vcomp,-composable\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import forkable\n\nopen import program-distance\nopen import diff\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","old_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import composable\nopen import vcomp\nopen import forkable\n\nopen import program-distance\nopen import diff\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8de3d97dc13a2bad5a52302e268a8745ecd57ea8","subject":"Inline trivial let bindings.","message":"Inline trivial let bindings.\n\nThese bindings used to be more complicated, but with the recent changes,\nwe can just inline them.\n\nOld-commit-hash: 1ab984cf29bc5dd53e7f1d6a9b36bc69fd00c2c5\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -})\n    (\u0394base : Base \u2192 Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type Base\nopen Context Type\n\nopen import Parametric.Change.Type \u0394base\nopen import Parametric.Syntax.Term Constant\n\nopen import Data.Product\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply :\n  (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff :\n  (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply :\n  (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394Type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -})\n    (\u0394base : Base \u2192 Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type Base\nopen Context Type\n\nopen import Parametric.Change.Type \u0394base\nopen import Parametric.Syntax.Term Constant\n\nopen import Data.Product\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply :\n  let\n    \u0394type = \u0394Type\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff :\n  let\n    \u0394type = \u0394Type\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply :\n  let\n    \u0394type = \u0394Type\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a3aad796be63ed1c7b63421a12219d2aaffd29c6","subject":"Added test for the conversion rules of subtraction.","message":"Added test for the conversion rules of subtraction.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/LTC-PCF\/Data\/Nat\/SubtractionRecCombinator.agda","new_file":"notes\/FOT\/LTC-PCF\/Data\/Nat\/SubtractionRecCombinator.agda","new_contents":"------------------------------------------------------------------------------\n-- Subtraction using the rec combinator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.LTC-PCF.Data.Nat.SubtractionRecCombinator where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat.Properties hiding ( \u2238-x0 ; \u2238-0x ; \u2238-0S ; \u2238-SS )\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.Equations\nopen import LTC-PCF.Data.Nat.Type\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u2238_\n\n------------------------------------------------------------------------------\n\n-- Subtraction\n_\u2238_ : D \u2192 D \u2192 D\nm \u2238 n = rec n m (lam (\u03bb _ \u2192 lam pred\u2081))\n\n-- Conversion rules (from the Agda standard library)\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S nzero =\n  rec (succ\u2081 zero) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) zero) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 predCong (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x nzero      = \u2238-x0 zero\n\u2238-0x (nsucc Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ nzero =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) zero) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS nzero (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb _ \u2192 lam pred\u2081))  \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS nzero Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (nsucc {m} Nm) (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS (nsucc Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb _ \u2192 lam pred\u2081) n)) \u27e9\n  (lam (\u03bb _ \u2192 lam pred\u2081)) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n------------------------------------------------------------------------------\n-- Conversion rules from the Agda standard library without totality\n-- hypotheses.\n\n-- We could not prove this property.\n-- \u2238-0S\u2081 : \u2200 n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n-- \u2238-0S\u2081 n =\n--   rec (succ\u2081 n) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n--     \u2261\u27e8 rec-S n zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 n \u00b7 (zero \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (zero \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (zero \u2238 n) \u27e9\n--   pred\u2081 (zero \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   zero \u220e\n\n-- We could not prove this property.\n-- \u2238-SS\u2081 : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n-- \u2238-SS\u2081 m n =\n--   rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n--   \u2261\u27e8 rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n--   pred\u2081 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   m \u2238 n \u220e\n\n------------------------------------------------------------------------------\n-- Coq conversion rules\n\n-- Fixpoint minus (n m:nat) : nat :=\n--   match n, m with\n--   | O, _ => n\n--   | S k, O => S k\n--   | S k, S l => k - l\n--   end\n\n\u2238-x0-coq : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0-coq n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n             n            \u220e\n\n\u2238-S0-coq : \u2200 n \u2192 succ\u2081 n \u2238 zero \u2261 succ\u2081 n\n\u2238-S0-coq n =\n  rec zero (succ\u2081 n) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-0 (succ\u2081 n) \u27e9\n  succ\u2081 n \u220e\n\n-- We could not prove this property.\n-- \u2238-SS-coq : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n-- \u2238-SS-coq m n =\n--   rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n--   \u2261\u27e8 rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n--   pred\u2081 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   m \u2238 n \u220e\n\n------------------------------------------------------------------------------\n-- Isabelle conversion rules (from Nat.thy)\n\n-- primrec minus_nat where\n--   diff_0 [code]: \"m - 0 = (m :: nat)\"\n-- | diff_Suc: \"m - Suc n = (case m - n of 0 => 0 | Suc k => k)\"\n\n\u2238-x0-isabelle : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0-isabelle n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n                  n            \u220e\n\n-- We could not prove this property.\n-- \u2238-xS-isabelle : \u2200 m n \u2192 m \u2238 succ\u2081 n \u2261\n--                         if (iszero\u2081 (m \u2238 n)) then zero else pred\u2081 (m \u2238 n)\n-- \u2238-xS-isabelle m n =\n--   rec (succ\u2081 n) m (lam (\u03bb _ \u2192 lam pred\u2081))\n--     \u2261\u27e8 rec-S n m (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (m \u2238 n) \u27e9\n--   pred\u2081 (m \u2238 n)\n--   \u2261\u27e8 {!!} \u27e9\n--   if (iszero\u2081 (m \u2238 n)) then zero else pred\u2081 (m \u2238 n) \u220e\n\n------------------------------------------------------------------------------\n-- Peter conversion rules\n\n-- The analogous situation for subtraction is that  given\n\n-- rec-0 : \u2200 a {f} \u2192 rec zero a f \u2261 a\n-- rec-S : \u2200 n a f \u2192 rec (succ\u2081 n) a f \u2261 f \u00b7 n \u00b7 (rec n a f)\n\n-- and\n\n-- _\u2238_ : D \u2192 D \u2192 D\n-- m \u2238 n = rec n m (lam (\u03bb _ \u2192 lam pred\u2081))\n\n-- you get the equations obtained by the special case (instantiate a and\n-- f according to the def of subtraction)! This has nothing to do a\n-- priori with Agda's standard library.\n\n\u2238-x0-peter : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0-peter n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n               n            \u220e\n\n\u2238-xS-peter : \u2200 m n \u2192 m \u2238 succ\u2081 n \u2261 pred\u2081 (m \u2238 n)\n\u2238-xS-peter m n =\n  rec (succ\u2081 n) m (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S n m (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n  lam pred\u2081 \u00b7 (m \u2238 n)\n    \u2261\u27e8 beta pred\u2081 (m \u2238 n) \u27e9\n  pred\u2081 (m \u2238 n) \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Subtraction using the rec combinator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.LTC-PCF.Data.Nat.SubtractionRecCombinator where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat.Properties hiding ( \u2238-x0 ; \u2238-0x ; \u2238-0S ; \u2238-SS )\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.Equations\nopen import LTC-PCF.Data.Nat.Type\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u2238_\n\n------------------------------------------------------------------------------\n\n-- Subtraction\n_\u2238_ : D \u2192 D \u2192 D\nm \u2238 n = rec n m (lam (\u03bb _ \u2192 lam pred\u2081))\n\n-- Conversion rules (from the Agda standard library)\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S nzero =\n  rec (succ\u2081 zero) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) zero) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 predCong (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x nzero      = \u2238-x0 zero\n\u2238-0x (nsucc Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ nzero =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) zero) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS nzero (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb _ \u2192 lam pred\u2081))  \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS nzero Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (nsucc {m} Nm) (nsucc {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n  lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) (succ\u2081 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (\u2238-SS (nsucc Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb _ \u2192 lam pred\u2081) n)) \u27e9\n  (lam (\u03bb _ \u2192 lam pred\u2081)) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n------------------------------------------------------------------------------\n-- Conversion rules from the Agda standard library without totality\n-- hypotheses.\n\n-- We could not prove this property.\n-- \u2238-0S\u2081 : \u2200 n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n-- \u2238-0S\u2081 n =\n--   rec (succ\u2081 n) zero (lam (\u03bb _ \u2192 lam pred\u2081))\n--     \u2261\u27e8 rec-S n zero (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb _ \u2192 lam pred\u2081) \u00b7 n \u00b7 (zero \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (zero \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (zero \u2238 n) \u27e9\n--   pred\u2081 (zero \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   zero \u220e\n\n-- We could not prove this property.\n-- \u2238-SS\u2081 : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n-- \u2238-SS\u2081 m n =\n--   rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n--   \u2261\u27e8 rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n--   pred\u2081 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   m \u2238 n \u220e\n\n------------------------------------------------------------------------------\n-- Coq conversion rules\n\n-- Fixpoint minus (n m:nat) : nat :=\n--   match n, m with\n--   | O, _ => n\n--   | S k, O => S k\n--   | S k, S l => k - l\n--   end\n\n\u2238-x0-coq : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0-coq n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n             n            \u220e\n\n\u2238-S0-coq : \u2200 n \u2192 succ\u2081 n \u2238 zero \u2261 succ\u2081 n\n\u2238-S0-coq n =\n  rec zero (succ\u2081 n) (lam (\u03bb _ \u2192 lam pred\u2081))\n    \u2261\u27e8 rec-0 (succ\u2081 n) \u27e9\n  succ\u2081 n \u220e\n\n-- We could not prove this property.\n-- \u2238-SS-coq : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n-- \u2238-SS-coq m n =\n--   rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081))\n--   \u2261\u27e8 rec-S n (succ\u2081 m) (lam (\u03bb _ \u2192 lam pred\u2081)) \u27e9\n--   lam (\u03bb x \u2192 lam pred\u2081) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam pred\u2081) n) \u27e9\n--   lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n--   pred\u2081 (succ\u2081 m \u2238 n)\n--     \u2261\u27e8 {!!} \u27e9\n--   {!!}\n--     \u2261\u27e8 {!!} \u27e9\n--   m \u2238 n \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"12e8fbbc6c3c0486a79028fdfe70ffb8d6a9e47f","subject":"Conat-coind and Conat-coind-wrong are not equivalents.","message":"Conat-coind and Conat-coind-wrong are not equivalents.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-unf-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e\n  --\n  -- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n  Conat-coind : \u2200 (A : D \u2192 Set) \u2192\n                (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho : \u2200 (A : D \u2192 Set) \u2192\n                   (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192\n                   \u2200 {n} \u2192 A n \u2192 Conat n\n\n  Conat-coind-wrong : \u2200 (A : D \u2192 Set) {n} \u2192\n                      -- A is post-fixed point of ConatF.\n                      (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                      -- Conat is greater than A.\n                      A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-unf and Conat-unf-ho are equivalents\n\nConat-unf' : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf' = Conat-unf-ho\n\nConat-unf-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-unf-ho' = Conat-unf\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind' : \u2200 (A : D \u2192 Set) \u2192\n               (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n               \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind' = Conat-coind-ho\n\nConat-coind-ho' : \u2200 (A : D \u2192 Set) \u2192\n                  (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192\n                  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-coind-wrong to Conat-coind-wrong\/Conat-coind\n\nConat-coind'' : \u2200 (A : D \u2192 Set) \u2192\n               (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n               \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-coind-wrong A h An\n\n-- 22 December 2013: We cannot prove Conat-coind-wrong'' using\n-- Conat-coind.\nConat-coind-wrong'' : \u2200 (A : D \u2192 Set) {n} \u2192\n                      (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                      A n \u2192 Conat n\nConat-coind-wrong'' A h An = Conat-coind A {!!} An\n","old_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-unf-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e\n  --\n  -- \u2200 P. P \u2264 NatF P \u21d2 P \u2264 Conat.\n  Conat-coind : \u2200 (A : D \u2192 Set) \u2192\n                (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho : \u2200 (A : D \u2192 Set) \u2192\n                   (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192\n                   \u2200 {n} \u2192 A n \u2192 Conat n\n\n  Conat-coind-wrong : \u2200 (A : D \u2192 Set) {n} \u2192\n                      -- A is post-fixed point of ConatF.\n                      (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n                      -- Conat is greater than A.\n                      A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-unf and Conat-unf-ho are equivalents\n\nConat-unf' : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-unf' = Conat-unf-ho\n\nConat-unf-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-unf-ho' = Conat-unf\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind' : \u2200 (A : D \u2192 Set) \u2192\n               (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n               \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind' = Conat-coind-ho\n\nConat-coind-ho' : \u2200 (A : D \u2192 Set) \u2192\n                  (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192\n                  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"497653b80374df298570525fff4ad3f3cbf70331","subject":"Remove obsolete TODO list 1, 2 are done. 3 is stuck due to the abstraction not being coarse enough. 4 shall never happen to NatBag.agda because of 3.","message":"Remove obsolete TODO list\n1, 2 are done.\n3 is stuck due to the abstraction not being coarse enough.\n4 shall never happen to NatBag.agda because of 3.\n\nOld-commit-hash: 129543756924f6c6745268dd0bc8f9dcebfe4892\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/NatBag.agda","new_file":"experimental\/NatBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = diff d b\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemmas\n\n-- diff-apply holds with propositional equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 sym (\u2205++b=b {{\u27e6 derive b \u27e7 \u03c1}}) \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 extract-\u0394equiv\n       (correctness-of-derive \u03c1 {consistency} b)\n       emptyBag tt tt \u27e9\n    emptyBag ++ (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 \u2205++b=b {{\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)}} \u27e9\n    \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Semantic brackets preserve \u2295 and \u229d\n\n\u2295-preservation : \u2200 {\u03c4 \u0393} \u2192\n  {t : Term \u0393 \u03c4} {dt : Term \u0393 (\u0394-Type \u03c4)} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u2295 dt \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1\n\n\u229d-preservation : \u2200 {\u03c4 \u0393} \u2192\n  {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u229d t \u27e7 \u03c1 \u2261 \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u27e7 \u27e6 t \u27e7 \u03c1\n\n\u2295-preservation {nats} {\u0393} {t} {dt} {\u03c1} = refl\n\u2295-preservation {bags} {\u0393} {t} {dt} {\u03c1} = refl\n\u2295-preservation {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {t} {dt} {\u03c1} = extensionality (\u03bb x \u2192\n  begin\n    \u27e6 app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) t) (var this) \u2295\n      app (app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) dt) (var this))\n      (var this \u229d var this) \u27e7 (x \u2022 \u03c1)\n  \u2261\u27e8 \u2295-preservation \u27e9\n    \u27e6 app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) t) (var this) \u27e7 (x \u2022 \u03c1)\n    \u27e6\u2295\u27e7\n    \u27e6 app (app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) dt) (var this))\n      (var this \u229d var this) \u27e7 (x \u2022 \u03c1)\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n       (trans (weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} t (x \u2022 \u03c1))\n              (cong \u27e6 t \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl)\n       (trans (weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} dt (x \u2022 \u03c1))\n              (cong \u27e6 dt \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl \u27e8$\u27e9 refl) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (\u27e6 (var this \u229d var this) \u27e7 (x \u2022 \u03c1))\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x hole)\n          (\u229d-preservation {\u03c4\u2081} {\u03c4\u2081 \u2022 \u0393} {var this} {var this}) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (x \u27e6\u229d\u27e7 x)\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x hole) (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f x) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (\u27e6derive\u27e7 x)\n  \u220e) where open \u2261-Reasoning\n\n\u229d-preservation {nats} {\u0393} {s} {t} {\u03c1} = extensionality (\u03bb f \u2192\n  cong\u2082 f\n    (trans (weaken-sound t (f \u2022 \u03c1)) (cong \u27e6 t \u27e7 identity-weakening))\n    (trans (weaken-sound s (f \u2022 \u03c1)) (cong \u27e6 s \u27e7 identity-weakening)))\n\u229d-preservation {bags} {\u0393} {s} {t} {\u03c1} = refl\n\u229d-preservation {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {s} {t} {\u03c1} =\n  extensionality (\u03bb x \u2192\n  extensionality (\u03bb dx \u2192\n  begin\n    \u27e6 app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n      (var (that this) \u2295 var this)\n      \u229d\n      app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)) \u27e7\n    (dx \u2022 x \u2022 \u03c1)\n  \u2261\u27e8 \u229d-preservation\n     {s = app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n              (var (that this) \u2295 var this)}\n     {t = app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t)\n               (var (that this))} \u27e9\n    \u27e6 weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s \u27e7 (dx \u2022 x \u2022 \u03c1)\n      (\u27e6 var (that this) \u2295 var this \u27e7 (dx \u2022 x \u2022 \u03c1))\n    \u27e6\u229d\u27e7\n    \u27e6 weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t \u27e7 (dx \u2022 x \u2022 \u03c1)\n      (\u27e6 (var (that this)) \u27e7 (dx \u2022 x \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (trans (weaken-sound s (dx \u2022 x \u2022 \u03c1))\n              (cong \u27e6 s \u27e7 identity-weakening)\n        \u27e8$\u27e9 \u2295-preservation {t = var (that this)} {dt = var this})\n       (trans (weaken-sound t (dx \u2022 x \u2022 \u03c1))\n              (cong \u27e6 t \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl) \u27e9\n    \u27e6 s \u27e7 \u03c1 (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 \u27e6 t \u27e7 \u03c1 x\n  \u220e)) where open \u2261-Reasoning\n\n-- Derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = validity-of-derive \u03c1 {consistency} t\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _++_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (sym ([a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} \n             {\u27e6 d \u27e7 (update \u03c1 {consistency})}\n             {\u27e6 b \u27e7 (ignore \u03c1)} {\u27e6 d \u27e7 (ignore \u03c1)})) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency})\n        ++ \u27e6 d \u27e7 (update \u03c1 {consistency}))\n        \\\\(\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (diff b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ \u27e6 derive b \u27e7 \u03c1 \\\\ \u27e6 derive d \u27e7 \u03c1\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _\\\\_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++  (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       \\\\ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       ([a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} {\u27e6 b \u27e7 (ignore \u03c1)}\n             {\u27e6 d \u27e7 (update \u03c1 {consistency})} {\u27e6 d \u27e7 (ignore \u03c1)}) \u27e9\n    c ++  (\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1))\n  \u220e) where open \u2261-Reasoning\n\n\ncorrectness-of-derive \u03c1 {consistency} (map f b) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u2295 derive f \u27e7 \u03c1)\n               (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1 ++ \u27e6 derive b \u27e7 \u03c1))\n       \\\\\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _\\\\_\n          (cong\u2082 mapBag\n            (\u2295-preservation { t = weaken \u0393\u227c\u0394\u0393 f}\n                            {dt = derive f} {\u03c1 = \u03c1})\n            (extract-\u0394equiv\n              (correctness-of-derive\u2032 \u03c1 {consistency} b)\n              (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1) tt tt))\n          (cong\u2082 mapBag\n            (weaken-sound f \u03c1) (weaken-sound b \u03c1))) \u27e9\n    c ++\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u03c1) -- \u2295-preservation\n               (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1 ++                 -- correctness\u2032\n                \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1))\n       \\\\\n       (mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1)))   -- weaken-sound\n  \u2261\u27e8 cong\u2082 (\u03bb hole-f hole-b \u2192 c ++ mapBag hole-f hole-b\n            \\\\ mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1)))\n       (trans\n         (extract-\u0394equiv\n           (correctness-of-derive\u2032 \u03c1 {consistency} f)\n           (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1)\n           (validity-of-derive\u2032 \u03c1 {consistency} f)\n           (R[f,g\u229df] (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 f \u27e7 (update \u03c1))))\n         (f\u2295[g\u229df]=g (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 f \u27e7 (update \u03c1))))\n       (b++[d\\\\b]=d {\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1} {\u27e6 b \u27e7 (update \u03c1)})\n        \u27e9\n    c ++  mapBag (\u27e6 f \u27e7 (update \u03c1)) (\u27e6 b \u27e7 (update \u03c1))\n       \\\\ mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1))\n  \u220e) where open \u2261-Reasoning\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","old_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   X. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   X. Introduce addition\n   X. Add MapBags and map\n2. Test it out with `inc` as primitive\n3. Finish ExplicitNils\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n4. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = diff d b\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemmas\n\n-- diff-apply holds with propositional equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 sym (\u2205++b=b {{\u27e6 derive b \u27e7 \u03c1}}) \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 extract-\u0394equiv\n       (correctness-of-derive \u03c1 {consistency} b)\n       emptyBag tt tt \u27e9\n    emptyBag ++ (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 \u2205++b=b {{\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)}} \u27e9\n    \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Semantic brackets preserve \u2295 and \u229d\n\n\u2295-preservation : \u2200 {\u03c4 \u0393} \u2192\n  {t : Term \u0393 \u03c4} {dt : Term \u0393 (\u0394-Type \u03c4)} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u2295 dt \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1\n\n\u229d-preservation : \u2200 {\u03c4 \u0393} \u2192\n  {s : Term \u0393 \u03c4} {t : Term \u0393 \u03c4} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 s \u229d t \u27e7 \u03c1 \u2261 \u27e6 s \u27e7 \u03c1 \u27e6\u229d\u27e7 \u27e6 t \u27e7 \u03c1\n\n\u2295-preservation {nats} {\u0393} {t} {dt} {\u03c1} = refl\n\u2295-preservation {bags} {\u0393} {t} {dt} {\u03c1} = refl\n\u2295-preservation {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {t} {dt} {\u03c1} = extensionality (\u03bb x \u2192\n  begin\n    \u27e6 app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) t) (var this) \u2295\n      app (app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) dt) (var this))\n      (var this \u229d var this) \u27e7 (x \u2022 \u03c1)\n  \u2261\u27e8 \u2295-preservation \u27e9\n    \u27e6 app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) t) (var this) \u27e7 (x \u2022 \u03c1)\n    \u27e6\u2295\u27e7\n    \u27e6 app (app (weaken (drop \u03c4\u2081 \u2022 \u0393\u227c\u0393) dt) (var this))\n      (var this \u229d var this) \u27e7 (x \u2022 \u03c1)\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n       (trans (weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} t (x \u2022 \u03c1))\n              (cong \u27e6 t \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl)\n       (trans (weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} dt (x \u2022 \u03c1))\n              (cong \u27e6 dt \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl \u27e8$\u27e9 refl) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (\u27e6 (var this \u229d var this) \u27e7 (x \u2022 \u03c1))\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x hole)\n          (\u229d-preservation {\u03c4\u2081} {\u03c4\u2081 \u2022 \u0393} {var this} {var this}) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (x \u27e6\u229d\u27e7 x)\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x hole) (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f x) \u27e9\n    \u27e6 t \u27e7 \u03c1 x \u27e6\u2295\u27e7 \u27e6 dt \u27e7 \u03c1 x (\u27e6derive\u27e7 x)\n  \u220e) where open \u2261-Reasoning\n\n\u229d-preservation {nats} {\u0393} {s} {t} {\u03c1} = extensionality (\u03bb f \u2192\n  cong\u2082 f\n    (trans (weaken-sound t (f \u2022 \u03c1)) (cong \u27e6 t \u27e7 identity-weakening))\n    (trans (weaken-sound s (f \u2022 \u03c1)) (cong \u27e6 s \u27e7 identity-weakening)))\n\u229d-preservation {bags} {\u0393} {s} {t} {\u03c1} = refl\n\u229d-preservation {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {s} {t} {\u03c1} =\n  extensionality (\u03bb x \u2192\n  extensionality (\u03bb dx \u2192\n  begin\n    \u27e6 app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n      (var (that this) \u2295 var this)\n      \u229d\n      app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)) \u27e7\n    (dx \u2022 x \u2022 \u03c1)\n  \u2261\u27e8 \u229d-preservation\n     {s = app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n              (var (that this) \u2295 var this)}\n     {t = app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t)\n               (var (that this))} \u27e9\n    \u27e6 weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s \u27e7 (dx \u2022 x \u2022 \u03c1)\n      (\u27e6 var (that this) \u2295 var this \u27e7 (dx \u2022 x \u2022 \u03c1))\n    \u27e6\u229d\u27e7\n    \u27e6 weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t \u27e7 (dx \u2022 x \u2022 \u03c1)\n      (\u27e6 (var (that this)) \u27e7 (dx \u2022 x \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (trans (weaken-sound s (dx \u2022 x \u2022 \u03c1))\n              (cong \u27e6 s \u27e7 identity-weakening)\n        \u27e8$\u27e9 \u2295-preservation {t = var (that this)} {dt = var this})\n       (trans (weaken-sound t (dx \u2022 x \u2022 \u03c1))\n              (cong \u27e6 t \u27e7 identity-weakening)\n        \u27e8$\u27e9 refl) \u27e9\n    \u27e6 s \u27e7 \u03c1 (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 \u27e6 t \u27e7 \u03c1 x\n  \u220e)) where open \u2261-Reasoning\n\n-- Derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = validity-of-derive \u03c1 {consistency} t\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _++_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (sym ([a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} \n             {\u27e6 d \u27e7 (update \u03c1 {consistency})}\n             {\u27e6 b \u27e7 (ignore \u03c1)} {\u27e6 d \u27e7 (ignore \u03c1)})) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency})\n        ++ \u27e6 d \u27e7 (update \u03c1 {consistency}))\n        \\\\(\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (diff b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ \u27e6 derive b \u27e7 \u03c1 \\\\ \u27e6 derive d \u27e7 \u03c1\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _\\\\_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++  (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       \\\\ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       ([a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} {\u27e6 b \u27e7 (ignore \u03c1)}\n             {\u27e6 d \u27e7 (update \u03c1 {consistency})} {\u27e6 d \u27e7 (ignore \u03c1)}) \u27e9\n    c ++  (\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1))\n  \u220e) where open \u2261-Reasoning\n\n\ncorrectness-of-derive \u03c1 {consistency} (map f b) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u2295 derive f \u27e7 \u03c1)\n               (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1 ++ \u27e6 derive b \u27e7 \u03c1))\n       \\\\\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _\\\\_\n          (cong\u2082 mapBag\n            (\u2295-preservation { t = weaken \u0393\u227c\u0394\u0393 f}\n                            {dt = derive f} {\u03c1 = \u03c1})\n            (extract-\u0394equiv\n              (correctness-of-derive\u2032 \u03c1 {consistency} b)\n              (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1) tt tt))\n          (cong\u2082 mapBag\n            (weaken-sound f \u03c1) (weaken-sound b \u03c1))) \u27e9\n    c ++\n       (mapBag (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u03c1) -- \u2295-preservation\n               (\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1 ++                 -- correctness\u2032\n                \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1))\n       \\\\\n       (mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1)))   -- weaken-sound\n  \u2261\u27e8 cong\u2082 (\u03bb hole-f hole-b \u2192 c ++ mapBag hole-f hole-b\n            \\\\ mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1)))\n       (trans\n         (extract-\u0394equiv\n           (correctness-of-derive\u2032 \u03c1 {consistency} f)\n           (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1)\n           (validity-of-derive\u2032 \u03c1 {consistency} f)\n           (R[f,g\u229df] (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 f \u27e7 (update \u03c1))))\n         (f\u2295[g\u229df]=g (\u27e6 weaken \u0393\u227c\u0394\u0393 f \u27e7 \u03c1) (\u27e6 f \u27e7 (update \u03c1))))\n       (b++[d\\\\b]=d {\u27e6 weaken \u0393\u227c\u0394\u0393 b \u27e7 \u03c1} {\u27e6 b \u27e7 (update \u03c1)})\n        \u27e9\n    c ++  mapBag (\u27e6 f \u27e7 (update \u03c1)) (\u27e6 b \u27e7 (update \u03c1))\n       \\\\ mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 b \u27e7 (ignore \u03c1))\n  \u220e) where open \u2261-Reasoning\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a478af099dcf5e8afcb3cfa1dbfb14e804463a3f","subject":"Refactor before adding bang changes","message":"Refactor before adding bang changes\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x ,  rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ _ _ _ (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nopen import Relation.Nullary\n\n-- Decidable equivalence for types and contexts :-|\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x ,  rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ _ _ _ (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2dd7d465cfed15f55c68dbd46e33c679b2fc640c","subject":"Organize imports.","message":"Organize imports.\n\nOld-commit-hash: 1e433ca7707664a3b07c401b8685fa51869876e1\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4}\n  (\u03c1\u2081 : \u27e6 \u0393prefix \u27e7) (\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7)\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\n\nlift-term-doWeakenOne-ignore\u2032 \u0393prefix \u2205 \u03c1\u2081 \u2205 t = refl\n--This step gives the proof idea.\nlift-term-doWeakenOne-ignore\u2032 \u2205 (\u03c4 \u2022 \u0393rest) \u2205 (dv \u2022 v \u2022 \u03c1\u2082) t = lift-term-doWeakenOne-ignore\u2032 (\u03c4 \u2022 \u2205) \u0393rest (v \u2022 \u2205) \u03c1\u2082 t\n\nlift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix) (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} (v\u2080 \u2022 \u03c1\u2081) (dv \u2022 v \u2022 \u03c1\u2082) t = {!!}\n--Look at C-c C-, for the normalized goal.\n-- The solution here should be something like:\n  --lift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest {\u03c4} ((v\u2080 \u2022 \u03c1\u2081) \u222a (v \u2022 \u2205)) \u03c1\u2082  ?\n\n-- accompanied by enough subst to make it typecheck, and maybe by\n-- weakenOne-ignore to account for weakenOne - or maybe not since\n-- weakenOne is a definitional equality. PG\n-- Not planning to finish this, it looks quite horrible.\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Denotational.Notation\nopen import Syntactic.Types\nopen import Denotational.Values\nopen import Changes\nopen import ChangeContexts\nopen import Syntactic.Contexts Type\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Syntactic.Terms.Total\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4}\n  (\u03c1\u2081 : \u27e6 \u0393prefix \u27e7) (\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7)\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\n\nlift-term-doWeakenOne-ignore\u2032 \u0393prefix \u2205 \u03c1\u2081 \u2205 t = refl\n--This step gives the proof idea.\nlift-term-doWeakenOne-ignore\u2032 \u2205 (\u03c4 \u2022 \u0393rest) \u2205 (dv \u2022 v \u2022 \u03c1\u2082) t = lift-term-doWeakenOne-ignore\u2032 (\u03c4 \u2022 \u2205) \u0393rest (v \u2022 \u2205) \u03c1\u2082 t\n\nlift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix) (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} (v\u2080 \u2022 \u03c1\u2081) (dv \u2022 v \u2022 \u03c1\u2082) t = {!!}\n--Look at C-c C-, for the normalized goal.\n-- The solution here should be something like:\n  --lift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest {\u03c4} ((v\u2080 \u2022 \u03c1\u2081) \u222a (v \u2022 \u2205)) \u03c1\u2082  ?\n\n-- accompanied by enough subst to make it typecheck, and maybe by\n-- weakenOne-ignore to account for weakenOne - or maybe not since\n-- weakenOne is a definitional equality. PG\n-- Not planning to finish this, it looks quite horrible.\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e065ee84b66b6135e70cd2a8a066687e5e197f2b","subject":"Added missing page.","message":"Added missing page.\n\nIgnore-this: bac04ca73cdae041f7072d82094ef472\n\ndarcs-hash:20100713233743-3bd4e-7fe2d60923eadc0461010e43403451384bcae909.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/SortList\/SortList.agda","new_file":"Examples\/SortList\/SortList.agda","new_contents":"------------------------------------------------------------------------------\n-- Sort a list\n------------------------------------------------------------------------------\n\n-- This module define the program which sorts a list by converting it\n-- into an ordered tree and then back to a list (Burstall, 1969, pp. 45-46).\n\n-- R. M. Burstall. Proving properties of programs by structural\n-- induction. The Computer Journal, 12(1):41\u201348, 1969.\n\nmodule Examples.SortList.SortList where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Bool\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.List\n\n------------------------------------------------------------------------------\n-- Tree terms\npostulate\n  nilTree  : D\n  tip      : D \u2192 D\n  node     : D \u2192 D \u2192 D \u2192 D\n\n-- The LTC tree data type\ndata Tree : D \u2192 Set where\n  nilT  : Tree nilTree\n  tipT  : {i : D} \u2192 N i \u2192 Tree (tip i)\n  nodeT : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192 Tree (node t\u2081 i t\u2082)\n{-# ATP hint nilT #-}\n{-# ATP hint tipT #-}\n{-# ATP hint nodeT #-}\n\n------------------------------------------------------------------------------\n-- Inequalites on lists and trees\n\n-- Note from Burstall (p. 46): \"The relation \u2264 between lists is only an\n-- ordering if nil is excluded, similarly for trees. This is untidy but\n-- will not cause trouble.\"\n\npostulate\n  \u2264-ItemList    : D \u2192 D \u2192 D\n  \u2264-ItemList-[] : (item : D) \u2192 \u2264-ItemList item [] \u2261 true\n  \u2264-ItemList-\u2237  : (item i is : D) \u2192\n                  \u2264-ItemList item (i \u2237 is) \u2261 item \u2264 i &&\n                                             \u2264-ItemList item is\n{-# ATP axiom \u2264-ItemList-[] #-}\n{-# ATP axiom \u2264-ItemList-\u2237 #-}\n\nLE-ItemList : D \u2192 D \u2192 Set\nLE-ItemList item is = \u2264-ItemList item is \u2261 true\n{-# ATP definition LE-ItemList #-}\n\npostulate\n  \u2264-Lists    : D \u2192 D \u2192 D\n  \u2264-Lists-[] : (is : D) \u2192 \u2264-Lists [] is            \u2261 true\n  \u2264-Lists-\u2237  : (i is js : D) \u2192 \u2264-Lists (i \u2237 is) js \u2261 \u2264-ItemList i is &&\n                                                     \u2264-Lists is js\n{-# ATP axiom \u2264-Lists-[] #-}\n{-# ATP axiom \u2264-Lists-\u2237 #-}\n\nLE-Lists : D \u2192 D \u2192 Set\nLE-Lists is js = \u2264-Lists is js \u2261 true\n{-# ATP definition LE-Lists #-}\n\npostulate\n  \u2264-ItemTree          : D \u2192 D \u2192 D\n  \u2264-ItemTree-nilTree  : (item : D) \u2192 \u2264-ItemTree item nilTree   \u2261 true\n  \u2264-ItemTree-tip      : (item i : D) \u2192 \u2264-ItemTree item (tip i) \u2261 item \u2264 i\n  \u2264-ItemTree-node     : (item t\u2081 i t\u2082 : D) \u2192\n                        \u2264-ItemTree item (node t\u2081 i t\u2082) \u2261 \u2264-ItemTree item t\u2081 &&\n                                                         \u2264-ItemTree item t\u2082\n{-# ATP axiom \u2264-ItemTree-nilTree #-}\n{-# ATP axiom \u2264-ItemTree-tip #-}\n{-# ATP axiom \u2264-ItemTree-node #-}\n\nLE-ItemTree : D \u2192 D \u2192 Set\nLE-ItemTree item t = \u2264-ItemTree item t \u2261 true\n{-# ATP definition LE-ItemTree #-}\n\n-- No defined by Burstall.\npostulate\n  \u2264-TreeItem         : D \u2192 D \u2192 D\n  \u2264-TreeItem-nilTree : (item : D) \u2192 \u2264-TreeItem nilTree item   \u2261 true\n  \u2264-TreeItem-tip     : (i item : D) \u2192 \u2264-TreeItem (tip i) item \u2261 i \u2264 item\n  \u2264-TreeItem-node    : (t\u2081 i t\u2082 item : D) \u2192\n                       \u2264-TreeItem (node t\u2081 i t\u2082) item \u2261 \u2264-TreeItem t\u2081 item &&\n                                                        \u2264-TreeItem t\u2082 item\n{-# ATP axiom \u2264-TreeItem-nilTree #-}\n{-# ATP axiom \u2264-TreeItem-tip #-}\n{-# ATP axiom \u2264-TreeItem-node #-}\n\nLE-TreeItem : D \u2192 D \u2192 Set\nLE-TreeItem t item = \u2264-TreeItem t item \u2261 true\n{-# ATP definition LE-TreeItem #-}\n\n------------------------------------------------------------------------------\n-- Ordering functions and predicates on lists and trees\n\npostulate\n  isListOrd    : D \u2192 D\n  isListOrd-[] : isListOrd [] \u2261 true\n  isListOrd-\u2237  : (i is : D) \u2192 isListOrd (i \u2237 is)  \u2261 \u2264-ItemList i is &&\n                                                    isListOrd is\n{-# ATP axiom isListOrd-[] #-}\n{-# ATP axiom isListOrd-\u2237 #-}\n\npostulate\n  isTreeOrd         : D \u2192 D\n  isTreeOrd-nilTree : isTreeOrd nilTree           \u2261 true\n  isTreeOrd-tip     : (i : D) \u2192 isTreeOrd (tip i) \u2261 true\n  isTreeOrd-node    : (t\u2081 i t\u2082 : D) \u2192\n                      isTreeOrd (node t\u2081 i t\u2082)    \u2261 isTreeOrd t\u2081     &&\n                                                    isTreeOrd t\u2082     &&\n                                                    \u2264-TreeItem t\u2081 i  &&\n                                                    \u2264-ItemTree i t\u2082\n{-# ATP axiom isTreeOrd-nilTree #-}\n{-# ATP axiom isTreeOrd-tip #-}\n{-# ATP axiom isTreeOrd-node #-}\n\nTreeOrd : D \u2192 Set\nTreeOrd t = isTreeOrd t \u2261 true\n{-# ATP definition TreeOrd #-}\n\nListOrd : D \u2192 Set\nListOrd is = isListOrd is \u2261 true\n{-# ATP definition ListOrd #-}\n\n------------------------------------------------------------------------------\n-- The program\n\n-- The function toTree adds an item to a tree.\n\n-- The items have an ordering \u2264 defined over them. The item held at a\n-- node of the tree is chosen so that the left subtree has items not\n-- greater than it and the right subtree has items not less than it.\n\npostulate\n  toTree          : D\n  toTree-nilTree  : (item : D) \u2192 toTree \u2219 item \u2219 nilTree \u2261 tip item\n  toTree-tip      : (item i : D) \u2192 toTree \u2219 item \u2219 (tip i) \u2261\n                    if (i \u2264 item)\n                      then (node (tip i) item (tip item))\n                      else (node (tip item) i (tip i))\n  toTree-node     : (item : D) \u2192 (t\u2081 i t\u2082 : D) \u2192\n                    toTree \u2219 item \u2219 (node t\u2081 i t\u2082) \u2261\n                       if (i \u2264 item)\n                         then (node t\u2081 i (toTree \u2219 item \u2219 t\u2082))\n                         else (node (toTree \u2219 item \u2219 t\u2081) i t\u2082)\n{-# ATP axiom toTree-nilTree #-}\n{-# ATP axiom toTree-tip #-}\n{-# ATP axiom toTree-node #-}\n\n-- The function makeTree converts a list to a tree.\nmakeTree : D \u2192 D\nmakeTree is = foldr toTree nilTree is\n{-# ATP definition makeTree #-}\n\n-- The function flatten converts a tree to a list.\npostulate\n  flatten         : D \u2192 D\n  flatten-nilTree : flatten nilTree           \u2261 []\n  flatten-tip     : (i : D) \u2192 flatten (tip i) \u2261 i \u2237 []\n  flatten-node    : (t\u2081 i t\u2082 : D) \u2192\n                    flatten (node t\u2081 i t\u2082)    \u2261 flatten t\u2081 ++ flatten t\u2082\n{-# ATP axiom flatten-nilTree #-}\n{-# ATP axiom flatten-tip #-}\n{-# ATP axiom flatten-node #-}\n\n-- The function which sorts the list\nsort : D \u2192 D\nsort is = flatten (makeTree is)\n{-# ATP definition sort #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Sort a list\n------------------------------------------------------------------------------\n\n-- This module define the program which sorts a list by converting it\n-- into an ordered tree and then back to a list (Burstall, 1969, pp. 45).\n\n-- R. M. Burstall. Proving properties of programs by structural\n-- induction. The Computer Journal, 12(1):41\u201348, 1969.\n\nmodule Examples.SortList.SortList where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Bool\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.List\n\n------------------------------------------------------------------------------\n-- Tree terms\npostulate\n  nilTree  : D\n  tip      : D \u2192 D\n  node     : D \u2192 D \u2192 D \u2192 D\n\n-- The LTC tree data type\ndata Tree : D \u2192 Set where\n  nilT  : Tree nilTree\n  tipT  : {i : D} \u2192 N i \u2192 Tree (tip i)\n  nodeT : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192 Tree (node t\u2081 i t\u2082)\n{-# ATP hint nilT #-}\n{-# ATP hint tipT #-}\n{-# ATP hint nodeT #-}\n\n------------------------------------------------------------------------------\n-- Inequalites on lists and trees\n\n-- Note from Burstall (p. 46): \"The relation \u2264 between lists is only an\n-- ordering if nil is excluded, similarly for trees. This is untidy but\n-- will not cause trouble.\"\n\npostulate\n  \u2264-ItemList    : D \u2192 D \u2192 D\n  \u2264-ItemList-[] : (item : D) \u2192 \u2264-ItemList item [] \u2261 true\n  \u2264-ItemList-\u2237  : (item i is : D) \u2192\n                  \u2264-ItemList item (i \u2237 is) \u2261 item \u2264 i &&\n                                             \u2264-ItemList item is\n{-# ATP axiom \u2264-ItemList-[] #-}\n{-# ATP axiom \u2264-ItemList-\u2237 #-}\n\nLE-ItemList : D \u2192 D \u2192 Set\nLE-ItemList item is = \u2264-ItemList item is \u2261 true\n{-# ATP definition LE-ItemList #-}\n\npostulate\n  \u2264-Lists    : D \u2192 D \u2192 D\n  \u2264-Lists-[] : (is : D) \u2192 \u2264-Lists [] is            \u2261 true\n  \u2264-Lists-\u2237  : (i is js : D) \u2192 \u2264-Lists (i \u2237 is) js \u2261 \u2264-ItemList i is &&\n                                                     \u2264-Lists is js\n{-# ATP axiom \u2264-Lists-[] #-}\n{-# ATP axiom \u2264-Lists-\u2237 #-}\n\nLE-Lists : D \u2192 D \u2192 Set\nLE-Lists is js = \u2264-Lists is js \u2261 true\n{-# ATP definition LE-Lists #-}\n\npostulate\n  \u2264-ItemTree          : D \u2192 D \u2192 D\n  \u2264-ItemTree-nilTree  : (item : D) \u2192 \u2264-ItemTree item nilTree   \u2261 true\n  \u2264-ItemTree-tip      : (item i : D) \u2192 \u2264-ItemTree item (tip i) \u2261 item \u2264 i\n  \u2264-ItemTree-node     : (item t\u2081 i t\u2082 : D) \u2192\n                        \u2264-ItemTree item (node t\u2081 i t\u2082) \u2261 \u2264-ItemTree item t\u2081 &&\n                                                         \u2264-ItemTree item t\u2082\n{-# ATP axiom \u2264-ItemTree-nilTree #-}\n{-# ATP axiom \u2264-ItemTree-tip #-}\n{-# ATP axiom \u2264-ItemTree-node #-}\n\nLE-ItemTree : D \u2192 D \u2192 Set\nLE-ItemTree item t = \u2264-ItemTree item t \u2261 true\n{-# ATP definition LE-ItemTree #-}\n\n-- No defined by Burstall.\npostulate\n  \u2264-TreeItem         : D \u2192 D \u2192 D\n  \u2264-TreeItem-nilTree : (item : D) \u2192 \u2264-TreeItem nilTree item   \u2261 true\n  \u2264-TreeItem-tip     : (i item : D) \u2192 \u2264-TreeItem (tip i) item \u2261 i \u2264 item\n  \u2264-TreeItem-node    : (t\u2081 i t\u2082 item : D) \u2192\n                       \u2264-TreeItem (node t\u2081 i t\u2082) item \u2261 \u2264-TreeItem t\u2081 item &&\n                                                        \u2264-TreeItem t\u2082 item\n{-# ATP axiom \u2264-TreeItem-nilTree #-}\n{-# ATP axiom \u2264-TreeItem-tip #-}\n{-# ATP axiom \u2264-TreeItem-node #-}\n\nLE-TreeItem : D \u2192 D \u2192 Set\nLE-TreeItem t item = \u2264-TreeItem t item \u2261 true\n{-# ATP definition LE-TreeItem #-}\n\n------------------------------------------------------------------------------\n-- Ordering functions and predicates on lists and trees\n\npostulate\n  isListOrd    : D \u2192 D\n  isListOrd-[] : isListOrd [] \u2261 true\n  isListOrd-\u2237  : (i is : D) \u2192 isListOrd (i \u2237 is)  \u2261 \u2264-ItemList i is &&\n                                                    isListOrd is\n{-# ATP axiom isListOrd-[] #-}\n{-# ATP axiom isListOrd-\u2237 #-}\n\npostulate\n  isTreeOrd         : D \u2192 D\n  isTreeOrd-nilTree : isTreeOrd nilTree           \u2261 true\n  isTreeOrd-tip     : (i : D) \u2192 isTreeOrd (tip i) \u2261 true\n  isTreeOrd-node    : (t\u2081 i t\u2082 : D) \u2192\n                      isTreeOrd (node t\u2081 i t\u2082)    \u2261 isTreeOrd t\u2081     &&\n                                                    isTreeOrd t\u2082     &&\n                                                    \u2264-TreeItem t\u2081 i  &&\n                                                    \u2264-ItemTree i t\u2082\n{-# ATP axiom isTreeOrd-nilTree #-}\n{-# ATP axiom isTreeOrd-tip #-}\n{-# ATP axiom isTreeOrd-node #-}\n\nTreeOrd : D \u2192 Set\nTreeOrd t = isTreeOrd t \u2261 true\n{-# ATP definition TreeOrd #-}\n\nListOrd : D \u2192 Set\nListOrd is = isListOrd is \u2261 true\n{-# ATP definition ListOrd #-}\n\n------------------------------------------------------------------------------\n-- The program\n\n-- The function toTree adds an item to a tree.\n\n-- The items have an ordering \u2264 defined over them. The item held at a\n-- node of the tree is chosen so that the left subtree has items not\n-- greater than it and the right subtree has items not less than it.\n\npostulate\n  toTree          : D\n  toTree-nilTree  : (item : D) \u2192 toTree \u2219 item \u2219 nilTree \u2261 tip item\n  toTree-tip      : (item i : D) \u2192 toTree \u2219 item \u2219 (tip i) \u2261\n                    if (i \u2264 item)\n                      then (node (tip i) item (tip item))\n                      else (node (tip item) i (tip i))\n  toTree-node     : (item : D) \u2192 (t\u2081 i t\u2082 : D) \u2192\n                    toTree \u2219 item \u2219 (node t\u2081 i t\u2082) \u2261\n                       if (i \u2264 item)\n                         then (node t\u2081 i (toTree \u2219 item \u2219 t\u2082))\n                         else (node (toTree \u2219 item \u2219 t\u2081) i t\u2082)\n{-# ATP axiom toTree-nilTree #-}\n{-# ATP axiom toTree-tip #-}\n{-# ATP axiom toTree-node #-}\n\n-- The function makeTree converts a list to a tree.\nmakeTree : D \u2192 D\nmakeTree is = foldr toTree nilTree is\n{-# ATP definition makeTree #-}\n\n-- The function flatten converts a tree to a list.\npostulate\n  flatten         : D \u2192 D\n  flatten-nilTree : flatten nilTree           \u2261 []\n  flatten-tip     : (i : D) \u2192 flatten (tip i) \u2261 i \u2237 []\n  flatten-node    : (t\u2081 i t\u2082 : D) \u2192\n                    flatten (node t\u2081 i t\u2082)    \u2261 flatten t\u2081 ++ flatten t\u2082\n{-# ATP axiom flatten-nilTree #-}\n{-# ATP axiom flatten-tip #-}\n{-# ATP axiom flatten-node #-}\n\n-- The function which sorts the list\nsort : D \u2192 D\nsort is = flatten (makeTree is)\n{-# ATP definition sort #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e41ee7bc6a42348d7050206b8746f460ef632c69","subject":"Move ValidChange next to Change.","message":"Move ValidChange next to Change.\n\nThis commit prepares for using ValidChange in the definition of Change.\n\nOld-commit-hash: a90d85385de56e35cc3aba5821ddc4a2d3448247\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- Change : Type \u2192 Set\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : Change \u03c3) \u2192 valid v dv \u2192 Change \u03c4\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : Change \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  ignore-valid-change (cons v _ _) = v\n\n  update-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  update-valid-change {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-valid-change {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-valid-change {\u03c4})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : Change \u03c3) \u2192 valid v dv \u2192 Change \u03c4\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : Change \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  ignore-valid-change (cons v _ _) = v\n\n  update-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  update-valid-change {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-valid-change {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-valid-change {\u03c4})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c636948066782d65906b0e4053d4924db64e1149","subject":"!-involutive","message":"!-involutive\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin.NP as Fin using (Fin; suc; zero; [zero:_,suc:_])\nopen import Data.Vec as Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\n\ncoe\u00d7= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081){x y}\n      \u2192 coe (\u00d7= A= B=) (x , y) \u2261 (coe A= x , coe B= y)\ncoe\u00d7= idp idp = idp\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n    private\n      module A\u2243 = Equiv A\u2243\n      A\u2192 = A\u2243.\u00b7\u2192\n      A\u2190 = A\u2243.\u00b7\u2190\n      module B\u2243 = Equiv B\u2243\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n\n    {-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 A\u2192 B\u2192) (map\u00d7 A\u2190 B\u2190)\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-r x) ({!coe-\u03b2!} \u2219 B\u2243.\u00b7\u2190-inv-r y) })\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-l x) {!!} })\n    -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e A\u2192 B\u2192) (map\u228e A\u2190 B\u2190)\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-r x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-r ]\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-l x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-l ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 B\u2192 \u2218 f \u2218 A\u2190)\n               (\u03bb f \u2192 B\u2190 \u2218 f \u2218 A\u2192)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-r _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-r x))) \n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-l _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-l x)))\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2243 B\u2081 x) where\n  private\n      module B\u2243 {x} = Equiv (B x)\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n  \u03a3\u2243-second : (\u03a3 A B\u2080) \u2243 (\u03a3 A B\u2081)\n  \u03a3\u2243-second = equiv (second B\u2192) (second B\u2190)\n                    (\u03bb { (x , y) \u2192 ap (_,_ x) (B\u2243.\u00b7\u2190-inv-r y) })\n                    (\u03bb { (x , y) \u2192 ap (_,_ x) (B\u2243.\u00b7\u2190-inv-l y) })\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : \u2605_ b}(B : B\u2080 \u2243 B\u2081) where\n  \u00d7\u2243-second : (A \u00d7 B\u2080) \u2243 (A \u00d7 B\u2081)\n  \u00d7\u2243-second = \u03a3\u2243-second A (\u03bb _ \u2192 B)\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n  !\u03a3=\u2032 : ! (\u03a3=\u2032 A B) \u2261 \u03a3=\u2032 A (!_ \u2218 B)\n  !\u03a3=\u2032 = !-ap _ (\u03bb= B) \u2219 ap (ap (\u03a3 A)) (!-\u03bb= B)\n\ncoe\u03a3=\u2032-aux : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (ap (\u03bb f \u2192 f x) (\u03bb= B)) y)\ncoe\u03a3=\u2032-aux A B with \u03bb= B\ncoe\u03a3=\u2032-aux A B | idp = idp\n\ncoe\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (B x) y)\ncoe\u03a3=\u2032 A B = coe\u03a3=\u2032-aux A B \u2219 ap (_,_ _) (coe-same (happly (happly-\u03bb= B) _) _)\n\ncoe!\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe! (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe! (B x) y)\ncoe!\u03a3=\u2032 A B {x}{y} = coe-same (!\u03a3=\u2032 A B) _ \u2219 coe\u03a3=\u2032 A (!_ \u2218 B)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-involutive (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2243Lift\ud835\udfd9\u228e : Maybe A \u2243 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n  Maybe\u2243Lift\ud835\udfd9\u228e = equiv (maybe inr (inl _))\n                        [inl: const nothing ,inr: just ]\n                        [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                        (maybe (\u03bb _ \u2192 idp) idp)\n\n  Vec0\u2243\ud835\udfd9 : Vec A 0 \u2243 \ud835\udfd9\n  Vec0\u2243\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec0\u2243Lift\ud835\udfd9 : Vec A 0 \u2243 Lift {\u2113 = a} \ud835\udfd9\n  Vec0\u2243Lift\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec\u2218suc\u2243\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2243 (A \u00d7 Vec A n)\n  Vec\u2218suc\u2243\u00d7 = equiv (\u03bb { (x \u2237 xs) \u2192 x , xs }) (\u03bb { (x , xs) \u2192 x \u2237 xs })\n                    (\u03bb { (x , xs) \u2192 idp }) (\u03bb { (x \u2237 xs) \u2192 idp })\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua Maybe\u2243Lift\ud835\udfd9\u228e\n\n    Vec0\u2261Lift\ud835\udfd9 : Vec A 0 \u2261 Lift {\u2113 = a} \ud835\udfd9\n    Vec0\u2261Lift\ud835\udfd9 = ua Vec0\u2243Lift\ud835\udfd9\n\n    Vec\u2218suc\u2261\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2261 (A \u00d7 Vec A n)\n    Vec\u2218suc\u2261\u00d7 = ua Vec\u2218suc\u2243\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}} where\n    Vec0\u2261\ud835\udfd9 : Vec A 0 \u2261 \ud835\udfd9\n    Vec0\u2261\ud835\udfd9 = ua Vec0\u2243\ud835\udfd9\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nFin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv [zero: inl _ ,suc: inr ] [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n  [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n  [zero: idp ,suc: (\u03bb _ \u2192 idp) ]\n\nFin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin.NP as Fin using (Fin; suc; zero; [zero:_,suc:_])\nopen import Data.Vec as Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\n\ncoe\u00d7= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081){x y}\n      \u2192 coe (\u00d7= A= B=) (x , y) \u2261 (coe A= x , coe B= y)\ncoe\u00d7= idp idp = idp\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n    private\n      module A\u2243 = Equiv A\u2243\n      A\u2192 = A\u2243.\u00b7\u2192\n      A\u2190 = A\u2243.\u00b7\u2190\n      module B\u2243 = Equiv B\u2243\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n\n    {-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 A\u2192 B\u2192) (map\u00d7 A\u2190 B\u2190)\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-r x) ({!coe-\u03b2!} \u2219 B\u2243.\u00b7\u2190-inv-r y) })\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-l x) {!!} })\n    -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e A\u2192 B\u2192) (map\u228e A\u2190 B\u2190)\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-r x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-r ]\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-l x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-l ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 B\u2192 \u2218 f \u2218 A\u2190)\n               (\u03bb f \u2192 B\u2190 \u2218 f \u2218 A\u2192)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-r _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-r x))) \n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-l _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-l x)))\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2243 B\u2081 x) where\n  private\n      module B\u2243 {x} = Equiv (B x)\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n  \u03a3\u2243-second : (\u03a3 A B\u2080) \u2243 (\u03a3 A B\u2081)\n  \u03a3\u2243-second = equiv (second B\u2192) (second B\u2190)\n                    (\u03bb { (x , y) \u2192 ap (_,_ x) (B\u2243.\u00b7\u2190-inv-r y) })\n                    (\u03bb { (x , y) \u2192 ap (_,_ x) (B\u2243.\u00b7\u2190-inv-l y) })\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : \u2605_ b}(B : B\u2080 \u2243 B\u2081) where\n  \u00d7\u2243-second : (A \u00d7 B\u2080) \u2243 (A \u00d7 B\u2081)\n  \u00d7\u2243-second = \u03a3\u2243-second A (\u03bb _ \u2192 B)\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n  !\u03a3=\u2032 : ! (\u03a3=\u2032 A B) \u2261 \u03a3=\u2032 A (!_ \u2218 B)\n  !\u03a3=\u2032 = !-ap _ (\u03bb= B) \u2219 ap (ap (\u03a3 A)) (!-\u03bb= B)\n\ncoe\u03a3=\u2032-aux : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (ap (\u03bb f \u2192 f x) (\u03bb= B)) y)\ncoe\u03a3=\u2032-aux A B with \u03bb= B\ncoe\u03a3=\u2032-aux A B | idp = idp\n\ncoe\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (B x) y)\ncoe\u03a3=\u2032 A B = coe\u03a3=\u2032-aux A B \u2219 ap (_,_ _) (coe-same (happly (happly-\u03bb= B) _) _)\n\ncoe!\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe! (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe! (B x) y)\ncoe!\u03a3=\u2032 A B {x}{y} = coe-same (!\u03a3=\u2032 A B) _ \u2219 coe\u03a3=\u2032 A (!_ \u2218 B)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2243Lift\ud835\udfd9\u228e : Maybe A \u2243 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n  Maybe\u2243Lift\ud835\udfd9\u228e = equiv (maybe inr (inl _))\n                        [inl: const nothing ,inr: just ]\n                        [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                        (maybe (\u03bb _ \u2192 idp) idp)\n\n  Vec0\u2243\ud835\udfd9 : Vec A 0 \u2243 \ud835\udfd9\n  Vec0\u2243\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec0\u2243Lift\ud835\udfd9 : Vec A 0 \u2243 Lift {\u2113 = a} \ud835\udfd9\n  Vec0\u2243Lift\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec\u2218suc\u2243\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2243 (A \u00d7 Vec A n)\n  Vec\u2218suc\u2243\u00d7 = equiv (\u03bb { (x \u2237 xs) \u2192 x , xs }) (\u03bb { (x , xs) \u2192 x \u2237 xs })\n                    (\u03bb { (x , xs) \u2192 idp }) (\u03bb { (x \u2237 xs) \u2192 idp })\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua Maybe\u2243Lift\ud835\udfd9\u228e\n\n    Vec0\u2261Lift\ud835\udfd9 : Vec A 0 \u2261 Lift {\u2113 = a} \ud835\udfd9\n    Vec0\u2261Lift\ud835\udfd9 = ua Vec0\u2243Lift\ud835\udfd9\n\n    Vec\u2218suc\u2261\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2261 (A \u00d7 Vec A n)\n    Vec\u2218suc\u2261\u00d7 = ua Vec\u2218suc\u2243\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}} where\n    Vec0\u2261\ud835\udfd9 : Vec A 0 \u2261 \ud835\udfd9\n    Vec0\u2261\ud835\udfd9 = ua Vec0\u2243\ud835\udfd9\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nFin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv [zero: inl _ ,suc: inr ] [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n  [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n  [zero: idp ,suc: (\u03bb _ \u2192 idp) ]\n\nFin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6156984d33b98f39610ee7244b167632999b9b41","subject":"call derive-correct-closed instead of derive-correct in thm:main","message":"call derive-correct-closed instead of derive-correct in thm:main\n\nOld-commit-hash: 9fa1103b0ac110498ae8ab0d0c0f3320ff4b264f\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Correctness.agda","new_file":"Parametric\/Change\/Correctness.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  derive-correct-closed : \u2200 {\u03c4} (t : Term \u2205 \u03c4) \u2192\n    \u27e6 t \u27e7\u0394 \u2205 \u2205 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 \u2205\n\n  derive-correct-closed t = derive-correct t \u2205 \u2205 \u2205 \u2205\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {s : Term \u2205 \u03c3} {ds : Term \u2205 (\u0394Type \u03c3)} \u2192\n    {dv : \u0394\u208d \u03c3 \u208e (\u27e6 s \u27e7 \u2205)} {erasure : dv \u2248\u208d \u03c3 \u208e (\u27e6 ds \u27e7 \u2205)} \u2192\n\n    \u27e6 app f (s \u2295\u208d \u03c3 \u208e ds) \u27e7 \u2261 \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {s} {ds} {dv} {erasure} =\n    let\n      g  = \u27e6 f \u27e7 \u2205\n      \u0394g = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394g\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 s \u27e7 \u2205\n      dv\u2032 = \u27e6 ds \u27e7 \u2205\n      u  = \u27e6 s \u2295\u208d \u03c3 \u208e ds \u27e7 \u2205\n      -- \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        g u\n      \u2261\u27e8 cong g (sym (meaning-\u2295 {t = s} {\u0394t = ds})) \u27e9\n        g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 dv\u2032)\n      \u2261\u27e8 cong g (sym (carry-over {\u03c3} dv erasure)) \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v dv \u27e9\n        g v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394g v dv\n      \u2261\u27e8 carry-over {\u03c4} (call-change {\u03c3} {\u03c4} \u0394g v dv)\n           (derive-correct-closed f v dv dv\u2032 erasure) \u27e9\n        g v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394g\u2032 v dv\u2032\n      \u2261\u27e8 meaning-\u2295 {t = app f s} {\u0394t = app (app (derive f) s) ds} \u27e9\n        \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  derive-correct-closed : \u2200 {\u03c4} (t : Term \u2205 \u03c4) \u2192\n    \u27e6 t \u27e7\u0394 \u2205 \u2205 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 \u2205\n\n  derive-correct-closed t = derive-correct t \u2205 \u2205 \u2205 \u2205\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {s : Term \u2205 \u03c3} {ds : Term \u2205 (\u0394Type \u03c3)} \u2192\n    {dv : \u0394\u208d \u03c3 \u208e (\u27e6 s \u27e7 \u2205)} {erasure : dv \u2248\u208d \u03c3 \u208e (\u27e6 ds \u27e7 \u2205)} \u2192\n\n    \u27e6 app f (s \u2295\u208d \u03c3 \u208e ds) \u27e7 \u2261 \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {s} {ds} {dv} {erasure} =\n    let\n      g  = \u27e6 f \u27e7 \u2205\n      \u0394g = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394g\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 s \u27e7 \u2205\n      dv\u2032 = \u27e6 ds \u27e7 \u2205\n      u  = \u27e6 s \u2295\u208d \u03c3 \u208e ds \u27e7 \u2205\n      -- \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        g u\n      \u2261\u27e8 cong g (sym (meaning-\u2295 {t = s} {\u0394t = ds})) \u27e9\n        g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 dv\u2032)\n      \u2261\u27e8 cong g (sym (carry-over {\u03c3} dv erasure)) \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v dv \u27e9\n        g v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394g v dv\n      \u2261\u27e8 carry-over {\u03c4} (call-change {\u03c3} {\u03c4} \u0394g v dv)\n           (derive-correct f \u2205 \u2205 \u2205 \u2205 v dv dv\u2032 erasure) \u27e9\n        g v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394g\u2032 v dv\u2032\n      \u2261\u27e8 meaning-\u2295 {t = app f s} {\u0394t = app (app (derive f) s) ds} \u27e9\n        \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4685f973de55e831581e37855f5ccf50d4510103","subject":"Removed duplicate local hint.","message":"Removed duplicate local hint.\n\nIgnore-this: 838f421d4dc03598b2756e6d265c972b\n\ndarcs-hash:20120525042628-3bd4e-465be1ef063ffe058d8595a1f9e7fc45dd94da44.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/SortList\/PropertiesATP.agda","new_file":"src\/FOTC\/Program\/SortList\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties stated in the Burstall's paper\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.SortList.PropertiesATP where\n\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.List.Type\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.List\nopen import FOTC.Program.SortList.Properties.Totality.BoolATP\nopen import FOTC.Program.SortList.Properties.Totality.ListN-ATP\nopen import FOTC.Program.SortList.Properties.Totality.OrdList.FlattenATP\nopen import FOTC.Program.SortList.Properties.Totality.OrdListATP\nopen import FOTC.Program.SortList.Properties.Totality.OrdTreeATP\nopen import FOTC.Program.SortList.Properties.Totality.TreeATP\nopen import FOTC.Program.SortList.SortList\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: If t is ordered then totree(i, t) is ordered.\ntoTree-OrdTree : \u2200 {item t} \u2192 N item \u2192 Tree t \u2192 OrdTree t \u2192\n                 OrdTree (toTree \u00b7 item \u00b7 t)\ntoTree-OrdTree {item} Nitem nilT OTt = prf\n  where\n  postulate prf : OrdTree (toTree \u00b7 item \u00b7 nilTree)\n  {-# ATP prove prf #-}\n\ntoTree-OrdTree {item} Nitem (tipT {i} Ni) OTt =\n  [ prf\u2081 , prf\u2082 ] (x>y\u2228x\u2264y Ni Nitem)\n  where\n  postulate prf\u2081 : GT i item \u2192 OrdTree (toTree \u00b7 item \u00b7 tip i)\n  {-# ATP prove prf\u2081 x\u2264x x<y\u2192x\u2264y x>y\u2192x\u2270y #-}\n\n  postulate prf\u2082 : LE i item \u2192 OrdTree (toTree \u00b7 item \u00b7 tip i)\n  {-# ATP prove prf\u2082 x\u2264x #-}\n\ntoTree-OrdTree {item} Nitem (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082) OTnodeT =\n  [ prf\u2081 (toTree-OrdTree Nitem Tt\u2081 (leftSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT))\n         (rightSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT)\n  , prf\u2082 (toTree-OrdTree Nitem Tt\u2082 (rightSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT))\n         (leftSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT)\n  ] (x>y\u2228x\u2264y Ni Nitem)\n  where\n  postulate prf\u2081 : ordTree (toTree \u00b7 item \u00b7 t\u2081) \u2261 true \u2192  -- IH.\n                   OrdTree t\u2082 \u2192\n                   GT i item \u2192\n                   OrdTree (toTree \u00b7 item \u00b7 node t\u2081 i t\u2082)\n  {-# ATP prove prf\u2081 \u2264-ItemTree-Bool \u2264-TreeItem-Bool ordTree-Bool\n                     x>y\u2192x\u2270y &&\u2083-proj\u2083 &&\u2083-proj\u2084\n                     toTree-OrdTree-helper\u2081\n  #-}\n\n  postulate prf\u2082 : ordTree (toTree \u00b7 item \u00b7 t\u2082) \u2261 true \u2192 -- IH.\n                   OrdTree t\u2081 \u2192\n                   LE i item \u2192\n                   OrdTree (toTree \u00b7 item \u00b7 node t\u2081 i t\u2082)\n  {-# ATP prove prf\u2082 \u2264-ItemTree-Bool \u2264-TreeItem-Bool ordTree-Bool\n                     &&\u2083-proj\u2083 &&\u2083-proj\u2084\n                     toTree-OrdTree-helper\u2082\n  #-}\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: ord(maketree(is)).\n\n-- makeTree-TreeOrd : \u2200 {is} \u2192 ListN is \u2192 OrdTree (makeTree is)\n-- makeTree-TreeOrd LNis =\n--   ind-lit OrdTree toTree nilTree LNis ordTree-nilTree\n--           (\u03bb Nx y TOy \u2192 toTree-OrdTree Nx {!!} TOy)\n\nmakeTree-OrdTree : \u2200 {is} \u2192 ListN is \u2192 OrdTree (makeTree is)\nmakeTree-OrdTree nilLN = prf\n  where\n  postulate prf : OrdTree (makeTree [])\n  {-# ATP prove prf #-}\n\nmakeTree-OrdTree (consLN {i} {is} Ni Lis) = prf $ makeTree-OrdTree Lis\n  where\n  postulate prf : OrdTree (makeTree is) \u2192  -- IH.\n                  OrdTree (makeTree (i \u2237 is))\n  {-# ATP prove prf makeTree-Tree toTree-OrdTree #-}\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: If ord(is1) and ord(is2) and is1 \u2264 is2 then\n-- ord(concat(is1, is2)).\n++-OrdList : \u2200 {is js} \u2192 ListN is \u2192 ListN js \u2192 OrdList is \u2192 OrdList js \u2192\n             LE-Lists is js \u2192 OrdList (is ++ js)\n\n++-OrdList {js = js} nilLN LNjs LOis LOjs is\u2264js =\n  subst (\u03bb t \u2192 OrdList t) (sym $ ++-[] js) LOjs\n\n++-OrdList {js = js} (consLN {i} {is} Ni LNis) LNjs OLi\u2237is OLjs i\u2237is\u2264js =\n  subst (\u03bb t \u2192 OrdList t)\n        (sym $ ++-\u2237 i is js)\n        (lemma (++-OrdList LNis LNjs\n                           (subList-OrdList Ni LNis OLi\u2237is)\n                           OLjs\n                           (&&-proj\u2082 (\u2264-ItemList-Bool Ni LNjs)\n                                     (\u2264-Lists-Bool LNis LNjs)\n                                     (trans (sym $ \u2264-Lists-\u2237 i is js) i\u2237is\u2264js))))\n  where\n  postulate lemma : OrdList (is ++ js) \u2192  -- IH\n                    OrdList (i \u2237 is ++ js)\n  {-# ATP prove lemma \u2264-ItemList-Bool \u2264-Lists-Bool ordList-Bool\n                      &&-proj\u2081 &&-proj\u2082\n                      ++-OrdList-helper\n  #-}\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: If t is ordered then (flatten t) is ordered.\nflatten-OrdList : \u2200 {t} \u2192 Tree t \u2192 OrdTree t \u2192 OrdList (flatten t)\nflatten-OrdList nilT OTt =\n  subst (\u03bb t \u2192 OrdList t) (sym flatten-nilTree) ordList-[]\n\nflatten-OrdList (tipT {i} Ni) OTt = prf\n  where\n  postulate prf : OrdList (flatten (tip i))\n  -- {-# ATP prove prf #-}\n\nflatten-OrdList (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082) OTt\n  = prf (++-OrdList (flatten-ListN Tt\u2081)\n                    (flatten-ListN Tt\u2082)\n                    (flatten-OrdList Tt\u2081 (leftSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTt)) -- IH.\n                    (flatten-OrdList Tt\u2082 (rightSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTt))-- IH.\n                    (flatten-OrdList-helper Tt\u2081 Ni Tt\u2082 OTt))\n  where\n  postulate prf : OrdList (flatten t\u2081 ++ flatten t\u2082) \u2192 -- Indirect IH.\n                  OrdList (flatten (node t\u2081 i t\u2082))\n  {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties stated in the Burstall's paper\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.SortList.PropertiesATP where\n\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.List.Type\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.List\nopen import FOTC.Program.SortList.Properties.Totality.BoolATP\nopen import FOTC.Program.SortList.Properties.Totality.ListN-ATP\nopen import FOTC.Program.SortList.Properties.Totality.OrdList.FlattenATP\nopen import FOTC.Program.SortList.Properties.Totality.OrdListATP\nopen import FOTC.Program.SortList.Properties.Totality.OrdTreeATP\nopen import FOTC.Program.SortList.Properties.Totality.TreeATP\nopen import FOTC.Program.SortList.SortList\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: If t is ordered then totree(i, t) is ordered.\ntoTree-OrdTree : \u2200 {item t} \u2192 N item \u2192 Tree t \u2192 OrdTree t \u2192\n                 OrdTree (toTree \u00b7 item \u00b7 t)\ntoTree-OrdTree {item} Nitem nilT OTt = prf\n  where\n  postulate prf : OrdTree (toTree \u00b7 item \u00b7 nilTree)\n  {-# ATP prove prf #-}\n\ntoTree-OrdTree {item} Nitem (tipT {i} Ni) OTt =\n  [ prf\u2081 , prf\u2082 ] (x>y\u2228x\u2264y Ni Nitem)\n  where\n  postulate prf\u2081 : GT i item \u2192 OrdTree (toTree \u00b7 item \u00b7 tip i)\n  {-# ATP prove prf\u2081 x\u2264x x<y\u2192x\u2264y x>y\u2192x\u2270y #-}\n\n  postulate prf\u2082 : LE i item \u2192 OrdTree (toTree \u00b7 item \u00b7 tip i)\n  {-# ATP prove prf\u2082 x\u2264x #-}\n\ntoTree-OrdTree {item} Nitem (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082) OTnodeT =\n  [ prf\u2081 (toTree-OrdTree Nitem Tt\u2081 (leftSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT))\n         (rightSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT)\n  , prf\u2082 (toTree-OrdTree Nitem Tt\u2082 (rightSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT))\n         (leftSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTnodeT)\n  ] (x>y\u2228x\u2264y Ni Nitem)\n  where\n  postulate prf\u2081 : ordTree (toTree \u00b7 item \u00b7 t\u2081) \u2261 true \u2192  -- IH.\n                   OrdTree t\u2082 \u2192\n                   GT i item \u2192\n                   OrdTree (toTree \u00b7 item \u00b7 node t\u2081 i t\u2082)\n  {-# ATP prove prf\u2081 \u2264-ItemTree-Bool \u2264-TreeItem-Bool ordTree-Bool\n                     x>y\u2192x\u2270y &&\u2083-proj\u2083 &&\u2083-proj\u2084\n                     ordTree-Bool toTree-OrdTree-helper\u2081\n  #-}\n\n  postulate prf\u2082 : ordTree (toTree \u00b7 item \u00b7 t\u2082) \u2261 true \u2192 -- IH.\n                   OrdTree t\u2081 \u2192\n                   LE i item \u2192\n                   OrdTree (toTree \u00b7 item \u00b7 node t\u2081 i t\u2082)\n  {-# ATP prove prf\u2082 \u2264-ItemTree-Bool \u2264-TreeItem-Bool ordTree-Bool\n                     &&\u2083-proj\u2083 &&\u2083-proj\u2084\n                     toTree-OrdTree-helper\u2082\n  #-}\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: ord(maketree(is)).\n\n-- makeTree-TreeOrd : \u2200 {is} \u2192 ListN is \u2192 OrdTree (makeTree is)\n-- makeTree-TreeOrd LNis =\n--   ind-lit OrdTree toTree nilTree LNis ordTree-nilTree\n--           (\u03bb Nx y TOy \u2192 toTree-OrdTree Nx {!!} TOy)\n\nmakeTree-OrdTree : \u2200 {is} \u2192 ListN is \u2192 OrdTree (makeTree is)\nmakeTree-OrdTree nilLN = prf\n  where\n  postulate prf : OrdTree (makeTree [])\n  {-# ATP prove prf #-}\n\nmakeTree-OrdTree (consLN {i} {is} Ni Lis) = prf $ makeTree-OrdTree Lis\n  where\n  postulate prf : OrdTree (makeTree is) \u2192  -- IH.\n                  OrdTree (makeTree (i \u2237 is))\n  {-# ATP prove prf makeTree-Tree toTree-OrdTree #-}\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: If ord(is1) and ord(is2) and is1 \u2264 is2 then\n-- ord(concat(is1, is2)).\n++-OrdList : \u2200 {is js} \u2192 ListN is \u2192 ListN js \u2192 OrdList is \u2192 OrdList js \u2192\n             LE-Lists is js \u2192 OrdList (is ++ js)\n\n++-OrdList {js = js} nilLN LNjs LOis LOjs is\u2264js =\n  subst (\u03bb t \u2192 OrdList t) (sym $ ++-[] js) LOjs\n\n++-OrdList {js = js} (consLN {i} {is} Ni LNis) LNjs OLi\u2237is OLjs i\u2237is\u2264js =\n  subst (\u03bb t \u2192 OrdList t)\n        (sym $ ++-\u2237 i is js)\n        (lemma (++-OrdList LNis LNjs\n                           (subList-OrdList Ni LNis OLi\u2237is)\n                           OLjs\n                           (&&-proj\u2082 (\u2264-ItemList-Bool Ni LNjs)\n                                     (\u2264-Lists-Bool LNis LNjs)\n                                     (trans (sym $ \u2264-Lists-\u2237 i is js) i\u2237is\u2264js))))\n  where\n  postulate lemma : OrdList (is ++ js) \u2192  -- IH\n                    OrdList (i \u2237 is ++ js)\n  {-# ATP prove lemma \u2264-ItemList-Bool \u2264-Lists-Bool ordList-Bool\n                      &&-proj\u2081 &&-proj\u2082\n                      ++-OrdList-helper\n  #-}\n\n------------------------------------------------------------------------------\n-- Burstall's lemma: If t is ordered then (flatten t) is ordered.\nflatten-OrdList : \u2200 {t} \u2192 Tree t \u2192 OrdTree t \u2192 OrdList (flatten t)\nflatten-OrdList nilT OTt =\n  subst (\u03bb t \u2192 OrdList t) (sym flatten-nilTree) ordList-[]\n\nflatten-OrdList (tipT {i} Ni) OTt = prf\n  where\n  postulate prf : OrdList (flatten (tip i))\n  -- {-# ATP prove prf #-}\n\nflatten-OrdList (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082) OTt\n  = prf (++-OrdList (flatten-ListN Tt\u2081)\n                    (flatten-ListN Tt\u2082)\n                    (flatten-OrdList Tt\u2081 (leftSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTt)) -- IH.\n                    (flatten-OrdList Tt\u2082 (rightSubTree-OrdTree Tt\u2081 Ni Tt\u2082 OTt))-- IH.\n                    (flatten-OrdList-helper Tt\u2081 Ni Tt\u2082 OTt))\n  where\n  postulate prf : OrdList (flatten t\u2081 ++ flatten t\u2082) \u2192 -- Indirect IH.\n                  OrdList (flatten (node t\u2081 i t\u2082))\n  {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7663dd877139afe7e0c37261dc74162d7ed7056c","subject":"IDesc: implement induction with all","message":"IDesc: implement induction with all","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const X\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> (xs : desc D X) -> desc (All D X xs) R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = k\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (All D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n{-\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (All (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = induction R P m i xs\n-}\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (All (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (All (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (All D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = refl\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (pil S) \n                                                                           (reflFun (\\ s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                              (reflFun (\u03bb s -> psi (phi (T s)))\n                                                                                       T\n                                                                                       hs)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> desc (R i) T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = induction R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : desc D (IMu R))\n                  (ms : desc (box D (IMu R) xs) (\\it -> T (fst it))) -> \n                  desc D T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Meta-language\n--********************************\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n\n\n--********************************\n-- Typed expressions\n--********************************\n\nexprFixMenu : (Type -> Set) -> FixMenu Type\nexprFixMenu constt = ( consE (consE nilE) , \n                       \\ty -> (const (constt ty),\n                              (prod (var bool) (prod (var ty) (var ty)), \n                               Void)))\n\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\nexpr : (Type -> Set) -> TagIDesc Type\nexpr constt = exprFixMenu constt , exprSensitiveMenu\n\nexprIDesc : (Type -> Set) -> ((Type -> Set) -> TagIDesc Type) -> (Type -> IDesc Type)\nexprIDesc constt D = toIDesc Type  (D constt)\n\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc Val expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : (ty : Type) -> IMu closeTerm ty -> Val ty\neval ty term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> desc (closeTerm ty) Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n\n\n--********************************\n-- Open terms\n--********************************\n\nVar : EnumU -> Type -> Set\nVar dom _ = EnumT dom\n\nopenTerm : EnumU -> Type -> IDesc Type\nopenTerm dom = exprIDesc (\\ty -> Val ty + Var dom ty) expr\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\ndischarge : (ty : Type)\n          (vars : EnumU)\n          (context : spi vars (\\_ -> IMu closeTerm ty)) ->\n          (Val ty + Var vars ty) -> \n          IMu closeTerm ty\ndischarge ty vars context (l value) = con ( EZe , value )\ndischarge ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm ty) context variable \n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        desc (toIDesc I (R ** X) i) (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), \n                           Void))\n\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (exprFree ** Val)\n\nopenTerm' : EnumU -> Type -> IDesc Type\nopenTerm' dom = toIDesc Type (exprFree ** (\\ty -> Val ty + Var dom ty))\n\n--********************************\n-- Evaluation of open terms'\n--********************************\n\ndischarge' : (ty : Type)\n            (vars : EnumU)\n            (context : spi vars (\\_ -> IMu closeTerm' ty)) ->\n            (Val ty + Var vars ty) -> \n            IMu closeTerm' ty\ndischarge' ty vars context (l value) = con (EZe , value) \ndischarge' ty vars context (r variable) = switch vars (\\_ -> IMu closeTerm' ty) context variable\n\nchooseDom : (ty : Type)(domNat domBool : EnumU) -> EnumU\nchooseDom nat domNat _ = domNat\nchooseDom bool _ domBool = domBool\n\nchooseGamma : (ty : Type)\n              (dom : EnumU)\n              (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n              (gammaBool : spi dom (\\_ -> IMu closeTerm' bool)) ->\n              spi dom (\\_ -> IMu closeTerm' ty)\nchooseGamma nat dom gammaNat gammaBool = gammaNat\nchooseGamma bool dom gammaNat gammaBool = gammaBool\n\nsubstExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (sigma : (dom : EnumU)\n                     (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n                     (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n                     (ty : Type) ->\n                     (Val ty + Var dom ty) ->\n                     IMu closeTerm' ty)\n            (ty : Type) ->\n            IMu (openTerm' dom) ty ->\n            IMu closeTerm' ty\nsubstExpr dom gammaNat gammaBool sig ty term = \n    substI (\\ty -> Val ty + Var dom ty)\n           Val\n           exprFree\n           (sig dom gammaNat gammaBool)\n           ty\n           term\n\n\nsigmaExpr : (dom : EnumU)\n            (gammaNat : spi dom (\\_ -> IMu closeTerm' nat))\n            (gammaBool : spi dom (\\_ -> IMu closeTerm' bool))\n            (ty : Type) ->\n            (Val ty + Var dom ty) ->\n            IMu closeTerm' ty\nsigmaExpr dom gammaNat gammaBool ty v = \n    discharge' ty\n               dom\n               (chooseGamma ty dom gammaNat gammaBool)\n               v\n\ntest : IMu (openTerm' (consE (consE nilE))) nat ->\n       IMu closeTerm' nat\ntest term = substExpr (consE (consE nilE)) \n                      ( con ( EZe , ze ) , ( con (EZe , su ze ) , Void )) \n                      ( con ( EZe , true ) , ( con ( EZe , false ) , Void ))\n                      sigmaExpr nat term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e85a9b2ad2f0774039fafcf09bd6c99f26c4cb12","subject":"bintree: more sorting...","message":"bintree: more sorting...\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nmodule Sorted {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : (x : A) \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low\u209c high\u209c low\u1d64 high\u1d64} \u2192\n             Sorted t low\u209c high\u209c \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             high\u209c \u2264\u1d2c low\u1d64 \u2192\n             Sorted (fork t u) low\u209c high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds (leaf ._)       = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high\u209c = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high\u209c = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n","old_contents":"module prefect-bintree where\n\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_)\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n\n{-\nmodule Sorting {a} {A : Set a} (sort\u1d2c : A \u00d7 A \u2192 A \u00d7 A) where\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    = case sort\u1d2c (x\u2080 , x\u2081) of (\u03bb { (y\u2080 , y\u2081) \u2192 fork (leaf y\u2080) (leaf y\u2081) })\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\nswap-\u00d7 : \u2200 {n} \u2192 Bits n \u00d7 Bits n \u2192 Bits n \u00d7 Bits n\nswap-\u00d7 (x , y) = ?\n\nmodule BitsSorting m where\n\n    module S = Sorting (swap-\u00d7 {m})\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    lem : \u2200 {n} (t : Tree (Bits n) n) \u2192 toFun (sort n t) \u2257 id\n    lem = ?\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fff6da15ad470038d849417d4765f8cb6ea827de","subject":"Fix the definition of [_\u2194_] using 0\u2194_","message":"Fix the definition of [_\u2194_] using 0\u2194_\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\n0\u2194\u2032_ : \u2200 {n} \u2192 Bits n \u2192 SwpOp n\n0\u2194\u2032_ {n} rewrite cong SwpOp (sym (\u2115\u00b0.+-comm n 0)) = 0\u2194_ {n} {0}\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = 0\u2194\u2032 p \u204f 0\u2194\u2032 q\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"88a46de4b9496c0c6c4f861d964d90e3ae3cfd27","subject":"Update qualified import for new module name.","message":"Update qualified import for new module name.\n\nOld-commit-hash: 2cb47a8d412768ed582e8af1a042b51c3ca781be\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Derive\/Plotkin.agda","new_file":"Syntax\/Derive\/Plotkin.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\nimport Syntax.Term.Plotkin as Term\nimport Syntax.DeltaContext as DeltaContext\nimport Parametric.Change.Type as ChangeType\n\nmodule Syntax.Derive.Plotkin\n    {Base : Set {- of base types -}}\n    {Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -}}\n    (\u0394Base : Base \u2192 Base)\n    (deriveConst :\n      \u2200 {\u0393 \u03a3 \u03c4} \u2192 Constant \u03a3 \u03c4 \u2192\n      Term.Terms\n        Constant\n        (DeltaContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u0393)\n        (DeltaContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u03a3) \u2192\n      Term.Term Constant (DeltaContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u0393) (ChangeType.\u0394Type \u0394Base \u03c4))\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen Type Base\nopen Context Type\n\nopen import Syntax.Context.Plotkin Base\n\nopen Term Constant\nopen ChangeType \u0394Base\nopen DeltaContext \u0394Type\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394Context \u0393) (\u0394Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) (\u0394Context \u03a3)\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\nderive-terms {\u2205}    \u2205 = \u2205\nderive-terms {\u03c4 \u2022 \u03a3} (t \u2022 ts) = derive t \u2022 fit t \u2022 derive-terms ts\n\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\nderive (const c ts) = deriveConst c (derive-terms ts)\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\nimport Syntax.Term.Plotkin as Term\nimport Syntax.DeltaContext as DeltaContext\nimport Parametric.Change.Type as DeltaType\n\nmodule Syntax.Derive.Plotkin\n    {Base : Set {- of base types -}}\n    {Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -}}\n    (\u0394Base : Base \u2192 Base)\n    (deriveConst :\n      \u2200 {\u0393 \u03a3 \u03c4} \u2192 Constant \u03a3 \u03c4 \u2192\n      Term.Terms\n        Constant\n        (DeltaContext.\u0394Context (DeltaType.\u0394Type \u0394Base) \u0393)\n        (DeltaContext.\u0394Context (DeltaType.\u0394Type \u0394Base) \u03a3) \u2192\n      Term.Term Constant (DeltaContext.\u0394Context (DeltaType.\u0394Type \u0394Base) \u0393) (DeltaType.\u0394Type \u0394Base \u03c4))\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen Type Base\nopen Context Type\n\nopen import Syntax.Context.Plotkin Base\n\nopen Term Constant\nopen DeltaType \u0394Base\nopen DeltaContext \u0394Type\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394Context \u0393) (\u0394Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) (\u0394Context \u03a3)\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\nderive-terms {\u2205}    \u2205 = \u2205\nderive-terms {\u03c4 \u2022 \u03a3} (t \u2022 ts) = derive t \u2022 fit t \u2022 derive-terms ts\n\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\nderive (const c ts) = deriveConst c (derive-terms ts)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"896361b9ae7fa70f12ddf791600acc61f394b002","subject":"Field","message":"Field\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_ 2*_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  2\u1da0 3\u1da0 -0\u1da0 -1\u1da0 : A\n  2\u1da0  = suc\u1da0 1\u1da0\n  3\u1da0  = suc\u1da0 2\u1da0\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0     ] = 0\u1da0\n  \u2115[ suc n ] = fold 1\u1da0 suc\u1da0 n\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0       = 1\u1da0\n  b ^\u2115 (suc e) = fold b (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\n  2+_ : A \u2192 A\n  2+ x = 2\u1da0 + x\n\n  2*_ : A \u2192 A\n  2* x = x + x\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x * x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 * x\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n  open \u2261-Reasoning\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : LeftInverseNonZero 0\u1da0 1\u1da0 _\u207b\u00b9 _*_\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse : RightInverseNonZero 0\u1da0 1\u1da0 _\u207b\u00b9 _*_\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  open FromOp\u2082 _+_ renaming ( op= to += )\n  open FromOp\u2082 _*_ renaming ( op= to *= )\n  open FromOp\u2082 _\u2212_ renaming ( op= to \u2212= )\n  open FromOp\u2082 _\/_ renaming ( op= to \/= )\n\n  open FromAssocComm _+_ +-assoc +-comm\n    renaming ( comm=       to +-comm=\n             ; assoc=      to +-assoc=\n             ; !assoc=     to +-!assoc=\n             ; assoc-comm  to +-assoc-comm\n             ; !assoc-comm to +-!assoc-comm\n             ; interchange to +-interchange\n             )\n\n  open FromAssocComm _*_ *-assoc *-comm\n    renaming ( comm=       to *-comm=\n             ; assoc=      to *-assoc=\n             ; !assoc=     to *-!assoc=\n             ; assoc-comm  to *-assoc-comm\n             ; !assoc-comm to *-!assoc-comm\n             ; interchange to *-interchange\n             )\n\n  0-= : \u2200 {x x'} \u2192 x \u2261 x' \u2192 0- x \u2261 0- x'\n  0-= = ap 0-_\n\n  +-*-distr : _*_ DistributesOver\u02b3 _+_\n  +-*-distr = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  +-\/-distr : _\/_ DistributesOver\u02b3 _+_\n  +-\/-distr = +-*-distr\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-!assoc= p \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  0\u2262-1 : 0\u1da0 \u2262 -1\u1da0\n  0\u2262-1 0\u2261-1 = 0\u22621 (! 0--inverse \u2219 +-comm \u2219 ! ap suc\u1da0 0\u2261-1 \u2219 +0-identity)\n\n  c\u207b\u00b9*c*x : \u2200 {c x} \u2192 c \u2262 0\u1da0 \u2192 c \u207b\u00b9 * c * x \u2261 x\n  c\u207b\u00b9*c*x c\u22620 = *= (\u207b\u00b9-inverse c\u22620) refl \u2219 1*-identity\n\n  *-left-cancel : LeftCancelNonZero 0\u1da0 _*_\n  *-left-cancel c\u22620 p = ! c\u207b\u00b9*c*x c\u22620 \u2219 *-!assoc= p \u2219 c\u207b\u00b9*c*x c\u22620\n\n  *-right-cancel : RightCancelNonZero 0\u1da0 _*_\n  *-right-cancel c\u22620 p = *-left-cancel c\u22620 (*-comm= p)\n\n  *0-zero : RightZero 0\u1da0 _*_\n  *0-zero = +-right-cancel  (+= refl (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= refl 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero 0\u1da0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  \u207b\u00b9-notZero : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 \u2262 0\u1da0\n  \u207b\u00b9-notZero x\/=0 = \u03bb p \u2192 0\u22621 (! 0*-zero \u2219 *= (! p) refl \u2219 \u207b\u00b9-inverse x\/=0)\n\n  0--*-distr : \u2200 {x y} \u2192 0- (x * y) \u2261 (0- x) * y\n  0--*-distr = +-right-cancel (0--inverse \u2219 ! 0*-zero \u2219 *= (! 0--inverse) refl \u2219 +-*-distr)\n\n  -1*-neg : \u2200 {x} \u2192 -1\u1da0 * x \u2261 0- x\n  -1*-neg = ! 0--*-distr \u2219 0-= 1*-identity\n\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n\n  1\u207b\u00b9\u22611 : 1\u1da0 \u207b\u00b9 \u2261 1\u1da0\n  1\u207b\u00b9\u22611 = ! *1-identity \u2219 \u207b\u00b9-inverse (\u03bb p \u2192 0\u22621 (! p))\n\n  noZeroDivisor : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 x * y \u2262 0\u1da0\n  noZeroDivisor nx ny x*y\u22610\u1da0 = ny (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse nx)\n                                   \u2219 *= refl *-comm \u2219 ! *-assoc\n                                   \u2219 *= (*-comm \u2219 x*y\u22610\u1da0) refl \u2219 0*-zero\n                                   )\n\n  2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u1da0 * n\n  2*-spec = ! += 1*-identity 1*-identity \u2219 ! +-*-distr\n\n  *-unique-inverse : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 x * y \u2261 1\u1da0 \u2192 y \u2261 x \u207b\u00b9\n  *-unique-inverse x\/=0 xy=1 = ! 1*-identity \u2219 *= (! \u207b\u00b9-inverse x\/=0) refl \u2219 *-assoc \u2219 *= refl xy=1 \u2219 *1-identity\n\n  \u207b\u00b9-involutive : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n  \u207b\u00b9-involutive x\/=0 = ! *-unique-inverse (\u207b\u00b9-notZero x\/=0) (\u207b\u00b9-inverse x\/=0)\n\n  \u207b\u00b9*-distr : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 (x * y)\u207b\u00b9 \u2261 x \u207b\u00b9  \/ y\n  \u207b\u00b9*-distr x\/=0 y\/=0 =\n    ! (*-unique-inverse (noZeroDivisor x\/=0 y\/=0) (*= refl *-comm \u2219 *-assoc \u2219 *= refl (! *-assoc \u2219 *= (\u207b\u00b9-right-inverse y\/=0) refl \u2219 1*-identity) \u2219 \u207b\u00b9-right-inverse x\/=0))\n\n  +-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192(a \/ b) + (a' \/ b') \u2261 (a * b' + a' * b) \/ (b * b')\n  +-quotient b b'\n    = += (*= (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse b') \u2219 *= refl *-comm \u2219 ! *-assoc) refl \u2219\n          *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b' b \u2219 ap _\u207b\u00b9 *-comm))\n         (*= (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse b ) \u2219 *= refl *-comm) refl \u2219\n          *-!assoc= *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b b'))\n    \u2219 ! +-\/-distr\n\n  \u2212-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192 (a \/ b) \u2212 (a' \/ b') \u2261 (a * b' \u2212 a' * b) \/ (b * b')\n  \u2212-quotient b b' = += refl 0--*-distr \u2219 +-quotient b b' \u2219 \/= (+= refl (! 0--*-distr)) refl\n\n  *-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192 (a \/ b) * (a' \/ b') \u2261 (a * a') \/ (b * b')\n  *-quotient b\/=0 b'\/=0 = *-interchange \u2219 *= refl (! \u207b\u00b9*-distr b\/=0 b'\/=0)\n\n  \/-quotient : \u2200 {a b a' b'} \u2192 a' \u2262 0\u1da0 \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n     \u2192 (a \/ b) \/ (a' \/ b') \u2261 (a * b') \/ (b * a')\n  \/-quotient a' b b' = *= refl (\u207b\u00b9*-distr a' (\u207b\u00b9-notZero b') \u2219 *= refl (\u207b\u00b9-involutive b') \u2219 *-comm)\n    \u2219 *-interchange \u2219 *= refl (! \u207b\u00b9*-distr b a')\n\n  0--*-distr' : \u2200 {x y} \u2192 0-(x * y) \u2261 x * (0- y)\n  0--*-distr' = ap 0-_ *-comm \u2219 0--*-distr \u2219 *-comm\n\n  *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n  *-\u2212-distr = *-+-distr \u2219 += refl (! 0--*-distr')\n\n  0--+-distr : \u2200 {x y} \u2192 0-(x + y) \u2261 0- x \u2212 y\n  0--+-distr =\n      +-left-cancel\n         (0--right-inverse\n          \u2219 (! 0+-identity \u2219 += (! 0--right-inverse)\n                                (! 0--right-inverse))\n                           \u2219 +-interchange)\n\n  \u00b2-*-distr : \u2200 {x y} \u2192 (x * y)\u00b2 \u2261 x \u00b2 * y \u00b2\n  \u00b2-*-distr = *-interchange\n\n  0--cancel : \u2200 {x y} \u2192 0- x \u2261 0- y \u2192 x \u2261 y\n  0--cancel {x} {y} p = *-left-cancel (\u03bb e \u2192 0\u2262-1 (! e)) (-1*-neg \u2219 p \u2219 ! -1*-neg)\n\n  \u00b2-0--distr : \u2200 {x} \u2192 (0- x)\u00b2 \u2261 x \u00b2\n  \u00b2-0--distr {x} = ! 0--*-distr \u2219 ap 0-_ (! 0--*-distr') \u2219 0--involutive\n\n  2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n  2*-*-distr = ! +-*-distr\n\n  \u00b2-+-distr : \u2200 {x y} \u2192 (x + y)\u00b2 \u2261 x \u00b2 + y \u00b2 + 2* x * y\n  \u00b2-+-distr {x} {y} = (x + y)\u00b2\n                    \u2261\u27e8 *-+-distr \u27e9\n                      (x + y) * x + (x + y) * y\n                    \u2261\u27e8 += +-*-distr +-*-distr \u27e9\n                      x \u00b2 + y * x + (x * y + y \u00b2)\n                    \u2261\u27e8 += (+= refl *-comm) +-comm \u2219 +-interchange \u27e9\n                      x \u00b2 + y \u00b2 + 2*(x * y)\n                    \u2261\u27e8 += refl 2*-*-distr \u27e9\n                       x \u00b2 + y \u00b2 + 2* x * y\n                    \u220e\n\n  \u00b2-\u2212-distr : \u2200 {x y} \u2192 (x \u2212 y)\u00b2 \u2261 x \u00b2 + y \u00b2 \u2212 2* x * y\n  \u00b2-\u2212-distr = \u00b2-+-distr \u2219 += (+= refl \u00b2-0--distr) (*-comm \u2219 ! 0--*-distr \u2219 ap 0-_ *-comm)\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_ 2*_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  2\u1da0 3\u1da0 -0\u1da0 -1\u1da0 : A\n  2\u1da0  = suc\u1da0 1\u1da0\n  3\u1da0  = suc\u1da0 2\u1da0\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0     ] = 0\u1da0\n  \u2115[ suc n ] = fold 1\u1da0 suc\u1da0 n\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0       = 1\u1da0\n  b ^\u2115 (suc e) = fold b (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\n  2+_ : A \u2192 A\n  2+ x = 2\u1da0 + x\n\n  2*_ : A \u2192 A\n  2* x = x + x\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x * x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 * x\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n  open \u2261-Reasoning\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : LeftInverseNonZero 0\u1da0 1\u1da0 _\u207b\u00b9 _*_\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse : RightInverseNonZero 0\u1da0 1\u1da0 _\u207b\u00b9 _*_\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  open FromOp\u2082 _\u2212_ renaming ( op= to \u2212= )\n  open FromOp\u2082 _\/_ renaming ( op= to \/= )\n\n  open FromAssocComm _+_ +-assoc +-comm\n    renaming ( op=         to +=\n             ; comm=       to +-comm=\n             ; assoc=      to +-assoc=\n             ; !assoc=     to +-!assoc=\n             ; assoc-comm  to +-assoc-comm\n             ; !assoc-comm to +-!assoc-comm\n             ; interchange to +-interchange\n             )\n\n  open FromAssocComm _*_ *-assoc *-comm\n    renaming ( op=         to *=\n             ; comm=       to *-comm=\n             ; assoc=      to *-assoc=\n             ; !assoc=     to *-!assoc=\n             ; assoc-comm  to *-assoc-comm\n             ; !assoc-comm to *-!assoc-comm\n             ; interchange to *-interchange\n             )\n\n  0-= : \u2200 {x x'} \u2192 x \u2261 x' \u2192 0- x \u2261 0- x'\n  0-= = ap 0-_\n\n  +-*-distr : _*_ DistributesOver\u02b3 _+_\n  +-*-distr = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  +-\/-distr : _\/_ DistributesOver\u02b3 _+_\n  +-\/-distr = +-*-distr\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-!assoc= p \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  0\u2262-1 : 0\u1da0 \u2262 -1\u1da0\n  0\u2262-1 0\u2261-1 = 0\u22621 (! 0--inverse \u2219 +-comm \u2219 ! ap suc\u1da0 0\u2261-1 \u2219 +0-identity)\n\n  c\u207b\u00b9*c*x : \u2200 {c x} \u2192 c \u2262 0\u1da0 \u2192 c \u207b\u00b9 * c * x \u2261 x\n  c\u207b\u00b9*c*x c\u22620 = *= (\u207b\u00b9-inverse c\u22620) refl \u2219 1*-identity\n\n  *-left-cancel : LeftCancelNonZero 0\u1da0 _*_\n  *-left-cancel c\u22620 p = ! c\u207b\u00b9*c*x c\u22620 \u2219 *-!assoc= p \u2219 c\u207b\u00b9*c*x c\u22620\n\n  *-right-cancel : RightCancelNonZero 0\u1da0 _*_\n  *-right-cancel c\u22620 p = *-left-cancel c\u22620 (*-comm= p)\n\n  *0-zero : RightZero 0\u1da0 _*_\n  *0-zero = +-right-cancel  (+= refl (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= refl 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero 0\u1da0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  \u207b\u00b9-notZero : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 \u2262 0\u1da0\n  \u207b\u00b9-notZero x\/=0 = \u03bb p \u2192 0\u22621 (! 0*-zero \u2219 *= (! p) refl \u2219 \u207b\u00b9-inverse x\/=0)\n\n  0--*-distr : \u2200 {x y} \u2192 0- (x * y) \u2261 (0- x) * y\n  0--*-distr = +-right-cancel (0--inverse \u2219 ! 0*-zero \u2219 *= (! 0--inverse) refl \u2219 +-*-distr)\n\n  -1*-neg : \u2200 {x} \u2192 -1\u1da0 * x \u2261 0- x\n  -1*-neg = ! 0--*-distr \u2219 0-= 1*-identity\n\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n\n  1\u207b\u00b9\u22611 : 1\u1da0 \u207b\u00b9 \u2261 1\u1da0\n  1\u207b\u00b9\u22611 = ! *1-identity \u2219 \u207b\u00b9-inverse (\u03bb p \u2192 0\u22621 (! p))\n\n  noZeroDivisor : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 x * y \u2262 0\u1da0\n  noZeroDivisor nx ny x*y\u22610\u1da0 = ny (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse nx)\n                                   \u2219 *= refl *-comm \u2219 ! *-assoc\n                                   \u2219 *= (*-comm \u2219 x*y\u22610\u1da0) refl \u2219 0*-zero\n                                   )\n\n  2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u1da0 * n\n  2*-spec = ! += 1*-identity 1*-identity \u2219 ! +-*-distr\n\n  *-unique-inverse : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 x * y \u2261 1\u1da0 \u2192 y \u2261 x \u207b\u00b9\n  *-unique-inverse x\/=0 xy=1 = ! 1*-identity \u2219 *= (! \u207b\u00b9-inverse x\/=0) refl \u2219 *-assoc \u2219 *= refl xy=1 \u2219 *1-identity\n\n  \u207b\u00b9-involutive : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n  \u207b\u00b9-involutive x\/=0 = ! *-unique-inverse (\u207b\u00b9-notZero x\/=0) (\u207b\u00b9-inverse x\/=0)\n\n  \u207b\u00b9*-distr : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 (x * y)\u207b\u00b9 \u2261 x \u207b\u00b9  \/ y\n  \u207b\u00b9*-distr x\/=0 y\/=0 =\n    ! (*-unique-inverse (noZeroDivisor x\/=0 y\/=0) (*= refl *-comm \u2219 *-assoc \u2219 *= refl (! *-assoc \u2219 *= (\u207b\u00b9-right-inverse y\/=0) refl \u2219 1*-identity) \u2219 \u207b\u00b9-right-inverse x\/=0))\n\n  +-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192(a \/ b) + (a' \/ b') \u2261 (a * b' + a' * b) \/ (b * b')\n  +-quotient b b'\n    = += (*= (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse b') \u2219 *= refl *-comm \u2219 ! *-assoc) refl \u2219\n          *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b' b \u2219 ap _\u207b\u00b9 *-comm))\n         (*= (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse b ) \u2219 *= refl *-comm) refl \u2219\n          *-!assoc= *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b b'))\n    \u2219 ! +-\/-distr\n\n  \u2212-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192 (a \/ b) \u2212 (a' \/ b') \u2261 (a * b' \u2212 a' * b) \/ (b * b')\n  \u2212-quotient b b' = += refl 0--*-distr \u2219 +-quotient b b' \u2219 \/= (+= refl (! 0--*-distr)) refl\n\n  *-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192 (a \/ b) * (a' \/ b') \u2261 (a * a') \/ (b * b')\n  *-quotient b\/=0 b'\/=0 = *-interchange \u2219 *= refl (! \u207b\u00b9*-distr b\/=0 b'\/=0)\n\n  \/-quotient : \u2200 {a b a' b'} \u2192 a' \u2262 0\u1da0 \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n     \u2192 (a \/ b) \/ (a' \/ b') \u2261 (a * b') \/ (b * a')\n  \/-quotient a' b b' = *= refl (\u207b\u00b9*-distr a' (\u207b\u00b9-notZero b') \u2219 *= refl (\u207b\u00b9-involutive b') \u2219 *-comm)\n    \u2219 *-interchange \u2219 *= refl (! \u207b\u00b9*-distr b a')\n\n  0--*-distr' : \u2200 {x y} \u2192 0-(x * y) \u2261 x * (0- y)\n  0--*-distr' = ap 0-_ *-comm \u2219 0--*-distr \u2219 *-comm\n\n  *-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n  *-\u2212-distr = *-+-distr \u2219 += refl (! 0--*-distr')\n\n  0--+-distr : \u2200 {x y} \u2192 0-(x + y) \u2261 0- x \u2212 y\n  0--+-distr =\n      +-left-cancel\n         (0--right-inverse\n          \u2219 (! 0+-identity \u2219 += (! 0--right-inverse)\n                                (! 0--right-inverse))\n                           \u2219 +-interchange)\n\n  \u00b2-*-distr : \u2200 {x y} \u2192 (x * y)\u00b2 \u2261 x \u00b2 * y \u00b2\n  \u00b2-*-distr = *-interchange\n\n  0--cancel : \u2200 {x y} \u2192 0- x \u2261 0- y \u2192 x \u2261 y\n  0--cancel {x} {y} p = *-left-cancel (\u03bb e \u2192 0\u2262-1 (! e)) (-1*-neg \u2219 p \u2219 ! -1*-neg)\n\n  \u00b2-0--distr : \u2200 {x} \u2192 (0- x)\u00b2 \u2261 x \u00b2\n  \u00b2-0--distr {x} = ! 0--*-distr \u2219 ap 0-_ (! 0--*-distr') \u2219 0--involutive\n\n  2*-*-distr : \u2200 {x y} \u2192 2*(x * y) \u2261 2* x * y\n  2*-*-distr = ! +-*-distr\n\n  \u00b2-+-distr : \u2200 {x y} \u2192 (x + y)\u00b2 \u2261 x \u00b2 + y \u00b2 + 2* x * y\n  \u00b2-+-distr {x} {y} = (x + y)\u00b2\n                    \u2261\u27e8 *-+-distr \u27e9\n                      (x + y) * x + (x + y) * y\n                    \u2261\u27e8 += +-*-distr +-*-distr \u27e9\n                      x \u00b2 + y * x + (x * y + y \u00b2)\n                    \u2261\u27e8 += (+= refl *-comm) +-comm \u2219 +-interchange \u27e9\n                      x \u00b2 + y \u00b2 + 2*(x * y)\n                    \u2261\u27e8 += refl 2*-*-distr \u27e9\n                       x \u00b2 + y \u00b2 + 2* x * y\n                    \u220e\n\n  \u00b2-\u2212-distr : \u2200 {x y} \u2192 (x \u2212 y)\u00b2 \u2261 x \u00b2 + y \u00b2 \u2212 2* x * y\n  \u00b2-\u2212-distr = \u00b2-+-distr \u2219 += (+= refl \u00b2-0--distr) (*-comm \u2219 ! 0--*-distr \u2219 ap 0-_ *-comm)\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e814bd396b003fe1f8c20487024de63756308a41","subject":"Avoid module name conflict","message":"Avoid module name conflict\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Raw.agda","new_file":"lib\/Algebra\/Raw.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type_)\nopen import Function.Base using (nest)\nopen import Data.Nat.Base using (\u2115) renaming (suc to suc\u2115)\nopen import Data.Integer.Base using (\u2124; +_; -[1+_])\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; ap; ap\u2082)\n\nmodule Algebra.Raw where\n\nrecord Magma {\u2113}(A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor \u27e8_\u27e9\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n\n  infix 8 _\u00b2 _\u00b3 _\u2074\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x \u2219 x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 \u2219 x\n\n  _\u2074 : A \u2192 A\n  x \u2074 = x \u00b3 \u2219 x\n\n  infix 8 _^\u00b9\u207a_\n  _^\u00b9\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u00b9\u207a n = nest n (_\u2219_ x) x\n\n  module _ {x x' y y'}(p : x \u2261 x')(q : y \u2261 y') where\n    \u2219= : x \u2219 y \u2261 x' \u2219 y'\n    \u2219= = ap\u2082 _\u2219_ p q\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Magma {\u2113}{M : Set \u2113} (mag : Magma M) where\n  private\n   module M = Magma mag\n    using    ()\n    renaming ( _\u2219_ to _+_\n             ; _\u00b2 to 2\u2297_\n             ; _\u00b3 to 3\u2297_\n             ; _\u2074 to 4\u2297_\n             ; _^\u00b9\u207a_ to _\u2297\u00b9\u207a_\n             ; \u2219= to +=\n             )\n  open M public using (+=)\n  infixl 6 _+_\n  infixl 7 _\u2297\u00b9\u207a_ 2\u2297_ 3\u2297_ 4\u2297_\n  _+_   = M._+_\n  _\u2297\u00b9\u207a_ = M._\u2297\u00b9\u207a_\n  2\u2297_   = M.2\u2297_\n  3\u2297_   = M.3\u2297_\n  4\u2297_   = M.4\u2297_\n\nrecord Monoid-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n    \u03b5   : A\n\n  \u2219-magma : Magma A\n  \u2219-magma = \u27e8 _\u2219_ \u27e9\n\n  open module \u2219-magma = Magma \u2219-magma public hiding (_\u2219_)\n\n  infix 8 _^\u207a_\n  _^\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u207a n = nest n (_\u2219_ x) \u03b5\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Monoid-Ops {\u2113}{M : Set \u2113} (mon : Monoid-Ops M) where\n  private\n   module M = Monoid-Ops mon\n    using ()\n    renaming ( \u03b5 to 0#\n             ; _^\u207a_ to _\u2297\u207a_\n             ; \u2219-magma to +-magma\n             )\n  open M public using (0#; +-magma)\n  open Additive-Magma +-magma public\n  infixl 7 _\u2297\u207a_\n  _\u2297\u207a_  = M._\u2297\u207a_\n\n-- A renaming of Monoid-Ops with multiplicative notation\nmodule Multiplicative-Monoid-Ops {\u2113}{M : Type \u2113} (mon-ops : Monoid-Ops M)\n  = Monoid-Ops mon-ops\n    renaming ( _\u2219_ to _*_\n             ; \u03b5 to 1#\n             ; \u2219= to *=\n             ; \u2219-magma to *-magma\n             ; module \u2219-magma to *-magma\n             )\n\nrecord Group-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n\n  field\n    mon-ops : Monoid-Ops A\n    _\u207b\u00b9     : A \u2192 A\n\n  open module mon-ops = Monoid-Ops mon-ops public\n\n  \u207b\u00b9= : \u2200 {x y} \u2192 x \u2261 y \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9\n  \u207b\u00b9= = ap _\u207b\u00b9\n\n  infixl 7 _\/_ _\/\u2032_\n\n  _\/_ : A \u2192 A \u2192 A\n  x \/ y = x \u2219 y \u207b\u00b9\n\n  \/-magma : Magma A\n  \/-magma = \u27e8 _\/_ \u27e9\n\n  open module \/-magma = Magma \/-magma public using () renaming (\u2219= to \/=)\n\n  _\/\u2032_ : A \u2192 A \u2192 A\n  x \/\u2032 y = y \u207b\u00b9 \u2219 x\n\n  \/\u2032-magma : Magma A\n  \/\u2032-magma = \u27e8 _\/\u2032_ \u27e9\n\n  open module \/\u2032-magma = Magma \/\u2032-magma public using () renaming (\u2219= to \/\u2032=)\n\n  _^\u207b_ _^\u207b\u2032_ : A \u2192 \u2115 \u2192 A\n  x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n  x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n  _^_ : A \u2192 \u2124 \u2192 A\n  x ^ -[1+ n ] = x ^\u207b(suc\u2115 n)\n  x ^ (+ n)    = x ^\u207a n\n\n-- A renaming of Group-Ops with additive notation\nmodule Additive-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) where\n  private\n   module M = Group-Ops grp\n    using    ()\n    renaming ( _\u207b\u00b9 to 0\u2212_\n             ; \u207b\u00b9= to 0\u2212=\n             ; _\/_ to _\u2212_\n             ; _\/\u2032_ to _\u2212\u2032_\n             ; _^\u207b_ to _\u2297\u207b_\n             ; _^_ to _\u2297_\n             ; mon-ops to +-mon-ops\n             ; \/= to \u2212=\n             ; \/\u2032= to \u2212\u2032=\n             )\n  open M public using (0\u2212_; +-mon-ops; \u2212=; 0\u2212=)\n  open Additive-Monoid-Ops +-mon-ops public\n  infixl 6 _\u2212_\n  infixl 7 _\u2297\u207b_ _\u2297_\n  _\u2212_   = M._\u2212_\n  _\u2297\u207b_  = M._\u2297\u207b_\n  _\u2297_   = M._\u2297_\n\n-- A renaming of Group-Ops with multiplicative notation\nmodule Multiplicative-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) = Group-Ops grp\n    using    ( _^\u207a_; _\u00b2; _\u00b3; _\u2074; _\u207b\u00b9; _\/_; \/=; _\/\u2032_; \/\u2032=\n             ; _^\u207b_; _^_\n             ; \/-magma; module \/-magma; \/\u2032-magma\n             ; module \/\u2032-magma; \u207b\u00b9= )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1#; mon-ops to *-mon-ops; \u2219= to *= )\n\nrecord Ring-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n  infixl 6 1+_ 2+_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-mon-ops : Monoid-Ops A\n\n  open module +-grp-ops = Additive-Group-Ops        +-grp-ops public\n    renaming (2\u2297_ to 2*_)\n  open module *-mon-ops = Multiplicative-Monoid-Ops *-mon-ops public\n\n  suc pred : A \u2192 A\n\n  suc = _+_ 1#\n  pred x = x \u2212 1#\n\n  1+_ = suc\n\n  2# 3# -0# -1# : A\n  2#  = suc 1#\n  3#  = suc 2#\n  -0# = 0\u2212 0#\n  -1# = 0\u2212 1#\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0    ] = 0#\n  \u2115[ suc\u2115 n ] = nest n suc 1#\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ + x      ] = \u2115[ x ]\n  \u2124[ -[1+ x ] ] = 0\u2212 \u2115[ suc\u2115 x ]\n\n  -- TODO use _^\u207a_ and _^_ from group\/monoid\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  _^\u2115_ = _^\u207a_\n  {-\n  b ^\u2115 0        = 1#\n  b ^\u2115 (suc\u2115 n) = nest n (_*_ b) b\n  -}\n\n  2+_ : A \u2192 A\n  2+ x = 2# + x\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-grp-ops : Group-Ops A\n\n  -- JS BUG this creates a conflict with *-grp-ops and shouldn't\n  open module *-grp-ops' = Multiplicative-Group-Ops *-grp-ops public\n    using ( _\u207b\u00b9; \u207b\u00b9=\n          ; _\/_; \/=; \/-magma; module \/-magma\n          ; _\/\u2032_; \/\u2032=; \/\u2032-magma; module \/\u2032-magma\n          ; _^\u207b_; _^_; *-mon-ops )\n\n  rng-ops : Ring-Ops A\n  rng-ops = +-grp-ops , *-mon-ops\n\n  open module rng-ops = Ring-Ops rng-ops public\n    hiding (+-grp-ops; *-mon-ops)\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  _^\u2124_ = _^_\n  {-\n  b ^\u2124 (+ n)    = b ^\u2115 n\n  b ^\u2124 -[1+ n ] = (b ^\u2115 (suc\u2115 n))\u207b\u00b9\n  -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type_)\nopen import Function.Base using (nest)\nopen import Data.Nat.Base using (\u2115) renaming (suc to suc\u2115)\nopen import Data.Integer.Base using (\u2124; +_; -[1+_])\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; ap; ap\u2082)\n\nmodule Algebra.Raw where\n\nrecord Magma {\u2113}(A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor \u27e8_\u27e9\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n\n  infix 8 _\u00b2 _\u00b3 _\u2074\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x \u2219 x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 \u2219 x\n\n  _\u2074 : A \u2192 A\n  x \u2074 = x \u00b3 \u2219 x\n\n  infix 8 _^\u00b9\u207a_\n  _^\u00b9\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u00b9\u207a n = nest n (_\u2219_ x) x\n\n  module _ {x x' y y'}(p : x \u2261 x')(q : y \u2261 y') where\n    \u2219= : x \u2219 y \u2261 x' \u2219 y'\n    \u2219= = ap\u2082 _\u2219_ p q\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Magma {\u2113}{M : Set \u2113} (mag : Magma M) where\n  private\n   module M = Magma mag\n    using    ()\n    renaming ( _\u2219_ to _+_\n             ; _\u00b2 to 2\u2297_\n             ; _\u00b3 to 3\u2297_\n             ; _\u2074 to 4\u2297_\n             ; _^\u00b9\u207a_ to _\u2297\u00b9\u207a_\n             ; \u2219= to +=\n             )\n  open M public using (+=)\n  infixl 6 _+_\n  infixl 7 _\u2297\u00b9\u207a_ 2\u2297_ 3\u2297_ 4\u2297_\n  _+_   = M._+_\n  _\u2297\u00b9\u207a_ = M._\u2297\u00b9\u207a_\n  2\u2297_   = M.2\u2297_\n  3\u2297_   = M.3\u2297_\n  4\u2297_   = M.4\u2297_\n\nrecord Monoid-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n    \u03b5   : A\n\n  \u2219-magma : Magma A\n  \u2219-magma = \u27e8 _\u2219_ \u27e9\n\n  open module \u2219-magma = Magma \u2219-magma public hiding (_\u2219_)\n\n  infix 8 _^\u207a_\n  _^\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u207a n = nest n (_\u2219_ x) \u03b5\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Monoid-Ops {\u2113}{M : Set \u2113} (mon : Monoid-Ops M) where\n  private\n   module M = Monoid-Ops mon\n    using ()\n    renaming ( \u03b5 to 0#\n             ; _^\u207a_ to _\u2297\u207a_\n             ; \u2219-magma to +-magma\n             )\n  open M public using (0#; +-magma)\n  open module +-magma = Additive-Magma +-magma public\n  infixl 7 _\u2297\u207a_\n  _\u2297\u207a_  = M._\u2297\u207a_\n\n-- A renaming of Monoid-Ops with multiplicative notation\nmodule Multiplicative-Monoid-Ops {\u2113}{M : Type \u2113} (mon-ops : Monoid-Ops M)\n  = Monoid-Ops mon-ops\n    renaming ( _\u2219_ to _*_\n             ; \u03b5 to 1#\n             ; \u2219= to *=\n             ; \u2219-magma to *-magma\n             ; module \u2219-magma to *-magma\n             )\n\nrecord Group-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n\n  field\n    mon-ops : Monoid-Ops A\n    _\u207b\u00b9     : A \u2192 A\n\n  open module mon-ops = Monoid-Ops mon-ops public\n\n  \u207b\u00b9= : \u2200 {x y} \u2192 x \u2261 y \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9\n  \u207b\u00b9= = ap _\u207b\u00b9\n\n  infixl 7 _\/_ _\/\u2032_\n\n  _\/_ : A \u2192 A \u2192 A\n  x \/ y = x \u2219 y \u207b\u00b9\n\n  \/-magma : Magma A\n  \/-magma = \u27e8 _\/_ \u27e9\n\n  open module \/-magma = Magma \/-magma public using () renaming (\u2219= to \/=)\n\n  _\/\u2032_ : A \u2192 A \u2192 A\n  x \/\u2032 y = y \u207b\u00b9 \u2219 x\n\n  \/\u2032-magma : Magma A\n  \/\u2032-magma = \u27e8 _\/\u2032_ \u27e9\n\n  open module \/\u2032-magma = Magma \/\u2032-magma public using () renaming (\u2219= to \/\u2032=)\n\n  _^\u207b_ _^\u207b\u2032_ : A \u2192 \u2115 \u2192 A\n  x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n  x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n  _^_ : A \u2192 \u2124 \u2192 A\n  x ^ -[1+ n ] = x ^\u207b(suc\u2115 n)\n  x ^ (+ n)    = x ^\u207a n\n\n-- A renaming of Group-Ops with additive notation\nmodule Additive-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) where\n  private\n   module M = Group-Ops grp\n    using    ()\n    renaming ( _\u207b\u00b9 to 0\u2212_\n             ; \u207b\u00b9= to 0\u2212=\n             ; _\/_ to _\u2212_\n             ; _\/\u2032_ to _\u2212\u2032_\n             ; _^\u207b_ to _\u2297\u207b_\n             ; _^_ to _\u2297_\n             ; mon-ops to +-mon-ops\n             ; \/= to \u2212=\n             ; \/\u2032= to \u2212\u2032=\n             )\n  open M public using (0\u2212_; +-mon-ops; \u2212=; 0\u2212=)\n  open module +-mon-ops = Additive-Monoid-Ops +-mon-ops public\n  infixl 6 _\u2212_\n  infixl 7 _\u2297\u207b_ _\u2297_\n  _\u2212_   = M._\u2212_\n  _\u2297\u207b_  = M._\u2297\u207b_\n  _\u2297_   = M._\u2297_\n\n-- A renaming of Group-Ops with multiplicative notation\nmodule Multiplicative-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) = Group-Ops grp\n    using    ( _^\u207a_; _\u00b2; _\u00b3; _\u2074; _\u207b\u00b9; _\/_; \/=; _\/\u2032_; \/\u2032=\n             ; _^\u207b_; _^_\n             ; \/-magma; module \/-magma; \/\u2032-magma\n             ; module \/\u2032-magma; \u207b\u00b9= )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1#; mon-ops to *-mon-ops; \u2219= to *= )\n\nrecord Ring-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n  infixl 6 1+_ 2+_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-mon-ops : Monoid-Ops A\n\n  open module +-grp-ops = Additive-Group-Ops        +-grp-ops public\n    renaming (2\u2297_ to 2*_)\n  open module *-mon-ops = Multiplicative-Monoid-Ops *-mon-ops public\n\n  suc pred : A \u2192 A\n\n  suc = _+_ 1#\n  pred x = x \u2212 1#\n\n  1+_ = suc\n\n  2# 3# -0# -1# : A\n  2#  = suc 1#\n  3#  = suc 2#\n  -0# = 0\u2212 0#\n  -1# = 0\u2212 1#\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0    ] = 0#\n  \u2115[ suc\u2115 n ] = nest n suc 1#\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ + x      ] = \u2115[ x ]\n  \u2124[ -[1+ x ] ] = 0\u2212 \u2115[ suc\u2115 x ]\n\n  -- TODO use _^\u207a_ and _^_ from group\/monoid\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  _^\u2115_ = _^\u207a_\n  {-\n  b ^\u2115 0        = 1#\n  b ^\u2115 (suc\u2115 n) = nest n (_*_ b) b\n  -}\n\n  2+_ : A \u2192 A\n  2+ x = 2# + x\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  inductive -- NO_ETA\n  constructor _,_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-grp-ops : Group-Ops A\n\n  -- JS BUG this creates a conflict with *-grp-ops and shouldn't\n  open module *-grp-ops' = Multiplicative-Group-Ops *-grp-ops public\n    using ( _\u207b\u00b9; \u207b\u00b9=\n          ; _\/_; \/=; \/-magma; module \/-magma\n          ; _\/\u2032_; \/\u2032=; \/\u2032-magma; module \/\u2032-magma\n          ; _^\u207b_; _^_; *-mon-ops )\n\n  rng-ops : Ring-Ops A\n  rng-ops = +-grp-ops , *-mon-ops\n\n  open module rng-ops = Ring-Ops rng-ops public\n    hiding (+-grp-ops; *-mon-ops)\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  _^\u2124_ = _^_\n  {-\n  b ^\u2124 (+ n)    = b ^\u2115 n\n  b ^\u2124 -[1+ n ] = (b ^\u2115 (suc\u2115 n))\u207b\u00b9\n  -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9e2eda40afa1c3517f37c6305aa4f0c8181e84b2","subject":"Adapt circuits to composable\/vcomposable\/forkable","message":"Adapt circuits to composable\/vcomposable\/forkable\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec.NP as Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nRewireTbl : CircuitType\nRewireTbl i o = Vec (Fin i) o\n\nmodule Rewire where\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunVComp : VComposable _+_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\nbitsFunVComp = mk _***_ where\n  C = _\u2192\u1d47_\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  (f *** g) xs with splitAt _ xs\n  ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Rewire.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec.NP as Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Rewire.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Rewire.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8744297b59448993f3cfd959cc60c8a20bbeeacd","subject":"Minor change in ZK\/SigmaProtocol\/KnownStatement","message":"Minor change in ZK\/SigmaProtocol\/KnownStatement\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_contents":"open import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\nimport ZK.SigmaProtocol.Types\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n  -- TODO Challenge should exp large wrt the security param\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- See the module ZK.Types for details about the types:\n--   * Prover-Interaction\n--   * Verifier\n--   * Transcript\n--   * Transcript\u00b2\nopen ZK.SigmaProtocol.Types Commitment Challenge Response public\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n_\u21c6_ : Verifier \u2192 Prover \u2192 (r : Randomness)(w : Witness)(c : Challenge) \u2192 Bool\n(verifier \u21c6 prover) r w c = verifier (run (prover r w) c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = \u2713 (verifier t)\n\n  prover-commitment : (r : Randomness)(w : Witness) \u2192 Commitment\n  prover-commitment r w = Prover-Interaction.get-A (prover r w)\n\n  prover-response : (r : Randomness)(w : Witness)(c : Challenge) \u2192 Response\n  prover-response r w = Prover-Interaction.get-f (prover r w)\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 w r c (w? : ValidWitness w) \u2192\n                              \u2713 ((verifier \u21c6 prover) r w c)\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713 (verifier (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator          : Simulator\n    .correct-simulator : Correct-simulator verifier simulator\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = (t\u00b2 : Transcript\u00b2 verifier) \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\n-- Also called 2-extractable\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor              : Extractor verifier\n    .extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    .correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\nimport ZK.SigmaProtocol.Types\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n  -- TODO Challenge should exp large wrt the security param\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- See the module ZK.Types for details about the types:\n--   * Prover-Interaction\n--   * Verifier\n--   * Transcript\n--   * Transcript\u00b2\nopen ZK.SigmaProtocol.Types Commitment Challenge Response public\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = \u2713 (verifier t)\n\n  prover-commitment : (r : Randomness)(w : Witness) \u2192 Commitment\n  prover-commitment r w = Prover-Interaction.get-A (prover r w)\n\n  prover-response : (r : Randomness)(w : Witness)(c : Challenge) \u2192 Response\n  prover-response r w = Prover-Interaction.get-f (prover r w)\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 w r c (w? : ValidWitness w) \u2192 \u2713 (verifier (run (prover r w) c))\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713 (verifier (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator          : Simulator\n    .correct-simulator : Correct-simulator verifier simulator\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = (t\u00b2 : Transcript\u00b2 verifier) \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor              : Extractor verifier\n    .extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    .correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f85089ef24284c1e88eea389d652b67e8f0a6785","subject":"Cleanup comments for deriveConst match","message":"Cleanup comments for deriveConst match\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Derive.agda","new_file":"New\/Derive.agda","new_contents":"module New.Derive where\nopen import New.Lang\nopen import New.Changes\nopen import New.LangOps\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Sum\nopen import Data.Product\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\n-- dplus = \u03bb m dm n dn \u2192 (m + dm) + (n + dn) - (m + n) = dm + dn\nderiveConst plus = abs (abs (abs (abs (app\u2082 (const plus) (var (that (that this))) (var this)))))\n-- minus = \u03bb m n \u2192 m - n\n-- dminus = \u03bb m dm n dn \u2192 (m + dm) - (n + dn) - (m - n) = dm - dn\nderiveConst minus = abs (abs (abs (abs (app\u2082 (const minus) (var (that (that this))) (var this)))))\nderiveConst cons = abs (abs (abs (abs (app (app (const cons) (var (that (that this)))) (var this)))))\nderiveConst fst = abs (abs (app (const fst) (var this)))\nderiveConst snd = abs (abs (app (const snd) (var this)))\nderiveConst linj = abs (abs (app (const linj) (app (const linj) (var this))))\nderiveConst rinj = abs (abs (app (const linj) (app (const rinj) (var this))))\nderiveConst (match {t1} {t2} {t3}) =\n  -- \u03bb s ds f df g dg \u2192\n  abs (abs (abs (abs (abs (abs\n    -- match ds\n    (app\u2083 (const match) (var (that (that (that (that this)))))\n      -- \u03bb ds\u2081 \u2192 match ds\u2081\n      (abs (app\u2083 (const match) (var this)\n        -- case inj\u2081 da \u2192 absV 1 (\u03bb da \u2192 match s\n        (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n          -- \u03bb a \u2192 app\u2082 df a da\n          (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (var (that this))))\n          -- absurd: \u03bb b \u2192 dg b (nil b)\n          (abs (app\u2082 (var (that (that (that this)))) (var this) (app (onil\u03c4o t2) (var this))))))\n        -- case inj\u2082 db \u2192 absV 1 (\u03bb db \u2192 match s\n        (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n          -- absurd: \u03bb a \u2192 df a (nil a)\n          (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (app (onil\u03c4o t1) (var this))))\n          -- \u03bb b \u2192 app\u2082 dg b db\n          (abs (app\u2082 (var (that (that (that this)))) (var this) (var (that this))))))))\n      -- recomputation branch:\n      -- \u03bb s2 \u2192 ominus (match s2 (f \u2295 df) (g \u2295 dg)) (match s f g)\n      (abs (app\u2082 (ominus\u03c4o t3)\n        -- (match s2 (f \u2295 df) (g \u2295 dg))\n        (app\u2083 (const match)\n          (var this)\n          (app\u2082 (oplus\u03c4o (t1 \u21d2 t3))\n            (var (that (that (that (that this)))))\n            (var (that (that (that this)))))\n          (app\u2082 (oplus\u03c4o (t2 \u21d2 t3))\n            (var (that (that this)))\n            (var (that this))))\n        -- (match s f g)\n        (app\u2083 (const match)\n          (var (that (that (that (that (that (that this)))))))\n          (var (that (that (that (that this)))))\n          (var (that (that this))))))))))))\n\n{-\nderive (\u03bb s f g \u2192 match s f g) =\n\u03bb s ds f df g dg \u2192\ncase ds of\n ch1 da \u2192\n   case s of\n     inj1 a \u2192 df a da\n     inj2 b \u2192 {- absurd -} dg b (nil b)\n  ch2 db \u2192\n   case s of\n     inj2 b \u2192 dg b db\n     inj1 a \u2192 {- absurd -} df a (nil a)\n  rp s2 \u2192\n    match (f \u2295 df) (g \u2295 dg) s2 \u229d\n    match f g s\n-}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n","old_contents":"module New.Derive where\nopen import New.Lang\nopen import New.Changes\nopen import New.LangOps\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Sum\nopen import Data.Product\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\n-- dplus = \u03bb m dm n dn \u2192 (m + dm) + (n + dn) - (m + n) = dm + dn\nderiveConst plus = abs (abs (abs (abs (app\u2082 (const plus) (var (that (that this))) (var this)))))\n-- minus = \u03bb m n \u2192 m - n\n-- dminus = \u03bb m dm n dn \u2192 (m + dm) - (n + dn) - (m - n) = dm - dn\nderiveConst minus = abs (abs (abs (abs (app\u2082 (const minus) (var (that (that this))) (var this)))))\nderiveConst cons = abs (abs (abs (abs (app (app (const cons) (var (that (that this)))) (var this)))))\nderiveConst fst = abs (abs (app (const fst) (var this)))\nderiveConst snd = abs (abs (app (const snd) (var this)))\nderiveConst linj = abs (abs (app (const linj) (app (const linj) (var this))))\nderiveConst rinj = abs (abs (app (const linj) (app (const rinj) (var this))))\nderiveConst (match {t1} {t2} {t3}) =\n  -- absV 6 (\u03bb s ds f df g dg \u2192\n  abs (abs (abs (abs (abs (abs\n  --   app\u2083 (const match) ds\n    (app\u2083 (const match) (var (that (that (that (that this)))))\n  --     (absV 1 (\u03bb ds\u2081 \u2192\n  --       app\u2083 (const match) ds\u2081\n      (abs (app\u2083 (const match) (var this)\n        -- case inj\u2081 da \u2192 absV 1 (\u03bb da \u2192 match s\n        (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n          -- \u03bb a \u2192 app\u2082 df a da\n          (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (var (that this))))\n          -- absurd\n          (abs (app\u2082 (var (that (that (that this)))) (var this) (app (onil\u03c4o t2) (var this)))))) -- \u03bb b \u2192 dg b (nil b)\n        -- case inj\u2082 db \u2192 absV 1 (\u03bb db \u2192 match s\n        (abs (app\u2083 (const match) (var (that (that (that (that (that (that (that this))))))))\n          -- absurd\n          (abs (app\u2082 (var (that (that (that (that (that this)))))) (var this) (app (onil\u03c4o t1) (var this)))) -- \u03bb a \u2192 df a (nil a)\n          (abs (app\u2082 (var (that (that (that this)))) (var this) (var (that this))))))))\n      -- recomputation branch\n      -- \u03bb s2 \u2192 ominus\n      (abs (app\u2082 (ominus\u03c4o t3)\n        -- match s2 (f \u2295 df) (g \u2295 df)\n        (app\u2083 (const match) (var this)\n          (app\u2082 (oplus\u03c4o (t1 \u21d2 t3)) (var (that (that (that (that this))))) (var (that (that (that this)))))\n          (app\u2082 (oplus\u03c4o (t2 \u21d2 t3)) (var (that (that this))) (var (that this))))\n        -- match s f g\n        (app\u2083 (const match) (var (that (that (that (that (that (that this)))))))\n          (var (that (that (that (that this)))))\n          (var (that (that this))))))))))))\n\n  -- absV 6 (\u03bb s ds f df g dg \u2192\n  --   app\u2083 (const match) ds\n  --     (absV 1 (\u03bb ds\u2081 \u2192\n  --       app\u2083 (const match) ds\u2081\n  --         (absV 1 {!\u03bb da \u2192 app\u2083 (const match) s (absV 1 (\u03bb a \u2192 app\u2082 df a da))!})\n  --         (absV 1 {!!})))\n  --     (absV 1 (\u03bb s\u2082 \u2192 app\u2083 (const match) s\u2082 (absV 1 {!!}) {!!})))\n\n-- abs (abs (abs (abs (abs (abs {!!})))))\n{-\nderive (\u03bb s f g \u2192 match s f g) =\n\u03bb s ds f df g dg \u2192\ncase ds of\n ch1 da \u2192\n   case s of\n     inj1 a \u2192 df a da\n     inj2 b \u2192 {- absurd -} dg b (nil b)\n  ch2 db \u2192\n   case s of\n     inj2 b \u2192 dg b db\n     inj1 a \u2192 {- absurd -} df a (nil a)\n\n\n-}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a7955542bb8a58e5f9fde45adb5211cb2bcf56d0","subject":"Properties of hom \u2248\u2141","message":"Properties of hom \u2248\u2141\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-implem.agda","new_file":"flipbased-implem.agda","new_contents":"module flipbased-implem where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Vec\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Data.Fin as Fin\nopen \u2261 using (_\u2257_; _\u2261_)\nopen Fin using (Fin; suc)\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run\u21ba : Bits n \u2192 A\n\nopen \u21ba public\n\nprivate\n  -- If you are not allowed to toss any coin, then you are deterministic.\n  Det : \u2200 {a} \u2192 Set a \u2192 Set a\n  Det = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet f = run\u21ba f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn\u21ba = mk \u2218 const\n\nmap\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap\u21ba f x = mk (f \u2218 run\u21ba x)\n-- map\u21ba f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin\u21ba {n\u2081} x = mk (\u03bb bs \u2192 run\u21ba (run\u21ba x (take _ bs)) (drop n\u2081 bs))\n-- join\u21ba x = x >>= id\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\n_\u2257\u21ba_ : \u2200 {c a} {A : Set a} (f g : \u21ba c A) \u2192 Set a\nf \u2257\u21ba g = run\u21ba f \u2257 run\u21ba g\n\n\u2141 : \u2115 \u2192 Set\n\u2141 n = \u21ba n Bit\n\n_\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c) \u2192 Set\n\u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n\u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n\u2257\u2141-trans p q b R = \u2261.trans (p b R) (q b R)\n\ncount\u21ba\u1da0 : \u2200 {c} \u2192 \u2141 c \u2192 Fin (suc (2^ c))\ncount\u21ba\u1da0 f = #\u27e8 run\u21ba f \u27e9\u1da0\n\ncount\u21ba : \u2200 {c} \u2192 \u2141 c \u2192 \u2115\ncount\u21ba = Fin.to\u2115 \u2218 count\u21ba\u1da0\n\n_\u223c[_]\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141_ {m} {n} f _\u223c_ g = \u27e82^ n \u27e9*( count\u21ba f ) \u223c \u27e82^ m \u27e9*( count\u21ba g )\n\n_\u223c[_]\u2141\u2032_ : \u2200 {n} \u2192 \u2141 n \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141\u2032_ {n} f _\u223c_ g = count\u21ba f \u223c count\u21ba g\n\n_\u2248\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\nf \u2248\u2141 g = f \u223c[ _\u2261_ ]\u2141 g\n\n_\u2248\u2141\u2032_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\nf \u2248\u2141\u2032 g = f \u223c[ _\u2261_ ]\u2141\u2032 g\n\n\u2248\u2141-refl : \u2200 {n} {f : \u2141 n} \u2192 f \u2248\u2141 f\n\u2248\u2141-refl = \u2261.refl\n\n\u2248\u2141-sym : \u2200 {n} \u2192 Symmetric {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-sym = \u2261.sym\n\n\u2248\u2141-trans : \u2200 {n} \u2192 Transitive {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-trans = \u2261.trans\n\n\u2257\u21d2\u2248\u2141 : \u2200 {c} {f g : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f \u2248\u2141 g\n\u2257\u21d2\u2248\u2141 {c} {f} {g} pf rewrite ext-# pf = \u2261.refl\n\n\u2248\u2141\u2032\u21d2\u2248\u2141 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141\u2032 g \u2192 f \u2248\u2141 g\n\u2248\u2141\u2032\u21d2\u2248\u2141 {n} eq rewrite eq = \u2261.refl\n\n\u2248\u2141\u21d2\u2248\u2141\u2032 : \u2200 {n} {f g : \u2141 n} \u2192 f \u2248\u2141 g \u2192 f \u2248\u2141\u2032 g\n\u2248\u2141\u21d2\u2248\u2141\u2032 {n} = 2^-inj n\n\n\u2248\u2141-cong : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141 f' \u2192 g \u2248\u2141 g'\n\u2248\u2141-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n\n\u2248\u2141\u2032-cong : \u2200 {c} {f g f' g' : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141\u2032 f' \u2192 g \u2248\u2141\u2032 g'\n\u2248\u2141\u2032-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n","old_contents":"module flipbased-implem where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Vec\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Data.Fin as Fin\nopen \u2261 using (_\u2257_; _\u2261_)\nopen Fin using (Fin; suc)\n\nimport flipbased\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\nrecord \u21ba {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run\u21ba : Bits n \u2192 A\n\nopen \u21ba public\n\nprivate\n  -- If you are not allowed to toss any coin, then you are deterministic.\n  Det : \u2200 {a} \u2192 Set a \u2192 Set a\n  Det = \u21ba 0\n\nrunDet : \u2200 {a} {A : Set a} \u2192 Det A \u2192 A\nrunDet f = run\u21ba f []\n\ntoss : \u21ba 1 Bit\ntoss = mk head\n\nreturn\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A\nreturn\u21ba = mk \u2218 const\n\nmap\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B\nmap\u21ba f x = mk (f \u2218 run\u21ba x)\n-- map\u21ba f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A\njoin\u21ba {n\u2081} x = mk (\u03bb bs \u2192 run\u21ba (run\u21ba x (take _ bs)) (drop n\u2081 bs))\n-- join\u21ba x = x >>= id\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 \u21ba m A \u2192 \u21ba n A\ncomap f (mk g) = mk (g \u2218 f)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 \u21ba m A \u2192 \u21ba n A\nweaken\u2264 p = comap (take\u2264 p)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\n_\u2257\u21ba_ : \u2200 {c a} {A : Set a} (f g : \u21ba c A) \u2192 Set a\nf \u2257\u21ba g = run\u21ba f \u2257 run\u21ba g\n\n\u2141 : \u2115 \u2192 Set\n\u2141 n = \u21ba n Bit\n\n_\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c) \u2192 Set\n\u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n\u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n\u2257\u2141-trans p q b R = \u2261.trans (p b R) (q b R)\n\ncount\u21ba\u1da0 : \u2200 {c} \u2192 \u2141 c \u2192 Fin (suc (2^ c))\ncount\u21ba\u1da0 f = #\u27e8 run\u21ba f \u27e9\u1da0\n\ncount\u21ba : \u2200 {c} \u2192 \u2141 c \u2192 \u2115\ncount\u21ba = Fin.to\u2115 \u2218 count\u21ba\u1da0\n\n_\u223c[_]\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2141 n \u2192 Set\n_\u223c[_]\u2141_ {m} {n} f _\u223c_ g = \u27e82^ n \u27e9*( count\u21ba f ) \u223c \u27e82^ m \u27e9*( count\u21ba g )\n\n_\u2248\u2141_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\nf \u2248\u2141 g = f \u223c[ _\u2261_ ]\u2141 g\n\n\u2248\u2141-refl : \u2200 {n} {f : \u2141 n} \u2192 f \u2248\u2141 f\n\u2248\u2141-refl = \u2261.refl\n\n\u2248\u2141-sym : \u2200 {n} \u2192 Symmetric {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-sym = \u2261.sym\n\n\u2248\u2141-trans : \u2200 {n} \u2192 Transitive {A = \u2141 n} _\u2248\u2141_\n\u2248\u2141-trans = \u2261.trans\n\n\u2257\u21d2\u2248\u2141 : \u2200 {c} {f g : \u2141 c} \u2192 f \u2257\u21ba g \u2192 f \u2248\u2141 g\n\u2257\u21d2\u2248\u2141 {c} {f} {g} pf rewrite ext-# pf = \u2261.refl\n\n\u2248\u2141-cong : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f \u2248\u2141 f' \u2192 g \u2248\u2141 g'\n\u2248\u2141-cong f\u2257g f'\u2257g' f\u2248f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f\u2248f'\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c29a27648b332ce83cfe1041c4a7e43ea99f6d47","subject":"Only white spaces.","message":"Only white spaces.\n\nIgnore-this: 6f362b12580bb6145ec36b58e23ec241\n\ndarcs-hash:20100723151524-3bd4e-63df6df78306a5446bc3908cc63b8bdbe667bc1c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/Add.agda","new_file":"Test\/Fail\/Add.agda","new_contents":"module Test.Fail.Add where\n\ninfix  4 _\u2261_\ninfixl 6 _+_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n  _+_    : D \u2192 D \u2192 D\n  +-0x   : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx   : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n-- The ATPs cannot prove this postulate from the axioms.\npostulate\n  +-comm : (d e : D) \u2192 d + e \u2261 e + d\n{-# ATP prove +-comm #-}\n","old_contents":"module Test.Fail.Add where\n\ninfix  4 _\u2261_\ninfixl 6 _+_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192   zero   + d \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n-- The ATPs cannot prove this postulate from the axioms.\npostulate\n  +-comm : (d e : D) \u2192 d + e \u2261 e + d\n{-# ATP prove +-comm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ad4c710bc6b0cc90f68602b20fe8298a3a483d51","subject":"patching canonical forms after rewrite","message":"patching canonical forms after rewrite\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"canonical-forms.agda","new_file":"canonical-forms.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule canonical-forms where\n  cf-lam : \u2200{\u0394 \u0393 d \u03c41 \u03c42 } \u2192\n            \u0394 , \u0393 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n            d val \u2192\n            \u03a3[ x \u2208 Nat ] \u03a3[ d' \u2208 dhexp ] (d == (\u00b7\u03bb_[_]_ x \u03c41 d'))\n  cf-lam (TAVar x\u2081) ()\n  cf-lam (TALam D) VLam = _ , _ , refl\n  cf-lam (TAAp D x) ()\n  cf-lam (TAEHole x x\u2081) ()\n  cf-lam (TANEHole x D x\u2081) ()\n  cf-lam (TACast D x) ()\n\n  cf-base : \u2200{ \u0394 d \u0393 } \u2192\n            \u0394 , \u0393 \u22a2 d :: b \u2192\n            d val \u2192\n            d == c\n  cf-base TAConst VConst = refl\n  cf-base (TAVar x\u2081) ()\n  cf-base (TAAp D x) ()\n  cf-base (TAEHole x x\u2081) ()\n  cf-base (TANEHole x D x\u2081) ()\n  cf-base (TACast D x) ()\n\n  invert-lam : \u2200{ \u0394 \u0393 x \u03c4 d \u03c4'} \u2192\n               \u0394 , \u0393 \u22a2 (\u00b7\u03bb_[_]_ x \u03c4 d) :: \u03c4' \u2192\n               \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] (\u03c4' == (\u03c41 ==> \u03c42))\n  invert-lam (TALam D) = _ , _ , refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule canonical-forms where\n  cf-lam : \u2200{\u0394 \u0393 d \u03c41 \u03c42 } \u2192\n            \u0394 , \u0393 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n            d val \u2192\n            \u03a3[ x \u2208 Nat ] \u03a3[ d' \u2208 dhexp ] (d == (\u00b7\u03bb_[_]_ x \u03c41 d'))\n  cf-lam (TAVar x\u2081) ()\n  cf-lam (TALam D) VLam = _ , _ , refl\n  cf-lam (TAAp D x D\u2081) ()\n  cf-lam (TAEHole x x\u2081) ()\n  cf-lam (TANEHole x D x\u2081) ()\n  cf-lam (TACast D x) ()\n\n  cf-base : \u2200{ \u0394 d \u0393 } \u2192\n            \u0394 , \u0393 \u22a2 d :: b \u2192\n            d val \u2192\n            d == c\n  cf-base TAConst VConst = refl\n  cf-base (TAVar x\u2081) ()\n  cf-base (TAAp D x D\u2081) ()\n  cf-base (TAEHole x x\u2081) ()\n  cf-base (TANEHole x D x\u2081) ()\n  cf-base (TACast D x) ()\n\n  invert-lam : \u2200{ \u0394 \u0393 x \u03c4 d \u03c4'} \u2192\n               \u0394 , \u0393 \u22a2 (\u00b7\u03bb_[_]_ x \u03c4 d) :: \u03c4' \u2192\n               \u03a3[ \u03c41 \u2208 htyp ] \u03a3[ \u03c42 \u2208 htyp ] (\u03c4' == (\u03c41 ==> \u03c42))\n  invert-lam (TALam D) = _ , _ , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"14981a02997c504e9762bb64b13adf1e715ecfd1","subject":"More refactorings before adding bang","message":"More refactorings before adding bang\n\nInference becomes weaker for some reason.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x ,  rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ _ _ _ (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"87fa662e214c117b12105c3a9f23a6999c8c215c","subject":"Forgotten an ATP pragma.","message":"Forgotten an ATP pragma.\n\nIgnore-this: 658d2892e85c2963744d89e294196d75\n\ndarcs-hash:20100513053134-3bd4e-59e8a25f6a0225ce10b842c45ffe0373292c06c6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\nopen import Postulates using ( trans-LT )\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate\n      prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sx\u22640 = \u22a5-elim prf\n  where\n    postulate\n      -- The proof uses the axiom true\u2260false\n      prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n  postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n  {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n  postulate prf : \u00ac (LT m m) \u2192 \u22a5\n  {-# ATP prove prf #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n  postulate prf : lt (m + n) m \u2261 false \u2192\n                  lt (succ m + n) (succ m) \u2261 false\n  {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n  postulate prf : lt (m - zero) (succ m) \u2261 true\n  {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n  postulate prf : lt (zero - succ n) (succ zero) \u2261 true\n  {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n  postulate prf : lt (m - n) (succ m) \u2261 true \u2192\n                  lt (succ m - succ n) (succ (succ m)) \u2261 true\n  {-# ATP prove prf trans-LT minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n  postulate prf : (succ m - zero) + zero \u2261 succ m\n  {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n  postulate m>n : GT m n\n  {-# ATP prove m>n #-}\n\n  postulate prf : (m - n) + n \u2261 m \u2192\n                  (succ m - succ n) + succ n \u2261 succ m\n  {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n  postulate prf : (n - zero) + zero \u2261 n\n  {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n  postulate m\u2264n : LE m n\n  {-# ATP prove m\u2264n #-}\n\n  postulate prf : (n - m) + m \u2261 n \u2192\n                  (succ n - succ m) + succ m \u2261 succ n\n  {-# ATP prove prf +-comm minus-N sN #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\nopen import Postulates using ( trans-LT )\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate\n      prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sx\u22640 = \u22a5-elim prf\n  where\n    postulate\n      -- The proof uses the axiom true\u2260false\n      prf : \u22a5\n    {-# ATP prove prf #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n  postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n  {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n  postulate prf : \u00ac (LT m m) \u2192 \u22a5\n  {-# ATP prove prf #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n  postulate prf : lt (m + n) m \u2261 false \u2192\n                  lt (succ m + n) (succ m) \u2261 false\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n  postulate prf : lt (m - zero) (succ m) \u2261 true\n  {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n  postulate prf : lt (zero - succ n) (succ zero) \u2261 true\n  {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n  postulate prf : lt (m - n) (succ m) \u2261 true \u2192\n                  lt (succ m - succ n) (succ (succ m)) \u2261 true\n  {-# ATP prove prf trans-LT minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n  postulate prf : (succ m - zero) + zero \u2261 succ m\n  {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n  postulate m>n : GT m n\n  {-# ATP prove m>n #-}\n\n  postulate prf : (m - n) + n \u2261 m \u2192\n                  (succ m - succ n) + succ n \u2261 succ m\n  {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n  postulate prf : (n - zero) + zero \u2261 n\n  {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n  postulate m\u2264n : LE m n\n  {-# ATP prove m\u2264n #-}\n\n  postulate prf : (n - m) + m \u2261 n \u2192\n                  (succ n - succ m) + succ m \u2261 succ n\n  {-# ATP prove prf +-comm minus-N sN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e8426806209e35743f83594be7ea99b0cd59901c","subject":"Changed a precedence.","message":"Changed a precedence.\n\nIgnore-this: 6ae16a5e6172de27107db32fe0fd9ed5\n\ndarcs-hash:20101102155828-3bd4e-dcc9de24ac92cdf8c53e21bab077d349d60fa7e0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Minimal.agda","new_file":"src\/LTC\/Minimal.agda","new_contents":"------------------------------------------------------------------------------\n-- The LTC core\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\nmodule LTC.Minimal where\n\ninfixl 6 _\u2219_\ninfixr 5 _\u2237_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain\n\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language of LTC correspond to the PCF terms.\n\n--   t ::= x    | t t    | \\x \u2192 t\n--      | true  | false  | if t then t else t\n--      | 0     | succ t | pred t             | isZero t\n--      | []    | _\u2237_    | head               | tail\n--      | fix t\n--      | error\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n\n  -- LTC abstraction.\n  lam : (D \u2192 D) \u2192 D\n\n  -- LTC application.\n  -- The Agda application has higher precedence level than LTC application.\n  _\u2219_ : D \u2192 D \u2192 D\n\n  -- LTC error.\n  error : D\n\n  -- LTC fixed point operator.\n  fix : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n  -- LTC lists.\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n  head : D \u2192 D\n  tail : D \u2192 D\n\n------------------------------------------------------------------------------\n-- The LTC equality is the propositional identity on 'D'.\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\n-- The substitution is defined in LTC.MinimalER.\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n\n-- The LTC logical constants are the type theory logical constants via\n-- the Curry-Howard isomorphism.  For the implication and the\n-- universal quantifier we use the Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import Lib.Data.Empty       public\nopen import Lib.Data.Product     public\nopen import Lib.Data.Sum         public\n-- open import Lib.Data.Unit        public\nopen import Lib.Relation.Nullary public\nopen import LTC.Data.Product     public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans.\n  if-true  : (d\u2081 : D){d\u2082 : D} \u2192  if true then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true #-}\n{-# ATP axiom if-false #-}\n\npostulate\n  -- Conversion rules for pred.\n  pred-0 :           pred zero     \u2261 zero\n  pred-S : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom pred-0 #-}\n{-# ATP axiom pred-S #-}\n\npostulate\n  -- Conversion rules for isZero.\n  isZero-0 :           isZero zero     \u2261 true\n  isZero-S : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom isZero-0 #-}\n{-# ATP axiom isZero-S #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  beta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom beta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  fix-f : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom fix-f #-}\n\npostulate\n  -- Conversion rule for head.\n  head-\u2237 : (x xs : D) \u2192 head (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\npostulate\n  -- Conversion rule for tail.\n  tail-\u2237 : (x xs : D) \u2192 tail (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The LTC core\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\nmodule LTC.Minimal where\n\ninfixl 6 _\u2219_\ninfixr 5 _\u2237_\ninfix  5 if_then_else_\ninfix  3 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain\n\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language of LTC correspond to the PCF terms.\n\n--   t ::= x    | t t    | \\x \u2192 t\n--      | true  | false  | if t then t else t\n--      | 0     | succ t | pred t             | isZero t\n--      | []    | _\u2237_    | head               | tail\n--      | fix t\n--      | error\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n\n  -- LTC abstraction.\n  lam : (D \u2192 D) \u2192 D\n\n  -- LTC application.\n  -- The Agda application has higher precedence level than LTC application.\n  _\u2219_ : D \u2192 D \u2192 D\n\n  -- LTC error.\n  error : D\n\n  -- LTC fixed point operator.\n  fix : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n  -- LTC lists.\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n  head : D \u2192 D\n  tail : D \u2192 D\n\n------------------------------------------------------------------------------\n-- The LTC equality is the propositional identity on 'D'.\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\n-- The substitution is defined in LTC.MinimalER.\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n\n-- The LTC logical constants are the type theory logical constants via\n-- the Curry-Howard isomorphism.  For the implication and the\n-- universal quantifier we use the Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import Lib.Data.Empty       public\nopen import Lib.Data.Product     public\nopen import Lib.Data.Sum         public\n-- open import Lib.Data.Unit        public\nopen import Lib.Relation.Nullary public\nopen import LTC.Data.Product     public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans.\n  if-true  : (d\u2081 : D){d\u2082 : D} \u2192  if true then d\u2081 else d\u2082 \u2261 d\u2081\n  if-false : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom if-true #-}\n{-# ATP axiom if-false #-}\n\npostulate\n  -- Conversion rules for pred.\n  pred-0 :           pred zero     \u2261 zero\n  pred-S : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom pred-0 #-}\n{-# ATP axiom pred-S #-}\n\npostulate\n  -- Conversion rules for isZero.\n  isZero-0 :           isZero zero     \u2261 true\n  isZero-S : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom isZero-0 #-}\n{-# ATP axiom isZero-S #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  beta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom beta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  fix-f : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom fix-f #-}\n\npostulate\n  -- Conversion rule for head.\n  head-\u2237 : (x xs : D) \u2192 head (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\npostulate\n  -- Conversion rule for tail.\n  tail-\u2237 : (x xs : D) \u2192 tail (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fcc8c8821a1a2e22034344dd795c48b09e426809","subject":"Small cleanups","message":"Small cleanups\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/FundamentalProperty.agda","new_file":"Thesis\/SIRelBigStep\/FundamentalProperty.agda","new_contents":"module Thesis.SIRelBigStep.FundamentalProperty where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb j j<k v1 dv v2 vv \u2192\n    -- If we replace abs t by rec t, here we would need to invoke (by\n    -- well-founded recursion) rfundamental3svv on (abs t) at a smaller j, as follows:\n    --   rfundamental3svv j (abs t) \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n    -- As you'd expect, Agda's termination checker does not accept that call directly.\n      rfundamental3 j t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3sv (suc (suc k)) vs \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3sv (suc (suc k)) vt \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrelV3\u2192\u2295 vtv1 dtv vtv2 dtvv) =\n        bang v2 , bangapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"module Thesis.SIRelBigStep.FundamentalProperty where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb j j<k v1 dv v2 vv \u2192\n    -- If we replace abs t by rec t, here we would need to invoke (by\n    -- well-founded recursion) rfundamental3svv on (abs t) at a smaller j, as follows:\n    --   rfundamental3svv j (abs t) \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n    -- As you'd expect, Agda's termination checker does not accept that call directly.\n      rfundamental3 j t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrelV3\u2192\u2295 vtv1 dtv vtv2 dtvv) =\n        bang v2 , bangapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f9ab5825ca4fde0b30fff9bd8c84140347d668c6","subject":"Bring proof up to Eq. 1 in PLDI'14","message":"Bring proof up to Eq. 1 in PLDI'14\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ]\u03c4 da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ]\u03c4 db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ]\u03c4 \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = sym (right-id-int n)\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ]\u03c4 da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ]\u03c4 da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ]\u03c4 da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ]\u03c4 dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ]\u03c4 df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n\ncorrectDeriveOplus : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 t \u27e7Term \u03c11) \u2295 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus \u03c4 t d\u03c1\u03c1 = fromto\u2192\u2295 _ _ _ (fromtoDerive \u03c4 t d\u03c1\u03c1)\n\n-- Getting to the original equation 1 from PLDI'14.\n\nopen import New.LangOps\n\ncorrectDeriveOplus\u03c4 : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4)\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit t) (derive t) \u27e7Term d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus\u03c4 \u03c4 t {d\u03c1 = d\u03c1} {\u03c11 = \u03c11} d\u03c1\u03c1\n  rewrite oplus\u03c4-equiv _ d\u03c1 _ (\u27e6 fit t \u27e7Term d\u03c1) (\u27e6 derive t \u27e7Term d\u03c1)\n  | sym (fit-sound t d\u03c1\u03c1)\n  = correctDeriveOplus \u03c4 t d\u03c1\u03c1\n\nderiveGivesDerivative : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3)\u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app f a \u27e7Term \u03c11) \u2295 (\u27e6 app f a \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus \u03c4 (app f a) d\u03c1\u03c1\n\nderiveGivesDerivative\u2082 : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) (derive a)) \u27e7Term d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative\u2082 \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus\u03c4 \u03c4 (app f a) d\u03c1\u03c1\n\n-- Proof of the original equation 1 from PLDI'14.\neq1 : \u2200 {\u0393} \u03c3 \u03c4 \u2192\n  {nil\u03c1 : eCh \u0393} {\u03c1 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 nil\u03c1 from \u03c1 to \u03c1 \u2192\n  \u2200 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) (da : Term (\u0394\u0393 \u0393) (\u0394t \u03c3)) \u2192\n  (daa : [ \u03c3 ]\u03c4 (\u27e6 da \u27e7Term nil\u03c1) from (\u27e6 a \u27e7Term \u03c1) to (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1)) \u2192\n    \u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1 \u2261 \u27e6 app (fit f) (app\u2082 (oplus\u03c4o \u03c3) (fit a) da) \u27e7Term nil\u03c1\neq1 \u03c3 \u03c4 {nil\u03c1} {\u03c1} d\u03c1\u03c1 f a da daa\n  rewrite\n    oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit (app f a) \u27e7Term nil\u03c1) (\u27e6 (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1)\n  | sym (fit-sound f d\u03c1\u03c1)\n  | oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit a \u27e7Term nil\u03c1) (\u27e6 da \u27e7Term nil\u03c1)\n  | sym (fit-sound a d\u03c1\u03c1)\n  = fromto\u2192\u2295 (\u27e6 f \u27e7\u0394Term \u03c1 nil\u03c1 (\u27e6 a \u27e7Term \u03c1) (\u27e6 da \u27e7Term nil\u03c1)) _ _\n    (fromtoDerive _ f d\u03c1\u03c1 (\u27e6 da \u27e7Term nil\u03c1) (\u27e6 a \u27e7Term \u03c1)\n      (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1) daa)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ]\u03c4 da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ]\u03c4 db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ]\u03c4 \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = sym (right-id-int n)\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ]\u03c4 da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ]\u03c4 da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ]\u03c4 da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ]\u03c4 dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ]\u03c4 df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"92f785054b5dd55703ecbd7d5aec15af5e0d04d2","subject":"Update a bit README.agda (#41)","message":"Update a bit README.agda (#41)\n\nREADME.agda was not up-to-date. Remaining issues are described inline.\n\nOld-commit-hash: 93469332bbc849a2a13486a1bfae7d21dc9eb399\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--\n-- XXX: fill in again other details.\n\n-- 41e0f95d7fda7244a258b67f8a4255e4128a42c3 declares that the main theorems are\n-- in these files:\n\nopen import Denotation.Derive.Canon-Popl14\nopen import Denotation.Derive.Optimized-Popl14\n\n-- TODO: this file is intended to load all maintained Agda code; otherwise, a\n-- better solution to issue #41 should be found.\n--\n-- Moreover, arguably it would be good if this file would list all relevant\n-- imports, to highlight the structure of the development, as it did previously.\n-- But I'm not sure what's the best structure.\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\n-- The formalization is split across the following files.\n-- TODO: document them more.\n\n-- Note that we postulate function extensionality (in\n-- `Denotational.ExtensionalityPostulate`), known to be consistent\n-- with intensional type theory.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.ChangeTypes.ChangesAreDerivatives\nopen import Syntactic.ChangeContexts\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.ExtensionalityPostulate\nopen import Denotational.EqualityLemmas\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Changes\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\nopen import Denotational.WeakValidChanges\n\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- This finishes the correctness proof of the above work.\nopen import total\n\n\n-- EXAMPLES\n\nopen import Examples.Examples1\n\n\n-- NATURAL semantics\n\nopen import Syntactic.Closures\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\nopen import Natural.Soundness\n\n\n-- Other calculi\n\nopen import partial\nopen import lambda\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9cdc324caee9ca0888f5f85d83d0727e04d6e654","subject":"Implement '\u0394-term' for variables.","message":"Implement '\u0394-term' for variables.\n\nOld-commit-hash: dd0c2916cc4b7472353b0073fa6538f60e7325f5\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- CHANGE TERMS\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n","old_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs nil\n -- \u03bbx. nil\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"79872753ccfb2b6bcac7e598e5bfcae5f893d2b3","subject":"a better go with clearer holes","message":"a better go with clearer holes\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/Data1.agda","new_file":"models\/Data1.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check --no-positivity-check\n #-}\n\nmodule Data1 where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Cx : Set where\n  E : Cx\n  _\/_ : Cx -> Set -> Cx\n\ndata _:>_ : Cx -> Set -> Set where\n  ze : {G : Cx}{S : Set} -> (G \/ S) :> S\n  su : {G : Cx}{S T : Set} -> G :> T -> (G \/ S) :> T\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (G : Cx) : Set where\n  ?? : {S : Set} -> G :> S -> S -> CON G\n  PrfC : PROP -> CON G\n  _*C_ : CON G -> CON G -> CON G\n  PiC : (S : Set) -> (S -> CON G) -> CON G\n  SiC : (S : Set) -> (S -> CON G) -> CON G\n  MuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n  NuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n\nBox : (G : Cx) -> ((S : Set) -> (G :> S) -> Set) -> Set\nBox E F = One\nBox (G \/ S) F = Box G (\\ S n -> F S (su n)) * F S ze\n\npr : {G : Cx}{S : Set}{F : (S : Set) -> (G :> S) -> Set} -> Box G F -> (n : G :> S) -> F S n\npr fsf ze = snd fsf\npr fsf (su n) = pr (fst fsf) n\n\nPay : Cx -> Set\nPay G = Box G (\\ I _ -> I -> Set)\n\nArr : (G : Cx) -> Pay G -> Pay G -> Set\nArr G X Y = Box G (\\ S n -> (s : S) -> pr X n s -> pr Y n s)\n\n\n\nmutual\n  [|_|] : {G : Cx} -> CON G -> Pay G -> Set\n  [| ?? n i |]     X = pr X n i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Mu C X ) -> Mu C X o\n  codata Nu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Nu C X ) -> Nu C X o\n\nnoc :  {G : Cx}{O : Set}{C : O -> CON (G \/ O)}{X : Pay G}{o : O} ->\n       Nu C X o -> [| C o |] ( X , Nu C X )\nnoc (con xs) = xs\n\nmutual\n  map : {G : Cx}{X Y : Pay G} -> (C : CON G) ->\n        Arr G X Y ->\n        [| C |] X -> [| C |] Y\n  map (?? n i)     f = pr f n i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = foldMap C (Mu C _) f (\\ o -> con) o\n  map (NuC O C o)  f = unfoldMap C (Nu C _) f (\\ o -> noc) o\n\n  foldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n          Arr G X Y ->\n          ((o : O) -> [| C o |] ( Y , Z ) -> Z o) ->\n          (o : O) -> Mu C X o -> Z o\n  foldMap C Z f alg o (con xs) = alg o (map (C o) (f , foldMap C Z f alg) xs)\n\n  unfoldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n            Arr G X Y ->\n            ((o : O) -> Z o -> [| C o |] ( X , Z )) ->\n            (o : O) -> Z o -> Nu C Y o\n  unfoldMap C Z f coalg o y = con (map (C o) (f , unfoldMap C Z f coalg) (coalg o y))\n\nida : (G : Cx) -> (X : Pay G) -> Arr G X X\nida E = id\nida (G \/ S) = sig \\ Xs X -> ida G Xs , \\ s x -> x\n\ncompa : (G : Cx) -> (X Y Z : Pay G) -> Arr G Y Z -> Arr G X Y -> Arr G X Z\ncompa E = \\ _ _ _ _ _ -> _\ncompa (G \/ S) = sig \\ Xs X -> sig \\ Ys Y -> sig \\ Zs Z -> sig \\ fs f -> sig \\ gs g ->\n  compa G Xs Ys Zs fs gs , \\ s x -> f s (g s x)\n\ndata _==_ {X : Set}(x : X) : {Y : Set} -> Y -> Set where\n  refl : x == x\n\nsubst : {X : Set}{P : X -> Set}{x y : X} -> x == y -> P x -> P y\nsubst refl p = p\n\n_=,=_ : {S : Set}{T : S -> Set}\n        {s0 : S}{s1 : S} -> s0 == s1 ->\n        {t0 : T s0}{t1 : T s1} -> t0 == t1 ->\n        _==_ {Sig S T} (s0 , t0) {Sig S T} (s1 , t1)\nrefl =,= refl = refl\n        \n\nmylem :  (G : Cx) -> (X : Pay G) -> compa G X X X (ida G X) (ida G X) == ida G X\nmylem E X = refl\nmylem (G \/ S) X = mylem G (fst X) =,= refl\n\nclaim : (G : Cx){O : Set}(C : O -> CON (G \/ O))(X : Pay G)\n        (P : (o : O) -> Mu C X o -> Set) -> (p : (o : O)(y : Mu C X o) -> P o y)\n        (o : O) ->\n        compa (G \/ O) (X , Mu C X) (X , (\\ o -> Sig (Mu C X o) (P o))) (X , Mu C X)\n         (ida G X , konst fst) (ida G X , (\\ o y -> (y , p o y)))\n        == ida (G \/ O) (X , Mu C X)\nclaim G C X P p o = mylem G X =,= refl\n\nidLem : {G : Cx}{S : Set}{X : Pay G}(n : G :> S) ->\n        pr (ida G X) n == (\\ (s : S)(x : pr X n s) -> x)\nidLem ze = refl\nidLem (su y) = idLem y\n\n_=$_ : {S : Set}{T : S -> Set}{f g : (s : S) -> T s} ->\n        f == g -> (s : S) -> f s == g s\nrefl =$ s = refl\n\nmutual  -- and too intensional\n  mapId : {G : Cx}{X : Pay G} -> (C : CON G) -> (xs : [| C |] X) ->\n          map C (ida G X) xs == xs\n  mapId (?? y y') xs = (idLem y =$ y') =$ xs\n  mapId (PrfC y) xs = refl\n  mapId (y *C y') xs = mapId y (fst xs) =,= mapId y' (snd xs)\n  mapId (PiC S y) xs = {!!} -- no chance\n  mapId (SiC S y) xs = refl =,= mapId (y (fst xs)) (snd xs)\n  mapId (MuC O y y') xs = foldMapId y y' xs\n  mapId (NuC O y y') xs = {!!}\n\n  foldMapId : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n              (o : O)(z : Mu C X o) ->\n              foldMap C (Mu C X) (ida G X) (\\ o -> con) o z == z\n  foldMapId C o (con xzs) = {!!} -- close but no banana\n\nmapComp :  {G : Cx}{X Y Z : Pay G} (f : Arr G Y Z)(g : Arr G X Y)\n           (C : CON G) -> (xs : [| C |] X) ->\n           map C f (map C g xs) == map C (compa G X Y Z f g) xs\nmapComp = {!!}\n\n\n_-=-_ : {X : Set}{x y z : X} -> x == y -> y == z -> x == z\nrefl -=- refl = refl\n\ninduction : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] (X , (\\ o -> Sig (Mu C X o) (P o)))) ->\n             P o (con (map (C o) (ida G X , konst fst) xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction {G} C {X} P p o (con xys) =\n  subst {P = \\ xys -> P o (con xys)}{y = xys}\n    ((mapComp \n     (ida G X , konst fst) (ida G X , (\\ o y -> (y , induction C P p o y))) (C o) xys\n     -=- {!!}) \n      -=- mapId (C o) xys)\n    (p o (map (C o) (ida G X , (\\ o y -> (y , induction C P p o y))) xys))\n","old_contents":"{-# OPTIONS --type-in-type --no-termination-check --no-positivity-check\n #-}\n\nmodule Data1 where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Cx : Set where\n  E : Cx\n  _\/_ : Cx -> Set -> Cx\n\ndata _:>_ : Cx -> Set -> Set where\n  ze : {G : Cx}{S : Set} -> (G \/ S) :> S\n  su : {G : Cx}{S T : Set} -> G :> T -> (G \/ S) :> T\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (G : Cx) : Set where\n  ?? : {S : Set} -> G :> S -> S -> CON G\n  PrfC : PROP -> CON G\n  _*C_ : CON G -> CON G -> CON G\n  PiC : (S : Set) -> (S -> CON G) -> CON G\n  SiC : (S : Set) -> (S -> CON G) -> CON G\n  MuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n  NuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n\nBox : (G : Cx) -> ((S : Set) -> (G :> S) -> Set) -> Set\nBox E F = One\nBox (G \/ S) F = Box G (\\ S n -> F S (su n)) * F S ze\n\npr : {G : Cx}{S : Set}{F : (S : Set) -> (G :> S) -> Set} -> Box G F -> (n : G :> S) -> F S n\npr fsf ze = snd fsf\npr fsf (su n) = pr (fst fsf) n\n\nPay : Cx -> Set\nPay G = Box G (\\ I _ -> I -> Set)\n\nArr : (G : Cx) -> Pay G -> Pay G -> Set\nArr G X Y = Box G (\\ S n -> (s : S) -> pr X n s -> pr Y n s)\n\n\n\nmutual\n  [|_|] : {G : Cx} -> CON G -> Pay G -> Set\n  [| ?? n i |]     X = pr X n i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Mu C X ) -> Mu C X o\n  codata Nu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Nu C X ) -> Nu C X o\n\nnoc :  {G : Cx}{O : Set}{C : O -> CON (G \/ O)}{X : Pay G}{o : O} ->\n       Nu C X o -> [| C o |] ( X , Nu C X )\nnoc (con xs) = xs\n\nmutual\n  map : {G : Cx}{X Y : Pay G} -> (C : CON G) ->\n        Arr G X Y ->\n        [| C |] X -> [| C |] Y\n  map (?? n i)     f = pr f n i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = foldMap C (Mu C _) f (\\ o -> con) o\n  map (NuC O C o)  f = unfoldMap C (Nu C _) f (\\ o -> noc) o\n\n  foldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n          Arr G X Y ->\n          ((o : O) -> [| C o |] ( Y , Z ) -> Z o) ->\n          (o : O) -> Mu C X o -> Z o\n  foldMap C Z f alg o (con xs) = alg o (map (C o) (f , foldMap C Z f alg) xs)\n\n  unfoldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n            Arr G X Y ->\n            ((o : O) -> Z o -> [| C o |] ( X , Z )) ->\n            (o : O) -> Z o -> Nu C Y o\n  unfoldMap C Z f coalg o y = con (map (C o) (f , unfoldMap C Z f coalg) (coalg o y))\n\nida : (G : Cx) -> (X : Pay G) -> Arr G X X\nida E = id\nida (G \/ S) = sig \\ Xs X -> ida G Xs , \\ s x -> x\n\ncompa : (G : Cx) -> (X Y Z : Pay G) -> Arr G Y Z -> Arr G X Y -> Arr G X Z\ncompa E = \\ _ _ _ _ _ -> _\ncompa (G \/ S) = sig \\ Xs X -> sig \\ Ys Y -> sig \\ Zs Z -> sig \\ fs f -> sig \\ gs g ->\n  compa G Xs Ys Zs fs gs , \\ s x -> f s (g s x)\n\ninduction : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] (X , (\\ o -> Sig (Mu C X o) (P o)))) ->\n             P o (con (map (C o) (ida G X , konst fst) xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction {G} C {X} P p o (con xys) =\n  p o (map (C o) (ida G X , (\\ o y -> (y , induction C P p o y))) xys)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0aa4f446dbfe0dd8614667c859b6901797605a41","subject":"par com proved","message":"par com proved\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _ where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    \u214b\u1d3e Q      = Q\n    P      \u214b\u1d3e end    = P\n    com P\u1d38 \u214b\u1d3e com P\u1d3f = \u03a3\u1d3e LR (P\u1d38 \u214b\u1d9c P\u1d3f)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : proj\u2081 x \u2261 proj\u2081 y) \u2192 subst B p (proj\u2082 x) \u2261 proj\u2082 y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com x) = id\n\n  \u214b\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR {P = end} m p = m , p\n  \u214b\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7 \u2192 \u27e6 com' Out M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-sendL {P = end}{Q} m p = m , \u214b\u1d3e-rend (Q m) p\n  \u214b\u1d3e-sendL {P = com x} m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR {P = end}   f = f\n  \u214b\u1d3e-recvR {P = com x} f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7) \u2192 \u27e6 com' In M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-recvL {P = end}{Q} f x = \u214b\u1d3e-rend (Q x) (f x)\n  \u214b\u1d3e-recvL {P = com x} f = `L , f\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL {P = P m} m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR {P = dual (P m)} m (\u214b\u1d3e-id (P m))\n\n  module _ (\u03a0-ext : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : \u2200 x \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (\u03a0-ext \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (\u03a0-ext \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (\u03a0-ext (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (\u03a0-ext (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end     Q   p q = q\n  comma\u1d3e (com x) end p q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  comma\u1d3e-equiv : \u2200 P Q \u2192 Equiv (comma\u1d3e P Q)\n  comma\u1d3e-equiv P Q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c (mk io M P) = \u03a3\u1d9c M \u03bb m \u2192 Trace (P m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end         = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk q M P) = mk (dual\u1d35\u1d3c q) M \u03bb m \u2192 dual (P m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com q M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com Out M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com Out M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n-}\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\n{-\nmodule V2 where\n  mutual\n    Sim\u1d3e : Proto \u2192 Proto \u2192 Proto\n    Sim\u1d3e end      Q         = Q\n    Sim\u1d3e (\u03a0\u1d3e M P) Q         = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e (P m) Q\n    Sim\u1d3e P        end       = P\n    Sim\u1d3e P        (\u03a0\u1d3e M  Q) = \u03a0\u1d3e M \u03bb m \u2192 Sim\u1d3e P (Q m)\n    Sim\u1d3e (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 Sim\u1d3e (P m) (\u03a3\u1d3e M' Q)) ,inr: (\u03bb m' \u2192 Sim\u1d3e (\u03a3\u1d3e M P) (Q m')) ]\n\n  Sim\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\n  Sim\u1d3e-assoc end      Q        R         s                 = s\n  Sim\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = Sim\u1d3e-assoc (P m) _ _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , Sim\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , Sim\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  Sim\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7 \u2192 (m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7\n  Sim\u1d3e-sendR P Q s m = {!!}\n\n  Sim\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 Sim\u1d3e P (Q m) \u27e7) \u2192 \u27e6 Sim\u1d3e P (\u03a0\u1d3e M Q) \u27e7\n  Sim\u1d3e-recvR P Q s = {!!}\n\n  Sim\u1d3e-! : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 Sim\u1d3e Q P \u27e7\n  Sim\u1d3e-! end Q p = {!!}\n  Sim\u1d3e-! (\u03a0\u1d3e M P) end p x = Sim\u1d3e-! (P x) end (p x)\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-recvR (Q m') P \u03bb m \u2192 Sim\u1d3e-sendR {!!} {!!} (p m) m'\n  Sim\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 Sim\u1d3e-! (P m) (com (mk Out M' Q)) (p m) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) end       p = p \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 Sim\u1d3e-! (com (mk Out M P)) (Q m') (p m') \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (Sim\u1d3e-! (P m) (com (mk Out M' Q)) p) \n  Sim\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (Sim\u1d3e-! (com (mk Out M P)) (Q m) p) \n\n  Sim\u1d3e-apply : \u2200 P Q \u2192 \u27e6 Sim\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  Sim\u1d3e-apply end      Q         s           p       = s\n  Sim\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = Sim\u1d3e-apply (P m) Q (s m) p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = Sim\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  Sim\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , Sim\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\nmodule _ where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    \u214b\u1d3e Q      = Q\n    P      \u214b\u1d3e end    = P\n    com P\u1d38 \u214b\u1d3e com P\u1d3f = \u03a3\u1d3e LR (P\u1d38 \u214b\u1d9c P\u1d3f)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com x) = id\n\n  \u214b\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR {P = end} m p = m , p\n  \u214b\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7 \u2192 \u27e6 com' Out M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-sendL {P = end}{Q} m p = m , \u214b\u1d3e-rend (Q m) p\n  \u214b\u1d3e-sendL {P = com x} m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR {P = end}   f = f\n  \u214b\u1d3e-recvR {P = com x} f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M P}{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 Q m \u214b\u1d3e P \u27e7) \u2192 \u27e6 com' In M Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-recvL {P = end}{Q} f x = \u214b\u1d3e-rend (Q x) (f x)\n  \u214b\u1d3e-recvL {P = com x} f = `L , f\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL {P = P m} m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR {P = dual (P m)} m (\u214b\u1d3e-id (P m))\n\n  \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm P Q = {!!}\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end     Q   p q = q\n  comma\u1d3e (com x) end p q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  comma\u1d3e-equiv : \u2200 P Q \u2192 Equiv (comma\u1d3e P Q)\n  comma\u1d3e-equiv P Q = {!!}\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = {!!}\n\n  \u2295\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map = {!!}\n\n  &\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map = {!!}\n\n  \u214b\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u214b\u1d3e R \u27e7 \u2192 \u27e6 Q \u214b\u1d3e S \u27e7\n  \u214b\u1d3e-map = {!!}\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map = {!!}\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = let z = switchL _ _ _ (comma\u1d3e (P \u214b\u1d3e Q) (dual Q \u214b\u1d3e R) pq qr) in {!z!}\n\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"10a7117b4f1a927aec6a7e89b93219addbf0b773","subject":"TaggedDeltaTypes: remove unused postulates","message":"TaggedDeltaTypes: remove unused postulates\n\nOld-commit-hash: 33a5915e3451b89be50eee6831fad85b983cee50\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n_is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-wrt \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-wrt \u03c1 = \u22a4\n\u0394bag db is-valid-wrt \u03c1 = \u22a4\n\u0394var x is-valid-wrt \u03c1 = \u22a4\n\n_is-valid-wrt_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-wrt_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-wrt \u03c1 = Quadruple\n  (ds is-valid-wrt \u03c1)\n  (\u03bb _ \u2192 dt is-valid-wrt \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n_is-valid-wrt_ (\u0394map\u2081 s db) \u03c1 = db is-valid-wrt \u03c1\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 v-dt = mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-wrt \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-wrt \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t}))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\nunrestricted {t = nat n} = tt\nunrestricted {t = bag b} = tt\nunrestricted {t = var x} {\u03c1} = tt\nunrestricted {t = abs t} {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted {t = t})\nunrestricted {t = app s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t})\n  (validity {t = t}) tt\nunrestricted {t = add s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\nunrestricted {t = map s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted {t = t})\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted {t = s})\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {b : Bag} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n_is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-wrt \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-wrt \u03c1 = \u22a4\n\u0394bag db is-valid-wrt \u03c1 = \u22a4\n\u0394var x is-valid-wrt \u03c1 = \u22a4\n\n_is-valid-wrt_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-wrt_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-wrt \u03c1 = Quadruple\n  (ds is-valid-wrt \u03c1)\n  (\u03bb _ \u2192 dt is-valid-wrt \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n_is-valid-wrt_ (\u0394map\u2081 s db) \u03c1 = db is-valid-wrt \u03c1\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 v-dt = mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-wrt \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-wrt \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t}))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\nunrestricted {t = nat n} = tt\nunrestricted {t = bag b} = tt\nunrestricted {t = var x} {\u03c1} = tt\nunrestricted {t = abs t} {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted {t = t})\nunrestricted {t = app s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t})\n  (validity {t = t}) tt\nunrestricted {t = add s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\nunrestricted {t = map s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted {t = t})\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted {t = s})\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b2970528f4d873fea089cade59a0c8924b93611e","subject":"Describe the purpose of meaning.agda (see #28).","message":"Describe the purpose of meaning.agda (see #28).\n\nOld-commit-hash: 84cbf4aaec6e61e796b1039fd152d3e0cd4c4d53\n","repos":"inc-lc\/ilc-agda","old_file":"meaning.agda","new_file":"meaning.agda","new_contents":"module meaning where\n\n-- OVERLOADING \u27e6_\u27e7\n--\n-- This module defines a general mechanism for overloading the\n-- \u27e6_\u27e7 notation, using Agda\u2019s instance arguments.\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\nopen Meaning public\n  using (\u27e8_\u27e9\u27e6_\u27e7)\n","old_contents":"module meaning where\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\nopen Meaning public\n  using (\u27e8_\u27e9\u27e6_\u27e7)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a286973d39308022a734c117931faeba4ccb12ee","subject":"Monoid no longer depends on Data.Nat.NP","message":"Monoid no longer depends on Data.Nat.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Monoid.agda","new_file":"lib\/Algebra\/Monoid.agda","new_contents":"open import Function.NP\nopen import Data.Product.NP\nopen import Data.Nat\n  using    (\u2115; fold; zero)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nopen import Data.Integer\n  using    (\u2124; +_; -[1+_]; _\u2296_)\n  renaming (_+_ to _+\u2124_; _*_ to _*\u2124_)\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nopen import Algebra.FunctionProperties.Eq\nopen \u2261-Reasoning\n\nmodule Algebra.Monoid where\n\nrecord Monoid-Ops {\u2113} (M : Set \u2113) : Set \u2113 where\n  constructor mk\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : M \u2192 M \u2192 M\n    \u03b5   : M\n\n  _^\u207a_ : M \u2192 \u2115 \u2192 M\n  x ^\u207a n = fold \u03b5 (_\u2219_ x) n\n\n  module FromInverseOp\n         (_\u207b\u00b9   : Op\u2081 M)\n         where\n    infixl 7 _\/_\n\n    _\/_ : M \u2192 M \u2192 M\n    x \/ y = x \u2219 y \u207b\u00b9\n\n    open FromOp\u2082 _\/_ public renaming (op= to \/=)\n\n    _^\u207b_ _^\u207b\u2032_ : M \u2192 \u2115 \u2192 M\n    x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n    x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n    _^_ : M \u2192 \u2124 \u2192 M\n    x ^ -[1+ n ] = x ^\u207b(1+ n)\n    x ^ (+ n)    = x ^\u207a n\n\nrecord Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n\n  -- laws\n  field\n    assoc    : Associative _\u2219_\n    identity : Identity  \u03b5 _\u2219_\n\n  open FromOp\u2082 _\u2219_ public renaming (op= to \u2219=)\n  open FromAssoc _\u2219_ assoc public\n\n  module _ {b} where\n    ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n    ^\u207a0-\u03b5 = idp\n\n    ^\u207a1-id : b ^\u207a 1 \u2261 b\n    ^\u207a1-id = snd identity\n\n    ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n    ^\u207a2-\u2219 = ap (_\u2219_ _) ^\u207a1-id\n\n    ^\u207a-+ : \u2200 m {n} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n    ^\u207a-+ 0      = ! fst identity\n    ^\u207a-+ (1+ m) = ap (_\u2219_ b) (^\u207a-+ m) \u2666 ! assoc\n\n    ^\u207a-* : \u2200 m {n} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n    ^\u207a-* 0 = idp\n    ^\u207a-* (1+ m) {n}\n      = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n        b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n        b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n  comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n  comm-\u03b5 = snd identity \u2666 ! fst identity\n\n  module _ {c x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n    elim-assoc= = assoc \u2666 \u2219= idp e \u2666 snd identity\n\n    elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n    elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 fst identity\n\n  module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n    elim-inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module FromLeftInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-left : LeftCancel _\u2219_\n    cancels-\u2219-left {c} {x} {y} e\n      = x              \u2261\u27e8 ! fst identity \u27e9\n        \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inv-l) idp \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inv-l idp \u27e9\n        \u03b5 \u2219 y          \u2261\u27e8 fst identity \u27e9\n        y \u220e\n\n    inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-r = cancels-\u2219-left (! assoc \u2666 \u2219= inv-l idp \u2666 ! comm-\u03b5)\n\n    \/-\u2219 : \u2200 {x y} \u2192 x \u2261 (x \/ y) \u2219 y\n    \/-\u2219 {x} {y}\n      = x               \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inv-l) \u27e9\n        x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n        (x \/ y) \u2219 y     \u220e\n\n  module FromRightInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-right : RightCancel _\u2219_\n    cancels-\u2219-right {c} {x} {y} e\n      = x              \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inv-r) \u27e9\n        x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n        y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inv-r \u27e9\n        y \u2219 \u03b5          \u2261\u27e8 snd identity \u27e9\n        y \u220e\n\n    inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-l = cancels-\u2219-right (assoc \u2666 \u2219= idp inv-r \u2666 comm-\u03b5)\n\n    module _ {x y} where\n      is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n      is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 fst identity\n\n      is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n      is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 snd identity\n\n      \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n      \u2219-\/\n        = x            \u2261\u27e8 ! snd identity \u27e9\n          x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inv-r) \u27e9\n          x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n          (x \u2219 y) \/ y  \u220e\n\n    module _ {x y} where\n      unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n      unique-\u03b5-left eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          y \/ y        \u2261\u27e8 inv-r \u27e9\n          \u03b5            \u220e\n\n      unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n      unique-\u03b5-right eq\n        = y               \u2261\u27e8 ! is-\u03b5-left inv-l \u27e9\n          x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n          x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n          x \u207b\u00b9 \u2219 x        \u2261\u27e8 inv-l \u27e9\n          \u03b5               \u220e\n\n      unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n      unique-\u207b\u00b9 eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          \u03b5 \/ y        \u2261\u27e8 fst identity \u27e9\n          y \u207b\u00b9         \u220e\n\n    open FromLeftInverse _\u207b\u00b9 inv-l hiding (inv-r)\n\n    \u03b5\u207b\u00b9-\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n    \u03b5\u207b\u00b9-\u03b5 = unique-\u03b5-left inv-l\n\n    involutive : Involutive _\u207b\u00b9\n    involutive {x}\n      = cancels-\u2219-right\n        (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inv-l \u27e9\n         \u03b5              \u2261\u27e8 ! inv-r \u27e9\n         x \u2219 x \u207b\u00b9 \u220e)\n\n    \u207b\u00b9-hom\u2032 : \u2200 {x y} \u2192 (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n    \u207b\u00b9-hom\u2032 {x} {y} = cancels-\u2219-left {x \u2219 y}\n       ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 inv-r \u27e9\n        \u03b5                       \u2261\u27e8 ! inv-r \u27e9\n        x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! fst identity) \u27e9\n        x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! inv-r) idp) \u27e9\n        x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n        x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n        (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n    elim-\u2219-left-\u207b\u00b9\u2219 : \u2200 {c x y} \u2192 (c \u2219 x)\u207b\u00b9 \u2219 (c \u2219 y) \u2261 x \u207b\u00b9 \u2219 y\n    elim-\u2219-left-\u207b\u00b9\u2219 {c} {x} {y}\n      = (c \u2219 x)\u207b\u00b9   \u2219 (c \u2219 y)  \u2261\u27e8 \u2219= \u207b\u00b9-hom\u2032 idp \u27e9\n        x \u207b\u00b9 \u2219 c \u207b\u00b9 \u2219 (c \u2219 y)  \u2261\u27e8 elim-inner= inv-l \u27e9\n        x \u207b\u00b9 \u2219 y               \u220e\n\n    elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n    elim-\u2219-right-\/ {c} {x} {y}\n      = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n        x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-inner= inv-r \u27e9\n        x \/ y \u220e \n\n    module _ {b} where\n      ^\u207asuc : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n      ^\u207asuc 0 = comm-\u03b5\n      ^\u207asuc (1+ n) = ap (_\u2219_ b) (^\u207asuc n) \u2666 ! assoc\n\n      ^\u207a-comm : \u2200 n \u2192 b \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b\n      ^\u207a-comm = ^\u207asuc\n\n      ^\u207bsuc : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n      ^\u207bsuc n = ap _\u207b\u00b9 (^\u207asuc n) \u2666 \u207b\u00b9-hom\u2032\n\n      ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n      ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9-\u03b5\n      ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                      \u2666 ! \u207b\u00b9-hom\u2032\n                      \u2666 ap _\u207b\u00b9 (! ^\u207asuc n)\n\n      ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n      ^\u207b\u20321-id = snd identity\n\n      ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n      ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n      ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n      ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\nrecord Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-struct : Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Monoid-Struct mon-struct public\n\nrecord Commutative-Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n  field\n    mon-struct : Monoid-Struct mon-ops\n    comm : Commutative _\u2219_\n  open Monoid-Struct mon-struct public\n  open FromAssocComm _\u2219_ assoc comm public\n    hiding (!assoc=; assoc=; inner=)\n\nrecord Commutative-Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-comm   : Commutative-Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Commutative-Monoid-Struct mon-comm public\n  mon : Monoid M\n  mon = record { mon-struct = mon-struct }\n\n-- A renaming of Monoid with additive notation\nmodule Additive-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             )\n\n-- A renaming of Monoid with multiplicative notation\nmodule Multiplicative-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             )\n\nmodule Additive-Commutative-Monoid {M} (mon-comm : Commutative-Monoid M)\n  = Commutative-Monoid mon-comm\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             ; assoc-comm to +-assoc-comm\n             ; interchange to +-interchange\n             ; outer= to +-outer=\n             )\n\nmodule Multiplicative-Commutative-Monoid {M} (mon : Commutative-Monoid M) = Commutative-Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             ; assoc-comm to *-assoc-comm\n             ; interchange to *-interchange\n             ; outer= to *-outer=\n             )\n\nrecord MonoidHomomorphism {A B : Set}\n                          (monA0+ : Monoid A)\n                          (monB1* : Monoid B)\n                          (f : A \u2192 B) : Set where\n  open Additive-Monoid monA0+\n  open Multiplicative-Monoid monB1*\n  field\n    0-hom-1 : f 0\u1d50 \u2261 1\u1d50\n    +-hom-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Function.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP\n  using    (\u2115; fold; zero; suc; 1+_)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_)\nopen import Data.Integer\n  using    (\u2124; +_; -[1+_]; _\u2296_)\n  renaming (_+_ to _+\u2124_; _*_ to _*\u2124_)\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nopen import Algebra.FunctionProperties.Eq\nopen \u2261-Reasoning\n\nmodule Algebra.Monoid where\n\nrecord Monoid-Ops {\u2113} (M : Set \u2113) : Set \u2113 where\n  constructor mk\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : M \u2192 M \u2192 M\n    \u03b5   : M\n\n  _^\u207a_ : M \u2192 \u2115 \u2192 M\n  x ^\u207a n = fold \u03b5 (_\u2219_ x) n\n\n  module FromInverseOp\n         (_\u207b\u00b9   : Op\u2081 M)\n         where\n    infixl 7 _\/_\n\n    _\/_ : M \u2192 M \u2192 M\n    x \/ y = x \u2219 y \u207b\u00b9\n\n    open FromOp\u2082 _\/_ public renaming (op= to \/=)\n\n    _^\u207b_ _^\u207b\u2032_ : M \u2192 \u2115 \u2192 M\n    x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n    x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n    _^_ : M \u2192 \u2124 \u2192 M\n    x ^ -[1+ n ] = x ^\u207b(1+ n)\n    x ^ (+ n)    = x ^\u207a n\n\nrecord Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n\n  -- laws\n  field\n    assoc    : Associative _\u2219_\n    identity : Identity  \u03b5 _\u2219_\n\n  open FromOp\u2082 _\u2219_ public renaming (op= to \u2219=)\n  open FromAssoc _\u2219_ assoc public\n\n  module _ {b} where\n    ^\u207a0-\u03b5 : b ^\u207a 0 \u2261 \u03b5\n    ^\u207a0-\u03b5 = idp\n\n    ^\u207a1-id : b ^\u207a 1 \u2261 b\n    ^\u207a1-id = snd identity\n\n    ^\u207a2-\u2219 : b ^\u207a 2 \u2261 b \u2219 b\n    ^\u207a2-\u2219 = ap (_\u2219_ _) ^\u207a1-id\n\n    ^\u207a-+ : \u2200 m {n} \u2192 b ^\u207a (m +\u2115 n) \u2261 b ^\u207a m \u2219 b ^\u207a n\n    ^\u207a-+ 0      = ! fst identity\n    ^\u207a-+ (1+ m) = ap (_\u2219_ b) (^\u207a-+ m) \u2666 ! assoc\n\n    ^\u207a-* : \u2200 m {n} \u2192 b ^\u207a(m *\u2115 n) \u2261 (b ^\u207a n)^\u207a m\n    ^\u207a-* 0 = idp\n    ^\u207a-* (1+ m) {n}\n      = b ^\u207a (n +\u2115 m *\u2115 n)     \u2261\u27e8 ^\u207a-+ n \u27e9\n        b ^\u207a n \u2219 b ^\u207a(m *\u2115 n)  \u2261\u27e8 ap (_\u2219_ (b ^\u207a n)) (^\u207a-* m) \u27e9\n        b ^\u207a n \u2219 (b ^\u207a n)^\u207a m  \u220e\n\n  comm-\u03b5 : \u2200 {x} \u2192 x \u2219 \u03b5 \u2261 \u03b5 \u2219 x\n  comm-\u03b5 = snd identity \u2666 ! fst identity\n\n  module _ {c x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-assoc= : (c \u2219 x) \u2219 y \u2261 c\n    elim-assoc= = assoc \u2666 \u2219= idp e \u2666 snd identity\n\n    elim-!assoc= : x \u2219 (y \u2219 c) \u2261 c\n    elim-!assoc= = ! assoc \u2666 \u2219= e idp \u2666 fst identity\n\n  module _ {c d x y} (e : (x \u2219 y) \u2261 \u03b5) where\n    elim-inner= : (c \u2219 x) \u2219 (y \u2219 d) \u2261 c \u2219 d\n    elim-inner= = assoc \u2666 ap (_\u2219_ c) (elim-!assoc= e)\n\n  module FromLeftInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-left : LeftCancel _\u2219_\n    cancels-\u2219-left {c} {x} {y} e\n      = x              \u2261\u27e8 ! fst identity \u27e9\n        \u03b5 \u2219 x          \u2261\u27e8 \u2219= (! inv-l) idp \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 x   \u2261\u27e8 !assoc= e \u27e9\n        c \u207b\u00b9 \u2219 c \u2219 y   \u2261\u27e8 \u2219= inv-l idp \u27e9\n        \u03b5 \u2219 y          \u2261\u27e8 fst identity \u27e9\n        y \u220e\n\n    inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-r = cancels-\u2219-left (! assoc \u2666 \u2219= inv-l idp \u2666 ! comm-\u03b5)\n\n    \/-\u2219 : \u2200 {x y} \u2192 x \u2261 (x \/ y) \u2219 y\n    \/-\u2219 {x} {y}\n      = x               \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! inv-l) \u27e9\n        x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n        (x \/ y) \u2219 y     \u220e\n\n  module FromRightInverse\n         (_\u207b\u00b9   : Op\u2081 M)\n         (inv-r : RightInverse \u03b5 _\u207b\u00b9 _\u2219_)\n         where\n    open FromInverseOp _\u207b\u00b9\n\n    cancels-\u2219-right : RightCancel _\u2219_\n    cancels-\u2219-right {c} {x} {y} e\n      = x              \u2261\u27e8 ! snd identity \u27e9\n        x \u2219 \u03b5          \u2261\u27e8 \u2219= idp (! inv-r) \u27e9\n        x \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 assoc= e \u27e9\n        y \u2219 (c \u2219 c \u207b\u00b9) \u2261\u27e8 \u2219= idp inv-r \u27e9\n        y \u2219 \u03b5          \u2261\u27e8 snd identity \u27e9\n        y \u220e\n\n    inv-l : LeftInverse \u03b5 _\u207b\u00b9 _\u2219_\n    inv-l = cancels-\u2219-right (assoc \u2666 \u2219= idp inv-r \u2666 comm-\u03b5)\n\n    module _ {x y} where\n      is-\u03b5-left : x \u2261 \u03b5 \u2192 x \u2219 y \u2261 y\n      is-\u03b5-left e = ap (\u03bb z \u2192 z \u2219 _) e \u2666 fst identity\n\n      is-\u03b5-right : y \u2261 \u03b5 \u2192 x \u2219 y \u2261 x\n      is-\u03b5-right e = ap (\u03bb z \u2192 _ \u2219 z) e \u2666 snd identity\n\n      \u2219-\/ : x \u2261 (x \u2219 y) \/ y\n      \u2219-\/\n        = x            \u2261\u27e8 ! snd identity \u27e9\n          x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! inv-r) \u27e9\n          x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n          (x \u2219 y) \/ y  \u220e\n\n    module _ {x y} where\n      unique-\u03b5-left : x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n      unique-\u03b5-left eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          y \/ y        \u2261\u27e8 inv-r \u27e9\n          \u03b5            \u220e\n\n      unique-\u03b5-right : x \u2219 y \u2261 x \u2192 y \u2261 \u03b5\n      unique-\u03b5-right eq\n        = y               \u2261\u27e8 ! is-\u03b5-left inv-l \u27e9\n          x \u207b\u00b9 \u2219  x \u2219 y   \u2261\u27e8 assoc \u27e9\n          x \u207b\u00b9 \u2219 (x \u2219 y)  \u2261\u27e8 \u2219= idp eq \u27e9\n          x \u207b\u00b9 \u2219 x        \u2261\u27e8 inv-l \u27e9\n          \u03b5               \u220e\n\n      unique-\u207b\u00b9 : x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n      unique-\u207b\u00b9 eq\n        = x            \u2261\u27e8 \u2219-\/ \u27e9\n          (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n          \u03b5 \/ y        \u2261\u27e8 fst identity \u27e9\n          y \u207b\u00b9         \u220e\n\n    open FromLeftInverse _\u207b\u00b9 inv-l hiding (inv-r)\n\n    \u03b5\u207b\u00b9-\u03b5 : \u03b5 \u207b\u00b9 \u2261 \u03b5\n    \u03b5\u207b\u00b9-\u03b5 = unique-\u03b5-left inv-l\n\n    involutive : Involutive _\u207b\u00b9\n    involutive {x}\n      = cancels-\u2219-right\n        (x \u207b\u00b9 \u207b\u00b9 \u2219 x \u207b\u00b9 \u2261\u27e8 inv-l \u27e9\n         \u03b5              \u2261\u27e8 ! inv-r \u27e9\n         x \u2219 x \u207b\u00b9 \u220e)\n\n    \u207b\u00b9-hom\u2032 : \u2200 {x y} \u2192 (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n    \u207b\u00b9-hom\u2032 {x} {y} = cancels-\u2219-left {x \u2219 y}\n       ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 inv-r \u27e9\n        \u03b5                       \u2261\u27e8 ! inv-r \u27e9\n        x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! fst identity) \u27e9\n        x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! inv-r) idp) \u27e9\n        x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n        x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n        (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\n    elim-\u2219-right-\/ : \u2200 {c x y} \u2192 (x \u2219 c) \/ (y \u2219 c) \u2261 x \/ y\n    elim-\u2219-right-\/ {c} {x} {y}\n      = x \u2219 c \u2219 (y \u2219 c)\u207b\u00b9  \u2261\u27e8 ap (_\u2219_ _) \u207b\u00b9-hom\u2032  \u27e9\n        x \u2219 c \u2219 (c \u207b\u00b9 \/ y) \u2261\u27e8 elim-inner= inv-r \u27e9\n        x \/ y \u220e \n\n    module _ {b} where\n      ^\u207asuc : \u2200 n \u2192 b ^\u207a(1+ n) \u2261 b ^\u207a n \u2219 b\n      ^\u207asuc 0 = comm-\u03b5\n      ^\u207asuc (1+ n) = ap (_\u2219_ b) (^\u207asuc n) \u2666 ! assoc\n\n      ^\u207a-comm : \u2200 n \u2192 b \u2219 b ^\u207a n \u2261 b ^\u207a n \u2219 b\n      ^\u207a-comm = ^\u207asuc\n\n      ^\u207bsuc : \u2200 n \u2192 b ^\u207b(1+ n) \u2261 b \u207b\u00b9 \u2219 b ^\u207b n\n      ^\u207bsuc n = ap _\u207b\u00b9 (^\u207asuc n) \u2666 \u207b\u00b9-hom\u2032\n\n      ^\u207b\u2032-spec : \u2200 n \u2192 b ^\u207b\u2032 n \u2261 b ^\u207b n\n      ^\u207b\u2032-spec 0 = ! \u03b5\u207b\u00b9-\u03b5\n      ^\u207b\u2032-spec (1+ n) = ap (_\u2219_ (b \u207b\u00b9)) (^\u207b\u2032-spec n)\n                      \u2666 ! \u207b\u00b9-hom\u2032\n                      \u2666 ap _\u207b\u00b9 (! ^\u207asuc n)\n\n      ^\u207b\u20321-id : b ^\u207b\u2032 1 \u2261 b \u207b\u00b9\n      ^\u207b\u20321-id = snd identity\n\n      ^\u207b1-id : b ^\u207b 1 \u2261 b \u207b\u00b9\n      ^\u207b1-id = ! ^\u207b\u2032-spec 1 \u2666 ^\u207b\u20321-id\n\n      ^\u207b\u20322-\u2219 : b ^\u207b\u2032 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b\u20322-\u2219 = ap (_\u2219_ _) ^\u207b\u20321-id\n\n      ^\u207b2-\u2219 : b ^\u207b 2 \u2261 b \u207b\u00b9 \u2219 b \u207b\u00b9\n      ^\u207b2-\u2219 = ! ^\u207b\u2032-spec 2 \u2666 ^\u207b\u20322-\u2219\n\nrecord Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-struct : Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Monoid-Struct mon-struct public\n\nrecord Commutative-Monoid-Struct {\u2113} {M : Set \u2113} (mon-ops : Monoid-Ops M) : Set \u2113 where\n  open Monoid-Ops mon-ops\n  field\n    mon-struct : Monoid-Struct mon-ops\n    comm : Commutative _\u2219_\n  open Monoid-Struct mon-struct public\n  open FromAssocComm _\u2219_ assoc comm public\n    hiding (!assoc=; assoc=; inner=)\n\nrecord Commutative-Monoid (M : Set) : Set where\n  field\n    mon-ops    : Monoid-Ops M\n    mon-comm   : Commutative-Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Commutative-Monoid-Struct mon-comm public\n  mon : Monoid M\n  mon = record { mon-struct = mon-struct }\n\n-- A renaming of Monoid with additive notation\nmodule Additive-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             )\n\n-- A renaming of Monoid with multiplicative notation\nmodule Multiplicative-Monoid {M} (mon : Monoid M) = Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             )\n\nmodule Additive-Commutative-Monoid {M} (mon-comm : Commutative-Monoid M)\n  = Commutative-Monoid mon-comm\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d50\n             ; assoc to +-assoc; identity to +-identity\n             ; \u2219= to +=\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             ; assoc-comm to +-assoc-comm\n             ; interchange to +-interchange\n             ; outer= to +-outer=\n             )\n\nmodule Multiplicative-Commutative-Monoid {M} (mon : Commutative-Monoid M) = Commutative-Monoid mon\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d50\n             ; assoc to *-assoc; identity to *-identity\n             ; \u2219= to *=\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             ; assoc-comm to *-assoc-comm\n             ; interchange to *-interchange\n             ; outer= to *-outer=\n             )\n\nrecord MonoidHomomorphism {A B : Set}\n                          (monA0+ : Monoid A)\n                          (monB1* : Monoid B)\n                          (f : A \u2192 B) : Set where\n  open Additive-Monoid monA0+\n  open Multiplicative-Monoid monB1*\n  field\n    0-hom-1 : f 0\u1d50 \u2261 1\u1d50\n    +-hom-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7f90c33e52d52f274265268aceabfba18d1a96cd","subject":"Desc stratified model: put implicits up to DescDConst. Troubles starting.","message":"Desc stratified model: put implicits up to DescDConst. Troubles starting.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : (l : Level) -> Set l -> DescDConst -> IDesc Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d8873683e4cab1b73cf9e9ab1b5a4a0b0e3598e1","subject":"Fixed a note.","message":"Fixed a note.\n\nIgnore-this: f347072c6b8284ab860fd1f642643ef5\n\ndarcs-hash:20120611210021-3bd4e-2465917dcf0f41f68d67bb666d8fc2554918628e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/List\/WellFoundedRelationsSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/List\/WellFoundedRelationsSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded relation on lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of the standard library on\n-- 11 June 2012.\n\nmodule WellFoundedRelationsSL {A : Set} where\n\nopen import Data.Product\nopen import Data.List hiding ( length )\nopen import Data.Nat\n\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n-- We use our own version of the function length because we have\n-- better reductions.\nlength : List A \u2192 \u2115\nlength []       = 0\nlength (x \u2237 xs) = 1 + length xs\n\nopen module II = Induction.WellFounded.Inverse-image {_<_ = _<\u2032_} length\n\n-- Well-founded relation on lists based on their length.\nLTL : List A \u2192 List A \u2192 Set\nLTL xs ys = length xs <\u2032 length ys\n\n-- The relation LTL is well-founded (using the image inverse\n-- combinator).\nwfLTL : Well-founded LTL\nwfLTL = II.well-founded <-well-founded\n\n-- Well-founded relation on lists based on their structure.\nLTC : List A \u2192 List A \u2192 Set\nLTC xs ys = \u03a3 A (\u03bb a \u2192 ys \u2261 a \u2237 xs)\n\nLTC\u2192LTL : \u2200 {xs ys} \u2192 LTC xs ys \u2192 LTL xs ys\nLTC\u2192LTL (x , ys\u2261x\u2237xs) = helper ys\u2261x\u2237xs\n  where\n    helper : \u2200 {x xs ys} \u2192 ys \u2261 x \u2237 xs \u2192 length xs <\u2032 length ys\n    helper refl = \u2264\u2032-refl\n\nopen module S = Induction.WellFounded.Subrelation {_<\u2081_ = LTC} LTC\u2192LTL\n\n-- The relation LTC is well-founded (using the subrelation combinator).\nwfLTC : Well-founded LTC\nwfLTC = S.well-founded wfLTL\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded relation on lists\n------------------------------------------------------------------------------\n\n-- Tested with the development version of the standard library on\n-- 05 March 2011.\n\nmodule WellFoundedRelationsSL {A : Set} where\n\nopen import Data.Product\nopen import Data.List hiding ( length )\nopen import Data.Nat\n\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n-- We use our own version of the function length because we have\n-- better reductions.\nlength : List A \u2192 \u2115\nlength []       = 0\nlength (x \u2237 xs) = 1 + length xs\n\nopen module II = Induction.WellFounded.Inverse-image length\n\n-- Well-founded relation on lists based on their length.\nLTL : List A \u2192 List A \u2192 Set\nLTL xs ys = length xs <\u2032 length ys\n\n-- The relation LTL is well-founded (using the image inverse\n-- combinator).\nwfLTL : Well-founded LTL\nwfLTL = II.well-founded <-well-founded\n\n-- Well-founded relation on lists based on their structure.\nLTC : List A \u2192 List A \u2192 Set\nLTC xs ys = \u03a3 A (\u03bb a \u2192 ys \u2261 a \u2237 xs)\n\nLTC\u2192LTL : \u2200 {xs ys} \u2192 LTC xs ys \u2192 LTL xs ys\nLTC\u2192LTL (x , ys\u2261x\u2237xs) = helper ys\u2261x\u2237xs\n  where\n    helper : \u2200 {x xs ys} \u2192 ys \u2261 x \u2237 xs \u2192 length xs <\u2032 length ys\n    helper refl = \u2264\u2032-refl\n\nopen module S = Induction.WellFounded.Subrelation LTC\u2192LTL\n\n-- The relation LTC is well-founded (using the subrelation combinator).\nwfLTC : Well-founded LTC\nwfLTC = S.well-founded wfLTL\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"777b708bc8ea63b2f5cdeecd44516800cff940a3","subject":"Desc model: varl, constl, prodl, sigmal, pil implicit.","message":"Desc model: varl, constl, prodl, sigmal, pil implicit.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"74f06b4a4756e88776a9abcc3bdb8a4f68873e41","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/ConversionRules.agda","new_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/ConversionRules.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion rules for inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for lt it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (lt-s\u2081,\n  -- lt-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q, e.g.\n  --\n  -- proof\u2081\u2192proof\u2082 : \u2200 m n \u2192 lt-s\u2082 m n \u2192 lt-s\u2083 m n.\n\n  -- The terms lt-00, lt-0S, lt-S0, and lt-SS show the use of the\n  -- states lt-s\u2081, lt-s\u2082, ..., and the proofs associated with the\n  -- execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt.\n\n  -- Initially, the conversion rule fix-eq is applied.\n  lt-s\u2081 : D \u2192 D \u2192 D\n  lt-s\u2081 m n = lth (fix lth) \u00b7 m \u00b7 n\n\n  -- First argument application.\n  lt-s\u2082 : D \u2192 D\n  lt-s\u2082 m = lam (\u03bb n \u2192\n               if (iszero\u2081 n)\n                  then false\n                  else (if (iszero\u2081 m)\n                           then true\n                           else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)))\n\n\n  -- Second argument application.\n  lt-s\u2083 : D \u2192 D \u2192 D\n  lt-s\u2083 m n = if (iszero\u2081 n)\n                 then false\n                 else (if (iszero\u2081 m)\n                          then true\n                          else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  lt-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2084 m n b = if b\n                   then false\n                   else (if (iszero\u2081 m)\n                            then true\n                            else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 true.\n  -- It should be\n  -- lt-s\u2085 : D \u2192 D \u2192 D\n  -- lt-s\u2085 m n = false\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  lt-s\u2085 : D \u2192 D \u2192 D\n  lt-s\u2085 m n = if (iszero\u2081 m)\n                 then true\n                 else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b.\n  lt-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2086 m n b = if b\n                   then true\n                   else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n  -- Reduction iszero\u2081 m \u2261 true.\n  -- It should be\n  -- lt-s\u2087 : D \u2192 D \u2192 D\n  -- lt-s\u2087 m n = true\n  -- but we do not give a name to this step.\n\n   -- Reduction iszero\u2081 m \u2261 false.\n  lt-s\u2087 : D \u2192 D \u2192 D\n  lt-s\u2087 m n = fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ m) \u2261 m.\n  lt-s\u2088 : D \u2192 D \u2192 D\n  lt-s\u2088 m n = fix lth \u00b7 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  lt-s\u2089 : D \u2192 D \u2192 D\n  lt-s\u2089 m n = fix lth \u00b7 m \u00b7 n\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n\n    To prove the execution steps, e.g.\n\n    proof\u2083\u2192proof\u2084 : \u2200 m n \u2192 lt-s\u2083 m n \u2192 lt-s\u2084 m n)\n\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n\n    We prove (1) using\n\n    subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\n    where\n      \u2022 P is given by \u03bb t \u2192 C [m] \u2261 C [t],\n      \u2022 x \u2261 y is given m \u2261 n, and\n      \u2022 P x is given by C [m] \u2261 C [m] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 m n \u2192 fix lth \u00b7 m \u00b7 n  \u2261 lt-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u00b7 m \u00b7 n  \u2261 lth (fix lth) \u00b7 m \u00b7 n)\n                       (sym (fix-eq lth))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 m n \u2192 lt-s\u2081 m n \u2261 lt-s\u2082 m \u00b7 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u00b7 n \u2261 lt-s\u2082 m \u00b7 n)\n                       (sym (beta lt-s\u2082 m))\n                       refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 m n \u2192 lt-s\u2082 m \u00b7 n \u2261 lt-s\u2083 m n\n  proof\u2082\u208b\u2083 m n = beta (lt-s\u2083 m) n\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n  --\n  -- Reduction iszero n \u2261 b using that proof.\n  proof\u2083\u208b\u2084 : \u2200 m n b \u2192 iszero\u2081 n \u2261 b \u2192 lt-s\u2083 m n \u2261 lt-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b h = subst (\u03bb x \u2192 lt-s\u2084 m n x \u2261 lt-s\u2084 m n b)\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 n \u2261 true using the conversion rule if-true.\n  proof\u2084\u208a : \u2200 m n \u2192 lt-s\u2084 m n true \u2261 false\n  proof\u2084\u208a m n = if-true false\n\n  -- Reduction of iszero\u2081 n \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2084\u208b\u2085 : \u2200 m n \u2192 lt-s\u2084 m n false \u2261 lt-s\u2085 m n\n  proof\u2084\u208b\u2085 m n = if-false (lt-s\u2085 m n)\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n\n  -- Reduction iszero\u2081 m \u2261 b using that proof.\n  proof\u2085\u208b\u2086 : \u2200 m n b \u2192 iszero\u2081 m \u2261 b \u2192 lt-s\u2085 m n \u2261 lt-s\u2086 m n b\n  proof\u2085\u208b\u2086 m n b h = subst (\u03bb x \u2192 lt-s\u2086 m n x \u2261 lt-s\u2086 m n b)\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 m \u2261 true using the conversion rule if-true.\n  proof\u2086\u208a : \u2200 m n \u2192 lt-s\u2086 m n true \u2261 true\n  proof\u2086\u208a m n = if-true true\n\n  -- Reduction of iszero\u2081 m \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2086\u208b\u2087 : \u2200 m n \u2192 lt-s\u2086 m n false \u2261 lt-s\u2087 m n\n  proof\u2086\u208b\u2087 m n = if-false (lt-s\u2087 m n)\n\n  -- Reduction pred (succ m) \u2261 m using the conversion rule pred-S.\n  proof\u2087\u208b\u2088 : \u2200 m n \u2192 lt-s\u2087 (succ\u2081 m) n \u2261 lt-s\u2088 m n\n  proof\u2087\u208b\u2088 m n = subst (\u03bb x \u2192 lt-s\u2088 x n \u2261 lt-s\u2088 m n)\n                       (sym (pred-S m))\n                       refl\n\n  -- Reduction pred (succ n) \u2261 n using the conversion rule pred-S.\n  proof\u2088\u208b\u2089 : \u2200 m n \u2192 lt-s\u2088 m (succ\u2081 n) \u2261 lt-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = subst (\u03bb x \u2192 lt-s\u2089 m x \u2261 lt-s\u2089 m n)\n                       (sym (pred-S n))\n                       refl\n\n------------------------------------------------------------------------------\n\nprivate\n  X\u226e0 : \u2200 n \u2192 n \u226e zero\n  X\u226e0 n = fix lth \u00b7 n \u00b7 zero  \u2261\u27e8 proof\u2080\u208b\u2081 n zero \u27e9\n          lt-s\u2081 n zero        \u2261\u27e8 proof\u2081\u208b\u2082 n zero \u27e9\n          lt-s\u2082 n \u00b7 zero      \u2261\u27e8 proof\u2082\u208b\u2083 n zero \u27e9\n          lt-s\u2083 n zero        \u2261\u27e8 proof\u2083\u208b\u2084 n zero true iszero-0 \u27e9\n          lt-s\u2084 n zero true   \u2261\u27e8 proof\u2084\u208a n zero \u27e9\n          false              \u220e\n\nlt-00 : zero \u226e zero\nlt-00 = X\u226e0 zero\n\nlt-0S : \u2200 n \u2192 zero < succ\u2081 n\nlt-0S n =\n  fix lth \u00b7 zero \u00b7 (succ\u2081 n)    \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ\u2081 n) \u27e9\n  lt-s\u2081 zero (succ\u2081 n)          \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ\u2081 n) \u27e9\n  lt-s\u2082 zero \u00b7 (succ\u2081 n)        \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ\u2081 n) \u27e9\n  lt-s\u2083 zero (succ\u2081 n)          \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 zero (succ\u2081 n) false    \u2261\u27e8 proof\u2084\u208b\u2085 zero (succ\u2081 n) \u27e9\n  lt-s\u2085 zero (succ\u2081 n)          \u2261\u27e8 proof\u2085\u208b\u2086 zero (succ\u2081 n) true iszero-0 \u27e9\n  lt-s\u2086 zero (succ\u2081 n) true     \u2261\u27e8 proof\u2086\u208a  zero (succ\u2081 n) \u27e9\n  true                         \u220e\n\nlt-S0 : \u2200 n \u2192 succ\u2081 n \u226e zero\nlt-S0 n = X\u226e0 (succ\u2081 n)\n\nlt-SS : \u2200 m n \u2192 lt (succ\u2081 m) (succ\u2081 n) \u2261 lt m n\nlt-SS m n =\n  fix lth \u00b7 (succ\u2081 m) \u00b7 (succ\u2081 n)  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2081 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2082 (succ\u2081 m) \u00b7 (succ\u2081 n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2083 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 m) (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2085 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 m) (succ\u2081 n) false (iszero-S m) \u27e9\n  lt-s\u2086 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2086\u208b\u2087 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2087 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2087\u208b\u2088 m (succ\u2081 n) \u27e9\n  lt-s\u2088 m (succ\u2081 n)                \u2261\u27e8 proof\u2088\u208b\u2089 m n \u27e9\n  lt-s\u2089 m n                        \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Conversion rules for inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for lt it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (lt-s\u2081,\n  -- lt-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q, e.g.\n  --\n  -- proof\u2081\u2192proof\u2082 : \u2200 m n \u2192 lt-s\u2082 m n \u2192 lt-s\u2083 m n.\n\n  -- The terms lt-00, lt-0S, lt-S0, and lt-SS show the use of the\n  -- states lt-s\u2081, lt-s\u2082, ..., and the proofs associated with the\n  -- execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt.\n\n  -- Initially, the conversion rule fix-eq is applied.\n  lt-s\u2081 : D \u2192 D \u2192 D\n  lt-s\u2081 m n = lth (fix lth) \u00b7 m \u00b7 n\n\n  -- First argument application.\n  lt-s\u2082 : D \u2192 D\n  lt-s\u2082 m = lam (\u03bb n \u2192\n               if (iszero\u2081 n)\n                  then false\n                  else (if (iszero\u2081 m)\n                           then true\n                           else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)))\n\n\n  -- Second argument application.\n  lt-s\u2083 : D \u2192 D \u2192 D\n  lt-s\u2083 m n = if (iszero\u2081 n)\n                 then false\n                 else (if (iszero\u2081 m)\n                          then true\n                          else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  lt-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2084 m n b = if b\n                   then false\n                   else (if (iszero\u2081 m)\n                            then true\n                            else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 true.\n  -- It should be\n  -- lt-s\u2085 : D \u2192 D \u2192 D\n  -- lt-s\u2085 m n = false\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  lt-s\u2085 : D \u2192 D \u2192 D\n  lt-s\u2085 m n = if (iszero\u2081 m)\n                 then true\n                 else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b.\n  lt-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2086 m n b = if b\n                   then true\n                   else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n  -- Reduction iszero\u2081 m \u2261 true.\n  -- It should be\n  -- lt-s\u2087 : D \u2192 D \u2192 D\n  -- lt-s\u2087 m n = true\n  -- but we do not give a name to this step.\n\n   -- Reduction iszero\u2081 m \u2261 false.\n  lt-s\u2087 : D \u2192 D \u2192 D\n  lt-s\u2087 m n = fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ m) \u2261 m.\n  lt-s\u2088 : D \u2192 D \u2192 D\n  lt-s\u2088 m n = fix lth \u00b7 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  lt-s\u2089 : D \u2192 D \u2192 D\n  lt-s\u2089 m n = fix lth \u00b7 m \u00b7 n\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n\n    To prove the execution steps, e.g.\n\n    proof\u2083\u2192proof\u2084 : \u2200 m n \u2192 lt-s\u2083 m n \u2192 lt-s\u2084 m n)\n\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n\n    We prove (1) using\n\n    subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\n    where\n      \u2022 P is given by \u03bb t \u2192 C [m] \u2261 C [t],\n      \u2022 x \u2261 y is given m \u2261 n, and\n      \u2022 P x is given by C [m] \u2261 C [m] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 m n \u2192 fix lth \u00b7 m \u00b7 n  \u2261 lt-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u00b7 m \u00b7 n  \u2261 lth (fix lth) \u00b7 m \u00b7 n )\n                       (sym (fix-eq lth ))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 m n \u2192 lt-s\u2081 m n \u2261 lt-s\u2082 m \u00b7 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u00b7 n \u2261 lt-s\u2082 m \u00b7 n)\n                       (sym (beta lt-s\u2082 m))\n                       refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 m n \u2192 lt-s\u2082 m \u00b7 n \u2261 lt-s\u2083 m n\n  proof\u2082\u208b\u2083 m n = beta (lt-s\u2083 m) n\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n  --\n  -- Reduction iszero n \u2261 b using that proof.\n  proof\u2083\u208b\u2084 : \u2200 m n b \u2192 iszero\u2081 n \u2261 b \u2192 lt-s\u2083 m n \u2261 lt-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b h = subst (\u03bb x \u2192 lt-s\u2084 m n x \u2261 lt-s\u2084 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 n \u2261 true using the conversion rule if-true.\n  proof\u2084\u208a : \u2200 m n \u2192 lt-s\u2084 m n true \u2261 false\n  proof\u2084\u208a m n = if-true false\n\n  -- Reduction of iszero\u2081 n \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2084\u208b\u2085 : \u2200 m n \u2192 lt-s\u2084 m n false \u2261 lt-s\u2085 m n\n  proof\u2084\u208b\u2085 m n = if-false (lt-s\u2085 m n)\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n  --\n  -- Reduction iszero\u2081 m \u2261 b using that proof.\n  proof\u2085\u208b\u2086 : \u2200 m n b \u2192 iszero\u2081 m \u2261 b \u2192 lt-s\u2085 m n \u2261 lt-s\u2086 m n b\n  proof\u2085\u208b\u2086 m n b h = subst (\u03bb x \u2192 lt-s\u2086 m n x \u2261 lt-s\u2086 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 m \u2261 true using the conversion rule if-true.\n  proof\u2086\u208a : \u2200 m n \u2192 lt-s\u2086 m n true \u2261 true\n  proof\u2086\u208a m n = if-true true\n\n  -- Reduction of iszero\u2081 m \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2086\u208b\u2087 : \u2200 m n \u2192 lt-s\u2086 m n false \u2261 lt-s\u2087 m n\n  proof\u2086\u208b\u2087 m n = if-false (lt-s\u2087 m n)\n\n  -- Reduction pred (succ m) \u2261 m using the conversion rule pred-S.\n  proof\u2087\u208b\u2088 : \u2200 m n \u2192 lt-s\u2087 (succ\u2081 m) n  \u2261 lt-s\u2088 m n\n  proof\u2087\u208b\u2088 m n = subst (\u03bb x \u2192 lt-s\u2088 x n \u2261 lt-s\u2088 m n)\n                       (sym (pred-S m ))\n                       refl\n\n  -- Reduction pred (succ n) \u2261 n using the conversion rule pred-S.\n  proof\u2088\u208b\u2089 : \u2200 m n \u2192 lt-s\u2088 m (succ\u2081 n)  \u2261 lt-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = subst (\u03bb x \u2192 lt-s\u2089 m x \u2261 lt-s\u2089 m n)\n                       (sym (pred-S n ))\n                       refl\n\n------------------------------------------------------------------------------\n\nprivate\n  X\u226e0 : \u2200 n \u2192 n \u226e zero\n  X\u226e0 n = fix lth \u00b7 n \u00b7 zero  \u2261\u27e8 proof\u2080\u208b\u2081 n zero \u27e9\n          lt-s\u2081 n zero        \u2261\u27e8 proof\u2081\u208b\u2082 n zero \u27e9\n          lt-s\u2082 n \u00b7 zero      \u2261\u27e8 proof\u2082\u208b\u2083 n zero \u27e9\n          lt-s\u2083 n zero        \u2261\u27e8 proof\u2083\u208b\u2084 n zero true iszero-0 \u27e9\n          lt-s\u2084 n zero true   \u2261\u27e8 proof\u2084\u208a n zero \u27e9\n          false              \u220e\n\nlt-00 : zero \u226e zero\nlt-00 = X\u226e0 zero\n\nlt-0S : \u2200 n \u2192 zero < succ\u2081 n\nlt-0S n =\n  fix lth \u00b7 zero \u00b7 (succ\u2081 n)    \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ\u2081 n) \u27e9\n  lt-s\u2081 zero (succ\u2081 n)          \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ\u2081 n) \u27e9\n  lt-s\u2082 zero \u00b7 (succ\u2081 n)        \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ\u2081 n) \u27e9\n  lt-s\u2083 zero (succ\u2081 n)          \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 zero (succ\u2081 n) false    \u2261\u27e8 proof\u2084\u208b\u2085 zero (succ\u2081 n) \u27e9\n  lt-s\u2085 zero (succ\u2081 n)          \u2261\u27e8 proof\u2085\u208b\u2086 zero (succ\u2081 n) true iszero-0 \u27e9\n  lt-s\u2086 zero (succ\u2081 n) true     \u2261\u27e8 proof\u2086\u208a  zero (succ\u2081 n) \u27e9\n  true                         \u220e\n\nlt-S0 : \u2200 n \u2192 succ\u2081 n \u226e zero\nlt-S0 n = X\u226e0 (succ\u2081 n)\n\nlt-SS : \u2200 m n \u2192 lt (succ\u2081 m) (succ\u2081 n) \u2261 lt m n\nlt-SS m n =\n  fix lth \u00b7 (succ\u2081 m) \u00b7 (succ\u2081 n)  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2081 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2082 (succ\u2081 m) \u00b7 (succ\u2081 n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2083 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 m) (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2085 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 m) (succ\u2081 n) false (iszero-S m) \u27e9\n  lt-s\u2086 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2086\u208b\u2087 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2087 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2087\u208b\u2088 m (succ\u2081 n) \u27e9\n  lt-s\u2088 m (succ\u2081 n)                \u2261\u27e8 proof\u2088\u208b\u2089 m n \u27e9\n  lt-s\u2089 m n                        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f9aeb58a90443e04e8fd3afd4951c3d768d88d29","subject":"Agda: update description of Plotkin's context","message":"Agda: update description of Plotkin's context\n\nReason: the previous description contains a modularization task\nthat is already partially carried out.\n\nReference:\nhttps:\/\/github.com\/ps-mr\/ilc\/commit\/249e99d807e2382a37fb35aee78c235b98e427aa#commitcomment-3659727\n\nOld-commit-hash: 3c52ba4a07600646db8de764b1460fa6b69c3014\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Context\/Plotkin.agda","new_file":"Syntax\/Context\/Plotkin.agda","new_contents":"module Syntax.Context.Plotkin where\n\n-- Context for Plotkin-stype language descriptions\n--\n-- This module will tend to duplicate Syntax.Context to a large\n-- extent. Consider having Syntax.Context specialize future\n-- content of this module to maintain its interface.\n\ninfixr 9 _\u2022_\n\ndata Context {Type : Set} : Set where\n  \u2205 : Context {Type}\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context {Type}) \u2192 Context {Type}\n\ndata Var {Type : Set} : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n","old_contents":"module Syntax.Context.Plotkin where\n\n-- Context for Plotkin-stype language descriptions\n--\n-- Duplicates Syntax.Context to a large extent.\n-- Consider having Syntax.Context import this module.\n\ninfixr 9 _\u2022_\n\ndata Context {Type : Set} : Set where\n  \u2205 : Context {Type}\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context {Type}) \u2192 Context {Type}\n\ndata Var {Type : Set} : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0ac7c7ef275b6e2f8d1e9bf5ad1b5030e778412a","subject":"Ternary validity: require matching typing contexts for closures","message":"Ternary validity: require matching typing contexts for closures\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n    }\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- The original relation allows unrelated environments. However, while that is\n  -- fine as a logical relation, it's not OK if we want to prove that validity\n  -- agrees with oplus. We want a finer relation.\n  -- Also: we still need to demand the actual environments to be related, and\n  -- the bodies to match. Haven't done that yet. On the other hand, since we do want\n  -- to allow for replacement changes, that would probably complicate the proof\n  -- elsewhere.\n  relT3 : \u2200 {\u03c4 \u0393 \u0394\u0393} (t1 : Term \u0393 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  -- Weakening in this definition is going to be annoying to use. And having to\n  -- construct terms is ugly.\n\n  -- Weakening could be avoided if we use a separate language of change terms\n  -- with two environments, and with a dclosure binding two variables at once,\n  -- and so on.\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure dt d\u03c1) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n      relV3 \u03c3 v1 dv v2 k \u2192\n      relT3 t1 (app (weaken (drop (\u0394\u03c4 \u03c3) \u2022 \u227c-refl) dt) (var this)) t2 (v1 \u2022 \u03c11) (dv \u2022 v1 \u2022 d\u03c1) (v2 \u2022 \u03c12) k\n    }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3[ \u2261\u0393 \u2208 \u03931 \u2261 \u03932 ]\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT\n        t1\n        (subst (\u03bb \u0393 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4) (sym \u2261\u0393) t2)\n        (v1 \u2022 \u03c11)\n        (subst (\u03bb \u0393 \u2192 \u27e6 \u03c3 \u2022 \u0393 \u27e7Context) (sym \u2261\u0393) (v2 \u2022 \u03c12))\n        k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus. Now relT demands\n  -- environments with matching contexts, we could do the same here.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  -- Weakening in this definition is going to be annoying to use. And having to\n  -- construct terms is ugly.\n\n  -- Weakening could be avoided if we use a separate language of change terms\n  -- with two environments, and with a dclosure binding two variables at once,\n  -- and so on.\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 (app (weaken (drop (\u0394\u03c4 \u03c3) \u2022 \u227c-refl) dt) (var this)) t2 (v1 \u2022 \u03c11) (dv \u2022 v1 \u2022 d\u03c1) (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"982272fd7b24b84fb858065b7fd527aa4aa1ef69","subject":"Remove ConsonantSound to allow for later combinations of consonants to indicate different sounds.","message":"Remove ConsonantSound to allow for later combinations of consonants to indicate different sounds.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda2\/Text\/Greek\/Smyth.agda","new_file":"agda2\/Text\/Greek\/Smyth.agda","new_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc : OlderLetter\n  \u03d8 : OlderLetter\n  \u03e1 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  \u1fb0 : VowelWithLength \u03b1 short\n  \u1fb1 : VowelWithLength \u03b1 long\n  \u03b5 : VowelWithLength \u03b5 short\n  \u03b7 : VowelWithLength \u03b7 long\n  \u1fd0 : VowelWithLength \u03b9 short\n  \u1fd1 : VowelWithLength \u03b9 long\n  \u03bf : VowelWithLength \u03bf short\n  \u1fe0 : VowelWithLength \u03c5 short\n  \u1fe1 : VowelWithLength \u03c5 long\n  \u03c9 : VowelWithLength \u03c9 long\n\n-- Smyth \u00a75,6\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9-gen : Diphthong \u03b5 \u03b9\n  \u03b5\u03b9-sp : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5-gen : Diphthong \u03bf \u03c5\n  \u03bf\u03c5-sp : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n  \u03c9\u03c5 : Diphthong \u03c9 \u03c5 -- Smyth \u00a75 D. New Ionic\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03c9 = open-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03c5 = closed\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5-gen = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b6 : Voiced \u03b6\n-- \u03b3-nasal : Voiced\n\n-- Smyth \u00a715 b\ndata Voiceless : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n--  Smyth \u00a715 a. \u03c1 with the rough breathing is voiceless\n\n-- Smyth \u00a716\ndata PartOfMouthClass : Set where\n  labial dental palatal : PartOfMouthClass\n\ndata Labial : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03c0 : Labial \u03c0\n  \u03b2 : Labial \u03b2\n  \u03c6 : Labial \u03c6\n\ndata Dental : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03c4 : Dental \u03c4\n  \u03b4 : Dental \u03b4\n  \u03b8 : Dental \u03b8\n\ndata Palatal : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03ba : Palatal \u03ba\n  \u03b3 : Palatal \u03b3\n  \u03c7 : Palatal \u03c7\n\ndata SmoothStop : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03c0 : SmoothStop \u03c0\n  \u03c4 : SmoothStop \u03c4\n  \u03ba : SmoothStop \u03ba\n\ndata MiddleStop : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03b2 : MiddleStop \u03b2\n  \u03b4 : MiddleStop \u03b4\n  \u03b3 : MiddleStop \u03b3\n\ndata RoughStop : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03c6 : RoughStop \u03c6\n  \u03b8 : RoughStop \u03b8\n  \u03c7 : RoughStop \u03c7\n\n-- Smyth \u00a717\ndata Spirant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03c3 : Spirant \u03c3\n\n-- Smyth \u00a718\ndata Liquid : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03bb\u2032 : Liquid \u03bb\u2032\n  \u03c1 : Liquid \u03c1\n\n-- Smyth \u00a719\ndata Nasal : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03bc : Nasal \u03bc\n  \u03bd : Nasal \u03bd\n--  \u03b3-nasal : Nasal\n\n-- Smyth \u00a719 a\ndata GammaNasalCombo : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Consonant \u2113\u2081 \u2192 Consonant \u2113\u2082 \u2192 Set where\n  \u03b3\u03ba : GammaNasalCombo \u03b3 \u03ba\n  \u03b3\u03b3 : GammaNasalCombo \u03b3 \u03b3\n  \u03b3\u03c7 : GammaNasalCombo \u03b3 \u03c7\n  \u03b3\u03be : GammaNasalCombo \u03b3 \u03be\n\n-- Smyth \u00a720\ndata Semivowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 Set where\n  \u03b9\u032f : Semivowel \u03b9\n  \u03c5\u032f : Semivowel \u03c5\n\n-- Smyth \u00a720 b\ndata Sonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03bb\u0325 : Sonant \u03bb\u2032\n  \u03bc\u0325 : Sonant \u03bc\n  \u03bd\u0325 : Sonant \u03b3\n  \u03c1\u0325 : Sonant \u03c1\n  \u03c3\u0325 : Sonant \u03c3\n\n-- Smyth \u00a721\ndata DoubleConsonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 Set where\n  \u03b6 : DoubleConsonant \u03b6\n  \u03be : DoubleConsonant \u03be\n  \u03c8 : DoubleConsonant \u03c8\n\ndata _+_\u21d2DoubleConsonant_ : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u2083} \u2192 Consonant \u2113\u2081 \u2192 Consonant \u2113\u2082 \u2192 Consonant \u2113\u2083 \u2192 Set where\n  \u03c3+\u03b4\u21d2DoubleConsonant\u03b6 : \u03c3 + \u03b4 \u21d2DoubleConsonant \u03b6 -- Smyth \u00a726 D. Aeolic has \u03c3\u03b4 for \u03b6\n  \u03b4+\u03c3\u21d2DoubleConsonant\u03b6 : \u03b4 + \u03c3 \u21d2DoubleConsonant \u03b6\n\n  \u03ba+\u03c3\u21d2DoubleConsonant\u03be : \u03ba + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03b3+\u03c3\u21d2DoubleConsonant\u03be : \u03b3 + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03c7+\u03c3\u21d2DoubleConsonant\u03be : \u03c7 + \u03c3 \u21d2DoubleConsonant \u03be\n\n  \u03c0+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c0 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03b2+\u03c3\u21d2DoubleConsonant\u03c8 : \u03b2 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03c6+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c6 + \u03c3 \u21d2DoubleConsonant \u03c8\n\n\n-- TODO (needs a unified model)\n\n-- Diphthong\n-- Smyth \u00a75 D. Ionic has \u03b7\u03c5 for Attic \u03b1\u03c5 in some words\n\n-- Accent\n-- Smyth \u00a74. All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78. Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n-- Smyth \u00a720 a. When \u03b9 and \u03c5 correspond to y and w (cp. minion, persuade) they are said to be unsyllabic; and, with a following vowel, make one syllable out of two.\n-- Smyth \u00a720 a. Initial \u03b9\u032f passed into \u0314 (h), as in \u1f27\u03c0\u03b1\u03c1 liver, Lat. jecur; and into \u03b6 in \u03b6\u03c5\u03b3\u03cc\u03bd yoke\n-- Smyth \u00a720 a. Initial \u03c5\u032f was written \u03dd\n-- Smyth \u00a720 a. Medial \u03b9\u032f, \u03c5\u032f before vowels were often lost, as in \u03c4\u1fd1\u03bc\u03ac-(\u03b9\u032f)\u03c9 I honour, \u03b2\u03bf (\u03c5\u032f)-\u03cc\u03c2, gen. of \u03b2\u03bf\u1fe6-\u03c2 ox, cow\n-- Smyth \u00a726. \u03c3 was usually like our sharp s; but before voiced consonants (15 a) it probably was soft, like z; thus we find both \u03ba\u03cc\u03b6\u03bc\u03bf\u03c2 and \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 on inscriptions.\n","old_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc : OlderLetter\n  \u03d8 : OlderLetter\n  \u03e1 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  \u1fb0 : VowelWithLength \u03b1 short\n  \u1fb1 : VowelWithLength \u03b1 long\n  \u03b5 : VowelWithLength \u03b5 short\n  \u03b7 : VowelWithLength \u03b7 long\n  \u1fd0 : VowelWithLength \u03b9 short\n  \u1fd1 : VowelWithLength \u03b9 long\n  \u03bf : VowelWithLength \u03bf short\n  \u1fe0 : VowelWithLength \u03c5 short\n  \u1fe1 : VowelWithLength \u03c5 long\n  \u03c9 : VowelWithLength \u03c9 long\n\n-- Smyth \u00a75,6\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9-gen : Diphthong \u03b5 \u03b9\n  \u03b5\u03b9-sp : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5-gen : Diphthong \u03bf \u03c5\n  \u03bf\u03c5-sp : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n  \u03c9\u03c5 : Diphthong \u03c9 \u03c5 -- Smyth \u00a75 D. New Ionic\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03c5 = closed\ntongue-position-vowel \u03c9 = open-medium\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5-gen = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata ConsonantSound : Letter \u2192 Set where\n  \u03b2 : ConsonantSound \u03b2\n  \u03b3 : ConsonantSound \u03b3\n  \u03b4 : ConsonantSound \u03b4\n  \u03b6 : ConsonantSound \u03b6\n  \u03b8 : ConsonantSound \u03b8\n  \u03ba : ConsonantSound \u03ba\n  \u03bb\u2032 : ConsonantSound \u03bb\u2032\n  \u03bc : ConsonantSound \u03bc\n  \u03bd : ConsonantSound \u03bd\n  \u03be : ConsonantSound \u03be\n  \u03c0 : ConsonantSound \u03c0\n  \u03c1 : ConsonantSound \u03c1\n  \u03c3 : ConsonantSound \u03c3\n  \u03c4 : ConsonantSound \u03c4\n  \u03c6 : ConsonantSound \u03c6\n  \u03c7 : ConsonantSound \u03c7\n  \u03c8 : ConsonantSound \u03c8\n  \u03b3-nasal : ConsonantSound \u03b3\n  \u03c1\u0314 : ConsonantSound \u03c1\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b3-nasal : Voiced \u03b3-nasal\n  \u03b6 : Voiced \u03b6\n\n-- Smyth \u00a715 a, b\ndata Voiceless : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c1\u0314 : Voiceless \u03c1\u0314\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n\n-- Smyth \u00a716\ndata PartOfMouthClass : Set where\n  labial dental palatal : PartOfMouthClass\n\ndata Labial : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : Labial \u03c0\n  \u03b2 : Labial \u03b2\n  \u03c6 : Labial \u03c6\n\ndata Dental : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c4 : Dental \u03c4\n  \u03b4 : Dental \u03b4\n  \u03b8 : Dental \u03b8\n\ndata Palatal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03ba : Palatal \u03ba\n  \u03b3 : Palatal \u03b3\n  \u03c7 : Palatal \u03c7\n\ndata SmoothStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : SmoothStop \u03c0\n  \u03c4 : SmoothStop \u03c4\n  \u03ba : SmoothStop \u03ba\n\ndata MiddleStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : MiddleStop \u03b2\n  \u03b4 : MiddleStop \u03b4\n  \u03b3 : MiddleStop \u03b3\n\ndata RoughStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c6 : RoughStop \u03c6\n  \u03b8 : RoughStop \u03b8\n  \u03c7 : RoughStop \u03c7\n\n-- Smyth \u00a717\ndata Spirant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c3 : Spirant \u03c3\n\n-- Smyth \u00a718\ndata Liquid : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u2032 : Liquid \u03bb\u2032\n  \u03c1 : Liquid \u03c1\n\n-- Smyth \u00a719\ndata Nasal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bc : Nasal \u03bc\n  \u03bd : Nasal \u03bd\n  \u03b3-nasal : Nasal \u03b3-nasal\n\n-- Smyth \u00a720\ndata Semivowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 Set where\n  \u03b9\u032f : Semivowel \u03b9\n  \u03c5\u032f : Semivowel \u03c5\n\n-- Smyth \u00a720 b\ndata Sonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u0325 : Sonant \u03bb\u2032\n  \u03bc\u0325 : Sonant \u03bc\n  \u03bd\u0325 : Sonant \u03b3\n  \u03c1\u0325 : Sonant \u03c1\n  \u03c3\u0325 : Sonant \u03c3\n\n-- Smyth \u00a721\ndata DoubleConsonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b6 : DoubleConsonant \u03b6\n  \u03be : DoubleConsonant \u03be\n  \u03c8 : DoubleConsonant \u03c8\n\ndata _+_\u21d2DoubleConsonant_ : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u2083} \u2192 ConsonantSound \u2113\u2081 \u2192 ConsonantSound \u2113\u2082 \u2192 ConsonantSound \u2113\u2083 \u2192 Set where\n  \u03c3+\u03b4\u21d2DoubleConsonant\u03b6 : \u03c3 + \u03b4 \u21d2DoubleConsonant \u03b6 -- Smyth \u00a726 D. Aeolic has \u03c3\u03b4 for \u03b6\n  \u03b4+\u03c3\u21d2DoubleConsonant\u03b6 : \u03b4 + \u03c3 \u21d2DoubleConsonant \u03b6\n  \n  \u03ba+\u03c3\u21d2DoubleConsonant\u03be : \u03ba + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03b3+\u03c3\u21d2DoubleConsonant\u03be : \u03b3 + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03c7+\u03c3\u21d2DoubleConsonant\u03be : \u03c7 + \u03c3 \u21d2DoubleConsonant \u03be\n\n  \u03c0+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c0 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03b2+\u03c3\u21d2DoubleConsonant\u03c8 : \u03b2 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03c6+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c6 + \u03c3 \u21d2DoubleConsonant \u03c8\n\n\n-- TODO (needs a unified model)\n\n-- Diphthong\n-- Smyth \u00a75 D. Ionic has \u03b7\u03c5 for Attic \u03b1\u03c5 in some words\n\n-- Accent\n-- Smyth \u00a74. All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78. Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n-- Smyth \u00a719 a. Gamma before \u03ba, \u03b3, \u03c7, \u03be is called \u03b3-nasal\n-- Smyth \u00a720 a. When \u03b9 and \u03c5 correspond to y and w (cp. minion, persuade) they are said to be unsyllabic; and, with a following vowel, make one syllable out of two.\n-- Smyth \u00a720 a. Initial \u03b9\u032f passed into \u0314 (h), as in \u1f27\u03c0\u03b1\u03c1 liver, Lat. jecur; and into \u03b6 in \u03b6\u03c5\u03b3\u03cc\u03bd yoke\n-- Smyth \u00a720 a. Initial \u03c5\u032f was written \u03dd\n-- Smyth \u00a720 a. Medial \u03b9\u032f, \u03c5\u032f before vowels were often lost, as in \u03c4\u1fd1\u03bc\u03ac-(\u03b9\u032f)\u03c9 I honour, \u03b2\u03bf (\u03c5\u032f)-\u03cc\u03c2, gen. of \u03b2\u03bf\u1fe6-\u03c2 ox, cow\n-- Smyth \u00a726. \u03c3 was usually like our sharp s; but before voiced consonants (15 a) it probably was soft, like z; thus we find both \u03ba\u03cc\u03b6\u03bc\u03bf\u03c2 and \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 on inscriptions.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"499fba7209cf2608618e78ee322fc34dd2696dda","subject":"Updated agda2atp: only the required ATP definitions are used.","message":"Updated agda2atp: only the required ATP definitions are used.\n\nIgnore-this: 26e6badd080349e9b24e6dc22c52a5ce\n\ndarcs-hash:20101220230343-3bd4e-a4b9968b14a2d376612957d3e2f545fdafb41b50.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/AxiomaticPA\/PropertiesATP.agda","new_file":"src\/AxiomaticPA\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- PA properties\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.PropertiesATP where\n\nopen import AxiomaticPA.Base\n\nopen import AxiomaticPA.Equality.Properties\n  using ()  -- We include this module due to its general hints.\n\n------------------------------------------------------------------------------\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2263 n\n+-rightIdentity = S\u2089 P P0 iStep\n  where\n    P : \u2115 \u2192 Set\n    P i = i + zero \u2263 i\n    {-# ATP definition P #-}\n\n    P0 : P zero\n    P0 = S\u2085 zero\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\nx+Sy\u2263S[x+y] : \u2200 m n \u2192 m + succ n \u2263 succ (m + n)\nx+Sy\u2263S[x+y] m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + succ n \u2263 succ (i + n)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\n+-comm : \u2200 m n \u2192 m + n \u2263 n + m\n+-comm m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n \u2263 n + i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 +-rightIdentity #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep x+Sy\u2263S[x+y] #-}\n\nx\u2263y\u2192x+z\u2263y+z : \u2200 {m n} o \u2192 m \u2263 n \u2192 m + o \u2263 n + o\nx\u2263y\u2192x+z\u2263y+z {m} {n} o m\u2263n = S\u2089 P P0 iStep o\n  where\n    P : \u2115 \u2192 Set\n    P i = m + i \u2263 n + i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 +-rightIdentity #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep x+Sy\u2263S[x+y] #-}\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2263 m + (n + o)\n+-asocc m n o = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n + o \u2263 i + (n + o)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 x\u2263y\u2192x+z\u2263y+z #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    -- E 1.2:                         No-success due to timeout (180).\n    -- Equinox 5.0alpha (2010-06-29): No-success due to timeout (180).\n    -- Metis 2.3 (release 20101019):  No-success due to timeout (180).\n    -- {-# ATP prove iStep x\u2263y\u2192x+z\u2263y+z #-}\n","old_contents":"------------------------------------------------------------------------------\n-- PA properties\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.PropertiesATP where\n\nopen import AxiomaticPA.Base\n\nopen import AxiomaticPA.Equality.Properties\n  using ()  -- We include this module due to its general hints.\n\n------------------------------------------------------------------------------\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2263 n\n+-rightIdentity = S\u2089 P P0 iStep\n  where\n    P : \u2115 \u2192 Set\n    P i = i + zero \u2263 i\n    {-# ATP definition P #-}\n\n    P0 : P zero\n    P0 = S\u2085 zero\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\nx+Sy\u2263S[x+y] : \u2200 m n \u2192 m + succ n \u2263 succ (m + n)\nx+Sy\u2263S[x+y] m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + succ n \u2263 succ (i + n)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-}\n\n+-comm : \u2200 m n \u2192 m + n \u2263 n + m\n+-comm m n = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n \u2263 n + i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 +-rightIdentity #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    -- TODO: The conjecture was proved in an isolate file.\n    -- {-# ATP prove iStep x+Sy\u2263S[x+y] #-}\n\nx\u2263y\u2192x+z\u2263y+z : \u2200 {m n} o \u2192 m \u2263 n \u2192 m + o \u2263 n + o\nx\u2263y\u2192x+z\u2263y+z {m} {n} o m\u2263n = S\u2089 P P0 iStep o\n  where\n    P : \u2115 \u2192 Set\n    P i = m + i \u2263 n + i\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 +-rightIdentity #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    -- TODO: The conjecture was proved in an isolate file.\n    -- {-# ATP prove iStep x+Sy\u2263S[x+y] #-}\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2263 m + (n + o)\n+-asocc m n o = S\u2089 P P0 iStep m\n  where\n    P : \u2115 \u2192 Set\n    P i = i + n + o \u2263 i + (n + o)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 x\u2263y\u2192x+z\u2263y+z #-}\n\n    postulate\n      iStep : \u2200 i \u2192 P i \u2192 P (succ i)\n    -- E 1.2:                         No-success due to timeout (180).\n    -- Equinox 5.0alpha (2010-06-29): No-success due to timeout (180).\n    -- Metis 2.3 (release 20101019):  No-success due to timeout (180).\n    -- {-# ATP prove iStep x\u2263y\u2192x+z\u2263y+z #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c55ea8563f4ea9ddb1f75a61e72fda9247450dbc","subject":"ermargerd! the Y case in subject expansion lemma finally looks doable!!!!","message":"ermargerd! the Y case in subject expansion lemma finally looks doable!!!!\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping.agda","new_file":"Agda\/ITyping.agda","new_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.List.Properties\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.Core\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing using (dom)\nopen import Typed-Core\nopen import Reduction using (_\u219d_)\n\ndata IType : Set where\n  o : IType\n  _~>_ : List IType -> List IType -> IType\n\n\n\u03c9 : List IType\n\u03c9 = []\n\n\u2229' : IType -> List IType\n\u2229' x = (x \u2237 [])\n\n_\u2229_ : IType -> IType -> List IType\nA \u2229 B = A \u2237 B \u2237 []\n\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n-- list-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> (x \u2237 \u03c4\u1d62) \u2261 (y \u2237 \u03c4\u2c7c) -> \u03c4\u1d62 \u2261 \u03c4\u2c7c\n-- list-inj-cons refl = refl\n--\n-- list-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> (x \u2237 \u03c4\u1d62) \u2261 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n-- \u2229-inj refl = refl\n--\n-- list-\u2237-\u2261 : \u2200 {x y : IType} {xs ys} -> x \u2261 y -> xs \u2261 ys -> (x \u2237 xs) \u2261 (y \u2237 ys)\n-- list-\u2237-\u2261 refl refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u2097_ : Decidable {A = List IType} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u2097 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u2097 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n[] \u225fTI\u2097 [] = yes refl\n[] \u225fTI\u2097 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 [] = no (\u03bb ())\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | \u03c4\u1d62 \u225fTI\u2097 \u03c4\u2c7c\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (proj\u2082 (\u2237-injective \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c)))\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (proj\u2081 (\u2237-injective \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c)))\n\n\n\nICtxt = List (Atom \u00d7 ((List IType) \u00d7 Type))\n\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u2097_ : List IType -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u2097 A -> \u03c4 \u2237'\u2097 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\ndata _\u2237'\u2097_ where\n  nil : \u2200 {A} -> \u03c9 \u2237'\u2097 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} -> \u03c4 \u2237' A -> \u03c4\u1d62 \u2237'\u2097 A -> (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u2097 A\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {A \u0393 x \u03c4} ->\n    (x\u2209 : x \u2209 dom \u0393) -> \u03c4 \u2237'\u2097 A -> Wf-ICtxt \u0393 ->\n    --------------------------------------------\n            Wf-ICtxt ((x , (\u03c4 , A)) \u2237 \u0393)\n\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u2097[_]_ : List IType -> Type -> List IType -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u2097[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u2097[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u2097[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u2097 A ->\n    -----------\n    \u03c9 \u2286\u2097[ A ] \u03c4\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> \u03c4'\u1d62 \u2286\u2097[ A ] \u03c4\u1d62 ->\n    -------------------------------------------------------\n                    (\u03c4' \u2237 \u03c4'\u1d62) \u2286\u2097[ A ] \u03c4\u1d62\n  ~>\u2229 : \u2200 {A B \u03c4 \u03c4\u1d62 \u03c4\u1d62' \u03c4\u2093} ->\n                ((\u03c4 ~> (\u03c4\u1d62 ++ \u03c4\u1d62')) \u2237 \u03c4\u2093) \u2237'\u2097 (A \u27f6 B) ->\n    ---------------------------------------------------------\n    ((\u03c4 ~> (\u03c4\u1d62 ++ \u03c4\u1d62')) \u2237 \u03c4\u2093) \u2286\u2097[ A \u27f6 B ] ((\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093)\n  \u2286\u2097-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n    \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u2097[ A ] \u03c4\u2096 ->\n    ---------------------------------\n              \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2096\n\n\n\u2237'\u2097-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2237'\u2097 A -> \u03c4 \u2237' A\n\u2237'\u2097-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u2097-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u2097-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u2097-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u2097-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\u2237'\u2097-++-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2237'\u2097 A -> \u03c4\u1d62 \u2237'\u2097 A\n\u2237'\u2097-++-l {\u03c4\u1d62 = []} \u03c4\u1d62++\u03c4\u2c7c\u2237A = nil\n\u2237'\u2097-++-l {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons x \u03c4\u1d62++\u03c4\u2c7c\u2237A) = cons x (\u2237'\u2097-++-l \u03c4\u1d62++\u03c4\u2c7c\u2237A)\n\n\u2237'\u2097-++-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2237'\u2097 A -> \u03c4\u2c7c \u2237'\u2097 A\n\u2237'\u2097-++-r {\u03c4\u1d62 = []} \u03c4\u1d62++\u03c4\u2c7c\u2237A = \u03c4\u1d62++\u03c4\u2c7c\u2237A\n\u2237'\u2097-++-r {A} {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons x \u03c4\u1d62++\u03c4\u2c7c\u2237A) = \u2237'\u2097-++-r {A} {\u03c4\u1d62} \u03c4\u1d62++\u03c4\u2c7c\u2237A\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u2097-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u2097 A -> \u03c4 \u2286\u2097[ A ] \u03c4\n\u2286\u2097-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u2097 A -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u2097-refl \u03c4\u2237\u1d62A) (\u2286\u2097-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u2097-refl {\u03c4 = []} nil = nil nil\n\u2286\u2097-refl {A} {\u03c4 \u2237 \u03c4\u1d62} \u03c4\u03c4\u1d62\u2237A = \u2286\u2097-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u2097-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u2097-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u2097-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u2097-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n\n-- \u2286\u2097-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u03c4\u2081 \u2286\u2097[ A ] \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n-- -- \u2286\u2097-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n-- -- \u2286\u2097-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u2097-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n-- \u2286\u2097-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n-- \u2286\u2097-\u2208-\u2203 (~>\u2229 x) (here refl) = {!   !}\n-- \u2286\u2097-\u2208-\u2203 (cons x\u2081 \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = {!   !}\n-- \u2286\u2097-\u2208-\u2203 (~>\u2229 x) (there \u03c4\u2208\u03c4\u2081) = {!   !}\n\n-- \u2208-\u2286\u2097-trans : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2c7c) \u00d7 \u03c4 \u2286[ A ] \u03c4')\n-- \u2208-\u2286\u2097-trans (here refl) (cons x _) = x\n-- \u2208-\u2286\u2097-trans (there \u03c4\u2208\u03c4\u1d62) (cons _ \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2208-\u2286\u2097-trans \u03c4\u2208\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\n-- \u2286\u2097-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u03c4 \u2237 \u03c4\u1d62) \u2286\u2097[ A ] \u03c9 -> \u22a5\n-- \u2286\u2097-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286-\u2237'-l : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4 \u2237' A\n\u2286-\u2237'-l base = base\n\u2286-\u2237'-l (arr _ _ x _) = x\n\n\u2286\u2097-\u2237'\u2097-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u2097 A\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = []} (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u1d62\u2286\u03c4\u2c7c\u2081) = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u2c7c\u2081\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = _ \u2237 \u03c4\u2093} (~>\u2229 {\u03c4\u1d62 = \u03c4\u1d62} (cons (arr x x\u2081) \u03c4\u2093\u2237')) =\n  cons (arr x (\u2237'\u2097-++-l x\u2081)) (cons (arr x (\u2237'\u2097-++-r {\u03c4\u1d62 = \u03c4\u1d62} x\u2081)) \u03c4\u2093\u2237')\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u1d62\u2286\u03c4\u2c7c\u2081) = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u2c7c\u2081\n\n\n\u2286\u2097-\u2237'\u2097-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u1d62 \u2237'\u2097 A\n\u2286\u2097-\u2237'\u2097-l (nil x) = nil\n\u2286\u2097-\u2237'\u2097-l (cons (_ , _ , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) = cons (\u2286-\u2237'-l \u03c4'\u2286\u03c4) (\u2286\u2097-\u2237'\u2097-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u2097-\u2237'\u2097-l (~>\u2229 x) = x\n\u2286\u2097-\u2237'\u2097-l (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096) = \u2286\u2097-\u2237'\u2097-l \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u2097-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u2097-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n-- \u2286\u2097-trans {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u2097-\u2237'\u2097-r \u03c4\u2c7c\u2286\u03c4\u2096)\n-- \u2286\u2097-trans {\u03c4\u1d62 = \u03c4' \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u2097-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n-- \u2286\u2097-trans {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} {\u03c4\u2c7c} {\u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n--   cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n--     where\n--     \u03c4'' = proj\u2081 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n--     \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n--     \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\n\n\n\u2286->\u2286\u2097 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u2097[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u2097 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u2097_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> List IType -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u03c4\u1d62 , A)) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u2097 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u2097[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u2097 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , (\u03c4 , A)) \u2237 \u0393) \u22a9\u2097 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> (\u03c4 ~> \u03c4') \u2237' (A \u27f6 B) ->\n    --------------------------------------------------------------------------------------------------------------\n                                                \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u2229' (\u03c4 ~> \u03c4)) \u2286\u2097[ A \u27f6 A ] \u03c4\u2081 -> \u03c4\u2082 \u2286\u2097[ A ] \u03c4 -> -- \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2081) \u00d7 ((\u03c4 ~> \u03c4) \u2286[ A \u27f6 A ] \u03c4')) -> \u03c4\u2081 \u2237'\u2097 (A \u27f6 A) ->\n    ----------------------------------------------------------------------\n                            \u0393 \u22a9 Y A \u2236 (\u03c4\u2081 ~> \u03c4\u2082)\n\ndata _\u22a9\u2097_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u2097 m\u2005 \u2236 \u03c9\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u2097 m\u2005 \u2236 \u03c4\u1d62 ->\n    --------------------------------\n          \u0393 \u22a9\u2097 m\u2005 \u2236 (\u03c4 \u2237 \u03c4\u1d62)\n  sub\u2097 : \u2200 {A \u0393 \u03c4 \u03c4'} {m : \u039b A} ->\n    \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4' \u2286\u2097[ A ] \u03c4 ->\n    -----------------------------\n            \u0393 \u22a9\u2097 m \u2236 \u03c4'\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n\ndata _\u2286\u0393_ : ICtxt -> ICtxt -> Set where\n  nil : \u2200 {\u0393} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          [] \u2286\u0393 \u0393\n  cons : \u2200 {A x \u03c4' \u0393 \u0393'} ->\n    \u2203(\u03bb \u03c4 -> ((x , (\u03c4 , A)) \u2208 \u0393') \u00d7 (\u03c4' \u2286\u2097[ A ] \u03c4)) -> \u0393 \u2286\u0393 \u0393' ->\n    ------------------------------------------------------------\n                      ((x , (\u03c4' , A)) \u2237 \u0393) \u2286\u0393 \u0393'\n\n\n\u22a9\u2097-wf-\u0393 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> Wf-ICtxt \u0393\n\u22a9\u2097-wf-\u0393 (nil wf-\u0393) = wf-\u0393\n\u22a9\u2097-wf-\u0393 (cons _ \u0393\u22a9\u2097m\u2236\u03c4) = \u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097m\u2236\u03c4\n\u22a9\u2097-wf-\u0393 (sub\u2097 x _) = \u22a9\u2097-wf-\u0393 x\n\n\u2286\u0393-wf\u0393' : \u2200 {\u0393 \u0393'} -> \u0393 \u2286\u0393 \u0393' -> Wf-ICtxt \u0393'\n\u2286\u0393-wf\u0393' (nil wf-\u0393') = wf-\u0393'\n\u2286\u0393-wf\u0393' (cons _ \u0393\u2286\u0393') = \u2286\u0393-wf\u0393' \u0393\u2286\u0393'\n\nwf-\u0393-\u2237'\u2097 : \u2200 {A x \u03c4 \u0393} -> (x , (\u03c4 , A)) \u2208 \u0393 -> Wf-ICtxt \u0393 -> \u03c4 \u2237'\u2097 A\nwf-\u0393-\u2237'\u2097 (here refl) (cons _ x\u2081 _) = x\u2081\nwf-\u0393-\u2237'\u2097 (there x\u03c4A\u2208\u0393) (cons _ _ wf-\u0393) = wf-\u0393-\u2237'\u2097 x\u03c4A\u2208\u0393 wf-\u0393\n\n\n\u2286\u0393-\u2286 : \u2200 {\u0393 \u0393'} -> Wf-ICtxt \u0393' -> \u0393 \u2286 \u0393' -> \u0393 \u2286\u0393 \u0393'\n\u2286\u0393-\u2286 {[]} wf-\u0393' \u0393\u2286\u0393' = nil wf-\u0393'\n\u2286\u0393-\u2286 {(x , \u03c4 , A) \u2237 \u0393} wf-\u0393' \u0393\u2286\u0393' = cons\n  (\u03c4 , ((\u0393\u2286\u0393' (here refl)) , \u2286\u2097-refl (wf-\u0393-\u2237'\u2097 (\u0393\u2286\u0393' (here refl)) wf-\u0393')))\n  (\u2286\u0393-\u2286 wf-\u0393' (\u03bb {x\u2081} z \u2192 \u0393\u2286\u0393' (there z)))\n\n\n\u2208-\u2286\u0393-trans : \u2200 {A x \u03c4\u1d62} {\u0393 \u0393'} -> (x , (\u03c4\u1d62 , A)) \u2208 \u0393 -> \u0393 \u2286\u0393 \u0393' -> \u2203(\u03bb \u03c4\u1d62' -> ((x , (\u03c4\u1d62' , A)) \u2208 \u0393') \u00d7 \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62')\n\u2208-\u2286\u0393-trans (here refl) (cons x _) = x\n\u2208-\u2286\u0393-trans (there x\u2208L) (cons _ L\u2286L') = \u2208-\u2286\u0393-trans x\u2208L L\u2286L'\n\n\n\u2286\u0393-\u2237 : \u2200 {A x \u03c4\u1d62 \u0393 \u0393'} -> x \u2209 dom \u0393' -> \u03c4\u1d62 \u2237'\u2097 A -> \u0393 \u2286\u0393 \u0393' -> \u0393 \u2286\u0393 ((x , \u03c4\u1d62 , A) \u2237 \u0393')\n\u2286\u0393-\u2237 {\u0393 = []} x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393' = nil (cons x\u2209\u0393' \u03c4\u1d62\u2237A (\u2286\u0393-wf\u0393' \u0393\u2286\u0393'))\n\u2286\u0393-\u2237 {\u0393 = (x , \u03c4\u1d62 , A) \u2237 \u0393} x\u2209\u0393' \u03c4\u1d62\u2237A (cons (proj\u2081 , proj\u2082 , proj\u2083) \u0393\u2286\u0393') =\n  cons (proj\u2081 , ((there proj\u2082) , proj\u2083)) (\u2286\u0393-\u2237 x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393')\n\n\n\u22a9-\u2237' : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4 \u2237' A\n\u22a9\u2097-\u2237'\u2097 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4 \u2237'\u2097 A\n\n\u22a9-\u2237' (var _ _ \u03c4\u2286\u03c4\u1d62) = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (app _ _ \u03c4\u2286\u03c4\u1d62 _) = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (abs _ _ x) = x\n\u22a9-\u2237' (Y _ \u03c4\u2286\u03c4~>\u03c4 \u03c4\u2082\u2286\u03c4) = arr (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4~>\u03c4) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2082\u2286\u03c4)\n\n\u22a9\u2097-\u2237'\u2097 (nil _) = nil\n\u22a9\u2097-\u2237'\u2097 (cons \u0393\u22a9m\u2236\u03c4 \u0393\u22a9\u2097m\u2236\u03c4\u1d62) = cons (\u22a9-\u2237' \u0393\u22a9m\u2236\u03c4) (\u22a9\u2097-\u2237'\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u1d62)\n\u22a9\u2097-\u2237'\u2097 (sub\u2097 _ x) = \u2286\u2097-\u2237'\u2097-l x\n\n\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\nsub\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4\nsub\u2097\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u2097 m \u2236 \u03c4\n\nsub\u0393 (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u0393\u2286\u0393' = var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u2097-trans \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub\u0393 (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u0393\u2286\u0393' = app (sub\u0393 \u0393\u22a9m\u2236\u03c4 \u0393\u2286\u0393') (sub\u2097\u0393 x \u0393\u2286\u0393') x\u2081 x\u2082\nsub\u0393 {\u0393' = \u0393'} (abs {\u03c4 = \u03c4} L cf (arr \u03c4\u2237A \u03c4'\u2237B)) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u2097\u0393\n    (cf (\u2209-cons-l _ _ x\u2209))\n    (cons\n      (\u03c4 , (here refl) , (\u2286\u2097-refl \u03c4\u2237A))\n      (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2237A \u03c4'\u2237B)\nsub\u0393 (Y x x\u2081 x\u2082) \u0393\u2286\u0393' = Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') x\u2081 x\u2082\n\nsub\u2097\u0393 (nil wf-\u0393) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u2097\u0393 (cons x \u0393\u22a9\u2097m\u2236\u03c4) \u0393\u2286\u0393' = cons (sub\u0393 x \u0393\u2286\u0393') (sub\u2097\u0393 \u0393\u22a9\u2097m\u2236\u03c4 \u0393\u2286\u0393')\nsub\u2097\u0393 (sub\u2097 x y) \u0393\u2286\u0393' = sub\u2097 (sub\u2097\u0393 x \u0393\u2286\u0393') y\n\nsub : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4'\nsub\u2097' : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4' \u2286\u2097[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u2097 m \u2236 \u03c4'\n\nsub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u2097-trans (\u2286\u2097-trans (\u2286->\u2286\u2097 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62) \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub (app \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u22a9\u2097t\u2236\u03c4\u2081 \u03c4\u2286\u03c4\u2082 \u03c4\u2237A) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = app\n  (sub\u0393 \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u2286\u0393')\n  (sub\u2097\u0393 \u0393\u22a9\u2097t\u2236\u03c4\u2081 \u0393\u2286\u0393')\n  (\u2286\u2097-trans (\u2286->\u2286\u2097 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u2082)\n  \u03c4\u2237A\nsub {_} {\u0393} {\u0393'} (abs {\u03c4 = \u03c4} {\u03c4'} L {t} cf \u03c4~>\u03c4'\u2237A\u27f6B) (arr {A} {\u03c4\u2081\u2081 = \u03c4\u2081\u2081} \u03c4\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4' (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B) x\u2083) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u2097'\n    (cf (\u2209-cons-l _ _ x\u2209))\n    \u03c4\u2081\u2082\u2286\u03c4'\n    (cons (\u03c4\u2081\u2081 , (here refl) , \u03c4\u2286\u03c4\u2081\u2081) (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2081\u2081\u2237A \u0393\u2286\u0393'))\n  )\n  (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B)\nsub (Y wf-\u0393 \u03c4\u2081\u2286\u03c4~>\u03c4 \u03c4\u2082\u2286\u03c4) (arr {\u03c4\u2081\u2081 = \u03c4\u2081'} \u03c4\u2081\u2286\u03c4\u2081' \u03c4\u2082'\u2286\u03c4\u2082 (arr \u2229\u03c4\u2081'\u2237A\u27f6A \u03c4\u2082'\u2237A) x\u2084) \u0393\u2286\u0393' =\n  Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') (\u2286\u2097-trans \u03c4\u2081\u2286\u03c4~>\u03c4 \u03c4\u2081\u2286\u03c4\u2081') (\u2286\u2097-trans \u03c4\u2082'\u2286\u03c4\u2082 \u03c4\u2082\u2286\u03c4)\n\nsub\u2097' \u0393\u22a9\u2097m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = sub\u2097\u0393 (sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4 \u03c4'\u2286\u03c4) \u0393\u2286\u0393'\n-- sub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4 (nil x) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\n-- sub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4'\u1d62\u2286\u03c4\u1d62) \u0393\u2286\u0393' with \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n-- ... | \u0393\u22a9m\u2236\u03c4 = cons (sub \u0393\u22a9m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393') (sub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4'\u1d62\u2286\u03c4\u1d62 \u0393\u2286\u0393')\n\n\n\n-- \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 : \u2200 {A \u03c4 \u03c4'} ->  \u00ac(\u03c4 \u2261 \u03c9) -> \u03c4 \u2286\u2097[ A ] \u03c4' -> \u00ac(\u03c4' \u2261 \u03c9)\n-- \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 {\u03c4 = []} \u03c4\u2260\u03c9 \u03c4\u2286\u03c4' x = \u03c4\u2260\u03c9 refl\n-- \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 {\u03c4 = x \u2237 \u03c4} {[]} \u03c4\u2260\u03c9 (cons (_ , () , _) \u03c4\u2286\u03c4') \u03c4'\u2261\u03c9\n-- \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 {\u03c4 = x \u2237 \u03c4} {[]} \u03c4\u2260\u03c9 (\u2286\u2097-trans {\u03c4\u2c7c = \u03c4''} x\u2237\u03c4\u2286\u03c4'' \u03c4''\u2286[]) \u03c4'\u2261\u03c9 = \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 \u03c4''\u2260\u03c9 \u03c4''\u2286[] refl\n--   where\n--   \u03c4''\u2260\u03c9 : \u00ac(\u03c4'' \u2261 \u03c9)\n--   \u03c4''\u2260\u03c9 = \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 \u03c4\u2260\u03c9 x\u2237\u03c4\u2286\u03c4''\n-- \u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 {\u03c4 = x \u2237 \u03c4} {x\u2081 \u2237 \u03c4'} \u03c4\u2260\u03c9 \u03c4\u2286\u03c4' ()\n--\n-- \u2229'\u2286\u03c9-imp-\u22a5 : \u2200 {A \u03c4} -> \u00ac(\u03c4 \u2261 \u03c9) -> \u03c4 \u2286\u2097[ A ] \u03c9 -> \u22a5\n-- \u2229'\u2286\u03c9-imp-\u22a5 \u03c4\u2260\u03c9 (nil x) = \u03c4\u2260\u03c9 refl\n-- \u2229'\u2286\u03c9-imp-\u22a5 \u03c4\u2260\u03c9 (cons (_ , () , _) \u03c4\u2286\u03c9)\n-- \u2229'\u2286\u03c9-imp-\u22a5 \u03c4\u2260\u03c9 (\u2286\u2097-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c9) = \u2229'\u2286\u03c9-imp-\u22a5 (\u03c4\u2260\u03c9-\u2286-\u03c4'\u2260\u03c9 \u03c4\u2260\u03c9 \u03c4\u2286\u03c4') \u03c4'\u2286\u03c9\n--\n--\n-- \u0393\u22a9Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u0393 \u22a9 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4')\n-- \u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app {\u03c4\u2081 = []} {\u03c4'}\n--   (Y {\u03c4 = \u03c4''} wf-\u0393 \u03c4''~>\u03c4''\u2286\u03c4\u2081 \u03c4'\u2286\u03c4'')\n--   \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n--   \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A) = \u22a5-elim (\u2229'\u2286\u03c9-imp-\u22a5 (\u03bb ()) \u03c4''~>\u03c4''\u2286\u03c4\u2081)\n--\n-- \u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app {\u03c4\u2081 = \u03c4\u2081~>\u03c4\u2082 \u2237 \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62} {\u03c4'}\n--   (Y {\u03c4 = \u03c4''} wf-\u0393 \u03c4''~>\u03c4''\u2286\u03c4\u2081 \u03c4'\u2286\u03c4'')\n--   \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n--   \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A) = {!   !}\n\n-- \u2208-\u2286\u2097-trans : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u1d62'} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62' -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u1d62') \u00d7 \u03c4' \u2286[ A ] \u03c4)\n-- \u2208-\u2286\u2097-trans () (nil x)\n-- \u2208-\u2286\u2097-trans (here refl) (cons x\u2081 \u03c4\u1d62\u2286\u03c4\u1d62') = {!   !}\n-- \u2208-\u2286\u2097-trans (there \u03c4\u2208\u03c4\u1d62) (cons x\u2081 \u03c4\u1d62\u2286\u03c4\u1d62') = \u2208-\u2286\u2097-trans \u03c4\u2208\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62'\n-- \u2208-\u2286\u2097-trans (here refl) (~>\u2229 x) = {!   !}\n-- \u2208-\u2286\u2097-trans (there \u03c4\u2208\u03c4\u1d62) (~>\u2229 x) = {!   !}\n-- \u2208-\u2286\u2097-trans \u03c4\u2208\u03c4\u1d62 (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u1d62' \u03c4\u1d62\u2286\u03c4\u1d62'') = {!   !}\n--\n-- \u22a9\u2097-\u2208-\u22a9 : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4\u1d62} -> \u0393 \u22a9\u2097 m \u2236 \u03c4\u1d62 -> \u03c4 \u2208 \u03c4\u1d62 -> \u2203(\u03bb \u03c4' -> \u03c4' \u2286[ A ] \u03c4 \u00d7 \u0393 \u22a9 m \u2236 \u03c4')\n-- \u22a9\u2097-\u2208-\u22a9 (nil wf-\u0393) ()\n-- \u22a9\u2097-\u2208-\u22a9 {\u03c4 = \u03c4} (cons x\u2081 \u0393\u22a9\u2097m\u2236\u03c4\u1d62) (here refl) = \u03c4 , (\u2286-refl (\u22a9-\u2237' x\u2081) , x\u2081)\n-- \u22a9\u2097-\u2208-\u22a9 (cons x\u2081 \u0393\u22a9\u2097m\u2236\u03c4\u1d62) (there \u03c4\u2208\u03c4\u1d62) = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n-- \u22a9\u2097-\u2208-\u22a9 {A} {\u0393} {m} {\u03c4} {\u03c4\u1d62} (sub\u2097 {\u03c4 = \u03c4\u1d62'} \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62') \u03c4\u2208\u03c4\u1d62 = proj\u2081 ih , (\u2286-trans \u03c4\u2286\u03c4' (proj\u2081 (proj\u2082 ih)) , proj\u2082 (proj\u2082 ih))\n--   where\n--   \u03c4' : IType\n--   \u03c4' = {!   !}\n--\n--   \u03c4\u2286\u03c4' : \u03c4 \u2286[ A ] \u03c4'\n--   \u03c4\u2286\u03c4' = {!   !}\n--\n--   ih : \u2203(\u03bb \u03c4'' -> \u03c4' \u2286[ A ] \u03c4'' \u00d7 \u0393 \u22a9 m \u2236 \u03c4'')\n--   ih = {!   !}\n\n\n-- aux1 : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4 : List IType} ->  \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4 ~> \u03c4) -> \u0393 \u22a9 m \u2236 (\u03c4 ~> \u03c4)\n-- aux1 (cons x \u0393\u22a9\u2097m\u2236\u03c4~>\u03c4) = x\n-- aux1 (sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4~>\u03c4 x) = {!   !}\n\n\n\u0393\u22a9Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u0393 \u22a9 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4' ~> \u03c4') \u00d7 (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4')\n\u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app {\u03c4\u2081 = \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62} {\u03c4'}\n  (Y {\u03c4 = \u03c4''} wf-\u0393 \u03c4''~>\u03c4''\u2286\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 \u03c4'\u2286\u03c4'')\n  \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n  \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A) = \u03c4'' , ((sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 \u03c4''~>\u03c4''\u2286\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62) , (\u2286\u2097-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))\n\n\n-- \u0393\u22a9Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u0393 \u22a9 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4')\n-- \u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app {\u03c4\u2081 = \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62} {\u03c4'}\n--   (Y {\u03c4 = \u03c4''} wf-\u0393 (cons (_ , \u03c4\u2081~>\u03c4\u2082\u2208\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 , arr {\u03c4\u2082\u2081 = \u03c4\u2081} {\u03c4\u2082} \u03c4\u2081\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082 x\u2082 x\u2083) []\u2286\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62) \u03c4'\u2286\u03c4'')\n--   \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n--   \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A) = {!   !}\n--   -- \u03c4\u2082 , \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 , \u03c4\u2286\u03c4\u2082\n--\n--   -- where\n--   -- \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z)\n--   -- \u03c4\u2286\u03c4'' = \u2286\u2097-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''\n--   -- \u03c4\u2286\u03c4\u2082 = \u2286\u2097-trans \u03c4\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082\n--   --\n--   -- \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u03c4\u2081 ~> \u03c4\u2082)\n--   -- \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 \u03c4\u2081~>\u03c4\u2082\u2208\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n--   --\n--   -- \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u03c4\u2082 ~> \u03c4\u2082)\n--   -- \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 = sub \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 (arr (\u2286\u2097-trans \u03c4\u2081\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082) (\u2286\u2097-refl \u03c4\u2082\u2237A) (arr \u03c4\u2082\u2237A \u03c4\u2082\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2082\u2237A)) \u0393\u2286\u0393-refl\n-- \u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app\n--   (Y wf-\u0393 (~>\u2229 x) \u03c4'\u2286\u03c4'')\n--   \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n--   \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A\u2082) = {!   !}\n--\n-- \u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app\n--   (Y wf-\u0393 (\u2286\u2097-trans x y) \u03c4'\u2286\u03c4'')\n--   \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n--   \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A\u2082) = {!   !}\n\n\n\u2237'\u2097-++ : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2237'\u2097 A -> \u03c4\u2c7c \u2237'\u2097 A -> (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2237'\u2097 A\n\u2237'\u2097-++ nil \u03c4\u2c7c\u2237A = \u03c4\u2c7c\u2237A\n\u2237'\u2097-++ (cons x \u03c4\u1d62\u2237A) \u03c4\u2c7c\u2237A = cons x (\u2237'\u2097-++ \u03c4\u1d62\u2237A \u03c4\u2c7c\u2237A)\n\n\n\u2286-++-comm : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4} -> (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2286\u2097[ A ] \u03c4 -> (\u03c4\u2c7c ++ \u03c4\u1d62) \u2286\u2097[ A ] \u03c4\n\u2286-++-comm {A} {\u03c4\u1d62} {\u03c4\u2c7c} \u03c4\u1d62++\u03c4\u2c7c\u2286\u2097\u03c4 = \u2286\u2097-trans (\u2286\u2097-\u2286 {!   !} (\u2237'\u2097-++ {A} {\u03c4\u1d62} {\u03c4\u2c7c} (\u2237'\u2097-++-l \u03c4\u1d62++\u03c4\u2c7c\u2237'A) (\u2237'\u2097-++-r {\u03c4\u1d62 = \u03c4\u1d62} \u03c4\u1d62++\u03c4\u2c7c\u2237'A))) \u03c4\u1d62++\u03c4\u2c7c\u2286\u2097\u03c4\n  where\n  \u03c4\u1d62++\u03c4\u2c7c\u2237'A = \u2286\u2097-\u2237'\u2097-l \u03c4\u1d62++\u03c4\u2c7c\u2286\u2097\u03c4\n\n\n\u2286\u2097++-r : \u2200 {A \u03c4\u1d62 \u03c4\u1d62' \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62' -> \u03c4\u2c7c \u2237'\u2097 A -> (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2286\u2097[ A ] (\u03c4\u1d62' ++ \u03c4\u2c7c)\n\u2286\u2097++-r {\u03c4\u1d62' = \u03c4\u1d62'} {\u03c4\u2c7c} (nil x) \u03c4\u2c7c\u2237'A = \u2286\u2097-\u2286 (\u03bb x\u2082 \u2192 \u2208-cons-r \u03c4\u1d62' x\u2082) (\u2237'\u2097-++ x \u03c4\u2c7c\u2237'A)\n\u2286\u2097++-r (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u1d62') \u03c4\u2c7c\u2237'A = cons (\u03c4 , (\u2208-cons-l _ \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4)) (\u2286\u2097++-r \u03c4\u1d62\u2286\u03c4\u1d62' \u03c4\u2c7c\u2237'A)\n\u2286\u2097++-r {\u03c4\u2c7c = \u03c4\u2c7c} (~>\u2229 x) \u03c4\u2c7c\u2237'A = ~>\u2229 (\u2237'\u2097-++ {\u03c4\u2c7c = \u03c4\u2c7c} x \u03c4\u2c7c\u2237'A)\n\u2286\u2097++-r (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u1d62'' \u03c4\u1d62''\u2286\u03c4\u1d62') \u03c4\u2c7c\u2237'A = \u2286\u2097-trans (\u2286\u2097++-r \u03c4\u1d62\u2286\u03c4\u1d62'' \u03c4\u2c7c\u2237'A) (\u2286\u2097++-r \u03c4\u1d62''\u2286\u03c4\u1d62' \u03c4\u2c7c\u2237'A)\n\n-- \u2286\u2097++-l : \u2200 {A \u03c4\u1d62 \u03c4\u1d62' \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62' -> \u03c4\u2c7c \u2237'\u2097 A -> (\u03c4\u2c7c ++ \u03c4\u1d62) \u2286\u2097[ A ] (\u03c4\u2c7c ++ \u03c4\u1d62')\n-- \u2286\u2097++-l = {!   !}\n\n\u22a9++ : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4\u1d62 \u03c4\u2c7c} -> \u0393 \u22a9\u2097 m \u2236 \u03c4\u1d62 -> \u0393 \u22a9\u2097 m \u2236 \u03c4\u2c7c -> \u0393 \u22a9\u2097 m \u2236 (\u03c4\u1d62 ++ \u03c4\u2c7c)\n\u22a9++ {\u03c4\u1d62 = []} \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u2c7c = \u0393\u22a9\u2097m\u2236\u03c4\u2c7c\n\u22a9++ {\u03c4\u1d62 = x \u2237 \u03c4\u1d62} (cons x\u2081 \u0393\u22a9\u2097m\u2236\u03c4\u1d62) \u0393\u22a9\u2097m\u2236\u03c4\u2c7c = cons x\u2081 (\u22a9++ \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u2c7c)\n\u22a9++ {\u03c4\u1d62 = x \u2237 \u03c4\u1d62} (sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 x\u2081) \u0393\u22a9\u2097m\u2236\u03c4\u2c7c = sub\u2097 (\u22a9++ \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u2c7c) (\u2286\u2097++-r x\u2081 (\u22a9\u2097-\u2237'\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u2c7c))\n\n\nglb : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4 -> \u03c4\u2c7c \u2286\u2097[ A ] \u03c4 -> (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2286\u2097[ A ] \u03c4\nglb (nil x) \u03c4\u2c7c\u2286\u03c4 = \u03c4\u2c7c\u2286\u03c4\nglb (cons x \u03c4\u1d62\u2286\u03c4) \u03c4\u2c7c\u2286\u03c4 = cons x (glb \u03c4\u1d62\u2286\u03c4 \u03c4\u2c7c\u2286\u03c4)\nglb {A \u27f6 B} {\u03c4\u2c7c = \u03c4\u2c7c} (~>\u2229 {\u03c4 = \u03c4} {\u03c4\u1d62} {\u03c4\u1d62'} {\u03c4\u2093} (cons (arr x x\u2081) x\u2082)) \u03c4\u2c7c\u2286\u03c4 = \u2286\u2097-trans {\u03c4\u2c7c = ((\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093 ++ \u03c4\u2c7c)}\n  (~>\u2229 (\u2237'\u2097-++ {\u03c4\u1d62 = (\u03c4 ~> (\u03c4\u1d62 ++ \u03c4\u1d62')) \u2237 \u03c4\u2093} {\u03c4\u2c7c} (cons (arr x x\u2081) x\u2082) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2c7c\u2286\u03c4)))\n  (\u2286-++-comm {\u03c4\u1d62 = \u03c4\u2c7c} {(\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093}\n    (\u2286\u2097-trans {\u03c4\u2c7c = ((\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093) ++ (\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093}\n      (\u2286\u2097++-r {\u03c4\u2c7c = ((\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093)} \u03c4\u2c7c\u2286\u03c4 \u2237'A\u27f6B)\n      (\u2286\u2097-\u2286 {!   !} \u2237'A\u27f6B)))\n  where\n  \u2237'A\u27f6B : ((\u03c4 ~> \u03c4\u1d62) \u2237 (\u03c4 ~> \u03c4\u1d62') \u2237 \u03c4\u2093) \u2237'\u2097 (A \u27f6 B)\n  \u2237'A\u27f6B = cons (arr x (\u2237'\u2097-++-l x\u2081)) (cons (arr x (\u2237'\u2097-++-r {\u03c4\u1d62 = \u03c4\u1d62} x\u2081)) x\u2082)\n\nglb (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u1d62' \u03c4\u1d62'\u2286\u03c4\u1d62) \u03c4\u2c7c\u2286\u03c4 = \u2286\u2097-trans (\u2286\u2097++-r \u03c4\u1d62\u2286\u03c4\u1d62' (\u2286\u2097-\u2237'\u2097-l \u03c4\u2c7c\u2286\u03c4)) (glb \u03c4\u1d62'\u2286\u03c4\u1d62 \u03c4\u2c7c\u2286\u03c4)\n\n\u2229-\u2286-imp-\u2286-\u2229 : \u2200 {A \u03c4 \u03c4' \u03c4\u1d62 \u03c4\u1d62'} -> \u03c4 \u2286\u2097[ A ] \u03c4' -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62' -> (\u03c4 ++ \u03c4\u1d62) \u2286\u2097[ A ] (\u03c4' ++ \u03c4\u1d62')\n\u2229-\u2286-imp-\u2286-\u2229 {\u03c4' = \u03c4'} \u03c4\u2286\u03c4' \u03c4\u1d62\u2286\u03c4\u1d62' = glb\n  (\u2286\u2097-trans \u03c4\u2286\u03c4' (\u2286\u2097-\u2286 (\u03bb x\u2081 \u2192 \u2208-cons-l _ x\u2081) (\u2237'\u2097-++ (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4') (\u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u1d62'))))\n  (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u1d62' (\u2286\u2097-\u2286 (\u03bb x\u2081 \u2192 \u2208-cons-r \u03c4' x\u2081) (\u2237'\u2097-++ (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4') (\u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u1d62'))))\n\n\u0393\u22a9\u2097Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u00ac(\u03c4 \u2261 \u03c9) -> \u0393 \u22a9\u2097 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4' ~> \u03c4') \u00d7 \u03c4 \u2286\u2097[ A ] \u03c4')\n\u0393\u22a9\u2097Ym-max \u03c4\u2260\u03c9 (nil wf-\u0393) = \u22a5-elim (\u03c4\u2260\u03c9 refl)\n\u0393\u22a9\u2097Ym-max {A} {\u0393} {m} {\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2260\u03c9 (cons x \u0393\u22a9\u2097Ym\u2236\u03c4) with \u03c4\u1d62 \u225fTI\u2097 \u03c9\n\u0393\u22a9\u2097Ym-max {A} {\u0393} {m} {\u03c4 \u2237 .[]} \u03c4\u2260\u03c9 (cons x \u0393\u22a9\u2097Ym\u2236\u03c4) | yes refl = \u0393\u22a9Ym-max x\n\u0393\u22a9\u2097Ym-max {A} {\u0393} {m} {\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2260\u03c9 (cons x \u0393\u22a9\u2097Ym\u2236\u03c4) | no \u03c4\u1d62\u2260\u03c9 =\n  \u03c4' ++ \u03c4\u1d62' ,\n  sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u03c4\u1d62'~>\u03c4'\u03c4\u03c4\u1d62'~>\u03c4\u1d62' (~>\u2229 (cons (arr (\u2237'\u2097-++ \u03c4'\u2237'A \u03c4\u1d62'\u2237'A) (\u2237'\u2097-++ \u03c4'\u2237'A \u03c4\u1d62'\u2237'A)) nil)) ,\n  \u2229-\u2286-imp-\u2286-\u2229 \u03c4\u2286\u03c4' \u03c4\u1d62\u2286\u03c4\u1d62'\n  where\n  ih : \u2203(\u03bb \u03c4\u1d62' -> \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4\u1d62' ~> \u03c4\u1d62') \u00d7 \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62')\n  ih = \u0393\u22a9\u2097Ym-max \u03c4\u1d62\u2260\u03c9 \u0393\u22a9\u2097Ym\u2236\u03c4\n\n  \u03c4\u1d62' = proj\u2081 ih\n  \u0393\u22a9\u2097m\u2236\u03c4\u1d62'~>\u03c4\u1d62' : \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4\u1d62' ~> \u03c4\u1d62')\n  \u0393\u22a9\u2097m\u2236\u03c4\u1d62'~>\u03c4\u1d62' = proj\u2081 (proj\u2082 ih)\n\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 ih)\n\n  body : \u2203(\u03bb \u03c4' -> \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4' ~> \u03c4') \u00d7 \u2229' \u03c4 \u2286\u2097[ A ] \u03c4')\n  body = \u0393\u22a9Ym-max x\n\n  \u03c4' = proj\u2081 body\n  \u0393\u22a9\u2097m\u2236\u03c4'~>\u03c4' : \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4' ~> \u03c4')\n  \u0393\u22a9\u2097m\u2236\u03c4'~>\u03c4' = proj\u2081 (proj\u2082 body)\n\n  \u03c4\u2286\u03c4' = proj\u2082 (proj\u2082 body)\n\n  \u03c4'\u2237'A = \u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4'\n  \u03c4\u1d62'\u2237'A = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u1d62'\n\n  \u0393\u22a9\u2097m\u2236\u03c4\u03c4\u1d62'~>\u03c4'\u03c4\u03c4\u1d62'~>\u03c4\u1d62' : \u0393 \u22a9\u2097 m \u2236 (\u2229' ((\u03c4' ++ \u03c4\u1d62') ~> \u03c4') ++ \u2229' ((\u03c4' ++ \u03c4\u1d62') ~> \u03c4\u1d62'))\n  \u0393\u22a9\u2097m\u2236\u03c4\u03c4\u1d62'~>\u03c4'\u03c4\u03c4\u1d62'~>\u03c4\u1d62' = \u22a9++ {\u03c4\u1d62 = \u2229' ((\u03c4' ++ \u03c4\u1d62') ~> \u03c4')} {\u2229' ((\u03c4' ++ \u03c4\u1d62') ~> \u03c4\u1d62')}\n    (sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4'~>\u03c4'\n      (cons\n        ( (\u03c4' ~> \u03c4') ,\n          ((here refl) ,\n          (arr (\u2286\u2097-\u2286 (\u03bb x\u2082 \u2192 \u2208-cons-l _ x\u2082) (\u2237'\u2097-++ \u03c4'\u2237'A \u03c4\u1d62'\u2237'A)) (\u2286\u2097-refl \u03c4'\u2237'A) (arr (\u2237'\u2097-++ \u03c4'\u2237'A \u03c4\u1d62'\u2237'A) \u03c4'\u2237'A) (arr \u03c4'\u2237'A \u03c4'\u2237'A))))\n        (nil (cons (arr \u03c4'\u2237'A \u03c4'\u2237'A) nil))))\n    (sub\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u1d62'~>\u03c4\u1d62'\n      (cons\n        ( (\u03c4\u1d62' ~> \u03c4\u1d62') ,\n          (here refl ,\n          arr (\u2286\u2097-\u2286 (\u03bb x\u2082 \u2192 \u2208-cons-r \u03c4' x\u2082) (\u2237'\u2097-++ \u03c4'\u2237'A \u03c4\u1d62'\u2237'A)) (\u2286\u2097-refl \u03c4\u1d62'\u2237'A) (arr (\u2237'\u2097-++ \u03c4'\u2237'A \u03c4\u1d62'\u2237'A) \u03c4\u1d62'\u2237'A) (arr \u03c4\u1d62'\u2237'A \u03c4\u1d62'\u2237'A)))\n        (nil (cons (arr \u03c4\u1d62'\u2237'A \u03c4\u1d62'\u2237'A) nil))))\n\n\u0393\u22a9\u2097Ym-max {A} {\u0393} {m} {\u03c4\u1d62} \u03c4\u2260\u03c9 (sub\u2097 {\u03c4 = \u03c4\u1d62'} \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62' x) = \u03c4\u1d62'' , \u0393\u22a9\u2097m\u2236\u03c4\u1d62''~>\u03c4\u1d62'' , (\u2286\u2097-trans x \u03c4\u1d62'\u2286\u03c4\u1d62'')\n  where\n  ih : \u2203(\u03bb \u03c4\u1d62'' -> \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4\u1d62'' ~> \u03c4\u1d62'') \u00d7 \u03c4\u1d62' \u2286\u2097[ A ] \u03c4\u1d62'')\n  ih = \u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62'\n\n  \u03c4\u1d62'' = proj\u2081 ih\n  \u0393\u22a9\u2097m\u2236\u03c4\u1d62''~>\u03c4\u1d62'' = proj\u2081 (proj\u2082 ih)\n  \u03c4\u1d62'\u2286\u03c4\u1d62'' = proj\u2082 (proj\u2082 ih)\n\n\n\u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n\u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u03c4\u1d62'} {\u03c4\u1d62} \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62' \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62'\u2237A) (Y _) = body \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62'\u2237A \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62'\n  where\n  body : \u2200 {\u03c4\u1d62 \u03c4\u1d62'} -> \u0393 \u22a9 m \u2236 (\u03c4\u1d62' ~> \u03c4\u1d62) -> \u2229' \u03c4 \u2286\u2097[ A ] \u03c4\u1d62 -> \u03c4\u1d62' \u2237'\u2097 A -> \u0393 \u22a9\u2097 app (Y A) m \u2236 \u03c4\u1d62' -> \u0393 \u22a9 app (Y A) m \u2236 \u03c4\n  body \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62'\u2237A (nil wf-\u0393) = app\n    (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n      wf-\u0393\n      (cons (([] ~> \u2229' \u03c4) , ((here refl) , (arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr \u03c4\u1d62'\u2237A \u03c4\u2237A)))) (nil (cons (arr \u03c4\u1d62'\u2237A \u03c4\u2237A) nil)))\n      (\u2286\u2097-refl \u03c4\u2237A))\n    (cons (sub \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 (arr (nil \u03c4\u1d62'\u2237A) \u03c4\u2286\u03c4\u1d62 (arr \u03c4\u1d62'\u2237A \u03c4\u2237A) (arr \u03c4\u1d62'\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n    (\u2286\u2097-refl \u03c4\u2237A)\n    (cons (arr \u03c4\u1d62'\u2237A \u03c4\u2237A) nil)\n\n    where\n    \u03c4\u2237A = \u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62\n  body {\u03c4\u1d62' = \u03c4' \u2237 \u03c4\u1d62'} \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62'\u2237A (cons x\u2081 \u0393\u22a9\u2097Ym\u2236\u03c4'\u1d62\u2081) = app\n    (Y {\u03c4 = [ \u03c4 ] ++ \u03c4''} {\u2229' (\u03c4'' ~> ([ \u03c4 ] ++ \u03c4''))} {\u2229' \u03c4}\n      (\u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097Ym\u2236\u03c4'\u1d62\u2081)\n      (cons\n        (\n          (\u03c4'' ~> ([ \u03c4 ] ++ \u03c4'')) ,\n          here refl ,\n          arr\n            (\u2286\u2097-\u2286 (\u03bb x\u2082 \u2192 \u2208-cons-r [ \u03c4 ] x\u2082) (cons \u03c4\u2237'A \u03c4''\u2237'A))\n            (\u2286\u2097-refl (cons \u03c4\u2237'A \u03c4''\u2237'A))\n            (arr (cons \u03c4\u2237'A \u03c4''\u2237'A) (cons \u03c4\u2237'A \u03c4''\u2237'A))\n            (arr \u03c4''\u2237'A (cons \u03c4\u2237'A \u03c4''\u2237'A)))\n        (nil (cons (arr \u03c4''\u2237'A (cons \u03c4\u2237'A \u03c4''\u2237'A)) nil)))\n      (cons (\u03c4 , (here refl , \u2286-refl \u03c4\u2237'A)) (nil (cons \u03c4\u2237'A \u03c4''\u2237'A))))\n    \u0393\u22a9\u2097m\u2236\u03c4''~>\u03c4++\u03c4''\n    (\u2286\u2097-refl (cons \u03c4\u2237'A nil))\n    (cons (arr \u03c4''\u2237'A (cons \u03c4\u2237'A \u03c4''\u2237'A)) nil)\n\n    where\n    \u03c4\u1d62'' = \u03c4' \u2237 \u03c4\u1d62'\n    body\u2081 : \u2203(\u03bb \u03c4'' -> \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4'' ~> \u03c4'') \u00d7 \u03c4\u1d62'' \u2286\u2097[ A ] \u03c4'')\n    body\u2081 = \u0393\u22a9\u2097Ym-max (\u03bb ()) (cons x\u2081 \u0393\u22a9\u2097Ym\u2236\u03c4'\u1d62\u2081)\n\n    \u03c4'' = proj\u2081 body\u2081\n\n    \u0393\u22a9\u2097m\u2236\u03c4''~>\u03c4'' : \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4'' ~> \u03c4'')\n    \u0393\u22a9\u2097m\u2236\u03c4''~>\u03c4'' = proj\u2081 (proj\u2082 body\u2081)\n\n    \u03c4\u1d62''\u2286\u03c4'' = proj\u2082 (proj\u2082 body\u2081)\n    \u03c4''\u2237'A = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62''\u2286\u03c4''\n    \u03c4\u1d62''\u2237'A = \u2286\u2097-\u2237'\u2097-l \u03c4\u1d62''\u2286\u03c4''\n\n    \u03c4\u2237'A : \u03c4 \u2237' A\n    \u03c4\u2237'A = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\n    \u0393\u22a9m\u2236\u03c4''~>\u03c4 : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u2229' \u03c4)\n    \u0393\u22a9m\u2236\u03c4''~>\u03c4 = sub \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 (arr \u03c4\u1d62''\u2286\u03c4'' \u03c4\u2286\u03c4\u1d62 (arr \u03c4''\u2237'A (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)) (arr \u03c4\u1d62''\u2237'A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 (\u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097Ym\u2236\u03c4'\u1d62\u2081) (\u03bb {x} z \u2192 z))\n\n    \u0393\u22a9\u2097m\u2236\u03c4''~>\u03c4++\u03c4'' : \u0393 \u22a9\u2097 m \u2236 \u2229' (\u03c4'' ~> ([ \u03c4 ] ++ \u03c4''))\n    \u0393\u22a9\u2097m\u2236\u03c4''~>\u03c4++\u03c4'' = sub\u2097 (cons \u0393\u22a9m\u2236\u03c4''~>\u03c4 \u0393\u22a9\u2097m\u2236\u03c4''~>\u03c4'') (~>\u2229 (cons (arr \u03c4''\u2237'A (cons \u03c4\u2237'A \u03c4''\u2237'A)) nil))\n\n  body \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62 \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62'\u2237A (sub\u2097 \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62'' \u03c4\u1d62'\u2286\u03c4\u1d62'') = body\n    (sub \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62\n      (arr \u03c4\u1d62'\u2286\u03c4\u1d62''\n        (\u2286\u2097-refl (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))\n        (arr (\u2286\u2097-\u2237'\u2097-r \u03c4\u1d62'\u2286\u03c4\u1d62'') (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))\n        (arr (\u2286\u2097-\u2237'\u2097-l \u03c4\u1d62'\u2286\u03c4\u1d62'') (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62)))\n      (\u2286\u0393-\u2286 (\u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62'') (\u03bb {x} z \u2192 z)))\n    \u03c4\u2286\u03c4\u1d62\n    (\u2286\u2097-\u2237'\u2097-r \u03c4\u1d62'\u2286\u03c4\u1d62'')\n    \u0393\u22a9\u2097Ym\u2236\u03c4\u1d62''\n\n\n\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (nil wf-\u0393) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y _) = app\n--   (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n--     wf-\u0393\n--     (cons (([] ~> \u2229' \u03c4) , ((here refl) , (arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr []\u2237A \u03c4\u2237A)))) (nil (cons (arr []\u2237A \u03c4\u2237A) nil)))\n--     (\u2286\u2097-refl \u03c4\u2237A))\n--   (cons (sub \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (arr (nil []\u2237A) \u03c4\u2286\u03c4\u1d62 (arr []\u2237A \u03c4\u2237A) (arr []\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n--   (\u2286\u2097-refl \u03c4\u2237A)\n--   (cons (arr []\u2237A \u03c4\u2237A) nil)\n--\n--   where\n--   \u03c4\u2237A = \u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62\n--\n-- \u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u03c4' \u2237 \u03c4'\u1d62} {\u03c4\u1d62} \u0393\u22a9m\u2236\u03c4'~>\u03c4\u1d62 \u0393\u22a9\u2097Ym\u2236\u03c4' \u03c4\u2286\u03c4\u1d62 \u03c4'\u2237A) (Y _) = {!   !}\n--   where\n--   \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 (\u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097Ym\u2236\u03c4') (\u03bb {x} z \u2192 z)\n--\n--   \u03c4'' = proj\u2081 (\u0393\u22a9\u2097Ym-max (\u03bb x \u2192 {!   !}) \u0393\u22a9\u2097Ym\u2236\u03c4')\n--\n--   \u0393\u22a9m\u2236\u03c4''~>\u03c4'' : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u03c4'')\n--   \u0393\u22a9m\u2236\u03c4''~>\u03c4'' = proj\u2081 (proj\u2082 (\u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4'))\n--\n--   \u03c4'\u03c4'\u1d62\u2286\u03c4'' : (\u03c4' \u2237 \u03c4'\u1d62) \u2286\u2097[ A ] \u03c4''\n--   \u03c4'\u03c4'\u1d62\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4'))\n--\n--   \u0393\u22a9m\u2236\u03c4''~>\u03c4\u1d62 : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u2229' \u03c4)\n--   \u0393\u22a9m\u2236\u03c4''~>\u03c4\u1d62 = sub \u0393\u22a9m\u2236\u03c4'~>\u03c4\u1d62 (arr \u03c4'\u03c4'\u1d62\u2286\u03c4'' \u03c4\u2286\u03c4\u1d62 (arr (\u2286\u2097-\u2237'\u2097-r \u03c4'\u03c4'\u1d62\u2286\u03c4'') (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)) (arr \u03c4'\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) \u0393\u2286\u0393-refl\n--\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (sub\u2097 x y) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y z) = {!   !}\n--\n--\n--\n\n\n\n\n\n\n\n-------------------------------------------\n\n\n\n\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (nil wf-\u0393) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y _) = app\n--   (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n--     wf-\u0393\n--     ((\u03c9 ~> \u2229' \u03c4) , here refl , arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr nil \u03c4\u2237A))\n--     (cons (arr []\u2237A \u03c4\u2237A) nil)\n--     (\u2286\u2097-refl \u03c4\u2237A))\n--   (cons (sub \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (arr (nil []\u2237A) \u03c4\u2286\u03c4\u1d62 (arr []\u2237A \u03c4\u2237A) (arr []\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n--   (\u2286\u2097-refl \u03c4\u2237A)\n--   (cons (arr []\u2237A \u03c4\u2237A) nil)\n--\n--\n-- -- \u22a9->\u03b2 (app {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} \u0393\u22a9m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082 (cons (app (Y wf-\u0393 (proj\u2081 , () , proj\u2083) x\u2081 x\u2082) (nil wf-\u0393\u2081) \u03c4\u2081\u2286\u03c4\u2081\u2082 x\u2085) \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62) \u03c4\u2286\u03c4\u2082 \u03c4\u2081\u2237A) (Y x)\n-- -- \u22a9->\u03b2 {\u03c4 = \u03c4} (app {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} {\u03c4\u2082} \u0393\u22a9m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082 (cons (app {\u03c4\u2081 = \u2229 (\u03c4\u2081\u2081 \u2237 \u03c4\u2081\u2081\u1d62)} {\u03c4\u2081\u2082} (Y {\u03c4 = \u03c4'} wf-\u0393 (\u03c4'' , \u03c4''\u2208 \u03c4\u2081\u2081, proj\u2083) x\u2081 x\u2082) (cons \u0393\u22a9m\u2236\u03c4\u2081\u2081 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081\u1d62) \u03c4\u2081\u2286\u03c4\u2081\u2082 x\u2085) \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62) \u03c4\u2286\u03c4\u2082 \u03c4\u2081\u2237A) (Y _) =\n-- \u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} {\u03c4\u2082}\n--   \u0393\u22a9m\u2236\u03c4\u2081\u1d62\u03c4\u2081\u1d62~>\u03c4\u2082\n--   (cons\n--     (app {\u03c4\u2081 = \u2229 (\u03c4\u2081\u2081)} {\u03c4\u2081\u2082}\n--       (Y {\u03c4 = \u03c4'} wf-\u0393 (_ , \u03c4''\u2208\u03c4\u2081\u2081 , arr {\u03c4\u2082\u2081 = \u03c4''\u2081} {\u03c4''\u2082} \u03c4''\u2081\u2286\u03c4' \u03c4'\u2286\u03c4''\u2082 \u03c4'~>\u03c4'\u2237A\u27f6A (arr \u03c4''\u2081\u2237A \u03c4''\u2082\u2237A)) \u03c4\u2081\u2081\u2237A\u27f6A \u03c4\u2081\u2082\u2286\u03c4')\n--       \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081 \u03c4\u2081\u2286\u03c4\u2081\u2082 \u03c4\u2081\u2081\u2237A\u27f6A\u2082)\n--     \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62)\n--   \u03c4\u2286\u03c4\u2082\n--   \u03c4\u2081\u2237A) (Y _) =\n--   {!   !}\n--\n--   where\n--   \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z)\n--\n--   \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 : \u0393 \u22a9 m \u2236 (\u03c4''\u2081 ~> \u03c4''\u2082)\n--   \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081 \u03c4''\u2208\u03c4\u2081\u2081\n--\n--   \u0393\u22a9m\u2236\u03c4''\u2082~>\u03c4''\u2082 : \u0393 \u22a9 m \u2236 (\u03c4''\u2082 ~> \u03c4''\u2082)\n--   \u0393\u22a9m\u2236\u03c4''\u2082~>\u03c4''\u2082 = sub \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 (arr (\u2286\u2097-trans \u03c4''\u2081\u2286\u03c4' \u03c4'\u2286\u03c4''\u2082) (\u2286\u2097-refl \u03c4''\u2082\u2237A) (arr \u03c4''\u2082\u2237A \u03c4''\u2082\u2237A) (arr \u03c4''\u2081\u2237A \u03c4''\u2082\u2237A)) \u0393\u2286\u0393-refl\n--\n--   \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u2229' \u03c4\u2081 ~> \u03c4\u2082)\n--   \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 = sub \u0393\u22a9m\u2236\u03c4\u2081\u1d62\u03c4\u2081\u1d62~>\u03c4\u2082 (arr {!   !} {!   !} {!   !} {!   !}) \u0393\u2286\u0393-refl\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u2097Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","old_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.List.Properties\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.Core\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing using (dom)\nopen import Typed-Core\nopen import Reduction using (_\u219d_)\n\ndata IType : Set where\n  o : IType\n  _~>_ : List IType -> List IType -> IType\n\n\n\u03c9 : List IType\n\u03c9 = []\n\n\u2229' : IType -> List IType\n\u2229' x = (x \u2237 [])\n\n_\u2229_ : IType -> IType -> List IType\nA \u2229 B = A \u2237 B \u2237 []\n\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n-- list-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> (x \u2237 \u03c4\u1d62) \u2261 (y \u2237 \u03c4\u2c7c) -> \u03c4\u1d62 \u2261 \u03c4\u2c7c\n-- list-inj-cons refl = refl\n--\n-- list-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> (x \u2237 \u03c4\u1d62) \u2261 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n-- \u2229-inj refl = refl\n--\n-- list-\u2237-\u2261 : \u2200 {x y : IType} {xs ys} -> x \u2261 y -> xs \u2261 ys -> (x \u2237 xs) \u2261 (y \u2237 ys)\n-- list-\u2237-\u2261 refl refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u2097_ : Decidable {A = List IType} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u2097 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u2097 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n[] \u225fTI\u2097 [] = yes refl\n[] \u225fTI\u2097 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 [] = no (\u03bb ())\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | \u03c4\u1d62 \u225fTI\u2097 \u03c4\u2c7c\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (proj\u2082 (\u2237-injective \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c)))\n(x \u2237 \u03c4\u1d62) \u225fTI\u2097 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (proj\u2081 (\u2237-injective \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c)))\n\n\n\nICtxt = List (Atom \u00d7 ((List IType) \u00d7 Type))\n\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u2097_ : List IType -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u2097 A -> \u03c4 \u2237'\u2097 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\ndata _\u2237'\u2097_ where\n  nil : \u2200 {A} -> \u03c9 \u2237'\u2097 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} -> \u03c4 \u2237' A -> \u03c4\u1d62 \u2237'\u2097 A -> (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u2097 A\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {A \u0393 x \u03c4} ->\n    (x\u2209 : x \u2209 dom \u0393) -> \u03c4 \u2237'\u2097 A -> Wf-ICtxt \u0393 ->\n    --------------------------------------------\n            Wf-ICtxt ((x , (\u03c4 , A)) \u2237 \u0393)\n\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u2097[_]_ : List IType -> Type -> List IType -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u2097[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u2097[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u2097[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u2097 A ->\n    -----------\n    \u03c9 \u2286\u2097[ A ] \u03c4\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> \u03c4'\u1d62 \u2286\u2097[ A ] \u03c4\u1d62 ->\n    -------------------------------------------------------\n                    (\u03c4' \u2237 \u03c4'\u1d62) \u2286\u2097[ A ] \u03c4\u1d62\n\n\n\u2237'\u2097-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2237'\u2097 A -> \u03c4 \u2237' A\n\u2237'\u2097-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u2097-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u2097-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u2097-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u2097-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u2097-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u2097 A -> \u03c4 \u2286\u2097[ A ] \u03c4\n\u2286\u2097-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u2097 A -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u2097-refl \u03c4\u2237\u1d62A) (\u2286\u2097-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u2097-refl {\u03c4 = []} nil = nil nil\n\u2286\u2097-refl {A} {\u03c4 \u2237 \u03c4\u1d62} \u03c4\u03c4\u1d62\u2237A = \u2286\u2097-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u2097-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u2097-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u2097-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u2097-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n\n\u2286\u2097-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u03c4\u2081 \u2286\u2097[ A ] \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n\u2286\u2097-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n\u2286\u2097-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u2097-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n\n\n\u2286\u2097-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u03c4 \u2237 \u03c4\u1d62) \u2286\u2097[ A ] \u03c9 -> \u22a5\n\u2286\u2097-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286-\u2237'-l : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4 \u2237' A\n\u2286-\u2237'-l base = base\n\u2286-\u2237'-l (arr _ _ x _) = x\n\n\u2286\u2097-\u2237'\u2097-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u2097 A\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u2097-\u2237'\u2097-r {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u2097-\u2237'\u2097-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\u2286\u2097-\u2237'\u2097-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u1d62 \u2237'\u2097 A\n\u2286\u2097-\u2237'\u2097-l (nil x) = nil\n\u2286\u2097-\u2237'\u2097-l (cons (_ , _ , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) = cons (\u2286-\u2237'-l \u03c4'\u2286\u03c4) (\u2286\u2097-\u2237'\u2097-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\n\n\u2286\u2097-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n  \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u2097[ A ] \u03c4\u2096 ->\n  ---------------------------------\n            \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2096\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\n\u2286\u2097-trans {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u2097-\u2237'\u2097-r \u03c4\u2c7c\u2286\u03c4\u2096)\n\u2286\u2097-trans {\u03c4\u1d62 = \u03c4' \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u2097-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u2097-trans {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} {\u03c4\u2c7c} {\u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n  cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u2097-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n    where\n    \u03c4'' = proj\u2081 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n    \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n    \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u2097-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u2097-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u2097-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n\n\n\u2286->\u2286\u2097 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u2097[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u2097 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u2097_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> List IType -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u03c4\u1d62 , A)) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u2097 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u2097[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u2097 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , (\u03c4 , A)) \u2237 \u0393) \u22a9\u2097 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> (\u03c4 ~> \u03c4') \u2237' (A \u27f6 B) ->\n    --------------------------------------------------------------------------------------------------------------\n                                                \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2081) \u00d7 ((\u03c4 ~> \u03c4) \u2286[ A \u27f6 A ] \u03c4')) -> \u03c4\u2081 \u2237'\u2097 (A \u27f6 A) -> \u03c4\u2082 \u2286\u2097[ A ] \u03c4 -> --  \u03c4\u2081 \u2286[ (A \u27f6 A) \u27f6 A ]((\u2229' (\u03c4 ~> \u03c4)) ~> \u03c4)\n    -----------------------------------------------------------------------------------------\n                                    \u0393 \u22a9 Y A \u2236 (\u03c4\u2081 ~> \u03c4\u2082)\n\ndata _\u22a9\u2097_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u2097 m\u2005 \u2236 \u03c9\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u2097 m\u2005 \u2236 \u03c4\u1d62 ->\n    --------------------------------\n          \u0393 \u22a9\u2097 m\u2005 \u2236 (\u03c4 \u2237 \u03c4\u1d62)\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n\ndata _\u2286\u0393_ : ICtxt -> ICtxt -> Set where\n  nil : \u2200 {\u0393} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          [] \u2286\u0393 \u0393\n  cons : \u2200 {A x \u03c4' \u0393 \u0393'} ->\n    \u2203(\u03bb \u03c4 -> ((x , (\u03c4 , A)) \u2208 \u0393') \u00d7 (\u03c4' \u2286\u2097[ A ] \u03c4)) -> \u0393 \u2286\u0393 \u0393' ->\n    ------------------------------------------------------------\n                      ((x , (\u03c4' , A)) \u2237 \u0393) \u2286\u0393 \u0393'\n\n\n\u22a9\u2097-wf-\u0393 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> Wf-ICtxt \u0393\n\u22a9\u2097-wf-\u0393 (nil wf-\u0393) = wf-\u0393\n\u22a9\u2097-wf-\u0393 (cons _ \u0393\u22a9\u2097m\u2236\u03c4) = \u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097m\u2236\u03c4\n\n\n\u2286\u0393-wf\u0393' : \u2200 {\u0393 \u0393'} -> \u0393 \u2286\u0393 \u0393' -> Wf-ICtxt \u0393'\n\u2286\u0393-wf\u0393' (nil wf-\u0393') = wf-\u0393'\n\u2286\u0393-wf\u0393' (cons _ \u0393\u2286\u0393') = \u2286\u0393-wf\u0393' \u0393\u2286\u0393'\n\nwf-\u0393-\u2237'\u2097 : \u2200 {A x \u03c4 \u0393} -> (x , (\u03c4 , A)) \u2208 \u0393 -> Wf-ICtxt \u0393 -> \u03c4 \u2237'\u2097 A\nwf-\u0393-\u2237'\u2097 (here refl) (cons _ x\u2081 _) = x\u2081\nwf-\u0393-\u2237'\u2097 (there x\u03c4A\u2208\u0393) (cons _ _ wf-\u0393) = wf-\u0393-\u2237'\u2097 x\u03c4A\u2208\u0393 wf-\u0393\n\n\n\u2286\u0393-\u2286 : \u2200 {\u0393 \u0393'} -> Wf-ICtxt \u0393' -> \u0393 \u2286 \u0393' -> \u0393 \u2286\u0393 \u0393'\n\u2286\u0393-\u2286 {[]} wf-\u0393' \u0393\u2286\u0393' = nil wf-\u0393'\n\u2286\u0393-\u2286 {(x , \u03c4 , A) \u2237 \u0393} wf-\u0393' \u0393\u2286\u0393' = cons\n  (\u03c4 , ((\u0393\u2286\u0393' (here refl)) , \u2286\u2097-refl (wf-\u0393-\u2237'\u2097 (\u0393\u2286\u0393' (here refl)) wf-\u0393')))\n  (\u2286\u0393-\u2286 wf-\u0393' (\u03bb {x\u2081} z \u2192 \u0393\u2286\u0393' (there z)))\n\n-- \u2286\u0393-refl {[]} _ = nil nil\n-- \u2286\u0393-refl {(x , \u03c4 , A) \u2237 \u0393} (cons _ \u03c4\u2237A wf-\u0393) = cons (\u03c4 , (here refl , \u2286\u2097-refl \u03c4\u2237A)) {!   !}\n\n\n\u22a9-\u2237' : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4 \u2237' A\n\u22a9\u2097-\u2237'\u2097 : \u2200 {A \u0393} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4 \u2237'\u2097 A\n\n\u22a9-\u2237' (var _ _ \u03c4\u2286\u03c4\u1d62) = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (app _ _ \u03c4\u2286\u03c4\u1d62 _) = \u2237'\u2097-\u2208 (here refl) (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)\n\u22a9-\u2237' (abs _ _ x) = x\n\u22a9-\u2237' (Y _ _ \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) = arr \u03c4\u2081\u2237A\u27f6A (\u2286\u2097-\u2237'\u2097-l \u03c4\u2082\u2286\u03c4)\n\n\u22a9\u2097-\u2237'\u2097 (nil _) = nil\n\u22a9\u2097-\u2237'\u2097 (cons \u0393\u22a9m\u2236\u03c4 \u0393\u22a9\u2097m\u2236\u03c4\u1d62) = cons (\u22a9-\u2237' \u0393\u22a9m\u2236\u03c4) (\u22a9\u2097-\u2237'\u2097 \u0393\u22a9\u2097m\u2236\u03c4\u1d62)\n\n\n\u22a9\u2097-\u2208-\u22a9 : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4\u1d62} -> \u0393 \u22a9\u2097 m \u2236 \u03c4\u1d62 -> \u03c4 \u2208 \u03c4\u1d62 -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9\u2097-\u2208-\u22a9 (nil _) ()\n\u22a9\u2097-\u2208-\u22a9 (cons \u0393\u22a9m\u2236\u03c4 x) (here refl) = \u0393\u22a9m\u2236\u03c4\n\u22a9\u2097-\u2208-\u22a9 (cons _ \u0393\u22a9\u2097m\u2236\u03c4\u1d62) (there \u03c4\u2208\u03c4\u1d62) = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n\u2208-\u2286\u0393-trans : \u2200 {A x \u03c4\u1d62} {\u0393 \u0393'} -> (x , (\u03c4\u1d62 , A)) \u2208 \u0393 -> \u0393 \u2286\u0393 \u0393' -> \u2203(\u03bb \u03c4\u1d62' -> ((x , (\u03c4\u1d62' , A)) \u2208 \u0393') \u00d7 \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u1d62')\n\u2208-\u2286\u0393-trans (here refl) (cons x _) = x\n\u2208-\u2286\u0393-trans (there x\u2208L) (cons _ L\u2286L') = \u2208-\u2286\u0393-trans x\u2208L L\u2286L'\n\n\n\u2286\u0393-\u2237 : \u2200 {A x \u03c4\u1d62 \u0393 \u0393'} -> x \u2209 dom \u0393' -> \u03c4\u1d62 \u2237'\u2097 A -> \u0393 \u2286\u0393 \u0393' -> \u0393 \u2286\u0393 ((x , \u03c4\u1d62 , A) \u2237 \u0393')\n\u2286\u0393-\u2237 {\u0393 = []} x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393' = nil (cons x\u2209\u0393' \u03c4\u1d62\u2237A (\u2286\u0393-wf\u0393' \u0393\u2286\u0393'))\n\u2286\u0393-\u2237 {\u0393 = (x , \u03c4\u1d62 , A) \u2237 \u0393} x\u2209\u0393' \u03c4\u1d62\u2237A (cons (proj\u2081 , proj\u2082 , proj\u2083) \u0393\u2286\u0393') =\n  cons (proj\u2081 , ((there proj\u2082) , proj\u2083)) (\u2286\u0393-\u2237 x\u2209\u0393' \u03c4\u1d62\u2237A \u0393\u2286\u0393')\n\n\nsub\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4\nsub\u1d62\u0393 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u2097 m \u2236 \u03c4\n\nsub\u0393 (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u0393\u2286\u0393' = var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u2097-trans \u03c4\u2286\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub\u0393 (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u0393\u2286\u0393' = app (sub\u0393 \u0393\u22a9m\u2236\u03c4 \u0393\u2286\u0393') (sub\u1d62\u0393 x \u0393\u2286\u0393') x\u2081 x\u2082\nsub\u0393 {\u0393' = \u0393'} (abs {\u03c4 = \u03c4} L cf (arr \u03c4\u2237A \u03c4'\u2237B)) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\u0393\n    (cf (\u2209-cons-l _ _ x\u2209))\n    (cons\n      (\u03c4 , (here refl) , (\u2286\u2097-refl \u03c4\u2237A))\n      (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2237A \u03c4'\u2237B)\nsub\u0393 (Y x x\u2081 x\u2082 x\u2083) \u0393\u2286\u0393' = Y (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') x\u2081 x\u2082 x\u2083\n\nsub\u1d62\u0393 (nil wf-\u0393) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62\u0393 (cons x \u0393\u22a9\u2097m\u2236\u03c4) \u0393\u2286\u0393' = cons (sub\u0393 x \u0393\u2286\u0393') (sub\u1d62\u0393 \u0393\u22a9\u2097m\u2236\u03c4 \u0393\u2286\u0393')\n\n\n\u2208-\u2286\u2097-trans : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2208 \u03c4\u1d62 -> \u03c4\u1d62 \u2286\u2097[ A ] \u03c4\u2c7c -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2c7c) \u00d7 \u03c4 \u2286[ A ] \u03c4')\n\u2208-\u2286\u2097-trans (here refl) (cons x _) = x\n\u2208-\u2286\u2097-trans (there \u03c4\u2208\u03c4\u1d62) (cons _ \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2208-\u2286\u2097-trans \u03c4\u2208\u03c4\u1d62 \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\nsub : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9 m \u2236 \u03c4'\nsub\u1d62 : \u2200 {A \u0393 \u0393'} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9\u2097 m \u2236 \u03c4 -> \u03c4' \u2286\u2097[ A ] \u03c4 -> \u0393 \u2286\u0393 \u0393' -> \u0393' \u22a9\u2097 m \u2236 \u03c4'\n\nsub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' =\n  var (\u2286\u0393-wf\u0393' \u0393\u2286\u0393') \u03c4\u1d62'\u2208 (\u2286\u2097-trans (\u2286\u2097-trans (\u2286->\u2286\u2097 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62) \u03c4\u1d62\u2286\u03c4\u1d62')\n  where\n  \u03c4\u1d62'\u2208 = proj\u2081 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n  \u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 (\u2208-\u2286\u0393-trans \u03c4\u1d62\u2208\u0393 \u0393\u2286\u0393'))\n\nsub (app \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u22a9\u2097t\u2236\u03c4\u2081 \u03c4\u2286\u03c4\u2082 \u03c4\u2237A) \u03c4'\u2286\u03c4 \u0393\u2286\u0393' = app\n  (sub\u0393 \u0393\u22a9s\u2236\u03c4\u2081~>\u03c4\u2082 \u0393\u2286\u0393')\n  (sub\u1d62\u0393 \u0393\u22a9\u2097t\u2236\u03c4\u2081 \u0393\u2286\u0393')\n  (\u2286\u2097-trans (\u2286->\u2286\u2097 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u2082)\n  \u03c4\u2237A\nsub {_} {\u0393} {\u0393'} (abs {\u03c4 = \u03c4} {\u03c4'} L {t} cf \u03c4~>\u03c4'\u2237A\u27f6B) (arr {A} {\u03c4\u2081\u2081 = \u03c4\u2081\u2081} \u03c4\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4' (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B) x\u2083) \u0393\u2286\u0393' = abs\n  (L ++ dom \u0393')\n  (\u03bb x\u2209 \u2192 sub\u1d62\n    (cf (\u2209-cons-l _ _ x\u2209))\n    \u03c4\u2081\u2082\u2286\u03c4'\n    (cons (\u03c4\u2081\u2081 , (here refl) , \u03c4\u2286\u03c4\u2081\u2081) (\u2286\u0393-\u2237 (\u2209-cons-r L _ x\u2209) \u03c4\u2081\u2081\u2237A \u0393\u2286\u0393')))\n  (arr \u03c4\u2081\u2081\u2237A \u03c4\u2081\u2082\u2237B)\nsub (Y wf-\u0393 (\u03c4' , \u03c4'\u2208\u03c4\u2081 , \u03c4~>\u03c4\u2286\u03c4') \u03c4\u2081\u2237A\u27f6A \u03c4\u2082\u2286\u03c4) (arr {\u03c4\u2081\u2081 = \u03c4\u2081'} \u03c4\u2081\u2286\u03c4\u2081' \u03c4\u2082'\u2286\u03c4\u2082 (arr \u2229\u03c4\u2081'\u2237A\u27f6A \u03c4\u2082'\u2237A) x\u2084) \u0393\u2286\u0393' =\n  Y ((\u2286\u0393-wf\u0393' \u0393\u2286\u0393')) (\u03c4'' , (\u03c4''\u2208\u03c4\u2081' , \u2286-trans \u03c4~>\u03c4\u2286\u03c4' \u03c4'\u2286\u03c4'')) \u2229\u03c4\u2081'\u2237A\u27f6A (\u2286\u2097-trans \u03c4\u2082'\u2286\u03c4\u2082 \u03c4\u2082\u2286\u03c4)\n  where\n  \u03c4'' = proj\u2081 (\u2208-\u2286\u2097-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081')\n  \u03c4''\u2208\u03c4\u2081' = proj\u2081 (proj\u2082 (\u2208-\u2286\u2097-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n  \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2208-\u2286\u2097-trans \u03c4'\u2208\u03c4\u2081 \u03c4\u2081\u2286\u03c4\u2081'))\n\nsub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4 (nil x) \u0393\u2286\u0393' = nil (\u2286\u0393-wf\u0393' \u0393\u2286\u0393')\nsub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4'\u1d62\u2286\u03c4\u1d62) \u0393\u2286\u0393' with \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62\n... | \u0393\u22a9m\u2236\u03c4 = cons (sub \u0393\u22a9m\u2236\u03c4 \u03c4'\u2286\u03c4 \u0393\u2286\u0393') (sub\u1d62 \u0393\u22a9\u2097m\u2236\u03c4\u1d62 \u03c4'\u1d62\u2286\u03c4\u1d62 \u0393\u2286\u0393')\n\n\n\n\n\n\n~>\u2229 : \u2200 {X A B C} -> (A ~> B) \u2237' X -> (A ~> C) \u2237' X ->\n  \u2229' (A ~> (B ++ C)) \u2286\u2097[ X ] ((A ~> B) \u2237 (A ~> C) \u2237 [])\n~>\u2229 A~>B\u2237X A~>C\u2237X = cons ({!   !} , ({!   !} , {!   !})) (nil (cons A~>B\u2237X (cons A~>C\u2237X nil)))\n\n\n\u0393\u22a9Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u0393 \u22a9 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 (\u2229' \u03c4) \u2286\u2097[ A ] \u03c4')\n\u0393\u22a9Ym-max {A} {\u0393} {m} {\u03c4} (app {\u03c4\u2081 = \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62} {\u03c4'}\n  (Y {\u03c4 = \u03c4''} wf-\u0393 (_ , \u03c4\u2081~>\u03c4\u2082\u2208\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 , arr {\u03c4\u2082\u2081 = \u03c4\u2081} {\u03c4\u2082} \u03c4\u2081\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082 \u03c4''~>\u03c4''\u2237A\u27f6A (arr \u03c4\u2081\u2237A \u03c4\u2082\u2237A)) \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A \u03c4'\u2286\u03c4'')\n  \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n  \u03c4\u2286\u03c4' \u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\u2237A\u27f6A\u2082) =\n  \u03c4\u2082 , \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 , \u03c4\u2286\u03c4\u2082\n\n  where\n  \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z)\n  \u03c4\u2286\u03c4'' = \u2286\u2097-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''\n  \u03c4\u2286\u03c4\u2082 = \u2286\u2097-trans \u03c4\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082\n\n  \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u03c4\u2081 ~> \u03c4\u2082)\n  \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62 \u03c4\u2081~>\u03c4\u2082\u2208\u03c4\u2081\u1d62~>\u03c4\u2082\u1d62\n\n  \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u03c4\u2082 ~> \u03c4\u2082)\n  \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2082 = sub \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 (arr (\u2286\u2097-trans \u03c4\u2081\u2286\u03c4'' \u03c4''\u2286\u03c4\u2082) (\u2286\u2097-refl \u03c4\u2082\u2237A) (arr \u03c4\u2082\u2237A \u03c4\u2082\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2082\u2237A)) \u0393\u2286\u0393-refl\n\n\n\n\u0393\u22a9\u2097Ym-max : \u2200 {A \u0393} {m : \u039b (A \u27f6 A)} {\u03c4} -> \u00ac(\u03c4 \u2261 \u03c9) -> \u0393 \u22a9\u2097 app (Y A) m \u2236 \u03c4 -> \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 \u03c4 \u2286\u2097[ A ] \u03c4')\n\u0393\u22a9\u2097Ym-max \u03c4\u2260\u03c9 (nil wf-\u0393) = \u22a5-elim (\u03c4\u2260\u03c9 refl)\n\u0393\u22a9\u2097Ym-max \u03c4\u2260\u03c9 (cons x (nil wf-\u0393)) = \u0393\u22a9Ym-max x\n\u0393\u22a9\u2097Ym-max {A} {\u0393} {m} \u03c4\u2260\u03c9 (cons {\u03c4 = \u03c4} x (cons {\u03c4 = \u03c4\u2082} {\u03c4\u1d62} x\u2081 \u0393\u22a9\u2097Ym\u2236\u03c4)) =\n  {!   !}\n  where\n  ih : \u2203(\u03bb \u03c4' -> \u0393 \u22a9 m \u2236 (\u03c4' ~> \u03c4') \u00d7 (\u03c4\u2082 \u2237 \u03c4\u1d62) \u2286\u2097[ A ] \u03c4')\n  ih = \u0393\u22a9\u2097Ym-max {!   !} (cons {\u03c4 = \u03c4\u2082} {\u03c4\u1d62} x\u2081 \u0393\u22a9\u2097Ym\u2236\u03c4)\n\n  \u03c4\u1d62' = proj\u2081 ih\n  \u0393\u22a9m\u2236\u03c4\u1d62'~>\u03c4\u1d62' = proj\u2081 (proj\u2082 ih)\n  \u03c4\u2082\u03c4\u1d62\u2286\u03c4\u1d62' = proj\u2082 (proj\u2082 ih)\n\n  \u03c4' = proj\u2081 (\u0393\u22a9Ym-max x)\n  \u0393\u22a9m\u2236\u03c4'~>\u03c4' = proj\u2081 (proj\u2082 (\u0393\u22a9Ym-max x))\n  \u03c4\u2286\u03c4' = proj\u2082 (proj\u2082 (\u0393\u22a9Ym-max x))\n\n\n\n\u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n\u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (nil wf-\u0393) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y _) = app\n  (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n    wf-\u0393\n    ((\u03c9 ~> \u2229' \u03c4) , here refl , arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr nil \u03c4\u2237A))\n    (cons (arr []\u2237A \u03c4\u2237A) nil)\n    (\u2286\u2097-refl \u03c4\u2237A))\n  (cons (sub \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (arr (nil []\u2237A) \u03c4\u2286\u03c4\u1d62 (arr []\u2237A \u03c4\u2237A) (arr []\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n  (\u2286\u2097-refl \u03c4\u2237A)\n  (cons (arr []\u2237A \u03c4\u2237A) nil)\n\n  where\n  \u03c4\u2237A = \u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62\n\n\u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u03c4' \u2237 \u03c4'\u1d62} {\u03c4\u1d62} \u0393\u22a9m\u2236\u03c4'~>\u03c4\u1d62 \u0393\u22a9\u2097Ym\u2236\u03c4' \u03c4\u2286\u03c4\u1d62 \u03c4'\u2237A) (Y _) = {!   !}\n  where\n  \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 (\u22a9\u2097-wf-\u0393 \u0393\u22a9\u2097Ym\u2236\u03c4') (\u03bb {x} z \u2192 z)\n\n  \u03c4'' = proj\u2081 (\u0393\u22a9\u2097Ym-max (\u03bb x \u2192 {!   !}) \u0393\u22a9\u2097Ym\u2236\u03c4')\n\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4'' : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u03c4'')\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4'' = proj\u2081 (proj\u2082 (\u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4'))\n\n  \u03c4'\u03c4'\u1d62\u2286\u03c4'' : (\u03c4' \u2237 \u03c4'\u1d62) \u2286\u2097[ A ] \u03c4''\n  \u03c4'\u03c4'\u1d62\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u0393\u22a9\u2097Ym-max {!   !} \u0393\u22a9\u2097Ym\u2236\u03c4'))\n\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4\u1d62 : \u0393 \u22a9 m \u2236 (\u03c4'' ~> \u2229' \u03c4)\n  \u0393\u22a9m\u2236\u03c4''~>\u03c4\u1d62 = sub \u0393\u22a9m\u2236\u03c4'~>\u03c4\u1d62 (arr \u03c4'\u03c4'\u1d62\u2286\u03c4'' \u03c4\u2286\u03c4\u1d62 (arr (\u2286\u2097-\u2237'\u2097-r \u03c4'\u03c4'\u1d62\u2286\u03c4'') (\u2286\u2097-\u2237'\u2097-l \u03c4\u2286\u03c4\u1d62)) (arr \u03c4'\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) \u0393\u2286\u0393-refl\n\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (nil wf-\u0393) \u03c4\u2286\u03c4\u1d62 []\u2237A) (Y _) = app\n--   (Y {\u03c4 = \u2229' \u03c4} {(\u03c9 ~> \u2229' \u03c4) \u2237 []} {\u2229' \u03c4}\n--     wf-\u0393\n--     ((\u03c9 ~> \u2229' \u03c4) , here refl , arr (nil \u03c4\u2237A) (\u2286\u2097-refl \u03c4\u2237A) (arr \u03c4\u2237A \u03c4\u2237A) (arr nil \u03c4\u2237A))\n--     (cons (arr []\u2237A \u03c4\u2237A) nil)\n--     (\u2286\u2097-refl \u03c4\u2237A))\n--   (cons (sub \u0393\u22a9m\u2236[]~>\u03c4\u1d62 (arr (nil []\u2237A) \u03c4\u2286\u03c4\u1d62 (arr []\u2237A \u03c4\u2237A) (arr []\u2237A (\u2286\u2097-\u2237'\u2097-r \u03c4\u2286\u03c4\u1d62))) (\u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z))) (nil wf-\u0393))\n--   (\u2286\u2097-refl \u03c4\u2237A)\n--   (cons (arr []\u2237A \u03c4\u2237A) nil)\n--\n--\n-- -- \u22a9->\u03b2 (app {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} \u0393\u22a9m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082 (cons (app (Y wf-\u0393 (proj\u2081 , () , proj\u2083) x\u2081 x\u2082) (nil wf-\u0393\u2081) \u03c4\u2081\u2286\u03c4\u2081\u2082 x\u2085) \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62) \u03c4\u2286\u03c4\u2082 \u03c4\u2081\u2237A) (Y x)\n-- -- \u22a9->\u03b2 {\u03c4 = \u03c4} (app {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} {\u03c4\u2082} \u0393\u22a9m\u2236\u03c4\u2081\u1d62~>\u03c4\u2082 (cons (app {\u03c4\u2081 = \u2229 (\u03c4\u2081\u2081 \u2237 \u03c4\u2081\u2081\u1d62)} {\u03c4\u2081\u2082} (Y {\u03c4 = \u03c4'} wf-\u0393 (\u03c4'' , \u03c4''\u2208 \u03c4\u2081\u2081, proj\u2083) x\u2081 x\u2082) (cons \u0393\u22a9m\u2236\u03c4\u2081\u2081 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081\u1d62) \u03c4\u2081\u2286\u03c4\u2081\u2082 x\u2085) \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62) \u03c4\u2286\u03c4\u2082 \u03c4\u2081\u2237A) (Y _) =\n-- \u22a9->\u03b2 {A} {\u0393} {\u03c4 = \u03c4} (app {s = m} {\u03c4\u2081 = \u2229 (\u03c4\u2081 \u2237 \u03c4\u2081\u1d62)} {\u03c4\u2082}\n--   \u0393\u22a9m\u2236\u03c4\u2081\u1d62\u03c4\u2081\u1d62~>\u03c4\u2082\n--   (cons\n--     (app {\u03c4\u2081 = \u2229 (\u03c4\u2081\u2081)} {\u03c4\u2081\u2082}\n--       (Y {\u03c4 = \u03c4'} wf-\u0393 (_ , \u03c4''\u2208\u03c4\u2081\u2081 , arr {\u03c4\u2082\u2081 = \u03c4''\u2081} {\u03c4''\u2082} \u03c4''\u2081\u2286\u03c4' \u03c4'\u2286\u03c4''\u2082 \u03c4'~>\u03c4'\u2237A\u27f6A (arr \u03c4''\u2081\u2237A \u03c4''\u2082\u2237A)) \u03c4\u2081\u2081\u2237A\u27f6A \u03c4\u2081\u2082\u2286\u03c4')\n--       \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081 \u03c4\u2081\u2286\u03c4\u2081\u2082 \u03c4\u2081\u2081\u2237A\u27f6A\u2082)\n--     \u0393\u22a9Ym\u2236\u03c4\u2081\u1d62)\n--   \u03c4\u2286\u03c4\u2082\n--   \u03c4\u2081\u2237A) (Y _) =\n--   {!   !}\n--\n--   where\n--   \u0393\u2286\u0393-refl = \u2286\u0393-\u2286 wf-\u0393 (\u03bb {x} z \u2192 z)\n--\n--   \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 : \u0393 \u22a9 m \u2236 (\u03c4''\u2081 ~> \u03c4''\u2082)\n--   \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 = \u22a9\u2097-\u2208-\u22a9 \u0393\u22a9\u2097m\u2236\u03c4\u2081\u2081 \u03c4''\u2208\u03c4\u2081\u2081\n--\n--   \u0393\u22a9m\u2236\u03c4''\u2082~>\u03c4''\u2082 : \u0393 \u22a9 m \u2236 (\u03c4''\u2082 ~> \u03c4''\u2082)\n--   \u0393\u22a9m\u2236\u03c4''\u2082~>\u03c4''\u2082 = sub \u0393\u22a9m\u2236\u03c4''\u2081~>\u03c4''\u2082 (arr (\u2286\u2097-trans \u03c4''\u2081\u2286\u03c4' \u03c4'\u2286\u03c4''\u2082) (\u2286\u2097-refl \u03c4''\u2082\u2237A) (arr \u03c4''\u2082\u2237A \u03c4''\u2082\u2237A) (arr \u03c4''\u2081\u2237A \u03c4''\u2082\u2237A)) \u0393\u2286\u0393-refl\n--\n--   \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 : \u0393 \u22a9 m \u2236 (\u2229' \u03c4\u2081 ~> \u03c4\u2082)\n--   \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4\u2082 = sub \u0393\u22a9m\u2236\u03c4\u2081\u1d62\u03c4\u2081\u1d62~>\u03c4\u2082 (arr {!   !} {!   !} {!   !} {!   !}) \u0393\u2286\u0393-refl\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u2097Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9e69e5a746afe927fd5ba1cc31079d6a70e34d20","subject":"the bultin pragram fix.","message":"the bultin pragram fix.\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat.agda","new_file":"agda\/Nat.agda","new_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115 #-}\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\nimport Relation.Binary.EqReasoning as EqR\nopen module EqNat = EqR (setoid \u2115)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b) =\n  begin\n    a + succ b\n      \u2248\u27e8 lemma a b \u27e9\n    succ (a + b)\n      \u2248\u27e8 cong succ (+-is-commutative a b) \u27e9\n    succ (b + a)\n      \u2248\u27e8 refl \u27e9\n    succ b + a\n  \u220e\n","old_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115 #-}\n{-# BUILTIN ZERO zero #-}\n{-# BUILTIN SUC  succ #-}\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\n_\u00d7_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   \u00d7 b = zero\nsucc a \u00d7 b = (a \u00d7 b) + b\n\n\nopen import Relation.Binary.PropositionalEquality\n\n0-is-right-identity-of-+ : \u2200 (n : \u2115) \u2192 n + zero \u2261 n\n0-is-right-identity-of-+ zero     = refl\n0-is-right-identity-of-+ (succ n) = cong succ (0-is-right-identity-of-+ n)\n\n\n+-is-associative : \u2200 (a b c : \u2115) \u2192 a + (b + c) \u2261 (a + b) + c\n+-is-associative zero     b c = refl\n+-is-associative (succ a) b c = cong succ (+-is-associative a b c)\n\n\nlemma : \u2200 (a b : \u2115) \u2192 a + succ b \u2261 succ (a + b)\nlemma zero     b = refl\nlemma (succ a) b = cong succ (lemma a b)\n\nimport Relation.Binary.EqReasoning as EqR\nopen module EqNat = EqR (setoid \u2115)\n\n+-is-commutative : \u2200 (a b : \u2115) \u2192 a + b \u2261 b + a\n+-is-commutative a zero     = 0-is-right-identity-of-+ a\n+-is-commutative a (succ b) =\n  begin\n    a + succ b\n      \u2248\u27e8 lemma a b \u27e9\n    succ (a + b)\n      \u2248\u27e8 cong succ (+-is-commutative a b) \u27e9\n    succ (b + a)\n      \u2248\u27e8 refl \u27e9\n    succ b + a\n  \u220e\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"63b0b983e562c4195da605874a1e5b1e379fa13c","subject":"Refactor UNDEFINED to simplify it (XXX untested)","message":"Refactor UNDEFINED to simplify it (XXX untested)\n\nInstance arguments changed behavior, so not every definition is\navailable anymore. Hence we can avoid the wrapping.\n\nIt typechecks, but not really tested on clients.\n","repos":"inc-lc\/ilc-agda","old_file":"UNDEFINED.agda","new_file":"UNDEFINED.agda","new_contents":"-- Allow holes in modules to import, by introducing a single general postulate.\n\nmodule UNDEFINED where\n\npostulate\n  UNDEFINED : \u2200 {\u2113} \u2192 {T : Set \u2113} \u2192 T\n","old_contents":"-- Allow holes in modules to import, by introducing a single general postulate.\n\nmodule UNDEFINED where\n\nopen import Data.Unit.NonEta using (Hidden)\nopen import Data.Unit.NonEta public using (reveal)\n\npostulate\n  -- If this postulate would produce a T, it could be used as instance argument, triggering many ambiguities.\n  -- This postulate is named in capitals because it introduces an inconsistency.\n  -- Hence, this must be used through `reveal UNDEFINED`.\n  UNDEFINED : \u2200 {\u2113} \u2192 {T : Set \u2113} \u2192 Hidden T\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"03b39e32079b3464f964cc88ad93cc6679dc3d90","subject":"minor changes","message":"minor changes\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"code\/GoaTest.agda","new_file":"code\/GoaTest.agda","new_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where -- finite type\n  true : Bool\n  false : Bool\n\nidBool : Bool \u2192 Bool -- lambda\nidBool x = x\n\nalwaysTrue : Bool \u2192 Bool\nalwaysTrue x = true\n\nnot : Bool \u2192 Bool -- case defn\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool -- lambda\nnotnot x = not(not(x))\n\n_&_ : Bool \u2192 Bool \u2192 Bool --curried function\ntrue & x = x\nfalse & _ = false\n\ndata \u2115 : Type where -- infinite type\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool -- recursive definition\neven zero = true\neven (succ x) = not (even x)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115 \nzero + y = y\nsucc x + y = x + succ y\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata \u2115List : Type where --list type \n  [] : \u2115List -- empty list \n  _::_ : \u2115 \u2192 \u2115List \u2192 \u2115List -- add number to head of list\n\nmylist : \u2115List \nmylist = 3 :: (4 :: (2 :: [])) -- the list [3, 4, 2]\n\ndata Vector : \u2115 \u2192 Type where -- type family\n  [] : Vector 0\n  _::_ : {n : \u2115} \u2192 \u2115 \u2192 Vector n \u2192 Vector (succ n) \n\nsum : {n : \u2115} \u2192 Vector n \u2192 \u2115\nsum [] = 0\nsum (x :: l) = x + sum l\n\n\ncountdown : (n : \u2115) \u2192 Vector n -- dependent function\ncountdown 0 = []\ncountdown (succ n) = (succ n) :: (countdown n)\n\nsumToN : \u2115 \u2192 \u2115 -- calculation\nsumToN n = sum(countdown n)\n\ndata isEven : \u2115 \u2192 Type where\n  0even : isEven 0\n  +2even : (n : \u2115) \u2192 isEven n \u2192 isEven (succ(succ(n)))\n\n4even : isEven 4\n4even = +2even _ (+2even _ 0even)\n\ndata False : Type where\n\n1odd : isEven 1 \u2192 False\n1odd ()\n\n3odd : isEven 3 \u2192 False\n3odd (+2even .1 ())\n\nhalf : (n : \u2115) \u2192 isEven n \u2192 \u2115\nhalf .0 0even = 0\nhalf .(succ (succ n)) (+2even n pf) = half n pf\n\ndouble : (n : \u2115) \u2192 \u2115\ndouble 0 = 0\ndouble (succ n) = succ(succ(double(n)))\n\nstep : (n : \u2115) \u2192 (isEven (double n)) \u2192 isEven (double(succ(n)))\nstep n pf = +2even _ pf\n\nthm : (n  : \u2115) \u2192 isEven (double n)\nthm zero = 0even\nthm (succ n) = step _ (thm n)\n\nhalfOfDouble : \u2115 \u2192 \u2115\nhalfOfDouble n = half (double n) (thm n)\n","old_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where -- finite type\n  true : Bool\n  false : Bool\n\nidBool : Bool \u2192 Bool -- lambda\nidBool x = x\n\nalwaysTrue : Bool \u2192 Bool\nalwaysTrue x = true\n\nnot : Bool \u2192 Bool -- case defn\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool -- lambda\nnotnot x = not(not(x))\n\n_&_ : Bool \u2192 Bool \u2192 Bool --curried function\ntrue & x = x\nfalse & _ = false\n\ndata \u2115 : Type where -- infinite type\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool -- recursive definition\neven zero = true\neven (succ x) = not (even x)\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115 \nzero + y = y\nsucc x + y = x + succ y\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata \u2115List : Type where --list type\n  [] : \u2115List\n  _::_ : \u2115 \u2192 \u2115List \u2192 \u2115List\n\n\ndata Vector : \u2115 \u2192 Type where -- type family\n  [] : Vector 0\n  _::_ : {n : \u2115} \u2192 \u2115 \u2192 Vector n \u2192 Vector (succ n) \n\nsum : {n : \u2115} \u2192 Vector n \u2192 \u2115\nsum [] = 0\nsum (x :: l) = x + sum l\n\n\ncountdown : (n : \u2115) \u2192 Vector n -- dependent function\ncountdown 0 = []\ncountdown (succ n) = (succ n) :: (countdown n)\n\nsumToN : \u2115 \u2192 \u2115 -- calculation\nsumToN n = sum(countdown n)\n\ndata isEven : \u2115 \u2192 Type where\n  0even : isEven 0\n  +2even : (n : \u2115) \u2192 isEven n \u2192 isEven (succ(succ(n)))\n\n4even : isEven 4\n4even = +2even _ (+2even _ 0even)\n\ndata False : Type where\n\n1odd : isEven 1 \u2192 False\n1odd ()\n\n3odd : isEven 3 \u2192 False\n3odd (+2even .1 ())\n\nhalf : (n : \u2115) \u2192 isEven n \u2192 \u2115\nhalf .0 0even = 0\nhalf .(succ (succ n)) (+2even n pf) = half n pf\n\ndouble : (n : \u2115) \u2192 \u2115\ndouble 0 = 0\ndouble (succ n) = succ(succ(double(n)))\n\nstep : (n : \u2115) \u2192 (isEven (double n)) \u2192 isEven (double(succ(n)))\nstep n pf = +2even _ pf\n\nthm : (n  : \u2115) \u2192 isEven (double n)\nthm zero = 0even\nthm (succ n) = step _ (thm n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"75e7c960c4fb73b5849965e752a1bde46c5f021f","subject":"Clarify comment","message":"Clarify comment\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Evaluation.agda","new_file":"Parametric\/Denotation\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.CBPVType as CBPVType\n  open CBPVType.Structure Base\n  open import Parametric.Syntax.CBPVTerm as CBPVTerm\n  open CBPVTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n  \u27e6 \u03c4\u2081 \u03a0 \u03c4\u2082 \u27e7CompType = \u27e6 \u03c4\u2081 \u27e7CompType \u00d7 \u27e6 \u03c4\u2082 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.>\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\u2081\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u03c4\u2081 )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6 \u03c4\u2081 \u03a0 \u03c4\u2082 \u27e7CompTypeHidCache = \u27e6 \u03c4\u2081 \u27e7CompTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\u2081\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values (where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things)!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cProduce v) \u03c1 = , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n\n  -- But if we alter _into_ as described above, composing the caches works!\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into2 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cProduce time, we just need the nested into2 to do their job, because as I\n  -- said intermediate results are a a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using into2 for the same reasons - which makes\n  -- sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into2 case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cProduce. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.CBPVType as CBPVType\n  open CBPVType.Structure Base\n  open import Parametric.Syntax.CBPVTerm as CBPVTerm\n  open CBPVTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n  \u27e6 \u03c4\u2081 \u03a0 \u03c4\u2082 \u27e7CompType = \u27e6 \u03c4\u2081 \u27e7CompType \u00d7 \u27e6 \u03c4\u2082 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.>\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\u2081\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = Lift Unit\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u03c4\u2081 )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6 \u03c4\u2081 \u03a0 \u03c4\u2082 \u27e7CompTypeHidCache = \u27e6 \u03c4\u2081 \u27e7CompTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\u2081\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values (where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things)!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cProduce v) \u03c1 = , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n\n  -- But if we alter _into_ as described above, composing the caches works!\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into2 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cProduce time, we just need the nested into2 to do their job, because as I\n  -- said intermediate results are a a writer monad. But then, do we still need\n  -- to do all the other stuff? IOW, do we still need to forbid (\u03bb x . y <- f\n  -- args; g args') and replace it with (\u03bb x . y <- f args; z <- g args'; z)?\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as we don't do (a, b) but\n  -- append a b (ahem, where?).\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cProduce. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9c628f8fc4514a3379c8bb109b925c9a196d5571","subject":"Drop extra parens","message":"Drop extra parens\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/MTerm.agda","new_file":"Parametric\/Syntax\/MTerm.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Syntax.MType as MType\n\nmodule Parametric.Syntax.MTerm\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n  where\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Term.Structure Base Const\n\n-- Our extension points are sets of primitives, indexed by the\n-- types of their arguments and their return type.\n\n-- We want different types of constants; some produce values, some produce\n-- computations. In all cases, arguments are assumed to be values (though\n-- individual primitives might require thunks).\n\nValConstStructure : Set\u2081\nValConstStructure = ValContext \u2192 ValType \u2192 Set\n\nCompConstStructure : Set\u2081\nCompConstStructure = ValContext \u2192 CompType \u2192 Set\n\nmodule Structure (ValConst : ValConstStructure) (CompConst : CompConstStructure) where\n  mutual\n    -- Analogues of Terms\n    Vals : ValContext \u2192 ValContext \u2192 Set\n    Vals \u0393 = DependentList (Val \u0393)\n\n    Comps : ValContext \u2192 List CompType \u2192 Set\n    Comps \u0393 = DependentList (Comp \u0393)\n\n    data Val \u0393 : (\u03c4 : ValType) \u2192 Set where\n      vVar : \u2200 {\u03c4} (x : ValVar \u0393 \u03c4) \u2192 Val \u0393 \u03c4\n      -- XXX Do we need thunks? The draft in the paper doesn't have them.\n      -- However, they will start being useful if we deal with CBN source\n      -- languages.\n      vThunk : \u2200 {\u03c4} \u2192 Comp \u0393 \u03c4 \u2192 Val \u0393 (U \u03c4)\n      vConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : ValConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Val \u0393 \u03c4\n\n    data Comp \u0393 : (\u03c4 : CompType) \u2192 Set where\n      cConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : CompConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Comp \u0393 \u03c4\n\n      cForce : \u2200 {\u03c4} \u2192 Val \u0393 (U \u03c4) \u2192 Comp \u0393 \u03c4\n      cReturn : \u2200 {\u03c4} (v : Val \u0393 \u03c4) \u2192 Comp \u0393 (F \u03c4)\n      {-\n      -- Originally, M to x in N. But here we have no names!\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 \u03c4\n      -}\n      -- The following constructor is the main difference between CBPV and this\n      -- monadic calculus. This is better for the caching transformation.\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) (F \u03c4)) \u2192\n        Comp \u0393 (F \u03c4)\n      cAbs : \u2200 {\u03c3 \u03c4} \u2192\n        (t : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 (\u03c3 \u21db \u03c4)\n      cApp : \u2200 {\u03c3 \u03c4} \u2192\n        (s : Comp \u0393 (\u03c3 \u21db \u03c4)) \u2192\n        (t : Val \u0393 \u03c3) \u2192\n        Comp \u0393 \u03c4\n\n  weaken-val : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Val \u0393\u2081 \u03c4 \u2192\n    Val \u0393\u2082 \u03c4\n\n  weaken-comp : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Comp \u0393\u2081 \u03c4 \u2192\n    Comp \u0393\u2082 \u03c4\n\n  weaken-vals : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Vals \u0393\u2081 \u03a3 \u2192\n    Vals \u0393\u2082 \u03a3\n\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vVar x) = vVar (weaken-val-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vThunk x) = vThunk (weaken-comp \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vConst c args) = vConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cConst c args) = cConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cForce x) = cForce (weaken-val \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cReturn v) = cReturn (weaken-val \u0393\u2081\u227c\u0393\u2082 v)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (_into_ {\u03c3} c c\u2081) = (weaken-comp \u0393\u2081\u227c\u0393\u2082 c) into (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c\u2081)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cAbs {\u03c3} c) = cAbs (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cApp s t) = cApp (weaken-comp \u0393\u2081\u227c\u0393\u2082 s) (weaken-val \u0393\u2081\u227c\u0393\u2082 t)\n\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 (px \u2022 ts) = (weaken-val \u0393\u2081\u227c\u0393\u2082 px) \u2022 (weaken-vals \u0393\u2081\u227c\u0393\u2082 ts)\n\n  fromCBN : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBNCtx \u0393) (cbnToCompType \u03c4)\n\n  fromCBNTerms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Vals (fromCBNCtx \u0393) (fromCBNCtx \u03a3)\n  fromCBNTerms \u2205 = \u2205\n  fromCBNTerms (px \u2022 ts) = vThunk (fromCBN px) \u2022 fromCBNTerms ts\n\n  open import UNDEFINED\n  -- This is really supposed to be part of the plugin interface.\n  cbnToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBNCtx \u03a3) (cbnToCompType \u03c4)\n  cbnToCompConst = reveal UNDEFINED\n\n  fromCBN (const c args) = cConst (cbnToCompConst c) (fromCBNTerms args)\n  fromCBN (var x) = cForce (vVar (fromVar cbnToValType x))\n  fromCBN (app s t) = cApp (fromCBN s) (vThunk (fromCBN t))\n  fromCBN (abs t) = cAbs (fromCBN t)\n\n  -- To satisfy termination checking, we'd need to inline fromCBV and weaken: fromCBV needs to produce a term in a bigger context.\n  -- But let's ignore that.\n  {-# NO_TERMINATION_CHECK #-}\n  fromCBV : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n\n  -- This is really supposed to be part of the plugin interface.\n  cbvToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBVCtx \u03a3) (cbvToCompType \u03c4)\n  cbvToCompConst = reveal UNDEFINED\n\n  cbvTermsToComps : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Comps (fromCBVCtx \u0393) (fromCBVToCompList \u03a3)\n  cbvTermsToComps \u2205 = \u2205\n  cbvTermsToComps (px \u2022 ts) = fromCBV px \u2022 cbvTermsToComps ts\n\n  module _ where\n\n    dequeValContexts : ValContext \u2192 ValContext \u2192 ValContext\n    dequeValContexts \u2205 \u0393 = \u0393\n    dequeValContexts (x \u2022 \u03a3) \u0393 = dequeValContexts \u03a3 (x \u2022 \u0393)\n\n    dequeValContexts\u227c\u227c : \u2200 \u03a3 \u0393 \u2192 \u0393 \u227c\u227c dequeValContexts \u03a3 \u0393\n    dequeValContexts\u227c\u227c \u2205 \u0393 = \u227c\u227c-refl\n    dequeValContexts\u227c\u227c (x \u2022 \u03a3) \u0393 = \u227c\u227c-trans (drop_\u2022\u2022_ x \u227c\u227c-refl) (dequeValContexts\u227c\u227c \u03a3 (x \u2022 \u0393)) --\n\n    dequeContexts : Context \u2192 Context \u2192 ValContext\n    dequeContexts \u03a3 \u0393 = dequeValContexts (fromCBVCtx \u03a3) (fromCBVCtx \u0393)\n\n    fromCBVArg : \u2200 {\u03c3 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 (Val (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToValType \u03c4) \u2192 Comp (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToCompType \u03c3)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c3)\n    fromCBVArg t k = (fromCBV t) into k (vVar vThis)\n\n    fromCBVArgs : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Terms \u0393 \u03a3 \u2192 (Vals (dequeContexts \u03a3 \u0393) (fromCBVCtx \u03a3) \u2192 Comp (dequeContexts \u03a3 \u0393) (cbvToCompType \u03c4)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    fromCBVArgs \u2205 k = k \u2205\n    fromCBVArgs {\u03c3 \u2022 \u03a3} {\u0393} (t \u2022 ts) k = fromCBVArg t (\u03bb v \u2192 fromCBVArgs (weaken-terms (drop_\u2022_ _ \u227c-refl) ts) (\u03bb vs \u2192 k (weaken-val (dequeValContexts\u227c\u227c (fromCBVCtx \u03a3) _) v \u2022 vs)))\n\n    -- Transform a constant application into\n    --\n    --   arg1 to x1 in (arg2 to x2 in ... (argn to xn in (cConst (cbvToCompConst c) (x1 :: x2 :: ... xn)))).\n\n    fromCBVConstCPSRoot : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 Terms \u0393 \u03a3 \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    -- pass that as a closure to compose with (_\u2022_ x) for each new variable.\n    fromCBVConstCPSRoot c ts = fromCBVArgs ts (\u03bb vs \u2192 cConst (cbvToCompConst c) vs) -- fromCBVConstCPSDo ts {!cConst (cbvToCompConst c)!}\n\n-- In the beginning, we should get a function that expects a whole set of\n-- arguments (Vals \u0393 \u03a3) for c, in the initial context \u0393.\n-- Later, each call should match:\n\n-- -  \u03a3 = \u03c4\u03a3 \u2022 \u03a3\u2032, then we recurse with \u0393\u2032 = \u03c4\u03a3 \u2022 \u0393 and \u03a3\u2032. Terms ought to be\n--    weakened. The result of f (or the arguments) also ought to be weakened! So f should weaken\n--    all arguments.\n\n-- - \u03a3 = \u2205.\n\n  fromCBV (const c args) = fromCBVConstCPSRoot c args\n  fromCBV (app s t) =\n    (fromCBV s) into\n      (fromCBV (weaken (drop _ \u2022 \u227c-refl) t) into\n        cApp (cForce (vVar (vThat vThis))) (vVar vThis))\n  -- Values\n  fromCBV (var x) = cReturn (vVar (fromVar cbvToValType x))\n  fromCBV (abs t) = cReturn (vThunk (cAbs (fromCBV t)))\n\n  -- This reflects the CBV conversion of values in TLCA '99, but we can't write\n  -- this because it's partial on the Term type. Instead, we duplicate thunking\n  -- at the call site.\n  {-\n  fromCBVToVal : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Val (fromCBVCtx \u0393) (cbvToValType \u03c4)\n  fromCBVToVal (var x) = vVar (fromVar cbvToValType x)\n  fromCBVToVal (abs t) = vThunk (cAbs (fromCBV t))\n\n  fromCBVToVal (const c args) = {!!}\n  fromCBVToVal (app t t\u2081) = {!!} -- Not a value\n  -}\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Syntax.MType as MType\n\nmodule Parametric.Syntax.MTerm\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n  where\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Term.Structure Base Const\n\n-- Our extension points are sets of primitives, indexed by the\n-- types of their arguments and their return type.\n\n-- We want different types of constants; some produce values, some produce\n-- computations. In all cases, arguments are assumed to be values (though\n-- individual primitives might require thunks).\n\nValConstStructure : Set\u2081\nValConstStructure = ValContext \u2192 ValType \u2192 Set\n\nCompConstStructure : Set\u2081\nCompConstStructure = ValContext \u2192 CompType \u2192 Set\n\nmodule Structure (ValConst : ValConstStructure) (CompConst : CompConstStructure) where\n  mutual\n    -- Analogues of Terms\n    Vals : ValContext \u2192 ValContext \u2192 Set\n    Vals \u0393 = DependentList (Val \u0393)\n\n    Comps : ValContext \u2192 List CompType \u2192 Set\n    Comps \u0393 = DependentList (Comp \u0393)\n\n    data Val \u0393 : (\u03c4 : ValType) \u2192 Set where\n      vVar : \u2200 {\u03c4} (x : ValVar \u0393 \u03c4) \u2192 Val \u0393 \u03c4\n      -- XXX Do we need thunks? The draft in the paper doesn't have them.\n      -- However, they will start being useful if we deal with CBN source\n      -- languages.\n      vThunk : \u2200 {\u03c4} \u2192 Comp \u0393 \u03c4 \u2192 Val \u0393 (U \u03c4)\n      vConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : ValConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Val \u0393 \u03c4\n\n    data Comp \u0393 : (\u03c4 : CompType) \u2192 Set where\n      cConst : \u2200 {\u03a3 \u03c4} \u2192\n        (c : CompConst \u03a3 \u03c4) \u2192\n        (args : Vals \u0393 \u03a3) \u2192\n        Comp \u0393 \u03c4\n\n      cForce : \u2200 {\u03c4} \u2192 Val \u0393 (U \u03c4) \u2192 Comp \u0393 \u03c4\n      cReturn : \u2200 {\u03c4} (v : Val \u0393 \u03c4) \u2192 Comp \u0393 (F \u03c4)\n      {-\n      -- Originally, M to x in N. But here we have no names!\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 \u03c4\n      -}\n      -- The following constructor is the main difference between CBPV and this\n      -- monadic calculus. This is better for the caching transformation.\n      _into_ : \u2200 {\u03c3 \u03c4} \u2192\n        (e\u2081 : Comp \u0393 (F \u03c3)) \u2192\n        (e\u2082 : Comp (\u03c3 \u2022\u2022 \u0393) (F \u03c4)) \u2192\n        Comp \u0393 (F \u03c4)\n      cAbs : \u2200 {\u03c3 \u03c4} \u2192\n        (t : Comp (\u03c3 \u2022\u2022 \u0393) \u03c4) \u2192\n        Comp \u0393 (\u03c3 \u21db \u03c4)\n      cApp : \u2200 {\u03c3 \u03c4} \u2192\n        (s : Comp \u0393 (\u03c3 \u21db \u03c4)) \u2192\n        (t : Val \u0393 \u03c3) \u2192\n        Comp \u0393 \u03c4\n\n  weaken-val : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Val \u0393\u2081 \u03c4 \u2192\n    Val \u0393\u2082 \u03c4\n\n  weaken-comp : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Comp \u0393\u2081 \u03c4 \u2192\n    Comp \u0393\u2082 \u03c4\n\n  weaken-vals : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c\u227c \u0393\u2082) \u2192\n    Vals \u0393\u2081 \u03a3 \u2192\n    Vals \u0393\u2082 \u03a3\n\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vVar x) = vVar (weaken-val-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vThunk x) = vThunk (weaken-comp \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-val \u0393\u2081\u227c\u0393\u2082 (vConst c args) = vConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cConst c args) = cConst c (weaken-vals \u0393\u2081\u227c\u0393\u2082 args)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cForce x) = cForce (weaken-val \u0393\u2081\u227c\u0393\u2082 x)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cReturn v) = cReturn (weaken-val \u0393\u2081\u227c\u0393\u2082 v)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (_into_ {\u03c3} c c\u2081) = (weaken-comp \u0393\u2081\u227c\u0393\u2082 c) into (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c\u2081)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cAbs {\u03c3} c) = cAbs (weaken-comp (keep \u03c3 \u2022\u2022 \u0393\u2081\u227c\u0393\u2082) c)\n  weaken-comp \u0393\u2081\u227c\u0393\u2082 (cApp s t) = cApp (weaken-comp \u0393\u2081\u227c\u0393\u2082 s) (weaken-val \u0393\u2081\u227c\u0393\u2082 t)\n\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-vals \u0393\u2081\u227c\u0393\u2082 (px \u2022 ts) = (weaken-val \u0393\u2081\u227c\u0393\u2082 px) \u2022 (weaken-vals \u0393\u2081\u227c\u0393\u2082 ts)\n\n  fromCBN : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBNCtx \u0393) (cbnToCompType \u03c4)\n\n  fromCBNTerms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Vals (fromCBNCtx \u0393) (fromCBNCtx \u03a3)\n  fromCBNTerms \u2205 = \u2205\n  fromCBNTerms (px \u2022 ts) = vThunk (fromCBN px) \u2022 fromCBNTerms ts\n\n  open import UNDEFINED\n  -- This is really supposed to be part of the plugin interface.\n  cbnToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBNCtx \u03a3) (cbnToCompType \u03c4)\n  cbnToCompConst = reveal UNDEFINED\n\n  fromCBN (const c args) = cConst (cbnToCompConst c) (fromCBNTerms args)\n  fromCBN (var x) = cForce (vVar (fromVar cbnToValType x))\n  fromCBN (app s t) = cApp (fromCBN s) (vThunk (fromCBN t))\n  fromCBN (abs t) = cAbs (fromCBN t)\n\n  -- To satisfy termination checking, we'd need to inline fromCBV and weaken: fromCBV needs to produce a term in a bigger context.\n  -- But let's ignore that.\n  {-# NO_TERMINATION_CHECK #-}\n  fromCBV : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n\n  -- This is really supposed to be part of the plugin interface.\n  cbvToCompConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 CompConst (fromCBVCtx \u03a3) (cbvToCompType \u03c4)\n  cbvToCompConst = reveal UNDEFINED\n\n  cbvTermsToComps : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 Comps (fromCBVCtx \u0393) (fromCBVToCompList \u03a3)\n  cbvTermsToComps \u2205 = \u2205\n  cbvTermsToComps (px \u2022 ts) = fromCBV px \u2022 cbvTermsToComps ts\n\n  module _ where\n\n    dequeValContexts : ValContext \u2192 ValContext \u2192 ValContext\n    dequeValContexts \u2205 \u0393 = \u0393\n    dequeValContexts (x \u2022 \u03a3) \u0393 = dequeValContexts \u03a3 (x \u2022 \u0393)\n\n    dequeValContexts\u227c\u227c : \u2200 \u03a3 \u0393 \u2192 \u0393 \u227c\u227c dequeValContexts \u03a3 \u0393\n    dequeValContexts\u227c\u227c \u2205 \u0393 = \u227c\u227c-refl\n    dequeValContexts\u227c\u227c (x \u2022 \u03a3) \u0393 = \u227c\u227c-trans (drop_\u2022\u2022_ x \u227c\u227c-refl) (dequeValContexts\u227c\u227c \u03a3 (x \u2022 \u0393)) --\n\n    dequeContexts : Context \u2192 Context \u2192 ValContext\n    dequeContexts \u03a3 \u0393 = dequeValContexts (fromCBVCtx \u03a3) (fromCBVCtx \u0393)\n\n    fromCBVArg : \u2200 {\u03c3 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 (Val (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToValType \u03c4) \u2192 Comp (cbvToValType \u03c4 \u2022 fromCBVCtx \u0393) (cbvToCompType \u03c3)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c3)\n    fromCBVArg t k = (fromCBV t) into k (vVar vThis)\n\n    fromCBVArgs : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Terms \u0393 \u03a3 \u2192 (Vals (dequeContexts \u03a3 \u0393) (fromCBVCtx \u03a3) \u2192 Comp (dequeContexts \u03a3 \u0393) (cbvToCompType \u03c4)) \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    fromCBVArgs \u2205 k = k \u2205\n    fromCBVArgs {\u03c3 \u2022 \u03a3} {\u0393} (t \u2022 ts) k = fromCBVArg t (\u03bb v \u2192 fromCBVArgs (weaken-terms (drop_\u2022_ _ \u227c-refl) ts) (\u03bb vs \u2192 k (weaken-val (dequeValContexts\u227c\u227c (fromCBVCtx \u03a3) _) v \u2022 vs)))\n\n    -- Transform a constant application into\n    --\n    --   arg1 to x1 in (arg2 to x2 in ... (argn to xn in (cConst (cbvToCompConst c) (x1 :: x2 :: ... xn)))).\n\n    fromCBVConstCPSRoot : \u2200 {\u03a3 \u0393 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 Terms \u0393 \u03a3 \u2192 Comp (fromCBVCtx \u0393) (cbvToCompType \u03c4)\n    -- pass that as a closure to compose with (_\u2022_ x) for each new variable.\n    fromCBVConstCPSRoot c ts = fromCBVArgs ts (\u03bb vs \u2192 cConst (cbvToCompConst c) vs) -- fromCBVConstCPSDo ts {!cConst (cbvToCompConst c)!}\n\n-- In the beginning, we should get a function that expects a whole set of\n-- arguments (Vals \u0393 \u03a3) for c, in the initial context \u0393.\n-- Later, each call should match:\n\n-- -  \u03a3 = \u03c4\u03a3 \u2022 \u03a3\u2032, then we recurse with \u0393\u2032 = \u03c4\u03a3 \u2022 \u0393 and \u03a3\u2032. Terms ought to be\n--    weakened. The result of f (or the arguments) also ought to be weakened! So f should weaken\n--    all arguments.\n\n-- - \u03a3 = \u2205.\n\n  fromCBV (const c args) = fromCBVConstCPSRoot c args\n  fromCBV (app s t) =\n    (fromCBV s) into\n      ((fromCBV (weaken (drop _ \u2022 \u227c-refl) t)) into\n        cApp (cForce (vVar (vThat vThis))) (vVar vThis))\n  -- Values\n  fromCBV (var x) = cReturn (vVar (fromVar cbvToValType x))\n  fromCBV (abs t) = cReturn (vThunk (cAbs (fromCBV t)))\n\n  -- This reflects the CBV conversion of values in TLCA '99, but we can't write\n  -- this because it's partial on the Term type. Instead, we duplicate thunking\n  -- at the call site.\n  {-\n  fromCBVToVal : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 Val (fromCBVCtx \u0393) (cbvToValType \u03c4)\n  fromCBVToVal (var x) = vVar (fromVar cbvToValType x)\n  fromCBVToVal (abs t) = vThunk (cAbs (fromCBV t))\n\n  fromCBVToVal (const c args) = {!!}\n  fromCBVToVal (app t t\u2081) = {!!} -- Not a value\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"950ea9231b6e638cc2a907deae7a1166a3938322","subject":"Proved N-ind\u2081 using N-least-pre-fixed","message":"Proved N-ind\u2081 using N-least-pre-fixed\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-ir-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-least-pre-fixed-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ir\/N-ir-ho.\n\n  N-ir\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-ir\u2081 = N-ir-ho\n\n  N-ir-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-ir-ho\u2081 = N-ir\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n\n  N-least-pre-fixed' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-ir (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-ir (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-least-pre-fixed B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-least-pre-fixed A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d      \u2261 d\n    +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-ir.\n\n  N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-ir {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N-least-pre-fixed\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed\u2082 A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n\n------------------------------------------------------------------------------\n-- References\n--\n-- \u00c9sik, Z. (2009). Fixed Point Theory. In: Handbook of Weighted\n-- Automata. Ed. by Droste, M., Kuich, W. and Vogler, H. Monographs in\n-- Theoretical Computer Science. An EATCS Series. Springer. Chap. 2.\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-ir-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-least-pre-fixed-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-ir\/N-ir-ho.\n\n  N-ir\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-ir\u2081 = N-ir-ho\n\n  N-ir-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-ir-ho\u2081 = N-ir\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n\n  N-least-pre-fixed' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-ir (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc Nn = N-ir (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed.\n\n  -- 22 December 2013. We couldn't prove N-ind\u2081 using\n  -- N-least-pre-fixed.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h {n} Nn = N-least-pre-fixed A h' Nn\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ]  m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610) = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = {!!}\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-least-pre-fixed A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d      \u2261 d\n    +-Sx : \u2200 d e \u2192 succ\u2081 d + e \u2261 succ\u2081 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-ir.\n\n  N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n  N-ir {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N-least-pre-fixed\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed\u2082 A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n\n------------------------------------------------------------------------------\n-- References\n--\n-- \u00c9sik, Z. (2009). Fixed Point Theory. In: Handbook of Weighted\n-- Automata. Ed. by Droste, M., Kuich, W. and Vogler, H. Monographs in\n-- Theoretical Computer Science. An EATCS Series. Springer. Chap. 2.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"06bff9f7999a50f6a3bf42e20c073918438e64a7","subject":"Add a projecting (curried) and substituing Mu eliminator in Agda model.","message":"Add a projecting (curried) and substituing Mu eliminator in Agda model.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : (i : I) (xs : El I D (\u03bc I D) i) \u2192 All I D (\u03bc I D) P i xs \u2192 P i (con xs))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El I D\u2081 (\u03bc I D\u2081) i) \u2192 All I D\u2081 (\u03bc I D\u2081) P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nUncurried : (I : Set) (D : Desc I) (X : I \u2192 Set) \u2192 Set\nUncurried I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurried : (I : Set) (D : Desc I) (X : I \u2192 Set) \u2192 Set\nCurried I (`End i) X = X i\nCurried I (`Rec j D) X = (x : X j) \u2192 Curried I D X\nCurried I (`Arg A B) X = (a : A) \u2192 Curried I (B a) X\nCurried I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 Curried I D X\n\ncurry : (I : Set) (D : Desc I) (X : I \u2192 Set)\n  (cn : Uncurried I D X) \u2192 Curried I D X\ncurry I (`End i) X cn = cn refl\ncurry I (`Rec i D) X cn = \u03bb x \u2192 curry I D X (\u03bb xs \u2192 cn (x , xs))\ncurry I (`Arg A B) X cn = \u03bb a \u2192 curry I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurry I (`RecFun A B D) X cn = \u03bb f \u2192 curry I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurry : (I : Set) (D : Desc I) (X : I \u2192 Set)\n  (cn : Curried I D X) \u2192 Uncurried I D X\nuncurry I (`End i) X cn refl = cn\nuncurry I (`Rec i D) X cn (x , xs) = uncurry I D X (cn x) xs\nuncurry I (`Arg A B) X cn (a , xs) = uncurry I (B a) X (cn a) xs\nuncurry I (`RecFun A B D) X cn (f , xs) = uncurry I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 Curried I D (\u03bc I D)\ncon2 I D = curry I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b27f598e144382567eee0927d0d2275c93489ef","subject":"more FOL tinkering","message":"more FOL tinkering\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/FOL.agda","new_file":"livecode\/FOL.agda","new_contents":"open import Base\n\nopen import Nat\n\nmodule FOL(Var Const : Set)(Func : \u2115 \u2192 Set)(Pred : \u2115 \u2192 Set) where\n\nopen import Fin\n\ndata Term : Type where\n  const : Const \u2192 Term\n  var : Var \u2192 Term\n  _apply_ : {n : \u2115} \u2192 Func n \u2192 (Fin n \u2192 Term) \u2192 Term\n\ndata AtomicFormula : Type where\n  _apply_ : {n : \u2115} \u2192 Pred n \u2192 (Fin n \u2192 Term) \u2192 AtomicFormula\n\ndata RecursiveFormula (Base : Set) : Type where\n  atom : Base \u2192 RecursiveFormula(Base)\n  \u00ac : RecursiveFormula Base \u2192 RecursiveFormula Base\n  _=>_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  _<=>_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  _\u2227_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  _\u2228_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  ForAll : Var \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  Exists : Var \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n\nFormula = RecursiveFormula AtomicFormula\n\nopen import Boolean\n  \ndata Interpretation (M : Set) : Type where\n  \u03c6 : (Const \u2192 M) \u2192 ({n : \u2115} \u2192 Func n \u2192 (Fin n \u2192 M) \u2192 M) \u2192 ((n : \u2115) \u2192 Pred n \u2192 (Fin n \u2192 M) \u2192 Bool) \u2192 Interpretation M \n\nterm : {M : Set} \u2192 Interpretation M \u2192 (Var \u2192 M) \u2192 Term \u2192 M\nterm (\u03c6 x x\u2081 x\u2082) \u03be (const x\u2083) = x x\u2083\nterm \u03b1 \u03be (var x) = \u03be x\nterm (\u03c6 x x\u2081 x\u2082) \u03be (x\u2083 apply x\u2084) = x\u2081 x\u2083 (\u03bb a \u2192 term (\u03c6 x x\u2081 x\u2082) \u03be (x\u2084 a))\n\nrecBool : {Base : Set} \u2192 (Base \u2192 Bool) \u2192 RecursiveFormula Base \u2192 Bool\nrecBool \u03b1 (atom x) = \u03b1 x\nrecBool \u03b1 (\u00ac fmla) = not (recBool \u03b1 fmla)\nrecBool \u03b1 (fmla => fmla\u2081) = {!!}\nrecBool \u03b1 (fmla <=> fmla\u2081) = {!!}\nrecBool \u03b1 (fmla \u2227 fmla\u2081) = recBool \u03b1 fmla & recBool \u03b1 fmla\u2081\nrecBool \u03b1 (fmla \u2228 fmla\u2081) = {!!}\nrecBool \u03b1 (ForAll x fmla) = {!!}\nrecBool \u03b1 (Exists x fmla) = {!!}\n\nAxioms = Formula \u2192 Bool\n\nisTrue : {M : Set} \u2192 Interpretation M \u2192 Formula \u2192 Bool\nisTrue (\u03c6 x x\u2081 x\u2082) (atom (x\u2083 apply x\u2084)) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (\u00ac fmla) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla => fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla <=> fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla \u2227 fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla \u2228 fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (ForAll x\u2083 fmla) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (Exists x\u2083 fmla) = {!!}\n\ndata Theory (axioms : Axioms) : Set\u2081 where\n  theory : {M : Set} \u2192 (interp : Interpretation M) \u2192 (fmla : Formula) \u2192 (axioms fmla) == true \u2192 (isTrue interp fmla) == true \u2192 Theory axioms \npostulate\n  \u22c0 : {I : Set} \u2192 (I \u2192 Bool) \u2192 Bool\n  \u22c0=> : {I : Set} \u2192 (p : I \u2192 Bool) \u2192 \u22c0 p == true \u2192 (i : I) \u2192 (p i) == true\n  =>\u22c0 : {I : Set} \u2192 (p : I \u2192 Bool) \u2192 ((i : I) \u2192 (p i) == true) \u2192 \u22c0 p == true\n\ndata Deduction (axiom : Formula -> Bool) : Formula -> Type where\n  assumption : (fmla : Formula) -> (axiom fmla) == true -> Deduction axiom fmla\n  modusPonens : {p q : Formula} -> Deduction axiom p \u2192 Deduction axiom q \u2192 Deduction axiom p -- cannot write p => q\n   \n","old_contents":"open import Base\n\nopen import Nat\n\nmodule FOL(Var Const : Set)(Func : \u2115 \u2192 Set)(Pred : \u2115 \u2192 Set) where\n\nopen import Fin\n\ndata Term : Type where\n  const : Const \u2192 Term\n  var : Var \u2192 Term\n  _apply_ : {n : \u2115} \u2192 Func n \u2192 (Fin n \u2192 Term) \u2192 Term\n\ndata AtomicFormula : Type where\n  _apply_ : {n : \u2115} \u2192 Pred n \u2192 (Fin n \u2192 Term) \u2192 AtomicFormula\n\ndata RecursiveFormula (Base : Set) : Type where\n  atom : Base \u2192 RecursiveFormula(Base)\n  \u00ac : RecursiveFormula Base \u2192 RecursiveFormula Base\n  _=>_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  _<=>_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  _\u2227_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  _\u2228_ : RecursiveFormula Base \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  ForAll : Var \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n  Exists : Var \u2192 RecursiveFormula Base \u2192 RecursiveFormula Base\n\nFormula = RecursiveFormula AtomicFormula\n\nopen import Boolean\n  \ndata Interpretation (M : Set) : Type where\n  \u03c6 : (Const \u2192 M) \u2192 ({n : \u2115} \u2192 Func n \u2192 (Fin n \u2192 M) \u2192 M) \u2192 ((n : \u2115) \u2192 Pred n \u2192 (Fin n \u2192 M) \u2192 Bool) \u2192 Interpretation M \n\nterm : {M : Set} \u2192 Interpretation M \u2192 (Var \u2192 M) \u2192 Term \u2192 M\nterm (\u03c6 x x\u2081 x\u2082) \u03be (const x\u2083) = x x\u2083\nterm \u03b1 \u03be (var x) = \u03be x\nterm (\u03c6 x x\u2081 x\u2082) \u03be (x\u2083 apply x\u2084) = x\u2081 x\u2083 (\u03bb a \u2192 term (\u03c6 x x\u2081 x\u2082) \u03be (x\u2084 a))\n\nrecBool : {Base : Set} \u2192 (Base \u2192 Bool) \u2192 RecursiveFormula Base \u2192 Bool\nrecBool \u03b1 (atom x) = \u03b1 x\nrecBool \u03b1 (\u00ac fmla) = not (recBool \u03b1 fmla)\nrecBool \u03b1 (fmla => fmla\u2081) = {!!}\nrecBool \u03b1 (fmla <=> fmla\u2081) = {!!}\nrecBool \u03b1 (fmla \u2227 fmla\u2081) = recBool \u03b1 fmla & recBool \u03b1 fmla\u2081\nrecBool \u03b1 (fmla \u2228 fmla\u2081) = {!!}\nrecBool \u03b1 (ForAll x fmla) = {!!}\nrecBool \u03b1 (Exists x fmla) = {!!}\n\nAxioms = Formula \u2192 Bool\n\nisTrue : {M : Set} \u2192 Interpretation M \u2192 Formula \u2192 Bool\nisTrue (\u03c6 x x\u2081 x\u2082) (atom (x\u2083 apply x\u2084)) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (\u00ac fmla) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla => fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla <=> fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla \u2227 fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (fmla \u2228 fmla\u2081) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (ForAll x\u2083 fmla) = {!!}\nisTrue (\u03c6 x x\u2081 x\u2082) (Exists x\u2083 fmla) = {!!}\n\ndata Theory (axioms : Axioms) : Set\u2081 where\n  theory : {M : Set} \u2192 (interp : Interpretation M) \u2192 (fmla : Formula) \u2192 (axioms fmla) == true \u2192 (isTrue interp fmla) == true \u2192 Theory axioms \n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1cb2e25384debdb1df6aa1785825938d4ab16548","subject":"IDesc model: modelled up to elimination, with no crazy flag but a mutual definition.","message":"IDesc model: modelled up to elimination, with no crazy flag but a mutual definition.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bf4c2374eee5aebf6d658c3c557a4233b3ba3172","subject":"flipbased: add choose","message":"flipbased: add choose\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP\nopen import Data.Bool\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\nchoose : \u2200 {n a} {A : Set a} \u2192 M n A \u2192 M n A \u2192 M (suc n) A\nchoose x y = \u27ea if_then_else_ \u00b7 toss \u00b7 x \u00b7 y \u27eb\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 M k (Bits n \u2192 A) \u2192 M (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 M (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 M k (Bits n \u2192 A) \u2192 M (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 M (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b2e5536ab47da9ade851c78e2d0e284fdaed9d6e","subject":"Break some lines.","message":"Break some lines.\n\nOld-commit-hash: f541af3f72f8831d3daba892d4230bda0bdb2cdf\n","repos":"inc-lc\/ilc-agda","old_file":"SymbolicDerivation.agda","new_file":"SymbolicDerivation.agda","new_contents":"module SymbolicDerivation where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Natural.Evaluation\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n     (diff-term\n       (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e))\n       (lift-term {{\u0393\u2032}} t))\n     (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n       (diff-term\n         (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t))\n         (lift-term {{\u0393\u2032}} e))\n       (if (lift-term {{\u0393\u2032}} c)\n         (derive-term {{\u0393\u2032}} t)\n         (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n","old_contents":"module SymbolicDerivation where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Natural.Evaluation\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"03ab8ab5205b986b2c5bf398644af64260b5afd1","subject":"Desc model: rename iso1 to phi.","message":"Desc model: rename iso1 to phi.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> phi (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (phi D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (phi D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : {I : Set} -> IDescl I -> IDesc I\niso1 {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"022f5cdcffc1bcc8957151837d6c1fbc3226b8d7","subject":"pisearch: more renamings","message":"pisearch: more renamings\n","repos":"crypto-agda\/crypto-agda","old_file":"pisearch.agda","new_file":"pisearch.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nTree : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nTree sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605) \u2192 (B : A \u2192 \u2605) \u2192 \u2605\n\u03a0 A B = (x : A) \u2192 B x\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Tree sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Tree sA B\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ toFunA toFunB t = uncurry (toFunB \u2218 toFunA t)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ fromFunA fromFunB f = fromFunA (fromFunB \u2218 curry f)\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB toFunA toFunB = toFun-\u03a3 sA sB toFunA toFunB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB fromFunA fromFunB = fromFun-\u03a3 sA sB fromFunA fromFunB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2081 x) = toFunA t x\ntoFun-\u228e sA sB toFunA toFunB (t , u) (inj\u2082 x) = toFunB u x\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB fromFunA fromFunB f = fromFunA (f \u2218 inj\u2081) , fromFunB (f \u2218 inj\u2082)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Tree s _) _,_\n","old_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\nTree : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\nTree sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605) \u2192 (B : A \u2192 \u2605) \u2192 \u2605\n\u03a0 A B = (x : A) \u2192 B x\n\nToFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nToFun {A} sA = \u2200 {B} \u2192 Tree sA B \u2192 \u03a0 A B\n\nFromFun : \u2200 {A} (sA : Search A) \u2192 \u2605\nFromFun {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 Tree sA B\n\ntoFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n          \u2192 ToFun sA\n          \u2192 (\u2200 {x} \u2192 ToFun (sB {x}))\n          \u2192 ToFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\ntoFun-\u03a3 _ _ PA PB t = uncurry (PB \u2218 PA t)\n\nfromFun-\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n            \u2192 FromFun sA\n            \u2192 (\u2200 {x} \u2192 FromFun (sB {x}))\n            \u2192 FromFun (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nfromFun-\u03a3 _ _ PA PB f = PA (PB \u2218 curry f)\n\ntoFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA \u00d7Search sB)\ntoFun-\u00d7 sA sB PA PB = toFun-\u03a3 sA sB PA PB\n\nfromFun-\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA \u00d7Search sB)\nfromFun-\u00d7 sA sB PA PB = fromFun-\u03a3 sA sB PA PB\n\ntoFun-Bit : ToFun (search \u03bcBit)\ntoFun-Bit (f , t) false = f\ntoFun-Bit (f , t) true  = t\n\nfromFun-Bit : FromFun (search \u03bcBit)\nfromFun-Bit f = f false , f true\n\ntoFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 ToFun sA \u2192 ToFun sB \u2192 ToFun (sA +Search sB)\ntoFun-\u228e sA sB PA PB (t , u) (inj\u2081 x) = PA t x\ntoFun-\u228e sA sB PA PB (t , u) (inj\u2082 x) = PB u x\n\nfromFun-\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 FromFun sA \u2192 FromFun sB \u2192 FromFun (sA +Search sB)\nfromFun-\u228e sA sB PA PB f = PA (f \u2218 inj\u2081) , PB (f \u2218 inj\u2082)\n\n-- toFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 ToFun sA\n-- toFun-searchInd {A} {sA} indA {B} t = ?\n\nfromFun-searchInd : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 FromFun sA\nfromFun-searchInd indA = indA (\u03bb s \u2192 Tree s _) _,_\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"88b905fb49012d44e79b5a8afa2cb027f9b9ab6f","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/co-data\/StreamSL.agda","new_file":"notes\/co-data\/StreamSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the co-induction principle for streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule StreamSL where\n\nopen import Data.Nat\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_\n\ndata Stream\u2115 : Set where\n  _\u2237_ : \u2115 \u2192 \u221e Stream\u2115 \u2192 Stream\u2115\n\n-- From\n-- https:\/\/groups.google.com\/forum\/#!topic\/homotopytypetheory\/Ev-9i2Va4_Q.\n--\n-- TODO (26 December 2013): We aren't using the function A \u2192 A.\nStream\u2115-coind : {A : Set} \u2192 (A \u2192 \u2115) \u2192 (A \u2192 A) \u2192 A \u2192 Stream\u2115\nStream\u2115-coind f g a = f a \u2237 \u266f (Stream\u2115-coind f g a)\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the co-induction principle for streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule StreamSL where\n\nopen import Data.Nat\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_\n\ndata Stream\u2115 : Set where\n  _\u2237_ : (x : \u2115) (xs : \u221e (Stream\u2115)) \u2192 Stream\u2115\n\n-- 26 December 2013. From\n-- https:\/\/groups.google.com\/forum\/#!topic\/homotopytypetheory\/Ev-9i2Va4_Q.\nStream-coind : {A : Set} \u2192 (A \u2192 \u2115) \u2192 (A \u2192 A) \u2192 A \u2192 Stream\u2115\nStream-coind f g a = f a \u2237 \u266f (Stream-coind f g a)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a65a43ed7f888f4a9c891d849553e523045eef34","subject":"defined \\bot and \\neg","message":"defined \\bot and \\neg\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/IPL.agda","new_file":"learyouanagda\/IPL.agda","new_contents":"module IPL where\n\n\ndata _\u2227_ (P : Set) (Q : Set) : Set where\n  \u2227-intro : P \u2192 Q \u2192 (P \u2227 Q)\n\n\nproof\u2081 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 P\nproof\u2081 (\u2227-intro p q) = p\n\nproof\u2082 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 Q\nproof\u2082 (\u2227-intro p q) = q\n\n\n_\u21d4_ : (P : Set) \u2192 (Q : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\n\u2227-comm\u2032 : (P Q : Set) \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm\u2032 _ _ (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm : (P Q : Set) \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm P Q = \u2227-intro (\u2227-comm\u2032 P Q) (\u2227-comm\u2032 Q P)\n\n\n\u2227-comm1\u2032 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm1\u2032 (\u2227-intro p q) = \u2227-intro q p\n\n\n\ndata _\u2228_ (P Q : Set) : Set where\n   \u2228-intro\u2081 : P \u2192 P \u2228 Q\n   \u2228-intro\u2082 : Q \u2192 P \u2228 Q\n\n\n\u2228-elim : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 (A \u2228 B) \u2192 C\n\u2228-elim ac bc (\u2228-intro\u2081 a) = ac a\n\u2228-elim ac bc (\u2228-intro\u2082 b) = bc b\n\n\n\u2228-comm\u2032 : {A B : Set} \u2192 (A \u2228 B) \u2192 (B \u2228 A)\n\u2228-comm\u2032 (\u2228-intro\u2081 a) = \u2228-intro\u2082 a\n\u2228-comm\u2032 (\u2228-intro\u2082 b) = \u2228-intro\u2081 b\n\n\u2228-comm : {A B : Set} \u2192 (A \u2228 B) \u21d4 (B \u2228 A)\n\u2228-comm = \u2227-intro \u2228-comm\u2032 \u2228-comm\u2032\n\n\ndata \u22a5 : Set where\n\n\u00ac : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n","old_contents":"module IPL where\n\n\ndata _\u2227_ (P : Set) (Q : Set) : Set where\n  \u2227-intro : P \u2192 Q \u2192 (P \u2227 Q)\n\n\nproof\u2081 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 P\nproof\u2081 (\u2227-intro p q) = p\n\nproof\u2082 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 Q\nproof\u2082 (\u2227-intro p q) = q\n\n\n_\u21d4_ : (P : Set) \u2192 (Q : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\n\u2227-comm\u2032 : (P Q : Set) \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm\u2032 _ _ (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm : (P Q : Set) \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm P Q = \u2227-intro (\u2227-comm\u2032 P Q) (\u2227-comm\u2032 Q P)\n\n\n\u2227-comm1\u2032 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm1\u2032 (\u2227-intro p q) = \u2227-intro q p\n\n\n\ndata _\u2228_ (P Q : Set) : Set where\n   \u2228-intro\u2081 : P \u2192 P \u2228 Q\n   \u2228-intro\u2082 : Q \u2192 P \u2228 Q\n\n\n\u2228-elim : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 (A \u2228 B) \u2192 C\n\u2228-elim ac bc (\u2228-intro\u2081 a) = ac a\n\u2228-elim ac bc (\u2228-intro\u2082 b) = bc b\n\n\n\u2228-comm\u2032 : {A B : Set} \u2192 (A \u2228 B) \u2192 (B \u2228 A)\n\u2228-comm\u2032 (\u2228-intro\u2081 a) = \u2228-intro\u2082 a\n\u2228-comm\u2032 (\u2228-intro\u2082 b) = \u2228-intro\u2081 b\n\n\u2228-comm : {A B : Set} \u2192 (A \u2228 B) \u21d4 (B \u2228 A)\n\u2228-comm = \u2227-intro \u2228-comm\u2032 \u2228-comm\u2032\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0f910b0d2fa3a2f33819d401c68b1d0874102f04","subject":"Control\/Protocol\/CoreOld.agda","message":"Control\/Protocol\/CoreOld.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/CoreOld.agda","new_file":"Control\/Protocol\/CoreOld.agda","new_contents":"{-# OPTIONS --without-K --copatterns #-}\nopen import Function\nopen import Type\nopen import Level\nopen import Data.Zero\nopen import Data.Product\n\nmodule Control.Protocol.CoreOld where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0 : \u2200 {\u2113 \u2113'}(A : \u2605_ \u2113)(B : A \u2192 \u2605_ \u2113') \u2192 \u2605_ (\u2113 \u2294 \u2113')\n\u03a0 A B = (x : A) \u2192 B x\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (A : \u2605)(B : A \u2192 Proto) \u2192 Proto\n  Client' Server' : (Q : \u2605)(R : Q \u2192 \u2605)(B : Proto) \u2192 Proto\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' A B) = \u03a3' A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' A B) = \u03a0' A (\u03bb x \u2192 dual (B x))\ndual (Client' Q R B) = Server' Q R (dual B)\ndual (Server' Q R B) = Client' Q R (dual B)\n\nmodule _ (Q : \u2605)(R : Q \u2192 \u2605)(A : \u2605) where\n  record Server : \u2605\n  record ServerResponse (q : Q) : \u2605\n\n  record Server where\n    coinductive\n    constructor _,_\n    field\n      srv-done : A\n      srv-ask  : \u2200 q \u2192 ServerResponse q\n\n  record ServerResponse q where\n    inductive\n    constructor _,_\n    field\n      srv-resp : R q\n      srv-cont : Server\nopen Server public\nopen ServerResponse public\n\nServer\u2032 : (Q R A : \u2605) \u2192 \u2605\nServer\u2032 Q R = Server Q (const R)\n\nmodule _ {Q A B : \u2605}{R : Q \u2192 \u2605} where\n  Server-Com : Server Q R A \u2192 Client Q R B \u2192 A \u00d7 B\n  Server-Com server (ask q? cont) = let sr = srv-ask server q? in Server-Com (srv-cont sr) (cont (srv-resp sr))\n  Server-Com server (done x) = srv-done server , x\n--\n\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\n  El (Client' Q R B) = Client Q R (El B)\n  El (Server' Q R B) = Server Q R (El B)\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El A P \u2192 El B (dual P) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (Client' Q R P) c s = let (x , y) = Server-Com s c in com P y x\n  com (Server' Q R P) s c = let (x , y) = Server-Com s c in com P x y\n\nun-client0 : \u2200 {R P} \u2192 Client \ud835\udfd8 R P \u2192 P\nun-client0 (done x) = x\nun-client0 (ask () _)\n\nserver0 : \u2200 {R P} \u2192 P \u2192 Server \ud835\udfd8 R P\nsrv-done (server0 p) = p\nsrv-ask  (server0 _) ()\n","old_contents":"{-# OPTIONS --without-K --copatterns #-}\nopen import Function\nopen import Type\nopen import Level\nopen import Data.Zero\nopen import Data.Product\n\nmodule Control.Protocol.CoreOld where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0 : \u2200 {\u2113 \u2113'}(A : \u2605_ \u2113)(B : A \u2192 \u2605_ \u2113') \u2192 \u2605_ (\u2113 \u2294 \u2113')\n\u03a0 A B = (x : A) \u2192 B x\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (A : \u2605)(B : A \u2192 Proto) \u2192 Proto\n  Client' Server' : (Q : \u2605)(R : Q \u2192 \u2605)(B : Proto) \u2192 Proto\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' A B) = \u03a3' A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' A B) = \u03a0' A (\u03bb x \u2192 dual (B x))\ndual (Client' Q R B) = Server' Q R (dual B)\ndual (Server' Q R B) = Client' Q R (dual B)\n\nmodule _ (Q : \u2605)(R : Q \u2192 \u2605)(A : \u2605) where\n  record Server : \u2605\n  record ServerResponse (q : Q) : \u2605\n\n  record Server where\n    coinductive\n    constructor _,_\n    field\n      srv-done : A\n      srv-ask  : \u2200 q \u2192 ServerResponse q\n\n  record ServerResponse q where\n    constructor _,_\n    field\n      srv-resp : R q\n      srv-cont : Server\nopen Server public\nopen ServerResponse public\n\nServer\u2032 : (Q R A : \u2605) \u2192 \u2605\nServer\u2032 Q R = Server Q (const R)\n\nmodule _ {Q A B : \u2605}{R : Q \u2192 \u2605} where\n  Server-Com : Server Q R A \u2192 Client Q R B \u2192 A \u00d7 B\n  Server-Com server (ask q? cont) = let sr = srv-ask server q? in Server-Com (srv-cont sr) (cont (srv-resp sr))\n  Server-Com server (done x) = srv-done server , x\n--\n\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\n  El (Client' Q R B) = Client Q R (El B)\n  El (Server' Q R B) = Server Q R (El B)\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El A P \u2192 El B (dual P) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (Client' Q R P) c s = let (x , y) = Server-Com s c in com P y x\n  com (Server' Q R P) s c = let (x , y) = Server-Com s c in com P x y\n\nun-client0 : \u2200 {R P} \u2192 Client \ud835\udfd8 R P \u2192 P\nun-client0 (done x) = x\nun-client0 (ask () _)\n\nserver0 : \u2200 {R P} \u2192 P \u2192 Server \ud835\udfd8 R P\nsrv-done (server0 p) = p\nsrv-ask  (server0 _) ()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3bf966d43e2d48c13fb8c15cf2c93967fb4387a8","subject":"Bits: from\u2115\/to\u2115","message":"Bits: from\u2115\/to\u2115\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {zero}  (_     `\u2237 _) _ = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | [ p ] = 2\u207f\u2270to\u2115 xs (\u2115<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |   _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x \u2115< 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | [ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (\u2115<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | [ p ] = to\u2115\u2218from\u2115 {n} x (\u2115<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x \u2115< 2^ n \u2192 y \u2115< 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : Bij\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : Bit \u2192 Bij\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : Bij \u2192 Bij \u2192 Bij\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : Bij \u2192 Bij\n    `if0 f = `if `id f\n\n    `if1 : Bij \u2192 Bij\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : Bij \u2192 Bij\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Bij\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Bij \u2192 Bij\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : Bij \u2192 Bij\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : Bij \u2192 Bij \u2192 Bij\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : Bit \u2192 Bij\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {zero}  (_     `\u2237 _) _ = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n <= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e6b5c67ec8697dca756840dfbf78cc0a359b87ff","subject":"Updated code to reflect changes to Agda.","message":"Updated code to reflect changes to Agda.\n\nIgnore-this: f9422e2d9df16073fcacf020811122d\n\ndarcs-hash:20110920143420-3bd4e-f209f1ab4d27e4a95e45d282c4e65d057fac75d9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC-PCF\/Data\/Nat\/Inequalities.agda","new_file":"src\/LTC-PCF\/Data\/Nat\/Inequalities.agda","new_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC-PCF.Data.Nat.Inequalities where\n\nopen import LTC-PCF.Base\n\n------------------------------------------------------------------------------\n-- Version using lambda-abstraction.\n-- lth : D \u2192 D\n-- lth lt = lam (\u03bb m \u2192 lam (\u03bb n \u2192\n--               if (isZero n) then false\n--                  else (if (isZero m) then true\n--                           else (lt \u00b7 (pred m) \u00b7 (pred n)))))\n\n-- Version using lambda lifting via super-combinators.\n-- (Hughes. Super-combinators. 1982)\n\nabstract\n  <-helper\u2081 : D \u2192 D \u2192 D \u2192 D\n  <-helper\u2081 d lt e = if (isZero e)\n                        then false\n                        else (if (isZero d)\n                                 then true\n                                 else (lt \u00b7 (pred d) \u00b7 (pred e)))\n  {-# ATP definition <-helper\u2081 #-}\n\n  <-helper\u2082 : D \u2192 D \u2192 D\n  <-helper\u2082 lt d = lam (<-helper\u2081 d lt)\n  {-# ATP definition <-helper\u2082 #-}\n\n  <-h : D \u2192 D\n  <-h lt = lam (<-helper\u2082 lt)\n  {-# ATP definition <-h #-}\n\n_<_ : D \u2192 D \u2192 D\nd < e = fix <-h \u00b7 d \u00b7 e\n{-# ATP definition _<_ #-}\n\n_\u2264_ : D \u2192 D \u2192 D\nd \u2264 e = d < succ e\n{-# ATP definition _\u2264_ #-}\n\n_>_ : D \u2192 D \u2192 D\nd > e = e < d\n{-# ATP definition _>_ #-}\n\n_\u2265_ : D \u2192 D \u2192 D\nd \u2265 e = e \u2264 d\n{-# ATP definition _\u2265_ #-}\n\n------------------------------------------------------------------------\n-- The data types\n\nGT : D \u2192 D \u2192 Set\nGT d e = d > e \u2261 true\n{-# ATP definition GT #-}\n\nNGT : D \u2192 D \u2192 Set\nNGT d e = d > e \u2261 false\n{-# ATP definition NGT #-}\n\nLT : D \u2192 D \u2192 Set\nLT d e = d < e \u2261 true\n{-# ATP definition LT #-}\n\nNLT : D \u2192 D \u2192 Set\nNLT d e = d < e \u2261 false\n{-# ATP definition NLT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = d \u2264 e \u2261 true\n{-# ATP definition LE #-}\n\nNLE : D \u2192 D \u2192 Set\nNLE d e = d \u2264 e \u2261 false\n{-# ATP definition NLE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = d \u2265 e \u2261 true\n{-# ATP definition GE #-}\n\nNGE : D \u2192 D \u2192 Set\nNGE d e = d \u2265 e \u2261 false\n{-# ATP definition NGE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order.\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC-PCF.Data.Nat.Inequalities where\n\nopen import LTC-PCF.Base\n\n------------------------------------------------------------------------------\n-- Version using lambda-abstraction.\n-- lth : D \u2192 D\n-- lth lt = lam (\u03bb m \u2192 lam (\u03bb n \u2192\n--               if (isZero n) then false\n--                  else (if (isZero m) then true\n--                           else (lt \u00b7 (pred m) \u00b7 (pred n)))))\n\n-- Version using lambda lifting via super-combinators.\n-- (Hughes. Super-combinators. 1982)\n\n<-helper\u2081 : D \u2192 D \u2192 D \u2192 D\n<-helper\u2081 d lt e = if (isZero e)\n                      then false\n                      else (if (isZero d)\n                               then true\n                               else (lt \u00b7 (pred d) \u00b7 (pred e)))\n{-# ATP definition <-helper\u2081 #-}\n\n<-helper\u2082 : D \u2192 D \u2192 D\n<-helper\u2082 lt d = lam (<-helper\u2081 d lt)\n{-# ATP definition <-helper\u2082 #-}\n\n<-h : D \u2192 D\n<-h lt = lam (<-helper\u2082 lt)\n{-# ATP definition <-h #-}\n\n_<_ : D \u2192 D \u2192 D\nd < e = fix <-h \u00b7 d \u00b7 e\n{-# ATP definition _<_ #-}\n\n_\u2264_ : D \u2192 D \u2192 D\nd \u2264 e = d < succ e\n{-# ATP definition _\u2264_ #-}\n\n_>_ : D \u2192 D \u2192 D\nd > e = e < d\n{-# ATP definition _>_ #-}\n\n_\u2265_ : D \u2192 D \u2192 D\nd \u2265 e = e \u2264 d\n{-# ATP definition _\u2265_ #-}\n\n------------------------------------------------------------------------\n-- The data types\n\nGT : D \u2192 D \u2192 Set\nGT d e = d > e \u2261 true\n{-# ATP definition GT #-}\n\nNGT : D \u2192 D \u2192 Set\nNGT d e = d > e \u2261 false\n{-# ATP definition NGT #-}\n\nLT : D \u2192 D \u2192 Set\nLT d e = d < e \u2261 true\n{-# ATP definition LT #-}\n\nNLT : D \u2192 D \u2192 Set\nNLT d e = d < e \u2261 false\n{-# ATP definition NLT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = d \u2264 e \u2261 true\n{-# ATP definition LE #-}\n\nNLE : D \u2192 D \u2192 Set\nNLE d e = d \u2264 e \u2261 false\n{-# ATP definition NLE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = d \u2265 e \u2261 true\n{-# ATP definition GE #-}\n\nNGE : D \u2192 D \u2192 Set\nNGE d e = d \u2265 e \u2261 false\n{-# ATP definition NGE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order.\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1fcd29534d391163990d255bd79591a69fe66032","subject":"Alternate: update","message":"Alternate: update\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Alternate.agda","new_file":"Control\/Protocol\/Alternate.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!; tr)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\nopen import Type\n\nopen import Control.Protocol.InOut\nopen import Control.Protocol.End\nimport Control.Protocol.Core as C\nopen C using (end;com)\n\nmodule Control.Protocol.Alternate where\n\ndata Proto : InOut \u2192 \u2605\u2081 where\n  end : \u2200 {io} \u2192 Proto io\n  com : (io : InOut){M : \u2605\u2080}(P : M \u2192 Proto (dual\u1d35\u1d3c io)) \u2192 Proto io\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 io {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 P\u2081}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 Proto.com io P\u2080 \u2261 com io P\u2081\n    com= io refl P= = ap (com _) (\u03bb= P=)\n\n    module _ io\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 P\u2081}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io P\u2080 \u2261 com io P\u2081\n        com\u2243 = com= io (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    send= = com= Out\n    send\u2243 = com\u2243 Out\n    recv= = com= In\n    recv\u2243 = com\u2243 In\n \n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= io refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\nP\u27e6_\u27e7 : \u2200 {io} \u2192 Proto io \u2192 C.Proto\nP\u27e6 end      \u27e7 = end\nP\u27e6 com io P \u27e7 = com io \u03bb m \u2192 P\u27e6 P m \u27e7\n\n\u27e6_\u27e7 : \u2200 {io} \u2192 Proto io \u2192 \u2605\n\u27e6 P \u27e7 = C.\u27e6 P\u27e6 P \u27e7 \u27e7\n\ndual : \u2200 {io} \u2192 Proto io \u2192 Proto (dual\u1d35\u1d3c io)\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nmodule _ {io}(P : Proto io) where\n    \u27e6_\u22a5\u27e7 : \u2605\n    \u27e6_\u22a5\u27e7 = \u27e6 dual P \u27e7\n\n    Log : \u2605\n    Log = C.\u27e6 C.source-of P\u27e6 P \u27e7 \u27e7\n\nmodule _ {{_ : FunExt}} where\n    P\u27e6_\u22a5\u27e7 : \u2200 {io}(P : Proto io) \u2192 C.dual P\u27e6 P \u27e7 \u2261 P\u27e6 dual P \u27e7\n    P\u27e6 end      \u22a5\u27e7 = refl\n    P\u27e6 com io P \u22a5\u27e7 = C.com= refl refl \u03bb m \u2192 P\u27e6 P m \u22a5\u27e7\n\n    telecom : \u2200 {io} (P : Proto io) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\n    telecom P t u = C.telecom P\u27e6 P \u27e7 t (tr C.\u27e6_\u27e7 (! P\u27e6 P \u22a5\u27e7) u)\n\nsend\u2032 : \u2605 \u2192 Proto In \u2192 Proto Out\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto Out \u2192 Proto In\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 {io} (P : Proto io) \u2192 tr Proto (dual\u1d35\u1d3c-involutive io) (dual (dual P)) \u2261 P\n    dual-involutive {In}  end      = refl\n    dual-involutive {Out} end      = refl\n    dual-involutive       (send P) = com=\u2032 _ \u03bb m \u2192 dual-involutive (P m)\n    dual-involutive       (recv P) = com=\u2032 _ \u03bb m \u2192 dual-involutive (P m)\n\n    {-\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_) renaming (proj\u2081 to fst)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!; subst)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\nopen import Type\n\nopen import Control.Protocol.InOut\nopen import Control.Protocol.End\nimport Control.Protocol.Core as C\nopen C using (end;com)\n\nmodule Control.Protocol.Alternate where\n\ndata Proto : InOut \u2192 \u2605\u2081 where\n  end : \u2200 {io} \u2192 Proto io\n  com : (io : InOut){M : \u2605\u2080}(P : M \u2192 Proto (dual\u1d35\u1d3c io)) \u2192 Proto io\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 io {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 P\u2081}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 Proto.com io P\u2080 \u2261 com io P\u2081\n    com= io refl P= = ap (com _) (\u03bb= P=)\n\n    module _ io\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 P\u2081}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io P\u2080 \u2261 com io P\u2081\n        com\u2243 = com= io (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    send= = com= Out\n    send\u2243 = com\u2243 Out\n    recv= = com= In\n    recv\u2243 = com\u2243 In\n \n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= io refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\nP\u27e6_\u27e7 : \u2200 {io} \u2192 Proto io \u2192 C.Proto\nP\u27e6 end      \u27e7 = end\nP\u27e6 com io P \u27e7 = com io \u03bb m \u2192 P\u27e6 P m \u27e7\n\n\u27e6_\u27e7 : \u2200 {io} \u2192 Proto io \u2192 \u2605\n\u27e6 P \u27e7 = C.\u27e6 P\u27e6 P \u27e7 \u27e7\n\ndual : \u2200 {io} \u2192 Proto io \u2192 Proto (dual\u1d35\u1d3c io)\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nmodule _ {io}(P : Proto io) where\n    \u27e6_\u22a5\u27e7 : \u2605\n    \u27e6_\u22a5\u27e7 = \u27e6 dual P \u27e7\n\n    Log : \u2605\n    Log = C.\u27e6 C.source-of P\u27e6 P \u27e7 \u27e7\n\nmodule _ {{_ : FunExt}} where\n    P\u27e6_\u22a5\u27e7 : \u2200 {io}(P : Proto io) \u2192 C.dual P\u27e6 P \u27e7 \u2261 P\u27e6 dual P \u27e7\n    P\u27e6 end      \u22a5\u27e7 = refl\n    P\u27e6 com io P \u22a5\u27e7 = C.com= refl refl \u03bb m \u2192 P\u27e6 P m \u22a5\u27e7\n\n    telecom : \u2200 {io} (P : Proto io) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\n    telecom P t u = C.telecom P\u27e6 P \u27e7 t (subst C.\u27e6_\u27e7 (! P\u27e6 P \u22a5\u27e7) u)\n\nsend\u2032 : \u2605 \u2192 Proto In \u2192 Proto Out\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto Out \u2192 Proto In\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 {io} (P : Proto io) \u2192 subst Proto (dual\u1d35\u1d3c-involutive io) (dual (dual P)) \u2261 P\n    dual-involutive {In}  end      = refl\n    dual-involutive {Out} end      = refl\n    dual-involutive       (send P) = com=\u2032 _ \u03bb m \u2192 dual-involutive (P m)\n    dual-involutive       (recv P) = com=\u2032 _ \u03bb m \u2192 dual-involutive (P m)\n\n    {-\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"789ee03b932a743e1b7f9832f09bd7efc37f4fa8","subject":"Split \u27e6_\u27e7Base from \u27e6_\u27e7Type (WIP).","message":"Split \u27e6_\u27e7Base from \u27e6_\u27e7Type (WIP).\n\nNeeds fixes to type inference failures!\n\nOld-commit-hash: 4f7d5e46e2d2a202dc2c672a08ae4deeebe589c2\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Denotation\/Value.agda","new_file":"Popl14\/Denotation\/Value.agda","new_contents":"module Popl14.Denotation.Value where\n\nopen import Popl14.Syntax.Type public\nopen import Popl14.Change.Type public\nopen import Base.Denotation.Notation public\n\nopen import Structure.Bag.Popl14\nopen import Data.Integer\n\n-- Values of Calculus Popl14\n--\n-- Contents\n-- - Domains associated with types\n-- - `diff` and `apply` in semantic domains\n\n\u27e6_\u27e7Base : Base \u2192 Set\n\u27e6 base-int \u27e7Base = \u2124\n\u27e6 base-bag \u27e7Base = Bag\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 base \u03b9 \u27e7Type = \u27e6 \u03b9 \u27e7Base\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n","old_contents":"module Popl14.Denotation.Value where\n\nopen import Popl14.Syntax.Type public\nopen import Popl14.Change.Type public\nopen import Base.Denotation.Notation public\n\nopen import Structure.Bag.Popl14\nopen import Data.Integer\n\n-- Values of Calculus Popl14\n--\n-- Contents\n-- - Domains associated with types\n-- - `diff` and `apply` in semantic domains\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 int \u27e7Type = \u2124\n\u27e6 bag \u27e7Type = Bag\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ca85296d99ae3900de72c22285a1ae7877faa8d4","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 670014f4732dfe4f76ae9b5a3d483201\n\ndarcs-hash:20100701203238-3bd4e-c887615167c98c78814c88d7a4765633c62f987d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List.agda","new_file":"LTC\/Data\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\n\npostulate\n  length : D \u2192 D\n  length-[] : length []                     \u2261 zero\n  length-\u2237  : ( d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D ) \u2192 map f []           \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\npostulate\n  replicate : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192 replicate zero d       \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","old_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\n\npostulate\n  length : D \u2192 D\n  length-[] : length []                     \u2261 zero\n  length-\u2237  : ( d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D ) \u2192 map f []           \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  replicate : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192 replicate zero d       \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3588eb2e3b8f2ea04acef8f2a94ddb1e0d6c6002","subject":"Added doc: Agda doesn't accept the proof of Conat \u221e.","message":"Added doc: Agda doesn't accept the proof of Conat \u221e.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\n-- TODO (06 January 2014). We couldn't prove Conat-coind.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h {n} An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221e.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\n-- TODO (06 January 2014). We couldn't prove Conat-coind.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h {n} An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\npostulate\n  inf    : D\n  inf-eq : inf \u2261 succ\u2081 inf\n{-# ATP axiom inf-eq #-}\n\n{-# NO_TERMINATION_CHECK #-}\ninf-Conat : Conat inf\ninf-Conat = subst Conat (sym inf-eq) (cosucc (\u266f inf-Conat))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"43bced6213bc8ce0a61bdd5f6e1f440f676da57e","subject":"and hey look at that, this case goes through now","message":"and hey look at that, this case goes through now\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , {!!} -- ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc x\u2081) x\u2082) = {!!}\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EASubsume (\u03bb u \u2192 \u03bb ()) (\u03bb e' u \u2192 \u03bb ()) (ESLam x\u2082 D) x\u2083\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAHole x\u2081) x\u2082) = {!!}\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = d' , \u0394' , ESAsc2 D\n    ... | Inr x = (< _ > d') , \u0394' , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , {!!} -- ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = (d1 \u2218 (< {!!} > d2)) , (\u03941 \u222a \u03942) , ESAp2 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = (d1 \u2218 d2) , (\u03941 \u222a \u03942) , ESAp3 {\u03941 = \u03941} {\u03942 = \u03942} {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc x\u2081) x\u2082) = {!!}\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083) with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EASubsume (\u03bb u \u2192 \u03bb ()) (\u03bb e' u \u2192 \u03bb ()) (ESLam x\u2082 D) x\u2083\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAHole x\u2081) x\u2082) = {!!}\n    expandability-ana {e = e1 \u2218 e\u2081} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' = {!!} , {!!} , {!!} , {!!}\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"22d73ff3da6bb71c6024afba8a436e020a5e5de5","subject":"Function: restore \u2026","message":"Function: restore \u2026\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/NP.agda","new_file":"lib\/Function\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary.PropositionalEquality.NP\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : Set) (n : \u2115) (B : Set) \u2192 Set\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : Set) (n : \u2115) (B : Set\u2081) \u2192 Set\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = refl\n  nest-+ (suc m) n = cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!} \n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very few code rely on it.\n\u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary.PropositionalEquality.NP\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : Set) (n : \u2115) (B : Set) \u2192 Set\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : Set) (n : \u2115) (B : Set\u2081) \u2192 Set\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = refl\n  nest-+ (suc m) n = cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!} \n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very few code rely on it.\n-- \u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n-- \u2026 \u2983 x \u2984 = x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3ef3b1f40e4b0b9e7308a67388ab552654f5feeb","subject":"Bits: use patterns 0\u2237 and 1\u2237","message":"Bits: use patterns 0\u2237 and 1\u2237\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Bits where\n\nopen import Algebra\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Fin.NP using (Fin; zero; suc; inject\u2081; inject+; raise; Fin\u25b9\u2115)\nopen import Data.Vec.NP\nopen import Function.NP\nimport Data.List.NP as L\n\n-- Re-export some vector functions, maybe they should be given\n-- less generic types.\nopen Data.Vec.NP public using ([]; _\u2237_; _++_; head; tail; map; replicate; RewireTbl; rewire; rewireTbl; on\u1d62)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Notice that all empty vectors are the same, hence 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\npattern 0\u2237_ xs = 0\u2082 \u2237 xs\npattern 1\u2237_ xs = 1\u2082 \u2237 xs\n{-\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n-}\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n-- see Data.Bits.Properties\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n\u2295-group : \u2115 \u2192 Group \u2080 \u2080\n\u2295-group = LiftGroup.group Xor\u00b0.+-group\n\nmodule \u2295-Group  (n : \u2115) = Group (\u2295-group n)\nmodule \u2295-Monoid (n : \u2115) = Monoid (\u2295-Group.monoid n)\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 k {n} \u2192 Bits (n + k) \u2192 Bits k\nlsb _ {n} = drop n\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 map not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2237 1\u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = map 0\u2237_ bs ++ map 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\n\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_\u2228\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2228\u00b0_ f g x = f x \u2228 g x\n\n_\u2227\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2227\u00b0_ f g x = f x \u2227 g x\n\nnot\u00b0 : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\nnot\u00b0 f = not \u2218 f\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []      = 0\u2237 []\nsucBCarry (0\u2237 xs) = 0\u2237 sucBCarry xs\nsucBCarry (1\u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2237 xs) = 1\u2237 xs\n  sucRB (1\u2237 xs) = 0\u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []      = zero\ntoFin         (0\u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []      = zero\nBits\u25b9\u2115         (0\u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []       = head\nlookupTbl         (0\u2237 key) = lookupTbl key \u2218 take _\nlookupTbl {suc n} (1\u2237 key) = lookupTbl key \u2218 drop (2^ n)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_)\n                    ++ tblFromFun {n} (f \u2218 1\u2237_)\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Bits where\n\nopen import Algebra\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Fin.NP using (Fin; zero; suc; inject\u2081; inject+; raise; Fin\u25b9\u2115)\nopen import Data.Vec.NP\nopen import Function.NP\nimport Data.List.NP as L\n\n-- Re-export some vector functions, maybe they should be given\n-- less generic types.\nopen Data.Vec.NP public using ([]; _\u2237_; _++_; head; tail; map; replicate; RewireTbl; rewire; rewireTbl; on\u1d62)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Notice that all empty vectors are the same, hence 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n-- see Data.Bits.Properties\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n\u2295-group : \u2115 \u2192 Group \u2080 \u2080\n\u2295-group = LiftGroup.group Xor\u00b0.+-group\n\nmodule \u2295-Group  (n : \u2115) = Group (\u2295-group n)\nmodule \u2295-Monoid (n : \u2115) = Monoid (\u2295-Group.monoid n)\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 k {n} \u2192 Bits (n + k) \u2192 Bits k\nlsb _ {n} = drop n\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2082 \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2082 \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 map not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2082 \u2237 1\u2082 \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = map 0\u2237_ bs ++ map 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\n\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_\u2228\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2228\u00b0_ f g x = f x \u2228 g x\n\n_\u2227\u00b0_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_\u2227\u00b0_ f g x = f x \u2227 g x\n\nnot\u00b0 : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\nnot\u00b0 f = not \u2218 f\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0\u2082 \u2237 []\nsucBCarry (0\u2082 \u2237 xs) = 0\u2082 \u2237 sucBCarry xs\nsucBCarry (1\u2082 \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2082 \u2237 xs) = 1\u2082 \u2237 xs\n  sucRB (1\u2082 \u2237 xs) = 0\u2082 \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0\u2082 \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2082 \u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []        = zero\nBits\u25b9\u2115         (0\u2082 \u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0\u2082 \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1\u2082 \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fb9b43055eca1cb6ade1a12b058040301d4fbfa7","subject":"switching args to implicit. another typo in theorem statement from copy and paste.","message":"switching args to implicit. another typo in theorem statement from copy and\npaste.\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"correspondence.agda","new_file":"correspondence.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule correspondence where\n  mutual\n    correspondence-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0393 \u22a2 e => \u03c4\n    correspondence-synth ESConst = SConst\n    correspondence-synth (ESVar x\u2081) = SVar x\u2081\n    correspondence-synth (ESLam ex) with correspondence-synth ex\n    ... | ih = SLam {!!} ih\n    correspondence-synth (ESAp1 x x\u2081 x\u2082 x\u2083) = {!!}\n    correspondence-synth (ESAp2 x ex x\u2081 x\u2082) = {!!}\n    correspondence-synth (ESAp3 x ex x\u2081) = {!!}\n    correspondence-synth ESEHole = SEHole\n    correspondence-synth (ESNEHole ex) = SNEHole (correspondence-synth ex)\n    correspondence-synth (ESAsc1 x x\u2081) = {!!}\n    correspondence-synth (ESAsc2 x) = {!!}\n\n    correspondence-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx}  \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          \u0393 \u22a2 e <= \u03c4\n    correspondence-ana (EALam ex) with correspondence-ana ex\n    ... | ih = ALam {!!} {!!} {!!}\n    correspondence-ana (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    correspondence-ana EAEHole = {!!}\n    correspondence-ana (EANEHole x x\u2081) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule correspondence where\n  mutual\n    correspondence-synth : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0393 \u22a2 e => \u03c4\n    correspondence-synth = {!!}\n\n    correspondence-ana : (\u0393 : tctx) (e : hexp) (\u03c4 \u03c4' : htyp) (d : dhexp) (\u0394 : hctx)  \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          \u0393 \u22a2 e => \u03c4\n    correspondence-ana = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"02fd5fbf60ed9294149438378c279b3e7ac6301c","subject":"Minor changes.","message":"Minor changes.\n\nIgnore-this: 529ad8704ff72d3193f5542bb233c112\n\ndarcs-hash:20110206134114-3bd4e-f473ba69ce27325b4fc34ad80c7bf75310040ede.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/SortList\/SortList.agda","new_file":"src\/LTC\/Program\/SortList\/SortList.agda","new_contents":"------------------------------------------------------------------------------\n-- Sort a list\n------------------------------------------------------------------------------\n\n-- This module define the program which sorts a list by converting it\n-- into an ordered tree and then back to a list (Burstall, 1969, pp. 45-46).\n\n-- R. M. Burstall. Proving properties of programs by structural\n-- induction. The Computer Journal, 12(1):41\u201348, 1969.\n\nmodule LTC.Program.SortList.SortList where\n\nopen import LTC.Base\n\nopen import LTC.Data.Bool using ( _&&_ )\nopen import LTC.Data.Nat.Inequalities using ( _\u2264_ )\nopen import LTC.Data.Nat.Type\n  using ( N  -- The LTC natural numbers type.\n        )\nopen import LTC.Data.List using ( _++_ )\n\n------------------------------------------------------------------------------\n-- Tree terms.\npostulate\n  nilTree  : D\n  tip      : D \u2192 D\n  node     : D \u2192 D \u2192 D \u2192 D\n\n-- The tree type.\ndata Tree : D \u2192 Set where\n  nilT  : Tree nilTree\n  tipT  : \u2200 {i} \u2192 N i \u2192 Tree (tip i)\n  nodeT : \u2200 {t\u2081 i t\u2082} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192 Tree (node t\u2081 i t\u2082)\n{-# ATP hint nilT #-}\n{-# ATP hint tipT #-}\n{-# ATP hint nodeT #-}\n\n------------------------------------------------------------------------------\n-- Inequalites on lists and trees\n\n-- Note from Burstall (p. 46): \"The relation \u2264 between lists is only an\n-- ordering if nil is excluded, similarly for trees. This is untidy but\n-- will not cause trouble.\"\n\npostulate\n  \u2264-ItemList    : D \u2192 D \u2192 D\n  \u2264-ItemList-[] : \u2200 item \u2192 \u2264-ItemList item [] \u2261 true\n  \u2264-ItemList-\u2237  : \u2200 item i is \u2192\n                  \u2264-ItemList item (i \u2237 is) \u2261 item \u2264 i && \u2264-ItemList item is\n{-# ATP axiom \u2264-ItemList-[] #-}\n{-# ATP axiom \u2264-ItemList-\u2237 #-}\n\nLE-ItemList : D \u2192 D \u2192 Set\nLE-ItemList item is = \u2264-ItemList item is \u2261 true\n{-# ATP definition LE-ItemList #-}\n\npostulate\n  \u2264-Lists    : D \u2192 D \u2192 D\n  \u2264-Lists-[] : \u2200 is \u2192 \u2264-Lists [] is \u2261 true\n  \u2264-Lists-\u2237  : \u2200 i is js \u2192\n               \u2264-Lists (i \u2237 is) js \u2261 \u2264-ItemList i js && \u2264-Lists is js\n{-# ATP axiom \u2264-Lists-[] #-}\n{-# ATP axiom \u2264-Lists-\u2237 #-}\n\nLE-Lists : D \u2192 D \u2192 Set\nLE-Lists is js = \u2264-Lists is js \u2261 true\n{-# ATP definition LE-Lists #-}\n\npostulate\n  \u2264-ItemTree          : D \u2192 D \u2192 D\n  \u2264-ItemTree-nilTree  : \u2200 item \u2192   \u2264-ItemTree item nilTree \u2261 true\n  \u2264-ItemTree-tip      : \u2200 item i \u2192 \u2264-ItemTree item (tip i) \u2261 item \u2264 i\n  \u2264-ItemTree-node     : \u2200 item t\u2081 i t\u2082 \u2192\n                          \u2264-ItemTree item (node t\u2081 i t\u2082) \u2261\n                          \u2264-ItemTree item t\u2081 && \u2264-ItemTree item t\u2082\n{-# ATP axiom \u2264-ItemTree-nilTree #-}\n{-# ATP axiom \u2264-ItemTree-tip #-}\n{-# ATP axiom \u2264-ItemTree-node #-}\n\nLE-ItemTree : D \u2192 D \u2192 Set\nLE-ItemTree item t = \u2264-ItemTree item t \u2261 true\n{-# ATP definition LE-ItemTree #-}\n\n-- This function is not defined in the paper.\npostulate\n  \u2264-TreeItem         : D \u2192 D \u2192 D\n  \u2264-TreeItem-nilTree : \u2200 item \u2192   \u2264-TreeItem nilTree item \u2261 true\n  \u2264-TreeItem-tip     : \u2200 i item \u2192 \u2264-TreeItem (tip i) item \u2261 i \u2264 item\n  \u2264-TreeItem-node    : \u2200 t\u2081 i t\u2082 item \u2192\n                         \u2264-TreeItem (node t\u2081 i t\u2082) item \u2261\n                         \u2264-TreeItem t\u2081 item && \u2264-TreeItem t\u2082 item\n{-# ATP axiom \u2264-TreeItem-nilTree #-}\n{-# ATP axiom \u2264-TreeItem-tip #-}\n{-# ATP axiom \u2264-TreeItem-node #-}\n\nLE-TreeItem : D \u2192 D \u2192 Set\nLE-TreeItem t item = \u2264-TreeItem t item \u2261 true\n{-# ATP definition LE-TreeItem #-}\n\n------------------------------------------------------------------------------\n-- Auxiliary functions\n\npostulate\n  -- The foldr function with the last two args flipped.\n  lit    : D \u2192 D \u2192 D \u2192 D\n  lit-[] : \u2200 (f : D) n \u2192      lit f []       n \u2261 n\n  lit-\u2237  : \u2200 (f : D) d ds n \u2192 lit f (d \u2237 ds) n \u2261 f \u00b7 d \u00b7 (lit f ds n)\n{-# ATP axiom lit-[] #-}\n{-# ATP axiom lit-\u2237 #-}\n\n------------------------------------------------------------------------------\n-- Ordering functions and predicates on lists and trees\n\npostulate\n  ordList    : D \u2192 D\n  ordList-[] :          ordList []       \u2261 true\n  ordList-\u2237  : \u2200 i is \u2192 ordList (i \u2237 is) \u2261 \u2264-ItemList i is && ordList is\n{-# ATP axiom ordList-[] #-}\n{-# ATP axiom ordList-\u2237 #-}\n\nOrdList : D \u2192 Set\nOrdList is = ordList is \u2261 true\n{-# ATP definition OrdList #-}\n\npostulate\n  ordTree         : D \u2192 D\n  ordTree-nilTree :       ordTree nilTree \u2261 true\n  ordTree-tip     : \u2200 i \u2192 ordTree (tip i) \u2261 true\n  ordTree-node    : \u2200 t\u2081 i t\u2082 \u2192\n                      ordTree (node t\u2081 i t\u2082) \u2261\n                      ordTree t\u2081 && ordTree t\u2082 && \u2264-TreeItem t\u2081 i && \u2264-ItemTree i t\u2082\n{-# ATP axiom ordTree-nilTree #-}\n{-# ATP axiom ordTree-tip #-}\n{-# ATP axiom ordTree-node #-}\n\nOrdTree : D \u2192 Set\nOrdTree t = ordTree t \u2261 true\n{-# ATP definition OrdTree #-}\n\n------------------------------------------------------------------------------\n-- The program\n\n-- The function toTree adds an item to a tree.\n\n-- The items have an ordering \u2264 defined over them. The item held at a\n-- node of the tree is chosen so that the left subtree has items not\n-- greater than it and the right subtree has items not less than it.\n\npostulate\n  toTree          : D\n  toTree-nilTree  : \u2200 item \u2192   toTree \u00b7 item \u00b7 nilTree \u2261 tip item\n  toTree-tip      : \u2200 item i \u2192 toTree \u00b7 item \u00b7 (tip i) \u2261\n                    if (i \u2264 item)\n                      then (node (tip i) item (tip item))\n                      else (node (tip item) i (tip i))\n  toTree-node     : \u2200 item t\u2081 i t\u2082 \u2192\n                    toTree \u00b7 item \u00b7 (node t\u2081 i t\u2082) \u2261\n                       if (i \u2264 item)\n                         then (node t\u2081 i (toTree \u00b7 item \u00b7 t\u2082))\n                         else (node (toTree \u00b7 item \u00b7 t\u2081) i t\u2082)\n{-# ATP axiom toTree-nilTree #-}\n{-# ATP axiom toTree-tip #-}\n{-# ATP axiom toTree-node #-}\n\n-- The function makeTree converts a list to a tree.\nmakeTree : D \u2192 D\nmakeTree is = lit toTree is nilTree\n{-# ATP definition makeTree #-}\n\n-- The function flatten converts a tree to a list.\npostulate\n  flatten         : D \u2192 D\n  flatten-nilTree :       flatten nilTree \u2261 []\n  flatten-tip     : \u2200 i \u2192 flatten (tip i) \u2261 i \u2237 []\n  flatten-node    : \u2200 t\u2081 i t\u2082 \u2192\n                    flatten (node t\u2081 i t\u2082)  \u2261 flatten t\u2081 ++ flatten t\u2082\n{-# ATP axiom flatten-nilTree #-}\n{-# ATP axiom flatten-tip #-}\n{-# ATP axiom flatten-node #-}\n\n-- The function which sorts the list\nsort : D \u2192 D\nsort is = flatten (makeTree is)\n{-# ATP definition sort #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Sort a list\n------------------------------------------------------------------------------\n\n-- This module define the program which sorts a list by converting it\n-- into an ordered tree and then back to a list (Burstall, 1969, pp. 45-46).\n\n-- R. M. Burstall. Proving properties of programs by structural\n-- induction. The Computer Journal, 12(1):41\u201348, 1969.\n\nmodule LTC.Program.SortList.SortList where\n\nopen import LTC.Base\n\nopen import LTC.Data.Bool using ( _&&_ )\nopen import LTC.Data.Nat.Inequalities using ( _\u2264_ )\nopen import LTC.Data.Nat.Type\n  using ( N  -- The LTC natural numbers type.\n        )\nopen import LTC.Data.List using ( _++_ )\n\n------------------------------------------------------------------------------\n-- Tree terms\npostulate\n  nilTree  : D\n  tip      : D \u2192 D\n  node     : D \u2192 D \u2192 D \u2192 D\n\n-- The LTC tree type.\ndata Tree : D \u2192 Set where\n  nilT  : Tree nilTree\n  tipT  : \u2200 {i} \u2192 N i \u2192 Tree (tip i)\n  nodeT : \u2200 {t\u2081 i t\u2082} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192 Tree (node t\u2081 i t\u2082)\n{-# ATP hint nilT #-}\n{-# ATP hint tipT #-}\n{-# ATP hint nodeT #-}\n\n------------------------------------------------------------------------------\n-- Inequalites on lists and trees\n\n-- Note from Burstall (p. 46): \"The relation \u2264 between lists is only an\n-- ordering if nil is excluded, similarly for trees. This is untidy but\n-- will not cause trouble.\"\n\npostulate\n  \u2264-ItemList    : D \u2192 D \u2192 D\n  \u2264-ItemList-[] : \u2200 item \u2192 \u2264-ItemList item [] \u2261 true\n  \u2264-ItemList-\u2237  : \u2200 item i is \u2192\n                  \u2264-ItemList item (i \u2237 is) \u2261 item \u2264 i && \u2264-ItemList item is\n{-# ATP axiom \u2264-ItemList-[] #-}\n{-# ATP axiom \u2264-ItemList-\u2237 #-}\n\nLE-ItemList : D \u2192 D \u2192 Set\nLE-ItemList item is = \u2264-ItemList item is \u2261 true\n{-# ATP definition LE-ItemList #-}\n\npostulate\n  \u2264-Lists    : D \u2192 D \u2192 D\n  \u2264-Lists-[] : \u2200 is \u2192 \u2264-Lists [] is \u2261 true\n  \u2264-Lists-\u2237  : \u2200 i is js \u2192\n               \u2264-Lists (i \u2237 is) js \u2261 \u2264-ItemList i js && \u2264-Lists is js\n{-# ATP axiom \u2264-Lists-[] #-}\n{-# ATP axiom \u2264-Lists-\u2237 #-}\n\nLE-Lists : D \u2192 D \u2192 Set\nLE-Lists is js = \u2264-Lists is js \u2261 true\n{-# ATP definition LE-Lists #-}\n\npostulate\n  \u2264-ItemTree          : D \u2192 D \u2192 D\n  \u2264-ItemTree-nilTree  : \u2200 item \u2192   \u2264-ItemTree item nilTree \u2261 true\n  \u2264-ItemTree-tip      : \u2200 item i \u2192 \u2264-ItemTree item (tip i) \u2261 item \u2264 i\n  \u2264-ItemTree-node     : \u2200 item t\u2081 i t\u2082 \u2192\n                          \u2264-ItemTree item (node t\u2081 i t\u2082) \u2261\n                          \u2264-ItemTree item t\u2081 && \u2264-ItemTree item t\u2082\n{-# ATP axiom \u2264-ItemTree-nilTree #-}\n{-# ATP axiom \u2264-ItemTree-tip #-}\n{-# ATP axiom \u2264-ItemTree-node #-}\n\nLE-ItemTree : D \u2192 D \u2192 Set\nLE-ItemTree item t = \u2264-ItemTree item t \u2261 true\n{-# ATP definition LE-ItemTree #-}\n\n-- This function is not defined in the paper.\npostulate\n  \u2264-TreeItem         : D \u2192 D \u2192 D\n  \u2264-TreeItem-nilTree : \u2200 item \u2192   \u2264-TreeItem nilTree item \u2261 true\n  \u2264-TreeItem-tip     : \u2200 i item \u2192 \u2264-TreeItem (tip i) item \u2261 i \u2264 item\n  \u2264-TreeItem-node    : \u2200 t\u2081 i t\u2082 item \u2192\n                         \u2264-TreeItem (node t\u2081 i t\u2082) item \u2261\n                         \u2264-TreeItem t\u2081 item && \u2264-TreeItem t\u2082 item\n{-# ATP axiom \u2264-TreeItem-nilTree #-}\n{-# ATP axiom \u2264-TreeItem-tip #-}\n{-# ATP axiom \u2264-TreeItem-node #-}\n\nLE-TreeItem : D \u2192 D \u2192 Set\nLE-TreeItem t item = \u2264-TreeItem t item \u2261 true\n{-# ATP definition LE-TreeItem #-}\n\n------------------------------------------------------------------------------\n-- Auxiliary functions\n\npostulate\n  -- The foldr function with the last two args flipped.\n  lit    : D \u2192 D \u2192 D \u2192 D\n  lit-[] : \u2200 (f : D) n \u2192      lit f []       n \u2261 n\n  lit-\u2237  : \u2200 (f : D) d ds n \u2192 lit f (d \u2237 ds) n \u2261 f \u00b7 d \u00b7 (lit f ds n)\n{-# ATP axiom lit-[] #-}\n{-# ATP axiom lit-\u2237 #-}\n\n------------------------------------------------------------------------------\n-- Ordering functions and predicates on lists and trees\n\npostulate\n  ordList    : D \u2192 D\n  ordList-[] :          ordList []       \u2261 true\n  ordList-\u2237  : \u2200 i is \u2192 ordList (i \u2237 is) \u2261 \u2264-ItemList i is && ordList is\n{-# ATP axiom ordList-[] #-}\n{-# ATP axiom ordList-\u2237 #-}\n\nOrdList : D \u2192 Set\nOrdList is = ordList is \u2261 true\n{-# ATP definition OrdList #-}\n\npostulate\n  ordTree         : D \u2192 D\n  ordTree-nilTree :       ordTree nilTree \u2261 true\n  ordTree-tip     : \u2200 i \u2192 ordTree (tip i) \u2261 true\n  ordTree-node    : \u2200 t\u2081 i t\u2082 \u2192\n                      ordTree (node t\u2081 i t\u2082) \u2261\n                      ordTree t\u2081 && ordTree t\u2082 && \u2264-TreeItem t\u2081 i && \u2264-ItemTree i t\u2082\n{-# ATP axiom ordTree-nilTree #-}\n{-# ATP axiom ordTree-tip #-}\n{-# ATP axiom ordTree-node #-}\n\nOrdTree : D \u2192 Set\nOrdTree t = ordTree t \u2261 true\n{-# ATP definition OrdTree #-}\n\n------------------------------------------------------------------------------\n-- The program\n\n-- The function toTree adds an item to a tree.\n\n-- The items have an ordering \u2264 defined over them. The item held at a\n-- node of the tree is chosen so that the left subtree has items not\n-- greater than it and the right subtree has items not less than it.\n\npostulate\n  toTree          : D\n  toTree-nilTree  : \u2200 item \u2192   toTree \u00b7 item \u00b7 nilTree \u2261 tip item\n  toTree-tip      : \u2200 item i \u2192 toTree \u00b7 item \u00b7 (tip i) \u2261\n                    if (i \u2264 item)\n                      then (node (tip i) item (tip item))\n                      else (node (tip item) i (tip i))\n  toTree-node     : \u2200 item t\u2081 i t\u2082 \u2192\n                    toTree \u00b7 item \u00b7 (node t\u2081 i t\u2082) \u2261\n                       if (i \u2264 item)\n                         then (node t\u2081 i (toTree \u00b7 item \u00b7 t\u2082))\n                         else (node (toTree \u00b7 item \u00b7 t\u2081) i t\u2082)\n{-# ATP axiom toTree-nilTree #-}\n{-# ATP axiom toTree-tip #-}\n{-# ATP axiom toTree-node #-}\n\n-- The function makeTree converts a list to a tree.\nmakeTree : D \u2192 D\nmakeTree is = lit toTree is nilTree\n{-# ATP definition makeTree #-}\n\n-- The function flatten converts a tree to a list.\npostulate\n  flatten         : D \u2192 D\n  flatten-nilTree :       flatten nilTree \u2261 []\n  flatten-tip     : \u2200 i \u2192 flatten (tip i) \u2261 i \u2237 []\n  flatten-node    : \u2200 t\u2081 i t\u2082 \u2192\n                    flatten (node t\u2081 i t\u2082)  \u2261 flatten t\u2081 ++ flatten t\u2082\n{-# ATP axiom flatten-nilTree #-}\n{-# ATP axiom flatten-tip #-}\n{-# ATP axiom flatten-node #-}\n\n-- The function which sorts the list\nsort : D \u2192 D\nsort is = flatten (makeTree is)\n{-# ATP definition sort #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"229d3b07c98fc8a701a979e34a5880b0fb876a0d","subject":"cleaner up funext","message":"cleaner up funext\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\npostulate\n    FunExt : \u2605\n    funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 {{fe : FunExt}} \u2192 f \u2261 g\n\nContractible : \u2200 {a}{A : \u2605_ a}(x : A) \u2192 \u2605_ a\nContractible x = \u2200 y \u2192 x \u2261 y\n\nmodule Equivalences where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  module _ {A B : \u2605}{f : A \u2192 B}(f\u1d31 : Equiv f) where\n    open Equiv f\u1d31\n    inv : B \u2192 A\n    inv = linv \u2218 f \u2218 rinv\n\n    inv-equiv : Equiv inv\n    inv-equiv = record { linv = f\n                       ; is-linv = \u03bb x \u2192 cong f (is-linv (rinv x)) \u2219 is-rinv x\n                       ; rinv = f\n                       ; is-rinv = \u03bb x \u2192 cong linv (is-rinv (f x)) \u2219 is-linv x }\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u2243-refl : Reflexive _\u2243_\n  \u2243-refl = _ , id\u1d31\n\n  \u2243-sym : Symmetric _\u2243_\n  \u2243-sym (_ , f\u1d31) = _ , inv-equiv f\u1d31\n\n  \u2243-trans : Transitive _\u2243_\n  \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n  module Contractible\u2192Equiv {A : \u2605}{x : A}(x-contr : Contractible x) where\n    const-equiv : Equiv {\ud835\udfd9} (\u03bb _ \u2192 x)\n    const-equiv = record { linv = _ ; is-linv = \u03bb _ \u2192 refl ; rinv = _ ; is-rinv = x-contr }\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-equiv\n  module Equiv\u2192Contractible {A : \u2605}(f : \ud835\udfd9 \u2192 A)(f-equiv : Equiv f) where\n    open Equiv f-equiv\n    A-contr : Contractible (f _)\n    A-contr = is-rinv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\nopen Equivalences\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com_ \u2113 : \u2605_(\u209b \u2113) where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605_ \u2113\n        P  : M \u2192 Proto_ \u2113\n\n    data Proto_ \u2113 : \u2605_(\u209b \u2113) where\n      end : Proto_ \u2113\n      com : Com_ \u2113 \u2192 Proto_ \u2113\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\nCom   : \u2605\u2081\nCom   = Com_ \u2080\n\n{-\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n-}\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e _ P) (\u03a0\u1d3e _ Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c (mk _ M P) = \u03a3\u1d9c M \u03bb m \u2192 source-of (P m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk io M P) = mk (dual\u1d35\u1d3c io) M \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113) (P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end        \u27e7 = End\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = End\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\ndata ViewProc {\u2113} : \u2200 (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u2605_(\u209b \u2113) where\n  send : \u2200 M(P : M \u2192 Proto_ \u2113)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto_ \u2113)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 {\u2113} (P : Proto_ \u2113) (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl M x) (com refl .M x\u2081) = com refl M (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\nmodule _ {{_ : FunExt}} where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end            = _\nsink (com' _ M P) x = sink (P x)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P \u2192 Contractible (sink P)\n    sink-contr end          s = refl\n    sink-contr (com' _ _ P) s = funExt \u03bb m \u2192 sink-contr (P m) (s m)\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Contractible\u2192Equiv.\ud835\udfd9\u2243 (sink-contr P)\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end         _        ys = ys\n++Log (com' q M P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n\nlift\u1d3e : \u2200 a {\u2113} \u2192 Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\nlift\u1d3e a end           = end\nlift\u1d3e a (com' io M P) = com' io (Lift {_} {a} M) \u03bb m \u2192 lift\u1d3e a (P (lower m))\n\nlift-proc : \u2200 a {\u2113} (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e a P \u27e7\nlift-proc a {\u2113} P0 p0 = lift-view (view-proc P0 p0)\n  where\n    lift-view : \u2200 {P : Proto_ \u2113}{p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 lift\u1d3e a P \u27e7\n    lift-view (send M P m p) = lift m , lift-proc _ (P m) p\n    lift-view (recv M P x)   = \u03bb { (lift m) \u2192 lift-proc _ (P m) (x m) }\n    lift-view end            = end\n\nmodule MonoMobility (P : Proto\u2080) where\n  Com\u1d3e : Proto\u2080\n  Com\u1d3e = \u03a0\u1d3e \u27e6 P  \u27e7  \u03bb p \u2192\n         \u03a0\u1d3e \u27e6 P \u22a5\u27e7  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Log P) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc p p\u22a5 = tele-com P p p\u22a5 , _\n\nmodule PolyMobility where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         lift\u1d3e \u2081 (MonoMobility.Com\u1d3e P)\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P = lift-proc \u2081 (MonoMobility.Com\u1d3e P) (MonoMobility.com-proc P)\n\nmodule PolyMobility' where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         \u03a0\u1d3e (Lift \u27e6 P  \u27e7)  \u03bb p \u2192\n         \u03a0\u1d3e (Lift \u27e6 P \u22a5\u27e7)  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Lift (Log P)) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P (lift p) (lift p\u22a5) = lift (tele-com P p p\u22a5) , _\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\n           {-\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n  \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind {{_ : FunExt}} where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Log P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = \u03a0\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n    StaticServer  = \u03a3\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ {{_ : FunExt}} where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end Q = \u2261\u1d3e-refl _\ndual->> (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\ndual->> (\u03a3\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 subst id (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (\u03a0\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (\u03a3\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _ {{_ : FunExt}} (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecv\u2610 : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecv\u2610 = recv \u2218\u2032 un\u2610\n\nsend\u2610 : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsend\u2610 m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\n{-\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n-}\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\nmodule _ {P Q R S} where\n    \u2295\u1d3e-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n    \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n    \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n    &\u1d3e-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n    &\u1d3e-map f g p `L = f (p `L)\n    &\u1d3e-map f g p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u1d3e\u2192\u228e : \u27e6 P \u2295\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u1d3e\u2192\u228e (`L , p) = inl p\n    \u2295\u1d3e\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295\u1d3e : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295\u1d3e Q \u27e7\n    \u228e\u2192\u2295\u1d3e (inl p) = `L , p\n    \u228e\u2192\u2295\u1d3e (inr q) = `R , q\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP\u1d9c >>\u1d9c S = record P\u1d9c { P = \u03bb m \u2192 S (P m) }\n  where open Com_ P\u1d9c\n\nmodule _ where\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' \u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m')\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  -- the terminology used for the constructor follows the behavior of the combined process\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    send   : \u2200 {M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end (m , p)\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end (m , p) = send P m p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end      = end\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n{- Useless\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL Q f x = f x\n-}\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) R (m : M)(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a3\u1d3e _ Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (\u03a3\u1d3e _ Q) R (inl m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7))(q : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) Q R p q\n    recvR-sendR : \u2200 {M M'}P(Q : M \u2192 Proto)(R : M' \u2192 Proto)(m : M')(p : \u27e6 com P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inr m , q)\n    recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n               (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : (m : MR) \u2192 \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a0\u1d3e _ R) p q\n    sendR-recvL : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)R(m : MQ)\n                    (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : (m : MQ) \u2192 \u27e6 dual (Q m) \u214b\u1d3e R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) R (inr m , p) q\n    recvR-sendL : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (p : (m : MQ) \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(m : MQ)(q : \u27e6 dual (Q m) \u214b\u1d3e \u03a3\u1d3e _ R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inl m , q)\n    endL : \u2200 Q R\n           \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n           \u2192 View-\u2218 end Q R q qr\n    sendLM : \u2200 {MP}(P : MP \u2192 Proto)R\n               (m : MP)(p : \u27e6 P m \u27e7)(r : \u27e6 R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) end R (m , p) r\n    recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                    (m : MQ)(p : \u2200 m \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : \u27e6 dual (Q m) \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) end p (m , q)\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) _                 = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = recvR-sendR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (send ._ _ _)     = recvL-sendR _ _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (send _ _ _)     _                 = sendLM _ _ _ _ _\n\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n\nmodule _ {{_ : FunExt}} where\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply (dual P) Q pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound (dual-involutive P))) p)\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)      = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (\u03a3\u1d3e _ Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) Q R (p m) qr\n    \u214b\u1d3e-\u2218-view (recvR-sendR P Q R m pq q) = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (\u03a0\u1d3e _ Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (R m) pq (q m)\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) R p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)           = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (sendLM P R m pq qr)       = \u214b\u1d3e-sendL R m (par (P m) R pq qr)\n    \u214b\u1d3e-\u2218-view (recvL-sendR P Q m pq qr)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) end (pq m) (\u214b\u1d3e-rend' (dual (Q m)) qr)\n \n  mutual\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm P Q p = \u214b\u1d3e-comm-view (view-\u214b P Q p)\n\n    \u214b\u1d3e-comm-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm-view (sendL' P Q m p)  = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-comm (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-comm-view (sendR' P Q m' p) = inl m' , \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') p\n    \u214b\u1d3e-comm-view (recvL' P Q pq)   = \u214b\u1d3e-recvR Q P \u03bb m \u2192 \u214b\u1d3e-comm (P m) Q (pq m)\n    \u214b\u1d3e-comm-view (recvR' P Q pq)   = \u03bb m' \u2192 \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') (pq m')\n    \u214b\u1d3e-comm-view (endL Q pq)       = \u214b\u1d3e-rend' Q pq\n    \u214b\u1d3e-comm-view (send P m pq)     = m , pq\n\n  \u214b\u1d3e-comm-equiv : \u2200 P Q \u2192 Equiv (\u214b\u1d3e-comm P Q)\n  \u214b\u1d3e-comm-equiv P Q = record { linv = \u214b\u1d3e-comm Q P ; is-linv = {!!} ; rinv = \u214b\u1d3e-comm Q P ; is-rinv = {!!} }\n  {-\n    where\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 \u214b\u1d3e-comm Q P (\u214b\u1d3e-comm P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n   -}\n  \u214b\u1d3e-comm-\u2248 : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm-\u2248 P Q = _ , \u214b\u1d3e-comm-equiv P Q\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n  comma\u1d3e P Q p q with view-proc P p | view-proc Q q\n  comma\u1d3e .(com (mk Out M P)) Q .(m , p) q | send M P m p | q' = m , comma\u1d3e (P m) Q p q\n  comma\u1d3e .(com (mk In M P)) .(com (mk Out M\u2081 P\u2081)) p .(m , p\u2081) | recv M P .p | send M\u2081 P\u2081 m p\u2081 = m , comma\u1d3e (\u03a0\u1d3e M P) (P\u2081 m) p p\u2081\n  comma\u1d3e ._ ._ ._ ._ | recv M P p | recv M\u2081 P\u2081 x = [inl: (\u03bb m \u2192 comma\u1d3e (P m) (\u03a0\u1d3e _ P\u2081) (p m) x)\n                                                   ,inr: (\u03bb m' \u2192 comma\u1d3e (\u03a0\u1d3e _ P) (P\u2081 m') p (x m') ) ]\n  comma\u1d3e ._ ._ ._ ._ | recv M P x | end = x\n  comma\u1d3e .end Q .end q | end | q' = q\n\n  \u2297\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297\u1d3e-fst end      Q        pq       = _\n  \u2297\u1d3e-fst (\u03a3\u1d3e M P) Q        (m , pq) = m , \u2297\u1d3e-fst (P m) Q pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) end      pq       = pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (_ , pq) = \u2297\u1d3e-fst (\u03a0\u1d3e M P) (Q _) pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-fst (P m) (\u03a0\u1d3e N Q) (pq (inl m))\n\n  \u2297\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  \u2297\u1d3e-snd end      Q        pq       = pq\n  \u2297\u1d3e-snd (\u03a3\u1d3e M P) Q        (_ , pq) = \u2297\u1d3e-snd (P _) Q pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) end      pq       = end\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (m , pq) = m , \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) (pq (inr m))\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\nmodule V4 {{_ : FunExt}} where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _ {{_ : FunExt}} where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _ {{_ : FunExt}} where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _ {{_ : FunExt}}\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\nContractible : \u2200 {a}{A : \u2605_ a}(x : A) \u2192 \u2605_ a\nContractible x = \u2200 y \u2192 x \u2261 y\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  module _ {A B : \u2605}{f : A \u2192 B}(f\u1d31 : Equiv f) where\n    open Equiv f\u1d31\n    inv : B \u2192 A\n    inv = linv \u2218 f \u2218 rinv\n\n    inv-equiv : Equiv inv\n    inv-equiv = record { linv = f\n                       ; is-linv = \u03bb x \u2192 cong f (is-linv (rinv x)) \u2219 is-rinv x\n                       ; rinv = f\n                       ; is-rinv = \u03bb x \u2192 cong linv (is-rinv (f x)) \u2219 is-linv x }\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  \u2243-refl : Reflexive _\u2243_\n  \u2243-refl = _ , id\u1d31\n\n  \u2243-sym : Symmetric _\u2243_\n  \u2243-sym (_ , f\u1d31) = _ , inv-equiv f\u1d31\n\n  \u2243-trans : Transitive _\u2243_\n  \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n  module Contractible\u2192Equiv {A : \u2605}{x : A}(x-contr : Contractible x) where\n    const-equiv : Equiv {\ud835\udfd9} (\u03bb _ \u2192 x)\n    const-equiv = record { linv = _ ; is-linv = \u03bb _ \u2192 refl ; rinv = _ ; is-rinv = x-contr }\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-equiv\n  module Equiv\u2192Contractible {A : \u2605}(f : \ud835\udfd9 \u2192 A)(f-equiv : Equiv f) where\n    open Equiv f-equiv\n    A-contr : Contractible (f _)\n    A-contr = is-rinv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com_ \u2113 : \u2605_(\u209b \u2113) where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605_ \u2113\n        P  : M \u2192 Proto_ \u2113\n\n    data Proto_ \u2113 : \u2605_(\u209b \u2113) where\n      end : Proto_ \u2113\n      com : Com_ \u2113 \u2192 Proto_ \u2113\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\nCom   : \u2605\u2081\nCom   = Com_ \u2080\n\n{-\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n-}\npattern com' q M P = com (mk q M P)\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (com refl M p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : \u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (\u03a0\u1d3e _ P) (\u03a0\u1d3e _ Q) (com refl M p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _ _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _ _) = \u2261-\u03a3 _\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    source-of : Proto \u2192 Proto\n    source-of end     = end\n    source-of (com c) = com (source-of\u1d9c c)\n\n    source-of\u1d9c : Com \u2192 Com\n    source-of\u1d9c (mk _ M P) = \u03a3\u1d9c M \u03bb m \u2192 source-of (P m)\n\n    {-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (\u03a3\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 dual (P m)\ndual (\u03a0\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 dual (P m)\n-}\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c (mk io M P) = mk (dual\u1d35\u1d3c io) M \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSource (P m)) \u2192 IsSource (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113) (P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end        \u27e7 = End\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = End\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\ndata ViewProc {\u2113} : \u2200 (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u2605_(\u209b \u2113) where\n  send : \u2200 M(P : M \u2192 Proto_ \u2113)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto_ \u2113)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 {\u2113} (P : Proto_ \u2113) (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl M x) (com refl .M x\u2081) = com refl M (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (cong f (\u2261\u1d3e-sound P\u2261Q))\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\ndual-inj : \u2200 P Q \u2192 dual P \u2261\u1d3e dual Q \u2192 P \u2261\u1d3e Q\ndual-inj end end end = end\ndual-inj end (com x) ()\ndual-inj (com x) end ()\ndual-inj (com (mk In M P\u2081)) (com (mk In .M P)) (ProtoRel.com q .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P\u2081 m) (P m) (x m))\ndual-inj (com (mk In M P\u2081)) (com (mk Out .M P)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk In .M Q)) (ProtoRel.com () .M x)\ndual-inj (com (mk Out M P)) (com (mk Out .M Q)) (ProtoRel.com refl .M x) = ProtoRel.com refl M (\u03bb m \u2192 dual-inj (P m) (Q m) (x m))\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end         = end\nsource-of-idempotent (com' _ M P) = com refl M \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end         = end\nsource-of-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end            = _\nsink (com' _ M P) x = sink (P x)\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    open Equivalences funExt\n    sink-contr : \u2200 P \u2192 Contractible (sink P)\n    sink-contr end          s = refl\n    sink-contr (com' _ _ P) s = funExt \u03bb m \u2192 sink-contr (P m) (s m)\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Contractible\u2192Equiv.\ud835\udfd9\u2243 (sink-contr P)\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end         _        ys = ys\n++Log (com' q M P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com refl M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com refl M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a3\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e (M \u228e A) [inl: (\u03bb m \u2192 add\u03a0\u1d3e (P m)) ,inr: A\u1d3e ]\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-add\u03a3\u1d3e : \u2200 P \u2192 dual (add\u03a3\u1d3e A\u1d3e P) \u2261\u1d3e add\u03a0\u1d3e (dual \u2218 A\u1d3e) (dual P)\n    dual-add\u03a3\u1d3e end      = end\n    dual-add\u03a3\u1d3e (\u03a0\u1d3e M P) = com refl M (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n    dual-add\u03a3\u1d3e (\u03a3\u1d3e M P) = com refl (M \u228e A) [inl: (\u03bb m \u2192 dual-add\u03a3\u1d3e (P m))\n                                           ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n\nlift\u1d3e : \u2200 a {\u2113} \u2192 Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\nlift\u1d3e a end           = end\nlift\u1d3e a (com' io M P) = com' io (Lift {_} {a} M) \u03bb m \u2192 lift\u1d3e a (P (lower m))\n\nlift-proc : \u2200 a {\u2113} (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e a P \u27e7\nlift-proc a {\u2113} P0 p0 = lift-view (view-proc P0 p0)\n  where\n    lift-view : \u2200 {P : Proto_ \u2113}{p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 lift\u1d3e a P \u27e7\n    lift-view (send M P m p) = lift m , lift-proc _ (P m) p\n    lift-view (recv M P x)   = \u03bb { (lift m) \u2192 lift-proc _ (P m) (x m) }\n    lift-view end            = end\n\nmodule MonoMobility (P : Proto\u2080) where\n  Com\u1d3e : Proto\u2080\n  Com\u1d3e = \u03a0\u1d3e \u27e6 P  \u27e7  \u03bb p \u2192\n         \u03a0\u1d3e \u27e6 P \u22a5\u27e7  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Log P) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc p p\u22a5 = tele-com P p p\u22a5 , _\n\nmodule PolyMobility where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         lift\u1d3e \u2081 (MonoMobility.Com\u1d3e P)\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P = lift-proc \u2081 (MonoMobility.Com\u1d3e P) (MonoMobility.com-proc P)\n\nmodule PolyMobility' where\n  Com\u1d3e : Proto_ \u2081\n  Com\u1d3e = \u03a0\u1d3e Proto\u2080         \u03bb P \u2192\n         \u03a0\u1d3e (Lift \u27e6 P  \u27e7)  \u03bb p \u2192\n         \u03a0\u1d3e (Lift \u27e6 P \u22a5\u27e7)  \u03bb p\u22a5 \u2192\n         \u03a3\u1d3e (Lift (Log P)) \u03bb log \u2192\n         end\n  com-proc : \u27e6 Com\u1d3e \u27e7\n  com-proc P (lift p) (lift p\u22a5) = lift (tele-com P p p\u22a5) , _\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\n           {-\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n  \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n-}\n\n{-\nEl : (P : Proto) \u2192 (Log P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Log P \u2192 Proto}{X : Log (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Log P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n-}\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = \u03a0\u1d3e Query \u03bb q \u2192 \u03a3\u1d3e (Resp q) \u03bb r \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = \u03a3\u1d3e Query \u03bb q \u2192 \u03a0\u1d3e (Resp q) \u03bb r \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = \u03a0\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n    StaticServer  = \u03a3\u1d3e \u2115 \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end Q = \u2261\u1d3e-refl _\ndual->> (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\ndual->> (\u03a3\u1d3e M P) Q = com refl M (\u03bb m \u2192 dual->> (P m) Q)\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 subst id (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (\u03a0\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (\u03a3\u1d3e M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong funExt (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n\u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n\u2295\u1d3e-map f g (`L , pr) = `L , f pr\n\u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n&\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n&\u1d3e-map f g p `L = f (p `L)\n&\u1d3e-map f g p `R = g (p `R)\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP\u1d9c >>\u1d9c S = record P\u1d9c { P = \u03bb m \u2192 S (P m) }\n  where open Com_ P\u1d9c\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' \u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m')\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  -- the terminology used for the constructor follows the behavior of the combined process\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    send   : \u2200 {M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end (m , p)\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end (m , p) = send P m p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-isendR : \u2200 {M'} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M' Q \u27e7 \u2192 (m' : M') \u2192 \u27e6 P \u214b\u1d3e Q m' \u27e7\n  \u214b\u1d3e-isendR end Q s m' = s m'\n  \u214b\u1d3e-isendR (\u03a0\u1d3e M P) Q s m' = \u03bb m \u2192 \u214b\u1d3e-isendR (P m) Q (s m) m'\n  \u214b\u1d3e-isendR (\u03a3\u1d3e M P) Q s m' = s m'\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL {P = P} (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m (\u214b\u1d3e-isendR (P m) _ p m')\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = inl m , p\n\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end      = end\n  \u214b\u1d3e-id (\u03a0\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendL (P x) x (\u214b\u1d3e-id (P x))\n  \u214b\u1d3e-id (\u03a3\u1d3e M P) = \u03bb x \u2192 \u214b\u1d3e-sendR (dual (P x)) x (\u214b\u1d3e-id (P x))\n\n{- Useless\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL Q f x = f x\n-}\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto) R (m : M)(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a3\u1d3e _ Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (\u03a3\u1d3e _ Q) R (inl m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7))(q : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) Q R p q\n    recvR-sendR : \u2200 {M M'}P(Q : M \u2192 Proto)(R : M' \u2192 Proto)(m : M')(p : \u27e6 com P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inr m , q)\n    recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n               (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e \u03a0\u1d3e _ Q \u27e7)(q : (m : MR) \u2192 \u27e6 dual (\u03a0\u1d3e _ Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a0\u1d3e _ R) p q\n    sendR-recvL : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)R(m : MQ)\n                    (p : \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : (m : MQ) \u2192 \u27e6 dual (Q m) \u214b\u1d3e R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) R (inr m , p) q\n    recvR-sendL : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (p : (m : MQ) \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(m : MQ)(q : \u27e6 dual (Q m) \u214b\u1d3e \u03a3\u1d3e _ R \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (\u03a3\u1d3e _ R) p (inl m , q)\n    endL : \u2200 Q R\n           \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n           \u2192 View-\u2218 end Q R q qr\n    sendLM : \u2200 {MP}(P : MP \u2192 Proto)R\n               (m : MP)(p : \u27e6 P m \u27e7)(r : \u27e6 R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e _ P) end R (m , p) r\n    recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                    (m : MQ)(p : \u2200 m \u2192 \u27e6 \u03a3\u1d3e _ P \u214b\u1d3e Q m \u27e7)(q : \u27e6 dual (Q m) \u27e7)\n                  \u2192 View-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) end p (m , q)\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) _                 = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = recvR-sendR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (send ._ _ _)     = recvL-sendR _ _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (send _ _ _)     _                 = sendLM _ _ _ _ _\n\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply (dual P) Q pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)      = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (\u03a3\u1d3e _ Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) Q R (p m) qr\n    \u214b\u1d3e-\u2218-view (recvR-sendR P Q R m pq q) = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (\u03a0\u1d3e _ Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)        = \u03bb m \u2192 \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) (R m) pq (q m)\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) R p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)           = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (sendLM P R m pq qr)       = \u214b\u1d3e-sendL R m (par (P m) R pq qr)\n    \u214b\u1d3e-\u2218-view (recvL-sendR P Q m pq qr)  = \u214b\u1d3e-\u2218 (\u03a3\u1d3e _ P) (Q m) end (pq m) (\u214b\u1d3e-rend' (dual (Q m)) qr)\n \n  mutual\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm P Q p = \u214b\u1d3e-comm-view (view-\u214b P Q p)\n\n    \u214b\u1d3e-comm-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm-view (sendL' P Q m p)  = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-comm (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-comm-view (sendR' P Q m' p) = inl m' , \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') p\n    \u214b\u1d3e-comm-view (recvL' P Q pq)   = \u214b\u1d3e-recvR Q P \u03bb m \u2192 \u214b\u1d3e-comm (P m) Q (pq m)\n    \u214b\u1d3e-comm-view (recvR' P Q pq)   = \u03bb m' \u2192 \u214b\u1d3e-comm (\u03a3\u1d3e _ P) (Q m') (pq m')\n    \u214b\u1d3e-comm-view (endL Q pq)       = \u214b\u1d3e-rend' Q pq\n    \u214b\u1d3e-comm-view (send P m pq)     = m , pq\n\n  \u214b\u1d3e-comm-equiv : \u2200 P Q \u2192 Equiv (\u214b\u1d3e-comm P Q)\n  \u214b\u1d3e-comm-equiv P Q = record { linv = \u214b\u1d3e-comm Q P ; is-linv = {!!} ; rinv = \u214b\u1d3e-comm Q P ; is-rinv = {!!} }\n  {-\n    where\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 \u214b\u1d3e-comm Q P (\u214b\u1d3e-comm P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n   -}\n  \u214b\u1d3e-comm-\u2248 : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-comm-\u2248 P Q = _ , \u214b\u1d3e-comm-equiv P Q\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297\u1d3e Q \u27e7\n  comma\u1d3e P Q p q with view-proc P p | view-proc Q q\n  comma\u1d3e .(com (mk Out M P)) Q .(m , p) q | send M P m p | q' = m , comma\u1d3e (P m) Q p q\n  comma\u1d3e .(com (mk In M P)) .(com (mk Out M\u2081 P\u2081)) p .(m , p\u2081) | recv M P .p | send M\u2081 P\u2081 m p\u2081 = m , comma\u1d3e (\u03a0\u1d3e M P) (P\u2081 m) p p\u2081\n  comma\u1d3e ._ ._ ._ ._ | recv M P p | recv M\u2081 P\u2081 x = [inl: (\u03bb m \u2192 comma\u1d3e (P m) (\u03a0\u1d3e _ P\u2081) (p m) x)\n                                                   ,inr: (\u03bb m' \u2192 comma\u1d3e (\u03a0\u1d3e _ P) (P\u2081 m') p (x m') ) ]\n  comma\u1d3e ._ ._ ._ ._ | recv M P x | end = x\n  comma\u1d3e .end Q .end q | end | q' = q\n\n  \u2297\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297\u1d3e-fst end      Q        pq       = _\n  \u2297\u1d3e-fst (\u03a3\u1d3e M P) Q        (m , pq) = m , \u2297\u1d3e-fst (P m) Q pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) end      pq       = pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (_ , pq) = \u2297\u1d3e-fst (\u03a0\u1d3e M P) (Q _) pq\n  \u2297\u1d3e-fst (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-fst (P m) (\u03a0\u1d3e N Q) (pq (inl m))\n\n  \u2297\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P \u2297\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  \u2297\u1d3e-snd end      Q        pq       = pq\n  \u2297\u1d3e-snd (\u03a3\u1d3e M P) Q        (_ , pq) = \u2297\u1d3e-snd (P _) Q pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) end      pq       = end\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a3\u1d3e _ Q) (m , pq) = m , \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) pq\n  \u2297\u1d3e-snd (\u03a0\u1d3e M P) (\u03a0\u1d3e N Q) pq       = \u03bb m \u2192 \u2297\u1d3e-snd (\u03a0\u1d3e M P) (Q m) (pq (inr m))\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Log P \u00d7 Log Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Log B \u00d7 Log E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ade754ba5af3c03fc37342609c06d2cb02add9d2","subject":"crypto-agda.agda type-checks for me","message":"crypto-agda.agda type-checks for me\n","repos":"crypto-agda\/crypto-agda","old_file":"crypto-agda.agda","new_file":"crypto-agda.agda","new_contents":"module crypto-agda where\nimport Cipher.ElGamal.Generic\nimport Composition.Forkable\nimport Composition.Horizontal\nimport Composition.Vertical\nimport Crypto.Schemes\n--import FiniteField.FinImplem\nimport FunUniverse.Agda\nimport FunUniverse.BinTree\nimport FunUniverse.Bits\nimport FunUniverse.Category\nimport FunUniverse.Category.Op\nimport FunUniverse.Circuit\nimport FunUniverse.Const\nimport FunUniverse.Core\nimport FunUniverse.Cost\nimport FunUniverse.Data\nimport FunUniverse.Defaults.FirstPart\n--import FunUniverse.ExnArrow\nimport FunUniverse.Fin\nimport FunUniverse.Fin.Op\nimport FunUniverse.Fin.Op.Abstract\n--import FunUniverse.FlatFunsProd\n--import FunUniverse.Interface.Bits\nimport FunUniverse.Interface.Two\nimport FunUniverse.Interface.Vec\nimport FunUniverse.Inverse\n--import FunUniverse.Loop\nimport FunUniverse.Nand\nimport FunUniverse.Nand.Function\nimport FunUniverse.Nand.Properties\nimport FunUniverse.README\nimport FunUniverse.Rewiring.Linear\n--import FunUniverse.State\nimport FunUniverse.Syntax\nimport FunUniverse.Types\nimport Game.DDH\nimport Game.EntropySmoothing\nimport Game.EntropySmoothing.WithKey\nimport Game.IND-CPA\nimport Game.Transformation.InternalExternal\nimport Language.Simple.Abstract\nimport Language.Simple.Interface\nimport Language.Simple.Two.Mux\nimport Language.Simple.Two.Mux.Normalise\nimport Language.Simple.Two.Nand\nimport README\nimport Solver.AddMax\nimport Solver.Linear\n--import Solver.Linear.Examples\nimport Solver.Linear.Parser\nimport Solver.Linear.Syntax\nimport adder\nimport alea.cpo\nimport bijection-syntax.Bijection-Fin\nimport bijection-syntax.Bijection\nimport bijection-syntax.README\n--import circuits.bytecode\nimport circuits.circuit\n--import elgamal\n--import sha1\n","old_contents":"module crypto-agda where\nimport Cipher.ElGamal.Generic\nimport Composition.Forkable\nimport Composition.Horizontal\nimport Composition.Vertical\nimport Crypto.Schemes\nimport FiniteField.FinImplem\nimport FunUniverse.Agda\nimport FunUniverse.BinTree\nimport FunUniverse.Bits\nimport FunUniverse.Category\nimport FunUniverse.Category.Op\nimport FunUniverse.Circuit\nimport FunUniverse.Const\nimport FunUniverse.Core\nimport FunUniverse.Cost\nimport FunUniverse.Data\nimport FunUniverse.Defaults.FirstPart\nimport FunUniverse.ExnArrow\nimport FunUniverse.Fin\nimport FunUniverse.Fin.Op\nimport FunUniverse.Fin.Op.Abstract\nimport FunUniverse.FlatFunsProd\nimport FunUniverse.Interface.Bits\nimport FunUniverse.Interface.Two\nimport FunUniverse.Interface.Vec\nimport FunUniverse.Inverse\nimport FunUniverse.Loop\nimport FunUniverse.Nand\nimport FunUniverse.Nand.Function\nimport FunUniverse.Nand.Properties\nimport FunUniverse.README\nimport FunUniverse.Rewiring.Linear\nimport FunUniverse.State\nimport FunUniverse.Syntax\nimport FunUniverse.Types\nimport Game.DDH\nimport Game.EntropySmoothing\nimport Game.EntropySmoothing.WithKey\nimport Game.IND-CPA\nimport Game.Transformation.InternalExternal\nimport Language.Simple.Abstract\nimport Language.Simple.Interface\nimport Language.Simple.Two.Mux\nimport Language.Simple.Two.Mux.Normalise\nimport Language.Simple.Two.Nand\nimport README\nimport Solver.AddMax\nimport Solver.Linear\nimport Solver.Linear.Examples\nimport Solver.Linear.Parser\nimport Solver.Linear.Syntax\nimport adder\nimport alea.cpo\nimport bijection-syntax.Bijection-Fin\nimport bijection-syntax.Bijection\nimport bijection-syntax.README\nimport circuits.bytecode\nimport circuits.circuit\nimport elgamal\nimport sha1\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1d673a6136319a02f9e2a0e131984caed3c227f5","subject":"Added missing comment.","message":"Added missing comment.\n\nIgnore-this: e94b7bc15f3a36f09bea4cb19c108816\n\ndarcs-hash:20120514153713-3bd4e-094c6eef9fb0ab388d0e5e524246803c7052297f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 14 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- you can get the instance\n\n  subst-aux : \u2200 x y \u2192 x \u2250 y \u2192 x \u2261 x \u2192 x \u2261 y\n  subst-aux x y h\u2081 h\u2082 = subst A x y h\u2081 refl\n    where A : D \u2192 Set\n          A z = x \u2261 z\n\n  -- hence you can prove\n\n  thm : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  thm {x} {y} h = subst-aux x y h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 14 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- you can get the instance\n\n  subst-aux : \u2200 x y \u2192 x \u2250 y \u2192 x \u2261 x \u2192 x \u2261 y\n  subst-aux x y h\u2081 h\u2082 = subst A x y h\u2081 refl\n    where A : D \u2192 Set\n          A z = x \u2261 z\n\n  -- hence you can prove\n\n  thm : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  thm {x} {y} h = subst-aux x y h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4cbed45974f3385d5c182dddefd1bb738b95c793","subject":"agda: document a function","message":"agda: document a function\n\nOld-commit-hash: c0550a6f2e84b20bc63a5c0f91a168988125b117\n","repos":"inc-lc\/ilc-agda","old_file":"ChangeContexts.agda","new_file":"ChangeContexts.agda","new_contents":"module ChangeContexts where\n\nopen import IlcModel\nopen import Changes\nopen import meaning\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n-- \u0394-Context\u2032: behaves like \u0394-Context, but has an extra argument \u0393\u2032, a\n-- prefix of its first argument which should not be touched.\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n","old_contents":"module ChangeContexts where\n\nopen import IlcModel\nopen import Changes\nopen import meaning\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dedea5ca29d9dacb3f6cc34810d20b89f8e9a8d2","subject":"Restore Parametric.Change.Validity","message":"Restore Parametric.Change.Validity\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d_\u208e {{change-algebra-base}} \u03b9\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d_\u208e {{environment-changes}} \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d_\u208e \u0393\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  instance\n    change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n    change-algebra (base \u03b9) = change-algebra\u208d_\u208e {{change-algebra-base}} \u03b9\n    change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  {-\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"daae99c1a563b1f44493f46d87b4592eadc6aabb","subject":"HoTT: +unique-idp-left","message":"HoTT: +unique-idp-left\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/HoTT.agda","new_file":"lib\/HoTT.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\nopen \u2261.\u2261-Reasoning\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  -- could be derived in any groupoid\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  module _ {x y : A} where\n\n    \u2219-assoc : (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n    \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n                 (\u03bb x q r \u2192 idp)\n\n    module _ (p : x \u2261 y) where\n      -- ! is a left-inverse for _\u2219_\n      !-\u2219 : ! p \u2219 p \u2261 idp_ y\n      !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp) p\n\n      -- ! is a right-inverse for _\u2219_\n      \u2219-! : p \u2219 ! p \u2261 idp_ x\n      \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp) p\n\n      -- ! is involutive\n      !-involutive : ! (! p) \u2261 p\n      !-involutive = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp) p\n\n      !p\u2219p = !-\u2219\n      p\u2219!p = \u2219-!\n\n      ==-refl-\u2219 : {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n      ==-refl-\u2219 = ap (flip _\u2219_ p)\n\n      \u2219-==-refl : {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n      \u2219-==-refl qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  module _ {x y : A} where\n    module _ (p : x \u2261 x)(q : x \u2261 y)(e : p \u2219 q \u2261 q) where\n      unique-idp-left : p \u2261 idp\n      unique-idp-left\n        = p              \u2261\u27e8 ! \u2219-refl p \u27e9\n          p \u2219 idp        \u2261\u27e8 ap (_\u2219_ p) (! \u2219-! q) \u27e9\n          p \u2219 (q \u2219 ! q)  \u2261\u27e8 \u2219-assoc p q (! q) \u27e9\n          (p \u2219 q) \u2219 ! q  \u2261\u27e8 ap (flip _\u2219_ (! q)) e \u27e9\n          q \u2219 ! q        \u2261\u27e8 \u2219-! q \u27e9\n          idp            \u220e\n\n  module _ {x y z : A}{p\u2080 p\u2081 : x \u2261 y}{q\u2080 q\u2081 : y \u2261 z}(p : p\u2080 \u2261 p\u2081)(q : q\u2080 \u2261 q\u2081) where\n      \u2219= : p\u2080 \u2219 q\u2080 \u2261 p\u2081 \u2219 q\u2081\n      \u2219= = ap (flip _\u2219_ q\u2080) p \u2219 ap (_\u2219_ p\u2081) q\n\n  module _ {x y z : A} where\n    module _ (p : x \u2261 y)(q : y \u2261 z) where\n      \u2219-\u2219-==-refl : {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n      \u2219-\u2219-==-refl rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n      !p\u2219p\u2219q : ! p \u2219 p \u2219 q \u2261 q\n      !p\u2219p\u2219q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n    p\u2219!p\u2219q : (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n    p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n    p\u2219!q\u2219q : (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n    p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n    p\u2219q\u2219!q : (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n    p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n    \u2219-cancel : {p q : x \u2261 y}(r : y \u2261 z) \u2192 p \u2219 r \u2261 q \u2219 r \u2192 p \u2261 q\n    \u2219-cancel {p = p} {q} r e\n      = p               \u2261\u27e8 ! p\u2219q\u2219!q p r \u27e9\n         p \u2219 r  \u2219 ! r   \u2261\u27e8 \u2219-assoc p r (! r) \u27e9\n        (p \u2219 r) \u2219 ! r   \u2261\u27e8 \u2219= e idp \u27e9\n        (q \u2219 r) \u2219 ! r   \u2261\u27e8 ! \u2219-assoc q r (! r) \u27e9\n        q \u2219 (r  \u2219 ! r)  \u2261\u27e8 p\u2219q\u2219!q q r \u27e9\n        q               \u220e\n\n!-ap : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B){x y}(p : x \u2261 y)\n       \u2192 ! (ap f p) \u2261 ap f (! p)\n!-ap f idp = idp\n\nap-id : \u2200 {a}{A : Set a}{x y : A}(p : x \u2261 y) \u2192 ap id p \u2261 p\nap-id idp = idp\n\nmodule _ {a b}{A : Set a}{B : Set b}{f g : A \u2192 B}(H : \u2200 x \u2192 f x \u2261 g x) where\n  ap-nat : \u2200 {x y}(q : x \u2261 y) \u2192 ap f q \u2219 H _ \u2261 H _ \u2219 ap g q\n  ap-nat idp = ! \u2219-refl _\n\nmodule _ {a}{A : Set a}{f : A \u2192 A}(H : \u2200 x \u2192 f x \u2261 x) where\n  ap-nat-id : \u2200 x \u2192 ap f (H x) \u2261 H (f x)\n  ap-nat-id x = \u2219-cancel (H x) (ap-nat H (H x) \u2219 ap (_\u2219_ (H (f x))) (ap-id (H x)))\n\ntr-\u2218 : \u2200 {a b p}{A : Set a}{B : Set b}(P : B \u2192 Set p)(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 tr (P \u2218 f) p \u2261 tr P (ap f p)\ntr-\u2218 P f idp = idp\n\nmodule _ {a}{A : \u2605_ a} where\n    tr-\u2219\u2032 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) \u2192\n             tr P (p \u2219 q) \u223c tr P q \u2218 tr P p\n    tr-\u2219\u2032 P idp _ _ = idp\n\n    tr-\u2219 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) (pq : x \u2261 z) \u2192\n             pq \u2261 p \u2219 q \u2192\n             tr P pq \u223c tr P q \u2218 tr P p\n    tr-\u2219 P p q ._ idp = tr-\u2219\u2032 P p q\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb _ p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) idp p\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605_(a \u2294 b)\n    HAE {f} f-qinv = \u2200 x \u2192 ap f (F.inv-is-linv x) \u2261 F.inv-is-rinv (f x)\n      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      f-g' : (x : B) \u2192 f (g x) \u2261 x\n      f-g' x = ! ap (f \u2218 g) (f-g x) \u2219 ap f (g-f (g x)) \u2219 f-g x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n\n      postulate hae : HAE (qinv g f-g' g-f)\n\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g' g-f\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module Equiv {a b}{A : \u2605_ a}{B : \u2605_ b}(e : A \u2243 B) where\n    \u00b7\u2192 : A \u2192 B\n    \u00b7\u2192 = fst e\n\n    open Is-equiv (snd e) public\n      renaming (linv to \u00b7\u2190; rinv to \u00b7\u2190\u2032; is-linv to \u00b7\u2190-inv-l; is-rinv to \u00b7\u2190-inv-r)\n\n    -- Equivalences are \"injective\"\n    equiv-inj : {x y : A} \u2192 (\u00b7\u2192 x \u2261 \u00b7\u2192 y \u2192 x \u2261 y)\n    equiv-inj p = ! \u00b7\u2190-inv-l _ \u2219 ap \u00b7\u2190 p \u2219 \u00b7\u2190-inv-l _\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n    <\u2013' = \u00b7\u2190\u2032\n\n    <\u2013-inv-l = \u00b7\u2190-inv-l\n    <\u2013-inv-r = \u00b7\u2190-inv-r\n\n  open Equiv public\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n        prop-has-all-paths : is-prop A \u2192 has-all-paths A\n        prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n        all-paths-is-prop : has-all-paths A \u2192 is-prop A\n        all-paths-is-prop c x y = c x y , canon-path\n          where\n          lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n          lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n          canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n          canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                          (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same p x = ap (\u03bb X \u2192 coe X x) p\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u00b7\u2192 e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u00b7\u2192 e a) (coe-equiv-\u03b2 e)\n\n  postulate\n    coe!-\u03b2 : (e : A \u2243 B) (b : B) \u2192 coe! (ua e) b \u2261 \u00b7\u2190 e b\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u00b7\u2192-paths-equiv : (x \u2261 y) \u2261 (\u00b7\u2192 e x \u2261 \u00b7\u2192 e y)\n    \u00b7\u2192-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n    \u2013>-paths-equiv = \u00b7\u2192-paths-equiv\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  !-inv : \u2200 {x y : A} (p : x \u2261 y) \u2192 ! (! p) \u2261 p\n  !-inv = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp)\n\n  !-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp_ y\n  !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp)\n\n  \u2219-! : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp_ x\n  \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp)\n\n  !p\u2219p = !-\u2219\n  p\u2219!p = \u2219-!\n\n  \u2219-assoc : \u2200 {x y : A} (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n  \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n               (\u03bb x q r \u2192 idp)\n\n  ==-refl-\u2219 :  {x y : A} (p : x \u2261 y) {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n  ==-refl-\u2219 p = ap (flip _\u2219_ p)\n\n  \u2219-==-refl :  {x y : A} (p : x \u2261 y) {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n  \u2219-==-refl p qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  \u2219-\u2219-==-refl :  {x y z : A} (p : x \u2261 y) (q : y \u2261 z) {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n  \u2219-\u2219-==-refl p q rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n  !p\u2219p\u2219q : {x y z : A} (p : x \u2261 y) (q : y \u2261 z) \u2192 ! p \u2219 p \u2219 q \u2261 q\n  !p\u2219p\u2219q p q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n  p\u2219!p\u2219q : {x y z : A} (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n  p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n  p\u2219!q\u2219q : {x y z : A} (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n  p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n  p\u2219q\u2219!q : {x y z : A} (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n  p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n  \u2219-cancel : {x y z : A}{p q : x \u2261 y}(r : y \u2261 z) \u2192 p \u2219 r \u2261 q \u2219 r \u2192 p \u2261 q\n  \u2219-cancel {p = p} {q} idp p\u2219id=q\u2219id = ! \u2219-refl p \u2219 p\u2219id=q\u2219id \u2219 \u2219-refl q\n\n!-ap : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 ! (ap f p) \u2261 ap f (! p)\n!-ap f idp = idp\n\nap-id : \u2200 {a}{A : Set a}{x y : A}(p : x \u2261 y)\n  \u2192 ap id p \u2261 p\nap-id idp = idp\n\nmodule _ {a b}{A : Set a}{B : Set b}{f g : A \u2192 B}(H : \u2200 x \u2192 f x \u2261 g x) where\n  ap-nat : \u2200 {x y}(q : x \u2261 y) \u2192 ap f q \u2219 H _ \u2261 H _ \u2219 ap g q\n  ap-nat idp = ! \u2219-refl _\n\nmodule _ {a}{A : Set a}{f : A \u2192 A}(H : \u2200 x \u2192 f x \u2261 x) where\n  ap-nat-id : \u2200 x \u2192 ap f (H x) \u2261 H (f x)\n  ap-nat-id x = \u2219-cancel (H x) (ap-nat H (H x) \u2219 ap (_\u2219_ (H (f x))) (ap-id (H x)))\n\ntr-\u2218 : \u2200 {a b p}{A : Set a}{B : Set b}(P : B \u2192 Set p)(f : A \u2192 B){x y}(p : x \u2261 y)\n  \u2192 tr (P \u2218 f) p \u2261 tr P (ap f p)\ntr-\u2218 P f idp = idp\n\nmodule _ {a}{A : \u2605_ a} where\n    tr-\u2219\u2032 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) \u2192\n             tr P (p \u2219 q) \u223c tr P q \u2218 tr P p\n    tr-\u2219\u2032 P idp _ _ = idp\n\n    tr-\u2219 : \u2200 {\u2113}(P : A \u2192 \u2605_ \u2113) {x y z} (p : x \u2261 y) (q : y \u2261 z) (pq : x \u2261 z) \u2192\n             pq \u2261 p \u2219 q \u2192\n             tr P pq \u223c tr P q \u2218 tr P p\n    tr-\u2219 P p q ._ idp = tr-\u2219\u2032 P p q\n\nmodule _ {k} {K : \u2605_ k} {a} {A : \u2605_ a} {x y : A} (p : x \u2261 y) where\n    tr-const : tr (const K) p \u2261 id\n    tr-const = J (\u03bb _ p\u2081 \u2192 tr (const K) p\u2081 \u2261 id) idp p\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605_(a \u2294 b)\n    HAE {f} f-qinv = \u2200 x \u2192 ap f (F.inv-is-linv x) \u2261 F.inv-is-rinv (f x)\n      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      f-g' : (x : B) \u2192 f (g x) \u2261 x\n      f-g' x = ! ap (f \u2218 g) (f-g x) \u2219 ap f (g-f (g x)) \u2219 f-g x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n\n      postulate hae : HAE (qinv g f-g' g-f)\n\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g' g-f\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module Equiv {a b}{A : \u2605_ a}{B : \u2605_ b}(e : A \u2243 B) where\n    \u00b7\u2192 : A \u2192 B\n    \u00b7\u2192 = fst e\n\n    open Is-equiv (snd e) public\n      renaming (linv to \u00b7\u2190; rinv to \u00b7\u2190\u2032; is-linv to \u00b7\u2190-inv-l; is-rinv to \u00b7\u2190-inv-r)\n\n    -- Equivalences are \"injective\"\n    equiv-inj : {x y : A} \u2192 (\u00b7\u2192 x \u2261 \u00b7\u2192 y \u2192 x \u2261 y)\n    equiv-inj p = ! \u00b7\u2190-inv-l _ \u2219 ap \u00b7\u2190 p \u2219 \u00b7\u2190-inv-l _\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n    <\u2013' = \u00b7\u2190\u2032\n\n    <\u2013-inv-l = \u00b7\u2190-inv-l\n    <\u2013-inv-r = \u00b7\u2190-inv-r\n\n  open Equiv public\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n        prop-has-all-paths : is-prop A \u2192 has-all-paths A\n        prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n        all-paths-is-prop : has-all-paths A \u2192 is-prop A\n        all-paths-is-prop c x y = c x y , canon-path\n          where\n          lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n          lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n          canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n          canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                          (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same p x = ap (\u03bb X \u2192 coe X x) p\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u00b7\u2192 e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u00b7\u2192 e a) (coe-equiv-\u03b2 e)\n\n  postulate\n    coe!-\u03b2 : (e : A \u2243 B) (b : B) \u2192 coe! (ua e) b \u2261 \u00b7\u2190 e b\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u00b7\u2192-paths-equiv : (x \u2261 y) \u2261 (\u00b7\u2192 e x \u2261 \u00b7\u2192 e y)\n    \u00b7\u2192-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n    \u2013>-paths-equiv = \u00b7\u2192-paths-equiv\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e77fe96ab0a8db76f34bfcf6db62efe444a9ee3e","subject":"Distance","message":"Distance\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Distance.agda","new_file":"lib\/Data\/Nat\/Distance.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm {x} {1} {x} | +-assoc-comm {y} {1} {y} = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm {x} {1} {k} | q | ! +-assoc-comm {x} {1} {k} | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm {1} {y} {k} | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e914d8244b295baa4ce16e97daa3fd784bc458e2","subject":"Formalize families of change algebras.","message":"Formalize families of change algebras.\n\nOld-commit-hash: 302fcf9d2fd70c553d0d8a7c60d274db43416a3d\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of change algebras\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n","old_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c93f0658eae864ce1393bedec2e525a5ccfd9579","subject":"Typo.","message":"Typo.\n\nIgnore-this: c3f2332a1570afe8b5821dc1e3510c66\n\ndarcs-hash:20110930182110-3bd4e-6ef85c673365d3fbc83ea36f7dfc0ad2c80b2d33.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/DataPostulate.agda","new_file":"Draft\/DataPostulate.agda","new_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulate predicate: Using the\n-- induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulate predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulate predicate to the data predicate\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulated inductive principle from the data one.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {n} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulated one.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {n} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- The predicates\n\n-- From the data predicate to the postulate predicate: Using the\n-- induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- From the data predicate to the postulate predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n-- From the postulate predicate to the data predicate\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n\n------------------------------------------------------------------------------\n-- The induction principles\n\n-- The postulate inductive principle from the data inductive principle.\nindD2P : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\nindD2P P P0 ih Mn = indN P P0 (\u03bb {n} Nn \u2192 ih (nat-D2P Nn)) (nat-P2D Mn)\n\n-- The data inductive principle from the postulate predicate.\nindP2D : (P : D \u2192 Set) \u2192 P zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 P n\nindP2D P P0 ih Nn = indM P P0 (\u03bb {n} Mn \u2192 ih (nat-P2D Mn)) (nat-D2P Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a59128e46464bb896ee26735529b6a11131119b9","subject":"Try further to make agda work","message":"Try further to make agda work\n","repos":"louisswarren\/hieretikz","old_file":"hierarchy.agda","new_file":"hierarchy.agda","new_contents":"module hierarchy where\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n\n_and_ : Bool \u2192 Bool \u2192 Bool\ntrue and true = true\ntrue and false = false\nfalse and true = false\nfalse and false = false\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A \u2192 List A \u2192 List A\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []      = false\nx \u2208 (y :: ys) with x == y\n...              | true  = true\n...              | false = x \u2208 ys\n\n_\u2282_ : List \u2115 \u2192 List \u2115 \u2192 Bool\n[] \u2282 ys      = true\n(x :: xs) \u2282 ys with x \u2208 ys\n...               | true  = xs \u2282 ys\n...               | false = false\n\n\n_++_ : List \u2115 \u2192 List \u2115 \u2192 List \u2115\n[] ++ ys        = ys\n(x :: xs) ++ ys with (x \u2208 ys)\n...               | true  = xs ++ ys\n...               | false = x :: (xs ++ ys)\n\nmap : {A : Set} \u2192 (A \u2192 A) \u2192 List A \u2192 List A\nmap f []        = []\nmap f (x :: xs) = (f x) :: (map f xs)\n\n\ndata Arrow : Set where\n  _\u21d2_ : List \u2115 \u2192 \u2115 \u2192 Arrow\n\n_\u21d4_ : Arrow \u2192 Arrow \u2192 Bool\n(at \u21d2 ah) \u21d4 (bt \u21d2 bh) = ((at \u2282 bt) and (bt \u2282 at)) and (ah == bh)\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []      = false\nx \u2208\u2208 (y :: ys) with x \u21d4 y\n...               | true  = true\n...               | false = x \u2208\u2208 ys\n\n_\u2282\u2282_ : List Arrow \u2192 List Arrow \u2192 Bool\n[] \u2282\u2282 ys      = true\n(x :: xs) \u2282\u2282 ys with x \u2208\u2208 ys\n...                | true  = xs \u2282\u2282 ys\n...                | false = false\n\n\n-- SingleClosure : Arrow \u2192 List \u2115 \u2192 List \u2115\n-- SingleClosure (ts \u21d2 h) ps with ts \u2282 ps\n-- ...                     | true  = h :: ps\n-- ...                     | false = ps\n--\n-- Closure' : List Arrow \u2192 List \u2115 \u2192 List \u2115\n-- Closure' [] roots = roots\n-- Closure' (c :: cs) roots = (SingleClosure c roots) ++ (Closure' cs roots)\n--\n-- Closure : List Arrow \u2192 List \u2115 \u2192 List \u2115\n-- Closure arrows roots with (Closure' arrows roots) \u2282 roots\n-- ...                     | true = roots\n-- ...                     | false = Closure arrows (Closure' arrows roots)\n--\n--\n-- _,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\n-- _ , [] \u22a2 _ = false\n-- thms , p :: premises \u22a2 conc = true\n--\n\nArrowIntro : \u2115 \u2192 Arrow \u2192 Arrow\nArrowIntro n (ts \u21d2 h) = (n :: ts) \u21d2 h\n\nArrowElim : \u2115 \u2192 Arrow \u2192 Arrow\nArrowElim _ ([] \u21d2 h) = [] \u21d2 h\nArrowElim n ((t :: ts) \u21d2 h) with (t == n)\n...                            | true  = ArrowElim  n (ts \u21d2 h)\n...                            | false = ArrowIntro t (ArrowElim n (ts \u21d2 h))\n\nReduce : List Arrow \u2192 List \u2115 \u2192 List Arrow\nReduce cs [] = cs\nReduce cs (n :: ns) with (map (ArrowElim n) cs) \u2282\u2282 cs\n...                    | true  = cs\n...                    | false = Reduce (map (ArrowElim n) cs) (n :: ns)\n","old_contents":"module hierarchy where\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n\n_and_ : Bool \u2192 Bool \u2192 Bool\ntrue and true = true\ntrue and false = false\nfalse and true = false\nfalse and false = false\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = true\n_ == _         = false\n\n\n\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A \u2192 List A \u2192 List A\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []      = false\nx \u2208 (y :: ys) with x == y\n...              | true  = true\n...              | false = x \u2208 ys\n\n_\u2282_ : List \u2115 \u2192 List \u2115 \u2192 Bool\n[] \u2282 ys      = true\n(x :: xs) \u2282 ys with x \u2208 ys\n...               | true  = xs \u2282 ys\n...               | false = false\n\n\n_++_ : List \u2115 \u2192 List \u2115 \u2192 List \u2115\n[] ++ ys        = ys\n(x :: xs) ++ ys with (x \u2208 ys)\n...               | true  = xs ++ ys\n...               | false = x :: (xs ++ ys)\n\n\n\ndata Arrow : Set where\n  _\u21d2_ : List \u2115 \u2192 \u2115 \u2192 Arrow\n\nSingleClosure : Arrow \u2192 List \u2115 \u2192 List \u2115\nSingleClosure (ts \u21d2 h) ps with ts \u2282 ps\n...                     | true  = h :: ps\n...                     | false = ps\n\nClosure' : List Arrow \u2192 List \u2115 \u2192 List \u2115\nClosure' [] roots = roots\nClosure' (c :: cs) roots = (SingleClosure c roots) ++ (Closure' cs roots)\n\nClosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nClosure arrows roots with (Closure' arrows roots) \u2282 roots\n...                     | true = roots\n...                     | false = Closure arrows (Closure' arrows roots)\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\n_ , [] \u22a2 _ = false\nthms , p :: premises \u22a2 conc = true\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70907b2ea28e2d6559f103705f74a8fefbca6f63","subject":"Move before and after out of the mutually dependent block.","message":"Move before and after out of the mutually dependent block.\n\nOld-commit-hash: 1d7a74aa7d62457de4936eae9faaeb605d739bf3\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : (\u03b9 : Base) \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 (v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 (u v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : (\u03c4 : Type) \u2192 \u27e6 \u03c4 \u27e7 \u2192 Set\n\n  nil-change : \u2200 \u03c4 v \u2192 Change \u03c4 v\n  apply-change : \u2200 \u03c4 \u2192 (v : \u27e6 \u03c4 \u27e7) (dv : Change \u03c4 v) \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) v = Change-base \u03b9 v\n  Change (\u03c3 \u21d2 \u03c4) f = Pair\n    (\u2200 v \u2192 Change \u03c3 v \u2192 Change \u03c4 (f v))\n    (\u03bb \u0394f \u2192 \u2200 v dv \u2192\n      f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e \u0394f (v \u229e\u208d \u03c3 \u208e dv) (nil-change \u03c3 (v \u229e\u208d \u03c3 \u208e dv)) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v dv)\n\n  open Pair public using () renaming\n    ( cdr to is-valid\n    ; car to call-change\n    )\n\n  nil-change \u03c4 v = diff-change \u03c4 v v\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  apply-change (base \u03b9) n \u0394n = apply-change-base \u03b9 n \u0394n\n  apply-change (\u03c3 \u21d2 \u03c4) f \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e call-change \u0394f v (nil-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  diff-change (base \u03b9) m n = diff-change-base \u03b9 m n\n  diff-change (\u03c3 \u21d2 \u03c4) g f = cons (\u03bb v dv \u2192 g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n    (\u03bb v dv \u2192\n      begin\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g ((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g \u25a1 \u229f\u208d \u03c4 \u208e (f (v \u229e\u208d \u03c3 \u208e dv))))\n              (v+[u-v]=u {\u03c3} {v \u229e\u208d \u03c3 \u208e dv} {v \u229e\u208d \u03c3 \u208e dv}) \u27e9\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f (v \u229e\u208d \u03c3 \u208e dv)} \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8  sym (v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f v} )  \u27e9\n        f v \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n      \u220e) where open \u2261-Reasoning\n\n  -- call this lemma \"replace\"?\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (apply-change (\u03c3 \u21d2 \u03c4) v (diff-change (\u03c3 \u21d2 \u03c4) u v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- abbrevitations\n  before : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  before {\u03c4} {v} _ = v\n\n  after : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  after {\u03c4} {v} dv = v \u229e\u208d \u03c4 \u208e dv\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  data \u0394Env : \u2200 (\u0393 : Context) \u2192 \u27e6 \u0393 \u27e7 \u2192 Set where\n    \u2205 : \u0394Env \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1} \u2192\n      (dv : Change \u03c4 v) \u2192\n      (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n      \u0394Env (\u03c4 \u2022 \u0393) (v \u2022 \u03c1)\n\n  ignore : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  ignore {\u0393} {\u03c1} _ = \u03c1\n\n  update : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  update \u2205 = \u2205\n  update {\u03c4 \u2022 \u0393} (dv \u2022 d\u03c1) = after {\u03c4} dv \u2022 update d\u03c1\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : (\u03b9 : Base) \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 (v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 (u v : \u27e6 \u03b9 \u27e7Base) \u2192 Change-base \u03b9 v\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : (\u03c4 : Type) \u2192 \u27e6 \u03c4 \u27e7 \u2192 Set\n\n  nil-change : \u2200 \u03c4 v \u2192 Change \u03c4 v\n  apply-change : \u2200 \u03c4 \u2192 (v : \u27e6 \u03c4 \u27e7) (dv : Change \u03c4 v) \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) v = Change-base \u03b9 v\n  Change (\u03c3 \u21d2 \u03c4) f = Pair\n    (\u2200 v \u2192 Change \u03c3 v \u2192 Change \u03c4 (f v))\n    (\u03bb \u0394f \u2192 \u2200 v dv \u2192\n      f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e \u0394f (v \u229e\u208d \u03c3 \u208e dv) (nil-change \u03c3 (v \u229e\u208d \u03c3 \u208e dv)) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v dv)\n\n  before : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  before {\u03c4} {v} _ = v\n\n  after : \u2200 {\u03c4 v} \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  after {\u03c4} {v} dv = v \u229e\u208d \u03c4 \u208e dv\n\n  open Pair public using () renaming\n    ( cdr to is-valid\n    ; car to call-change\n    )\n\n  nil-change \u03c4 v = diff-change \u03c4 v v\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  apply-change (base \u03b9) n \u0394n = apply-change-base \u03b9 n \u0394n\n  apply-change (\u03c3 \u21d2 \u03c4) f \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e call-change \u0394f v (nil-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  diff-change (base \u03b9) m n = diff-change-base \u03b9 m n\n  diff-change (\u03c3 \u21d2 \u03c4) g f = cons (\u03bb v dv \u2192 g (after {\u03c3} dv) \u229f\u208d \u03c4 \u208e f v)\n    (\u03bb v dv \u2192\n      begin\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g ((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g \u25a1 \u229f\u208d \u03c4 \u208e (f (v \u229e\u208d \u03c3 \u208e dv))))\n              (v+[u-v]=u {\u03c3} {v \u229e\u208d \u03c3 \u208e dv} {v \u229e\u208d \u03c3 \u208e dv}) \u27e9\n        f (v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f (v \u229e\u208d \u03c3 \u208e dv))\n      \u2261\u27e8 v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f (v \u229e\u208d \u03c3 \u208e dv)} \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8  sym (v+[u-v]=u {\u03c4} {g (v \u229e\u208d \u03c3 \u208e dv)} {f v} )  \u27e9\n        f v \u229e\u208d \u03c4 \u208e (g (v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c4 \u208e f v)\n      \u220e) where open \u2261-Reasoning\n\n  -- call this lemma \"replace\"?\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (apply-change (\u03c3 \u21d2 \u03c4) v (diff-change (\u03c3 \u21d2 \u03c4) u v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 (u v : \u27e6 \u03c4 \u27e7) \u2192 Change \u03c4 v\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  data \u0394Env : \u2200 (\u0393 : Context) \u2192 \u27e6 \u0393 \u27e7 \u2192 Set where\n    \u2205 : \u0394Env \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1} \u2192\n      (dv : Change \u03c4 v) \u2192\n      (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n      \u0394Env (\u03c4 \u2022 \u0393) (v \u2022 \u03c1)\n\n  ignore : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  ignore {\u0393} {\u03c1} _ = \u03c1\n\n  update : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  update \u2205 = \u2205\n  update {\u03c4 \u2022 \u0393} (dv \u2022 d\u03c1) = after {\u03c4} dv \u2022 update d\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b92d284c9d89a7befebbf3db0fa5f09c051584a5","subject":"Modify validity, so that it agrees with \u2295","message":"Modify validity, so that it agrees with \u2295\n\nNew here:\n- decidable equality on types and contexts\n- \u2295, on values and environments, using decidable equality\n- restrictions on validity that require more things to be propositionally equal\n  (using a judgement for validity might allow less ceremony here, if that works)\n- a proof that \u2295 and validity (as fixed here) agree.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 0 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 0 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 0 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nopen import Relation.Nullary\n\n-- Decidable equivalence for types and contexts :-|\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , refl , p , refl , refl , dvv) with \u0393 \u225fCtx \u0393\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) | yes refl = refl\n... | no \u00acq = \u22a5-elim (\u00acq refl)\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .0 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x , \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .0 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv ,  rrel\u03c13-mono k\u2081 (suc k) (lt1 k<n) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .0 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 0 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 0 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 0 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , refl , vv) = (refl  , refl) , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .0 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x , \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .0 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv ,  rrel\u03c13-mono k\u2081 (suc k) (lt1 k<n) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .0 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a42f97175fbdf6967b776f0214135b3c4d920cca","subject":"+-finable","message":"+-finable\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-properties.agda","new_file":"sum-properties.agda","new_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Empty using (\u22a5)\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit using (\u22a4)\n\nopen import Function.NP\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nrecord Finable {A : Set}(\u03bcA : SumProp A) : Set where\n  constructor mk\n  field\n    FinCard  : \u2115\n\n  FIN = Fin (suc FinCard)\n\n  field\n    iso      : A \u2194 Fin (suc FinCard)\n    \n  toFin : A \u2192 FIN\n  toFin x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  fromFin : FIN \u2192 A\n  fromFin x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  field\n    sums-ok  : \u2200 f \u2192 sum \u03bcA f \u2261 sum (\u03bcFinSuc FinCard) (f \u2218 fromFin)\n\n  from-inj : Injective fromFin\n  from-inj = Inv.Inverse.injective (Inv.sym iso)\n\n  to-inj : Injective toFin\n  to-inj = Inv.Inverse.injective iso\n\n  iso' : fromFin \u2218 toFin \u2257 id\n  iso' = Inv.Inverse.left-inverse-of iso\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = mk 0 iso sums-ok where\n\n  iso : _\n  iso = \u22a4 \u2194\u27e8 sym A\u228e\u22a5\u2194A \u27e9\n        (\u22a4 \u228e \u22a5) \u2194\u27e8 Inv.id \u228e-cong sym Fin0\u2194\u22a5 \u27e9\n        (\u22a4 \u228e Fin 0) \u2194\u27e8 sym Fin\u2218suc\u2194\u22a4\u228eFin \u27e9\n        Fin (suc 0) \u220e\n    where open EquationalReasoning\n          open import Relation.Binary.Sum\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable {A}{B} \u03bcA \u03bcB finA finB = mk FinCard iso sums-ok where\n\n  |A| |B| : \u2115\n  |A| = suc (Finable.FinCard finA)\n  |B| = suc (Finable.FinCard finB)\n\n  \u03bcFinA = \u03bcFinSuc (Finable.FinCard finA)\n  \u03bcFinB = \u03bcFinSuc (Finable.FinCard finB)\n\n  FinCard : _\n  FinCard = Finable.FinCard finA + |B|\n\n  iso : _\n  iso = (A \u228e B) \n      \u2194\u27e8 (Finable.iso finA) \u228e-cong (Finable.iso finB) \u27e9 \n        (Fin |A| \u228e Fin |B|)\n      \u2194\u27e8 Fin-\u228e-+ |A| |B| \u27e9\n        Fin (|A| + |B|)\n      \u220e\n    where \n      open import Relation.Binary.Sum\n      open EquationalReasoning\n\n  from : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 B \u2192 A\n  from iso x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  to : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 A \u2192 B\n  to iso x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  fromFin : Fin (|A| + |B|) \u2192 A \u228e B\n  fromFin x = from iso x\n  \n  open import Data.Vec.NP using (Vec ; foldr\u2081 ; tabulate-ext)\n  vsum : \u2200 {n} \u2192 Vec \u2115 (suc n) \u2192 \u2115\n  vsum = foldr\u2081 _+_\n\n  fin-proof : \u2200 n m (f : Fin (suc n) \u228e Fin (suc m) \u2192 \u2115) \u2192 \n                      sum (\u03bcFinSuc n) (f \u2218 inj\u2081) \n                    + sum (\u03bcFinSuc m) (f \u2218 inj\u2082) \n                    \u2261 sum (\u03bcFinSuc (n + suc m)) (f \u2218 from  (Fin-\u228e-+ (suc n) (suc m)))\n  fin-proof zero m f    rewrite tabulate-ext (\u2261.cong (f \u2218 inj\u2082 \u2218 suc) \u2218 \u2261.sym \u2218 Inv.Inverse.right-inverse-of (Maybe^\u22a5\u2194Fin m)) = \u2261.refl\n  fin-proof (suc n) m f = sum (\u03bcFinSuc (suc n)) (f \u2218 inj\u2081) + sum (\u03bcFinSuc m) (f \u2218 inj\u2082) \n                        \u2261\u27e8 \u2115\u00b0.+-assoc (f (inj\u2081 zero)) (sum (\u03bcFinSuc n) (f \u2218 inj\u2081 \u2218 suc)) (sum (\u03bcFinSuc m) (f \u2218 inj\u2082)) \u27e9 \n                          f (inj\u2081 zero) + (sum (\u03bcFinSuc n) (f \u2218 inj\u2081 \u2218 suc) + sum (\u03bcFinSuc m) (f \u2218 inj\u2082))\n                        \u2261\u27e8 \u2261.cong (_+_ (f (inj\u2081 zero))) (fin-proof n m (F f)) \u27e9\n                          f (inj\u2081 zero) + sum (\u03bcFinSuc (n + suc m)) (F f \u2218 from (Fin-\u228e-+ (suc n) (suc m)))\n                        \u2261\u27e8 \u2261.cong (_+_ (f (inj\u2081 zero))) (sum-ext (\u03bcFinSuc (n + suc m)) (F-proof (suc n) (suc m) f)) \u27e9\n                          f (inj\u2081 zero) + sum (\u03bcFinSuc (n + suc m)) (f \u2218 from (Fin-\u228e-+ (suc (suc n)) (suc m)) \u2218 suc)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          sum (\u03bcFinSuc (suc n + suc m))\n                                     (f \u2218 from (Fin-\u228e-+ (suc (suc n)) (suc m))) \n                        \u220e\n    where \n      open \u2261.\u2261-Reasoning\n      F : \u2200 {n m} \u2192 (Fin (suc n) \u228e Fin m \u2192 \u2115) \u2192 Fin n \u228e Fin m \u2192 \u2115\n      F f = [ (f \u2218 inj\u2081 \u2218 suc) , (f \u2218 inj\u2082) ]\n\n\n      F-proof : \u2200 n m f \u2192 F f \u2218 from (Fin-\u228e-+ n m) \u2257 f \u2218 from (Fin-\u228e-+ (suc n) m) \u2218 suc\n      F-proof n m f x with (Inv.Inverse.from (Maybe^-\u228e-+ n m) \u27e8$\u27e9 (Inv.Inverse.from (Maybe^\u22a5\u2194Fin (n + m)) \u27e8$\u27e9 x))\n      ... | inj\u2081 y = \u2261.refl\n      ... | inj\u2082 y = \u2261.refl\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = sum (\u03bcA +\u03bc \u03bcB) f\n            \u2261\u27e8 \u2261.cong\u2082 _+_ (Finable.sums-ok finA (f \u2218 inj\u2081)) (Finable.sums-ok finB (f \u2218 inj\u2082)) \u27e9\n              sum \u03bcFinA (f \u2218 inj\u2081 \u2218 from (Finable.iso finA))\n            + sum \u03bcFinB (f \u2218 inj\u2082 \u2218 from (Finable.iso finB))\n            \u2261\u27e8 \u2261.cong\u2082 _+_ (sum-ext \u03bcFinA {f = f \u2218 inj\u2081 \u2218 from (Finable.iso finA)} (\u03bb x \u2192 \u2261.refl)) \n                           (sum-ext \u03bcFinB {f = f \u2218 inj\u2082 \u2218 from (Finable.iso finB)} (\u03bb _ \u2192 \u2261.refl)) \u27e9 \n              sum \u03bcFinA (f \u2218 from F \u2218 inj\u2081)\n            + sum \u03bcFinB (f \u2218 from F \u2218 inj\u2082)\n            \u2261\u27e8 fin-proof _ _ (f \u2218 from F) \u27e9\n              sum (\u03bcFinSuc FinCard) (f \u2218 from F \u2218 from (Fin-\u228e-+ |A| |B|))\n            \u2261\u27e8 \u2261.refl \u27e9\n              sum (\u03bcFinSuc FinCard) (f \u2218 from iso) \n            \u220e\n    where\n      open \u2261.\u2261-Reasoning\n      open import Relation.Binary.Sum\n      F :  (A \u228e B) \u2194 (Fin |A| \u228e Fin |B|)\n      F = Finable.iso finA \u228e-cong Finable.iso finB\n\n\u00d7-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA \u00d7\u03bc \u03bcB)\n\u00d7-Finable {A} {B} \u03bcA \u03bcB finA finB = mk FinCard iso sums-ok where\n  |A| |B| : \u2115\n  |A| = suc (Finable.FinCard finA)\n  |B| = suc (Finable.FinCard finB)\n\n  \u03bcFinA = \u03bcFinSuc (Finable.FinCard finA)\n  \u03bcFinB = \u03bcFinSuc (Finable.FinCard finB)\n\n  FinCard = Finable.FinCard finB + Finable.FinCard finA * |B|\n\n  prf : suc FinCard \u2261 |A| * |B|\n  prf = \u2261.refl\n\n  iso : _\n  iso = (A \u00d7 B) \u2194\u27e8 (Finable.iso finA) \u00d7-cong (Finable.iso finB) \u27e9 \n        (Fin |A| \u00d7 Fin |B|) \u2194\u27e8 {!!} \u27e9\n        Fin (|A| * |B|) \u220e\n    where open EquationalReasoning\n          open import Relation.Binary.Product.Pointwise\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = {!!}\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA fin f p p-inj\n  = sum \u03bcA f\n  \u2261\u27e8 sums-ok f \u27e9\n    sum (\u03bcFinSuc FinCard) (f \u2218 fromFin)\n  \u2261\u27e8 \u03bcFinSUI (f \u2218 fromFin) (toFin \u2218 p \u2218 fromFin) (from-inj \u2218 p-inj \u2218 to-inj) \u27e9\n    sum (\u03bcFinSuc FinCard) (f \u2218 fromFin \u2218 toFin \u2218 p \u2218 fromFin)\n  \u2261\u27e8 sum-ext (\u03bcFinSuc FinCard) (\u03bb x \u2192 \u2261.cong f (iso' (p (fromFin x)))) \u27e9\n    sum (\u03bcFinSuc FinCard) (f \u2218 p \u2218 fromFin)\n  \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 p)) \u27e9\n    sum \u03bcA (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open Finable fin\n\n","old_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Function.NP\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nrecord Finable {A : Set}(\u03bcA : SumProp A) : Set where\n  constructor mk\n  field\n    FinCard  : \u2115\n    toFin    : A \u2192 Fin FinCard\n    fromFin  : Fin FinCard \u2192 A\n    to-inj   : Injective toFin\n    from-inj : Injective fromFin\n    iso      : fromFin \u2218 toFin \u2257 id\n    sums-ok  : \u2200 f \u2192 sum \u03bcA f \u2261 sumFin FinCard (f \u2218 fromFin)\n\npostulate\n  Fin-Stab : \u2200 n \u2192 StableUnderInjection (\u03bcFin n)\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = mk 1 (\u03bb x \u2192 zero) (\u03bb x \u2192 _) (\u03bb x\u2081 \u2192 \u2261.refl) (\u03bb x\u2081 \u2192 help) (\u03bb x \u2192 \u2261.refl) (\u03bb f \u2192 ?)\n  where help : {x y : Fin 1} \u2192 x \u2261 y\n        help {zero} {zero} = \u2261.refl\n        help {zero} {suc ()}\n        help {suc ()}\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable {A}{B} \u03bcA \u03bcB finA finB = mk FinCard toFin fromFin to-inj from-inj iso sums-ok where\n    open import Data.Sum\n    open import Data.Empty\n\n    |A| = Finable.FinCard finA\n    |B| = Finable.FinCard finB\n\n    Fsuc-injective : \u2200 {n} {i j : Fin n} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\n    Fsuc-injective \u2261.refl = \u2261.refl\n    \n    inj\u2081-inj : \u2200 {A B : Set} {x y : A} \u2192 inj\u2081 {B = B} x \u2261 inj\u2081 y \u2192 x \u2261 y\n    inj\u2081-inj \u2261.refl = \u2261.refl\n\n    inj\u2082-inj : \u2200 {A B : Set} {x y : B} \u2192 inj\u2082 {A = A} x \u2261 inj\u2082 y \u2192 x \u2261 y\n    inj\u2082-inj \u2261.refl = \u2261.refl\n\n    fin\u2081 : \u2200 n {m} \u2192 Fin n \u2192 Fin (n + m)\n    fin\u2081 zero ()\n    fin\u2081 (suc n) zero = zero\n    fin\u2081 (suc n) (suc i) = suc (fin\u2081 n i)\n\n    fin\u2082 : \u2200 n {m} \u2192 Fin m \u2192 Fin (n + m)\n    fin\u2082 zero i = i\n    fin\u2082 (suc n) i = suc (fin\u2082 n i)\n\n    toFin' : \u2200 {n} {m} \u2192 Fin n \u228e Fin m \u2192 Fin (n + m)\n    toFin' {n} (inj\u2081 x) = fin\u2081 n x\n    toFin' {n} (inj\u2082 y) = fin\u2082 n y\n\n    fin\u2081-inj : \u2200 n {m} \u2192 Injective (fin\u2081 n {m})\n    fin\u2081-inj .(suc n) {m} {zero {n}} {zero} x\u2261y = \u2261.refl\n    fin\u2081-inj .(suc n) {m} {zero {n}} {suc y} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {zero} ()\n    fin\u2081-inj .(suc n) {m} {suc {n} x} {suc y} x\u2261y = \u2261.cong suc (fin\u2081-inj n (Fsuc-injective x\u2261y))\n\n    fin\u2082-inj : \u2200 n {m} \u2192 Injective (fin\u2082 n {m})\n    fin\u2082-inj zero eq = eq\n    fin\u2082-inj (suc n) eq = fin\u2082-inj n (Fsuc-injective eq)\n\n    fin\u2081\u2260fin\u2082 : \u2200 n {m} x y \u2192 (fin\u2081 n {m} x \u2261 fin\u2082 n y) \u2192 \u22a5\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) zero ()\n    fin\u2081\u2260fin\u2082 .(suc n) (zero {n}) (suc y) ()\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) zero eq = fin\u2081\u2260fin\u2082 n x zero (Fsuc-injective eq)\n    fin\u2081\u2260fin\u2082 .(suc n) (suc {n} x) (suc y) eq = fin\u2081\u2260fin\u2082 n x (suc y) (Fsuc-injective eq)\n\n    fromFin' : \u2200 n {m} \u2192 Fin (n + m) \u2192 Fin n \u228e Fin m\n    fromFin' zero k = inj\u2082 k\n    fromFin' (suc n) zero = inj\u2081 zero\n    fromFin' (suc n) (suc k) with fromFin' n k\n    ... | inj\u2081 x = inj\u2081 (suc x)\n    ... | inj\u2082 y = inj\u2082 y\n\n    fromFin'-inj : \u2200 n {m} x y \u2192 fromFin' n {m} x \u2261 fromFin' n y \u2192 x \u2261 y\n    fromFin'-inj zero x y eq = inj\u2082-inj eq\n    fromFin'-inj (suc n) zero zero eq = \u2261.refl\n    fromFin'-inj (suc n) zero (suc y) eq = {!!}\n    fromFin'-inj (suc n) (suc x) zero eq = {!!}\n    fromFin'-inj (suc n) (suc x) (suc y) eq = {!fromFin'-inj n x y!}\n\n    fromFin'-inj\u2081 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2081 n x) \u2261 inj\u2081 x\n    fromFin'-inj\u2081 zero ()\n    fromFin'-inj\u2081 (suc n) zero = \u2261.refl\n    fromFin'-inj\u2081 (suc n) {m} (suc x) rewrite fromFin'-inj\u2081 n {m} x  = \u2261.refl\n\n    fromFin'-inj\u2082 : \u2200 n {m} x \u2192 fromFin' n {m} (fin\u2082 n x) \u2261 inj\u2082 x\n    fromFin'-inj\u2082 zero x = \u2261.refl\n    fromFin'-inj\u2082 (suc n) x rewrite fromFin'-inj\u2082 n x = \u2261.refl\n\n    FinCard  : \u2115\n    FinCard = Finable.FinCard finA + Finable.FinCard finB\n\n    toFin    : A \u228e B \u2192 Fin FinCard\n    toFin (inj\u2081 x) = toFin' (inj\u2081 (Finable.toFin finA x))\n    toFin (inj\u2082 y) = toFin' {n = |A|} (inj\u2082 (Finable.toFin finB y))\n\n    fromFin  : Fin FinCard \u2192 A \u228e B\n    fromFin x+y with fromFin' |A| x+y\n    ... | inj\u2081 x = inj\u2081 (Finable.fromFin finA x)\n    ... | inj\u2082 y = inj\u2082 (Finable.fromFin finB y)\n\n    to-inj   : Injective toFin\n    to-inj {inj\u2081 x} {inj\u2081 x\u2081} tx\u2261ty = \u2261.cong inj\u2081 (Finable.to-inj finA (fin\u2081-inj |A| tx\u2261ty))\n    to-inj {inj\u2081 x} {inj\u2082 y} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ tx\u2261ty)\n    to-inj {inj\u2082 y} {inj\u2081 x} tx\u2261ty = \u22a5-elim (fin\u2081\u2260fin\u2082 |A| _ _ (\u2261.sym tx\u2261ty))\n    to-inj {inj\u2082 y} {inj\u2082 y\u2081} tx\u2261ty = \u2261.cong inj\u2082 (Finable.to-inj finB (fin\u2082-inj |A| tx\u2261ty))\n\n    from-inj : Injective fromFin\n    from-inj {x} {y} fx\u2261fy = {!!}\n\n    iso      : fromFin \u2218 toFin \u2257 id\n    iso (inj\u2081 x) rewrite fromFin'-inj\u2081 |A| {|B|} (Finable.toFin finA x) | Finable.iso finA x = \u2261.refl\n    iso (inj\u2082 y) rewrite fromFin'-inj\u2082 |A| {|B|} (Finable.toFin finB y) | Finable.iso finB y = \u2261.refl\n\n    fin-proof : \u2200 n m (f : Fin n \u228e Fin m \u2192 \u2115) \u2192 sumFin n (f \u2218 inj\u2081) + sumFin m (f \u2218 inj\u2082) \u2261 sumFin (n + m) (f \u2218 fromFin' n)\n    fin-proof zero    m f = \u2261.refl\n    fin-proof (suc n) m f = sumFin (suc n) (f \u2218 inj\u2081) + sumFin m (f \u2218 inj\u2082)\n                          \u2261\u27e8 \u2115\u00b0.+-assoc (f (inj\u2081 zero)) (sumFin n (f' f \u2218 inj\u2081))\n                               (sumFin m (f \u2218 inj\u2082)) \u27e9\n                            f (inj\u2081 zero) + (sumFin n (f' f \u2218 inj\u2081) + sumFin m (f \u2218 inj\u2082))\n                          \u2261\u27e8 \u2261.cong (_+_ (f (inj\u2081 zero))) (fin-proof n m (f' f)) \u27e9\n                            f (inj\u2081 zero) + (sumFin (n + m) (f' f \u2218 fromFin' n))\n                          \u2261\u27e8 \u2261.cong (\u03bb p \u2192 f (inj\u2081 zero) + vsum p)\n                               (tabulate\u2257 (\u03bb x \u2192 \u2261.sym (f'-proof n x f))) \u27e9\n                            sumFin (suc n + m) (f \u2218 fromFin' (suc n)) \n                          \u220e\n     where open \u2261.\u2261-Reasoning\n           open import Data.Vec renaming (sum to vsum)\n\n           tabulate\u2257 : \u2200 {n} {A : Set}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n           tabulate\u2257 {zero} eq = \u2261.refl\n           tabulate\u2257 {suc n\u2081} eq rewrite eq zero | tabulate\u2257 (eq \u2218 suc) = \u2261.refl\n\n           f' : \u2200 {n m} \u2192 (Fin (suc n) \u228e Fin m \u2192 \u2115) \u2192 Fin n \u228e Fin m \u2192 \u2115\n           f' f (inj\u2081 x) = f (inj\u2081 (suc x))\n           f' f (inj\u2082 x) = f (inj\u2082 x)\n\n           f'-proof : \u2200 n {m} x f \u2192 f (fromFin' (suc n) (suc x)) \u2261 f' {m = m} f (fromFin' n x)\n           f'-proof n x f with fromFin' n x\n           ... | inj\u2081 p = \u2261.refl\n           ... | inj\u2082 p = \u2261.refl\n\n    sums-ok  : \u2200 f \u2192 sum (\u03bcA +\u03bc \u03bcB) f \u2261 sumFin FinCard (f \u2218 fromFin)\n    sums-ok f =\n        sum (\u03bcA +\u03bc \u03bcB) f\n      \u2261\u27e8 \u2261.cong\u2082 _+_ (Finable.sums-ok finA (f \u2218 inj\u2081))\n           (Finable.sums-ok finB (f \u2218 inj\u2082)) \u27e9\n        sumFin (Finable.FinCard finA) (f \u2218 inj\u2081 \u2218 Finable.fromFin finA)\n      + sumFin (Finable.FinCard finB) (f \u2218 inj\u2082 \u2218 Finable.fromFin finB)\n      \u2261\u27e8 {!!} \u27e9\n        sumFin FinCard (f \u2218 fromFin)\n      \u220e\n      where open \u2261.\u2261-Reasoning\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA fin f p p-inj\n  = sum \u03bcA f\n  \u2261\u27e8 sums-ok f \u27e9\n    sumFin FinCard (f \u2218 fromFin)\n  \u2261\u27e8 Fin-Stab FinCard (f \u2218 fromFin) (toFin \u2218 p \u2218 fromFin) (from-inj \u2218 p-inj \u2218 to-inj) \u27e9\n    sumFin FinCard (f \u2218 fromFin \u2218 toFin \u2218 p \u2218 fromFin)\n  \u2261\u27e8 sum-ext (\u03bcFin FinCard) (\u03bb x \u2192 \u2261.cong f (iso (p (fromFin x)))) \u27e9\n    sumFin FinCard (f \u2218 p \u2218 fromFin)\n  \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 p)) \u27e9\n    sum \u03bcA (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open Finable fin\n\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = {!sumFin-spec n (f \u2218 p)!} -- {!sumFinSUI (suc n) f p p-inj!}\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6dc0ba9bdf23ef1a1cb8952c769afd60782c5101","subject":"Simplified a proof (following Ana Bove's observation).","message":"Simplified a proof (following Ana Bove's observation).\n\nIgnore-this: f3500a5ff85f1e145d5976f0edf694c5\n\ndarcs-hash:20120220141424-3bd4e-4266a16af4fbfa7d0212a25f9979594037b2d3fd.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/Commutator\/PropertiesI.agda","new_file":"src\/GroupTheory\/Commutator\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with the group commutator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Commutator.PropertiesI where\n\nopen import GroupTheory.Base\nopen import GroupTheory.Commutator\nopen import GroupTheory.PropertiesI\nopen import GroupTheory.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- From: A. G. Kurosh. The Theory of Groups, vol. 1. Chelsea Publising\n-- Company, 2nd edition, 1960. p. 99.\ncommutatorInverse : \u2200 a b \u2192 \u27e6 a , b \u27e7 \u00b7 \u27e6 b , a \u27e7 \u2261 \u03b5\ncommutatorInverse a b =\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a) b (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (\u00b7-rightCong (assoc (b \u207b\u00b9 \u00b7 a \u207b\u00b9) b a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 \u00b7-rightCong (\u00b7-rightCong (assoc (b \u207b\u00b9) (a \u207b\u00b9) (b \u00b7 a))) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n    \u2261\u27e8 \u00b7-rightCong (sym (assoc b (b \u207b\u00b9) (a \u207b\u00b9 \u00b7 (b \u00b7 a)))) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (rightInverse b)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 \u00b7-rightCong (leftIdentity (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9) a (a \u207b\u00b9 \u00b7 (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 \u00b7-rightCong (sym (assoc a (a \u207b\u00b9) (b \u00b7 a))) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (rightInverse a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (leftIdentity (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9) (b \u207b\u00b9) (b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 (b \u207b\u00b9 \u00b7 (b \u00b7 a))\n     \u2261\u27e8 \u00b7-rightCong (sym (assoc (b \u207b\u00b9) b a)) \u27e9\n  a \u207b\u00b9 \u00b7 ((b \u207b\u00b9 \u00b7 b) \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (leftInverse b)) \u27e9\n  a \u207b\u00b9 \u00b7 (\u03b5 \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (leftIdentity a) \u27e9\n  a \u207b\u00b9 \u00b7 a\n     \u2261\u27e8 leftInverse a \u27e9\n  \u03b5 \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with the group commutator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Commutator.PropertiesI where\n\nopen import GroupTheory.Base\nopen import GroupTheory.Commutator\nopen import GroupTheory.PropertiesI\nopen import GroupTheory.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- From: A. G. Kurosh. The Theory of Groups, vol. 1. Chelsea Publising\n-- Company, 2nd edition, 1960. p. 99.\ncommutatorInverse : \u2200 a b \u2192 \u27e6 a , b \u27e7 \u00b7 \u27e6 b , a \u27e7 \u2261 \u03b5\ncommutatorInverse a b =\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a) b (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 t))\n             (assoc (b \u207b\u00b9 \u00b7 a \u207b\u00b9) b a)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 t))\n             (assoc (b \u207b\u00b9) (a \u207b\u00b9) (b \u00b7 a))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 t)\n             (sym (assoc b (b \u207b\u00b9) (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (t \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n             (rightInverse b)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 t)\n             (leftIdentity (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9) a (a \u207b\u00b9 \u00b7 (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 t)\n             (sym (assoc a (a \u207b\u00b9) (b \u00b7 a)))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (rightInverse a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (leftIdentity (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9) (b \u207b\u00b9) (b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 (b \u207b\u00b9 \u00b7 (b \u00b7 a))\n     \u2261\u27e8 \u00b7-rightCong (sym (assoc (b \u207b\u00b9) b a)) \u27e9\n  a \u207b\u00b9 \u00b7 ((b \u207b\u00b9 \u00b7 b) \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (leftInverse b)) \u27e9\n  a \u207b\u00b9 \u00b7 (\u03b5 \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (leftIdentity a) \u27e9\n  a \u207b\u00b9 \u00b7 a\n     \u2261\u27e8 leftInverse a \u27e9\n  \u03b5 \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"65b8c33e714dbb8c31888eb1e3663efb6a12f8d8","subject":"Additive: update","message":"Additive: update\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Additive.agda","new_file":"Control\/Protocol\/Additive.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Type.Identities\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol.Core\n\nmodule Control.Protocol.Additive where\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto) where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    module _ {{_ : FunExt}}{{_ : UA}} where\n        P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n        P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n        P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n        P\u22a4-uniq = \u03a0\ud835\udfd8-uniq\u2080 _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} {P Q} where\n    dual-\u2295 : dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} P Q where\n    &-comm : P & Q \u2261 Q & P\n    &-comm = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm = send\u2243 LR!-equiv [L: refl R: refl ]\n\n    -- additive mix (left-biased)\n    amixL : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixL pq = (`L , pq `L)\n\n    amixR : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixR pq = (`R , pq `R)\n\nmodule _ {P Q R S}(f : \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7)(g : \u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) where\n    \u2295-map : \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map (`L , pr) = `L , f pr\n    \u2295-map (`R , pr) = `R , g pr\n\n    &-map : \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map p `L = f (p `L)\n    &-map p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 = [inl: _,_ `L ,inr: _,_ `R ]\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = equiv \u2295\u2192\u228e \u228e\u2192\u2295 [inl: (\u03bb _ \u2192 refl) ,inr: (\u03bb _ \u2192 refl) ] \u228e\u2192\u2295\u2192\u228e\n      where\n        \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n        \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n        \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = equiv &\u2192\u00d7 \u00d7\u2192& &\u2192\u00d7\u2192& \u00d7\u2192&\u2192\u00d7\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ P {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\nmodule _ P Q R {{_ : FunExt}}{{_ : UA}} where\n    &-assoc : \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (_\u00d7_; _,_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol.Core\n\nmodule Control.Protocol.Additive where\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto) where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    module _ {{_ : FunExt}}{{_ : UA}} where\n        P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n        P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n        P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n        P\u22a4-uniq = \u03a0\ud835\udfd8-uniq _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} {P Q} where\n    dual-\u2295 : dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} P Q where\n    &-comm : P & Q \u2261 Q & P\n    &-comm = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm = send\u2243 LR!-equiv [L: refl R: refl ]\n\n    -- additive mix (left-biased)\n    amixL : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixL pq = (`L , pq `L)\n\n    amixR : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixR pq = (`R , pq `R)\n\nmodule _ {P Q R S}(f : \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7)(g : \u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) where\n    \u2295-map : \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map (`L , pr) = `L , f pr\n    \u2295-map (`R , pr) = `R , g pr\n\n    &-map : \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map p `L = f (p `L)\n    &-map p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 = [inl: _,_ `L ,inr: _,_ `R ]\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = equiv \u2295\u2192\u228e \u228e\u2192\u2295 [inl: (\u03bb _ \u2192 refl) ,inr: (\u03bb _ \u2192 refl) ] \u228e\u2192\u2295\u2192\u228e\n      where\n        \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n        \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n        \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = &\u2192\u00d7 , record { linv = \u00d7\u2192& ; is-linv = \u00d7\u2192&\u2192\u00d7 ; rinv = \u00d7\u2192& ; is-rinv = &\u2192\u00d7\u2192& }\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ P {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\nmodule _ P Q R {{_ : FunExt}}{{_ : UA}} where\n    &-assoc : \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bfdbacd72e32e7f5c2aeacb13b760c5baf2d55fe","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 63bc13d9462d2ea02aba142e6e57f1cc\n\ndarcs-hash:20110218144205-3bd4e-f456c27be1fa9df61e809d09b1187d7358a11415.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/MCR\/LT2MCR-ATP.agda","new_file":"Draft\/McCarthy91\/MCR\/LT2MCR-ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Property LT2MCR which proves that the recursive calls of the\n-- McCarthy 91 functions are on smaller arguments.\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.MCR.LT2MCR-ATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.MCR\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\nLT2MCR-helper : \u2200 {n m k} \u2192 N n \u2192 N m \u2192 N k \u2192\n                LT m n \u2192 LT (succ n) k \u2192 LT (succ m) k \u2192\n                LT (k \u2238 n) (k \u2238 m) \u2192 LT (k \u2238 succ n) (k \u2238 succ m)\nLT2MCR-helper zN Nm Nk p qn qm h = \u22a5-elim (x<0\u2192\u22a5 Nm p)\nLT2MCR-helper (sN Nn) Nm zN p qn qm h = \u22a5-elim (x<0\u2192\u22a5 (sN Nm) qm)\nLT2MCR-helper (sN {n} Nn) zN (sN {k} Nk) p qn qm h = prfS0S\n  where\n    postulate k\u2265Sn   : GE k (succ n)\n              k\u2238Sn<k : LT (k \u2238 (succ n)) k\n              prfS0S : LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ zero)\n    {-# ATP prove k\u2265Sn x<y\u2192x\u2264y #-}\n    {-# ATP prove k\u2238Sn<k k\u2265Sn x\u2265y\u2192y>0\u2192x\u2238y<x #-}\n    {-# ATP prove prfS0S k\u2238Sn<k #-}\n\nLT2MCR-helper (sN {n} Nn) (sN {m} Nm) (sN {k} Nk) p qn qm h =\n  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm (LT2MCR-helper Nn Nm Nk m<n Sn<k Sm<k k\u2238n<k\u2238m)\n  where\n    postulate\n      k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm : LT (k \u2238 succ n) (k \u2238 succ m) \u2192\n                                LT (succ k \u2238 succ (succ n))\n                                   (succ k \u2238 succ (succ m))\n    {-# ATP prove  k\u2238Sn<k\u2238Sm\u2192Sk\u2238SSn<Sk\u2238SSm #-}\n\n    postulate\n      m<n     : LT m n\n      Sn<k    : LT (succ n) k\n      Sm<k    : LT (succ m) k\n      k\u2238n<k\u2238m : LT (k \u2238 n) (k \u2238 m)\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn<k #-}\n    {-# ATP prove Sm<k #-}\n    {-# ATP prove k\u2238n<k\u2238m #-}\n\nLT2MCR : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE m one-hundred \u2192 LT m n \u2192 MCR n m\nLT2MCR zN Nm p h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nLT2MCR (sN {n} Nn) zN p h = prfS0\n  where\n    postulate prfS0 : MCR (succ n) zero\n    {-# ATP prove prfS0 x\u2238y<Sx #-}\n\nLT2MCR (sN {n} Nn) (sN {m} Nm) p h with x<y\u2228x\u2265y Nn N100\n... | inj\u2081 n<100 = LT2MCR-helper Nn Nm N101 m<n Sn\u2264101 Sm\u2264101\n                                 (LT2MCR Nn Nm m\u2264100 m<n)\n  where\n    postulate m\u2264100  : LE m one-hundred\n              m<n    : LT m n\n              Sn\u2264101 : LT (succ n) hundred-one\n              Sm\u2264101 : LT (succ m) hundred-one\n    {-# ATP prove m\u2264100 x<y\u2192x\u2264y #-}\n    {-# ATP prove m<n #-}\n    {-# ATP prove Sn\u2264101 #-}\n    {-# ATP prove Sm\u2264101 #-}\n... | inj\u2082 n\u2265199 = prf-n\u2265100\n  where\n    postulate\n      101\u2238Sn\u22610  : hundred-one \u2238 succ n \u2261 zero\n      0<101\u2238Sm  : LT zero (hundred-one \u2238 succ m)\n      prf-n\u2265100 : MCR (succ n) (succ m)\n    {-# ATP prove 101\u2238Sn\u22610 x\u2264y\u2192x-y\u22610 #-}\n    {-# ATP prove 0<101\u2238Sm x<y\u21920<x\u2238y #-}\n    {-# ATP prove prf-n\u2265100 101\u2238Sn\u22610 0<101\u2238Sm #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Property LT2MCR which proves that the recursive calls of the\n-- McCarthy 91 functions are on smaller arguments.\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.MCR.LT2MCR-ATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.MCR\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\nLT2MCR-aux : \u2200 {n m k} \u2192 N n \u2192 N m \u2192 N k \u2192\n             LT m n \u2192 LT (succ n) k \u2192 LT (succ m) k \u2192\n             LT (k \u2238 n) (k \u2238 m) \u2192 LT (k \u2238 succ n) (k \u2238 succ m)\nLT2MCR-aux zN Nm Nk p qn qm h = \u22a5-elim (x<0\u2192\u22a5 Nm p)\nLT2MCR-aux (sN Nn) Nm zN p qn qm h = \u22a5-elim (x<0\u2192\u22a5 (sN Nm) qm)\nLT2MCR-aux (sN {n} Nn) zN (sN {k} Nk) p qn qm h = prfS0S\n  where postulate k\u2265Sn : GE k (succ n)\n                  k-Sn<k : LT (k \u2238 (succ n)) k\n                  prfS0S : LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ zero)\n        {-# ATP prove k\u2265Sn x<y\u2192x\u2264y #-}\n        {-# ATP prove k-Sn<k k\u2265Sn x\u2265y\u2192y>0\u2192x\u2238y<x #-}\n        {-# ATP prove prfS0S k-Sn<k #-}\nLT2MCR-aux (sN {n} Nn) (sN {m} Nm) (sN {k} Nk) p qn qm h =\n          k-Sn<k-Sm\u2192Sk-SSn<Sk-SSm (LT2MCR-aux Nn Nm Nk m<n Sn<k Sm<k k-n<k-m)\n  where postulate k-Sn<k-Sm\u2192Sk-SSn<Sk-SSm : LT (k \u2238 succ n) (k \u2238 succ m) \u2192\n                            LT (succ k \u2238 succ (succ n)) (succ k \u2238 succ (succ m))\n                  m<n : LT m n\n                  Sn<k : LT (succ n) k\n                  Sm<k : LT (succ m) k\n                  k-n<k-m : LT (k \u2238 n) (k \u2238 m)\n        {-# ATP prove  k-Sn<k-Sm\u2192Sk-SSn<Sk-SSm #-}\n        {-# ATP prove m<n #-}\n        {-# ATP prove Sn<k #-}\n        {-# ATP prove Sm<k #-}\n        {-# ATP prove k-n<k-m #-}\n\nLT2MCR : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE m one-hundred \u2192 LT m n \u2192 MCR n m\nLT2MCR zN Nm p h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nLT2MCR (sN {n} Nn) zN p h = prfS0\n  where postulate prfS0 : MCR (succ n) zero\n        {-# ATP prove prfS0 x\u2238y<Sx #-}\nLT2MCR (sN {n} Nn) (sN {m} Nm) p h with x<y\u2228x\u2265y Nn N100\n... | inj\u2081 n<100 = LT2MCR-aux Nn Nm N101 m<n Sn\u2264101 Sm\u2264101\n                             (LT2MCR Nn Nm m\u2264100 m<n)\n  where postulate m\u2264100 : LE m one-hundred\n                  m<n : LT m n\n                  Sn\u2264101 : LT (succ n) hundred-one\n                  Sm\u2264101 : LT (succ m) hundred-one\n        {-# ATP prove m\u2264100 x<y\u2192x\u2264y #-}\n        {-# ATP prove m<n #-}\n        {-# ATP prove Sn\u2264101 #-}\n        {-# ATP prove Sm\u2264101 #-}\n... | inj\u2082 n\u2265199 = prf-n\u2265100\n  where postulate 101-Sn=0 : hundred-one \u2238 succ n \u2261 zero\n                  0<101-Sm : LT zero (hundred-one \u2238 succ m)\n                  prf-n\u2265100 : MCR (succ n) (succ m)\n        {-# ATP prove 101-Sn=0 x\u2264y\u2192x-y\u22610 #-}\n        {-# ATP prove 0<101-Sm x<y\u21920<x\u2238y #-}\n        {-# ATP prove prf-n\u2265100 101-Sn=0 0<101-Sm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d0cf36a3218331192a7a115a77bcf4de12e20fa6","subject":"Two: using (_\u225f_)","message":"Two: using (_\u225f_)\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Two.agda","new_file":"lib\/Data\/Two.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two where\n\nopen import Data.Two.Base public\nopen import Data.Bool using (_\u225f_)\nopen import Data.Bool.Properties\n  public\n  using (isCommutativeSemiring-\u2228-\u2227\n        ;commutativeSemiring-\u2228-\u2227\n        ;module RingSolver\n        ;isCommutativeSemiring-\u2227-\u2228\n        ;commutativeSemiring-\u2227-\u2228\n        ;isBooleanAlgebra\n        ;booleanAlgebra\n        ;commutativeRing-xor-\u2227\n        ;module XorRingSolver\n        ;not-involutive\n        ;not-\u00ac\n        ;\u00ac-not\n        ;\u21d4\u2192\u2261\n        ;proof-irrelevance)\n  renaming\n        (T-\u2261 to \u2713-\u2261\n        ;T-\u2227 to \u2713-\u2227\n        ;T-\u2228 to \u2713-\u2228)\n\nopen import Type using (\u2605_)\n\nopen import Algebra\n  using (module CommutativeRing; module CommutativeSemiring)\n\nopen import Function.Equivalence using (module Equivalence)\nopen import Function.NP          using (id; _\u2218_; _\u27e8_\u27e9\u00b0_; Op\u2081; Op\u2082)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; refl; _\u2219_)\n\nopen import HoTT\nopen Equivalences\n\nopen Equivalence using (to; from)\n\nmodule Xor\u00b0 = CommutativeRing     commutativeRing-xor-\u2227\nmodule \ud835\udfda\u00b0   = CommutativeSemiring commutativeSemiring-\u2227-\u2228\n\n\ud835\udfda-is-set : is-set \ud835\udfda\n\ud835\udfda-is-set = dec-eq-is-set _\u225f_\n\ntwist-equiv : \ud835\udfda \u2243 \ud835\udfda\ntwist-equiv = self-inv-equiv not not-involutive\n\nmodule _ {{_ : UA}} where\n    twist : \ud835\udfda \u2261 \ud835\udfda\n    twist = ua twist-equiv\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not 0\u2082 y = not-involutive y\nxor-not-not 1\u2082 _ = refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 (x xor y) \u2261 (x xor z) \u2192 y \u2261 z\nxor-inj\u2081 0\u2082 = id\nxor-inj\u2081 1\u2082 = not-inj\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 (y xor x) \u2261 (z xor x) \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} p = xor-inj\u2081 x (Xor\u00b0.+-comm x y \u2219 p \u2219 Xor\u00b0.+-comm z x)\n\nmodule Indexed {a} {A : \u2605 a} where\n\n    infixr 6 _\u2227\u00b0_\n    infixr 5 _\u2228\u00b0_ _xor\u00b0_\n    infix  5 _==\u00b0_\n\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 \ud835\udfda)\n    not\u00b0 f = not \u2218 f\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two where\n\nopen import Data.Two.Base public\nopen import Data.Bool.Properties\n  public\n  using (isCommutativeSemiring-\u2228-\u2227\n        ;commutativeSemiring-\u2228-\u2227\n        ;module RingSolver\n        ;isCommutativeSemiring-\u2227-\u2228\n        ;commutativeSemiring-\u2227-\u2228\n        ;isBooleanAlgebra\n        ;booleanAlgebra\n        ;commutativeRing-xor-\u2227\n        ;module XorRingSolver\n        ;not-involutive\n        ;not-\u00ac\n        ;\u00ac-not\n        ;\u21d4\u2192\u2261\n        ;proof-irrelevance)\n  renaming\n        (T-\u2261 to \u2713-\u2261\n        ;T-\u2227 to \u2713-\u2227\n        ;T-\u2228 to \u2713-\u2228)\n\nopen import Type using (\u2605_)\n\nopen import Algebra\n  using (module CommutativeRing; module CommutativeSemiring)\n\nopen import Function.Equivalence using (module Equivalence)\nopen import Function.NP          using (id; _\u2218_; _\u27e8_\u27e9\u00b0_; Op\u2081; Op\u2082)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; refl; _\u2219_)\n\nopen import HoTT\nopen Equivalences\n\nopen Equivalence using (to; from)\n\nmodule Xor\u00b0 = CommutativeRing     commutativeRing-xor-\u2227\nmodule \ud835\udfda\u00b0   = CommutativeSemiring commutativeSemiring-\u2227-\u2228\n\n\ud835\udfda-is-set : is-set \ud835\udfda\n\ud835\udfda-is-set = dec-eq-is-set _\u225f_\n\ntwist-equiv : \ud835\udfda \u2243 \ud835\udfda\ntwist-equiv = self-inv-equiv not not-involutive\n\nmodule _ {{_ : UA}} where\n    twist : \ud835\udfda \u2261 \ud835\udfda\n    twist = ua twist-equiv\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not 0\u2082 y = not-involutive y\nxor-not-not 1\u2082 _ = refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 (x xor y) \u2261 (x xor z) \u2192 y \u2261 z\nxor-inj\u2081 0\u2082 = id\nxor-inj\u2081 1\u2082 = not-inj\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 (y xor x) \u2261 (z xor x) \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} p = xor-inj\u2081 x (Xor\u00b0.+-comm x y \u2219 p \u2219 Xor\u00b0.+-comm z x)\n\nmodule Indexed {a} {A : \u2605 a} where\n\n    infixr 6 _\u2227\u00b0_\n    infixr 5 _\u2228\u00b0_ _xor\u00b0_\n    infix  5 _==\u00b0_\n\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 \ud835\udfda)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 \ud835\udfda)\n    not\u00b0 f = not \u2218 f\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d775b5531c606b881e0d0121ad6e4fa8f59edea3","subject":"comments","message":"comments\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import dom-eq\n\nmodule disjointness where\n  -- if a hole name is new in a term, then the resultant context is\n  -- disjoint from any singleton context with that hole name\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  -- dual of the above: if expanding a term produces a context that's\n  -- disjoint with a singleton context, it must be that the index is a new\n  -- hole name in the original term\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  -- collect up the hole names of a term as the indices of a trivial contex\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  -- the above judgement has mode (\u2200,\u2203). this doesn't prove uniqueness; any\n  -- contex that extends the one computed here will be indistinguishable\n  -- but we'll treat this one as canonical\n  find-holes : (e : hexp) \u2192 \u03a3[ H \u2208 \u22a4 ctx ](holes e H)\n  find-holes c = \u2205 , HConst\n  find-holes (e \u00b7: x) with find-holes e\n  ... | (h , d)= h , (HAsc d)\n  find-holes (X x) = \u2205 , HVar\n  find-holes (\u00b7\u03bb x e) with find-holes e\n  ... | (h , d) = h , HLam1 d\n  find-holes (\u00b7\u03bb x [ x\u2081 ] e) with find-holes e\n  ... | (h , d) = h , HLam2 d\n  find-holes \u2987\u2988[ x ] = (\u25a0 (x , <>)) , HEHole\n  find-holes \u2987 e \u2988[ x ] with find-holes e\n  ... | (h , d) = h ,, (x , <>) , HNEHole d\n  find-holes (e1 \u2218 e2) with find-holes e1 | find-holes e2\n  ... | (h1 , d1) | (h2 , d2)  = (h1 \u222a h2 ) , (HAp d1 d2)\n\n  -- if a hole name is new then it's apart from the collection of hole\n  -- names\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  -- if the holes of two expressions are disjoint, so are their collections\n  -- of hole names\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-sing-disj (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-sing-disj (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  -- the holes of an expression have the same domain as the context\n  -- produced during expansion; that is, we don't add anything we don't\n  -- find in the term during expansion.\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) =\n                                  dom-union (##-comm (lem-apart-sing-disj (lem-apart-new h (expand-disjoint-new-synth x\u2081 x))))\n                                            (holes-delta-synth h x\u2081)\n                                            (dom-single u)\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union ((##-comm (lem-apart-sing-disj (lem-apart-new h (expand-disjoint-new-synth exp x)))))\n                                                                       (holes-delta-synth h exp)\n                                                                       (dom-single u)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) = dom-union (holes-disjoint-disjoint h h\u2081 x) (holes-delta-ana h x\u2084) (holes-delta-ana h\u2081 x\u2085)\n\n  -- this is the main result of this file:\n  --\n  -- if you expand two hole-disjoint expressions analytically, the \u0394s\n  -- produced are disjoint.\n  --\n  -- note that this is likely true for synthetic expansions in much the\n  -- same way, but we only prove half of the usual pair here because that's\n  -- all we need to establish expansion generality and expandability. the\n  -- proof technique here is explcitly *not* structurally inductive on the\n  -- expansion judgement, because that approach relies on weakening of\n  -- expansion, which is false because of the substitution contexts. giving\n  -- expansion weakning would take away unicity, so we avoid the whole\n  -- question.\n  expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n  expand-ana-disjoint {e1} {e2} hd ana1 ana2\n    with find-holes e1 | find-holes e2\n  ... | (_ , he1) | (_ , he2) = dom-eq-disj (holes-disjoint-disjoint he1 he2 hd)\n                                            (holes-delta-ana he1 ana1)\n                                            (holes-delta-ana he2 ana2)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import dom-eq\n\nmodule disjointness where\n  -- if a hole name is new in a term, then the resultant context is\n  -- disjoint from any singleton context with that hole name\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  -- dual of the above: if expanding a term produces a context that's\n  -- disjoint with a singleton context, it must be that the index is a new\n  -- hole name in the original term\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  -- collect up the hole names of a term as the indices of a trivial contex\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  -- the above judgement has mode (\u2200,\u2203). this doesn't prove uniqueness; any\n  -- contex that extends the one computed here will be indistinguishable\n  -- but we'll treat this one as canonical\n  find-holes : (e : hexp) \u2192 \u03a3[ H \u2208 \u22a4 ctx ](holes e H)\n  find-holes c = \u2205 , HConst\n  find-holes (e \u00b7: x) with find-holes e\n  ... | (h , d)= h , (HAsc d)\n  find-holes (X x) = \u2205 , HVar\n  find-holes (\u00b7\u03bb x e) with find-holes e\n  ... | (h , d) = h , HLam1 d\n  find-holes (\u00b7\u03bb x [ x\u2081 ] e) with find-holes e\n  ... | (h , d) = h , HLam2 d\n  find-holes \u2987\u2988[ x ] = (\u25a0 (x , <>)) , HEHole\n  find-holes \u2987 e \u2988[ x ] with find-holes e\n  ... | (h , d) = h ,, (x , <>) , HNEHole d\n  find-holes (e1 \u2218 e2) with find-holes e1 | find-holes e2\n  ... | (h1 , d1) | (h2 , d2)  = (h1 \u222a h2 ) , (HAp d1 d2)\n\n  -- if a hole name is new then it's apart from the collection of hole\n  -- names\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  -- if the holes of two expressions are disjoint, so are their collections\n  -- of hole names\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-sing-disj (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-sing-disj (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  -- the holes of an expression have the same domain as the context\n  -- produced during expansion; that is, we don't add anything we don't\n  -- find in the term during expansion.\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) =\n                                  dom-union (##-comm (lem-apart-sing-disj (lem-apart-new h (expand-disjoint-new-synth x\u2081 x))))\n                                            (holes-delta-synth h x\u2081)\n                                            (dom-single u)\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union ((##-comm (lem-apart-sing-disj (lem-apart-new h (expand-disjoint-new-synth exp x)))))\n                                                                       (holes-delta-synth h exp)\n                                                                       (dom-single u)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) = dom-union (holes-disjoint-disjoint h h\u2081 x) (holes-delta-ana h x\u2084) (holes-delta-ana h\u2081 x\u2085)\n\n  -- this is the main result of this file:\n  --\n  -- if you expand two hole-disjoint expressions analytically, the \u0394s\n  -- produced are disjoint.\n  --\n  -- note that this is likely true for synthetic expansions in much the\n  -- same way, but we only prove half of the usual pair here because that's\n  -- all we need to establish expansion generality and expandability. the\n  -- proof technique here is explcitly *not* structurally inductive on the\n  -- expansion judgement, because that approach relies on weakening of\n  -- expansion, which is false because of the substitution contexts. giving\n  -- expansion weakning would take away unicity, so we avoid it.\n  expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n  expand-ana-disjoint {e1} {e2} hd ana1 ana2\n    with find-holes e1 | find-holes e2\n  ... | (_ , he1) | (_ , he2) = dom-eq-disj (holes-disjoint-disjoint he1 he2 hd)\n                                            (holes-delta-ana he1 ana1)\n                                            (holes-delta-ana he2 ana2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ec75838ee84956dfcf8cf5c9c6cd26c9a3bc1396","subject":"flat-funs: fix","message":"flat-funs: fix\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n  open Composable isComposable\n  open VComposable isVComposable\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk \u22a4 Bit _\u00d7_ Vec (\u03bb A B \u2192 A \u2192 B)\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk 0 1 _+_ _*_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_ proj\u2081 proj\u2082\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id bitsFunComp bitsFunVComp _&&&_ (\u03bb {A} \u2192 take A) (\u03bb {A} \u2192 drop A)\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk \u22a4 Bit _\u00d7_ Vec (\u03bb A B \u2192 A \u2192 B)\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk 0 1 _+_ _*_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_ proj\u2081 proj\u2082\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id bitsFunComp bitsFunVComp _&&&_ (\u03bb {A} \u2192 take A) (\u03bb {A} \u2192 drop A)\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1b48f4da6589d7ea75b7933b4916af6df367596b","subject":"Update ZK\/Disjunctive","message":"Update ZK\/Disjunctive\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Disjunctive.agda","new_file":"ZK\/Disjunctive.agda","new_contents":"open import Level.NP\nopen import Type\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen import Data.Zero\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat.NP as \u2115 hiding (_\u225f_)\nopen import Data.Product renaming (proj\u2081 to fst) hiding (map)\nopen import Data.Fin.NP as Fin\nopen import Data.Vec\nopen import Data.Vec.Properties\nimport Data.List as L\nopen L using (List; []; _\u2237_)\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP\nimport ZK.SigmaProtocol\nimport ZK.ChaumPedersen\n\nmodule ZK.Disjunctive\n    (G \u2124q : \u2605)\n    (g    : G)\n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_\/_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_-_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_==_ : (x y : G) \u2192 \ud835\udfda)\n    (_==\u2124q_ : (x y : \u2124q) \u2192 \ud835\udfda)\n    (max  : \u2115)\n    (Fin\u25b9\u2124q : Fin (suc max) \u2192 \u2124q)\n    where\n\n  0q : \u2124q\n  0q = Fin\u25b9\u2124q zero\n\n  record Structure : Set where\n    field\n      \u2713-==\u2124q : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==\u2124q y)\n      \u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n      ^-+  : \u2200 {b x y} \u2192 b ^(x + y) \u2261 (b ^ x) \u00b7 (b ^ y)\n      ^-*  : \u2200 {b x y} \u2192 b ^(x * y) \u2261 (b ^ x) ^ y\n      \u00b7-\/  : \u2200 {P Q} \u2192 P \u2261 (P \u00b7 Q) \/ Q\n      \/-\u00b7  : \u2200 {P Q} \u2192 P \u2261 (P \/ Q) \u00b7 Q\n      -+   : \u2200 {x y} \u2192 (x - y) + y \u2261 x\n      +-comm     : Commutative _+_\n      +-assoc    : Associative _+_\n      0-+-identity : Identity 0q _+_\n\n  _\u203c_ : \u2200 {n} {A : \u2605} \u2192 Vec A n \u2192 Fin n \u2192 A\n  v \u203c i = lookup i v\n\n  tabulate\u2032 : \u2200 {A : \u2605} (n : \u2115) (f : \u2115 \u2192 A) \u2192 Vec A n\n  tabulate\u2032 n f = tabulate {n = n} (f \u2218 Fin\u25b9\u2115)\n\n  tabulateL : \u2200 {A : \u2605} (n : \u2115) (f : \u2115 \u2192 A) \u2192 List A\n  tabulateL n f = toList (tabulate\u2032 n f)\n\n  tubulate :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : Fin n \u2192 A)\n      (f\u2261m : A)\n    \u2192 Vec A n\n  tubulate zero    f x = x  \u2237 tabulate (f \u2218 suc)\n  tubulate (suc m) f x = fz \u2237 tubulate m (f \u2218 suc) x\n     where fz = f zero\n\n  tubulate-tabulate[]\u2254 :\n      \u2200 {n} {A : \u2605} (m : Fin n)\n        {f\u2262m : Fin n \u2192 A}\n        {f\u2261m : A}\n      \u2192 tubulate m f\u2262m f\u2261m \u2261 (tabulate f\u2262m)[ m ]\u2254 f\u2261m\n  tubulate-tabulate[]\u2254 zero    = refl\n  tubulate-tabulate[]\u2254 (suc m) = ap (_\u2237_ _) (tubulate-tabulate[]\u2254 m)\n\n  tubulate\u203cm :\n      \u2200 {n} {A : \u2605} (m : Fin n)\n        (f\u2262m : Fin n \u2192 A)\n        (f\u2261m : A)\n      \u2192 tubulate m f\u2262m f\u2261m \u203c m \u2261 f\u2261m\n  tubulate\u203cm zero    _ _ = refl\n  tubulate\u203cm (suc m) _ _ = tubulate\u203cm m _ _\n\n  tubulate\u203c\u2262m :\n      \u2200 {n} {A : \u2605} (m : Fin n)\n        (f\u2262m : Fin n \u2192 A)\n        (f\u2261m : A)\n        i (m\u2262i : m \u2262 i) \u2192\n        tubulate m f\u2262m f\u2261m \u203c i \u2261 f\u2262m i\n  tubulate\u203c\u2262m m f\u2262m f\u2261m i m\u2262i\n    rewrite tubulate-tabulate[]\u2254 m {f\u2262m} {f\u2261m}\n          | lookup\u2218update\u2032 (m\u2262i \u2218 !_) (tabulate f\u2262m) f\u2261m\n          = lookup\u2218tabulate f\u2262m i\n  {-\n  tubulate\u203c\u2262m zero       zero    i\u2262m = \ud835\udfd8-elim (i\u2262m refl)\n  tubulate\u203c\u2262m (suc m)    zero    i\u2262m = refl\n  tubulate\u203c\u2262m zero {f\u2262m} (suc i) i\u2262m = lookup\u2218tabulate f\u2262m (suc i)\n  tubulate\u203c\u2262m (suc m)    (suc i) i\u2262m = tubulate\u203c\u2262m m i (i\u2262m \u2218 ap suc)\n  -}\n\n  module _ {A : \u2605} where\n    _\u27e6_\u27e7\u2254_ : \u2200 {n} (f : Fin n \u2192 A) (i : Fin n) (z : A) \u2192 Fin n \u2192 A\n    (f \u27e6 zero  \u27e7\u2254 z) zero    = z\n    (f \u27e6 zero  \u27e7\u2254 z) (suc j) = f (suc j)\n    (f \u27e6 suc i \u27e7\u2254 z) zero    = f zero\n    (f \u27e6 suc i \u27e7\u2254 z) (suc j) = ((f \u2218 suc) \u27e6 i \u27e7\u2254 z) j\n\n    tabulate-\u27e6\u27e7\u2254 : \u2200 {n} (f : Fin n \u2192 A) (i : Fin n) (z : A) \u2192 tabulate (f \u27e6 i \u27e7\u2254 z) \u2261 tabulate f [ i ]\u2254 z\n    tabulate-\u27e6\u27e7\u2254 f zero z = refl\n    tabulate-\u27e6\u27e7\u2254 f (suc i) z = ap (_\u2237_ _) (tabulate-\u27e6\u27e7\u2254 (f \u2218 suc) i z)\n\n  {-\n  module _ {A : \u2605} where\n    [_\u2254_]_ : \u2200 {n} (m : Fin n) (z : A) (xs : Vec A n) \u2192 Vec A n\n    [ zero  \u2254 z ] (x \u2237 xs) = z \u2237 xs\n    [ suc m \u2254 z ] (x \u2237 xs) = x \u2237 [ m \u2254 z ] xs\n\n    [\u2254]-\u203c-m : \u2200 {n} (m : Fin n) (z : A) (xs : Vec A n) \u2192 [ m \u2254 z ] xs \u203c m \u2261 z\n    [\u2254]-\u203c-m m z xs = {!!}\n    [\u2254]-\u203c-\u2262m : \u2200 {n} (m : Fin n) (z : A) (xs : Vec A n) j (j\u2262m : j \u2262 m) \u2192 [ m \u2254 z ] xs \u203c j \u2261 xs \u203c j\n    -}\n\n  -- split : \u2200 {A : \u2605} {n} (m : Fin n) \u2192 \n\n  {-\n  module _ {A : \u2605}\n           (B : \u2115 \u2192 \u2605)\n           (_\u25c5\u2032_ _\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n           (\u03b5   : B zero) where\n    fuldr : \u2200 {n} (m : Fin n) (f : Fin n \u2192 A) \u2192 B n\n    fuldr zero    f = f zero \u25c5\u2032 iterate B _\u25c5_ \u03b5 (f \u2218 suc)\n    fuldr (suc m) f = f zero \u25c5 fuldr m (f \u2218 suc)\n  -}\n\n  {-\n  xs [ m ]\u2254 z \u2261 take xs ++ z \u2237 drop xs\n\n  foldr B _\u25c5_ \u03b5 (xs [ m ]\u2254 z)\n  \u2261\n  (foldr B _\u25c5_ \u03b5 (xs [ m ]\u2254 \u03b5))\n  -}\n\n  module _ {A : \u2605}\n           (_+_ : A \u2192 A \u2192 A)\n           (+-comm     : Commutative _+_)\n           (+-assoc    : Associative _+_)\n           (z z' : A)\n           where\n     +-exch-top : \u2200 {a b c} \u2192 a + (b + c) \u2261 b + (a + c)\n     +-exch-top = ! +-assoc \u2219 ap\u2082 _+_ +-comm refl \u2219 +-assoc\n     +-exch : \u2200 {a b c d e} \u2192 a + c \u2261 d + e \u2192 a + (b + c) \u2261 d + (b + e)\n     +-exch p = +-exch-top \u2219 ap (_+_ _) p \u2219 ! +-exch-top\n     foldr-exch : \u2200 {n} (xs : Vec A n) \u2192 z + foldr _ _+_ z' xs \u2261 z' + foldr _ _+_ z xs \n     foldr-exch [] = +-comm\n     foldr-exch (x \u2237 xs) = +-exch (foldr-exch xs)\n     foldr-[]\u2254-exch : \u2200 {n} (xs : Vec A n) (m : Fin n)\n                         \u2192 foldr _ _+_ z' (xs [ m ]\u2254 z)\n                         \u2261  foldr _ _+_ z  (xs [ m ]\u2254 z')\n     foldr-[]\u2254-exch (x \u2237 xs) zero    = foldr-exch xs\n     foldr-[]\u2254-exch (x \u2237 xs) (suc m) = ap (_+_ x) (foldr-[]\u2254-exch xs m)\n\n     foldr-[]\u2254-exch\u2032 : \u2200 {n \u03b5} (xs : Vec A n) (m : Fin n)\n                         \u2192 z' + foldr _ _+_ \u03b5 (xs [ m ]\u2254 z)\n                         \u2261  z  + foldr _ _+_ \u03b5 (xs [ m ]\u2254 z')\n     foldr-[]\u2254-exch\u2032 (x \u2237 xs) zero    = +-exch-top\n     foldr-[]\u2254-exch\u2032 (x \u2237 xs) (suc m) = +-exch (foldr-[]\u2254-exch\u2032 xs m)\n\n     -- foldr _ _+_ z \u2261 foldr _ _+_ \u03b5 + z\n\n\n  {-\n  fuldr-as-tubulate :\n    \u2200 {A : \u2605} {n} (m : Fin n) \u2192\n    fuldr (Vec A) ? _\u2237_ [] m xs\n  -}\n --   x \u25c5\u2032 (y \u25c5 z) \u2261 y \u25c5\u2032 \n\n {-\n  module _ {A : \u2605}\n           (B : \u2115 \u2192 \u2605)\n           (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n           (\u03b5   : B zero)\n           where\n\n    foldr-tubulate : \u2200 {n} (m : Fin n)\n                       {f\u2262m : Fin n \u2192 A}\n                       {f\u2261m : A}\n                  \u2192 foldr B _\u25c5_ \u03b5 (tubulate m f\u2262m f\u2261m) \u2261 foldr B (\u03bb _ b \u2192 f\u2261m \u25c5 b) _\u25c5_ \u03b5 m f\u2262m\n    foldr-tubulate zero    = ap (_\u25c5_ _) (! iterate-foldr\u2218tabulate B _\u25c5_ \u03b5 _)\n    foldr-tubulate (suc m) = ap (_\u25c5_ _) (foldr-tubulate m) \n\n    foldr-tubulate : \u2200 {n} (m : Fin n)\n                       {f\u2262m : Fin n \u2192 A}\n                       {f\u2261m : A}\n                  \u2192 foldr B _\u25c5_ \u03b5 (tubulate m f\u2262m f\u2261m) \u2261 fuldr B (\u03bb _ b \u2192 f\u2261m \u25c5 b) _\u25c5_ \u03b5 m f\u2262m\n    foldr-tubulate zero    = ap (_\u25c5_ _) (! iterate-foldr\u2218tabulate B _\u25c5_ \u03b5 _)\n    foldr-tubulate (suc m) = ap (_\u25c5_ _) (foldr-tubulate m) \n-}\n{-\n  module _ {A : \u2605}\n           (B : \u2115 \u2192 \u2605)\n           (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n           (\u03b5   : B zero) where\n    tribulate :\n        \u2200 {n} (m : Fin n)\n          (f\u2262m : Fin n \u2192 A) -- (f\u2262m : (i : Fin n) \u2192 i \u2262 m \u2192 A)\n          (f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A)\n        \u2192 Vec A n\n    tribulate zero f x = x \u03b5 (foldr B _\u25c5_ \u03b5 tl) \u2237 tl\n         where tl = tabulate (f \u2218 suc)\n    tribulate (suc m) f x = fz \u2237 tribulate m (f \u2218 suc) (x \u2218 _\u25c5_ fz)\n         where fz = f zero\n\n    module _\n        (P : A \u2192 \u2605)\n        (Q : \u2200 {n} \u2192 B n \u2192 \u2605)\n        (Q[_\u25c5_] : \u2200 {n x} {y : B n} \u2192 P x \u2192 Q y \u2192 Q (x \u25c5 y))\n        (Q\u03b5 : Q \u03b5) where\n        tabulate-spec :\n          \u2200 {n} {f : Fin n \u2192 A}\n            (Pf : \u2200 i \u2192 P (f i))\n          \u2192 Q (foldr B _\u25c5_ \u03b5 (tabulate f))\n        tabulate-spec {zero}  Pf = Q\u03b5\n        tabulate-spec {suc n} Pf = Q[ Pf zero \u25c5 tabulate-spec (Pf \u2218 suc) ]\n\n        tribulate-spec :\n          \u2200 {n} (m : Fin n)\n            {f\u2262m : Fin n \u2192 A}\n            {f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A}\n            (P\u2262m : \u2200 i \u2192 i \u2262 m \u2192 P (f\u2262m i))\n            (P\u2261m : \u2200 {pr po} \u2192 Q pr \u2192 Q po \u2192 P (f\u2261m pr po))\n          \u2192 \u2200 i \u2192 P (tribulate m f\u2262m f\u2261m \u203c i)\n        tribulate-spec zero       P\u2262m P\u2261m zero\n          = P\u2261m Q\u03b5 (tabulate-spec (\u03bb i \u2192 P\u2262m (suc i) (\u03bb())))\n        tribulate-spec (suc m)    P\u2262m P\u2261m zero\n          = P\u2262m zero (\u03bb())\n        tribulate-spec zero {f\u2262m} P\u2262m P\u2261m (suc i)\n          rewrite lookup\u2218tabulate (f\u2262m \u2218 suc) i\n          = P\u2262m (suc i) (\u03bb())\n        tribulate-spec (suc m) P\u2262m P\u2261m (suc i)\n          = tribulate-spec m (\u03bb j j\u2262m \u2192 P\u2262m (suc j) (j\u2262m \u2218 Fin.suc-injective))\n                             (\u03bb Qpr Qpo \u2192 P\u2261m Q[ P\u2262m zero (\u03bb()) \u25c5 Qpr ] Qpo) i\n\n        tribulate\u2262m :\n          \u2200 {n} (m : Fin n)\n            {f\u2262m : Fin n \u2192 A}\n            {f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A}\n            i (i\u2262m : i \u2262 m) \u2192\n            tribulate m f\u2262m f\u2261m \u203c i \u2261 f\u2262m i\n        tribulate\u2262m = {!!}\n\n    tribulate-naive :\n      \u2200 {n} (m : Fin n)\n        (f\u2262m : \u2115 \u2192 A)\n        (f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A)\n      \u2192 Vec A (Fin\u25b9\u2115 m \u2115.+ suc (n \u2115-\u2115 suc m))\n    tribulate-naive {n = n} m f\u2262m f\u2261m\n       = prefix ++ f\u2261m fprefix fpostfix \u2237 postfix\n       where\n         prefix   = tabulate\u2032 (Fin\u25b9\u2115 m)      f\u2262m\n         postfix  = tabulate\u2032 (n \u2115-\u2115 suc m) (f\u2262m \u2218 \u2115._+_ (Fin\u25b9\u2115 (suc m)))\n         fprefix  = foldr B _\u25c5_ \u03b5 prefix\n         fpostfix = foldr B _\u25c5_ \u03b5 postfix\n\n    tribulate-correct :\n      \u2200 {n} (m : Fin n)\n        (f\u2262m : \u2115 \u2192 A)\n        (f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A)\n      \u2192 toList (tribulate m (f\u2262m \u2218 Fin\u25b9\u2115) f\u2261m)\n       \u2261 toList (tribulate-naive m f\u2262m f\u2261m)\n    tribulate-correct zero    f\u2262m f\u2261m = refl\n    tribulate-correct (suc m) f\u2262m f\u2261m =\n      ap (_\u2237_ (f\u2262m 0)) (tribulate-correct m (f\u2262m \u2218 suc) (f\u2261m \u2218 _\u25c5_ (f\u2262m 0)))\n\n  test-tribulate1 :\n    tribulate (Vec \u2115) _\u2237_ [] (# 4) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 [])\n  test-tribulate1 = refl\n\n  test-tribulate2 :\n    tribulate _ L._++_ L.[]\n              (# 4) (L.[_] \u2218 Fin\u25b9\u2115)\n              (\u03bb xs ys \u2192 xs L.++ 42 \u2237 ys)\n      \u2261\n    (0 \u2237 []) \u2237 (1 \u2237 []) \u2237 (2 \u2237 []) \u2237 (3 \u2237 []) \u2237 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 []) \u2237 (5 \u2237 []) \u2237 []\n  test-tribulate2 = refl\n\n  test-tribulate3 :\n    tribulate (Vec \u2115) _\u2237_ [] (# 3) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 42 \u2237 [])\n  test-tribulate3 = refl\n\n  test-tribulate4 :\n    tribulate (const \u2115) \u2115._+_ 0 (# 3) Fin\u25b9\u2115 (\u03bb x y \u2192 (x \u2115.\u2238 1) \u2115.+ 10 \u2115.* y) \u2261 (0 \u2237 1 \u2237 2 \u2237 42 \u2237 4 \u2237 [])\n  test-tribulate4 = refl\n\n     {-\n  tribulate :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : Fin n \u2192 A)\n      (f\u2261m : Vec A (Fin\u25b9\u2115 m) \u2192 Vec A (n \u2115-\u2115 suc m) \u2192 A)\n    \u2192 Vec A n\n  tribulate zero f x = x [] tl \u2237 tl\n     where tl = tabulate (f \u2218 suc)\n  tribulate (suc m) f x = fz \u2237 tribulate m (f \u2218 suc) (x \u2218 _\u2237_ fz)\n     where fz = f zero\n\n  test-tribulate1 :\n    tribulate (# 4) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 [])\n  test-tribulate1 = refl\n\n  test-tribulate2 :\n    tribulate (# 4) (L.[_] \u2218 Fin\u25b9\u2115)\n              (\u03bb xs ys \u2192 L.concat (toList (xs ++ L.[ 42 ] \u2237 ys)))\n      \u2261\n    (0 \u2237 []) \u2237 (1 \u2237 []) \u2237 (2 \u2237 []) \u2237 (3 \u2237 []) \u2237 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 []) \u2237 (5 \u2237 []) \u2237 []\n  test-tribulate2 = refl\n\n  test-tribulate3 :\n    tribulate (# 3) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 42 \u2237 [])\n  test-tribulate3 = refl\n\n  tribulate-naive :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : \u2115 \u2192 A)\n      (f\u2261m : List A \u2192 List A \u2192 A)\n    \u2192 List A\n  tribulate-naive {n = n} m f\u2262m f\u2261m\n     = prefix L.++ f\u2261m prefix postfix \u2237 postfix\n     where\n       prefix  = tabulateL (Fin\u25b9\u2115 m)      f\u2262m\n       postfix = tabulateL (n \u2115-\u2115 suc m) (f\u2262m \u2218 \u2115._+_ (Fin\u25b9\u2115 (suc m)))\n\n  tribulate-correct :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : \u2115 \u2192 A)\n      (f\u2261m : List A \u2192 List A \u2192 A)\n    \u2192 toList (tribulate m (f\u2262m \u2218 Fin\u25b9\u2115) (\u03bb xs ys \u2192 f\u2261m (toList xs) (toList ys)))\n     \u2261 tribulate-naive m f\u2262m f\u2261m\n  tribulate-correct zero    f\u2262m f\u2261m = refl\n  tribulate-correct (suc m) f\u2262m f\u2261m =\n    ap (_\u2237_ (f\u2262m 0)) (tribulate-correct m (f\u2262m \u2218 suc) (f\u2261m \u2218 _\u2237_ (f\u2262m 0)))\n    -}\n-}\n  and : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 \ud835\udfda\n  and = foldr _ _\u2227_ 1\u2082\n\n  \u2713-and : \u2200 {n} {v : Vec \ud835\udfda n} (\u2713p : \u2200 i \u2192 \u2713(v \u203c i)) \u2192 \u2713(and v)\n  \u2713-and {v = []}     \u2713p = _\n  \u2713-and {v = x \u2237 xs} \u2713p = \u2713\u2227 (\u2713p zero) (\u2713-and (\u2713p \u2218 suc))\n\n  module _ {A B C D : \u2605} {n}\n           (f : Fin n \u2192 A \u2192 B \u2192 C \u2192 D)\n           (va : Vec A n) (vb : Vec B n) (vc : Vec C n) where\n    mapi3 : Vec D n\n    mapi3 = replicate f \u229b allFin _ \u229b va \u229b vb \u229b vc\n\n  open import Category.Applicative.Indexed -- .Morphism\n  module _ {A B C D : \u2605} {n}\n           {f : Fin n \u2192 A \u2192 B \u2192 C \u2192 D}\n           {va : Vec A n} {vb : Vec B n} {vc : Vec C n}\n           i where\n    -- Surely this could be done better...\n    open Morphism (lookup-morphism {\u2080} {n} i)\n    mapi3\u203c : mapi3 f va vb vc \u203c i \u2261 f i (va \u203c i) (vb \u203c i) (vc \u203c i)\n    mapi3\u203c = op-\u229b (replicate f \u229b allFin _ \u229b va \u229b vb) vc\n           \u2219 ap (\u03bb g \u2192 g (vc \u203c i))\n                (op-\u229b (replicate f \u229b allFin _ \u229b va) vb)\n           \u2219 ap (\u03bb g \u2192 g (vb \u203c i) (vc \u203c i)) (op-\u229b (replicate f \u229b allFin _) va)\n           \u2219 ap (\u03bb g \u2192 g (va \u203c i) (vb \u203c i) (vc \u203c i)) (op-\u229b (replicate f) (allFin _))\n           \u2219 ap (\u03bb g \u2192 g (allFin _ \u203c i) (va \u203c i) (vb \u203c i) (vc \u203c i)) (op-pure f)\n           \u2219 ap (\u03bb x \u2192 f x (va \u203c i) (vb \u203c i) (vc \u203c i)) (lookup\u2218tabulate id i)\n\n  module _ {A B C : \u2605} {n}\n           (p : Fin n \u2192 A \u2192 B \u2192 C \u2192 \ud835\udfda)\n           (va : Vec A n) (vb : Vec B n) (vc : Vec C n) where\n    all3 : \ud835\udfda\n    all3 = and (mapi3 p va vb vc)\n\n  module _ {A B C : \u2605} where\n    \u2713-all3 : \u2200 {n}\n           {p  : Fin n \u2192 A \u2192 B \u2192 C \u2192 \ud835\udfda}\n           {va : Vec A n} {vb : Vec B n} {vc : Vec C n}\n           (\u2713p : \u2200 i \u2192 \u2713(p i (va \u203c i) (vb \u203c i) (vc \u203c i)))\n           \u2192 \u2713(all3 p va vb vc)\n    \u2713-all3 {p = p} {va} {vb} {vc} \u2713p = \u2713-and {v = mapi3 p va vb vc} (\u03bb i \u2192 tr \u2713 (! mapi3\u203c i) (\u2713p i))\n\n  sum\u2124q : \u2200 {n} \u2192 Vec \u2124q n \u2192 \u2124q\n  sum\u2124q = foldr _ _+_ 0q\n\n  sum\u2124q-fun : \u2200 {n} (f : Fin n \u2192 \u2124q) \u2192 \u2124q\n  sum\u2124q-fun = iterate _ _+_ 0q\n\n  module CP = ZK.ChaumPedersen {G} {\u2124q} g _^_ _\u00b7_ _\/_ _+_ _*_ _==_\n  open CP.ElGamal-encryption using (PubKey; CipherText; EncRnd; enc)\n\n  VecCommitments = Vec CP.Commitment (suc max)\n  VecChallenges  = Vec CP.Challenge  (suc max)\n  VecResponses   = Vec CP.Response   (suc max)\n\n  module Interactive (y : PubKey) where\n    Response   = VecChallenges \u00d7 VecResponses\n\n    open ZK.SigmaProtocol VecCommitments \u2124q Response public\n\n    private\n      M = \u03bb i \u2192 g ^ Fin\u25b9\u2124q i\n\n    module _ (ct : CipherText) where\n      verifier1 = \u03bb i A c s \u2192 CP.verifier y (M i) ct (CP.mk A c s)\n\n      -- H-commitments is the actual challenge for this \u03a3-protocol\n      verifier : Verifier\n      verifier (mk commitments H-commitments (challenges , responses))\n          = all3 verifier1 commitments challenges responses\n          \u2227 sum\u2124q challenges ==\u2124q H-commitments\n\n    module Prover-and-correctness\n             (i : Fin (suc max))\n             (rnd-challenge  : Fin (suc max) \u2192 CP.Challenge) -- the i\u1d57\u02b0 slot is not used\n             (rnd-response   : Fin (suc max) \u2192 CP.Response)  -- the i\u1d57\u02b0 slot is not used\n             (r : EncRnd) (w : \u2124q)\n             where\n\n      ct = enc y r (M i)\n\n      commitment-i = CP.prover-commitment y r w\n\n      simulated-commitment-j : (j : Fin (suc max)) \u2192 CP.Commitment\n      simulated-commitment-j j\n        = CP.simulate-commitment y (M j) ct (rnd-challenge j) (rnd-response j)\n\n      commitments = tubulate i simulated-commitment-j commitment-i\n\n      sum\u2262i = sum\u2124q-fun (rnd-challenge \u27e6 i \u27e7\u2254 0q)\n\n      prover-response : \u2124q \u2192 Response\n      prover-response Hcommitments = (challenges , responses)\n        module Prover-response where\n          challenge-i = Hcommitments - sum\u2262i\n\n          challenges = tubulate i rnd-challenge challenge-i\n\n          response-i = CP.prover-response y r w (challenges \u203c i)\n\n          responses = tubulate i rnd-response response-i\n\n      prover : Prover\n      prover = commitments , prover-response\n\n      module Correctness-proof (structure : Structure) where\n        open Structure structure\n        CPcs = \u03bb M c s \u2192 CP.Correct-simulation.correct-simulation y M ct c s \u2713-== \/-\u00b7\n        CPcp = CP.Correctness-proof.correctness \u2713-== ^-+ ^-* \u00b7-\/ y r w\n        correctness : Correct (prover , verifier ct)\n        correctness Hcommitments = \u2713\u2227 (\u2713-all3 {p = verifier1 ct} helper) helper'\n          where\n            open Prover-response Hcommitments\n            open \u2261-Reasoning\n            c-i-e : challenges \u203c i \u2261 challenge-i\n            c-i-e = tubulate\u203cm i _ _\n            W-i-e : commitments \u203c i \u2261 commitment-i\n            W-i-e = tubulate\u203cm i _ _\n            s-i-e : responses \u203c i \u2261 response-i\n            s-i-e = tubulate\u203cm i _ _\n            s-j-e : \u2200 j \u2192 i \u2262 j \u2192 responses \u203c j \u2261 rnd-response j\n            s-j-e = tubulate\u203c\u2262m i _ _\n            W-j-e : \u2200 j \u2192 i \u2262 j \u2192 commitments \u203c j \u2261 simulated-commitment-j j\n            W-j-e = tubulate\u203c\u2262m i _ _\n            c-j-e : \u2200 j \u2192 i \u2262 j \u2192 challenges \u203c j \u2261 rnd-challenge j\n            c-j-e = tubulate\u203c\u2262m i _ _\n            helper : \u2200 j \u2192 \u2713(verifier1 ct j (commitments \u203c j) (challenges \u203c j) (responses \u203c j))\n            helper j with Fin._\u225f_ i j\n            helper .i | yes refl\n              rewrite s-i-e | W-i-e | tubulate\u203cm i rnd-challenge challenge-i\n                    = CPcp (M i) challenge-i\n            helper  j | no \u00acp rewrite s-j-e j \u00acp | W-j-e j \u00acp | c-j-e j \u00acp = CPcs _ _ _\n            sum\u2262i-def : sum\u2262i \u2261 sum\u2124q (tabulate rnd-challenge [ i ]\u2254 0q)\n            sum\u2262i-def = iterate-foldr\u2218tabulate _ _+_ 0q (rnd-challenge \u27e6 i \u27e7\u2254 0q)\n                      \u2219 ap sum\u2124q (tabulate-\u27e6\u27e7\u2254 rnd-challenge i 0q)\n            helper' : \u2713(sum\u2124q challenges ==\u2124q Hcommitments)\n            helper' = \u2713-==\u2124q (sum\u2124q challenges\n                             \u2261\u27e8 ap sum\u2124q (tubulate-tabulate[]\u2254 i) \u27e9\n                              sum\u2124q (tabulate rnd-challenge [ i ]\u2254 challenge-i)\n                             \u2261\u27e8 ! fst 0-+-identity \u27e9\n                              0q + sum\u2124q (tabulate rnd-challenge [ i ]\u2254 challenge-i)\n                             \u2261\u27e8 foldr-[]\u2254-exch\u2032 _+_ +-comm +-assoc challenge-i 0q (tabulate rnd-challenge) i \u27e9\n                              challenge-i + sum\u2124q (tabulate rnd-challenge [ i ]\u2254 0q)\n                             \u2261\u27e8  ap (_+_ challenge-i) (! sum\u2262i-def)  \u27e9\n                              challenge-i + sum\u2262i\n                             \u2261\u27e8  -+  \u27e9\n                              Hcommitments\n                             \u220e)\n\n    -- Like Prover-and-correctness but avoids wasting randomness for r and w\n    -- This is maybe not a good idea as this is rather brittle\n    module Prover-and-correctness\u2032\n             (i : Fin (suc max))\n             (rnd-challenge  : Fin (suc max) \u2192 CP.Challenge)\n             (rnd-response   : Fin (suc max) \u2192 CP.Response)\n      = Prover-and-correctness i rnd-challenge rnd-response (rnd-challenge i) (rnd-response i)\n\n  module Non-interactive\n    (y : PubKey)\n    (H : VecCommitments \u2192 \u2124q)\n    where\n    module I = Interactive y\n\n    open ZK.SigmaProtocol VecCommitments VecChallenges VecResponses public\n      using (Transcript; mk; Verifier)\n\n    module _ (ct : CipherText) where\n        verifier : Verifier\n        verifier (mk commitments challenges responses)\n          = I.verifier ct (I.mk commitments (H commitments) (challenges , responses))\n\n    module Prover-and-correctness\n             (i : Fin (suc max))\n             (rnd-challenge  : Fin (suc max) \u2192 CP.Challenge) -- the i\u1d57\u02b0 slot is not used\n             (rnd-response   : Fin (suc max) \u2192 CP.Response)  -- the i\u1d57\u02b0 slot is not used\n             (r : EncRnd) (w : \u2124q)\n             where\n      module IPC = I.Prover-and-correctness i rnd-challenge rnd-response r w\n      open IPC using (commitments; ct)\n      open IPC.Prover-response (H commitments)\n      transcript : Transcript\n      transcript = mk commitments challenges responses\n      module Correctness-proof (structure : Structure) where\n        correctness : \u2713 (verifier ct transcript)\n        correctness = IPC.Correctness-proof.correctness structure (H commitments)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Level.NP\nopen import Type\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen import Data.Zero\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat.NP as \u2115 hiding (_\u225f_)\nopen import Data.Product renaming (proj\u2081 to fst) hiding (map)\nopen import Data.Fin.NP as Fin\nopen import Data.Vec\nopen import Data.Vec.Properties\nimport Data.List as L\nopen L using (List; []; _\u2237_)\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP\nimport ZK.Sigma-Protocol\nimport ZK.Chaum-Pedersen\n\nmodule ZK.Disjunctive\n    (G \u2124q : \u2605)\n    (g    : G)\n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_\/_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_-_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_==_ : (x y : G) \u2192 \ud835\udfda)\n    (_==\u2124q_ : (x y : \u2124q) \u2192 \ud835\udfda)\n    (max  : \u2115)\n    (Fin\u25b9\u2124q : Fin (suc max) \u2192 \u2124q)\n    where\n\n  0q : \u2124q\n  0q = Fin\u25b9\u2124q zero\n\n  record Structure : Set where\n    field\n      \u2713-==\u2124q : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==\u2124q y)\n      \u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n      ^-+  : \u2200 {b x y} \u2192 b ^(x + y) \u2261 (b ^ x) \u00b7 (b ^ y)\n      ^-*  : \u2200 {b x y} \u2192 b ^(x * y) \u2261 (b ^ x) ^ y\n      \u00b7-\/  : \u2200 {P Q} \u2192 P \u2261 (P \u00b7 Q) \/ Q\n      \/-\u00b7  : \u2200 {P Q} \u2192 P \u2261 (P \/ Q) \u00b7 Q\n      -+   : \u2200 {x y} \u2192 (x - y) + y \u2261 x\n      +-comm     : Commutative _+_\n      +-assoc    : Associative _+_\n      0-+-identity : Identity 0q _+_\n\n  _\u203c_ : \u2200 {n} {A : \u2605} \u2192 Vec A n \u2192 Fin n \u2192 A\n  v \u203c i = lookup i v\n\n  tabulate\u2032 : \u2200 {A : \u2605} (n : \u2115) (f : \u2115 \u2192 A) \u2192 Vec A n\n  tabulate\u2032 n f = tabulate {n = n} (f \u2218 Fin\u25b9\u2115)\n\n  tabulateL : \u2200 {A : \u2605} (n : \u2115) (f : \u2115 \u2192 A) \u2192 List A\n  tabulateL n f = toList (tabulate\u2032 n f)\n\n  tubulate :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : Fin n \u2192 A)\n      (f\u2261m : A)\n    \u2192 Vec A n\n  tubulate zero    f x = x  \u2237 tabulate (f \u2218 suc)\n  tubulate (suc m) f x = fz \u2237 tubulate m (f \u2218 suc) x\n     where fz = f zero\n\n  tubulate-tabulate[]\u2254 :\n      \u2200 {n} {A : \u2605} (m : Fin n)\n        {f\u2262m : Fin n \u2192 A}\n        {f\u2261m : A}\n      \u2192 tubulate m f\u2262m f\u2261m \u2261 (tabulate f\u2262m)[ m ]\u2254 f\u2261m\n  tubulate-tabulate[]\u2254 zero    = refl\n  tubulate-tabulate[]\u2254 (suc m) = ap (_\u2237_ _) (tubulate-tabulate[]\u2254 m)\n\n  tubulate\u203cm :\n      \u2200 {n} {A : \u2605} (m : Fin n)\n        (f\u2262m : Fin n \u2192 A)\n        (f\u2261m : A)\n      \u2192 tubulate m f\u2262m f\u2261m \u203c m \u2261 f\u2261m\n  tubulate\u203cm zero    _ _ = refl\n  tubulate\u203cm (suc m) _ _ = tubulate\u203cm m _ _\n\n  tubulate\u203c\u2262m :\n      \u2200 {n} {A : \u2605} (m : Fin n)\n        (f\u2262m : Fin n \u2192 A)\n        (f\u2261m : A)\n        i (m\u2262i : m \u2262 i) \u2192\n        tubulate m f\u2262m f\u2261m \u203c i \u2261 f\u2262m i\n  tubulate\u203c\u2262m m f\u2262m f\u2261m i m\u2262i\n    rewrite tubulate-tabulate[]\u2254 m {f\u2262m} {f\u2261m}\n          | lookup\u2218update\u2032 (m\u2262i \u2218 !_) (tabulate f\u2262m) f\u2261m\n          = lookup\u2218tabulate f\u2262m i\n  {-\n  tubulate\u203c\u2262m zero       zero    i\u2262m = \ud835\udfd8-elim (i\u2262m refl)\n  tubulate\u203c\u2262m (suc m)    zero    i\u2262m = refl\n  tubulate\u203c\u2262m zero {f\u2262m} (suc i) i\u2262m = lookup\u2218tabulate f\u2262m (suc i)\n  tubulate\u203c\u2262m (suc m)    (suc i) i\u2262m = tubulate\u203c\u2262m m i (i\u2262m \u2218 ap suc)\n  -}\n\n  module _ {A : \u2605} where\n    _\u27e6_\u27e7\u2254_ : \u2200 {n} (f : Fin n \u2192 A) (i : Fin n) (z : A) \u2192 Fin n \u2192 A\n    (f \u27e6 zero  \u27e7\u2254 z) zero    = z\n    (f \u27e6 zero  \u27e7\u2254 z) (suc j) = f (suc j)\n    (f \u27e6 suc i \u27e7\u2254 z) zero    = f zero\n    (f \u27e6 suc i \u27e7\u2254 z) (suc j) = ((f \u2218 suc) \u27e6 i \u27e7\u2254 z) j\n\n    tabulate-\u27e6\u27e7\u2254 : \u2200 {n} (f : Fin n \u2192 A) (i : Fin n) (z : A) \u2192 tabulate (f \u27e6 i \u27e7\u2254 z) \u2261 tabulate f [ i ]\u2254 z\n    tabulate-\u27e6\u27e7\u2254 f zero z = refl\n    tabulate-\u27e6\u27e7\u2254 f (suc i) z = ap (_\u2237_ _) (tabulate-\u27e6\u27e7\u2254 (f \u2218 suc) i z)\n\n  {-\n  module _ {A : \u2605} where\n    [_\u2254_]_ : \u2200 {n} (m : Fin n) (z : A) (xs : Vec A n) \u2192 Vec A n\n    [ zero  \u2254 z ] (x \u2237 xs) = z \u2237 xs\n    [ suc m \u2254 z ] (x \u2237 xs) = x \u2237 [ m \u2254 z ] xs\n\n    [\u2254]-\u203c-m : \u2200 {n} (m : Fin n) (z : A) (xs : Vec A n) \u2192 [ m \u2254 z ] xs \u203c m \u2261 z\n    [\u2254]-\u203c-m m z xs = {!!}\n    [\u2254]-\u203c-\u2262m : \u2200 {n} (m : Fin n) (z : A) (xs : Vec A n) j (j\u2262m : j \u2262 m) \u2192 [ m \u2254 z ] xs \u203c j \u2261 xs \u203c j\n    -}\n\n  -- split : \u2200 {A : \u2605} {n} (m : Fin n) \u2192 \n\n  {-\n  module _ {A : \u2605}\n           (B : \u2115 \u2192 \u2605)\n           (_\u25c5\u2032_ _\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n           (\u03b5   : B zero) where\n    fuldr : \u2200 {n} (m : Fin n) (f : Fin n \u2192 A) \u2192 B n\n    fuldr zero    f = f zero \u25c5\u2032 iterate B _\u25c5_ \u03b5 (f \u2218 suc)\n    fuldr (suc m) f = f zero \u25c5 fuldr m (f \u2218 suc)\n  -}\n\n  {-\n  xs [ m ]\u2254 z \u2261 take xs ++ z \u2237 drop xs\n\n  foldr B _\u25c5_ \u03b5 (xs [ m ]\u2254 z)\n  \u2261\n  (foldr B _\u25c5_ \u03b5 (xs [ m ]\u2254 \u03b5))\n  -}\n\n  module _ {A : \u2605}\n           (_+_ : A \u2192 A \u2192 A)\n           (+-comm     : Commutative _+_)\n           (+-assoc    : Associative _+_)\n           (z z' : A)\n           where\n     +-exch-top : \u2200 {a b c} \u2192 a + (b + c) \u2261 b + (a + c)\n     +-exch-top = ! +-assoc \u2219 ap\u2082 _+_ +-comm refl \u2219 +-assoc\n     +-exch : \u2200 {a b c d e} \u2192 a + c \u2261 d + e \u2192 a + (b + c) \u2261 d + (b + e)\n     +-exch p = +-exch-top \u2219 ap (_+_ _) p \u2219 ! +-exch-top\n     foldr-exch : \u2200 {n} (xs : Vec A n) \u2192 z + foldr _ _+_ z' xs \u2261 z' + foldr _ _+_ z xs \n     foldr-exch [] = +-comm\n     foldr-exch (x \u2237 xs) = +-exch (foldr-exch xs)\n     foldr-[]\u2254-exch : \u2200 {n} (xs : Vec A n) (m : Fin n)\n                         \u2192 foldr _ _+_ z' (xs [ m ]\u2254 z)\n                         \u2261  foldr _ _+_ z  (xs [ m ]\u2254 z')\n     foldr-[]\u2254-exch (x \u2237 xs) zero    = foldr-exch xs\n     foldr-[]\u2254-exch (x \u2237 xs) (suc m) = ap (_+_ x) (foldr-[]\u2254-exch xs m)\n\n     foldr-[]\u2254-exch\u2032 : \u2200 {n \u03b5} (xs : Vec A n) (m : Fin n)\n                         \u2192 z' + foldr _ _+_ \u03b5 (xs [ m ]\u2254 z)\n                         \u2261  z  + foldr _ _+_ \u03b5 (xs [ m ]\u2254 z')\n     foldr-[]\u2254-exch\u2032 (x \u2237 xs) zero    = +-exch-top\n     foldr-[]\u2254-exch\u2032 (x \u2237 xs) (suc m) = +-exch (foldr-[]\u2254-exch\u2032 xs m)\n\n     -- foldr _ _+_ z \u2261 foldr _ _+_ \u03b5 + z\n\n\n  {-\n  fuldr-as-tubulate :\n    \u2200 {A : \u2605} {n} (m : Fin n) \u2192\n    fuldr (Vec A) ? _\u2237_ [] m xs\n  -}\n --   x \u25c5\u2032 (y \u25c5 z) \u2261 y \u25c5\u2032 \n\n {-\n  module _ {A : \u2605}\n           (B : \u2115 \u2192 \u2605)\n           (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n           (\u03b5   : B zero)\n           where\n\n    foldr-tubulate : \u2200 {n} (m : Fin n)\n                       {f\u2262m : Fin n \u2192 A}\n                       {f\u2261m : A}\n                  \u2192 foldr B _\u25c5_ \u03b5 (tubulate m f\u2262m f\u2261m) \u2261 foldr B (\u03bb _ b \u2192 f\u2261m \u25c5 b) _\u25c5_ \u03b5 m f\u2262m\n    foldr-tubulate zero    = ap (_\u25c5_ _) (! iterate-foldr\u2218tabulate B _\u25c5_ \u03b5 _)\n    foldr-tubulate (suc m) = ap (_\u25c5_ _) (foldr-tubulate m) \n\n    foldr-tubulate : \u2200 {n} (m : Fin n)\n                       {f\u2262m : Fin n \u2192 A}\n                       {f\u2261m : A}\n                  \u2192 foldr B _\u25c5_ \u03b5 (tubulate m f\u2262m f\u2261m) \u2261 fuldr B (\u03bb _ b \u2192 f\u2261m \u25c5 b) _\u25c5_ \u03b5 m f\u2262m\n    foldr-tubulate zero    = ap (_\u25c5_ _) (! iterate-foldr\u2218tabulate B _\u25c5_ \u03b5 _)\n    foldr-tubulate (suc m) = ap (_\u25c5_ _) (foldr-tubulate m) \n-}\n{-\n  module _ {A : \u2605}\n           (B : \u2115 \u2192 \u2605)\n           (_\u25c5_ : \u2200 {n} \u2192 A \u2192 B n \u2192 B (suc n))\n           (\u03b5   : B zero) where\n    tribulate :\n        \u2200 {n} (m : Fin n)\n          (f\u2262m : Fin n \u2192 A) -- (f\u2262m : (i : Fin n) \u2192 i \u2262 m \u2192 A)\n          (f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A)\n        \u2192 Vec A n\n    tribulate zero f x = x \u03b5 (foldr B _\u25c5_ \u03b5 tl) \u2237 tl\n         where tl = tabulate (f \u2218 suc)\n    tribulate (suc m) f x = fz \u2237 tribulate m (f \u2218 suc) (x \u2218 _\u25c5_ fz)\n         where fz = f zero\n\n    module _\n        (P : A \u2192 \u2605)\n        (Q : \u2200 {n} \u2192 B n \u2192 \u2605)\n        (Q[_\u25c5_] : \u2200 {n x} {y : B n} \u2192 P x \u2192 Q y \u2192 Q (x \u25c5 y))\n        (Q\u03b5 : Q \u03b5) where\n        tabulate-spec :\n          \u2200 {n} {f : Fin n \u2192 A}\n            (Pf : \u2200 i \u2192 P (f i))\n          \u2192 Q (foldr B _\u25c5_ \u03b5 (tabulate f))\n        tabulate-spec {zero}  Pf = Q\u03b5\n        tabulate-spec {suc n} Pf = Q[ Pf zero \u25c5 tabulate-spec (Pf \u2218 suc) ]\n\n        tribulate-spec :\n          \u2200 {n} (m : Fin n)\n            {f\u2262m : Fin n \u2192 A}\n            {f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A}\n            (P\u2262m : \u2200 i \u2192 i \u2262 m \u2192 P (f\u2262m i))\n            (P\u2261m : \u2200 {pr po} \u2192 Q pr \u2192 Q po \u2192 P (f\u2261m pr po))\n          \u2192 \u2200 i \u2192 P (tribulate m f\u2262m f\u2261m \u203c i)\n        tribulate-spec zero       P\u2262m P\u2261m zero\n          = P\u2261m Q\u03b5 (tabulate-spec (\u03bb i \u2192 P\u2262m (suc i) (\u03bb())))\n        tribulate-spec (suc m)    P\u2262m P\u2261m zero\n          = P\u2262m zero (\u03bb())\n        tribulate-spec zero {f\u2262m} P\u2262m P\u2261m (suc i)\n          rewrite lookup\u2218tabulate (f\u2262m \u2218 suc) i\n          = P\u2262m (suc i) (\u03bb())\n        tribulate-spec (suc m) P\u2262m P\u2261m (suc i)\n          = tribulate-spec m (\u03bb j j\u2262m \u2192 P\u2262m (suc j) (j\u2262m \u2218 Fin.suc-injective))\n                             (\u03bb Qpr Qpo \u2192 P\u2261m Q[ P\u2262m zero (\u03bb()) \u25c5 Qpr ] Qpo) i\n\n        tribulate\u2262m :\n          \u2200 {n} (m : Fin n)\n            {f\u2262m : Fin n \u2192 A}\n            {f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A}\n            i (i\u2262m : i \u2262 m) \u2192\n            tribulate m f\u2262m f\u2261m \u203c i \u2261 f\u2262m i\n        tribulate\u2262m = {!!}\n\n    tribulate-naive :\n      \u2200 {n} (m : Fin n)\n        (f\u2262m : \u2115 \u2192 A)\n        (f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A)\n      \u2192 Vec A (Fin\u25b9\u2115 m \u2115.+ suc (n \u2115-\u2115 suc m))\n    tribulate-naive {n = n} m f\u2262m f\u2261m\n       = prefix ++ f\u2261m fprefix fpostfix \u2237 postfix\n       where\n         prefix   = tabulate\u2032 (Fin\u25b9\u2115 m)      f\u2262m\n         postfix  = tabulate\u2032 (n \u2115-\u2115 suc m) (f\u2262m \u2218 \u2115._+_ (Fin\u25b9\u2115 (suc m)))\n         fprefix  = foldr B _\u25c5_ \u03b5 prefix\n         fpostfix = foldr B _\u25c5_ \u03b5 postfix\n\n    tribulate-correct :\n      \u2200 {n} (m : Fin n)\n        (f\u2262m : \u2115 \u2192 A)\n        (f\u2261m : B (Fin\u25b9\u2115 m) \u2192 B (n \u2115-\u2115 suc m) \u2192 A)\n      \u2192 toList (tribulate m (f\u2262m \u2218 Fin\u25b9\u2115) f\u2261m)\n       \u2261 toList (tribulate-naive m f\u2262m f\u2261m)\n    tribulate-correct zero    f\u2262m f\u2261m = refl\n    tribulate-correct (suc m) f\u2262m f\u2261m =\n      ap (_\u2237_ (f\u2262m 0)) (tribulate-correct m (f\u2262m \u2218 suc) (f\u2261m \u2218 _\u25c5_ (f\u2262m 0)))\n\n  test-tribulate1 :\n    tribulate (Vec \u2115) _\u2237_ [] (# 4) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 [])\n  test-tribulate1 = refl\n\n  test-tribulate2 :\n    tribulate _ L._++_ L.[]\n              (# 4) (L.[_] \u2218 Fin\u25b9\u2115)\n              (\u03bb xs ys \u2192 xs L.++ 42 \u2237 ys)\n      \u2261\n    (0 \u2237 []) \u2237 (1 \u2237 []) \u2237 (2 \u2237 []) \u2237 (3 \u2237 []) \u2237 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 []) \u2237 (5 \u2237 []) \u2237 []\n  test-tribulate2 = refl\n\n  test-tribulate3 :\n    tribulate (Vec \u2115) _\u2237_ [] (# 3) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 42 \u2237 [])\n  test-tribulate3 = refl\n\n  test-tribulate4 :\n    tribulate (const \u2115) \u2115._+_ 0 (# 3) Fin\u25b9\u2115 (\u03bb x y \u2192 (x \u2115.\u2238 1) \u2115.+ 10 \u2115.* y) \u2261 (0 \u2237 1 \u2237 2 \u2237 42 \u2237 4 \u2237 [])\n  test-tribulate4 = refl\n\n     {-\n  tribulate :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : Fin n \u2192 A)\n      (f\u2261m : Vec A (Fin\u25b9\u2115 m) \u2192 Vec A (n \u2115-\u2115 suc m) \u2192 A)\n    \u2192 Vec A n\n  tribulate zero f x = x [] tl \u2237 tl\n     where tl = tabulate (f \u2218 suc)\n  tribulate (suc m) f x = fz \u2237 tribulate m (f \u2218 suc) (x \u2218 _\u2237_ fz)\n     where fz = f zero\n\n  test-tribulate1 :\n    tribulate (# 4) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 [])\n  test-tribulate1 = refl\n\n  test-tribulate2 :\n    tribulate (# 4) (L.[_] \u2218 Fin\u25b9\u2115)\n              (\u03bb xs ys \u2192 L.concat (toList (xs ++ L.[ 42 ] \u2237 ys)))\n      \u2261\n    (0 \u2237 []) \u2237 (1 \u2237 []) \u2237 (2 \u2237 []) \u2237 (3 \u2237 []) \u2237 (0 \u2237 1 \u2237 2 \u2237 3 \u2237 42 \u2237 5 \u2237 []) \u2237 (5 \u2237 []) \u2237 []\n  test-tribulate2 = refl\n\n  test-tribulate3 :\n    tribulate (# 3) Fin\u25b9\u2115 (\u03bb _ _ \u2192 42) \u2261 (0 \u2237 1 \u2237 2 \u2237 42 \u2237 [])\n  test-tribulate3 = refl\n\n  tribulate-naive :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : \u2115 \u2192 A)\n      (f\u2261m : List A \u2192 List A \u2192 A)\n    \u2192 List A\n  tribulate-naive {n = n} m f\u2262m f\u2261m\n     = prefix L.++ f\u2261m prefix postfix \u2237 postfix\n     where\n       prefix  = tabulateL (Fin\u25b9\u2115 m)      f\u2262m\n       postfix = tabulateL (n \u2115-\u2115 suc m) (f\u2262m \u2218 \u2115._+_ (Fin\u25b9\u2115 (suc m)))\n\n  tribulate-correct :\n    \u2200 {A : \u2605} {n} (m : Fin n)\n      (f\u2262m : \u2115 \u2192 A)\n      (f\u2261m : List A \u2192 List A \u2192 A)\n    \u2192 toList (tribulate m (f\u2262m \u2218 Fin\u25b9\u2115) (\u03bb xs ys \u2192 f\u2261m (toList xs) (toList ys)))\n     \u2261 tribulate-naive m f\u2262m f\u2261m\n  tribulate-correct zero    f\u2262m f\u2261m = refl\n  tribulate-correct (suc m) f\u2262m f\u2261m =\n    ap (_\u2237_ (f\u2262m 0)) (tribulate-correct m (f\u2262m \u2218 suc) (f\u2261m \u2218 _\u2237_ (f\u2262m 0)))\n    -}\n-}\n  and : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 \ud835\udfda\n  and = foldr _ _\u2227_ 1\u2082\n\n  \u2713-and : \u2200 {n} {v : Vec \ud835\udfda n} (\u2713p : \u2200 i \u2192 \u2713(v \u203c i)) \u2192 \u2713(and v)\n  \u2713-and {v = []}     \u2713p = _\n  \u2713-and {v = x \u2237 xs} \u2713p = \u2713\u2227 (\u2713p zero) (\u2713-and (\u2713p \u2218 suc))\n\n  module _ {A B C D : \u2605} {n}\n           (f : Fin n \u2192 A \u2192 B \u2192 C \u2192 D)\n           (va : Vec A n) (vb : Vec B n) (vc : Vec C n) where\n    mapi3 : Vec D n\n    mapi3 = replicate f \u229b allFin _ \u229b va \u229b vb \u229b vc\n\n  open import Category.Applicative.Indexed -- .Morphism\n  module _ {A B C D : \u2605} {n}\n           {f : Fin n \u2192 A \u2192 B \u2192 C \u2192 D}\n           {va : Vec A n} {vb : Vec B n} {vc : Vec C n}\n           i where\n    -- Surely this could be done better...\n    open Morphism (lookup-morphism {\u2080} {n} i)\n    mapi3\u203c : mapi3 f va vb vc \u203c i \u2261 f i (va \u203c i) (vb \u203c i) (vc \u203c i)\n    mapi3\u203c = op-\u229b (replicate f \u229b allFin _ \u229b va \u229b vb) vc\n           \u2219 ap (\u03bb g \u2192 g (vc \u203c i))\n                (op-\u229b (replicate f \u229b allFin _ \u229b va) vb)\n           \u2219 ap (\u03bb g \u2192 g (vb \u203c i) (vc \u203c i)) (op-\u229b (replicate f \u229b allFin _) va)\n           \u2219 ap (\u03bb g \u2192 g (va \u203c i) (vb \u203c i) (vc \u203c i)) (op-\u229b (replicate f) (allFin _))\n           \u2219 ap (\u03bb g \u2192 g (allFin _ \u203c i) (va \u203c i) (vb \u203c i) (vc \u203c i)) (op-pure f)\n           \u2219 ap (\u03bb x \u2192 f x (va \u203c i) (vb \u203c i) (vc \u203c i)) (lookup\u2218tabulate id i)\n\n  module _ {A B C : \u2605} {n}\n           (p : Fin n \u2192 A \u2192 B \u2192 C \u2192 \ud835\udfda)\n           (va : Vec A n) (vb : Vec B n) (vc : Vec C n) where\n    all3 : \ud835\udfda\n    all3 = and (mapi3 p va vb vc)\n\n  module _ {A B C : \u2605} where\n    \u2713-all3 : \u2200 {n}\n           {p  : Fin n \u2192 A \u2192 B \u2192 C \u2192 \ud835\udfda}\n           {va : Vec A n} {vb : Vec B n} {vc : Vec C n}\n           (\u2713p : \u2200 i \u2192 \u2713(p i (va \u203c i) (vb \u203c i) (vc \u203c i)))\n           \u2192 \u2713(all3 p va vb vc)\n    \u2713-all3 {p = p} {va} {vb} {vc} \u2713p = \u2713-and {v = mapi3 p va vb vc} (\u03bb i \u2192 tr \u2713 (! mapi3\u203c i) (\u2713p i))\n\n  sum\u2124q : \u2200 {n} \u2192 Vec \u2124q n \u2192 \u2124q\n  sum\u2124q = foldr _ _+_ 0q\n\n  sum\u2124q-fun : \u2200 {n} (f : Fin n \u2192 \u2124q) \u2192 \u2124q\n  sum\u2124q-fun = iterate _ _+_ 0q\n\n  module CP = ZK.Chaum-Pedersen G \u2124q g _^_ _\u00b7_ _\/_ _+_ _*_ _==_\n  open CP using (PubKey; CipherText; EncRnd; enc)\n\n  VecCommitments = Vec CP.Commitment (suc max)\n  VecChallenges  = Vec CP.Challenge  (suc max)\n  VecResponses   = Vec CP.Response   (suc max)\n\n  module Interactive (y : PubKey) where\n    Response   = VecChallenges \u00d7 VecResponses\n\n    open ZK.Sigma-Protocol VecCommitments \u2124q Response public\n\n    private\n      M = \u03bb i \u2192 g ^ Fin\u25b9\u2124q i\n\n    module _ (ct : CipherText) where\n      verifier1 = \u03bb i A c s \u2192 CP.verifier y (M i) ct (CP.mk A c s)\n\n      -- H-commitments is the actual challenge for this \u03a3-protocol\n      verifier : Verifier\n      verifier (mk commitments H-commitments (challenges , responses))\n          = all3 verifier1 commitments challenges responses\n          \u2227 sum\u2124q challenges ==\u2124q H-commitments\n\n    module Prover-and-correctness\n             (i : Fin (suc max))\n             (rnd-challenge  : Fin (suc max) \u2192 CP.Challenge) -- the i\u1d57\u02b0 slot is not used\n             (rnd-response   : Fin (suc max) \u2192 CP.Response)  -- the i\u1d57\u02b0 slot is not used\n             (r : EncRnd) (w : \u2124q)\n             where\n\n      ct = enc y r (M i)\n\n      commitment-i = CP.prover-commitment y r w\n\n      simulated-commitment-j : (j : Fin (suc max)) \u2192 CP.Commitment\n      simulated-commitment-j j\n        = CP.simulate-commitment y (M j) ct (rnd-challenge j) (rnd-response j)\n\n      commitments = tubulate i simulated-commitment-j commitment-i\n\n      sum\u2262i = sum\u2124q-fun (rnd-challenge \u27e6 i \u27e7\u2254 0q)\n\n      prover-response : \u2124q \u2192 Response\n      prover-response Hcommitments = (challenges , responses)\n        module Prover-response where\n          challenge-i = Hcommitments - sum\u2262i\n\n          challenges = tubulate i rnd-challenge challenge-i\n\n          response-i = CP.prover-response y r w (challenges \u203c i)\n\n          responses = tubulate i rnd-response response-i\n\n      prover : Prover\n      prover = commitments , prover-response\n\n      module Correctness-proof (structure : Structure) where\n        open Structure structure\n        CPcs = \u03bb M c s \u2192 CP.Correct-simulation.correct-simulation y M ct c s \u2713-== \/-\u00b7\n        CPcp = CP.Correctness-proof.correctness \u2713-== ^-+ ^-* \u00b7-\/ y r w\n        correctness : Correctness prover (verifier ct)\n        correctness Hcommitments = \u2713\u2227 (\u2713-all3 {p = verifier1 ct} helper) helper'\n          where\n            open Prover-response Hcommitments\n            open \u2261-Reasoning\n            c-i-e : challenges \u203c i \u2261 challenge-i\n            c-i-e = tubulate\u203cm i _ _\n            W-i-e : commitments \u203c i \u2261 commitment-i\n            W-i-e = tubulate\u203cm i _ _\n            s-i-e : responses \u203c i \u2261 response-i\n            s-i-e = tubulate\u203cm i _ _\n            s-j-e : \u2200 j \u2192 i \u2262 j \u2192 responses \u203c j \u2261 rnd-response j\n            s-j-e = tubulate\u203c\u2262m i _ _\n            W-j-e : \u2200 j \u2192 i \u2262 j \u2192 commitments \u203c j \u2261 simulated-commitment-j j\n            W-j-e = tubulate\u203c\u2262m i _ _\n            c-j-e : \u2200 j \u2192 i \u2262 j \u2192 challenges \u203c j \u2261 rnd-challenge j\n            c-j-e = tubulate\u203c\u2262m i _ _\n            helper : \u2200 j \u2192 \u2713(verifier1 ct j (commitments \u203c j) (challenges \u203c j) (responses \u203c j))\n            helper j with Fin._\u225f_ i j\n            helper .i | yes refl\n              rewrite s-i-e | W-i-e | tubulate\u203cm i rnd-challenge challenge-i\n                    = CPcp (M i) challenge-i\n            helper  j | no \u00acp rewrite s-j-e j \u00acp | W-j-e j \u00acp | c-j-e j \u00acp = CPcs _ _ _\n            sum\u2262i-def : sum\u2262i \u2261 sum\u2124q (tabulate rnd-challenge [ i ]\u2254 0q)\n            sum\u2262i-def = iterate-foldr\u2218tabulate _ _+_ 0q (rnd-challenge \u27e6 i \u27e7\u2254 0q)\n                      \u2219 ap sum\u2124q (tabulate-\u27e6\u27e7\u2254 rnd-challenge i 0q)\n            helper' : \u2713(sum\u2124q challenges ==\u2124q Hcommitments)\n            helper' = \u2713-==\u2124q (sum\u2124q challenges\n                             \u2261\u27e8 ap sum\u2124q (tubulate-tabulate[]\u2254 i) \u27e9\n                              sum\u2124q (tabulate rnd-challenge [ i ]\u2254 challenge-i)\n                             \u2261\u27e8 ! fst 0-+-identity \u27e9\n                              0q + sum\u2124q (tabulate rnd-challenge [ i ]\u2254 challenge-i)\n                             \u2261\u27e8 foldr-[]\u2254-exch\u2032 _+_ +-comm +-assoc challenge-i 0q (tabulate rnd-challenge) i \u27e9\n                              challenge-i + sum\u2124q (tabulate rnd-challenge [ i ]\u2254 0q)\n                             \u2261\u27e8  ap (_+_ challenge-i) (! sum\u2262i-def)  \u27e9\n                              challenge-i + sum\u2262i\n                             \u2261\u27e8  -+  \u27e9\n                              Hcommitments\n                             \u220e)\n\n    -- Like Prover-and-correctness but avoids wasting randomness for r and w\n    -- This is maybe not a good idea as this is rather brittle\n    module Prover-and-correctness\u2032\n             (i : Fin (suc max))\n             (rnd-challenge  : Fin (suc max) \u2192 CP.Challenge)\n             (rnd-response   : Fin (suc max) \u2192 CP.Response)\n      = Prover-and-correctness i rnd-challenge rnd-response (rnd-challenge i) (rnd-response i)\n\n  module Non-interactive\n    (y : PubKey)\n    (H : VecCommitments \u2192 \u2124q)\n    where\n    module I = Interactive y\n\n    open ZK.Sigma-Protocol VecCommitments VecChallenges VecResponses public\n      using (Transcript; mk; Verifier)\n\n    module _ (ct : CipherText) where\n        verifier : Verifier\n        verifier (mk commitments challenges responses)\n          = I.verifier ct (I.mk commitments (H commitments) (challenges , responses))\n\n    module Prover-and-correctness\n             (i : Fin (suc max))\n             (rnd-challenge  : Fin (suc max) \u2192 CP.Challenge) -- the i\u1d57\u02b0 slot is not used\n             (rnd-response   : Fin (suc max) \u2192 CP.Response)  -- the i\u1d57\u02b0 slot is not used\n             (r : EncRnd) (w : \u2124q)\n             where\n      module IPC = I.Prover-and-correctness i rnd-challenge rnd-response r w\n      open IPC using (commitments; ct)\n      open IPC.Prover-response (H commitments)\n      transcript : Transcript\n      transcript = mk commitments challenges responses\n      module Correctness-proof (structure : Structure) where\n        correctness : \u2713 (verifier ct transcript)\n        correctness = IPC.Correctness-proof.correctness structure (H commitments)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"84b371ec5816eae99ae8d016e62df94a0c70c2a6","subject":"Plug Atlas.Change.Derive into Parametric.C.D.","message":"Plug Atlas.Change.Derive into Parametric.C.D.\n\nOld-commit-hash: 2fb657275ba8085c801b6c81ccd68d6b09f16075\n","repos":"inc-lc\/ilc-agda","old_file":"Atlas\/Change\/Derive.agda","new_file":"Atlas\/Change\/Derive.agda","new_contents":"module Atlas.Change.Derive where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\nopen import Atlas.Change.Term\n\nimport Parametric.Change.Derive Const \u0394Base as Derive\n\n\u0394Const : Derive.Structure\n\n\u0394Const true  \u2205 = false!\n\u0394Const false \u2205 = false!\n\n\u0394Const xor (\u0394x \u2022 x \u2022 \u0394y \u2022 y \u2022 \u2205) = xor! \u0394x \u0394y\n\n\u0394Const empty \u2205 = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\n\u0394Const (update {\u03ba} {\u03b9}) (\u0394k \u2022 k \u2022 \u0394v \u2022 v \u2022 \u0394m \u2022 m \u2022 \u2205) =\n  insert (apply {base \u03ba} \u0394k k) (apply {base \u03b9} \u0394v v) (delete k v \u0394m)\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\n\u0394Const (lookup {\u03ba} {\u03b9}) (\u0394k \u2022 k \u2022 \u0394m \u2022 m \u2022 \u2205) =\n  let\n    k\u2032 = apply {base \u03ba} \u0394k k\n  in\n    (diff (apply {base _} (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n          (lookup! k m))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\n\u0394Const zip (\u0394f \u2022 f \u2022 \u0394m\u2081 \u2022 m\u2081 \u2022 \u0394m\u2082 \u2022 m\u2082 \u2022 \u2205) =\n  let\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\n\u0394Const (fold {\u03ba} {\u03b1} {\u03b2}) (\u0394f\u2032 \u2022 f\u2032 \u2022 \u0394z \u2022 z \u2022 \u0394m \u2022 m \u2022 \u2205) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        f\u2032))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        \u0394f\u2032))\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))\n\nopen Derive.Structure \u0394Const public\n","old_contents":"module Atlas.Change.Derive where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\nopen import Atlas.Change.Term\n\n\u0394Const : \u2200 {\u0393 \u03a3 \u03c4} \u2192\n  Const \u03a3 \u03c4 \u2192\n  Terms (\u0394Context \u0393) (\u0394Context \u03a3) \u2192\n  Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\n\u0394Const true  \u2205 = false!\n\u0394Const false \u2205 = false!\n\n\u0394Const xor (\u0394x \u2022 x \u2022 \u0394y \u2022 y \u2022 \u2205) = xor! \u0394x \u0394y\n\n\u0394Const empty \u2205 = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\n\u0394Const (update {\u03ba} {\u03b9}) (\u0394k \u2022 k \u2022 \u0394v \u2022 v \u2022 \u0394m \u2022 m \u2022 \u2205) =\n  insert (apply {base \u03ba} \u0394k k) (apply {base \u03b9} \u0394v v) (delete k v \u0394m)\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\n\u0394Const (lookup {\u03ba} {\u03b9}) (\u0394k \u2022 k \u2022 \u0394m \u2022 m \u2022 \u2205) =\n  let\n    k\u2032 = apply {base \u03ba} \u0394k k\n  in\n    (diff (apply {base _} (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n          (lookup! k m))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\n\u0394Const zip (\u0394f \u2022 f \u2022 \u0394m\u2081 \u2022 m\u2081 \u2022 \u0394m\u2082 \u2022 m\u2082 \u2022 \u2205) =\n  let\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\n\u0394Const (fold {\u03ba} {\u03b1} {\u03b2}) (\u0394f\u2032 \u2022 f\u2032 \u2022 \u0394z \u2022 z \u2022 \u0394m \u2022 m \u2022 \u2205) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        f\u2032))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        \u0394f\u2032))\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6f7fde98554f01770bfdbd43711ab8e45eab8b9b","subject":"Split Control.Protocol","message":"Split Control.Protocol\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol.agda","new_file":"Control\/Protocol.agda","new_contents":"module Control.Protocol where\n\nopen import Control.Protocol.InOut public\nopen import Control.Protocol.End   public\nopen import Control.Protocol.Core  public\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_) renaming (proj\u2081 to fst)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nmodule Control.Protocol where\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual\u1d35\u1d3c-equiv : Is-equiv dual\u1d35\u1d3c\ndual\u1d35\u1d3c-equiv = self-inv-is-equiv dual\u1d35\u1d3c-involutive\n\ndual\u1d35\u1d3c-inj : \u2200 {x y} \u2192 dual\u1d35\u1d3c x \u2261 dual\u1d35\u1d3c y \u2192 x \u2261 y\ndual\u1d35\u1d3c-inj = Is-equiv.injective dual\u1d35\u1d3c-equiv\n\n{-\nmodule UniversalProtocols \u2113 {U : \u2605_(\u209b \u2113)}(U\u27e6_\u27e7 : U \u2192 \u2605_ \u2113) where\n-}\nmodule _ \u2113 where\n  U = \u2605_ \u2113\n  U\u27e6_\u27e7 = id\n  data Proto_ : \u2605_(\u209b \u2113) where\n    end : Proto_\n    com : (io : InOut){M : U}(P : U\u27e6 M \u27e7 \u2192 Proto_) \u2192 Proto_\n{-\nmodule U\u2605 \u2113 = UniversalProtocols \u2113 {\u2605_ \u2113} id\nopen U\u2605\n-}\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n    com= refl refl P= = ap (com _) (\u03bb= P=)\n\n    module _ {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n        com\u2243 = com= io= (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    module _ io {M N}(P : M \u00d7 N \u2192 Proto)\n             where\n        com\u2082 : Proto\n        com\u2082 = com io \u03bb m \u2192 com io \u03bb n \u2192 P (m , n)\n\n    {- Proving this would be awesome...\n    module _ io\n             {M\u2080 M\u2081 N\u2080 N\u2081 : \u2605}\n             (M\u00d7N\u2243 : (M\u2080 \u00d7 N\u2080) \u2243 (M\u2081 \u00d7 N\u2081))\n             {P\u2080 : M\u2080 \u00d7 N\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u00d7 N\u2081 \u2192 Proto}\n             (P= : \u2200 m,n\u2080 \u2192 P\u2080 m,n\u2080 \u2261 P\u2081 (\u2013> M\u00d7N\u2243 m,n\u2080))\n             {{_ : UA}} where\n        com\u2082\u2243 : com\u2082 io P\u2080 \u2261 com\u2082 io P\u2081\n        com\u2082\u2243 = {!com=!}\n    -}\n\n    send= = com= {Out} refl\n    send\u2243 = com\u2243 {Out} refl\n    recv= = com= {In} refl\n    recv\u2243 = com\u2243 {In} refl\n\n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= refl refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\npattern recvE M P = com In  {M} P\npattern sendE M P = com Out {M} P\n\nrecv\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nrecv\u2610 P = recv (\u03bb { [ m ] \u2192 P m })\n\nsend\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nsend\u2610 P = send (\u03bb { [ m ] \u2192 P m })\n\n{-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (send P) = recv \u03bb m \u2192 dual (P m)\ndual (recv P) = send \u03bb m \u2192 dual (P m)\n-}\n\ndual : Proto \u2192 Proto\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nsource-of : Proto \u2192 Proto\nsource-of end       = end\nsource-of (com _ P) = send \u03bb m \u2192 source-of (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end   : IsSource end\n  send' : \u2200 {M P} (PT : (m : M) \u2192 IsSource (P m)) \u2192 IsSource (send P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end   : IsSink end\n  recv' : \u2200 {M P} (PT : (m : M) \u2192 IsSink (P m)) \u2192 IsSink (recv P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 io {M P} (P\u2610 : \u2200 (m : \u2610 M) \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com io P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\n\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\nEnd-equiv : End \u2243 \ud835\udfd9\nEnd-equiv = equiv {\u2080} _ _ (\u03bb _ \u2192 refl) (\u03bb _ \u2192 refl)\n\nEnd-uniq : {{_ : UA}} \u2192 End \u2261 \ud835\udfd9\nEnd-uniq = ua End-equiv\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113)(P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end      \u27e7 = End\n\u27e6 com io P \u27e7 = \u27e6 io \u27e7\u1d35\u1d3c _ \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end         = _\nsink (com _ P) x = sink (P x)\n\ntelecom : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntelecom end      _       _       = _\ntelecom (recv P) p       (m , q) = m , telecom (P m) (p m) q\ntelecom (send P) (m , p) q       = m , telecom (P m) p (q m)\n\nsend\u2032 : \u2605 \u2192 Proto \u2192 Proto\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto \u2192 Proto\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 P \u2192 dual (dual P) \u2261 P\n    dual-involutive end        = refl\n    dual-involutive (com io P) = com= (dual\u1d35\u1d3c-involutive io) refl \u03bb m \u2192 dual-involutive (P m)\n\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n\n    source-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261 source-of P\n    source-of-idempotent end       = refl\n    source-of-idempotent (com _ P) = com= refl refl \u03bb m \u2192 source-of-idempotent (P m)\n\n    source-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261 source-of P\n    source-of-dual-oblivious end       = refl\n    source-of-dual-oblivious (com _ P) = com= refl refl \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P s \u2192 sink P \u2261 s\n    sink-contr end       s = refl\n    sink-contr (com _ P) s = \u03bb= \u03bb m \u2192 sink-contr (P m) (s m)\n\n    Sink-is-contr : \u2200 P \u2192 Is-contr (Sink P)\n    Sink-is-contr P = sink P , sink-contr P\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (Sink-is-contr P)\n\n    dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n    dual-Log = ap \u27e6_\u27e7 \u2218 source-of-dual-oblivious\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ac37fb1c5faf1fb71f7ff82f6cdf182514293a98","subject":"#14 scraps","message":"#14 scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    -- todo: also being used below\n    lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    --- todo: okay this is now getting reused here and in the case of ie\n    --- below, so this is apparently more of an interesting property than\n    --- i'd thought\n    lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n    lem (CECastFail x\u2081 x\u2082 () x\u2084)\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVVal x) (CECong FHOuter er) = ve (BVVal x) er\n  ve (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  ve (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  ve (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  ve (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  vs (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  vs (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole () ind) (ITCastID x\u2081)\n  lem2 (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  lem2 (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  lem2 (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  lem2 (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  lem2 (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail cases\n  es (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  es (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  es (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  es (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!} -- es ? (_ , Step x\u2084 x\u2085 x\u2086)\n\n    -- congruence cases -- es ce\n  es (CECong x ce) (_ , Step FHOuter x\u2082 FHOuter) = {!!}\n  es (CECong x ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = {!x!}\n  es (CECong x ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  es (CECong x ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = {!!}\n  es (CECong x ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!}\n\n\n  -- es (CECong FHOuter er) (d0' , Step FHOuter x\u2082 FHOuter) = es er (_ , Step FHOuter x\u2082 FHOuter)\n  -- es (CECong (FHAp1 x) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!er!}\n  -- es (CECong (FHAp2 x x\u2081) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!!}\n  -- es (CECong (FHNEHole x) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!!}\n  -- es (CECong (FHCast x) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  -- es (CECong FHOuter er) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = {!!}\n  -- es (CECong (FHAp1 x) er) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = {!!}\n  -- es (CECong (FHAp2 x x\u2081) er) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = fs x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  -- es (CECong FHOuter er) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  -- es (CECong (FHAp1 x) er) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  -- es (CECong (FHAp2 x x\u2081) er) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n  -- --     with lem {!!}\n  -- --     where\n  -- --     lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n  -- --     lem (CECong FHOuter ce) = lem ce\n  -- --     lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n  -- --     lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n  -- -- ... | Inl d1err = {!!}\n  -- -- ... | Inr d2err = {!!} -- = es {!!} (_ , Step x\u2082 x\u2083 x\u2085)\n  -- es (CECong FHOuter er) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = {!!}\n  -- es (CECong (FHNEHole x) er) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = {!!}\n\n  -- es (CECong FHOuter er) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!}\n  -- es (CECong (FHCast x) er) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    -- todo: also being used below\n    lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong FHOuter er) = ve bv (lem er)\n    where\n    --- todo: okay this is now getting reused here and in the case of ie\n    --- below, so this is apparently more of an interesting property than\n    --- i'd thought\n    lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n    lem (CECastFail x\u2081 x\u2082 () x\u2084)\n    lem (CECong FHOuter ce) = lem ce\n    lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n  ve (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVVal x) (CECong FHOuter er) = ve (BVVal x) er\n  ve (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  ve (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  ve (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  ve (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  vs (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  vs (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole () ind) (ITCastID x\u2081)\n  lem2 (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  lem2 (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  lem2 (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  lem2 (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  lem2 (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  es (CECastFail x x\u2081 x\u2082 x\u2083) (d' , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  es (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  es (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!} -- cyrus\n\n  es (CECong FHOuter er) (d0' , Step FHOuter x\u2082 FHOuter) = es er (_ , Step FHOuter x\u2082 FHOuter)\n  es (CECong (FHAp1 x) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!er!}\n  es (CECong (FHAp2 x x\u2081) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!!}\n  es (CECong (FHNEHole x) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!!}\n  es (CECong (FHCast x) er) (d0' , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  es (CECong FHOuter er) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = {!!}\n  es (CECong (FHAp1 x) er) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = {!!}\n  es (CECong (FHAp2 x x\u2081) er) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = fs x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  es (CECong FHOuter er) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  es (CECong (FHAp1 x) er) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  es (CECong (FHAp2 x x\u2081) er) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n  --     with lem {!!}\n  --     where\n  --     lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n  --     lem (CECong FHOuter ce) = lem ce\n  --     lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n  --     lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n  -- ... | Inl d1err = {!!}\n  -- ... | Inr d2err = {!!} -- = es {!!} (_ , Step x\u2082 x\u2083 x\u2085)\n  es (CECong FHOuter er) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = {!!}\n  es (CECong (FHNEHole x) er) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = {!!}\n\n  es (CECong FHOuter er) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!}\n  es (CECong (FHCast x) er) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b5f69aca79e43e9a66e567fb8183fde160f76f35","subject":"cleaning up preservation pre ae, rel to #52 and #49","message":"cleaning up preservation pre ae, rel to #52 and #49\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import binders-disjoint-checks\n\nopen import lemmas-subst-ta\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            binders-unique d \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans bd TAConst ()\n  preserve-trans bd (TAVar x\u2081) ()\n  preserve-trans bd (TALam _ ta) ()\n  preserve-trans (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2082 x\u2083)) (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt x\u2082 bd\u2081 ta ta\u2081\n  preserve-trans bd (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans bd (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans bd (TAEHole x x\u2081) ()\n  preserve-trans bd (TANEHole x ta x\u2081) ()\n  preserve-trans bd (TACast ta x) (ITCastID) = ta\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans bd (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans bd (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans bd (TAFailedCast x y z q) ()\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             binders-unique d \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preservation bd D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans (lem-bd-\u03b51 x bd) wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import binders-disjoint-checks\n\nopen import lemmas-subst-ta\n\nmodule preservation where\n  -- if d and d' both result from filling the hole in \u03b5 with terms of the\n  -- same type, they too have the same type.\n  wt-different-fill : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d' \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d' == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  wt-different-fill FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  wt-different-fill (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (wt-different-fill eps D1 D3 D4 D5) D2\n  wt-different-fill (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (wt-different-fill eps D2 D3 D4 D5)\n  wt-different-fill (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (wt-different-fill eps D1 D2 D3 D4) x\u2081\n  wt-different-fill (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (wt-different-fill eps D1 D2 D3 D4) x\n  wt-different-fill (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (wt-different-fill x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- if a well typed term results from filling the hole in \u03b5, then the term\n  -- that filled the hole is also well typed\n  wt-filling : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  wt-filling TAConst FHOuter = _ , TAConst\n  wt-filling (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  wt-filling (TALam f ta) FHOuter = _ , TALam f ta\n\n  wt-filling (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  wt-filling (TAAp ta ta\u2081) (FHAp1 eps) = wt-filling ta eps\n  wt-filling (TAAp ta ta\u2081) (FHAp2 eps) = wt-filling ta\u2081 eps\n\n  wt-filling (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  wt-filling (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  wt-filling (TANEHole x ta x\u2081) (FHNEHole eps) = wt-filling ta eps\n  wt-filling (TACast ta x) FHOuter = _ , TACast ta x\n  wt-filling (TACast ta x) (FHCast eps) = wt-filling ta eps\n  wt-filling (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  wt-filling (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = wt-filling x y\n\n  -- instruction transitions preserve type\n  preserve-trans : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            binders-unique d \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  preserve-trans bd TAConst ()\n  preserve-trans bd (TAVar x\u2081) ()\n  preserve-trans bd (TALam _ ta) ()\n  preserve-trans (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2082 x\u2083)) (TAAp (TALam apt ta) ta\u2081) ITLam = lem-subst apt x\u2082 bd\u2081 ta ta\u2081\n  preserve-trans bd (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  preserve-trans bd (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  preserve-trans bd (TAEHole x x\u2081) ()\n  preserve-trans bd (TANEHole x ta x\u2081) ()\n  preserve-trans bd (TACast ta x) (ITCastID) = ta\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  preserve-trans bd (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  preserve-trans bd (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  preserve-trans bd (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  preserve-trans bd (TAFailedCast x y z q) ()\n\n  -- binder uniqueness is preserved by hole filling\n  lem-bd-\u03b51 : \u2200{ d \u03b5 d0} \u2192 d == \u03b5 \u27e6 d0 \u27e7 \u2192 binders-unique d \u2192 binders-unique d0\n  lem-bd-\u03b51 FHOuter bd = bd\n  lem-bd-\u03b51 (FHAp1 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHAp2 eps) (BUAp bd bd\u2081 x) = lem-bd-\u03b51 eps bd\u2081\n  lem-bd-\u03b51 (FHNEHole eps) (BUNEHole bd x) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHCast eps) (BUCast bd) = lem-bd-\u03b51 eps bd\n  lem-bd-\u03b51 (FHFailedCast eps) (BUFailedCast bd) = lem-bd-\u03b51 eps bd\n\n\n  -- mutual\n  --   bu-subst : \u2200{d1 d2 x} \u2192 binders-unique d1 \u2192 binders-unique d2 \u2192 unbound-in x d1 \u2192 binders-unique ([ d2 \/ x ] d1)\n  --   bu-subst BUHole bu2 ub = BUHole\n  --   bu-subst {x = x} (BUVar {x = y}) bu2 UBVar with natEQ y x\n  --   bu-subst BUVar bu2 UBVar | Inl refl = bu2\n  --   bu-subst BUVar bu2 UBVar | Inr x\u2082 = BUVar\n  --   bu-subst {x = x} (BULam {x = y}  bu1 x\u2082) bu2 (UBLam2 x\u2083 ub) with natEQ y x\n  --   bu-subst (BULam bu1 x\u2083) bu2 (UBLam2 x\u2084 ub) | Inl refl = BULam bu1 ub -- can also abort here; odd\n  --   bu-subst (BULam bu1 x\u2083) bu2 (UBLam2 x\u2084 ub) | Inr x\u2082 = BULam (bu-subst bu1 bu2 ub) {!!}\n  --   bu-subst (BUEHole x\u2081) bu2 (UBHole x\u2082) = BUEHole (BU\u03c3Subst bu2 {!!})\n  --   bu-subst (BUNEHole bu1 x\u2081) bu2 (UBNEHole x\u2082 ub) = BUNEHole (bu-subst bu1 bu2 ub) (BU\u03c3Subst bu2 {!!})\n  --   bu-subst (BUAp bu1 bu2 x\u2081) bu3 (UBAp ub ub\u2081) = BUAp (bu-subst bu1 bu3 ub) (bu-subst bu2 bu3 ub\u2081) {!!}\n  --   bu-subst (BUCast bu1) bu2 (UBCast ub) = BUCast (bu-subst bu1 bu2 ub)\n  --   bu-subst (BUFailedCast bu1) bu2 (UBFailedCast ub) = BUFailedCast (bu-subst bu1 bu2 ub)\n\n  -- bu-trans : \u2200{d0 d0'} \u2192 binders-unique d0 \u2192 d0 \u2192> d0' \u2192 binders-unique d0'\n  -- bu-trans BUHole ()\n  -- bu-trans BUVar ()\n  -- bu-trans (BULam bu x\u2081) ()\n  -- bu-trans (BUEHole x) ()\n  -- bu-trans (BUNEHole bu x) ()\n  -- bu-trans (BUAp (BULam bu x\u2081) bu\u2081 (BDLam x\u2082 x\u2083)) ITLam = bu-subst bu bu\u2081 x\u2081\n  -- bu-trans (BUAp (BUCast bu) bu\u2081 (BDCast x)) ITApCast = BUCast (BUAp bu (BUCast bu\u2081) (lem-bd-into-cast x))\n  -- bu-trans (BUCast bu) ITCastID = bu\n  -- bu-trans (BUCast (BUCast bu)) (ITCastSucceed x) = bu\n  -- bu-trans (BUCast (BUCast bu)) (ITCastFail x x\u2081 x\u2082) = BUFailedCast bu\n  -- bu-trans (BUCast bu) (ITGround x) = BUCast (BUCast bu)\n  -- bu-trans (BUCast bu) (ITExpand x) = BUCast (BUCast bu)\n  -- bu-trans (BUFailedCast bu) ()\n\n  -- subst-disjoint : \u2200{d1 d2 d3 x} \u2192 binders-disjoint d1 d2 \u2192 unbound-in x d2 \u2192 binders-disjoint d3 d2 \u2192  binders-disjoint ([ d3 \/ x ] d1) d2\n  -- subst-disjoint = {!!}\n\n  -- lem3 : \u2200{d1 d2 d3} \u2192 d1 \u2192> d3 \u2192 binders-disjoint d1 d2 \u2192 binders-disjoint d3 d2\n  -- lem3 ITLam (BDAp (BDLam disj x\u2081) disj\u2081) = subst-disjoint disj x\u2081 disj\u2081\n  -- lem3 ITCastID (BDCast disj) = disj\n  -- lem3 (ITCastSucceed x) (BDCast (BDCast disj)) = disj\n  -- lem3 (ITCastFail x x\u2081 x\u2082) (BDCast (BDCast disj)) = BDFailedCast disj\n  -- lem3 ITApCast (BDAp (BDCast disj) disj\u2081) = BDCast (BDAp disj (BDCast disj\u2081))\n  -- lem3 (ITGround x) (BDCast disj) = BDCast (BDCast disj)\n  -- lem3 (ITExpand x) (BDCast disj) = BDCast (BDCast disj)\n\n  -- decomp1 : \u2200{d0 d1 d2 \u03b5} \u2192 d1 == \u03b5 \u27e6 d0 \u27e7 \u2192 binders-disjoint d1 d2 \u2192 binders-disjoint d0 d2\n  -- decomp1 = {!!}\n\n  -- lem2 : \u2200{d1 d2 d3} \u2192 d1 \u21a6 d3 \u2192 binders-disjoint d1 d2 \u2192 binders-disjoint d3 d2\n  -- lem2 (Step x y z) disj = {!(lem3 y (decomp x disj))!} --\n  -- lem2 (Step FHOuter () FHOuter) BDConst\n  -- lem2 (Step FHOuter () x\u2083) BDVar\n  -- lem2 (Step FHOuter () x\u2083) (BDLam disj x\u2084)\n  -- lem2 (Step FHOuter () FHOuter) (BDHole x\u2083)\n  -- lem2 (Step FHOuter () FHOuter) (BDNEHole x\u2083 disj)\n  -- lem2 (Step (FHNEHole x) x\u2081 (FHNEHole x\u2082)) (BDNEHole x\u2083 disj) = BDNEHole x\u2083 (lem2 (Step x x\u2081 x\u2082) disj)\n  -- lem2 (Step FHOuter ITLam FHOuter) (BDAp (BDLam disj x\u2081) disj\u2081) = subst-disjoint disj x\u2081 disj\u2081\n  -- lem2 (Step FHOuter ITApCast FHOuter) (BDAp (BDCast disj) disj\u2081) = BDCast (BDAp disj (BDCast disj\u2081))\n\n  -- lem2 (Step (FHAp1 x\u2081) ITLam (FHAp1 x\u2082)) (BDAp disj disj\u2081) = BDAp {!!} disj\u2081\n\n  -- lem2 (Step (FHAp1 x) ITCastID (FHAp1 x\u2082)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 (Step (FHAp1 x) (ITCastSucceed x\u2081) (FHAp1 x\u2082)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 (Step (FHAp1 x) (ITCastFail x\u2081 x\u2082 x\u2083) (FHAp1 x\u2084)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 (Step (FHAp1 x) ITApCast (FHAp1 x\u2082)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 (Step (FHAp1 x) (ITGround x\u2081) (FHAp1 x\u2082)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 (Step (FHAp1 x) (ITExpand x\u2081) (FHAp1 x\u2082)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 (Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) (BDAp disj disj\u2081) = {!!}\n  -- lem2 step (BDCast disj) = {!!}\n  -- lem2 step (BDFailedCast disj) = {!!}\n\n\n  -- lem' : \u2200{d d'} \u2192 d \u21a6 d' \u2192 binders-unique d \u2192 binders-unique d'\n  -- lem' (Step FHOuter () FHOuter) BUHole\n  -- lem' (Step FHOuter () FHOuter) BUVar\n  -- lem' (Step FHOuter () FHOuter) (BULam bd x\u2084)\n  -- lem' (Step FHOuter () FHOuter) (BUEHole x\u2083)\n\n  -- lem' (Step FHOuter () FHOuter) (BUNEHole bd x\u2083)\n  -- lem' (Step (FHNEHole x) x\u2081 (FHNEHole x\u2082)) (BUNEHole bd x\u2083) = BUNEHole (lem' (Step x x\u2081 x\u2082) bd) x\u2083\n\n  -- lem' (Step FHOuter ITLam FHOuter) (BUAp (BULam bd x\u2081) bd\u2081 (BDLam x\u2083 x\u2082)) = bu-subst bd bd\u2081 x\u2081\n  -- lem' (Step FHOuter ITApCast FHOuter) (BUAp (BUCast bd) bd\u2081 (BDCast x\u2083)) = BUCast (BUAp bd (BUCast bd\u2081) {!lem-bd-cast!}) -- easy lemma, i think?\n  -- lem' (Step (FHAp1 x) x\u2081 (FHAp1 x\u2082)) (BUAp bd bd\u2081 x\u2083) = BUAp (lem' (Step x x\u2081 x\u2082) bd) bd\u2081 {!Step x x\u2081 x\u2082!}\n  -- lem' (Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) (BUAp bd bd\u2081 x\u2083) = BUAp bd (lem' (Step x x\u2081 x\u2082) bd\u2081) {!(Step x x\u2081 x\u2082)!}\n\n  -- lem' (Step FHOuter ITCastID FHOuter) (BUCast bd) = bd\n  -- lem' (Step FHOuter (ITCastSucceed x) FHOuter) (BUCast (BUCast bd)) = bd\n  -- lem' (Step FHOuter (ITCastFail x x\u2081 x\u2082) FHOuter) (BUCast (BUCast bd)) = BUFailedCast bd\n  -- lem' (Step FHOuter (ITGround x) FHOuter) (BUCast bd) = BUCast (BUCast bd)\n  -- lem' (Step FHOuter (ITExpand x) FHOuter) (BUCast bd) = BUCast (BUCast bd)\n  -- lem' (Step (FHCast x) x\u2081 (FHCast x\u2082)) (BUCast bd) = BUCast (lem' (Step x x\u2081 x\u2082) bd)\n\n  -- lem' (Step FHOuter () FHOuter) (BUFailedCast bd)\n  -- lem' (Step (FHFailedCast x) x\u2081 (FHFailedCast x\u2082)) (BUFailedCast bd) = BUFailedCast (lem' (Step x x\u2081 x\u2082) bd)\n\n\n  -- this is the main preservation theorem, gluing together the above\n  preservation : {\u0394 : hctx} {d d' : ihexp} {\u03c4 : htyp} {\u0393 : tctx} \u2192\n             binders-unique d \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u0393 \u22a2 d' :: \u03c4 -- \u00d7 binders-unique d'\n  preservation bd D (Step x x\u2081 x\u2082)\n    with wt-filling D x\n  ... | (_ , wt) = wt-different-fill x D wt (preserve-trans (lem-bd-\u03b51 x bd) wt x\u2081) x\u2082\n                 -- {!bu-trans (lem-bd-\u03b51 x bd) x\u2081 !}\n\n-- lem' (Step x x\u2081 x\u2082) bd --\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"787757916378c6f45458fa51c08c64f58490ddb8","subject":"Solver.Linear: fixity","message":"Solver.Linear: fixity\n","repos":"crypto-agda\/crypto-agda","old_file":"Solver\/Linear.agda","new_file":"Solver\/Linear.agda","new_contents":"module Solver.Linear where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'   : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'  : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  infixr 5 _,_\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n  infix 4 _\u21db_\n\n  record Eq : Set where\n    constructor _\u21db_\n    field\n      LHS RHS : Syn\n\n  open import Data.Vec.N-ary using (N-ary ; _$\u207f_)\n  open import Data.Vec using (allFin) renaming (map to vmap)\n\n  rewire' : (f : N-ary nrVars Syn Eq) \u2192 let (S\u2081 \u21db S\u2082) = f $\u207f (vmap Syn.var (allFin nrVars))\n          in CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire' f eq = let S \u21db S' = f $\u207f vmap Syn.var (allFin  nrVars)\n         in rewire S S' eq\n \n         {- move them to sub modules\nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire' (\u03bb a b c \u2192 (a , b) , c \u21db (b , a) , c) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2083 where\n\n  open import Data.Unit\n  open import Data.Product\n  open import Data.Vec\n\n  open import Function using (flip ; const)\n  \n  open import Function.Inverse\n  open import Function.Related.TypeIsomorphisms.NP\n\n  open \u00d7-CMon using () renaming (\u2219-cong to \u00d7-cong ; assoc to \u00d7-assoc)\n\n  module STest n M = Syntax Set _\u00d7_ \u22a4 _\u2194_ id (flip _\u2218_) A\u00d7\u22a4\u2194A (sym A\u00d7\u22a4\u2194A) (A\u00d7\u22a4\u2194A \u2218 swap-iso) (swap-iso \u2218 sym A\u00d7\u22a4\u2194A)\n                            \u00d7-cong first-iso (\u03bb x \u2192 second-iso (const x))\n                            (sym (\u00d7-assoc _ _ _)) (\u00d7-assoc _ _ _) swap-iso n M\n\n  test : \u2200 A B C \u2192 ((A \u00d7 B) \u00d7 C) \u2194 (C \u00d7 (B \u00d7 A))\n  test A B C = rewire ((# 0 , # 1) , # 2) (# 2 , (# 1 , # 0)) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n-}\n","old_contents":"module Solver.Linear where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'   : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'  : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n  infix 4 _\u21db_\n\n  record Eq : Set where\n    constructor _\u21db_\n    field\n      LHS RHS : Syn\n\n  open import Data.Vec.N-ary using (N-ary ; _$\u207f_)\n  open import Data.Vec using (allFin) renaming (map to vmap)\n\n  rewire' : (f : N-ary nrVars Syn Eq) \u2192 let (S\u2081 \u21db S\u2082) = f $\u207f (vmap Syn.var (allFin nrVars))\n          in CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire' f eq = let S \u21db S' = f $\u207f vmap Syn.var (allFin  nrVars)\n         in rewire S S' eq\n \nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n  test2 : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire' (\u03bb a b c \u2192 (a , b) , c \u21db (b , a) , c) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n\nmodule example\u2083 where\n\n  open import Data.Unit\n  open import Data.Product\n  open import Data.Vec\n\n  open import Function using (flip ; const)\n  \n  open import Function.Inverse\n  open import Function.Related.TypeIsomorphisms.NP\n\n  open \u00d7-CMon using () renaming (\u2219-cong to \u00d7-cong ; assoc to \u00d7-assoc)\n\n  module STest n M = Syntax Set _\u00d7_ \u22a4 _\u2194_ id (flip _\u2218_) A\u00d7\u22a4\u2194A (sym A\u00d7\u22a4\u2194A) (A\u00d7\u22a4\u2194A \u2218 swap-iso) (swap-iso \u2218 sym A\u00d7\u22a4\u2194A)\n                            \u00d7-cong first-iso (\u03bb x \u2192 second-iso (const x))\n                            (sym (\u00d7-assoc _ _ _)) (\u00d7-assoc _ _ _) swap-iso n M\n\n  test : \u2200 A B C \u2192 ((A \u00d7 B) \u00d7 C) \u2194 (C \u00d7 (B \u00d7 A))\n  test A B C = rewire ((# 0 , # 1) , # 2) (# 2 , (# 1 , # 0)) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6f81adff41c15d4dd8a27927b2cd33d64b57d965","subject":"Updated doc.","message":"Updated doc.\n\nIgnore-this: 2b06fbd1abe6a6e7f68c1fd718655c6a\n\ndarcs-hash:20100921153238-3bd4e-a05cc8fbd5079de38e29928fc6b7f7f296f33f4b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/Add.agda","new_file":"Test\/Fail\/Add.agda","new_contents":"module Test.Fail.Add where\n\ninfix  4 _\u2261_\ninfixl 6 _+_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n  _+_    : D \u2192 D \u2192 D\n  +-0x   : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx   : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n-- The ATPs should not prove this postulate.\npostulate\n  +-comm : (d e : D) \u2192 d + e \u2261 e + d\n-- E 1.2 success (SZS status CounterSatisfiable).\n-- Equinox 5.0alpha (2010-03-29) no-success due to timeout.\n-- Metis 2.3 (release 20100920) success (SZS status CounterSatisfiable).\n{-# ATP prove +-comm #-}\n","old_contents":"module Test.Fail.Add where\n\ninfix  4 _\u2261_\ninfixl 6 _+_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n  _+_    : D \u2192 D \u2192 D\n  +-0x   : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx   : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n-- The ATPs should not prove this postulate.\npostulate\n  +-comm : (d e : D) \u2192 d + e \u2261 e + d\n-- Equinox 5.0alpha (2010-03-29) no-success due to timeout.\n-- E 1.2 no-success due to timeout.\n-- TODO: Metis 2.2 (release 20100825) success (SZS status CounterSatisfiable).\n{-# ATP prove +-comm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"81eecc922c3dc4fc227770d378988bd4c8881616","subject":"GroupHom: split the verifier function","message":"GroupHom: split the verifier function\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom.agda","new_file":"ZK\/GroupHom.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Type.Eq using (Eq?; _==_; \u2261\u21d2==; ==\u21d2\u2261)\nopen import Function using (case_of_)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nimport ZK.SigmaProtocol.KnownStatement\n\nmodule ZK.GroupHom\n    {G+ G* : Type} (grp-G+ : Group G+) (grp-G* : Group G*)\n    {{eq?-G* : Eq? G*}}\n\n    (open Additive-Group grp-G+ hiding (_\u2297_))\n    (open Multiplicative-Group grp-G* hiding (_^_))\n\n    {C : Type} -- e.g. C \u2286 \u2115\n    (_>_ : C \u2192 C \u2192 Type)\n    (swap? : {c\u2080 c\u2081 : C} \u2192 c\u2080 \u2262 c\u2081 \u2192 (c\u2080 > c\u2081) \u228e (c\u2081 > c\u2080))\n    (_\u2297_ : G+ \u2192 C \u2192 G+)\n    (_^_ : G* \u2192 C \u2192 G*)\n    (_\u2212\u1d9c_ : C \u2192 C \u2192 C)\n    (1\/ : C \u2192 C)\n\n    (\u03c6 : G+ \u2192 G*) -- For instance \u03c6(x) = g^x\n    (\u03c6-hom : GroupHomomorphism grp-G+ grp-G* \u03c6)\n    (\u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297 c) \u2261 \u03c6 x ^ c)\n\n    (Y : G*)\n\n    (^-\u2212\u1d9c  : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 Y ^(c\u2080 \u2212\u1d9c c\u2081) \u2261 (Y ^ c\u2080) \/ (Y ^ c\u2081))\n    (^-^-1\/ : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 (Y ^(c\u2080 \u2212\u1d9c c\u2081))^(1\/(c\u2080 \u2212\u1d9c c\u2081)) \u2261 Y)\n    where\n\n  Commitment = G*\n  Challenge  = C\n  Response   = G+\n  Witness    = G+\n  Randomness = G+\n  _\u2208y : Witness \u2192 Type\n  x \u2208y = \u03c6 x \u2261 Y\n\n  open ZK.SigmaProtocol.KnownStatement Commitment Challenge Response Randomness Witness _\u2208y public\n\n  prover : Prover\n  prover a x = \u03c6 a , response\n     where response : Challenge \u2192 Response\n           response c = (x \u2297 c) + a\n\n  verifier' : (A : Commitment)(c : Challenge)(s : Response) \u2192 _\n  verifier' A c s = \u03c6 s == ((Y ^ c) * A)\n\n  verifier : Verifier\n  verifier (mk A c s) = verifier' A c s\n\n  -- We still call it Schnorr since essentially not much changed.\n  -- The scheme abstracts away g^ as an homomorphism called \u03c6\n  -- and one get to distinguish between the types C vs G+ and\n  -- their operations _\u2297_ vs _*_.\n  Schnorr : \u03a3-Protocol\n  Schnorr = (prover , verifier)\n\n  -- The simulator shows why it is so important for the\n  -- challenge to be picked once the commitment is known.\n  -- To fake a transcript, the challenge and response can\n  -- be arbitrarily chosen. However to be indistinguishable\n  -- from a valid proof they need to be picked at random.\n  simulator : Simulator\n  simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (Y ^ c)\u207b\u00b9 * \u03c6 s\n\n  swap-t\u00b2? : Transcript\u00b2[ _\u2262_ ] verifier \u2192 Transcript\u00b2[ _>_ ] verifier\n  swap-t\u00b2? t\u00b2 = t\u00b2'\n      module Swap? where\n        open Transcript\u00b2 t\u00b2 public\n          using (verify\u2080; verify\u2081; c\u2080\u2262c\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   )\n        t\u00b2' = case swap? c\u2080\u2262c\u2081 of \u03bb\n              { (inl c\u2080>c\u2081) \u2192 mk A c\u2080 c\u2081 c\u2080>c\u2081 r\u2080 r\u2081 verify\u2080 verify\u2081\n              ; (inr c\u2080<c\u2081) \u2192 mk A c\u2081 c\u2080 c\u2080<c\u2081 r\u2081 r\u2080 verify\u2081 verify\u2080\n              }\n\n  -- The extractor shows the importance of never reusing a\n  -- commitment. If the prover answers to two different challenges\n  -- comming from the same commitment then the knowledge of the\n  -- prover (the witness) can be extracted.\n  extractor : Extractor verifier\n  extractor t\u00b2 = x'\n      module Witness-extractor where\n        open Transcript\u00b2 (swap-t\u00b2? t\u00b2) public\n          using (verify\u2080; verify\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   ; c\u2080\u2262c\u2081 to c\u2080>c\u2081\n                   )\n        rd   = r\u2080 \u2212 r\u2081\n        cd   = c\u2080 \u2212\u1d9c c\u2081\n        1\/cd = 1\/ cd\n        x'   = rd \u2297 1\/cd\n\n  open \u2261-Reasoning\n  open GroupHomomorphism \u03c6-hom\n\n  correct : Correct Schnorr\n  correct x a c w\n    = \u2261\u21d2== (\u03c6((x \u2297 c) + a)  \u2261\u27e8 hom \u27e9\n            \u03c6(x \u2297 c)  * A   \u2261\u27e8 *= \u03c6-hom-iterated idp \u27e9\n            (\u03c6 x ^ c) * A   \u2261\u27e8 ap (\u03bb z \u2192 (z ^ c) * A) w \u27e9\n            (Y ^ c)   * A   \u220e)\n    where\n      A = \u03c6 a\n\n  shvzk : Special-Honest-Verifier-Zero-Knowledge Schnorr\n  shvzk = record { simulator = simulator\n                 ; correct-simulator =\n                    \u03bb _ _ \u2192 \u2261\u21d2== (! elim-*-!assoc= (snd \u207b\u00b9-inverse)) }\n\n  module _ (t\u00b2 : Transcript\u00b2 verifier) where\n    open Witness-extractor t\u00b2\n\n    \u03c6rd\u2261ycd : _\n    \u03c6rd\u2261ycd\n      = \u03c6 rd                        \u2261\u27e8by-definition\u27e9\n        \u03c6 (r\u2080 \u2212 r\u2081)                 \u2261\u27e8 \u2212-\/ \u27e9\n        \u03c6 r\u2080 \/ \u03c6 r\u2081                 \u2261\u27e8 \/= (==\u21d2\u2261 verify\u2080) (==\u21d2\u2261 verify\u2081) \u27e9\n        (Y ^ c\u2080 * A) \/ (Y ^ c\u2081 * A) \u2261\u27e8 elim-*-right-\/ \u27e9\n        Y ^ c\u2080 \/ Y ^ c\u2081             \u2261\u27e8 ! ^-\u2212\u1d9c c\u2080>c\u2081 \u27e9\n        Y ^(c\u2080 \u2212\u1d9c c\u2081)               \u2261\u27e8by-definition\u27e9\n        Y ^ cd                      \u220e\n\n    -- The extracted x is correct\n    extractor-ok : \u03c6 x' \u2261 Y\n    extractor-ok\n      = \u03c6(rd \u2297 1\/cd)    \u2261\u27e8 \u03c6-hom-iterated \u27e9\n        \u03c6 rd ^ 1\/cd     \u2261\u27e8 ap\u2082 _^_ \u03c6rd\u2261ycd idp \u27e9\n        (Y ^ cd)^ 1\/cd  \u2261\u27e8 ^-^-1\/ c\u2080>c\u2081 \u27e9\n        Y               \u220e\n\n  special-soundness : Special-Soundness Schnorr\n  special-soundness = record { extractor = extractor\n                             ; extract-valid-witness = extractor-ok }\n\n  special-\u03a3-protocol : Special-\u03a3-Protocol\n  special-\u03a3-protocol = record { \u03a3-protocol = Schnorr ; correct = correct ; shvzk = shvzk ; ssound = special-soundness }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Type.Eq using (Eq?; _==_; \u2261\u21d2==; ==\u21d2\u2261)\nopen import Function using (case_of_)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nimport ZK.SigmaProtocol.KnownStatement\n\nmodule ZK.GroupHom\n    {G+ G* : Type} (grp-G+ : Group G+) (grp-G* : Group G*)\n    {{eq?-G* : Eq? G*}}\n\n    (open Additive-Group grp-G+ hiding (_\u2297_))\n    (open Multiplicative-Group grp-G* hiding (_^_))\n\n    {C : Type} -- e.g. C \u2286 \u2115\n    (_>_ : C \u2192 C \u2192 Type)\n    (swap? : {c\u2080 c\u2081 : C} \u2192 c\u2080 \u2262 c\u2081 \u2192 (c\u2080 > c\u2081) \u228e (c\u2081 > c\u2080))\n    (_\u2297_ : G+ \u2192 C \u2192 G+)\n    (_^_ : G* \u2192 C \u2192 G*)\n    (_\u2212\u1d9c_ : C \u2192 C \u2192 C)\n    (1\/ : C \u2192 C)\n\n    (\u03c6 : G+ \u2192 G*) -- For instance \u03c6(x) = g^x\n    (\u03c6-hom : GroupHomomorphism grp-G+ grp-G* \u03c6)\n    (\u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297 c) \u2261 \u03c6 x ^ c)\n\n    (Y : G*)\n\n    (^-\u2212\u1d9c  : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 Y ^(c\u2080 \u2212\u1d9c c\u2081) \u2261 (Y ^ c\u2080) \/ (Y ^ c\u2081))\n    (^-^-1\/ : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 (Y ^(c\u2080 \u2212\u1d9c c\u2081))^(1\/(c\u2080 \u2212\u1d9c c\u2081)) \u2261 Y)\n    where\n\n  Commitment = G*\n  Challenge  = C\n  Response   = G+\n  Witness    = G+\n  Randomness = G+\n  _\u2208y : Witness \u2192 Type\n  x \u2208y = \u03c6 x \u2261 Y\n\n  open ZK.SigmaProtocol.KnownStatement Commitment Challenge Response Randomness Witness _\u2208y public\n\n  prover : Prover\n  prover a x = \u03c6 a , response\n     where response : Challenge \u2192 Response\n           response c = (x \u2297 c) + a\n\n  verifier : Verifier\n  verifier (mk A c s) = \u03c6 s == ((Y ^ c) * A)\n\n  -- We still call it Schnorr since essentially not much changed.\n  -- The scheme abstracts away g^ as an homomorphism called \u03c6\n  -- and one get to distinguish between the types C vs G+ and\n  -- their operations _\u2297_ vs _*_.\n  Schnorr : \u03a3-Protocol\n  Schnorr = (prover , verifier)\n\n  -- The simulator shows why it is so important for the\n  -- challenge to be picked once the commitment is known.\n  -- To fake a transcript, the challenge and response can\n  -- be arbitrarily chosen. However to be indistinguishable\n  -- from a valid proof they need to be picked at random.\n  simulator : Simulator\n  simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (Y ^ c)\u207b\u00b9 * \u03c6 s\n\n  swap-t\u00b2? : Transcript\u00b2[ _\u2262_ ] verifier \u2192 Transcript\u00b2[ _>_ ] verifier\n  swap-t\u00b2? t\u00b2 = t\u00b2'\n      module Swap? where\n        open Transcript\u00b2 t\u00b2 public\n          using (verify\u2080; verify\u2081; c\u2080\u2262c\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   )\n        t\u00b2' = case swap? c\u2080\u2262c\u2081 of \u03bb\n              { (inl c\u2080>c\u2081) \u2192 mk A c\u2080 c\u2081 c\u2080>c\u2081 r\u2080 r\u2081 verify\u2080 verify\u2081\n              ; (inr c\u2080<c\u2081) \u2192 mk A c\u2081 c\u2080 c\u2080<c\u2081 r\u2081 r\u2080 verify\u2081 verify\u2080\n              }\n\n  -- The extractor shows the importance of never reusing a\n  -- commitment. If the prover answers to two different challenges\n  -- comming from the same commitment then the knowledge of the\n  -- prover (the witness) can be extracted.\n  extractor : Extractor verifier\n  extractor t\u00b2 = x'\n      module Witness-extractor where\n        open Transcript\u00b2 (swap-t\u00b2? t\u00b2) public\n          using (verify\u2080; verify\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   ; c\u2080\u2262c\u2081 to c\u2080>c\u2081\n                   )\n        rd   = r\u2080 \u2212 r\u2081\n        cd   = c\u2080 \u2212\u1d9c c\u2081\n        1\/cd = 1\/ cd\n        x'   = rd \u2297 1\/cd\n\n  open \u2261-Reasoning\n  open GroupHomomorphism \u03c6-hom\n\n  correct : Correct Schnorr\n  correct x a c w\n    = \u2261\u21d2== (\u03c6((x \u2297 c) + a)  \u2261\u27e8 hom \u27e9\n            \u03c6(x \u2297 c)  * A   \u2261\u27e8 *= \u03c6-hom-iterated idp \u27e9\n            (\u03c6 x ^ c) * A   \u2261\u27e8 ap (\u03bb z \u2192 (z ^ c) * A) w \u27e9\n            (Y ^ c)   * A   \u220e)\n    where\n      A = \u03c6 a\n\n  shvzk : Special-Honest-Verifier-Zero-Knowledge Schnorr\n  shvzk = record { simulator = simulator\n                 ; correct-simulator =\n                    \u03bb _ _ \u2192 \u2261\u21d2== (! elim-*-!assoc= (snd \u207b\u00b9-inverse)) }\n\n  module _ (t\u00b2 : Transcript\u00b2 verifier) where\n    open Witness-extractor t\u00b2\n\n    \u03c6rd\u2261ycd : _\n    \u03c6rd\u2261ycd\n      = \u03c6 rd                        \u2261\u27e8by-definition\u27e9\n        \u03c6 (r\u2080 \u2212 r\u2081)                 \u2261\u27e8 \u2212-\/ \u27e9\n        \u03c6 r\u2080 \/ \u03c6 r\u2081                 \u2261\u27e8 \/= (==\u21d2\u2261 verify\u2080) (==\u21d2\u2261 verify\u2081) \u27e9\n        (Y ^ c\u2080 * A) \/ (Y ^ c\u2081 * A) \u2261\u27e8 elim-*-right-\/ \u27e9\n        Y ^ c\u2080 \/ Y ^ c\u2081             \u2261\u27e8 ! ^-\u2212\u1d9c c\u2080>c\u2081 \u27e9\n        Y ^(c\u2080 \u2212\u1d9c c\u2081)               \u2261\u27e8by-definition\u27e9\n        Y ^ cd                      \u220e\n\n    -- The extracted x is correct\n    extractor-ok : \u03c6 x' \u2261 Y\n    extractor-ok\n      = \u03c6(rd \u2297 1\/cd)    \u2261\u27e8 \u03c6-hom-iterated \u27e9\n        \u03c6 rd ^ 1\/cd     \u2261\u27e8 ap\u2082 _^_ \u03c6rd\u2261ycd idp \u27e9\n        (Y ^ cd)^ 1\/cd  \u2261\u27e8 ^-^-1\/ c\u2080>c\u2081 \u27e9\n        Y               \u220e\n\n  special-soundness : Special-Soundness Schnorr\n  special-soundness = record { extractor = extractor\n                             ; extract-valid-witness = extractor-ok }\n\n  special-\u03a3-protocol : Special-\u03a3-Protocol\n  special-\u03a3-protocol = record { \u03a3-protocol = Schnorr ; correct = correct ; shvzk = shvzk ; ssound = special-soundness }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6635d49027cc0b54fa3b226c05ed130b0f7764b2","subject":"spatie tussen Koopa en Troopa","message":"spatie tussen Koopa en Troopa\n","repos":"toonn\/popartt","old_file":"koopa.agda","new_file":"koopa.agda","new_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_; _<_ to _F<_; suc to fsuc;\n    zero to fzero)\n  open import Data.Vec renaming (map to vmap; lookup to vlookup;\n                                     replicate to vreplicate)\n  open import Data.Unit\n  open import Data.Empty\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa : Color \u2192 Set where\n    _KT : (c : Color) \u2192 KoopaTroopa c\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  data Clearance : Set where\n    Low  : Clearance\n    High : Clearance\n    God  : Clearance\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n      clr : Clearance\n\n  data _c>_ : Color \u2192 Clearance \u2192 Set where\n    <red>   : \u2200 {c} \u2192 c c> Low\n    <green> : Green c> High\n\n  data _follows_\u27e8_\u27e9 : Position \u2192 Position \u2192 Color \u2192 Set where\n    stay  : \u2200 {c x y} \u2192 pos x y gas Low follows pos x y gas Low \u27e8 c \u27e9\n    next  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos (suc x) y gas cl follows pos x y gas Low \u27e8 c \u27e9\n    back  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos x y gas cl follows pos (suc x) y gas Low \u27e8 c \u27e9\n    -- jump  : \u2200 {c x y} \u2192 pos x (suc y) gas High follows pos x y gas Low \u27e8 c \u27e9\n    fall  : \u2200 {c cl x y} \u2192 pos x y gas cl follows pos x (suc y) gas High \u27e8 c \u27e9\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path {c : Color} (Koopa : KoopaTroopa c) :\n       Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p \u27e8 c \u27e9\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 gas Low) (pos 0 0 gas Low)\n  ex_path = pos 0 0 gas Low \u21a0\u27e8 next \u27e9\n            pos 1 0 gas Low \u21a0\u27e8 back \u27e9\n            pos 0 0 gas Low \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192\n                   Vec Position n\n  matToPosVec []           []        _ _ = []\n  matToPosVec (mat \u2237 mats) (under \u2237 unders) x y =\n    pos x y mat cl \u2237 matToPosVec mats unders (x + 1) y\n      where\n        clearance : Material \u2192 Material \u2192 Clearance\n        clearance gas   gas   = High\n        clearance gas   solid = Low\n        clearance solid _     = God\n        cl = clearance mat under\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs [] = []\n  matToPosVecs (_\u2237_ {y} mats matss) =\n    matToPosVec mats (unders matss gas) 0 y \u2237 matToPosVecs matss\n    where\n      unders : \u2200 {m n \u2113}{A : Set \u2113} \u2192 Vec (Vec A m) n \u2192 A \u2192 Vec A m\n      unders [] fallback = vreplicate fallback\n      unders (us \u2237 _) _ = us\n\n  matsToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matsToMat matss = Mat (reverse (matToPosVecs matss))\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = matsToMat (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 [])\n\n  _<'_ : \u2115 \u2192 \u2115 \u2192 Set\n  m <' zero = \u22a5\n  zero <' suc n = \u22a4\n  suc m <' suc n = m <' n\n\n  fromNat : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  fromNat {zero} k {}\n  fromNat {suc n} zero = fzero\n  fromNat {suc n} (suc k) {p} = fsuc (fromNat k {p})\n \n  f : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  f = fromNat\n\n  p : (x : Fin 10) \u2192 (y : Fin 7) \u2192 Position\n  p x y = lookup y x example_level\n\n  red_path_one : Path (Red KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  red_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                 p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  red_path_two : Path (Red KT) (p (f 2) (f 1)) (p (f 3) (f 1))\n  red_path_two = p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 4) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 stay \u27e9\n                 []\n\n  -- -- Type error shows up 'late' because 'cons' is right associative\n  -- red_nopath_one : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- red_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9\n  --                  p (f 0) (f 1) \u21a0\u27e8 stay \u27e9\n  --                  []\n\n  -- -- Red KoopaTroopa can't step into a wall\n  -- red_nopath_two : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- red_nopath_two = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n\n  -- -- Red KoopaTroopa can't step into air\n  -- red_nopath_three : Path (Red KT) (p (f 4) (f 1)) (p (f 5) (f 1))\n  -- red_nopath_three = p (f 4) (f 1) \u21a0\u27e8 next \u27e9 []\n\n  -- Any path that is valid for red KoopaTroopas, is also valid for green\n  -- KoopaTroopas because we did not constrain KoopaTroopas to only turn\n  -- When there is an obstacle\n  green_path_one : Path (Green KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  green_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                   p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                   p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  green_path_two : Path (Green KT) (p (f 7) (f 6)) (p (f 5) (f 0))\n  green_path_two = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 6) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 5) (f 6) \u21a0\u27e8 fall \u27e9\n                   p (f 5) (f 5) \u21a0\u27e8 fall \u27e9\n                   p (f 5) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 4) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 3) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 2) (f 4) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 3) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 2) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                   p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 4) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 5) (f 1) \u21a0\u27e8 fall \u27e9\n                   []\n\n  -- -- Green KoopaTroopa can't step into a wall\n  -- green_nopath_one : Path (Green KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- green_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n","old_contents":"{-\n\n    Verified KoopaTroopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_; _<_ to _F<_; suc to fsuc;\n    zero to fzero)\n  open import Data.Vec renaming (map to vmap; lookup to vlookup;\n                                     replicate to vreplicate)\n  open import Data.Unit\n  open import Data.Empty\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa : Color \u2192 Set where\n    _KT : (c : Color) \u2192 KoopaTroopa c\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  data Clearance : Set where\n    Low  : Clearance\n    High : Clearance\n    God  : Clearance\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n      clr : Clearance\n\n  data _c>_ : Color \u2192 Clearance \u2192 Set where\n    <red>   : \u2200 {c} \u2192 c c> Low\n    <green> : Green c> High\n\n  data _follows_\u27e8_\u27e9 : Position \u2192 Position \u2192 Color \u2192 Set where\n    stay  : \u2200 {c x y} \u2192 pos x y gas Low follows pos x y gas Low \u27e8 c \u27e9\n    next  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos (suc x) y gas cl follows pos x y gas Low \u27e8 c \u27e9\n    back  : \u2200 {c cl x y}{{_ : c c> cl}} \u2192\n            pos x y gas cl follows pos (suc x) y gas Low \u27e8 c \u27e9\n    -- jump  : \u2200 {c x y} \u2192 pos x (suc y) gas High follows pos x y gas Low \u27e8 c \u27e9\n    fall  : \u2200 {c cl x y} \u2192 pos x y gas cl follows pos x (suc y) gas High \u27e8 c \u27e9\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path {c : Color} (Koopa : KoopaTroopa c) :\n       Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p \u27e8 c \u27e9\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 gas Low) (pos 0 0 gas Low)\n  ex_path = pos 0 0 gas Low \u21a0\u27e8 next \u27e9\n            pos 1 0 gas Low \u21a0\u27e8 back \u27e9\n            pos 0 0 gas Low \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192\n                   Vec Position n\n  matToPosVec []           []        _ _ = []\n  matToPosVec (mat \u2237 mats) (under \u2237 unders) x y =\n    pos x y mat cl \u2237 matToPosVec mats unders (x + 1) y\n      where\n        clearance : Material \u2192 Material \u2192 Clearance\n        clearance gas   gas   = High\n        clearance gas   solid = Low\n        clearance solid _     = God\n        cl = clearance mat under\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs [] = []\n  matToPosVecs (_\u2237_ {y} mats matss) =\n    matToPosVec mats (unders matss gas) 0 y \u2237 matToPosVecs matss\n    where\n      unders : \u2200 {m n \u2113}{A : Set \u2113} \u2192 Vec (Vec A m) n \u2192 A \u2192 Vec A m\n      unders [] fallback = vreplicate fallback\n      unders (us \u2237 _) _ = us\n\n  matsToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matsToMat matss = Mat (reverse (matToPosVecs matss))\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = matsToMat (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 [])\n\n  _<'_ : \u2115 \u2192 \u2115 \u2192 Set\n  m <' zero = \u22a5\n  zero <' suc n = \u22a4\n  suc m <' suc n = m <' n\n\n  fromNat : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  fromNat {zero} k {}\n  fromNat {suc n} zero = fzero\n  fromNat {suc n} (suc k) {p} = fsuc (fromNat k {p})\n \n  f : \u2200 {n}(k : \u2115){_ : k <' n} \u2192 Fin n\n  f = fromNat\n\n  p : (x : Fin 10) \u2192 (y : Fin 7) \u2192 Position\n  p x y = lookup y x example_level\n\n  red_path_one : Path (Red KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  red_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                 p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                 p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  red_path_two : Path (Red KT) (p (f 2) (f 1)) (p (f 3) (f 1))\n  red_path_two = p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                 p (f 4) (f 1) \u21a0\u27e8 back \u27e9\n                 p (f 3) (f 1) \u21a0\u27e8 stay \u27e9\n                 []\n\n  -- -- Type error shows up 'late' because 'cons' is right associative\n  -- red_nopath_one : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- red_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9\n  --                  p (f 0) (f 1) \u21a0\u27e8 stay \u27e9\n  --                  []\n\n  -- -- Red KoopaTroopa can't step into a wall\n  -- red_nopath_two : Path (Red KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- red_nopath_two = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n\n  -- -- Red KoopaTroopa can't step into air\n  -- red_nopath_three : Path (Red KT) (p (f 4) (f 1)) (p (f 5) (f 1))\n  -- red_nopath_three = p (f 4) (f 1) \u21a0\u27e8 next \u27e9 []\n\n  -- Any path that is valid for red KoopaTroopas, is also valid for green\n  -- KoopaTroopas because we did not constrain KoopaTroopas to only turn\n  -- When there is an obstacle\n  green_path_one : Path (Green KT) (p (f 7) (f 6)) (p (f 8) (f 6))\n  green_path_one = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 6) (f 6) \u21a0\u27e8 next \u27e9\n                   p (f 7) (f 6) \u21a0\u27e8 next \u27e9\n                   p (f 8) (f 6) \u21a0\u27e8 stay \u27e9 []\n\n  green_path_two : Path (Green KT) (p (f 7) (f 6)) (p (f 5) (f 0))\n  green_path_two = p (f 7) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 6) (f 6) \u21a0\u27e8 back \u27e9\n                   p (f 5) (f 6) \u21a0\u27e8 fall \u27e9\n                   p (f 5) (f 5) \u21a0\u27e8 fall \u27e9\n                   p (f 5) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 4) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 3) (f 4) \u21a0\u27e8 back \u27e9\n                   p (f 2) (f 4) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 3) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 2) \u21a0\u27e8 fall \u27e9\n                   p (f 2) (f 1) \u21a0\u27e8 back \u27e9\n                   p (f 1) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 2) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 3) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 4) (f 1) \u21a0\u27e8 next \u27e9\n                   p (f 5) (f 1) \u21a0\u27e8 fall \u27e9\n                   []\n\n  -- -- Green KoopaTroopa can't step into a wall\n  -- green_nopath_one : Path (Green KT) (p (f 1) (f 1)) (p (f 0) (f 1))\n  -- green_nopath_one = p (f 1) (f 1) \u21a0\u27e8 back \u27e9 []\n","returncode":0,"stderr":"","license":"bsd-2-clause","lang":"Agda"}
{"commit":"a7de1a448948e2f543a1a3d8c684eaed65d53c30","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0 where\n\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  exb0      : Val 0 \u21d3 0\n  exb0      = \u21d3Base\n\n  exb3      : Val 3 \u21d3 3\n  exb3      = \u21d3Base\n\n  exs331    : Div (Val 3) (Val 3) \u21d3 1\n  exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n  exs931    : Div (Val 9) (Val 3) \u21d3 3\n  exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n  exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n  exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n  exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n  exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MPT where\n\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WPP where\n\n    exwpp  : Expr -> Set\n    exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp' = refl\n\n    wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1  = \u21d3Base\n    wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd   = \u21d3Step \u21d3Base \u21d3Base\n    wppd'  : Set\n    wppd'  = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given e : Expr, satisfaction of predicate means\n  - 'e' evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define safety criteria that we believe in\n  - prove every expression that satisfies that criteria satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - so SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SD where\n\n    exsdv : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv = refl\n    exsdd : SafeDiv (Div (Val 3) (Val 3))\n          \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsdd = refl\n\n    exsdx : SafeDiv (Div (Val 3) (Val 0))\n          \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (   _ , (sdel , sder)) with \u27e6 el \u27e7 | \u27e6 er \u27e7 | correct el sdel | correct er sder\n  correct (Div  _  _) (ernz , (   _ ,    _)) | Pure _       | Pure Zero     |     _ | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Pure (Succ _) | el\u21d3vl | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Step Abort _  |     _ | ()\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Step Abort _ | _             | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7 | sound el | sound er\n  sound (Div el er) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div el er) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div el er) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div el er) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7\n  inDom (Div el er) () | Pure v1      | Pure Zero\n  inDom (Div el er) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div el er) () | Pure _       | Step Abort _\n  inDom (Div el er) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ el\n  wpPartial1 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 {el} {er} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {el} {er} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux el x eq\n  wpPartial1 {el} {er} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {el} {er} () | Step Abort x | eq         | ver\n\n  wpPartial2 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ er\n  wpPartial2 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n  wpPartial2 {el} {er} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux er y eqy\n  wpPartial2 {el} {er} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | ser           | eq2\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n      | complete el (wpPartial1 {el} {er} h)\n      | complete er (wpPartial2 {el} {er} h)\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div el er) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div el er) h | Step Abort x | eq1         | ser           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  exb0 : Val 0 \u21d3 0\n  exb0 = \u21d3Base\n\n  exb3 : Val 3 \u21d3 3\n  exb3 = \u21d3Base\n\n  exs331 : Div (Val 3) (Val 3) \u21d3 1\n  exs331 = \u21d3Step \u21d3Base \u21d3Base\n\n  exs931 : Div (Val 9) (Val 3) \u21d3 3\n  exs931 = \u21d3Step \u21d3Base \u21d3Base\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  relate the two definitions:\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, define this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder)) with \u27e6 el \u27e7 | \u27e6 er \u27e7 | correct el sdel | correct er sder\n  correct (Div el er) (ernz , (sdel , sder)) | Pure v1      | Pure Zero         | el\u21d3v1 | er\u21d3Z   = magic (ernz er\u21d3Z)\n  correct (Div el er) (ernz , (sdel , sder)) | Pure v1      | Pure (Succ v2)    | el\u21d3v1 | er\u21d3Sv2 = \u21d3Step el\u21d3v1 er\u21d3Sv2\n  correct (Div el er) (ernz , (sdel , sder)) | Pure v1      | Step Abort \u22a5\u2192FCRN | el\u21d3v1 | ()\n  correct (Div el er) (ernz , (sdel , sder)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7 | sound el | sound er\n  sound (Div el er) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div el er) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div el er) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div el er) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7\n  inDom (Div el er) () | Pure v1      | Pure Zero\n  inDom (Div el er) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div el er) () | Pure _       | Step Abort _\n  inDom (Div el er) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ el\n  wpPartial1 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 {el} {er} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {el} {er} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux el x eq\n  wpPartial1 {el} {er} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {el} {er} () | Step Abort x | eq         | ver\n\n  wpPartial2 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ er\n  wpPartial2 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n  wpPartial2 {el} {er} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux er y eqy\n  wpPartial2 {el} {er} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | ser           | eq2\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n      | complete el (wpPartial1 {el} {er} h)\n      | complete er (wpPartial2 {el} {er} h)\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div el er) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div el er) h | Step Abort x | eq1         | ser           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"2a3161c1ad3d322da083c544df002cad0b9a74ab","subject":"Relate different definitions of validity","message":"Relate different definitions of validity\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n-- XXX This would be more elegant as a datatype \u2014 that would avoid the need for\n-- an equality proof.\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1' \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n   [ \u03c4 ] dv from v1 to v2 \u00d7 v1 \u2261 v1' \u00d7 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} {\u2205} tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {_ \u2022 _} {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} {dv \u2022 .v1 \u2022 d\u03c1} (dvv , refl , d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {\u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive \u03c4 t (daa , refl , d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 v1 v2 dv \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb v1 v2 dv x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) (da , db) (daa , dbb) = (fromto\u2192valid  _ _ _ daa) , (fromto\u2192valid _ _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} f1 f2 df dff = \u03bb a da ada \u2192\n  fromto\u2192valid _ _ _ (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto _ _ ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid _ _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 _ _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\n-- XXX This would be more elegant as a datatype \u2014 that would avoid the need for\n-- an equality proof.\n[_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\n[ \u2205 ]\u0393 \u2205 from \u2205 to \u2205 = \u22a4\n[ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1' \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12) =\n   [ \u03c4 ] dv from v1 to v2 \u00d7 v1 \u2261 v1' \u00d7 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} {\u2205} tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {_ \u2022 _} {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} {dv \u2022 .v1 \u2022 d\u03c1} (dvv , refl , d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {\u03c11 \u03c12 d\u03c1} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : eCh \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dv \u2022 .v1 \u2022 d\u03c1) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv , refl , d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x _ _ _ d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb da a1 a2 daa \u2192\n  fromtoDerive \u03c4 t (daa , refl , d\u03c1\u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6a8bcf946435b96446026676119f966462075f6e","subject":"Only white spaces.","message":"Only white spaces.\n\nIgnore-this: 7e6768261b39a8b4436e08638e9598f1\n\ndarcs-hash:20100909160311-3bd4e-cb2365f5ecefa9aa12d7517199d998a8c52233f8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Minimal.agda","new_file":"LTC\/Minimal.agda","new_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\nmodule LTC.Minimal where\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain.\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n\n  -- LTC error\n  error  : D\n\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n------------------------------------------------------------------------------\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties.\n\n-- The substitution is defined in LTC.MinimalER\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Empty       public\nopen import LTC.Data.Product     public\nopen import LTC.Data.Sum         public\nopen import LTC.Data.Unit        public\nopen import LTC.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom cB\u2081 #-}\n{-# ATP axiom cB\u2082 #-}\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom cZ\u2081 #-}\n{-# ATP axiom cZ\u2082 #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom cBeta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom cFix #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Agda as a logical framework for LTC\n------------------------------------------------------------------------------\n{-\n\nLTC                              Agda\n* Logical constants              * Curry-Howard isomorphism\n* Equality                       * Identity type\n* Term language                  * Postulates\n* Inductive predicates           * Inductive families\n-}\n\nmodule LTC.Minimal where\n\ninfixl 6 _\u2219_\ninfix  5 if_then_else_\ninfix  4 _\u2261_\n\n------------------------------------------------------------------------------\n-- The universal domain.\n-- N.B. The following module is exported by this module.\nopen import LTC.Minimal.Core public\n\n------------------------------------------------------------------------------\n-- The term language of LTC correspond to the PCF terms\n\n--   t ::= x | t t | \\x -> t\n--           | true | false | if t then t else t\n--           | 0 | succ t | pred t | isZero t\n--           | error\n--           | fix t\n\npostulate\n\n  -- LTC partial booleans.\n  true          : D\n  false         : D\n  if_then_else_ : D \u2192 D \u2192 D \u2192 D\n  -- LTC partial natural numbers.\n  zero   : D\n  succ   : D \u2192 D\n  pred   : D \u2192 D\n  isZero : D \u2192 D\n  -- LTC abstraction.\n  lam    : (D \u2192 D) \u2192 D\n  -- LTC application\n  -- Left associative aplication operator\n  -- The Agda application has higher precedence level than LTC application\n  _\u2219_    : D \u2192 D \u2192 D\n  -- LTC error\n  error  : D\n  -- LTC fixed point operator\n  fix    : (D \u2192 D) \u2192 D\n  -- fixFO  : D\n\n------------------------------------------------------------------------------\n-- The LTC's equality is the propositional identity on 'D'.\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties.\n\n-- The substitution is defined in LTC.MinimalER\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\ntrans : {x y z : D} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl y\u2261z = y\u2261z\n\n------------------------------------------------------------------------------\n-- Logical constants: Curry-Howard isomorphism\n\n-- The LTC's logical constants are the type theory's logical\n-- constants via the Curry-Howard isomorphism.\n-- For the implication and the universal quantifier\n-- we use Agda (dependent) function type.\n\n-- N.B. The following modules are exported by this module.\nopen import LTC.Data.Empty       public\nopen import LTC.Data.Product     public\nopen import LTC.Data.Sum         public\nopen import LTC.Data.Unit        public\nopen import LTC.Relation.Nullary public\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\npostulate\n  -- Conversion rules for booleans.\n  cB\u2081 : (d\u2081 : D){d\u2082 : D} \u2192 if true  then d\u2081 else d\u2082 \u2261 d\u2081\n  cB\u2082 : {d\u2081 : D}(d\u2082 : D) \u2192 if false then d\u2081 else d\u2082 \u2261 d\u2082\n{-# ATP axiom cB\u2081 #-}\n{-# ATP axiom cB\u2082 #-}\n\npostulate\n  -- Conversion rules for pred.\n  cP\u2081 : pred zero               \u2261 zero\n  cP\u2082 : (n : D) \u2192 pred (succ n) \u2261 n\n{-# ATP axiom cP\u2081 #-}\n{-# ATP axiom cP\u2082 #-}\n\npostulate\n  -- Conversion rules for isZero\n  cZ\u2081 : isZero zero               \u2261 true\n  cZ\u2082 : (n : D) \u2192 isZero (succ n) \u2261 false\n{-# ATP axiom cZ\u2081 #-}\n{-# ATP axiom cZ\u2082 #-}\n\npostulate\n  -- Conversion rule for the abstraction and the application.\n  cBeta : (f : D \u2192 D) \u2192 (a : D) \u2192 (lam f) \u2219 a \u2261 f a\n{-# ATP axiom cBeta #-}\n\npostulate\n  -- Conversion rule for the fixed pointed operator.\n  cFix   : (f : D \u2192 D) \u2192 fix f \u2261 f (fix  f)\n  -- cFixFO : (f : D) \u2192 fixFO \u2219 f  \u2261 f \u2219 (fixFO \u2219 f)\n{-# ATP axiom cFix #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\npostulate\n  true\u2260false : \u00ac (true \u2261 false)\n  0\u2260S        : {d : D} \u2192 \u00ac (zero \u2261 succ d)\n{-# ATP axiom true\u2260false #-}\n{-# ATP axiom 0\u2260S #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"865bb64f85f95816502c5a4fc895a8847c15e1c7","subject":"IDesc model: finer grain universe control","message":"IDesc model: finer grain universe control","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc zero I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc zero I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc zero I)(P : I -> Set) -> desc I D P -> IDesc zero (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc zero I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc zero I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc zero I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set1 where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc (suc zero) Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc (suc zero) Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b6630b986761233dedd2055977f44ad966db82eb","subject":"Strategy: runT and runS","message":"Strategy: runT and runS\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Strategy.agda","new_file":"Control\/Strategy.agda","new_contents":"module Control.Strategy where\n\nopen import Function.NP\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f)\n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n\n  private\n    Transcript = List (\u03a3 Q R)\n  module TranscriptRun (Oracle : Transcript \u2192 \u03a0 Q R){A} where\n    runT : M A \u2192 Transcript \u2192 A \u00d7 Transcript\n    runT (ask q? cont) t = let r = Oracle t q? in runT (cont r) ((q? , r) \u2237 t)\n    runT (done x)      t = x , t\n\n    open StatefulRun\n    -- runT is runS where the state is the transcript and the Oracle cannot change it\n    runT-runS : \u2200 (str : M A) t \u2192 runT str t \u2261 runS (\u03bb q s \u2192 let r = Oracle s q in r , (q , r) \u2237 s) str t\n    runT-runS (ask q? cont) t = let r = Oracle t q? in runT-runS (cont r) ((q? , r) \u2237 t)\n    runT-runS (done x)      t = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Control.Strategy where\n\nopen import Function.NP\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f)\n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n\n  private\n    Transcript = List (\u03a3 Q R)\n  module TranscriptRun (Oracle : Transcript \u2192 \u03a0 Q R){A} where\n    runT : M A \u2192 Transcript \u2192 A \u00d7 Transcript\n    runT (ask q? cont) t = let r = Oracle t q? in runT (cont r) ((q? , r) \u2237 t)\n    runT (done x)      t = x , t\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"90aff21d03b068f50fef74f2c22666b5ea68c76b","subject":"Improve Strategy to have a the response type depend on the query type","message":"Improve Strategy to have a the response type depend on the query type\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Strategy.agda","new_file":"Control\/Strategy.agda","new_contents":"module Control.Strategy where\n\nopen import Function\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f) \n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Control.Strategy where\n\nopen import Function\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q R A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 {k\u2080 k\u2081} \u2192 (q? : Q) (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q R : \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : Q \u2192 E R) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : Q \u2192 R) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f) \n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6d326f0aa2418112a658ff1d5bf5637c314550c6","subject":"Specify subcontext relation (see #29).","message":"Specify subcontext relation (see #29).\n\nOld-commit-hash: 84902a7b394ef9bf693b3ccdfe7f582349e93d2e\n","repos":"inc-lc\/ilc-agda","old_file":"Syntactic\/Contexts.agda","new_file":"Syntactic\/Contexts.agda","new_contents":"module Syntactic.Contexts\n    (Type : Set)\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Specialized congruence rules\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation.\n\nmodule Subcontexts where\n  infix 8 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  lift (keep \u03c4 \u2022 \u227c\u2081) this = this\n  lift (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (lift \u227c\u2081 x)\n  lift (drop \u03c4 \u2022 \u227c\u2081) x = that (lift \u227c\u2081 x)\n\n-- Currently, we export context prefixes, but that might change\n-- in the future\n\nopen Prefixes public\n","old_contents":"module Syntactic.Contexts\n    (Type : Set)\n  where\n\n-- CONTEXTS\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Specialized congruence rules\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Currently, we export context prefixes, but that might change\n-- in the future\n\nopen Prefixes public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f36f16e21da6b8cf325b4a80a5a673d43920fa78","subject":"Defined the exponential of-course \"!\".","message":"Defined the exponential of-course \"!\".\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = let h = \u03bb r \u2192 fst (f (fst r)) (snd r)\n        H = \u03bb r z \u2192 F (fst r) (snd r , z) , snd (f (fst r)) (snd r) z\n     in h , (H , cond)\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Z} \u2192\n      (\u03b1 \u2297\u1d63 \u03b2) u (F (fst u) (snd u , y) , snd (f (fst u)) (snd u) y) \u2192\n      \u03b3 (fst (f (fst u)) (snd u)) y\n  cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u} {v , z}\n  ... | p\u2084 with f u\n  ... | i , I = p\u2084 p\u2082 p\u2083\n  \ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   aux\u2082 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 (\u03bb z \u2192 fst (F (fst a , snd a) z) , snd (F (fst a , snd a) z)) \u2261 F a\n   aux\u2082 {u , v} = ext-set aux\u2083\n    where\n      aux\u2083 : {a : Z} \u2192 (fst (F (u , v) a) , snd (F (u , v) a)) \u2261 F (u , v) a\n      aux\u2083 {z} with F (u , v) z\n      ... | x , y = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = (ext-set aux) , ext-set (ext-set aux')\n where\n  aux : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb v z \u2192 snd (g a) v z)) \u2261 g a\n  aux {u} with g u\n  ... | i , I = refl\n  aux' : {u : U}{r : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G u (fst r , snd r) \u2261 G u r\n  aux' {u}{v , z} = refl\n\n\n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set} \u2192 (\u03b1 : U \u2192 X \u2192 Set) \u2192 U \u2192 \ud835\udd43 X \u2192 Set  \n!\u2092-cond {U}{X} \u03b1 u [] = \u22a4\n!\u2092-cond {U}{X} \u03b1 u (x :: xs) = (\u03b1 u x) \u00d7 (!\u2092-cond \u03b1 u xs)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U ,  X * , !\u2092-cond \u03b1\n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n{-\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = let h = \u03bb r \u2192 fst (f (fst r)) (snd r)\n        H = \u03bb r z \u2192 F (fst r) (snd r , z) , snd (f (fst r)) (snd r) z\n     in h , (H , cond)\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Z} \u2192\n      (\u03b1 \u2297\u1d63 \u03b2) u (F (fst u) (snd u , y) , snd (f (fst u)) (snd u) y) \u2192\n      \u03b3 (fst (f (fst u)) (snd u)) y\n  cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u} {v , z}\n  ... | p\u2084 with f u\n  ... | i , I = p\u2084 p\u2082 p\u2083\n  \ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   aux\u2082 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 (\u03bb z \u2192 fst (F (fst a , snd a) z) , snd (F (fst a , snd a) z)) \u2261 F a\n   aux\u2082 {u , v} = ext-set aux\u2083\n    where\n      aux\u2083 : {a : Z} \u2192 (fst (F (u , v) a) , snd (F (u , v) a)) \u2261 F (u , v) a\n      aux\u2083 {z} with F (u , v) z\n      ... | x , y = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = (ext-set aux) , ext-set (ext-set aux')\n where\n  aux : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb v z \u2192 snd (g a) v z)) \u2261 g a\n  aux {u} with g u\n  ... | i , I = refl\n  aux' : {u : U}{r : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G u (fst r , snd r) \u2261 G u r\n  aux' {u}{v , z} = refl\n{-\n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b54ca8e0e2d8cc08555efc44dba3c001cc549cbd","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val _) _ = \u21d3Base\n  correct (Div el er) (er\u21d30\u2192\u22a5 , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d30   = magic (er\u21d30\u2192\u22a5 er\u21d30)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb n -> Pure 3 >>= _\u00f7 n))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val _) _ = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Znr = magic (ernz er\u21d3Znr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                  -- Goal\n                  -- mustPT (\u03bb _ _ \u2192 \u22a4' zero)\n                  --                (Div el er) (Pure nl >>= (\u03bb nl' \u2192 Pure Zero >>= _\u00f7_ nl'))\n                  -- \u22a5\n                  -- Context\n                  -- h : mustPT _\u21d3_ (Div el er)  (\u27e6 el \u27e7 >>= (\u03bb nl' \u2192 \u27e6 er \u27e7    >>= _\u00f7_ nl'))\n    rewrite\n      \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                      (\u27e6 er \u27e7    >>= _\u00f7_ nl)\n    | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = h -- magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"a38ef1f5fbb3b394b35713c09035e0c2b6b74166","subject":"progress progress #3","message":"progress progress #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus (had an idea for a lemma here but it's false; see bottom)\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus (see bottom)\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FIndet i)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FBoxed v\u2082)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter (ITCastID (FIndet x)) FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter (ITCastID (FBoxed x)) FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n\n\n\n  -- this would fill two above, but it's false: \u2987\u2988 indet but its type, \u2987\u2988, is not ground\n  postulate\n    lem-groundindet : \u2200{ \u0394 d \u03c4} \u2192 \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 d indet \u2192 \u03c4 ground\n\n  -- this is also false, but tempting looking in two holes above from the ap case\n  postulate\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n\n  counter : \u03a3[ d \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0394 \u2208 hctx ]\n               ((d indet) \u00d7 (\u0394 , \u2205 \u22a2 d :: \u03c4 ==> \u03c4'))\n  counter = (\u2987\u2988\u27e8 Z , \u2205 \u27e9 \u27e8 b ==> b \u21d2 b ==> \u2987\u2988 \u27e9) ,\n            b , \u2987\u2988 ,  \u25a0 (Z , \u2205 , b ==> b ) ,\n            ICastArr (\u03bb ()) IEHole , TACast (TAEHole refl (\u03bb x d \u2192 \u03bb ())) (TCArr TCRefl TCHole1)\n\n  oops : \u22a5\n  oops = lem (\u03c02 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter)))))\n                (\u03c01 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter))))) b b b \u2987\u2988 \u2987\u2988\u27e8 Z , \u2205 \u27e9 refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import Agda.Primitive using (Level; lzero; lsuc) renaming (_\u2294_ to lmax)\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus (had an idea for a lemma here but it's false; see bottom)\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus (see bottom)\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FIndet i)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter (ITLam (FBoxed v\u2082)) FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n    where\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter (ITCastID (FIndet x)) FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter (ITCastID (FBoxed x)) FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n\n  -- this would fill two above, but it's false: \u2987\u2988 indet but its type, \u2987\u2988, is not ground\n  -- lem-groundindet : \u2200{ \u0394 d \u03c4} \u2192 \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 d indet \u2192 \u03c4 ground\n\n\n  counter : \u03a3[ d \u2208 dhexp ] \u03a3[ \u03c4 \u2208 htyp ] \u03a3[ \u03c4' \u2208 htyp ] \u03a3[ \u0394 \u2208 hctx ]\n               ((d indet) \u00d7 (\u0394 , \u2205 \u22a2 d :: \u03c4 ==> \u03c4'))\n  counter = (\u2987\u2988\u27e8 Z , \u2205 \u27e9 \u27e8 b ==> b \u21d2 b ==> \u2987\u2988 \u27e9) ,\n            b , \u2987\u2988 ,  \u25a0 (Z , \u2205 , b ==> b ) ,\n            ICastArr (\u03bb ()) IEHole , TACast (TAEHole refl (\u03bb x d \u2192 \u03bb ())) (TCArr TCRefl TCHole1)\n\n  postulate\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n\n  problem : \u22a5\n  problem = lem (\u03c02 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter)))))\n                (\u03c01 (\u03c02 (\u03c02 (\u03c02 (\u03c02 counter))))) b b b \u2987\u2988 \u2987\u2988\u27e8 Z , \u2205 \u27e9 refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1a5ecfaa043009dc412dfcb4e94cd305374d9e8c","subject":"(FIXUP) Move ignore-valid-change and update-valid-change.","message":"(FIXUP) Move ignore-valid-change and update-valid-change.\n\nOld-commit-hash: a3daf9914185d1d8c20a4512d8e75d1b42252370\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- Change : Type \u2192 Set\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : Change \u03c3) \u2192 valid v dv \u2192 Change \u03c4\n\n  ignore-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  ignore-valid-change (cons v _ _) = v\n\n  update-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  update-valid-change {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : Change \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-valid-change {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-valid-change {\u03c4})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  -- Change : Type \u2192 Set\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : Change \u03c3) \u2192 valid v dv \u2192 Change \u03c4\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3})\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u0394v R[v,\u0394v] \u2192 g (v \u229e\u208d \u03c3 \u208e \u0394v) \u229f\u208d \u03c4 \u208e f v\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : \u27e6 \u03c3 \u27e7) (\u0394v : Change \u03c3) (R[v,\u0394v] : valid v \u0394v)\n    \u2192 valid {\u03c4} (f v) (\u0394f v \u0394v R[v,\u0394v])\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (v \u229e\u208d \u03c3 \u208e \u0394v) \u2261 f v \u229e\u208d \u03c4 \u208e \u0394f v \u0394v R[v,\u0394v]\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u0394w R[w,\u0394w] \u2192\n    let\n      w\u2032 = w \u229e\u208d \u03c3 \u208e \u0394w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v w}) \u27e9\n        v w \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w \u0394w R[w,\u0394w]\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  ignore-valid-change (cons v _ _) = v\n\n  update-valid-change : ValidChange \u2286 \u27e6_\u27e7\n  update-valid-change {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 ignore-valid-change {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 update-valid-change {\u03c4})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ed2b8f7c86f336c9f4b7dd96a5e07d486970ac10","subject":"Adapt Base.Denotation.Environment to Agda master","message":"Adapt Base.Denotation.Environment to Agda master\n\nmeaningOf\u227c stopped working, essentially, and prevents even meaningOfVar\nfrom being inferred.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Denotation\/Environment.agda","new_file":"Base\/Denotation\/Environment.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Environments\n--\n-- This module defines the meaning of contexts, that is,\n-- the type of environments that fit a context, together\n-- with operations and properties of these operations.\n--\n-- This module is parametric in the syntax and semantics\n-- of types, so it can be reused for different calculi\n-- and models.\n------------------------------------------------------------------------\n\nmodule Base.Denotation.Environment\n    (Type : Set)\n    {\u2113}\n    (\u27e6_\u27e7Type : Type \u2192 Set \u2113)\n  where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Base.Syntax.Context Type\nopen import Base.Denotation.Notation\nopen import Base.Data.DependentList as DependentList\n\nprivate\n  instance\n    meaningOfType : Meaning Type\n    meaningOfType = meaning \u27e6_\u27e7Type\n\n\u27e6_\u27e7Context : Context \u2192 Set \u2113\n\u27e6_\u27e7Context = DependentList \u27e6_\u27e7Type\n\ninstance\n  meaningOfContext : Meaning Context\n  meaningOfContext = meaning \u27e6_\u27e7Context\n\n-- VARIABLES\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\ninstance\n  meaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\n  meaningOfVar = meaning \u27e6_\u27e7Var\n\n-- WEAKENING\n\n-- Remove a variable from an environment\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\ninstance\n  meaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\n  meaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\n-- Properties\n\n\u27e6\u27e7-\u227c-trans : \u2200 {\u0393\u2083 \u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (\u0393\u2033 : \u0393\u2082 \u227c \u0393\u2083) \u2192\n   \u2200 (\u03c1 : \u27e6 \u0393\u2083 \u27e7) \u2192 \u27e6_\u27e7 {{meaningOf\u227c}} (\u227c-trans \u0393\u2032 \u0393\u2033) \u03c1 \u2261 \u27e6_\u27e7 {{meaningOf\u227c}} \u0393\u2032 (\u27e6_\u27e7 {{meaningOf\u227c}} \u0393\u2033 \u03c1)\n\u27e6\u27e7-\u227c-trans \u0393\u2032 \u2205 \u2205 = refl\n\u27e6\u27e7-\u227c-trans (keep \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n\u27e6\u27e7-\u227c-trans (drop \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\u27e6\u27e7-\u227c-trans \u0393\u2032 (drop \u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\n\u27e6\u27e7-\u227c-refl : \u2200 {\u0393 : Context} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6_\u27e7 {{meaningOf\u227c}} \u227c-refl \u03c1 \u2261 \u03c1\n\u27e6\u27e7-\u227c-refl {\u2205} \u2205 = refl\n\u27e6\u27e7-\u227c-refl {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-refl \u03c1)\n\n-- SOUNDNESS of variable lifting\n\nweaken-var-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6_\u27e7 {{meaningOfVar}} (weaken-var \u0393\u2032 x) \u03c1 \u2261 \u27e6_\u27e7 {{meaningOfVar}} x ( \u27e6_\u27e7 {{meaningOf\u227c}} \u0393\u2032  \u03c1)\nweaken-var-sound \u2205 () \u03c1\nweaken-var-sound (keep \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = refl\nweaken-var-sound (keep \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = weaken-var-sound \u0393\u2032 x \u03c1\nweaken-var-sound (drop \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = weaken-var-sound \u0393\u2032 this \u03c1\nweaken-var-sound (drop \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = weaken-var-sound \u0393\u2032 (that x) \u03c1\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Environments\n--\n-- This module defines the meaning of contexts, that is,\n-- the type of environments that fit a context, together\n-- with operations and properties of these operations.\n--\n-- This module is parametric in the syntax and semantics\n-- of types, so it can be reused for different calculi\n-- and models.\n------------------------------------------------------------------------\n\nmodule Base.Denotation.Environment\n    (Type : Set)\n    {\u2113}\n    (\u27e6_\u27e7Type : Type \u2192 Set \u2113)\n  where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Base.Syntax.Context Type\nopen import Base.Denotation.Notation\nopen import Base.Data.DependentList as DependentList\n\nprivate\n  instance\n    meaningOfType : Meaning Type\n    meaningOfType = meaning \u27e6_\u27e7Type\n\n\u27e6_\u27e7Context : Context \u2192 Set \u2113\n\u27e6_\u27e7Context = DependentList \u27e6_\u27e7Type\n\ninstance\n  meaningOfContext : Meaning Context\n  meaningOfContext = meaning \u27e6_\u27e7Context\n\n-- VARIABLES\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\ninstance\n  meaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\n  meaningOfVar = meaning \u27e6_\u27e7Var\n\n-- WEAKENING\n\n-- Remove a variable from an environment\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 \u0393\u2032 \u27e7\u227c (v \u2022 \u03c1) = \u27e6 \u0393\u2032 \u27e7\u227c \u03c1\n\ninstance\n  meaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\n  meaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\n-- Properties\n\n\u27e6\u27e7-\u227c-trans : \u2200 {\u0393\u2083 \u0393\u2081 \u0393\u2082} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (\u0393\u2033 : \u0393\u2082 \u227c \u0393\u2083) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2083 \u27e7) \u2192 \u27e6 \u227c-trans \u0393\u2032 \u0393\u2033 \u27e7 \u03c1 \u2261 \u27e6 \u0393\u2032 \u27e7 (\u27e6 \u0393\u2033 \u27e7 \u03c1)\n\u27e6\u27e7-\u227c-trans \u0393\u2032 \u2205 \u2205 = refl\n\u27e6\u27e7-\u227c-trans (keep \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n\u27e6\u27e7-\u227c-trans (drop \u03c4 \u2022 \u0393\u2032) (keep .\u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\u27e6\u27e7-\u227c-trans \u0393\u2032 (drop \u03c4 \u2022 \u0393\u2033) (v \u2022 \u03c1) = \u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1\n\n\u27e6\u27e7-\u227c-refl : \u2200 {\u0393 : Context} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u27e6 \u227c-refl \u27e7 \u03c1 \u2261 \u03c1\n\u27e6\u27e7-\u227c-refl {\u2205} \u2205 = refl\n\u27e6\u27e7-\u227c-refl {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u27e7-\u227c-refl \u03c1)\n\n-- SOUNDNESS of variable lifting\n\nweaken-var-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken-var \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-var-sound \u2205 () \u03c1\nweaken-var-sound (keep \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = refl\nweaken-var-sound (keep \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = weaken-var-sound \u0393\u2032 x \u03c1\nweaken-var-sound (drop \u03c4 \u2022 \u0393\u2032) this (v \u2022 \u03c1) = weaken-var-sound \u0393\u2032 this \u03c1\nweaken-var-sound (drop \u03c4 \u2022 \u0393\u2032) (that x) (v \u2022 \u03c1) = weaken-var-sound \u0393\u2032 (that x) \u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b27cffe2bb955ab44e19350854a6de6ba952f8e5","subject":"updated completeness to be fully judgmental","message":"updated completeness to be fully judgmental\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes anywhere\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those expressions without holes anywhere\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n    DCFailedCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) dcomplete\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"722fbe4b4328cee66844c32f4bc3a509981a7b3d","subject":"progress but still stuck","message":"progress but still stuck\n","repos":"crypto-agda\/crypto-agda","old_file":"isos-examples.agda","new_file":"isos-examples.agda","new_contents":"module isos-examples where\n\nopen import Function\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nimport Function.Related as FR\nopen import Type hiding (\u2605)\nopen import Data.Product.NP\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality as \u2261\n\n_\u2248\u2082_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 Bit) \u2192 \u2605 _\n_\u2248\u2082_ {A = A} f g = \u03a3 A (T \u2218 f) \u2194 \u03a3 A (T \u2218 g)\n\nmodule _ {a r} {A : \u2605 a} {R : \u2605 r} where\n  _\u2248_ : (f g : A \u2192 R) \u2192 \u2605 _\n  f \u2248 g = \u2200 (O : R \u2192 \u2605 r) \u2192 \u03a3 A (O \u2218 f) \u2194 \u03a3 A (O \u2218 g)\n\n  \u2248-refl : Reflexive {A = A \u2192 R} _\u2248_\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : Transitive {A = A \u2192 R} _\u2248_\n  \u2248-trans p q O = q O FI.\u2218 p O\n\n  \u2248-sym : Symmetric {A = A \u2192 R} _\u2248_\n  \u2248-sym p O = FI.sym (p O)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = FI.Inverse.left-inverse-of f\n    right-f = FI.Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (left-f x))\n    coe' : (xp : \u03a3 A C) \u2192 C (from f (to f (proj\u2081 xp)))\n    coe' (x , p) = coe x p\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C \u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C \u2218 from f) \u2192 \u03a3 A C\n    \u21d0 (x , p) = from f x , p\n    left : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    left (x , p) rewrite left-f x = refl\n    mk\u03a3\u2261 : \u2200 {a b} {A : \u2605 a} {x y : A} (B : A \u2192 \u2605 b) {p : B x} {q : B y} (xy : x \u2261 y) \u2192 subst B xy p \u2261 q \u2192 (x , p) \u2261 (y , q)\n    mk\u03a3\u2261 _ xy h rewrite xy | h = refl\n    right : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    right p = mk\u03a3\u2261 (C \u2218 from f) (right-f (proj\u2081 p)) (helper p)\n            where\n                helper : \u2200 p \u2192 subst (C \u2218 from f) (right-f (proj\u2081 p)) (coe (proj\u2081 (\u21d0 p)) (proj\u2082 (\u21d0 p))) \u2261 proj\u2082 p\n                helper p with left-f (from f (proj\u2081 p)) | right-f (proj\u2081 p)\n                ... | q | r = {!!}\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C \u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) left right\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n  _\u224b_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b g = (f \u2218 proj\u2081) \u2248 (g \u2218 proj\u2082)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C \u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n\n  _\u224b\u2032_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b\u2032 g = \u2200 (O : R \u2192 \u2605 r) \u2192\n            (B \u00d7 \u03a3 A (O \u2218 f)) \u2194 (A \u00d7 \u03a3 B (O \u2218 g))\n\n  module _ {f : A \u2192 R} {g : B \u2192 R} where\n    open FR.EquationalReasoning\n\n    \u224b\u2032\u2192\u224b : f \u224b\u2032 g \u2192 f \u224b g\n    \u224b\u2032\u2192\u224b h O = \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 h O \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u220e\n\n    \u224b\u2192\u224b\u2032 : f \u224b g \u2192 f \u224b\u2032 g\n    \u224b\u2192\u224b\u2032 h O = (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 h O \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u220e\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n","old_contents":"module isos-examples where\n\nopen import Function\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nimport Function.Related as FR\nopen import Type hiding (\u2605)\nopen import Data.Product.NP\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality as \u2261\n\n_\u2248\u2082_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 Bit) \u2192 \u2605 _\n_\u2248\u2082_ {A = A} f g = \u03a3 A (T \u2218 f) \u2194 \u03a3 A (T \u2218 g)\n\nmodule _ {a r} {A : \u2605 a} {R : \u2605 r} where\n  _\u2248_ : (f g : A \u2192 R) \u2192 \u2605 _\n  f \u2248 g = \u2200 (O : R \u2192 \u2605 r) \u2192 \u03a3 A (O \u2218 f) \u2194 \u03a3 A (O \u2218 g)\n\n  \u2248-refl : Reflexive {A = A \u2192 R} _\u2248_\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : Transitive {A = A \u2192 R} _\u2248_\n  \u2248-trans p q O = q O FI.\u2218 p O\n\n  \u2248-sym : Symmetric {A = A \u2192 R} _\u2248_\n  \u2248-sym p O = FI.sym (p O)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.subst C (\u2261.sym (FI.Inverse.left-inverse-of f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C \u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C \u2218 from f) \u2192 \u03a3 A C\n    \u21d0 (x , p) = from f x , p\n    left : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    left (x , p) rewrite FI.Inverse.left-inverse-of f x = refl\n    right : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    right (x , p) = ?\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C \u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) left right\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n  _\u224b_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b g = (f \u2218 proj\u2081) \u2248 (g \u2218 proj\u2082)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C \u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\nmodule _ {a b r} {A : \u2605 a} {B : \u2605 b} {R : \u2605 r} where\n\n  _\u224b\u2032_ : (f : A \u2192 R) (g : B \u2192 R) \u2192 \u2605 _\n  f \u224b\u2032 g = \u2200 (O : R \u2192 \u2605 r) \u2192\n            (B \u00d7 \u03a3 A (O \u2218 f)) \u2194 (A \u00d7 \u03a3 B (O \u2218 g))\n\n  module _ {f : A \u2192 R} {g : B \u2192 R} where\n    open FR.EquationalReasoning\n\n    \u224b\u2032\u2192\u224b : f \u224b\u2032 g \u2192 f \u224b g\n    \u224b\u2032\u2192\u224b h O = \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 h O \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u220e\n\n    \u224b\u2192\u224b\u2032 : f \u224b g \u2192 f \u224b\u2032 g\n    \u224b\u2192\u224b\u2032 h O = (B \u00d7 \u03a3 A (O \u2218 f))\n             \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n               \u03a3 (B \u00d7 A) (O \u2218 f \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3\u00d7-swap \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 f \u2218 proj\u2081)\n             \u2194\u27e8 h O \u27e9\n               \u03a3 (A \u00d7 B) (O \u2218 g \u2218 proj\u2082)\n             \u2194\u27e8 \u03a3-assoc \u27e9\n               (A \u00d7 \u03a3 B (O \u2218 g))\n             \u220e\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n             -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"993f123a9cf6d810b51345d39e404babdbca0b0a","subject":"List: +find, update to \ud835\udfda","message":"List: +find, update to \ud835\udfda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/List\/NP.agda","new_file":"lib\/Data\/List\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.List.NP where\n\nopen import Type hiding (\u2605)\nopen import Category.Monad\nopen import Category.Applicative.NP\nopen import Data.List  public\nopen import Data.Two   using (\ud835\udfda; not; case_0:_1:_)\nopen import Data.Nat   using (\u2115; zero; suc)\nopen import Data.Maybe using (Maybe; just; nothing)\nopen import Function   using (_\u2218_)\n\nmodule Monad {\u2113} where\n  open RawMonad (monad {\u2113}) public\n  open RawApplicative rawIApplicative public using (replicateM)\n\n-- Move this\nEq : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nEq A = (x\u2081 x\u2082 : A) \u2192 \ud835\udfda\n\nuniqBy : \u2200 {a} {A : \u2605 a} \u2192 Eq A \u2192 List A \u2192 List A\nuniqBy _==_ []       = []\nuniqBy _==_ (x \u2237 xs) = x \u2237 filter (not \u2218 _==_ x) (uniqBy _==_ xs)\n                    -- x \u2237 uniqBy _==_ (filter (not \u2218 _==_ x) xs)\n\nlistToMaybe : \u2200 {a} {A : \u2605 a} \u2192 List A \u2192 Maybe A\nlistToMaybe []      = nothing\nlistToMaybe (x \u2237 _) = just x\n\nfindIndices : \u2200 {a} {A : \u2605 a} \u2192 (A \u2192 \ud835\udfda) \u2192 List A \u2192 List \u2115\n-- findIndices p xs = [ i | (x,i) <- zip xs [0..], p x ]\nfindIndices {A = A} p = go 0 where\n  go : \u2115 \u2192 List A \u2192 List \u2115\n  go _ []       = []\n  go i (x \u2237 xs) = case p x 0: go (suc i) xs\n                           1: (i \u2237 go (suc i) xs)\n\nfindIndex : \u2200 {a} {A : \u2605 a} \u2192 (A \u2192 \ud835\udfda) \u2192 List A \u2192 Maybe \u2115\nfindIndex p = listToMaybe \u2218 findIndices p\n\nindex : \u2200 {a} {A : \u2605 a} \u2192 Eq A \u2192 A \u2192 List A \u2192 Maybe \u2115\nindex _==_ x = findIndex (_==_ x)\n\nfind : \u2200 {a}{A : \u2605_ a}(p : A \u2192 \ud835\udfda) \u2192 List A \u2192 Maybe A\nfind p = listToMaybe \u2218 filter p\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.List.NP where\n\nopen import Type hiding (\u2605)\nopen import Category.Monad\nopen import Category.Applicative.NP\nopen import Data.List  public\nopen import Data.Bool  using (Bool; not; if_then_else_)\nopen import Data.Nat   using (\u2115; zero; suc)\nopen import Data.Maybe using (Maybe; just; nothing)\nopen import Function   using (_\u2218_)\n\nmodule Monad {\u2113} where\n  open RawMonad (monad {\u2113}) public\n  open RawApplicative rawIApplicative public using (replicateM)\n\n-- Move this\nEq : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nEq A = (x\u2081 x\u2082 : A) \u2192 Bool\n\nuniqBy : \u2200 {a} {A : \u2605 a} \u2192 Eq A \u2192 List A \u2192 List A\nuniqBy _==_ []       = []\nuniqBy _==_ (x \u2237 xs) = x \u2237 filter (not \u2218 _==_ x) (uniqBy _==_ xs)\n                    -- x \u2237 uniqBy _==_ (filter (not \u2218 _==_ x) xs)\n\nlistToMaybe : \u2200 {a} {A : \u2605 a} \u2192 List A \u2192 Maybe A\nlistToMaybe []      = nothing\nlistToMaybe (x \u2237 _) = just x\n\nfindIndices : \u2200 {a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List \u2115\n-- findIndices p xs = [ i | (x,i) <- zip xs [0..], p x ]\nfindIndices {A = A} p = go 0 where\n  go : \u2115 \u2192 List A \u2192 List \u2115\n  go _ []       = []\n  go i (x \u2237 xs) = if p x then i \u2237 go (suc i) xs else go (suc i) xs\n\nfindIndex : \u2200 {a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Maybe \u2115\nfindIndex p = listToMaybe \u2218 findIndices p\n\nindex : \u2200 {a} {A : \u2605 a} \u2192 Eq A \u2192 A \u2192 List A \u2192 Maybe \u2115\nindex _==_ x = findIndex (_==_ x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0e47ba2fc8222dd41f79c3723f7eed97bbf5b2c5","subject":"Tmp definition of fold in Data.Nat.NP","message":"Tmp definition of fold in Data.Nat.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d0ba0de309f84d2b7951097eeac213b99e3b0501","subject":"Removed unnecessary postulate.","message":"Removed unnecessary postulate.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOL\/SchemataATP.agda","new_file":"src\/fot\/FOL\/SchemataATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Examples of translation of logical schemata\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOL.SchemataATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n\nmodule NonSchemas where\n\n  postulate\n    A B C    : Set\n    A\u00b9       : D \u2192 Set\n    A\u00b2 B\u00b2    : D \u2192 D \u2192 Set\n    A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set\n\n  postulate id : A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : A \u2228 B \u2192 B \u2228 A\n  {-# ATP prove \u2228-comm #-}\n\n  postulate \u2228-comm\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate\n    \u2227\u2228-dist\u00b3 : \u2200 {x y z} \u2192\n               (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z)) \u2194\n               (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule Schemas where\n\n  postulate id : {A : Set} \u2192 A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : {A\u00b9 : D \u2192 Set} \u2192 \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : {A\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 A \u2228 B\n  {-# ATP prove \u2228-comm #-}\n\n  postulate \u2228-comm\u00b2 : {A\u00b2 B\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192\n                      A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : {A B C : Set} \u2192 A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate \u2227\u2228-dist\u00b3 : {A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set} \u2192 \u2200 {x y z} \u2192\n                       (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z)) \u2194\n                       (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule SchemaInstances where\n\n  -- A schema\n  -- Current translation: \u2200 p q x. app(p,x) \u2192 app(q,x).\n  postulate schema : (A B : D \u2192 Set) \u2192 \u2200 {x} \u2192 A x \u2192 B x\n\n  -- Using the current translation, the ATPs can prove an instance of\n  -- the schema.\n  postulate\n    d         : D\n    A B       : D \u2192 Set\n    instanceC : A d \u2192 B d\n  {-# ATP prove instanceC schema #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Examples of translation of logical schemata\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOL.SchemataATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n\npostulate _\u2261_ : D \u2192 D \u2192 Set\n\nmodule NonSchemas where\n\n  postulate\n    A B C    : Set\n    A\u00b9       : D \u2192 Set\n    A\u00b2 B\u00b2    : D \u2192 D \u2192 Set\n    A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set\n\n  postulate id : A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : A \u2228 B \u2192 B \u2228 A\n  {-# ATP prove \u2228-comm #-}\n\n  postulate \u2228-comm\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate\n    \u2227\u2228-dist\u00b3 : \u2200 {x y z} \u2192\n               (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z)) \u2194\n               (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule Schemas where\n\n  postulate id : {A : Set} \u2192 A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : {A\u00b9 : D \u2192 Set} \u2192 \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : {A\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 A \u2228 B\n  {-# ATP prove \u2228-comm #-}\n\n  postulate \u2228-comm\u00b2 : {A\u00b2 B\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192\n                      A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : {A B C : Set} \u2192 A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate \u2227\u2228-dist\u00b3 : {A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set} \u2192 \u2200 {x y z} \u2192\n                       (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z)) \u2194\n                       (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule SchemaInstances where\n\n  -- A schema\n  -- Current translation: \u2200 p q x. app(p,x) \u2192 app(q,x).\n  postulate schema : (A B : D \u2192 Set) \u2192 \u2200 {x} \u2192 A x \u2192 B x\n\n  -- Using the current translation, the ATPs can prove an instance of\n  -- the schema.\n  postulate\n    d         : D\n    A B       : D \u2192 Set\n    instanceC : A d \u2192 B d\n  {-# ATP prove instanceC schema #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f1f55119854a27617ba6d8786230e7ca527ff616","subject":"Add more Smyth encoding.","message":"Add more Smyth encoding.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda2\/Text\/Greek\/Smyth.agda","new_file":"agda2\/Text\/Greek\/Smyth.agda","new_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc : OlderLetter\n  \u03d8 : OlderLetter\n  \u03e1 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata AlwaysShort : Letter \u2192 Set where\n  \u03b5 : AlwaysShort \u03b5\n  \u03bf : AlwaysShort \u03bf\n\ndata AlwaysLong : Letter \u2192 Set where\n  \u03b7 : AlwaysLong \u03b7\n  \u03c9 : AlwaysLong \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  always-short : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysShort \u2113 \u2192 VowelWithLength v short\n  always-long : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysLong \u2113 \u2192 VowelWithLength v long\n  \u03b1-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b1 vl\n  \u03b9-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b9 vl\n  \u03c5-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03c5 vl\n\nalwaysShort-Vowel : \u2200 {\u2113} \u2192 AlwaysShort \u2113 \u2192 Vowel \u2113\nalwaysShort-Vowel \u03b5 = \u03b5\nalwaysShort-Vowel \u03bf = \u03bf\n\nalwaysLong-Vowel : \u2200 {\u2113} \u2192 AlwaysLong \u2113 \u2192 Vowel \u2113\nalwaysLong-Vowel \u03b7 = \u03b7\nalwaysLong-Vowel \u03c9 = \u03c9\n\n-- Smyth \u00a75\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9 : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5 : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n\n-- Smyth \u00a76\ndata SpuriousDiphthong : \u2200 {v\u2081 v\u2082} \u2192 Diphthong v\u2081 v\u2082 \u2192 Set where\n  \u03b5\u03b9-sp : SpuriousDiphthong \u03b5\u03b9\n  \u03bf\u03c5-sp : SpuriousDiphthong \u03bf\u03c5\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03c5 = closed\ntongue-position-vowel \u03c9 = open-medium\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5 = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata ConsonantSound : Letter \u2192 Set where\n  \u03b2 : ConsonantSound \u03b2\n  \u03b3 : ConsonantSound \u03b3\n  \u03b4 : ConsonantSound \u03b4\n  \u03b6 : ConsonantSound \u03b6\n  \u03b8 : ConsonantSound \u03b8\n  \u03ba : ConsonantSound \u03ba\n  \u03bb\u2032 : ConsonantSound \u03bb\u2032\n  \u03bc : ConsonantSound \u03bc\n  \u03bd : ConsonantSound \u03bd\n  \u03be : ConsonantSound \u03be\n  \u03c0 : ConsonantSound \u03c0\n  \u03c1 : ConsonantSound \u03c1\n  \u03c3 : ConsonantSound \u03c3\n  \u03c4 : ConsonantSound \u03c4\n  \u03c6 : ConsonantSound \u03c6\n  \u03c7 : ConsonantSound \u03c7\n  \u03c8 : ConsonantSound \u03c8\n  \u03b3-nasal : ConsonantSound \u03b3\n  \u03c1\u0314 : ConsonantSound \u03c1\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b3-nasal : Voiced \u03b3-nasal\n  \u03b6 : Voiced \u03b6\n\n-- Smyth \u00a715 a, b\ndata Voiceless : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c1\u0314 : Voiceless \u03c1\u0314\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n\n-- Smyth \u00a716\ndata PartOfMouthClass : Set where\n  labial dental palatal : PartOfMouthClass\n\ndata Labial : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : Labial \u03c0\n  \u03b2 : Labial \u03b2\n  \u03c6 : Labial \u03c6\n\ndata Dental : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c4 : Dental \u03c4\n  \u03b4 : Dental \u03b4\n  \u03b8 : Dental \u03b8\n\ndata Palatal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03ba : Palatal \u03ba\n  \u03b3 : Palatal \u03b3\n  \u03c7 : Palatal \u03c7\n\ndata SmoothStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : SmoothStop \u03c0\n  \u03c4 : SmoothStop \u03c4\n  \u03ba : SmoothStop \u03ba\n\ndata MiddleStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : MiddleStop \u03b2\n  \u03b4 : MiddleStop \u03b4\n  \u03b3 : MiddleStop \u03b3\n\ndata RoughStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c6 : RoughStop \u03c6\n  \u03b8 : RoughStop \u03b8\n  \u03c7 : RoughStop \u03c7\n\n-- Smyth \u00a717\ndata Spirant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c3 : Spirant \u03c3\n\n-- Smyth \u00a718\ndata Liquid : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u2032 : Liquid \u03bb\u2032\n  \u03c1 : Liquid \u03c1\n\n-- Smyth \u00a719\ndata Nasal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bc : Nasal \u03bc\n  \u03bd : Nasal \u03bd\n  \u03b3-nasal : Nasal \u03b3-nasal\n\n-- Smyth \u00a720\ndata Semivowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 Set where\n  \u03b9\u032f : Semivowel \u03b9\n  \u03c5\u032f : Semivowel \u03c5\n\n-- Smyth \u00a720 b\ndata Sonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u0325 : Sonant \u03bb\u2032\n  \u03bc\u0325 : Sonant \u03bc\n  \u03bd\u0325 : Sonant \u03b3\n  \u03c1\u0325 : Sonant \u03c1\n  \u03c3\u0325 : Sonant \u03c3\n\n-- Smyth \u00a721\ndata DoubleConsonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b6 : DoubleConsonant \u03b6\n  \u03be : DoubleConsonant \u03be\n  \u03c8 : DoubleConsonant \u03c8\n\ndata _+_\u21d2DoubleConsonant_ : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u2083} \u2192 ConsonantSound \u2113\u2081 \u2192 ConsonantSound \u2113\u2082 \u2192 ConsonantSound \u2113\u2083 \u2192 Set where\n  \u03c3+\u03b4\u21d2DoubleConsonant\u03b6 : \u03c3 + \u03b4 \u21d2DoubleConsonant \u03b6 -- Smyth \u00a726 D. Aeolic has \u03c3\u03b4 for \u03b6\n  \u03b4+\u03c3\u21d2DoubleConsonant\u03b6 : \u03b4 + \u03c3 \u21d2DoubleConsonant \u03b6\n  \n  \u03ba+\u03c3\u21d2DoubleConsonant\u03be : \u03ba + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03b3+\u03c3\u21d2DoubleConsonant\u03be : \u03b3 + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03c7+\u03c3\u21d2DoubleConsonant\u03be : \u03c7 + \u03c3 \u21d2DoubleConsonant \u03be\n\n  \u03c0+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c0 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03b2+\u03c3\u21d2DoubleConsonant\u03c8 : \u03b2 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03c6+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c6 + \u03c3 \u21d2DoubleConsonant \u03c8\n\n\n-- TODO (needs a unified model)\n\n-- Accent\n-- Smyth \u00a74. All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78. Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n-- Smyth \u00a719 a. Gamma before \u03ba, \u03b3, \u03c7, \u03be is called \u03b3-nasal\n-- Smyth \u00a720 a. When \u03b9 and \u03c5 correspond to y and w (cp. minion, persuade) they are said to be unsyllabic; and, with a following vowel, make one syllable out of two.\n-- Smyth \u00a720 a. Initial \u03b9\u032f passed into \u0314 (h), as in \u1f27\u03c0\u03b1\u03c1 liver, Lat. jecur; and into \u03b6 in \u03b6\u03c5\u03b3\u03cc\u03bd yoke\n-- Smyth \u00a720 a. Initial \u03c5\u032f was written \u03dd\n-- Smyth \u00a720 a. Medial \u03b9\u032f, \u03c5\u032f before vowels were often lost, as in \u03c4\u1fd1\u03bc\u03ac-(\u03b9\u032f)\u03c9 I honour, \u03b2\u03bf (\u03c5\u032f)-\u03cc\u03c2, gen. of \u03b2\u03bf\u1fe6-\u03c2 ox, cow\n-- Smyth \u00a726. \u03c3 was usually like our sharp s; but before voiced consonants (15 a) it probably was soft, like z; thus we find both \u03ba\u03cc\u03b6\u03bc\u03bf\u03c2 and \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 on inscriptions.\n","old_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc\u2032 : OlderLetter\n  \u03d8\u2032 : OlderLetter\n  \u03e1\u2032 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata AlwaysShort : Letter \u2192 Set where\n  \u03b5 : AlwaysShort \u03b5\n  \u03bf : AlwaysShort \u03bf\n\ndata AlwaysLong : Letter \u2192 Set where\n  \u03b7 : AlwaysLong \u03b7\n  \u03c9 : AlwaysLong \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  always-short : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysShort \u2113 \u2192 VowelWithLength v short\n  always-long : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysLong \u2113 \u2192 VowelWithLength v long\n  \u03b1-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b1 vl\n  \u03b9-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b9 vl\n  \u03c5-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03c5 vl\n\nalwaysShort-Vowel : \u2200 {\u2113} \u2192 AlwaysShort \u2113 \u2192 Vowel \u2113\nalwaysShort-Vowel \u03b5 = \u03b5\nalwaysShort-Vowel \u03bf = \u03bf\n\nalwaysLong-Vowel : \u2200 {\u2113} \u2192 AlwaysLong \u2113 \u2192 Vowel \u2113\nalwaysLong-Vowel \u03b7 = \u03b7\nalwaysLong-Vowel \u03c9 = \u03c9\n\n-- Smyth \u00a75\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9 : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5 : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n\n-- Smyth \u00a76\ndata SpuriousDiphthong : \u2200 {v\u2081 v\u2082} \u2192 Diphthong v\u2081 v\u2082 \u2192 Set where\n  \u03b5\u03b9-sp : SpuriousDiphthong \u03b5\u03b9\n  \u03bf\u03c5-sp : SpuriousDiphthong \u03bf\u03c5\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03c5 = closed\ntongue-position-vowel \u03c9 = open-medium\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5 = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata ConsonantSound : Letter \u2192 Set where\n  \u03b2 : ConsonantSound \u03b2\n  \u03b3 : ConsonantSound \u03b3\n  \u03b4 : ConsonantSound \u03b4\n  \u03b6 : ConsonantSound \u03b6\n  \u03b8 : ConsonantSound \u03b8\n  \u03ba : ConsonantSound \u03ba\n  \u03bb\u2032 : ConsonantSound \u03bb\u2032\n  \u03bc : ConsonantSound \u03bc\n  \u03bd : ConsonantSound \u03bd\n  \u03be : ConsonantSound \u03be\n  \u03c0 : ConsonantSound \u03c0\n  \u03c1 : ConsonantSound \u03c1\n  \u03c3 : ConsonantSound \u03c3\n  \u03c4 : ConsonantSound \u03c4\n  \u03c6 : ConsonantSound \u03c6\n  \u03c7 : ConsonantSound \u03c7\n  \u03c8 : ConsonantSound \u03c8\n  \u03b3-nasal : ConsonantSound \u03b3\n  \u03c1\u0314 : ConsonantSound \u03c1\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b3-nasal : Voiced \u03b3-nasal\n  \u03b6 : Voiced \u03b6\n\n-- Smyth \u00a715 a, b\ndata Voiceless : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c1\u0314 : Voiceless \u03c1\u0314\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n\n\n\n-- TODO (needs a unified model)\n\n-- Accent\n-- Smyth \u00a74 All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78 Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"960e1e698808fe20b662b54ed062d6995c6c5c23","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: f8de50703b41b6359dc492733b5b50b8\n\ndarcs-hash:20110513130039-3bd4e-3120ca397ff80a722eb022ef496ec3ffd0a06789.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC-PCF\/Program\/GCD\/EquationsATP.agda","new_file":"src\/LTC-PCF\/Program\/GCD\/EquationsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Equations for the greatest common divisor\n------------------------------------------------------------------------------\n\nmodule LTC-PCF.Program.GCD.EquationsATP where\n\nopen import LTC-PCF.Base\n\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\n\nopen import LTC-PCF.Program.GCD.GCD\n\n------------------------------------------------------------------------------\n\npostulate\n  gcd-00 : gcd zero zero \u2261 error\n-- E 1.2: CPU time limit exceeded (180 sec).\n-- Equinox 5.0alpha (2010-06-29): TIMEOUT (180 seconds).\n{-# ATP prove gcd-00 #-}\n\npostulate\n gcd-S0 : \u2200 n \u2192 gcd (succ n) zero \u2261 succ n\n-- E 1.2: CPU time limit exceeded (180 sec).\n{-# ATP prove gcd-S0 #-}\n\npostulate\n  gcd-0S : \u2200 n \u2192 gcd zero (succ n) \u2261 succ n\n-- E 1.2: CPU time limit exceeded (180 sec).\n{-# ATP prove gcd-0S #-}\n\npostulate\n  gcd-S>S : \u2200 m n \u2192 GT (succ m) (succ n) \u2192\n            gcd (succ m) (succ n) \u2261 gcd (succ m \u2238 succ n) (succ n)\n-- E 1.2: CPU time limit exceeded (180 sec).\n-- E 1.2: CPU time limit exceeded (300 sec).\n-- Equinox 5.0alpha (2010-06-29): TIMEOUT (180 seconds).\n-- Equinox 5.0alpha (2010-06-29): TIMEOUT (300 seconds).\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 300 sec).\n{-# ATP prove gcd-S>S #-}\n\n-- Because we use the hypothesis x\u2264y\u2192x\u226fy, we need the extra hypotheses\n-- N m and N n.\npostulate\n  gcd-S\u2264S : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) (succ n) \u2192\n            gcd (succ m) (succ n) \u2261 gcd (succ m) (succ n \u2238 succ m)\n{-# ATP prove gcd-S\u2264S x\u2264y\u2192x\u226fy #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Equations for the greatest common divisor\n------------------------------------------------------------------------------\n\nmodule LTC-PCF.Program.GCD.EquationsATP where\n\nopen import LTC-PCF.Base\n\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\n\nopen import LTC-PCF.Program.GCD.GCD\n\n------------------------------------------------------------------------------\n\npostulate\n  gcd-00 : gcd zero zero \u2261 error\n-- E 1.2: CPU time limit exceeded (180 sec).\n-- Equinox 5.0alpha (2010-06-29): TIMEOUT (180 seconds).\n{-# ATP prove gcd-00 #-}\n\npostulate\n gcd-S0 : \u2200 n \u2192 gcd (succ n) zero \u2261 succ n\n-- E 1.2: CPU time limit exceeded (180 sec).\n{-# ATP prove gcd-S0 #-}\n\npostulate\n  gcd-0S : \u2200 n \u2192 gcd zero (succ n) \u2261 succ n\n-- E 1.2: CPU time limit exceeded (180 sec).\n{-# ATP prove gcd-0S #-}\n\npostulate\n  gcd-S>S : \u2200 m n \u2192 GT (succ m) (succ n) \u2192\n            gcd (succ m) (succ n) \u2261 gcd (succ m \u2238 succ n) (succ n)\n-- E 1.2: CPU time limit exceeded (180 sec).\n-- E 1.2: CPU time limit exceeded (300 sec).\n-- Equinox 5.0alpha (2010-06-29): TIMEOUT (180 seconds).\n-- Equinox 5.0alpha (2010-06-29): TIMEOUT (300 seconds).\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 300 sec).\n{-# ATP prove gcd-S>S #-}\n\n-- Because we define LE on terms of LT, we need the extra hypotheses\n-- N m and N n.\npostulate\n  gcd-S\u2264S : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) (succ n) \u2192\n            gcd (succ m) (succ n) \u2261 gcd (succ m) (succ n \u2238 succ m)\n{-# ATP prove gcd-S\u2264S x\u2264y\u2192x\u226fy #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9172ee5584064b8e34a7e40649a05a8d4e6a3dc1","subject":"Changed doc.","message":"Changed doc.\n\nIgnore-this: e3c08f5a87582ef4868ac60bb2f35c0\n\ndarcs-hash:20110120221856-3bd4e-6000da7a8748eb21ca3f730305120ff87b2905c9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/AxiomaticPA\/Base.agda","new_file":"src\/AxiomaticPA\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfix  7  _\u2263_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the tool agda2atp as the ATPs equality.\npostulate\n  _\u2263_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- N.B. We make the recursion in the first argument for _+_ and _*_.\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. 0 + n = n\n-- S\u2086. succ m + n = succ (m + n)\n-- S\u2087. 0 * n = 0\n-- S\u2088. succ m * n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2263 n \u2192 m \u2263 o \u2192 n \u2263 o\n{-# ATP axiom S\u2081 #-}\n\npostulate\n  S\u2082 : \u2200 {m n} \u2192 m \u2263 n \u2192 succ m \u2263 succ n\n{-# ATP axiom S\u2082 #-}\n\npostulate\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2263 succ n)\n{-# ATP axiom S\u2083 #-}\n\npostulate\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2263 succ n \u2192 m \u2263 n\n{-# ATP axiom S\u2084 #-}\n\npostulate\n  S\u2085 : \u2200 n \u2192   zero   + n \u2263 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2263 succ (m + n)\n{-# ATP axiom S\u2085 #-}\n{-# ATP axiom S\u2086 #-}\n\npostulate\n  S\u2087 : \u2200 n \u2192   zero   * n \u2263 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2263 n + m * n\n{-# ATP axiom S\u2087 #-}\n{-# ATP axiom S\u2088 #-}\n\npostulate\n  -- The axiom S\u2089 is a higher-order one, therefore we do not translate it\n  -- as an ATP axiom.\n  S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfix  7  _\u2263_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import Common.Universe public renaming ( D to \u2115 )\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Non-logical constants\npostulate\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n  _+_  : \u2115 \u2192 \u2115 \u2192 \u2115\n  _*_  : \u2115 \u2192 \u2115 \u2192 \u2115\n\n-- The PA equality.\n-- N.B. The symbol _\u2261_ should not be used because it is hard-coded by\n-- the tool agda2atp as the ATPs equality.\npostulate\n  _\u2263_ : \u2115 \u2192 \u2115 \u2192 Set\n\n-- Proper axioms\n-- (From Elliott Mendelson. Introduction to mathematical\n-- logic. Chapman & Hall, 4th edition, 1997, p. 155)\n\n-- S\u2081. m = n \u2192 m = o \u2192 n = o\n-- S\u2082. m = n \u2192 succ m = succ n\n-- S\u2083. 0 \u2260 succ n\n-- S\u2084. succ m = succ n \u2192 m = n\n-- S\u2085. n + 0 = n\n-- S\u2086. m + succ n = succ (m + n)\n-- S\u2087. n * 0 = 0\n-- S\u2088. m * succ n = (m * n) + m\n-- S\u2089. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wf P(n) of PA.\n\npostulate\n  S\u2081 : \u2200 {m n o} \u2192 m \u2263 n \u2192 m \u2263 o \u2192 n \u2263 o\n{-# ATP axiom S\u2081 #-}\n\npostulate\n  S\u2082 : \u2200 {m n} \u2192 m \u2263 n \u2192 succ m \u2263 succ n\n{-# ATP axiom S\u2082 #-}\n\npostulate\n  S\u2083 : \u2200 {n} \u2192 \u00ac (zero \u2263 succ n)\n{-# ATP axiom S\u2083 #-}\n\npostulate\n  S\u2084 : \u2200 {m n} \u2192 succ m \u2263 succ n \u2192 m \u2263 n\n{-# ATP axiom S\u2084 #-}\n\n-- N.B. We make the recursion in the first argument.\npostulate\n  S\u2085 : \u2200 n \u2192   zero   + n \u2263 n\n  S\u2086 : \u2200 m n \u2192 succ m + n \u2263 succ (m + n)\n{-# ATP axiom S\u2085 #-}\n{-# ATP axiom S\u2086 #-}\n\n-- N.B. We make the recursion on the first argument.\npostulate\n  S\u2087 : \u2200 n \u2192   zero   * n \u2263 zero\n  S\u2088 : \u2200 m n \u2192 succ m * n \u2263 n + m * n\n{-# ATP axiom S\u2087 #-}\n{-# ATP axiom S\u2088 #-}\n\npostulate\n  -- The axiom S\u2089 is a higher-order one, therefore we do not translate it\n  -- as an ATP axiom.\n  S\u2089 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0a6d64cbb3f30ec3fc6223f68bcc6cc54b2bbb86","subject":"Game: Re-org the Beh type","message":"Game: Re-org the Beh type\n","repos":"crypto-agda\/crypto-agda","old_file":"guessing-game.agda","new_file":"guessing-game.agda","new_contents":"module guessing-game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Fin hiding ({-Ordering; compare;-} _+_)\nopen import Data.Bool -- hiding (_\u225f_)\n\nrecord Beh (MyMove OpMove : Set) : Set where\n  constructor _,_\n  field\n    myMove : MyMove\n    opMove : OpMove \u2192 Beh MyMove OpMove\n\nmodule OrderingTags where\n  data OrderingTag : Set where\n    LT EQ GT : OrderingTag\n\n  tag : \u2200 {m n} \u2192 Ordering m n \u2192 OrderingTag\n  tag (less _ _)    = LT\n  tag (equal _)     = EQ\n  tag (greater _ _) = GT\n\n  cmp : (m n : \u2115) \u2192 OrderingTag\n  cmp m n = tag (compare m n)\n\nopen OrderingTags\n\ndata Trace : Set where\n  say : (lastguess : \u2115) \u2192 Trace\n  ask : (guess : \u2115) (ordering : OrderingTag) \u2192 Trace \u2192 Trace\n\ndata Win answer : Trace \u2192 Set where\n  win  : Win answer (say answer)\n  step : \u2200 {guess trace} \u2192 Win answer trace \u2192 Win answer (ask guess (cmp guess answer) trace)\n\ndata ChalMove : Set where\n  chalStop : ChalMove\n  chalStep : OrderingTag \u2192 ChalMove\n\ndata AdvMove : Set where\n  advStop : (guess : \u2115) \u2192 AdvMove\n  advStep : (guess : \u2115) \u2192 AdvMove\n\nadvGuess : AdvMove \u2192 \u2115\nadvGuess (advStop guess) = guess\nadvGuess (advStep guess) = guess\n\nChal : Set\nChal = Beh ChalMove AdvMove\n\nAdv : Set\nAdv = Beh AdvMove ChalMove\n\nchalBeh : (answer : \u2115) \u2192 AdvMove \u2192 Chal\nchalBeh answer (advStop lastguess) = chalStop , chalBeh answer\nchalBeh answer (advStep guess) = chalStep (cmp guess answer) , chalBeh answer\n\nmiddle : (lo hi : \u2115) \u2192 \u2115\nmiddle lo hi = lo + \u230a hi \u2238 lo \/2\u230b\n\nadvStop' : \u2115 \u2192 Adv\nadvStop' g = advStop g , const (advStop' g)\n\nadv : \u2200 (lo g hi : \u2115) \u2192 ChalMove \u2192 Adv\n\nadvStep' : \u2200 (lo g hi : \u2115) \u2192 Adv\nadvStep' lo g hi = advStep g , adv lo g hi\n\nadv lo g hi chalStop = advStop' g\nadv lo g hi (chalStep LT) = advStep' g (middle g hi) hi\nadv lo g hi (chalStep EQ) = advStop' g\nadv lo g hi (chalStep GT) = advStep' lo (middle lo g) g\n\nplay : \u2115 \u2192 (AdvMove \u2192 Chal) \u2192 Adv \u2192 Trace\nplay (suc n) chal (advStep g , adv) with chal (advStep g)\n... | chalStop , chalNext = say g\n... | chalStep o , chalNext = ask g o (play n chalNext (adv (chalStep o)))\nplay _ _ (advMove , _) = say (advGuess advMove)\n\nplay100 : Fin 100 \u2192 Trace\nplay100 a = play 100 (chalBeh (to\u2115 a)) (advStep' 0 50 100)\n\nex : Trace\nex = play100 (# 5)\n\nex-wining : Win 5 ex\nex-wining = step (step (step (step (step (step (step win))))))\n\n-- winner's-win : \u2200 answer \u2192 Win (to\u2115 answer) (play100 answer)\n-- winner's-win answer = {!!}\n","old_contents":"module guessing-game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Fin hiding (Ordering; compare; _+_)\nopen import Data.Bool -- hiding (_\u225f_)\n\nrecord Beh (ChalMove AdvMove : Set) : Set where\n  constructor mk\n  field run : AdvMove \u2192 ChalMove \u00d7 Beh ChalMove AdvMove\n\nmodule OrderingTags where\n  data OrderingTag : Set where\n    LT EQ GT : OrderingTag\n\n  tag : \u2200 {m n} \u2192 Ordering m n \u2192 OrderingTag\n  tag (less _ _)    = LT\n  tag (equal _)     = EQ\n  tag (greater _ _) = GT\n\n  cmp : (m n : \u2115) \u2192 OrderingTag\n  cmp m n = tag (compare m n)\n\nopen OrderingTags\n\ndata Trace : Set where\n  say : (lastguess : \u2115) \u2192 Trace\n  ask : (guess : \u2115) (ordering : OrderingTag) \u2192 Trace \u2192 Trace\n\ndata Win answer : Trace \u2192 Set where\n  win  : Win answer (say answer)\n  step : \u2200 {guess trace} \u2192 Win answer trace \u2192 Win answer (ask guess (cmp guess answer) trace)\n\ndata ChalMove : Set where\n  chalStop : ChalMove\n  chalStep : OrderingTag \u2192 ChalMove\n\ndata AdvMove : Set where\n  advStop : (guess : \u2115) \u2192 AdvMove\n  advStep : (guess : \u2115) \u2192 AdvMove\n\nadvGuess : AdvMove \u2192 \u2115\nadvGuess (advStop guess) = guess\nadvGuess (advStep guess) = guess\n\nChal : Set\nChal = Beh ChalMove AdvMove\n\nAdv : Set\nAdv = Beh AdvMove ChalMove\n\nchalBeh : (answer : \u2115) \u2192 Chal\nchalBeh answer = chal where\n  chal : Chal\n  go : AdvMove \u2192 ChalMove \u00d7 Chal\n  go (advStop lastguess) = chalStop , chal\n  go (advStep guess) = chalStep (cmp guess answer) , chal\n  chal = mk go\n\nmiddle : (lo hi : \u2115) \u2192 \u2115\nmiddle lo hi = lo + \u230a hi \u2238 lo \/2\u230b\n\nadvStop' : \u2115 \u2192 AdvMove \u00d7 Adv\nadvStop' g = advStop g , mk (const (advStop' g))\n\nadv : \u2200 (lo g hi : \u2115) \u2192 ChalMove \u2192 AdvMove \u00d7 Adv\n\nadvStep' : \u2200 (lo g hi : \u2115) \u2192 AdvMove \u00d7 Adv\nadvStep' lo g hi = advStep g , mk (adv lo g hi)\n\nadv lo g hi chalStop = advStop' g\nadv lo g hi (chalStep LT) = advStep' g (middle g hi) hi\nadv lo g hi (chalStep EQ) = advStop' g\nadv lo g hi (chalStep GT) = advStep' lo (middle lo g) g\n\nplay : \u2115 \u2192 Chal \u2192 AdvMove \u00d7 Adv \u2192 Trace\nplay (suc n) (mk chal) (advStep g , mk adv) with chal (advStep g)\n... | chalStop , chalNext = say g\n... | chalStep o , chalNext = ask g o (play n chalNext (adv (chalStep o)))\nplay _ _ (advMove , _) = say (advGuess advMove)\n\nplay100 : Fin 100 \u2192 Trace\nplay100 a = play 100 (chalBeh (to\u2115 a)) (advStep' 0 50 100)\n\nex : Trace\nex = play100 (# 5)\n\nex-wining : Win 5 ex\nex-wining = step (step (step (step (step (step (step win))))))\n\n--  winner's-win : \u2200 answer \u2192 Win (to\u2115 answer) (play100 answer)\n-- winner's-win answer = {!!}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1df19261bfcb448257f52539b75fe4ef4a6dcca2","subject":"Re-export parts of Data.Vec in Data.Bits","message":"Re-export parts of Data.Vec in Data.Bits\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u207f) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u207f (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u207f) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u207f (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"06bde513bfacb788678960df6d1fcccdfd8b44a8","subject":"[ mirror ] Removed auxiliary reverse function","message":"[ mirror ] Removed auxiliary reverse function\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/Mirror\/MirrorListTerminatingSL.agda","new_file":"notes\/FOT\/FOTC\/Program\/Mirror\/MirrorListTerminatingSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Terminating mirror function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.Mirror.MirrorListTerminatingSL where\n\nopen import Data.List\n\n------------------------------------------------------------------------------\n-- The rose tree type.\ndata Tree (A : Set) : Set where\n  tree : A \u2192 List (Tree A) \u2192 Tree A\n\n------------------------------------------------------------------------------\n-- An alternative and terminating definition of mirror. Adapted from\n-- http:\/\/stackoverflow.com\/questions\/9146928\/termination-of-structural-induction\n\n-- The mirror function.\nmirror       : {A : Set} \u2192 Tree A \u2192 Tree A\nmirrorBranch : {A : Set} \u2192 List (Tree A) \u2192 List (Tree A)\n\nmirror       (tree a ts) = tree a (reverse (mirrorBranch ts))\nmirrorBranch []          = []\nmirrorBranch (t \u2237 ts)    = mirror t \u2237 mirrorBranch ts\n","old_contents":"------------------------------------------------------------------------------\n-- Terminating mirror function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.Mirror.MirrorListTerminatingSL where\n\nopen import Data.List as List hiding ( reverse )\n\n------------------------------------------------------------------------------\n-- Auxiliary functions\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse []       = []\nreverse (x \u2237 xs) = reverse xs ++ x \u2237 []\n\n------------------------------------------------------------------------------\n-- The rose tree type.\ndata Tree (A : Set) : Set where\n  tree : A \u2192 List (Tree A) \u2192 Tree A\n\n------------------------------------------------------------------------------\n-- An alternative and terminating definition of mirror. Adapted from\n-- http:\/\/stackoverflow.com\/questions\/9146928\/termination-of-structural-induction\n\n-- The mirror function.\nmirror       : {A : Set} \u2192 Tree A \u2192 Tree A\nmirrorBranch : {A : Set} \u2192 List (Tree A) \u2192 List (Tree A)\n\nmirror       (tree a ts) = tree a (reverse (mirrorBranch ts))\nmirrorBranch []          = []\nmirrorBranch (t \u2237 ts)    = mirror t \u2237 mirrorBranch ts\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"73e5a43e1ef316fbe76bfb0af21ad73acf0175e6","subject":"Cosmetic change.","message":"Cosmetic change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/Collatz\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/Collatz\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesATP using ( \u2238-N )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP using ( 2-N )\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\nhelper : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ succ\u2081 n) \u2261 collatz (2' ^ n)\nhelper nzero = prf\n  where postulate prf : collatz (2' ^ succ\u2081 zero) \u2261 collatz (2' ^ zero)\n        {-# ATP prove prf #-}\nhelper (nsucc {n} Nn) = prf\n  where postulate prf : collatz (2' ^ succ\u2081 (succ\u2081 n)) \u2261 collatz (2' ^ succ\u2081 n)\n        {-# ATP prove prf +\u22382 ^-N 2-N 2^x\u22620 2^[x+1]\u22621 div-2^[x+1]-2\u22612^x\n                      x-Even\u2192SSx-Even \u2238-N \u2238-Even 2^[x+1]-Even 2-Even\n        #-}\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ n) \u2261 1'\ncollatz-2^x nzero = prf\n  where postulate prf : collatz (2' ^ 0') \u2261 1'\n        {-# ATP prove prf #-}\n\ncollatz-2^x (nsucc {n} Nn) = prf (collatz-2^x Nn)\n  where postulate prf : collatz (2' ^ n) \u2261 1' \u2192 collatz (2' ^ succ\u2081 n) \u2261 1'\n        {-# ATP prove prf helper #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the Collatz function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Program.Collatz.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesATP using ( \u2238-N )\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP using ( 2-N )\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\nopen import FOTC.Program.Collatz.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\nhelper : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ succ\u2081 n) \u2261 collatz (2' ^ n)\nhelper nzero = prf\n  where postulate prf : collatz (2' ^ succ\u2081 zero) \u2261 collatz (2' ^ zero)\n        {-# ATP prove prf #-}\nhelper (nsucc {n} Nn) = prf\n  where postulate prf : collatz (2' ^ succ\u2081 (succ\u2081 n)) \u2261 collatz (2' ^ succ\u2081 n)\n        {-# ATP prove prf +\u22382 ^-N 2-N 2^x\u22620 2^[x+1]\u22621 div-2^[x+1]-2\u22612^x x-Even\u2192SSx-Even \u2238-N \u2238-Even 2^[x+1]-Even 2-Even #-}\n\ncollatz-2^x : \u2200 {n} \u2192 N n \u2192 collatz (2' ^ n) \u2261 1'\ncollatz-2^x nzero = prf\n  where postulate prf : collatz (2' ^ 0') \u2261 1'\n        {-# ATP prove prf #-}\n\ncollatz-2^x (nsucc {n} Nn) = prf (collatz-2^x Nn)\n  where postulate prf : collatz (2' ^ n) \u2261 1' \u2192 collatz (2' ^ succ\u2081 n) \u2261 1'\n        {-# ATP prove prf helper #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f6544cdf2b41aa0730418c4d5d83b346e2b74f2e","subject":"Base.Change.Equivalence: add variant of delta-ext","message":"Base.Change.Equivalence: add variant of delta-ext\n\nSince I was confused myself for a while on the correct statement for\ndelta-ext, clarify it explicitly.\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- You could think that the function should relate equivalent changes, but\n  -- that's a stronger hypothesis, which doesn't give you extra guarantees. But\n  -- here's the statement and proof, for completeness:\n\n  delta-ext\u2082 : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx\u2081 dx\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df x dx\u2081 \u2259 apply dg x dx\u2082) \u2192 df \u2259 dg\n  delta-ext\u2082 {f} {df} {dg} df-x-dx\u2259dg-x-dx = delta-ext (\u03bb x dx \u2192 df-x-dx\u2259dg-x-dx x dx dx \u2259-refl)\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    (f : A \u2192 B) \u2192 (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192 Derivative f df \u2192\n    \u2200 v \u2192 df v (nil v) \u2259 nil (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil v)\n        \u2261\u27e8 proof v (nil v) \u27e9\n          f (v \u229e (nil v))\n        \u2261\u27e8 cong f (update-nil v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil (f v)) \u27e9\n          f v \u229e nil (f v) \u220e\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\nopen import Function\n\nopen import Base.Change.Equivalence.Base public\nopen import Base.Change.Equivalence.EqReasoning public\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 proof dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x = doe $\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Let's show that nil x is d.o.e. to x \u229f x\n  nil-x-is-x\u229fx : nil x \u2259 x \u229f x\n  nil-x-is-x\u229fx = \u2259-sym (\u229e-unit-is-nil (x \u229f x) (update-diff x x))\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = doe lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = doe lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 \u2259-cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 \u2259-cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = doe lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 x dx = proof $ df-x-dx\u2259dg-x-dx x dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 \u2259-sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n\n  -- This is Lemma 2.5 in the paper. Note that the statement in the paper uses\n  -- (incorrectly) normal equality instead of delta-observational equivalence.\n  deriv-zero :\n    (f : A \u2192 B) \u2192 (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192 Derivative f df \u2192\n    \u2200 v \u2192 df v (nil v) \u2259 nil (f v)\n  deriv-zero f df proof v = doe lemma\n    where\n      open \u2261-Reasoning\n      lemma : f v \u229e df v (nil v) \u2261 f v \u229e nil (f v)\n      lemma =\n        begin\n          f v \u229e df v (nil v)\n        \u2261\u27e8 proof v (nil v) \u27e9\n          f (v \u229e (nil v))\n        \u2261\u27e8 cong f (update-nil v) \u27e9\n          f v\n        \u2261\u27e8 sym (update-nil (f v)) \u27e9\n          f v \u229e nil (f v) \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d28f502ee28c40e08c0b9b87608b45508e528301","subject":"FunUniverse.Core: comments","message":"FunUniverse.Core: comments\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Core.agda","new_file":"FunUniverse\/Core.agda","new_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nopen import FunUniverse.Types public\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\nopen import FunUniverse.Rewiring.Linear\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    -- bijFork f\u2080 f\u2081 (0' , x) = 0' , f\u2080 x\n    -- bijFork f\u2080 f\u2081 (1' , x) = 1' , f\u2081 x\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  -- bijFork\u2032 f\u2080 f\u2081 (0' , x) = 0' , f\u2080 0' x\n  -- bijFork\u2032 f\u2080 f\u2081 (1' , x) = 1' , f\u2081 1' x\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- cond (0' , x\u2080 , x\u2081) = x\u2080\n    -- cond (1' , x\u2080 , x\u2081) = x\u2081\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- fork f\u2080 f\u2081 (0' , x) = f\u2080 x\n    -- fork f\u2080 f\u2081 (1' , x) = f\u2081 x\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  -- Bad idea\u2122\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <0b> <1b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","old_contents":"module FunUniverse.Core where\n\nopen import Type\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nimport Data.Bool.NP as B\nopen B using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bit using (Bit; 0b; 1b)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport FunUniverse.BinTree as Tree\nopen Tree using (Tree)\nopen import FunUniverse.Data\n\nopen import FunUniverse.Types public\nimport FunUniverse.Defaults.FirstPart as Defaults\u27e8first-part\u27e9\nopen import FunUniverse.Rewiring.Linear\n\nrecord HasBijFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    bijFork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n\n  bijFork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  bijFork\u2032 f = bijFork (f 0b) (f 1b)\n\nrecord HasFork {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor _,_\n  open FunUniverse funU\n  field\n    -- See Defaults.DefaultCond\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n\n    -- See Defaults.DefaultFork\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n  fork\u2032 : \u2200 {A B} \u2192 (Bit \u2192 A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u2032 f = fork (f 0b) (f 1b)\n\nrecord HasXor {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n  field\n    xor : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = xor 1\u207f\n\n  \u27e8\u2295_\u27e9 : \u2200 {n} \u2192 Bits n \u2192 `Endo (`Bits n)\n  \u27e8\u2295 xs \u27e9 = xor xs\n\nrecord Bijective {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n\n  field\n    linRewiring : LinRewiring funU\n    hasBijFork  : HasBijFork  funU\n    hasXor      : HasXor      funU\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit (ignoring its argument)\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n    hasFork  : HasFork  funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n  open HasFork  hasFork  public\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  not : `Bit `\u2192 `Bit\n  not = <id,tt> \u204f fork <0b> <1b>\n\n  -- We might want it to be part of the interface\n  hasXor : HasXor funU\n  hasXor = mk (DefaultXor.xor id not <_\u229b>)\n\n  hasBijFork : HasBijFork funU\n  hasBijFork = mk (DefaultBijForkFromFork.bijFork <_,_> fst fork)\n\n  bijective : Bijective funU\n  bijective = mk linRewiring hasBijFork hasXor\n\n  open HasXor     hasXor public\n  open HasBijFork hasBijFork public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  -- vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  -- vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  half-adder : `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  half-adder = < <xor> , <and> >\n\n  full-adder : `Bit `\u00d7 `Bit `\u00d7 `Bit `\u2192 `Bit `\u00d7 `Bit\n  full-adder = < (sc\u2082 \u204f fst) , < (sc\u2081 \u204f snd) , (sc\u2082 \u204f snd) > \u204f <or> >\n    where a   = snd \u204f fst\n          b   = snd \u204f snd\n          c\u1d62\u2099 = fst\n          sc\u2081 = < a , b > \u204f half-adder\n          sc\u2082 = < (sc\u2081 \u204f fst) , c\u1d62\u2099 > \u204f half-adder\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8a922ee7afbf46e3b1fc5fa01fe326e9e1998bc7","subject":"Added map-List.","message":"Added map-List.\n\nIgnore-this: b35c24d87b48f9caac63345afe728821\n\ndarcs-hash:20100915163947-3bd4e-a5e7cb7e78c04b0c728879c1515a0ad436dc7c72.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List\/Properties.agda","new_file":"LTC\/Data\/List\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat.Type using\n  ( N ; sN ; zN -- The LTC natural numbers type\n  )\nopen import LTC.Data.List using\n  ( _++_ ; ++-[]\n  ; List ; consL ; nilL -- The LTC list type\n  ; length\n  ; map\n  ; replicate\n  ; reverse\n  )\n\n------------------------------------------------------------------------------\n-- Closure properties\n\n++-List : {xs ys : D} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL ysL = prf\n  where\n    postulate prf : List ([] ++ ys)\n    {-# ATP prove prf #-}\n\n++-List {ys = ys} (consL x {xs} xsL) ysL = prf (++-List xsL ysL)\n  where\n    postulate prf : List (xs ++ ys) \u2192  -- IH.\n                    List ((x \u2237 xs) ++ ys)\n    {-# ATP prove prf consL #-}\n\nmap-List : {xs : D}(f : D) \u2192 List xs \u2192 List (map f xs)\nmap-List f nilL = prf\n  where\n    postulate prf : List (map f [])\n    {-# ATP prove prf nilL #-}\nmap-List f (consL x {xs} xsL) = prf (map-List f xsL)\n  where\n    postulate prf : List (map f xs) \u2192 -- IH.\n                    List (map f (x \u2237 xs))\n    {-# ATP prove prf consL #-}\n\nreverse-List : {xs : D} \u2192 List xs \u2192 List (reverse xs)\nreverse-List nilL = prf\n  where\n    postulate prf : List (reverse [])\n    {-# ATP prove prf nilL #-}\n\nreverse-List (consL x {xs} xsL) = prf (reverse-List xsL)\n  where\n    postulate prf : List (reverse xs) \u2192  -- IH.\n                    List (reverse (x \u2237 xs))\n    {-# ATP prove prf consL nilL ++-List #-}\n\n++-leftIdentity : {xs : D} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : {xs : D} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL = prf\n  where\n    postulate prf : [] ++ [] \u2261 []\n    {-# ATP prove prf #-}\n\n++-rightIdentity (consL x {xs} xsL) = prf (++-rightIdentity xsL)\n  where\n    postulate prf : xs ++ [] \u2261 xs \u2192  -- IH.\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n    {-# ATP prove prf #-}\n\n++-assoc : {as bs cs : D} \u2192 List as \u2192 List bs \u2192 List cs \u2192\n           (as ++ bs) ++ cs \u2261 as ++ (bs ++ cs)\n++-assoc .{[]} {bs} {cs} nilL bsL csL = prf\n  where\n    postulate prf : ([] ++ bs) ++ cs \u2261 [] ++ bs ++ cs\n    {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} {bs} {cs} (consL x {xs} xsL) bsL csL =\n  prf (++-assoc xsL bsL csL)\n  where\n    postulate prf : (xs ++ bs) ++ cs \u2261 xs ++ bs ++ cs \u2192  -- IH.\n                    ((x \u2237 xs) ++ bs) ++ cs \u2261 (x \u2237 xs) ++ bs ++ cs\n    {-# ATP prove prf #-}\n\npostulate\n  reverse-[x]\u2261[x] : (x : D) \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nreverse-++ : {xs ys : D} \u2192 List xs \u2192 List ys \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ {ys = ys} nilL ysL = prf\n  where\n    postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n    {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++ {ys = ys} (consL x {xs} xsL) ysL = prf (reverse-++ xsL ysL)\n  where\n    postulate prf : reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs \u2192  -- IH.\n                    reverse ((x \u2237 xs) ++ ys) \u2261 reverse ys ++ reverse (x \u2237 xs)\n    -- E 1.2 no-success due to timeout (180).\n    {-# ATP prove prf ++-assoc nilL consL reverse-List ++-List #-}\n\nreverse\u00b2 : {xs : D} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse\u00b2 nilL = prf\n  where\n    postulate prf : reverse (reverse []) \u2261 []\n    {-# ATP prove prf #-}\n\nreverse\u00b2 (consL x {xs} xsL) = prf (reverse\u00b2 xsL)\n  where\n    postulate prf : reverse (reverse xs) \u2261 xs \u2192  -- IH.\n                    reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n    {-# ATP prove prf reverse-++ consL nilL reverse-List ++-List #-}\n\nmap-++ : (f : D){xs ys : D} \u2192 List xs \u2192 List ys \u2192\n         map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++ f {ys = ys} nilL ysL = prf\n  where\n    postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n    {-# ATP prove prf #-}\nmap-++ f {ys = ys} (consL x {xs} xsL) ysL = prf (map-++ f xsL ysL)\n  where\n    postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192  -- IH.\n                    map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n    -- E 1.2 no-success due to timeout (180).\n    {-# ATP prove prf #-}\n\nlength-replicate : {n : D} \u2192 N n \u2192 (d : D) \u2192 length (replicate n d) \u2261 n\nlength-replicate zN d = prf\n  where\n    postulate prf : length (replicate zero d) \u2261 zero\n    {-# ATP prove prf #-}\nlength-replicate (sN {n} Nn) d = prf (length-replicate Nn d)\n  where\n    postulate prf : length (replicate n d) \u2261 n \u2192  -- IH.\n                    length (replicate (succ n) d) \u2261 succ n\n    {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat.Type using\n  ( N ; sN ; zN -- The LTC natural numbers type\n  )\nopen import LTC.Data.List using\n  ( _++_ ; ++-[]\n  ; List ; consL ; nilL -- The LTC list type\n  ; length\n  ; map\n  ; replicate\n  ; reverse\n  )\n\n------------------------------------------------------------------------------\n-- Closure properties\n\n++-List : {xs ys : D} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} nilL ysL = prf\n  where\n    postulate prf : List ([] ++ ys)\n    {-# ATP prove prf #-}\n\n++-List {ys = ys} (consL x {xs} xsL) ysL = prf (++-List xsL ysL)\n  where\n    postulate prf : List (xs ++ ys) \u2192\n                    List ((x \u2237 xs) ++ ys)\n    {-# ATP prove prf consL #-}\n\nreverse-List : {xs : D} \u2192 List xs \u2192 List (reverse xs)\nreverse-List nilL = prf\n  where\n    postulate prf : List (reverse [])\n    {-# ATP prove prf nilL #-}\n\nreverse-List (consL x {xs} xsL) = prf (reverse-List xsL)\n  where\n    postulate prf : List (reverse xs) \u2192\n                    List (reverse (x \u2237 xs))\n    {-# ATP prove prf consL nilL ++-List #-}\n\n++-leftIdentity : {xs : D} \u2192 List xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity {xs} _ = ++-[] xs\n\n++-rightIdentity : {xs : D} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity nilL = prf\n  where\n    postulate prf : [] ++ [] \u2261 []\n    {-# ATP prove prf #-}\n\n++-rightIdentity (consL x {xs} xsL) = prf (++-rightIdentity xsL)\n  where\n    postulate prf : xs ++ [] \u2261 xs \u2192\n                    (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n    {-# ATP prove prf #-}\n\n++-assoc : {as bs cs : D} \u2192 List as \u2192 List bs \u2192 List cs \u2192\n           (as ++ bs) ++ cs \u2261 as ++ (bs ++ cs)\n++-assoc .{[]} {bs} {cs} nilL bsL csL = prf\n  where\n    postulate prf : ([] ++ bs) ++ cs \u2261 [] ++ bs ++ cs\n    {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} {bs} {cs} (consL x {xs} xsL) bsL csL =\n  prf (++-assoc xsL bsL csL)\n  where\n    postulate prf : (xs ++ bs) ++ cs \u2261 xs ++ bs ++ cs \u2192\n                    ((x \u2237 xs) ++ bs) ++ cs \u2261 (x \u2237 xs) ++ bs ++ cs\n    {-# ATP prove prf #-}\n\npostulate\n  reverse-[x]\u2261[x] : (x : D) \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nreverse-++ : {xs ys : D} \u2192 List xs \u2192 List ys \u2192\n             reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++ {ys = ys} nilL ysL = prf\n  where\n    postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n    {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++ {ys = ys} (consL x {xs} xsL) ysL = prf (reverse-++ xsL ysL)\n  where\n    postulate prf : reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs \u2192\n                    reverse ((x \u2237 xs) ++ ys) \u2261 reverse ys ++ reverse (x \u2237 xs)\n    -- E 1.2 no-success due to timeout (180).\n    {-# ATP prove prf ++-assoc nilL consL reverse-List ++-List #-}\n\nreverse\u00b2 : {xs : D} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse\u00b2 nilL = prf\n  where\n    postulate prf : reverse (reverse []) \u2261 []\n    {-# ATP prove prf #-}\n\nreverse\u00b2 (consL x {xs} xsL) = prf (reverse\u00b2 xsL)\n  where\n    postulate prf : reverse (reverse xs) \u2261 xs \u2192\n                    reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n    {-# ATP prove prf reverse-++ consL nilL reverse-List ++-List #-}\n\nmap-++ : (f : D){xs ys : D} \u2192 List xs \u2192 List ys \u2192\n         map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++ f {ys = ys} nilL ysL = prf\n  where\n    postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n    {-# ATP prove prf #-}\nmap-++ f {ys = ys} (consL x {xs} xsL) ysL = prf (map-++ f xsL ysL)\n  where\n    postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n                    map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n    -- E 1.2 no-success due to timeout (180).\n    {-# ATP prove prf #-}\n\nlength-replicate : {n : D} \u2192 N n \u2192 (d : D) \u2192 length (replicate n d) \u2261 n\nlength-replicate zN d = prf\n  where\n    postulate prf : length (replicate zero d) \u2261 zero\n    {-# ATP prove prf #-}\nlength-replicate (sN {n} Nn) d = prf (length-replicate Nn d)\n  where\n    postulate prf : length (replicate n d) \u2261 n \u2192\n                    length (replicate (succ n) d) \u2261 succ n\n    {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7a02010d7669652c2bf6aeed9436c65f0d41e465","subject":"Testing Agda primitives.","message":"Testing Agda primitives.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing normalisation on arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Inequalities.ArithmeticPropertiesSL where\n\nmodule NonPrimitives where\n\n  open import Data.Nat\n  open import Relation.Binary.PropositionalEquality\n\n  101\u2261100+11\u223810 : 101 \u2261 100 + 11 \u2238 10\n  101\u2261100+11\u223810 = refl\n\n  91\u2261100+11\u223810\u223810 : 91 \u2261 100 + 11 \u2238 10 \u2238 10\n  91\u2261100+11\u223810\u223810 = refl\n\n  -- Agsy failed using a timeout of 5 minutes and it succeed with a\n  -- timeout of 10 minutes.\n  100+11>100 : 100 + 11 > 100\n  100+11>100 = s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      (s\u2264s\n                                                       (s\u2264s\n                                                        (s\u2264s\n                                                         (s\u2264s\n                                                          (s\u2264s\n                                                           (s\u2264s\n                                                            (s\u2264s\n                                                             (s\u2264s\n                                                              (s\u2264s\n                                                               (s\u2264s\n                                                                (s\u2264s\n                                                                 (s\u2264s\n                                                                  (s\u2264s\n                                                                   (s\u2264s\n                                                                    (s\u2264s\n                                                                     (s\u2264s\n                                                                      (s\u2264s\n                                                                       (s\u2264s\n                                                                        (s\u2264s\n                                                                         (s\u2264s\n                                                                          (s\u2264s\n                                                                           (s\u2264s\n                                                                            (s\u2264s\n                                                                             (s\u2264s\n                                                                              (s\u2264s\n                                                                               (s\u2264s\n                                                                                (s\u2264s\n                                                                                 (s\u2264s\n                                                                                  (s\u2264s\n                                                                                   (s\u2264s\n                                                                                    (s\u2264s\n                                                                                     (s\u2264s\n                                                                                      (s\u2264s\n                                                                                       (s\u2264s\n                                                                                        (s\u2264s\n                                                                                         (s\u2264s\n                                                                                          (s\u2264s\n                                                                                           (s\u2264s\n                                                                                            (s\u2264s\n                                                                                             (s\u2264s\n                                                                                              (s\u2264s\n                                                                                               (s\u2264s\n                                                                                                (s\u2264s\n                                                                                                 (s\u2264s\n                                                                                                  (s\u2264s\n                                                                                                   (s\u2264s\n                                                                                                    (s\u2264s\n                                                                                                     (s\u2264s\n                                                                                                      (s\u2264s\n                                                                                                       (s\u2264s\n                                                                                                        (s\u2264s\n                                                                                                         (s\u2264s\n                                                                                                          (s\u2264s\n                                                                                                           (s\u2264s\n                                                                                                            (s\u2264s\n                                                                                                             (s\u2264s\n                                                                                                              (s\u2264s\n                                                                                                               (s\u2264s\n                                                                                                                (s\u2264s\n                                                                                                                 (s\u2264s\n                                                                                                                  (s\u2264s\n                                                                                                                   z\u2264n))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n\nmodule Primitives where\n\n  open import Data.Bool\n  open import Data.Nat hiding ( _<_ )\n  open import Relation.Binary.PropositionalEquality\n\n  primitive primNatLess : \u2115 \u2192 \u2115 \u2192 Bool\n\n  _<_ : \u2115 \u2192 \u2115 \u2192 Bool\n  _<_ = primNatLess\n\n  100<100+11 : 100 < (100 + 11) \u2261 true\n  100<100+11 = refl\n","old_contents":"------------------------------------------------------------------------------\n-- Testing normalisation on arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Inequalities.ArithmeticPropertiesSL where\n\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\n\n101\u2261100+11\u223810 : 101 \u2261 100 + 11 \u2238 10\n101\u2261100+11\u223810 = refl\n\n91\u2261100+11\u223810\u223810 : 91 \u2261 100 + 11 \u2238 10 \u2238 10\n91\u2261100+11\u223810\u223810 = refl\n\n-- Agsy failed using a timeout of 5 minutes and it succeed with a\n-- timeout of 10 minutes.\n100+11>100 : 100 + 11 > 100\n100+11>100 = s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      (s\u2264s\n                                                       (s\u2264s\n    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             (s\u2264s\n                                                                      (s\u2264s\n                                                                       (s\u2264s\n                                                                        (s\u2264s\n                                                                         (s\u2264s\n                                                                          (s\u2264s\n                                                                           (s\u2264s\n                                                                            (s\u2264s\n                                                                             (s\u2264s\n                                                                              (s\u2264s\n                                                                               (s\u2264s\n                                                                                (s\u2264s\n                               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                        (s\u2264s\n                                                                                            (s\u2264s\n                                                                                             (s\u2264s\n                                                                                              (s\u2264s\n                                                                                               (s\u2264s\n                                                                                                (s\u2264s\n                                                                                                 (s\u2264s\n                                                                                                  (s\u2264s\n                                                                                                   (s\u2264s\n                                                                                                    (s\u2264s\n                                                                                                     (s\u2264s\n                                                                                                      (s\u2264s\n                                                                                                       (s\u2264s\n                                                                                                        (s\u2264s\n                                                                                                         (s\u2264s\n                                                                                                          (s\u2264s\n                                                                                                           (s\u2264s\n                                                                                                            (s\u2264s\n                                                                                                             (s\u2264s\n                                                                                                              (s\u2264s\n                                                                                                               (s\u2264s\n                                                                                                                (s\u2264s\n                                                                                                                 (s\u2264s\n                                                                                                                  (s\u2264s\n                                                                                                                   z\u2264n))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8511033c35f76d26fa4bf8fbf4a17d7b036fd3d0","subject":"Spelling","message":"Spelling\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/Matrix.agda","new_file":"FLABloM\/Matrix.agda","new_contents":"-- Matrices indexed by shape\nmodule Matrix where\n\nopen import Shape\n\n-- Matrix representation\ndata M (a : Set) : (rows cols : S) \u2192 Set where\n  -- 1x1 matrices\n  One\n    : a \u2192 M a L L\n\n  -- 1xn matrices\n  Row\n    : \u2200 {c\u2081 c\u2082} \u2192 M a L c\u2081 \u2192 M a L c\u2082 \u2192 M a L (B c\u2081 c\u2082)\n\n  -- nx1 matrices\n  Col\n    : \u2200 {r\u2081 r\u2082} \u2192 M a r\u2081 L \u2192 M a r\u2082 L \u2192 M a (B r\u2081 r\u2082) L\n\n  -- nxm matrices\n  Q\n    : \u2200 {r\u2081 r\u2082 c\u2081 c\u2082} \u2192\n      M a r\u2081 c\u2081 \u2192 M a r\u2081 c\u2082 \u2192\n      M a r\u2082 c\u2081 \u2192 M a r\u2082 c\u2082 \u2192\n      M a (B r\u2081 r\u2082) (B c\u2081 c\u2082)\n\n\n-- Matrix addition and multiplication need the corresponding\n-- operations on the underlying type\nmodule Operations (T : Set) (_*T_ : T \u2192 T \u2192 T) (_+T_ : T \u2192 T \u2192 T) where\n\n  infixr 60 _*_\n  infixr 50 _+_\n\n  _+_ : \u2200 {r c} \u2192 M T r c \u2192 M T r c \u2192 M T r c\n  One x + One x\u2081 = One (x +T x\u2081)\n  Row m m\u2081 + Row n n\u2081 = Row (m + n) (m\u2081 + n\u2081)\n  Col m m\u2081 + Col n n\u2081 = Col (m + n) (m\u2081 + n\u2081)\n  Q m00 m01 m10 m11 + Q n00 n01 n10 n11 = Q (m00 + n00) (m01 + n01) (m10 + n10) (m11 + n11)\n\n  _*_ : \u2200 {r\u2081 c\u2082 x} \u2192 M T r\u2081 x \u2192 M T x c\u2082 \u2192 M T r\u2081 c\u2082\n  One x * One x\u2081 = One (x *T x\u2081)\n  One x * Row n n\u2081 = Row (One x * n) (One x * n\u2081)\n  Row m m\u2081 * Col n n\u2081 = m * n + m\u2081 * n\u2081\n  Row m m\u2081 * Q n00 n01 n10 n11 = Row (m * n00 + m\u2081 * n10) (m * n01 + m\u2081 * n11)\n  Col m m\u2081 * One x = Col (m * One x) (m\u2081 * One x)\n  Col m m\u2081 * Row n n\u2081 = Q (m * n) (m * n\u2081) (m\u2081 * n) (m\u2081 * n\u2081)\n  Q m00 m01 m10 m11 * Col n n\u2081 = Col (m00 * n + m01 * n\u2081) (m10 * n + m11 * n\u2081)\n  Q m00 m01 m10 m11 * Q n00 n01 n10 n11\n    = Q (m00 * n00 + m01 * n10) (m00 * n01 + m01 * n11)\n        (m10 * n00 + m11 * n10) (m10 * n01 + m11 * n11)\n\n  private\n    postulate t : T\n\n    -- 3x3 matrix\n    mat : M T (B L (B L L)) (B (B L L) L)\n    mat = Q (Row (One t) (One t)) (One t)\n               (Q (One t) (One t) (One t) (One t)) (Col (One t) (One t))\n\n    -- 3x2 matrix\n    mat\u2081 : M T (B L (B L L)) (B L L)\n    mat\u2081 = Q (One t) (One t) (Col (One t) (One t)) (Col (One t) (One t))\n","old_contents":"-- Matrices indexed by shape\nmodule Matrix where\n\nopen import Shape\n\n-- Matrix representation\ndata M (a : Set) : (rows cols : S) \u2192 Set where\n  -- 1x1 matrices\n  One\n    : a \u2192 M a L L\n\n  -- 1xn matrices\n  Row\n    : \u2200 {c\u2081 c\u2082} \u2192 M a L c\u2081 \u2192 M a L c\u2082 \u2192 M a L (B c\u2081 c\u2082)\n\n  -- nx1 matrices\n  Col\n    : \u2200 {r\u2081 r\u2082} \u2192 M a r\u2081 L \u2192 M a r\u2082 L \u2192 M a (B r\u2081 r\u2082) L\n\n  -- nxm matrices\n  Q\n    : \u2200 {r\u2081 r\u2082 c\u2081 c\u2082} \u2192\n      M a r\u2081 c\u2081 \u2192 M a r\u2081 c\u2082 \u2192\n      M a r\u2082 c\u2081 \u2192 M a r\u2082 c\u2082 \u2192\n      M a (B r\u2081 r\u2082) (B c\u2081 c\u2082)\n\n\n-- For matrix addition and multiplication wee need the corresponding\n-- operations on the underlying type\nmodule Operations (T : Set) (_*T_ : T \u2192 T \u2192 T) (_+T_ : T \u2192 T \u2192 T) where\n\n  infixr 60 _*_\n  infixr 50 _+_\n\n  _+_ : \u2200 {r c} \u2192 M T r c \u2192 M T r c \u2192 M T r c\n  One x + One x\u2081 = One (x +T x\u2081)\n  Row m m\u2081 + Row n n\u2081 = Row (m + n) (m\u2081 + n\u2081)\n  Col m m\u2081 + Col n n\u2081 = Col (m + n) (m\u2081 + n\u2081)\n  Q m00 m01 m10 m11 + Q n00 n01 n10 n11 = Q (m00 + n00) (m01 + n01) (m10 + n10) (m11 + n11)\n\n  _*_ : \u2200 {r\u2081 c\u2082 x} \u2192 M T r\u2081 x \u2192 M T x c\u2082 \u2192 M T r\u2081 c\u2082\n  One x * One x\u2081 = One (x *T x\u2081)\n  One x * Row n n\u2081 = Row (One x * n) (One x * n\u2081)\n  Row m m\u2081 * Col n n\u2081 = m * n + m\u2081 * n\u2081\n  Row m m\u2081 * Q n00 n01 n10 n11 = Row (m * n00 + m\u2081 * n10) (m * n01 + m\u2081 * n11)\n  Col m m\u2081 * One x = Col (m * One x) (m\u2081 * One x)\n  Col m m\u2081 * Row n n\u2081 = Q (m * n) (m * n\u2081) (m\u2081 * n) (m\u2081 * n\u2081)\n  Q m00 m01 m10 m11 * Col n n\u2081 = Col (m00 * n + m01 * n\u2081) (m10 * n + m11 * n\u2081)\n  Q m00 m01 m10 m11 * Q n00 n01 n10 n11\n    = Q (m00 * n00 + m01 * n10) (m00 * n01 + m01 * n11)\n        (m10 * n00 + m11 * n10) (m10 * n01 + m11 * n11)\n\n  private\n    postulate t : T\n\n    -- 3x3 matrix\n    mat : M T (B L (B L L)) (B (B L L) L)\n    mat = Q (Row (One t) (One t)) (One t)\n               (Q (One t) (One t) (One t) (One t)) (Col (One t) (One t))\n\n    -- 3x2 matrix\n    mat\u2081 : M T (B L (B L L)) (B L L)\n    mat\u2081 = Q (One t) (One t) (Col (One t) (One t)) (Col (One t) (One t))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ffe1fe1414bdba693f1527305efaf63e85fe5324","subject":"Defined div using lambda-lifting.","message":"Defined div using lambda-lifting.\n\nIgnore-this: 94f88552defac3f726456630aa18fe9e\n\ndarcs-hash:20100604025539-3bd4e-2b743fc3019b955d3b4cf809b1ed9a2a55ae58f1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Division.agda","new_file":"Examples\/Division.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of division using repeated subtraction\n------------------------------------------------------------------------------\n\nmodule Examples.Division where\n\nopen import LTC.Minimal\n\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Relation.InequalitiesPCF\n\n------------------------------------------------------------------------------\n\n-- Version using lambda-abstraction.\n-- divh : D \u2192 D\n-- divh g = lam (\u03bb i \u2192 lam (\u03bb j \u2192\n--              if (lt i j)\n--                then zero\n--                else (succ (g \u2219 (i - j) \u2219 j))))\n\n-- Version using lambda lifting via super-combinators.\n-- (Hughes. Super-combinators. 1982)\n\ndiv-aux\u2081 : D \u2192 D \u2192 D \u2192 D\ndiv-aux\u2081 i g j = if (lt i j) then zero else (succ (g \u2219 (i - j) \u2219 j))\n\ndiv-aux\u2082 : D \u2192 D \u2192 D\ndiv-aux\u2082 g i = lam (div-aux\u2081 i g)\n\ndivh : D \u2192 D\ndivh g = lam (div-aux\u2082 g)\n\ndiv : D \u2192 D \u2192 D\ndiv i j = fix divh \u2219 i \u2219 j\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of division using repeated subtraction\n------------------------------------------------------------------------------\n\nmodule Examples.Division where\n\nopen import LTC.Minimal\n\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Relation.InequalitiesPCF\n\n------------------------------------------------------------------------------\n\ndivh : D \u2192 D\ndivh g = lam (\u03bb i \u2192 lam (\u03bb j \u2192\n             if (lt i j)\n               then zero\n               else (succ (g \u2219 (i - j) \u2219 j))))\n\ndiv : D \u2192 D \u2192 D\ndiv i j = fix divh \u2219 i \u2219 j\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"accc20b535c408faf3e1f4d9a0b23227b2ecf8b0","subject":"Added TODO about the introduction rule for the for all.","message":"Added TODO about the introduction rule for the for all.\n\nIgnore-this: 23c3712981b3fe1e4e05c429a8dfa7d1\n\ndarcs-hash:20120403203537-3bd4e-a8e8e957e2baa844bcbbd8705e3a246470c995be.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/TheoremsATP.agda","new_file":"src\/FOL\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- FOL theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module contains some examples showing the use of the ATPs for\n-- proving FOL theorems.\n\nmodule FOL.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A     : Set\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n      \u03c6(x)\n  -----------  \u2200-intro\n    \u2200x.\u03c6(x)\n\n    \u2200x.\u03c6(x)\n  -----------  \u2200-elim\n     \u03c6(t)\n\n      \u03c6(t)\n  -----------  \u2203-intro\n    \u2203x.\u03c6(x)\n\n   \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n ----------------------  \u2203-elim\n           \u03c8\n-}\n\n-- TODO: 2012-04-03\n-- postulate\n--   \u2200-intro : (x : D) \u2192 A\u00b9 x \u2192 \u2200 x \u2192 A\u00b9 x\n-- {-# ATP prove \u2200-intro #-}\n\npostulate\n  \u2200-elim  : ((x : D) \u2192 A\u00b9 x) \u2192 (t : D) \u2192 A\u00b9 t\n  \u2203-intro : (t : D) \u2192 A\u00b9 t \u2192 \u2203 A\u00b9\n  \u2203-elim' : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n{-# ATP prove \u2200-elim #-}\n{-# ATP prove \u2203-intro #-}\n{-# ATP prove \u2203-elim' #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200 x \u2192 A\u00b9 x) \u2194 (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 A\u00b9)       \u2194 (\u2200 x \u2192 \u00ac (A\u00b9 x))\n  gDM\u2083 : (\u2200 x \u2192 A\u00b9 x)   \u2194 \u00ac (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2084 : \u2203 A\u00b9           \u2194 \u00ac (\u2200 x \u2192 \u00ac (A\u00b9 x))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : (\u2200 x y \u2192 A\u00b2 x y) \u2194 (\u2200 y x \u2192 A\u00b2 x y)\n  \u2203-ord : (\u2203[ x ] \u2203[ y ] A\u00b2 x y) \u2194 (\u2203[ y ] \u2203[ x ] A\u00b2 x y)\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate \u2203-erase-add : (\u2203[ x ] A \u2227 A\u00b9 x) \u2194 A \u2227 (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2203-erase-add #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : (\u2200 x \u2192 A\u00b9 x \u2227 B\u00b9 x) \u2194 ((\u2200 x \u2192 A\u00b9 x) \u2227 (\u2200 x \u2192 B\u00b9 x))\n  \u2203-dist : (\u2203[ x ] A\u00b9 x \u2228 B\u00b9 x) \u2194 (\u2203 A\u00b9 \u2228 \u2203 B\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n\n-- Interchange of quantifiers.\n-- The related theorem \u2200x\u2203y.Axy \u2192 \u2203y\u2200x.Axy is not (classically) valid.\npostulate \u2203\u2200 : \u2203[ x ] (\u2200 y \u2192 A\u00b2 x y) \u2192 \u2200 y \u2192 \u2203[ x ] A\u00b2 x y\n{-# ATP prove \u2203\u2200 #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate\n  \u2203\u2192\u00ac\u2200\u00ac : \u2203[ x ] A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 \u00ac A\u00b9 x)\n  \u2203\u00ac\u2192\u00ac\u2200 : \u2203[ x ] \u00ac A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 A\u00b9 x)\n{-# ATP prove \u2203\u2192\u00ac\u2200\u00ac #-}\n{-# ATP prove \u2203\u00ac\u2192\u00ac\u2200 #-}\n\n-- \u2200 in terms of \u2203 and \u00ac.\npostulate\n  \u2200\u2192\u00ac\u2203\u00ac : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u00ac (\u2203[ x ] \u00ac A\u00b9 x)\n  \u2200\u00ac\u2192\u00ac\u2203 : (\u2200 {x} \u2192 \u00ac A\u00b9 x) \u2192 \u00ac (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200\u2192\u00ac\u2203\u00ac #-}\n{-# ATP prove \u2200\u00ac\u2192\u00ac\u2203 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- FOL theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module contains some examples showing the use of the ATPs for\n-- proving FOL theorems.\n\nmodule FOL.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A     : Set\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n      \u03c6(x)\n  -----------  \u2200-intro\n    \u2200x.\u03c6(x)\n\n    \u2200x.\u03c6(x)\n  -----------  \u2200-elim\n     \u03c6(t)\n\n      \u03c6(t)\n  -----------  \u2203-intro\n    \u2203x.\u03c6(x)\n\n   \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n ----------------------  \u2203-elim\n           \u03c8\n-}\n\npostulate\n  \u2200-intro : (x : D) \u2192 A\u00b9 x \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200-elim  : ((x : D) \u2192 A\u00b9 x) \u2192 (t : D) \u2192 A\u00b9 t\n  \u2203-intro : (t : D) \u2192 A\u00b9 t \u2192 \u2203 A\u00b9\n  \u2203-elim' : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n{-# ATP prove \u2200-intro #-}\n{-# ATP prove \u2200-elim #-}\n{-# ATP prove \u2203-intro #-}\n{-# ATP prove \u2203-elim' #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200 x \u2192 A\u00b9 x) \u2194 (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 A\u00b9)       \u2194 (\u2200 x \u2192 \u00ac (A\u00b9 x))\n  gDM\u2083 : (\u2200 x \u2192 A\u00b9 x)   \u2194 \u00ac (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2084 : \u2203 A\u00b9           \u2194 \u00ac (\u2200 x \u2192 \u00ac (A\u00b9 x))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : (\u2200 x y \u2192 A\u00b2 x y) \u2194 (\u2200 y x \u2192 A\u00b2 x y)\n  \u2203-ord : (\u2203[ x ] \u2203[ y ] A\u00b2 x y) \u2194 (\u2203[ y ] \u2203[ x ] A\u00b2 x y)\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate \u2203-erase-add : (\u2203[ x ] A \u2227 A\u00b9 x) \u2194 A \u2227 (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2203-erase-add #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : (\u2200 x \u2192 A\u00b9 x \u2227 B\u00b9 x) \u2194 ((\u2200 x \u2192 A\u00b9 x) \u2227 (\u2200 x \u2192 B\u00b9 x))\n  \u2203-dist : (\u2203[ x ] A\u00b9 x \u2228 B\u00b9 x) \u2194 (\u2203 A\u00b9 \u2228 \u2203 B\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n\n-- Interchange of quantifiers.\n-- The related theorem \u2200x\u2203y.Axy \u2192 \u2203y\u2200x.Axy is not (classically) valid.\npostulate \u2203\u2200 : \u2203[ x ] (\u2200 y \u2192 A\u00b2 x y) \u2192 \u2200 y \u2192 \u2203[ x ] A\u00b2 x y\n{-# ATP prove \u2203\u2200 #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate\n  \u2203\u2192\u00ac\u2200\u00ac : \u2203[ x ] A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 \u00ac A\u00b9 x)\n  \u2203\u00ac\u2192\u00ac\u2200 : \u2203[ x ] \u00ac A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 A\u00b9 x)\n{-# ATP prove \u2203\u2192\u00ac\u2200\u00ac #-}\n{-# ATP prove \u2203\u00ac\u2192\u00ac\u2200 #-}\n\n-- \u2200 in terms of \u2203 and \u00ac.\npostulate\n  \u2200\u2192\u00ac\u2203\u00ac : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u00ac (\u2203[ x ] \u00ac A\u00b9 x)\n  \u2200\u00ac\u2192\u00ac\u2203 : (\u2200 {x} \u2192 \u00ac A\u00b9 x) \u2192 \u00ac (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200\u2192\u00ac\u2203\u00ac #-}\n{-# ATP prove \u2200\u00ac\u2192\u00ac\u2203 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cb1ae5239a92d6b5a162643c972805fee252b07a","subject":"agda: Missing comment","message":"agda: Missing comment\n\nOld-commit-hash: 3b3398b8ac36ce60d1d04406bd3f7411b03c6761\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality using (sym)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality using (sym)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8f017c6136524d1546c71e1484d6011f82180f52","subject":"Cut out power","message":"Cut out power\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\nmodule Guess (Power : Set) (coins : Power \u2192 Coins) (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Power \u2192 Set\n  Oracle power = \u2203 (\u03bb (A : GuessAdv (coins power)) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {t} {T : Set t}\n             {\u266dFuns : FlatFuns T}\n             (\u266dops : FlatFunsOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red : (post-E : C `\u2192 C) \u2192 SemSecReduction _\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {c} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {c} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take c R)) b))\n                                       (drop c R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess Coins id prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","old_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule FunctionExtra where\n  _***_ : \u2200 {A B C D : Set} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n  -- Fanout\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n  _>>>_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n            (A \u2192 B) \u2192 (B \u2192 C) \u2192 (A \u2192 C)\n  f >>> g = g \u2218 f\n  infixr 1 _>>>_\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\nmodule Guess (Power : Set) (coins : Power \u2192 Coins) (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Power \u2192 Set\n  Oracle power = \u2203 (\u03bb (A : GuessAdv (coins power)) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nrecord FlatFuns (Power : Set) {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits   : \u2115 \u2192 T\n    `Bit    : T\n    _`\u00d7_    : T \u2192 T \u2192 T\n    \u27e8_\u27e9_\u219d_  : Power \u2192 T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 \u27e8_\u27e9_\u219d_\n\nrecord PowerOps (Power : Set) : Set where\n  constructor mk\n  infixr 1 _>>>\u1d56_\n  infixr 3 _***\u1d56_\n  field\n    id\u1d56 : Power\n    _>>>\u1d56_ _***\u1d56_ : Power \u2192 Power \u2192 Power\n\nconstPowerOps : PowerOps \u22a4\nconstPowerOps = _\n\nrecord FlatFunsOps {P t} {T : Set t} (powerOps : PowerOps P) (\u266dFuns : FlatFuns P T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  open PowerOps powerOps\n  field\n    idO : \u2200 {A p} \u2192 \u27e8 p \u27e9 A \u219d A\n    isIComposable  : IComposable  {I = \u22a4} _>>>\u1d56_ \u27e8_\u27e9_\u219d_\n    isVIComposable : VIComposable {I = \u22a4} _***\u1d56_ _`\u00d7_ \u27e8_\u27e9_\u219d_\n  open FlatFuns \u266dFuns public\n  open PowerOps powerOps public\n  open IComposable isIComposable public\n  open VIComposable isVIComposable public\n\nfun\u266dFuns : FlatFuns \u22a4 Set\nfun\u266dFuns = mk Bits Bit _\u00d7_ (\u03bb _ A B \u2192 A \u2192 B)\n\nfun\u266dOps : FlatFunsOps _ fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp\n\nmodule AbsSemSec (|M| |C| : \u2115) {FunPower : Set} {t} {T : Set t}\n                 (\u266dFuns : FlatFuns FunPower T) where\n\n  record Power : Set where\n    constructor mk\n    field\n      p\u2080 p\u2081 : FunPower\n      |R| : Coins\n  coins = Power.|R|\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (p : Power) : Set where\n    constructor mk\n\n    open Power p\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Power \u2192 Power) \u2192 Set\n  SemSecReduction f = \u2200 {p} \u2192 AbsSemSecAdv p \u2192 AbsSemSecAdv (f p)\n\n-- Here we use Agda functions for FlatFuns and \u22a4 for power.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open PowerOps constPowerOps\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv |R| = AbsSemSecAdv (mk _ _ |R|)\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n    where open FunctionExtra using (_&&&_)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Power \u2192 Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {p} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {p} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\nmodule PostCompSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  module PostCompRed {FunPower : Set} {t} {T : Set t}\n             {\u266dFuns : FlatFuns FunPower T}\n             {funPowerOps : PowerOps FunPower}\n             (\u266dops : FlatFunsOps funPowerOps \u266dFuns) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec |M| |C| \u266dFuns\n\n    post-comp-red-power : FunPower \u2192 Power \u2192 Power\n    post-comp-red-power p (mk p\u2080 p\u2081 |R|) = mk p\u2080 (p ***\u1d56 id\u1d56 >>>\u1d56 p\u2081) |R|\n\n    post-comp-red : \u2200 {p} (post-E : \u27e8 p \u27e9 C \u219d C) \u2192 SemSecReduction (post-comp-red-power p)\n    post-comp-red post-E (mk A\u2080 A\u2081) = mk A\u2080 (post-E *** idO >>> A\u2081)\n\n  open PrgDist prgDist\n  open PostCompRed fun\u266dOps\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist |M| |C|\n  open AbsSemSec |M| |C| fun\u266dFuns\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr id (post-comp {cc} post-E)\n  post-comp-pres-sem-sec post-E = post-comp-red post-E , (\u03bb _ \u2192 id)\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-comp post-E E)\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {E} = red , helper where\n    E' = post-comp post-E E\n    red : SemSecReduction id\n    red = post-comp-red post-E\u207b\u00b9\n    helper : \u2200 {p} A \u2192 (E \u21c4 A) \u21d3 (E' \u21c4 red {p} A)\n    helper {p} A A-breaks-E = A'-breaks-E'\n     where open FunSemSecAdv A renaming (step\u2080F to A\u2080F)\n           A' = red {p} A\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (A\u2080F (take (coins p) R)) b))\n                                       (drop (coins p) R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = extensional-reduction same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg = post-comp vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr id (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {E} = post-comp-pres-sem-sec vnot {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc} {E : Enc cc}\n                            \u2192 SafeSemSecReduction id E (post-neg E)\n  post-neg-pres-sem-sec' {cc} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess Coins id prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module PostCompSec' = PostCompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"261730b7c65d25769fe56a01367ca7528345de57","subject":"bit-guessing-game: all we need is antisym of ]-[","message":"bit-guessing-game: all we need is antisym of ]-[\n","repos":"crypto-agda\/crypto-agda","old_file":"bit-guessing-game.agda","new_file":"bit-guessing-game.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\nopen import Function\nopen import Data.Bool.NP\n\nopen import Data.Bits hiding (#\u27e8_\u27e9)\n\nopen import flipbased-implem using (Coins; \u21ba; \u2141; return\u21ba; toss; Rat; _\/_; Pr[_\u22611]) renaming (count\u21ba to #\u27e8_\u27e9)\nopen import program-distance using (PrgDist; module PrgDist)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n\nopen PrgDist prgDist\n\nGuessAdv : Coins \u2192 Set\nGuessAdv = \u2141\n\nrunGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u2141 ca\nrunGuess\u2141 A _ = A\n\n-- An oracle: an adversary who can break the guessing game.\nOracle : Coins \u2192 Set\nOracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n-- The guessing game is secure iff all adversaries have\n-- only a neglible advantage over the game.\n-- Meaning that no adversary can be behave differently when\n-- the game is to guess 0 than when the game is to guess 1.\nGuessSec : Coins \u2192 Set\nGuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\n-- The adversary actually has to behave the same way in both\n-- situation give the definition of the runGuess\u2141.\n-- Agda reduces to two sides of the equation to same thing.\nsame-behavior : \u2200 {ca} (A : GuessAdv ca) \u2192 runGuess\u2141 A 0b \u2261 runGuess\u2141 A 1b\nsame-behavior A = \u2261.refl\n\n-- The guessing game is actually secure since any adversary\n-- has an advantage of zero.\n-- This a direct application of the fact that the \"far-apart\" _]-[_\n-- relation is antisymetric, meaning that any point cannot be \"far-apart\"\n-- from itself. The definiton of \"breaks \u2141\" being that \u2141 0b is \"far-apart\"\n-- from \u2141 1b.\nguessSec : \u2200 {ca} \u2192 GuessSec ca\nguessSec = ]-[-antisym\n\n-- Let's look at three possible strategies:\n\n-- Always returning 1 will allow to win all the time when\n-- one has to guess 1 but lose all the time when one has to\n-- guess 0.\ncount\u21ba-return\u21ba-1b : Pr[ return\u21ba {0} 1b \u22611] \u2261 1 \/ 1\ncount\u21ba-return\u21ba-1b = \u2261.refl\n\n-- Similarily returning always 0 is not better.\ncount\u21ba-return\u21ba-0b : Pr[ return\u21ba {0} 0b \u22611] \u2261 0 \/ 1\ncount\u21ba-return\u21ba-0b = \u2261.refl\n\n-- Tossing a coin will allow to win half the time in\n-- both situtations, but still once we look at the distance\n-- between the two situtations these two halfs cancel each\n-- other.\ncount\u21ba-toss : Pr[ toss \u22611] \u2261 1 \/ 2\ncount\u21ba-toss = \u2261.refl\n\n-- In the end, at the pure guessing game it is as hard to be\n-- consitently good than to be consitently bad.\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (\u2203)\nopen import Relation.Nullary using (\u00ac_)\nopen import Function\nopen import Data.Bool.NP\n\nopen import Data.Bits\n\nopen import flipbased-implem using (Coins; \u21ba; \u2141) -- ; #\u27e8_\u27e9) renaming (count\u21ba to #\u27e8_\u27e9)\nopen import program-distance using (PrgDist; module PrgDist)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n\nopen PrgDist prgDist\n\nGuessAdv : Coins \u2192 Set\nGuessAdv = \u2141\n\nrunGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u2141 ca\nrunGuess\u2141 A _ = A\n\n-- An oracle: an adversary who can break the guessing game.\nOracle : Coins \u2192 Set\nOracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n-- Any adversary cannot do better than a random guess.\nGuessSec : Coins \u2192 Set\nGuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f97ff6f67a4cac4360ae09731c7f0ab22eca83b7","subject":"Add a new test","message":"Add a new test\n","repos":"bch29\/agda-holes","old_file":"src\/Holes\/Test\/Limited.agda","new_file":"src\/Holes\/Test\/Limited.agda","new_contents":"module Holes.Test.Limited where\n\nopen import Holes.Prelude hiding (_\u229b_)\nopen import Holes.Util\nopen import Holes.Term\nopen import Holes.Cong.Limited\n\nopen import Agda.Builtin.Equality using (_\u2261_; refl)\n\nmodule Contrived where\n  infixl 5 _\u2295_\n  infixl 6 _\u229b_\n  infix 4 _\u2248_\n\n  -- A type of expression trees for natural number calculations\n\n  data Expr : Set where\n    zero : Expr\n    succ : Expr \u2192 Expr\n    _\u2295_ _\u229b_ : Expr \u2192 Expr \u2192 Expr\n\n  -- An unsophisticated 'equality' relation on the expression trees\n\n  data _\u2248_ : Expr \u2192 Expr \u2192 Set where\n    zero-cong : zero \u2248 zero\n    succ-cong : \u2200 {m n} \u2192 m \u2248 n \u2192 succ m \u2248 succ n\n    \u2295-cong : \u2200 {a a\u2032 b b\u2032} \u2192 a \u2248 a\u2032 \u2192 b \u2248 b\u2032 \u2192 a \u2295 b \u2248 a\u2032 \u2295 b\u2032\n    \u229b-cong : \u2200 {a a\u2032 b b\u2032} \u2192 a \u2248 a\u2032 \u2192 b \u2248 b\u2032 \u2192 a \u229b b \u2248 a\u2032 \u229b b\u2032\n\n    \u2295-comm : \u2200 a b \u2192 a \u2295 b \u2248 b \u2295 a\n\n  -- Some lemmas that are necessary to proceed\n\n  \u2248-refl : \u2200 {n} \u2192 n \u2248 n\n  \u2248-refl {zero} = zero-cong\n  \u2248-refl {succ n} = succ-cong \u2248-refl\n  \u2248-refl {m \u2295 n} = \u2295-cong \u2248-refl \u2248-refl\n  \u2248-refl {m \u229b n} = \u229b-cong \u2248-refl \u2248-refl\n\n  {-\n  Each congruence in the database must have a type with the same shape as the\n  below, following these rules:\n\n  - A congruence `c` is for a single _top-level_ function `f`\n  - `c` may only be a congruence for _one_ of `f`'s arguments\n  - Each explicit argument to `f` must be an explicit argument to `c`\n  - The 'rewritten' version of the changing argument must follow as an implicit\n    argument to `c`\n  - The equation to do the local rewriting must be the next explicit argument\n  -}\n\n  open CongSplit _\u2248_ \u2248-refl\n\n  \u2295-cong\u2081 = two\u2192one\u2081 \u2295-cong\n  \u2295-cong\u2082 = two\u2192one\u2082 \u2295-cong\n  \u229b-cong\u2081 = two\u2192one\u2081 \u229b-cong\n  \u229b-cong\u2082 = two\u2192one\u2082 \u229b-cong\n\n  succ-cong\u2032 : \u2200 n {n\u2032} \u2192 n \u2248 n\u2032 \u2192 succ n \u2248 succ n\u2032\n  succ-cong\u2032 _ = succ-cong\n\n  {-\n  Create the database of congruences.\n  - This is a list of (Name \u00d7 ArgPlace \u00d7 Congruence) triples.\n  - The `Name` is the (reflected) name of the function to which the congruence\n    applies.\n  - The `ArgPlace` is the index of the argument place that the congruence can\n    rewrite at.\n  - The `Congruence` is a reflected copy of the congruence function, of type `Term`\n  -}\n\n  database\n    = (quote _\u2295_ , 0 , quote \u2295-cong\u2081)\n    \u2237 (quote _\u2295_ , 1 , quote \u2295-cong\u2082)\n    \u2237 (quote _\u229b_ , 0 , quote \u229b-cong\u2081)\n    \u2237 (quote _\u229b_ , 1 , quote \u229b-cong\u2082)\n    \u2237 (quote succ , 0 , quote succ-cong\u2032)\n    \u2237 []\n\n  open AutoCong database\n\n  test1 : \u2200 a b \u2192 succ \u231e a \u2295 b \u231f \u2248 succ (b \u2295 a)\n  test1 a b = cong! (\u2295-comm a b)\n\n  test2 : \u2200 a b c \u2192 succ (\u231e a \u2295 b \u231f \u2295 c) \u2248 succ (b \u2295 a \u2295 c)\n  test2 a b c = cong! (\u2295-comm a b)\n\n  test3 : \u2200 a b c \u2192 succ (a \u2295 \u231e b \u2295 c \u231f) \u2248 succ (a \u2295 (c \u2295 b))\n  test3 a b c = cong! (\u2295-comm b c)\n\n  -- We can use a proof obtained by pattern matching for `cong!`\n  test5 : \u2200 a b c \u2192 a \u2248 b \u2295 c \u00d7 b \u2248 succ c \u2192 succ (\u231e b \u231f \u2295 c \u229b a) \u2248 succ (succ c \u2295 c \u229b a)\n  test5 a b c (eq1 , eq2) = cong! eq2\n\n  record \u2203 {a p} {A : Set a} (P : A \u2192 Set p) : Set (a \u2294 p) where\n    constructor _,_\n\n    field\n      evidence : A\n      proof : P evidence\n\n  -- We can use `cong!` to prove some part of the result even when it depends on\n  -- other parts of the result\n  test6 : \u2200 a b c \u2192 b \u2295 c \u2248 a \u00d7 b \u2248 succ c \u2192 \u2203 \u03bb x \u2192 succ (\u231e b \u2295 x \u231f \u2295 a) \u2248 succ (a \u2295 a)\n  test6 a b c (eq1 , eq2) = c , cong! eq1\n\n  -- Deep nesting\n  test7 : \u2200 a b c\n    \u2192 succ (a \u2295 succ (b \u2295 succ (c \u2295 succ (a \u2295 succ \u231e b \u2295 c \u231f))))\n    \u2248 succ (a \u2295 succ (b \u2295 succ (c \u2295 succ (a \u2295 succ \u231e c \u2295 b \u231f))))\n  test7 _ b c = cong! (\u2295-comm b c)\n\nmodule Propositional (+-comm : \u2200 a b \u2192 a + b \u2261 b + a) where\n  +-cong\u2081 : \u2200 a b {a\u2032} \u2192 a \u2261 a\u2032 \u2192 a + b \u2261 a\u2032 + b\n  +-cong\u2081 _ _ refl = refl\n\n  +-cong\u2082 : \u2200 a b {b\u2032} \u2192 b \u2261 b\u2032 \u2192 a + b \u2261 a + b\u2032\n  +-cong\u2082 _ _ refl = refl\n\n  suc-cong : \u2200 n {n\u2032} \u2192 n \u2261 n\u2032 \u2192 suc n \u2261 suc n\u2032\n  suc-cong _ refl = refl\n\n  database\n    = (quote _+_ , 0 , quote +-cong\u2081)\n    \u2237 (quote _+_ , 1 , quote +-cong\u2082)\n    \u2237 (quote suc , 0 , quote suc-cong)\n    \u2237 []\n\n  open AutoCong database\n\n  test1 : \u2200 a b \u2192 suc \u231e a + b \u231f \u2261 suc (b + a)\n  test1 a b = cong! (+-comm a b)\n\n  test2 : \u2200 a b c \u2192 suc (\u231e a + b \u231f + c) \u2261 suc (b + a + c)\n  test2 a b c = cong! (+-comm a b)\n\n  test3 : \u2200 a b c \u2192 suc (a + \u231e b + c \u231f) \u2261 suc (a + (c + b))\n  test3 a b c = cong! (+-comm b c)\n\n  test4 : \u2200 a b c a\u2032 \u2192 a \u2261 a\u2032 \u2192 suc (\u231e a \u231f + b + c) \u2261 suc (a\u2032 + b + c)\n  test4 _ _ _ _ eq = cong! eq\n","old_contents":"module Holes.Test.Limited where\n\nopen import Holes.Prelude hiding (_\u229b_)\nopen import Holes.Util\nopen import Holes.Term\nopen import Holes.Cong.Limited\n\nopen import Agda.Builtin.Equality using (_\u2261_; refl)\n\nmodule Contrived where\n  infixl 5 _\u2295_\n  infixl 6 _\u229b_\n  infix 4 _\u2248_\n\n  -- A type of expression trees for natural number calculations\n\n  data Expr : Set where\n    zero : Expr\n    succ : Expr \u2192 Expr\n    _\u2295_ _\u229b_ : Expr \u2192 Expr \u2192 Expr\n\n  -- An unsophisticated 'equality' relation on the expression trees\n\n  data _\u2248_ : Expr \u2192 Expr \u2192 Set where\n    zero-cong : zero \u2248 zero\n    succ-cong : \u2200 {m n} \u2192 m \u2248 n \u2192 succ m \u2248 succ n\n    \u2295-cong : \u2200 {a a\u2032 b b\u2032} \u2192 a \u2248 a\u2032 \u2192 b \u2248 b\u2032 \u2192 a \u2295 b \u2248 a\u2032 \u2295 b\u2032\n    \u229b-cong : \u2200 {a a\u2032 b b\u2032} \u2192 a \u2248 a\u2032 \u2192 b \u2248 b\u2032 \u2192 a \u229b b \u2248 a\u2032 \u229b b\u2032\n\n    \u2295-comm : \u2200 a b \u2192 a \u2295 b \u2248 b \u2295 a\n\n  -- Some lemmas that are necessary to proceed\n\n  \u2248-refl : \u2200 {n} \u2192 n \u2248 n\n  \u2248-refl {zero} = zero-cong\n  \u2248-refl {succ n} = succ-cong \u2248-refl\n  \u2248-refl {m \u2295 n} = \u2295-cong \u2248-refl \u2248-refl\n  \u2248-refl {m \u229b n} = \u229b-cong \u2248-refl \u2248-refl\n\n  {-\n  Each congruence in the database must have a type with the same shape as the\n  below, following these rules:\n\n  - A congruence `c` is for a single _top-level_ function `f`\n  - `c` may only be a congruence for _one_ of `f`'s arguments\n  - Each explicit argument to `f` must be an explicit argument to `c`\n  - The 'rewritten' version of the changing argument must follow as an implicit\n    argument to `c`\n  - The equation to do the local rewriting must be the next explicit argument\n  -}\n\n  open CongSplit _\u2248_ \u2248-refl\n\n  \u2295-cong\u2081 = two\u2192one\u2081 \u2295-cong\n  \u2295-cong\u2082 = two\u2192one\u2082 \u2295-cong\n  \u229b-cong\u2081 = two\u2192one\u2081 \u229b-cong\n  \u229b-cong\u2082 = two\u2192one\u2082 \u229b-cong\n\n  succ-cong\u2032 : \u2200 n {n\u2032} \u2192 n \u2248 n\u2032 \u2192 succ n \u2248 succ n\u2032\n  succ-cong\u2032 _ = succ-cong\n\n  {-\n  Create the database of congruences.\n  - This is a list of (Name \u00d7 ArgPlace \u00d7 Congruence) triples.\n  - The `Name` is the (reflected) name of the function to which the congruence\n    applies.\n  - The `ArgPlace` is the index of the argument place that the congruence can\n    rewrite at.\n  - The `Congruence` is a reflected copy of the congruence function, of type `Term`\n  -}\n\n  database\n    = (quote _\u2295_ , 0 , quote \u2295-cong\u2081)\n    \u2237 (quote _\u2295_ , 1 , quote \u2295-cong\u2082)\n    \u2237 (quote _\u229b_ , 0 , quote \u229b-cong\u2081)\n    \u2237 (quote _\u229b_ , 1 , quote \u229b-cong\u2082)\n    \u2237 (quote succ , 0 , quote succ-cong\u2032)\n    \u2237 []\n\n  open AutoCong database\n\n  test1 : \u2200 a b \u2192 succ \u231e a \u2295 b \u231f \u2248 succ (b \u2295 a)\n  test1 a b = cong! (\u2295-comm a b)\n\n  test2 : \u2200 a b c \u2192 succ (\u231e a \u2295 b \u231f \u2295 c) \u2248 succ (b \u2295 a \u2295 c)\n  test2 a b c = cong! (\u2295-comm a b)\n\n  test3 : \u2200 a b c \u2192 succ (a \u2295 \u231e b \u2295 c \u231f) \u2248 succ (a \u2295 (c \u2295 b))\n  test3 a b c = cong! (\u2295-comm b c)\n\n  -- We can use a proof obtained by pattern matching for `cong!`\n  test5 : \u2200 a b c \u2192 a \u2248 b \u2295 c \u00d7 b \u2248 succ c \u2192 succ (\u231e b \u231f \u2295 c \u229b a) \u2248 succ (succ c \u2295 c \u229b a)\n  test5 a b c (eq1 , eq2) = cong! eq2\n\n  record \u2203 {a p} {A : Set a} (P : A \u2192 Set p) : Set (a \u2294 p) where\n    constructor _,_\n\n    field\n      evidence : A\n      proof : P evidence\n\n  -- We can use `cong!` to prove some part of the result even when it depends on\n  -- other parts of the result\n  test6 : \u2200 a b c \u2192 b \u2295 c \u2248 a \u00d7 b \u2248 succ c \u2192 \u2203 \u03bb x \u2192 succ (\u231e b \u2295 x \u231f \u2295 a) \u2248 succ (a \u2295 a)\n  test6 a b c (eq1 , eq2) = c , cong! eq1\n\nmodule Propositional (+-comm : \u2200 a b \u2192 a + b \u2261 b + a) where\n  +-cong\u2081 : \u2200 a b {a\u2032} \u2192 a \u2261 a\u2032 \u2192 a + b \u2261 a\u2032 + b\n  +-cong\u2081 _ _ refl = refl\n\n  +-cong\u2082 : \u2200 a b {b\u2032} \u2192 b \u2261 b\u2032 \u2192 a + b \u2261 a + b\u2032\n  +-cong\u2082 _ _ refl = refl\n\n  suc-cong : \u2200 n {n\u2032} \u2192 n \u2261 n\u2032 \u2192 suc n \u2261 suc n\u2032\n  suc-cong _ refl = refl\n\n  database\n    = (quote _+_ , 0 , quote +-cong\u2081)\n    \u2237 (quote _+_ , 1 , quote +-cong\u2082)\n    \u2237 (quote suc , 0 , quote suc-cong)\n    \u2237 []\n\n  open AutoCong database\n\n  test1 : \u2200 a b \u2192 suc \u231e a + b \u231f \u2261 suc (b + a)\n  test1 a b = cong! (+-comm a b)\n\n  test2 : \u2200 a b c \u2192 suc (\u231e a + b \u231f + c) \u2261 suc (b + a + c)\n  test2 a b c = cong! (+-comm a b)\n\n  test3 : \u2200 a b c \u2192 suc (a + \u231e b + c \u231f) \u2261 suc (a + (c + b))\n  test3 a b c = cong! (+-comm b c)\n\n  test4 : \u2200 a b c a\u2032 \u2192 a \u2261 a\u2032 \u2192 suc (\u231e a \u231f + b + c) \u2261 suc (a\u2032 + b + c)\n  test4 _ _ _ _ eq = cong! eq\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3534fc2dfd19ab386ec2829236fbfe919683468e","subject":"Added info about the translation of projection-like functions.","message":"Added info about the translation of projection-like functions.\n\nIgnore-this: 98dc5ffe6bd78b412551655b8d1fade4\n\ndarcs-hash:20110923235943-3bd4e-28930d976bb4e282ef682856830eda4f0a6b8ee5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/ProjectionLikeFunction.agda","new_file":"Test\/Fail\/ProjectionLikeFunction.agda","new_contents":"------------------------------------------------------------------------------\n-- The translation of projection-like functions is not implemented\n\n-- 2011-09-23: From the Agda mailing list\n-- Subject: Compiler internals for projection functions\n-- There is some additional information about this issue.\n\n-- 2011-09-19: Maybe the information in the issue 465 (Missing\n-- patterns in funClauses) is useful.\n\n------------------------------------------------------------------------------\n\nmodule Test.Fail.ProjectionLikeFunction where\n\n-- Error: The translation of projection-like functions Test.Fail.ProjectionLikeFunction._.P is not implemented.\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  N    : D \u2192 Set\n\nfoo : \u2200 {n} \u2192 N n \u2192 D\nfoo {n} Nn = n\n  where\n  P : D \u2192 Set\n  P i = i \u2261 i\n  {-# ATP definition P #-}\n\n  postulate bar : P n\n  {-# ATP prove bar #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The translation of projection-like functions is not implemented\n\n-- 2011-09-19: Maybe the information in the issue 465 (Missing\n-- patterns in funClauses) is useful.\n\n------------------------------------------------------------------------------\n\nmodule Test.Fail.ProjectionLikeFunction where\n\n-- Error: The translation of projection-like functions Test.Fail.ProjectionLikeFunction._.P is not implemented.\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  N    : D \u2192 Set\n\nfoo : \u2200 {n} \u2192 N n \u2192 D\nfoo {n} Nn = n\n  where\n  P : D \u2192 Set\n  P i = i \u2261 i\n  {-# ATP definition P #-}\n\n  postulate bar : P n\n  {-# ATP prove bar #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a8a13708e0de6a37f57b9446c19146127d7f093b","subject":"Added test for the negation.","message":"Added test for the negation.\n\nIgnore-this: eef0e5ab2be59aeb7ea611443bfb12a6\n\ndarcs-hash:20100406000752-3bd4e-e69c88b0955957b64e82ad003e721c1e8b836091.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  4 _\u2261_\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\nmodule Conjunction where\n\n  infixr 4 _,_\n  infixr 2 _\u00d7_\n\n  -- We want to use the product type to define the conjunction type\n  record _\u00d7_ (A B : Set) : Set where\n    constructor _,_\n    field\n      proj\u2081 : A\n      proj\u2082 : B\n\n  -- N.B. It is not necessary to add the constructor _,_, nor the fields\n  -- proj\u2081, proj\u2082 as hints, because the ATP implements it.\n\n  _\u2227_ : (A B : Set) \u2192 Set\n  A \u2227 B = A \u00d7 B\n\n  postulate\n    pred : D \u2192 D \u2192 Set\n\n  -- Testing the conjunction data constructor\n  postulate\n    testAndDataConstructor : {m n : D} \u2192\n                             (pred m m) \u2192\n                             (pred n n) \u2192\n                             (pred m m ) \u2227 (pred n n)\n  {-# ATP prove testAndDataConstructor #-}\n\n  -- Testing the first projection\n  postulate\n    testProj\u2081 : {m n : D} \u2192\n                (pred m m ) \u2227 (pred n n) \u2192\n                (pred m m)\n  {-# ATP prove testProj\u2081 #-}\n\n-- Testing the second projection\n  postulate\n    testProj\u2082 : {m n : D} \u2192\n                (pred m m ) \u2227 (pred n n) \u2192\n                (pred n n)\n  {-# ATP prove testProj\u2082 #-}\n\n------------------------------------------------------------------------------\n-- The negation\n\nmodule Negation where\n\n  infix 3 \u00ac\n\n  data \u22a5 : Set where\n\n  \u00ac : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  postulate\n    true  : D\n    false : D\n\n  postulate\n    true\u2260false : \u00ac (true \u2261 false)\n  {-# ATP axiom true\u2260false #-}\n\n  postulate\n    testContradiction : (d : D) \u2192 true \u2261 false \u2192 d \u2261 true\n  {-# ATP prove testContradiction #-}\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfixr 4 _,_\ninfixr 2 _\u00d7_\n\npostulate\n  D     : Set\n  equal : D \u2192 D \u2192 Set\n\n------------------------------------------------------------------------------\n-- The conjuction data type\n\n-- We want to use the product type to define the conjunction type\nrecord _\u00d7_ (A B : Set) : Set where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B\n\n-- N.B. It is not necessary to add the constructor _,_, nor the fields\n-- proj\u2081, proj\u2082 as hints, because the ATP implements it.\n\n_\u2227_ : (A B : Set) \u2192 Set\nA \u2227 B = A \u00d7 B\n\n-- Testing the conjunction data constructor\npostulate\n testAndDataConstructor : {m n : D} \u2192\n                        (equal m m) \u2192\n                        (equal n n) \u2192\n                        (equal m m ) \u2227 (equal n n)\n{-# ATP prove testAndDataConstructor #-}\n\n-- Testing the first projection\npostulate\n  testProj\u2081 : {m n : D} \u2192\n              (equal m m ) \u2227 (equal n n) \u2192\n              (equal m m)\n{-# ATP prove testProj\u2081 #-}\n\n-- Testing the second projection\npostulate\n  testProj\u2082 : {m n : D} \u2192\n              (equal m m ) \u2227 (equal n n) \u2192\n              (equal n n)\n{-# ATP prove testProj\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"14638777254cda4cf79d2fa340674206e2c6c050","subject":"Fixed note related to the inductive approach.","message":"Fixed note related to the inductive approach.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/InductiveApproach\/Recursion.agda","new_file":"notes\/FOT\/FOTC\/InductiveApproach\/Recursion.agda","new_contents":"------------------------------------------------------------------------------\n-- Discussion about the inductive approach\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\n-- Andr\u00e9s: From our discussion about the inductive approach, can I\n-- conclude that it is possible to rewrite the proofs using pattern\n-- matching on _\u2261_, by proofs using only subst, because the types\n-- associated with these proofs haven't proof terms?\n\n-- Peter: Yes, provided the RHS of the definition does not refer to the\n-- function defined, i e, there is no recursion.\n\nmodule FOT.FOTC.InductiveApproach.Recursion where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.PropertiesI\n\n------------------------------------------------------------------------------\n\n-- foo is recursive and we pattern matching on _\u2261_.\nfoo : \u2200 {m n} \u2192 N m \u2192 m \u2261 n \u2192 N (m + n)\nfoo nzero refl = subst N (sym (+-leftIdentity zero)) nzero\nfoo (nsucc {m} Nm) refl = subst N helper (nsucc (nsucc (foo Nm refl)))\n  where\n  helper : succ\u2081 (succ\u2081 (m + m)) \u2261 succ\u2081 m + succ\u2081 m\n  helper =\n    succ\u2081 (succ\u2081 (m + m)) \u2261\u27e8 succCong (sym (+-Sx m m))  \u27e9\n    succ\u2081 (succ\u2081 m + m)   \u2261\u27e8 succCong (+-comm (nsucc Nm) Nm) \u27e9\n    succ\u2081 (m + succ\u2081 m)   \u2261\u27e8 sym (+-Sx m (succ\u2081 m)) \u27e9\n    succ\u2081 m + succ\u2081 m     \u220e\n\n-- foo' is recursive and we only use subst.\nfoo' : \u2200 {m n} \u2192 N m \u2192 m \u2261 n \u2192 N (m + n)\nfoo' {n = n} nzero h = subst N (sym (+-leftIdentity n)) (subst N h nzero)\nfoo' {n = n} (nsucc {m} Nm) h = subst N helper (nsucc (nsucc (foo' Nm refl)))\n  where\n  helper : succ\u2081 (succ\u2081 (m + m)) \u2261 succ\u2081 m + n\n  helper =\n    succ\u2081 (succ\u2081 (m + m)) \u2261\u27e8 succCong (sym (+-Sx m m))  \u27e9\n    succ\u2081 (succ\u2081 m + m)   \u2261\u27e8 succCong (+-comm (nsucc Nm) Nm) \u27e9\n    succ\u2081 (m + succ\u2081 m)   \u2261\u27e8 sym (+-Sx m (succ\u2081 m)) \u27e9\n    succ\u2081 m + succ\u2081 m     \u2261\u27e8 +-rightCong h \u27e9\n    succ\u2081 m + n           \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Discussion about the inductive approach\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Andr\u00e9s: From our discussion about the inductive approach, can I\n-- conclude that it is possible to rewrite the proofs using pattern\n-- matching on _\u2261_, by proofs using only subst, because the types\n-- associated with these proofs haven't proof terms?\n\n-- Peter: Yes, provided the RHS of the definition does not refer to the\n-- function defined, i e, there is no recursion.\n\nmodule FOT.FOTC.InductiveApproach.Recursion where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\nfoo : \u2200 {m n} \u2192 m \u2261 n \u2192 m + n \u2261 m + m\nfoo refl = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"79741c8ce11e0b5f81dd8c628d307143a242940d","subject":"Data.Vec.N-ary.NP.lift\u2083","message":"Data.Vec.N-ary.NP.lift\u2083\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/N-ary\/NP.agda","new_file":"lib\/Data\/Vec\/N-ary\/NP.agda","new_contents":"module Data.Vec.N-ary.NP where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Vec.N-ary public renaming (curry\u207f to curry\u207f\u2032; _$\u207f_ to _$\u207f\u2032_)\nopen import Relation.Binary.PropositionalEquality\n\ncurry\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192\n         ((xs : Vec A n) \u2192 B $\u207f\u2032 xs) \u2192 \u2200\u207f n B\ncurry\u207f {zero}  f = f []\ncurry\u207f {suc n} f = \u03bb x \u2192 curry\u207f (f \u2218 _\u2237_ x)\n\n_$\u207f_ : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192 \u2200\u207f n B \u2192 ((xs : Vec A n) \u2192 B $\u207f\u2032 xs)\nf $\u207f []       = f\nf $\u207f (x \u2237 xs) = f x $\u207f xs\n\ncurry-$\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} (f : (xs : Vec A n) \u2192 B $\u207f\u2032 xs) (xs : Vec A n)\n           \u2192 curry\u207f f $\u207f xs \u2261 f xs\ncurry-$\u207f f []       = refl\ncurry-$\u207f f (x \u2237 xs) = curry-$\u207f (f \u2218 _\u2237_ x) xs\n\ncurry-$\u207f\u2032 : \u2200 {n a b} {A : Set a} {B : Set b} (f : Vec A n \u2192 B) (xs : Vec A n)\n           \u2192 curry\u207f\u2032 f $\u207f\u2032 xs \u2261 f xs\ncurry-$\u207f\u2032 f []       = refl\ncurry-$\u207f\u2032 f (x \u2237 xs) = curry-$\u207f\u2032 (f \u2218 _\u2237_ x) xs\n\nconst\u207f : \u2200 n {a b} {A : Set a} {B : Set b} \u2192 B \u2192 N-ary n A B\nconst\u207f zero x = x\nconst\u207f (suc n) x = const (const\u207f n x)\n\nlift\u2081 : \u2200 {a b} {A : Set a} {B : Set b} (f g : A \u2192 B) \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 \u2200 {n} (xs : Vec A n) \u2192 map f xs \u2261 map g xs\nlift\u2081 f g pf [] = refl\nlift\u2081 f g pf (x \u2237 xs) rewrite lift\u2081 f g pf xs | pf x = refl\n\nlift\u2083 : \u2200 {a b} {A : Set a} {B : Set b} (f g : N-ary 3 A B) \u2192 (\u2200 x y z \u2192 f x y z \u2261 g x y z)\n        \u2192 \u2200 {n} (xs ys zs : Vec A n)\n        \u2192 replicate f \u229b xs \u229b ys \u229b zs \u2261 replicate g \u229b xs \u229b ys \u229b zs\nlift\u2083 f g pf [] [] [] = refl\nlift\u2083 f g pf (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite lift\u2083 f g pf xs ys zs | pf x y z = refl\n\n-- move it\ntranspose : \u2200 {m n a} {A : Set a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap' : \u2200 {m a b} {A : Set a} {B : Set b} (f : Vec A m \u2192 B)\n       \u2192 \u2200 {n} \u2192 Vec (Vec A m) n \u2192 Vec B n\nvap' f [] = []\nvap' f (xs \u2237 xs\u2081) = f xs \u2237 vap' f xs\u2081\n\nvap'' : \u2200 {m a b} {A : Set a} {B : Set b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap'' f = vap' f \u2218 transpose\n\nvap : \u2200 {m a b} {A : Set a} {B : Set b} (f : N-ary m A B)\n       \u2192 \u2200 {n} \u2192 N-ary m (Vec A n) (Vec B n)\nvap {m} f = curry\u207f\u2032 {m} (\u03bb xs \u2192 vap'' (_$\u207f\u2032_ f) xs)\n\nlift' : \u2200 {m a b} {A : Set a} {B : Set b} (f g : Vec A m \u2192 B)\n       \u2192 (\u2200 (xs : Vec A m) \u2192 f xs \u2261 g xs)\n       \u2192 \u2200 {n : \u2115} (xs : Vec (Vec A m) n) \u2192 vap' f xs \u2261 vap' g xs\nlift' f g pf [] = refl\nlift' f g pf (x \u2237 xs) rewrite lift' f g pf xs | pf x = refl\n\nlift : \u2200 {m a b} {A : Set a} {B : Set b} (f g : N-ary m A B)\n       \u2192 (\u2200\u207f m (curry\u207f\u2032 (\u03bb (xs : Vec A m) \u2192 f $\u207f\u2032 xs \u2261 g $\u207f\u2032 xs)))\n       \u2192 \u2200 {n} \u2192 \u2200\u207f m (curry\u207f\u2032 (\u03bb (xs : Vec (Vec A n) m) \u2192 vap {m} f $\u207f\u2032 xs \u2261 vap {m} g $\u207f\u2032 xs))\nlift {m} f g pf = curry\u207f helper\n  where f' = vap'' {m} (_$\u207f\u2032_ f)\n        g' = vap'' {m} (_$\u207f\u2032_ g)\n        h  = \u03bb (xs : Vec _ m) \u2192 curry\u207f\u2032 f' $\u207f\u2032 xs \u2261 curry\u207f\u2032 g' $\u207f\u2032 xs\n        hh = \u03bb (xs : Vec _ m) \u2192 f $\u207f\u2032 xs \u2261 g $\u207f\u2032 xs\n        pf' : \u2200 xs \u2192 hh xs\n        pf' xs rewrite sym (curry-$\u207f\u2032 hh xs) = pf $\u207f xs\n        helper : \u2200 xs \u2192 curry\u207f\u2032 h $\u207f\u2032 xs\n        helper xs rewrite curry-$\u207f\u2032 h xs | curry-$\u207f\u2032 f' xs | curry-$\u207f\u2032 g' xs\n          = lift' (_$\u207f\u2032_ f) (_$\u207f\u2032_ g) pf' (transpose xs)\n","old_contents":"module Data.Vec.N-ary.NP where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Vec.N-ary public renaming (curry\u207f to curry\u207f\u2032; _$\u207f_ to _$\u207f\u2032_)\nopen import Relation.Binary.PropositionalEquality\n\ncurry\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192\n         ((xs : Vec A n) \u2192 B $\u207f\u2032 xs) \u2192 \u2200\u207f n B\ncurry\u207f {zero}  f = f []\ncurry\u207f {suc n} f = \u03bb x \u2192 curry\u207f (f \u2218 _\u2237_ x)\n\n_$\u207f_ : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192 \u2200\u207f n B \u2192 ((xs : Vec A n) \u2192 B $\u207f\u2032 xs)\nf $\u207f []       = f\nf $\u207f (x \u2237 xs) = f x $\u207f xs\n\ncurry-$\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} (f : (xs : Vec A n) \u2192 B $\u207f\u2032 xs) (xs : Vec A n)\n           \u2192 curry\u207f f $\u207f xs \u2261 f xs\ncurry-$\u207f f []       = refl\ncurry-$\u207f f (x \u2237 xs) = curry-$\u207f (f \u2218 _\u2237_ x) xs\n\ncurry-$\u207f\u2032 : \u2200 {n a b} {A : Set a} {B : Set b} (f : Vec A n \u2192 B) (xs : Vec A n)\n           \u2192 curry\u207f\u2032 f $\u207f\u2032 xs \u2261 f xs\ncurry-$\u207f\u2032 f []       = refl\ncurry-$\u207f\u2032 f (x \u2237 xs) = curry-$\u207f\u2032 (f \u2218 _\u2237_ x) xs\n\nconst\u207f : \u2200 n {a b} {A : Set a} {B : Set b} \u2192 B \u2192 N-ary n A B\nconst\u207f zero x = x\nconst\u207f (suc n) x = const (const\u207f n x)\n\nlift\u2081 : \u2200 {a b} {A : Set a} {B : Set b} (f g : A \u2192 B) \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 \u2200 {n} (xs : Vec A n) \u2192 map f xs \u2261 map g xs\nlift\u2081 f g pf [] = refl\nlift\u2081 f g pf (x \u2237 xs) rewrite lift\u2081 f g pf xs | pf x = refl\n\n-- move it\ntranspose : \u2200 {m n a} {A : Set a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap' : \u2200 {m a b} {A : Set a} {B : Set b} (f : Vec A m \u2192 B)\n       \u2192 \u2200 {n} \u2192 Vec (Vec A m) n \u2192 Vec B n\nvap' f [] = []\nvap' f (xs \u2237 xs\u2081) = f xs \u2237 vap' f xs\u2081\n\nvap'' : \u2200 {m a b} {A : Set a} {B : Set b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap'' f = vap' f \u2218 transpose\n\nvap : \u2200 {m a b} {A : Set a} {B : Set b} (f : N-ary m A B)\n       \u2192 \u2200 {n} \u2192 N-ary m (Vec A n) (Vec B n)\nvap {m} f = curry\u207f\u2032 {m} (\u03bb xs \u2192 vap'' (_$\u207f\u2032_ f) xs)\n\nlift' : \u2200 {m a b} {A : Set a} {B : Set b} (f g : Vec A m \u2192 B)\n       \u2192 (\u2200 (xs : Vec A m) \u2192 f xs \u2261 g xs)\n       \u2192 \u2200 {n : \u2115} (xs : Vec (Vec A m) n) \u2192 vap' f xs \u2261 vap' g xs\nlift' f g pf [] = refl\nlift' f g pf (x \u2237 xs) rewrite lift' f g pf xs | pf x = refl\n\nlift : \u2200 {m a b} {A : Set a} {B : Set b} (f g : N-ary m A B)\n       \u2192 (\u2200\u207f m (curry\u207f\u2032 (\u03bb (xs : Vec A m) \u2192 f $\u207f\u2032 xs \u2261 g $\u207f\u2032 xs)))\n       \u2192 \u2200 {n} \u2192 \u2200\u207f m (curry\u207f\u2032 (\u03bb (xs : Vec (Vec A n) m) \u2192 vap {m} f $\u207f\u2032 xs \u2261 vap {m} g $\u207f\u2032 xs))\nlift {m} f g pf = curry\u207f helper\n  where f' = vap'' {m} (_$\u207f\u2032_ f)\n        g' = vap'' {m} (_$\u207f\u2032_ g)\n        h  = \u03bb (xs : Vec _ m) \u2192 curry\u207f\u2032 f' $\u207f\u2032 xs \u2261 curry\u207f\u2032 g' $\u207f\u2032 xs\n        hh = \u03bb (xs : Vec _ m) \u2192 f $\u207f\u2032 xs \u2261 g $\u207f\u2032 xs\n        pf' : \u2200 xs \u2192 hh xs\n        pf' xs rewrite sym (curry-$\u207f\u2032 hh xs) = pf $\u207f xs\n        helper : \u2200 xs \u2192 curry\u207f\u2032 h $\u207f\u2032 xs\n        helper xs rewrite curry-$\u207f\u2032 h xs | curry-$\u207f\u2032 f' xs | curry-$\u207f\u2032 g' xs\n          = lift' (_$\u207f\u2032_ f) (_$\u207f\u2032_ g) pf' (transpose xs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e449e00f2c1e586ddf5a78a508185eb30347d0fc","subject":"Import bool","message":"Import bool\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"open import Agda.Builtin.Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n{-# BUILTIN LIST List #-}\n{-# BUILTIN NIL  \u2218    #-}\n{-# BUILTIN CONS _\u2237_  #-}\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n\n\nmain = (closure proofs (5 \u2237 \u2218))\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1f3095d1b5024994ffe58632d8ca3bf9b314efd3","subject":"Bool: de-morgan","message":"Bool: de-morgan\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e2dd69c1e2812c42a43534da791ac11bdd768153","subject":"update cong\u2082 to ap\u2082","message":"update cong\u2082 to ap\u2082\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_) \nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = \u2261.ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f g : A \u2192 A \u2192 A}\n                 (f-g : \u2200 x y z \u2192 f (g x y) z \u2248 f x (g y z)) where\n    zipWith-assoc :\n      \u2200 {n} (xs ys zs : Vec A n) \u2192\n      zipWith f (zipWith g xs ys) zs \u2248\u1d5b zipWith f xs (zipWith g ys zs)\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = f-g x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-left [] = []-cong\n    zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-right [] = []-cong\n    zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n    zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n    zipWith-comm [] [] = []-cong\n    zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-left-inverse [] = []-cong\n     zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-right-inverse [] = []-cong\n     zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                                 ; identity = zipWith-id-left (proj\u2081 M.identity)\n                                                 , zipWith-id-right (proj\u2082 M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (proj\u2081 G.inverse))\n                                                , (zipWith-right-inverse (proj\u2082 G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = idp\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = idp\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\n\nmodule Here {a} {A : \u2605 a} where\n  open Data.Vec.Equality.Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = \u2261.ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = idp\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_) \nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = \u2261.cong\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f g : A \u2192 A \u2192 A}\n                 (f-g : \u2200 x y z \u2192 f (g x y) z \u2248 f x (g y z)) where\n    zipWith-assoc :\n      \u2200 {n} (xs ys zs : Vec A n) \u2192\n      zipWith f (zipWith g xs ys) zs \u2248\u1d5b zipWith f xs (zipWith g ys zs)\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = f-g x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-left [] = []-cong\n    zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-right [] = []-cong\n    zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n    zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n    zipWith-comm [] [] = []-cong\n    zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-left-inverse [] = []-cong\n     zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-right-inverse [] = []-cong\n     zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                                 ; identity = zipWith-id-left (proj\u2081 M.identity)\n                                                 , zipWith-id-right (proj\u2082 M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (proj\u2081 G.inverse))\n                                                , (zipWith-right-inverse (proj\u2082 G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = idp\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = idp\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\n\nmodule Here {a} {A : \u2605 a} where\n  open Data.Vec.Equality.Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = \u2261.cong\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = idp\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ee97a3e6749a3dd5f7faa559198900ab683eb47d","subject":"Revise walkthrough in README.agda (for #275).","message":"Revise walkthrough in README.agda (for #275).\n\nOld-commit-hash: 97b77b8c0bb96fc3d4df29ee914a158019dd3cdd\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n--\n-- The formalization has four parts:\n--\n--   1. A formalization of change structures. This lives in\n--   Base.Change.Algebra. (What we call \"change structure\" in the\n--   paper, we call \"change algebra\" in the Agda code. We changed\n--   the name when writing the paper, and never got around to\n--   updating the name in the Agda code).\n--\n--   2. Incrementalization framework for first-class functions,\n--   with extension points for plugging in data types and their\n--   incrementalization. This lives in the Parametric.*\n--   hierarchy.\n--\n--   3. An example plugin that provides integers and bags\n--   with negative multiplicity. This lives in the Nehemiah.*\n--   hierarchy. (For some reason, we choose to call this\n--   particular incarnation of the plugin Nehemiah).\n--\n--   4. Other material that is unrelated to the framework\/plugin\n--   distinction. This is all other files.\n\n\n\n-- FORMALIZATION OF CHANGE STRUCTURES\n-- ==================================\n--\n-- Section 2 of the paper, and Base.Change.Algebra in Agda.\n\nimport Base.Change.Algebra\n\n\n\n-- INCREMENTALIZATION FRAMEWORK\n-- ============================\n--\n-- Section 3 of the paper, and Parametric.* hierarchy in Agda.\n--\n-- The extension points are implemented as module parameters. See\n-- detailed explanation in Parametric.Syntax.Type and\n-- Parametric.Syntax.Term. Some extension points are for types,\n-- some for values, and some for proof fragments. In Agda, these\n-- three kinds of entities are unified anyway, so we can encode\n-- all of them as module parameters.\n--\n-- Every module in the Parametric.* hierarchy adds at least one\n-- extension point, so the module hierarchy of a plugin will\n-- typically mirror the Parametric.* hierarchy, defining the\n-- values for these extension points.\n--\n-- Firstly, we have the syntax of types and terms, the erased\n-- change structure for function types and incrementalization as\n-- a term-to-term transformation. The contents of these modules\n-- formalizes Sec. 3.1 and 3.2 of the paper, except for the\n-- second and third line of Figure 3, which is formalized further\n-- below.\n\nimport Parametric.Syntax.Type   -- syntax of types\nimport Parametric.Syntax.Term   -- syntax of terms\nimport Parametric.Change.Type   -- simply-typed changes\nimport Parametric.Change.Derive -- incrementalization\n\n-- Secondly, we define the usual denotational semantics of the\n-- simply-typed lambda calculus in terms of total functions, and\n-- the change semantics in terms of a change structure on total\n-- functions. The contents of these modules formalize Sec. 3.3,\n-- 3.4 and 3.5 of the paper.\n\nimport Parametric.Denotation.Value      -- standard values\nimport Parametric.Denotation.Evaluation -- standard evaluation\nimport Parametric.Change.Validity       -- dependently-typed changes\nimport Parametric.Change.Specification  -- change evaluation\n\n-- Thirdly, we define terms that operate on simply-typed changes,\n-- and connect them to their values. The concents of these\n-- modules formalize the second and third line of Figure 3, as\n-- well as the semantics of these lines.\n\nimport Parametric.Change.Term        -- terms that operate on simply-typed changes\nimport Parametric.Change.Value       -- the values of these terms\nimport Parametric.Change.Evaluation  -- connecting the terms and their values\n\n-- Finally, we prove correctness by connecting the (syntactic)\n-- incrementalization to the (semantic) change evaluation by a\n-- logical relation, and a proof that the values of terms\n-- produced by the incrementalization transformation are related\n-- to the change values of the original terms. The contents of\n-- these modules formalize Sec. 3.6.\n\nimport Parametric.Change.Implementation -- logical relation\nimport Parametric.Change.Correctness    -- main correctness proof\n\n\n\n-- EXAMPLE PLUGIN\n-- ==============\n--\n-- Sec. 3.7 in the paper, and the Nehemiah.* hierarchy in Agda.\n--\n-- The module structure of the plugin follows the structure of\n-- the Parametric.* hierarchy.  For example, the extension point\n-- defined in Parametric.Syntax.Term is instantiated in\n-- Nehemiah.Syntax.Term.\n--\n-- As discussed in Sec. 3.7 of the paper, the point of this\n-- plugin is not to speed up any real programs, but to show \"that\n-- the interface for proof plugins can be implemented\". As a\n-- first step towards proving correctness of the more complicated\n-- plugin with integers, bags and finite maps we implement in\n-- Scala, we choose to define plugin with integers and bags in\n-- Agda. Instead of implementing bags (with negative\n-- multiplicities, like in the paper) in Agda, though, we\n-- postulate that a group of such bags exist. Note that integer\n-- bags with integer multiplicities are actually the free group\n-- given a singleton operation `Integer -> Bag`, so this should\n-- be easy to formalize in principle.\n\n-- Before we start with the plugin, we postulate an abstract data\n-- type for integer bags.\nimport Postulate.Bag-Nehemiah\n\n-- Firstly, we extend the syntax of types and terms, the erased\n-- change structure for function types, and incrementalization as\n-- a term-to-term transformation to account for the data types of\n-- the Nehemiah language. The contents of these modules\n-- instantiate the extension points in Sec. 3.1 and 3.2 of the\n-- paper, except for the second and third line of Figure 3, which\n-- is instantiated further below.\n\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\nimport Nehemiah.Change.Type\nimport Nehemiah.Change.Derive\n\n-- Secondly, we extend the usual denotational semantics and the\n-- change semantics to account for the data types of the Nehemiah\n-- language. The contents of these modules instantiate the\n-- extension points in Sec. 3.3, 3.4 and 3.5 of the paper.\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\n\n-- Thirdly, we extend the terms that operate on simply-typed\n-- changes, and the connection to their values to account for the\n-- data types of the Nehemiah language. The concents of these\n-- modules instantiate the extension points in the second and\n-- third line of Figure 3.\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- Finally, we extend the logical relation and the main\n-- correctness proof to account for the data types in the\n-- Nehemiah language. The contents of these modules instantiate\n-- the extension points defined in Sec. 3.6.\n\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n\n\n-- OTHER MATERIAL\n-- ==============\n--\n-- We postulate extensionality of Agda functions. This postulate\n-- is well known to be compatible with Agda's type theory.\n\nimport Postulate.Extensionality\n\n-- For historical reasons, we reexport Data.List.All from the\n-- standard library under the name DependentList.\n\nimport Base.Data.DependentList\n\n-- This module supports overloading the \u27e6_\u27e7 notation based on\n-- Agda's instance arguments.\n\nimport Base.Denotation.Notation\n\n-- These modules implement contexts including typed de Bruijn\n-- indices to represent bound variables, sets of bound variables,\n-- and environments. These modules are parametric in the set of\n-- types (that are stored in contexts) and the set of values\n-- (that are stored in environments). So these modules are even\n-- more parametric than the Parametric.* hierarchy.\n\nimport Base.Syntax.Context\nimport Base.Syntax.Vars\nimport Base.Denotation.Environment\n\n-- This module contains some helper definitions to merge a\n-- context of values and a context of changes.\nimport Base.Change.Context\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n-- We claim that this formalization\n--\n--   (1) proves every lemma and theorem in Sec. 2 and 3 of the paper,\n--   (2) formally specifies the interface between our incrementalization\n--       framework and potential plugins, and\n--   (3) shows that the plugin interface can be instantiated.\n--\n-- The first claim is the main reason for a machine-checked\n-- proof: We want to be sure that we got the proofs right.\n--\n-- The second claim is about reusability and applicability: Only\n-- a clearly defined interface allows other researchers to\n-- provide plugins for our framework.\n--\n-- The third claim is to show that the plugin interface is\n-- consistent: An inconsistent plugin interface would allow to\n-- prove arbitrary results in the framework.\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n-- (This import is also important to ensure that if we typecheck\n-- README.agda, we also typecheck every other file of our\n-- formalization).\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"373cad035653a2909a9543dcd31b513eb83efd83","subject":"judgemental forms, no rules","message":"judgemental forms, no rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    num   : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2295_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2297_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- expressions, prefixed with a \u00b7 to distinguish name clashes with agda\n  -- built-ins\n  data e\u0307 : Set where\n    _\u00b7:_   : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X      : Nat \u2192 e\u0307\n    \u00b7\u03bb     : Nat \u2192 e\u0307 \u2192 e\u0307\n    N      : Nat \u2192 e\u0307\n    _\u00b7+_   : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]  : Nat \u2192 e\u0307\n    \u2987_\u2988[_] : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_    : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    inl    : e\u0307 \u2192 e\u0307\n    inr    : e\u0307 \u2192 e\u0307\n    case   : e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 e\u0307\n    [_,_]  : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    fst    : e\u0307 \u2192 e\u0307\n    snd    : e\u0307 \u2192 e\u0307\n\n  data subst : Set where -- todo\n\n  data \u00eb : Set where\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb       : Nat \u2192 \u00eb \u2192 \u00eb\n    N        : Nat \u2192 \u00eb\n    _\u00b7+_     : \u00eb \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_,_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_,_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    inl      : \u00eb \u2192 \u00eb\n    inr      : \u00eb \u2192 \u00eb\n    case     : \u00eb \u2192 Nat \u2192 \u00eb \u2192 Nat \u2192 \u00eb \u2192 \u00eb\n    [_,_]    : \u00eb \u2192 \u00eb \u2192 \u00eb\n    fst      : \u00eb \u2192 \u00eb\n    snd      : \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 ==> t2) ~ (t1' ==> t2')\n    TCSum : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2295 t2) ~ (t1' \u2295 t2')\n    TCPro : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2297 t2) ~ (t1' \u2297 t2')\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICNumArr1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 ==> t2)\n    ICNumArr2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 num\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n\n    ICNumSum1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2295 t2)\n    ICNumSum2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 num\n    ICSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSumArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 ==> t4)\n    ICSumArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2295 t4)\n\n    ICNumPro1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2297 t2)\n    ICNumPro2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 num\n    ICPro1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICPro2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICProArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 ==> t4)\n    ICProArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2297 t4)\n\n    ICProSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 \u2295 t4)\n    ICProSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 \u2297 t4)\n\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr (\u2987\u2988 ==> \u2987\u2988)\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) \u25b8arr (t1 ==> t2)\n\n  data _\u25b8sum_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MSHole  : \u2987\u2988 \u25b8sum (\u2987\u2988 \u2295 \u2987\u2988)\n    MSPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) \u25b8sum (t1 \u2295 t2)\n\n  data _\u25b8pro_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MPHole  : \u2987\u2988 \u25b8pro (\u2987\u2988 \u2297 \u2987\u2988)\n    MPPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) \u25b8pro (t1 \u2297 t2)\n\n  -- matching produces unique answers for arrows, sums, and products\n  \u25b8arr-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8arr t2 \u2192\n                 t \u25b8arr t3 \u2192\n                 t2 == t3\n  \u25b8arr-unicity MAHole MAHole = refl\n  \u25b8arr-unicity MAArr MAArr = refl\n\n  \u25b8sum-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8sum t2 \u2192\n                 t \u25b8sum t3 \u2192\n                 t2 == t3\n  \u25b8sum-unicity MSHole MSHole = refl\n  \u25b8sum-unicity MSPlus MSPlus = refl\n\n  \u25b8pro-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8pro t2 \u2192\n                 t \u25b8pro t3 \u2192\n                 t2 == t3\n  \u25b8pro-unicity MPHole MPHole = refl\n  \u25b8pro-unicity MPPlus MPPlus = refl\n\n\n  -- if an arrow matches, then it's consistent with the least restrictive\n  -- function type\n  matchconsist : \u2200{t t'} \u2192\n                 t \u25b8arr t' \u2192\n                 t ~ (\u2987\u2988 ==> \u2987\u2988)\n  matchconsist MAHole = TCHole2\n  matchconsist MAArr = TCArr TCHole1 TCHole1\n\n  matchnotnum : \u2200{t1 t2} \u2192 num \u25b8arr (t1 ==> t2) \u2192 \u22a5\n  matchnotnum ()\n\n  ---- contexts and some operations on them\n\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  \u00b7ctx : Set\n  \u00b7ctx = Nat \u2192 Maybe \u03c4\u0307\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : \u00b7ctx\n  \u2205 _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : \u00b7ctx \u2192 (Nat \u00d7 \u03c4\u0307) \u2192 \u00b7ctx\n  (\u0393 ,, (x , t)) y with natEQ x y\n  (\u0393 ,, (x , t)) .x | Inl refl = Some t\n  (\u0393 ,, (x , t)) y  | Inr neq  = \u0393 y\n\n  -- membership, or presence, in a context\n  _\u2208_ : (p : Nat \u00d7 \u03c4\u0307) \u2192 (\u0393 : \u00b7ctx) \u2192 Set\n  (x , t) \u2208 \u0393 = (\u0393 x) == Some t\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : (n : Nat) \u2192 (\u0393 : \u00b7ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- without: remove a variable from a context\n  _\/_ : \u00b7ctx \u2192 Nat \u2192 \u00b7ctx\n  (\u0393 \/ x) y with natEQ x y\n  (\u0393 \/ x) .x | Inl refl = None\n  (\u0393 \/ x) y  | Inr neq  = \u0393 y\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : \u00b7ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SAsc    : {\u0393 : \u00b7ctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : \u00b7ctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : \u00b7ctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr (t2 ==> t') \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SNum    :  {\u0393 : \u00b7ctx} {n : Nat} \u2192\n                 \u0393 \u22a2 N n => num\n      SPlus   : {\u0393 : \u00b7ctx} {e1 e2 : e\u0307}  \u2192\n                 \u0393 \u22a2 e1 <= num \u2192\n                 \u0393 \u22a2 e2 <= num \u2192\n                 \u0393 \u22a2 (e1 \u00b7+ e2) => num\n      SEHole  : {\u0393 : \u00b7ctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : \u00b7ctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n\n    --todo: add rules for products\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u00b7ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : \u00b7ctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : \u00b7ctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr (t1 ==> t2) \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n      AInl : {\u0393 : \u00b7ctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t1 \u2192\n                 \u0393 \u22a2 inl e <= t+\n      AInr : {\u0393 : \u00b7ctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 inr e <= t+\n      ACase : {\u0393 : \u00b7ctx} {e e1 e2 : e\u0307} {t t+ t1 t2 : \u03c4\u0307} {x y : Nat} \u2192\n                 x # \u0393 \u2192\n                 y # \u0393 \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e => t+ \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e1 <= t \u2192\n                 (\u0393 ,, (y , t2)) \u22a2 e2 <= t \u2192\n                 \u0393 \u22a2 case e x e1 y e2 <= t\n\n  ----- some theorems about the rules and judgement presented so far.\n\n  -- a variable is apart from any context from which it is removed\n  aar : (\u0393 : \u00b7ctx) (x : Nat) \u2192 x # (\u0393 \/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {\u0393 : \u00b7ctx} {n : Nat} {t t' : \u03c4\u0307} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- type consistency is symmetric\n  ~sym : {t1 t2 : \u03c4\u0307} \u2192 t1 ~ t2 \u2192 t2 ~ t1\n  ~sym TCRefl = TCRefl\n  ~sym TCHole1 = TCHole2\n  ~sym TCHole2 = TCHole1\n  ~sym (TCArr p1 p2) = TCArr (~sym p1) (~sym p2)\n  ~sym (TCSum p1 p2) = TCSum (~sym p1) (~sym p2)\n  ~sym (TCPro p1 p2) = TCPro (~sym p1) (~sym p2)\n\n  -- type consistency isn't transitive\n  not-trans : ((t1 t2 t3 : \u03c4\u0307) \u2192 t1 ~ t2 \u2192 t2 ~ t3 \u2192 t1 ~ t3) \u2192 \u22a5\n  not-trans t with t (num ==> num) \u2987\u2988 num TCHole1 TCHole2\n  ... | ()\n\n  --  every pair of types is either consistent or not consistent\n  ~dec : (t1 t2 : \u03c4\u0307) \u2192 ((t1 ~ t2) + (t1 ~\u0338 t2))\n    -- this takes care of all hole cases, so we don't consider them below\n  ~dec _ \u2987\u2988 = Inl TCHole1\n  ~dec \u2987\u2988 _ = Inl TCHole2\n    -- num cases\n  ~dec num num = Inl TCRefl\n  ~dec num (t2 ==> t3) = Inr ICNumArr1\n    -- arrow cases\n  ~dec (t1 ==> t2) num = Inr ICNumArr2\n  ~dec (t1 ==> t2) (t3 ==> t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCArr x y)\n  ... | Inl _ | Inr y = Inr (ICArr2 y)\n  ... | Inr x | _     = Inr (ICArr1 x)\n    -- plus cases\n  ~dec (t1 \u2295 t2) num = Inr ICNumSum2\n  ~dec (t1 \u2295 t2) (t3 ==> t4) = Inr ICSumArr1\n  ~dec (t1 \u2295 t2) (t3 \u2295 t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCSum x y)\n  ... | _     | Inr x = Inr (ICSum2 x)\n  ... | Inr x | Inl _ = Inr (ICSum1 x)\n  ~dec num (y \u2295 y\u2081) = Inr ICNumSum1\n  ~dec (x ==> x\u2081) (y \u2295 y\u2081) = Inr ICSumArr2\n  ~dec num (t2 \u2297 t3) = Inr ICNumPro1\n  ~dec (t1 ==> t2) (t3 \u2297 t4) = Inr ICProArr2\n  ~dec (t1 \u2295 t2) (t3 \u2297 t4) = Inr ICProSum2\n  ~dec (t1 \u2297 t2) num = Inr ICNumPro2\n  ~dec (t1 \u2297 t2) (t3 ==> t4) = Inr ICProArr1\n  ~dec (t1 \u2297 t2) (t3 \u2295 t4) = Inr ICProSum1\n  ~dec (t1 \u2297 t2) (t3 \u2297 t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCPro x y)\n  ... | _     | Inr x = Inr (ICPro2 x)\n  ... | Inr x | Inl _ = Inr (ICPro1 x)\n\n  -- no pair of types is both consistent and not consistent\n  ~apart : {t1 t2 : \u03c4\u0307} \u2192 (t1 ~\u0338 t2) \u2192 (t1 ~ t2) \u2192 \u22a5\n  ~apart ICNumArr1 ()\n  ~apart ICNumArr2 ()\n  ~apart (ICArr1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICArr1 x) (TCArr y y\u2081) = ~apart x y\n  ~apart (ICArr2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICArr2 x) (TCArr y y\u2081) = ~apart x y\u2081\n  ~apart ICNumSum1 ()\n  ~apart ICNumSum2 ()\n  ~apart (ICSum1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICSum1 x) (TCSum y y\u2081) = ~apart x y\n  ~apart (ICSum2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICSum2 x) (TCSum y y\u2081) = ~apart x y\u2081\n  ~apart ICSumArr1 ()\n  ~apart ICSumArr2 ()\n  ~apart ICNumPro1 ()\n  ~apart ICNumPro2 ()\n  ~apart (ICPro1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICPro1 x) (TCPro y y\u2081) = ~apart x y\n  ~apart (ICPro2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICPro2 x) (TCPro y y\u2081) = ~apart x y\u2081\n  ~apart ICProArr1 ()\n  ~apart ICProArr2 ()\n  ~apart ICProSum1 ()\n  ~apart ICProSum2 ()\n\n  -- synthesis only produces equal types. note that there is no need for an\n  -- analagous theorem for analytic positions because we think of\n  -- the type as an input\n  synthunicity : {\u0393 : \u00b7ctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                  (\u0393 \u22a2 e => t)\n                \u2192 (\u0393 \u22a2 e => t')\n                \u2192 t == t'\n  synthunicity (SAsc _) (SAsc _) = refl\n  synthunicity {\u0393 = G} (SVar in1) (SVar in2) = ctxunicity {\u0393 = G} in1 in2\n  synthunicity (SAp D1 MAHole b) (SAp D2 MAHole y) = refl\n  synthunicity (SAp D1 MAHole b) (SAp D2 MAArr y) with synthunicity D1 D2\n  ... | ()\n  synthunicity (SAp D1 MAArr b) (SAp D2 MAHole y) with synthunicity D1 D2\n  ... | ()\n  synthunicity (SAp D1 MAArr b) (SAp D2 MAArr y) with synthunicity D1 D2\n  ... | refl = refl\n  synthunicity SNum SNum = refl\n  synthunicity (SPlus _ _ ) (SPlus _ _ ) = refl\n  synthunicity SEHole SEHole = refl\n  synthunicity (SNEHole _) (SNEHole _) = refl\n\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete num         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2295 t2)   = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2297 t2)   = tcomplete t1 \u00d7 tcomplete t2\n\n  -- similarly to the complete types, the complete expressions\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete (N x)      = \u22a4\n  ecomplete (e1 \u00b7+ e2) = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (inl e)    = ecomplete e\n  ecomplete (inr e)    = ecomplete e\n  ecomplete (case e x e1 y e2)  = ecomplete e \u00d7 ecomplete e1 \u00d7 ecomplete e2\n  ecomplete [ e1 , e2 ] = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (fst e) = ecomplete e\n  ecomplete (snd e) = ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : \u00b7ctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : \u00b7ctx) \u2192 Set where\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : \u00b7ctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : \u00b7ctx) \u2192 Set where\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0393 : \u00b7ctx) (\u0394 : \u00b7ctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n\n  -- value\n  data _value : \u00eb \u2192 Set where\n\n  -- indeterminate\n  data _indet : \u00eb \u2192 Set where\n\n  -- error\n  data _err[_] : \u00eb \u2192 \u00b7ctx \u2192 Set where -- todo not a context\n\n  -- final\n  data _final : \u00eb \u2192 Set where\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","old_contents":"open import Nat\nopen import Prelude\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    num   : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2295_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2297_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- expressions, prefixed with a \u00b7 to distinguish name clashes with agda\n  -- built-ins\n  data e\u0307 : Set where\n    _\u00b7:_   : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X      : Nat \u2192 e\u0307\n    \u00b7\u03bb     : Nat \u2192 e\u0307 \u2192 e\u0307\n    N      : Nat \u2192 e\u0307\n    _\u00b7+_   : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]  : Nat \u2192 e\u0307\n    \u2987_\u2988[_] : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_    : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    inl    : e\u0307 \u2192 e\u0307\n    inr    : e\u0307 \u2192 e\u0307\n    case   : e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 e\u0307\n    [_,_]  : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    fst    : e\u0307 \u2192 e\u0307\n    snd    : e\u0307 \u2192 e\u0307\n\n  data subst : Set where -- todo\n\n  data \u00eb : Set where\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb       : Nat \u2192 \u00eb \u2192 \u00eb\n    N        : Nat \u2192 \u00eb\n    _\u00b7+_     : \u00eb \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_,_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_,_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    inl      : \u00eb \u2192 \u00eb\n    inr      : \u00eb \u2192 \u00eb\n    case     : \u00eb \u2192 Nat \u2192 \u00eb \u2192 Nat \u2192 \u00eb \u2192 \u00eb\n    [_,_]    : \u00eb \u2192 \u00eb \u2192 \u00eb\n    fst      : \u00eb \u2192 \u00eb\n    snd      : \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 ==> t2) ~ (t1' ==> t2')\n    TCSum : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2295 t2) ~ (t1' \u2295 t2')\n    TCPro : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2297 t2) ~ (t1' \u2297 t2')\n\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICNumArr1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 ==> t2)\n    ICNumArr2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 num\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n\n    ICNumSum1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2295 t2)\n    ICNumSum2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 num\n    ICSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSumArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 ==> t4)\n    ICSumArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2295 t4)\n\n    ICNumPro1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2297 t2)\n    ICNumPro2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 num\n    ICPro1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICPro2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICProArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 ==> t4)\n    ICProArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2297 t4)\n\n    ICProSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 \u2295 t4)\n    ICProSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 \u2297 t4)\n\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr (\u2987\u2988 ==> \u2987\u2988)\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) \u25b8arr (t1 ==> t2)\n\n  data _\u25b8sum_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MSHole  : \u2987\u2988 \u25b8sum (\u2987\u2988 \u2295 \u2987\u2988)\n    MSPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) \u25b8sum (t1 \u2295 t2)\n\n  data _\u25b8pro_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MPHole  : \u2987\u2988 \u25b8pro (\u2987\u2988 \u2297 \u2987\u2988)\n    MPPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) \u25b8pro (t1 \u2297 t2)\n\n  -- matching produces unique answers for arrows, sums, and products\n  \u25b8arr-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8arr t2 \u2192\n                 t \u25b8arr t3 \u2192\n                 t2 == t3\n  \u25b8arr-unicity MAHole MAHole = refl\n  \u25b8arr-unicity MAArr MAArr = refl\n\n  \u25b8sum-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8sum t2 \u2192\n                 t \u25b8sum t3 \u2192\n                 t2 == t3\n  \u25b8sum-unicity MSHole MSHole = refl\n  \u25b8sum-unicity MSPlus MSPlus = refl\n\n  \u25b8pro-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8pro t2 \u2192\n                 t \u25b8pro t3 \u2192\n                 t2 == t3\n  \u25b8pro-unicity MPHole MPHole = refl\n  \u25b8pro-unicity MPPlus MPPlus = refl\n\n\n  -- if an arrow matches, then it's consistent with the least restrictive\n  -- function type\n  matchconsist : \u2200{t t'} \u2192\n                 t \u25b8arr t' \u2192\n                 t ~ (\u2987\u2988 ==> \u2987\u2988)\n  matchconsist MAHole = TCHole2\n  matchconsist MAArr = TCArr TCHole1 TCHole1\n\n  matchnotnum : \u2200{t1 t2} \u2192 num \u25b8arr (t1 ==> t2) \u2192 \u22a5\n  matchnotnum ()\n\n  ---- contexts and some operations on them\n\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  \u00b7ctx : Set\n  \u00b7ctx = Nat \u2192 Maybe \u03c4\u0307\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : \u00b7ctx\n  \u2205 _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : \u00b7ctx \u2192 (Nat \u00d7 \u03c4\u0307) \u2192 \u00b7ctx\n  (\u0393 ,, (x , t)) y with natEQ x y\n  (\u0393 ,, (x , t)) .x | Inl refl = Some t\n  (\u0393 ,, (x , t)) y  | Inr neq  = \u0393 y\n\n  -- membership, or presence, in a context\n  _\u2208_ : (p : Nat \u00d7 \u03c4\u0307) \u2192 (\u0393 : \u00b7ctx) \u2192 Set\n  (x , t) \u2208 \u0393 = (\u0393 x) == Some t\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : (n : Nat) \u2192 (\u0393 : \u00b7ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- without: remove a variable from a context\n  _\/_ : \u00b7ctx \u2192 Nat \u2192 \u00b7ctx\n  (\u0393 \/ x) y with natEQ x y\n  (\u0393 \/ x) .x | Inl refl = None\n  (\u0393 \/ x) y  | Inr neq  = \u0393 y\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : \u00b7ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SAsc    : {\u0393 : \u00b7ctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : \u00b7ctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : \u00b7ctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr (t2 ==> t') \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SNum    :  {\u0393 : \u00b7ctx} {n : Nat} \u2192\n                 \u0393 \u22a2 N n => num\n      SPlus   : {\u0393 : \u00b7ctx} {e1 e2 : e\u0307}  \u2192\n                 \u0393 \u22a2 e1 <= num \u2192\n                 \u0393 \u22a2 e2 <= num \u2192\n                 \u0393 \u22a2 (e1 \u00b7+ e2) => num\n      SEHole  : {\u0393 : \u00b7ctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : \u00b7ctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n\n    --todo: add rules for products\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u00b7ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : \u00b7ctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : \u00b7ctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr (t1 ==> t2) \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n      AInl : {\u0393 : \u00b7ctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t1 \u2192\n                 \u0393 \u22a2 inl e <= t+\n      AInr : {\u0393 : \u00b7ctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 inr e <= t+\n      ACase : {\u0393 : \u00b7ctx} {e e1 e2 : e\u0307} {t t+ t1 t2 : \u03c4\u0307} {x y : Nat} \u2192\n                 x # \u0393 \u2192\n                 y # \u0393 \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e => t+ \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e1 <= t \u2192\n                 (\u0393 ,, (y , t2)) \u22a2 e2 <= t \u2192\n                 \u0393 \u22a2 case e x e1 y e2 <= t\n\n  ----- some theorems about the rules and judgement presented so far.\n\n  -- a variable is apart from any context from which it is removed\n  aar : (\u0393 : \u00b7ctx) (x : Nat) \u2192 x # (\u0393 \/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {\u0393 : \u00b7ctx} {n : Nat} {t t' : \u03c4\u0307} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- type consistency is symmetric\n  ~sym : {t1 t2 : \u03c4\u0307} \u2192 t1 ~ t2 \u2192 t2 ~ t1\n  ~sym TCRefl = TCRefl\n  ~sym TCHole1 = TCHole2\n  ~sym TCHole2 = TCHole1\n  ~sym (TCArr p1 p2) = TCArr (~sym p1) (~sym p2)\n  ~sym (TCSum p1 p2) = TCSum (~sym p1) (~sym p2)\n  ~sym (TCPro p1 p2) = TCPro (~sym p1) (~sym p2)\n\n  -- type consistency isn't transitive\n  not-trans : ((t1 t2 t3 : \u03c4\u0307) \u2192 t1 ~ t2 \u2192 t2 ~ t3 \u2192 t1 ~ t3) \u2192 \u22a5\n  not-trans t with t (num ==> num) \u2987\u2988 num TCHole1 TCHole2\n  ... | ()\n\n  --  every pair of types is either consistent or not consistent\n  ~dec : (t1 t2 : \u03c4\u0307) \u2192 ((t1 ~ t2) + (t1 ~\u0338 t2))\n    -- this takes care of all hole cases, so we don't consider them below\n  ~dec _ \u2987\u2988 = Inl TCHole1\n  ~dec \u2987\u2988 _ = Inl TCHole2\n    -- num cases\n  ~dec num num = Inl TCRefl\n  ~dec num (t2 ==> t3) = Inr ICNumArr1\n    -- arrow cases\n  ~dec (t1 ==> t2) num = Inr ICNumArr2\n  ~dec (t1 ==> t2) (t3 ==> t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCArr x y)\n  ... | Inl _ | Inr y = Inr (ICArr2 y)\n  ... | Inr x | _     = Inr (ICArr1 x)\n    -- plus cases\n  ~dec (t1 \u2295 t2) num = Inr ICNumSum2\n  ~dec (t1 \u2295 t2) (t3 ==> t4) = Inr ICSumArr1\n  ~dec (t1 \u2295 t2) (t3 \u2295 t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCSum x y)\n  ... | _     | Inr x = Inr (ICSum2 x)\n  ... | Inr x | Inl _ = Inr (ICSum1 x)\n  ~dec num (y \u2295 y\u2081) = Inr ICNumSum1\n  ~dec (x ==> x\u2081) (y \u2295 y\u2081) = Inr ICSumArr2\n  ~dec num (t2 \u2297 t3) = Inr ICNumPro1\n  ~dec (t1 ==> t2) (t3 \u2297 t4) = Inr ICProArr2\n  ~dec (t1 \u2295 t2) (t3 \u2297 t4) = Inr ICProSum2\n  ~dec (t1 \u2297 t2) num = Inr ICNumPro2\n  ~dec (t1 \u2297 t2) (t3 ==> t4) = Inr ICProArr1\n  ~dec (t1 \u2297 t2) (t3 \u2295 t4) = Inr ICProSum1\n  ~dec (t1 \u2297 t2) (t3 \u2297 t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCPro x y)\n  ... | _     | Inr x = Inr (ICPro2 x)\n  ... | Inr x | Inl _ = Inr (ICPro1 x)\n\n  -- no pair of types is both consistent and not consistent\n  ~apart : {t1 t2 : \u03c4\u0307} \u2192 (t1 ~\u0338 t2) \u2192 (t1 ~ t2) \u2192 \u22a5\n  ~apart ICNumArr1 ()\n  ~apart ICNumArr2 ()\n  ~apart (ICArr1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICArr1 x) (TCArr y y\u2081) = ~apart x y\n  ~apart (ICArr2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICArr2 x) (TCArr y y\u2081) = ~apart x y\u2081\n  ~apart ICNumSum1 ()\n  ~apart ICNumSum2 ()\n  ~apart (ICSum1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICSum1 x) (TCSum y y\u2081) = ~apart x y\n  ~apart (ICSum2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICSum2 x) (TCSum y y\u2081) = ~apart x y\u2081\n  ~apart ICSumArr1 ()\n  ~apart ICSumArr2 ()\n  ~apart ICNumPro1 ()\n  ~apart ICNumPro2 ()\n  ~apart (ICPro1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICPro1 x) (TCPro y y\u2081) = ~apart x y\n  ~apart (ICPro2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICPro2 x) (TCPro y y\u2081) = ~apart x y\u2081\n  ~apart ICProArr1 ()\n  ~apart ICProArr2 ()\n  ~apart ICProSum1 ()\n  ~apart ICProSum2 ()\n\n  -- synthesis only produces equal types. note that there is no need for an\n  -- analagous theorem for analytic positions because we think of\n  -- the type as an input\n  synthunicity : {\u0393 : \u00b7ctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                  (\u0393 \u22a2 e => t)\n                \u2192 (\u0393 \u22a2 e => t')\n                \u2192 t == t'\n  synthunicity (SAsc _) (SAsc _) = refl\n  synthunicity {\u0393 = G} (SVar in1) (SVar in2) = ctxunicity {\u0393 = G} in1 in2\n  synthunicity (SAp D1 MAHole b) (SAp D2 MAHole y) = refl\n  synthunicity (SAp D1 MAHole b) (SAp D2 MAArr y) with synthunicity D1 D2\n  ... | ()\n  synthunicity (SAp D1 MAArr b) (SAp D2 MAHole y) with synthunicity D1 D2\n  ... | ()\n  synthunicity (SAp D1 MAArr b) (SAp D2 MAArr y) with synthunicity D1 D2\n  ... | refl = refl\n  synthunicity SNum SNum = refl\n  synthunicity (SPlus _ _ ) (SPlus _ _ ) = refl\n  synthunicity SEHole SEHole = refl\n  synthunicity (SNEHole _) (SNEHole _) = refl\n\n\n  ----- the zippered form of the forms above and the rules for actions on them\n\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete num         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2295 t2)   = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2297 t2)   = tcomplete t1 \u00d7 tcomplete t2\n\n  -- similarly to the complete types, the complete expressions\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete (N x)      = \u22a4\n  ecomplete (e1 \u00b7+ e2) = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (inl e)    = ecomplete e\n  ecomplete (inr e)    = ecomplete e\n  ecomplete (case e x e1 y e2)  = ecomplete e \u00d7 ecomplete e1 \u00d7 ecomplete e2\n  ecomplete [ e1 , e2 ] = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (fst e) = ecomplete e\n  ecomplete (snd e) = ecomplete e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e4e49db6ddcf10a09517397c6ef66a28a90558ee","subject":"details on a few more cases; switching some things to implicit","message":"details on a few more cases; switching some things to implicit\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec {!!} \u03c4'\n    expandability-synth (SAsc x\u2081) | d' , \u0394' , _ , D | Inl refl = d' , \u0394' , ESAsc2 D\n    expandability-synth (SAsc x\u2081) | d' , \u0394' , \u03c4' , D | Inr x = (< {!!} > d') , \u0394' , ESAsc1 D {!!}\n    expandability-synth (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth (SAp D MAHole x\u2081)\n      with expandability-synth D | expandability-ana x\u2081\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = ((< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2) , (\u03941 \u222a \u03942) , ESAp1 {!!} D D2 {!!}\n    expandability-synth (SAp D MAArr x\u2081) = {!!} , {!!} , {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , \u2205 , ESLam wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana (ASubsume wt x\u2081) with expandability-synth wt\n    ... | d' , \u0394' , D' = d' , \u0394' , _ , EASubsume (\u03bb x \u2192 {!!}) (\u03bb x \u2192 {!!}) D' x\u2081\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D'  = {!!} , {!!} , {!!} , {!!}\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , {!!} , {!!} , EALam {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth \u0393 .c .b SConst = c , \u2205 , ESConst\n    expandability-synth \u0393 _ \u03c4 (SAsc wt)\n      with expandability-ana _ _ _ wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    expandability-synth \u0393 _ \u03c4 (SAsc x\u2081) | d' , \u0394' , .\u03c4 , D | Inl refl = d' , \u0394' , ESAsc2 D\n    expandability-synth \u0393 _ \u03c4 (SAsc x\u2081) | d' , \u0394' , \u03c4' , D | Inr x = (< \u03c4 > d') , \u0394' , ESAsc1 D x\n    expandability-synth \u0393 _ \u03c4 (SVar {n = n} x) = X n , \u2205 , ESVar x\n    expandability-synth \u0393 _ \u03c4 (SAp D x x\u2081) = {!!}\n    expandability-synth \u0393 _ .\u2987\u2988 SEHole = _ , _ , ESEHole\n    expandability-synth \u0393 _ .\u2987\u2988 (SNEHole D) with expandability-synth _ _ _  D\n    ... | d' , \u0394' , D' = _ , _ , ESNEHole D'\n    expandability-synth \u0393 _ _ (SLam x\u2081 D) = {!!}\n\n    expandability-ana : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana \u0393 e \u03c4 (ASubsume x x\u2081) = {!!} , {!!} , {!!}\n    expandability-ana \u0393 _ \u03c4 (ALam x\u2081 x\u2082 wt) = {!!} , {!!} , {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5a703e3eb686770fc4b49aecd39c330743563cf9","subject":"bintree: more proofs in the easy direction","message":"bintree: more proofs in the easy direction\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b4ba02ffe9a1a6356c94398a8bb6e356c60e18fc","subject":"Hutton's razor: nop, don't try to compute context, life's too short for that.","message":"Hutton's razor: nop, don't try to compute context, life's too short for that.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/RevisedHutton.agda","new_file":"models\/RevisedHutton.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\n-- Open term: holes are either values or variables in a context\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = \n    subst (cong (IMu closeTerm) (snd variable)) \n          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata List (A : Set) : Set where\n  [] : List A\n  _::_ : A -> List A -> List A\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\ndata Comp : Set where\n  LT : Comp\n  EQ : Comp\n  GT : Comp\n\ncomp : {n : Nat} -> Fin n -> Fin n -> Comp\ncomp fze fze = EQ\ncomp fze _ = LT\ncomp _ fze = GT\ncomp (fsu n) (fsu n') = comp n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : Nat -> Type -> Set\nVar n ty = Fin n\n\n-- Open term: holes are either values or variables in a context\nopenTerm : Nat -> Type -> IDesc Type\nopenTerm n = toIDesc Type (exprFree ** (\\ty -> Val ty + Var n ty))\n\ndata Maybe (A : Set) : Set where\n  Nothing : Maybe A\n  Just : A -> Maybe A\n\nmergeVars : {n : Nat} ->\n            Maybe (List (Fin n * Type)) ->\n            Maybe (List (Fin n * Type)) ->\n            Maybe (List (Fin n * Type))\nmergeVars Nothing _ = Nothing\nmergeVars _ Nothing = Nothing\nmergeVars (Just []) (Just y) = Just y\nmergeVars (Just x) (Just []) = Just x\nmergeVars (Just (x :: xs)) (Just (y :: ys)) with comp (fst x) (fst y) \n                                              | mergeVars (Just xs) (Just (y :: ys)) \n                                              | mergeVars (Just (x :: xs)) (Just ys)\n... | LT | _        | Nothing   = Nothing\n... | LT | _        | Just xys  = Just (y :: xys)\n... | EQ | _        | _         = Nothing\n... | GT | Nothing  | _         = Nothing\n... | GT | Just xys | _         = Just (x :: xys)\n\ncollectVars : {ty : Type}{n : Nat} -> IMu (openTerm n) ty -> Maybe (List (Fin n * Type))\ncollectVars {ty} {n} t = cata Type (openTerm n) (\\_ -> Maybe (List (Fin n * Type))) collectVarsHelp ty t\n  where collectVarsHelp : (i : Type) -> [| openTerm n i |] (\\ _ -> Maybe (List (Fin n * Type))) -> Maybe (List (Fin n * Type))\n        collectVarsHelp ty (EZe , r n) = Just ((n , ty) :: [])\n        collectVarsHelp ty (EZe , l _) = Just []\n        collectVarsHelp ty (ESu EZe , (t1 , ( t2 , t3))) = mergeVars (mergeVars t1 t2) t3\n        collectVarsHelp nat (ESu (ESu EZe) , (x , y)) = mergeVars x y\n        collectVarsHelp nat (ESu (ESu (ESu ())) , _) \n        collectVarsHelp bool (ESu (ESu EZe) , (x , y) ) = mergeVars x y\n        collectVarsHelp bool (ESu (ESu (ESu ())) , _) \n        collectVarsHelp (pair x y) (ESu (ESu ()) , _)\n        \n\nValidContext : {ty : Type}{n : Nat}(c : Context n)(tm : IMu (openTerm n) ty) -> Set\nValidContext {ty} {n} c tm  with collectVars tm\n... | Nothing = Zero\n... | (Just y) = checkContext y\n  where checkContext : List (Fin n * Type) -> Set\n        checkContext [] = Unit\n        checkContext ((x , ty) :: xs) = ((typeAt c x) == ty) * checkContext xs \n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var n ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = {!lookup c variable!}\n--          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var n ty) ->\n                     IMu closeTerm ty) ->\n            (tm : IMu (openTerm n) ty) ->\n            ValidContext context tm ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term valid = \n    substI (\\ty -> Val ty + Var n ty) Val exprFree sig ty term\n\n\n\n{-\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n-}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d4d6ce085b8c14776cda265fcef70a5f18f8fb63","subject":"sha1.agda","message":"sha1.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"sha1.agda","new_file":"sha1.agda","new_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\nopen FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\nopen FunOps funOps hiding (_\u2218_)\n\nWord : T\nWord = `Bits 32\n\nmap\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\nmap\u00b2\u02b7 = zipWith\n\nmap\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\nmap\u02b7 = map\n\nlift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\nlift f g = g \u204f f\n\nlift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\nlift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n`not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n`not = lift (map\u02b7 not)\n\ninfixr 3 _`\u2295_\n_`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\ninfixr 3 _`\u2227_\n_`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\ninfixr 2 _`\u2228_\n_`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\nopen import Solver.Linear\nopen import Data.Vec as Vec using ([]; _\u2237_)\n\nmodule STest n M = Syntax T _`\u00d7_ `\ud835\udfd9 _`\u2192_ id _\u204f_ fst <id,tt> snd <tt,id> <_\u00d7_> first second assoc\u2032 assoc swap n M\n\n--iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\niter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\niter {zero}  F = <[]>\niter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n               \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n               \u204f <\u2237>\n\n  where\n    helper : \u2200 {A B D VA VB} \u2192 (D `\u00d7 (A `\u00d7 VA) `\u00d7 (B `\u00d7 VB)) `\u2192 (D `\u00d7 A `\u00d7 B) `\u00d7 (VA `\u00d7 VB)\n    helper {A} {B} {D} {VA} {VB} = solve (! 2 , (! 0 , ! 3) , (! 1 , ! 4))\n                                 ((! 2 , ! 0 , ! 1) , (! 3 , ! 4)) _ where\n      open STest 5 (\u03bb i \u2192 Vec.lookup i (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])) renaming (rewire to solve; #_ to !_)\n\n<\u229e> adder : Word `\u00d7 Word `\u2192 Word\nadder = <tt\u204f <0b> , id > \u204f iter full-adder\n<\u229e> = adder\n\ninfixl 4 _`\u229e_\n_`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n_`\u229e_ = lift\u2082 <\u229e>\n\n<_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n< f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n<_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                              \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                              \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n< f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n<<<\u2085 : `Endo Word\n<<<\u2085 = rot-left 5\n\n<<<\u2083\u2080 : `Endo Word\n<<<\u2083\u2080 = rot-left 30\n\nWord\u00b2 = Word `\u00d7 Word\nWord\u00b3 = Word `\u00d7 Word\u00b2\nWord\u2074 = Word `\u00d7 Word\u00b3\nWord\u2075 = Word `\u00d7 Word\u2074\n\niterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\niterate\u207f zero    f = id\niterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n_\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n_\u00b2\u2070 = iterate\u207f 20\n\nmodule _ (#\u02b7 : \u2115 \u2192 `\ud835\udfd9 `\u2192 Word) where\n\n  K\u2080 = #\u02b7 0x5A827999\n  K\u2082 = #\u02b7 0x6ED9EBA1\n  K\u2084 = #\u02b7 0x8F1BBCDC\n  K\u2086 = #\u02b7 0xCA62C1D6\n\n  H0 = #\u02b7 0x67452301\n  H1 = #\u02b7 0xEFCDAB89\n  H2 = #\u02b7 0x98BADCFE\n  H3 = #\u02b7 0x10325476\n  H4 = #\u02b7 0xC3D2E1F0\n\n  A B C D E : Word\u2075 `\u2192 Word\n  A = fst\n  B = snd \u204f fst\n  C = snd \u204f snd \u204f fst\n  D = snd \u204f snd \u204f snd \u204f fst\n  E = snd \u204f snd \u204f snd \u204f snd\n\n  F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n  F\u2082 = B `\u2295 C `\u2295 D\n  F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n  F\u2086 = F\u2082\n\n  module _ (F : Word\u2075 `\u2192 Word)\n           (K : `\ud835\udfd9  `\u2192 Word)\n           (W : `\ud835\udfd9  `\u2192 Word) where\n    Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n      where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n  module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n    W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n    W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n    W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n    W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n    W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n    Iteration\u2078\u2070 : `Endo Word\u2075\n    Iteration\u2078\u2070 =\n      (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n      (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n      (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n      (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n    -- WIP\n    SHA1 : `\ud835\udfd9 `\u2192 Word\u2075\n    SHA1 = < H0 , < H1 , < H2 , < H3 , H4 > > > > \u204f Iterations.Iteration\u2078\u2070\n\n-- -}\n","old_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\nopen FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\nopen FunOps funOps hiding (_\u2218_)\n\nWord : T\nWord = `Bits 32\n\nmap\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\nmap\u00b2\u02b7 = zipWith\n\nmap\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\nmap\u02b7 = map\n\nlift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\nlift f g = g \u204f f\n\nlift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\nlift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n`not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n`not = lift (map\u02b7 not)\n\ninfixr 3 _`\u2295_\n_`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\ninfixr 3 _`\u2227_\n_`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\ninfixr 2 _`\u2228_\n_`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\nopen import Solver.Linear\nopen import Data.Vec as Vec using ([]; _\u2237_)\n\nmodule STest n M = Syntax T _`\u00d7_ `\ud835\udfd9 _`\u2192_ id _\u204f_ fst <id,tt> snd <tt,id> <_\u00d7_> first second assoc\u2032 assoc swap n M\n\niter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\niter {zero}  F = <[]>\niter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n               \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n               \u204f <\u2237>\n\n  where\n    helper : \u2200 {A B D VA VB} \u2192 (D `\u00d7 (A `\u00d7 VA) `\u00d7 (B `\u00d7 VB)) `\u2192 (D `\u00d7 A `\u00d7 B) `\u00d7 (VA `\u00d7 VB)\n    helper {A} {B} {D} {VA} {VB} = solve (! 2 , (! 0 , ! 3) , (! 1 , ! 4))\n                                 ((! 2 , ! 0 , ! 1) , (! 3 , ! 4)) _ where\n      open STest 5 (\u03bb i \u2192 Vec.lookup i (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])) renaming (rewire to solve; #_ to !_)\n\n<\u229e> adder : Word `\u00d7 Word `\u2192 Word\nadder = <tt\u204f <0b> , id > \u204f iter full-adder\n<\u229e> = adder\n\ninfixl 4 _`\u229e_\n_`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n_`\u229e_ = lift\u2082 <\u229e>\n  \n<_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n< f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n<_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                              \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                              \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n< f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n<<<\u2085 : Word `\u2192 Word\n<<<\u2085 = rot-left 5\n\n<<<\u2083\u2080 : Word `\u2192 Word\n<<<\u2083\u2080 = rot-left 30\n\nWord\u00b2 = Word `\u00d7 Word\nWord\u00b3 = Word `\u00d7 Word\u00b2\nWord\u2074 = Word `\u00d7 Word\u00b3\nWord\u2075 = Word `\u00d7 Word\u2074\n\niterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 A `\u2192 A) \u2192 A `\u2192 A\niterate\u207f zero    f = id\niterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n_\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 A `\u2192 A) \u2192 (A `\u2192 A)\n_\u00b2\u2070 = iterate\u207f 20\n\nA-E : T\nA-E = Word\u2075\n\nmodule _ (#\u02b7 : \u2115 \u2192 `\ud835\udfd9 `\u2192 Word) where\n\n  module Iterations\n    (A B C D E : A-E `\u2192 Word)\n    where\n\n    module _ (F : A-E `\u2192 Word)\n             (K : `\ud835\udfd9  `\u2192 Word)\n             (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n         where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n    module _ \n        (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n\n        Iteration\u2078\u2070 =\n              (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n              (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n              (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n              (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n          where\n            F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n            F\u2082 = B `\u2295 C `\u2295 D\n            F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n            F\u2086 = F\u2082\n\n            K\u2080 = #\u02b7 0x5A827999\n            K\u2082 = #\u02b7 0x6ED9EBA1\n            K\u2084 = #\u02b7 0x8F1BBCDC\n            K\u2086 = #\u02b7 0xCA62C1D6\n\n            W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n            W\u2080 = W \u2218 inject+ 60\n            W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n            W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n            W\u2086 = W              \u2218 raise 60\n\n  SHA1 : (Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 A-E `\u2192 A-E\n  SHA1 = Iterations.Iteration\u2078\u2070 H0 H1 H2 H3 H4\n   where\n    H0 = tt \u204f #\u02b7 0x67452301\n    H1 = tt \u204f #\u02b7 0xEFCDAB89\n    H2 = tt \u204f #\u02b7 0x98BADCFE\n    H3 = tt \u204f #\u02b7 0x10325476\n    H4 = tt \u204f #\u02b7 0xC3D2E1F0\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1f9135b378a01c73d41d78b6f2f272bd1886f9d8","subject":"Rename to avoid name clashes in spire implementation.","message":"Rename to avoid name clashes in spire implementation.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (Ds : BranchesD I E) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) (\u03bc E Ds)\n\ninj : {I : Set} (E : Enum) (Ds : BranchesD I E) (t : Tag E) \u2192 CurriedEl (caseD Ds t) (\u03bc E Ds)\ninj E Ds t = curryEl (caseD Ds t) (\u03bc E Ds) (init t)\n\n----------------------------------------------------------------------\n\nprove : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nprove (End j) X P \u03b1 i q = tt\nprove (Rec j D) X P \u03b1 i (x , xs) = \u03b1 j x , prove D X P \u03b1 i xs\nprove (Arg A B) X P \u03b1 i (a , xs) = prove (B a) X P \u03b1 i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nind E Ds P \u03b1 i (init t xs) = \u03b1 t i xs $\n  prove (caseD Ds t) (\u03bc E Ds) P (ind E Ds P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nindCurried E Ds P f i x = ind E Ds P (\u03bb t \u2192 uncurryHyps (caseD Ds t) (\u03bc E Ds) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E Ds X cn P t = CurriedHyps (caseD Ds t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E Ds P t = Summer E Ds (\u03bc E Ds) init P t\n\nelimUncurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E Ds P)\n  \u2192 (i : I) (x : \u03bc E Ds i) \u2192 P i x\nelimUncurried E Ds P cs i x =\n  indCurried E Ds P\n    (case (SumCurriedHyps E Ds P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E Ds P)\n      ((i : I) (x : \u03bc E Ds i) \u2192 P i x)\nelim E Ds P = curryBranches (elimUncurried E Ds P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115Ds : BranchesD \u22a4 \u2115E\n\u2115Ds =\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T (caseD \u2115Ds)\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115Ds\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecDs : (A : Set) \u2192 BranchesD (\u2115 tt) VecE\nVecDs A =\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (caseD (VecDs A))\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecDs A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecDs A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecDs A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecDs A) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs A) t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecDs (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs (Vec A m)) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecDs A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecDs (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (Ds : BranchesD I E) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) (\u03bc E Ds)\n\ninj : {I : Set} (E : Enum) (Ds : BranchesD I E) (t : Tag E) \u2192 CurriedEl (caseD Ds t) (\u03bc E Ds)\ninj E Ds t = curryEl (caseD Ds t) (\u03bc E Ds) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i x,xs = \u03b1 j (proj\u2081 x,xs) , hyps D X P \u03b1 i (proj\u2082 x,xs)\nhyps (Arg A B) X P \u03b1 i a,xs = hyps (B (proj\u2081 a,xs)) X P \u03b1 i (proj\u2082 a,xs)\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nind E Ds P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (caseD Ds t) (\u03bc E Ds) P (ind E Ds P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nindCurried E Ds P f i x = ind E Ds P (\u03bb t \u2192 uncurryHyps (caseD Ds t) (\u03bc E Ds) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E Ds X cn P t = CurriedHyps (caseD Ds t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E Ds P t = Summer E Ds (\u03bc E Ds) init P t\n\nelimUncurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E Ds P)\n  \u2192 (i : I) (x : \u03bc E Ds i) \u2192 P i x\nelimUncurried E Ds P cs i x =\n  indCurried E Ds P\n    (case (SumCurriedHyps E Ds P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E Ds P)\n      ((i : I) (x : \u03bc E Ds i) \u2192 P i x)\nelim E Ds P = curryBranches (elimUncurried E Ds P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115Ds : BranchesD \u22a4 \u2115E\n\u2115Ds =\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T (caseD \u2115Ds)\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115Ds\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecDs : (A : Set) \u2192 BranchesD (\u2115 tt) VecE\nVecDs A =\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (caseD (VecDs A))\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecDs A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecDs A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecDs A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecDs A) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs A) t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecDs (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs (Vec A m)) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecDs A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecDs (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e46c6d08f63ee4d72e852c1b4faa9eb44996f465","subject":"Added x<y\u21920<x-y (by Ana Bove).","message":"Added x<y\u21920<x-y (by Ana Bove).\n\nIgnore-this: 7fcc6a14cfb022d4a8318e89adf5cd7e\n\ndarcs-hash:20110210165128-3bd4e-263f526351a75ce3371f498b3976f42c3447f026.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx<y\u21920<x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x-y Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u21920<x-y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x-y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x-y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx<y\u2192Sx-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx-y\u22610 Nm zN h = \u22a5-elim (\u00acx<0 Nm h)\nx<y\u2192Sx-y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx-y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx-y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx-y\u22610 #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"11413469714bba635b972f40607b808e31925650","subject":"Remove spurious imports","message":"Remove spurious imports\n\nOld-commit-hash: fc57d2d6ee3ff6f548c16e7073aeb6ae12646d85\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nuncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\nuncurriedConst constant = const constant\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\ncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurriedConst constant = curryTermConstructor (uncurriedConst constant)\n","old_contents":"import Syntax.Type.Plotkin as Type\nimport Syntax.Context as Context\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Context.Context {Type.Type B} \u2192 Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type B\nopen Context {Type}\n\nopen import Denotation.Environment Type\nopen import Syntax.Context.Plotkin B\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nuncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\nuncurriedConst constant = const constant\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\ncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurriedConst constant = curryTermConstructor (uncurriedConst constant)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"30936b7dbc0661507a17615dcc9b8b2115954abf","subject":"Prototype proof that static caching is type-correct","message":"Prototype proof that static caching is type-correct\n\nSee comments for extra info.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Evaluation.agda","new_file":"Parametric\/Denotation\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Unit\n\n  -- A semantics for empty caching semantics\n  \u27e6_\u27e7Type2 : (\u03c4 : Type) \u2192 Set\n  \u27e6 base \u03b9 \u27e7Type2 = \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type2 = \u27e6 \u03c3 \u27e7Type \u2192 (\u27e6 \u03c4 \u27e7Type \u00d7 \u22a4)\n\n  open import Level\n\n  {-\n  -- Hidden cache semantics, try 1.\n  --\n  -- Wrong: even the inputs should be extended.\n  -- It's just that extension on base values does nothing interesting.\n\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache =  \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n  -- This type would be cooler, but it also places \"excessive\" requirements on\n  -- the object language compared to what we formalize - e.g., unit types.\n  --\n  --\u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7Type \u2192 (\u03a3[ \u03c4\u2081 \u2208 Type ] \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u03c4\u2081 \u27e7Type )\n\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v x) , tt)\n\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v x)))\n\n  -- Wrong type signature:\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = {!!}\n  -- Oh. We don't even know what to return without matching on the type. That's\n  -- also true in our term transformation: we need to eta-expand also\n  -- non-primitives (higher-order arguments) for everything to possibly work.\n  -- Hm... or we need to have extended values in the environment - where the extension is again just for functions.\n  \u27e6_\u27e7TermCache (var x) \u03c1 = extend (\u27e6 var x \u27e7 \u03c1)\n  -- XXX: this delegates to standard evaluation, I'm not sure whether we should do that.\n  -- In fact, we should use our evaluation and select from the result.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7Term \u03c1)))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n\n  -- Wrong type definitions: functions get argument of transformed types.\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = {!!}\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (dropCache (\u27e6 t \u27e7TermCache2 \u03c1))))\n  -- Provide an empty cache :-)!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 ((extend x) \u2022 \u03c1) , tt)\n  -}\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n\n  -- The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 \u22a4 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = \u22a4 , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n  \n  meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n  meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1832c6af3f90a6b9fc8c06181cccfcb95932c5fc","subject":"Fix Preorder-Reasoning and Setoid-Reasoning","message":"Fix Preorder-Reasoning and Setoid-Reasoning\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/NP.agda","new_file":"lib\/Relation\/Binary\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Trans-Reasoning _\u223c_ trans public hiding (finally) renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_)\n  open Equivalence-Reasoning isEquivalence public renaming (_\u220e to _\u2610)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _ \u220e = refl\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Equivalence-Reasoning (Setoid.isEquivalence s) public\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Relation.Binary.NP where\n\nopen import Level\nopen import Relation.Binary public\nimport Relation.Binary.PropositionalEquality as PropEq\n\n-- could this be moved in place of Trans is Relation.Binary.Core\nTrans' : \u2200 {a b c \u2113\u2081 \u2113\u2082 \u2113\u2083} {A : Set a} {B : Set b} {C : Set c} \u2192\n        REL A B \u2113\u2081 \u2192 REL B C \u2113\u2082 \u2192 REL A C \u2113\u2083 \u2192 Set _\nTrans' P Q R = \u2200 {i j k} (x : P i j) (xs : Q j k) \u2192 R i k\n\nsubstitutive-to-reflexive : \u2200 {a \u2113 \u2113'} {A : Set a} {\u2248 : Rel A \u2113} {\u2261 : Rel A \u2113'}\n                            \u2192 Substitutive \u2261 \u2113 \u2192 Reflexive \u2248 \u2192 \u2261 \u21d2 \u2248\nsubstitutive-to-reflexive {\u2248 = \u2248} \u2261-subst \u2248-refl x\u1d63 = \u2261-subst (\u2248 _) x\u1d63 \u2248-refl\n\nsubstitutive\u21d2\u2261 : \u2200 {a \u2113} {A : Set a} {\u2261 : Rel A \u2113} \u2192 Substitutive \u2261 a \u2192 \u2261 \u21d2 PropEq._\u2261_\nsubstitutive\u21d2\u2261 subst = substitutive-to-reflexive {\u2248 = PropEq._\u2261_} subst PropEq.refl\n\nrecord Equality {a \u2113} {A : Set a} (_\u2261_ : Rel A \u2113) : Set (suc a \u2294 \u2113) where\n  field\n    isEquivalence : IsEquivalence _\u2261_\n    subst : Substitutive _\u2261_ a\n\n  open IsEquivalence isEquivalence public\n\n  to-reflexive : \u2200 {\u2248} \u2192 Reflexive \u2248 \u2192 _\u2261_ \u21d2 \u2248\n  to-reflexive = substitutive-to-reflexive subst\n\n  to-propositional : _\u2261_ \u21d2 PropEq._\u2261_\n  to-propositional = substitutive\u21d2\u2261 subst\n\n-- Equational reasoning combinators.\n\nmodule Trans-Reasoning {a \u2113} {A : Set a} (_\u2248_ : Rel A \u2113) (trans : Transitive _\u2248_) where\n\n  infix  2 finally\n  infixr 2 _\u2248\u27e8_\u27e9_\n\n  _\u2248\u27e8_\u27e9_ : \u2200 x {y z : A} \u2192 x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z\n  _ \u2248\u27e8 x\u2248y \u27e9 y\u2248z = trans x\u2248y y\u2248z\n\n  -- When there is no reflexivty available this\n  -- combinator can be used to end the reasoning.\n  finally : \u2200 (x y : A) \u2192 x \u2248 y \u2192 x \u2248 y\n  finally _ _ x\u2248y = x\u2248y\n\n  syntax finally x y x\u2248y = x \u2248\u27e8 x\u2248y \u27e9\u220e y \u220e\n\nmodule Equivalence-Reasoning\n         {a \u2113} {A : Set a} {_\u2248_ : Rel A \u2113}\n         (E : IsEquivalence _\u2248_) where\n  open IsEquivalence E\n  open Trans-Reasoning _\u2248_ trans public hiding (finally)\n\n  infix  2 _\u220e\n\n  _\u220e : \u2200 x \u2192 x \u2248 x\n  _ \u220e = refl\n\nmodule Preorder-Reasoning\n         {p\u2081 p\u2082 p\u2083} (P : Preorder p\u2081 p\u2082 p\u2083) where\n  open Preorder P\n  open Equivalence-Reasoning isEquivalence public renaming (_\u2248\u27e8_\u27e9_ to _\u223c\u27e8_\u27e9_)\n\nmodule Setoid-Reasoning {a \u2113} (s : Setoid a \u2113) where\n  open Preorder-Reasoning (Setoid.preorder s) public renaming (_\u223c\u27e8_\u27e9_ to _\u2248\u27e8_\u27e9_)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ca844f383fa743a9d42a6d2c2f33719838947bcf","subject":"rephrasing lemma to match paper text per  #43","message":"rephrasing lemma to match paper text per  #43\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-progress.agda","new_file":"complete-progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import progress\nopen import htype-decidable\n\nmodule complete-progress where\n\n  data okc : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 okc d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 okc d \u0394\n\n  -- no term is both complete and indeterminate\n  lem-ind-comp : \u2200{d} \u2192 d dcomplete \u2192 d indet \u2192 \u22a5\n  lem-ind-comp DCVar ()\n  lem-ind-comp DCConst ()\n  lem-ind-comp (DCLam comp x\u2081) ()\n  lem-ind-comp (DCAp comp comp\u2081) (IAp x ind x\u2081) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastArr x\u2082 ind) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastGroundHole x\u2082 ind) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastHoleGround x\u2082 ind x\u2083) = lem-ind-comp comp ind\n\n  -- if an arrow is disequal, it disagrees in the first or second argument\n  ne-factor : \u2200{\u03c41 \u03c42 \u03c43 \u03c44} \u2192 (\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44) \u2192 (\u03c41 \u2260 \u03c43) + (\u03c42 \u2260 \u03c44)\n  ne-factor {\u03c41} {\u03c42} {\u03c43} {\u03c44} ne with htype-dec \u03c41 \u03c43 | htype-dec \u03c42 \u03c44\n  ne-factor ne | Inl refl | Inl refl = Inl (\u03bb x \u2192 ne refl)\n  ne-factor ne | Inl x | Inr x\u2081 = Inr x\u2081\n  ne-factor ne | Inr x | Inl x\u2081 = Inl x\n  ne-factor ne | Inr x | Inr x\u2081 = Inl x\n\n  -- complete types that are consistent are equal\n  eq-complete-consist : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 \u03c41 ~ \u03c42 \u2192 \u03c41 == \u03c42\n  eq-complete-consist TCBase TCBase consis = refl\n  eq-complete-consist TCBase (TCArr tc2 tc3) ()\n  eq-complete-consist (TCArr tc1 tc2) TCBase ()\n  eq-complete-consist (TCArr tc1 tc2) (TCArr tc3 tc4) TCRefl = refl\n  eq-complete-consist (TCArr tc1 tc2) (TCArr tc3 tc4) (TCArr consis1 consis2)\n    with eq-complete-consist tc1 tc3 consis1\n  ... | refl with eq-complete-consist tc2 tc4 consis2\n  ... | refl = refl\n\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n                       \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                       d dcomplete \u2192\n                       okc d \u0394\n  complete-progress wt comp with progress wt\n  complete-progress wt comp | I x = abort (lem-ind-comp comp x)\n  complete-progress wt comp | S x = S x\n  complete-progress wt comp | BV (BVVal x) = V x\n  complete-progress wt (DCCast comp x\u2082 ()) | BV (BVHoleCast x x\u2081)\n  complete-progress (TACast wt x) (DCCast comp x\u2083 x\u2084) | BV (BVArrCast x\u2081 x\u2082) = abort (x\u2081 (eq-complete-consist x\u2083 x\u2084 x))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import progress\nopen import htype-decidable\n\nmodule complete-progress where\n\n  data okc : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 okc d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 okc d \u0394\n\n  -- no term is both complete and indeterminate\n  lem-ind-comp : \u2200{d} \u2192 d dcomplete \u2192 d indet \u2192 \u22a5\n  lem-ind-comp DCVar ()\n  lem-ind-comp DCConst ()\n  lem-ind-comp (DCLam comp x\u2081) ()\n  lem-ind-comp (DCAp comp comp\u2081) (IAp x ind x\u2081) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastArr x\u2082 ind) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastGroundHole x\u2082 ind) = lem-ind-comp comp ind\n  lem-ind-comp (DCCast comp x x\u2081) (ICastHoleGround x\u2082 ind x\u2083) = lem-ind-comp comp ind\n\n  -- if an arrow is disequal, it disagrees in the first or second argument\n  ne-factor : \u2200{\u03c41 \u03c42 \u03c43 \u03c44} \u2192 (\u03c41 ==> \u03c42) \u2260 (\u03c43 ==> \u03c44) \u2192 (\u03c41 \u2260 \u03c43) + (\u03c42 \u2260 \u03c44)\n  ne-factor {\u03c41} {\u03c42} {\u03c43} {\u03c44} ne with htype-dec \u03c41 \u03c43 | htype-dec \u03c42 \u03c44\n  ne-factor ne | Inl refl | Inl refl = Inl (\u03bb x \u2192 ne refl)\n  ne-factor ne | Inl x | Inr x\u2081 = Inr x\u2081\n  ne-factor ne | Inr x | Inl x\u2081 = Inl x\n  ne-factor ne | Inr x | Inr x\u2081 = Inl x\n\n  -- disequal complete types are not consistent\n  eq-com-con : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 \u03c41 \u2260 \u03c42 \u2192 \u03c41 ~ \u03c42 \u2192 \u22a5\n  eq-com-con tc1 tc2 ne TCRefl = ne refl\n  eq-com-con tc1 () ne TCHole1\n  eq-com-con () tc2 ne TCHole2\n  eq-com-con (TCArr tc1 tc2) (TCArr tc3 tc4) ne (TCArr con con\u2081) with ne-factor ne\n  eq-com-con (TCArr tc1 tc2) (TCArr tc3 tc4) ne (TCArr con con\u2081) | Inl x = eq-com-con tc1 tc3 x con\n  eq-com-con (TCArr tc1 tc2) (TCArr tc3 tc4) ne (TCArr con con\u2081) | Inr x = eq-com-con tc2 tc4 x con\u2081\n\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n                       \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                       d dcomplete \u2192\n                       okc d \u0394\n  complete-progress wt comp with progress wt\n  complete-progress wt comp | I x = abort (lem-ind-comp comp x)\n  complete-progress wt comp | S x = S x\n  complete-progress wt comp | BV (BVVal x) = V x\n  complete-progress wt (DCCast comp x\u2082 ()) | BV (BVHoleCast x x\u2081)\n  complete-progress (TACast wt x) (DCCast comp x\u2083 x\u2084) | BV (BVArrCast x\u2081 x\u2082) = abort (eq-com-con x\u2083 x\u2084 x\u2081 x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c3166171f7003e914434743ea4c5bedaa0d82c67","subject":"Working dependent evaluator, but need to remove explicit weakening.","message":"Working dependent evaluator, but need to remove explicit weakening.\n","repos":"spire\/spire","old_file":"src\/Spire\/DependentEvaluator.agda","new_file":"src\/Spire\/DependentEvaluator.agda","new_contents":"module Spire.DependentEvaluator where\n\n----------------------------------------------------------------------\n\ndata Con : Set\ndata Type : Con \u2192 Set\ndata NType : Con \u2192 Set\ndata Val : (\u0393 : Con) \u2192 Type \u0393 \u2192 Set\ndata NVal : (\u0393 : Con) \u2192 Type \u0393 \u2192 Set\ndata Var : (\u0393 : Con) \u2192 Type \u0393 \u2192 Set\nstrC : \u2200{\u0393 A} \u2192 Var \u0393 A \u2192 Con\nstrT : \u2200{\u0393 A} (i : Var \u0393 A) \u2192 Type (strC i)\nSub : \u2200{\u0393 A} \u2192 Var \u0393 A \u2192 Set\nsubC : \u2200{\u0393 A} (i : Var \u0393 A) \u2192 Sub i \u2192 Con\nsubT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Type (subC i x)\nsubNT : \u2200{\u0393 A} \u2192 NType \u0393 \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Type (subC i x)\nsubV : \u2200{\u0393 A B} \u2192 Val \u0393 B \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Val (subC i x) (subT B i x)\nsubNV : \u2200{\u0393 A B} \u2192 NVal \u0393 B \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Val (subC i x) (subT B i x)\n\n----------------------------------------------------------------------\n\ninfixl 3 _,_\ndata Con where\n  \u2205 : Con\n  _,_ : (\u0393 : Con) (A : Type \u0393) \u2192 Con\n\ndata Type where\n  `wkn : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  `\u22a4 `Bool : \u2200{\u0393} \u2192 Type \u0393\n  `neutral : \u2200{\u0393} \u2192 NType \u0393 \u2192 Type \u0393\n\ndata NType where\n  `if_then_else_ : \u2200{\u0393} \u2192 NVal \u0393 `Bool\n    \u2192 Type \u0393 \u2192 Type \u0393 \u2192 NType \u0393\n\n----------------------------------------------------------------------\n\ndata Val where\n  `wkn : \u2200{\u0393 A B} \u2192 Val \u0393 A \u2192 Val (\u0393 , B) (`wkn A)\n  `tt : \u2200{\u0393} \u2192 Val \u0393 `\u22a4\n  `true `false : \u2200{\u0393} \u2192 Val \u0393 `Bool\n  `neutral : \u2200{\u0393 A} \u2192 NVal \u0393 A \u2192 Val \u0393 A\n\ndata NVal where\n  `var : \u2200{\u0393 A} \u2192 Var \u0393 A \u2192 NVal \u0393 A\n  `not : \u2200{\u0393} \u2192 NVal \u0393 `Bool \u2192 NVal \u0393 `Bool\n\ndata Var where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (`wkn A)\n  there : \u2200{\u0393 A B} (i : Var \u0393 A) \u2192 Var (\u0393 , B) (`wkn A)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393} \u2192 Val \u0393 `Bool \u2192 Type \u0393 \u2192 Type \u0393 \u2192 Type \u0393\nif `true then A else B = A\nif `false then A else B = B\nif `neutral b then A else B = `neutral (`if b then A else B)\n\nnot : \u2200{\u0393} \u2192 Val \u0393 `Bool \u2192 Val \u0393 `Bool\nnot `true = `false\nnot `false = `true\nnot (`neutral b) = `neutral (`not b)\n\n----------------------------------------------------------------------\n\nstrC (here {\u0393}) = \u0393\nstrC (there i) = strC i\nstrT (here {\u0393} {A}) = A\nstrT (there i) = strT i\nSub i = Val (strC i) (strT i)\nsubC (here {\u0393}) x = \u0393\nsubC (there {\u0393} {A} {B} i) x = subC i x , subT B i x\nsubT (`wkn A) here x = A\nsubT (`wkn A) (there i) x = `wkn (subT A i x)\nsubT `\u22a4 i x = `\u22a4\nsubT `Bool i x = `Bool\nsubT (`neutral n) i x = subNT n i x\nsubNT (`if b then A else B) i x = if (subNV b i x) then subT A i x else subT B i x\nsubV (`wkn a) here x = a\nsubV (`wkn a) (there i) x = `wkn (subV a i x)\nsubV `tt i x = `tt\nsubV `true i x = `true\nsubV `false i x = `false\nsubV (`neutral n) i x = subNV n i x\nsubNV (`var here) here x = x\nsubNV (`var here) (there i) x = `neutral (`var here)\nsubNV (`var (there j)) here x = `neutral (`var j)\nsubNV (`var (there j)) (there i) x = `wkn (subNV (`var j) i x)\nsubNV (`not b) i x = not (subNV b i x)\n\n----------------------------------------------------------------------\n\ndata Term : (\u0393 : Con) \u2192 Type \u0393 \u2192 Set where\n  `var : \u2200{\u0393 A} (i : Var \u0393 A) \u2192 Term \u0393 A\n  `tt : \u2200{\u0393} \u2192 Term \u0393 `\u22a4\n  `true `false : \u2200{\u0393} \u2192 Term \u0393 `Bool\n  `not : \u2200{\u0393} (b : Term \u0393 `Bool) \u2192 Term \u0393 `Bool\n\ndata TermType : (\u0393 : Con) \u2192 Set where\n  `\u22a4 `Bool : \u2200{\u0393} \u2192 TermType \u0393\n  `if_then_else_ : \u2200{\u0393}\n    \u2192 Term \u0393 `Bool\n    \u2192 TermType \u0393 \u2192 TermType \u0393\n    \u2192 TermType \u0393\n\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Val \u0393 A\neval (`var i) = `neutral (`var i)\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (`not b) = not (eval b)\n\nevalType : \u2200{\u0393} \u2192 TermType \u0393 \u2192 Type \u0393\nevalType `\u22a4 = `\u22a4\nevalType `Bool = `Bool\nevalType (`if b then A else B) = if eval b then evalType A else evalType B\n\n----------------------------------------------------------------------\n","old_contents":"module Spire.DependentEvaluator where\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type : Context \u2192 Set\ndata NType : Context \u2192 Set\ndata Val : (\u0393 : Context) \u2192 Type \u0393 \u2192 Set\ndata NVal : (\u0393 : Context) \u2192 Type \u0393 \u2192 Set\ndata Var : (\u0393 : Context) \u2192 Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ninfixl 3 _,_\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) (A : Type \u0393) \u2192 Context\n\n----------------------------------------------------------------------\n\ndata Type where\n  `wkn : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  `\u22a4 `Bool : \u2200{\u0393} \u2192 Type \u0393\n  `neutral : \u2200{\u0393} \u2192 NType \u0393 \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\ndata NType where\n  `if_then_else_ : \u2200{\u0393} \u2192 NVal \u0393 `Bool\n    \u2192 Type \u0393 \u2192 Type \u0393 \u2192 NType \u0393\n\n----------------------------------------------------------------------\n\ndata Val where\n  `tt : \u2200{\u0393} \u2192 Val \u0393 `\u22a4\n  `true `false : \u2200{\u0393} \u2192 Val \u0393 `Bool\n  `neutral : \u2200{\u0393 A} \u2192 NVal \u0393 A \u2192 Val \u0393 A\n\n----------------------------------------------------------------------\n\ndata NVal where\n  `var : \u2200{\u0393 A} \u2192 Var \u0393 A \u2192 NVal \u0393 A\n  `not : \u2200{\u0393} \u2192 NVal \u0393 `Bool \u2192 NVal \u0393 `Bool\n\n----------------------------------------------------------------------\n\ndata Var where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (`wkn A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (`wkn A)\n\n----------------------------------------------------------------------\n\nstrC : \u2200{\u0393 A} \u2192 Var \u0393 A \u2192 Context\nstrC (here {\u0393}) = \u0393\nstrC (there i) = strC i\n\nstrT : \u2200{\u0393 A} (i : Var \u0393 A) \u2192 Type (strC i)\nstrT (here {\u0393} {A}) = A\nstrT (there i) = strT i\n\nSub : \u2200{\u0393 A} \u2192 Var \u0393 A \u2192 Set\nSub i = Val (strC i) (strT i)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393} \u2192 Val \u0393 `Bool \u2192 Type \u0393 \u2192 Type \u0393 \u2192 Type \u0393\nif `true then A else B = A\nif `false then A else B = B\nif `neutral b then A else B = `neutral (`if b then A else B)\n\nnot : \u2200{\u0393} \u2192 Val \u0393 `Bool \u2192 Val \u0393 `Bool\nnot `true = `false\nnot `false = `true\nnot (`neutral b) = `neutral (`not b)\n\nsubC : \u2200{\u0393 A} (i : Var \u0393 A) \u2192 Sub i \u2192 Context\nsubT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Type (subC i x)\nsubNT : \u2200{\u0393 A} \u2192 NType \u0393 \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Type (subC i x)\nsubV : \u2200{\u0393 A B} \u2192 Val \u0393 B \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Val (subC i x) (subT B i x)\nsubNV : \u2200{\u0393 A B} \u2192 NVal \u0393 B \u2192 (i : Var \u0393 A) (x : Sub i) \u2192 Val (subC i x) (subT B i x)\n\nsubC (here {\u0393}) x = \u0393\nsubC (there {\u0393} {A} {B} i) x = subC i x , subT B i x\n\nsubT (`wkn B) i x = {!!}\nsubT `\u22a4 i x = `\u22a4\nsubT `Bool i x = `Bool\nsubT (`neutral n) i x = subNT n i x\n\npostulate\n  wknVal : \u2200{\u0393 A} (i : Var \u0393 A) (x : Sub i) \u2192 Val (subC i x) (subT A i x)\n\nwknVar : \u2200{\u0393 A B} (i : Var \u0393 A) (x : Sub i) \u2192 Var (subC i x) (subT B i x) \u2192 Var \u0393 B\nwknVar i x j = {!!}\n\ndata Compare {\u0393} {A : Type \u0393} (i : Var \u0393 A) : {B : Type \u0393} \u2192 Var \u0393 B \u2192 Sub i \u2192 Set where\n  equal : (x : Val (strC i) (strT i)) \u2192 Compare i i x\n  diff : \u2200{B}\n    (x : Val (strC i) (strT i))\n    (j : Var (subC i x) (subT B i x))\n    \u2192 Compare i {B = B} (wknVar i x j) x\n\npostulate\n  compare : \u2200{\u0393 A B} (i : Var \u0393 A) (j : Var \u0393 B) (x : Sub i) \u2192 Compare i j x\n\nsubNT (`if b then A else B) i x = if (subNV b i x) then subT A i x else subT B i x\n\nsubV `tt i x = `tt\nsubV `true i x = `true\nsubV `false i x = `false\nsubV (`neutral n) i x = subNV n i x\n\nsubNV (`var j) i x = {!!}\n-- with compare i j x\n-- subNV (`var .i) i .x | equal x = wknVal i x\n-- subNV (`var .(wknVar i x j)) i .x | diff x j = `neutral (`var j)\nsubNV (`not b) i x = not (subNV b i x)\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e37cbfe41044e98fc5d27a557d83f57dc106858d","subject":"Helios: setup the intermediate games","message":"Helios: setup the intermediate games\n","repos":"crypto-agda\/crypto-agda","old_file":"Helios.agda","new_file":"Helios.agda","new_contents":"open import Type\nopen import Function\nopen import Data.List\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\n{-\nopen import Data.Zero\nopen import Data.One\n-}\nopen import Data.Two\nopen import Data.Nat.NP hiding (_==_; _<_)\nopen import Control.Strategy\n\nmodule Helios\n  (VoterId : \u2605)\n  (PublicKey : \u2605)\n  (SecretKey : \u2605)\n  (NIZKSecretKey : PublicKey \u2192 \u2605)\n  (Vote   : \u2605)\n  (Ballot : \u2605)\n  (Decryption-share : \u2605)\n  (PublicInfo : \u2605)\n  (R-setup : \u2605)\n  (R-vote : \u2605)\n  (vote : PublicKey \u2192 R-vote \u2192 VoterId \u2192 Vote \u2192 Ballot)\n  (let Ballots = List (VoterId \u00d7 Ballot)\n       Decryption-shares = List Decryption-share\n  )\n  (validate-ballots : PublicKey \u2192 Ballots \u2192 \ud835\udfda)\n  (Tally : \u2605) -- could be a valid tally or a special symbol \u22a5\n  (fake-decrytion-share : Tally \u2192 Ballots \u2192 Decryption-share)\n  (R-adversary : \u2605)\n  where\n\nrecord KeyPair : \u2605 where\n  constructor mk\n  field\n    pk : PublicKey\n    sk : SecretKey\n    zk : NIZKSecretKey pk\n\n\nrecord BB : \u2605 where\n  constructor <_,,_,,_,,_>\n  field\n    Y-pk              : PublicKey\n    \u03c0-pk              : NIZKSecretKey Y-pk\n    ballots           : Ballots\n    decryption-shares : Decryption-shares\nopen BB public\n\nMake-decryption-shares = KeyPair \u2192 Ballots \u2192 Decryption-shares\nSimulate-decryption-shares = (bs\u2080 : Ballots) \u2192 Make-decryption-shares\n\nvalidate : PublicKey \u2192 Ballots \u2192 VoterId \u00d7 Ballot \u2192 \ud835\udfda\nvalidate pk bs vb = validate-ballots pk (vb \u2237 bs)\n\ndata Q : \u2605 where\n  ask-vote   : (v_ : Vote \u00b2) \u2192 Q\n  ask-ballot : (b  : Ballot) \u2192 Q\n\nVotingPhase = List (Ballots \u2192 VoterId \u00d7 Q)\n\nmodule Tallying\n  (decryption-shares : Make-decryption-shares)\n  (result : BB \u2192 Tally)\n  where\n\n  tally-bb : KeyPair \u2192 Ballots \u2192 BB\n  tally-bb (mk Y sk zk) bs = < Y ,, zk ,, bs ,, decryption-shares (mk Y sk zk) bs >\n\n  tally : KeyPair \u2192 Ballots \u2192 Tally\n  tally kp bs = result (tally-bb kp bs)\n\nmodule Run-common\n           (pk : PublicKey)\n           (r-votes : VoterId \u2192 R-vote \u00b2) where\n    run-vote-query\u00b9 : R-vote \u2192 VoterId \u2192 Vote \u2192 Ballots \u2192 Ballots\n    run-vote-query\u00b9 r-vote vid v bs = case validate pk bs (vid , b)\n                                        0: bs\n                                        1: ((vid , b) \u2237 bs)\n      where\n        b : Ballot\n        b = vote pk r-vote vid v\n\n    run-vote-query : (hack-run-vote-query : \ud835\udfda \u2192 \ud835\udfda) \u2192 VoterId \u2192 Vote \u00b2 \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-vote-query hack-run-vote-query vid v_ bs_ i = run-vote-query\u00b9 (r-votes vid i) vid (v_ (hack-run-vote-query i)) (bs_ i)\n\n    run-ballot-query : (\u03b2 : \ud835\udfda) \u2192 VoterId \u2192 Ballot \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-ballot-query \u03b2 vid b bs_ i =\n        case validate pk (bs_ \u03b2) (vid , b)\n          0: bs_ i\n          1: case validate pk (bs_ (not \u03b2)) (vid , b)\n               0: case i\n                    0: ((vid , b) \u2237 bs_ 0\u2082)\n                    1: bs_ 1\u2082\n               1: ((vid , b) \u2237 bs_ i)\n\nHack-run-vote-query = \u2115 \u2192 \ud835\udfda \u2192 \ud835\udfda\n\nmodule Run (\u03b2 : \ud835\udfda)\n           (hack-run-vote-query : Hack-run-vote-query)\n           (pk : PublicKey)\n           (r-votes : VoterId \u2192 R-vote \u00b2) where\n    open Run-common pk r-votes\n\n    run-query : (hack-run-vote-query : \ud835\udfda \u2192 \ud835\udfda) \u2192 VoterId \u2192 Q \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-query hack-run-vote-query vid (ask-vote v_)  = run-vote-query hack-run-vote-query vid v_\n    run-query hack-run-vote-query vid (ask-ballot b) = run-ballot-query \u03b2 vid b\n\n    run-voting-phase : VotingPhase \u2192 Ballots \u00b2 \u2192 Ballots \u00b2\n    run-voting-phase []       bs_ = bs_\n    run-voting-phase (q \u2237 qs) bs_ = run-voting-phase qs (uncurry (run-query (hack-run-vote-query (length qs))) (q (bs_ \u03b2)) bs_)\n\nmodule _ where\n  Adversary = R-adversary \u2192\n              PublicKey \u2192\n              VotingPhase \u00d7\n              (BB \u2192\n              \ud835\udfda)\n\n  module CommonChallenger\n              (setup : R-setup \u2192 KeyPair)\n              (r-setup : R-setup)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              where\n    kp = setup r-setup\n    open KeyPair kp public\n\n    A-setup = A r-adversary pk\n\n    A-voting-phase : VotingPhase\n    A-voting-phase = fst A-setup\n\n  -- The real challenger (has access to \u03b2)\n  module Game (setup : R-setup \u2192 KeyPair)\n              (decryption-shares : Make-decryption-shares)\n              (hack-run-vote-query : Hack-run-vote-query)\n\n              (r-setup : R-setup)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (simulate-decryption-shares : Simulate-decryption-shares)\n              (\u03b2 : \ud835\udfda)\n              where\n    open CommonChallenger setup r-setup r-adversary A public\n\n    open Run \u03b2 hack-run-vote-query pk r-votes\n\n    bs_ : Ballots \u00b2\n    bs_ = run-voting-phase A-voting-phase (const [])\n\n    decryption-shares-\u03b2 =\n      (case \u03b2\n         0: decryption-shares\n         1: simulate-decryption-shares (bs_ 0\u2082))\n      kp (bs_ \u03b2)\n\n    bb : BB\n    bb = record { Y-pk = pk ; \u03c0-pk = zk ; ballots = bs_ \u03b2 ; decryption-shares = decryption-shares-\u03b2 }\n\n    exp : \ud835\udfda\n    exp = snd A-setup bb\n\n    game = exp == \u03b2\n\n  module Unused-simulate-decryption-shares\n              (decryption-shares : Make-decryption-shares)\n              (result : BB \u2192 Tally)\n              (kp  : KeyPair)\n              (bs_ : Ballots \u00b2)\n           where\n    open KeyPair kp\n    open Tallying decryption-shares result\n    simulated-decryption-shares : Decryption-shares\n    simulated-decryption-shares = fake-decrytion-share (tally kp (bs_ 0\u2082)) (bs_ 1\u2082) \u2237 []\n\n    {-\n  module Simulated-Challenger\n              (r-setup : R-setup)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n            where\n    open CommonChallenger r-setup r-adversary A public\n-}\n  module Proof (setup : R-setup \u2192 KeyPair)\n               (result : BB \u2192 Tally)\n               (decryption-shares  decryption-shares\u2090 : Make-decryption-shares)\n               (r-setup : R-setup)\n               (r-adversary : R-adversary)\n               (A : Adversary)\n               (r-votes : VoterId \u2192 R-vote \u00b2)\n               (simulate-decryption-shares : Simulate-decryption-shares)\n               (simulate-zksecret : (pk : PublicKey) \u2192 NIZKSecretKey pk)\n               where\n     hack-no-queries : Hack-run-vote-query\n     hack-no-queries = const id\n     module EXP = Game setup decryption-shares hack-no-queries r-setup r-adversary A r-votes simulate-decryption-shares\n     G\u2080 = EXP.exp 0\u2082\n     G\u2081 = EXP.exp 1\u2082\n\n     setup\u2090 : R-setup \u2192 KeyPair\n     setup\u2090 r-setup = record (setup r-setup) { zk = simulate-zksecret _ }\n\n     module EXP\u2090 = Game setup\u2090 decryption-shares\u2090 hack-no-queries r-setup r-adversary A r-votes simulate-decryption-shares 0\u2082\n\n     G\u2090 = EXP\u2090.exp\n\n     simulate-decryption-shares-B : Make-decryption-shares\n     simulate-decryption-shares-B kp bs = fake-decrytion-share t bs \u2237 []\n       where\n         open Tallying decryption-shares\u2090 result\n         t  = tally kp bs\n\n     module EXP-B = Game setup\u2090 simulate-decryption-shares-B hack-no-queries r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     G-B = EXP-B.exp\n\n     hack-all-queries : Hack-run-vote-query\n     hack-all-queries = const not\n\n     Seq : \u2605 \u2192 \u2605\n     Seq A = \u2115 \u2192 A\n\n     _<_ : \u2115 \u2192 \u2115 \u2192 \ud835\udfda\n     x < y = suc x <= y\n\n     replicateSeq : {A : \u2605} \u2192 \u2115 \u2192 A \u2192 A \u2192 Seq A\n     replicateSeq n x y i = case (i < n) 0: x 1: y\n\n     hack-queries-up-to : \u2115 \u2192 Hack-run-vote-query\n     hack-queries-up-to n i = replicateSeq (n id not i\n\n     module EXP-C-up-to (n : \u2115) = Game setup\u2090 simulate-decryption-shares-B (hack-queries-up-to n) r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     module EXP-C = Game setup\u2090 simulate-decryption-shares-B hack-all-queries r-setup r-adversary A r-votes (\u03bb _ _ _ \u2192 {- unused -} []) 0\u2082\n\n     G-C = EXP-C.exp\n\n     {- EXP-C-up-to zero -}\n\n     -- G\u2080\u223cG\u2090\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Data.List\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\n{-\nopen import Data.Zero\nopen import Data.One\nopen import Data.Nat\n-}\nopen import Data.Two\nopen import Control.Strategy\n\nmodule Helios\n  (VoterId : \u2605)\n  (PublicKey : \u2605)\n  (SecretKey : \u2605)\n  (Vote   : \u2605)\n  (Ballot : \u2605)\n  (Decryption-share : \u2605)\n  (PublicInfo : \u2605)\n  where\n\nBallots = List (VoterId \u00d7 Ballot)\nDecryption-shares = List Decryption-share\n\nrecord BB : \u2605 where\n  constructor <_,,_,,_>\n  field\n    Y-pk              : PublicKey\n    ballots           : Ballots\n    decryption-shares : Decryption-shares\nopen BB public\n\nmodule Nested\n  (R-setup : \u2605)\n  (setup : R-setup \u2192 PublicKey \u00d7 {-List-} SecretKey)\n  (R-vote : \u2605)\n\n  (vote : PublicKey \u2192 R-vote \u2192 VoterId \u2192 Vote \u2192 Ballot)\n  (validate : BB \u2192 VoterId \u00d7 Ballot \u2192 \ud835\udfda)\n\n  (Tally : \u2605) -- could be a valid tally or a special symbol \u22a5\n  (dec-shares : PublicKey \u2192 Ballots \u2192 {-List-} SecretKey \u2192 Decryption-shares)\n  (result : BB \u2192 Tally)\n\n  (R-adversary : \u2605)\n  where\n\n  tally : BB \u2192 {-List-} SecretKey \u2192 BB\n  tally < Y ,, bs ,, ds > sks = < Y ,, bs ,, dec-shares Y bs sks >\n\n  data Q : \u2605 where\n    ask-vote   : (v_ : Vote \u00b2) \u2192 Q\n    ask-ballot : (b  : Ballot) \u2192 Q\n\n  VotingPhase = List (BB \u2192 VoterId \u00d7 Q)\n\n  add-ballot : VoterId \u2192 Ballot \u2192 BB \u2192 BB\n  add-ballot vid b bb = record bb { ballots = (vid , b) \u2237 ballots bb }\n\n  module Run (\u03b2 : \ud835\udfda)\n             (pk : PublicKey)\n             (r-votes : VoterId \u2192 R-vote \u00b2)\n           where\n\n    run-query : VoterId \u2192 Q \u2192 BB \u00b2 \u2192 BB \u00b2\n    run-query vid (ask-vote v_)  bb_ i = case validate (bb_ i) (vid , b)\n                                          0: bb_ i\n                                          1: add-ballot vid b (bb_ i)\n      where\n        b : Ballot\n        b = vote pk (r-votes vid i) vid (v i)\n\n    run-query vid (ask-ballot b) bb_ i =\n        case validate (bb_ \u03b2) (vid , b)\n          0: bb_ i\n          1: case validate (bb_ (not \u03b2)) (vid , b)\n               0: case i\n                    0: add-ballot vid b (bb_ 0\u2082)\n                    1: bb_ 1\u2082\n               1: add-ballot vid b (bb_ i)\n\n    run-voting-phase : VotingPhase \u2192 BB \u00b2 \u2192 BB \u00b2\n    run-voting-phase []       bb_ = bb_\n    run-voting-phase (q \u2237 qs) bb_ = run-voting-phase qs (uncurry run-query (q (bb_ \u03b2)) bb_)\n\n  Adversary = R-adversary \u2192\n              PublicKey \u2192\n              VotingPhase \u00d7\n              (BB \u2192\n              \ud835\udfda)\n\n  module Game (\u03b2 : \ud835\udfda)\n              (r-setup : R-setup)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              where\n    pksk = setup r-setup\n    pk = fst pksk\n    sk = snd pksk\n\n    BB-setup : BB\n    BB-setup = < pk ,, [] ,, [] >\n\n    BB\u00b2-setup : BB \u00b2\n    BB\u00b2-setup _ = BB-setup\n\n    A-setup = A r-adversary pk\n\n    A-voting-phase : VotingPhase\n    A-voting-phase = fst A-setup\n\n    open Run \u03b2 pk r-votes\n\n    exp : \ud835\udfda\n    exp = snd A-setup (run-voting-phase A-voting-phase BB\u00b2-setup \u03b2)\n\n    game = exp == \u03b2\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7334d3b57f3c60e93d102fac10adb8d500fad0c6","subject":"Added doc related to projection-like functions.","message":"Added doc related to projection-like functions.\n\nIgnore-this: 1691c2b03146dd4bef59f0eb004fe936\n\ndarcs-hash:20110919213152-3bd4e-9c46d3f7c4ca039d9b04c80d6bf54f98d2cf3dee.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/ProjectionLikeFunction.agda","new_file":"Test\/Fail\/ProjectionLikeFunction.agda","new_contents":"------------------------------------------------------------------------------\n-- The translation of projection-like functions is not implemented\n\n-- 2011-09-19: Maybe the information in the issue 465 (Missing\n-- patterns in funClauses) is useful.\n\n------------------------------------------------------------------------------\n\nmodule Test.Fail.ProjectionLikeFunction where\n\n-- Error: The translation of projection-like functions Test.Fail.ProjectionLikeFunction._.P is not implemented.\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  N    : D \u2192 Set\n\nfoo : \u2200 {n} \u2192 N n \u2192 D\nfoo {n} Nn = n\n  where\n  P : D \u2192 Set\n  P i = i \u2261 i\n  {-# ATP definition P #-}\n\n  postulate bar : P n\n  {-# ATP prove bar #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The translation of projection-like functions is not implemented\n------------------------------------------------------------------------------\n\nmodule Test.Fail.ProjectionLikeFunction where\n\n-- Error: The translation of projection-like functions Test.Fail.ProjectionLikeFunction._.P is not implemented.\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  N    : D \u2192 Set\n\nfoo : \u2200 {n} \u2192 N n \u2192 D\nfoo {n} Nn = n\n  where\n  P : D \u2192 Set\n  P i = i \u2261 i\n  {-# ATP definition P #-}\n\n  postulate bar : P n\n  {-# ATP prove bar #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6392efbbe853410d12c6c0d9e5f385090b757644","subject":"Minor changes.","message":"Minor changes.\n\nIgnore-this: 80b6d52ceb4e05f8daca0c4feaf390e6\n\ndarcs-hash:20110518144052-3bd4e-876879e98b56ce1879cc4bae32deec0b8c804064.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/NestBC.agda","new_file":"Draft\/FOTC\/Program\/Nest\/NestBC.agda","new_contents":"------------------------------------------------------------------------------\n-- Example of a nested recursive function using the Bove-Capretta\n-- method\n------------------------------------------------------------------------------\n\n-- Tested with the development version of the standard library on\n-- 18 May 2011.\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. vol 2152 LNCS. 2001\n\nmodule NestBC where\n\nopen import Data.Nat\nopen import Data.Nat.Properties\n\nopen import Induction\nopen import Induction.Nat\n\nopen import Relation.Binary\n\nmodule NDTO = DecTotalOrder decTotalOrder\n\n------------------------------------------------------------------------------\n\n\u2264\u2032-trans : Transitive _\u2264\u2032_\n\u2264\u2032-trans i\u2264\u2032j j\u2264\u2032k = \u2264\u21d2\u2264\u2032 (NDTO.trans (\u2264\u2032\u21d2\u2264 i\u2264\u2032j) (\u2264\u2032\u21d2\u2264 j\u2264\u2032k))\n\nmutual\n  -- The domain predicate of the nest function.\n  data NestDom : \u2115 \u2192 Set where\n    nestDom0 : NestDom 0\n    nestDomS : \u2200 {n} \u2192 (h\u2081 : NestDom n) \u2192\n               (h\u2082 : NestDom (nestD n h\u2081)) \u2192\n               NestDom (suc n)\n\n  -- The nest function by structural recursion on the domain predicate.\n  nestD : \u2200 n \u2192 NestDom n \u2192 \u2115\n  nestD .0       nestDom0             = 0\n  nestD .(suc n) (nestDomS {n} h\u2081 h\u2082) = nestD (nestD n h\u2081) h\u2082\n\nnestD-\u2264\u2032 : \u2200 n \u2192 (h : NestDom n) \u2192 nestD n h \u2264\u2032 n\nnestD-\u2264\u2032 .0       nestDom0             = \u2264\u2032-refl\nnestD-\u2264\u2032 .(suc n) (nestDomS {n} h\u2081 h\u2082) =\n  \u2264\u2032-trans (\u2264\u2032-trans (nestD-\u2264\u2032 (nestD n h\u2081) h\u2082) (nestD-\u2264\u2032 n h\u2081))\n           (\u2264\u2032-step \u2264\u2032-refl)\n\n-- The nest function is total.\nallNestDom : \u2200 n \u2192 NestDom n\nallNestDom = build <-rec-builder P ih\n  where\n    P : \u2115 \u2192 Set\n    P = NestDom\n\n    ih : \u2200 y \u2192 <-Rec P y \u2192 P y\n    ih zero    rec = nestDom0\n    ih (suc y) rec = nestDomS nd-y (rec (nestD y nd-y) (s\u2264\u2032s (nestD-\u2264\u2032 y nd-y)))\n        where\n          helper : \u2200 x \u2192 x <\u2032 y \u2192 P x\n          helper x Sx\u2264\u2032y = rec x (\u2264\u2032-step Sx\u2264\u2032y)\n\n          nd-y : NestDom y\n          nd-y = ih y helper\n\n-- The nest function.\nnest : \u2115 \u2192 \u2115\nnest n = nestD n (allNestDom n)\n","old_contents":"------------------------------------------------------------------------------\n-- Example of a nested recursive function using the Bove-Capretta\n-- method\n------------------------------------------------------------------------------\n\n-- Tested with the development version of the standard library on 04\n-- April 2011.\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. vol 2152 LNCS. 2001\n\nmodule NestBC where\n\nopen import Data.Nat\nopen import Data.Nat.Properties\n\nopen import Induction.Nat\nopen import Induction\n\n------------------------------------------------------------------------------\n\n\u2264\u2032-trans : \u2200 {m n o} \u2192 m \u2264\u2032 n \u2192 n \u2264\u2032 o \u2192 m \u2264\u2032 o\n\u2264\u2032-trans \u2264\u2032-refl            n\u2264\u2032o                = n\u2264\u2032o\n\u2264\u2032-trans (\u2264\u2032-step {n} m\u2264\u2032n) \u2264\u2032-refl             = \u2264\u2032-step m\u2264\u2032n\n\u2264\u2032-trans (\u2264\u2032-step {n} m\u2264\u2032n) (\u2264\u2032-step {o} Sn\u2264\u2032o) =\n  \u2264\u2032-step (\u2264\u2032-trans m\u2264\u2032n (\u2264\u2032-trans (\u2264\u2032-step \u2264\u2032-refl) Sn\u2264\u2032o))\n\nmutual\n  -- The domain predicate of the nest function.\n  data DomNest : \u2115 \u2192 Set where\n    dom0 : DomNest 0\n    domS : \u2200 {n} \u2192 (h\u2081 : DomNest n) \u2192\n                   (h\u2082 : DomNest (nestD n h\u2081)) \u2192\n                   DomNest (suc n)\n\n  -- The nest function by structural recursion on the domain predicate.\n  nestD : \u2200 n \u2192 DomNest n \u2192 \u2115\n  nestD .0       dom0             = 0\n  nestD .(suc n) (domS {n} h\u2081 h\u2082) = nestD (nestD n h\u2081) h\u2082\n\nnestD-\u2264\u2032 : \u2200 n \u2192 (h : DomNest n) \u2192 nestD n h \u2264\u2032 n\nnestD-\u2264\u2032 .0       dom0             = \u2264\u2032-refl\nnestD-\u2264\u2032 .(suc n) (domS {n} h\u2081 h\u2082) =\n  \u2264\u2032-trans (\u2264\u2032-trans (nestD-\u2264\u2032 (nestD n h\u2081) h\u2082) (nestD-\u2264\u2032 n h\u2081))\n           (\u2264\u2032-step \u2264\u2032-refl)\n\n-- The nest function is total.\nallDomNest : \u2200 n \u2192 DomNest n\nallDomNest = build <-rec-builder P ih\n  where\n    P : \u2115 \u2192 Set\n    P = DomNest\n\n    ih : \u2200 y \u2192 <-Rec P y \u2192 P y\n    ih zero    rec = dom0\n    ih (suc y) rec = domS dn-y (rec (nestD y dn-y) (s\u2264\u2032s (nestD-\u2264\u2032 y dn-y)))\n        where\n          helper : (x : \u2115) \u2192 x <\u2032 y \u2192 P x\n          helper x Sx\u2264\u2032y = rec x (\u2264\u2032-step Sx\u2264\u2032y)\n\n          dn-y : DomNest y\n          dn-y = ih y helper\n\n-- The nest function.\nnest : \u2115 \u2192 \u2115\nnest n = nestD n (allDomNest n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ffc4d7de6c9c565efebc92e8284dcc9fa8c517c9","subject":"Added x<x+Sy (by Ana Bove).","message":"Added x<x+Sy (by Ana Bove).\n\nIgnore-this: 1b7829d941cd8b1cf318d5579418b1a0\n\ndarcs-hash:20110210163141-3bd4e-05b6623c2d6425ca20325ece755c6603f568ee14.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        )\n\n------------------------------------------------------------------------------\n\n0<0-elim : LT zero zero \u2192 \u22a5\n0<0-elim 0<0 = true\u2260false $ trans (sym 0<0) <-00\n\nS<0-elim : {d : D} \u2192 LT (succ d) zero \u2192 \u22a5\nS<0-elim {d} Sd<0 = true\u2260false $ trans (sym Sd<0) (<-S0 d)\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN      0<0  = 0<0-elim 0<0\n\u00acx<0 (sN Nn) Sn<0 = S<0-elim Sn<0\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\npostulate\n  \u00ac0>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x 0\u226fx #-}\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\n\u00acS\u22640 : \u2200 {n} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : \u2200 {n} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\n\u00acx<x : \u2200 {n} \u2192 N n \u2192 \u00ac (LT n n)\n\u00acx<x zN           0<0  = 0<0-elim 0<0\n\u00acx<x (sN {n} Nn) Sn<Sn = \u22a5-elim $ \u00acx<x Nn (trans (sym $ <-SS n n) Sn<Sn)\n\n\u00acx>x : \u2200 {n} \u2192 N n \u2192 \u00ac (GT n n)\n\u00acx>x Nn = \u00acx<x Nn\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ \u00ac0\u2265S Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ \u00ac0>x Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0-elim 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0-elim Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ \u00acx<0 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0-elim Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ \u00acS\u22640 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx-y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x-y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx-Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx \u2238-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ \u00ac0>x Nn 0>n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x-y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y-x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ \u00acS\u22640 Nm Sm\u22640\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ \u00acx<0 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ \u00acx<0 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x-y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim $ \u00acx>x zN 0>0\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ \u00acS\u22640 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x-y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 aux m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n  where\n    aux : \u2200 {a b c} \u2192 LT a b \u2192 b \u2261 c \u2192 LT a c\n    aux a<b refl = a<b\n\nx\u2264y+x-y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x-y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x-y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x-y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x-y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  \u00ac0Sx<00 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy-Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x-y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6629f5816805b71b7477c3fb4f8c3e6a910c6a43","subject":"Added automatic proof of \u2261\u2192\u2248N.","message":"Added automatic proof of \u2261\u2192\u2248N.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Conat\/Equality\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Data\/Conat\/Equality\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the equality on Conat\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- \u2022 Herbert P. Sander. A logic of functional programs with an\n--   application to concurrency. PhD thesis, Chalmers University of\n--   Technology and University of Gothenburg, Department of Computer\n--   Sciences, 1992.\n\nmodule FOTC.Data.Conat.Equality.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\n\n------------------------------------------------------------------------------\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} Cn = \u2248N-coind R prf\u2081 prf\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n  {-# ATP definition R #-}\n\n  postulate\n    prf\u2081 : \u2200 {a b} \u2192 R a b \u2192\n           a \u2261 zero \u2227 b \u2261 zero\n           \u2228 (\u2203[ a' ] \u2203[ b' ] a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b' \u2227 R a' b')\n  {-# ATP prove prf\u2081 #-}\n\n  postulate prf\u2082 : Conat n \u2227 Conat n \u2227 n \u2261 n\n  {-# ATP prove prf\u2082 #-}\n\npostulate \u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n{-# ATP prove \u2261\u2192\u2248N \u2248N-refl #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for the equality on Conat\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- \u2022 Herbert P. Sander. A logic of functional programs with an\n--   application to concurrency. PhD thesis, Chalmers University of\n--   Technology and University of Gothenburg, Department of Computer\n--   Sciences, 1992.\n\nmodule FOTC.Data.Conat.Equality.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\n\n------------------------------------------------------------------------------\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} Cn = \u2248N-coind R prf\u2081 prf\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n  {-# ATP definition R #-}\n\n  postulate\n    prf\u2081 : \u2200 {a b} \u2192 R a b \u2192\n           a \u2261 zero \u2227 b \u2261 zero\n           \u2228 (\u2203[ a' ] \u2203[ b' ] a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b' \u2227 R a' b')\n  {-# ATP prove prf\u2081 #-}\n\n  postulate prf\u2082 : Conat n \u2227 Conat n \u2227 n \u2261 n\n  {-# ATP prove prf\u2082 #-}\n\n\u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n\u2261\u2192\u2248N h _ refl = \u2248N-refl h\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6ca1fd5063bb7c6918676514704cd5dda52673fb","subject":"Removed unnecesary implicit arguments.","message":"Removed unnecesary implicit arguments.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n-----------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS h with (Stream-unf h)\n... | x' , xs' , Sxs' , h\u2081 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2081))) Sxs'\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n\n-- Requires K.\nlengthStream : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nlengthStream {xs} Sxs = \u2248N-coind _R_ h\u2081 h\u2082\n  where\n  _R_ : D \u2192 D \u2192 Set\n  m R n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 m R n \u2192\n       m \u2261 zero \u2227 n \u2261 zero \u2228\n       (\u2203[ m' ] \u2203[ n' ] m' R n' \u2227 m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n')\n  h\u2081 (_ , Sxs' , _ ) with Stream-unf Sxs'\n  h\u2081 {m} {n} (.(x'' \u2237 xs'') , Sxs' , aux , n\u2261\u221e) | x'' , xs'' , Sxs'' , refl =\n    inj\u2082 (length xs'' , \u221e , (xs'' , Sxs'' , refl , refl) , helper\u2081 , helper\u2082)\n    where\n    helper\u2081 : m \u2261 succ\u2081 (length xs'')\n    helper\u2081 = trans aux (length-\u2237 x'' xs'')\n\n    helper\u2082 : n \u2261 succ\u2081 \u221e\n    helper\u2082 = trans n\u2261\u221e \u221e-eq\n\n  h\u2082 : length xs R \u221e\n  h\u2082 = xs , Sxs , refl , refl\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n-----------------------------------------------------------------------------\n\ntailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\ntailS {x} {xs} h with (Stream-unf h)\n... | x' , xs' , Sxs' , h\u2081 = subst Stream (sym (\u2227-proj\u2082 (\u2237-injective h\u2081))) Sxs'\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n\n-- Requires K.\nlengthStream : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nlengthStream {xs} Sxs = \u2248N-coind _R_ h\u2081 h\u2082\n  where\n  _R_ : D \u2192 D \u2192 Set\n  m R n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 m R n \u2192\n       m \u2261 zero \u2227 n \u2261 zero \u2228\n       (\u2203[ m' ] \u2203[ n' ] m' R n' \u2227 m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n')\n  h\u2081 (_ , Sxs' , _ ) with Stream-unf Sxs'\n  h\u2081 {m} {n} (.(x'' \u2237 xs'') , Sxs' , aux , n\u2261\u221e) | x'' , xs'' , Sxs'' , refl =\n    inj\u2082 (length xs'' , \u221e , (xs'' , Sxs'' , refl , refl) , helper\u2081 , helper\u2082)\n    where\n    helper\u2081 : m \u2261 succ\u2081 (length xs'')\n    helper\u2081 = trans aux (length-\u2237 x'' xs'')\n\n    helper\u2082 : n \u2261 succ\u2081 \u221e\n    helper\u2082 = trans n\u2261\u221e \u221e-eq\n\n  h\u2082 : length xs R \u221e\n  h\u2082 = xs , Sxs , refl , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b8daf1e6e26a3ddbf57276e2bc2997e2fc93d22b","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: f3b9a4830249b8abbba002764a454e10\n\ndarcs-hash:20110912041947-3bd4e-a8e4c2508e86ade13b770c69c61835a022a773da.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/ABP\/Lemma1ATP.agda","new_file":"src\/FOTC\/Program\/ABP\/Lemma1ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n-- From the paper: The first lemma states that given a start state Abp\n-- (of the alternating bit protocol) we will arrive at a state Abp',\n-- where the message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOTC.Program.ABP.Lemma1ATP where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.List\n\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\n-- TODO: These variables are outside the where clause due to an issue\n-- in the translation.\n\nas\u2075 : \u2200 b i' is' ds \u2192 D\nas\u2075 b i' is' ds = await b i' is' ds\n{-# ATP definition as\u2075 #-}\n\nbs\u2075 : \u2200 b i' is' ds os\u2080\u2075 \u2192 D\nbs\u2075 b i' is' ds os\u2080\u2075 = corrupt \u00b7 os\u2080\u2075 \u00b7 (as\u2075 b i' is' ds)\n{-# ATP definition bs\u2075 #-}\n\ncs\u2075 : \u2200 b i' is' ds os\u2080\u2075 \u2192 D\ncs\u2075 b i' is' ds os\u2080\u2075 = abpack \u00b7 b \u00b7 (bs\u2075 b i' is' ds os\u2080\u2075)\n{-# ATP definition cs\u2075 #-}\n\nds\u2075 : \u2200 b i' is' ds os\u2080\u2075 os\u2081\u2075 \u2192 D\nds\u2075 b i' is' ds os\u2080\u2075 os\u2081\u2075 = corrupt \u00b7 os\u2081\u2075 \u00b7 cs\u2075 b i' is' ds os\u2080\u2075\n{-# ATP definition ds\u2075 #-}\n\nos\u2080\u2075 : \u2200 os\u2080' ol\u2080\u2075 \u2192 D\nos\u2080\u2075 os\u2080' ol\u2080\u2075 = ol\u2080\u2075 ++ os\u2080'\n{-# ATP definition os\u2080\u2075 #-}\n\nos\u2081\u2075 : \u2200 os\u2081 \u2192 D\nos\u2081\u2075 os\u2081 = tail os\u2081\n{-# ATP definition os\u2081\u2075 #-}\n\nlemma\u2081-helper : \u2200 {b i' is' os\u2080 os\u2081 as bs cs ds js} \u2192\n                Bit b \u2192\n                Fair os\u2081 \u2192\n                Abp b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js \u2192\n                \u2200 {ol\u2080} \u2192 O*L ol\u2080 \u2192\n                \u2200 {os\u2080'-aux} \u2192 Fair os\u2080'-aux \u2192 os\u2080 \u2261 ol\u2080 ++ os\u2080'-aux \u2192\n                \u2203 \u03bb os\u2080' \u2192 \u2203 \u03bb os\u2081' \u2192\n                \u2203 \u03bb as' \u2192 \u2203 \u03bb bs' \u2192 \u2203 \u03bb cs' \u2192 \u2203 \u03bb ds' \u2192 \u2203 \u03bb js' \u2192\n                Fair os\u2080'\n                \u2227 Fair os\u2081'\n                \u2227 Abp' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                \u2227 js \u2261 i' \u2237 js'\nlemma\u2081-helper {b} {i'} {is'} {js = js} Bb Fos\u2081 h nilO*L Fos\u2080' os\u2080-eq = prf\n  where\n  postulate\n    prf : \u2203 (\u03bb os\u2080' \u2192 \u2203 (\u03bb os\u2081' \u2192\n          \u2203 (\u03bb as' \u2192 \u2203 (\u03bb bs' \u2192 \u2203 (\u03bb cs' \u2192 \u2203 (\u03bb ds' \u2192 \u2203 (\u03bb js' \u2192\n          Fair os\u2080'\n          \u2227 Fair os\u2081'\n          \u2227 (ds' \u2261 corrupt \u00b7 os\u2081' \u00b7 (b \u2237 cs')\n             \u2227 as' \u2261 await b i' is' ds'\n             \u2227 bs' \u2261 corrupt \u00b7 os\u2080' \u00b7 as'\n             \u2227 cs' \u2261 abpack \u00b7 (not b) \u00b7 bs'\n             \u2227 js' \u2261 abpout \u00b7 (not b) \u00b7 bs')\n          \u2227 js \u2261 i' \u2237 js')))))))\n  {-# ATP prove prf #-}\n\nlemma\u2081-helper {b} {i'} {is'} {os\u2080} {os\u2081} {as} {bs} {cs} {ds} {js}\n              Bb Fos\u2081 abp\n              (consO*L ol\u2080\u2075 OLol\u2080\u2075)\n              {os\u2080'-aux = os\u2080'} Fos\u2080' os\u2080-eq =\n                lemma\u2081-helper Bb (tail-Fair Fos\u2081) AbpIH OLol\u2080\u2075 Fos\u2080' refl\n  where\n  postulate os\u2080-eq-helper : os\u2080 \u2261 O \u2237 os\u2080\u2075 os\u2080' ol\u2080\u2075\n  {-# ATP prove os\u2080-eq-helper #-}\n\n  postulate as-eq : as \u2261 < i' , b > \u2237 (as\u2075 b i' is' ds)\n  {-# ATP prove as-eq #-}\n\n  postulate bs-eq : bs \u2261 error \u2237 (bs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075))\n  {-# ATP prove bs-eq os\u2080-eq-helper as-eq #-}\n\n  postulate cs-eq : cs \u2261 not b \u2237 cs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075)\n  {-# ATP prove cs-eq bs-eq #-}\n\n  postulate\n    ds-eq-helper\u2081 : os\u2081 \u2261 L \u2237 tail os\u2081 \u2192\n                    ds \u2261 ok (not b) \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n  postulate\n    ds-eq-helper\u2082 : os\u2081 \u2261 O \u2237 tail os\u2081 \u2192\n                    ds \u2261 error \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n  ds-eq : ds \u2261 ok (not b) \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n          \u2228 ds \u2261 error \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  ds-eq = [ (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n          , (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n          ] (head-tail-Fair Fos\u2081)\n\n  postulate\n    as\u2075-eq-helper\u2081 : ds \u2261 ok (not b) \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081) \u2192\n                     as\u2075 b i' is' ds \u2261\n                     abpsend \u00b7 b \u00b7 (i' \u2237 is') \u00b7 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {- ATP prove as\u2075-eq-helper\u2081 #-}\n\n  postulate\n    as\u2075-eq-helper\u2082 : ds \u2261 error \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081) \u2192\n                     as\u2075 b i' is' ds \u2261\n                     abpsend \u00b7 b \u00b7 (i' \u2237 is') \u00b7 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {- ATP prove as\u2075-eq-helper\u2082 #-}\n\n  as\u2075-eq : as\u2075 b i' is' ds \u2261\n           abpsend \u00b7 b \u00b7 (i' \u2237 is') \u00b7 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  as\u2075-eq = [ as\u2075-eq-helper\u2081 , as\u2075-eq-helper\u2082 ] ds-eq\n\n  postulate js-eq : js \u2261 abpout \u00b7 b \u00b7 bs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075)\n  {-# ATP prove js-eq bs-eq #-}\n\n  AbpIH : Abp b\n              (i' \u2237 is')\n              (os\u2080\u2075 os\u2080' ol\u2080\u2075)\n              (os\u2081\u2075 os\u2081)\n              (as\u2075 b i' is' ds)\n              (bs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075))\n              (cs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075))\n              (ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081))\n              js\n  AbpIH = as\u2075-eq , refl , refl , refl , js-eq\n\n-- From the paper: From the assumption that os\u2080 \u2208 Fair, and hence by\n-- unfolding Fair we conclude that there are ol\u2080 : O*L and os\u2080' :\n-- Fair, such that\n--\n-- os\u2080 = ol\u2080 ++ os\u2080'.\n--\n-- We proceed by induction on ol\u2080 : O*L using lemma\u2081-helper.\nlemma\u2081 : \u2200 {b i' is' os\u2080 os\u2081 as bs cs ds js} \u2192\n         Bit b \u2192\n         Fair os\u2080 \u2192\n         Fair os\u2081 \u2192\n         Abp b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js \u2192\n         \u2203 \u03bb os\u2080' \u2192 \u2203 \u03bb os\u2081' \u2192\n         \u2203 \u03bb as' \u2192 \u2203 \u03bb bs' \u2192 \u2203 \u03bb cs' \u2192 \u2203 \u03bb ds' \u2192 \u2203 \u03bb js' \u2192\n         Fair os\u2080'\n         \u2227 Fair os\u2081'\n         \u2227 Abp' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n         \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 {os\u2080 = os\u2080} Bb Fos\u2080 Fos\u2081 h = lemma\u2081-helper Bb Fos\u2081 h OLol\u2080 Fos\u2080' os\u2080-eq\n  where\n  unfold-os\u2080 : \u2203 \u03bb ol\u2080 \u2192 \u2203 \u03bb os\u2080' \u2192 O*L ol\u2080 \u2227 Fair os\u2080' \u2227 os\u2080 \u2261 ol\u2080 ++ os\u2080'\n  unfold-os\u2080 = Fair-gfp\u2081 Fos\u2080\n\n  ol\u2080 : D\n  ol\u2080 = \u2203-proj\u2081 unfold-os\u2080\n\n  os\u2080' : D\n  os\u2080' = \u2203-proj\u2081 (\u2203-proj\u2082 unfold-os\u2080)\n\n  OLol\u2080 : O*L ol\u2080\n  OLol\u2080 = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-os\u2080))\n\n  Fos\u2080' : Fair os\u2080'\n  Fos\u2080' = \u2227-proj\u2081 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-os\u2080)))\n\n  os\u2080-eq : os\u2080 \u2261 ol\u2080 ++ os\u2080'\n  os\u2080-eq = \u2227-proj\u2082 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-os\u2080)))\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n-- From the paper: The first lemma states that given a start state Abp\n-- (of the alternating bit protocol) we will arrive at a state Abp',\n-- where the message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOTC.Program.ABP.Lemma1ATP where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.List\n\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\n-- TODO: These variables are outside the where clause due to an issue\n-- in the translation.\n\nas\u2075 : \u2200 b i' is' ds \u2192 D\nas\u2075 b i' is' ds = await b i' is' ds\n{-# ATP definition as\u2075 #-}\n\nbs\u2075 : \u2200 b i' is' ds os\u2080\u2075 \u2192 D\nbs\u2075 b i' is' ds os\u2080\u2075 = corrupt \u00b7 os\u2080\u2075 \u00b7 (as\u2075 b i' is' ds)\n{-# ATP definition bs\u2075 #-}\n\ncs\u2075 : \u2200 b i' is' ds os\u2080\u2075 \u2192 D\ncs\u2075 b i' is' ds os\u2080\u2075 = abpack \u00b7 b \u00b7 (bs\u2075 b i' is' ds os\u2080\u2075)\n{-# ATP definition cs\u2075 #-}\n\nds\u2075 : \u2200 b i' is' ds os\u2080\u2075 os\u2081\u2075 \u2192 D\nds\u2075 b i' is' ds os\u2080\u2075 os\u2081\u2075 = corrupt \u00b7 os\u2081\u2075 \u00b7 cs\u2075 b i' is' ds os\u2080\u2075\n{-# ATP definition ds\u2075 #-}\n\nos\u2080\u2075 : \u2200 os\u2080' ol\u2080\u2075 \u2192 D\nos\u2080\u2075 os\u2080' ol\u2080\u2075 = ol\u2080\u2075 ++ os\u2080'\n{-# ATP definition os\u2080\u2075 #-}\n\nos\u2081\u2075 : \u2200 os\u2081 \u2192 D\nos\u2081\u2075 os\u2081 = tail os\u2081\n{-# ATP definition os\u2081\u2075 #-}\n\nlemma\u2081-helper : \u2200 {b i' is' os\u2080 os\u2081 as bs cs ds js} \u2192\n                Bit b \u2192\n                Fair os\u2081 \u2192\n                Abp b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js \u2192\n                \u2200 {ol\u2080} \u2192 O*L ol\u2080 \u2192\n                \u2200 {os\u2080'-aux} \u2192 Fair os\u2080'-aux \u2192 os\u2080 \u2261 ol\u2080 ++ os\u2080'-aux \u2192\n                \u2203 \u03bb os\u2080' \u2192 \u2203 \u03bb os\u2081' \u2192\n                \u2203 \u03bb as' \u2192 \u2203 \u03bb bs' \u2192 \u2203 \u03bb cs' \u2192 \u2203 \u03bb ds' \u2192 \u2203 \u03bb js' \u2192\n                Fair os\u2080'\n                \u2227 Fair os\u2081'\n                \u2227 Abp' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                \u2227 js \u2261 i' \u2237 js'\nlemma\u2081-helper {b} {i'} {is'} {js = js} Bb Fos\u2081 h nilO*L Fos\u2080' os\u2080-eq = prf\n  where\n  postulate\n    prf : \u2203 (\u03bb os\u2080' \u2192 \u2203 (\u03bb os\u2081' \u2192\n          \u2203 (\u03bb as' \u2192 \u2203 (\u03bb bs' \u2192 \u2203 (\u03bb cs' \u2192 \u2203 (\u03bb ds' \u2192 \u2203 (\u03bb js' \u2192\n          Fair os\u2080'\n          \u2227 Fair os\u2081'\n          \u2227 (ds' \u2261 corrupt \u00b7 os\u2081' \u00b7 (b \u2237 cs')\n             \u2227 as' \u2261 await b i' is' ds'\n             \u2227 bs' \u2261 corrupt \u00b7 os\u2080' \u00b7 as'\n             \u2227 cs' \u2261 abpack \u00b7 (not b) \u00b7 bs'\n             \u2227 js' \u2261 abpout \u00b7 (not b) \u00b7 bs')\n          \u2227 js \u2261 i' \u2237 js')))))))\n  {-# ATP prove prf #-}\n\nlemma\u2081-helper {b} {i'} {is'} {os\u2080} {os\u2081} {as} {bs} {cs} {ds} {js}\n              Bb Fos\u2081 abp\n              (consO*L ol\u2080\u2075 OLol\u2080\u2075)\n              {os\u2080'-aux = os\u2080'} Fos\u2080' os\u2080-eq =\n                lemma\u2081-helper Bb (tail-Fair Fos\u2081) AbpIH OLol\u2080\u2075 Fos\u2080' refl\n  where\n  postulate os\u2080-eq-helper : os\u2080 \u2261 O \u2237 os\u2080\u2075 os\u2080' ol\u2080\u2075\n  {-# ATP prove os\u2080-eq-helper #-}\n\n  postulate as-eq : as \u2261 < i' , b > \u2237 (as\u2075 b i' is' ds)\n  {-# ATP prove as-eq #-}\n\n  postulate bs-eq : bs \u2261 error \u2237 (bs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075))\n  {-# ATP prove bs-eq os\u2080-eq-helper as-eq #-}\n\n  postulate cs-eq : cs \u2261 not b \u2237 cs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075)\n  {-# ATP prove cs-eq bs-eq #-}\n\n  postulate\n    ds-eq-helper\u2081 : os\u2081 \u2261 L \u2237 tail os\u2081 \u2192\n                    ds \u2261 ok (not b) \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n  postulate\n    ds-eq-helper\u2082 : os\u2081 \u2261 O \u2237 tail os\u2081 \u2192\n                    ds \u2261 error \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n  ds-eq : ds \u2261 ok (not b) \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n          \u2228 ds \u2261 error \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  ds-eq = [ (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n          , (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n          ] (head-tail-Fair Fos\u2081)\n\n  postulate\n    as\u2075-eq-helper\u2081 : ds \u2261 ok (not b) \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081) \u2192\n                     as\u2075 b i' is' ds \u2261\n                     abpsend \u00b7 b \u00b7 (i' \u2237 is') \u00b7 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  {- ATP prove as\u2075-eq-helper\u2081 #-}\n\n  postulate\n    as\u2075-eq-helper\u2082 : ds \u2261 error \u2237 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081) \u2192\n                     as\u2075 b i' is' ds \u2261\n                     abpsend \u00b7 b \u00b7 (i' \u2237 is') \u00b7 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075 )(os\u2081\u2075 os\u2081)\n  {- ATP prove as\u2075-eq-helper\u2082 #-}\n\n  as\u2075-eq : as\u2075 b i' is' ds \u2261\n           abpsend \u00b7 b \u00b7 (i' \u2237 is') \u00b7 ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081)\n  as\u2075-eq = [ as\u2075-eq-helper\u2081 , as\u2075-eq-helper\u2082 ] ds-eq\n\n  postulate js-eq : js \u2261 abpout \u00b7 b \u00b7 bs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075)\n  {-# ATP prove js-eq bs-eq #-}\n\n  AbpIH : Abp b\n              (i' \u2237 is')\n              (os\u2080\u2075 os\u2080' ol\u2080\u2075)\n              (os\u2081\u2075 os\u2081)\n              (as\u2075 b i' is' ds)\n              (bs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075))\n              (cs\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075))\n              (ds\u2075 b i' is' ds (os\u2080\u2075 os\u2080' ol\u2080\u2075) (os\u2081\u2075 os\u2081))\n              js\n  AbpIH = as\u2075-eq , refl , refl , refl , js-eq\n\n-- From the paper: From the assumption that os\u2080 \u2208 Fair, and hence by\n-- unfolding Fair we conclude that there are ol\u2080 : O*L and os\u2080' :\n-- Fair, such that\n--\n-- os\u2080 = ol\u2080 ++ os\u2080'.\n--\n-- We proceed by induction on ol\u2080 : O*L using lemma\u2081-helper.\nlemma\u2081 : \u2200 {b i' is' os\u2080 os\u2081 as bs cs ds js} \u2192\n         Bit b \u2192\n         Fair os\u2080 \u2192\n         Fair os\u2081 \u2192\n         Abp b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js \u2192\n         \u2203 \u03bb os\u2080' \u2192 \u2203 \u03bb os\u2081' \u2192\n         \u2203 \u03bb as' \u2192 \u2203 \u03bb bs' \u2192 \u2203 \u03bb cs' \u2192 \u2203 \u03bb ds' \u2192 \u2203 \u03bb js' \u2192\n         Fair os\u2080'\n         \u2227 Fair os\u2081'\n         \u2227 Abp' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n         \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 {os\u2080 = os\u2080} Bb Fos\u2080 Fos\u2081 h = lemma\u2081-helper Bb Fos\u2081 h OLol\u2080 Fos\u2080' os\u2080-eq\n  where\n  unfold-os\u2080 : \u2203 \u03bb ol\u2080 \u2192 \u2203 \u03bb os\u2080' \u2192 O*L ol\u2080 \u2227 Fair os\u2080' \u2227 os\u2080 \u2261 ol\u2080 ++ os\u2080'\n  unfold-os\u2080 = Fair-gfp\u2081 Fos\u2080\n\n  ol\u2080 : D\n  ol\u2080 = \u2203-proj\u2081 unfold-os\u2080\n\n  os\u2080' : D\n  os\u2080' = \u2203-proj\u2081 (\u2203-proj\u2082 unfold-os\u2080)\n\n  OLol\u2080 : O*L ol\u2080\n  OLol\u2080 = \u2227-proj\u2081 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-os\u2080))\n\n  Fos\u2080' : Fair os\u2080'\n  Fos\u2080' = \u2227-proj\u2081 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-os\u2080)))\n\n  os\u2080-eq : os\u2080 \u2261 ol\u2080 ++ os\u2080'\n  os\u2080-eq = \u2227-proj\u2082 (\u2227-proj\u2082 (\u2203-proj\u2082 (\u2203-proj\u2082 unfold-os\u2080)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a55145a5a9255d14cef0ac7a49e915b86c600358","subject":"Args is fused with \u0394-Type and Vars is fused with \u0394-Context.","message":"Args is fused with \u0394-Type and Vars is fused with \u0394-Context.\n\nOld-commit-hash: 4c73a170c8dc1dc6e527cbd13ce42874fe858a39\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag renaming\n--   (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {b : Bag} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  nats : \u0394-Type\n  bags : \u0394-Type\n  alter_\u21d2_ : \u0394-Type \u2192 \u0394-Type \u2192 \u0394-Type\n  abide_\u21d2_ : \u0394-Type \u2192 \u0394-Type \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u2205 : \u0394-Context\n  alter_\u2022_ : \u0394-Type \u2192 \u0394-Context \u2192 \u0394-Context\n  abide_\u2022_ : \u0394-Type \u2192 \u0394-Context \u2192 \u0394-Context\n\n-- Convert a type\/context to a \u0394-type\/\u0394-context without any\n-- assumption about arguments\n\n\u0394-type : Type \u2192 \u0394-Type\n\u0394-type nats = nats\n\u0394-type bags = bags\n\u0394-type (\u03c4\u2081 \u21d2 \u03c4\u2082) = alter \u0394-type \u03c4\u2081 \u21d2 \u0394-type \u03c4\u2082\n\n\u0394-context : Context \u2192 \u0394-Context\n\u0394-context \u2205 = \u2205\n\u0394-context (\u03c4 \u2022 \u0393) = alter \u0394-type \u03c4 \u2022 \u0394-context \u0393\n\n{-\n\n\u27e6_\u27e7\u0394Type : \u2200 {\u03c4 args} \u2192 \u0394-Type \u03c4 {args} \u2192 Set\n\ndata \u0394-Env : (\u0393 : Context) \u2192 {vars : Vars \u0393} \u2192 Set\ndata \u0394-Term : (\u0393 : Context) \u2192 Type \u2192 {vars : Vars \u0393} \u2192 Set\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u03c4 : Type} {\u0393 : Context} {vars}\n    \u2192 \u0394-Term \u0393 \u03c4 {vars} \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} {vars} \u2192 (\u03c1 : \u0394-Env \u0393 {vars}) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} {vars} \u2192 (\u03c1 : \u0394-Env \u0393 {vars}) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393 vars} \u2192\n    (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term \u0393 nats {vars}\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393 vars} \u2192\n    (db : Bag) \u2192 \u0394-Term \u0393 bags {vars}\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393 vars} \u2192\n    (x : Var \u0393 \u03c4) \u2192 \u0394-Term \u0393 \u03c4 {vars}\n  -- changes to abstractions are binders of x and dx\n  -- There are two kinds of those: One who expects the argument\n  -- to change, and one who does not.\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} \u2192 (t : \u0394-Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082 {alter vars}) \u2192\n         \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars}\n  \u0394abs\u2080 : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} \u2192 (t : \u0394-Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082 {abide vars}) \u2192\n          \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars}\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars}\n    (ds : \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars})\n    ( t :   Term \u0393 \u03c4\u2081)\n    (dt : \u0394-Term \u0393 \u03c4\u2081 {vars})\n    (R[t,dt] : {\u03c1 : \u0394-Env \u0393 {-vars-}} \u2192\n      valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1)) \u2192\n    \u0394-Term \u0393 \u03c4\u2082 {vars}\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393 vars}\n    (ds : \u0394-Term \u0393 nats {vars})\n    (dt : \u0394-Term \u0393 nats {vars}) \u2192\n    \u0394-Term \u0393 nats {vars}\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats))\n    (df : \u0394-Term \u0393 (nats \u21d2 nats) {vars})\n    ( b :   Term \u0393 bags)\n    (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n  \u0394map\u2081 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats)) (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n\n\u27e6 \u0394 nats \u27e7\u0394Type = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394Type = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) {alter args} \u27e7\u0394Type =\n  \u2200 {args\u2081} \u2192\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 {args\u2081} \u27e7\u0394Type) \u2192 valid v dv \u2192\n  \u27e6 \u0394 \u03c4\u2082 {args} \u27e7\u0394Type\n\nmeaning-\u0394Type : Meaning\u0394 Type\nmeaning-\u0394Type = meaning\u0394 \u27e6_\u27e7\u0394Type\n\nrecord Meaning\u0394\n  (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n    constructor\n      meaning\u0394\n    field\n      {\u0394-Semantics} : Set \u2113\n      \u27e8_\u27e9\u27e6_\u27e7\u0394 : Syntax \u2192 \u0394-Semantics\n\nopen Meaning\u0394 {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7\u0394 to \u27e6_\u27e7\u0394)\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u03c4\u2081 \u27e7\u0394Type) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency and honesty.\n-- \u0394-Env : (\u0393 : Context) \u2192 {vars : Vars \u0393} \u2192 Set\ndata \u0394-Env where\n  \u2205 : \u0394-Env \u2205 {\u2205}\n  abide : \u2200 {\u03c4 \u0393 vars} \u2192\n    (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u03c4 \u27e7\u0394) \u2192 valid v dv \u2192 v \u2295 dv \u2261 v \u2192\n    \u0394-Env \u0393 {vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {abide vars}\n  alter : \u2200 {\u03c4 \u0393 vars} \u2192\n    (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u03c4 \u27e7\u0394) \u2192 valid v dv \u2192\n    \u0394-Env \u0393 {vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {alter vars}\n\n\u2016_\u2016 : \u2200 {\u03c4 \u0393 vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {vars} \u2192\n  Quadruple \u27e6 \u03c4 \u27e7 (\u03bb _ \u2192 \u27e6 \u03c4 \u27e7\u0394) (\u03bb v dv \u2192 valid v dv)\n    (\u03bb _ _ _ \u2192 \u0394-Env \u0393 {cdr vars})\n\n\u2016 abide v dv R[v,dv] _ \u03c1 \u2016 = cons v dv R[v,dv] \u03c1\n\u2016 alter v dv R[v,dv]   \u03c1 \u2016 = cons v dv R[v,dv] \u03c1\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393 vars} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6 this   \u27e7\u0394Var \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\nmeaning-\u0394Var : \u2200 {\u03c4 \u0393} {vars : Vars \u0393} \u2192 Meaning\u0394 (Var \u0393 \u03c4)\nmeaning-\u0394Var {\u03c4} {\u0393} {vars} = meaning\u0394 (\u27e6_\u27e7\u0394Var {\u03c4} {\u0393} {vars})\n\n-- \u27e6_\u27e7\u0394Term : \u2200 {\u03c4 \u0393 vars} \u2192\n--   \u0394-Term \u0393 \u03c4 {vars} \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6 \u0394nat old new \u27e7\u0394Term \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394Term \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394Term \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv]          \u2192 \u27e6 t \u27e7\u0394Term (alter v dv R[v,dv] \u03c1)\n\u27e6 \u0394abs\u2080 t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv] {-v\u2295dv=v-} \u2192 \u27e6 t \u27e7\u0394Term (abide v dv R[v,dv] {!v\u2295dv=v!} \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394Term \u03c1 =\n  \u27e6 ds \u27e7\u0394Term \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1) R[dt,t]\n  --(R[dt,t] {\u03c1} {honesty = {!!}})\n\u27e6 \u0394add ds dt \u27e7\u0394Term \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394Term \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394Term \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394Term \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394Term \u03c1\n    dh = \u27e6 df \u27e7\u0394Term \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394Term \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394Term \u03c1)\n\nmeaning-\u0394Term-0 : \u2200 {\u03c4 \u0393 vars} \u2192 Meaning (\u0394-Term \u0393 \u03c4 {vars})\nmeaning-\u0394Term-0 = meaning \u27e6_\u27e7\u0394Term\n\nmeaning-\u0394Term-1 : \u2200 {\u03c4 \u0393 vars} \u2192 Meaning\u0394 (\u0394-Term \u0393 \u03c4 {vars})\nmeaning-\u0394Term-1 = meaning\u0394 \u27e6_\u27e7\u0394Term\n\n{-\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7 \u03c1\n    db = \u27e6 derive t \u27e7 \u03c1\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394-Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7 \u03c1\u2032\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n-}\n-}\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag renaming\n--   (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {b : Bag} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\n-- Descriptions of whether free variables or future arguments\n-- are expected to change.\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nfickle-args : {\u03c4 : Type} \u2192 Args \u03c4\nfickle-args {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter fickle-args\nfickle-args {nats} = \u2205-nat\nfickle-args {bags} = \u2205-bag\n\nfickle-vars : {\u0393 : Context} \u2192 Vars \u0393\nfickle-vars {\u2205} = \u2205\nfickle-vars {\u03c4 \u2022 \u0393} = alter fickle-vars\n\ncdr : \u2200 {\u03c4 \u0393} \u2192 Vars (\u03c4 \u2022 \u0393) \u2192 Vars \u0393\ncdr (abide vars) = vars\ncdr (alter vars) = vars\n\n{-\n\ndata \u0394-Type : (\u03c4 : Type) \u2192 {args : Args \u03c4} \u2192 Set where\n  \u0394 : (\u03c4 : Type) \u2192 {args : Args \u03c4} \u2192 \u0394-Type \u03c4 {args}\n\n\u27e6_\u27e7\u0394Type : \u2200 {\u03c4 args} \u2192 \u0394-Type \u03c4 {args} \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394Type = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394Type = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) {alter args} \u27e7\u0394Type =\n  \u2200 {args\u2081} \u2192\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 {args\u2081} \u27e7\u0394Type) \u2192 valid v dv \u2192\n  \u27e6 \u0394 \u03c4\u2082 {args} \u27e7\u0394Type\n\nmeaning-\u0394Type : Meaning\u0394 Type\nmeaning-\u0394Type = meaning\u0394 \u27e6_\u27e7\u0394Type\n\n\ndata \u0394-Env : (\u0393 : Context) \u2192 {vars : Vars \u0393} \u2192 Set\ndata \u0394-Term : (\u0393 : Context) \u2192 Type \u2192 {vars : Vars \u0393} \u2192 Set\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u03c4 : Type} {\u0393 : Context} {vars}\n    \u2192 \u0394-Term \u0393 \u03c4 {vars} \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} {vars} \u2192 (\u03c1 : \u0394-Env \u0393 {vars}) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} {vars} \u2192 (\u03c1 : \u0394-Env \u0393 {vars}) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393 vars} \u2192\n    (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term \u0393 nats {vars}\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393 vars} \u2192\n    (db : Bag) \u2192 \u0394-Term \u0393 bags {vars}\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393 vars} \u2192\n    (x : Var \u0393 \u03c4) \u2192 \u0394-Term \u0393 \u03c4 {vars}\n  -- changes to abstractions are binders of x and dx\n  -- There are two kinds of those: One who expects the argument\n  -- to change, and one who does not.\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} \u2192 (t : \u0394-Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082 {alter vars}) \u2192\n         \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars}\n  \u0394abs\u2080 : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} \u2192 (t : \u0394-Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082 {abide vars}) \u2192\n          \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars}\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars}\n    (ds : \u0394-Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082) {vars})\n    ( t :   Term \u0393 \u03c4\u2081)\n    (dt : \u0394-Term \u0393 \u03c4\u2081 {vars})\n    (R[t,dt] : {\u03c1 : \u0394-Env \u0393 {-vars-}} \u2192\n      valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1)) \u2192\n    \u0394-Term \u0393 \u03c4\u2082 {vars}\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393 vars}\n    (ds : \u0394-Term \u0393 nats {vars})\n    (dt : \u0394-Term \u0393 nats {vars}) \u2192\n    \u0394-Term \u0393 nats {vars}\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats))\n    (df : \u0394-Term \u0393 (nats \u21d2 nats) {vars})\n    ( b :   Term \u0393 bags)\n    (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n  \u0394map\u2081 : \u2200 {\u0393 vars} \u2192\n    ( f :   Term \u0393 (nats \u21d2 nats)) (db : \u0394-Term \u0393 bags {vars}) \u2192\n    \u0394-Term \u0393 bags {vars}\n\nrecord Meaning\u0394\n  (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n    constructor\n      meaning\u0394\n    field\n      {\u0394-Semantics} : Set \u2113\n      \u27e8_\u27e9\u27e6_\u27e7\u0394 : Syntax \u2192 \u0394-Semantics\n\nopen Meaning\u0394 {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7\u0394 to \u27e6_\u27e7\u0394)\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\u0394Type \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u03c4\u2081 \u27e7\u0394Type) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency and honesty.\n-- \u0394-Env : (\u0393 : Context) \u2192 {vars : Vars \u0393} \u2192 Set\ndata \u0394-Env where\n  \u2205 : \u0394-Env \u2205 {\u2205}\n  abide : \u2200 {\u03c4 \u0393 vars} \u2192\n    (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u03c4 \u27e7\u0394) \u2192 valid v dv \u2192 v \u2295 dv \u2261 v \u2192\n    \u0394-Env \u0393 {vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {abide vars}\n  alter : \u2200 {\u03c4 \u0393 vars} \u2192\n    (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u03c4 \u27e7\u0394) \u2192 valid v dv \u2192\n    \u0394-Env \u0393 {vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {alter vars}\n\n\u2016_\u2016 : \u2200 {\u03c4 \u0393 vars} \u2192 \u0394-Env (\u03c4 \u2022 \u0393) {vars} \u2192\n  Quadruple \u27e6 \u03c4 \u27e7 (\u03bb _ \u2192 \u27e6 \u03c4 \u27e7\u0394) (\u03bb v dv \u2192 valid v dv)\n    (\u03bb _ _ _ \u2192 \u0394-Env \u0393 {cdr vars})\n\n\u2016 abide v dv R[v,dv] _ \u03c1 \u2016 = cons v dv R[v,dv] \u03c1\n\u2016 alter v dv R[v,dv]   \u03c1 \u2016 = cons v dv R[v,dv] \u03c1\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393 vars} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6 this   \u27e7\u0394Var \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var \u03c1\u2032 with \u2016 \u03c1\u2032 \u2016\n... | (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\nmeaning-\u0394Var : \u2200 {\u03c4 \u0393} {vars : Vars \u0393} \u2192 Meaning\u0394 (Var \u0393 \u03c4)\nmeaning-\u0394Var {\u03c4} {\u0393} {vars} = meaning\u0394 (\u27e6_\u27e7\u0394Var {\u03c4} {\u0393} {vars})\n\n-- \u27e6_\u27e7\u0394Term : \u2200 {\u03c4 \u0393 vars} \u2192\n--   \u0394-Term \u0393 \u03c4 {vars} \u2192 \u0394-Env \u0393 {vars} \u2192 \u27e6 \u03c4 \u27e7\u0394Type\n\u27e6 \u0394nat old new \u27e7\u0394Term \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394Term \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394Term \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv]          \u2192 \u27e6 t \u27e7\u0394Term (alter v dv R[v,dv] \u03c1)\n\u27e6 \u0394abs\u2080 t \u27e7\u0394Term \u03c1 =\n  \u03bb v dv R[v,dv] {-v\u2295dv=v-} \u2192 \u27e6 t \u27e7\u0394Term (abide v dv R[v,dv] {!v\u2295dv=v!} \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394Term \u03c1 =\n  \u27e6 ds \u27e7\u0394Term \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394Term \u03c1) R[dt,t]\n  --(R[dt,t] {\u03c1} {honesty = {!!}})\n\u27e6 \u0394add ds dt \u27e7\u0394Term \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394Term \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394Term \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394Term \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394Term \u03c1\n    dh = \u27e6 df \u27e7\u0394Term \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394Term \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394Term \u03c1)\n\nmeaning-\u0394Term-0 : \u2200 {\u03c4 \u0393 vars} \u2192 Meaning (\u0394-Term \u0393 \u03c4 {vars})\nmeaning-\u0394Term-0 = meaning \u27e6_\u27e7\u0394Term\n\nmeaning-\u0394Term-1 : \u2200 {\u03c4 \u0393 vars} \u2192 Meaning\u0394 (\u0394-Term \u0393 \u03c4 {vars})\nmeaning-\u0394Term-1 = meaning\u0394 \u27e6_\u27e7\u0394Term\n\n{-\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term \u0393 \u03c4\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7 \u03c1\n    db = \u27e6 derive t \u27e7 \u03c1\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394-Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7 \u03c1\u2032\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n-}\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e1966728884ba74a349c4c3d76eaa915eab0d670","subject":"Minor change.","message":"Minor change.\n\nIgnore-this: ab8dfcfec3f2a04fc460edbac7099718\n\ndarcs-hash:20120223161424-3bd4e-6baacd366d268a3dbbc2fc1c856157257f5252a3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Axiomatic\/Standard\/Base.agda","new_file":"src\/PA\/Axiomatic\/Standard\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Standard.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe.\n-- We chose the symbol M because there are non-standard models of\n-- Peano Arithmetic, where the domain is not the set of natural\n-- numbers.\nopen import FOL.Universe public renaming ( D to M )\n\n-- FOL with equality.\nopen import FOL.FOL-Eq public\n\n-- Non-logical constants\npostulate\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- Proper axioms (see [Machover, 1977, p. 263], [H\u00e1jeck and Pudl\u00e1k,\n  1998, p. 28])\n--\n-- * Mosh\u00e9 Machover. Set theory, logic and their\n--   limitations. Cambridge University Press, 1996.\n--\n-- * Petr H\u00e1jeck and Pavel Pudl\u00e1k. Metamathematics of First-Order\n--   Arithmetic. Springer, 1998. 2nd printing.\n\n-- A\u2081. 0 \u2260 succ n\n-- A\u2082. succ m = succ n \u2192 m = n\n-- A\u2083. 0 + n = n\n-- A\u2084. succ m + n = succ (m + n)\n-- A\u2085. 0 * n = 0\n-- A\u2086. succ m * n = (m * n) + n\n-- A\u2087. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  A\u2081 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\n  A\u2082 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n  A\u2083 : \u2200 n \u2192 zero + n \u2261 n\n  A\u2084 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  A\u2085 : \u2200 n \u2192 zero * n \u2261 zero\n  A\u2086 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n{-# ATP axiom A\u2081 A\u2082 A\u2083 A\u2084 A\u2085 A\u2086 #-}\n\n-- A\u2087 is an axiom schema, therefore we do not translate it to TPTP.\npostulate A\u2087 : (P : M \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","old_contents":"------------------------------------------------------------------------------\n-- Axiomatic Peano arithmetic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule PA.Axiomatic.Standard.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe.\n-- We chose the symbol M because there are non-standard models of\n-- Peano Arithmetic, where the domain is not the set of natural\n-- numbers.\nopen import FOL.Universe public renaming ( D to M )\n\n-- FOL with equality.\nopen import FOL.FOL-Eq public\n\n-- Non-logical constants\npostulate\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- Proper axioms (see [p. 263, 1], [p. 28, 2])\n--\n-- [1] Mosh\u00e9 Machover. Set theory, logic and their\n--     limitations. Cambridge University Press, 1996.\n--\n-- [2] Petr H\u00e1jeck and Pavel Pudl\u00e1k. Metamathematics of First-Order\n--     Arithmetic. Springer, 1998. 2nd printing.\n\n-- A\u2081. 0 \u2260 succ n\n-- A\u2082. succ m = succ n \u2192 m = n\n-- A\u2083. 0 + n = n\n-- A\u2084. succ m + n = succ (m + n)\n-- A\u2085. 0 * n = 0\n-- A\u2086. succ m * n = (m * n) + n\n-- A\u2087. P(0) \u2192 (\u2200n.P(n) \u2192 P(succ n)) \u2192 \u2200n.P(n), for any wff P(n) of PA.\n\npostulate\n  A\u2081 : \u2200 {n} \u2192 \u00ac (zero \u2261 succ n)\n  A\u2082 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n  A\u2083 : \u2200 n \u2192 zero + n \u2261 n\n  A\u2084 : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  A\u2085 : \u2200 n \u2192 zero * n \u2261 zero\n  A\u2086 : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n{-# ATP axiom A\u2081 A\u2082 A\u2083 A\u2084 A\u2085 A\u2086 #-}\n\n-- A\u2087 is an axiom schema, therefore we do not translate it to TPTP.\npostulate A\u2087 : (P : M \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"da3a75e49d5db615e779ce11158026cc7a7d5fb6","subject":"Bright idea","message":"Bright idea\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n--------------------\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n--------------------\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218 = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n--------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\nmodusponens : \u2115 \u2192 Arrow \u2192 Arrow\nmodusponens n (\u21d2 q)   = (\u21d2 q)\nmodusponens n (p \u21d2 q) with (n \u2261 p)\n...                      | true  = modusponens n q\n...                      | false = p \u21d2 (modusponens n q)\n\n\nreduce : \u2115 \u2192 List Arrow \u2192 List Arrow\nreduce n \u2218 = \u2218\nreduce n (arr \u2237 rst) = (modusponens n arr) \u2237 (reduce n rst)\n\n\nsearch : List Arrow \u2192 List \u2115\nsearch \u2218 = \u2218\nsearch ((\u21d2 n) \u2237 rst)   = n \u2237 (search (reduce n rst))\nsearch ((n \u21d2 q) \u2237 rst) with (n \u2208 (search rst))\n...                       | true  = search (q \u2237 rst)\n...                       | false = search rst\n\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n--------------------\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n--------------------\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218 = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n--------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_trivialIn_ : \u2115 \u2192 List Arrow \u2192 Bool\nn trivialIn \u2218 = false\nn trivialIn ((\u21d2 m) \u2237 cs) with n \u2261 m\n...                         | true  = true\n...                         | false = n trivialIn cs\nn trivialIn (_ \u2237 cs) = n trivialIn cs\n\n--------------------\n\nSimplify : Arrow \u2192 List \u2115 \u2192 Arrow\nSimplify (\u21d2 q) _ = \u21d2 q\nSimplify (p \u21d2 q) cs with p \u2208 cs\n...                    | true  = Simplify q cs\n...                    | false = p \u21d2 (Simplify q cs)\n\n\nClosure\u2081 : List Arrow \u2192 List Arrow \u2192 List Arrow\nClosure\u2081 ((p \u21d2 q) \u2237 cs) ds with (p trivialIn cs)\n...                           | true  = Closure\u2082 \n\n\nClosure : List Arrow \u2192 List Arrow\nClosure cs = Closure\u2081 cs \u2218\n\n\n--_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\n--cs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n--cs \u22a2 (\u21d2 q) = false\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"02baa476d130e50ce8bf3d227e5685db5ae9c929","subject":"README","message":"README\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"67c4403a44eaa990cd27ff950ca7232f5ba48405","subject":"fussing with variable names","message":"fussing with variable names\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (_ , Step (FHFinal _) () (FHFinal _))\n  vs VConst (_ , Step (FHFinal _) () FHEHole)\n  vs VConst (_ , Step (FHFinal _) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal _) () (FHNEHoleFinal _))\n  vs VConst (_ , Step (FHFinal _) () (FHCastFinal _))\n  vs VLam (_ , Step (FHFinal _) () (FHFinal _))\n  vs VLam (_ , Step (FHFinal _) () FHEHole)\n  vs VLam (_ , Step (FHFinal _) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal _) () (FHNEHoleFinal _))\n  vs VLam (_ , Step (FHFinal _) () (FHCastFinal _))\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    ie IEHole ()\n    ie (INEHole x) (ENEHole e) = fe x e\n    ie (IAp i x) (EAp1 e) = ie i e\n    ie (IAp i x) (EAp2 e) = fe x e\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    fe (FVal x) er = ve x er\n    fe (FIndet x) er = ie x er\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole _) ()\n  lem2 (IAp () _) (ITLam _)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal _))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal _))\n  is IEHole (_ , Step FHEHole () (FHCastFinal _))\n  is (INEHole _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole _) (_ , Step FHNEHoleEvaled () (FHFinal _))\n  is (INEHole _) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole _) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole _) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal _))\n  is (INEHole _) (_ , Step FHNEHoleEvaled () (FHCastFinal _))\n  is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (d' , Step FHNEHoleEvaled () x\u2082)\n  es (ENEHole er) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i (FVal x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = vs x (_ , Step p q r)\n  is (IAp i (FIndet x)) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n\n\n  -- final expressions are not errors (not one of the 6 cases for progress)\n  fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  fe (FVal x) er = ve x er\n  fe (FIndet x) er = ie x er\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (d' , Step FHNEHoleEvaled () x\u2082)\n  es (ENEHole er) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a142fb0c90cbca00caedff3475c5f7fd1655731e","subject":"Data.Bits: search, #\u27e8\u27e9\u1da0, #\u27e8\u27e9, find? and findB","message":"Data.Bits: search, #\u27e8\u27e9\u1da0, #\u27e8\u27e9, find? and findB\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {zero}  _   f = f []\nsearch {suc n} _\u00b7_ f = search {n} _\u00b7_ (f \u2218 0\u2237_) \u00b7 search {n} _\u00b7_ (f \u2218 1\u2237_)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = search _+_ (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Maybe (Bits n)\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cbfc74728f14529427faece77d2f178f648acf57","subject":"FunctionSorting","message":"FunctionSorting\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bits hiding (replicate)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Data.Sum using (_\u228e_; inj\u2081; inj\u2082)\nopen Alternative-Reverse\nopen import Relation.Binary\nimport Level as L\nopen import Data.Bool\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\ndata _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n  here  : x \u2208 leaf x\n  left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n  right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x)   = leaf (f x)\nmap f (fork t u) = fork (map f t) (map f u)\n\nprivate\n  module Dummy {a} {A : Set a} where\n    replicate : \u2200 n \u2192 A \u2192 Tree A n\n    replicate zero    x = leaf x\n    replicate (suc n) x = fork t t where t = replicate n x\n\n    fromFun : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Tree A n\n    fromFun {zero}  f = leaf (f [])\n    fromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\n    toFun : \u2200 {n} \u2192 Tree A n \u2192 Bits n \u2192 A\n    toFun (leaf x)   _        = x\n    toFun (fork t u) (b \u2237 bs) = toFun (if b then u else t) bs\n\n    toFun\u2218fromFun : \u2200 {n} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\n    toFun\u2218fromFun {zero}  f [] = \u2261.refl\n    toFun\u2218fromFun {suc n} f (false \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\n    toFun\u2218fromFun {suc n} f (true \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\n    fromFun\u2218toFun : \u2200 {n} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\n    fromFun\u2218toFun (leaf x) = \u2261.refl\n    fromFun\u2218toFun (fork t\u2080 t\u2081)\n      rewrite fromFun\u2218toFun t\u2080\n            | fromFun\u2218toFun t\u2081 = \u2261.refl\n\n    toFun\u2192fromFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\n    toFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\n    toFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n      rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n            | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\n    fromFun\u2192toFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\n    fromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\n    fromFun-\u2257 : \u2200 {n} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\n    fromFun-\u2257 {zero}  f\u2257g\n      rewrite f\u2257g [] = \u2261.refl\n    fromFun-\u2257 {suc n} f\u2257g\n      rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n            | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n            = \u2261.refl\n\n    lookup : \u2200 {n} \u2192 Bits n \u2192 Tree A n \u2192 A\n    lookup = flip toFun\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\n\n    first : \u2200 {n} \u2192 Tree A n \u2192 A\n    first (leaf x)   = x\n    first (fork t _) = first t\n\n    last : \u2200 {n} \u2192 Tree A n \u2192 A\n    last (leaf x)   = x\n    last (fork _ t) = last t\n\n    -- Returns the flat vector of leaves underlying the perfect binary tree.\n    toVec : \u2200 {n} \u2192 Tree A n \u2192 Vec A (2^ n)\n    toVec (leaf x)     = x \u2237 []\n    toVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\n    lookup' : \u2200 {m n} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\n    lookup' [] t = t\n    lookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n    update' : \u2200 {m n} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\n    update' []        val t = val\n    update' (b \u2237 key) val (fork t u) = if b then fork t (update' key val u)\n                                            else fork (update' key val t) u\n\n    \u2208-toBits : \u2200 {x n} {t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n    \u2208-toBits here = []\n    \u2208-toBits (left key) = 0b \u2237 \u2208-toBits key\n    \u2208-toBits (right key) = 1b \u2237 \u2208-toBits key\n\n    \u2208-lookup : \u2200 {x n} {t : Tree A n} (path : x \u2208 t) \u2192 lookup (\u2208-toBits path) t \u2261 x\n    \u2208-lookup here = \u2261.refl\n    \u2208-lookup (left path) = \u2208-lookup path\n    \u2208-lookup (right path) = \u2208-lookup path\n\n    lookup-\u2208 : \u2200 {n} key (t : Tree A n) \u2192 lookup key t \u2208 t\n    lookup-\u2208 [] (leaf x) = here\n    lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n    lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\nopen Dummy public\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n = n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                              (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = \u2208-toBits p \u2264\u1d2e \u2208-toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                                (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting-\u2293-\u2294 {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x , y) = (x \u2293\u1d2c y , x \u2294\u1d2c y)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc _} = merge \u2218 map-outer sort sort\n\nmodule \u2293-\u2294\u1d2c {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n    _\u2293\u1d2c_ : A \u2192 A \u2192 A\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n    _\u2294\u1d2c_ : A \u2192 A \u2192 A\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\nmodule Sorting-<= {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n    open \u2293-\u2294\u1d2c _<=\u1d2c_\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_ public\n\nmodule EvalTree {a} {A : Set a} where\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {n} \u2192 Bij n \u2192 Endo (Tree A n)\n    evalTree `id          = id\n    evalTree (f `\u204f g)      = evalTree g \u2218 evalTree f\n    evalTree (`id   `\u2237 f) = map-outer (evalTree (f 0b)) (evalTree (f 1b))\n    evalTree (`not\u1d2e `\u2237 f) = map-outer (evalTree (f 1b)) (evalTree (f 0b)) \u2218 swap\n    evalTree `0\u21941         = interchange\n\n    evalTree-eval : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id  _ _ = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval (f `\u204f g) t xs\n      rewrite evalTree-eval f t xs\n            | evalTree-eval g (evalTree f t) (eval f xs)\n            = \u2261.refl\n    evalTree-eval (`id   `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id   `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule BijSpec {a} {A : Set a} where\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n\n    record Bij[_\u21a6_] {n} (t u : Tree A n) : Set a where\n      constructor _,_\n      field\n        bij   : Bij n\n        proof : evalTree bij t \u2261 u\n    open Bij[_\u21a6_] public\n\n    Bij[\u2257_] : \u2200 {n} (f : Endo (Tree A n)) \u2192 Set a\n    Bij[\u2257 f ] = \u2200 t \u2192 Bij[ t \u21a6 f t ]\n\n    evalB : \u2200 {n} {f : Endo (Tree A n)} (b : Bij[\u2257 f ]) \u2192 Endo (Tree A n)\n    evalB b t = evalTree (bij (b t)) t\n\n    bij-evalB-spec : \u2200 {n} {f : Endo (Tree A n)} (b : Bij[\u2257 f ]) \u2192 evalB b \u2257 f\n    bij-evalB-spec b = proof \u2218 b\n\n    id-bij : \u2200 {n} \u2192 Bij[\u2257 id {A = Tree A n} ]\n    id-bij _ = `id , \u2261.refl\n\n    infixr 9 _\u2218-bij_\n    _\u2218-bij_ : \u2200 {n} {f g : Endo (Tree A n)} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 f \u2218 g ]\n    _\u2218-bij_ {f = f} {g} `f `g t\n      = `bij , helper\n      where `bij = bij (`g t) `\u204f bij (`f (g t))\n            helper : evalTree `bij t \u2261 f (g t)\n            helper rewrite proof (`g t) = proof (`f (g t))\n\n    swap-bij : \u2200 {n} \u2192 Bij[\u2257 swap {n = n} ]\n    swap-bij (fork _ _) = `not , \u2261.refl\n\n    map-outer-bij : \u2200 {n} {f g : Endo (Tree A n)}\n                      \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 map-outer f g ]\n    map-outer-bij `f `g (fork t u)\n      = `map-outer (bij (`f t)) (bij (`g u))\n      , \u2261.cong\u2082 fork (proof (`f t)) (proof (`g u))\n\n    map-inner-bij : \u2200 {n} {f : Endo (Tree A (1 + n))} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 map-inner f ]\n    map-inner-bij {f = f} `f t = map-inner-perm , helper\n       where map-inner-perm = `map-inner (bij (`f (inner t)))\n             helper : evalTree map-inner-perm t \u2261 map-inner f t\n             helper with t\n             ... | fork (fork a b) (fork c d) with f (fork b c) | proof (`f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\n    interchange-bij : \u2200 {n} \u2192 Bij[\u2257 interchange {n = n} ]\n    interchange-bij = map-inner-bij swap-bij\n\nmodule SortingBijSpec {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool)\n                      (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    open Sorting-<= _<=\u1d2c_\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open BijSpec\n\n    `sort\u2081 : Tree A 1 \u2192 Bij 1\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    sort\u2081-bij : Bij[\u2257 sort\u2081 ]\n    sort\u2081-bij t = `sort\u2081 t , helper t\n      where helper : \u2200 t \u2192 evalTree (`sort\u2081 t) t \u2261 sort\u2081 t\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-bij : \u2200 {n} \u2192 Bij[\u2257 merge {n} ]\n    merge-bij {zero}  = sort\u2081-bij\n    merge-bij {suc _} = map-inner-bij merge-bij\n                  \u2218-bij map-outer-bij merge-bij merge-bij\n                  \u2218-bij interchange-bij\n\n    sort-bij : \u2200 {n} \u2192 Bij[\u2257 sort {n} ]\n    sort-bij {zero}  = id-bij\n    sort-bij {suc _} = merge-bij \u2218-bij map-outer-bij sort-bij sort-bij\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule FunctionSorting {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n  _\u2264\u1d2c_ = \u03bb x y \u2192 T (x <=\u1d2c y)\n  open \u2293-\u2294\u1d2c _<=\u1d2c_\n  module S = Sorting-<= _<=\u1d2c_\n  open BijSpec\n\n  sort : \u2200 {n} \u2192 Endo (Bits n \u2192 A)\n  sort = toFun \u2218 S.sort \u2218 fromFun\n\n  module Properties (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n                    (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                    (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                    (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                    (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                    (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                    (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                    (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                    (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z) where\n\n    module B = SortingBijSpec _<=\u1d2c_  isTotalOrder\n    open IsTotalOrder isTotalOrder\n    open SortedMonotonicFunctions _\u2264\u1d2c_ isPreorder\n    module SP = SortingProperties _\u2264\u1d2c_ _\u2293\u1d2c_ _\u2294\u1d2c_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open OperationSyntax using (eval)\n    open EvalTree\n\n    sort-is-sorting : \u2200 {n} (f : Bits n \u2192 A) \u2192 Monotone (sort f)\n    sort-is-sorting = DataSorted\u2192Sorted \u2218 SP.sort-spec \u2218 fromFun\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting-\u2293-\u2294 _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    open SortingBijSpec _<=_ isTotalOrder public\n    open EvalTree public using (evalTree)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bits hiding (replicate)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Data.Sum using (_\u228e_; inj\u2081; inj\u2082)\nopen Alternative-Reverse\nopen import Relation.Binary\nimport Level as L\nopen import Data.Bool\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\ndata _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n  here  : x \u2208 leaf x\n  left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n  right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x)   = leaf (f x)\nmap f (fork t u) = fork (map f t) (map f u)\n\nprivate\n  module Dummy {a} {A : Set a} where\n    replicate : \u2200 n \u2192 A \u2192 Tree A n\n    replicate zero    x = leaf x\n    replicate (suc n) x = fork t t where t = replicate n x\n\n    fromFun : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Tree A n\n    fromFun {zero}  f = leaf (f [])\n    fromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\n    toFun : \u2200 {n} \u2192 Tree A n \u2192 Bits n \u2192 A\n    toFun (leaf x)   _        = x\n    toFun (fork t u) (b \u2237 bs) = toFun (if b then u else t) bs\n\n    toFun\u2218fromFun : \u2200 {n} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\n    toFun\u2218fromFun {zero}  f [] = \u2261.refl\n    toFun\u2218fromFun {suc n} f (false \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\n    toFun\u2218fromFun {suc n} f (true \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\n    fromFun\u2218toFun : \u2200 {n} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\n    fromFun\u2218toFun (leaf x) = \u2261.refl\n    fromFun\u2218toFun (fork t\u2080 t\u2081)\n      rewrite fromFun\u2218toFun t\u2080\n            | fromFun\u2218toFun t\u2081 = \u2261.refl\n\n    toFun\u2192fromFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\n    toFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\n    toFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n      rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n            | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\n    fromFun\u2192toFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\n    fromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\n    fromFun-\u2257 : \u2200 {n} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\n    fromFun-\u2257 {zero}  f\u2257g\n      rewrite f\u2257g [] = \u2261.refl\n    fromFun-\u2257 {suc n} f\u2257g\n      rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n            | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n            = \u2261.refl\n\n    lookup : \u2200 {n} \u2192 Bits n \u2192 Tree A n \u2192 A\n    lookup = flip toFun\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\n\n    first : \u2200 {n} \u2192 Tree A n \u2192 A\n    first (leaf x)   = x\n    first (fork t _) = first t\n\n    last : \u2200 {n} \u2192 Tree A n \u2192 A\n    last (leaf x)   = x\n    last (fork _ t) = last t\n\n    -- Returns the flat vector of leaves underlying the perfect binary tree.\n    toVec : \u2200 {n} \u2192 Tree A n \u2192 Vec A (2^ n)\n    toVec (leaf x)     = x \u2237 []\n    toVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\n    lookup' : \u2200 {m n} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\n    lookup' [] t = t\n    lookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n    update' : \u2200 {m n} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\n    update' []        val t = val\n    update' (b \u2237 key) val (fork t u) = if b then fork t (update' key val u)\n                                            else fork (update' key val t) u\n\n    \u2208-toBits : \u2200 {x n} {t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n    \u2208-toBits here = []\n    \u2208-toBits (left key) = 0b \u2237 \u2208-toBits key\n    \u2208-toBits (right key) = 1b \u2237 \u2208-toBits key\n\n    \u2208-lookup : \u2200 {x n} {t : Tree A n} (path : x \u2208 t) \u2192 lookup (\u2208-toBits path) t \u2261 x\n    \u2208-lookup here = \u2261.refl\n    \u2208-lookup (left path) = \u2208-lookup path\n    \u2208-lookup (right path) = \u2208-lookup path\n\n    lookup-\u2208 : \u2200 {n} key (t : Tree A n) \u2192 lookup key t \u2208 t\n    lookup-\u2208 [] (leaf x) = here\n    lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n    lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\nopen Dummy public\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n = n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                              (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = \u2208-toBits p \u2264\u1d2e \u2208-toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                                (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting-\u2293-\u2294 {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x , y) = (x \u2293\u1d2c y , x \u2294\u1d2c y)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc _} = merge \u2218 map-outer sort sort\n\nmodule Sorting-<= {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n\n    _\u2293\u1d2c_ : A \u2192 A \u2192 A\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n\n    _\u2294\u1d2c_ : A \u2192 A \u2192 A\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_ public\n\nmodule EvalTree {a} {A : Set a} where\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {n} \u2192 Bij n \u2192 Endo (Tree A n)\n    evalTree `id          = id\n    evalTree (f `\u204f g)      = evalTree g \u2218 evalTree f\n    evalTree (`id   `\u2237 f) = map-outer (evalTree (f 0b)) (evalTree (f 1b))\n    evalTree (`not\u1d2e `\u2237 f) = map-outer (evalTree (f 1b)) (evalTree (f 0b)) \u2218 swap\n    evalTree `0\u21941         = interchange\n\n    evalTree-eval : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id  _ _ = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval (f `\u204f g) t xs\n      rewrite evalTree-eval f t xs\n            | evalTree-eval g (evalTree f t) (eval f xs)\n            = \u2261.refl\n    evalTree-eval (`id   `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id   `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule BijSpec {a} {A : Set a} where\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n\n    record Bij[_\u21a6_] {n} (t u : Tree A n) : Set a where\n      constructor _,_\n      field\n        bij   : Bij n\n        proof : t \u2261 evalTree bij u\n    open Bij[_\u21a6_] public\n\n    Bij[\u2257_] : \u2200 {n} (f : Endo (Tree A n)) \u2192 Set a\n    Bij[\u2257 f ] = \u2200 t \u2192 Bij[ t \u21a6 f t ]\n\n    id-bij : \u2200 {n} \u2192 Bij[\u2257 id {A = Tree A n} ]\n    id-bij _ = `id , \u2261.refl\n\n    infixr 9 _\u2218-bij_\n    _\u2218-bij_ : \u2200 {n} {f g : Endo (Tree A n)} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 f \u2218 g ]\n    _\u2218-bij_ {f = f} {g} `f `g t\n      = (bij (`f (g t)) `\u204f bij (`g t))\n      , \u2261.trans (proof (`g t)) (\u2261.cong (evalTree (bij (`g t))) (proof (`f (g t))))\n\n    swap-bij : \u2200 {n} \u2192 Bij[\u2257 swap {n = n} ]\n    swap-bij (fork _ _) = `not , \u2261.refl\n\n    map-outer-bij : \u2200 {n} {f g : Endo (Tree A n)}\n                      \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 map-outer f g ]\n    map-outer-bij `f `g (fork t u)\n      = `map-outer (bij (`f t)) (bij (`g u))\n      , \u2261.cong\u2082 fork (proof (`f t)) (proof (`g u))\n\n    map-inner-bij : \u2200 {n} {f : Endo (Tree A (1 + n))} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 map-inner f ]\n    map-inner-bij {f = f} `f t = map-inner-perm , helper\n       where map-inner-perm = `map-inner (bij (`f (inner t)))\n             helper : t \u2261 evalTree map-inner-perm (map-inner f t)\n             helper with t\n             ... | fork (fork a b) (fork c d) with f (fork b c) | \u2261.sym (proof (`f (fork b c)))\n             ... | fork B C | p rewrite p = \u2261.refl\n\n    interchange-bij : \u2200 {n} \u2192 Bij[\u2257 interchange {n = n} ]\n    interchange-bij = map-inner-bij swap-bij\n\nmodule Sorting-Perm-Properties {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool)\n                               (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    open Sorting-<= _<=\u1d2c_\n    open EvalTree\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open BijSpec\n\n    `sort\u2081 : Tree A 1 \u2192 Bij 1\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    sort\u2081-bij : Bij[\u2257 sort\u2081 ]\n    sort\u2081-bij t = `sort\u2081 t , helper t\n      where helper : \u2200 t \u2192 t \u2261 evalTree (`sort\u2081 t) (sort\u2081 t)\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-bij : \u2200 {n} \u2192 Bij[\u2257 merge {n} ]\n    merge-bij {zero}  = sort\u2081-bij\n    merge-bij {suc _} = map-inner-bij merge-bij\n                  \u2218-bij map-outer-bij merge-bij merge-bij\n                  \u2218-bij interchange-bij\n\n    sort-bij : \u2200 {n} \u2192 Bij[\u2257 sort {n} ]\n    sort-bij {zero}  = id-bij\n    sort-bij {suc _} = merge-bij \u2218-bij map-outer-bij sort-bij sort-bij\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting-\u2293-\u2294 _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    module SPP = Sorting-Perm-Properties _<=_ isTotalOrder\n    open EvalTree public using (evalTree)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0f955beaa8113f257ed083139272dfa44950620b","subject":"adding a todo","message":"adding a todo\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192 -- todo: why does \u0394 get thrown away here?\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u2205\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u2205\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 \u0394 , (\u0393 ,, (x , \u03c4)) \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"690abd76c7d1e71a729a29216af9b048173d3e65","subject":"Improved some proofs.","message":"Improved some proofs.\n\nIgnore-this: 1b0023619da1f13f7994e29414318322\n\ndarcs-hash:20100705031855-3bd4e-d02afc857c6f6ec97cc69a8c9d6e369231f1468f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/NatPCF\/InequalitiesPCF\/PropertiesPCF-ER.agda","new_file":"LTC\/Data\/NatPCF\/InequalitiesPCF\/PropertiesPCF-ER.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Data.NatPCF.InequalitiesPCF.PropertiesPCF-ER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.NatPCF\nopen import LTC.Data.NatPCF.InequalitiesPCF\nopen import LTC.Data.NatPCF.PropertiesPCF-ER\nopen import LTC.Relation.Equalities.PropertiesER\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for 'lt i j' it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states ('<-s\u2081',\n  -- '<-s\u2082', ...). Then we write down the proof for\n  -- the execution step from the state 'p' to the state 'q'\n  -- (e.g 's\u2081\u2192s\u2082 : (m n : D) \u2192 <-s\u2082 m n \u2192 <-s\u2083 m n' ).\n\n  -- The terms 'lt-00', 'lt-0S', 'lt-S0', and 'lt-S>S' show the use of\n  -- the states '<-s\u2081', '<-s\u2082', ..., and the proofs associated\n  -- with the execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt\n\n  -- The conversion rule 'cFix' is applied.\n  <-s\u2081 : D \u2192 D \u2192 D\n  <-s\u2081 d e = lth (fix lth) \u2219 d \u2219 e\n\n  -- Definition of lth.\n  <-s\u2082 : D \u2192 D \u2192 D\n  <-s\u2082 d e = lam (lt-aux\u2082 (fix lth)) \u2219 d \u2219 e\n\n  -- cBeta application.\n  <-s\u2083 : D \u2192 D \u2192 D\n  <-s\u2083 d e = lt-aux\u2082 (fix lth) d \u2219 e\n\n  -- Definition of lt-aux\u2082.\n  <-s\u2084 : D \u2192 D \u2192 D\n  <-s\u2084 d e = lam (lt-aux\u2081 d (fix lth)) \u2219 e\n\n  -- cBeta application.\n  <-s\u2085 : D \u2192 D \u2192 D\n  <-s\u2085 d e = lt-aux\u2081 d (fix lth) e\n\n  -- Definition lt-aux\u2081.\n  <-s\u2086 : D \u2192 D \u2192 D\n  <-s\u2086 d e = if (isZero e) then false\n                else (if (isZero d) then true\n                         else ((fix lth) \u2219 (pred d) \u2219 (pred e)))\n\n  -- Reduction 'isZero e \u2261 b'.\n  <-s\u2087 : D \u2192 D \u2192 D \u2192 D\n  <-s\u2087 d e b = if b then false\n                  else (if (isZero d) then true\n                           else ((fix lth) \u2219 (pred d) \u2219 (pred e)))\n\n  -- Reduction 'isZero e \u2261 false'.\n  <-s\u2088 : D \u2192 D \u2192 D\n  <-s\u2088 d e = if (isZero d) then true\n                 else ((fix lth) \u2219 (pred d) \u2219 (pred e))\n\n  -- Reduction 'isZero d \u2261 b'.\n  <-s\u2089 : D \u2192 D \u2192 D \u2192 D\n  <-s\u2089 d e b = if b then true\n                  else ((fix lth) \u2219 (pred d) \u2219 (pred e))\n\n  -- Reduction 'isZero d \u2261 false'.\n  <-s\u2081\u2080 : D \u2192 D \u2192 D\n  <-s\u2081\u2080 d e = (fix lth) \u2219 (pred d) \u2219 (pred e)\n\n  -- Reduction 'pred (succ d) \u2261 d'.\n  <-s\u2081\u2081 : D \u2192 D \u2192 D\n  <-s\u2081\u2081 d e = (fix lth) \u2219 d \u2219 (pred e)\n\n  -- Reduction 'pred (succ e) \u2261 e'.\n  <-s\u2081\u2082 : D \u2192 D \u2192 D\n  <-s\u2081\u2082 d e = (fix lth) \u2219 d \u2219 e\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n    To prove the execution steps (e.g. s\u2083\u2192s\u2084 : (m n : D) \u2192 <-s\u2083 m n \u2192 <-s\u2084 m n),\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n    We prove (1) using\n    'subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y'\n    where\n    'P' is given by '\u03bb t \u2192 C [m] \u2261 C [t]',\n    'x \u2261 y' is given 'm \u2261 n', and\n    'P x' is given by 'C [m] \u2261 C [m]' (i.e. 'refl').\n  -}\n\n  -- Application of the conversion rule 'cFix'.\n  initial\u2192s\u2081 : (d e : D) \u2192 fix lth \u2219 d \u2219 e \u2261 <-s\u2081 d e\n  initial\u2192s\u2081 d e = subst (\u03bb t \u2192 fix lth \u2219 d \u2219 e \u2261 t \u2219 d \u2219 e ) (cFix lth) refl\n\n  -- The definition of lth.\n  s\u2081\u2192s\u2082 : (d e : D) \u2192 <-s\u2081 d e \u2261 <-s\u2082 d e\n  s\u2081\u2192s\u2082 d e = refl\n\n  -- cBeta application.\n  s\u2082\u2192s\u2083 : (d e : D) \u2192 <-s\u2082 d e \u2261 <-s\u2083 d e\n  s\u2082\u2192s\u2083 d e = subst (\u03bb t \u2192 lam (lt-aux\u2082 (fix lth)) \u2219 d \u2219 e \u2261 t \u2219 e)\n                    (cBeta (lt-aux\u2082 (fix lth)) d)\n                    refl\n\n  -- Definition of lt-aux\u2082\n  s\u2083\u2192s\u2084 : (d e : D) \u2192 <-s\u2083 d e \u2261 <-s\u2084 d e\n  s\u2083\u2192s\u2084 d e = refl\n\n  -- cBeta application.\n  s\u2084\u2192s\u2085 : (d e : D) \u2192 <-s\u2084 d e \u2261 <-s\u2085 d e\n  s\u2084\u2192s\u2085 d e = cBeta (lt-aux\u2081 d (fix lth)) e\n\n  -- Definition of lt-aux\u2081.\n  s\u2085\u2192s\u2086 : (d e : D) \u2192 <-s\u2085 d e \u2261 <-s\u2086 d e\n  s\u2085\u2192s\u2086 d e = refl\n\n  -- Reduction 'isZero e \u2261 b' using that proof.\n  s\u2086\u2192s\u2087 : (d e b : D) \u2192 isZero e \u2261 b \u2192 <-s\u2086 d e \u2261 <-s\u2087 d e b\n  s\u2086\u2192s\u2087 d e b prf = subst (\u03bb t \u2192 <-s\u2086 d e \u2261 <-s\u2087 d e t) prf refl\n\n  -- Reduction of 'isZero e \u2261 true' using the conversion rule cB\u2081.\n  s\u2087\u2192end : (d e : D) \u2192 <-s\u2087 d e true \u2261 false\n  s\u2087\u2192end _ _ = cB\u2081 false\n\n  -- Reduction of 'isZero e \u2261 false ...' using the conversion rule cB\u2082.\n  s\u2087\u2192s\u2088 : (d e : D) \u2192 <-s\u2087 d e false \u2261 <-s\u2088 d e\n  s\u2087\u2192s\u2088 d e = cB\u2082 (<-s\u2088 d e)\n\n  -- Reduction 'isZero d \u2261 b' using that proof.\n  s\u2088\u2192s\u2089 : (d e b : D) \u2192 isZero d \u2261 b \u2192 <-s\u2088 d e \u2261 <-s\u2089 d e b\n  s\u2088\u2192s\u2089 d e b prf = subst (\u03bb t \u2192 <-s\u2088 d e \u2261 <-s\u2089 d e t ) prf refl\n\n  -- Reduction of 'isZero d \u2261 true' using the conversion rule cB\u2081.\n  s\u2089\u2192end : (d e : D) \u2192 <-s\u2089 d e true \u2261 true\n  s\u2089\u2192end _ _ = cB\u2081 true\n\n  -- Reduction of 'isZero d \u2261 false ...' using the conversion rule cB\u2082.\n  s\u2089\u2192s\u2081\u2080 : (d e : D) \u2192 <-s\u2089 d e false \u2261 <-s\u2081\u2080 d e\n  s\u2089\u2192s\u2081\u2080 d e = cB\u2082 (<-s\u2081\u2080 d e)\n\n  -- Reduction 'pred (succ d) \u2261 d' using the conversion rule cP\u2082.\n  s\u2081\u2080\u2192s\u2081\u2081 : (d e : D) \u2192 <-s\u2081\u2080 (succ d) e  \u2261 <-s\u2081\u2081 d e\n  s\u2081\u2080\u2192s\u2081\u2081 d e = subst (\u03bb t \u2192 <-s\u2081\u2080 (succ d) e \u2261 <-s\u2081\u2081 t e) (cP\u2082 d) refl\n\n  -- Reduction 'pred (succ e) \u2261 e' using the conversion rule 'cP\u2082'.\n  s\u2081\u2081\u2192s\u2081\u2082 : (d e : D) \u2192 <-s\u2081\u2081 d (succ e)  \u2261 <-s\u2081\u2082 d e\n  s\u2081\u2081\u2192s\u2081\u2082 d e = subst (\u03bb t \u2192 <-s\u2081\u2081 d (succ e) \u2261 <-s\u2081\u2082 d t) (cP\u2082 e) refl\n\n------------------------------------------------------------------------------\n\nlt-00 : NLT zero zero\nlt-00 =\n  begin\n    fix lth \u2219 zero \u2219 zero \u2261\u27e8 initial\u2192s\u2081 zero zero \u27e9\n    <-s\u2081 zero zero        \u2261\u27e8 s\u2081\u2192s\u2082 zero zero \u27e9\n    <-s\u2082 zero zero        \u2261\u27e8 s\u2082\u2192s\u2083 zero zero \u27e9\n    <-s\u2083 zero zero        \u2261\u27e8 s\u2083\u2192s\u2084 zero zero \u27e9\n    <-s\u2084 zero zero        \u2261\u27e8 s\u2084\u2192s\u2085 zero zero \u27e9\n    <-s\u2085 zero zero        \u2261\u27e8 s\u2085\u2192s\u2086 zero zero \u27e9\n    <-s\u2086 zero zero        \u2261\u27e8 s\u2086\u2192s\u2087 zero zero true cZ\u2081 \u27e9\n    <-s\u2087 zero zero true   \u2261\u27e8 s\u2087\u2192end zero zero \u27e9\n    false\n    \u220e\n\nlt-0S : (d : D) \u2192 LT zero (succ d)\nlt-0S d =\n  begin\n    fix lth \u2219 zero \u2219 (succ d) \u2261\u27e8 initial\u2192s\u2081 zero (succ d) \u27e9\n    <-s\u2081 zero (succ d)        \u2261\u27e8 s\u2081\u2192s\u2082 zero (succ d) \u27e9\n    <-s\u2082 zero (succ d)        \u2261\u27e8 s\u2082\u2192s\u2083 zero (succ d) \u27e9\n    <-s\u2083 zero (succ d)        \u2261\u27e8 s\u2083\u2192s\u2084 zero (succ d) \u27e9\n    <-s\u2084 zero (succ d)        \u2261\u27e8 s\u2084\u2192s\u2085 zero (succ d) \u27e9\n    <-s\u2085 zero (succ d)        \u2261\u27e8 s\u2085\u2192s\u2086 zero (succ d) \u27e9\n    <-s\u2086 zero (succ d)        \u2261\u27e8 s\u2086\u2192s\u2087 zero (succ d) false (cZ\u2082 d) \u27e9\n    <-s\u2087 zero (succ d) false  \u2261\u27e8 s\u2087\u2192s\u2088 zero (succ d) \u27e9\n    <-s\u2088 zero (succ d)        \u2261\u27e8 s\u2088\u2192s\u2089 zero (succ d) true cZ\u2081 \u27e9\n    <-s\u2089 zero (succ d) true   \u2261\u27e8 s\u2089\u2192end zero (succ d) \u27e9\n    true\n  \u220e\n\nlt-S0 : (d : D) \u2192 NLT (succ d) zero\nlt-S0 d =\n  begin\n    fix lth \u2219 (succ d) \u2219 zero \u2261\u27e8 initial\u2192s\u2081 (succ d) zero \u27e9\n    <-s\u2081 (succ d) zero        \u2261\u27e8 s\u2081\u2192s\u2082 (succ d) zero \u27e9\n    <-s\u2082 (succ d) zero        \u2261\u27e8 s\u2082\u2192s\u2083 (succ d) zero \u27e9\n    <-s\u2083 (succ d) zero        \u2261\u27e8 s\u2083\u2192s\u2084 (succ d) zero \u27e9\n    <-s\u2084 (succ d) zero        \u2261\u27e8 s\u2084\u2192s\u2085 (succ d) zero \u27e9\n    <-s\u2085 (succ d) zero        \u2261\u27e8 s\u2085\u2192s\u2086 (succ d) zero \u27e9\n    <-s\u2086 (succ d) zero        \u2261\u27e8 s\u2086\u2192s\u2087 (succ d) zero true cZ\u2081 \u27e9\n    <-s\u2087 (succ d) zero true   \u2261\u27e8 s\u2087\u2192end (succ d) zero \u27e9\n    false\n  \u220e\n\nlt-SS : (d e : D) \u2192 lt (succ d) (succ e) \u2261 lt d e\nlt-SS d e =\n  begin\n    fix lth \u2219 (succ d) \u2219 (succ e) \u2261\u27e8 initial\u2192s\u2081 (succ d) (succ e) \u27e9\n    <-s\u2081 (succ d) (succ e)        \u2261\u27e8 s\u2081\u2192s\u2082 (succ d) (succ e) \u27e9\n    <-s\u2082 (succ d) (succ e)        \u2261\u27e8 s\u2082\u2192s\u2083 (succ d) (succ e) \u27e9\n    <-s\u2083 (succ d) (succ e)        \u2261\u27e8 s\u2083\u2192s\u2084 (succ d) (succ e) \u27e9\n    <-s\u2084 (succ d) (succ e)        \u2261\u27e8 s\u2084\u2192s\u2085 (succ d) (succ e) \u27e9\n    <-s\u2085 (succ d) (succ e)        \u2261\u27e8 s\u2085\u2192s\u2086 (succ d) (succ e) \u27e9\n    <-s\u2086 (succ d) (succ e)        \u2261\u27e8 s\u2086\u2192s\u2087 (succ d) (succ e) false (cZ\u2082 e) \u27e9\n    <-s\u2087 (succ d) (succ e) false  \u2261\u27e8 s\u2087\u2192s\u2088 (succ d) (succ e) \u27e9\n    <-s\u2088 (succ d) (succ e)        \u2261\u27e8 s\u2088\u2192s\u2089 (succ d) (succ e) false (cZ\u2082 d) \u27e9\n    <-s\u2089 (succ d) (succ e) false  \u2261\u27e8 s\u2089\u2192s\u2081\u2080 (succ d) (succ e) \u27e9\n    <-s\u2081\u2080 (succ d) (succ e)       \u2261\u27e8 s\u2081\u2080\u2192s\u2081\u2081 d (succ e) \u27e9\n    <-s\u2081\u2081 d (succ e)              \u2261\u27e8 s\u2081\u2081\u2192s\u2081\u2082 d e \u27e9\n    <-s\u2081\u2082 d e                     \u2261\u27e8 refl \u27e9\n    lt d e\n  \u220e\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-0S zero\nx\u22650 (sN {n} Nn) = lt-0S (succ n)\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0           = true\u2260false (trans (sym 0<0) (lt-00))\n\u00acx<0 (sN {n} Nn) Sn<0 = true\u2260false (trans (sym Sn<0) (lt-S0 n))\n\n0\u226fx : {n : D} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = lt-00\n0\u226fx (sN {n} Nn) = lt-S0 n\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ 0\u226fx Nn\n\nx\u2270x : {n : D} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = lt-00\nx\u2270x (sN {n} Nn) = trans (lt-SS n n) (x\u2270x Nn)\n\nS\u22700 : {n : D} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (lt-SS (succ n) zero) (lt-S0 n)\n\n\u00acS\u22640 : {n : D} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : {n : D} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = trans (lt-SS n (succ n)) (x<Sx Nn)\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\n\u00acx>x : {m : D} \u2192 N m \u2192 \u00ac (GT m m)\n\u00acx>x Nm = \u00acx<x Nm\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-0S zero\nx\u2264x (sN {m} Nm) = trans (lt-SS m (succ m)) (x\u2264x Nm)\n\nx\u2265y\u2192x\u226ey : {m n : D} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim (\u00ac0\u2265S Nn 0\u2265Sn)\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = lt-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (lt-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym (lt-SS n (succ m))) Sm\u2265Sn))\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (lt-SS n m) m>n) )\n  , (\u03bb m\u2264n \u2192 inj\u2082 (trans (lt-SS m (succ n)) m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 {Nm : N m} \u2192 {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn) _              = lt-0S (succ n)\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  begin\n    lt (succ m) (succ (succ n)) \u2261\u27e8 lt-SS m (succ n) \u27e9\n    lt m (succ n)               \u2261\u27e8 x<y\u2192x\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn) \u27e9\n    true\n  \u220e\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN               m<0       = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192Sx\u2264y zN (sN zN)          _         = x\u2264x (sN zN)\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _         = trans (lt-SS zero (succ (succ n)))\n                                               (0\u2264x (sN Nn))\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (lt-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Nm Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  trans (lt-SS m n) (Sx\u2264y\u2192x<y Nm Nn (trans (sym (lt-SS (succ m) (succ n)))\n                                           SSm\u2264Sn))\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  begin\n    lt (succ m) (succ o) \u2261\u27e8 lt-SS m o \u27e9\n    lt m o \u2261\u27e8 <-trans Nm Nn No\n                       (trans (sym (lt-SS m n)) Sm<Sn)\n                       (trans (sym (lt-SS n o)) Sn<So) \u27e9\n    true\n  \u220e\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Nm Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Nn Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  trans (lt-SS m (succ o))\n        (\u2264-trans Nm Nn No\n                 (trans (sym (lt-SS m (succ n))) Sm\u2264Sn)\n                 (trans (sym (lt-SS n (succ o))) Sn\u2264So))\n\nSx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS m (succ n))) Sm\u2264Sn\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m) (succ (succ m + n)) \u2261\u27e8 lt-SS m (succ m + n) \u27e9\n    lt m (succ m + n)               \u2261\u27e8 subst (\u03bb t \u2192 lt m (succ m + n) \u2261 lt m t)\n                                             (+-Sx m n)\n                                             refl\n                                    \u27e9\n    lt m (succ (m + n))             \u2261\u27e8 refl \u27e9\n    le m (m + n)                    \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    true\n  \u220e\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN =\n  begin\n    lt (m - zero) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (m - zero) (succ m) \u2261\n                                           lt t (succ m))\n                                    (minus-x0 m)\n                                    refl\n                           \u27e9\n    lt m (succ m)          \u2261\u27e8 x<Sx Nm \u27e9\n    true\n  \u220e\n\nx-y<Sx zN (sN {n} Nn) =\n  begin\n    lt (zero - succ n) (succ zero)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (zero - succ n) (succ zero) \u2261 lt t (succ zero))\n               (minus-0S Nn)\n               refl\n      \u27e9\n    lt zero (succ zero) \u2261\u27e8 lt-0S zero \u27e9\n    true\n  \u220e\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) =\n  begin\n    lt (succ m - succ n) (succ (succ m))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ (succ m)) \u2261\n                      lt t (succ (succ m)))\n               (minus-SS Nm Nn)\n               refl\n      \u27e9\n    lt (m - n) (succ (succ m))\n      \u2261\u27e8 <-trans (minus-N Nm Nn) (sN Nm) (sN (sN Nm))\n                 (x-y<Sx Nm Nn) (x<Sx (sN Nm))\n      \u27e9\n    true\n  \u220e\n\nSx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\nSx-Sy<Sx {m} {n} Nm Nn =\n  begin\n    lt (succ m - succ n) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n)\n                                                     (succ m) \u2261\n                                                  lt t (succ m))\n                                           (minus-SS Nm Nn)\n                                           refl\n                                  \u27e9\n    lt (m - n) (succ m)           \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n    \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS Nm Nn)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN      Nn 0\u2264n  = trans (+-rightIdentity (minus-N Nn zN))\n                                            (minus-x0 n)\nx\u2264y\u2192y-x+x\u2261y         (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Nm Sm\u22640)\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  begin\n    (succ n - succ m) + succ m \u2261\u27e8 subst (\u03bb t \u2192 (succ n - succ m) + succ m \u2261\n                                               t + succ m)\n                                        (minus-SS Nn Nm)\n                                        refl\n                               \u27e9\n    (n - m) + succ m           \u2261\u27e8 +-comm (minus-N Nn Nm) (sN Nm) \u27e9\n    succ m + (n - m)           \u2261\u27e8 +-Sx m (n - m) \u27e9\n    succ (m + (n - m))         \u2261\u27e8 subst (\u03bb t \u2192 succ (m + (n - m)) \u2261 succ t )\n                                        (+-comm Nm (minus-N Nn Nm))\n                                        refl\n                               \u27e9\n    succ ((n - m) + m)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((n - m) + m) \u2261 succ t )\n                                        (x\u2264y\u2192y-x+x\u2261y Nm Nn\n                                             (trans (sym (lt-SS m (succ n)))\n                                                    Sm\u2264Sn) )\n                                        refl\n                               \u27e9\n    succ n\n  \u220e\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS m (succ n)) (x<y\u2192x<Sy Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim (\u00acx<0 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ n))) Sm<SSn)\n\nx<y\u2192y\u2261z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\nx<y\u2192y\u2261z\u2192x<z {m} {n} {o} Nm Nn No m<n n\u2261o =\n  begin\n    lt m o \u2261\u27e8 subst (\u03bb t \u2192 lt m o \u2261 lt m t)\n                    (sym n\u2261o)\n                    refl\n           \u27e9\n    lt m n \u2261\u27e8 m<n \u27e9\n    true\n  \u220e\n\nx\u2261y\u2192y<z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\nx\u2261y\u2192y<z\u2192x<z {m} {n} {o} Nm Nn No m\u2261n n<o =\n  begin\n    lt m o \u2261\u27e8 subst (\u03bb t \u2192 lt m o \u2261 lt t o)\n                    m\u2261n\n                    refl\n           \u27e9\n    lt n o \u2261\u27e8 n<o \u27e9\n    true\n  \u220e\n\nx\u2265y\u2192y>0\u2192x-y<x : {m n : D} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m - n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim (\u00acx>x zN 0>0)\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim (\u00acS\u22640 Nn 0\u2265Sn)\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 =\n  begin\n    lt (succ m - succ n) (succ m)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ m) \u2261 lt t (succ m))\n               (minus-SS Nm Nn)\n               refl\n      \u27e9\n    lt (m - n) (succ m) \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n  \u220e\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\n\u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n\u00acxy<00 Nm Nn mn<00 =\n  [ (\u03bb m<0     \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n  ]\n  mn<00\n\n\u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n\u00ac0Sx<00 Nm 0Sm<00 =\n  [ (\u03bb 0<0      \u2192 \u22a5-elim (\u00acx<0 zN 0<0))\n  , (\u03bb 0\u22610\u2227Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) (\u2227-proj\u2082 0\u22610\u2227Sm<0)))\n  ]\n  0Sm<00\n\n\u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n\u00acSxy\u2081<0y\u2082 Nm Nn\u2081 Nn\u2082 Smn\u2081<0n\u2082 =\n  [ (\u03bb Sm<0       \u2192 \u22a5-elim (\u00acx<0 (sN Nm) Sm<0))\n  , (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2260S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n  ]\n  Smn\u2081<0n\u2082\n\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 Nm\u2081 Nn Nm\u2082 m\u2081n<m\u2082zero =\n  [ (\u03bb m\u2081<n\u2081      \u2192 m\u2081<n\u2081)\n  , (\u03bb m\u2081\u2261n\u2081\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u2081\u2261n\u2081\u2227n<0)))\n  ]\n  m\u2081n<m\u2082zero\n\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                    m \u2261 zero \u2227 LT n\u2081 n\u2082\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 Nm Nn\u2081 Nn\u2082 mn\u2081<0n\u2082 =\n  [ (\u03bb m<0          \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u2261zero\u2227n\u2081<n\u2082 \u2192 m\u2261zero\u2227n\u2081<n\u2082)\n  ]\n  mn\u2081<0n\u2082\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = inj\u2081 (Sx-Sy<Sx Nm Nn)\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = inj\u2082 (refl , Sx-Sy<Sx Nn Nm)\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Data.NatPCF.InequalitiesPCF.PropertiesPCF-ER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.NatPCF\nopen import LTC.Data.NatPCF.InequalitiesPCF\nopen import LTC.Data.NatPCF.PropertiesPCF-ER\nopen import LTC.Relation.Equalities.PropertiesER\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for 'lt i j' it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states ('<-s\u2081',\n  -- '<-s\u2082', ...). Then we write down the proof for\n  -- the execution step from the state 'p' to the state 'q'\n  -- (e.g 's\u2081\u2192s\u2082 : (m n : D) \u2192 <-s\u2082 m n \u2192 <-s\u2083 m n' ).\n\n  -- The terms 'lt-00', 'lt-0S', 'lt-S0', and 'lt-S>S' show the use of\n  -- the states '<-s\u2081', '<-s\u2082', ..., and the proofs associated\n  -- with the execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt\n\n  -- The conversion rule 'cFix' is applied.\n  <-s\u2081 : D \u2192 D \u2192 D\n  <-s\u2081 d e = lth (fix lth) \u2219 d \u2219 e\n\n  -- Definition of lth.\n  <-s\u2082 : D \u2192 D \u2192 D\n  <-s\u2082 d e = lam (lt-aux\u2082 (fix lth)) \u2219 d \u2219 e\n\n  -- cBeta application.\n  <-s\u2083 : D \u2192 D \u2192 D\n  <-s\u2083 d e = lt-aux\u2082 (fix lth) d \u2219 e\n\n  -- Definition of lt-aux\u2082.\n  <-s\u2084 : D \u2192 D \u2192 D\n  <-s\u2084 d e = lam (lt-aux\u2081 d (fix lth)) \u2219 e\n\n  -- cBeta application.\n  <-s\u2085 : D \u2192 D \u2192 D\n  <-s\u2085 d e = lt-aux\u2081 d (fix lth) e\n\n  -- Definition lt-aux\u2081.\n  <-s\u2086 : D \u2192 D \u2192 D\n  <-s\u2086 d e = if (isZero e) then false\n                else (if (isZero d) then true\n                         else ((fix lth) \u2219 (pred d) \u2219 (pred e)))\n\n  -- Reduction 'isZero e \u2261 b'.\n  <-s\u2087 : D \u2192 D \u2192 D \u2192 D\n  <-s\u2087 d e b = if b then false\n                  else (if (isZero d) then true\n                           else ((fix lth) \u2219 (pred d) \u2219 (pred e)))\n\n  -- Reduction 'isZero e \u2261 false'.\n  <-s\u2088 : D \u2192 D \u2192 D\n  <-s\u2088 d e = if (isZero d) then true\n                 else ((fix lth) \u2219 (pred d) \u2219 (pred e))\n\n  -- Reduction 'isZero d \u2261 b'.\n  <-s\u2089 : D \u2192 D \u2192 D \u2192 D\n  <-s\u2089 d e b = if b then true\n                  else ((fix lth) \u2219 (pred d) \u2219 (pred e))\n\n  -- Reduction 'isZero d \u2261 false'.\n  <-s\u2081\u2080 : D \u2192 D \u2192 D\n  <-s\u2081\u2080 d e = (fix lth) \u2219 (pred d) \u2219 (pred e)\n\n  -- Reduction 'pred (succ d) \u2261 d'.\n  <-s\u2081\u2081 : D \u2192 D \u2192 D\n  <-s\u2081\u2081 d e = (fix lth) \u2219 d \u2219 (pred e)\n\n  -- Reduction 'pred (succ e) \u2261 e'.\n  <-s\u2081\u2082 : D \u2192 D \u2192 D\n  <-s\u2081\u2082 d e = (fix lth) \u2219 d \u2219 e\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n    To prove the execution steps (e.g. s\u2083\u2192s\u2084 : (m n : D) \u2192 <-s\u2083 m n \u2192 <-s\u2084 m n),\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n    We prove (1) using\n    'subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y'\n    where\n    'P' is given by '\u03bb m \u2192 C [m ] \u2261 C [n]',\n    'x \u2261 y' is given 'n \u2261 m' (actually, we use '-sym (m \u2261 n)'), and\n    'P x' is given by 'C [n] \u2261 C [n]' (i.e. 'refl').\n  -}\n\n  -- Application of the conversion rule 'cFix'.\n  initial\u2192s\u2081 : (d e : D) \u2192 fix lth \u2219 d \u2219 e  \u2261 <-s\u2081 d e\n  initial\u2192s\u2081 d e = subst (\u03bb t \u2192 t \u2219 d \u2219 e  \u2261 lth (fix lth) \u2219 d \u2219 e )\n                         (sym (cFix lth ))\n                         refl\n\n  -- The definition of lth.\n  s\u2081\u2192s\u2082 : (d e : D) \u2192 <-s\u2081 d e \u2261 <-s\u2082 d e\n  s\u2081\u2192s\u2082 d e = refl\n\n  -- cBeta application.\n  s\u2082\u2192s\u2083 : (d e : D) \u2192 <-s\u2082 d e \u2261 <-s\u2083 d e\n  s\u2082\u2192s\u2083 d e = subst (\u03bb t \u2192 lam (lt-aux\u2082 (fix lth)) \u2219 d \u2219 e \u2261 t \u2219 e)\n                    (cBeta (lt-aux\u2082 (fix lth)) d)\n                    refl\n\n  -- Definition of lt-aux\u2082\n  s\u2083\u2192s\u2084 : (d e : D) \u2192 <-s\u2083 d e \u2261 <-s\u2084 d e\n  s\u2083\u2192s\u2084 d e = refl\n\n  -- cBeta application.\n  s\u2084\u2192s\u2085 : (d e : D) \u2192 <-s\u2084 d e \u2261 <-s\u2085 d e\n  s\u2084\u2192s\u2085 d e = cBeta (lt-aux\u2081 d (fix lth)) e\n\n  -- Definition of lt-aux\u2081.\n  s\u2085\u2192s\u2086 : (d e : D) \u2192 <-s\u2085 d e \u2261 <-s\u2086 d e\n  s\u2085\u2192s\u2086 d e = refl\n\n  -- Reduction 'isZero e \u2261 b' using that proof.\n  s\u2086\u2192s\u2087 : (d e b : D) \u2192 isZero e \u2261 b \u2192 <-s\u2086 d e \u2261 <-s\u2087 d e b\n  s\u2086\u2192s\u2087 d e b prf = subst (\u03bb t \u2192 <-s\u2087 d e t \u2261 <-s\u2087 d e b )\n                          (sym prf)\n                          refl\n\n  -- Reduction of 'isZero e \u2261 true' using the conversion rule cB\u2081.\n  s\u2087\u2192end : (d e : D) \u2192 <-s\u2087 d e true \u2261 false\n  s\u2087\u2192end _ _ = cB\u2081 false\n\n  -- Reduction of 'isZero e \u2261 false ...' using the conversion rule cB\u2082.\n  s\u2087\u2192s\u2088 : (d e : D) \u2192 <-s\u2087 d e false \u2261 <-s\u2088 d e\n  s\u2087\u2192s\u2088 d e = cB\u2082 (<-s\u2088 d e)\n\n  -- Reduction 'isZero d \u2261 b' using that proof.\n  s\u2088\u2192s\u2089 : (d e b : D) \u2192 isZero d \u2261 b \u2192 <-s\u2088 d e \u2261 <-s\u2089 d e b\n  s\u2088\u2192s\u2089 d e b prf = subst (\u03bb t \u2192 <-s\u2089 d e t \u2261 <-s\u2089 d e b )\n                          (sym prf)\n                          refl\n\n  -- Reduction of 'isZero d \u2261 true' using the conversion rule cB\u2081.\n  s\u2089\u2192end : (d e : D) \u2192 <-s\u2089 d e true \u2261 true\n  s\u2089\u2192end _ _ = cB\u2081 true\n\n  -- Reduction of 'isZero d \u2261 false ...' using the conversion rule cB\u2082.\n  s\u2089\u2192s\u2081\u2080 : (d e : D) \u2192 <-s\u2089 d e false \u2261 <-s\u2081\u2080 d e\n  s\u2089\u2192s\u2081\u2080 d e = cB\u2082 (<-s\u2081\u2080 d e)\n\n  -- Reduction 'pred (succ d) \u2261 d' using the conversion rule cP\u2082.\n  s\u2081\u2080\u2192s\u2081\u2081 : (d e : D) \u2192 <-s\u2081\u2080 (succ d) e  \u2261 <-s\u2081\u2081 d e\n  s\u2081\u2080\u2192s\u2081\u2081 d e = subst (\u03bb t \u2192 <-s\u2081\u2081 t e \u2261 <-s\u2081\u2081 d e)\n                      (sym (cP\u2082 d ))\n                      refl\n\n  -- Reduction 'pred (succ e) \u2261 e' using the conversion rule 'cP\u2082'.\n  s\u2081\u2081\u2192s\u2081\u2082 : (d e : D) \u2192 <-s\u2081\u2081 d (succ e)  \u2261 <-s\u2081\u2082 d e\n  s\u2081\u2081\u2192s\u2081\u2082 d e = subst (\u03bb t \u2192 <-s\u2081\u2082 d t \u2261 <-s\u2081\u2082 d e)\n                      (sym (cP\u2082 e ))\n                      refl\n\n------------------------------------------------------------------------------\n\nlt-00 : NLT zero zero\nlt-00 =\n  begin\n    fix lth \u2219 zero \u2219 zero \u2261\u27e8 initial\u2192s\u2081 zero zero \u27e9\n    <-s\u2081 zero zero        \u2261\u27e8 s\u2081\u2192s\u2082 zero zero \u27e9\n    <-s\u2082 zero zero        \u2261\u27e8 s\u2082\u2192s\u2083 zero zero \u27e9\n    <-s\u2083 zero zero        \u2261\u27e8 s\u2083\u2192s\u2084 zero zero \u27e9\n    <-s\u2084 zero zero        \u2261\u27e8 s\u2084\u2192s\u2085 zero zero \u27e9\n    <-s\u2085 zero zero        \u2261\u27e8 s\u2085\u2192s\u2086 zero zero \u27e9\n    <-s\u2086 zero zero        \u2261\u27e8 s\u2086\u2192s\u2087 zero zero true cZ\u2081 \u27e9\n    <-s\u2087 zero zero true   \u2261\u27e8 s\u2087\u2192end zero zero \u27e9\n    false\n    \u220e\n\nlt-0S : (d : D) \u2192 LT zero (succ d)\nlt-0S d =\n  begin\n    fix lth \u2219 zero \u2219 (succ d) \u2261\u27e8 initial\u2192s\u2081 zero (succ d) \u27e9\n    <-s\u2081 zero (succ d)        \u2261\u27e8 s\u2081\u2192s\u2082 zero (succ d) \u27e9\n    <-s\u2082 zero (succ d)        \u2261\u27e8 s\u2082\u2192s\u2083 zero (succ d) \u27e9\n    <-s\u2083 zero (succ d)        \u2261\u27e8 s\u2083\u2192s\u2084 zero (succ d) \u27e9\n    <-s\u2084 zero (succ d)        \u2261\u27e8 s\u2084\u2192s\u2085 zero (succ d) \u27e9\n    <-s\u2085 zero (succ d)        \u2261\u27e8 s\u2085\u2192s\u2086 zero (succ d) \u27e9\n    <-s\u2086 zero (succ d)        \u2261\u27e8 s\u2086\u2192s\u2087 zero (succ d) false (cZ\u2082 d) \u27e9\n    <-s\u2087 zero (succ d) false  \u2261\u27e8 s\u2087\u2192s\u2088 zero (succ d) \u27e9\n    <-s\u2088 zero (succ d)        \u2261\u27e8 s\u2088\u2192s\u2089 zero (succ d) true cZ\u2081 \u27e9\n    <-s\u2089 zero (succ d) true   \u2261\u27e8 s\u2089\u2192end zero (succ d) \u27e9\n    true\n  \u220e\n\nlt-S0 : (d : D) \u2192 NLT (succ d) zero\nlt-S0 d =\n  begin\n    fix lth \u2219 (succ d) \u2219 zero \u2261\u27e8 initial\u2192s\u2081 (succ d) zero \u27e9\n    <-s\u2081 (succ d) zero        \u2261\u27e8 s\u2081\u2192s\u2082 (succ d) zero \u27e9\n    <-s\u2082 (succ d) zero        \u2261\u27e8 s\u2082\u2192s\u2083 (succ d) zero \u27e9\n    <-s\u2083 (succ d) zero        \u2261\u27e8 s\u2083\u2192s\u2084 (succ d) zero \u27e9\n    <-s\u2084 (succ d) zero        \u2261\u27e8 s\u2084\u2192s\u2085 (succ d) zero \u27e9\n    <-s\u2085 (succ d) zero        \u2261\u27e8 s\u2085\u2192s\u2086 (succ d) zero \u27e9\n    <-s\u2086 (succ d) zero        \u2261\u27e8 s\u2086\u2192s\u2087 (succ d) zero true cZ\u2081 \u27e9\n    <-s\u2087 (succ d) zero true   \u2261\u27e8 s\u2087\u2192end (succ d) zero \u27e9\n    false\n  \u220e\n\nlt-SS : (d e : D) \u2192 lt (succ d) (succ e) \u2261 lt d e\nlt-SS d e =\n  begin\n    fix lth \u2219 (succ d) \u2219 (succ e) \u2261\u27e8 initial\u2192s\u2081 (succ d) (succ e) \u27e9\n    <-s\u2081 (succ d) (succ e)        \u2261\u27e8 s\u2081\u2192s\u2082 (succ d) (succ e) \u27e9\n    <-s\u2082 (succ d) (succ e)        \u2261\u27e8 s\u2082\u2192s\u2083 (succ d) (succ e) \u27e9\n    <-s\u2083 (succ d) (succ e)        \u2261\u27e8 s\u2083\u2192s\u2084 (succ d) (succ e) \u27e9\n    <-s\u2084 (succ d) (succ e)        \u2261\u27e8 s\u2084\u2192s\u2085 (succ d) (succ e) \u27e9\n    <-s\u2085 (succ d) (succ e)        \u2261\u27e8 s\u2085\u2192s\u2086 (succ d) (succ e) \u27e9\n    <-s\u2086 (succ d) (succ e)        \u2261\u27e8 s\u2086\u2192s\u2087 (succ d) (succ e) false (cZ\u2082 e) \u27e9\n    <-s\u2087 (succ d) (succ e) false  \u2261\u27e8 s\u2087\u2192s\u2088 (succ d) (succ e) \u27e9\n    <-s\u2088 (succ d) (succ e)        \u2261\u27e8 s\u2088\u2192s\u2089 (succ d) (succ e) false (cZ\u2082 d) \u27e9\n    <-s\u2089 (succ d) (succ e) false  \u2261\u27e8 s\u2089\u2192s\u2081\u2080 (succ d) (succ e) \u27e9\n    <-s\u2081\u2080 (succ d) (succ e)       \u2261\u27e8 s\u2081\u2080\u2192s\u2081\u2081 d (succ e) \u27e9\n    <-s\u2081\u2081 d (succ e)              \u2261\u27e8 s\u2081\u2081\u2192s\u2081\u2082 d e \u27e9\n    <-s\u2081\u2082 d e                     \u2261\u27e8 refl \u27e9\n    lt d e\n  \u220e\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-0S zero\nx\u22650 (sN {n} Nn) = lt-0S (succ n)\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0           = true\u2260false (trans (sym 0<0) (lt-00))\n\u00acx<0 (sN {n} Nn) Sn<0 = true\u2260false (trans (sym Sn<0) (lt-S0 n))\n\n0\u226fx : {n : D} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = lt-00\n0\u226fx (sN {n} Nn) = lt-S0 n\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ 0\u226fx Nn\n\nx\u2270x : {n : D} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = lt-00\nx\u2270x (sN {n} Nn) = trans (lt-SS n n) (x\u2270x Nn)\n\nS\u22700 : {n : D} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (lt-SS (succ n) zero) (lt-S0 n)\n\n\u00acS\u22640 : {n : D} \u2192 N n \u2192 \u00ac (LE (succ n) zero)\n\u00acS\u22640 {d} Nn Sn\u22640 = true\u2260false $ trans (sym Sn\u22640) (S\u22700 Nn)\n\n\u00ac0\u2265S : {n : D} \u2192 N n \u2192 \u00ac (GE zero (succ n))\n\u00ac0\u2265S Nn 0\u2265Sn = \u00acS\u22640 Nn 0\u2265Sn\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = trans (lt-SS n (succ n)) (x<Sx Nn)\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\n\u00acx>x : {m : D} \u2192 N m \u2192 \u00ac (GT m m)\n\u00acx>x Nm = \u00acx<x Nm\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-0S zero\nx\u2264x (sN {m} Nm) = trans (lt-SS m (succ m)) (x\u2264x Nm)\n\nx\u2265y\u2192x\u226ey : {m n : D} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim (\u00ac0\u2265S Nn 0\u2265Sn)\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = lt-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (lt-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym (lt-SS n (succ m))) Sm\u2265Sn))\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (lt-SS n m) m>n) )\n  , (\u03bb m\u2264n \u2192 inj\u2082 (trans (lt-SS m (succ n)) m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 {Nm : N m} \u2192 {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn) _              = lt-0S (succ n)\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  begin\n    lt (succ m) (succ (succ n)) \u2261\u27e8 lt-SS m (succ n) \u27e9\n    lt m (succ n)               \u2261\u27e8 x<y\u2192x\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn) \u27e9\n    true\n  \u220e\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN               m<0       = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192Sx\u2264y zN (sN zN)          _         = x\u2264x (sN zN)\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _         = trans (lt-SS zero (succ (succ n)))\n                                               (0\u2264x (sN Nn))\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (lt-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Nm Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  trans (lt-SS m n) (Sx\u2264y\u2192x<y Nm Nn (trans (sym (lt-SS (succ m) (succ n)))\n                                           SSm\u2264Sn))\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  begin\n    lt (succ m) (succ o) \u2261\u27e8 lt-SS m o \u27e9\n    lt m o \u2261\u27e8 <-trans Nm Nn No\n                       (trans (sym (lt-SS m n)) Sm<Sn)\n                       (trans (sym (lt-SS n o)) Sn<So) \u27e9\n    true\n  \u220e\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Nm Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Nn Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  trans (lt-SS m (succ o))\n        (\u2264-trans Nm Nn No\n                 (trans (sym (lt-SS m (succ n))) Sm\u2264Sn)\n                 (trans (sym (lt-SS n (succ o))) Sn\u2264So))\n\nSx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS m (succ n))) Sm\u2264Sn\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m) (succ (succ m + n)) \u2261\u27e8 lt-SS m (succ m + n) \u27e9\n    lt m (succ m + n)               \u2261\u27e8 subst (\u03bb t \u2192 lt m (succ m + n) \u2261 lt m t)\n                                             (+-Sx m n)\n                                             refl\n                                    \u27e9\n    lt m (succ (m + n))             \u2261\u27e8 refl \u27e9\n    le m (m + n)                    \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    true\n  \u220e\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN =\n  begin\n    lt (m - zero) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (m - zero) (succ m) \u2261\n                                           lt t (succ m))\n                                    (minus-x0 m)\n                                    refl\n                           \u27e9\n    lt m (succ m)          \u2261\u27e8 x<Sx Nm \u27e9\n    true\n  \u220e\n\nx-y<Sx zN (sN {n} Nn) =\n  begin\n    lt (zero - succ n) (succ zero)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (zero - succ n) (succ zero) \u2261 lt t (succ zero))\n               (minus-0S Nn)\n               refl\n      \u27e9\n    lt zero (succ zero) \u2261\u27e8 lt-0S zero \u27e9\n    true\n  \u220e\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) =\n  begin\n    lt (succ m - succ n) (succ (succ m))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ (succ m)) \u2261\n                      lt t (succ (succ m)))\n               (minus-SS Nm Nn)\n               refl\n      \u27e9\n    lt (m - n) (succ (succ m))\n      \u2261\u27e8 <-trans (minus-N Nm Nn) (sN Nm) (sN (sN Nm))\n                 (x-y<Sx Nm Nn) (x<Sx (sN Nm))\n      \u27e9\n    true\n  \u220e\n\nSx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\nSx-Sy<Sx {m} {n} Nm Nn =\n  begin\n    lt (succ m - succ n) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n)\n                                                     (succ m) \u2261\n                                                  lt t (succ m))\n                                           (minus-SS Nm Nn)\n                                           refl\n                                  \u27e9\n    lt (m - n) (succ m)           \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n    \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS Nm Nn)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN      Nn 0\u2264n  = trans (+-rightIdentity (minus-N Nn zN))\n                                            (minus-x0 n)\nx\u2264y\u2192y-x+x\u2261y         (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Nm Sm\u22640)\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  begin\n    (succ n - succ m) + succ m \u2261\u27e8 subst (\u03bb t \u2192 (succ n - succ m) + succ m \u2261\n                                               t + succ m)\n                                        (minus-SS Nn Nm)\n                                        refl\n                               \u27e9\n    (n - m) + succ m           \u2261\u27e8 +-comm (minus-N Nn Nm) (sN Nm) \u27e9\n    succ m + (n - m)           \u2261\u27e8 +-Sx m (n - m) \u27e9\n    succ (m + (n - m))         \u2261\u27e8 subst (\u03bb t \u2192 succ (m + (n - m)) \u2261 succ t )\n                                        (+-comm Nm (minus-N Nn Nm))\n                                        refl\n                               \u27e9\n    succ ((n - m) + m)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((n - m) + m) \u2261 succ t )\n                                        (x\u2264y\u2192y-x+x\u2261y Nm Nn\n                                             (trans (sym (lt-SS m (succ n)))\n                                                    Sm\u2264Sn) )\n                                        refl\n                               \u27e9\n    succ n\n  \u220e\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS m (succ n)) (x<y\u2192x<Sy Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim (\u00acx<0 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ n))) Sm<SSn)\n\nx<y\u2192y\u2261z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\nx<y\u2192y\u2261z\u2192x<z {m} {n} {o} Nm Nn No m<n n\u2261o =\n  begin\n    lt m o \u2261\u27e8 subst (\u03bb t \u2192 lt m o \u2261 lt m t)\n                    (sym n\u2261o)\n                    refl\n           \u27e9\n    lt m n \u2261\u27e8 m<n \u27e9\n    true\n  \u220e\n\nx\u2261y\u2192y<z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\nx\u2261y\u2192y<z\u2192x<z {m} {n} {o} Nm Nn No m\u2261n n<o =\n  begin\n    lt m o \u2261\u27e8 subst (\u03bb t \u2192 lt m o \u2261 lt t o)\n                    m\u2261n\n                    refl\n           \u27e9\n    lt n o \u2261\u27e8 n<o \u27e9\n    true\n  \u220e\n\nx\u2265y\u2192y>0\u2192x-y<x : {m n : D} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m - n) m\nx\u2265y\u2192y>0\u2192x-y<x Nm          zN          _     0>0  = \u22a5-elim (\u00acx>x zN 0>0)\nx\u2265y\u2192y>0\u2192x-y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim (\u00acS\u22640 Nn 0\u2265Sn)\nx\u2265y\u2192y>0\u2192x-y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 =\n  begin\n    lt (succ m - succ n) (succ m)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ m) \u2261 lt t (succ m))\n               (minus-SS Nm Nn)\n               refl\n      \u27e9\n    lt (m - n) (succ m) \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n  \u220e\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\n\u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n\u00acxy<00 Nm Nn mn<00 =\n  [ (\u03bb m<0     \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n  ]\n  mn<00\n\n\u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n\u00ac0Sx<00 Nm 0Sm<00 =\n  [ (\u03bb 0<0      \u2192 \u22a5-elim (\u00acx<0 zN 0<0))\n  , (\u03bb 0\u22610\u2227Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) (\u2227-proj\u2082 0\u22610\u2227Sm<0)))\n  ]\n  0Sm<00\n\n\u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n\u00acSxy\u2081<0y\u2082 Nm Nn\u2081 Nn\u2082 Smn\u2081<0n\u2082 =\n  [ (\u03bb Sm<0       \u2192 \u22a5-elim (\u00acx<0 (sN Nm) Sm<0))\n  , (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2260S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n  ]\n  Smn\u2081<0n\u2082\n\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 Nm\u2081 Nn Nm\u2082 m\u2081n<m\u2082zero =\n  [ (\u03bb m\u2081<n\u2081      \u2192 m\u2081<n\u2081)\n  , (\u03bb m\u2081\u2261n\u2081\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u2081\u2261n\u2081\u2227n<0)))\n  ]\n  m\u2081n<m\u2082zero\n\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                    m \u2261 zero \u2227 LT n\u2081 n\u2082\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 Nm Nn\u2081 Nn\u2082 mn\u2081<0n\u2082 =\n  [ (\u03bb m<0          \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u2261zero\u2227n\u2081<n\u2082 \u2192 m\u2261zero\u2227n\u2081<n\u2082)\n  ]\n  mn\u2081<0n\u2082\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = inj\u2081 (Sx-Sy<Sx Nm Nn)\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = inj\u2082 (refl , Sx-Sy<Sx Nn Nm)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7257505604e7df0478b8f19eb563905418431bcb","subject":"Document P.C.Specification.","message":"Document P.C.Specification.\n\nOld-commit-hash: e6e682df0d2f25bc8c50a5872acd4540f0116289\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Specification.agda","new_file":"Parametric\/Change\/Specification.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- This module defines two extension points, so to avoid some of\n-- the boilerplate, we use record notation.\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    -- Extension point 1: Derivatives of constants.\n    \u27e6_\u27e7\u0394Const : \u2200 {\u03a3 \u03c4} \u2192 (c  : Const \u03a3 \u03c4) (\u03c1 : \u27e6 \u03a3 \u27e7) \u2192 \u0394\u208d \u03a3 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 c \u27e7Const \u03c1)\n\n    -- Extension point 2: Correctness of the instantiation of extension point 1.\n    correctness-const : \u2200 {\u03a3 \u03c4} (c : Const \u03a3 \u03c4) \u2192\n      Derivative\u208d \u03a3 , \u03c4 \u208e \u27e6 c \u27e7Const \u27e6 c \u27e7\u0394Const\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms and lists of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n  \u27e6_\u27e7\u0394Terms : \u2200 {\u03a3 \u0393} \u2192 (ts : Terms \u0393 \u03a3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03a3 \u208e (\u27e6 ts \u27e7Terms \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    Derivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  correctness-terms : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3) \u2192\n    Derivative\u208d \u0393 , \u03a3 \u208e \u27e6 ts \u27e7Terms \u27e6 ts \u27e7\u0394Terms\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const c ts) \u03c1 d\u03c1 = \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = cons\n    (\u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1))\n    (\u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e) where open \u2261-Reasoning\n\n  \u27e6 \u2205 \u27e7\u0394Terms \u03c1 d\u03c1 = \u2205\n  \u27e6 t \u2022 ts \u27e7\u0394Terms \u03c1 d\u03c1 = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2022 \u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    Derivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness-terms \u2205 \u03c1 d\u03c1 = refl\n  correctness-terms (t \u2022 ts) \u03c1 d\u03c1 =\n    cong\u2082 _\u2022_\n      (correctness t \u03c1 d\u03c1)\n      (correctness-terms ts \u03c1 d\u03c1)\n\n  correctness (const {\u03a3} {\u03c4} c ts) \u03c1 d\u03c1 =\n    begin\n      after\u208d \u03c4 \u208e (\u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1))\n    \u2261\u27e8 correctness-const c (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u27e9\n      \u27e6 c \u27e7Const (after-env (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1))\n    \u2261\u27e8 cong \u27e6 c \u27e7Const (correctness-terms ts \u03c1 d\u03c1) \u27e9\n      \u27e6 c \u27e7Const (\u27e6 ts \u27e7Terms (after-env d\u03c1))\n    \u220e where open \u2261-Reasoning\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f = \u27e6 s \u27e7 \u03c1\n      g = \u27e6 s \u27e7 (after-env d\u03c1)\n      u = \u27e6 t \u27e7 \u03c1\n      v = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f u \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u \u0394u) \u27e9\n        (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        g v\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\n\n-- Denotation-as-specification\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\n-- open import Data.Integer\n-- open import Theorem.Groups-Nehemiah\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    \u27e6_\u27e7\u0394Const : \u2200 {\u03a3 \u03c4} \u2192 (c  : Const \u03a3 \u03c4) (\u03c1 : \u27e6 \u03a3 \u27e7) \u2192 \u0394\u208d \u03a3 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 c \u27e7Const \u03c1)\n\n    correctness-const : \u2200 {\u03a3 \u03c4} (c : Const \u03a3 \u03c4) \u2192\n      Derivative\u208d \u03a3 , \u03c4 \u208e \u27e6 c \u27e7Const \u27e6 c \u27e7\u0394Const\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n  \u27e6_\u27e7\u0394Terms : \u2200 {\u03a3 \u0393} \u2192 (ts : Terms \u0393 \u03a3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03a3 \u208e (\u27e6 ts \u27e7Terms \u03c1)\n\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    Derivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  correctness-terms : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3) \u2192\n    Derivative\u208d \u0393 , \u03a3 \u208e \u27e6 ts \u27e7Terms \u27e6 ts \u27e7\u0394Terms\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const c ts) \u03c1 d\u03c1 = \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = cons\n    (\u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1))\n    (\u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e) where open \u2261-Reasoning\n\n  \u27e6 \u2205 \u27e7\u0394Terms \u03c1 d\u03c1 = \u2205\n  \u27e6 t \u2022 ts \u27e7\u0394Terms \u03c1 d\u03c1 = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2022 \u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    Derivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness-terms \u2205 \u03c1 d\u03c1 = refl\n  correctness-terms (t \u2022 ts) \u03c1 d\u03c1 =\n    cong\u2082 _\u2022_\n      (correctness t \u03c1 d\u03c1)\n      (correctness-terms ts \u03c1 d\u03c1)\n\n  correctness (const {\u03a3} {\u03c4} c ts) \u03c1 d\u03c1 =\n    begin\n      after\u208d \u03c4 \u208e (\u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1))\n    \u2261\u27e8 correctness-const c (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u27e9\n      \u27e6 c \u27e7Const (after-env (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1))\n    \u2261\u27e8 cong \u27e6 c \u27e7Const (correctness-terms ts \u03c1 d\u03c1) \u27e9\n      \u27e6 c \u27e7Const (\u27e6 ts \u27e7Terms (after-env d\u03c1))\n    \u220e where open \u2261-Reasoning\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f = \u27e6 s \u27e7 \u03c1\n      g = \u27e6 s \u27e7 (after-env d\u03c1)\n      u = \u27e6 t \u27e7 \u03c1\n      v = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f u \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u \u0394u) \u27e9\n        (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        g v\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"63695b71b63d310d06cc3681643756b6ed05941d","subject":"Defined curry.","message":"Defined curry.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n{-\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\n{-\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) , (\u03bb p \u2192  fst (F (snd p)) (fst p)) , cur-cond\n where\n   cur-cond : \u2200{u : U}{y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (snd y)) (fst y))\n     \u2192 \u22b8-cond \u03b2 \u03b3 ((\u03bb v \u2192 f (u , v)) , (\u03bb z \u2192 snd (F z) u)) y\n   cur-cond {u}{v , z} p\u2082 p\u2083 with p\u2081 {u , v}{z} \n   ... | p\u2081' with F z\n   ... | (j\u2081 , j\u2082) = p\u2081' (p\u2082 , p\u2083)\n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb p \u2192 fst (f (fst p)) (snd p)) , (\u03bb z \u2192 (\u03bb v \u2192 F (v , z)) , (\u03bb u \u2192 snd (f u) z)) , uncur-cond\n  where\n    uncur-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : Z}\n      \u2192 (\u03b1 \u2297\u1d63 \u03b2) u ((\u03bb v \u2192 F (v , y)) , (\u03bb u\u2081 \u2192 snd (f u\u2081) y))\n      \u2192 \u03b3 (fst (f (fst u)) (snd u)) y\n    uncur-cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u}{v , z} p\u2082\n    ... | p\u2081' with f u\n    ... | (j\u2081 , j\u2082) = p\u2081' p\u2083\n\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"67aa5099ae5a8d395971d87275e7a19e03681c4b","subject":"elgamal updates","message":"elgamal updates\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\nopen import sum\n\nmodule elgamal where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\n    \u03bcU : SumProp X \u2192 \u2200 u \u2192 SumProp (El u)\n    \u03bcU \u03bcX `\u22a4         = \u03bc\u22a4\n    \u03bcU \u03bcX `X         = \u03bcX\n    \u03bcU \u03bcX (u\u2080 `\u00d7 u\u2081) = \u03bcU \u03bcX u\u2080 \u00d7\u03bc \u03bcU \u03bcX u\u2081\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  where\n\n  -- open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum (\u03bcU \u03bc\u2124q u) (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = count \u03bc\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  -- open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  \u03bc\u2124q : SumProp \u2124q\n  \u03bc\u2124q = \u03bcFin q\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  {-\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n  -}\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum \u03bc\u2124q (f \u2218 [suc]) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ m) \u2261 sum \u03bc\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n  {-\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum \u03bc\u2124q (f \u2218 _\u229e_ x) \u2261 sum \u03bc\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 D (r , x , y , _) = D r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 D (r , x , y , z) = D r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba D) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (D b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2219_ : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_ : Message \u2192 G \u2192 Message)\n\n  -- Required for the correctness proof\n  (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n  (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n  -- Required for the security proof\n  (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n  (\u03bc\u2124q : SumProp \u2124q)\n  (R\u2090 : \u2605)\n  (\u03bcR\u2090 : SumProp R\u2090)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2219 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R\u2090 = (R\u2090 \u2192 PubKey \u2192 Bit \u2192 M)\n              \u00d7 (R\u2090 \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , D) b (r\u2090 , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      D r\u2090 pk (Enc pk (m r\u2090 pk b) y)\n\n      -- Unused\n    Game : (i : Bit) \u2192 \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 (Bit \u00d7 R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , D) (b , r\u2090 , x , y , z) = b ==\u1d47 D r\u2090 g\u02e3 (g\u02b8 , \u03b6)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n            \u03b6  = \u03b4 \u2219 m r\u2090 g\u02e3 b\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R\u2090} \u2192 Game 0b \u2261 Game0 {R\u2090}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M D (r , x , y , z) = D r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R\u2090} \u2192 Bit \u2192 EncAdv R\u2090 \u2192 DDHAdv R\u2090\n    TrA b (m , D) r\u2090 g\u02e3 g\u02b8 g\u02e3\u02b8 = D r\u2090 g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r\u2090 g\u02e3 b)\n\n    projM : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 Message\n    projM (m , _) b r\u2090 g\u02e3 = m r\u2090 g\u02e3 b\n\n    projD : \u2200 {R\u2090} \u2192 EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 Message \u2192 Bit\n    projD (_ , D) r\u2090 g\u02e3 g\u02b8 g\u1dbb\u2219M = D r\u2090 g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n    like-SS\u2141 : \u2200 {R\u2090 _I : \u2605} \u2192 EncAdv R\u2090 \u2192 Bit \u2192 (R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , D) b (r\u2090 , x , y , _z) =\n      D r\u2090 g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r\u2090 g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    -- open Sum\n\n    R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n    \u03bcR : SumProp R\n    \u03bcR = \u03bcR\u2090 \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q \u00d7\u03bc \u03bc\u2124q\n\n    #R_ : Count R\n    #R_ = count \u03bcR\n\n    #q_ : Count \u2124q\n    #q_ = count \u03bc\u2124q\n\n    _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n    f \u2248q g = #q f \u2261 #q g\n\n    _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n    f \u2248R g = #R f \u2261 #R g\n\n    functional-correctness : \u2200 x y m \u2192 Dec x (Enc (g^ x) m y) \u2261 m\n    functional-correctness x y m rewrite comm-^ g x y | \/-\u2219 (g^ y ^ x) m = refl\n\n    module Proof\n        (ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n        (otp-lem : \u2200 (A : Message \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081)))\n        (A : EncAdv R\u2090) (b : Bit)\n      where\n\n        OTP\u2141-lem : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248R OTP\u2141 M\u2081 D\n        OTP\u2141-lem D M\u2080 M\u2081 = sum-ext \u03bcR\u2090 (\u03bb r \u2192\n                             sum-ext \u03bc\u2124q (\u03bb x \u2192\n                               sum-ext \u03bc\u2124q (\u03bb y \u2192\n                                 pf r x y)))\n          where\n          f0 = \u03bb M r x y z \u2192 OTP\u2141 M D (r , x , y , z)\n          f1 = \u03bb M r x y \u2192 sum \u03bc\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n          f2 = \u03bb M r x \u2192 sum \u03bc\u2124q (f1 M r x)\n          f3 = \u03bb M r \u2192 sum \u03bc\u2124q (f2 M r)\n          pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n          pf r x y = pf'\n            where g\u02e3 = g^ x\n                  g\u02b8 = g^ y\n                  m\u2080 = M\u2080 r g\u02e3\n                  m\u2081 = M\u2081 r g\u02e3\n                  f5 = \u03bb M z \u2192 D r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                  pf' : f5 m\u2080 \u2248q f5 m\u2081\n                  pf' rewrite otp-lem (D r g\u02e3 g\u02b8) m\u2080 m\u2081 = refl\n\n        A\u1d47 = TrA b A\n        A\u00ac\u1d47 = TrA (not b) A\n\n        pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n        pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n        pf1 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf0,5)\n\n        pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n        pf2 = ddh-hyp A\u1d47\n\n        pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n        pf2,5 = refl\n\n        pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n        pf3 = OTP\u2141-lem (projD A) (projM A b) (projM A (not b))\n\n        pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n        pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n        pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n        pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n        pf5 = sum-ext \u03bcR (cong Bool.to\u2115 \u2218 pf4,5)\n\n        final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n        final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n\n    -- Required for decryption\n    (_\/_ : G \u2192 G \u2192 G)\n\n    -- Required for the correctness proof\n    (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    \n    {-\n    (_\u207b\u00b9 : G \u2192 G)\n    (\u207b\u00b9-inverse : \u2200 x \u2192 x \u207b\u00b9 \u2219 x \u2261 1G)\n    -}\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_\n           _\/_ \/-\u2219 comm-^\n           dist-^-\u22a0 \u03bc\u2124q R\u2090 \u03bcR\u2090 public\n\n    module OTP\u2141-LEM\n            (otp-lem1 : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n      where\n\n        otp-lem : \u2200 (A : G \u2192 Bit) m\u2080 m\u2081 \u2192 (\u03bb x \u2192 A (g^ x \u2219 m\u2080)) \u2248q (\u03bb x \u2192 A (g^ x \u2219 m\u2081))\n        otp-lem A m\u2080 m\u2081 rewrite otp-lem1 A m\u2080 | otp-lem1 A m\u2081 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|)\n\n    -- (\/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y)\n    (comm-^   : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n\n    -- Required for the proof\n    (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n    (\u03bc\u2124q : SumProp \u2124q)\n    (R\u2090 : \u2605)\n    (\u03bcR\u2090 : SumProp R\u2090)\n    where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    _\/_ : Message \u2192 G \u2192 Message\n    _\/_ = {!!}\n\n    {-\n    \/-\u2219 : \u2200 x y \u2192 (x \u2219 y) \/ x \u2261 y\n    \/-\u2219 x y = ?\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_ _\/_ {!!} {!!}\n           dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext public\n           -}\n\n           {-\n    OTP\u2141-lem : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248R OTP\u2141 M\u2081 D\n    OTP\u2141-lem = ?\n    -}\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  open \u2124q-count \u2124q _\u229e_ \u03bc\u2124q sum\u2124q-\u229e-lem\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum `R\u2090\n  sumR\u2090-ext = sum-ext `R\u2090\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u2219_ _\u207b\u00b9 dist-^-\u22a0 sum\u2124q sum\u2124q-ext R\u2090 sumR\u2090 sumR\u2090-ext\n  open EB hiding (g^_)\n\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n  open OTP\u2141-LEM otp-lem\n\n  {-\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = Proof.final ddh-hyp OTP\u2141-lem A 0b\n  -}\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\nimport cont as cont\nopen cont using (Cont; ContA)\n\nmodule elgamal where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bit \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bit \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\nmodule EntropySmoothing\n  (M    : \u2605)        -- Message\n  (Hash : \u2605)\n  (\u210b   : M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)        -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (m , _ , r\u2090) = A r\u2090 (\u210b m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (_ , h , r\u2090) = A r\u2090 h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule EntropySmoothingWithKey\n  (M    : \u2605)\n  (Key  : \u2605)\n  (Hash : \u2605)\n  (\u210b   : Key \u2192 M \u2192 Hash) -- Hashing function\n  (R\u2090   : \u2605)              -- Adversary randomness\n  where\n\n  -- Entropy smoothing adversary\n  ESAdv : \u2605\n  ESAdv = R\u2090 \u2192 Key \u2192 Hash \u2192 Bit\n\n  -- The randomness universe needed for the following games\n  R : \u2605\n  R = Key \u00d7 M \u00d7 Hash \u00d7 R\u2090\n\n  -- In this game we always use \u210b on a random message\n  ES\u2141\u2080 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2080 A (k , m , _ , r\u2090) = A r\u2090 k (\u210b k m)\n\n  -- In this game we just retrun a random Hash value\n  ES\u2141\u2081 : ESAdv \u2192 R \u2192 Bit\n  ES\u2141\u2081 A (k , _ , h , r\u2090) = A r\u2090 k h\n\n  ES\u2141 : ESAdv \u2192 Bit \u2192 R \u2192 Bit\n  ES\u2141 A b r = (case b 0\u2192 ES\u2141\u2080 1\u2192 ES\u2141\u2081) A r\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum.Sum \u2124q)\n  (sum\u2124q-ext : Sum.SumExt sum\u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  where\n\n  open Sum\n  open Univ \u2124q public\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  sum : \u2200 u \u2192 Sum (El u)\n  sum `\u22a4         = sum\u22a4\n  sum `X         = sum\u2124q\n  sum (u\u2080 `\u00d7 u\u2081) = sumProd (sum u\u2080) (sum u\u2081)\n\n  sum-ext : \u2200 u \u2192 SumExt (sum u)\n  sum-ext `\u22a4         = sum\u22a4-ext\n  sum-ext `X         = sum\u2124q-ext\n  sum-ext (u\u2080 `\u00d7 u\u2081) = sumProd-ext (sum-ext u\u2080) (sum-ext u\u2081)\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum u (Bool.to\u2115 \u2218 run\u21ba f)\n\n  #q_ : Count \u2124q\n  #q_ = sumToCount sum\u2124q\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _\n  lem x Adv = sum\u2124q-\u229e-lem x (Bool.to\u2115 \u2218 Adv)\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047)\n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  open Sum\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum\u2124q (f \u2218 [suc]) \u2261 sum\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\n  vmap-ext : \u2200 {A B : \u2605} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 vmap f \u2257 vmap {n = n} g\n  vmap-ext f\u2257g [] = refl\n  vmap-ext f\u2257g (x \u2237 xs) rewrite f\u2257g x | vmap-ext f\u2257g xs = refl\n\n  sum\u2124q-ext : SumExt sum\u2124q\n  sum\u2124q-ext f\u2257g rewrite vmap-ext f\u2257g all\u2124q = refl\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  {-\n  g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n  g^-inj = {!!}\n  -}\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : Sum.Sum \u2124q)\n  (sum\u2124q-ext : Sum.SumExt sum\u2124q)\n  (sum\u2124q-\u229e-lem : \u2200 x f \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2219_ : G \u2192 G \u2192 G)\n  where\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  open Sum\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-ext sum\u2124q-\u229e-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g^ x\n\n  #G : Count G\n  #G f = #q (f \u2218 g^_)\n\n  {-\n  #G-\u2219 : \u2200 f m \u2192 #G (f \u2218 _\u2219_ m) \u2261 #G f\n  #G-\u2219 f m = {!!}\n  -}\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = R \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n    DDH\u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    DDH\u2141\u2080 D (r , x , y , _) = D r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n    DDH\u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141\u2081 D (r , x , y , z) = D r (g^ x) (g^ y) (g^ z)\n\n    DDH\u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    DDH\u2141 D b = (case b 0\u2192 DDH\u2141\u2080 1\u2192 DDH\u2141\u2081) D\n\n    -- DDH\u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bit \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bit\n    -- DDH\u2141\u2032 D (b , x , y , z , r) = DDH\u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bit\n        DDH\u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (R `\u00d7 `\u2124q `\u00d7 `\u2124q `\u00d7 _I) Bit\n        run\u21ba (DDH\u2141\u2080\u21ba D) = DDH\u2141\u2080 (\u03bb a b c d \u2192 run\u21ba (D b c d) a)\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2219_ : G \u2192 Message \u2192 Message)\n  (_\u2219\u207b\u00b9_ : G \u2192 Message \u2192 Message)\n  where\n\n    g^_ : \u2124q \u2192 G\n    g^ x = g ^ x\n\n    -- g\u02e3 is the pk\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc g\u02e3 m y = g\u02b8 , \u03b6 where\n      g\u02b8 = g^ y\n      \u03b4 = g\u02e3 ^ y\n      \u03b6 = \u03b4 \u2219 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba g\u02e3 m = mk (Enc g\u02e3 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (g\u02b8 , \u03b6) = (g\u02b8 ^ x) \u2219\u207b\u00b9 \u03b6\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R = (R \u2192 PubKey \u2192 Bit \u2192 M) \u00d7 (R \u2192 PubKey \u2192 C \u2192 Bit)\n\n    SS\u2141 : \u2200 {R _I : \u2605} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    SS\u2141 (m , D) b (r , x , y , z) =\n      let pk = proj\u2081 (KeyGen x) in\n      D r pk (Enc pk (m r pk b) y)\n\n    Game : (i : Bit) \u2192 \u2200 {R} \u2192 EncAdv R \u2192 (Bit \u00d7 R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    Game i (m , D) (b , r , x , y , z) =\n      let g\u02e3 = g^ x\n          g\u02b8 = g^ y\n          \u03b4  = g\u02e3 ^ case i 0\u2192 y 1\u2192 z\n          \u03b6  = \u03b4 \u2219 m r g\u02e3 b\n      in b ==\u1d47 D r g\u02e3 (g\u02b8 , \u03b6)\n\n    {-\n    Game-0b\u2261Game0 : \u2200 {R} \u2192 Game 0b \u2261 Game0 {R}\n    Game-0b\u2261Game0 = refl\n      -}\n\n    open DDH \u2124q _\u22a0_ G g^_ public\n\n    -- Game0 \u2248 Game 0b\n    -- Game1 = Game 1b\n\n    -- Game0 \u2264 \u03b5\n    -- Game1 \u2261 0\n\n    OTP\u2141 : \u2200 {R : \u2605} \u2192 (R \u2192 G \u2192 Message) \u2192 (R \u2192 G \u2192 G \u2192 Message \u2192 Bit)\n                     \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q) \u2192 Bit\n    OTP\u2141 M D (r , x , y , z) = D r g\u02e3 g\u02b8 (g\u1dbb \u2219 M r g\u02e3)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n            g\u1dbb = g^ z\n\n    TrA : \u2200 {R} \u2192 Bit \u2192 EncAdv R \u2192 DDHAdv R\n    TrA b (m , D) r g\u02e3 g\u02b8 g\u02e3\u02b8 = D r g\u02e3 (g\u02b8 , g\u02e3\u02b8 \u2219 m r g\u02e3 b)\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u207b\u00b9 : G \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n    where\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_ (flip _\/_) public\n\n    like-SS\u2141 : \u2200 {R _I : \u2605} \u2192 EncAdv R \u2192 Bit \u2192 (R \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I) \u2192 Bit\n    like-SS\u2141 (m , D) b (r , x , y , _z) =\n      D r g\u02e3 (g\u02b8 , (g\u02e3 ^ y) \u2219 m r g\u02e3 b)\n      where g\u02e3 = g^ x\n            g\u02b8 = g^ y\n\n    SS\u2141\u2261like-SS\u2141 : \u2200 {R _I} \u2192 SS\u2141 {R} {_I} \u2261 like-SS\u2141\n    SS\u2141\u2261like-SS\u2141 = refl\n\n    open Sum\n    module WithSumRAnd\u2124qProps\n        (R\u2090 : \u2605)\n        (sumR\u2090 : Sum R\u2090)\n        (sumR\u2090-ext : SumExt sumR\u2090)\n        (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n        (sum\u2124q : Sum \u2124q)\n        (sum\u2124q-ext : SumExt sum\u2124q)\n        where\n\n        module SumR where\n          R = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n          sumR : Sum R\n          sumR = sumProd sumR\u2090 (sumProd sum\u2124q (sumProd sum\u2124q sum\u2124q))\n          #R_ : Count R\n          #R_ = sumToCount sumR\n          sumR-ext : SumExt sumR\n          sumR-ext = sumProd-ext sumR\u2090-ext (sumProd-ext sum\u2124q-ext (sumProd-ext sum\u2124q-ext sum\u2124q-ext))\n        private\n            #q_ : Count \u2124q\n            #q_ = sumToCount sum\u2124q\n\n        open SumR\n\n        _\u2248q_ : (f g : \u2124q \u2192 Bit) \u2192 \u2605\n        f \u2248q g = #q f \u2261 #q g\n\n        _\u2248R_ : (f g : R \u2192 Bit) \u2192 \u2605\n        f \u2248R g = #R f \u2261 #R g\n\n        module EvenMoreProof\n            (ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b)\n            (otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x)))\n            where\n\n                otp-lem' : \u2200 D M\u2080 M\u2081 \u2192 OTP\u2141 M\u2080 D \u2248R OTP\u2141 M\u2081 D\n                otp-lem' D M\u2080 M\u2081 = sumR\u2090-ext (\u03bb r \u2192\n                                     sum\u2124q-ext (\u03bb x \u2192\n                                       sum\u2124q-ext (\u03bb y \u2192\n                                         pf r x y)))\n                  where\n                  f0 = \u03bb M r x y z \u2192 OTP\u2141 M D (r , x , y , z)\n                  f1 = \u03bb M r x y \u2192 sum\u2124q (Bool.to\u2115 \u2218 f0 M r x y)\n                  f2 = \u03bb M r x \u2192 sum\u2124q (f1 M r x)\n                  f3 = \u03bb M r \u2192 sum\u2124q (f2 M r)\n                  pf : \u2200 r x y \u2192 f1 M\u2080 r x y \u2261 f1 M\u2081 r x y\n                  pf r x y = pf'\n                    where g\u02e3 = g^ x\n                          g\u02b8 = g^ y\n                          m\u2080 = M\u2080 r g\u02e3\n                          m\u2081 = M\u2081 r g\u02e3\n                          f5 = \u03bb M z \u2192 D r g\u02e3 g\u02b8 (g^ z \u2219 M)\n                          pf' : f5 m\u2080 \u2248q f5 m\u2081\n                          pf' rewrite otp-lem (D r g\u02e3 g\u02b8) m\u2080\n                                    | otp-lem (D r g\u02e3 g\u02b8) m\u2081 = refl\n\n                projM : EncAdv R\u2090 \u2192 Bit \u2192 R\u2090 \u2192 G \u2192 G\n                projM (m , _) b r g\u02e3 = m r g\u02e3 b\n\n                projD : EncAdv R\u2090 \u2192 R\u2090 \u2192 G \u2192 G \u2192 G \u2192 Bit\n                projD (_ , D) r g\u02e3 g\u02b8 g\u1dbb\u2219M = D r g\u02e3 (g\u02b8 , g\u1dbb\u2219M)\n\n                module WithAdversary (A : EncAdv R\u2090) b where\n                    A\u1d47 = TrA b A\n                    A\u00ac\u1d47 = TrA (not b) A\n\n                    pf0,5 : SS\u2141 A b \u2257 DDH\u2141 A\u1d47 0b\n                    pf0,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf1 : SS\u2141 A b \u2248R DDH\u2141 A\u1d47 0b\n                    pf1 = sumR-ext (cong Bool.to\u2115 \u2218 pf0,5)\n\n                    pf2 : DDH\u2141 A\u1d47 0b \u2248R DDH\u2141 A\u1d47 1b\n                    pf2 = ddh-hyp A\u1d47\n\n                    pf2,5 : DDH\u2141 A\u1d47 1b \u2261 OTP\u2141 (projM A b) (projD A)\n                    pf2,5 = refl\n\n                    pf3 : DDH\u2141 A\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 1b\n                    pf3 = otp-lem' (projD A) (projM A b) (projM A (not b))\n\n                    pf4 : DDH\u2141 A\u00ac\u1d47 1b \u2248R DDH\u2141 A\u00ac\u1d47 0b\n                    pf4 = \u2261.sym (ddh-hyp A\u00ac\u1d47)\n\n                    pf4,5 : SS\u2141 A (not b) \u2257 DDH\u2141 A\u00ac\u1d47 0b\n                    pf4,5 (r , x , y , z) rewrite dist-^-\u22a0 g x y = refl\n\n                    pf5 : SS\u2141 A (not b) \u2248R DDH\u2141 A\u00ac\u1d47 0b\n                    pf5 = sumR-ext (cong Bool.to\u2115 \u2218 pf4,5)\n\n                    final : SS\u2141 A b \u2248R SS\u2141 A (not b)\n                    final rewrite pf1 | pf2 | pf3 | pf4 | pf5 = refl\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|) where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_\n\nmodule \u27e8\u2124p\u27e9\u2605 p-3 {- p is prime -} (`R\u2090 : `\u2605) where\n  p : \u2115\n  p = 3 + p-3\n\n  q : \u2115\n  q = p \u2238 1\n\n  module G = G-implem p q (## 2) (## 0) (## 1) (## 0) (## 1)\n  open G\n\n  postulate\n    _\u207b\u00b9 : G \u2192 G\n    dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y\n\n  module EB = El-Gamal-Base _ _\u22a0_ G g _^_ _\u207b\u00b9 _\u2219_\n  open EB hiding (g^_)\n\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-ext sum\u2124q-\u229e-lem\n\n  R\u2090 = El `R\u2090\n  sumR\u2090 = sum `R\u2090\n  sumR\u2090-ext = sum-ext `R\u2090\n\n  open WithSumRAnd\u2124qProps R\u2090 sumR\u2090 sumR\u2090-ext dist-^-\u22a0 sum\u2124q sum\u2124q-ext\n\n  open SumR\n  postulate\n    ddh-hyp : (A : DDHAdv R\u2090) \u2192 DDH\u2141 A 0b \u2248R DDH\u2141 A 1b\n    otp-lem : \u2200 (A : G \u2192 Bit) m \u2192 (\u03bb x \u2192 A (g^ x \u2219 m)) \u2248q (\u03bb x \u2192 A (g^ x))\n\n  open EvenMoreProof ddh-hyp otp-lem\n\n  final : \u2200 A \u2192 SS\u2141 A 0b \u2248R SS\u2141 A 1b\n  final A = WithAdversary.final A 0b\n\nmodule \u27e8\u212411\u27e9\u2605 = \u27e8\u2124p\u27e9\u2605 (11 \u2238 3)\n                   `X -- the amount of adversary randomness\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"351737b9010e4a14a8b06675017e628dfb7d6269","subject":"Mark the Translation module as unfinished","message":"Mark the Translation module as unfinished\n","repos":"crypto-agda\/protocols","old_file":"PTT\/Translation.agda","new_file":"PTT\/Translation.agda","new_contents":"-- UNFINISHED\n\nopen import Function\nopen import Data.Product hiding (zip)\n                         renaming (_,_ to \u27e8_,_\u27e9; proj\u2081 to fst; proj\u2082 to snd;\n                                   map to \u00d7map)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\n\nopen import PTT.Dom\nimport PTT.Session as Session\nimport PTT.Map as Map\nimport PTT.Env as Env\nimport PTT.Proto as Proto\nopen Session hiding (Ended)\nopen Env     hiding (_\/\u2080_; _\/\u2081_; Ended)\nopen Proto   hiding ()\nopen import PTT.Term\nopen import PTT.Split\nopen import Relation.Binary.PropositionalEquality.NP hiding (J)\n\nS-\u214b-inp : \u2200 {c c\u2080 c\u2081 \u03b4I S\u2080 S\u2081}{I : Proto \u03b4I}(l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n  \u2192 S\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9 \u2192 S\u27e8 I \u27e9\nS-\u214b-inp l P = {!!}\n\ntranslate : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 \u27e8 I \u27e9 \u2192 S\u27e8 I \u27e9\ntranslate (end x) = S-T (TC-end x)\ntranslate (\u214b-inp l P) = S-\u214b-inp l (translate (P c\u2080 c\u2081))\n  where postulate c\u2080 c\u2081 : URI\ntranslate (\u2297-out l P) = {!!}\ntranslate (split \u03c3s A1 P P\u2081) = {!!}\ntranslate (nu D P) = {!!}\n\n{-\nmodule PTT.Translation where\nmodule Translation\n {t}\n (T\u27e8_\u27e9 : \u2200 {\u03b4I} \u2192 Proto \u03b4I \u2192 Set t)\n (T-\u2297-out :\n    \u2200 {\u03b4I I c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 \u00ab S\u2080 \u00bb ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 \u00ab S\u2081 \u00bb ] \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-\u214b-inp :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}{c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-end :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}\n      (E : Ended I)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-split :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : T\u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : T\u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-cut :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}{S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (\u03c3s : Selections \u03b4I)\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/\u2080 \u03c3s ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/\u2081 \u03c3s ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-\u2297-reorg :\n    \u2200 {\u03b4I \u03b4E c c\u2080 c\u2081 S\u2080 S\u2081}{J : Proto \u03b4I}{E : Env \u03b4E}\n      (l  : [ E ]\u2208 J)\n      (l\u2080 : c\u2080 \u21a6 \u00ab S\u2080 \u00bb \u2208 E)\n      (l\u2081 : c\u2081 \u21a6 \u00ab S\u2081 \u00bb \u2208 E)\n      (P : T\u27e8 J \u27e9)\n    \u2192 T\u27e8 J Proto.\/ l ,[ (E Env.\/' l\u2080 \/D (\u21a6\u2208.lA l\u2081) , c \u21a6 \u00ab S\u2080 \u2297 S\u2081 \u00bb) ] \u27e9)\n\n (T-conv : \u2200 {\u03b4I \u03b4J}{I : Proto \u03b4I}{J : Proto \u03b4J} \u2192 I \u2248 J \u2192 T\u27e8 I \u27e9 \u2192 T\u27e8 J \u27e9)\n\n  where\n\n  T-fwd : \u2200 {S\u2080 S\u2081} (S : Dual S\u2080 S\u2081) c\u2080 c\u2081 \u2192 T\u27e8 \u00b7 ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9\n  T-fwd \ud835\udfd9\u22a5 c\u2080 c\u2081 = {!!}\n  T-fwd \u22a5\ud835\udfd9 c\u2080 c\u2081 = {!!}\n  T-fwd (act (?! S S\u2081)) c\u2080 c\u2081 = {!!}\n  T-fwd (act (!? S S\u2081)) c\u2080 c\u2081 = {!!}\n  T-fwd (\u2297\u214b S\u2080 S\u2081 S\u2082 S\u2083) c\u2080 c\u2081 =\n    T-\u214b-inp here[]' \u03bb c\u2082 c\u2083 \u2192\n      T-\u2297-out (there\u2026' (there\u2026' (there\u2026' here\u2026')))\n              ((((\u00b7 ,[ (\u03b5 , c\u2080 \u21a6 0\u2082) ]) ,[ (\u03b5 , c\u2081 \u21a6 0\u2082) ]) ,[ (\u03b5 , c\u2082 \u21a6 0\u2082) ]) ,[ (\u03b5 , c\u2083 \u21a6 1\u2082) ])\n              (\u03b5 , c\u2080 \u21a6 0\u2082)\n              ((((\u00b7 ,[ {!!} ]) ,[ {!!} ]) ,[ {!!} ]) ,[ {!!} ])\n              (\u03bb c\u2084 \u2192 T-conv (\u2248,[] (\u2248-! (\u2248,[swap] \u2248-\u2219 {!\u2248,[] \u2248-refl ?!})) (\u223c,\u21a6 (\u223c-! \u223c,\u21a6end))) (T-fwd S\u2081 c\u2083 c\u2084))\n              (\u03bb c\u2084 \u2192 T-conv (\u2248,[] (\u2248,[] (\u2248-! (\u2248,[end] _ \u2248-\u2219 (\u2248,[end] _ \u2248-\u2219 \u2248,[end] _))) \u223c-refl) (\u223c,\u21a6 (\u223c-! \u223c,\u21a6end))) (T-fwd S\u2083 c\u2083 c\u2084))\n  T-fwd (\u214b\u2297 S S\u2081 S\u2082 S\u2083) c\u2080 c\u2081 = {!!}\n\n  go : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 \u27e8 I \u27e9 \u2192 T\u27e8 I \u27e9\n  go (end x) = T-end x\n  go (\u214b-inp l P) = T-\u214b-inp l (\u03bb c\u2080 c\u2081 \u2192 go (P c\u2080 c\u2081))\n  go {I = I}(\u2297-out {c} {S\u2080} {S\u2081} l P) = T-conv e rPP\n    where postulate c0 c1 : URI\n          open [\u21a6\u2026]\u2208 l\n          F = E Env.\/' lE , c0 \u21a6 \u00ab S\u2080 \u00bb , c1 \u21a6 \u00ab S\u2081 \u00bb\n          J = I Proto.\/ lI ,[ F ]\n          G = F \/D there here \/D here , c \u21a6 \u00ab S\u2080 \u2297 S\u2081 \u00bb\n          K = J Proto.\/ heRe[] ,[ G ]\n          rPP : T\u27e8 K \u27e9\n          rPP = T-\u2297-reorg heRe[] (theRe here) heRe (go (P c0 c1))\n          e : K \u2248 I\n          e = \u2248,[] (\u2248,[end] (Ended-\/* F)) (\u223c,\u21a6 (\u223c,\u21a6end \u223c-\u2219 \u223c,\u21a6end)) \u2248-\u2219 (\u2248-! (\u2248-\/\u2026,[\u2026] l))\n  go {I = I}(nu {S\u2080} {S\u2081} D P) = T-conv {!!} (T-cut {I = I'} D (sel\u2080 _ ,[ (\u03b5 , c \u21a6 0\u2082) ] ,[ (\u03b5 , c' \u21a6 1\u2082) ]) {!!} (\u03bb c\u2080' \u2192 {!cPP!}) (\u03bb c\u2081' \u2192 {!T-fwd!}))\n    where postulate c c' c0 c1 : URI\n          E   = \u03b5 , c0 \u21a6 \u00ab S\u2080 \u00bb , c1 \u21a6 \u00ab S\u2081 \u00bb\n          E\/* = \u03b5 , c0 \u21a6 end    , c1 \u21a6 end\n          J = I ,[ E ]\n          -- K = J \/ here ,[ E\/* , c \u21a6 S\u2080 \u2297 S\u2081 ]\n          K = I ,[ E\/* ] ,[ E\/* , c \u21a6 \u00ab S\u2080 \u2297 S\u2081 \u00bb ]\n          gP : T\u27e8 J \u27e9\n          gP = go (P c0 c1)\n          rPP : T\u27e8 K \u27e9\n          rPP = T-\u2297-reorg {c = c} heRe[] (theRe here) heRe gP\n          e : K \u2248 I ,[ c \u21a6 S\u2080 \u2297 S\u2081 ]\n          e = \u2248,[] (\u2248,[end] _) (\u223c,\u21a6 (\u223c,\u21a6end \u223c-\u2219 \u223c,\u21a6end))\n          cPP : T\u27e8 I ,[ c \u21a6 S\u2080 \u2297 S\u2081 ] \u27e9\n          cPP = T-conv e rPP\n          I' = I ,[ c \u21a6 S\u2080 \u2297 S\u2081 ] ,[ c' \u21a6 {!!} ]\n  go (split \u03c3s A P\u2080 P\u2081) = T-split \u03c3s A (go P\u2080) (go P\u2081)\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Function\nopen import Data.Product hiding (zip)\n                         renaming (_,_ to \u27e8_,_\u27e9; proj\u2081 to fst; proj\u2082 to snd;\n                                   map to \u00d7map)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\n\nopen import PTT.Dom\nimport PTT.Session as Session\nimport PTT.Map as Map\nimport PTT.Env as Env\nimport PTT.Proto as Proto\nopen Session hiding (Ended)\nopen Env     hiding (_\/\u2080_; _\/\u2081_; Ended)\nopen Proto   hiding ()\nopen import PTT.Term\nopen import PTT.Split\nopen import Relation.Binary.PropositionalEquality.NP hiding (J)\n\nS-\u214b-inp : \u2200 {c c\u2080 c\u2081 \u03b4I S\u2080 S\u2081}{I : Proto \u03b4I}(l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n  \u2192 S\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9 \u2192 S\u27e8 I \u27e9\nS-\u214b-inp l P = {!!}\n\ntranslate : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 \u27e8 I \u27e9 \u2192 S\u27e8 I \u27e9\ntranslate (end x) = S-T (TC-end x)\ntranslate (\u214b-inp l P) = S-\u214b-inp l (translate (P c\u2080 c\u2081))\n  where postulate c\u2080 c\u2081 : URI\ntranslate (\u2297-out l P) = {!!}\ntranslate (split \u03c3s A1 P P\u2081) = {!!}\ntranslate (nu D P) = {!!}\n\n{-\nmodule PTT.Translation where\nmodule Translation\n {t}\n (T\u27e8_\u27e9 : \u2200 {\u03b4I} \u2192 Proto \u03b4I \u2192 Set t)\n (T-\u2297-out :\n    \u2200 {\u03b4I I c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u2297 S\u2081 \u2026]\u2208 I)\n      (\u03c3s : Selections \u03b4I)\n      (\u03c3E : Selection ([\u21a6\u2026]\u2208.\u03b4E l))\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I [\/\u2026] l \/\u2080 \u03c3s ,[ E\/ l Env.\/\u2080 \u03c3E , c\u2080 \u21a6 \u00ab S\u2080 \u00bb ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I [\/\u2026] l \/\u2081 \u03c3s ,[ E\/ l Env.\/\u2081 \u03c3E , c\u2081 \u21a6 \u00ab S\u2081 \u00bb ] \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-\u214b-inp :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}{c S\u2080 S\u2081}\n      (l : [ c \u21a6 S\u2080 \u214b S\u2081 ]\u2208 I)\n      (P : \u2200 c\u2080 c\u2081 \u2192 T\u27e8 I [\/] l ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-end :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}\n      (E : Ended I)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-split :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}\n      (\u03c3s : Selections \u03b4I)\n      (A1 : AtMost 1 \u03c3s)\n      (P\u2080 : T\u27e8 I \/\u2080 \u03c3s \u27e9)\n      (P\u2081 : T\u27e8 I \/\u2081 \u03c3s \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-cut :\n    \u2200 {\u03b4I}{I : Proto \u03b4I}{S\u2080 S\u2081}\n      (D : Dual S\u2080 S\u2081)\n      (\u03c3s : Selections \u03b4I)\n      (A0 : AtMost 0 \u03c3s)\n      (P\u2080 : \u2200 c\u2080 \u2192 T\u27e8 I \/\u2080 \u03c3s ,[ c\u2080 \u21a6 S\u2080 ] \u27e9)\n      (P\u2081 : \u2200 c\u2081 \u2192 T\u27e8 I \/\u2081 \u03c3s ,[ c\u2081 \u21a6 S\u2081 ] \u27e9)\n    \u2192 T\u27e8 I \u27e9)\n\n (T-\u2297-reorg :\n    \u2200 {\u03b4I \u03b4E c c\u2080 c\u2081 S\u2080 S\u2081}{J : Proto \u03b4I}{E : Env \u03b4E}\n      (l  : [ E ]\u2208 J)\n      (l\u2080 : c\u2080 \u21a6 \u00ab S\u2080 \u00bb \u2208 E)\n      (l\u2081 : c\u2081 \u21a6 \u00ab S\u2081 \u00bb \u2208 E)\n      (P : T\u27e8 J \u27e9)\n    \u2192 T\u27e8 J Proto.\/ l ,[ (E Env.\/' l\u2080 \/D (\u21a6\u2208.lA l\u2081) , c \u21a6 \u00ab S\u2080 \u2297 S\u2081 \u00bb) ] \u27e9)\n\n (T-conv : \u2200 {\u03b4I \u03b4J}{I : Proto \u03b4I}{J : Proto \u03b4J} \u2192 I \u2248 J \u2192 T\u27e8 I \u27e9 \u2192 T\u27e8 J \u27e9)\n\n  where\n\n  T-fwd : \u2200 {S\u2080 S\u2081} (S : Dual S\u2080 S\u2081) c\u2080 c\u2081 \u2192 T\u27e8 \u00b7 ,[ c\u2080 \u21a6 S\u2080 ] ,[ c\u2081 \u21a6 S\u2081 ] \u27e9\n  T-fwd \ud835\udfd9\u22a5 c\u2080 c\u2081 = {!!}\n  T-fwd \u22a5\ud835\udfd9 c\u2080 c\u2081 = {!!}\n  T-fwd (act (?! S S\u2081)) c\u2080 c\u2081 = {!!}\n  T-fwd (act (!? S S\u2081)) c\u2080 c\u2081 = {!!}\n  T-fwd (\u2297\u214b S\u2080 S\u2081 S\u2082 S\u2083) c\u2080 c\u2081 =\n    T-\u214b-inp here[]' \u03bb c\u2082 c\u2083 \u2192\n      T-\u2297-out (there\u2026' (there\u2026' (there\u2026' here\u2026')))\n              ((((\u00b7 ,[ (\u03b5 , c\u2080 \u21a6 0\u2082) ]) ,[ (\u03b5 , c\u2081 \u21a6 0\u2082) ]) ,[ (\u03b5 , c\u2082 \u21a6 0\u2082) ]) ,[ (\u03b5 , c\u2083 \u21a6 1\u2082) ])\n              (\u03b5 , c\u2080 \u21a6 0\u2082)\n              ((((\u00b7 ,[ {!!} ]) ,[ {!!} ]) ,[ {!!} ]) ,[ {!!} ])\n              (\u03bb c\u2084 \u2192 T-conv (\u2248,[] (\u2248-! (\u2248,[swap] \u2248-\u2219 {!\u2248,[] \u2248-refl ?!})) (\u223c,\u21a6 (\u223c-! \u223c,\u21a6end))) (T-fwd S\u2081 c\u2083 c\u2084))\n              (\u03bb c\u2084 \u2192 T-conv (\u2248,[] (\u2248,[] (\u2248-! (\u2248,[end] _ \u2248-\u2219 (\u2248,[end] _ \u2248-\u2219 \u2248,[end] _))) \u223c-refl) (\u223c,\u21a6 (\u223c-! \u223c,\u21a6end))) (T-fwd S\u2083 c\u2083 c\u2084))\n  T-fwd (\u214b\u2297 S S\u2081 S\u2082 S\u2083) c\u2080 c\u2081 = {!!}\n{-\n  -}\n\n  go : \u2200 {\u03b4I}{I : Proto \u03b4I} \u2192 \u27e8 I \u27e9 \u2192 T\u27e8 I \u27e9\n  go (end x) = T-end x\n  go (\u214b-inp l P) = T-\u214b-inp l (\u03bb c\u2080 c\u2081 \u2192 go (P c\u2080 c\u2081))\n  go {I = I}(\u2297-out {c} {S\u2080} {S\u2081} l P) = T-conv e rPP\n    where postulate c0 c1 : URI\n          open [\u21a6\u2026]\u2208 l\n          F = E Env.\/' lE , c0 \u21a6 \u00ab S\u2080 \u00bb , c1 \u21a6 \u00ab S\u2081 \u00bb\n          J = I Proto.\/ lI ,[ F ]\n          G = F \/D there here \/D here , c \u21a6 \u00ab S\u2080 \u2297 S\u2081 \u00bb\n          K = J Proto.\/ heRe[] ,[ G ]\n          rPP : T\u27e8 K \u27e9\n          rPP = T-\u2297-reorg heRe[] (theRe here) heRe (go (P c0 c1))\n          e : K \u2248 I\n          e = \u2248,[] (\u2248,[end] (Ended-\/* F)) (\u223c,\u21a6 (\u223c,\u21a6end \u223c-\u2219 \u223c,\u21a6end)) \u2248-\u2219 (\u2248-! (\u2248-\/\u2026,[\u2026] l))\n  go {I = I}(nu {S\u2080} {S\u2081} D P) = T-conv {!!} (T-cut {I = I'} D (sel\u2080 _ ,[ (\u03b5 , c \u21a6 0\u2082) ] ,[ (\u03b5 , c' \u21a6 1\u2082) ]) {!!} (\u03bb c\u2080' \u2192 {!cPP!}) (\u03bb c\u2081' \u2192 {!T-fwd!}))\n    where postulate c c' c0 c1 : URI\n          E   = \u03b5 , c0 \u21a6 \u00ab S\u2080 \u00bb , c1 \u21a6 \u00ab S\u2081 \u00bb\n          E\/* = \u03b5 , c0 \u21a6 end    , c1 \u21a6 end\n          J = I ,[ E ]\n          -- K = J \/ here ,[ E\/* , c \u21a6 S\u2080 \u2297 S\u2081 ]\n          K = I ,[ E\/* ] ,[ E\/* , c \u21a6 \u00ab S\u2080 \u2297 S\u2081 \u00bb ]\n          gP : T\u27e8 J \u27e9\n          gP = go (P c0 c1)\n          rPP : T\u27e8 K \u27e9\n          rPP = T-\u2297-reorg {c = c} heRe[] (theRe here) heRe gP\n          e : K \u2248 I ,[ c \u21a6 S\u2080 \u2297 S\u2081 ]\n          e = \u2248,[] (\u2248,[end] _) (\u223c,\u21a6 (\u223c,\u21a6end \u223c-\u2219 \u223c,\u21a6end))\n          cPP : T\u27e8 I ,[ c \u21a6 S\u2080 \u2297 S\u2081 ] \u27e9\n          cPP = T-conv e rPP\n          I' = I ,[ c \u21a6 S\u2080 \u2297 S\u2081 ] ,[ c' \u21a6 {!!} ]\n  go (split \u03c3s A P\u2080 P\u2081) = T-split \u03c3s A (go P\u2080) (go P\u2081)\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3633b31c59ac3f2acc22ea1d80a49688b67de0ec","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/InstanceArguments.agda","new_file":"notes\/InstanceArguments.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: On the Bright Side of Type Classes: Instances Arguments in\n-- Agda (ICFP'11).\n\nmodule InstanceArguments where\n\nopen import Data.Bool\nopen import Data.Nat hiding ( equal )\n\n-- Note: Agda doesn't have a primitive function primBoolEquality.\nprimBoolEquality : Bool \u2192 Bool \u2192 Bool\nprimBoolEquality true  true  = true\nprimBoolEquality false false = true\nprimBoolEquality _     _     = false\n\nprimitive\n  primNatEquality : \u2115 \u2192 \u2115 \u2192 Bool\n\nrecord Eq (t : Set) : Set where\n  field equal : t \u2192 t \u2192 Bool\n\neqBool : Eq Bool\neqBool = record { equal = primBoolEquality }\n\neq\u2115 : Eq \u2115\neq\u2115 = record { equal = primNatEquality }\n\nequal : {t : Set} \u2192 {{eqT : Eq t}} \u2192 t \u2192 t \u2192 Bool\nequal {{eqT}} = Eq.equal eqT\n\ntest : Bool\ntest = equal 5 3 \u2228 equal true false\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: On the Bright Side of Type Classes: Instances Arguments in\n-- Agda (ICFP'11).\n\nmodule InstanceArguments where\n\nopen import Data.Bool\nopen import Data.Nat hiding ( equal )\n\n-- Note: Agda doesn't have a primitive function primBoolEquality\nprimBoolEquality : Bool \u2192 Bool \u2192 Bool\nprimBoolEquality true  true  = true\nprimBoolEquality false false = true\nprimBoolEquality _     _     = false\n\nprimitive\n  primNatEquality : \u2115 \u2192 \u2115 \u2192 Bool\n\nrecord Eq (t : Set) : Set where\n  field equal : t \u2192 t \u2192 Bool\n\neqBool : Eq Bool\neqBool = record { equal = primBoolEquality }\n\neq\u2115 : Eq \u2115\neq\u2115 = record { equal = primNatEquality }\n\nequal : {t : Set} \u2192 {{eqT : Eq t}} \u2192 t \u2192 t \u2192 Bool\nequal {{eqT}} = Eq.equal eqT\n\ntest : Bool\ntest = equal 5 3 \u2228 equal true false\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c5b9953cdc3f8a48882f01493714cec1f0d16c61","subject":"Revert \"Import bool\" - The agda stdlib is too slow to be worthwhile","message":"Revert \"Import bool\" - The agda stdlib is too slow to be worthwhile\n\nThis reverts commit e449e00f2c1e586ddf5a78a508185eb30347d0fc.\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","old_contents":"open import Agda.Builtin.Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n{-# BUILTIN LIST List #-}\n{-# BUILTIN NIL  \u2218    #-}\n{-# BUILTIN CONS _\u2237_  #-}\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n\n\nmain = (closure proofs (5 \u2237 \u2218))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b6299ea01c57073875d673ebc967d87341d4ff4c","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: bac32ab4f5fe0e1bf251fec79f4cbf09\n\ndarcs-hash:20110424201526-3bd4e-93f4fc80361388a84c929edec36dcaa21fe1e201.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PredicateLogic\/TheoremsATP.agda","new_file":"src\/PredicateLogic\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Predicate logic theorems\n------------------------------------------------------------------------------\n\n-- This module contains some examples showing the use of the ATPs to\n-- prove theorems from predicate logic.\n\nmodule PredicateLogic.TheoremsATP where\n\nopen import PredicateLogic.Constants\n\n------------------------------------------------------------------------------\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n        \u03c6(x)          \u2200x.\u03c6(x)          \u03c6(t)           \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n  \u2200I ---------    \u2200E ---------    \u2203I ---------    \u2203E --------------------\n      \u2200x.\u03c6(x)          \u03c6(t)           \u2203x.\u03c6(x)               \u03c8\n-}\n\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 \u22c0 P\u00b9\n  \u2200E : \u22c0 P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- It is necessary postulate a non-empty domain. See\n  -- PredicateLogic.NonEmptyDomain.TheoremsATP.\u2203I.\n  \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203 P\u00b9\n  \u2203E : \u2203 P\u00b9 \u2192 ((x : D) \u2192 P\u00b9 x \u2192 P\u2070) \u2192 P\u2070\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203E #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u22c0 P\u00b9) \u2194 \u2203 (\u03bb x \u2192 \u00ac (P\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 P\u00b9) \u2194 \u22c0 (\u03bb x \u2192 \u00ac (P\u00b9 x))\n  gDM\u2083 : \u22c0 P\u00b9     \u2194 \u00ac (\u2203 (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n  gDM\u2084 : \u2203 P\u00b9     \u2194 \u00ac (\u22c0 (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : \u22c0 (\u03bb x \u2192 \u22c0 (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u22c0 (\u03bb y \u2192 \u22c0 (\u03bb x \u2192 P\u00b2 x y))\n  \u2203-ord : \u2203 (\u03bb x \u2192 \u2203 (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u2203 (\u03bb y \u2192 \u2203 (\u03bb x \u2192 P\u00b2 x y))\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be delete.\npostulate\n  \u2200-erase : \u22c0 (\u03bb _ \u2192 P\u2070) \u2194 P\u2070\n  \u2203-erase : \u2203 (\u03bb x \u2192 P\u2070 \u2227 P\u00b9 x) \u2194 P\u2070 \u2227 \u2203 (\u03bb x \u2192 P\u00b9 x)\n{-# ATP prove \u2200-erase #-}\n{-# ATP prove \u2203-erase #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : \u22c0 (\u03bb x \u2192 P\u00b9 x \u2227 Q\u00b9 x) \u2194 (\u22c0 P\u00b9 \u2227 \u22c0 Q\u00b9)\n  \u2203-dist : \u2203 (\u03bb x \u2192 P\u00b9 x \u2228 Q\u00b9 x) \u2194 (\u2203 P\u00b9 \u2228 \u2203 Q\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Predicate logic theorems\n------------------------------------------------------------------------------\n\n-- This module contains some examples showing the use of the ATPs to\n-- prove theorems from predicate logic.\n\nmodule PredicateLogic.TheoremsATP where\n\nopen import PredicateLogic.Constants\n\n------------------------------------------------------------------------------\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n        \u03c6(x)          \u2200x.\u03c6(x)          \u03c6(t)           \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n  \u2200I ---------    \u2200E ---------    \u2203I ---------    \u2203E --------------------\n      \u2200x.\u03c6(x)          \u03c6(t)           \u2203x.\u03c6(x)               \u03c8\n-}\n\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 \u22c0 P\u00b9\n  \u2200E : \u22c0 P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- It is necessary postulate a non-empty domain.\n  -- See PredicateLogic.NonEmptyDomain.TheoremsATP.\n  -- \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203 P\u00b9\n  \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203 P\u00b9\n  \u2203E : \u2203 P\u00b9 \u2192 ((x : D) \u2192 P\u00b9 x \u2192 P\u2070) \u2192 P\u2070\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203E #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u22c0 P\u00b9) \u2194 \u2203 (\u03bb x \u2192 \u00ac (P\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 P\u00b9) \u2194 \u22c0 (\u03bb x \u2192 \u00ac (P\u00b9 x))\n  gDM\u2083 : \u22c0 P\u00b9     \u2194 \u00ac (\u2203 (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n  gDM\u2084 : \u2203 P\u00b9     \u2194 \u00ac (\u22c0 (\u03bb x \u2192 \u00ac (P\u00b9 x)))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : \u22c0 (\u03bb x \u2192 \u22c0 (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u22c0 (\u03bb y \u2192 \u22c0 (\u03bb x \u2192 P\u00b2 x y))\n  \u2203-ord : \u2203 (\u03bb x \u2192 \u2203 (\u03bb y \u2192 P\u00b2 x y)) \u2194 \u2203 (\u03bb y \u2192 \u2203 (\u03bb x \u2192 P\u00b2 x y))\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be delete.\npostulate\n  \u2200-erase  : \u22c0 (\u03bb _ \u2192 P\u2070) \u2194 P\u2070\n  \u2203-erase : \u2203 (\u03bb x \u2192 P\u2070 \u2227 P\u00b9 x) \u2194 P\u2070 \u2227 \u2203 (\u03bb x \u2192 P\u00b9 x)\n{-# ATP prove \u2200-erase #-}\n{-# ATP prove \u2203-erase #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : \u22c0 (\u03bb x \u2192 P\u00b9 x \u2227 Q\u00b9 x) \u2194 (\u22c0 P\u00b9 \u2227 \u22c0 Q\u00b9)\n  \u2203-dist : \u2203 (\u03bb x \u2192 P\u00b9 x \u2228 Q\u00b9 x) \u2194 (\u2203 P\u00b9 \u2228 \u2203 Q\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"935ce7286949f7c0f7752fd9e46cb8147aadf6ae","subject":"redate list programming","message":"redate list programming\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"code\/ListLogic.agda","new_file":"code\/ListLogic.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"a053e56f4c9eaa90d93033cec0f2cb52ebd1756f","subject":"agda: try and give up proving lift-term-doWeakenOne-ignore","message":"agda: try and give up proving lift-term-doWeakenOne-ignore\n\nSee comments in the proof. Tillmann plans to rewrite the whole\nhandling of contexts.\n\nOld-commit-hash: c120bb673f9dc0bbb6bdf96a9f0756063f874702\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4}\n  (\u03c1\u2081 : \u27e6 \u0393prefix \u27e7) (\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7)\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\n\nlift-term-doWeakenOne-ignore\u2032 \u0393prefix \u2205 \u03c1\u2081 \u2205 t = refl\n--This step gives the proof idea.\nlift-term-doWeakenOne-ignore\u2032 \u2205 (\u03c4 \u2022 \u0393rest) \u2205 (dv \u2022 v \u2022 \u03c1\u2082) t = lift-term-doWeakenOne-ignore\u2032 (\u03c4 \u2022 \u2205) \u0393rest (v \u2022 \u2205) \u03c1\u2082 t\n\nlift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix) (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} (v\u2080 \u2022 \u03c1\u2081) (dv \u2022 v \u2022 \u03c1\u2082) t = {!!}\n--Look at C-c C-, for the normalized goal.\n-- The solution here should be something like:\n  --lift-term-doWeakenOne-ignore\u2032 (\u03c4\u2080 \u2022 \u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest {\u03c4} ((v\u2080 \u2022 \u03c1\u2081) \u222a (v \u2022 \u2205)) \u03c1\u2082  ?\n\n-- accompanied by enough subst to make it typecheck, and maybe by\n-- weakenOne-ignore to account for weakenOne - or maybe not since\n-- weakenOne is a definitional equality. PG\n-- Not planning to finish this, it looks quite horrible.\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  {\u03c1\u2081 : \u27e6 \u0393prefix \u27e7} {\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7}\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\nlift-term-doWeakenOne-ignore\u2032 = {!!}\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c6df8251b159a6d5d0490726b27d486890c82882","subject":"Update to Neglible","message":"Update to Neglible\n","repos":"crypto-agda\/crypto-agda","old_file":"Neglible.agda","new_file":"Neglible.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (fN k) (gD k) (f'D k) (g'D k) \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange (gN k) (fD k) (g'D k) (f'D k))\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (f'N k) (fD k) (g'D k) (gD k))\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange (g'N k) (gD k) (f'D k) (fD k)\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} fg gh) = \u2264-NB (~dist-sum f g h) (_~_.~ fg +NB _~_.~ gh)\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\n\nmodule _ (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9258aaf7976f2eb8d3f2bb85051604b9b9fc3a7d","subject":"Added difference lemma","message":"Added difference lemma\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-counting.agda","new_file":"flipbased-counting.agda","new_contents":"open import Function\nopen import Data.Nat.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_; _\u2261_)\n\nopen import Data.Bits\nimport flipbased\n\nmodule flipbased-counting\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n   (count\u21ba : \u2200 {n} \u2192 \u21ba n Bit \u2192 \u2115)\n where\n\nopen flipbased \u21ba toss return\u21ba map\u21ba join\u21ba\n\ninfix 4 _\u2248\u21ba_ _\u224b\u21ba_ _\u2248\u2141?_ _\u2248\u1d2c_ _\u2248\u1d2c\u2032_\n\n-- f \u2248\u21ba g when f and g return the same number of 1 (and 0).\n_\u2248\u21ba_ : \u2200 {n} (f g : EXP n) \u2192 Set\n_\u2248\u21ba_ = _\u2261_ on count\u21ba\n\n_\u2248\u2141?_ : \u2200 {c} (g\u2080 g\u2081 : \u2141? c) \u2192 Set\ng\u2080 \u2248\u2141? g\u2081 = \u2200 b \u2192 g\u2080 b \u2248\u21ba g\u2081 b\n\n_\u2248\u1d2c_ : \u2200 {n a} {A : Set a} (f g : \u21ba n A) \u2192 Set _\n_\u2248\u1d2c_ {n} {A = A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n_\u2248\u1d2c\u2032_ : \u2200 {n a} {A : Set a} (f g : \u21ba n A) \u2192 Set _\n_\u2248\u1d2c\u2032_ {n} {A = A} f g = \u2200 {ca} (Adv : A \u2192 \u21ba ca Bit) \u2192 f >>= Adv \u2248\u21ba g >>= Adv\n\n{- Restore if we find more use of it\nRelatedEXP : (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\nRelatedEXP _\u223c_ {m} {n} f g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\nmodule ... z where\n  x \u223c y = dist x y > ... z ...\n-}\n\n_\u224b\u21ba_ : \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\n-- _\u224b\u21ba_ = RelatedEXP _\u2261_\n_\u224b\u21ba_ {m} {n} f g = \u27e82^ n * count\u21ba f \u27e9 \u2261 \u27e82^ m * count\u21ba g \u27e9\n\nSafe\u2141? : \u2200 {c} (f : \u2141? c) \u2192 Set\nSafe\u2141? f = f 0b \u2248\u21ba f 1b\n\nReversible\u2141? : \u2200 {c} (g : \u2141? c) \u2192 Set\nReversible\u2141? g = g \u2248\u2141? g \u2218 not\n\n\u2248\u21d2\u224b\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u2248\u21ba g \u2192 f \u224b\u21ba g\n\u2248\u21d2\u224b\u21ba eq rewrite eq = \u2261.refl\n\n\u224b\u21d2\u2248\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u224b\u21ba g \u2192 f \u2248\u21ba g\n\u224b\u21d2\u2248\u21ba {n} = 2^-inj n\n\n\ndifference-lemma\u21ba : \u2200 {n}(A B F : EXP n)\n  \u2192 #\u27e8 |not| (run\u21ba F) |\u2227| run\u21ba A \u27e9 \u2261 #\u27e8 |not| (run\u21ba F) |\u2227| run\u21ba B \u27e9\n  \u2192 dist #\u27e8 run\u21ba A \u27e9 #\u27e8 run\u21ba B \u27e9 \u2264 #\u27e8 run\u21ba F \u27e9\ndifference-lemma\u21ba A B F = difference-lemma (run\u21ba A) (run\u21ba B) (run\u21ba F)\n\n\nmodule \u2248\u2141? {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u2141? n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u2141?_\n\n        refl : Reflexive \u211b\n        refl b = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p b = \u2261.sym (p b)\n\n        trans : Transitive \u211b\n        trans p q b = \u2261.trans (p b) (q b)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u224b\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u224b\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u1d2c {n} {a} {A : Set a} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u21ba n A\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u1d2c_\n\n        refl : Reflexive \u211b\n        refl A = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p A = \u2261.sym (p A)\n\n        trans : Transitive \u211b\n        trans p q A = \u2261.trans (p A) (q A)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid\n  open Setoid setoid public\n\nmodule \u2248\u1d2c\u2032 {n} {a} {A : Set a} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u21ba n A\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u1d2c\u2032_\n\n        refl : Reflexive \u211b\n        refl A = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p A = \u2261.sym (p A)\n\n        trans : Transitive \u211b\n        trans p q A = \u2261.trans (p A) (q A)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid\n  open Setoid setoid public\n\nmodule \u2141? {n} where\n  join : EXP n \u2192 EXP n \u2192 \u2141? n\n  join f g b = if b then f else g\n\n  safe-sym : \u2200 {g : \u2141? n} \u2192 Safe\u2141? g \u2192 Safe\u2141? (g \u2218 not)\n  safe-sym {g} g-safe = \u2248\u21ba.sym {n} {g 0b} {g 1b} g-safe\n\n  Reversible\u21d2Safe : \u2200 {c} (g : \u2141? c) \u2192 Reversible\u2141? g \u2192 Safe\u2141? g\n  Reversible\u21d2Safe g g\u2248g\u2218not = g\u2248g\u2218not 0b\n\ndata Rat : Set where _\/_ : (num denom : \u2115) \u2192 Rat\n\nPr[_\u22611] : \u2200 {n} (f : EXP n) \u2192 Rat\nPr[_\u22611] {n} f = count\u21ba f \/ 2^ n\n","old_contents":"open import Function\nopen import Data.Nat.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_; _\u2261_)\n\nopen import Data.Bits\nimport flipbased\n\nmodule flipbased-counting\n   (\u21ba : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a)\n   (toss : \u21ba 1 Bit)\n   (return\u21ba : \u2200 {n a} {A : Set a} \u2192 A \u2192 \u21ba n A)\n   (map\u21ba : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba n A \u2192 \u21ba n B)\n   (join\u21ba : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192 \u21ba n\u2081 (\u21ba n\u2082 A) \u2192 \u21ba (n\u2081 + n\u2082) A)\n   (count\u21ba : \u2200 {n} \u2192 \u21ba n Bit \u2192 \u2115)\n where\n\nopen flipbased \u21ba toss return\u21ba map\u21ba join\u21ba\n\ninfix 4 _\u2248\u21ba_ _\u224b\u21ba_ _\u2248\u2141?_ _\u2248\u1d2c_ _\u2248\u1d2c\u2032_\n\n-- f \u2248\u21ba g when f and g return the same number of 1 (and 0).\n_\u2248\u21ba_ : \u2200 {n} (f g : EXP n) \u2192 Set\n_\u2248\u21ba_ = _\u2261_ on count\u21ba\n\n_\u2248\u2141?_ : \u2200 {c} (g\u2080 g\u2081 : \u2141? c) \u2192 Set\ng\u2080 \u2248\u2141? g\u2081 = \u2200 b \u2192 g\u2080 b \u2248\u21ba g\u2081 b\n\n_\u2248\u1d2c_ : \u2200 {n a} {A : Set a} (f g : \u21ba n A) \u2192 Set _\n_\u2248\u1d2c_ {n} {A = A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n_\u2248\u1d2c\u2032_ : \u2200 {n a} {A : Set a} (f g : \u21ba n A) \u2192 Set _\n_\u2248\u1d2c\u2032_ {n} {A = A} f g = \u2200 {ca} (Adv : A \u2192 \u21ba ca Bit) \u2192 f >>= Adv \u2248\u21ba g >>= Adv\n\n{- Restore if we find more use of it\nRelatedEXP : (\u2115 \u2192 \u2115 \u2192 Set) \u2192 \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\nRelatedEXP _\u223c_ {m} {n} f g = \u27e82^ n * count\u21ba f \u27e9 \u223c \u27e82^ m * count\u21ba g \u27e9\n\nmodule ... z where\n  x \u223c y = dist x y > ... z ...\n-}\n\n_\u224b\u21ba_ : \u2200 {m n} \u2192 EXP m \u2192 EXP n \u2192 Set\n-- _\u224b\u21ba_ = RelatedEXP _\u2261_\n_\u224b\u21ba_ {m} {n} f g = \u27e82^ n * count\u21ba f \u27e9 \u2261 \u27e82^ m * count\u21ba g \u27e9\n\nSafe\u2141? : \u2200 {c} (f : \u2141? c) \u2192 Set\nSafe\u2141? f = f 0b \u2248\u21ba f 1b\n\nReversible\u2141? : \u2200 {c} (g : \u2141? c) \u2192 Set\nReversible\u2141? g = g \u2248\u2141? g \u2218 not\n\n\u2248\u21d2\u224b\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u2248\u21ba g \u2192 f \u224b\u21ba g\n\u2248\u21d2\u224b\u21ba eq rewrite eq = \u2261.refl\n\n\u224b\u21d2\u2248\u21ba : \u2200 {n} {f g : EXP n} \u2192 f \u224b\u21ba g \u2192 f \u2248\u21ba g\n\u224b\u21d2\u2248\u21ba {n} = 2^-inj n\n\nmodule \u2248\u2141? {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u2141? n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u2141?_\n\n        refl : Reflexive \u211b\n        refl b = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p b = \u2261.sym (p b)\n\n        trans : Transitive \u211b\n        trans p q b = \u2261.trans (p b) (q b)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u224b\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u224b\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u21ba {n} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = EXP n\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u21ba_\n\n        refl : Reflexive \u211b\n        refl = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym = \u2261.sym\n\n        trans : Transitive \u211b\n        trans = \u2261.trans\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid \n  open Setoid setoid public\n\nmodule \u2248\u1d2c {n} {a} {A : Set a} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u21ba n A\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u1d2c_\n\n        refl : Reflexive \u211b\n        refl A = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p A = \u2261.sym (p A)\n\n        trans : Transitive \u211b\n        trans p q A = \u2261.trans (p A) (q A)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid\n  open Setoid setoid public\n\nmodule \u2248\u1d2c\u2032 {n} {a} {A : Set a} where\n  setoid : Setoid _ _\n  setoid = record { Carrier = C; _\u2248_ = \u211b; isEquivalence = isEquivalence }\n      where\n        C : Set _\n        C = \u21ba n A\n\n        \u211b : C \u2192 C \u2192 Set _\n        \u211b = _\u2248\u1d2c\u2032_\n\n        refl : Reflexive \u211b\n        refl A = \u2261.refl\n\n        sym : Symmetric \u211b\n        sym p A = \u2261.sym (p A)\n\n        trans : Transitive \u211b\n        trans p q A = \u2261.trans (p A) (q A)\n\n        isEquivalence : IsEquivalence \u211b\n        isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                               ; sym = \u03bb {x y} \u2192 sym {x} {y}\n                               ; trans = \u03bb {x y z} \u2192 trans {x} {y} {z} }\n\n  module Reasoning = Setoid-Reasoning setoid\n  open Setoid setoid public\n\nmodule \u2141? {n} where\n  join : EXP n \u2192 EXP n \u2192 \u2141? n\n  join f g b = if b then f else g\n\n  safe-sym : \u2200 {g : \u2141? n} \u2192 Safe\u2141? g \u2192 Safe\u2141? (g \u2218 not)\n  safe-sym {g} g-safe = \u2248\u21ba.sym {n} {g 0b} {g 1b} g-safe\n\n  Reversible\u21d2Safe : \u2200 {c} (g : \u2141? c) \u2192 Reversible\u2141? g \u2192 Safe\u2141? g\n  Reversible\u21d2Safe g g\u2248g\u2218not = g\u2248g\u2218not 0b\n\ndata Rat : Set where _\/_ : (num denom : \u2115) \u2192 Rat\n\nPr[_\u22611] : \u2200 {n} (f : EXP n) \u2192 Rat\nPr[_\u22611] {n} f = count\u21ba f \/ 2^ n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3a9443c065ba78ee40b830e311341f8abe6915bc","subject":"Fin: modulo","message":"Fin: modulo\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 Set where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : Set a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : Set a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : Set a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  lookup-sucmod-rot\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n)\n                 \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n  lookup-sucmod-rot\u2081 {zero} () xs\n  lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n  lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n  lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                 | Vec.lookup-allFin (sucmod i)\n                                 | lookup-map sucmod i (allFin n)\n                                 | Vec.lookup\u2218tabulate id i\n                                 = refl\n\n  rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n  rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n","old_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nopen import Data.Sum as Sum\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse zero    = from\u2115 _\nreverse (suc x) = inject\u2081 (reverse x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e34e65b4afbd9cf3d2ac0e45b0e836fa098cc8d8","subject":"Nat: add dist-sum or the triangular inequality","message":"Nat: add dist-sum or the triangular inequality\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"96b578369f621424f7eb1a223aa10ac902b8ff2d","subject":"Used succ instead of suc in the Agsy modules.","message":"Used succ instead of suc in the Agsy modules.\n\nIgnore-this: 8720611d205851bdea094f7b5aa76b7a\n\ndarcs-hash:20110122145421-3bd4e-f8bb2ddcb603afe7c58e95486707f3705889cb9b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Agsy\/PA\/Inductive\/Properties.agda","new_file":"src\/Agsy\/PA\/Inductive\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Inductive PA arithmetic properties using Agsy\n------------------------------------------------------------------------------\n\nmodule Agsy.PA.Inductive.Properties where\n\nopen import Data.Nat renaming (suc to succ)\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n  -- via Agsy {-c}\n+-rightIdentity zero     = refl\n+-rightIdentity (succ n) = cong succ (+-rightIdentity n)\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero     n o = refl\n+-assoc (succ m) n o = cong succ (+-assoc m n o)\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)  -- via Agsy {-c}\nx+Sy\u2261S[x+y] zero     n = refl\nx+Sy\u2261S[x+y] (succ m) n = cong succ (x+Sy\u2261S[x+y] m n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = sym (+-rightIdentity n)\n+-comm (succ m) n =\n  begin\n    succ (m + n) \u2261\u27e8 cong succ (+-comm m n) \u27e9\n    succ (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n    n + succ m\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Inductive PA arithmetic properties using Agsy\n------------------------------------------------------------------------------\n\nmodule Agsy.PA.Inductive.Properties where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n  -- via Agsy {-c}\n+-rightIdentity zero    = refl\n+-rightIdentity (suc n) = cong suc (+-rightIdentity n)\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero    n o = refl\n+-assoc (suc m) n o = cong suc (+-assoc m n o)\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + suc n \u2261 suc (m + n)  -- via Agsy {-c}\nx+Sy\u2261S[x+y] zero    n = refl\nx+Sy\u2261S[x+y] (suc m) n = cong suc (x+Sy\u2261S[x+y] m n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = sym (+-rightIdentity n)\n+-comm (suc m) n =\n  begin\n    suc (m + n) \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n    n + suc m\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b43c1cfa49381a6cbe3d4b31b8e99a98463b4999","subject":"Added injectivity proof for modq","message":"Added injectivity proof for modq\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Empty\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 Set where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n\n  sucmod-inj : \u2200 {q}(i j : Fin q) \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {q} i j eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj i j eq | just x | just x\u2081 | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj i j eq | just x | nothing | p | ni | nj = \u22a5-elim (ni (cong Maybe.just eq))\n  sucmod-inj i j eq | nothing | just x | p | ni | nj = \u22a5-elim (nj (cong Maybe.just (sym eq)))\n  sucmod-inj i j eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : Set a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : Set a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : Set a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  lookup-sucmod-rot\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n)\n                 \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n  lookup-sucmod-rot\u2081 {zero} () xs\n  lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n  lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n  lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                 | Vec.lookup-allFin (sucmod i)\n                                 | lookup-map sucmod i (allFin n)\n                                 | Vec.lookup\u2218tabulate id i\n                                 = refl\n\n  rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n  rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n","old_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 Set where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : Set a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : Set a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : Set a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  lookup-sucmod-rot\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n)\n                 \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n  lookup-sucmod-rot\u2081 {zero} () xs\n  lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n  lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n  lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                 | Vec.lookup-allFin (sucmod i)\n                                 | lookup-map sucmod i (allFin n)\n                                 | Vec.lookup\u2218tabulate id i\n                                 = refl\n\n  rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n  rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"905379564a50f91a3435bb015a582178b9bc39e2","subject":"Revise agda\/README.md (for #275).","message":"Revise agda\/README.md (for #275).\n\nOld-commit-hash: 88f72736101147e9754ea68565ed89d6702452bb\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Machine-checked formalization of the theoretical results presented\n-- in the paper:\n--\n--   Yufei Cai, Paolo G. Giarrusso, Tillmann Rendel, Klaus Ostermann.\n--   A Theory of Changes for Higher-Order Languages:\n--   Incrementalizing \u03bb-Calculi by Static Differentiation.\n--   To appear at PLDI, ACM 2014.\n--\n------------------------------------------------------------------------\n\nmodule README where\n\n-- We know two good ways to read this code base. You can either\n-- use Emacs with agda2-mode to interact with the source files,\n-- or you can use a web browser to view the pretty-printed and\n-- hyperlinked source files. For Agda power users, we also\n-- include basic setup information for your own machine.\n\n\n-- IF YOU WANT TO USE A BROWSER\n-- ============================\n--\n-- Start with *HTML* version of this readme. On the AEC\n-- submission website (online or inside the VM), follow the link\n-- \"view the Agda code in their browser\" to the file\n-- agda\/README.html.\n--\n-- The source code is syntax highlighted and hyperlinked. You can\n-- click on module names to open the corresponding files, or you\n-- can click on identifiers to jump to their definition. In\n-- general, a Agda file with name `Foo\/Bar\/Baz.agda` contains a\n-- module `Foo.Bar.Baz` and is shown in an HTML file\n-- `Foo.Bar.Baz.html`.\n--\n-- Note that we also include the HTML files generated for our\n-- transitive dependencies from the Agda standard library. This\n-- allows you to follow hyperlinks to the Agda standard\n-- library. It is the default behavior of `agda --html` which we\n-- used to generate the HTML.\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE EMACS\n-- ========================\n--\n-- Open this file in Emacs with agda2-mode installed. See below\n-- for which Agda version you need to install. On the VM image,\n-- everything is setup for you and you can just open this file in\n-- Emacs.\n--\n--   C-c C-l   Load code. Type checks and syntax highlights. Use\n--             this if the code is not syntax highlighted, or\n--             after you changed something that you want to type\n--             check.\n--\n--   M-.       Jump to definition of identifier at the point.\n--   M-*       Jump back.\n--\n-- Note that README.agda imports Everything.agda which imports\n-- every Agda file in our formalization. So if README.agda type\n-- checks successfully, everything does. If you want to type\n-- check everything from scratch, delete the *.agdai files to\n-- disable separate compilation. You can use \"find . -name\n-- '*.agdai' | xargs rm\" to do that.\n--\n-- More information on the Agda mode is available on the Agda wiki:\n--\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Main.QuickGuideToEditingTypeCheckingAndCompilingAgdaCode\n-- http:\/\/wiki.portal.chalmers.se\/agda\/pmwiki.php?n=Docs.EmacsModeKeyCombinations\n--\n-- To get started, continue below on \"Where to start reading?\".\n\n\n\n-- IF YOU WANT TO USE YOUR OWN SETUP\n-- =================================\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n--\n-- If you're not an Agda power user, it is probably easier to use\n-- the VM image or look at the pretty-printed and hyperlinked\n-- HTML files, see above.\n\n\n\n-- WHERE TO START READING?\n-- =======================\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--   Good if you want to get a broad overview.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda\n--   formalization.  Good if you want to begin at the beginning\n--   and understand the structure of our code.\n--\n-- PLDI14-List-of-Theorems.agda\n--   Pointers to the Agda formalizations of all theorems, lemmas\n--   or definitions in the PLDI paper. Good if you want to read\n--   the paper and the Agda code side by side.\n--\n--   Here is an import of this file, so you can jump to it\n--   directly (use M-. in Emacs or click the module name in the\n--   Browser):\n\nimport PLDI14-List-of-Theorems\n\n-- Everything.agda\n--   Imports every Agda module in our formalization. Good if you\n--   want to make sure you don't miss anything.\n--\n--   Again, here's is an import of this file so you can navigate\n--   there:\n\nimport Everything\n\n\n\n-- THE AGDA CODE\n-- =============\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","old_contents":"module README where\n\n----------------------------\n-- INCREMENTAL \u03bb-CALCULUS --\n----------------------------\n--\n--\n-- IMPORTANT FILES\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda formalization.\n--\n-- PLDI14-List-of-Theorems.agda\n--   For each theorem, lemma or definition in the PLDI 2014 submission,\n--   it points to the corresponding Agda object.\n--\n--\n-- LOCATION OF AGDA MODULES\n--\n-- To find the file containing an Agda module, replace the dots in its\n-- full name by directory separators. The result is the file's path relative\n-- to this directory. For example, Parametric.Syntax.Type is defined in\n-- Parametric\/Syntax\/Type.agda.\n--\n--\n-- HOW TO TYPE CHECK EVERYTHING\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n-- Import everything else. This ensures that typechecking README.agda typechecks\n-- the entire codebase, because Everything.agda is auto-generated.\nimport Everything\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"876790ac48ef4ba5ada82e623ffe072f02c6143c","subject":"Reported Agda issue 1014.","message":"Reported Agda issue 1014.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 \u221e (Conat n) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\nConat-coind-helper :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (\u2200 {m} \u2192 A m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m')) \u2192\n  A n \u2192 Conat n\nConat-coind-helper A h An with h An\n... | inj\u2081 n\u22610 = subst Conat (sym n\u22610) cozero\n... | inj\u2082 (n' , prf , An') =\n  subst Conat (sym prf) (cosucc (\u266f (Conat-coind-helper A h An')))\n\n-- 07 January 2014. Agda bug? We couldn't directly prove Conat-coind\n-- because Agda's termination checker doesn't accept the proof, but it\n-- accepts the proof of Conat-coind-helper. The only difference is the\n-- *position* of the universal quantifier \u2200 {n}. See Agda issue 1014.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h An = Conat-coind-helper A h An\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221e.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 \u221e (Conat n) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\nConat-coind-helper :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (\u2200 {m} \u2192 A m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m')) \u2192\n  A n \u2192 Conat n\nConat-coind-helper A h An with h An\n... | inj\u2081 n\u22610 = subst Conat (sym n\u22610) cozero\n... | inj\u2082 (n' , prf , An') =\n  subst Conat (sym prf) (cosucc (\u266f (Conat-coind-helper A h An')))\n\n-- 07 January 2014. Agda bug? We couldn't directly prove Conat-coind\n-- because Agda's termination checker doesn't accept the proof, but it\n-- accepts the proof of Conat-coind-helper. The only difference is the\n-- *position* of the universal quantifier \u2200 {n}.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h An = Conat-coind-helper A h An\n\npostulate\n  \u221eD    : D\n  \u221e-eq : \u221eD \u2261 succ\u2081 \u221eD\n\n-- TODO (06 January 2014): Agda doesn't accept the proof of Conat \u221e.\n{-# NO_TERMINATION_CHECK #-}\n\u221e-Conat : Conat \u221eD\n\u221e-Conat = subst Conat (sym \u221e-eq) (cosucc (\u266f \u221e-Conat))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"25965a51f1727cc628e747ce1a13dff3ae6246a3","subject":"Test derivation (see #50).","message":"Test derivation (see #50).\n\nOld-commit-hash: 584a9768a06e8d22637c93652317a598f6c81162\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\nsame-var : \u2200 {\u0393 \u03c4} \u2192 (x\u2081 x\u2082 : Var \u0393 \u03c4) \u2192 Bool\nsame-var this this = true\nsame-var this (that x\u2082) = false\nsame-var (that x\u2081) this = false\nsame-var (that x\u2081) (that x\u2082) = same-var x\u2081 x\u2082\n\nsame-term : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 Bool\nsame-term (abs t\u2081) (abs t\u2082) = same-term t\u2081 t\u2082\nsame-term (abs t\u2081) _ = false\nsame-term (app {\u0393} {\u03c4\u2081} t\u2081 t\u2082) (app .{\u0393} {\u03c4\u2082} t\u2083 t\u2084) with \u03c4\u2081 \u225f \u03c4\u2082\nsame-term (app t\u2081 t\u2082) (app t\u2083 t\u2084) | yes refl = same-term t\u2081 t\u2083 \u2227 same-term t\u2082 t\u2084\nsame-term (app t\u2081 t\u2082) (app t\u2083 t\u2084) | no \u00acp = false\nsame-term (app t\u2081 t\u2082) _ = false\nsame-term (var x\u2081) (var x\u2082) = same-var x\u2081 x\u2082\nsame-term (var x\u2081) _ = false\nsame-term true true = true\nsame-term true _ = false\nsame-term false false = true\nsame-term false _ = false\nsame-term (if t\u2081 t\u2082 t\u2083) (if t\u2084 t\u2085 t\u2086) = same-term t\u2081 t\u2084 \u2227 same-term t\u2082 t\u2085 \u2227 same-term t\u2083 t\u2086\nsame-term (if t\u2081 t\u2082 t\u2083) _ = false\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nif\u2032 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nif\u2032 true t\u2082 t\u2083 = t\u2082\nif\u2032 false t\u2082 t\u2083 = t\u2083\nif\u2032 t\u2081 true false = t\u2081\nif\u2032 t\u2081 t\u2082 t\u2083 = if same-term t\u2082 t\u2083 then t\u2082 else if t\u2081 t\u2082 t\u2083\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if\u2032 t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n\n-- SYMBOLIC DERIVATION\n--\n-- compare derive with _\u27ea_\u27eb_Term\n-- and derive-var with \u27e6_\u27e7Var\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = bool\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\nxor\u2083 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nxor\u2083 t\u2081 t\u2082 t\u2083\n  = if\u2032 t\u2081 (if\u2032 t\u2082 t\u2083 (if\u2032 t\u2083 false true)) (if\u2032 t\u2082 (if\u2032 t\u2083 false true) t\u2083)\n\nderive-if : \u2200 {\u03c4 \u0393\u2081} \u2192\n  (t dt : Term \u0393\u2081 bool) \u2192\n  (v\u2081 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2081 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  (v\u2082 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2082 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-if {\u03c4\u2081 \u21d2 \u03c4\u2082} t dt v\u2081 dv\u2081 v\u2082 dv\u2082 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 let \u227c\u2083 = \u227c-trans \u227c\u2081 \u227c\u2082 in\n    derive-if (weaken \u227c\u2083 t) (weaken \u227c\u2083 dt)\n              (v\u2081 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2081 \u227c\u2081 v \u227c\u2082 dv)\n              (v\u2082 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2082 \u227c\u2081 v \u227c\u2082 dv)\nderive-if {bool} t\u2081 dt\u2081 t\u2082 dt\u2082 t\u2083 dt\u2083 =\n  if\u2032 dt\u2081 (if\u2032 t\u2081 (xor\u2083 dt\u2083 t\u2082 t\u2083) (xor\u2083 dt\u2082 t\u2082 t\u2083)) (if\u2032 t\u2081 dt\u2083 dt\u2082)\n\nderive-var : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-var this (dv SymEnv.\u2022 d\u03c1) = dv\nderive-var (that x) (dv SymEnv.\u2022 d\u03c1) = derive-var x d\u03c1\n\nderive : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive (abs t) \u03c1 d\u03c1 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 derive t (liftVal \u227c\u2082 v SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) \u03c1) (dv SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) d\u03c1)\nderive (app t\u2081 t\u2082) \u03c1 d\u03c1 =\n  derive t\u2081 \u03c1 d\u03c1 \u227c-refl (_ \u27ea t\u2082 \u27ebTerm \u03c1) \u227c-refl (derive t\u2082 \u03c1 d\u03c1)\nderive (var x) \u03c1 d\u03c1 = derive-var x d\u03c1\nderive true \u03c1 d\u03c1 = false\nderive false \u03c1 d\u03c1 = false\nderive {\u0393\u2081} (if t\u2081 t\u2082 t\u2083) \u03c1 d\u03c1 =\n  derive-if (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (derive t\u2081 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (derive t\u2082 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2083 \u27ebTerm \u03c1) (derive t\u2083 \u03c1 d\u03c1)\n\nderive-closed : \u2200 {\u03c4} \u2192 Term \u2205 \u03c4 \u2192 Term \u2205 (\u0394-Type \u03c4)\nderive-closed {\u03c4} t = \u2193 (\u0394-Type \u03c4) (derive t SymEnv.\u2205 SymEnv.\u2205)\n\n-- TESTS\n\nid-term : \u2200 \u03c4 \u2192 Term \u2205 (\u03c4 \u21d2 \u03c4)\nid-term \u03c4 = abs (var this)\n\nnot-term : Term \u2205 (bool \u21d2 bool)\nnot-term = abs (if (var this) false true)\n\nxor-term : Term \u2205 (bool \u21d2 bool \u21d2 bool)\nxor-term = abs (abs (if (var this)\n                        (if (var (that this)) false true)\n                        (if (var (that this)) true false)))\n\n-- derive (\u03bb x \u2192 x) \u2261 \u03bb x dx \u2192 dx\ntest\u2081 : derive-closed (id-term bool) \u2261 abs (abs (var this))\ntest\u2081 = refl\n\n-- derive (\u03bb x \u2192 not x) \u2261 \u03bb x dx \u2192 dx\ntest\u2082 : derive-closed not-term \u2261 abs (abs (var this))\ntest\u2082 = refl\n\n-- derive (\u03bb a b \u2192 a xor b) \u2261 \u03bb a da b db \u2192 db xor da\ntest\u2083 :\n  derive-closed xor-term \u2261\n  abs (abs (abs (abs\n    (if (var this)\n        (if (var (that (that this))) false true)\n        (var (that (that this)))))))\ntest\u2083 = refl\n","old_contents":"module lambda where\n\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\nsame-var : \u2200 {\u0393 \u03c4} \u2192 (x\u2081 x\u2082 : Var \u0393 \u03c4) \u2192 Bool\nsame-var this this = true\nsame-var this (that x\u2082) = false\nsame-var (that x\u2081) this = false\nsame-var (that x\u2081) (that x\u2082) = same-var x\u2081 x\u2082\n\nsame-term : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 Bool\nsame-term (abs t\u2081) (abs t\u2082) = same-term t\u2081 t\u2082\nsame-term (abs t\u2081) _ = false\nsame-term (app {\u0393} {\u03c4\u2081} t\u2081 t\u2082) (app .{\u0393} {\u03c4\u2082} t\u2083 t\u2084) with \u03c4\u2081 \u225f \u03c4\u2082\nsame-term (app t\u2081 t\u2082) (app t\u2083 t\u2084) | yes refl = same-term t\u2081 t\u2083 \u2227 same-term t\u2082 t\u2084\nsame-term (app t\u2081 t\u2082) (app t\u2083 t\u2084) | no \u00acp = false\nsame-term (app t\u2081 t\u2082) _ = false\nsame-term (var x\u2081) (var x\u2082) = same-var x\u2081 x\u2082\nsame-term (var x\u2081) _ = false\nsame-term true true = true\nsame-term true _ = false\nsame-term false false = true\nsame-term false _ = false\nsame-term (if t\u2081 t\u2082 t\u2083) (if t\u2084 t\u2085 t\u2086) = same-term t\u2081 t\u2084 \u2227 same-term t\u2082 t\u2085 \u2227 same-term t\u2083 t\u2086\nsame-term (if t\u2081 t\u2082 t\u2083) _ = false\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n\n-- SYMBOLIC EXECUTION\n--\n-- Naming Convention:\n-- \u0393 \u27ea_\u27ebX is like \u27e6_\u27e7X but for symbolic execution in context \u0393.\n\n_\u27ea_\u27ebType : Context \u2192 Type \u2192 Set\n\u0393\u2081 \u27ea \u03c4\u2081 \u21d2 \u03c4\u2082 \u27ebType = \u2200 {\u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2082 \u27ea \u03c4\u2081 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4\u2082 \u27ebType\n\u0393\u2081 \u27ea bool \u27ebType = Term \u0393\u2081 bool\n\nmodule _ (\u0393 : Context) where\n  import Denotational.Environments\n  module SymEnv = Denotational.Environments Type (\u03bb \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType)\n\n  open SymEnv public using ()\n    renaming (\u27e6_\u27e7Context to _\u27ea_\u27ebContext; \u27e6_\u27e7Var to _\u27ea_\u27ebVar_)\n\nliftVal : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType \u2192 \u0393\u2082 \u27ea \u03c4 \u27ebType\nliftVal {\u03c4\u2081 \u21d2 \u03c4\u2082} \u227c\u2081 v\u2081 = \u03bb \u227c\u2082 v\u2082 \u2192 v\u2081 (\u227c-trans \u227c\u2081 \u227c\u2082) v\u2082\nliftVal {bool} \u227c\u2081 v = weaken \u227c\u2081 v\n\nliftEnv : \u2200 {\u0393 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2082 \u27ea \u0393 \u27ebContext\nliftEnv {\u2205} \u227c\u2081 \u2205 = SymEnv.\u2205\nliftEnv {\u03c4 \u2022 \u0393} \u227c\u2081 (v \u2022 \u03c1) = liftVal \u227c\u2081 v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1\n\nif\u2032 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nif\u2032 true t\u2082 t\u2083 = t\u2082\nif\u2032 false t\u2082 t\u2083 = t\u2083\nif\u2032 t\u2081 true false = t\u2081\nif\u2032 t\u2081 t\u2082 t\u2083 = if same-term t\u2082 t\u2083 then t\u2082 else if t\u2081 t\u2082 t\u2083\n\nmixed-if : \u2200 {\u0393\u2081} \u03c4 \u2192 (t\u2081 : Term \u0393\u2081 bool) (v\u2082 v\u2083 : \u0393\u2081 \u27ea \u03c4 \u27ebType) \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\nmixed-if (\u03c4\u2081 \u21d2 \u03c4\u2082) t\u2081 v\u2082 v\u2083 = \u03bb \u227c\u2081 v \u2192 mixed-if \u03c4\u2082 (weaken \u227c\u2081 t\u2081) (v\u2082 \u227c\u2081 v) (v\u2083 \u227c\u2081 v)\nmixed-if bool t\u2081 t\u2082 t\u2083 = if\u2032 t\u2081 t\u2082 t\u2083\n\n_\u27ea_\u27ebTerm_ : \u2200 \u0393\u2081 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u03c4 \u27ebType\n\u0393\u2081 \u27ea abs t \u27ebTerm \u03c1 = \u03bb {\u0393\u2082} \u227c\u2081 v \u2192 \u0393\u2082 \u27ea t \u27ebTerm (v SymEnv.\u2022 liftEnv \u227c\u2081 \u03c1)\n\u0393\u2081 \u27ea app t\u2081 t\u2082 \u27ebTerm \u03c1 = (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) \u227c-refl (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\u0393\u2081 \u27ea var x \u27ebTerm \u03c1 = \u0393\u2081 \u27ea x \u27ebVar \u03c1\n\u0393\u2081 \u27ea true \u27ebTerm \u03c1 = true\n\u0393\u2081 \u27ea false \u27ebTerm \u03c1 = false\n\u0393\u2081 \u27ea if t\u2081 t\u2082 t\u2083 \u27ebTerm \u03c1 = mixed-if _ (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1)\n\n\u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType \u2192 Term \u0393 \u03c4\n\u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u27ea \u03c4 \u27ebType\n\n\u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n\u2193 bool v = v\n\n\u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u227c\u2081 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u227c\u2081 t) (\u2193 \u03c4\u2081 v))\n\u2191 bool t = t\n\n\u2191-Context : \u2200 {\u0393} \u2192 \u0393 \u27ea \u0393 \u27ebContext\n\u2191-Context {\u2205} = SymEnv.\u2205\n\u2191-Context {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) SymEnv.\u2022 liftEnv (drop \u03c4 \u2022 \u227c-refl) \u2191-Context\n\nnorm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\nnorm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u27ea t \u27ebTerm \u2191-Context)\n\n-- SYMBOLIC DERIVATION\n--\n-- compare derive with _\u27ea_\u27eb_Term\n-- and derive-var with \u27e6_\u27e7Var\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = bool\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\nxor\u2083 : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\nxor\u2083 t\u2081 t\u2082 t\u2083\n  = if\u2032 t\u2081 (if\u2032 t\u2082 t\u2083 (if\u2032 t\u2083 false true)) (if\u2032 t\u2082 (if\u2032 t\u2083 false true) t\u2083)\n\nderive-if : \u2200 {\u03c4 \u0393\u2081} \u2192\n  (t dt : Term \u0393\u2081 bool) \u2192\n  (v\u2081 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2081 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  (v\u2082 : \u0393\u2081 \u27ea \u03c4 \u27ebType) (dv\u2082 : \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType) \u2192\n  \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-if {\u03c4\u2081 \u21d2 \u03c4\u2082} t dt v\u2081 dv\u2081 v\u2082 dv\u2082 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 let \u227c\u2083 = \u227c-trans \u227c\u2081 \u227c\u2082 in\n    derive-if (weaken \u227c\u2083 t) (weaken \u227c\u2083 dt)\n              (v\u2081 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2081 \u227c\u2081 v \u227c\u2082 dv)\n              (v\u2082 \u227c\u2083 (liftVal \u227c\u2082 v)) (dv\u2082 \u227c\u2081 v \u227c\u2082 dv)\nderive-if {bool} t\u2081 dt\u2081 t\u2082 dt\u2082 t\u2083 dt\u2083 =\n  if\u2032 dt\u2081 (if\u2032 t\u2081 (xor\u2083 dt\u2083 t\u2082 t\u2083) (xor\u2083 dt\u2082 t\u2082 t\u2083)) (if\u2032 t\u2081 dt\u2083 dt\u2082)\n\nderive-var : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive-var this (dv SymEnv.\u2022 d\u03c1) = dv\nderive-var (that x) (dv SymEnv.\u2022 d\u03c1) = derive-var x d\u03c1\n\nderive : \u2200 {\u0393\u2081 \u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u0393\u2081 \u27ea \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Context \u0393 \u27ebContext \u2192 \u0393\u2081 \u27ea \u0394-Type \u03c4 \u27ebType\nderive (abs t) \u03c1 d\u03c1 =\n  \u03bb \u227c\u2081 v \u227c\u2082 dv \u2192 derive t (liftVal \u227c\u2082 v SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) \u03c1) (dv SymEnv.\u2022 liftEnv (\u227c-trans \u227c\u2081 \u227c\u2082) d\u03c1)\nderive (app t\u2081 t\u2082) \u03c1 d\u03c1 =\n  derive t\u2081 \u03c1 d\u03c1 \u227c-refl (_ \u27ea t\u2082 \u27ebTerm \u03c1) \u227c-refl (derive t\u2082 \u03c1 d\u03c1)\nderive (var x) \u03c1 d\u03c1 = derive-var x d\u03c1\nderive true \u03c1 d\u03c1 = false\nderive false \u03c1 d\u03c1 = false\nderive {\u0393\u2081} (if t\u2081 t\u2082 t\u2083) \u03c1 d\u03c1 =\n  derive-if (\u0393\u2081 \u27ea t\u2081 \u27ebTerm \u03c1) (derive t\u2081 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2082 \u27ebTerm \u03c1) (derive t\u2082 \u03c1 d\u03c1)\n            (\u0393\u2081 \u27ea t\u2083 \u27ebTerm \u03c1) (derive t\u2083 \u03c1 d\u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7fd85ab97d034036dc625ca2a1b7aae27fa9c224","subject":"Core: update","message":"Core: update\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Core.agda","new_file":"Control\/Protocol\/Core.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nopen import Control.Protocol.InOut\n\nmodule Control.Protocol.Core where\n\nopen import Control.Protocol.End    public\n\n{-\nmodule UniversalProtocols \u2113 {U : \u2605_(\u209b \u2113)}(U\u27e6_\u27e7 : U \u2192 \u2605_ \u2113) where\n-}\nmodule _ \u2113 where\n  U = \u2605_ \u2113\n  U\u27e6_\u27e7 = id\n  data Proto_ : \u2605_(\u209b \u2113) where\n    end : Proto_\n    com : (io : InOut){M : U}(P : U\u27e6 M \u27e7 \u2192 Proto_) \u2192 Proto_\n{-\nmodule U\u2605 \u2113 = UniversalProtocols \u2113 {\u2605_ \u2113} id\nopen U\u2605\n-}\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081 : \u2605}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 P\u2081}(P= : (m\u2080 : M\u2080) \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n    com= refl refl P= = ap (com _) (\u03bb= P=)\n\n    module _ {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n        com\u2243 = com= io= (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    module _ io {M N}(P : M \u00d7 N \u2192 Proto)\n             where\n        com\u2082 : Proto\n        com\u2082 = com io \u03bb m \u2192 com io \u03bb n \u2192 P (m , n)\n\n    {- Proving this would be awesome...\n    module _ io\n             {M\u2080 M\u2081 N\u2080 N\u2081 : \u2605}\n             (M\u00d7N\u2243 : (M\u2080 \u00d7 N\u2080) \u2243 (M\u2081 \u00d7 N\u2081))\n             {P\u2080 : M\u2080 \u00d7 N\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u00d7 N\u2081 \u2192 Proto}\n             (P= : \u2200 m,n\u2080 \u2192 P\u2080 m,n\u2080 \u2261 P\u2081 (\u2013> M\u00d7N\u2243 m,n\u2080))\n             {{_ : UA}} where\n        com\u2082\u2243 : com\u2082 io P\u2080 \u2261 com\u2082 io P\u2081\n        com\u2082\u2243 = {!com=!}\n    -}\n\n    send= = com= {Out} refl\n    send\u2243 = com\u2243 {Out} refl\n    recv= = com= {In} refl\n    recv\u2243 = com\u2243 {In} refl\n\n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= refl refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\npattern recvE M P = com In  {M} P\npattern sendE M P = com Out {M} P\n\nrecv\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nrecv\u2610 P = recv (\u03bb { [ m ] \u2192 P m })\n\nsend\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nsend\u2610 P = send (\u03bb { [ m ] \u2192 P m })\n\n{-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (send P) = recv \u03bb m \u2192 dual (P m)\ndual (recv P) = send \u03bb m \u2192 dual (P m)\n-}\n\ndual : Proto \u2192 Proto\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nsource-of : Proto \u2192 Proto\nsource-of end       = end\nsource-of (com _ P) = send \u03bb m \u2192 source-of (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end   : IsSource end\n  send' : \u2200 {M P} (PT : (m : M) \u2192 IsSource (P m)) \u2192 IsSource (send P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end   : IsSink end\n  recv' : \u2200 {M P} (PT : (m : M) \u2192 IsSink (P m)) \u2192 IsSink (recv P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 io {M P} (P\u2610 : \u2200 (m : \u2610 M) \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com io P)\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end      \u27e7 = End\n\u27e6 com io P \u27e7 = \u27e6 io \u27e7\u1d35\u1d3c _ \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end         = _\nsink (com _ P) x = sink (P x)\n\ntelecom : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntelecom end      _       _       = _\ntelecom (recv P) p       (m , q) = m , telecom (P m) (p m) q\ntelecom (send P) (m , p) q       = m , telecom (P m) p (q m)\n\nsend\u2032 : \u2605 \u2192 Proto \u2192 Proto\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto \u2192 Proto\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 P \u2192 dual (dual P) \u2261 P\n    dual-involutive end        = refl\n    dual-involutive (com io P) = com= (dual\u1d35\u1d3c-involutive io) refl \u03bb m \u2192 dual-involutive (P m)\n\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv _ dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n\n    source-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261 source-of P\n    source-of-idempotent end       = refl\n    source-of-idempotent (com _ P) = com= refl refl \u03bb m \u2192 source-of-idempotent (P m)\n\n    source-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261 source-of P\n    source-of-dual-oblivious end       = refl\n    source-of-dual-oblivious (com _ P) = com= refl refl \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P s \u2192 sink P \u2261 s\n    sink-contr end       s = refl\n    sink-contr (com _ P) s = \u03bb= \u03bb m \u2192 sink-contr (P m) (s m)\n\n    Sink-is-contr : \u2200 P \u2192 Is-contr (Sink P)\n    Sink-is-contr P = sink P , sink-contr P\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (Sink-is-contr P)\n\n    dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n    dual-Log = ap \u27e6_\u27e7 \u2218 source-of-dual-oblivious\n\n\u2713\u1d3e : \ud835\udfda \u2192 Proto\n\u2713\u1d3e 0\u2082 = send\u2032 \ud835\udfd8 end\n\u2713\u1d3e 1\u2082 = end\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_) renaming (proj\u2081 to fst)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nopen import Control.Protocol.InOut\n\nmodule Control.Protocol.Core where\n\nopen import Control.Protocol.End    public\n\n{-\nmodule UniversalProtocols \u2113 {U : \u2605_(\u209b \u2113)}(U\u27e6_\u27e7 : U \u2192 \u2605_ \u2113) where\n-}\nmodule _ \u2113 where\n  U = \u2605_ \u2113\n  U\u27e6_\u27e7 = id\n  data Proto_ : \u2605_(\u209b \u2113) where\n    end : Proto_\n    com : (io : InOut){M : U}(P : U\u27e6 M \u27e7 \u2192 Proto_) \u2192 Proto_\n{-\nmodule U\u2605 \u2113 = UniversalProtocols \u2113 {\u2605_ \u2113} id\nopen U\u2605\n-}\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081 : \u2605}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 P\u2081}(P= : (m\u2080 : M\u2080) \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n    com= refl refl P= = ap (com _) (\u03bb= P=)\n\n    module _ {io\u2080 io\u2081}(io= : io\u2080 \u2261 io\u2081)\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io\u2080 P\u2080 \u2261 com io\u2081 P\u2081\n        com\u2243 = com= io= (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    module _ io {M N}(P : M \u00d7 N \u2192 Proto)\n             where\n        com\u2082 : Proto\n        com\u2082 = com io \u03bb m \u2192 com io \u03bb n \u2192 P (m , n)\n\n    {- Proving this would be awesome...\n    module _ io\n             {M\u2080 M\u2081 N\u2080 N\u2081 : \u2605}\n             (M\u00d7N\u2243 : (M\u2080 \u00d7 N\u2080) \u2243 (M\u2081 \u00d7 N\u2081))\n             {P\u2080 : M\u2080 \u00d7 N\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u00d7 N\u2081 \u2192 Proto}\n             (P= : \u2200 m,n\u2080 \u2192 P\u2080 m,n\u2080 \u2261 P\u2081 (\u2013> M\u00d7N\u2243 m,n\u2080))\n             {{_ : UA}} where\n        com\u2082\u2243 : com\u2082 io P\u2080 \u2261 com\u2082 io P\u2081\n        com\u2082\u2243 = {!com=!}\n    -}\n\n    send= = com= {Out} refl\n    send\u2243 = com\u2243 {Out} refl\n    recv= = com= {In} refl\n    recv\u2243 = com\u2243 {In} refl\n\n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= refl refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\npattern recvE M P = com In  {M} P\npattern sendE M P = com Out {M} P\n\nrecv\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nrecv\u2610 P = recv (\u03bb { [ m ] \u2192 P m })\n\nsend\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nsend\u2610 P = send (\u03bb { [ m ] \u2192 P m })\n\n{-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (send P) = recv \u03bb m \u2192 dual (P m)\ndual (recv P) = send \u03bb m \u2192 dual (P m)\n-}\n\ndual : Proto \u2192 Proto\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nsource-of : Proto \u2192 Proto\nsource-of end       = end\nsource-of (com _ P) = send \u03bb m \u2192 source-of (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end   : IsSource end\n  send' : \u2200 {M P} (PT : (m : M) \u2192 IsSource (P m)) \u2192 IsSource (send P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end   : IsSink end\n  recv' : \u2200 {M P} (PT : (m : M) \u2192 IsSink (P m)) \u2192 IsSink (recv P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 io {M P} (P\u2610 : \u2200 (m : \u2610 M) \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com io P)\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end      \u27e7 = End\n\u27e6 com io P \u27e7 = \u27e6 io \u27e7\u1d35\u1d3c _ \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end         = _\nsink (com _ P) x = sink (P x)\n\ntelecom : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntelecom end      _       _       = _\ntelecom (recv P) p       (m , q) = m , telecom (P m) (p m) q\ntelecom (send P) (m , p) q       = m , telecom (P m) p (q m)\n\nsend\u2032 : \u2605 \u2192 Proto \u2192 Proto\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto \u2192 Proto\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 P \u2192 dual (dual P) \u2261 P\n    dual-involutive end        = refl\n    dual-involutive (com io P) = com= (dual\u1d35\u1d3c-involutive io) refl \u03bb m \u2192 dual-involutive (P m)\n\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv _ dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n\n    source-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261 source-of P\n    source-of-idempotent end       = refl\n    source-of-idempotent (com _ P) = com= refl refl \u03bb m \u2192 source-of-idempotent (P m)\n\n    source-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261 source-of P\n    source-of-dual-oblivious end       = refl\n    source-of-dual-oblivious (com _ P) = com= refl refl \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P s \u2192 sink P \u2261 s\n    sink-contr end       s = refl\n    sink-contr (com _ P) s = \u03bb= \u03bb m \u2192 sink-contr (P m) (s m)\n\n    Sink-is-contr : \u2200 P \u2192 Is-contr (Sink P)\n    Sink-is-contr P = sink P , sink-contr P\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (Sink-is-contr P)\n\n    dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n    dual-Log = ap \u27e6_\u27e7 \u2218 source-of-dual-oblivious\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7fd51b2647e644185cc4572997c6dea6c2177a26","subject":"Modernize weaken.","message":"Modernize weaken.\n\nOld-commit-hash: 83dbf89ee312f8567b4e2ab716f0970f76fff9e1\n","repos":"inc-lc\/ilc-agda","old_file":"lambda.agda","new_file":"lambda.agda","new_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken \u227c\u2081 (abs t) = abs (weaken (keep _ \u2022 \u227c\u2081) t)\nweaken \u227c\u2081 (app t\u2081 t\u2082) = app (weaken \u227c\u2081 t\u2081) (weaken \u227c\u2081 t\u2082)\nweaken \u227c\u2081 (var x) = var (lift \u227c\u2081 x)\nweaken \u227c\u2081 true = true\nweaken \u227c\u2081 false = false\nweaken \u227c\u2081 (if e\u2081 e\u2082 e\u2083) = if (weaken \u227c\u2081 e\u2081) (weaken \u227c\u2081 e\u2082) (weaken \u227c\u2081 e\u2083)\n","old_contents":"module lambda where\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types public\nopen import Syntactic.Contexts Type public hiding (lift)\n\nopen Prefixes\n\nopen import Denotational.Notation\nopen import Denotational.Values public\nopen import Denotational.Environments Type \u27e6_\u27e7Type public\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- NATURAL SEMANTICS\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  true : Val bool\n  false : Val bool\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n  if-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  if-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n    \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n    \u03c1 \u22a2 if t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 true \u27e7Val = true\n\u27e6 false \u27e7Val = false\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) = trans (cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082)) (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\u2193-sound (if-true \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2082\n\u2193-sound (if-false \u2193\u2081 \u2193\u2083) rewrite \u2193-sound \u2193\u2081 = \u2193-sound \u2193\u2083\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} true = true\nweaken {\u0393\u2081} {\u0393\u2082} false = false\nweaken {\u0393\u2081} {\u0393\u2082} (if e\u2081 e\u2082 e\u2083) = if (weaken {\u0393\u2081} {\u0393\u2082} e\u2081) (weaken {\u0393\u2081} {\u0393\u2082} e\u2082) (weaken {\u0393\u2081} {\u0393\u2082} e\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"95d209d155fcf3703b96d3d4c7d655a1f9f355ec","subject":"+Lift","message":"+Lift\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol.agda","new_file":"Control\/Protocol.agda","new_contents":"module Control.Protocol where\n\nopen import Control.Protocol.InOut          public\nopen import Control.Protocol.End            public\nopen import Control.Protocol.Core           public\nopen import Control.Protocol.Sequence       public\nopen import Control.Protocol.Additive       public\nopen import Control.Protocol.Multiplicative public\nopen import Control.Protocol.MultiParty     public\nopen import Control.Protocol.Lift           public\n","old_contents":"module Control.Protocol where\n\nopen import Control.Protocol.InOut          public\nopen import Control.Protocol.End            public\nopen import Control.Protocol.Core           public\nopen import Control.Protocol.Sequence       public\nopen import Control.Protocol.Additive       public\nopen import Control.Protocol.Multiplicative public\nopen import Control.Protocol.MultiParty     public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bd95efc7be0d61b666c1a3c53bcf760cec36e3d4","subject":"goofing around with some lemmas","message":"goofing around with some lemmas\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub {\u0393 = \u0393} x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!}\n  lem-idsub x d () | None\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x) = {!lem-weaken\u03941!}\n  lem-weaken\u03941 (TANEHole D x) = {!!}\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-##eq : {\u03941 \u03942 : hctx} (n : Nat) \u2192 \u03941 ## \u03942 \u2192 \u03941 n == (\u03941 \u222a \u03942) n\n  lem-##eq {\u03941} {\u03942} n apart with \u03941 n\n  lem-##eq n apart | Some x = refl\n  lem-##eq {\u03941} {\u03942} n apart | None with \u03942 n\n  lem-##eq n (\u03c01 , \u03c02) | None | Some x = {!\u03c01 n!}\n  lem-##eq n apart | None | None = refl\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4}  D = tr (\u03bb x \u2192 x , \u0393 \u22a2 d :: \u03c4) (funext (\u03bb x \u2192 {!!})) D\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth ESEHole = TAEHole lem-idsub\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana EAEHole = TCRefl , TAEHole lem-idsub\n    typed-expansion-ana (EANEHole x) = TCRefl , TANEHole (typed-expansion-synth x) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!} -- this hole should be refl from pattern matching but it isn't\n  lem-idsub x d xin       | None   = abort (somenotnone (! xin))\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole x) = {!!}\n  lem-weaken\u03941 (TANEHole D x) = {!!}\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 = {!!}\n  -- lem-weaken\u03942 TAConst = TAConst\n  -- lem-weaken\u03942 (TAVar x\u2081) = TAVar x\u2081\n  -- lem-weaken\u03942 (TALam D) = TALam (lem-weaken\u03942 D)\n  -- lem-weaken\u03942 (TAAp D x D\u2081) = TAAp (lem-weaken\u03942 D) x (lem-weaken\u03942 D\u2081)\n  -- lem-weaken\u03942 (TAEHole x) = {!!}\n  -- lem-weaken\u03942 (TANEHole D x) = {!!}\n  -- lem-weaken\u03942 (TACast D x) = TACast (lem-weaken\u03942 D) x\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth ESEHole = TAEHole lem-idsub\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana EAEHole = TCRefl , TAEHole lem-idsub\n    typed-expansion-ana (EANEHole x) = TCRefl , TANEHole (typed-expansion-synth x) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5e28e95336edd7da4ef18ee12d86f53112e719a2","subject":"Use if_then_else in \u27e6_\u27e7Term.","message":"Use if_then_else in \u27e6_\u27e7Term.\n\nOld-commit-hash: f469e7f3aa20a7cdade4976d8bea10180b56dedb\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Evaluation\/Total.agda","new_file":"Denotational\/Evaluation\/Total.agda","new_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n","old_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"87895cda38378abac828e0969230e4b54e274b23","subject":"whitespace and alpha renaming #3","message":"whitespace and alpha renaming #3\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | V x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | V y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | V y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | V y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = {!!} -- I {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (_ , Step x a q) = S (_ , Step {!!} (ITCastFail y z w) {!!} )\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | V x = I (IFailedCast (FBoxed x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFACast (d' , \u03c41 , \u03c42 , \u03c43 , \u03c44 , refl , d'' , ne ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAEHole (_ , _ , _ , refl , _)           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFANEHole (_ , _ , _ , _ , _ , refl , _)  = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAAp (_ , _ , _ , _ , _ , refl , _)      = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFACastHole (_ , refl , refl , refl , _ ) = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n  progress (TAAp wt1 wt2) | I x | I x\u2082 | CIFAFailedCast (_ , _ , refl , _ )         = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet x\u2082))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | V x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFACast (d' , \u03c41 , \u03c42 , \u03c43 , \u03c44 , refl , d'' , ne ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAEHole (_ , _ , _ , refl , _)           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFANEHole (_ , _ , _ , _ , _ , refl , _)  = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAAp (_ , _ , _ , _ , _ , refl , _)      = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFACastHole (_ , refl , refl , refl , _ ) = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n  progress (TAAp wt1 wt2) | I x | V x\u2082 | CIFAFailedCast (_ , _ , refl , _ )         = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed x\u2082))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = {!!} -- I {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (_ , Step x a q) = S (_ , Step {!!} (ITCastFail y z w) {!!} )\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | V x = I (IFailedCast (FBoxed x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3218eb705abbdcb5492c6db11f79892be5130af3","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"canonical-value-forms.agda","new_file":"canonical-value-forms.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\n\nmodule canonical-value-forms where\n  canonical-value-forms-b : \u2200{\u0394 d} \u2192\n                            \u0394 , \u2205 \u22a2 d :: b \u2192\n                            d val \u2192\n                            d == c\n  canonical-value-forms-b TAConst VConst = refl\n  canonical-value-forms-b (TAVar x\u2081) ()\n  canonical-value-forms-b (TAAp wt wt\u2081) ()\n  canonical-value-forms-b (TAEHole x x\u2081) ()\n  canonical-value-forms-b (TANEHole x wt x\u2081) ()\n  canonical-value-forms-b (TACast wt x) ()\n\n  canonical-value-forms-arr : \u2200{\u0394 d \u03c41 \u03c42} \u2192\n                              \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                              d val \u2192\n                              \u03a3[ x \u2208 Nat ] \u03a3[ d' \u2208 dhexp ]\n                                ((d == (\u00b7\u03bb x [ \u03c41 ] d')) \u00d7 (\u0394 , \u25a0 (x , \u03c41) \u22a2 d' :: \u03c42))\n  canonical-value-forms-arr (TAVar x\u2081) ()\n  canonical-value-forms-arr (TALam wt) VLam = _ , _ , refl , wt\n  canonical-value-forms-arr (TAAp wt wt\u2081) ()\n  canonical-value-forms-arr (TAEHole x x\u2081) ()\n  canonical-value-forms-arr (TANEHole x wt x\u2081) ()\n  canonical-value-forms-arr (TACast wt x) ()\n\n\n\n  -- this argues (somewhat informally) that we didn't miss any cases above;\n  -- this intentionally will make this file fail to typecheck if we added\n  -- more types, hopefully forcing us to remember to add canonical forms\n  -- lemmas as appropriate\n  canonical-value-forms-coverage1 : \u2200{\u0394 d \u03c4} \u2192\n                                   \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                   d val \u2192\n                                   \u03c4 \u2260 b \u2192\n                                   ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                   \u22a5\n  canonical-value-forms-coverage1 TAConst VConst nb na = nb refl\n  canonical-value-forms-coverage1 (TAVar x\u2081) () nb na\n  canonical-value-forms-coverage1 (TALam wt) VLam nb na = na _ _ refl\n  canonical-value-forms-coverage1 (TAAp wt wt\u2081) () nb na\n  canonical-value-forms-coverage1 (TAEHole x x\u2081) () nb na\n  canonical-value-forms-coverage1 (TANEHole x wt x\u2081) () nb na\n  canonical-value-forms-coverage1 (TACast wt x) () nb na\n\n  canonical-value-forms-coverage2 : \u2200{\u0394 d} \u2192\n                                   \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                   d val \u2192\n                                   \u22a5\n  canonical-value-forms-coverage2 (TAVar x\u2081) ()\n  canonical-value-forms-coverage2 (TAAp wt wt\u2081) ()\n  canonical-value-forms-coverage2 (TAEHole x x\u2081) ()\n  canonical-value-forms-coverage2 (TANEHole x wt x\u2081) ()\n  canonical-value-forms-coverage2 (TACast wt x) ()\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\nopen import core\n\nmodule canonical-value-forms where\n  canonical-value-forms-b : \u2200{\u0394 d} \u2192\n                            \u0394 , \u2205 \u22a2 d :: b \u2192\n                            d val \u2192\n                            d == c\n  canonical-value-forms-b TAConst VConst = refl\n  canonical-value-forms-b (TAVar x\u2081) ()\n  canonical-value-forms-b (TAAp wt wt\u2081) ()\n  canonical-value-forms-b (TAEHole x x\u2081) ()\n  canonical-value-forms-b (TANEHole x wt x\u2081) ()\n  canonical-value-forms-b (TACast wt x) ()\n\n  canonical-value-forms-arr : \u2200{\u0394 d \u03c41 \u03c42} \u2192\n                              \u0394 , \u2205 \u22a2 d :: (\u03c41 ==> \u03c42) \u2192\n                              d val \u2192\n                              \u03a3[ x \u2208 Nat ] \u03a3[ d' \u2208 dhexp ]\n                                (d == (\u00b7\u03bb x [ \u03c41 ] d') \u00d7 \u0394 , \u25a0 (x , \u03c41) \u22a2 d' :: \u03c42)\n  canonical-value-forms-arr (TAVar x\u2081) ()\n  canonical-value-forms-arr (TALam wt) VLam = _ , _ , refl , wt\n  canonical-value-forms-arr (TAAp wt wt\u2081) ()\n  canonical-value-forms-arr (TAEHole x x\u2081) ()\n  canonical-value-forms-arr (TANEHole x wt x\u2081) ()\n  canonical-value-forms-arr (TACast wt x) ()\n\n\n\n  -- this argues (somewhat informally) that we didn't miss any cases above;\n  -- this intentionally will make this file fail to typecheck if we added\n  -- more types, hopefully forcing us to remember to add canonical forms\n  -- lemmas as appropriate\n  canonical-value-forms-coverage1 : \u2200{\u0394 d \u03c4} \u2192\n                                   \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n                                   d val \u2192\n                                   \u03c4 \u2260 b \u2192\n                                   ((\u03c41 : htyp) (\u03c42 : htyp) \u2192 \u03c4 \u2260 (\u03c41 ==> \u03c42)) \u2192\n                                   \u22a5\n  canonical-value-forms-coverage1 TAConst VConst nb na = nb refl\n  canonical-value-forms-coverage1 (TAVar x\u2081) () nb na\n  canonical-value-forms-coverage1 (TALam wt) VLam nb na = na _ _ refl\n  canonical-value-forms-coverage1 (TAAp wt wt\u2081) () nb na\n  canonical-value-forms-coverage1 (TAEHole x x\u2081) () nb na\n  canonical-value-forms-coverage1 (TANEHole x wt x\u2081) () nb na\n  canonical-value-forms-coverage1 (TACast wt x) () nb na\n\n  canonical-value-forms-coverage2 : \u2200{\u0394 d} \u2192\n                                   \u0394 , \u2205 \u22a2 d :: \u2987\u2988 \u2192\n                                   d val \u2192\n                                   \u22a5\n  canonical-value-forms-coverage2 (TAVar x\u2081) ()\n  canonical-value-forms-coverage2 (TAAp wt wt\u2081) ()\n  canonical-value-forms-coverage2 (TAEHole x x\u2081) ()\n  canonical-value-forms-coverage2 (TANEHole x wt x\u2081) ()\n  canonical-value-forms-coverage2 (TACast wt x) ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c22cf8909f1e4ed57da03a5d59c9c031d5a9494c","subject":"proved!","message":"proved!\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expansion-unicity.agda","new_file":"expansion-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import correspondence\nopen import synth-unicity\n\nmodule expansion-unicity where\n  \u2987\u2988\u2260arr : \u2200{t1 t2} \u2192 t1 ==> t2 == \u2987\u2988 \u2192 \u22a5\n  \u2987\u2988\u2260arr ()\n\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , ih3\n    expansion-unicity-synth (ESAp1 x\u2081 x\u2082 x\u2083) (ESAp1 x\u2085 x\u2086 x\u2087)\n       with expansion-unicity-ana x\u2083 x\u2087\n    ... | refl , refl , refl\n      with expansion-unicity-ana x\u2082 x\u2086\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAp1 x\u2081 x\u2082 x\u2083) (ESAp2 d5 x\u2085 x\u2086) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2081))\n    expansion-unicity-synth (ESAp1 x\u2081 x\u2082 x\u2083) (ESAp3 d5 x\u2085)    = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2081))\n    expansion-unicity-synth (ESAp2 d5 x\u2081 x\u2082) (ESAp1 x\u2084 x\u2085 x\u2086) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2084))\n    expansion-unicity-synth (ESAp2 d5 x\u2081 x\u2082) (ESAp2 d6 x\u2084 x\u2085)\n      with expansion-unicity-synth d5 d6\n    ... | refl , refl , refl\n      with expansion-unicity-ana x\u2084 x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAp2 d5 x\u2081 x\u2082) (ESAp3 d6 x\u2084)\n      with expansion-unicity-synth d5 d6\n    ...| refl , refl , refl\n      with expansion-unicity-ana x\u2084 x\u2081\n    ... | refl , refl , refl = abort (x\u2082 refl)\n    expansion-unicity-synth (ESAp3 d5 x\u2081) (ESAp1 x\u2083 x\u2084 x\u2085) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2083))\n    expansion-unicity-synth (ESAp3 d5 x\u2081) (ESAp2 d6 x\u2083 x\u2084)\n      with expansion-unicity-synth d5 d6\n    ...| refl , refl , refl\n      with expansion-unicity-ana x\u2081 x\u2083\n    ... | refl , refl , refl = abort (x\u2084 refl)\n    expansion-unicity-synth (ESAp3 d5 x\u2081) (ESAp3 d6 x\u2083)\n      with expansion-unicity-synth d5 d6\n    ...| refl , refl , refl\n      with expansion-unicity-ana x\u2081 x\u2083\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083)\n      with expansion-unicity-ana x x\u2082\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAsc1 {\u0393} {e} {\u03c4} {d} {\u03c4'} x x\u2081) (ESAsc2 x\u2082)\n      with expansion-unicity-ana x x\u2082\n    ... | refl , contr , refl = abort (x\u2081 (! contr))\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , contr , refl = abort (x\u2082 contr)\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 \u03c42' : htyp} {d d' : dhexp} {\u0394 \u0394' : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d  :: \u03c42  \u22a3 \u0394  \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d' :: \u03c42' \u22a3 \u0394' \u2192\n                          d == d' \u00d7 \u03c42 == \u03c42' \u00d7 \u0394 == \u0394'\n    expansion-unicity-ana (EALam x\u2081 D1) (EALam x\u2082 D2)\n      with expansion-unicity-ana D1 D2\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EALam x\u2081 D1) (EASubsume x\u2082 x\u2083 () x\u2085)\n    expansion-unicity-ana (EALamHole x\u2081 D1) (EALamHole x\u2082 D2)\n      with expansion-unicity-ana D1 D2\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EALamHole x\u2081 D1) (EASubsume x\u2082 x\u2083 () x\u2085)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALam x\u2085 D2)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALamHole x\u2085 D2)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087)\n      with expansion-unicity-synth x\u2082 x\u2086\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = abort (x _ refl)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084) = abort (x\u2081 _ _ refl)\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = abort (x _ refl)\n    expansion-unicity-ana EAEHole EAEHole = refl , refl , refl\n    expansion-unicity-ana (EANEHole x) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = abort (x\u2082 _ _ refl)\n    expansion-unicity-ana (EANEHole x) (EANEHole x\u2081)\n      with expansion-unicity-synth x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import correspondence\nopen import synth-unicity\n\nmodule expansion-unicity where\n  \u2987\u2988\u2260arr : \u2200{t1 t2} \u2192 t1 ==> t2 == \u2987\u2988 \u2192 \u22a5\n  \u2987\u2988\u2260arr ()\n\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , ih3\n    expansion-unicity-synth (ESAp1 x\u2081 x\u2082 x\u2083) (ESAp1 x\u2085 x\u2086 x\u2087)\n       with expansion-unicity-ana x\u2083 x\u2087\n    ... | refl , refl , refl\n      with expansion-unicity-ana x\u2082 x\u2086\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAp1 x\u2081 x\u2082 x\u2083) (ESAp2 d5 x\u2085 x\u2086) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2081))\n    expansion-unicity-synth (ESAp1 x\u2081 x\u2082 x\u2083) (ESAp3 d5 x\u2085)    = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2081))\n    expansion-unicity-synth (ESAp2 d5 x\u2081 x\u2082) (ESAp1 x\u2084 x\u2085 x\u2086) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2084))\n    expansion-unicity-synth (ESAp2 d5 x\u2081 x\u2082) (ESAp2 d6 x\u2084 x\u2085)\n      with expansion-unicity-synth d5 d6 -- | expansion-unicity-ana x\u2084 x\u2081\n    ... | refl , refl , refl = {!expansion-unicity-ana x\u2084 x\u2081!} -- | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAp2 d5 x\u2081 x\u2082) (ESAp3 d6 x\u2084) = {!!}\n    --   with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2084 x\u2081\n    -- ...| refl , refl , refl | refl , refl , refl , refl  = abort (x\u2082 refl)\n    expansion-unicity-synth (ESAp3 d5 x\u2081) (ESAp1 x\u2083 x\u2084 x\u2085) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2083))\n    expansion-unicity-synth (ESAp3 d5 x\u2081) (ESAp2 d6 x\u2083 x\u2084) = {!!}\n    --   with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2083 x\u2081\n    -- ...| refl , refl , refl | refl , refl , refl = abort (x\u2084 refl)\n    expansion-unicity-synth (ESAp3 d5 x\u2081) (ESAp3 d6 x\u2083) = {!!}\n    --   with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2083 x\u2081\n    -- ... | refl , refl , refl | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083)\n      with expansion-unicity-ana x x\u2082\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAsc1 {\u0393} {e} {\u03c4} {d} {\u03c4'} x x\u2081) (ESAsc2 x\u2082)\n      with expansion-unicity-ana x x\u2082\n    ... | refl , contr , refl = abort (x\u2081 (! contr))\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , contr , refl = abort (x\u2082 contr)\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 \u03c42' : htyp} {d d' : dhexp} {\u0394 \u0394' : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d  :: \u03c42  \u22a3 \u0394  \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d' :: \u03c42' \u22a3 \u0394' \u2192\n                          d == d' \u00d7 \u03c42 == \u03c42' \u00d7 \u0394 == \u0394'\n    expansion-unicity-ana (EALam x\u2081 D1) (EALam x\u2082 D2)\n      with expansion-unicity-ana D1 D2\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EALam x\u2081 D1) (EASubsume x\u2082 x\u2083 () x\u2085)\n    expansion-unicity-ana (EALamHole x\u2081 D1) (EALamHole x\u2082 D2)\n      with expansion-unicity-ana D1 D2\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EALamHole x\u2081 D1) (EASubsume x\u2082 x\u2083 () x\u2085)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALam x\u2085 D2)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALamHole x\u2085 D2)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087)\n      with expansion-unicity-synth x\u2082 x\u2086\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = abort (x _ refl)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084) = abort (x\u2081 _ _ refl)\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = abort (x _ refl)\n    expansion-unicity-ana EAEHole EAEHole = refl , refl , refl\n    expansion-unicity-ana (EANEHole x) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = abort (x\u2082 _ _ refl)\n    expansion-unicity-ana (EANEHole x) (EANEHole x\u2081)\n      with expansion-unicity-synth x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"89a5dcaaf80b8ea4f12e541e0a0679b136ac8194","subject":"Desc model: replace All with Pi. It's all Set now.","message":"Desc model: replace All with Pi. It's all Set now.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  ((y : H) -> p (proj\u2081 v y))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v = (\\y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> Set) ->\n         ((x : descOp d (Mu d)) -> (boxOp d (Mu d) bp x) -> bp (Con x)) ->\n         (v : Mu d) -> bp v\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","old_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n-- Bringing in an hand-made Prop universe\n\ndata PROP : Set where\n  All : (S : Set) -> (S -> PROP) -> PROP\n  And : PROP -> PROP -> PROP\n  Trivial : PROP\n  Absurd : PROP\n\nPrf : PROP -> Set\nPrf Absurd = \u22a5\nPrf Trivial = \u22a4\nPrf (All S P) = (x : S) -> Prf (P x)\nPrf (And A B) = \u03a3 (Prf A)  (\\ _ -> Prf B)\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  Prf ( All H (\\y -> p (proj\u2081 v y)))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) ->\n           (Prf (All D (\\y -> bp y))) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n-- On the Ind case, the Pig code is saying something along those lines:\n-- mapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (p?! (proj\u2082 v))\n\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> PROP) ->\n         (Prf (All (descOp d (Mu d)) (\\ x -> (All (boxOp d (Mu d) bp x) (\\ _ -> bp (Con x)))))) ->\n         (v : Mu d) -> Prf (bp v)\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"98b7114876708cb2087e6480b96eaa8682e6152a","subject":"correctness-of-derive on `add`, `bag`, and part of `union` Pending: `diff`, `map` Caution: Derivative of `map` is implemented by recomputation in          this file.","message":"correctness-of-derive on `add`, `bag`, and part of `union`\nPending: `diff`, `map`\nCaution: Derivative of `map` is implemented by recomputation in\n         this file.\n\nOld-commit-hash: 97d6340178c48d60d13422ad5fb5fcc5fe67fc1b\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/NatBag.agda","new_file":"experimental\/NatBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\nopen import NatBag\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemma: diff-apply holds with propositional\n-- equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9 -- extensional equivalence as changes applied to emptyBag\n    emptyBag ++ \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    identity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\n    identity-weakening {\u2205} {\u2205} = refl\n    identity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n      cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) ++ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","old_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e36904bccaa06002f3fa9632a4ed510f746a6049","subject":"Vec: permutations, syntax and properties","message":"Vec: permutations, syntax and properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs \n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      `tl : Perm \u2192 Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    `tl \u03c0 \u207b\u00b9 = `tl (\u03c0 \u207b\u00b9)\n    (\u03c0\u2080 `\u204f \u03c0\u2081) \u207b\u00b9 = \u03c0\u2081 \u207b\u00b9 `\u204f \u03c0\u2080 \u207b\u00b9\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `tl \u03c0 \u2219 []       = []\n    `tl \u03c0 \u2219 (x \u2237 xs) = x \u2237 \u03c0 \u2219 xs\n    (\u03c0 `\u204f \u03c0\u2081) \u2219 xs = \u03c0\u2081 \u2219 \u03c0 \u2219 xs\n\n    _\u2257\u03c0_ : Perm \u2192 Perm \u2192 Set _\n    \u03c0\u2080 \u2257\u03c0 \u03c0\u2081 = \u2200 {n} (xs : Vec A n) \u2192 \u03c0\u2080 \u2219 xs \u2261 \u03c0\u2081 \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 \u03c0 \u2192 (\u03c0 `\u204f \u03c0 \u207b\u00b9) \u2257\u03c0 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    (`tl \u03c0 \u207b\u00b9-inverse) [] = refl\n    (`tl \u03c0 \u207b\u00b9-inverse) (x \u2237 xs) rewrite (\u03c0 \u207b\u00b9-inverse) xs = refl\n    ((\u03c0\u2080 `\u204f \u03c0\u2081) \u207b\u00b9-inverse) xs rewrite (\u03c0\u2081 \u207b\u00b9-inverse) (\u03c0\u2080 \u2219 xs) | (\u03c0\u2080 \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 \u03c0 \u2192 (\u03c0 \u207b\u00b9) \u207b\u00b9 \u2261 \u03c0\n    `id \u207b\u00b9-involutive = refl\n    `0\u21941 \u207b\u00b9-involutive = refl\n    `tl \u03c0 \u207b\u00b9-involutive rewrite \u03c0 \u207b\u00b9-involutive = refl\n    (\u03c0\u2080 `\u204f \u03c0\u2081) \u207b\u00b9-involutive rewrite \u03c0\u2080 \u207b\u00b9-involutive | \u03c0\u2081 \u207b\u00b9-involutive = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 \u03c0 \u2192 (\u03c0 \u207b\u00b9 `\u204f \u03c0) \u2257\u03c0 `id\n    (\u03c0 \u207b\u00b9-inverse\u2032) xs with ((\u03c0 \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite \u03c0 \u207b\u00b9-involutive = p\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    \u2219-replicate : \u2200 n (x : A) \u03c0 \u2192 \u03c0 \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl \u03c0) = refl\n    \u2219-replicate (suc n) x (`tl \u03c0) rewrite \u2219-replicate n x \u03c0 = refl\n    \u2219-replicate n x (\u03c0 `\u204f \u03c0\u2081) rewrite \u2219-replicate n x \u03c0 | \u2219-replicate n x \u03c0\u2081 = refl\n\n    \u229b-dist-0\u21941 : \u2200 {n} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n    \u229b-dist-0\u21941 _           []          = refl\n    \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationProperties {a} (A : Set a) where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} (fs : Vec (Endo A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 fs      `id        xs = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u229b-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u229b-dist-\u2219 fs \u03c0 xs = refl\n    \u229b-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u229b-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u229b-dist-\u2219 fs \u03c0\u2080 xs = refl\n\n    private\n        lem : \u2200 {n} (fs : Vec (Endo A) n) \u03c0 xs \u2192 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (\u03c0 \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs \u03c0 xs rewrite sym (\u229b-dist-\u2219 (\u03c0 \u207b\u00b9 \u2219 fs) \u03c0 xs) | (\u03c0 \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} (f : Endo A) \u03c0 (xs : Vec A n) \u2192 map f (\u03c0 \u2219 xs) \u2261 \u03c0 \u2219 map f xs\n    \u2219-map {n} f \u03c0 xs rewrite sym (\u229b-dist-\u2219 (replicate f) \u03c0 xs) | \u2219-replicate n f \u03c0 = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4747897fd7bf429d38df7ce3badc93a51987a0cc","subject":"Vec: lifting alg structures to vectors","message":"Vec: lifting alg structures to vectors\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_) \nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = \u2261.cong\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f g : A \u2192 A \u2192 A}\n                 (f-g : \u2200 x y z \u2192 f (g x y) z \u2248 f x (g y z)) where\n    zipWith-assoc :\n      \u2200 {n} (xs ys zs : Vec A n) \u2192\n      zipWith f (zipWith g xs ys) zs \u2248\u1d5b zipWith f xs (zipWith g ys zs)\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = f-g x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-left [] = []-cong\n    zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-right [] = []-cong\n    zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n    zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n    zipWith-comm [] [] = []-cong\n    zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-left-inverse [] = []-cong\n     zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-right-inverse [] = []-cong\n     zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                                 ; identity = zipWith-id-left (proj\u2081 M.identity)\n                                                 , zipWith-id-right (proj\u2082 M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (proj\u2081 G.inverse))\n                                                , (zipWith-right-inverse (proj\u2082 G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = idp\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = idp\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\n\nmodule Here {a} {A : \u2605 a} where\n  open Data.Vec.Equality.Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = \u2261.cong\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = idp\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_) \nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = cong\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : \u2605 a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = !(++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = cong\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(proj\u2081 \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = refl\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = refl\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"eeea21e61380cecdd38d82a52529910f4125bc0f","subject":"Two: +\u228e-\u2713\u2228","message":"Two: +\u228e-\u2713\u2228\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Two\/Base.agda","new_file":"lib\/Data\/Two\/Base.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two.Base where\n\nopen import Data.Zero\n\nopen import Data.Bool\n  public\n  hiding (if_then_else_)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\n\nopen import Data.Product\n  renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nopen import Data.Sum\n  using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr)\n\nopen import Data.Nat.Base\n  using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\n\nopen import Type using (\u2605_)\n\nopen import Relation.Nullary using (\u00ac_; Dec; yes; no)\n\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; _\u2262_; refl)\n\n0\u22621\u2082 : 0\u2082 \u2262 1\u2082\n0\u22621\u2082 ()\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: fst 1: snd ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\ninfix 4 _==_\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n-- See also \u2713-\u2227\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 {0\u2082} p q = p\n\u2713\u2227 {1\u2082} p q = q\n\n-- See also \u2713-\u2227\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 {0\u2082} ()\n\u2713\u2227\u2081 {1\u2082} _ = _\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {0\u2082} ()\n\u2713\u2227\u2082 {1\u2082} p = p\n\n\u2713\u2227\u00d7 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081 \u00d7 \u2713 b\u2082\n\u2713\u2227\u00d7 {0\u2082} ()\n\u2713\u2227\u00d7 {1\u2082} p = _ , p\n\n-- Similar to \u2713-\u2228\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {0\u2082} = inr\n\u2713\u2228-\u228e {1\u2082} = inl\n\n\u228e-\u2713\u2228 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u228e-\u2713\u2228 {0\u2082} (inl ())\n\u228e-\u2713\u2228 {0\u2082} (inr y) = y\n\u228e-\u2713\u2228 {1\u2082} x = _\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 {1\u2082} _ = _\n\u2713\u2228\u2081 {0\u2082} ()\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {0\u2082} p = p\n\u2713\u2228\u2082 {1\u2082} _ = _\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n-- This particular implementation has been\n-- chosen for computational content.\n-- Namely the proof is \"re-created\" when b is 1\u2082.\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \u2713 b\ncheck 0\u2082 {}\ncheck 1\u2082 {_} = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Two.Base where\n\nopen import Data.Zero\n\nopen import Data.Bool\n  public\n  hiding (if_then_else_)\n  renaming (Bool to \ud835\udfda; false to 0\u2082; true to 1\u2082; T to \u2713)\n\nopen import Data.Product\n  renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nopen import Data.Sum\n  using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr)\n\nopen import Data.Nat.Base\n  using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\n\nopen import Type using (\u2605_)\n\nopen import Relation.Nullary using (\u00ac_; Dec; yes; no)\n\nopen import Relation.Binary.PropositionalEquality\n  using (_\u2261_; _\u2262_; refl)\n\n0\u22621\u2082 : 0\u2082 \u2262 1\u2082\n0\u22621\u2082 ()\n\n_\u00b2 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nA \u00b2 = \ud835\udfda \u2192 A\n\nmodule _ {p} {P : \ud835\udfda \u2192 \u2605 p} where\n\n    [0:_1:_] : P 0\u2082 \u2192 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    [0: e\u2080 1: e\u2081 ] 0\u2082 = e\u2080\n    [0: e\u2080 1: e\u2081 ] 1\u2082 = e\u2081\n\n    tabulate\u2082 : ((b : \ud835\udfda) \u2192 P b) \u2192 P 0\u2082 \u00d7 P 1\u2082\n    tabulate\u2082 f = f 0\u2082 , f 1\u2082\n\n    \u03b7-[0:1:] : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 [0: f 0\u2082 1: f 1\u2082 ] b \u2261 f b\n    \u03b7-[0:1:] f 0\u2082 = refl\n    \u03b7-[0:1:] f 1\u2082 = refl\n\n    proj : P 0\u2082 \u00d7 P 1\u2082 \u2192 (b : \ud835\udfda) \u2192 P b\n    proj = uncurry [0:_1:_]\n\n    proj-tabulate\u2082 : \u2200 (f : (b : \ud835\udfda) \u2192 P b) b \u2192 proj (tabulate\u2082 f) b \u2261 f b\n    proj-tabulate\u2082 = \u03b7-[0:1:]\n\nmodule _ {a} {A : \u2605 a} where\n\n    [0:_1:_]\u2032 : A \u2192 A \u2192 A \u00b2\n    [0:_1:_]\u2032 = [0:_1:_]\n\n    case_0:_1:_ : \ud835\udfda \u2192 A \u2192 A \u2192 A\n    case b 0: e\u2080 1: e\u2081 = [0: e\u2080\n                          1: e\u2081 ] b\n\n    proj\u2032 : A \u00d7 A \u2192 A \u00b2\n    proj\u2032 = proj\n\n    proj[_] : \ud835\udfda \u2192 A \u00d7 A \u2192 A\n    proj[_] = [0: fst 1: snd ]\n\n    mux : \ud835\udfda \u00d7 (A \u00d7 A) \u2192 A\n    mux (s , e\u1d62) = proj e\u1d62 s\n\nnor : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnor b\u2080 b\u2081 = not (b\u2080 \u2228 b\u2081)\n\nnand : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nnand b\u2080 b\u2081 = not (b\u2080 \u2227 b\u2081)\n\ninfix 4 _==_\n\n-- For properties about _==_ see Data.Two.Equality\n_==_ : (b\u2080 b\u2081 : \ud835\udfda) \u2192 \ud835\udfda\nb\u2080 == b\u2081 = (not b\u2080) xor b\u2081\n\n\u2261\u2192\u2713 : \u2200 {b} \u2192 b \u2261 1\u2082 \u2192 \u2713 b\n\u2261\u2192\u2713 refl = _\n\n\u2261\u2192\u2713not : \u2200 {b} \u2192 b \u2261 0\u2082 \u2192 \u2713 (not b)\n\u2261\u2192\u2713not refl = _\n\n\u2713\u2192\u2261 : \u2200 {b} \u2192 \u2713 b \u2192 b \u2261 1\u2082\n\u2713\u2192\u2261 {1\u2082} _ = refl\n\u2713\u2192\u2261 {0\u2082} ()\n\n\u2713not\u2192\u2261 : \u2200 {b} \u2192 \u2713 (not b) \u2192 b \u2261 0\u2082\n\u2713not\u2192\u2261 {0\u2082} _ = refl\n\u2713not\u2192\u2261 {1\u2082} ()\n\n-- See also \u2713-\u2227\n\u2713\u2227 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2227 b\u2082)\n\u2713\u2227 {0\u2082} p q = p\n\u2713\u2227 {1\u2082} p q = q\n\n-- See also \u2713-\u2227\n\u2713\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081\n\u2713\u2227\u2081 {0\u2082} ()\n\u2713\u2227\u2081 {1\u2082} _ = _\n\n\u2713\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2082\n\u2713\u2227\u2082 {0\u2082} ()\n\u2713\u2227\u2082 {1\u2082} p = p\n\n\u2713\u2227\u00d7 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2227 b\u2082) \u2192 \u2713 b\u2081 \u00d7 \u2713 b\u2082\n\u2713\u2227\u00d7 {0\u2082} ()\n\u2713\u2227\u00d7 {1\u2082} p = _ , p\n\n-- Similar to \u2713-\u2228\n\u2713\u2228-\u228e : \u2200 {b\u2081 b\u2082} \u2192 \u2713 (b\u2081 \u2228 b\u2082) \u2192 \u2713 b\u2081 \u228e \u2713 b\u2082\n\u2713\u2228-\u228e {0\u2082} = inr\n\u2713\u2228-\u228e {1\u2082} = inl\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2081 {1\u2082} _ = _\n\u2713\u2228\u2081 {0\u2082} ()\n\n-- Similar to \u2713-\u2228\n\u2713\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2082 \u2192 \u2713 (b\u2081 \u2228 b\u2082)\n\u2713\u2228\u2082 {0\u2082} p = p\n\u2713\u2228\u2082 {1\u2082} _ = _\n\n\u2713-not-\u00ac : \u2200 {b} \u2192 \u2713 (not b) \u2192 \u00ac (\u2713 b)\n\u2713-not-\u00ac {0\u2082} _ = \u03bb()\n\u2713-not-\u00ac {1\u2082} ()\n\n\u2713-\u00ac-not : \u2200 {b} \u2192 \u00ac (\u2713 b) \u2192 \u2713 (not b)\n\u2713-\u00ac-not {0\u2082} _ = _\n\u2713-\u00ac-not {1\u2082} f = f _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2192 \u2713 (x \u2228 y)\n\u2227\u21d2\u2228 0\u2082 _ = \u03bb ()\n\u2227\u21d2\u2228 1\u2082 _ = _\n\n-- This particular implementation has been\n-- chosen for computational content.\n-- Namely the proof is \"re-created\" when b is 1\u2082.\ncheck : \u2200 b \u2192 {pf : \u2713 b} \u2192 \u2713 b\ncheck 0\u2082 {}\ncheck 1\u2082 {_} = _\n\n\u2713dec : \u2200 b \u2192 Dec (\u2713 b)\n\u2713dec = [0: no (\u03bb())\n        1: yes _ ]\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan 0\u2082 _ = refl\nde-morgan 1\u2082 _ = refl\n\n\u22620\u2192\u22611 : \u2200 {x} \u2192 x \u2262 0\u2082 \u2192 x \u2261 1\u2082\n\u22620\u2192\u22611 {1\u2082} p = refl\n\u22620\u2192\u22611 {0\u2082} p = \ud835\udfd8-elim (p refl)\n\n\u22621\u2192\u22610 : \u2200 {x} \u2192 x \u2262 1\u2082 \u2192 x \u2261 0\u2082\n\u22621\u2192\u22610 {0\u2082} p = refl\n\u22621\u2192\u22610 {1\u2082} p = \ud835\udfd8-elim (p refl)\n\n-- 0\u2082 is 0 and 1\u2082 is 1\n\ud835\udfda\u25b9\u2115 : \ud835\udfda \u2192 \u2115\n\ud835\udfda\u25b9\u2115 = [0: 0\n       1: 1 ]\n\n\ud835\udfda\u25b9\u2115\u22641 : \u2200 b \u2192 \ud835\udfda\u25b9\u2115 b \u2264 1\n\ud835\udfda\u25b9\u2115\u22641 = [0: z\u2264n\n         1: s\u2264s z\u2264n ]\n\n\ud835\udfda\u25b9\u2115-\u2293 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2293 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 b)\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2293 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2294 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2294 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2228 b)\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 1\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2294 0\u2082 _  = refl\n\n\ud835\udfda\u25b9\u2115-\u2238 : \u2200 a b \u2192 \ud835\udfda\u25b9\u2115 a \u2238 \ud835\udfda\u25b9\u2115 b \u2261 \ud835\udfda\u25b9\u2115 (a \u2227 not b)\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 0\u2082 1\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 0\u2082 = refl\n\ud835\udfda\u25b9\u2115-\u2238 1\u2082 1\u2082 = refl\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {0\u2082} {0\u2082} _ = refl\nnot-inj {1\u2082} {1\u2082} _ = refl\nnot-inj {0\u2082} {1\u2082} ()\nnot-inj {1\u2082} {0\u2082} ()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"56b968c2a1e0d04fcd0e8ca3bf8fe919266e2568","subject":"Data.Nat.Distance: ap\u2082","message":"Data.Nat.Distance: ap\u2082\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Distance.agda","new_file":"lib\/Data\/Nat\/Distance.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (cong\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"56a5c69d9cd849b0dd6d389929fdf9c7e1a49f31","subject":"FunctionProperties.Eq: Injective, Conflict","message":"FunctionProperties.Eq: Injective, Conflict\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Like Algebra.FunctionProperties but specialized to _\u2261_ and using implict arguments\n-- Moreover there is some extensions such as:\n-- * interchange\n-- * non-zero inversions\n-- * cancelation and non-zero cancelation\n\n-- These properties can (for instance) be used to define algebraic\n-- structures.\n\nopen import Level\nopen import Function using (flip)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\n-- The properties are specified using the following relation as\n-- \"equality\".\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\n------------------------------------------------------------------------\n-- Unary and binary operations\n\nopen import Algebra.FunctionProperties.Core public\n\n------------------------------------------------------------------------\n-- Properties of operations\n\nAssociative : Op\u2082 A \u2192 Set _\nAssociative _\u00b7_ = \u2200 {x y z} \u2192 ((x \u00b7 y) \u00b7 z) \u2261 (x \u00b7 (y \u00b7 z))\n\nCommutative : Op\u2082 A \u2192 Set _\nCommutative _\u00b7_ = \u2200 {x y} \u2192 (x \u00b7 y) \u2261 (y \u00b7 x)\n\nLeftIdentity : A \u2192 Op\u2082 A \u2192 Set _\nLeftIdentity e _\u00b7_ = \u2200 {x} \u2192 (e \u00b7 x) \u2261 x\n\nRightIdentity : A \u2192 Op\u2082 A \u2192 Set _\nRightIdentity e _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 e) \u2261 x\n\nIdentity : A \u2192 Op\u2082 A \u2192 Set _\nIdentity e \u00b7 = LeftIdentity e \u00b7 \u00d7 RightIdentity e \u00b7\n\nLeftZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftZero z _\u00b7_ = \u2200 {x} \u2192 (z \u00b7 x) \u2261 z\n\nRightZero : A \u2192 Op\u2082 A \u2192 Set _\nRightZero z _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 z) \u2261 z\n\nZero : A \u2192 Op\u2082 A \u2192 Set _\nZero z \u00b7 = LeftZero z \u00b7 \u00d7 RightZero z \u00b7\n\nLeftInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nLeftInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nRightInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nRightInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverse e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\nInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverseNonZero zero e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\n_DistributesOver\u02e1_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02e1 _+_ =\n  \u2200 {x y z} \u2192 (x * (y + z)) \u2261 ((x * y) + (x * z))\n\n_DistributesOver\u02b3_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02b3 _+_ =\n  \u2200 {x y z} \u2192 ((y + z) * x) \u2261 ((y * x) + (z * x))\n\n_DistributesOver_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n* DistributesOver + = (* DistributesOver\u02e1 +) \u00d7 (* DistributesOver\u02b3 +)\n\n_IdempotentOn_ : Op\u2082 A \u2192 A \u2192 Set _\n_\u00b7_ IdempotentOn x = (x \u00b7 x) \u2261 x\n\nIdempotent : Op\u2082 A \u2192 Set _\nIdempotent \u00b7 = \u2200 {x} \u2192 \u00b7 IdempotentOn x\n\nIdempotentFun : Op\u2081 A \u2192 Set _\nIdempotentFun f = \u2200 {x} \u2192 f (f x) \u2261 f x\n\n_Absorbs_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_\u00b7_ Absorbs _\u2218_ = \u2200 {x y} \u2192 (x \u00b7 (x \u2218 y)) \u2261 x\n\nAbsorptive : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nAbsorptive \u00b7 \u2218 = (\u00b7 Absorbs \u2218) \u00d7 (\u2218 Absorbs \u00b7)\n\nInvolutive : Op\u2081 A \u2192 Set _\nInvolutive f = \u2200 {x} \u2192 f (f x) \u2261 x\n\nInterchange : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nInterchange _\u00b7_ _\u2218_ = \u2200 {x y z t} \u2192 ((x \u00b7 y) \u2218 (z \u00b7 t)) \u2261 ((x \u2218 z) \u00b7 (y \u2218 t))\n\nLeftCancel : Op\u2082 A \u2192 Set _\nLeftCancel _\u00b7_ = \u2200 {c x y} \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancel : Op\u2082 A \u2192 Set _\nRightCancel _\u00b7_ = \u2200 {c x y} \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nLeftCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nRightCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nmodule InterchangeFromAssocComm\n         (_\u00b7_     : Op\u2082 A)\n         (\u00b7-assoc : Associative _\u00b7_)\n         (\u00b7-comm  : Commutative _\u00b7_)\n         where\n\n    open \u2261-Reasoning\n\n    \u00b7= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 (x \u00b7 y) \u2261 (x' \u00b7 y')\n    \u00b7= refl refl = refl\n\n    \u00b7-interchange : Interchange _\u00b7_ _\u00b7_\n    \u00b7-interchange {x} {y} {z} {t}\n                = (x \u00b7 y) \u00b7 (z \u00b7 t)\n                \u2261\u27e8 \u00b7-assoc \u27e9\n                  x \u00b7 (y \u00b7 (z \u00b7 t))\n                \u2261\u27e8 \u00b7= refl (! \u00b7-assoc) \u27e9\n                  x \u00b7 ((y \u00b7 z) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl (\u00b7= \u00b7-comm refl) \u27e9\n                  x \u00b7 ((z \u00b7 y) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl \u00b7-assoc \u27e9\n                  x \u00b7 (z \u00b7 (y \u00b7 t))\n                \u2261\u27e8 ! \u00b7-assoc \u27e9\n                  (x \u00b7 z) \u00b7 (y \u00b7 t)\n                \u220e\n\nmodule _ {b} {B : Set b} (f : A \u2192 B) where\n\n    Injective : Set (b \u2294 a)\n    Injective = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n    Conflict : Set (b \u2294 a)\n    Conflict = \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 (x \u2262 y) \u00d7 f x \u2261 f y\n\nmodule _ {b} {B : Set b} {f : A \u2192 B} where\n    Injective-\u00acConflict : Injective f \u2192 \u00ac (Conflict f)\n    Injective-\u00acConflict inj (x , y , x\u2262y , fx\u2261fy) = x\u2262y (inj fx\u2261fy)\n\n    Conflict-\u00acInjective : Conflict f \u2192 \u00ac (Injective f)\n    Conflict-\u00acInjective = flip Injective-\u00acConflict\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Like Algebra.FunctionProperties but specialized to _\u2261_ and using implict arguments\n-- Moreover there is some extensions such as:\n-- * interchange\n-- * non-zero inversions\n-- * cancelation and non-zero cancelation\n\n-- These properties can (for instance) be used to define algebraic\n-- structures.\n\nopen import Level\nopen import Data.Product\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\n-- The properties are specified using the following relation as\n-- \"equality\".\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\n------------------------------------------------------------------------\n-- Unary and binary operations\n\nopen import Algebra.FunctionProperties.Core public\n\n------------------------------------------------------------------------\n-- Properties of operations\n\nAssociative : Op\u2082 A \u2192 Set _\nAssociative _\u00b7_ = \u2200 {x y z} \u2192 ((x \u00b7 y) \u00b7 z) \u2261 (x \u00b7 (y \u00b7 z))\n\nCommutative : Op\u2082 A \u2192 Set _\nCommutative _\u00b7_ = \u2200 {x y} \u2192 (x \u00b7 y) \u2261 (y \u00b7 x)\n\nLeftIdentity : A \u2192 Op\u2082 A \u2192 Set _\nLeftIdentity e _\u00b7_ = \u2200 {x} \u2192 (e \u00b7 x) \u2261 x\n\nRightIdentity : A \u2192 Op\u2082 A \u2192 Set _\nRightIdentity e _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 e) \u2261 x\n\nIdentity : A \u2192 Op\u2082 A \u2192 Set _\nIdentity e \u00b7 = LeftIdentity e \u00b7 \u00d7 RightIdentity e \u00b7\n\nLeftZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftZero z _\u00b7_ = \u2200 {x} \u2192 (z \u00b7 x) \u2261 z\n\nRightZero : A \u2192 Op\u2082 A \u2192 Set _\nRightZero z _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 z) \u2261 z\n\nZero : A \u2192 Op\u2082 A \u2192 Set _\nZero z \u00b7 = LeftZero z \u00b7 \u00d7 RightZero z \u00b7\n\nLeftInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nLeftInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nRightInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nRightInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverseNonZero zero e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 x \u2262 zero \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverse e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\nInverseNonZero : (zero e : A) \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverseNonZero zero e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\n_DistributesOver\u02e1_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02e1 _+_ =\n  \u2200 {x y z} \u2192 (x * (y + z)) \u2261 ((x * y) + (x * z))\n\n_DistributesOver\u02b3_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02b3 _+_ =\n  \u2200 {x y z} \u2192 ((y + z) * x) \u2261 ((y * x) + (z * x))\n\n_DistributesOver_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n* DistributesOver + = (* DistributesOver\u02e1 +) \u00d7 (* DistributesOver\u02b3 +)\n\n_IdempotentOn_ : Op\u2082 A \u2192 A \u2192 Set _\n_\u00b7_ IdempotentOn x = (x \u00b7 x) \u2261 x\n\nIdempotent : Op\u2082 A \u2192 Set _\nIdempotent \u00b7 = \u2200 {x} \u2192 \u00b7 IdempotentOn x\n\nIdempotentFun : Op\u2081 A \u2192 Set _\nIdempotentFun f = \u2200 {x} \u2192 f (f x) \u2261 f x\n\n_Absorbs_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_\u00b7_ Absorbs _\u2218_ = \u2200 {x y} \u2192 (x \u00b7 (x \u2218 y)) \u2261 x\n\nAbsorptive : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nAbsorptive \u00b7 \u2218 = (\u00b7 Absorbs \u2218) \u00d7 (\u2218 Absorbs \u00b7)\n\nInvolutive : Op\u2081 A \u2192 Set _\nInvolutive f = \u2200 {x} \u2192 f (f x) \u2261 x\n\nInterchange : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nInterchange _\u00b7_ _\u2218_ = \u2200 {x y z t} \u2192 ((x \u00b7 y) \u2218 (z \u00b7 t)) \u2261 ((x \u2218 z) \u00b7 (y \u2218 t))\n\nLeftCancel : Op\u2082 A \u2192 Set _\nLeftCancel _\u00b7_ = \u2200 {c x y} \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancel : Op\u2082 A \u2192 Set _\nRightCancel _\u00b7_ = \u2200 {c x y} \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nLeftCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancelNonZero : A \u2192 Op\u2082 A \u2192 Set _\nRightCancelNonZero zero _\u00b7_ = \u2200 {c x y} \u2192 c \u2262 zero \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nmodule InterchangeFromAssocComm\n         (_\u00b7_     : Op\u2082 A)\n         (\u00b7-assoc : Associative _\u00b7_)\n         (\u00b7-comm  : Commutative _\u00b7_)\n         where\n\n    open \u2261-Reasoning\n\n    \u00b7= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 (x \u00b7 y) \u2261 (x' \u00b7 y')\n    \u00b7= refl refl = refl\n\n    \u00b7-interchange : Interchange _\u00b7_ _\u00b7_\n    \u00b7-interchange {x} {y} {z} {t}\n                = (x \u00b7 y) \u00b7 (z \u00b7 t)\n                \u2261\u27e8 \u00b7-assoc \u27e9\n                  x \u00b7 (y \u00b7 (z \u00b7 t))\n                \u2261\u27e8 \u00b7= refl (! \u00b7-assoc) \u27e9\n                  x \u00b7 ((y \u00b7 z) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl (\u00b7= \u00b7-comm refl) \u27e9\n                  x \u00b7 ((z \u00b7 y) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl \u00b7-assoc \u27e9\n                  x \u00b7 (z \u00b7 (y \u00b7 t))\n                \u2261\u27e8 ! \u00b7-assoc \u27e9\n                  (x \u00b7 z) \u00b7 (y \u00b7 t)\n                \u220e\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"503ea3511701eaf5aaf641a525f84d728e6b4fd6","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 41bb3024527f44ef6b3ff3211487a27d\n\ndarcs-hash:20110315233243-3bd4e-3d26d21d646b4dca78b52b0b769dc3be08ea85ab.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/PA\/Inductive\/Base.agda","new_file":"src\/PA\/Inductive\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Inductive Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule PA.Inductive.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import PA.Inductive.Base.Core public\n\n-- The induction principle on the PA universe\nind\u2115 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\nind\u2115 P P0 h zero     = P0\nind\u2115 P P0 h (succ n) = h n (ind\u2115 P P0 h n)\n\n-- The identity type on the PA universe\nopen import PA.Inductive.Relation.Binary.PropositionalEquality public\n\n-- PA primitive recursive functions\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   * n = zero\nsucc m * n = n + m * n\n\n------------------------------------------------------------------------------\n-- ATPs helper\n-- We don't traslate the body of functions, only the types. Therefore\n-- we need to feed the ATPs with the functions' equations.\n\n-- Addition axioms\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = refl\n-- {-# ATP hint +-0x #-}\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\n-- Multiplication axioms\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = refl\n-- {-# ATP hint *-0x #-}\n\n*-Sx : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n*-Sx m n = refl\n-- {-# ATP hint *-Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Inductive Peano arithmetic base\n------------------------------------------------------------------------------\n\nmodule PA.Inductive.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n-- PA universe\nopen import PA.Inductive.Base.Core public\n\n-- The induction principle on the PA universe\nind\u2115 : (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 n \u2192 P n \u2192 P (succ n)) \u2192 \u2200 n \u2192 P n\nind\u2115 P P0 h zero     = P0\nind\u2115 P P0 h (succ n) = h n (ind\u2115 P P0 h n)\n\n-- The identity type on the PA universe\nopen import PA.Inductive.Relation.Binary.PropositionalEquality public\n\n-- PA primitive recursive functions\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + n = n\nsucc m + n = succ (m + n)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   * n = zero\nsucc m * n = n + m * n\n\n------------------------------------------------------------------------------\n-- ATPs helper\n-- We don't traslate the body of functions, only the types. Therefore\n-- we need to feed the ATPs with the functions' equations.\n\n-- Addition\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = refl\n-- {-# ATP hint +-0x #-}\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\n-- Multiplication\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = refl\n-- {-# ATP hint *-0x #-}\n\n*-Sx : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n*-Sx m n = refl\n-- {-# ATP hint *-Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9bba73dc3aabe72030acd568feaa61cc4b485a70","subject":"Added type signatures.","message":"Added type signatures.\n\nIgnore-this: f2f84215be297949e5c81897085eb7fe\n\ndarcs-hash:20110120134248-3bd4e-cffab5d437a347611a4de357606331cdac079ee6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/DistributiveLaws\/TaskB-I.agda","new_file":"src\/DistributiveLaws\/TaskB-I.agda","new_contents":"------------------------------------------------------------------------------\n-- Example using distributive laws on a binary operation\n------------------------------------------------------------------------------\n\nmodule DistributiveLaws.TaskB-I where\n\nopen import DistributiveLaws.Base\n\nopen import DistributiveLaws.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\ntaskB : \u2200 u x y z \u2192 (x \u00b7 y \u00b7 (z \u00b7 u)) \u00b7\n                    (( x \u00b7 y \u00b7 ( z \u00b7 u)) \u00b7 (x \u00b7 z \u00b7 (y \u00b7 u))) \u2261\n                    x \u00b7 z \u00b7 (y \u00b7 u)\ntaskB u x y z =\n  begin\n    xy\u00b7zu \u00b7 (xy\u00b7zu \u00b7 xz\u00b7yu)                                         \u2261\u27e8 j\u2081 \u27e9\n\n    xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 y\u00b7zu \u00b7 xz\u00b7yu)                                   \u2261\u27e8 j\u2082 \u27e9\n\n    xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                         \u2261\u27e8 j\u2083 \u27e9\n\n    xy\u00b7zu \u00b7 (xz\u00b7xu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                        \u2261\u27e8 j\u2084 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xu\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                           \u2261\u27e8 j\u2085 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                             \u2261\u27e8 j\u2086 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7yu \u00b7 xz\u00b7yu))                            \u2261\u27e8 j\u2087 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7xz \u00b7 yu))                               \u2261\u27e8 j\u2088 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yxz \u00b7 yu))                                 \u2261\u27e8 j\u2089 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yx\u00b7yu \u00b7 z\u00b7yu))                             \u2261\u27e8 j\u2081\u2080 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                              \u2261\u27e8 j\u2081\u2081 \u27e9\n\n    xyz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                          \u2261\u27e8 j\u2081\u2082 \u27e9\n\n    xz\u00b7yz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                        \u2261\u27e8 j\u2081\u2083 \u27e9\n\n    xz \u00b7 xyu \u00b7 (yz \u00b7 xyu) \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))              \u2261\u27e8 j\u2081\u2084 \u27e9\n\n    xz \u00b7 xyu \u00b7 (yz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                           \u2261\u27e8 j\u2081\u2085 \u27e9\n\n    xz \u00b7 xyu \u00b7 (yz \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                         \u2261\u27e8 j\u2081\u2086 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu) \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))            \u2261\u27e8 j\u2081\u2087 \u27e9\n\n    xz \u00b7 xyu \u00b7\n    (y \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))       \u2261\u27e8 j\u2081\u2088 \u27e9\n\n    xz \u00b7 xyu \u00b7\n    (y\u00b7xu \u00b7 y\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))     \u2261\u27e8 j\u2081\u2089 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 (y\u00b7yu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\u27e8 j\u2082\u2080 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 yz\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))         \u2261\u27e8 j\u2082\u2081 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 y\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))          \u2261\u27e8 j\u2082\u2082 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))            \u2261\u27e8 j\u2082\u2083 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 z\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))          \u2261\u27e8 j\u2082\u2084 \u27e9\n\n    (xz \u00b7 xyu) \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 y\u00b7xu \u00b7 z\u00b7yu))                 \u2261\u27e8 j\u2082\u2085 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 z\u00b7yu))                         \u2261\u27e8 j\u2082\u2086 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 zy\u00b7zu))                        \u2261\u27e8 j\u2082\u2087 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy \u00b7 xu\u00b7zu))                           \u2261\u27e8 j\u2082\u2088 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xu\u00b7zu)                                       \u2261\u27e8 j\u2082\u2089 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xzu)                                         \u2261\u27e8 j\u2083\u2080 \u27e9\n\n    xz\u00b7xy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)                                      \u2261\u27e8 j\u2083\u2081 \u27e9\n\n    x\u00b7zy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)                                       \u2261\u27e8 j\u2083\u2082 \u27e9\n\n    x\u00b7zy \u00b7 y\u00b7zy \u00b7 xzu                                               \u2261\u27e8 j\u2083\u2083 \u27e9\n\n    xy\u00b7zy \u00b7 xzu                                                     \u2261\u27e8 j\u2083\u2084 \u27e9\n\n    xzy \u00b7 xzu                                                       \u2261\u27e8 j\u2083\u2085 \u27e9\n\n    xz\u00b7yu\n  \u220e\n  where\n    -- Two variables abbreviations\n\n    xz = x \u00b7 z\n    yu = y \u00b7 u\n    yz = y \u00b7 z\n    zy = z \u00b7 y\n\n    -- Three variables abbreviations\n\n    xyu = x \u00b7 y \u00b7 u\n    xyz = x \u00b7 y \u00b7 z\n    xzu = x \u00b7 z \u00b7 u\n    xzy = x \u00b7 z \u00b7 y\n    yxz = y \u00b7 x \u00b7 z\n\n    x\u00b7yu = x \u00b7 (y \u00b7 u)\n    x\u00b7zu = x \u00b7 (z \u00b7 u)\n    x\u00b7zy = x \u00b7 (z \u00b7 y)\n\n    y\u00b7xu = y \u00b7 (x \u00b7 u)\n    y\u00b7yu = y \u00b7 (y \u00b7 u)\n    y\u00b7zu = y \u00b7 (z \u00b7 u)\n    y\u00b7zy = y \u00b7 (z \u00b7 y)\n\n    z\u00b7xu = z \u00b7 (x \u00b7 u)\n    z\u00b7yu = z \u00b7 (y \u00b7 u)\n\n    -- Four variables abbreviations\n\n    xu\u00b7yu = x \u00b7 u \u00b7 (y \u00b7 u)\n    xu\u00b7zu = x \u00b7 u \u00b7 (z \u00b7 u)\n\n    xy\u00b7yz = x \u00b7 y \u00b7 (y \u00b7 z)\n    xy\u00b7zu = x \u00b7 y \u00b7 (z \u00b7 u)\n    xy\u00b7zy = x \u00b7 y \u00b7 (z \u00b7 y)\n\n    xz\u00b7xu = x \u00b7 z \u00b7 (x \u00b7 u)\n    xz\u00b7xy = x \u00b7 z \u00b7 (x \u00b7 y)\n    xz\u00b7yu = x \u00b7 z \u00b7 (y \u00b7 u)\n    xz\u00b7yz = x \u00b7 z \u00b7 (y \u00b7 z)\n\n    yx\u00b7yu = y \u00b7 x \u00b7 (y \u00b7 u)\n\n    yz\u00b7xz = y \u00b7 z \u00b7 (x \u00b7 z)\n    yz\u00b7yu = y \u00b7 z \u00b7 (y \u00b7 u)\n\n    zy\u00b7xu = z \u00b7 y \u00b7 (x \u00b7 u)\n    zy\u00b7zu = z \u00b7 y \u00b7 (z \u00b7 u)\n\n    -- Steps justifications\n\n    j\u2081 : xy\u00b7zu \u00b7 (xy\u00b7zu \u00b7 xz\u00b7yu) \u2261\n         xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 y\u00b7zu \u00b7 xz\u00b7yu)\n    j\u2081 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xy\u00b7zu \u00b7 xz\u00b7yu) \u2261 xy\u00b7zu \u00b7 (t \u00b7 xz\u00b7yu))\n               (rightDistributive x y (z \u00b7 u))\n               refl\n\n    j\u2082 : xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 y\u00b7zu \u00b7 xz\u00b7yu) \u2261\n         xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))\n    j\u2082 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 y\u00b7zu \u00b7 xz\u00b7yu) \u2261 xy\u00b7zu \u00b7 t)\n               (rightDistributive x\u00b7zu y\u00b7zu xz\u00b7yu)\n               refl\n\n    j\u2083 : xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n         xy\u00b7zu \u00b7 (xz\u00b7xu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))\n    j\u2083 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (t \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)))\n               (leftDistributive x z u)\n               refl\n\n    j\u2084 : xy\u00b7zu \u00b7 (xz\u00b7xu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n         xy\u00b7zu \u00b7 (xz \u00b7 xu\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))\n    j\u2084 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz\u00b7xu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (t \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)))\n               (sym (leftDistributive (x \u00b7 z) (x \u00b7 u) (y \u00b7 u)))\n               refl\n\n    j\u2085 : xy\u00b7zu \u00b7 (xz \u00b7 xu\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n         xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))\n    j\u2085 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xu\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 t \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)))\n               (sym (rightDistributive x y u))\n               refl\n\n    j\u2086 : xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n         xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7yu \u00b7 xz\u00b7yu))\n    j\u2086 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (t \u00b7 xz\u00b7yu)))\n               (leftDistributive y z u)\n               refl\n\n    j\u2087 : xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7yu \u00b7 xz\u00b7yu)) \u2261\n         xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7xz \u00b7 yu))\n    j\u2087 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7yu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 t))\n               (sym (rightDistributive (y \u00b7 z) (x \u00b7 z) (y \u00b7 u)))\n               refl\n\n    j\u2088 : xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7xz \u00b7 yu)) \u2261\n         xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yxz \u00b7 yu))\n    j\u2088 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7xz \u00b7 yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (t \u00b7 yu)))\n               (sym (rightDistributive y x z))\n               refl\n\n    j\u2089 : xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yxz \u00b7 yu)) \u2261\n         xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yx\u00b7yu \u00b7 z\u00b7yu))\n    j\u2089 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yxz \u00b7 yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 t))\n               (rightDistributive (y \u00b7 x) z yu)\n               refl\n\n    j\u2081\u2080 : xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yx\u00b7yu \u00b7 z\u00b7yu)) \u2261\n          xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2080 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yx\u00b7yu \u00b7 z\u00b7yu)) \u2261\n                       xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (t \u00b7 z\u00b7yu)))\n                (sym (leftDistributive y x u))\n                refl\n\n    j\u2081\u2081 : xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xyz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2081 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       t \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (leftDistributive (x \u00b7 y) z u)\n                refl\n\n    j\u2081\u2082 : xyz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz\u00b7yz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2082 = subst (\u03bb t \u2192 xyz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       t \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive x y z)\n                refl\n\n    j\u2081\u2083 : xz\u00b7yz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7 (yz \u00b7 xyu) \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2083 = subst (\u03bb t \u2192 xz\u00b7yz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       t \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive (x \u00b7 z) (y \u00b7 z) xyu)\n                refl\n\n    j\u2081\u2084 : xz \u00b7 xyu \u00b7 (yz \u00b7 xyu) \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7 (yz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2084 = sym (leftDistributive (xz \u00b7 xyu) (yz \u00b7 xyu) (y\u00b7xu \u00b7 z\u00b7yu))\n\n    j\u2081\u2085 : xz \u00b7 xyu \u00b7 (yz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7 (yz \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2085 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (yz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (yz \u00b7 t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive x y u)\n                refl\n\n    j\u2081\u2086 : xz \u00b7 xyu \u00b7 (yz \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu) \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2081\u2086 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (yz \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive y z xu\u00b7yu)\n                refl\n\n    j\u2081\u2087 : xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu) \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7\n          (y \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2081\u2087 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu) \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 t)\n                (rightDistributive (y \u00b7 xu\u00b7yu) (z \u00b7 xu\u00b7yu) (y\u00b7xu \u00b7 z\u00b7yu))\n                refl\n\n    j\u2081\u2088 : xz \u00b7 xyu \u00b7\n          (y \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          xz \u00b7 xyu \u00b7\n          (y\u00b7xu \u00b7 y\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2081\u2088 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (leftDistributive y (x \u00b7 u) (y \u00b7 u))\n                refl\n\n    j\u2081\u2089 : xz \u00b7 xyu \u00b7\n          (y\u00b7xu \u00b7 y\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 (y\u00b7yu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2081\u2089 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 y\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7 (t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (leftDistributive y\u00b7xu y\u00b7yu z\u00b7yu))\n                refl\n\n    j\u2082\u2080 : xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 (y\u00b7yu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 yz\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2082\u2080 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 (y\u00b7yu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (rightDistributive y z (y \u00b7 u)))\n                refl\n\n    j\u2082\u2081 : xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 yz\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 y\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2082\u2081 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 yz\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (leftDistributive y z u))\n                refl\n\n    j\u2082\u2082 : xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 y\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2082\u2082 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 y\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (leftDistributive y (x \u00b7 u) (z \u00b7 u)))\n                refl\n\n    j\u2082\u2083 : xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 z\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n    j\u2082\u2083 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (leftDistributive z (x \u00b7 u) (y \u00b7 u))\n                refl\n\n    j\u2082\u2084 : xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 z\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n          (xz \u00b7 xyu) \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 y\u00b7xu \u00b7 z\u00b7yu))\n    j\u2082\u2084 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 z\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 t))\n                (sym (rightDistributive z\u00b7xu y\u00b7xu z\u00b7yu))\n                refl\n\n    j\u2082\u2085 : (xz \u00b7 xyu) \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 z\u00b7yu))\n    j\u2082\u2085 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (t \u00b7 z\u00b7yu)))\n                (sym (rightDistributive z y (x \u00b7 u)))\n                refl\n\n    j\u2082\u2086 : xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 z\u00b7yu)) \u2261\n          xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 zy\u00b7zu))\n    j\u2082\u2086 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 t)))\n                (leftDistributive z y u)\n                refl\n\n    j\u2082\u2087 : xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 zy\u00b7zu)) \u2261\n          xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy \u00b7 xu\u00b7zu))\n    j\u2082\u2087 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 zy\u00b7zu)) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 t))\n                (sym (leftDistributive (z \u00b7 y) (x \u00b7 u) (z \u00b7 u)))\n                refl\n\n    j\u2082\u2088 : xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy \u00b7 xu\u00b7zu)) \u2261\n          xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xu\u00b7zu)\n    j\u2082\u2088 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy \u00b7 xu\u00b7zu)) \u2261 xz \u00b7 xyu \u00b7 t)\n                (sym (rightDistributive y zy xu\u00b7zu))\n                refl\n\n    j\u2082\u2089 : xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xu\u00b7zu) \u2261\n          xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xzu)\n    j\u2082\u2089 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xu\u00b7zu) \u2261 xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 t))\n                (sym (rightDistributive x z u))\n                refl\n\n    j\u2083\u2080 : xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261\n          xz\u00b7xy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)\n    j\u2083\u2080 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261 t \u00b7 (y\u00b7zy \u00b7 xzu))\n                (leftDistributive xz (x \u00b7 y) u)\n                refl\n\n    j\u2083\u2081 : xz\u00b7xy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261\n          x\u00b7zy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)\n    j\u2083\u2081 = subst (\u03bb t \u2192 xz\u00b7xy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261 (t \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)))\n                (sym (leftDistributive x z y))\n                refl\n\n    j\u2083\u2082 : x\u00b7zy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261\n          x\u00b7zy \u00b7 y\u00b7zy \u00b7 xzu\n    j\u2083\u2082 = sym (rightDistributive x\u00b7zy y\u00b7zy xzu)\n\n    j\u2083\u2083 : x\u00b7zy \u00b7 y\u00b7zy \u00b7 xzu \u2261\n          xy\u00b7zy \u00b7 xzu\n    j\u2083\u2083 = subst (\u03bb t \u2192 x\u00b7zy \u00b7 y\u00b7zy \u00b7 xzu \u2261 t \u00b7 xzu)\n                (sym (rightDistributive x y zy))\n                refl\n\n    j\u2083\u2084 : xy\u00b7zy \u00b7 xzu \u2261\n          xzy \u00b7 xzu\n    j\u2083\u2084 = subst (\u03bb t \u2192 xy\u00b7zy \u00b7 xzu \u2261 t \u00b7 xzu)\n                (sym (rightDistributive x z y))\n                refl\n\n    j\u2083\u2085 : xzy \u00b7 xzu \u2261\n          xz\u00b7yu\n    j\u2083\u2085 = sym (leftDistributive xz y u)\n","old_contents":"------------------------------------------------------------------------------\n-- Example using distributive laws on a binary operation\n------------------------------------------------------------------------------\n\nmodule DistributiveLaws.TaskB-I where\n\nopen import DistributiveLaws.Base\n\nopen import DistributiveLaws.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\ntaskB : \u2200 u x y z \u2192 (x \u00b7 y \u00b7 (z \u00b7 u)) \u00b7\n                    (( x \u00b7 y \u00b7 ( z \u00b7 u)) \u00b7 (x \u00b7 z \u00b7 (y \u00b7 u))) \u2261\n                    x \u00b7 z \u00b7 (y \u00b7 u)\ntaskB u x y z =\n  begin\n    xy\u00b7zu \u00b7 (xy\u00b7zu \u00b7 xz\u00b7yu)                                         \u2261\u27e8 j\u2081 \u27e9\n\n    xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 y\u00b7zu \u00b7 xz\u00b7yu)                                   \u2261\u27e8 j\u2082 \u27e9\n\n    xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                         \u2261\u27e8 j\u2083 \u27e9\n\n    xy\u00b7zu \u00b7 (xz\u00b7xu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                        \u2261\u27e8 j\u2084 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xu\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                           \u2261\u27e8 j\u2085 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu))                             \u2261\u27e8 j\u2086 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7yu \u00b7 xz\u00b7yu))                            \u2261\u27e8 j\u2087 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7xz \u00b7 yu))                               \u2261\u27e8 j\u2088 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yxz \u00b7 yu))                                 \u2261\u27e8 j\u2089 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yx\u00b7yu \u00b7 z\u00b7yu))                             \u2261\u27e8 j\u2081\u2080 \u27e9\n\n    xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                              \u2261\u27e8 j\u2081\u2081 \u27e9\n\n    xyz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                          \u2261\u27e8 j\u2081\u2082 \u27e9\n\n    xz\u00b7yz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                        \u2261\u27e8 j\u2081\u2083 \u27e9\n\n    xz \u00b7 xyu \u00b7 (yz \u00b7 xyu) \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))              \u2261\u27e8 j\u2081\u2084 \u27e9\n\n    xz \u00b7 xyu \u00b7 (yz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                           \u2261\u27e8 j\u2081\u2085 \u27e9\n\n    xz \u00b7 xyu \u00b7 (yz \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))                         \u2261\u27e8 j\u2081\u2086 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu) \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))            \u2261\u27e8 j\u2081\u2087 \u27e9\n\n    xz \u00b7 xyu \u00b7\n    (y \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))       \u2261\u27e8 j\u2081\u2088 \u27e9\n\n    xz \u00b7 xyu \u00b7\n    (y\u00b7xu \u00b7 y\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))     \u2261\u27e8 j\u2081\u2089 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 (y\u00b7yu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\u27e8 j\u2082\u2080 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 yz\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))         \u2261\u27e8 j\u2082\u2081 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 y\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))          \u2261\u27e8 j\u2082\u2082 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))            \u2261\u27e8 j\u2082\u2083 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 z\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))          \u2261\u27e8 j\u2082\u2084 \u27e9\n\n    (xz \u00b7 xyu) \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 y\u00b7xu \u00b7 z\u00b7yu))                 \u2261\u27e8 j\u2082\u2085 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 z\u00b7yu))                         \u2261\u27e8 j\u2082\u2086 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 zy\u00b7zu))                        \u2261\u27e8 j\u2082\u2087 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy \u00b7 xu\u00b7zu))                           \u2261\u27e8 j\u2082\u2088 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xu\u00b7zu)                                       \u2261\u27e8 j\u2082\u2089 \u27e9\n\n    xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xzu)                                         \u2261\u27e8 j\u2083\u2080 \u27e9\n\n    xz\u00b7xy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)                                      \u2261\u27e8 j\u2083\u2081 \u27e9\n\n    x\u00b7zy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)                                       \u2261\u27e8 j\u2083\u2082 \u27e9\n\n    x\u00b7zy \u00b7 y\u00b7zy \u00b7 xzu                                               \u2261\u27e8 j\u2083\u2083 \u27e9\n\n    xy\u00b7zy \u00b7 xzu                                                     \u2261\u27e8 j\u2083\u2084 \u27e9\n\n    xzy \u00b7 xzu                                                       \u2261\u27e8 j\u2083\u2085 \u27e9\n\n    xz\u00b7yu\n  \u220e\n  where\n    -- Two variables abbreviations\n\n    xz = x \u00b7 z\n    yu = y \u00b7 u\n    yz = y \u00b7 z\n    zy = z \u00b7 y\n\n    -- Three variables abbreviations\n\n    xyu = x \u00b7 y \u00b7 u\n    xyz = x \u00b7 y \u00b7 z\n    xzu = x \u00b7 z \u00b7 u\n    xzy = x \u00b7 z \u00b7 y\n    yxz = y \u00b7 x \u00b7 z\n\n    x\u00b7yu = x \u00b7 (y \u00b7 u)\n    x\u00b7zu = x \u00b7 (z \u00b7 u)\n    x\u00b7zy = x \u00b7 (z \u00b7 y)\n\n    y\u00b7xu = y \u00b7 (x \u00b7 u)\n    y\u00b7yu = y \u00b7 (y \u00b7 u)\n    y\u00b7zu = y \u00b7 (z \u00b7 u)\n    y\u00b7zy = y \u00b7 (z \u00b7 y)\n\n    z\u00b7xu = z \u00b7 (x \u00b7 u)\n    z\u00b7yu = z \u00b7 (y \u00b7 u)\n\n    -- Four variables abbreviations\n\n    xu\u00b7yu = x \u00b7 u \u00b7 (y \u00b7 u)\n    xu\u00b7zu = x \u00b7 u \u00b7 (z \u00b7 u)\n\n    xy\u00b7yz = x \u00b7 y \u00b7 (y \u00b7 z)\n    xy\u00b7zu = x \u00b7 y \u00b7 (z \u00b7 u)\n    xy\u00b7zy = x \u00b7 y \u00b7 (z \u00b7 y)\n\n    xz\u00b7xu = x \u00b7 z \u00b7 (x \u00b7 u)\n    xz\u00b7xy = x \u00b7 z \u00b7 (x \u00b7 y)\n    xz\u00b7yu = x \u00b7 z \u00b7 (y \u00b7 u)\n    xz\u00b7yz = x \u00b7 z \u00b7 (y \u00b7 z)\n\n    yx\u00b7yu = y \u00b7 x \u00b7 (y \u00b7 u)\n\n    yz\u00b7xz = y \u00b7 z \u00b7 (x \u00b7 z)\n    yz\u00b7yu = y \u00b7 z \u00b7 (y \u00b7 u)\n\n    zy\u00b7xu = z \u00b7 y \u00b7 (x \u00b7 u)\n    zy\u00b7zu = z \u00b7 y \u00b7 (z \u00b7 u)\n\n    -- Steps justifications\n\n    j\u2081 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xy\u00b7zu \u00b7 xz\u00b7yu) \u2261 xy\u00b7zu \u00b7 (t \u00b7 xz\u00b7yu))\n               (rightDistributive x y (z \u00b7 u))\n               refl\n\n    j\u2082 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 y\u00b7zu \u00b7 xz\u00b7yu) \u2261 xy\u00b7zu \u00b7 t)\n               (rightDistributive x\u00b7zu y\u00b7zu xz\u00b7yu)\n               refl\n\n    j\u2083 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (x\u00b7zu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (t \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)))\n               (leftDistributive x z u)\n               refl\n\n    j\u2084 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz\u00b7xu \u00b7 xz\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (t \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)))\n               (sym (leftDistributive (x \u00b7 z) (x \u00b7 u) (y \u00b7 u)))\n               refl\n\n    j\u2085 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xu\u00b7yu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 t \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)))\n               (sym (rightDistributive x y u))\n               refl\n\n    j\u2086 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7zu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (t \u00b7 xz\u00b7yu)))\n               (leftDistributive y z u)\n               refl\n\n    j\u2087 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7yu \u00b7 xz\u00b7yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 t))\n               (sym (rightDistributive (y \u00b7 z) (x \u00b7 z) (y \u00b7 u)))\n               refl\n\n    j\u2088 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yz\u00b7xz \u00b7 yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (t \u00b7 yu)))\n               (sym (rightDistributive y x z))\n               refl\n\n    j\u2089 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yxz \u00b7 yu)) \u2261\n                      xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 t))\n               (rightDistributive (y \u00b7 x) z yu)\n               refl\n\n    j\u2081\u2080 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (yx\u00b7yu \u00b7 z\u00b7yu)) \u2261\n                       xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (t \u00b7 z\u00b7yu)))\n                (sym (leftDistributive y x u))\n                refl\n\n    j\u2081\u2081 = subst (\u03bb t \u2192 xy\u00b7zu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       t \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (leftDistributive (x \u00b7 y) z u)\n                refl\n\n    j\u2081\u2082 = subst (\u03bb t \u2192 xyz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       t \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive x y z)\n                refl\n\n    j\u2081\u2083 = subst (\u03bb t \u2192 xz\u00b7yz \u00b7 xyu \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       t \u00b7 (xz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive (x \u00b7 z) (y \u00b7 z) xyu)\n                refl\n\n    j\u2081\u2084 = sym (leftDistributive (xz \u00b7 xyu) (yz \u00b7 xyu) (y\u00b7xu \u00b7 z\u00b7yu))\n\n    j\u2081\u2085 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (yz \u00b7 xyu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (yz \u00b7 t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive x y u)\n                refl\n\n    j\u2081\u2086 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (yz \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                (rightDistributive y z xu\u00b7yu)\n                refl\n\n    j\u2081\u2087 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu) \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 t)\n                (rightDistributive (y \u00b7 xu\u00b7yu) (z \u00b7 xu\u00b7yu) (y\u00b7xu \u00b7 z\u00b7yu))\n                refl\n\n    j\u2081\u2088 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (leftDistributive y (x \u00b7 u) (y \u00b7 u))\n                refl\n\n    j\u2081\u2089 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 y\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7 (t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (leftDistributive y\u00b7xu y\u00b7yu z\u00b7yu))\n                refl\n\n    j\u2082\u2080 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 (y\u00b7yu \u00b7 z\u00b7yu) \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (rightDistributive y z (y \u00b7 u)))\n                refl\n\n    j\u2082\u2081 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 yz\u00b7yu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (leftDistributive y z u))\n                refl\n\n    j\u2082\u2082 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7\n                       (y\u00b7xu \u00b7 y\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu)))\n                       \u2261\n                       xz \u00b7 xyu \u00b7\n                       (t \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (sym (leftDistributive y (x \u00b7 u) (z \u00b7 u)))\n                refl\n\n    j\u2082\u2083 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z \u00b7 xu\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (t \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))))\n                (leftDistributive z (x \u00b7 u) (y \u00b7 u))\n                refl\n\n    j\u2082\u2084 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 z\u00b7yu \u00b7 (y\u00b7xu \u00b7 z\u00b7yu))) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 t))\n                (sym (rightDistributive z\u00b7xu y\u00b7xu z\u00b7yu))\n                refl\n\n    j\u2082\u2085 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (z\u00b7xu \u00b7 y\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (t \u00b7 z\u00b7yu)))\n                (sym (rightDistributive z y (x \u00b7 u)))\n                refl\n\n    j\u2082\u2086 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 z\u00b7yu)) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 t)))\n                (leftDistributive z y u)\n                refl\n\n    j\u2082\u2087 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy\u00b7xu \u00b7 zy\u00b7zu)) \u2261\n                       xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 t))\n                (sym (leftDistributive (z \u00b7 y) (x \u00b7 u) (z \u00b7 u)))\n                refl\n\n    j\u2082\u2088 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y \u00b7 xu\u00b7zu \u00b7 (zy \u00b7 xu\u00b7zu)) \u2261 xz \u00b7 xyu \u00b7 t)\n                (sym (rightDistributive y zy xu\u00b7zu))\n                refl\n\n    j\u2082\u2089 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xu\u00b7zu) \u2261 xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 t))\n                (sym (rightDistributive x z u))\n                refl\n\n    j\u2083\u2080 = subst (\u03bb t \u2192 xz \u00b7 xyu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261 t \u00b7 (y\u00b7zy \u00b7 xzu))\n                (leftDistributive xz (x \u00b7 y) u)\n                refl\n\n    j\u2083\u2081 = subst (\u03bb t \u2192 xz\u00b7xy \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu) \u2261 (t \u00b7 xzu \u00b7 (y\u00b7zy \u00b7 xzu)))\n                (sym (leftDistributive x z y))\n                refl\n\n    j\u2083\u2082 = sym (rightDistributive x\u00b7zy y\u00b7zy xzu)\n\n    j\u2083\u2083 = subst (\u03bb t \u2192 x\u00b7zy \u00b7 y\u00b7zy \u00b7 xzu \u2261 t \u00b7 xzu)\n                (sym (rightDistributive x y zy))\n                refl\n\n    j\u2083\u2084 = subst (\u03bb t \u2192 xy\u00b7zy \u00b7 xzu \u2261 t \u00b7 xzu)\n                (sym (rightDistributive x z y))\n                refl\n\n    j\u2083\u2085 = sym (leftDistributive xz y u)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e8b0a2064d7927030a67be90fa6130d3a7099cf3","subject":"Maintain \u27e6_\u27e7CompTypeHidCacheWrong","message":"Maintain \u27e6_\u27e7CompTypeHidCacheWrong\n\nThis got out of sync with the other version. Probably this belongs to\nhistory though.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingMValue.agda","new_file":"Parametric\/Denotation\/CachingMValue.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for caching evaluation of MTerm\n------------------------------------------------------------------------\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.MType as MType\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.MValue as MValue\n\nmodule Parametric.Denotation.CachingMValue\n    (Base : Type.Structure)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\nopen MValue.Structure Base \u27e6_\u27e7Base\n\nopen import Data.Product hiding (map)\nopen import Data.Sum hiding (map)\nopen import Data.Unit\nopen import Level\nopen import Function hiding (const)\n\nmodule Structure where\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCacheWrong )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Values for caching evaluation of MTerm\n------------------------------------------------------------------------\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.MType as MType\n\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.MValue as MValue\n\nmodule Parametric.Denotation.CachingMValue\n    (Base : Type.Structure)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen import Base.Denotation.Notation\n\nopen Type.Structure Base\nopen MType.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\nopen MValue.Structure Base \u27e6_\u27e7Base\n\nopen import Data.Product hiding (map)\nopen import Data.Sum hiding (map)\nopen import Data.Unit\nopen import Level\nopen import Function hiding (const)\n\nmodule Structure where\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2b11a9538823cbdebe98b43f91df0701efad84e1","subject":"patching","message":"patching\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-preservation.agda","new_file":"complete-preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-preservation where\n  -- TODO: convert this into a module imports once proven\n  postulate\n      preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n      typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n\n\n  complete-preservation : {\u0394 : hctx} {e : hexp} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             e ecomplete \u2192\n             \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  complete-preservation _ exp step = preservation (typed-expansion-synth exp) step\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-preservation where\n  -- TODO: convert this into a module imports once proven\n  postulate\n      preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n      typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n\n\n  complete-preservation : {\u0394 : hctx} {e : hexp} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             ecomplete e \u2192\n             \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  complete-preservation comp exp step = preservation (typed-expansion-synth exp) step\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fe3de5ab89cfe95bf571ddcf352a495db09be63f","subject":"Updated a draft.","message":"Updated a draft.\n\nIgnore-this: c1d9b58a804d2b621ab7a0f8215b3c57\n\ndarcs-hash:20120308052436-3bd4e-6c689ede5d9e06f2c191cf1695e75c3e0ebc40c8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n-- Tested with FOT and agda2atp on 07 March 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 ih Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    ih {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n-- Combined proof using an instance of the induction principle.\nN-ind-instance : \u2200 n \u2192\n                N (zero + n) \u2192\n                (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n                \u2200 {m} \u2192 N m \u2192 N (m + n)\nN-ind-instance n = N-ind (\u03bb i \u2192 N (i + n))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 N-ind-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 N-ind #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n-- Tested with FOT and agda2atp on 02 March 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 ih Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    ih {i} ih = subst N (sym (+-Sx i n)) (sN ih)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : \u2200 x \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N (n + x) \u2192 N (succ\u2081 n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = N-ind (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n-- {-# ATP prove +-N\u2082 indN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5281a6ad4e11b3b2086c2287bb5e98a387a7bb48","subject":"New assoc proof, and new trace","message":"New assoc proof, and new trace\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndualInOut : InOut \u2192 InOut\ndualInOut In  = Out\ndualInOut Out = In\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  com  : (q : InOut)(M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com q M P \u2261\u1d3e com q M Q\n\npattern \u03a0' M P = com In  M P\npattern \u03a3' M P = com Out M P\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0S' M P = \u03a0' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3S' M P = \u03a3' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u27e6_\u27e7\u03a0\u03a3 : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u03a0\u03a3 In  = \u03a0\n\u27e6_\u27e7\u03a0\u03a3 Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com q M P \u27e7 = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 \u27e6 P x \u27e7\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\n_-[_]\u2192\u00f8\u204f_ : \u2200 {I}(A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\nA -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  I\u03a3D : \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  I\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n--  I\u03a0S : \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 \u03a0' (\u2610 M) \u2102C ]\n  \u2605\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  \u2605\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n  --\u00f8\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ A \u2261 \u03a3' S M \u2102A ]\n  --\u00f8\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ B \u2261 \u03a0' S M \u2102B ]\n  end : \u2200 {A} \u2192 [ end \/ A \u2261 end ]\n\nTrace : Proto \u2192 Proto\nTrace end          = end\nTrace (com _ A B) = \u03a3' A \u03bb m \u2192 Trace (B m)\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (com q A B) = com (dualInOut q) A (\u03bb x \u2192 dual (B x))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com q M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com q M P) = com q M (\u03bb m \u2192 \u2261\u1d3e-refl (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end = end\nTrace-idempotent (com q M P) = \u03a3' M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end = end\nTrace-dual-oblivious (com q M P) = \u03a3' M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\ncom \u03c0 A B >>\u2261 Q = com \u03c0 A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>\u2261 Q)\n++Tele end         _        ys = ys\n++Tele (com q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\nright-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261\u1d3e P\nright-unit end = end\nright-unit (com q M P) = com q M \u03bb m \u2192 right-unit (P m)\n\nassoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>\u2261 Q) >>\u2261 R)\nassoc end         Q R = \u2261\u1d3e-refl (Q _ >>\u2261 R)\nassoc (com q M P) Q R = com q M \u03bb m \u2192 assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndata DualInOut : InOut \u2192 InOut \u2192 \u2605 where\n  DInOut : DualInOut In  Out\n  DOutIn : DualInOut Out In\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' A B) (\u03a3' A B')\n  \u03a3\u00b7\u03a0 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' A B) (\u03a0' A B')\n\ndata Sing {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  sing : \u2200 x \u2192 Sing x\n\n-- Commit = \u03bb M R \u2192 \u03a3' D (\u2610 M) \u03bb { [ m ] \u2192 \u03a0' D R \u03bb r \u2192 \u03a3' D (Sing m) (\u03bb _ \u2192 end) }\nCommit = \u03bb M R \u2192 \u03a3S' M \u03bb m \u2192 \u03a0' R \u03bb r \u2192 \u03a3' (Sing m) (\u03bb _ \u2192 end)\n\ncommit : \u2200 M R (m : M)  \u2192 \u27e6 Commit M R \u27e7\ncommit M R m = [ m ] , (\u03bb x \u2192 (sing m) , _)\n\ndecommit : \u2200 M R (r : R) \u2192 \u27e6 dual (Commit M R) \u27e7\ndecommit M R r = \u03bb { [ m ] \u2192 r , (\u03bb x \u2192 0\u2081) }\n\ndata [_&_\u2261_]InOut : InOut \u2192 InOut \u2192 InOut \u2192 \u2605\u2081 where\n  \u03a0XX : \u2200 {X} \u2192 [ In & X  \u2261 X ]InOut\n  X\u03a0X : \u2200 {X} \u2192 [ X  & In \u2261 X ]InOut\n\n&InOut-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]InOut \u2192 [ Q & P \u2261 R ]InOut\n&InOut-comm \u03a0XX = X\u03a0X\n&InOut-comm X\u03a0X = \u03a0XX\n\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end& : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  &end : \u2200 {P} \u2192 [ P & end \u2261 P ]\n  D&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q m     \u2261 R m ]) \u2192 [ com qP    M  P & com qQ    M  Q \u2261 com qR M R ]\n  S&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P [ m ] & Q m     \u2261 R m ]) \u2192 [ com qP (\u2610 M) P & com qQ    M  Q \u2261 com qR M R ]\n  D&S  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q [ m ] \u2261 R m ]) \u2192 [ com qP    M  P & com qQ (\u2610 M) Q \u2261 com qR M R ]\n\n&-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n&-comm end& = &end\n&-comm &end = end&\n&-comm (D&D q P&) = D&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (S&D q P&) = D&S (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (D&S q P&) = S&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n\nDualInOut-sym : \u2200 {P Q} \u2192 DualInOut P Q \u2192 DualInOut Q P\nDualInOut-sym DInOut = DOutIn\nDualInOut-sym DOutIn = DInOut\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDualInOut-spec : \u2200 P \u2192 DualInOut P (dualInOut P)\nDualInOut-spec In  = DInOut\nDualInOut-spec Out = DOutIn\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (com In M P) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-spec (P x))\nDual-spec (com Out M P) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-spec (P x))\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com q M P) X = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>\u2261 Q) \u2192 \u2605} \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  bind-spec end         = refl\n  bind-spec (com q M P) = cong (\u27e6 q \u27e7\u03a0\u03a3 M) (funExt \u03bb m \u2192 bind-spec (P m))\n\n\nmodule _ {A B} where\n  run-com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  run-com end      a       b       = a , b\n  run-com (\u03a0' M P) p       (m , q) = run-com (P m) (p m) q\n  run-com (\u03a3' M P) (m , p) q       = run-com (P m) p (q m)\n\ncom-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\ncom-cont end p q = (_ , p)  , (_ , q)\ncom-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\ncom-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0' (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3' (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q M P Q} \u2192 SimL (com q M P) Q \u2192 Sim (com q M P) Q\n    right : \u2200 {P q M Q} \u2192 SimR P (com q M Q) \u2192 Sim P (com q M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recv PQA)   (send m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (send m PQB) = send m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (send m PQA) (recv PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (send x\u2081 x\u2082) (recv x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = send m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = ? -- send m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb x \u2192 left (send x (sim-id (B x)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb x \u2192 right (send x (sim-id (B x)))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = left (recv (\u03bb m \u2192 right (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = right (recv (\u03bb m \u2192 left (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u00b7\u03a3 x\u2081) (recv x) (send x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u00b7\u03a0 x)  (send x\u2081 x\u2082) (recv x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recv PQ) QR = recv (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recv P)) = do (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (send m P)) = do (send m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule 3-way-trace where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (send x x\u2082)) (left (recv x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (right (recv x)) (left (send x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (right (send x x\u2082)) (left (recv x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end} {end} end = _\n  trace {end} {com q M P} (right (send m x)) = mod\u2082 (_,_ m) (trace x)\n  trace {com q M P} {end} (left (send m x)) = mod\u2081 (_,_ m) (trace x)\n  trace {com q M P} {com q\u2081 M\u2081 P\u2081} (left (send m x)) = mod\u2081 (_,_ m) (trace {_} {com (q\u2081) _ _} x)\n  trace {com q M P} {com q\u2081 M\u2081 P\u2081} (right (send m x)) = mod\u2082 (_,_ m) (trace {com q _ _} x)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (left (recv x)) PQ QR = cong (left \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (left (send m x)) PQ QR = cong (left \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (recv x)) (left (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (send m x)) (left (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (right x) (right (recv x\u2081)) (left (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (right x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (right x) (right (send m x\u2081)) (left (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (right x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) (right (recv x\u2082)) = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) (right (send m x\u2082)) = cong (right \u2218 send m) (sim-comp-assoc P-P' Q-Q' (right x) (right x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ A where\n    Test : Proto\n    Test = com Out A (const end)\n\n    s : A \u2192 Sim Test Test\n    s m = right (send m (left (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = right (recv (\u03bb m \u2192 left (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (left (recv x)) = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (left (send m x)) = cong (left \u2218 send m) (sim-comp-id x)\n  sim-comp-id (right (recv x)) = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (right (send m x)) = {!cong (right \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recv x))    = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (send x x\u2081)) = cong (left \u2218 send x) (sim-!! x\u2081)\n  sim-!! (right (recv x))    = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (send x x\u2081)) = cong (right \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (right (recv x)) (left (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (right (send x\u2081 x\u2082)) (left (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recv x\u2081))\n    = cong (left \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (send x\u2081 x\u2082))\n    = cong (left \u2218 send x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end (right (recv x)) = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (right (send m x)) = cong (left \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndualInOut : InOut \u2192 InOut\ndualInOut In  = Out\ndualInOut Out = In\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  com  : (q : InOut)(M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com q M P \u2261\u1d3e com q M Q\n\npattern \u03a0' M P = com In  M P\npattern \u03a3' M P = com Out M P\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0S' M P = \u03a0' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3S' M P = \u03a3' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u27e6_\u27e7\u03a0\u03a3 : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u03a0\u03a3 In  = \u03a0\n\u27e6_\u27e7\u03a0\u03a3 Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end       \u27e7 = \ud835\udfd9\n\u27e6 com q M P \u27e7 = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 \u27e6 P x \u27e7\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\n_-[_]\u2192\u00f8\u204f_ : \u2200 {I}(A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\nA -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  I\u03a3D : \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  I\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n--  I\u03a0S : \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 \u03a0' (\u2610 M) \u2102C ]\n  \u2605\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  \u2605\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n  --\u00f8\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ A \u2261 \u03a3' S M \u2102A ]\n  --\u00f8\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ B \u2261 \u03a0' S M \u2102B ]\n  end : \u2200 {A} \u2192 [ end \/ A \u2261 end ]\n\nTrace : Proto \u2192 Proto\nTrace end          = end\nTrace (com _ A B) = \u03a3' A \u03bb m \u2192 Trace (B m)\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (com q A B) = com (dualInOut q) A (\u03bb x \u2192 dual (B x))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end           = refl\n    \u2261\u1d3e-sound (com q M P\u2261Q) = cong (com q M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com q M P) = com q M (\u03bb m \u2192 \u2261\u1d3e-refl (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end = end\nTrace-idempotent (com q M P) = \u03a3' M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end = end\nTrace-dual-oblivious (com q M P) = \u03a3' M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\ncom \u03c0 A B >>\u2261 Q = com \u03c0 A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>\u2261 Q)\n++Tele end         _        ys = ys\n++Tele (com q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\nright-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261\u1d3e P\nright-unit end = end\nright-unit (com q M P) = com q M \u03bb m \u2192 right-unit (P m)\n\nassoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>\u2261 Q) >>\u2261 R)\nassoc end         Q R = \u2261\u1d3e-refl (Q _ >>\u2261 R)\nassoc (com q M P) Q R = com q M \u03bb m \u2192 assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndata DualInOut : InOut \u2192 InOut \u2192 \u2605 where\n  DInOut : DualInOut In  Out\n  DOutIn : DualInOut Out In\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' A B) (\u03a3' A B')\n  \u03a3\u00b7\u03a0 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' A B) (\u03a0' A B')\n\ndata Sing {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  sing : \u2200 x \u2192 Sing x\n\n-- Commit = \u03bb M R \u2192 \u03a3' D (\u2610 M) \u03bb { [ m ] \u2192 \u03a0' D R \u03bb r \u2192 \u03a3' D (Sing m) (\u03bb _ \u2192 end) }\nCommit = \u03bb M R \u2192 \u03a3S' M \u03bb m \u2192 \u03a0' R \u03bb r \u2192 \u03a3' (Sing m) (\u03bb _ \u2192 end)\n\ncommit : \u2200 M R (m : M)  \u2192 \u27e6 Commit M R \u27e7\ncommit M R m = [ m ] , (\u03bb x \u2192 (sing m) , _)\n\ndecommit : \u2200 M R (r : R) \u2192 \u27e6 dual (Commit M R) \u27e7\ndecommit M R r = \u03bb { [ m ] \u2192 r , (\u03bb x \u2192 0\u2081) }\n\ndata [_&_\u2261_]InOut : InOut \u2192 InOut \u2192 InOut \u2192 \u2605\u2081 where\n  \u03a0XX : \u2200 {X} \u2192 [ In & X  \u2261 X ]InOut\n  X\u03a0X : \u2200 {X} \u2192 [ X  & In \u2261 X ]InOut\n\n&InOut-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]InOut \u2192 [ Q & P \u2261 R ]InOut\n&InOut-comm \u03a0XX = X\u03a0X\n&InOut-comm X\u03a0X = \u03a0XX\n\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end& : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  &end : \u2200 {P} \u2192 [ P & end \u2261 P ]\n  D&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q m     \u2261 R m ]) \u2192 [ com qP    M  P & com qQ    M  Q \u2261 com qR M R ]\n  S&D  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P [ m ] & Q m     \u2261 R m ]) \u2192 [ com qP (\u2610 M) P & com qQ    M  Q \u2261 com qR M R ]\n  D&S  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m     & Q [ m ] \u2261 R m ]) \u2192 [ com qP    M  P & com qQ (\u2610 M) Q \u2261 com qR M R ]\n\n&-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n&-comm end& = &end\n&-comm &end = end&\n&-comm (D&D q P&) = D&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (S&D q P&) = D&S (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (D&S q P&) = S&D (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n\nDualInOut-sym : \u2200 {P Q} \u2192 DualInOut P Q \u2192 DualInOut Q P\nDualInOut-sym DInOut = DOutIn\nDualInOut-sym DOutIn = DInOut\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDualInOut-spec : \u2200 P \u2192 DualInOut P (dualInOut P)\nDualInOut-spec In  = DInOut\nDualInOut-spec Out = DOutIn\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (com In M P) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-spec (P x))\nDual-spec (com Out M P) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-spec (P x))\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com q M P) X = \u27e6 q \u27e7\u03a0\u03a3 M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>\u2261 Q) \u2192 \u2605} \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  bind-spec end         = refl\n  bind-spec (com q M P) = cong (\u27e6 q \u27e7\u03a0\u03a3 M) (funExt \u03bb m \u2192 bind-spec (P m))\n\n\nmodule _ {A B} where\n  run-com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  run-com end      a       b       = a , b\n  run-com (\u03a0' M P) p       (m , q) = run-com (P m) (p m) q\n  run-com (\u03a3' M P) (m , p) q       = run-com (P m) p (q m)\n\ncom-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\ncom-cont end p q = (_ , p)  , (_ , q)\ncom-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\ncom-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0' (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3' (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q M P Q} \u2192 SimL (com q M P) Q \u2192 Sim (com q M P) Q\n    right : \u2200 {P q M Q} \u2192 SimR P (com q M Q) \u2192 Sim P (com q M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recv PQA)   (send m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (send m PQB) = send m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (send m PQA) (recv PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (send x\u2081 x\u2082) (recv x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = send m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = ? -- send m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb x \u2192 left (send x (sim-id (B x)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb x \u2192 right (send x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u00b7\u03a3 x\u2081) (recv x) (send x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u00b7\u03a0 x)  (send x\u2081 x\u2082) (recv x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recv PQ) QR = recv (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recv P)) = do (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (send m P)) = do (send m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (send x x\u2082)) (left (recv x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (right (recv x)) (left (send x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (right (send x x\u2082)) (left (recv x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u00b7\u03a0 x) Q-Q' (right (send x\u2081 x\u2082)) (left (recv x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' (\u03a0\u00b7\u03a3 x\u2081) (right \u00f8P) (right (recv x)) (left (send x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u00b7\u03a0 x) (right \u00f8P) (right (send x\u2081 x\u2082)) (left (recv x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recv x\u2081))\n    = cong (right \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (send x\u2081 x\u2082))\n    = cong (right \u2218 send x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recv x))    = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (send x x\u2081)) = cong (left \u2218 send x) (sim-!! x\u2081)\n  sim-!! (right (recv x))    = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (send x x\u2081)) = cong (right \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (right (recv x)) (left (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (right (send x\u2081 x\u2082)) (left (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recv x\u2081))\n    = cong (left \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (send x\u2081 x\u2082))\n    = cong (left \u2218 send x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"24e8583332cb12030d6c6af750c856927fc3cc65","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: d33440a423653161d00f300d40c20753\n\ndarcs-hash:20100604042307-3bd4e-d23ab22b2266165e563892c3232c552392571930.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Division\/EquationsER.agda","new_file":"Examples\/Division\/EquationsER.agda","new_contents":"------------------------------------------------------------------------------\n-- Equations for the division\n------------------------------------------------------------------------------\n\nmodule Examples.Division.EquationsER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.InequalitiesPCF\n\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Division properties\n\n-- Note: This module was written for the version of div using the\n-- lambda abstractions, but we can use it with the version of div\n-- using super-combinators.\n\nprivate\n    -- Before to prove some properties for 'div i j' it is convenient\n    -- to have a proof for each possible execution step.\n\n    -- Initially, we define the possible states ('div-s\u2081',\n    -- 'dih-s\u2082', ...) and after that, we write down the proof for\n    -- the execution step from the state 'p' to the state 'q'\n    -- (i.e. (i j : D) \u2192 div-sp i j \u2192 div-sq i j).\n\n    -- Initially, 'cFix' conversion rule is applied\n    div-s\u2081 : D \u2192 D \u2192 D\n    div-s\u2081 i j = divh (fix divh) \u2219 i \u2219 j\n\n    -- First argument application\n    div-s\u2082 : D \u2192 D \u2192 D\n    div-s\u2082 i j = fun \u2219 j\n      where fun : D\n            fun = lam (\u03bb j \u2192 if (lt i j)\n                                then zero\n                                else (succ (fix divh \u2219 (i - j) \u2219 j)))\n\n    -- Second argument application\n    div-s\u2083 : D \u2192 D \u2192 D\n    div-s\u2083 i j = if (lt i j)\n                    then zero\n                    else (succ (fix divh \u2219 (i - j) \u2219 j))\n\n    -- 'lt i j \u2261 true'\n    div-s\u2084 : D \u2192 D \u2192 D\n    div-s\u2084 i j = if true\n                    then zero\n                    else (succ (fix divh \u2219 (i - j) \u2219 j))\n\n    -- 'lt i j \u2261 false'\n    div-s\u2085 : D \u2192 D \u2192 D\n    div-s\u2085 i j = if false\n                    then zero\n                    else (succ (fix divh \u2219 (i - j) \u2219 j))\n\n    -- The conditional is 'true'\n    div-s\u2086 : D\n    div-s\u2086 = zero\n\n    -- The conditional is 'false'\n    div-s\u2087 : D \u2192 D \u2192 D\n    div-s\u2087 i j = succ (fix divh \u2219 (i - j) \u2219 j)\n\n    {-\n    To prove the execution steps\n    (e.g. proof\u2083\u208b\u2084 : (i j : D) \u2192 div-s\u2083 i j \u2192 divh_s\u2084 i j),\n    we usually need to prove that\n\n                             '... m ... \u2261 ... n ...'    (1)\n\n    given that\n                                  'm \u2261 n',              (2)\n\n    where (2) is a conversion rule usually.\n    We prove (1) using\n    '\u2261-subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y'\n    where\n    'P' is given by '\\m \u2192 ... m ... \u2261 ... n ...',\n    'x \u2261 y' is given 'n \u2261 m' (actually, we use '\u2261-sym (m \u2261 n)'), and\n    'P x' is given by '... n ... \u2261 ... n ...' (i.e. '\u2261-refl')\n\n    -}\n\n    -- From 'div \u2219 i \u2219 j' to 'div-s\u2081' using the conversion rule 'Cfix'\n    proof\u2080\u208b\u2081 : (i j : D) \u2192 fix divh \u2219 i \u2219 j  \u2261 div-s\u2081 i j\n    proof\u2080\u208b\u2081 i j = subst (\u03bb t \u2192 t \u2219 i \u2219 j \u2261 divh (fix divh) \u2219 i \u2219 j )\n                         (sym (cFix divh ))\n                         refl\n\n    -- From 'div-s\u2081' to 'div-s\u2082' using the conversion rule 'beta'\n    proof\u2081\u208b\u2082 : (i j : D) \u2192 div-s\u2081 i j  \u2261 div-s\u2082 i j\n    proof\u2081\u208b\u2082 i j =\n      subst (\u03bb t \u2192 t \u2219 j \u2261 fun i \u2219 j )\n               (sym (cBeta fun i ))\n               refl\n         where\n          -- The function 'fun' is the same that the 'fun' part\n          -- of 'div-s\u2082', except that we need a fresh\n          -- variable 'y' to avoid the clashing of the variable 'i' in\n          -- the application of the 'beta' rule.\n          fun : D \u2192 D\n          fun y = lam (\u03bb j \u2192 if (lt y j)\n                                then zero\n                                else (succ (fix divh \u2219 (y - j) \u2219 j)))\n\n    -- From 'div-s\u2082' to 'div-s\u2083' using the conversion rule 'beta'\n    proof\u2082\u208b\u2083 : (i j : D) \u2192 div-s\u2082 i j  \u2261 div-s\u2083 i j\n    proof\u2082\u208b\u2083 i j  = cBeta fun j\n      where\n        -- The function 'fun' is the same that 'div-s\u2083', except that we\n        -- need a fresh 'y' to avoid the clashing of the variable 'j' in\n        -- the application of the 'beta' rule.\n        fun : D \u2192 D\n        fun y = if (lt i y)\n                   then zero\n                   else (succ ((fix divh) \u2219 (i - y) \u2219 y))\n\n    -- From 'div-s\u2083' to 'div-s\u2084' using the proof 'i<j'\n    proof\u2083_\u2084 : (i j : D) \u2192 LT i j \u2192 div-s\u2083 i j \u2261 div-s\u2084 i j\n    proof\u2083_\u2084 i j i<j =\n      subst (\u03bb t \u2192 if t\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n                      \u2261\n                   if true\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n               )\n               (sym i<j )\n               refl\n\n    -- From 'div-s\u2083' to 'div-s\u2085' using the proof  'i\u2265j'\n    proof\u2083\u208b\u2085 : (i j : D) \u2192 GE i j \u2192 div-s\u2083 i j  \u2261 div-s\u2085 i j\n    proof\u2083\u208b\u2085 i j i\u2265j =\n      subst (\u03bb t \u2192 if t\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n                      \u2261\n                   if false\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n               )\n               (sym i\u2265j )\n               refl\n\n\n    -- From 'div-s\u2084' to 'div-s\u2086' using the conversion rule 'CB1'\n    -- ToDo: Why we need to use 'div-s\u2084 {i} {j}' instead of 'div-s\u2084'\n    proof\u2084\u208b\u2086 : (i j : D) \u2192 div-s\u2084 i j \u2261 div-s\u2086\n    proof\u2084\u208b\u2086 i j = cB\u2081 zero\n\n    -- From 'div-s\u2085' to 'div-s\u2087' using the conversion rule 'CB2'\n    proof\u2085\u208b\u2087 : (i j : D) \u2192 div-s\u2085 i j \u2261 div-s\u2087 i j\n    proof\u2085\u208b\u2087 i j = cB\u2082 (succ (fix divh \u2219 (i - j) \u2219 j))\n\n----------------------------------------------------------------------\n-- The division result when the dividend is minor than the\n-- the divisor\n\ndiv-x<y : {i j : D} \u2192 LT i j \u2192 div i j \u2261 zero\ndiv-x<y {i} {j} i<j =\n  begin\n    div i j    \u2261\u27e8 proof\u2080\u208b\u2081 i j \u27e9\n    div-s\u2081 i j \u2261\u27e8 proof\u2081\u208b\u2082 i j \u27e9\n    div-s\u2082 i j \u2261\u27e8 proof\u2082\u208b\u2083 i j \u27e9\n    div-s\u2083 i j \u2261\u27e8 proof\u2083_\u2084 i j i<j \u27e9\n    div-s\u2084 i j \u2261\u27e8 proof\u2084\u208b\u2086 i j \u27e9\n    div-s\u2086\n  \u220e\n\n----------------------------------------------------------------------\n-- The division result when the dividend is greater or equal than the\n-- the divisor\n\ndiv-x\u2265y : {i j : D} \u2192 GE i j \u2192 div i j \u2261 succ (div (i - j) j)\ndiv-x\u2265y {i} {j} i\u2265j =\n  begin\n    div i j    \u2261\u27e8 proof\u2080\u208b\u2081 i j \u27e9\n    div-s\u2081 i j \u2261\u27e8 proof\u2081\u208b\u2082 i j \u27e9\n    div-s\u2082 i j \u2261\u27e8 proof\u2082\u208b\u2083 i j \u27e9\n    div-s\u2083 i j \u2261\u27e8 proof\u2083\u208b\u2085 i j i\u2265j \u27e9\n    div-s\u2085 i j \u2261\u27e8 proof\u2085\u208b\u2087 i j \u27e9\n    div-s\u2087 i j\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Equations for the division\n------------------------------------------------------------------------------\n\nmodule Examples.Division.EquationsER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.InequalitiesPCF\n\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Division properties\n\nprivate\n    -- Before to prove some properties for 'div i j' it is convenient\n    -- to have a proof for each possible execution step.\n\n    -- Initially, we define the possible states ('div-s\u2081',\n    -- 'dih-s\u2082', ...) and after that, we write down the proof for\n    -- the execution step from the state 'p' to the state 'q'\n    -- (i.e. (i j : D) \u2192 div-sp i j \u2192 div-sq i j).\n\n    -- Initially, 'cFix' conversion rule is applied\n    div-s\u2081 : D \u2192 D \u2192 D\n    div-s\u2081 i j = divh (fix divh) \u2219 i \u2219 j\n\n    -- First argument application\n    div-s\u2082 : D \u2192 D \u2192 D\n    div-s\u2082 i j = fun \u2219 j\n      where fun : D\n            fun = lam (\u03bb j \u2192 if (lt i j)\n                                then zero\n                                else (succ (fix divh \u2219 (i - j) \u2219 j)))\n\n    -- Second argument application\n    div-s\u2083 : D \u2192 D \u2192 D\n    div-s\u2083 i j = if (lt i j)\n                    then zero\n                    else (succ (fix divh \u2219 (i - j) \u2219 j))\n\n    -- 'lt i j \u2261 true'\n    div-s\u2084 : D \u2192 D \u2192 D\n    div-s\u2084 i j = if true\n                    then zero\n                    else (succ (fix divh \u2219 (i - j) \u2219 j))\n\n    -- 'lt i j \u2261 false'\n    div-s\u2085 : D \u2192 D \u2192 D\n    div-s\u2085 i j = if false\n                    then zero\n                    else (succ (fix divh \u2219 (i - j) \u2219 j))\n\n    -- The conditional is 'true'\n    div-s\u2086 : D\n    div-s\u2086 = zero\n\n    -- The conditional is 'false'\n    div-s\u2087 : D \u2192 D \u2192 D\n    div-s\u2087 i j = succ (fix divh \u2219 (i - j) \u2219 j)\n\n    {-\n    To prove the execution steps\n    (e.g. proof\u2083\u208b\u2084 : (i j : D) \u2192 div-s\u2083 i j \u2192 divh_s\u2084 i j),\n    we usually need to prove that\n\n                             '... m ... \u2261 ... n ...'    (1)\n\n    given that\n                                  'm \u2261 n',              (2)\n\n    where (2) is a conversion rule usually.\n    We prove (1) using\n    '\u2261-subst : {A : Set}(P : A \u2192 Set){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y'\n    where\n    'P' is given by '\\m \u2192 ... m ... \u2261 ... n ...',\n    'x \u2261 y' is given 'n \u2261 m' (actually, we use '\u2261-sym (m \u2261 n)'), and\n    'P x' is given by '... n ... \u2261 ... n ...' (i.e. '\u2261-refl')\n\n    -}\n\n    -- From 'div \u2219 i \u2219 j' to 'div-s\u2081' using the conversion rule 'Cfix'\n    proof\u2080\u208b\u2081 : (i j : D) \u2192 fix divh \u2219 i \u2219 j  \u2261 div-s\u2081 i j\n    proof\u2080\u208b\u2081 i j = subst (\u03bb t \u2192 t \u2219 i \u2219 j \u2261 divh (fix divh) \u2219 i \u2219 j )\n                         (sym (cFix divh ))\n                         refl\n\n    -- From 'div-s\u2081' to 'div-s\u2082' using the conversion rule 'beta'\n    proof\u2081\u208b\u2082 : (i j : D) \u2192 div-s\u2081 i j  \u2261 div-s\u2082 i j\n    proof\u2081\u208b\u2082 i j =\n      subst (\u03bb t \u2192 t \u2219 j \u2261 fun i \u2219 j )\n               (sym (cBeta fun i ))\n               refl\n         where\n          -- The function 'fun' is the same that the 'fun' part\n          -- of 'div-s\u2082', except that we need a fresh\n          -- variable 'y' to avoid the clashing of the variable 'i' in\n          -- the application of the 'beta' rule.\n          fun : D \u2192 D\n          fun y = lam (\u03bb j \u2192 if (lt y j)\n                                then zero\n                                else (succ (fix divh \u2219 (y - j) \u2219 j)))\n\n    -- From 'div-s\u2082' to 'div-s\u2083' using the conversion rule 'beta'\n    proof\u2082\u208b\u2083 : (i j : D) \u2192 div-s\u2082 i j  \u2261 div-s\u2083 i j\n    proof\u2082\u208b\u2083 i j  = cBeta fun j\n      where\n        -- The function 'fun' is the same that 'div-s\u2083', except that we\n        -- need a fresh 'y' to avoid the clashing of the variable 'j' in\n        -- the application of the 'beta' rule.\n        fun : D \u2192 D\n        fun y = if (lt i y)\n                   then zero\n                   else (succ ((fix divh) \u2219 (i - y) \u2219 y))\n\n    -- From 'div-s\u2083' to 'div-s\u2084' using the proof 'i<j'\n    proof\u2083_\u2084 : (i j : D) \u2192 LT i j \u2192 div-s\u2083 i j \u2261 div-s\u2084 i j\n    proof\u2083_\u2084 i j i<j =\n      subst (\u03bb t \u2192 if t\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n                      \u2261\n                   if true\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n               )\n               (sym i<j )\n               refl\n\n    -- From 'div-s\u2083' to 'div-s\u2085' using the proof  'i\u2265j'\n    proof\u2083\u208b\u2085 : (i j : D) \u2192 GE i j \u2192 div-s\u2083 i j  \u2261 div-s\u2085 i j\n    proof\u2083\u208b\u2085 i j i\u2265j =\n      subst (\u03bb t \u2192 if t\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n                      \u2261\n                   if false\n                      then zero\n                      else (succ ((fix divh) \u2219 (i - j) \u2219 j))\n               )\n               (sym i\u2265j )\n               refl\n\n\n    -- From 'div-s\u2084' to 'div-s\u2086' using the conversion rule 'CB1'\n    -- ToDo: Why we need to use 'div-s\u2084 {i} {j}' instead of 'div-s\u2084'\n    proof\u2084\u208b\u2086 : (i j : D) \u2192 div-s\u2084 i j \u2261 div-s\u2086\n    proof\u2084\u208b\u2086 i j = cB\u2081 zero\n\n    -- From 'div-s\u2085' to 'div-s\u2087' using the conversion rule 'CB2'\n    proof\u2085\u208b\u2087 : (i j : D) \u2192 div-s\u2085 i j \u2261 div-s\u2087 i j\n    proof\u2085\u208b\u2087 i j = cB\u2082 (succ (fix divh \u2219 (i - j) \u2219 j))\n\n----------------------------------------------------------------------\n-- The division result when the dividend is minor than the\n-- the divisor\n\ndiv-x<y : {i j : D} \u2192 LT i j \u2192 div i j \u2261 zero\ndiv-x<y {i} {j} i<j =\n  begin\n    div i j    \u2261\u27e8 proof\u2080\u208b\u2081 i j \u27e9\n    div-s\u2081 i j \u2261\u27e8 proof\u2081\u208b\u2082 i j \u27e9\n    div-s\u2082 i j \u2261\u27e8 proof\u2082\u208b\u2083 i j \u27e9\n    div-s\u2083 i j \u2261\u27e8 proof\u2083_\u2084 i j i<j \u27e9\n    div-s\u2084 i j \u2261\u27e8 proof\u2084\u208b\u2086 i j \u27e9\n    div-s\u2086\n  \u220e\n\n----------------------------------------------------------------------\n-- The division result when the dividend is greater or equal than the\n-- the divisor\n\ndiv-x\u2265y : {i j : D} \u2192 GE i j \u2192 div i j \u2261 succ (div (i - j) j)\ndiv-x\u2265y {i} {j} i\u2265j =\n  begin\n    div i j    \u2261\u27e8 proof\u2080\u208b\u2081 i j \u27e9\n    div-s\u2081 i j \u2261\u27e8 proof\u2081\u208b\u2082 i j \u27e9\n    div-s\u2082 i j \u2261\u27e8 proof\u2082\u208b\u2083 i j \u27e9\n    div-s\u2083 i j \u2261\u27e8 proof\u2083\u208b\u2085 i j i\u2265j \u27e9\n    div-s\u2085 i j \u2261\u27e8 proof\u2085\u208b\u2087 i j \u27e9\n    div-s\u2087 i j\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dd9c11f4cb25f9dd85e9a41c3133a7daf9557b63","subject":"Import more appropriate module.","message":"Import more appropriate module.\n\nOld-commit-hash: aea8365dbe457bc010a8d71fdb5a66491ee75223\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/FreeVars\/Popl14.agda","new_file":"Syntax\/FreeVars\/Popl14.agda","new_contents":"module Syntax.FreeVars.Popl14 where\n\n-- Free variables of Calculus Popl14\n--\n-- Contents\n-- FV: get the free variables of a term\n-- closed?: test if a term is closed, producing a witness if yes.\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Base.Syntax.Vars Type public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Sum\n\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\nFV (const \u03b9 ts) = FV-terms ts\nFV (var x) = singleton x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV s \u222a FV t\n\nFV-terms \u2205 = none\nFV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\nclosed? t = empty? (FV t)\n","old_contents":"module Syntax.FreeVars.Popl14 where\n\n-- Free variables of Calculus Popl14\n--\n-- Contents\n-- FV: get the free variables of a term\n-- closed?: test if a term is closed, producing a witness if yes.\n\nopen import Popl14.Syntax.Type\nopen import Popl14.Change.Term\nopen import Base.Syntax.Vars Type public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Sum\n\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\nFV (const \u03b9 ts) = FV-terms ts\nFV (var x) = singleton x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV s \u222a FV t\n\nFV-terms \u2205 = none\nFV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\nclosed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\nclosed? t = empty? (FV t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5c486ce205d3253f2a7cbd822bf697c83bb45c20","subject":"Indentation","message":"Indentation\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) : ISet I where\n  init : UncurriedEl D (\u03bc D)\n\n----------------------------------------------------------------------\n\nData : Set\nData =\n  \u03a3 Set \u03bb I \u2192\n  \u03a3 Enum \u03bb E \u2192\n  Branches E (\u03bb _ \u2192 Desc I)\n\nDataI : Data \u2192 Set\nDataI (I , E , B) = I\n\nDataE : Data \u2192 Enum\nDataE (I , E , B) = E\n\nDataT : Data \u2192 Set\nDataT R = Tag (DataE R)\n\nDataB : (R : Data) \u2192 Branches (DataE R) (\u03bb _ \u2192 Desc (DataI R))\nDataB (I , E , B) = B\n\nDataD : (R : Data) \u2192 Desc (DataI R)\nDataD R = Arg (DataT R) (case (\u03bb _ \u2192 Desc (DataI R)) (DataB R))\n\ncaseR : (R : Data) \u2192 DataT R \u2192 Desc (DataI R)\ncaseR R = case (\u03bb _ \u2192 Desc (DataI R)) (DataB R)\n\n----------------------------------------------------------------------\n\ninj : (R : Data)\n  \u2192 let D = DataD R\n  in CurriedEl D (\u03bc D)\ninj R = let D = DataD R\n  in curryEl D (\u03bc D) init\n\n----------------------------------------------------------------------\n\nprove : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nprove (End j) X P \u03b1 i q = tt\nprove (Rec j D) X P \u03b1 i (x , xs) = \u03b1 j x , prove D X P \u03b1 i xs\nprove (Arg A B) X P \u03b1 i (a , xs) = prove (B a) X P \u03b1 i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : UncurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nind D P \u03b1 i (init xs) = \u03b1 i xs $\n  prove D (\u03bc D) P (ind D P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nindCurried D P f i x = ind D P (uncurryHyps D (\u03bc D) P init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 DataT R \u2192 Set\nSumCurriedHyps R P t =\n  CurriedHyps (caseR R t) (\u03bc (DataD R)) P (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 Branches (DataE R) (SumCurriedHyps R P)\n  \u2192 \u2200 i (x : \u03bc D i) \u2192 P i x\nelimUncurried R P cs i x =\n  indCurried (DataD R) P\n    (case (SumCurriedHyps R P) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 CurriedBranches (DataE R)\n      (SumCurriedHyps R P)\n      (\u2200 i (x : \u03bc D i) \u2192 P i x)\nelim R P = curryBranches (elimUncurried R P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115R : Data\n\u2115R = \u22a4 , \u2115E\n  , End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = DataD \u2115R\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115D\n\nzero : \u2115 tt\nzero = init (zeroT , refl)\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init (sucT , n , refl)\n\nVecR : (A : Set) \u2192 Data\nVecR A = (\u2115 tt) , VecE\n  , End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = DataD (VecR A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc (VecD A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init (nilT , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init (consT , n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj (VecR A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj (VecR A) consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nadd = elim \u2115R _\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n  tt\n\nmult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nmult = elim \u2115R _\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n  tt\n\nappend : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\nappend A = elim (VecR A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\nconcat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m = elim (VecR (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n  (nil A)\n  (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) : ISet I where\n  init : UncurriedEl D (\u03bc D)\n\n----------------------------------------------------------------------\n\nData : Set\nData =\n  \u03a3 Set \u03bb I \u2192\n  \u03a3 Enum \u03bb E \u2192\n  Branches E (\u03bb _ \u2192 Desc I)\n\nDataI : Data \u2192 Set\nDataI (I , E , B) = I\n\nDataE : Data \u2192 Enum\nDataE (I , E , B) = E\n\nDataT : Data \u2192 Set\nDataT R = Tag (DataE R)\n\nDataB : (R : Data) \u2192 Branches (DataE R) (\u03bb _ \u2192 Desc (DataI R))\nDataB (I , E , B) = B\n\nDataD : (R : Data) \u2192 Desc (DataI R)\nDataD R = Arg (DataT R) (case (\u03bb _ \u2192 Desc (DataI R)) (DataB R))\n\ncaseR : (R : Data) \u2192 DataT R \u2192 Desc (DataI R)\ncaseR R = case (\u03bb _ \u2192 Desc (DataI R)) (DataB R)\n\n----------------------------------------------------------------------\n\ninj : (R : Data)\n  \u2192 let D = DataD R\n  in CurriedEl D (\u03bc D)\ninj R = let D = DataD R\n  in curryEl D (\u03bc D) init\n\n----------------------------------------------------------------------\n\nprove : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nprove (End j) X P \u03b1 i q = tt\nprove (Rec j D) X P \u03b1 i (x , xs) = \u03b1 j x , prove D X P \u03b1 i xs\nprove (Arg A B) X P \u03b1 i (a , xs) = prove (B a) X P \u03b1 i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : UncurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nind D P \u03b1 i (init xs) = \u03b1 i xs $\n  prove D (\u03bc D) P (ind D P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nindCurried D P f i x = ind D P (uncurryHyps D (\u03bc D) P init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 DataT R \u2192 Set\nSumCurriedHyps R P t =\n  CurriedHyps (caseR R t) (\u03bc (DataD R)) P (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 Branches (DataE R) (SumCurriedHyps R P)\n  \u2192 \u2200 i (x : \u03bc D i) \u2192 P i x\nelimUncurried R P cs i x =\n  indCurried (DataD R) P\n    (case (SumCurriedHyps R P) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 CurriedBranches (DataE R)\n      (SumCurriedHyps R P)\n      (\u2200 i (x : \u03bc D i) \u2192 P i x)\nelim R P = curryBranches (elimUncurried R P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115R : Data\n\u2115R = \u22a4 , \u2115E\n  , End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = DataD \u2115R\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115D\n\nzero : \u2115 tt\nzero = init (zeroT , refl)\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init (sucT , n , refl)\n\nVecR : (A : Set) \u2192 Data\nVecR A = (\u2115 tt) , VecE\n  , End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = DataD (VecR A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc (VecD A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init (nilT , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init (consT , n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj (VecR A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj (VecR A) consT\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115R _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115R _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim (VecR A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim (VecR (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bce471b3be844f987aa2bd3e748f542683de4829","subject":"Correct wrong comment on changes of products","message":"Correct wrong comment on changes of products\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Products.agda","new_file":"Base\/Change\/Products.agda","new_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  -- An interesting alternative definition allows omitting the nil change of a\n  -- component when that nil change can be computed from the type. For instance, the nil change for integers is always the same.\n\n  -- However, the nil change for function isn't always the same (unless we\n  -- defunctionalize them first), so nil changes for functions can't be omitted.\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-nil : (v : A \u00d7 B) \u2192 PChange v\n  p-nil (a , b) = (nil a , nil b)\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  p-update-nil : (v : A \u00d7 B) \u2192 v \u2295 (p-nil v) \u2261 v\n  p-update-nil (a , b) =\n    let v = (a , b)\n    in\n      begin\n        v \u2295 (p-nil v)\n      \u2261\u27e8\u27e9\n        (a \u229e nil a , b \u229e nil b)\n      \u2261\u27e8 cong\u2082 _,_ (update-nil a) (update-nil b)\u27e9\n         (a , b)\n      \u2261\u27e8\u27e9\n        v\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = p-nil\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      ; update-nil = p-update-nil\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  -- We should do the same for uncurry instead.\n\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032 a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of its own validity, and because after application it does not contain a proof\n\n  -- As a consequence, proving that's a derivative seems too insanely hard. We\n  -- might want to provide a proof schema for curried functions at once,\n  -- starting from the right correctness equation.\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n\n  _,_\u2032-real : \u0394 _,_\n  _,_\u2032-real = nil _,_\n  _,_\u2032-real-Derivative : Derivative {{CA}} {{B\u2192A\u00d7B}} _,_ (\u0394A\u2192B\u2192A\u00d7B.apply _,_\u2032-real)\n  _,_\u2032-real-Derivative =\n    FunctionChanges.nil-is-derivative A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}} _,_\n\n  _,_\u2032\u2032 :  (a : A) \u2192 \u0394 a \u2192\n      \u0394 {{B\u2192A\u00d7B}} (\u03bb b \u2192 (a , b))\n  _,_\u2032\u2032 a da = record\n    { apply = _,_\u2032 a da\n    ; correct = \u03bb b db \u2192\n      begin\n        (a , b \u229e db) \u229e (_,_\u2032 a da) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n        a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9\n        (a , b) \u229e (_,_\u2032 a da) b db\n      \u220e\n    }\n{-\n  _,_\u2032\u2032\u2032 : \u0394 {{A\u2192B\u2192A\u00d7B}} _,_\n  _,_\u2032\u2032\u2032 = record\n    { apply = _,_\u2032\u2032\n    ; correct = \u03bb a da \u2192\n      begin\n        update\n        (_,_ (a \u229e da))\n        (_,_\u2032\u2032 (a \u229e da) (nil (a \u229e da)))\n      \u2261\u27e8 {!!} \u27e9\n        update (_,_ a) (_,_\u2032\u2032 a da)\n      \u220e\n    }\n    where\n      -- This is needed to use update above.\n      -- Passing the change structure seems hard with the given operators; maybe I'm just using them wrongly.\n      open ChangeAlgebra B\u2192A\u00d7B hiding (nil)\n{-\n  {!\n      begin\n        (_,_ (a \u229e da)) \u229e _,_\u2032\u2032 (a \u229e da) (nil (a \u229e da))\n      {- \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db)) -}\n      \u2261\u27e8 ? \u27e9\n        {-a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9-}\n       (_,_ a) \u229e (_,_\u2032\u2032 a da)\n      \u220e!}\n    }\n-}\n  open import Postulate.Extensionality\n\n\n  _,_\u2032Derivative :\n    Derivative {{CA}} {{B\u2192A\u00d7B}} _,_  _,_\u2032\u2032\n  _,_\u2032Derivative a da =\n    begin\n      _\u229e_ {{B\u2192A\u00d7B}} (_,_ a) (_,_\u2032\u2032 a da)\n    \u2261\u27e8\u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    --ext (\u03bb b \u2192 cong (\u03bb \u25a1 \u2192  (a , b) \u229e \u25a1) (update-nil {{?}} b))\n    \u2261\u27e8 {!!} \u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    \u2261\u27e8 sym {!\u0394A\u2192B\u2192A\u00d7B.incrementalization _,_ _,_\u2032\u2032\u2032 a da!} \u27e9\n  --FunctionChanges.incrementalization A (B \u2192 A \u00d7 B) {{CA}} {{{!B\u2192A\u00d7B!}}} _,_ {!!} {!!} {!!}\n       _,_ (a \u229e da)\n    \u220e\n-}\n","old_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  -- The following is probably bullshit:\n  -- Does not handle products of functions - more accurately, writing the\n  -- derivative of fst and snd for products of functions is hard: fst' p dp must return the change of fst p\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-nil : (v : A \u00d7 B) \u2192 PChange v\n  p-nil (a , b) = (nil a , nil b)\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  p-update-nil : (v : A \u00d7 B) \u2192 v \u2295 (p-nil v) \u2261 v\n  p-update-nil (a , b) =\n    let v = (a , b)\n    in\n      begin\n        v \u2295 (p-nil v)\n      \u2261\u27e8\u27e9\n        (a \u229e nil a , b \u229e nil b)\n      \u2261\u27e8 cong\u2082 _,_ (update-nil a) (update-nil b)\u27e9\n         (a , b)\n      \u2261\u27e8\u27e9\n        v\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = p-nil\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      ; update-nil = p-update-nil\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  -- We should do the same for uncurry instead.\n\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032 a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of its own validity, and because after application it does not contain a proof\n\n  -- As a consequence, proving that's a derivative seems too insanely hard. We\n  -- might want to provide a proof schema for curried functions at once,\n  -- starting from the right correctness equation.\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n\n  _,_\u2032-real : \u0394 _,_\n  _,_\u2032-real = nil _,_\n  _,_\u2032-real-Derivative : Derivative {{CA}} {{B\u2192A\u00d7B}} _,_ (\u0394A\u2192B\u2192A\u00d7B.apply _,_\u2032-real)\n  _,_\u2032-real-Derivative =\n    FunctionChanges.nil-is-derivative A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}} _,_\n\n  _,_\u2032\u2032 :  (a : A) \u2192 \u0394 a \u2192\n      \u0394 {{B\u2192A\u00d7B}} (\u03bb b \u2192 (a , b))\n  _,_\u2032\u2032 a da = record\n    { apply = _,_\u2032 a da\n    ; correct = \u03bb b db \u2192\n      begin\n        (a , b \u229e db) \u229e (_,_\u2032 a da) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n        a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9\n        (a , b) \u229e (_,_\u2032 a da) b db\n      \u220e\n    }\n{-\n  _,_\u2032\u2032\u2032 : \u0394 {{A\u2192B\u2192A\u00d7B}} _,_\n  _,_\u2032\u2032\u2032 = record\n    { apply = _,_\u2032\u2032\n    ; correct = \u03bb a da \u2192\n      begin\n        update\n        (_,_ (a \u229e da))\n        (_,_\u2032\u2032 (a \u229e da) (nil (a \u229e da)))\n      \u2261\u27e8 {!!} \u27e9\n        update (_,_ a) (_,_\u2032\u2032 a da)\n      \u220e\n    }\n    where\n      -- This is needed to use update above.\n      -- Passing the change structure seems hard with the given operators; maybe I'm just using them wrongly.\n      open ChangeAlgebra B\u2192A\u00d7B hiding (nil)\n{-\n  {!\n      begin\n        (_,_ (a \u229e da)) \u229e _,_\u2032\u2032 (a \u229e da) (nil (a \u229e da))\n      {- \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db)) -}\n      \u2261\u27e8 ? \u27e9\n        {-a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9-}\n       (_,_ a) \u229e (_,_\u2032\u2032 a da)\n      \u220e!}\n    }\n-}\n  open import Postulate.Extensionality\n\n\n  _,_\u2032Derivative :\n    Derivative {{CA}} {{B\u2192A\u00d7B}} _,_  _,_\u2032\u2032\n  _,_\u2032Derivative a da =\n    begin\n      _\u229e_ {{B\u2192A\u00d7B}} (_,_ a) (_,_\u2032\u2032 a da)\n    \u2261\u27e8\u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    --ext (\u03bb b \u2192 cong (\u03bb \u25a1 \u2192  (a , b) \u229e \u25a1) (update-nil {{?}} b))\n    \u2261\u27e8 {!!} \u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.apply (_,_\u2032\u2032 a da) b (nil b))\n    \u2261\u27e8 sym {!\u0394A\u2192B\u2192A\u00d7B.incrementalization _,_ _,_\u2032\u2032\u2032 a da!} \u27e9\n  --FunctionChanges.incrementalization A (B \u2192 A \u00d7 B) {{CA}} {{{!B\u2192A\u00d7B!}}} _,_ {!!} {!!} {!!}\n       _,_ (a \u229e da)\n    \u220e\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"072206563afa62d16a638999da07fe562cc30a82","subject":"s\/ColistF\/ListF.","message":"s\/ColistF\/ListF.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GFPs\/Colist.agda","new_file":"notes\/fixed-points\/GFPs\/Colist.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GFPs.Colist where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Colist is a greatest fixed-point of a functor\n\n-- The functor.\nListF : (D \u2192 Set) \u2192 D \u2192 Set\nListF A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n\n-- Colist is the greatest fixed-point of ListF.\npostulate\n  Colist : D \u2192 Set\n\n  -- Colist is a post-fixed point of ListF, i.e.\n  --\n  -- Colist \u2264 ListF Colist.\n  Colist-out-ho : \u2200 {n} \u2192 Colist n \u2192 ListF Colist n\n\n  -- Colist is the greatest post-fixed point of ListF, i.e.\n  --\n  -- \u2200 A. A \u2264 ListF A \u21d2 A \u2264 Colist.\n  Colist-coind-ho :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ListF.\n    (\u2200 {xs} \u2192 A xs \u2192 ListF A xs) \u2192\n    -- Colist is greater than A.\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n------------------------------------------------------------------------------\n-- First-order versions\n\nColist-out : \u2200 {xs} \u2192\n             Colist xs \u2192\n             xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\nColist-out = Colist-out-ho\n\nColist-coind :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind = Colist-coind-ho\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Colist predicate is also a pre-fixed point of the functional ListF,\n-- i.e.\n--\n-- ListF Colist \u2264 Colist.\nColist-in-ho : \u2200 {xs} \u2192 ListF Colist xs \u2192 Colist xs\nColist-in-ho h = Colist-coind-ho A h' h\n  where\n  A : D \u2192 Set\n  A xs = ListF Colist xs\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 ListF A xs\n  h' (inj\u2081 xs\u22610) = inj\u2081 xs\u22610\n  h' (inj\u2082 (x' , xs' , prf , CLxs' )) =\n    inj\u2082 (x' , xs' , prf , Colist-out CLxs')\n\n-- The first-order version.\nColist-in : \u2200 {xs} \u2192\n            xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs') \u2192\n            Colist xs\nColist-in = Colist-in-ho\n\n------------------------------------------------------------------------------\n-- A stronger co-induction principle\n--\n-- From (Paulson, 1997. p. 16).\n\npostulate\n  Colist-coind-stronger-ho :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 A xs \u2192 ListF A xs \u2228 Colist xs) \u2192\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\nColist-coind-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ListF A xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-ho' A h Axs =\n  Colist-coind-stronger-ho A (\u03bb Ays \u2192 inj\u2081 (h Ays)) Axs\n\n-- The first-order version.\nColist-coind-stronger :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192\n    (xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'))\n    \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger = Colist-coind-stronger-ho\n\n-- 13 January 2014. As expected, we cannot prove\n-- Colist-coind-stronger-ho from Colist-coind-ho.\nColist-coind-stronger-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ListF A xs \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger-ho' A h {xs} Axs = case prf (\u03bb h' \u2192 h') (h Axs)\n  where\n  prf : ListF A xs \u2192 Colist xs\n  prf h' = Colist-coind-ho A {!!} Axs\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. In: Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","old_contents":"------------------------------------------------------------------------------\n-- Co-lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GFPs.Colist where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n-- Colist is a greatest fixed-point of a functor\n\n-- The functor.\nColistF : (D \u2192 Set) \u2192 D \u2192 Set\nColistF A xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n\n-- Colist is the greatest fixed-point of ColistF.\npostulate\n  Colist : D \u2192 Set\n\n  -- Colist is a post-fixed point of ColistF, i.e.\n  --\n  -- Colist \u2264 ColistF Colist.\n  Colist-out-ho : \u2200 {n} \u2192 Colist n \u2192 ColistF Colist n\n\n  -- Colist is the greatest post-fixed point of ColistF, i.e.\n  --\n  -- \u2200 A. A \u2264 ColistF A \u21d2 A \u2264 Colist.\n  Colist-coind-ho :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ColistF.\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n    -- Colist is greater than A.\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\n------------------------------------------------------------------------------\n-- First-order versions\n\nColist-out : \u2200 {xs} \u2192\n             Colist xs \u2192\n             xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs')\nColist-out = Colist-out-ho\n\nColist-coind :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind = Colist-coind-ho\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Colist predicate is also a pre-fixed point of the functional\n-- ColistF, i.e.\n--\n-- ColistF Colist \u2264 Colist.\nColist-in-ho : \u2200 {xs} \u2192 ColistF Colist xs \u2192 Colist xs\nColist-in-ho h = Colist-coind-ho A h' h\n  where\n  A : D \u2192 Set\n  A xs = ColistF Colist xs\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 ColistF A xs\n  h' (inj\u2081 xs\u22610) = inj\u2081 xs\u22610\n  h' (inj\u2082 (x' , xs' , prf , CLxs' )) =\n    inj\u2082 (x' , xs' , prf , Colist-out CLxs')\n\n-- The first-order version.\nColist-in : \u2200 {xs} \u2192\n            xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Colist xs') \u2192\n            Colist xs\nColist-in = Colist-in-ho\n\n------------------------------------------------------------------------------\n-- A stronger co-induction principle\n--\n-- From (Paulson, 1997. p. 16).\n\npostulate\n  Colist-coind-stronger-ho :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs \u2228 Colist xs) \u2192\n    \u2200 {xs} \u2192 A xs \u2192 Colist xs\n\nColist-coind-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-ho' A h Axs =\n  Colist-coind-stronger-ho A (\u03bb Ays \u2192 inj\u2081 (h Ays)) Axs\n\n-- The first-order version.\nColist-coind-stronger :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192\n    (xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'))\n    \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger = Colist-coind-stronger-ho\n\n-- 13 January 2014. As expected, we cannot prove\n-- Colist-coind-stronger-ho from Colist-coind-ho.\nColist-coind-stronger-ho' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {xs} \u2192 A xs \u2192 ColistF A xs \u2228 Colist xs) \u2192\n  \u2200 {xs} \u2192 A xs \u2192 Colist xs\nColist-coind-stronger-ho' A h {xs} Axs = case prf (\u03bb h' \u2192 h') (h Axs)\n  where\n  prf : ColistF A xs \u2192 Colist xs\n  prf h' = Colist-coind-ho A {!!} Axs\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. In: Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c94191eefcde0b3f090664eadcb7d62a4e048ccc","subject":"Adapt the compression attack to Message \u00d7 Message interface","message":"Adapt the compression attack to Message \u00d7 Message interface\n","repos":"crypto-agda\/crypto-agda","old_file":"Attack\/Compression.agda","new_file":"Attack\/Compression.agda","new_contents":"-- Compression can be used an an Oracle to defeat encryption.\n-- Here we show how compressing before encrypting lead to a\n-- NOT semantically secure construction (IND-CPA).\nmodule Attack.Compression where\n\nopen import Type using (\u2605)\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Two hiding (_==_)\nopen import Data.Product\nopen import Data.Zero\nopen import Relation.Binary.PropositionalEquality.NP\n\nimport Game.IND-CPA\n\nrecord Sized (A : \u2605) : \u2605 where\n  field\n    size  : A \u2192 \u2115\n\nopen Sized {{...}}\n\nmodule _ {A B} {{_ : Sized A}} {{_ : Sized B}} where\n    -- Same size\n    _==\u02e2_ : A \u2192 B \u2192 \ud835\udfda\n    x ==\u02e2 y = size x == size y\n\n    -- Same size\n    _\u2261\u02e2_ : A \u2192 B \u2192 \u2605\n    x \u2261\u02e2 y = size x \u2261 size y\n\n    \u2261\u02e2\u2192==\u02e2 : \u2200 {x y} \u2192 x \u2261\u02e2 y \u2192 (x ==\u02e2 y) \u2261 1\u2082\n    \u2261\u02e2\u2192==\u02e2 {x} {y} x\u2261\u02e2y rewrite x\u2261\u02e2y = \u2713\u2192\u2261 (==.refl {size y})\n\n    ==\u02e2\u2192\u2261\u02e2 : \u2200 {x y} \u2192 (x ==\u02e2 y) \u2261 1\u2082 \u2192 x \u2261\u02e2 y\n    ==\u02e2\u2192\u2261\u02e2 p = ==.sound _ _ (\u2261\u2192\u2713 p)\n\nmodule _ {PubKey Message CipherText R\u2091 : \u2605}\n         (Enc  : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n         {{_ : Sized Message}}\n         {{_ : Sized CipherText}}\n  where\n    -- Encryption size is independant of the randomness\n    EncSizeRndInd =\n      \u2200 {pk m r\u2080 r\u2081} \u2192 Enc pk m r\u2080 \u2261\u02e2 Enc pk m r\u2081\n\n    -- Encrypted ciphertexts of the same size, will lead to messages of the same size\n    EncLeakSize =\n      \u2200 {pk m\u2080 m\u2081 r\u2080 r\u2081} \u2192 Enc pk m\u2080 r\u2080 \u2261\u02e2 Enc pk m\u2081 r\u2081 \u2192 m\u2080 \u2261\u02e2 m\u2081\n\nmodule M\n  {Message CompressedMessage : \u2605}\n  {{_ : Sized CompressedMessage}}\n\n  (compress : Message \u2192 CompressedMessage)\n\n  -- 2 messages which have different size after compression\n  (m\u2080 m\u2081 : Message)\n  (different-compression\n     : size (compress m\u2080) \u2262 size (compress m\u2081))\n\n  (PubKey     : \u2605)\n  (SecKey     : \u2605)\n  (CipherText : \u2605)\n  {{_ : Sized CipherText}}\n  (R\u2091 R\u2096 R\u2093 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc\u2080 : PubKey \u2192 CompressedMessage \u2192 R\u2091 \u2192 CipherText)\n  (Enc\u2080SizeRndInd : EncSizeRndInd Enc\u2080)\n  (Enc\u2080LeakSize : EncLeakSize Enc\u2080)\n  where\n\n  -- Our adversary runs one encryption\n  R\u2090 = R\u2091\n\n  Enc\u2081 : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\n  Enc\u2081 pk m r\u2091 = Enc\u2080 pk (compress m) r\u2091\n\n  module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText\n                                R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\u2081\n\n  adv : IND-CPA.Adv\n  adv = (\u03bb { _  _    \u2192 [0: m\u2080 1: m\u2081 ] })\n      , (\u03bb { r\u2091 pk c \u2192 c ==\u02e2 Enc\u2081 pk m\u2081 r\u2091 })\n\n  -- The adversary adv is always winning.\n  adv-win : \u2200 {r} b \u2192 IND-CPA.\u2141 b adv r \u2261 b\n  adv-win 0\u2082 = \u22621\u2192\u22610 (different-compression \u2218 Enc\u2080LeakSize \u2218 ==\u02e2\u2192\u2261\u02e2)\n  adv-win 1\u2082 = \u2261\u02e2\u2192==\u02e2 Enc\u2080SizeRndInd\n","old_contents":"-- Compression can be used an an Oracle to defeat encryption.\n-- Here we show how compressing before encrypting lead to a\n-- NOT semantically secure construction (IND-CPA).\nmodule Attack.Compression where\n\nopen import Type using (\u2605)\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Two hiding (_==_)\nopen import Data.Product\nopen import Data.Zero\nopen import Relation.Binary.PropositionalEquality.NP\n\nimport Game.IND-CPA\n\nrecord Sized (A : \u2605) : \u2605 where\n  field\n    size  : A \u2192 \u2115\n\nopen Sized {{...}}\n\nmodule _ {A B} {{_ : Sized A}} {{_ : Sized B}} where\n    -- Same size\n    _==\u02e2_ : A \u2192 B \u2192 \ud835\udfda\n    x ==\u02e2 y = size x == size y\n\n    -- Same size\n    _\u2261\u02e2_ : A \u2192 B \u2192 \u2605\n    x \u2261\u02e2 y = size x \u2261 size y\n\n    \u2261\u02e2\u2192==\u02e2 : \u2200 {x y} \u2192 x \u2261\u02e2 y \u2192 (x ==\u02e2 y) \u2261 1\u2082\n    \u2261\u02e2\u2192==\u02e2 {x} {y} x\u2261\u02e2y rewrite x\u2261\u02e2y = \u2713\u2192\u2261 (==.refl {size y})\n\n    ==\u02e2\u2192\u2261\u02e2 : \u2200 {x y} \u2192 (x ==\u02e2 y) \u2261 1\u2082 \u2192 x \u2261\u02e2 y\n    ==\u02e2\u2192\u2261\u02e2 p = ==.sound _ _ (\u2261\u2192\u2713 p)\n\nmodule _ {PubKey Message CipherText R\u2091 : \u2605}\n         (Enc  : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n         {{_ : Sized Message}}\n         {{_ : Sized CipherText}}\n  where\n    -- Encryption size is independant of the randomness\n    EncSizeRndInd =\n      \u2200 {pk m r\u2080 r\u2081} \u2192 Enc pk m r\u2080 \u2261\u02e2 Enc pk m r\u2081\n\n    -- Encrypted ciphertexts of the same size, will lead to messages of the same size\n    EncLeakSize =\n      \u2200 {pk m\u2080 m\u2081 r\u2080 r\u2081} \u2192 Enc pk m\u2080 r\u2080 \u2261\u02e2 Enc pk m\u2081 r\u2081 \u2192 m\u2080 \u2261\u02e2 m\u2081\n\nmodule M\n  {Message CompressedMessage : \u2605}\n  {{_ : Sized CompressedMessage}}\n\n  (compress : Message \u2192 CompressedMessage)\n\n  -- 2 messages which have different size after compression\n  (m : \ud835\udfda \u2192 Message)\n  (different-compression\n     : size (compress (m 0\u2082)) \u2262 size (compress (m 1\u2082)))\n\n  (PubKey     : \u2605)\n  (SecKey     : \u2605)\n  (CipherText : \u2605)\n  {{_ : Sized CipherText}}\n  (R\u2091 R\u2096 R\u2093 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc\u2080 : PubKey \u2192 CompressedMessage \u2192 R\u2091 \u2192 CipherText)\n  (Enc\u2080SizeRndInd : EncSizeRndInd Enc\u2080)\n  (Enc\u2080LeakSize : EncLeakSize Enc\u2080)\n  where\n\n  -- Our adversary runs one encryption\n  R\u2090 = R\u2091\n\n  Enc\u2081 : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\n  Enc\u2081 pk m r\u2091 = Enc\u2080 pk (compress m) r\u2091\n\n  module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText\n                                R\u2091 R\u2096 R\u2090 R\u2093 KeyGen Enc\u2081\n\n  adv : IND-CPA.Adv\n  adv = (\u03bb { _  _    \u2192 m })\n      , (\u03bb { r\u2091 pk c \u2192 c ==\u02e2 Enc\u2081 pk (m 1\u2082) r\u2091 })\n\n  -- The adversary adv is always winning.\n  adv-win : \u2200 {r} b \u2192 IND-CPA.\u2141 b adv r \u2261 b\n  adv-win 0\u2082 = \u22621\u2192\u22610 (different-compression \u2218 Enc\u2080LeakSize \u2218 ==\u02e2\u2192\u2261\u02e2)\n  adv-win 1\u2082 = \u2261\u02e2\u2192==\u02e2 Enc\u2080SizeRndInd\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dc7ee6eecdcbd16750848959161250e98cb8ee72","subject":"Lamport: renamings","message":"Lamport: renamings\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/Lamport.agda","new_file":"Crypto\/Sig\/Lamport.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH = hash-secret\n\n#signkey1   = OTS1.#signkey\n#verifkey1  = OTS1.#verifkey\n#signature1 = OTS1.#signature\n#signkey    = #message * #signkey1\n#verifkey   = #message * #verifkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #verifkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #signkey1 sk\n  vk   = verif-key sk\n  open verifkey vk public\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and (map OTS1.verify vk1s \u229b m \u229b sig1s)\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig = lemma\n  where\n    module lemma {#m} sk b m where\n      skL  = take #signkey1 sk\n      skH  = drop #signkey1 sk\n      vkL  = OTS1.verif-key skL\n      vkH  = map* #m OTS1.verif-key skH\n      sigL = OTS1.sign skL b\n      sigH = concat (map OTS1.sign (group #m #signkey1 skH) \u229b m)\n\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #verifkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                               (concat (map OTS1.sign (group #m #signkey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #verifkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH = hash-secret\n\n#seckey1    = 2* #secret\n#pubkey1    = 2* #digest\n#signature1 = #secret\n#seckey     = #message * #seckey1\n#pubkey     = #message * #pubkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #seckey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #pubkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #pubkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #seckey1 sk\n  vk   = verif-key sk\n  open verifkey vk public\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and (map OTS1.verify vk1s \u229b m \u229b sig1s)\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig = lemma\n  where\n    module lemma {#m} sk b m where\n      skL  = take #seckey1 sk\n      skH  = drop #seckey1 sk\n      vkL  = OTS1.verif-key skL\n      vkH  = map* #m OTS1.verif-key skH\n      sigL = OTS1.sign skL b\n      sigH = concat (map OTS1.sign (group #m #seckey1 skH) \u229b m)\n\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #pubkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                               (concat (map OTS1.sign (group #m #seckey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #pubkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #pubkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"30936886bb410fbf699a91bd25fbf51d2878e09f","subject":"Fix argument order for fromto\u2192valid","message":"Fix argument order for fromto\u2192valid\n\nAlso make some arguments explicit where it's going to be required soon.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 v1 v2 dv \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb v1 v2 dv x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) (da , db) (daa , dbb) = (fromto\u2192valid  _ _ _ daa) , (fromto\u2192valid _ _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} f1 f2 df dff = \u03bb a da ada \u2192\n  fromto\u2192valid _ _ _ (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto _ _ ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid _ _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 _ _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9806377d17063141d1be0d07fda712feb086d8b0","subject":"agda: clarify implementation of diff-term by adding pseudocode","message":"agda: clarify implementation of diff-term by adding pseudocode\n\nOld-commit-hash: 531f469c648998b9b89cff66da12e2dbc01a3aea\n","repos":"inc-lc\/ilc-agda","old_file":"Syntactic\/Changes.agda","new_file":"Syntactic\/Changes.agda","new_contents":"module Syntactic.Changes where\n\n-- TERMS that operate on CHANGES\n--\n-- This module defines terms that correspond to the basic change\n-- operations, as well as some other helper terms.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Changes\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\n-- Other problem: in fact, \u0394 t is not the nil change of t, in this context. That's a problem.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs (abs (var this xor-term var (that this)))\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) (diff-term (var this) (var this)))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n-- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb df f \u2192 app (app apply-pure-term df) f\napply-term {bool} = _xor-term_\n\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} =\n  \u03bb f\u2081 f\u2082 \u2192\n  -- The following can be written as:\n  -- * Using Unicode symbols for change operations:\n  -- \u03bb x dx. (f1 (dx \u229d x)) \u229d (f2 x)\n  -- * In lambda calculus with names:\n  --\u03bbx.  \u03bbdx. diff           (     f\u2081                   (apply      dx         x))                 (f\u2082                        x)\n  -- * As a deBrujin-encoded term in Agda:\n    abs (abs (diff-term {\u03c4\u2081} (app (weaken \u227c-in-body f\u2081) (apply-term (var this) (var (that this)))) (app (weaken \u227c-in-body f\u2082) (var (that this)))))\n  where\n    \u227c-in-body = drop (\u0394-Type \u03c4) \u2022 (drop \u03c4 \u2022 \u227c-refl)\n\ndiff-term {bool} = _xor-term_\n\n","old_contents":"module Syntactic.Changes where\n\n-- TERMS that operate on CHANGES\n--\n-- This module defines terms that correspond to the basic change\n-- operations, as well as some other helper terms.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Changes\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\n-- Other problem: in fact, \u0394 t is not the nil change of t, in this context. That's a problem.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs (abs (var this xor-term var (that this)))\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) (diff-term (var this) (var this)))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n-- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb df f \u2192 app (app apply-pure-term df) f\napply-term {bool} = _xor-term_\n\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} =\n  \u03bb f\u2081 f\u2082 \u2192\n  --\u03bbx.  \u03bbdx. diff           (     f\u2081                   (apply      dx         x))                 (f\u2082                        x)\n    abs (abs (diff-term {\u03c4\u2081} (app (weaken \u227c-in-body f\u2081) (apply-term (var this) (var (that this)))) (app (weaken \u227c-in-body f\u2082) (var (that this)))))\n  where\n    \u227c-in-body = drop (\u0394-Type \u03c4) \u2022 (drop \u03c4 \u2022 \u227c-refl)\n\ndiff-term {bool} = _xor-term_\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"23ba387dd8ddfc68c89da7cc893881061b5bd06a","subject":"Make the Adequacy module universe polymorphic","message":"Make the Adequacy module universe polymorphic\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Properties.agda","new_file":"lib\/Explore\/Properties.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- The core properties behind exploration functions\nmodule Explore.Properties where\n\nopen import Level.NP using (_\u2294_; \u2080; \u2081; \u209b; Lift)\nopen import Type using (\u2605\u2080; \u2605\u2081; \u2605_)\nopen import Function.NP using (id; _\u2218\u2032_; _\u2218_; flip; const; \u03a0; Cmp)\nopen import Algebra using (Semigroup; module Semigroup; Monoid; module Monoid; CommutativeMonoid; module CommutativeMonoid)\nimport      Algebra.FunctionProperties.NP\nimport      Algebra.FunctionProperties.Derived as FPD\nopen import Algebra.FunctionProperties.Core using (Op\u2082)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits using (Injective)\nopen import Data.Nat.NP using (_+_; _*_; _\u2264_; _+\u00b0_)\nopen import Data.Product using (\u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nopen import Data.Sum  using (_\u228e_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One  using (\ud835\udfd9)\nopen import Data.Two  using (\ud835\udfda; \u2713)\nopen import Data.Fin  using (Fin)\nopen import Relation.Binary.NP using (module Setoid-Reasoning; _Preserves\u2082_\u27f6_\u27f6_)\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; _\u2257_)\n\nopen import Explore.Core\n\nmodule FP = Algebra.FunctionProperties.NP \u03a0\n\nmodule SgrpExtra {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605 _\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 \u2605 _\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n  !_ = sym\n\nmodule Sgrp {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule RawMon {c} {C : \u2605 c} (rawMon : C \u00d7 Op\u2082 C) where\n  \u03b5    = proj\u2081 rawMon\n  _\u2219_  = proj\u2082 rawMon\n\nmodule Mon {c \u2113} (m : Monoid c \u2113) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  RawMon = C \u00d7 Op\u2082 C\n  rawMon : RawMon\n  rawMon = \u03b5 , _\u2219_\n\nmodule CMon {c \u2113} (cm : CommutativeMonoid c \u2113) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n  open FP _\u2248_\n\n  \u2219-interchange : Interchange _\u2219_ _\u2219_\n  \u2219-interchange = FPD.FromAssocCommCong.interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nmodule _ {\u2113 a} {A : \u2605 a} where\n    ExploreInd : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 (a \u2294 \u209b (\u2113 \u2294 p))\n    ExploreInd p exp =\n      \u2200 (P  : Explore \u2113 A \u2192 \u2605 p)\n        (P\u03b5 : P empty-explore)\n        (P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081))\n        (Pf : \u2200 x \u2192 P (point-explore x))\n      \u2192 P exp\n\n    module _ {p} where\n        point-explore-ind : (x : A) \u2192 ExploreInd p (point-explore x)\n        point-explore-ind x _ _ _ Pf = Pf x\n\n        empty-explore-ind : ExploreInd p empty-explore\n        empty-explore-ind _ P\u03b5 _ _ = P\u03b5\n\n        merge-explore-ind : \u2200 {e\u2080 e\u2081 : Explore \u2113 A}\n                            \u2192 ExploreInd p e\u2080 \u2192 ExploreInd p e\u2081\n                            \u2192 ExploreInd p (merge-explore e\u2080 e\u2081)\n        merge-explore-ind Pe\u2080 Pe\u2081 P P\u03b5 _P\u2219_ Pf = (Pe\u2080 P P\u03b5 _P\u2219_ Pf) P\u2219 (Pe\u2081 P P\u03b5 _P\u2219_ Pf)\n\n    ExploreInd\u2080 : Explore \u2113 A \u2192 \u2605 _\n    ExploreInd\u2080 = ExploreInd \u2080\n\n    ExploreInd\u2081 : Explore \u2113 A \u2192 \u2605 _\n    ExploreInd\u2081 = ExploreInd \u2081\n\n    BigOpMonInd : \u2200 p {c} (M : Monoid c \u2113) \u2192 BigOpMon M A \u2192 \u2605 _\n    BigOpMonInd p {c} M exp =\n      \u2200 (P  : BigOpMon M A \u2192 \u2605 p)\n        (P\u03b5 : P (const \u03b5))\n        (P\u2219 : \u2200 {e\u2080 e\u2081 : BigOpMon M A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (\u03bb f \u2192 e\u2080 f \u2219 e\u2081 f))\n        (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n        (P\u2248 : \u2200 {e e'} \u2192 e \u2248\u00b0 e' \u2192 P e \u2192 P e')\n      \u2192 P exp\n      where open Mon M\n\n    module _ (e\u1d2c : Explore {a} (\u209b \u2113) A) where\n        \u03a0\u1d49 : (A \u2192 \u2605 \u2113) \u2192 \u2605 \u2113\n        \u03a0\u1d49 = e\u1d2c (Lift \ud835\udfd9) _\u00d7_\n\n        \u03a3\u1d49 : (A \u2192 \u2605 \u2113) \u2192 \u2605 \u2113\n        \u03a3\u1d49 = e\u1d2c (Lift \ud835\udfd8) _\u228e_\n\nmodule _ {\u2113 a} {A : \u2605 a} where\n    module _ (e\u1d2c : Explore (\u209b \u2113) A) where\n        Lookup : \u2605 (\u209b \u2113 \u2294 a)\n        Lookup = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a0\u1d49 e\u1d2c P \u2192 \u03a0 A P\n\n        -- alternative name suggestion: tabulate\n        Reify : \u2605 (a \u2294 \u209b \u2113)\n        Reify = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a0 A P \u2192 \u03a0\u1d49 e\u1d2c P\n\n        Unfocus : \u2605 (a \u2294 \u209b \u2113)\n        Unfocus = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3\u1d49 e\u1d2c P \u2192 \u03a3 A P\n\n        -- alternative name suggestion: inject\n        Focus : \u2605 (a \u2294 \u209b \u2113)\n        Focus = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3 A P \u2192 \u03a3\u1d49 e\u1d2c P\n\n    Adequate-\u03a3 : ((A \u2192 \u2605 \u2113) \u2192 \u2605 _) \u2192 \u2605 _\n    Adequate-\u03a3 \u03a3\u1d2c = \u2200 F \u2192 \u03a3\u1d2c F \u2261 \u03a3 A F\n\n    Adequate-\u03a0 : ((A \u2192 \u2605 \u2113) \u2192 \u2605 _) \u2192 \u2605 _\n    Adequate-\u03a0 \u03a0\u1d2c = \u2200 F \u2192 \u03a0\u1d2c F \u2261 \u03a0 A F\n\n-- This module could be parameterised by the relation on types, here _\u2261_\nmodule Universal-Adequacy {\u2113u \u2113e \u2113r \u2113a}\n                          (U : \u2605_ \u2113u)(El : U \u2192 \u2605_ \u2113e)\n                          (_\u2248_ : \u2605_ \u2113e \u2192 \u2605_ (\u2113a \u2294 \u2113e) \u2192 \u2605_ \u2113r){A : \u2605_ \u2113a} where\n    Adequate-univ-sum : ((A \u2192 U) \u2192 U) \u2192 \u2605_ (\u2113a \u2294 (\u2113r \u2294 \u2113u))\n    Adequate-univ-sum sum\u1d2c = \u2200 f \u2192 El (sum\u1d2c f) \u2248 \u03a3 A (El \u2218 f)\n\n    Adequate-univ-product : ((A \u2192 U) \u2192 U) \u2192 \u2605_ (\u2113a \u2294 (\u2113r \u2294 \u2113u))\n    Adequate-univ-product product\u1d2c = \u2200 f \u2192 El (product\u1d2c f) \u2248 \u03a0 A (El \u2218 f)\n\nmodule Adequacy {\u2113a \u2113r}(_\u2248_ : \u2605\u2080 \u2192 \u2605_ \u2113a \u2192 \u2605_ \u2113r){A : \u2605_ \u2113a} where\n\n    -- Universal-Adequacy.Adequate-univ-sum \u2115 Fin _\u2261_\n    Adequate-sum : Sum A \u2192 \u2605_(\u2113a \u2294 \u2113r)\n    Adequate-sum sum\u1d2c = \u2200 f \u2192 Fin (sum\u1d2c f) \u2248 \u03a3 A (Fin \u2218 f)\n\n    -- Universal-Adequacy.Adequate-univ-product \u2115 Fin _\u2261_\n    Adequate-product : Product A \u2192 \u2605_(\u2113a \u2294 \u2113r)\n    Adequate-product product\u1d2c = \u2200 f \u2192 Fin (product\u1d2c f) \u2248 \u03a0 A (Fin \u2218 f)\n\n    -- Universal-Adequacy.Adequate-univ-product \ud835\udfda \u2713 _\u2261_\n    Adequate-any : (any : BigOp \ud835\udfda A) \u2192 \u2605_(\u2113a \u2294 \u2113r)\n    Adequate-any any\u1d2c = \u2200 f \u2192 \u2713 (any\u1d2c f) \u2248 \u03a3 A (\u2713 \u2218 f)\n\n    -- Universal-Adequacy.Adequate-univ-product \ud835\udfda \u2713 _\u2261_\n    Adequate-all : (all : BigOp \ud835\udfda A) \u2192 \u2605_(\u2113a \u2294 \u2113r)\n    Adequate-all all\u1d2c = \u2200 f \u2192 \u2713 (all\u1d2c f) \u2248 \u03a0 A (\u2713 \u2218 f)\n\nmodule _ {m a}{M : \u2605 m}{A : \u2605 a}([\u2295] : BigOp M A) where\n    BigOpStableUnder : (p : A \u2192 A) \u2192 \u2605 _\n    BigOpStableUnder p = \u2200 f \u2192 [\u2295] f \u2261 [\u2295] (f \u2218 p)\n\n    -- Extensionality of a big-operator\n    BigOp= : \u2605 _\n    BigOp= = {f g : A \u2192 M} \u2192 f \u2257 g \u2192 [\u2295] f \u2261 [\u2295] g\n\nmodule _ {\u2113 a} {A : \u2605 a} (e\u1d2c : Explore \u2113 A) where\n    StableUnder : (A \u2192 A) \u2192 \u2605 _\n    StableUnder p = \u2200 {M}(\u03b5 : M) op \u2192 BigOpStableUnder (e\u1d2c \u03b5 op) p\n\n    -- Extensionality of an exploration function\n    Explore= : \u2605 _\n    Explore= = \u2200 {M}(\u03b5 : M) op \u2192 BigOp= (e\u1d2c \u03b5 op)\n    ExploreExt = Explore=\n\nmodule _ {\u2113 a} {A : \u2605 a} r (e\u1d2c : Explore \u2113 A) where\n    ExploreMono : \u2605 _\n    ExploreMono = \u2200 {C} (_\u2286_ : C \u2192 C \u2192 \u2605 r)\n                    {z\u2080 z\u2081} (z\u2080\u2286z\u2081 : z\u2080 \u2286 z\u2081)\n                    {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                    {f g} \u2192\n                    (\u2200 x \u2192 f x \u2286 g x) \u2192 e\u1d2c z\u2080 _\u2219_ f \u2286 e\u1d2c z\u2081 _\u2219_ g\n\n    ExploreMonExt : \u2605 _\n    ExploreMonExt =\n      \u2200 (m : Monoid \u2113 r) {f g}\n      \u2192 let open Mon m\n            bigop = e\u1d2c \u03b5 _\u2219_\n        in\n        f \u2248\u00b0 g \u2192 bigop f \u2248 bigop g\n\n    Explore\u03b5 : \u2605 _\n    Explore\u03b5 = \u2200 (m : Monoid \u2113 r) \u2192\n                 let open Mon m in\n                 e\u1d2c \u03b5 _\u2219_ (const \u03b5) \u2248 \u03b5\n\n    ExploreLin\u02e1 : \u2605 _\n    ExploreLin\u02e1 =\n      \u2200 m _\u25ce_ f k \u2192\n        let open Mon {\u2113} {r} m\n            open FP _\u2248_ in\n        k \u25ce \u03b5 \u2248 \u03b5 \u2192\n        _\u25ce_ DistributesOver\u02e1 _\u2219_ \u2192\n        e\u1d2c \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e\u1d2c \u03b5 _\u2219_ f\n\n    ExploreLin\u02b3 : \u2605 _\n    ExploreLin\u02b3 =\n      \u2200 m _\u25ce_ f k \u2192\n        let open Mon {\u2113} {r} m\n            open FP _\u2248_ in\n        \u03b5 \u25ce k \u2248 \u03b5 \u2192\n        _\u25ce_ DistributesOver\u02b3 _\u2219_ \u2192\n        e\u1d2c \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e\u1d2c \u03b5 _\u2219_ f \u25ce k\n\n    ExploreHom : \u2605 _\n    ExploreHom =\n      \u2200 cm f g \u2192\n        let open CMon {\u2113} {r} cm in\n        e\u1d2c \u03b5 _\u2219_ (f \u2219\u00b0 g) \u2248 e\u1d2c \u03b5 _\u2219_ f \u2219 e\u1d2c \u03b5 _\u2219_ g\n\n        {-\n    ExploreSwap'' : \u2200 {b} \u2192 \u2605 _\n    ExploreSwap'' {b}\n                = \u2200 (monM : Monoid _) (monN : Monoid _) \u2192\n                    let module M = Mon {_} {r} monM in\n                    let module N = Mon {_} {r} monN in\n                  \u2200 {h : M.C \u2192 N.C}\n                    (h-\u03b5 : h M.\u03b5 \u2248 N.\u03b5)\n                    (h-\u2219 : \u2200 x y \u2192 h (x M.\u2219 y) \u2248 h x N.\u2219 h y)\n                    f\n                  \u2192 e\u1d2c \u03b5 _\u2219_ (h \u2218 f) \u2248 h (e\u1d2c \u03b5 _\u2219_ f)\n-}\n\n-- derived from lift-hom with the source monoid being (a \u2192 m)\n    ExploreSwap : \u2200 {b} \u2192 \u2605 _\n    ExploreSwap {b}\n                = \u2200 {B : \u2605 b} mon \u2192\n                    let open Mon {_} {r} mon in\n                  \u2200 {op\u1d2e : (B \u2192 C) \u2192 C}\n                    (op\u1d2e-\u03b5 : op\u1d2e (const \u03b5) \u2248 \u03b5)\n                    (hom : \u2200 f g \u2192 op\u1d2e (f \u2219\u00b0 g) \u2248 op\u1d2e f \u2219 op\u1d2e g)\n                    f\n                  \u2192 e\u1d2c \u03b5 _\u2219_ (op\u1d2e \u2218 f) \u2248 op\u1d2e (e\u1d2c \u03b5 _\u2219_ \u2218 flip f)\n\nmodule _ {a} {A : \u2605 a} (sum\u1d2c : Sum A) where\n    SumStableUnder : (A \u2192 A) \u2192 \u2605 _\n    SumStableUnder p = \u2200 f \u2192 sum\u1d2c f \u2261 sum\u1d2c (f \u2218 p)\n\n    SumStableUnderInjection : \u2605 _\n    SumStableUnderInjection = \u2200 p \u2192 Injective p \u2192 SumStableUnder p\n\n    SumInd : \u2605(\u2081 \u2294 a)\n    SumInd = \u2200 (P  : Sum A \u2192 \u2605\u2080)\n               (P0 : P (\u03bb f \u2192 0))\n               (P+ : \u2200 {s\u2080 s\u2081 : Sum A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f))\n               (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n             \u2192 P sum\u1d2c\n\n    SumExt : \u2605 _\n    SumExt = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\n    SumZero : \u2605 _\n    SumZero = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\n    SumLin : \u2605 _\n    SumLin = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\n    SumHom : \u2605 _\n    SumHom = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\n    SumMono : \u2605 _\n    SumMono = \u2200 {f g} \u2192 (\u2200 x \u2192 f x \u2264 g x) \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\n    SumConst : \u2605 _\n    SumConst = \u2200 x \u2192 sum\u1d2c (const x) \u2261 sum\u1d2c (const 1) * x\n\n    SumSwap : \u2605 _\n    SumSwap = \u2200 {B : \u2605\u2080}\n                {sum\u1d2e : Sum B}\n                (sum\u1d2e-0 : sum\u1d2e (const 0) \u2261 0)\n                (hom : \u2200 f g \u2192 sum\u1d2e (f +\u00b0 g) \u2261 sum\u1d2e f + sum\u1d2e g)\n                f\n              \u2192 sum\u1d2c (sum\u1d2e \u2218 f) \u2261 sum\u1d2e (sum\u1d2c \u2218 flip f)\n\nmodule _ {a} {A : \u2605 a} (count\u1d2c : Count A) where\n    CountStableUnder : (A \u2192 A) \u2192 \u2605 _\n    CountStableUnder p = \u2200 f \u2192 count\u1d2c f \u2261 count\u1d2c (f \u2218 p)\n\n    CountExt : \u2605 _\n    CountExt = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\n    Unique : Cmp A \u2192 \u2605 _\n    Unique cmp = \u2200 x \u2192 count\u1d2c (cmp x) \u2261 1\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- The core properties behind exploration functions\nmodule Explore.Properties where\n\nopen import Level.NP using (_\u2294_; \u2080; \u2081; \u209b; Lift)\nopen import Type using (\u2605\u2080; \u2605\u2081; \u2605_)\nopen import Function.NP using (id; _\u2218\u2032_; _\u2218_; flip; const; \u03a0; Cmp)\nopen import Algebra using (Semigroup; module Semigroup; Monoid; module Monoid; CommutativeMonoid; module CommutativeMonoid)\nimport      Algebra.FunctionProperties.NP\nimport      Algebra.FunctionProperties.Derived as FPD\nopen import Algebra.FunctionProperties.Core using (Op\u2082)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits using (Injective)\nopen import Data.Nat.NP using (_+_; _*_; _\u2264_; _+\u00b0_)\nopen import Data.Product using (\u03a3; _\u00d7_; _,_; proj\u2081; proj\u2082)\nopen import Data.Sum  using (_\u228e_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One  using (\ud835\udfd9)\nopen import Data.Two  using (\ud835\udfda; \u2713)\nopen import Data.Fin  using (Fin)\nopen import Relation.Binary.NP using (module Setoid-Reasoning; _Preserves\u2082_\u27f6_\u27f6_)\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; _\u2257_)\n\nopen import Explore.Core\n\nmodule FP = Algebra.FunctionProperties.NP \u03a0\n\nmodule SgrpExtra {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605 _\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 \u2605 _\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {a} {A : \u2605 a} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n  !_ = sym\n\nmodule Sgrp {c \u2113} (sg : Semigroup c \u2113) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule RawMon {c} {C : \u2605 c} (rawMon : C \u00d7 Op\u2082 C) where\n  \u03b5    = proj\u2081 rawMon\n  _\u2219_  = proj\u2082 rawMon\n\nmodule Mon {c \u2113} (m : Monoid c \u2113) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  RawMon = C \u00d7 Op\u2082 C\n  rawMon : RawMon\n  rawMon = \u03b5 , _\u2219_\n\nmodule CMon {c \u2113} (cm : CommutativeMonoid c \u2113) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n  open FP _\u2248_\n\n  \u2219-interchange : Interchange _\u2219_ _\u2219_\n  \u2219-interchange = FPD.FromAssocCommCong.interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nmodule _ {\u2113 a} {A : \u2605 a} where\n    ExploreInd : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 (a \u2294 \u209b (\u2113 \u2294 p))\n    ExploreInd p exp =\n      \u2200 (P  : Explore \u2113 A \u2192 \u2605 p)\n        (P\u03b5 : P empty-explore)\n        (P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081))\n        (Pf : \u2200 x \u2192 P (point-explore x))\n      \u2192 P exp\n\n    module _ {p} where\n        point-explore-ind : (x : A) \u2192 ExploreInd p (point-explore x)\n        point-explore-ind x _ _ _ Pf = Pf x\n\n        empty-explore-ind : ExploreInd p empty-explore\n        empty-explore-ind _ P\u03b5 _ _ = P\u03b5\n\n        merge-explore-ind : \u2200 {e\u2080 e\u2081 : Explore \u2113 A}\n                            \u2192 ExploreInd p e\u2080 \u2192 ExploreInd p e\u2081\n                            \u2192 ExploreInd p (merge-explore e\u2080 e\u2081)\n        merge-explore-ind Pe\u2080 Pe\u2081 P P\u03b5 _P\u2219_ Pf = (Pe\u2080 P P\u03b5 _P\u2219_ Pf) P\u2219 (Pe\u2081 P P\u03b5 _P\u2219_ Pf)\n\n    ExploreInd\u2080 : Explore \u2113 A \u2192 \u2605 _\n    ExploreInd\u2080 = ExploreInd \u2080\n\n    ExploreInd\u2081 : Explore \u2113 A \u2192 \u2605 _\n    ExploreInd\u2081 = ExploreInd \u2081\n\n    BigOpMonInd : \u2200 p {c} (M : Monoid c \u2113) \u2192 BigOpMon M A \u2192 \u2605 _\n    BigOpMonInd p {c} M exp =\n      \u2200 (P  : BigOpMon M A \u2192 \u2605 p)\n        (P\u03b5 : P (const \u03b5))\n        (P\u2219 : \u2200 {e\u2080 e\u2081 : BigOpMon M A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (\u03bb f \u2192 e\u2080 f \u2219 e\u2081 f))\n        (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n        (P\u2248 : \u2200 {e e'} \u2192 e \u2248\u00b0 e' \u2192 P e \u2192 P e')\n      \u2192 P exp\n      where open Mon M\n\n    module _ (e\u1d2c : Explore {a} (\u209b \u2113) A) where\n        \u03a0\u1d49 : (A \u2192 \u2605 \u2113) \u2192 \u2605 \u2113\n        \u03a0\u1d49 = e\u1d2c (Lift \ud835\udfd9) _\u00d7_\n\n        \u03a3\u1d49 : (A \u2192 \u2605 \u2113) \u2192 \u2605 \u2113\n        \u03a3\u1d49 = e\u1d2c (Lift \ud835\udfd8) _\u228e_\n\nmodule _ {\u2113 a} {A : \u2605 a} where\n    module _ (e\u1d2c : Explore (\u209b \u2113) A) where\n        Lookup : \u2605 (\u209b \u2113 \u2294 a)\n        Lookup = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a0\u1d49 e\u1d2c P \u2192 \u03a0 A P\n\n        -- alternative name suggestion: tabulate\n        Reify : \u2605 (a \u2294 \u209b \u2113)\n        Reify = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a0 A P \u2192 \u03a0\u1d49 e\u1d2c P\n\n        Unfocus : \u2605 (a \u2294 \u209b \u2113)\n        Unfocus = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3\u1d49 e\u1d2c P \u2192 \u03a3 A P\n\n        -- alternative name suggestion: inject\n        Focus : \u2605 (a \u2294 \u209b \u2113)\n        Focus = \u2200 {P : A \u2192 \u2605 \u2113} \u2192 \u03a3 A P \u2192 \u03a3\u1d49 e\u1d2c P\n\n    Adequate-\u03a3 : ((A \u2192 \u2605 \u2113) \u2192 \u2605 _) \u2192 \u2605 _\n    Adequate-\u03a3 \u03a3\u1d2c = \u2200 F \u2192 \u03a3\u1d2c F \u2261 \u03a3 A F\n\n    Adequate-\u03a0 : ((A \u2192 \u2605 \u2113) \u2192 \u2605 _) \u2192 \u2605 _\n    Adequate-\u03a0 \u03a0\u1d2c = \u2200 F \u2192 \u03a0\u1d2c F \u2261 \u03a0 A F\n\n-- This module could be parameterised by the relation on types, here _\u2261_\nmodule Universal-Adequacy {\u2113u \u2113e \u2113r \u2113a}\n                          (U : \u2605_ \u2113u)(El : U \u2192 \u2605_ \u2113e)\n                          (_\u2248_ : \u2605_ \u2113e \u2192 \u2605_ (\u2113a \u2294 \u2113e) \u2192 \u2605_ \u2113r){A : \u2605_ \u2113a} where\n    Adequate-univ-sum : ((A \u2192 U) \u2192 U) \u2192 \u2605_ (\u2113a \u2294 (\u2113r \u2294 \u2113u))\n    Adequate-univ-sum sum\u1d2c = \u2200 f \u2192 El (sum\u1d2c f) \u2248 \u03a3 A (El \u2218 f)\n\n    Adequate-univ-product : ((A \u2192 U) \u2192 U) \u2192 \u2605_ (\u2113a \u2294 (\u2113r \u2294 \u2113u))\n    Adequate-univ-product product\u1d2c = \u2200 f \u2192 El (product\u1d2c f) \u2248 \u03a0 A (El \u2218 f)\n\nmodule Adequacy {\u2113r}(_\u2248_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605_ \u2113r){A : \u2605\u2080} where\n\n    -- Universal-Adequacy.Adequate-univ-sum \u2115 Fin _\u2261_\n    Adequate-sum : Sum A \u2192 \u2605_ \u2113r\n    Adequate-sum sum\u1d2c = \u2200 f \u2192 Fin (sum\u1d2c f) \u2248 \u03a3 A (Fin \u2218 f)\n\n    -- Universal-Adequacy.Adequate-univ-product \u2115 Fin _\u2261_\n    Adequate-product : Product A \u2192 \u2605_ \u2113r\n    Adequate-product product\u1d2c = \u2200 f \u2192 Fin (product\u1d2c f) \u2248 \u03a0 A (Fin \u2218 f)\n\n    -- Universal-Adequacy.Adequate-univ-product \ud835\udfda \u2713 _\u2261_\n    Adequate-any : (any : BigOp \ud835\udfda A) \u2192 \u2605_ \u2113r\n    Adequate-any any\u1d2c = \u2200 f \u2192 \u2713 (any\u1d2c f) \u2248 \u03a3 A (\u2713 \u2218 f)\n\n    -- Universal-Adequacy.Adequate-univ-product \ud835\udfda \u2713 _\u2261_\n    Adequate-all : (all : BigOp \ud835\udfda A) \u2192 \u2605_ \u2113r\n    Adequate-all all\u1d2c = \u2200 f \u2192 \u2713 (all\u1d2c f) \u2248 \u03a0 A (\u2713 \u2218 f)\n\nmodule _ {m a}{M : \u2605 m}{A : \u2605 a}([\u2295] : BigOp M A) where\n    BigOpStableUnder : (p : A \u2192 A) \u2192 \u2605 _\n    BigOpStableUnder p = \u2200 f \u2192 [\u2295] f \u2261 [\u2295] (f \u2218 p)\n\n    -- Extensionality of a big-operator\n    BigOp= : \u2605 _\n    BigOp= = {f g : A \u2192 M} \u2192 f \u2257 g \u2192 [\u2295] f \u2261 [\u2295] g\n\nmodule _ {\u2113 a} {A : \u2605 a} (e\u1d2c : Explore \u2113 A) where\n    StableUnder : (A \u2192 A) \u2192 \u2605 _\n    StableUnder p = \u2200 {M}(\u03b5 : M) op \u2192 BigOpStableUnder (e\u1d2c \u03b5 op) p\n\n    -- Extensionality of an exploration function\n    Explore= : \u2605 _\n    Explore= = \u2200 {M}(\u03b5 : M) op \u2192 BigOp= (e\u1d2c \u03b5 op)\n    ExploreExt = Explore=\n\nmodule _ {\u2113 a} {A : \u2605 a} r (e\u1d2c : Explore \u2113 A) where\n    ExploreMono : \u2605 _\n    ExploreMono = \u2200 {C} (_\u2286_ : C \u2192 C \u2192 \u2605 r)\n                    {z\u2080 z\u2081} (z\u2080\u2286z\u2081 : z\u2080 \u2286 z\u2081)\n                    {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                    {f g} \u2192\n                    (\u2200 x \u2192 f x \u2286 g x) \u2192 e\u1d2c z\u2080 _\u2219_ f \u2286 e\u1d2c z\u2081 _\u2219_ g\n\n    ExploreMonExt : \u2605 _\n    ExploreMonExt =\n      \u2200 (m : Monoid \u2113 r) {f g}\n      \u2192 let open Mon m\n            bigop = e\u1d2c \u03b5 _\u2219_\n        in\n        f \u2248\u00b0 g \u2192 bigop f \u2248 bigop g\n\n    Explore\u03b5 : \u2605 _\n    Explore\u03b5 = \u2200 (m : Monoid \u2113 r) \u2192\n                 let open Mon m in\n                 e\u1d2c \u03b5 _\u2219_ (const \u03b5) \u2248 \u03b5\n\n    ExploreLin\u02e1 : \u2605 _\n    ExploreLin\u02e1 =\n      \u2200 m _\u25ce_ f k \u2192\n        let open Mon {\u2113} {r} m\n            open FP _\u2248_ in\n        k \u25ce \u03b5 \u2248 \u03b5 \u2192\n        _\u25ce_ DistributesOver\u02e1 _\u2219_ \u2192\n        e\u1d2c \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e\u1d2c \u03b5 _\u2219_ f\n\n    ExploreLin\u02b3 : \u2605 _\n    ExploreLin\u02b3 =\n      \u2200 m _\u25ce_ f k \u2192\n        let open Mon {\u2113} {r} m\n            open FP _\u2248_ in\n        \u03b5 \u25ce k \u2248 \u03b5 \u2192\n        _\u25ce_ DistributesOver\u02b3 _\u2219_ \u2192\n        e\u1d2c \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e\u1d2c \u03b5 _\u2219_ f \u25ce k\n\n    ExploreHom : \u2605 _\n    ExploreHom =\n      \u2200 cm f g \u2192\n        let open CMon {\u2113} {r} cm in\n        e\u1d2c \u03b5 _\u2219_ (f \u2219\u00b0 g) \u2248 e\u1d2c \u03b5 _\u2219_ f \u2219 e\u1d2c \u03b5 _\u2219_ g\n\n        {-\n    ExploreSwap'' : \u2200 {b} \u2192 \u2605 _\n    ExploreSwap'' {b}\n                = \u2200 (monM : Monoid _) (monN : Monoid _) \u2192\n                    let module M = Mon {_} {r} monM in\n                    let module N = Mon {_} {r} monN in\n                  \u2200 {h : M.C \u2192 N.C}\n                    (h-\u03b5 : h M.\u03b5 \u2248 N.\u03b5)\n                    (h-\u2219 : \u2200 x y \u2192 h (x M.\u2219 y) \u2248 h x N.\u2219 h y)\n                    f\n                  \u2192 e\u1d2c \u03b5 _\u2219_ (h \u2218 f) \u2248 h (e\u1d2c \u03b5 _\u2219_ f)\n-}\n\n-- derived from lift-hom with the source monoid being (a \u2192 m)\n    ExploreSwap : \u2200 {b} \u2192 \u2605 _\n    ExploreSwap {b}\n                = \u2200 {B : \u2605 b} mon \u2192\n                    let open Mon {_} {r} mon in\n                  \u2200 {op\u1d2e : (B \u2192 C) \u2192 C}\n                    (op\u1d2e-\u03b5 : op\u1d2e (const \u03b5) \u2248 \u03b5)\n                    (hom : \u2200 f g \u2192 op\u1d2e (f \u2219\u00b0 g) \u2248 op\u1d2e f \u2219 op\u1d2e g)\n                    f\n                  \u2192 e\u1d2c \u03b5 _\u2219_ (op\u1d2e \u2218 f) \u2248 op\u1d2e (e\u1d2c \u03b5 _\u2219_ \u2218 flip f)\n\nmodule _ {a} {A : \u2605 a} (sum\u1d2c : Sum A) where\n    SumStableUnder : (A \u2192 A) \u2192 \u2605 _\n    SumStableUnder p = \u2200 f \u2192 sum\u1d2c f \u2261 sum\u1d2c (f \u2218 p)\n\n    SumStableUnderInjection : \u2605 _\n    SumStableUnderInjection = \u2200 p \u2192 Injective p \u2192 SumStableUnder p\n\n    SumInd : \u2605(\u2081 \u2294 a)\n    SumInd = \u2200 (P  : Sum A \u2192 \u2605\u2080)\n               (P0 : P (\u03bb f \u2192 0))\n               (P+ : \u2200 {s\u2080 s\u2081 : Sum A} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f))\n               (Pf : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n             \u2192 P sum\u1d2c\n\n    SumExt : \u2605 _\n    SumExt = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\n    SumZero : \u2605 _\n    SumZero = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\n    SumLin : \u2605 _\n    SumLin = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\n    SumHom : \u2605 _\n    SumHom = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\n    SumMono : \u2605 _\n    SumMono = \u2200 {f g} \u2192 (\u2200 x \u2192 f x \u2264 g x) \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\n    SumConst : \u2605 _\n    SumConst = \u2200 x \u2192 sum\u1d2c (const x) \u2261 sum\u1d2c (const 1) * x\n\n    SumSwap : \u2605 _\n    SumSwap = \u2200 {B : \u2605\u2080}\n                {sum\u1d2e : Sum B}\n                (sum\u1d2e-0 : sum\u1d2e (const 0) \u2261 0)\n                (hom : \u2200 f g \u2192 sum\u1d2e (f +\u00b0 g) \u2261 sum\u1d2e f + sum\u1d2e g)\n                f\n              \u2192 sum\u1d2c (sum\u1d2e \u2218 f) \u2261 sum\u1d2e (sum\u1d2c \u2218 flip f)\n\nmodule _ {a} {A : \u2605 a} (count\u1d2c : Count A) where\n    CountStableUnder : (A \u2192 A) \u2192 \u2605 _\n    CountStableUnder p = \u2200 f \u2192 count\u1d2c f \u2261 count\u1d2c (f \u2218 p)\n\n    CountExt : \u2605 _\n    CountExt = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\n    Unique : Cmp A \u2192 \u2605 _\n    Unique cmp = \u2200 x \u2192 count\u1d2c (cmp x) \u2261 1\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e0c37498879bfcc4052f0ed3094c8354f6118a11","subject":"Desc model: rename iso2 to psi.","message":"Desc model: rename iso2 to psi.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : (I : Set) -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi I (var i) = refl\nproof-phi-psi I (const x) = refl\nproof-phi-psi I (prod D D') with proof-phi-psi I D | proof-phi-psi I D'\n...                             | p | q = cong2 prod p q\nproof-phi-psi I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi I (T s)))\nproof-phi-psi I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                              T\n                                              (\\s -> proof-phi-psi I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> phi (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (phi D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (phi D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9df1056695e49ba8d199081681450dbf409ca429","subject":"Rename lift-apply to apply-term.","message":"Rename lift-apply to apply-term.\n\nOld-commit-hash: b02f9113ba8fab93ce132d5a8abe88b8d4ce471c\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x)\n      ,\n      abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  diff-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  diff-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  apply-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  apply-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x)\n      ,\n      abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  diff-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  diff-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 lift-apply\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"07ee99089637bbb5349f32952f8c881c684fd40f","subject":"Boring.","message":"Boring.\n\nIgnore-this: 388f14f883c989b9b05f0318e6a8750\n\ndarcs-hash:20100509150614-3bd4e-798fd3f93aeb4bf23f39bbb36ed0147ae3c78f00.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/OnlyAxioms\/EraseQuantification.agda","new_file":"Test\/Succeed\/OnlyAxioms\/EraseQuantification.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the removal of the quantification\n------------------------------------------------------------------------------\n\nmodule Test.Succeed.OnlyAxioms.EraseQuantification where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  succ : D \u2192 D\n\nmodule EraseQuantificationOnSet where\n\n  postulate a : D\n\n  -- The translation must erase the quantification on Set (i.e. {A : Set}),\n  -- the translation must be 'a = a'.\n  postulate foo : {A : Set} \u2192 a \u2261 a\n  {-# ATP axiom foo #-}\n\nmodule EraseQuantificationOnProofs where\n\n  -- The data constructors sN\u2081 and sN\u2082 must have the same translation,\n  -- i.e. we must erase the quantification of the variable Nn on N n.\n  -- The translation of sN\u2082 must be '\u2200 x. n(x) \u2192 n(succ(x))\n\n  data N : D \u2192 Set where\n    -- zN : N zero\n    sN\u2081 : {n : D} \u2192  N n \u2192 N (succ n)\n    sN\u2082 : {n : D} \u2192 (Nn : N n) \u2192 N (succ n)\n  {-# ATP hint sN\u2082 #-}\n\n  -- We need to remove the quantification over proofs inside a where\n  -- clause. The translation of P must be '\u2200 x. \u2200 y. p(x, y) \u2194 y = x',\n  -- i.e. we must erase the quantification on N n.\n  foo : (n : D) \u2192 N n \u2192 Set\n  foo n Nn = P n\n    where\n      P : D \u2192 Set\n      P m = m \u2261 n\n      {-# ATP definition P #-}\n\n-- module EraseQuantificationOverFunctionsAndPiTerm\n\n-- -- Possible test\n\n-- data A : Set where\n--   a : A\n\n-- foo : {e : D } \u2192 (N e \u2192 D) \u2192 A \u2192 e \u2261 e\n-- foo {e} fn a = prf\n--   where\n--   postulate prf : e \u2261 e\n--   {-# ATP prove prf #-}\n\n-- bar : {e : D } \u2192 ((d : D) \u2192 N d \u2192 D) \u2192 A \u2192 e \u2261 e\n-- bar {e} fn a = prf\n--   where\n--   postulate prf : e \u2261 e\n--   {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the removal of the quantification\n------------------------------------------------------------------------------\n\nmodule Test.Succeed.OnlyAxioms.EraseQuantification where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  succ : D \u2192 D\n\nmodule EraseQuantificationOnSet where\n\n  postulate a : D\n\n  -- The translation must erase the quantification on Set (i.e. {A : Set}),\n  -- the translation must be 'a = a'.\n  postulate foo : {A : Set} \u2192 a \u2261 a\n  {-# ATP axiom foo #-}\n\nmodule EraseQuantificationOverProofs where\n\n  -- The data constructors sN\u2081 and sN\u2082 must have the same translation,\n  -- i.e. we must erase the quantification of the variable Nn on N n.\n  -- The translation of sN\u2082 must be '\u2200 x. n(x) \u2192 n(succ(x))\n\n  data N : D \u2192 Set where\n    -- zN : N zero\n    sN\u2081 : {n : D} \u2192  N n \u2192 N (succ n)\n    sN\u2082 : {n : D} \u2192 (Nn : N n) \u2192 N (succ n)\n  {-# ATP hint sN\u2082 #-}\n\n\n  -- We need to remove the quantification over proofs inside a where\n  -- clause. The translation of P must be '\u2200 x. \u2200 y. p(x, y) \u2194 y = x',\n  -- i.e. we must erase the quantification on N n.\n  foo : (n : D) \u2192 N n \u2192 Set\n  foo n Nn = P n\n    where\n      P : D \u2192 Set\n      P m = m \u2261 n\n      {-# ATP definition P #-}\n\n-- module EraseQuantificationOverFunctionsAndPiTerm\n\n-- -- Possible test\n\n-- data A : Set where\n--   a : A\n\n-- foo : {e : D } \u2192 (N e \u2192 D) \u2192 A \u2192 e \u2261 e\n-- foo {e} fn a = prf\n--   where\n--   postulate prf : e \u2261 e\n--   {-# ATP prove prf #-}\n\n-- bar : {e : D } \u2192 ((d : D) \u2192 N d \u2192 D) \u2192 A \u2192 e \u2261 e\n-- bar {e} fn a = prf\n--   where\n--   postulate prf : e \u2261 e\n--   {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"deef53c65c59080266e5b542fcf378f75dbc5a65","subject":"Added doc: We couldn't prove Conat-coind.","message":"Added doc: We couldn't prove Conat-coind.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/TypeSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\n-- TODO (06 January 2014). We couldn't prove Conat-coind.\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h {n} An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\npostulate\n  inf    : D\n  inf-eq : inf \u2261 succ\u2081 inf\n{-# ATP axiom inf-eq #-}\n\n{-# NO_TERMINATION_CHECK #-}\ninf-Conat : Conat inf\ninf-Conat = subst Conat (sym inf-eq) (cosucc (\u266f inf-Conat))\n","old_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.TypeSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out cozero          = inj\u2081 refl\nConat-out (cosucc {n} Cn) = inj\u2082 (n , refl , \u266d Cn)\n\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in (inj\u2081 n\u22610)              = subst Conat (sym n\u22610) cozero\nConat-in (inj\u2082 (n' , prf , Cn')) = subst Conat (sym prf) (cosucc (\u266f Cn'))\n\nConat-coind : (A : D \u2192 Set) \u2192\n              (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n              \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind A h {n} An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\npostulate\n  inf    : D\n  inf-eq : inf \u2261 succ\u2081 inf\n{-# ATP axiom inf-eq #-}\n\n{-# NO_TERMINATION_CHECK #-}\ninf-Conat : Conat inf\ninf-Conat = subst Conat (sym inf-eq) (cosucc (\u266f inf-Conat))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a6476334a09b39aec3077a9122d4f7f8ca511173","subject":"README.agda: commit the version generated without YAML","message":"README.agda: commit the version generated without YAML\n\nOld-commit-hash: 50638b3bf74b27250b21099c86b61824c9113a06\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\n{- Terms of a calculus described in Plotkin style\n- types are parametric in base types\n- terms are parametric in constants\nThis style of language description is employed in:\nG. D. Plotkin. \"LCF considered as a programming language.\"\nTheoretical Computer Science 5(3) pp. 223--255, 1997.\nhttp:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5\n -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a1b1e5915214625b81e1399a850d92f16cf11d05","subject":"Delete copy-n-pasted-by-mistake comment","message":"Delete copy-n-pasted-by-mistake comment\n\nThis comes directly from Relation.Binary.PropositionalEquality, and is\ntotally irrelevant here.\n\nOld-commit-hash: d8c5accf1966a913db36fefe149720366811965c\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n{-\n      lemma : nil f \u2259 dg\n      -- Goes through\n      --lemma = sym (derivative-is-nil dg fdg)\n      -- Generates tons of crazy yellow ambiguities of equality type. Apparently, unifying against \u2259 does not work so well.\n      lemma = \u2259-symm {{changeAlgebra}} {f} {dg} {nil f} (derivative-is-nil dg fdg)\n-}\n\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  -- Relation.Binary.EqReasoning is more convenient to use with _\u2261_ if\n  -- the combinators take the type argument (a) as a hidden argument,\n  -- instead of being locked to a fixed type at module instantiation\n  -- time.\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\n  -- This results pairs with update-diff.\n  diff-update : \u2200 {dx} \u2192 (x \u229e dx) \u229f x \u2259 dx\n  diff-update {dx} = lemma\n    where\n      lemma : x \u229e (x \u229e dx \u229f x) \u2261 x \u229e dx\n      lemma = update-diff (x \u229e dx) x\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n{-\n      lemma : nil f \u2259 dg\n      -- Goes through\n      --lemma = sym (derivative-is-nil dg fdg)\n      -- Generates tons of crazy yellow ambiguities of equality type. Apparently, unifying against \u2259 does not work so well.\n      lemma = \u2259-symm {{changeAlgebra}} {f} {dg} {nil f} (derivative-is-nil dg fdg)\n-}\n\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4b7484be14712a8127872b15fea304c655011a93","subject":"Level: \u2115\u2192Level","message":"Level: \u2115\u2192Level\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Level\/NP.agda","new_file":"lib\/Level\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Level.NP where\n\nimport Level\nopen import Data.Nat\n\nopen Level public renaming (zero to \u2080; suc to \u209b)\n\n\u2081 \u2082 \u2083 \u2084 : Level\n\u2081 = \u209b \u2080\n\u2082 = \u209b \u2081\n\u2083 = \u209b \u2082\n\u2084 = \u209b \u2083\n\n\u2115\u2192Level : \u2115 \u2192 Level\n\u2115\u2192Level zero    = \u2080\n\u2115\u2192Level (suc n) = \u209b (\u2115\u2192Level n)\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Level.NP where\n\nimport Level\n\nopen Level public renaming (zero to \u2080; suc to \u209b)\n\n\u2081 \u2082 \u2083 \u2084 : Level\n\u2081 = \u209b \u2080\n\u2082 = \u209b \u2081\n\u2083 = \u209b \u2082\n\u2084 = \u209b \u2083\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"93bade3cafd677ea87a758695671d5a909c92d06","subject":"We only need one point and decidability on objects.","message":"We only need one point and decidability on objects.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DCBSets.agda","new_file":"dialectica-cats\/DCBSets.agda","new_contents":"module DCBSets where\n\nopen import prelude\nopen import relations\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (\u03a3[ \u03b1 \u2208 (U \u2192 X \u2192 Set) ]( \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x) ))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , \u03b1 , _) (V , Y , _ , \u03b2 , _) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , n\u2081 , \u03b1 , _)} {(V , Y , n\u2082 , \u03b2 , _)} {(W , Z , n\u2083 , \u03b3 , _)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , n , \u03b1 , _)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , \u03b1 , _)}{(V , Y , _ , \u03b2 , _)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , \u03b1 , _}\n         {V , Y , _ , \u03b2 , _}\n         {W , Z , _ , \u03b3 , _}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , \u03b1 , _}{V , Y , _ , \u03b2 , _}{W , Z , _ , \u03b3 , _}{S , T , _ , \u03b9 , _}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n\u2297d : \u2200{U V X Y}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2  : V \u2192 Y \u2192 Set}{d\u2081 : (u : U) (x : X) \u2192 Dec (\u03b1 u x)}{d\u2082 : (u : V) (x : Y) \u2192 Dec (\u03b2 u x)} \u2192 (u : \u03a3 U (\u03bb x \u2192 V)) (x : \u03a3 X (\u03bb x\u2081 \u2192 Y)) \u2192 Dec ((\u03b1 \u2297\u1d63 \u03b2) u x)\n\u2297d {U}{V}{X}{Y}{\u03b1}{\u03b2}{d\u2081}{d\u2082} (u , v) (x , y) with d\u2081 u x | d\u2082 v y\n... | yes z | yes z' = yes (z , z')\n... | yes z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n... | no z | yes z' = no (\u03bb z'' \u2192 z (fst z''))\n... | no z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2 , \u2297d {\u03b1 = \u03b1} {\u03b2}{d\u2081}{d\u2082})\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , \u03b1 , _)}{(V , Y , _ , \u03b2 , _)}{(W , Z , _ , \u03b3 , _)}{(S , T , _ , \u03b4 , _)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}\n    \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n    \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}\n    \u2192 Hom ((U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082)) (U , X , n\u2081 , \u03b1 , d\u2081)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}\n    \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n    \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}    \n    \u2192 Hom ((U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082)) (V , Y , n\u2082 , \u03b2 , d\u2082)\n\u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n\ncart-ar : {U X V Y W Z : Set}\n  \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n  \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n  \u2192 {n\u2081 : \u22a4 \u2192 X}\n  \u2192 {n\u2082 : \u22a4 \u2192 Y}\n  \u2192 {n\u2083 : \u22a4 \u2192 Z}\n  \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n  \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}\n  \u2192 {d\u2083 : \u2200(w : W)(z : Z) \u2192 Dec (\u03b3 w z)}  \n  \u2192 Hom (W , Z , n\u2083 , \u03b3 , d\u2083) (U , X , n\u2081 , \u03b1 , d\u2081)\n  \u2192 Hom (W , Z , n\u2083 , \u03b3 , d\u2083) (V , Y , n\u2082 , \u03b2 , d\u2082)\n  \u2192 Hom (W , Z , n\u2083 , \u03b3 , d\u2083) ((U , X , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , n\u2082 , \u03b2 , d\u2082))\ncart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3}{n\u2081}{n\u2082}{n\u2083}{d\u2081}{d\u2082}{d\u2083} (f , F , p\u2081) (g , G , p\u2082) = trans-\u00d7 f g ,  cart-ar-d , cond\n where\n   cart-ar-d : W \u2192 \u03a3 X (\u03bb x \u2192 Y) \u2192 Z\n   cart-ar-d w (x , y) with d\u2082 (g w) y\n   ... | yes t = F w x\n   ... | no t = G w y\n\n   cond : {u : W} {y : \u03a3 X (\u03bb x \u2192 Y)} \u2192 \u03b3 u (cart-ar-d u y) \u2192 (\u03b1 \u2297\u1d63 \u03b2) (f u , g u) y\n   cond {w}{x , y} p\u2083 with d\u2082 (g w) y\n   ... | yes t = p\u2081 p\u2083 , t\n   ... | no t = \u22a5-elim (t (p\u2082 p\u2083)) , p\u2082  p\u2083\n\n\n","old_contents":"module DCBSets where\n\nopen import prelude\nopen import relations\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ m \u2208 (\u22a4 \u2192 U) ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (\u03a3[ \u03b1 \u2208 (U \u2192 X \u2192 Set) ]( \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x) )))))\n\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , _ , \u03b1 , _) (V , Y , _ , _ , \u03b2 , _) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , m\u2081 , n\u2081 , \u03b1 , _)} {(V , Y , m\u2082 , n\u2082 , \u03b2 , _)} {(W , Z , m\u2083 , n\u2083 , \u03b3 , _)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , m , n , \u03b1 , _)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , _ , \u03b1 , _)}{(V , Y , _ , _ , \u03b2 , _)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , _ , \u03b1 , _}{V , Y , _ , _ , \u03b2 , _}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , _ , \u03b1 , _}{V , Y , _ , _ , \u03b2 , _}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , _ , \u03b1 , _}{V , Y , _ , _ , \u03b2 , _}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , _ , \u03b1 , _}\n         {V , Y , _ , _ , \u03b2 , _}\n         {W , Z , _ , _ , \u03b3 , _}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , _ , \u03b1 , _}{V , Y , _ , _ , \u03b2 , _}{W , Z , _ , _ , \u03b3 , _}{S , T , _ , _ , \u03b9 , _}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _ , _}{V , Y , _ , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _ , _}{V , Y , _ , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n\u2297d : \u2200{U V X Y}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2  : V \u2192 Y \u2192 Set}{d\u2081 : (u : U) (x : X) \u2192 Dec (\u03b1 u x)}{d\u2082 : (u : V) (x : Y) \u2192 Dec (\u03b2 u x)} \u2192 (u : \u03a3 U (\u03bb x \u2192 V)) (x : \u03a3 X (\u03bb x\u2081 \u2192 Y)) \u2192 Dec ((\u03b1 \u2297\u1d63 \u03b2) u x)\n\u2297d {U}{V}{X}{Y}{\u03b1}{\u03b2}{d\u2081}{d\u2082} (u , v) (x , y) with d\u2081 u x | d\u2082 v y\n... | yes z | yes z' = yes (z , z')\n... | yes z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n... | no z | yes z' = no (\u03bb z'' \u2192 z (fst z''))\n... | no z | no z' = no (\u03bb z'' \u2192 z' (snd z''))\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , m\u2081 , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2 , d\u2082) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 m\u2081 m\u2082  , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2 , \u2297d {\u03b1 = \u03b1} {\u03b2}{d\u2081}{d\u2082})\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , _ , \u03b1 , _)}{(V , Y , _ , _ , \u03b2 , _)}{(W , Z , _ , _ , \u03b3 , _)}{(S , T , _ , _ , \u03b4 , _)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}\n    \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n    \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}\n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2 , d\u2082)) (U , X , m\u2081 , n\u2081 , \u03b1 , d\u2081)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}\n    \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n    \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2 , d\u2082)) (V , Y , m\u2082 , n\u2082 , \u03b2 , d\u2082)\n\u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n\ncart-ar : {U X V Y W Z : Set}\n  \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n  \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n  \u2192 {m\u2081 : \u22a4 \u2192 U}\n  \u2192 {n\u2081 : \u22a4 \u2192 X}\n  \u2192 {m\u2082 : \u22a4 \u2192 V}\n  \u2192 {n\u2082 : \u22a4 \u2192 Y}\n  \u2192 {m\u2083 : \u22a4 \u2192 W}\n  \u2192 {n\u2083 : \u22a4 \u2192 Z}\n  \u2192 {d\u2081 : \u2200(u : U)(x : X) \u2192 Dec (\u03b1 u x)}\n  \u2192 {d\u2082 : \u2200(v : V)(y : Y) \u2192 Dec (\u03b2 v y)}\n  \u2192 {d\u2083 : \u2200(w : W)(z : Z) \u2192 Dec (\u03b3 w z)}  \n  \u2192 Hom (W , Z , m\u2083 , n\u2083 , \u03b3 , d\u2083) (U , X , m\u2081 , n\u2081 , \u03b1 , d\u2081)\n  \u2192 Hom (W , Z , m\u2083 , n\u2083 , \u03b3 , d\u2083) (V , Y , m\u2082 , n\u2082 , \u03b2 , d\u2082)\n  \u2192 Hom (W , Z , m\u2083 , n\u2083 , \u03b3 , d\u2083) ((U , X , m\u2081 , n\u2081 , \u03b1 , d\u2081) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2 , d\u2082))\ncart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3}{m\u2081}{n\u2081}{m\u2082}{n\u2082}{m\u2083}{n\u2083}{d\u2081}{d\u2082}{d\u2083} (f , F , p\u2081) (g , G , p\u2082) = trans-\u00d7 f g ,  cart-ar-d , cond\n where\n   cart-ar-d : W \u2192 \u03a3 X (\u03bb x \u2192 Y) \u2192 Z\n   cart-ar-d w (x , y) with d\u2082 (g w) y\n   ... | yes t = F w x\n   ... | no t = G w y\n\n   cond : {u : W} {y : \u03a3 X (\u03bb x \u2192 Y)} \u2192 \u03b3 u (cart-ar-d u y) \u2192 (\u03b1 \u2297\u1d63 \u03b2) (f u , g u) y\n   cond {w}{x , y} p\u2083 with d\u2082 (g w) y\n   ... | yes t = p\u2081 p\u2083 , t\n   ... | no t = \u22a5-elim (t (p\u2082 p\u2083)) , p\u2082  p\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4167595dec03fe26576a71cefda2115fcb6f4563","subject":" scraps","message":" scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n\n  lem : \u2200{\u0394 d d' u \u03c3 m} \u2192 \u0394 \u22a2 d \u21a6 d' \u2192 \u0394 \u22a2 \u2987 d \u2988\u27e8 (u , \u03c3 , \u2717) \u27e9 \u21a6 \u2987 d' \u2988\u27e8 (u , \u03c3 , \u2717) \u27e9\n  lem (Step x x\u2081 x\u2082) = {!!}\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V VConst\n\n  -- variables\n  progress (TAVar x) = abort (somenotnone (! x))\n\n  -- lambdas\n  progress (TALam D) = V VLam\n\n  -- applications\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n    -- left applicand value\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n    -- errors propagate\n  progress (TAAp D1 x\u2082 D2) | _   | E x\u2081 = E (EAp2 {!!} x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n    -- indeterminates\n  progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n    -- either applicand steps\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , {!!})\n  progress (TAAp {d2 = d2} D1 x\u2082 D2) | S (\u03c01 , \u03c02) | _ = S ((\u03c01 \u2218 d2) , {!!})\n\n  -- empty holes\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {m = \u2717} x x\u2081) = S (_ , Step FHEHole ITEHole FHEHole)\n\n  -- non-empty holes\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | V v = S ( _ , Step (FHNEHoleFinal (FVal v)) (ITNEHole (FVal v)) FHNEHoleEvaled)\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | I x = S (_ , Step (FHNEHoleFinal (FIndet x)) (ITNEHole (FIndet x)) FHNEHoleEvaled )\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  -- progress (TANEHole {d = d} {u = u} {\u03c3 = \u03c3} {m = m} x\u2083 D x\u2084) | S (d' , Step x x\u2081 x\u2082) = S (_ , lem (Step x x\u2081 x\u2082) ) -- S ( \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 , {!!}) -- maybe depends on m\n  progress (TANEHole {d = d} {u = u} {\u03c3 = \u03c3} {m = \u2713} x\u2083 D x\u2084) | S (d' , Step x x\u2081 x\u2082) =  S ( \u2987 d' \u2988\u27e8 u , \u03c3 , {!!} \u27e9 ,\n                                                                                             Step FHNEHoleEvaled {!!} FHNEHoleEvaled)\n  progress (TANEHole {d = d} {u = u} {\u03c3 = \u03c3} {m = \u2717} x\u2083 D x\u2084) | S (d' , Step x x\u2081 x\u2082) = S(_ , lem (Step x x\u2081 x\u2082)) -- S ( \u2987 d' \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 , {!!})\n\n  -- casts\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  progress (TACast D m) | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  progress (TACast D x\u2081) | I x = S {!!}\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V VConst\n\n  -- variables\n  progress (TAVar x) = abort (somenotnone (! x))\n\n  -- lambdas\n  progress (TALam D) = V VLam\n\n  -- applications\n  progress (TAAp D1 x D2)\n    with progress D1 | progress D2\n    -- left applicand value\n  progress (TAAp TAConst () D2) | V VConst | _\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n    -- errors propagate\n  progress (TAAp D1 x\u2082 D2) | _   | E x\u2081 = E (EAp2 x\u2081)\n  progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n    -- indeterminates\n  progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n    -- either applicand steps\n  progress (TAAp {d1 = d1} D1 x\u2084 D2) | _ | S (d , Step x\u2081 x\u2082 x\u2083) = S ((d1 \u2218 d) , {!!})\n  progress (TAAp {d2 = d2} D1 x\u2082 D2) | S (\u03c01 , \u03c02) | _ = S ((\u03c01 \u2218 d2) , {!!})\n\n  -- empty holes\n  progress (TAEHole {m = \u2713} x x\u2081) = I IEHole\n  progress (TAEHole {m = \u2717} x x\u2081) = S (_ , Step FHEHole ITEHole FHEHole)\n\n  -- non-empty holes\n  progress (TANEHole x D x\u2081)\n    with progress D\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | V v = S ( _ , Step (FHNEHoleFinal (FVal v)) (ITNEHole (FVal v)) FHNEHoleEvaled)\n  progress (TANEHole {m = \u2713} x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  progress (TANEHole {m = \u2717} x\u2081 D x\u2082) | I x = S (_ , Step (FHNEHoleFinal (FIndet x)) (ITNEHole (FIndet x)) FHNEHoleEvaled )\n  progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  progress (TANEHole {d = d} {u = u} {\u03c3 = \u03c3} {m = m} x\u2083 D x\u2084) | S (d' , Step x x\u2081 x\u2082) = S ( \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 , {!!}) -- maybe depends on m\n\n  -- casts\n  progress (TACast D x)\n    with progress D\n  progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  progress (TACast D m) | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  progress (TACast D x\u2081) | I x = S {!!}\n  progress (TACast D x\u2081) | E x = E (ECastProp x)\n  progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ec848c5bccace6dff1be1e34e44cac5dfdb9e5d6","subject":"Agda: remove experimental\/ModalLogic.agda (its existence on branch master is a misunderstanding)","message":"Agda: remove experimental\/ModalLogic.agda\n(its existence on branch master is a misunderstanding)\n\nOld-commit-hash: 57203332fc449b7188df1085e5ff637de929dbc0\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ModalLogic.agda","new_file":"experimental\/ModalLogic.agda","new_contents":"","old_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotation.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntax.Context Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n--\n-- Attempt 1 at weakening, subcontext-based. This works, but it is not what we\n-- need later (because of the way substitution is stated in the paper).\n--\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be\n-- fixed and lift introduced by hand.\n\nweaken : \u2200 {\u0393 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393 \u0394\u2081 \u03c4 \u2192 Term \u0393 \u0394\u2082 \u03c4\nweaken \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (app t t\u2081) = app (weaken \u0394\u2032 t) (weaken \u0394\u2032 t\u2081)\nweaken \u0394\u2032 (ovar x) = ovar x\nweaken \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0394\u2032 (box t) = box (weaken \u0394\u2032 t)\nweaken \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0394\u2032 t in-\n          weaken (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n\nopen import Relation.Binary.PropositionalEquality\n\n-- These two lemmas could be merged.\n\nweaken-\u227c : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u0394 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u0394 \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u0394 \u03c4\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} {\u0393\u2083} (abs {\u03c4\u2081} t) =  abs (weaken-\u227c {\u03c4\u2081 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (app t t\u2081) = app (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t) (weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081)\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (ovar x) = ovar (Prefixes.lift {\u0393\u2081} {\u0393\u2082} x)\nweaken-\u227c (mvar u) = mvar u\nweaken-\u227c (box t) = box t\nweaken-\u227c {\u0393\u2081} {\u0393\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c {\u0393\u2081} {\u0393\u2082} t in- weaken-\u227c {\u0393\u2081} {\u0393\u2082} t\u2081\n\n\nweaken-\u227c-\u0394 : \u2200 {\u0394\u2081 \u0394\u2082 \u0394\u2083 \u0393 \u03c4} \u2192 Term \u0393 (\u0394\u2081 \u22ce \u0394\u2083) \u03c4 \u2192 Term \u0393 (\u0394\u2081 \u22ce \u0394\u2082 \u22ce \u0394\u2083) \u03c4\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (abs t) = abs (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (app t t\u2081) = app (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t) (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t\u2081)\nweaken-\u227c-\u0394 (ovar x) = ovar x\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (mvar u) = mvar (Prefixes.lift {\u0394\u2081} {\u0394\u2082} u)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (box t) = box (weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t)\nweaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} (let-box t in- t\u2081) = let-box weaken-\u227c-\u0394 {\u0394\u2081} {\u0394\u2082} t in- weaken-\u227c-\u0394 {_ \u2022 \u0394\u2081} {\u0394\u2082} t\u2081 \n\nsubstVar : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (v : Var (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstVar {\u2205} t  this = t\nsubstVar {\u2205} t (that v) = ovar v\nsubstVar {\u03c4 \u2022 \u0393} t  this = ovar this\nsubstVar {\u03c4 \u2022 \u0393} t (that v) = weaken-\u227c {\u2205} {\u03c4 \u2022 \u2205} (substVar {\u0393} t v)\n\nsubstTerm : \u2200 {\u0393 \u0393\u2032 \u0394 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393\u2032 \u0394 \u03c4\u2081) \u2192 (t\u2082 : Term (\u0393 \u22ce (\u03c4\u2081 \u2022 \u0393\u2032)) \u0394 \u03c4\u2082) \u2192 Term (\u0393 \u22ce \u0393\u2032) \u0394 \u03c4\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (abs {\u03c4\u2082} t\u2082) = abs ( substTerm {\u03c4\u2082 \u2022 \u0393} {\u0393\u2032} t\u2081 t\u2082 )\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (app t\u2082 t\u2083) =  app (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082) (substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2083) \nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (ovar v) =  substVar {\u0393} {\u0393\u2032} t\u2081 v\nsubstTerm t\u2081 (mvar u) = mvar u\nsubstTerm t\u2081 (box t\u2082) = box t\u2082\nsubstTerm {\u0393} {\u0393\u2032} t\u2081 (let-box_in-_ {\u03c4\u2083} t\u2082 t\u2083) =\n  let-box substTerm {\u0393} {\u0393\u2032} t\u2081 t\u2082 in-\n    substTerm {\u0393} {\u0393\u2032}\n      (weaken-\u227c-\u0394 {\u2205} {\u03c4\u2083 \u2022 \u2205} t\u2081)\n      t\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d1ae746feb957be6626080e1c96643de72039293","subject":"Boring.","message":"Boring.\n\nIgnore-this: feb10a12a97f1bd478ce9f1f568aa588\n\ndarcs-hash:20100701135430-3bd4e-902af82059cc21b42a2427dcaa2509ae48223f83.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List.agda","new_file":"LTC\/Data\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\npostulate\n  head : D \u2192 D\n  cHead : (d ds : D) \u2192 head (d \u2237 ds) \u2261 d\n\npostulate\n  tail : D \u2192 D\n  cTail : (d ds : D) \u2192 tail (d \u2237 ds) \u2261 ds\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","old_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\npostulate\n  head : D \u2192 D\n  cHead : (d ds : D) \u2192 head (d \u2237 ds) \u2261 d\n\npostulate\n  tail : D \u2192 D\n  cTail : (d ds : D) \u2192 tail (d \u2237 ds) \u2261 ds\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192      []       ++ ds \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] :              reverse []       \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"53c71826c649a1f6e88c8e74de9240bddaa63b7e","subject":"Formalize nil changes (as syntactic sugar).","message":"Formalize nil changes (as syntactic sugar).\n\nPossibly to be replaced with an axiomatic formalization.\n\nOld-commit-hash: 2b0add1effb48cccdfff962349ae9b80bc0ac14e\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\n  nil : \u2200 v \u2192 Change v\n  nil v = diff v v\n\n  update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n  update-nil v = update-diff v v\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n","old_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0202539e3a0699578d52a30e6a7e5a51fd9e9103","subject":"Reflection.NP: many improvements","message":"Reflection.NP: many improvements\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/NP.agda","new_file":"lib\/Reflection\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Reflection.NP where\n\nopen import Type\nopen import Level.NP\nopen import Data.Nat using (\u2115; zero; suc) renaming (_\u2294_ to _\u2294\u2115_)\nopen import Data.Maybe.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9; 0\u2081)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; [0:_1:_])\nopen import Data.List\nopen import Data.Vec using (Vec) -- ; []; _\u2237_)\nopen import Data.Product.NP\nopen import Function.NP hiding (\u03a0)\n\nopen import Reflection public\n\nArgs : \u2605\nArgs = List (Arg Term)\n\n-- lam\u1d5b : Term \u2192 Term\npattern lam\u1d5b = lam visible\n\n-- lam\u02b0 : Term \u2192 Term\npattern lam\u02b0 = lam hidden\n\n-- argi\u1d5b\u02b3 : Arg-info\npattern argi\u1d5b\u02b3 = arg-info visible relevant\n\n-- argi\u02b0\u02b3 : Arg-info\npattern argi\u02b0\u02b3 = arg-info hidden relevant\n\n-- arg\u1d5b\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\npattern arg\u1d5b\u02b3 v = arg argi\u1d5b\u02b3 v\n\n-- arg\u02b0\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\npattern arg\u02b0\u02b3 v = arg argi\u02b0\u02b3 v\n\nArg-infos : \u2605\nArg-infos = List Arg-info\n\napp` : (Args \u2192 Term) \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\napp` f = go [] where\n  go : Args \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\n  go args []         = f (reverse args)\n  go args (ai \u2237 ais) = \u03bb t \u2192 go (arg ai t \u2237 args) ais\n\npattern con\u1d5b\u02b3 n t = con n (arg\u1d5b\u02b3 t \u2237 [])\npattern con\u02b0\u02b3 n t = con n (arg\u02b0\u02b3 t \u2237 [])\npattern def\u1d5b\u02b3 n t = def n (arg\u1d5b\u02b3 t \u2237 [])\npattern def\u02b0\u02b3 n t = def n (arg\u02b0\u02b3 t \u2237 [])\n\ncon` : Name \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\ncon` x = app` (con x)\n\ndef` : Name \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\ndef` x = app` (def x)\n\nvar` : \u2115 \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\nvar` x = app` (var x)\n\ncoe : \u2200 {A : \u2605} {z : A} n \u2192 (Term \u2192\u27e8 length (replicate n z) \u27e9 Term) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncoe zero    t = t\ncoe (suc n) f = \u03bb t \u2192 coe n (f t)\n\ncon`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncon`\u207f\u02b3 x n = coe n (app` (con x) (replicate n argi\u1d5b\u02b3))\n\ndef`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ndef`\u207f\u02b3 x n = coe n (app` (def x) (replicate n argi\u1d5b\u02b3))\n\nvar`\u207f\u02b3 : \u2115 \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\nvar`\u207f\u02b3 x n = coe n (app` (var x) (replicate n argi\u1d5b\u02b3))\n\n-- sort\u2080 : Sort\npattern sort\u2080 = lit 0\n\n-- sort\u2081 : Sort\npattern sort\u2081 = lit 1\n\n-- `\u2605\u2080 : Term\npattern `\u2605\u2080 = sort sort\u2080\n\n-- el\u2080 : Term \u2192 Type\npattern el\u2080 t = el sort\u2080 t\n\n-- Builds a type variable (of type \u2605\u2080)\n``var\u2080 : \u2115 \u2192 Args \u2192 Type\n``var\u2080 n args = el\u2080 (var n args)\n\n-- ``\u2605\u2080 : Type\npattern ``\u2605\u2080 = el sort\u2081 `\u2605\u2080\n\nunEl : Type \u2192 Term\nunEl (el _ tm) = tm\n\ngetSort : Type \u2192 Sort\ngetSort (el s _) = s\n\nunArg : \u2200 {A} \u2192 Arg A \u2192 A\nunArg (arg _ a) = a\n\n-- `Level : Term\npattern `Level = def (quote Level) []\n\n-- ``Level : Type\npattern ``Level = el\u2080 `Level\n\npattern `\u2080 = def (quote \u2080) []\n\n-- `\u209b_ : Term \u2192 Term\n-- `\u209b_ = def`\u207f\u02b3 (quote L.suc) 1\npattern `\u209b_ v = def (quote \u209b) (arg\u1d5b\u02b3 v \u2237 [])\n\n-- sucSort : Sort \u2192 Sort\npattern sucSort s = set (`\u209b (sort s))\n\npattern `set\u2080 = set `\u2080\npattern `set\u209b_ s = set (`\u209b s)\npattern `set_ s = set (sort s)\n\ndecode-Sort : Sort \u2192 Maybe \u2115\ndecode-Sort `set\u2080 = just zero\ndecode-Sort (`set\u209b_ s) = map? suc (decode-Sort (set s))\ndecode-Sort (`set_ s) = decode-Sort s\ndecode-Sort (set _) = nothing\ndecode-Sort (lit n) = just n\ndecode-Sort unknown = nothing\n\npattern _`\u2294_ s\u2081 s\u2082 = set (def (quote _\u2294_) (arg\u1d5b\u02b3 (sort s\u2081) \u2237 arg\u1d5b\u02b3 (sort s\u2082) \u2237 []))\n\n_`\u2294`_ : Sort \u2192 Sort \u2192 Sort\ns\u2081 `\u2294` s\u2082 with decode-Sort s\u2081 | decode-Sort s\u2082\n...          | just n\u2081        | just n\u2082        = lit (n\u2081 \u2294\u2115 n\u2082)\n...          | _              | _              = s\u2081 `\u2294 s\u2082\n\npattern pi\u1d5b\u02b3 t u = pi (arg\u1d5b\u02b3 t) u\npattern pi\u02b0\u02b3 t u = pi (arg\u02b0\u02b3 t) u\n\n\u03a0 : Arg Type \u2192 Type \u2192 Type\n\u03a0 t u = el (getSort (unArg t) `\u2294` getSort u) (pi t u)\n\n\u03a0\u1d5b\u02b3 : Type \u2192 Type \u2192 Type\n\u03a0\u1d5b\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u1d5b\u02b3 t u)\n\n\u03a0\u02b0\u02b3 : Type \u2192 Type \u2192 Type\n\u03a0\u02b0\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u02b0\u02b3 t u)\n\n-- \u03b7 vs mk: performs no shifting of the result of mk.\n-- Safe values of mk are def and con for instance\n\u03b7 : List Arg-info \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7 ais\u2080 mk = go ais\u2080 where\n  vars : List Arg-info \u2192 Args\n  vars []         = []\n  vars (ai \u2237 ais) = arg ai (var (length ais) []) \u2237 vars ais\n  go : List Arg-info \u2192 Term\n  go []                   = mk (vars ais\u2080)\n  go (arg-info v _ \u2237 ais) = lam v (go ais)\n\n\u03b7\u02b0 : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u02b0 n = \u03b7 (replicate n argi\u02b0\u02b3)\n\n\u03b7\u1d5b : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u1d5b n = \u03b7 (replicate n argi\u1d5b\u02b3)\n\n\u03b7\u02b0\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u02b0\u207f n = \u03b7\u02b0 n \u2218 def\n\n\u03b7\u1d5b\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u1d5b\u207f n = \u03b7\u1d5b n \u2218 def\n\narityOfTerm : Term \u2192 List Arg-info\narityOfType : Type \u2192 List Arg-info\n\narityOfType (el _ u) = arityOfTerm u\narityOfTerm (pi (arg ai _) t) = ai \u2237 arityOfType t\n\n-- no more arguments\narityOfTerm (var _ _) = []\n\n-- TODO\narityOfTerm (con c args) = []\narityOfTerm (def f args) = []\narityOfTerm (sort s)     = []\n\n-- fail\narityOfTerm unknown = []\n\n-- absurd cases\narityOfTerm (lam _ _) = []\n\n\u03b7\u207f : Name \u2192 Term\n\u03b7\u207f nm = \u03b7 (arityOfType (type nm)) (def nm)\n\nDecode-Term : \u2200 {a} \u2192 \u2605_ a \u2192 \u2605_ a\nDecode-Term A = Term \u2192 Maybe A\n\npattern `\ud835\udfd8 = def (quote \ud835\udfd8) []\n\npattern `\ud835\udfd9  = def (quote \ud835\udfd9) []\npattern `0\u2081 = con (quote 0\u2081) []\n\ndecode-\ud835\udfd9 : Decode-Term \ud835\udfd9\ndecode-\ud835\udfd9 `0\u2081 = just 0\u2081\ndecode-\ud835\udfd9 _   = nothing\n\npattern `\ud835\udfda  = def (quote \ud835\udfda) []\npattern `0\u2082 = con (quote 0\u2082) []\npattern `1\u2082 = con (quote 1\u2082) []\n\ndecode-\ud835\udfda : Decode-Term \ud835\udfda\ndecode-\ud835\udfda `0\u2082 = just 0\u2082\ndecode-\ud835\udfda `1\u2082 = just 1\u2082\ndecode-\ud835\udfda _   = nothing\n\npattern `\u2115     = def (quote \u2115) []\npattern `zero  = con (quote zero) []\npattern `suc t = con\u1d5b\u02b3 (quote suc) t\n\ndecode-\u2115 : Decode-Term \u2115\ndecode-\u2115 `zero    = just 0\ndecode-\u2115 (`suc t) = map? suc (decode-\u2115 t)\ndecode-\u2115 _        = nothing\n\npattern `Maybe t = def\u1d5b\u02b3 (quote Maybe) t\npattern `nothing = con (quote Maybe.nothing) []\npattern `just  t = con\u1d5b\u02b3 (quote Maybe.just) t\n\ndecode-Maybe : \u2200 {a} {A : \u2605_ a} \u2192 Decode-Term A \u2192 Decode-Term (Maybe A)\ndecode-Maybe decode-A `nothing  = just nothing\ndecode-Maybe decode-A (`just t) = map? just (decode-A t)\ndecode-Maybe decode-A _         = nothing\n\npattern `List t = def\u1d5b\u02b3 (quote List) t\npattern `[] = con (quote List.[]) []\npattern _`\u2237_ t u = con (quote List._\u2237_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n\ndecode-List : \u2200 {a} {A : \u2605_ a} \u2192 Decode-Term A \u2192 Decode-Term (List A)\ndecode-List decode-A `[]      = just []\ndecode-List decode-A (t `\u2237 u) = \u27ea _\u2237_ \u00b7 decode-A t \u00b7 decode-List decode-A u \u27eb?\ndecode-List decode-A _        = nothing\n\npattern `Vec t u = def (quote Vec) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern `v[]      = con (quote Vec.[]) []\npattern _`v\u2237_ t u = con (quote Vec._\u2237_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n\n{-\ndecode-Vec : \u2200 {a} {A : \u2605_ a} {n} \u2192 Decode-Term A \u2192 Decode-Term (Vec A n)\ndecode-Vec decode-A `[]      = just []\ndecode-Vec decode-A (t `\u2237 u) = \u27ea _\u2237_ \u00b7 decode-A t \u00b7 decode-Vec decode-A u \u27eb?\ndecode-Vec decode-A _        = nothing\n-}\n\npattern `\u03a3 t u = def (quote \u03a3) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern _`,_ t u = con (quote _,_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern `fst t = def\u1d5b\u02b3 (quote fst) t\npattern `snd t = def\u1d5b\u02b3 (quote snd) t\n\nmodule _ {a b} {A : \u2605_ a} {B : A \u2192 \u2605_ b}\n         (decode-A : Decode-Term A)\n         (decode-B : (x : A) \u2192 Decode-Term (B x))\n         where\n    decode-\u03a3 : Decode-Term (\u03a3 A B)\n    decode-\u03a3 (t `, u) = decode-A t   >>=? \u03bb x \u2192\n                        decode-B x u >>=? \u03bb y \u2192\n                        just (x , y)\n    decode-\u03a3 _        = nothing\n\nmodule Ex where\n  open import Relation.Binary.PropositionalEquality\n  postulate\n    f : \u2115 \u2192 \u2115 \u2192 \u2115\n    g : {x y : \u2115} \u2192 \u2115\n    h : {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  H : \u2605\n  H = {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  postulate\n    h\u2082 : H\n  test\u2081 : unquote (\u03b7\u1d5b\u207f 2 (quote f)) \u2261 f\n  test\u2081 = refl\n  test\u2082 : unquote (\u03b7\u02b0\u207f 2 (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2082 = refl\n  test\u2083 : unquote (\u03b7\u207f (quote f)) \u2261 f\n  test\u2083 = refl\n  test\u2084 : unquote (\u03b7\u207f (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2084 = refl\n  \u03b7h = \u03b7\u207f (quote h)\n  -- this test passes but leave an undecided instance argument\n  -- test\u2085 : unquote \u03b7h \u2261 \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  -- test\u2085 = refl\n  \u03b7h\u2082 : Term\n  \u03b7h\u2082 = \u03b7\u207f (quote h\u2082)\n  {-\n  test\u2086 : unquote \u03b7h\u2082 \u2261 {!unquote \u03b7h\u2082!} -- \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  test\u2086 = refl\n  -}\n  test\u2087 : decode-\u2115 (quoteTerm (suc (suc zero))) \u2261 just 2\n  test\u2087 = refl\n  test\u2088 : decode-\u2115 (quoteTerm (suc (suc 3))) \u2261 just 5\n  test\u2088 = refl\n  test\u2089 : decode-Maybe decode-\ud835\udfda (quoteTerm (Maybe.just 0\u2082)) \u2261 just (just 0\u2082)\n  test\u2089 = refl\n  test\u2081\u2080 : decode-List decode-\u2115 (quoteTerm (0 \u2237 1 \u2237 2 \u2237 [])) \u2261 just (0 \u2237 1 \u2237 2 \u2237 [])\n  test\u2081\u2080 = refl\n  test\u2081\u2081 : quoteTerm (_,\u2032_ 0\u2082 1\u2082) \u2261 `0\u2082 `, `1\u2082\n  test\u2081\u2081 = refl\n  test\u2081\u2081' : decode-List (decode-\u03a3 {A = \ud835\udfda} {B = [0: \ud835\udfda 1: \u2115 ]} decode-\ud835\udfda [0: decode-\ud835\udfda 1: decode-\u2115 ])\n                        (quoteTerm ((_,_ {B = [0: \ud835\udfda 1: \u2115 ]} 0\u2082 1\u2082) \u2237 (1\u2082 , 4) \u2237 [])) \u2261 just ((0\u2082 , 1\u2082) \u2237 (1\u2082 , 4) \u2237 [])\n  test\u2081\u2081' = refl\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Reflection.NP where\n\nopen import Type\nimport Level as L\nopen L using (Level; _\u2294_)\nopen import Data.Nat using (\u2115; zero; suc) renaming (_\u2294_ to _\u2294\u2115_)\nopen import Data.Maybe as May\nopen import Data.List\nopen import Data.Bool\nopen import Data.Product\nopen import Reflection public\nopen import Function\n\nArgs : \u2605\nArgs = List (Arg Term)\n\nprivate\n  primitive\n    primQNameEquality : Name \u2192 Name \u2192 Bool\n\n_==_ : Name \u2192 Name \u2192 Bool\n_==_ = primQNameEquality\n\nprivate\n  _\u2192\u27e8_\u27e9_ : \u2200 (A : \u2605) (n : \u2115) (B : \u2605) \u2192 \u2605\n  A \u2192\u27e8 zero  \u27e9 B = B\n  A \u2192\u27e8 suc n \u27e9 B = A \u2192 A \u2192\u27e8 n \u27e9 B\n\n  when : \u2200 {A : \u2605} \u2192 Bool \u2192 Maybe A \u2192 Maybe A\n  when true  x = x\n  when false _ = nothing\n\n  mapMaybe : \u2200 {A B : \u2605} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n  mapMaybe f (just x) = just (f x)\n  mapMaybe f nothing  = nothing\n\nlam\u1d5b : Term \u2192 Term\nlam\u1d5b = lam visible\n\nlam\u02b0 : Term \u2192 Term\nlam\u02b0 = lam hidden\n\narg\u1d5b\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\narg\u1d5b\u02b3 = arg visible relevant\n\narg\u02b0\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\narg\u02b0\u02b3 = arg hidden relevant\n\nFlags : \u2605\nFlags = List (Visibility \u00d7 Relevance)\n\napp` : (Args \u2192 Term) \u2192 (fs : Flags) \u2192 Term \u2192\u27e8 length fs \u27e9 Term\napp` f = go [] where\n  go : Args \u2192 (fs : Flags) \u2192 Term \u2192\u27e8 length fs \u27e9 Term\n  go args []             = f (reverse args)\n  go args ((h , r) \u2237 hs) = \u03bb t \u2192 go (arg h r t \u2237 args) hs\n\ncon` : Name \u2192 (fs : Flags) \u2192 Term \u2192\u27e8 length fs \u27e9 Term\ncon` x = app` (con x)\n\ndef` : Name \u2192 (fs : Flags) \u2192 Term \u2192\u27e8 length fs \u27e9 Term\ndef` x = app` (def x)\n\nvar` : \u2115 \u2192 (fs : Flags) \u2192 Term \u2192\u27e8 length fs \u27e9 Term\nvar` x = app` (var x)\n\ncoe : \u2200 {A : \u2605} {z : A} n \u2192 (Term \u2192\u27e8 length (replicate n z) \u27e9 Term) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncoe zero    t = t\ncoe (suc n) f = \u03bb t \u2192 coe n (f t)\n\ncon`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncon`\u207f\u02b3 x n = coe n (app` (con x) (replicate n (visible , relevant)))\n\ndef`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ndef`\u207f\u02b3 x n = coe n (app` (def x) (replicate n (visible , relevant)))\n\nvar`\u207f\u02b3 : \u2115 \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\nvar`\u207f\u02b3 x n = coe n (app` (var x) (replicate n (visible , relevant)))\n\nsort\u2080 : Sort\nsort\u2080 = lit 0\n\nsort\u2081 : Sort\nsort\u2081 = lit 1\n\n`\u2605\u2080 : Term\n`\u2605\u2080 = sort sort\u2080\n\nel\u2080 : Term \u2192 Type\nel\u2080 = el sort\u2080\n\n-- Builds a type variable (of type \u2605\u2080)\n``var\u2080 : \u2115 \u2192 Args \u2192 Type\n``var\u2080 n args = el\u2080 (var n args)\n\n``\u2605\u2080 : Type\n``\u2605\u2080 = el sort\u2081 `\u2605\u2080\n\nunEl : Type \u2192 Term\nunEl (el _ tm) = tm\n\ngetSort : Type \u2192 Sort\ngetSort (el s _) = s\n\nunArg : \u2200 {A} \u2192 Arg A \u2192 A\nunArg (arg _ _ a) = a\n\n`Level : Term\n`Level = def (quote Level) []\n\n``Level : Type\n``Level = el\u2080 `Level\n\n`sucLevel : Term \u2192 Term\n`sucLevel = def`\u207f\u02b3 (quote L.suc) 1\n\nsucSort : Sort \u2192 Sort\nsucSort s = set (`sucLevel (sort s))\n\n\u2115\u2192Level : \u2115 \u2192 Level\n\u2115\u2192Level zero    = L.zero\n\u2115\u2192Level (suc n) = L.suc (\u2115\u2192Level n)\n\ndecodeSort : Sort \u2192 Maybe \u2115\ndecodeSort (set (con c [])) = when (quote L.zero == c) (just zero)\ndecodeSort (set (con c (arg visible relevant s \u2237 [])))\n    = when (quote L.suc == c) (mapMaybe suc (decodeSort (set s)))\ndecodeSort (set (sort s)) = decodeSort s\ndecodeSort (set _) = nothing\ndecodeSort (lit n) = just n\ndecodeSort unknown = nothing\n\n_`\u2294`_ : Sort \u2192 Sort \u2192 Sort\ns\u2081 `\u2294` s\u2082 with decodeSort s\u2081 | decodeSort s\u2082\n...          | just n\u2081       | just n\u2082        = lit (n\u2081 \u2294\u2115 n\u2082)\n...          | _             | _              = set (def (quote _\u2294_) (arg\u1d5b\u02b3 (sort s\u2081) \u2237 arg\u1d5b\u02b3 (sort s\u2082) \u2237 []))\n\n\u03a0 : Arg Type \u2192 Type \u2192 Type\n\u03a0 t u = el (getSort (unArg t) `\u2294` getSort u) (pi t u)\n\n\u03a0\u1d5b\u02b3 : Type \u2192 Type \u2192 Type\n\u03a0\u1d5b\u02b3 t u = el (getSort t `\u2294` getSort u) (pi (arg visible relevant t) u)\n\n\u03a0\u02b0\u02b3 : Type \u2192 Type \u2192 Type\n\u03a0\u02b0\u02b3 t u = el (getSort t `\u2294` getSort u) (pi (arg hidden relevant t) u)\n\n-- \u03b7 vs mk: performs no shifting of the result of mk.\n-- Safe values of mk are def and con for instance\n\u03b7 : List Visibility \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7 vs\u2080 mk = go vs\u2080 where\n  vars : List Visibility \u2192 Args\n  vars []       = []\n  vars (v \u2237 vs) = arg v relevant (var (length vs) []) \u2237 vars vs\n  go : List Visibility \u2192 Term\n  go []       = mk (vars vs\u2080)\n  go (v \u2237 vs) = lam v (go vs)\n\n\u03b7\u02b0 : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u02b0 n = \u03b7 (replicate n hidden)\n\n\u03b7\u1d5b : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u1d5b n = \u03b7 (replicate n visible)\n\n\u03b7\u02b0\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u02b0\u207f n = \u03b7\u02b0 n \u2218 def\n\n\u03b7\u1d5b\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u1d5b\u207f n = \u03b7\u1d5b n \u2218 def\n\narityOfType : Type \u2192 List Visibility\narityOfType (el _ u) with u\n... | pi (arg v _ _) t = v \u2237 arityOfType t\n\n-- no more arguments\n... | var _ _ = []\n\n-- TODO\n... | con c args = []\n... | def f args = []\n... | sort s = []\n\n-- fail\n... | unknown = []\n\n-- absurd cases\n... | lam _ _ = []\n\n\u03b7\u207f : Name \u2192 Term\n\u03b7\u207f nm = \u03b7 (arityOfType (type nm)) (def nm)\n\nmodule Ex where\n  open import Relation.Binary.PropositionalEquality\n  postulate\n    f : \u2115 \u2192 \u2115 \u2192 \u2115\n    g : {x y : \u2115} \u2192 \u2115\n    h : {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  H : \u2605\n  H = {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  postulate\n    h\u2082 : H\n  test\u2081 : unquote (\u03b7\u1d5b\u207f 2 (quote f)) \u2261 f\n  test\u2081 = refl\n  test\u2082 : unquote (\u03b7\u02b0\u207f 2 (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2082 = refl\n  test\u2083 : unquote (\u03b7\u207f (quote f)) \u2261 f\n  test\u2083 = refl\n  test\u2084 : unquote (\u03b7\u207f (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2084 = refl\n  \u03b7h = \u03b7\u207f (quote h)\n  -- this test passes but leave an undecided instance argument\n  -- test\u2085 : unquote \u03b7h \u2261 \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  -- test\u2085 = refl\n{-\n  \u03b7h\u2082 = \u03b7\u207f (quote h\u2082)\n  test\u2086 : unquote \u03b7h\u2082 \u2261 {!!} -- \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  test\u2086 = refl\n\n  here = {!!}\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0ae4ad830c9d6c7f0b169d4d1be77e6b69514988","subject":"Follow the stdlib change of argument order of Data.Maybe.Param.Binary","message":"Follow the stdlib change of argument order of Data.Maybe.Param.Binary\n","repos":"np\/agda-parametricity","old_file":"lib\/Data\/Maybe\/Param\/Binary.agda","new_file":"lib\/Data\/Maybe\/Param\/Binary.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Function\nopen import Data.Maybe renaming (map to map?)\n\nopen import Type.Param.Binary\nopen import Function.Param.Binary\n\nmodule Data.Maybe.Param.Binary where\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {_} {b}) (map? {a} {_} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (\u27e6just\u27e7 x\u1d63) = \u27e6just\u27e7 (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 \u27e6nothing\u27e7   = \u27e6nothing\u27e7\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (\u27e6just\u27e7 x\u1d63) = \u27e6just\u27e7 x\u1d63\n\u27e6map?-id\u27e7 _ \u27e6nothing\u27e7   = \u27e6nothing\u27e7\n\nprivate\n    infixr 0 _\u2192?_\n    _\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\n    A \u2192? B = A \u2192 Maybe B\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Function\nopen import Data.Maybe renaming (map to map?)\n\nopen import Type.Param.Binary\nopen import Function.Param.Binary\n\nmodule Data.Maybe.Param.Binary where\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (\u27e6just\u27e7 x\u1d63) = \u27e6just\u27e7 (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 \u27e6nothing\u27e7   = \u27e6nothing\u27e7\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (\u27e6just\u27e7 x\u1d63) = \u27e6just\u27e7 x\u1d63\n\u27e6map?-id\u27e7 _ \u27e6nothing\u27e7   = \u27e6nothing\u27e7\n\nprivate\n    infixr 0 _\u2192?_\n    _\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\n    A \u2192? B = A \u2192 Maybe B\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7b74bdbab467964d34794623d08cf51d3df67932","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: f4311e3c634ecccf6a0fcd67a601ba76\n\ndarcs-hash:20120302234006-3bd4e-9e8ce7689ece434d6dcc145f4e901f0ade004319.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/Base.agda","new_file":"src\/FOL\/Base.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"2adbf2cb061a57bf6bb0de6ac998076371d239d6","subject":"Explorable: re-org to depend less on induction principles","message":"Explorable: re-org to depend less on induction principles\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Explorable.agda","new_file":"lib\/Explore\/Explorable.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two\nopen import Data.Nat.NP hiding (_^_; _\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product\nopen import Data.Sum\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Binary\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nopen import Explore.Type\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule FromExplore\n    {m A}\n    (explore : Explore m A) where\n\n  exploreMon : \u2200 {\u2113} (M : Monoid m \u2113) \u2192 ExploreMon M A\n  exploreMon M = explore _\u2219_\n    where open Mon M\n\n  explore\u2218 : \u2200 {M} \u2192 (M \u2192 M \u2192 M) \u2192 (A \u2192 M) \u2192 (M \u2192 M)\n  explore\u2218 = explore\u2218FromExplore explore\n\n  exploreMon\u2218 : \u2200 {\u2113} (M : Monoid m \u2113) \u2192 ExploreMon M A\n  exploreMon\u2218 M f = explore\u2218 _\u2219_ f \u03b5 where open Mon M\n\nmodule FromExplore\u2080 {A} (explore : Explore\u2080 A) where\n  open FromExplore explore\n\n  sum : Sum A\n  sum = explore _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore _\u2227_\n  and   = big-\u2227\n\n  big-\u2228 = explore _\u2228_\n  or    = big-\u2228\n\n  big-xor = explore _xor_\n\n  toBinTree : BinTree A\n  toBinTree = explore fork leaf\n\n  toList : List A\n  toList = explore _++_ List.[_]\n\n  toList\u2218 : List A\n  toList\u2218 = explore\u2218 _++_ List.[_] List.[]\n\n  find? : Find? A\n  find? = explore (M?._\u2223_ _)\n\n  findKey : FindKey A\n  findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule Explorable\u2098\u209a\n    {m p A}\n    {explore     : Explore m A}\n    (explore-ind : ExploreInd p explore) where\n\n  open FromExplore explore public\n\n  explore-sg-ext : ExploreSgExt _ explore\n  explore-sg-ext sg {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ f \u2248 s _ g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  explore-mono : ExploreMono _ explore\n  explore-mono _\u2286_ _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n    explore-ind (\u03bb s \u2192 s _ f \u2286 s _ g) _\u2219-mono_ f\u2286\u00b0g\n\n  explore-swap : ExploreSwap _ explore\n  explore-swap sg f {s\u1d2e} pf =\n    explore-ind (\u03bb s \u2192 s _ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s _ \u2218 flip f))\n               (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _)))\n               (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\nplugKit : \u2200 {m p A} (M : Monoid m p) \u2192 ExploreIndKit _ {A = A} (ExplorePlug M)\nplugKit M = (\u03bb Ps Ps' _ x \u2192\n                  trans (\u2219-cong (sym (Ps _ _)) refl)\n                        (trans (assoc _ _ _)\n                               (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n            , (\u03bb x f _ \u2192 \u2219-cong (proj\u2082 identity (f x)) refl)\n     where open Mon M\n\nmodule Explorable\u2098\n    {m A}\n    {explore     : Explore m A}\n    (explore-ind : ExploreInd m explore) where\n  open Explorable\u2098\u209a explore-ind\n\n  explore\u2218-plug : (M : Monoid m m) \u2192 ExplorePlug M explore\n  explore\u2218-plug M = explore-ind $kit plugKit M\n\n  exploreMon\u2218-spec : \u2200 (M : Monoid _ m) \u2192\n                      let open Mon M in\n                      (f : A \u2192 C) \u2192 exploreMon M f \u2248 exploreMon\u2218 M f\n  exploreMon\u2218-spec M f = proj\u2082 (explore-ind\n                     (\u03bb s \u2192 ExplorePlug M s \u00d7 s _\u2219_ f \u2248 explore\u2218FromExplore s _\u2219_ f \u03b5)\n                     (\u03bb {s} {s'} Ps Ps' \u2192\n                        P\u2219 {s} {s'} (proj\u2081 Ps) (proj\u2081 Ps')\n                      , trans (\u2219-cong (proj\u2082 Ps) (proj\u2082 Ps')) (proj\u2081 Ps f _))\n                     (\u03bb x \u2192 Pf x , sym (proj\u2082 identity _)))\n                        where open Mon M\n                              open ExploreIndKit (plugKit M)\n\n  explore\u2218-ind : \u2200 (M : Monoid m m) \u2192 ExploreMonInd m M (exploreMon\u2218 M)\n  explore\u2218-ind M P P\u2219 Pf P\u2248 =\n    proj\u2082 (explore-ind (\u03bb s \u2192 ExplorePlug M s \u00d7 P (\u03bb f \u2192 s _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n               (\u03bb {s} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {s} {s'} (proj\u2081 Ps) (proj\u2081 Ps')\n                                  , P\u2248 (\u03bb f \u2192 proj\u2081 Ps f _) (P\u2219 (proj\u2082 Ps) (proj\u2082 Ps')))\n               (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                    , P\u2248 (\u03bb f \u2192 sym (proj\u2082 identity _)) (Pf x)))\n    where open Mon M\n\n  explore-ext : ExploreExt explore\n  explore-ext op = explore-ind (\u03bb s \u2192 s _ _ \u2261 s _ _) (\u2261.cong\u2082 op)\n\n  \u27e6explore\u27e7\u1d64 : \u2200 {A\u1d63 : A \u2192 A \u2192 \u2605_ _}\n               (A\u1d63-refl : Reflexive A\u1d63)\n              \u2192 \u27e6Explore\u27e7\u1d64 A\u1d63 explore explore\n  \u27e6explore\u27e7\u1d64 A\u1d63-refl M\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb s \u2192 M\u1d63 (s _ _) (s _ _)) (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n  explore-\u03b5 : Explore\u03b5 m m explore\n  explore-\u03b5 M = explore-ind (\u03bb s \u2192 s _ (const \u03b5) \u2248 \u03b5)\n                          (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5))\n                          (\u03bb _ \u2192 refl)\n    where open Mon M\n\n  explore-hom : ExploreMonHom m m explore\n  explore-hom cm f g = explore-ind (\u03bb s \u2192 s _ (f \u2219\u00b0 g) \u2248 s _ f \u2219 s _ g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  explore-lin\u02e1 : ExploreLin\u02e1 m m explore\n  explore-lin\u02e1 m _\u25ce_ f k dist = explore-ind (\u03bb s \u2192 s _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce s _\u2219_ f) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (sym (dist k _ _))) (\u03bb x \u2192 refl)\n    where open Mon m\n\n  explore-lin\u02b3 : ExploreLin\u02b3 m m explore\n  explore-lin\u02b3 m _\u25ce_ f k dist = explore-ind (\u03bb s \u2192 s _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 s _\u2219_ f \u25ce k) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (sym (dist k _ _))) (\u03bb x \u2192 refl)\n    where open Mon m\n\n  module _\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 m)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (_+_ : Op\u2082 S)\n       (_*_ : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom : \u2200 x y \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore _+_ g) \u2248 explore _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb s \u2192 f (s _+_ g) \u2248 s _*_ (f \u2218 g))\n                               (\u03bb p q \u2192 \u2248-trans (hom _ _) (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  lift-hom-\u2261 :\n      \u2200 {S T}\n        (_+_ : Op\u2082 S)\n        (_*_ : Op\u2082 T)\n        (f   : S \u2192 T)\n        (g   : A \u2192 S)\n        (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore _+_ g) \u2261 explore _*_ (f \u2218 g)\n  lift-hom-\u2261 _+_ _*_ = lift-hom _\u2261_ \u2261.refl \u2261.trans _+_ _*_ (\u2261.cong\u2082 _*_)\n\nmodule Explorable\u2080\n    {A}\n    {explore     : Explore\u2080 A}\n    (explore-ind : ExploreInd\u2080 explore) where\n  open Explorable\u2098\u209a explore-ind public\n  open Explorable\u2098  explore-ind public\n  open FromExplore\u2080 explore     public\n\n  \u27e6explore\u27e7 : \u2200 {A\u1d63 : A \u2192 A \u2192 \u2605_ _}\n               (A\u1d63-refl : Reflexive A\u1d63)\n              \u2192 \u27e6Explore\u27e7 A\u1d63 explore explore\n  \u27e6explore\u27e7 = \u27e6explore\u27e7\u1d64\n\n  sum-ind : SumInd sum\n  sum-ind P P+ Pf = explore-ind (\u03bb s \u2192 P (s _+_)) P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  sumStableUnder SU-p = SU-p _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  toList\u2261toList\u2218 : toList \u2261 toList\u2218\n  toList\u2261toList\u2218 = exploreMon\u2218-spec (List.monoid A) List.[_]\n\nmodule Explorable\u2081\u2080 {A} {explore\u2081 : Explore\u2081 A}\n                    (explore-ind\u2080 : ExploreInd \u2080 explore\u2081) where\n\n  reify : Reify explore\u2081\n  reify = explore-ind\u2080 (\u03bb s \u2192 Data\u03a0 s _) _,_\n\nmodule Explorable\u2081\u2081 {A} {explore\u2081 : Explore\u2081 A}\n                    (explore-ind\u2081 : ExploreInd \u2081 explore\u2081) where\n\n  unfocus : Unfocus explore\u2081\n  unfocus = explore-ind\u2081 Unfocus (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\nrecord Explorable A : \u2605\u2081 where\n  constructor mk\n  field\n    explore     : Explore\u2080 A\n    explore-ind : ExploreInd\u2080 explore\n\n  open Explorable\u2080 explore-ind\n  field\n    adequate-sum     : AdequateSum sum\n--  adequate-product : AdequateProduct product\n\n  open Explorable\u2080 explore-ind public\n\nopen Explorable public\n\nExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\nExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\nrecord Funable A : \u2605\u2082 where\n  constructor _,_\n  field\n    explorable : Explorable A\n    negative   : ExploreForFun A\n\nmodule DistFun {A} (\u03bcA : Explorable A)\n                   (\u03bcA\u2192 : ExploreForFun A)\n                   {B} (\u03bcB : Explorable B){X}\n                   (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                   (\u2219 : X \u2192 X \u2192 X)\n                   (\u25ce : X \u2192 X \u2192 X) where\n\n  \u03a0' = explore \u03bcA \u25ce\n  \u03a3\u1d2e = explore \u03bcB \u2219\n  \u03a3' = explore (\u03bcA\u2192 \u03bcB) \u2219\n\n  DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\nDistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\nDistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                   \u2200 \u25ce \u2192 _DistributesOver_ _\u2248_ \u25ce _\u2219_ \u2192 \u25ce Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                   \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ _\u2219_ \u25ce\n\nDistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\nDistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 Explorable A \u2192 Explorable B\n\u03bc-iso {A}{B} A\u2194B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n  where\n    A\u2192B = to A\u2194B\n    ade = \u03bb f \u2192 sym-first-iso A\u2194B FI.\u2218 adequate-sum \u03bcA (f \u2218 A\u2192B)\n\n-- I guess this could be more general\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 from A\u2194B)\n\u03bc-iso-preserve A\u2194B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 \u2261.sym \u2218 Inverse.left-inverse-of A\u2194B)\n\n\u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n\u03bcLift = \u03bc-iso (FI.sym Lift\u2194id)\n\nexplore-swap' : \u2200 {A B} cm (\u03bcA : Explorable A) (\u03bcB : Explorable B) f \u2192\n               let open CMon cm\n                   s\u1d2c = explore \u03bcA _\u2219_\n                   s\u1d2e = explore \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nexplore-swap' cm \u03bcA \u03bcB f = explore-swap \u03bcA sg f (explore-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : Explorable A) (\u03bcB : Explorable B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two\nopen import Data.Nat.NP hiding (_^_; _\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product\nopen import Data.Sum\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Binary\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nopen import Explore.Type\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule Explorable\u2098\u209a\n    {m p A}\n    {explore     : Explore m A}\n    (explore-ind : ExploreInd p explore) where\n\n  explore-sg-ext : ExploreSgExt _ explore\n  explore-sg-ext sg {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ f \u2248 s _ g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  explore-mono : ExploreMono _ explore\n  explore-mono _\u2286_ _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n    explore-ind (\u03bb s \u2192 s _ f \u2286 s _ g) _\u2219-mono_ f\u2286\u00b0g\n\n  explore-swap : ExploreSwap _ explore\n  explore-swap sg f {s\u1d2e} pf =\n    explore-ind (\u03bb s \u2192 s _ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s _ \u2218 flip f))\n               (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _)))\n               (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\n  exploreMon : \u2200 {\u2113} (M : Monoid m \u2113) \u2192 ExploreMon M A\n  exploreMon M = explore _\u2219_\n    where open Mon M\n\n  explore\u2218 : \u2200 {M} \u2192 (M \u2192 M \u2192 M) \u2192 (A \u2192 M) \u2192 (M \u2192 M)\n  explore\u2218 = explore\u2218FromExplore explore\n\n  exploreMon\u2218 : \u2200 {\u2113} (M : Monoid m \u2113) \u2192 ExploreMon M A\n  exploreMon\u2218 M f = explore\u2218 _\u2219_ f \u03b5 where open Mon M\n\nplugKit : \u2200 {m p A} (M : Monoid m p) \u2192 ExploreIndKit _ {A = A} (ExplorePlug M)\nplugKit M = (\u03bb Ps Ps' _ x \u2192\n                  trans (\u2219-cong (sym (Ps _ _)) refl)\n                        (trans (assoc _ _ _)\n                               (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n            , (\u03bb x f _ \u2192 \u2219-cong (proj\u2082 identity (f x)) refl)\n     where open Mon M\n\nmodule Explorable\u2098\n    {m A}\n    {explore     : Explore m A}\n    (explore-ind : ExploreInd m explore) where\n  open Explorable\u2098\u209a explore-ind\n\n  explore\u2218-plug : (M : Monoid m m) \u2192 ExplorePlug M explore\n  explore\u2218-plug M = explore-ind $kit plugKit M\n\n  exploreMon\u2218-spec : \u2200 (M : Monoid _ m) \u2192\n                      let open Mon M in\n                      (f : A \u2192 C) \u2192 exploreMon M f \u2248 exploreMon\u2218 M f\n  exploreMon\u2218-spec M f = proj\u2082 (explore-ind\n                     (\u03bb s \u2192 ExplorePlug M s \u00d7 s _\u2219_ f \u2248 explore\u2218FromExplore s _\u2219_ f \u03b5)\n                     (\u03bb {s} {s'} Ps Ps' \u2192\n                        P\u2219 {s} {s'} (proj\u2081 Ps) (proj\u2081 Ps')\n                      , trans (\u2219-cong (proj\u2082 Ps) (proj\u2082 Ps')) (proj\u2081 Ps f _))\n                     (\u03bb x \u2192 Pf x , sym (proj\u2082 identity _)))\n                        where open Mon M\n                              open ExploreIndKit (plugKit M)\n\n  explore\u2218-ind : \u2200 (M : Monoid m m) \u2192 ExploreMonInd m M (exploreMon\u2218 M)\n  explore\u2218-ind M P P\u2219 Pf P\u2248 =\n    proj\u2082 (explore-ind (\u03bb s \u2192 ExplorePlug M s \u00d7 P (\u03bb f \u2192 s _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n               (\u03bb {s} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {s} {s'} (proj\u2081 Ps) (proj\u2081 Ps')\n                                  , P\u2248 (\u03bb f \u2192 proj\u2081 Ps f _) (P\u2219 (proj\u2082 Ps) (proj\u2082 Ps')))\n               (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                    , P\u2248 (\u03bb f \u2192 sym (proj\u2082 identity _)) (Pf x)))\n    where open Mon M\n\n  explore-ext : ExploreExt explore\n  explore-ext op = explore-ind (\u03bb s \u2192 s _ _ \u2261 s _ _) (\u2261.cong\u2082 op)\n\n  \u27e6explore\u27e7\u1d64 : \u2200 {A\u1d63 : A \u2192 A \u2192 \u2605_ _}\n               (A\u1d63-refl : Reflexive A\u1d63)\n              \u2192 \u27e6Explore\u27e7\u1d64 A\u1d63 explore explore\n  \u27e6explore\u27e7\u1d64 A\u1d63-refl M\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb s \u2192 M\u1d63 (s _ _) (s _ _)) (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n  explore-\u03b5 : Explore\u03b5 m m explore\n  explore-\u03b5 M = explore-ind (\u03bb s \u2192 s _ (const \u03b5) \u2248 \u03b5)\n                          (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5))\n                          (\u03bb _ \u2192 refl)\n    where open Mon M\n\n  explore-hom : ExploreMonHom m m explore\n  explore-hom cm f g = explore-ind (\u03bb s \u2192 s _ (f \u2219\u00b0 g) \u2248 s _ f \u2219 s _ g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  explore-lin\u02e1 : ExploreLin\u02e1 m m explore\n  explore-lin\u02e1 m _\u25ce_ f k dist = explore-ind (\u03bb s \u2192 s _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce s _\u2219_ f) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (sym (dist k _ _))) (\u03bb x \u2192 refl)\n    where open Mon m\n\n  explore-lin\u02b3 : ExploreLin\u02b3 m m explore\n  explore-lin\u02b3 m _\u25ce_ f k dist = explore-ind (\u03bb s \u2192 s _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 s _\u2219_ f \u25ce k) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (sym (dist k _ _))) (\u03bb x \u2192 refl)\n    where open Mon m\n\n  module _\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 m)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (_+_ : Op\u2082 S)\n       (_*_ : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom : \u2200 x y \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore _+_ g) \u2248 explore _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb s \u2192 f (s _+_ g) \u2248 s _*_ (f \u2218 g))\n                               (\u03bb p q \u2192 \u2248-trans (hom _ _) (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  lift-hom-\u2261 :\n      \u2200 {S T}\n        (_+_ : Op\u2082 S)\n        (_*_ : Op\u2082 T)\n        (f   : S \u2192 T)\n        (g   : A \u2192 S)\n        (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore _+_ g) \u2261 explore _*_ (f \u2218 g)\n  lift-hom-\u2261 _+_ _*_ = lift-hom _\u2261_ \u2261.refl \u2261.trans _+_ _*_ (\u2261.cong\u2082 _*_)\n\nmodule Explorable\u2080\n    {A}\n    {explore     : Explore\u2080 A}\n    (explore-ind : ExploreInd\u2080 explore) where\n  open Explorable\u2098\u209a explore-ind public\n  open Explorable\u2098  explore-ind public\n\n  sum : Sum A\n  sum = explore _+_\n\n  \u27e6explore\u27e7 : \u2200 {A\u1d63 : A \u2192 A \u2192 \u2605_ _}\n               (A\u1d63-refl : Reflexive A\u1d63)\n              \u2192 \u27e6Explore\u27e7 A\u1d63 explore explore\n  \u27e6explore\u27e7 = \u27e6explore\u27e7\u1d64\n\n  sum-ind : SumInd sum\n  sum-ind P P+ Pf = explore-ind (\u03bb s \u2192 P (s _+_)) P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  sumStableUnder SU-p = SU-p _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore _\u2227_\n  and   = big-\u2227\n\n  big-\u2228 = explore _\u2228_\n  or    = big-\u2228\n\n  big-xor = explore _xor_\n\n  toBinTree : BinTree A\n  toBinTree = explore fork leaf\n\n  toList : List A\n  toList = explore _++_ List.[_]\n\n  toList\u2218 : List A\n  toList\u2218 = explore\u2218 _++_ List.[_] List.[]\n\n  toList\u2261toList\u2218 : toList \u2261 toList\u2218\n  toList\u2261toList\u2218 = exploreMon\u2218-spec (List.monoid A) List.[_]\n\n  find? : Find? A\n  find? = explore (M?._\u2223_ _)\n\n  findKey : FindKey A\n  findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule Explorable\u2081\u2080 {A} {explore\u2081 : Explore\u2081 A}\n                    (explore-ind\u2080 : ExploreInd \u2080 explore\u2081) where\n\n  reify : Reify explore\u2081\n  reify = explore-ind\u2080 (\u03bb s \u2192 Data\u03a0 s _) _,_\n\nmodule Explorable\u2081\u2081 {A} {explore\u2081 : Explore\u2081 A}\n                    (explore-ind\u2081 : ExploreInd \u2081 explore\u2081) where\n\n  unfocus : Unfocus explore\u2081\n  unfocus = explore-ind\u2081 Unfocus (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\nrecord Explorable A : \u2605\u2081 where\n  constructor mk\n  field\n    explore     : Explore\u2080 A\n    explore-ind : ExploreInd\u2080 explore\n\n  open Explorable\u2080 explore-ind\n  field\n    adequate-sum     : AdequateSum sum\n--  adequate-product : AdequateProduct product\n\n  open Explorable\u2080 explore-ind public\n\nopen Explorable public\n\nExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\nExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\nrecord Funable A : \u2605\u2082 where\n  constructor _,_\n  field\n    explorable : Explorable A\n    negative   : ExploreForFun A\n\nmodule DistFun {A} (\u03bcA : Explorable A)\n                   (\u03bcA\u2192 : ExploreForFun A)\n                   {B} (\u03bcB : Explorable B){X}\n                   (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                   (\u2219 : X \u2192 X \u2192 X)\n                   (\u25ce : X \u2192 X \u2192 X) where\n\n  \u03a0' = explore \u03bcA \u25ce\n  \u03a3\u1d2e = explore \u03bcB \u2219\n  \u03a3' = explore (\u03bcA\u2192 \u03bcB) \u2219\n\n  DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\nDistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\nDistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                   \u2200 \u25ce \u2192 _DistributesOver_ _\u2248_ \u25ce _\u2219_ \u2192 \u25ce Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                   \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ _\u2219_ \u25ce\n\nDistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\nDistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 Explorable A \u2192 Explorable B\n\u03bc-iso {A}{B} A\u2194B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n  where\n    A\u2192B = to A\u2194B\n    ade = \u03bb f \u2192 sym-first-iso A\u2194B FI.\u2218 adequate-sum \u03bcA (f \u2218 A\u2192B)\n\n-- I guess this could be more general\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 from A\u2194B)\n\u03bc-iso-preserve A\u2194B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 \u2261.sym \u2218 Inverse.left-inverse-of A\u2194B)\n\n\u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n\u03bcLift = \u03bc-iso (FI.sym Lift\u2194id)\n\nexplore-swap' : \u2200 {A B} cm (\u03bcA : Explorable A) (\u03bcB : Explorable B) f \u2192\n               let open CMon cm\n                   s\u1d2c = explore \u03bcA _\u2219_\n                   s\u1d2e = explore \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nexplore-swap' cm \u03bcA \u03bcB f = explore-swap \u03bcA sg f (explore-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : Explorable A) (\u03bcB : Explorable B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"337ab7d5b181baeff495d34d16ef3ba4ab017882","subject":"Data.Bits.vnot\u2218vnot\u2257id","message":"Data.Bits.vnot\u2218vnot\u2257id\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"71d3cf829a84e65c7c028a11e8b02c35297f9f49","subject":"make the Agda specification of `coerce` useful for humans","message":"make the Agda specification of `coerce` useful for humans\n\nType-safety is sacrificed to describe `coerce` without prohibitive\nadministrative overhead. Unfortunately, `coerce` calls a postulate\non every encounter of a recursive type. It is only good for humans\nto read right now; it is not executable.\n","repos":"yfcai\/CREG","old_file":"experiments\/Coerce.agda","new_file":"experiments\/Coerce.agda","new_contents":"{-# OPTIONS --guardedness-preserving-type-constructors #-}\n\n-- Agda specification of `coerce`\n--\n-- Incomplete: too much work just to make type system happy.\n-- Conversions between morally identical types obscure\n-- the operational content.\n\nopen import Function\nopen import Data.Product\nopen import Data.Maybe\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Sum\nopen import Data.Nat\nimport Data.Fin as Fin\nopen Fin using (Fin ; zero ; suc)\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Coinduction renaming (Rec to Fix ; fold to roll ; unfold to unroll)\n\n-- Things fail sometimes\npostulate fail : \u2200 {\u2113} {A : Set \u2113} \u2192 A\n\n-- I know what I'm doing\npostulate unsafeCast : \u2200 {A B : Set} \u2192 A \u2192 B\n\ndata Data : Set where\n  base : \u2115 \u2192 Data\n  pair : Data \u2192 Data \u2192 Data\n  plus : Data \u2192 Data \u2192 Data\n  var  : \u2115 \u2192 Data\n  fix  : Data \u2192 Data\n\n-- infinite environments\nEnv : \u2200 {\u2113} \u2192 Set \u2113 \u2192 Set \u2113\nEnv A = \u2115 \u2192 A\n\ninfixr 0 decide_then_else_\n\ndecide_then_else_ : \u2200 {p \u2113} {P : Set p} {A : Set \u2113} \u2192 Dec P \u2192 A \u2192 A \u2192 A\ndecide yes p then x else y = x\ndecide no \u00acp then x else y = y\n\n--update : \u2200 {\u2113} {A : Set \u2113} \u2192 \u2115 \u2192 A \u2192 Env A \u2192 Env A\n--update n v env = \u03bb m \u2192 decide n \u225f m then v else env m\n\nprepend : \u2200 {\u2113} {A : Set \u2113} \u2192 A \u2192 Env A \u2192 Env A\nprepend v env zero = v\nprepend v env (suc m) = env m\n\n-- universe construction; always terminating.\n{-# NO_TERMINATION_CHECK #-}\nuc : Data \u2192 Env Set \u2192 Set\nuc (base x) \u03c1 = Fin x\nuc (pair S T) \u03c1 = uc S \u03c1 \u00d7 uc T \u03c1\nuc (plus S T) \u03c1 = uc S \u03c1 \u228e uc T \u03c1\nuc (var x) \u03c1 = \u03c1 x\nuc (fix D) \u03c1 = set where set = Fix (\u266f (uc D (prepend set \u03c1)))\n\n\u2205 : \u2200 {\u2113} {A : Set \u2113} \u2192 Env A\n\u2205 = fail\n\nmy-nat-data : Data\nmy-nat-data = fix (plus (base 1) (var zero))\n\nMy-nat : Set\nMy-nat = uc my-nat-data \u2205\n\nmy-zero : My-nat\nmy-zero = roll (inj\u2081 zero)\n\nmy-one : My-nat\nmy-one = roll (inj\u2082 (roll (inj\u2081 zero)))\n\n-- nonterminating stuff\nbotData : Data\nbotData = fix (var zero)\n\nbotType : Set\nbotType = uc botData \u2205\n\n{-# NON_TERMINATING #-}\n\u22ef : botType\n\u22ef = roll (roll (roll (roll (roll (roll \u22ef)))))\n\nshift : \u2115 \u2192 \u2115 \u2192 Data \u2192 Data\nshift c d (base x) = base x\nshift c d (pair S T) = pair (shift c d S) (shift c d T)\nshift c d (plus S T) = plus (shift c d S) (shift c d T)\nshift c d (var k) = decide suc k \u2264? c then var k else var (k + d)\nshift c d (fix D) = fix (shift c d D)\n\n_[_\u21a6_] : Data \u2192 \u2115 \u2192 Data \u2192 Data\nbase x [ n \u21a6 S ] = base x\npair T T\u2081 [ n \u21a6 S ] = pair (T [ n \u21a6 S ]) (T\u2081 [ n \u21a6 S ])\nplus T T\u2081 [ n \u21a6 S ] = plus (T [ n \u21a6 S ]) (T\u2081 [ n \u21a6 S ])\nvar k [ n \u21a6 S ] = decide k \u225f n then S else var k\nfix T [ n \u21a6 S ] = fix (T [ n + 1 \u21a6 shift 0 1 S ])\n\n\nimport Level\nopen import Category.Monad\nopen RawMonad {Level.zero} monad\n\n{-# NON_TERMINATING #-}\ncoerce : (S : Data) \u2192 (T : Data) \u2192 Maybe (uc S \u2205 \u2192 (uc T \u2205))\n\ncoerce (base m) (base n) with m \u2264? n\n... | yes m\u2264n = just (\u03bb s \u2192 Fin.inject\u2264 s m\u2264n)\n... | no  m>n = nothing\n\ncoerce (pair S\u2081 S\u2082) (pair T\u2081 T\u2082) =\n  coerce S\u2081 T\u2081 >>= (\u03bb f\u2081 \u2192\n  coerce S\u2082 T\u2082 >>= (\u03bb f\u2082 \u2192\n  return (\u03bb s \u2192 f\u2081 (proj\u2081 s) , f\u2082 (proj\u2082 s))))\n\ncoerce (plus S\u2081 S\u2082) (plus T\u2081 T\u2082) =\n  coerce S\u2081 T\u2081 >>= (\u03bb f\u2081 \u2192\n  coerce S\u2082 T\u2082 >>= (\u03bb f\u2082 \u2192\n  return [ inj\u2081 \u2218 f\u2081 , inj\u2082 \u2218 f\u2082 ]))\n\ncoerce (var j) T =\n  nothing -- should not happen\n\ncoerce (fix S) T =\n  let\n    S\u2032 = S [ 0 \u21a6 fix S ]\n  in\n    coerce S\u2032 T >>= (\u03bb f \u2192\n    return (\u03bb { (roll x) \u2192 f (unsafeCast x) }))\n\ncoerce S (fix T) =\n  let\n    T\u2032 = T [ 0 \u21a6 fix T ]\n  in\n    coerce S T\u2032 >>= (\u03bb f \u2192\n    return (\u03bb x \u2192 (roll (unsafeCast (f x)))))\n\ncoerce _ _ = nothing\n","old_contents":"{-# OPTIONS --guardedness-preserving-type-constructors #-}\n\n-- Agda specification of `coerce`\n--\n-- Incomplete: too much work just to make type system happy.\n-- Conversions between morally identical types obscure\n-- the operational content.\n\nopen import Data.Product\nopen import Data.Maybe\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec\nimport Data.Nat as \u2115\nopen \u2115 using (\u2115 ; suc ; zero)\nopen import Coinduction renaming (Rec to Fix ; fold to roll ; unfold to unroll)\n\ndata Data (\u2113 : \u2115) : Set where\n  base : \u2115 \u2192 Data \u2113\n  pair : Data \u2113 \u2192 Data \u2113 \u2192 Data \u2113\n  plus : Data \u2113 \u2192 Data \u2113 \u2192 Data \u2113\n  var  : Fin \u2113 \u2192 Data \u2113 -- de Bruijn index\n  fix  : Data (suc \u2113) \u2192 Data \u2113\n\ntry : \u2200 {\u2113 m} {A : Set \u2113} {B : Set m} \u2192 B \u2192 B \u2192 A \u2192 A\ntry a b b\u2082 = b\u2082\n\nweaken-fin : \u2200{n} (m : \u2115) \u2192 Fin n \u2192 Fin (m \u2115.+ n)\nweaken-fin zero x = x\nweaken-fin (suc m) x = suc (weaken-fin m x)\n\n-- this terminates for sure\n{-# NO_TERMINATION_CHECK #-}\nweaken : \u2200 {n} \u2192 (m : \u2115) \u2192 Data n \u2192 Data (m \u2115.+ n)\nweaken 0 t = t\nweaken m (base x) = base x\nweaken m (pair S T) = pair (weaken m S) (weaken m T)\nweaken m (plus S T) = plus (weaken m S) (weaken m T)\nweaken m (var x) = var (weaken-fin m x)\nweaken 1 (fix T) = fix (weaken 1 T)\nweaken (suc m) (fix T) = weaken 1 (weaken m (fix T))\n\n-- universe construction; always terminating.\n{-# NO_TERMINATION_CHECK #-}\nuc : \u2200 {\u2113} \u2192 Data \u2113 \u2192 Vec Set \u2113 \u2192 Set\nuc (base n) _ = Fin n\nuc (pair \u03c3 \u03c4) \u03c1 = uc \u03c3 \u03c1 \u00d7 uc \u03c4 \u03c1\nuc (plus \u03c3 \u03c4) \u03c1 = uc \u03c3 \u03c1 \u228e uc \u03c4 \u03c1\nuc (var i) \u03c1 = lookup i \u03c1\nuc (fix \u03c4) \u03c1 = set where set = Fix (\u266f (uc \u03c4 (set \u2237 \u03c1)))\n\nmy-nat-data : Data 0\nmy-nat-data = fix (plus (base 1) (var zero))\n\nMy-nat : Set\nMy-nat = uc my-nat-data []\n\nmy-zero : My-nat\nmy-zero = roll (inj\u2081 zero)\n\nmy-one : My-nat\nmy-one = roll (inj\u2082 (roll (inj\u2081 zero)))\n\n-- nonterminating stuff\nbotData : Data 0\nbotData = fix (var zero)\n\nbotType : Set\nbotType = uc botData []\n\n{-# NON_TERMINATING #-}\n\u22ef : botType\n\u22ef = roll (roll (roll (roll (roll (roll \u22ef)))))\n\n-- object language: STLC with fix\n\ninfixr 3 _\u21d2_\n\n-- is a coinductive datatype\ndata Type : Set\u2081 where\n  base : (\u03b9 : Set) \u2192 Type\n  _\u21d2_  : (\u03c3 : Type) \u2192 (\u03c4 : Type) \u2192 Type\n  _*_  : \u221e Type \u2192 \u221e Type \u2192 Type\n  _+_  : \u221e Type \u2192 \u221e Type \u2192 Type\n\n-- is generative on coinductive data\n{-# NO_TERMINATION_CHECK #-}\nval : Type \u2192 Set\nval (base \u03b9) = \u03b9\nval (\u03c3 \u21d2 \u03c4) = val \u03c3 \u2192 val \u03c4\nval (\u03c3 * \u03c4) = val (\u266d \u03c3) \u00d7 val (\u266d \u03c4)\nval (\u03c3 + \u03c4) = val (\u266d \u03c3) \u228e val (\u266d \u03c4)\n\nContext : \u2115 \u2192 Set\u2081\nContext = Vec Type\n\ndata Term {n} (\u0393 : Context n) : Type \u2192 Set\u2081 where\n  var : (i : Fin n) \u2192 Term \u0393 (lookup i \u0393)\n  app : \u2200 {\u03c3 \u03c4} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192 (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2237 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  fix : \u2200 {\u03c4} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- intro\/elim products\n  pair : \u2200 {\u03c3 \u03c4} \u2192 Term \u0393 \u03c3 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u266f \u03c3 * \u266f \u03c4)\n  uncr : \u2200 {\u03c3 \u03c4 \u03b2} \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4 \u21d2 \u03b2) \u2192 Term \u0393 (\u266f \u03c3 * \u266f \u03c4) \u2192 Term \u0393 \u03b2\n\n  -- intro\/elim sums\n  left : \u2200 {\u03c3 \u03c4} \u2192 Term \u0393 \u03c3 \u2192 Term \u0393 (\u266f \u03c3 + \u266f \u03c4)\n  rite : \u2200 {\u03c3 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u266f \u03c3 + \u266f \u03c4)\n  mach : \u2200 {\u03c3 \u03c4 \u03b2} \u2192\n           Term \u0393 (\u03c3 \u21d2 \u03b2) \u2192 Term \u0393 (\u03c4 \u21d2 \u03b2) \u2192 Term \u0393 (\u266f \u03c3 + \u266f \u03c4) \u2192 Term \u0393 \u03b2\n\n  -- intro\/elim base types\n  base : \u2200 {S : Set} \u2192 S \u2192 Term \u0393 (base S)\n  call : \u2200 {S T : Set} \u2192 Term \u0393 (base (S \u2192 T)) \u2192 Term \u0393 (base S) \u2192 Term \u0393 (base T)\n\ndata Env : \u2200 {n} (\u0393 : Context n) \u2192 Set\u2081 where\n  [] : Env []\n  _\u2237_ : \u2200 {n \u03c4} {\u0393 : Context n} \u2192 val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2237 \u0393)\n\nseek : \u2200 {n} {\u0393 : Context n} \u2192 (i : Fin n) \u2192 Env \u0393 \u2192 val (lookup i \u0393)\nseek zero    (v \u2237 _) = v\nseek (suc i) (_ \u2237 \u03c1) = seek i \u03c1\n\n{-# NON_TERMINATING #-}\neval : \u2200 {n} {\u0393 : Context n} {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env \u0393 \u2192 val \u03c4\neval (var x) \u03c1   = seek x \u03c1\neval (app s t) \u03c1 = eval s \u03c1 (eval t \u03c1)\neval (abs t) \u03c1   = \u03bb v \u2192 eval t (v \u2237 \u03c1)\neval (fix f) \u03c1   = eval (app f (fix f)) \u03c1\n\neval (pair s t) \u03c1 = eval s \u03c1 , eval t \u03c1\neval (uncr f p) \u03c1 = uncurry (eval f \u03c1) (eval p \u03c1)\n\neval (left x) \u03c1 = inj\u2081 (eval x \u03c1)\neval (rite y) \u03c1 = inj\u2082 (eval y \u03c1)\neval (mach f g t) \u03c1 = [ eval f \u03c1 , eval g \u03c1 ]\u2032 (eval t \u03c1)\n\neval (base v) \u03c1 = v\neval (call f x) \u03c1 = eval f \u03c1 (eval x \u03c1)\n\nnat-eq : \u2115 \u2192 \u2115 \u2192 Bool\nnat-eq  zero    zero   = true\nnat-eq (suc m) (suc n) = nat-eq m n\nnat-eq  _       _      = false\n\nfin-eq : \u2200 {n} \u2192 Fin n \u2192 Fin n \u2192 Bool\nfin-eq m n = nat-eq (to\u2115 m) (to\u2115 n)\n\ninfix 7 _==_\n_==_ : \u2200 {n} \u2192 Data n \u2192 Data n \u2192 Bool\nbase x == base y = nat-eq x y\npair \u03c3\u2081 \u03c4\u2081 == pair \u03c3\u2082 \u03c4\u2082 = \u03c3\u2081 == \u03c3\u2082 \u2227 \u03c4\u2081 == \u03c4\u2082\nplus \u03c3\u2081 \u03c4\u2081 == plus \u03c3\u2082 \u03c4\u2082 = \u03c3\u2081 == \u03c3\u2082 \u2227 \u03c4\u2081 == \u03c4\u2082\nvar x == var y = fin-eq x y\nfix \u03c3 == fix \u03c4 = \u03c3 == \u03c4\n_ == _ = false\n\n-- this is nonterminating for sure (because we use equirecursion)\n-- but it's generative (as in coinduction)\n-- we want it to expand a couple times during type checking\n{-# NO_TERMINATION_CHECK #-}\ntype-of : \u2200 {n} \u2192 Context n \u2192 Data n \u2192 Type\ntype-of _ (base n)   = base (Fin n)\ntype-of \u0393 (pair \u03c3 \u03c4) = \u266f (type-of \u0393 \u03c3) * \u266f (type-of \u0393 \u03c4)\ntype-of \u0393 (plus \u03c3 \u03c4) = \u266f (type-of \u0393 \u03c3) + \u266f (type-of \u0393 \u03c4)\ntype-of \u0393 (var x)    = lookup x \u0393\ntype-of \u0393 (fix \u03c4)    = type-of (type-of \u0393 (fix \u03c4) \u2237 \u0393) \u03c4\n\ninject-fin : \u2200 {m n} \u2192 m \u2115.\u2264 n \u2192 Fin m \u2192 Fin n\ninject-fin {zero} {zero} pf ()\ninject-fin {zero} {suc n} pf ()\ninject-fin {suc m} {zero} () i\ninject-fin {suc m} {suc n} pf zero = zero\ninject-fin {suc m} {suc n} (\u2115.s\u2264s pf) (suc i) = suc (inject-fin pf i)\n\nmapMaybe : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmapMaybe f (just x) = just (f x)\nmapMaybe f nothing = nothing\n\n{- have to do weakening properly...\nweak : \u2200 {n} {\u0393 : Context n} {\u03c3 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2237 \u0393) \u03c4\nweak (var i) = var (suc i)\nweak (app t t\u2081) = app (weak t) (weak t\u2081)\nweak {\u0393 = []} (abs t) = abs (try t {!!} {!!})\nweak {\u0393 = x \u2237 \u0393} (abs t) = {!!}\nweak (fix t) = fix (weak t)\nweak (pair t t\u2081) = pair (weak t) (weak t\u2081)\nweak (uncr t t\u2081) = uncr (weak t) (weak t\u2081)\nweak (left t) = left (weak t)\nweak (rite t) = rite (weak t)\nweak (mach t t\u2081 t\u2082) = mach (weak t) (weak t\u2081) (weak t\u2082)\nweak (base x) = base x\nweak (call t t\u2081) = call (weak t) (weak t\u2081)\n-}\n\nSrc : \u2200 {n} \u2192 Context n \u2192 Data n \u2192 Data n \u2192 Set\u2081\nSrc \u0393 S T = Term \u0393 (type-of \u0393 S \u21d2 type-of \u0393 T)\n\nWitness : \u2200 {n} \u2192 Context n \u2192 Set\u2081\nWitness {n} \u0393 = (S : Data n) \u2192 (T : Data n) \u2192 Maybe (Src \u0393 S T)\n\nResult : \u2200 {n} \u2192 Data n \u2192 Data n \u2192 Set\u2081\nResult {n} S T =\n  \u03a3 \u2115\n    (\u03bb m\u2032 \u2192 let m = m\u2032 \u2115.+ n in\n      \u03a3 (Context m) (\u03bb \u0393\u2032 \u2192 Witness \u0393\u2032 \u00d7 Src \u0393\u2032 (weaken m\u2032 S) (weaken m\u2032 T)))\n\ncoerce-src : \u2200 {n} {\u0393 : Context n} \u2192\n             Witness \u0393 \u2192 (S : Data n) \u2192 (T : Data n) \u2192\n             Maybe (Result S T)\n\ncoerce-src {n} {\u0393} A S T = match (A S T) where\n  \u03c3\u2192\u03c4 : Type\n  \u03c3\u2192\u03c4 = type-of \u0393 S \u21d2 type-of \u0393 T\n\n  A\u2080 : Witness (\u03c3\u2192\u03c4 \u2237 \u0393)\n  A\u2080 S' T' =\n    if S' == weaken 1 S \u2227 T' == weaken 1 T then\n      {!!} -- just (var zero) -- error due to lack of awareness of equality\n    else\n      weaken-witness A S' T'\n      where\n        weaken-witness : Witness \u0393 \u2192 Witness (\u03c3\u2192\u03c4 \u2237 \u0393)\n        weaken-witness = {!!} -- some form of fmap'ed weeakening\n\n  inspect : (S : Data n) \u2192 (T : Data n) \u2192 Maybe (Result S T)\n\n  inspect (base m) (base n) = {!!}\n  {-\n    inj (m \u2115.\u2264? n) where\n    open import Relation.Nullary.Core\n    inj : Dec (m \u2115.\u2264 n) \u2192 Maybe (Witness \u0393 \u00d7 Src \u0393 (base m) (base n))\n    inj (yes m\u2264n) = just ({!!} , abs (call (base (inject-fin m\u2264n)) (var zero)))\n    inj (no _) = nothing\n  -}\n\n  inspect (pair S\u2081 S\u2082) (pair T\u2081 T\u2082) = {!!}\n  inspect (plus S\u2081 S\u2082) (plus T\u2081 T\u2082) = {!!}\n  inspect (var x) T = {!!}\n  inspect (fix S) T = {!!}\n  inspect _ _ = nothing\n\n  match : Maybe (Src \u0393 S T) \u2192 Maybe (Result S T)\n  match (just t) = just (0 , \u0393 , A , t)\n  match nothing = inspect S T\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"62c403042d4523ae7adb26be8f70414ea55e6955","subject":"Data.Bits","message":"Data.Bits\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u2026 : \u2200 {n} \u2192 Bits n\n0\u2026 = replicate 0b\n\n-- Warning: 0\u2026 {0} \u2261 1\u2026 {0}\n1\u2026 : \u2200 {n} \u2192 Bits n\n1\u2026 = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u2026\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u2026 \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u2026 x) \u27e9\n         x \u2295 (x \u2295 1\u2026) \u2261\u27e8 sym (\u2295-assoc x x 1\u2026) \u27e9\n         (x \u2295 x) \u2295 1\u2026 \u2261\u27e8 cong (flip _\u2295_ 1\u2026) (\u2295-\u2261 x) \u27e9\n         0\u2026 \u2295 1\u2026       \u2261\u27e8 \u2295-left-identity 1\u2026 \u27e9\n         1\u2026 \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u2026) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u2026 (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u2026 sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u2026 : \u2200 {n} \u2192 Bits n\n0\u2026 = replicate 0b\n\n-- Warning: 0\u2026 {0} \u2261 1\u2026 {0}\n1\u2026 : \u2200 {n} \u2192 Bits n\n1\u2026 = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u2026\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u2026 \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u2026 x) \u27e9\n         x \u2295 (x \u2295 1\u2026) \u2261\u27e8 sym (\u2295-assoc x x 1\u2026) \u27e9\n         (x \u2295 x) \u2295 1\u2026 \u2261\u27e8 cong (flip _\u2295_ 1\u2026) (\u2295-\u2261 x) \u27e9\n         0\u2026 \u2295 1\u2026       \u2261\u27e8 \u2295-left-identity 1\u2026 \u27e9\n         1\u2026 \u220e where open \u2261-Reasoning\n\nmsb : \u2200 {n} k \u2192 Bits (k + n) \u2192 Bits k\nmsb zero    bs       = []\nmsb (suc k) (b \u2237 bs) = b \u2237 msb k bs\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Maybe (Bits n)\nsucBCarry [] = nothing\nsucBCarry (0b \u2237 xs) = just (maybe\u2032 0\u2237_ (1b \u2237 0\u2026) (sucBCarry xs))\nsucBCarry (1b \u2237 xs) = maybe\u2032 (just \u2218\u2032 1\u2237_) nothing (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB xs = maybe\u2032 id 0\u2026 (sucBCarry xs)\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u2026 sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fb32be1b44d22f7733965008ac2f4f8845c1922e","subject":"example of export from module","message":"example of export from module\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/SemiGroups.agda","new_file":"livecode\/SemiGroups.agda","new_contents":"open import Base\n\nopen import Id\n\nmodule SemiGroups where\n\ntest : Type \u2192 Type\ntest A = (a : A) \u2192 (a == a)\n\nassoc : {A : Type} \u2192 (A \u2192 A \u2192 A) \u2192 Type\nassoc {A} _*_ = (x : A) \u2192 (y : A) \u2192 (z : A) \u2192 ((x * y) * z) == (x * (y * z))\n\nsemiGroup : (A : Type) \u2192 Type\nsemiGroup A = \u03a3 (A \u2192 A \u2192 A) (\u03bb op \u2192 assoc op) \n\nmodule SemiGroup(A : Type)(_\u00b7_ : A \u2192 A \u2192 A)(ass : (assoc _\u00b7_)) where\n  leftId : A \u2192 Type\n  leftId x = (y : A) \u2192 ((x \u00b7 y) == y)\n\n  rightId : A \u2192 Type\n  rightId x = (y : A) \u2192 ((y \u00b7 x) == y)\n\n  Id = \u03bb a \u2192 (leftId a) \u00d7 (rightId a)\n\n  idUnique : (x y : A) \u2192 leftId x \u2192 rightId y \u2192 y == x\n  idUnique x y lidx ridy = sym (lidx y) && ridy x\n\nopen import Nat\n\nass+ : assoc _+_\nass+ zero y z = refl (y + z)\nass+ (succ x) y z = succ # ass+ x y z\n \nmodule AddNatSG = SemiGroup \u2115 _+_ ass+\n\nlftId : {A : Type} \u2192 semiGroup A \u2192 A \u2192 Type\nlftId {A} [ a , x ] a\u2081 = lfid a\u2081 where\n  module SG = SemiGroup A a x\n  lfid = SG.leftId\n\n\n","old_contents":"open import Base\n\nopen import Id\n\nmodule SemiGroups where\n\ntest : Type \u2192 Type\ntest A = (a : A) \u2192 (a == a)\n\nassoc : {A : Type} \u2192 (A \u2192 A \u2192 A) \u2192 Type\nassoc {A} _*_ = (x : A) \u2192 (y : A) \u2192 (z : A) \u2192 ((x * y) * z) == (x * (y * z))\n\nsemiGroup : (A : Type) \u2192 Type\nsemiGroup A = \u03a3 (A \u2192 A \u2192 A) (\u03bb op \u2192 assoc op) \n\nmodule SemiGroup(A : Type)(_\u00b7_ : A \u2192 A \u2192 A)(ass : (assoc _\u00b7_)) where\n  leftId : A \u2192 Type\n  leftId x = (y : A) \u2192 ((x \u00b7 y) == y)\n\n  rightId : A \u2192 Type\n  rightId x = (y : A) \u2192 ((y \u00b7 x) == y)\n\n  Id = \u03bb a \u2192 (leftId a) \u00d7 (rightId a)\n\n  idUnique : (x y : A) \u2192 leftId x \u2192 rightId y \u2192 y == x\n  idUnique x y lidx ridy = sym (lidx y) && ridy x\n\nopen import Nat\n\nass+ : assoc _+_\nass+ zero y z = refl (y + z)\nass+ (succ x) y z = succ # ass+ x y z\n\nmodule AddNatSG = SemiGroup \u2115 _+_ ass+\n\n\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e2c5d8c15c6f96eba991fdceb281940176988501","subject":"Add \u03a3\u2243-second and \u00d7\u2243-second","message":"Add \u03a3\u2243-second and \u00d7\u2243-second\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Identities.agda","new_file":"lib\/Type\/Identities.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin.NP as Fin using (Fin; suc; zero; [zero:_,suc:_])\nopen import Data.Vec as Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\n\ncoe\u00d7= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081){x y}\n      \u2192 coe (\u00d7= A= B=) (x , y) \u2261 (coe A= x , coe B= y)\ncoe\u00d7= idp idp = idp\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n    private\n      module A\u2243 = Equiv A\u2243\n      A\u2192 = A\u2243.\u00b7\u2192\n      A\u2190 = A\u2243.\u00b7\u2190\n      module B\u2243 = Equiv B\u2243\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n\n    {-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 A\u2192 B\u2192) (map\u00d7 A\u2190 B\u2190)\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-r x) ({!coe-\u03b2!} \u2219 B\u2243.\u00b7\u2190-inv-r y) })\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-l x) {!!} })\n    -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e A\u2192 B\u2192) (map\u228e A\u2190 B\u2190)\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-r x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-r ]\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-l x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-l ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 B\u2192 \u2218 f \u2218 A\u2190)\n               (\u03bb f \u2192 B\u2190 \u2218 f \u2218 A\u2192)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-r _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-r x))) \n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-l _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-l x)))\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2243 B\u2081 x) where\n  private\n      module B\u2243 {x} = Equiv (B x)\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n  \u03a3\u2243-second : (\u03a3 A B\u2080) \u2243 (\u03a3 A B\u2081)\n  \u03a3\u2243-second = equiv (second B\u2192) (second B\u2190)\n                    (\u03bb { (x , y) \u2192 ap (_,_ x) (B\u2243.\u00b7\u2190-inv-r y) })\n                    (\u03bb { (x , y) \u2192 ap (_,_ x) (B\u2243.\u00b7\u2190-inv-l y) })\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : \u2605_ b}(B : B\u2080 \u2243 B\u2081) where\n  \u00d7\u2243-second : (A \u00d7 B\u2080) \u2243 (A \u00d7 B\u2081)\n  \u00d7\u2243-second = \u03a3\u2243-second A (\u03bb _ \u2192 B)\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n  !\u03a3=\u2032 : ! (\u03a3=\u2032 A B) \u2261 \u03a3=\u2032 A (!_ \u2218 B)\n  !\u03a3=\u2032 = !-ap _ (\u03bb= B) \u2219 ap (ap (\u03a3 A)) (!-\u03bb= B)\n\ncoe\u03a3=\u2032-aux : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (ap (\u03bb f \u2192 f x) (\u03bb= B)) y)\ncoe\u03a3=\u2032-aux A B with \u03bb= B\ncoe\u03a3=\u2032-aux A B | idp = idp\n\ncoe\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (B x) y)\ncoe\u03a3=\u2032 A B = coe\u03a3=\u2032-aux A B \u2219 ap (_,_ _) (coe-same (happly (happly-\u03bb= B) _) _)\n\ncoe!\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe! (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe! (B x) y)\ncoe!\u03a3=\u2032 A B {x}{y} = coe-same (!\u03a3=\u2032 A B) _ \u2219 coe\u03a3=\u2032 A (!_ \u2218 B)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2243Lift\ud835\udfd9\u228e : Maybe A \u2243 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n  Maybe\u2243Lift\ud835\udfd9\u228e = equiv (maybe inr (inl _))\n                        [inl: const nothing ,inr: just ]\n                        [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                        (maybe (\u03bb _ \u2192 idp) idp)\n\n  Vec0\u2243\ud835\udfd9 : Vec A 0 \u2243 \ud835\udfd9\n  Vec0\u2243\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec0\u2243Lift\ud835\udfd9 : Vec A 0 \u2243 Lift {\u2113 = a} \ud835\udfd9\n  Vec0\u2243Lift\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec\u2218suc\u2243\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2243 (A \u00d7 Vec A n)\n  Vec\u2218suc\u2243\u00d7 = equiv (\u03bb { (x \u2237 xs) \u2192 x , xs }) (\u03bb { (x , xs) \u2192 x \u2237 xs })\n                    (\u03bb { (x , xs) \u2192 idp }) (\u03bb { (x \u2237 xs) \u2192 idp })\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua Maybe\u2243Lift\ud835\udfd9\u228e\n\n    Vec0\u2261Lift\ud835\udfd9 : Vec A 0 \u2261 Lift {\u2113 = a} \ud835\udfd9\n    Vec0\u2261Lift\ud835\udfd9 = ua Vec0\u2243Lift\ud835\udfd9\n\n    Vec\u2218suc\u2261\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2261 (A \u00d7 Vec A n)\n    Vec\u2218suc\u2261\u00d7 = ua Vec\u2218suc\u2243\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}} where\n    Vec0\u2261\ud835\udfd9 : Vec A 0 \u2261 \ud835\udfd9\n    Vec0\u2261\ud835\udfd9 = ua Vec0\u2243\ud835\udfd9\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nFin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv [zero: inl _ ,suc: inr ] [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n  [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n  [zero: idp ,suc: (\u03bb _ \u2192 idp) ]\n\nFin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\n\nopen import Level.NP\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\n\nopen import Data.Maybe.NP using (Maybe ; just ; nothing ; maybe ; maybe\u2032 ; just-injective; Maybe^)\nopen import Data.Zero using (\ud835\udfd8 ; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two\nopen import Data.Fin.NP as Fin using (Fin; suc; zero; [zero:_,suc:_])\nopen import Data.Vec as Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.NP as \u2115 using (\u2115 ; suc ; zero; _+_)\nopen import Data.Product.NP renaming (map to map\u00d7)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]; map to map\u228e)\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_ ; ap; coe; coe!; !_; _\u2219_; J ; inspect ; [_]; tr; ap\u2082; apd) renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nmodule Type.Identities where\n\nopen Equivalences\n\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\n    \u2192= : (A\u2080 \u2192 B\u2080) \u2261 (A\u2081 \u2192 B\u2081)\n    \u2192= = ap\u2082 -\u2192- A= B=\n\n\ncoe\u00d7= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081){x y}\n      \u2192 coe (\u00d7= A= B=) (x , y) \u2261 (coe A= x , coe B= y)\ncoe\u00d7= idp idp = idp\n\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A\u2243 : A\u2080 \u2243 A\u2081)(B\u2243 : B\u2080 \u2243 B\u2081) where\n    private\n      module A\u2243 = Equiv A\u2243\n      A\u2192 = A\u2243.\u00b7\u2192\n      A\u2190 = A\u2243.\u00b7\u2190\n      module B\u2243 = Equiv B\u2243\n      B\u2192 = B\u2243.\u00b7\u2192\n      B\u2190 = B\u2243.\u00b7\u2190\n\n    {-\n    \u00d7\u2243 : (A\u2080 \u00d7 B\u2080) \u2243 (A\u2081 \u00d7 B\u2081)\n    \u00d7\u2243 = equiv (map\u00d7 A\u2192 B\u2192) (map\u00d7 A\u2190 B\u2190)\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-r x) ({!coe-\u03b2!} \u2219 B\u2243.\u00b7\u2190-inv-r y) })\n               (\u03bb { (x , y) \u2192 pair= (A\u2243.\u00b7\u2190-inv-l x) {!!} })\n    -}\n\n    \u228e\u2243 : (A\u2080 \u228e B\u2080) \u2243 (A\u2081 \u228e B\u2081)\n    \u228e\u2243 = equiv (map\u228e A\u2192 B\u2192) (map\u228e A\u2190 B\u2190)\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-r x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-r ]\n               [inl: (\u03bb x \u2192 ap inl (A\u2243.\u00b7\u2190-inv-l x)) ,inr: ap inr \u2218 B\u2243.\u00b7\u2190-inv-l ]\n\n    \u2192\u2243 : {{_ : FunExt}} \u2192 (A\u2080 \u2192 B\u2080) \u2243 (A\u2081 \u2192 B\u2081)\n    \u2192\u2243 = equiv (\u03bb f \u2192 B\u2192 \u2218 f \u2218 A\u2190)\n               (\u03bb f \u2192 B\u2190 \u2218 f \u2218 A\u2192)\n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-r _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-r x))) \n               (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 B\u2243.\u00b7\u2190-inv-l _ \u2219 ap f (A\u2243.\u00b7\u2190-inv-l x)))\n\nmodule _ {{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x) where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {a b}{A\u2080 : \u2605_ a}{B\u2080 : A\u2080 \u2192 \u2605_ b}{{_ : FunExt}} where\n    \u03a3= : {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n         {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n       \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= = J (\u03bb A\u2081 A= \u2192 {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n                    \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081) (\u03a3=\u2032 _)\n    -- \u03a3= idp B= = \u03a3=\u2032 _ B=\n\n    \u03a0= : \u2200 {A\u2081 : \u2605_ a}(A= : A\u2080 \u2261 A\u2081)\n           {B\u2081 : A\u2081 \u2192 \u2605_ b}(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A= x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B= = \u03a0=\u2032 _ B=\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n  !\u03a3=\u2032 : ! (\u03a3=\u2032 A B) \u2261 \u03a3=\u2032 A (!_ \u2218 B)\n  !\u03a3=\u2032 = !-ap _ (\u03bb= B) \u2219 ap (ap (\u03a3 A)) (!-\u03bb= B)\n\ncoe\u03a3=\u2032-aux : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (ap (\u03bb f \u2192 f x) (\u03bb= B)) y)\ncoe\u03a3=\u2032-aux A B with \u03bb= B\ncoe\u03a3=\u2032-aux A B | idp = idp\n\ncoe\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe (B x) y)\ncoe\u03a3=\u2032 A B = coe\u03a3=\u2032-aux A B \u2219 ap (_,_ _) (coe-same (happly (happly-\u03bb= B) _) _)\n\ncoe!\u03a3=\u2032 : \u2200{{_ : FunExt}}{a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){x y}\n  \u2192 coe! (\u03a3=\u2032 A B) (x , y) \u2261 (x , coe! (B x) y)\ncoe!\u03a3=\u2032 A B {x}{y} = coe-same (!\u03a3=\u2032 A B) _ \u2219 coe\u03a3=\u2032 A (!_ \u2218 B)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A\u2243 : A\u2080 \u2243 A\u2081)(B= : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A\u2243 x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A\u2243 B= = \u03a3= (ua A\u2243) \u03bb x \u2192 B= x \u2219 ap B\u2081 (! coe-\u03b2 A\u2243 x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    {-\nmodule _ {{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}(A : A\u2080 \u2243 A\u2081)(B : (x : A\u2081) \u2192 B\u2080 (<\u2013 A x) \u2243 B\u2081 x) where\n    \u03a0\u2243' : (\u03a0 A\u2080 B\u2080) \u2243 (\u03a0 A\u2081 B\u2081)\n    \u03a0\u2243' = equiv (\u03bb f x \u2192 \u2013> (B x) (f (<\u2013 A x)))\n                (\u03bb f x \u2192 tr B\u2080 (<\u2013-inv-l A x) (<\u2013 (B (\u2013> A x)) (f (\u2013> A x))))\n                (\u03bb f \u2192 \u03bb= (\u03bb x \u2192 {!apd (<\u2013-inv-l A (<\u2013 A x))!}))\n                {!\u03bb f \u2192 \u03bb= (\u03bb x \u2192 <\u2013-inv-l B _ \u2219 ap f (<\u2013-inv-l A x))!}\n    -}\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b} where\n    \u03a3-fst\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst\u2243 A B = \u03a3\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a3-fst= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2080 (B \u2218 coe A) \u2261 \u03a3 A\u2081 B\n    \u03a3-fst= A = \u03a3-fst\u2243 (coe-equiv A)\n\n    \u03a0-dom\u2243 : \u2200 (A : A\u2080 \u2243 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 \u2013> A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom\u2243 A B = \u03a0\u2243 A (\u03bb x \u2192 idp)\n\n    \u03a0-dom= : \u2200 (A : A\u2080 \u2261 A\u2081)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2080 (B \u2218 coe A) \u2261 \u03a0 A\u2081 B\n    \u03a0-dom= A = \u03a0-dom\u2243 (coe-equiv A)\n\n    -- variations where the equiv is transported backward on the right side\n\n    \u03a3-fst\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 <\u2013 A)\n    \u03a3-fst\u2243\u2032 A B = ! \u03a3-fst\u2243 (\u2243-sym A) B\n\n    \u03a3-fst=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a3 A\u2081 B \u2261 \u03a3 A\u2080 (B \u2218 coe! A)\n    \u03a3-fst=\u2032 A = \u03a3-fst\u2243\u2032 (coe-equiv A)\n\n    \u03a0-dom\u2243\u2032 : (A : A\u2081 \u2243 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 <\u2013 A)\n    \u03a0-dom\u2243\u2032 A B = ! \u03a0-dom\u2243 (\u2243-sym A) B\n\n    \u03a0-dom=\u2032 : (A : A\u2081 \u2261 A\u2080)(B : A\u2081 \u2192 \u2605_ b) \u2192 \u03a0 A\u2081 B \u2261 \u03a0 A\u2080 (B \u2218 coe! A)\n    \u03a0-dom=\u2032 A = \u03a0-dom\u2243\u2032 (coe-equiv A)\n\nmodule _ {a b c} {A : \u2605_ a} {B : A \u2192 \u2605_ b} {C : \u03a3 A B \u2192 \u2605_ c} where\n    \u03a0\u03a3-curry-equiv : \u03a0 (\u03a3 A B) C \u2243 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry-equiv = equiv curry uncurry (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a3-curry : {{_ : UA}} \u2192 \u03a0 (\u03a3 A B) C \u2261 ((x : A) (y : B x) \u2192 C (x , y))\n    \u03a0\u03a3-curry = ua \u03a0\u03a3-curry-equiv\n\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2243 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 = equiv [0: inl _ 1: inr _ ]\n              [inl: const 0\u2082 ,inr: const 1\u2082 ]\n              [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n              [0: idp 1: idp ]\n\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 : {{_ : UA}} \u2192 \ud835\udfda \u2261 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2261\ud835\udfd9\u228e\ud835\udfd9 = ua \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9\n\nmodule _ {{_ : UA}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{a b c}{A : \u2605_ a}{B : \u2605_ b}{C : A \u228e B \u2192 \u2605_ c} where\n    dist-\u228e-\u03a3-equiv : \u03a3 (A \u228e B) C \u2243 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u03a3 : \u03a3 (A \u228e B) C \u2261 (\u03a3 A (C \u2218 inl) \u228e \u03a3 B (C \u2218 inr))\n    dist-\u228e-\u03a3 = ua dist-\u228e-\u03a3-equiv\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0-equiv : \u03a0 (A \u228e B) C \u2243 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u03a0 : \u03a0 (A \u228e B) C \u2261 (\u03a0 A (C \u2218 inl) \u00d7 \u03a0 B (C \u2218 inr))\n      dist-\u00d7-\u03a0 = ua dist-\u00d7-\u03a0-equiv\n\nmodule _ {{_ : UA}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3[] : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3[] = dist-\u228e-\u03a3\n\n    module _{{_ : FunExt}} where\n      dist-\u00d7-\u03a0[] : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n      dist-\u00d7-\u03a0[] = dist-\u00d7-\u03a0\n\nmodule _ {{_ : UA}}{A B C : \u2605} where\n    dist-\u228e-\u00d7-equiv : ((A \u228e B) \u00d7 C) \u2243 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7-equiv = equiv (\u03bb { (inl x , y) \u2192 inl (x , y)\n                              ; (inr x , y) \u2192 inr (x , y) })\n                           [inl: (\u03bb x \u2192 inl (fst x) , snd x)\n                           ,inr: (\u03bb x \u2192 inr (fst x) , snd x) ]\n                           [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                           (\u03bb { (inl x , y) \u2192 idp\n                              ; (inr x , y) \u2192 idp })\n\n    dist-\u228e-\u00d7 : ((A \u228e B) \u00d7 C) \u2261 (A \u00d7 C \u228e B \u00d7 C)\n    dist-\u228e-\u00d7 = ua dist-\u228e-\u00d7-equiv\n\n    module _ {{_ : FunExt}} where\n      dist-\u00d7-\u2192-equiv : ((A \u228e B) \u2192 C ) \u2243 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192-equiv = equiv (\u03bb f \u2192 f \u2218 inl , f \u2218 inr) (\u03bb fg \u2192 [inl: fst fg ,inr: snd fg ])\n                             (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ])\n\n      dist-\u00d7-\u2192 : ((A \u228e B) \u2192 C) \u2261 ((A \u2192 C) \u00d7 (B \u2192 C))\n      dist-\u00d7-\u2192 = ua dist-\u00d7-\u2192-equiv\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : (x : A) \u2192 B x \u2192 \u2605_ c} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {a b}{A : \u2605_ a} {B : \u2605_ b} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {a b c} {A : \u2605_ a} {B : \u2605_ b} {C : \u2605_ c} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}(A : \ud835\udfd8 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 Lift \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2080 : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2080 = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}{a}(A : \ud835\udfd9 \u2192 \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}{a}(A : \u2605_ a) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605\u2080) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq\u2080 (\u03bb _ \u2192 A)\n\n    \ud835\udfd8\u2192A\u2261\ud835\udfd9 = \u03a0\ud835\udfd8-uniq\u2032\n\nmodule _ {{_ : FunExt}}{\u2113}(F G : \ud835\udfd8 \u2192 \u2605_ \u2113) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {a \u2113} {A : \ud835\udfd8 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv : \u03a3 \ud835\udfd8 A \u2243 Lift {\u2113 = \u2113} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst-equiv = equiv (lift \u2218 fst) (\u03bb { (lift ()) }) (\u03bb { (lift ()) }) (\u03bb { (() , _) })\n\nmodule _ {a} {A : \ud835\udfd8 \u2192 \u2605_ a} {{_ : UA}} where\n    \u03a3\ud835\udfd8-lift\u2218fst : \u03a3 \ud835\udfd8 A \u2261 Lift {\u2113 = a} \ud835\udfd8\n    \u03a3\ud835\udfd8-lift\u2218fst = ua \u03a3\ud835\udfd8-lift\u2218fst-equiv\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {a}{A : \ud835\udfd9 \u2192 \u2605_ a} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    module _ {{_ : UA}} where\n        \u228e\ud835\udfd8-inl : A \u2261 (A \u228e \ud835\udfd8)\n        \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n\n        \ud835\udfd8\u228e-inr : A \u2261 (\ud835\udfd8 \u228e A)\n        \ud835\udfd8\u228e-inr = \u228e\ud835\udfd8-inl \u2219 \u228e-comm\n\n        \ud835\udfd9\u00d7-snd : (\ud835\udfd9 \u00d7 A) \u2261 A\n        \ud835\udfd9\u00d7-snd = \u03a3\ud835\udfd9-snd\n\n        \u00d7\ud835\udfd9-fst : (A \u00d7 \ud835\udfd9) \u2261 A\n        \u00d7\ud835\udfd9-fst = \u00d7-comm \u2219 \ud835\udfd9\u00d7-snd\n\n        \ud835\udfd8\u00d7-fst : (\ud835\udfd8 \u00d7 A) \u2261 \ud835\udfd8\n        \ud835\udfd8\u00d7-fst = \u03a3\ud835\udfd8-fst\n\n        \u00d7\ud835\udfd8-snd : (A \u00d7 \ud835\udfd8) \u2261 \ud835\udfd8\n        \u00d7\ud835\udfd8-snd = \u00d7-comm \u2219 \u03a3\ud835\udfd8-fst\n\n        -- old names\n        A\u00d7\ud835\udfd9\u2261A = \u00d7\ud835\udfd9-fst\n        \ud835\udfd9\u00d7A\u2261A = \ud835\udfd9\u00d7-snd\n        \ud835\udfd8\u228eA\u2261A = \ud835\udfd8\u228e-inr\n        A\u228e\ud835\udfd8\u2261A = \u228e\ud835\udfd8-inl\n        \ud835\udfd8\u00d7A\u2261\ud835\udfd8 = \ud835\udfd8\u00d7-fst\n        A\u00d7\ud835\udfd8\u2261\ud835\udfd8 = \u00d7\ud835\udfd8-snd\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : \u03a3 A B \u2192 \u2605} where\n    -- AC: Dependent axiom of choice\n    -- In Type Theory it happens to be neither an axiom nor to be choosing anything.\n    \u03a0\u03a3-comm-equiv : (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2243 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm-equiv = equiv (\u03bb H \u2192 fst \u2218 H , snd \u2218 H) (\u03bb H \u2192 < fst H , snd H >) (\u03bb H \u2192 idp) (\u03bb H \u2192 idp)\n\n    \u03a0\u03a3-comm : {{_ : UA}}\n            \u2192 (\u2200 (x : A) \u2192 \u2203 \u03bb (y : B x) \u2192 C (x , y)) \u2261 (\u2203 \u03bb (f : \u03a0 A B) \u2192 \u2200 (x : A) \u2192 C (x , f x))\n    \u03a0\u03a3-comm = ua \u03a0\u03a3-comm-equiv\n\nmodule _ {\u2113}{A : \ud835\udfda \u2192 \u2605_ \u2113}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e : \u03a3 \ud835\udfda A \u2261 (A 0\u2082 \u228e A 1\u2082)\n  \u03a3\ud835\udfda-\u228e = \u03a3-fst\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u228e-\u03a3 \u2219 \u228e= \u03a3\ud835\udfd9-snd \u03a3\ud835\udfd9-snd\n\n  \u03a0\ud835\udfda-\u00d7 : \u03a0 \ud835\udfda A \u2261 (A 0\u2082 \u00d7 A 1\u2082)\n  \u03a0\ud835\udfda-\u00d7 = \u03a0-dom\u2243\u2032 \ud835\udfda\u2243\ud835\udfd9\u228e\ud835\udfd9 _ \u2219 dist-\u00d7-\u03a0 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) (\u03a0\ud835\udfd9-uniq _)\n\n  \u03a0\ud835\udfdaF\u2261F\u2080\u00d7F\u2081 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}}{{_ : FunExt}} where\n  \u03a3\ud835\udfda-\u228e\u2032 : (\ud835\udfda \u00d7 A) \u2261 (A \u228e A)\n  \u03a3\ud835\udfda-\u228e\u2032 = \u03a3\ud835\udfda-\u228e\n\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 : (\ud835\udfda \u2192 A) \u2261 (A \u00d7 A)\n  \u03a0\ud835\udfda\u2192\u00d7\u2032 = \u03a0\ud835\udfda-\u00d7\n\nmodule _ {a}{A : \u2605_ a} where\n\n  \u03a3\u2261x\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (_\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3\u2261x\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192 pair= (snd p) (tr-r\u2261 (snd p) idp))\n\n  \u03a3x\u2261\u2243\ud835\udfd9 : \u2200 x \u2192 (\u03a3 A (flip _\u2261_ x)) \u2243 \ud835\udfd9\n  \u03a3x\u2261\u2243\ud835\udfd9 x = equiv (\u03bb _ \u2192 _) (\u03bb _ \u2192 x , idp) (\u03bb _ \u2192 idp) (\u03bb p \u2192  pair= (! snd p)  ( tr-l\u2261 (! snd p) idp \u2219\n    \u2219-refl (! (! (snd p))) \u2219 !-inv (snd p)))\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : A \u2192 B \u2192 \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n\n  \u03a0\u228e-equiv : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2243 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e-equiv = equiv (\u03bb f \u2192 [0: (\u03bb x y \u2192 f (inl x) y) 1: ((\u03bb x y \u2192 f (inr y) x)) ])\n                   (\u03bb f \u2192 [inl: f 0\u2082 ,inr: flip (f 1\u2082) ])\n                   (\u03bb f \u2192 \u03bb= [0: idp 1: idp ])\n                   (\u03bb f \u2192 \u03bb= [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ])\n\n  \u03a0\u228e : (\u03a0 (A \u228e B) [inl: (\u03bb x \u2192 \u2200 y \u2192 C x y) ,inr: (\u03bb y \u2192 \u2200 x \u2192 C x y) ]) \u2261 ((t : \ud835\udfda)(x : A)(y : B) \u2192 C x y)\n  \u03a0\u228e = ua \u03a0\u228e-equiv\n\nmodule _ {ab c}{A B : \u2605_ ab}{C : \u2605_ c}{{_ : UA}}{{_ : FunExt}} where\n  \u03a0\u228e\u2032 : (\u03a0 (A \u228e B) [inl: const (B \u2192 C) ,inr: const (A \u2192 C) ]) \u2261 (\ud835\udfda \u2192 A \u2192 B \u2192 C)\n  \u03a0\u228e\u2032 = \u03a0\u228e\n\nmodule _ where\n\n  private\n    maybe-elim : {X : Set}(m : Maybe X) \u2192 m \u2262 nothing \u2192 X\n    maybe-elim {X} m = maybe {B = \u03bb m' \u2192 m' \u2262 _ \u2192 _} (\u03bb x _ \u2192 x) (\u03bb e \u2192 \ud835\udfd8-elim (e idp)) m\n\n    maybe-elim-just : {X : Set}(m : Maybe X)(p : m \u2262 nothing)\n      \u2192 just (maybe-elim m p) \u2261 m\n    maybe-elim-just (just x) p = idp\n    maybe-elim-just nothing p = \ud835\udfd8-elim (p idp)\n\n    maybe-elim-cong : \u2200 {X : Set}{m m' : Maybe X}{p p'} \u2192 m \u2261 m'\n      \u2192 maybe-elim m p \u2261 maybe-elim m' p'\n    maybe-elim-cong {m = just x} idp = idp\n    maybe-elim-cong {m = nothing} {p = p} idp = \ud835\udfd8-elim (p idp)\n\n    not-just : {X : Set}{x : X} \u2192 _\u2262_ {A = Maybe X} (just x) nothing\n    not-just ()\n\n    !!-p : \u2200 {x}{X : Set x}{x y : X}(p : x \u2261 y) \u2192 ! (! p) \u2261 p\n    !!-p idp = idp\n\n    !\u2219 : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp\n    !\u2219 idp = idp\n\n    \u2219! : \u2200 {\u2113}{A : Set \u2113}{x y : A}(p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp\n    \u2219! idp = idp\n\n\n    record PreservePoint {A B : Set}(eq : Maybe A \u2261 Maybe B) : Set where\n      field\n        coe-p : \u2200 x \u2192 coe eq (just x) \u2262 nothing\n        coe!-p : \u2200 x \u2192 coe! eq (just x) \u2262 nothing\n    open PreservePoint\n\n    nothing-to-nothing : {A B : Set}(eq : Maybe A \u2261 Maybe B)\n      \u2192 coe eq nothing \u2261 nothing \u2192 PreservePoint eq\n    nothing-to-nothing eq n2n = record { coe-p = \u03bb x p \u2192 not-just (coe-inj eq (p \u2219 ! n2n))\n                                       ; coe!-p = \u03bb x p \u2192 not-just (coe-same (! !\u2219 eq) (just x)\n                                                 \u2219 ! (coe\u2218coe eq (! eq) (just x)) \u2219 ap (coe eq) p \u2219 n2n) }\n\n    !-PP : \u2200 {A B : Set} {e : Maybe A \u2261 Maybe B} \u2192 PreservePoint e \u2192 PreservePoint (! e)\n    !-PP {e = e} e-pp = record { coe-p = coe!-p e-pp\n                               ; coe!-p = \u03bb x p \u2192 coe-p e-pp x ( ap (\u03bb X \u2192 coe X (just x)) (! !!-p e ) \u2219 p) }\n\n    to : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e \u2192 A \u2192 B\n    to e e-pp x = maybe-elim (coe e (just x)) (coe-p e-pp x)\n\n    module _{A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}{ep ep'}(e'\u2219e=id : e' \u2219 e \u2261 idp)(x : B) where\n      to-to : to e ep (to e' ep' x) \u2261 x\n      to-to = just-injective (ap just (maybe-elim-cong\n                                                {m = coe e (just (maybe-elim (coe e' (just x)) _))}\n                                                {m' = coe e (coe e' (just x))}\n                                                {p' = \u03bb q \u2192 not-just (! p \u2219 q)} (ap (coe e) (maybe-elim-just _ _)))\n                                      \u2219 maybe-elim-just (coe e (coe e' (just x))) (\u03bb q \u2192 not-just (! p \u2219 q))\n                                      \u2219 p)\n        where\n          p : coe e (coe e' (just x)) \u2261 just x\n          p = coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n\n    cto : \u2200 {A B : Set}(e : Maybe A \u2261 Maybe B) \u2192 Maybe A \u2192 Maybe B\n    cto e = maybe (maybe\u2032 just (coe e nothing) \u2218 coe e \u2218 just) nothing\n\n    module _ {A B : Set}{e : Maybe A \u2261 Maybe B}{e' : Maybe B \u2261 Maybe A}(e'\u2219e=id : e' \u2219 e \u2261 idp) where\n      cto-cto : (x : Maybe B) \u2192 cto e (cto e' x) \u2261 x\n      cto-cto nothing = idp\n      cto-cto (just x) with coe e' (just x) | coe\u2218coe e e' (just x) \u2219 coe-same e'\u2219e=id (just x)\n      cto-cto (just x) | just x\u2081 | p = ap (maybe just (coe e nothing)) p\n      cto-cto (just x) | nothing | p with coe e' nothing\n                                        | coe\u2218coe e e' nothing \u2219 coe-same e'\u2219e=id nothing\n      cto-cto (just x) | nothing | p | just y  | q = ap (maybe just (coe e nothing)) q \u2219 p\n      cto-cto (just x) | nothing | p | nothing | q = ! q \u2219 p\n\n  module _ {A B : Set} where\n    create-PP-\u2243 : Maybe A \u2261 Maybe B \u2192 Maybe A \u2243 Maybe B\n    create-PP-\u2243 e = equiv (cto e)\n                          (cto (! e))\n                          (cto-cto {e = e} {e' = ! e} (!\u2219 e))\n                          (cto-cto {e = ! e}{e' = e} (\u2219! e))\n\n    create-PP : {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 \u03a3 (Maybe A \u2261 Maybe B) PreservePoint\n    create-PP eq = ua (create-PP-\u2243 eq) , nothing-to-nothing _ (coe-\u03b2 (create-PP-\u2243 eq) nothing)\n\n    Maybe-injective-PP : {{_ : UA}} \u2192 (e : Maybe A \u2261 Maybe B) \u2192 PreservePoint e\n      \u2192 A \u2261 B\n    Maybe-injective-PP e e-pp = ua\n      (equiv (to e e-pp)\n             (to (! e) (!-PP e-pp))\n             (to-to {e = e} { ! e} {e-pp}{ !-PP e-pp} (!\u2219 e))\n             (to-to {e = ! e} {e} { !-PP e-pp} {e-pp} (\u2219! e)))\n\n    Maybe-injective : \u2200 {{_ : UA}} \u2192 Maybe A \u2261 Maybe B \u2192 A \u2261 B\n    Maybe-injective e = let (e' , e'-pp) = create-PP e in Maybe-injective-PP e' e'-pp\n\n\nmodule _ {a}{A : \u2605_ a} where\n  Maybe\u2243\ud835\udfd9\u228e : Maybe A \u2243 (\ud835\udfd9 \u228e A)\n  Maybe\u2243\ud835\udfd9\u228e = equiv (maybe inr (inl _)) [inl: const nothing ,inr: just ]\n     [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n     (maybe (\u03bb x \u2192 idp) idp)\n\n  Maybe\u2243Lift\ud835\udfd9\u228e : Maybe A \u2243 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n  Maybe\u2243Lift\ud835\udfd9\u228e = equiv (maybe inr (inl _))\n                        [inl: const nothing ,inr: just ]\n                        [inl: (\u03bb _ \u2192 idp)   ,inr: (\u03bb _ \u2192 idp) ]\n                        (maybe (\u03bb _ \u2192 idp) idp)\n\n  Vec0\u2243\ud835\udfd9 : Vec A 0 \u2243 \ud835\udfd9\n  Vec0\u2243\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec0\u2243Lift\ud835\udfd9 : Vec A 0 \u2243 Lift {\u2113 = a} \ud835\udfd9\n  Vec0\u2243Lift\ud835\udfd9 = equiv _ (const []) (\u03bb _ \u2192 idp) (\u03bb { [] \u2192 idp })\n\n  Vec\u2218suc\u2243\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2243 (A \u00d7 Vec A n)\n  Vec\u2218suc\u2243\u00d7 = equiv (\u03bb { (x \u2237 xs) \u2192 x , xs }) (\u03bb { (x , xs) \u2192 x \u2237 xs })\n                    (\u03bb { (x , xs) \u2192 idp }) (\u03bb { (x \u2237 xs) \u2192 idp })\n\n  module _ {{_ : UA}} where\n\n    Maybe\u2261\ud835\udfd9\u228e : Maybe A \u2261 (\ud835\udfd9 \u228e A)\n    Maybe\u2261\ud835\udfd9\u228e = ua Maybe\u2243\ud835\udfd9\u228e\n\n    Maybe\u2261Lift\ud835\udfd9\u228e : Maybe A \u2261 (Lift {\u2113 = a} \ud835\udfd9 \u228e A)\n    Maybe\u2261Lift\ud835\udfd9\u228e = ua Maybe\u2243Lift\ud835\udfd9\u228e\n\n    Vec0\u2261Lift\ud835\udfd9 : Vec A 0 \u2261 Lift {\u2113 = a} \ud835\udfd9\n    Vec0\u2261Lift\ud835\udfd9 = ua Vec0\u2243Lift\ud835\udfd9\n\n    Vec\u2218suc\u2261\u00d7 : \u2200 {n} \u2192 Vec A (suc n) \u2261 (A \u00d7 Vec A n)\n    Vec\u2218suc\u2261\u00d7 = ua Vec\u2218suc\u2243\u00d7\n\nmodule _ {A : \u2605}{{_ : UA}} where\n    Vec0\u2261\ud835\udfd9 : Vec A 0 \u2261 \ud835\udfd9\n    Vec0\u2261\ud835\udfd9 = ua Vec0\u2243\ud835\udfd9\n\nFin0\u2243\ud835\udfd8 : Fin 0 \u2243 \ud835\udfd8\nFin0\u2243\ud835\udfd8 = equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ())\n\nFin1\u2243\ud835\udfd9 : Fin 1 \u2243 \ud835\udfd9\nFin1\u2243\ud835\udfd9 = equiv _ (\u03bb _ \u2192 zero) (\u03bb _ \u2192 idp) \u03b2\n  where \u03b2 : (_ : Fin 1) \u2192 _\n        \u03b2 zero = idp\n        \u03b2 (suc ())\n\nmodule _ {{_ : UA}} where\n    Fin0\u2261\ud835\udfd8 : Fin 0 \u2261 \ud835\udfd8\n    Fin0\u2261\ud835\udfd8 = ua Fin0\u2243\ud835\udfd8\n\n    Fin1\u2261\ud835\udfd9 : Fin 1 \u2261 \ud835\udfd9\n    Fin1\u2261\ud835\udfd9 = ua Fin1\u2243\ud835\udfd9\n\nFin\u2218suc\u2243\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2243 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2243\ud835\udfd9\u228eFin = equiv [zero: inl _ ,suc: inr ] [inl: (\u03bb _ \u2192 zero) ,inr: suc ]\n  [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb _ \u2192 idp) ]\n  [zero: idp ,suc: (\u03bb _ \u2192 idp) ]\n\nFin\u2218suc\u2261\ud835\udfd9\u228eFin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2261\ud835\udfd9\u228eFin = ua Fin\u2218suc\u2243\ud835\udfd9\u228eFin\n\nFin-\u228e-+ : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u228e Fin n) \u2261 Fin (m \u2115.+ n)\nFin-\u228e-+ {zero}  = \u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\nFin-\u228e-+ {suc m} = \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ap (_\u228e_ \ud835\udfd9) Fin-\u228e-+ \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\n\nFin-\u00d7-* : \u2200 {{_ : UA}} {m n} \u2192 (Fin m \u00d7 Fin n) \u2261 Fin (m \u2115.* n)\nFin-\u00d7-* {zero}  = \u00d7= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a3\ud835\udfd8-fst \u2219 ! Fin0\u2261\ud835\udfd8\nFin-\u00d7-* {suc m} = \u00d7= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u228e-\u00d7 \u2219 \u228e= \u03a3\ud835\udfd9-snd Fin-\u00d7-* \u2219 Fin-\u228e-+\n\nFin-\u2192-^ : \u2200 {{_ : UA}}{{_ : FunExt}}{m n} \u2192 (Fin m \u2192 Fin n) \u2261 Fin (n \u2115.^ m)\nFin-\u2192-^ {zero}  = \u2192= Fin0\u2261\ud835\udfd8 idp \u2219 \u03a0\ud835\udfd8-uniq\u2032 _ \u2219 \u228e\ud835\udfd8-inl \u2219 ap (_\u228e_ \ud835\udfd9) (! Fin0\u2261\ud835\udfd8)\n                  \u2219 ! Fin\u2218suc\u2261\ud835\udfd9\u228eFin\nFin-\u2192-^ {suc m} = \u2192= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 dist-\u00d7-\u2192 \u2219 \u00d7= (\u03a0\ud835\udfd9-uniq _) Fin-\u2192-^ \u2219 Fin-\u00d7-*\n\nFin\u2218suc\u2261Maybe\u2218Fin : \u2200 {{_ : UA}}{n} \u2192 Fin (suc n) \u2261 Maybe (Fin n)\nFin\u2218suc\u2261Maybe\u2218Fin = Fin\u2218suc\u2261\ud835\udfd9\u228eFin \u2219 ! Maybe\u2261\ud835\udfd9\u228e\n\nFin-injective : \u2200 {{_ : UA}}{m n} \u2192 Fin m \u2261 Fin n \u2192 m \u2261 n\nFin-injective {zero} {zero} Finm\u2261Finn = idp\nFin-injective {zero} {suc n} Finm\u2261Finn = \ud835\udfd8-elim (coe (! Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {zero} Finm\u2261Finn = \ud835\udfd8-elim (coe (Finm\u2261Finn \u2219 Fin0\u2261\ud835\udfd8) zero)\nFin-injective {suc m} {suc n} Finm\u2261Finn\n  = ap suc (Fin-injective (Maybe-injective\n            (! Fin\u2218suc\u2261Maybe\u2218Fin \u2219 Finm\u2261Finn \u2219 Fin\u2218suc\u2261Maybe\u2218Fin)))\n\nFin\u228e-injective : \u2200 {{_ : UA}}{A B : Set} n \u2192 (Fin n \u228e A) \u2261 (Fin n \u228e B) \u2192 A \u2261 B\nFin\u228e-injective zero    f = \u228e\ud835\udfd8-inl \u2219 \u228e-comm \u2219  \u228e= (! Fin0\u2261\ud835\udfd8) idp\n                       \u2219 f \u2219 (\u228e= Fin0\u2261\ud835\udfd8 idp \u2219 \u228e-comm) \u2219 ! \u228e\ud835\udfd8-inl\nFin\u228e-injective (suc n) f = Fin\u228e-injective n (Maybe-injective\n   (Maybe\u2261\ud835\udfd9\u228e \u2219 \u228e-assoc \u2219 \u228e= (! Fin\u2218suc\u2261\ud835\udfd9\u228eFin) idp \u2219 f\n   \u2219 \u228e= Fin\u2218suc\u2261\ud835\udfd9\u228eFin idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261\ud835\udfd9\u228e))\n\nLift\u2243id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2243 A\nLift\u2243id = equiv lower lift (\u03bb _ \u2192 idp) (\u03bb { (lift x) \u2192 idp })\n\nmodule _ {{_ : UA}} where\n    Fin-\u2261-\u22611\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 b) \u2261 (b \u2261 1\u2082)\n    Fin-\u2261-\u22611\u2082 1\u2082 = Fin1\u2261\ud835\udfd9 \u2219 ua (Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (\u03a9\u2081-set-to-contr \ud835\udfda-is-set 1\u2082))\n    Fin-\u2261-\u22611\u2082 0\u2082 = ua (equiv (\u03bb ()) (\u03bb ()) (\u03bb ()) (\u03bb ()))\n\n    Fin-\u2261-\u22610\u2082 : \u2200 b \u2192 Fin (\ud835\udfda\u25b9\u2115 (not b)) \u2261 (b \u2261 0\u2082)\n    Fin-\u2261-\u22610\u2082 b = Fin-\u2261-\u22611\u2082 (not b) \u2219 ! \u2013>-paths-equiv twist-equiv\n\n    \u2713-\u2227-\u00d7 : \u2200 x y \u2192 \u2713 (x \u2227 y) \u2261 (\u2713 x \u00d7 \u2713 y)\n    \u2713-\u2227-\u00d7 1\u2082 y = ! \ud835\udfd9\u00d7-snd\n    \u2713-\u2227-\u00d7 0\u2082 y = ! \ud835\udfd8\u00d7-fst\n\n    count-\u2261 : \u2200 {a} {A : \u2605_ a} (p : A \u2192 \ud835\udfda) x \u2192 Fin (\ud835\udfda\u25b9\u2115 (p x)) \u2261 (p x \u2261 1\u2082)\n    count-\u2261 p x = Fin-\u2261-\u22611\u2082 (p x)\n\n    Lift\u2261id : \u2200 {a} {A : \u2605_ a} \u2192 Lift {a} {a} A \u2261 A\n    Lift\u2261id = ua Lift\u2243id\n\n    \u03a0\ud835\udfd9F\u2261F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2261 F _\n    \u03a0\ud835\udfd9F\u2261F = ua (equiv (\u03bb x \u2192 x _) const (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\n    \ud835\udfd9\u2192A\u2261A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2261 A\n    \ud835\udfd9\u2192A\u2261A = \u03a0\ud835\udfd9F\u2261F\n\n    not-\ud835\udfda\u2261\ud835\udfda : \ud835\udfda \u2261 \ud835\udfda\n    not-\ud835\udfda\u2261\ud835\udfda = twist\n\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 : Maybe \ud835\udfd8 \u2261 \ud835\udfd9\n    Maybe\ud835\udfd8\u2261\ud835\udfd9 = Maybe\u2261\ud835\udfd9\u228e \u2219 ! \u228e\ud835\udfd8-inl\n\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe : \u2200 {a} {A : \u2605_ a} n \u2192 Maybe (Maybe^ n A) \u2261 Maybe^ n (Maybe A)\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe zero    = idp\n    Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe (suc n) = ap Maybe (Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n)\n\n    Maybe^\ud835\udfd8\u2261Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2261 Fin n\n    Maybe^\ud835\udfd8\u2261Fin zero    = ! Fin0\u2261\ud835\udfd8\n    Maybe^\ud835\udfd8\u2261Fin (suc n) = ap Maybe (Maybe^\ud835\udfd8\u2261Fin n) \u2219 ! Fin\u2218suc\u2261Maybe\u2218Fin\n\n    Maybe^\ud835\udfd9\u2261Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2261 Fin (suc n)\n    Maybe^\ud835\udfd9\u2261Fin1+ n = ap (Maybe^ n) (! Maybe\ud835\udfd8\u2261\ud835\udfd9) \u2219 ! Maybe\u2218Maybe^\u2261Maybe^\u2218Maybe n \u2219 Maybe^\ud835\udfd8\u2261Fin (suc n)\n\n    Maybe-\u228e : \u2200 {a} {A B : \u2605_ a} \u2192 (Maybe A \u228e B) \u2261 Maybe (A \u228e B)\n    Maybe-\u228e {a} = \u228e= Maybe\u2261Lift\ud835\udfd9\u228e idp \u2219 ! \u228e-assoc \u2219 ! Maybe\u2261Lift\ud835\udfd9\u228e\n\n    Maybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2261 Maybe^ (m + n) A\n    Maybe^-\u228e-+ zero    n = ! \ud835\udfd8\u228e-inr\n    Maybe^-\u228e-+ (suc m) n = Maybe-\u228e \u2219 ap Maybe (Maybe^-\u228e-+ m n)\n\nmodule _ {A : Set} {p q : A \u2192 \ud835\udfda} where\n    \u03a3AP\u00acQ : Set\n    \u03a3AP\u00acQ = \u03a3 A (\u03bb x \u2192 p x \u2261 1\u2082 \u00d7 q x \u2261 0\u2082)\n\n    \u03a3A\u00acPQ : Set\n    \u03a3A\u00acPQ = \u03a3 A (\u03bb x \u2192 p x \u2261 0\u2082 \u00d7 q x \u2261 1\u2082)\n\n    module EquivalentSubsets (e : \u03a3AP\u00acQ \u2261 \u03a3A\u00acPQ) where\n\n      f' : \u03a3AP\u00acQ \u2192 \u03a3A\u00acPQ\n      f' = coe e\n\n      f-1' : \u03a3A\u00acPQ \u2192 \u03a3AP\u00acQ\n      f-1' = coe! e\n\n      f-1f' : \u2200 x \u2192 f-1' (f' x) \u2261 x\n      f-1f' = coe!-inv-l e\n\n      ff-1' : \u2200 x \u2192 f' (f-1' x) \u2261 x\n      ff-1' = coe!-inv-r e\n\n      f   : (x : A) \u2192 p x \u2261 1\u2082 \u2192 q x \u2261 0\u2082 \u2192 \u03a3A\u00acPQ\n      f x px qx = f' (x , (px , qx))\n\n      f-1 : (x : A) \u2192 p x \u2261 0\u2082 \u2192 q x \u2261 1\u2082 \u2192 \u03a3AP\u00acQ\n      f-1 x px qx = f-1' (x , (px , qx))\n\n      f-1f : \u2200 x px nqx \u2192\n             let y = snd (f x px nqx) in fst (f-1 (fst (f x px nqx)) (fst y) (snd y)) \u2261 x\n      f-1f x px nqx = \u2261.cong fst (f-1f' (x , (px , nqx)))\n\n      ff-1 : \u2200 x px nqx \u2192\n             let y = snd (f-1 x px nqx) in fst (f (fst (f-1 x px nqx)) (fst y) (snd y)) \u2261 x\n      ff-1 x px nqx = \u2261.cong fst (ff-1' (x , (px , nqx)))\n\n      \u03c0' : (x : A) (px qx : \ud835\udfda) \u2192 p x \u2261 px \u2192 q x \u2261 qx \u2192 A\n      \u03c0' x 1\u2082 1\u2082 px qx = x\n      \u03c0' x 1\u2082 0\u2082 px qx = fst (f x px qx)\n      \u03c0' x 0\u2082 1\u2082 px qx = fst (f-1 x px qx)\n      \u03c0' x 0\u2082 0\u2082 px qx = x\n\n      \u03c0 : A \u2192 A\n      \u03c0 x = \u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      0\u22621 : 0\u2082 \u2262 1\u2082\n      0\u22621 ()\n\n      \u03c001 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px0 : p x \u2261 0\u2082) (qx1 : q x \u2261 1\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 0\u2082 1\u2082 px0 qx1\n      \u03c001 x 1\u2082 _  ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym px0) ppx))\n      \u03c001 x 0\u2082 1\u2082 ppx qqx px0 qx1 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f-1 x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px0) (UIP-set \ud835\udfda-is-set qqx qx1)\n      \u03c001 x 0\u2082 0\u2082 ppx qqx px0 qx1 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qqx) qx1))\n\n      \u03c010 : \u2200 x px qx (ppx : p x \u2261 px) (qqx : q x \u2261 qx) (px1 : p x \u2261 1\u2082) (qx0 : q x \u2261 0\u2082) \u2192 \u03c0' x px qx ppx qqx \u2261 \u03c0' x 1\u2082 0\u2082 px1 qx0\n      \u03c010 x 0\u2082 _  ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym ppx) px1))\n      \u03c010 x 1\u2082 0\u2082 ppx qqx px1 qx0 = \u2261.ap\u2082 (\u03bb z1 z2 \u2192 fst (f x z1 z2)) (UIP-set \ud835\udfda-is-set ppx px1) (UIP-set \ud835\udfda-is-set qqx qx0)\n      \u03c010 x 1\u2082 1\u2082 ppx qqx px1 qx0 = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym qx0) qqx))\n\n      \u03c0'bb : \u2200 {b} x (px : p x \u2261 b) (qx : q x \u2261 b) ppx qqx ([ppx] : p x \u2261 ppx) ([qqx] : q x \u2261 qqx) \u2192 \u03c0' x ppx qqx [ppx] [qqx] \u2261 x\n      \u03c0'bb x px qx 1\u2082 1\u2082 [ppx] [qqx] = \u2261.refl\n      \u03c0'bb x px qx 1\u2082 0\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [qqx]) (\u2261.trans qx (\u2261.trans (\u2261.sym px) [ppx]))))\n      \u03c0'bb x px qx 0\u2082 1\u2082 [ppx] [qqx] = \ud835\udfd8-elim (0\u22621 (\u2261.trans (\u2261.sym [ppx]) (\u2261.trans px (\u2261.trans (\u2261.sym qx) [qqx]))))\n      \u03c0'bb x px qx 0\u2082 0\u2082 [ppx] [qqx] = \u2261.refl\n\n      \u03c0\u03c0' : \u2200 x px qx [px] [qx] \u2192 let y = (\u03c0' x px qx [px] [qx]) in \u03c0' y (p y) (q y) \u2261.refl \u2261.refl \u2261 x\n      \u03c0\u03c0' x 1\u2082 1\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n      \u03c0\u03c0' x 1\u2082 0\u2082 px qx = let fx = f x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c001 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (f-1f x px qx)\n      \u03c0\u03c0' x 0\u2082 1\u2082 px qx = let fx = f-1 x px qx in let pfx = fst (snd fx) in let qfx = snd (snd fx) in \u2261.trans (\u03c010 (fst fx) (p (fst fx)) (q (fst fx)) \u2261.refl \u2261.refl pfx qfx) (ff-1 x px qx)\n      \u03c0\u03c0' x 0\u2082 0\u2082 px qx = \u03c0'bb x px qx (p x) (q x) \u2261.refl \u2261.refl\n\n      \u03c0\u03c0 : \u2200 x \u2192 \u03c0 (\u03c0 x) \u2261 x\n      \u03c0\u03c0 x = \u03c0\u03c0' x (p x) (q x) \u2261.refl \u2261.refl\n\n      prop' : \u2200 px qx x ([px] : p x \u2261 px) ([qx] : q x \u2261 qx) \u2192 q (\u03c0' x px qx [px] [qx]) \u2261 px\n      prop' 1\u2082 1\u2082 x px qx = qx\n      prop' 1\u2082 0\u2082 x px qx = snd (snd (f x px qx))\n      prop' 0\u2082 1\u2082 x px qx = snd (snd (f-1 x px qx))\n      prop' 0\u2082 0\u2082 x px qx = qx\n\n      prop'' : \u2200 x \u2192 p x \u2261 q (\u03c0 x)\n      prop'' x = \u2261.sym (prop' (p x) (q x) x \u2261.refl \u2261.refl)\n\n      prop : {{_ : FunExt}} \u2192 p \u2261 q \u2218 \u03c0\n      prop = \u03bb= prop''\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a67bf548acefea35ca58a616530a71b06abe270b","subject":"First order version of the domain predicate of the nest function (by Peter Dybjer).","message":"First order version of the domain predicate of the nest function (by Peter Dybjer).\n\nIgnore-this: 818878ef09d0216d5d695103783d6d6e\n\ndarcs-hash:20110405114655-3bd4e-a08888227d710d86cf36a682e5ca57ed4880e327.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\n-- The nest function is total (via structural recursion in the domain\n-- predicate).\nnest-N : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-N dom0           = subst N (sym nest-0) zN\nnest-N (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-N h\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\nmutual\n  -- The domain predicate of the nest function.\n  data DomNest : (n : D) \u2192 N n \u2192 Set where\n    dom0 : DomNest zero zN\n    domS : \u2200 {n} \u2192 (Nn : N n) \u2192\n                   (h\u2081 : DomNest n Nn) \u2192\n                   (h\u2082 : DomNest (nest n) (nest-N Nn h\u2081)) \u2192\n                   DomNest (succ n) (sN Nn)\n\n  -- The nest function is total (via structural recursion in the\n  -- domain predicate).\n  nest-N : \u2200 {n} \u2192 (Nn : N n) \u2192 DomNest n Nn \u2192 N (nest n)\n  nest-N .zN dom0                     = subst N (sym nest-0) zN\n  nest-N .(sN Nn) (domS {n} Nn h\u2081 h\u2082) =\n    subst N (sym (nest-S n)) (nest-N (nest-N Nn h\u2081) h\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fe3a762ad40dd426bceb16d75cdb4ec6a0bc985e","subject":"Data.Bits: restore some parts","message":"Data.Bits: restore some parts\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"-- NOTE with-K\nmodule Data.Bits where\n\nopen import Type hiding (\u2605)\n-- cleanup\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nimport Data.Two.Equality as ==\u1d47\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise; Fin\u25b9\u2115) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen import Data.Vec.Bijection\nopen import Data.Vec.Permutation\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[]         == []         = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (y \u2237 ys) rewrite ==\u1d47.comm x y | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1\u2082\n==-refl [] = refl\n==-refl (1\u2082 \u2237 xs) = ==-refl xs\n==-refl (0\u2082 \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n        x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n        x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n        (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n        0\u207f \u2295 1\u207f      \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n        1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2082 \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2082 \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2082 \u2237 1\u2082 \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n|not| : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n|not| f = not \u2218 f\n\n|\u2227|-comm : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2227| g \u2257 g |\u2227| f\n|\u2227|-comm f g x with f x | g x\n... | 0\u2082 | 0\u2082 = refl\n... | 0\u2082 | 1\u2082 = refl\n... | 1\u2082 | 0\u2082 = refl\n... | 1\u2082 | 1\u2082 = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u00b7\n    \u2295-dist-\u00b7 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u00b7 pad \u2295 \u03c0 \u00b7 xs \u2261 \u03c0 \u00b7 (pad \u2295 xs)\n    \u2295-dist-\u00b7 fs      `id        xs = refl\n    \u2295-dist-\u00b7 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u00b7 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u00b7 fs \u03c0 xs = refl\n    \u2295-dist-\u00b7 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u00b7 (\u03c0\u2080 \u00b7 fs) \u03c0\u2081 (\u03c0\u2080 \u00b7 xs)\n                                         | \u2295-dist-\u00b7 fs \u03c0\u2080 xs = refl\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0\u2082 \u2237 []\nsucBCarry (0\u2082 \u2237 xs) = 0\u2082 \u2237 sucBCarry xs\nsucBCarry (1\u2082 \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2082 \u2237 xs) = 1\u2082 \u2237 xs\n  sucRB (1\u2082 \u2237 xs) = 0\u2082 \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0\u2082 \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2082 \u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []        = zero\nBits\u25b9\u2115         (0\u2082 \u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\nBits\u25b9\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 Bits\u25b9\u2115 xs < 2^ n \nBits\u25b9\u2115-bound         [] = s\u2264s z\u2264n\nBits\u25b9\u2115-bound {suc n} (1\u2082 \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (Bits\u25b9\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono Bits\u25b9\u2115-bound xs\nBits\u25b9\u2115-bound {suc n} (0\u2082 \u2237 xs) = \u2264-steps (2^ n) (Bits\u25b9\u2115-bound xs)\n\nBits\u25b9\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 Bits\u25b9\u2115 {n} x \u2264 2^ n + y\nBits\u25b9\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (Bits\u25b9\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270Bits\u25b9\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 Bits\u25b9\u2115 {n} y\n2\u207f+\u2270Bits\u25b9\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (Bits\u25b9\u2115-bound y)\n\nBits\u25b9\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 Bits\u25b9\u2115 x \u2261 Bits\u25b9\u2115 y \u2192 x \u2261 y\nBits\u25b9\u2115-inj         []        []        _ = refl\nBits\u25b9\u2115-inj         (0\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = cong 0\u2237_ (Bits\u25b9\u2115-inj xs ys p)\nBits\u25b9\u2115-inj {suc n} (1\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = cong 1\u2237_ (Bits\u25b9\u2115-inj xs ys (cancel-+-left (2^ n) p))\nBits\u25b9\u2115-inj {suc n} (0\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = \ud835\udfd8-elim (2\u207f+\u2270Bits\u25b9\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nBits\u25b9\u2115-inj {suc n} (1\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = \ud835\udfd8-elim (2\u207f+\u2270Bits\u25b9\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 \u2605\u2080 where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 \u2713 (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0\u2082 pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1\u2082 pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 \u2713 (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1\u2082 \u2237 p) (0\u2082 \u2237 q) ()\n<=\u2192\u2264\u1d2e (0\u2082 \u2237 p) (1\u2082 \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0\u2082 \u2237 p) (0\u2082 \u2237 q) pf = there 0\u2082 (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1\u2082 \u2237 p) (1\u2082 \u2237 q) pf = there 1\u2082 (<=\u2192\u2264\u1d2e p q pf)\n\nBits\u25b9\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 Bits\u25b9\u2115 x \u2264 Bits\u25b9\u2115 y \u2192 x \u2264\u1d2e y\nBits\u25b9\u2115-\u2264-inj     [] [] p = []\nBits\u25b9\u2115-\u2264-inj         (0\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = there 0\u2082 (Bits\u25b9\u2115-\u2264-inj xs ys p)\nBits\u25b9\u2115-\u2264-inj         (0\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = 0-1 _ _\nBits\u25b9\u2115-\u2264-inj {suc n} (1\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = \ud835\udfd8-elim (2\u207f+\u2270Bits\u25b9\u2115 ys p)\nBits\u25b9\u2115-\u2264-inj {suc n} (1\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = there 1\u2082 (Bits\u25b9\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = [0: 0\u2237 \u2115\u25b9Bits x\n                    1: 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n)\n                   ]\u2032 (2^ n \u2115<= x)\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0\u2082 \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1\u2082 \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero}  f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : \u2605 a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0\u2082 \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1\u2082 \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : \u2605 a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 Bits\u25b9\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (Bits\u25b9\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270Bits\u25b9\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 Bits\u25b9\u2115 xs\n2\u207f\u2270Bits\u25b9\u2115 xs p = \u00acn\u2264x<n _ p (Bits\u25b9\u2115-bound xs)\n\n\u2713not2\u207f<=Bits\u25b9\u2115 : \u2200 {n} (xs : Bits n) \u2192 \u2713 (not (2^ n \u2115<= (Bits\u25b9\u2115 xs)))\n\u2713not2\u207f<=Bits\u25b9\u2115 {n} xs with (2^ n) \u2115<= (Bits\u25b9\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (Bits\u25b9\u2115 xs)\n... | 1\u2082 | \u2261.[ p ] = 2\u207f\u2270Bits\u25b9\u2115 xs (<=.sound (2^ n) (Bits\u25b9\u2115 xs) (\u2261\u2192\u2713 p))\n... | 0\u2082 |     _   = _\n\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 : \u2200 {n} (x : Bits n) \u2192 \u2115\u25b9Bits (Bits\u25b9\u2115 x) \u2261 x\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 [] = \u2261.refl\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs)\n  rewrite \u2713\u2192\u2261 (<=-steps\u2032 {2^ n} (Bits\u25b9\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (Bits\u25b9\u2115 xs)\n        | m+n\u2238n\u2261m (Bits\u25b9\u2115 xs) (2^ n)\n        | \u2115\u25b9Bits\u2218Bits\u25b9\u2115 xs\n        = \u2261.refl\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 (0\u2082 \u2237 xs)\n  rewrite \u2713not\u2192\u2261 (\u2713not2\u207f<=Bits\u25b9\u2115 xs)\n        | \u2115\u25b9Bits\u2218Bits\u25b9\u2115 xs\n        = \u2261.refl\n\nBits\u25b9\u2115\u2218\u2115\u25b9Bits : \u2200 {n} x \u2192 x < 2^ n \u2192 Bits\u25b9\u2115 {n} (\u2115\u25b9Bits x) \u2261 x\nBits\u25b9\u2115\u2218\u2115\u25b9Bits {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nBits\u25b9\u2115\u2218\u2115\u25b9Bits {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | 1\u2082 | \u2261.[ p ] rewrite Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192\u2713 p))\n... | 0\u2082 | \u2261.[ p ] = Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192\u2713not p)))\n\n\u2115\u25b9Bits-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 \u2115\u25b9Bits {n} x \u2261 \u2115\u25b9Bits y \u2192 x \u2261 y\n\u2115\u25b9Bits-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} x x<) \u27e9\n    Bits\u25b9\u2115 (\u2115\u25b9Bits {n} x)\n  \u2261\u27e8 \u2261.cong Bits\u25b9\u2115 fx\u2261fy \u27e9\n    Bits\u25b9\u2115 (\u2115\u25b9Bits {n} y)\n  \u2261\u27e8 Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n","old_contents":"-- NOTE with-K\nmodule Data.Bits where\n\nopen import Type hiding (\u2605)\n-- cleanup\nimport Level\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nopen import Data.Bit using (Bit)\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nimport Data.Two.Equality as ==\u1d47\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen import Data.Vec.Bijection\nopen import Data.Vec.Permutation\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBits : \u2115 \u2192 \u2605\u2080\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0\u2082\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1\u2082\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0\u2082 \u2237 xs\n\n{-\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1\u2082 \u2237 xs\n\n_!_ : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bit\n[] == [] = 1\u2082\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (y \u2237 ys) rewrite ==\u1d47.comm x y | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1\u2082\n==-refl [] = refl\n==-refl (1\u2082 \u2237 xs) = ==-refl xs\n==-refl (0\u2082 \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bit\n[]        <= []        = 1\u2082\n(1\u2082 \u2237 xs) <= (0\u2082 \u2237 ys) = 0\u2082\n(0\u2082 \u2237 xs) <= (1\u2082 \u2237 ys) = 1\u2082\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n        x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n        x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n        (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n        0\u207f \u2295 1\u207f      \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n        1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0\u2082 \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1\u2082 \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0\u2082 \u2237 1\u2082 \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1\u2082\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0\u2082\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n|not| : \u2200 {n} (f : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n|not| f = not \u2218 f\n\n|\u2227|-comm : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2227| g \u2257 g |\u2227| f\n|\u2227|-comm f g x with f x | g x\n... | 0\u2082 | 0\u2082 = refl\n... | 0\u2082 | 1\u2082 = refl\n... | 1\u2082 | 0\u2082 = refl\n... | 1\u2082 | 1\u2082 = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u00b7\n    \u2295-dist-\u00b7 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u00b7 pad \u2295 \u03c0 \u00b7 xs \u2261 \u03c0 \u00b7 (pad \u2295 xs)\n    \u2295-dist-\u00b7 fs      `id        xs = refl\n    \u2295-dist-\u00b7 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u00b7 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u00b7 fs \u03c0 xs = refl\n    \u2295-dist-\u00b7 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u00b7 (\u03c0\u2080 \u00b7 fs) \u03c0\u2081 (\u03c0\u2080 \u00b7 xs)\n                                         | \u2295-dist-\u00b7 fs \u03c0\u2080 xs = refl\n\nview\u2237 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Vec A n \u2192 B) \u2192 Vec A (suc n) \u2192 B\nview\u2237 f (x \u2237 xs) = f x xs\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry []        = 0\u2082 \u2237 []\nsucBCarry (0\u2082 \u2237 xs) = 0\u2082 \u2237 sucBCarry xs\nsucBCarry (1\u2082 \u2237 xs) = view\u2237 (\u03bb x xs \u2192 x \u2237 not x \u2237 xs) (sucBCarry xs)\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n--_[mod_] : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n--a [mod b ] = DivMod' a b\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0\u2082 \u2237 xs) = 1\u2082 \u2237 xs\n  sucRB (1\u2082 \u2237 xs) = 0\u2082 \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0\u2082 \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1\u2082 \u2237 xs) = raise (2^ n) (toFin xs)\n\nBits\u25b9\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nBits\u25b9\u2115         []        = zero\nBits\u25b9\u2115         (0\u2082 \u2237 xs) = Bits\u25b9\u2115 xs\nBits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs) = 2^ n + Bits\u25b9\u2115 xs\n\nBits\u25b9\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 Bits\u25b9\u2115 xs < 2^ n \nBits\u25b9\u2115-bound         [] = s\u2264s z\u2264n\nBits\u25b9\u2115-bound {suc n} (1\u2082 \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (Bits\u25b9\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono Bits\u25b9\u2115-bound xs\nBits\u25b9\u2115-bound {suc n} (0\u2082 \u2237 xs) = \u2264-steps (2^ n) (Bits\u25b9\u2115-bound xs)\n\nBits\u25b9\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 Bits\u25b9\u2115 {n} x \u2264 2^ n + y\nBits\u25b9\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (Bits\u25b9\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270Bits\u25b9\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 Bits\u25b9\u2115 {n} y\n2\u207f+\u2270Bits\u25b9\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (Bits\u25b9\u2115-bound y)\n\nBits\u25b9\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 Bits\u25b9\u2115 x \u2261 Bits\u25b9\u2115 y \u2192 x \u2261 y\nBits\u25b9\u2115-inj         []        []        _ = refl\nBits\u25b9\u2115-inj         (0\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = cong 0\u2237_ (Bits\u25b9\u2115-inj xs ys p)\nBits\u25b9\u2115-inj {suc n} (1\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = cong 1\u2237_ (Bits\u25b9\u2115-inj xs ys (cancel-+-left (2^ n) p))\nBits\u25b9\u2115-inj {suc n} (0\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = \ud835\udfd8-elim (2\u207f+\u2270Bits\u25b9\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nBits\u25b9\u2115-inj {suc n} (1\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = \ud835\udfd8-elim (2\u207f+\u2270Bits\u25b9\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 \u2605\u2080 where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 \u2713 (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0\u2082 pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1\u2082 pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 \u2713 (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1\u2082 \u2237 p) (0\u2082 \u2237 q) ()\n<=\u2192\u2264\u1d2e (0\u2082 \u2237 p) (1\u2082 \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0\u2082 \u2237 p) (0\u2082 \u2237 q) pf = there 0\u2082 (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1\u2082 \u2237 p) (1\u2082 \u2237 q) pf = there 1\u2082 (<=\u2192\u2264\u1d2e p q pf)\n\nBits\u25b9\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 Bits\u25b9\u2115 x \u2264 Bits\u25b9\u2115 y \u2192 x \u2264\u1d2e y\nBits\u25b9\u2115-\u2264-inj     [] [] p = []\nBits\u25b9\u2115-\u2264-inj         (0\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = there 0\u2082 (Bits\u25b9\u2115-\u2264-inj xs ys p)\nBits\u25b9\u2115-\u2264-inj         (0\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = 0-1 _ _\nBits\u25b9\u2115-\u2264-inj {suc n} (1\u2082 \u2237 xs) (0\u2082 \u2237 ys) p = \ud835\udfd8-elim (2\u207f+\u2270Bits\u25b9\u2115 ys p)\nBits\u25b9\u2115-\u2264-inj {suc n} (1\u2082 \u2237 xs) (1\u2082 \u2237 ys) p = there 1\u2082 (Bits\u25b9\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\n\u2115\u25b9Bits : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits {zero}  _ = []\n\u2115\u25b9Bits {suc n} x = if 2^ n \u2115<= x then 1\u2237 \u2115\u25b9Bits (x \u2238 2^ n) else 0\u2237 \u2115\u25b9Bits x\n\n\u2115\u25b9Bits\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\n\u2115\u25b9Bits\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = \u2115\u25b9Bits \u2218 Fin.Bits\u25b9\u2115\n\nlookupTbl : \u2200 {n a} {A : \u2605 a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0\u2082 \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1\u2082 \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : \u2605 a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : \u2605 a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0\u2082 \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1\u2082 \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : \u2605 a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 Bits\u25b9\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (Bits\u25b9\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270Bits\u25b9\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 Bits\u25b9\u2115 xs\n2\u207f\u2270Bits\u25b9\u2115 xs p = \u00acn\u2264x<n _ p (Bits\u25b9\u2115-bound xs)\n\n\u2713not2\u207f<=Bits\u25b9\u2115 : \u2200 {n} (xs : Bits n) \u2192 \u2713 (not (2^ n \u2115<= (Bits\u25b9\u2115 xs)))\n\u2713not2\u207f<=Bits\u25b9\u2115 {n} xs with (2^ n) \u2115<= (Bits\u25b9\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (Bits\u25b9\u2115 xs)\n... | 1\u2082 | \u2261.[ p ] = 2\u207f\u2270Bits\u25b9\u2115 xs (<=.sound (2^ n) (Bits\u25b9\u2115 xs) (\u2261\u2192\u2713 p))\n... | 0\u2082 |     _   = _\n\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 : \u2200 {n} (x : Bits n) \u2192 \u2115\u25b9Bits (Bits\u25b9\u2115 x) \u2261 x\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 [] = \u2261.refl\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 {suc n} (1\u2082 \u2237 xs)\n  rewrite \u2713\u2192\u2261 (<=-steps\u2032 {2^ n} (Bits\u25b9\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (Bits\u25b9\u2115 xs)\n        | m+n\u2238n\u2261m (Bits\u25b9\u2115 xs) (2^ n)\n        | \u2115\u25b9Bits\u2218Bits\u25b9\u2115 xs\n        = \u2261.refl\n\u2115\u25b9Bits\u2218Bits\u25b9\u2115 (0\u2082 \u2237 xs)\n  rewrite \u2713not\u2192\u2261 (\u2713not2\u207f<=Bits\u25b9\u2115 xs)\n        | \u2115\u25b9Bits\u2218Bits\u25b9\u2115 xs\n        = \u2261.refl\n\nBits\u25b9\u2115\u2218\u2115\u25b9Bits : \u2200 {n} x \u2192 x < 2^ n \u2192 Bits\u25b9\u2115 {n} (\u2115\u25b9Bits x) \u2261 x\nBits\u25b9\u2115\u2218\u2115\u25b9Bits {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nBits\u25b9\u2115\u2218\u2115\u25b9Bits {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | 1\u2082 | \u2261.[ p ] rewrite Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192\u2713 p))\n... | 0\u2082 | \u2261.[ p ] = Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192\u2713not p)))\n\n\u2115\u25b9Bits-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 \u2115\u25b9Bits {n} x \u2261 \u2115\u25b9Bits y \u2192 x \u2261 y\n\u2115\u25b9Bits-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} x x<) \u27e9\n    Bits\u25b9\u2115 (\u2115\u25b9Bits {n} x)\n  \u2261\u27e8 \u2261.cong Bits\u25b9\u2115 fx\u2261fy \u27e9\n    Bits\u25b9\u2115 (\u2115\u25b9Bits {n} y)\n  \u2261\u27e8 Bits\u25b9\u2115\u2218\u2115\u25b9Bits {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"59d3e7c3f356759a10716755510c2c7489802115","subject":"AddTotality.agda: Minor changes.","message":"AddTotality.agda: Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\nmodule InductionPrinciple where\n  -- Interactive proof using the induction principle for natural numbers.\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {m} {n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n  -- Combined proof using the induction principle.\n  --\n  -- The translation is\n  -- \u2200 p. app\u2081(p,zero) \u2192\n  --      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- N-ind\n  --      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n  ----------------------------------------------------------------\n  -- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\n\n  -- Because the ATPs don't handle induction, them cannot prove this\n  -- postulate.\n  postulate +-N' : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  -- {-# ATP prove +-N' N-ind #-}\n\nmodule Instance where\n  -- Interactive proof using an instance of the induction principle.\n  +-N-ind : \u2200 {n} \u2192\n            N (zero + n) \u2192\n            (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n            \u2200 {m} \u2192 N m \u2192 N (m + n)\n  +-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = +-N-ind A0 is Nm\n    where\n    A0 : N (zero + n)\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n    is {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\n  -- Combined proof using an instance of the induction principle.\n  postulate +-N' : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  {-# ATP prove +-N' +-N-ind #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Totality of natural numbers addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = N-ind A A0 is Nm\n  where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    A0 : A zero\n    A0 = subst N (sym (+-0x n)) Nn\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n-- Interactive proof using an instance of the induction principle.\n+-N-ind : \u2200 {n} \u2192\n          N (zero + n) \u2192\n          (\u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)) \u2192\n          \u2200 {m} \u2192 N m \u2192 N (m + n)\n+-N-ind {n} = N-ind (\u03bb i \u2192 N (i + n))\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = +-N-ind A0 is Nm\n  where\n  A0 : N (zero + n)\n  A0 = subst N (sym (+-0x n)) Nn\n\n  is : \u2200 {m} \u2192 N (m + n) \u2192 N (succ\u2081 m + n)\n  is {m} ih = subst N (sym (+-Sx m n)) (nsucc ih)\n\n-- Combined proof using an instance of the induction principle.\npostulate +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2082 +-N-ind #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- N-ind\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate +-N\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- Because the ATPs don't handle induction, them cannot prove this\n-- postulate.\n-- {-# ATP prove +-N\u2083 N-ind #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4185720722e52ed8a66dd8db258f4850cf2c2fb1","subject":"Fixed doc.","message":"Fixed doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers inductive predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base hiding ( succ\u2081 )\n\n------------------------------------------------------------------------------\n-- We define succ\u2081 outside an abstract block.\n\nsucc\u2081 : D \u2192 D\nsucc\u2081 n = succ \u00b7 n\n\nmodule ConstantAndUnaryFunction where\n\n  -- N using the constant succ.\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  N-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\n  N-ind A A0 h nzero      = A0\n  N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n  -- N using the unary function succ\u2081.\n  data N\u2081 : D \u2192 Set where\n    nzero\u2081 : N\u2081 zero\n    nsucc\u2081 : \u2200 {n} \u2192 N\u2081 n \u2192 N\u2081 (succ\u2081 n)\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N\u2081 n \u2192 A n\n  N-ind\u2081 A A0 h nzero\u2081      = A0\n  N-ind\u2081 A A0 h (nsucc\u2081 Nn) = h (N-ind\u2081 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- From N\/N\u2081 to N\u2081\/N.\n\n  N\u2192N\u2081 : \u2200 {n} \u2192 N n \u2192 N\u2081 n\n  N\u2192N\u2081 nzero          = nzero\u2081\n  N\u2192N\u2081 (nsucc {n} Nn) = nsucc\u2081 (N\u2192N\u2081 Nn)\n\n  N\u2081\u2192N : \u2200 {n} \u2192 N\u2081 n \u2192 N n\n  N\u2081\u2192N nzero\u2081           = nzero\n  N\u2081\u2192N (nsucc\u2081 {n} N\u2081n) = nsucc (N\u2081\u2192N N\u2081n)\n\n  ----------------------------------------------------------------------------\n  -- From N-ind \u2192 N-ind\u2081.\n  N-ind\u2081' : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N\u2081 n \u2192 A n\n  N-ind\u2081' A A0 h Nn\u2081 = N-ind A A0 h' (N\u2081\u2192N Nn\u2081)\n    where\n    h' : \u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)\n    h' {n} An = h An\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2081 \u2192 N-ind.\n  N-ind' : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A A0 h Nn = N-ind\u2081 A A0 h' (N\u2192N\u2081 Nn)\n    where\n    h' : \u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)\n    h' {n} An = h An\n\n------------------------------------------------------------------------------\n\nmodule AdditionalHypotheis where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle generated by Coq 8.4pl3 when we define\n  -- the data type N in Prop.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle removing the hypothesis N n from the\n  -- inductive step (see Martin-L\u00f6f 1971, p. 190).\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n            A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-ind\u2082 from N-ind\u2081.\n  N-ind\u2082' : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n  ----------------------------------------------------------------------------\n  -- N-ind\u2081 from N-ind\u2082.\n  N-ind\u2081' : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081' A A0 h {n} Nn = \u2227-proj\u2082 (N-ind\u2082 B B0 h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    B0 : B zero\n    B0 = nzero , A0\n\n    h' : \u2200 {m} \u2192 B m \u2192 B (succ\u2081 m)\n    h' (Nm , Am) = nsucc Nm , h Nm Am\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Martin-L\u00f6f, P. (1971). Hauptsatz for the Intuitionistic Theory of\n-- Iterated Inductive De\ufb01nitions. In: Proceedings of the Second\n-- Scandinavian Logic Symposium. Ed. by Fenstad,\n-- J. E. Vol. 63. Studies in Logic and the Foundations of\n-- Mathematics. North-Holland Publishing Company, pp. 179\u2013216.\n","old_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers inductive predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base hiding ( succ\u2081 )\n\n------------------------------------------------------------------------------\n-- We define succ\u2081 outside an abstract block.\n\nsucc\u2081 : D \u2192 D\nsucc\u2081 n = succ \u00b7 n\n\nmodule ConstantAndUnaryFunction where\n\n  -- N using 0-ary constant (i.e. succ)\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  N-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\n  N-ind A A0 h nzero      = A0\n  N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n  -- N using unary function (i.e. succ\u2081)\n  data N\u2081 : D \u2192 Set where\n    nzero\u2081 : N\u2081 zero\n    nsucc\u2081 : \u2200 {n} \u2192 N\u2081 n \u2192 N\u2081 (succ\u2081 n)\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N\u2081 n \u2192 A n\n  N-ind\u2081 A A0 h nzero\u2081      = A0\n  N-ind\u2081 A A0 h (nsucc\u2081 Nn) = h (N-ind\u2081 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- From N\/N\u2081 to N\u2081\/N.\n\n  N\u2192N\u2081 : \u2200 {n} \u2192 N n \u2192 N\u2081 n\n  N\u2192N\u2081 nzero          = nzero\u2081\n  N\u2192N\u2081 (nsucc {n} Nn) = nsucc\u2081 (N\u2192N\u2081 Nn)\n\n  N\u2081\u2192N : \u2200 {n} \u2192 N\u2081 n \u2192 N n\n  N\u2081\u2192N nzero\u2081           = nzero\n  N\u2081\u2192N (nsucc\u2081 {n} N\u2081n) = nsucc (N\u2081\u2192N N\u2081n)\n\n  ----------------------------------------------------------------------------\n  -- From N-ind \u2192 N-ind\u2081.\n  N-ind\u2081' : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N\u2081 n \u2192 A n\n  N-ind\u2081' A A0 h Nn\u2081 = N-ind A A0 h' (N\u2081\u2192N Nn\u2081)\n    where\n    h' : \u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)\n    h' {n} An = h An\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2081 \u2192 N-ind.\n  N-ind' : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A A0 h Nn = N-ind\u2081 A A0 h' (N\u2192N\u2081 Nn)\n    where\n    h' : \u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)\n    h' {n} An = h An\n\n------------------------------------------------------------------------------\n\nmodule AdditionalHypotheis where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- The induction principle generated by Coq 8.4pl3 when we define\n  -- the data type N in Prop.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle removing the hypothesis N n from the\n  -- inductive step (see Martin-L\u00f6f 1971, p. 190).\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n            A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-ind\u2082 from N-ind\u2081.\n  N-ind\u2082' : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n  ----------------------------------------------------------------------------\n  -- N-ind\u2081 from N-ind\u2082.\n  N-ind\u2081' : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081' A A0 h {n} Nn = \u2227-proj\u2082 (N-ind\u2082 B B0 h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    B0 : B zero\n    B0 = nzero , A0\n\n    h' : \u2200 {m} \u2192 B m \u2192 B (succ\u2081 m)\n    h' (Nm , Am) = nsucc Nm , h Nm Am\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Martin-L\u00f6f, P. (1971). Hauptsatz for the Intuitionistic Theory of\n-- Iterated Inductive De\ufb01nitions. In: Proceedings of the Second\n-- Scandinavian Logic Symposium. Ed. by Fenstad,\n-- J. E. Vol. 63. Studies in Logic and the Foundations of\n-- Mathematics. North-Holland Publishing Company, pp. 179\u2013216.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"690cafdee6fee2ac524e0f1da23868ad14235b31","subject":"Working on the equivalence of the induction principles of N.","message":"Working on the equivalence of the induction principles of N.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Induction principle generated by Coq when we define the data type N\n-- in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n------------------------------------------------------------------------------\n-- N-ind\u2082 from N-ind\u2081.\n\nN-ind\u2082' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n------------------------------------------------------------------------------\n-- We cannot prove N-ind\u2081 from N-ind\u2082.\n\nN\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN\u21920\u2228S = N-ind\u2082 A A0 is\n  where\n  A : D \u2192 Set\n  A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n  A0 : A zero\n  A0 = inj\u2081 refl\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = case prf\u2081 prf\u2082 Ai\n    where\n    prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n    prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n           succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\nN-ind\u2081' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081' A A0 h {n} Nn = case prf\u2081 prf\u2082 (N\u21920\u2228S Nn)\n  where\n  prf\u2081 : n \u2261 zero \u2192 A n\n  prf\u2081 n\u22610 = subst A (sym n\u22610) A0\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 A n\n  prf\u2082 (n' , prf , Nn') = subst A (sym prf) (h Nn' {!!})\n","old_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Induction principle generated by Coq when we define the data type N\n-- in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n-- N-ind\u2082 from N-ind\u2081.\nN-ind\u2082' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n-- We cannot prove N-ind\u2081 from N-ind\u2082.\nN-ind\u2081' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081' A A0 h = N-ind\u2082 A A0 {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"797cf2e55e59bad2fbba168038b1da6e55a21f34","subject":"Added TODO about a regression.","message":"Added TODO about a regression.\n\nIgnore-this: f00d93af876ed7be1aa3d2f9ea68964b\n\ndarcs-hash:20120303194504-3bd4e-ab63034841ffbf499f5b1ebf18b7519923fb5590.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/TheoremsATP.agda","new_file":"src\/FOL\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- FOL theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module contains some examples showing the use of the ATPs for\n-- proving FOL theorems.\n\nmodule FOL.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A     : Set\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n      \u03c6(x)\n  -----------  \u2200-intro\n    \u2200x.\u03c6(x)\n\n    \u2200x.\u03c6(x)\n  -----------  \u2200-elim\n     \u03c6(t)\n\n      \u03c6(t)\n  -----------  \u2203-intro\n    \u2203x.\u03c6(x)\n\n   \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n ----------------------  \u2203-elim\n           \u03c8\n-}\n\npostulate\n  -- TODO: 2012-03-02. Fix? the universal introduction\n  \u2200-intro : (\u2200 x \u2192 A\u00b9 x) \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200-elim  : (\u2200 x \u2192 A\u00b9 x) \u2192 (t : D) \u2192 A\u00b9 t\n  -- It is necessary to assume a non-empty domain. See\n  -- FOL.NonEmptyDomain.TheoremsI\/ATP.\u2203I.\n  --\n  -- TODO: 2012-02-28. Fix the existential introduction rule.\n  -- \u2203-intro : ((t : D) \u2192 A\u00b9 t) \u2192 \u2203 A\u00b9\n  \u2203-elim'  : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n-- {-# ATP prove \u2200-intro #-}\n{-# ATP prove \u2200-elim #-}\n-- {-# ATP prove \u2203-intro #-}\n{-# ATP prove \u2203-elim' #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200 x \u2192 A\u00b9 x) \u2194 (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 A\u00b9)       \u2194 (\u2200 x \u2192 \u00ac (A\u00b9 x))\n  gDM\u2083 : (\u2200 x \u2192 A\u00b9 x)   \u2194 \u00ac (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2084 : \u2203 A\u00b9           \u2194 \u00ac (\u2200 x \u2192 \u00ac (A\u00b9 x))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : (\u2200 x y \u2192 A\u00b2 x y) \u2194 (\u2200 y x \u2192 A\u00b2 x y)\n  \u2203-ord : (\u2203[ x ] \u2203[ y ] A\u00b2 x y) \u2194 (\u2203[ y ] \u2203[ x ] A\u00b2 x y)\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate\n  -- TODO: 2012-02-03. The ATPs are not proving the theorem.\n  -- \u2200-erase-add : ((x : D) \u2192 A) \u2194 A\n  \u2203-erase-add : (\u2203[ x ] A \u2227 A\u00b9 x) \u2194 A \u2227 (\u2203[ x ] A\u00b9 x)\n-- {-# ATP prove \u2200-erase-add #-}\n{-# ATP prove \u2203-erase-add #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : \u2200 x \u2192 A\u00b9 x \u2227 B\u00b9 x \u2194 ((\u2200 x \u2192 A\u00b9 x) \u2227 (\u2200 x \u2192 B\u00b9 x))\n  \u2203-dist : (\u2203[ x ] A\u00b9 x \u2228 B\u00b9 x) \u2194 (\u2203 A\u00b9 \u2228 \u2203 B\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n\n-- Interchange of quantifiers.\n-- The related theorem \u2200x\u2203y.Axy \u2192 \u2203y\u2200x.Axy is not (classically) valid.\npostulate \u2203\u2200 : \u2203[ x ] (\u2200 y \u2192 A\u00b2 x y) \u2192 \u2200 y \u2192 \u2203[ x ] A\u00b2 x y\n{-# ATP prove \u2203\u2200 #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate\n  \u2203\u2192\u00ac\u2200\u00ac : \u2203[ x ] A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 \u00ac A\u00b9 x)\n  \u2203\u00ac\u2192\u00ac\u2200 : \u2203[ x ] \u00ac A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 A\u00b9 x)\n{-# ATP prove \u2203\u2192\u00ac\u2200\u00ac #-}\n{-# ATP prove \u2203\u00ac\u2192\u00ac\u2200 #-}\n\n-- \u2200 in terms of \u2203 and \u00ac.\npostulate\n  \u2200\u2192\u00ac\u2203\u00ac : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u00ac (\u2203[ x ] \u00ac A\u00b9 x)\n  \u2200\u00ac\u2192\u00ac\u2203 : (\u2200 {x} \u2192 \u00ac A\u00b9 x) \u2192 \u00ac (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200\u2192\u00ac\u2203\u00ac #-}\n{-# ATP prove \u2200\u00ac\u2192\u00ac\u2203 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- FOL theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module contains some examples showing the use of the ATPs for\n-- proving FOL theorems.\n\nmodule FOL.TheoremsATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n-- We postulate some formulae and propositional functions.\npostulate\n  A     : Set\n  A\u00b9 B\u00b9 : D \u2192 Set\n  A\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\n{-\n      \u03c6(x)\n  -----------  \u2200-intro\n    \u2200x.\u03c6(x)\n\n    \u2200x.\u03c6(x)\n  -----------  \u2200-elim\n     \u03c6(t)\n\n      \u03c6(t)\n  -----------  \u2203-intro\n    \u2203x.\u03c6(x)\n\n   \u2203x.\u03c6(x)   \u03c6(x) \u2192 \u03c8\n ----------------------  \u2203-elim\n           \u03c8\n-}\n\npostulate\n  -- TODO: 2012-03-02. Fix? the universal introduction\n  \u2200-intro : (\u2200 x \u2192 A\u00b9 x) \u2192 \u2200 x \u2192 A\u00b9 x\n  \u2200-elim  : (\u2200 x \u2192 A\u00b9 x) \u2192 (t : D) \u2192 A\u00b9 t\n  -- It is necessary to assume a non-empty domain. See\n  -- FOL.NonEmptyDomain.TheoremsI\/ATP.\u2203I.\n  --\n  -- TODO: 2012-02-28. Fix the existential introduction rule.\n  -- \u2203-intro : ((t : D) \u2192 A\u00b9 t) \u2192 \u2203 A\u00b9\n  \u2203-elim'  : \u2203 A\u00b9 \u2192 ((x : D) \u2192 A\u00b9 x \u2192 A) \u2192 A\n-- {-# ATP prove \u2200-intro #-}\n{-# ATP prove \u2200-elim #-}\n-- {-# ATP prove \u2203-intro #-}\n{-# ATP prove \u2203-elim' #-}\n\n-- Generalization of De Morgan's laws.\npostulate\n  gDM\u2081 : \u00ac (\u2200 x \u2192 A\u00b9 x) \u2194 (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2082 : \u00ac (\u2203 A\u00b9)       \u2194 (\u2200 x \u2192 \u00ac (A\u00b9 x))\n  gDM\u2083 : (\u2200 x \u2192 A\u00b9 x)   \u2194 \u00ac (\u2203[ x ] \u00ac (A\u00b9 x))\n  gDM\u2084 : \u2203 A\u00b9           \u2194 \u00ac (\u2200 x \u2192 \u00ac (A\u00b9 x))\n{-# ATP prove gDM\u2081 #-}\n{-# ATP prove gDM\u2082 #-}\n{-# ATP prove gDM\u2083 #-}\n{-# ATP prove gDM\u2084 #-}\n\n-- The order of quantifiers of the same sort is irrelevant.\npostulate\n  \u2200-ord : (\u2200 x y \u2192 A\u00b2 x y) \u2194 (\u2200 y x \u2192 A\u00b2 x y)\n  \u2203-ord : (\u2203[ x ] \u2203[ y ] A\u00b2 x y) \u2194 (\u2203[ y ] \u2203[ x ] A\u00b2 x y)\n{-# ATP prove \u2200-ord #-}\n{-# ATP prove \u2203-ord #-}\n\n-- Quantification over a variable that does not occur can be erased or\n-- added.\npostulate\n  \u2200-erase-add : ((_ : D) \u2192 A) \u2194 A\n  \u2203-erase-add : (\u2203[ x ] A \u2227 A\u00b9 x) \u2194 A \u2227 (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200-erase-add #-}\n{-# ATP prove \u2203-erase-add #-}\n\n-- Distributes laws for the quantifiers.\npostulate\n  \u2200-dist : \u2200 x \u2192 A\u00b9 x \u2227 B\u00b9 x \u2194 ((\u2200 x \u2192 A\u00b9 x) \u2227 (\u2200 x \u2192 B\u00b9 x))\n  \u2203-dist : (\u2203[ x ] A\u00b9 x \u2228 B\u00b9 x) \u2194 (\u2203 A\u00b9 \u2228 \u2203 B\u00b9)\n{-# ATP prove \u2200-dist #-}\n{-# ATP prove \u2203-dist #-}\n\n-- Interchange of quantifiers.\n-- The related theorem \u2200x\u2203y.Axy \u2192 \u2203y\u2200x.Axy is not (classically) valid.\npostulate \u2203\u2200 : \u2203[ x ] (\u2200 y \u2192 A\u00b2 x y) \u2192 \u2200 y \u2192 \u2203[ x ] A\u00b2 x y\n{-# ATP prove \u2203\u2200 #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate\n  \u2203\u2192\u00ac\u2200\u00ac : \u2203[ x ] A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 \u00ac A\u00b9 x)\n  \u2203\u00ac\u2192\u00ac\u2200 : \u2203[ x ] \u00ac A\u00b9 x \u2192 \u00ac (\u2200 {x} \u2192 A\u00b9 x)\n{-# ATP prove \u2203\u2192\u00ac\u2200\u00ac #-}\n{-# ATP prove \u2203\u00ac\u2192\u00ac\u2200 #-}\n\n-- \u2200 in terms of \u2203 and \u00ac.\npostulate\n  \u2200\u2192\u00ac\u2203\u00ac : (\u2200 {x} \u2192 A\u00b9 x) \u2192 \u00ac (\u2203[ x ] \u00ac A\u00b9 x)\n  \u2200\u00ac\u2192\u00ac\u2203 : (\u2200 {x} \u2192 \u00ac A\u00b9 x) \u2192 \u00ac (\u2203[ x ] A\u00b9 x)\n{-# ATP prove \u2200\u2192\u00ac\u2203\u00ac #-}\n{-# ATP prove \u2200\u00ac\u2192\u00ac\u2203 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39b0ed06ce81e7a55bbf0054b176322db6ad4008","subject":"agda: *old* failed experiment with other change definition","message":"agda: *old* failed experiment with other change definition\n\nThis was lying around on my old laptop.\n\nOld-commit-hash: f8ce36e22c6a0d4d2a65bddf007e16ac35ad881f\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/PureChanges.agda","new_file":"experimental\/PureChanges.agda","new_contents":"module PureChanges where\n\nopen import Data.Empty\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.List\nopen import Algebra\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl; cong\u2082; cong; sym; module \u2261-Reasoning)\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n--import Relation.Binary.HeterogeneousEquality as H\n\nopen import DecidableEq \u2115 _\u225f_\n\nBase : Set\nBase = \u2115\n\nChange\u2081 : Set\nChange\u2081 = \u2115 \u00d7 \u2115\n\nChange : Set\nChange = List Change\u2081\n\napply\u2081 : Change\u2081 \u2192 Base \u2192 Base\napply\u2081 (proj\u2081 , proj\u2082) v = if \u230a proj\u2081 \u225f v \u230b then proj\u2082 else v\n\napply : Change \u2192 Base \u2192 Base\napply = flip (foldr apply\u2081)\n\ndiff : Base \u2192 Base \u2192 Change\ndiff new old = [ old , new ]\n\ncompose : Change \u2192 Change \u2192 Change\ncompose = _++_\n\nopen import Data.List.Properties\ncompose-IsFold : \u2200 {xs ys} \u2192 compose xs ys \u2261 foldr _\u2237_ ys xs\ncompose-IsFold {xs} {ys} = ++IsFold xs ys\n\n_\u2295_ = apply\n_\u229d_ = diff\n_\u229a_ = compose\n\ninfixl 6 _\u2295_ _\u229d_ _\u229a_ -- as with + - in GHC.Num\n\n\n\u2261-cong\u2082 = cong\u2082\n\n\u2261-diff : \u2200 {v\u2081 v\u2082 v\u2083 v\u2084} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {dv\u2081 dv\u2082} {v\u2081 v\u2082} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\napply-diff : \u2200 {v\u2081 v\u2082} \u2192 apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\napply-diff {v\u2081} {v\u2082} = \n  begin\n    apply (diff v\u2082 v\u2081) v\u2081\n  \u2261\u27e8\u27e9\n    (if \u230a v\u2081 \u225f v\u2081 \u230b then v\u2082 else v\u2081)\n  \u2261\u27e8 cong (\u03bb x \u2192 if x then v\u2082 else v\u2081) (v\u225fv-true v\u2081) \u27e9 \n    (if true then v\u2082 else v\u2081)\n  \u2261\u27e8\u27e9\n    v\u2082\n  \u220e\n  where\n    open \u2261-Reasoning\n\ndiff-apply : \u2200 ds s \u2192 ds \u2295 s \u229d s \u2261 ds\ndiff-apply [] s = {!!}\ndiff-apply ((proj\u2081 , proj\u2082) \u2237 ds) s rewrite diff-apply ds s = {!refl!}\n\napply-compose : \u2200 v dv\u2081 dv\u2082 \u2192\n  (dv\u2081 \u229a dv\u2082) \u2295 v \u2261 dv\u2081 \u2295 (dv\u2082 \u2295 v)\napply-compose v [] dv\u2082 = refl\napply-compose v (x \u2237 dv\u2081) dv\u2082 rewrite apply-compose v dv\u2081 dv\u2082 = refl\n\ncompose-assoc : \u2200 dv\u2081 dv\u2082 dv\u2083 \u2192 (dv\u2081 \u229a dv\u2082) \u229a dv\u2083 \u2261 dv\u2081 \u229a (dv\u2082 \u229a dv\u2083)\ncompose-assoc = let open Monoid (monoid Change\u2081) in assoc\n\n-- compose-assoc (old\u2081 , new\u2081) (old\u2082 , new\u2082) (old\u2083 , new\u2083) with new\u2081 \u225f old\u2082 | new\u2082 \u225f old\u2083\n-- compose-assoc (old\u2081 , new\u2081) (.new\u2081 , new\u2082) (.new\u2082 , new\u2083) | yes refl | yes refl = let\n--     open \u2261-Reasoning in\n--   begin\n--     (if \u230a new\u2082 \u225f new\u2082 \u230b then old\u2081 , new\u2083 else old\u2081 , new\u2082)\n--   \u2261\u27e8 cong (\u03bb b \u2192 if b then old\u2081 , new\u2083 else old\u2081 , new\u2082) (v\u225fv-true new\u2082) \u27e9\n--     old\u2081 , new\u2083\n--   \u2261\u27e8 sym (cong (\u03bb b \u2192 if b then old\u2081 , new\u2083 else old\u2081 , new\u2081) (v\u225fv-true new\u2081)) \u27e9\n--     (if \u230a new\u2081 \u225f new\u2081 \u230b then old\u2081 , new\u2083 else old\u2081 , new\u2081)\n--   \u220e\n-- compose-assoc (old\u2081 , new\u2081) (.new\u2081 , new\u2082) (old\u2083 , new\u2083) | yes refl | no \u00acp = {!!}\n-- compose-assoc (old\u2081 , new\u2081) (old\u2082 , new\u2082) (old\u2083 , new\u2083) | no \u00acp | yes p = {!!}\n-- compose-assoc (old\u2081 , new\u2081) (old\u2082 , new\u2082) (old\u2083 , new\u2083) | no \u00acp | no \u00acp\u2081 = let\n--     open \u2261-Reasoning in\n--   begin\n--     (if \u230a new\u2081 \u225f old\u2083 \u230b then old\u2081 , new\u2083 else old\u2081 , new\u2081)\n--   \u2261\u27e8 cong (\u03bb b \u2192 if b then old\u2081 , new\u2083 else old\u2081 , new\u2081) (\u2262\u2192\u225ffalse {new\u2081} {old\u2083} {!\n--   {- This hole cannot be filled - one can easily construct a counterexample. -}\n-- !}) \u27e9\n--     old\u2081 , new\u2081\n--   \u2261\u27e8 sym (cong (\u03bb b \u2192 if b then old\u2081 , new\u2082 else old\u2081 , new\u2081) (\u2262\u2192\u225ffalse \u00acp)) \u27e9\n--     (if \u230a new\u2081 \u225f old\u2082 \u230b then old\u2081 , new\u2082 else old\u2081 , new\u2081)\n--   \u220e\n","old_contents":"module PureChanges where\n\nopen import Data.Empty\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Bool hiding (_\u225f_)\n\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n--import Relation.Binary.HeterogeneousEquality as H\n\nopen import DecidableEq \u2115 _\u225f_\n\nBase : Set\nBase = \u2115\n\nChange : Set\nChange = \u2115 \u00d7 \u2115\n\napply : Change \u2192 Base \u2192 Base\napply (proj\u2081 , proj\u2082) v = if \u230a proj\u2081 \u225f v \u230b then proj\u2082 else v\n\ndiff : Base \u2192 Base \u2192 Change\ndiff new old = old , new\n\ncompose : Change \u2192 Change \u2192 Change\ncompose (proj\u2081\u2081 , proj\u2082\u2081) (proj\u2081\u2082 , proj\u2082\u2082) = if \u230a proj\u2082\u2081 \u225f proj\u2081\u2082 \u230b then proj\u2081\u2081 , proj\u2082\u2082 else proj\u2081\u2081 , proj\u2082\u2081\n\n_\u2295_ = apply\n_\u229d_ = diff\n_\u229a_ = compose\n\ninfixl 6 _\u2295_ _\u229d_ _\u229a_ -- as with + - in GHC.Num\n\n\n\u2261-cong\u2082 = cong\u2082\n\n\u2261-diff : \u2200 {v\u2081 v\u2082 v\u2083 v\u2084} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {dv\u2081 dv\u2082} {v\u2081 v\u2082} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\napply-diff : \u2200 {v\u2081 v\u2082} \u2192 apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\napply-diff {v\u2081} {v\u2082} = \n  begin\n    apply (diff v\u2082 v\u2081) v\u2081\n  \u2261\u27e8\u27e9\n    (if \u230a v\u2081 \u225f v\u2081 \u230b then v\u2082 else v\u2081)\n  \u2261\u27e8 cong (\u03bb x \u2192 if x then v\u2082 else v\u2081) (v\u225fv-true v\u2081) \u27e9 \n    (if true then v\u2082 else v\u2081)\n  \u2261\u27e8\u27e9\n    v\u2082\n  \u220e\n  where\n    open \u2261-Reasoning\n\ncompose-assoc : \u2200 dv\u2081 dv\u2082 dv\u2083 \u2192 (dv\u2081 \u229a dv\u2082) \u229a dv\u2083 \u2261 dv\u2081 \u229a (dv\u2082 \u229a dv\u2083)\ncompose-assoc (old\u2081 , new\u2081) (old\u2082 , new\u2082) (old\u2083 , new\u2083) with new\u2081 \u225f old\u2082 | new\u2082 \u225f old\u2083\ncompose-assoc (old\u2081 , new\u2081) (.new\u2081 , new\u2082) (.new\u2082 , new\u2083) | yes refl | yes refl = let\n    open \u2261-Reasoning in\n  begin\n    (if \u230a new\u2082 \u225f new\u2082 \u230b then old\u2081 , new\u2083 else old\u2081 , new\u2082)\n  \u2261\u27e8 cong (\u03bb b \u2192 if b then old\u2081 , new\u2083 else old\u2081 , new\u2082) (v\u225fv-true new\u2082) \u27e9\n    old\u2081 , new\u2083\n  \u2261\u27e8 sym (cong (\u03bb b \u2192 if b then old\u2081 , new\u2083 else old\u2081 , new\u2081) (v\u225fv-true new\u2081)) \u27e9\n    (if \u230a new\u2081 \u225f new\u2081 \u230b then old\u2081 , new\u2083 else old\u2081 , new\u2081)\n  \u220e\ncompose-assoc (old\u2081 , new\u2081) (.new\u2081 , new\u2082) (old\u2083 , new\u2083) | yes refl | no \u00acp = {!!}\ncompose-assoc (old\u2081 , new\u2081) (old\u2082 , new\u2082) (old\u2083 , new\u2083) | no \u00acp | yes p = {!!}\ncompose-assoc (old\u2081 , new\u2081) (old\u2082 , new\u2082) (old\u2083 , new\u2083) | no \u00acp | no \u00acp\u2081 = let\n    open \u2261-Reasoning in\n  begin\n    (if \u230a new\u2081 \u225f old\u2083 \u230b then old\u2081 , new\u2083 else old\u2081 , new\u2081)\n  \u2261\u27e8 cong (\u03bb b \u2192 if b then old\u2081 , new\u2083 else old\u2081 , new\u2081) (\u2262\u2192\u225ffalse {new\u2081} {old\u2083} {!\n  {- This hole cannot be filled - one can easily construct a counterexample. -}\n!}) \u27e9\n    old\u2081 , new\u2081\n  \u2261\u27e8 sym (cong (\u03bb b \u2192 if b then old\u2081 , new\u2082 else old\u2081 , new\u2081) (\u2262\u2192\u225ffalse \u00acp)) \u27e9\n    (if \u230a new\u2081 \u225f old\u2082 \u230b then old\u2081 , new\u2082 else old\u2081 , new\u2081)\n  \u220e \n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"30eff1b7c2acf3c57ac9266790528bffb5dbb5ce","subject":"changed the proof of \\top from true to obvious","message":"changed the proof of \\top from true to obvious\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Logic.agda","new_file":"agda\/Logic.agda","new_contents":"module Logic where\n\n-- The true proposition.\ndata \u22a4 : Set where\n  obvious : \u22a4 -- The proof of truth.\n\n-- The false proposition.\ndata \u22a5 : Set where\n  -- There is nothing here so one can never prove false.\n\n-- The AND of two statments.\ndata _\u2227_ (A B : Set)  : Set where\n  -- The only way to construct a proof of A \u2227 B is by pairing a a\n  -- proof of A with a proof of and B.\n  \u27e8_,_\u27e9 : (a : A)  -- Proof of A\n        \u2192 (b : B)  -- Proof of B\n        \u2192 A \u2227 B    -- Proof of A \u2227 B\n\n-- The OR of two statements.\ndata _\u2228_ (A B : Set) : Set where\n  -- There are two ways of constructing a proof of A \u2228 B.\n  inl : (a : A) \u2192  A \u2228 B   -- From a proof of A by left introduction\n  inr : (b : B) \u2192  A \u2228 B   -- From a proof of B by right introduction\n\n-- The not of statement A\n\u00ac_ : (A : Set) \u2192 Set\n\u00ac A = A \u2192 \u22a5  -- Given a proof of A one should be able to get a proof\n             -- of \u22a5.\n\n-- The statement A \u2194 B are equivalent.\n_\u2194_ : (A B : Set) \u2192 Set\nA \u2194 B = (A \u2192 B) -- If\n           \u2227    -- and\n        (B \u2192 A) -- only if\n\n\ninfixr 1 _\u2227_\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\ninfix  2 \u00ac_\n\n-- Function composition\n_\u2218_   : {A B C : Set} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n(f \u2218 g) x = f (g x)\n\n-- Double negation\ndoubleNegation : \u2200 {A : Set} \u2192 A \u2192 \u00ac (\u00ac A)\ndoubleNegation a negNegA = negNegA a\n\n{-\n\ndoubleNegation' : \u2200 {A : Set} \u2192 \u00ac ( \u00ac (\u00ac A)) \u2192 \u00ac A\n\n-}\n\n\ndeMorgan1 : \u2200 (A B : Set) \u2192 \u00ac (A \u2228 B) \u2192 \u00ac A \u2227 \u00ac B\ndeMorgan1 A B notAorB = \u27e8 notAorB \u2218 inl , notAorB \u2218 inr \u27e9\n\ndeMorgan2 : \u2200 (A B : Set) \u2192 \u00ac A \u2227 \u00ac B \u2192 \u00ac (A \u2228 B)\ndeMorgan2 A B \u27e8 notA , notB \u27e9 (inl a) = notA a\ndeMorgan2 A B \u27e8 notA , notB \u27e9 (inr b) = notB b\n\n\ndeMorgan : \u2200 (A B : Set) \u2192 \u00ac (A \u2228 B) \u2194 \u00ac A \u2227 \u00ac B\ndeMorgan A B = \u27e8 deMorgan1 A B , deMorgan2 A B \u27e9\n","old_contents":"module Logic where\n\n-- The true proposition.\ndata \u22a4 : Set where\n  true : \u22a4 -- The proof of truth.\n\n-- The false proposition.\ndata \u22a5 : Set where\n  -- There is nothing here so one can never prove false.\n\n-- The AND of two statments.\ndata _\u2227_ (A B : Set)  : Set where\n  -- The only way to construct a proof of A \u2227 B is by pairing a a\n  -- proof of A with a proof of and B.\n  \u27e8_,_\u27e9 : (a : A)  -- Proof of A\n        \u2192 (b : B)  -- Proof of B\n        \u2192 A \u2227 B    -- Proof of A \u2227 B\n\n-- The OR of two statements.\ndata _\u2228_ (A B : Set) : Set where\n  -- There are two ways of constructing a proof of A \u2228 B.\n  inl : (a : A) \u2192  A \u2228 B   -- From a proof of A by left introduction\n  inr : (b : B) \u2192  A \u2228 B   -- From a proof of B by right introduction\n\n-- The not of statement A\n\u00ac_ : (A : Set) \u2192 Set\n\u00ac A = A \u2192 \u22a5  -- Given a proof of A one should be able to get a proof\n             -- of \u22a5.\n\n-- The statement A \u2194 B are equivalent.\n_\u2194_ : (A B : Set) \u2192 Set\nA \u2194 B = (A \u2192 B) -- If\n           \u2227    -- and\n        (B \u2192 A) -- only if\n\n\ninfixr 1 _\u2227_\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\ninfix  2 \u00ac_\n\n-- Function composition\n_\u2218_   : {A B C : Set} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n(f \u2218 g) x = f (g x)\n\n-- Double negation\ndoubleNegation : \u2200 {A : Set} \u2192 A \u2192 \u00ac (\u00ac A)\ndoubleNegation a negNegA = negNegA a\n\n{-\n\ndoubleNegation' : \u2200 {A : Set} \u2192 \u00ac ( \u00ac (\u00ac A)) \u2192 \u00ac A\n\n-}\n\n\ndeMorgan1 : \u2200 (A B : Set) \u2192 \u00ac (A \u2228 B) \u2192 \u00ac A \u2227 \u00ac B\ndeMorgan1 A B notAorB = \u27e8 notAorB \u2218 inl , notAorB \u2218 inr \u27e9\n\ndeMorgan2 : \u2200 (A B : Set) \u2192 \u00ac A \u2227 \u00ac B \u2192 \u00ac (A \u2228 B)\ndeMorgan2 A B \u27e8 notA , notB \u27e9 (inl a) = notA a\ndeMorgan2 A B \u27e8 notA , notB \u27e9 (inr b) = notB b\n\n\ndeMorgan : \u2200 (A B : Set) \u2192 \u00ac (A \u2228 B) \u2194 \u00ac A \u2227 \u00ac B\ndeMorgan A B = \u27e8 deMorgan1 A B , deMorgan2 A B \u27e9\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"d83a2135709fc013dd2eab82389d77e97bd12aa4","subject":"Simplify and complete stability on variables","message":"Simplify and complete stability on variables\n\nOld-commit-hash: 67bac93b9f4260305c74a19d6593a0ad14481f72\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-- Debug tool\nabsurd! : \u2200 {A B : Set} \u2192 B \u2192 A \u2192 A \u2192 B\nabsurd! b _ _ = b\n\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\n{-\n-- Reformulating stableness as a decidable relation\n-- Not sure if it is necessary or not.\ndata StableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  abide-this : \u2200 {\u03c4 \u0393} \u2192 {vars : Vars \u0393} \u2192 StableVar this (abide {\u03c4} vars)\n  abide-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (abide vars)\n  alter-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (alter vars)\n\ndata Stable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  stable-nat : \u2200 {\u0393 n vars} \u2192 Stable {nats} {\u0393} (nat n) vars\n  stable-bag : \u2200 {\u0393 b vars} \u2192 Stable {bags} {\u0393} (bag b) vars\n  stable-var : \u2200 {\u03c4 \u0393 x vars} \u2192\n               StableVar {\u03c4} {\u0393} x vars \u2192 Stable (var x) vars\n  stable-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n               Stable t (abide vars) \u2192 Stable (abs t) vars\n  -- TODO app add map diff union\n-}\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\nvolatility\u21d2identity :\n  \u2200 {\u03c4} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg WRT (expect-volatility {\u03c4}) \u2192 df \u2261 dg\n\nvolatility\u21d2identity {nats} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {bags} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {dg} df\u2248dg =\n  extensionality (\u03bb x \u2192 extensionality (\u03bb dx \u2192\n    volatility\u21d2identity {\u03c4\u2082} (df\u2248dg {x} {dx})))\n\ndf\u2248df : \u2200 {\u03c4} {df : \u27e6 \u0394-Type \u03c4 \u27e7} {args : Args \u03c4} \u2192 df \u2248 df WRT args\ndf\u2248df {nats} = refl\ndf\u2248df {bags} = refl\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {abide args} =\n  \u03bb {x} {dx} _ \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {alter args} =\n  \u03bb {x} {dx}   \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {\u03c4}{_} {args} {df} {dg} {\u03c1} df=dg\n  rewrite df=dg = df\u2248df {\u03c4} {\u27e6 dg \u27e7 \u03c1} {args}\n\n-- A variable does not change if its value is unchanging.\nstabilityVar : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stableVar x vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 deriveVar x \u27e7 \u03c1 \u2261 \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1\n\nstabilityVar this (alter vars) () (alter honesty)\nstabilityVar this (abide vars) refl (abide proof honesty) = proof\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (abide vars) truth (abide _ honesty)\n  = stabilityVar x vars truth honesty\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (alter vars) truth (alter honesty)\n  = stabilityVar x vars truth honesty\n\n-- A term does not change if its free variables are unchanging.\nstability : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stable t vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\n\nstability (nat n) vars truth {\u03c1} _ = refl\n\nstability (bag b) vars truth {\u03c1} _ = b++\u2205=b\n\nstability (var x) vars truth {\u03c1} honesty =\n  stabilityVar x vars truth honesty\n\nstability (abs t) vars truth {\u03c1} honesty = {!!}\nstability (app t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (add t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (map t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (diff t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (union t t\u2081) vars truth {\u03c1} honesty = {!!}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\n-- If both the environment and the future arguments are honest\n-- about nil changes, then the optimized derivation delivers\n-- the same result as the original derivation.\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy (app f s) args vars \u03c1 honesty\n  with stable s vars | inspect (stable s) vars\n... | true  | [ truth ] = {!!}\n--  absurd! {!!} (stability s vars truth {\u03c1}) {!!}\n\n... | false | [ falsehood ] = {!!}\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) (abide args) vars \u03c1 honesty =\n  \u03bb {x} {dx} x\u2295dx=x \u2192 honesty-is-the-best-policy\n    t args (abide vars) (dx \u2022 x \u2022 \u03c1) (abide x\u2295dx=x honesty)\n\nhonesty-is-the-best-policy (abs t) (alter args) vars \u03c1 honesty =\n  \u03bb {x} {dx} \u2192 honesty-is-the-best-policy\n    t args (alter vars) (dx \u2022 x \u2022 \u03c1) (alter honesty)\n\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\n-- Debug tool\nabsurd! : \u2200 {A B : Set} \u2192 B \u2192 A \u2192 A \u2192 B\nabsurd! b _ _ = b\n\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\n{-\n-- Reformulating stableness as a decidable relation\n-- Not sure if it is necessary or not.\ndata StableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  abide-this : \u2200 {\u03c4 \u0393} \u2192 {vars : Vars \u0393} \u2192 StableVar this (abide {\u03c4} vars)\n  abide-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (abide vars)\n  alter-that : \u2200 {\u03c4 \u0393 \u03c4\u2032} \u2192 {x : Var \u0393 \u03c4} \u2192 {vars : Vars \u0393} \u2192\n    StableVar x vars \u2192 StableVar (that {\u03c4} {\u03c4\u2032} x) (alter vars)\n\ndata Stable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Set where\n  stable-nat : \u2200 {\u0393 n vars} \u2192 Stable {nats} {\u0393} (nat n) vars\n  stable-bag : \u2200 {\u0393 b vars} \u2192 Stable {bags} {\u0393} (bag b) vars\n  stable-var : \u2200 {\u03c4 \u0393 x vars} \u2192\n               StableVar {\u03c4} {\u0393} x vars \u2192 Stable (var x) vars\n  stable-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 vars} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n               Stable t (abide vars) \u2192 Stable (abs t) vars\n  -- TODO app add map diff union\n-}\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u0393} vars \u03c1 \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\nvolatility\u21d2identity :\n  \u2200 {\u03c4} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg WRT (expect-volatility {\u03c4}) \u2192 df \u2261 dg\n\nvolatility\u21d2identity {nats} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {bags} df\u2248dg = df\u2248dg\nvolatility\u21d2identity {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {dg} df\u2248dg =\n  extensionality (\u03bb x \u2192 extensionality (\u03bb dx \u2192\n    volatility\u21d2identity {\u03c4\u2082} (df\u2248dg {x} {dx})))\n\ndf\u2248df : \u2200 {\u03c4} {df : \u27e6 \u0394-Type \u03c4 \u27e7} {args : Args \u03c4} \u2192 df \u2248 df WRT args\ndf\u2248df {nats} = refl\ndf\u2248df {bags} = refl\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {abide args} =\n  \u03bb {x} {dx} _ \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\ndf\u2248df {\u03c4\u2081 \u21d2 \u03c4\u2082} {df} {alter args} =\n  \u03bb {x} {dx}   \u2192 df\u2248df {\u03c4\u2082} {df x dx} {args}\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {\u03c4}{_} {args} {df} {dg} {\u03c1} df=dg\n  rewrite df=dg = df\u2248df {\u03c4} {\u27e6 dg \u27e7 \u03c1} {args}\n\n-- A variable does not change if its value is unchanging.\nstabilityVar : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stableVar x vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 deriveVar x \u27e7 \u03c1 \u2261 \u27e6 weakenVar \u0393\u227c\u0394\u0393 x \u27e7 \u03c1\n\nstabilityVar this (alter vars) () (alter honesty)\nstabilityVar this (abide vars) refl (abide proof honesty) = proof\n\nstabilityVar {\u03c4} {\u03c4\u2032 \u2022 \u0393 } (that x) (abide vars) truth (abide _ honesty) =\n  stabilityVar x vars (trans eq2 truth) honesty\n  where\n    eq2 : stableVar x vars \u2261 stableVar (that {\u03c4} {\u03c4\u2032} {\u0393} x) (abide vars)\n    eq2 = refl\n\nstabilityVar (that x) (alter vars) truth honesty = {!ditto!}\n\n-- A term does not change if its free variables are unchanging.\nstability : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (vars : Vars \u0393) \u2192\n  stable t vars \u2261 true \u2192\n  \u2200 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192 Honest vars \u03c1 \u2192\n    \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1 \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\n\nstability (nat n) vars truth {\u03c1} _ = refl\n\nstability (bag b) vars truth {\u03c1} _ = b++\u2205=b\n\nstability (var x) vars truth {\u03c1} honesty =\n  {!!}\n  where\n    eq1  : stableVar x vars \u2261 stable (var x) vars\n    eq1  = refl\n    tVar : stableVar x vars \u2261 true\n    tVar = trans eq1 truth\n\nstability (abs t) vars truth {\u03c1} honesty = {!!}\nstability (app t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (add t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (map t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (diff t t\u2081) vars truth {\u03c1} honesty = {!!}\nstability (union t t\u2081) vars truth {\u03c1} honesty = {!!}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\n-- If both the environment and the future arguments are honest\n-- about nil changes, then the optimized derivation delivers\n-- the same result as the original derivation.\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy (app f s) args vars \u03c1 honesty\n  with stable s vars | inspect (stable s) vars\n... | true  | [ truth ] = {!!}\n--  absurd! {!!} (stability s vars truth {\u03c1}) {!!}\n\n... | false | [ falsehood ] = {!!}\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) (abide args) vars \u03c1 honesty =\n  \u03bb {x} {dx} x\u2295dx=x \u2192 honesty-is-the-best-policy\n    t args (abide vars) (dx \u2022 x \u2022 \u03c1) (abide x\u2295dx=x honesty)\n\nhonesty-is-the-best-policy (abs t) (alter args) vars \u03c1 honesty =\n  \u03bb {x} {dx} \u2192 honesty-is-the-best-policy\n    t args (alter vars) (dx \u2022 x \u2022 \u03c1) (alter honesty)\n\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f99b5149b16b777c7f389739c05c835a06df3b48","subject":"Remove inappropriate exec. permission","message":"Remove inappropriate exec. permission\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Context.agda","new_file":"Base\/Change\/Context.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change contexts\n--\n-- This module specifies how a context of values is merged\n-- together with the corresponding context of changes.\n--\n-- In the PLDI paper, instead of merging the contexts together, we just\n-- concatenate them. For example, in the \"typing rule\" for Derive\n-- in Sec. 3.2 of the paper, we write in the conclusion of the rule:\n-- \"\u0393, \u0394\u0393 \u22a2 Derive(t) : \u0394\u03c4\". Simple concatenation is possible because\n-- the paper uses named variables and assumes that no user variables\n-- start with \"d\". In this formalization, we use de Bruijn indices, so\n-- it is easier to alternate values and their changes in the context.\n------------------------------------------------------------------------\n\nmodule Base.Change.Context\n    {Type : Set}\n    (\u0394Type : Type \u2192 Type) where\n\nopen import Base.Syntax.Context Type\n\n-- Transform a context of values into a context of values and\n-- changes.\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- like \u0394Context, but \u0394Type \u03c4 and \u03c4 are swapped\n\u0394Context\u2032 : Context \u2192 Context\n\u0394Context\u2032 \u2205 = \u2205\n\u0394Context\u2032 (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394Type \u03c4 \u2022 \u0394Context\u2032 \u0393\n\n-- This sub-context relationship explains how to go back from\n-- \u0394Context \u0393 to \u0393: You have to drop every other binding.\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change contexts\n--\n-- This module specifies how a context of values is merged\n-- together with the corresponding context of changes.\n--\n-- In the PLDI paper, instead of merging the contexts together, we just\n-- concatenate them. For example, in the \"typing rule\" for Derive\n-- in Sec. 3.2 of the paper, we write in the conclusion of the rule:\n-- \"\u0393, \u0394\u0393 \u22a2 Derive(t) : \u0394\u03c4\". Simple concatenation is possible because\n-- the paper uses named variables and assumes that no user variables\n-- start with \"d\". In this formalization, we use de Bruijn indices, so\n-- it is easier to alternate values and their changes in the context.\n------------------------------------------------------------------------\n\nmodule Base.Change.Context\n    {Type : Set}\n    (\u0394Type : Type \u2192 Type) where\n\nopen import Base.Syntax.Context Type\n\n-- Transform a context of values into a context of values and\n-- changes.\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- like \u0394Context, but \u0394Type \u03c4 and \u03c4 are swapped\n\u0394Context\u2032 : Context \u2192 Context\n\u0394Context\u2032 \u2205 = \u2205\n\u0394Context\u2032 (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394Type \u03c4 \u2022 \u0394Context\u2032 \u0393\n\n-- This sub-context relationship explains how to go back from\n-- \u0394Context \u0393 to \u0393: You have to drop every other binding.\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7a8020cb32ca2429346dc37af611cdb3906409f1","subject":"change is consistent with other things and seems ok","message":"change is consistent with other things and seems ok\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988[_]   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e u e' \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 e' \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988[ u , id \u0393 ] :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988[ u , id \u0393 ] :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4 -- todo: made a change here\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988[ u , \u03c3 ] :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988[ u , \u03c3 ] :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988[ u , \u03c3 ] \u21a6  \u2987 d' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6f88a08b26bb5ac07254a9f0f89b68d42e15c064","subject":"layout","message":"layout\n","repos":"crypto-agda\/crypto-agda","old_file":"composition-sem-sec-reduction.agda","new_file":"composition-sem-sec-reduction.agda","new_contents":"module composition-sem-sec-reduction where\n\nopen import Data.Nat.NP using (\u2115)\nopen import Data.Bits using (vnot; vnot\u2218vnot\u2257id)\nopen import Data.Product using (_,_)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import flipbased-implem using (run\u21ba; map\u21ba)\nopen import program-distance using (module PrgDist; PrgDist)\nopen import flat-funs\nopen import flat-funs-implem\nopen import one-time-semantic-security\nimport bit-guessing-game\n\n-- One focuses on a particular kind of transformation\n-- over a cipher, namely pre and post composing functions\n-- to a given cipher E.\n-- This transformation can change the size of input\/output\n-- of the given cipher.\nmodule Comp (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  open EncParams\u00b2-same-|R|\u1d49 ep\u2080 |M|\u2081 |C|\u2081 using (M\u2080; M\u2081; C\u2080; C\u2081; Tr)\n  open FunOps\n\n  comp : (pre-E : M\u2081 \u2192 M\u2080) (post-E : C\u2080 \u2192 C\u2081) \u2192 Tr\n  comp pre-E post-E E = pre-E \u204f E \u204f map\u21ba post-E\n\n-- This module shows how to adapt an adversary that is supposed to break\n-- one time semantic security of some cipher E enhanced by pre and post\n-- composition to an adversary breaking one time semantic security of the\n-- original cipher E.\n--\n-- The adversary transformation works for any notion of function space\n-- (FlatFuns) and the corresponding operations (FlatFunsOps).\n-- Because of this abstraction the following construction can be safely\n-- instantiated for at least three uses:\n--   * When provided with a concrete notion of function like circuits\n--     or at least functions over bit-vectors, one get a concrete circuit\n--     transformation process.\n--   * When provided with a high-level function space with real products\n--     then proving properties about the code is made easy.\n--   * When provided with a notion of cost such as time or space then\n--     one get a the cost of the resulting adversary given the cost of\n--     the inputs (adversary...).\nmodule CompRed {t} {T : Set t}\n               {\u266dFuns   : FlatFuns T}\n               (\u266dops    : FlatFunsOps \u266dFuns)\n               (ep\u2080 ep\u2081 : EncParams) where\n  open FlatFuns \u266dFuns\n  open FlatFunsOps \u266dops\n  open AbsSemSec \u266dFuns\n  open \u266dEncParams\u00b2 \u266dFuns ep\u2080 ep\u2081 using (`M\u2080; `C\u2080; `M\u2081; `C\u2081)\n\n  comp-red : (pre-E  : `M\u2081 `\u2192 `M\u2080)\n             (post-E : `C\u2080 `\u2192 `C\u2081)\n           \u2192 SemSecReduction ep\u2081 ep\u2080 _\n  comp-red pre-E post-E (mk A\u2080 A\u2081 A\u2082 A\u2083) =\n    mk A\u2080\n      (A\u2081 \u204f first < pre-E \u00d7 pre-E >)\n      (first post-E \u204f A\u2082)\n       A\u2083\n\nmodule CompSec (prgDist : PrgDist) (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  open PrgDist prgDist\n  open FunSemSec prgDist\n  open FunOps\n  open AbsSemSec fun\u266dFuns\n  open EncParams\u00b2-same-|R|\u1d49 ep\u2080 |M|\u2081 |C|\u2081\n  open SemSecReductions ep\u2080 ep\u2081 id\n  open CompRed fun\u266dOps ep\u2080 ep\u2081\n\n  private\n    module Comp-implicit {x y z} = Comp x y z\n  open Comp-implicit public\n\n  comp-pres-sem-sec : \u2200 pre-E post-E \u2192 SemSecTr (comp pre-E post-E)\n  comp-pres-sem-sec pre-E post-E = comp-red pre-E post-E , \u03bb E A \u2192 id\n\n  comp-pres-sem-sec\u207b\u00b9 : \u2200 pre-E pre-E\u207b\u00b9\n                          (pre-E-right-inv : pre-E \u2218 pre-E\u207b\u00b9 \u2257 id)\n                          post-E post-E\u207b\u00b9\n                          (post-E-left-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                        \u2192 SemSecTr\u207b\u00b9 (comp pre-E post-E)\n  comp-pres-sem-sec\u207b\u00b9 pre-E pre-E\u207b\u00b9 pre-E-inv post-E post-E\u207b\u00b9 post-E-inv =\n    SemSecTr\u2192SemSecTr\u207b\u00b9\n      (comp pre-E\u207b\u00b9 post-E\u207b\u00b9)\n      (comp pre-E post-E)\n      (\u03bb E m R \u2192 \u2261.trans (post-E-inv _)\n      (\u2261.cong (\u03bb x \u2192 run\u21ba (E x) R) (pre-E-inv _)))\n      (comp-pres-sem-sec pre-E\u207b\u00b9 post-E\u207b\u00b9)\n\n-- As a concrete example this module post-compose a negation of all the\n-- bits.\nmodule PostNegSec (prgDist : PrgDist) ep where\n  open EncParams ep\n  open EncParams\u00b2 ep ep using (Tr)\n  open CompSec prgDist ep |M| |C|\n  open FunSemSec prgDist\n  open FunOps\n  open SemSecReductions ep ep id\n\n  post-neg : Tr\n  post-neg = comp id vnot\n\n  post-neg-pres-sem-sec : SemSecTr post-neg\n  post-neg-pres-sem-sec = comp-pres-sem-sec id vnot\n\n  post-neg-pres-sem-sec\u207b\u00b9 : SemSecTr\u207b\u00b9 post-neg\n  post-neg-pres-sem-sec\u207b\u00b9 = comp-pres-sem-sec\u207b\u00b9 id id (\u03bb _ \u2192 \u2261.refl)\n                                               vnot vnot vnot\u2218vnot\u2257id\n\n-- Here we apply a concrete prgDist but still with an abstract precision\n-- bound.\nmodule ConcretePrgDist (k : \u2115) where\n  open program-distance.Implem k\n  module Guess-prgDist     = bit-guessing-game prgDist\n  module FunSemSec-prgDist = FunSemSec prgDist\n  module CompSec-prgDist   = CompSec prgDist\n","old_contents":"module composition-sem-sec-reduction where\n\nopen import Data.Nat.NP using (\u2115)\nopen import Data.Bits using (vnot; vnot\u2218vnot\u2257id)\nopen import Data.Product using (_,_)\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2257_)\n\nopen import flipbased-implem using (run\u21ba; map\u21ba)\nopen import program-distance using (module PrgDist; PrgDist)\nopen import flat-funs\nopen import flat-funs-implem\nopen import one-time-semantic-security\nimport bit-guessing-game\n\n-- One focuses on a particular kind of transformation\n-- over a cipher, namely pre and post composing functions\n-- to a given cipher E.\n-- This transformation can change the size of input\/output\n-- of the given cipher.\nmodule Comp (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  open EncParams\u00b2-same-|R|\u1d49 ep\u2080 |M|\u2081 |C|\u2081 using (M\u2080; M\u2081; C\u2080; C\u2081; Tr)\n  open FunOps\n\n  comp : (pre-E : M\u2081 \u2192 M\u2080) (post-E : C\u2080 \u2192 C\u2081) \u2192 Tr\n  comp pre-E post-E E = pre-E \u204f E \u204f map\u21ba post-E\n\n-- This module shows how to adapt an adversary that is supposed to break\n-- one time semantic security of some cipher E enhanced by pre and post\n-- composition to an adversary breaking one time semantic security of the\n-- original cipher E.\n--\n-- The adversary transformation works for any notion of function space\n-- (FlatFuns) and the corresponding operations (FlatFunsOps).\n-- Because of this abstraction the following construction can be safely\n-- instantiated for at least three uses:\n--   * When provided with a concrete notion of function like circuits\n--     or at least functions over bit-vectors, one get a concrete circuit\n--     transformation process.\n--   * When provided with a high-level function space with real products\n--     then proving properties about the code is made easy.\n--   * When provided with a notion of cost such as time or space then\n--     one get a the cost of the resulting adversary given the cost of\n--     the inputs (adversary...).\nmodule CompRed {t} {T : Set t}\n               {\u266dFuns   : FlatFuns T}\n               (\u266dops    : FlatFunsOps \u266dFuns)\n               (ep\u2080 ep\u2081 : EncParams) where\n  open FlatFuns \u266dFuns\n  open FlatFunsOps \u266dops\n  open AbsSemSec \u266dFuns\n  open \u266dEncParams\u00b2 \u266dFuns ep\u2080 ep\u2081 using (`M\u2080; `C\u2080; `M\u2081; `C\u2081)\n\n  comp-red : (pre-E  : `M\u2081 `\u2192 `M\u2080)\n             (post-E : `C\u2080 `\u2192 `C\u2081)\n           \u2192 SemSecReduction ep\u2081 ep\u2080 _\n  comp-red pre-E post-E (mk A\u2080 A\u2081 A\u2082 A\u2083) =\n    mk A\u2080 (A\u2081 \u204f first < pre-E \u00d7 pre-E >) (first post-E \u204f A\u2082) A\u2083\n\nmodule CompSec (prgDist : PrgDist) (ep\u2080 : EncParams) (|M|\u2081 |C|\u2081 : \u2115) where\n  open PrgDist prgDist\n  open FunSemSec prgDist\n  open FunOps\n  open AbsSemSec fun\u266dFuns\n  open EncParams\u00b2-same-|R|\u1d49 ep\u2080 |M|\u2081 |C|\u2081\n  open SemSecReductions ep\u2080 ep\u2081 id\n  open CompRed fun\u266dOps ep\u2080 ep\u2081\n\n  private\n    module Comp-implicit {x y z} = Comp x y z\n  open Comp-implicit public\n\n  comp-pres-sem-sec : \u2200 pre-E post-E \u2192 SemSecTr (comp pre-E post-E)\n  comp-pres-sem-sec pre-E post-E = comp-red pre-E post-E , \u03bb E A \u2192 id\n\n  comp-pres-sem-sec\u207b\u00b9 : \u2200 pre-E pre-E\u207b\u00b9\n                          (pre-E-right-inv : pre-E \u2218 pre-E\u207b\u00b9 \u2257 id)\n                          post-E post-E\u207b\u00b9\n                          (post-E-left-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                        \u2192 SemSecTr\u207b\u00b9 (comp pre-E post-E)\n  comp-pres-sem-sec\u207b\u00b9 pre-E pre-E\u207b\u00b9 pre-E-inv post-E post-E\u207b\u00b9 post-E-inv =\n    SemSecTr\u2192SemSecTr\u207b\u00b9\n      (comp pre-E\u207b\u00b9 post-E\u207b\u00b9)\n      (comp pre-E post-E)\n      (\u03bb E m R \u2192 \u2261.trans (post-E-inv _)\n      (\u2261.cong (\u03bb x \u2192 run\u21ba (E x) R) (pre-E-inv _)))\n      (comp-pres-sem-sec pre-E\u207b\u00b9 post-E\u207b\u00b9)\n\n-- As a concrete example this module post-compose a negation of all the\n-- bits.\nmodule PostNegSec (prgDist : PrgDist) ep where\n  open EncParams ep\n  open EncParams\u00b2 ep ep using (Tr)\n  open CompSec prgDist ep |M| |C|\n  open FunSemSec prgDist\n  open FunOps\n  open SemSecReductions ep ep id\n\n  post-neg : Tr\n  post-neg = comp id vnot\n\n  post-neg-pres-sem-sec : SemSecTr post-neg\n  post-neg-pres-sem-sec = comp-pres-sem-sec id vnot\n\n  post-neg-pres-sem-sec\u207b\u00b9 : SemSecTr\u207b\u00b9 post-neg\n  post-neg-pres-sem-sec\u207b\u00b9 = comp-pres-sem-sec\u207b\u00b9 id id (\u03bb _ \u2192 \u2261.refl)\n                                               vnot vnot vnot\u2218vnot\u2257id\n\n-- Here we apply a concrete prgDist but still with an abstract precision\n-- bound.\nmodule ConcretePrgDist (k : \u2115) where\n  open program-distance.Implem k\n  module Guess-prgDist     = bit-guessing-game prgDist\n  module FunSemSec-prgDist = FunSemSec prgDist\n  module CompSec-prgDist   = CompSec prgDist\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c4286bde265ec051c1ed0b2dc4b3f95ba028710d","subject":"Support type arguments for infix \u2248.","message":"Support type arguments for infix \u2248.\n\nOld-commit-hash: e96c855b72f3792e2bc0279596338ed2c82c4c8b\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Implementation\/Popl14.agda","new_file":"Denotation\/Implementation\/Popl14.agda","new_contents":"module Denotation.Implementation.Popl14 where\n\n-- Notions of programs being implementations of specifications\n-- for Calculus Popl14\n\nopen import Denotation.Specification.Canon-Popl14 public\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Popl14.Change.Derive\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Postulate.Extensionality\nopen import Popl14.Denotation.WeakenSound\n\ninfix 4 implements\nsyntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\nimplements : \u2200 (\u03c4 : Type) \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\nu \u2248\u208d int \u208e v = u \u2261 v\nu \u2248\u208d bag \u208e v = u \u2261 v\nu \u2248\u208d \u03c3 \u21d2 \u03c4 \u208e v =\n  (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394Val \u03c3) (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7)\n  (R[w,\u0394w] : valid {\u03c3} w \u0394w) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 \u0394w \u0394w\u2032) \u2192\n  implements \u03c4 (u w \u0394w R[w,\u0394w]) (v w \u0394w\u2032)\n\ninfix 4 _\u2248_\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {\u03c4} = implements \u03c4\n\nmodule Disambiguation (\u03c4 : Type) where\n  _\u271a_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a_ _\u2212_\n  _\u2243_ : \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243_\n\nmodule FunctionDisambiguation (\u03c3 : Type) (\u03c4 : Type) where\n  open Disambiguation (\u03c3 \u21d2 \u03c4) public\n  _\u271a\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212\u2081_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a\u2081_ _\u2212\u2081_\n  _\u2243\u2081_ : \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243\u2081_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243\u2081_\n  _\u271a\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7\n  _\u271a\u2080_ = _\u27e6\u2295\u27e7_ {\u03c3}\n  _\u2212\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7\n  _\u2212\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n  infixl 6 _\u271a\u2080_ _\u2212\u2080_\n  _\u2243\u2080_ : \u0394Val \u03c3 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 Set\n  _\u2243\u2080_ = _\u2248_ {\u03c3}\n  infix 4 _\u2243\u2080_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = \u22a4\ncompatible {\u03c4 \u2022 \u0393} (cons v \u0394v _ \u03c1) (\u0394v\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  Triple (v \u2261 v\u2032) (\u03bb _ \u2192 \u0394v \u2248 \u0394v\u2032) (\u03bb _ _ \u2192 compatible \u03c1 \u03c1\u2032)\n\n-- If a program implements a specification, then certain things\n-- proven about the specification carry over to the programs.\ncarry-over : \u2200 {\u03c4}\n  {v : \u27e6 \u03c4 \u27e7} {\u0394v : \u0394Val \u03c4} {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7}\n (R[v,\u0394v] : valid v \u0394v) (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248 \u0394v\u2032) \u2192\n let open Disambiguation \u03c4 in\n   v \u229e \u0394v \u2261 v \u271a \u0394v\u2032\n\nu\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let open Disambiguation \u03c4 in\n    u \u229f v \u2243 u \u2212 v\nu\u229fv\u2248u\u229dv {int} = refl\nu\u229fv\u2248u\u229dv {bag} = refl\nu\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n  open FunctionDisambiguation \u03c3 \u03c4\n  result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394Val \u03c3) (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192\n    valid w \u0394w \u2192 \u0394w \u2248 \u0394w\u2032 \u2192\n    g (w \u229e \u0394w) \u229f f w \u2243\u2081 g (w \u271a\u2080 \u0394w\u2032) \u2212\u2081 f w\n  result w \u0394w \u0394w\u2032 R[w,\u0394w] \u0394w\u2248\u0394w\u2032\n    rewrite carry-over {\u03c3} {w} R[w,\u0394w] \u0394w\u2248\u0394w\u2032 =\n    u\u229fv\u2248u\u229dv {\u03c4} {g (w \u271a\u2080 \u0394w\u2032)} {f w}\ncarry-over {int} {v} tt \u0394v\u2248\u0394v\u2032 = cong (_+_  v) \u0394v\u2248\u0394v\u2032\ncarry-over {bag} {v} tt \u0394v\u2248\u0394v\u2032 = cong (_++_ v) \u0394v\u2248\u0394v\u2032\ncarry-over {\u03c3 \u21d2 \u03c4} {f} {\u0394f} {\u0394f\u2032} R[f,\u0394f] \u0394f\u2248\u0394f\u2032 =\n  ext (\u03bb v \u2192\n  let\n    open FunctionDisambiguation \u03c3 \u03c4\n    V = R[v,u-v] {\u03c3} {v} {v}\n    S = u\u229fv\u2248u\u229dv {\u03c3} {v} {v}\n  in\n    carry-over {\u03c4} {f v}\n      {\u0394f v (v \u229f v) V} {\u0394f\u2032 v (v \u2212\u2080 v)}\n      (proj\u2081 (R[f,\u0394f] v (v \u229f v) V))\n      (\u0394f\u2248\u0394f\u2032 v (v \u229f v) (v \u2212\u2080 v) V S))\n\n-- A property relating `ignore` and the subcontext relation \u0393\u227c\u0394\u0393\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} _ = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032}\n  (cons v\u2261v\u2032 _ C) = cong\u2082 _\u2022_ v\u2261v\u2032 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)\n\n-- A specialization of the soundness of weakening\n\u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 fit t \u27e7 \u03c1\u2032\n\u27e6fit\u27e7 t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)) (sym (weaken-sound t \u03c1\u2032))\n","old_contents":"module Denotation.Implementation.Popl14 where\n\n-- Notions of programs being implementations of specifications\n-- for Calculus Popl14\n\nopen import Denotation.Specification.Canon-Popl14 public\nopen import Popl14.Syntax.Type\nopen import Popl14.Syntax.Term\nopen import Popl14.Change.Derive\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Postulate.Extensionality\nopen import Popl14.Denotation.WeakenSound\n\n-- Better name for v \u2248 \u27e6 t \u27e7:\n-- t implements v.\ninfix 4 _\u2248_\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {int} u v = u \u2261 v\n_\u2248_ {bag} u v = u \u2261 v\n_\u2248_ {\u03c3 \u21d2 \u03c4} u v = \n  (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394Val \u03c3) (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7)\n  (R[w,\u0394w] : valid w \u0394w) (\u0394w\u2248\u0394w\u2032 : \u0394w \u2248 \u0394w\u2032) \u2192\n  u w \u0394w R[w,\u0394w] \u2248 v w \u0394w\u2032\n\nmodule Disambiguation (\u03c4 : Type) where\n  _\u271a_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a_ _\u2212_\n  _\u2243_ : \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243_\n\nmodule FunctionDisambiguation (\u03c3 : Type) (\u03c4 : Type) where\n  open Disambiguation (\u03c3 \u21d2 \u03c4) public\n  _\u271a\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u271a\u2081_ = _\u27e6\u2295\u27e7_ {\u03c4}\n  _\u2212\u2081_ : \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u2212\u2081_ = _\u27e6\u229d\u27e7_ {\u03c4}\n  infixl 6 _\u271a\u2081_ _\u2212\u2081_\n  _\u2243\u2081_ : \u0394Val \u03c4 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2243\u2081_ = _\u2248_ {\u03c4}\n  infix 4 _\u2243\u2081_\n  _\u271a\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7\n  _\u271a\u2080_ = _\u27e6\u2295\u27e7_ {\u03c3}\n  _\u2212\u2080_ : \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u03c3 \u27e7 \u2192 \u27e6 \u0394Type \u03c3 \u27e7\n  _\u2212\u2080_ = _\u27e6\u229d\u27e7_ {\u03c3}\n  infixl 6 _\u271a\u2080_ _\u2212\u2080_\n  _\u2243\u2080_ : \u0394Val \u03c3 \u2192 \u27e6 \u0394Type \u03c3 \u27e7 \u2192 Set\n  _\u2243\u2080_ = _\u2248_ {\u03c3}\n  infix 4 _\u2243\u2080_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = \u22a4\ncompatible {\u03c4 \u2022 \u0393} (cons v \u0394v _ \u03c1) (\u0394v\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  Triple (v \u2261 v\u2032) (\u03bb _ \u2192 \u0394v \u2248 \u0394v\u2032) (\u03bb _ _ \u2192 compatible \u03c1 \u03c1\u2032)\n\n-- If a program implements a specification, then certain things\n-- proven about the specification carry over to the programs.\ncarry-over : \u2200 {\u03c4}\n  {v : \u27e6 \u03c4 \u27e7} {\u0394v : \u0394Val \u03c4} {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7}\n (R[v,\u0394v] : valid v \u0394v) (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248 \u0394v\u2032) \u2192\n let open Disambiguation \u03c4 in\n   v \u229e \u0394v \u2261 v \u271a \u0394v\u2032\n\nu\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n  let open Disambiguation \u03c4 in\n    u \u229f v \u2243 u \u2212 v\nu\u229fv\u2248u\u229dv {int} = refl\nu\u229fv\u2248u\u229dv {bag} = refl\nu\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n  open FunctionDisambiguation \u03c3 \u03c4\n  result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394Val \u03c3) (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192\n    valid w \u0394w \u2192 \u0394w \u2248 \u0394w\u2032 \u2192\n    g (w \u229e \u0394w) \u229f f w \u2243\u2081 g (w \u271a\u2080 \u0394w\u2032) \u2212\u2081 f w\n  result w \u0394w \u0394w\u2032 R[w,\u0394w] \u0394w\u2248\u0394w\u2032\n    rewrite carry-over {\u03c3} {w} R[w,\u0394w] \u0394w\u2248\u0394w\u2032 =\n    u\u229fv\u2248u\u229dv {\u03c4} {g (w \u271a\u2080 \u0394w\u2032)} {f w}\ncarry-over {int} {v} tt \u0394v\u2248\u0394v\u2032 = cong (_+_  v) \u0394v\u2248\u0394v\u2032\ncarry-over {bag} {v} tt \u0394v\u2248\u0394v\u2032 = cong (_++_ v) \u0394v\u2248\u0394v\u2032\ncarry-over {\u03c3 \u21d2 \u03c4} {f} {\u0394f} {\u0394f\u2032} R[f,\u0394f] \u0394f\u2248\u0394f\u2032 =\n  ext (\u03bb v \u2192\n  let\n    open FunctionDisambiguation \u03c3 \u03c4\n    V = R[v,u-v] {\u03c3} {v} {v}\n    S = u\u229fv\u2248u\u229dv {\u03c3} {v} {v}\n  in\n    carry-over {\u03c4} {f v}\n      {\u0394f v (v \u229f v) V} {\u0394f\u2032 v (v \u2212\u2080 v)}\n      (proj\u2081 (R[f,\u0394f] v (v \u229f v) V))\n      (\u0394f\u2248\u0394f\u2032 v (v \u229f v) (v \u2212\u2080 v) V S))\n\n-- A property relating `ignore` and the subcontext relation \u0393\u227c\u0394\u0393\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u2205} {\u2205} {\u2205} _ = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032}\n  (cons v\u2261v\u2032 _ C) = cong\u2082 _\u2022_ v\u2261v\u2032 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)\n\n-- A specialization of the soundness of weakening\n\u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 fit t \u27e7 \u03c1\u2032\n\u27e6fit\u27e7 t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 C)) (sym (weaken-sound t \u03c1\u2032))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"be5009e736f000abc308cda35f51ca8c075da427","subject":"Use subst instead of rewrite.","message":"Use subst instead of rewrite.\n\nFor unknown reasons, this seems to play better with type inference.\n\nOld-commit-hash: 55d911c0be0ad7f2ae6e77af9d5250bf4e4932b2\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Correctness.agda","new_file":"Parametric\/Change\/Correctness.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {y : Term \u2205 \u03c3}\n    \u2192 \u27e6 app f y \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {x} {y} =\n    let\n      h  = \u27e6 f \u27e7 \u2205\n      \u0394h = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394h\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 x \u27e7 \u2205\n      u  = \u27e6 y \u27e7 \u2205\n      \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        h u\n      \u2261\u27e8 cong h (sym (v+[u-v]=u {\u03c3})) \u27e9\n        h (v \u229e\u208d \u03c3 \u208e (u \u229f\u208d \u03c3 \u208e v))\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v (u \u229f\u208d \u03c3 \u208e v) \u27e9\n        h v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v)\n      \u2261\u27e8 carry-over {\u03c4}\n         (call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v))\n         (derive-correct f\n           \u2205 \u2205 \u2205 \u2205 v (u \u229f\u208d \u03c3 \u208e v) (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) (u\u229fv\u2248u\u229dv {\u03c3} {u} {v})) \u27e9\n         h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 trans\n         (cong (\u03bb hole \u2192 h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v hole) (meaning-\u229d {\u03c3} {s = y} {x}))\n         (meaning-\u2295 {t = app f x} {\u0394t = \u0394output-term}) \u27e9\n         \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n    rewrite sym (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032) =\n      derive-correct s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {x : Term \u2205 \u03c3} {y : Term \u2205 \u03c3}\n    \u2192 \u27e6 app f y \u27e7\n    \u2261 \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {x} {y} =\n    let\n      h  = \u27e6 f \u27e7 \u2205\n      \u0394h = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394h\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 x \u27e7 \u2205\n      u  = \u27e6 y \u27e7 \u2205\n      \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        h u\n      \u2261\u27e8 cong h (sym (v+[u-v]=u {\u03c3})) \u27e9\n        h (v \u229e\u208d \u03c3 \u208e (u \u229f\u208d \u03c3 \u208e v))\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v (u \u229f\u208d \u03c3 \u208e v) \u27e9\n        h v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v)\n      \u2261\u27e8 carry-over {\u03c4}\n         (call-change {\u03c3} {\u03c4} \u0394h v (u \u229f\u208d \u03c3 \u208e v))\n         (derive-correct f\n           \u2205 \u2205 \u2205 \u2205 v (u \u229f\u208d \u03c3 \u208e v) (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) (u\u229fv\u2248u\u229dv {\u03c3} {u} {v})) \u27e9\n         h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v (u \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n      \u2261\u27e8 trans\n         (cong (\u03bb hole \u2192 h v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394h\u2032 v hole) (meaning-\u229d {\u03c3} {s = y} {x}))\n         (meaning-\u2295 {t = app f x} {\u0394t = \u0394output-term}) \u27e9\n         \u27e6 app f x \u2295\u208d \u03c4 \u208e app (app (derive f) x) (y \u229d x) \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a16a9c7be091c37cf12be701c152a7f5b795e4e3","subject":"simplified otp-lem by factor it a little","message":"simplified otp-lem by factor it a little\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-lem.agda","new_file":"otp-lem.agda","new_contents":"module otp-lem where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\n\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Search.Type\nopen import Search.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)(f : A \u2192 B)(searchA : Search zero A)(f-homo : GroupHomomorphism GA GB f)([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n              (sui : \u2200 k \u2192 StableUnder searchA (flip (Group._\u2219_ GA) k))(search-ext : SearchExt searchA) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the search (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the search function respects extensionality\n  -}\n  \n  {-\n   I had some problems with using the standard library definiton of Groups\n   so I rolled my own, therefor I need some boring proofs first\n\n  -}\n  \n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n  -}\n\n               \n  -- this proof isn't actually any hard..\n  thm : \u2200 {X}(op : X \u2192 X \u2192 X)(O : B \u2192 X) m\u2080 m\u2081 \u2192 searchA op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 searchA op (\u03bb x \u2192 O (f x * m\u2081))\n  thm op O m\u2080 m\u2081 = searchA op (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) op (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (lemma1 x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) op (\u03bb x \u2192 O (f x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (lemma2 x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n    where\n      open \u2261-Reasoning\n\n      lemma1 : \u2200 x \u2192 f (x + - [f] m\u2080) * m\u2080 \u2261 f x\n      lemma1 x rewrite f-homo x (- [f] m\u2080)\n                     | f-pres-inv ([f] m\u2080)\n                     | f-sur m\u2080\n                     | Group.assoc GB (f x) (1\/ m\u2080) m\u2080\n                     | proj\u2081 (Group.inverse GB) m\u2080\n                     | proj\u2082 (Group.identity GB) (f x) = refl\n\n      lemma2 : \u2200 x \u2192 f (x + [f] m\u2081) \u2261 f x * m\u2081\n      lemma2 x rewrite f-homo x ([f] m\u2081)\n                     | f-sur m\u2081 = refl\n","old_contents":"module otp-lem where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\n\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Search.Type\nopen import Search.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)(f : A \u2192 B)(searchA : Search zero A)(f-homo : GroupHomomorphism GA GB f)([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n              (sui : \u2200 k \u2192 StableUnder searchA (flip (Group._\u2219_ GA) k))(search-ext : SearchExt searchA) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the search (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the search function respects extensionality\n  -}\n  \n  {-\n   I had some problems with using the standard library definiton of Groups\n   so I rolled my own, therefor I need some boring proofs first\n\n  -}\n  \n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n    Should be possible to simplify a little, or atleast name some parts betweeen calls of sui..\n  -}\n\n               \n  -- this proof isn't actually any hard..\n  thm : \u2200 {X}(op : X \u2192 X \u2192 X)(O : B \u2192 X) m\u2080 m\u2081 \u2192 searchA op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 searchA op (\u03bb x \u2192 O (f x * m\u2081))\n  thm op O m\u2080 m\u2081 = searchA op (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) op (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong (\u03bb p \u2192 O (p * m\u2080)) (f-homo x (- [f] m\u2080))) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * f(- [f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O ((f x * p) * m\u2080))) (f-pres-inv ([f] m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * 1\/ f([f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (Group.assoc GB (f x) (1\/ f ([f] m\u2080)) m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ f([f] m\u2080)  * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * (1\/ p * m\u2080)))) (f-sur m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ m\u2080 * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (proj\u2081 (Group.inverse GB) m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * 1g))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (proj\u2082 (Group.identity GB) (f x))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) op (\u03bb x \u2192 O (f x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (f-homo x ([f] m\u2081))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * f ([f] m\u2081)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (f-sur m\u2081) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n    where\n      open \u2261-Reasoning\n      \n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5c780ccbd5c829ffabcb3aa2ec8905291fcbc87a","subject":"Convert Base.Change.Algebra to Agda 2.4.2","message":"Convert Base.Change.Algebra to Agda 2.4.2\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  {-\n  instance\n    change-algebra-extractor : \u2200 {x} \u2192 ChangeAlgebra \u2113 (P x)\n    change-algebra-extractor {x} = change-algebra x\n  -}\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\n-- XXX not clear this is ever used\ninstance\n  change-algebra-family-inst = change-algebra\u208d_\u208e\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of Derivative for change algebra families.\nDerivative\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : (px : P x) (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)) \u2192\n  Set (p \u2294 q \u2294 c)\nDerivative\u208d x , y \u208e f df = Derivative {{change-algebra\u208d _ \u208e}} {{change-algebra\u208d _ \u208e}} f df where\n  CPx = change-algebra\u208d x \u208e\n  CQy = change-algebra\u208d y \u208e\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        -- Definition 2.6a\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n    funNil = \u03bb f \u2192 funDiff f f\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      instance\n        changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n\n      changeAlgebra = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e nil a) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d _ \u208e px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    instance\n      changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\u208d _ \u208e\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change Structures.\n--\n-- This module defines the notion of change structures,\n-- as well as change structures for groups, functions and lists.\n--\n-- This module corresponds to Section 2 of the PLDI paper.\n------------------------------------------------------------------------\n\nmodule Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n-- ===============\n--\n-- In the paper, change algebras are called \"change structures\"\n-- and they are described in Section 2.1. We follow the design of\n-- the Agda standard library and define change algebras as two\n-- records, so Definition 2.1 from the PLDI paper maps to the\n-- records IsChangeAlgebra and ChangeAlgebra.\n--\n-- A value of type (IsChangeAlgebra Change update diff) proves that\n-- Change, update and diff together form a change algebra.\n--\n-- A value of type (ChangeAlgebra Carrier) contains the necessary\n-- ingredients to create a change algebra for a carrier set.\n--\n-- In the paper, Carrier is called V (Def. 2.1a),\n-- Change is called \u0394 (Def. 2.1b),\n-- update is written as infix \u2295 (Def. 2.1c),\n-- diff is written as infix \u229d (Def. 2.1d),\n-- and update-diff is specified in Def. 2.1e.\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v)\n    (nil : (u : Carrier) \u2192 Change u) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n    -- This corresponds to Lemma 2.3 from the paper.\n    update-nil : \u2200 v \u2192 update v (nil v) \u2261 v\n\n  -- abbreviations\n  before : \u2200 {v} \u2192 Change v \u2192 Carrier\n  before {v} _ = v\n\n  after : \u2200 {v} \u2192 Change v \u2192 Carrier\n  after {v} dv = update v dv\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    -- This generalizes Def. 2.2. from the paper.\n    nil : (u : Carrier) \u2192 Change u\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff nil\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\n-- The following open ... public statement lets us use infix \u229e\n-- and \u229f for update and diff. In the paper, we use infix \u2295 and\n-- \u229d.\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    ; update-nil\n    ; nil\n    ; before\n    ; after\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n\n-- Families of Change Algebras\n-- ===========================\n--\n-- This is some Agda machinery to allow subscripting change\n-- algebra operations to avoid ambiguity. In the paper,\n-- we simply write (in the paragraph after Def. 2.1):\n--\n--   We overload operators \u2206, \u229d and \u2295 to refer to the\n--   corresponding operations of different change structures; we\n--   will subscript these symbols when needed to prevent\n--   ambiguity.\n--\n-- The following definitions implement this idea formally.\n\nrecord ChangeAlgebraFamily {a p} \u2113 {A : Set a} (P : A \u2192 Set p): Set (suc \u2113 \u2294 p \u2294 a) where\n  constructor\n    family\n  field\n    change-algebra : \u2200 x \u2192 ChangeAlgebra \u2113 (P x)\n\n  module _ x where\n    open ChangeAlgebra (change-algebra x) public\n\nmodule Family = ChangeAlgebraFamily {{...}}\n\nopen Family public\n  using\n    (\n    )\n  renaming\n    ( Change to \u0394\u208d_\u208e\n    ; nil to nil\u208d_\u208e\n    ; update-diff to update-diff\u208d_\u208e\n    ; update-nil to update-nil\u208d_\u208e\n    ; change-algebra to change-algebra\u208d_\u208e\n    ; before to before\u208d_\u208e\n    ; after to after\u208d_\u208e\n    )\n\ninfixl 6 update\u2032 diff\u2032\n\nupdate\u2032 = Family.update\nsyntax update\u2032 x v dv = v \u229e\u208d x \u208e dv\n\ndiff\u2032 = Family.diff\nsyntax diff\u2032 x u v = u \u229f\u208d x \u208e v\n\n-- Derivatives\n-- ===========\n--\n-- This corresponds to Def. 2.4 in the paper.\n\nDerivative : \u2200 {a b c d} {A : Set a} {B : Set b} \u2192\n  {{CA : ChangeAlgebra c A}} \u2192\n  {{CB : ChangeAlgebra d B}} \u2192\n  (f : A \u2192 B) \u2192\n  (df : (a : A) (da : \u0394 a) \u2192 \u0394 (f a)) \u2192\n  Set (a \u2294 b \u2294 c)\nDerivative f df = \u2200 a da \u2192 f a \u229e df a da \u2261 f (a \u229e da)\n\n-- This is a variant of Derivative for change algebra families.\nDerivative\u208d_,_\u208e : \u2200 {a b p q c d} {A : Set a} {B : Set b} {P : A \u2192 Set p} {Q : B \u2192 Set q} \u2192\n  {{CP : ChangeAlgebraFamily c P}} \u2192\n  {{CQ : ChangeAlgebraFamily d Q}} \u2192\n  (x : A) \u2192\n  (y : B) \u2192\n  (f : P x \u2192 Q y) \u2192\n  (df : (px : P x) (dpx : \u0394\u208d x \u208e px) \u2192 \u0394\u208d y \u208e (f px)) \u2192\n  Set (p \u2294 q \u2294 c)\nDerivative\u208d x , y \u208e f df = Derivative f df where\n  CPx = change-algebra\u208d x \u208e\n  CQy = change-algebra\u208d y \u208e\n\n-- Lemma 2.5 appears in Base.Change.Equivalence.\n\n-- Abelian Groups Induce Change Algebras\n-- =====================================\n--\n-- In the paper, as the first example for change structures after\n-- Def. 2.1, we mention that \"each abelian group ... induces a\n-- change structure\". This is the formalization of this result.\n--\n-- The module GroupChanges below takes a proof that A forms an\n-- abelian group and provides a changeAlgebra for A. The proof of\n-- Def 2.1e is by equational reasoning using the group axioms, in\n-- the definition of changeAlgebra.isChangeAlgebra.update-diff.\n\nopen import Algebra.Structures\nopen import Data.Product\nopen import Function\n\nmodule GroupChanges\n    {a} (A : Set a) {_\u2295_} {\u03b5} {_\u207b\u00b9}\n    {{isAbelianGroup : IsAbelianGroup {A = A} _\u2261_ _\u2295_ \u03b5 _\u207b\u00b9}}\n  where\n    open IsAbelianGroup isAbelianGroup\n      hiding\n        ( refl\n        )\n      renaming\n        ( _-_ to _\u229d_\n        ; \u2219-cong to _\u27e8\u2295\u27e9_\n        )\n\n    open \u2261-Reasoning\n\n    changeAlgebra : ChangeAlgebra a A\n    changeAlgebra = record\n      { Change = const A\n      ; update = _\u2295_\n      ; diff = _\u229d_\n      ; nil = const \u03b5\n      ; isChangeAlgebra = record\n        { update-diff = \u03bb u v \u2192\n            begin\n              v \u2295 (u \u229d v)\n            \u2261\u27e8 comm _ _ \u27e9\n              (u \u229d v ) \u2295 v\n            \u2261\u27e8\u27e9\n              (u \u2295 (v \u207b\u00b9)) \u2295 v\n            \u2261\u27e8 assoc _ _ _ \u27e9\n              u \u2295 ((v \u207b\u00b9) \u2295 v)\n            \u2261\u27e8  refl \u27e8\u2295\u27e9 proj\u2081 inverse v  \u27e9\n              u \u2295 \u03b5\n            \u2261\u27e8  proj\u2082 identity u \u27e9\n              u\n            \u220e\n        ; update-nil = proj\u2082 identity\n        }\n      }\n\n-- Function Changes\n-- ================\n--\n-- This is one of our most important results: Change structures\n-- can be lifted to function spaces. We formalize this as a module\n-- FunctionChanges that takes the change algebras for A and B as\n-- arguments, and provides a changeAlgebra for (A \u2192 B). The proofs\n-- are by equational reasoning using 2.1e for A and B.\n\nmodule FunctionChanges\n    {a} {b} {c} {d} (A : Set a) (B : Set b) {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}}\n  where\n    -- This corresponds to Definition 2.6 in the paper.\n    record FunctionChange (f : A \u2192 B) : Set (a \u2294 b \u2294 c \u2294 d) where\n      constructor\n        cons\n      field\n        -- Definition 2.6a\n        apply : (a : A) (da : \u0394 a) \u2192\n          \u0394 (f a)\n\n        -- Definition 2.6b.\n        -- (for some reason, the version in the paper has the arguments of \u2261\n        -- flipped. Since \u2261 is symmetric, this doesn't matter).\n        correct : (a : A) (da : \u0394 a) \u2192\n          f (a \u229e da) \u229e apply (a \u229e da) (nil (a \u229e da)) \u2261 f a \u229e apply a da\n\n    open FunctionChange public\n    open \u2261-Reasoning\n    open import Postulate.Extensionality\n\n    funDiff : (g f : A \u2192 B) \u2192 FunctionChange f\n    funDiff = \u03bb g f \u2192 record\n        { apply = \u03bb a da \u2192 g (a \u229e da) \u229f f a\n          -- the proof of correct is the first half of what we\n          -- have to prove for Theorem 2.7.\n        ; correct = \u03bb a da \u2192\n          begin\n            f (a \u229e da) \u229e (g ((a \u229e da) \u229e nil (a \u229e da)) \u229f f (a \u229e da))\n          \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u229e da) \u229e (g \u25a1 \u229f f (a \u229e da)))\n               (update-nil (a \u229e da)) \u27e9\n            f (a \u229e da) \u229e (g (a \u229e da) \u229f f (a \u229e da))\n          \u2261\u27e8 update-diff (g (a \u229e da)) (f (a \u229e da)) \u27e9\n            g (a \u229e da)\n          \u2261\u27e8 sym (update-diff (g (a \u229e da)) (f a)) \u27e9\n            f a \u229e (g (a \u229e da) \u229f f a)\n          \u220e\n        }\n    funUpdate : \u2200 (f : A \u2192 B) (df : FunctionChange f) \u2192 A \u2192 B\n    funUpdate = \u03bb f df a \u2192 f a \u229e apply df a (nil a)\n    funNil = \u03bb f \u2192 funDiff f f\n\n    mutual\n      -- I have to write the type of funUpdateDiff without using changeAlgebra,\n      -- so I just use the underlying implementations.\n      funUpdateDiff : \u2200 u v \u2192 funUpdate v (funDiff u v) \u2261 u\n      changeAlgebra : ChangeAlgebra (a \u2294 b \u2294 c \u2294 d) (A \u2192 B)\n\n      changeAlgebra = record\n        { Change = FunctionChange\n          -- in the paper, update and diff below are in Def. 2.7\n        ; update = funUpdate\n        ; diff = funDiff\n        ; nil = funNil\n        ; isChangeAlgebra = record\n            -- the proof of update-diff is the second half of what\n            -- we have to prove for Theorem 2.8.\n          { update-diff = funUpdateDiff\n          ; update-nil = \u03bb v \u2192 funUpdateDiff v v\n          }\n        }\n      -- XXX remove mutual recursion by inlining the algebra in here?\n      funUpdateDiff = \u03bb g f \u2192 ext (\u03bb a \u2192\n        begin\n          f a \u229e (g (a \u229e nil a) \u229f f a)\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f a \u229e (g \u25a1 \u229f f a)) (update-nil a) \u27e9\n          f a \u229e (g a \u229f f a)\n        \u2261\u27e8 update-diff (g a) (f a) \u27e9\n          g a\n        \u220e)\n\n    -- This is Theorem 2.9 in the paper.\n    incrementalization : \u2200 (f : A \u2192 B) df a da \u2192\n      (f \u229e df) (a \u229e da) \u2261 f a \u229e apply df a da\n    incrementalization f df a da = correct df a da\n\n    -- This is Theorem 2.10 in the paper.\n    nil-is-derivative : \u2200 (f : A \u2192 B) \u2192\n      Derivative f (apply (nil f))\n    nil-is-derivative f a da =\n      begin\n        f a \u229e apply (nil f) a da\n      \u2261\u27e8 sym (incrementalization f (nil f) a da) \u27e9\n        (f \u229e nil f) (a \u229e da)\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u229e da))\n           (update-nil f) \u27e9\n        f (a \u229e da)\n      \u220e\n\n-- List (== Environment) Changes\n-- =============================\n--\n-- Here, we define a change structure on environments, given a\n-- change structure on the values in the environments. In the\n-- paper, we describe this in Definition 3.5. But note that this\n-- Agda formalization uses de Bruijn indices instead of names, so\n-- environments are just lists. Therefore, when we use Definition\n-- 3.5 in the paper, in this formalization, we use the list-like\n-- change structure defined here.\n\nopen import Data.List\nopen import Data.List.All\n\ndata All\u2032 {a p q} {A : Set a}\n    {P : A \u2192 Set p}\n    (Q : {x : A} \u2192 P x \u2192 Set q)\n  : {xs : List A} (pxs : All P xs) \u2192 Set (p \u2294 q \u2294 a) where\n    []  : All\u2032 Q []\n    _\u2237_ : \u2200 {x xs} {px : P x} {pxs : All P xs} (qx : Q px) (qxs : All\u2032 Q pxs) \u2192 All\u2032 Q (px \u2237 pxs)\n\nmodule ListChanges\n    {a} {c} {A : Set a} (P : A \u2192 Set) {{C : ChangeAlgebraFamily c P}}\n  where\n    update-all : \u2200 {xs} \u2192 (pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs \u2192 All P xs\n    update-all {[]} [] [] = []\n    update-all {x \u2237 xs} (px \u2237 pxs) (dpx \u2237 dpxs) = (px \u229e\u208d x \u208e dpx) \u2237 update-all pxs dpxs\n\n    diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 All\u2032 (\u0394\u208d _ \u208e) pxs\n    diff-all [] [] = []\n    diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = (px\u2032 \u229f\u208d _ \u208e px) \u2237 diff-all pxs\u2032 pxs\n\n    update-diff-all : \u2200 {xs} \u2192 (pxs\u2032 pxs : All P xs) \u2192 update-all pxs (diff-all pxs\u2032 pxs) \u2261 pxs\u2032\n    update-diff-all [] [] = refl\n    update-diff-all (px\u2032 \u2237 pxs\u2032) (px \u2237 pxs) = cong\u2082 _\u2237_ (update-diff\u208d _ \u208e px\u2032 px) (update-diff-all pxs\u2032 pxs)\n\n    changeAlgebra : ChangeAlgebraFamily (c \u2294 a) (All P)\n    changeAlgebra = record\n      { change-algebra = \u03bb xs \u2192 record\n        { Change = All\u2032 \u0394\u208d _ \u208e\n        ; update = update-all\n        ; diff = diff-all\n        ; nil = \u03bb xs \u2192 diff-all xs xs\n        ; isChangeAlgebra = record\n          { update-diff = update-diff-all\n          ; update-nil = \u03bb xs\u2081 \u2192 update-diff-all xs\u2081 xs\u2081\n          }\n        }\n      }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"94e9f25080c2facdc59f9c43578f7269de872983","subject":"Removing subst from Conat\u2192N.","message":"Removing subst from Conat\u2192N.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/PropertiesI.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/PropertiesI.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.PropertiesI\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- 09 January 2014. Doesn't type check with --without-K.\n--\n{-# NO_TERMINATION_CHECK #-}\nConat\u2192N : \u2200 {n} \u2192 Conat n \u2192 N n\nConat\u2192N Cn with Conat-out Cn\n... | inj\u2081 prf               = subst N (sym prf) nzero\n... | inj\u2082 (n' , refl , Cn') = nsucc (Conat\u2192N Cn')\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.PropertiesI\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n{-# NO_TERMINATION_CHECK #-}\nConat\u2192N : \u2200 {n} \u2192 Conat n \u2192 N n\nConat\u2192N Cn with Conat-out Cn\n... | inj\u2081 prf              = subst N (sym prf) nzero\n... | inj\u2082 (n' , prf , Cn') = subst N (sym prf) (nsucc (Conat\u2192N Cn'))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a162e2849ac2bc8c773ce1e19756b28861685f73","subject":"Minor changes.","message":"Minor changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/Predicates.agda","new_file":"notes\/fixed-points\/Predicates.agda","new_contents":"------------------------------------------------------------------------------\n-- The FOTC types without using data, i.e. using Agda as a logical framework\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Predicates where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- The existential proyections.\n\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using postulates.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N     : D \u2192 Set\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  [1] : D\n  [1] = succ\u2081 zero\n\n  1-N : N [1]\n  1-N = nsucc nzero\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    lnil  : List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    lnnil  : ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 [1] \u2237 [])\n  ln = lncons nzero (lncons 1-N lnnil)\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  [1] : D\n  [1] = succ\u2081 zero\n\n  1-N : N [1]\n  1-N = nsucc nzero\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    lnil  : List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    lnnil  : ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 [1] \u2237 [])\n  ln = lncons nzero (lncons 1-N lnnil)\n\n------------------------------------------------------------------------------\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined types.\n    --\n    -- N.B. We cannot write \u03bc in first-order logic. We should use an\n    -- instance instead.\n    \u03bc : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (\u03bc f) \u2261 \u03bc f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    -- LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 \u03bc f d \u2192 f (\u03bc f) d\n    \u03bc-in : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (\u03bc f) d \u2192 \u03bc f d\n\n------------------------------------------------------------------------------\n-- The greatest fixed-point operator.\n\n-- N.B. At the moment, the definitions of \u03bc and \u03bd are the same.\n\nmodule GFP where\n\n  postulate\n    -- Greatest fixed-points correspond to co-inductively defined\n    -- types.\n\n    \u03bd : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    \u03bd-out : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 \u03bd f d \u2192 f (\u03bd f) d\n    GFP\u2082  : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (\u03bd f) d \u2192 \u03bd f d\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type.\n  --\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero      = \u22a4\n  -- NatF X (succ\u2081 n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 X n')\n\n  -- The FOTC natural numbers type using \u03bc.\n  N : D \u2192 Set\n  N = \u03bc NatF\n\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = \u03bc-in NatF zero (inj\u2081 refl)\n\n  nsucc : {n : D} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc {n} Nn = \u03bc-in NatF (succ\u2081 n) (inj\u2082 (n , refl , Nn))\n\n  -- Example.\n  [1] : D\n  [1] = succ\u2081 zero\n\n  1-N : N [1]\n  1-N = nsucc nzero\n\n------------------------------------------------------------------------------\n-- The FOTC list type as the least fixed-point of a functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 X xs')\n\n  -- The FOTC list type using \u03bc.\n  List : D \u2192 Set\n  List = \u03bc ListF\n\n  -- The data constructors of List.\n  lnil : List []\n  lnil = \u03bc-in ListF [] (inj\u2081 refl)\n\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  lcons x {xs} Lxs = \u03bc-in ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ns = ns \u2261 [] \u2228 (\u2203[ n' ] \u2203[ ns' ] ns \u2261 n' \u2237 ns' \u2227 N n' \u2227 X ns')\n\n  -- The FOTC list type using \u03bc.\n  ListN : D \u2192 Set\n  ListN = \u03bc ListNF\n\n  -- The data constructors of ListN.\n  lnnil : ListN []\n  lnnil = \u03bc-in ListNF [] (inj\u2081 refl)\n\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  lncons {n} {ns} Nn LNns =\n    \u03bc-in ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 [1] \u2237 [])\n  ln = lncons nzero (lncons 1-N lnnil)\n\n------------------------------------------------------------------------------\n-- The FOTC Colist type as the greatest fixed-point of a functor.\n\nmodule CoList where\n\n  open GFP\n\n  -- Functor for the FOTC Colists type (the same functor that for the\n  -- List type).\n  ColistF : (D \u2192 Set) \u2192 D \u2192 Set\n  ColistF X xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 X xs')\n\n  -- The FOTC Colist type using \u03bd.\n  Colist : D \u2192 Set\n  Colist = \u03bd ColistF\n\n  -- The data constructors of Colist.\n  nilCL : Colist []\n  nilCL = GFP\u2082 ColistF [] (inj\u2081 refl)\n\n  consCL : \u2200 x xs \u2192 Colist xs \u2192 Colist (x \u2237 xs)\n  consCL x xs CLxs = GFP\u2082 ColistF (x \u2237 xs) (inj\u2082 (x , xs , refl , CLxs))\n\n  -- Example (finite colist).\n  l : Colist (zero \u2237 true \u2237 [])\n  l = consCL zero (true \u2237 []) (consCL true [] nilCL)\n\n  -- TODO: Example (infinite colist).\n  -- zerosCL : Colist {!!}\n  -- zerosCL = consCL zero {!!} zerosCL\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X xs = \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 X xs'\n\n  -- The FOTC Stream type using \u03bd.\n  Stream : D \u2192 Set\n  Stream = \u03bd StreamF\n\n  -- The data constructor of Stream.\n  consS : \u2200 x {xs} \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x {xs} Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , refl , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} Sx\u2237xs = Sxs\n    where\n    unfoldS : StreamF (\u03bd StreamF) (x \u2237 xs)\n    unfoldS = \u03bd-out StreamF (x \u2237 xs) Sx\u2237xs\n\n    e : D\n    e = \u2203-proj\u2081 unfoldS\n\n    Pe : \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 \u03bd StreamF es\n    Pe = \u2203-proj\u2082 unfoldS\n\n    es : D\n    es = \u2203-proj\u2081 Pe\n\n    Pes : x \u2237 xs \u2261 e \u2237 es \u2227 \u03bd StreamF es\n    Pes = \u2203-proj\u2082 Pe\n\n    xs\u2261es : xs \u2261 es\n    xs\u2261es = \u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2081 Pes))\n\n    Sxs : \u03bd StreamF xs\n    Sxs = subst (\u03bd StreamF) (sym xs\u2261es) (\u2227-proj\u2082 Pes)\n","old_contents":"------------------------------------------------------------------------------\n-- The FOTC types without use data, i.e. using Agda as a logical framework\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Predicates where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- The existential proyections.\n\u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n\u2203-proj\u2082 (_ , Ax) = Ax\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using postulates.\n\nmodule Pure where\n\n  -- The FOTC natural numbers type.\n  postulate\n    N     : D \u2192 Set\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  [1] : D\n  [1] = succ\u2081 zero\n\n  1-N : N [1]\n  1-N = nsucc nzero\n\n  -- The FOTC lists type.\n  postulate\n    List  : D \u2192 Set\n    lnil  : List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  postulate\n    ListN  : D \u2192 Set\n    lnnil  : ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 [1] \u2237 [])\n  ln = lncons nzero (lncons 1-N lnnil)\n\n------------------------------------------------------------------------------\n-- The inductive FOTC types using data.\n\nmodule Inductive where\n\n  -- The FOTC natural numbers type.\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n  -- Example.\n  [1] : D\n  [1] = succ\u2081 zero\n\n  1-N : N [1]\n  1-N = nsucc nzero\n\n  -- The FOTC lists type.\n  data List : D \u2192 Set where\n    lnil  :                      List []\n    lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n  -- The FOTC list of natural numbers type.\n  data ListN : D \u2192 Set where\n    lnnil  :                             ListN []\n    lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n  -- Example.\n  ln : ListN (zero \u2237 [1] \u2237 [])\n  ln = lncons nzero (lncons 1-N lnnil)\n\n------------------------------------------------------------------------------\n-- The least fixed-point operator.\n\nmodule LFP where\n\n  postulate\n    -- Least fixed-points correspond to inductively defined types.\n    --\n    -- N.B. We cannot write LFP in first-order logic. We should use an\n    -- instance instead.\n    LFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    -- In FOTC, we cannot use the equality on predicates, i.e. we\n    -- cannot write\n    --\n    -- (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 f (LFP f) \u2261 LFP f  (1)\n    --\n    -- because the type of the equality is\n    --\n    -- _\u2261_ : D \u2192 D \u2192 Set,\n    --\n    -- therefore we postulate both directions of the conversion rule (1).\n\n    LFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 LFP f d \u2192 f (LFP f) d\n    LFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (LFP f) d \u2192 LFP f d\n\n------------------------------------------------------------------------------\n-- The greatest fixed-point operator.\n\n-- N.B. At the moment, the definitions of LFP and GFP are the same.\n\nmodule GFP where\n\n  postulate\n    -- Greatest fixed-points correspond to co-inductively defined\n    -- types.\n\n    GFP : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n\n    GFP\u2081 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 GFP f d \u2192 f (GFP f) d\n    GFP\u2082 : (f : (D \u2192 Set) \u2192 D \u2192 Set)(d : D) \u2192 f (GFP f) d \u2192 GFP f d\n\n------------------------------------------------------------------------------\n-- The FOTC natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule NLFP where\n\n  open LFP\n\n  -- Functor for the FOTC natural numbers type.\n\n  -- From Peter: NatF if D was an inductive type\n  -- NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  -- NatF X zero      = \u22a4\n  -- NatF X (succ\u2081 n) = X n\n\n  -- From Peter: NatF in pure predicate logic.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF X n = n \u2261 zero \u2228 (\u2203 \u03bb m \u2192 n \u2261 succ\u2081 m \u2227 X m)\n\n  -- The FOTC natural numbers type using LFP.\n  N : D \u2192 Set\n  N = LFP NatF\n\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = LFP\u2082 NatF zero (inj\u2081 refl)\n\n  nsucc : {n : D} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc {n} Nn = LFP\u2082 NatF (succ\u2081 n) (inj\u2082 (n , (refl , Nn)))\n\n  -- Example.\n  [1] : D\n  [1] = succ\u2081 zero\n\n  1-N : N [1]\n  1-N = nsucc nzero\n\n------------------------------------------------------------------------------\n-- The FOTC list type as the least fixed-point of a functor.\n\nmodule ListLFT where\n\n  open LFP\n\n  -- Functor for the FOTC lists type.\n  ListF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  List : D \u2192 Set\n  List = LFP ListF\n\n  -- The data constructors of List.\n  lnil : List []\n  lnil = LFP\u2082 ListF [] (inj\u2081 refl)\n\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n  lcons x {xs} Lxs = LFP\u2082 ListF (x \u2237 xs) (inj\u2082 (x , xs , refl , Lxs))\n\n  -- Example.\n  l : List (zero \u2237 true \u2237 [])\n  l = lcons zero (lcons true lnil)\n\n------------------------------------------------------------------------------\n-- The FOTC list of natural numbers type as the least fixed-point of a\n-- functor.\n\nmodule ListNLFT where\n\n  open LFP\n  open NLFP\n\n  -- Functor for the FOTC list of natural numbers type.\n  ListNF : (D \u2192 Set) \u2192 D \u2192 Set\n  ListNF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 N e \u2227 X es)\n\n  -- The FOTC list type using LFP.\n  ListN : D \u2192 Set\n  ListN = LFP ListNF\n\n  -- The data constructors of ListN.\n  lnnil : ListN []\n  lnnil = LFP\u2082 ListNF [] (inj\u2081 refl)\n\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n  lncons {n} {ns} Nn LNns =\n    LFP\u2082 ListNF (n \u2237 ns) (inj\u2082 (n , ns , refl , Nn , LNns))\n\n  -- Example.\n  ln : ListN (zero \u2237 [1] \u2237 [])\n  ln = lncons nzero (lncons 1-N lnnil)\n\n------------------------------------------------------------------------------\n-- The FOTC Colist type as the greatest fixed-point of a functor.\n\nmodule CoList where\n\n  open GFP\n\n  -- Functor for the FOTC Colists type (the same functor that for the\n  -- List type).\n  ColistF : (D \u2192 Set) \u2192 D \u2192 Set\n  ColistF X ds = ds \u2261 [] \u2228 (\u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es)\n\n  -- The FOTC Colist type using GFP.\n  Colist : D \u2192 Set\n  Colist = GFP ColistF\n\n  -- The data constructors of Colist.\n  nilCL : Colist []\n  nilCL = GFP\u2082 ColistF [] (inj\u2081 refl)\n\n  consCL : \u2200 x xs \u2192 Colist xs \u2192 Colist (x \u2237 xs)\n  consCL x xs CLxs = GFP\u2082 ColistF (x \u2237 xs) (inj\u2082 (x , xs , refl , CLxs))\n\n  -- Example (finite colist).\n  l : Colist (zero \u2237 true \u2237 [])\n  l = consCL zero (true \u2237 []) (consCL true [] nilCL)\n\n  -- TODO: Example (infinite colist).\n  -- zerosCL : Colist {!!}\n  -- zerosCL = consCL zero {!!} zerosCL\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2081 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 X ds\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- Using StreamF we cannot define this data constructor\n  -- consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  -- consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) {!!}\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2082 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} S = GFP\u2082 StreamF xs (x , x \u2237 xs , S)\n\n  -- The functor StreamF does not link together the parts of the\n  -- stream, so I can get a stream from any stream.\n  bad : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys\n  bad {xs} {ys} S = GFP\u2082 StreamF ys (ys , xs , S)\n\n------------------------------------------------------------------------------\n-- The FOTC Stream type as the greatest fixed-point of a functor.\n\nmodule Stream\u2083 where\n\n  open GFP\n\n  -- Functor for the FOTC Stream type.\n  StreamF : (D \u2192 Set) \u2192 D \u2192 Set\n  StreamF X ds = \u2203 \u03bb e \u2192 \u2203 \u03bb es \u2192 ds \u2261 e \u2237 es \u2227 X es\n\n  -- The FOTC Stream type using GFP.\n  Stream : D \u2192 Set\n  Stream = GFP StreamF\n\n  -- The data constructor of Stream.\n  -- TODO: To use implicit arguments.\n  consS : \u2200 x xs \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  consS x xs Sxs = GFP\u2082 StreamF (x \u2237 xs) (x , xs , refl , Sxs)\n\n  -- TODO: Example\n  -- zerosS : Stream {!!}\n  -- zerosS = consS zero {!!} zerosS\n\n  headS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 D\n  headS {x} _ = x\n\n  tailS : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n  tailS {x} {xs} Sx\u2237xs = Sxs\n    where\n    unfoldS : StreamF (GFP StreamF) (x \u2237 xs)\n    unfoldS = GFP\u2081 StreamF (x \u2237 xs) Sx\u2237xs\n\n    e : D\n    e = \u2203-proj\u2081 unfoldS\n\n    Pe : \u2203 \u03bb es \u2192 x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pe = \u2203-proj\u2082 unfoldS\n\n    es : D\n    es = \u2203-proj\u2081 Pe\n\n    Pes : x \u2237 xs \u2261 e \u2237 es \u2227 GFP StreamF es\n    Pes = \u2203-proj\u2082 Pe\n\n    xs\u2261es : xs \u2261 es\n    xs\u2261es = \u2227-proj\u2082 (\u2237-injective (\u2227-proj\u2081 Pes))\n\n    Sxs : GFP StreamF xs\n    Sxs = subst (GFP StreamF) (sym xs\u2261es) (\u2227-proj\u2082 Pes)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1fd25d4aff13f116be831b1d538e1df362ba14e5","subject":"Agda: refactor proof of embed (#62)","message":"Agda: refactor proof of embed (#62)\n\nThis refactoring makes the code of embed readable to humans (say, me).\n(One should remember that programming should be about communicating with\nhumans).\n\nOld-commit-hash: bf8e0402fab99cc5be02c0d14d407dd6c8123e06\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/Embed62.agda","new_file":"experimental\/Embed62.agda","new_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import ExplicitNil using (ext\u00b3)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Data.Nat\n\nnatPairVisitor : Type\nnatPairVisitor = nats \u21d2 nats \u21d2 nats\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = natPairVisitor \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd! : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd! _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\noldFrom newFrom : \u2200 {\u0393} \u2192 Term \u0393 (\u0394-Type nats) \u2192 Term \u0393 nats\noldFrom d = app d fst\nnewFrom d = app d snd\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) s)) (weaken (drop _ \u2022 \u0393\u227c\u0393) t))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nnatPair : \u2200 {\u0393} \u2192 (old new : Term (natPairVisitor \u2022 \u0393) nats) \u2192 Term \u0393 (\u0394-Type nats)\nnatPair old new = abs (app (app (var this) old) new)\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = natPair (nat old) (nat new)\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = natPair\n  (add (oldFrom (embedWeaken ds))\n       (oldFrom (embedWeaken dt)))\n  (add (newFrom (embedWeaken ds))\n       (newFrom (embedWeaken dt)))\n  where\n    embedWeaken : \u2200 {\u0393 \u03c4} \u2192 (d : \u0394Term \u0393 nats) \u2192\n      Term (\u03c4 \u2022 \u0394-Context \u0393) (natPairVisitor \u21d2 nats)\n    embedWeaken d = (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed d))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {nats} (old , new) v = old \u2261 v \u27e6fst\u27e7 \u00d7 new \u2261 v \u27e6snd\u27e7\n_\u2248_ {bags} u v = u \u2261 v\n_\u2248_ {\u03c3 \u21d2 \u03c4} u v = \n  (w : \u27e6 \u03c3 \u27e7) (dw : \u0394Val \u03c3) (dw\u2032 : \u27e6 \u0394-Type \u03c3 \u27e7)\n  (R[w,dw] : valid w dw) (eq : dw \u2248 dw\u2032) \u2192\n  u w dw R[w,dw] \u2248 v w dw\u2032\n\ninfix 4 _\u2248_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = EmptySet\ncompatible {\u03c4 \u2022 \u0393} (cons v dv _ \u03c1) (dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  triple (v \u2261 v\u2032) (dv \u2248 dv\u2032) (compatible \u03c1 \u03c1\u2032)\n\nderiveVar-correct : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 x \u27e7\u0394Var \u03c1 \u2248 \u27e6 deriveVar x \u27e7 \u03c1\u2032\n\nderiveVar-correct {x = this} -- pattern-matching on \u03c1, \u03c1\u2032 NOT optional\n  {cons _ _ _ _} {_ \u2022 _ \u2022 _} {cons _ dv\u2248dv\u2032 _ _} = dv\u2248dv\u2032\nderiveVar-correct {x = that y}\n  {cons _ _ _ \u03c1} {_ \u2022 _ \u2022 \u03c1\u2032} {cons _ _ C _} =\n  deriveVar-correct {x = y} {\u03c1} {\u03c1\u2032} {C}\n\nweak-id : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nweak-id {\u2205} {\u2205} = refl\nweak-id {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} = cong\u2082 _\u2022_ {x = v} refl (weak-id {\u0393} {\u03c1})\n\nweak-eq : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\nweak-eq {\u2205} {\u2205} {\u2205} _ = refl\nweak-eq {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032} (cons v\u2261v\u2032 _ C _) =\n  cong\u2082 _\u2022_ v\u2261v\u2032 (weak-eq C)\n\nweak-eq-term : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\u2032\n\nweak-eq-term t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (weak-eq C)) (sym (weaken-sound t \u03c1\u2032))\n\nweaken-once : \u2200 {\u03c4 \u0393 \u03c3} {t : Term \u0393 \u03c4} {v : \u27e6 \u03c3 \u27e7} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) t \u27e7 (v \u2022 \u03c1)\n\nweaken-once {t = t} {v} {\u03c1} = trans\n  (cong \u27e6 t \u27e7 (sym weak-id))\n  (sym (weaken-sound t (v \u2022 \u03c1)))\n\nembed-correct : \u2200 {\u03c4 \u0393}\n  {dt : \u0394Term \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {V : dt is-valid-for \u03c1}\n  {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 dt \u27e7\u0394 \u03c1 V \u2248 \u27e6 embed dt \u27e7 \u03c1\u2032\n\nembed-correct {dt = \u0394nat old new} = refl , refl\n\nembed-correct {dt = \u0394bag db} = refl\n\nembed-correct {dt = \u0394var x} {\u03c1} {\u03c1\u2032 = \u03c1\u2032} {C} =\n  deriveVar-correct {x = x} {\u03c1} {\u03c1\u2032} {C}\n\nembed-correct {dt = \u0394add ds dt} {\u03c1} {V} {\u03c1\u2032} {C} =\n  let\n    s00 , s01 = \u27e6 ds \u27e7\u0394 \u03c1 (car V)\n    t00 , t01 = \u27e6 dt \u27e7\u0394 \u03c1 (cadr V)\n    s10 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    s11 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    t10 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    t11 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    rec-s = embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n    rec-t = embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C}\n  in\n    \n    (begin\n      s00 + t00\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2081 rec-s) (proj\u2081 rec-t) \u27e9\n      s10 + t10\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6fst\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6fst\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed ds) (\u27e6fst\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed dt) (\u27e6fst\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7\n    \u220e)\n    ,\n    (begin\n      s01 + t01\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2082 rec-s) (proj\u2082 rec-t) \u27e9\n      s11 + t11\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6snd\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6snd\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed ds) (\u27e6snd\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed dt) (\u27e6snd\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7\n    \u220e)\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394union ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _++_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394diff ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2081 s dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 mapBag\n  (weak-eq-term s C)\n  (embed-correct {dt = dt} {\u03c1} {car V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2080 s ds t dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (cong\u2082 mapBag\n    (extensionality (\u03bb v \u2192\n    begin\n      proj\u2082 (\u27e6 ds \u27e7\u0394 \u03c1 (car V) v (v , v) refl)\n    \u2261\u27e8 proj\u2082 (embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n        v (v , v) (\u03bb f \u2192 f v v) refl (refl , refl)) \u27e9\n      \u27e6 embed ds \u27e7 \u03c1\u2032 v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole v (\u03bb f \u2192 f v v) \u27e6snd\u27e7)\n        (weaken-once {t = embed ds} {v} {\u03c1\u2032}) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (v \u2022 \u03c1\u2032) v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u220e))\n    (cong\u2082 _++_\n      (weak-eq-term t C)\n      (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})))\n  (cong\u2082 mapBag (weak-eq-term s C) (weak-eq-term t C))\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394app ds t dt} {\u03c1} {cons vds vdt R[t,dt] _} {\u03c1\u2032} {C}\n  rewrite sym (weak-eq-term t C) =\n  embed-correct {dt = ds} {\u03c1} {vds} {\u03c1\u2032} {C}\n    (\u27e6 t \u27e7Term (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 vdt) (\u27e6 embed dt \u27e7Term \u03c1\u2032)\n    R[t,dt] (embed-correct {dt = dt} {\u03c1} {vdt} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394abs dt} {\u03c1} {V} {\u03c1\u2032} {C} = \u03bb w dw dw\u2032 R[w,dw] dw\u2248dw\u2032 \u2192\n  embed-correct {dt = dt}\n    {cons w dw R[w,dw] \u03c1} {V w dw R[w,dw]}\n    {dw\u2032 \u2022 w \u2022 \u03c1\u2032} {cons refl dw\u2248dw\u2032 C tt}\n","old_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import ExplicitNil using (ext\u00b3)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Data.Nat\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd! : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd! _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) s)) (weaken (drop _ \u2022 \u0393\u227c\u0393) t))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = abs (app (app (var this) (nat old)) (nat new))\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = abs (app (app (var this)\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) fst)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) fst)))\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) snd)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) snd)))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {nats} (old , new) v = old \u2261 v \u27e6fst\u27e7 \u00d7 new \u2261 v \u27e6snd\u27e7\n_\u2248_ {bags} u v = u \u2261 v\n_\u2248_ {\u03c3 \u21d2 \u03c4} u v = \n  (w : \u27e6 \u03c3 \u27e7) (dw : \u0394Val \u03c3) (dw\u2032 : \u27e6 \u0394-Type \u03c3 \u27e7)\n  (R[w,dw] : valid w dw) (eq : dw \u2248 dw\u2032) \u2192\n  u w dw R[w,dw] \u2248 v w dw\u2032\n\ninfix 4 _\u2248_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = EmptySet\ncompatible {\u03c4 \u2022 \u0393} (cons v dv _ \u03c1) (dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  triple (v \u2261 v\u2032) (dv \u2248 dv\u2032) (compatible \u03c1 \u03c1\u2032)\n\nderiveVar-correct : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 x \u27e7\u0394Var \u03c1 \u2248 \u27e6 deriveVar x \u27e7 \u03c1\u2032\n\nderiveVar-correct {x = this} -- pattern-matching on \u03c1, \u03c1\u2032 NOT optional\n  {cons _ _ _ _} {_ \u2022 _ \u2022 _} {cons _ dv\u2248dv\u2032 _ _} = dv\u2248dv\u2032\nderiveVar-correct {x = that y}\n  {cons _ _ _ \u03c1} {_ \u2022 _ \u2022 \u03c1\u2032} {cons _ _ C _} =\n  deriveVar-correct {x = y} {\u03c1} {\u03c1\u2032} {C}\n\nweak-id : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nweak-id {\u2205} {\u2205} = refl\nweak-id {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} = cong\u2082 _\u2022_ {x = v} refl (weak-id {\u0393} {\u03c1})\n\nweak-eq : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\nweak-eq {\u2205} {\u2205} {\u2205} _ = refl\nweak-eq {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032} (cons v\u2261v\u2032 _ C _) =\n  cong\u2082 _\u2022_ v\u2261v\u2032 (weak-eq C)\n\nweak-eq-term : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\u2032\n\nweak-eq-term t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (weak-eq C)) (sym (weaken-sound t \u03c1\u2032))\n\nweaken-once : \u2200 {\u03c4 \u0393 \u03c3} {t : Term \u0393 \u03c4} {v : \u27e6 \u03c3 \u27e7} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) t \u27e7 (v \u2022 \u03c1)\n\nweaken-once {t = t} {v} {\u03c1} = trans\n  (cong \u27e6 t \u27e7 (sym weak-id))\n  (sym (weaken-sound t (v \u2022 \u03c1)))\n\nembed-correct : \u2200 {\u03c4 \u0393}\n  {dt : \u0394Term \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {V : dt is-valid-for \u03c1}\n  {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 dt \u27e7\u0394 \u03c1 V \u2248 \u27e6 embed dt \u27e7 \u03c1\u2032\n\nembed-correct {dt = \u0394nat old new} = refl , refl\n\nembed-correct {dt = \u0394bag db} = refl\n\nembed-correct {dt = \u0394var x} {\u03c1} {\u03c1\u2032 = \u03c1\u2032} {C} =\n  deriveVar-correct {x = x} {\u03c1} {\u03c1\u2032} {C}\n\nembed-correct {dt = \u0394add ds dt} {\u03c1} {V} {\u03c1\u2032} {C} =\n  let\n    s00 , s01 = \u27e6 ds \u27e7\u0394 \u03c1 (car V)\n    t00 , t01 = \u27e6 dt \u27e7\u0394 \u03c1 (cadr V)\n    s10 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    s11 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    t10 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    t11 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    rec-s = embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n    rec-t = embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C}\n  in\n    \n    (begin\n      s00 + t00\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2081 rec-s) (proj\u2081 rec-t) \u27e9\n      s10 + t10\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6fst\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6fst\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed ds) (\u27e6fst\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed dt) (\u27e6fst\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7\n    \u220e)\n    ,\n    (begin\n      s01 + t01\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2082 rec-s) (proj\u2082 rec-t) \u27e9\n      s11 + t11\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6snd\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6snd\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed ds) (\u27e6snd\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed dt) (\u27e6snd\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7\n    \u220e)\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394union ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _++_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394diff ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2081 s dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 mapBag\n  (weak-eq-term s C)\n  (embed-correct {dt = dt} {\u03c1} {car V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2080 s ds t dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (cong\u2082 mapBag\n    (extensionality (\u03bb v \u2192\n    begin\n      proj\u2082 (\u27e6 ds \u27e7\u0394 \u03c1 (car V) v (v , v) refl)\n    \u2261\u27e8 proj\u2082 (embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n        v (v , v) (\u03bb f \u2192 f v v) refl (refl , refl)) \u27e9\n      \u27e6 embed ds \u27e7 \u03c1\u2032 v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole v (\u03bb f \u2192 f v v) \u27e6snd\u27e7)\n        (weaken-once {t = embed ds} {v} {\u03c1\u2032}) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (v \u2022 \u03c1\u2032) v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u220e))\n    (cong\u2082 _++_\n      (weak-eq-term t C)\n      (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})))\n  (cong\u2082 mapBag (weak-eq-term s C) (weak-eq-term t C))\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394app ds t dt} {\u03c1} {cons vds vdt R[t,dt] _} {\u03c1\u2032} {C}\n  rewrite sym (weak-eq-term t C) =\n  embed-correct {dt = ds} {\u03c1} {vds} {\u03c1\u2032} {C}\n    (\u27e6 t \u27e7Term (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 vdt) (\u27e6 embed dt \u27e7Term \u03c1\u2032)\n    R[t,dt] (embed-correct {dt = dt} {\u03c1} {vdt} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394abs dt} {\u03c1} {V} {\u03c1\u2032} {C} = \u03bb w dw dw\u2032 R[w,dw] dw\u2248dw\u2032 \u2192\n  embed-correct {dt = dt}\n    {cons w dw R[w,dw] \u03c1} {V w dw R[w,dw]}\n    {dw\u2032 \u2022 w \u2022 \u03c1\u2032} {cons refl dw\u2248dw\u2032 C tt}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8540181bade2c5e04e93bbc3714aac21ac894778","subject":"Postulate for MapBag introduces inconsistency.","message":"Postulate for MapBag introduces inconsistency.\n\nRemark. This implementation of bag is unnecessarily complicated\nand is not adapted to the needs of Agda. Another development is\nunderway, which shall be imported here later.\n\nOld-commit-hash: bcf9dc795ad99a2c6a79e846d507267ef93c2288\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/MapBag.agda","new_file":"experimental\/MapBag.agda","new_contents":"{-\n\nIntroducing the MapBag, which can be considered a bag\nof \u2124 with negative multiplicities, or a map from\n\u2124 to \u2124 with default value 0.\n\nThe initial goal of this file is to make the 5th example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    -- Haskell's Data.Map.map is Scala's Map.mapValues\n    map :: (a -> b) -> Map k a -> Map k b\n    \n    incVal :: Map Integer Integer -> Map Integer Integer\n    incVal = map (+1)\n\n    old = fromAscList  [(1, 1), (2, 2) .. (n, n)]\n    res = incVal old = [(1, 2), (2, 3) .. (n, n + 1)]\n\n\nTODO\nX. Stop not getting why hole in half\u2081-WF can't be filled\nX. Prove well-foundedness of half\u2081\nX. Fix singleton\n3. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n\n4. Finish ExplicitNils\n5. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule MapBag where\n\nopen import Data.Unit using\n  (\u22a4)\nopen import Data.Nat using\n  (\u2115 ; suc ; _\u2264\u2032_ ; \u2264\u2032-refl ; \u2264\u2032-step)\nopen import Induction.Nat using\n  (<-rec)\nopen import Data.Integer using\n  (\u2124 ; +_ ; -[1+_] ; _+_ ; _-_)\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n-- Map ints to ints with default 0\ndata MapBag : Set where\n  \u2205 : MapBag\n  btree : (v : \u2124) \u2192 (left : MapBag) \u2192 (right : MapBag) \u2192 MapBag\n\n-- Bag's extensional equivalence:\n-- Remove Quinean \"identity without identity\" on bags\n-- (keys with 0 multiplicities and no subkeys are as good\n-- as not to exist).\n\ndata Empty : MapBag \u2192 Set where\n  \u2205-is-empty : Empty \u2205\n  vacant : \u2200 {left right} \u2192 Empty left \u2192 Empty right \u2192\n             Empty (btree (+ 0) left right)\n\n_\\\\_ : MapBag \u2192 MapBag \u2192 MapBag -- Bag difference\n\n-- THIS introduces inconsistency haha\npostulate ext-bag : \u2200 {b\u2081 b\u2082} \u2192 Empty (b\u2081 \\\\ b\u2082) \u2192 b\u2081 \u2261 b\u2082\n\n-- To understand why there must be an empty NatMap,\n-- observe the termination checker's complaint upon\n-- seeing Haskell's empty NatMap:\n--\n--     emptyMap : NatMap\n--     emptyMap = \u2115tree (+ 0) emptyMap emptyMap\n\ndata Oddity : Set where\n  odd  : Oddity\n  even : Oddity\n\noddity : \u2115 \u2192 Oddity\noddity 0 = even\noddity 1 = odd\noddity (suc (suc n)) = oddity n\n\n-- Elimination form of Oddity to please termination checker\n-- ... eventually.\n\nif-odd_then_else_ : \u2200 {A : Set} \u2192 \u2115 \u2192 A \u2192 A \u2192 A\nif-odd k then thenBranch else elseBranch with oddity k\n... | odd  = thenBranch\n... | even = elseBranch\n\n-- oddity-index k = (oddity k , (k - 1) \/ 2)\n-- Here to please termination checker so that \u2115lookup\n-- doesn't have to pass well-foundedness-proofs around\noddity-index : \u2115 \u2192 Oddity \u00d7 \u2115\noddity-index 0 = (even , 0)\noddity-index 1 = (odd  , 0)\noddity-index 2 = (even , 0)\noddity-index (suc (suc k)) = (oddity-k , suc [k-1]\/2) where\n  oddity-index-k = oddity-index k\n  oddity-k = proj\u2081 oddity-index-k\n  [k-1]\/2  = proj\u2082 oddity-index-k\n\n-- Computes next key after going down 1 level of a nat tree\nhalf\u2081 : \u2115 \u2192 \u2115\nhalf\u2081 0 = 0\nhalf\u2081 1 = 0\nhalf\u2081 2 = 0\nhalf\u2081 (suc (suc n)) = suc (half\u2081 n)\n\n-- Nonzero n holds a proof that n != 0.\ndata Nonzero : \u2115 \u2192 Set where\n  suc : (n : \u2115) \u2192 Nonzero (suc n)\n\n\u2264\u2032-suc : \u2200 {m n : \u2115} \u2192 m \u2264\u2032 n \u2192 (suc m) \u2264\u2032 (suc n)\n\u2264\u2032-suc \u2264\u2032-refl = \u2264\u2032-refl\n\u2264\u2032-suc (\u2264\u2032-step le) = \u2264\u2032-step (\u2264\u2032-suc le)\n\nhalf\u2081-WF : \u2200 (n : \u2115) \u2192 Nonzero n \u2192 suc (half\u2081 n) \u2264\u2032 n\nhalf\u2081-WF 0 ()\nhalf\u2081-WF 1 _ = \u2264\u2032-refl\nhalf\u2081-WF 2 _ = \u2264\u2032-step \u2264\u2032-refl\nhalf\u2081-WF (suc (suc (suc n))) _ =\n  \u2264\u2032-suc {suc (half\u2081 (suc n))} {suc (suc n)}\n         (\u2264\u2032-step (half\u2081-WF (suc n) (suc n)))\n\n-- Here to please the termination checker\n\u2115singleton : \u2115 \u2192 \u2124 \u2192 MapBag\n\u2115singleton k v = loop k where\n  loop : \u2115 \u2192 MapBag\n  loop = <-rec _ \u03bb\n    { 0 _ \u2192 btree v \u2205 \u2205\n    ; (suc n\u2080) rec \u2192\n      let\n        n = suc n\u2080\n        next = rec (half\u2081 n) (half\u2081-WF n (suc n\u2080))\n      in\n        if-odd n\n        then btree (+ 0) next \u2205\n        else btree (+ 0) \u2205 next\n    }\n\n\u2115lookup : \u2115 \u2192 MapBag \u2192 \u2124\n\u2115lookup _ \u2205 = (+ 0)\n\u2115lookup 0 (btree v _ _) = v\n\u2115lookup k (btree _ left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115lookup [k-1]\/2 left\n... | (even , [k-1]\/2) = \u2115lookup [k-1]\/2 right\n\n-- `update` in the sense of\n-- Data.Sequence.update : Int \u2192 a \u2192 Seq a \u2192 Seq a\n\u2115update : \u2115 \u2192 \u2124 \u2192 MapBag \u2192 MapBag\n\u2115update k v \u2205 = \u2115singleton k v\n\u2115update 0 v (btree v\u2080 left right) = btree v left right\n\u2115update k v (btree v\u2080 left right) with oddity-index k\n... | (odd  , [k-1]\/2) = btree v\u2080 (\u2115update [k-1]\/2 v left) right\n... | (even , [k-1]\/2) = btree v\u2080 left (\u2115update [k-1]\/2 v right)\n\n-- Inject \u2124 into \u2115\n\ndouble : \u2115 \u2192 \u2115\ndouble 0 = 0\ndouble (suc n) = suc (suc (double n))\n\n\u2124\u21d2\u2115 : \u2124 \u2192 \u2115\n\u2124\u21d2\u2115  ( + n ) = double n\n\u2124\u21d2\u2115 -[1+ n ] = suc (double n)\n\n_\u25e6_ : {A : Set} {B : A -> Set} {C : (x : A) -> B x -> Set}\n      (f : {x : A} (y : B x) -> C x y) (g : (x : A) -> B x)\n      (x : A) -> C x (g x)\n(f \u25e6 g) = \\ x -> f (g x)\n\nlookup : \u2124 \u2192 MapBag \u2192 \u2124\nlookup = \u2115lookup \u25e6 \u2124\u21d2\u2115\n\nupdate : \u2124 \u2192 \u2124 \u2192 MapBag \u2192 MapBag\nupdate = \u2115update \u25e6 \u2124\u21d2\u2115\n\n-- We implement nothing but the necessary NatMap operations to save time:\n-- union, difference, mapValues.\n\nmapValues : (\u2124 \u2192 \u2124) \u2192 MapBag \u2192 MapBag\nmapValues _ \u2205 = \u2205\nmapValues f (btree v left right) =\n  btree (f v) (mapValues f left) (mapValues f right)\n\n-- Associativity of ++ follows Scala (left), fixity follows Haskell (5).\n_++_ : MapBag \u2192 MapBag \u2192 MapBag\n\u2205 ++ b = b\nb ++ \u2205 = b\n(btree v\u2081 left\u2081 right\u2081) ++ (btree v\u2082 left\u2082 right\u2082) =\n  btree (v\u2081 + v\u2082) (left\u2081 ++ left\u2082) (right\u2081 ++ right\u2082)\n\ninfixl 5 _++_\n\n-- Data.Map.(\\\\) : Map k a \u2192 Map k b \u2192 Map k a (where b = a = \u2124)\ninfixl 9 _\\\\_\n\n\u2205 \\\\ b = \u2205\nb \\\\ \u2205 = b\n(btree v\u2081 left\u2081 right\u2081) \\\\ (btree v\u2082 left\u2082 right\u2082) =\n  btree (v\u2081 - v\u2082) (left\u2081 \\\\ left\u2082) (right\u2081 \\\\ right\u2082)\n\n-- Here begins the language definition.\n\ndata Type : Set where\n  ints : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  int : \u2200 {\u0393} \u2192 (n : \u2124) \u2192 Term \u0393 ints\n  bag : \u2200 {\u0393} \u2192 (b : MapBag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 ints) \u2192 (t\u2082 : Term \u0393 ints) \u2192 Term \u0393 ints\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (ints \u21d2 ints)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to ints = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (int x) = int x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 ints \u27e7Type = \u2124\n\u27e6 bags \u27e7Type = MapBag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 int n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapValues (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (int n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 = cong\u2082 mapValues (weaken-sound f \u03c1)\n                                           (weaken-sound b \u03c1)\n\n-- Changes to \u2124 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on mapbags are mapbags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type ints = (ints \u21d2 ints \u21d2 ints) \u21d2 ints\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\u27e6snd\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\n\u27e6derive\u27e7 {ints} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = \u2205\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {ints} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {ints} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Cool lemmas\n\nn-n=0 : \u2200 {n : \u2124} \u2192 n - n \u2261 (+ 0)\nn-n=0 { -[1+     0 ]} = refl\nn-n=0 { -[1+ suc n ]} = n-n=0 { -[1+ n ]} -- 'ware that comment symbol\nn-n=0 { +      0 } = refl\nn-n=0 { + (suc n)} = n-n=0 { -[1+ n ]}\n\nb\\\\\u2205=b : \u2200 {b} \u2192 b \\\\ \u2205 \u2261 b\nb\\\\\u2205=b {\u2205} = refl\nb\\\\\u2205=b {btree v b b\u2081} = refl\n\nb\\\\b=\u2205 : \u2200 {b} \u2192 b \\\\ b \u2261 \u2205\nb\\\\b=\u2205 {b} = ext-bag void where\n  loop : (b : MapBag) \u2192 Empty (b \\\\ b)\n  loop b = {!!}\n  void : Empty (b \\\\ b \\\\ \u2205)\n  void rewrite b\\\\\u2205=b {b \\\\ b} = loop b\n\n\n{-\n  go rewrite n-n=0 {v}\n  ext-bag (vacant\u2081 (b\\\\b=\u2205 {left}) (b\\\\b=\u2205 {right}))\n-}\n\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {ints} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = {!!}\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n{-\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {ints} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = {!!}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {ints} n = refl\nf\u2295\u0394f=f {bags} b = {!!}\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {ints} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = {!!}\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {ints} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = {!!}\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (int n) = abs (app (app (var this) (int n)) (int n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\nderive _ = {!!}\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case int: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {ints} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {bags} b db valid-b-db = {!!}\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (int n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {ints} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = {!!}\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (int n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n-}\n\n","old_contents":"{-\n\nIntroducing the MapBag, which can be considered a bag\nof \u2124 with negative multiplicities, or a map from\n\u2124 to \u2124 with default value 0.\n\nThe initial goal of this file is to make the 5th example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    -- Haskell's Data.Map.map is Scala's Map.mapValues\n    map :: (a -> b) -> Map k a -> Map k b\n    \n    incVal :: Map Integer Integer -> Map Integer Integer\n    incVal = map (+1)\n\n    old = fromAscList  [(1, 1), (2, 2) .. (n, n)]\n    res = incVal old = [(1, 2), (2, 3) .. (n, n + 1)]\n\n\nTODO\nX. Stop not getting why hole in half\u2081-WF can't be filled\nX. Prove well-foundedness of half\u2081\nX. Fix singleton\n3. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n\n4. Finish ExplicitNils\n5. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule MapBag where\n\nopen import Data.Unit using\n  (\u22a4)\nopen import Data.Nat using\n  (\u2115 ; suc ; _\u2264\u2032_ ; \u2264\u2032-refl ; \u2264\u2032-step)\nopen import Induction.Nat using\n  (<-rec)\nopen import Data.Integer using\n  (\u2124 ; +_ ; -[1+_] ; _+_ ; _-_)\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n-- Map nats to ints with default 0\ndata NatMap : Set where\n  \u2205 : NatMap\n  \u2115tree : (v : \u2124) \u2192 (left : NatMap) \u2192 (right : NatMap) \u2192 NatMap\n\n-- To understand why there must be an empty NatMap,\n-- observe the termination checker's complaint upon\n-- seeing Haskell's empty NatMap:\n--\n--     emptyMap : NatMap\n--     emptyMap = \u2115tree (+ 0) emptyMap emptyMap\n\ndata Oddity : Set where\n  odd  : Oddity\n  even : Oddity\n\noddity : \u2115 \u2192 Oddity\noddity 0 = even\noddity 1 = odd\noddity (suc (suc n)) = oddity n\n\n-- Elimination form of Oddity to please termination checker\n-- ... eventually.\n\nif-odd_then_else_ : \u2200 {A : Set} \u2192 \u2115 \u2192 A \u2192 A \u2192 A\nif-odd k then thenBranch else elseBranch with oddity k\n... | odd  = thenBranch\n... | even = elseBranch\n\n-- oddity-index k = (oddity k , (k - 1) \/ 2)\n-- Here to please termination checker so that \u2115lookup\n-- doesn't have to pass well-foundedness-proofs around\noddity-index : \u2115 \u2192 Oddity \u00d7 \u2115\noddity-index 0 = (even , 0)\noddity-index 1 = (odd  , 0)\noddity-index 2 = (even , 0)\noddity-index (suc (suc k)) = (oddity-k , suc [k-1]\/2) where\n  oddity-index-k = oddity-index k\n  oddity-k = proj\u2081 oddity-index-k\n  [k-1]\/2  = proj\u2082 oddity-index-k\n\n-- Computes next key after going down 1 level of a nat tree\nhalf\u2081 : \u2115 \u2192 \u2115\nhalf\u2081 0 = 0\nhalf\u2081 1 = 0\nhalf\u2081 2 = 0\nhalf\u2081 (suc (suc n)) = suc (half\u2081 n)\n\n-- Nonzero n holds a proof that n != 0.\ndata Nonzero : \u2115 \u2192 Set where\n  suc : (n : \u2115) \u2192 Nonzero (suc n)\n\n\u2264\u2032-suc : \u2200 {m n : \u2115} \u2192 m \u2264\u2032 n \u2192 (suc m) \u2264\u2032 (suc n)\n\u2264\u2032-suc \u2264\u2032-refl = \u2264\u2032-refl\n\u2264\u2032-suc (\u2264\u2032-step le) = \u2264\u2032-step (\u2264\u2032-suc le)\n\nhalf\u2081-WF : \u2200 (n : \u2115) \u2192 Nonzero n \u2192 suc (half\u2081 n) \u2264\u2032 n\nhalf\u2081-WF 0 ()\nhalf\u2081-WF 1 _ = \u2264\u2032-refl\nhalf\u2081-WF 2 _ = \u2264\u2032-step \u2264\u2032-refl\nhalf\u2081-WF (suc (suc (suc n))) _ =\n  \u2264\u2032-suc {suc (half\u2081 (suc n))} {suc (suc n)}\n         (\u2264\u2032-step (half\u2081-WF (suc n) (suc n)))\n\n-- Here to please the termination checker\nsingleton : \u2115 \u2192 \u2124 \u2192 NatMap\nsingleton k v = loop k where\n  loop : \u2115 \u2192 NatMap\n  loop = <-rec _ \u03bb\n    { 0 _ \u2192 \u2115tree v \u2205 \u2205\n    ; (suc n\u2080) rec \u2192\n      let\n        n = suc n\u2080\n        next = rec (half\u2081 n) (half\u2081-WF n (suc n\u2080))\n      in\n        if-odd n\n        then \u2115tree (+ 0) next \u2205\n        else \u2115tree (+ 0) \u2205 next\n    }\n\n\u2115lookup : \u2115 \u2192 NatMap \u2192 \u2124\n\u2115lookup _ \u2205 = (+ 0)\n\u2115lookup 0 (\u2115tree v _ _) = v\n\u2115lookup k (\u2115tree _ left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115lookup [k-1]\/2 left\n... | (even , [k-1]\/2) = \u2115lookup [k-1]\/2 right\n\n-- `update` in the sense of\n-- Data.Sequence.update : Int \u2192 a \u2192 Seq a \u2192 Seq a\n\u2115update : \u2115 \u2192 \u2124 \u2192 NatMap \u2192 NatMap\n\u2115update k v \u2205 = singleton k v\n\u2115update 0 v (\u2115tree v\u2080 left right) = \u2115tree v left right\n\u2115update k v (\u2115tree v\u2080 left right) with oddity-index k\n... | (odd  , [k-1]\/2) = \u2115tree v\u2080 (\u2115update [k-1]\/2 v left) right\n... | (even , [k-1]\/2) = \u2115tree v\u2080 left (\u2115update [k-1]\/2 v right)\n\nMapBag = NatMap \u00d7 NatMap\n\nemptyBag : MapBag\nemptyBag = (\u2205 , \u2205)\n\nlookup : \u2124 \u2192 MapBag \u2192 \u2124\nlookup -[1+ k ] (negative , nonnegative) = \u2115lookup k negative\nlookup   (+ k)  (negative , nonnegative) = \u2115lookup k nonnegative\n\nupdate : \u2124 \u2192 \u2124 \u2192 MapBag \u2192 MapBag\nupdate -[1+ k ] v (neg , pos) = (\u2115update k v neg , pos)\nupdate  ( + k ) v (neg , pos) = (neg , \u2115update k v pos)\n\n-- We implement nothing but the necessary NatMap operations to save time:\n-- union, difference, mapValues.\n\n\u2115mapValues : (\u2124 \u2192 \u2124) \u2192 NatMap \u2192 NatMap\n\u2115mapValues _ \u2205 = \u2205\n\u2115mapValues f (\u2115tree v left right) =\n  \u2115tree (f v) (\u2115mapValues f left) (\u2115mapValues f right)\n\nmapValues : (\u2124 \u2192 \u2124) \u2192 MapBag \u2192 MapBag\nmapValues f (neg , pos) = (\u2115mapValues f neg , \u2115mapValues f pos)\n\n\u2115union : NatMap \u2192 NatMap \u2192 NatMap\n\u2115union \u2205 b = b\n\u2115union b \u2205 = b\n\u2115union (\u2115tree v\u2081 left\u2081 right\u2081) (\u2115tree v\u2082 left\u2082 right\u2082) =\n  \u2115tree (v\u2081 + v\u2082) (\u2115union left\u2081 left\u2082) (\u2115union right\u2081 right\u2082)\n\n-- Prelude.(++) : [a] \u2192 [a] \u2192 [a]\n_++_ : MapBag \u2192 MapBag \u2192 MapBag\nb\u2081 ++ b\u2082 = Product.zip \u2115union \u2115union b\u2081 b\u2082\n\ninfixr 5 _++_\n\n\u2115diff : NatMap \u2192 NatMap \u2192 NatMap\n\u2115diff \u2205 b = \u2205\n\u2115diff b \u2205 = b\n\u2115diff (\u2115tree v\u2081 left\u2081 right\u2081) (\u2115tree v\u2082 left\u2082 right\u2082) =\n  \u2115tree (v\u2081 - v\u2082) (\u2115diff left\u2081 left\u2082) (\u2115diff right\u2081 right\u2082)\n\n-- Data.Map.(\\\\) : Map k a \u2192 Map k b \u2192 Map k a (where b = a = \u2124)\n_\\\\_ : MapBag \u2192 MapBag \u2192 MapBag\nb\u2081 \\\\ b\u2082 = Product.zip \u2115diff \u2115diff b\u2081 b\u2082\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n-- Bag's extensional equivalence:\n-- Remove Quinean \"identity without identity\" on bags\n-- (keys with 0 multiplicities and no subkeys are as good\n-- as not to exist).\n\ndata \u2115Empty : NatMap \u2192 Set where\n  is-empty : \u2115Empty \u2205\n  0-mult : \u2200 {left right} \u2192 \u2115Empty left \u2192 \u2115Empty right \u2192\n             \u2115Empty (\u2115tree (+ 0) left right)\n\nEmpty : MapBag \u2192 Set\nEmpty (neg , pos) = \u2115Empty neg \u00d7 \u2115Empty pos\n\npostulate\n  ext-bag : \u2200 {b\u2081 b\u2082} \u2192 Empty (b\u2081 \\\\ b\u2082) \u2192 b\u2081 \u2261 b\u2082\n\ninfixl 9 _\\\\_\n\ndata Type : Set where\n  ints : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  int : \u2200 {\u0393} \u2192 (n : \u2124) \u2192 Term \u0393 ints\n  bag : \u2200 {\u0393} \u2192 (b : MapBag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 ints) \u2192 (t\u2082 : Term \u0393 ints) \u2192 Term \u0393 ints\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (ints \u21d2 ints)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to ints = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (int x) = int x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 ints \u27e7Type = \u2124\n\u27e6 bags \u27e7Type = MapBag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 int n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapValues (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (int n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 = cong\u2082 mapValues (weaken-sound f \u03c1)\n                                           (weaken-sound b \u03c1)\n\n-- Changes to \u2124 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on mapbags are mapbags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type ints = (ints \u21d2 ints \u21d2 ints) \u21d2 ints\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\u27e6snd\u27e7 : \u2124 \u2192 \u2124 \u2192 \u2124\n\n\u27e6derive\u27e7 {ints} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {ints} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {ints} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n{-\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {ints} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = {!!}\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {ints} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = {!!}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {ints} n = refl\nf\u2295\u0394f=f {bags} b = {!!}\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {ints} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = {!!}\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {ints} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = {!!}\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (int n) = abs (app (app (var this) (int n)) (int n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\nderive _ = {!!}\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case int: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {ints} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {bags} b db valid-b-db = {!!}\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (int n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {ints} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = {!!}\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\nvalidity-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (int n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fc95d4b35aac03ede8b530c2fa18e42a73a3177b","subject":"CBPV ValType: Name members","message":"CBPV ValType: Name members\n\nTo get nicer names in generated pattern matches.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/CBPVType.agda","new_file":"Parametric\/Syntax\/CBPVType.agda","new_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Syntax.CBPVType where\n\nmodule Structure (Base : Type.Structure) where\n  open Type.Structure Base\n\n  mutual\n    --  CBPV\n    data ValType : Set where\n      U : (c : CompType) \u2192 ValType\n      B : (\u03b9 : Base) \u2192 ValType\n      vUnit : ValType\n      _v\u00d7_ : (\u03c4\u2081 : ValType) \u2192 (\u03c4\u2082 : ValType) \u2192 ValType\n      _v+_ : (\u03c4\u2081 : ValType) \u2192 (\u03c4\u2082 : ValType) \u2192 ValType\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  open import Data.List\n  open Data.List using (List) public\n  fromCBVToCompList : Context \u2192 List CompType\n  fromCBVToCompList \u0393 = mapValCtx cbvToCompType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","old_contents":"import Parametric.Syntax.Type as Type\n\nmodule Parametric.Syntax.CBPVType where\n\nmodule Structure (Base : Type.Structure) where\n  open Type.Structure Base\n\n  mutual\n    --  CBPV\n    data ValType : Set where\n      U : CompType \u2192 ValType\n      B : Base \u2192 ValType\n      vUnit : ValType\n      _v\u00d7_ : ValType \u2192 ValType \u2192 ValType\n      _v+_ : ValType \u2192 ValType \u2192 ValType\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  open import Data.List\n  open Data.List using (List) public\n  fromCBVToCompList : Context \u2192 List CompType\n  fromCBVToCompList \u0393 = mapValCtx cbvToCompType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1207acebbca238b4041372986370848ca90c91c9","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"structural-assumptions.agda","new_file":"structural-assumptions.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule structural-assumptions where\n\n-- this module contains the signatures of theorems that we intend to\n-- prove in the finished artefact. they are all either canonical\n-- structural properties of the hypothetical judgements or common\n-- properties of contexts. we strongly believe that they are true,\n-- especially since they're all standard, but have not yet proven them\n-- for our specific choices of implementations.\n\n-- the only assumptions not in this file are 1) for function\n-- extensionality in Prelude.agda; funext is known to be independent\n-- of Agda, and we have no intension of removing it, and 2) the\n-- signature of the disjointness judgements in core.agda used below to\n-- break a circular module dependency, and 3) the signature of\n-- applying a substitition to a hole in core.agda\n\npostulate\n  \u222acomm : {A : Set} \u2192 (C1 C2 : A ctx) \u2192 C1 ## C2 \u2192 (C1 \u222a C2) == (C2 \u222a C1)\n-- postulate\n--   gcomp-extend : \u2200{\u0393 \u03c4 x} \u2192 \u0393 gcomplete \u2192 \u03c4 tcomplete \u2192 (\u0393 ,, (x , \u03c4)) gcomplete\npostulate\n  subst-weaken : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 ## \u0394' \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\npostulate\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192 \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192 \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192 \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n\n\n----- former assumptions that have now been proven, at least up to the\n----- assumptions above, but not placed in their rightful files\n----- because of cyclic dependency junk\n\n\n-- todo: this belongs in contexts once the above are proven and the\n-- cyclic dependency gets broken\nx\u2208\u222ar : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 \u0393' ## \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\nx\u2208\u222ar \u0393 \u0393' n x nx\u2208 disj = tr (\u03bb qq \u2192 (n , x) \u2208 qq) (\u222acomm _ _ disj) (x\u2208\u222al \u0393' \u0393 n x nx\u2208)\n\n-- todo : see below; this is copied code\nlem-deleteme : {A : Set} (y : A) (n m : Nat) \u2192 dom (\u25a0 (m , y)) n \u2192 n == m\nlem-deleteme y n m (\u03c01 , \u03c02) with natEQ m n\nlem-deleteme y n .n (\u03c01 , \u03c02) | Inl refl = refl\nlem-deleteme y n m (\u03c01 , \u03c02) | Inr x = abort (somenotnone (! \u03c02))\n\n-- todo: this lemma belongs in lemmas-disjointness, which already\n-- contains lem-deleteme under disjoint-singles; remove one and\n-- promote the other once you break the cyclic dependency\nlem-apart-sing-disj : {A : Set} {n : Nat} {a : A} {\u0393 : A ctx} \u2192 n # \u0393 \u2192 (\u25a0 (n , a)) ## \u0393\nlem-apart-sing-disj {A} {n} {a} {\u0393} apt = asd1 , asd2\n  where\n    asd1 : (n\u2081 : Nat) \u2192 dom (\u25a0 (n , a)) n\u2081 \u2192 n\u2081 # \u0393\n    asd1 m d with lem-deleteme _ _ _ d\n    asd1 .n d | refl = apt\n\n    asd2 : (n\u2081 : Nat) \u2192 dom \u0393 n\u2081 \u2192 n\u2081 # (\u25a0 (n , a))\n    asd2 m (\u03c01 , \u03c02) with natEQ n m\n    asd2 .n (\u03c01 , \u03c02) | Inl refl = abort (somenotnone (! \u03c02 \u00b7 apt ))\n    asd2 m (\u03c01 , \u03c02) | Inr x = refl\n\n-- todo: belongs in contexts\nctx-top : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n # \u0393) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\nctx-top \u0393 n a apt = x\u2208\u222ar \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a) (lem-apart-sing-disj apt)\n\n-- todo: belongs in contexts\nlem-insingeq : {A : Set} {x x' : Nat} {\u03c4 \u03c4' : A} \u2192 (\u25a0 (x , \u03c4)) x' == Some \u03c4' \u2192 \u03c4 == \u03c4'\nlem-insingeq {A} {x} {x'} {\u03c4} {\u03c4'} eq with lem-deleteme \u03c4 x' x (\u03c4' , eq)\nlem-insingeq {A} {x} {.x} {\u03c4} {\u03c4'} eq | refl with natEQ x x\nlem-insingeq refl | refl | Inl refl = refl\nlem-insingeq eq | refl | Inr x\u2081 = abort (somenotnone (! eq))\n\nlem-apart-union-eq : {A : Set} {\u0393 : A ctx} {x x' : Nat} {\u03c4 \u03c4' : A} \u2192 x' # \u0393 \u2192 (\u0393 \u222a \u25a0 (x , \u03c4)) x' == Some \u03c4' \u2192 \u03c4 == \u03c4'\nlem-apart-union-eq {A} {\u0393} {x} {x'} {\u03c4} {\u03c4'} apart eq with \u0393 x'\nlem-apart-union-eq apart eq | Some x = abort (somenotnone apart)\nlem-apart-union-eq apart eq | None = lem-insingeq eq\n\nlem-neq-union-eq : {A : Set} {\u0393 : A ctx} {x x' : Nat} {\u03c4 \u03c4' : A} \u2192 x' \u2260 x \u2192 (\u0393 \u222a \u25a0 (x , \u03c4)) x' == Some \u03c4' \u2192 \u0393 x' == Some \u03c4'\nlem-neq-union-eq {A} {\u0393} {x} {x'} {\u03c4} {\u03c4'} neq eq with \u0393 x'\nlem-neq-union-eq neq eq | Some x = eq\nlem-neq-union-eq {A} {\u0393} {x} {x'} {\u03c4} {\u03c4'} neq eq | None with natEQ x x'\nlem-neq-union-eq neq eq | None | Inl x\u2081 = abort ((flip neq) x\u2081)\nlem-neq-union-eq neq eq | None | Inr x\u2081 = abort (somenotnone (! eq))\n\n-- we need this apartness premise because it's false otherwise, given\n-- the particular implementation of ,, which is backed by \u222a, which\n-- queries one side first.\ngcomp-extend : \u2200{\u0393 \u03c4 x} \u2192 \u0393 gcomplete \u2192 \u03c4 tcomplete \u2192 x # \u0393 \u2192 (\u0393 ,, (x , \u03c4)) gcomplete\ngcomp-extend {\u0393} {\u03c4} {x} gc tc apart x_query \u03c4_query x\u2081 with natEQ x x_query\ngcomp-extend {\u0393} {\u03c4} {x} gc tc apart .x \u03c4_query x\u2082 | Inl refl = tr (\u03bb qq \u2192 qq tcomplete) (lem-apart-union-eq {\u0393 = \u0393} apart x\u2082) tc\ngcomp-extend {\u0393} {\u03c4} {x} gc tc apart x_query \u03c4_query x\u2082 | Inr x\u2081 = gc x_query \u03c4_query (lem-neq-union-eq {\u0393 = \u0393} (flip x\u2081) x\u2082 )\n\n-- gcomp-extend {\u0393} {\u03c4} {x} gc tc aprt x\u2081 \u03c4\u2081 x\u2082 with \u0393 x\n-- gcomp-extend gc tc aprt x\u2082 \u03c4\u2081 x\u2083 | Some x\u2081 = abort (somenotnone aprt)\n-- gcomp-extend {\u0393} {\u03c4} {x} gc tc aprt x\u2081 \u03c4\u2081 x\u2082 | None with ctxindirect' \u0393 x\u2081\n-- gcomp-extend {\u0393} {\u03c4} {x} gc tc aprt x\u2081 \u03c4\u2081 x\u2083 | None | Inl (a , eq , inj) with lem-apart-union-eq {\u0393 = \u0393} {!!}  x\u2083\n-- gcomp-extend gc tc aprt x\u2081 \u03c4\u2081 x\u2083 | None | Inl (a , eq , inj) | refl = tc\n-- gcomp-extend gc tc aprt x\u2081 \u03c4\u2081 x\u2083 | None | Inr x = {!!}\n\n-- gcomp-extend {\u0393} {\u03c4} {x} gc apt tc x' \u03c4' mem with \u0393 x\n-- gcomp-extend gc apt tc x' \u03c4' mem | Some x\u2081 = abort (somenotnone tc)\n-- gcomp-extend {\u0393} gc apt refl x' \u03c4' mem | None with ctxindirect \u0393 x'\n-- gcomp-extend {\u0393} {x = y} gc tc refl x' \u03c4' mem | None | Inl (a , x) = tr (\u03bb qq \u2192 qq tcomplete) (ctxunicity x {!!}) (gc x' a x)\n-- gcomp-extend {\u0393} gc tc refl x' \u03c4' mem | None | Inr neq with \u0393 x'\n-- gcomp-extend gc tc refl x' \u03c4' mem | None | Inr neq | Some x = abort (somenotnone neq)\n-- gcomp-extend gc tc refl x' \u03c4' mem | None | Inr refl | None with lem-insingeq mem\n-- gcomp-extend gc tc refl x' \u03c4' mem | None | Inr refl | None | refl = tc\n\n-- (\u0393 \u222a \u25a0 (x , \u03c4) x\u2081 | \u0393 x\u2081) == Some \u03c4\u2081\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\n\nmodule structural-assumptions where\n\n-- this module contains the signatures of theorems that we intend to\n-- prove in the finished artefact. they are all either canonical\n-- structural properties of the hypothetical judgements or common\n-- properties of contexts. we strongly believe that they are true,\n-- especially since they're all standard, but have not yet proven them\n-- for our specific choices of implementations.\n\n-- the only assumptions not in this file are 1) for function\n-- extensionality in Prelude.agda; funext is known to be independent\n-- of Agda, and we have no intension of removing it, and 2) the\n-- signature of the disjointness judgements in core.agda used below to\n-- break a circular module dependency, and 3) the signature of\n-- applying a substitition to a hole in core.agda\n\npostulate\n  \u222acomm : {A : Set} \u2192 (C1 C2 : A ctx) \u2192 C1 ## C2 \u2192 (C1 \u222a C2) == (C2 \u222a C1)\n-- postulate\n--   gcomp-extend : \u2200{\u0393 \u03c4 x} \u2192 \u0393 gcomplete \u2192 \u03c4 tcomplete \u2192 (\u0393 ,, (x , \u03c4)) gcomplete\npostulate\n  subst-weaken : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 ## \u0394' \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\npostulate\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192 \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192 \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192 \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n\n\n----- former assumptions that have now been proven, at least up to the\n----- assumptions above, but not placed in their rightful files\n----- because of cyclic dependency junk\n\n\n-- todo: this belongs in contexts once the above are proven and the\n-- cyclic dependency gets broken\nx\u2208\u222ar : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 \u0393' ## \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\nx\u2208\u222ar \u0393 \u0393' n x nx\u2208 disj = tr (\u03bb qq \u2192 (n , x) \u2208 qq) (\u222acomm _ _ disj) (x\u2208\u222al \u0393' \u0393 n x nx\u2208)\n\n-- todo : see below; this is copied code\nlem-deleteme : {A : Set} (y : A) (n m : Nat) \u2192 dom (\u25a0 (m , y)) n \u2192 n == m\nlem-deleteme y n m (\u03c01 , \u03c02) with natEQ m n\nlem-deleteme y n .n (\u03c01 , \u03c02) | Inl refl = refl\nlem-deleteme y n m (\u03c01 , \u03c02) | Inr x = abort (somenotnone (! \u03c02))\n\n-- todo: this lemma belongs in lemmas-disjointness, which already\n-- contains lem-deleteme under disjoint-singles; remove one and\n-- promote the other once you break the cyclic dependency\nlem-apart-sing-disj : {A : Set} {n : Nat} {a : A} {\u0393 : A ctx} \u2192 n # \u0393 \u2192 (\u25a0 (n , a)) ## \u0393\nlem-apart-sing-disj {A} {n} {a} {\u0393} apt = asd1 , asd2\n  where\n    asd1 : (n\u2081 : Nat) \u2192 dom (\u25a0 (n , a)) n\u2081 \u2192 n\u2081 # \u0393\n    asd1 m d with lem-deleteme _ _ _ d\n    asd1 .n d | refl = apt\n\n    asd2 : (n\u2081 : Nat) \u2192 dom \u0393 n\u2081 \u2192 n\u2081 # (\u25a0 (n , a))\n    asd2 m (\u03c01 , \u03c02) with natEQ n m\n    asd2 .n (\u03c01 , \u03c02) | Inl refl = abort (somenotnone (! \u03c02 \u00b7 apt ))\n    asd2 m (\u03c01 , \u03c02) | Inr x = refl\n\n-- todo: belongs in contexts\nctx-top : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n # \u0393) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\nctx-top \u0393 n a apt = x\u2208\u222ar \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a) (lem-apart-sing-disj apt)\n\n-- todo: belongs in contexts\nlem-insingeq : {A : Set} {x x' : Nat} {\u03c4 \u03c4' : A} \u2192 (\u25a0 (x , \u03c4)) x' == Some \u03c4' \u2192 \u03c4 == \u03c4'\nlem-insingeq {A} {x} {x'} {\u03c4} {\u03c4'} eq with lem-deleteme \u03c4 x' x (\u03c4' , eq)\nlem-insingeq {A} {x} {.x} {\u03c4} {\u03c4'} eq | refl with natEQ x x\nlem-insingeq refl | refl | Inl refl = refl\nlem-insingeq eq | refl | Inr x\u2081 = abort (somenotnone (! eq))\n\nlem-apart-union-eq : {A : Set} {\u0393 : A ctx} {x x' : Nat} {\u03c4 \u03c4' : A} \u2192 x' # \u0393 \u2192 (\u0393 \u222a \u25a0 (x , \u03c4)) x' == Some \u03c4' \u2192 \u03c4 == \u03c4'\nlem-apart-union-eq {A} {\u0393} {x} {x'} {\u03c4} {\u03c4'} apart eq with \u0393 x'\nlem-apart-union-eq apart eq | Some x = abort (somenotnone apart)\nlem-apart-union-eq apart eq | None = lem-insingeq eq\n\nlem-neq-union-eq : {A : Set} {\u0393 : A ctx} {x x' : Nat} {\u03c4 \u03c4' : A} \u2192 x' \u2260 x \u2192 (\u0393 \u222a \u25a0 (x , \u03c4)) x' == Some \u03c4' \u2192 \u0393 x' == Some \u03c4'\nlem-neq-union-eq {A} {\u0393} {x} {x'} {\u03c4} {\u03c4'} neq eq with \u0393 x'\nlem-neq-union-eq neq eq | Some x = eq\nlem-neq-union-eq {A} {\u0393} {x} {x'} {\u03c4} {\u03c4'} neq eq | None with natEQ x x'\nlem-neq-union-eq neq eq | None | Inl x\u2081 = abort ((flip neq) x\u2081)\nlem-neq-union-eq neq eq | None | Inr x\u2081 = abort (somenotnone (! eq))\n\n-- we need this apartness premise because it's false otherwise, given\n-- the particular implementation of ,, which is backed by \u222a, which\n-- queries one side first.\ngcomp-extend : \u2200{\u0393 \u03c4 x} \u2192 \u0393 gcomplete \u2192 \u03c4 tcomplete \u2192 x # \u0393 \u2192 (\u0393 ,, (x , \u03c4)) gcomplete\ngcomp-extend {\u0393} {\u03c4} {x} gc tc apart x_query \u03c4_query x\u2081 with natEQ x x_query\ngcomp-extend {\u0393} {\u03c4} {x} gc tc apart .x \u03c4_query x\u2082 | Inl refl = tr (\u03bb qq \u2192 qq tcomplete) (lem-apart-union-eq {\u0393 = \u0393} apart x\u2082) tc\ngcomp-extend {\u0393} {\u03c4} {x} gc tc apart x_query \u03c4_query x\u2082 | Inr x\u2081 = gc x_query \u03c4_query (lem-neq-union-eq {\u0393 = \u0393} (flip x\u2081) x\u2082 )\n\n-- gcomp-extend {\u0393} {\u03c4} {x} gc tc aprt x\u2081 \u03c4\u2081 x\u2082 with \u0393 x\n-- gcomp-extend gc tc aprt x\u2082 \u03c4\u2081 x\u2083 | Some x\u2081 = abort (somenotnone aprt)\n-- gcomp-extend {\u0393} {\u03c4} {x} gc tc aprt x\u2081 \u03c4\u2081 x\u2082 | None with ctxindirect' \u0393 x\u2081\n-- gcomp-extend {\u0393} {\u03c4} {x} gc tc aprt x\u2081 \u03c4\u2081 x\u2083 | None | Inl (a , eq , inj) with lem-poop {\u0393 = \u0393} {!!}  x\u2083\n-- gcomp-extend gc tc aprt x\u2081 \u03c4\u2081 x\u2083 | None | Inl (a , eq , inj) | refl = tc\n-- gcomp-extend gc tc aprt x\u2081 \u03c4\u2081 x\u2083 | None | Inr x = {!!}\n\n-- gcomp-extend {\u0393} {\u03c4} {x} gc apt tc x' \u03c4' mem with \u0393 x\n-- gcomp-extend gc apt tc x' \u03c4' mem | Some x\u2081 = abort (somenotnone tc)\n-- gcomp-extend {\u0393} gc apt refl x' \u03c4' mem | None with ctxindirect \u0393 x'\n-- gcomp-extend {\u0393} {x = y} gc tc refl x' \u03c4' mem | None | Inl (a , x) = tr (\u03bb qq \u2192 qq tcomplete) (ctxunicity x {!!}) (gc x' a x)\n-- gcomp-extend {\u0393} gc tc refl x' \u03c4' mem | None | Inr neq with \u0393 x'\n-- gcomp-extend gc tc refl x' \u03c4' mem | None | Inr neq | Some x = abort (somenotnone neq)\n-- gcomp-extend gc tc refl x' \u03c4' mem | None | Inr refl | None with lem-insingeq mem\n-- gcomp-extend gc tc refl x' \u03c4' mem | None | Inr refl | None | refl = tc\n\n-- (\u0393 \u222a \u25a0 (x , \u03c4) x\u2081 | \u0393 x\u2081) == Some \u03c4\u2081\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d489dae230dd423a0087c0da7ca812bb1beb89e3","subject":"Fixed map-iterate-Stream.","message":"Fixed map-iterate-Stream.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/MapIterate\/MapIterateI.agda","new_file":"src\/fot\/FOTC\/Program\/MapIterate\/MapIterateI.agda","new_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using co-induction\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The map-iterate property (Gibbons and Hutton, 2005):\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\nmodule FOTC.Program.MapIterate.MapIterateI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\nopen import FOTC.Relation.Binary.Bisimilarity.Type\n\n------------------------------------------------------------------------------\n\nunfoldMapIterate : \u2200 f x \u2192\n                   map f (iterate f x) \u2261 f \u00b7 x \u2237 map f (iterate f (f \u00b7 x))\nunfoldMapIterate f x =\n  map f (iterate f x)               \u2261\u27e8 mapCong\u2082 (iterate-eq f x) \u27e9\n  map f (x \u2237 iterate f (f \u00b7 x))     \u2261\u27e8 map-\u2237 f x (iterate f (f \u00b7 x)) \u27e9\n  f \u00b7 x \u2237 map f (iterate f (f \u00b7 x)) \u220e\n\nmap-iterate-Stream : \u2200 f x \u2192 Stream (map f (iterate f x))\nmap-iterate-Stream f x = Stream-coind A h (x , refl)\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ y ] xs \u2261 map f (iterate f y)\n\n  h : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h (y , prf) = f \u00b7 y\n                , map f (iterate f (f \u00b7 y))\n                , (trans prf ((unfoldMapIterate f y)))\n                , f \u00b7 y , refl\n\n-- TODO (23 December 2013).\n-- map-iterate-Stream\u2082 : \u2200 f x \u2192 Stream (iterate f (f \u00b7 x))\n\n-- The map-iterate property.\n\u2248-map-iterate : \u2200 f x \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-coind B h (x , refl , refl)\n  where\n  -- Based on the relation used by (Gim\u00e9nez and Cast\u00e9ran, 2007).\n  B : D \u2192 D \u2192 Set\n  B xs ys = \u2203[ y ] xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y)\n\n  h : \u2200 {xs} {ys} \u2192 B xs ys \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 B xs' ys'\n  h (y , prf\u2081 , prf\u2082) =\n      f \u00b7 y\n    , map f (iterate f (f \u00b7 y))\n    , iterate f (f \u00b7 (f \u00b7 y))\n    , trans prf\u2081 (unfoldMapIterate f y)\n    , trans prf\u2082 (iterate-eq f (f \u00b7 y))\n    , ((f \u00b7 y) , refl , refl)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Gim\u00e9nez, Eduardo and Caster\u00e1n, Pierre (2007). A Tutorial on\n-- [Co-]Inductive Types in Coq.\n--\n-- Gibbons, Jeremy and Hutton, Graham (2005). Proof Methods for\n-- Corecursive Programs. In: Fundamenta Informaticae XX, pp. 1\u201314.\n","old_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using co-induction\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The map-iterate property (Gibbons and Hutton, 2005):\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\nmodule FOTC.Program.MapIterate.MapIterateI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Relation.Binary.Bisimilarity.Type\n\n------------------------------------------------------------------------------\n\nunfoldMapIterate : \u2200 f x \u2192\n                   map f (iterate f x) \u2261 f \u00b7 x \u2237 map f (iterate f (f \u00b7 x))\nunfoldMapIterate f x =\n  map f (iterate f x)               \u2261\u27e8 mapCong\u2082 (iterate-eq f x) \u27e9\n  map f (x \u2237 iterate f (f \u00b7 x))     \u2261\u27e8 map-\u2237 f x (iterate f (f \u00b7 x)) \u27e9\n  f \u00b7 x \u2237 map f (iterate f (f \u00b7 x)) \u220e\n\n-- TODO (23 December 2013).\n-- map-iterate-Stream\u2081 : \u2200 f x \u2192 Stream (map f (iterate f x))\n\n-- TODO (23 December 2013).\n-- map-iterate-Stream\u2082 : \u2200 f x \u2192 Stream (iterate f (f \u00b7 x))\n\n-- The map-iterate property.\n\u2248-map-iterate : \u2200 f x \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-coind B h (x , refl , refl)\n  where\n  -- Based on the relation used by (Gim\u00e9nez and Cast\u00e9ran, 2007).\n  B : D \u2192 D \u2192 Set\n  B xs ys = \u2203[ y ] xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y)\n\n  h : \u2200 {xs} {ys} \u2192 B xs ys \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 B xs' ys'\n  h (y , prf\u2081 , prf\u2082) =\n      f \u00b7 y\n    , map f (iterate f (f \u00b7 y))\n    , iterate f (f \u00b7 (f \u00b7 y))\n    , trans prf\u2081 (unfoldMapIterate f y)\n    , trans prf\u2082 (iterate-eq f (f \u00b7 y))\n    , ((f \u00b7 y) , refl , refl)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Gim\u00e9nez, Eduardo and Caster\u00e1n, Pierre (2007). A Tutorial on\n-- [Co-]Inductive Types in Coq.\n--\n-- Gibbons, Jeremy and Hutton, Graham (2005). Proof Methods for\n-- Corecursive Programs. In: Fundamenta Informaticae XX, pp. 1\u201314.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"94c528d33647e49a67e021b9fcca5af7dc882a4c","subject":"example","message":"example\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"contraction.agda","new_file":"contraction.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nmodule contraction where\n  -- in the same style as the proofs of exchange, this argument along with\n  -- trasnport allows you to prove contraction for all the hypothetical\n  -- judgements uniformly. we never explicitly use contraction anywhere, so\n  -- we omit any of the specific instances for concision; they are entirely\n  -- mechanical, as are the specific instances of exchange. one is shown\n  -- below as an example.\n  contract : {A : Set} {x : Nat} {\u03c4 : A} (\u0393 : A ctx) \u2192\n         ((\u0393 ,, (x , \u03c4)) ,, (x , \u03c4)) == (\u0393 ,, (x , \u03c4))\n  contract {A} {x} {\u03c4} \u0393 = funext guts\n    where\n      guts : (y : Nat) \u2192 (\u0393 ,, (x , \u03c4) ,, (x , \u03c4)) y == (\u0393 ,, (x , \u03c4)) y\n      guts y with natEQ x y\n      guts .x | Inl refl with \u0393 x\n      guts .x | Inl refl | Some x\u2081 = refl\n      guts .x | Inl refl | None with natEQ x x\n      guts .x | Inl refl | None | Inl refl = refl\n      guts .x | Inl refl | None | Inr x\u2081 = abort (x\u2081 refl)\n      guts y | Inr x\u2081 with \u0393 y\n      guts y | Inr x\u2082 | Some x\u2081 = refl\n      guts y | Inr x\u2081 | None with natEQ x y\n      guts .x | Inr x\u2082 | None | Inl refl = refl\n      guts y | Inr x\u2082 | None | Inr x\u2081 with natEQ x y\n      guts .x | Inr x\u2083 | None | Inr x\u2082 | Inl refl = abort (x\u2083 refl)\n      guts y | Inr x\u2083 | None | Inr x\u2082 | Inr x\u2081 = refl\n\n  contract-synth : \u2200{ \u0393 x \u03c4 e \u03c4'}\n                  \u2192 (\u0393 ,, (x , \u03c4) ,, (x , \u03c4)) \u22a2 e => \u03c4'\n                  \u2192 (\u0393 ,, (x , \u03c4)) \u22a2 e => \u03c4'\n  contract-synth {\u0393 = \u0393} {e = e} {\u03c4' = \u03c4'} =\n    tr (\u03bb qq \u2192 qq \u22a2 e => \u03c4') (contract \u0393)\n\n  -- as an aside, this also establishes the other direction which is rarely\n  -- mentioned, since equality is symmetric\n  expand-synth : \u2200{ \u0393 x \u03c4 e \u03c4'}\n                  \u2192 (\u0393 ,, (x , \u03c4)) \u22a2 e => \u03c4'\n                  \u2192 (\u0393 ,, (x , \u03c4) ,, (x , \u03c4)) \u22a2 e => \u03c4'\n  expand-synth {\u0393 = \u0393} {e = e} {\u03c4' = \u03c4'} =\n    tr (\u03bb qq \u2192 qq \u22a2 e => \u03c4') (! (contract \u0393))\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\n\nmodule contraction where\n  -- in the same style as the proofs of exchange, this argument along with\n  -- trasnport allows you to prove contraction for all the hypothetical\n  -- judgements uniformly. we never explicitly use contraction anywhere, so\n  -- we omit any of the specific instances for concision; they are entirely\n  -- mechanical, as are the specific instances of exchange.\n  contract : {A : Set} {x : Nat} {\u03c4 : A} (\u0393 : A ctx) \u2192\n         ((\u0393 ,, (x , \u03c4)) ,, (x , \u03c4)) == (\u0393 ,, (x , \u03c4))\n  contract {A} {x} {\u03c4} \u0393 = funext guts\n    where\n      guts : (y : Nat) \u2192 (\u0393 ,, (x , \u03c4) ,, (x , \u03c4)) y == (\u0393 ,, (x , \u03c4)) y\n      guts y with natEQ x y\n      guts .x | Inl refl with \u0393 x\n      guts .x | Inl refl | Some x\u2081 = refl\n      guts .x | Inl refl | None with natEQ x x\n      guts .x | Inl refl | None | Inl refl = refl\n      guts .x | Inl refl | None | Inr x\u2081 = abort (x\u2081 refl)\n      guts y | Inr x\u2081 with \u0393 y\n      guts y | Inr x\u2082 | Some x\u2081 = refl\n      guts y | Inr x\u2081 | None with natEQ x y\n      guts .x | Inr x\u2082 | None | Inl refl = refl\n      guts y | Inr x\u2082 | None | Inr x\u2081 with natEQ x y\n      guts .x | Inr x\u2083 | None | Inr x\u2082 | Inl refl = abort (x\u2083 refl)\n      guts y | Inr x\u2083 | None | Inr x\u2082 | Inr x\u2081 = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"46ea55a1cff1a55f5e77fd87b7af72362e358405","subject":"Update qualified import for new module name.","message":"Update qualified import for new module name.\n\nOld-commit-hash: d14bfe564331d1fd643bdd11a759bb2a5c9338dc\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Derive\/Plotkin.agda","new_file":"Syntax\/Derive\/Plotkin.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\nimport Parametric.Syntax.Term as Term\nimport Base.Change.Context as ChangeContext\nimport Parametric.Change.Type as ChangeType\n\nmodule Syntax.Derive.Plotkin\n    {Base : Set {- of base types -}}\n    {Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -}}\n    (\u0394Base : Base \u2192 Base)\n    (deriveConst :\n      \u2200 {\u0393 \u03a3 \u03c4} \u2192 Constant \u03a3 \u03c4 \u2192\n      Term.Terms\n        Constant\n        (ChangeContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u0393)\n        (ChangeContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u03a3) \u2192\n      Term.Term Constant (ChangeContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u0393) (ChangeType.\u0394Type \u0394Base \u03c4))\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen Type Base\nopen Context Type\nopen Term Constant\nopen ChangeType \u0394Base\nopen ChangeContext \u0394Type\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394Context \u0393) (\u0394Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) (\u0394Context \u03a3)\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\nderive-terms {\u2205}    \u2205 = \u2205\nderive-terms {\u03c4 \u2022 \u03a3} (t \u2022 ts) = derive t \u2022 fit t \u2022 derive-terms ts\n\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\nderive (const c ts) = deriveConst c (derive-terms ts)\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\nimport Parametric.Syntax.Term as Term\nimport Base.Change.Context as DeltaContext\nimport Parametric.Change.Type as ChangeType\n\nmodule Syntax.Derive.Plotkin\n    {Base : Set {- of base types -}}\n    {Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -}}\n    (\u0394Base : Base \u2192 Base)\n    (deriveConst :\n      \u2200 {\u0393 \u03a3 \u03c4} \u2192 Constant \u03a3 \u03c4 \u2192\n      Term.Terms\n        Constant\n        (DeltaContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u0393)\n        (DeltaContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u03a3) \u2192\n      Term.Term Constant (DeltaContext.\u0394Context (ChangeType.\u0394Type \u0394Base) \u0393) (ChangeType.\u0394Type \u0394Base \u03c4))\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen Type Base\nopen Context Type\nopen Term Constant\nopen ChangeType \u0394Base\nopen DeltaContext \u0394Type\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394Context \u0393) (\u0394Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Terms (\u0394Context \u0393) (\u0394Context \u03a3)\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\nderive-terms {\u2205}    \u2205 = \u2205\nderive-terms {\u03c4 \u2022 \u03a3} (t \u2022 ts) = derive t \u2022 fit t \u2022 derive-terms ts\n\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\nderive (const c ts) = deriveConst c (derive-terms ts)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"388c73ec814498c8a13653701108da88b76a31e2","subject":"Algebra.Group.Homorphism: nicer notation with less explicit arguments","message":"Algebra.Group.Homorphism: nicer notation with less explicit arguments\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_; id)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n\nopen GroupHomomorphism\n\nmodule Identity\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n\n  id-hom : GroupHomomorphism \ud835\udd38 \ud835\udd38 id\n  id-hom = mk refl\n\nmodule Compose\n  {a}{A : Type a}\n  {b}{B : Type b}\n  {c}{C : Type c}\n  (\ud835\udd38 : Group A)\n  (\ud835\udd39 : Group B)\n  (\u2102 : Group C)\n  (\u03c8 : A \u2192 B)\n  (\u03c8-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39 \u03c8)\n  (\u03c6 : B \u2192 C)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd39 \u2102 \u03c6)\n  where\n\n  compose-hom : GroupHomomorphism \ud835\udd38 \u2102 (\u03c6 \u2218 \u03c8)\n  compose-hom = mk (ap \u03c6 (hom \u03c8-hom) \u2219 hom \u03c6-hom)\n\n-- A nicer version of Compose.compose-hom: more implicit arguments\n-- AND reverse order to fit with _\u2218_ order.\nmodule _\n  {a}{A : Type a}\n  {b}{B : Type b}\n  {c}{C : Type c}\n  {\ud835\udd38 : Group A}\n  {\ud835\udd39 : Group B}\n  {\u2102 : Group C}\n  {\u03c6 : B \u2192 C}\n  (\u03c6-hom : GroupHomomorphism \ud835\udd39 \u2102 \u03c6)\n  {\u03c8 : A \u2192 B}\n  (\u03c8-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39 \u03c8)\n  where\n\n  _\u2218-hom_ : GroupHomomorphism \ud835\udd38 \u2102 (\u03c6 \u2218 \u03c8)\n  _\u2218-hom_ = Compose.compose-hom _ _ _ _ \u03c8-hom _ \u03c6-hom\n\nmodule Delta\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n  open Algebra.Group.Constructions.Product\n\n  \u0394-hom : GroupHomomorphism \ud835\udd38 (\u00d7-grp \ud835\udd38 \ud835\udd38) (\u03bb x \u2192 x , x)\n  \u0394-hom = mk refl\n\nmodule Zip\n  {a\u2080}{A\u2080 : Type a\u2080}\n  {a\u2081}{A\u2081 : Type a\u2081}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38\u2080 : Group A\u2080)\n  (\ud835\udd38\u2081 : Group A\u2081)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A\u2080 \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38\u2080 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A\u2081 \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38\u2081 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  open Algebra.Group.Constructions.Product\n\n  zip-hom : GroupHomomorphism (\u00d7-grp \ud835\udd38\u2080 \ud835\udd38\u2081) (\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) (map \u03c6\u2080 \u03c6\u2081)\n  zip-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n\n-- A nicer version of Zip.zip-hom: more implicit arguments\nmodule _\n  {a\u2080}{A\u2080 : Type a\u2080}\n  {a\u2081}{A\u2081 : Type a\u2081}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  {\ud835\udd38\u2080 : Group A\u2080}\n  {\ud835\udd38\u2081 : Group A\u2081}\n  {\ud835\udd39\u2080 : Group B\u2080}\n  {\ud835\udd39\u2081 : Group B\u2081}\n  {\u03c6\u2080 : A\u2080 \u2192 B\u2080}\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38\u2080 \ud835\udd39\u2080 \u03c6\u2080)\n  {\u03c6\u2081 : A\u2081 \u2192 B\u2081}\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38\u2081 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  <_\u00d7_>-hom = Zip.zip-hom _ _ _ _ _ \u03c6\u2080-hom _ \u03c6\u2081-hom\n\nmodule Pair\n  {a}{A   : Type a}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38  : Group A)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n\n  -- pair = zip \u2218 \u0394\n  pair-hom : GroupHomomorphism \ud835\udd38 (Product.\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) < \u03c6\u2080 , \u03c6\u2081 >\n  pair-hom = < \u03c6\u2080-hom \u00d7 \u03c6\u2081-hom >-hom \u2218-hom Delta.\u0394-hom _\n  -- OR:\n  -- pair-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n\n-- A nicer version of Pair.pair-hom: more implicit arguments\nmodule _\n  {a}{A   : Type a}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  {\ud835\udd38  : Group A}\n  {\ud835\udd39\u2080 : Group B\u2080}\n  {\ud835\udd39\u2081 : Group B\u2081}\n  {\u03c6\u2080 : A \u2192 B\u2080}\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2080 \u03c6\u2080)\n  {\u03c6\u2081 : A \u2192 B\u2081}\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  <_,_>-hom = Pair.pair-hom _ _ _ _ \u03c6\u2080-hom _ \u03c6\u2081-hom\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_; id)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n\nopen GroupHomomorphism\n\nmodule Identity\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n\n  id-hom : GroupHomomorphism \ud835\udd38 \ud835\udd38 id\n  id-hom = mk refl\n\nmodule Compose\n  {a}{A : Type a}\n  {b}{B : Type b}\n  {c}{C : Type c}\n  (\ud835\udd38 : Group A)\n  (\ud835\udd39 : Group B)\n  (\u2102 : Group C)\n  (\u03c8 : A \u2192 B)\n  (\u03c8-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39 \u03c8)\n  (\u03c6 : B \u2192 C)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd39 \u2102 \u03c6)\n  where\n\n  \u2218-hom : GroupHomomorphism \ud835\udd38 \u2102 (\u03c6 \u2218 \u03c8)\n  \u2218-hom = mk (ap \u03c6 (hom \u03c8-hom) \u2219 hom \u03c6-hom)\n\nmodule Delta\n  {a}{A : Type a}\n  (\ud835\udd38 : Group A)\n  where\n  open Algebra.Group.Constructions.Product\n\n  \u0394-hom : GroupHomomorphism \ud835\udd38 (\u00d7-grp \ud835\udd38 \ud835\udd38) (\u03bb x \u2192 x , x)\n  \u0394-hom = mk refl\n\nmodule Zip\n  {a\u2080}{A\u2080 : Type a\u2080}\n  {a\u2081}{A\u2081 : Type a\u2081}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38\u2080 : Group A\u2080)\n  (\ud835\udd38\u2081 : Group A\u2081)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A\u2080 \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38\u2080 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A\u2081 \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38\u2081 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n  open Algebra.Group.Constructions.Product\n\n  zip-hom : GroupHomomorphism (\u00d7-grp \ud835\udd38\u2080 \ud835\udd38\u2081) (\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) (map \u03c6\u2080 \u03c6\u2081)\n  zip-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n\nmodule Pair\n  {a}{A   : Type a}\n  {b\u2080}{B\u2080 : Type b\u2080}\n  {b\u2081}{B\u2081 : Type b\u2081}\n  (\ud835\udd38  : Group A)\n  (\ud835\udd39\u2080 : Group B\u2080)\n  (\ud835\udd39\u2081 : Group B\u2081)\n  (\u03c6\u2080 : A \u2192 B\u2080)\n  (\u03c6\u2080-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2080 \u03c6\u2080)\n  (\u03c6\u2081 : A \u2192 B\u2081)\n  (\u03c6\u2081-hom : GroupHomomorphism \ud835\udd38 \ud835\udd39\u2081 \u03c6\u2081)\n  where\n\n  -- pair = zip \u2218 \u0394\n  pair-hom : GroupHomomorphism \ud835\udd38 (Product.\u00d7-grp \ud835\udd39\u2080 \ud835\udd39\u2081) < \u03c6\u2080 , \u03c6\u2081 >\n  pair-hom = Compose.\u2218-hom _ _ _\n               _ (Delta.\u0394-hom \ud835\udd38)\n               _ (Zip.zip-hom _ _ _ _ _ \u03c6\u2080-hom _ \u03c6\u2081-hom)\n  -- OR:\n  -- pair-hom = mk (ap\u2082 _,_ (hom \u03c6\u2080-hom) (hom \u03c6\u2081-hom))\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ab90ea46e880bc19715ac996cbd99528d7521474","subject":"Renamed some stuff.","message":"Renamed some stuff.\n\nIgnore-this: e2f54884bcaca61788941d5e933768be\n\ndarcs-hash:20100604023904-3bd4e-d676bf2fd9d4f0894a26c11b8d10462bae861a86.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Division\/IsCorrectER.agda","new_file":"Examples\/Division\/IsCorrectER.agda","new_contents":"------------------------------------------------------------------------------\n-- The division result is correct\n------------------------------------------------------------------------------\n\nmodule Examples.Division.IsCorrectER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import Examples.Division.EquationsER\nopen import Examples.Division.Specification\n\nopen import LTC.Data.N\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF-ER\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.InequalitiesPCF\n\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IsNER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\n-- The division result is correct when the dividend is less than\n-- the divisor.\n\ndiv-x<y-aux : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192 i \u2261 j * (div i j) + i\ndiv-x<y-aux {i} {j} Ni Nj i<j = sym\n    ( begin\n        j * (div i j) + i \u2261\u27e8 prf\u2081 \u27e9\n        j * zero + i      \u2261\u27e8 prf\u2082 \u27e9\n        zero + i          \u2261\u27e8 +-leftIdentity Ni \u27e9\n        i\n      \u220e\n    )\n    where\n      prf\u2081 : j * (div i j) + i \u2261 j * zero + i\n      prf\u2081 = subst (\u03bb x \u2192 j * x + i \u2261 j * zero + i )\n                   (sym (div-x<y i<j ))\n                   refl\n\n      prf\u2082 : j * zero + i \u2261 zero + i\n      prf\u2082 = subst (\u03bb x \u2192 x + i \u2261 zero + i)\n                   (sym (*-rightZero Nj))\n                   refl\n\ndiv-x<y-correct : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192\n                  (\u2203D (\u03bb r \u2192 N r \u2227 LT r j \u2227 i \u2261 j * (div i j) + r))\ndiv-x<y-correct {i} Ni Nj i<j =\n  i , (Ni , (i<j , (div-x<y-aux Ni Nj i<j)))\n\n\n-- The division result is correct when the dividend is greater or equal\n-- than the divisor.\n-- Using the inductive hypothesis 'ih' we know that\n-- 'i - j = j * (div (i - j) j) + r'. From\n-- that we get 'i = j * (succ (div (i - j) j)) + r', and\n-- we know 'div i j = succ (div (i - j) j); therefore we\n-- get 'i = j * div i j + r'.\n\npostulate\n  aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n        i - j \u2261 j * (div (i - j) j) + r \u2192\n        i \u2261 j * succ (div (i - j) j) + r\n\ndiv-x\u2265y-aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n              GE i j \u2192\n              i - j \u2261 j * (div (i - j) j) + r \u2192\n              i \u2261 j * (div i j) + r\ndiv-x\u2265y-aux {i} {j} {r} Ni Nj Nr i\u2265j auxH =\n  begin\n    i \u2261\u27e8 aux Ni Nj Nr auxH \u27e9\n    j * succ (div (i - j) j) + r \u2261\u27e8 prf \u27e9\n    j * (div i j) + r\n  \u220e\n\n  where prf : j * succ (div (i - j) j) + r \u2261 j * (div i j) + r\n        prf = subst (\u03bb x \u2192 j * x + r \u2261 j * (div i j) + r )\n                    (div-x\u2265y i\u2265j  )\n                    refl\n\ndiv-x\u2265y-correct : {i j : D} \u2192 N i \u2192 N j \u2192\n                  (ih : DIV (i - j) j (div (i - j) j)) \u2192\n                  GE i j \u2192\n                  \u2203D (\u03bb r \u2192 (N r) \u2227 (LT r j) \u2227 (i \u2261 j * (div i j) + r))\ndiv-x\u2265y-correct {i} {j} Ni Nj ih i\u2265j =\n  r , (Nr , (r<j , (div-x\u2265y-aux Ni Nj Nr i\u2265j auxH)))\n\n    where\n      -- The parts of the inductive hipothesis 'ih-i-j'\n      r : D\n      r = \u2203D-proj\u2081 (\u2227-proj\u2082 ih)\n\n      r-correct : N r \u2227 LT r j \u2227 i - j \u2261 j * (div (i - j) j) + r\n      r-correct = \u2203D-proj\u2082 (\u2227-proj\u2082 ih)\n\n      Nr : N r\n      Nr = \u2227-proj\u2081 r-correct\n\n      r<j : LT r j\n      r<j = \u2227-proj\u2081 (\u2227-proj\u2082 r-correct)\n\n      auxH : i - j \u2261 j * (div (i - j) j) + r\n      auxH = \u2227-proj\u2082 (\u2227-proj\u2082 r-correct)\n","old_contents":"------------------------------------------------------------------------------\n-- The division result is correct\n------------------------------------------------------------------------------\n\nmodule Examples.Division.IsCorrectER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import Examples.Division\nopen import Examples.Division.EquationsER\nopen import Examples.Division.Specification\n\nopen import LTC.Data.N\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF-ER\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.InequalitiesPCF\n\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IsNER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\n-- The division result is correct when the dividend is less than\n-- the divisor.\n\ndiv-x<y-aux : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192 i \u2261 j * (div i j) + i\ndiv-x<y-aux {i} {j} Ni Nj i<j = sym\n    ( begin\n        j * (div i j) + i \u2261\u27e8 prf\u2081 \u27e9\n        j * zero + i      \u2261\u27e8 prf\u2082 \u27e9\n        zero + i          \u2261\u27e8 +-leftIdentity Ni \u27e9\n        i\n      \u220e\n    )\n    where\n      prf\u2081 : j * (div i j) + i \u2261 j * zero + i\n      prf\u2081 = subst (\u03bb x \u2192 j * x + i \u2261 j * zero + i )\n                   (sym (div-x<y i<j ))\n                   refl\n\n      prf\u2082 : j * zero + i \u2261 zero + i\n      prf\u2082 = subst (\u03bb x \u2192 x + i \u2261 zero + i)\n                   (sym (*-rightZero Nj))\n                   refl\n\ndiv-x<y-correct : {i j : D} \u2192 N i \u2192 N j \u2192 LT i j \u2192\n                  (\u2203D (\u03bb r \u2192 N r \u2227 LT r j \u2227 i \u2261 j * (div i j) + r))\ndiv-x<y-correct {i} Ni Nj i<j =\n  i , (Ni , (i<j , (div-x<y-aux Ni Nj i<j)))\n\n\n-- The division result is correct when the dividend is greater or equal\n-- than the divisor.\n-- Using the inductive hypothesis 'ih' we know that\n-- 'i - j = j * (div (i - j) j) + r'. From\n-- that we get 'i = j * (succ (div (i - j) j)) + r', and\n-- we know 'div i j = succ (div (i - j) j); therefore we\n-- get 'i = j * div i j + r'.\n\npostulate\n    ih-eq : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n            i - j \u2261 j * (div (i - j) j) + r \u2192\n            i \u2261 j * succ (div (i - j) j) + r\n\ndiv-x\u2265y-aux : {i j r : D} \u2192 N i \u2192 N j \u2192 N r \u2192\n              GE i j \u2192\n              i - j \u2261 j * (div (i - j) j) + r \u2192\n              i \u2261 j * (div i j) + r\ndiv-x\u2265y-aux {i} {j} {r} Ni Nj Nr i\u2265j DIV-eq =\n  begin\n    i \u2261\u27e8 ih-eq Ni Nj Nr DIV-eq \u27e9\n    j * succ (div (i - j) j) + r \u2261\u27e8 prf \u27e9\n    j * (div i j) + r\n  \u220e\n\n  where prf : j * succ (div (i - j) j) + r \u2261 j * (div i j) + r\n        prf = subst (\u03bb x \u2192 j * x + r \u2261 j * (div i j) + r )\n                    (div-x\u2265y i\u2265j  )\n                    refl\n\ndiv-x\u2265y-correct : {i j : D} \u2192 N i \u2192 N j \u2192\n                  (ih : DIV (i - j) j (div (i - j) j)) \u2192\n                  GE i j \u2192\n                  \u2203D (\u03bb r \u2192 (N r) \u2227 (LT r j) \u2227 (i \u2261 j * (div i j) + r))\ndiv-x\u2265y-correct {i} {j} Ni Nj ih i\u2265j =\n  r , (Nr , (r<j , (div-x\u2265y-aux Ni Nj Nr i\u2265j DIV-eq)))\n\n    where\n      -- The parts of the inductive hipothesis 'ih-i-j'\n      r : D\n      r = \u2203D-proj\u2081 (\u2227-proj\u2082 ih)\n\n      r-correct : N r \u2227 LT r j \u2227 i - j \u2261 j * (div (i - j) j) + r\n      r-correct = \u2203D-proj\u2082 (\u2227-proj\u2082 ih)\n\n      Nr : N r\n      Nr = \u2227-proj\u2081 r-correct\n\n      r<j : LT r j\n      r<j = \u2227-proj\u2081 (\u2227-proj\u2082 r-correct)\n\n      DIV-eq : i - j \u2261 j * (div (i - j) j) + r\n      DIV-eq = \u2227-proj\u2082 (\u2227-proj\u2082 r-correct)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7efde702867deae6eb19e9244a20af89d9eb68b1","subject":"Added missing definition.","message":"Added missing definition.\n\nIgnore-this: d856635bcd908756de7bd5fb0d97adef\n\ndarcs-hash:20100703041106-3bd4e-6a27085b48360f2388c10fe42bd1ac2d3a427103.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Nat\/Inequalities.agda","new_file":"LTC\/Data\/Nat\/Inequalities.agda","new_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\npostulate\n  lt : D \u2192 D \u2192 D\n\n  lt-00 : lt zero zero                     \u2261 false\n\n  lt-0S : (d : D) \u2192 lt zero (succ d)       \u2261 true\n\n  lt-S0 : (d : D) \u2192 lt (succ d) zero       \u2261 false\n\n  lt-SS : (d e : D) \u2192 lt (succ d) (succ e) \u2261 lt d e\n\n{-# ATP axiom lt-00 #-}\n{-# ATP axiom lt-0S #-}\n{-# ATP axiom lt-S0 #-}\n{-# ATP axiom lt-SS #-}\n\nle : D \u2192 D \u2192 D\nle d e = lt d (succ e)\n{-# ATP definition le #-}\n\n------------------------------------------------------------------------\n-- The data types\n\n-- infix 4 _\u2264_ _<_ _\u2265_ _>_\n\nGT : D \u2192 D \u2192 Set\nGT d e = lt e d \u2261 true\n{-# ATP definition GT #-}\n\nLT : D \u2192 D  \u2192 Set\nLT d e = lt d e \u2261 true\n{-# ATP definition LT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = lt e d \u2261 false\n{-# ATP definition LE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = lt d e \u2261 false\n{-# ATP definition GE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n\n","old_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\npostulate\n  lt : D \u2192 D \u2192 D\n\n  lt-00 : lt zero zero                     \u2261 false\n\n  lt-0S : (d : D) \u2192 lt zero (succ d)       \u2261 true\n\n  lt-S0 : (d : D) \u2192 lt (succ d) zero       \u2261 false\n\n  lt-SS : (d e : D) \u2192 lt (succ d) (succ e) \u2261 lt d e\n\n{-# ATP axiom lt-00 #-}\n{-# ATP axiom lt-0S #-}\n{-# ATP axiom lt-S0 #-}\n{-# ATP axiom lt-SS #-}\n\nle : D \u2192 D \u2192 D\nle d e = lt d (succ e)\n\n------------------------------------------------------------------------\n-- The data types\n\n-- infix 4 _\u2264_ _<_ _\u2265_ _>_\n\nGT : D \u2192 D \u2192 Set\nGT d e = lt e d \u2261 true\n{-# ATP definition GT #-}\n\nLT : D \u2192 D  \u2192 Set\nLT d e = lt d e \u2261 true\n{-# ATP definition LT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = lt e d \u2261 false\n{-# ATP definition LE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = lt d e \u2261 false\n{-# ATP definition GE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d5a77340f08eb933adfd19c9895b2e9ee3266caa","subject":"Make-over of the Desc model, including natural numbers and doc.","message":"Make-over of the Desc model, including natural numbers and doc.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\nrecord One : Set where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\n-- Inductive types are implemented as a Universe. Hence, in this\n-- section, we implement their code.\n\n-- We can read this code as follow (see Conor's \"Ornamental\n-- algebras\"): a description in |Desc| is a program to read one node\n-- of the described tree.\n\ndata Desc : Set where\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  -- Read a field in |X|; continue, given its value\n\n  -- Often, |X| is an |EnumT|, hence allowing to choose a constructor\n  -- among a finite set of constructors\n\n  Ind : (H : Set) -> Desc -> Desc\n  -- Read a field in H; read a recursive subnode given the field,\n  -- continue regarless of the subnode\n\n  -- Often, |H| is |1|, hence |Ind| simplifies to |Inf : Desc -> Desc|,\n  -- meaning: read a recursive subnode and continue regardless\n\n  Done : Desc\n  -- Stop reading\n\n\n--********************************************\n-- Desc decoder\n--********************************************\n\n-- Provided the type of the recursive subnodes |R|, we decode a\n-- description as a record describing the node.\n\n[|_|]_ : Desc -> Set -> Set\n[| Arg A D |] R = Sigma A (\\ a -> [| D a |] R)\n[| Ind H D |] R = (H -> R) * [| D |] R\n[| Done |] R = One\n\n\n--********************************************\n-- Functions on codes\n--********************************************\n\n-- Saying that a \"predicate\" |p| holds everywhere in |v| amounts to\n-- write something of the following type:\n\nEverywhere : (d : Desc) (D : Set) (bp : D -> Set) (V : [| d |] D) -> Set\nEverywhere (Arg A f) d p v =  Everywhere (f (fst v)) d p (snd v)\n-- It must hold for this constructor\n\nEverywhere (Ind H x) d p v =  ((y : H) -> p (fst v y)) * Everywhere x d p (snd v)\n-- It must hold for the subtrees \n\nEverywhere Done _ _ _ =  _\n-- It trivially holds at endpoints\n\n-- Then, we can build terms that inhabits this type. That is, a\n-- function that takes a \"predicate\" |bp| and makes it hold everywhere\n-- in the data-structure. It is the equivalent of a \"map\", but in a\n-- dependently-typed setting.\n\neverywhere : (d : Desc) (D : Set) (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : [| d |] D) ->\n           Everywhere d D bp v\neverywhere (Arg a f) d bp p v =  everywhere (f (fst v)) d bp p (snd v)\n-- It holds everywhere on this constructor\n\neverywhere (Ind H x) d bp p v = (\\y -> p (fst v y)) ,  everywhere x d bp p (snd v)\n-- It holds here, and down in the recursive subtrees\n\neverywhere Done _ _ _ _ =  One\n-- Nothing needs to be done on endpoints\n\n\n-- Looking at the decoder, a natural thing to do is to define its\n-- fixpoint |Mu D|, hence instantiating |R| with |Mu D| itself.\n\ndata Mu (D : Desc) : Set where\n  Con : [| D |] (Mu D) -> Mu D\n\n-- Using the \"map\" defined by |everywhere|, we can implement a \"fold\"\n-- over the |Mu| fixpoint:\n\nfoldDesc : (D : Desc) (bp : Mu D -> Set) ->\n         ((x : [| D |] (Mu D)) -> Everywhere D (Mu D) bp x -> bp (Con x)) ->\n         (v : Mu D) ->\n         bp v\nfoldDesc D bp p (Con v) =  p v (everywhere D (Mu D) bp (\\x -> foldDesc D bp p x) v) \n\n\n--********************************************\n-- Nat\n--********************************************\n\ndata NatConst : Set where\n  ZE : NatConst\n  SU : NatConst\n\nnatc : NatConst -> Desc\nnatc ZE = Done\nnatc SU = Ind One Done\n\nnatd : Desc\nnatd = Arg NatConst natc\n\nnat : Set\nnat = Mu natd\n\nzero : nat\nzero = Con ( ZE ,  _ )\n\nsuc : nat -> nat\nsuc n = Con ( SU , ( (\\_ -> n) ,  _ ) )\n\ntwo : nat\ntwo = suc (suc zero)\n\nfour : nat\nfour = suc (suc (suc (suc zero)))\n\nsum : nat -> \n      ((x : Sigma NatConst (\\ a -> [| natc a |] Mu (Arg NatConst natc))) ->\n      Everywhere (natc (fst x)) (Mu (Arg NatConst natc)) (\\ _ -> Mu (Arg NatConst natc)) (snd x) ->\n      Mu (Arg NatConst natc))\nsum n2 (ZE , _) p =  n2\nsum n2 (SU , f) p =  suc ( fst p _) \n\n\nplus : nat -> nat -> nat\nplus n1 n2 = foldDesc natd (\\_ -> nat) (sum n2) n1 \n\nx : nat\nx = plus two two\n\n\n\n","old_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  ((y : H) -> p (proj\u2081 v y))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v = (\\y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> Set) ->\n         ((x : descOp d (Mu d)) -> (boxOp d (Mu d) bp x) -> bp (Con x)) ->\n         (v : Mu d) -> bp v\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"39e5ce683c40ffc5f54dbc2507f3f3e1f048e0e9","subject":"Parser: minor changes","message":"Parser: minor changes\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Text\/Parser.agda","new_file":"lib\/Text\/Parser.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (\u2605; \u2605_)\nopen import Data.Maybe.NP using (_\u2192?_; module M?; just; maybe\u2032; nothing; map?; is-nothing)\nopen import Data.Nat using (zero; suc)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Char.NP renaming (_==_ to _==\u1d9c_)\nopen import Data.List as L using (List; []; _\u2237_; null; filter)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.BoundedVec using (BoundedVec) renaming ([] to []\u1d47\u1d5b; _\u2237_ to _\u2237\u1d47\u1d5b_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bool using (Bool; if_then_else_; false)\nopen import Data.String using (String) renaming (toList to S\u25b9L; fromList to L\u25b9S)\nopen import Function.NP\nopen import Level.NP\n\nmodule Text.Parser where\n\nopen M? \u2080 using () renaming (_=<<_ to _=<<?_; _\u229b_ to _\u229b?_; join to join?)\n\nParser : \u2605 \u2192 \u2605\nParser A = List Char \u2192? (A \u00d7 List Char)\n\npure : \u2200 {A} \u2192 A \u2192 Parser A\npure a s = just (a , s)\n\ninfixr 3 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : Parser A) \u2192 Parser A\n(f \u27e8|\u27e9 g) s = maybe\u2032 just (g s) (f s)\n\nempty : \u2200 {A} \u2192 Parser A\nempty _ = nothing\n\nsat : (Char \u2192 Bool) \u2192 Parser Char\nsat pred (x \u2237 xs) = (if pred x then pure x else empty) xs\nsat _    []       = nothing\n\nlookSat : (List Char \u2192 Bool) \u2192 Parser \ud835\udfd9\nlookSat p xs = if p xs then just (_ , xs) else nothing\n\nnotFollowedBy : \u2200 {A} \u2192 Parser A \u2192 Parser \ud835\udfd9\nnotFollowedBy m = lookSat (is-nothing \u2218 m)\n\nlookAhead : \u2200 {A} \u2192 Parser A \u2192 Parser A\nlookAhead p xs = map? (second (const xs)) (p xs)\n\neof : Parser \ud835\udfd9\neof = lookSat null\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 Parser A \u2192 List Char \u2192? A\nrunParser p s = map? fst (p s)\n\nrunParser\u02e2 : \u2200 {A} \u2192 Parser A \u2192 String \u2192? A\nrunParser\u02e2 p = runParser p \u2218 S\u25b9L\n\ninfixr 0 _>>=_\n_>>=_ : \u2200 {A B} \u2192 Parser A \u2192 (A \u2192 Parser B) \u2192 Parser B\n_>>=_ pa f = maybe\u2032 (uncurry f) nothing \u2218 pa\n\ninfixl 0 _=<<_\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 Parser B) \u2192 Parser A \u2192 Parser B\n_=<<_ f pa = pa >>= f\n\njoin : \u2200 {A} \u2192 Parser (Parser A) \u2192 Parser A\njoin = _=<<_ id\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\nmap f pa = pa >>= pure \u2218 f\n\ninfixl 4 _\u229b_ _*>_ _<*_\n_\u229b_ : \u2200 {A B} \u2192 Parser (A \u2192 B) \u2192 Parser A \u2192 Parser B\n_\u229b_ pf pa = pf >>= \u03bb f \u2192 pa >>= \u03bb a \u2192 pure (f a)\n\n\u27ea_\u00b7_\u27eb : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192\n            (A \u2192 B \u2192 C) \u2192 Parser A \u2192 Parser B \u2192 Parser C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D} \u2192 (A \u2192 B \u2192 C \u2192 D)\n            \u2192 Parser A \u2192 Parser B \u2192 Parser C \u2192 Parser D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_*>_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser B\np\u2081 *> p\u2082 = p\u2081 >>= const p\u2082\n\n_<*_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser A\np\u2081 <* p\u2082 = p\u2081 >>= \u03bb x \u2192 p\u2082 *> pure x\n\nmapM : \u2200 {a} {A : \u2605_ a} {B} \u2192 (A \u2192 Parser B) \u2192 List A \u2192 Parser (List B)\nmapM f []       = pure []\nmapM f (x \u2237 xs) = \u27ea _\u2237_ \u00b7 f x \u00b7 mapM f xs \u27eb\n\nmapMvoid : \u2200 {a} {A : \u2605_ a} {B} \u2192 (A \u2192 Parser B) \u2192 List A \u2192 Parser \ud835\udfd9\nmapMvoid f []       = pure _\nmapMvoid f (x \u2237 xs) = f x *> mapMvoid f xs\n\nchoices : \u2200 {A} \u2192 List (Parser A) \u2192 Parser A\nchoices = L.foldr _\u27e8|\u27e9_ empty\n\nvec : \u2200 {A} n \u2192 Parser A \u2192 Parser (Vec A n)\nvec zero    p = pure []\nvec (suc n) p = \u27ea _\u2237_ \u00b7 p \u00b7 vec n p \u27eb\n\n-- These are producing bounded vectors mainly for termination\n-- reasons.\nmanyBV : \u2200 {A} n \u2192 Parser A \u2192 Parser (BoundedVec A n)\nsomeBV : \u2200 {A} n \u2192 Parser A \u2192 Parser (BoundedVec A (suc n))\n\nmanyBV 0       _ = empty\nmanyBV (suc n) p = someBV n p \u27e8|\u27e9 pure []\u1d47\u1d5b\n\nsomeBV n p = \u27ea _\u2237\u1d47\u1d5b_ \u00b7 p \u00b7 manyBV n p \u27eb\n\nmany : \u2200 {A} n \u2192 Parser A \u2192 Parser (List A)\nsome : \u2200 {A} n \u2192 Parser A \u2192 Parser (List A)\n\nmany 0       _ = empty\nmany (suc n) p = some n p \u27e8|\u27e9 pure []\n\nsome n p = \u27ea _\u2237_ \u00b7 p \u00b7 many n p \u27eb\n\nmanySat : (Char \u2192 Bool) \u2192 Parser (List Char)\nmanySat p []       = pure [] []\nmanySat p (x \u2237 xs) = if p x then map? (first (_\u2237_ x)) (manySat p xs) else pure [] (x \u2237 xs)\n\nmanySat\u02e2 : (Char \u2192 Bool) \u2192 Parser String\nmanySat\u02e2 = map L\u25b9S \u2218 manySat\n\nsomeSat : (Char \u2192 Bool) \u2192 Parser (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nsomeSat\u02e2 : (Char \u2192 Bool) \u2192 Parser String\nsomeSat\u02e2 = map L\u25b9S \u2218 someSat\n\nchar : Char \u2192 Parser \ud835\udfd9\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nstring : String \u2192 Parser \ud835\udfd9\nstring = mapMvoid char \u2218 S\u25b9L\n\noneOf : List Char \u2192 Parser Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\noneOf\u02e2 : String \u2192 Parser Char\noneOf\u02e2 = oneOf \u2218 S\u25b9L\n\nnoneOf : List Char \u2192 Parser Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nnoneOf\u02e2 : String \u2192 Parser Char\nnoneOf\u02e2 = noneOf \u2218 S\u25b9L\n\nsomeOneOf : List Char \u2192 Parser (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeOneOf\u02e2 : String \u2192 Parser String\nsomeOneOf\u02e2 = map L\u25b9S \u2218 someOneOf \u2218 S\u25b9L\n\nsomeNoneOf : List Char \u2192 Parser (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nsomeNoneOf\u02e2 : String \u2192 Parser String\nsomeNoneOf\u02e2 = map L\u25b9S \u2218 someNoneOf \u2218 S\u25b9L\n\nmanyOneOf : List Char \u2192 Parser (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyOneOf\u02e2 : String \u2192 Parser String\nmanyOneOf\u02e2 = map L\u25b9S \u2218 manyOneOf \u2218 S\u25b9L\n\nmanyNoneOf : List Char \u2192 Parser (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n\nmanyNoneOf\u02e2 : String \u2192 Parser String\nmanyNoneOf\u02e2 = map L\u25b9S \u2218 manyNoneOf \u2218 S\u25b9L\n\nlookSatHead : (Char \u2192 Bool) \u2192 Parser \ud835\udfd9\nlookSatHead p\u1d9c = lookSat p\n    where p : List Char \u2192 Bool\n          p (x \u2237 xs) = p\u1d9c x\n          p []       = false\n\nlookHeadNoneOf : List Char \u2192 Parser \ud835\udfd9\nlookHeadNoneOf cs = lookSatHead (\u03bb c \u2192 c `notElem` cs)\n\nlookHeadNoneOf\u02e2 : String \u2192 Parser \ud835\udfd9\nlookHeadNoneOf\u02e2 = lookHeadNoneOf \u2218 S\u25b9L\n\nlookHeadSomeOf : List Char \u2192 Parser \ud835\udfd9\nlookHeadSomeOf cs = lookSatHead (\u03bb c \u2192 c `elem` cs)\n\nlookHeadSomeOf\u02e2 : String \u2192 Parser \ud835\udfd9\nlookHeadSomeOf\u02e2 = lookHeadSomeOf \u2218 S\u25b9L\n\nmodule _ {A} where\n    between : \u2200 {_V} (begin end : Parser _V) \u2192 Endo (Parser A)\n    between begin end p = begin *> p <* end\n\n    between\u1d9c : (begin end : Char) \u2192 Endo (Parser A)\n    between\u1d9c begin end = between (char begin) (char end)\n\n    between\u02e2 : (begin end : String) \u2192 Endo (Parser A)\n    between\u02e2 begin end = between (string begin) (string end)\n\n    parens   = between\u1d9c '(' ')'\n    braces   = between\u1d9c '{' '}'\n    brackets = between\u1d9c '[' ']'\n    angles   = between\u1d9c '<' '>'\n    oxford   = between\u1d9c '\u27e6' '\u27e7'\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (\u2605; \u2605_)\nopen import Data.Maybe.NP using (_\u2192?_; module M?; just; maybe\u2032; nothing; map?)\nopen import Data.Nat using (zero; suc)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Char.NP renaming (_==_ to _==\u1d9c_)\nopen import Data.List as L using (List; []; _\u2237_; null; filter)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.BoundedVec using (BoundedVec) renaming ([] to []\u1d47\u1d5b; _\u2237_ to _\u2237\u1d47\u1d5b_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bool using (Bool; if_then_else_; false)\nopen import Data.String using (String) renaming (toList to S\u25b9L; fromList to L\u25b9S)\nopen import Function.NP\nopen import Level.NP\n\nmodule Text.Parser where\n\nopen M? \u2080 using () renaming (_=<<_ to _=<<?_; _\u229b_ to _\u229b?_; join to join?)\n\nParser : \u2605 \u2192 \u2605\nParser A = List Char \u2192? (A \u00d7 List Char)\n\npure : \u2200 {A} \u2192 A \u2192 Parser A\npure a s = just (a , s)\n\ninfixr 3 _\u27e8|\u27e9_\n_\u27e8|\u27e9_ : \u2200 {A} \u2192 (f g : Parser A) \u2192 Parser A\n_\u27e8|\u27e9_ f g s = maybe\u2032 just (g s) $ f s\n\nempty : \u2200 {A} \u2192 Parser A\nempty _ = nothing\n\nsat : (Char \u2192 Bool) \u2192 Parser Char\nsat pred (x \u2237 xs) = (if pred x then pure x else empty) xs\nsat _    []       = nothing\n\neof : Parser \ud835\udfd9\neof s = if null s then pure _ [] else empty []\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 Parser A \u2192 List Char \u2192? A\nrunParser p s = map? fst (p s)\n\nrunParser\u02e2 : \u2200 {A} \u2192 Parser A \u2192 String \u2192? A\nrunParser\u02e2 p = runParser p \u2218 S\u25b9L\n\ninfixr 0 _>>=_\n_>>=_ : \u2200 {A B} \u2192 Parser A \u2192 (A \u2192 Parser B) \u2192 Parser B\n_>>=_ pa f = maybe\u2032 (uncurry f) nothing \u2218 pa\n\ninfixl 0 _=<<_\n_=<<_ : \u2200 {A B} \u2192 (A \u2192 Parser B) \u2192 Parser A \u2192 Parser B\n_=<<_ f pa = pa >>= f\n\njoin : \u2200 {A} \u2192 Parser (Parser A) \u2192 Parser A\njoin = _=<<_ id\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\nmap f pa = pa >>= pure \u2218 f\n\ninfixl 4 _\u229b_ _*>_ _<*_\n_\u229b_ : \u2200 {A B} \u2192 Parser (A \u2192 B) \u2192 Parser A \u2192 Parser B\n_\u229b_ pf pa = pf >>= \u03bb f \u2192 pa >>= \u03bb a \u2192 pure (f a)\n\n\u27ea_\u00b7_\u27eb : \u2200 {A B} \u2192 (A \u2192 B) \u2192 Parser A \u2192 Parser B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192\n            (A \u2192 B \u2192 C) \u2192 Parser A \u2192 Parser B \u2192 Parser C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D} \u2192 (A \u2192 B \u2192 C \u2192 D)\n            \u2192 Parser A \u2192 Parser B \u2192 Parser C \u2192 Parser D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_*>_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser B\np\u2081 *> p\u2082 = p\u2081 >>= const p\u2082\n\n_<*_ : \u2200 {A B} \u2192 Parser A \u2192 Parser B \u2192 Parser A\np\u2081 <* p\u2082 = p\u2081 >>= \u03bb x \u2192 p\u2082 *> pure x\n\nmapM : \u2200 {a} {A : \u2605_ a} {B} \u2192 (A \u2192 Parser B) \u2192 List A \u2192 Parser (List B)\nmapM f []       = pure []\nmapM f (x \u2237 xs) = \u27ea _\u2237_ \u00b7 f x \u00b7 mapM f xs \u27eb\n\nmapMvoid : \u2200 {a} {A : \u2605_ a} {B} \u2192 (A \u2192 Parser B) \u2192 List A \u2192 Parser \ud835\udfd9\nmapMvoid f []       = pure _\nmapMvoid f (x \u2237 xs) = f x *> mapMvoid f xs\n\nchoices : \u2200 {A} \u2192 List (Parser A) \u2192 Parser A\nchoices = L.foldr _\u27e8|\u27e9_ empty\n\nvec : \u2200 {A} n \u2192 Parser A \u2192 Parser (Vec A n)\nvec zero    p = pure []\nvec (suc n) p = \u27ea _\u2237_ \u00b7 p \u00b7 vec n p \u27eb\n\n-- These are producing bounded vectors mainly for termination\n-- reasons.\nmanyBV : \u2200 {A} n \u2192 Parser A \u2192 Parser (BoundedVec A n)\nsomeBV : \u2200 {A} n \u2192 Parser A \u2192 Parser (BoundedVec A (suc n))\n\nmanyBV 0       _ = empty\nmanyBV (suc n) p = someBV n p \u27e8|\u27e9 pure []\u1d47\u1d5b\n\nsomeBV n p = \u27ea _\u2237\u1d47\u1d5b_ \u00b7 p \u00b7 manyBV n p \u27eb\n\nmany : \u2200 {A} n \u2192 Parser A \u2192 Parser (List A)\nsome : \u2200 {A} n \u2192 Parser A \u2192 Parser (List A)\n\nmany 0       _ = empty\nmany (suc n) p = some n p \u27e8|\u27e9 pure []\n\nsome n p = \u27ea _\u2237_ \u00b7 p \u00b7 many n p \u27eb\n\nmanySat : (Char \u2192 Bool) \u2192 Parser (List Char)\nmanySat p []       = pure [] []\nmanySat p (x \u2237 xs) = if p x then map? (first (_\u2237_ x)) (manySat p xs) else pure [] (x \u2237 xs)\n\nmanySat\u02e2 : (Char \u2192 Bool) \u2192 Parser String\nmanySat\u02e2 = map L\u25b9S \u2218 manySat\n\nsomeSat : (Char \u2192 Bool) \u2192 Parser (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nsomeSat\u02e2 : (Char \u2192 Bool) \u2192 Parser String\nsomeSat\u02e2 = map L\u25b9S \u2218 someSat\n\nchar : Char \u2192 Parser \ud835\udfd9\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nstring : String \u2192 Parser \ud835\udfd9\nstring = mapMvoid char \u2218 S\u25b9L\n\noneOf : List Char \u2192 Parser Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\noneOf\u02e2 : String \u2192 Parser Char\noneOf\u02e2 = oneOf \u2218 S\u25b9L\n\nnoneOf : List Char \u2192 Parser Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nnoneOf\u02e2 : String \u2192 Parser Char\nnoneOf\u02e2 = noneOf \u2218 S\u25b9L\n\nsomeOneOf : List Char \u2192 Parser (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeOneOf\u02e2 : String \u2192 Parser String\nsomeOneOf\u02e2 = map L\u25b9S \u2218 someOneOf \u2218 S\u25b9L\n\nsomeNoneOf : List Char \u2192 Parser (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nsomeNoneOf\u02e2 : String \u2192 Parser String\nsomeNoneOf\u02e2 = map L\u25b9S \u2218 someNoneOf \u2218 S\u25b9L\n\nmanyOneOf : List Char \u2192 Parser (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyOneOf\u02e2 : String \u2192 Parser String\nmanyOneOf\u02e2 = map L\u25b9S \u2218 manyOneOf \u2218 S\u25b9L\n\nmanyNoneOf : List Char \u2192 Parser (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n\nmanyNoneOf\u02e2 : String \u2192 Parser String\nmanyNoneOf\u02e2 = map L\u25b9S \u2218 manyNoneOf \u2218 S\u25b9L\n\nlookSat : (List Char \u2192 Bool) \u2192 Parser \ud835\udfd9\nlookSat p xs = if p xs then just (_ , xs) else nothing\n\nlookSatHead : (Char \u2192 Bool) \u2192 Parser \ud835\udfd9\nlookSatHead p\u1d9c = lookSat p\n    where p : List Char \u2192 Bool\n          p (x \u2237 xs) = p\u1d9c x\n          p []       = false\n\nlookHeadNoneOf : List Char \u2192 Parser \ud835\udfd9\nlookHeadNoneOf cs = lookSatHead (\u03bb c \u2192 c `notElem` cs)\n\nlookHeadNoneOf\u02e2 : String \u2192 Parser \ud835\udfd9\nlookHeadNoneOf\u02e2 = lookHeadNoneOf \u2218 S\u25b9L\n\nlookHeadSomeOf : List Char \u2192 Parser \ud835\udfd9\nlookHeadSomeOf cs = lookSatHead (\u03bb c \u2192 c `elem` cs)\n\nlookHeadSomeOf\u02e2 : String \u2192 Parser \ud835\udfd9\nlookHeadSomeOf\u02e2 = lookHeadSomeOf \u2218 S\u25b9L\n\nmodule _ {A} where\n    between : \u2200 {_V} (begin end : Parser _V) \u2192 Endo (Parser A)\n    between begin end p = begin *> p <* end\n\n    between\u1d9c : (begin end : Char) \u2192 Endo (Parser A)\n    between\u1d9c begin end = between (char begin) (char end)\n\n    between\u02e2 : (begin end : String) \u2192 Endo (Parser A)\n    between\u02e2 begin end = between (string begin) (string end)\n\n    parens   = between\u1d9c '(' ')'\n    braces   = between\u1d9c '{' '}'\n    brackets = between\u1d9c '[' ']'\n    angles   = between\u1d9c '<' '>'\n    oxford   = between\u1d9c '\u27e6' '\u27e7'\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d6c01d7ff4aeee011a0fd395b2d05817d609e6cc","subject":"FLABloM: completing the Square","message":"FLABloM: completing the Square\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product hiding (swap)\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  infixr 60 _\u2219S_\n  infixr 50 _+S_\n\n\n  _+S_ : \u2200 {r c} \u2192 M s r c \u2192 M s r c \u2192 M s r c\n  One x     +S One x\u2081    = One (x +\u209b x\u2081)\n  Row m m\u2081  +S Row n n\u2081  = Row (m +S n) (m\u2081 +S n\u2081)\n  Col m m\u2081  +S Col n n\u2081  = Col (m +S n) (m\u2081 +S n\u2081)\n  Q m00 m01\n    m10 m11 +S Q n00 n01\n                n10 n11 = Q (m00 +S n00) (m01 +S n01)\n                            (m10 +S n10) (m11 +S n11)\n\n  _\u2219S_ : \u2200 {r m c} \u2192 M s r m \u2192 M s m c \u2192 M s r c\n  One x     \u2219S One x\u2081    = One (x \u2219\u209b x\u2081)\n  One x     \u2219S Row n n\u2081  = Row (One x \u2219S n) (One x \u2219S n\u2081)\n  Row m m\u2081  \u2219S Col n n\u2081  = m \u2219S n +S m\u2081 \u2219S n\u2081\n  Row m m\u2081  \u2219S Q n00 n01\n                n10 n11 = Row (m \u2219S n00 +S m\u2081 \u2219S n10) (m \u2219S n01 +S m\u2081 \u2219S n11)\n  Col m m\u2081  \u2219S One x     = Col (m \u2219S One x) (m\u2081 \u2219S One x)\n  Col m m\u2081  \u2219S Row n n\u2081  = Q (m \u2219S n)   (m \u2219S n\u2081)\n                            (m\u2081 \u2219S n)  (m\u2081 \u2219S n\u2081)\n  Q m00 m01\n    m10 m11 \u2219S Col n n\u2081  = Col (m00 \u2219S n +S m01 \u2219S n\u2081) (m10 \u2219S n +S m11 \u2219S n\u2081)\n  Q m00 m01\n    m10 m11 \u2219S Q n00 n01\n                n10 n11 = Q (m00 \u2219S n00 +S m01 \u2219S n10) (m00 \u2219S n01 +S m01 \u2219S n11)\n                            (m10 \u2219S n00 +S m11 \u2219S n10) (m10 \u2219S n01 +S m11 \u2219S n11)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  _\u2243S_ : {r c : Shape} \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  _\u2243S_ {L} {L} (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  _\u2243S_ {L} {(B c\u2081 c\u2082)} (Row m m\u2081) (Row n n\u2081) =\n    _\u2243S_ m n \u00d7 _\u2243S_ m\u2081 n\u2081\n  _\u2243S_ {(B r\u2081 r\u2082)} {L} (Col m m\u2081) (Col n n\u2081) =\n    _\u2243S_ m n \u00d7 _\u2243S_ m\u2081 n\u2081\n  _\u2243S_ {(B r\u2081 r\u2082)} {(B c\u2081 c\u2082)} (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    _\u2243S_ m00 n00 \u00d7\n    _\u2243S_ m01 n01 \u00d7\n    _\u2243S_ m10 n10 \u00d7\n    _\u2243S_ m11 n11\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S j \u2192 j \u2243S i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S j \u2192 j \u2243S k \u2192 i \u2243S k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  assocS : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S (x +S (y +S z))\n  assocS L L (One x) (One y) (One z) = assoc\u209b x y z\n  assocS L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    assocS L c x y z  , assocS L c\u2081 x\u2081 y\u2081 z\u2081\n  assocS (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    assocS r L x y z , assocS r\u2081 L x\u2081 y\u2081 z\u2081\n  assocS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (assocS r c x y z) , (assocS r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (assocS r\u2081 c x\u2082 y\u2082 z\u2082) , (assocS r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S y \u2192 u \u2243S v \u2192 (x +S u) \u2243S (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = assocS shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  commS : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S (y +S x)\n  commS L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  commS L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (commS L c x y) , (commS L c\u2081 x\u2081 y\u2081)\n  commS (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (commS r L x y) , (commS r\u2081 L x\u2081 y\u2081)\n  commS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    commS r c x y , commS r c\u2081 x\u2081 y\u2081 ,\n    commS r\u2081 c x\u2082 y\u2082 , commS r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = commS shape shape }\n\n\n  setoidS : {r c : Shape} \u2192 Setoid _ _\n  setoidS {r} {c} =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = _\u2243S_ {r} {c}\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) , (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 L b L x\n      ih\u2081 = zeroS\u02b3 L b\u2081 L x\u2081\n    in begin\n      Row x x\u2081 \u2219S Col (0S b L) (0S b\u2081 L)\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (x \u2219S 0S b L) +S (x\u2081 \u2219S 0S b\u2081 L)\n    \u2248\u27e8 <+S> L L {x \u2219S 0S b L} {0S L L} {x\u2081 \u2219S 0S b\u2081 L} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 L b c x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n      0S L c +S 0S L c\n    \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n      0S L c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 L b c\u2081 x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 (0S L c\u2081) \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) = zeroS\u02b3 a L L x , zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a b L x\n      ih\u2081 = zeroS\u02b3 a b\u2081 L x\u2081\n    in begin\n      x \u2219S 0S b L +S x\u2081 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L (0S _ _) \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a\u2081 b L x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 L x\u2083\n    in begin\n      x\u2082 \u2219S 0S b L +S x\u2083 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 L (0S _ _) \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a b c x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c (0S _ _) \u27e9\n      0S _ _\n    \u220e) ,\n    (let\n      ih = zeroS\u02b3 a b c\u2081 x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c\u2081 x\u2081\n    in transS a c\u2081 (<+S> a c\u2081 ih ih\u2081) (identS\u02e1 a c\u2081 (0S _ _))\n    ) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a\u2081 b c x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c +S x\u2083 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c (0S _ _) \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning setoidS\n      ih = zeroS\u02b3 a\u2081 b c\u2081 x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c\u2081 +S x\u2083 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 (0S _ _) \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  <\u2219S> : (a b c : Shape) {x y : M s a b} {u v : M s b c} \u2192\n    x \u2243S y \u2192 u \u2243S v \u2192 (x \u2219S u) \u2243S (y \u2219S v)\n  <\u2219S> L L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <\u2219> q\n  <\u2219S> L L (B c c\u2081) {One x} {One x\u2081} {Row u u\u2081} {Row v v\u2081} p (q , q\u2081) =\n    (<\u2219S> L L c {One x} {One x\u2081} {u} {v} p q) ,\n    <\u2219S> L L c\u2081 {One x} {One x\u2081} {u\u2081} {v\u2081} p q\u2081\n  <\u2219S> L (B b b\u2081) L {Row x x\u2081} {Row y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    -- Row x x\u2081 \u2219S Col u u\u2081 \u2243S Row y y\u2081 \u2219S Col v v\u2081\n    let\n      open EqReasoning setoidS\n      ih = <\u2219S> _ _ _ {x} {y} {u} {v} p q\n      ih\u2081 = <\u2219S> _ _ _ {x\u2081} {y\u2081} {u\u2081} {v\u2081} p\u2081 q\u2081\n    in begin\n      Row x x\u2081 \u2219S Col u u\u2081\n    \u2261\u27e8 refl-\u2261 \u27e9\n      x \u2219S u +S x\u2081 \u2219S u\u2081\n    \u2248\u27e8 <+S> L L {x \u2219S u} {y \u2219S v} {x\u2081 \u2219S u\u2081} {y\u2081 \u2219S v\u2081} ih ih\u2081 \u27e9\n      y \u2219S v +S y\u2081 \u2219S v\u2081\n    \u220e\n  <\u2219S> L (B b b\u2081) (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081) (q , q\u2081 , q\u2082 , q\u2083) =\n    (let\n      ih = <\u2219S> L b c p q\n      ih\u2081 = <\u2219S> L b\u2081 c p\u2081 q\u2082\n    in <+S> L c ih ih\u2081) ,\n    <+S> L c\u2081 (<\u2219S> L b c\u2081 p q\u2081) (<\u2219S> L b\u2081 c\u2081 p\u2081 q\u2083)\n  <\u2219S> (B a a\u2081) L L {Col x x\u2081} {Col y y\u2081} {One x\u2082} {One x\u2083} (p , p\u2081) q =\n    <\u2219S> a L L p q ,\n    <\u2219S> a\u2081 L L p\u2081 q\n  <\u2219S> (B a a\u2081) L (B c c\u2081) {Col x x\u2081} {Col y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <\u2219S> a L c p q ,\n    <\u2219S> a L c\u2081 p q\u2081  ,\n    <\u2219S> a\u2081 L c p\u2081 q ,\n    <\u2219S> a\u2081 L c\u2081 p\u2081 q\u2081\n  <\u2219S> (B a a\u2081) (B b b\u2081) L {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Col u u\u2081} {Col v v\u2081} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081) =\n    <+S> a L (<\u2219S> a b L p q) (<\u2219S> a b\u2081 L p\u2081 q\u2081) ,\n    <+S> a\u2081 L (<\u2219S> a\u2081 b L p\u2082 q) (<\u2219S> a\u2081 b\u2081 L p\u2083 q\u2081 )\n  <\u2219S> (B a a\u2081) (B b b\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> a c (<\u2219S> a b c p q) (<\u2219S> a b\u2081 c p\u2081 q\u2082) ,\n    <+S> a c\u2081 (<\u2219S> a b c\u2081 p q\u2081) (<\u2219S> a b\u2081 c\u2081 p\u2081 q\u2083) ,\n    <+S> a\u2081 c (<\u2219S> a\u2081 b c p\u2082 q) (<\u2219S> a\u2081 b\u2081 c p\u2083 q\u2082) ,\n    <+S> a\u2081 c\u2081 (<\u2219S> a\u2081 b c\u2081 p\u2082 q\u2081) (<\u2219S> a\u2081 b\u2081 c\u2081 p\u2083 q\u2083)\n\n  idemS : (r c : Shape) (x : M s r c) \u2192 x +S x \u2243S x\n  idemS L L (One x) = idem x\n  idemS L (B c c\u2081) (Row x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) L (Col x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (idemS _ _ x) , (idemS _ _ x\u2081 , (idemS _ _ x\u2082  , idemS _ _ x\u2083))\n\n  swapMid : {r c : Shape} (x y z w : M s r c) \u2192\n    (x +S y) +S (z +S w) \u2243S (x +S z) +S (y +S w)\n  swapMid {r} {c} x y z w =\n    let open EqReasoning setoidS\n    in begin\n      (x +S y) +S (z +S w)\n    \u2248\u27e8 assocS _ _ x y (z +S w) \u27e9\n      x +S y +S z +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (symS r c (assocS r c y z w)) \u27e9\n      x +S (y +S z) +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (<+S> r c (commS r c y z) (reflS r c)) \u27e9\n      x +S (z +S y) +S w\n    \u2248\u27e8 <+S> r c (reflS r c) (assocS r c z y w) \u27e9\n      x +S z +S y +S w\n    \u2248\u27e8 symS r c (assocS r c x z (y +S w)) \u27e9\n      (x +S z) +S (y +S w)\n    \u220e\n\n  distlHelp : \u2200 {a b b\u2081 c\u2081}\n              (x : M s a b)\n              (y z : M s b c\u2081)\n              (x\u2081 : M s a b\u2081)\n              (y\u2081 z\u2081 : M s b\u2081 c\u2081) \u2192\n            (x \u2219S (y +S z)) \u2243S (x \u2219S y +S x \u2219S z) \u2192\n            (x\u2081 \u2219S (y\u2081 +S z\u2081)) \u2243S (x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081) \u2192\n            (x \u2219S (y +S z) +S x\u2081 \u2219S (y\u2081 +S z\u2081))\n            \u2243S ((x \u2219S y +S x\u2081 \u2219S y\u2081) +S x \u2219S z +S x\u2081 \u2219S z\u2081)\n  distlHelp x y z x\u2081 y\u2081 z\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      x \u2219S (y +S z) +S x\u2081 \u2219S (y\u2081 +S z\u2081)\n    \u2248\u27e8 <+S> _ _ {x \u2219S (y +S z)} {x \u2219S y +S x \u2219S z}\n                {x\u2081 \u2219S (y\u2081 +S z\u2081)} {x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081} p q \u27e9\n      (x \u2219S y +S x \u2219S z) +S x\u2081 \u2219S y\u2081 +S x\u2081 \u2219S z\u2081\n    \u2248\u27e8 swapMid (x \u2219S y) (x \u2219S z) (x\u2081 \u2219S y\u2081) (x\u2081 \u2219S z\u2081) \u27e9\n      (x \u2219S y +S x\u2081 \u2219S y\u2081) +S x \u2219S z +S x\u2081 \u2219S z\u2081\n    \u220e\n\n  distlS : {a b c : Shape} (x : M s a b) (y z : M s b c) \u2192\n    (x \u2219S (y +S z)) \u2243S ((x \u2219S y) +S (x \u2219S z))\n  distlS {L} {L} {L} (One x) (One y) (One z) = distl x y z\n  distlS {L} {L} {B c c\u2081} (One x) (Row y y\u2081) (Row z z\u2081) =\n    distlS (One x) y z ,\n    distlS (One x) y\u2081 z\u2081\n  distlS {L} {(B b b\u2081)} {L} (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    distlHelp x y z x\u2081 y\u2081 z\u2081 (distlS x y z) (distlS x\u2081 y\u2081 z\u2081)\n  distlS {L} {(B b b\u2081)} {(B c c\u2081)} (Row x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distlHelp x y z x\u2081 y\u2082 z\u2082 (distlS x y z) (distlS x\u2081 y\u2082 z\u2082) ,\n    distlHelp x y\u2081 z\u2081 x\u2081 y\u2083 z\u2083  (distlS x y\u2081 z\u2081) (distlS x\u2081 y\u2083 z\u2083)\n  distlS {(B a a\u2081)} {L} {L} (Col x x\u2081) (One x\u2082) (One x\u2083) =\n    distlS x (One x\u2082) (One x\u2083) ,\n    distlS x\u2081 (One x\u2082) (One x\u2083)\n  distlS {(B a a\u2081)} {L} {(B c c\u2081)} (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    distlS x y z ,\n    distlS x y\u2081 z\u2081 ,\n    distlS x\u2081 y z ,\n    distlS x\u2081 y\u2081 z\u2081\n  distlS {(B a a\u2081)} {(B b b\u2081)} {L} (Q x x\u2081 x\u2082 x\u2083) (Col y y\u2081) (Col z z\u2081) =\n    distlHelp x y z x\u2081 y\u2081 z\u2081 (distlS x y z) (distlS x\u2081 y\u2081 z\u2081) ,\n    distlHelp x\u2082 y z x\u2083 y\u2081 z\u2081 (distlS x\u2082 y z) (distlS x\u2083 y\u2081 z\u2081)\n  distlS {(B a a\u2081)} {(B b b\u2081)} {(B c c\u2081)} (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distlHelp x y z x\u2081 y\u2082 z\u2082 (distlS x y z) (distlS x\u2081 y\u2082 z\u2082) ,\n    distlHelp x y\u2081 z\u2081 x\u2081 y\u2083 z\u2083 (distlS x y\u2081 z\u2081) (distlS x\u2081 y\u2083 z\u2083) ,\n    distlHelp x\u2082 y z x\u2083 y\u2082 z\u2082 (distlS x\u2082 y z) (distlS x\u2083 y\u2082 z\u2082)  ,\n    distlHelp x\u2082 y\u2081 z\u2081 x\u2083 y\u2083 z\u2083 (distlS x\u2082 y\u2081 z\u2081) (distlS x\u2083 y\u2083 z\u2083)\n\n  distrHelp : \u2200 {r m m\u2081 c : Shape}\n              (x : M s m c)\n              (y z : M s r m)\n              (x\u2081 : M s m\u2081 c)\n              (y\u2081 z\u2081 : M s r m\u2081) \u2192\n            ((y +S z) \u2219S x) \u2243S (y \u2219S x +S z \u2219S x) \u2192\n            ((y\u2081 +S z\u2081) \u2219S x\u2081) \u2243S (y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081) \u2192\n            ((y +S z) \u2219S x +S (y\u2081 +S z\u2081) \u2219S x\u2081)\n            \u2243S ((y \u2219S x +S y\u2081 \u2219S x\u2081) +S z \u2219S x +S z\u2081 \u2219S x\u2081)\n  distrHelp x y z x\u2081 y\u2081 z\u2081 p q =\n    let open EqReasoning setoidS\n    in begin\n      (y +S z) \u2219S x +S (y\u2081 +S z\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> _ _ {(y +S z) \u2219S x} {y \u2219S x +S z \u2219S x}\n                {(y\u2081 +S z\u2081) \u2219S x\u2081} {y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081} p q \u27e9\n      (y \u2219S x +S z \u2219S x) +S y\u2081 \u2219S x\u2081 +S z\u2081 \u2219S x\u2081\n    \u2248\u27e8 swapMid (y \u2219S x) (z \u2219S x) (y\u2081 \u2219S x\u2081) (z\u2081 \u2219S x\u2081) \u27e9\n      (y \u2219S x +S y\u2081 \u2219S x\u2081) +S z \u2219S x +S z\u2081 \u2219S x\u2081\n    \u220e\n\n  distrS : {r m c : Shape} (x : M s m c) (y z : M s r m) \u2192\n    ((y +S z) \u2219S x) \u2243S ((y \u2219S x) +S (z \u2219S x))\n  distrS {L} {L} {L} (One x) (One y) (One z) =\n    distr x y z\n  distrS {L} {L} {B c c\u2081} (Row x x\u2081) (One x\u2082) (One x\u2083) =\n    (distrS x (One x\u2082) (One x\u2083)) ,\n    (distrS x\u2081 (One x\u2082) (One x\u2083))\n  distrS {L} {B m m\u2081} {L} (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    distrHelp x y z x\u2081 y\u2081 z\u2081 (distrS x y z) (distrS x\u2081 y\u2081 z\u2081)\n  distrS {L} {B m m\u2081} {B c c\u2081} (Q x x\u2081 x\u2082 x\u2083) (Row y y\u2081) (Row z z\u2081) =\n    (distrHelp x y z x\u2082 y\u2081 z\u2081 (distrS x y z) (distrS x\u2082 y\u2081 z\u2081)) ,\n    (distrHelp x\u2081 y z x\u2083 y\u2081 z\u2081 (distrS x\u2081 y z) (distrS x\u2083 y\u2081 z\u2081))\n  distrS {B r r\u2081} {L} {L} (One x) (Col y y\u2081) (Col z z\u2081) =\n    distrS (One x) y z ,\n    distrS (One x) y\u2081 z\u2081\n  distrS {B r r\u2081} {L} {B c c\u2081} (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    (distrS x y z) ,\n    (distrS x\u2081 y z) ,\n    (distrS x y\u2081 z\u2081) ,\n    (distrS x\u2081 y\u2081 z\u2081)\n  distrS {B r r\u2081} {B m m\u2081} {L} (Col x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (distrHelp x y z x\u2081 y\u2081 z\u2081 (distrS x y z) (distrS x\u2081 y\u2081 z\u2081)) ,\n    (distrHelp x y\u2082 z\u2082 x\u2081 y\u2083 z\u2083 (distrS x y\u2082 z\u2082) (distrS x\u2081 y\u2083 z\u2083))\n  distrS {B r r\u2081} {B m m\u2081} {B c c\u2081} (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    distrHelp x y z x\u2082 y\u2081 z\u2081 (distrS x y z) (distrS x\u2082 y\u2081 z\u2081) ,\n    distrHelp x\u2081 y z x\u2083 y\u2081 z\u2081 (distrS x\u2081 y z) (distrS x\u2083 y\u2081 z\u2081) ,\n    distrHelp x y\u2082 z\u2082 x\u2082 y\u2083 z\u2083 (distrS x y\u2082 z\u2082) (distrS x\u2082 y\u2083 z\u2083) ,\n    distrHelp x\u2081 y\u2082 z\u2082 x\u2083 y\u2083 z\u2083 (distrS x\u2081 y\u2082 z\u2082) (distrS x\u2083 y\u2083 z\u2083)\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = _\u2243S_ {shape} {shape}\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = <\u2219S> shape shape shape\n      ; idem = idemS shape shape\n      ; distl = distlS {shape} {shape}\n      ; distr = distrS {shape} {shape}\n      }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  setoidS : (r c : Shape) \u2192 Setoid _ _\n  setoidS r c =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = \u2243S r c\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S' 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning (setoidS a L)\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 L)\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS a c)\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a c\u2081)\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c)\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c\u2081)\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S' 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) , (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02b3 L b L x\n      ih\u2081 = zeroS\u02b3 L b\u2081 L x\u2081\n    in begin\n      Row x x\u2081 \u2219S Col (0S b L) (0S b\u2081 L)\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (x \u2219S 0S b L) +S (x\u2081 \u2219S 0S b\u2081 L)\n    \u2248\u27e8 <+S> L L {x \u2219S 0S b L} {0S L L} {x\u2081 \u2219S 0S b\u2081 L} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02b3 L b c x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n      0S L c +S 0S L c\n    \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n      0S L c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02b3 L b c\u2081 x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 (0S L c\u2081) \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) = zeroS\u02b3 a L L x , zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b L x\n      ih\u2081 = zeroS\u02b3 a b\u2081 L x\u2081\n    in begin\n      x \u2219S 0S b L +S x\u2081 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L (0S _ _) \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b L x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 L x\u2083\n    in begin\n      x\u2082 \u2219S 0S b L +S x\u2083 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 L (0S _ _) \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c (0S _ _) \u27e9\n      0S _ _\n    \u220e) ,\n    (let\n      -- open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c\u2081 x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c\u2081 x\u2081\n    in transS a c\u2081 (<+S> a c\u2081 ih ih\u2081) (identS\u02e1 a c\u2081 (0S _ _))\n    -- begin\n    --   x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    -- \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n    --   0S _ _ +S 0S _ _\n    -- \u2248\u27e8 identS\u02e1 a c\u2081 (0S _ _) \u27e9\n    --   0S a c\u2081\n    -- \u220e\n    ) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c +S x\u2083 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c (0S _ _) \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c\u2081 x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c\u2081 +S x\u2083 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 (0S _ _) \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  <\u2219S> : (a b c : Shape) {x y : M s a b} {u v : M s b c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x \u2219S u) \u2243S' (y \u2219S v)\n  <\u2219S> L L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <\u2219> q\n  <\u2219S> L L (B c c\u2081) {One x} {One x\u2081} {Row u u\u2081} {Row v v\u2081} p (q , q\u2081) =\n    (<\u2219S> L L c {One x} {One x\u2081} {u} {v} p q) ,\n    <\u2219S> L L c\u2081 {One x} {One x\u2081} {u\u2081} {v\u2081} p q\u2081\n  <\u2219S> L (B b b\u2081) L {Row x x\u2081} {Row y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    -- Row x x\u2081 \u2219S Col u u\u2081 \u2243S' Row y y\u2081 \u2219S Col v v\u2081\n    let\n      open EqReasoning (setoidS _ _)\n      ih = <\u2219S> _ _ _ {x} {y} {u} {v} p q\n      ih\u2081 = <\u2219S> _ _ _ {x\u2081} {y\u2081} {u\u2081} {v\u2081} p\u2081 q\u2081\n    in begin\n      Row x x\u2081 \u2219S Col u u\u2081\n    \u2261\u27e8 refl-\u2261 \u27e9\n      x \u2219S u +S x\u2081 \u2219S u\u2081\n    \u2248\u27e8 <+S> L L {x \u2219S u} {y \u2219S v} {x\u2081 \u2219S u\u2081} {y\u2081 \u2219S v\u2081} ih ih\u2081 \u27e9\n      y \u2219S v +S y\u2081 \u2219S v\u2081\n    \u220e\n  <\u2219S> L (B b b\u2081) (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081) (q , q\u2081 , q\u2082 , q\u2083) =\n    (let\n      ih = <\u2219S> L b c p q\n      ih\u2081 = <\u2219S> L b\u2081 c p\u2081 q\u2082\n    in <+S> L c ih ih\u2081) ,\n    <+S> L c\u2081 (<\u2219S> L b c\u2081 p q\u2081) (<\u2219S> L b\u2081 c\u2081 p\u2081 q\u2083)\n  <\u2219S> (B a a\u2081) L L {Col x x\u2081} {Col y y\u2081} {One x\u2082} {One x\u2083} (p , p\u2081) q =\n    <\u2219S> a L L p q ,\n    <\u2219S> a\u2081 L L p\u2081 q\n  <\u2219S> (B a a\u2081) L (B c c\u2081) {Col x x\u2081} {Col y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <\u2219S> a L c p q ,\n    <\u2219S> a L c\u2081 p q\u2081  ,\n    <\u2219S> a\u2081 L c p\u2081 q ,\n    <\u2219S> a\u2081 L c\u2081 p\u2081 q\u2081\n  <\u2219S> (B a a\u2081) (B b b\u2081) L {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Col u u\u2081} {Col v v\u2081} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081) =\n    <+S> a L (<\u2219S> a b L p q) (<\u2219S> a b\u2081 L p\u2081 q\u2081) ,\n    <+S> a\u2081 L (<\u2219S> a\u2081 b L p\u2082 q) (<\u2219S> a\u2081 b\u2081 L p\u2083 q\u2081 )\n  <\u2219S> (B a a\u2081) (B b b\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083} (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> a c (<\u2219S> a b c p q) (<\u2219S> a b\u2081 c p\u2081 q\u2082) ,\n    <+S> a c\u2081 (<\u2219S> a b c\u2081 p q\u2081) (<\u2219S> a b\u2081 c\u2081 p\u2081 q\u2083) ,\n    <+S> a\u2081 c (<\u2219S> a\u2081 b c p\u2082 q) (<\u2219S> a\u2081 b\u2081 c p\u2083 q\u2082) ,\n    <+S> a\u2081 c\u2081 (<\u2219S> a\u2081 b c\u2081 p\u2082 q\u2081) (<\u2219S> a\u2081 b\u2081 c\u2081 p\u2083 q\u2083)\n\n  idemS : (r c : Shape) (x : M s r c) \u2192 x +S x \u2243S' x\n  idemS L L (One x) = idem x\n  idemS L (B c c\u2081) (Row x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) L (Col x x\u2081) = (idemS _ _ x) , (idemS _ _ x\u2081)\n  idemS (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (idemS _ _ x) , (idemS _ _ x\u2081 , (idemS _ _ x\u2082  , idemS _ _ x\u2083))\n\n  distlS : (a b c : Shape) (x : M s a b) (y z : M s b c) \u2192\n    (x \u2219S (y +S z)) \u2243S' ((x \u2219S y) +S (x \u2219S z))\n  distlS L L L (One x) (One y) (One z) = distl x y z\n  distlS L L (B c c\u2081) (One x) (Row y y\u2081) (Row z z\u2081) =\n    distlS L L c (One x) y z ,\n    distlS L L c\u2081 (One x) y\u2081 z\u2081\n  distlS L (B b b\u2081) L (Row x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    {!!}\n  distlS L (B b b\u2081) (B c c\u2081) (Row x x\u2081) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    {!!} ,\n    {!!}\n  distlS (B a a\u2081) L L (Col x x\u2081) (One x\u2082) (One x\u2083) = {!!}\n  distlS (B a a\u2081) L (B c c\u2081) (Col x x\u2081) (Row y y\u2081) (Row z z\u2081) = {!!}\n  distlS (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) (Col y y\u2081) (Col z z\u2081) = {!!}\n  distlS (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) = {!!}\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = <\u2219S> shape shape shape\n      ; idem = idemS shape shape\n      ; distl = {!!}\n      ; distr = {!!}\n      }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fb3086ee3e4a4e80e15116dad738330bf3812c8a","subject":"Refactor proof","message":"Refactor proof\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nopen import Relation.Nullary\n\n-- Decidable equivalence for types and contexts :-|\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x ,  rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ _ _ _ (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nopen import Relation.Nullary\n\n-- Decidable equivalence for types and contexts :-|\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , refl , p , refl , refl , dvv) with \u0393 \u225fCtx \u0393\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) | yes refl = refl\n... | no \u00acq = \u22a5-elim (\u00acq refl)\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x ,  rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ _ _ _ (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8660f3954fd983db3890db111c47c5add113b49c","subject":"Distance: \u2238-dist","message":"Distance: \u2238-dist\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Distance.agda","new_file":"lib\/Data\/Nat\/Distance.agda","new_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-comm : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-comm zero zero = idp\ndist-comm zero (suc y) = idp\ndist-comm (suc x) zero = idp\ndist-comm (suc x) (suc y) = dist-comm x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-comm (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-comm (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-comm x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-comm x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\ndist\u22610 : \u2200 x y \u2192 dist x y \u2261 0 \u2192 x \u2261 y\ndist\u22610 zero zero d\u22610 = refl\ndist\u22610 zero (suc y) ()\ndist\u22610 (suc x) zero ()\ndist\u22610 (suc x) (suc y) d\u22610 = ap suc (dist\u22610 x y d\u22610)\n\n\u2238-dist : \u2200 x y \u2192 y \u2264 x \u2192 x \u2238 y \u2261 dist x y\n\u2238-dist x .0 z\u2264n = ! dist-comm x 0\n\u2238-dist ._ ._ (s\u2264s y\u2264x) = \u2238-dist _ _ y\u2264x\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","old_contents":"open import Data.Nat.NP\nopen import Data.Nat.Properties as Props\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.Distance where\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = idp\ndist-refl (suc x) rewrite dist-refl x = idp\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = idp\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = idp\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = idp\ndist-sym zero (suc y) = idp\ndist-sym (suc x) zero = idp\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = idp\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = idp\ndist-2* (suc x) zero = idp\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = idp\ndist-asym-def (s\u2264s pf) = ap suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | ! +-assoc-comm x 1 k | dist-x-x+y\u2261y x (suc k) = idp\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = dist-sum x zero z\n                                \u2219\u2264 \u2115\u2264.reflexive (ap\u2082 _+_ (dist-sym x 0) idp)\n                                \u2219\u2264 \u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\ndist\u22610 : \u2200 x y \u2192 dist x y \u2261 0 \u2192 x \u2261 y\ndist\u22610 zero zero d\u22610 = refl\ndist\u22610 zero (suc y) ()\ndist\u22610 (suc x) zero ()\ndist\u22610 (suc x) (suc y) d\u22610 = ap suc (dist\u22610 x y d\u22610)\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f0143fea39cd9ef10ecc18c39c4bcb147cf3dc0f","subject":"agda: implement apply-term (see #25)","message":"agda: implement apply-term (see #25)\n\nThe mutual recursion is accepted by Agda.\n\nOld-commit-hash: 22f8cd2f280a41e9e8e4578726267f3f2a199d3f\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {T : Type} \u2192 \u27e6 T \u27e7 \u2192 \u27e6 \u0394-Type T \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\n-- Other problem: in fact, \u0394 t is not the nil change of t, in this context. That's a problem.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs (abs (var this xor-term var (that this)))\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) (diff-term (var this) (var this)))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n-- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb df f \u2192 app (app apply-pure-term df) f\napply-term {bool} = _xor-term_\n\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} =\n  \u03bb f\u2081 f\u2082 \u2192\n  --\u03bbx.  \u03bbdx. diff           (     f\u2081                   (apply      dx         x))                 (f\u2082                        x)\n    abs (abs (diff-term {\u03c4\u2081} (app (weaken \u227c-in-body f\u2081) (apply-term (var this) (var (that this)))) (app (weaken \u227c-in-body f\u2082) (var (that this)))))\n  where\n    \u227c-in-body = drop (\u0394-Type \u03c4) \u2022 (drop \u03c4 \u2022 \u227c-refl)\n\ndiff-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {T : Type} \u2192 \u27e6 T \u27e7 \u2192 \u27e6 \u0394-Type T \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\n-- Other problem: in fact, \u0394 t is not the nil change of t, in this context. That's a problem.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs (abs (var this xor-term var (that this)))\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n-- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb df f \u2192 app (app apply-pure-term df) f\napply-term {bool} = _xor-term_\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} =\n  \u03bb f\u2081 f\u2082 \u2192\n  --\u03bbx.  \u03bbdx. diff           (     f\u2081                   (apply      dx         x))                 (f\u2082                        x)\n    abs (abs (diff-term {\u03c4\u2081} (app (weaken \u227c-in-body f\u2081) (apply-term (var this) (var (that this)))) (app (weaken \u227c-in-body f\u2082) (var (that this)))))\n  where\n    \u227c-in-body = drop (\u0394-Type \u03c4) \u2022 (drop \u03c4 \u2022 \u227c-refl)\n\ndiff-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a9c82edfc0c23425c7a08a6da96173e46e11b460","subject":"Data.Nat: _+\u1d43_...","message":"Data.Nat: _+\u1d43_...\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits\nopen import Data.Two.Base hiding (_==_; _\u225f_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple public using (+-suc; +-*-suc)\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\npattern 5+_ x = 4+ suc x\npattern 6+_ x = 5+ suc x\npattern 7+_ x = 6+ suc x\npattern 8+_ x = 7+ suc x\npattern 9+_ x = 8+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromOp\u2082 _+_\n  renaming ( op= to += )\n  public\n\nopen FromAssocComm _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm\n  renaming ( assoc-comm to +-assoc-comm\n           ; assocs to +-assocs\n           ; interchange to +-interchange\n           ; !assoc-comm to +-!assoc-comm\n           ; comm= to +-comm=\n           ; assoc= to +-assoc=\n           ; !assoc= to +-!assoc=\n           ; inner= to +-inner=\n           ; outer= to +-outer=\n           )\n  public\n\nopen FromAssocComm _*_ \u2115\u00b0.*-assoc \u2115\u00b0.*-comm\n  renaming ( assoc-comm to *-assoc-comm\n           ; assocs to *-assocs\n           ; interchange to *-interchange\n           ; !assoc-comm to *-!assoc-comm\n           ; comm= to *-comm=\n           ; inner= to *-inner=\n           ; outer= to *-outer=\n           ; assoc= to *-assoc=\n           ; !assoc= to *-!assoc=\n           )\n  public\n\nno-<-> : \u2200 {x y} \u2192 x < y \u2192 x > y \u2192 \ud835\udfd8\nno-<-> (s\u2264s p) (s\u2264s q) = no-<-> p q\n\n\u22610\u2192\u225f0 : \u2200 n \u2192 n \u2261 0 \u2192 True (0 \u225f n)\n\u22610\u2192\u225f0 .0 idp = _\n\n\u225f0\u2192\u22610 : \u2200 n \u2192 True (0 \u225f n) \u2192 n \u2261 0\n\u225f0\u2192\u22610 zero p = idp\n\u225f0\u2192\u22610 (suc n) ()\n\n\u22620\u21d20< : \u2200 n \u2192 n \u2262 0 \u2192 0 < n\n\u22620\u21d20< zero x = \ud835\udfd8-elim (x idp)\n\u22620\u21d20< (suc n) x = s\u2264s z\u2264n\n\n\u2264\u2262\u2192< : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 y \u2192 x < y\n\u2264\u2262\u2192< z\u2264n     q = \u22620\u21d20< _ (q \u2218 !_)\n\u2264\u2262\u2192< (s\u2264s p) q = s\u2264s (\u2264\u2262\u2192< p (q \u2218 ap suc))\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n>\u2192\u2265 : \u2200 {m n} \u2192 m > n \u2192 m \u2265 n\n>\u2192\u2265 i = \u2264-pred (\u2115\u2264.trans i (\u2264-steps 1 \u2115\u2264.refl))\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\n+\u2264\u2192\u2264\u2238 : \u2200 {x} y {z} \u2192 y + x \u2264 z \u2192 x \u2264 z \u2238 y\n+\u2264\u2192\u2264\u2238 {x} y i with \u2264\u21d2\u2203 i\n+\u2264\u2192\u2264\u2238 {x} y i | k , idp =\n  x             \u2264\u27e8 \u2264-steps\u2032 k \u27e9\n  x + k         \u2261\u27e8 ! m+n\u2238n\u2261m _ y \u27e9\n  x + k + y \u2238 y \u2261\u27e8 ap (\u03bb z \u2192 z \u2238 y) (+-!assoc= {x} (\u2115\u00b0.+-comm k y)) \u27e9\n  x + y + k \u2238 y \u2261\u27e8 ap (\u03bb z \u2192 z + k \u2238 y) (\u2115\u00b0.+-comm x y) \u27e9\n  y + x + k \u2238 y \u220e\n  where open \u2264-Reasoning\n\n+-\u2238 : \u2200 x y z \u2192 x \u2261 y + z \u2192 y \u2261 x \u2238 z\n+-\u2238 .(y + z) y z idp =\n  y \u2261\u27e8 \u2115\u00b0.+-comm 0 y \u27e9\n  y + 0 \u2261\u27e8 ap (_+_ y) (! n\u2238n\u22610 z) \u27e9\n  y + (z \u2238 z) \u2261\u27e8 ! +-\u2238-assoc y (\u2115\u2264.refl {z}) \u27e9\n  (y + z) \u2238 z \u220e\n  where open \u2261-Reasoning\n\n\n+-\u2238' : \u2200 x y z \u2192 x + y \u2261 z \u2192 y \u2261 z \u2238 x\n+-\u2238' x y z e = +-\u2238 z y x (! e \u2219 \u2115\u00b0.+-comm x y)\n\n\u2261+-\u2265 : \u2200 x y z \u2192 x \u2261 y + z \u2192 x \u2265 z\n\u2261+-\u2265 .(y + z) y z idp = \u2264-steps y \u2115\u2264.refl\n\n+\u2264 : \u2200 x {y z} \u2192 x + y \u2264 z \u2192 x \u2264 z\n+\u2264 zero    i = z\u2264n\n+\u2264 (suc x) (s\u2264s i) = s\u2264s (+\u2264 x i)\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = snd \u2115\u00b0.*-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = ! snd \u2115\u00b0.+-identity (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! \u2115\u00b0.*-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\ninfixl 6 _+\u1d43_\n\n-- Accumulator based addition\n_+\u1d43_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u1d43 acc = acc\nsuc n +\u1d43 acc = n +\u1d43 suc acc\n\nopen FromOp\u2082 _+\u1d43_\n  renaming ( op= to +\u1d43= )\n  public\n\n+\u1d43-+ : \u2200 m n \u2192 m +\u1d43 n \u2261 m + n\n+\u1d43-+ zero n = idp\n+\u1d43-+ (suc m) n = +\u1d43-+ m (suc n) \u2219 +-suc m n\n\n+\u1d43-+= : \u2200 m m' {n n'} \u2192 m + n \u2261 m' + n' \u2192 m +\u1d43 n \u2261 m' +\u1d43 n'\n+\u1d43-+= m m' e = +\u1d43-+ m _ \u2219 e \u2219 ! +\u1d43-+ m' _\n\n+\u1d43-comm : Commutative _+\u1d43_\n+\u1d43-comm x y = +\u1d43-+= x y (\u2115\u00b0.+-comm x y)\n\n+\u1d43-assoc : Associative _+\u1d43_\n+\u1d43-assoc x y z = +\u1d43-+= (x +\u1d43 y) x (+= (+\u1d43-+ x y) idp \u2219 \u2115\u00b0.+-assoc x y z \u2219 ap (_+_ x) (! +\u1d43-+ y z))\n\n+\u1d430-identity : \u2200 x \u2192 x +\u1d43 0 \u2261 x\n+\u1d430-identity x = +\u1d43-comm x 0\n\nopen FromAssocComm _+\u1d43_ +\u1d43-assoc +\u1d43-comm\n  renaming ( assoc-comm to +\u1d43-assoc-comm\n           ; assocs to +\u1d43-assocs\n           ; interchange to +\u1d43-interchange\n           ; !assoc-comm to +\u1d43-!assoc-comm\n           ; comm= to +\u1d43-comm=\n           ; assoc= to +\u1d43-assoc=\n           ; !assoc= to +\u1d43-!assoc=\n           ; inner= to +\u1d43-inner=\n           ; outer= to +\u1d43-outer=\n           )\n  public\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromAssocComm _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm\n  renaming ( assoc-comm to +-assoc-comm\n           ; assocs to +-assocs\n           ; interchange to +-interchange\n           ; !assoc-comm to +-!assoc-comm\n           ; comm= to +-comm=\n           ; assoc= to +-assoc=\n           ; !assoc= to +-!assoc=\n           ; inner= to +-inner=\n           ; outer= to +-outer=\n           )\n  public\n\nopen FromAssocComm _*_ \u2115\u00b0.*-assoc \u2115\u00b0.*-comm\n  renaming ( assoc-comm to *-assoc-comm\n           ; assocs to *-assocs\n           ; interchange to *-interchange\n           ; !assoc-comm to *-!assoc-comm\n           ; comm= to *-comm=\n           ; inner= to *-inner=\n           ; outer= to *-outer=\n           ; assoc= to *-assoc=\n           ; !assoc= to *-!assoc=\n           )\n  public\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = snd \u2115\u00b0.*-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = ! snd \u2115\u00b0.+-identity (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! \u2115\u00b0.*-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n>\u2192\u2265 : \u2200 {m n} \u2192 m > n \u2192 m \u2265 n\n>\u2192\u2265 i = \u2264-pred (\u2115\u2264.trans i (\u2264-steps 1 \u2115\u2264.refl))\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e7050744d59161794231c3e29774193901898756","subject":"Nat: rename assoc-comm to +-assoc-comm","message":"Nat: rename assoc-comm to +-assoc-comm\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"daf35b78d46aec84fc693a8662b22f7a94eabbe7","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (_ , Step (FHFinal _) () _)\n  vs VLam (_ , Step (FHFinal _) () _)\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    ie IEHole ()\n    ie (INEHole x) (ENEHole e) = fe x e\n    ie (IAp i x) (EAp1 e) = ie i e\n    ie (IAp i x) (EAp2 y e) = fe x e\n    ie (ICast i) (ECastError x x\u2081) = {!!} -- todo: this is evidence that casts are busted\n    ie (ICast i) (ECastProp x) = ie i x\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    fe (FVal x) er = ve x er\n    fe (FIndet x) er = ie x er\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole f) ()\n  lem2 (IAp () _) (ITLam _)\n  lem2 (ICast x) (ITCast (FVal x\u2081) x\u2082 x\u2083) = vi x\u2081 x\n  lem2 (ICast x) (ITCast (FIndet x\u2081) x\u2082 x\u2083) = {!!} -- todo: this is evidence that casts are busted\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 f (FHFinal x) = x\n  lem4 (FVal ()) (FHAp1 x\u2082 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp1 x\u2083 sub) = lem4 x\u2082 sub\n  lem4 (FVal ()) (FHAp2 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp2 sub) = lem4 (FIndet x\u2081) sub\n  lem4 f FHEHole = f\n  lem4 f (FHNEHoleFinal x) = f\n  lem4 (FVal ()) (FHCast sub)\n  lem4 (FIndet (ICast i)) (FHCast sub) = lem4 (FIndet i) sub\n  lem4 f (FHCastFinal x) = f\n  lem4 (FVal ()) (FHNEHole y)\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole y) = lem4 x\u2081 y\n\n  lem5 : \u2200{d \u0394 d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 \u0394 \u22a2 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () _)\n  is (INEHole _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () FHEHole)\n  is (INEHole x) (d , Step (FHNEHoleFinal x\u2081) () (FHFinal x\u2083))\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHCastFinal x\u2083))\n  is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n  is (ICast a) (d' , Step (FHFinal x) q x\u2084) = lem1 x q\n  is (ICast a) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = is a (_ , Step x x\u2081 x\u2082)\n  is (ICast a) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = lem5 (FIndet (ICast a)) (FHCastFinal x) x\u2081 -- todo: this feels odd\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!} -- todo: this is evidence that casts are busted\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 a er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 a er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 a er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = lem5 a x x\u2081\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (_ , Step (FHNEHole a) x (FHNEHole x\u2082)) = es er (_ , Step a x x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (_ , Step (FHFinal _) () _)\n  vs VLam (_ , Step (FHFinal _) () _)\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    ie IEHole ()\n    ie (INEHole x) (ENEHole e) = fe x e\n    ie (IAp i x) (EAp1 e) = ie i e\n    ie (IAp i x) (EAp2 y e) = fe x e\n    ie (ICast i) (ECastError x x\u2081) = {!!} -- todo: this is evidence that casts are busted\n    ie (ICast i) (ECastProp x) = ie i x\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d \u0394} \u2192 d final \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n    fe (FVal x) er = ve x er\n    fe (FIndet x) er = ie x er\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole f) ()\n  lem2 (IAp () _) (ITLam _)\n  lem2 (ICast x) (ITCast (FVal x\u2081) x\u2082 x\u2083) = vi x\u2081 x\n  lem2 (ICast x) (ITCast (FIndet x\u2081) x\u2082 x\u2083) = {!!} -- todo: this is evidence that casts are busted\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 f (FHFinal x) = x\n  lem4 (FVal ()) (FHAp1 x\u2082 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp1 x\u2083 sub) = lem4 x\u2082 sub\n  lem4 (FVal ()) (FHAp2 sub)\n  lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp2 sub) = lem4 (FIndet x\u2081) sub\n  lem4 f FHEHole = f\n  lem4 f (FHNEHoleFinal x) = f\n  lem4 (FVal ()) (FHCast sub)\n  lem4 (FIndet (ICast i)) (FHCast sub) = lem4 (FIndet i) sub\n  lem4 f (FHCastFinal x) = f\n  lem4 (FVal ()) (FHNEHole y)\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole y) = lem4 x\u2081 y\n\n  lem5 : \u2200{d \u0394 d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 \u0394 \u22a2 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () _)\n  is (INEHole _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () FHEHole)\n  is (INEHole x) (d , Step (FHNEHoleFinal x\u2081) () (FHFinal x\u2083))\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHCastFinal x\u2083))\n  is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n  is (ICast a) (d' , Step (FHFinal x) q x\u2084) = lem1 x q\n  is (ICast a) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = is a (_ , Step x x\u2081 x\u2082)\n  is (ICast a) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = lem5 (FIndet (ICast a)) (FHCastFinal x) x\u2081 -- todo: this feels odd\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  -- cast error cases\n  es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n\n  -- ap1 cases\n  es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- ap2 cases\n  es (EAp2 a er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (EAp2 a er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  es (EAp2 a er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = lem5 a x x\u2081\n\n  -- nehole cases\n  es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ENEHole er) (_ , Step (FHNEHole a) x (FHNEHole x\u2082)) = es er (_ , Step a x x\u2082)\n  es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- castprop cases\n  es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d70a41c8626fd4c749319072995861fea914c3c6","subject":"agda\/total.agda: tinker with congruence","message":"agda\/total.agda: tinker with congruence\n\nI just managed to analyze a few more cases and solve 1-2 trivial ones,\nbut this isn't much progress.\n\nOld-commit-hash: e587758a1c8f3e4ae193e82c0c1302fa1779b70b\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n--\u2261-cong {bool} {bool} {v\u2081} f bool = bool\n\u2261-cong {bool} {bool} f bool = bool\n\u2261-cong {bool} {\u03c4\u2083 \u21d2 \u03c4\u2084} {v\u2081} {v\u2082} f (ext \u2261\u2081) = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} {bool} {bool} f bool bool = bool\n\u2261-cong\u2082 {bool} {bool} {\u03c4\u2082 \u21d2 \u03c4\u2083} f bool (ext \u2261\u2082) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {bool} f (ext \u2261\u2081) (bool) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u03c4\u2083 \u21d2 \u03c4\u2084} f (ext \u2261\u2081) (ext \u2261\u2082) = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n\u2261-cong {bool} f \u2261 = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} f \u2261\u2081 \u2261\u2082 = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cbe1ca6e8fe8ab815052c5862ecc57d1c11fd7f7","subject":"Document P.C.Validity.","message":"Document P.C.Validity.\n\nOld-commit-hash: 35c83b38382aa41cf8d8162e7cb4e14ec2cd75f0\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Dependently typed changes (Def 3.4 and 3.5, Fig. 4b and 4e)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\nopen import Level\n\n-- Extension Point: Change algebras for base types\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- We provide: change algebra for every type\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- We also provide: change environments (aka. environment changes).\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\nopen import Level\n\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- change algebra for every type\n\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- environment changes\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"613ff2cc16e4ef46a6ec57a4cb085744119d6533","subject":"fun-universe: more defaults","message":"fun-universe: more defaults\n","repos":"crypto-agda\/crypto-agda","old_file":"fun-universe.agda","new_file":"fun-universe.agda","new_contents":"module fun-universe where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nopen import Data.Bool using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bits using (Bit; Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport bintree as Tree\nopen Tree using (Tree)\nopen import data-universe\n\nrecord FunUniverse {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n\n  infix 0 _`\u2192_\n  open Universe universe public\n\n  _`\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\n  i `\u2192\u1d47 o = `Bits i `\u2192 `Bits o\n\n  `Endo : T \u2192 Set\n  `Endo A = A `\u2192 A\n\nmodule Defaults\u27e8first-part\u27e9 {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultSecondFromFirstSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n  module Default<\u00d7>FromFirstSecond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n  module DefaultFstFromSndSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module DefaultsGroup1\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open Default<\u00d7>FromFirstSecond _\u2218_ first second public\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    open DefaultFstFromSndSwap _\u2218_ snd swap public\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\n    (_\u2218_   : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    open CompositionNotations _\u2218_\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u204f fst , < fst \u204f snd , snd > >\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\n  module DefaultRewire\n    (rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o) where\n\n    rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewire fun = rewireTbl (V.tabulate fun)\n\n  module DefaultRewireTbl\n    (rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o\n    rewireTbl tbl = rewire (flip V.lookup tbl)\n\n  module LinDefaults\n    (_\u2218_   : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))) where\n\n    open CompositionNotations _\u2218_ public\n    open <\u00d7>Default _\u2218_ first swap public\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\nrecord LinRewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  infixr 9 _\u2218_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    tt\u2192[] : \u2200 {A} \u2192 `\u22a4 `\u2192 `Vec A 0\n    []\u2192tt : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\u22a4\n    <\u2237>    : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons  : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open CompositionNotations _\u2218_ public\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  infixr 3 _***_\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  fst<,tt> : \u2200 {A} \u2192 A `\u00d7 `\u22a4 `\u2192 A\n  fst<,tt> = swap \u204f snd<tt,>\n\n  fst<,[]> : \u2200 {A B} \u2192 A `\u00d7 `Vec B 0 `\u2192 A\n  fst<,[]> = second []\u2192tt \u204f fst<,tt>\n\n  snd<[],> : \u2200 {A B} \u2192 `Vec A 0 `\u00d7 B `\u2192 B\n  snd<[],> = first []\u2192tt \u204f snd<tt,>\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  swap-top : \u2200 {A B C} \u2192 A `\u00d7 B `\u00d7 C `\u2192 B `\u00d7 A `\u00d7 C\n  swap-top = assoc-first swap\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_\u00d7\u2081_> : \u2200 {A B C D E F} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 (C `\u2192 F) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 F\n  < f \u00d7\u2081 g > = assoc\u2032 \u204f < f \u00d7 g > \u204f assoc\n\n  <_\u00d7\u2082_> : \u2200 {A B C D E F} \u2192 (A `\u2192 D) \u2192 (B `\u00d7 C `\u2192 E `\u00d7 F) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (D `\u00d7 E) `\u00d7 F\n  < f \u00d7\u2082 g > = assoc \u204f < f \u00d7 g > \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f <\u2237>\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f <\u2237>\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f <\u2237>\n\n  <_\u2237[]> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237[]> = < f \u2237\u2032tt\u204f tt\u2192[] >\n\n  <\u2237[]> : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  <\u2237[]> = < id \u2237[]>\n\n  <[],_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <[], f > = <tt\u204f tt\u2192[] , f >\n\n  <_,[]> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,[]> = < f ,tt\u204f tt\u2192[] >\n\n  head<\u2237> : \u2200 {A} \u2192 `Vec A 1 `\u2192 A\n  head<\u2237> = uncons \u204f fst<,[]>\n\n  constVec\u22a4 : \u2200 {n a} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 `\u22a4 `\u2192 `Vec B n\n  constVec\u22a4 f [] = tt\u2192[]\n  constVec\u22a4 f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec\u22a4 f xs >\n\n  []\u2192[] : \u2200 {A B} \u2192 `Vec A 0 `\u2192 `Vec B 0\n  []\u2192[] = []\u2192tt \u204f tt\u2192[]\n\n  <[],[]>\u2192[] : \u2200 {A B C} \u2192 (`Vec A 0 `\u00d7 `Vec B 0) `\u2192 `Vec C 0\n  <[],[]>\u2192[] = fst<,[]> \u204f []\u2192[]\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = []\u2192[]\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst<,[]>\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd<[],>\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = <[],[]>\u2192[]\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f <\u2237>\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd<[],> \u2237[]>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f <\u2237>\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = id\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd<[],>\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f <\u2237>\n\n  <_++_> : \u2200 {m n A} \u2192 (`\u22a4 `\u2192 `Vec A m) \u2192 (`\u22a4 `\u2192 `Vec A n) \u2192\n                         `\u22a4 `\u2192 `Vec A (m + n)\n  < f ++ g > = <tt\u204f f , g > \u204f append\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <[], id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first <\u2237>\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head<\u2237>\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = []\u2192[]\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = []\u2192[]\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f <\u2237>\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate\u22a4 : \u2200 n \u2192 `\u22a4 `\u2192 `Vec `\u22a4 n\n  replicate\u22a4 _ = constVec\u22a4 (\u03bb _ \u2192 id) (V.replicate {A = \u22a4} _)\n\n  loop : \u2200 {A} \u2192 \u2115 \u2192 (A `\u2192 A) \u2192 (A `\u2192 A)\n  loop zero    _ = id\n  loop (suc n) f = f \u204f loop n f\n  -- or based on fold:\n  -- loop n f = < id ,tt\u204f replicate\u22a4 n > \u204f foldl (fst<,tt> \u204f f)\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  -- this is problematic if the space cost is 1 unit\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults\u27e8first-part\u27e9 funU\n  open Rewiring rewiring public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  -- this is problematic if the space cost is 1 unit\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  not : `Bit `\u2192 `Bit\n  not = <if id then <0b> else <1b> >\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  open Defaults\u27e8first-part\u27e9 funU public\n\n  module RewiringDefaults\n    (linRewiring : LinRewiring funU)\n    (tt       : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (dup      : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (rewire   : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    open LinRewiring linRewiring\n    open DefaultsGroup1 _\u2218_ tt snd<tt,> dup swap first public\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    <[]> = tt \u204f tt\u2192[]\n    open DefaultRewireTbl rewire public\n","old_contents":"module fun-universe where\n\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_)\nopen import Data.Bool using (if_then_else_; true; false)\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Function using (_\u2218\u2032_; flip)\nimport Data.Vec.NP as V\nimport Level as L\nopen V using (Vec; []; _\u2237_)\n\nopen import Data.Bits using (Bit; Bits; _\u2192\u1d47_; RewireTbl; 0\u207f; 1\u207f)\n\nimport bintree as Tree\nopen Tree using (Tree)\nopen import data-universe\n\nrecord FunUniverse {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n\n  infix 0 _`\u2192_\n  open Universe universe public\n\n  _`\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\n  i `\u2192\u1d47 o = `Bits i `\u2192 `Bits o\n\n  `Endo : T \u2192 Set\n  `Endo A = A `\u2192 A\n\nmodule Defaults {t} {T : Set t} (funU : FunUniverse T) where\n  open FunUniverse funU\n\n  module CompositionNotations\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)) where\n\n    infixr 1 _\u204f_\n    infixr 1 _>>>_\n\n    _\u204f_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f \u204f g = g \u2218 f\n\n    _>>>_ : \u2200 {a b c} \u2192 (a `\u2192 b) \u2192 (b `\u2192 c) \u2192 (a `\u2192 c)\n    f >>> g = f \u204f g\n\n  module DefaultsFirstSecond\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (<_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)) where\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = < id \u00d7 f >\n\n  module DefaultsGroup2\n    (id : \u2200 {A} \u2192 A `\u2192 A)\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = < f \u2218 fst , g \u2218 snd >\n\n    open DefaultsFirstSecond id <_\u00d7_> public\n\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n    dup = < id , id >\n\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap = < snd , fst >\n\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc = < fst \u2218 fst , first snd >\n\n    <tt,id> : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    <tt,id> = < tt , id >\n\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n    snd<tt,> = snd\n\n  module DefaultSecondFromFirstSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second f = swap \u204f first f \u204f swap\n\n  module Default<\u00d7>FromFirstSecond\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f second g\n\n  module DefaultFstFromSndSwap\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    fst = swap \u204f snd\n\n  module DefaultsGroup1\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4)\n    (snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A)\n    (dup   : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A)\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)) where\n\n    open CompositionNotations _\u2218_\n    open DefaultSecondFromFirstSwap _\u2218_ first swap public\n    open Default<\u00d7>FromFirstSecond _\u2218_ first second public\n\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    < f , g > = dup \u204f < f \u00d7 g >\n\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n    snd = first tt \u204f snd<tt,>\n\n    open DefaultFstFromSndSwap _\u2218_ snd swap public\n\n  module <\u00d7>Default\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)) where\n\n    open CompositionNotations _\u2218_\n\n    <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    < f \u00d7 g > = first f \u204f swap \u204f first g \u204f swap\n\n  module DefaultAssoc\u2032\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C)))\n    (swap  : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A))\n    (first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B))\n    where\n    open CompositionNotations _\u2218_\n\n    -- This definition would cost 1 unit of space instead of 0.\n    -- assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    -- assoc\u2032 = < second fst , snd \u204f snd >\n\n    assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n    assoc\u2032 = swap \u204f first swap \u204f assoc \u204f swap \u204f first swap\n\n  module DefaultCond\n    (fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B)\n    (fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A)\n    (snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B) where\n\n    cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    cond = fork fst snd\n\n   -- This definition cost 2 units of space instead of 1.\n  module DefaultFork\n    (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n    (second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C))\n    (<_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C)\n    (cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A) where\n\n    open CompositionNotations _\u2218_\n\n    fork : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n    fork f g = second < f , g > \u204f cond\n\n  module DefaultRewire\n    (rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o) where\n\n    rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewire fun = rewireTbl (V.tabulate fun)\n\n  module DefaultRewireTbl\n    (rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o) where\n\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i `\u2192\u1d47 o\n    rewireTbl tbl = rewire (flip V.lookup tbl)\n\nrecord LinRewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  infixr 9 _\u2218_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n    -- Products (group 2 primitive functions or derived from group 1)\n    first    : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n    swap     : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    assoc    : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    <tt,id>  : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n    snd<tt,> : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n\n    -- Products (derived from group 1 or 2)\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n\n    -- Vectors\n    tt\u2192[] : \u2200 {A} \u2192 `\u22a4 `\u2192 `Vec A 0\n    []\u2192tt : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\u22a4\n    <\u2237>    : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons  : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  open Defaults funU\n  open CompositionNotations _\u2218_ public\n  open DefaultAssoc\u2032 _\u2218_ assoc swap first public\n\n  infixr 3 _***_\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  <tt\u204f_,_> : \u2200 {A B C} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <tt\u204f f , g > = <tt,id> \u204f < f \u00d7 g >\n\n  <_,tt\u204f_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f ,tt\u204f g > = <tt\u204f g , f > \u204f swap\n\n  fst<,tt> : \u2200 {A} \u2192 A `\u00d7 `\u22a4 `\u2192 A\n  fst<,tt> = swap \u204f snd<tt,>\n\n  fst<,[]> : \u2200 {A B} \u2192 A `\u00d7 `Vec B 0 `\u2192 A\n  fst<,[]> = second []\u2192tt \u204f fst<,tt>\n\n  snd<[],> : \u2200 {A B} \u2192 `Vec A 0 `\u00d7 B `\u2192 B\n  snd<[],> = first []\u2192tt \u204f snd<tt,>\n\n  -- Like first, but applies on a triple associated the other way\n  assoc-first : \u2200 {A B C D E} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 C\n  assoc-first f = assoc\u2032 \u204f first f \u204f assoc\n\n  -- Like assoc-first but for second\n  assoc-second : \u2200 {A B C D E} \u2192 (B `\u00d7 C `\u2192 E `\u00d7 D) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (A `\u00d7 E) `\u00d7 D\n  assoc-second f = assoc \u204f second f \u204f assoc\u2032\n\n  <_\u00d7\u2081_> : \u2200 {A B C D E F} \u2192 (A `\u00d7 B `\u2192 D `\u00d7 E) \u2192 (C `\u2192 F) \u2192 A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E `\u00d7 F\n  < f \u00d7\u2081 g > = assoc\u2032 \u204f < f \u00d7 g > \u204f assoc\n\n  <_\u00d7\u2082_> : \u2200 {A B C D E F} \u2192 (A `\u2192 D) \u2192 (B `\u00d7 C `\u2192 E `\u00d7 F) \u2192 (A `\u00d7 B) `\u00d7 C `\u2192 (D `\u00d7 E) `\u00d7 F\n  < f \u00d7\u2082 g > = assoc \u204f < f \u00d7 g > \u204f assoc\u2032\n\n  <_`zip`_> : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  < f `zip` g > = assoc-first (assoc-second swap) \u204f < f \u00d7 g >\n\n{- This one use one unit of space\n  < f `zip` g > = < < fst \u00d7 fst > \u204f f ,\n                    < snd \u00d7 snd > \u204f g >\n-}\n\n  <_\u2237\u2032_> : \u2200 {n A B C} \u2192 (A `\u2192 C) \u2192 (B `\u2192 `Vec C n)\n                       \u2192 A `\u00d7 B `\u2192 `Vec C (1 + n)\n  < f \u2237\u2032 g > = < f \u00d7 g > \u204f <\u2237>\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons \u204f < f \u2237\u2032 g >\n\n  <tt\u204f_\u2237\u2032_> : \u2200 {n A B} \u2192 (`\u22a4 `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                       \u2192 A `\u2192 `Vec B (1 + n)\n  <tt\u204f f \u2237\u2032 g > = <tt\u204f f , g > \u204f <\u2237>\n\n  <_\u2237\u2032tt\u204f_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`\u22a4 `\u2192 `Vec B n)\n                        \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032tt\u204f g > = < f ,tt\u204f g > \u204f <\u2237>\n\n  <_\u2237[]> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec B 1\n  < f \u2237[]> = < f \u2237\u2032tt\u204f tt\u2192[] >\n\n  <[],_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Vec C 0 `\u00d7 B\n  <[], f > = <tt\u204f tt\u2192[] , f >\n\n  <_,[]> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 A `\u2192 B `\u00d7 `Vec C 0\n  < f ,[]> = < f ,tt\u204f tt\u2192[] >\n\n  head<\u2237> : \u2200 {A} \u2192 `Vec A 1 `\u2192 A\n  head<\u2237> = uncons \u204f fst<,[]>\n\n  constVec\u22a4 : \u2200 {n a} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 `\u22a4 `\u2192 `Vec B n\n  constVec\u22a4 f [] = tt\u2192[]\n  constVec\u22a4 f (x \u2237 xs) = <tt\u204f f x \u2237\u2032 constVec\u22a4 f xs >\n\n  []\u2192[] : \u2200 {A B} \u2192 `Vec A 0 `\u2192 `Vec B 0\n  []\u2192[] = []\u2192tt \u204f tt\u2192[]\n\n  <[],[]>\u2192[] : \u2200 {A B C} \u2192 (`Vec A 0 `\u00d7 `Vec B 0) `\u2192 `Vec C 0\n  <[],[]>\u2192[] = fst<,[]> \u204f []\u2192[]\n\n  <_\u229b> : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  <_\u229b> []       = []\u2192[]\n  <_\u229b> (f \u2237 fs) = < f \u2237 < fs \u229b> >\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst<,[]>\n  foldl {suc n} f = second uncons\n                  \u204f assoc\u2032\n                  \u204f first f\n                  \u204f foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons \u204f foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd<[],>\n  foldr {suc n} f = first uncons\n                  \u204f assoc\n                  \u204f second (foldr f)\n                  \u204f f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons \u204f swap \u204f foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map f = < V.replicate f \u229b>\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = <[],[]>\u2192[]\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                    \u204f < f `zip` (zipWith f) >\n                    \u204f <\u2237>\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = < snd<[],> \u2237[]>\n  snoc {suc n} = first uncons \u204f assoc \u204f second snoc \u204f <\u2237>\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = id\n  reverse {suc n} = uncons \u204f swap \u204f first reverse \u204f snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd<[],>\n  append {suc m} = first uncons\n                 \u204f assoc\n                 \u204f second append\n                 \u204f <\u2237>\n\n  <_++_> : \u2200 {m n A} \u2192 (`\u22a4 `\u2192 `Vec A m) \u2192 (`\u22a4 `\u2192 `Vec A n) \u2192\n                         `\u22a4 `\u2192 `Vec A (m + n)\n  < f ++ g > = <tt\u204f f , g > \u204f append\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = <[], id >\n  splitAt (suc m) = uncons\n                  \u204f second (splitAt m)\n                  \u204f assoc\u2032\n                  \u204f first <\u2237>\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head<\u2237>\n  folda (suc n) f = splitAt (2^ n) \u204f < folda n f \u00d7 folda n f > \u204f f\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = []\u2192[]\n  concat {n = suc n} = uncons \u204f second concat \u204f append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = []\u2192[]\n  group (suc n) k = splitAt k \u204f second (group n k) \u204f <\u2237>\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f \u204f concat\n\n  replicate\u22a4 : \u2200 n \u2192 `\u22a4 `\u2192 `Vec `\u22a4 n\n  replicate\u22a4 _ = constVec\u22a4 (\u03bb _ \u2192 id) (V.replicate {A = \u22a4} _)\n\n  loop : \u2200 {A} \u2192 \u2115 \u2192 (A `\u2192 A) \u2192 (A `\u2192 A)\n  loop zero    _ = id\n  loop (suc n) f = f \u204f loop n f\n  -- or based on fold:\n  -- loop n f = < id ,tt\u204f replicate\u22a4 n > \u204f foldl (fst<,tt> \u204f f)\n\nrecord Rewiring {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    linRewiring : LinRewiring funU\n\n    -- Unit\n    tt : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n\n    -- Products (all that comes from LinRewiring)\n    dup : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n\n    -- Vectors\n    <[]> : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n    -- * <\u2237> and uncons come from LinRewiring\n\n    -- Products (group 1 primitive functions or derived from group 2)\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd   : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    rewire    : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i `\u2192\u1d47 o\n    rewireTbl : \u2200 {i o} \u2192 RewireTbl i o    \u2192 i `\u2192\u1d47 o\n\n  open LinRewiring linRewiring public\n\n  proj : \u2200 {A} \u2192 Bit \u2192 (A `\u00d7 A) `\u2192 A\n  proj true  = fst\n  proj false = snd\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons \u204f fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons \u204f snd\n\n  constVec : \u2200 {n a _\u22a4} {A : Set a} {B} \u2192 (A \u2192 `\u22a4 `\u2192 B) \u2192 Vec A n \u2192 _\u22a4 `\u2192 `Vec B n\n  constVec f vec = tt \u204f constVec\u22a4 f vec\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = <[]>\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail \u204f drop m\n\n  msb : \u2200 m {n} \u2192 (m + n) `\u2192\u1d47 m\n  msb m = take m\n\n  lsb : \u2200 {n} k \u2192 (n + k) `\u2192\u1d47 k\n  lsb {n} _ = drop n\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = <[]>\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail \u204f last\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = <[]>\n  replicate {suc n} = < id , replicate > \u204f <\u2237>\n\n  -- this is problematic if the space cost is 1 unit\n  constBits\u2032 : \u2200 {n A} \u2192 Bits n \u2192 (A `\u00d7 A) `\u2192 `Vec A n\n  constBits\u2032 [] = <[]>\n  constBits\u2032 (b \u2237 xs) = dup \u204f < proj b \u2237\u2032 constBits\u2032 xs >\n\nrecord FunOps {t} {T : Set t} (funU : FunUniverse T) : Set t where\n  constructor mk\n  open FunUniverse funU\n\n  field\n    rewiring : Rewiring funU\n\n    -- Bit\n    <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n    cond      : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n    fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n\n    -- Products\n    -- * <_\u00d7_>; first; second; swap; assoc; <tt,id>; snd<tt,> come from LinRewiring\n    -- * dup; <_,_>; fst; snd come from Rewiring\n\n    -- Vectors\n    -- <[]>; <\u2237>; uncons come from Rewiring\n\n  open Defaults funU\n  open Rewiring rewiring public\n\n  infixr 3 _&&&_\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  constBit : \u2200 {_\u22a4} \u2192 Bit \u2192 _\u22a4 `\u2192 `Bit\n  constBit b = if b then <1b> else <0b>\n\n  <if_then_else_> : \u2200 {A B} (b : A `\u2192 `Bit) (f g : A `\u2192 B) \u2192 A `\u2192 B\n  <if b then if-1 else if-0 > = < b , id > \u204f fork if-0 if-1\n\n  -- Notice that this one costs 1 unit of space.\n  dup\u204f<_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                         \u2192 A `\u2192 `Vec B (1 + n)\n  dup\u204f< f \u2237\u2032 g > = dup \u204f < f \u2237\u2032 g >\n\n  <0,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <0, f > = <tt\u204f <0b> , f >\n\n  <1,_> : \u2200 {A B} \u2192 (A `\u2192 B) \u2192 A `\u2192 `Bit `\u00d7 B\n  <1, f > = <tt\u204f <1b> , f >\n\n  <0,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <0,> = <0, id >\n\n  <1,> : \u2200 {A} \u2192 A `\u2192 `Bit `\u00d7 A\n  <1,> = <1, id >\n\n  <0,1> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit `\u00d7 `Bit\n  <0,1> = <0, <1b> >\n\n  <0\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <0\u2237 f > = <tt\u204f <0b> \u2237\u2032 f >\n\n  <1\u2237_> : \u2200 {n A} \u2192 (A `\u2192 `Bits n) \u2192 A `\u2192 `Bits (1 + n)\n  <1\u2237 f > = <tt\u204f <1b> \u2237\u2032 f >\n\n  <0\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <0\u2237> = <0\u2237 id >\n\n  <1\u2237> : \u2200 {n} \u2192 n `\u2192\u1d47 (1 + n)\n  <1\u2237> = <1\u2237 id >\n\n  constBits : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits = constVec constBit\n\n  <0\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <0\u207f> = constBits 0\u207f\n\n  <1\u207f> : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Bits n\n  <1\u207f> = constBits 1\u207f\n\n  -- this is problematic if the space cost is 1 unit\n  constBits\u2032\u2032 : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 `\u2192 `Bits n\n  constBits\u2032\u2032 bs = <0,1> \u204f constBits\u2032 bs\n\n  `Maybe : T \u2192 T\n  `Maybe A = `Bit `\u00d7 A\n\n  <nothing> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <nothing> = <0,>\n\n  <just> : \u2200 {A} \u2192 A `\u2192 `Maybe A\n  <just> = <1,>\n\n  <is-just?_\u2236_> : \u2200 {A B C} \u2192 (f : A `\u00d7 B `\u2192 C) (g : B `\u2192 C) \u2192 `Maybe A `\u00d7 B `\u2192 C\n  <is-just? f \u2236 g > = <if fst \u204f fst then first snd \u204f f else snd \u204f g >\n\n  _\u2223?_ : \u2200 {A} \u2192 `Maybe A `\u00d7 `Maybe A `\u2192 `Maybe A\n  _\u2223?_ = <is-just? fst \u204f <just> \u2236 id >\n\n  _`\u2192?_ : T \u2192 T \u2192 Set\n  A `\u2192? B = A `\u2192 `Maybe B\n\n  search : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 A\n  search {zero}  _  f = <[]> \u204f f\n  search {suc n} op f = <tt\u204f search op (f \u2218 <0\u2237>) , search op (f \u2218 <1\u2237>) > \u204f op\n\n  find? : \u2200 {n A} \u2192 (`Bits n `\u2192? A) \u2192 `\u22a4 `\u2192? A\n  find? = search _\u2223?_\n\n  findB : \u2200 {n} \u2192 (`Bits n `\u2192 `Bit) \u2192 `\u22a4 `\u2192? `Bits n\n  findB pred = find? <if pred then <just> else <nothing> >\n\n  fromTree : \u2200 {n A} \u2192 Tree (`\u22a4 `\u2192 A) n \u2192 `Bits n `\u2192 A\n  fromTree (Tree.leaf x) = tt \u204f x\n  fromTree (Tree.fork t\u2080 t\u2081) = uncons \u204f fork (fromTree t\u2080) (fromTree t\u2081)\n\n  fromFun : \u2200 {n A} \u2192 (Bits n \u2192 `\u22a4 `\u2192 A) \u2192 `Bits n `\u2192 A\n  fromFun = fromTree \u2218\u2032 Tree.fromFun\n\n  fromBitsFun : \u2200 {i o} \u2192 (i \u2192\u1d47 o) \u2192 i `\u2192\u1d47 o\n  fromBitsFun f = fromFun (constBits \u2218\u2032 f)\n\n  not : `Bit `\u2192 `Bit\n  not = <if id then <0b> else <1b> >\n\n  <xor> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <xor> = fork id not\n\n  <or> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <or> = fork id <1b>\n\n  <and> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <and> = fork <0b> id\n\n  <==\u1d47> : `Bit `\u00d7 `Bit `\u2192 `Bit\n  <==\u1d47> = <xor> \u204f not\n\n  <==> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bit\n  <==> {zero}  = <1b>\n  <==> {suc n} = < uncons \u00d7 uncons > \u204f < <==\u1d47> `zip` <==> {n} > \u204f <or>\n\n  <\u2295> : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n  <\u2295> = zipWith <xor>\n\n  vnot : \u2200 {n} \u2192 `Endo (`Bits n)\n  vnot = map not\n\n  allBits : \u2200 n \u2192 `\u22a4 `\u2192 `Vec (`Bits n) (2^ n)\n  allBits zero    = < <[]> \u2237[]>\n  allBits (suc n) = < bs \u204f map <0\u2237> ++ bs \u204f map <1\u2237> >\n    where bs = allBits n\n\n  sucBCarry : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (1 + n)\n  sucBCarry {zero}  = < <0b> \u2237[]>\n  sucBCarry {suc n} = uncons\n                    \u204f fork <0\u2237 sucBCarry >\n                          (sucBCarry \u204f uncons \u204f fork <0\u2237 <1\u2237> > <1\u2237 <0\u2237> >)\n\n  sucB : \u2200 {n} \u2192 `Bits n `\u2192 `Bits n\n  sucB = sucBCarry \u204f tail\n\n  lookupTbl : \u2200 {n A} \u2192 `Bits n `\u00d7 `Vec A (2^ n) `\u2192 A\n  lookupTbl {zero} = snd \u204f head\n  lookupTbl {suc n}\n    = first uncons\n    \u204f assoc\n    \u204f fork (second (take (2^ n)) \u204f lookupTbl)\n           (second (drop (2^ n)) \u204f lookupTbl)\n\n  funFromTbl : \u2200 {n A} \u2192 Vec (`\u22a4 `\u2192 A) (2^ n) \u2192 (`Bits n `\u2192 A)\n  funFromTbl {zero} (x \u2237 []) = tt \u204f x\n  funFromTbl {suc n} tbl\n    = uncons \u204f fork (funFromTbl (V.take (2^ n) tbl))\n                    (funFromTbl (V.drop (2^ n) tbl))\n\n  tblFromFun : \u2200 {n A} \u2192 (`Bits n `\u2192 A) \u2192 `\u22a4 `\u2192 `Vec A (2^ n)\n  tblFromFun {zero}  f = < <[]> \u204f f \u2237[]>\n  tblFromFun {suc n} f = < tblFromFun (<0\u2237> \u204f f) ++\n                           tblFromFun (<1\u2237> \u204f f) >\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = <[]>\n    tabulate {suc n} f = <tt\u204f fz \u204f f \u2237\u2032 tabulate (fs \u204f f) >\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst \u204f elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail \u204f lookup)\n\n    allFin : \u2200 {n _\u22a4} \u2192 _\u22a4 `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"50819fad2a9bd8093c7722e7d88f99deb1ca7fc2","subject":"Move parameter to module level.","message":"Move parameter to module level.\n\nOld-commit-hash: 46e0ad74c26c053c33e0317365ac3c7047809529\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  lift-apply :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff-base apply-base {base \u03b9} = diff-base , apply-base\n  lift-diff-apply diff-base apply-base {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply-base {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply-base {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff-base apply-base = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n\n  lift-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff-base apply-base = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5df1a818f107f4e8eea6cba0bb984f2ed984ef92","subject":"Desc stratified model: descDChoice implicit.","message":"Desc stratified model: descDChoice implicit.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : (l : Level) -> Set l -> DescDConst -> IDesc Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc Unit\ndescD x I = sigma DescDConst (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d109d6a26c2ede68c53dd2c7f83ec7f6e6508725","subject":"Agda: remove redundant 'open' in Syntax.Context.Popl14","message":"Agda: remove redundant 'open' in Syntax.Context.Popl14\n\nhttps:\/\/github.com\/ps-mr\/ilc\/commit\/02bb5ef8bf31b80d9b640a440b4ec1801aa500fe#commitcomment-3645919\n\nOld-commit-hash: 97db5813bea3cc8a973f051efb3b6d3f075681d0\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Context\/Popl14.agda","new_file":"Syntax\/Context\/Popl14.agda","new_contents":"module Syntax.Context.Popl14 where\n\n-- Context specific to Popl14 version of the calculus\n--\n-- This module exports Syntax.Context specialized to\n-- Syntax.Type.Popl14.Type and declares \u0394Context appropriate\n-- to Popl14 version of the incrementalization system.\n-- This \u0394Context may not make sense for other systems.\n\nopen import Syntax.Type.Popl14 public\nimport Syntax.Context Type as Ctx\nopen Ctx public hiding (lift)\n\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- Aliasing of lemmas in Calculus Popl14\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 = \u227c-refl\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\n-- Aliasing of weakening of variables\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar = Ctx.lift\n","old_contents":"module Syntax.Context.Popl14 where\n\n-- Context specific to Popl14 version of the calculus\n--\n-- This module exports Syntax.Context specialized to\n-- Syntax.Type.Popl14.Type and declares \u0394Context appropriate\n-- to Popl14 version of the incrementalization system.\n-- This \u0394Context may not make sense for other systems.\n\nopen import Syntax.Type.Popl14 public\nopen import Syntax.Context Type as Ctx\nopen Ctx public hiding (lift)\n\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- Aliasing of lemmas in Calculus Popl14\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 = \u227c-refl\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\n-- Aliasing of weakening of variables\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar = Ctx.lift\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"921ed6233ae3c1c6f762c1de19bc602bf7730f05","subject":"Fewer object types in ANormalUntyped","message":"Fewer object types in ANormalUntyped\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/ANormalUntyped.agda","new_file":"Thesis\/ANormalUntyped.agda","new_contents":"module Thesis.ANormalUntyped where\n\nopen import Data.Nat.Base\nopen import Data.Integer.Base\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Using a record gives an eta rule saying that all types are equal.\nrecord Type : Set where\n  constructor Uni\n\nopen import Base.Syntax.Context Type public\ndata Term (\u0393 : Context) : Set where\n  var : (x : Var \u0393 Uni) \u2192\n    Term \u0393\n  lett : (f : Var \u0393 Uni) \u2192 (x : Var \u0393 Uni) \u2192 Term (Uni \u2022 \u0393) \u2192 Term \u0393\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u2192\n  Term \u0393\u2082\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Fun (\u0393 : Context) : Set where\n  term : Term \u0393 \u2192 Fun \u0393\n  abs : \u2200 {\u03c3} \u2192 Fun (\u03c3 \u2022 \u0393) \u2192 Fun \u0393\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u2192\n  Fun \u0393\u2082\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393} \u2192 (t : Fun (Uni \u2022 \u0393)) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val Uni\n  intV : \u2200 (n : \u2124) \u2192 Val Uni\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\ndata ErrVal : Set where\n  Done : (v : Val Uni) \u2192 ErrVal\n  Error : ErrVal\n  TimeOut : ErrVal\n\n_>>=_ : ErrVal \u2192 (Val Uni \u2192 ErrVal) \u2192 ErrVal\nDone v >>= f = f v\nError >>= f = Error\nTimeOut >>= f = TimeOut\n\nRes : Set\nRes = \u2115 \u2192 ErrVal\n\neval-fun : \u2200 {\u0393} \u2192 Fun \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\neval-term : \u2200 {\u0393} \u2192 Term \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res\n\napply : Val Uni \u2192 Val Uni \u2192 Res\napply (closure f \u03c1) a n = eval-fun f (a \u2022 \u03c1) n\napply (intV _) a n = Error\n\neval-term t \u03c1 zero = TimeOut\neval-term (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval-term (lett f x t) \u03c1 (suc n) = apply (\u27e6 f \u27e7Var \u03c1) (\u27e6 x \u27e7Var \u03c1) n >>= (\u03bb v \u2192 eval-term t (v \u2022 \u03c1) n)\n\neval-fun (term t) \u03c1 n = eval-term t \u03c1 n\neval-fun (abs f) \u03c1 n = Done (closure f \u03c1)\n\n-- Erasure from typed to untyped values.\nimport Thesis.ANormalBigStep as T\n\nerase-type : T.Type \u2192 Type\nerase-type _ = Uni\n\nerase-val : \u2200 {\u03c4} \u2192 T.Val \u03c4 \u2192 Val (erase-type \u03c4)\n\nerase-errVal : \u2200 {\u03c4} \u2192 T.ErrVal \u03c4 \u2192 ErrVal\nerase-errVal (T.Done v) = Done (erase-val v)\nerase-errVal T.Error = Error\nerase-errVal T.TimeOut = TimeOut\n\nerase-res : \u2200 {\u03c4} \u2192 T.Res \u03c4 \u2192 Res\nerase-res r n = erase-errVal (r n)\n\nerase-ctx : T.Context \u2192 Context\nerase-ctx \u2205 = \u2205\nerase-ctx (\u03c4 \u2022 \u0393) = erase-type \u03c4 \u2022 (erase-ctx \u0393)\n\nerase-env : \u2200 {\u0393} \u2192 T.Op.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 erase-ctx \u0393 \u27e7Context\nerase-env \u2205 = \u2205\nerase-env (v \u2022 \u03c1) = erase-val v \u2022 erase-env \u03c1\n\nerase-var : \u2200 {\u0393 \u03c4} \u2192 T.Var \u0393 \u03c4 \u2192 Var (erase-ctx \u0393) (erase-type \u03c4)\nerase-var T.this = this\nerase-var (T.that x) = that (erase-var x)\n\nerase-term : \u2200 {\u0393 \u03c4} \u2192 T.Term \u0393 \u03c4 \u2192 Term (erase-ctx \u0393)\nerase-term (T.var x) = var (erase-var x)\nerase-term (T.lett f x t) = lett (erase-var f) (erase-var x) (erase-term t)\n\nerase-fun : \u2200 {\u0393 \u03c4} \u2192 T.Fun \u0393 \u03c4 \u2192 Fun (erase-ctx \u0393)\nerase-fun (T.term x) = term (erase-term x)\nerase-fun (T.abs f) = abs (erase-fun f)\n\nerase-val (T.closure t \u03c1) = closure (erase-fun t) (erase-env \u03c1)\nerase-val (T.intV n) = intV n\n\n-- Different erasures commute.\nerasure-commute-var : \u2200 {\u0393 \u03c4} (x : T.Var \u0393 \u03c4) \u03c1 \u2192\n  erase-val (T.Op.\u27e6 x \u27e7Var \u03c1) \u2261 \u27e6 erase-var x \u27e7Var (erase-env \u03c1)\nerasure-commute-var T.this (v \u2022 \u03c1) = refl\nerasure-commute-var (T.that x) (v \u2022 \u03c1) = erasure-commute-var x \u03c1\n\nerase-bind : \u2200 {\u03c3 \u03c4 \u0393} a (t : T.Term (\u03c3 \u2022 \u0393) \u03c4) \u03c1 n \u2192 erase-errVal (a T.>>= (\u03bb v \u2192 T.eval-term t (v \u2022 \u03c1) n)) \u2261 erase-errVal a >>= (\u03bb v \u2192 eval-term (erase-term t) (v \u2022 erase-env \u03c1) n)\n\nerasure-commute-fun : \u2200 {\u0393 \u03c4} (t : T.Fun \u0393 \u03c4) \u03c1 n \u2192\n  erase-errVal (T.eval-fun t \u03c1 n) \u2261 eval-fun (erase-fun t) (erase-env \u03c1) n\n\nerasure-commute-apply : \u2200 {\u03c3 \u03c4} (f : T.Val (\u03c3 T.\u21d2 \u03c4)) a n \u2192 erase-errVal (T.apply f a n) \u2261 apply (erase-val f) (erase-val a) n\nerasure-commute-apply {\u03c3} (T.closure t \u03c1) a n = erasure-commute-fun t (a \u2022 \u03c1) n\n\nerasure-commute-term : \u2200 {\u0393 \u03c4} (t : T.Term \u0393 \u03c4) \u03c1 n \u2192\n  erase-errVal (T.eval-term t \u03c1 n) \u2261 eval-term (erase-term t) (erase-env \u03c1) n\n\nerasure-commute-fun (T.term t) \u03c1 n = erasure-commute-term t \u03c1 n\nerasure-commute-fun (T.abs t) \u03c1 n = refl\n\nerasure-commute-term t \u03c1 zero = refl\nerasure-commute-term (T.var x) \u03c1 (\u2115.suc n) = cong Done (erasure-commute-var x \u03c1)\nerasure-commute-term (T.lett f x t) \u03c1 (\u2115.suc n) rewrite erase-bind (T.apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n) t \u03c1 n | erasure-commute-apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n | erasure-commute-var f \u03c1 | erasure-commute-var x \u03c1 = refl\n\nerase-bind (T.Done v) t \u03c1 n = erasure-commute-term t (v \u2022 \u03c1) n\nerase-bind T.Error t \u03c1 n = refl\nerase-bind T.TimeOut t \u03c1 n = refl\n","old_contents":"module Thesis.ANormalUntyped where\n\nopen import Data.Nat.Base\nopen import Data.Integer.Base\nopen import Relation.Binary.PropositionalEquality\n\n{- Typed deBruijn indexes for untyped languages. -}\n\n-- Make this abstract again *and* add needed equations.\nabstract\n  data Type : Set where\n    Uni : Type\n  uni = Uni\n\n  -- Allow exposing that there's a single type where needed.\n  isUni : \u2200 \u03c4 \u2192 \u03c4 \u2261 uni\n  isUni Uni = refl\n\n-- This equation allows bypassing more checks. Otherwise we'd need to use isUni\n-- even more often.\n_\u21d2_ : Type \u2192 Type \u2192 Type\n_\u21d2_ \u03c3 \u03c4 = \u03c4\n\nopen import Base.Syntax.Context Type public\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set where\n  var : (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3 \u03c4\u2081} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4 \u2192 Term \u0393 \u03c4\n\n{-\nderiveC \u0394 (lett f x t) = dlett df x dx\n-}\n\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n-- data \u0394CTerm (\u0393 : Context) (\u03c4 : Type) (\u0394 : Context) : Set where\n--   cvar : (x : Var \u0393 \u03c4) \u0394 \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n--   clett : \u2200 {\u03c3 \u03c4\u2081 \u03ba} \u2192 (f : Var \u0393 (\u03c3 \u21d2 \u03c4\u2081)) \u2192 (x : Var \u0393 \u03c3) \u2192\n--     \u0394CTerm (\u03c4\u2081 \u2022 \u0393) \u03c4 (? \u2022 \u0394) \u2192\n--     \u0394CTerm \u0393 \u03c4 \u0394\n\nweaken-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken-term \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken-term \u0393\u2081\u227c\u0393\u2082 (lett f x t) = lett (weaken-var \u0393\u2081\u227c\u0393\u2082 f) (weaken-var \u0393\u2081\u227c\u0393\u2082 x) (weaken-term (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Fun (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  term : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Fun \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4} \u2192 Fun (\u03c3 \u2022 \u0393) \u03c4 \u2192 Fun \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken-fun : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Fun \u0393\u2081 \u03c4 \u2192\n  Fun \u0393\u2082 \u03c4\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (term x) = term (weaken-term \u0393\u2081\u227c\u0393\u2082 x)\nweaken-fun \u0393\u2081\u227c\u0393\u2082 (abs f) = abs (weaken-fun (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) f)\n\ndata Val : Type \u2192 Set\n-- data Val (\u03c4 : Type) : Set\n\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\n-- data Val (\u03c4 : Type) where\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Fun (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 {\u03c4} (n : \u2124) \u2192 Val \u03c4\n-- \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n-- \u27e6 lett f x t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (\u27e6 f \u27e7Var \u03c1 (\u27e6 x \u27e7Var \u03c1) \u2022 \u03c1)\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 ErrVal \u03c3 \u2192 (Val \u03c3 \u2192 ErrVal \u03c4) \u2192 ErrVal \u03c4\nDone v >>= f = f v\nError >>= f = Error\nTimeOut >>= f = TimeOut\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\neval-fun : \u2200 {\u0393 \u03c4} \u2192 Fun \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\neval-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply {\u03c3} (closure {\u03c3 = \u03c3\u2081} f \u03c1) a n with isUni \u03c3 | isUni \u03c3\u2081\napply {.uni} (closure {_} {.uni} f \u03c1) a n | refl | refl = eval-fun f (a \u2022 \u03c1) n\napply (intV _) a n = Error\n\neval-term t \u03c1 zero = TimeOut\neval-term (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval-term (lett f x t) \u03c1 (suc n) = apply (\u27e6 f \u27e7Var \u03c1) (\u27e6 x \u27e7Var \u03c1) n >>= (\u03bb v \u2192 eval-term t (v \u2022 \u03c1) n)\n\neval-fun (term t) \u03c1 n = eval-term t \u03c1 n\neval-fun (abs f) \u03c1 n = Done (closure f \u03c1)\n\n-- Erasure from typed to untyped values.\nimport Thesis.ANormalBigStep as T\n\nerase-type : T.Type \u2192 Type\nerase-type _ = uni\n\nerase-val : \u2200 {\u03c4} \u2192 T.Val \u03c4 \u2192 Val (erase-type \u03c4)\n\nerase-errVal : \u2200 {\u03c4} \u2192 T.ErrVal \u03c4 \u2192 ErrVal (erase-type \u03c4)\nerase-errVal (T.Done v) = Done (erase-val v)\nerase-errVal T.Error = Error\nerase-errVal T.TimeOut = TimeOut\n\nerase-res : \u2200 {\u03c4} \u2192 T.Res \u03c4 \u2192 Res (erase-type \u03c4)\nerase-res r n = erase-errVal (r n)\n\nerase-ctx : T.Context \u2192 Context\nerase-ctx \u2205 = \u2205\nerase-ctx (\u03c4 \u2022 \u0393) = erase-type \u03c4 \u2022 (erase-ctx \u0393)\n\nerase-env : \u2200 {\u0393} \u2192 T.Op.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 erase-ctx \u0393 \u27e7Context\nerase-env \u2205 = \u2205\nerase-env (v \u2022 \u03c1) = erase-val v \u2022 erase-env \u03c1\n\nerase-var : \u2200 {\u0393 \u03c4} \u2192 T.Var \u0393 \u03c4 \u2192 Var (erase-ctx \u0393) (erase-type \u03c4)\nerase-var T.this = this\nerase-var (T.that x) = that (erase-var x)\n\nerase-term : \u2200 {\u0393 \u03c4} \u2192 T.Term \u0393 \u03c4 \u2192 Term (erase-ctx \u0393) (erase-type \u03c4)\nerase-term (T.var x) = var (erase-var x)\nerase-term (T.lett f x t) = lett (erase-var f) (erase-var x) (erase-term t)\n\nerase-fun : \u2200 {\u0393 \u03c4} \u2192 T.Fun \u0393 \u03c4 \u2192 Fun (erase-ctx \u0393) (erase-type \u03c4)\nerase-fun (T.term x) = term (erase-term x)\nerase-fun (T.abs f) = abs (erase-fun f)\n\nerase-val (T.closure t \u03c1) = closure (erase-fun t) (erase-env \u03c1)\nerase-val (T.intV n) = intV n\n\n-- Different erasures commute.\nerasure-commute-var : \u2200 {\u0393 \u03c4} (x : T.Var \u0393 \u03c4) \u03c1 \u2192\n  erase-val (T.Op.\u27e6 x \u27e7Var \u03c1) \u2261 \u27e6 erase-var x \u27e7Var (erase-env \u03c1)\nerasure-commute-var T.this (v \u2022 \u03c1) = refl\nerasure-commute-var (T.that x) (v \u2022 \u03c1) = erasure-commute-var x \u03c1\n\nerase-bind : \u2200 {\u03c3 \u03c4 \u0393} a (t : T.Term (\u03c3 \u2022 \u0393) \u03c4) \u03c1 n \u2192 erase-errVal (a T.>>= (\u03bb v \u2192 T.eval-term t (v \u2022 \u03c1) n)) \u2261 erase-errVal a >>= (\u03bb v \u2192 eval-term (erase-term t) (v \u2022 erase-env \u03c1) n)\n\nerasure-commute-fun : \u2200 {\u0393 \u03c4} (t : T.Fun \u0393 \u03c4) \u03c1 n \u2192\n  erase-errVal (T.eval-fun t \u03c1 n) \u2261 eval-fun (erase-fun t) (erase-env \u03c1) n\n\nerasure-commute-apply : \u2200 {\u03c3 \u03c4} (f : T.Val (\u03c3 T.\u21d2 \u03c4)) a n \u2192 erase-errVal (T.apply f a n) \u2261 apply (erase-val f) (erase-val a) n\nerasure-commute-apply {\u03c3} (T.closure t \u03c1) a n with isUni (erase-type \u03c3)\nerasure-commute-apply {\u03c3} (T.closure t \u03c1) a n | refl = erasure-commute-fun t (a \u2022 \u03c1) n\n\nerasure-commute-term : \u2200 {\u0393 \u03c4} (t : T.Term \u0393 \u03c4) \u03c1 n \u2192\n  erase-errVal (T.eval-term t \u03c1 n) \u2261 eval-term (erase-term t) (erase-env \u03c1) n\n\nerasure-commute-fun (T.term t) \u03c1 n = erasure-commute-term t \u03c1 n\nerasure-commute-fun (T.abs t) \u03c1 n = refl\n\nerasure-commute-term t \u03c1 zero = refl\nerasure-commute-term (T.var x) \u03c1 (\u2115.suc n) = cong Done (erasure-commute-var x \u03c1)\nerasure-commute-term (T.lett f x t) \u03c1 (\u2115.suc n) rewrite erase-bind (T.apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n) t \u03c1 n | erasure-commute-apply (T.Op.\u27e6 f \u27e7Var \u03c1) (T.Op.\u27e6 x \u27e7Var \u03c1) n | erasure-commute-var f \u03c1 | erasure-commute-var x \u03c1 = refl\n\nerase-bind (T.Done v) t \u03c1 n = erasure-commute-term t (v \u2022 \u03c1) n\nerase-bind T.Error t \u03c1 n = refl\nerase-bind T.TimeOut t \u03c1 n = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b1880f17f5e597a6f515f2e5f0a807b456ab1805","subject":"Updated a draft.","message":"Updated a draft.\n\nIgnore-this: d19bf16a7f9fa8bcde036ed98abcdd17\n\ndarcs-hash:20120320165307-3bd4e-2abe5f33e84622e1ff73dfdc8d456e1b8ceaa175.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Conat\/PropertiesI.agda","new_file":"Draft\/FOTC\/Data\/Conat\/PropertiesI.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT 20 March 2012.\n\nmodule Draft.FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.Nat\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} h = \u2248N-gfp\u2082 P helper\u2081 helper\u2082\n  where\n  P : D \u2192 D \u2192 Set\n  P a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n\n  helper\u2081 : \u2200 {a b} \u2192 P a b \u2192 a \u2261 zero \u2227 b \u2261 zero\n            \u2228 (\u2203 (\u03bb a' \u2192 \u2203 (\u03bb b' \u2192 P a' b' \u2227 a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b')))\n  helper\u2081 {a} {.a} (Ca , Cb , refl) with Conat-gfp\u2081 Ca\n  ... | inj\u2081 prf = inj\u2081 (prf , prf)\n  ... | inj\u2082 (a' , Ca' , prf) = inj\u2082 (a' , a' , (Ca' , Ca' , refl) , (prf , prf))\n\n  helper\u2082 : Conat n \u2227 Conat n \u2227 n \u2261 n\n  helper\u2082 = h , h , refl\n\n\u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n\u2261\u2192\u2248N h\u2081 h\u2082 refl = \u2248N-refl h\u2081\n\n{-# NO_TERMINATION_CHECK #-}\nConat\u2192N : \u2200 {n} \u2192 Conat n \u2192 N n\nConat\u2192N Cn with Conat-gfp\u2081 Cn\n... | inj\u2081 prf = subst N (sym prf) zN\n... | inj\u2082 (n' , Cn' , prf) = subst N (sym prf) (sN {!Conat\u2192N Cn'!})\n-- Agda error: Failed to give (Conat\u2192N Cn')\n\n-- TODO: 2012-03-20. Agda bug?\n-- N\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\n-- N\u2192Conat zN          = Conat-gfp\u2083 (inj\u2081 refl)\n-- N\u2192Conat (sN {n} Nn) = Conat-gfp\u2083 (inj\u2082 (n , ({!N\u2192Conat Nn!} , refl)))\n-- Agda error: Failed to give N\u2192Conat Nn\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat Nn = Conat-gfp\u2082 N helper Nn\n  where\n  helper : \u2200 {m} \u2192 N m \u2192 m \u2261 zero \u2228 \u2203 (\u03bb m' \u2192 N m' \u2227 m \u2261 succ\u2081 m')\n  helper zN          = inj\u2081 refl\n  helper (sN {m} Nm) = inj\u2082 (m , Nm , refl)\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT 20 March 2012.\n\nmodule Draft.FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.Nat\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} h = \u2248N-gfp\u2082 P helper\u2081 helper\u2082\n  where\n  P : D \u2192 D \u2192 Set\n  P a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n\n  helper\u2081 : \u2200 {a b} \u2192 P a b \u2192 a \u2261 zero \u2227 b \u2261 zero\n            \u2228 (\u2203 (\u03bb a' \u2192 \u2203 (\u03bb b' \u2192 P a' b' \u2227 a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b')))\n  helper\u2081 {a} {.a} (Ca , Cb , refl) with Conat-gfp\u2081 Ca\n  ... | inj\u2081 prf = inj\u2081 (prf , prf)\n  ... | inj\u2082 (a' , Ca' , prf) = inj\u2082 (a' , a' , (Ca' , Ca' , refl) , (prf , prf))\n\n  helper\u2082 : Conat n \u2227 Conat n \u2227 n \u2261 n\n  helper\u2082 = h , h , refl\n\n\u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n\u2261\u2192\u2248N h\u2081 h\u2082 refl = \u2248N-refl h\u2081\n\n{-# NO_TERMINATION_CHECK #-}\nConat\u2192N : \u2200 {n} \u2192 Conat n \u2192 N n\nConat\u2192N Cn with Conat-gfp\u2081 Cn\n... | inj\u2081 prf = subst N (sym prf) zN\n... | inj\u2082 (n' , Cn' , prf) = subst N (sym prf) (sN {!Conat\u2192N Cn'!})\n-- Agda error: Failed to give (Conat\u2192N Cn')\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat zN          = Conat-gfp\u2083 (inj\u2081 refl)\nN\u2192Conat (sN {n} Nn) = Conat-gfp\u2083 (inj\u2082 (n , ({!N\u2192Conat Nn!} , refl)))\n-- Agda error: Failed to give N\u2192Conat Nn\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b824a784090dc91abf40d6f7330c5bf546797f92","subject":"IDesc model: polymorphize the Prelude","message":"IDesc model: polymorphize the Prelude\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i : Level}(A B : Set i) -> Set i\nA * B = Sigma A \\_ -> B\n\nfst : {i : Level}{A : Set i}{B : A -> Set i} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i : Level}{A : Set i}{B : A -> Set i} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i : Level}(A B : Set i) : Set i where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d91ad665b4773ae05df5699edcc04d979b0f786b","subject":"IDesc model: remove IMu junk","message":"IDesc model: remove IMu junk","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ac5970524f01ba1d4dfda134c9486fe83efe0e35","subject":"Use more the stdlib","message":"Use more the stdlib\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nimport Data.Bool.Properties\nopen import Data.Unit using (\u22a4)\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Xor\u00b0 = Algebra.CommutativeRing Data.Bool.Properties.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring Data.Bool.Properties.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 {true}   {true}  _   _  = _\nT\u2227 {false}  {_}     ()  _\nT\u2227 {true}   {false} _   ()\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 {true} {true} _ = _\nT\u2227\u2081 {false} {_}   ()\nT\u2227\u2081 {true} {false}   ()\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {true} {true} _ = _\nT\u2227\u2082 {false} {_}   ()\nT\u2227\u2082 {true} {false}   ()\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {true} _ = inj\u2081 _\nT\u2228'\u228e {false} {true} _ = inj\u2082 _\nT\u2228'\u228e {false} {false} ()\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 {true} _ = _\nT\u2228\u2081 {false} {true} _ = _\nT\u2228\u2081 {false} {false} ()\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {true} _ = _\nT\u2228\u2082 {false} {true} _ = _\nT\u2228\u2082 {false} {false} ()\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7e6f4d5766eea1ce4cf2d6a8d58451541548e932","subject":"IDesc model: make that damned fix-point. Stuck with a dead lift to make most constructors.","message":"IDesc model: make that damned fix-point. Stuck with a dead lift to make most constructors.","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\n{-\nvarl : (l : Level)(I : Set l) -> IDescl l I\nvarl x I = con (lvar , {!!})\n-}\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lconst , X)\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lprod , (D , D'))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"90837080559e56d5a2bbb81a85c0f6e1752f0d40","subject":"Added  xx\u2261\u03b5\u2192comm.","message":"Added  xx\u2261\u03b5\u2192comm.\n\nIgnore-this: a7e32cd63e173ed60034dbf8f2aacb0e\n\ndarcs-hash:20101208184750-3bd4e-4a8fced75462b9013e4bce7738377aa36747f7e2.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/Properties.agda","new_file":"src\/GroupTheory\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Group theory properties\n------------------------------------------------------------------------------\n\nmodule GroupTheory.Properties where\n\nopen import GroupTheory.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  rightIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 x \u2219 u \u2261 x) \u2227\n                                 (\u2200 u' \u2192 (\u2200 x \u2192 x \u2219 u' \u2261 x) \u2192 u \u2261 u')\n{-# ATP prove rightIdentityUnique #-}\n\npostulate\n  leftIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 u \u2219 x \u2261 x) \u2227\n                                (\u2200 u' \u2192 (\u2200 x \u2192 u' \u2219 x \u2261 x) \u2192 u \u2261 u')\n{-# ATP prove leftIdentityUnique #-}\n\npostulate\n  rightCancellation : \u2200 {x y z} \u2192 y \u2219 x \u2261 z \u2219 x \u2192 y \u2261 z\n{-# ATP prove rightCancellation #-}\n\npostulate\n  leftCancellation : \u2200 {x y z} \u2192 x \u2219 y \u2261 x \u2219 z \u2192 y \u2261 z\n{-# ATP prove leftCancellation #-}\n\npostulate\n  rightInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb r \u2192 (x \u2219 r \u2261 \u03b5) \u2227\n                                        (\u2200 r' \u2192 x \u2219 r' \u2261 \u03b5 \u2192 r \u2261 r')\n{-# ATP prove rightInverseUnique #-}\n\n-- A more appropiate version to be used in the proofs.\npostulate\n  rightInverseUnique' : \u2200 {x r} \u2192 x \u2219 r \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 r\n{-# ATP prove rightInverseUnique' #-}\n\npostulate\n  leftInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb l \u2192 (l \u2219 x \u2261 \u03b5) \u2227\n                                       (\u2200 l' \u2192 l' \u2219 x \u2261 \u03b5 \u2192 l \u2261 l')\n{-# ATP prove leftInverseUnique #-}\n\n-- A more appropiate version to be used in the proofs.\npostulate\n  leftInverseUnique' : \u2200 {x l} \u2192 l \u2219 x \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 l\n{-# ATP prove leftInverseUnique' #-}\n\npostulate\n  \u207b\u00b9-involutive : \u2200 x \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n{-# ATP prove \u207b\u00b9-involutive #-}\n\npostulate\n  identityInverse : \u03b5 \u207b\u00b9 \u2261 \u03b5\n{-# ATP prove identityInverse #-}\n\npostulate\n  inverseDistribution : \u2200 x y \u2192 (x \u2219 y) \u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n{-# ATP prove inverseDistribution #-}\n\n-- The equation xa = b has an unique solution.\npostulate\n  xa\u2261b-uniqueSolution : \u2200 a b \u2192 \u2203D \u03bb x \u2192 (x \u2219 a \u2261 b) \u2227\n                                         (\u2200 x' \u2192 x' \u2219 a \u2261 b \u2192 x \u2261 x')\n{-# ATP prove xa\u2261b-uniqueSolution #-}\n\n-- The equation ax = b has an unique solution.\npostulate\n  ax\u2261b-uniqueSolution : \u2200 a b \u2192 \u2203D \u03bb x \u2192 (a \u2219 x \u2261 b) \u2227\n                                         (\u2200 x' \u2192 a \u2219 x' \u2261 b \u2192 x \u2261 x')\n{-# ATP prove ax\u2261b-uniqueSolution #-}\n\n-- If the square of every element is the identity, the system is commutative.\n-- From: TPTP (v5.0.0). File: Problems\/GRP\/GRP001-2.\npostulate\n  xx\u2261\u03b5\u2192comm : \u2200 x a b c \u2192 x \u2219 x \u2261 \u03b5 \u2192 a \u2219 b \u2261 c \u2192 b \u2219 a \u2261 c\n\n-- Paper proof:\n-- 1. c(ab)  = cc  (Hypothesis ab = c).\n-- 2. c(ab)  = \u03b5   (Hypothesis cc = \u03b5).\n-- 3. c(ab)b = b   (By 2).\n-- 4. ca(bb) = b   (Associativity).\n-- 5. ca     = b   (Hypothesis bb = \u03b5).\n-- 6. (ca)a  = ba  (By 5).\n-- 7. c(aa)  = ba  (Associativity).\n-- 6. c      = ba  (Hypothesis aa = \u03b5).\n\n-- E 1.2 no-success due to timeout (300 sec).\n-- Equinox 5.0alpha (2010-06-29) no-success due to timeout (300 sec).\n-- Metis 2.3 (release 20101019) no-success due to timeout (300 sec).\n-- {-# ATP prove xx\u2261\u03b5\u2192comm #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory properties\n------------------------------------------------------------------------------\n\nmodule GroupTheory.Properties where\n\nopen import GroupTheory.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  rightIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 x \u2219 u \u2261 x) \u2227\n                                 (\u2200 u' \u2192 (\u2200 x \u2192 x \u2219 u' \u2261 x) \u2192 u \u2261 u')\n{-# ATP prove rightIdentityUnique #-}\n\npostulate\n  leftIdentityUnique : \u2203D \u03bb u \u2192 (\u2200 x \u2192 u \u2219 x \u2261 x) \u2227\n                                (\u2200 u' \u2192 (\u2200 x \u2192 u' \u2219 x \u2261 x) \u2192 u \u2261 u')\n{-# ATP prove leftIdentityUnique #-}\n\npostulate\n  rightCancellation : \u2200 {x y z} \u2192 y \u2219 x \u2261 z \u2219 x \u2192 y \u2261 z\n{-# ATP prove rightCancellation #-}\n\npostulate\n  leftCancellation : \u2200 {x y z} \u2192 x \u2219 y \u2261 x \u2219 z \u2192 y \u2261 z\n{-# ATP prove leftCancellation #-}\n\npostulate\n  rightInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb r \u2192 (x \u2219 r \u2261 \u03b5) \u2227\n                                        (\u2200 r' \u2192 x \u2219 r' \u2261 \u03b5 \u2192 r \u2261 r')\n{-# ATP prove rightInverseUnique #-}\n\n-- A more appropiate version to be used in the proofs.\npostulate\n  rightInverseUnique' : \u2200 {x r} \u2192 x \u2219 r \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 r\n{-# ATP prove rightInverseUnique' #-}\n\npostulate\n  leftInverseUnique : \u2200 {x} \u2192 \u2203D \u03bb l \u2192 (l \u2219 x \u2261 \u03b5) \u2227\n                                       (\u2200 l' \u2192 l' \u2219 x \u2261 \u03b5 \u2192 l \u2261 l')\n{-# ATP prove leftInverseUnique #-}\n\n-- A more appropiate version to be used in the proofs.\npostulate\n  leftInverseUnique' : \u2200 {x l} \u2192 l \u2219 x \u2261 \u03b5 \u2192 x \u207b\u00b9 \u2261 l\n{-# ATP prove leftInverseUnique' #-}\n\npostulate\n  \u207b\u00b9-involutive : \u2200 x \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n{-# ATP prove \u207b\u00b9-involutive #-}\n\npostulate\n  identityInverse : \u03b5 \u207b\u00b9 \u2261 \u03b5\n{-# ATP prove identityInverse #-}\n\npostulate\n  inverseDistribution : \u2200 x y \u2192 (x \u2219 y) \u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n{-# ATP prove inverseDistribution #-}\n\n-- The equation xa = b has an unique solution.\npostulate\n  xa\u2261b-uniqueSolution : \u2200 a b \u2192 \u2203D \u03bb x \u2192 (x \u2219 a \u2261 b) \u2227\n                                         (\u2200 x' \u2192 x' \u2219 a \u2261 b \u2192 x \u2261 x')\n{-# ATP prove xa\u2261b-uniqueSolution #-}\n\n-- The equation ax = b has an unique solution.\npostulate\n  ax\u2261b-uniqueSolution : \u2200 a b \u2192 \u2203D \u03bb x \u2192 (a \u2219 x \u2261 b) \u2227\n                                         (\u2200 x' \u2192 a \u2219 x' \u2261 b \u2192 x \u2261 x')\n{-# ATP prove ax\u2261b-uniqueSolution #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8001545ebac0c90e1eedb6d2f68606144c10521a","subject":"Updated the McCarthy 91 function example (by Ana Bove).","message":"Updated the McCarthy 91 function example (by Ana Bove).\n\nIgnore-this: 4415846248f61a58bb63c0a22b669b29\n\ndarcs-hash:20110208222623-3bd4e-4c574bc56ced1aea427b8b178cbb339c18c01753.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/McCarthy91.agda","new_file":"Draft\/McCarthy91\/McCarthy91.agda","new_contents":"------------------------------------------------------------------------------\n-- McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.McCarthy91 where\n\nopen import Draft.McCarthy91.Numbers\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\n\n------------------------------------------------------------------------------\n\n-- McCarthy 91 function\npostulate\n  mc91     : D \u2192 D\n  mc91-eq\u2081 : \u2200 n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\n  mc91-eq\u2082 : \u2200 n \u2192 LE n one-hundred \u2192 mc91 n \u2261 mc91 (mc91 (n + eleven))\n{-# ATP axiom mc91-eq\u2081 #-}\n{-# ATP axiom mc91-eq\u2082 #-}\n\npostulate N0   : N zero\n          N1   : N one\n          N10  : N ten\n          N91  : N ninety-one\n          N100 : N one-hundred\n          N101 : N hundred-one\n          N111 : N hundred-eleven\n{-# ATP prove N0 #-}\n{-# ATP prove N1 #-}\n{-# ATP prove N10 #-}\n{-# ATP prove N91 #-}\n{-# ATP prove N100 #-}\n{-# ATP prove N101 #-}\n{-# ATP prove N111 #-}\n\npostulate n111' : eleven + one-hundred \u2261 hundred-eleven\n          n111  : one-hundred + eleven \u2261 hundred-eleven\n          n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n          n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n          n91'  : hundred-one \u2238 ten \u2261 ninety-one\n          n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n          n102' : eleven + ninety-one \u2261 hundred-two\n          n102  : ninety-one + eleven \u2261 hundred-two\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n\npostulate 101>100' : GT hundred-one one-hundred\n          101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n          111>100' : GT hundred-eleven one-hundred\n          111>100  : GT (one-hundred + eleven) one-hundred\n          100<102' : LT one-hundred hundred-two\n          100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate N1+n  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n          N11+n : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n          Nn-10 : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n{-# ATP prove N1+n #-}\n{-# ATP prove N11+n #-}\n{-# ATP prove Nn-10 \u2238-N N10 #-}\n\npostulate n<n+1 : \u2200 {n} \u2192 N n \u2192 LT n (n + one)\n{-# ATP prove n<n+1 x<Sx +-comm #-}\n\nn<=m+n-m : \u2200 {n m} \u2192 N n \u2192 N m \u2192 LE n (m + (n \u2238 m))\nn<=m+n-m {m = m} zN Nm = prf0\n  where postulate prf0 : LE zero (m + (zero \u2238 m))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nn<=m+n-m (sN {n} Nn) zN = prfx0\n  where postulate prfx0 : LE (succ n) (zero + (succ n \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nn<=m+n-m (sN {n} Nn) (sN {m} Nm) = prfSS (n<=m+n-m Nn Nm)\n  where postulate prfSS : LE n (m + (n \u2238 m)) \u2192\n                        LE (succ n) (succ m + (succ n \u2238 succ m))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\npostulate n+1<=n-10+11 : \u2200 {n} \u2192 N n \u2192  LE (n + one) ((n \u2238 ten) + eleven)\n{-# ATP prove n+1<=n-10+11 n<=m+n-m N10 N1+n N11+n Nn-10 +-comm #-}\n\n---- Case n > 100\npostulate term>100     : \u2200 n \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n          f>100\u2261n-10   : \u2200 n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\n          >100-n<fn+11 : \u2200 n \u2192 N n \u2192 GT n one-hundred \u2192 LT n (mc91 n + eleven)\n{-# ATP prove term>100 Nn-10 #-}\n{-# ATP prove f>100\u2261n-10 #-}\n{-# ATP prove >100-n<fn+11 x<y\u2192y\u2264z\u2192x<z n<n+1 n+1<=n-10+11 N1+n N11+n Nn-10\n              +-comm\n#-}\n\n---- Case n \u2261 100\npostulate term\u2261100     : N (mc91 one-hundred)\n          f100\u226191      : mc91 one-hundred \u2261 ninety-one\n          \u2261100-n<fn+11 : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove term\u2261100 111>100 101>100 n101 n91 term>100 N111 N101 #-}\n{-# ATP prove f100\u226191 111>100 101>100 n101 n91 #-}\n{-# ATP prove term\u2261100 f100\u226191 111>100 101>100 n101 n91 N91 #-}\n{-# ATP prove \u2261100-n<fn+11 f100\u226191 n102 100<102 #-}\n\n-- $ agda -isrc -i. Draft\/McCarthy91\/McCarthy91.agda\n-- $ agda2atp -isrc -i. --atp=e --atp=equinox Draft\/McCarthy91\/McCarthy91.agda\n","old_contents":"------------------------------------------------------------------------------\n-- McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.McCarthy91 where\n\nopen import Draft.McCarthy91.Numbers\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\n-- McCarthy 91 function\npostulate\n  mc91     : D \u2192 D\n  mc91-eq\u2081 : \u2200 n \u2192 GT n one-hundred \u2192 mc91 n \u2261 n \u2238 ten\n  mc91-eq\u2082 : \u2200 n \u2192 LE n one-hundred \u2192 mc91 n \u2261 mc91 (mc91 (n + eleven))\n{-# ATP axiom mc91-eq\u2081 #-}\n{-# ATP axiom mc91-eq\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"66d20c910a8fbd90f2103ef297fab4468f5bcdc7","subject":"Minor change.","message":"Minor change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"notes\/FOT\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Stream properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality.Type\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.PropertiesI using ( Stream-pre-fixed )\nopen import FOTC.Data.Stream.Type\n\n------------------------------------------------------------------------------\n\npostulate\n  zeros    : D\n  zeros-eq : zeros \u2261 zero \u2237 zeros\n\nzeros-Stream : Stream zeros\nzeros-Stream = Stream-coind A h refl\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 zeros\n\n  h : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h Axs = zero , zeros , trans Axs zeros-eq , refl\n\n-- ++-Stream with a diferent type.\n++-Stream : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 Stream (xs ++ ys)\n++-Stream {xs} {ys} Sxs Sys with Stream-unf Sxs\n... | x' , xs' , prf , Sxs' = subst Stream prf\u2081 prf\u2082\n  where\n  prf\u2081 : x' \u2237 (xs' ++ ys) \u2261 xs ++ ys\n  prf\u2081 = trans (sym (++-\u2237 x' xs' ys)) (++-leftCong (sym prf))\n\n  -- TODO (15 December 2013): Why the termination checker accepts the\n  -- recursive called ++-Stream_Sxs'_Sys?\n  prf\u2082 : Stream (x' \u2237 xs' ++ ys)\n  prf\u2082 = Stream-pre-fixed (x' , xs' ++ ys , refl , ++-Stream Sxs' Sys)\n","old_contents":"------------------------------------------------------------------------------\n-- Stream properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality.Type\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\nopen import FOTC.Data.Stream.PropertiesI using ( Stream-pre-fixed )\n\n------------------------------------------------------------------------------\n\npostulate\n  zeros    : D\n  zeros-eq : zeros \u2261 zero \u2237 zeros\n\nzeros-Stream : Stream zeros\nzeros-Stream = Stream-coind A h refl\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 zeros\n\n  h : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h Axs = zero , zeros , trans Axs zeros-eq , refl\n\n-- ++-Stream with a diferent type.\n++-Stream : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 Stream (xs ++ ys)\n++-Stream {xs} {ys} Sxs Sys with Stream-unf Sxs\n... | x' , xs' , prf , Sxs' = subst Stream prf\u2081 prf\u2082\n  where\n  prf\u2081 : x' \u2237 (xs' ++ ys) \u2261 xs ++ ys\n  prf\u2081 = trans (sym (++-\u2237 x' xs' ys)) (++-leftCong (sym prf))\n\n  -- TODO (15 December 2013): Why the termination checker accepts the\n  -- recursive called ++-Stream_Sxs'_Sys?\n  prf\u2082 : Stream (x' \u2237 xs' ++ ys)\n  prf\u2082 = Stream-pre-fixed (x' , xs' ++ ys , refl , ++-Stream Sxs' Sys)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c3e60f6c99d95ba12feaea576278af55d225c0a","subject":"Desc@ICFP: add \\update to model","message":"Desc@ICFP: add \\update to model\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDescTT.agda","new_file":"models\/IDescTT.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\nupdate : {ty : Type} -> IMu closeTerm ty -> IMu closeTerm' ty\nupdate {ty} tm = cata Type closeTerm (IMu closeTerm') (\\ _ tagTm -> con (ESu (fst tagTm) , (snd tagTm))) ty tm\n\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            Var context ty ->\n            IMu closeTerm' ty\ndischarge ctxt env ty variable = con (ESu EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     Var context ty ->\n                     IMu closeTerm' ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm' ty\nsubstExpr {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval' (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu (ESu EZe)) , (con (ESu EZe , su ze) , con ( EZe , l (r refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , l (l (r refl))) ,\n                             (con (EZe , l (r refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe ,  r refl  )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDescTT where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\nall : {I : Set}(D : IDesc I)(X : I -> Set)\n      (R : Sigma I X -> Set)(P : (x : Sigma I X) -> R x) -> \n      (xs : [| D |] X) -> [| All D X xs |] R\nall (var i) X R P x = P (i , x)\nall (const K) X R P k = Void\nall (prod D D') X R P (x , y) = ( all D X R P x , all D' X R P y )\nall (sigma S T) X R P (a , b) = all (T a) X R P b\nall (pi S T) X R P f = \\a -> all (T a) X R P (f a)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\nindI : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\nindI {I} R P m i (con xs) = m i xs (all (R i) (IMu R) P induct xs)\n  where induct : (x : Sigma I (IMu R)) -> P x\n        induct (i , xs) = indI R P m i xs\n-}\n\n-- But the termination-checker complains, so here we go\n-- inductive-recursive:\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc Unit\nnatCases Ze = const Unit\nnatCases Suc = var Void\n\nNatD : Unit -> IDesc Unit\nNatD Void = sigma NatConst  natCases\n\nNatd : Unit -> Set\nNatd x = IMu NatD x\n\nzed : Natd Void\nzed = con (Ze , Void)\n\nsud : Natd Void -> Natd Void\nsud n = con (Su , n)\n\n\n--****************\n-- Desc\n--****************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl0 : (I : Set) -> Unit -> Set\nIDescl0 I = IMu (\\_ -> descD I) \n\nIDescl : (I : Set) -> Set\nIDescl I = IDescl0 I _\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n--****************\n-- Vec (constraints)\n--****************\n\ndata VecConst : Set where\n  Vnil : VecConst\n  Vcons : VecConst\n\nvecDChoice : Set -> Nat -> VecConst -> IDesc Nat\nvecDChoice X n Vnil = const (n == ze)\nvecDChoice X n Vcons = sigma Nat (\\m -> prod (var m) (const (n == su m)))\n\nvecD : Set -> Nat -> IDesc Nat\nvecD X n = sigma VecConst (vecDChoice X n)\n\nvec : Set -> Nat -> Set\nvec X n = IMu (vecD X) n\n\n--****************\n-- Vec (de-tagged, forced)\n--****************\n\ndata VecConst2 : Nat -> Set where\n  Vnil : VecConst2 ze\n  Vcons : {n : Nat} -> VecConst2 (su n)\n\nvecDChoice2 : Set -> (n : Nat) -> VecConst2 n -> IDesc Nat\nvecDChoice2 X ze Vnil = const Unit\nvecDChoice2 X (su n) Vcons = prod (const X) (var n)\n\nvecD2 : Set -> Nat -> IDesc Nat\nvecD2 X n = sigma (VecConst2 n) (vecDChoice2 X n)\n\nvec2 : Set -> Nat -> Set\nvec2 X n = IMu (vecD2 X) n\n\n--****************\n-- Fin (de-tagged)\n--****************\n\ndata FinConst : Nat -> Set where\n  Fz : {n : Nat} -> FinConst (su n)\n  Fs : {n : Nat} -> FinConst (su n)\n\nfinDChoice : (n : Nat) -> FinConst n -> IDesc Nat\nfinDChoice ze ()\nfinDChoice (su n) Fz = const Unit\nfinDChoice (su n) Fs = var n\n\nfinD : Nat -> IDesc Nat\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : Nat -> Set\nfin n = IMu finD n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n--********************************\n-- Typed expressions\n--********************************\n\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\n\n-- Fix menu:\nexprFixMenu : FixMenu Type\nexprFixMenu = ( consE (consE nilE) , \n                \\ty -> (const (Val ty),                           -- Val t\n                       (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                       Void)))\n\n-- Indexed menu:\nchoiceMenu : Type -> EnumU\nchoiceMenu nat = consE nilE\nchoiceMenu bool = consE nilE\nchoiceMenu (pair x y) = nilE\n\nchoiceDessert : (ty : Type) -> spi (choiceMenu ty) (\\ _ -> IDesc Type)\nchoiceDessert nat = (prod (var nat) (var nat) , Void)\nchoiceDessert bool = (prod (var nat) (var nat) , Void )\nchoiceDessert (pair x y) = Void\n\nexprSensitiveMenu : SensitiveMenu Type\nexprSensitiveMenu = ( choiceMenu ,  choiceDessert )\n\n\n-- Expression:\nexpr : TagIDesc Type\nexpr = exprFixMenu , exprSensitiveMenu\n\nexprIDesc : TagIDesc Type -> (Type -> IDesc Type)\nexprIDesc D = toIDesc Type D\n\n\n--********************************\n-- Closed terms\n--********************************\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = exprIDesc expr\n\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\nEmpty : Type -> Set\nEmpty _ = Zero\n\ncloseTerm' : Type -> IDesc Type\ncloseTerm' = toIDesc Type (expr ** Empty)\n\n--********************************\n-- Closed term' evaluation\n--********************************\n\neval' : {ty : Type} -> IMu closeTerm' ty -> Val ty\neval' {ty} term = cata Type closeTerm' Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm' ty |] Val -> Val ty\n              evalOneStep _ (EZe , ())\n              evalOneStep _ (ESu EZe , t) = t\n              evalOneStep _ ((ESu (ESu EZe)) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu (ESu EZe)) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu (ESu EZe))) , (x , y)) = plus x y\n              evalOneStep nat (((ESu (ESu (ESu (ESu ()))))) , t) \n              evalOneStep bool ((ESu (ESu (ESu EZe))) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu (ESu ())))) , _) \n              evalOneStep (pair x y) (ESu (ESu (ESu ())) , _)\n\n\n--********************************\n-- Open terms\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (expr ** (Var c))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            Var context ty ->\n            IMu closeTerm' ty\ndischarge ctxt env ty variable = con (ESu EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     Var context ty ->\n                     IMu closeTerm' ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm' ty\nsubstExpr {ty} c sig term = \n    substI (Var c) Empty expr sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval' (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm' nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu (ESu EZe)) , (con (ESu EZe , su ze) , con ( EZe , l (r refl) )) )\n\ntestSubst2 : IMu closeTerm' nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu (ESu EZe) , (con (EZe , l (l (r refl))) ,\n                             (con (EZe , l (r refl)) ,\n                              con (ESu EZe , ze))))\n\ntestSubst3 : IMu closeTerm' nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe ,  r refl  )\n\ntestSubst4 : IMu closeTerm' (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"49d928e9b0d1b47d023c588ec997d5e4d76c5966","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: 92e96c490ccb652eb5b12bd173ef1902\n\ndarcs-hash:20110405134352-3bd4e-96dd1569b8d456cc6a7d595c3c68e0aee7c7de1a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-N : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-N dom0           = subst N (sym nest-0) zN\nnest-N (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-N h\u2082)\n\npostulate\n  nest-\u2264 : \u2200 {n} \u2192 N n \u2192 LE (nest n) n\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h = domS x dn-x (h (nest-N dn-x)\n                                      (x\u2264y\u2192x<Sy (nest-N dn-x) Nx (nest-\u2264 Nx)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\n-- The nest function is total (via structural recursion in the domain\n-- predicate).\nnest-N : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-N dom0           = subst N (sym nest-0) zN\nnest-N (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-N h\u2082)\n\npostulate\n  nest-\u2264 : \u2200 {n} \u2192 N n \u2192 LE (nest n) n\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h = domS x dn-x (h (nest-N dn-x)\n                                      (x\u2264y\u2192x<Sy (nest-N dn-x) Nx (nest-\u2264 Nx)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"23b3e5376bd66a7bc1239a94dff9d2749e9a1aee","subject":"FunExt: change argument order","message":"FunExt: change argument order\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Extensionality.agda","new_file":"lib\/Function\/Extensionality.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Relation.Binary.PropositionalEquality renaming (subst to tr)\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n           (f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}{{fe : FunExt}}(fg : (x : A) \u2192 f x \u2261 g x)\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Relation.Binary.PropositionalEquality renaming (subst to tr)\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}(f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x){{fe : FunExt}} \u2192 f\u2080 \u2261 f\u2081\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}(fg : (x : A) \u2192 f x \u2261 g x){{fe : FunExt}}\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"42cc20cfd8b4d8e6f4506d5e9389a05e8456e2c0","subject":"Only white space.","message":"Only white space.\n\nIgnore-this: 45cb23b8ad3cb68f4c6b44c58cbfea03\n\ndarcs-hash:20100623211545-3bd4e-359e979cb855f818a3dbc2ac8521763ff17cd8db.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic.agda","new_file":"LTC\/Function\/Arithmetic.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic functions on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic where\n\nopen import LTC.Minimal\nopen import LTC.Data.N\n\n------------------------------------------------------------------------------\ninfixl 7 _*_\ninfixl 6 _+_ _-_\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192   zero   + d \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\npostulate\n  _-_      : D \u2192 D \u2192 D\n  minus-x0 : (d : D) \u2192   d      - zero   \u2261 d\n  minus-0S : (d : D) \u2192   zero   - succ d \u2261 zero\n  minus-SS : (d e : D) \u2192 succ d - succ e \u2261 d - e\n{-# ATP axiom minus-x0 #-}\n{-# ATP axiom minus-0S #-}\n{-# ATP axiom minus-SS #-}\n\npostulate\n  _*_ : D \u2192 D \u2192 D\n  *-0x : (d : D)   \u2192 zero   * d \u2261 zero\n  *-Sx : (d e : D) \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x #-}\n{-# ATP axiom *-Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic functions on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic where\n\nopen import LTC.Minimal\nopen import LTC.Data.N\n\n------------------------------------------------------------------------------\ninfixl 7 _*_\ninfixl 6 _+_ _-_\n\npostulate\n  _+_    : D \u2192 D \u2192 D\n  +-0x : (d : D) \u2192   zero   + d \u2261 d\n  +-Sx : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\npostulate\n  _-_      : D \u2192 D \u2192 D\n  minus-x0 : (d : D) \u2192   d      - zero   \u2261 d\n  minus-0S : (d : D) \u2192   zero   - succ d \u2261 zero\n  minus-SS : (d e : D) \u2192 succ d - succ e \u2261 d - e\n{-# ATP axiom minus-x0 #-}\n{-# ATP axiom minus-0S #-}\n{-# ATP axiom minus-SS #-}\n\npostulate\n  _*_ : D \u2192 D \u2192 D\n  *-0x : (d : D)   \u2192 zero   * d \u2261 zero\n  *-Sx : (d e : D) \u2192 succ d * e \u2261 e + d * e\n{-# ATP axiom *-0x #-}\n{-# ATP axiom *-Sx #-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"89f99238ec73533604712e27553bd981c818d86a","subject":"Remove lots of duplication across abs\u2081 .. abs\u2086","message":"Remove lots of duplication across abs\u2081 .. abs\u2086\n\nThe new code is much more clever. Still not perfect, with varying\narities we can probably do better.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr)\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    -- A better type for absN but a mess to use due to the proofs (which aren't synthesized, even though maybe they should be?)\n    -- absN : (n : \u2115) \u2192 {_ : n > 0} \u2192 absNType n\n    -- XXX See \"how to keep your neighbours in order\" for tricks.\n\n    -- Please the termination checker by keeping this case separate.\n    absNBase : \u2200 {\u03c4\u2081} \u2192 absNType (\u03c4\u2081 \u2237 [])\n    absNBase {\u03c4\u2081} f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    -- Otherwise, the recursive step of absN would invoke absN twice, and the\n    -- termination checker does not figure out that the calls are in fact\n    -- terminating.\n\n  open AbsNHelpers using (absNType; absNBase)\n\n  -- XXX: could we be using the same trick as above to take the N implicit\n  -- arguments individually, rather than in a vector? I can't figure out how,\n  -- at least not without trying it. But it seems that's what's shown in Agda 2.4.0 release notes!\n  absN : {n : \u2115} \u2192 (\u03c4s : Vec _ (suc n)) \u2192 absNType \u03c4s\n  absN {zero}  (\u03c4\u2081 \u2237 []) = absNBase\n  absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n    absNBase (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  -- What I'd like to write, avoiding the need for absNBase, but can't because of the termination checker.\n  {-\n  absN {zero}  (\u03c4\u2081 \u2237 []) f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n  absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n    absN (\u03c4\u2081 \u2237 []) (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n  -}\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n  module _ {\u03c4\u2081 : Type} where\n    \u03c4s\u2081 =  \u03c4\u2081 \u2237 []\n    abs\u2081 : absNType \u03c4s\u2081\n    abs\u2081 = absN \u03c4s\u2081\n    module _ {\u03c4\u2082 : Type} where\n      \u03c4s\u2082 = \u03c4\u2082 \u2237 \u03c4s\u2081\n      abs\u2082 : absNType \u03c4s\u2082\n      abs\u2082 = absN \u03c4s\u2082\n      module _ {\u03c4\u2083 : Type} where\n        \u03c4s\u2083 = \u03c4\u2083 \u2237 \u03c4s\u2082\n        abs\u2083 : absNType \u03c4s\u2083\n        abs\u2083 = absN \u03c4s\u2083\n        module _ {\u03c4\u2084 : Type} where\n          \u03c4s\u2084 = \u03c4\u2084 \u2237 \u03c4s\u2083\n          abs\u2084 : absNType \u03c4s\u2084\n          abs\u2084 = absN \u03c4s\u2084\n          module _ {\u03c4\u2085 : Type} where\n            \u03c4s\u2085 = \u03c4\u2085 \u2237 \u03c4s\u2084\n            abs\u2085 : absNType \u03c4s\u2085\n            abs\u2085 = absN \u03c4s\u2085\n            module _ {\u03c4\u2086 : Type} where\n              \u03c4s\u2086 = \u03c4\u2086 \u2237 \u03c4s\u2085\n              abs\u2086 : absNType \u03c4s\u2086\n              abs\u2086 = absN \u03c4s\u2086\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  abs\u2081 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 (x : Term \u0393\u2032 \u03c4\u2081) \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n  abs\u2081 {\u0393} {\u03c4\u2081} =  \u03bb f \u2192 abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n\n  abs\u2082 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4))\n  abs\u2082 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2081 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2083 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4))\n  abs\u2083 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2082 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2084 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4))\n  abs\u2084 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2083 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2085 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4))\n  abs\u2085 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2084 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n  abs\u2086 :\n    \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4\u2086 \u03c4} \u2192\n      (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4\u2086 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4\u2086 \u21d2 \u03c4))\n  abs\u2086 f =\n    abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n      abs\u2085 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n        f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"30e1935a87b6ee45bd8266999805c80b80dd4bd0","subject":"add map-Dec","message":"add map-Dec\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Nullary\/NP.agda","new_file":"lib\/Relation\/Nullary\/NP.agda","new_contents":"module Relation.Nullary.NP where\n\nopen import Type\nopen import Data.Sum\nopen import Relation.Nullary public\n\nmodule _ {a b} {A : \u2605_ a} {B : \u2605_ b} where\n    Dec-\u228e : Dec A \u2192 Dec B \u2192 Dec (A \u228e B)\n    Dec-\u228e (yes p) _       = yes (inj\u2081 p)\n    Dec-\u228e (no \u00acp) (yes p) = yes (inj\u2082 p)\n    Dec-\u228e (no \u00acp) (no \u00acq) = no [ \u00acp , \u00acq ]\n\n    module _ (to : A \u2192 B)(from : B \u2192 A) where\n\n      map-Dec : Dec A \u2192 Dec B\n      map-Dec (yes p) = yes (to p)\n      map-Dec (no \u00acp) = no (\u03bb z \u2192 \u00acp (from z))\n","old_contents":"module Relation.Nullary.NP where\n\nopen import Type\nopen import Data.Sum\nopen import Relation.Nullary public\n\nmodule _ {a b} {A : \u2605_ a} {B : \u2605_ b} where\n    Dec-\u228e : Dec A \u2192 Dec B \u2192 Dec (A \u228e B)\n    Dec-\u228e (yes p) _       = yes (inj\u2081 p)\n    Dec-\u228e (no \u00acp) (yes p) = yes (inj\u2082 p)\n    Dec-\u228e (no \u00acp) (no \u00acq) = no [ \u00acp , \u00acq ]\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fe3b2e3e4244647bac4e1390e292f3a50e6550fe","subject":"Added x<Sy\u2192x<y\u2228x\u2261y (ER version).","message":"Added x<Sy\u2192x<y\u2228x\u2261y (ER version).\n\nIgnore-this: 83e09323662f63fdbea33f75c37a6f1e\n\ndarcs-hash:20100524233743-3bd4e-a6375dc6c522873d3d2b3dd8a9766e9034d81c9f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_file":"LTC\/Relation\/Inequalities\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Function.Arithmetic.PropertiesER\nopen import LTC.Relation.Equalities.PropertiesER\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0           = true\u2260false (trans (sym 0<0) (lt-00))\n\u00acx<0 (sN {n} Nn) Sn<0 = true\u2260false (trans (sym Sn<0) (lt-S0 n))\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sd\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sd\u22640\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = trans (lt-SS n (succ n)) (x<Sx Nn)\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-00\nx\u2264x (sN {m} Nm) = trans (lt-SS m m) (x\u2264x Nm)\n\n-- TODO: Why not a dot pattern?\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn) _              = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  begin\n    lt (succ n) (succ m) \u2261\u27e8 lt-SS n m \u27e9\n    lt n m \u2261\u27e8 x<y\u2192x\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn) \u27e9\n    false\n  \u220e\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN               m<0       = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192Sx\u2264y zN (sN zN)          _         = trans (lt-SS zero zero) lt-00\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _         = trans (lt-SS (succ n) zero) (lt-S0 n)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS n (succ m)) (x<y\u2192Sx\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  trans (lt-SS m n) (Sx\u2264y\u2192x<y Nm Nn (trans (sym (lt-SS n (succ m))) SSm\u2264Sn))\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  begin\n    lt (succ m) (succ o) \u2261\u27e8 lt-SS m o \u27e9\n    lt m o \u2261\u27e8 <-trans Nm Nn No\n                       (trans (sym (lt-SS m n)) Sm<Sn)\n                       (trans (sym (lt-SS n o)) Sn<So) \u27e9\n    true\n  \u220e\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  trans (lt-SS o m)\n        (\u2264-trans Nm Nn No\n                 (trans (sym (lt-SS n m)) Sm\u2264Sn)\n                 (trans (sym (lt-SS o n)) Sn\u2264So))\n\nSx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS n m)) Sm\u2264Sn\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m + n) (succ m)   \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m + n) (succ m) \u2261\n                                               lt t (succ m))\n                                        (+-Sx m n)\n                                        refl\n                               \u27e9\n    lt (succ (m + n)) (succ m) \u2261\u27e8 lt-SS (m + n) m \u27e9\n    lt (m + n) m               \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    false\n  \u220e\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN =\n  begin\n    lt (m - zero) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (m - zero) (succ m) \u2261\n                                           lt t (succ m))\n                                    (minus-x0 m)\n                                    refl\n                           \u27e9\n    lt m (succ m)          \u2261\u27e8 x<Sx Nm \u27e9\n    true\n  \u220e\n\nx-y<Sx zN (sN {n} Nn) =\n  begin\n    lt (zero - succ n) (succ zero)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (zero - succ n) (succ zero) \u2261 lt t (succ zero))\n               (minus-0S n)\n               refl\n      \u27e9\n    lt zero (succ zero) \u2261\u27e8 lt-0S zero \u27e9\n    true\n  \u220e\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) =\n  begin\n    lt (succ m - succ n) (succ (succ m))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ (succ m)) \u2261\n                      lt t (succ (succ m)))\n               (minus-SS m n)\n               refl\n      \u27e9\n    lt (m - n) (succ (succ m))\n      \u2261\u27e8 <-trans (minus-N Nm Nn) (sN Nm) (sN (sN Nm))\n                 (x-y<Sx Nm Nn) (x<Sx (sN Nm))\n      \u27e9\n    true\n  \u220e\n\nSx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\nSx-Sy<Sx {m} {n} Nm Nn =\n  begin\n    lt (succ m - succ n) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n)\n                                                     (succ m) \u2261\n                                                  lt t (succ m))\n                                           (minus-SS m n)\n                                           refl\n                                  \u27e9\n    lt (m - n) (succ m)           \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n    \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS m n)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN      Nn 0\u2264n  = trans (+-rightIdentity (minus-N Nn zN))\n                                            (minus-x0 n)\nx\u2264y\u2192y-x+x\u2261y         (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  begin\n    (succ n - succ m) + succ m \u2261\u27e8 subst (\u03bb t \u2192 (succ n - succ m) + succ m \u2261\n                                               t + succ m)\n                                        (minus-SS n m)\n                                        refl\n                               \u27e9\n    (n - m) + succ m           \u2261\u27e8 +-comm (minus-N Nn Nm) (sN Nm) \u27e9\n    succ m + (n - m)           \u2261\u27e8 +-Sx m (n - m) \u27e9\n    succ (m + (n - m))         \u2261\u27e8 subst (\u03bb t \u2192 succ (m + (n - m)) \u2261 succ t )\n                                        (+-comm Nm (minus-N Nn Nm))\n                                        refl\n                               \u27e9\n    succ ((n - m) + m)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((n - m) + m) \u2261 succ t )\n                                        (x\u2264y\u2192y-x+x\u2261y Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm\u2264Sn) )\n                                        refl\n                               \u27e9\n    succ n\n  \u220e\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS m (succ n)) (x<y\u2192x<Sy Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim (\u00acx<0 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ n))) Sm<SSn)\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\n\u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n\u00acxy<00 Nm Nn mn<00 =\n  [ (\u03bb m<0     \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n  ]\n  mn<00\n\n\u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n\u00ac0Sx<00 Nm 0Sm<00 =\n  [ (\u03bb 0<0      \u2192 \u22a5-elim (\u00acx<0 zN 0<0))\n  , (\u03bb 0\u22610\u2227Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) (\u2227-proj\u2082 0\u22610\u2227Sm<0)))\n  ]\n  0Sm<00\n\n\u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n\u00acSxy\u2081<0y\u2082 Nm Nn\u2081 Nn\u2082 Smn\u2081<0n\u2082 =\n  [ (\u03bb Sm<0       \u2192 \u22a5-elim (\u00acx<0 (sN Nm) Sm<0))\n  , (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2260S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n  ]\n  Smn\u2081<0n\u2082\n\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 Nm\u2081 Nn Nm\u2082 m\u2081n<m\u2082zero =\n  [ (\u03bb m\u2081<n\u2081      \u2192 m\u2081<n\u2081)\n  , (\u03bb m\u2081\u2261n\u2081\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u2081\u2261n\u2081\u2227n<0)))\n  ]\n  m\u2081n<m\u2082zero\n\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                    m \u2261 zero \u2227 LT n\u2081 n\u2082\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 Nm Nn\u2081 Nn\u2082 mn\u2081<0n\u2082 =\n  [ (\u03bb m<0          \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u2261zero\u2227n\u2081<n\u2082 \u2192 m\u2261zero\u2227n\u2081<n\u2082)\n  ]\n  mn\u2081<0n\u2082\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = inj\u2081 (Sx-Sy<Sx Nm Nn)\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = inj\u2082 (refl , Sx-Sy<Sx Nn Nm)\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities (using equational reasoning)\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties using ( +-comm )\nopen import LTC.Function.Arithmetic.PropertiesER\nopen import LTC.Relation.Inequalities\nopen import LTC.Data.N\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module IPER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0           = true\u2260false (trans (sym 0<0) (lt-00))\n\u00acx<0 (sN {n} Nn) Sn<0 = true\u2260false (trans (sym Sn<0) (lt-S0 n))\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = true\u2260false $ trans (sym 0>n ) $ x\u22650 Nn\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 {d} Sd\u22640 = true\u2260false $ trans (sym $ lt-0S d ) Sd\u22640\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = trans (lt-SS n (succ n)) (x<Sx Nn)\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  subst (\u03bb a \u2192 a \u2261 true \u2228 a \u2261 false)\n        (sym $ lt-SS n m)\n        (x>y\u2228x\u2264y Nm Nn )\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN          0<0   = \u22a5-elim (true\u2260false (trans (sym 0<0) lt-00))\n\u00acx<x (sN {m} Nm) Sm<Sm = \u22a5-elim (\u00acx<x Nm (trans (sym (lt-SS m m)) Sm<Sm))\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-00\nx\u2264x (sN {m} Nm) = trans (lt-SS m m) (x\u2264x Nm)\n\n-- TODO: Why not a dot pattern?\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn) _              = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  begin\n    lt (succ n) (succ m) \u2261\u27e8 lt-SS n m \u27e9\n    lt n m \u2261\u27e8 x<y\u2192x\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn) \u27e9\n    false\n  \u220e\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN               m<0       = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192Sx\u2264y zN (sN zN)          _         = trans (lt-SS zero zero) lt-00\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _         = trans (lt-SS (succ n) zero) (lt-S0 n)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS n (succ m)) (x<y\u2192Sx\u2264y Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  trans (lt-SS m n) (Sx\u2264y\u2192x<y Nm Nn (trans (sym (lt-SS n (succ m))) SSm\u2264Sn))\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  begin\n    lt (succ m) (succ o) \u2261\u27e8 lt-SS m o \u27e9\n    lt m o \u2261\u27e8 <-trans Nm Nn No\n                       (trans (sym (lt-SS m n)) Sm<Sn)\n                       (trans (sym (lt-SS n o)) Sn<So) \u27e9\n    true\n  \u220e\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  trans (lt-SS o m)\n        (\u2264-trans Nm Nn No\n                 (trans (sym (lt-SS n m)) Sm\u2264Sn)\n                 (trans (sym (lt-SS o n)) Sn\u2264So))\n\nSx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\nSx\u2264Sy\u2192x\u2264y {m} {n} Sm\u2264Sn = trans (sym (lt-SS n m)) Sm\u2264Sn\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn =\n  begin\n    lt (succ m + n) (succ m)   \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m + n) (succ m) \u2261\n                                               lt t (succ m))\n                                        (+-Sx m n)\n                                        refl\n                               \u27e9\n    lt (succ (m + n)) (succ m) \u2261\u27e8 lt-SS (m + n) m \u27e9\n    lt (m + n) m               \u2261\u27e8 x\u2264x+y Nm Nn \u27e9\n    false\n  \u220e\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN =\n  begin\n    lt (m - zero) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (m - zero) (succ m) \u2261\n                                           lt t (succ m))\n                                    (minus-x0 m)\n                                    refl\n                           \u27e9\n    lt m (succ m)          \u2261\u27e8 x<Sx Nm \u27e9\n    true\n  \u220e\n\nx-y<Sx zN (sN {n} Nn) =\n  begin\n    lt (zero - succ n) (succ zero)\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (zero - succ n) (succ zero) \u2261 lt t (succ zero))\n               (minus-0S n)\n               refl\n      \u27e9\n    lt zero (succ zero) \u2261\u27e8 lt-0S zero \u27e9\n    true\n  \u220e\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) =\n  begin\n    lt (succ m - succ n) (succ (succ m))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n) (succ (succ m)) \u2261\n                      lt t (succ (succ m)))\n               (minus-SS m n)\n               refl\n      \u27e9\n    lt (m - n) (succ (succ m))\n      \u2261\u27e8 <-trans (minus-N Nm Nn) (sN Nm) (sN (sN Nm))\n                 (x-y<Sx Nm Nn) (x<Sx (sN Nm))\n      \u27e9\n    true\n  \u220e\n\nSx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\nSx-Sy<Sx {m} {n} Nm Nn =\n  begin\n    lt (succ m - succ n) (succ m) \u2261\u27e8 subst (\u03bb t \u2192 lt (succ m - succ n)\n                                                     (succ m) \u2261\n                                                  lt t (succ m))\n                                           (minus-SS m n)\n                                           refl\n                                  \u27e9\n    lt (m - n) (succ m)           \u2261\u27e8 x-y<Sx Nm Nn \u27e9\n    true\n    \u220e\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN          Nn 0>n  = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = trans (+-rightIdentity (minus-N (sN Nm) zN))\n                                        (minus-x0 (succ m))\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  begin\n    (succ m - succ n) + succ n \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) + succ n \u2261\n                                               t + succ n)\n                                        (minus-SS m n)\n                                        refl\n                               \u27e9\n    (m - n) + succ n           \u2261\u27e8 +-comm (minus-N Nm Nn) (sN Nn) \u27e9\n    succ n + (m - n)           \u2261\u27e8 +-Sx n (m - n) \u27e9\n    succ (n + (m - n))         \u2261\u27e8 subst (\u03bb t \u2192 succ (n + (m - n)) \u2261 succ t )\n                                        (+-comm Nn (minus-N Nm Nn))\n                                        refl\n                               \u27e9\n    succ ((m - n) + n)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((m - n) + n) \u2261 succ t )\n                                        (x>y\u2192x-y+y\u2261x Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm>Sn) )\n                                        refl\n                               \u27e9\n    succ m\n  \u220e\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN      Nn 0\u2264n  = trans (+-rightIdentity (minus-N Nn zN))\n                                            (minus-x0 n)\nx\u2264y\u2192y-x+x\u2261y         (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  begin\n    (succ n - succ m) + succ m \u2261\u27e8 subst (\u03bb t \u2192 (succ n - succ m) + succ m \u2261\n                                               t + succ m)\n                                        (minus-SS n m)\n                                        refl\n                               \u27e9\n    (n - m) + succ m           \u2261\u27e8 +-comm (minus-N Nn Nm) (sN Nm) \u27e9\n    succ m + (n - m)           \u2261\u27e8 +-Sx m (n - m) \u27e9\n    succ (m + (n - m))         \u2261\u27e8 subst (\u03bb t \u2192 succ (m + (n - m)) \u2261 succ t )\n                                        (+-comm Nm (minus-N Nn Nm))\n                                        refl\n                               \u27e9\n    succ ((n - m) + m)         \u2261\u27e8 subst (\u03bb t \u2192 succ ((n - m) + m) \u2261 succ t )\n                                        (x\u2264y\u2192y-x+x\u2261y Nm Nn\n                                             (trans (sym (lt-SS n m)) Sm\u2264Sn) )\n                                        refl\n                               \u27e9\n    succ n\n  \u220e\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  trans (lt-SS m (succ n)) (x<y\u2192x<Sy Nm Nn (trans (sym (lt-SS m n)) Sm<Sn))\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\n\u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n\u00acxy<00 Nm Nn mn<00 =\n  [ (\u03bb m<0     \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u22610\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u22610\u2227n<0)))\n  ]\n  mn<00\n\n\u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n\u00ac0Sx<00 Nm 0Sm<00 =\n  [ (\u03bb 0<0      \u2192 \u22a5-elim (\u00acx<0 zN 0<0))\n  , (\u03bb 0\u22610\u2227Sm<0 \u2192 \u22a5-elim (\u00acx<0 (sN Nm) (\u2227-proj\u2082 0\u22610\u2227Sm<0)))\n  ]\n  0Sm<00\n\n\u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n\u00acSxy\u2081<0y\u2082 Nm Nn\u2081 Nn\u2082 Smn\u2081<0n\u2082 =\n  [ (\u03bb Sm<0       \u2192 \u22a5-elim (\u00acx<0 (sN Nm) Sm<0))\n  , (\u03bb Sm\u22610\u2227n\u2081<n\u2082 \u2192 \u22a5-elim (0\u2260S (sym (\u2227-proj\u2081 Sm\u22610\u2227n\u2081<n\u2082))))\n  ]\n  Smn\u2081<0n\u2082\n\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\nx\u2081y<x\u20820\u2192x\u2081<x\u2082 Nm\u2081 Nn Nm\u2082 m\u2081n<m\u2082zero =\n  [ (\u03bb m\u2081<n\u2081      \u2192 m\u2081<n\u2081)\n  , (\u03bb m\u2081\u2261n\u2081\u2227n<0 \u2192 \u22a5-elim (\u00acx<0 Nn (\u2227-proj\u2082 m\u2081\u2261n\u2081\u2227n<0)))\n  ]\n  m\u2081n<m\u2082zero\n\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                    m \u2261 zero \u2227 LT n\u2081 n\u2082\nxy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 Nm Nn\u2081 Nn\u2082 mn\u2081<0n\u2082 =\n  [ (\u03bb m<0          \u2192 \u22a5-elim (\u00acx<0 Nm m<0))\n  , (\u03bb m\u2261zero\u2227n\u2081<n\u2082 \u2192 m\u2261zero\u2227n\u2081<n\u2082)\n  ]\n  mn\u2081<0n\u2082\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = inj\u2081 (Sx-Sy<Sx Nm Nn)\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = inj\u2082 (refl , Sx-Sy<Sx Nn Nm)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"05cf6d36fb099224db91e27ee05ee8c9f2ee7efe","subject":"Nat: hiding -> using","message":"Nat: hiding -> using\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits\nopen import Data.Two.Base using (\ud835\udfda; 0\u2082; 1\u2082; \u2713; not; \u2713-not-\u00ac)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum.NP renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.Base\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.Base\n  using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe; isEquivalence)\n  renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple public\n  using (+-suc; +-*-suc; +-comm; *-comm; +-assoc; *-assoc)\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\npattern 5+_ x = 4+ suc x\npattern 6+_ x = 5+ suc x\npattern 7+_ x = 6+ suc x\npattern 8+_ x = 7+ suc x\npattern 9+_ x = 8+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 x       = \u27e80\u21941\u27e9 x\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0       x       = \u27e80\u21941\u27e9-involutive x\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\n\ninfixr 8 _\u2219\u2264_\n_\u2219<_ = <-trans\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inl z\u2264n\ntotal-\u2264 (suc a) zero = inr z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inl p = inl (s\u2264s p)\n... | inr p = inr (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite +-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromOp\u2082 _+_\n  renaming ( op= to += )\n  public\n\nopen FromAssocComm _+_ +-assoc +-comm\n  renaming ( assoc-comm to +-assoc-comm\n           ; assocs to +-assocs\n           ; interchange to +-interchange\n           ; !assoc-comm to +-!assoc-comm\n           ; comm= to +-comm=\n           ; assoc= to +-assoc=\n           ; !assoc= to +-!assoc=\n           ; inner= to +-inner=\n           ; outer= to +-outer=\n           )\n  public\n\nopen FromAssocComm _*_ *-assoc *-comm\n  renaming ( assoc-comm to *-assoc-comm\n           ; assocs to *-assocs\n           ; interchange to *-interchange\n           ; !assoc-comm to *-!assoc-comm\n           ; comm= to *-comm=\n           ; inner= to *-inner=\n           ; outer= to *-outer=\n           ; assoc= to *-assoc=\n           ; !assoc= to *-!assoc=\n           )\n  public\n\nno-<-> : \u2200 {x y} \u2192 x < y \u2192 x > y \u2192 \ud835\udfd8\nno-<-> (s\u2264s p) (s\u2264s q) = no-<-> p q\n\n\u22610\u2192\u225f0 : \u2200 n \u2192 n \u2261 0 \u2192 True (0 \u225f n)\n\u22610\u2192\u225f0 .0 idp = _\n\n\u225f0\u2192\u22610 : \u2200 n \u2192 True (0 \u225f n) \u2192 n \u2261 0\n\u225f0\u2192\u22610 zero p = idp\n\u225f0\u2192\u22610 (suc n) ()\n\n\u22620\u21d20< : \u2200 n \u2192 n \u2262 0 \u2192 0 < n\n\u22620\u21d20< zero x = \ud835\udfd8-elim (x idp)\n\u22620\u21d20< (suc n) x = s\u2264s z\u2264n\n\n\u2264\u2262\u2192< : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 y \u2192 x < y\n\u2264\u2262\u2192< z\u2264n     q = \u22620\u21d20< _ (q \u2218 !_)\n\u2264\u2262\u2192< (s\u2264s p) q = s\u2264s (\u2264\u2262\u2192< p (q \u2218 ap suc))\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite +-comm x y = \u2264-steps y \u2115\u2264.refl\n\n>\u2192\u2265 : \u2200 {m n} \u2192 m > n \u2192 m \u2265 n\n>\u2192\u2265 i = \u2264-pred (\u2115\u2264.trans i (\u2264-steps 1 \u2115\u2264.refl))\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\n+\u2264\u2192\u2264\u2238 : \u2200 {x} y {z} \u2192 y + x \u2264 z \u2192 x \u2264 z \u2238 y\n+\u2264\u2192\u2264\u2238 {x} y i with \u2264\u21d2\u2203 i\n+\u2264\u2192\u2264\u2238 {x} y i | k , idp =\n  x             \u2264\u27e8 \u2264-steps\u2032 k \u27e9\n  x + k         \u2261\u27e8 ! m+n\u2238n\u2261m _ y \u27e9\n  x + k + y \u2238 y \u2261\u27e8 ap (\u03bb z \u2192 z \u2238 y) (+-!assoc= {x} (+-comm k y)) \u27e9\n  x + y + k \u2238 y \u2261\u27e8 ap (\u03bb z \u2192 z + k \u2238 y) (+-comm x y) \u27e9\n  y + x + k \u2238 y \u220e\n  where open \u2264-Reasoning\n\n+-\u2238 : \u2200 x y z \u2192 x \u2261 y + z \u2192 y \u2261 x \u2238 z\n+-\u2238 .(y + z) y z idp =\n  y \u2261\u27e8 +-comm 0 y \u27e9\n  y + 0 \u2261\u27e8 ap (_+_ y) (! n\u2238n\u22610 z) \u27e9\n  y + (z \u2238 z) \u2261\u27e8 ! +-\u2238-assoc y (\u2115\u2264.refl {z}) \u27e9\n  (y + z) \u2238 z \u220e\n  where open \u2261-Reasoning\n\n\n+-\u2238' : \u2200 x y z \u2192 x + y \u2261 z \u2192 y \u2261 z \u2238 x\n+-\u2238' x y z e = +-\u2238 z y x (! e \u2219 +-comm x y)\n\n\u2261+-\u2265 : \u2200 x y z \u2192 x \u2261 y + z \u2192 x \u2265 z\n\u2261+-\u2265 .(y + z) y z idp = \u2264-steps y \u2115\u2264.refl\n\n+\u2264 : \u2200 x {y z} \u2192 x + y \u2264 z \u2192 x \u2264 z\n+\u2264 zero    i = z\u2264n\n+\u2264 (suc x) (s\u2264s i) = s\u2264s (+\u2264 x i)\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite +-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite *-comm (suc k) i\n        | *-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | +-comm x (1 + x) | +-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b          rewrite +-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n           rewrite +-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b) rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | +-comm b (suc b)\n                                              | +-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | +-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\nprivate\n  *1-identity : \u2200 n \u2192 n * 1 \u2261 n\n  *1-identity n = *-comm n 1 \u2219 +-comm n 0\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = +-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = *-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! *1-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = *1-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = +-comm 0 (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! *-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite *1-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite +-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | *-assoc 2 (2 ^ m) n\n                         | +-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\ninfixl 6 _+\u1d43_\n\n-- Accumulator based addition\n_+\u1d43_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u1d43 acc = acc\nsuc n +\u1d43 acc = n +\u1d43 suc acc\n\nopen FromOp\u2082 _+\u1d43_\n  renaming ( op= to +\u1d43= )\n  public\n\n+\u1d43-+ : \u2200 m n \u2192 m +\u1d43 n \u2261 m + n\n+\u1d43-+ zero n = idp\n+\u1d43-+ (suc m) n = +\u1d43-+ m (suc n) \u2219 +-suc m n\n\n+\u1d43-+= : \u2200 m m' {n n'} \u2192 m + n \u2261 m' + n' \u2192 m +\u1d43 n \u2261 m' +\u1d43 n'\n+\u1d43-+= m m' e = +\u1d43-+ m _ \u2219 e \u2219 ! +\u1d43-+ m' _\n\n+\u1d43-comm : Commutative _+\u1d43_\n+\u1d43-comm x y = +\u1d43-+= x y (+-comm x y)\n\n+\u1d43-assoc : Associative _+\u1d43_\n+\u1d43-assoc x y z = +\u1d43-+= (x +\u1d43 y) x (+= (+\u1d43-+ x y) idp \u2219 +-assoc x y z \u2219 ap (_+_ x) (! +\u1d43-+ y z))\n\n+\u1d430-identity : \u2200 x \u2192 x +\u1d43 0 \u2261 x\n+\u1d430-identity x = +\u1d43-comm x 0\n\nopen FromAssocComm _+\u1d43_ +\u1d43-assoc +\u1d43-comm\n  renaming ( assoc-comm to +\u1d43-assoc-comm\n           ; assocs to +\u1d43-assocs\n           ; interchange to +\u1d43-interchange\n           ; !assoc-comm to +\u1d43-!assoc-comm\n           ; comm= to +\u1d43-comm=\n           ; assoc= to +\u1d43-assoc=\n           ; !assoc= to +\u1d43-!assoc=\n           ; inner= to +\u1d43-inner=\n           ; outer= to +\u1d43-outer=\n           )\n  public\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n\nsplit-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\nsplit-\u2264 {zero} {zero} p = inl idp\nsplit-\u2264 {zero} {suc y} p = inr (s\u2264s z\u2264n)\nsplit-\u2264 {suc x} {zero} ()\nsplit-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n... | inl q rewrite q = inl idp\n... | inr q = inr (s\u2264s q)\n\n<\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n<\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits\nopen import Data.Two.Base hiding (_==_; _\u225f_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum.NP renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.Base\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.Base\n  using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe; isEquivalence)\n  renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple public\n  using (+-suc; +-*-suc; +-comm; *-comm; +-assoc; *-assoc)\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\npattern 5+_ x = 4+ suc x\npattern 6+_ x = 5+ suc x\npattern 7+_ x = 6+ suc x\npattern 8+_ x = 7+ suc x\npattern 9+_ x = 8+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 x       = \u27e80\u21941\u27e9 x\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0       x       = \u27e80\u21941\u27e9-involutive x\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\n\ninfixr 8 _\u2219\u2264_\n_\u2219<_ = <-trans\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inl z\u2264n\ntotal-\u2264 (suc a) zero = inr z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inl p = inl (s\u2264s p)\n... | inr p = inr (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite +-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromOp\u2082 _+_\n  renaming ( op= to += )\n  public\n\nopen FromAssocComm _+_ +-assoc +-comm\n  renaming ( assoc-comm to +-assoc-comm\n           ; assocs to +-assocs\n           ; interchange to +-interchange\n           ; !assoc-comm to +-!assoc-comm\n           ; comm= to +-comm=\n           ; assoc= to +-assoc=\n           ; !assoc= to +-!assoc=\n           ; inner= to +-inner=\n           ; outer= to +-outer=\n           )\n  public\n\nopen FromAssocComm _*_ *-assoc *-comm\n  renaming ( assoc-comm to *-assoc-comm\n           ; assocs to *-assocs\n           ; interchange to *-interchange\n           ; !assoc-comm to *-!assoc-comm\n           ; comm= to *-comm=\n           ; inner= to *-inner=\n           ; outer= to *-outer=\n           ; assoc= to *-assoc=\n           ; !assoc= to *-!assoc=\n           )\n  public\n\nno-<-> : \u2200 {x y} \u2192 x < y \u2192 x > y \u2192 \ud835\udfd8\nno-<-> (s\u2264s p) (s\u2264s q) = no-<-> p q\n\n\u22610\u2192\u225f0 : \u2200 n \u2192 n \u2261 0 \u2192 True (0 \u225f n)\n\u22610\u2192\u225f0 .0 idp = _\n\n\u225f0\u2192\u22610 : \u2200 n \u2192 True (0 \u225f n) \u2192 n \u2261 0\n\u225f0\u2192\u22610 zero p = idp\n\u225f0\u2192\u22610 (suc n) ()\n\n\u22620\u21d20< : \u2200 n \u2192 n \u2262 0 \u2192 0 < n\n\u22620\u21d20< zero x = \ud835\udfd8-elim (x idp)\n\u22620\u21d20< (suc n) x = s\u2264s z\u2264n\n\n\u2264\u2262\u2192< : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 y \u2192 x < y\n\u2264\u2262\u2192< z\u2264n     q = \u22620\u21d20< _ (q \u2218 !_)\n\u2264\u2262\u2192< (s\u2264s p) q = s\u2264s (\u2264\u2262\u2192< p (q \u2218 ap suc))\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite +-comm x y = \u2264-steps y \u2115\u2264.refl\n\n>\u2192\u2265 : \u2200 {m n} \u2192 m > n \u2192 m \u2265 n\n>\u2192\u2265 i = \u2264-pred (\u2115\u2264.trans i (\u2264-steps 1 \u2115\u2264.refl))\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\n+\u2264\u2192\u2264\u2238 : \u2200 {x} y {z} \u2192 y + x \u2264 z \u2192 x \u2264 z \u2238 y\n+\u2264\u2192\u2264\u2238 {x} y i with \u2264\u21d2\u2203 i\n+\u2264\u2192\u2264\u2238 {x} y i | k , idp =\n  x             \u2264\u27e8 \u2264-steps\u2032 k \u27e9\n  x + k         \u2261\u27e8 ! m+n\u2238n\u2261m _ y \u27e9\n  x + k + y \u2238 y \u2261\u27e8 ap (\u03bb z \u2192 z \u2238 y) (+-!assoc= {x} (+-comm k y)) \u27e9\n  x + y + k \u2238 y \u2261\u27e8 ap (\u03bb z \u2192 z + k \u2238 y) (+-comm x y) \u27e9\n  y + x + k \u2238 y \u220e\n  where open \u2264-Reasoning\n\n+-\u2238 : \u2200 x y z \u2192 x \u2261 y + z \u2192 y \u2261 x \u2238 z\n+-\u2238 .(y + z) y z idp =\n  y \u2261\u27e8 +-comm 0 y \u27e9\n  y + 0 \u2261\u27e8 ap (_+_ y) (! n\u2238n\u22610 z) \u27e9\n  y + (z \u2238 z) \u2261\u27e8 ! +-\u2238-assoc y (\u2115\u2264.refl {z}) \u27e9\n  (y + z) \u2238 z \u220e\n  where open \u2261-Reasoning\n\n\n+-\u2238' : \u2200 x y z \u2192 x + y \u2261 z \u2192 y \u2261 z \u2238 x\n+-\u2238' x y z e = +-\u2238 z y x (! e \u2219 +-comm x y)\n\n\u2261+-\u2265 : \u2200 x y z \u2192 x \u2261 y + z \u2192 x \u2265 z\n\u2261+-\u2265 .(y + z) y z idp = \u2264-steps y \u2115\u2264.refl\n\n+\u2264 : \u2200 x {y z} \u2192 x + y \u2264 z \u2192 x \u2264 z\n+\u2264 zero    i = z\u2264n\n+\u2264 (suc x) (s\u2264s i) = s\u2264s (+\u2264 x i)\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite +-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite *-comm (suc k) i\n        | *-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | +-comm x (1 + x) | +-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b          rewrite +-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n           rewrite +-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b) rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | +-comm b (suc b)\n                                              | +-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | +-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\nprivate\n  *1-identity : \u2200 n \u2192 n * 1 \u2261 n\n  *1-identity n = *-comm n 1 \u2219 +-comm n 0\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = +-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = *-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! *1-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = *1-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = +-comm 0 (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! *-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite *1-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite +-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | *-assoc 2 (2 ^ m) n\n                         | +-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\ninfixl 6 _+\u1d43_\n\n-- Accumulator based addition\n_+\u1d43_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u1d43 acc = acc\nsuc n +\u1d43 acc = n +\u1d43 suc acc\n\nopen FromOp\u2082 _+\u1d43_\n  renaming ( op= to +\u1d43= )\n  public\n\n+\u1d43-+ : \u2200 m n \u2192 m +\u1d43 n \u2261 m + n\n+\u1d43-+ zero n = idp\n+\u1d43-+ (suc m) n = +\u1d43-+ m (suc n) \u2219 +-suc m n\n\n+\u1d43-+= : \u2200 m m' {n n'} \u2192 m + n \u2261 m' + n' \u2192 m +\u1d43 n \u2261 m' +\u1d43 n'\n+\u1d43-+= m m' e = +\u1d43-+ m _ \u2219 e \u2219 ! +\u1d43-+ m' _\n\n+\u1d43-comm : Commutative _+\u1d43_\n+\u1d43-comm x y = +\u1d43-+= x y (+-comm x y)\n\n+\u1d43-assoc : Associative _+\u1d43_\n+\u1d43-assoc x y z = +\u1d43-+= (x +\u1d43 y) x (+= (+\u1d43-+ x y) idp \u2219 +-assoc x y z \u2219 ap (_+_ x) (! +\u1d43-+ y z))\n\n+\u1d430-identity : \u2200 x \u2192 x +\u1d43 0 \u2261 x\n+\u1d430-identity x = +\u1d43-comm x 0\n\nopen FromAssocComm _+\u1d43_ +\u1d43-assoc +\u1d43-comm\n  renaming ( assoc-comm to +\u1d43-assoc-comm\n           ; assocs to +\u1d43-assocs\n           ; interchange to +\u1d43-interchange\n           ; !assoc-comm to +\u1d43-!assoc-comm\n           ; comm= to +\u1d43-comm=\n           ; assoc= to +\u1d43-assoc=\n           ; !assoc= to +\u1d43-!assoc=\n           ; inner= to +\u1d43-inner=\n           ; outer= to +\u1d43-outer=\n           )\n  public\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n\nsplit-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\nsplit-\u2264 {zero} {zero} p = inl idp\nsplit-\u2264 {zero} {suc y} p = inr (s\u2264s z\u2264n)\nsplit-\u2264 {suc x} {zero} ()\nsplit-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n... | inl q rewrite q = inl idp\n... | inr q = inr (s\u2264s q)\n\n<\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n<\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bd3c7e9ede5e1077663b478e2d90ba337c971567","subject":"FFI.JS.BigI: add Eq? instance, thus depends on nplib","message":"FFI.JS.BigI: add Eq? instance, thus depends on nplib\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS\/BigI.agda","new_file":"lib\/FFI\/JS\/BigI.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import FFI.JS hiding (toString)\nopen import Type.Eq using (Eq?)\nopen import Data.Bool.Base using (Bool; true; false) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; refl)\nopen import Relation.Binary.PropositionalEquality.TrustMe using (trustMe)\n\nmodule FFI.JS.BigI where\n\npostulate BigI : Set\n\npostulate BigI\u25b9JSValue : BigI \u2192 JSValue\n{-# COMPILED_JS BigI\u25b9JSValue function(x) { return x; } #-}\n\npostulate unsafe-JSValue\u25b9BigI : JSValue \u2192 BigI\n{-# COMPILED_JS unsafe-JSValue\u25b9BigI function(x) { return x; } #-}\n\npostulate bigI     : (x base : String) \u2192 BigI\n{-# COMPILED_JS bigI       function(x) { return function (y) { return require(\"bigi\")(x,y); }; } #-}\n\npostulate negate   : (x : BigI) \u2192 BigI\n{-# COMPILED_JS negate     function(x) { return x.negate(); } #-}\n\npostulate add      : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS add        function(x) { return function (y) { return x.add(y); }; } #-}\n\npostulate subtract : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS subtract   function(x) { return function (y) { return x.subtract(y); }; } #-}\n\npostulate multiply : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS multiply   function(x) { return function (y) { return x.multiply(y); }; } #-}\n\npostulate divide   : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS divide     function(x) { return function (y) { return x.divide(y); }; } #-}\n\npostulate remainder : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS remainder  function(x) { return function (y) { return x.remainder(y); }; } #-}\n\npostulate mod      : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS mod        function(x) { return function (y) { return x.mod(y); }; } #-}\n\npostulate modPow   : (this e m : BigI) \u2192 BigI\n{-# COMPILED_JS modPow     function(x) { return function (y) { return function (z) { return x.modPow(y,z); }; }; } #-}\n\npostulate modInv   : (this m   : BigI) \u2192 BigI\n{-# COMPILED_JS modInv     function(x) { return function (y) { return x.modInverse(y); }; } #-}\n\npostulate gcd      : (this m   : BigI) \u2192 BigI\n{-# COMPILED_JS gcd        function(x) { return function (y) { return x.gcd(y); }; } #-}\n\n-- test primality with certainty >= 1-.5^t\npostulate isProbablePrime : (this : BigI)(t : Number) \u2192 Bool\n{-# COMPILED_JS isProbablePrime function(x) { return function (y) { return x.isProbablePrime(y); }; } #-}\n\npostulate signum   : (this : BigI) \u2192 Number\n{-# COMPILED_JS signum function(x) { return x.signum(); } #-}\n\npostulate equals   : (x y : BigI) \u2192 Bool\n{-# COMPILED_JS equals     function(x) { return function (y) { return x.equals(y); }; } #-}\n\npostulate compareTo : (this x : BigI) \u2192 Number\n{-# COMPILED_JS compareTo  function(x) { return function (y) { return x.compareTo(y); }; } #-}\n\npostulate toString : (x : BigI) \u2192 String\n{-# COMPILED_JS toString   function(x) { return x.toString(); } #-}\n\npostulate fromHex  : String \u2192 BigI\n{-# COMPILED_JS fromHex    require(\"bigi\").fromHex #-}\n\npostulate toHex    : BigI \u2192 String\n{-# COMPILED_JS toHex      function(x) { return x.toHex(); } #-}\n\npostulate byteLength : BigI \u2192 Number\n{-# COMPILED_JS byteLength function(x) { return x.byteLength(); } #-}\n\n0I 1I 2I : BigI\n0I = bigI \"0\" \"10\"\n1I = bigI \"1\" \"10\"\n2I = bigI \"2\" \"10\"\n\n_\u2264I_ : BigI \u2192 BigI \u2192 Bool\nx \u2264I y = compareTo x y \u2264Number 0N\n\n_<I_ : BigI \u2192 BigI \u2192 Bool\nx <I y = compareTo x y <Number 0N\n\n_>I_ : BigI \u2192 BigI \u2192 Bool\nx >I y = compareTo x y >Number 0N\n\n_\u2265I_ : BigI \u2192 BigI \u2192 Bool\nx \u2265I y = compareTo x y \u2265Number 0N\n\n-- This is a postulates which computes as much as it needs\nequals-refl : \u2200 {x : BigI} \u2192 \u2713 (equals x x)\nequals-refl {x} with equals x x\n... | true  = _\n... | false = STUCK where postulate STUCK : _\n\ninstance\n  BigI-Eq? : Eq? BigI\n  BigI-Eq? = record\n    { _==_ = _==_\n    ; \u2261\u21d2== = \u2261\u21d2==\n    ; ==\u21d2\u2261 = ==\u21d2\u2261 }\n    module BigI-Eq? where\n      _==_ : BigI \u2192 BigI \u2192 Bool\n      _==_ = equals\n      \u2261\u21d2== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n      \u2261\u21d2== refl = equals-refl\n      ==\u21d2\u2261 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y\n      ==\u21d2\u2261 _ = trustMe\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import FFI.JS hiding (toString)\n\nmodule FFI.JS.BigI where\n\npostulate BigI : Set\n\npostulate BigI\u25b9JSValue : BigI \u2192 JSValue\n{-# COMPILED_JS BigI\u25b9JSValue function(x) { return x; } #-}\n\npostulate unsafe-JSValue\u25b9BigI : JSValue \u2192 BigI\n{-# COMPILED_JS unsafe-JSValue\u25b9BigI function(x) { return x; } #-}\n\npostulate bigI     : (x base : String) \u2192 BigI\n{-# COMPILED_JS bigI       function(x) { return function (y) { return require(\"bigi\")(x,y); }; } #-}\n\npostulate negate   : (x : BigI) \u2192 BigI\n{-# COMPILED_JS negate     function(x) { return x.negate(); } #-}\n\npostulate add      : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS add        function(x) { return function (y) { return x.add(y); }; } #-}\n\npostulate subtract : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS subtract   function(x) { return function (y) { return x.subtract(y); }; } #-}\n\npostulate multiply : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS multiply   function(x) { return function (y) { return x.multiply(y); }; } #-}\n\npostulate divide   : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS divide     function(x) { return function (y) { return x.divide(y); }; } #-}\n\npostulate remainder : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS remainder  function(x) { return function (y) { return x.remainder(y); }; } #-}\n\npostulate mod      : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS mod        function(x) { return function (y) { return x.mod(y); }; } #-}\n\npostulate modPow   : (this e m : BigI) \u2192 BigI\n{-# COMPILED_JS modPow     function(x) { return function (y) { return function (z) { return x.modPow(y,z); }; }; } #-}\n\npostulate modInv   : (this m   : BigI) \u2192 BigI\n{-# COMPILED_JS modInv     function(x) { return function (y) { return x.modInverse(y); }; } #-}\n\npostulate gcd      : (this m   : BigI) \u2192 BigI\n{-# COMPILED_JS gcd        function(x) { return function (y) { return x.gcd(y); }; } #-}\n\n-- test primality with certainty >= 1-.5^t\npostulate isProbablePrime : (this : BigI)(t : Number) \u2192 Bool\n{-# COMPILED_JS isProbablePrime function(x) { return function (y) { return x.isProbablePrime(y); }; } #-}\n\npostulate signum   : (this : BigI) \u2192 Number\n{-# COMPILED_JS signum function(x) { return x.signum(); } #-}\n\npostulate equals   : (x y : BigI) \u2192 Bool\n{-# COMPILED_JS equals     function(x) { return function (y) { return x.equals(y); }; } #-}\n\npostulate compareTo : (this x : BigI) \u2192 Number\n{-# COMPILED_JS compareTo  function(x) { return function (y) { return x.compareTo(y); }; } #-}\n\npostulate toString : (x : BigI) \u2192 String\n{-# COMPILED_JS toString   function(x) { return x.toString(); } #-}\n\npostulate fromHex  : String \u2192 BigI\n{-# COMPILED_JS fromHex    require(\"bigi\").fromHex #-}\n\npostulate toHex    : BigI \u2192 String\n{-# COMPILED_JS toHex      function(x) { return x.toHex(); } #-}\n\npostulate byteLength : BigI \u2192 Number\n{-# COMPILED_JS byteLength function(x) { return x.byteLength(); } #-}\n\n0I 1I 2I : BigI\n0I = bigI \"0\" \"10\"\n1I = bigI \"1\" \"10\"\n2I = bigI \"2\" \"10\"\n\n_\u2264I_ : BigI \u2192 BigI \u2192 Bool\nx \u2264I y = compareTo x y \u2264Number 0N\n\n_<I_ : BigI \u2192 BigI \u2192 Bool\nx <I y = compareTo x y <Number 0N\n\n_>I_ : BigI \u2192 BigI \u2192 Bool\nx >I y = compareTo x y >Number 0N\n\n_\u2265I_ : BigI \u2192 BigI \u2192 Bool\nx \u2265I y = compareTo x y \u2265Number 0N\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a3dfe39d1cf1f696b4da9f2b3347315c15fe79e5","subject":"Vec: tabulate-\u2218","message":"Vec: tabulate-\u2218\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3208436839812a87ab544bee42ac9a6668713748","subject":"Removed $ from a module.","message":"Removed $ from a module.\n\nIgnore-this: 4538e6f19fd38a4038b09471605febf6\n\ndarcs-hash:20120602034612-3bd4e-1e9935da91bfd44d3818eef64dbc3da563f75109.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC-PCF\/Data\/Nat\/PropertiesI.agda","new_file":"src\/LTC-PCF\/Data\/Nat\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n+-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n+-Sx m n =\n  rec (succ\u2081 m) n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)))\n    \u2261\u27e8 rec-S m n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u00b7 m \u00b7 (m + n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m \u00b7 (m + n)\n    \u2261\u27e8 refl \u27e9\n  lam succ\u2081 \u00b7 (m + n)\n    \u2261\u27e8 beta succ\u2081 (m + n) \u27e9\n  succ\u2081 (m + n) \u220e\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S zN =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 cong pred\u2081 (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN      = \u2238-x0 zero\n\u2238-0x (sN Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ zN =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS zN (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS zN Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (sN {m} Nm) (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS (sN Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (beta pred\u2081 (succ\u2081 m \u2238 n)) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 refl \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n)) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = rec zero zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u2261\u27e8 rec-0 zero \u27e9\n         zero                                          \u220e\n\n*-Sx : \u2200 m n \u2192 succ\u2081 m * n \u2261 n + m * n\n*-Sx m n =\n  rec (succ\u2081 m) zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)))\n    \u2261\u27e8 rec-S m zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u27e9\n  (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u00b7 m \u00b7 (m * n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m) \u27e9\n  (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m \u00b7 (m * n)\n    \u2261\u27e8 refl \u27e9\n  lam (\u03bb y \u2192 n + y) \u00b7 (m * n)\n    \u2261\u27e8 beta (\u03bb y \u2192 n + y) (m * n) \u27e9\n  (\u03bb y \u2192 n + y) (m * n)\n    \u2261\u27e8 refl \u27e9\n  n + (m * n) \u220e\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym (+-0x n)) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm       zN     = subst N (sym (\u2238-x0 m)) Nm\n\u2238-N     zN      (sN Nn) = subst N (sym (\u2238-0S Nn)) zN\n\u2238-N     (sN Nm) (sN Nn) = subst N (sym (\u2238-SS Nm Nn)) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t) (+-rightIdentity Nn) refl)\n\n\n+-assoc : \u2200 {m} \u2192 N m \u2192  \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zN n o =\n  zero + n + o   \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (sN {m} Nm) n o =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm n o) refl \u27e9\n  succ\u2081 (m + (n + o))\n    \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] zN n =\n  zero + succ\u2081 n \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym (+-leftIdentity n)) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (sN {m} Nm) n =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] Nm n) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym (+-Sx m n)) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n    n \u2238 (zero + o)\n      \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n    n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN          _  = subst N (sym (*-0x n)) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero n) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n              \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = sym\n  ( zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero n) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym (*-leftZero (succ\u2081 n)) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym (+-assoc Nn m (m * n)))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n             refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym (*-Sx m n))\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero n) (sym (*-rightZero Nn))\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n  \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m    \u2261\u27e8 sym (x*Sy\u2261x+xy Nn Nm) \u27e9\n  n * succ\u2081 m  \u220e\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o            \u2261\u27e8 sym (\u2238-x0 (m * o)) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym (*-0x o)) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S Nn) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0x (*-N (sN Nn) No)) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o)\n             (sym (*-0x o))\n             refl\n    \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym (\u2238-0x (*-N (sN Nn) zN)) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym (*-0x (succ\u2081 m)))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (\u2238-SS Nm Nn)\n             refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n    \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym ([x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No))) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym (*-Sx m (succ\u2081 o)))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym (*-Sx n (succ\u2081 o)))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n  zero                 \u2261\u27e8 sym (*-0x m) \u27e9\n  zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero             \u2261\u27e8 sym (+-rightIdentity (*-N Nm zN)) \u27e9\n  m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                (trans (sym (*-0x n)) (*-comm zN Nn))\n                                  refl\n                       \u27e9\n  m * zero + n * zero  \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym (+-leftIdentity (n * succ\u2081 o)) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym (*-0x (succ\u2081 o)))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym (+-rightIdentity (*-N (sN Nm) (sN No))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym (*-leftZero (succ\u2081 o)))\n             refl\n    \u27e9\n    succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym (+-assoc (sN No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym (*-Sx m (succ\u2081 o)))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Base.Properties\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Rec\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\n+-0x : \u2200 n \u2192 zero + n \u2261 n\n+-0x n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n+-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n+-Sx m n =\n  rec (succ\u2081 m) n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)))\n    \u2261\u27e8 rec-S m n (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y))) \u00b7 m \u00b7 (m + n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 succ\u2081 y)) m \u00b7 (m + n)\n    \u2261\u27e8 refl \u27e9\n  lam succ\u2081 \u00b7 (m + n)\n    \u2261\u27e8 beta succ\u2081 (m + n) \u27e9\n  succ\u2081 (m + n) \u220e\n\n\u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n\u2238-x0 n = rec zero n _ \u2261\u27e8 rec-0 n \u27e9\n         n            \u220e\n\n\u2238-0S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n\u2238-0S zN =\n  rec (succ\u2081 zero) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 zero) \u27e9\n  pred\u2081 (zero \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 zero) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0S (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) zero (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  (succ\u2081 n) \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 beta pred\u2081 (zero \u2238 (succ\u2081 n)) \u27e9\n  pred\u2081 (zero \u2238 (succ\u2081 n))\n    \u2261\u27e8 cong pred\u2081 (\u2238-0S Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero \u220e\n\n\u2238-0x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n\u2238-0x zN      = \u2238-x0 zero\n\u2238-0x (sN Nn) = \u2238-0S Nn\n\n\u2238-SS : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n\u2238-SS {m} _ zN =\n  rec (succ\u2081 zero) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S zero (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))  \u00b7 zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) zero \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 m \u2238 zero) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 cong pred\u2081 (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym $ \u2238-x0 m \u27e9\n  m \u2238 zero \u220e\n\n\u2238-SS zN (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 zero) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))  \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 zero \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS zN Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-0x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0S Nn \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\n\u2238-SS (sN {m} Nm) (sN {n} Nn) =\n  rec (succ\u2081 (succ\u2081 n)) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 rec-S (succ\u2081 n) (succ\u2081 (succ\u2081 m)) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) \u00b7 (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n)) \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) (succ\u2081 n) \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 refl \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 beta pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 cong pred\u2081 (\u2238-SS (sN Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym $ beta pred\u2081 (succ\u2081 m \u2238 n) \u27e9\n  lam pred\u2081 \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 refl \u27e9\n  (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 \u00b7-leftCong (sym $ beta (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)) n) \u27e9\n  (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u00b7 n \u00b7 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym $ rec-S n (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y))) \u27e9\n  rec (succ\u2081 n) (succ\u2081 m) (lam (\u03bb x \u2192 lam (\u03bb y \u2192 pred\u2081 y)))\n    \u2261\u27e8 refl \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\n*-0x : \u2200 n \u2192 zero * n \u2261 zero\n*-0x n = rec zero zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u2261\u27e8 rec-0 zero \u27e9\n         zero                                          \u220e\n\n*-Sx : \u2200 m n \u2192 succ\u2081 m * n \u2261 n + m * n\n*-Sx m n =\n  rec (succ\u2081 m) zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)))\n    \u2261\u27e8 rec-S m zero (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u27e9\n  (lam (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y))) \u00b7 m \u00b7 (m * n)\n    \u2261\u27e8 \u00b7-leftCong (beta (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m) \u27e9\n  (\u03bb _ \u2192 lam (\u03bb y \u2192 n + y)) m \u00b7 (m * n)\n    \u2261\u27e8 refl \u27e9\n  lam (\u03bb y \u2192 n + y) \u00b7 (m * n)\n    \u2261\u27e8 beta (\u03bb y \u2192 n + y) (m * n) \u27e9\n  (\u03bb y \u2192 n + y) (m * n)\n    \u2261\u27e8 refl \u27e9\n  n + (m * n) \u220e\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} zN          Nn = subst N (sym $ +-0x n) Nn\n+-N {n = n} (sN {m} Nm) Nn = subst N (sym $ +-Sx m n) (sN (+-N Nm Nn))\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm       zN     = subst N (sym $ \u2238-x0 m) Nm\n\u2238-N     zN      (sN Nn) = subst N (sym $ \u2238-0S Nn) zN\n\u2238-N     (sN Nm) (sN Nn) = subst N (sym $ \u2238-SS Nm Nn) (\u2238-N Nm Nn)\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity zN          = +-leftIdentity zero\n+-rightIdentity (sN {n} Nn) =\n  trans (+-Sx n zero)\n        (subst (\u03bb t \u2192 succ\u2081 (n + zero) \u2261 succ\u2081 t) (+-rightIdentity Nn) refl)\n\n\n+-assoc : \u2200 {m} \u2192 N m \u2192  \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc zN n o =\n  zero + n + o   \u2261\u27e8 subst (\u03bb t \u2192 zero + n + o \u2261 t + o) (+-leftIdentity n) refl \u27e9\n  n + o          \u2261\u27e8 sym $ +-leftIdentity (n + o) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (sN {m} Nm) n o =\n  succ\u2081 m + n + o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n + o \u2261 t + o) (+-Sx m n) refl \u27e9\n  succ\u2081 (m + n) + o\n    \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + o) \u2261 succ\u2081 t) (+-assoc Nm n o) refl \u27e9\n  succ\u2081 (m + (n + o))\n    \u2261\u27e8 sym $ +-Sx m (n + o) \u27e9\n  succ\u2081 m + (n + o) \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] zN n =\n  zero + succ\u2081 n \u2261\u27e8 +-0x (succ\u2081 n) \u27e9\n  succ\u2081 n \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n \u2261 succ\u2081 t) (sym $ +-leftIdentity n) refl \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (sN {m} Nm) n =\n  succ\u2081 m + succ\u2081 n\n    \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + succ\u2081 n) \u2261 succ\u2081 t) (x+Sy\u2261S[x+y] Nm n) refl \u27e9\n  succ\u2081 (succ\u2081 (m + n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (succ\u2081 (m + n)) \u2261 succ\u2081 t) (sym $ +-Sx m n) refl \u27e9\n  succ\u2081 (succ\u2081 m + n) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} zN _ _ =\n  (zero + n) \u2238 (zero + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) \u2238 (zero + o) \u2261 t \u2238 (zero + o)) (+-0x n) refl \u27e9\n    n \u2238 (zero + o)\n      \u2261\u27e8 subst (\u03bb t \u2192 n \u2238 (zero + o) \u2261 n \u2238 t) (+-0x o) refl \u27e9\n    n \u2238 o \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (sN {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + n \u2238 (succ\u2081 m + o) \u2261 t \u2238 (succ\u2081 m + o))\n             (+-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261 succ\u2081 (m + n) \u2238 t)\n             (+-Sx m o)\n             refl\n    \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o)\n    \u2261\u27e8 \u2238-SS (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)\n    \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} zN Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym $ +-rightIdentity Nn \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 cong succ\u2081 (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym $ x+Sy\u2261S[x+y] Nn m \u27e9\n  n + succ\u2081 m   \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} zN          _  = subst N (sym $ *-0x n) zN\n*-N {n = n} (sN {m} Nm) Nn = subst N (sym $ *-Sx m n) (+-N Nn (*-N Nm Nn))\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero zN          = *-leftZero zero\n*-rightZero (sN {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 succ\u2081 zero * n \u2261 n\n*-leftIdentity {n} Nn =\n  succ\u2081 zero * n \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n   \u2261\u27e8 subst (\u03bb t \u2192 n + zero * n \u2261 n + t) (*-leftZero n) refl \u27e9\n  n + zero       \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n              \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} zN _ = sym\n  ( zero + zero * n\n      \u2261\u27e8 subst (\u03bb t \u2192 zero + zero * n \u2261 zero + t) (*-leftZero n) refl \u27e9\n    zero + zero\n      \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero\n      \u2261\u27e8 sym $ *-leftZero (succ\u2081 n) \u27e9\n    zero * succ\u2081 n \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 n + m * succ\u2081 n \u2261 succ\u2081 n + t)\n             (x*Sy\u2261x+xy Nm Nn)\n             refl\n    \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + (m + m * n)) \u2261 succ\u2081 t)\n             (sym $ +-assoc Nn m (m * n))\n             refl\n    \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (n + m + m * n) \u2261 succ\u2081 (t + m * n))\n             (+-comm Nn Nm)\n             refl\n    \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n             refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym $ +-Sx m (n + m * n) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m + (n + m * n) \u2261 succ\u2081 m + t)\n             (sym $ *-Sx m n)\n             refl\n    \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} zN Nn          = trans (*-leftZero n) (sym $ *-rightZero Nn)\n*-comm {n = n} (sN {m} Nm) Nn =\n  succ\u2081 m * n  \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n    \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m    \u2261\u27e8 sym $ x*Sy\u2261x+xy Nn Nm \u27e9\n  n * succ\u2081 m  \u220e\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ zN _ =\n  (m \u2238 zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m \u2238 zero) * o \u2261 t * o) (\u2238-x0 m) refl \u27e9\n  m * o            \u2261\u27e8 sym $ \u2238-x0 (m * o) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o \u2238 zero \u2261 m * o \u2238 t) (sym $ *-0x o) refl \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} zN (sN {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero \u2238 succ\u2081 n) * o \u2261 t * o) (\u2238-0S Nn) refl \u27e9\n  zero * o\n    \u2261\u27e8 *-0x o \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) No) \u27e9\n  zero \u2238 succ\u2081 n * o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * o \u2261 t \u2238 succ\u2081 n * o)\n             (sym $ *-0x o)\n             refl\n    \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) zN =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (sN Nm) (sN Nn)) zN \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-0x (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym $ \u2238-0x (*-N (sN Nn) zN) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (sym $ *-0x (succ\u2081 m))\n             refl\n    \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 subst (\u03bb t \u2192 zero * succ\u2081 m \u2238 succ\u2081 n * zero \u2261 t \u2238 succ\u2081 n * zero)\n             (*-comm zN (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (\u2238-SS Nm Nn)\n             refl\n    \u27e9\n  (m \u2238 n) * succ\u2081 o\n    \u2261\u27e8 *\u2238-leftDistributive Nm Nn (sN No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym $ [x+y]\u2238[x+z]\u2261y\u2238z (sN No) (*-N Nm (sN No)) (*-N Nn (sN No)) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    t \u2238 (succ\u2081 o + n * succ\u2081 o))\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o) \u2261\n                    (succ\u2081 m * succ\u2081 o) \u2238 t)\n             (sym $ *-Sx n (succ\u2081 o))\n             refl\n    \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn zN =\n  (m + n) * zero       \u2261\u27e8 *-comm (+-N Nm Nn) zN \u27e9\n  zero * (m + n)       \u2261\u27e8 *-0x (m + n) \u27e9\n  zero                 \u2261\u27e8 sym $ *-0x m \u27e9\n  zero * m             \u2261\u27e8 *-comm zN Nm \u27e9\n  m * zero             \u2261\u27e8 sym $ +-rightIdentity (*-N Nm zN) \u27e9\n  m * zero + zero      \u2261\u27e8 subst (\u03bb t \u2192 m * zero + zero \u2261 m * zero + t)\n                                (trans (sym $ *-0x n) (*-comm zN Nn))\n                                  refl\n                       \u27e9\n  m * zero + n * zero  \u220e\n\n*+-leftDistributive {n = n} zN Nn (sN {o} No) =\n  (zero + n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-leftIdentity n)\n             refl\n    \u27e9\n  n * succ\u2081 o\n    \u2261\u27e8 sym $ +-leftIdentity (n * succ\u2081 o) \u27e9\n  zero + n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 zero + n * succ\u2081 o \u2261 t +  n * succ\u2081 o)\n             (sym $ *-0x (succ\u2081 o))\n             refl\n    \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) zN (sN {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + zero) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-rightIdentity (sN Nm))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym $ +-rightIdentity (*-N (sN Nm) (sN No)) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 m * succ\u2081 o + zero \u2261 succ\u2081 m * succ\u2081 o + t)\n             (sym $ *-leftZero (succ\u2081 o))\n             refl\n    \u27e9\n    succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 (succ\u2081 m + succ\u2081 n) * succ\u2081 o \u2261 t * succ\u2081 o)\n             (+-Sx m (succ\u2081 n))\n             refl\n    \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o \u2261 succ\u2081 o + t)\n             (*+-leftDistributive Nm (sN Nn) (sN No))\n             refl\n    \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym $ +-assoc (sN No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u2261\n                    t + succ\u2081 n * succ\u2081 o)\n             (sym $ *-Sx m (succ\u2081 o))\n             refl\n    \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7f6d69823a2e8343796f9f591afdfafbe467f5fe","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (d0' , Step (FHFinal x) () (FHFinal x\u2081))\n  is IEHole (d0' , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step (FHFinal x) () FHEHole)\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step (FHFinal x) () (FHCastFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (d , Step (FHFinal x\u2081) () (FHFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHEHole)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2082))\n  is (INEHole x) (d , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (d' , Step p q r) = {!!}\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n--  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n\n  --es (ENEHole e) x = {!e!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step FHNEHoleEvaled x\u2082 x\u2083) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleInside x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  -- es (ENEHole e) (d' , Step (FHFinal (FVal ())) x\u2081 x\u2082)\n  -- es (ENEHole e) (d' , Step (FHFinal (FIndet (INEHole x))) () x\u2082)\n  -- es (ENEHole e) (d' , Step FHNEHoleEvaled () x\u2082)\n  -- es (ENEHole e) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FVal ())))\n  -- es (ENEHole (ECastError x x\u2081)) (_ , Step (FHNEHoleFinal x\u2082) (ITNEHole x\u2083) (FHFinal (FIndet (INEHole x\u2084)))) = {!x\u2081!}\n  -- es (ENEHole (EAp1 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (EAp2 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ENEHole e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ECastProp e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) FHNEHoleEvaled) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (d0' , Step (FHFinal x) () (FHFinal x\u2081))\n  is IEHole (d0' , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step (FHFinal x) () FHEHole)\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step (FHFinal x) () (FHCastFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (d , Step (FHFinal x\u2081) () (FHFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHEHole)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2082))\n  is (INEHole x) (d , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (d' , Step p q r) = {!p r!}\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2083!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n  es (ENEHole e) s = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4ff6f529455268506c893740d821d44eb486d87c","subject":"FunUniverse.Category: layout","message":"FunUniverse.Category: layout\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Category.agda","new_file":"FunUniverse\/Category.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Level.NP using (_\u2294_)\nopen import Type hiding (\u2605)\nopen import Function.NP using (Endo)\nopen import Data.Nat using (\u2115; zero; suc)\n\nmodule FunUniverse.Category {t \u2113} {T : \u2605 t} (_`\u2192_ : T \u2192 T \u2192 \u2605 \u2113) where\n\nmodule CompositionNotations\n  (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n  {A B C} where\n\n  infixr 1 _\u204f_\n  infixr 1 _>>>_\n\n  _\u204f_ : (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f \u204f g = g \u2218 f\n\n  _<<<_ : (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n  g <<< f = g \u2218 f\n\n  _>>>_ : (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f >>> g = f \u204f g\n\nrecord Category : \u2605 (\u2113 \u2294 t) where\n  constructor _,_\n\n  infixr 9 _\u2218_\n  field\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n  private\n    `Endo : T \u2192 Set \u2113\n    `Endo A = A `\u2192 A\n\n  iterate : \u2200 {A} \u2192 \u2115 \u2192 Endo (`Endo A)\n  iterate zero    _ = id\n  iterate (suc n) f = iterate n f \u2218 f\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Level.NP using (_\u2294_)\nopen import Type hiding (\u2605)\nopen import Function.NP using (Endo)\nopen import Data.Nat using (\u2115; zero; suc)\n\nmodule FunUniverse.Category\n  {t \u2113} {T : \u2605 t}\n  (_`\u2192_ : T \u2192 T \u2192 \u2605 \u2113)\n  where\n\nmodule CompositionNotations\n  (_\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C))\n  {A B C} where\n\n  infixr 1 _\u204f_\n  infixr 1 _>>>_\n\n  _\u204f_ : (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f \u204f g = g \u2218 f\n\n  _<<<_ : (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n  g <<< f = g \u2218 f\n\n  _>>>_ : (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n  f >>> g = f \u204f g\n\nrecord Category : \u2605 (\u2113 \u2294 t) where\n  constructor _,_\n\n  infixr 9 _\u2218_\n  field\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _\u2218_ : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u2192 B) \u2192 (A `\u2192 C)\n\n  private\n    `Endo : T \u2192 Set \u2113\n    `Endo A = A `\u2192 A\n\n  iterate : \u2200 {A} \u2192 \u2115 \u2192 Endo (`Endo A)\n  iterate zero    _ = id\n  iterate (suc n) f = iterate n f \u2218 f\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"33b3e52bdba9377441342cc7cc04c8a9b4464674","subject":"Added 91>100\u2192\u22a5.","message":"Added 91>100\u2192\u22a5.\n\nIgnore-this: c6c081ad69301a024e7cac361fe06542\n\ndarcs-hash:20110211220224-3bd4e-0a3b532bdac6d8621101a096859b8f946e7125dc.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n  n100' : eleven + eighty-nine \u2261 one-hundred\n  n100  : eighty-nine + eleven \u2261 one-hundred\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n\npostulate 91>100\u2192\u22a5 : GT ninety-one one-hundred \u2192 \u22a5\n{-# ATP prove 91>100\u2192\u22a5 #-}","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n  n100' : eleven + eighty-nine \u2261 one-hundred\n  n100  : eighty-nine + eleven \u2261 one-hundred\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9896f7d78362c6a07aced627b1fd4cc3a66567ad","subject":"Added n100' and n100.","message":"Added n100' and n100.\n\nIgnore-this: e44aad0771a0721e01719e3a2fdca522\n\ndarcs-hash:20110211215613-3bd4e-212ecd0be383d9a032cc44bf046374145d6cc141.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_file":"Draft\/McCarthy91\/ArithmeticATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n  n100' : eleven + eighty-nine \u2261 one-hundred\n  n100  : eighty-nine + eleven \u2261 one-hundred\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n{-# ATP prove n100' #-}\n{-# ATP prove n100 n100' +-comm #-}\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic stuff used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.ArithmeticATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.IsN-ATP\n\n------------------------------------------------------------------------------\n\npostulate\n  1+x-N  : \u2200 {n} \u2192 N n \u2192 N (one + n)\n  11+x-N : \u2200 {n} \u2192 N n \u2192 N (eleven + n)\n  x\u223810-N : \u2200 {n} \u2192 N n \u2192 N (n \u2238 ten)\n  x+11-N : \u2200 {n} \u2192 N n \u2192 N (n + eleven)\n{-# ATP prove 1+x-N #-}\n{-# ATP prove 11+x-N #-}\n{-# ATP prove x\u223810-N \u2238-N N10 #-}\n{-# ATP prove x+11-N 11+x-N +-comm #-}\n\npostulate\n  n111' : eleven + one-hundred \u2261 hundred-eleven\n  n111  : one-hundred + eleven \u2261 hundred-eleven\n  n101' : hundred-eleven \u2238 ten \u2261 hundred-one\n  n101  : (one-hundred + eleven) \u2238 ten \u2261 hundred-one\n  n91'  : hundred-one \u2238 ten \u2261 ninety-one\n  n91   : ((one-hundred + eleven) \u2238 ten) \u2238 ten \u2261 ninety-one\n  n102' : eleven + ninety-one \u2261 hundred-two\n  n102  : ninety-one + eleven \u2261 hundred-two\n{-# ATP prove n111' #-}\n{-# ATP prove n111 +-comm n111' #-}\n{-# ATP prove n101' #-}\n{-# ATP prove n101 n111 n101' #-}\n{-# ATP prove n91' #-}\n{-# ATP prove n91 n101 n91' #-}\n{-# ATP prove n102' #-}\n{-# ATP prove n102 n102' +-comm #-}\n\n\npostulate\n  101>100' : GT hundred-one one-hundred\n  101>100  : GT ((one-hundred + eleven) \u2238 ten) one-hundred\n  111>100' : GT hundred-eleven one-hundred\n  111>100  : GT (one-hundred + eleven) one-hundred\n  100<102' : LT one-hundred hundred-two\n  100<102  : LT one-hundred (ninety-one + eleven)\n{-# ATP prove 101>100' #-}\n{-# ATP prove 101>100 101>100' n101 #-}\n{-# ATP prove 111>100' #-}\n{-# ATP prove 111>100 111>100' n111 #-}\n{-# ATP prove 100<102' #-}\n{-# ATP prove 100<102 100<102' n102 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"99f38eb61e08f84e20adc47de913075b0b00a2c4","subject":"Clarify","message":"Clarify\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\n\nmodule _ {t1 t2 t3 : Type}\n  (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (df : Cht (t1 \u21d2 t3))\n  (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (dg : Cht (t2 \u21d2 t3))\n  where\n  private\n    \u0393 = sum t1 t2 \u2022\n      (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n      (t2 \u21d2 t3) \u2022\n      (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n      (t1 \u21d2 t3) \u2022\n      sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n      sum t1 t2 \u2022 \u2205\n  module _ where\n    private\n      \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n    changeMatchSem-lem1 :\n      \u2200 a1 b2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n      \u2261\n        g b2 \u2295 dg b2 (nil b2) \u229d f a1\n    changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n  module _ where\n    private\n      \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n    changeMatchSem-lem2 :\n      \u2200 b1 a2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n      \u2261\n        f a2 \u2295 df a2 (nil a2) \u229d g b1\n    changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\ncorrectDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\ncorrectDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\ncorrectDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n  where\n    lemma : \u2200 s f g \u2192\n      \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n      (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n    lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n    lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nopen import New.Equivalence\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\nvalidDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\nvalidDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\nvalidDeriveConst (match {t1} {t2} {t3}) =\n  ternary-valid dmatch-valid dmatch-eq\n  where\n    open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n    dmatch-valid : ternary-valid-preserve-hp\n    dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n    dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n    dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n    dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n    dmatch-eq : ternary-valid-eq-hp\n    dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n      rewrite update-nil (a \u2295 da)\n      | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n    dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n      rewrite update-nil (b \u2295 db)\n      | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n    dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      | update-nil b2\n      | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n      | update-nil (g b2 \u2295 dg b2 (nil b2))\n      = refl\n    dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      | update-nil a2\n      | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n      | update-nil (f a2 \u2295 df a2 (nil a2))\n      = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    fv = \u27e6 t \u27e7Term \u03c12\n    dfvdv = \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term \u03c12 \u2295 \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","old_contents":"module New.Correctness where\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\n\nmodule _ {t1 t2 t3 : Type}\n  (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (df : Cht (t1 \u21d2 t3))\n  (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (dg : Cht (t2 \u21d2 t3))\n  where\n  private\n    \u0393 = sum t1 t2 \u2022\n      (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n      (t2 \u21d2 t3) \u2022\n      (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n      (t1 \u21d2 t3) \u2022\n      sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n      sum t1 t2 \u2022 \u2205\n  module _ where\n    private\n      \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n    changeMatchSem-lem1 :\n      \u2200 a1 b2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n      \u2261\n        g b2 \u2295 dg b2 (nil b2) \u229d f a1\n    changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n  module _ where\n    private\n      \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n    changeMatchSem-lem2 :\n      \u2200 b1 a2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n      \u2261\n        f a2 \u2295 df a2 (nil a2) \u229d g b1\n    changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\ncorrectDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\ncorrectDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\ncorrectDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n  where\n    lemma : \u2200 s f g \u2192\n      \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n      (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n    lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n    lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nopen import New.Equivalence\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\nvalidDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\nvalidDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\nvalidDeriveConst (match {t1} {t2} {t3}) =\n  ternary-valid dmatch-valid dmatch-eq\n  where\n    open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n    dmatch-valid : ternary-valid-preserve-hp\n    dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n    dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n    dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n    dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n    dmatch-eq : ternary-valid-eq-hp\n    dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n      rewrite update-nil (a \u2295 da)\n      | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n    dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n      rewrite update-nil (b \u2295 db)\n      | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n    dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      | update-nil b2\n      | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n      | update-nil (g b2 \u2295 dg b2 (nil b2))\n      = refl\n    dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      | update-nil a2\n      | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n      | update-nil (f a2 \u2295 df a2 (nil a2))\n      = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    fv = \u27e6 t \u27e7Term (a \u2022 \u03c1)\n    dfvdv = \u27e6 derive t \u27e7Term (da \u2022 a \u2022 alternate \u03c1 d\u03c1)\n    rdr = validDerive t (a \u2022 \u03c1) (da \u2022 d\u03c1) (ada , \u03c1d\u03c1)\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term \u03c12 \u2295 \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4a9744d474288be4e0c19d9e18d92b6ba0b0b1f2","subject":"getting stuck in the predictable places #2","message":"getting stuck in the predictable places #2\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import htype-decidable\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (pres-lem eps D1 D3 D4 D5) D2\n  pres-lem (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (pres-lem eps D2 D3 D4 D5)\n  pres-lem (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (pres-lem eps D1 D2 D3 D4) x\u2081\n  pres-lem (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (pres-lem eps D1 D2 D3 D4) x\n  pres-lem (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (pres-lem x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = pres-lem2 x y\n\n  -- this is the literal contents of the hole in lem3; it might not go\n  -- through exactly like this.\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst TAConst D2 = TAConst\n  lem-subst {\u0393 = \u0393} {x = x'} (TAVar {x = x} x\u2082) D2\n    with \u0393 x\n  lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | Some x\u2081\n    with natEQ x' x\n  lem-subst (TAVar xin) D2 | Some x\u2083 | Inl refl = {!!}\n  lem-subst (TAVar refl) D2 | Some x\u2083 | Inr x\u2082 = {!!}\n  lem-subst {x = x} (TAVar {x = x'} x\u2082) D2 | None with natEQ x' x\n  lem-subst {x = x} (TAVar x\u2083) D2 | None | Inl refl with natEQ x x\n  lem-subst (TAVar refl) D2 | None | Inl refl | Inl refl = D2\n  lem-subst (TAVar x\u2083) D2 | None | Inl refl | Inr x\u2081 = abort (somenotnone (! x\u2083))\n  lem-subst {x = x} (TAVar {x = x'} x\u2083) D2 | None | Inr x\u2082 with natEQ x x'\n  lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inl x\u2082 = abort ((flip x\u2083) x\u2082)\n  lem-subst (TAVar x\u2084) D2 | None | Inr x\u2083 | Inr x\u2082 = abort (somenotnone (! x\u2084))\n  lem-subst (TALam D1) D2 = {!!}\n  lem-subst (TAAp D1 D2) D3 = TAAp (lem-subst D1 D3) (lem-subst D2 D3)\n  lem-subst (TAEHole x\u2081 x\u2082) D2 = TAEHole x\u2081 {!!}\n  lem-subst (TANEHole x\u2081 D1 x\u2082) D2 = TANEHole x\u2081 (lem-subst D1 D2) {!!}\n  lem-subst (TACast D1 x\u2081) D2 = TACast (lem-subst D1 D2) x\u2081\n  lem-subst (TAFailedCast D1 x\u2081 x\u2082 x\u2083) D2 = TAFailedCast (lem-subst D1 D2) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) (TAAp D1 D2) D3 D4 (FHAp1 D5) = TAAp (pres-lem eps D1 D3 D4 D5) D2\n  pres-lem (FHAp2 eps) (TAAp D1 D2) D3 D4 (FHAp2 D5) = TAAp D1 (pres-lem eps D2 D3 D4 D5)\n  pres-lem (FHNEHole eps) (TANEHole x D1 x\u2081) D2 D3 (FHNEHole D4) = TANEHole x (pres-lem eps D1 D2 D3 D4) x\u2081\n  pres-lem (FHCast eps) (TACast D1 x) D2 D3 (FHCast D4) = TACast (pres-lem eps D1 D2 D3 D4) x\n  pres-lem (FHFailedCast x) (TAFailedCast y x\u2081 x\u2082 x\u2083) D3 D4 (FHFailedCast eps) = TAFailedCast (pres-lem x y D3 D4 eps) x\u2081 x\u2082 x\u2083\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) FHOuter = _ , TAFailedCast x y z w\n  pres-lem2 (TAFailedCast x x\u2081 x\u2082 x\u2083) (FHFailedCast y) = pres-lem2 x y\n\n  -- this is the literal contents of the hole in lem3; it might not go\n  -- through exactly like this.\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n              \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n              \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n              \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst D1 D2 = {!!}\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp (TALam ta) ta\u2081) ITLam = lem-subst ta ta\u2081\n  pres-lem3 (TAAp (TACast ta TCRefl) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 TCRefl)) TCRefl\n  pres-lem3 (TAAp (TACast ta (TCArr x x\u2081)) ta\u2081) ITApCast = TACast (TAAp ta (TACast ta\u2081 (~sym x))) x\u2081\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) ()\n  pres-lem3 (TACast ta x) (ITCastID) = ta\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastSucceed x\u2082) = ta\n  pres-lem3 (TACast ta x) (ITGround (MGArr x\u2081)) = TACast (TACast ta (TCArr TCHole1 TCHole1)) TCHole1\n  pres-lem3 (TACast ta TCHole2) (ITExpand (MGArr x\u2081)) = TACast (TACast ta TCHole2) (TCArr TCHole2 TCHole2)\n  pres-lem3 (TACast (TACast ta x) x\u2081) (ITCastFail w y z) = TAFailedCast ta w y z\n  pres-lem3 (TAFailedCast x y z q) ()\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fce53eeb8e89f3591a587ee1cf5e94373772c9a3","subject":"Cause proof-embedded interpretation function to act on less envs","message":"Cause proof-embedded interpretation function to act on less envs\n\nOld-commit-hash: bf9b5052c169db7e91ed78dfebbea792854bb573\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- Used to signal free variables of a term.\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393)\n\n-- Declare everything in \u0393 to be volatile.\nselect-none-in : (\u0393 : Context) \u2192 Vars \u0393\nselect-none-in \u2205 = \u2205\nselect-none-in (\u03c4 \u2022 \u0393) = alter (select-none-in \u0393)\n\nselect-just : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393\nselect-just {\u0393 = \u03c4 \u2022 \u0393\u2080} this = abide (select-none-in \u0393\u2080)\nselect-just (that x) = alter (select-just x)\n\n-- De-facto union of free variables\nFV-union : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 Vars \u0393 \u2192 Vars \u0393\nFV-union \u2205 \u2205 = \u2205\nFV-union (alter us) (alter vs) = alter (FV-union us vs)\nFV-union (alter us) (abide vs) = abide (FV-union us vs)\nFV-union (abide us) (alter vs) = abide (FV-union us vs)\nFV-union (abide us) (abide vs) = abide (FV-union us vs)\n\ntail : \u2200 {\u03c4 \u0393} \u2192 Vars (\u03c4 \u2022 \u0393) \u2192 Vars \u0393\ntail (abide vars) = vars\ntail (alter vars) = vars\n\n-- Free variables of a term.\n-- Free variables are marked as abiding, bound variables altering.\nFV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\nFV {\u0393 = \u0393} (nat n) = select-none-in \u0393\nFV {\u0393 = \u0393} (bag b) = select-none-in \u0393\nFV (var x) = select-just x\nFV (abs t) = tail (FV t)\nFV (app s t) = FV-union (FV s) (FV t)\nFV (add s t) = FV-union (FV s) (FV t)\nFV (map s t) = FV-union (FV s) (FV t)\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 Vars \u0393 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (alter vars)\n  abide : \u2200 {\u0393 \u03c4} {v : \u27e6 \u03c4 \u27e7} {dv R[v,dv] vars \u03c1} \u2192\n          v \u2295 dv \u2261 v \u2192\n          Honest {\u0393} \u03c1 vars \u2192\n          Honest {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) (abide vars)\n\n_is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-for \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-for \u03c1 = \u22a4\n\u0394bag db is-valid-for \u03c1 = \u22a4\n\u0394var x is-valid-for \u03c1 = \u22a4\n\n_is-valid-for_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-for_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-for \u03c1 = Quadruple\n  (ds is-valid-for \u03c1)\n  (\u03bb _ \u2192 dt is-valid-for \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2081 s db is-valid-for \u03c1 = couple\n  (db is-valid-for \u03c1)\n  (Honest \u03c1 (FV s))\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 (cons v-dt honesty _ _) =\n  mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-for \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-for \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- Corollary: (f \u2295 df) (v \u2295 dv) = f v \u2295 df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore \u03c1) \u2295 \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s))\n    (\u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n  \u2261  \u27e6 s \u27e7 (ignore \u03c1) (\u27e6 t \u27e7 (ignore \u03c1)) \u2295\n     \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s) (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t})\n\ncorollary s t {\u03c1} = proj\u2082\n  (validity {t = s} (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t}))\n\nunrestricted (nat n) = tt\nunrestricted (bag b) = tt\nunrestricted (var x) {\u03c1} = tt\nunrestricted (abs t) {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted t)\nunrestricted (app s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t))\n  (validity {t = t}) tt\nunrestricted (add s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\nunrestricted (map s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted t)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n_is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-for \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-for_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-for \u03c1 = \u22a4\n\u0394bag db is-valid-for \u03c1 = \u22a4\n\u0394var x is-valid-for \u03c1 = \u22a4\n\n_is-valid-for_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-for_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-for \u03c1 = Quadruple\n  (ds is-valid-for \u03c1)\n  (\u03bb _ \u2192 dt is-valid-for \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-for \u03c1 = couple\n  (ds is-valid-for \u03c1)\n  (dt is-valid-for \u03c1)\n_is-valid-for_ (\u0394map\u2081 s db) \u03c1 = db is-valid-for \u03c1\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 v-dt = mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-for \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-for \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- Corollary: (f \u2295 df) (v \u2295 dv) = f v \u2295 df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore \u03c1) \u2295 \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s))\n    (\u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t))\n  \u2261  \u27e6 s \u27e7 (ignore \u03c1) (\u27e6 t \u27e7 (ignore \u03c1)) \u2295\n     \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s) (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t})\n\ncorollary s t {\u03c1} = proj\u2082\n  (validity {t = s} (\u27e6 t \u27e7 (ignore \u03c1))\n    (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)) (validity {t = t}))\n\nunrestricted (nat n) = tt\nunrestricted (bag b) = tt\nunrestricted (var x) {\u03c1} = tt\nunrestricted (abs t) {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted t)\nunrestricted (app s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t))\n  (validity {t = t}) tt\nunrestricted (add s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\nunrestricted (map s t) {\u03c1} = cons\n  (unrestricted (s))\n  (unrestricted (t)) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted t)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted s)\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted t)\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted t)\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1ff9d738e42e37f5e770372fa1ca14a175f1f8ab","subject":"stab at some judgements in subst","message":"stab at some judgements in subst\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  -- todo: add a premise from a new judgement like holes disjoint that\n  -- classifies pairs of dhexps that share no variable names\n  -- whatsoever. that should imply freshness here. then propagate that\n  -- change to preservation. this is morally what \u03b1-equiv lets us do in a\n  -- real setting.\n\n  -- the variable name x does not appear in the term d\n  data var-name-new : (x : Nat) (d : dhexp) \u2192 Set where\n    VNNConst : \u2200{x} \u2192 var-name-new x c\n    VNNVar : \u2200{x y} \u2192 x \u2260 y \u2192 var-name-new x (X y)\n    VNNLam2 : \u2200{x d y \u03c4} \u2192 x \u2260 y\n                         \u2192 var-name-new x d\n                         \u2192 var-name-new x (\u00b7\u03bb_[_]_ y \u03c4 d)\n    VNNHole : \u2200{x u \u03c3} \u2192 var-name-new x (\u2987\u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNNNEHole : \u2200{x u \u03c3 d } \u2192\n                var-name-new x d \u2192\n                var-name-new x (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNNAp : \u2200{ x d1 d2 } \u2192\n           var-name-new x d1 \u2192\n           var-name-new x d2 \u2192\n           var-name-new x (d1 \u2218 d2)\n    VNNCast : \u2200{x d \u03c41 \u03c42} \u2192 var-name-new x d \u2192 var-name-new x (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n    VNNFailedCast : \u2200{x d \u03c41 \u03c42} \u2192 var-name-new x d \u2192 var-name-new x (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  -- two terms that do not share any hole names\n  data var-names-disjoint : (d1 : dhexp) \u2192 (d2 : dhexp) \u2192 Set where\n    VNDConst : \u2200{d} \u2192 var-names-disjoint c d\n    VNDVar : \u2200{x d} \u2192 var-names-disjoint (X x) d\n    VNDLam : \u2200{x \u03c4 d1 d2} \u2192 var-names-disjoint d1 d2\n                          \u2192 var-names-disjoint (\u00b7\u03bb_[_]_ x \u03c4 d1) d2\n    VNDHole : \u2200{u \u03c3 d2} \u2192 var-name-new u d2\n                        \u2192 var-names-disjoint (\u2987\u2988\u27e8 u , \u03c3 \u27e9) d2 -- todo something about \u03c3?\n    VNDNEHole : \u2200{u \u03c3 d1 d2} \u2192 var-name-new u d2\n                             \u2192 var-names-disjoint d1 d2\n                             \u2192 var-names-disjoint (\u2987 d1 \u2988\u27e8 u , \u03c3 \u27e9) d2 -- todo something about \u03c3?\n    VNDAp :  \u2200{d1 d2 d3} \u2192 var-names-disjoint d1 d3\n                         \u2192 var-names-disjoint d2 d3\n                         \u2192 var-names-disjoint (d1 \u2218 d2) d3\n\n  -- all the variable names in the term are unique\n  data var-names-unique : dhexp \u2192 Set where\n    VNUHole : var-names-unique c\n    VNUVar : \u2200{x} \u2192 var-names-unique (X x)\n    VNULam : {x : Nat} {\u03c4 : htyp} {d : dhexp} \u2192 var-names-unique d\n                                              \u2192 var-name-new x d\n                                              \u2192 var-names-unique (\u00b7\u03bb_[_]_ x \u03c4 d)\n    VNUEHole : \u2200{u \u03c3} \u2192 var-names-unique (\u2987\u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNUNEHole : \u2200{u \u03c3 d} \u2192 var-names-unique d\n                         \u2192 var-names-unique (\u2987 d \u2988\u27e8 u , \u03c3 \u27e9) -- todo something about \u03c3?\n    VNUAp : \u2200{d1 d2} \u2192 var-names-unique d1\n                     \u2192 var-names-unique d2\n                     \u2192 var-names-disjoint d1 d2\n                     \u2192 var-names-unique (d1 \u2218 d2)\n    VNUCast : \u2200{d \u03c41 \u03c42} \u2192 var-names-unique d\n                         \u2192 var-names-unique (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n    VNUFailedCast : \u2200{d \u03c41 \u03c42} \u2192 var-names-unique d\n                               \u2192 var-names-unique (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","old_contents":"open import Prelude\nopen import Nat\nopen import List\nopen import core\nopen import contexts\nopen import weakening\nopen import exchange\nopen import lemmas-disjointness\n\nmodule lemmas-subst-ta where\n  -- todo: add a premise from a new judgement like holes disjoint that\n  -- classifies pairs of dhexps that share no variable names\n  -- whatsoever. that should imply freshness here. then propagate that\n  -- change to preservation. this is morally what \u03b1-equiv lets us do in a\n  -- real setting.\n\n  -- the variable name x does not appear in the term d\n  data var-name-new : (d : dhexp) (x : Nat) \u2192 Set where\n    VNConst : \u2200{x} \u2192 var-name-new c x\n    VNVar : \u2200{x y} \u2192 x \u2260 y \u2192 var-name-new (X y) x\n    VNLam2 : \u2200{x e y \u03c4} \u2192\n             x \u2260 y \u2192\n             var-name-new e x \u2192\n             var-name-new (\u00b7\u03bb_[_]_ y \u03c4 e) x\n    -- VNHole : \u2200{x u \u03c3} \u2192\n    --           -- \u2192\n    --          var-name-new (\u2987\u2988[ u , \u03c3 ]) x\n    -- VNNEHole : \u2200{u u' e} \u2192\n    --            u' \u2260 u \u2192\n    --            var-name-new e u \u2192\n    --            var-name-new (\u2987 e \u2988[ u' ]) u\n    -- VNAp : \u2200{ u e1 e2 } \u2192\n    --        var-name-new e1 u \u2192\n    --        var-name-new e2 u \u2192\n    --        var-name-new (e1 \u2218 e2) u\n\n  -- two terms that do not share any hole names\n  data vars-disjoint : (e1 : hexp) \u2192 (e2 : hexp) \u2192 Set where\n    -- HDConst : \u2200{e} \u2192 holes-disjoint c e\n    -- HDAsc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (e1 \u00b7: \u03c4) e2\n    -- HDVar : \u2200{x e} \u2192 holes-disjoint (X x) e\n    -- HDLam1 : \u2200{x e1 e2} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x e1) e2\n    -- HDLam2 : \u2200{x e1 e2 \u03c4} \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u00b7\u03bb x [ \u03c4 ] e1) e2\n    -- HDHole : \u2200{u e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint (\u2987\u2988[ u ]) e2\n    -- HDNEHole : \u2200{u e1 e2} \u2192 hole-name-new e2 u \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint (\u2987 e1 \u2988[ u ]) e2\n    -- HDAp :  \u2200{e1 e2 e3} \u2192 holes-disjoint e1 e3 \u2192 holes-disjoint e2 e3 \u2192 holes-disjoint (e1 \u2218 e2) e3\n\n\n  lem-subst : \u2200{\u0394 \u0393 x \u03c41 d1 \u03c4 d2 } \u2192\n                  x # \u0393 \u2192\n                  \u0394 , \u0393 ,, (x , \u03c41) \u22a2 d1 :: \u03c4 \u2192\n                  \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n                  \u0394 , \u0393 \u22a2 [ d2 \/ x ] d1 :: \u03c4\n  lem-subst apt TAConst wt2 = TAConst\n  lem-subst {x = x} apt (TAVar {x = x'} x\u2082) wt2 with natEQ x' x\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inl refl with lem-apart-union-eq {\u0393 = \u0393} apt x\u2083\n  lem-subst apt (TAVar x\u2083) wt2 | Inl refl | refl = wt2\n  lem-subst {\u0393 = \u0393} apt (TAVar x\u2083) wt2 | Inr x\u2082 = TAVar (lem-neq-union-eq {\u0393 = \u0393} x\u2082 x\u2083)\n  lem-subst {\u0394 = \u0394} {\u0393 = \u0393} {x = x} {d2 = d2} x#\u0393 (TALam {x = y} {\u03c41 = \u03c41} {d = d} {\u03c42 = \u03c42} x\u2082 wt1) wt2\n    with lem-union-none {\u0393 = \u0393} x\u2082\n  ... |  x\u2260y , y#\u0393 with natEQ y x\n  ... | Inl eq = abort (x\u2260y (! eq))\n  ... | Inr _  = TALam y#\u0393 (lem-subst {\u0394 = \u0394} {\u0393 = \u0393 ,, (y , \u03c41)} {x = x} {d1 = d} (apart-extend1 \u0393 x\u2260y x#\u0393) (exchange-ta-\u0393 {\u0393 = \u0393} x\u2260y wt1) (weaken-ta {!!} wt2))\n  lem-subst apt (TAAp wt1 wt2) wt3 = TAAp (lem-subst apt wt1 wt3) (lem-subst apt wt2 wt3)\n  lem-subst apt (TAEHole in\u0394 sub) wt2 = TAEHole in\u0394 (STASubst sub wt2)\n  lem-subst apt (TANEHole x\u2081 wt1 x\u2082) wt2 = TANEHole x\u2081 (lem-subst apt wt1 wt2) (STASubst x\u2082 wt2)\n  lem-subst apt (TACast wt1 x\u2081) wt2 = TACast (lem-subst apt wt1 wt2) x\u2081\n  lem-subst apt (TAFailedCast wt1 x\u2081 x\u2082 x\u2083) wt2 = TAFailedCast (lem-subst apt wt1 wt2) x\u2081 x\u2082 x\u2083\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f0b332cdbd5ef3bb4c7a345b7bfff18c5a0e81e0","subject":"BigI: more bindings","message":"BigI: more bindings\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS\/BigI.agda","new_file":"lib\/FFI\/JS\/BigI.agda","new_contents":"open import FFI.JS hiding (toString)\n\nmodule FFI.JS.BigI where\n\nabstract\n  BigI : Set\n  BigI = JSValue\n\n  BigI\u25b9JSValue : BigI \u2192 JSValue\n  BigI\u25b9JSValue x = x\n\n  unsafe-JSValue\u25b9BigI : JSValue \u2192 BigI\n  unsafe-JSValue\u25b9BigI x = x\n\npostulate bigI     : (x base : String) \u2192 BigI\n{-# COMPILED_JS bigI       function(x) { return function (y) { return require(\"bigi\")(x,y); }; } #-}\n\npostulate negate   : (x : BigI) \u2192 BigI\n{-# COMPILED_JS negate     function(x) { return x.negate(); } #-}\n\npostulate add      : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS add        function(x) { return function (y) { return x.add(y); }; } #-}\n\npostulate subtract : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS subtract   function(x) { return function (y) { return x.subtract(y); }; } #-}\n\npostulate multiply : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS multiply   function(x) { return function (y) { return x.multiply(y); }; } #-}\n\npostulate divide   : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS divide     function(x) { return function (y) { return x.divide(y); }; } #-}\n\npostulate remainder : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS remainder  function(x) { return function (y) { return x.remainder(y); }; } #-}\n\npostulate mod      : (x y : BigI) \u2192 BigI\n{-# COMPILED_JS mod        function(x) { return function (y) { return x.mod(y); }; } #-}\n\npostulate modPow   : (this e m : BigI) \u2192 BigI\n{-# COMPILED_JS modPow     function(x) { return function (y) { return function (z) { return x.modPow(y,z); }; }; } #-}\n\npostulate modInv   : (this m   : BigI) \u2192 BigI\n{-# COMPILED_JS modInv     function(x) { return function (y) { return x.modInverse(y); }; } #-}\n\npostulate gcd      : (this m   : BigI) \u2192 BigI\n{-# COMPILED_JS gcd        function(x) { return function (y) { return x.gcd(y); }; } #-}\n\n-- test primality with certainty >= 1-.5^t\npostulate isProbablePrime : (this : BigI)(t : Number) \u2192 Bool\n{-# COMPILED_JS isProbablePrime function(x) { return function (y) { return x.isProbablePrime(y); }; } #-}\n\npostulate signum   : (this : BigI) \u2192 Number\n{-# COMPILED_JS signum function(x) { return x.signum(); } #-}\n\npostulate equals   : (x y : BigI) \u2192 Bool\n{-# COMPILED_JS equals     function(x) { return function (y) { return x.equals(y); }; } #-}\n\npostulate compareTo : (this x : BigI) \u2192 Number\n{-# COMPILED_JS compareTo  function(x) { return function (y) { return x.compareTo(y); }; } #-}\n\npostulate toString : (x : BigI) \u2192 String\n{-# COMPILED_JS toString   function(x) { return x.toString(); } #-}\n\npostulate fromHex  : String \u2192 BigI\n{-# COMPILED_JS fromHex    require(\"bigi\").fromHex #-}\n\npostulate toHex    : BigI \u2192 String\n{-# COMPILED_JS toHex      function(x) { return x.toHex(); } #-}\n\npostulate byteLength : BigI \u2192 Number\n{-# COMPILED_JS byteLength function(x) { return x.byteLength(); } #-}\n\n0I 1I 2I : BigI\n0I = bigI \"0\" \"10\"\n1I = bigI \"1\" \"10\"\n2I = bigI \"2\" \"10\"\n\n_\u2264I_ : BigI \u2192 BigI \u2192 Bool\nx \u2264I y = compareTo x y \u2264Number 0N\n\n_<I_ : BigI \u2192 BigI \u2192 Bool\nx <I y = compareTo x y <Number 0N\n\n_>I_ : BigI \u2192 BigI \u2192 Bool\nx >I y = compareTo x y >Number 0N\n\n_\u2265I_ : BigI \u2192 BigI \u2192 Bool\nx \u2265I y = compareTo x y \u2265Number 0N\n","old_contents":"open import FFI.JS hiding (toString)\n\nmodule FFI.JS.BigI where\n\nabstract\n  BigI : Set\n  BigI = JSValue\n\n  BigI\u25b9JSValue : BigI \u2192 JSValue\n  BigI\u25b9JSValue x = x\n\n  unsafe-JSValue\u25b9BigI : BigI \u2192 JSValue\n  unsafe-JSValue\u25b9BigI x = x\n\npostulate\n  bigI     : (x base : String) \u2192 BigI\n  add      : (x y : BigI) \u2192 BigI\n  multiply : (x y : BigI) \u2192 BigI\n  mod      : (x y : BigI) \u2192 BigI\n  modPow   : (this e m : BigI) \u2192 BigI\n  modInv   : (this m   : BigI) \u2192 BigI\n  equals   : (x y : BigI) \u2192 Bool\n  toString : (x : BigI) \u2192 String\n  fromHex  : String \u2192 BigI\n  toHex    : BigI \u2192 String\n\n{-# COMPILED_JS bigI       function(x) { return function (y) { return require(\"bigi\")(x,y); }; } #-}\n{-# COMPILED_JS add        function(x) { return function (y) { return x.add(y); }; } #-}\n{-# COMPILED_JS multiply   function(x) { return function (y) { return x.multiply(y); }; } #-}\n{-# COMPILED_JS mod        function(x) { return function (y) { return x.mod(y); }; } #-}\n{-# COMPILED_JS modPow     function(x) { return function (y) { return function (z) { return x.modPow(y,z); }; }; } #-}\n{-# COMPILED_JS modInv     function(x) { return function (y) { return x.modInverse(y); }; } #-}\n{-# COMPILED_JS equals     function(x) { return function (y) { return x.equals(y); }; } #-}\n{-# COMPILED_JS toString   function(x) { return x.toString(); } #-}\n{-# COMPILED_JS fromHex    require(\"bigi\").fromHex #-}\n{-# COMPILED_JS toHex      function(x) { return x.toHex(); } #-}\n\n0I 1I 2I : BigI\n0I = bigI \"0\" \"10\"\n1I = bigI \"1\" \"10\"\n2I = bigI \"2\" \"10\"\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5ac17d3260e3331d96ba03e39c63c839241ef250","subject":"fussing with whitespace","message":"fussing with whitespace\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | Inl (_ , _ , _ , refl , f)                             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _))       = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inr (_ , \u03c4 , refl , _ , _)))\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (_ , ._ , refl , gnd , _))) | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (_ , _ , refl , gnd , _)))  | Inr neq = S (_ , Step FHOuter (ITCastFail gnd GBase neq) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u2987\u2988})) | I x = S (_ , Step FHOuter ITCastID FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inl (_ , _ , _ , refl , _)                             = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _))       = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (_ , ._ , refl , GBase , _))) = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (_ , ._ , refl , GHole , _))) = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n\n\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | BV x = {!canonical-boxed-forms!} -- BV (BVHoleCast {!!} x) -- cyrus\n\n  -- this is the case i was working on on friday; this part seems ok if maybe redundant\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with ground-decidable \u03c4 | canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inl x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter )\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x    = S (_ , Step FHOuter (ITCastFail gnd x\u2082 x) FHOuter )\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inr x  | d' , \u03c4' , refl , gnd , wt' -- this case goes off the rails here\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = abort (x\u2082 gnd)\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x = {!!}\n\n  -- progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .b , refl , GBase , wt' | Inr x = {!!}\n  -- progress {\u03c4 = b} (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , wt' | Inr x = abort (x\u2082 GBase)\n  -- progress {\u03c4 = \u2987\u2988} (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , wt' | Inr x = {!!} -- S (_ , Step (FHCast FHOuter) {!ground-arr-lem _ x\u2082 !} (FHCast FHOuter))\n  -- progress {\u03c4 = \u03c41 ==> \u03c42} (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , wt' | Inr x = {!!}\n\n  -- this is the beginning of the next case -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | BV x = {!!}\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S(_ , Step (FHFailedCast (FHCast x)) a (FHFailedCast (FHCast q)))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxed x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (_ , _ , refl , _)         = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | Inl (_ , _ , _ , refl , f) = I (ICastHoleGround (\u03bb d' \u03c4' \u2192 \u03bb ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _)) = I (ICastHoleGround (\u03bb d' \u03c4' \u2192 \u03bb ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb d' \u03c4' \u2192 \u03bb ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | Inr (Inr (Inr (d' , \u03c4 , refl , gnd , ind )))\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (d' , .b , refl , gnd , ind))) | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | Inr (Inr (Inr (d' , \u03c4 , refl , gnd , ind)))  | Inr x = S (_ , Step FHOuter (ITCastFail gnd GBase x) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u2987\u2988})) | I x = S (_ , Step FHOuter ITCastID FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inl (_ , _ , _ , refl , f) = I (ICastHoleGround (\u03bb d' \u03c4' \u2192 \u03bb ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _)) = I (ICastHoleGround (\u03bb d' \u03c4' \u2192 \u03bb ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = I (ICastHoleGround (\u03bb d' \u03c4' \u2192 \u03bb ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (d' , .b , refl , GBase , ind))) = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | Inr (Inr (Inr (d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , ind))) = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n\n  -- progress (TACast wt TCHole2) | I x | Inl (_ , _ , _ , refl , f) = {!!}\n  -- progress (TACast wt TCHole2) | I x | Inr (Inl (_ , _ , _ , _ , _ , refl , _ , _ , _)) = {!!}\n  -- progress (TACast wt TCHole2) | I x | Inr (Inr (Inl (_ , _ , _ , refl , _ , _ , _ , _ , _))) = {!!}\n  -- progress (TACast wt TCHole2) | I x | Inr (Inr (Inr (_ , _ , refl , _ , _ ))) = {!!}\n\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n\n\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | BV x = {!canonical-boxed-value!} -- BV (BVHoleCast {!!} x) -- cyrus\n\n  -- this is the case i was working on on friday; this part seems ok if maybe redundant\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with ground-decidable \u03c4 | canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inl x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = S (_ , Step FHOuter (ITCastSucceed gnd) FHOuter )\n  progress (TACast wt TCHole2) | BV x\u2081 | Inl x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x    = S (_ , Step FHOuter (ITCastFail gnd x\u2082 x) FHOuter )\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | Inr x  | d' , \u03c4' , refl , gnd , wt' -- this case goes off the rails here\n    with htype-dec \u03c4' \u03c4\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4 , refl , gnd , wt'  | Inl refl = abort (x\u2082 gnd)\n  progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x = {!!}\n\n  -- progress (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .b , refl , GBase , wt' | Inr x = {!!}\n  -- progress {\u03c4 = b} (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , wt' | Inr x = abort (x\u2082 GBase)\n  -- progress {\u03c4 = \u2987\u2988} (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , wt' | Inr x = {!!} -- S (_ , Step (FHCast FHOuter) {!ground-arr-lem _ x\u2082 !} (FHCast FHOuter))\n  -- progress {\u03c4 = \u03c41 ==> \u03c42} (TACast wt TCHole2) | BV x\u2081 | Inr x\u2082 | d' , .(\u2987\u2988 ==> \u2987\u2988) , refl , GHole , wt' | Inr x = {!!}\n\n  -- this is the beginning of the next case -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | BV x = {!!}\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S(_ , Step (FHFailedCast (FHCast x)) a (FHFailedCast (FHCast q)))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxed x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bb9b0d2a8ea2f038b890dd2e90fa0dbeaa9f6111","subject":" [ doc ] Fixed documentation related to a TPTP problem","message":" [ doc ] Fixed documentation related to a TPTP problem\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/GroupTheory\/Commutator.agda","new_file":"src\/fot\/GroupTheory\/Commutator.agda","new_contents":"------------------------------------------------------------------------------\n-- The group commutator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Commutator where\n\nopen import GroupTheory.Base\n\n------------------------------------------------------------------------------\n-- The group commutator\n--\n-- There are two definitions for the group commutator:\n--\n-- [a,b] = aba\u207b\u00b9b\u207b\u00b9 (Mac Lane and Garret 1999, p. 418), or\n--\n-- [a,b] = a\u207b\u00b9b\u207b\u00b9ab (Kurosh 1960, p. 99).\n--\n-- We use Kurosh's definition, because this is the definition used by\n-- the TPTP 6.0.0 problem GRP\/GRP024-5.p. (Actually the problem uses\n-- the definition\n--\n-- [a,b] = a\u207b\u00b9(b\u207b\u00b9(ab)).\n\n[_,_] : G \u2192 G \u2192 G\n[ a , b ] = a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 b\n{-# ATP definition [_,_] #-}\n\ncommutatorAssoc : G \u2192 G \u2192 G \u2192 Set\ncommutatorAssoc a b c = [ [ a , b ] , c ] \u2261 [ a , [ b , c ] ]\n{-# ATP definition commutatorAssoc #-}\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Mac Lane, S. and Birkhof, G. (1999). Algebra. 3rd ed. AMS Chelsea\n-- Publishing.\n--\n-- Kurosh, A. G. (1960). The Theory of Groups. 2nd\n-- ed. Vol. 1. Translated and edited by K. A. Hirsch. Chelsea\n-- Publising Company.\n","old_contents":"------------------------------------------------------------------------------\n-- The group commutator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Commutator where\n\nopen import GroupTheory.Base\n\n------------------------------------------------------------------------------\n-- The group commutator\n--\n-- There are two definitions for the group commutator:\n--\n-- \u2022 [a,b] = aba\u207b\u00b9b\u207b\u00b9 (Mac Lane and Garret 1999, p. 418).\n--\n-- \u2022 [a,b] = a\u207b\u00b9b\u207b\u00b9ab (Kurosh 1960, p. 99).\n--\n-- We use Kurosh's definition, because this is the definition used\n-- by the TPTP 6.0.0 problem GRP\/GRP024-5.p.\n\n[_,_] : G \u2192 G \u2192 G\n[ a , b ] = a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 b\n{-# ATP definition [_,_] #-}\n\ncommutatorAssoc : G \u2192 G \u2192 G \u2192 Set\ncommutatorAssoc a b c = [ [ a , b ] , c ] \u2261 [ a , [ b , c ] ]\n{-# ATP definition commutatorAssoc #-}\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Mac Lane, S. and Birkhof, G. (1999). Algebra. 3rd ed. AMS Chelsea\n-- Publishing.\n--\n-- Kurosh, A. G. (1960). The Theory of Groups. 2nd\n-- ed. Vol. 1. Translated and edited by K. A. Hirsch. Chelsea\n-- Publising Company.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f2c7f4cc1e067d7947b146373a458b3331daecf2","subject":"Bits: some cleanups and more props on #","message":"Bits: some cleanups and more props on #\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\nsearch\u2032 : \u2200 {n a} {A : \u2115 \u2192 Set a} \u2192 (\u2200 {m} \u2192 A m \u2192 A m \u2192 A (2* m)) \u2192 (Bits n \u2192 A 1) \u2192 A (2^ n)\nsearch\u2032 {n} {a} {A} op f = Search.search 1 2*_ {a} {A} (\u03bb {m} \u2192 op {m}) f\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {n} {a} {A} _\u00b7_ f = search\u2032 {n} {a} {const A} _\u00b7_ f\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\nsearch-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\nsearch-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n  where\n    go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n    go zero = refl\n    go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^ n = sum-const n 1\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n#-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n#-bound zero    f = Bool.to\u2115\u22641 (f [])\n#-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n#-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n#-\u2295 [] f = \u2261.refl\n#-\u2295 (b \u2237 bs) f\n  rewrite #-\u2295 bs (f \u2218 0\u2237_)\n        | #-\u2295 bs (f \u2218 1\u2237_)\n  with b\n... | false = refl\n... | true  = \u2115\u00b0.+-comm #\u27e8 f \u2218 0\u2237_ \u2218 _\u2295_ bs \u27e9 _\n\n#-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n#-\u2218vnot _ f = #-\u2295 1\u207f f\n\n#-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n#-\u2218xor\u1d62 i f b = pf\n  where pad = on\u1d62 (_xor_ b) i 0\u207f\n        pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n        pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n#-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n#-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n#-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n#-not\u2218 zero f with f []\n... | true  = \u2261.refl\n... | false = \u2261.refl\n#-not\u2218 (suc c) f\n  rewrite #-not\u2218 c (f \u2218 0\u2237_)\n        | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op (|de-morgan| f g)\n\nsearch-comm :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-comm {zero} _+_ _*_ f p hom = refl\nsearch-comm {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-comm {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-comm _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (_\u229b_; rewire; rewireTbl; sum) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u00b7_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u00b7 search (f \u2218 1\u2237_)\n\nsearch\u2032 : \u2200 {n a} {A : \u2115 \u2192 Set a} \u2192 (\u2200 {m} \u2192 A m \u2192 A m \u2192 A (2* m)) \u2192 (Bits n \u2192 A 1) \u2192 A (2^ n)\nsearch\u2032 {n} {a} {A} op f = Search.search 1 2*_ {a} {A} (\u03bb {m} \u2192 op {m}) f\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {n} {a} {A} _\u00b7_ f = search\u2032 {n} {a} {const A} _\u00b7_ f\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nsum : \u2200 {n} \u2192 (Bits n \u2192 \u2115) \u2192 \u2115\nsum = search _+_\n\nsum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\nsum-const zero    _ = refl\nsum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = sum (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\nsearch-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                 (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\nsearch-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n  where\n    go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n    go zero = refl\n    go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n#never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n#never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n#always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n#always\u22612^_ zero = refl\n#always\u22612^_ (suc n) = cong\u2082 _+_ pf pf where pf = #always\u22612^ n\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\ncount\u1d47 : Bit \u2192 \u2115\ncount\u1d47 b = if b then 1 else 0\n\n#\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n#\u27e8== [] \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n#\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n\u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n\u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n\u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                        (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n\u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n\u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n#-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n#-+ f f0 f1 rewrite f0 | f1 = refl\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\n#-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-take-drop zero n f g with f []\n... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n... | false = #never\u22610 n\n#-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                       (#-take-drop m n (f \u2218 0\u2237_) g))\n                                  (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                       (#-take-drop m n (f \u2218 1\u2237_) g)))\n                           (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n#-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n#-drop-take m n f g =\n           #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n         \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n           #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n         \u2261\u27e8 #-take-drop m n g f \u27e9\n           #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n         \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n           #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n         \u220e\n  where open \u2261-Reasoning\n\n#-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n             \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n#-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n             \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n               #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n               2^ n * #\u27e8 f \u27e9\n             \u220e\n  where open \u2261-Reasoning\n\n#\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n#\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n#\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n#\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n... | true  | true  | _ = s\u2264s z\u2264n\n... | true  | false | p = \u22a5-elim (p _)\n... | false | _     | _ = z\u2264n\n#\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n#-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n#-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n#-\u2227-\u2228\u1d47 false y = refl\n\n#-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n#-\u2227-\u2228 {suc n} f g =\n  trans\n    (trans\n       (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n               #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n               #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n       (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                  (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n    (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n#\u2228' {zero} f g with f []\n... | true  = s\u2264s z\u2264n\n... | false = \u2115\u2264.refl\n#\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                             #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                    (\u2115\u2264.reflexive\n                      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n    where open SemiringSolver\n          helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n          helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n#\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n#\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\n#\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n#\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = \u2257-cong-search op {-(f |\u2228| g) (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))-} (|de-morgan| f g)\n\nsearch-comm :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-comm {zero} _+_ _*_ f p hom = refl\nsearch-comm {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-comm {n} _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-comm _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n[0\u2194_] : \u2200 {n} \u2192 Fin n \u2192 Bits n \u2192 Bits n\n[0\u2194_] {zero}  i xs = xs\n[0\u2194_] {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n[0\u21941] : Bits 2 \u2192 Bits 2\n[0\u21941] = [0\u2194 suc zero ]\n\n[0\u21941]-spec : [0\u21941] \u2257 (\u03bb { (x \u2237 y \u2237 []) \u2192 y \u2237 x \u2237 [] })\n[0\u21941]-spec (x \u2237 y \u2237 []) = refl\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d32c76033519bcdbadfa664dd8d72f814b0a1899","subject":"+bitsfun","message":"+bitsfun\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits : \u2115 \u2192 T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n  open FlatFuns \u266dFuns public\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Bits Bit _\u00d7_ (\u03bb A B \u2192 A \u2192 B)\n\nbitsfun\u266dFuns : FlatFuns \u2115\nbitsfun\u266dFuns = mk id 1 _+_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `Bits : \u2115 \u2192 T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n  infixr 2 _`\u00d7_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n  open FlatFuns \u266dFuns public\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Bits Bit _\u00d7_ (\u03bb A B \u2192 A \u2192 B)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"633358202df2414ee41c961672407df087271369","subject":"WIP implement composition for sum changes STUCK","message":"WIP implement composition for sum changes STUCK\n\nGoal: implement change composition without using \u229d for any datatype, so\nthat we can also drop the base value index for compose.\n\nProblem: if I compose a change from inj\u2081 a1 to inj\u2082 b2, and a change\nfrom inj\u2082 b2 to inj\u2081 a3, I need to produce a change from inj\u2081 a1 to inj\u2081\na3. I can either use a3 \u229d a1, and we don't want, or produce a\nreplacement change from a inj\u2081 to inj\u2081, which is not valid. So I'll need\nto widen validity.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/Changes.agda","new_file":"Thesis\/Changes.agda","new_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  valid : \u2200 (a : A) (da : ChA) \u2192 Set\n  valid a da = ch da from a to (a \u2295 da)\n  \u0394 : (a : A) \u2192 Set\n  \u0394 a = \u03a3[ da \u2208 ChA ] valid a da\n  \u0394\u2082 : (a1 : A) (a2 : A) \u2192 Set\n  \u0394\u2082 a1 a2 = \u03a3[ da \u2208 ChA ] ch da from a1 to a2\n\n  _\u2295'_ : (a1 : A) -> {a2 : A} -> (da : \u0394\u2082 a1 a2) -> A\n  a1 \u2295' (da , daa) = a1 \u2295 da\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  _s\u229a2[_]_ : SumChange2 \u2192 A \u228e B \u2192 SumChange2 \u2192 SumChange2\n  ch\u2081 da1 s\u229a2[ inj\u2081 a ] ch\u2081 da2 = ch\u2081 (da1 \u229a[ a ] da2)\n  ch\u2081 da1 s\u229a2[ inj\u2082 b ] ch\u2081 da2 = {!!}\n  ch\u2082 db s\u229a2[ s ] ch\u2081 da = {!!}\n  rp (inj\u2081 a2) s\u229a2[ s ] ch\u2081 da2 = rp (inj\u2081 (a2 \u2295 da2))\n  rp (inj\u2082 y) s\u229a2[ s ] ch\u2081 da2 = {!!}\n  ch\u2081 da s\u229a2[ s ] ch\u2082 db = {!!}\n  ch\u2082 db1 s\u229a2[ inj\u2081 a ] ch\u2082 db2 = ch\u2082 {!!}\n  ch\u2082 db1 s\u229a2[ inj\u2082 b ] ch\u2082 db2 = ch\u2082 (db1 \u229a[ b ] db2)\n  rp (inj\u2081 a2) s\u229a2[ s ] ch\u2082 db2 = {!!}\n  rp (inj\u2082 b2) s\u229a2[ s ] ch\u2082 db2 = rp (inj\u2082 (b2 \u2295 db2))\n  ch\u2081 da s\u229a2[ s ] rp s\u2081 = rp s\u2081\n  ch\u2082 db s\u229a2[ s ] rp s\u2081 = rp s\u2081\n  rp s2 s\u229a2[ s1 ] rp s3 = rp s3\n\n  _s\u229a[_]_ : SumChange \u2192 A \u228e B \u2192 SumChange \u2192 SumChange\n  ds1 s\u229a[ s ] ds2 = convert\u2081 (convert ds1 s\u229a2[ s ] convert ds2)\n\n  s\u229a-fromto : (s1 s2 s3 : A \u228e B) (ds1 ds2 : SumChange) \u2192\n      sch ds1 from s1 to s2 \u2192\n      sch ds2 from s2 to s3 \u2192 sch ds1 s\u229a[ s1 ] ds2 from s1 to s3\n  s\u229a-fromto (inj\u2081 a1) (inj\u2081 a2) (inj\u2081 a3) (inj\u2081 (inj\u2081 da1)) (inj\u2081 (inj\u2081 da2)) (sft\u2081 daa1) (sft\u2081 daa2) = sft\u2081 (\u229a-fromto a1 a2 a3 _ _ daa1 daa2)\n  s\u229a-fromto .(inj\u2082 b1) .(inj\u2081 a2) (inj\u2081 a3) .(inj\u2082 (inj\u2081 a2)) (inj\u2081 (inj\u2081 da2)) (sftrp\u2082 b1 a2) (sft\u2081 daa2) with sfromto\u2192\u2295 (inj\u2081 (inj\u2081 da2)) _ _ (sft\u2081 daa2)\n  s\u229a-fromto .(inj\u2082 b1) .(inj\u2081 a2) (inj\u2081 .(a2 \u2295 da2)) .(inj\u2082 (inj\u2081 a2)) (inj\u2081 (inj\u2081 da2)) (sftrp\u2082 b1 a2) (sft\u2081 daa2) | refl = sftrp\u2082 b1 (a2 \u2295 da2)\n  s\u229a-fromto (inj\u2082 b1) (inj\u2082 b2) (inj\u2082 b3) (inj\u2081 (inj\u2082 db1)) (inj\u2081 (inj\u2082 db2)) (sft\u2082 dbb1) (sft\u2082 dbb2) = sft\u2082 (\u229a-fromto b1 b2 b3 _ _ dbb1 dbb2)\n  s\u229a-fromto .(inj\u2081 a1) .(inj\u2082 b2) .(inj\u2082 _) .(inj\u2082 (inj\u2082 b2)) (inj\u2081 (inj\u2082 db2)) (sftrp\u2081 a1 b2) (sft\u2082 dbb2) with sfromto\u2192\u2295 (inj\u2081 (inj\u2082 db2)) _ _ (sft\u2082 dbb2)\n  s\u229a-fromto .(inj\u2081 a1) .(inj\u2082 b2) .(inj\u2082 (b2 \u2295 db2)) .(inj\u2082 (inj\u2082 b2)) (inj\u2081 (inj\u2082 db2)) (sftrp\u2081 a1 b2) (sft\u2082 dbb2) | refl = sftrp\u2081 a1 (b2 \u2295 db2)\n  s\u229a-fromto (inj\u2081 a1) .(inj\u2081 a2) .(inj\u2082 b2) .(inj\u2081 (inj\u2081 _)) .(inj\u2082 (inj\u2082 b2)) (sft\u2081 daa) (sftrp\u2081 a2 b2) = sftrp\u2081 a1 b2\n  s\u229a-fromto .(inj\u2082 b1) .(inj\u2081 a1) .(inj\u2082 b2) .(inj\u2082 (inj\u2081 a1)) .(inj\u2082 (inj\u2082 b2)) (sftrp\u2082 b1 .a1) (sftrp\u2081 a1 b2) = {!!}\n  s\u229a-fromto .(inj\u2082 _) .(inj\u2082 b2) .(inj\u2081 a2) .(inj\u2081 (inj\u2082 _)) .(inj\u2082 (inj\u2081 a2)) (sft\u2082 dbb2) (sftrp\u2082 b2 a2) = sftrp\u2082 _ _\n  s\u229a-fromto .(inj\u2081 a1) .(inj\u2082 b1) .(inj\u2081 a2) .(inj\u2082 (inj\u2082 b1)) .(inj\u2082 (inj\u2081 a2)) (sftrp\u2081 a1 .b1) (sftrp\u2082 b1 a2) = {!!}\n\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      }\n      ; _\u229a[_]_ = _s\u229a[_]_\n      ; \u229a-fromto = s\u229a-fromto\n      }\n    }\n\ninstance\n  unitCS : ChangeStructure \u22a4\n\n  unitCS = record\n    { Ch = \u22a4\n    ; ch_from_to_ = \u03bb dv v1 v2 \u2192 \u22a4\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n      { _\u2295_ = \u03bb _ _ \u2192 tt\n      ; fromto\u2192\u2295 = \u03bb { _ _ tt _ \u2192 refl }\n      ; _\u229d_ = \u03bb _ _ \u2192 tt\n      ; \u229d-fromto = \u03bb a b \u2192 tt\n      }\n      ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 tt\n      ; \u229a-fromto = \u03bb a1 a2 a3 da1 da2 daa1 daa2 \u2192 tt\n      }\n    }\n","old_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  valid : \u2200 (a : A) (da : ChA) \u2192 Set\n  valid a da = ch da from a to (a \u2295 da)\n  \u0394 : (a : A) \u2192 Set\n  \u0394 a = \u03a3[ da \u2208 ChA ] valid a da\n  \u0394\u2082 : (a1 : A) (a2 : A) \u2192 Set\n  \u0394\u2082 a1 a2 = \u03a3[ da \u2208 ChA ] ch da from a1 to a2\n\n  _\u2295'_ : (a1 : A) -> {a2 : A} -> (da : \u0394\u2082 a1 a2) -> A\n  a1 \u2295' (da , daa) = a1 \u2295 da\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\ninstance\n  unitCS : ChangeStructure \u22a4\n\n  unitCS = record\n    { Ch = \u22a4\n    ; ch_from_to_ = \u03bb dv v1 v2 \u2192 \u22a4\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n      { _\u2295_ = \u03bb _ _ \u2192 tt\n      ; fromto\u2192\u2295 = \u03bb { _ _ tt _ \u2192 refl }\n      ; _\u229d_ = \u03bb _ _ \u2192 tt\n      ; \u229d-fromto = \u03bb a b \u2192 tt\n      }\n      ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 tt\n      ; \u229a-fromto = \u03bb a1 a2 a3 da1 da2 daa1 daa2 \u2192 tt\n      }\n    }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"23a95f7587bd0f26c567388e9114c092ece0a98d","subject":"Just fixed some comments","message":"Just fixed some comments\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\nimport ZK.SigmaProtocol.Types\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n  -- TODO Challenge should exp large wrt the security param\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- See the module ZK.Types for details about the types:\n--   * Prover-Interaction\n--   * Verifier\n--   * Transcript\n--   * Transcript\u00b2\nopen ZK.SigmaProtocol.Types Commitment Challenge Response public\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n_\u21c6_ : Verifier \u2192 Prover \u2192 (r : Randomness)(w : Witness)(c : Challenge) \u2192 Bool\n(verifier \u21c6 prover) r w c = verifier (run (prover r w) c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = \u2713 (verifier t)\n\n  prover-commitment : (r : Randomness)(w : Witness) \u2192 Commitment\n  prover-commitment r w = Prover-Interaction.get-A (prover r w)\n\n  prover-response : (r : Randomness)(w : Witness)(c : Challenge) \u2192 Response\n  prover-response r w = Prover-Interaction.get-f (prover r w)\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the honest prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 w r c (w? : ValidWitness w) \u2192\n                              \u2713 ((verifier \u21c6 prover) r w c)\n\n-- A simulator takes a challenge and a response and returns a commitment (to be used to prove the Special Honest Verifier Zero-Knowledge property).\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713 (verifier (mk A c s))\n\n{-\n  A \u03a3-protocol is said to be Special Honest Verifier Zero-Knowledge if there is a correct simulator.\n  This property is one of the necessary conditions to apply the Strong Fiat Shamir\n  transformation.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the honest prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator          : Simulator\n    .correct-simulator : Correct-simulator verifier simulator\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = (t\u00b2 : Transcript\u00b2 verifier) \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct knowledge extractor is said to have the Special Soundness property.\n-- This property is one of the conditions needed to apply the Strong Fiat Shamir\n-- transformation.\n-- Also called 2-extractable\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor              : Extractor verifier\n    .extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    .correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\nimport ZK.SigmaProtocol.Types\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n  -- TODO Challenge should exp large wrt the security param\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- See the module ZK.Types for details about the types:\n--   * Prover-Interaction\n--   * Verifier\n--   * Transcript\n--   * Transcript\u00b2\nopen ZK.SigmaProtocol.Types Commitment Challenge Response public\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n_\u21c6_ : Verifier \u2192 Prover \u2192 (r : Randomness)(w : Witness)(c : Challenge) \u2192 Bool\n(verifier \u21c6 prover) r w c = verifier (run (prover r w) c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = \u2713 (verifier t)\n\n  prover-commitment : (r : Randomness)(w : Witness) \u2192 Commitment\n  prover-commitment r w = Prover-Interaction.get-A (prover r w)\n\n  prover-response : (r : Randomness)(w : Witness)(c : Challenge) \u2192 Response\n  prover-response r w = Prover-Interaction.get-f (prover r w)\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 w r c (w? : ValidWitness w) \u2192\n                              \u2713 ((verifier \u21c6 prover) r w c)\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713 (verifier (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator          : Simulator\n    .correct-simulator : Correct-simulator verifier simulator\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = (t\u00b2 : Transcript\u00b2 verifier) \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\n-- Also called 2-extractable\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor              : Extractor verifier\n    .extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    .correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"68af677d56d4f0a4c1aaa95134ae06f0813500c1","subject":"Convert to stdlib","message":"Convert to stdlib\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"open import Data.Bool\nopen import Data.List\nopen import Data.Nat\n\n\n----------------------------------------\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n----------------------------------------\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 []))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 []\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 [])) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (5 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (6 \u2237 3 \u2237 [])) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (6 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 []) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 []) (3 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 [])) \u2237\n  (model (10 \u2237 9 \u2237 []) (1 \u2237 [])) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 []) (4 \u2237 11 \u2237 8 \u2237 [])) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 []) (11 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 []) (11 \u2237 3 \u2237 [])) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 []) ([])) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 []) (12 \u2237 [])) \u2237 []\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n\n\nmain = (closure proofs (5 \u2237 []))\n","old_contents":"open import Agda.Builtin.Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\ninfixr 5 _\u2237_\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n{-# BUILTIN LIST List #-}\n{-# BUILTIN NIL  \u2218    #-}\n{-# BUILTIN CONS _\u2237_  #-}\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 \u2218\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 \u2218)) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (5 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 \u2218) (6 \u2237 3 \u2237 \u2218)) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 \u2218) (6 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 \u2218) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 \u2218) (3 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218)) \u2237\n  (model (10 \u2237 9 \u2237 \u2218) (1 \u2237 \u2218)) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 \u2218) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 \u2218)) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 \u2218) (4 \u2237 11 \u2237 8 \u2237 \u2218)) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 \u2218) (11 \u2237 \u2218)) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 \u2218) (11 \u2237 3 \u2237 \u2218)) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 \u2218) (\u2218)) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 \u2218) (12 \u2237 \u2218)) \u2237 \u2218\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n\n\nmain = (closure proofs (5 \u2237 \u2218))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fed88a10935d042bd79da68f2e7233104f1a0536","subject":"distance: upgrade count, and add antisym","message":"distance: upgrade count, and add antisym\n","repos":"crypto-agda\/crypto-agda","old_file":"program-distance.agda","new_file":"program-distance.agda","new_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nprivate\n  2^_\u22651 : \u2200 n \u2192 2^ n \u2265 1\n  2^ zero \u22651  = s\u2264s z\u2264n\n  2^ suc n \u22651 = z\u2264n {2^ n} +-mono 2^ n \u22651\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n  ]-[-antisym {n} f f]-[g rewrite dist-refl (count\u21ba f) with \u2115\u2264.trans (2^ (n \u2238 k) \u22651) f]-[g\n  ... | ()\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist \u27e82^ n * count\u21ba f \u27e9 \u27e82^ m * count\u21ba g \u27e9 \u2265 2^((m + n) \u2238 k)\n\n  ]-[-antisym : \u2200 {n} (f : \u2141 n) \u2192 \u00ac (f ]-[ f)\n  ]-[-antisym {n} f f]-[g rewrite dist-refl \u27e82^ n * count\u21ba f \u27e9 with \u2115\u2264.trans (2^ (n + n \u2238 k) \u22651) f]-[g\n  ... | ()\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym \u27e82^ n * count\u21ba f \u27e9 \u27e82^ m * count\u21ba g \u27e9 | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 m n o k \u2192 m + ((n + o) \u2238 k) \u2261 n + ((m + o) \u2238 k)\n      helper\u2032 : \u2200 m n o k \u2192 \u27e82^ m * (2^((n + o) \u2238 k))\u27e9 \u2261 \u27e82^ n * (2^((m + o) \u2238 k))\u27e9\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n           -- 2\u1d50(dist 2\u1d52#g 2\u207f#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n  ... | q rewrite sym (dist-2^* m \u27e82^ o * count\u21ba g \u27e9 \u27e82^ n * count\u21ba h \u27e9)\n           -- dist 2\u1d502\u1d52#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m o (count\u21ba g)\n           -- dist 2\u1d522\u1d50#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | sym f\u2248g\n           -- dist 2\u1d522\u207f#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm o n (count\u21ba f)\n           -- dist 2\u207f2\u1d52#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m n (count\u21ba h)\n           -- dist 2\u207f2\u1d52#f 2\u207f2\u1d50#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | dist-2^* n \u27e82^ o * count\u21ba f \u27e9 \u27e82^ m * count\u21ba h \u27e9\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-+ m (n + o \u2238 k) 1\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d50\u207a\u207f\u207a\u1d52\u207b\u1d4f\n                | helper m n o k\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f\u207a\u1d50\u207a\u1d52\u207b\u1d4f\n                | sym (2^-+ n (m + o \u2238 k) 1)\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f2\u1d50\u207a\u1d52\u207b\u1d4f\n                = 2^*-mono\u2032 n q\n           -- dist 2\u1d52#f 2\u1d50#h \u2264 2\u1d50\u207a\u1d52\u207b\u1d4f\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_\n               ]-[-antisym\n               (\u03bb {m n f g} \u2192 ]-[-sym {f = f} {g})\n               (\u03bb {m n o f g h} \u2192 ]-[-cong-left-\u2248\u21ba {f = f} {g} {h})\n","old_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) \u2265 2^((m + n) \u2238 k)\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 x y z t \u2192 x + ((y + z) \u2238 t) \u2261 y + ((x + z) \u2238 t)\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n           -- 2\u1d50(dist 2\u1d52#g 2\u207f#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n  ... | q rewrite sym (dist-2^* m (\u27e82^ o \u27e9* (count\u21ba g)) (\u27e82^ n \u27e9* (count\u21ba h)))\n           -- dist 2\u1d502\u1d52#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m o (count\u21ba g)\n           -- dist 2\u1d522\u1d50#g 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | sym f\u2248g\n           -- dist 2\u1d522\u207f#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm o n (count\u21ba f)\n           -- dist 2\u207f2\u1d52#f 2\u1d502\u207f#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-comm m n (count\u21ba h)\n           -- dist 2\u207f2\u1d52#f 2\u207f2\u1d50#h \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | dist-2^* n (\u27e82^ o \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba h))\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d502\u207f\u207a\u1d52\u207b\u1d4f\n                | 2^-+ m (n + o \u2238 k) 1\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u1d50\u207a\u207f\u207a\u1d52\u207b\u1d4f\n                | helper m n o k\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f\u207a\u1d50\u207a\u1d52\u207b\u1d4f\n                | sym (2^-+ n (m + o \u2238 k) 1)\n           -- 2\u207f(dist 2\u1d52#f 2\u1d50#h) \u2264 2\u207f2\u1d50\u207a\u1d52\u207b\u1d4f\n                = 2^*-mono\u2032 n q\n           -- dist 2\u1d52#f 2\u1d50#h \u2264 2\u1d50\u207a\u1d52\u207b\u1d4f\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_ (\u03bb {m n f g} x \u2192 ]-[-sym {m} {n} {f} {g} x) (\u03bb {m n o f g h} x y \u2192 ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} x y)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"74ad4a9a6900f8062300b3ca1cc4b318038c4af5","subject":"Use HOAS-enabled helpers for nested abstractions.","message":"Use HOAS-enabled helpers for nested abstractions.\n\nOld-commit-hash: 4efbb3abc5e02e14285a2e9c6998ee6f663cbc2d\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Change\/Term.agda","new_file":"Popl14\/Change\/Term.agda","new_contents":"module Popl14.Change.Term where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Data.Integer\n\nopen import Popl14.Syntax.Type public\nopen import Popl14.Syntax.Term public\nopen import Popl14.Change.Type public\n\nimport Parametric.Change.Term Const \u0394Base as ChangeTerm\n\ndiff-base : ChangeTerm.DiffStructure\ndiff-base {base-int} = abs\u2082 (\u03bb x y \u2192 add x (minus y))\ndiff-base {base-bag} = abs\u2082 (\u03bb x y \u2192 union x (negate y))\n\napply-base : ChangeTerm.ApplyStructure\napply-base {base-int} = abs\u2082 (\u03bb \u0394x x \u2192 add x \u0394x)\napply-base {base-bag} = abs\u2082 (\u03bb \u0394x x \u2192 union x \u0394x)\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-term {base \u03b9} = apply-base {\u03b9}\napply-term {\u03c3 \u21d2 \u03c4} =\n    abs\u2083 (\u03bb \u0394f f x \u2192\n      app f x \u2295 app (app \u0394f x) (x \u229d x))\n\ndiff-term {base \u03b9} = diff-base {\u03b9}\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  abs\u2084 (\u03bb g f x \u0394x \u2192\n    app g (x \u2295 \u0394x) \u229d app f x)\n","old_contents":"module Popl14.Change.Term where\n\n-- Terms Calculus Popl14\n--\n-- Contents\n-- - Term constructors\n-- - Weakening on terms\n-- - `fit`: weaken a term to its \u0394Context\n-- - diff-term, apply-term and their syntactic sugars\n\nopen import Data.Integer\n\nopen import Popl14.Syntax.Type public\nopen import Popl14.Syntax.Term public\nopen import Popl14.Change.Type public\n\nimport Parametric.Change.Term Const \u0394Base as ChangeTerm\n\ndiff-base : ChangeTerm.DiffStructure\ndiff-base {base-int} = abs\u2082 (\u03bb x y \u2192 add x (minus y))\ndiff-base {base-bag} = abs\u2082 (\u03bb x y \u2192 union x (negate y))\n\napply-base : ChangeTerm.ApplyStructure\napply-base {base-int} = abs\u2082 (\u03bb \u0394x x \u2192 add x \u0394x)\napply-base {base-bag} = abs\u2082 (\u03bb \u0394x x \u2192 union x \u0394x)\n\ndiff-term  : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n-- Sugars for diff-term and apply-term\ninfixl 6 _\u2295_ _\u229d_\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394Type \u03c4)\nt \u2295 \u0394t = app (app apply-term \u0394t) t\ns \u229d t  = app (app  diff-term  s) t\n\napply-term {base \u03b9} = apply-base {\u03b9}\napply-term {\u03c3 \u21d2 \u03c4} =\n  let\n    \u0394f = var (that (that this))\n    f  = var (that this)\n    x  = var this\n  in\n  -- \u0394f   f    x\n    abs (abs (abs\n      (app f x \u2295 app (app \u0394f x) (x \u229d x))))\n\ndiff-term {base \u03b9} = diff-base {\u03b9}\ndiff-term {\u03c3 \u21d2 \u03c4} =\n  let\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n  in\n  -- g    f    x    \u0394x\n    abs (abs (abs (abs\n      (app g (x \u2295 \u0394x) \u229d app f x))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"400e2cbe466aab63f8821e6edb152a0a594ce192","subject":"We are not using the induction principle for N.","message":"We are not using the induction principle for N.\n\nIgnore-this: d4db0e016b44ac1278d071501fae50e2\n\ndarcs-hash:20100503135901-3bd4e-f27604c8ecc5865e2dfb311c3790883a9f9c24f0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/N.agda","new_file":"LTC\/Data\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Inductive predicate for the total natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.N where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\n-- The inductive predicate 'N' represents the type of the natural\n-- numbers. They are a subset of 'D'.\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n-- {-# ATP hint zN #-}\n-- {-# ATP hint sN #-}\n\n-- Induction principle for N (elimination rule).\n-- indN : (P : D \u2192 Set) \u2192\n--        P zero \u2192\n--        ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n--        {n : D} \u2192 N n \u2192 P n\n-- indN P p0 h zN      = p0\n-- indN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n\n-- -- Other version\n-- indN : (P : {m : D} \u2192 N m \u2192 Set) \u2192\n--        P zN \u2192\n--        ({n : D} \u2192 (Nn : N n) \u2192 P Nn \u2192 P (sN Nn)) \u2192\n--        {n : D} \u2192 (Nn : N n) \u2192 P Nn\n-- indN P p0 h zN      = p0\n-- indN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Inductive predicate for the total natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.N where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\n-- The inductive predicate 'N' represents the type of the natural\n-- numbers. They are a subset of 'D'.\n\n-- The LTC natural numbers type.\ndata N : D \u2192 Set where\n  zN : N zero\n  sN : {n : D} \u2192 N n \u2192 N (succ n)\n-- {-# ATP hint zN #-}\n-- {-# ATP hint sN #-}\n\n-- Induction principle for N (elimination rule).\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       ({n : D} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       {n : D} \u2192 N n \u2192 P n\nindN P p0 h zN      = p0\nindN P p0 h (sN Nn) = h Nn (indN P p0 h Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c23799f5988bf5caaa7f47dd75171d0ea3eb183","subject":"Possibly useful dead code","message":"Possibly useful dead code\n","repos":"inc-lc\/ilc-agda","old_file":"New\/LangOps.agda","new_file":"New\/LangOps.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule New.LangOps where\n\nopen import Data.List.All\nopen import Data.Unit\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Integer\n\nopen import New.Lang\nopen import New.Changes\nopen import New.LangChanges\n\noplus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4 \u21d2 \u03c4)\nominus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394t \u03c4)\n\nonil\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4)\nonil\u03c4o \u03c4 = abs (app\u2082 (ominus\u03c4o \u03c4) (var this) (var this))\n\noplus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs\n  (app\u2082 (oplus\u03c4o \u03c4)\n    (app (var (that (that this))) (var this))\n    (app\u2082 (var (that this)) (var this) (app (onil\u03c4o \u03c3) (var this))))))\noplus\u03c4o int = const plus\noplus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (oplus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (oplus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\noplus\u03c4o (sum \u03c3 \u03c4) = {!!}\n\nominus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs (abs (app\u2082 (ominus\u03c4o \u03c4)\n  (app (var (that (that (that this)))) (app\u2082 (oplus\u03c4o \u03c3) (var (that this)) (var this)))\n  (app (var (that (that this))) (var (that this)))))))\nominus\u03c4o int = const minus\nominus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (ominus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (ominus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\nominus\u03c4o (sum \u03c3 \u03c4) = {!!}\n\nopen import Postulate.Extensionality\n\nsyntax oplus\u03c4o t a da = a \u229e[ t ] da\n\noplus\u03c4 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\noplus\u03c4 \u03c4 = \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u2205\nominus\u03c4 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\nominus\u03c4 \u03c4 = \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u2205\n\n\u27e6oplus\u03c4\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6ominus\u03c4\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n\u27e6oplus\u03c4\u27e7 \u03c4 = _\u2295_\n\u27e6ominus\u03c4\u27e7 \u03c4 = _\u229d_\n\noplus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 a da \u2192 \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1 a da \u2261 \u27e6oplus\u03c4\u27e7 \u03c4 a da\nominus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 a da \u2192 \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u03c1 a da \u2261 \u27e6ominus\u03c4\u27e7 \u03c4 a da\n\noplus\u03c4-equiv-ext : \u2200 \u03c4 \u2192 oplus\u03c4 \u03c4 \u2261 \u27e6oplus\u03c4\u27e7 \u03c4\noplus\u03c4-equiv-ext \u03c4 = ext (\u03bb a \u2192 ext (oplus\u03c4-equiv \u2205 \u2205 \u03c4 a))\nominus\u03c4-equiv-ext : \u2200 \u03c4 \u2192 ominus\u03c4 \u03c4 \u2261 \u27e6ominus\u03c4\u27e7 \u03c4\nominus\u03c4-equiv-ext \u03c4 = ext (\u03bb a \u2192 ext (ominus\u03c4-equiv \u2205 \u2205 \u03c4 a))\n\noplus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) f df = ext (\u03bb a \u2192 lemma a)\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) where\n      \u03c1\u2032 : \u27e6 \u03c3 \u2022 \u0394t (\u03c3 \u21d2 \u03c4) \u2022 (\u03c3 \u21d2 \u03c4) \u2022 \u0393 \u27e7Context\n      \u03c1\u2032 = a \u2237 df \u2237 f \u2237 \u03c1\n      \u03c1\u2032\u2032 = a \u2237 \u03c1\u2032\n      lemma : \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1\u2032 (f a)\n         (df a (\u27e6 ominus\u03c4o \u03c3 \u27e7Term \u03c1\u2032\u2032 a a))\n         \u2261 \u27e6oplus\u03c4\u27e7 \u03c4 (f a) (df a (\u27e6ominus\u03c4\u27e7 \u03c3 a a))\n      lemma\n        rewrite ominus\u03c4-equiv _ \u03c1\u2032\u2032 \u03c3 a a\n        | oplus\u03c4-equiv _ \u03c1\u2032 \u03c4 (f a) (df a (\u27e6ominus\u03c4\u27e7 \u03c3 a a))\n        = refl\noplus\u03c4-equiv \u0393 \u03c1 int a da = refl\noplus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a , b) (da , db)\n  rewrite oplus\u03c4-equiv _ ((da , db) \u2237 (a , b) \u2237 \u03c1) \u03c3 a da\n  | oplus\u03c4-equiv _ ((da , db) \u2237 (a , b) \u2237 \u03c1) \u03c4 b db\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) s ds = {!!}\n\nominus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) g f = ext (\u03bb a \u2192 ext (lemma a))\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) (da : Cht \u03c3) where\n      \u03c1\u2032 = da \u2237 a \u2237 f \u2237 g \u2237 \u03c1\n      lemma : \u27e6 ominus\u03c4o \u03c4 \u27e7Term (da \u2237 a \u2237 f \u2237 g \u2237 \u03c1)\n        (g (\u27e6 oplus\u03c4o \u03c3 \u27e7Term (da \u2237 a \u2237 f \u2237 g \u2237 \u03c1) a da)) (f a)\n        \u2261 \u27e6ominus\u03c4\u27e7 \u03c4 (g (\u27e6oplus\u03c4\u27e7 \u03c3 a da)) (f a)\n      lemma\n        rewrite oplus\u03c4-equiv _ \u03c1\u2032 \u03c3 a da\n        | ominus\u03c4-equiv _ \u03c1\u2032 \u03c4 (g (\u27e6oplus\u03c4\u27e7 \u03c3 a da)) (f a) = refl\nominus\u03c4-equiv \u0393 \u03c1 int b a = refl\nominus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a2 , b2) (a1 , b1)\n  rewrite ominus\u03c4-equiv _ ((a1 , b1) \u2237 (a2 , b2) \u2237 \u03c1) \u03c3 a2 a1\n  | ominus\u03c4-equiv _ ((a1 , b1) \u2237 (a2 , b2) \u2237 \u03c1) \u03c4 b2 b1\n  = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) s2 s1 = {!!}\n\n-- synIsChAlg\u03c4 : (\u03c4 : Type) \u2192 IsChAlg \u27e6 \u03c4 \u27e7Type \u27e6 \u0394t \u03c4 \u27e7Type (oplus\u03c4 \u03c4) (ominus\u03c4 \u03c4)\n\n-- synChAlgt : (\u03c4 : Type) \u2192 ChAlg \u27e6 \u03c4 \u27e7Type\n-- synChAlgt \u03c4 = record { Ch = Cht \u03c4 ; _\u2295_ = oplus\u03c4 \u03c4 ; _\u229d_ = ominus\u03c4 \u03c4 ; isChAlg = isChAlg\u03c4 \u03c4}\n-- private\n--   instance\n--     synIchAlgt : \u2200 {\u03c4} \u2192 ChAlg \u27e6 \u03c4 \u27e7Type\n-- synIchAlgt {\u03c4} = chAlgt \u03c4\n\n-- synIsChAlg\u03c4 \u03c4 rewrite oplus\u03c4-equiv-ext \u03c4 | ominus\u03c4-equiv-ext \u03c4 = \u27e6isChAlg\u03c4\u27e7 \u03c4\n\n-- oplus\u03c4o-invariant : \u2200 \u03c4 {\u0393\u2081 \u0393\u2082} (\u03c1\u2081 : \u27e6 \u0393\u2081 \u27e7Context) (\u03c1\u2082 : \u27e6 \u0393\u2082 \u27e7Context) \u2192\n--   \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1\u2081 \u2261 \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1\u2082\n-- ominus\u03c4o-invariant : \u2200 \u03c4 {\u0393\u2081 \u0393\u2082} (\u03c1\u2081 : \u27e6 \u0393\u2081 \u27e7Context) (\u03c1\u2082 : \u27e6 \u0393\u2082 \u27e7Context) \u2192\n--   \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u03c1\u2081 \u2261 \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u03c1\u2082\n-- ominus\u03c4o-invariant (\u03c3 \u21d2 \u03c4) \u03c1\u2081 \u03c1\u2082 = {!!}\n-- ominus\u03c4o-invariant int \u03c1\u2081 \u03c1\u2082 = refl\n-- ominus\u03c4o-invariant (pair \u03c3 \u03c4) \u03c1\u2081 \u03c1\u2082 = {!!}\n-- ominus\u03c4o-invariant (sum \u03c3 \u03c4) \u03c1\u2081 \u03c1\u2082 = {!!}\n-- oplus\u03c4o-invariant (\u03c3 \u21d2 \u03c4) \u03c1\u2081 \u03c1\u2082 = ext\u00b3 lemma\n--   where\n--     module _ (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) (df : Cht (\u03c3 \u21d2 \u03c4)) (a : \u27e6 \u03c3 \u27e7Type) where\n--       \u03c1\u2081\u2032 = a \u2237 df \u2237 f \u2237 \u03c1\u2081\n--       \u03c1\u2081\u2032\u2032 = a \u2237 \u03c1\u2081\u2032\n--       \u03c1\u2082\u2032 = a \u2237 df \u2237 f \u2237 \u03c1\u2082\n--       \u03c1\u2082\u2032\u2032 = a \u2237 \u03c1\u2082\u2032\n--       lemma : \u27e6 oplus\u03c4o \u03c4 \u27e7Term (a \u2237 df \u2237 f \u2237 \u03c1\u2081) (f a)\n--         (df a (\u27e6 ominus\u03c4o \u03c3 \u27e7Term (a \u2237 a \u2237 df \u2237 f \u2237 \u03c1\u2081) a a))\n--         \u2261\n--         \u27e6 oplus\u03c4o \u03c4 \u27e7Term (a \u2237 df \u2237 f \u2237 \u03c1\u2082) (f a)\n--         (df a (\u27e6 ominus\u03c4o \u03c3 \u27e7Term (a \u2237 a \u2237 df \u2237 f \u2237 \u03c1\u2082) a a))\n--       lemma  rewrite oplus\u03c4o-invariant \u03c4 \u03c1\u2081\u2032 \u03c1\u2082\u2032 | ominus\u03c4o-invariant \u03c3 \u03c1\u2081\u2032\u2032 \u03c1\u2082\u2032\u2032 = refl\n-- oplus\u03c4o-invariant int \u03c1\u2081 \u03c1\u2082 = refl\n-- oplus\u03c4o-invariant (pair \u03c3 \u03c4) \u03c1\u2081 \u03c1\u2082 = ext (\u03bb p \u2192 ext (lemma p))\n--   where\n--     module _ (p : \u27e6 pair \u03c3 \u03c4 \u27e7Type) (dp : Cht (pair \u03c3 \u03c4)) where\n--       \u03c1\u2081\u2032 = dp \u2237 p \u2237 \u03c1\u2081\n--       \u03c1\u2082\u2032 = dp \u2237 p \u2237 \u03c1\u2082\n--       lemma :\n--         (\u27e6 oplus\u03c4o \u03c3 \u27e7Term (dp \u2237 p \u2237 \u03c1\u2081) (proj\u2081 p) (proj\u2081 dp) ,\n--          \u27e6 oplus\u03c4o \u03c4 \u27e7Term (dp \u2237 p \u2237 \u03c1\u2081) (proj\u2082 p) (proj\u2082 dp))\n--         \u2261\n--         (\u27e6 oplus\u03c4o \u03c3 \u27e7Term (dp \u2237 p \u2237 \u03c1\u2082) (proj\u2081 p) (proj\u2081 dp) ,\n--          \u27e6 oplus\u03c4o \u03c4 \u27e7Term (dp \u2237 p \u2237 \u03c1\u2082) (proj\u2082 p) (proj\u2082 dp))\n--       lemma rewrite oplus\u03c4o-invariant \u03c3 \u03c1\u2081\u2032 \u03c1\u2082\u2032 | oplus\u03c4o-invariant \u03c4 \u03c1\u2081\u2032 \u03c1\u2082\u2032 = refl\n-- oplus\u03c4o-invariant (sum \u03c3 \u03c4) \u03c1\u2081 \u03c1\u2082 = {!!}\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule New.LangOps where\n\nopen import Data.List.All\nopen import Data.Unit\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Integer\n\nopen import New.Lang\nopen import New.Changes\nopen import New.LangChanges\n\noplus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4 \u21d2 \u03c4)\nominus\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394t \u03c4)\n\nonil\u03c4o : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 (\u03c4 \u21d2 \u0394t \u03c4)\nonil\u03c4o \u03c4 = abs (app\u2082 (ominus\u03c4o \u03c4) (var this) (var this))\n\noplus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs\n  (app\u2082 (oplus\u03c4o \u03c4)\n    (app (var (that (that this))) (var this))\n    (app\u2082 (var (that this)) (var this) (app (onil\u03c4o \u03c3) (var this))))))\noplus\u03c4o int = const plus\noplus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (oplus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (oplus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\noplus\u03c4o (sum \u03c3 \u03c4) = {!!}\n\nominus\u03c4o (\u03c3 \u21d2 \u03c4) = abs (abs (abs (abs (app\u2082 (ominus\u03c4o \u03c4)\n  (app (var (that (that (that this)))) (app\u2082 (oplus\u03c4o \u03c3) (var (that this)) (var this)))\n  (app (var (that (that this))) (var (that this)))))))\nominus\u03c4o int = const minus\nominus\u03c4o (pair \u03c3 \u03c4) = abs (abs (app\u2082 (const cons)\n  (app\u2082 (ominus\u03c4o \u03c3) (app (const fst) (var (that this))) (app (const fst) (var this)))\n  (app\u2082 (ominus\u03c4o \u03c4) (app (const snd) (var (that this))) (app (const snd) (var this)))))\nominus\u03c4o (sum \u03c3 \u03c4) = {!!}\n\nopen import Postulate.Extensionality\n\nsyntax oplus\u03c4o t a da = a \u229e[ t ] da\n\noplus\u03c4 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\noplus\u03c4 \u03c4 = \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u2205\nominus\u03c4 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\nominus\u03c4 \u03c4 = \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u2205\n\n\u27e6oplus\u03c4\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6ominus\u03c4\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type \u2192 \u27e6 \u0394t \u03c4 \u27e7Type\n\u27e6oplus\u03c4\u27e7 \u03c4 = _\u2295_\n\u27e6ominus\u03c4\u27e7 \u03c4 = _\u229d_\n\noplus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 a da \u2192 \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1 a da \u2261 \u27e6oplus\u03c4\u27e7 \u03c4 a da\nominus\u03c4-equiv : \u2200 \u0393 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u03c4 a da \u2192 \u27e6 ominus\u03c4o \u03c4 \u27e7Term \u03c1 a da \u2261 \u27e6ominus\u03c4\u27e7 \u03c4 a da\n\noplus\u03c4-equiv-ext : \u2200 \u03c4 \u2192 oplus\u03c4 \u03c4 \u2261 \u27e6oplus\u03c4\u27e7 \u03c4\noplus\u03c4-equiv-ext \u03c4 = ext (\u03bb a \u2192 ext (oplus\u03c4-equiv \u2205 \u2205 \u03c4 a))\nominus\u03c4-equiv-ext : \u2200 \u03c4 \u2192 ominus\u03c4 \u03c4 \u2261 \u27e6ominus\u03c4\u27e7 \u03c4\nominus\u03c4-equiv-ext \u03c4 = ext (\u03bb a \u2192 ext (ominus\u03c4-equiv \u2205 \u2205 \u03c4 a))\n\noplus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) f df = ext (\u03bb a \u2192 lemma a)\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) where\n      \u03c1\u2032 : \u27e6 \u03c3 \u2022 \u0394t (\u03c3 \u21d2 \u03c4) \u2022 (\u03c3 \u21d2 \u03c4) \u2022 \u0393 \u27e7Context\n      \u03c1\u2032 = a \u2237 df \u2237 f \u2237 \u03c1\n      \u03c1\u2032\u2032 = a \u2237 \u03c1\u2032\n      lemma : \u27e6 oplus\u03c4o \u03c4 \u27e7Term \u03c1\u2032 (f a)\n         (df a (\u27e6 ominus\u03c4o \u03c3 \u27e7Term \u03c1\u2032\u2032 a a))\n         \u2261 \u27e6oplus\u03c4\u27e7 \u03c4 (f a) (df a (\u27e6ominus\u03c4\u27e7 \u03c3 a a))\n      lemma\n        rewrite ominus\u03c4-equiv _ \u03c1\u2032\u2032 \u03c3 a a\n        | oplus\u03c4-equiv _ \u03c1\u2032 \u03c4 (f a) (df a (\u27e6ominus\u03c4\u27e7 \u03c3 a a))\n        = refl\noplus\u03c4-equiv \u0393 \u03c1 int a da = refl\noplus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a , b) (da , db)\n  rewrite oplus\u03c4-equiv _ ((da , db) \u2237 (a , b) \u2237 \u03c1) \u03c3 a da\n  | oplus\u03c4-equiv _ ((da , db) \u2237 (a , b) \u2237 \u03c1) \u03c4 b db\n  = refl\noplus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) s ds = {!!}\n\nominus\u03c4-equiv \u0393 \u03c1 (\u03c3 \u21d2 \u03c4) g f = ext (\u03bb a \u2192 ext (lemma a))\n  where\n    module _ (a : \u27e6 \u03c3 \u27e7Type) (da : Cht \u03c3) where\n      \u03c1\u2032 = da \u2237 a \u2237 f \u2237 g \u2237 \u03c1\n      lemma : \u27e6 ominus\u03c4o \u03c4 \u27e7Term (da \u2237 a \u2237 f \u2237 g \u2237 \u03c1)\n        (g (\u27e6 oplus\u03c4o \u03c3 \u27e7Term (da \u2237 a \u2237 f \u2237 g \u2237 \u03c1) a da)) (f a)\n        \u2261 \u27e6ominus\u03c4\u27e7 \u03c4 (g (\u27e6oplus\u03c4\u27e7 \u03c3 a da)) (f a)\n      lemma\n        rewrite oplus\u03c4-equiv _ \u03c1\u2032 \u03c3 a da\n        | ominus\u03c4-equiv _ \u03c1\u2032 \u03c4 (g (\u27e6oplus\u03c4\u27e7 \u03c3 a da)) (f a) = refl\nominus\u03c4-equiv \u0393 \u03c1 int b a = refl\nominus\u03c4-equiv \u0393 \u03c1 (pair \u03c3 \u03c4) (a2 , b2) (a1 , b1)\n  rewrite ominus\u03c4-equiv _ ((a1 , b1) \u2237 (a2 , b2) \u2237 \u03c1) \u03c3 a2 a1\n  | ominus\u03c4-equiv _ ((a1 , b1) \u2237 (a2 , b2) \u2237 \u03c1) \u03c4 b2 b1\n  = refl\nominus\u03c4-equiv \u0393 \u03c1 (sum \u03c3 \u03c4) s2 s1 = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"928dbc42fd4d6ac6895ee7c73fb723a32f864f1c","subject":"update text 2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects","message":"update text 2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f g wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er n1 n2}\n         ->     el    \u21d3  n1\n         ->        er \u21d3         (Succ n2)\n         -> Div el er \u21d3 (n1 div (Succ n2))\n\n  exb : Val 3 \u21d3 3\n  exb = \u21d3Base\n\n  exs : Div (Val 3) (Val 3) \u21d3 1\n  exs = \u21d3Step \u21d3Base \u21d3Base\n\n  -- alternatively, semantics specified by an INTERPRETER\n  -- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- define operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  How to relate the two definitions:\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign a weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set _ (Pure ba)      = a\u2192ba\u2192Set _ ba\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, define this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f g wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  Note that responses to Abort command are empty;\n  abort smart constructor abort uses this to discharge the continuation\n  in the second argument of Step constructor\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er n1 n2}\n         ->     el    \u21d3  n1\n         ->        er \u21d3         (Succ n2)\n         -> Div el er \u21d3 (n1 div (Succ n2))\n\n  exb : Val 3 \u21d3 3\n  exb = \u21d3Base\n\n  exs : Div (Val 3) (Val 3) \u21d3 1\n  exs = \u21d3Step \u21d3Base \u21d3Base\n\n  -- alternatively, semantics specified by an INTERPRETER\n  -- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- define operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  How to relate two definitions:\n  - std lib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign a weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set _ (Pure bx)      = a\u2192ba\u2192Set _ bx\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBx a\u2192ba\u2192Set = wp a\u2192partialBx (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by def mustPT\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, consider defining this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"4dcd68aaeb71a04ee5734108448c063b58410dcd","subject":"[ Agda ] Fixed issue 1264","message":"[ Agda ] Fixed issue 1264\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/ABP-SL.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/ABP-SL.agda","new_contents":"------------------------------------------------------------------------------\n-- The ABP using the Agda standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.ABP-SL where\n\nopen import Coinduction\nopen import Data.Bool\nopen import Data.Product\nopen import Data.Stream\nopen import Relation.Nullary\n\n------------------------------------------------------------------------------\n\nBit : Set\nBit = Bool\n\n-- Data type used to model the fair unreliable transmission channel.\ndata Err (A : Set) : Set where\n  ok    : (x : A) \u2192 Err A\n  error : Err A\n\n-- The mutual sender functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nsendA  : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bool)\nawaitA : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bit)\n\nsendA b (i \u2237 is) ds = (i , b) \u2237 \u266f awaitA b (i \u2237 is) ds\n\nawaitA b (i \u2237 is) (ok b' \u2237 ds) with b \u225f b'\n... | yes p = sendA (not b) (\u266d is) (\u266d ds)\n... | no \u00acp = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\nawaitA b (i \u2237 is) (error \u2237 ds) = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\n\n-- The receiver functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nackA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream Bit\nackA b (ok (_ , b') \u2237 bs) with b \u225f b'\n... | yes p = b \u2237 \u266f (ackA (not b) (\u266d bs))\n... | no \u00acp = not b \u2237 \u266f (ackA b (\u266d bs))\nackA b (error \u2237 bs) = not b \u2237 \u266f (ackA b (\u266d bs))\n\n{-# NO_TERMINATION_CHECK #-}\noutA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream A\noutA b (ok (i , b') \u2237 bs) with b \u225f b'\n... | yes p = i \u2237 \u266f (outA (not b) (\u266d bs))\n... | no \u00acp = outA b (\u266d bs)\noutA b (error \u2237 bs) = outA b (\u266d bs)\n\n-- Model the fair unreliable tranmission channel.\ncorruptA : {A : Set} \u2192 Stream Bit \u2192 Stream A \u2192 Stream (Err A)\ncorruptA (true \u2237 os)  (_ \u2237 xs) = error \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\ncorruptA (false \u2237 os) (x \u2237 xs) = ok x \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\n\n-- The ABP transfer function.\n{-# NO_TERMINATION_CHECK #-}\nabpTransA : {A : Set} \u2192 Bit \u2192 Stream Bit \u2192 Stream Bit \u2192 Stream A \u2192 Stream A\nabpTransA {A} b os\u2081 os\u2082 is = outA b bs\n  where\n  as : Stream (A \u00d7 Bit)\n  bs : Stream (Err (A \u00d7 Bit))\n  cs : Stream Bit\n  ds : Stream (Err Bit)\n\n  as = sendA b is ds\n  bs = corruptA os\u2081 as\n  cs = ackA b bs\n  ds = corruptA os\u2082 cs\n","old_contents":"------------------------------------------------------------------------------\n-- The ABP using the Agda standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.ABP-SL where\n\nopen import Coinduction\nopen import Data.Bool\nopen import Data.Product\nopen import Data.Stream\nopen import Relation.Nullary\n\n------------------------------------------------------------------------------\n\nBit : Set\nBit = Bool\n\n-- Data type used to model the fair unreliable transmission channel.\ndata Err (A : Set) : Set where\n  ok    : (x : A) \u2192 Err A\n  error : Err A\n\n-- The mutual sender functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nsendA  : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bool)\nawaitA : {A : Set} \u2192 Bit \u2192 Stream A \u2192 Stream (Err Bit) \u2192 Stream (A \u00d7 Bit)\n\nsendA b (i \u2237 is) ds = (i , b) \u2237 \u266f awaitA b (i \u2237 is) ds\n\nawaitA b (i \u2237 is) (ok b' \u2237 ds) with b \u225f b'\n... | yes p = sendA (not b) (\u266d is) (\u266d ds)\n... | no \u00acp = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\nawaitA b (i \u2237 is) (error \u2237 ds) = (i , b) \u2237 \u266f (awaitA b (i \u2237 is) (\u266d ds))\n\n-- The receiver functions.\n--\n-- 25 June 2014. Requires the non-termination flag when using\n-- --without-K. See Agda issue 1214.\n{-# NO_TERMINATION_CHECK #-}\nackA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream Bit\nackA b (ok (_ , b') \u2237 bs) with b \u225f b'\n... | yes p = b \u2237 \u266f (ackA (not b) (\u266d bs))\n... | no \u00acp = not b \u2237 \u266f (ackA b (\u266d bs))\nackA b (error \u2237 bs) = not b \u2237 \u266f (ackA b (\u266d bs))\n\n{-# NO_TERMINATION_CHECK #-}\noutA : {A : Set} \u2192 Bit \u2192 Stream (Err (A \u00d7 Bit)) \u2192 Stream A\noutA b (ok (i , b') \u2237 bs) with b \u225f b'\n... | yes p = i \u2237 \u266f (outA (not b) (\u266d bs))\n... | no \u00acp = outA b (\u266d bs)\noutA b (error \u2237 bs) = outA b (\u266d bs)\n\n-- Model the fair unreliable tranmission channel.\n--\n-- 29 August 2014 Requires the non-termination flag when using\n-- --without-K after the release of Agda 2.4.2. See issue 1264.\n{-# NO_TERMINATION_CHECK #-}\ncorruptA : {A : Set} \u2192 Stream Bit \u2192 Stream A \u2192 Stream (Err A)\ncorruptA (true \u2237 os)  (_ \u2237 xs) = error \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\ncorruptA (false \u2237 os) (x \u2237 xs) = ok x \u2237 \u266f (corruptA (\u266d os) (\u266d xs))\n\n-- The ABP transfer function.\n{-# NO_TERMINATION_CHECK #-}\nabpTransA : {A : Set} \u2192 Bit \u2192 Stream Bit \u2192 Stream Bit \u2192 Stream A \u2192 Stream A\nabpTransA {A} b os\u2081 os\u2082 is = outA b bs\n  where\n  as : Stream (A \u00d7 Bit)\n  bs : Stream (Err (A \u00d7 Bit))\n  cs : Stream Bit\n  ds : Stream (Err Bit)\n\n  as = sendA b is ds\n  bs = corruptA os\u2081 as\n  cs = ackA b bs\n  ds = corruptA os\u2082 cs\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57b799a8a5f32809a9c23460be5e1342baac59f6","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Nat\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Nat\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- M. J. Beeson. Proving programs and programming proofs. In Ruth\n-- Barcan Marcus, George J. W. Dorn, and Paul Weingartner,\n-- editors. Logic, Methodology and Philosophy of Science VII (1983),\n-- volume 114 of Studies in Logic and the Foundations of\n-- Mathematics. North-Holland Publishing Company, 1988, pages 51\u201382.\n\nmodule FOTC.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\nSx\u2262x : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2262 n\nSx\u2262x nzero h      = \u22a5-elim (0\u2262S (sym h))\nSx\u2262x (nsucc Nn) h = \u22a5-elim (Sx\u2262x Nn (succInjective h))\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity nzero          = +-leftIdentity zero\n+-rightIdentity (nsucc {n} Nn) =\n  trans (+-Sx n zero) (succCong (+-rightIdentity Nn))\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N nzero          = subst N (sym pred-0) nzero\npred-N (nsucc {n} Nn) = subst N (sym (pred-S n)) Nn\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} nzero          Nn = subst N (sym (+-leftIdentity n)) Nn\n+-N {n = n} (nsucc {m} Nm) Nn = subst N (sym (+-Sx m n)) (nsucc (+-N Nm Nn))\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm nzero          = subst N (sym (\u2238-x0 m)) Nm\n\u2238-N {m} Nm (nsucc {n} Nn) = subst N (sym (\u2238-xS m n)) (pred-N (\u2238-N Nm Nn))\n\n+-assoc : \u2200 {m} \u2192 N m \u2192 \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc nzero n o =\n  zero + n + o   \u2261\u27e8 +-leftCong (+-leftIdentity n) \u27e9\n  n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (nsucc {m} Nm) n o =\n  succ\u2081 m + n + o     \u2261\u27e8 +-leftCong (+-Sx m n) \u27e9\n  succ\u2081 (m + n) + o   \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)   \u2261\u27e8 succCong (+-assoc Nm n o) \u27e9\n  succ\u2081 (m + (n + o)) \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n  succ\u2081 m + (n + o)   \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] nzero n =\n  zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n  succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (nsucc {m} Nm) n =\n  succ\u2081 m + succ\u2081 n     \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)   \u2261\u27e8 succCong (x+Sy\u2261S[x+y] Nm n) \u27e9\n  succ\u2081 (succ\u2081 (m + n)) \u2261\u27e8 succCong (sym (+-Sx m n)) \u27e9\n  succ\u2081 (succ\u2081 m + n)   \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} nzero Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\nprivate\n  0\u2238S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n  0\u2238S nzero =\n    zero \u2238 [1]          \u2261\u27e8 \u2238-xS zero zero \u27e9\n    pred\u2081 (zero \u2238 zero) \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n    pred\u2081 zero          \u2261\u27e8 pred-0 \u27e9\n    zero                \u220e\n\n  0\u2238S (nsucc {n} Nn) =\n    zero \u2238 succ\u2081 (succ\u2081 n)   \u2261\u27e8 \u2238-xS zero (succ\u2081 n) \u27e9\n    pred\u2081 (zero \u2238 (succ\u2081 n)) \u2261\u27e8 predCong (0\u2238S Nn) \u27e9\n    pred\u2081 zero               \u2261\u27e8 pred-0 \u27e9\n    zero                     \u220e\n\n0\u2238x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n0\u2238x nzero      = \u2238-x0 zero\n0\u2238x (nsucc Nn) = 0\u2238S Nn\n\nS\u2238S : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\nS\u2238S {m} _ nzero =\n  succ\u2081 m \u2238 [1]\n    \u2261\u27e8 \u2238-xS (succ\u2081 m) zero \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\nS\u2238S nzero (nsucc {n} Nn) =\n  [1] \u2238 succ\u2081 (succ\u2081 n)\n    \u2261\u27e8 \u2238-xS [1] (succ\u2081 n) \u27e9\n  pred\u2081 ([1] \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (S\u2238S nzero Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (0\u2238x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (0\u2238S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\nS\u2238S (nsucc {m} Nm) (nsucc {n} Nn) =\n  succ\u2081 (succ\u2081 m) \u2238 succ\u2081 (succ\u2081 n)\n    \u2261\u27e8 \u2238-xS (succ\u2081 (succ\u2081 m)) (succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (S\u2238S (nsucc Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (\u2238-xS (succ\u2081 m) n) \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\nx\u2238x\u22610 : \u2200 {n} \u2192 N n \u2192 n \u2238 n \u2261 zero\nx\u2238x\u22610 nzero      = \u2238-x0 zero\nx\u2238x\u22610 (nsucc Nn) = trans (S\u2238S Nn Nn) (x\u2238x\u22610 Nn)\n\nSx\u2238x\u22611 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2238 n \u2261 [1]\nSx\u2238x\u22611 nzero      = \u2238-x0 [1]\nSx\u2238x\u22611 (nsucc Nn) = trans (S\u2238S (nsucc Nn) Nn) (Sx\u2238x\u22611 Nn)\n\n[x+Sy]\u2238y\u2261Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + succ\u2081 n \u2238 n \u2261 succ\u2081 m\n[x+Sy]\u2238y\u2261Sx {n = n} nzero Nn =\n  zero + succ\u2081 n \u2238 n \u2261\u27e8 \u2238-leftCong (+-leftIdentity (succ\u2081 n)) \u27e9\n  succ\u2081 n \u2238 n        \u2261\u27e8 Sx\u2238x\u22611 Nn \u27e9\n  [1]                \u220e\n\n[x+Sy]\u2238y\u2261Sx (nsucc {m} Nm) nzero =\n  succ\u2081 m + [1] \u2238 zero     \u2261\u27e8 \u2238-leftCong (+-Sx m [1]) \u27e9\n  succ\u2081 (m + [1]) \u2238 zero   \u2261\u27e8 \u2238-x0 (succ\u2081 (m + [1])) \u27e9\n  succ\u2081 (m + [1])          \u2261\u27e8 succCong (+-comm Nm 1-N) \u27e9\n  succ\u2081 ([1] + m)          \u2261\u27e8 succCong (+-Sx zero m) \u27e9\n  succ\u2081 (succ\u2081 (zero + m)) \u2261\u27e8 succCong (succCong (+-leftIdentity m)) \u27e9\n  succ\u2081 (succ\u2081 m)          \u220e\n\n[x+Sy]\u2238y\u2261Sx (nsucc {m} Nm) (nsucc {n} Nn) =\n  succ\u2081 m + succ\u2081 (succ\u2081 n) \u2238 succ\u2081 n\n    \u2261\u27e8 \u2238-leftCong (+-Sx m (succ\u2081 (succ\u2081 n))) \u27e9\n  succ\u2081 (m + succ\u2081 (succ\u2081 n)) \u2238 succ\u2081 n\n    \u2261\u27e8 S\u2238S (+-N Nm (nsucc (nsucc Nn))) Nn \u27e9\n  m + succ\u2081 (succ\u2081 n) \u2238 n\n    \u2261\u27e8 \u2238-leftCong (x+Sy\u2261S[x+y] Nm (succ\u2081 n)) \u27e9\n  succ\u2081 (m + succ\u2081 n) \u2238 n\n    \u2261\u27e8 \u2238-leftCong (sym (+-Sx m (succ\u2081 n))) \u27e9\n  succ\u2081 m + succ\u2081 n \u2238 n\n    \u2261\u27e8 [x+Sy]\u2238y\u2261Sx (nsucc Nm) Nn \u27e9\n  succ\u2081 (succ\u2081 m) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} nzero _ _ =\n  (zero + n) \u2238 (zero + o) \u2261\u27e8 \u2238-leftCong (+-leftIdentity n) \u27e9\n    n \u2238 (zero + o)        \u2261\u27e8 \u2238-rightCong (+-leftIdentity o) \u27e9\n    n \u2238 o                 \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (nsucc {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o) \u2261\u27e8 \u2238-leftCong (+-Sx m n) \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261\u27e8 \u2238-rightCong (+-Sx m o) \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o) \u2261\u27e8 S\u2238S (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)             \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o                         \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero nzero          = *-leftZero zero\n*-rightZero (nsucc {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} nzero          Nn = subst N (sym (*-leftZero n)) nzero\n*-N {n = n} (nsucc {m} Nm) Nn = subst N (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 [1] * n \u2261 n\n*-leftIdentity {n} Nn =\n  [1] * n      \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n \u2261\u27e8 +-rightCong (*-leftZero n) \u27e9\n  n + zero     \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n            \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} nzero Nn = sym\n  (\n    zero + zero * n \u2261\u27e8 +-rightCong (*-leftZero n) \u27e9\n    zero + zero     \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero            \u2261\u27e8 sym (*-leftZero (succ\u2081 n)) \u27e9\n    zero * succ\u2081 n  \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 +-rightCong (x*Sy\u2261x+xy Nm Nn) \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 succCong (sym (+-assoc Nn m (m * n))) \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 succCong (+-leftCong (+-comm Nn Nm)) \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n               refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 +-rightCong (sym (*-Sx m n)) \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} nzero Nn          = trans (*-leftZero n) (sym (*-rightZero Nn))\n*-comm {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m * n \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n   \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m   \u2261\u27e8 sym (x*Sy\u2261x+xy Nn Nm) \u27e9\n  n * succ\u2081 m \u220e\n\n*-rightIdentity : \u2200 {n} \u2192 N n \u2192 n * [1] \u2261 n\n*-rightIdentity {n} Nn = trans (*-comm Nn (nsucc nzero)) (*-leftIdentity Nn)\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ nzero _ =\n  (m \u2238 zero) * o   \u2261\u27e8 *-leftCong (\u2238-x0 m) \u27e9\n  m * o            \u2261\u27e8 sym (\u2238-x0 (m * o)) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 \u2238-rightCong (sym (*-leftZero o)) \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} nzero (nsucc {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o   \u2261\u27e8 *-leftCong (0\u2238S Nn) \u27e9\n  zero * o               \u2261\u27e8 *-leftZero o \u27e9\n  zero                   \u2261\u27e8 sym (0\u2238x (*-N (nsucc Nn) No)) \u27e9\n  zero \u2238 succ\u2081 n * o     \u2261\u27e8 \u2238-leftCong (sym (*-leftZero o)) \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (nsucc {m} Nm) (nsucc {n} Nn) nzero =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (nsucc Nm) (nsucc Nn)) nzero \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-leftZero (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym (0\u2238x (*-N (nsucc Nn) nzero)) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 \u2238-leftCong (sym (*-leftZero (succ\u2081 m))) \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 \u2238-leftCong (*-comm nzero (nsucc Nm)) \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-leftCong (S\u2238S Nm Nn) \u27e9\n  (m \u2238 n) * succ\u2081 o\n     \u2261\u27e8 *\u2238-leftDistributive Nm Nn (nsucc No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym ([x+y]\u2238[x+z]\u2261y\u2238z (nsucc No) (*-N Nm (nsucc No)) (*-N Nn (nsucc No))) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 \u2238-leftCong (sym (*-Sx m (succ\u2081 o))) \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 \u2238-rightCong (sym (*-Sx n (succ\u2081 o))) \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn nzero =\n  (m + n) * zero\n    \u2261\u27e8 *-comm (+-N Nm Nn) nzero \u27e9\n  zero * (m + n)\n    \u2261\u27e8 *-leftZero (m + n) \u27e9\n  zero\n    \u2261\u27e8 sym (*-leftZero m) \u27e9\n  zero * m\n    \u2261\u27e8 *-comm nzero Nm \u27e9\n  m * zero\n    \u2261\u27e8 sym (+-rightIdentity (*-N Nm nzero)) \u27e9\n  m * zero + zero\n    \u2261\u27e8 +-rightCong (trans (sym (*-leftZero n)) (*-comm nzero Nn)) \u27e9\n  m * zero + n * zero \u220e\n\n*+-leftDistributive {n = n} nzero Nn (nsucc {o} No) =\n  (zero + n) * succ\u2081 o         \u2261\u27e8 *-leftCong (+-leftIdentity n) \u27e9\n  n * succ\u2081 o                  \u2261\u27e8 sym (+-leftIdentity (n * succ\u2081 o)) \u27e9\n  zero + n * succ\u2081 o           \u2261\u27e8 +-leftCong (sym (*-leftZero (succ\u2081 o))) \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (nsucc {m} Nm) nzero (nsucc {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 *-leftCong (+-rightIdentity (nsucc Nm)) \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym (+-rightIdentity (*-N (nsucc Nm) (nsucc No))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 +-rightCong (sym (*-leftZero (succ\u2081 o))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-leftCong (+-Sx m (succ\u2081 n)) \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 +-rightCong (*+-leftDistributive Nm (nsucc Nn) (nsucc No)) \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym (+-assoc (nsucc No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 +-leftCong (sym (*-Sx m (succ\u2081 o))) \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u2228 n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 nzero          _              _      = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (nsucc Nm)     nzero          _      = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (nsucc {m} Nm) (nsucc {n} Nn) SmSn\u22610 = \u22a5-elim (0\u2262S prf)\n  where\n  prf : zero \u2261 succ\u2081 (n + m * succ\u2081 n)\n  prf = zero                    \u2261\u27e8 sym SmSn\u22610 \u27e9\n        succ\u2081 m * succ\u2081 n       \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n        succ\u2081 n + m * succ\u2081 n   \u2261\u27e8 +-Sx n (m * succ\u2081 n) \u27e9\n        succ\u2081 (n + m * succ\u2081 n) \u220e\n\nxy\u22611\u2192x\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 [1] \u2192 m \u2261 [1]\nxy\u22611\u2192x\u22611 {n = n} nzero Nn h = \u22a5-elim (0\u2262S (trans (sym (*-leftZero n)) h))\nxy\u22611\u2192x\u22611 (nsucc nzero) Nn h = refl\nxy\u22611\u2192x\u22611 (nsucc (nsucc {m} Nm)) nzero h =\n  \u22a5-elim (0\u2262S (trans (sym (*-rightZero (nsucc (nsucc Nm)))) h))\nxy\u22611\u2192x\u22611 (nsucc (nsucc {m} Nm)) (nsucc {n} Nn) h = \u22a5-elim (0\u2262S prf\u2082)\n  where\n  prf\u2081 : [1] \u2261 succ\u2081 (succ\u2081 (m + n * succ\u2081 (succ\u2081 m)))\n  prf\u2081 = [1]\n           \u2261\u27e8 sym h \u27e9\n         succ\u2081 (succ\u2081 m) * succ\u2081 n\n           \u2261\u27e8 *-comm (nsucc (nsucc Nm)) (nsucc Nn) \u27e9\n         succ\u2081 n * succ\u2081 (succ\u2081 m)\n           \u2261\u27e8 *-Sx n (succ\u2081 (succ\u2081 m)) \u27e9\n         succ\u2081 (succ\u2081 m) + n * succ\u2081 (succ\u2081 m)\n           \u2261\u27e8 +-Sx (succ\u2081 m) (n * succ\u2081 (succ\u2081 m)) \u27e9\n         succ\u2081 (succ\u2081 m + n * succ\u2081 (succ\u2081 m))\n           \u2261\u27e8 succCong (+-Sx m (n * succ\u2081 (succ\u2081 m))) \u27e9\n         succ\u2081 (succ\u2081 (m + n * succ\u2081 (succ\u2081 m))) \u220e\n\n  prf\u2082 : zero \u2261 succ\u2081 (m + n * succ\u2081 (succ\u2081 m))\n  prf\u2082 = succInjective prf\u2081\n\nxy\u22611\u2192y\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 [1] \u2192 n \u2261 [1]\nxy\u22611\u2192y\u22611 Nm Nn h = xy\u22611\u2192x\u22611 Nn Nm (trans (*-comm Nn Nm) h)\n\n-- Feferman's axiom as presented by (Beeson 1986, p. 74).\nsuccOnto : \u2200 {n} \u2192 N n \u2192 n \u2262 zero \u2192 succ\u2081 (pred\u2081 n) \u2261 n\nsuccOnto nzero          h = \u22a5-elim (h refl)\nsuccOnto (nsucc {n} Nn) h = succCong (pred-S n)\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- M. J. Beeson. Proving programs and programming proofs. In Ruth\n-- Barcan Marcus, George J. W. Dorn, and Paul Weingartner,\n-- editors. Logic, Methodology and Philosophy of Science VII (1983),\n-- volume 114 of Studies in Logic and the Foundations of\n-- Mathematics. North-Holland Publishing Company, 1988, pages 51\u201382.\n\nmodule FOTC.Data.Nat.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityI\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m + o \u2261 n + o\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m + n \u2261 m + o\n+-rightCong refl = refl\n\n\u2238-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m \u2238 o \u2261 n \u2238 o\n\u2238-leftCong refl = refl\n\n\u2238-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m \u2238 n \u2261 m \u2238 o\n\u2238-rightCong refl = refl\n\n*-leftCong : \u2200 {m n o} \u2192 m \u2261 n \u2192 m * o \u2261 n * o\n*-leftCong refl = refl\n\n*-rightCong : \u2200 {m n o} \u2192 n \u2261 o \u2192 m * n \u2261 m * o\n*-rightCong refl = refl\n\n------------------------------------------------------------------------------\n-- Some proofs are based on the proofs in the standard library.\n\nSx\u2262x : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2262 n\nSx\u2262x nzero h      = \u22a5-elim (0\u2262S (sym h))\nSx\u2262x (nsucc Nn) h = \u22a5-elim (Sx\u2262x Nn (succInjective h))\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = +-0x\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity nzero          = +-leftIdentity zero\n+-rightIdentity (nsucc {n} Nn) =\n  trans (+-Sx n zero) (succCong (+-rightIdentity Nn))\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N nzero =         subst N (sym pred-0) nzero\npred-N (nsucc {n} Nn) = subst N (sym (pred-S n)) Nn\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} nzero          Nn = subst N (sym (+-leftIdentity n)) Nn\n+-N {n = n} (nsucc {m} Nm) Nn = subst N (sym (+-Sx m n)) (nsucc (+-N Nm Nn))\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm nzero          = subst N (sym (\u2238-x0 m)) Nm\n\u2238-N {m} Nm (nsucc {n} Nn) = subst N (sym (\u2238-xS m n)) (pred-N (\u2238-N Nm Nn))\n\n+-assoc : \u2200 {m} \u2192 N m \u2192 \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc nzero n o =\n  zero + n + o   \u2261\u27e8 +-leftCong (+-leftIdentity n) \u27e9\n  n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n  zero + (n + o) \u220e\n\n+-assoc (nsucc {m} Nm) n o =\n  succ\u2081 m + n + o     \u2261\u27e8 +-leftCong (+-Sx m n) \u27e9\n  succ\u2081 (m + n) + o   \u2261\u27e8 +-Sx (m + n) o \u27e9\n  succ\u2081 (m + n + o)   \u2261\u27e8 succCong (+-assoc Nm n o) \u27e9\n  succ\u2081 (m + (n + o)) \u2261\u27e8 sym (+-Sx m (n + o)) \u27e9\n  succ\u2081 m + (n + o)   \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] nzero n =\n  zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n  succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n  succ\u2081 (zero + n) \u220e\n\nx+Sy\u2261S[x+y] (nsucc {m} Nm) n =\n  succ\u2081 m + succ\u2081 n     \u2261\u27e8 +-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 (m + succ\u2081 n)   \u2261\u27e8 succCong (x+Sy\u2261S[x+y] Nm n) \u27e9\n  succ\u2081 (succ\u2081 (m + n)) \u2261\u27e8 succCong (sym (+-Sx m n)) \u27e9\n  succ\u2081 (succ\u2081 m + n)   \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} nzero Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\nprivate\n  0\u2238S : \u2200 {n} \u2192 N n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n  0\u2238S nzero =\n    zero \u2238 [1]          \u2261\u27e8 \u2238-xS zero zero \u27e9\n    pred\u2081 (zero \u2238 zero) \u2261\u27e8 predCong (\u2238-x0 zero) \u27e9\n    pred\u2081 zero          \u2261\u27e8 pred-0 \u27e9\n    zero                \u220e\n\n  0\u2238S (nsucc {n} Nn) =\n    zero \u2238 succ\u2081 (succ\u2081 n)   \u2261\u27e8 \u2238-xS zero (succ\u2081 n) \u27e9\n    pred\u2081 (zero \u2238 (succ\u2081 n)) \u2261\u27e8 predCong (0\u2238S Nn) \u27e9\n    pred\u2081 zero               \u2261\u27e8 pred-0 \u27e9\n    zero                     \u220e\n\n0\u2238x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n0\u2238x nzero      = \u2238-x0 zero\n0\u2238x (nsucc Nn) = 0\u2238S Nn\n\nS\u2238S : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\nS\u2238S {m} _ nzero =\n  succ\u2081 m \u2238 [1]\n    \u2261\u27e8 \u2238-xS (succ\u2081 m) zero \u27e9\n  pred\u2081 (succ\u2081 m \u2238 zero)\n    \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n  pred\u2081 (succ\u2081 m)\n    \u2261\u27e8 pred-S m \u27e9\n  m\n    \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n  m \u2238 zero \u220e\n\nS\u2238S nzero (nsucc {n} Nn) =\n  [1] \u2238 succ\u2081 (succ\u2081 n)\n    \u2261\u27e8 \u2238-xS [1] (succ\u2081 n) \u27e9\n  pred\u2081 ([1] \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (S\u2238S nzero Nn) \u27e9\n  pred\u2081 (zero \u2238 n)\n    \u2261\u27e8 predCong (0\u2238x Nn) \u27e9\n  pred\u2081 zero\n    \u2261\u27e8 pred-0 \u27e9\n  zero\n    \u2261\u27e8 sym (0\u2238S Nn) \u27e9\n  zero \u2238 succ\u2081 n \u220e\n\nS\u2238S (nsucc {m} Nm) (nsucc {n} Nn) =\n  succ\u2081 (succ\u2081 m) \u2238 succ\u2081 (succ\u2081 n)\n    \u2261\u27e8 \u2238-xS (succ\u2081 (succ\u2081 m)) (succ\u2081 n) \u27e9\n  pred\u2081 (succ\u2081 (succ\u2081 m) \u2238 succ\u2081 n)\n    \u2261\u27e8 predCong (S\u2238S (nsucc Nm) Nn) \u27e9\n  pred\u2081 (succ\u2081 m \u2238 n)\n    \u2261\u27e8 sym (\u2238-xS (succ\u2081 m) n) \u27e9\n  succ\u2081 m \u2238 succ\u2081 n \u220e\n\nx\u2238x\u22610 : \u2200 {n} \u2192 N n \u2192 n \u2238 n \u2261 zero\nx\u2238x\u22610 nzero      = \u2238-x0 zero\nx\u2238x\u22610 (nsucc Nn) = trans (S\u2238S Nn Nn) (x\u2238x\u22610 Nn)\n\nSx\u2238x\u22611 : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2238 n \u2261 [1]\nSx\u2238x\u22611 nzero      = \u2238-x0 [1]\nSx\u2238x\u22611 (nsucc Nn) = trans (S\u2238S (nsucc Nn) Nn) (Sx\u2238x\u22611 Nn)\n\n[x+Sy]\u2238y\u2261Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + succ\u2081 n \u2238 n \u2261 succ\u2081 m\n[x+Sy]\u2238y\u2261Sx {n = n} nzero Nn =\n  zero + succ\u2081 n \u2238 n \u2261\u27e8 \u2238-leftCong (+-leftIdentity (succ\u2081 n)) \u27e9\n  succ\u2081 n \u2238 n        \u2261\u27e8 Sx\u2238x\u22611 Nn \u27e9\n  [1]                \u220e\n\n[x+Sy]\u2238y\u2261Sx (nsucc {m} Nm) nzero =\n  succ\u2081 m + [1] \u2238 zero     \u2261\u27e8 \u2238-leftCong (+-Sx m [1]) \u27e9\n  succ\u2081 (m + [1]) \u2238 zero   \u2261\u27e8 \u2238-x0 (succ\u2081 (m + [1])) \u27e9\n  succ\u2081 (m + [1])          \u2261\u27e8 succCong (+-comm Nm 1-N) \u27e9\n  succ\u2081 ([1] + m)          \u2261\u27e8 succCong (+-Sx zero m) \u27e9\n  succ\u2081 (succ\u2081 (zero + m)) \u2261\u27e8 succCong (succCong (+-leftIdentity m)) \u27e9\n  succ\u2081 (succ\u2081 m)          \u220e\n\n[x+Sy]\u2238y\u2261Sx (nsucc {m} Nm) (nsucc {n} Nn) =\n  succ\u2081 m + succ\u2081 (succ\u2081 n) \u2238 succ\u2081 n\n    \u2261\u27e8 \u2238-leftCong (+-Sx m (succ\u2081 (succ\u2081 n))) \u27e9\n  succ\u2081 (m + succ\u2081 (succ\u2081 n)) \u2238 succ\u2081 n\n    \u2261\u27e8 S\u2238S (+-N Nm (nsucc (nsucc Nn))) Nn \u27e9\n  m + succ\u2081 (succ\u2081 n) \u2238 n\n    \u2261\u27e8 \u2238-leftCong (x+Sy\u2261S[x+y] Nm (succ\u2081 n)) \u27e9\n  succ\u2081 (m + succ\u2081 n) \u2238 n\n    \u2261\u27e8 \u2238-leftCong (sym (+-Sx m (succ\u2081 n))) \u27e9\n  succ\u2081 m + succ\u2081 n \u2238 n\n    \u2261\u27e8 [x+Sy]\u2238y\u2261Sx (nsucc Nm) Nn \u27e9\n  succ\u2081 (succ\u2081 m) \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) \u2238 (m + o) \u2261 n \u2238 o\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} nzero _ _ =\n  (zero + n) \u2238 (zero + o) \u2261\u27e8 \u2238-leftCong (+-leftIdentity n) \u27e9\n    n \u2238 (zero + o)        \u2261\u27e8 \u2238-rightCong (+-leftIdentity o) \u27e9\n    n \u2238 o                 \u220e\n\n[x+y]\u2238[x+z]\u2261y\u2238z {n = n} {o} (nsucc {m} Nm) Nn No =\n  (succ\u2081 m + n) \u2238 (succ\u2081 m + o) \u2261\u27e8 \u2238-leftCong (+-Sx m n) \u27e9\n  succ\u2081 (m + n) \u2238 (succ\u2081 m + o) \u2261\u27e8 \u2238-rightCong (+-Sx m o) \u27e9\n  succ\u2081 (m + n) \u2238 succ\u2081 (m + o) \u2261\u27e8 S\u2238S (+-N Nm Nn) (+-N Nm No) \u27e9\n  (m + n) \u2238 (m + o)             \u2261\u27e8 [x+y]\u2238[x+z]\u2261y\u2238z Nm Nn No \u27e9\n  n \u2238 o                         \u220e\n\n*-leftZero : \u2200 n \u2192 zero * n \u2261 zero\n*-leftZero = *-0x\n\n*-rightZero : \u2200 {n} \u2192 N n \u2192 n * zero \u2261 zero\n*-rightZero nzero          = *-leftZero zero\n*-rightZero (nsucc {n} Nn) =\n  trans (*-Sx n zero)\n        (trans (+-leftIdentity (n * zero)) (*-rightZero Nn))\n\n*-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N {n = n} nzero          Nn = subst N (sym (*-leftZero n)) nzero\n*-N {n = n} (nsucc {m} Nm) Nn = subst N (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n*-leftIdentity : \u2200 {n} \u2192 N n \u2192 [1] * n \u2261 n\n*-leftIdentity {n} Nn =\n  [1] * n      \u2261\u27e8 *-Sx zero n \u27e9\n  n + zero * n \u2261\u27e8 +-rightCong (*-leftZero n) \u27e9\n  n + zero     \u2261\u27e8 +-rightIdentity Nn \u27e9\n  n            \u220e\n\nx*Sy\u2261x+xy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * succ\u2081 n \u2261 m + m * n\nx*Sy\u2261x+xy {n = n} nzero Nn = sym\n  (\n    zero + zero * n \u2261\u27e8 +-rightCong (*-leftZero n) \u27e9\n    zero + zero     \u2261\u27e8 +-leftIdentity zero \u27e9\n    zero            \u2261\u27e8 sym (*-leftZero (succ\u2081 n)) \u27e9\n    zero * succ\u2081 n  \u220e\n  )\n\nx*Sy\u2261x+xy {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m * succ\u2081 n\n    \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n  succ\u2081 n + m * succ\u2081 n\n    \u2261\u27e8 +-rightCong (x*Sy\u2261x+xy Nm Nn) \u27e9\n  succ\u2081 n + (m + m * n)\n    \u2261\u27e8 +-Sx n (m + m * n) \u27e9\n  succ\u2081 (n + (m + m * n))\n    \u2261\u27e8 succCong (sym (+-assoc Nn m (m * n))) \u27e9\n  succ\u2081 (n + m + m * n)\n    \u2261\u27e8 succCong (+-leftCong (+-comm Nn Nm)) \u27e9\n  succ\u2081 (m + n + m * n)\n    \u2261\u27e8 subst (\u03bb t \u2192 succ\u2081 (m + n + m * n) \u2261 succ\u2081 t)\n             (+-assoc Nm n (m * n))\n               refl\n    \u27e9\n  succ\u2081 (m + (n + m * n))\n    \u2261\u27e8 sym (+-Sx m (n + m * n)) \u27e9\n  succ\u2081 m + (n + m * n)\n    \u2261\u27e8 +-rightCong (sym (*-Sx m n)) \u27e9\n  succ\u2081 m + succ\u2081 m * n \u220e\n\n*-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 n * m\n*-comm {n = n} nzero Nn          = trans (*-leftZero n) (sym (*-rightZero Nn))\n*-comm {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m * n \u2261\u27e8 *-Sx m n \u27e9\n  n + m * n   \u2261\u27e8 subst (\u03bb t \u2192 n + m * n \u2261 n + t) (*-comm Nm Nn) refl \u27e9\n  n + n * m   \u2261\u27e8 sym (x*Sy\u2261x+xy Nn Nm) \u27e9\n  n * succ\u2081 m \u220e\n\n*-rightIdentity : \u2200 {n} \u2192 N n \u2192 n * [1] \u2261 n\n*-rightIdentity {n} Nn = trans (*-comm Nn (nsucc nzero)) (*-leftIdentity Nn)\n\n*\u2238-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m \u2238 n) * o \u2261 m * o \u2238 n * o\n*\u2238-leftDistributive {m} {o = o} _ nzero _ =\n  (m \u2238 zero) * o   \u2261\u27e8 *-leftCong (\u2238-x0 m) \u27e9\n  m * o            \u2261\u27e8 sym (\u2238-x0 (m * o)) \u27e9\n  m * o \u2238 zero     \u2261\u27e8 \u2238-rightCong (sym (*-leftZero o)) \u27e9\n  m * o \u2238 zero * o \u220e\n\n*\u2238-leftDistributive {o = o} nzero (nsucc {n} Nn) No =\n  (zero \u2238 succ\u2081 n) * o   \u2261\u27e8 *-leftCong (0\u2238S Nn) \u27e9\n  zero * o               \u2261\u27e8 *-leftZero o \u27e9\n  zero                   \u2261\u27e8 sym (0\u2238x (*-N (nsucc Nn) No)) \u27e9\n  zero \u2238 succ\u2081 n * o     \u2261\u27e8 \u2238-leftCong (sym (*-leftZero o)) \u27e9\n  zero * o \u2238 succ\u2081 n * o \u220e\n\n*\u2238-leftDistributive (nsucc {m} Nm) (nsucc {n} Nn) nzero =\n  (succ\u2081 m \u2238 succ\u2081 n) * zero\n    \u2261\u27e8 *-comm (\u2238-N (nsucc Nm) (nsucc Nn)) nzero \u27e9\n  zero * (succ\u2081 m \u2238 succ\u2081 n)\n    \u2261\u27e8 *-leftZero (succ\u2081 m \u2238 succ\u2081 n) \u27e9\n  zero\n    \u2261\u27e8 sym (0\u2238x (*-N (nsucc Nn) nzero)) \u27e9\n  zero \u2238 succ\u2081 n * zero\n    \u2261\u27e8 \u2238-leftCong (sym (*-leftZero (succ\u2081 m))) \u27e9\n  zero * succ\u2081 m \u2238 succ\u2081 n * zero\n    \u2261\u27e8 \u2238-leftCong (*-comm nzero (nsucc Nm)) \u27e9\n  succ\u2081 m * zero \u2238 succ\u2081 n * zero \u220e\n\n*\u2238-leftDistributive (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) =\n  (succ\u2081 m \u2238 succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-leftCong (S\u2238S Nm Nn) \u27e9\n  (m \u2238 n) * succ\u2081 o\n     \u2261\u27e8 *\u2238-leftDistributive Nm Nn (nsucc No) \u27e9\n  m * succ\u2081 o \u2238 n * succ\u2081 o\n    \u2261\u27e8 sym ([x+y]\u2238[x+z]\u2261y\u2238z (nsucc No) (*-N Nm (nsucc No)) (*-N Nn (nsucc No))) \u27e9\n  (succ\u2081 o + m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 \u2238-leftCong (sym (*-Sx m (succ\u2081 o))) \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 o + n * succ\u2081 o)\n    \u2261\u27e8 \u2238-rightCong (sym (*-Sx n (succ\u2081 o))) \u27e9\n  (succ\u2081 m * succ\u2081 o) \u2238 (succ\u2081 n * succ\u2081 o) \u220e\n\n*+-leftDistributive : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) * o \u2261 m * o + n * o\n*+-leftDistributive {m} {n} Nm Nn nzero =\n  (m + n) * zero\n    \u2261\u27e8 *-comm (+-N Nm Nn) nzero \u27e9\n  zero * (m + n)\n    \u2261\u27e8 *-leftZero (m + n) \u27e9\n  zero\n    \u2261\u27e8 sym (*-leftZero m) \u27e9\n  zero * m\n    \u2261\u27e8 *-comm nzero Nm \u27e9\n  m * zero\n    \u2261\u27e8 sym (+-rightIdentity (*-N Nm nzero)) \u27e9\n  m * zero + zero\n    \u2261\u27e8 +-rightCong (trans (sym (*-leftZero n)) (*-comm nzero Nn)) \u27e9\n  m * zero + n * zero \u220e\n\n*+-leftDistributive {n = n} nzero Nn (nsucc {o} No) =\n  (zero + n) * succ\u2081 o         \u2261\u27e8 *-leftCong (+-leftIdentity n) \u27e9\n  n * succ\u2081 o                  \u2261\u27e8 sym (+-leftIdentity (n * succ\u2081 o)) \u27e9\n  zero + n * succ\u2081 o           \u2261\u27e8 +-leftCong (sym (*-leftZero (succ\u2081 o))) \u27e9\n  zero * succ\u2081 o + n * succ\u2081 o \u220e\n\n*+-leftDistributive (nsucc {m} Nm) nzero (nsucc {o} No) =\n  (succ\u2081 m + zero) * succ\u2081 o\n    \u2261\u27e8 *-leftCong (+-rightIdentity (nsucc Nm)) \u27e9\n  succ\u2081 m * succ\u2081 o\n    \u2261\u27e8 sym (+-rightIdentity (*-N (nsucc Nm) (nsucc No))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero\n    \u2261\u27e8 +-rightCong (sym (*-leftZero (succ\u2081 o))) \u27e9\n  succ\u2081 m * succ\u2081 o + zero * succ\u2081 o \u220e\n\n*+-leftDistributive (nsucc {m} Nm) (nsucc {n} Nn) (nsucc {o} No) =\n  (succ\u2081 m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-leftCong (+-Sx m (succ\u2081 n)) \u27e9\n  succ\u2081 (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 *-Sx (m + succ\u2081 n) (succ\u2081 o) \u27e9\n  succ\u2081 o + (m + succ\u2081 n) * succ\u2081 o\n    \u2261\u27e8 +-rightCong (*+-leftDistributive Nm (nsucc Nn) (nsucc No)) \u27e9\n  succ\u2081 o + (m * succ\u2081 o + succ\u2081 n * succ\u2081 o)\n    \u2261\u27e8 sym (+-assoc (nsucc No) (m * succ\u2081 o) (succ\u2081 n * succ\u2081 o)) \u27e9\n  succ\u2081 o + m * succ\u2081 o + succ\u2081 n * succ\u2081 o\n    \u2261\u27e8 +-leftCong (sym (*-Sx m (succ\u2081 o))) \u27e9\n  succ\u2081 m * succ\u2081 o + succ\u2081 n * succ\u2081 o \u220e\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u2228 n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 nzero          _              _      = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (nsucc Nm)     nzero          _      = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (nsucc {m} Nm) (nsucc {n} Nn) SmSn\u22610 = \u22a5-elim (0\u2262S prf)\n  where\n  prf : zero \u2261 succ\u2081 (n + m * succ\u2081 n)\n  prf = zero                    \u2261\u27e8 sym SmSn\u22610 \u27e9\n        succ\u2081 m * succ\u2081 n       \u2261\u27e8 *-Sx m (succ\u2081 n) \u27e9\n        succ\u2081 n + m * succ\u2081 n   \u2261\u27e8 +-Sx n (m * succ\u2081 n) \u27e9\n        succ\u2081 (n + m * succ\u2081 n) \u220e\n\nxy\u22611\u2192x\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 [1] \u2192 m \u2261 [1]\nxy\u22611\u2192x\u22611 {n = n} nzero Nn h = \u22a5-elim (0\u2262S (trans (sym (*-leftZero n)) h))\nxy\u22611\u2192x\u22611 (nsucc nzero) Nn h = refl\nxy\u22611\u2192x\u22611 (nsucc (nsucc {m} Nm)) nzero h =\n  \u22a5-elim (0\u2262S (trans (sym (*-rightZero (nsucc (nsucc Nm)))) h))\nxy\u22611\u2192x\u22611 (nsucc (nsucc {m} Nm)) (nsucc {n} Nn) h = \u22a5-elim (0\u2262S prf\u2082)\n  where\n  prf\u2081 : [1] \u2261 succ\u2081 (succ\u2081 (m + n * succ\u2081 (succ\u2081 m)))\n  prf\u2081 = [1]\n           \u2261\u27e8 sym h \u27e9\n         succ\u2081 (succ\u2081 m) * succ\u2081 n\n           \u2261\u27e8 *-comm (nsucc (nsucc Nm)) (nsucc Nn) \u27e9\n         succ\u2081 n * succ\u2081 (succ\u2081 m)\n           \u2261\u27e8 *-Sx n (succ\u2081 (succ\u2081 m)) \u27e9\n         succ\u2081 (succ\u2081 m) + n * succ\u2081 (succ\u2081 m)\n           \u2261\u27e8 +-Sx (succ\u2081 m) (n * succ\u2081 (succ\u2081 m)) \u27e9\n         succ\u2081 (succ\u2081 m + n * succ\u2081 (succ\u2081 m))\n           \u2261\u27e8 succCong (+-Sx m (n * succ\u2081 (succ\u2081 m))) \u27e9\n         succ\u2081 (succ\u2081 (m + n * succ\u2081 (succ\u2081 m))) \u220e\n\n  prf\u2082 : zero \u2261 succ\u2081 (m + n * succ\u2081 (succ\u2081 m))\n  prf\u2082 = succInjective prf\u2081\n\nxy\u22611\u2192y\u22611 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m * n \u2261 [1] \u2192 n \u2261 [1]\nxy\u22611\u2192y\u22611 Nm Nn h = xy\u22611\u2192x\u22611 Nn Nm (trans (*-comm Nn Nm) h)\n\n-- Feferman's axiom as presented by (Beeson 1986, p. 74).\nsuccOnto : \u2200 {n} \u2192 N n \u2192 n \u2262 zero \u2192 succ\u2081 (pred\u2081 n) \u2261 n\nsuccOnto nzero          h = \u22a5-elim (h refl)\nsuccOnto (nsucc {n} Nn) h = succCong (pred-S n)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3fc2c408017ffa13e058bb25b2b82531a9a3aa22","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 80d926b9e4dbcac1a79e75c92af756e4\n\ndarcs-hash:20100910030406-3bd4e-315bf9d30a56b926e882dc4e4bf2d9510b894407.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Fail\/Add.agda","new_file":"Test\/Fail\/Add.agda","new_contents":"module Test.Fail.Add where\n\ninfix  4 _\u2261_\ninfixl 6 _+_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n  _+_    : D \u2192 D \u2192 D\n  +-0x   : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx   : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n-- The ATPs should not prove this postulate.\npostulate\n  +-comm : (d e : D) \u2192 d + e \u2261 e + d\n-- Equinox 5.0alpha (2010-03-29) no-success due to timeout.\n-- E 1.2 no-success due to timeout.\n-- TODO: Metis 2.2 (release 20100825) success (SZS status CounterSatisfiable).\n{-# ATP prove +-comm #-}\n","old_contents":"module Test.Fail.Add where\n\ninfix  4 _\u2261_\ninfixl 6 _+_\n\npostulate\n  D      : Set\n  zero   : D\n  succ   : D \u2192 D\n  _\u2261_    : D \u2192 D \u2192 Set\n  _+_    : D \u2192 D \u2192 D\n  +-0x   : (d : D) \u2192 zero + d     \u2261 d\n  +-Sx   : (d e : D) \u2192 succ d + e \u2261 succ (d + e)\n{-# ATP axiom +-0x #-}\n{-# ATP axiom +-Sx #-}\n\n-- The ATPs cannot prove this postulate from the axioms.\npostulate\n  +-comm : (d e : D) \u2192 d + e \u2261 e + d\n{-# ATP prove +-comm #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a9bd1671df1dc4abec5bce195456bfb6fd69cdad","subject":"FFI.JS.Type: more types and some notes","message":"FFI.JS.Type: more types and some notes\n","repos":"crypto-agda\/agda-libjs,audreyt\/agda-libjs","old_file":"lib\/FFI\/JS\/Type.agda","new_file":"lib\/FFI\/JS\/Type.agda","new_contents":"module FFI.JS.Type where\n\nopen import FFI.JS using (JSValue; String)\n\n-- https:\/\/javascriptweblog.wordpress.com\/2011\/08\/08\/fixing-the-javascript-typeof-operator\/\n\n-- typeof [1,2,3] === 'object'\n-- typeof null    === 'object'\n\n-- typeof NaN === \"number\" :)\n\n-- Maybe there should be a JSONType with: array object number string bool null\n\ndata JSType : Set where\n  array object number string bool null undefined function : JSType\n{-# COMPILED_JS JSType function (x,v) { return require(\"libagda\").readProp(v)(x)(); } #-}\n{-# COMPILED_JS array     \"array\"  #-}\n{-# COMPILED_JS object    \"object\" #-}\n{-# COMPILED_JS number    \"number\" #-}\n{-# COMPILED_JS string    \"string\" #-}\n{-# COMPILED_JS bool      \"bool\"   #-}\n{-# COMPILED_JS null      \"null\"   #-}\n{-# COMPILED_JS undefined \"undefined\" #-}\n{-# COMPILED_JS function  \"function\"  #-}\n\npostulate typeof : JSValue \u2192 JSType\n{-# COMPILED_JS typeof function(x) { return typeof(x); } #-}\n\npostulate showJSType : JSType \u2192 String\n{-# COMPILED_JS showJSType String #-}\n\n-- TODO dyn check?\npostulate castJSType : String \u2192 JSType\n{-# COMPILED_JS castJSType String #-}\n","old_contents":"module FFI.JS.Type where\n\nopen import FFI.JS using (JSValue; String)\n\ndata JSType : Set where\n  array object number string bool null : JSType\n{-# COMPILED_JS JSType function (x,v) { return require(\"libagda\").readProp(v)(x)(); } #-}\n{-# COMPILED_JS array  \"array\"  #-}\n{-# COMPILED_JS object \"object\" #-}\n{-# COMPILED_JS number \"number\" #-}\n{-# COMPILED_JS string \"string\" #-}\n{-# COMPILED_JS bool   \"bool\"   #-}\n{-# COMPILED_JS null   \"null\"   #-}\n\npostulate typeof : JSValue \u2192 JSType\n{-# COMPILED_JS typeof function(x) { return typeof(x); } #-}\n\npostulate showJSType : JSType \u2192 String\n{-# COMPILED_JS showJSType String #-}\n\n-- TODO dyn check?\npostulate castJSType : String \u2192 JSType\n{-# COMPILED_JS castJSType String #-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f47ade85d162ddf7df4b0244f81dd1f6a48889df","subject":"Typo.","message":"Typo.\n\nIgnore-this: 7a1ad958638531dfb96e1595d378bb26\n\ndarcs-hash:20120601191753-3bd4e-9b05ece3a07b7f4c240900179c8f58d8a48a3749.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/FOL\/TermToFOLTerm\/ConTerm.agda","new_file":"Test\/Succeed\/FOL\/TermToFOLTerm\/ConTerm.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the function FOL.Translation.Terms.termToFOLTerm: Con term\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test.Succeed.FOL.TermToFOLTerm.ConTerm where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _+_\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n\ndata M : Set where\n  zero :     M\n  succ : M \u2192 M\n\ndata _\u2261_ (m : M) : M \u2192 Set where\n  refl : m \u2261 m\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n_+_ : M \u2192 M \u2192 M\nzero   + n = n\nsucc m + n = succ (m + n)\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  x+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm zero     n = sym (+-rightIdentity n)\n+-comm (succ m) n = prf (+-comm m n)\n  where\n  postulate prf : m + n \u2261 n + m \u2192  -- IH.\n                  succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the function FOL.Translation.Terms.termToFOLTerm : Con term\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test.Succeed.FOL.TermToFOLTerm.ConTerm where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _+_\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n\ndata M : Set where\n  zero :     M\n  succ : M \u2192 M\n\ndata _\u2261_ (m : M) : M \u2192 Set where\n  refl : m \u2261 m\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n_+_ : M \u2192 M \u2192 M\nzero   + n = n\nsucc m + n = succ (m + n)\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  x+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm zero     n = sym (+-rightIdentity n)\n+-comm (succ m) n = prf (+-comm m n)\n  where\n  postulate prf : m + n \u2261 n + m \u2192  -- IH.\n                  succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"71a13a5d9aa66fbb03a2e0c6e37ec26561c12541","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 80903b2cf7fe900d44f42d86dbdcea46\n\ndarcs-hash:20101201234212-3bd4e-27546a12f5c2b6f6c478b64bb34f66a05030b62b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Relation\/Binary\/EqReasoning.agda","new_file":"src\/Common\/Relation\/Binary\/EqReasoning.agda","new_contents":"------------------------------------------------------------------------------\n-- Equality reasoning on the universe of discourse.\n------------------------------------------------------------------------------\n\nmodule Common.Relation.Binary.EqReasoning where\n\nopen import Common.Relation.Binary.PropositionalEquality using ( _\u2261_ ; refl )\nopen import Common.Relation.Binary.PropositionalEquality.Properties\n  using ( trans )\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2243_\ninfix  4 begin_\ninfixr 5 _\u2261\u27e8_\u27e9_\ninfix  5 _\u220e\n\n------------------------------------------------------------------------------\n-- Adapted from the standard library (Relation.Binary.PreorderReasoning).\nprivate\n  data _\u2243_ (x y : D) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\nbegin_ : {x y : D} \u2192 x \u2243 y \u2192 x \u2261 y\nbegin prf x\u2261y = x\u2261y\n\n_\u2261\u27e8_\u27e9_ : (x : D){y z : D} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n_ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n_\u220e : (x : D) \u2192 x \u2243 x\n_\u220e _ = prf refl\n","old_contents":"------------------------------------------------------------------------------\n-- Equality reasoning on the universe of discourse.\n------------------------------------------------------------------------------\n\nmodule Common.Relation.Binary.EqReasoning where\n\nopen import Common.Relation.Binary.PropositionalEquality using ( _\u2261_ ; refl )\nopen import Common.Relation.Binary.PropositionalEquality.Properties\n  using ( trans )\nopen import Common.Universe using ( D )\n\ninfix 7 _\u2243_\n\n------------------------------------------------------------------------------\n\nprivate\n  data _\u2243_ (x y : D) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\n-- Adapted from the Ulf's thesis.\n-- infix  5 chain>_\n-- infixl 5 _===_by_\n-- infix  4 _qed\n\n-- chain>_ : (x : D) \u2192 x \u2243 x\n-- chain> x = prf (refl {x})\n\n-- _===_by_ : {x y : D} \u2192 x \u2243 y \u2192 (z : D) \u2192 y \u2261 z \u2192 x \u2243 z\n-- prf p === z by q = prf (trans {_} {_} {_} p q)\n\n-- _qed : {x y : D} \u2192 x \u2243 y \u2192 x \u2261 y\n-- prf p qed = p\n\n-- Adapted from the standard library (Relation.Binary.PreorderReasoning).\n\n-- We add 3 to the fixities of the standard library.\ninfix  4 begin_\ninfixr 5 _\u2261\u27e8_\u27e9_\ninfix  5 _\u220e\n\nbegin_ : {x y : D} \u2192 x \u2243 y \u2192 x \u2261 y\nbegin prf x\u2261y = x\u2261y\n\n_\u2261\u27e8_\u27e9_ : (x : D){y z : D} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n_ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n_\u220e : (x : D) \u2192 x \u2243 x\n_\u220e _ = prf refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9e784e0aa6d51fcbd636ced9b6c1d1d8bc3fdcab","subject":"Cleaning.","message":"Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/N.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base hiding ( pred-0 ; pred-S )\nopen import FOTC.Base.PropertiesI hiding ( predCong )\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-least-pre-fixed-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n\n  N-least-pre-fixed' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-least-pre-fixed B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-least-pre-fixed A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d         \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ \u00b7 d) + e \u2261 succ \u00b7 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ \u00b7 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  postulate\n    pred-0 : pred \u00b7 zero             \u2261 zero\n    pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n\n  predCong : \u2200 {m n} \u2192 m \u2261 n \u2192 pred \u00b7 m \u2261 pred \u00b7 n\n  predCong refl = refl\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred \u00b7 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred \u00b7 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N (pred \u00b7 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ \u00b7 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N-least-pre-fixed\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed\u2082 A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalence: N as the least fixed-point and N using Agda's data constructor\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LeastFixedPoints.N where\n\nopen import FOTC.Base hiding ( pred-0 ; pred-S )\nopen import FOTC.Base.PropertiesI hiding ( predCong )\n\n------------------------------------------------------------------------------\n-- Basic definitions\n\n-- Pre-fixed point  : d is a pre-fixed point of f if f d \u2264 d\n\n-- Post-fixed point : d is a post-fixed point of f if d \u2264 f d\n\n-- Fixed-point      : d is a fixed-point of f if f d = d\n\n-- Least pre-fixed point : d is the least pre-fixed point of f if\n-- 1. f d \u2264 d  -- d is a pre-fixed point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Least fixed-point : d is the least fixed-point of f if\n-- 1. f d = d  -- d is a fixed-point of f\n-- 2. \u2200 e. f e \u2264 e \u21d2 d \u2264 e\n\n-- Thm: If d is the least pre-fixed point of f, then d is the least\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n-- Thm: If d is the greatest post-fixed point of f, then d is the greatest\n-- fixed-point of f (\u00c9sik, 2009, p. 31).\n\n------------------------------------------------------------------------------\n-- Auxiliary definitions and properties\n\nflip : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\nflip f b a = f a b\n\ncong : (f : D \u2192 D) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 f x \u2261 f y\ncong f refl = refl\n\n------------------------------------------------------------------------------\nmodule LFP where\n\n  -- N is a least fixed-point of a functor\n\n  -- The functor.\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n  -- The natural numbers are the least fixed-point of NatF.\n  postulate\n    N : D \u2192 Set\n\n    -- N is a pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the introduction rules.\n    N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n\n    -- The higher-order version.\n    N-in-ho : \u2200 {n} \u2192 NatF N n \u2192 N n\n\n    -- N is the least pre-fixed point of NatF.\n    --\n    -- Peter: It corresponds to the elimination rule of an inductively\n    -- defined predicate.\n    N-least-pre-fixed :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n\n    -- Higher-order version.\n    N-least-pre-fixed-ho :\n      (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-in\/N-in-ho.\n\n  N-in\u2081 : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-in\u2081 = N-in-ho\n\n  N-in-ho\u2081 : \u2200 {n} \u2192 NatF N n \u2192 N n\n  N-in-ho\u2081 = N-in\u2081\n\n  ----------------------------------------------------------------------------\n  -- From\/to N-least-pre-fixed\/N-least-pre-fixed-ho\n\n  N-least-pre-fixed' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed' = N-least-pre-fixed-ho\n\n  N-least-pre-fixed-ho' :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed-ho' = N-least-pre-fixed\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N.\n  nzero : N zero\n  nzero = N-in (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n  nsucc Nn = N-in (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- Because N is the least pre-fixed point of NatF (i.e. N-in and\n  -- N-ind), we can proof that N is also a post-fixed point of NatF.\n\n  -- N is a post-fixed point of NatF.\n  N-post-fixed : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n')\n  N-post-fixed = N-least-pre-fixed A h\n    where\n    A : D \u2192 Set\n    A m = m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 N m')\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610)              = inj\u2081 m\u22610\n    h (inj\u2082 (m' , prf , Am')) = inj\u2082 (m' , prf , helper Am')\n      where\n      helper : A m' \u2192 N m'\n      helper (inj\u2081 prf')                = subst N (sym prf') nzero\n      helper (inj\u2082 (m'' , prf' , Am'')) = subst N (sym prf') (nsucc Am'')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed.\n\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h Nn = \u2227-proj\u2082 (N-least-pre-fixed B h' Nn)\n    where\n    B : D \u2192 Set\n    B n = N n \u2227 A n\n\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 B m') \u2192 B m\n    h' (inj\u2081 m\u22610) = subst B (sym m\u22610) (nzero , A0)\n    h' (inj\u2082 (m' , prf , Nm' , Am')) =\n      (subst N (sym prf) (nsucc Nm')) , subst A (sym prf) (h Nm' Am')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step using N-least-pre-fixed\n\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h = N-least-pre-fixed A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  -- Example: We will use N-least-pre-fixed as the induction\n  -- principle on N.\n\n  postulate\n    _+_  : D \u2192 D \u2192 D\n    +-0x : \u2200 d \u2192 zero + d         \u2261 d\n    +-Sx : \u2200 d e \u2192 (succ \u00b7 d) + e \u2261 succ \u00b7 (d + e)\n\n  +-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n  +-leftIdentity n = +-0x n\n\n  +-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n  +-N {n = n} Nm Nn = N-least-pre-fixed A h Nm\n    where\n    A : D \u2192 Set\n    A i = N (i + n)\n\n    h : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h (inj\u2081 m\u22610) = subst N (cong (flip _+_ n) (sym m\u22610)) A0\n      where\n      A0 : A zero\n      A0 = subst N (sym (+-leftIdentity n)) Nn\n    h (inj\u2082 (m' , prf , Am')) =\n      subst N (cong (flip _+_ n) (sym prf)) (is Am')\n      where\n      is : \u2200 {i} \u2192 A i \u2192 A (succ \u00b7 i)\n      is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n  ----------------------------------------------------------------------------\n  -- Example: A proof using N-post-fixed.\n\n  postulate\n    pred-0 : pred \u00b7 zero             \u2261 zero\n    pred-S : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n\n  predCong : \u2200 {m n} \u2192 m \u2261 n \u2192 pred \u00b7 m \u2261 pred \u00b7 n\n  predCong refl = refl\n\n  pred-N : \u2200 {n} \u2192 N n \u2192 N (pred \u00b7 n)\n  pred-N {n} Nn = case h\u2081 h\u2082 (N-post-fixed Nn)\n    where\n    h\u2081 : n \u2261 zero \u2192 N (pred \u00b7 n)\n    h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n    h\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N (pred \u00b7 n)\n    h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n  ----------------------------------------------------------------------------\n  -- From\/to N as a least fixed-point to\/from N as data type.\n\n  data N' : D \u2192 Set where\n    nzero' : N' zero\n    nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ \u00b7 n)\n\n  N'\u2192N : \u2200 {n} \u2192 N' n \u2192 N n\n  N'\u2192N nzero'      = nzero\n  N'\u2192N (nsucc' Nn) = nsucc (N'\u2192N Nn)\n\n  -- Using N-ind\u2081.\n  N\u2192N' : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N' = N-ind\u2081 N' nzero' (\u03bb _ \u2192 nsucc')\n\n  -- Using N-ind\u2082.\n  N\u2192N'\u2081 : \u2200 {n} \u2192 N n \u2192 N' n\n  N\u2192N'\u2081 = N-ind\u2082 N' nzero' nsucc'\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  -- The induction principle for N *with* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2081 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2081 A A0 h nzero      = A0\n  N-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n  -- The induction principle for N *without* the hypothesis N n in the\n  -- induction step.\n  N-ind\u2082 : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind\u2082 A A0 h nzero      = A0\n  N-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- N-in.\n\n  N-in : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-in {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- From N-ind\u2082 to N-least-pre-fixed.\n\n  N-least-pre-fixed\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-least-pre-fixed\u2082 A h = N-ind\u2082 A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n\n------------------------------------------------------------------------------\n-- References\n--\n-- \u00c9sik, Z. (2009). Fixed Point Theory. In: Handbook of Weighted\n-- Automata. Ed. by Droste, M., Kuich, W. and Vogler, H. Monographs in\n-- Theoretical Computer Science. An EATCS Series. Springer. Chap. 2.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2a43f10fc65e57c762dc9cda227c06ae8f23601a","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/List\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Data\/List\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.PropertiesATP where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Data.Conat\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} lnil Lys = prf\n  where postulate prf : List ([] ++ ys)\n        {-# ATP prove prf #-}\n\n++-List {ys = ys} (lcons x {xs} Lxs) Lys = prf (++-List Lxs Lys)\n  where postulate prf : List (xs ++ ys) \u2192 List ((x \u2237 xs) ++ ys)\n        {-# ATP prove prf #-}\n\nmap-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List f lnil = prf\n  where postulate prf : List (map f [])\n        {-# ATP prove prf #-}\nmap-List f (lcons x {xs} Lxs) = prf (map-List f Lxs)\n  where postulate prf : List (map f xs) \u2192 List (map f (x \u2237 xs))\n        {-# ATP prove prf #-}\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} lnil Lys = prf\n  where postulate prf : List (rev [] ys)\n        {-# ATP prove prf #-}\nrev-List {ys = ys} (lcons x {xs} Lxs) Lys = prf (rev-List Lxs (lcons x Lys))\n  where postulate prf : List (rev xs (x \u2237 ys)) \u2192 List (rev (x \u2237 xs) ys)\n        {-# ATP prove prf #-}\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs lnil\n\n-- Length properties\n\nlength-replicate : \u2200 x {n} \u2192 N n \u2192 length (replicate n x) \u2261 n\nlength-replicate x nzero = prf\n  where postulate prf : length (replicate zero x) \u2261 zero\n        {-# ATP prove prf #-}\nlength-replicate x (nsucc {n} Nn) = prf (length-replicate x Nn)\n  where postulate prf : length (replicate n x) \u2261 n \u2192\n                        length (replicate (succ\u2081 n) x) \u2261 succ\u2081 n\n        {-# ATP prove prf #-}\n\npostulate lg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e : \u2200 x xs \u2192 length xs \u2261 \u221e \u2192 length (x \u2237 xs) \u2261 \u221e\n{-# ATP prove lg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e #-}\n\n-- Append properties\n\n++-leftIdentity : \u2200 xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity = ++-[]\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity lnil = prf\n  where postulate prf : [] ++ [] \u2261 []\n        {-# ATP prove prf #-}\n\n++-rightIdentity (lcons x {xs} Lxs) = prf (++-rightIdentity Lxs)\n  where postulate prf : xs ++ [] \u2261 xs \u2192 (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n        {-# ATP prove prf #-}\n\n++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc .{[]} lnil ys zs = prf\n  where postulate prf : ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs\n        {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} (lcons x {xs} Lxs) ys zs = prf (++-assoc Lxs ys zs)\n  where postulate prf : (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192\n                        ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs\n        {-# ATP prove prf #-}\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f lnil ys = prf\n  where postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n        {-# ATP prove prf #-}\n\nmap-++-commute f (lcons x {xs} Lxs) ys = prf (map-++-commute f Lxs ys)\n  where postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n                        map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n        {-# ATP prove prf #-}\n\n-- Reverse properties\n\npostulate reverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nrev-++-commute : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute lnil ys = prf\n  where postulate prf : rev [] ys \u2261 rev [] [] ++ ys\n        {-# ATP prove prf #-}\nrev-++-commute (lcons x {xs} Lxs) ys =\n  prf (rev-++-commute Lxs (x \u2237 ys)) (rev-++-commute Lxs (x \u2237 []))\n  where postulate prf : rev xs (x \u2237 ys) \u2261 rev xs [] ++ x \u2237 ys \u2192\n                        rev xs (x \u2237 []) \u2261 rev xs [] ++ x \u2237 [] \u2192\n                        rev (x \u2237 xs) ys \u2261 rev (x \u2237 xs) [] ++ ys\n        {-# ATP prove prf ++-assoc rev-List #-}\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} lnil Lys = prf\n  where postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n        {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++-commute (lcons x {xs} Lxs) lnil = prf\n  where\n  postulate prf : reverse ((x \u2237 xs) ++ []) \u2261 reverse [] ++ reverse (x \u2237 xs)\n  {-# ATP prove prf ++-rightIdentity #-}\n\nreverse-++-commute (lcons x {xs} Lxs) (lcons y {ys} Lys) =\n  prf (reverse-++-commute Lxs (lcons y Lys))\n  where\n  postulate prf : reverse (xs ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                           reverse xs \u2192\n                  reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                                 reverse (x \u2237 xs)\n  {-# ATP prove prf reverse-List ++-List rev-++-commute ++-assoc #-}\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x lnil = prf\n  where postulate prf : reverse (x \u2237 []) \u2261 reverse [] ++ x \u2237 []\n        {-# ATP prove prf #-}\n\nreverse-\u2237 x (lcons y {ys} Lys) = prf\n  where postulate prf : reverse (x \u2237 y \u2237 ys) \u2261 reverse (y \u2237 ys) ++ x \u2237 []\n        {-# ATP prove prf reverse-[x]\u2261[x] reverse-++-commute #-}\n\nreverse-involutive : \u2200 {xs} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse-involutive lnil = prf\n  where postulate prf : reverse (reverse []) \u2261 []\n        {-# ATP prove prf #-}\n\nreverse-involutive (lcons x {xs} Lxs) = prf (reverse-involutive Lxs)\n  where\n  postulate prf : reverse (reverse xs) \u2261 xs \u2192\n                  reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n  {-# ATP prove prf rev-List reverse-++-commute ++-List ++-rightIdentity #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.PropertiesATP where\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Data.Conat\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} lnil Lys = prf\n  where postulate prf : List ([] ++ ys)\n        {-# ATP prove prf #-}\n\n++-List {ys = ys} (lcons x {xs} Lxs) Lys = prf (++-List Lxs Lys)\n  where postulate prf : List (xs ++ ys) \u2192 List ((x \u2237 xs) ++ ys)\n        {-# ATP prove prf #-}\n\nmap-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List f lnil = prf\n  where postulate prf : List (map f [])\n        {-# ATP prove prf #-}\nmap-List f (lcons x {xs} Lxs) = prf (map-List f Lxs)\n  where postulate prf : List (map f xs) \u2192 List (map f (x \u2237 xs))\n        {-# ATP prove prf #-}\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} lnil Lys = prf\n  where postulate prf : List (rev [] ys)\n        {-# ATP prove prf #-}\nrev-List {ys = ys} (lcons x {xs} Lxs) Lys = prf (rev-List Lxs (lcons x Lys))\n  where postulate prf : List (rev xs (x \u2237 ys)) \u2192 List (rev (x \u2237 xs) ys)\n        {-# ATP prove prf #-}\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs lnil\n\n-- Length properties\n\nlength-replicate : \u2200 x {n} \u2192 N n \u2192 length (replicate n x) \u2261 n\nlength-replicate x nzero = prf\n  where postulate prf : length (replicate zero x) \u2261 zero\n        {-# ATP prove prf #-}\nlength-replicate x (nsucc {n} Nn) = prf (length-replicate x Nn)\n  where postulate prf : length (replicate n x) \u2261 n \u2192\n                        length (replicate (succ\u2081 n) x) \u2261 succ\u2081 n\n        {-# ATP prove prf #-}\n\npostulate lg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e : \u2200 x xs \u2192 length xs \u2261 \u221e \u2192 length (x \u2237 xs) \u2261 \u221e\n{-# ATP prove lg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e #-}\n\n-- Append properties\n\n++-leftIdentity : \u2200 xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity = ++-[]\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity lnil = prf\n  where postulate prf : [] ++ [] \u2261 []\n        {-# ATP prove prf #-}\n\n++-rightIdentity (lcons x {xs} Lxs) = prf (++-rightIdentity Lxs)\n  where postulate prf : xs ++ [] \u2261 xs \u2192 (x \u2237 xs) ++ [] \u2261 x \u2237 xs\n        {-# ATP prove prf #-}\n\n++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc .{[]} lnil ys zs = prf\n  where postulate prf : ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs\n        {-# ATP prove prf #-}\n\n++-assoc .{x \u2237 xs} (lcons x {xs} Lxs) ys zs = prf (++-assoc Lxs ys zs)\n  where postulate prf : (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192\n                        ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs\n        {-# ATP prove prf #-}\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f lnil ys = prf\n  where postulate prf : map f ([] ++ ys) \u2261 map f [] ++ map f ys\n        {-# ATP prove prf #-}\n\nmap-++-commute f (lcons x {xs} Lxs) ys =\n  prf (map-++-commute f Lxs ys)\n  where postulate prf : map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n                        map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys\n        {-# ATP prove prf #-}\n\n-- Reverse properties\n\npostulate reverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\n{-# ATP prove reverse-[x]\u2261[x] #-}\n\nrev-++-commute : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute lnil ys = prf\n  where postulate prf : rev [] ys \u2261 rev [] [] ++ ys\n        {-# ATP prove prf #-}\nrev-++-commute (lcons x {xs} Lxs) ys =\n  prf (rev-++-commute Lxs (x \u2237 ys)) (rev-++-commute Lxs (x \u2237 []))\n  where postulate prf : rev xs (x \u2237 ys) \u2261 rev xs [] ++ x \u2237 ys \u2192\n                        rev xs (x \u2237 []) \u2261 rev xs [] ++ x \u2237 [] \u2192\n                        rev (x \u2237 xs) ys \u2261 rev (x \u2237 xs) [] ++ ys\n        {-# ATP prove prf ++-assoc rev-List #-}\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} lnil Lys = prf\n  where postulate prf : reverse ([] ++ ys) \u2261 reverse ys ++ reverse []\n        {-# ATP prove prf ++-rightIdentity reverse-List #-}\n\nreverse-++-commute (lcons x {xs} Lxs) lnil = prf\n  where\n  postulate prf : reverse ((x \u2237 xs) ++ []) \u2261 reverse [] ++ reverse (x \u2237 xs)\n  {-# ATP prove prf ++-rightIdentity #-}\n\nreverse-++-commute (lcons x {xs} Lxs) (lcons y {ys} Lys) =\n  prf (reverse-++-commute Lxs (lcons y Lys))\n  where\n  postulate prf : reverse (xs ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                           reverse xs \u2192\n                  reverse ((x \u2237 xs) ++ y \u2237 ys) \u2261 reverse (y \u2237 ys) ++\n                                                 reverse (x \u2237 xs)\n  {-# ATP prove prf reverse-List ++-List rev-++-commute ++-assoc #-}\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x lnil = prf\n  where postulate prf : reverse (x \u2237 []) \u2261 reverse [] ++ x \u2237 []\n        {-# ATP prove prf #-}\n\nreverse-\u2237 x (lcons y {ys} Lys) = prf\n  where postulate prf : reverse (x \u2237 y \u2237 ys) \u2261 reverse (y \u2237 ys) ++ x \u2237 []\n        {-# ATP prove prf reverse-[x]\u2261[x] reverse-++-commute #-}\n\nreverse-involutive : \u2200 {xs} \u2192 List xs \u2192 reverse (reverse xs) \u2261 xs\nreverse-involutive lnil = prf\n  where postulate prf : reverse (reverse []) \u2261 []\n        {-# ATP prove prf #-}\n\nreverse-involutive (lcons x {xs} Lxs) = prf (reverse-involutive Lxs)\n  where\n  postulate prf : reverse (reverse xs) \u2261 xs \u2192\n                  reverse (reverse (x \u2237 xs)) \u2261 x \u2237 xs\n  {-# ATP prove prf rev-List reverse-++-commute ++-List ++-rightIdentity #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"37fb93d614c44b62579bfe9876ea6cff70b1c87c","subject":"Finish script rules.","message":"Finish script rules.\n\nScript doesn't have length except with macron and breve marks.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"src\/Language\/Greek\/Script.agda","new_file":"src\/Language\/Greek\/Script.agda","new_contents":"module Language.Greek.Script where\n\nopen import Relation.Nullary using (\u00ac_)\n\ndata Vowel : Set where\n  \u03b1 \u03b5 \u03b7 \u03b9 \u03bf \u03c5 \u03c9 : Vowel\n\ndata Consonant : Set where\n  \u03b2 \u03b3 \u03b4 \u03b6 \u03b8 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03c0 \u03c1 \u03c3 \u03c4 \u03c6 \u03c7 \u03c8 : Consonant\n\ndata Letter : Set where\n  vowel : Vowel \u2192 Letter\n  consonant : Consonant \u2192 Letter\n\ndata LengthMark : Set where\n  breve macron : LengthMark\n\ndata VowelLengthMark : Vowel \u2192 LengthMark \u2192 Set where\n  either-\u03b1 : \u2200 {l} \u2192 VowelLengthMark \u03b1 l\n  either-\u03b9 : \u2200 {l} \u2192 VowelLengthMark \u03b9 l\n  either-\u03c5 : \u2200 {l} \u2192 VowelLengthMark \u03c5 l\n\ndata Case : Set where\n  upper lower : Case\n\ndata LetterCase : Letter \u2192 Case \u2192 Set where\n  any-case : \u2200 {l c} \u2192 LetterCase l c\n\ndata VowelIotaSubscript : Vowel \u2192 Set where\n  \u1fb3 : VowelIotaSubscript \u03b1\n  \u1fc3 : VowelIotaSubscript \u03b7\n  \u1ff3 : VowelIotaSubscript \u03c9\n\ndata VowelDiaeresis : Vowel \u2192 Set where\n  diaeresis : \u2200 {v} \u2192 VowelDiaeresis v\n\ndata Accent : Set where\n  acute grave circumflex : Accent\n\ndata VowelAccent : Vowel \u2192 Accent \u2192 Set where\n  vowelAcute : \u2200 {v} \u2192 VowelAccent v acute\n  vowelGrave : \u2200 {v} \u2192 VowelAccent v grave\n  vowelCircumflex-\u03b1 : VowelAccent \u03b1 circumflex\n  vowelCircumflex-\u03b7 : VowelAccent \u03b7 circumflex\n  vowelCircumflex-\u03b9 : VowelAccent \u03b9 circumflex\n  vowelCircumflex-\u03c5 : VowelAccent \u03c5 circumflex\n  vowelCircumflex-\u03c9 : VowelAccent \u03c9 circumflex\n\ndata Breathing : Set where\n  smooth rough : Breathing\n\ndata LetterBreathing : Letter \u2192 Breathing \u2192 Set where\n  vowelBreathing : \u2200 {v b} \u2192 LetterBreathing v b\n  rhoBreathing : \u2200 {b} \u2192 LetterBreathing (consonant \u03c1) b\n\ndata Final : Letter \u2192 Set where\n  \u03c2 : Final (consonant \u03c3)\n\n\n-- proofs about iota subscript\n\n\u03b5-no-subscript : \u00ac VowelIotaSubscript \u03b5\n\u03b5-no-subscript ()\n\n\u03b9-no-subscript : \u00ac VowelIotaSubscript \u03b9\n\u03b9-no-subscript ()\n\n\u03bf-no-subscript : \u00ac VowelIotaSubscript \u03bf\n\u03bf-no-subscript ()\n\n\u03c5-no-subscript : \u00ac VowelIotaSubscript \u03c5\n\u03c5-no-subscript ()\n\n\n","old_contents":"module Language.Greek.Script where\n\nopen import Relation.Nullary using (\u00ac_)\n\ndata Vowel : Set where\n  \u03b1 \u03b5 \u03b7 \u03b9 \u03bf \u03c5 \u03c9 : Vowel\n\ndata Consonant : Set where\n  \u03b2 \u03b3 \u03b4 \u03b6 \u03b8 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03c0 \u03c1 \u03c3 \u03c4 \u03c6 \u03c7 \u03c8 : Consonant\n\ndata Letter : Set where\n  vowel : Vowel \u2192 Letter\n  consonant : Consonant \u2192 Letter\n\ndata Length : Set where\n  short long : Length\n\ndata VowelLength : Vowel \u2192 Length \u2192 Set where\n  either-\u03b1 : \u2200 {l} \u2192 VowelLength \u03b1 l\n  either-\u03b9 : \u2200 {l} \u2192 VowelLength \u03b9 l\n  either-\u03c5 : \u2200 {l} \u2192 VowelLength \u03c5 l\n  short-\u03b5 : VowelLength \u03b5 short\n  short-\u03bf : VowelLength \u03bf short\n  long-\u03b7 : VowelLength \u03b7 long\n  long-\u03c9 : VowelLength \u03c9 long\n\ndata Case : Set where\n  upper lower : Case\n\ndata LetterCase : Letter \u2192 Case \u2192 Set where\n  any-case : \u2200 {l c} \u2192 LetterCase l c\n\ndata IotaSubscript : Vowel \u2192 Set where\n  \u03b1_\u03b9 : IotaSubscript \u03b1\n  \u03b7_\u03b9 : IotaSubscript \u03b7\n  \u03c9_\u03b9 : IotaSubscript \u03c9\n\n\n-- proofs about length\n\n\u03b5-not-long : \u00ac VowelLength \u03b5 long\n\u03b5-not-long ()\n\n\u03bf-not-long : \u00ac VowelLength \u03bf long\n\u03bf-not-long ()\n\n\u03b7-not-short : \u00ac VowelLength \u03b7 short\n\u03b7-not-short ()\n\n\u03c9-not-short : \u00ac VowelLength \u03c9 short\n\u03c9-not-short ()\n\n\n-- proofs about iota subscript\n\n\u03b5-no-subscript : \u00ac IotaSubscript \u03b5\n\u03b5-no-subscript ()\n\n\u03b9-no-subscript : \u00ac IotaSubscript \u03b9\n\u03b9-no-subscript ()\n\n\u03bf-no-subscript : \u00ac IotaSubscript \u03bf\n\u03bf-no-subscript ()\n\n\u03c5-no-subscript : \u00ac IotaSubscript \u03c5\n\u03c5-no-subscript ()\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b1fdb10a5e323a77a174af8dfdcaab062c73405f","subject":"Upgrade flat-funs-implem","message":"Upgrade flat-funs-implem\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs-implem.agda","new_file":"flat-funs-implem.agda","new_contents":"module flat-funs-implem where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Data.Nat.NP using (\u2115)\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bits using (_\u2192\u1d47_; 0b; 1b)\n\nopen import data-universe\nopen import flat-funs\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfunLin : LinRewiring fun\u266dFuns\nfunLin = mk F.id _\u2218\u2032_\n            (\u03bb f \u2192 \u00d7.map f F.id)\n            \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n            (\u03bb f g \u2192 \u00d7.map f g) (\u03bb f \u2192 \u00d7.map F.id f)\n            (F.const []) _ (uncurry _\u2237_) V.uncons\n\nfunRewiring : Rewiring fun\u266dFuns\nfunRewiring = mk funLin _ (\u03bb x \u2192 x , x) (F.const []) \u00d7.<_,_> proj\u2081 proj\u2082\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk funRewiring (F.const 0b) (F.const 1b) (\u03bb { (b , x , y) \u2192 if b then x else y })\n             (\u03bb { f g (b , x) \u2192 (if b then f else g) x })\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk rewiring (const [ 0b ]) (const [ 1b ]) cond fork\u1d47\n  where\n  open BitsFunTypes\n  open FunOps using (id; _\u2218_)\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = V.take A\n  snd\u1d47 : \u2200 {B} A \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 A = V.drop A\n  <_,_>\u1d47 : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <_,_>\u1d47 f g x = f x ++ g x\n  fork\u1d47 : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u1d47 f g (b \u2237 xs) = (if b then f else g) xs\n  open Defaults bitsFun\u266dFuns\n  open DefaultsGroup2 id _\u2218_ (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n  open DefaultCond fork\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n\n  lin : LinRewiring bitsFun\u266dFuns\n  lin = mk id _\u2218_ first (\u03bb {x} \u2192 swap {x}) (\u03bb {x} \u2192 assoc {x}) id id\n           <_\u00d7_> (\u03bb {x} \u2192 second {x}) id id id id\n\n  rewiring : Rewiring bitsFun\u266dFuns\n  rewiring = mk lin (const []) dup (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n\nbitsFunRewiring : Rewiring bitsFun\u266dFuns\nbitsFunRewiring = FlatFunsOps.rewiring bitsFun\u266dOps\n\nbitsFunLin : LinRewiring bitsFun\u266dFuns\nbitsFunLin = Rewiring.linRewiring bitsFunRewiring\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n\n\u22a4-FunOps : FlatFunsOps (constFuns \u22a4)\n\u22a4-FunOps = _\n","old_contents":"module flat-funs-implem where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Fin using (Fin)\nopen import Data.Nat.NP using (\u2115)\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bits using (_\u2192\u1d47_; 0b; 1b)\n\nopen import data-universe\nopen import flat-funs\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nmodule FunTypes = FlatFuns fun\u266dFuns\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nmodule BitsFunTypes = FlatFuns bitsFun\u266dFuns\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nmodule FinFunTypes = FlatFuns finFun\u266dFuns\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id _\u2218\u2032_\n             (F.const 0b) (F.const 1b) (\u03bb { (b , x , y) \u2192 if b then x else y })\n             (\u03bb { f g (b , x) \u2192 (if b then f else g) x }) _\n             \u00d7.<_,_> proj\u2081 proj\u2082 (\u03bb x \u2192 x , x) (\u03bb f \u2192 \u00d7.map f F.id)\n             \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n             (\u03bb f g \u2192 \u00d7.map f g) (\u03bb f \u2192 \u00d7.map F.id f)\n             (F.const []) (uncurry _\u2237_) V.uncons\n\nmodule FunOps = FlatFunsOps fun\u266dOps\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _\u2218_\n                 (const [ 0b ]) (const [ 1b ]) cond fork\u1d47 (const [])\n                 <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x) dup first\n                 (\u03bb {x} \u2192 swap {x}) (\u03bb {x} \u2192 assoc {x}) id id\n                 <_\u00d7_> (\u03bb {x} \u2192 second {x}) (const []) id id\n  where\n  open BitsFunTypes\n  open FunOps using (id; _\u2218_)\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = V.take A\n  snd\u1d47 : \u2200 {B} A \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 A = V.drop A\n  <_,_>\u1d47 : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <_,_>\u1d47 f g x = f x ++ g x\n  fork\u1d47 : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u1d47 f g (b \u2237 xs) = (if b then f else g) xs\n  open Defaults bitsFun\u266dFuns\n  open DefaultsGroup2 id _\u2218_ (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n  open DefaultCond fork\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n\nmodule BitsFunOps = FlatFunsOps bitsFun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\nmodule ConstFunTypes A = FlatFuns (constFuns A)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"092b540f41b26ea73eb656be06181975aba5a05d","subject":"Removed unnecessary import.","message":"Removed unnecessary import.\n\nIgnore-this: 26a6cc88934801a8cf0d09f6f5d6bef\n\ndarcs-hash:20111208203353-3bd4e-576bb8e2e9180af6a7f4e5fbe64f7112429b79c3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/papers\/FoSSaCS-2012\/BottomBottom.agda","new_file":"notes\/papers\/FoSSaCS-2012\/BottomBottom.agda","new_contents":"-- Tested on 08 December 2011.\n\nmodule BottomBottom where\n\nopen import Common.Universe\nopen import Common.Data.Empty\n\npostulate bot\u2081 bot\u2082 : \u22a5\n{-# ATP prove bot\u2081 bot\u2082 #-}\n{-# ATP prove bot\u2082 bot\u2081 #-}\n\n-- $ agda2atp -i src -i notes\/papers\/FoSSaCS-2012\/ notes\/papers\/FoSSaCS-2012\/BottomBottom.agda\n-- Proving the conjecture in \/tmp\/BottomBottom.bot1_8.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot1_8.tptp\n-- Proving the conjecture in \/tmp\/BottomBottom.bot2_8.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot2_8.tptp\n\n","old_contents":"-- Tested on 08 December 2011.\n\nmodule BottomBottom where\n\nopen import Common.Universe\nopen import Common.Data.Empty\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.Inductive\n\npostulate\n  bot\u2081 : \u22a5\n  bot\u2082 : \u22a5\n{-# ATP prove bot\u2081 bot\u2082 #-}\n{-# ATP prove bot\u2082 bot\u2081 #-}\n\n-- $ agda2atp -isrc -inotes\/papers\/FoSSaCS-2012\/  notes\/papers\/FoSSaCS-2012\/BottomBottom.agda\n-- Proving the conjecture in \/tmp\/BottomBottom.bot1_12.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot1_12.tptp\n-- Proving the conjecture in \/tmp\/BottomBottom.bot2_13.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/BottomBottom.bot2_13.tptp\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"01ff47d3d4a672288a0fab47674caedd10efd390","subject":"Added examples of polimorphic lists.","message":"Added examples of polimorphic lists.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool.Type\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Heterogeneous lists\ndata List : D \u2192 Set where\n  lnil  :                      List []\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\nxs : D\nxs = [0] \u2237 true \u2237 [1] \u2237 false \u2237 []\n\nxs-List : List xs\nxs-List = lcons [0] (lcons true (lcons [1] (lcons false lnil)))\n\n-- Lists of total natural numbers\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\nys : D\nys = [0] \u2237 [1] \u2237 [2] \u2237 []\n\nys-ListN : ListN ys\nys-ListN =\n  lncons nzero (lncons (nsucc nzero) (lncons (nsucc (nsucc nzero)) lnnil))\n\n-- Lists of total Booleans\ndata ListB : D \u2192 Set where\n  lbnil  :                                ListB []\n  lbcons : \u2200 {b bs} \u2192 Bool b \u2192 ListB bs \u2192 ListB (b \u2237 bs)\n\nzs : D\nzs = true \u2237 false \u2237 true \u2237 []\n\nzs-ListB : ListB zs\nzs-ListB = lbcons btrue (lbcons bfalse (lbcons btrue lbnil))\n\n-- Polymorphic lists.\n-- NB. The data type list is in *Set\u2081*.\ndata ListP (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                               ListP A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n\nList\u2081 : D \u2192 Set\u2081\nList\u2081 = ListP (\u03bb d \u2192 d \u2261 d)\n\nxs-List\u2081 : List\u2081 xs\nxs-List\u2081 = lcons refl (lcons refl (lcons refl (lcons refl lnil)))\n\nListN\u2081 : D \u2192 Set\u2081\nListN\u2081 = ListP N\n\nys-ListN\u2081 : ListN\u2081 ys\nys-ListN\u2081 = lcons nzero (lcons (nsucc nzero) (lcons (nsucc (nsucc nzero)) lnil))\n\nListB\u2081 : D \u2192 Set\u2081\nListB\u2081 = ListP Bool\n\nzs-ListB\u2081 : ListB\u2081 zs\nzs-ListB\u2081 = lcons btrue (lcons bfalse (lcons btrue lnil))\n","old_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool.Type\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Heterogeneous lists\ndata List : D \u2192 Set where\n  lnil  :                      List []\n  lcons : \u2200 x {xs} \u2192 List xs \u2192 List (x \u2237 xs)\n\n-- Lists of total natural numbers\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n\n-- Lists of total Booleans\ndata ListB : D \u2192 Set where\n  lbnil  :                                ListB []\n  lbcons : \u2200 {b bs} \u2192 Bool b \u2192 ListB bs \u2192 ListB (b \u2237 bs)\n\n-- Polymorphic lists.\n-- NB. The data type list is in *Set\u2081*.\ndata ListP (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                               ListP A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n\nList\u2081 : D \u2192 Set\u2081\nList\u2081 = ListP (\u03bb d \u2192 d \u2261 d)\n\nListN\u2081 : D \u2192 Set\u2081\nListN\u2081 = ListP N\n\nListB\u2081 : D \u2192 Set\u2081\nListB\u2081 = ListP Bool\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c8a0ccc95094fb62b3b7908063dd383f917bfd80","subject":"Simplified some proofs.","message":"Simplified some proofs.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/List\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/List\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Data.Conat\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n\u2237-leftCong : \u2200 {x y xs} \u2192 x \u2261 y \u2192 x \u2237 xs \u2261 y \u2237 xs\n\u2237-leftCong refl = refl\n\n\u2237-rightCong : \u2200 {x xs ys} \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 x \u2237 ys\n\u2237-rightCong refl = refl\n\n++-leftCong : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 xs ++ zs \u2261 ys ++ zs\n++-leftCong refl = refl\n\n++-rightCong : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 xs ++ ys \u2261 xs ++ zs\n++-rightCong refl = refl\n\nrevLeftCong : \u2200 {xs ys zs} \u2192 xs \u2261 ys \u2192 rev xs zs \u2261 rev ys zs\nrevLeftCong refl = refl\n\nrevRightCong : \u2200 {xs ys zs} \u2192 ys \u2261 zs \u2192 rev xs ys \u2261 rev xs zs\nrevRightCong refl = refl\n\nreverseCong : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 reverse xs \u2261 reverse ys\nreverseCong refl = refl\n\n------------------------------------------------------------------------------\n-- Totality properties\n\nlengthList-N : \u2200 {xs} \u2192 List xs \u2192 N (length xs)\nlengthList-N lnil               = subst N (sym length-[]) nzero\nlengthList-N (lcons x {xs} Lxs) =\n  subst N (sym (length-\u2237 x xs)) (nsucc (lengthList-N Lxs))\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} lnil               Lys = subst List (sym (++-[] ys)) Lys\n++-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (++-\u2237 x xs ys)) (lcons x (++-List Lxs Lys))\n\nmap-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List f lnil               = subst List (sym (map-[] f)) lnil\nmap-List f (lcons x {xs} Lxs) =\n  subst List (sym (map-\u2237 f x xs)) (lcons (f \u00b7 x) (map-List f Lxs))\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} lnil               Lys = subst List (sym (rev-[] ys)) Lys\nrev-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (rev-\u2237 x xs ys)) (rev-List Lxs (lcons x Lys))\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs lnil\n\n-- Length properties\n\nlg-x<lg-x\u2237xs : \u2200 x {xs} \u2192 List xs \u2192 LT (length xs) (length (x \u2237 xs))\nlg-x<lg-x\u2237xs x lnil =\n  length [] < length (x \u2237 [])\n    \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 length [] < length (x \u2237 []) \u2261 t\u2081 < t\u2082)\n              length-[]\n              (length-\u2237 x [])\n              refl\n    \u27e9\n  zero < succ\u2081 (length [])\n    \u2261\u27e8 <-0S (length []) \u27e9\n  true \u220e\n\nlg-x<lg-x\u2237xs x (lcons y {xs} Lxs) =\n  length (y \u2237 xs) < length (x \u2237 y \u2237 xs)\n    \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 length (y \u2237 xs) < length (x \u2237 y \u2237 xs) \u2261 t\u2081 < t\u2082)\n              (length-\u2237 y xs)\n              (length-\u2237 x (y \u2237 xs))\n              refl\n    \u27e9\n  succ\u2081 (length xs) < succ\u2081 (length (y \u2237 xs))\n    \u2261\u27e8 <-SS (length xs) (length (y \u2237 xs)) \u27e9\n  length xs < length (y \u2237 xs)\n    \u2261\u27e8 lg-x<lg-x\u2237xs y Lxs \u27e9\n  true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 : \u2200 {xs} \u2192 List xs \u2192 \u00ac (LT (length xs) (length []))\nlg-xs<lg-[]\u2192\u22a5 lnil lg-[]<lg-[] = \u22a5-elim (0<0\u2192\u22a5 helper)\n  where\n  helper : zero < zero \u2261 true\n  helper =\n    zero < zero\n      \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 zero < zero \u2261 t\u2081 < t\u2082)\n                (sym length-[])\n                (sym length-[])\n                refl\n      \u27e9\n    length [] < length []\n      \u2261\u27e8 lg-[]<lg-[] \u27e9\n    true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 (lcons x {xs} Lxs) lg-x\u2237xs<lg-[] = \u22a5-elim (S<0\u2192\u22a5 helper)\n  where\n  helper : succ\u2081 (length xs) < zero \u2261 true\n  helper =\n    succ\u2081 (length xs) < zero\n      \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 succ\u2081 (length xs) < zero \u2261 t\u2081 < t\u2082)\n                (sym (length-\u2237 x xs))\n                (sym length-[])\n                refl\n      \u27e9\n    length (x \u2237 xs) < length []\n      \u2261\u27e8 lg-x\u2237xs<lg-[] \u27e9\n    true \u220e\n\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e : \u2200 x xs \u2192 length xs \u2261 \u221e \u2192 length (x \u2237 xs) \u2261 \u221e\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e x xs h =\n  length (x \u2237 xs)   \u2261\u27e8 length-\u2237 x xs \u27e9\n  succ\u2081 (length xs) \u2261\u27e8 succCong h \u27e9\n  succ\u2081 \u221e           \u2261\u27e8 sym \u221e-eq \u27e9\n  \u221e                 \u220e\n\n-- Append properties\n\n++-leftIdentity : \u2200 xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity = ++-[]\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity lnil               = ++-[] []\n++-rightIdentity (lcons x {xs} Lxs) =\n  (x \u2237 xs) ++ [] \u2261\u27e8 ++-\u2237 x xs [] \u27e9\n  x \u2237 (xs ++ []) \u2261\u27e8 \u2237-rightCong (++-rightIdentity Lxs) \u27e9\n  x \u2237 xs         \u220e\n\n++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc lnil ys zs =\n  ([] ++ ys) ++ zs \u2261\u27e8 ++-leftCong (++-[] ys) \u27e9\n  ys ++ zs         \u2261\u27e8 sym (++-leftIdentity (ys ++ zs)) \u27e9\n  [] ++ ys ++ zs   \u220e\n\n++-assoc (lcons x {xs} Lxs) ys zs =\n  ((x \u2237 xs) ++ ys) ++ zs \u2261\u27e8 ++-leftCong (++-\u2237 x xs ys) \u27e9\n  (x \u2237 (xs ++ ys)) ++ zs \u2261\u27e8 ++-\u2237 x (xs ++ ys) zs \u27e9\n  x \u2237 ((xs ++ ys) ++ zs) \u2261\u27e8 \u2237-rightCong (++-assoc Lxs ys zs) \u27e9\n  x \u2237 (xs ++ ys ++ zs)   \u2261\u27e8 sym (++-\u2237 x xs (ys ++ zs)) \u27e9\n  (x \u2237 xs) ++ ys ++ zs   \u220e\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f lnil ys =\n  map f ([] ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 map f ([] ++ ys) \u2261 map f t) (++-[] ys) refl \u27e9\n  map f ys\n    \u2261\u27e8 sym (++-leftIdentity (map f ys)) \u27e9\n  [] ++ map f ys\n     \u2261\u27e8 subst (\u03bb t \u2192 [] ++ map f ys \u2261 t ++ map f ys) (sym (map-[] f)) refl\n     \u27e9\n  map f [] ++ map f ys \u220e\n\nmap-++-commute f (lcons x {xs} Lxs) ys =\n  map f ((x \u2237 xs) ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 map f ((x \u2237 xs) ++ ys) \u2261 map f t) (++-\u2237 x xs ys) refl \u27e9\n  map f (x \u2237 xs ++ ys)\n    \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n  f \u00b7 x \u2237 map f (xs ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 f \u00b7 x \u2237 map f (xs ++ ys) \u2261 f \u00b7 x \u2237 t)\n             (map-++-commute f Lxs ys)\n             refl\n    \u27e9\n  f \u00b7 x \u2237 (map f xs ++ map f ys)\n    \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n  (f \u00b7 x \u2237 map f xs) ++ map f ys\n     \u2261\u27e8 subst (\u03bb t \u2192 (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261 t ++ map f ys)\n              (sym (map-\u2237 f x xs))\n              refl\n     \u27e9\n  map f (x \u2237 xs) ++ map f ys \u220e\n\nmap\u2261[] : \u2200 {f xs} \u2192 List xs \u2192 map f xs \u2261 [] \u2192 xs \u2261 []\nmap\u2261[] lnil                   h = refl\nmap\u2261[] {f} (lcons x {xs} Lxs) h = \u22a5-elim ([]\u2262cons (trans (sym h) (map-\u2237 f x xs)))\n\n-- Reverse properties\n\nreverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\nreverse-[x]\u2261[x] x =\n  rev (x \u2237 []) [] \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 []) \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []          \u220e\n\nrev-++-commute : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute lnil ys =\n  rev [] ys       \u2261\u27e8 rev-[] ys \u27e9\n  ys              \u2261\u27e8 sym $ ++-leftIdentity ys \u27e9\n  [] ++ ys        \u2261\u27e8 ++-leftCong (sym $ rev-[] []) \u27e9\n  rev [] [] ++ ys \u220e\n\nrev-++-commute (lcons x {xs} Lxs) ys =\n  rev (x \u2237 xs) ys\n    \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n  rev xs (x \u2237 ys)\n    \u2261\u27e8 rev-++-commute Lxs (x \u2237 ys) \u27e9\n  rev xs [] ++ x \u2237 ys\n    \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n             (sym\n               ( (x \u2237 []) ++ ys\n                    \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                 x \u2237 ([] ++ ys)\n                   \u2261\u27e8 \u2237-rightCong (++-leftIdentity ys) \u27e9\n                 x \u2237 ys \u220e\n               )\n             )\n             refl\n    \u27e9\n  rev xs [] ++ (x \u2237 []) ++ ys\n    \u2261\u27e8 sym $ ++-assoc (rev-List Lxs lnil) (x \u2237 []) ys \u27e9\n  (rev xs [] ++ (x \u2237 [])) ++ ys\n    \u2261\u27e8 ++-leftCong (sym $ rev-++-commute Lxs (x \u2237 [])) \u27e9\n  rev xs (x \u2237 []) ++ ys\n    \u2261\u27e8 ++-leftCong (sym $ rev-\u2237 x xs []) \u27e9\n  rev (x \u2237 xs) [] ++ ys \u220e\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} lnil Lys =\n  reverse ([] ++ ys)       \u2261\u27e8 reverseCong (++-[] ys) \u27e9\n  reverse ys               \u2261\u27e8 sym (++-rightIdentity (reverse-List Lys)) \u27e9\n  reverse ys ++ []         \u2261\u27e8 ++-rightCong (sym (rev-[] [])) \u27e9\n  reverse ys ++ reverse [] \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) lnil =\n  reverse ((x \u2237 xs) ++ [])\n    \u2261\u27e8 reverseCong (++-rightIdentity (lcons x Lxs)) \u27e9\n  reverse (x \u2237 xs)\n    \u2261\u27e8 sym (++-[] (reverse (x \u2237 xs))) \u27e9\n  [] ++ reverse (x \u2237 xs)\n     \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  reverse [] ++ reverse (x \u2237 xs) \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) (lcons y {ys} Lys) =\n  reverse ((x \u2237 xs) ++ y \u2237 ys)\n    \u2261\u27e8 refl \u27e9\n  rev ((x \u2237 xs) ++ y \u2237 ys) []\n    \u2261\u27e8 revLeftCong (++-\u2237 x xs (y \u2237 ys)) \u27e9\n  rev (x \u2237 (xs ++ y \u2237 ys)) []\n    \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n  rev (xs ++ y \u2237 ys) (x \u2237 [])\n    \u2261\u27e8 rev-++-commute (++-List Lxs (lcons y Lys)) (x \u2237 []) \u27e9\n  rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n    \u2261\u27e8 refl \u27e9\n  reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n    \u2261\u27e8 ++-leftCong (reverse-++-commute Lxs (lcons y Lys)) \u27e9\n  (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n    \u2261\u27e8 ++-assoc (reverse-List (lcons y Lys)) (reverse xs) (x \u2237 []) \u27e9\n  reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n    \u2261\u27e8 refl \u27e9\n  reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n    \u2261\u27e8 ++-rightCong (sym $ rev-++-commute Lxs (x \u2237 [])) \u27e9\n  reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n    \u2261\u27e8 ++-rightCong (sym $ rev-\u2237 x xs []) \u27e9\n  reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n    \u2261\u27e8 refl \u27e9\n  reverse (y \u2237 ys) ++ reverse (x \u2237 xs) \u220e\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x lnil =\n  rev (x \u2237 []) []     \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 [])     \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []              \u2261\u27e8 sym (++-leftIdentity (x \u2237 [])) \u27e9\n  [] ++ x \u2237 []        \u2261\u27e8 ++-leftCong (sym (rev-[] [])) \u27e9\n  rev [] [] ++ x \u2237 [] \u220e\n\nreverse-\u2237 x (lcons y {ys} Lys) = sym prf\n  where\n  prf : reverse (y \u2237 ys) ++ x \u2237 [] \u2261 reverse (x \u2237 y \u2237 ys)\n  prf =\n    reverse (y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 ++-rightCong (sym (reverse-[x]\u2261[x] x)) \u27e9\n    (reverse (y \u2237 ys) ++ reverse (x \u2237 []))\n      \u2261\u27e8 sym (reverse-++-commute (lcons x lnil) (lcons y Lys)) \u27e9\n    reverse ((x \u2237 []) ++ (y \u2237 ys))\n      \u2261\u27e8 reverseCong (++-\u2237 x [] (y \u2237 ys)) \u27e9\n    reverse (x \u2237 ([] ++ (y \u2237 ys)))\n      \u2261\u27e8 reverseCong (\u2237-rightCong (++-leftIdentity (y \u2237 ys))) \u27e9\n    reverse (x \u2237 y \u2237 ys) \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\nopen import Common.Function\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Data.Conat\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\nlengthList-N : \u2200 {xs} \u2192 List xs \u2192 N (length xs)\nlengthList-N lnil               = subst N (sym length-[]) nzero\nlengthList-N (lcons x {xs} Lxs) =\n  subst N (sym (length-\u2237 x xs)) (nsucc (lengthList-N Lxs))\n\n++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n++-List {ys = ys} lnil               Lys = subst List (sym (++-[] ys)) Lys\n++-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (++-\u2237 x xs ys)) (lcons x (++-List Lxs Lys))\n\nmap-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List f lnil               = subst List (sym (map-[] f)) lnil\nmap-List f (lcons x {xs} Lxs) =\n  subst List (sym (map-\u2237 f x xs)) (lcons (f \u00b7 x) (map-List f Lxs))\n\nrev-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (rev xs ys)\nrev-List {ys = ys} lnil               Lys = subst List (sym (rev-[] ys)) Lys\nrev-List {ys = ys} (lcons x {xs} Lxs) Lys =\n  subst List (sym (rev-\u2237 x xs ys)) (rev-List Lxs (lcons x Lys))\n\nreverse-List : \u2200 {xs} \u2192 List xs \u2192 List (reverse xs)\nreverse-List Lxs = rev-List Lxs lnil\n\n-- Length properties\n\nlg-x<lg-x\u2237xs : \u2200 x {xs} \u2192 List xs \u2192 LT (length xs) (length (x \u2237 xs))\nlg-x<lg-x\u2237xs x lnil =\n  length [] < length (x \u2237 [])\n    \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 length [] < length (x \u2237 []) \u2261 t\u2081 < t\u2082)\n              length-[]\n              (length-\u2237 x [])\n              refl\n    \u27e9\n  zero < succ\u2081 (length [])\n    \u2261\u27e8 <-0S (length []) \u27e9\n  true \u220e\n\nlg-x<lg-x\u2237xs x (lcons y {xs} Lxs) =\n  length (y \u2237 xs) < length (x \u2237 y \u2237 xs)\n    \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 length (y \u2237 xs) < length (x \u2237 y \u2237 xs) \u2261 t\u2081 < t\u2082)\n              (length-\u2237 y xs)\n              (length-\u2237 x (y \u2237 xs))\n              refl\n    \u27e9\n  succ\u2081 (length xs) < succ\u2081 (length (y \u2237 xs))\n    \u2261\u27e8 <-SS (length xs) (length (y \u2237 xs)) \u27e9\n  length xs < length (y \u2237 xs)\n    \u2261\u27e8 lg-x<lg-x\u2237xs y Lxs \u27e9\n  true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 : \u2200 {xs} \u2192 List xs \u2192 \u00ac (LT (length xs) (length []))\nlg-xs<lg-[]\u2192\u22a5 lnil lg-[]<lg-[] = \u22a5-elim (0<0\u2192\u22a5 helper)\n  where\n  helper : zero < zero \u2261 true\n  helper =\n    zero < zero\n      \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 zero < zero \u2261 t\u2081 < t\u2082)\n                (sym length-[])\n                (sym length-[])\n                refl\n      \u27e9\n    length [] < length []\n      \u2261\u27e8 lg-[]<lg-[] \u27e9\n    true \u220e\n\nlg-xs<lg-[]\u2192\u22a5 (lcons x {xs} Lxs) lg-x\u2237xs<lg-[] = \u22a5-elim (S<0\u2192\u22a5 helper)\n  where\n  helper : succ\u2081 (length xs) < zero \u2261 true\n  helper =\n    succ\u2081 (length xs) < zero\n      \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 succ\u2081 (length xs) < zero \u2261 t\u2081 < t\u2082)\n                (sym (length-\u2237 x xs))\n                (sym length-[])\n                refl\n      \u27e9\n    length (x \u2237 xs) < length []\n      \u2261\u27e8 lg-x\u2237xs<lg-[] \u27e9\n    true \u220e\n\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e : \u2200 x xs \u2192 length xs \u2261 \u221e \u2192 length (x \u2237 xs) \u2261 \u221e\nlg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e x xs h =\n  length (x \u2237 xs) \u2261\u27e8 length-\u2237 x xs \u27e9\n  succ\u2081 (length xs) \u2261\u27e8 succCong h \u27e9\n  succ\u2081 \u221e \u2261\u27e8 sym \u221e-eq \u27e9\n  \u221e \u220e\n\n-- Append properties\n\n++-leftIdentity : \u2200 xs \u2192 [] ++ xs \u2261 xs\n++-leftIdentity = ++-[]\n\n++-rightIdentity : \u2200 {xs} \u2192 List xs \u2192 xs ++ [] \u2261 xs\n++-rightIdentity lnil               = ++-[] []\n++-rightIdentity (lcons x {xs} Lxs) =\n  (x \u2237 xs) ++ []\n     \u2261\u27e8 ++-\u2237 x xs [] \u27e9\n  x \u2237 (xs ++ [])\n    \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 (xs ++ []) \u2261 x \u2237 t) (++-rightIdentity Lxs) refl \u27e9\n  x \u2237 xs \u220e\n\n++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ (ys ++ zs)\n++-assoc lnil ys zs =\n  ([] ++ ys) ++ zs\n    \u2261\u27e8 subst (\u03bb t \u2192 ([] ++ ys) ++ zs \u2261 t ++ zs) (++-[] ys) refl \u27e9\n  ys ++ zs\n     \u2261\u27e8 sym (++-leftIdentity (ys ++ zs)) \u27e9\n  [] ++ ys ++ zs \u220e\n\n++-assoc (lcons x {xs} Lxs) ys zs =\n  ((x \u2237 xs) ++ ys) ++ zs\n    \u2261\u27e8 subst (\u03bb t \u2192 ((x \u2237 xs) ++ ys) ++ zs \u2261 t ++ zs) (++-\u2237 x xs ys) refl \u27e9\n  (x \u2237 (xs ++ ys)) ++ zs\n     \u2261\u27e8 ++-\u2237 x (xs ++ ys) zs \u27e9\n  x \u2237 ((xs ++ ys) ++ zs)\n    \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ((xs ++ ys) ++ zs) \u2261 x \u2237 t)\n             (++-assoc Lxs ys zs)\n             refl\n    \u27e9\n  x \u2237 (xs ++ ys ++ zs)\n    \u2261\u27e8 sym (++-\u2237 x xs (ys ++ zs)) \u27e9\n  (x \u2237 xs) ++ ys ++ zs \u220e\n\n-- Map properties\n\nmap-++-commute : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n                 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-commute f lnil ys =\n  map f ([] ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 map f ([] ++ ys) \u2261 map f t) (++-[] ys) refl \u27e9\n  map f ys\n    \u2261\u27e8 sym (++-leftIdentity (map f ys)) \u27e9\n  [] ++ map f ys\n     \u2261\u27e8 subst (\u03bb t \u2192 [] ++ map f ys \u2261 t ++ map f ys) (sym (map-[] f)) refl\n     \u27e9\n  map f [] ++ map f ys \u220e\n\nmap-++-commute f (lcons x {xs} Lxs) ys =\n  map f ((x \u2237 xs) ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 map f ((x \u2237 xs) ++ ys) \u2261 map f t) (++-\u2237 x xs ys) refl \u27e9\n  map f (x \u2237 xs ++ ys)\n    \u2261\u27e8 map-\u2237 f x (xs ++ ys) \u27e9\n  f \u00b7 x \u2237 map f (xs ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 f \u00b7 x \u2237 map f (xs ++ ys) \u2261 f \u00b7 x \u2237 t)\n             (map-++-commute f Lxs ys)\n             refl\n    \u27e9\n  f \u00b7 x \u2237 (map f xs ++ map f ys)\n    \u2261\u27e8 sym (++-\u2237 (f \u00b7 x) (map f xs) (map f ys)) \u27e9\n  (f \u00b7 x \u2237 map f xs) ++ map f ys\n     \u2261\u27e8 subst (\u03bb t \u2192 (f \u00b7 x \u2237 map f xs) ++ map f ys \u2261 t ++ map f ys)\n              (sym (map-\u2237 f x xs))\n              refl\n     \u27e9\n  map f (x \u2237 xs) ++ map f ys \u220e\n\nmap\u2261[] : \u2200 {f xs} \u2192 List xs \u2192 map f xs \u2261 [] \u2192 xs \u2261 []\nmap\u2261[] lnil                   h = refl\nmap\u2261[] {f} (lcons x {xs} Lxs) h = \u22a5-elim ([]\u2262cons (trans (sym h) (map-\u2237 f x xs)))\n\n-- Reverse properties\n\nreverse-[x]\u2261[x] : \u2200 x \u2192 reverse (x \u2237 []) \u2261 x \u2237 []\nreverse-[x]\u2261[x] x =\n  rev (x \u2237 []) [] \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 []) \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []          \u220e\n\nrev-++-commute : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys \u2192 rev xs ys \u2261 rev xs [] ++ ys\nrev-++-commute lnil ys =\n  rev [] ys       \u2261\u27e8 rev-[] ys \u27e9\n  ys              \u2261\u27e8 sym $ ++-leftIdentity ys \u27e9\n  [] ++ ys        \u2261\u27e8 subst (\u03bb t \u2192 [] ++ ys \u2261 t ++ ys) (sym $ rev-[] []) refl \u27e9\n  rev [] [] ++ ys \u220e\n\nrev-++-commute (lcons x {xs} Lxs) ys =\n  rev (x \u2237 xs) ys\n    \u2261\u27e8 rev-\u2237 x xs ys \u27e9\n  rev xs (x \u2237 ys)\n    \u2261\u27e8 rev-++-commute Lxs (x \u2237 ys) \u27e9\n  rev xs [] ++ x \u2237 ys\n    \u2261\u27e8 subst (\u03bb t \u2192 rev xs [] ++ x \u2237 ys \u2261 rev xs [] ++ t)\n             (sym\n               ( (x \u2237 []) ++ ys\n                    \u2261\u27e8 ++-\u2237 x [] ys \u27e9\n                 x \u2237 ([] ++ ys)\n                   \u2261\u27e8 subst (\u03bb t \u2192 x \u2237 ([] ++ ys) \u2261 x \u2237 t)\n                            (++-leftIdentity ys)\n                            refl\n                   \u27e9\n                 x \u2237 ys \u220e\n               )\n             )\n             refl\n    \u27e9\n  rev xs [] ++ (x \u2237 []) ++ ys\n    \u2261\u27e8 sym $ ++-assoc (rev-List Lxs lnil) (x \u2237 []) ys \u27e9\n  (rev xs [] ++ (x \u2237 [])) ++ ys\n    \u2261\u27e8 subst (\u03bb t \u2192 (rev xs [] ++ (x \u2237 [])) ++ ys \u2261 t ++ ys)\n             (sym $ rev-++-commute Lxs (x \u2237 []))\n             refl\n    \u27e9\n  rev xs (x \u2237 []) ++ ys\n    \u2261\u27e8 subst (\u03bb t \u2192 rev xs (x \u2237 []) ++ ys \u2261 t ++ ys)\n             (sym $ rev-\u2237 x xs [])\n             refl\n    \u27e9\n  rev (x \u2237 xs) [] ++ ys \u220e\n\nreverse-++-commute : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192\n                     reverse (xs ++ ys) \u2261 reverse ys ++ reverse xs\nreverse-++-commute {ys = ys} lnil Lys =\n  reverse ([] ++ ys)\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse ([] ++ ys) \u2261 reverse t) (++-[] ys) refl \u27e9\n  reverse ys\n    \u2261\u27e8 sym (++-rightIdentity (reverse-List Lys)) \u27e9\n  reverse ys ++ []\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse ys ++ [] \u2261 reverse ys ++ t) (sym (rev-[] [])) refl \u27e9\n    reverse ys ++ reverse [] \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) lnil =\n  reverse ((x \u2237 xs) ++ [])\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse ((x \u2237 xs) ++ []) \u2261 reverse t)\n             (++-rightIdentity (lcons x Lxs))\n             refl\n    \u27e9\n  reverse (x \u2237 xs)\n    \u2261\u27e8 sym (++-[] (reverse (x \u2237 xs))) \u27e9\n  [] ++ reverse (x \u2237 xs)\n     \u2261\u27e8 subst (\u03bb t \u2192 [] ++ reverse (x \u2237 xs) \u2261 t ++ reverse (x \u2237 xs))\n              (sym (rev-[] []))\n              refl\n     \u27e9\n  reverse [] ++ reverse (x \u2237 xs) \u220e\n\nreverse-++-commute (lcons x {xs} Lxs) (lcons y {ys} Lys) =\n  reverse ((x \u2237 xs) ++ y \u2237 ys)\n    \u2261\u27e8 refl \u27e9\n  rev ((x \u2237 xs) ++ y \u2237 ys) []\n    \u2261\u27e8 subst (\u03bb t \u2192 rev ((x \u2237 xs) ++ y \u2237 ys) [] \u2261 rev t [])\n             (++-\u2237 x xs (y \u2237 ys))\n             refl\n    \u27e9\n  rev (x \u2237 (xs ++ y \u2237 ys)) []\n    \u2261\u27e8 rev-\u2237 x (xs ++ y \u2237 ys) [] \u27e9\n  rev (xs ++ y \u2237 ys) (x \u2237 [])\n    \u2261\u27e8 rev-++-commute (++-List Lxs (lcons y Lys)) (x \u2237 []) \u27e9\n  rev (xs ++ y \u2237 ys) [] ++ (x \u2237 [])\n    \u2261\u27e8 subst (\u03bb t \u2192 rev (xs ++ y \u2237 ys) [] ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n             refl\n             refl\n    \u27e9\n  reverse (xs ++ y \u2237 ys) ++ (x \u2237 [])\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse (xs ++ y \u2237 ys) ++ (x \u2237 []) \u2261 t ++ (x \u2237 []))\n             (reverse-++-commute Lxs (lcons y Lys))\n             refl\n    \u27e9\n  (reverse (y \u2237 ys) ++ reverse xs) ++ x \u2237 []\n    \u2261\u27e8 ++-assoc (reverse-List (lcons y Lys)) (reverse xs) (x \u2237 []) \u27e9\n  reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 []\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ reverse xs ++ x \u2237 [] \u2261\n                    reverse (y \u2237 ys) ++ t ++ x \u2237 [])\n             refl\n             refl\n    \u27e9\n  reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 []\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs [] ++ x \u2237 [] \u2261\n                    reverse (y \u2237 ys) ++ t)\n             (sym $ rev-++-commute Lxs (x \u2237 []))\n             refl\n    \u27e9\n  reverse (y \u2237 ys) ++ rev xs (x \u2237 [])\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev xs (x \u2237 []) \u2261\n                    reverse (y \u2237 ys) ++ t)\n             (sym $ rev-\u2237 x xs [])\n             refl\n    \u27e9\n  reverse (y \u2237 ys) ++ rev (x \u2237 xs) []\n    \u2261\u27e8 subst (\u03bb t \u2192 reverse (y \u2237 ys) ++ rev (x \u2237 xs) [] \u2261\n                    reverse (y \u2237 ys) ++ t)\n             refl\n             refl\n    \u27e9\n  reverse (y \u2237 ys) ++ reverse (x \u2237 xs) \u220e\n\nreverse-\u2237 : \u2200 x {ys} \u2192 List ys \u2192\n            reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\nreverse-\u2237 x lnil =\n  rev (x \u2237 []) []\n    \u2261\u27e8 rev-\u2237 x [] [] \u27e9\n  rev [] (x \u2237 [])\n    \u2261\u27e8 rev-[] (x \u2237 []) \u27e9\n  x \u2237 []\n    \u2261\u27e8 sym (++-leftIdentity (x \u2237 [])) \u27e9\n  [] ++ x \u2237 []\n    \u2261\u27e8 subst (\u03bb p \u2192 [] ++ x \u2237 [] \u2261 p ++ x \u2237 []) (sym (rev-[] [])) refl \u27e9\n  rev [] [] ++ x \u2237 [] \u220e\n\nreverse-\u2237 x (lcons y {ys} Lys) = sym\n  ( reverse (y \u2237 ys) ++ x \u2237 []\n      \u2261\u27e8 subst (\u03bb p \u2192 reverse (y \u2237 ys) ++ x \u2237 [] \u2261 reverse (y \u2237 ys) ++ p)\n               (sym (reverse-[x]\u2261[x] x))\n               refl\n      \u27e9\n    (reverse (y \u2237 ys) ++ reverse (x \u2237 []))\n      \u2261\u27e8 sym (reverse-++-commute (lcons x lnil) (lcons y Lys)) \u27e9\n    reverse ((x \u2237 []) ++ (y \u2237 ys))\n      \u2261\u27e8 subst (\u03bb p \u2192 reverse ((x \u2237 []) ++ (y \u2237 ys)) \u2261 reverse p)\n               (++-\u2237 x [] (y \u2237 ys))\n               refl\n      \u27e9\n    reverse (x \u2237 ([] ++ (y \u2237 ys)))\n      \u2261\u27e8 subst (\u03bb p \u2192 reverse (x \u2237 ([] ++ (y \u2237 ys))) \u2261 reverse (x \u2237 p))\n               (++-leftIdentity (y \u2237 ys))\n               refl\n      \u27e9\n    reverse (x \u2237 y \u2237 ys) \u220e\n  )\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b1f077a5b7a2cfcde79aee15b99e3aa6de9e0c42","subject":"Maybe monads and applicative","message":"Maybe monads and applicative\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n            {A : Set a} {B : Set b} {C : Set c} \u2192\n            (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = M?.join _\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {A} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 m n A \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\njust?\u2192IsJust : \u2200 {A : Set} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\nAny-join? : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : Set a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {A : Set} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : Set} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : Set a} {B : Set b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n         (f g : A \u2192? B) \u2192 Set _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\nf \u2218? g = join? \u2218 map? f \u2218 g\n\n\u2218?-just : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 Set (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 Set (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : Set a} {B : Set b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 Set _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : Set a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : Set a} {B : Set b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : Set a} {B : Set b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : Set a} {B : Set b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 Set \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \u22a5-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : Set a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : Set a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 Set \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : Set a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Maybe.NP where\n\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Binary.Logical\nopen import Function using (type-signature;_$_;flip;id)\nopen import Data.Empty using (\u22a5)\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : Set a) (B : A \u2192 Set b) \u2192 Set _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\n_\u2192?_ : \u2200 {a b} \u2192 Set a \u2192 Set b \u2192 Set _\nA \u2192? B = A \u2192 Maybe B\n\nmap? : \u2200 {a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n-- map? = M?._<$>_ _ <= not universe-polymorphic enough\n\n-- more universe-polymorphic than M?.join\njoin? : \u2200 {a} {A : Set a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? nothing  = nothing\njoin? (just x) = x\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B : Set \u2113} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n  \u27ea f \u00b7 x \u27eb = map? f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C : Set \u2113} \u2192\n              (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map? f x \u229b y\n\n  \u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D : Set \u2113} \u2192 (A \u2192 B \u2192 C \u2192 D)\n              \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n  \u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map? f x \u229b y \u229b z\n\nMaybe^ : \u2115 \u2192 Set \u2192 Set\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {A} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 m n A \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : Set a} {x y : A}\n                 \u2192 (just x \u2236 Maybe A) \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : Set a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\napplicative : \u2200 {f} \u2192 RawApplicative {f} Maybe\napplicative = record { pure = just ; _\u229b_  = _\u229b_ }\n  where\n    _\u229b_ : \u2200 {a b}{A : Set a}{B : Set b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n    just f  \u229b just x = just (f x)\n    _       \u229b _      = nothing\n\n_\u2261JAll_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : Set a} (x y : Maybe A) \u2192 Set a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : Set a} \u2192 Maybe A \u2192 Set\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\njust?\u2192IsJust : \u2200 {A : Set} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata \u27e6Maybe\u27e7 {a b r} {A : Set a} {B : Set b} (_\u223c_ : A \u2192 B \u2192 Set r) : Maybe A \u2192 Maybe B \u2192 Set (a \u2294 b \u2294 r) where\n  \u27e6just\u27e7 : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  \u27e6nothing\u27e7 : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (\u27e6just\u27e7 x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 \u27e6nothing\u27e7   = nothing\u1d63\n\nAny-join? : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {A : Set} {P : A \u2192 Set} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : Set a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {A : Set} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : Set} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : Set a} {B : Set b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n         (f g : A \u2192? B) \u2192 Set _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\nf \u2218? g = join? \u2218 map? f \u2218 g\n\n\u2218?-just : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 Set (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 Set (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : Set a} {B : Set b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : Set a} {B : Set b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 Set _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : Set a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : Set a} {B : Set b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : Set a} {B : Set b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : Set a} {B : Set b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a b c d} \u2192 (\u27e6Set\u27e7 {a} {b} c \u27e6\u2192\u27e7 \u27e6Set\u27e7 {a} {b} d \u27e6\u2192\u27e7 \u27e6Set\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a} {A : Set a} {_\u223c_ : A \u2192 A \u2192 Set} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = \u27e6just\u27e7 (refl-A x)\n  refl refl-A nothing = \u27e6nothing\u27e7\n\n  sym : \u2200 {a b} {A : Set a} {B : Set b} {_\u223c\u2081_ : A \u2192 B \u2192 Set} {_\u223c\u2082_ : B \u2192 A \u2192 Set}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (\u27e6just\u27e7 x\u223c\u2081y) = \u27e6just\u27e7 (sym-A x\u223c\u2081y)\n  sym sym-A \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  trans : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 Set}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 Set}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 Set}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (\u27e6just\u27e7 x\u223cy) (\u27e6just\u27e7 y\u223cz) = \u27e6just\u27e7 (trans' x\u223cy y\u223cz)\n  trans trans' \u27e6nothing\u27e7 \u27e6nothing\u27e7 = \u27e6nothing\u27e7\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b} {A : Set a} {B : Set b}\n                 (P : Maybe A \u2192 Set)\n                 (Q : Maybe B \u2192 Set)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 Set)\n                 (subst-\u27e6AB\u27e7 : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7 _ (\u27e6just\u27e7 x\u223cy) Pmx = subst-\u27e6AB\u27e7 x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _          f \u27e6nothing\u27e7    Pnothing = f Pnothing\n\n  subst : \u2200 {a} {A : Set a}\n            (P : Maybe A \u2192 Set)\n            (A\u1d63 : A \u2192 A \u2192 Set)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : Set a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\nmodule MonadLemmas where\n\n  open M? L.zero public\n --  open RawApplicative applicative public\n  cong-Maybe : \u2200 {A B : Set}\n  -- cong-Maybe : \u2200 {a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C : Set}\n               (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule FunctorLemmas where\n  open M? L.zero\n\n  <$>-injective\u2081 : \u2200 {A B : Set}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C : Set} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 Set \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \u22a5-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : Set a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : Set a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 Set \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : Set a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dc6885dcf621e8d07fba99bdff744ddbde19c1d3","subject":"Added Sx-Sy<Sx.","message":"Added Sx-Sy<Sx.\n\nIgnore-this: 66e252433433d7075bf5a827ba62277b\n\ndarcs-hash:20100531151012-3bd4e-81274b7f2bccd69b20906abccbe29eb363f4007a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Equalities.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\npostulate\n  \u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x x\u22650 #-}\n\npostulate\n  \u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n{-# ATP prove \u00acS\u22640 #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-00\nx\u2264x (sN {m} Nm) = prf (x\u2264x Nm)\n  where\n    postulate prf : LE m m \u2192 LE (succ m) (succ m)\n    {-# ATP prove prf #-}\n\n-- TODO: Why not a dot pattern?\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  prf (\u2264-trans Nm Nn No m\u2264n n\u2264o)\n    where\n      postulate m\u2264n : LE m n\n      {-# ATP prove m\u2264n #-}\n\n      postulate n\u2264o : LE n o\n      {-# ATP prove n\u2264o #-}\n\n      postulate prf : LE m o \u2192 LE (succ m) (succ o)\n      {-# ATP prove prf #-}\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\npostulate\n  Sx-Sy<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (succ m - succ n) (succ m)\n{-# ATP prove Sx-Sy<Sx minus-SS x-y<Sx #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x<Sy Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LT m (succ n) \u2192 LT (succ m) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx<Sy\u2192x<y\u2228x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim (\u00acx<0 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ n))) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 sN #-}\n\npostulate\n  \u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 sN #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Equalities.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n0\u2264x : {n : D} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n\u00acx<0 : {n : D} \u2192 N n \u2192 \u00ac (LT n zero)\n\u00acx<0 zN 0<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<0 (sN Nn) Sn<0 = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\npostulate\n  \u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n{-# ATP prove \u00ac0>x x\u22650 #-}\n\npostulate\n  \u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n{-# ATP prove \u00acS\u22640 #-}\n\nx<Sx : {n : D} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = lt-0S zero\nx<Sx (sN {n} Nn) = prf (x<Sx Nn)\n  where\n    postulate prf : LT n (succ n) \u2192 LT (succ n) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n    postulate prf : \u00ac (LT m m) \u2192 \u22a5\n    {-# ATP prove prf #-}\n\nx\u2264x : {m : D} \u2192 N m \u2192 LE m m\nx\u2264x zN          = lt-00\nx\u2264x (sN {m} Nm) = prf (x\u2264x Nm)\n  where\n    postulate prf : LE m m \u2192 LE (succ m) (succ m)\n    {-# ATP prove prf #-}\n\n-- TODO: Why not a dot pattern?\nx\u2261y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y Nm Nn refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = lt-S0 n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE m n \u2192\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx<y\u2192Sx\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm zN m<0 = \u22a5-elim (\u00acx<0 Nm m<0)\n\nx<y\u2192Sx\u2264y zN (sN zN) _ = S0\u2264S0\n  where\n    postulate S0\u2264S0 : LE (succ zero) (succ zero)\n    {-# ATP prove S0\u2264S0 #-}\n\nx<y\u2192Sx\u2264y zN (sN (sN {n} Nn)) _ = S0\u2264SSn\n  where\n    postulate S0\u2264SSn : LE (succ zero) (succ (succ n))\n    {-# ATP prove S0\u2264SSn #-}\n\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192Sx\u2264y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LE (succ m) n \u2192 LE (succ (succ m)) (succ n)\n    {-# ATP prove prf #-}\n\nSx\u2264y\u2192x<y : {m n : D} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim (\u00acS\u22640 Sm\u22640)\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = lt-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn = prf (Sx\u2264y\u2192x<y Nm Nn Sm\u2264n)\n  where\n    postulate Sm\u2264n : LE (succ m) n\n    {-# ATP prove Sm\u2264n #-}\n\n    postulate prf : LT m n \u2192 LT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN           _          0<0   _    = \u22a5-elim (\u00acx<0 zN 0<0)\n<-trans zN          (sN Nn)     zN          _     Sn<0 = \u22a5-elim (\u00acx<0 (sN Nn) Sn<0)\n<-trans zN          (sN Nn)     (sN {o} No) _     _    = lt-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0  = \u22a5-elim (\u00acx<0 Nn n<0)\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _    = \u22a5-elim (\u00acx<0 (sN Nm) Sm<0)\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  prf (<-trans Nm Nn No m<n n<o)\n\n  where\n    postulate prf : LT m o \u2192 LT (succ m) (succ o)\n    {-# ATP prove prf #-}\n\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate n<o : LT n o\n    {-# ATP prove n<o #-}\n\n\u2264-trans : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim (\u00acS\u22640 Sn\u22640)\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  prf (\u2264-trans Nm Nn No m\u2264n n\u2264o)\n    where\n      postulate m\u2264n : LE m n\n      {-# ATP prove m\u2264n #-}\n\n      postulate n\u2264o : LE n o\n      {-# ATP prove n\u2264o #-}\n\n      postulate prf : LE m o \u2192 LE (succ m) (succ o)\n      {-# ATP prove prf #-}\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : {m n : D} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n    postulate prf : LE m (m + n) \u2192 LE (succ m) (succ m + n)\n    {-# ATP prove prf #-}\n\nx-y<Sx : {m n : D} \u2192 N m \u2192 N n \u2192 LT (m - n) (succ m)\nx-y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m - zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx-y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero - succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx-y<Sx (sN {m} Nm) (sN {n} Nn) = prf (x-y<Sx Nm Nn)\n  where\n    postulate prf : LT (m - n) (succ m) \u2192\n                    LT (succ m - succ n) (succ (succ m))\n    {-# ATP prove prf <-trans minus-N x<Sx sN #-}\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m - zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m - n) + n \u2261 m \u2192\n                    (succ m - succ n) + succ n \u2261 succ m\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx\u2264y\u2192y-x+x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n - m) + m \u2261 n\nx\u2264y\u2192y-x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n - zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx\u2264y\u2192y-x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim (\u00acS\u22640 Sm\u22640)\n\nx\u2264y\u2192y-x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y-x+x\u2261y Nm Nn m\u2264n )\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n - m) + m \u2261 n \u2192\n                    (succ n - succ m) + succ m \u2261 succ n\n    {-# ATP prove prf +-comm minus-N sN #-}\n\nx<y\u2192x<Sy : {m n : D} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim (\u00acx<0 Nm m<0)\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = lt-0S (succ n)\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn = prf (x<y\u2192x<Sy Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prf : LT m (succ n) \u2192 LT (succ m) (succ (succ n))\n    {-# ATP prove prf #-}\n\nx<Sy\u2192x<y\u2228x\u2261y : {m n : D} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (lt-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim (\u00acx<0 Nm (trans (sym (lt-SS m zero)) Sm<S0))\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (lt-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym (lt-SS m (succ n))) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  \u00acxy<00 : {m n : D} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n{-# ATP prove \u00acxy<00 \u00acx<0 #-}\n\npostulate\n  \u00acSxy\u2081<0y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n{-# ATP prove \u00acSxy\u2081<0y\u2082 \u00acx<0 sN #-}\n\npostulate\n  \u00ac0Sx<00 : {m : D} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n{-# ATP prove \u00ac0Sx<00 \u00acx<0 sN #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : {m\u2081 n m\u2082 : D} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 \u00acx<0 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : {m n\u2081 n\u2082 : D} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 \u00acx<0 #-}\n\n[Sx-Sy,Sy]<[Sx,Sy] :\n  {m n : D} \u2192 N m \u2192 N n \u2192\n  LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n[Sx-Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m - succ n) (succ n) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n\n[Sx,Sy-Sx]<[Sx,Sy] : {m n : D} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n[Sx,Sy-Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n - succ m) (succ m) (succ n)\n    {-# ATP prove prf sN x-y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9b763493d067a9d574644c9fdbeec64fc16872f0","subject":"ZK.JSChecker.Proof: Fix imports","message":"ZK.JSChecker.Proof: Fix imports\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker\/Proof.agda","new_file":"ZK\/JSChecker\/Proof.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import ZK.Types using (PoK-CP-EG-rnd; module PoK-CP-EG-rnd)\nopen import FFI.JS.BigI using (BigI)\nopen import Data.Product using (_,_)\nopen import Crypto.JS.BigI.FiniteField\nopen import Crypto.JS.BigI.CyclicGroup\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; refl)\nimport ZK.JSChecker\nimport ZK.GroupHom.ElGamal.JS\n\nmodule ZK.JSChecker.Proof (p q : BigI) (pok : PoK-CP-EG-rnd \u2124[ q ] \u2124[ p ]\u2605) where\n\nopen module ^-h = ^-hom p {q} using (_^_)\nopen module pok = PoK-CP-EG-rnd pok\n\npf : ZK.GroupHom.ElGamal.JS.known-enc-rnd.verify-transcript p q g y (g ^ m) (\u03b1 , \u03b2) (A , B) (\u2124q\u25b9BigI q c) s\n   \u2261 ZK.JSChecker.verify-PoK-CP-EG-rnd p q pok\npf = refl\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import ZK.JSChecker using (verify-PoK-CP-EG-rnd; PoK-CP-EG-rnd; module PoK-CP-EG-rnd)\nopen import FFI.JS.BigI using (BigI)\nopen import Data.Product using (_,_)\nopen import Crypto.JS.BigI.FiniteField\nopen import Crypto.JS.BigI.CyclicGroup\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; refl)\nimport ZK.GroupHom.ElGamal.JS\n\nmodule ZK.JSChecker.Proof (p q : BigI) (pok : PoK-CP-EG-rnd \u2124[ q ] \u2124[ p ]\u2605) where\n\nopen module ^-h = ^-hom p {q} using (_^_)\nopen module pok = PoK-CP-EG-rnd pok\n\npf : ZK.GroupHom.ElGamal.JS.known-enc-rnd.verify-transcript p q g y (g ^ m) (\u03b1 , \u03b2) (A , B) (\u2124q\u25b9BigI q c) s\n   \u2261 verify-PoK-CP-EG-rnd p q pok\npf = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"74013d8d747c5699c1d1c971faebd303e0d9e1c3","subject":"Made prettier syntax for constructing the equations","message":"Made prettier syntax for constructing the equations\n","repos":"crypto-agda\/agda-nplib","old_file":"experiment\/liftVec.agda","new_file":"experiment\/liftVec.agda","new_contents":"open import Function\nopen import Data.Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin using (Fin)\nopen import Data.Vec\nopen import Data.Vec.N-ary.NP\nopen import Relation.Binary.PropositionalEquality\n\nmodule liftVec where\nmodule single (A : Set)(_+_ : A \u2192 A \u2192 A)where\n\n  data Tm : Set where\n    var : Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  eval : Tm \u2192 A \u2192 A\n  eval var x = x\n  eval (cst c) x = c\n  eval (fun tm tm') x = eval tm x + eval tm' x\n\n  veval : {m : \u2115} \u2192 Tm \u2192 Vec A m \u2192 Vec A m\n  veval var xs = xs\n  veval (cst x) xs = replicate x\n  veval (fun tm tm') xs = zipWith _+_ (veval tm xs) (veval tm' xs)\n\n  record Fm : Set where\n    constructor _=='_\n    field\n      lhs : Tm\n      rhs : Tm\n\n  s[_] : Fm \u2192 Set\n  s[ lhs ==' rhs ] = (x : A) \u2192 eval lhs x \u2261 eval rhs x\n\n  v[_] : Fm \u2192 Set\n  v[ lhs ==' rhs ] = {m : \u2115}(xs : Vec A m) \u2192 veval lhs xs \u2261 veval rhs xs\n\n  lemma1 : (t : Tm) \u2192 veval t [] \u2261 []\n  lemma1 var = refl\n  lemma1 (cst x) = refl\n  lemma1 (fun t t') rewrite lemma1 t | lemma1 t' = refl\n\n  lemma2 : {m : \u2115}(x : A)(xs : Vec A m)(t : Tm) \u2192 veval t (x \u2237 xs) \u2261 eval t x \u2237 veval t xs\n  lemma2 _ _ var = refl\n  lemma2 _ _ (cst c) = refl\n  lemma2 x xs (fun t t') rewrite lemma2 x xs t | lemma2 x xs t' = refl\n\n  prf : (t : Fm) \u2192 s[ t ] \u2192 v[ t ]\n  prf (lhs ==' rhs) pr [] rewrite lemma1 lhs | lemma1 rhs = refl\n  prf (l ==' r) pr (x \u2237 xs) rewrite lemma2 x xs l | lemma2 x xs r = cong\u2082 (_\u2237_) (pr x) (prf (l ==' r) pr xs)\n\nmodule MultiDataType {A : Set} {nrVar : \u2115} where\n  data Tm : Set where\n    var : Fin nrVar \u2192 Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  infixl 6 _+'_\n  _+'_ : Tm \u2192 Tm \u2192 Tm\n  _+'_ = fun  \n\n  infix 4 _\u2261'_\n  record TmEq : Set where\n    constructor _\u2261'_ \n    field \n      lhs rhs : Tm\n\nmodule multi (A : Set)(_+_ : A \u2192 A \u2192 A)(nrVar : \u2115) where\n  open MultiDataType {A} {nrVar} public\n\n  sEnv : Set\n  sEnv = Vec A nrVar\n\n  vEnv : \u2115 \u2192 Set\n  vEnv m = Vec (Vec A m) nrVar\n\n  Var : Set\n  Var = Fin nrVar\n\n  _!_ : {X : Set} \u2192 Vec X nrVar \u2192 Var \u2192 X\n  E ! v = lookup v E\n\n  eval : Tm \u2192 sEnv \u2192 A\n  eval (var x) G = G ! x\n  eval (cst c) G = c\n  eval (fun t t') G = eval t G + eval t' G\n\n  veval : {m : \u2115} \u2192 Tm \u2192 vEnv m \u2192 Vec A m\n  veval (var x) G = G ! x\n  veval (cst c) G = replicate c\n  veval (fun t t') G = zipWith _+_ (veval t G) (veval t' G)\n\n  lemma1 : {xs : Vec (Vec A 0) nrVar} (t : Tm) \u2192 veval t xs \u2261 []\n  lemma1 {xs} (var x) with xs ! x\n  ... | [] = refl\n  lemma1 (cst x) = refl\n  lemma1 {xs} (fun t t') rewrite lemma1 {xs} t | lemma1 {xs} t' = refl\n\n  lemVar : {m n : \u2115}(G : Vec (Vec A (suc m)) n)(i : Fin n) \u2192 lookup i G \u2261 lookup i (map head G) \u2237 lookup i (map tail G)\n  lemVar [] ()\n  lemVar ((x \u2237 G) \u2237 G\u2081) Data.Fin.zero = refl\n  lemVar (G \u2237 G') (Data.Fin.suc i) = lemVar G' i\n\n  lemma2 : {n : \u2115}(xs : vEnv (suc n))(t : Tm) \u2192 veval t xs \u2261 eval t (map head xs) \u2237 veval t (map tail xs)\n  lemma2 G (var x) = lemVar G x\n  lemma2 _ (cst x) = refl\n  lemma2 G (fun t t') rewrite lemma2 G t | lemma2 G t' = refl\n\n  s[_==_] : Tm \u2192 Tm \u2192 Set\n  s[ l == r ] = (G : Vec A nrVar) \u2192 eval l G \u2261 eval r G\n\n  v[_==_] : Tm \u2192 Tm \u2192 Set\n  v[ l == r ] = {m : \u2115}(G : Vec (Vec A m) nrVar) \u2192 veval l G \u2261 veval r G\n\n  prf : (l r : Tm) \u2192 s[ l == r ] \u2192 v[ l == r ]\n  prf l r pr {zero} G rewrite lemma1 {G} l | lemma1 {G} r = refl\n  prf l r pr {suc m} G rewrite lemma2 G l | lemma2 G r\n     = cong\u2082 _\u2237_ (pr (replicate head \u229b G)) (prf l r pr (map tail G))\n\nmodule Full (A : Set)(_+_ : A \u2192 A \u2192 A) (m : \u2115) n where\n  open multi A _+_ n public\n\n  solve : \u2200 (l r : Tm) \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  solve l r x = curry\u207f {n} {A = Vec A m} {B = curry\u207f\u2032 f}\n        (\u03bb xs \u2192 subst id (sym (curry-$\u207f\u2032 f xs))\n                   (prf l r (\u03bb G \u2192 subst id (curry-$\u207f\u2032 g G) (x $\u207f G)) xs))\n    where f = \u03bb G \u2192 veval {m} l G \u2261 veval r G\n          g = \u03bb G \u2192 eval l G \u2261 eval r G\n\n  mkTm : N-ary n Tm TmEq \u2192 TmEq\n  mkTm = go (tabulate id) where\n    go : {m : \u2115} -> Vec (Fin n) m \u2192 N-ary m Tm TmEq \u2192 TmEq\n    go [] f         = f\n    go (x \u2237 args) f = go args (f (var x))\n\n  prove :  \u2200 (t : N-ary n Tm TmEq) \u2192 \n           let \n             l = TmEq.lhs (mkTm t)\n             r = TmEq.rhs (mkTm t)\n           in \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  prove t x = solve (TmEq.lhs (mkTm t)) (TmEq.rhs (mkTm t)) x\n\nopen import Data.Bool\n\nmodule example where\n\n  open import Data.Fin\n  open import Data.Bool.NP\n  open import Data.Product using (\u03a3 ; proj\u2081)\n\n  module VecBoolXorProps {m} = Full Bool _xor_ m\n  open MultiDataType\n\n  coolTheorem : {m : \u2115} \u2192 (xs ys : Vec Bool m) \u2192 zipWith _xor_ xs ys \u2261 zipWith _xor_ ys xs\n  coolTheorem = VecBoolXorProps.prove 2 (\u03bb x y \u2192 x +' y \u2261' y +' x) Xor\u00b0.+-comm \n\n  coolTheorem2 : {m : \u2115} \u2192 (xs : Vec Bool m) \u2192 _\n  coolTheorem2 = VecBoolXorProps.prove 1 (\u03bb x \u2192 (x +' x) \u2261' (cst false)) (proj\u2081 Xor\u00b0.-\u203finverse) \n \ntest = example.coolTheorem (true \u2237 false \u2237 []) (false \u2237 false \u2237 [])","old_contents":"open import Function\nopen import Data.Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin using (Fin)\nopen import Data.Vec\nopen import Data.Vec.N-ary.NP\nopen import Relation.Binary.PropositionalEquality\n\nmodule liftVec where\nmodule single (A : Set)(_+_ : A \u2192 A \u2192 A)where\n\n  data Tm : Set where\n    var : Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\n  eval : Tm \u2192 A \u2192 A\n  eval var x = x\n  eval (cst c) x = c\n  eval (fun tm tm') x = eval tm x + eval tm' x\n\n  veval : {m : \u2115} \u2192 Tm \u2192 Vec A m \u2192 Vec A m\n  veval var xs = xs\n  veval (cst x) xs = replicate x\n  veval (fun tm tm') xs = zipWith _+_ (veval tm xs) (veval tm' xs)\n\n  record Fm : Set where\n    constructor _=='_\n    field\n      lhs : Tm\n      rhs : Tm\n\n  s[_] : Fm \u2192 Set\n  s[ lhs ==' rhs ] = (x : A) \u2192 eval lhs x \u2261 eval rhs x\n\n  v[_] : Fm \u2192 Set\n  v[ lhs ==' rhs ] = {m : \u2115}(xs : Vec A m) \u2192 veval lhs xs \u2261 veval rhs xs\n\n  lemma1 : (t : Tm) \u2192 veval t [] \u2261 []\n  lemma1 var = refl\n  lemma1 (cst x) = refl\n  lemma1 (fun t t') rewrite lemma1 t | lemma1 t' = refl\n\n  lemma2 : {m : \u2115}(x : A)(xs : Vec A m)(t : Tm) \u2192 veval t (x \u2237 xs) \u2261 eval t x \u2237 veval t xs\n  lemma2 _ _ var = refl\n  lemma2 _ _ (cst c) = refl\n  lemma2 x xs (fun t t') rewrite lemma2 x xs t | lemma2 x xs t' = refl\n\n  prf : (t : Fm) \u2192 s[ t ] \u2192 v[ t ]\n  prf (lhs ==' rhs) pr [] rewrite lemma1 lhs | lemma1 rhs = refl\n  prf (l ==' r) pr (x \u2237 xs) rewrite lemma2 x xs l | lemma2 x xs r = cong\u2082 (_\u2237_) (pr x) (prf (l ==' r) pr xs)\n\nmodule MultiDataType (A : Set) (nrVar : \u2115) where\n  data Tm : Set where\n    var : Fin nrVar \u2192 Tm\n    cst : A \u2192 Tm\n    fun : Tm \u2192 Tm \u2192 Tm\n\nmodule multi (A : Set)(_+_ : A \u2192 A \u2192 A)(nrVar : \u2115) where\n  open MultiDataType A nrVar public\n\n  sEnv : Set\n  sEnv = Vec A nrVar\n\n  vEnv : \u2115 \u2192 Set\n  vEnv m = Vec (Vec A m) nrVar\n\n  Var : Set\n  Var = Fin nrVar\n\n  _!_ : {X : Set} \u2192 Vec X nrVar \u2192 Var \u2192 X\n  E ! v = lookup v E\n\n  eval : Tm \u2192 sEnv \u2192 A\n  eval (var x) G = G ! x\n  eval (cst c) G = c\n  eval (fun t t') G = eval t G + eval t' G\n\n  veval : {m : \u2115} \u2192 Tm \u2192 vEnv m \u2192 Vec A m\n  veval (var x) G = G ! x\n  veval (cst c) G = replicate c\n  veval (fun t t') G = zipWith _+_ (veval t G) (veval t' G)\n\n  lemma1 : {xs : Vec (Vec A 0) nrVar} (t : Tm) \u2192 veval t xs \u2261 []\n  lemma1 {xs} (var x) with xs ! x\n  ... | [] = refl\n  lemma1 (cst x) = refl\n  lemma1 {xs} (fun t t') rewrite lemma1 {xs} t | lemma1 {xs} t' = refl\n\n  lemVar : {m n : \u2115}(G : Vec (Vec A (suc m)) n)(i : Fin n) \u2192 lookup i G \u2261 lookup i (map head G) \u2237 lookup i (map tail G)\n  lemVar [] ()\n  lemVar ((x \u2237 G) \u2237 G\u2081) Data.Fin.zero = refl\n  lemVar (G \u2237 G') (Data.Fin.suc i) = lemVar G' i\n\n  lemma2 : {n : \u2115}(xs : vEnv (suc n))(t : Tm) \u2192 veval t xs \u2261 eval t (map head xs) \u2237 veval t (map tail xs)\n  lemma2 G (var x) = lemVar G x\n  lemma2 _ (cst x) = refl\n  lemma2 G (fun t t') rewrite lemma2 G t | lemma2 G t' = refl\n\n  s[_==_] : Tm \u2192 Tm \u2192 Set\n  s[ l == r ] = (G : Vec A nrVar) \u2192 eval l G \u2261 eval r G\n\n  v[_==_] : Tm \u2192 Tm \u2192 Set\n  v[ l == r ] = {m : \u2115}(G : Vec (Vec A m) nrVar) \u2192 veval l G \u2261 veval r G\n\n  prf : (l r : Tm) \u2192 s[ l == r ] \u2192 v[ l == r ]\n  prf l r pr {zero} G rewrite lemma1 {G} l | lemma1 {G} r = refl\n  prf l r pr {suc m} G rewrite lemma2 G l | lemma2 G r\n     = cong\u2082 _\u2237_ (pr (replicate head \u229b G)) (prf l r pr (map tail G))\n\nmodule Full (A : Set)(_+_ : A \u2192 A \u2192 A) (m : \u2115) n where\n  open multi A _+_ n public\n\n  solve : \u2200 (l r : Tm) \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 eval l G \u2261 eval r G))\n         \u2192 \u2200\u207f n (curry\u207f\u2032 (\u03bb G \u2192 veval {m} l G \u2261 veval r G))\n  solve l r x = curry\u207f {n} {A = Vec A m} {B = curry\u207f\u2032 f}\n        (\u03bb xs \u2192 subst id (sym (curry-$\u207f\u2032 f xs))\n                   (prf l r (\u03bb G \u2192 subst id (curry-$\u207f\u2032 g G) (x $\u207f G)) xs))\n    where f = \u03bb G \u2192 veval {m} l G \u2261 veval r G\n          g = \u03bb G \u2192 eval l G \u2261 eval r G\n\nopen import Data.Bool\n\nmodule example where\n\n  open import Data.Fin\n  open import Data.Bool.NP\n\n  module VecBoolXorProps m n = Full Bool _xor_ m n\n  open MultiDataType\n\n  coolTheorem : {m : \u2115} \u2192 (xs ys : Vec Bool m) \u2192 zipWith _xor_ xs ys \u2261 zipWith _xor_ ys xs\n  coolTheorem {m} = VecBoolXorProps.solve m 2 l r Xor\u00b0.+-comm where\n    l = fun (var zero) (var (suc zero))\n    r = fun (var (suc zero)) (var zero)\n\ntest = example.coolTheorem (true \u2237 false \u2237 []) (false \u2237 false \u2237 [])\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"319d6c9830ece047a3ae1b3049650092546e4a34","subject":"Push let bindings down.","message":"Push let bindings down.\n\nOld-commit-hash: 4efeded1434c1c2c008cb76c7971482c7c66fff7\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    (let\n       diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n       apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n       _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n       _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n     in\n       abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))\n    ,\n    (let\n       diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n       apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n       _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n       _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n     in\n       abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y)))\n\n  diff-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  diff-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  apply-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  apply-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure\n    (diff-base : DiffStructure)\n    (apply-base : ApplyStructure)\n  where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply {base \u03b9} = diff-base , apply-base\n  lift-diff-apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c3 {\u0393}) s t\n      _\u229d\u03c4_ = \u03bb {\u0393} s t  \u2192 app\u2082 (diff\u03c4 {\u0393}) s t\n      _\u2295\u03c3_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c3 {\u0393}) \u0394t t\n      _\u2295\u03c4_ = \u03bb {\u0393} t \u0394t \u2192 app\u2082 (apply\u03c4 {\u0393}) \u0394t t\n    in\n      abs\u2084 (\u03bb g f x \u0394x \u2192 app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x)\n      ,\n      abs\u2083 (\u03bb \u0394h h y \u2192 app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))\n\n  diff-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  diff-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply {\u03c4} {\u0393})\n\n  apply-term :\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  apply-term = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply {\u03c4} {\u0393})\n\n  diff : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 (\u0394Type \u03c4)\n  diff = app\u2082 diff-term\n\n  apply : \u2200 {\u03c4 \u0393} \u2192\n    Term \u0393 (\u0394Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  apply = app\u2082 apply-term\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c26185b3c8e3115573f5a605f121b66be94df89c","subject":"one-time-semantic-security: Cleanup","message":"one-time-semantic-security: Cleanup\n","repos":"crypto-agda\/crypto-agda","old_file":"one-time-semantic-security.agda","new_file":"one-time-semantic-security.agda","new_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import flat-funs\nopen import Data.Vec using (splitAt)\nopen import program-distance\nopen import Data.Bits\nopen import flipbased-implem\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n","old_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import flat-funs\nopen import Data.Vec using (splitAt)\nopen import program-distance\nopen import Data.Bits\nopen import flipbased-implem\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nCoins = \u2115\n\n-- module AbsSemSec (|M| |C| : \u2115) {FunPower : Set} {t} {T : Set t}\n--                  (\u266dFuns : PoweredFlatFuns FunPower T) where\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n  -- runAdv is not the good name\n  -- runAdv : \u2200 {p} (A : AbsSemSecAdv p) \u2192 \u27e8 ? \u27e9 (C `\u00d7 AbsSemSecAdv.R A) \u219d (M `\u00d7 M) `\u00d7 Bit\n  -- runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n{-\nmodule FFF |M| |C| {\u266dFuns : FlatFuns \u22a4 Set}\n               (\u266dops : FlatFunsOps _ \u266dFuns)\n               (run : \u2200 {i o} \u2192 FlatFuns.\u27e8_\u27e9_\u219d_ \u266dFuns _ i o \u2192 i \u2192 o) where\n  open AbsSemSec |M| |C| \u266dFuns\n  module FunSemSecAdv {|R|} (A : AbsSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n    open FlatFunsOps \u266dops\n    step\u2080O : \u27e8 _ \u27e9 R \u219d ((\u27e8 _ \u27e9 Bit \u219d M) \u00d7 S)\n    step\u2080O = {!run step\u2080!} >>> {!proj!} *** idO\n-}\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  -- FunSemSecAdv |R| = AbsSemSecAdv (mk _ _ |R|)\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  module Unused (Power : Set) (coins : Power \u2192 Coins) where\n  -- NP: not sure about these two. such a function has acess to the two encryption\n\n    SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n    SemBroken E power = \u2203 (\u03bb (A : FunSemSecAdv (coins power)) \u2192 breaks (E \u21c4 A))\n\n    -- functions.\n    -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n    --   security of E\u2080 reduces to security of E\u2081\n    --   breaking E\u2081 reduces to breaking E\u2080\n    SemSecReduction' : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n    SemSecReduction' p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\n  -- SemSecReductionToOracle : \u2200 (p\u2080 p\u2081 : Power) {cc} (E : Enc cc) \u2192 Set\n  -- SemSecReductionToOracle = SemBroken E p\u2080 \u2192 Oracle p\u2081\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"635b7946ea3c70427f8905228267b18e65128730","subject":"Cleaning.","message":"Cleaning.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/co-data\/Stream\/PropertiesSL.agda","new_file":"notes\/co-data\/Stream\/PropertiesSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Stream properties using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Stream.PropertiesSL where\n\nopen import Data.Nat\nopen import Data.Stream\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\nzeros : Stream \u2115\nzeros = 0 \u2237 \u266f zeros\n\nzeros\u2261zeros : zeros \u2261 zeros\nzeros\u2261zeros = refl\n","old_contents":"------------------------------------------------------------------------------\n-- Stream properties using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Stream.PropertiesSL where\n\nopen import Data.Nat\nopen import Data.Stream\nopen import Coinduction\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\nzeros : Stream \u2115\nzeros = 0 \u2237 \u266f zeros\n\n-- TODO (07 November 2013): Why the identity type works with infinite\n-- lists?\nzeros\u2261zeros : zeros \u2261 zeros\nzeros\u2261zeros = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"28efdb89df293c0268cbfeb25de2d7375a9d6627","subject":"Use more unified data types.","message":"Use more unified data types.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda2\/Text\/Greek\/Smyth.agda","new_file":"agda2\/Text\/Greek\/Smyth.agda","new_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc : OlderLetter\n  \u03d8 : OlderLetter\n  \u03e1 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  \u1fb0 : VowelWithLength \u03b1 short\n  \u1fb1 : VowelWithLength \u03b1 long\n  \u03b5 : VowelWithLength \u03b5 short\n  \u03b7 : VowelWithLength \u03b7 long\n  \u1fd0 : VowelWithLength \u03b9 short\n  \u1fd1 : VowelWithLength \u03b9 long\n  \u03bf : VowelWithLength \u03bf short\n  \u1fe0 : VowelWithLength \u03c5 short\n  \u1fe1 : VowelWithLength \u03c5 long\n  \u03c9 : VowelWithLength \u03c9 long\n\n-- Smyth \u00a75,6\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9-gen : Diphthong \u03b5 \u03b9\n  \u03b5\u03b9-sp : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5-gen : Diphthong \u03bf \u03c5\n  \u03bf\u03c5-sp : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n  \u03c9\u03c5 : Diphthong \u03c9 \u03c5 -- Smyth \u00a75 D. New Ionic\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03c5 = closed\ntongue-position-vowel \u03c9 = open-medium\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5-gen = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata ConsonantSound : Letter \u2192 Set where\n  \u03b2 : ConsonantSound \u03b2\n  \u03b3 : ConsonantSound \u03b3\n  \u03b4 : ConsonantSound \u03b4\n  \u03b6 : ConsonantSound \u03b6\n  \u03b8 : ConsonantSound \u03b8\n  \u03ba : ConsonantSound \u03ba\n  \u03bb\u2032 : ConsonantSound \u03bb\u2032\n  \u03bc : ConsonantSound \u03bc\n  \u03bd : ConsonantSound \u03bd\n  \u03be : ConsonantSound \u03be\n  \u03c0 : ConsonantSound \u03c0\n  \u03c1 : ConsonantSound \u03c1\n  \u03c3 : ConsonantSound \u03c3\n  \u03c4 : ConsonantSound \u03c4\n  \u03c6 : ConsonantSound \u03c6\n  \u03c7 : ConsonantSound \u03c7\n  \u03c8 : ConsonantSound \u03c8\n  \u03b3-nasal : ConsonantSound \u03b3\n  \u03c1\u0314 : ConsonantSound \u03c1\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b3-nasal : Voiced \u03b3-nasal\n  \u03b6 : Voiced \u03b6\n\n-- Smyth \u00a715 a, b\ndata Voiceless : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c1\u0314 : Voiceless \u03c1\u0314\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n\n-- Smyth \u00a716\ndata PartOfMouthClass : Set where\n  labial dental palatal : PartOfMouthClass\n\ndata Labial : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : Labial \u03c0\n  \u03b2 : Labial \u03b2\n  \u03c6 : Labial \u03c6\n\ndata Dental : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c4 : Dental \u03c4\n  \u03b4 : Dental \u03b4\n  \u03b8 : Dental \u03b8\n\ndata Palatal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03ba : Palatal \u03ba\n  \u03b3 : Palatal \u03b3\n  \u03c7 : Palatal \u03c7\n\ndata SmoothStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : SmoothStop \u03c0\n  \u03c4 : SmoothStop \u03c4\n  \u03ba : SmoothStop \u03ba\n\ndata MiddleStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : MiddleStop \u03b2\n  \u03b4 : MiddleStop \u03b4\n  \u03b3 : MiddleStop \u03b3\n\ndata RoughStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c6 : RoughStop \u03c6\n  \u03b8 : RoughStop \u03b8\n  \u03c7 : RoughStop \u03c7\n\n-- Smyth \u00a717\ndata Spirant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c3 : Spirant \u03c3\n\n-- Smyth \u00a718\ndata Liquid : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u2032 : Liquid \u03bb\u2032\n  \u03c1 : Liquid \u03c1\n\n-- Smyth \u00a719\ndata Nasal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bc : Nasal \u03bc\n  \u03bd : Nasal \u03bd\n  \u03b3-nasal : Nasal \u03b3-nasal\n\n-- Smyth \u00a720\ndata Semivowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 Set where\n  \u03b9\u032f : Semivowel \u03b9\n  \u03c5\u032f : Semivowel \u03c5\n\n-- Smyth \u00a720 b\ndata Sonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u0325 : Sonant \u03bb\u2032\n  \u03bc\u0325 : Sonant \u03bc\n  \u03bd\u0325 : Sonant \u03b3\n  \u03c1\u0325 : Sonant \u03c1\n  \u03c3\u0325 : Sonant \u03c3\n\n-- Smyth \u00a721\ndata DoubleConsonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b6 : DoubleConsonant \u03b6\n  \u03be : DoubleConsonant \u03be\n  \u03c8 : DoubleConsonant \u03c8\n\ndata _+_\u21d2DoubleConsonant_ : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u2083} \u2192 ConsonantSound \u2113\u2081 \u2192 ConsonantSound \u2113\u2082 \u2192 ConsonantSound \u2113\u2083 \u2192 Set where\n  \u03c3+\u03b4\u21d2DoubleConsonant\u03b6 : \u03c3 + \u03b4 \u21d2DoubleConsonant \u03b6 -- Smyth \u00a726 D. Aeolic has \u03c3\u03b4 for \u03b6\n  \u03b4+\u03c3\u21d2DoubleConsonant\u03b6 : \u03b4 + \u03c3 \u21d2DoubleConsonant \u03b6\n  \n  \u03ba+\u03c3\u21d2DoubleConsonant\u03be : \u03ba + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03b3+\u03c3\u21d2DoubleConsonant\u03be : \u03b3 + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03c7+\u03c3\u21d2DoubleConsonant\u03be : \u03c7 + \u03c3 \u21d2DoubleConsonant \u03be\n\n  \u03c0+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c0 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03b2+\u03c3\u21d2DoubleConsonant\u03c8 : \u03b2 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03c6+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c6 + \u03c3 \u21d2DoubleConsonant \u03c8\n\n\n-- TODO (needs a unified model)\n\n-- Diphthong\n-- Smyth \u00a75 D. Ionic has \u03b7\u03c5 for Attic \u03b1\u03c5 in some words\n\n-- Accent\n-- Smyth \u00a74. All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78. Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n-- Smyth \u00a719 a. Gamma before \u03ba, \u03b3, \u03c7, \u03be is called \u03b3-nasal\n-- Smyth \u00a720 a. When \u03b9 and \u03c5 correspond to y and w (cp. minion, persuade) they are said to be unsyllabic; and, with a following vowel, make one syllable out of two.\n-- Smyth \u00a720 a. Initial \u03b9\u032f passed into \u0314 (h), as in \u1f27\u03c0\u03b1\u03c1 liver, Lat. jecur; and into \u03b6 in \u03b6\u03c5\u03b3\u03cc\u03bd yoke\n-- Smyth \u00a720 a. Initial \u03c5\u032f was written \u03dd\n-- Smyth \u00a720 a. Medial \u03b9\u032f, \u03c5\u032f before vowels were often lost, as in \u03c4\u1fd1\u03bc\u03ac-(\u03b9\u032f)\u03c9 I honour, \u03b2\u03bf (\u03c5\u032f)-\u03cc\u03c2, gen. of \u03b2\u03bf\u1fe6-\u03c2 ox, cow\n-- Smyth \u00a726. \u03c3 was usually like our sharp s; but before voiced consonants (15 a) it probably was soft, like z; thus we find both \u03ba\u03cc\u03b6\u03bc\u03bf\u03c2 and \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 on inscriptions.\n","old_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\nopen import Relation.Nullary using (\u00ac_)\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1 capital = '\u0391'\nform \u03b1 small = '\u03b1'\nform \u03b2 capital = '\u0392'\nform \u03b2 small = '\u03b2'\nform \u03b3 capital = '\u0393'\nform \u03b3 small = '\u03b3'\nform \u03b4 capital = '\u0394'\nform \u03b4 small = '\u03b4'\nform \u03b5 capital = '\u0395'\nform \u03b5 small = '\u03b5'\nform \u03b6 capital = '\u0396'\nform \u03b6 small = '\u03b6'\nform \u03b7 capital = '\u0397'\nform \u03b7 small = '\u03b7'\nform \u03b8 capital = '\u0398'\nform \u03b8 small = '\u03b8'\nform \u03b9 capital = '\u0399'\nform \u03b9 small = '\u03b9'\nform \u03ba capital = '\u039a'\nform \u03ba small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc capital = '\u039c'\nform \u03bc small = '\u03bc'\nform \u03bd capital = '\u039d'\nform \u03bd small = '\u03bd'\nform \u03be capital = '\u039e'\nform \u03be small = '\u03be'\nform \u03bf capital = '\u039f'\nform \u03bf small = '\u03bf'\nform \u03c0 capital = '\u03a0'\nform \u03c0 small = '\u03c0'\nform \u03c1 capital = '\u03a1'\nform \u03c1 small = '\u03c1'\nform \u03c3 capital = '\u03a3'\nform \u03c3 small = '\u03c3'\nform \u03c4 capital = '\u03a4'\nform \u03c4 small = '\u03c4'\nform \u03c5 capital = '\u03a5'\nform \u03c5 small = '\u03c5'\nform \u03c6 capital = '\u03a6'\nform \u03c6 small = '\u03c6'\nform \u03c7 capital = '\u03a7'\nform \u03c7 small = '\u03c7'\nform \u03c8 capital = '\u03a8'\nform \u03c8 small = '\u03c8'\nform \u03c9 capital = '\u03a9'\nform \u03c9 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  start-of-word : PositionInWord\n  middle-of-word : PositionInWord\n  end-of-word : PositionInWord\n\ndata FinalForm : Set where\n  final-form not-final-form : FinalForm\n\nget-final-form : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 FinalForm\nget-final-form \u03c3\u2032 small end-of-word = final-form\nget-final-form _ _ _ = not-final-form\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc : OlderLetter\n  \u03d8 : OlderLetter\n  \u03e1 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1 : Vowel \u03b1\n  \u03b5 : Vowel \u03b5\n  \u03b7 : Vowel \u03b7\n  \u03b9 : Vowel \u03b9\n  \u03bf : Vowel \u03bf\n  \u03c5 : Vowel \u03c5\n  \u03c9 : Vowel \u03c9\n\ndata AlwaysShort : Letter \u2192 Set where\n  \u03b5 : AlwaysShort \u03b5\n  \u03bf : AlwaysShort \u03bf\n\ndata AlwaysLong : Letter \u2192 Set where\n  \u03b7 : AlwaysLong \u03b7\n  \u03c9 : AlwaysLong \u03c9\n\ndata VowelLength : Set where\n  short long : VowelLength\n\ndata VowelWithLength : \u2200 {v} \u2192 Vowel v \u2192 VowelLength \u2192 Set where\n  always-short : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysShort \u2113 \u2192 VowelWithLength v short\n  always-long : \u2200 {\u2113} \u2192 (v : Vowel \u2113) \u2192 AlwaysLong \u2113 \u2192 VowelWithLength v long\n  \u03b1-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b1 vl\n  \u03b9-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03b9 vl\n  \u03c5-with-length : (vl : VowelLength) \u2192 VowelWithLength \u03c5 vl\n\nalwaysShort-Vowel : \u2200 {\u2113} \u2192 AlwaysShort \u2113 \u2192 Vowel \u2113\nalwaysShort-Vowel \u03b5 = \u03b5\nalwaysShort-Vowel \u03bf = \u03bf\n\nalwaysLong-Vowel : \u2200 {\u2113} \u2192 AlwaysLong \u2113 \u2192 Vowel \u2113\nalwaysLong-Vowel \u03b7 = \u03b7\nalwaysLong-Vowel \u03c9 = \u03c9\n\n-- Smyth \u00a75\ndata Diphthong : Letter \u2192 Letter \u2192 Set where\n  \u03b1\u03b9 : Diphthong \u03b1 \u03b9\n  \u03b5\u03b9 : Diphthong \u03b5 \u03b9\n  \u03bf\u03b9 : Diphthong \u03bf \u03b9\n  \u1fb1\u0345 : Diphthong \u03b1 \u03b9\n  \u1fc3 : Diphthong \u03b7 \u03b9\n  \u1ff3 : Diphthong \u03c9 \u03b9\n  \u03b1\u03c5 : Diphthong \u03b1 \u03c5\n  \u03b5\u03c5 : Diphthong \u03b5 \u03c5\n  \u03bf\u03c5 : Diphthong \u03bf \u03c5\n  \u03b7\u03c5 : Diphthong \u03b7 \u03c5\n  \u03c5\u03b9 : Diphthong \u03c5 \u03b9\n\n-- Smyth \u00a76\ndata SpuriousDiphthong : \u2200 {v\u2081 v\u2082} \u2192 Diphthong v\u2081 v\u2082 \u2192 Set where\n  \u03b5\u03b9-sp : SpuriousDiphthong \u03b5\u03b9\n  \u03bf\u03c5-sp : SpuriousDiphthong \u03bf\u03c5\n\n-- Smyth \u00a77\ndata TonguePosition : Set where\n  closed closed-medium open-medium open\u2032 closing opening : TonguePosition\n\ntongue-position-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 TonguePosition\ntongue-position-vowel \u03b1 = open\u2032\ntongue-position-vowel \u03b5 = closed-medium\ntongue-position-vowel \u03b7 = open-medium\ntongue-position-vowel \u03b9 = closed\ntongue-position-vowel \u03bf = closed-medium\ntongue-position-vowel \u03c5 = closed\ntongue-position-vowel \u03c9 = open-medium\n\ntongue-position-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 TonguePosition\ntongue-position-diphthong d = closing\n\ndata RoundedLips : Set where\n  rounded-lips not-rounded-lips : RoundedLips\n\nrounded-lips-vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 RoundedLips\nrounded-lips-vowel \u03bf = rounded-lips\nrounded-lips-vowel \u03c5 = rounded-lips\nrounded-lips-vowel \u03c9 = rounded-lips\nrounded-lips-vowel _ = not-rounded-lips\n\nrounded-lips-diphthong : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 Diphthong \u2113\u2081 \u2113\u2082 \u2192 RoundedLips\nrounded-lips-diphthong \u03bf\u03c5 = rounded-lips\nrounded-lips-diphthong _ = not-rounded-lips\n\n-- Smyth \u00a79\ndata Breathing : Set where\n  \u1ffe \u1fbf : Breathing\n\n-- Smyth \u00a715\ndata Consonant : Letter \u2192 Set where\n  \u03b2 : Consonant \u03b2\n  \u03b3 : Consonant \u03b3\n  \u03b4 : Consonant \u03b4\n  \u03b6 : Consonant \u03b6\n  \u03b8 : Consonant \u03b8\n  \u03ba : Consonant \u03ba\n  \u03bb\u2032 : Consonant \u03bb\u2032\n  \u03bc : Consonant \u03bc\n  \u03bd : Consonant \u03bd\n  \u03be : Consonant \u03be\n  \u03c0 : Consonant \u03c0\n  \u03c1 : Consonant \u03c1\n  \u03c3 : Consonant \u03c3\n  \u03c4 : Consonant \u03c4\n  \u03c6 : Consonant \u03c6\n  \u03c7 : Consonant \u03c7\n  \u03c8 : Consonant \u03c8\n\ndata LetterCategory : Letter \u2192 Set where\n  vowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 LetterCategory \u2113\n  consonant : \u2200 {\u2113} \u2192 Consonant \u2113 \u2192 LetterCategory \u2113\n\nLetter-to-LetterCategory : (\u2113 : Letter) \u2192 LetterCategory \u2113\nLetter-to-LetterCategory \u03b1 = vowel \u03b1\nLetter-to-LetterCategory \u03b2 = consonant \u03b2\nLetter-to-LetterCategory \u03b3 = consonant \u03b3\nLetter-to-LetterCategory \u03b4 = consonant \u03b4\nLetter-to-LetterCategory \u03b5 = vowel \u03b5\nLetter-to-LetterCategory \u03b6 = consonant \u03b6\nLetter-to-LetterCategory \u03b7 = vowel \u03b7\nLetter-to-LetterCategory \u03b8 = consonant \u03b8\nLetter-to-LetterCategory \u03b9 = vowel \u03b9\nLetter-to-LetterCategory \u03ba = consonant \u03ba\nLetter-to-LetterCategory \u03bb\u2032 = consonant \u03bb\u2032\nLetter-to-LetterCategory \u03bc = consonant \u03bc\nLetter-to-LetterCategory \u03bd = consonant \u03bd\nLetter-to-LetterCategory \u03be = consonant \u03be\nLetter-to-LetterCategory \u03bf = vowel \u03bf\nLetter-to-LetterCategory \u03c0 = consonant \u03c0\nLetter-to-LetterCategory \u03c1 = consonant \u03c1\nLetter-to-LetterCategory \u03c3 = consonant \u03c3\nLetter-to-LetterCategory \u03c4 = consonant \u03c4\nLetter-to-LetterCategory \u03c5 = vowel \u03c5\nLetter-to-LetterCategory \u03c6 = consonant \u03c6\nLetter-to-LetterCategory \u03c7 = consonant \u03c7\nLetter-to-LetterCategory \u03c8 = consonant \u03c8\nLetter-to-LetterCategory \u03c9 = vowel \u03c9\n\nLetterCategory-to-Letter : \u2200 {\u2113} \u2192 LetterCategory \u2113 \u2192 Letter\nLetterCategory-to-Letter {\u2113} _ = \u2113\n\ndata ConsonantSound : Letter \u2192 Set where\n  \u03b2 : ConsonantSound \u03b2\n  \u03b3 : ConsonantSound \u03b3\n  \u03b4 : ConsonantSound \u03b4\n  \u03b6 : ConsonantSound \u03b6\n  \u03b8 : ConsonantSound \u03b8\n  \u03ba : ConsonantSound \u03ba\n  \u03bb\u2032 : ConsonantSound \u03bb\u2032\n  \u03bc : ConsonantSound \u03bc\n  \u03bd : ConsonantSound \u03bd\n  \u03be : ConsonantSound \u03be\n  \u03c0 : ConsonantSound \u03c0\n  \u03c1 : ConsonantSound \u03c1\n  \u03c3 : ConsonantSound \u03c3\n  \u03c4 : ConsonantSound \u03c4\n  \u03c6 : ConsonantSound \u03c6\n  \u03c7 : ConsonantSound \u03c7\n  \u03c8 : ConsonantSound \u03c8\n  \u03b3-nasal : ConsonantSound \u03b3\n  \u03c1\u0314 : ConsonantSound \u03c1\n\ndata VocalChords : Set where\n  voiced voiceless : VocalChords\n\n-- Smyth \u00a715 a\ndata Voiced : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : Voiced \u03b2\n  \u03b4 : Voiced \u03b4\n  \u03b3 : Voiced \u03b3\n  \u03bb\u2032 : Voiced \u03bb\u2032\n  \u03c1 : Voiced \u03c1\n  \u03bc : Voiced \u03bc\n  \u03bd : Voiced \u03bd\n  \u03b3-nasal : Voiced \u03b3-nasal\n  \u03b6 : Voiced \u03b6\n\n-- Smyth \u00a715 a, b\ndata Voiceless : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c1\u0314 : Voiceless \u03c1\u0314\n  \u03c0 : Voiceless \u03c0\n  \u03c4 : Voiceless \u03c4\n  \u03ba : Voiceless \u03ba\n  \u03c6 : Voiceless \u03c6\n  \u03b8 : Voiceless \u03b8\n  \u03c7 : Voiceless \u03c7\n  \u03c3 : Voiceless \u03c3\n  \u03c8 : Voiceless \u03c8\n  \u03be : Voiceless \u03be\n\n-- Smyth \u00a716\ndata PartOfMouthClass : Set where\n  labial dental palatal : PartOfMouthClass\n\ndata Labial : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : Labial \u03c0\n  \u03b2 : Labial \u03b2\n  \u03c6 : Labial \u03c6\n\ndata Dental : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c4 : Dental \u03c4\n  \u03b4 : Dental \u03b4\n  \u03b8 : Dental \u03b8\n\ndata Palatal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03ba : Palatal \u03ba\n  \u03b3 : Palatal \u03b3\n  \u03c7 : Palatal \u03c7\n\ndata SmoothStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c0 : SmoothStop \u03c0\n  \u03c4 : SmoothStop \u03c4\n  \u03ba : SmoothStop \u03ba\n\ndata MiddleStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b2 : MiddleStop \u03b2\n  \u03b4 : MiddleStop \u03b4\n  \u03b3 : MiddleStop \u03b3\n\ndata RoughStop : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c6 : RoughStop \u03c6\n  \u03b8 : RoughStop \u03b8\n  \u03c7 : RoughStop \u03c7\n\n-- Smyth \u00a717\ndata Spirant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03c3 : Spirant \u03c3\n\n-- Smyth \u00a718\ndata Liquid : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u2032 : Liquid \u03bb\u2032\n  \u03c1 : Liquid \u03c1\n\n-- Smyth \u00a719\ndata Nasal : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bc : Nasal \u03bc\n  \u03bd : Nasal \u03bd\n  \u03b3-nasal : Nasal \u03b3-nasal\n\n-- Smyth \u00a720\ndata Semivowel : \u2200 {\u2113} \u2192 Vowel \u2113 \u2192 Set where\n  \u03b9\u032f : Semivowel \u03b9\n  \u03c5\u032f : Semivowel \u03c5\n\n-- Smyth \u00a720 b\ndata Sonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03bb\u0325 : Sonant \u03bb\u2032\n  \u03bc\u0325 : Sonant \u03bc\n  \u03bd\u0325 : Sonant \u03b3\n  \u03c1\u0325 : Sonant \u03c1\n  \u03c3\u0325 : Sonant \u03c3\n\n-- Smyth \u00a721\ndata DoubleConsonant : \u2200 {\u2113} \u2192 ConsonantSound \u2113 \u2192 Set where\n  \u03b6 : DoubleConsonant \u03b6\n  \u03be : DoubleConsonant \u03be\n  \u03c8 : DoubleConsonant \u03c8\n\ndata _+_\u21d2DoubleConsonant_ : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u2083} \u2192 ConsonantSound \u2113\u2081 \u2192 ConsonantSound \u2113\u2082 \u2192 ConsonantSound \u2113\u2083 \u2192 Set where\n  \u03c3+\u03b4\u21d2DoubleConsonant\u03b6 : \u03c3 + \u03b4 \u21d2DoubleConsonant \u03b6 -- Smyth \u00a726 D. Aeolic has \u03c3\u03b4 for \u03b6\n  \u03b4+\u03c3\u21d2DoubleConsonant\u03b6 : \u03b4 + \u03c3 \u21d2DoubleConsonant \u03b6\n  \n  \u03ba+\u03c3\u21d2DoubleConsonant\u03be : \u03ba + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03b3+\u03c3\u21d2DoubleConsonant\u03be : \u03b3 + \u03c3 \u21d2DoubleConsonant \u03be\n  \u03c7+\u03c3\u21d2DoubleConsonant\u03be : \u03c7 + \u03c3 \u21d2DoubleConsonant \u03be\n\n  \u03c0+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c0 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03b2+\u03c3\u21d2DoubleConsonant\u03c8 : \u03b2 + \u03c3 \u21d2DoubleConsonant \u03c8\n  \u03c6+\u03c3\u21d2DoubleConsonant\u03c8 : \u03c6 + \u03c3 \u21d2DoubleConsonant \u03c8\n\n\n-- TODO (needs a unified model)\n\n-- Accent\n-- Smyth \u00a74. All vowels with the circumflex (149) are long.\n\n-- Diaeresis\n-- Smyth \u00a78. Diaeresis.\u2014A double dot, the mark of diaeresis, may be written over \u03b9 or \u03c5 when these do not form a diphthong with the preceding vowel\n\n-- Breathing\n-- Smyth \u00a79. Every initial vowel or diphthong has either the rough (\u1ffe) or the smooth (\u1fbf) breathing.\n-- Smyth \u00a710. Initial \u03c5 (\u1fe0 and \u1fe1) always has the rough breathing.\n-- Smyth \u00a711. Diphthongs take the breathing, as the accent (152), over the second vowel: \u03b1\u1f31\u03c1\u03ad\u03c9 hair\u00e9o I seize, \u03b1\u1f34\u03c1\u03c9 a\u00edro I lift.\n-- Smyth \u00a711. But \u1fb3, \u1fc3, \u1ff3 take both the breathing and the accent on the first vowel, even when \u03b9 is written in the line (5): \u1f84\u03b4\u03c9 = \u1f0c\u03b9\u03b4\u03c9 I sing\u2026 The writing \u1f00\u03af\u03b4\u03b7\u03bb\u03bf\u03c2 (\u1f08\u03af\u03b4\u03b7\u03bb\u03bf\u03c2) destroying shows that \u03b1\u03b9 does not here form a diphthong\n-- Smyth \u00a712. In compound words (as in \u03c0\u03c1\u03bf\u03bf\u03c1\u1fb6\u03bd to foresee, from \u03c0\u03c1\u03cc + \u1f41\u03c1\u1fb6\u03bd) the rough breathing is not written, though it must often have been pronounced\n-- Smyth \u00a713. Every initial \u03c1 has the rough breathing\n-- Smyth \u00a713. Medial \u03c1\u03c1 is written \u1fe4\u1fe5 in some texts\n\n-- Consonants\n-- Smyth \u00a715 c. \u03b9\u032f \u03c5\u032f\n-- Smyth \u00a719 a. Gamma before \u03ba, \u03b3, \u03c7, \u03be is called \u03b3-nasal\n-- Smyth \u00a720 a. When \u03b9 and \u03c5 correspond to y and w (cp. minion, persuade) they are said to be unsyllabic; and, with a following vowel, make one syllable out of two.\n-- Smyth \u00a720 a. Initial \u03b9\u032f passed into \u0314 (h), as in \u1f27\u03c0\u03b1\u03c1 liver, Lat. jecur; and into \u03b6 in \u03b6\u03c5\u03b3\u03cc\u03bd yoke\n-- Smyth \u00a720 a. Initial \u03c5\u032f was written \u03dd\n-- Smyth \u00a720 a. Medial \u03b9\u032f, \u03c5\u032f before vowels were often lost, as in \u03c4\u1fd1\u03bc\u03ac-(\u03b9\u032f)\u03c9 I honour, \u03b2\u03bf (\u03c5\u032f)-\u03cc\u03c2, gen. of \u03b2\u03bf\u1fe6-\u03c2 ox, cow\n-- Smyth \u00a726. \u03c3 was usually like our sharp s; but before voiced consonants (15 a) it probably was soft, like z; thus we find both \u03ba\u03cc\u03b6\u03bc\u03bf\u03c2 and \u03ba\u03cc\u03c3\u03bc\u03bf\u03c2 on inscriptions.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8279740c03d222eb6089ba357173cb8440a0331d","subject":"cleanup of ChurchRosser.agda","message":"cleanup of ChurchRosser.agda\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ChurchRosser.agda","new_file":"Agda\/ChurchRosser.agda","new_contents":"module ChurchRosser where\nopen import Data.Empty\nopen import Data.Product\nopen import Data.List\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality using (_\u2261_)\n\nopen import Core\nopen import Core-Lemmas\nopen import Reduction\nopen import Typing\n\n\ndata _* (R : PTerm \u219d PTerm) : PTerm \u219d PTerm where\n  base : \u2200 {a b} -> R a b -> (R *) a b\n  refl : \u2200 {a} -> (R *) a a\n  trans : \u2200 {a b c} -> (R *) a b -> (R *) b c -> (R *) a c\n\n\n->\u03b2*-\u22a2 : \u2200 {\u0393 m m' \u03c4} -> \u0393 \u22a2 m \u2236 \u03c4 -> (_->\u03b2_ *) m m' -> \u0393 \u22a2 m' \u2236 \u03c4\n->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 (base m->\u03b2m') = ->\u03b2-\u22a2 \u0393\u22a2m\u2236\u03c4 m->\u03b2m'\n->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 refl = \u0393\u22a2m\u2236\u03c4\n->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 (trans m->\u03b2*m' m->\u03b2*m'') = ->\u03b2*-\u22a2 (->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 m->\u03b2*m') m->\u03b2*m''\n\n\nDP : (R : PTerm \u219d PTerm)(T : PTerm \u219d PTerm) -> Set\nDP R T = \u2200 {a b c} -> R a b -> T a c -> \u2203 (\u03bb d \u2192 (T b d) \u00d7 (R c d))\n\n\nDP->||->||-imp-DP->||->||* : DP (_->||_) (_->||_) -> DP (_->||_) (_->||_ *)\nDP->||->||-imp-DP->||->||* DP->||->|| {b = b} a->||b (base a->||c) = d , (base b->||d , c->||d)\n  where\n  d = proj\u2081 (DP->||->|| a->||b a->||c)\n  b->||d = proj\u2081 (proj\u2082 (DP->||->|| a->||b a->||c))\n  c->||d = proj\u2082 (proj\u2082 (DP->||->|| a->||b a->||c))\n\nDP->||->||-imp-DP->||->||* DP->||->|| {b = b} a->||b refl = b , (refl , a->||b)\nDP->||->||-imp-DP->||->||* DP->||->|| {a} {b} {c} a->||b (trans {b = g} a->||*g g->||*c) =\n  d , ((trans b->||*g' g'->||*d) , c->||d)\n  where\n  left-IH : \u2203 (\u03bb g' \u2192 (_->||_ *) b g' \u00d7 (g ->|| g'))\n  left-IH = DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*g\n\n  g' = proj\u2081 left-IH\n  b->||*g' = proj\u2081 (proj\u2082 left-IH)\n  g->||g' = proj\u2082 (proj\u2082 left-IH)\n\n  right-IH : \u2203 (\u03bb d \u2192 (_->||_ *) g' d \u00d7 (c ->|| d))\n  right-IH = DP->||->||-imp-DP->||->||* DP->||->|| g->||g' g->||*c\n\n  d = proj\u2081 right-IH\n  g'->||*d = proj\u2081 (proj\u2082 right-IH)\n  c->||d = proj\u2082 (proj\u2082 right-IH)\n\n\nDP->||->||-imp-DP->||*->||* : DP (_->||_) (_->||_) -> DP (_->||_ *) (_->||_ *)\nDP->||->||-imp-DP->||*->||* DP->||->|| (base a->||b) a->||*c = d , (b->||*d , base c->||d)\n  where\n  d = proj\u2081 (DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*c)\n  b->||*d = proj\u2081 (proj\u2082 (DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*c))\n  c->||d = proj\u2082 (proj\u2082 (DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*c))\n\nDP->||->||-imp-DP->||*->||* DP->||->|| refl a->||*c = _ , (a->||*c , refl)\nDP->||->||-imp-DP->||*->||* DP->||->|| {a} {b} {c} (trans {b = g} a->||*g g->||*b) a->||*c =\n  d , b->||*d , (trans c->||*g' g'->||*d)\n  where\n  left-IH : \u2203 (\u03bb g' \u2192 (_->||_ *) g g' \u00d7 (_->||_ *) c g')\n  left-IH = DP->||->||-imp-DP->||*->||* DP->||->|| a->||*g a->||*c\n\n  g' = proj\u2081 left-IH\n  c->||*g' = proj\u2082 (proj\u2082 left-IH)\n  g->||*g' = proj\u2081 (proj\u2082 left-IH)\n\n  right-IH : \u2203 (\u03bb d \u2192 (_->||_ *) b d \u00d7 (_->||_ *) g' d)\n  right-IH = DP->||->||-imp-DP->||*->||* DP->||->|| g->||*b g->||*g'\n\n  d = proj\u2081 right-IH\n  g'->||*d = proj\u2082 (proj\u2082 right-IH)\n  b->||*d = proj\u2081 (proj\u2082 right-IH)\n\n\n->||-refl : \u2200 {s} -> Term s -> s ->|| s\n->||-refl var = refl\n->||-refl (lam L cf) = abs L (\u03bb {x} z \u2192 ->||-refl (cf z))\n->||-refl (app trm-s trm-s\u2081) = app (->||-refl trm-s) (->||-refl trm-s\u2081)\n->||-refl Y = reflY\n\n\n->\u03b2-imp->|| : \u2200 {m m'} -> m ->\u03b2 m' -> m ->|| m'\n->\u03b2-imp->|| (redL trm-n m->\u03b2m') = app (->\u03b2-imp->|| m->\u03b2m') (->||-refl trm-n)\n->\u03b2-imp->|| (redR trm-m n->\u03b2n') = app (->||-refl trm-m) (->\u03b2-imp->|| n->\u03b2n')\n->\u03b2-imp->|| (abs L x) = abs L (\u03bb {x\u2081} z \u2192 ->\u03b2-imp->|| (x z))\n->\u03b2-imp->|| (beta {m} (lam L cf) trm-n) =\n  beta L {m' = m} (\u03bb {x} x\u2209L \u2192 ->||-refl (cf x\u2209L)) (->||-refl trm-n)\n->\u03b2-imp->|| (Y trm-m) = Y (->||-refl trm-m)\n\n\n->\u03b2*-imp->||* : \u2200 {m m'} -> (_->\u03b2_ *) m m' -> (_->||_ *) m m'\n->\u03b2*-imp->||* (base m->\u03b2m') = base (->\u03b2-imp->|| m->\u03b2m')\n->\u03b2*-imp->||* refl = refl\n->\u03b2*-imp->||* (trans m->\u03b2*m' m->\u03b2*m'') =\n  trans (->\u03b2*-imp->||* m->\u03b2*m') (->\u03b2*-imp->||* m->\u03b2*m'')\n\n\nredL* : \u2200 {m m' n} -> (_->\u03b2_ *) m m' -> Term n -> (_->\u03b2_ *) (app m n) (app m' n)\nredL* (base x) trm-n = base (redL trm-n x)\nredL* refl trm-n = refl\nredL* (trans m->\u03b2*o o->\u03b2*m') trm-n = trans (redL* m->\u03b2*o trm-n) (redL* o->\u03b2*m' trm-n)\n\n\nredR* : \u2200 {m n n'} -> (_->\u03b2_ *) n n' -> Term m -> (_->\u03b2_ *) (app m n) (app m n')\nredR* (base x) trm-m = base (redR trm-m x)\nredR* refl trm-m = refl\nredR* (trans n->\u03b2*o o->\u03b2*n) trm-m = trans (redR* n->\u03b2*o trm-m) (redR* o->\u03b2*n trm-m)\n\n\nabs' : \u2200 {m m' x} -> m ->\u03b2 m' -> (lam (* x ^ m)) ->\u03b2 (lam (* x ^ m'))\nabs' {m} {m'} {x} m->\u03b2m' = abs [] body\n  where\n  body : \u2200 {y} -> y \u2209 [] -> ((* x ^ m) ^' y) ->\u03b2 ((* x ^ m') ^' y)\n  body {y} y\u2209 rewrite\n    *^-^\u2261subst m x y {0} (->\u03b2-Term-l m->\u03b2m') |\n    *^-^\u2261subst m' x y {0} (->\u03b2-Term-r m->\u03b2m') = lem2-5-1->\u03b2 m m' x y m->\u03b2m'\n\n\nabs*' : \u2200 {m m' x} -> (_->\u03b2_ *) m m' -> (_->\u03b2_ *) (lam (* x ^ m)) (lam (* x ^ m'))\nabs*' (base m->\u03b2m') = base (abs' m->\u03b2m')\nabs*' refl = refl\nabs*' (trans m->\u03b2*m' m->\u03b2*m'') = trans (abs*' m->\u03b2*m') (abs*' m->\u03b2*m'')\n\n\nabs* : \u2200 {m m'} L -> (cf : \u2200 {x} -> x \u2209 L -> (_->\u03b2_ *) (m ^' x) (m' ^' x)) -> (_->\u03b2_ *) (lam m) (lam m')\nabs* {m} {m'} L cf = body\n  where\n  x = \u2203fresh (L ++ FV m ++ FV m')\n\n  x\u2209 : x \u2209 (L ++ FV m ++ FV m')\n  x\u2209 = \u2203fresh-spec (L ++ FV m ++ FV m')\n\n  subst : \u2200 m -> x \u2209 FV m -> _\u2261_ {_} {PTerm} (lam m) (lam (* x ^ (m ^' x)))\n  subst m x\u2209m rewrite\n    fv-^-*^-refl x m {0} x\u2209m = _\u2261_.refl\n\n  body : (_->\u03b2_ *) (lam m) (lam m')\n  body rewrite\n    subst m (\u2209-cons-l _ _ (\u2209-cons-r L _ x\u2209)) |\n    subst m' (\u2209-cons-r (FV m) _ (\u2209-cons-r L _ x\u2209)) = abs*' (cf (\u2209-cons-l _ _ x\u2209))\n\n\n->||-imp->\u03b2* : \u2200 {m m'} -> m ->|| m' -> (_->\u03b2_ *) m m'\n->||-imp->\u03b2* refl = refl\n->||-imp->\u03b2* reflY = refl\n->||-imp->\u03b2* (app {m} {m'} {n} {n'} m->||m' n->||n') =\n  trans {b = (app m n')}\n    (redR* (->||-imp->\u03b2* n->||n') (->||-Term-l m->||m'))\n    (redL* (->||-imp->\u03b2* m->||m') (->||-Term-r n->||n'))\n->||-imp->\u03b2* (abs L cf) = abs* L (\u03bb x\u2209 \u2192 ->||-imp->\u03b2* (cf x\u2209))\n->||-imp->\u03b2* (beta L {m} {m'} {n} {n'} cf n->||n') =\n  trans {b = app (lam m') n}\n    (redL* (abs* L (\u03bb {x} z \u2192 ->||-imp->\u03b2* (cf z))) (->||-Term-l n->||n'))\n    (trans {b = app (lam m') n'}\n      (redR* (->||-imp->\u03b2* n->||n') (lam L (\u03bb x\u2209L \u2192 ->||-Term-r (cf x\u2209L))))\n      (base (beta (lam L (\u03bb x\u2209L \u2192 ->||-Term-r (cf x\u2209L))) (->||-Term-r n->||n'))))\n->||-imp->\u03b2* (Y {m} {m'} {\u03c4} m->||m') =\n  trans {b = app m (app (Y \u03c4) m)}\n    (base (Y (->||-Term-l m->||m')))\n    (trans {b = app m' (app (Y \u03c4) m)}\n      (redL* (->||-imp->\u03b2* m->||m') (app Y (->||-Term-l m->||m')))\n      (redR* (redR* (->||-imp->\u03b2* m->||m') Y) (->||-Term-r m->||m')))\n\n\n->||*-imp->\u03b2* : \u2200 {m m'} -> (_->||_ *) m m' -> (_->\u03b2_ *) m m'\n->||*-imp->\u03b2* (base m->||m') = ->||-imp->\u03b2* m->||m'\n->||*-imp->\u03b2* refl = refl\n->||*-imp->\u03b2* (trans m->||*m' n->||*n') = trans (->||*-imp->\u03b2* m->||*m') (->||*-imp->\u03b2* n->||*n')\n\n\nchurch-rosser-\u22a2 : \u2200 {\u0393 \u03c4 a b c} -> \u0393 \u22a2 a \u2236 \u03c4 -> (_->\u03b2_ *) a b -> (_->\u03b2_ *) a c ->\n  \u2203(\u03bb d -> ((_->\u03b2_ *) b d \u00d7 (_->\u03b2_ *) c d) \u00d7 \u0393 \u22a2 d \u2236 \u03c4)\nchurch-rosser-\u22a2 {\u0393} {\u03c4} {a} {b} {c} \u0393\u22a2a\u2236\u03c4 a->\u03b2*b a->\u03b2*c =\n  d , (b->\u03b2*d , c->\u03b2*d) , ->\u03b2*-\u22a2 \u0393\u22a2a\u2236\u03c4 a->\u03b2*d\n  where\n  a->||*b = ->\u03b2*-imp->||* a->\u03b2*b\n  a->||*c = ->\u03b2*-imp->||* a->\u03b2*c\n\n  d-spec : \u2203 (\u03bb d \u2192 ((_->||_ *) b d) \u00d7 ((_->||_ *) c d))\n  d-spec = DP->||->||-imp-DP->||*->||* lem2-5-2 a->||*b a->||*c\n\n  d = proj\u2081 d-spec\n  b->\u03b2*d = ->||*-imp->\u03b2* (proj\u2081 (proj\u2082 d-spec))\n  c->\u03b2*d = ->||*-imp->\u03b2* (proj\u2082 (proj\u2082 d-spec))\n  a->\u03b2*d = trans a->\u03b2*b b->\u03b2*d\n","old_contents":"module ChurchRosser where\nopen import Data.Empty\nopen import Data.Product\nopen import Data.List\nopen import Data.List.Any as Any\nopen Any.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality using (_\u2261_)\n\nopen import Core\nopen import Core-Lemmas\nopen import Reduction\nopen import Typing\n\ndata _* (R : PTerm \u219d PTerm) : PTerm \u219d PTerm where\n  base : \u2200 {a b} -> R a b -> (R *) a b\n  refl : \u2200 {a} -> (R *) a a\n  trans : \u2200 {a b c} -> (R *) a b -> (R *) b c -> (R *) a c\n\n->\u03b2*-\u22a2 : \u2200 {\u0393 m m' \u03c4} -> \u0393 \u22a2 m \u2236 \u03c4 -> (_->\u03b2_ *) m m' -> \u0393 \u22a2 m' \u2236 \u03c4\n->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 (base m->\u03b2m') = ->\u03b2-\u22a2 \u0393\u22a2m\u2236\u03c4 m->\u03b2m'\n->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 refl = \u0393\u22a2m\u2236\u03c4\n->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 (trans m->\u03b2*m' m->\u03b2*m'') = ->\u03b2*-\u22a2 (->\u03b2*-\u22a2 \u0393\u22a2m\u2236\u03c4 m->\u03b2*m') m->\u03b2*m''\n\nDP : (R : PTerm \u219d PTerm)(T : PTerm \u219d PTerm) -> Set\nDP R T = \u2200 {a b c} -> R a b -> T a c -> \u2203 (\u03bb d \u2192 (T b d) \u00d7 (R c d))\n\nDP->||->||-imp-DP->||->||* : DP (_->||_) (_->||_) -> DP (_->||_) (_->||_ *)\nDP->||->||-imp-DP->||->||* DP->||->|| {b = b} a->||b (base a->||c) = d , (base b->||d , c->||d)\n  where\n  d = proj\u2081 (DP->||->|| a->||b a->||c)\n  b->||d = proj\u2081 (proj\u2082 (DP->||->|| a->||b a->||c))\n  c->||d = proj\u2082 (proj\u2082 (DP->||->|| a->||b a->||c))\n\nDP->||->||-imp-DP->||->||* DP->||->|| {b = b} a->||b refl = b , (refl , a->||b)\nDP->||->||-imp-DP->||->||* DP->||->|| {a} {b} {c} a->||b (trans {b = g} a->||*g g->||*c) =\n  d , ((trans b->||*g' g'->||*d) , c->||d)\n  where\n  left-IH : \u2203 (\u03bb g' \u2192 (_->||_ *) b g' \u00d7 (g ->|| g'))\n  left-IH = DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*g\n\n  g' = proj\u2081 left-IH\n  b->||*g' = proj\u2081 (proj\u2082 left-IH)\n  g->||g' = proj\u2082 (proj\u2082 left-IH)\n\n  right-IH : \u2203 (\u03bb d \u2192 (_->||_ *) g' d \u00d7 (c ->|| d))\n  right-IH = DP->||->||-imp-DP->||->||* DP->||->|| g->||g' g->||*c\n\n  d = proj\u2081 right-IH\n  g'->||*d = proj\u2081 (proj\u2082 right-IH)\n  c->||d = proj\u2082 (proj\u2082 right-IH)\n\n\nDP->||->||-imp-DP->||*->||* : DP (_->||_) (_->||_) -> DP (_->||_ *) (_->||_ *)\nDP->||->||-imp-DP->||*->||* DP->||->|| (base a->||b) a->||*c = d , (b->||*d , base c->||d)\n  where\n  d = proj\u2081 (DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*c)\n  b->||*d = proj\u2081 (proj\u2082 (DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*c))\n  c->||d = proj\u2082 (proj\u2082 (DP->||->||-imp-DP->||->||* DP->||->|| a->||b a->||*c))\n\nDP->||->||-imp-DP->||*->||* DP->||->|| refl a->||*c = _ , (a->||*c , refl)\nDP->||->||-imp-DP->||*->||* DP->||->|| {a} {b} {c} (trans {b = g} a->||*g g->||*b) a->||*c =\n  d , b->||*d , (trans c->||*g' g'->||*d)\n  where\n  left-IH : \u2203 (\u03bb g' \u2192 (_->||_ *) g g' \u00d7 (_->||_ *) c g')\n  left-IH = DP->||->||-imp-DP->||*->||* DP->||->|| a->||*g a->||*c\n\n  g' = proj\u2081 left-IH\n  c->||*g' = proj\u2082 (proj\u2082 left-IH)\n  g->||*g' = proj\u2081 (proj\u2082 left-IH)\n\n  right-IH : \u2203 (\u03bb d \u2192 (_->||_ *) b d \u00d7 (_->||_ *) g' d)\n  right-IH = DP->||->||-imp-DP->||*->||* DP->||->|| g->||*b g->||*g'\n\n  d = proj\u2081 right-IH\n  g'->||*d = proj\u2082 (proj\u2082 right-IH)\n  b->||*d = proj\u2081 (proj\u2082 right-IH)\n\n\n->||-refl : \u2200 {s} -> Term s -> s ->|| s\n->||-refl var = refl\n->||-refl (lam L cf) = abs L (\u03bb {x} z \u2192 ->||-refl (cf z))\n->||-refl (app trm-s trm-s\u2081) = app (->||-refl trm-s) (->||-refl trm-s\u2081)\n->||-refl Y = reflY\n\n->\u03b2-imp->|| : \u2200 {m m'} -> m ->\u03b2 m' -> m ->|| m'\n->\u03b2-imp->|| (redL trm-n m->\u03b2m') = app (->\u03b2-imp->|| m->\u03b2m') (->||-refl trm-n)\n->\u03b2-imp->|| (redR trm-m n->\u03b2n') = app (->||-refl trm-m) (->\u03b2-imp->|| n->\u03b2n')\n->\u03b2-imp->|| (abs L x) = abs L (\u03bb {x\u2081} z \u2192 ->\u03b2-imp->|| (x z))\n->\u03b2-imp->|| (beta {m} (lam L cf) trm-n) =\n  beta L {m' = m} (\u03bb {x} x\u2209L \u2192 ->||-refl (cf x\u2209L)) (->||-refl trm-n)\n->\u03b2-imp->|| (Y trm-m) = Y (->||-refl trm-m)\n\n->\u03b2*-imp->||* : \u2200 {m m'} -> (_->\u03b2_ *) m m' -> (_->||_ *) m m'\n->\u03b2*-imp->||* (base m->\u03b2m') = base (->\u03b2-imp->|| m->\u03b2m')\n->\u03b2*-imp->||* refl = refl\n->\u03b2*-imp->||* (trans m->\u03b2*m' m->\u03b2*m'') =\n  trans (->\u03b2*-imp->||* m->\u03b2*m') (->\u03b2*-imp->||* m->\u03b2*m'')\n\nredL* : \u2200 {m m' n} -> (_->\u03b2_ *) m m' -> Term n -> (_->\u03b2_ *) (app m n) (app m' n)\nredL* (base x) trm-n = base (redL trm-n x)\nredL* refl trm-n = refl\nredL* (trans m->\u03b2*o o->\u03b2*m') trm-n = trans (redL* m->\u03b2*o trm-n) (redL* o->\u03b2*m' trm-n)\n\nredR* : \u2200 {m n n'} -> (_->\u03b2_ *) n n' -> Term m -> (_->\u03b2_ *) (app m n) (app m n')\nredR* (base x) trm-m = base (redR trm-m x)\nredR* refl trm-m = refl\nredR* (trans n->\u03b2*o o->\u03b2*n) trm-m = trans (redR* n->\u03b2*o trm-m) (redR* o->\u03b2*n trm-m)\n\nabs' : \u2200 {m m' x} -> m ->\u03b2 m' -> (lam (* x ^ m)) ->\u03b2 (lam (* x ^ m'))\nabs' {m} {m'} {x} m->\u03b2m' = abs [] body\n  where\n  body : \u2200 {y} -> y \u2209 [] -> ((* x ^ m) ^' y) ->\u03b2 ((* x ^ m') ^' y)\n  body {y} y\u2209 rewrite\n    *^-^\u2261subst m x y {0} (->\u03b2-Term-l m->\u03b2m') |\n    *^-^\u2261subst m' x y {0} (->\u03b2-Term-r m->\u03b2m') = lem2-5-1->\u03b2 m m' x y m->\u03b2m'\n\nabs*' : \u2200 {m m' x} -> (_->\u03b2_ *) m m' -> (_->\u03b2_ *) (lam (* x ^ m)) (lam (* x ^ m'))\nabs*' (base m->\u03b2m') = base (abs' m->\u03b2m')\nabs*' refl = refl\nabs*' (trans m->\u03b2*m' m->\u03b2*m'') = trans (abs*' m->\u03b2*m') (abs*' m->\u03b2*m'')\n\nabs* : \u2200 {m m'} L -> (cf : \u2200 {x} -> x \u2209 L -> (_->\u03b2_ *) (m ^' x) (m' ^' x)) -> (_->\u03b2_ *) (lam m) (lam m')\nabs* {m} {m'} L cf = body\n  where\n  x = \u2203fresh (L ++ FV m ++ FV m')\n\n  x\u2209 : x \u2209 (L ++ FV m ++ FV m')\n  x\u2209 = \u2203fresh-spec (L ++ FV m ++ FV m')\n\n  subst : \u2200 m -> x \u2209 FV m -> _\u2261_ {_} {PTerm} (lam m) (lam (* x ^ (m ^' x)))\n  subst m x\u2209m rewrite\n    fv-^-*^-refl x m {0} x\u2209m = _\u2261_.refl\n\n  body : (_->\u03b2_ *) (lam m) (lam m')\n  body rewrite\n    subst m (\u2209-cons-l _ _ (\u2209-cons-r L _ x\u2209)) |\n    subst m' (\u2209-cons-r (FV m) _ (\u2209-cons-r L _ x\u2209)) = abs*' (cf (\u2209-cons-l _ _ x\u2209))\n\n\n->||-imp->\u03b2* : \u2200 {m m'} -> m ->|| m' -> (_->\u03b2_ *) m m'\n->||-imp->\u03b2* refl = refl\n->||-imp->\u03b2* reflY = refl\n->||-imp->\u03b2* (app {m} {m'} {n} {n'} m->||m' n->||n') =\n  trans {b = (app m n')}\n    (redR* (->||-imp->\u03b2* n->||n') (->||-Term-l m->||m'))\n    (redL* (->||-imp->\u03b2* m->||m') (->||-Term-r n->||n'))\n->||-imp->\u03b2* (abs L cf) = abs* L (\u03bb x\u2209 \u2192 ->||-imp->\u03b2* (cf x\u2209))\n->||-imp->\u03b2* (beta L {m} {m'} {n} {n'} cf n->||n') =\n  trans {b = app (lam m') n}\n    (redL* (abs* L (\u03bb {x} z \u2192 ->||-imp->\u03b2* (cf z))) (->||-Term-l n->||n'))\n    (trans {b = app (lam m') n'}\n      (redR* (->||-imp->\u03b2* n->||n') (lam L (\u03bb x\u2209L \u2192 ->||-Term-r (cf x\u2209L))))\n      (base (beta (lam L (\u03bb x\u2209L \u2192 ->||-Term-r (cf x\u2209L))) (->||-Term-r n->||n'))))\n->||-imp->\u03b2* (Y {m} {m'} {\u03c4} m->||m') =\n  trans {b = app m (app (Y \u03c4) m)}\n    (base (Y (->||-Term-l m->||m')))\n    (trans {b = app m' (app (Y \u03c4) m)}\n      (redL* (->||-imp->\u03b2* m->||m') (app Y (->||-Term-l m->||m')))\n      (redR* (redR* (->||-imp->\u03b2* m->||m') Y) (->||-Term-r m->||m')))\n\n\n->||*-imp->\u03b2* : \u2200 {m m'} -> (_->||_ *) m m' -> (_->\u03b2_ *) m m'\n->||*-imp->\u03b2* (base m->||m') = ->||-imp->\u03b2* m->||m'\n->||*-imp->\u03b2* refl = refl\n->||*-imp->\u03b2* (trans m->||*m' n->||*n') = trans (->||*-imp->\u03b2* m->||*m') (->||*-imp->\u03b2* n->||*n')\n\n\nchurch-rosser-\u22a2 : \u2200 {\u0393 \u03c4 a b c} -> \u0393 \u22a2 a \u2236 \u03c4 -> (_->\u03b2_ *) a b -> (_->\u03b2_ *) a c ->\n  \u2203(\u03bb d -> ((_->\u03b2_ *) b d \u00d7 (_->\u03b2_ *) c d) \u00d7 \u0393 \u22a2 d \u2236 \u03c4)\nchurch-rosser-\u22a2 {\u0393} {\u03c4} {a} {b} {c} \u0393\u22a2a\u2236\u03c4 a->\u03b2*b a->\u03b2*c =\n  d , (b->\u03b2*d , c->\u03b2*d) , ->\u03b2*-\u22a2 \u0393\u22a2a\u2236\u03c4 a->\u03b2*d\n  where\n  a->||*b = ->\u03b2*-imp->||* a->\u03b2*b\n  a->||*c = ->\u03b2*-imp->||* a->\u03b2*c\n\n  d-spec : \u2203 (\u03bb d \u2192 ((_->||_ *) b d) \u00d7 ((_->||_ *) c d))\n  d-spec = DP->||->||-imp-DP->||*->||* lem2-5-2 a->||*b a->||*c\n\n  d = proj\u2081 d-spec\n  b->||*d = proj\u2081 (proj\u2082 d-spec)\n  c->||*d = proj\u2082 (proj\u2082 d-spec)\n  a->||*d = trans a->||*b b->||*d\n  a->\u03b2*d = ->||*-imp->\u03b2* a->||*d\n  b->\u03b2*d = ->||*-imp->\u03b2* b->||*d\n  c->\u03b2*d = ->||*-imp->\u03b2* c->||*d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9b23b0aa187cf242bf6074fdc58bbcac60a313f4","subject":"Final sugared generic eliminator for Desc\/Mu in Agda model.","message":"Final sugared generic eliminator for Desc\/Mu in Agda model.\n\nThis combines the case eliminator for tags with the\nprojecting + substituting eliminator for Mu.\n\nThe result is specialized to tagged descriptions, which result from\nelaborating datatype declarations.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\nelim :\n  (I : Set)\n  (E : Enum)\n  (Cs : Tag E \u2192 Desc I)\n  \u2192 let D = `Arg (Tag E) Cs in\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (cs : Cases E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nelim I E Cs P cs i x =\n  let D = `Arg (Tag E) Cs in\n  ind2 I D P (case E (\u03bb t \u2192 CurriedAll I (Cs t) (\u03bc I D) P (\u03bb xs \u2192 con (t , xs))) cs) i x\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim \u22a4 \u2115T \u2115C _\n      ( (\u03bb n \u2192 n)\n      , (\u03bb m ih n \u2192 suc (ih n))\n      , tt\n      )\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim \u22a4 \u2115T \u2115C _\n      ( (\u03bb n \u2192 zero)\n      , (\u03bb m ih n \u2192 add n (ih n))\n      , tt\n      )\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elim (\u2115 tt) VecT (VecC A) _\n      ( (\u03bb n ys \u2192 ys)\n      , (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n      , tt\n      )\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elim (\u2115 tt) VecT (VecC (Vec A m)) _\n      ( (nil A)\n      , (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n      , tt\n      )\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind2 \u22a4 \u2115D _\n      (case \u2115T _\n        ( (\u03bb n \u2192 n)\n        , (\u03bb m ih n \u2192 suc (ih n))\n        , tt\n        )\n      )\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind2 \u22a4 \u2115D _\n      (case \u2115T _\n        ( (\u03bb n \u2192 zero)\n        , (\u03bb m ih n \u2192 add n (ih n))\n        , tt\n        )\n      )\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = ind2 (\u2115 tt) (VecD A) _\n      (case VecT _\n        ( (\u03bb n ys \u2192 ys)\n        , (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n        , tt\n        )\n      )\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind2 (\u2115 tt) (VecD (Vec A m)) _\n      (case VecT _\n        ( (nil A)\n        , (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n        , tt\n        )\n      )\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1a0b2e5649127e93fe70a4d7e9b90cbb45b2394d","subject":"Added a different version of wellFoundedInd-N\u2082.","message":"Added a different version of wellFoundedInd-N\u2082.\n\nIgnore-this: c1a3d7e996665a522e734367d7df7b1a\n\ndarcs-hash:20100505162359-3bd4e-fe9a22b379062c77ef1c3bd5d50151e422c0942b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/N\/Induction\/Lexicographic.agda","new_file":"LTC\/Data\/N\/Induction\/Lexicographic.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the lexicographic order on N\n------------------------------------------------------------------------------\n\nmodule LTC.Data.N.Induction.Lexicographic where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import Postulates using ( wellFoundedLT )\nopen import LTC.Relation.Inequalities\n\nimport MyStdLib.Induction.Lexicographic\nopen module LexicographicLT\u2082 = MyStdLib.Induction.Lexicographic LT LT\nopen import MyStdLib.Induction.WellFounded\n\n------------------------------------------------------------------------------\n\n-- Well-founded induction on the lexicographic order on N.\nwellFoundedInd-N\u2082 :\n  (P : D \u00d7 D \u2192 Set) \u2192\n  ((mn : D \u00d7 D) \u2192\n       (( op : D \u00d7 D) \u2192 LT\u2082 op mn \u2192 N\u2082 op \u2192 P op) \u2192\n       N\u2082 mn \u2192 P mn) \u2192\n  {mn : D \u00d7 D } \u2192 N\u2082 mn \u2192 P mn\nwellFoundedInd-N\u2082 P accH {mn} =\n  wellFoundedInd {D \u00d7 D}\n                 {LT\u2082}\n                 {\u03bb xy \u2192 N\u2082 xy \u2192 P xy}\n                 (wellFoundedLexi wellFoundedLT wellFoundedLT)\n                 accH\n                 mn\n\n-- A version without use the data type N\u2082.\nwellFoundedInd-N\u2082' :\n  (P : D \u00d7 D \u2192 Set) \u2192\n  ((mn : D \u00d7 D) \u2192\n       (( op : D \u00d7 D) \u2192 LT\u2082 op mn \u2192 N (\u00d7-proj\u2081 op) \u2192 N (\u00d7-proj\u2082 op) \u2192 P op) \u2192\n       N (\u00d7-proj\u2081 mn) \u2192 N (\u00d7-proj\u2082 mn) \u2192 P mn) \u2192\n  {mn : D \u00d7 D } \u2192 N (\u00d7-proj\u2081 mn) \u2192 N (\u00d7-proj\u2082 mn) \u2192 P mn\nwellFoundedInd-N\u2082' P accH {mn} =\n  wellFoundedInd {D \u00d7 D}\n                 {LT\u2082}\n                 {\u03bb xy \u2192 N (\u00d7-proj\u2081 xy) \u2192 N (\u00d7-proj\u2082 xy) \u2192 P xy}\n                 (wellFoundedLexi wellFoundedLT wellFoundedLT)\n                 accH\n                 mn\n","old_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the lexicographic order on N\n------------------------------------------------------------------------------\n\nmodule LTC.Data.N.Induction.Lexicographic where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import Postulates using ( wellFoundedLT )\nopen import LTC.Relation.Inequalities\n\nimport MyStdLib.Induction.Lexicographic\nopen module LexicographicLT\u2082 = MyStdLib.Induction.Lexicographic LT LT\nopen import MyStdLib.Induction.WellFounded\n\n\n------------------------------------------------------------------------------\n\nwellFoundedInd-N\u2082 :\n  (P : D \u00d7 D \u2192 Set) \u2192\n  ((mn : D \u00d7 D) \u2192\n       (( op : D \u00d7 D) \u2192 LT\u2082 op mn \u2192 N\u2082 op \u2192 P op) \u2192\n       N\u2082 mn \u2192 P mn) \u2192\n  {mn : D \u00d7 D } \u2192 N\u2082 mn \u2192 P mn\nwellFoundedInd-N\u2082 P accH {mn} =\n  wellFoundedInd {D \u00d7 D}\n                 {LT\u2082}\n                 {\u03bb xy \u2192 N\u2082 xy \u2192 P xy}\n                 (wellFoundedLexi wellFoundedLT wellFoundedLT)\n                 accH\n                 mn\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ce5d7dcd38e313c528faabaed95e8e7e5b4365ca","subject":"Only white spaces.","message":"Only white spaces.\n\nIgnore-this: 7e7ce7ddc8733882ff7663f6213fc31b\n\ndarcs-hash:20100916165334-3bd4e-4af66f15aeaee3ced653775d9a17dea2f52ab444.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Consistency\/LTC\/Data\/List\/Bisimulation.agda","new_file":"Test\/Consistency\/LTC\/Data\/List\/Bisimulation.agda","new_contents":"------------------------------------------------------------------------------\n-- Test the consistency of LTC.Data.List.Bisimulation\n------------------------------------------------------------------------------\n\nmodule Test.Consistency.LTC.Data.List.Bisimulation where\n\nopen import LTC.Minimal\nopen import LTC.Data.List.Bisimulation\n\n------------------------------------------------------------------------------\n\npostulate\n  impossible : ( d e : D) \u2192 d \u2261 e\n{-# ATP prove impossible #-}\n\n","old_contents":"------------------------------------------------------------------------------\n-- Test the consistency of LTC.Data.List.Bisimulation\n------------------------------------------------------------------------------\n\nmodule Test.Consistency.LTC.Data.List.Bisimulation where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.List.Bisimulation\n\n------------------------------------------------------------------------------\n\npostulate\n  impossible : ( d e : D) \u2192 d \u2261 e\n{-# ATP prove impossible #-}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b527b8e4c5bfde2a32d75e8226bf5c668e506fc6","subject":"example of export from module","message":"example of export from module\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/SemiGroups.agda","new_file":"livecode\/SemiGroups.agda","new_contents":"open import Base\n\nopen import Id\n\nmodule SemiGroups where\n\ntest : Type \u2192 Type\ntest A = (a : A) \u2192 (a == a)\n\nassoc : {A : Type} \u2192 (A \u2192 A \u2192 A) \u2192 Type\nassoc {A} _*_ = (x : A) \u2192 (y : A) \u2192 (z : A) \u2192 ((x * y) * z) == (x * (y * z))\n\nsemiGroup : (A : Type) \u2192 Type\nsemiGroup A = \u03a3 (A \u2192 A \u2192 A) (\u03bb op \u2192 assoc op) \n\nmodule SemiGroup(A : Type)(_\u00b7_ : A \u2192 A \u2192 A)(ass : (assoc _\u00b7_)) where\n  leftId : A \u2192 Type\n  leftId x = (y : A) \u2192 ((x \u00b7 y) == y)\n\n  rightId : A \u2192 Type\n  rightId x = (y : A) \u2192 ((y \u00b7 x) == y)\n\n  Id = \u03bb a \u2192 (leftId a) \u00d7 (rightId a)\n\n  idUnique : (x y : A) \u2192 leftId x \u2192 rightId y \u2192 y == x\n  idUnique x y lidx ridy = sym (lidx y) && ridy x\n\nopen import Nat\n\nass+ : assoc _+_\nass+ zero y z = refl (y + z)\nass+ (succ x) y z = succ # ass+ x y z\n\nmodule AddNatSG = SemiGroup \u2115 _+_ ass+\n\nlftId : {A : Type} \u2192 semiGroup A \u2192 A \u2192 Type\nlftId {A} [ a , x ] a\u2081 = lfid a\u2081 where\n  module SG = SemiGroup A a x\n  lfid = SG.leftId\n\n\n","old_contents":"open import Base\n\nopen import Id\n\nmodule SemiGroups where\n\ntest : Type \u2192 Type\ntest A = (a : A) \u2192 (a == a)\n\nassoc : {A : Type} \u2192 (A \u2192 A \u2192 A) \u2192 Type\nassoc {A} _*_ = (x : A) \u2192 (y : A) \u2192 (z : A) \u2192 ((x * y) * z) == (x * (y * z))\n\nsemiGroup : (A : Type) \u2192 Type\nsemiGroup A = \u03a3 (A \u2192 A \u2192 A) (\u03bb op \u2192 assoc op) \n\nmodule SemiGroup(A : Type)(_\u00b7_ : A \u2192 A \u2192 A)(ass : (assoc _\u00b7_)) where\n  leftId : A \u2192 Type\n  leftId x = (y : A) \u2192 ((x \u00b7 y) == y)\n\n  rightId : A \u2192 Type\n  rightId x = (y : A) \u2192 ((y \u00b7 x) == y)\n\n  Id = \u03bb a \u2192 (leftId a) \u00d7 (rightId a)\n\n  idUnique : (x y : A) \u2192 leftId x \u2192 rightId y \u2192 y == x\n  idUnique x y lidx ridy = sym (lidx y) && ridy x\n\nopen import Nat\n\nass+ : assoc _+_\nass+ zero y z = refl (y + z)\nass+ (succ x) y z = succ # ass+ x y z\n\nmodule AddNatSG = SemiGroup \u2115 _+_ ass+\n\n\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c7e56a795604834b0ad2be2a11fc053093a6bb4f","subject":"fussing with unicity","message":"fussing with unicity\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expansion-unicity.agda","new_file":"expansion-unicity.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"8c93fbb0f2fd6e40c236c773d2287dad6a38f425","subject":"Product: cleanup","message":"Product: cleanup\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Product\/NP.agda","new_file":"lib\/Data\/Product\/NP.agda","new_contents":"-- move this to product\n{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Product.NP where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Data.Product public hiding (\u2203)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nopen import Function.Injection using (Injection; module Injection)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\n\n\u2203 : \u2200 {a b} {A : \u2605 a} \u2192 (A \u2192 \u2605 b) \u2192 \u2605 (a \u2294 b)\n\u2203 = \u03a3 _\n\nfirst : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 A \u00d7 C \u2192 B \u00d7 C\nfirst f (x , y) = (f x , y)\n\nsecond : \u2200 {a p q} {A : \u2605 a} {P : A \u2192 \u2605 p} {Q : A \u2192 \u2605 q} \u2192\n           (\u2200 {x} \u2192 P x \u2192 Q x) \u2192 \u03a3 A P \u2192 \u03a3 A Q\nsecond f (x , y) = (x , f y)\n\nsecond\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n           (B \u2192 C) \u2192 A \u00d7 B \u2192 A \u00d7 C\nsecond\u2032 f = second f\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\nrecord [\u03a3] {a b a\u209a b\u209a}\n           {A : \u2605 a}\n           {B : A \u2192 \u2605 b}\n           (A\u209a : A \u2192 \u2605 a\u209a)\n           (B\u209a : {x : A} (x\u209a : A\u209a x)\n                 \u2192 B x \u2192 \u2605 b\u209a)\n           (p : \u03a3 A B) : \u2605 (a\u209a \u2294 b\u209a) where\n  constructor _[,]_\n  field\n    [proj\u2081] : A\u209a (proj\u2081 p)\n    [proj\u2082] : B\u209a [proj\u2081] (proj\u2082 p)\nopen [\u03a3] public\ninfixr 4 _[,]_\n\nsyntax [\u03a3] A\u209a (\u03bb x\u209a \u2192 e) = [ x\u209a \u2236 A\u209a ][\u00d7][ e ]\n\n[\u2203] : \u2200 {a a\u209a b b\u209a} \u2192\n        (\u2200i\u27e8 A\u209a \u2236 [\u2605] {a} a\u209a \u27e9[\u2192] ((A\u209a [\u2192] [\u2605] {b} b\u209a) [\u2192] [\u2605] _)) \u2203\n[\u2203] {x\u209a = A\u209a} = [\u03a3] A\u209a\n\nsyntax [\u2203] (\u03bb A\u209a \u2192 f) = [\u2203][ A\u209a ] f\n\n_[\u00d7]_ : \u2200 {a b a\u209a b\u209a} \u2192 ([\u2605] {a} a\u209a [\u2192] [\u2605] {b} b\u209a [\u2192] [\u2605] _) _\u00d7_\n_[\u00d7]_ A\u209a B\u209a = [\u03a3] A\u209a (\u03bb _ \u2192 B\u209a)\n\nprivate\n\n  Dec\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n               (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n             \u2192 \u2605 _\n  Dec\u27e6\u2605\u27e7 A\u1d63 = \u2200 x\u2081 x\u2082 \u2192 Dec (A\u1d63 x\u2081 x\u2082)\n\n  Dec\u27e6Pred\u27e7 : \u2200 {a\u2081 a\u2082 p\u2081 p\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n                (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n                {P\u2081 : Pred p\u2081 A\u2081} {P\u2082 : Pred p\u2082 A\u2082} {p\u1d63}\n              \u2192 (P\u1d63 : \u27e6Pred\u27e7 p\u1d63 A\u1d63 P\u2081 P\u2082) \u2192 \u2605 _\n  Dec\u27e6Pred\u27e7 {A\u2081 = A\u2081} {A\u2082} A\u1d63 {P\u2081} {P\u2082} P\u1d63 = \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) y\u2081 y\u2082 \u2192 Dec (P\u1d63 x\u1d63 y\u2081 y\u2082) -- Dec\u27e6\u2605\u27e7 A\u1d63 \u21d2 Dec\u27e6\u2605\u27e7 P\u1d63\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n           {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : \u2605 (a\u1d63 \u2294 b\u1d63) where\n  constructor _\u27e6,\u27e7_\n  field\n    \u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082)\n    \u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _\u27e6,\u27e7_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u27e6\u00d7\u27e7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63}\n        {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n        {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 \u2605 _\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n          {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082) \u00d7\n                        B\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082))\n       (\u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\ndec\u27e6\u03a3\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63 A\u2081 A\u2082 B\u2081 B\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 {b\u2081} {b\u2082} b\u1d63 A\u1d63 B\u2081 B\u2082}\n       (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n       -- (substA\u1d63 : \u2200 {x y \u2113} {p q} (P : A\u1d63 x y \u2192 \u2605 \u2113) \u2192 P p \u2192 P q)\n       (uniqA\u1d63 : \u2200 {x y} (p q : A\u1d63 x y) \u2192 p \u2261 q)\n       (decB\u1d63 : Dec\u27e6Pred\u27e7 A\u1d63 {_} {_} {b\u1d63 {-BUG: with _ here Agda loops-}} B\u1d63)\n     \u2192 Dec\u27e6\u2605\u27e7 (\u27e6\u03a3\u27e7 A\u1d63 B\u1d63)\ndec\u27e6\u03a3\u27e7 {B\u1d63 = B\u1d63} decA\u1d63 uniqA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 x\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 f \u2218 \u27e6proj\u2082\u27e7)\n  where f : \u2200 {x\u1d63'} \u2192 B\u1d63 x\u1d63' y\u2081 y\u2082 \u2192 B\u1d63 x\u1d63 y\u2081 y\u2082\n        f {x\u1d63'} y\u1d63 rewrite uniqA\u1d63 x\u1d63' x\u1d63 = y\u1d63\n\nmodule Dec\u2082 where\n  map\u2082\u2032 : \u2200 {p\u2081 p\u2082 q} {P\u2081 : \u2605 p\u2081} {P\u2082 : \u2605 p\u2082} {Q : \u2605 q}\n          \u2192 (P\u2081 \u2192 P\u2082 \u2192 Q) \u2192 (Q \u2192 P\u2081) \u2192 (Q \u2192 P\u2082) \u2192 Dec P\u2081 \u2192 Dec P\u2082 \u2192 Dec Q\n  map\u2082\u2032 _   \u03c0\u2081  _   (no \u00acp\u2081)  _         = no (\u00acp\u2081 \u2218 \u03c0\u2081)\n  map\u2082\u2032 _   _   \u03c0\u2082  _         (no \u00acp\u2082)  = no (\u00acp\u2082 \u2218 \u03c0\u2082)\n  map\u2082\u2032 mk  _   _   (yes p\u2081)  (yes p\u2082)  = yes (mk p\u2081 p\u2082)\n\ndec\u27e6\u00d7\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n           {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n           {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082}\n           (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n           (decB\u1d63 : Dec\u27e6\u2605\u27e7 B\u1d63)\n         \u2192 Dec\u27e6\u2605\u27e7 (A\u1d63 \u27e6\u00d7\u27e7 B\u1d63)\ndec\u27e6\u00d7\u27e7 decA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 \u27e6proj\u2082\u27e7)\n\n\n\u03a3,-injective\u2082 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x : A} {y z : B x} \u2192 ((x , y) \u2236 \u03a3 A B) \u2261 (x , z) \u2192 y \u2261 z\n\u03a3,-injective\u2082 refl = refl\n\n\u03a3,-injective\u2081 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x\u2081 x\u2082 : A} {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} \u2192 ((x\u2081 , y\u2081) \u2236 \u03a3 A B) \u2261 (x\u2082 , y\u2082) \u2192 x\u2081 \u2261 x\u2082\n\u03a3,-injective\u2081 refl = refl\n\nproj\u2081-injective : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x y : \u03a3 A B}\n                    (B-uniq : \u2200 {z} (p\u2081 p\u2082 : B z) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 proj\u2081 x \u2261 proj\u2081 y \u2192 x \u2261 y\nproj\u2081-injective {x = (a , p\u2081)} {y = (_ , p\u2082)} B-uniq eq rewrite sym eq\n  = cong (\u03bb p \u2192 (a , p)) (B-uniq p\u2081 p\u2082)\n\nproj\u2082-irrelevance : \u2200 {a b} {A : \u2605 a} {B C : A \u2192 \u2605 b} {x\u2081 x\u2082 : A}\n                      {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} {z\u2081 : C x\u2081} {z\u2082 : C x\u2082}\n                    \u2192 (C-uniq : \u2200 {z} (p\u2081 p\u2082 : C z) \u2192 p\u2081 \u2261 p\u2082)\n                    \u2192 ((x\u2081 , y\u2081) \u2236 \u03a3 A B) \u2261 (x\u2082 , y\u2082)\n                    \u2192 ((x\u2081 , z\u2081) \u2236 \u03a3 A C) \u2261 (x\u2082 , z\u2082)\nproj\u2082-irrelevance C-uniq = proj\u2081-injective C-uniq \u2218 \u03a3,-injective\u2081\n\n\u225f\u03a3 : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (decP : \u2200 x \u2192 Decidable {A = P x} _\u2261_)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl with decP w p\u2081 p\u2082\n\u225f\u03a3 decA decP (w  , p)  (.w , .p) | yes refl | yes refl = yes refl\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl | no  p\u2262\n    = no (p\u2262 \u2218 \u03a3,-injective\u2082)\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\n\u225f\u03a3' : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (uniqP : \u2200 {x} (p q : P x) \u2192 p \u2261 q)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3' decA uniqP (w  , p\u2081) (.w , p\u2082) | yes refl\n  = yes (cong (\u03bb p \u2192 (w , p)) (uniqP p\u2081 p\u2082))\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\nproj\u2081-Injection : \u2200 {a b} {A : \u2605 a} {B : A \u2192  \u2605 b}\n                  \u2192 (\u2200 {x} (p\u2081 p\u2082 : B x) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 Injection (setoid (\u03a3 A B))\n                              (setoid A)\nproj\u2081-Injection {B = B} B-uniq\n     = record { to        = \u2192-to-\u27f6 (proj\u2081 {B = B})\n              ; injective = proj\u2081-injective B-uniq\n              }\n","old_contents":"-- move this to product\n{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Product.NP where\n\nopen import Level\nopen import Data.Product public hiding (\u2203)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nopen import Function.Injection using (Injection; module Injection)\nopen import Relation.Binary.Logical\n\n\u2203 : \u2200 {a b} {A : Set a} \u2192 (A \u2192 Set b) \u2192 Set (a \u2294 b)\n\u2203 = \u03a3 _\n\nfirst : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n          (f : A \u2192 B) \u2192 A \u00d7 C \u2192 B \u00d7 C\nfirst f (x , y) = (f x , y)\n\nsecond : \u2200 {a p q} {A : Set a} {P : A \u2192 Set p} {Q : A \u2192 Set q} \u2192\n           (\u2200 {x} \u2192 P x \u2192 Q x) \u2192 \u03a3 A P \u2192 \u03a3 A Q\nsecond f (x , y) = (x , f y)\n\nsecond\u2032 : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n           (B \u2192 C) \u2192 A \u00d7 B \u2192 A \u00d7 C\nsecond\u2032 f = second f\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\nprivate\n\n  Dec\u27e6Set\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n               (A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082)\n             \u2192 Set _\n  Dec\u27e6Set\u27e7 A\u1d63 = \u2200 x\u2081 x\u2082 \u2192 Dec (A\u1d63 x\u2081 x\u2082)\n\n  Dec\u27e6Pred\u27e7 : \u2200 {a\u2081 a\u2082 p\u2081 p\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n                (A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082)\n                {P\u2081 : Pred p\u2081 A\u2081} {P\u2082 : Pred p\u2082 A\u2082} {p\u1d63}\n              \u2192 (P\u1d63 : \u27e6Pred\u27e7 p\u1d63 A\u1d63 P\u2081 P\u2082) \u2192 Set _\n  Dec\u27e6Pred\u27e7 {A\u2081 = A\u2081} {A\u2082} A\u1d63 {P\u2081} {P\u2082} P\u1d63 = \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) y\u2081 y\u2082 \u2192 Dec (P\u1d63 x\u1d63 y\u2081 y\u2082) -- Dec\u27e6Set\u27e7 A\u1d63 \u21d2 Dec\u27e6Set\u27e7 P\u1d63\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n           {B\u2081 : A\u2081 \u2192 Set b\u2081}\n           {B\u2082 : A\u2082 \u2192 Set b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : Set (a\u1d63 \u2294 b\u1d63) where\n  constructor _\u27e6,\u27e7_\n  field\n    \u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082)\n    \u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _\u27e6,\u27e7_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u00d7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63}\n        {B\u2081 : A\u2081 \u2192 Set b\u2081}\n        {B\u2082 : A\u2082 \u2192 Set b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 Set _ -- (a\u1d63 \u2294 b\u1d63)\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n          {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 Set (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 Set _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082) \u00d7\n                        B\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n       {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082}\n       {A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n       {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082}\n       {A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082))\n       (\u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\ndec\u27e6\u03a3\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63 A\u2081 A\u2082 B\u2081 B\u2082}\n       {A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 {b\u2081} {b\u2082} b\u1d63 A\u1d63 B\u2081 B\u2082}\n       (decA\u1d63 : Dec\u27e6Set\u27e7 A\u1d63)\n       -- (substA\u1d63 : \u2200 {x y \u2113} {p q} (P : A\u1d63 x y \u2192 Set \u2113) \u2192 P p \u2192 P q)\n       (uniqA\u1d63 : \u2200 {x y} (p q : A\u1d63 x y) \u2192 p \u2261 q)\n       (decB\u1d63 : Dec\u27e6Pred\u27e7 A\u1d63 {_} {_} {b\u1d63 {-BUG: with _ here Agda loops-}} B\u1d63)\n     \u2192 Dec\u27e6Set\u27e7 (\u27e6\u03a3\u27e7 A\u1d63 B\u1d63)\ndec\u27e6\u03a3\u27e7 {B\u1d63 = B\u1d63} decA\u1d63 uniqA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 x\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 f \u2218 \u27e6proj\u2082\u27e7)\n  where f : \u2200 {x\u1d63'} \u2192 B\u1d63 x\u1d63' y\u2081 y\u2082 \u2192 B\u1d63 x\u1d63 y\u2081 y\u2082\n        f {x\u1d63'} y\u1d63 rewrite uniqA\u1d63 x\u1d63' x\u1d63 = y\u1d63\n\nmodule Dec\u2082 where\n  map\u2082\u2032 : \u2200 {p\u2081 p\u2082 q} {P\u2081 : Set p\u2081} {P\u2082 : Set p\u2082} {Q : Set q}\n          \u2192 (P\u2081 \u2192 P\u2082 \u2192 Q) \u2192 (Q \u2192 P\u2081) \u2192 (Q \u2192 P\u2082) \u2192 Dec P\u2081 \u2192 Dec P\u2082 \u2192 Dec Q\n  map\u2082\u2032 _   \u03c0\u2081  _   (no \u00acp\u2081)  _         = no (\u00acp\u2081 \u2218 \u03c0\u2081)\n  map\u2082\u2032 _   _   \u03c0\u2082  _         (no \u00acp\u2082)  = no (\u00acp\u2082 \u2218 \u03c0\u2082)\n  map\u2082\u2032 mk  _   _   (yes p\u2081)  (yes p\u2082)  = yes (mk p\u2081 p\u2082)\n\ndec\u27e6\u00d7\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n           {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n           {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082}\n           {A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082}\n           {B\u1d63 : \u27e6Set\u27e7 b\u1d63 B\u2081 B\u2082}\n           (decA\u1d63 : Dec\u27e6Set\u27e7 A\u1d63)\n           (decB\u1d63 : Dec\u27e6Set\u27e7 B\u1d63)\n         \u2192 Dec\u27e6Set\u27e7 (A\u1d63 \u27e6\u00d7\u27e7 B\u1d63)\ndec\u27e6\u00d7\u27e7 decA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 \u27e6proj\u2082\u27e7)\n\n\n\u03a3,-injective\u2082 : \u2200 {a b} {A : Set a} {B : A \u2192 Set b} {x : A} {y z : B x} \u2192 ((x , y) \u2236 \u03a3 A B) \u2261 (x , z) \u2192 y \u2261 z\n\u03a3,-injective\u2082 refl = refl\n\n\u03a3,-injective\u2081 : \u2200 {a b} {A : Set a} {B : A \u2192 Set b} {x\u2081 x\u2082 : A} {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} \u2192 ((x\u2081 , y\u2081) \u2236 \u03a3 A B) \u2261 (x\u2082 , y\u2082) \u2192 x\u2081 \u2261 x\u2082\n\u03a3,-injective\u2081 refl = refl\n\nproj\u2081-injective : \u2200 {a b} {A : Set a} {B : A \u2192 Set b} {x y : \u03a3 A B}\n                    (B-uniq : \u2200 {z} (p\u2081 p\u2082 : B z) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 proj\u2081 x \u2261 proj\u2081 y \u2192 x \u2261 y\nproj\u2081-injective {x = (a , p\u2081)} {y = (_ , p\u2082)} B-uniq eq rewrite sym eq\n  = cong (\u03bb p \u2192 (a , p)) (B-uniq p\u2081 p\u2082)\n\nproj\u2082-irrelevance : \u2200 {a b} {A : Set a} {B C : A \u2192 Set b} {x\u2081 x\u2082 : A}\n                      {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} {z\u2081 : C x\u2081} {z\u2082 : C x\u2082}\n                    \u2192 (C-uniq : \u2200 {z} (p\u2081 p\u2082 : C z) \u2192 p\u2081 \u2261 p\u2082)\n                    \u2192 ((x\u2081 , y\u2081) \u2236 \u03a3 A B) \u2261 (x\u2082 , y\u2082)\n                    \u2192 ((x\u2081 , z\u2081) \u2236 \u03a3 A C) \u2261 (x\u2082 , z\u2082)\nproj\u2082-irrelevance C-uniq = proj\u2081-injective C-uniq \u2218 \u03a3,-injective\u2081\n\n\u225f\u03a3 : \u2200 {A : Set} {P : A \u2192 Set}\n       (decA : Decidable {A = A} _\u2261_)\n       (decP : \u2200 x \u2192 Decidable {A = P x} _\u2261_)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl with decP w p\u2081 p\u2082\n\u225f\u03a3 decA decP (w  , p)  (.w , .p) | yes refl | yes refl = yes refl\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl | no  p\u2262\n    = no (p\u2262 \u2218 \u03a3,-injective\u2082)\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\n\u225f\u03a3' : \u2200 {A : Set} {P : A \u2192 Set}\n       (decA : Decidable {A = A} _\u2261_)\n       (uniqP : \u2200 {x} (p q : P x) \u2192 p \u2261 q)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3' decA uniqP (w  , p\u2081) (.w , p\u2082) | yes refl\n  = yes (cong (\u03bb p \u2192 (w , p)) (uniqP p\u2081 p\u2082))\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\nproj\u2081-Injection : \u2200 {a b} {A : Set a} {B : A \u2192  Set b}\n                  \u2192 (\u2200 {x} (p\u2081 p\u2082 : B x) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 Injection (setoid (\u03a3 A B))\n                              (setoid A)\nproj\u2081-Injection {B = B} B-uniq\n     = record { to        = \u2192-to-\u27f6 (proj\u2081 {B = B})\n              ; injective = proj\u2081-injective B-uniq\n              }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ce1224ca15777f0fc4fe2d907807a557fb1bd21d","subject":"restructuring the ana proof to avoid redoing all the work of synth.","message":"restructuring the ana proof to avoid redoing all the work of synth.\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081) with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {\u0393} {c} (ASubsume D x\u2081)             | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {e \u00b7: x} (ASubsume D x\u2081)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {X x} (ASubsume D x\u2081)           | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {\u00b7\u03bb x e} (ASubsume D x\u2081)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {\u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082) | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {\u0393} {e \u2218 e\u2081} (ASubsume D x\u2081)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {\u0393} {\u2987\u2988[ x ]} (ASubsume _ _ )       | _ , _ , _ = _ , _ , _ , EAEHole\n    expandability-ana {\u0393} {\u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _ = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume wt x\u2081) =  _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e' \u2988[ x ]} (ASubsume (SNEHole wt) x\u2081)\n      with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EANEHole D\n    expandability-ana {e = c} (ASubsume SConst x\u2081) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ())  ESConst x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume (SAsc wt) x\u2082)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' with htype-dec x \u03c4'\n    expandability-ana {\u0393} {e \u00b7: .x} (ASubsume (SAsc wt) x\u2082) | d' , \u0394' , x , D' | Inl refl = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAsc2 D') x\u2082\n    expandability-ana {\u0393} {e \u00b7: x} (ASubsume (SAsc wt) x\u2082) | d' , \u0394' , \u03c4' , D' | Inr x\u2081 = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAsc1 D' x\u2081) x\u2082\n    expandability-ana {e = X x} (ASubsume (SVar x\u2081) x\u2082) = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESVar x\u2081) x\u2082\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume () x\u2081)\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume (SLam x\u2082 wt) x\u2083)\n      with expandability-synth wt\n    ... | d , \u0394 , D = _ , _ , _ , EASubsume (\u03bb u \u2192 \u03bb ()) (\u03bb e' u \u2192 \u03bb ()) (ESLam x\u2082 D) x\u2083\n    expandability-ana {e = e1 \u2218 e2} (ASubsume (SAp wt1 MAHole wt2) x\u2082)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2 = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) (ESAp1 {!!} wt1 {!!} D2) x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume (SAp wt MAArr x\u2081) x\u2082) = {!!}\n    expandability-ana (ALam x\u2081 MAHole wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt)\n      with expandability-ana wt\n    ... | d' , \u0394'\u202f, \u03c4' , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bd055de40203e0fece52da219f7f957370bdf167","subject":"Proved that curry and uncurry from a bijection.","message":"Proved that curry and uncurry from a bijection.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = let h = \u03bb r \u2192 fst (f (fst r)) (snd r)\n        H = \u03bb r z \u2192 F (fst r) (snd r , z) , snd (f (fst r)) (snd r) z\n     in h , (H , cond)\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Z} \u2192\n      (\u03b1 \u2297\u1d63 \u03b2) u (F (fst u) (snd u , y) , snd (f (fst u)) (snd u) y) \u2192\n      \u03b3 (fst (f (fst u)) (snd u)) y\n  cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u} {v , z}\n  ... | p\u2084 with f u\n  ... | i , I = p\u2084 p\u2082 p\u2083\n  \ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   aux\u2082 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 (\u03bb z \u2192 fst (F (fst a , snd a) z) , snd (F (fst a , snd a) z)) \u2261 F a\n   aux\u2082 {u , v} = ext-set aux\u2083\n    where\n      aux\u2083 : {a : Z} \u2192 (fst (F (u , v) a) , snd (F (u , v) a)) \u2261 F (u , v) a\n      aux\u2083 {z} with F (u , v) z\n      ... | x , y = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = (ext-set aux) , ext-set (ext-set aux')\n where\n  aux : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb v z \u2192 snd (g a) v z)) \u2261 g a\n  aux {u} with g u\n  ... | i , I = refl\n  aux' : {u : U}{r : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G u (fst r , snd r) \u2261 G u r\n  aux' {u}{v , z} = refl\n{-\n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = h , (H , cond)\n  where\n  h : \u03a3 U (\u03bb x \u2192 V) \u2192 W\n  h (u , v) with f u\n  ... | i , I = i v\n  H : \u03a3 U (\u03bb x \u2192 V) \u2192 Z \u2192 \u03a3 X (\u03bb x \u2192 Y)\n  H (u , v) z with f u\n  ... | i , I = F u (v , z) , I v z\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {z : Z} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (H u z) \u2192 \u03b3 (h u) z\n  cond {u , v}{z} r with (p\u2081 {u}{v , z})\n  ... | p\u2084 with f u\n  ... | i , I with r\n  ... | p\u2082 , p\u2083 = p\u2084 p\u2082 p\u2083\n  \n{-\ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   \n   aux\u2082 : {a : Z} \u2192 ((\u03bb v \u2192 fst (F a) v) , (\u03bb u \u2192 snd (F a) u)) \u2261 F a\n   aux\u2082 {z} with F z\n   ... | j\u2081 , j\u2082 = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb z \u2192 snd (g a) z)) \u2261 g a\n   aux\u2081 {u} with g u\n   ... | (j\u2081 , j\u2082) = refl\n\n   aux\u2082 : {a : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G (fst a , snd a) \u2261 G a\n   aux\u2082 {v , z} = refl\n   \n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set}\n  \u2192 (U \u2192 X \u2192 Set)\n  \u2192 U\n  \u2192 (U \u2192 X *)\n  \u2192 Set\n!\u2092-cond \u03b1 u f = all-pred (\u03b1 u) (f u)\n   \n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U , (U \u2192 X *) , !\u2092-cond \u03b1\n\n!-cta : {V Y U X : Set}\n  \u2192 (Y \u2192 X)\n  \u2192 (U \u2192 V)\n  \u2192 (V \u2192 Y *)\n  \u2192 (U \u2192 X *)\n!-cta F f g = \u03bb u \u2192 list-funct F (g (f u))\n\n!\u2090-cond : \u2200{U V Y X : Set}{F : Y \u2192 X}{f : U \u2192 V}\n  \u2192 (\u03b1 : U \u2192 X \u2192 Set)\n  \u2192 (\u03b2 : V \u2192 Y \u2192 Set)\n  \u2192 (p : {u : U} {y : Y} \u2192 \u03b1 u (F y) \u2192 \u03b2 (f u) y)\n    {u : U}{l : Y *}\n  \u2192 all-pred (\u03b1 u) (list-funct F l)\n  \u2192 all-pred (\u03b2 (f u)) l\n!\u2090-cond _ _ _ {l = []} _ = triv\n!\u2090-cond \u03b1 \u03b2 p {u}{x :: xs} (p' , p'') = p p' , !\u2090-cond \u03b1 \u03b2 p p''     \n      \n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , !-cta F f , !\u2090-cond \u03b1 \u03b2 p\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb x y \u2192 [ x ]) , fst\n\n\u03b4-cta : {U X : Set} \u2192 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X)) \u2192 U \u2192 \ud835\udd43 X\n\u03b4-cta g u = foldr (\u03bb f rest \u2192 (f u) ++ rest) [] (g u)\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-cta , \u03b4-cond\n  where\n   \u03b4-cond : {u : U} {l : \ud835\udd43 (U \u2192 \ud835\udd43 X)}\n     \u2192 all-pred (\u03b1 u) (foldr (\u03bb f \u2192 _++_ (f u)) [] l)\n     \u2192 all-pred (\u03bb f\n     \u2192 all-pred (\u03b1 u) (f u)) l\n   \u03b4-cond {l = []} _ = triv\n   \u03b4-cond {u}{l = x :: l'} p with\n     all-pred-append {X}{\u03b1 u}\n                     {x u}\n                     {foldr (\u03bb f \u2192 _++_ (f u)) [] l'}\n                     \u2227-unit \u2227-assoc\n   ... | p' rewrite p' = fst p , \u03b4-cond {u} {l'} (snd p)\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {a} \u2192 ext-set (\u03bb {a\u2081} \u2192 aux {a\u2081}{a a\u2081}))\n where\n   aux : \u2200{a\u2081 : U}{l : \ud835\udd43 (U \u2192 \ud835\udd43 (U \u2192 \ud835\udd43 X))} \u2192\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) []\n      (map (\u03bb g u \u2192 foldr (\u03bb f \u2192 _++_ (f u)) [] (g u)) l)\n      \u2261\n      foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] (foldr (\u03bb f \u2192 _++_ (f a\u2081)) [] l)\n   aux {a}{[]} = refl  \n   aux {a}{x :: l} rewrite\n     sym (foldr-append {l\u2081 = x a}{foldr (\u03bb f \u2192 _++_ (f a)) [] l}{a})\n     = cong2 {a = foldr (\u03bb f \u2192 _++_ (f a)) [] (x a)}\n             _++_\n             refl\n             (aux {a}{l})\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1fde24fb24ccef8f08e65f7497a7ec3cd7eee960","subject":"Ensure README.agda imports everything (fix #41)","message":"Ensure README.agda imports everything (fix #41)\n\nThis was broken in 49e8b38c23e9c250b738a0ddba6384e6268034bd, so add a\ncomment explaining why that line needs to be there.\n\nI verified that the code still typechecks on my machine.\n\nOld-commit-hash: 36ff398aab5149e6b6845d2c6ea1f05ab8d97bf8\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n----------------------------\n-- INCREMENTAL \u03bb-CALCULUS --\n----------------------------\n--\n--\n-- IMPORTANT FILES\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda formalization.\n--\n-- PLDI14-List-of-Theorems.agda\n--   For each theorem, lemma or definition in the PLDI 2014 submission,\n--   it points to the corresponding Agda object.\n--\n--\n-- LOCATION OF AGDA MODULES\n--\n-- To find the file containing an Agda module, replace the dots in its\n-- full name by directory separators. The result is the file's path relative\n-- to this directory. For example, Parametric.Syntax.Type is defined in\n-- Parametric\/Syntax\/Type.agda.\n--\n--\n-- HOW TO TYPE CHECK EVERYTHING\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n\n-- Import everything else. This ensures that typechecking README.agda typechecks\n-- the entire codebase, because Everything.agda is auto-generated.\nimport Everything\n","old_contents":"module README where\n\n----------------------------\n-- INCREMENTAL \u03bb-CALCULUS --\n----------------------------\n--\n--\n-- IMPORTANT FILES\n--\n-- modules.pdf\n--   The graph of dependencies between Agda modules.\n--\n-- README.agda\n--   This file. A coarse-grained introduction to the Agda formalization.\n--\n-- PLDI14-List-of-Theorems.agda\n--   For each theorem, lemma or definition in the PLDI 2014 submission,\n--   it points to the corresponding Agda object.\n--\n--\n-- LOCATION OF AGDA MODULES\n--\n-- To find the file containing an Agda module, replace the dots in its\n-- full name by directory separators. The result is the file's path relative\n-- to this directory. For example, Parametric.Syntax.Type is defined in\n-- Parametric\/Syntax\/Type.agda.\n--\n--\n-- HOW TO TYPE CHECK EVERYTHING\n--\n-- To typecheck this formalization, you need to install the appropriate version\n-- of Agda, the Agda standard library (version 0.7), generate Everything.agda\n-- with the attached Haskell helper, and finally run Agda on this file.\n--\n-- Given a Unix-like environment (including Cygwin), running the .\/agdaCheck.sh\n-- script and following instructions given on output will eventually generate\n-- Everything.agda and proceed to type check everything on command line.\n--\n-- We use Agda HEAD from September 2013; Agda 2.3.2.1 might happen to work, but\n-- has some bugs with serialization of code using some recent syntactic sugar\n-- which we use (https:\/\/code.google.com\/p\/agda\/issues\/detail?id=756), so it\n-- might work or not. When it does not, removing Agda caches (.agdai files)\n-- appears to often help. You can use \"find . -name '*.agdai' | xargs rm\" to do\n-- that.\n\nimport Postulate.Extensionality\n\nimport Base.Data.DependentList\n\n-- Variables and contexts\nimport Base.Syntax.Context\n\n-- Sets of variables\nimport Base.Syntax.Vars\n\nimport Base.Denotation.Notation\n\n-- Environments\nimport Base.Denotation.Environment\n\n-- Change contexts\nimport Base.Change.Context\n\n-- # Base, parametric proof.\n--\n-- This is for a parametric calculus where:\n-- types are parametric in base types\n-- terms are parametric in constants\n--\n--\n-- Modules are ordered and grouped according to what they represent.\n\n-- ## Definitions\n\nimport Parametric.Syntax.Type\nimport Parametric.Syntax.Term\n\nimport Parametric.Denotation.Value\nimport Parametric.Denotation.Evaluation\n\nimport Parametric.Change.Type\nimport Parametric.Change.Term\n\nimport Parametric.Change.Derive\n\nimport Parametric.Change.Value\nimport Parametric.Change.Evaluation\n\n-- ## Proofs\n\nimport Parametric.Change.Validity\nimport Parametric.Change.Specification\nimport Parametric.Change.Implementation\nimport Parametric.Change.Correctness\n\n-- # Nehemiah plugin\n--\n-- The structure is the same as the parametric proof (down to the\n-- order and the grouping of modules), except for the postulate module.\n\n-- Postulate an abstract data type for integer Bags.\nimport Postulate.Bag-Nehemiah\n\n-- ## Definitions\nimport Nehemiah.Syntax.Type\nimport Nehemiah.Syntax.Term\n\nimport Nehemiah.Denotation.Value\nimport Nehemiah.Denotation.Evaluation\n\nimport Nehemiah.Change.Term\nimport Nehemiah.Change.Type\n\nimport Nehemiah.Change.Derive\n\nimport Nehemiah.Change.Value\nimport Nehemiah.Change.Evaluation\n\n-- ## Proofs\n\nimport Nehemiah.Change.Validity\nimport Nehemiah.Change.Specification\nimport Nehemiah.Change.Implementation\nimport Nehemiah.Change.Correctness\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6b29ddad85f077eb4bde6c708ffa6d5fcb4b1838","subject":"\u2026 is now called it","message":"\u2026 is now called it\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/InstanceArguments.agda","new_file":"lib\/Function\/InstanceArguments.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Type hiding (\u2605)\n\nmodule Function.InstanceArguments where\n\nexplicitize : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} \u2192\n                (\u2983 x : A \u2984 \u2192 B x) \u2192 (x : A) \u2192 B x\nexplicitize f x = f \u2983 x \u2984\n\nit : \u2200 {a} {A : \u2605 a} \u2983 x : A \u2984 \u2192 A\nit \u2983 x \u2984 = x\n\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Type hiding (\u2605)\n\nmodule Function.InstanceArguments where\n\nexplicitize : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} \u2192\n                (\u2983 x : A \u2984 \u2192 B x) \u2192 (x : A) \u2192 B x\nexplicitize f x = f \u2983 x \u2984\n\n\u2026 : \u2200 {a} {A : \u2605 a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3c17168aff6b71912f30f7d6a6e212b10de62c23","subject":"working on contextual dynamics transcription","message":"working on contextual dynamics transcription\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               (\u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9) ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  -- d is result of filling the hole in \u03b5 with d'\n  data _==_[_] : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 [ d ]\n    -- FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n    --        d1 == \u03b5 [ d ] \u2192\n    --        (\u00b7\u03bb x [ \u03c4 ] d1) == ?\n    -- FAp1 :\n    -- FAp2 :\n    -- FNEHole : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n    --        d1 == \u03b5 [ d ] \u2192\n    --        (\u2987 d1 \u2988\u27e8 u , \u03c3 , m\u27e9) == \u2987 d \u2988\u27e8 u , \u03c3 , m\u27e9\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n           d1 == \u03b5 [ d ] \u2192\n           (< \u03c4 > d1) == < \u03c4 > \u03b5 [ d ]\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 [ d0 ] \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 [ d0' ] \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7e41bea4c8fdf5515c5d163b86c4b01ce7e0e399","subject":"Add classification","message":"Add classification\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or _      = true\n_ or true      = true\nfalse or false = false\n\n_and_ : Bool \u2192 Bool \u2192 Bool\nfalse and _   = false\n_ and false   = false\ntrue and true = true\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\napply : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\napply _ \u2218        = \u2218\napply f (x \u2237 xs) = (f x) \u2237 (apply f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q \u2261 s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p \u2261 r) and (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 \u2218        = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) and (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea \u2218 \u27eb arr = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) or (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue or true   = true\ntrue or false  = true\nfalse or true  = true\nfalse or false = false\n\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_\u2261_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero \u2261 zero   = true\nsuc n \u2261 suc m = n \u2261 m\n_ \u2261 _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  \u2218   : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ \u2218        = false\nany f (x \u2237 xs) = (f x) or (any f xs)\n\n\napply : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\napply _ \u2218        = \u2218\napply f (x \u2237 xs) = (f x) \u2237 (apply f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 \u2218        = false\nx \u2208 (y \u2237 ys) with x \u2261 y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure \u2218 found                = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) or (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs \u2218)\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs (n \u2237 \u2218))) fails\n\n\n_,_,_\u00ac\u22a8_ : List Separation \u2192 List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\n--(m \u2237 ms) , cs , ps \u00ac\u22a8 n    with (modelsupports m cs \n\n\n_,_\u00ac\u22a8_ : List Separation \u2192 List Arrow \u2192 Arrow \u2192 Bool\n\n\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"60f1b914630d73e6a30d358e9a1a6e68a3214f78","subject":"Added an example.","message":"Added an example.\n\nIgnore-this: 985eb6502f1fc33826fd4223f222a769\n\ndarcs-hash:20100804170305-3bd4e-7fc9f285102f43cb8694ff2077dff0badde0cd3f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Nat\/PropertiesByInduction.agda","new_file":"LTC\/Data\/Nat\/PropertiesByInduction.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the LTC natural numbers)\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.PropertiesByInduction where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the LTC natural numbers. The following\n-- example shows a proof using it.\n\n-- Proof without use definitions\n+-assocND : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assocND {n = n} {o} Nm _ _ = indN P P0 iStep Nm\n  where\n    P : D \u2192 Set\n    P i = i + n + o \u2261 i + (n + o)\n\n    postulate\n      P0 : zero + n + o \u2261 zero + (n + o)\n    {-# ATP prove P0 #-} -- We use the ATP systems to prove the base case.\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192\n              i + n + o \u2261 i + (n + o) \u2192 -- IH.\n              succ i + n + o \u2261 succ i + (n + o)\n    {-# ATP prove iStep #-} -- We use the ATP systems to prove the\n                            -- induction step.\n\n-- Proof using definitions\n+-assocD : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 m + n + o \u2261 m + (n + o)\n+-assocD {n = n} {o} Nm _ _ = indN P P0 iStep Nm\n  where\n    P : D \u2192 Set\n    P i = i + n + o \u2261 i + (n + o)\n    {-# ATP definition P #-}\n\n    postulate\n      P0 : P zero\n    {-# ATP prove P0 #-} -- We use the ATP systems to prove the base case.\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192 P i \u2192 P (succ i)\n    {-# ATP prove iStep #-} -- We use the ATP systems to prove the\n                            -- induction step.\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the LTC natural numbers)\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.PropertiesByInduction where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the LTC natural numbers. The following\n-- example shows a proof using it.\n+-assoc : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m + n) + o \u2261 m + (n + o)\n+-assoc {m} {n} {o} Nm Nn No = indN P P0 iStep Nm\n  where\n    P : D \u2192 Set\n    P i = i + n + o \u2261 i + (n + o)\n\n    postulate\n      P0 : zero + n + o \u2261 zero + (n + o)\n    {-# ATP prove P0 #-} -- We use the ATP systems to prove the base case.\n\n    postulate\n      iStep : {i : D} \u2192 N i \u2192\n              i + n + o \u2261 i + (n + o) \u2192 -- IH.\n              succ i + n + o \u2261 succ i + (n + o)\n    {-# ATP prove iStep #-} -- We use the ATP systems to prove the\n                            -- induction step.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5e583450a3104dc6cb3ffb9b3876a297c6a9baf8","subject":"Document P.C.Value.","message":"Document P.C.Value.\n\nOld-commit-hash: 1a3643a6990f5380e0a80e97aa83d76b6a2e2244\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Value.agda","new_file":"Parametric\/Change\/Value.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The values of terms in Parametric.Change.Term.\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Value\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen ChangeType.Structure Base \u0394Base\n\nopen import Base.Denotation.Notation\n\n-- Extension point 1: The value of \u2295 for base types.\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base\n\n-- Extension point 2: The value of \u229d for base types.\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base\n\nmodule Structure\n    (\u27e6apply-base\u27e7 : ApplyStructure)\n    (\u27e6diff-base\u27e7 : DiffStructure)\n  where\n\n  -- We provide: The value of \u2295 and \u229d for arbitrary types.\n  \u27e6apply\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  \u27e6diff\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n  infixl 6 \u27e6apply\u27e7 \u27e6diff\u27e7\n  syntax \u27e6apply\u27e7 \u03c4 v dv = v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 dv\n  syntax \u27e6diff\u27e7 \u03c4 u v = u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v = \u27e6apply-base\u27e7 \u03b9 v \u0394v\n  f \u27e6\u2295\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 \u0394f = \u03bb v \u2192 f v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394f v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n\n  u \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 v = \u27e6diff-base\u27e7 \u03b9 u v\n  g \u27e6\u229d\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 f = \u03bb v \u0394v \u2192 (g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v)) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 (f v)\n\n  _\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u27e6\u2295\u27e7_ {\u03c4} = \u27e6apply\u27e7 \u03c4\n\n  _\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u27e6\u229d\u27e7_ {\u03c4} = \u27e6diff\u27e7 \u03c4\n\n  alternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 \u27e6 \u0394Context \u0393 \u27e7\n  alternate {\u2205} \u2205 \u2205 = \u2205\n  alternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Value\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen ChangeType.Structure Base \u0394Base\n\nopen import Base.Denotation.Notation\n\n-- `diff` and `apply`, without validity proofs\n\nApplyStructure : Set\nApplyStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base\n\nDiffStructure : Set\nDiffStructure = \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base\n\nmodule Structure\n    (\u27e6apply-base\u27e7 : ApplyStructure)\n    (\u27e6diff-base\u27e7 : DiffStructure)\n  where\n\n  \u27e6apply\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  \u27e6diff\u27e7 : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n\n  infixl 6 \u27e6apply\u27e7 \u27e6diff\u27e7\n  syntax \u27e6apply\u27e7 \u03c4 v dv = v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 dv\n  syntax \u27e6diff\u27e7 \u03c4 u v = u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v = \u27e6apply-base\u27e7 \u03b9 v \u0394v\n  f \u27e6\u2295\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 \u0394f = \u03bb v \u2192 f v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394f v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)\n\n  u \u27e6\u229d\u208d base \u03b9 \u208e\u27e7 v = \u27e6diff-base\u27e7 \u03b9 u v\n  g \u27e6\u229d\u208d \u03c3 \u21d2 \u03c4 \u208e\u27e7 f = \u03bb v \u0394v \u2192 (g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394v)) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 (f v)\n\n  _\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n  _\u27e6\u2295\u27e7_ {\u03c4} = \u27e6apply\u27e7 \u03c4\n\n  _\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394Type \u03c4 \u27e7\n  _\u27e6\u229d\u27e7_ {\u03c4} = \u27e6diff\u27e7 \u03c4\n\n  alternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 \u27e6 \u0394Context \u0393 \u27e7\n  alternate {\u2205} \u2205 \u2205 = \u2205\n  alternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5648f104edb43aa82ca314438ba07e63d10e54be","subject":"Butcher it so it compiles","message":"Butcher it so it compiles\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"open import Data.Bool.Base using (Bool)\n------------------------------------------------------------------------\n-- The Agda standard library\n--\n-- IO\n------------------------------------------------------------------------\n\nopen import Coinduction\nopen import Data.Unit\nopen import Data.String\nopen import Data.Colist\nopen import Function\nimport IO.Primitive as Prim\nopen import Level\n\n------------------------------------------------------------------------\n-- The IO monad\n\n-- One cannot write \"infinitely large\" computations with the\n-- postulated IO monad in IO.Primitive without turning off the\n-- termination checker (or going via the FFI, or perhaps abusing\n-- something else). The following coinductive deep embedding is\n-- introduced to avoid this problem. Possible non-termination is\n-- isolated to the run function below.\n\ninfixl 1 _>>=_ _>>_\n\ndata IO {a} (A : Set a) : Set (suc a) where\n  lift   : (m : Prim.IO A) \u2192 IO A\n  return : (x : A) \u2192 IO A\n  _>>=_  : {B : Set a} (m : \u221e (IO B)) (f : (x : B) \u2192 \u221e (IO A)) \u2192 IO A\n  _>>_   : {B : Set a} (m\u2081 : \u221e (IO B)) (m\u2082 : \u221e (IO A)) \u2192 IO A\n\n{-# NON_TERMINATING #-}\n\nrun : \u2200 {a} {A : Set a} \u2192 IO A \u2192 Prim.IO A\nrun (lift m)   = m\nrun (return x) = Prim.return x\nrun (m  >>= f) = Prim._>>=_ (run (\u266d m )) \u03bb x \u2192 run (\u266d (f x))\nrun (m\u2081 >> m\u2082) = Prim._>>=_ (run (\u266d m\u2081)) \u03bb _ \u2192 run (\u266d m\u2082)\n\nputStr\u221e : Costring \u2192 IO \u22a4\nputStr\u221e s =\n  \u266f lift (Prim.putStr s) >>\n  \u266f return _\n\nputStr : String \u2192 IO \u22a4\nputStr s = putStr\u221e (toCostring s)\n\nputStrLn\u221e : Costring \u2192 IO \u22a4\nputStrLn\u221e s =\n  \u266f lift (Prim.putStrLn s) >>\n  \u266f return _\n\nputStrLn : String \u2192 IO \u22a4\nputStrLn s = putStrLn\u221e (toCostring s)\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\nBool.true \u2228 _  = Bool.true\nBool.false \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nBool.false \u2227 _ = Bool.false\nBool.true \u2227 b  = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif Bool.true  then a else _ = a\nif Bool.false then _ else b = b\n\n\n----------------------------------------\n\n_>>>_ : String \u2192 String \u2192 String\n_>>>_ = primStringAppend\n\ninfixl 1 _>>>_\n\n\n\n----------------------------------------\n\ndata Principle : Set where\n  # : String \u2192 Principle\n\n\n_=p=_ : Principle \u2192 Principle \u2192 Bool\n(# a) =p= (# b) = a == b\n\n----------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[[_]] : {A : Set} \u2192 A \u2192 List A\n[[ x ]] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = Bool.false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208\u2208_ : Principle \u2192 List Principle \u2192 Bool\nx \u2208\u2208 []       = Bool.false\nx \u2208\u2208 (y \u2237 ys) with x =p= y\n...             | Bool.true  = Bool.true\n...             | Bool.false = x \u2208\u2208 ys\n\n\n_\u220b\u220b_ : List Principle \u2192 Principle \u2192 Bool\nxs \u220b\u220b y = y \u2208\u2208 xs\n\n\n\n----------------------------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : Principle \u2192 Arrow\n  _\u21d2_ : Principle \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q =p= s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p =p= r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = Bool.false\n\n\n_\u2208\u2208\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208\u2208\u2208 []       = Bool.false\nx \u2208\u2208\u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | Bool.true  = Bool.true\n...              | Bool.false = x \u2208\u2208\u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List Principle \u2192 List Principle\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208\u2208 found) \u2228 (n \u2208\u2208 (closure rest found))\n...                               | Bool.true  = closure (q \u2237 rest) found\n...                               | Bool.false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List Principle \u2192 Principle \u2192 Bool\ncs , ps \u22a2 q = q \u2208\u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208\u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List Principle \u2192 List Principle \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 Principle \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 Principle \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b\u220b_ (closure cs ([[ n ]]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = Bool.false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208\u2208\u2208 cs)\n...                   | Bool.true  = Proved\n...                   | Bool.false with (cs \u22a2 arr)\n...                              | Bool.true  = Derivable\n...                              | Bool.false with (cs \u27ea ms \u27eb arr)\n...                                         | Bool.true  = Separated\n...                                         | Bool.false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   ((# \"3\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n--   ((# \"5\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"10\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"11\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"11\"))) \u2237\n   ((# \"11\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"8\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"5\") \u21d2 ((# \"3\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"6\") \u21d2 ((# \"7\") \u21d2 (\u21d2 (# \"10\")))) \u2237\n   ((# \"6\") \u21d2 ((# \"3\") \u21d2 (\u21d2 (# \"3\")))) \u2237\n   ((# \"7\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"7\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"7\") \u21d2 ((# \"8\") \u21d2 (\u21d2 (# \"10\")))) \u2237\n   ((# \"3\") \u21d2 ((# \"10\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"1\"))) \u2237\n   ((# \"3\") \u21d2 ((# \"1\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"1\") \u21d2 (\u21d2 (# \"2\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"2\"))) \u2237 []\n\ncms : List Separation\ncms =\n  (model ((# \"12\") \u2237 (# \"6\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"1\") \u2237 []) ((# \"5\") \u2237 (# \"3\") \u2237 (# \"7\") \u2237 (# \"7\") \u2237 [])) \u2237\n  (model ((# \"6\") \u2237 (# \"3\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"3\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"5\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"5\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"1\") \u2237 []) ((# \"6\") \u2237 (# \"3\") \u2237 [])) \u2237\n  (model ((# \"5\") \u2237 (# \"3\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"3\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"6\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 []) ((# \"5\") \u2237 (# \"6\") \u2237 (# \"3\") \u2237 (# \"8\") \u2237 (# \"9\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 (# \"1\") \u2237 []) ((# \"3\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 (# \"7\") \u2237 []) ((# \"9\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"10\") \u2237 (# \"8\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"10\") \u2237 (# \"9\") \u2237 []) ((# \"1\") \u2237 [])) \u2237\n  (model ((# \"3\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 []) ((# \"9\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"10\") \u2237 (# \"7\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"7\") \u2237 (# \"1\") \u2237 []) ((# \"4\") \u2237 (# \"11\") \u2237 (# \"8\") \u2237 [])) \u2237\n  (model ((# \"9\") \u2237 (# \"3\") \u2237 (# \"10\") \u2237 (# \"8\") \u2237 (# \"1\") \u2237 []) ((# \"11\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"11\") \u2237 (# \"3\") \u2237 [])) \u2237\n  (model ((# \"3\") \u2237 (# \"6\") \u2237 (# \"5\") \u2237 []) ([])) \u2237\n  (model ((# \"1\") \u2237 (# \"2\") \u2237 (# \"3\") \u2237 (# \"4\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"11\") \u2237 []) ((# \"12\") \u2237 [])) \u2237 []\n\ntestp : Relation\ntestp = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"10\")))\n\ntestd : Relation\ntestd = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"4\")))\n\ntests : Relation\ntests = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"3\")))\n\ntestu : Relation\ntestu = consider proofs cms ((# \"6\") \u21d2 (\u21d2 (# \"1\")))\n\ntestcl : List Principle\ntestcl = closure proofs ((# \"5\") \u2237 [])\n\n\n\nstringifylp : List Principle \u2192 String\nstringifylp [] = \"[]\"\nstringifylp ((# s) \u2237 xs) = s >>> \" \u2237 \" >>> (stringifylp xs)\n\n\nmain = run (putStrLn (stringifylp testcl))\n","old_contents":"open import IO\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n-- {-# BUILTIN BOOL  Bool  #-}\n-- {-# BUILTIN TRUE  true  #-}\n-- {-# BUILTIN FALSE false #-}\n\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\ntrue \u2228 _  = true\nfalse \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nfalse \u2227 _ = false\ntrue \u2227 b  = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then a else _ = a\nif false then _ else b = b\n\n\n----------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\nprimitive\n  primStringAppend   : String \u2192 String \u2192 String\n  primStringEquality : String \u2192 String \u2192 Bool\n  primShowString     : String \u2192 String\n\n_>>>_ : String \u2192 String \u2192 String\n_>>>_ = primStringAppend\n\n_===_ : String \u2192 String \u2192 Bool\n_===_ = primStringEquality\n\ninfixl 1 _>>>_\n\n\n\n----------------------------------------\n\ndata Principle : Set where\n  # : String \u2192 Principle\n\n\n_==_ : Principle \u2192 Principle \u2192 Bool\n(# a) == (# b) = a === b\n\n----------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[_] : {A : Set} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208_ : Principle \u2192 List Principle \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x == y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List Principle \u2192 Principle \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n\n\n----------------------------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : Principle \u2192 Arrow\n  _\u21d2_ : Principle \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q == s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p == r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List Principle \u2192 List Principle\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List Principle \u2192 Principle \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List Principle \u2192 List Principle \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 Principle \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 Principle \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs ([ n ]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   ((# \"3\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n--   ((# \"5\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"10\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"11\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"11\"))) \u2237\n   ((# \"11\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"8\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"5\") \u21d2 ((# \"3\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"6\") \u21d2 ((# \"7\") \u21d2 (\u21d2 (# \"10\")))) \u2237\n   ((# \"6\") \u21d2 ((# \"3\") \u21d2 (\u21d2 (# \"3\")))) \u2237\n   ((# \"7\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"7\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"7\") \u21d2 ((# \"8\") \u21d2 (\u21d2 (# \"10\")))) \u2237\n   ((# \"3\") \u21d2 ((# \"10\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"1\"))) \u2237\n   ((# \"3\") \u21d2 ((# \"1\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"1\") \u21d2 (\u21d2 (# \"2\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"2\"))) \u2237 []\n\ncms : List Separation\ncms =\n  (model ((# \"12\") \u2237 (# \"6\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"1\") \u2237 []) ((# \"5\") \u2237 (# \"3\") \u2237 (# \"7\") \u2237 (# \"7\") \u2237 [])) \u2237\n  (model ((# \"6\") \u2237 (# \"3\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"3\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"5\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"5\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"1\") \u2237 []) ((# \"6\") \u2237 (# \"3\") \u2237 [])) \u2237\n  (model ((# \"5\") \u2237 (# \"3\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"3\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"6\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 []) ((# \"5\") \u2237 (# \"6\") \u2237 (# \"3\") \u2237 (# \"8\") \u2237 (# \"9\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 (# \"1\") \u2237 []) ((# \"3\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 (# \"7\") \u2237 []) ((# \"9\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"10\") \u2237 (# \"8\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"10\") \u2237 (# \"9\") \u2237 []) ((# \"1\") \u2237 [])) \u2237\n  (model ((# \"3\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 []) ((# \"9\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"10\") \u2237 (# \"7\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"7\") \u2237 (# \"1\") \u2237 []) ((# \"4\") \u2237 (# \"11\") \u2237 (# \"8\") \u2237 [])) \u2237\n  (model ((# \"9\") \u2237 (# \"3\") \u2237 (# \"10\") \u2237 (# \"8\") \u2237 (# \"1\") \u2237 []) ((# \"11\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"11\") \u2237 (# \"3\") \u2237 [])) \u2237\n  (model ((# \"3\") \u2237 (# \"6\") \u2237 (# \"5\") \u2237 []) ([])) \u2237\n  (model ((# \"1\") \u2237 (# \"2\") \u2237 (# \"3\") \u2237 (# \"4\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"11\") \u2237 []) ((# \"12\") \u2237 [])) \u2237 []\n\ntestp : Relation\ntestp = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"10\")))\n\ntestd : Relation\ntestd = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"4\")))\n\ntests : Relation\ntests = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"3\")))\n\ntestu : Relation\ntestu = consider proofs cms ((# \"6\") \u21d2 (\u21d2 (# \"1\")))\n\ntestcl : List Principle\ntestcl = closure proofs ((# \"5\") \u2237 [])\n\n\n\nstringifylp : List Principle \u2192 String\nstringifylp [] = \"[]\"\nstringifylp ((# s) \u2237 xs) = s >>> \" \u2237 \" >>> (stringifylp xs)\n\n\nmain = run (putStrLn (stringifylp testcl))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"973dbd23f4d5565473a04f7c9302bde7992022f1","subject":"Desc model: followed by the generic proof that map (f . g) = map f . map g","message":"Desc model: followed by the generic proof that map (f . g) = map f . map g","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A : Set}{B : A -> Set}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\nproof-map-compos : (D : Desc)(X Y Z : Set)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A : Set}{B : A -> Set}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"55d18c66660476d348fa1767e4aeffbf94dd8b38","subject":"Using a type.","message":"Using a type.\n\nIgnore-this: c9e2e45698c2765b84e15b7737ef6002\n\ndarcs-hash:20100714232148-3bd4e-e3832e16a375b271655fd4dc2b4e2a756a14722c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/SortList\/Closures\/ListOrd.agda","new_file":"Examples\/SortList\/Closures\/ListOrd.agda","new_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to ListOrd\n------------------------------------------------------------------------------\n\nmodule Examples.SortList.Closures.ListOrd where\n\nopen import LTC.Minimal\n\nopen import Examples.SortList.Closures.Bool using\n  ( \u2264-ItemList-Bool ; \u2264-Lists-Bool ; isListOrd-Bool\n  ; \u2264-ItemTree-Bool ; \u2264-TreeItem-Bool ; isTreeOrd-Bool\n  )\nopen import Examples.SortList.Closures.List using ( flatten-List )\nopen import Examples.SortList.Closures.TreeOrd\n  using ( leftSubTree-TreeOrd ; rightSubTree-TreeOrd )\nopen import Examples.SortList.Properties\n  using ( subList-ListOrd ; xs\u2264[] ; listOrd-xs++ys\u2192ys\u2264zs\u2192xs++ys\u2264zs )\nopen import Examples.SortList.SortList\n\nopen import LTC.Data.Bool.Properties using\n  ( \u2264-Bool\n  ; x&&y\u2261true\u2192x\u2261true ; x&&y\u2261true\u2192y\u2261true\n  ; w&&x&&y&&z\u2261true\u2192y\u2261true ; w&&x&&y&&z\u2261true\u2192z\u2261true\n  )\nopen import LTC.Data.Nat.List.Type\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.List\n\nopen import Postulates using ( ++-ListOrd-aux\u2081 )\n\n------------------------------------------------------------------------------\n-- Auxiliar functions\n\n++-ListOrd-aux\u2082 : {item is js : D} \u2192 N item \u2192 ListN is \u2192 ListN js \u2192\n                  LE-ItemList item is \u2192\n                  LE-Lists is js \u2192\n                  LE-ItemList item (is ++ js)\n++-ListOrd-aux\u2082 {item} {js = js} Nitem nilLN LNjs item\u2264niL niL\u2264js = prf\n  where\n    postulate prf : LE-ItemList item ([] ++ js)\n    {-# ATP prove prf ++-ListOrd-aux\u2081 #-}\n\n++-ListOrd-aux\u2082 {item} {js = js} Nitem\n               (consLN {i} {is} Ni LNis) LNjs item\u2264i\u2237is i\u2237is\u2264js =\n  prf (++-ListOrd-aux\u2082 Nitem LNis LNjs\n        (x&&y\u2261true\u2192y\u2261true (\u2264-Bool Nitem Ni)\n                          (\u2264-ItemList-Bool Nitem LNis)\n                          (trans (sym (\u2264-ItemList-\u2237 item i is)) item\u2264i\u2237is))\n        (x&&y\u2261true\u2192y\u2261true (\u2264-ItemList-Bool Ni LNis)\n                          (\u2264-Lists-Bool LNis LNjs)\n                          (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : LE-ItemList item (is ++ js) \u2192 -- IH.\n                    LE-ItemList item ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-Bool \u2264-ItemList-Bool x&&y\u2261true\u2192x\u2261true #-}\n\n------------------------------------------------------------------------------\n-- Append preserves the order.\n++-ListOrd : {is js : D} \u2192 ListN is \u2192 ListN js \u2192 ListOrd is \u2192 ListOrd js \u2192\n             LE-Lists is js \u2192\n             ListOrd (is ++ js)\n\n++-ListOrd {js = js} nilLN LNjs LOis LOjs is\u2264js = prf\n  where\n    postulate prf : ListOrd ([] ++ js)\n    {-# ATP prove prf #-}\n\n++-ListOrd {js = js} (consLN {i} {is} Ni LNis) LNjs LOi\u2237is LOjs i\u2237is\u2264js =\n  prf (++-ListOrd LNis LNjs (subList-ListOrd Ni LNis LOi\u2237is) LOjs\n                  (x&&y\u2261true\u2192y\u2261true\n                    (\u2264-ItemList-Bool Ni LNis)\n                    (\u2264-Lists-Bool LNis LNjs)\n                    (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : ListOrd (is ++ js) \u2192 -- IH.\n                    ListOrd ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-ItemList-Bool \u2264-Lists-Bool\n                      x&&y\u2261true\u2192x\u2261true x&&y\u2261true\u2192y\u2261true\n                      ++-ListOrd-aux\u2082\n    #-}\n\n------------------------------------------------------------------------------\nmutual\n  -- If t is ordered then (flatten t) is ordered\n  flatten-ListOrd : {t : D} \u2192 Tree t \u2192 TreeOrd t \u2192 ListOrd (flatten t)\n  flatten-ListOrd nilT TOnilT = prf\n    where\n      postulate prf : ListOrd (flatten nilTree)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd (tipT {i} Ni ) TOtipT = prf\n    where\n      postulate prf : ListOrd (flatten (tip i))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082 ) TOnodeT =\n    prf (flatten-ListOrd Tt\u2081 (leftSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n        (flatten-ListOrd Tt\u2082 (rightSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n        (flatten-ListOrd-aux Tt\u2081 Ni Tt\u2082 TOnodeT)\n    where\n      postulate prf : ListOrd (flatten t\u2081) \u2192 -- IH.\n                      ListOrd (flatten t\u2082) \u2192 -- IH.\n                      LE-Lists (flatten t\u2081) (flatten t\u2082) \u2192\n                      ListOrd (flatten (node t\u2081 i t\u2082))\n      {-# ATP prove prf ++-ListOrd flatten-List\n                        leftSubTree-TreeOrd rightSubTree-TreeOrd\n      #-}\n\n  flatten-ListOrd-aux : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192\n                        TreeOrd (node t\u2081 i t\u2082) \u2192\n                        LE-Lists (flatten t\u2081) (flatten t\u2082)\n  flatten-ListOrd-aux {t\u2082 = t\u2082} nilT Niaux Tt\u2082 _ = prf\n    where\n      postulate prf : LE-Lists (flatten nilTree) (flatten t\u2082)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni nilT TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten nilTree)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni (tipT {k} Nk) TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten (tip k))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni (nodeT {ta} {k} {tb} Tta Nk Ttb )\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten (node ta k tb))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb) Ni nilT TOnode =\n    prf (flatten-List (nodeT Tta Nj Ttb))\n        (flatten-ListOrd (nodeT Tta Nj Ttb)\n        (leftSubTree-TreeOrd (nodeT Tta Nj Ttb) Ni nilT TOnode))\n    where\n      postulate prf : ListN (flatten (node ta j tb)) \u2192\n                      ListOrd (flatten (node ta j tb)) \u2192 -- mutual IH.\n                      LE-Lists (flatten (node ta j tb)) (flatten nilTree)\n      {-# ATP prove prf xs\u2264[] #-}\n\n  flatten-ListOrd-aux {i = i} (nodeT {ta} {j} {tb} Tta Nj Ttb)\n                      Ni\n                      (tipT {k} Nk)\n                      TOnode =\n    prf (flatten-List Tta)\n        (flatten-List Ttb)\n        (flatten-List (tipT Nk))\n        (++-ListOrd (flatten-List Tta)\n                    (flatten-List Ttb)\n                    (flatten-ListOrd Tta\n                                     (leftSubTree-TreeOrd Tta Nj Ttb\n                                                          treeOrd-ta-j-tb))\n                    (flatten-ListOrd Ttb\n                                     (rightSubTree-TreeOrd Tta Nj Ttb\n                                                           treeOrd-ta-j-tb))\n                    (flatten-ListOrd-aux Tta Nj Ttb treeOrd-ta-j-tb))\n        (flatten-ListOrd-aux Ttb Ni (tipT Nk) (treeOrd-tb-i-k tb\u2264-i))\n    where\n      postulate prf : ListN (flatten ta) \u2192\n                      ListN (flatten tb) \u2192\n                      ListN (flatten (tip k)) \u2192\n                      ListOrd (flatten ta ++ flatten tb) \u2192 -- Indirect mutual\n                                                           -- IH and IH.\n                      LE-Lists (flatten tb) (flatten (tip k)) \u2192 -- IH.\n                      LE-Lists (flatten (node ta j tb)) (flatten (tip k))\n      {-# ATP prove prf listOrd-xs++ys\u2192ys\u2264zs\u2192xs++ys\u2264zs #-}\n\n      treeOrd-ta-j-tb : TreeOrd (node ta j tb)\n      treeOrd-ta-j-tb = leftSubTree-TreeOrd (nodeT Tta Nj Ttb) Ni (tipT Nk)\n                                            TOnode\n\n      postulate\n        tb\u2264-i : LE-TreeItem tb i\n      {-# ATP prove tb\u2264-i \u2264-ItemTree-Bool \u2264-TreeItem-Bool isTreeOrd-Bool\n                          x&&y\u2261true\u2192y\u2261true w&&x&&y&&z\u2261true\u2192y\u2261true\n      #-}\n\n      postulate\n        treeOrd-tb-i-k : LE-TreeItem tb i \u2192 -- NB. We need prove this\n                                            -- hypothesis in a separate\n                                            -- postulate.\n                         TreeOrd (node tb i (tip k))\n      {-# ATP prove treeOrd-tb-i-k rightSubTree-TreeOrd treeOrd-ta-j-tb\n                                   \u2264-ItemTree-Bool \u2264-TreeItem-Bool\n                                   isTreeOrd-Bool w&&x&&y&&z\u2261true\u2192z\u2261true\n      #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb)\n                      Ni\n                      (nodeT {tc} {k} {td} Ttc Nk Ttd)\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (node ta j tb))\n                               (flatten (node tc k td))\n","old_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to ListOrd\n------------------------------------------------------------------------------\n\nmodule Examples.SortList.Closures.ListOrd where\n\nopen import LTC.Minimal\n\nopen import Examples.SortList.Closures.Bool using\n  ( \u2264-ItemList-Bool ; \u2264-Lists-Bool ; isListOrd-Bool\n  ; \u2264-ItemTree-Bool ; \u2264-TreeItem-Bool ; isTreeOrd-Bool\n  )\nopen import Examples.SortList.Closures.List using ( flatten-List )\nopen import Examples.SortList.Closures.TreeOrd\n  using ( leftSubTree-TreeOrd ; rightSubTree-TreeOrd )\nopen import Examples.SortList.Properties\n  using ( subList-ListOrd ; xs\u2264[] ; listOrd-xs++ys\u2192ys\u2264zs\u2192xs++ys\u2264zs )\nopen import Examples.SortList.SortList\n\nopen import LTC.Data.Bool.Properties using\n  ( \u2264-Bool\n  ; x&&y\u2261true\u2192x\u2261true ; x&&y\u2261true\u2192y\u2261true\n  ; w&&x&&y&&z\u2261true\u2192y\u2261true ; w&&x&&y&&z\u2261true\u2192z\u2261true\n  )\nopen import LTC.Data.Nat.List.Type\nopen import LTC.Data.Nat.Type\nopen import LTC.Data.List\n\nopen import Postulates using ( ++-ListOrd-aux\u2081 )\n\n------------------------------------------------------------------------------\n-- Auxiliar functions\n\n++-ListOrd-aux\u2082 : {item is js : D} \u2192 N item \u2192 ListN is \u2192 ListN js \u2192\n                  LE-ItemList item is \u2192\n                  LE-Lists is js \u2192\n                  LE-ItemList item (is ++ js)\n++-ListOrd-aux\u2082 {item} {js = js} Nitem nilLN LNjs item\u2264niL niL\u2264js = prf\n  where\n    postulate prf : LE-ItemList item ([] ++ js)\n    {-# ATP prove prf ++-ListOrd-aux\u2081 #-}\n\n++-ListOrd-aux\u2082 {item} {js = js} Nitem\n               (consLN {i} {is} Ni LNis) LNjs item\u2264i\u2237is i\u2237is\u2264js =\n  prf (++-ListOrd-aux\u2082 Nitem LNis LNjs\n        (x&&y\u2261true\u2192y\u2261true (\u2264-Bool Nitem Ni)\n                          (\u2264-ItemList-Bool Nitem LNis)\n                          (trans (sym (\u2264-ItemList-\u2237 item i is)) item\u2264i\u2237is))\n        (x&&y\u2261true\u2192y\u2261true (\u2264-ItemList-Bool Ni LNis)\n                          (\u2264-Lists-Bool LNis LNjs)\n                          (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : LE-ItemList item (is ++ js) \u2192 -- IH.\n                    LE-ItemList item ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-Bool \u2264-ItemList-Bool x&&y\u2261true\u2192x\u2261true #-}\n\n------------------------------------------------------------------------------\n-- Append preserves the order.\n++-ListOrd : {is js : D} \u2192 ListN is \u2192 ListN js \u2192 ListOrd is \u2192 ListOrd js \u2192\n             LE-Lists is js \u2192\n             ListOrd (is ++ js)\n\n++-ListOrd {js = js} nilLN LNjs LOis LOjs is\u2264js = prf\n  where\n    postulate prf : ListOrd ([] ++ js)\n    {-# ATP prove prf #-}\n\n++-ListOrd {js = js} (consLN {i} {is} Ni LNis) LNjs LOi\u2237is LOjs i\u2237is\u2264js =\n  prf (++-ListOrd LNis LNjs (subList-ListOrd Ni LNis LOi\u2237is) LOjs\n                  (x&&y\u2261true\u2192y\u2261true\n                    (\u2264-ItemList-Bool Ni LNis)\n                    (\u2264-Lists-Bool LNis LNjs)\n                    (trans (sym (\u2264-Lists-\u2237 i is js)) i\u2237is\u2264js)))\n  where\n    postulate prf : ListOrd (is ++ js) \u2192 -- IH.\n                    ListOrd ((i \u2237 is) ++ js)\n    {-# ATP prove prf \u2264-ItemList-Bool \u2264-Lists-Bool\n                      x&&y\u2261true\u2192x\u2261true x&&y\u2261true\u2192y\u2261true\n                      ++-ListOrd-aux\u2082\n    #-}\n\n------------------------------------------------------------------------------\nmutual\n  -- If t is ordered then (flatten t) is ordered\n  flatten-ListOrd : {t : D} \u2192 Tree t \u2192 TreeOrd t \u2192 ListOrd (flatten t)\n  flatten-ListOrd nilT TOnilT = prf\n    where\n      postulate prf : ListOrd (flatten nilTree)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd (tipT {i} Ni ) TOtipT = prf\n    where\n      postulate prf : ListOrd (flatten (tip i))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd (nodeT {t\u2081} {i} {t\u2082} Tt\u2081 Ni Tt\u2082 ) TOnodeT =\n    prf (flatten-ListOrd Tt\u2081 (leftSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n        (flatten-ListOrd Tt\u2082 (rightSubTree-TreeOrd Tt\u2081 Ni Tt\u2082 TOnodeT))\n        (flatten-ListOrd-aux Tt\u2081 Ni Tt\u2082 TOnodeT)\n    where\n      postulate prf : ListOrd (flatten t\u2081) \u2192 -- IH.\n                      ListOrd (flatten t\u2082) \u2192 -- IH.\n                      LE-Lists (flatten t\u2081) (flatten t\u2082) \u2192\n                      ListOrd (flatten (node t\u2081 i t\u2082))\n      {-# ATP prove prf ++-ListOrd flatten-List\n                        leftSubTree-TreeOrd rightSubTree-TreeOrd\n      #-}\n\n  flatten-ListOrd-aux : {t\u2081 i t\u2082 : D} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192\n                        TreeOrd (node t\u2081 i t\u2082) \u2192\n                        LE-Lists (flatten t\u2081) (flatten t\u2082)\n  flatten-ListOrd-aux {t\u2082 = t\u2082} nilT Niaux Tt\u2082 _ = prf\n    where\n      postulate prf : LE-Lists (flatten nilTree) (flatten t\u2082)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni nilT TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten nilTree)\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni (tipT {k} Nk) TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten (tip k))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (tipT {j} Nj) Ni (nodeT {ta} {k} {tb} Tta Nk Ttb )\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (tip j)) (flatten (node ta k tb))\n      {-# ATP prove prf #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb) Ni nilT TOnode =\n    prf (flatten-List (nodeT Tta Nj Ttb))\n        (flatten-ListOrd (nodeT Tta Nj Ttb)\n        (leftSubTree-TreeOrd (nodeT Tta Nj Ttb) Ni nilT TOnode))\n    where\n      postulate prf : ListN (flatten (node ta j tb)) \u2192\n                      ListOrd (flatten (node ta j tb)) \u2192 -- mutual IH.\n                      LE-Lists (flatten (node ta j tb)) (flatten nilTree)\n      {-# ATP prove prf xs\u2264[] #-}\n\n  flatten-ListOrd-aux {i = i} (nodeT {ta} {j} {tb} Tta Nj Ttb)\n                      Ni\n                      (tipT {k} Nk)\n                      TOnode =\n    prf (flatten-List Tta)\n        (flatten-List Ttb)\n        (flatten-List (tipT Nk))\n        (++-ListOrd (flatten-List Tta)\n                    (flatten-List Ttb)\n                    (flatten-ListOrd Tta\n                                     (leftSubTree-TreeOrd Tta Nj Ttb\n                                                          treeOrd-ta-j-tb))\n                    (flatten-ListOrd Ttb\n                                     (rightSubTree-TreeOrd Tta Nj Ttb\n                                                           treeOrd-ta-j-tb))\n                    (flatten-ListOrd-aux Tta Nj Ttb treeOrd-ta-j-tb))\n        (flatten-ListOrd-aux Ttb Ni (tipT Nk) (treeOrd-tb-i-k tb\u2264-i))\n    where\n      postulate prf : ListN (flatten ta) \u2192\n                      ListN (flatten tb) \u2192\n                      ListN (flatten (tip k)) \u2192\n                      ListOrd (flatten ta ++ flatten tb) \u2192 -- Indirect mutual\n                                                           -- IH and IH.\n                      LE-Lists (flatten tb) (flatten (tip k)) \u2192 -- IH.\n                      LE-Lists (flatten (node ta j tb)) (flatten (tip k))\n      {-# ATP prove prf listOrd-xs++ys\u2192ys\u2264zs\u2192xs++ys\u2264zs #-}\n\n      treeOrd-ta-j-tb : TreeOrd (node ta j tb)\n      treeOrd-ta-j-tb = leftSubTree-TreeOrd (nodeT Tta Nj Ttb) Ni (tipT Nk)\n                                            TOnode\n\n      postulate\n        tb\u2264-i : LE-TreeItem tb i\n      {-# ATP prove tb\u2264-i \u2264-ItemTree-Bool \u2264-TreeItem-Bool isTreeOrd-Bool\n                          x&&y\u2261true\u2192y\u2261true w&&x&&y&&z\u2261true\u2192y\u2261true\n      #-}\n\n      postulate\n        treeOrd-tb-i-k : \u2264-TreeItem tb i \u2261 true \u2192 -- NB. We need prove this\n                                                  -- hypothesis in a separate\n                                                  -- postulate.\n                         TreeOrd (node tb i (tip k))\n      {-# ATP prove treeOrd-tb-i-k rightSubTree-TreeOrd treeOrd-ta-j-tb\n                                   \u2264-ItemTree-Bool \u2264-TreeItem-Bool\n                                   isTreeOrd-Bool w&&x&&y&&z\u2261true\u2192z\u2261true\n      #-}\n\n  flatten-ListOrd-aux (nodeT {ta} {j} {tb} Tta Nj Ttb)\n                      Ni\n                      (nodeT {tc} {k} {td} Ttc Nk Ttd)\n                      TOnode = prf\n    where\n      postulate prf : LE-Lists (flatten (node ta j tb))\n                               (flatten (node tc k td))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"820f19fdef2b22d6a0b220e2ab84ca3be2f8f943","subject":"making rule names match, adding missing rules","message":"making rule names match, adding missing rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- substitution\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 = \u2987\u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] (d1 \u2218 d2) =  ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2081 \u03b5) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2082 d) evalctx\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 evalctx \u2192\n             (< \u03c4 > \u03b5 ) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHConst : c == \u2299 \u27e6 c \u27e7\n    FHAp1 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d2' \u27e7\n    FHAp2 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d1' \u27e7\n    FHEHole : \u2200{u \u03c3 m} \u2192 \u2987\u2988\u27e8 (u , \u03c3 , m) \u27e9 == \u2299 \u27e6 \u2987\u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e7\n    FHNEHoleEvaled : \u2200{ d u \u03c3} \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 , \u2713) \u27e9 ==  \u2299 \u27e6 \u2987 d \u2988\u27e8 (u , \u03c3 , \u2713) \u27e9 \u27e7\n    FHNEHoleInside : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 , \u2717) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , \u2717) \u27e9 \u27e6 d' \u27e7\n    FHNEHoleFinal : \u2200{ d u \u03c3} \u2192\n              d final \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 , \u2717) \u27e9 ==  \u2299 \u27e6 \u2987 d \u2988\u27e8 (u , \u03c3 , \u2717) \u27e9 \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c4 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (< \u03c4 > d) == < \u03c4 > \u03b5 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d'\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- substitution\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 = \u2987\u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] (d1 \u2218 d2) =  ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 evalctx \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2081 \u03b5) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2082 d) evalctx\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 evalctx \u2192\n             (< \u03c4 > \u03b5 ) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FDot : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d' \u03b5 x \u03c4 } \u2192\n           d == \u03b5 \u27e6 d' \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d' \u27e7\n    FAp1 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d2' \u27e7\n    FAp2 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d1' \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d' \u03b5 \u03c4 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (< \u03c4 > d) == < \u03c4 > \u03b5 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8f0eb59f1abe6886d8f35bd2b3e915bb24e4ba7d","subject":"Drop standard big-step semantics","message":"Drop standard big-step semantics\n\nNot useful.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192 eval (app (app ds t) dt) \u03c1 (suc (suc n)) \u2261 dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Failed variants, the step-indexes are wrong:\n-- dapply-equiv-2 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc n)) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-2 (suc n) t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 (suc zero) (const c) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (var x) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (app t t\u2081) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (abs t) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc (suc n)) t d\u03c1 \u03c11 da a1 = refl\n\n-- Worse:\n-- dapply-equiv-0 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-0 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc zero) t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc (suc n)) t d\u03c1 \u03c11 da a1 = {!!}\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\n-- Standard relational big-step semantics.\ndata _\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 closure t \u03c1\n  app : \u2200 {\u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 closure t\u2032 \u03c1\u2032 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    (v\u2082 \u2022 \u03c1\u2032) \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n    \u03c1 \u22a2 var x \u2193 (\u27e6 x \u27e7Var \u03c1)\n  lit : \u2200 n \u2192\n    \u03c1 \u22a2 const (lit n) \u2193 intV n\n  plus : \u2200 {t1 t2 v1 v2} \u2192\n    \u03c1 \u22a2 t1 \u2193 (intV v1) \u2192\n    \u03c1 \u22a2 t2 \u2193 (intV v2) \u2192\n    \u03c1 \u22a2 app (app (const plus) t1) t2 \u2193 intV (v1 + v2)\n  minus : \u2200 {t1 t2 v1 v2} \u2192\n    \u03c1 \u22a2 t1 \u2193 (intV v1) \u2192\n    \u03c1 \u22a2 t2 \u2193 (intV v2) \u2192\n    \u03c1 \u22a2 app (app (const minus) t1) t2 \u2193 intV (v1 - v2)\n  -- cond-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n  --   \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n  --   \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n  --   \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  -- cond-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n  --   \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n  --   \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n  --   \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n\u2193-sound (var x) = \u21a6-sound _ x\n\u2193-sound (lit n) = refl\n\u2193-sound (plus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\u2193-sound (minus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192 eval (app (app ds t) dt) \u03c1 (suc (suc n)) \u2261 dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Failed variants, the step-indexes are wrong:\n-- dapply-equiv-2 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc n)) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-2 (suc n) t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 (suc zero) (const c) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (var x) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (app t t\u2081) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (abs t) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc (suc n)) t d\u03c1 \u03c11 da a1 = refl\n\n-- Worse:\n-- dapply-equiv-0 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-0 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc zero) t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc (suc n)) t d\u03c1 \u03c11 da a1 = {!!}\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1d37d5dc7cf104062e0fa00000b3b1486d05ea34","subject":"Rename C to Const.","message":"Rename C to Const.\n\nOld-commit-hash: ed6d806588b71d4c3272db594c241e23d0c12a79\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\n\nmodule Parametric.Syntax.Term\n    {Base : Set}\n    (Constant : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -})\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type Base\nopen Context Type\n\nopen import Syntax.Context.Plotkin Base\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : Constant \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nuncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Constant \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\nuncurriedConst constant = const constant\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\ncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Constant \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n-- HOAS-like smart constructors for lambdas, for different arities.\n\nabs\u2081 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 (x : Term \u0393\u2032 \u03c4\u2081) \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\nabs\u2081 {\u0393} {\u03c4\u2081} =  \u03bb f \u2192 abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n\nabs\u2082 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4))\nabs\u2082 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2081 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2083 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4))\nabs\u2083 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2082 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2084 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4))\nabs\u2084 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2083 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2085 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4))\nabs\u2085 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2084 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2086 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4\u2086 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4\u2086 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4\u2086 \u21d2 \u03c4))\nabs\u2086 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2085 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Base.Syntax.Context as Context\n\nmodule Parametric.Syntax.Term\n    {Base : Set}\n    (C : Context.Context (Type.Type Base) \u2192 Type.Type Base \u2192 Set {- of constants -})\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\n\nopen Type Base\nopen Context Type\n\nopen import Syntax.Context.Plotkin Base\n\n-- Declarations of Term and Terms to enable mutual recursion\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n\ndata Terms\n  (\u0393 : Context) :\n  (\u03a3 : Context) \u2192 Set\n\n-- (Term \u0393 \u03c4) represents a term of type \u03c4\n-- with free variables bound in \u0393.\ndata Term \u0393 where\n  const : \u2200 {\u03a3 \u03c4} \u2192\n    (c : C \u03a3 \u03c4) \u2192\n    Terms \u0393 \u03a3 \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n-- with free variables bound in \u0393.\ndata Terms \u0393 where\n  \u2205 : Terms \u0393 \u2205\n  _\u2022_ : \u2200 {\u03c4 \u03a3} \u2192\n    Term \u0393 \u03c4 \u2192\n    Terms \u0393 \u03a3 \u2192\n    Terms \u0393 (\u03c4 \u2022 \u03a3)\n\ninfixr 9 _\u2022_\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : Base \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\n\nweakenAll : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Terms \u0393\u2081 \u03a3 \u2192\n  Terms \u0393\u2082 \u03a3\n\nweaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weakenAll \u0393\u2081\u227c\u0393\u2082 ts)\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nweakenAll \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\nweakenAll \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weakenAll \u0393\u2081\u227c\u0393\u2082 ts\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n\nUncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nUncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\nuncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\nuncurriedConst constant = const constant\n\nCurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\nCurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\nCurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\ncurryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurryTermConstructor {\u2205} k = k \u2205\ncurryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\ncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 C \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\ncurriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n-- HOAS-like smart constructors for lambdas, for different arities.\n\nabs\u2081 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 (x : Term \u0393\u2032 \u03c4\u2081) \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\nabs\u2081 {\u0393} {\u03c4\u2081} =  \u03bb f \u2192 abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n\nabs\u2082 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4))\nabs\u2082 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2081 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2083 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4))\nabs\u2083 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2082 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2084 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4))\nabs\u2084 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2083 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2085 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4))\nabs\u2085 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2084 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\nabs\u2086 :\n  \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c4\u2083 \u03c4\u2084 \u03c4\u2085 \u03c4\u2086 \u03c4} \u2192\n    (\u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2083 \u2192 Term \u0393\u2032 \u03c4\u2084 \u2192 Term \u0393\u2032 \u03c4\u2085 \u2192 Term \u0393\u2032 \u03c4\u2086 \u2192 Term \u0393\u2032 \u03c4) \u2192\n    (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082 \u21d2 \u03c4\u2083 \u21d2 \u03c4\u2084 \u21d2 \u03c4\u2085 \u21d2 \u03c4\u2086 \u21d2 \u03c4))\nabs\u2086 f =\n  abs\u2081 (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n    abs\u2085 (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n      f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3d721fa8ec4e3e2ec1bf00e4dc87ad1cc5a00225","subject":"Update to stdlib","message":"Update to stdlib\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","old_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Maybe\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2 ^ n)\nallBits zero = [] \u2237 []\nallBits (suc n) rewrite \u2115\u00b0.+-comm (2 ^ n) 0 = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2 ^ n))\n#\u27e8 pred \u27e9 = count pred (allBits _)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e5751b75f2b7d3b073e6112f65d2ed4cadb2ac23","subject":"agda\/...: misc (to revert)","message":"agda\/...: misc (to revert)\n\nThis commit simply adds some experiments which turned out to be\nprobably useless. Given the time invested, saving them might be\nuseful. But they are going to be reverted.\n\nOld-commit-hash: cfca67fe7667584365fce6095ebdbeb09d59b816\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-var-3 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context (\u0394-Context \u0393)) \u03c4\nlift-var-3 this = that (that (that this))\nlift-var-3 (that x) = that (that (that (that (lift-var-3 x))))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : --\u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\n{-\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u2205 \u2205 t\u2081 = \u0394 t\u2081\n    weakenMore2 (\u03c4\u2081 \u2022 \u0393\u2081) \u2205 t\u2081 =\n      weakenOne \u2205 (\u0394-Type (\u0394-Type \u03c4\u2081))\n        (weakenOne (\u0394-Type \u03c4\u2081 \u2022 \u2205) (\u0394-Type \u03c4\u2081) {!\n          weakenMore2\n            (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} {\u03c4} t\u2081)!})\n          --(weakenMore2 {\u03c4\u2081 \u2022 \u0393\u2081} (\u0394-Type \u03c4\u2081 \u2022 \u03c4\u2081 \u2022 \u2205) t\u2081 ))\n\n--        (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} {\u03c4} t\u2081)))\n\n    weakenMore2 (\u03c4\u2081 \u2022 \u0393\u2081) (.(\u0394-Type \u03c4\u2081) \u2022 \u0393\u2032\u2081) t\u2081 = {!!}\n-}\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : --\u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a112770b324164fbf03d56704d244f21d4586d15","subject":"Agda: replace some holes with postulates","message":"Agda: replace some holes with postulates\n\nIssue #24 requires splitting up total.agda in different modules.\nHowever, some of the code to move away contains holes, and Agda\nrejects importing modules with holes (with an error complaining about\nunsolved metas). Hence we need to replace the holes by postulates\nbefore moving the code in another module.\n\nSome of these postulates are clearly wrong and need to be\nrestricted a bit, so they should be replaced with proofs.\n\nOld-commit-hash: 2012bd3208ab4d269dcab3a436530872e88a8830\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\npostulate\n  \u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n    v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n  \u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n    v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n{-\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n--\u2261-cong {bool} {bool} {v\u2081} f bool = bool\n\u2261-cong {bool} {bool} f bool = bool\n\u2261-cong {bool} {\u03c4\u2083 \u21d2 \u03c4\u2084} {v\u2081} {v\u2082} f (ext \u2261\u2081) = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} {bool} {bool} f bool bool = bool\n\u2261-cong\u2082 {bool} {bool} {\u03c4\u2082 \u21d2 \u03c4\u2083} f bool (ext \u2261\u2082) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {bool} f (ext \u2261\u2081) (bool) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u03c4\u2083 \u21d2 \u03c4\u2084} f (ext \u2261\u2081) (ext \u2261\u2082) = {!!}\n-}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\n-- XXX: as given, this is false!\npostulate\n  diff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    diff (apply dv v) v \u2261 dv\n\n{-\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n-}\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = ?\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = ?\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n--\u2261-cong {bool} {bool} {v\u2081} f bool = bool\n\u2261-cong {bool} {bool} f bool = bool\n\u2261-cong {bool} {\u03c4\u2083 \u21d2 \u03c4\u2084} {v\u2081} {v\u2082} f (ext \u2261\u2081) = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} {bool} {bool} f bool bool = bool\n\u2261-cong\u2082 {bool} {bool} {\u03c4\u2082 \u21d2 \u03c4\u2083} f bool (ext \u2261\u2082) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {bool} f (ext \u2261\u2081) (bool) = {!!}\n\u2261-cong\u2082 {bool} {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u03c4\u2083 \u21d2 \u03c4\u2084} f (ext \u2261\u2081) (ext \u2261\u2082) = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n\u2261-consistent : \u00ac (\u2200 (\u03c4 : Type) \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 v\u2081 \u2261 v\u2082)\n\u2261-consistent H with H bool true false\n... | ()\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = ?\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = ?\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext x with x \u2205\n... | ()\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"88702ef746d0c6b0ff60553b14069166fe1b024b","subject":"Profunctor: some updates","message":"Profunctor: some updates\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Profunctor.agda","new_file":"lib\/Category\/Profunctor.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Based on profunctors 4.0.4 haskell package\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Sum.NP renaming (map to \u228e-map)\nopen import Data.Product.NP hiding (second) renaming (first to \u00d7-first; second\u2032 to \u00d7-second)\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n  dimap : \u2200 {A B C D} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 B \u219d C \u2192 A \u219d D\n  dimap f g = lmap f \u2218 rmap g\n\n\u2192-Profunctor : Profunctor {\u2113} -\u2192-\n\u2192-Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\n------------------------------------------------------------------------------\n-- Strong\n------------------------------------------------------------------------------\n\nrecord Strong {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor mk\n  field\n    profunctor : Profunctor _\u219d_\n    first      : \u2200 {A B C : Set i} \u2192 A \u219d B \u2192 (A \u00d7 C) \u219d (B \u00d7 C)\n    second     : \u2200 {A B C : Set i} \u2192 A \u219d B \u2192 (C \u00d7 A) \u219d (C \u00d7 B)\n  open Profunctor profunctor public\n\n  module _ {A B C : Set i} (second : A \u219d B \u2192 (C \u00d7 A) \u219d (C \u00d7 B)) where\n    first-from-second : A \u219d B \u2192 (A \u00d7 C) \u219d (B \u00d7 C)\n    first-from-second = dimap swap swap \u2218 second\n\n  module _ {A B C : Set i} (first : A \u219d B \u2192 (A \u00d7 C) \u219d (B \u00d7 C)) where\n    second-from-first : A \u219d B \u2192 (C \u00d7 A) \u219d (C \u00d7 B)\n    second-from-first = dimap swap swap \u2218 first\n\n\u2192-Strong : Strong {\u2113} -\u2192-\n\u2192-Strong = mk \u2192-Profunctor \u00d7-first \u00d7-second\n\n------------------------------------------------------------------------------\n-- Choice\n------------------------------------------------------------------------------\n\nrecord Choice {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor mk\n  field\n    profunctor : Profunctor _\u219d_\n    left       : \u2200 {A B C : Set i} \u2192 A \u219d B \u2192 (A \u228e C) \u219d (B \u228e C)\n    right      : \u2200 {A B C : Set i} \u2192 A \u219d B \u2192 (C \u228e A) \u219d (C \u228e B)\n  open Profunctor profunctor public\n\n  module _ {A B C : Set i} (right : A \u219d B \u2192 (C \u228e A) \u219d (C \u228e B)) where\n    left-from-right : A \u219d B \u2192 (A \u228e C) \u219d (B \u228e C)\n    left-from-right = dimap twist twist \u2218 right\n\n  module _ {A B C : Set i} (left : A \u219d B \u2192 (A \u228e C) \u219d (B \u228e C)) where\n    right-from-left : A \u219d B \u2192 (C \u228e A) \u219d (C \u228e B)\n    right-from-left = dimap twist twist \u2218 left\n\n\u2192-Choice : Choice {\u2113} -\u2192-\n\u2192-Choice = mk \u2192-Profunctor (flip \u228e-map id) (\u228e-map id)\n\n------------------------------------------------------------------------------\n-- UpStar\n------------------------------------------------------------------------------\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nmodule _ {F : \u2605 \u2113 \u2192 \u2605 \u2113} (funF : RawFunctor F) where\n    open RawFunctor funF\n\n    upStar-RawFunctor : \u2200 {A} \u2192 RawFunctor (UpStar F A)\n    upStar-RawFunctor = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n\n    upStar-Profunctor : Profunctor (UpStar F)\n    upStar-Profunctor = flip _\u2218\u2032_\n                          , RawFunctor._<$>_ upStar-RawFunctor\n\n    upStar-Strong : Strong (UpStar F)\n    upStar-Strong = mk upStar-Profunctor\n                       (\u03bb f x \u2192 flip _,_ (snd x) <$> f (fst x))\n                       (\u03bb f x \u2192 _,_ (fst x) <$> f (snd x))\n\nmodule _ {F : \u2605 \u2113 \u2192 \u2605 \u2113} (appF : RawApplicative F) where\n    open RawApplicative appF\n\n    upStar-Choice : Choice (UpStar F)\n    upStar-Choice = mk (upStar-Profunctor rawFunctor)\n                       (\u03bb f \u2192 [inl: _<$>_ inl \u2218 f    ,inr: _<$>_ inr \u2218 pure ])\n                       (\u03bb f \u2192 [inl: _<$>_ inl \u2218 pure ,inr: _<$>_ inr \u2218 f    ])\n\n------------------------------------------------------------------------------\n-- DownStar\n------------------------------------------------------------------------------\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStar-RawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStar-RawFunctor = record { _<$>_ = _\u2218\u2032_ }\n\ndownStar-Profunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStar-Profunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\n------------------------------------------------------------------------------\n-- Forget\n------------------------------------------------------------------------------\n\nForget : (R A B : Set \u2113) \u2192 Set \u2113\nForget R A B = A \u2192 R\n\nforget-RawFunctor : \u2200 {R A} \u2192 RawFunctor (Forget R A)\nforget-RawFunctor = record { _<$>_ = const id }\n\nmodule _ {R} where\n    forget-Profunctor : Profunctor (Forget R)\n    forget-Profunctor = flip _\u2218\u2032_ , const id\n\n    forget-Strong : Strong (Forget R)\n    forget-Strong = mk forget-Profunctor (\u03bb f \u2192 f \u2218 fst) (\u03bb f \u2192 f \u2218 snd)\n\n-- DEPRECATED\nrecord Lenticular {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor mk\n  field\n    profunctor : Profunctor _\u219d_\n    lenticular : \u2200 {A B} \u2192 A \u219d B \u2192 A \u219d (A \u00d7 B)\n  open Profunctor profunctor public\n\n\u2192-Lenticular : Lenticular {\u2113} -\u2192-\n\u2192-Lenticular = mk \u2192-Profunctor (\u03bb f x \u2192 x , f x)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Based on profunctors 3.1 haskell package\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\nopen import Data.Product\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n  dimap : \u2200 {A B C D} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 B \u219d C \u2192 A \u219d D\n  dimap f g = lmap f \u2218 rmap g\n\nrecord Lenticular {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor mk\n  field\n    profunctor : Profunctor _\u219d_\n    lenticular : \u2200 {A B} \u2192 A \u219d B \u2192 A \u219d (A \u00d7 B)\n  open Profunctor profunctor public\n\n\u2192Profunctor : Profunctor {\u2113} -\u2192-\n\u2192Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\n\u2192Lenticular : Lenticular {\u2113} -\u2192-\n\u2192Lenticular = mk \u2192Profunctor (\u03bb f x \u2192 x , f x)\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nupStarRawFunctor : \u2200 {F A} \u2192 RawFunctor F \u2192 RawFunctor (UpStar F A)\nupStarRawFunctor fun = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n  where open RawFunctor fun\n\nupStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (UpStar F)\nupStarProfunctor fun = flip _\u2218\u2032_\n                     , RawFunctor._<$>_ (upStarRawFunctor fun)\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStarProfunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\ndownStarRawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStarRawFunctor = record { _<$>_ = _\u2218\u2032_ }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4c58b009165ce674a87c1d45177df0c30d306d41","subject":"Make Explore.Summable typecheck","message":"Make Explore.Summable typecheck\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Summable.agda","new_file":"lib\/Explore\/Summable.agda","new_contents":"module Explore.Summable where\n\nopen import Type\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\nopen import Explore.Type\nopen import Explore.Product\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Bool.NP renaming (Bool to \ud835\udfda; true to 1b; false to 0b; to\u2115 to \ud835\udfda\u25b9\u2115)\nopen Data.Bool.NP.Indexed\n\nmodule FromSum {A : \u2605} (sum : Sum A) where\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\nmodule FromSumInd {A : \u2605}\n                  {sum : Sum A}\n                  (sum-ind : SumInd sum) where\n  open FromSum sum public\n\n  sum-ext : SumExt sum\n  sum-ext = sum-ind (\u03bb s \u2192 s _ \u2261 s _) (\u2261.cong\u2082 _+_)\n\n  sum-zero : SumZero sum\n  sum-zero = sum-ind (\u03bb s \u2192 s (const 0) \u2261 0) (\u2261.cong\u2082 _+_) (\u03bb _ \u2192 \u2261.refl)\n\n  sum-hom : SumHom sum\n  sum-hom f g = sum-ind (\u03bb s \u2192 s (f +\u00b0 g) \u2261 s f + s g)\n                        (\u03bb {s\u2080} {s\u2081} p\u2080 p\u2081 \u2192 \u2261.trans (\u2261.cong\u2082 _+_ p\u2080 p\u2081) (+-interchange (s\u2080 _) (s\u2080 _) _ _))\n                        (\u03bb _ \u2192 \u2261.refl)\n\n  sum-mono : SumMono sum\n  sum-mono = sum-ind (\u03bb s \u2192 s _ \u2264 s _) _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  module _ (f g : A \u2192 \u2115) where\n    open \u2261.\u2261-Reasoning\n\n    sum-\u2293-\u2238 : sum f \u2261 sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g)\n    sum-\u2293-\u2238 = sum f                          \u2261\u27e8 sum-ext (f \u27e8 a\u2261a\u2293b+a\u2238b \u27e9\u00b0 g) \u27e9\n              sum ((f \u2293\u00b0 g) +\u00b0 (f \u2238\u00b0 g))     \u2261\u27e8 sum-hom (f \u2293\u00b0 g) (f \u2238\u00b0 g) \u27e9\n              sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g) \u220e\n\n    sum-\u2294-\u2293 : sum f + sum g \u2261 sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g)\n    sum-\u2294-\u2293 = sum f + sum g               \u2261\u27e8 \u2261.sym (sum-hom f g) \u27e9\n              sum (f +\u00b0 g)                \u2261\u27e8 sum-ext (f \u27e8 a+b\u2261a\u2294b+a\u2293b \u27e9\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g +\u00b0 f \u2293\u00b0 g)      \u2261\u27e8 sum-hom (f \u2294\u00b0 g) (f \u2293\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g) \u220e\n\n    sum-\u2294 : sum (f \u2294\u00b0 g) \u2264 sum f + sum g\n    sum-\u2294 = \u2115\u2264.trans (sum-mono (f \u27e8 \u2294\u2264+ \u27e9\u00b0 g)) (\u2115\u2264.reflexive (sum-hom f g))\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sum-const : \u2200 k \u2192 sum (const k) \u2261 Card * k\n  sum-const k\n      rewrite \u2115\u00b0.*-comm Card k\n            | \u2261.sym (sum-lin (const 1) k)\n            | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\n  module _ f g where\n    count-\u2227-not : count f \u2261 count (f \u2227\u00b0 g) + count (f \u2227\u00b0 not\u00b0 g)\n    count-\u2227-not rewrite sum-\u2293-\u2238 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2238 \u27e9\u00b0 g)\n                      = \u2261.refl\n\n    count-\u2228-\u2227 : count f + count g \u2261 count (f \u2228\u00b0 g) + count (f \u2227\u00b0 g)\n    count-\u2228-\u2227 rewrite sum-\u2294-\u2293 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                    = \u2261.refl\n\n    count-\u2228\u2264+ : count (f \u2228\u00b0 g) \u2264 count f + count g\n    count-\u2228\u2264+ = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext (\u2261.sym \u2218 (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g))))\n                         (sum-\u2294 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g))\n\nmodule FromSum\u00d7\n         {A B}\n         {sum\u1d2c     : Sum A}\n         (sum-ind\u1d2c : SumInd sum\u1d2c)\n         {sum\u1d2e     : Sum B}\n         (sum-ind\u1d2e : SumInd sum\u1d2e) where\n\n  module |A| = FromSumInd sum-ind\u1d2c\n  module |B| = FromSumInd sum-ind\u1d2e\n  open Operators\n  \n  sum\u1d2c\u1d2e = sum\u1d2c \u00d7\u02e2 sum\u1d2e\n\n  sum-\u2218proj\u2081\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 |B|.Card * sum\u1d2c f\n  sum-\u2218proj\u2081\u2261Card* f\n      rewrite |A|.sum-ext (|B|.sum-const \u2218 f)\n            = |A|.sum-lin f |B|.Card\n\n  sum-\u2218proj\u2082\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 |A|.Card * sum\u1d2e f\n  sum-\u2218proj\u2082\u2261Card* = |A|.sum-const \u2218 sum\u1d2e\n\n  sum-\u2218proj\u2081 : \u2200 {f} {g} \u2192 sum\u1d2c f \u2261 sum\u1d2c g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2081)\n  sum-\u2218proj\u2081 {f} {g} sumf\u2261sumg\n      rewrite sum-\u2218proj\u2081\u2261Card* f\n            | sum-\u2218proj\u2081\u2261Card* g\n            | sumf\u2261sumg = \u2261.refl\n\n  sum-\u2218proj\u2082 : \u2200 {f} {g} \u2192 sum\u1d2e f \u2261 sum\u1d2e g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2082)\n  sum-\u2218proj\u2082 sumf\u2261sumg = |A|.sum-ext (const sumf\u2261sumg)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Explore.Summable where\n\nopen import Type\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\nopen import Explore.Type\nopen import Explore.Product\nopen import Data.Product\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Bool.NP renaming (Bool to \ud835\udfda; true to 1b; false to 0b; to\u2115 to \ud835\udfda\u25b9\u2115)\nopen Data.Bool.NP.Indexed\n\nmodule FromSum {A : \u2605} (sum : Sum A) where\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\nmodule FromSumInd {A : \u2605}\n                  {sum : Sum A}\n                  (sum-ind : SumInd sum) where\n  open FromSum sum public\n\n  sum-ext : SumExt sum\n  sum-ext = sum-ind (\u03bb s \u2192 s _ \u2261 s _) (\u2261.cong\u2082 _+_)\n\n  sum-zero : SumZero sum\n  sum-zero = sum-ind (\u03bb s \u2192 s (const 0) \u2261 0) (\u2261.cong\u2082 _+_) (\u03bb _ \u2192 \u2261.refl)\n\n  sum-hom : SumHom sum\n  sum-hom f g = sum-ind (\u03bb s \u2192 s (f +\u00b0 g) \u2261 s f + s g)\n                        (\u03bb {s\u2080} {s\u2081} p\u2080 p\u2081 \u2192 \u2261.trans (\u2261.cong\u2082 _+_ p\u2080 p\u2081) (+-interchange (s\u2080 _) (s\u2080 _) _ _))\n                        (\u03bb _ \u2192 \u2261.refl)\n\n  sum-mono : SumMono sum\n  sum-mono = sum-ind (\u03bb s \u2192 s _ \u2264 s _) _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  module _ (f g : A \u2192 \u2115) where\n    open \u2261.\u2261-Reasoning\n\n    sum-\u2293-\u2238 : sum f \u2261 sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g)\n    sum-\u2293-\u2238 = sum f                          \u2261\u27e8 sum-ext (f \u27e8 a\u2261a\u2293b+a\u2238b \u27e9\u00b0 g) \u27e9\n              sum ((f \u2293\u00b0 g) +\u00b0 (f \u2238\u00b0 g))     \u2261\u27e8 sum-hom (f \u2293\u00b0 g) (f \u2238\u00b0 g) \u27e9\n              sum (f \u2293\u00b0 g) + sum (f \u2238\u00b0 g) \u220e\n\n    sum-\u2294-\u2293 : sum f + sum g \u2261 sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g)\n    sum-\u2294-\u2293 = sum f + sum g               \u2261\u27e8 \u2261.sym (sum-hom f g) \u27e9\n              sum (f +\u00b0 g)                \u2261\u27e8 sum-ext (f \u27e8 a+b\u2261a\u2294b+a\u2293b \u27e9\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g +\u00b0 f \u2293\u00b0 g)      \u2261\u27e8 sum-hom (f \u2294\u00b0 g) (f \u2293\u00b0 g) \u27e9\n              sum (f \u2294\u00b0 g) + sum (f \u2293\u00b0 g) \u220e\n\n    sum-\u2294 : sum (f \u2294\u00b0 g) \u2264 sum f + sum g\n    sum-\u2294 = \u2115\u2264.trans (sum-mono (f \u27e8 \u2294\u2264+ \u27e9\u00b0 g)) (\u2115\u2264.reflexive (sum-hom f g))\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sum-const : \u2200 k \u2192 sum (const k) \u2261 Card * k\n  sum-const k\n      rewrite \u2115\u00b0.*-comm Card k\n            | \u2261.sym (sum-lin (const 1) k)\n            | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\n  module _ f g where\n    count-\u2227-not : count f \u2261 count (f \u2227\u00b0 g) + count (f \u2227\u00b0 not\u00b0 g)\n    count-\u2227-not rewrite sum-\u2293-\u2238 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                      | sum-ext (f \u27e8 to\u2115-\u2238 \u27e9\u00b0 g)\n                      = \u2261.refl\n\n    count-\u2228-\u2227 : count f + count g \u2261 count (f \u2228\u00b0 g) + count (f \u2227\u00b0 g)\n    count-\u2228-\u2227 rewrite sum-\u2294-\u2293 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g)\n                    | sum-ext (f \u27e8 to\u2115-\u2293 \u27e9\u00b0 g)\n                    = \u2261.refl\n\n    count-\u2228\u2264+ : count (f \u2228\u00b0 g) \u2264 count f + count g\n    count-\u2228\u2264+ = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext (\u2261.sym \u2218 (f \u27e8 to\u2115-\u2294 \u27e9\u00b0 g))))\n                         (sum-\u2294 (\ud835\udfda\u25b9\u2115 \u2218 f) (\ud835\udfda\u25b9\u2115 \u2218 g))\n\nmodule FromSum\u00d7\n         {A B}\n         {sum\u1d2c     : Sum A}\n         (sum-ind\u1d2c : SumInd sum\u1d2c)\n         {sum\u1d2e     : Sum B}\n         (sum-ind\u1d2e : SumInd sum\u1d2e) where\n\n  module |A| = FromSumInd sum-ind\u1d2c\n  module |B| = FromSumInd sum-ind\u1d2e\n\n  sum\u1d2c\u1d2e = sum\u1d2c \u00d7-sum sum\u1d2e\n\n  sum-\u2218proj\u2081\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 |B|.Card * sum\u1d2c f\n  sum-\u2218proj\u2081\u2261Card* f\n      rewrite |A|.sum-ext (|B|.sum-const \u2218 f)\n            = |A|.sum-lin f |B|.Card\n\n  sum-\u2218proj\u2082\u2261Card* : \u2200 f \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 |A|.Card * sum\u1d2e f\n  sum-\u2218proj\u2082\u2261Card* = |A|.sum-const \u2218 sum\u1d2e\n\n  sum-\u2218proj\u2081 : \u2200 {f} {g} \u2192 sum\u1d2c f \u2261 sum\u1d2c g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2081) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2081)\n  sum-\u2218proj\u2081 {f} {g} sumf\u2261sumg\n      rewrite sum-\u2218proj\u2081\u2261Card* f\n            | sum-\u2218proj\u2081\u2261Card* g\n            | sumf\u2261sumg = \u2261.refl\n\n  sum-\u2218proj\u2082 : \u2200 {f} {g} \u2192 sum\u1d2e f \u2261 sum\u1d2e g \u2192 sum\u1d2c\u1d2e (f \u2218 proj\u2082) \u2261 sum\u1d2c\u1d2e (g \u2218 proj\u2082)\n  sum-\u2218proj\u2082 sumf\u2261sumg = |A|.sum-ext (const sumf\u2261sumg)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0f2cc6b467502bed861d0b9855cc3afda202e631","subject":"+ Data.Nat.NP.dist","message":"+ Data.Nat.NP.dist\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\n{-\npostulate\n  dist-refl  : \u2200 x \u2192 dist x x \u2261 0\n  dist-sym   : \u2200 x y \u2192 dist x y \u2261 dist y x\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-x+    : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n  dist-x*  : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ zero  = 1\n2^ suc n = 2* 2^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ zero  = 1\n2^ suc n = 2* 2^ n\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4b7555bf5ad22ddf79e254cfec3161115b7a36f6","subject":"Fixed setoid equality.","message":"Fixed setoid equality.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-positivity-check #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\nmodule FOTC where\n\nmodule Aczel-CA where\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    \u2250-refl  : \u2200 {x} \u2192 x \u2250 x\n    \u2250-sym   : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    \u2250-trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    \u2250-cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 y\u2081 \u2192 x\u2082 \u2250 y\u2082 \u2192 x\u2081 \u00b7 x\u2082 \u2250 y\u2081 \u00b7 y\u2082\n    K-eq    : \u2200 x y \u2192 K \u00b7 x \u00b7 y \u2250 x\n    S-eq    : \u2200 x y z \u2192 S \u00b7 x \u00b7 y \u00b7 z \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate \u2250-subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- we can proof\n\n  \u2250\u2192\u2261 : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  \u2250\u2192\u2261 {x} h = \u2250-subst (\u03bb z \u2192 x \u2261 z) h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule FOTC where\n\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    \u2250-refl   : \u2200 {x} \u2192 x \u2250 x\n    \u2250-sym    : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    \u2250-trans  : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    \u2250-cong   : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 y\u2081 \u2192 x\u2082 \u2250 y\u2082 \u2192 x\u2081 \u00b7 x\u2082 \u2250 y\u2081 \u00b7 y\u2082\n    if-true  : \u2200 d\u2081 d\u2082 \u2192 if \u00b7 true \u00b7 d\u2081 \u00b7 d\u2082 \u2250 d\u2081\n    if-false : \u2200 d\u2081 d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2250 d\u2082\n    pred-0   : pred \u00b7 zero \u2250 zero\n    pred-S   : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2250 n\n    iszero-0 : iszero \u00b7 zero \u2250 true\n    iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2250 false\n    beta     : \u2200 f a \u2192 lam f \u00b7 a \u2250 f a\n    fix-eq   : \u2200 f \u2192 fix f \u2250 f (fix f)\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-positivity-check #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\nmodule FOTC where\n\nmodule Aczel-CA where\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 {x} \u2192 x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    K-eq  : \u2200 x y \u2192 K \u00b7 x \u00b7 y \u2250 x\n    S-eq  : \u2200 x y z \u2192 S \u00b7 x \u00b7 y \u00b7 z \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- we can proof\n\n  \u2250\u2192\u2261 : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  \u2250\u2192\u2261 {x} h = subst (\u03bb z \u2192 x \u2261 z) h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n  --\n  -- The point is that \u2250 is a non-trivial equivalence relation, and\n  -- not all properties preserve it. However, all properties are\n  -- preserved by \u2261.\n\n------------------------------------------------------------------------------\n\nmodule FOTC where\n\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl     : \u2200 {x} \u2192 x \u2250 x\n    sym      : \u2200 {x y} \u2192 x \u2250 y \u2192 y \u2250 x\n    trans    : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192 x \u2250 z\n    if-true  : \u2200 d\u2081 d\u2082 \u2192 if \u00b7 true \u00b7 d\u2081 \u00b7 d\u2082 \u2250 d\u2081\n    if-false : \u2200 d\u2081 d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2250 d\u2082\n    pred-0   : pred \u00b7 zero \u2250 zero\n    pred-S   : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2250 n\n    iszero-0 : iszero \u00b7 zero \u2250 true\n    iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2250 false\n    beta     : \u2200 f a \u2192 lam f \u00b7 a \u2250 f a\n    fix-eq   : \u2200 f \u2192 fix f \u2250 f (fix f)\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    _\u00b7_ : D \u2192 D \u2192 D\n    lam fix : (D \u2192 D) \u2192 D\n    true false if zero succ pred iszero : D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"818dfccaa5b7e626b635720fbf80318263d4c8d9","subject":"Change strucure for products","message":"Change strucure for products\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Unit\nopen import Data.Product\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  infixl 6 _\u2295_ _\u229d_\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n      ; \u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 Ch A \u00d7 Ch B\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = Ch A \u00d7 Ch B\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    } }\n\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b} } }\n","old_contents":"module New.Changes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Unit\nopen import Data.Product\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  infixl 6 _\u2295_ _\u229d_\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n      ; \u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  -- instance\n  --   pairCA : ChAlg (A \u00d7 B)\n  -- pairCA = record\n  --   { Ch = Ch CA \u00d7 Ch CB ; isChAlg = record { _\u2295_ = {!!} ; _\u229d_ = {!!} ; valid = {!!} ; \u229d-valid = {!!} ; \u2295-\u229d = {!!} } }\n\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b} } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cceb613c5e0c4f4a72385d9f7bb5814dc8f6019d","subject":"Desc stratified model: IDescl implicit.","message":"Desc stratified model: IDescl implicit.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl I) -> IDescl I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"da063111bdead7bc9476b14520d957b8057cba93","subject":"Add Description introductions to operational semantics.","message":"Add Description introductions to operational semantics.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Operational.agda","new_file":"formalization\/agda\/Spire\/Operational.agda","new_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Desc `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u03bc : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Neutral \u0393 (`Desc \u2113) \u2192 Type \u0393 \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\u27e6_\u27e7\u1d48 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Type \u0393 \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Desc `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u03bc : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type \u2113)\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Value \u0393 (`Type \u2113))\n    (B : Value (\u0393 , \u27e6 A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Value \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `con : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Value \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D)) \u2192 Value \u0393 (`\u03bc D)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elim\u22a5 : \u2200{A} \u2192 Neutral \u0393 `\u22a5 \u2192 Neutral \u0393 A\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `\u03bc D \u27e7 = `\u03bc D\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `Desc {\u2113} \u27e7 = `Desc \u2113\n\u27e6 `\u27e6 D \u27e7\u1d48 X \u27e7 = \u27e6 D \u27e7\u1d48 \u27e6 X \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\n\u27e6 `\u22a4\u1d48 \u27e7\u1d48 X = `\u22a4\n\u27e6 `X\u1d48 \u27e7\u1d48 X = X\n\u27e6 `\u03a0\u1d48 A D \u27e7\u1d48 X = `\u03a0 \u27e6 A \u27e7 (\u27e6 D \u27e7\u1d48 (wknT X))\n\u27e6 `\u03a3\u1d48 A D \u27e7\u1d48 X = `\u03a3 \u27e6 A \u27e7 (\u27e6 D \u27e7\u1d48 (wknT X))\n\u27e6 `neut D \u27e7\u1d48 X = `\u27e6 D \u27e7\u1d48 X\n\n----------------------------------------------------------------------\n\nelim\u22a5 : \u2200{\u0393 A} \u2192 Value \u0393 `\u22a5 \u2192 Value \u0393 A\nelim\u22a5 (`neut bot) = `neut (`elim\u22a5 bot)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u03bc : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113) \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type \u2113)\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Term \u0393 (`Type \u2113))\n    (D : Term (\u0393 , \u27e6 eval A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Term \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B}\n    (b : Term (\u0393 , A) B)\n    \u2192 Term \u0393 (`\u03a0 A B)\n  `con : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Term \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D)) \u2192 Term \u0393 (`\u03bc D)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elim\u22a5 : \u2200{A} \u2192 Term \u0393 `\u22a5 \u2192 Term \u0393 A\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval (`\u03bc D) = `\u03bc (eval D)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\neval (`\u27e6 D \u27e7\u1d48 X) = `\u27e6 eval D \u27e7\u1d48 (eval X)\n\n{- Desc introduction -}\neval `\u22a4\u1d48 = `\u22a4\u1d48\neval `X\u1d48 = `X\u1d48\neval (`\u03a0\u1d48 A D) = `\u03a0\u1d48 (eval A) (eval D)\neval (`\u03a3\u1d48 A D) = `\u03a3\u1d48 (eval A) (eval D)\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\neval (`\u03bb b) = `\u03bb (eval b)\neval (`con x) = `con (eval x)\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elim\u22a5 bot) = elim\u22a5 (eval bot)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\n\n----------------------------------------------------------------------\n\n","old_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Desc `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Desc `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Value \u0393 (`Type \u2113))\n    (B : Value (\u0393 , \u27e6 A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Value \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elim\u22a5 : \u2200{A} \u2192 Neutral \u0393 `\u22a5 \u2192 Neutral \u0393 A\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `Desc {\u2113} \u27e7 = `Desc \u2113\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nelim\u22a5 : \u2200{\u0393 A} \u2192 Value \u0393 `\u22a5 \u2192 Value \u0393 A\nelim\u22a5 (`neut bot) = `neut (`elim\u22a5 bot)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Term \u0393 (`Type \u2113))\n    (D : Term (\u0393 , \u27e6 eval A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Term \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B}\n    (b : Term (\u0393 , A) B)\n    \u2192 Term \u0393 (`\u03a0 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elim\u22a5 : \u2200{A} \u2192 Term \u0393 `\u22a5 \u2192 Term \u0393 A\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Desc introduction -}\neval `\u22a4\u1d48 = `\u22a4\u1d48\neval `X\u1d48 = `X\u1d48\neval (`\u03a0\u1d48 A D) = `\u03a0\u1d48 (eval A) (eval D)\neval (`\u03a3\u1d48 A D) = `\u03a3\u1d48 (eval A) (eval D)\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\neval (`\u03bb b) = `\u03bb (eval b)\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elim\u22a5 bot) = elim\u22a5 (eval bot)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3d4fe54e867b2816ee1d1189b968548884a9fa1c","subject":"Nicer sum combinators and properties","message":"Nicer sum combinators and properties\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"open import Function\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Relation.Binary.PropositionalEquality\n\nmodule sum where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nrecord SumProp {A} (sum\u1d2c : Sum A) : \u2605 where\n  constructor mk\n  field\n    sum-ext : SumExt sum\u1d2c\n    sum-lin : SumLin sum\u1d2c\n\n  sum : Sum A\n  sum = sum\u1d2c\n\n  Card : \u2115\n  Card = sum\u1d2c (const 1)\n\n  count : Count A\n  count f = sum\u1d2c (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\nsum\u22a4 : Sum \u22a4\nsum\u22a4 f = f _\n\n\u03bc\u22a4 : SumProp sum\u22a4\n\u03bc\u22a4 = mk sum\u22a4-ext sum\u22a4-lin\n  where\n    sum\u22a4-ext : SumExt sum\u22a4\n    sum\u22a4-ext f\u2257g = f\u2257g _\n\n    sum\u22a4-lin : SumLin sum\u22a4\n    sum\u22a4-lin f k = refl\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\n\u03bcBit : SumProp sumBit\n\u03bcBit = mk sumBit-ext sumBit-lin\n  where\n    sumBit-ext : SumExt sumBit\n    sumBit-ext f\u2257g rewrite f\u2257g 0b | f\u2257g 1b = refl\n\n    sumBit-lin : SumLin sumBit\n    sumBit-lin f k\n      rewrite \u2115\u00b0.*-comm k (f 0b)\n            | \u2115\u00b0.*-comm k (f 1b)\n            | \u2115\u00b0.*-comm k (f 0b + f 1b)\n            | \u2115\u00b0.distrib\u02b3 k (f 0b) (f 1b)\n            = refl\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n_\u00d7Sum-ext_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} \u2192\n              SumExt sum\u1d2c \u2192 SumExt sum\u1d2e \u2192 SumExt (sum\u1d2c \u00d7Sum sum\u1d2e)\n(sumA-ext \u00d7Sum-ext sumB-ext) f\u2257g = sumA-ext (\u03bb x \u2192 sumB-ext (\u03bb y \u2192 f\u2257g (x , y)))\n\n_\u00d7\u03bc_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B}\n              \u2192 SumProp sum\u1d2c\n              \u2192 SumProp sum\u1d2e\n              \u2192 SumProp (sum\u1d2c \u00d7Sum sum\u1d2e)\n(\u03bcA \u00d7\u03bc \u03bcB)\n   = mk (sum-ext \u03bcA \u00d7Sum-ext sum-ext \u03bcB) lin\n   where\n     lin : SumLin (sum \u03bcA \u00d7Sum sum \u03bcB)\n     lin f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-lin \u03bcB (\u03bb y \u2192 f (x , y)) k) = sum-lin \u03bcA (\u03bb x \u2192 sum \u03bcB (\u03bb y \u2192 f (x , y))) k\n\nswapS : \u2200 {A B} \u2192 Sum (A \u00d7 B) \u2192 Sum (B \u00d7 A)\nswapS sumA\u00d7B f = sumA\u00d7B (f \u2218 swap)\n\nsum-const : \u2200 {A} {sum\u1d2c : Sum A} (\u03bcA : SumProp sum\u1d2c) \u2192 \u2200 k \u2192 sum\u1d2c (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = refl\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B}\n               (\u03bcA : SumProp sum\u1d2c)\n               (\u03bcB : SumProp sum\u1d2e)\n               f \u2192\n               (sum\u1d2c \u00d7Sum sum\u1d2e) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum\u1d2c f\n_\u00d7Sum-proj\u2081_ {sum\u1d2e = sum\u1d2e} (mk sum\u1d2c-ext sum\u1d2c-lin) \u03bcB f\n  rewrite sum\u1d2c-ext (sum-const \u03bcB \u2218 f)\n        = sum\u1d2c-lin f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B}\n               (\u03bcA : SumProp sum\u1d2c)\n               (\u03bcB : SumProp sum\u1d2e)\n               f \u2192\n               (sum\u1d2c \u00d7Sum sum\u1d2e) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum\u1d2e f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B}\n                  (\u03bcA : SumProp sum\u1d2c) (\u03bcB : SumProp sum\u1d2e)\n                  {f} {g} \u2192\n                  sum\u1d2c f \u2261 sum\u1d2c g \u2192\n                  (sum\u1d2c \u00d7Sum sum\u1d2e) (f \u2218 proj\u2081) \u2261 (sum\u1d2c \u00d7Sum sum\u1d2e) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B}\n                  (\u03bcA : SumProp sum\u1d2c) (\u03bcB : SumProp sum\u1d2e)\n                  {f} {g} \u2192\n                  sum\u1d2e f \u2261 sum\u1d2e g \u2192\n                  (sum\u1d2c \u00d7Sum sum\u1d2e) (f \u2218 proj\u2082) \u2261 (sum\u1d2c \u00d7Sum sum\u1d2e) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n","old_contents":"open import Function\nopen import Data.Nat\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Relation.Binary.PropositionalEquality\n\nmodule sum where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sumA = \u2200 {f g} \u2192 f \u2257 g \u2192 sumA f \u2261 sumA g\n\nsumToCount : \u2200 {A} \u2192 Sum A \u2192 Count A\nsumToCount sumA f = sumA (Bool.to\u2115 \u2218 f)\n\nsum\u22a4 : Sum \u22a4\nsum\u22a4 f = f _\n\nsum\u22a4-ext : SumExt sum\u22a4\nsum\u22a4-ext f\u2257g = f\u2257g _\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\nsumBit-ext : SumExt sumBit\nsumBit-ext f\u2257g rewrite f\u2257g 0b | f\u2257g 1b = refl\n\n-- liftM2 _,_ in the continuation monad\nsumProd : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsumProd sumA sumB f = sumA (\u03bb x\u2080 \u2192\n                        sumB (\u03bb x\u2081 \u2192\n                          f (x\u2080 , x\u2081)))\n\nsumProd-ext : \u2200 {A B} {sumA : Sum A} {sumB : Sum B} \u2192\n              SumExt sumA \u2192 SumExt sumB \u2192 SumExt (sumProd sumA sumB)\nsumProd-ext sumA-ext sumB-ext f\u2257g = sumA-ext (\u03bb x \u2192 sumB-ext (\u03bb y \u2192 f\u2257g (x , y)))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e869902c24c1d79f55a5b9786a3aa5b860916883","subject":"Updated setoids note.","message":"Updated setoids note.\n\nIgnore-this: b2195213e40354b7581f2b5296058599\n\ndarcs-hash:20120514153433-3bd4e-445a63bf03c50e3c97e90940ceba07c74ac2a202.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/setoids\/FOTC.agda","new_file":"notes\/setoids\/FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 14 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n  ----------------------------------------------------------------------------\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- Adapted from Peter's email:\n\n  -- Given\n  postulate subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- you can get the instance\n\n  subst-aux : \u2200 x y \u2192 x \u2250 y \u2192 x \u2261 x \u2192 x \u2261 y\n  subst-aux x y h\u2081 h\u2082 = subst A x y h\u2081 refl\n    where A : D \u2192 Set\n          A z = x \u2261 z\n\n  -- hence you can prove\n\n  thm : \u2200 {x y} \u2192 x \u2250 y \u2192 x \u2261 y\n  thm {x} {y} h = subst-aux x y h refl\n\n  -- but this doesn't hold because \"x \u2261 y\" (propositional equality)\n  -- means identical expressions. We do NOT have K \u00b7 x \u00b7 y \u2261 x.\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","old_contents":"------------------------------------------------------------------------------\n-- Using setoids for formalizing the FOTC\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of Agda on 14 May 2012.\n\n{-\n\nFrom Peter emails:\n\nAt that time I was considering an inductive data type D and an\ninductively defined equality on D. But I think what we are doing now\nis better.\n\nFor =: The reason is that with the propositional identity we have\nidentity elimination which lets us replace equals by equals. We cannot\ndo that in general if we have setoid equality.\n\nFor D: the reason why I prefer to postulate D is that we have no use\nfor the inductive structure of D, and this inductive structure would\nmake e g 0 + 0 different from 0. So in that case we need setoid\nequality.\n\n-}\n\nmodule FOTC where\n\n-- References:\n--\n-- \u2022 Gilles Barthe, Venanzio Capretta, and Olivier Pons. Setoids in\n--   type theory. Journal of Functional Programming, 13(2):261\u2013293,\n--   2003.\n\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n  -- From Peter's slides\n  -- http:\/\/www.cse.chalmers.se\/~peterd\/slides\/Amagasaki.pdf\n\n  -- We add 3 to the fixities of the standard library.\n  infixl 9 _\u00b7_  -- The symbol is '\\cdot'.\n  infix 7  _\u2250_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- The setoid equality.\n  data _\u2250_ : D \u2192 D \u2192 Set where\n    refl  : \u2200 x \u2192                                       x \u2250 x\n    sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n    trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n    cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n    Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n    Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n  -- 14 May 2012. We cannot define the identity elimination using the\n  -- setoid equality.\n  --\n  -- subst : (A : D \u2192 Set) \u2192 \u2200 x y \u2192 x \u2250 y \u2192 A x \u2192 A y\n\n  -- The identity type.\n  data _\u2261_ (x : D) : D \u2192 Set where\n    refl : x \u2261 x\n\n  -- 14 May 2012: Using the inductive structure we cannot prove\n  --\n  -- K \u00b7 x \u00b7 y \u2261 x,\n  --\n  -- we need the setoid equality.\n  -- K-eq : \u2200 {x y} \u2192 (K \u00b7 x \u00b7 y) \u2261 x\n\n------------------------------------------------------------------------------\n\nmodule LeibnizEquality where\n\n  infix 7 _\u2261_\n\n  data D : Set where\n    K S : D\n    _\u00b7_ : D \u2192 D \u2192 D\n\n  -- (Barthe et al. 2003, p. 262) use the Leibniz equality when\n  -- they talk about setoids.\n\n  -- Using the Leibniz equality (adapted from\n  -- Agda\/examples\/lib\/Logic\/Leibniz.agda)\n\n  _\u2261_ : D \u2192 D \u2192 Set\u2081\n  x \u2261 y = (A : D \u2192 Set) \u2192 A x \u2192 A y\n\n  -- we can prove the setoids properties\n\n  refl : \u2200 x \u2192 x \u2261 x\n  refl x A Ax = Ax\n\n  sym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\n  sym {x} h A Ay = h (\u03bb z \u2192 A z \u2192 A x) (\u03bb Ax \u2192 Ax) Ay\n\n  trans : \u2200 x y z \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\n  trans x y z h\u2081 h\u2082 A Ax = h\u2082 A (h\u2081 A Ax)\n\n  -- and the identity elimination\n\n  subst : (A : D \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A x\u2261y = x\u2261y A\n\n  -- and the congruency\n\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 x\u2082 \u2192 y\u2081 \u2261 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2261 x\u2082 \u00b7 y\u2082\n  cong {x\u2081} {x\u2082} {y\u2081} {y\u2082} h\u2081 h\u2082 A Ax\u2081x\u2082 =\n    h\u2082 (\u03bb z \u2192 A (x\u2082 \u00b7 z)) (h\u2081 (\u03bb z \u2192 A (z \u00b7 y\u2081)) Ax\u2081x\u2082)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5d7647a9747da1890100da0dfd670ed16f37961a","subject":"Rename apply to apply-base.","message":"Rename apply to apply-base.\n\nOld-commit-hash: d8eae9079296e08a6b1db1db97ac5c9722947b28\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Term.agda","new_file":"Parametric\/Change\/Term.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff-base apply-base {base \u03b9} = diff-base , apply-base\n  lift-diff-apply diff-base apply-base {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply-base {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply-base {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff-base apply-base = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n\n  lift-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff-base apply-base = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff-base apply-base {\u03c4} {\u0393})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Change.Type as ChangeType\n\nmodule Parametric.Change.Term\n    {Base : Set}\n    (Constant : Term.Structure Base)\n    (\u0394Base : ChangeType.Structure Base)\n  where\n\n-- Terms that operate on changes\n\nopen Type.Structure Base\nopen Term.Structure Base Constant\nopen ChangeType.Structure Base \u0394Base\n\nopen import Data.Product\n\nDiffStructure : Set\nDiffStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 base (\u0394Base \u03b9))\n\nApplyStructure : Set\nApplyStructure = \u2200 {\u03b9 \u0393} \u2192 Term \u0393 (\u0394Type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\n\nmodule Structure where\n\n  -- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n  -- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\n  lift-diff-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192\n      Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4) \u00d7 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-diff-apply diff-base apply {base \u03b9} = diff-base , apply\n  lift-diff-apply diff-base apply {\u03c3 \u21d2 \u03c4} =\n    let\n      -- for diff\n      g  = var (that (that (that this)))\n      f  = var (that (that this))\n      x  = var (that this)\n      \u0394x = var this\n      -- for apply\n      \u0394h = var (that (that this))\n      h  = var (that this)\n      y  = var this\n      -- syntactic sugars\n      diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply {\u03c3} {\u0393})\n      diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n      apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply {\u03c3} {\u0393})\n      apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n      _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n      _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n      _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n      _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n    in\n      abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n      ,\n      abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\n  lift-diff :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394Type \u03c4)\n\n  lift-diff diff-base apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2081 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n\n  lift-apply :\n    DiffStructure \u2192\n    ApplyStructure \u2192\n    \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\n  lift-apply diff-base apply = \u03bb {\u03c4 \u0393} \u2192\n    proj\u2082 (lift-diff-apply diff-base apply {\u03c4} {\u0393})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1cd2e5da496e2586bbfd89e1cc85a64458588a3b","subject":"Pick correct base case for recursive functions","message":"Pick correct base case for recursive functions\n\nA little duplication remained, because absN and absV do not have as base\ncase arity 0 but arity 1, just because the case for arity 0 is not very\nuseful per se. Hence, abs{N,V} need to unroll one recursive invocation\ninto the base case, and need to have an helper (absNBase) to factor out\nthe ensuing duplication.\n\nThis commit fixes that; next commit will inline absNBase.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_])\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    -- A better type for absN but a mess to use due to the proofs (which aren't synthesized, even though maybe they should be?)\n    -- absN : (n : \u2115) \u2192 {_ : n > 0} \u2192 absNType n\n    -- XXX See \"how to keep your neighbours in order\" for tricks.\n\n    -- Please the termination checker by keeping this case separate.\n    absNBase : \u2200 {\u03c4\u2081} \u2192 absNType [ \u03c4\u2081 ]\n    absNBase {\u03c4\u2081} f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    -- Otherwise, the recursive step of absN would invoke absN twice, and the\n    -- termination checker does not figure out that the calls are in fact\n    -- terminating.\n\n    -- What I'd like to write, avoiding the need for absNBase, but can't because of the termination checker.\n    {-\n    absN {zero}  (\u03c4\u2081 \u2237 []) f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n      absN (\u03c4\u2081 \u2237 []) (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n    -}\n    --What I have to write instead:\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN {zero}  [] f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4s) f =\n      absNBase (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n    -- Using a similar trick, we can declare absV which takes the N implicit\n    -- type arguments individually, collects them and passes them on to absN.\n    -- This is inspired by what's shown in the Agda 2.4.0 release notes, and\n    -- relies critically on support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType 0       \u03c4s = absNType \u03c4s\n    absVType (suc n) \u03c4s = {\u03c4\u1d62 : Type} \u2192 absVType n (\u03c4\u1d62 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN \u03c4s\n    absVAux \u03c4s (suc n) = \u03bb {\u03c4\u1d62} \u2192 absVAux (\u03c4\u1d62 \u2237 \u03c4s) n\n\n    absV = absVAux []\n\n  open AbsNHelpers using (absV) public\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n\n  abs\u2081 = absV 1\n  abs\u2082 = absV 2\n  abs\u2083 = absV 3\n  abs\u2084 = absV 4\n  abs\u2085 = absV 5\n  abs\u2086 = absV 6\n\n  {-\n  Example types:\n  abs\u2081:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n        ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n        Term \u0393 (\u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n  abs\u2082:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n        ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n         Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n        Term \u0393 (\u03c4\u2082 Type.Structure.\u21d2 \u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_])\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    -- A better type for absN but a mess to use due to the proofs (which aren't synthesized, even though maybe they should be?)\n    -- absN : (n : \u2115) \u2192 {_ : n > 0} \u2192 absNType n\n    -- XXX See \"how to keep your neighbours in order\" for tricks.\n\n    -- Please the termination checker by keeping this case separate.\n    absNBase : \u2200 {\u03c4\u2081} \u2192 absNType [ \u03c4\u2081 ]\n    absNBase {\u03c4\u2081} f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    -- Otherwise, the recursive step of absN would invoke absN twice, and the\n    -- termination checker does not figure out that the calls are in fact\n    -- terminating.\n\n    -- What I'd like to write, avoiding the need for absNBase, but can't because of the termination checker.\n    {-\n    absN {zero}  (\u03c4\u2081 \u2237 []) f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n      absN (\u03c4\u2081 \u2237 []) (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n    -}\n    --What I have to write instead:\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ (suc n)) \u2192 absNType \u03c4s\n    absN {zero}  (\u03c4\u2081 \u2237 []) = absNBase\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n      absNBase (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n    -- Using a similar trick, we can declare absV which takes the N implicit\n    -- type arguments individually, collects them and passes them on to absN.\n    -- This is inspired by what's shown in the Agda 2.4.0 release notes, and\n    -- relies critically on support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType 0       \u03c4s = absNType \u03c4s\n    absVType (suc n) \u03c4s = {\u03c4\u1d62 : Type} \u2192 absVType n (\u03c4\u1d62 \u2237 \u03c4s)\n\n    -- XXX\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType (suc n) \u03c4s\n    absVAux \u03c4s zero    {\u03c4\u1d62} = absN (\u03c4\u1d62 \u2237 \u03c4s)\n    absVAux \u03c4s (suc n) {\u03c4\u1d62} = absVAux (\u03c4\u1d62 \u2237 \u03c4s) n\n\n    absV = absVAux []\n\n  open AbsNHelpers using (absV) public\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n\n  abs\u2081 = absV 0\n  abs\u2082 = absV 1\n  abs\u2083 = absV 2\n  abs\u2084 = absV 3\n  abs\u2085 = absV 4\n  abs\u2086 = absV 5\n\n{-\nabs\u2081\nHave: {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      Term \u0393 (\u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\nabs\u2082\nHave: {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n      ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n       Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n      Term \u0393 (\u03c4\u2082 Type.Structure.\u21d2 \u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"183052671becc3f5336e73837cf344a6ebea2787","subject":"removing an unused lemma, killing a todo","message":"removing an unused lemma, killing a todo\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expansion-unicity.agda","new_file":"expansion-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import synth-unicity\nopen import lemmas-matching\n\nmodule expansion-unicity where\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2)\n      with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , ih3\n    expansion-unicity-synth (ESAp _ _ x x\u2081 x\u2082 x\u2083) (ESAp _ _ x\u2084 x\u2085 x\u2086 x\u2087)\n      with synthunicity x x\u2084\n    ... | refl with match-unicity x\u2081 x\u2085\n    ... | refl with expansion-unicity-ana x\u2082 x\u2086\n    ... | refl , refl , refl with expansion-unicity-ana x\u2083 x\u2087\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole _ d1) (ESNEHole _ d2)\n      with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc x) (ESAsc x\u2081)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 \u03c42' : htyp} {d d' : dhexp} {\u0394 \u0394' : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d  :: \u03c42  \u22a3 \u0394  \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d' :: \u03c42' \u22a3 \u0394' \u2192\n                          d == d' \u00d7 \u03c42 == \u03c42' \u00d7 \u0394 == \u0394'\n    expansion-unicity-ana (EALam x\u2081 m D1) (EALam x\u2082 m2 D2)\n      with match-unicity m m2\n    ... | refl with expansion-unicity-ana D1 D2\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EALam x\u2081 m D1) (EASubsume x\u2082 x\u2083 () x\u2085)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALam x\u2085 m D2)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087)\n      with expansion-unicity-synth x\u2082 x\u2086\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = abort (x _ refl)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole _ x\u2084) = abort (x\u2081 _ _ refl)\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = abort (x _ refl)\n    expansion-unicity-ana EAEHole EAEHole = refl , refl , refl\n    expansion-unicity-ana (EANEHole _ x) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = abort (x\u2082 _ _ refl)\n    expansion-unicity-ana (EANEHole _ x) (EANEHole _ x\u2081)\n      with expansion-unicity-synth x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import synth-unicity\nopen import lemmas-matching\n\nmodule expansion-unicity where\n  -- todo: move to a lemmas file\n  \u2987\u2988\u2260arr : \u2200{t1 t2} \u2192 t1 ==> t2 == \u2987\u2988 \u2192 \u22a5\n  \u2987\u2988\u2260arr ()\n\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2)\n      with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , ih3\n    expansion-unicity-synth (ESAp _ _ x x\u2081 x\u2082 x\u2083) (ESAp _ _ x\u2084 x\u2085 x\u2086 x\u2087)\n      with synthunicity x x\u2084\n    ... | refl with match-unicity x\u2081 x\u2085\n    ... | refl with expansion-unicity-ana x\u2082 x\u2086\n    ... | refl , refl , refl with expansion-unicity-ana x\u2083 x\u2087\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole _ d1) (ESNEHole _ d2)\n      with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc x) (ESAsc x\u2081)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 \u03c42' : htyp} {d d' : dhexp} {\u0394 \u0394' : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d  :: \u03c42  \u22a3 \u0394  \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d' :: \u03c42' \u22a3 \u0394' \u2192\n                          d == d' \u00d7 \u03c42 == \u03c42' \u00d7 \u0394 == \u0394'\n    expansion-unicity-ana (EALam x\u2081 m D1) (EALam x\u2082 m2 D2)\n      with match-unicity m m2\n    ... | refl with expansion-unicity-ana D1 D2\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EALam x\u2081 m D1) (EASubsume x\u2082 x\u2083 () x\u2085)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALam x\u2085 m D2)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087)\n      with expansion-unicity-synth x\u2082 x\u2086\n    ... | refl , refl , refl = refl , refl , refl\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = abort (x _ refl)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole _ x\u2084) = abort (x\u2081 _ _ refl)\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = abort (x _ refl)\n    expansion-unicity-ana EAEHole EAEHole = refl , refl , refl\n    expansion-unicity-ana (EANEHole _ x) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = abort (x\u2082 _ _ refl)\n    expansion-unicity-ana (EANEHole _ x) (EANEHole _ x\u2081)\n      with expansion-unicity-synth x x\u2081\n    ... | refl , refl , refl = refl , refl , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5cdc145c60ac3a135d4ee0d9a34a4c17c8943e5a","subject":"Some more semsec proofs","message":"Some more semsec proofs\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP\nopen import circuit\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule FunctionExtra where\n  _***_ : \u2200 {A B C D : Set} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n  _>>>_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n            (A \u2192 B) \u2192 (B \u2192 C) \u2192 (A \u2192 C)\n  f >>> g = g \u2218 f\n  infixr 1 _>>>_\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\nPorts = \u2115\nSize  = \u2115\nTime  = \u2115\n\nrecord Power : Set where\n  constructor mk\n  field\n    coins : Coins\n    size  : Size\n    time  : Time\nopen Power public\n\nsame-power : Power \u2192 Power\nsame-power = id\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Power \u2192 Set\n  Oracle power = \u2203 (\u03bb (A : GuessAdv (coins power)) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule FunAdv (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n\n  M = Bits |M|\n  C = Bits |C|\n\n  M\u00d7M = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n{-\n  record SemSecAdv' power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c)\n    field\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n      beh : \u21ba c\u2080 (M\u00d7M \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    R' = subst Bits c\u2261c\u2080+c\u2081\n    R\u2080 = take c\u2080 \u2218 R'\n    R\u2081 = drop c\u2080 \u2218 R'\n-}\n\n  SemSecAdv : Coins \u2192 Set\n  SemSecAdv c = \u21ba c (M\u00d7M \u00d7 (C \u2192 Bit))\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : SemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A >>= \u03bb { (m , kont) \u2192 map\u21ba kont (E (m b)) }\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : SemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  _\u2257A_ : \u2200 {ca} (A\u2081 A\u2082 : SemSecAdv ca) \u2192 Set\n  _\u2257A_ A\u2081 A\u2082 = (proj\u2081 \u2218 run\u21ba A\u2081 \u2257 proj\u2081 \u2218 run\u21ba A\u2082)\n             \u00d7 (\u2200 R \u2192 proj\u2082 (run\u21ba A\u2081 R) \u2257 proj\u2082 (run\u21ba A\u2082 R))\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : SemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {cc} {ca} {E} {A\u2081} {A\u2082} (pf1 , pf2) b R with splitAt ca R\n  change-adv {cc} {ca} {E} {A\u2081} {A\u2082} (pf1 , pf2) b ._ | pre , post , refl =\n      proj\u2082 (run\u21ba A\u2081 pre) (run\u21ba (E (proj\u2081 (run\u21ba A\u2081 pre) b)) post) \u2261\u27e8 cong (\u03bb A \u2192 proj\u2082 (run\u21ba A\u2081 pre) (run\u21ba (E (A b)) post)) (pf1 pre) \u27e9\n      proj\u2082 (run\u21ba A\u2081 pre) (run\u21ba (E (proj\u2081 (run\u21ba A\u2082 pre) b)) post) \u2261\u27e8 pf2 pre (run\u21ba (E (proj\u2081 (run\u21ba A\u2082 pre) b)) post) \u27e9\n      proj\u2082 (run\u21ba A\u2082 pre) (run\u21ba (E (proj\u2081 (run\u21ba A\u2082 pre) b)) post) \u220e\n\n  module SplitSemSecAdv {c} (A : SemSecAdv c) where\n    R = Bits c\n    |S| = c\n    S = R\n\n    beh\u2081 : R \u2192 M\u00d7M -- \u00d7 S\n    beh\u2081 R = proj\u2081 (run\u21ba A R) -- , R\n\n    beh\u2082 : S \u2192 C \u2192 Bit\n    beh\u2082 R = proj\u2082 (run\u21ba A R)\n\n    open FunctionExtra\n\n    beh : R \u2192 M\u00d7M \u00d7 (C \u2192 Bit)\n    beh = beh\u2081 &&& beh\u2082\n\n    beh\u21ba : SemSecAdv c\n    beh\u21ba = mk beh\n\n    coh\u2081 : proj\u2081 \u2218 run\u21ba A \u2257 proj\u2081 \u2218 beh\n    coh\u2081 _ = refl\n\n    coh\u2082 : \u2200 R \u2192 proj\u2082 (run\u21ba A R) \u2257 proj\u2082 (beh R)\n    coh\u2082 _ _ = refl\n\n    coh : A \u2257A beh\u21ba\n    coh = coh\u2081 , coh\u2082\n\n  ext-as-broken : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                  \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n  ext-as-broken same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\n  SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n  SemBroken E power = \u2203 (\u03bb (A : SemSecAdv (coins power)) \u2192 breaks (E \u21c4 A))\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n  --   security of E\u2080 reduces to security of E\u2081\n  --   breaking E\u2081 reduces to breaking E\u2080\n  SemSecReduction : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} {cc\u2081} f tr = \u2200 {p} {E : Enc cc\u2080} \u2192 SemSecReduction (f p) p E (tr E)\n\n  -- SemSecReductionToOracle : \u2200 (p\u2080 p\u2081 : Power) {cc} (E : Enc cc) \u2192 Set\n  -- SemSecReductionToOracle = SemBroken E p\u2080 \u2192 Oracle p\u2081\n\n  open FunctionExtra\n\n  -- post-comp : \u2200 {cc k} (post-E : C \u2192 \u21ba k C) \u2192 Tr cc (cc + k)\n  -- post-comp post-E E = E >=> post-E\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : C \u2192 C)\n                           \u2192 SemSecTr same-power (post-comp {cc} post-E)\n  post-comp-pres-sem-sec {cc} post-E {p} {E} (A' , A'-breaks-E') = A , A-breaks-E\n     where E' : Enc cc\n           E' = post-comp post-E E\n           pre-post-E : (C \u2192 Bit) \u2192 (C \u2192 Bit)\n           pre-post-E kont = post-E >>> kont\n           A : SemSecAdv (coins p)\n           A = mk (run\u21ba A' >>> id *** pre-post-E)\n           A-breaks-E : breaks (E \u21c4 A)\n           A-breaks-E = A'-breaks-E'\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc p} {E : Enc cc}\n                            \u2192 SemSecReduction p p (post-comp post-E E) E\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {p} {E} (A , A-breaks-E)\n       = A' , A'-breaks-E'\n     where E' : Enc cc\n           E' = post-comp post-E E\n           open Power p renaming (coins to ca)\n           pre-post-E\u207b\u00b9 : (C \u2192 Bit) \u2192 (C \u2192 Bit)\n           pre-post-E\u207b\u00b9 kont = post-E\u207b\u00b9 >>> kont\n           A' : SemSecAdv ca \n           A' = mk (run\u21ba A >>> id *** pre-post-E\u207b\u00b9)\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (run\u21ba A (take ca R)) b)) (drop ca R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = ext-as-broken same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg E = E >>> map\u21ba vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr same-power (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {p} {E} = post-comp-pres-sem-sec vnot {p} {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc p} {E : Enc cc}\n                            \u2192 SemSecReduction p p (post-neg E) E\n  post-neg-pres-sem-sec' {cc} {p} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {p} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\n{-\nmodule F''\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (Pr-ext : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 Pr[ f \u22611] \u2261 Pr[ g \u22611])\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n  (non-negligible : Proba \u2192 Set)\n\n  where\n  advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Proba\n  advantage EXP = dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = non-negligible (advantage \u2141)\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible (advantage (E \u21c4 A))\n-}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecReduction (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec = ?\n-}\n\nrecord Cp-kit : Set\u2081 where\n  constructor mk\n  field\n    Cp : Ports \u2192 Ports \u2192 Set\n    builder : CircuitBuilder Cp\n    runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o\n  open CircuitBuilder builder hiding (idC-spec)\n  field\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 runC idC bs \u2261 bs\n    >>>-spec : \u2200 {i m o} (c\u2080 : Cp i m) (c\u2081 : Cp m o) xs \u2192 runC (c\u2080 >>> c\u2081) xs \u2261 runC c\u2081 (runC c\u2080 xs)\n    ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : Cp i\u2080 o\u2080) (c\u2081 : Cp i\u2081 o\u2081) xs {ys}\n               \u2192 runC (c\u2080 *** c\u2081) (xs ++ ys) \u2261 runC c\u2080 xs ++ runC c\u2081 ys\n\nfunBits-kit : Cp-kit\nfunBits-kit = mk _\u2192\u1d47_ bitsFunCircuitBuilder id idC-spec >>>-spec ***-spec where\n  open CircuitBuilder bitsFunCircuitBuilder\n  open BitsFunExtras\n\nmodule CpAdv\n  (cp-kit : Cp-kit)\n  -- (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  -- (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n\n  (prgDist : PrgDist)\n  (|M| |C| : \u2115)\n\n  where\n  open Cp-kit cp-kit\n\n  open PrgDist prgDist\n  module FunAdv' = FunAdv prgDist |M| |C|\n  open FunAdv' public using (module SplitSemSecAdv; M; C; M\u00d7M; Enc; Tr; breaks; ext-as-broken; change-adv; _\u2257A_; _\u2257\u2141_; \u2257\u2141-trans) renaming (_\u21c4_ to _\u21c4\u21ba_; SemSecAdv to SemSecAdv\u21ba)\n\n{-\n  record SemSecAdv' power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n-}\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n\n    R = Bits c\n\n    open FunctionExtra\n\n    |S| : \u2115\n    |S| = c\n\n    field\n      -- |S| : \u2115\n\n      cp\u2081 : Cp c (|M| + |M| {- + |S| -})\n\n      cp\u2082 : Cp (|S| + |C|) 1\n\n    S = Bits |S|\n\n    beh\u2081 : R \u2192 M\u00d7M -- \u00d7 S\n    beh\u2081 = runC cp\u2081 >>>\n           -- splitAt\u2032 (|M| + |M|) >>>\n           (splitAt\u2032 |M| >>> proj) -- *** id\n\n    beh\u2082 : S \u2192 C \u2192 Bit\n    beh\u2082 S C = head (runC cp\u2082 (S ++ C))\n\n    beh : R \u2192 M\u00d7M \u00d7 (C \u2192 Bit)\n    -- beh = beh\u2081 >>> id *** beh\u2082\n    beh = beh\u2081 &&& beh\u2082\n\n    beh\u21ba : \u21ba c (M\u00d7M \u00d7 (C \u2192 Bit))\n    beh\u21ba = mk beh\n\n    adv\u21ba : SemSecAdv\u21ba c\n    adv\u21ba = beh\u21ba\n\n  record SemSecAdv+FunBeh power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n\n    field\n      Acp : SemSecAdv power\n      A\u21ba : SemSecAdv\u21ba c\n\n    open SemSecAdv Acp renaming (beh\u21ba to Acp-beh\u21ba; beh to Acp-beh; beh\u2082 to Acp-beh\u2082; beh\u2081 to Acp-beh\u2081)\n    A\u21ba-beh : R \u2192 M\u00d7M \u00d7 (C \u2192 Bit)\n    A\u21ba-beh = run\u21ba A\u21ba\n\n    module SplitA\u21ba = SplitSemSecAdv A\u21ba\n    open SplitA\u21ba using () renaming (beh\u2081 to As-beh\u2081; beh\u2082 to As-beh\u2082)\n\n    field\n      coh\u2081 : Acp-beh\u2081 \u2257 As-beh\u2081\n      coh\u2082 : \u2200 R \u2192 Acp-beh\u2082 R \u2257 As-beh\u2082 R\n\n    coh : SemSecAdv.adv\u21ba Acp \u2257A A\u21ba\n    coh = coh\u2081 , coh\u2082\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  _\u21c4_ : \u2200 {cc p} (E : Enc cc) (A : SemSecAdv p) b \u2192 \u21ba (coins p + cc) Bit\n  E \u21c4 A = E \u21c4\u21ba SemSecAdv.beh\u21ba A\n\n  SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n  SemBroken E power = \u2203 (\u03bb (A : SemSecAdv power) \u2192 breaks (E \u21c4 A))\n\n  -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n  --   security of E\u2080 reduces to security of E\u2081\n  --   breaking E\u2081 reduces to breaking E\u2080\n  SemSecReduction : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} {cc\u2081} f tr = \u2200 {p} {E : Enc cc\u2080} \u2192 SemSecReduction (f p) p E (tr E)\n\n  module FE = FunctionExtra\n\n  post-comp-cp : \u2200 {cc} post-E \u2192 Tr cc cc\n  post-comp-cp post-E E = E >>> map\u21ba (runC post-E)\n    where open FunctionExtra\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : Cp |C| |C|) \u2192 SemSecTr same-power (post-comp-cp {cc} post-E)\n  post-comp-pres-sem-sec {cc} post-E {p} {E} (A' , A'-breaks-E') = A , A-breaks-E\n     where E' : Enc cc\n           E' = post-comp-cp post-E E\n           open Power p renaming (coins to c)\n           open SemSecAdv A' using (R; S; |S|; cp\u2081) renaming (beh to A'-beh; beh\u21ba to A'-beh\u21ba; cp\u2082 to A'-cp\u2082)\n           pre-post-E-spec : (C \u2192 Bit) \u2192 (C \u2192 Bit)\n           pre-post-E-spec kont = runC post-E >>> kont\n             where open FunctionExtra\n           A-beh-spec = A'-beh >>> id *** pre-post-E-spec\n             where open FunctionExtra\n           A-beh\u21ba-spec : SemSecAdv\u21ba (coins p)\n           A-beh\u21ba-spec = mk A-beh-spec\n           A-cp\u2082 = idC {|S|} *** post-E >>> A'-cp\u2082\n             where open CircuitBuilder builder\n           A : SemSecAdv p\n           A = mk {-|S|-} cp\u2081 A-cp\u2082\n           open SemSecAdv A using () renaming (beh\u21ba to A-beh\u21ba; beh to A-beh)\n\n           open CircuitBuilder builder hiding (idC-spec)\n           pf1 : \u2200 R \u2192 SemSecAdv.beh\u2082 A R \u2257 pre-post-E-spec (SemSecAdv.beh\u2082 A' R)\n           pf1 R C rewrite >>>-spec (idC {c} *** post-E) A'-cp\u2082 (R ++ C)\n                         | ***-spec (idC {c}) post-E R {C}\n                         | idC-spec R = refl\n           pf2 : \u2200 R \u2192 proj\u2082 (A-beh-spec R) \u2257 proj\u2082 (A-beh R)\n           pf2 R C rewrite pf1 R C = refl\n           pf3 : A-beh\u21ba-spec \u2257A A-beh\u21ba\n           pf3 = (\u03bb _ \u2192 refl) , pf2\n           same-games : (E' \u21c4 A') \u2257\u2141 (E \u21c4 A)\n           same-games = change-adv {E = E} pf3\n           A-breaks-E : breaks (E \u21c4 A)\n           A-breaks-E = ext-as-broken same-games A'-breaks-E'\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecReduction (_\u2218_ (_\u2295_ mask))\n-}\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess prgDist\n  module FunAdv' = FunAdv prgDist\n  module CpAdv' = CpAdv funBits-kit prgDist\n\n\nmodule OTP (prgDist : PrgDist) where\n  open FunAdv prgDist 1 1\n  open Guess prgDist renaming (breaks to reads-your-mind)\n\n  -- OTP\u2081 : \u2200 {c} (k : Bit) \u2192 Enc c\n  -- OTP\u2081 k m = return\u21ba ((k xor (head m)) \u2237 [])\n\n  OTP\u2081 : Enc 1\n  OTP\u2081 m = map\u21ba (\u03bb k \u2192 (k xor (head m)) \u2237 []) toss\n\n  open FunctionExtra\n\n  -- \u2200 k \u2192 SemSecReductionToOracle no-power no-power (OTP\u2081 k)\n  foo : \u2200 p \u2192 SemBroken OTP\u2081 p \u2192 Oracle p\n  foo p (A , A-breaks-OTP) = O , {!!}\n     where b : Bit\n           b = {!!}\n           k : Bit\n           k = {!!}\n           O : \u21ba (coins p) Bit\n           O = mk (run\u21ba A >>> \u03bb { (m , kont) \u2192 (head (m b)) xor (kont (run\u21ba (OTP\u2081 (m b)) (k \u2237 []))) })\n           O-reads-your-mind : reads-your-mind (runGuess\u2141 O)\n           O-reads-your-mind = ?\n","old_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP\nopen import circuit\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import flipbased-implem\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule FunctionExtra where\n  _***_ : \u2200 {A B C D : Set} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n  _>>>_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n            (A \u2192 B) \u2192 (B \u2192 C) \u2192 (A \u2192 C)\n  f >>> g = g \u2218 f\n  infixr 1 _>>>_\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\nCoins = \u2115\nPorts = \u2115\nSize  = \u2115\nTime  = \u2115\n\nrecord Power : Set where\n  constructor mk\n  field\n    coins : Coins\n    size  : Size\n    time  : Time\nopen Power public\n\nsame-power : Power \u2192 Power\nsame-power = id\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\nmodule Guess (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Power \u2192 Set\n  GuessSec power = \u2200 (A : GuessAdv (coins power)) \u2192 \u00ac(breaks (runGuess\u2141 A))\n\nmodule F' (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n\n{-\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n  open Power public\n-}\n{-\n  Power = Coins\n  module Power (p : Power) where\n    coins : Coins\n    coins = p\n  open Power\n-}\n\n  M = Bits |M|\n  C = Bits |C|\n\n  M\u00d7M = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n{-\n  record SemSecAdv' power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c)\n    field\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n      beh : \u21ba c\u2080 (M\u00d7M \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    R' = subst Bits c\u2261c\u2080+c\u2081\n    R\u2080 = take c\u2080 \u2218 R'\n    R\u2081 = drop c\u2080 \u2218 R'\n-}\n\n  SemSecAdv : Coins \u2192 Set\n  SemSecAdv c = \u21ba c (M\u00d7M \u00d7 (C \u2192 Bit))\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : SemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A >>= \u03bb { (m , kont) \u2192 map\u21ba kont (E (m b)) }\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : SemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  _\u2257A_ : \u2200 {ca} (A\u2081 A\u2082 : SemSecAdv ca) \u2192 Set\n  _\u2257A_ A\u2081 A\u2082 = \u2200 R \u2192 proj\u2081 (run\u21ba A\u2081 R) \u2261 proj\u2081 (run\u21ba A\u2082 R) \u00d7 proj\u2082 (run\u21ba A\u2081 R) \u2257 proj\u2082 (run\u21ba A\u2082 R) \n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : SemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv A\u2081\u2257A\u2082 = {!!}\n\n  ext-as-broken : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                  \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n  ext-as-broken same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\n  SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n  SemBroken E power = \u2203 (\u03bb (A : SemSecAdv (coins power)) \u2192 breaks (E \u21c4 A))\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n  --   security of E\u2080 reduces to security of E\u2081\n  --   breaking E\u2081 reduces to breaking E\u2080\n  SemSecReduction : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} {cc\u2081} f tr = \u2200 {p} {E : Enc cc\u2080} \u2192 SemSecReduction (f p) p E (tr E)\n\n  open FunctionExtra\n\n  -- post-comp : \u2200 {cc k} (post-E : C \u2192 \u21ba k C) \u2192 Tr cc (cc + k)\n  -- post-comp post-E E = E >=> post-E\n\n  post-comp : \u2200 {cc} (post-E : C \u2192 C) \u2192 Tr cc cc\n  post-comp post-E E = E >>> map\u21ba post-E\n\n  post-comp-pres-sem-sec : \u2200 {cc {-k-}} (post-E : C \u2192 {-\u21ba k-} C)\n                           \u2192 SemSecTr same-power (post-comp {cc} post-E)\n  post-comp-pres-sem-sec {cc} {-k-} post-E {p} {E} (A' , A'-breaks-E') = A , A-breaks-E\n     where E' : Enc cc {-(cc + k)-}\n           E' = post-comp post-E E\n           pre-post-E : (C \u2192 Bit) \u2192 (C \u2192 Bit)\n           pre-post-E kont = post-E >>> kont\n           A : SemSecAdv (coins p)\n           A = mk (run\u21ba A' >>> id *** pre-post-E)\n           A-breaks-E : breaks (E \u21c4 A)\n           A-breaks-E = A'-breaks-E'\n\n  post-comp-pres-sem-sec' : \u2200 (post-E post-E\u207b\u00b9 : C \u2192 C)\n                              (post-E-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                              {cc p} {E : Enc cc}\n                            \u2192 SemSecReduction p p (post-comp post-E E) E\n  post-comp-pres-sem-sec' post-E post-E\u207b\u00b9 post-E-inv {cc} {p} {E} (A , A-breaks-E)\n       = A' , A'-breaks-E'\n     where E' : Enc cc\n           E' = post-comp post-E E\n           open Power p renaming (coins to ca)\n           pre-post-E\u207b\u00b9 : (C \u2192 Bit) \u2192 (C \u2192 Bit)\n           pre-post-E\u207b\u00b9 kont = post-E\u207b\u00b9 >>> kont\n           A' : SemSecAdv ca \n           A' = mk (run\u21ba A >>> id *** pre-post-E\u207b\u00b9)\n           same-games : (E \u21c4 A) \u2257\u2141 (E' \u21c4 A')\n           same-games b R\n             rewrite post-E-inv (run\u21ba (E (proj\u2081 (run\u21ba A (take ca R)) b)) (drop ca R)) = refl\n           A'-breaks-E' : breaks (E' \u21c4 A')\n           A'-breaks-E' = ext-as-broken same-games A-breaks-E\n\n  post-neg : \u2200 {cc} \u2192 Tr cc cc\n  post-neg E = E >>> map\u21ba vnot\n\n  post-neg-pres-sem-sec : \u2200 {cc} \u2192 SemSecTr same-power (post-neg {cc})\n  post-neg-pres-sem-sec {cc} {p} {E} = post-comp-pres-sem-sec vnot {p} {E}\n\n  post-neg-pres-sem-sec' : \u2200 {cc p} {E : Enc cc}\n                            \u2192 SemSecReduction p p (post-neg E) E\n  post-neg-pres-sem-sec' {cc} {p} {E} = post-comp-pres-sem-sec' vnot vnot vnot\u2218vnot {cc} {p} {E}\n\nopen import diff\nimport Data.Fin as Fin\n_]-_-[_ : \u2200 {c} (f : \u21ba c Bit) k (g : \u21ba c Bit) \u2192 Set\n_]-_-[_ {c} f k g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n\nopen import Data.Vec.NP using (count)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\n]-[-cong : \u2200 {k c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]- k -[ f' \u2192 g ]- k -[ g'\n]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\nmodule Concrete k where\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ f g = f ]- k -[ g\n  cong' : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  cong' = ]-[-cong {k}\n  prgDist : PrgDist\n  prgDist = mk _]-[_ cong'\n  module Guess' = Guess prgDist\n  module F'' = F' prgDist\n\n{-\nmodule F''\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (Pr-ext : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 Pr[ f \u22611] \u2261 Pr[ g \u22611])\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n  (non-negligible : Proba \u2192 Set)\n\n  where\n  advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Proba\n  advantage EXP = dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = non-negligible (advantage \u2141)\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible (advantage (E \u21c4 A))\n-}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecReduction (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec = ?\n-}\n\nmodule F\n  -- (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  (Cp : Ports \u2192 Ports \u2192 Set)\n  (builder : CircuitBuilder Cp)\n  -- (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n  (runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o)\n\n  (prgDist : PrgDist)\n  (|M| |C| : \u2115)\n\n  where\n\n  open PrgDist prgDist\n  module FF' = F' prgDist |M| |C|\n  open FF' public using (M; C; M\u00d7M; Enc; Tr; breaks; ext-as-broken; _\u2257\u2141_) renaming (_\u21c4_ to _\u21c4\u21ba_; SemSecAdv to SemSecAdv\u21ba)\n\n{-\n  record SemSecAdv' power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n-}\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n\n    R = Bits c\n\n    open FunctionExtra\n\n    field\n      |S| : \u2115\n\n      cp\u2081 : Cp c (|M| + |M| + |S|)\n\n      cp\u2082 : Cp (|S| + |C|) 1\n\n    S = Bits |S|\n\n    beh\u2081 : R \u2192 M\u00d7M \u00d7 S\n    beh\u2081 = runC cp\u2081 >>>\n           splitAt\u2032 (|M| + |M|) >>>\n           (splitAt\u2032 |M| >>> proj) *** id\n\n    beh\u2082 : S \u2192 C \u2192 Bit\n    beh\u2082 S C = head (runC cp\u2082 (S ++ C))\n\n    beh : R \u2192 M\u00d7M \u00d7 (C \u2192 Bit)\n    beh = beh\u2081 >>> id *** beh\u2082\n\n    beh\u21ba : \u21ba c (M\u00d7M \u00d7 (C \u2192 Bit))\n    beh\u21ba = mk beh\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  _\u21c4_ : \u2200 {cc p} (E : Enc cc) (A : SemSecAdv p) b \u2192 \u21ba (coins p + cc) Bit\n  E \u21c4 A = E \u21c4\u21ba SemSecAdv.beh\u21ba A\n\n  SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n  SemBroken E power = \u2203 (\u03bb (A : SemSecAdv power) \u2192 breaks (E \u21c4 A))\n\n  -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n  --   security of E\u2080 reduces to security of E\u2081\n  --   breaking E\u2081 reduces to breaking E\u2080\n  SemSecReduction : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Power \u2192 Power) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} {cc\u2081} f tr = \u2200 {p} {E : Enc cc\u2080} \u2192 SemSecReduction (f p) p E (tr E)\n\n  module FE = FunctionExtra\n\n  post-comp-cp : \u2200 {cc} post-E \u2192 Tr cc cc\n  post-comp-cp post-E E = E >>> map\u21ba (runC post-E)\n    where open FunctionExtra\n\n  post-comp-pres-sem-sec : \u2200 {cc} (post-E : Cp |C| |C|) \u2192 SemSecTr same-power (post-comp-cp {cc} post-E)\n  post-comp-pres-sem-sec {cc} post-E {p} {E} (A' , A'-breaks-E') = A , A-breaks-E\n     where E' : Enc cc\n           E' = post-comp-cp post-E E\n           open Power p renaming (coins to c)\n           open SemSecAdv A' using (|S|; cp\u2081) renaming (beh to A'-beh; beh\u21ba to A'-beh\u21ba; cp\u2082 to A'-cp\u2082)\n           pre-post-E-spec : (C \u2192 Bit) \u2192 (C \u2192 Bit)\n           pre-post-E-spec kont = runC post-E >>> kont\n             where open FunctionExtra\n           A-beh-spec = A'-beh >>> id *** pre-post-E-spec\n             where open FunctionExtra\n           A-beh\u21ba-spec : SemSecAdv\u21ba (coins p)\n           A-beh\u21ba-spec = mk A-beh-spec\n           A : SemSecAdv p\n           A = mk |S| cp\u2081 (idC {|S|} *** post-E >>> A'-cp\u2082)\n             where open CircuitBuilder builder\n           open SemSecAdv A using () renaming (beh\u21ba to A-beh\u21ba; beh to A-beh; cp\u2082 to A-cp\u2082)\n           pf1 : (E \u21c4 A) \u2257\u2141 (E \u21c4\u21ba A-beh\u21ba)\n           pf1 b R = refl\n           pf2 : (E' \u21c4 A') \u2257\u2141 (E' \u21c4\u21ba A'-beh\u21ba)\n           pf2 b R = refl\n           pf8 : \u2200 R \u2192 proj\u2082 (A-beh-spec R) \u2257 pre-post-E-spec (proj\u2082 (A'-beh R))\n           pf8 R C = refl\n           open FunctionExtra\n           pf9 : \u2200 S \u2192 SemSecAdv.beh\u2082 A S \u2257 pre-post-E-spec (SemSecAdv.beh\u2082 A' S)\n           pf9 S C = {!refl!}\n           pf7 : \u2200 R \u2192 proj\u2082 (A-beh-spec R) \u2257 proj\u2082 (A-beh R)\n           pf7 R C = {!refl!}\n           pf5 : A-beh\u21ba-spec FF'.\u2257A A-beh\u21ba\n           pf5 R = refl , pf7 R\n           pf4 : (E \u21c4\u21ba A-beh\u21ba-spec) \u2257\u2141 (E \u21c4 A)\n           pf4 = FF'.change-adv {E = E} pf5\n           same-games' : (E' \u21c4 A') \u2257\u2141 (E \u21c4\u21ba A-beh\u21ba-spec)\n           same-games' b R = refl\n           same-games : (E' \u21c4 A') \u2257\u2141 (E \u21c4 A)\n           same-games = FF'.\u2257\u2141-trans same-games' pf4\n           A-breaks-E : breaks (E \u21c4 A)\n           A-breaks-E = ext-as-broken same-games A'-breaks-E'\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecReduction (_\u2218_ (_\u2295_ mask))\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"23a1872a5364849a3933d8bff9994451fc2e8af0","subject":"adding error rules","message":"adding error rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error -- todo\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error -- todo\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ \u25a0 (x , d2) ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d -- is that the right thing to step to?\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"369a950d7d4409bef26003a7886c5953168f7c32","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 815f23479d55496bdf66c71607acc9dd\n\ndarcs-hash:20120216141238-3bd4e-04648c2f32e1b3ebc6fae9dbabbdc8746a64cfe7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Common\/Data\/Product.agda","new_file":"Draft\/Common\/Data\/Product.agda","new_contents":"-----------------------------------------------------------------------------\n-- Classical existential elimination\n-----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.Common.Data.Product where\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\n-----------------------------------------------------------------------------\n-- Existential elimination\n--     \u2203x.P(x)   P(x) \u2192 Q\n--  ------------------------\n--             Q\n\npostulate\n  D : Set\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n-- The existential elimination.\n\n-- NB. We do not use the usual type theory elimination with two\n-- projections because we are working in first-order logic where we\n-- cannot extract a term from an existence proof.\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n","old_contents":"-----------------------------------------------------------------------------\n-- Classical existential elimination\n-----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.Common.Data.Product where\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\n-----------------------------------------------------------------------------\n-- Existential elimination\n--     \u2203x.P(x)   P(x) \u2192 Q\n--  ------------------------\n--             Q\n\npostulate\n  D : Set\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6abb26d7c78f3a4cd9e7b2488dcdbeae868eb2d7","subject":"JS: more bindings","message":"JS: more bindings\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS.agda","new_file":"lib\/FFI\/JS.agda","new_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber Number #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString String #-}\n\n-- TODO dyn check of length 1?\npostulate castChar : JSValue \u2192 Char\n{-# COMPILED_JS castChar String #-}\n\n-- TODO dyn check of length 1?\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char String #-}\n\n-- TODO dyn check?\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray function(x) { return x; } #-}\n\n-- TODO dyn check?\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject function(x) { return x; } #-}\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\npostulate onJSArray : {A : Set} (f : JSArray JSValue \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onJSArray require(\"libagda\").onJSArray #-}\n\npostulate onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onString require(\"libagda\").onString #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = castString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback0 : Set\nCallback0 = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 Callback0\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ncheck : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\ncheck true  errmsg x = x\ncheck false errmsg x = throw (errmsg _) x\n\nwarn-check : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\nwarn-check true  errmsg x = x\nwarn-check false errmsg x = trace (\"Warning: \" ++ errmsg _) x id\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\ndata JS! : Set\u2081 where\n  end  : JS!\n  _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n  _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n\n_>>_ : Callback0 \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate zero : Number\n{-# COMPILED_JS zero 0 #-}\n\npostulate one : Number\n{-# COMPILED_JS one 1 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber Number #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString String #-}\n\n-- TODO dyn check of length 1?\npostulate castChar : JSValue \u2192 Char\n{-# COMPILED_JS castChar String #-}\n\n-- TODO dyn check of length 1?\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char String #-}\n\n-- TODO dyn check?\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray function(x) { return x; } #-}\n\n-- TODO dyn check?\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject function(x) { return x; } #-}\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\npostulate onJSArray : {A : Set} (f : JSArray JSValue \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onJSArray require(\"libagda\").onJSArray #-}\n\npostulate onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onString require(\"libagda\").onString #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = castString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback0 : Set\nCallback0 = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 Callback0\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\ndata JS! : Set\u2081 where\n  end  : JS!\n  _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n  _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n\n_>>_ : Callback0 \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"144768b93a323280507e81e1fe33d1c7b17af57d","subject":"Added eta-beta equivalence, which finishes the model of F-omega.","message":"Added eta-beta equivalence, which finishes the model of F-omega.\n","repos":"agda\/agda-frp-js,agda\/agda-frp-js","old_file":"src\/agda\/FRP\/JS\/Model.agda","new_file":"src\/agda\/FRP\/JS\/Model.agda","new_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.Bool using ( true ; false )\nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\nsym : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 (a \u2261 b) \u2192 (b \u2261 a)\nsym refl = refl\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\nstruct-sym : \u2200 K {A B C D \u2111 \u211c} \u2192 (A\u2261B : A \u2261 B) \u2192 (C\u2261D : C \u2261 D) \u2192\n  (\u2111 \u2261 struct K A\u2261B \u211c C\u2261D) \u2192 \n    (\u211c \u2261 struct K (sym A\u2261B) \u2111 (sym C\u2261D))\nstruct-sym K refl refl refl = refl\n\nstruct-trans : \u2200 K {A B C D E F}\n  (A\u2261B : A \u2261 B) (B\u2261C : B \u2261 C) (\u211c : K \u220b C \u2194 F) (E\u2261F : E \u2261 F) (D\u2261E : D \u2261 E) \u2192\n    struct K A\u2261B (struct K B\u2261C \u211c E\u2261F) D\u2261E \u2261\n      struct K (trans A\u2261B B\u2261C) \u211c (trans D\u2261E E\u2261F)\nstruct-trans K refl refl \u211c refl refl = refl\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Concatenation of type contexts\n\n_++_ : Kinds \u2192 Kinds \u2192 Kinds\n[]      ++ \u03a5 = \u03a5\n(K \u2237 \u03a3) ++ \u03a5 = K \u2237 (\u03a3 ++ \u03a5)\n\n_\u220b_++_\u220b_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u2200 \u03a5 \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 ++ \u03a5 \u27e7\n[]      \u220b tt       ++ \u03a5 \u220b Bs = Bs\n(K \u2237 \u03a3) \u220b (A , As) ++ \u03a5 \u220b Bs = (A , (\u03a3 \u220b As ++ \u03a5 \u220b Bs))\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03a3 {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 \u2200 \u03a5 {Cs Ds} \u2192 (\u03a5 \u220b Cs \u2194* Ds) \u2192 \n  ((\u03a3 ++ \u03a5) \u220b (\u03a3 \u220b As ++ \u03a5 \u220b Cs) \u2194* (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds))\n[]      \u220b tt       ++\u00b2 \u03a5 \u220b \u2111s = \u2111s\n(K \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s = (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq lt : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 lt \u27e7     = \u03bb t u \u2192 True (t < u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 lt \u27e7\u00b2     = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n_\u227a_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227a_ = capp\u2082 lt\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time (app tvar\u2081 tvar\u2080))\n\ninterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\ninterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227c tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\nconstreq : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nconstreq {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (\u03a0 time \n  ((tvar\u2081 \u227c tvar\u2080) \u22b8 (app (app (app interval tvar\u2083) tvar\u2081) tvar\u2080 \u22b8 app tvar\u2082 tvar\u2080)))))\n\n_\u22b5_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b5_ = app\u2082 constreq\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Susbtitution on type variables under a context\n\n\u03c4substn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 TVar K (\u03a3 ++ (L \u2237 \u03a5)) \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\n\u03c4substn+ []      zero    U = U\n\u03c4substn+ []      (suc \u03c4) U = var \u03c4\n\u03c4substn+ (K \u2237 \u03a3) zero    U = var zero\n\u03c4substn+ (K \u2237 \u03a3) (suc \u03c4) U = weaken (skip K id) (\u03c4substn+ \u03a3 \u03c4 U)\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7 (A , As) Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Bs = trans \n  (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs) \n  (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip K id) (A , (\u03a3 \u220b As ++ _ \u220b Bs)))\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ (J \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s = refl\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 (J \u2237 \u03a3) {\u03a5} {L} {K} (suc \u03c4) U {A , As} {B , Bs} {Cs} {Ds} (\u211c , \u211cs) \u2111s = \n  begin\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) X (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (skip J id) (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n      (struct K\n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs))) \n        (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))) \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 struct-trans K \n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs)))\n       (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)))\n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))\n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 \u03c4substn+ (J \u2237 \u03a3) (suc \u03c4) U \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) )\n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (B , Bs) Ds)    \n  \u220e\n\n-- Substitution on types under a context\n\nsubstn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 Typ (\u03a3 ++ (L \u2237 \u03a5)) K \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\nsubstn+ \u03a3 (const C) U = const C\nsubstn+ \u03a3 (abs K T) U = abs K (substn+ (K \u2237 \u03a3) T U)\nsubstn+ \u03a3 (app S T) U = app (substn+ \u03a3 S U) (substn+ \u03a3 T U)\nsubstn+ \u03a3 (var \u03c4)   U = \u03c4substn+ \u03a3 \u03c4 U\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    T\u27e6 T \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 substn+ \u03a3 T U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7 As Bs = refl\nsubstn+ \u03a3 \u27e6 abs K T \u27e7\u27e6 U \u27e7 As Bs = ext (\u03bb A \u2192 substn+ K \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Bs)\nsubstn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Bs = apply (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Bs) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Bs)\nsubstn+ \u03a3 \u27e6 var \u03c4   \u27e7\u27e6 U \u27e7 As Bs = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = refl\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (abs J {K} T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs J T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u211c\n  \u2261\u27e8 substn+ (J \u2237 \u03a3) \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s \u27e9\n    struct K \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds)\n  \u2261\u27e8 struct-ext J K \n       (\u03bb A \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n       (\u03bb \u211c \u2192 T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b \u211c , \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (\u03bb B \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds) \u211c \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 (abs J T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 Bs Ds) \u211c\n  \u220e)))\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (app {J} {K} S T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  begin\n    T\u27e6 app S T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 cong (T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct J \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 struct-apply J K \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n       (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 substn+ \u03a3 (app S T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u220e\nsubstn+ \u03a3 \u27e6 var \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s\n\n-- Substitution on types\n\nsubstn : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 \u03a3) K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 K\nsubstn = substn+ []\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) (As : \u03a3\u27e6 \u03a3 \u27e7)\u2192\n  T\u27e6 T \u27e7 (T\u27e6 U \u27e7 As , As) \u2261 T\u27e6 substn T U \u27e7 As\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7 tt\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192\n  T\u27e6 T \u27e7\u00b2 (T\u27e6 U \u27e7\u00b2 \u211cs , \u211cs) \u2261 \n    struct K (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 Bs)\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 tt\n\n-- Eta-beta equivalence on types\n\ndata _\u220b_\u2263_ {\u03a3} : \u2200 K \u2192 Typ \u03a3 K \u2192 Typ \u03a3 K \u2192 Set where\n  abs : \u2200 K {L T U} \u2192 (L \u220b T \u2263 U) \u2192 ((K \u21d2 L) \u220b abs K T \u2263 abs K U)\n  app : \u2200 {K L F G T U} \u2192 ((K \u21d2 L) \u220b F \u2263 G) \u2192 (K \u220b T \u2263 U) \u2192 (L \u220b app F T \u2263 app G U)\n  beta : \u2200 {K L} T U \u2192 (L \u220b app (abs K T) U \u2263 substn T U)\n  eta : \u2200 {K L} T \u2192 ((K \u21d2 L) \u220b T \u2263 abs K (app (weaken (skip K id) T) tvar\u2080))\n  \u2263-refl : \u2200 {K T} \u2192 (K \u220b T \u2263 T)\n  \u2263-sym : \u2200 {K T U} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 T)\n  \u2263-trans : \u2200 {K T U V} \u2192 (K \u220b T \u2263 U) \u2192 (K \u220b U \u2263 V) \u2192 (K \u220b T \u2263 V)\n\n\u2263\u27e6_\u27e7 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} \u2192 (K \u220b T \u2263 U) \u2192 \u2200 As \u2192 T\u27e6 T \u27e7 As \u2261 T\u27e6 U \u27e7 As\n\u2263\u27e6 abs K T\u2263U \u27e7       As = ext (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As))\n\u2263\u27e6 app F\u2263G T\u2263U \u27e7     As = apply (\u2263\u27e6 F\u2263G \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 beta T U \u27e7        As = substn\u27e6 T \u27e7\u27e6 U \u27e7 As\n\u2263\u27e6 eta {K} T \u27e7       As = ext (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n\u2263\u27e6 \u2263-refl \u27e7          As = refl\n\u2263\u27e6 \u2263-sym T\u2263U \u27e7       As = sym (\u2263\u27e6 T\u2263U \u27e7 As)\n\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As = trans (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As)\n\n\u2263\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} {T U : Typ \u03a3 K} (T\u2263U : K \u220b T \u2263 U) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 \u211cs \u2261 struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n\u2263\u27e6 abs K {L} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 (\u211c , \u211cs) \u27e9\n    struct L (\u2263\u27e6 T\u2263U \u27e7 (A , As)) (T\u27e6 U \u27e7\u00b2 (\u211c , \u211cs)) (\u2263\u27e6 T\u2263U \u27e7 (B , Bs))\n  \u2261\u27e8 struct-ext K L (\u03bb A \u2192 \u2263\u27e6 T\u2263U \u27e7 (A , As)) (\u03bb \u211c' \u2192 T\u27e6 U \u27e7\u00b2 (\u211c' , \u211cs)) (\u03bb B \u2192 \u2263\u27e6 T\u2263U \u27e7 (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 abs K T\u2263U \u27e7 As) (T\u27e6 abs K U \u27e7\u00b2 \u211cs) (\u2263\u27e6 abs K T\u2263U \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 app {K} {L} {F} {G} {T} {U} F\u2263G T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app F T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 cong (T\u27e6 F \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) \u27e9\n    T\u27e6 F \u27e7\u00b2 \u211cs (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)))\n       (\u2263\u27e6 F\u2263G \u27e7\u00b2 \u211cs) \u27e9\n    struct (K \u21d2 L) (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n      (struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs))\n  \u2261\u27e8 struct-apply K L\n       (\u2263\u27e6 F\u2263G \u27e7 As) (T\u27e6 G \u27e7\u00b2 \u211cs) (\u2263\u27e6 F\u2263G \u27e7 Bs)\n       (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct L (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 As) (T\u27e6 app G U \u27e7\u00b2 \u211cs) (\u2263\u27e6 app F\u2263G T\u2263U \u27e7 Bs)\n  \u220e\n\u2263\u27e6 beta T U \u27e7\u00b2 \u211cs = substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs\n\u2263\u27e6 eta {K} {L} T \u27e7\u00b2 {As} {Bs} \u211cs = iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 \n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs \u211c\n  \u2261\u27e8 cong (\u03bb X \u2192 X \u211c) (weaken\u27e6 T \u27e7\u00b2 (skip K id) (\u211c , \u211cs)) \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 T \u27e7 (skip K id) (A , As))\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs))\n      (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) \u211c\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 (skip K id) (A , As)) \n       (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl \u211c refl \u27e9\n    struct L \n      (apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl)\n      (T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c)\n      (apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl)\n  \u2261\u27e8 struct-ext K L\n       (\u03bb A \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (A , As)) refl) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (skip K id) T \u27e7\u00b2 (\u211c , \u211cs) \u211c) \n       (\u03bb B \u2192 apply (weaken\u27e6 T \u27e7 (skip K id) (B , Bs)) refl) \u211c \u27e9\n    struct (K \u21d2 L) \n      (\u2263\u27e6 eta T \u27e7 As)\n      (T\u27e6 abs K (app (weaken (skip K id) T) (var zero)) \u27e7\u00b2 \u211cs)\n      (\u2263\u27e6 eta T \u27e7 Bs) \u211c\n  \u220e)))\n\u2263\u27e6 \u2263-refl \u27e7\u00b2 \u211cs = refl\n\u2263\u27e6 \u2263-sym {K} {T} {U} T\u2263U \u27e7\u00b2 {As} {Bs} \u211cs = \n  struct-sym K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs)\n\u2263\u27e6 \u2263-trans {K} {T} {U} {V} T\u2263U U\u2263V \u27e7\u00b2 {As} {Bs} \u211cs =\n  begin\n    T\u27e6 T \u27e7\u00b2 \u211cs\n  \u2261\u27e8 \u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u2263\u27e6 T\u2263U \u27e7 As) X (\u2263\u27e6 T\u2263U \u27e7 Bs)) (\u2263\u27e6 U\u2263V \u27e7\u00b2 \u211cs) \u27e9\n    struct K (\u2263\u27e6 T\u2263U \u27e7 As) (struct K (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs)) (\u2263\u27e6 T\u2263U \u27e7 Bs)\n  \u2261\u27e8 struct-trans K (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 U\u2263V \u27e7 Bs) (\u2263\u27e6 T\u2263U \u27e7 Bs) \u27e9\n    struct K (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 As) (T\u27e6 V \u27e7\u00b2 \u211cs) (\u2263\u27e6 \u2263-trans T\u2263U U\u2263V \u27e7 Bs)\n  \u220e\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  leq-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  leq-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7      As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7       As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7       As = \u03bb A B \u2192 proj\u2082\nc\u27e6 leq-refl \u27e7  As = \u2264-refl\nc\u27e6 leq-trans \u27e7 As = \u2264-trans\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2      \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 leq-refl \u27e7\u00b2  \u211cs = _\nc\u27e6 leq-trans \u27e7\u00b2 \u211cs = _\n\n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n  tapp : \u2200 {K \u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} \u2192 Exp \u0393 (\u03a0 K T) \u2192 \u2200 U \u2192 Exp \u0393 (substn T U)\n  eq : \u2200 {\u03b1 T U} \u2192 (set \u03b1 \u220b T \u2263 U) \u2192 (Exp \u0393 T) \u2192 (Exp \u0393 U)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\nM\u27e6 tapp {T = T} M U \u27e7 As as = \n  cast (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (M\u27e6 M \u27e7 As as (T\u27e6 U \u27e7 As))\nM\u27e6 eq T\u2263U M \u27e7 As as = cast (\u2263\u27e6 T\u2263U \u27e7 As) (M\u27e6 M \u27e7 As as)\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\nM\u27e6 tapp {T = T} M U \u27e7\u00b2 \u211cs as\u211cbs = \n  struct-cast (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _)\n    (cast\u00b2 (substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 U \u27e7\u00b2 \u211cs)))\nM\u27e6 eq {\u03b1} {T} {U} T\u2263U M \u27e7\u00b2 {As} {Bs} \u211cs as\u211cbs = \n  struct-cast (T\u27e6 U \u27e7\u00b2 \u211cs) (\u2263\u27e6 T\u2263U \u27e7 As) (\u2263\u27e6 T\u2263U \u27e7 Bs) (cast\u00b2 (\u2263\u27e6 T\u2263U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs))\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app taut \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar\u2080 \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsignal : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsignal T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsignal\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsignal\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsignal-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (signal T s (signal\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsignal-iso T s \u03c3 = refl\n\nsignal-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (signal\u207b\u00b9 T s (signal T s \u03c3) \u2261 \u03c3)\nsignal-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = signal U s (f (signal\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = signal\u207b\u00b9 U s (f (signal T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _)\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 (T at s \u22a8 \u03c3 s \u2248[ u ] \u03c4 s)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192 T\u27ea T \u21d2 U \u27eb s \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U s f = \u2200 t \u03c3 \u03c4 \u2192 \n  (T at s \u22a8 \u03c3 \u2248[ t ] \u03c4) \u2192 (U at s \u22a8 f \u03c3 \u2248[ t ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u a c a\u211cc , \u211c-impl-\u2248 U at s u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u a b a\u2248b))\n  \u211c-impl-\u2248 (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb t s\u2264t t\u2264u \u2192 \u211c-impl-\u2248 T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u a c a\u2248c , \u2248-impl-\u211c U at s u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt t\u2264u with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt t\u2264u | refl = \n      \u2248-impl-\u211c T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u2248\u03c4 t s\u2264t (t\u2264u refl {s\u2264t} {s\u2264t} tt {\u2264-refl t} {\u2264-refl t} tt))\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s \u2192 \u211c-impl-\u2248 T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u211c\u03c4 refl)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl = \u2248-impl-\u211c T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u2248\u03c4 s)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) (M : Exp [] \u27ea [ T \u21d2 U ] \u27eb) s \u2192 \n  Causal T U s (M\u27e6 M \u27e7 (World , tt) tt s)\ncausality T U M s t \n  = \u211c-impl-\u2248 T \u21d2 U at s t \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ t ] , _) tt refl)\n\n","old_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.Bool using ( true ; false )\nopen import FRP.JS.True using ( True ; tt )\n\nmodule FRP.JS.Model where\n\n-- This model is essentially System F-omega with a kind time\n-- together with a type for the partial order on time,\n-- and expressions for reflexivity and transitivity.\n-- We prove parametricity, and then show that parametricity implies causality.\n\n-- Note that this is a \"deep\" notion of causality, not the \"shallow\"\n-- causality usually used in FRP. The pragmatic upshot of this is that\n-- there is only one time model: nested signals are in the same time\n-- model, not a simulated time model. This fits with the JS implementation,\n-- which uses wall clock time for all signals.\n\n-- Propositional equality\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\ntrans : \u2200 {\u03b1} {A : Set \u03b1} {a b c : A} \u2192 (a \u2261 b) \u2192 (b \u2261 c) \u2192 (a \u2261 c)\ntrans refl refl = refl\n\ncong : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} (f : A \u2192 B) {a\u2081 a\u2082 : A} \u2192\n  (a\u2081 \u2261 a\u2082) \u2192 (f a\u2081 \u2261 f a\u2082)\ncong f refl = refl\n\napply : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192 (F \u2261 G) \u2192 \n  \u2200 {a b} \u2192 (a \u2261 b) \u2192 (F a \u2261 G b)\napply refl refl = refl\n\ncast : \u2200 {\u03b1} {A B : Set \u03b1} \u2192 (A \u2261 B) \u2192 A \u2192 B\ncast refl a = a\n\ncast\u00b2 : \u2200 {\u03b1} {A B : Set \u03b1} {\u211c \u2111 : A \u2192 B \u2192 Set \u03b1} \u2192 (\u211c \u2261 \u2111) \u2192 \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 a b\ncast\u00b2 refl a\u211cb = a\u211cb\n\nirrel : \u2200 b \u2192 (b\u2081 b\u2082 : True b) \u2192 (b\u2081 \u2261 b\u2082)\nirrel true  tt tt = refl\nirrel false () ()\n\n-- Postulates (including dependent extensionality)\n\npostulate\n  \u2264-refl : \u2200 t \u2192 True (t \u2264 t)\n  \u2264-trans : \u2200 t u v \u2192 True (t \u2264 u) \u2192 True (u \u2264 v) \u2192 True (t \u2264 v)\n  dext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 a \u2192 B a} \u2192 (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\n\next : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : Set \u03b2} {F G : A \u2192 B} \u2192\n  (\u2200 a \u2192 F a \u2261 G a) \u2192 (F \u2261 G)\next = dext\n\niext : \u2200 {\u03b1 \u03b2} {A : Set \u03b1} {B : A \u2192 Set \u03b2} {F G : \u2200 {a} \u2192 B a} \u2192 \n  (\u2200 a \u2192 F {a} \u2261 G {a}) \u2192 ((\u03bb {a} \u2192 F {a}) \u2261 (\u03bb {a} \u2192 G {a}))\niext F\u2248G = cong (\u03bb X {a} \u2192 X a) (dext F\u2248G)\n\n-- Finite products\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\n_\u00d7\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u00d7 B) \u2192 (C \u00d7 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u00d7\u00b2 \u2111) (a , b) (c , d) = (\u211c a c \u00d7 \u2111 b d)\n\n_\u2192\u00b2_ : \u2200 {\u03b1 \u03b2} {A C : Set \u03b1} {B D : Set \u03b2} \u2192 \n  (A \u2192 C \u2192 Set \u03b1) \u2192 (B \u2192 D \u2192 Set \u03b2) \u2192 ((A \u2192 B) \u2192 (C \u2192 D) \u2192 Set (\u03b1 \u2294 \u03b2))\n(\u211c \u2192\u00b2 \u2111) f g = \u2200 {a b} \u2192 \u211c a b \u2192 \u2111 (f a) (g b)\n\n-- Reactive sets\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\n-- Equalitional reasoning\n\ninfix  4 _IsRelatedTo_\ninfix  2 _\u220e\ninfixr 2 _\u2261\u27e8_\u27e9_\ninfix  1 begin_\n\ndata _IsRelatedTo_ {\u03b1} {A : Set \u03b1} (a b : A) : Set \u03b1 where\n  relTo : (a\u2261b : a \u2261 b) \u2192 a IsRelatedTo b\n\nbegin_ : \u2200 {\u03b1} {A : Set \u03b1} {a b : A} \u2192 a IsRelatedTo b \u2192 a \u2261 b\nbegin relTo a\u2261b = a\u2261b\n\n_\u2261\u27e8_\u27e9_ : \u2200 {\u03b1} {A : Set \u03b1} a {b c : A} \u2192 a \u2261 b \u2192 b IsRelatedTo c \u2192 a IsRelatedTo c\n_ \u2261\u27e8 a\u2261b \u27e9 relTo b\u2261c = relTo (trans a\u2261b b\u2261c)\n\n_\u220e : \u2200 {\u03b1} {A : Set \u03b1} (a : A) \u2192 a IsRelatedTo a\n_\u220e _ = relTo refl\n\n-- Kinds\n\ndata Kind : Set where\n  time : Kind\n  set : Level \u2192 Kind\n  _\u21d2_ : Kind \u2192 Kind \u2192 Kind\n\nlevel : Kind \u2192 Level\nlevel time    = o\nlevel (set \u03b1) = \u2191 \u03b1\nlevel (K \u21d2 L) = level K \u2294 level L\n\nK\u27e6_\u27e7 : \u2200 K \u2192 Set (level K)\nK\u27e6 time \u27e7  = Time\nK\u27e6 set \u03b1 \u27e7 = Set \u03b1\nK\u27e6 K \u21d2 L \u27e7 = K\u27e6 K \u27e7 \u2192 K\u27e6 L \u27e7\n\n_\u220b_\u2194_ : \u2200 K \u2192 K\u27e6 K \u27e7 \u2192 K\u27e6 K \u27e7 \u2192 Set (level K)\ntime    \u220b t \u2194 u = (t \u2261 u)\nset \u03b1   \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n(K \u21d2 L) \u220b F \u2194 G = \u2200 {A B} \u2192 (K \u220b A \u2194 B) \u2192 (L \u220b F A \u2194 G B)\n\n-- \u2261 can be used as a structural equivalence on relations.\n\nstruct : \u2200 K {A B C D} \u2192 (A \u2261 B) \u2192 (K \u220b B \u2194 D) \u2192 (C \u2261 D) \u2192 (K \u220b A \u2194 C)\nstruct K refl \u211c refl = \u211c\n\nstruct-ext : \u2200 K L {A B} {F G H I : K\u27e6 K \u21d2 L \u27e7} \n  (F\u2248G : \u2200 A \u2192 F A \u2261 G A) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2248I : \u2200 B \u2192 H B \u2261 I B) (\u2111 : K \u220b A \u2194 B) \u2192\n    struct L (F\u2248G A) (\u211c \u2111) (H\u2248I B) \u2261 struct (K \u21d2 L) (ext F\u2248G) \u211c (ext H\u2248I) \u2111\nstruct-ext K L {A} {B} F\u2248G \u211c H\u2248I \u2111 \n with ext F\u2248G | ext H\u2248I | F\u2248G A | H\u2248I B\n... | refl    | refl    | refl  | refl = refl\n\nstruct-apply : \u2200 K L {F G H I A B C D} \u2192 \n  (F\u2261G : F \u2261 G) (\u211c : (K \u21d2 L) \u220b G \u2194 I) (H\u2261I : H \u2261 I) \u2192 \n  (A\u2261B : A \u2261 B) (\u2111 : K \u220b B \u2194 D) (C\u2261D : C \u2261 D) \u2192 \n    struct (K \u21d2 L) F\u2261G \u211c H\u2261I (struct K A\u2261B \u2111 C\u2261D)\n      \u2261 struct L (apply F\u2261G A\u2261B) (\u211c \u2111) (apply H\u2261I C\u2261D)\nstruct-apply K L refl \u211c refl refl \u2111 refl = refl\n\nstruct-cast : \u2200 {\u03b1 A B C D} (\u211c : set \u03b1 \u220b B \u2194 D) (A\u2261B : A \u2261 B) (C\u2261D : C \u2261 D) {a c} \u2192\n  struct (set \u03b1) A\u2261B \u211c C\u2261D a c \u2192 \u211c (cast A\u2261B a) (cast C\u2261D c)\nstruct-cast \u211c refl refl a\u211cc = a\u211cc\n\nstruct-trans : \u2200 K {A B C D E F}\n  (A\u2261B : A \u2261 B) (B\u2261C : B \u2261 C) (\u211c : K \u220b C \u2194 F) (E\u2261F : E \u2261 F) (D\u2261E : D \u2261 E) \u2192\n    struct K A\u2261B (struct K B\u2261C \u211c E\u2261F) D\u2261E \u2261\n      struct K (trans A\u2261B B\u2261C) \u211c (trans D\u2261E E\u2261F)\nstruct-trans K refl refl \u211c refl refl = refl\n\n-- Type contexts\n\ninfixr 4 _\u2237_\n\ndata Kinds : Set where\n  [] : Kinds\n  _\u2237_ : Kind \u2192 Kinds \u2192 Kinds\n\nlevels : Kinds \u2192 Level\nlevels []      = o\nlevels (K \u2237 \u03a3) = level K \u2294 levels \u03a3\n\n\u03a3\u27e6_\u27e7 : \u2200 \u03a3 \u2192 Set (levels \u03a3)\n\u03a3\u27e6 [] \u27e7    = \u22a4\n\u03a3\u27e6 K \u2237 \u03a3 \u27e7 = K\u27e6 K \u27e7 \u00d7 \u03a3\u27e6 \u03a3 \u27e7\n\n_\u220b_\u2194*_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (levels \u03a3)\n[]      \u220b tt       \u2194* tt       = \u22a4\n(K \u2237 \u03a3) \u220b (A , As) \u2194* (B , Bs) = (K \u220b A \u2194 B) \u00d7 (\u03a3 \u220b As \u2194* Bs)\n\n-- Inclusion order on type contexts.\n-- Credited by Randy Pollack to Geuvers and Nederhof, JAR 1991.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/3259\/focus=3267\n\ndata _\u2291_ : Kinds \u2192 Kinds \u2192 Set where\n  id : \u2200 {\u03a3} \u2192 \u03a3 \u2291 \u03a3\n  keep : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 ((K \u2237 \u03a3) \u2291 (K \u2237 \u03a5))\n  skip : \u2200 K {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 (\u03a3 \u2291 (K \u2237 \u03a5))\n\n\u2291\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 \u27e7\n\u2291\u27e6 id \u27e7         As       = As\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7 (A , As) = (A , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7 (A , As) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As\n\n\u2291\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) \u2192 \u2200 {As Bs} \u2192 (\u03a5 \u220b As \u2194* Bs) \u2192 (\u03a3 \u220b \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As \u2194* \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 Bs)\n\u2291\u27e6 id \u27e7\u00b2         \u211cs       = \u211cs\n\u2291\u27e6 keep K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = (\u211c , \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n\u2291\u27e6 skip K \u03a3\u2291\u03a5 \u27e7\u00b2 (\u211c , \u211cs) = \u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs\n\n-- Concatenation of type contexts\n\n_++_ : Kinds \u2192 Kinds \u2192 Kinds\n[]      ++ \u03a5 = \u03a5\n(K \u2237 \u03a3) ++ \u03a5 = K \u2237 (\u03a3 ++ \u03a5)\n\n_\u220b_++_\u220b_ : \u2200 \u03a3 \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 \u2200 \u03a5 \u2192 \u03a3\u27e6 \u03a5 \u27e7 \u2192 \u03a3\u27e6 \u03a3 ++ \u03a5 \u27e7\n[]      \u220b tt       ++ \u03a5 \u220b Bs = Bs\n(K \u2237 \u03a3) \u220b (A , As) ++ \u03a5 \u220b Bs = (A , (\u03a3 \u220b As ++ \u03a5 \u220b Bs))\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03a3 {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 \u2200 \u03a5 {Cs Ds} \u2192 (\u03a5 \u220b Cs \u2194* Ds) \u2192 \n  ((\u03a3 ++ \u03a5) \u220b (\u03a3 \u220b As ++ \u03a5 \u220b Cs) \u2194* (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds))\n[]      \u220b tt       ++\u00b2 \u03a5 \u220b \u2111s = \u2111s\n(K \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s = (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n\n-- Type variables\n\ndata TVar (K : Kind) : Kinds \u2192 Set where\n  zero : \u2200 {\u03a3} \u2192 TVar K (K \u2237 \u03a3)\n  suc : \u2200 {L \u03a3} \u2192 TVar K \u03a3 \u2192 TVar K (L \u2237 \u03a3)\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\n\u03c4\u27e6 zero \u27e7  (A , As)  = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (A , As)  = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (\u03c4 : TVar K \u03a3) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u2194 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211c , \u211cs)  = \u211c\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211c , \u211cs)  = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type constants\n\ndata TConst : Kind \u2192 Set where\n  prod fun : \u2200 {\u03b1 \u03b2} \u2192 TConst (set \u03b1 \u21d2 (set \u03b2 \u21d2 set (\u03b1 \u2294 \u03b2)))\n  leq lt : TConst (time \u21d2 (time \u21d2 set o))\n  univ : \u2200 K {\u03b1} \u2192 TConst ((K \u21d2 set \u03b1) \u21d2 set (level K \u2294 \u03b1))\n\nC\u27e6_\u27e7 : \u2200 {K} \u2192 (TConst K) \u2192 K\u27e6 K \u27e7\nC\u27e6 prod \u27e7   = \u03bb A B \u2192 (A \u00d7 B)\nC\u27e6 fun \u27e7    = \u03bb A B \u2192 (A \u2192 B)\nC\u27e6 leq \u27e7    = \u03bb t u \u2192 True (t \u2264 u)\nC\u27e6 lt \u27e7     = \u03bb t u \u2192 True (t < u)\nC\u27e6 univ K \u27e7 = \u03bb F \u2192 \u2200 A \u2192 F A\n\nC\u27e6_\u27e7\u00b2 : \u2200 {K} (C : TConst K) \u2192 (K \u220b C\u27e6 C \u27e7 \u2194 C\u27e6 C \u27e7)\nC\u27e6 prod \u27e7\u00b2   = \u03bb \u211c \u2111 \u2192 (\u211c \u00d7\u00b2 \u2111)\nC\u27e6 fun \u27e7\u00b2    = \u03bb \u211c \u2111 \u2192 (\u211c \u2192\u00b2 \u2111)\nC\u27e6 leq \u27e7\u00b2    = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 lt \u27e7\u00b2     = \u03bb _ _ _ _ \u2192 \u22a4\nC\u27e6 univ K \u27e7\u00b2 = \u03bb \u211c f g \u2192 \u2200 {a b} \u2111 \u2192 \u211c \u2111 (f a) (g b)\n\n-- Types\n\ndata Typ (\u03a3 : Kinds) : Kind \u2192 Set where\n  const : \u2200 {K} \u2192 TConst K \u2192 Typ \u03a3 K\n  abs : \u2200 K {L} \u2192 Typ (K \u2237 \u03a3) L \u2192 Typ \u03a3 (K \u21d2 L)\n  app : \u2200 {K L} \u2192 Typ \u03a3 (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\n  var : \u2200 {K} \u2192 TVar K \u03a3 \u2192 Typ \u03a3 K\n\ntlevel : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Level\ntlevel {\u03a3} {\u03b1} T = \u03b1\n\nT\u27e6_\u27e7 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 K\u27e6 K \u27e7\nT\u27e6 const C \u27e7 As  = C\u27e6 C \u27e7\nT\u27e6 abs K T \u27e7 As  = \u03bb A \u2192 T\u27e6 T \u27e7 (A , As)\nT\u27e6 app T U \u27e7 As  = T\u27e6 T \u27e7 As (T\u27e6 U \u27e7 As)\nT\u27e6 var \u03c4 \u27e7 As    = \u03c4\u27e6 \u03c4 \u27e7 As\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K} (T : Typ \u03a3 K) {As Bs} \u2192 (\u03a3 \u220b As \u2194* Bs) \u2192 (K \u220b T\u27e6 T \u27e7 As \u2194 T\u27e6 T \u27e7 Bs)\nT\u27e6 const C \u27e7\u00b2 \u211cs  = C\u27e6 C \u27e7\u00b2\nT\u27e6 abs K T \u27e7\u00b2 \u211cs  = \u03bb \u211c \u2192 T\u27e6 T \u27e7\u00b2 (\u211c , \u211cs)\nT\u27e6 app T U \u27e7\u00b2 \u211cs  = T\u27e6 T \u27e7\u00b2 \u211cs (T\u27e6 U \u27e7\u00b2 \u211cs)\nT\u27e6 var \u03c4 \u27e7\u00b2 \u211cs    = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs\n\n-- Type shorthands\n\napp\u2082 :  \u2200 {\u03a3 K L M} \u2192 Typ \u03a3 (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\napp\u2082 T U V = app (app T U) V\n\ncapp :  \u2200 {\u03a3 K L} \u2192 TConst (K \u21d2 L) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L\ncapp C = app (const C)\n\ncapp\u2082 :  \u2200 {\u03a3 K L M} \u2192 TConst (K \u21d2 (L \u21d2 M)) \u2192 Typ \u03a3 K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 M\ncapp\u2082 C = app\u2082 (const C)\n\n_\u2297_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u2297_ = capp\u2082 prod\n\n_\u22b8_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (set \u03b1) \u2192 Typ \u03a3 (set \u03b2) \u2192 Typ \u03a3 (set (\u03b1 \u2294 \u03b2))\n_\u22b8_ = capp\u2082 fun\n\n_\u227c_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227c_ = capp\u2082 leq\n\n_\u227a_ : \u2200 {\u03a3} \u2192 Typ \u03a3 time \u2192 Typ \u03a3 time \u2192 Typ \u03a3 (set o)\n_\u227a_ = capp\u2082 lt\n\n\u03a0 : \u2200 {\u03a3 \u03b1} K \u2192 Typ (K \u2237 \u03a3) (set \u03b1) \u2192 Typ \u03a3 (set (level K \u2294 \u03b1))\n\u03a0 K T = capp (univ K) (abs K T)\n\ntvar\u2080 : \u2200 {\u03a3 K} \u2192 Typ (K \u2237 \u03a3) K\ntvar\u2080 = var zero\n\ntvar\u2081 : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 K \u2237 \u03a3) K\ntvar\u2081 = var (suc zero)\n\ntvar\u2082 : \u2200 {\u03a3 K L M} \u2192 Typ (M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2082 = var (suc (suc zero))\n\ntvar\u2083 : \u2200 {\u03a3 K L M N} \u2192 Typ (N \u2237 M \u2237 L \u2237 K \u2237 \u03a3) K\ntvar\u2083 = var (suc (suc (suc zero)))\n\nrset : Level \u2192 Kind\nrset \u03b1 = time \u21d2 set \u03b1\n\nrset\u2080 : Kind\nrset\u2080 = rset o\n\nprod\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nprod\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u2297 app tvar\u2081 tvar\u2080)))\n\n_\u2297\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u2297\u02b3_ = app\u2082 prod\u02b3\n\nfun\u02b3 : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nfun\u02b3 {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (app tvar\u2082 tvar\u2080 \u22b8 app tvar\u2081 tvar\u2080)))\n\n_\u22b8\u02b3_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b8\u02b3_ = app\u2082 fun\u02b3\n\nalways : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (set \u03b1 \u21d2 rset \u03b1)\nalways {\u03a3} {\u03b1} = abs (set \u03b1) (abs time tvar\u2081)\n\ntaut : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 set \u03b1)\ntaut {\u03a3} {\u03b1} = abs (rset \u03b1) (\u03a0 time (app tvar\u2081 tvar\u2080))\n\ninterval : \u2200 {\u03a3 \u03b1} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (time \u21d2 (time \u21d2 set \u03b1)))\ninterval {\u03a3} {\u03b1} = abs (rset \u03b1) (abs time (abs time (\u03a0 time \n  ((tvar\u2082 \u227c tvar\u2080) \u22b8 ((tvar\u2080 \u227c tvar\u2081) \u22b8 app tvar\u2083 tvar\u2080)))))\n\nconstreq : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1 \u21d2 (rset \u03b2 \u21d2 rset (\u03b1 \u2294 \u03b2)))\nconstreq {\u03a3} {\u03b1} {\u03b2} = abs (rset \u03b1) (abs (rset \u03b2) (abs time (\u03a0 time \n  ((tvar\u2081 \u227c tvar\u2080) \u22b8 (app (app (app interval tvar\u2083) tvar\u2081) tvar\u2080 \u22b8 app tvar\u2082 tvar\u2080)))))\n\n_\u22b5_ : \u2200 {\u03a3 \u03b1 \u03b2} \u2192 Typ \u03a3 (rset \u03b1) \u2192 Typ \u03a3 (rset \u03b2) \u2192 Typ \u03a3 (rset (\u03b1 \u2294 \u03b2))\n_\u22b5_ = app\u2082 constreq\n\n-- Contexts\n\ndata Typs (\u03a3 : Kinds) : Set where\n  [] : Typs \u03a3\n  _\u2237_ : \u2200 {\u03b1} \u2192 (Typ \u03a3 (set \u03b1)) \u2192 Typs \u03a3 \u2192 Typs \u03a3\n\ntlevels : \u2200 {\u03a3} \u2192 Typs \u03a3 \u2192 Level\ntlevels []       = o\ntlevels (T \u2237 \u0393) = tlevel T \u2294 tlevels \u0393\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) \u2192 \u03a3\u27e6 \u03a3 \u27e7 \u2192 Set (tlevels \u0393)\n\u0393\u27e6 [] \u27e7    As = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7 As = T\u27e6 T \u27e7 As \u00d7 \u0393\u27e6 \u0393 \u27e7 As\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} (\u0393 : Typs \u03a3) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u0393\u27e6 \u0393 \u27e7 As \u2192 \u0393\u27e6 \u0393 \u27e7 Bs \u2192 Set (tlevels \u0393))\n\u0393\u27e6 [] \u27e7\u00b2    \u211cs tt       tt       = \u22a4\n\u0393\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u211cs (a , as) (b , bs) = T\u27e6 T \u27e7\u00b2 \u211cs a b \u00d7 \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 TVar K \u03a3 \u2192 TVar K \u03a5\n\u03c4weaken id           x       = x\n\u03c4weaken (keep K \u03a3\u2291\u03a5) zero    = zero\n\u03c4weaken (keep K \u03a3\u2291\u03a5) (suc x) = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\u03c4weaken (skip K \u03a3\u2291\u03a5) x       = suc (\u03c4weaken \u03a3\u2291\u03a5 x)\n\n\u03c4weaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 \u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     id           As       = refl\n\u03c4weaken\u27e6 zero \u27e7  (keep K \u03a3\u2291\u03a5) (A , As) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7 (keep K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\u03c4weaken\u27e6 \u03c4 \u27e7     (skip K \u03a3\u2291\u03a5) (A , As) = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n\n\u03c4weaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (\u03c4 : TVar K \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 \n    struct K (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As) (\u03c4\u27e6 \u03c4weaken \u03a3\u2291\u03a5 \u03c4 \u27e7\u00b2 \u211cs) (\u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 Bs)\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     id           \u211cs       = refl\n\u03c4weaken\u27e6 zero \u27e7\u00b2  (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = refl\n\u03c4weaken\u27e6 suc \u03c4 \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\u03c4weaken\u27e6 \u03c4 \u27e7\u00b2     (skip K \u03a3\u2291\u03a5) (\u211c , \u211cs) = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening of types\n\nweaken : \u2200 {\u03a3 \u03a5 K} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typ \u03a3 K \u2192 Typ \u03a5 K\nweaken \u03a3\u2291\u03a5 (const C)  = const C\nweaken \u03a3\u2291\u03a5 (abs K T)  = abs K (weaken (keep K \u03a3\u2291\u03a5) T)\nweaken \u03a3\u2291\u03a5 (app T U)  = app (weaken \u03a3\u2291\u03a5 T) (weaken \u03a3\u2291\u03a5 U)\nweaken \u03a3\u2291\u03a5 (var \u03c4)    = var (\u03c4weaken \u03a3\u2291\u03a5 \u03c4)\n\nweaken\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  T\u27e6 T \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2261 T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7 As\nweaken\u27e6 const C \u27e7 \u03a3\u2291\u03a5 As = refl\nweaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As = ext (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As))\nweaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As = apply (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) \nweaken\u27e6 var \u03c4 \u27e7   \u03a3\u2291\u03a5 As = \u03c4weaken\u27e6 \u03c4 \u27e7 \u03a3\u2291\u03a5 As\n  \nweaken\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5 K} (T : Typ \u03a3 K) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) \u2192 \n  T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u2261 struct K (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs)\nweaken\u27e6 const C \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = refl\nweaken\u27e6 abs K {L} T \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs =\n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs K T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) \u211c\n  \u2261\u27e8 weaken\u27e6 T \u27e7\u00b2 (keep K \u03a3\u2291\u03a5) (\u211c , \u211cs) \u27e9\n    struct L \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n      (T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n      (weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs))\n  \u2261\u27e8 struct-ext K L \n       (\u03bb A \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (A , As)) \n       (\u03bb \u211c \u2192 T\u27e6 weaken (keep K \u03a3\u2291\u03a5) T \u27e7\u00b2 (\u211c , \u211cs)) \n       (\u03bb B \u2192 weaken\u27e6 T \u27e7 (keep K \u03a3\u2291\u03a5) (B , Bs)) \u211c \u27e9\n    struct (K \u21d2 L) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (abs K T) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 abs K T \u27e7 \u03a3\u2291\u03a5 Bs) \u211c\n  \u220e)))\nweaken\u27e6 app {K} {L} T U \u27e7\u00b2 \u03a3\u2291\u03a5 {As} {Bs} \u211cs = \n  begin\n    T\u27e6 app T U \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n  \u2261\u27e8 cong (T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)) (weaken\u27e6 U \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    T\u27e6 T \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs)\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs)))\n       (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) \u27e9\n    (struct (K \u21d2 L) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs))\n      (struct K (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs))\n  \u2261\u27e8 struct-apply K L \n       (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 Bs) \n       (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 As) (T\u27e6 weaken \u03a3\u2291\u03a5 U \u27e7\u00b2 \u211cs) (weaken\u27e6 U \u27e7 \u03a3\u2291\u03a5 Bs) \u27e9\n    struct L\n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 As)\n      (T\u27e6 weaken \u03a3\u2291\u03a5 (app T U) \u27e7\u00b2 \u211cs) \n      (weaken\u27e6 app T U \u27e7 \u03a3\u2291\u03a5 Bs)\n  \u220e\nweaken\u27e6 var \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs = \u03c4weaken\u27e6 \u03c4 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs\n\n-- Weakening on type contexts\n\nweakens : \u2200 {\u03a3 \u03a5} \u2192 (\u03a3 \u2291 \u03a5) \u2192 Typs \u03a3 \u2192 Typs \u03a5\nweakens \u03a3\u2291\u03a5 []      = []\nweakens \u03a3\u2291\u03a5 (T \u2237 \u0393) = weaken \u03a3\u2291\u03a5 T \u2237 weakens \u03a3\u2291\u03a5 \u0393\n\nweakens\u27e6_\u27e7 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) (As : \u03a3\u27e6 \u03a5 \u27e7) \u2192 \n  \u0393\u27e6 \u0393 \u27e7 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7 As) \u2192 \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7 As\nweakens\u27e6 [] \u27e7    \u03a3\u2291\u03a5 As tt       = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7 \u03a3\u2291\u03a5 As (B , Bs) = (cast (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 As) B , weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As Bs)\n\nweakens\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 \u03a5} (\u0393 : Typs \u03a3) (\u03a3\u2291\u03a5 : \u03a3 \u2291 \u03a5) {As Bs} (\u211cs : \u03a5 \u220b As \u2194* Bs) {as bs} \u2192 \n  \u0393\u27e6 \u0393 \u27e7\u00b2 (\u2291\u27e6 \u03a3\u2291\u03a5 \u27e7\u00b2 \u211cs) as bs \u2192 \n  \u0393\u27e6 weakens \u03a3\u2291\u03a5 \u0393 \u27e7\u00b2 \u211cs (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 As as) (weakens\u27e6 \u0393 \u27e7 \u03a3\u2291\u03a5 Bs bs)\nweakens\u27e6 [] \u27e7\u00b2    \u03a3\u2291\u03a5 \u211cs tt\n  = tt\nweakens\u27e6 T \u2237 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs (a\u211cb , as\u211cbs) \n  = ( struct-cast (T\u27e6 weaken \u03a3\u2291\u03a5 T \u27e7\u00b2 \u211cs) \n        (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (weaken\u27e6 T \u27e7 \u03a3\u2291\u03a5 _) (cast\u00b2 (weaken\u27e6 T \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs) a\u211cb)\n    , weakens\u27e6 \u0393 \u27e7\u00b2 \u03a3\u2291\u03a5 \u211cs as\u211cbs)\n\n-- Susbtitution on type variables under a context\n\n\u03c4substn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 TVar K (\u03a3 ++ (L \u2237 \u03a5)) \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\n\u03c4substn+ []      zero    U = U\n\u03c4substn+ []      (suc \u03c4) U = var \u03c4\n\u03c4substn+ (K \u2237 \u03a3) zero    U = var zero\n\u03c4substn+ (K \u2237 \u03a3) (suc \u03c4) U = weaken (skip K id) (\u03c4substn+ \u03a3 \u03c4 U)\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 tt       Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7 (A , As) Bs = refl\n\u03c4substn+ (K \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Bs = trans \n  (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs) \n  (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip K id) (A , (\u03a3 \u220b As ++ _ \u220b Bs)))\n\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (\u03c4 : TVar K (\u03a3 ++ (L \u2237 \u03a5))) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n\u03c4substn+ []      \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ []      \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7\u00b2 tt       \u2111s = refl\n\u03c4substn+ (J \u2237 \u03a3) \u27e6 zero  \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s = refl\n\u03c4substn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 (J \u2237 \u03a3) {\u03a5} {L} {K} (suc \u03c4) U {A , As} {B , Bs} {Cs} {Ds} (\u211c , \u211cs) \u2111s = \n  begin\n    \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 cong (\u03bb X \u2192 struct K (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) X (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7\u00b2 (skip J id) (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \u27e9\n    struct K \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n      (struct K\n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs))) \n        (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))) \n        (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))) \n      (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u2261\u27e8 struct-trans K \n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Cs) \n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (A , (\u03a3 \u220b As ++ \u03a5 \u220b Cs)))\n       (T\u27e6 weaken (skip J id) (\u03c4substn+ \u03a3 \u03c4 U) \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)))\n       (weaken\u27e6 \u03c4substn+ \u03a3 \u03c4 U \u27e7 (skip J id) (B , (\u03a3 \u220b Bs ++ \u03a5 \u220b Ds)))\n       (\u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 \u03c4substn+ (J \u2237 \u03a3) (suc \u03c4) U \u27e7\u00b2 (\u211c , (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) )\n      (\u03c4substn+ (J \u2237 \u03a3) \u27e6 suc \u03c4 \u27e7\u27e6 U \u27e7 (B , Bs) Ds)    \n  \u220e\n\n-- Substitution on types under a context\n\nsubstn+ : \u2200 \u03a3 {\u03a5 K L} \u2192 Typ (\u03a3 ++ (L \u2237 \u03a5)) K \u2192 Typ \u03a5 L \u2192 Typ (\u03a3 ++ \u03a5) K\nsubstn+ \u03a3 (const C) U = const C\nsubstn+ \u03a3 (abs K T) U = abs K (substn+ (K \u2237 \u03a3) T U)\nsubstn+ \u03a3 (app S T) U = app (substn+ \u03a3 S U) (substn+ \u03a3 T U)\nsubstn+ \u03a3 (var \u03c4)   U = \u03c4substn+ \u03a3 \u03c4 U\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7 : \u2200 \u03a3 {\u03a5 K L} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) \n  (As : \u03a3\u27e6 \u03a3 \u27e7) (Bs : \u03a3\u27e6 \u03a5 \u27e7) \u2192\n    T\u27e6 T \u27e7 (\u03a3 \u220b As ++ (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7 Bs , Bs)) \u2261 \n      T\u27e6 substn+ \u03a3 T U \u27e7 (\u03a3 \u220b As ++ \u03a5 \u220b Bs)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7 As Bs = refl\nsubstn+ \u03a3 \u27e6 abs K T \u27e7\u27e6 U \u27e7 As Bs = ext (\u03bb A \u2192 substn+ K \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Bs)\nsubstn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Bs = apply (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Bs) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Bs)\nsubstn+ \u03a3 \u27e6 var \u03c4   \u27e7\u27e6 U \u27e7 As Bs = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7 As Bs\n\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 \u03a3 {\u03a5 L K} (T : Typ (\u03a3 ++ (L \u2237 \u03a5)) K) (U : Typ \u03a5 L) {As Bs Cs Ds} \n  (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 (\u2111s : \u03a5 \u220b Cs \u2194* Ds) \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u2261 \n      struct K \n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs) \n        (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s) )\n        (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds)\nsubstn+ \u03a3 \u27e6 const C \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = refl\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (abs J {K} T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  iext (\u03bb A \u2192 iext (\u03bb B \u2192 ext (\u03bb \u211c \u2192 begin\n    T\u27e6 abs J T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 (L \u2237 \u03a5) \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s)) \u211c\n  \u2261\u27e8 substn+ (J \u2237 \u03a3) \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 (\u211c , \u211cs) \u2111s \u27e9\n    struct K \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n      (T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b (\u211c , \u211cs) ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds)\n  \u2261\u27e8 struct-ext J K \n       (\u03bb A \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (A , As) Cs) \n       (\u03bb \u211c \u2192 T\u27e6 substn+ (J \u2237 \u03a3) T U \u27e7\u00b2 ((J \u2237 \u03a3) \u220b \u211c , \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (\u03bb B \u2192 substn+ J \u2237 \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 (B , Bs) Ds) \u211c \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 (abs J T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n      (substn+ \u03a3 \u27e6 abs J T \u27e7\u27e6 U \u27e7 Bs Ds) \u211c\n  \u220e)))\nsubstn+_\u27e6_\u27e7\u27e6_\u27e7\u00b2 \u03a3 {\u03a5} {L} (app {J} {K} S T) U {As} {Bs} {Cs} {Ds} \u211cs \u2111s = \n  begin\n    T\u27e6 app S T \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n  \u2261\u27e8 cong (T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))) (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    T\u27e6 S \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 L \u2237 \u03a5 \u220b (T\u27e6 U \u27e7\u00b2 \u2111s , \u2111s))\n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 cong (\u03bb X \u2192 X (struct J \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s) \u27e9\n    struct (J \u21d2 K) \n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n      (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n      (struct J \n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n         (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n         (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds))\n  \u2261\u27e8 struct-apply J K \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 As Cs) \n       (T\u27e6 substn+ \u03a3 S U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s)) \n       (substn+ \u03a3 \u27e6 S \u27e7\u27e6 U \u27e7 Bs Ds) \n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 As Cs)\n       (T\u27e6 substn+ \u03a3 T U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n       (substn+ \u03a3 \u27e6 T \u27e7\u27e6 U \u27e7 Bs Ds) \u27e9\n    struct K \n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 As Cs)\n      (T\u27e6 substn+ \u03a3 (app S T) U \u27e7\u00b2 (\u03a3 \u220b \u211cs ++\u00b2 \u03a5 \u220b \u2111s))\n      (substn+ \u03a3 \u27e6 app S T \u27e7\u27e6 U \u27e7 Bs Ds)\n  \u220e\nsubstn+ \u03a3 \u27e6 var \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s = \u03c4substn+ \u03a3 \u27e6 \u03c4 \u27e7\u27e6 U \u27e7\u00b2 \u211cs \u2111s\n\n-- Substitution on types\n\nsubstn : \u2200 {\u03a3 K L} \u2192 Typ (L \u2237 \u03a3) K \u2192 Typ \u03a3 L \u2192 Typ \u03a3 K\nsubstn = substn+ []\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) (As : \u03a3\u27e6 \u03a3 \u27e7)\u2192\n  T\u27e6 T \u27e7 (T\u27e6 U \u27e7 As , As) \u2261 T\u27e6 substn T U \u27e7 As\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7 tt\n\nsubstn\u27e6_\u27e7\u27e6_\u27e7\u00b2 : \u2200 {\u03a3 K L} (T : Typ (L \u2237 \u03a3) K) (U : Typ \u03a3 L) {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192\n  T\u27e6 T \u27e7\u00b2 (T\u27e6 U \u27e7\u00b2 \u211cs , \u211cs) \u2261 \n    struct K (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 Bs)\nsubstn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 = substn+ [] \u27e6 T \u27e7\u27e6 U \u27e7\u00b2 tt\n\n-- Variables\n\ndata Var {\u03a3 : Kinds} {\u03b1} (T : Typ \u03a3 (set \u03b1)) : Typs \u03a3 \u2192 Set where\n  zero : \u2200 {\u0393} \u2192 Var T (T \u2237 \u0393)\n  suc : \u2200 {\u03b2 \u0393} {U : Typ \u03a3 (set \u03b2)} \u2192 Var T \u0393 \u2192 Var T (U \u2237 \u0393)\n\nx\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Var T \u0393 \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nx\u27e6 zero \u27e7  As (a , as) = a\nx\u27e6 suc x \u27e7 As (a , as) = x\u27e6 x \u27e7 As as\n\nx\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (x : Var T \u0393) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (x\u27e6 x \u27e7 As as) (x\u27e6 x \u27e7 Bs bs))\nx\u27e6 zero \u27e7\u00b2  \u211cs (a\u211cb , as\u211cbs) = a\u211cb\nx\u27e6 suc x \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs) = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Constants\n\ndata Const {\u03a3 : Kinds} : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  pair : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) (tvar\u2081 \u22b8 (tvar\u2080 \u22b8 (tvar\u2081 \u2297 tvar\u2080)))))\n  fst : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2081)))\n  snd : \u2200 {\u03b1 \u03b2} \u2192 Const (\u03a0 (set \u03b1) (\u03a0 (set \u03b2) ((tvar\u2081 \u2297 tvar\u2080) \u22b8 tvar\u2080)))\n  leq-refl : Const (\u03a0 time (tvar\u2080 \u227c tvar\u2080))\n  leq-trans : Const (\u03a0 time (\u03a0 time (\u03a0 time ((tvar\u2082 \u227c tvar\u2081) \u22b8 ((tvar\u2081 \u227c tvar\u2080) \u22b8 (tvar\u2082 \u227c tvar\u2080))))))\n\nc\u27e6_\u27e7 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Const T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (T\u27e6 T \u27e7 As)\nc\u27e6 pair \u27e7      As = \u03bb A B a b \u2192 (a , b)\nc\u27e6 fst \u27e7       As = \u03bb A B \u2192 proj\u2081\nc\u27e6 snd \u27e7       As = \u03bb A B \u2192 proj\u2082\nc\u27e6 leq-refl \u27e7  As = \u2264-refl\nc\u27e6 leq-trans \u27e7 As = \u2264-trans\n\nc\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (c : Const T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) \u2192 \n    (T\u27e6 T \u27e7\u00b2 \u211cs (c\u27e6 c \u27e7 As) (c\u27e6 c \u27e7 Bs))\nc\u27e6 pair \u27e7\u00b2      \u211cs = \u03bb \u211c \u2111 a\u211cb c\u2111d \u2192 (a\u211cb , c\u2111d)\nc\u27e6 fst \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2081\nc\u27e6 snd \u27e7\u00b2       \u211cs = \u03bb \u211c \u2111 \u2192 proj\u2082\nc\u27e6 leq-refl \u27e7\u00b2  \u211cs = _\nc\u27e6 leq-trans \u27e7\u00b2 \u211cs = _\n\n-- Expressions\n\ndata Exp {\u03a3 : Kinds} (\u0393 : Typs \u03a3) : \u2200 {\u03b1} \u2192 Typ \u03a3 (set \u03b1) \u2192 Set where\n  const : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Const T \u2192 Exp \u0393 T\n  abs : \u2200 {\u03b1 \u03b2} (T : Typ \u03a3 (set \u03b1)) {U : Typ \u03a3 (set \u03b2)} (M : Exp (T \u2237 \u0393) U) \u2192 Exp \u0393 (T \u22b8 U)\n  app : \u2200 {\u03b1 \u03b2} {T : Typ \u03a3 (set \u03b1)} {U : Typ \u03a3 (set \u03b2)} (M : Exp \u0393 (T \u22b8 U)) (N : Exp \u0393 T) \u2192 Exp \u0393 U\n  var : \u2200 {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 Var T \u0393 \u2192 Exp \u0393 T\n  tabs : \u2200 K {\u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} (M : Exp (weakens (skip K id) \u0393) T) \u2192 Exp \u0393 (\u03a0 K T)\n  tapp : \u2200 {K \u03b1} {T : Typ (K \u2237 \u03a3) (set \u03b1)} \u2192 Exp \u0393 (\u03a0 K T) \u2192 \u2200 U \u2192 Exp \u0393 (substn T U)\n\nctxt : \u2200 {\u03a3 \u0393 \u03b1 T} \u2192 Exp {\u03a3} \u0393 {\u03b1} T \u2192 Typs \u03a3\nctxt {\u03a3} {\u0393} M = \u0393\n\nM\u27e6_\u27e7 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} \u2192 \n  Exp \u0393 T \u2192 (As : \u03a3\u27e6 \u03a3 \u27e7) \u2192 (as : \u0393\u27e6 \u0393 \u27e7 As) \u2192 (T\u27e6 T \u27e7 As)\nM\u27e6 const c \u27e7  As as = c\u27e6 c \u27e7 As\nM\u27e6 abs T M \u27e7  As as = \u03bb a \u2192 M\u27e6 M \u27e7 As (a , as)\nM\u27e6 app M N \u27e7  As as = M\u27e6 M \u27e7 As as (M\u27e6 N \u27e7 As as)\nM\u27e6 var x \u27e7    As as = x\u27e6 x \u27e7 As as\nM\u27e6 tabs K M \u27e7 As as = \u03bb A \u2192 \n  M\u27e6 M \u27e7 (A , As) (weakens\u27e6 ctxt (tabs K M) \u27e7 (skip K id) (A , As) as)\nM\u27e6 tapp {T = T} M U \u27e7 As as = \n  cast (substn\u27e6 T \u27e7\u27e6 U \u27e7 As) (M\u27e6 M \u27e7 As as (T\u27e6 U \u27e7 As))\n\nM\u27e6_\u27e7\u00b2 : \u2200 {\u03a3} {\u0393 : Typs \u03a3} {\u03b1} {T : Typ \u03a3 (set \u03b1)} (M : Exp \u0393 T) \u2192 \n  \u2200 {As Bs} (\u211cs : \u03a3 \u220b As \u2194* Bs) {as bs} \u2192 \n    (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs as bs) \u2192 (T\u27e6 T \u27e7\u00b2 \u211cs (M\u27e6 M \u27e7 As as) (M\u27e6 M \u27e7 Bs bs))\nM\u27e6 const c \u27e7\u00b2  \u211cs as\u211cbs = c\u27e6 c \u27e7\u00b2 \u211cs\nM\u27e6 abs T M \u27e7\u00b2  \u211cs as\u211cbs = \u03bb a\u211cb \u2192 M\u27e6 M \u27e7\u00b2 \u211cs (a\u211cb , as\u211cbs)\nM\u27e6 app M N \u27e7\u00b2  \u211cs as\u211cbs = M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (M\u27e6 N \u27e7\u00b2 \u211cs as\u211cbs)\nM\u27e6 var x  \u27e7\u00b2   \u211cs as\u211cbs = x\u27e6 x \u27e7\u00b2 \u211cs as\u211cbs\nM\u27e6 tabs K M \u27e7\u00b2 \u211cs as\u211cbs = \u03bb \u211c \u2192 \n  M\u27e6 M \u27e7\u00b2 (\u211c , \u211cs) (weakens\u27e6 ctxt (tabs K M) \u27e7\u00b2 (skip K id) (\u211c , \u211cs) as\u211cbs)\nM\u27e6 tapp {T = T} M U \u27e7\u00b2 \u211cs as\u211cbs = \n  struct-cast (T\u27e6 substn T U \u27e7\u00b2 \u211cs) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _) (substn\u27e6 T \u27e7\u27e6 U \u27e7 _)\n    (cast\u00b2 (substn\u27e6 T \u27e7\u27e6 U \u27e7\u00b2 \u211cs) (M\u27e6 M \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 U \u27e7\u00b2 \u211cs)))\n\n-- Types with a chosen free world variable\n\n_\u2237\u02b3_ : Kinds \u2192 Kind \u2192 Kinds\n[]      \u2237\u02b3 K = K \u2237 []\n(T \u2237 \u03a3) \u2237\u02b3 K = T \u2237 (\u03a3 \u2237\u02b3 K)\n\nTVar+ : Kind \u2192 Kinds \u2192 Set\nTVar+ K \u03a3 = TVar K (\u03a3 \u2237\u02b3 rset\u2080)\n\nTyp+ : Kinds \u2192 Kind \u2192 Set\nTyp+ \u03a3 = Typ (\u03a3 \u2237\u02b3 rset\u2080)\n\nwvar : \u2200 \u03a3 \u2192 TVar+ rset\u2080 \u03a3\nwvar []      = zero\nwvar (K \u2237 \u03a3) = suc (wvar \u03a3)\n\nworld : \u2200 {\u03a3} \u2192 Typ+ \u03a3 rset\u2080\nworld {\u03a3} = var (wvar \u03a3)\n\nWorld : Time \u2192 Set\nWorld t = \u22a4\n\n-- Surface types\n\ndata STyp : Kind \u2192 Set where\n  \u27e8_\u27e9 : \u2200 {\u03b1} \u2192 STyp (set \u03b1) \u2192 STyp (rset \u03b1)\n  [_] : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (set \u03b1)\n  _\u22a0_ _\u21a6_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (set \u03b1) \u2192 STyp (set \u03b2) \u2192 STyp (set (\u03b1 \u2294 \u03b2))\n  _\u2227_ _\u21d2_ : \u2200 {\u03b1 \u03b2} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b2) \u2192 STyp (rset (\u03b1 \u2294 \u03b2))\n  \u25a1 : \u2200 {\u03b1} \u2192 STyp (rset \u03b1) \u2192 STyp (rset \u03b1)\n\n\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 Typ+ [] K\n\u27ea \u27e8 T \u27e9 \u27eb = app always \u27ea T \u27eb\n\u27ea [ T ] \u27eb = app taut \u27ea T \u27eb\n\u27ea T \u22a0 U \u27eb = \u27ea T \u27eb \u2297 \u27ea U \u27eb\n\u27ea T \u21a6 U \u27eb = \u27ea T \u27eb \u22b8 \u27ea U \u27eb\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2297\u02b3 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u22b8\u02b3 \u27ea U \u27eb\n\u27ea \u25a1 T \u27eb   = tvar\u2080 \u22b5 \u27ea T \u27eb\n\nT\u27ea_\u27eb : \u2200 {K} \u2192 STyp K \u2192 K\u27e6 K \u27e7\nT\u27ea T \u27eb = T\u27e6 \u27ea T \u27eb \u27e7 (World , tt)\n\n-- Signals of T are iso to \u25a1 T\n\nSignal : \u2200 {\u03b1} \u2192 RSet \u03b1 \u2192 RSet \u03b1\nSignal A s = \u2200 t \u2192 True (s \u2264 t) \u2192 A t\n\nsignal : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  T\u27ea \u25a1 T \u27eb s \u2192 Signal T\u27ea T \u27eb s\nsignal T s \u03c3 t s\u2264t = \u03c3 t s\u2264t _\n\nsignal\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 \n  Signal T\u27ea T \u27eb s \u2192 T\u27ea \u25a1 T \u27eb s\nsignal\u207b\u00b9 T s \u03c3 t s\u2264t _ = \u03c3 t s\u2264t\n\nsignal-iso : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192 \n  (signal T s (signal\u207b\u00b9 T s \u03c3) \u2261 \u03c3)\nsignal-iso T s \u03c3 = refl\n\nsignal-iso\u207b\u00b9 : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u03c3 \u2192\n  (signal\u207b\u00b9 T s (signal T s \u03c3) \u2261 \u03c3)\nsignal-iso\u207b\u00b9 T s \u03c3 = refl\n\n-- Signal functions from T to U are iso to \u25a1 T \u21d2 \u25a1 U\n\nSF : \u2200 {\u03b1 \u03b2} \u2192 RSet \u03b1 \u2192 RSet \u03b2 \u2192 RSet (\u03b1 \u2294 \u03b2)\nSF A B s = Signal A s \u2192 Signal B s\n\nsf : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n  T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s \u2192 SF T\u27ea T \u27eb T\u27ea U \u27eb s\nsf T U s f \u03c3 = signal U s (f (signal\u207b\u00b9 T s \u03c3))\n\nsf\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192\n   SF T\u27ea T \u27eb T\u27ea U \u27eb s \u2192 T\u27ea \u25a1 T \u21d2 \u25a1 U \u27eb s\nsf\u207b\u00b9 T U s f \u03c3 = signal\u207b\u00b9 U s (f (signal T s \u03c3))\n\nsf-iso : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf T U s (sf\u207b\u00b9 T U s f) \u2261 f)\nsf-iso T U s f = refl\n\nsf-iso\u207b\u00b9 : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s f \u2192 \n  (sf\u207b\u00b9 T U s (sf T U s f) \u2261 f)\nsf-iso\u207b\u00b9 T U s f = refl\n\n-- Causality\n\nmutual\n\n  _at_\u22a8_\u2248[_]_ : \u2200 {\u03b1} (T : STyp (rset \u03b1)) s \u2192 T\u27ea T \u27eb s \u2192 Time \u2192 T\u27ea T \u27eb s \u2192 Set \u03b1\n  \u27e8 T \u27e9   at s \u22a8 a       \u2248[ u ] b       = T \u22a8 a \u2248[ u ] b\n  (T \u2227 U) at s \u22a8 (a , b) \u2248[ u ] (c , d) = (T at s \u22a8 a \u2248[ u ] c) \u00d7 (U at s \u22a8 b \u2248[ u ] d)\n  (T \u21d2 U) at s \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T at s \u22a8 a \u2248[ u ] b) \u2192 (U at s \u22a8 f a \u2248[ u ] g b)\n  \u25a1 T     at s \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 t s\u2264t \u2192 True (t \u2264 u) \u2192 (T at t \u22a8 \u03c3 t s\u2264t _ \u2248[ u ] \u03c4 t s\u2264t _)\n\n  _\u22a8_\u2248[_]_ : \u2200 {\u03b1} \u2192 (T : STyp (set \u03b1)) \u2192 T\u27ea T \u27eb \u2192 Time \u2192 T\u27ea T \u27eb \u2192 Set \u03b1\n  [ T ]   \u22a8 \u03c3       \u2248[ u ] \u03c4       = \u2200 s \u2192 (T at s \u22a8 \u03c3 s \u2248[ u ] \u03c4 s)\n  (T \u22a0 U) \u22a8 (a , b) \u2248[ u ] (c , d) = (T \u22a8 a \u2248[ u ] c) \u00d7 (U \u22a8 b \u2248[ u ] d)\n  (T \u21a6 U) \u22a8 f       \u2248[ u ] g       = \u2200 a b \u2192 (T \u22a8 a \u2248[ u ] b) \u2192 (U \u22a8 f a \u2248[ u ] g b)\n\nCausal : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) s \u2192 T\u27ea T \u21d2 U \u27eb s \u2192 Set (\u03b1 \u2294 \u03b2)\nCausal T U s f = \u2200 t \u03c3 \u03c4 \u2192 \n  (T at s \u22a8 \u03c3 \u2248[ t ] \u03c4) \u2192 (U at s \u22a8 f \u03c3 \u2248[ t ] f \u03c4)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (rset o \u220b World \u2194 World)\n\u211c[ u ] {t} s\u2261t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b) \u2192 (T at s \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 \u27e8 T \u27e9   at s u a b a\u211cb\n    = \u211c-impl-\u2248 T u a b a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) at s u (a , b) (c , d) (a\u211cc , b\u211cd) \n    = (\u211c-impl-\u2248 T at s u a c a\u211cc , \u211c-impl-\u2248 U at s u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) at s u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U at s u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T at s u a b a\u2248b))\n  \u211c-impl-\u2248 (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb t s\u2264t t\u2264u \u2192 \u211c-impl-\u2248 T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u211c\u03c4 refl tt (\u03bb {r} _ _ {r\u2264t} _ \u2192 \u2264-trans r t u r\u2264t t\u2264u))\n\n  \u2248-impl-\u211c_at : \u2200 {\u03b1} (T : STyp (rset \u03b1)) (s u : Time) (a b : T\u27ea T \u27eb s) \u2192\n     (T at s \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) refl a b)\n  \u2248-impl-\u211c \u27e8 T \u27e9   at s u a b a\u2248b\n    = \u2248-impl-\u211c T u a b a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) at s u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T at s u a c a\u2248c , \u2248-impl-\u211c U at s u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) at s u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U at s u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T at s u a b a\u211cb))\n  \u2248-impl-\u211c (\u25a1 T)   at s u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea \u25a1 T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) {s} refl \u03c3 \u03c4\n    lemma {t} refl {s\u2264t} {s\u2264t\u2032} tt t\u2264u with irrel (s \u2264 t) s\u2264t s\u2264t\u2032\n    lemma {t} refl {s\u2264t}        tt t\u2264u | refl = \n      \u2248-impl-\u211c T at t u (\u03c3 t s\u2264t _) (\u03c4 t s\u2264t _) \n        (\u03c3\u2248\u03c4 t s\u2264t (t\u2264u refl {s\u2264t} {s\u2264t} tt {\u2264-refl t} {\u2264-refl t} tt))\n\n  \u211c-impl-\u2248 : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b) \u2192 (T \u22a8 a \u2248[ u ] b)\n  \u211c-impl-\u2248 (T \u22a0 U) u (a , b) (c , d) (a\u211cc , b\u211cd)\n    = (\u211c-impl-\u2248 T u a c a\u211cc , \u211c-impl-\u2248 U u b d b\u211cd)\n  \u211c-impl-\u2248 (T \u21a6 U) u f g f\u211cg\n    = \u03bb a b a\u2248b \u2192 \u211c-impl-\u2248 U u (f a) (g b) (f\u211cg (\u2248-impl-\u211c T u a b a\u2248b))\n  \u211c-impl-\u2248 [ T ] u \u03c3 \u03c4 \u03c3\u211c\u03c4\n    = \u03bb s \u2192 \u211c-impl-\u2248 T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u211c\u03c4 refl)\n\n  \u2248-impl-\u211c : \u2200 {\u03b1} (T : STyp (set \u03b1)) (u : Time) (a b : T\u27ea T \u27eb) \u2192\n    (T \u22a8 a \u2248[ u ] b) \u2192 (T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) a b)\n  \u2248-impl-\u211c (T \u22a0 U) u (a , b) (c , d) (a\u2248c , b\u2248d)\n    = (\u2248-impl-\u211c T u a c a\u2248c , \u2248-impl-\u211c U u b d b\u2248d)\n  \u2248-impl-\u211c (T \u21a6 U) u f g f\u2248g\n    = \u03bb {a} {b} a\u211cb \u2192 \u2248-impl-\u211c U u (f a) (g b) (f\u2248g a b (\u211c-impl-\u2248 T u a b a\u211cb))\n  \u2248-impl-\u211c [ T ] u \u03c3 \u03c4 \u03c3\u2248\u03c4 = lemma where\n    lemma : T\u27e6 \u27ea [ T ] \u27eb \u27e7\u00b2 (\u211c[ u ] , tt) \u03c3 \u03c4\n    lemma {s} refl = \u2248-impl-\u211c T at s u (\u03c3 s) (\u03c4 s) (\u03c3\u2248\u03c4 s)\n\n-- Every F-omega function is causal\n\ncausality : \u2200 {\u03b1 \u03b2} (T : STyp (rset \u03b1)) (U : STyp (rset \u03b2)) (M : Exp [] \u27ea [ T \u21d2 U ] \u27eb) s \u2192 \n  Causal T U s (M\u27e6 M \u27e7 (World , tt) tt s)\ncausality T U M s t \n  = \u211c-impl-\u2248 T \u21d2 U at s t \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7 (World , tt) tt s) \n      (M\u27e6 M \u27e7\u00b2 (\u211c[ t ] , _) tt refl)\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"33bf8a91f464b543294643f0a34a74d18336b197","subject":"correctness-of-derive on union via diff-apply specialized to bags.","message":"correctness-of-derive on union via diff-apply specialized to bags.\n\nOld-commit-hash: 66c9bf647ea2671843dba31c125bda1646123f31\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/NatBag.agda","new_file":"experimental\/NatBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemma: diff-apply holds with propositional\n-- equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 sym (\u2205++b=b {{\u27e6 derive b \u27e7 \u03c1}}) \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 extract-\u0394equiv\n       (correctness-of-derive \u03c1 {consistency} b)\n       emptyBag tt tt \u27e9\n    emptyBag ++ (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 \u2205++b=b {{\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)}} \u27e9\n    \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    identity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\n    identity-weakening {\u2205} {\u2205} = refl\n    identity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n      cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (cong\u2082 _++_\n         (db=b\u2295db\u229db b)\n         (db=b\u2295db\u229db d)) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       (sym ([a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d]\n             {\u27e6 b \u27e7 (update \u03c1 {consistency})} \n             {\u27e6 d \u27e7 (update \u03c1 {consistency})}\n             {\u27e6 b \u27e7 (ignore \u03c1)} {\u27e6 d \u27e7 (ignore \u03c1)})) \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1 {consistency})\n        ++ \u27e6 d \u27e7 (update \u03c1 {consistency}))\n        \\\\(\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","old_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemma: diff-apply holds with propositional\n-- equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9 -- extensional equivalence as changes applied to emptyBag\n    emptyBag ++ \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    identity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\n    identity-weakening {\u2205} {\u2205} = refl\n    identity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n      cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) ++ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5943b7724caeba33c76ba50a60ed1252e294dd0c","subject":"Fixed doc.","message":"Fixed doc.\n\nIgnore-this: 553c22852000c127f219f23b6637febd\n\ndarcs-hash:20101209104359-3bd4e-9c16f4157e4ebaae8af970aca9d17770a19ec048.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Consistency\/README.agda","new_file":"Test\/Consistency\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- Test the consistency of the axioms\n------------------------------------------------------------------------------\n-- In some modules, we declare Agda postulates as FOL axioms. We\n-- test in these modules if it is possible to prove an unprovable\n-- theorem from these axioms (see the rule all_consistency in\n-- ..\/..\/Makefile).\n\nmodule Test.Consistency.README where\n\nopen import Test.Consistency.AxiomaticPA.Base.Impossible\n\nopen import Test.Consistency.GroupTheory.Base.Impossible\n\nopen import Test.Consistency.LTC.Base.Impossible\nopen import Test.Consistency.LTC.Data.Bool.Impossible\nopen import Test.Consistency.LTC.Data.List.Impossible\nopen import Test.Consistency.LTC.Data.Nat.Impossible\nopen import Test.Consistency.LTC.Data.Nat.Inequalities.Impossible\nopen import Test.Consistency.LTC.Data.Stream.Bisimilarity.Impossible\nopen import Test.Consistency.LTC.Program.GCD.GCD.Impossible\nopen import Test.Consistency.LTC.Program.SortList.SortList.Impossible\n","old_contents":"------------------------------------------------------------------------------\n-- Test the consistency of the axioms\n------------------------------------------------------------------------------\n-- In some modules, we declare Agda postulates as FOL axioms. We\n-- test in these modules if it is possible to prove an unprovable\n-- theorem from these axioms (see the rule all_consistency in\n-- src\/Makefile).\n\nmodule Test.Consistency.README where\n\nopen import Test.Consistency.AxiomaticPA.Base.Impossible\n\nopen import Test.Consistency.GroupTheory.Base.Impossible\n\nopen import Test.Consistency.LTC.Base.Impossible\nopen import Test.Consistency.LTC.Data.Bool.Impossible\nopen import Test.Consistency.LTC.Data.List.Impossible\nopen import Test.Consistency.LTC.Data.Nat.Impossible\nopen import Test.Consistency.LTC.Data.Nat.Inequalities.Impossible\nopen import Test.Consistency.LTC.Data.Stream.Bisimilarity.Impossible\nopen import Test.Consistency.LTC.Program.GCD.GCD.Impossible\nopen import Test.Consistency.LTC.Program.SortList.SortList.Impossible\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c9577b28ad7df88259d2e6d37f39c96b8b8e6dd6","subject":"Products","message":"Products\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Product\/NP.agda","new_file":"lib\/Data\/Product\/NP.agda","new_contents":"-- move this to product\n{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Product.NP where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Data.Product public hiding (\u2203)\nopen import Relation.Binary.PropositionalEquality as \u2261\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nopen import Function.Injection using (Injection; module Injection)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\n\n\u2203 : \u2200 {a b} {A : \u2605 a} \u2192 (A \u2192 \u2605 b) \u2192 \u2605 (a \u2294 b)\n\u2203 = \u03a3 _\n\nfirst : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 \u03a3 A (C \u2218 f) \u2192 \u03a3 B C\nfirst f = map f id -- f (x , y) = (f x , y)\n\n-- generalized first\u2032 but differently than first\nfirst' : \u2200 {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} \u2192\n          (f : (x : A) \u2192 B x) (p : \u03a3 A C) \u2192 B (proj\u2081 p) \u00d7 C (proj\u2081 p)\nfirst' f (x , y) = (f x , y)\n\nfirst\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 A \u00d7 C \u2192 B \u00d7 C\nfirst\u2032 = first\n\nsecond : \u2200 {a p q} {A : \u2605 a} {P : A \u2192 \u2605 p} {Q : A \u2192 \u2605 q} \u2192\n           (\u2200 {x} \u2192 P x \u2192 Q x) \u2192 \u03a3 A P \u2192 \u03a3 A Q\nsecond = map id\n\nsecond\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n           (B \u2192 C) \u2192 A \u00d7 B \u2192 A \u00d7 C\nsecond\u2032 f = second f\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\nrecord [\u03a3] {a b a\u209a b\u209a}\n           {A : \u2605 a}\n           {B : A \u2192 \u2605 b}\n           (A\u209a : A \u2192 \u2605 a\u209a)\n           (B\u209a : {x : A} (x\u209a : A\u209a x)\n                 \u2192 B x \u2192 \u2605 b\u209a)\n           (p : \u03a3 A B) : \u2605 (a\u209a \u2294 b\u209a) where\n  constructor _[,]_\n  field\n    [proj\u2081] : A\u209a (proj\u2081 p)\n    [proj\u2082] : B\u209a [proj\u2081] (proj\u2082 p)\nopen [\u03a3] public\ninfixr 4 _[,]_\n\nsyntax [\u03a3] A\u209a (\u03bb x\u209a \u2192 e) = [ x\u209a \u2236 A\u209a ][\u00d7][ e ]\n\n[\u2203] : \u2200 {a a\u209a b b\u209a} \u2192\n        (\u2200i\u27e8 A\u209a \u2236 [\u2605] {a} a\u209a \u27e9[\u2192] ((A\u209a [\u2192] [\u2605] {b} b\u209a) [\u2192] [\u2605] _)) \u2203\n[\u2203] {x\u209a = A\u209a} = [\u03a3] A\u209a\n\nsyntax [\u2203] (\u03bb A\u209a \u2192 f) = [\u2203][ A\u209a ] f\n\n_[\u00d7]_ : \u2200 {a b a\u209a b\u209a} \u2192 ([\u2605] {a} a\u209a [\u2192] [\u2605] {b} b\u209a [\u2192] [\u2605] _) _\u00d7_\n_[\u00d7]_ A\u209a B\u209a = [\u03a3] A\u209a (\u03bb _ \u2192 B\u209a)\n\nprivate\n\n  Dec\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n               (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n             \u2192 \u2605 _\n  Dec\u27e6\u2605\u27e7 A\u1d63 = \u2200 x\u2081 x\u2082 \u2192 Dec (A\u1d63 x\u2081 x\u2082)\n\n  Dec\u27e6Pred\u27e7 : \u2200 {a\u2081 a\u2082 p\u2081 p\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n                (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n                {P\u2081 : Pred p\u2081 A\u2081} {P\u2082 : Pred p\u2082 A\u2082} {p\u1d63}\n              \u2192 (P\u1d63 : \u27e6Pred\u27e7 p\u1d63 A\u1d63 P\u2081 P\u2082) \u2192 \u2605 _\n  Dec\u27e6Pred\u27e7 {A\u2081 = A\u2081} {A\u2082} A\u1d63 {P\u2081} {P\u2082} P\u1d63 = \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) y\u2081 y\u2082 \u2192 Dec (P\u1d63 x\u1d63 y\u2081 y\u2082) -- Dec\u27e6\u2605\u27e7 A\u1d63 \u21d2 Dec\u27e6\u2605\u27e7 P\u1d63\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n           {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : \u2605 (a\u1d63 \u2294 b\u1d63) where\n  constructor _\u27e6,\u27e7_\n  field\n    \u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082)\n    \u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _\u27e6,\u27e7_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u27e6\u00d7\u27e7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63}\n        {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n        {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 \u2605 _\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n          {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082) \u00d7\n                        B\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082))\n       (\u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\ndec\u27e6\u03a3\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63 A\u2081 A\u2082 B\u2081 B\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 {b\u2081} {b\u2082} b\u1d63 A\u1d63 B\u2081 B\u2082}\n       (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n       -- (substA\u1d63 : \u2200 {x y \u2113} {p q} (P : A\u1d63 x y \u2192 \u2605 \u2113) \u2192 P p \u2192 P q)\n       (uniqA\u1d63 : \u2200 {x y} (p q : A\u1d63 x y) \u2192 p \u2261 q)\n       (decB\u1d63 : Dec\u27e6Pred\u27e7 A\u1d63 {_} {_} {b\u1d63 {-BUG: with _ here Agda loops-}} B\u1d63)\n     \u2192 Dec\u27e6\u2605\u27e7 (\u27e6\u03a3\u27e7 A\u1d63 B\u1d63)\ndec\u27e6\u03a3\u27e7 {B\u1d63 = B\u1d63} decA\u1d63 uniqA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 x\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 f \u2218 \u27e6proj\u2082\u27e7)\n  where f : \u2200 {x\u1d63'} \u2192 B\u1d63 x\u1d63' y\u2081 y\u2082 \u2192 B\u1d63 x\u1d63 y\u2081 y\u2082\n        f {x\u1d63'} y\u1d63 rewrite uniqA\u1d63 x\u1d63' x\u1d63 = y\u1d63\n\nmodule Dec\u2082 where\n  map\u2082\u2032 : \u2200 {p\u2081 p\u2082 q} {P\u2081 : \u2605 p\u2081} {P\u2082 : \u2605 p\u2082} {Q : \u2605 q}\n          \u2192 (P\u2081 \u2192 P\u2082 \u2192 Q) \u2192 (Q \u2192 P\u2081) \u2192 (Q \u2192 P\u2082) \u2192 Dec P\u2081 \u2192 Dec P\u2082 \u2192 Dec Q\n  map\u2082\u2032 _   \u03c0\u2081  _   (no \u00acp\u2081)  _         = no (\u00acp\u2081 \u2218 \u03c0\u2081)\n  map\u2082\u2032 _   _   \u03c0\u2082  _         (no \u00acp\u2082)  = no (\u00acp\u2082 \u2218 \u03c0\u2082)\n  map\u2082\u2032 mk  _   _   (yes p\u2081)  (yes p\u2082)  = yes (mk p\u2081 p\u2082)\n\ndec\u27e6\u00d7\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n           {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n           {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082}\n           (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n           (decB\u1d63 : Dec\u27e6\u2605\u27e7 B\u1d63)\n         \u2192 Dec\u27e6\u2605\u27e7 (A\u1d63 \u27e6\u00d7\u27e7 B\u1d63)\ndec\u27e6\u00d7\u27e7 decA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 \u27e6proj\u2082\u27e7)\n\nmk\u03a3\u2261 : \u2200 {a b} {A : \u2605 a} {x y : A} (B : A \u2192 \u2605 b) {p : B x} {q : B y} (xy : x \u2261 y) \u2192 subst B xy p \u2261 q \u2192 (x , p) \u2261 (y , q)\nmk\u03a3\u2261 _ xy h rewrite xy | h = \u2261.refl\n\n\u03a3,-injective\u2082 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x : A} {y z : B x} \u2192 (_,_ {B = B} x y) \u2261 (x , z) \u2192 y \u2261 z\n\u03a3,-injective\u2082 refl = refl\n\n\u03a3,-injective\u2081 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x\u2081 x\u2082 : A} {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} \u2192 (x\u2081 \u03a3., y\u2081) \u2261 (x\u2082 , y\u2082) \u2192 x\u2081 \u2261 x\u2082\n\u03a3,-injective\u2081 refl = refl\n\nproj\u2081-injective : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x y : \u03a3 A B}\n                    (B-uniq : \u2200 {z} (p\u2081 p\u2082 : B z) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 proj\u2081 x \u2261 proj\u2081 y \u2192 x \u2261 y\nproj\u2081-injective {x = (a , p\u2081)} {y = (_ , p\u2082)} B-uniq eq rewrite sym eq\n  = cong (\u03bb p \u2192 (a , p)) (B-uniq p\u2081 p\u2082)\n\nproj\u2082-irrelevance : \u2200 {a b} {A : \u2605 a} {B C : A \u2192 \u2605 b} {x\u2081 x\u2082 : A}\n                      {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} {z\u2081 : C x\u2081} {z\u2082 : C x\u2082}\n                    \u2192 (C-uniq : \u2200 {z} (p\u2081 p\u2082 : C z) \u2192 p\u2081 \u2261 p\u2082)\n                    \u2192 (x\u2081 \u03a3., y\u2081) \u2261 (x\u2082 , y\u2082)\n                    \u2192 (x\u2081 \u03a3., z\u2081) \u2261 (x\u2082 , z\u2082)\nproj\u2082-irrelevance C-uniq = proj\u2081-injective C-uniq \u2218 \u03a3,-injective\u2081\n\n\u225f\u03a3 : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (decP : \u2200 x \u2192 Decidable {A = P x} _\u2261_)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl with decP w p\u2081 p\u2082\n\u225f\u03a3 decA decP (w  , p)  (.w , .p) | yes refl | yes refl = yes refl\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl | no  p\u2262\n    = no (p\u2262 \u2218 \u03a3,-injective\u2082)\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\n\u225f\u03a3' : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (uniqP : \u2200 {x} (p q : P x) \u2192 p \u2261 q)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3' decA uniqP (w  , p\u2081) (.w , p\u2082) | yes refl\n  = yes (cong (\u03bb p \u2192 (w , p)) (uniqP p\u2081 p\u2082))\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\nproj\u2081-Injection : \u2200 {a b} {A : \u2605 a} {B : A \u2192  \u2605 b}\n                  \u2192 (\u2200 {x} (p\u2081 p\u2082 : B x) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 Injection (setoid (\u03a3 A B))\n                              (setoid A)\nproj\u2081-Injection {B = B} B-uniq\n     = record { to        = \u2192-to-\u27f6 (proj\u2081 {B = B})\n              ; injective = proj\u2081-injective B-uniq\n              }\n","old_contents":"-- move this to product\n{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Product.NP where\n\nopen import Type hiding (\u2605)\nopen import Level\nopen import Data.Product public hiding (\u2203)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Unary.NP hiding (Decidable)\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Function\nopen import Function.Injection using (Injection; module Injection)\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\n\n\u2203 : \u2200 {a b} {A : \u2605 a} \u2192 (A \u2192 \u2605 b) \u2192 \u2605 (a \u2294 b)\n\u2203 = \u03a3 _\n\nfirst : \u2200 {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} \u2192\n          (f : (x : A) \u2192 B x) (p : \u03a3 A C) \u2192 B (proj\u2081 p) \u00d7 C (proj\u2081 p)\nfirst f (x , y) = (f x , y)\n\nfirst\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n          (f : A \u2192 B) \u2192 A \u00d7 C \u2192 B \u00d7 C\nfirst\u2032 = first\n\nsecond : \u2200 {a p q} {A : \u2605 a} {P : A \u2192 \u2605 p} {Q : A \u2192 \u2605 q} \u2192\n           (\u2200 {x} \u2192 P x \u2192 Q x) \u2192 \u03a3 A P \u2192 \u03a3 A Q\nsecond f (x , y) = (x , f y)\n\nsecond\u2032 : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n           (B \u2192 C) \u2192 A \u00d7 B \u2192 A \u00d7 C\nsecond\u2032 f = second f\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\nrecord [\u03a3] {a b a\u209a b\u209a}\n           {A : \u2605 a}\n           {B : A \u2192 \u2605 b}\n           (A\u209a : A \u2192 \u2605 a\u209a)\n           (B\u209a : {x : A} (x\u209a : A\u209a x)\n                 \u2192 B x \u2192 \u2605 b\u209a)\n           (p : \u03a3 A B) : \u2605 (a\u209a \u2294 b\u209a) where\n  constructor _[,]_\n  field\n    [proj\u2081] : A\u209a (proj\u2081 p)\n    [proj\u2082] : B\u209a [proj\u2081] (proj\u2082 p)\nopen [\u03a3] public\ninfixr 4 _[,]_\n\nsyntax [\u03a3] A\u209a (\u03bb x\u209a \u2192 e) = [ x\u209a \u2236 A\u209a ][\u00d7][ e ]\n\n[\u2203] : \u2200 {a a\u209a b b\u209a} \u2192\n        (\u2200i\u27e8 A\u209a \u2236 [\u2605] {a} a\u209a \u27e9[\u2192] ((A\u209a [\u2192] [\u2605] {b} b\u209a) [\u2192] [\u2605] _)) \u2203\n[\u2203] {x\u209a = A\u209a} = [\u03a3] A\u209a\n\nsyntax [\u2203] (\u03bb A\u209a \u2192 f) = [\u2203][ A\u209a ] f\n\n_[\u00d7]_ : \u2200 {a b a\u209a b\u209a} \u2192 ([\u2605] {a} a\u209a [\u2192] [\u2605] {b} b\u209a [\u2192] [\u2605] _) _\u00d7_\n_[\u00d7]_ A\u209a B\u209a = [\u03a3] A\u209a (\u03bb _ \u2192 B\u209a)\n\nprivate\n\n  Dec\u27e6\u2605\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n               (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n             \u2192 \u2605 _\n  Dec\u27e6\u2605\u27e7 A\u1d63 = \u2200 x\u2081 x\u2082 \u2192 Dec (A\u1d63 x\u2081 x\u2082)\n\n  Dec\u27e6Pred\u27e7 : \u2200 {a\u2081 a\u2082 p\u2081 p\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n                (A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082)\n                {P\u2081 : Pred p\u2081 A\u2081} {P\u2082 : Pred p\u2082 A\u2082} {p\u1d63}\n              \u2192 (P\u1d63 : \u27e6Pred\u27e7 p\u1d63 A\u1d63 P\u2081 P\u2082) \u2192 \u2605 _\n  Dec\u27e6Pred\u27e7 {A\u2081 = A\u2081} {A\u2082} A\u1d63 {P\u2081} {P\u2082} P\u1d63 = \u2200 {x\u2081 x\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082) y\u2081 y\u2082 \u2192 Dec (P\u1d63 x\u1d63 y\u2081 y\u2082) -- Dec\u27e6\u2605\u27e7 A\u1d63 \u21d2 Dec\u27e6\u2605\u27e7 P\u1d63\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n           {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 \u2605 b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : \u2605 (a\u1d63 \u2294 b\u1d63) where\n  constructor _\u27e6,\u27e7_\n  field\n    \u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082)\n    \u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _\u27e6,\u27e7_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u27e6\u00d7\u27e7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63}\n        {B\u2081 : A\u2081 \u2192 \u2605 b\u2081}\n        {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 \u2605 _\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n          {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 \u2605 a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 \u2605 b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 \u2605 _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082) \u00d7\n                        B\u1d63 (proj\u2082 p\u2081) (proj\u2082 p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n       {B\u2081 : A\u2081 \u2192 \u2605 b\u2081} {B\u2082 : A\u2082 \u2192 \u2605 b\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6proj\u2081\u27e7 : A\u1d63 (proj\u2081 p\u2081) (proj\u2081 p\u2082))\n       (\u27e6proj\u2082\u27e7 : B\u1d63 \u27e6proj\u2081\u27e7 (proj\u2082 p\u2081) (proj\u2082 p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\ndec\u27e6\u03a3\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63 A\u2081 A\u2082 B\u2081 B\u2082}\n       {A\u1d63 : \u27e6\u2605\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 {b\u2081} {b\u2082} b\u1d63 A\u1d63 B\u2081 B\u2082}\n       (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n       -- (substA\u1d63 : \u2200 {x y \u2113} {p q} (P : A\u1d63 x y \u2192 \u2605 \u2113) \u2192 P p \u2192 P q)\n       (uniqA\u1d63 : \u2200 {x y} (p q : A\u1d63 x y) \u2192 p \u2261 q)\n       (decB\u1d63 : Dec\u27e6Pred\u27e7 A\u1d63 {_} {_} {b\u1d63 {-BUG: with _ here Agda loops-}} B\u1d63)\n     \u2192 Dec\u27e6\u2605\u27e7 (\u27e6\u03a3\u27e7 A\u1d63 B\u1d63)\ndec\u27e6\u03a3\u27e7 {B\u1d63 = B\u1d63} decA\u1d63 uniqA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 x\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 f \u2218 \u27e6proj\u2082\u27e7)\n  where f : \u2200 {x\u1d63'} \u2192 B\u1d63 x\u1d63' y\u2081 y\u2082 \u2192 B\u1d63 x\u1d63 y\u2081 y\u2082\n        f {x\u1d63'} y\u1d63 rewrite uniqA\u1d63 x\u1d63' x\u1d63 = y\u1d63\n\nmodule Dec\u2082 where\n  map\u2082\u2032 : \u2200 {p\u2081 p\u2082 q} {P\u2081 : \u2605 p\u2081} {P\u2082 : \u2605 p\u2082} {Q : \u2605 q}\n          \u2192 (P\u2081 \u2192 P\u2082 \u2192 Q) \u2192 (Q \u2192 P\u2081) \u2192 (Q \u2192 P\u2082) \u2192 Dec P\u2081 \u2192 Dec P\u2082 \u2192 Dec Q\n  map\u2082\u2032 _   \u03c0\u2081  _   (no \u00acp\u2081)  _         = no (\u00acp\u2081 \u2218 \u03c0\u2081)\n  map\u2082\u2032 _   _   \u03c0\u2082  _         (no \u00acp\u2082)  = no (\u00acp\u2082 \u2218 \u03c0\u2082)\n  map\u2082\u2032 mk  _   _   (yes p\u2081)  (yes p\u2082)  = yes (mk p\u2081 p\u2082)\n\ndec\u27e6\u00d7\u27e7 : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n           {A\u2081 : \u2605 a\u2081} {A\u2082 : \u2605 a\u2082}\n           {B\u2081 : \u2605 b\u2081} {B\u2082 : \u2605 b\u2082}\n           {A\u1d63 : \u27e6\u2605\u27e7 a\u1d63 A\u2081 A\u2082}\n           {B\u1d63 : \u27e6\u2605\u27e7 b\u1d63 B\u2081 B\u2082}\n           (decA\u1d63 : Dec\u27e6\u2605\u27e7 A\u1d63)\n           (decB\u1d63 : Dec\u27e6\u2605\u27e7 B\u1d63)\n         \u2192 Dec\u27e6\u2605\u27e7 (A\u1d63 \u27e6\u00d7\u27e7 B\u1d63)\ndec\u27e6\u00d7\u27e7 decA\u1d63 decB\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082) with decA\u1d63 x\u2081 x\u2082\n... | no \u00acx\u1d63 = no (\u00acx\u1d63 \u2218 \u27e6proj\u2081\u27e7)\n... | yes x\u1d63 with decB\u1d63 y\u2081 y\u2082\n...           | yes y\u1d63 = yes (x\u1d63 \u27e6,\u27e7 y\u1d63)\n...           | no \u00acy\u1d63 = no (\u00acy\u1d63 \u2218 \u27e6proj\u2082\u27e7)\n\n\n\u03a3,-injective\u2082 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x : A} {y z : B x} \u2192 ((x , y) \u2236 \u03a3 A B) \u2261 (x , z) \u2192 y \u2261 z\n\u03a3,-injective\u2082 refl = refl\n\n\u03a3,-injective\u2081 : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x\u2081 x\u2082 : A} {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} \u2192 ((x\u2081 , y\u2081) \u2236 \u03a3 A B) \u2261 (x\u2082 , y\u2082) \u2192 x\u2081 \u2261 x\u2082\n\u03a3,-injective\u2081 refl = refl\n\nproj\u2081-injective : \u2200 {a b} {A : \u2605 a} {B : A \u2192 \u2605 b} {x y : \u03a3 A B}\n                    (B-uniq : \u2200 {z} (p\u2081 p\u2082 : B z) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 proj\u2081 x \u2261 proj\u2081 y \u2192 x \u2261 y\nproj\u2081-injective {x = (a , p\u2081)} {y = (_ , p\u2082)} B-uniq eq rewrite sym eq\n  = cong (\u03bb p \u2192 (a , p)) (B-uniq p\u2081 p\u2082)\n\nproj\u2082-irrelevance : \u2200 {a b} {A : \u2605 a} {B C : A \u2192 \u2605 b} {x\u2081 x\u2082 : A}\n                      {y\u2081 : B x\u2081} {y\u2082 : B x\u2082} {z\u2081 : C x\u2081} {z\u2082 : C x\u2082}\n                    \u2192 (C-uniq : \u2200 {z} (p\u2081 p\u2082 : C z) \u2192 p\u2081 \u2261 p\u2082)\n                    \u2192 ((x\u2081 , y\u2081) \u2236 \u03a3 A B) \u2261 (x\u2082 , y\u2082)\n                    \u2192 ((x\u2081 , z\u2081) \u2236 \u03a3 A C) \u2261 (x\u2082 , z\u2082)\nproj\u2082-irrelevance C-uniq = proj\u2081-injective C-uniq \u2218 \u03a3,-injective\u2081\n\n\u225f\u03a3 : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (decP : \u2200 x \u2192 Decidable {A = P x} _\u2261_)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl with decP w p\u2081 p\u2082\n\u225f\u03a3 decA decP (w  , p)  (.w , .p) | yes refl | yes refl = yes refl\n\u225f\u03a3 decA decP (w  , p\u2081) (.w , p\u2082) | yes refl | no  p\u2262\n    = no (p\u2262 \u2218 \u03a3,-injective\u2082)\n\u225f\u03a3 decA decP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\n\u225f\u03a3' : \u2200 {A : \u2605\u2080} {P : A \u2192 \u2605\u2080}\n       (decA : Decidable {A = A} _\u2261_)\n       (uniqP : \u2200 {x} (p q : P x) \u2192 p \u2261 q)\n     \u2192 Decidable {A = \u03a3 A P} _\u2261_\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) with decA w\u2081 w\u2082\n\u225f\u03a3' decA uniqP (w  , p\u2081) (.w , p\u2082) | yes refl\n  = yes (cong (\u03bb p \u2192 (w , p)) (uniqP p\u2081 p\u2082))\n\u225f\u03a3' decA uniqP (w\u2081 , p\u2081) (w\u2082 , p\u2082) | no w\u2262 = no (w\u2262 \u2218 cong proj\u2081)\n\nproj\u2081-Injection : \u2200 {a b} {A : \u2605 a} {B : A \u2192  \u2605 b}\n                  \u2192 (\u2200 {x} (p\u2081 p\u2082 : B x) \u2192 p\u2081 \u2261 p\u2082)\n                  \u2192 Injection (setoid (\u03a3 A B))\n                              (setoid A)\nproj\u2081-Injection {B = B} B-uniq\n     = record { to        = \u2192-to-\u27f6 (proj\u2081 {B = B})\n              ; injective = proj\u2081-injective B-uniq\n              }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8d4965d37f6b555d7d430f418e267dd62b4bfbb6","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er n1 n2}\n         ->     el    \u21d3  n1\n         ->        er \u21d3         (Succ n2) -- divisor is non-zero\n         -> Div el er \u21d3 (n1 div (Succ n2))\n\n  exb0 : Val 0 \u21d3 0\n  exb0 = \u21d3Base\n\n  exb3 : Val 3 \u21d3 3\n  exb3 = \u21d3Base\n\n  exs331 : Div (Val 3) (Val 3) \u21d3 1\n  exs331 = \u21d3Step \u21d3Base \u21d3Base\n\n  exs931 : Div (Val 9) (Val 3) \u21d3 3\n  exs931 = \u21d3Step \u21d3Base \u21d3Base\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  relate the two definitions:\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, define this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- - function f : a -> b, and\n  -- - desired postcondition on function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g -> (x : a)\n         ->    f x == g x\n  \u2291-eq f g wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er n1 n2}\n         ->     el    \u21d3  n1\n         ->        er \u21d3         (Succ n2)\n         -> Div el er \u21d3 (n1 div (Succ n2))\n\n  exb : Val 3 \u21d3 3\n  exb = \u21d3Base\n\n  exs : Div (Val 3) (Val 3) \u21d3 1\n  exs = \u21d3Step \u21d3Base \u21d3Base\n\n  -- alternatively, semantics specified by an INTERPRETER\n  -- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- define operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  How to relate the two definitions:\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign a weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                    Set\n  mustPT a\u2192ba\u2192Set _ (Pure ba)      = a\u2192ba\u2192Set _ ba\n  mustPT _        _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, define this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3))\n        \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0))\n        \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"175f180bb610ca89aa9a0b78b1ddbaa15c272192","subject":"NaPa: Avoid ambiguity","message":"NaPa: Avoid ambiguity\n","repos":"np\/NomPa","old_file":"lib\/NaPa\/LogicalRelation.agda","new_file":"lib\/NaPa\/LogicalRelation.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule NaPa.LogicalRelation where\n\nimport Level as L\nopen import Irrelevance\nopen import Data.Nat.NP hiding (_\u2294_) renaming (_+_ to _+\u2115_; _\u2264_ to _\u2264\u2115_; _\u2264?_ to _\u2264?\u2115_; _<_ to _<\u2115_\n                                              ;module == to ==\u2115; _==_ to _==\u2115_)\nopen import Data.Nat.Logical renaming (_\u225f_ to _\u225f\u2115_; _\u27e6+\u27e7_ to _\u27e6+\u2115\u27e7_)\nopen import Data.Nat.Properties as Nat\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Maybe.NP\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Sum.NP\nopen import Data.Product.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality as \u27f6\u2261 using (_\u27f6_; _\u27e8$\u27e9_)\nopen import Function.Injection.NP using (Injection; Injective; module\n    Injection; _\u2208_)\nopen import NaPa hiding (_\u2208_)\n\nprivate\n  module Inj = Injection\n  module \u2115s = Setoid \u27e6\u2115\u27e7-setoid\n  module ==\u2115s = Setoid ==\u2115.setoid\n  module \u2115e = Equality \u27e6\u2115\u27e7-equality\n\n_\u27e8$\u27e9\u1d62_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Injection From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\n_\u27e8$\u27e9\u1d62_ = \u27f6\u2261._\u27e8$\u27e9_ \u2218 Inj.to -- inj x = Inj.to inj \u27e8$\u27e9 x\n\n==\u2115-dec : \u2200 x y \u2192 \u27e6\ud835\udfda\u27e7 \u230a x \u225f\u2115 y \u230b (x ==\u2115 y)\n==\u2115-dec x y with x \u225f\u2115 y\n... | yes p = \u27e61\u2082\u27e7\u2032 (==\u2115s.reflexive (\u2115e.to-propositional p))\n... | no \u00acp = \u27e60\u2082\u27e7\u2032 (T'\u00ac'not (\u00acp \u2218 \u2115s.reflexive \u2218 ==\u2115.sound _ _))\n\n\u27e6dec\u27e7 : \u2200 {P Q : Set} (decP : Dec P) (decQ : Dec Q) \u2192 (P \u2192 Q) \u2192 (Q \u2192 P) \u2192 \u27e6\ud835\udfda\u27e7 \u230a decP \u230b \u230a decQ \u230b\n\u27e6dec\u27e7 (yes _) (yes _) _ _ = \u27e61\u2082\u27e7\n\u27e6dec\u27e7 (no  _) (no  _) _ _ = \u27e60\u2082\u27e7\n\u27e6dec\u27e7 (yes p) (no \u00acq) f _ = \u22a5-elim (\u00acq (f p))\n\u27e6dec\u27e7 (no \u00acp) (yes q) _ g = \u22a5-elim (\u00acp (g q))\n\n\u27e6World\u27e7 : (\u03b1\u2081 \u03b1\u2082 : World) \u2192 Set\n\u27e6World\u27e7 = Injection on Nm\n\n-- dup with NotSoFresh.LogicalRelation\n\u27e6WorldPred\u27e7 : (P\u2081 P\u2082 : Pred L.zero World) \u2192 Set\u2081\n\u27e6WorldPred\u27e7 = \u27e6Pred\u27e7 _ \u27e6World\u27e7\n\n\u27e6WorldRel\u27e7 : (R\u2081 R\u2082 : Rel World L.zero) \u2192 Set\u2081\n\u27e6WorldRel\u27e7 = \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = to-inj } where\n  to : Nm (\u03b1\u2081 \u21911) \u27f6 Nm (\u03b1\u2082 \u21911)\n  to = Name\u2192-to-Nm\u27f6 (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63))\n  to-inj : Injective to\n  to-inj {zero  , _} {zero  , _} zero    = zero\n  to-inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  to-inj {suc _ , _} {zero  , _} ()\n  to-inj {zero  , _} {suc _ , _} ()\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = inj } where\n  to : Nm (\u03b1\u2081 +1) \u27f6 Nm (\u03b1\u2082 +1)\n  to = Name\u2192-to-Nm\u27f6 (\u03bb x \u2192 suc\u1d3a (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (pred\u1d3a x)))\n  inj : Injective to\n  inj {zero  , _} {zero  , _} _       = zero\n  inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  inj {suc _ , _} {zero  , ()} _\n  inj {zero  , ()} {suc _ , _} _\n\n_\u27e6\u2191_\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 (suc k) = \u03b1\u1d63 \u27e6\u2191 k \u27e7 \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082 = ( x\u2081 , x\u2082 ) \u2208 \u03b1\u1d63\n\n\u27e6Name\u27e7\u2261 : \u27e6WorldPred\u27e7 Name Name\n\u27e6Name\u27e7\u2261 \u03b1\u1d63 x\u2081 x\u2082 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\u2081 \u2261 x\u2082\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = record { to = Name\u2192-to-Nm\u27f6 Name\u00f8-elim\n             ; injective = \u03bb{x} \u2192 Name\u00f8-elim x }\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 {_} {x\u2082} x\u1d63 {_} {y\u2082} y\u1d63 = helper (\u2261\u1d3a\u21d2\u2261 {_} {_} {x\u2082} x\u1d63) (\u2261\u1d3a\u21d2\u2261 {_} {_} {y\u2082} y\u1d63) where\n  helper : (\u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n  helper {x} \u2261.refl {y} \u2261.refl =\n       x ==\u1d3a y       \u2248\u27e8 sym (==\u2115-dec _ _) \u27e9\n       \u230a x \u225f\u1d3a y \u230b   \u2248\u27e8 \u27e6dec\u27e7 (x \u225f\u1d3a y) (x\u2032 \u225f\u1d3a y\u2032) (\u27f6\u2261.cong (Inj.to \u03b1\u1d63)) (Inj.injective \u03b1\u1d63) \u27e9\n       \u230a x\u2032 \u225f\u1d3a y\u2032 \u230b \u2248\u27e8 ==\u2115-dec _ _ \u27e9\n       x\u2032 ==\u1d3a y\u2032 \u220e\n    where\n      x\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\n      y\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 y\n      open \u27e6\ud835\udfda\u27e7-Props using (sym)\n      open \u27e6\ud835\udfda\u27e7-Reasoning\n\n\u27e6zero\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) zero\u1d3a zero\u1d3a\n\u27e6zero\u1d3a\u27e7 _ = zero\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 _ = suc\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 _ = suc\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 _ = \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _ zero x = x\n\u27e6add\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\n   = \u2115.suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-p-})))\n   \u2248\u27e8 \u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 x))) \u2115e.refl \u27e9\n     suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-q-})))\n   \u2248\u27e8 suc (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x) \u27e9\n     suc (k\u2082 +\u2115 name x\u2082)\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 suc k\u2081 , _{-p-}))\n   \u2248\u27e8 \u2261-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) (name-injective (\u2261.sym (\u2238-+-assoc (name x\u2081) 1 k\u2081))) \u27e9\n      name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 1 \u2238 k\u2081 , _{-r-}))\n   \u2248\u27e8 \u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {pred\u1d3a x\u2081} {pred\u1d3a x\u2082} (\u27e6pred\u1d3a\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) {x\u2081} {x\u2082} x\u1d63)  \u27e9\n      (name x\u2082 \u2238 1) \u2238 k\u2082\n   \u2248\u27e8 \u2115e.reflexive (\u2238-+-assoc (name x\u2082) 1 k\u2082) \u27e9\n      name x\u2082 \u2238 suc k\u2082\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n  helper zero                   \u2261.refl  = \u27e60\u2082\u27e7\n  helper (suc k\u1d63) {zero  , _ }  \u2261.refl  = \u27e61\u2082\u27e7\n  helper (suc k\u1d63) {suc x , pf\u2081} \u2261.refl  = helper k\u1d63 {x , pf\u2081} \u2261.refl\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) easy-cmp\u1d3a easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) easy-cmp\u1d3a easy-cmp\u1d3a\n  helper zero                             \u2261.refl = inj\u2082 (\u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) \u2115e.refl)\n  helper (suc k\u1d63)           {zero  , _ }  \u2261.refl = inj\u2081 zero\n  helper (suc {k\u2081} {k\u2082} k\u1d63) {suc x , pf\u2081} \u2261.refl\n    = \u27e6map\u27e7 _ _ _ _ suc suc (helper k\u1d63 {x , pf\u2081} \u2261.refl)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6protect\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n           (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) \u27e6\u2192\u27e7 (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7))) protect\u21911 protect\u21911\n\u27e6protect\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {f\u2081} {f\u2082} f\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n   helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (protect\u21911 f\u2081) (protect\u21911 f\u2082)\n   helper {zero ,  _}  \u2261.refl = zero\n   helper {suc x , pf} \u2261.refl = suc (f\u1d63 \u2115e.refl)\n\n_\u27e6\u2286\u27e7b_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk\u27e6\u2286\u27e7 : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x : \u03b1\u2081 \u2286 \u03b2\u2081} {y : \u03b1\u2082 \u2286 \u03b2\u2082} \u2192\n  _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y\nun\u27e6\u2286\u27e7 (mk\u27e6\u2286\u27e7 x) {a} {b} c = x {a} {b} c\n\n-- this is the start of an experimentation to see if the issue with \u27e6dropName\u27e7 is due\n-- to implicit lambdas.\n_\u27e6\u2286\u27e7e_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7e_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7e \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk\u27e6\u2286\u27e7 (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {_} {f\u2082} f g = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {_} {coerce\u1d3a f\u2082 x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 {_ , ()}\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , _} \u2261.refl = zero\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a \u2286-+1\u21911) (coerce\u1d3a \u2286-+1\u21911)\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc \u2115e.refl\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7) where\n   helper : (\u27e6\u00f8\u27e7 \u27e6\u2286\u27e7b \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n   helper {_ , ()}\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ _ \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ _ \u03b1+\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8 \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 _ {\u03b1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1\u2286\u00f8 (pred\u1d3a x))\n\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8 \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 _ {\u03b1+1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1+1\u2286\u00f8 (suc\u1d3a x))\n\n-- not primitive\n\n\u27e6pred\u1d3a?\u27e7-core : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) x\n                    \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x) (pred\u1d3a? (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x))\n\u27e6pred\u1d3a?\u27e7-core _  (zero  , _) = \u27e6nothing\u27e7\n\u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 (suc _ , _) = \u27e6just\u27e7 (\u27f6\u2261.cong (Inj.to \u03b1\u1d63) \u2115e.refl)\n\n\u27e6pred\u1d3a?\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) pred\u1d3a? pred\u1d3a?\n\u27e6pred\u1d3a?\u27e7 {_} {\u03b1\u2082} \u03b1\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)\n    -- BUG: the 'with' notation failed here\n  where helper : \u2200 {x\u2082} (eq : protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x\u2081 \u2261 x\u2082) \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x\u2081) (pred\u1d3a? x\u2082)\n        helper \u2261.refl = \u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 x\u2081\n\n\u27e6\u2286-ctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-ctx-+1\u21911 \u2286-ctx-+1\u21911\n\u27e6\u2286-ctx-+1\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u00acName\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u03b1\u1d63))) \u00acName \u00acName\n\u27e6\u00acName\u27e7 _ {\u03b1\u2286\u00f8\u2081} _ {x\u2081} _ = Name-elim \u03b1\u2286\u00f8\u2081 x\u2081\n\n\u27e6\u2286-cong-+\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)))) \u2286-cong-+ \u2286-cong-+\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 zero    = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 (suc k\u1d63) = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 k\u1d63)\n\n\u27e6\u2286-+-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+-inj \u2286-+-inj\n\u27e6\u2286-+-inj\u27e7 _ _ zero    \u03b1\u2286\u03b2\u1d63 = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-+-inj\u27e7 _ _ (suc k\u1d63) \u03b1\u2286\u03b2\u1d63 = \u27e6\u2286-+-inj\u27e7 _ _ k\u1d63 (\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-assoc-+\u2032\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u2081\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e8 k\u2082\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 (k\u2082\u1d63 \u27e6+\u2115\u27e7 k\u2081\u1d63)) \u27e6\u2286\u27e7 ((\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u2081\u1d63) \u27e6+\u1d42\u27e7 k\u2082\u1d63)))) \u2286-assoc-+\u2032 \u2286-assoc-+\u2032\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 zero      = \u27e6\u2286-cong-+\u27e7 _ _ base k\u2081\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 (suc k\u2082)  = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 k\u2082)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule NaPa.LogicalRelation where\n\nimport Level as L\nopen import Irrelevance\nopen import Data.Nat.NP hiding (_\u2294_) renaming (_+_ to _+\u2115_; _\u2264_ to _\u2264\u2115_; _\u2264?_ to _\u2264?\u2115_; _<_ to _<\u2115_\n                                              ;module == to ==\u2115; _==_ to _==\u2115_)\nopen import Data.Nat.Logical renaming (_\u225f_ to _\u225f\u2115_; _\u27e6+\u27e7_ to _\u27e6+\u2115\u27e7_)\nopen import Data.Nat.Properties as Nat\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Maybe.NP\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Sum.NP\nopen import Data.Product.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen \u2261.\u2261-Reasoning\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality as \u27f6\u2261 using (_\u27f6_; _\u27e8$\u27e9_)\nopen import Function.Injection.NP using (Injection; Injective; module\n    Injection; _\u2208_)\nopen import NaPa hiding (_\u2208_)\n\nprivate\n  module Inj = Injection\n  module \u2115s = Setoid \u27e6\u2115\u27e7-setoid\n  module ==\u2115s = Setoid ==\u2115.setoid\n  module \u2115e = Equality \u27e6\u2115\u27e7-equality\n\n_\u27e8$\u27e9\u1d62_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Injection From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\n_\u27e8$\u27e9\u1d62_ = \u27f6\u2261._\u27e8$\u27e9_ \u2218 Inj.to -- inj x = Inj.to inj \u27e8$\u27e9 x\n\n==\u2115-dec : \u2200 x y \u2192 \u27e6\ud835\udfda\u27e7 \u230a x \u225f\u2115 y \u230b (x ==\u2115 y)\n==\u2115-dec x y with x \u225f\u2115 y\n... | yes p = \u27e61\u2082\u27e7\u2032 (==\u2115s.reflexive (\u2115e.to-propositional p))\n... | no \u00acp = \u27e60\u2082\u27e7\u2032 (T'\u00ac'not (\u00acp \u2218 \u2115s.reflexive \u2218 ==\u2115.sound _ _))\n\n\u27e6dec\u27e7 : \u2200 {P Q : Set} (decP : Dec P) (decQ : Dec Q) \u2192 (P \u2192 Q) \u2192 (Q \u2192 P) \u2192 \u27e6\ud835\udfda\u27e7 \u230a decP \u230b \u230a decQ \u230b\n\u27e6dec\u27e7 (yes _) (yes _) _ _ = \u27e61\u2082\u27e7\n\u27e6dec\u27e7 (no  _) (no  _) _ _ = \u27e60\u2082\u27e7\n\u27e6dec\u27e7 (yes p) (no \u00acq) f _ = \u22a5-elim (\u00acq (f p))\n\u27e6dec\u27e7 (no \u00acp) (yes q) _ g = \u22a5-elim (\u00acp (g q))\n\n\u27e6World\u27e7 : (\u03b1\u2081 \u03b1\u2082 : World) \u2192 Set\n\u27e6World\u27e7 = Injection on Nm\n\n-- dup with NotSoFresh.LogicalRelation\n\u27e6WorldPred\u27e7 : (P\u2081 P\u2082 : Pred L.zero World) \u2192 Set\u2081\n\u27e6WorldPred\u27e7 = \u27e6Pred\u27e7 _ \u27e6World\u27e7\n\n\u27e6WorldRel\u27e7 : (R\u2081 R\u2082 : Rel World L.zero) \u2192 Set\u2081\n\u27e6WorldRel\u27e7 = \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = to-inj } where\n  to : Nm (\u03b1\u2081 \u21911) \u27f6 Nm (\u03b1\u2082 \u21911)\n  to = Name\u2192-to-Nm\u27f6 (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63))\n  to-inj : Injective to\n  to-inj {zero  , _} {zero  , _} zero    = zero\n  to-inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  to-inj {suc _ , _} {zero  , _} ()\n  to-inj {zero  , _} {suc _ , _} ()\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = inj } where\n  to : Nm (\u03b1\u2081 +1) \u27f6 Nm (\u03b1\u2082 +1)\n  to = Name\u2192-to-Nm\u27f6 (\u03bb x \u2192 suc\u1d3a (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (pred\u1d3a x)))\n  inj : Injective to\n  inj {zero  , _} {zero  , _} _       = zero\n  inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  inj {suc _ , _} {zero  , ()} _\n  inj {zero  , ()} {suc _ , _} _\n\n_\u27e6\u2191_\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 (suc k) = \u03b1\u1d63 \u27e6\u2191 k \u27e7 \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082 = ( x\u2081 , x\u2082 ) \u2208 \u03b1\u1d63\n\n\u27e6Name\u27e7\u2261 : \u27e6WorldPred\u27e7 Name Name\n\u27e6Name\u27e7\u2261 \u03b1\u1d63 x\u2081 x\u2082 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\u2081 \u2261 x\u2082\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = record { to = Name\u2192-to-Nm\u27f6 Name\u00f8-elim\n             ; injective = \u03bb{x} \u2192 Name\u00f8-elim x }\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 {_} {x\u2082} x\u1d63 {_} {y\u2082} y\u1d63 = helper (\u2261\u1d3a\u21d2\u2261 {_} {_} {x\u2082} x\u1d63) (\u2261\u1d3a\u21d2\u2261 {_} {_} {y\u2082} y\u1d63) where\n  helper : (\u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n  helper {x} \u2261.refl {y} \u2261.refl =\n       x ==\u1d3a y       \u2248\u27e8 sym (==\u2115-dec _ _) \u27e9\n       \u230a x \u225f\u1d3a y \u230b   \u2248\u27e8 \u27e6dec\u27e7 (x \u225f\u1d3a y) (x\u2032 \u225f\u1d3a y\u2032) (\u27f6\u2261.cong (Inj.to \u03b1\u1d63)) (Inj.injective \u03b1\u1d63) \u27e9\n       \u230a x\u2032 \u225f\u1d3a y\u2032 \u230b \u2248\u27e8 ==\u2115-dec _ _ \u27e9\n       x\u2032 ==\u1d3a y\u2032 \u220e\n    where\n      x\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\n      y\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 y\n      open \u27e6\ud835\udfda\u27e7-Props using (sym)\n      open \u27e6\ud835\udfda\u27e7-Reasoning\n\n\u27e6zero\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) zero\u1d3a zero\u1d3a\n\u27e6zero\u1d3a\u27e7 _ = zero\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 _ = suc\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 _ = suc\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 _ = \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _ zero x = x\n\u27e6add\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\n   = \u2115.suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-p-})))\n   \u2248\u27e8 \u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 x))) \u2115e.refl \u27e9\n     suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-q-})))\n   \u2248\u27e8 suc (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x) \u27e9\n     suc (k\u2082 +\u2115 name x\u2082)\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 suc k\u2081 , _{-p-}))\n   \u2248\u27e8 \u2261-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) (name-injective (\u2261.sym (\u2238-+-assoc (name x\u2081) 1 k\u2081))) \u27e9\n      name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 1 \u2238 k\u2081 , _{-r-}))\n   \u2248\u27e8 \u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {pred\u1d3a x\u2081} {pred\u1d3a x\u2082} (\u27e6pred\u1d3a\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) {x\u2081} {x\u2082} x\u1d63)  \u27e9\n      (name x\u2082 \u2238 1) \u2238 k\u2082\n   \u2248\u27e8 \u2115e.reflexive (\u2238-+-assoc (name x\u2082) 1 k\u2082) \u27e9\n      name x\u2082 \u2238 suc k\u2082\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n  helper zero                   \u2261.refl  = \u27e60\u2082\u27e7\n  helper (suc k\u1d63) {zero  , _ }  \u2261.refl  = \u27e61\u2082\u27e7\n  helper (suc k\u1d63) {suc x , pf\u2081} \u2261.refl  = helper k\u1d63 {x , pf\u2081} \u2261.refl\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) easy-cmp\u1d3a easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) easy-cmp\u1d3a easy-cmp\u1d3a\n  helper zero                             \u2261.refl = inj\u2082 (\u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) \u2115e.refl)\n  helper (suc k\u1d63)           {zero  , _ }  \u2261.refl = inj\u2081 zero\n  helper (suc {k\u2081} {k\u2082} k\u1d63) {suc x , pf\u2081} \u2261.refl\n    = \u27e6map\u27e7 _ _ _ _ suc suc (helper k\u1d63 {x , pf\u2081} \u2261.refl)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6protect\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n           (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) \u27e6\u2192\u27e7 (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7))) protect\u21911 protect\u21911\n\u27e6protect\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {f\u2081} {f\u2082} f\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n   helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (protect\u21911 f\u2081) (protect\u21911 f\u2082)\n   helper {zero ,  _}  \u2261.refl = zero\n   helper {suc x , pf} \u2261.refl = suc (f\u1d63 \u2115e.refl)\n\n_\u27e6\u2286\u27e7b_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk\u27e6\u2286\u27e7 : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x : \u03b1\u2081 \u2286 \u03b2\u2081} {y : \u03b1\u2082 \u2286 \u03b2\u2082} \u2192\n  _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y\nun\u27e6\u2286\u27e7 (mk\u27e6\u2286\u27e7 x) {a} {b} c = x {a} {b} c\n\n-- this is the start of an experimentation to see if the issue with \u27e6dropName\u27e7 is due\n-- to implicit lambdas.\n_\u27e6\u2286\u27e7e_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7e_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7e \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk\u27e6\u2286\u27e7 (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {_} {f\u2082} f g = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {_} {coerce\u1d3a f\u2082 x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 {_ , ()}\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , _} \u2261.refl = zero\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a \u2286-+1\u21911) (coerce\u1d3a \u2286-+1\u21911)\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc \u2115e.refl\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7) where\n   helper : (\u27e6\u00f8\u27e7 \u27e6\u2286\u27e7b \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n   helper {_ , ()}\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ _ \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ _ \u03b1+\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8 \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 _ {\u03b1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1\u2286\u00f8 (pred\u1d3a x))\n\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8 \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 _ {\u03b1+1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1+1\u2286\u00f8 (suc\u1d3a x))\n\n-- not primitive\n\n\u27e6pred\u1d3a?\u27e7-core : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) x\n                    \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x) (pred\u1d3a? (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x))\n\u27e6pred\u1d3a?\u27e7-core _  (zero  , _) = \u27e6nothing\u27e7\n\u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 (suc _ , _) = \u27e6just\u27e7 (\u27f6\u2261.cong (Inj.to \u03b1\u1d63) \u2115e.refl)\n\n\u27e6pred\u1d3a?\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) pred\u1d3a? pred\u1d3a?\n\u27e6pred\u1d3a?\u27e7 {_} {\u03b1\u2082} \u03b1\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)\n    -- BUG: the 'with' notation failed here\n  where helper : \u2200 {x\u2082} (eq : protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x\u2081 \u2261 x\u2082) \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x\u2081) (pred\u1d3a? x\u2082)\n        helper \u2261.refl = \u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 x\u2081\n\n\u27e6\u2286-ctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-ctx-+1\u21911 \u2286-ctx-+1\u21911\n\u27e6\u2286-ctx-+1\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u00acName\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u03b1\u1d63))) \u00acName \u00acName\n\u27e6\u00acName\u27e7 _ {\u03b1\u2286\u00f8\u2081} _ {x\u2081} _ = Name-elim \u03b1\u2286\u00f8\u2081 x\u2081\n\n\u27e6\u2286-cong-+\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)))) \u2286-cong-+ \u2286-cong-+\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 zero    = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 (suc k\u1d63) = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 k\u1d63)\n\n\u27e6\u2286-+-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+-inj \u2286-+-inj\n\u27e6\u2286-+-inj\u27e7 _ _ zero    \u03b1\u2286\u03b2\u1d63 = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-+-inj\u27e7 _ _ (suc k\u1d63) \u03b1\u2286\u03b2\u1d63 = \u27e6\u2286-+-inj\u27e7 _ _ k\u1d63 (\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-assoc-+\u2032\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u2081\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e8 k\u2082\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 (k\u2082\u1d63 \u27e6+\u2115\u27e7 k\u2081\u1d63)) \u27e6\u2286\u27e7 ((\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u2081\u1d63) \u27e6+\u1d42\u27e7 k\u2082\u1d63)))) \u2286-assoc-+\u2032 \u2286-assoc-+\u2032\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 zero      = \u27e6\u2286-cong-+\u27e7 _ _ base k\u2081\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 (suc k\u2082)  = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 k\u2082)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c977519bd3f18fabed3a76d701f953e7f67ed60c","subject":"Amend horrendous circular import","message":"Amend horrendous circular import\n\nOld-commit-hash: 2659d1d9b6e5eb45e9cbea66319b0265b12a08b2\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/NatBag.agda","new_file":"experimental\/NatBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemma: diff-apply holds with propositional\n-- equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9 -- extensional equivalence as changes applied to emptyBag\n    emptyBag ++ \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    identity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\n    identity-weakening {\u2205} {\u2205} = refl\n    identity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n      cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) ++ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","old_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast:\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n\nTODO\n1. Make sure this file has no hole\n   X. Replace \u2115 by \u2124\n   0. Replace \u2124 by \u2115 -- our bags are bags of nats now.\n   0. Introduce addition\n   0. Add MapBags and map\n   0. Figure out a way to communicate to a derivative that\n      certain changes are always nil (in this case, `+1`).\n2. Finish ExplicitNils\n3. Consider appending ExplicitNils\n\n\nChecklist: Adding syntactic constructs\n\n- weaken\n- \u27e6_\u27e7Term\n- weaken-sound\n- derive (symbolic derivation)\n- validity-of-derive\n- correctness-of-derive\n\nChecklist: Adding types\n\n- \u27e6_\u27e7Type\n- \u0394-Type\n- \u27e6derive\u27e7\n- _\u27e6\u229d\u27e7_\n- _\u27e6\u2295\u27e7_\n- f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f\n- f\u2295[g\u229df]=g\n- f\u2295\u0394f=f\n- valid-\u0394\n- R[f,g\u229df]\n- df=f\u2295df\u229df\n- R (inside validity-of-derive)\n\n-}\n\nmodule NatBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\n\n-- postulate a\\\\[b++c]=a\\\\b\\\\c : \u2200 {a b c} \u2192 a \\\\ (b ++ c) \u2261 a \\\\ b \\\\ c\n-- postulate [a++b]\\\\c=a\\\\c++b : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a \\\\ c ++ b\n-- postulate [a++b]\\\\c=a++b\\\\c : \u2200 {a b c} \u2192 (a ++ b) \\\\ c \u2261 a ++ b \\\\ c\n-- \n-- and consequently:\n\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\n\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  add : \u2200 {\u0393} \u2192 (t\u2081 : Term \u0393 nats) \u2192 (t\u2082 : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n  union : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n  diff : \u2200 {\u0393} \u2192 (b\u2081 : Term \u0393 bags) \u2192 (b\u2082 : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\nweaken subctx (union b\u2081 b\u2082) = union (weaken subctx b\u2081) (weaken subctx b\u2082)\nweaken subctx (diff b\u2081 b\u2082) = diff (weaken subctx b\u2081) (weaken subctx b\u2082)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\u27e6 union b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 ++ \u27e6 d \u27e7Term \u03c1\n\u27e6 diff b d \u27e7Term \u03c1 = \u27e6 b \u27e7Term \u03c1 \\\\ \u27e6 d \u27e7Term \u03c1\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\ninfix 0 _\u27e8$\u27e9_ -- infix 0 $ in Haskell\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\nweaken-sound (union b d) \u03c1 =\n  cong\u2082 _++_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\nweaken-sound (diff b d) \u03c1 =\n  cong\u2082 _\\\\_ (weaken-sound b \u03c1) (weaken-sound d \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n_\u2295_ : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\ninfixl 6 _\u229d_\ninfixl 6 _\u2295_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n-- m \u229d n = \u03bb f. f n m\n_\u229d_ {nats} m n =\n  abs (app (app (var this)\n    (weaken (drop _ \u2022 \u0393\u227c\u0393) n)) (weaken (drop _ \u2022 \u0393\u227c\u0393) m))\n-- d \u229d b = d \\\\ b\n_\u229d_ {bags} d b = weaken \u0393\u227c\u0393 (diff d b)\n-- g \u229d f = \u03bb x. \u03bb dx. g (x \u2295 dx) \u229d f x -- Incurs recomputation!\n_\u229d_ {\u03c4 \u21d2 \u03c4\u2081} g f =\n  abs (abs ((app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) g)\n            (var (that this) \u2295 var this))\n          \u229d app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) f) (var (that this))))\n\n_\u2295_ {nats} n dn = app dn snd\n_\u2295_ {bags} b db = union b db\n-- f \u2295 df = \u03bb x. f x \u2295 df x (x \u229d x) -- Incurs recomputation!\n_\u2295_ {\u03c4 \u21d2 \u03c4\u2081} f df =\n  abs (app (weaken (drop _ \u2022 \u0393\u227c\u0393) f) (var this)\n    \u2295 app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) df)\n                (var this)) ((var this) \u229d (var this)))\n\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive (add x y) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) x) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) y)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive x)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive y)) snd)))\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(map f b) = map (f \u2295 df) (b \u2295 db) \u229d map f b\nderive (map f b) = map ((weaken \u0393\u227c\u0394\u0393 f) \u2295 derive f)\n                       ((weaken \u0393\u227c\u0394\u0393 b) \u2295 derive b)\n                 \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\n-- derive(b ++ d) = derive(b) ++ derive(d)\nderive (union b d) = union (derive b) (derive d)\n\n-- derive(b \\\\ d) = derive(b) \\\\ derive(d)\nderive (diff b d) = diff (derive b) (derive d)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\nopen import NatBag\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\n\nopen import Data.Unit using\n  (\u22a4 ; tt)\n\nimport Data.Integer as \u2124\n\nopen import Data.Product using\n  (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\nimport Level\nimport Data.Product as Product\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemma: diff-apply holds with propositional\n-- equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 {!!} \u27e9 -- extensional equivalence as changes applied to emptyBag\n    emptyBag ++ \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Mutually recursive lemma: derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (map f b) = tt\n\nvalidity-of-derive \u03c1 (union b d) = tt\n\nvalidity-of-derive \u03c1 (diff d b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive \u03c1 {consistency} (add m n) =\n  begin\n    \u27e6 m \u27e7 (ignore \u03c1) + \u27e6 n \u27e7 (ignore \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (sym (weaken-sound m (\u27e6fst\u27e7 \u2022 \u03c1)))\n       (sym (weaken-sound n (\u27e6fst\u27e7 \u2022 \u03c1))) \u27e9\n    (\u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1) +\n     \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 refl \u27e9\n    \u27e6 abs (app (app (var this)\n        (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n        (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n          (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd)))\n    \u27e7 \u03c1 \u27e6fst\u27e7\n  \u2261\u27e8 refl \u27e9\n    \u27e6 derive (add m n) \u27e7 \u03c1 \u27e6fst\u27e7\n  \u220e where\n    open \u2261-Reasoning\n    blah : \u2115\n    blah =\n      \u27e6 add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m)) snd)\n        (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n)) snd) \u27e7\n      (\u27e6fst\u27e7 \u2022 \u03c1)\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (add m n) = ext-\u0394 (\u03bb _ _ _ \u2192\n  begin\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n    +\n    \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1) \u27e6snd\u27e7\n  \u2261\u27e8 refl \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive m) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive n) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1))\n  \u2261\u27e8 cong\u2082 _+_\n       (cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 m \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = m} {\u03c1 = \u03c1}))\n       ((cong\u2082 _\u27e6\u2295\u27e7_ {x = \u27e6 n \u27e7 (ignore \u03c1)} refl\n         (weaken-derivative {t = n} {\u03c1 = \u03c1}))) \u27e9\n    (\u27e6 m \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive m \u27e7 \u03c1)\n    +\n    (\u27e6 n \u27e7 (ignore \u03c1) \u27e6\u2295\u27e7 \u27e6 derive n \u27e7 \u03c1)\n  \u2261\u27e8 cong\u2082 _+_\n       (extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} m)\n         (\u27e6 m \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} m)\n         (R[f,g\u229df] (\u27e6 m \u27e7 (ignore \u03c1)) (\u27e6 m \u27e7 (update \u03c1 {consistency}))))\n       ((extract-\u0394equiv (correctness-of-derive \u03c1 {consistency} n)\n         (\u27e6 n \u27e7 (ignore \u03c1))\n         (validity-of-derive \u03c1 {consistency} n)\n         (R[f,g\u229df] (\u27e6 n \u27e7 (ignore \u03c1)) (\u27e6 n \u27e7 (update \u03c1 {consistency}))))) \u27e9\n    \u27e6 m \u27e7 (update \u03c1) + \u27e6 n \u27e7 (update \u03c1)\n  \u220e\n  ) where\n    open \u2261-Reasoning\n    identity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\n    identity-weakening {\u2205} {\u2205} = refl\n    identity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n      cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n    weaken-derivative :\n      \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7}\n        \u2192 \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (derive t) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1)\n        \u2261 \u27e6 derive t \u27e7 \u03c1\n    weaken-derivative {t = t} {\u03c1 = \u03c1}\n      rewrite weaken-sound {subctx = drop _ \u2022 \u0393\u227c\u0393} (derive t) (\u27e6snd\u27e7 \u2022 \u03c1)\n            | identity-weakening {\u03c1 = \u03c1}\n      = refl\n\ncorrectness-of-derive \u03c1 {consistency} (union b d) = ext-\u0394 (\u03bb c _ _ \u2192\n  begin\n    c ++ (\u27e6 derive b \u27e7 \u03c1 ++ \u27e6 derive d \u27e7 \u03c1)\n  \u2261\u27e8 {!!} \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n       ++ (\u27e6 d \u27e7 (update \u03c1) \\\\ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 cong\u2082 _++_\n       {x = c} refl\n       [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] \u27e9\n    c ++ ((\u27e6 b \u27e7 (update \u03c1) ++ \u27e6 d \u27e7 (update \u03c1))\n       \\\\ (\u27e6 b \u27e7 (ignore \u03c1) ++ \u27e6 d \u27e7 (ignore \u03c1)))\n  \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} _ = {!!}\n\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"893d0a2c51c61ffda507ee4b53dffd6d64ca8ef0","subject":"Updated README.agda.","message":"Updated README.agda.\n","repos":"asr\/fotc,asr\/fotc","old_file":"examples\/README.agda","new_file":"examples\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Paper, prerequisites, tested versions of the ATPS, and use\n\n-- See https:\/\/github.com\/asr\/agda2atp\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Paper, prerequisites, tested versions of the ATPS, and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fossacs-2012\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"23cc753582f5e13bfde70b71ec9ce2527ed5dcf4","subject":"Try using strings","message":"Try using strings\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"open import IO\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n-- {-# BUILTIN BOOL  Bool  #-}\n-- {-# BUILTIN TRUE  true  #-}\n-- {-# BUILTIN FALSE false #-}\n\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\ntrue \u2228 _  = true\nfalse \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nfalse \u2227 _ = false\ntrue \u2227 b  = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then a else _ = a\nif false then _ else b = b\n\n\n----------------------------------------\n\npostulate String : Set\n{-# BUILTIN STRING String #-}\n\nprimitive\n  primStringAppend   : String \u2192 String \u2192 String\n  primStringEquality : String \u2192 String \u2192 Bool\n  primShowString     : String \u2192 String\n\n_>>>_ : String \u2192 String \u2192 String\n_>>>_ = primStringAppend\n\n_===_ : String \u2192 String \u2192 Bool\n_===_ = primStringEquality\n\ninfixl 1 _>>>_\n\n\n\n----------------------------------------\n\ndata Principle : Set where\n  # : String \u2192 Principle\n\n\n_==_ : Principle \u2192 Principle \u2192 Bool\n(# a) == (# b) = a === b\n\n----------------------------------------\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[_] : {A : Set} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208_ : Principle \u2192 List Principle \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x == y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List Principle \u2192 Principle \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n\n\n----------------------------------------\n\n\ndata Arrow : Set where\n  \u21d2_  : Principle \u2192 Arrow\n  _\u21d2_ : Principle \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q == s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p == r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List Principle \u2192 List Principle\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List Principle \u2192 Principle \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List Principle \u2192 List Principle \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 Principle \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 Principle \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs ([ n ]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   ((# \"3\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n--   ((# \"5\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"3\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"10\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"11\"))) \u2237\n   ((# \"6\") \u21d2 (\u21d2 (# \"11\"))) \u2237\n   ((# \"11\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"8\") \u21d2 (\u21d2 (# \"4\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"9\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"3\") \u21d2 (\u21d2 (# \"8\"))) \u2237\n   ((# \"5\") \u21d2 ((# \"3\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"6\") \u21d2 ((# \"7\") \u21d2 (\u21d2 (# \"10\")))) \u2237\n   ((# \"6\") \u21d2 ((# \"3\") \u21d2 (\u21d2 (# \"3\")))) \u2237\n   ((# \"7\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"7\") \u21d2 (\u21d2 (# \"7\"))) \u2237\n   ((# \"7\") \u21d2 ((# \"8\") \u21d2 (\u21d2 (# \"10\")))) \u2237\n   ((# \"3\") \u21d2 ((# \"10\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"5\") \u21d2 (\u21d2 (# \"1\"))) \u2237\n   ((# \"3\") \u21d2 ((# \"1\") \u21d2 (\u21d2 (# \"9\")))) \u2237\n   ((# \"1\") \u21d2 (\u21d2 (# \"2\"))) \u2237\n   ((# \"10\") \u21d2 (\u21d2 (# \"2\"))) \u2237 []\n\ncms : List Separation\ncms =\n  (model ((# \"12\") \u2237 (# \"6\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"1\") \u2237 []) ((# \"5\") \u2237 (# \"3\") \u2237 (# \"7\") \u2237 (# \"7\") \u2237 [])) \u2237\n  (model ((# \"6\") \u2237 (# \"3\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"3\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"5\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"5\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"1\") \u2237 []) ((# \"6\") \u2237 (# \"3\") \u2237 [])) \u2237\n  (model ((# \"5\") \u2237 (# \"3\") \u2237 (# \"11\") \u2237 (# \"4\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"3\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"6\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 []) ((# \"5\") \u2237 (# \"6\") \u2237 (# \"3\") \u2237 (# \"8\") \u2237 (# \"9\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 (# \"1\") \u2237 []) ((# \"3\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 (# \"7\") \u2237 []) ((# \"9\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"10\") \u2237 (# \"8\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"10\") \u2237 (# \"9\") \u2237 []) ((# \"1\") \u2237 [])) \u2237\n  (model ((# \"3\") \u2237 (# \"4\") \u2237 (# \"11\") \u2237 []) ((# \"9\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"10\") \u2237 (# \"7\") \u2237 (# \"1\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"7\") \u2237 (# \"1\") \u2237 []) ((# \"4\") \u2237 (# \"11\") \u2237 (# \"8\") \u2237 [])) \u2237\n  (model ((# \"9\") \u2237 (# \"3\") \u2237 (# \"10\") \u2237 (# \"8\") \u2237 (# \"1\") \u2237 []) ((# \"11\") \u2237 [])) \u2237\n  (model ((# \"12\") \u2237 (# \"4\") \u2237 (# \"10\") \u2237 (# \"1\") \u2237 []) ((# \"11\") \u2237 (# \"3\") \u2237 [])) \u2237\n  (model ((# \"3\") \u2237 (# \"6\") \u2237 (# \"5\") \u2237 []) ([])) \u2237\n  (model ((# \"1\") \u2237 (# \"2\") \u2237 (# \"3\") \u2237 (# \"4\") \u2237 (# \"5\") \u2237 (# \"6\") \u2237 (# \"7\") \u2237 (# \"8\") \u2237 (# \"9\") \u2237 (# \"10\") \u2237 (# \"11\") \u2237 []) ((# \"12\") \u2237 [])) \u2237 []\n\ntestp : Relation\ntestp = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"10\")))\n\ntestd : Relation\ntestd = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"4\")))\n\ntests : Relation\ntests = consider proofs cms ((# \"5\") \u21d2 (\u21d2 (# \"3\")))\n\ntestu : Relation\ntestu = consider proofs cms ((# \"6\") \u21d2 (\u21d2 (# \"1\")))\n\ntestcl : List Principle\ntestcl = closure proofs ((# \"5\") \u2237 [])\n\n\n\nstringifylp : List Principle \u2192 String\nstringifylp [] = \"[]\"\nstringifylp ((# s) \u2237 xs) = s >>> \" \u2237 \" >>> (stringifylp xs)\n\n\nmain = run (putStrLn (stringifylp testcl))\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\ntrue \u2228 _  = true\nfalse \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nfalse \u2227 _ = false\ntrue \u2227 b  = b\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then a else _ = a\nif false then _ else b = b\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[_] : {A : Set} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x == y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q == s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p == r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs ([ n ]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 []\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 [])) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (5 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (6 \u2237 3 \u2237 [])) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (6 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 []) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 []) (3 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 [])) \u2237\n  (model (10 \u2237 9 \u2237 []) (1 \u2237 [])) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 []) (4 \u2237 11 \u2237 8 \u2237 [])) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 []) (11 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 []) (11 \u2237 3 \u2237 [])) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 []) ([])) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 []) (12 \u2237 [])) \u2237 []\n\ntestp : Relation\ntestp = consider proofs cms (5 \u21d2 (\u21d2 10))\n\ntestd : Relation\ntestd = consider proofs cms (5 \u21d2 (\u21d2 4))\n\ntests : Relation\ntests = consider proofs cms (5 \u21d2 (\u21d2 3))\n\ntestu : Relation\ntestu = consider proofs cms (6 \u21d2 (\u21d2 1))\n\ntestcl : List \u2115\ntestcl = closure proofs (5 \u2237 [])\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"668f3fcb3e3409416efd4888c2246ad3fac68a93","subject":"Added simple {anti,mono}-tone proof for subtraction","message":"Added simple {anti,mono}-tone proof for subtraction\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 zero y z = \u2261.refl\n+-inj-over-\u2238 (suc x) y z = +-inj-over-\u2238 x y z\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x <= y)) \u2192 T (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 <=.complete))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"53ab65ded43659f648d15f20246757eb96f2f280","subject":"Vector bijections","message":"Vector bijections\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nimport Level as L\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Function.Bijection.SyntaxKit\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : Set a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = refl\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n      `tl : Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id     \u2219 xs       = xs\n    (f `\u204f g) \u2219 xs       = g \u2219 f \u2219 xs\n    `0\u21941    \u2219 xs       = 0\u21941 xs\n    (`tl f) \u2219 []       = []\n    (`tl f) \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : Perm \u2192 Perm \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) [] = refl\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) [] = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u2219 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : L.Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) (f : Perm) xs \u2192 (f \u2219 fs \u229b f \u2219 xs) \u2261 f \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 _       `id        _  = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl _)   [] = refl\n    \u229b-dist-\u2219 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u2219 fs f xs = refl\n    \u229b-dist-\u2219 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u2219 (f \u2219 fs) g (f \u2219 xs)\n                                         | \u229b-dist-\u2219 fs f xs = refl\n\n    \u2219-replicate : \u2200 n {A : Set a} (x : A) f \u2192 f \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl _) = refl\n    \u2219-replicate (suc n) x (`tl f) rewrite \u2219-replicate n x f = refl\n    \u2219-replicate n x (f `\u204f g) rewrite \u2219-replicate n x f | \u2219-replicate n x g = refl\n\n    private\n        lem : \u2200 {n} {A : Set a} (fs : Vec (Endo A) n) f xs \u2192 fs \u229b f \u2219 xs \u2261 f \u2219 (f \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u2219 (f \u207b\u00b9 \u2219 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} {A : Set a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u2219 xs) \u2261 g \u2219 map f xs\n    \u2219-map {n} f g xs rewrite sym (\u229b-dist-\u2219 (replicate f) g xs) | \u2219-replicate n f g = refl\n\nmodule BijectionSyntax {a b} (A : Set a) (Bij\u1d2c : Set b) where\n    infixr 1 _`\u204f_\n    data Bij : Set (a L.\u2294 b) where\n      `id `0\u21941 : Bij\n      _`\u204f_ : Bij \u2192 Bij \u2192 Bij\n      _`\u2237_ : Bij\u1d2c \u2192 (A \u2192 Bij) \u2192 Bij\n\nmodule BijectionLib where\n    open BijectionSyntax\n    mapBij : \u2200 {a b\u1d2c} {A : Set a} {Bij\u1d2c : Set b\u1d2c}\n               {b b\u1d2e} {B : Set b} {Bij\u1d2e : Set b\u1d2e}\n             \u2192 (B \u2192 A)\n             \u2192 (Bij\u1d2c \u2192 Bij\u1d2e)\n             \u2192 Bij A Bij\u1d2c \u2192 Bij B Bij\u1d2e\n    mapBij f\u1d2e\u1d2c f `id = `id\n    mapBij f\u1d2e\u1d2c f `0\u21941 = `0\u21941\n    mapBij f\u1d2e\u1d2c f (`g `\u204f `h) = mapBij f\u1d2e\u1d2c f `g `\u204f mapBij f\u1d2e\u1d2c f `h\n    mapBij f\u1d2e\u1d2c f (`f\u1d2c `\u2237 `g) = f `f\u1d2c `\u2237 \u03bb x \u2192 mapBij f\u1d2e\u1d2c f (`g (f\u1d2e\u1d2c x))\n\nmodule BijectionSemantics {a b} {A : Set a} (bijKit\u1d2c : BijKit b A) where\n    open BijKit bijKit\u1d2c renaming (Bij to Bij\u1d2c; eval to eval\u1d2c; _\u207b\u00b9 to _\u207b\u00b9\u1d2c;\n                                  idBij to id\u1d2c; _\u2257Bij_ to _\u2257\u1d2c_;\n                                  _\u207b\u00b9-inverse to _\u207b\u00b9-inverse\u1d2c;\n                                  _\u207b\u00b9-involutive to _\u207b\u00b9-involutive\u1d2c;\n                                  id-spec to id\u1d2c-spec)\n    open BijectionSyntax A Bij\u1d2c\n\n    _\u207b\u00b9 : Endo Bij\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (f\u1d2c `\u2237 f) \u207b\u00b9 = f\u1d2c\u207b\u00b9 `\u2237 \u03bb x \u2192 (f (eval\u1d2c f\u1d2c\u207b\u00b9 x))\u207b\u00b9 where f\u1d2c\u207b\u00b9 = f\u1d2c \u207b\u00b9\u1d2c\n\n    infixr 9 _\u2219_\n    _\u2219_ : Bij \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id \u2219 xs             = xs\n    (f `\u204f g)   \u2219 xs       = g \u2219 f \u2219 xs\n    `0\u21941      \u2219 xs       = 0\u21941 xs\n    (f\u1d2c `\u2237 f) \u2219 []       = []\n    (f\u1d2c `\u2237 f) \u2219 (x \u2237 xs) = eval\u1d2c f\u1d2c x \u2237 f x \u2219 xs\n\n    _\u2257\u2032_ : Bij \u2192 Bij \u2192 Set _\n    f \u2257\u2032 g = \u2200 {n} (xs : Vec A n) \u2192 f \u2219 xs \u2261 g \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u2219 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = \u27e8\u2194\u27e9.0\u21941\u00b2-cancel _ xs\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-inverse\u1d2c) x | (f x \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u2219 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) [] = refl\n    ((f\u1d2c `\u2237 f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f\u1d2c \u207b\u00b9-involutive\u1d2c) x\n            | (f\u1d2c \u207b\u00b9-inverse\u1d2c) x\n            | (f x \u207b\u00b9-involutive) xs\n            = refl\n\n    Vec-bijKit : \u2200 n \u2192 BijKit _ (Vec A n)\n    Vec-bijKit n = mk Bij (\u03bb f xs \u2192 _\u2219_ f {n} xs) _\u207b\u00b9 `id _`\u204f_ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl)\n                (\u03bb f x \u2192 _\u207b\u00b9-inverse f x) (\u03bb f x \u2192 _\u207b\u00b9-involutive f x)\n\n    module VecBijKit n = BijKit (Vec-bijKit n)\n\n    `tl : Endo Bij\n    `tl f = id\u1d2c `\u2237 const f\n\n    module P where\n        open PermutationSyntax public\n        open PermutationSemantics {A = A} public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n    fromPerm : Perm \u2192 Bij\n    fromPerm `id = `id\n    fromPerm `0\u21941 = `0\u21941\n    fromPerm (\u03c0\u2080 `\u204f \u03c0\u2081) = fromPerm \u03c0\u2080 `\u204f fromPerm \u03c0\u2081\n    fromPerm (P.`tl \u03c0) = `tl (fromPerm \u03c0)\n\n    fromPerm-spec : \u2200 \u03c0 {n} (xs : Vec A n) \u2192 \u03c0 P.\u2219 xs \u2261 fromPerm \u03c0 \u2219 xs\n    fromPerm-spec `id xs = refl\n    fromPerm-spec `0\u21941 xs = refl\n    fromPerm-spec (\u03c0 `\u204f \u03c0\u2081) xs rewrite fromPerm-spec \u03c0 xs | fromPerm-spec \u03c0\u2081 (fromPerm \u03c0 \u2219 xs) = refl\n    fromPerm-spec (P.`tl \u03c0) [] = refl\n    fromPerm-spec (P.`tl \u03c0) (x \u2237 xs) rewrite id\u1d2c-spec x | fromPerm-spec \u03c0 xs = refl\n\n    module Unused where\n        `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u21941+ i \u27e9 = fromPerm P.`\u27e80\u21941+ i \u27e9\n\n        `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n        `\u27e80\u21941+ i \u27e9-spec xs rewrite sym (P.`\u27e80\u21941+ i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u21941+ i \u27e9 xs = refl\n\n        `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij\n        `\u27e80\u2194 i \u27e9 = fromPerm P.`\u27e80\u2194 i \u27e9\n\n        `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n        `\u27e80\u2194 i \u27e9-spec xs rewrite sym (P.`\u27e80\u2194 i \u27e9-spec xs) | fromPerm-spec P.`\u27e80\u2194 i \u27e9 xs = refl\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij\n    `\u27e8 i \u2194 j \u27e9 = fromPerm P.`\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8 i \u2194 j \u27e9-spec xs rewrite sym (P.`\u27e8 i \u2194 j \u27e9-spec xs) | fromPerm-spec P.`\u27e8 i \u2194 j \u27e9 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","old_contents":"module Data.Vec.NP where\n\nopen import Category.Applicative\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap)\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Set a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = refl\n    tabulate-\u2218 {suc n} f g =\n      \u2261.cong (_\u2237_ (f (g zero))) (tabulate-\u2218 f (g \u2218 suc))\n\nopen waiting-for-a-fix-in-the-stdlib public\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-aux : \u2200 {a} {A : Set a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nswap : \u2200 m {n} {a} {A : Set a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys rewrite drop-++ m xs ys | take-++ m xs ys = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : Set a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = refl\n\nmap-tail-\u2257 : \u2200 {m n a} {A : Set a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) rewrite f\u2257g xs = refl\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n_\u00b2 : \u2200 {a} {A : Set a} \u2192 Endo (Endo A)\nf \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : Set a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = \u2261.refl\n    comm zero    (suc _) _ = \u2261.refl\n    comm (suc _) zero    _ = \u2261.refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = \u2261.refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs \n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = \u2261.refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : Set a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n    {-\n    lem1+ : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9 \u2257 map-tail \u27e8 i \u2194 j \u27e9\n    lem1+ zero zero xs = {!!}\n    lem1+ zero (suc j) xs = {!!}\n    lem1+ (suc i) zero xs = {!!}\n    lem1+ (suc i) (suc j) xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ j \u27e9 (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs)))\n            | map-tail\u2218map-tail \u27e80\u21941+ j \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 (\u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) (0\u21941 xs)\n            | map-tail-\u2257 _ _ (lem1+ i j) (0\u21941 xs)\n            | lem01maptail2 \u27e8 i \u2194 j \u27e9 xs\n            = refl\n\n    lem : \u2200 {n} (i j : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9 \u2257 \u27e8 i \u2194 j \u27e9\n    lem zero j xs = refl\n    lem (suc i) zero xs = {!\u27e80\u21941+ i \u27e9\u00b2-cancel xs!}\n    lem (suc i) (suc j) xs = (\u27e80\u21941+ i \u27e9 \u2218 \u27e80\u21941+ j \u27e9 \u2218 \u27e80\u21941+ i \u27e9) xs\n                 \u2261\u27e8 lem1+ i j xs \u27e9\n                   \u27e8 suc i \u2194 suc j \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n    test = {!!}\n    -}\n{-\n    lem : \u2200 {n} (i j k : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 k \u27e9\n    lem i j k xs = (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 id \u2218 \u27e8 i \u2194 j \u27e9) xs\n                   (\u27e8 i \u2194 j \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 k \u27e9 \u2218 \u27e8 i \u2194 j \u27e9) xs\n                 \u2261\u27e8 {!!} \u27e9\n                   \u27e8 j \u2194 k \u27e9 xs\n                 \u220e where open \u2261-Reasoning\n-}\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : Set where\n      `id `0\u21941 : Perm\n      `tl : Perm \u2192 Perm\n      _`\u204f_ : Perm \u2192 Perm \u2192 Perm\n\n    _\u207b\u00b9 : Endo Perm\n    `id \u207b\u00b9 = `id\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    `tl \u03c0 \u207b\u00b9 = `tl (\u03c0 \u207b\u00b9)\n    (\u03c0\u2080 `\u204f \u03c0\u2081) \u207b\u00b9 = \u03c0\u2081 \u207b\u00b9 `\u204f \u03c0\u2080 \u207b\u00b9\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : Set a} where\n    open PermutationSyntax\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    infixr 9 _\u2219_\n    _\u2219_ : Perm \u2192 \u2200 {n} \u2192 Endo (Vec A n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `tl \u03c0 \u2219 []       = []\n    `tl \u03c0 \u2219 (x \u2237 xs) = x \u2237 \u03c0 \u2219 xs\n    (\u03c0 `\u204f \u03c0\u2081) \u2219 xs = \u03c0\u2081 \u2219 \u03c0 \u2219 xs\n\n    _\u2257\u03c0_ : Perm \u2192 Perm \u2192 Set _\n    \u03c0\u2080 \u2257\u03c0 \u03c0\u2081 = \u2200 {n} (xs : Vec A n) \u2192 \u03c0\u2080 \u2219 xs \u2261 \u03c0\u2081 \u2219 xs\n\n    _\u207b\u00b9-inverse : \u2200 \u03c0 \u2192 (\u03c0 `\u204f \u03c0 \u207b\u00b9) \u2257\u03c0 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    (`tl \u03c0 \u207b\u00b9-inverse) [] = refl\n    (`tl \u03c0 \u207b\u00b9-inverse) (x \u2237 xs) rewrite (\u03c0 \u207b\u00b9-inverse) xs = refl\n    ((\u03c0\u2080 `\u204f \u03c0\u2081) \u207b\u00b9-inverse) xs rewrite (\u03c0\u2081 \u207b\u00b9-inverse) (\u03c0\u2080 \u2219 xs) | (\u03c0\u2080 \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 \u03c0 \u2192 (\u03c0 \u207b\u00b9) \u207b\u00b9 \u2261 \u03c0\n    `id \u207b\u00b9-involutive = refl\n    `0\u21941 \u207b\u00b9-involutive = refl\n    `tl \u03c0 \u207b\u00b9-involutive rewrite \u03c0 \u207b\u00b9-involutive = refl\n    (\u03c0\u2080 `\u204f \u03c0\u2081) \u207b\u00b9-involutive rewrite \u03c0\u2080 \u207b\u00b9-involutive | \u03c0\u2081 \u207b\u00b9-involutive = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 \u03c0 \u2192 (\u03c0 \u207b\u00b9 `\u204f \u03c0) \u2257\u03c0 `id\n    (\u03c0 \u207b\u00b9-inverse\u2032) xs with ((\u03c0 \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite \u03c0 \u207b\u00b9-involutive = p\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    \u2219-replicate : \u2200 n (x : A) \u03c0 \u2192 \u03c0 \u2219 replicate {n = n} x \u2261 replicate x\n    \u2219-replicate n x `id = refl\n    \u2219-replicate zero x `0\u21941 = refl\n    \u2219-replicate (suc zero) x `0\u21941 = refl\n    \u2219-replicate (suc (suc n)) x `0\u21941 = refl\n    \u2219-replicate zero x (`tl \u03c0) = refl\n    \u2219-replicate (suc n) x (`tl \u03c0) rewrite \u2219-replicate n x \u03c0 = refl\n    \u2219-replicate n x (\u03c0 `\u204f \u03c0\u2081) rewrite \u2219-replicate n x \u03c0 | \u2219-replicate n x \u03c0\u2081 = refl\n\n    \u229b-dist-0\u21941 : \u2200 {n} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n    \u229b-dist-0\u21941 _           []          = refl\n    \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationProperties {a} (A : Set a) where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u2219 : \u2200 {n} (fs : Vec (Endo A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u229b-dist-\u2219 fs      `id        xs = refl\n    \u229b-dist-\u2219 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u229b-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u229b-dist-\u2219 fs \u03c0 xs = refl\n    \u229b-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u229b-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u229b-dist-\u2219 fs \u03c0\u2080 xs = refl\n\n    private\n        lem : \u2200 {n} (fs : Vec (Endo A) n) \u03c0 xs \u2192 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (\u03c0 \u207b\u00b9 \u2219 fs \u229b xs)\n        lem fs \u03c0 xs rewrite sym (\u229b-dist-\u2219 (\u03c0 \u207b\u00b9 \u2219 fs) \u03c0 xs) | (\u03c0 \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u2219-map : \u2200 {n} (f : Endo A) \u03c0 (xs : Vec A n) \u2192 map f (\u03c0 \u2219 xs) \u2261 \u03c0 \u2219 map f xs\n    \u2219-map {n} f \u03c0 xs rewrite sym (\u229b-dist-\u2219 (replicate f) \u03c0 xs) | \u2219-replicate n f \u03c0 = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b0125641f2cc236052975151c09c48f2990c4006","subject":"Removed unused functions.","message":"Removed unused functions.\n\nIgnore-this: cde3e29fa3e2fd39ade0109bdef5ba09\n\ndarcs-hash:20100701203132-3bd4e-51db1b5f863816d281905ee995e602105455987f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List.agda","new_file":"LTC\/Data\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\n\npostulate\n  length : D \u2192 D\n  length-[] : length []                     \u2261 zero\n  length-\u2237  : ( d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D ) \u2192 map f []           \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  replicate : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192 replicate zero d       \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","old_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule LTC.Data.List where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\ninfixr 5 _\u2237_ _++_\n\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- Basic functions\npostulate\n  head : D \u2192 D\n  cHead : (d ds : D) \u2192 head (d \u2237 ds) \u2261 d\n\npostulate\n  tail : D \u2192 D\n  cTail : (d ds : D) \u2192 tail (d \u2237 ds) \u2261 ds\n\npostulate\n  length : D \u2192 D\n  length-[] : length []                     \u2261 zero\n  length-\u2237  : ( d ds : D) \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : (ds : D) \u2192 [] ++ ds            \u2261 ds\n  ++-\u2237  : (d ds es : D) \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\npostulate\n  reverse    : D \u2192 D\n  reverse-[] : reverse []                    \u2261 []\n  reverse-\u2237  : (d ds : D) \u2192 reverse (d \u2237 ds) \u2261 reverse ds ++ d \u2237 []\n{-# ATP axiom reverse-[] #-}\n{-# ATP axiom reverse-\u2237 #-}\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D ) \u2192 map f []           \u2261 []\n  map-\u2237  : (f d ds : D) \u2192 map f (d \u2237 ds) \u2261 f \u2219 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  replicate : D \u2192 D \u2192 D\n  replicate-0 : (d : D) \u2192 replicate zero d       \u2261 []\n  replicate-S : (d e : D) \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n------------------------------------------------------------------------------\n\n-- The LTC list data type\ndata List : D \u2192 Set where\n  nil  : List []\n  cons : (d : D){ds : D} \u2192 (dsL : List ds) \u2192 List (d \u2237 ds)\n\n-- Induction principle for List\nindList : (P : D \u2192 Set) \u2192\n          P [] \u2192\n          ((d : D){ds : D} \u2192 List ds \u2192 P ds \u2192 P (d \u2237 ds)) \u2192\n          {ds : D} \u2192 List ds \u2192 P ds\nindList P p[] iStep nil               = p[]\nindList P p[] iStep (cons d {ds} dsL) = iStep d dsL (indList P p[] iStep dsL)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8bee65ab328a4888ab4cd5668968df81a964a927","subject":"New names for equiv projections \u00b7\u2192\/\u00b7\u2190","message":"New names for equiv projections \u00b7\u2192\/\u00b7\u2190\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/HoTT.agda","new_file":"lib\/HoTT.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  !-inv : \u2200 {x y : A} (p : x \u2261 y) \u2192 ! (! p) \u2261 p\n  !-inv = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp)\n\n  !-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp_ y\n  !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp)\n\n  \u2219-! : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp_ x\n  \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp)\n\n  !p\u2219p = !-\u2219\n  p\u2219!p = \u2219-!\n\n  \u2219-assoc : \u2200 {x y : A} (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n  \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n               (\u03bb x q r \u2192 idp)\n\n  ==-refl-\u2219 :  {x y : A} (p : x \u2261 y) {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n  ==-refl-\u2219 p = ap (flip _\u2219_ p)\n\n  \u2219-==-refl :  {x y : A} (p : x \u2261 y) {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n  \u2219-==-refl p qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  \u2219-\u2219-==-refl :  {x y z : A} (p : x \u2261 y) (q : y \u2261 z) {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n  \u2219-\u2219-==-refl p q rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n  !p\u2219p\u2219q : {x y z : A} (p : x \u2261 y) (q : y \u2261 z) \u2192 ! p \u2219 p \u2219 q \u2261 q\n  !p\u2219p\u2219q p q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n  p\u2219!p\u2219q : {x y z : A} (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n  p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n  p\u2219!q\u2219q : {x y z : A} (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n  p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n  p\u2219q\u2219!q : {x y z : A} (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n  p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    postulate\n        HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605\u2080\n--    HAE {f} f-qinv = {!F.is-linv!}\n--      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      postulate\n        g-f' : (x : A) \u2192 g (f x) \u2261 x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n        hae : HAE (qinv g f-g g-f')\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g g-f'\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    \u00b7\u2192 : (e : A \u2243 B) \u2192 (A \u2192 B)\n    \u00b7\u2192 e = fst e\n\n    \u00b7\u2190 : (e : A \u2243 B) \u2192 (B \u2192 A)\n    \u00b7\u2190 e = Is-equiv.linv (snd e)\n\n    \u2013> = \u00b7\u2192\n    <\u2013 = \u00b7\u2190\n\n    <\u2013' : (e : A \u2243 B) \u2192 (B \u2192 A)\n    <\u2013' e = Is-equiv.rinv (snd e)\n\n    <\u2013-inv-l : (e : A \u2243 B) (a : A)\n              \u2192 (<\u2013 e (\u2013> e a) \u2261 a)\n    <\u2013-inv-l = Is-equiv.is-linv \u2218 snd\n\n    <\u2013-inv-r : (e : A \u2243 B) (b : B)\n                \u2192 (\u2013> e (<\u2013' e b) \u2261 b)\n    <\u2013-inv-r = Is-equiv.is-rinv \u2218 snd\n\n    -- Equivalences are \"injective\"\n    equiv-inj : (e : A \u2243 B) {x y : A}\n                \u2192 (\u2013> e x \u2261 \u2013> e y \u2192 x \u2261 y)\n    equiv-inj e p = ! <\u2013-inv-l e _ \u2219 ap (<\u2013 e) p \u2219 <\u2013-inv-l e _\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n        prop-has-all-paths : is-prop A \u2192 has-all-paths A\n        prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n        all-paths-is-prop : has-all-paths A \u2192 is-prop A\n        all-paths-is-prop c x y = c x y , canon-path\n          where\n          lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n          lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n          canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n          canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                          (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same idp x = idp\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u2013> e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u2013> e a) (coe-equiv-\u03b2 e)\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u2013>-paths-equiv : (x \u2261 y) \u2261 (\u2013> e x \u2261 \u2013> e y)\n    \u2013>-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Nullary.NP\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J; ap\u2193; PathOver; tr; ap\u2082)\n       renaming (refl to idp; _\u2257_ to _\u223c_; J-orig to J')\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nmodule _ {a} {A : \u2605_ a} where\n  idp_ : (x : A) \u2192 x \u2261 x\n  idp_ _ = idp\n\n  refl-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 idp_ x \u2219 p \u2261 p\n  refl-\u2219 _ = idp\n\n  \u2219-refl : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 idp_ y \u2261 p\n  \u2219-refl = J' (\u03bb (x y : A) (p : x \u2261 y) \u2192 (p \u2219 idp_ y) \u2261 p) (\u03bb x \u2192 idp)\n\n  hom-!-\u2219 : \u2200 {x y z : A} (p : x \u2261 y)(q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p\n  hom-!-\u2219 p q = J' (\u03bb x y p \u2192 \u2200 z \u2192 (q : y \u2261 z) \u2192 !(p \u2219 q) \u2261 ! q \u2219 ! p) (\u03bb x z q \u2192 ! \u2219-refl (! q)) p _ q\n\n  !-inv : \u2200 {x y : A} (p : x \u2261 y) \u2192 ! (! p) \u2261 p\n  !-inv = J' (\u03bb x y p \u2192 ! (! p) \u2261 p) (\u03bb x \u2192 idp)\n\n  !-\u2219 : \u2200 {x y : A} (p : x \u2261 y) \u2192 ! p \u2219 p \u2261 idp_ y\n  !-\u2219 = J' (\u03bb x y p \u2192 (! p \u2219 p) \u2261 idp_ y) (\u03bb x \u2192 idp)\n\n  \u2219-! : \u2200 {x y : A} (p : x \u2261 y) \u2192 p \u2219 ! p \u2261 idp_ x\n  \u2219-! = J' (\u03bb x y p \u2192 (p \u2219 ! p) \u2261 idp_ x) (\u03bb x \u2192 idp)\n\n  !p\u2219p = !-\u2219\n  p\u2219!p = \u2219-!\n\n  \u2219-assoc : \u2200 {x y : A} (p : x \u2261 y) {z : A} (q : y \u2261 z) {t : A} (r : z \u2261 t) \u2192 p \u2219 q \u2219 r \u2261 (p \u2219 q) \u2219 r\n  \u2219-assoc = J' (\u03bb x y p \u2192 \u2200 {z} (q : y \u2261 z) {t} (r : z \u2261 t) \u2192 p \u2219 (q \u2219 r) \u2261 (p \u2219 q) \u2219 r)\n               (\u03bb x q r \u2192 idp)\n\n  ==-refl-\u2219 :  {x y : A} (p : x \u2261 y) {q : x \u2261 x} \u2192 q \u2261 idp_ x \u2192 q \u2219 p \u2261 p\n  ==-refl-\u2219 p = ap (flip _\u2219_ p)\n\n  \u2219-==-refl :  {x y : A} (p : x \u2261 y) {q : y \u2261 y} \u2192 q \u2261 idp_ y \u2192 p \u2219 q \u2261 p\n  \u2219-==-refl p qr = ap (_\u2219_ p) qr \u2219 \u2219-refl p\n\n  \u2219-\u2219-==-refl :  {x y z : A} (p : x \u2261 y) (q : y \u2261 z) {r : z \u2261 z} \u2192 r \u2261 idp_ z \u2192 p \u2219 q \u2219 r \u2261 p \u2219 q\n  \u2219-\u2219-==-refl p q rr = \u2219-assoc p q _ \u2219 \u2219-==-refl (p \u2219 q) rr\n\n  !p\u2219p\u2219q : {x y z : A} (p : x \u2261 y) (q : y \u2261 z) \u2192 ! p \u2219 p \u2219 q \u2261 q\n  !p\u2219p\u2219q p q = \u2219-assoc (! p) p q \u2219 ==-refl-\u2219 q (!-\u2219 p)\n\n  p\u2219!p\u2219q : {x y z : A} (p : y \u2261 x) (q : y \u2261 z) \u2192 p \u2219 ! p \u2219 q \u2261 q\n  p\u2219!p\u2219q p q = \u2219-assoc p _ q \u2219 ==-refl-\u2219 q (\u2219-! p)\n\n  p\u2219!q\u2219q : {x y z : A} (p : x \u2261 y) (q : z \u2261 y) \u2192 p \u2219 ! q \u2219 q \u2261 p\n  p\u2219!q\u2219q p q = \u2219-==-refl p (!-\u2219 q)\n\n  p\u2219q\u2219!q : {x y z : A} (p : x \u2261 y) (q : y \u2261 z) \u2192 p \u2219 q \u2219 ! q \u2261 p\n  p\u2219q\u2219!q p q = \u2219-==-refl p (\u2219-! q)\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\n\n    tr-snd= : \u2200 {p}(P : \u03a3 A B \u2192 \u2605_ p){x}{y\u2080 y\u2081 : B x}(y= : y\u2080 \u2261 y\u2081)\n            \u2192 tr P (snd= {x = x} y=) \u223c tr (P \u2218 _,_ x) y=\n    tr-snd= P idp p = idp\n\nmodule _ {a}{A : \u2605_ a} where\n  tr-r\u2261 : {x y z : A}(p : y \u2261 z)(q : x \u2261 y) \u2192 tr (\u03bb v \u2192 x \u2261 v) p q \u2261 q \u2219 p\n  tr-r\u2261 idp q = ! \u2219-refl q\n\n  tr-l\u2261 : {x y z : A}(p : x \u2261 y)(q : x \u2261 z) \u2192 tr (\u03bb v \u2192 v \u2261 z) p q \u2261 ! p \u2219 q\n  tr-l\u2261 idp q = idp\n\nmodule _ {A : \u2605}(f g : A \u2192 \u2605){x y : A}(p : x \u2261 y)(h : f x \u2192 g x) where\n    tr-\u2192 : tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x)))\n    tr-\u2192 = J' (\u03bb x y p \u2192 (h : f x \u2192 g x) \u2192 tr (\u03bb x \u2192 f x \u2192 g x) p h \u2261 (\u03bb x \u2192 tr g p (h (tr f (! p) x))))\n             (\u03bb _ _ \u2192 idp) p h\n\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a b c}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{x\u2080 : A}{y\u2080 : B x\u2080}{C : \u2605_ c}\n         (f : (x : A) (y : B x) \u2192 C) where\n    ap\u2082\u2193 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : y\u2080 \u2261 y\u2081 [ B \u2193 x= ])\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    ap\u2082\u2193 idp = ap (f x\u2080)\n    {- Or with J\n    ap\u2082\u2193 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : y\u2080 \u2261 y\u2081 [ _ \u2193 x=' ])\n                          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\n    apd\u2082 : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n           {y\u2081 : B x\u2081}(y= : tr B x= y\u2080 \u2261 y\u2081)\n         \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    -- apd\u2082 idp = ap (f x\u2080)\n    -- {- Or with J\n    apd\u2082 x= = J (\u03bb x\u2081' x=' \u2192 {y\u2081 : B x\u2081'}(y= : tr B x=' y\u2080 \u2261 y\u2081) \u2192 f x\u2080 y\u2080 \u2261 f x\u2081' y\u2081)\n                (\u03bb y= \u2192 ap (f x\u2080) y=) x=\n    -- -}\n\nmodule _ {a b c d}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{C : \u2605_ c}{x\u2080 : A}{y\u2080 : B x\u2080 \u2192 C}{D : \u2605_ d}\n         {{_ : FunExt}}\n         (f : (x : A) (y : B x \u2192 C) \u2192 D) where\n    apd\u2082\u207b : {x\u2081 : A}(x= : x\u2080 \u2261 x\u2081)\n            {y\u2081 : B x\u2081 \u2192 C}(y= : \u2200 x \u2192 y\u2080 x \u2261 y\u2081 (tr B x= x))\n          \u2192 f x\u2080 y\u2080 \u2261 f x\u2081 y\u2081\n    apd\u2082\u207b idp y= = ap (f x\u2080) (\u03bb= y=)\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv    : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective p = ! is-linv _ \u2219 ap linv p \u2219 is-linv _\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv    : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n    record Biinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-linv : Linv f\n        has-rinv : Rinv f\n\n      open Linv has-linv public\n      open Rinv has-rinv public\n\n    module _ {f : A \u2192 B}\n             (g : B \u2192 A)(g-f : (x : A) \u2192 g (f x) \u2261 x)\n             (h : B \u2192 A)(f-h : (y : B) \u2192 f (h y) \u2261 y) where\n      biinv : Biinv f\n      biinv = record { has-linv = record { linv = g ; is-linv = g-f }\n                     ; has-rinv = record { rinv = h ; is-rinv = f-h } }\n\n    record Qinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        inv : B \u2192 A\n        inv-is-linv : \u2200 x \u2192 inv (f x) \u2261 x\n        inv-is-rinv : \u2200 x \u2192 f (inv x) \u2261 x\n\n      has-biinv : Biinv f\n      has-biinv = record { has-linv = record { linv = inv\n                                             ; is-linv = inv-is-linv }\n                         ; has-rinv = record { rinv = inv\n                                             ; is-rinv = inv-is-rinv } }\n\n      open Biinv has-biinv public\n\n    postulate\n        HAE : {f : A \u2192 B} \u2192 Qinv f \u2192 \u2605\u2080\n--    HAE {f} f-qinv = {!F.is-linv!}\n--      where module F = Qinv f-qinv\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        has-qinv : Qinv f\n        is-hae   : HAE has-qinv\n      open Qinv has-qinv public\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      qinv : Qinv f\n      qinv = record\n            { inv = g\n            ; inv-is-linv = g-f\n            ; inv-is-rinv = f-g }\n\n    module _ {f : A \u2192 B}(g : B \u2192 A)\n             (f-g : (y : B) \u2192 f (g y) \u2261 y)\n             (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n      postulate\n        g-f' : (x : A) \u2192 g (f x) \u2261 x\n      -- g-f' x = ap g {!f-g ?!} \u2219 {!!}\n        hae : HAE (qinv g f-g g-f')\n      is-equiv : Is-equiv f\n      is-equiv = record\n        { has-qinv = qinv g f-g g-f'\n        ; is-hae   = hae }\n\n  module Biinv-inv {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n                   (f\u1d2e : Biinv f) where\n      open Biinv f\u1d2e\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-biinv : Biinv inv\n      inv-biinv =\n        biinv f (\u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x)\n              f (\u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x)\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}\n           (f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = is-equiv f is-linv is-rinv\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {a}{A : \u2605_ a}\n           (f : A \u2192 A)(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = is-equiv f f-inv f-inv\n\n      self-inv-equiv : A \u2243 A\n      self-inv-equiv = f , self-inv-is-equiv\n\n      self-inv-biinv : Biinv f\n      self-inv-biinv = biinv f f-inv f f-inv\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv _ \u03bb _ \u2192 idp\n\n    id\u1d2e : Biinv {A = A} id\n    id\u1d2e = self-inv-biinv _ \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = is-equiv (F.inv \u2218 G.inv)\n                        (\u03bb x \u2192 ap g (F.inv-is-rinv _) \u2219 G.inv-is-rinv _)\n                        (\u03bb x \u2192 ap F.inv (G.inv-is-linv _) \u2219 F.inv-is-linv _)\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n    _\u2218\u1d2e_ : Biinv g \u2192 Biinv f \u2192 Biinv (g \u2218 f)\n    g\u1d2e \u2218\u1d2e f\u1d2e =\n      biinv (F.linv \u2218 G.linv)\n            (\u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x)\n            (F.rinv \u2218 G.rinv)\n            (\u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x)\n      where\n        module G = Biinv g\u1d2e\n        module F = Biinv f\u1d2e\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    \u2013> : (e : A \u2243 B) \u2192 (A \u2192 B)\n    \u2013> e = fst e\n\n    <\u2013 : (e : A \u2243 B) \u2192 (B \u2192 A)\n    <\u2013 e = Is-equiv.linv (snd e)\n\n    <\u2013' : (e : A \u2243 B) \u2192 (B \u2192 A)\n    <\u2013' e = Is-equiv.rinv (snd e)\n\n    <\u2013-inv-l : (e : A \u2243 B) (a : A)\n              \u2192 (<\u2013 e (\u2013> e a) \u2261 a)\n    <\u2013-inv-l = Is-equiv.is-linv \u2218 snd\n\n    <\u2013-inv-r : (e : A \u2243 B) (b : B)\n                \u2192 (\u2013> e (<\u2013' e b) \u2261 b)\n    <\u2013-inv-r = Is-equiv.is-rinv \u2218 snd\n\n    -- Equivalences are \"injective\"\n    equiv-inj : (e : A \u2243 B) {x y : A}\n                \u2192 (\u2013> e x \u2261 \u2013> e y \u2192 x \u2261 y)\n    equiv-inj e p = ! <\u2013-inv-l e _ \u2219 ap (<\u2013 e) p \u2219 <\u2013-inv-l e _\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}(A : \u2605_ a) where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n  module _ {a}{A : \u2605_ a} where\n    id-path : A \u2192 Paths A\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb y p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = is-equiv fst fst-rinv-id-path (\u03bb x \u2192 idp)\n\n    \u2243-Paths : A \u2243 Paths A\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = is-equiv _ (snd A-contr) (\u03bb _ \u2192 idp)\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , is-equiv (from iso) (Inverse.right-inverse-of iso) (Inverse.left-inverse-of iso)\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\ndata T-level : \u2605\u2080 where\n  \u27e8-2\u27e9 : T-level\n  \u27e8S_\u27e9 : (n : T-level) \u2192 T-level\n\n\u27e8-1\u27e9 \u27e80\u27e9 : T-level\n\u27e8-1\u27e9 = \u27e8S \u27e8-2\u27e9 \u27e9\n\u27e80\u27e9  = \u27e8S \u27e8-1\u27e9 \u27e9\n\u27e81\u27e9  = \u27e8S \u27e80\u27e9  \u27e9\n\u27e82\u27e9  = \u27e8S \u27e81\u27e9  \u27e9\n\n\u2115\u208b\u2082 = T-level\n\nmodule _ {a} where\n    private\n      U = \u2605_ a\n\n    has-level : T-level \u2192 U \u2192 U\n    has-level \u27e8-2\u27e9   A = Is-contr A\n    has-level \u27e8S n \u27e9 A = (x y : A) \u2192 has-level n (x \u2261 y)\n\n    is-prop : U \u2192 U\n    is-prop A = has-level \u27e8-1\u27e9 A\n\n    is-set : U \u2192 U\n    is-set A = has-level \u27e80\u27e9 A\n\n    has-all-paths : U \u2192 U\n    has-all-paths A = (x y : A) \u2192 x \u2261 y\n\n    UIP : U \u2192 U\n    UIP A = {x y : A} (p q : x \u2261 y) \u2192 p \u2261 q\n\n    private\n      UIP-check : {A : U} \u2192 UIP A \u2261 ({x y : A} \u2192 has-all-paths (x \u2261 y))\n      UIP-check = idp\n\n    module _ {A : U} where\n        prop-has-all-paths : is-prop A \u2192 has-all-paths A\n        prop-has-all-paths A-prop x y = fst (A-prop x y)\n\n        all-paths-is-prop : has-all-paths A \u2192 is-prop A\n        all-paths-is-prop c x y = c x y , canon-path\n          where\n          lemma : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p \u2219 c y y\n          lemma = J' (\u03bb x y p \u2192 c x y \u2261 p \u2219 c y y) (\u03bb x \u2192 idp)\n\n          canon-path : {x y : A} (p : x \u2261 y) \u2192 c x y \u2261 p\n          canon-path = J' (\u03bb x y p \u2192 c x y \u2261 p)\n                          (\u03bb x \u2192 lemma (! c x x) \u2219 !-\u2219 (c x x))\n\n\nmodule _ {a} (A : \u2605_ a) where\n    has-dec-eq : \u2605_ a\n    has-dec-eq = (x y : A) \u2192 Dec (x \u2261 y)\n\nmodule _ {a} {A : \u2605_ a} (d : has-dec-eq A) where\n    private\n        Code' : {x y : A} (dxy : Dec (x \u2261 y)) (dxx : Dec (x \u2261 x)) \u2192 x \u2261 y \u2192 \u2605_ a\n        Code' {x} {y} dxy dxx p = case dxy of \u03bb\n          { (no  _) \u2192 Lift \ud835\udfd8\n          ; (yes b) \u2192 case dxx of \u03bb\n                      { (no   _) \u2192 Lift \ud835\udfd8\n                      ; (yes b') \u2192 p \u2261 ! b' \u2219 b\n                      }\n          }\n\n        Code : {x y : A} \u2192 x \u2261 y \u2192 \u2605_ a\n        Code {x} {y} p = Code' (d x y) (d x x) p\n\n        encode : {x y : A} \u2192 (p : x \u2261 y) -> Code p\n        encode {x} = J (\u03bb y (p : x \u2261 y) \u2192 Code p) (elim-Dec (\u03bb d \u2192 Code' d d idp) (!_ \u2218 !p\u2219p) (\u03bb x\u2081 \u2192 lift (x\u2081 idp)) (d x x))\n\n    UIP-dec : UIP A\n    UIP-dec {x} idp q with d x x | encode q\n    UIP-dec     idp q    | yes a | p' = ! !p\u2219p a \u2219 ! p'\n    UIP-dec     idp q    | no  r | _  = \ud835\udfd8-elim (r idp)\n\n    dec-eq-is-set : is-set A\n    dec-eq-is-set _ _ = all-paths-is-prop UIP-dec\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    UIP-set : is-set A \u2192 UIP A\n    UIP-set A-is-set p q = fst (A-is-set _ _ p q)\n\n    UIP\u2192is-set : UIP A \u2192 is-set A\n    UIP\u2192is-set A-is-set' x y = all-paths-is-prop A-is-set'\n\n    \u03a9\u2081-set-to-contr : is-set A \u2192 (x : A) \u2192 Is-contr (x \u2261 x)\n    \u03a9\u2081-set-to-contr A-is-set x = idp , UIP-set A-is-set idp\n\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\n    coe\u2218coe : \u2200 {B C}(p : B \u2261 C)(q : A \u2261 B)(m : A) \u2192 coe p (coe q m) \u2261 coe (q \u2219 p) m\n    coe\u2218coe p idp m = idp\n\n    coe-same : \u2200 {B}{p q : A \u2261 B}(e : p \u2261 q)(x : A) \u2192 coe p x \u2261 coe q x\n    coe-same idp x = idp\n\n    coe-inj : \u2200 {B}{x y : A}(p : A \u2261 B) \u2192 coe p x \u2261 coe p y \u2192 x \u2261 y\n    coe-inj idp = id\n\n    module _ {B : \u2605_ \u2113}(p : A \u2261 B){x y : A} where\n        coe-paths-equiv : (x \u2261 y) \u2261 (coe p x \u2261 coe p y)\n        coe-paths-equiv = J (\u03bb B (p : A \u2261 B) \u2192 (x \u2261 y) \u2261 (coe p x \u2261 coe p y)) idp p\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u2013> e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u2013> e a) (coe-equiv-\u03b2 e)\n\n  module _ (e : A \u2243 B){x y : A} where\n    \u2013>-paths-equiv : (x \u2261 y) \u2261 (\u2013> e x \u2261 \u2013> e y)\n    \u2013>-paths-equiv = coe-paths-equiv (ua e) \u2219 ap\u2082 _\u2261_ (coe-\u03b2 e x) (coe-\u03b2 e y)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cdf12c26d0f2e2933d628446765c91b45d58cfa3","subject":"More fiddling","message":"More fiddling\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\nopen import Data.Nat.Base using (_<_)\n\n-- XXX: This is a lie for now, but all papers agree we'll want this in the end.\n_[_]\u03c4mono_from_to : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 (dv : Val (\u0394t \u03c4)) \u2192\n  \u2200 v1 v2 n' \u2192\n  n' < n \u2192\n  n [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  n' [ \u03c4 ]\u03c4 dv from v1 to v2\n_[_]\u03c4mono_from_to = {!!}\n\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 (suc n) \u2261 Done v1 \u2192 apply f2 a2 (suc n) \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nopen import Relation.Binary hiding (_\u21d2_)\nfoo : \u2200 {\u0393 \u03c3 \u03c4 v1 v2 dv} n (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 a2 : Val \u03c3) \u2192\n  (eqv1 : eval t (a1 \u2022 \u03c11) (suc (suc n)) \u2261 Done v1) \u2192\n  (eqv2 : eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2261 Done v2) \u2192\n  (eqvv : eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) \u2261 Done dv) \u2192\n  suc (suc n) [ \u03c4 ]\u03c4' eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) from\n      eval t (a1 \u2022 \u03c11) (suc (suc n)) to eval t (a2 \u2022 \u03c12) (suc (suc n)) \u2192\n  suc n [ \u03c4 ]\u03c4 dv from v1 to v2\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv dvv with eval t (a1 \u2022 \u03c11) (suc (suc n)) | eval t (a2 \u2022 \u03c12) (suc (suc n)) | eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl refl dvv | Done v1 | (Done v2) | (Done dv) = _[_]\u03c4mono_from_to (suc (suc n)) _ dv v1 v2 (suc n) (DecTotalOrder.reflexive Data.Nat.decTotalOrder refl) dvv\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl refl () dvv | Done v\u2081 | (Done v) | TimeOut\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 refl () eqvv dvv | Done v | TimeOut | s\nfoo n t d\u03c1 \u03c11 \u03c12 da a1 a2 () eqv2 eqvv dvv | TimeOut | r | s\n-- q r s eqv1 eqv2 eqvv\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!!}\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 with\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n)) | inspect (eval (derive t) (da \u2022 a1 \u2022 d\u03c1)) (suc (suc n))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | Done dv | [ eqvv ] = dv , refl , foo n t  d\u03c1 \u03c11 \u03c12 da a1 a2 eqv1 eqv2 eqvv (fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1))\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 | TimeOut | eq = {!!}\n-- {!eval (derive t)!} , {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"abeda4b342c437d506e6e55cef2e20b34d09e7dd","subject":"Convert \u0394Const to new derivation interface.","message":"Convert \u0394Const to new derivation interface.\n\nOld-commit-hash: 7b59661ce496b2c2659e20c6f9c4f528ddbd6102\n","repos":"inc-lc\/ilc-agda","old_file":"Atlas\/Change\/Derive.agda","new_file":"Atlas\/Change\/Derive.agda","new_contents":"module Atlas.Change.Derive where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\nopen import Atlas.Change.Term\n\n\u0394Const : \u2200 {\u0393 \u03a3 \u03c4} \u2192\n  Const \u03a3 \u03c4 \u2192\n  Terms (\u0394Context \u0393) (\u0394Context \u03a3) \u2192\n  Term (\u0394Context \u0393) (\u0394Type \u03c4)\n\n\u0394Const true  \u2205 = false!\n\u0394Const false \u2205 = false!\n\n\u0394Const xor (\u0394x \u2022 x \u2022 \u0394y \u2022 y \u2022 \u2205) = xor! \u0394x \u0394y\n\n\u0394Const empty \u2205 = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\n\u0394Const (update {\u03ba} {\u03b9}) (\u0394k \u2022 k \u2022 \u0394v \u2022 v \u2022 \u0394m \u2022 m \u2022 \u2205) =\n  insert (apply {base \u03ba} \u0394k k) (apply {base \u03b9} \u0394v v) (delete k v \u0394m)\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\n\u0394Const (lookup {\u03ba} {\u03b9}) (\u0394k \u2022 k \u2022 \u0394m \u2022 m \u2022 \u2205) =\n  let\n    k\u2032 = apply {base \u03ba} \u0394k k\n  in\n    (diff (apply {base _} (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n          (lookup! k m))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\n\u0394Const zip (\u0394f \u2022 f \u2022 \u0394m\u2081 \u2022 m\u2081 \u2022 \u0394m\u2082 \u2022 m\u2082 \u2022 \u2205) =\n  let\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\n\u0394Const (fold {\u03ba} {\u03b1} {\u03b2}) (\u0394f\u2032 \u2022 f\u2032 \u2022 \u0394z \u2022 z \u2022 \u0394m \u2022 m \u2022 \u2205) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        f\u2032))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        \u0394f\u2032))\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))\n","old_contents":"module Atlas.Change.Derive where\n\nopen import Atlas.Syntax.Type\nopen import Atlas.Syntax.Term\nopen import Atlas.Change.Type\nopen import Atlas.Change.Term\n\n\u0394Const : \u2200 {\u0393 \u03a3 \u03c4} \u2192 (c : Const \u03a3 \u03c4) \u2192\n  Term \u0393 (internalizeContext (\u0394Context\u2032 \u03a3) (\u0394Type \u03c4))\n\n\u0394Const true  = false!\n\u0394Const false = false!\n\n\u0394Const xor = abs\u2084 (\u03bb x \u0394x y \u0394y \u2192 xor! \u0394x \u0394y)\n\n\u0394Const empty = empty!\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\n\u0394Const update = abs\u2086 (\u03bb k \u0394k v \u0394v m \u0394m \u2192\n  insert (apply \u0394k k) (apply \u0394v v) (delete k v \u0394m))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\n\u0394Const lookup = abs\u2084 (\u03bb k \u0394k m \u0394m \u2192\n  let\n    k\u2032 = apply \u0394k k\n  in\n    (diff (apply {base _} (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n          (lookup! k m)))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\n\u0394Const zip = abs\u2086 (\u03bb f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 \u2192\n  let\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082)\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\n\u0394Const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0f8e84bbfdbbbb46939ebfb26a19342b0164c2a0","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: b3263208f4a36371cda767221bbdfa8a\n\ndarcs-hash:20110930164503-3bd4e-20fed55b995059fa3f2ec5feb994ad7cc26b1339.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/DataPostulate.agda","new_file":"Draft\/DataPostulate.agda","new_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulates the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data.\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model').\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- From data to postulates\n\n-- The predicate: Using the induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- The predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n------------------------------------------------------------------------------\n-- From postulates to data\n\n-- The predicate.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n","old_contents":"------------------------------------------------------------------------------\n-- Data and postulates\n------------------------------------------------------------------------------\n\n-- Are the FOTC natural numbers defined by data and postulate the\n-- same?\n\nmodule DataPostulate where\n\n------------------------------------------------------------------------------\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\n-- The FOTC natural numbers using data\n\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\nindN : (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\nindN P P0 h zN      = P0\nindN P P0 h (sN Nn) = h Nn (indN P P0 h Nn)\n\n-- The FOTC natural numbers using postulates (we choose 'M' by 'Model')\npostulate\n  M    : D \u2192 Set\n  zM   : M zero\n  sM   : \u2200 {n} \u2192 M n \u2192 M (succ n)\n  indM : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {n} \u2192 M n \u2192 P n \u2192 P (succ n)) \u2192\n         \u2200 {n} \u2192 M n \u2192 P n\n\n------------------------------------------------------------------------------\n-- From data to postulates\n\n-- The predicate: Using the induction principle.\nnat-D2P : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P = indN M zM (\u03bb _ Mn \u2192 sM Mn)\n\n-- The predicate: Using pattern matching.\nnat-D2P' : \u2200 {n} \u2192 N n \u2192 M n\nnat-D2P' zN      = zM\nnat-D2P' (sN Nn) = sM (nat-D2P' Nn)\n\n------------------------------------------------------------------------------\n-- From postulate to data\n\n-- The predicate.\nnat-P2D : \u2200 {n} \u2192 M n \u2192 N n\nnat-P2D = indM N zN (\u03bb _ Nn \u2192 sN Nn)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64cd852b2260c1166274fdf0a621510fce2939f3","subject":"ReceiptFreeness: various changes","message":"ReceiptFreeness: various changes\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/ReceiptFreeness.agda","new_file":"Game\/ReceiptFreeness.agda","new_contents":"-- {-# OPTIONS --without-K #-}\n\n\nopen import Function\nopen import Type\nopen import Data.Maybe.NP\nopen import Data.Fin.NP as Fin hiding (_+_; _==_)\nopen import Data.Product renaming (zip to zip-\u00d7)\nopen import Data.One\nopen import Data.Two\nopen import Data.Vec\nopen import Data.List as L\n\nopen import Data.Nat.NP renaming (_==_ to _==\u2115_)\n\n{-\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\n-}\nopen import Relation.Binary.PropositionalEquality\n--open import Control.Strategy renaming (run to runStrategy)\n\n\nmodule Game.ReceiptFreeness\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : (checked? : \ud835\udfda) \u2192 \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (#q : \u2115) (max#q : Fin #q)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText 1\u2082)\n  (forget : CipherText 1\u2082 \u2192 CipherText 0\u2082)\n  (check  : CipherText 0\u2082 \u2192 Maybe (CipherText 1\u2082))\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CipherText 1\u2082 \u2192 Message)\n\nwhere\n\nChecked? = \ud835\udfda\nchecked unchecked : Checked?\nunchecked = 0\u2082\nchecked = 1\u2082\n\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} (Oracle : (q : Q) \u2192 R q) where\n  private\n    M = Strategy Q R\n  run : \u2200 {A} \u2192 M A \u2192 A\n  run (ask q? cont) = run (cont (Oracle q?))\n  run (done x)      = x\n\nState : (S A : \u2605) \u2192 \u2605\nState S A = S \u2192 A \u00d7 S\n\nmodule _ {S Q : \u2605} {R : Q \u2192 \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n  private\n    M = Strategy Q R\n  runS : \u2200 {A} \u2192 M A \u2192 State S A\n  runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n  runS (done x)      s = x , s\n\n  evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n  evalS x s = proj\u2081 (runS x s)\n\n  execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n  execS x s = proj\u2082 (runS x s)\n\nCandidate : \u2605\nCandidate = \ud835\udfda -- as in the paper: \"for simplicity\"\n\nalice bob : Candidate\nalice = 0\u2082\nbob   = 1\u2082\n\n-- candidate order\n-- also known as LHS or Message\n-- represented by who is the first candidate\nCO : \u2605\nCO = \ud835\udfda\n\nalice-then-bob bob-then-alice : CO\nalice-then-bob = alice\nbob-then-alice = bob\n\ndata CO-spec : CO \u2192 Candidate \u2192 Candidate \u2192 \u2605 where\n  alice-then-bob-spec : CO-spec alice-then-bob alice bob\n  bob-then-alice-spec : CO-spec bob-then-alice bob alice\n\nMarkedReceipt : \u2605\nMarkedReceipt = \ud835\udfda\n\nmarked-on-first-cell marked-on-second-cell : MarkedReceipt\nmarked-on-first-cell  = 0\u2082\nmarked-on-second-cell = 1\u2082\n\ndata MarkedReceipt-spec : CO \u2192 MarkedReceipt \u2192 Candidate \u2192 \u2605 where\n  m1 : MarkedReceipt-spec alice-then-bob marked-on-first-cell  alice\n  m2 : MarkedReceipt-spec alice-then-bob marked-on-second-cell bob\n  m3 : MarkedReceipt-spec bob-then-alice marked-on-first-cell  bob\n  m4 : MarkedReceipt-spec bob-then-alice marked-on-second-cell alice\n\ndata MarkedReceipt? : \u2605 where\n  not-marked : MarkedReceipt?\n  marked     : MarkedReceipt \u2192 MarkedReceipt?\n\n-- Receipt or also called RHS\n-- Made of a potential mark, a serial number, and an encrypted candidate order\nReceipt : (checked? : \ud835\udfda) \u2192 \u2605\nReceipt checked? = MarkedReceipt? \u00d7 SerialNumber \u00d7 CipherText checked?\n\nforgetReceipt : Receipt checked \u2192 Receipt unchecked\nforgetReceipt (m? , sn , enc-co) = m? , sn , forget enc-co\n\ncheckReceipt : Receipt unchecked \u2192 Maybe (Receipt checked)\ncheckReceipt (m? , sn , ck-enc-co) = map? (\u03bb x \u2192 m? , sn , x) (check ck-enc-co)\n\nmarkedReceipt? : \u2200 {checked?} \u2192 Receipt checked? \u2192 MarkedReceipt?\nmarkedReceipt? = proj\u2081\n\n-- Marked when there is a 1\nmarked? : MarkedReceipt? \u2192 \ud835\udfda\nmarked? not-marked = 0\u2082\nmarked? (marked _) = 1\u2082\n\nmarked-on-first-cell? : MarkedReceipt? \u2192 \ud835\udfda\nmarked-on-first-cell? not-marked = 0\u2082\nmarked-on-first-cell? (marked x) = x == 0\u2082\n\nmarked-on-second-cell? : MarkedReceipt? \u2192 \ud835\udfda\nmarked-on-second-cell? not-marked = 0\u2082\nmarked-on-second-cell? (marked x) = x == 1\u2082\n\nBallot : (checked? : \ud835\udfda) \u2192 \u2605\nBallot checked? = CO \u00d7 Receipt checked?\n\n-- co or also called LHS\nco : \u2200 {checked?} \u2192 Ballot checked? \u2192 CO\nco = proj\u2081\n\n-- receipt or also called RHS\nreceipt : \u2200 {checked?} \u2192 Ballot checked? \u2192 Receipt checked?\nreceipt = proj\u2082\n\n-- randomness for genBallot\nRgb : \u2605\nRgb = CO \u00d7 SerialNumber \u00d7 R\u2091\n\ngenBallot : PubKey \u2192 Rgb \u2192 Ballot checked\ngenBallot pk (r-co , sn , r\u2091) = r-co , not-marked , sn , Enc pk r-co r\u2091\n\nforgetBallot : Ballot checked \u2192 Ballot unchecked\nforgetBallot (co , m? , sn , enc-co) = co , m? , sn , forget enc-co\n\nmark : CO \u2192 Candidate \u2192 MarkedReceipt\nmark co c = co xor c\n\nmark-ok : \u2200 co c \u2192 MarkedReceipt-spec co (mark co c) c\nmark-ok 1\u2082 1\u2082 = m3\nmark-ok 1\u2082 0\u2082 = m4\nmark-ok 0\u2082 1\u2082 = m2\nmark-ok 0\u2082 0\u2082 = m1\n\nfillBallot : \u2200 {checked?} \u2192 Candidate \u2192 Ballot checked? \u2192 Ballot checked?\nfillBallot c (co , _ , sn , enc-co) = co , marked (mark co c) , sn , enc-co\n\n-- TODO Ballot-spec c (fillBallot b)\n\nBB : \u2605\nBB = List (Receipt checked)\n\nTally : \u2605\nTally = \u2115 \u00d7 \u2115\n\nalice-score : Tally \u2192 \u2115\nalice-score = proj\u2081\n\nbob-score : Tally \u2192 \u2115\nbob-score = proj\u2082\n\n1:alice-0:bob 0:alice-1:bob : Tally\n1:alice-0:bob = 1 , 0\n0:alice-1:bob = 0 , 1\n\ndata TallyMarkedReceipt-spec : CO \u2192 MarkedReceipt \u2192 Tally \u2192 \u2605 where\n  c1 : TallyMarkedReceipt-spec alice-then-bob marked-on-first-cell  1:alice-0:bob\n  c2 : TallyMarkedReceipt-spec alice-then-bob marked-on-second-cell 0:alice-1:bob\n  c3 : TallyMarkedReceipt-spec bob-then-alice marked-on-first-cell  0:alice-1:bob\n  c4 : TallyMarkedReceipt-spec bob-then-alice marked-on-second-cell 1:alice-0:bob\n\nmarked-for-alice? : CO \u2192 MarkedReceipt \u2192 \ud835\udfda\nmarked-for-alice? co m = co == m\n\nmarked-for-bob? : CO \u2192 MarkedReceipt \u2192 \ud835\udfda\nmarked-for-bob? co m = not (marked-for-alice? co m)\n\ntallyMarkedReceipt : CO \u2192 MarkedReceipt \u2192 Tally\ntallyMarkedReceipt co m = \ud835\udfda\u25b9\u2115 for-alice , \ud835\udfda\u25b9\u2115 (not for-alice)\n  where for-alice = marked-for-alice? co m\n\ntallyMarkedReceipt-ok : \u2200 co m \u2192 TallyMarkedReceipt-spec co m (tallyMarkedReceipt co m)\ntallyMarkedReceipt-ok 1\u2082 1\u2082 = c4\ntallyMarkedReceipt-ok 1\u2082 0\u2082 = c3\ntallyMarkedReceipt-ok 0\u2082 1\u2082 = c2\ntallyMarkedReceipt-ok 0\u2082 0\u2082 = c1\n\ntallyMarkedReceipt? : CO \u2192 MarkedReceipt? \u2192 Tally\ntallyMarkedReceipt? co not-marked    = 0 , 0\ntallyMarkedReceipt? co (marked mark) = tallyMarkedReceipt co mark\n\ntallyCheckedReceipt : SecKey \u2192 Receipt checked \u2192 Tally\ntallyCheckedReceipt sk (marked? , _ , enc-co) = tallyMarkedReceipt? (Dec sk enc-co) marked?\n\n-- Not taking advantage of any homomorphic encryption\ntally : SecKey \u2192 BB \u2192 Tally\ntally sk bb = L.foldr (zip-\u00d7 _+_ _+_) (0 , 0) (L.map (tallyCheckedReceipt sk) bb)\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\n\n-- In the paper RBB is the returning the Tally and we\n-- return the BB, here RTally is returning the Tally\ndata Q : \u2605 where\n  REB RBB RTally : Q\n  RCO            : Receipt checked \u2192 Q\n  Vote           : Receipt unchecked \u2192 Q\n\nResp : Q \u2192 \u2605\nResp REB = Ballot unchecked\nResp (RCO x) = CO\nResp (Vote x) = Accept?\nResp RBB = BB\nResp RTally = Tally\n\nPhase : \u2605 \u2192 \u2605\nPhase = Strategy Q Resp\n\n-- How to read types as protocols:\n-- A \u00d7 B   sends A, then behave as B\n-- A \u2192 B   receives A, then behave as B\n\nRFChallenge : \u2605 \u2192 \u2605\nRFChallenge Next = (\ud835\udfda \u2192 Receipt checked) \u2192 Next\n\nAdversary : \u2605\nAdversary = R\u2090 \u2192 PubKey \u2192 Phase -- Phase[I]\n                           (RFChallenge\n                             (Phase -- Phase[II]\n                               \ud835\udfda)) -- Adversary guess of whether the vote is for alice\n\n-- TODO adversary validity\n-- Valid-Adversary : Adversary \u2192 \u2605\n\nmodule Oracle (sk : SecKey) (pk : PubKey) (rgb : Rgb) (bb : BB) where\n    resp : (q : Q) \u2192 Resp q\n    resp REB = forgetBallot (genBallot pk rgb)\n    resp RBB = bb\n    resp RTally = tally sk bb\n    resp (RCO (_ , _ , receipt)) = Dec sk receipt\n    -- do we check if the sn is already here?\n    resp (Vote (m? , sn , receipt)) = case check receipt of \u03bb { (just x) \u2192 accept ; nothing \u2192 reject }\n\n    newBB : Q \u2192 BB\n    newBB (Vote (m? , sn , receipt)) = case check receipt of \u03bb { (just x) \u2192 (m? , sn , x) \u2237 bb ; nothing \u2192 bb }\n    newBB _ = bb\n\nPhaseNumber = \ud835\udfda\nmodule EXP (b : \ud835\udfda) (A : Adversary) (pk : PubKey) (sk : SecKey)\n           (r\u2090 : R\u2090) (rgb : Candidate \u2192 Rgb)\n           (v : PhaseNumber \u2192 Vec Rgb #q) where\n  BBsetup : BB\n  BBsetup = []\n\n  Aphase[I] : Phase _\n  Aphase[I] = A r\u2090 pk\n\n  S = BB \u00d7 Fin #q\n\n  -- When the adversary runs out of allowed queries (namely\n  -- when i becomes zero), then all successive queries will\n  -- be deterministic. The only case asking for randomness\n  -- is ballot creation.\n  OracleS : (phase# : \ud835\udfda) (q : Q) \u2192 State S (Resp q)\n  OracleS phase# q (bb , i) = O.resp q , O.newBB q , Fin.pred i\n    where module O = Oracle sk pk (lookup i (v phase#)) bb\n\n  phase[I] = runS (OracleS 0\u2082) Aphase[I] (BBsetup , max#q)\n\n  AdversaryRFChallenge : RFChallenge _\n  AdversaryRFChallenge = proj\u2081 phase[I]\n\n  BBphase[I] : BB\n  BBphase[I] = proj\u2081 (proj\u2082 phase[I])\n\n  ballots : Candidate \u2192 Ballot checked\n  ballots c = fillBallot c (genBallot pk (rgb c))\n\n  ballot-for-alice = ballots alice\n  ballot-for-bob   = ballots bob\n\n  randomly-swapped-ballots : Candidate \u2192 Ballot checked\n  randomly-swapped-ballots = ballots \u2218 _xor_ b\n\n  randomly-swapped-receipts : Candidate \u2192 Receipt checked\n  randomly-swapped-receipts = receipt \u2218 randomly-swapped-ballots\n\n  BBrfc : BB\n  BBrfc = randomly-swapped-receipts 0\u2082 \u2237 randomly-swapped-receipts 1\u2082 \u2237 BBphase[I]\n\n  Aphase[II] : Phase _\n  Aphase[II] = AdversaryRFChallenge randomly-swapped-receipts\n\n  phase[II] = runS (OracleS 1\u2082) Aphase[II] (BBrfc , max#q)\n\n  -- adversary guess\n  b\u2032 = proj\u2081 phase[II]\n\n_\u00b2 : \u2605 \u2192 \u2605\nA \u00b2 = A \u00d7 A\n\nR : \u2605\nR = R\u2096 \u00d7 R\u2090 \u00d7 \ud835\udfda \u00d7 (Vec Rgb (1{-for the challenge-} + #q))\u00b2\n\ngame : Adversary \u2192 R \u2192 \ud835\udfda\ngame A (r\u2096 , r\u2090 , b , (rgb\u2080 \u2237 rgbs\u2080) , (rgb\u2081 \u2237 rgbs\u2081)) =\n  case KeyGen r\u2096 of \u03bb\n  { (pk , sk) \u2192\n    b == EXP.b\u2032 b A pk sk r\u2090 [0: rgb\u2080 1: rgb\u2081 ] [0: rgbs\u2080 1: rgbs\u2081 ]\n  }\n\n-- Winning condition\nWin : Adversary \u2192 R \u2192 \u2605\nWin A r = game A r \u2261 1\u2082\n\n{- Not all adversaries of the Adversary type are valid.\n\n   First, we do not forbid the challenge in the 2nd step of the Oracle.\n   Second, no complexity analysis is done.\n-}\n\nmodule Cheating1 where\n    cheatingA : Adversary\n    cheatingA r\u2090 pk = done (\u03bb m \u2192 ask (RCO (m 1\u2082)) (\u03bb co \u2192 done (co == (marked-on-second-cell? (markedReceipt? (m 1\u2082))))))\n\n    module _\n     (DecEnc : \u2200 r\u2096 r\u2091 m \u2192 let (pk , sk) = KeyGen r\u2096 in\n                           Dec sk (Enc pk m r\u2091) \u2261 m) where\n\n        cheatingA-wins : \u2200 r \u2192 game cheatingA r \u2261 1\u2082\n        cheatingA-wins (r\u2096 , _ , 0\u2082 , ((co\u2080 , _ , r\u2091\u2080) \u2237 _) , ((co\u2081 , _ , r\u2091\u2081) \u2237 _))\n          rewrite DecEnc r\u2096 r\u2091\u2081 co\u2081 with co\u2081\n        ... | 0\u2082 = refl\n        ... | 1\u2082 = refl\n        cheatingA-wins (r\u2096 , _ , 1\u2082 , ((co\u2080 , _ , r\u2091\u2080) \u2237 _) , ((_ , _ , r\u2091\u2081) \u2237 _))\n          rewrite DecEnc r\u2096 r\u2091\u2080 co\u2080 with co\u2080\n        ... | 0\u2082 = refl\n        ... | 1\u2082 = refl\n\nmodule Cheating2 where\n    cheatingA : Adversary\n    cheatingA r\u2090 pk = done \u03bb m \u2192 ask (Vote (forgetReceipt (m 1\u2082)))\n                                     (\u03bb { accept \u2192 ask RTally (\u03bb { (x , y) \u2192\n                                     done (x ==\u2115 2) }) ; reject \u2192 done 1\u2082 })\n\n    module _\n     (DecEnc : \u2200 r\u2096 r\u2091 m \u2192 let (pk , sk) = KeyGen r\u2096 in\n                           Dec sk (Enc pk m r\u2091) \u2261 m)\n     (check-forget : \u2200 x \u2192 check (forget x) \u2261 just x)\n                       where\n\n        cheatingA-wins : \u2200 r \u2192 game cheatingA r \u2261 1\u2082\n        cheatingA-wins (r\u2096 , _ , 0\u2082 , ((co\u2080 , _ , r\u2091\u2080) \u2237 _) , ((co\u2081 , _ , r\u2091\u2081) \u2237 _))\n           rewrite check-forget (Enc (proj\u2081 (KeyGen r\u2096)) co\u2081 r\u2091\u2081)\n                 | DecEnc r\u2096 r\u2091\u2080 co\u2080\n                 | DecEnc r\u2096 r\u2091\u2081 co\u2081 with co\u2080 | co\u2081\n        ... | 0\u2082 | 0\u2082 = refl\n        ... | 0\u2082 | 1\u2082 = refl\n        ... | 1\u2082 | 0\u2082 = refl\n        ... | 1\u2082 | 1\u2082 = refl\n        cheatingA-wins (r\u2096 , _ , 1\u2082 , ((co\u2080 , _ , r\u2091\u2080) \u2237 _) , ((co\u2081 , _ , r\u2091\u2081) \u2237 _)) -- = {!!}\n           rewrite check-forget (Enc (proj\u2081 (KeyGen r\u2096)) co\u2080 r\u2091\u2080)\n                 | DecEnc r\u2096 r\u2091\u2080 co\u2080\n                 | DecEnc r\u2096 r\u2091\u2081 co\u2081 with co\u2080 | co\u2081\n        ... | 0\u2082 | 0\u2082 = refl\n        ... | 0\u2082 | 1\u2082 = refl\n        ... | 1\u2082 | 0\u2082 = refl\n        ... | 1\u2082 | 1\u2082 = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\n\n\nopen import Function\nopen import Type\nopen import Data.Maybe\nopen import Data.Fin.NP as Fin hiding (_+_; _==_)\nopen import Data.Product renaming (zip to zip-\u00d7)\nopen import Data.One\nopen import Data.Two\nopen import Data.Vec\nopen import Data.List as L\n\nopen import Data.Nat.NP renaming (_==_ to _==\u2115_)\n\n{-\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\n-}\nopen import Relation.Binary.PropositionalEquality\n--open import Control.Strategy renaming (run to runStrategy)\n\n\nmodule Game.ReceiptFreeness\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n  (CheckedCipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (#q : \u2115) (max#q : Fin #q)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CheckedCipherText)\n  (forget : CheckedCipherText \u2192 CipherText)\n  (check  : CipherText \u2192 Maybe CheckedCipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CheckedCipherText \u2192 Message)\n\nwhere\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} (Oracle : (q : Q) \u2192 R q) where\n  private\n    M = Strategy Q R\n  run : \u2200 {A} \u2192 M A \u2192 A\n  run (ask q? cont) = run (cont (Oracle q?))\n  run (done x)      = x\n\nState : (S A : \u2605) \u2192 \u2605\nState S A = S \u2192 A \u00d7 S\n\nmodule _ {S Q : \u2605} {R : Q \u2192 \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n  private\n    M = Strategy Q R\n  runS : \u2200 {A} \u2192 M A \u2192 State S A\n  runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n  runS (done x)      s = x , s\n\n  evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n  evalS x s = proj\u2081 (runS x s)\n\n  execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n  execS x s = proj\u2082 (runS x s)\n\nCandidate : \u2605\nCandidate = \ud835\udfda -- as in the paper: \"for simplicity\"\n\nalice bob : Candidate\nalice = 0\u2082\nbob   = 1\u2082\n\n-- candidate order\n-- also known as LHS or Message\n-- represented by who is the first candidate\nCO : \u2605\nCO = \ud835\udfda\n\nalice-then-bob bob-then-alice : CO\nalice-then-bob = 0\u2082\nbob-then-alice = 1\u2082\n\ndata CO-spec : CO \u2192 Candidate \u2192 Candidate \u2192 \u2605 where\n  alice-then-bob-spec : CO-spec alice-then-bob alice bob\n  bob-then-alice-spec : CO-spec bob-then-alice bob alice\n\nMarkedRHS : \u2605\nMarkedRHS = \ud835\udfda\n\nmarked-on-first-cell marked-on-second-cell : MarkedRHS\nmarked-on-first-cell  = 0\u2082\nmarked-on-second-cell = 1\u2082\n\ndata MarkedRHS-spec : MarkedRHS \u2192 \u2605 where\n  marked-on-first-cell-spec  : MarkedRHS-spec marked-on-first-cell\n  marked-on-second-cell-spec : MarkedRHS-spec marked-on-second-cell\n\n-- CipherText is the encrypted candidate order\nReceipt : \u2605\nReceipt = SerialNumber \u00d7 CipherText\n\nCheckedReceipt : \u2605\nCheckedReceipt = SerialNumber \u00d7 CheckedCipherText\n\nforgetReceipt : CheckedReceipt \u2192 Receipt\nforgetReceipt (sn , encCO) = sn , forget encCO\n\ndata MarkedRHS? : \u2605 where\n  not-marked : MarkedRHS?\n  marked     : MarkedRHS \u2192 MarkedRHS?\n  \nRHS : \u2605\nRHS = Receipt \u00d7 MarkedRHS?\n\nCheckedRHS : \u2605\nCheckedRHS = CheckedReceipt \u00d7 MarkedRHS?\n\nreceipt : RHS \u2192 Receipt\nreceipt = proj\u2081\n\nmarkedRHS? : RHS \u2192 MarkedRHS?\nmarkedRHS? = proj\u2082\n\n-- Marked when there is a 1\nmarked? : MarkedRHS? \u2192 \ud835\udfda\nmarked? not-marked = 0\u2082\nmarked? (marked _) = 1\u2082\n\nmarked-on-first-cell? : MarkedRHS? \u2192 \ud835\udfda\nmarked-on-first-cell? not-marked = 0\u2082\nmarked-on-first-cell? (marked x) = x == 0\u2082\n\nmarked-on-second-cell? : MarkedRHS? \u2192 \ud835\udfda\nmarked-on-second-cell? not-marked = 0\u2082\nmarked-on-second-cell? (marked x) = x == 1\u2082\n\nBallot : \u2605\nBallot = CO \u00d7 RHS\n\n-- unused\nCheckedBallot : \u2605\nCheckedBallot = CO \u00d7 CheckedRHS\n\n-- randomness for genBallot\nRgb : \u2605\nRgb = CO \u00d7 SerialNumber \u00d7 R\u2091\n\ngenBallot : PubKey \u2192 Rgb \u2192 Ballot\ngenBallot pk (rCO , sn , r\u2091) = rCO , (sn , forget (Enc pk rCO r\u2091)) , not-marked\n\nlhs : Ballot \u2192 CO\nlhs = proj\u2081\n\nrhs : Ballot \u2192 RHS\nrhs = proj\u2082\n\nBB : \u2605\nBB = List CheckedRHS\n\nTally : \u2605\nTally = \u2115 \u00d7 \u2115\n\nalice-score : Tally \u2192 \u2115\nalice-score = proj\u2081\n\nbob-score : Tally \u2192 \u2115\nbob-score = proj\u2082\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\n\n-- In the paper RBB is the returning the Tally and we\n-- return the BB, here RTally is returning the Tally\ndata Q : \u2605 where\n  REB RBB RTally : Q\n  RCO            : CheckedRHS \u2192 Q\n  Vote           : RHS \u2192 Q\n\nResp : Q \u2192 \u2605\nResp REB = Ballot\nResp (RCO x) = CO\nResp (Vote x) = Accept?\nResp RBB = BB\nResp RTally = Tally\n\nPhase : \u2605 \u2192 \u2605\nPhase = Strategy Q Resp\n\n-- How to read types as protocols:\n-- A \u00d7 B   sends A, then behave as B\n-- A \u2192 B   receives A, then behave as B\n\nRFChallenge : \u2605 \u2192 \u2605\nRFChallenge Next = (\ud835\udfda \u2192 CheckedReceipt) \u2192 Next\n\nAdversary : \u2605\nAdversary = R\u2090 \u2192 PubKey \u2192 Phase -- Phase[I]\n                           (RFChallenge\n                             (Phase -- Phase[II]\n                               \ud835\udfda)) -- Adversary guess of whether the vote is for alice\n\n-- TODO adversary validity\n\n1:alice-0:bob 0:alice-1:bob : Tally\n1:alice-0:bob = 1 , 0\n0:alice-1:bob = 0 , 1\n\ndata TallyMarkedRHS-spec : CO \u2192 MarkedRHS \u2192 Tally \u2192 \u2605 where\n  c1 : TallyMarkedRHS-spec alice-then-bob marked-on-first-cell  1:alice-0:bob\n  c2 : TallyMarkedRHS-spec alice-then-bob marked-on-second-cell 0:alice-1:bob\n  c3 : TallyMarkedRHS-spec bob-then-alice marked-on-first-cell  0:alice-1:bob\n  c4 : TallyMarkedRHS-spec bob-then-alice marked-on-second-cell 1:alice-0:bob\n\ntallyMarkedRHS : CO \u2192 MarkedRHS \u2192 Tally\ntallyMarkedRHS co m = \ud835\udfda\u25b9\u2115 for-alice , \ud835\udfda\u25b9\u2115 (not for-alice)\n  where for-alice = co == m\n\ntallyMarkedRHS-ok : \u2200 co m \u2192 TallyMarkedRHS-spec co m (tallyMarkedRHS co m)\ntallyMarkedRHS-ok 1\u2082 1\u2082 = c4\ntallyMarkedRHS-ok 1\u2082 0\u2082 = c3\ntallyMarkedRHS-ok 0\u2082 1\u2082 = c2\ntallyMarkedRHS-ok 0\u2082 0\u2082 = c1\n\ntallyCheckedRHS : SecKey \u2192 CheckedRHS \u2192 Tally\ntallyCheckedRHS sk (_ , not-marked) = 0 , 0\ntallyCheckedRHS sk ((_ , encCO) , marked mark) = tallyMarkedRHS (Dec sk encCO) mark\n\n-- Not taking advantage of any homomorphic encryption\ntally : SecKey \u2192 BB \u2192 Tally\ntally sk bb = L.foldr (zip-\u00d7 _+_ _+_) (0 , 0) (L.map (tallyCheckedRHS sk) bb)\n\nmodule _ (sk : SecKey) (pk : PubKey) (rgb : Rgb) (bb : BB) where\n    Oracle : (q : Q) \u2192 Resp q\n    Oracle REB = genBallot pk rgb\n    Oracle RBB = bb\n    Oracle RTally = tally sk bb\n    Oracle (RCO ((_ , receipt) , _)) = Dec sk receipt\n    -- do we check if the sn is already here?\n    Oracle (Vote ((sn , receipt) , m?)) = case check receipt of \u03bb { (just x) \u2192 accept ; nothing \u2192 reject }\n\n    OracleBB : Q \u2192 BB\n    OracleBB REB = bb\n    OracleBB RBB = bb\n    OracleBB RTally = bb\n    OracleBB (RCO _) = bb\n    OracleBB (Vote ((sn , receipt) , m?)) = case check receipt of \u03bb { (just x) \u2192 ((sn , x) , m?) \u2237 bb ; nothing \u2192 bb }\n\nmodule EXP (b : \ud835\udfda) (A : Adversary) (pk : PubKey) (sk : SecKey)\n           (r\u2090 : R\u2090) (cr\u2091 : \ud835\udfda \u2192 R\u2091) (csn : \ud835\udfda \u2192 SerialNumber)\n           (v : \ud835\udfda \u2192 Vec Rgb #q) where\n  BBsetup : BB\n  BBsetup = []\n\n  Aphase[I] : Phase _\n  Aphase[I] = A r\u2090 pk\n\n  S = BB \u00d7 Fin #q\n\n  OracleS : (phase# : \ud835\udfda) (q : Q) \u2192 State S (Resp q)\n  OracleS phase# q (bb , i) = Oracle sk pk rgb bb q , OracleBB sk pk rgb bb q , Fin.pred i\n    where rgb = lookup i (v phase#)\n\n  phase[I] = runS (OracleS 0\u2082) Aphase[I] (BBsetup , max#q)\n\n  ARFChallenge : RFChallenge _\n  ARFChallenge = proj\u2081 phase[I]\n\n  BBphase[I] : BB\n  BBphase[I] = proj\u2081 (proj\u2082 phase[I])\n\n  Aphase[II] : Phase _\n  Aphase[II] = ARFChallenge (\u03bb i \u2192 csn i , Enc pk (i == b) (cr\u2091 i))\n\n  phase[II] = runS (OracleS 1\u2082) Aphase[II] (BBphase[I] , max#q)\n\n  -- adversary guess\n  b\u2032 = proj\u2081 phase[II]\n\n_\u00b2 : \u2605 \u2192 \u2605\nA \u00b2 = A \u00d7 A\n\nR : \u2605\nR = R\u2096 \u00d7 R\u2090 \u00d7 \ud835\udfda \u00d7 (R\u2091 \u00d7 SerialNumber \u00d7 Vec Rgb #q)\u00b2\n\ngame : Adversary \u2192 R \u2192 \ud835\udfda\ngame A (r\u2096 , r\u2090 , b , (r\u2091\u2080 , sn\u2080 , rgbs\u2080) , (r\u2091\u2081 , sn\u2081 , rgbs\u2081)) =\n  case KeyGen r\u2096 of \u03bb\n  { (pk , sk) \u2192\n    b == EXP.b\u2032 b A pk sk r\u2090 [0: r\u2091\u2080 1: r\u2091\u2081 ] [0: sn\u2080 1: sn\u2081 ] [0: rgbs\u2080 1: rgbs\u2081 ]\n  }\n\n-- Winning condition\nWin : Adversary \u2192 R \u2192 \u2605\nWin A r = game A r \u2261 1\u2082\n\n{- Not all adversaries of the Adversary type are valid.\n\n   First, we do not forbid the challenge in the 2nd step of the Oracle.\n   Second, no complexity analysis is done.\n-}\n\nmodule Cheating1 where\n    cheatingA : Adversary\n    cheatingA r\u2090 pk = done (\u03bb m \u2192 ask (RCO (m 1\u2082 , marked 1\u2082)) done)\n\n    module _\n     (DecEnc : \u2200 r\u2096 r\u2091 m \u2192 let (pk , sk) = KeyGen r\u2096 in\n                           Dec sk (Enc pk m r\u2091) \u2261 m) where\n\n        cheatingA-wins : \u2200 r \u2192 game cheatingA r \u2261 1\u2082\n        cheatingA-wins (r\u2096 , _ , b , (r\u2091\u2080 , _) , (r\u2091\u2081 , _))\n          rewrite DecEnc r\u2096 r\u2091\u2080 b | DecEnc r\u2096 r\u2091\u2081 b = \u2713\u2192\u2261 (==-refl {b})\n\nmodule Cheating2 where\n    cheatingA : Adversary\n    cheatingA r\u2090 pk = done \u03bb m \u2192 ask (Vote (forgetReceipt (m 1\u2082) , marked 1\u2082))\n                                     (\u03bb { accept \u2192 ask RTally (\u03bb { (x , y) \u2192\n                                     done (x ==\u2115 1) }) ; reject \u2192 done 1\u2082 })\n\n    module _\n     (DecEnc : \u2200 r\u2096 r\u2091 m \u2192 let (pk , sk) = KeyGen r\u2096 in\n                           Dec sk (Enc pk m r\u2091) \u2261 m)\n     (check-forget : \u2200 x \u2192 check (forget x) \u2261 just x)\n                       where\n\n        cheatingA-wins : \u2200 r \u2192 game cheatingA r \u2261 1\u2082\n        cheatingA-wins (r\u2096 , r\u2090 , b , (r\u2091\u2080 , proj\u2082) , r\u2091\u2081 , proj\u2083)\n           rewrite check-forget (Enc (proj\u2081 (KeyGen r\u2096)) b r\u2091\u2081)\n                 | DecEnc r\u2096 r\u2091\u2081 b with b\n        ... | 0\u2082 = {!!}\n        ... | 1\u2082 = {!!}\n                 {-\n        cheatingA-wins (r\u2096 , r\u2090 , 1\u2082 , (r\u2091\u2080 , proj\u2082) , r\u2091\u2081 , proj\u2083)\n           rewrite check-forget (Enc (proj\u2081 (KeyGen r\u2096)) 1\u2082 r\u2091\u2081)\n                 | DecEnc r\u2096 r\u2091\u2081 1\u2082 = refl\n        cheatingA-wins (r\u2096 , r\u2090 , 0\u2082 , (r\u2091\u2080 , proj\u2082) , r\u2091\u2081 , proj\u2083)\n           rewrite check-forget (Enc (proj\u2081 (KeyGen r\u2096)) 0\u2082 r\u2091\u2081)\n                 | DecEnc r\u2096 r\u2091\u2081 0\u2082 = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ba6e90556968a93a02eb4217748c057be3436e9e","subject":"Organize imports.","message":"Organize imports.\n\nOld-commit-hash: 3a650cd7e771b0a7c77cdd494a648adcc4b57fa5\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Equivalence.agda","new_file":"Denotational\/Equivalence.agda","new_contents":"module Denotational.Equivalence where\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext-t :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open import Relation.Binary.EqReasoning (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n","old_contents":"module Denotational.Equivalence where\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Relation.Binary.PropositionalEquality as P\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext-t :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"26d77add3aa27ceac665cf1dbd492ffbfae04aa2","subject":"Removed non FOL schemata.","message":"Removed non FOL schemata.\n\nIgnore-this: c3dec63812f856a7e0919b463a09088\n\ndarcs-hash:20120429184726-3bd4e-4d1aaa5d3be0a981adafeb32fdfa611f65d1b48e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/SchemataATP.agda","new_file":"src\/FOL\/SchemataATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Examples of translation of logical schemata\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOL.SchemataATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n\npostulate _\u2261_ : D \u2192 D \u2192 Set\n\nmodule NonSchemas where\n\n  postulate\n    A B C    : Set\n    A\u00b9       : D \u2192 Set\n    A\u00b2 B\u00b2    : D \u2192 D \u2192 Set\n    A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set\n\n  postulate id : A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : A \u2228 B \u2192 B \u2228 A\n  {-# ATP prove \u2228-comm #-}\n\n  postulate \u2228-comm\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate\n    \u2227\u2228-dist\u00b3 : \u2200 {x y z} \u2192\n               (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z )) \u2194\n               (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule Schemas where\n\n  postulate id : {A : Set} \u2192 A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : {A\u00b9 : D \u2192 Set} \u2192 \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : {A\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 A \u2228 B\n  {-# ATP prove \u2228-comm #-}\n\n  postulate\n    \u2228-comm\u00b2 : {A\u00b2 B\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192\n              A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : {A B C : Set} \u2192 A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate\n    \u2227\u2228-dist\u00b3 : {A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set} \u2192 \u2200 {x y z} \u2192\n               (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z)) \u2194\n               (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule SchemaInstances where\n\n  -- A schema\n  -- Current translation: \u2200 p q x. app(p,x) \u2192 app(q,x).\n  postulate schema : (A B : D \u2192 Set) \u2192 \u2200 {x} \u2192 A x \u2192 B x\n\n  -- Using the current translation, the ATPs can prove an instance of\n  -- the schema.\n  postulate\n    d         : D\n    A B       : D \u2192 Set\n    instanceC : A d \u2192 B d\n  {-# ATP prove instanceC schema #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Examples of translation of logical schemata\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOL.SchemataATP where\n\nopen import FOL.Base\n\n------------------------------------------------------------------------------\n\npostulate _\u2261_ : D \u2192 D \u2192 Set\n\nmodule NonSchemas where\n\n  postulate\n    A B C    : Set\n    A\u00b9       : D \u2192 Set\n    A\u00b2 B\u00b2    : D \u2192 D \u2192 Set\n    A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set\n    f\u00b9       : D \u2192 D\n    f\u00b2       : D \u2192 D \u2192 D\n    f\u00b3       : D \u2192 D \u2192 D \u2192 D\n\n  postulate f\u00b9-refl : \u2200 x \u2192 f\u00b9 x \u2261 f\u00b9 x\n  {-# ATP prove f\u00b9-refl #-}\n\n  postulate f\u00b2-refl : \u2200 x y \u2192 f\u00b2 x y \u2261 f\u00b2 x y\n  {-# ATP prove f\u00b2-refl #-}\n\n  postulate f\u00b3-refl : \u2200 x y z  \u2192 f\u00b3 x y z \u2261 f\u00b3 x y z\n  {-# ATP prove f\u00b3-refl #-}\n\n  postulate id : A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : A \u2228 B \u2192 B \u2228 A\n  {-# ATP prove \u2228-comm #-}\n\n  postulate \u2228-comm\u00b2 : \u2200 {x y} \u2192 A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate\n    \u2227\u2228-dist\u00b3 : \u2200 {x y z} \u2192\n               (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z )) \u2194\n               (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule Schemas where\n\n  postulate f\u00b9-refl : (f\u00b9 : D \u2192 D) \u2192 \u2200 x \u2192 f\u00b9 x \u2261 f\u00b9 x\n  {-# ATP prove f\u00b9-refl #-}\n\n  postulate f\u00b2-refl : (f\u00b2 : D \u2192 D \u2192 D) \u2192 \u2200 x y \u2192 f\u00b2 x y \u2261 f\u00b2 x y\n  {-# ATP prove f\u00b2-refl #-}\n\n  postulate f\u00b3-refl : (f\u00b3 : D \u2192 D \u2192 D \u2192 D) \u2192 \u2200 x y z \u2192 f\u00b3 x y z \u2261 f\u00b3 x y z\n  {-# ATP prove f\u00b3-refl #-}\n\n  postulate id : {A : Set} \u2192 A \u2192 A\n  {-# ATP prove id #-}\n\n  postulate id\u00b9 : {A\u00b9 : D \u2192 Set} \u2192 \u2200 {x} \u2192 A\u00b9 x \u2192 A\u00b9 x\n  {-# ATP prove id\u00b9 #-}\n\n  postulate id\u00b2 : {A\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192 A\u00b2 x y \u2192 A\u00b2 x y\n  {-# ATP prove id\u00b2 #-}\n\n  postulate \u2228-comm : {A B : Set} \u2192 A \u2228 B \u2192 A \u2228 B\n  {-# ATP prove \u2228-comm #-}\n\n  postulate\n    \u2228-comm\u00b2 : {A\u00b2 B\u00b2 : D \u2192 D \u2192 Set} \u2192 \u2200 {x y} \u2192\n              A\u00b2 x y \u2228 B\u00b2 x y \u2192 B\u00b2 x y \u2228 A\u00b2 x y\n  {-# ATP prove \u2228-comm\u00b2 #-}\n\n  postulate \u2227\u2228-dist : {A B C : Set} \u2192 A \u2227 (B \u2228 C) \u2194 A \u2227 B \u2228 A \u2227 C\n  {-# ATP prove \u2227\u2228-dist #-}\n\n  postulate\n    \u2227\u2228-dist\u00b3 : {A\u00b3 B\u00b3 C\u00b3 : D \u2192 D \u2192 D \u2192 Set} \u2192 \u2200 {x y z} \u2192\n               (A\u00b3 x y z \u2227 (B\u00b3 x y z \u2228 C\u00b3 x y z)) \u2194\n               (A\u00b3 x y z \u2227 B\u00b3 x y z \u2228 A\u00b3 x y z \u2227 C\u00b3 x y z)\n  {-# ATP prove \u2227\u2228-dist\u00b3 #-}\n\nmodule SchemaInstances where\n\n  -- A schema\n  -- Current translation: \u2200 p q x. app(p,x) \u2192 app(q,x).\n  postulate schema : (A B : D \u2192 Set) \u2192 \u2200 {x} \u2192 A x \u2192 B x\n\n  -- Using the current translation, the ATPs can prove an instance of\n  -- the schema.\n  postulate\n    d         : D\n    A B       : D \u2192 Set\n    instanceC : A d \u2192 B d\n  {-# ATP prove instanceC schema #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6d4ce38ab949f2a79c7caa46d5283b91888d5b61","subject":"FunUniverse.Fin: add finCat","message":"FunUniverse.Fin: add finCat\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Fin.agda","new_file":"FunUniverse\/Fin.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.Fin where\n\nopen import Type\nopen import Function\nopen import Data.Nat using (\u2115)\nopen import Data.Fin using (Fin)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Core\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 \u2605\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nfinFunU : FunUniverse \u2115\nfinFunU = Fin-U , _\u2192\u1da0_\n\nmodule FinFunUniverse = FunUniverse finFunU\n\nfinCat : Category _\u2192\u1da0_\nfinCat = id , _\u2218\u2032_\n\n{-\nfinLin : LinRewiring finFunU\nfinLin = mk finCat {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!}\n-}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.Fin where\n\nopen import Type\nopen import Data.Nat using (\u2115)\nopen import Data.Fin using (Fin)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Core\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 \u2605\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nfinFunU : FunUniverse \u2115\nfinFunU = Fin-U , _\u2192\u1da0_\n\nmodule FinFunUniverse = FunUniverse finFunU\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f53c070dcba6e92a79746f5762bb453a510cf059","subject":"agda: comment fix","message":"agda: comment fix\n\nOld-commit-hash: d5ff611d7aaf67a4b7821fedb255e43d07c53504\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\n-- The formalization is split across the following files.\n-- TODO: document them more.\n\n-- Note that we use two postulates currently:\n\n-- 1. function extensionality (in\n--    `Denotational.ExtensionalityPostulate`), known to be consistent\n--    with intensional type theory.\n--\n-- 2. Temporarily, we use `diff-apply` as a postulate, which is only\n--    true in a slightly weaker form, `diff-apply-proof` - we are\n--    adapting the proofs to use the latter.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.ExtensionalityPostulate\nopen import Denotational.EqualityLemmas\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- This finishes the correctness proof of the above work.\nopen import total\n\n\n-- EXAMPLES\n\nopen import Examples.Examples1\n\n\n-- NATURAL semantics\n\nopen import Syntactic.Closures\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\nopen import Natural.Soundness\n\n\n-- Other calculi\n\nopen import partial\nopen import lambda\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\n-- The formalization is split across the following files.\n\n-- Note that we use two postulates currently:\n\n-- 1. function extensionality (in\n--    `Denotational.ExtensionalityPostulate`), known to be consistent\n--    with intensional type theory.\n--\n-- 2. Temporarily, we use `diff-apply` as a postulate, which is only\n--    true in a slightly weaker form, `diff-apply-proof` - we are\n--    adapting the proofs to use the latter.\n\n-- TODO: document them more.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.ExtensionalityPostulate\nopen import Denotational.EqualityLemmas\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- This finishes the correctness proof of the above work.\nopen import total\n\n\n-- EXAMPLES\n\nopen import Examples.Examples1\n\n\n-- NATURAL semantics\n\nopen import Syntactic.Closures\nopen import Denotational.Closures\n\nopen import Natural.Lookup\nopen import Natural.Evaluation\n\nopen import Natural.Soundness\n\n\n-- Other calculi\n\nopen import partial\nopen import lambda\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"011451feda614507a6230b82f90c935611d41cf6","subject":"README import prg","message":"README import prg\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import prg\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import Data.Bits\n\nopen import data-universe\nopen import fun-universe\nopen import agda-fun-universe\nopen import bits-fun-universe\nopen import fin-fun-universe\nopen import const-fun-universe\nopen import cost-fun-universe\n\n-- Fix & restore the product of universes and ops\n\nopen import program-distance\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","old_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import Data.Bits\n\nopen import data-universe\nopen import fun-universe\nopen import agda-fun-universe\nopen import bits-fun-universe\nopen import fin-fun-universe\nopen import const-fun-universe\nopen import cost-fun-universe\n\n-- Fix & restore the product of universes and ops\n\nopen import program-distance\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0e27c3917b606cc0f1c46e8d38f391bbced1f066","subject":"Manually reorder imports in README.agda","message":"Manually reorder imports in README.agda\n\nDebatable, but I don't like that the imports are de facto randomized\nanyway. README.agda shouldn't be generated.\n\nOld-commit-hash: d718d74f6eb82413aef66792cd10b570ec84c25a\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\n\nimport Syntax.Constant.Popl14\nimport Syntax.Context.Plotkin\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.DeltaContext\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\nimport Syntax.Language.Popl14\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Atlas\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\nimport Syntax.Constant.Popl14\nimport Syntax.Context.Plotkin\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.DeltaContext\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\nimport Syntax.Language.Popl14\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Atlas\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2cf425c915364c989a28b50e75f8d096d98a2b30","subject":"Prepare for using \u2248-Reasoning to prove derive-term-correct (see #25).","message":"Prepare for using \u2248-Reasoning to prove derive-term-correct (see #25).\n\nOld-commit-hash: b0ca315e1dcaa44e39fdfa67bb29e1da512641a6\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083) =\n  begin\n    \u0394 (if t\u2081 t\u2082 t\u2083)\n  \u2248\u27e8 {!!} \u27e9\n    if (derive-term t\u2081 and lift-term {{\u0393\u2032}} t\u2081)\n       (diff-term\n         (apply-term (derive-term t\u2083)\n         (lift-term {{\u0393\u2032}} t\u2083)) (lift-term {{\u0393\u2032}} t\u2082))\n       (if (derive-term t\u2081 and (lift-term {{\u0393\u2032}} (! t\u2081)))\n           (diff-term\n             (apply-term (derive-term t\u2082)\n             (lift-term {{\u0393\u2032}} t\u2082)) (lift-term {{\u0393\u2032}} t\u2083))\n           (if (lift-term {{\u0393\u2032}} t\u2081)\n             (derive-term t\u2082)\n             (derive-term t\u2083)))\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083)\n  \u220e where open \u2248-Reasoning\n\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\nopen import Denotational.ValidChanges\n\nopen import Changes\nopen import ChangeContexts\nopen import ChangeContextLifting\nopen import PropsDelta\nopen import SymbolicDerivation\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct {{\u0393\u2032}} (if t\u2081 t\u2082 t\u2083) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 (if t\u2081 t\u2082 t\u2083) \u27e7Term \u03c1\n  \u2261\u27e8\u27e9\n     diff\n       (\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n       (\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 derive-term (if t\u2081 t\u2082 t\u2083) \u27e7 \u03c1\n  \u220e) where open \u2261-Reasoning\n\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"013595048b496691ac0767b6f0fc83f0d02727a2","subject":"agda\/...: inline substTerm-based lemmas","message":"agda\/...: inline substTerm-based lemmas\n\nBefore using substTerm, these lemmas were useful to help type\ninference, but now this is not needed.\n\nOld-commit-hash: adffd6a90ebb103bdbc9718a03bb579ffbf5ac85\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = {!!}\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = {!!}\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  context-cast-2 \u0393\u2081 \u0393\u2083\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (context-cast \u0393\u2081 \u0393\u2083 (derive-term t))))\n\n  where\n    open import Relation.Binary.PropositionalEquality using (subst)\n\n    context-cast :\n      \u2200 \u0393\u2081 \u0393\u2082 {\u03c4} \u2192\n        Term (\u0394-Context (\u0393\u2081 \u22ce \u0393\u2082)) \u03c4 \u2192 Term (\u0394-Context \u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\n    context-cast \u0393\u2081 \u0393\u2082 {\u03c4} =\n      substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2082)\n\n    context-cast-2 :\n      \u2200 \u0393\u2081 \u0393\u2082 {\u03c4 \u03c4\u2082} \u2192\n        Term (\u0394-Context \u0393\u2081 \u22ce (\u0394-Type \u03c4\u2082 \u2022 \u03c4\u2082 \u2022 \u0394-Context \u0393\u2082)) \u03c4 \u2192\n        Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2082))) \u03c4\n    context-cast-2 \u0393\u2081 \u0393\u2082 {\u03c4} {\u03c4\u2082} =\n      substTerm\n      (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2082)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"587ce47a7cfe86417c43770125889a6ad128c5a6","subject":"circuit: more combinators and specs","message":"circuit: more combinators and specs\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin) renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : RunCircuit RewireFun\n  runRewireFun = rewire\n\n  runRewireTbl : RunCircuit RewireTbl\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup : \u2200 o \u2192 C 1 o\n  dup _ = rewire (const zero)\n-- dup-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2082 : C 1 2\n  dup\u2082 = dup 2\n-- dup\u2082-spec : (b \u2237 []) =[ dup\u2082 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2261 i\u2081 \u2192 o\u2080 \u2261 o\u2081 \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081\n  coerce refl refl = id\n\n  dup\u207f : \u2200 {n} k \u2192 C n (k * n)\n  dup\u207f {n} k = coerce (proj\u2082 \u2115\u00b0.*-identity n) (\u2115\u00b0.*-comm n k) (vcat {n = n} (replicate (dup _)))\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  takeC : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) o\n  takeC {k} {_} {o} c = coerce refl (\u2115\u00b0.+-comm o 0) (c *** sink k)\n\n  takeC' : \u2200 {i} k \u2192 C (i + k) i\n  takeC' k = takeC {k} idC\n\n  dropC : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) o\n  dropC {k} c = sink k *** c\n\n  dropC' : \u2200 {i} k \u2192 C (k + i) i\n  dropC' k = dropC {k} idC\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = idC {1} *** sink _\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC' 1\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    forkC : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n\n  open RewiringBuilder isRewiringBuilder public\n\n  bit : Bit \u2192 C 0 1\n  bit b = arr (const (b \u2237 []))\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Rewire.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_ (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    open Rewire\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Rewire.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Rewire.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_ id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Rewire.rewire r ]= Rewire.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) (\u03bb { f g (b \u2237 bs) \u2192 (if b then f else g) bs })\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_ (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Rewire.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Data.Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Data.Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr leaf fork\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2082 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin) renaming (map to vmap)\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : \u2200 {i o} \u2192 RewireFun i o \u2192 Bits i \u2192 Bits o\n  runRewireFun = rewire\n\n  runRewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Bits i \u2192 Bits o\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idC : \u2200 {i} \u2192 C i i\n  idC = rewire id\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n\n  dup : \u2200 o \u2192 C 1 o\n  dup _ = rewire (const zero)\n\n  dup\u2082 : C 1 2\n  dup\u2082 = dup 2\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2261 i\u2081 \u2192 o\u2080 \u2261 o\u2081 \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081\n  coerce refl refl = id\n\n  dup\u207f : \u2200 {n} k \u2192 C n (k * n)\n  dup\u207f {n} k = coerce (proj\u2082 \u2115\u00b0.*-identity n) (\u2115\u00b0.*-comm n k) (vcat {n = n} (replicate (dup _)))\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  dropC : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) o\n  dropC {k} c = sink k *** c\n\n  dropC' : \u2200 {k i} \u2192 C (k + i) i\n  dropC' {k} = dropC {k} idC\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\nrecord CircuitBuilder (C : CircuitType) : Set where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n\n  open RewiringBuilder isRewiringBuilder public\n\n  bit : Bit \u2192 C 0 1\n  bit b = arr (const (b \u2237 []))\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_\n  where\n    C = _\u2192\u1da0_\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_\n  where\n    open Rewire\n    C = RewireTbl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_\n  where\n    C = _\u2192\u1d47_\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2082 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4746a75aa9bf2b7cdbb1fbd7e01d9183850f463e","subject":"Nat.Modulo: sucmod-inj","message":"Nat.Modulo: sucmod-inj\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Empty\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 Set where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n  sucmod-inj : \u2200 {q}{i j : Fin q} \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {i = i} {j} eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj eq | just _  | just _  | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n  sucmod-inj eq | just _  | nothing | _ | p | _ = \u22a5-elim (p (cong Maybe.just eq))\n  sucmod-inj eq | nothing | just _  | _ | _ | p = \u22a5-elim (p (cong Maybe.just (sym eq)))\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : Set a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : Set a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : Set a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  -- most likely this is subsumed by the StableUnderInjection parts\n  private\n     module Unused where\n        lookup-sucmod-rot\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n)\n                       \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n        lookup-sucmod-rot\u2081 {zero} () xs\n        lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n        lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n        lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                       | Vec.lookup-allFin (sucmod i)\n                                       | lookup-map sucmod i (allFin n)\n                                       | Vec.lookup\u2218tabulate id i\n                                       = refl\n\n        rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n        rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n\n  {-\n\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","old_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Empty\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup; rot\u2081) renaming (map to vmap)\nimport Data.Vec.Properties as Vec\nopen import Data.Maybe.NP\nopen import Data.Sum as Sum\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nsuc-injective : \u2200 {m}{i j : Fin m} \u2192 Fin.suc i \u2261 suc j \u2192 i \u2261 j\nsuc-injective refl = refl\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\nmax : \u2200 n \u2192 Fin (suc n)\nmax = from\u2115\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse {suc n} zero    = from\u2115 n\nreverse {suc n} (suc x) = inject\u2081 (reverse {n} x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n\ndata FinView {n} : Fin (suc n) \u2192 Set where\n  `from\u2115   : FinView (from\u2115 n)\n  `inject\u2081 : \u2200 x \u2192 FinView (inject\u2081 x)\n\nsucFinView : \u2200 {n} {i : Fin (suc n)} \u2192 FinView i \u2192 FinView (suc i)\nsucFinView `from\u2115 = `from\u2115\nsucFinView (`inject\u2081 x) = `inject\u2081 (suc x)\n\nfinView : \u2200 {n} \u2192 (i : Fin (suc n)) \u2192 FinView i\nfinView {zero}  zero    = `from\u2115\nfinView {suc n} zero    = `inject\u2081 zero\nfinView {suc n} (suc i) = sucFinView (finView i)\nfinView {zero}  (suc ())\n\nmodule Modulo where\n  modq : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq {zero}  _       = nothing\n  modq {suc q} zero    = just zero\n  modq {suc q} (suc x) = map? suc (modq x)\n\n  modq-inj : \u2200 {q} (i j : Fin (suc q)) \u2192 modq i \u2261 modq j \u2192 i \u2261 j\n  modq-inj {zero} zero zero eq = refl\n  modq-inj {zero} zero (suc ()) eq\n  modq-inj {zero} (suc ()) zero eq\n  modq-inj {zero} (suc ()) (suc ()) eq\n  modq-inj {suc q} zero zero eq = refl\n  modq-inj {suc q} zero (suc j) eq with modq j\n  modq-inj {suc q} zero (suc j) () | nothing\n  modq-inj {suc q} zero (suc j) () | just j'\n  modq-inj {suc q} (suc i) zero eq with modq i\n  modq-inj {suc q} (suc i) zero () | just x\n  modq-inj {suc q} (suc i) zero () | nothing\n  modq-inj {suc q} (suc i) (suc j) eq with modq i | modq j | modq-inj i j\n  modq-inj {suc q} (suc i) (suc j) eq | just x | just x\u2081 | p = cong suc (p (cong just (suc-injective (just-injective eq))))\n  modq-inj {suc q} (suc i) (suc j) () | just x | nothing | p\n  modq-inj {suc q} (suc i) (suc j) () | nothing | just x | p\n  modq-inj {suc q} (suc i) (suc j) eq | nothing | nothing | p = cong suc (p refl)\n\n  modq\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Fin (suc q)\n  modq\u2032 {zero}  _       = zero\n  modq\u2032 {suc q} zero    = suc zero\n  modq\u2032 {suc q} (suc x) = lift 1 suc (modq\u2032 x)\n\n  modqq : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  modqq {zero}  x = x\n  modqq {suc q} x = modq\u2032 x\n\n  -- Maybe (Fin n) \u2245 Fin (suc n)\n\n  modq\u2032\u2032 : \u2200 {q} \u2192 Fin (suc q) \u2192 Maybe (Fin q)\n  modq\u2032\u2032 x with modq\u2032 x\n  ... | zero  = nothing\n  ... | suc y = just y\n\n  zero\u2203 : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  zero\u2203 {zero}  ()\n  zero\u2203 {suc q} _  = zero\n\n  sucmod : \u2200 {q} \u2192 Fin q \u2192 Fin q\n  sucmod x with modq (suc x)\n  ... | nothing = zero\u2203 x\n  ... | just y  = y\n\n  modq-suc : \u2200 {q} (i j : Fin q) \u2192 modq (suc i) \u2262 just (zero\u2203 j)\n  modq-suc {zero} () () eq\n  modq-suc {suc q} i j eq with modq i\n  modq-suc {suc q} i j () | just x\n  modq-suc {suc q} i j () | nothing\n\n\n  sucmod-inj : \u2200 {q}(i j : Fin q) \u2192 sucmod i \u2261 sucmod j \u2192 i \u2261 j\n  sucmod-inj {q} i j eq with modq (suc i) | modq (suc j) | modq-inj (suc i) (suc j) | modq-suc i j | modq-suc j i\n  sucmod-inj i j eq | just x | just x\u2081 | p | _ | _ = suc-injective (p (cong just eq))\n  sucmod-inj i j eq | just x | nothing | p | ni | nj = \u22a5-elim (ni (cong Maybe.just eq))\n  sucmod-inj i j eq | nothing | just x | p | ni | nj = \u22a5-elim (nj (cong Maybe.just (sym eq)))\n  sucmod-inj i j eq | nothing | nothing | p | _ | _ = suc-injective (p refl)\n\n  modq-from\u2115 : \u2200 q \u2192 modq (from\u2115 q) \u2261 nothing\n  modq-from\u2115 zero = refl\n  modq-from\u2115 (suc q) rewrite modq-from\u2115 q = refl\n\n  modq-inject\u2081 : \u2200 {q} (i : Fin q) \u2192 modq (inject\u2081 i) \u2261 just i\n  modq-inject\u2081 zero = refl\n  modq-inject\u2081 (suc i) rewrite modq-inject\u2081 i = refl\n\n  sucmod-from\u2115 : \u2200 q \u2192 sucmod (from\u2115 q) \u2261 zero\n  sucmod-from\u2115 q rewrite modq-from\u2115 q = refl\n\n  sucmod-inject\u2081 : \u2200 {n} (i : Fin n) \u2192 sucmod (inject\u2081 i) \u2261 suc i\n  sucmod-inject\u2081 i rewrite modq-inject\u2081 i = refl\n\n  lem-inject\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n) x \u2192 lookup (inject\u2081 i) (xs \u2237\u02b3 x) \u2261 lookup i xs\n  lem-inject\u2081 zero    (x\u2080 \u2237 xs) x\u2081 = refl\n  lem-inject\u2081 (suc i) (x\u2080 \u2237 xs) x\u2081 = lem-inject\u2081 i xs x\u2081\n\n  lem-from\u2115 : \u2200 {n a} {A : Set a} (xs : Vec A n) x \u2192 lookup (from\u2115 n) (xs \u2237\u02b3 x) \u2261 x\n  lem-from\u2115 {zero}  []       x = refl\n  lem-from\u2115 {suc n} (_ \u2237 xs) x = lem-from\u2115 xs x\n\n  lookup-sucmod : \u2200 {n a} {A : Set a} (i : Fin (suc n)) (x : A) xs\n                 \u2192 lookup i (xs \u2237\u02b3 x) \u2261 lookup (sucmod i) (x \u2237 xs)\n  lookup-sucmod i x xs with finView i\n  lookup-sucmod {n} .(from\u2115 n) x xs | `from\u2115 rewrite sucmod-from\u2115 n = lem-from\u2115 xs x\n  lookup-sucmod .(inject\u2081 x) x\u2081 xs | `inject\u2081 x rewrite sucmod-inject\u2081 x = lem-inject\u2081 x xs x\u2081\n\n  lookup-map : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) i (xs : Vec A n)\n               \u2192 lookup i (vmap f xs) \u2261 f (lookup i xs)\n  lookup-map f zero    (x \u2237 xs) = refl\n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n\n  vec\u2257\u21d2\u2261 : \u2200 {n a} {A : Set a} (xs ys : Vec A n) \u2192 flip lookup xs \u2257 flip lookup ys \u2192 xs \u2261 ys\n  vec\u2257\u21d2\u2261 []       []       _ = refl\n  vec\u2257\u21d2\u2261 (x \u2237 xs) (y \u2237 ys) p rewrite vec\u2257\u21d2\u2261 xs ys (p \u2218 suc) | p zero = refl\n\n  lookup-sucmod-rot\u2081 : \u2200 {n a} {A : Set a} (i : Fin n) (xs : Vec A n)\n                 \u2192 lookup i (rot\u2081 xs) \u2261 lookup (sucmod i) xs\n  lookup-sucmod-rot\u2081 {zero} () xs\n  lookup-sucmod-rot\u2081 {suc n} i (x \u2237 xs) = lookup-sucmod i x xs\n\n  lookup-rot\u2081-allFin : \u2200 {n} i \u2192 lookup i (rot\u2081 (allFin n)) \u2261 lookup i (vmap sucmod (allFin n))\n  lookup-rot\u2081-allFin {n} i rewrite lookup-sucmod-rot\u2081 i (allFin _)\n                                 | Vec.lookup-allFin (sucmod i)\n                                 | lookup-map sucmod i (allFin n)\n                                 | Vec.lookup\u2218tabulate id i\n                                 = refl\n\n  rot\u2081-map-sucmod : \u2200 n \u2192 rot\u2081 (allFin n) \u2261 vmap sucmod (allFin n)\n  rot\u2081-map-sucmod _ = vec\u2257\u21d2\u2261 _ _ lookup-rot\u2081-allFin\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5fb098d972ab83b5c70bb4bf21de3d96a0b53bec","subject":"More imports in the README","message":"More imports in the README\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import flat-funs-implem\nopen import flat-funs-prod\n\nopen import program-distance\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","old_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\n\nopen import program-distance\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c12c385c45ca08bbde8f84c62b42e7fd3b90a15d","subject":"Simpler","message":"Simpler\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a (suc n) = eval t (a \u2022 \u03c1) n\napply (closure t \u03c1) a zero = TimeOut\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 = eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\", POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n-- Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) zero = TimeOut\nevalConst (lit v) (suc n) = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a (suc n) = eval t (a \u2022 \u03c1) n\napply (closure t \u03c1) a zero = TimeOut\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\n-- eval (app s t) \u03c1 = eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)\neval (app s t) \u03c1 n0 with eval s \u03c1 n0\n... | Error = Error\n... | TimeOut = TimeOut\n... | Done sv n1 with eval t \u03c1 n1\n... | Done tv n2 = apply sv tv n2\n... | Error = Error\n... | TimeOut = TimeOut\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b581524cae764a6e80535458364e1289e2f686dd","subject":"Spacing fix","message":"Spacing fix\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/DeriveCorrect.agda","new_file":"Thesis\/DeriveCorrect.agda","new_contents":"module Thesis.DeriveCorrect where\n\nopen import Thesis.Lang\nopen import Thesis.Changes\nopen import Thesis.LangChanges\nopen import Thesis.Derive\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Theorem.Groups-Nehemiah\n\nfromtoDeriveConst : \u2200 {\u03c4 : Type} (c : Const \u03c4) \u2192\n  ch \u27e6 c \u27e7\u0394Const from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = right-id-int n\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db})\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb | sym (-m\u00b7-n=-mn {b1} {db}) = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db})\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = sft\u2081 daa\nfromtoDeriveConst rinj db b1 b2 dbb = sft\u2082 dbb\nfromtoDeriveConst match .(inj\u2081 (inj\u2081 _)) .(inj\u2081 _) .(inj\u2081 _) (sft\u2081 daa) df f1 f2 dff dg g1 g2 dgg = dff _ _ _ daa\nfromtoDeriveConst match .(inj\u2081 (inj\u2082 _)) .(inj\u2082 _) .(inj\u2082 _) (sft\u2082 dbb) df f1 f2 dff dg g1 g2 dgg = dgg _ _ _ dbb\nfromtoDeriveConst match .(inj\u2082 (inj\u2082 b2)) .(inj\u2081 a1) .(inj\u2082 b2) (sftrp\u2081 a1 b2) df f1 f2 dff dg g1 g2 dgg\n rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2\n | sym (fromto\u2192\u2295 dg g1 g2 dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match .(inj\u2082 (inj\u2081 a2)) .(inj\u2082 b1) .(inj\u2081 a2) (sftrp\u2082 b1 a2) df f1 f2 dff dg g1 g2 dgg\n  rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2\n  | sym (fromto\u2192\u2295 df f1 f2 dff)\n   = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveBase : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  ch \u27e6 t \u27e7\u0394Term from \u27e6 t \u27e7Term to \u27e6 t \u27e7Term\nfromtoDeriveBase \u03c4 (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDeriveBase \u03c4 (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDeriveBase \u03c4 (app {\u03c3} s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDeriveBase (\u03c3 \u21d2 \u03c4) s _ _ _ d\u03c1\u03c1\n  in let fromToB = fromtoDeriveBase \u03c3 t _ _ _ d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDeriveBase (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n   fromtoDeriveBase \u03c4 t _ _ _ (dvv v\u2022 d\u03c1\u03c1)\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 t d\u03c1\u03c1 = fromtoDeriveBase \u03c4 t _ _ _ d\u03c1\u03c1\n\n-- Getting to the original equation 1 from PLDI'14.\n\ncorrectDeriveOplus : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 t \u27e7Term \u03c11) \u2295 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus \u03c4 t d\u03c1\u03c1 = fromto\u2192\u2295 _ _ _ (fromtoDerive \u03c4 t d\u03c1\u03c1)\n\nopen import Thesis.LangOps\n\ncorrectDeriveOplus\u03c4 : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4)\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit t) (derive t) \u27e7Term d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus\u03c4 \u03c4 t {d\u03c1 = d\u03c1} {\u03c11 = \u03c11} d\u03c1\u03c1\n  rewrite oplus\u03c4-equiv _ d\u03c1 _ (\u27e6 fit t \u27e7Term d\u03c1) (\u27e6 derive t \u27e7Term d\u03c1)\n  | sym (fit-sound t d\u03c1\u03c1)\n  = correctDeriveOplus \u03c4 t d\u03c1\u03c1\n\nderiveGivesDerivative : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3)\u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app f a \u27e7Term \u03c11) \u2295 (\u27e6 app f a \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus \u03c4 (app f a) d\u03c1\u03c1\n\nderiveGivesDerivative\u2082 : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) (derive a)) \u27e7Term d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative\u2082 \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus\u03c4 \u03c4 (app f a) d\u03c1\u03c1\n\n-- Proof of the original equation 1 from PLDI'14. The original was restricted to\n-- closed terms. This is a generalization, because it holds also for open terms,\n-- *as long as* the environment change is a nil change.\neq1 : \u2200 {\u0393} \u03c3 \u03c4 \u2192\n  {nil\u03c1 : Ch\u0393 \u0393} {\u03c1 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 nil\u03c1 from \u03c1 to \u03c1 \u2192\n  \u2200 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) (da : Term (\u0394\u0393 \u0393) (\u0394t \u03c3)) \u2192\n  (daa : [ \u03c3 ]\u03c4 (\u27e6 da \u27e7Term nil\u03c1) from (\u27e6 a \u27e7Term \u03c1) to (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1)) \u2192\n    \u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1 \u2261 \u27e6 app (fit f) (app\u2082 (oplus\u03c4o \u03c3) (fit a) da) \u27e7Term nil\u03c1\neq1 \u03c3 \u03c4 {nil\u03c1} {\u03c1} d\u03c1\u03c1 f a da daa\n  rewrite\n    oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit (app f a) \u27e7Term nil\u03c1) (\u27e6 (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1)\n  | sym (fit-sound f d\u03c1\u03c1)\n  | oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit a \u27e7Term nil\u03c1) (\u27e6 da \u27e7Term nil\u03c1)\n  | sym (fit-sound a d\u03c1\u03c1)\n  = fromto\u2192\u2295 (\u27e6 f \u27e7\u0394Term \u03c1 nil\u03c1 (\u27e6 a \u27e7Term \u03c1) (\u27e6 da \u27e7Term nil\u03c1)) _ _\n    (fromtoDerive _ f d\u03c1\u03c1 (\u27e6 da \u27e7Term nil\u03c1) (\u27e6 a \u27e7Term \u03c1)\n      (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1) daa)\n","old_contents":"module Thesis.DeriveCorrect where\n\nopen import Thesis.Lang\nopen import Thesis.Changes\nopen import Thesis.LangChanges\nopen import Thesis.Derive\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Theorem.Groups-Nehemiah\n\nfromtoDeriveConst : \u2200 {\u03c4 : Type} (c : Const \u03c4) \u2192\n  ch \u27e6 c \u27e7\u0394Const from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = right-id-int n\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db})\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite sym daa | sym dbb | sym (-m\u00b7-n=-mn {b1} {db}) = sym (mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db})\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = sft\u2081 daa\nfromtoDeriveConst rinj db b1 b2 dbb = sft\u2082 dbb\nfromtoDeriveConst match .(inj\u2081 (inj\u2081 _)) .(inj\u2081 _) .(inj\u2081 _) (sft\u2081 daa) df f1 f2 dff dg g1 g2 dgg = dff _ _ _ daa\nfromtoDeriveConst match .(inj\u2081 (inj\u2082 _)) .(inj\u2082 _) .(inj\u2082 _) (sft\u2082 dbb) df f1 f2 dff dg g1 g2 dgg = dgg _ _ _ dbb\nfromtoDeriveConst match .(inj\u2082 (inj\u2082 b2)) .(inj\u2081 a1) .(inj\u2082 b2) (sftrp\u2081 a1 b2) df f1 f2 dff dg g1 g2 dgg\n rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2\n | sym (fromto\u2192\u2295 dg g1 g2 dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match .(inj\u2082 (inj\u2081 a2)) .(inj\u2082 b1) .(inj\u2081 a2) (sftrp\u2082 b1 a2) df f1 f2 dff dg g1 g2 dgg\n  rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2\n  | sym (fromto\u2192\u2295 df f1 f2 dff)\n   = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveBase : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  ch \u27e6 t \u27e7\u0394Term from \u27e6 t \u27e7Term to \u27e6 t \u27e7Term\nfromtoDeriveBase \u03c4 (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDeriveBase \u03c4 (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDeriveBase \u03c4 (app {\u03c3} s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDeriveBase (\u03c3 \u21d2 \u03c4) s _ _ _ d\u03c1\u03c1\n  in let fromToB = fromtoDeriveBase \u03c3 t _ _ _ d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDeriveBase (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n   fromtoDeriveBase \u03c4 t _ _ _ (dvv v\u2022 d\u03c1\u03c1)\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 t d\u03c1\u03c1 = fromtoDeriveBase \u03c4 t _ _ _ d\u03c1\u03c1\n\n-- Getting to the original equation 1 from PLDI'14.\n\ncorrectDeriveOplus : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 t \u27e7Term \u03c11) \u2295 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus \u03c4 t d\u03c1\u03c1 = fromto\u2192\u2295 _ _ _ (fromtoDerive \u03c4 t d\u03c1\u03c1)\n\nopen import Thesis.LangOps\n\ncorrectDeriveOplus\u03c4 : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4)\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit t) (derive t) \u27e7Term d\u03c1) \u2261 (\u27e6 t \u27e7Term \u03c12)\ncorrectDeriveOplus\u03c4 \u03c4 t {d\u03c1 = d\u03c1} {\u03c11 = \u03c11} d\u03c1\u03c1\n  rewrite oplus\u03c4-equiv _ d\u03c1 _ (\u27e6 fit t \u27e7Term d\u03c1) (\u27e6 derive t \u27e7Term d\u03c1)\n  | sym (fit-sound t d\u03c1\u03c1)\n  = correctDeriveOplus \u03c4 t d\u03c1\u03c1\n\nderiveGivesDerivative : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3)\u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app f a \u27e7Term \u03c11) \u2295 (\u27e6 app f a \u27e7\u0394Term \u03c11 d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus \u03c4 (app f a) d\u03c1\u03c1\n\nderiveGivesDerivative\u2082 : \u2200 {\u0393} \u03c3 \u03c4 \u2192 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) \u2192\n  {d\u03c1 : Ch\u0393 \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n     (\u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) (derive a)) \u27e7Term d\u03c1) \u2261 (\u27e6 app f a \u27e7Term \u03c12)\nderiveGivesDerivative\u2082 \u03c3 \u03c4 f a d\u03c1\u03c1 = correctDeriveOplus\u03c4 \u03c4 (app f a) d\u03c1\u03c1\n\n-- Proof of the original equation 1 from PLDI'14. The original was restricted to\n-- closed terms. This is a generalization, because it holds also for open terms,\n-- *as long as* the environment change is a nil change.\neq1 : \u2200 {\u0393} \u03c3 \u03c4 \u2192\n  {nil\u03c1 : Ch\u0393 \u0393} {\u03c1 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 nil\u03c1 from \u03c1 to \u03c1 \u2192\n  \u2200 (f : Term \u0393 (\u03c3 \u21d2 \u03c4)) (a : Term \u0393 \u03c3) (da : Term (\u0394\u0393 \u0393) (\u0394t \u03c3)) \u2192\n  (daa : [ \u03c3 ]\u03c4 (\u27e6 da \u27e7Term nil\u03c1) from (\u27e6 a \u27e7Term \u03c1) to (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1)) \u2192\n    \u27e6 app\u2082 (oplus\u03c4o \u03c4) (fit (app f a)) (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1 \u2261 \u27e6 app (fit f) (app\u2082 (oplus\u03c4o \u03c3) (fit a) da) \u27e7Term nil\u03c1\neq1 \u03c3 \u03c4 {nil\u03c1} {\u03c1} d\u03c1\u03c1 f a da daa\n  rewrite\n    oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit (app f a) \u27e7Term nil\u03c1) (\u27e6 (app\u2082 (derive f) (fit a) da) \u27e7Term nil\u03c1)\n  | sym (fit-sound f d\u03c1\u03c1)\n  | oplus\u03c4-equiv _ nil\u03c1 _ (\u27e6 fit a \u27e7Term nil\u03c1) (\u27e6 da \u27e7Term nil\u03c1)\n  | sym (fit-sound a d\u03c1\u03c1)\n  = fromto\u2192\u2295 (\u27e6 f \u27e7\u0394Term \u03c1 nil\u03c1 (\u27e6 a \u27e7Term \u03c1) (\u27e6 da \u27e7Term nil\u03c1)) _ _\n    (fromtoDerive _ f d\u03c1\u03c1 (\u27e6 da \u27e7Term nil\u03c1) (\u27e6 a \u27e7Term \u03c1)\n      (\u27e6 a \u27e7Term \u03c1 \u2295 \u27e6 da \u27e7Term nil\u03c1) daa)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"af276c3574aaed52b27173042b8d4081744fbbd3","subject":"More level-polymorphism","message":"More level-polymorphism\n\nI'm guessing that Agda defaults levels to zero - which is fine, as long\nas you don't try to use the resulting definition in a more\nlevel-polymorphic statement (which I did).\n\nOld-commit-hash: a2a8030a427f6f81ce80429db1805b7bd0608bf2\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Equivalence.agda","new_file":"Parametric\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Parametric.Change.Equivalence where\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set a\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely. That should be true for functions\n    -- using changes parametrically, for derivatives and function changes, and\n    -- for functions using only the interface to changes (including the fact\n    -- that function changes are functions). Stating the general result, though,\n    -- seems hard, we should rather have lemmas proving that certain classes of\n    -- functions respect this equivalence.\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Parametric.Change.Equivalence where\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {\u2113} {A} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely. That should be true for functions\n    -- using changes parametrically, for derivatives and function changes, and\n    -- for functions using only the interface to changes (including the fact\n    -- that function changes are functions). Stating the general result, though,\n    -- seems hard, we should rather have lemmas proving that certain classes of\n    -- functions respect this equivalence.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4bfe8bf28bee269955e79053d110a9bfab5ea19d","subject":"perfect-bintree,Swp: have two congruence instead of one","message":"perfect-bintree,Swp: have two congruence instead of one\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n{-\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' = ?\n-}\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n{-\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' = ?\n-}\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (fork s\u2080 s\u2081) = fork (Swp-sym s\u2080) (Swp-sym s\u2081)\nSwp-sym swp\u2081         = swp\u2081\nSwp-sym swp\u2082         = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"111378aa88d8af5b042d93bce5d5569fbcf5bfd3","subject":"Updated note about the existential quantifier syntax.","message":"Updated note about the existential quantifier syntax.\n\nIgnore-this: 7d1568f1b17c6b4fcfea070a0706bd3b\n\ndarcs-hash:20111201124629-3bd4e-3ac12a2e7b246e27322cd0be8343ab9ec866e9cd.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/ExistentialSyntax.agda","new_file":"notes\/ExistentialSyntax.agda","new_contents":"-- Tested on 01 December 2011.\n\nmodule ExistentialSyntax where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 {d} \u2192 d \u2261 d\n  d    : D\n\nmodule \u2203\u2081 where\n  infixr 4 _,_\n\n  data \u2203 (P : D \u2192 Set) : Set where\n    _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n  syntax \u2203 (\u03bb x \u2192 e)  = \u2203[ x ] e\n\n  t\u2081 : \u2203 \u03bb x \u2192 x \u2261 x\n  t\u2081 = d , refl\n\n  t\u2082 : \u2203[ x ] x \u2261 x\n  t\u2082 = d , refl\n\n  t\u2083 : \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 x \u2261 y\n  t\u2083 = d , d , refl\n\n  t\u2084 : \u2203[ x ] \u2203[ y ] x \u2261 y\n  t\u2084 = d , d ,  refl\n\nmodule \u2203\u2082 where\n  infixr 4 _,_,_\n\n  data \u2203\u2082 (P : D \u2192 D \u2192 Set) : Set where\n    _,_,_ : (x y : D) \u2192 P x y \u2192 \u2203\u2082 P\n\n  -- Parse error\n  -- syntax \u2203\u2082 (\u03bb x y \u2192 e) = \u2203\u2082[ x y ] e\n\n  t\u2081 : \u2203\u2082 \u03bb x y \u2192 x \u2261 y\n  t\u2081 = d , d , refl\n","old_contents":"module ExistentialSyntax where\n\ninfixr 4 _,_\n\npostulate\n  D : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  refl : \u2200 {d} \u2192 d \u2261 d\n  d : D\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (d : D) \u2192 P d \u2192 \u2203 P\n\nsyntax \u2203 (\u03bb x \u2192 e)  = \u2203[ x ] e\n\nt1 : \u2203 \u03bb x \u2192 x \u2261 x\nt1 = d , refl\n\nt2 : \u2203[ x ] (x \u2261 x)\nt2 = d , refl\n\nt3 : \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 x \u2261 y\nt3 = d , d , refl\n\nt4 : \u2203[ x ] \u2203[ y ] x \u2261 y\nt4 = d , d ,  refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"08121492d6878116d031b64ad182babff9b77d75","subject":"FunBigStepSILR.agda: Allow unsolved metas","message":"FunBigStepSILR.agda: Allow unsolved metas\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 with svv n \u2264-refl tv1 tv2 {! tvv !} v1 {! eqv1 !}\n... | v2 , eqv2 , final-v = v2 , {! eqv2 !} , {! final-v !}\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 with svv n \u2264-refl tv1 tv2 {! tvv !} v1 {! eqv1 !}\n... | v2 , eqv2 , final-v = v2 , {! eqv2 !} , {! final-v !}\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"71d02150437583370bda1e049f45d9c69598c1e0","subject":"IDesc model: fix termination bug. I was kidding myself.","message":"IDesc model: fix termination bug. I was kidding myself.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : (A : Set)(B : Set)(f : A -> B)(x y : A) -> x == y -> f x == f y\ncong A B f x .x refl = refl\n\ncong2 : (A B C : Set)(f : A -> B -> C)(x y : A)(z t : B) -> \n        x == y -> z == t -> f x z == f y t\ncong2 A B C f x .x z .z refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : (A : Set)(B : Set)(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc Unit (descD I) (IMu Unit (\u03bb _ -> descD I)))\n        (hs : desc (Sigma Unit (IMu Unit (\u03bb _ -> descD I)))\n              (box Unit (descD I) (IMu Unit (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 (IDesc I) \n                                                (IDesc I)\n                                                (IDesc I) \n                                                prod \n                                                (iso1 I (iso2 I D))\n                                                D \n                                                (iso1 I (iso2 I D'))\n                                                D' p q\nproof-iso1-iso2 I (pi S T) = cong (S \u2192 IDesc I)\n                                  (IDesc I)\n                                  (pi S) \n                                  (\\x -> iso1 I (iso2 I (T x)))\n                                  T \n                                  (reflFun S (IDesc I)\n                                             (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                             T\n                                             (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (S \u2192 IDesc I)\n                                     (IDesc I)\n                                     (sigma S) \n                                     (\\x -> iso1 I (iso2 I (T x)))\n                                     T \n                                     (reflFun S \n                                              (IDesc I)\n                                              (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- induction : (I : Set)\n--             (R : I -> IDesc I)\n--             (P : Sigma I (IMu I R) -> Set)\n--             (m : (i : I)\n--                  (xs : desc I (R i) (IMu I R))\n--                  (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n--                  P ( i , con xs)) ->\n--             (i : I)(x : IMu I R i) -> P ( i , x )\n\n\nP : (I : Set) -> Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (IDescl I) \n                                          (IDescl I) \n                                          (IDescl I)\n                                          (prodl I) \n                                          (iso2 I (iso1 I D))\n                                          D \n                                          (iso2 I (iso1 I D'))\n                                          D' p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                          (IDescl I)\n                                                                          (pil I S) \n                                                                          (\\x -> iso2 I (iso1 I (T x)))\n                                                                          T \n                                                                          (reflFun S \n                                                                                   (IDescl I)\n                                                                                   (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                             (IDescl I)\n                                                                             (sigmal I S) \n                                                                             (\\x -> iso2 I (iso1 I (T x)))\n                                                                             T \n                                                                             (reflFun S \n                                                                                      (IDescl I)\n                                                                                      (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : (A : Set)(B : Set)(f : A -> B)(x y : A) -> x == y -> f x == f y\ncong A B f x .x refl = refl\n\ncong2 : (A B C : Set)(f : A -> B -> C)(x y : A)(z t : B) -> \n        x == y -> z == t -> f x z == f y t\ncong2 A B C f x .x z .z refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : (A : Set)(B : Set)(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\ncases : (I : Set) \n        (xs : desc Unit (descD I) (IMu Unit (\u03bb _ -> descD I)))\n        (hs : desc (Sigma Unit (IMu Unit (\u03bb _ -> descD I)))\n              (box Unit (descD I) (IMu Unit (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction Unit (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 (IDesc I) \n                                                (IDesc I)\n                                                (IDesc I) \n                                                prod \n                                                (iso1 I (iso2 I D))\n                                                D \n                                                (iso1 I (iso2 I D'))\n                                                D' p q\nproof-iso1-iso2 I (pi S T) = cong (S \u2192 IDesc I)\n                                  (IDesc I)\n                                  (pi S) \n                                  (\\x -> iso1 I (iso2 I (T x)))\n                                  T \n                                  (reflFun S (IDesc I)\n                                             (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                             T\n                                             (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (S \u2192 IDesc I)\n                                     (IDesc I)\n                                     (sigma S) \n                                     (\\x -> iso1 I (iso2 I (T x)))\n                                     T \n                                     (reflFun S \n                                              (IDesc I)\n                                              (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- induction : (I : Set)\n--             (R : I -> IDesc I)\n--             (P : Sigma I (IMu I R) -> Set)\n--             (m : (i : I)\n--                  (xs : desc I (R i) (IMu I R))\n--                  (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n--                  P ( i , con xs)) ->\n--             (i : I)(x : IMu I R i) -> P ( i , x )\n\n\nP : (I : Set) -> Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction Unit \n                                 (\u03bb x \u2192 sigma DescDConst (descDChoice I))\n                                 (P I)\n                                 ( proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                    (box Unit (sigma DescDConst (descDChoice I))\n                                                    (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) hs with proof-iso2-iso1 I D | proof-iso2-iso1 I D' \n                      ... | p | q = cong2 (IDescl I) \n                                          (IDescl I) \n                                          (IDescl I)\n                                          (prodl I) \n                                          (iso2 I (iso1 I D))\n                                          D \n                                          (iso2 I (iso1 I D'))\n                                          D' p q\n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                          (IDescl I)\n                                                                          (pil I S) \n                                                                          (\\x -> iso2 I (iso1 I (T x)))\n                                                                          T \n                                                                          (reflFun S \n                                                                                   (IDescl I)\n                                                                                   (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   (\\s -> proof-iso2-iso1 I (T s)))\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (S \u2192 IDescl I)\n                                                                             (IDescl I)\n                                                                             (sigmal I S) \n                                                                             (\\x -> iso2 I (iso1 I (T x)))\n                                                                             T \n                                                                             (reflFun S \n                                                                                      (IDescl I)\n                                                                                      (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      (\\s -> proof-iso2-iso1 I (T s)))\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc Unit (descDChoice I s)\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc (Sigma Unit (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))))\n                                                     (box Unit (sigma DescDConst (descDChoice I))\n                                                     (IMu Unit (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"331feb7fa6410abd3dc625cf80d95330ec71d0fa","subject":"IDesc model: remove IMu junk","message":"IDesc model: remove IMu junk\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fd99b86b8dd46e8fe410240cc3eea353d099ae97","subject":"Maybe: restore <$>-injective\u2081","message":"Maybe: restore <$>-injective\u2081\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"-- NOTE with-K\nmodule Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Relation.Unary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Product\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\ninfixr 0 _\u2192?_\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\ninfixl 1 _>>=?_\n\n-- More universe-polymorphic than M?._>>=_\n_>>=?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 Maybe A \u2192 (A \u2192 Maybe B) \u2192 Maybe B\nmx >>=? f = maybe f nothing mx\n\ninfixr 1 _=<<?_\n\n-- More universe-polymorphic than M?._=<<_\n_=<<?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Maybe B) \u2192 Maybe A \u2192 Maybe B\nf =<<? mx = mx >>=? f\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f mx = mx >>=? (just \u2218 f)\n\n_<$?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2192 Maybe B \u2192 Maybe A\n_<$?_ x = map? (const x)\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = _=<<?_ id\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \ud835\udfd8\njust? (just _) = \ud835\udfd9\n\njust?\u2192Is-just : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 Is-just x\njust?\u2192Is-just {x = just _}  p = just _\njust?\u2192Is-just {x = nothing} ()\n\nAny\u2192just? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata [Maybe] {a p} {A : \u2605 a} (A\u209a : A \u2192 \u2605 p) : Maybe A \u2192 \u2605 (a \u2294 p) where\n  nothing : [Maybe] A\u209a nothing\n  just    : (A\u209a [\u2192] [Maybe] A\u209a) just\n\n[maybe] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] (\u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] ((A\u209a [\u2192] B\u209a) [\u2192] (B\u209a [\u2192] ([Maybe] A\u209a [\u2192] B\u209a)))))\n                     (maybe {a} {b})\n[maybe] _ _ just\u209a nothing\u209a (just x\u209a) = just\u209a x\u209a\n[maybe] _ _ just\u209a nothing\u209a nothing   = nothing\u209a\n\n[map?] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] \u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] (A\u209a [\u2192] B\u209a) [\u2192] [Maybe] A\u209a [\u2192] [Maybe] B\u209a) (map? {a} {b})\n[map?] _ _ f\u209a (just x\u209a) = just (f\u209a x\u209a)\n[map?] _ _ f\u209a nothing   = nothing\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  just    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  nothing : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (just x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 nothing   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (just x\u1d63) = just (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 nothing   = nothing\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (just x\u1d63) = just x\u1d63\n\u27e6map?-id\u27e7 _ nothing   = nothing\n\nAny-join? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192Is-just : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P x \u2192 Is-just x\nAny\u2192Is-just (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192Is-just\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\n(f \u2218? g) x = g x >>=? f\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 Is-just (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_[\u2192?]_ : \u2200 {a b pa pb} \u2192 ([\u2605] {a} pa [\u2192] [\u2605] {b} pb [\u2192] [\u2605] _) _\u2192?_\nA\u209a [\u2192?] B\u209a = A\u209a [\u2192] [Maybe] B\u209a\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = just (refl-A x)\n  refl refl-A nothing  = nothing\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (just x\u223c\u2081y) = just (sym-A x\u223c\u2081y)\n  sym sym-A nothing     = nothing\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (just x\u223cy) (just y\u223cz) = just (trans' x\u223cy y\u223cz)\n  trans trans' nothing    nothing    = nothing\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (just x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f nothing    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIs-nothing-\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 Is-nothing x \u2192 x \u2261 nothing\nIs-nothing-\u2261nothing nothing = \u2261.refl\nIs-nothing-\u2261nothing (just ())\n\n\u2261nothing-Is-nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 Is-nothing x\n\u2261nothing-Is-nothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \ud835\udfd8-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n-- -}\n","old_contents":"-- NOTE with-K\nmodule Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Relation.Unary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Product\nopen import Data.Zero using (\ud835\udfd8; \ud835\udfd8-elim)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\ninfixr 0 _\u2192?_\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\ninfixl 1 _>>=?_\n\n-- More universe-polymorphic than M?._>>=_\n_>>=?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 Maybe A \u2192 (A \u2192 Maybe B) \u2192 Maybe B\nmx >>=? f = maybe f nothing mx\n\ninfixr 1 _=<<?_\n\n-- More universe-polymorphic than M?._=<<_\n_=<<?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Maybe B) \u2192 Maybe A \u2192 Maybe B\nf =<<? mx = mx >>=? f\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f mx = mx >>=? (just \u2218 f)\n\n_<$?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2192 Maybe B \u2192 Maybe A\n_<$?_ x = map? (const x)\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = _=<<?_ id\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \ud835\udfd8\njust? (just _) = \ud835\udfd9\n\njust?\u2192Is-just : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 Is-just x\njust?\u2192Is-just {x = just _}  p = just _\njust?\u2192Is-just {x = nothing} ()\n\nAny\u2192just? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata [Maybe] {a p} {A : \u2605 a} (A\u209a : A \u2192 \u2605 p) : Maybe A \u2192 \u2605 (a \u2294 p) where\n  nothing : [Maybe] A\u209a nothing\n  just    : (A\u209a [\u2192] [Maybe] A\u209a) just\n\n[maybe] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] (\u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] ((A\u209a [\u2192] B\u209a) [\u2192] (B\u209a [\u2192] ([Maybe] A\u209a [\u2192] B\u209a)))))\n                     (maybe {a} {b})\n[maybe] _ _ just\u209a nothing\u209a (just x\u209a) = just\u209a x\u209a\n[maybe] _ _ just\u209a nothing\u209a nothing   = nothing\u209a\n\n[map?] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] \u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] (A\u209a [\u2192] B\u209a) [\u2192] [Maybe] A\u209a [\u2192] [Maybe] B\u209a) (map? {a} {b})\n[map?] _ _ f\u209a (just x\u209a) = just (f\u209a x\u209a)\n[map?] _ _ f\u209a nothing   = nothing\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  just    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  nothing : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (just x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 nothing   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (just x\u1d63) = just (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 nothing   = nothing\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (just x\u1d63) = just x\u1d63\n\u27e6map?-id\u27e7 _ nothing   = nothing\n\nAny-join? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192Is-just : \u2200 {a p} {A : \u2605 a} {P : A \u2192 \u2605 p} {x} \u2192 Any P x \u2192 Is-just x\nAny\u2192Is-just (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192Is-just\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\n(f \u2218? g) x = g x >>=? f\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 Is-just (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_[\u2192?]_ : \u2200 {a b pa pb} \u2192 ([\u2605] {a} pa [\u2192] [\u2605] {b} pb [\u2192] [\u2605] _) _\u2192?_\nA\u209a [\u2192?] B\u209a = A\u209a [\u2192] [Maybe] B\u209a\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = just (refl-A x)\n  refl refl-A nothing  = nothing\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (just x\u223c\u2081y) = just (sym-A x\u223c\u2081y)\n  sym sym-A nothing     = nothing\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (just x\u223cy) (just y\u223cz) = just (trans' x\u223cy y\u223cz)\n  trans trans' nothing    nothing    = nothing\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (just x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f nothing    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIs-nothing-\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 Is-nothing x \u2192 x \u2261 nothing\nIs-nothing-\u2261nothing nothing = \u2261.refl\nIs-nothing-\u2261nothing (just ())\n\n\u2261nothing-Is-nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 Is-nothing x\n\u2261nothing-Is-nothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  {-\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {f = f} {x}  {y} f-inj =\n    maybe {B = {!\u03bb x \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y!} {!!} x\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n  -}\n  \n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \ud835\udfd8-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \ud835\udfd8-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \ud835\udfd8-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"de2dcbb465aa72c27c8516fc918571cd6713b73e","subject":"circuit: fix","message":"circuit: fix\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits hiding (rewire; rewireTbl)\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp;\n                tabulate-\u2218)\n         renaming (map to vmap)\nimport Data.Vec.Properties as VecProps\nopen VecProps using (tabulate\u2218lookup)\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Vec.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Vec.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Vec.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Vec.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Vec.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = Vec.rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Vec.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Vec.rewire r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Vec.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Vec.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits hiding (rewire; rewireTbl)\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp)\n         renaming (map to vmap)\nopen import Data.Vec.Properties\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Vec.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Vec.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Vec.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Vec.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Vec.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = Vec.rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Vec.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Vec.rewire r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Vec.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Vec.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1858a22c8a4a22d4ecf7a8423d9cbfa2cf6b0430","subject":"Trivial cases of correctness of optimized derivation","message":"Trivial cases of correctness of optimized derivation\n\nOld-commit-hash: dac6ff0b569595a84da55bf797b2027e4252d864\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/ExplicitNil.agda","new_file":"experimental\/ExplicitNil.agda","new_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\ntoo-close : \u2200 {\u03c4 \u0393} {args : Args \u03c4} \u2192\n  {df dg : Term (\u0394-Context \u0393) (\u0394-Type \u03c4)} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n  df \u2261 dg \u2192 \u27e6 df \u27e7 \u03c1 \u2248 \u27e6 dg \u27e7 \u03c1 WRT args\n\ntoo-close {nats}{_}{_} {df} {dg} {\u03c1} = cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1)\ntoo-close {bags}{_}{_} {df} {dg} {\u03c1} = {!!}\ntoo-close {\u03c4\u2081 \u21d2 \u03c4\u2082}{_}{_} {df} {dg} {\u03c1} = {!!}\n\nhonestyVar : {\u03c4 : Type} \u2192 {\u0393 : Context} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (x : Var \u0393 \u03c4) \u2192 derive (var x) \u2261 derive' args vars (var x)\nhonestyVar \u2205-nat vars x = refl\nhonestyVar \u2205-bag vars x = refl\nhonestyVar (abide args) vars x = refl\nhonestyVar (alter args) vars x = refl\n\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy {nats} {\u0393} (nat n) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (nat n) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (nat n) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args nats) (vars : Vars \u0393) \u2192\n      derive (nat n) \u2261 derive' {nats} {\u0393} args vars (nat n)\n    lemma \u2205 \u2205-nat \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-nat (alter vars) = refl\n\nhonesty-is-the-best-policy {bags} {\u0393} (bag b) args vars \u03c1 honesty =\n  begin\n    \u27e6 derive (bag b) \u27e7 \u03c1\n  \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 hole \u27e7 \u03c1) (lemma \u0393 args vars) \u27e9\n    \u27e6 derive' args vars (bag b) \u27e7 \u03c1\n  \u220e where\n    open \u2261-Reasoning\n    lemma : (\u0393 : Context) (args : Args bags) (vars : Vars \u0393) \u2192\n      derive (bag b) \u2261 derive' {bags} {\u0393} args vars (bag b)\n    lemma \u2205 \u2205-bag \u2205 = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (abide vars) = refl\n    lemma (\u03c4 \u2022 \u0393) \u2205-bag (alter vars) = refl\n\nhonesty-is-the-best-policy {\u03c4} {\u0393} (var x) args vars \u03c1 honesty =\n  too-close {\u03c4} {\u0393} (honestyVar args vars x)\n\nhonesty-is-the-best-policy (abs t) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (app f x) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (add m n) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (map f b) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (diff b d) args vars \u03c1 honesty = {!!}\nhonesty-is-the-best-policy (union b d) args vars \u03c1 honesty = {!!}\n\n","old_contents":"{-\nCommunicate to derivatives that changes to certain arguments\nare always nil (i. e., certain arguments are stable).\n-}\n\nmodule ExplicitNil where\n\nopen import NatBag\n\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\ndata Args : (\u03c4 : Type) \u2192 Set where\n  \u2205-nat : Args nats\n  \u2205-bag : Args bags\n  abide : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  alter : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192 (args : Args \u03c4\u2082) \u2192 Args (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\n-- Totally like subcontext relation _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set\ndata Vars : Context \u2192 Set where\n  \u2205 : Vars \u2205\n  abide : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is in the set\n  alter : \u2200 {\u03c4 \u0393} \u2192 Vars \u0393 \u2192 Vars (\u03c4 \u2022 \u0393) -- is out of the set\n\nstableVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstableVar this (abide _) = true\nstableVar this (alter _) = false\nstableVar (that x) (abide vars) = stableVar x vars\nstableVar (that x) (alter vars) = stableVar x vars\n\n-- A term is stable if all its free variables are unchanging\n-- Alternative definition:\n--\n--     stable t vars = isNil t (derive t)\n--\nstable : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393 \u2192 Bool\nstable (nat n) vars = true\nstable (bag b) vars = true\nstable (var x) vars = stableVar x vars\nstable (abs t) vars = stable t (abide vars)\nstable (app f x) vars = stable f vars \u2227 stable x vars\nstable (add m n) vars = stable m vars \u2227 stable n vars\nstable (map f b) vars = stable f vars \u2227 stable b vars\nstable (diff b d) vars = stable b vars \u2227 stable d vars\nstable (union b d) vars = stable b vars \u2227 stable d vars\n\nexpect-volatility : {\u03c4 : Type} \u2192 Args \u03c4\nexpect-volatility {\u03c4\u2081 \u21d2 \u03c4\u2082} = alter expect-volatility\nexpect-volatility {nats} = \u2205-nat\nexpect-volatility {bags} = \u2205-bag\n\n-- Type of `derive'`\nderive' : \u2200 {\u03c4 \u0393} \u2192 Args \u03c4 \u2192 Vars \u0393 \u2192\n                    Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nderive' {\u03c4\u2081 \u21d2 \u03c4\u2082} (abide args) vars (abs t) =\n  abs (abs (derive' args (abide vars) t))\n\nderive' (alter args) vars (abs t) =\n  abs (abs (derive' args (alter vars) t))\n\n-- Assume, for safety, that all arguments that `t` will\n-- eventually receive in `s` or receive curried out of `s`\n-- are volatile.\nderive' args vars (app s t) =\n  if stable t vars\n  then app (app (derive' (abide args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n  else app (app (derive' (alter args) vars s) (weaken \u0393\u227c\u0394\u0393 t))\n                (derive' expect-volatility vars t)\n\nderive' args vars (map f b) =\n  if stable f vars\n  then map (weaken \u0393\u227c\u0394\u0393 f) (derive' args vars b)\n  else map (weaken \u0393\u227c\u0394\u0393 f \u2295 derive' (abide \u2205-nat) vars f)\n           (weaken \u0393\u227c\u0394\u0393 b \u2295 derive' args vars b)\n       \u229d weaken \u0393\u227c\u0394\u0393 (map f b)\n\nderive' args vars (diff b d) =\n  diff (derive' args vars b) (derive' args vars d)\n\nderive' args vars (union b d) =\n  union (derive' args vars b) (derive' args vars d)\n\n-- derive(x + y) = replace (x + y) by (dx snd + dy snd)\n--               = \u03bb f. f (x + y) (dx snd + dy snd)\nderive' args vars (add m n) =\n  abs (app (app (var this)\n    (add (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) m) (weaken (drop _ \u2022 \u0393\u227c\u0394\u0393) n)))\n    (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars m)) snd)\n         (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (derive' args vars n)) snd)))\n\nderive' args vars constant-or-variable = derive constant-or-variable\n\n-- A description of variables is honest w.r.t. a \u0394-environment\n-- if every variable described as stable receives the nil change.\ndata Honest : \u2200 {\u0393} \u2192 Vars \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set where\n  clearly : Honest \u2205 \u2205\n  alter : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          Honest {\u03c4 \u2022 \u0393} (alter vars) (dv \u2022 v \u2022 \u03c1)\n  abide : \u2200 {\u0393 \u03c4 v dv} \u2192 {vars : Vars \u0393} \u2192 {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n          v \u27e6\u2295\u27e7 dv \u2261 v \u2192\n          Honest {\u03c4 \u2022 \u0393} (abide vars) (dv \u2022 v \u2022 \u03c1)\n\n-- Two \u0394-values are close enough w.r.t. a set of arguments if they\n-- behave the same when fully applied (cf. extensionality) given\n-- that each argument declared stable receives the nil change.\nclose-enough : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Args \u03c4 \u2192 Set\nclose-enough {nats} df dg args = df \u2261 dg -- extensionally\nclose-enough {bags} df dg args = df \u2261 dg -- literally\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (alter args) =\n  \u2200 {x dx} \u2192 close-enough (df x dx) (dg x dx) args\nclose-enough {\u03c4\u2081 \u21d2 \u03c4\u2082} df dg (abide args) =\n  \u2200 {x dx} \u2192 x \u27e6\u2295\u27e7 dx \u2261 x \u2192 close-enough (df x dx) (dg x dx) args\n\nsyntax close-enough df dg args = df \u2248 dg WRT args\n\nhonesty-is-the-best-policy : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} \u2192\n  (args : Args \u03c4) \u2192 (vars : Vars \u0393) \u2192\n  (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Honest vars \u03c1 \u2192\n  \u27e6 derive t \u27e7 \u03c1 \u2248 \u27e6 derive' args vars t \u27e7 \u03c1 WRT args\n\nhonesty-is-the-best-policy = {!!}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4de0c55b9a5201bdc47a9e82a128b4811456e1aa","subject":"Desc stratified model: painful implicitification of IDescl constructors","message":"Desc stratified model: painful implicitification of IDescl constructors\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst  {l = suc x}\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl I\nconstl x I X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl I) -> IDescl I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl I\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"877d01d17d7760064bb77e08cd9117b683f6f50d","subject":"Comment on validity definition","message":"Comment on validity definition\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n\n-- This definition violates all tips from Pitts tutorial (Step-Indexed\n-- Biorthogonality: a Tutorial Example). Compare with Definition 4.1,\n-- equation (10), and remark 4.4.\n--\n-- For extra complication, the setting is rather different, so a direct\n-- comparison doesn't quite work.\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"83616db66fdaf50e339a0a7534de4653f7c02caf","subject":"Implement change composition directly for unit","message":"Implement change composition directly for unit\n\nWould like to avoid the generic implementation.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/Changes.agda","new_file":"Thesis\/Changes.agda","new_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  valid : \u2200 (a : A) (da : ChA) \u2192 Set\n  valid a da = ch da from a to (a \u2295 da)\n  \u0394 : (a : A) \u2192 Set\n  \u0394 a = \u03a3[ da \u2208 ChA ] valid a da\n  \u0394\u2082 : (a1 : A) (a2 : A) \u2192 Set\n  \u0394\u2082 a1 a2 = \u03a3[ da \u2208 ChA ] ch da from a1 to a2\n\n  _\u2295'_ : (a1 : A) -> {a2 : A} -> (da : \u0394\u2082 a1 a2) -> A\n  a1 \u2295' (da , daa) = a1 \u2295 da\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\ninstance\n  unitCS : ChangeStructure \u22a4\n\n  unitCS = record\n    { Ch = \u22a4\n    ; ch_from_to_ = \u03bb dv v1 v2 \u2192 \u22a4\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n      { _\u2295_ = \u03bb _ _ \u2192 tt\n      ; fromto\u2192\u2295 = \u03bb { _ _ tt _ \u2192 refl }\n      ; _\u229d_ = \u03bb _ _ \u2192 tt\n      ; \u229d-fromto = \u03bb a b \u2192 tt\n      }\n      ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 tt\n      ; \u229a-fromto = \u03bb a1 a2 a3 da1 da2 daa1 daa2 \u2192 tt\n      }\n    }\n","old_contents":"module Thesis.Changes where\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord IsChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  infixl 6 _\u2295_ _\u229d_\n  field\n    _\u2295_ : A \u2192 ChA \u2192 A\n    fromto\u2192\u2295 : \u2200 dv v1 v2 \u2192\n      ch dv from v1 to v2 \u2192\n      v1 \u2295 dv \u2261 v2\n    _\u229d_ : A \u2192 A \u2192 ChA\n    \u229d-fromto : \u2200 (a b : A) \u2192 ch (b \u229d a) from a to b\n\n  update-diff : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  update-diff b a = fromto\u2192\u2295 (b \u229d a) a b (\u229d-fromto a b)\n  nil : A \u2192 ChA\n  nil a = a \u229d a\n  nil-fromto : (a : A) \u2192 ch (nil a) from a to a\n  nil-fromto a = \u229d-fromto a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  valid : \u2200 (a : A) (da : ChA) \u2192 Set\n  valid a da = ch da from a to (a \u2295 da)\n  \u0394 : (a : A) \u2192 Set\n  \u0394 a = \u03a3[ da \u2208 ChA ] valid a da\n  \u0394\u2082 : (a1 : A) (a2 : A) \u2192 Set\n  \u0394\u2082 a1 a2 = \u03a3[ da \u2208 ChA ] ch da from a1 to a2\n\n  _\u2295'_ : (a1 : A) -> {a2 : A} -> (da : \u0394\u2082 a1 a2) -> A\n  a1 \u2295' (da , daa) = a1 \u2295 da\n\nrecord IsCompChangeStructure (A : Set) (ChA : Set) (ch_from_to_ : (dv : ChA) \u2192 (v1 v2 : A) \u2192 Set) : Set\u2081 where\n  field\n    isChangeStructure : IsChangeStructure A ChA ch_from_to_\n    _\u229a[_]_ : ChA \u2192 A \u2192 ChA \u2192 ChA\n    \u229a-fromto : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n      ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192 ch da1 \u229a[ a1 ] da2 from a1 to a3\n\n  open IsChangeStructure isChangeStructure public\n  \u229a-correct : \u2200 (a1 a2 a3 : A) (da1 da2 : ChA) \u2192\n    ch da1 from a1 to a2 \u2192 ch da2 from a2 to a3 \u2192\n    a1 \u2295 (da1 \u229a[ a1 ] da2) \u2261 a3\n  \u229a-correct a1 a2 a3 da1 da2 daa1 daa2 = fromto\u2192\u2295 _ _ _ (\u229a-fromto _ _ _ da1 da2 daa1 daa2)\n\nIsChangeStructure\u2192IsCompChangeStructure : \u2200 {A ChA ch_from_to_} \u2192 IsChangeStructure A ChA ch_from_to_ \u2192 IsCompChangeStructure A ChA ch_from_to_\nIsChangeStructure\u2192IsCompChangeStructure {A} {ChA} {ch_from_to_} isCS = record\n  { isChangeStructure = isCS\n  ; _\u229a[_]_ = \u03bb da1 a da2 \u2192 a \u2295 da1 \u2295 da2 \u229d a\n  ; \u229a-fromto = body\n  }\n  where\n    _\u2295_ = IsChangeStructure._\u2295_ isCS\n    _\u229d_ = IsChangeStructure._\u229d_ isCS\n    fromto\u2192\u2295 = IsChangeStructure.fromto\u2192\u2295 isCS\n    \u229d-fromto = IsChangeStructure.\u229d-fromto isCS\n    infixl 6 _\u2295_ _\u229d_\n    body : \u2200 (a1 a2 a3 : A) da1 da2 \u2192\n      ch da1 from a1 to a2 \u2192\n      ch da2 from a2 to a3 \u2192 ch a1 \u2295 da1 \u2295 da2 \u229d a1 from a1 to a3\n    body a1 a2 a3 da1 da2 daa1 daa2 rewrite fromto\u2192\u2295 _ _ _ daa1 | fromto\u2192\u2295 _ _ _ daa2 =\n      \u229d-fromto a1 a3\n\n\nrecord ChangeStructure (A : Set) : Set\u2081 where\n  field\n    Ch : Set\n    ch_from_to_ : (dv : Ch) \u2192 (v1 v2 : A) \u2192 Set\n    isCompChangeStructure : IsCompChangeStructure A Ch ch_from_to_\n  open IsCompChangeStructure isCompChangeStructure public\n\nopen ChangeStructure {{...}} public hiding (Ch)\nCh : \u2200 (A : Set) \u2192 {{CA : ChangeStructure A}} \u2192 Set\nCh A {{CA}} = ChangeStructure.Ch CA\n\n{-# DISPLAY IsChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChangeStructure._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChangeStructure._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY IsChangeStructure.nil x = nil #-}\n{-# DISPLAY ChangeStructure.nil x = nil #-}\n{-# DISPLAY IsCompChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure._\u229a[_]_ x = _\u229a[_]_ #-}\n{-# DISPLAY ChangeStructure.ch_from_to_ x = ch_from_to_ #-}\n\nmodule _ {A B : Set} {{CA : ChangeStructure A}} {{CB : ChangeStructure B}} where\n\n  -- In this module, given change structures CA and CB for A and B, we define\n  -- change structures for A \u2192 B, A \u00d7 B and A \u228e B.\n\n  open import Postulate.Extensionality\n\n  -- Functions\n  instance\n    funCS : ChangeStructure (A \u2192 B)\n\n  infixl 6 _f\u2295_ _f\u229d_\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n\n    fCh_from_to_ : (df : fCh) \u2192 (f1 f2 : A \u2192 B) \u2192 Set\n    fCh_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n    f\u229d-fromto : \u2200 (f1 f2 : A \u2192 B) \u2192 fCh (f2 f\u229d f1) from f1 to f2\n    f\u229d-fromto f1 f2 da a1 a2 daa\n      rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\n    _f\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f\u229a[_]_ df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n\n    _f2\u229a[_]_ : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    _f2\u229a[_]_ df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n\n    f\u229a-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f\u229a[ f1 ] df2 from f1 to f3\n    f\u229a-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a1) (f3 a2)\n        (df1 a1 (nil a1))\n        (df2 a1 da)\n        (dff1 (nil a1) a1 a1 (nil-fromto a1))\n        (dff2 da a1 a2 daa)\n\n    f\u229a2-fromto : \u2200 (f1 f2 f3 : A \u2192 B) (df1 df2 : fCh) \u2192 fCh df1 from f1 to f2 \u2192 fCh df2 from f2 to f3 \u2192\n      fCh df1 f2\u229a[ f1 ] df2 from f1 to f3\n    f\u229a2-fromto f1 f2 f3 df1 df2 dff1 dff2 da a1 a2 daa rewrite fromto\u2192\u2295 da a1 a2 daa =\n      \u229a-fromto (f1 a1) (f2 a2) (f3 a2)\n        (df1 a1 da)\n        (df2 a2 (nil a2))\n        (dff1 da a1 a2 daa)\n        (dff2 (nil a2) a2 a2 (nil-fromto a2))\n\n  funCS = record\n    { Ch = fCh\n    ; ch_from_to_ =\n      \u03bb df f1 f2 \u2192 \u2200 da (a1 a2 : A) (daa : ch da from a1 to a2) \u2192\n        ch df a1 da from f1 a1 to f2 a2\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _f\u2295_\n        ; fromto\u2192\u2295 = \u03bb df f1 f2 dff \u2192\n          ext (\u03bb v \u2192\n            fromto\u2192\u2295 (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\n        ; _\u229d_ = _f\u229d_\n        ; \u229d-fromto = f\u229d-fromto\n        }\n      ; _\u229a[_]_ = _f\u229a[_]_\n      ; \u229a-fromto = f\u229a-fromto\n      }\n    }\n\n  -- Products\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pch_from_to_ : pCh \u2192 A \u00d7 B \u2192 A \u00d7 B \u2192 Set\n    pch (da , db) from (a1 , b1) to (a2 , b2) = ch da from a1 to a2 \u00d7 ch db from b1 to b2\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    pfromto\u2192\u2295 : \u2200 dp p1 p2 \u2192\n      pch dp from p1 to p2 \u2192 p1 p\u2295 dp \u2261 p2\n    pfromto\u2192\u2295 (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n      cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n    p\u229d-fromto : \u2200 (p1 p2 : A \u00d7 B) \u2192 pch p2 p\u229d p1 from p1 to p2\n    p\u229d-fromto (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n    p\u229a-fromto : \u2200 p1 p2 p3 dp1 dp2 \u2192\n      pch dp1 from p1 to p2 \u2192 (pch dp2 from p2 to p3) \u2192 pch dp1 p\u229a[ p1 ] dp2 from p1 to p3\n    p\u229a-fromto (a1 , b1) (a2 , b2) (a3 , b3) (da1 , db1) (da2 , db2)\n      (daa1 , dbb1) (daa2 , dbb2) =\n        \u229a-fromto a1 a2 a3 da1 da2 daa1 daa2 , \u229a-fromto b1 b2 b3 db1 db2 dbb1 dbb2\n\n  instance\n    pairCS : ChangeStructure (A \u00d7 B)\n  pairCS = record\n    { Ch = pCh\n    ; ch_from_to_ = pch_from_to_\n    ; isCompChangeStructure = record\n      { isChangeStructure = record\n        { _\u2295_ = _p\u2295_\n        ; fromto\u2192\u2295 = pfromto\u2192\u2295\n        ; _\u229d_ = _p\u229d_\n        ; \u229d-fromto = p\u229d-fromto\n        }\n      ; _\u229a[_]_ = _p\u229a[_]_\n      ; \u229a-fromto = p\u229a-fromto\n      }\n    }\n\n  -- Sums\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    _s\u22952_ : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    _s\u22952_ (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    _s\u22952_ (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    _s\u22952_ (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    _s\u22952_ (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    _s\u22952_ s (rp s\u2081) = s\u2081\n\n    _s\u2295_ : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s s\u2295 ds = s s\u22952 (convert ds)\n\n    _s\u229d2_ : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    _s\u229d2_ (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    _s\u229d2_ (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    _s\u229d2_ s2 s1 = rp s2\n\n    _s\u229d_ : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s2 s\u229d s1 = convert\u2081 (s2 s\u229d2 s1)\n\n  data sch_from_to_ : SumChange \u2192 A \u228e B \u2192 A \u228e B \u2192 Set where\n    -- sft = Sum From To\n    sft\u2081 : \u2200 {da a1 a2} (daa : ch da from a1 to a2) \u2192 sch (convert\u2081 (ch\u2081 da)) from (inj\u2081 a1) to (inj\u2081 a2)\n    sft\u2082 : \u2200 {db b1 b2} (dbb : ch db from b1 to b2) \u2192 sch (convert\u2081 (ch\u2082 db)) from (inj\u2082 b1) to (inj\u2082 b2)\n    sftrp\u2081 : \u2200 a1 b2 \u2192 sch (convert\u2081 (rp (inj\u2082 b2))) from (inj\u2081 a1) to (inj\u2082 b2)\n    sftrp\u2082 : \u2200 b1 a2 \u2192 sch (convert\u2081 (rp (inj\u2081 a2))) from (inj\u2082 b1) to (inj\u2081 a2)\n\n  sfromto\u2192\u22952 : (ds : SumChange2) (s1 s2 : A \u228e B) \u2192\n    sch convert\u2081 ds from s1 to s2 \u2192 s1 s\u22952 ds \u2261 s2\n  sfromto\u2192\u22952 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) (sft\u2081 daa) = cong inj\u2081 (fromto\u2192\u2295 _ _ _ daa)\n  sfromto\u2192\u22952 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) (sft\u2082 dbb) = cong inj\u2082 (fromto\u2192\u2295 _ _ _ dbb)\n  sfromto\u2192\u22952 (rp .(inj\u2082 y)) (inj\u2081 x) (inj\u2082 y) (sftrp\u2081 .x .y) = refl\n  sfromto\u2192\u22952 (rp .(inj\u2081 x)) (inj\u2082 y) (inj\u2081 x) (sftrp\u2082 .y .x) = refl\n\n  sfromto\u2192\u2295 : (ds : SumChange) (s1 s2 : A \u228e B) \u2192\n    sch ds from s1 to s2 \u2192 s1 s\u2295 ds \u2261 s2\n  sfromto\u2192\u2295 ds s1 s2 dss =\n    sfromto\u2192\u22952 (convert ds) s1 s2\n      (subst (sch_from s1 to s2) (sym (inv1 ds))\n        dss)\n  s\u229d-fromto : (s1 s2 : A \u228e B) \u2192 sch s2 s\u229d s1 from s1 to s2\n  s\u229d-fromto (inj\u2081 a1) (inj\u2081 a2) = sft\u2081 (\u229d-fromto a1 a2)\n  s\u229d-fromto (inj\u2081 a1) (inj\u2082 b2) = sftrp\u2081 a1 b2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2081 a2) = sftrp\u2082 b1 a2\n  s\u229d-fromto (inj\u2082 b1) (inj\u2082 b2) = sft\u2082 (\u229d-fromto b1 b2)\n  instance\n    sumCS : ChangeStructure (A \u228e B)\n  sumCS = record\n    { Ch = SumChange\n    ; ch_from_to_ = sch_from_to_\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = _s\u2295_\n      ; fromto\u2192\u2295 = sfromto\u2192\u2295\n      ; _\u229d_ = _s\u229d_\n      ; \u229d-fromto = s\u229d-fromto\n      })\n    }\n\ninstance\n  unitCS : ChangeStructure \u22a4\n\n  unitCS = record\n    { Ch = \u22a4\n    ; ch_from_to_ = \u03bb dv v1 v2 \u2192 \u22a4\n    ; isCompChangeStructure = IsChangeStructure\u2192IsCompChangeStructure (record\n      { _\u2295_ = \u03bb _ _ \u2192 tt\n      ; fromto\u2192\u2295 = \u03bb { _ _ tt _ \u2192 refl }\n      ; _\u229d_ = \u03bb _ _ \u2192 tt\n      ; \u229d-fromto = \u03bb a b \u2192 tt\n      })\n    }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5084db0b3e01eca253b247da48daf55234039968","subject":"Update derivation.agda","message":"Update derivation.agda\n\ntypo\n\nOld-commit-hash: 719b7f355a6ddb5cacf3f0d3a90f563748f921a0\n","repos":"inc-lc\/ilc-agda","old_file":"derivation.agda","new_file":"derivation.agda","new_contents":"module derivation where\n\nopen import lambda\n\n-- KO: Is it correct that this file is supposed to contain the syntactic \n-- variants of the operations in Syntactic.Changes.agda?\n-- If so, why does it have a different (and rather strange) \u0394-Type definition?\n-- Why are the base cases (bool) all missing? The inductive cases look\n-- boring since they don't actually do anything (which explains why compose and apply are the same)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","old_contents":"module derivation where\n\nopen import lambda\n\n-- KO: Is it correct that this file is supposed to contain the syntactic \n-- variants of the operations in Changes.agda?\n-- If so, why does it have a different (and rather strange) \u0394-Type definition?\n-- Why are the base cases (bool) all missing? The inductive cases look\n-- boring since they don't actually do anything (which explains why compose and apply are the same)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e4f79cd37331028ce9e00507f9b5332149c887d9","subject":"documenting Equality module","message":"documenting Equality module\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Equality.agda","new_file":"agda\/Equality.agda","new_contents":"-- This is an implementation of the equality type for Sets. Agda's\n-- standard equality is more powerful. The main idea here is to\n-- illustrate the equality type.\nmodule Equality where\n\n-- The equality of two elements of type A. The type a \u2261 b is a family\n-- of types which captures the statement of equality in A. Not all of\n-- the types are inhabited as in general not all elements are equal\n-- here. The only type that are inhabited are a \u2261 a by the element\n-- that we call definition. This is called refl (for reflection in\n-- agda).\ndata _\u2261_  {A : Set} : (a b : A) \u2192 Set where\n  definition : {x : A} \u2192 x \u2261 x\n\n-- \u2261 is a symmetric relation.\nsym : {A : Set} {a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\nsym definition = definition\n\n-- \u2261 is a transitive relation.\ntrans : {A : Set} {a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\ntrans pAB definition = pAB\n-- trans definition pBC = pBC  -- alternate proof of transitivity.\n\n-- Congruence. If we apply f to equals the result are also equal.\ncong : {A B : Set} {a\u2080 a\u2081 : A} \u2192 (f : A \u2192 B) \u2192 a\u2080 \u2261 a\u2081 \u2192 f a\u2080 \u2261 f a\u2081\ncong f definition = definition\n\n-- Pretty way of doing equational reasoning.  If we want to prove a\u2080 \u2261\n-- b through an intermediate set of equations use this. The general\n-- form will look like.\n--\n-- begin a  \u2248 a\u2080 by p\u2080\n--          \u2248 a\u2081 by p\u2081\n--          ...\n--          \u2248 b  by p\n-- \u220e\n\nbegin : {A : Set} (a : A) \u2192  a \u2261 a\n_\u2248_by_ : {A : Set} {a b : A} \u2192 a \u2261 b \u2192 (c : A) \u2192 b \u2261 c \u2192 a \u2261 c\n_\u220e : {A : Set} (a : A) \u2192 A\n\nbegin a = definition\naEb \u2248 c by bEc = trans aEb bEc\nx \u220e = x\n\ninfixl 1 _\u2248_by_\ninfixl 1 _\u2261_\n","old_contents":"module Equality where\n\ndata _\u2261_  {A : Set} : (a b : A) \u2192 Set where\n  definition : {x : A} \u2192 x \u2261 x\n\nsym : {A : Set} {a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\nsym definition = definition\n\ntrans : {A : Set} {a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\ntrans pAB definition = pAB\n-- trans definition pBC = pBC\n\ncong : {A B : Set} {a\u2080 a\u2081 : A} \u2192 (f : A \u2192 B) \u2192 a\u2080 \u2261 a\u2081 \u2192 f a\u2080 \u2261 f a\u2081\ncong f definition = definition\n\nbegin : {A : Set} (a : A) \u2192  a \u2261 a\nbegin a = definition\n\n\n_\u2248_by_ : {A : Set} {a b : A} \u2192 a \u2261 b \u2192 (c : A) \u2192 b \u2261 c \u2192 a \u2261 c\naEb \u2248 c by bEc = trans aEb bEc\n\n\n_\u220e : {A : Set} (a : A) \u2192 A\nx \u220e = x\n\ninfixl 1 _\u2248_by_\ninfixl 1 _\u2261_\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"0ea931fffedcc73acbb0013981dff55d1c0d69ce","subject":"Revert \"Show we can define e\u2295 using base values in changes\"","message":"Revert \"Show we can define e\u2295 using base values in changes\"\n\nThat was a demonstration, but the old definition is the one I prefer.\nThis reverts commit 47190645cc58ddea24ef59e6877fda0473ae4818.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/LangChanges.agda","new_file":"Thesis\/LangChanges.agda","new_contents":"module Thesis.LangChanges where\n\nopen import Thesis.Changes\nopen import Thesis.IntChanges\nopen import Thesis.Lang\nopen import Relation.Binary.PropositionalEquality\n\nCh\u03c4 : (\u03c4 : Type) \u2192 Set\nCh\u03c4 \u03c4 = \u27e6 \u0394t \u03c4 \u27e7Type\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Ch\u03c4 \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\nisCompChangeStructure\u03c4 : \u2200 \u03c4 \u2192 IsCompChangeStructure \u27e6 \u03c4 \u27e7Type \u27e6 \u0394t \u03c4 \u27e7Type [ \u03c4 ]\u03c4_from_to_\n\nchangeStructure\u03c4 : \u2200 \u03c4 \u2192 ChangeStructure \u27e6 \u03c4 \u27e7Type\nchangeStructure\u03c4 \u03c4 = record { isCompChangeStructure = isCompChangeStructure\u03c4 \u03c4 }\n\ninstance\n  ichangeStructure\u03c4 : \u2200 {\u03c4} \u2192 ChangeStructure \u27e6 \u03c4 \u27e7Type\n  ichangeStructure\u03c4 {\u03c4} = changeStructure\u03c4 \u03c4\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Ch\u03c4 \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v1 + dv \u2261 v2\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sch_from_to_ dv v1 v2\n\nisCompChangeStructure\u03c4 (\u03c3 \u21d2 \u03c4) = ChangeStructure.isCompChangeStructure (funCS {{changeStructure\u03c4 \u03c3}} {{changeStructure\u03c4 \u03c4}})\nisCompChangeStructure\u03c4 int = ChangeStructure.isCompChangeStructure intCS\nisCompChangeStructure\u03c4 (pair \u03c3 \u03c4) = ChangeStructure.isCompChangeStructure (pairCS {{changeStructure\u03c4 \u03c3}} {{changeStructure\u03c4 \u03c4}})\nisCompChangeStructure\u03c4 (sum \u03c3 \u03c4) = ChangeStructure.isCompChangeStructure ((sumCS {{changeStructure\u03c4 \u03c3}} {{changeStructure\u03c4 \u03c4}}))\n\nopen import Data.Unit\n-- We can define an alternative semantics for contexts, that defines environments as nested products:\n\u27e6_\u27e7Context-prod : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context-prod = \u22a4\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context-prod = \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u0393 \u27e7Context-prod\n\n-- And define a change structure for such environments, reusing the change\n-- structure constructor for pairs. However, this change structure does not\n-- duplicate base values in the environment. We probably *could* define a\n-- different change structure for pairs that gives the right result.\nchangeStructure\u0393-old : \u2200 \u0393 \u2192 ChangeStructure \u27e6 \u0393 \u27e7Context-prod\nchangeStructure\u0393-old \u2205 = unitCS\nchangeStructure\u0393-old (\u03c4 \u2022 \u0393) = pairCS {{changeStructure\u03c4 \u03c4}}  {{changeStructure\u0393-old \u0393}}\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nmodule _ where\n  _e\u2295_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0393 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n  _e\u2295_ \u2205 \u2205 = \u2205\n  _e\u2295_ (v \u2022 \u03c1) (dv \u2022 _ \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c1 e\u2295 d\u03c1\n  _e\u229d_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0393 \u0393\n  _e\u229d_ \u2205 \u2205 = \u2205\n  _e\u229d_ (v\u2082 \u2022 \u03c1\u2082) (v\u2081 \u2022 \u03c1\u2081) = v\u2082 \u229d v\u2081 \u2022 v\u2081 \u2022 \u03c1\u2082 e\u229d \u03c1\u2081\n\n  isEnvCompCS : \u2200 \u0393 \u2192 IsCompChangeStructure \u27e6 \u0393 \u27e7Context (Ch\u0393 \u0393) [ \u0393 ]\u0393_from_to_\n\n  efromto\u2192\u2295 : \u2200 {\u0393} (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192\n      [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192 (\u03c11 e\u2295 d\u03c1) \u2261 \u03c12\n  efromto\u2192\u2295 .\u2205 .\u2205 .\u2205 v\u2205 = refl\n  efromto\u2192\u2295 .(_ \u2022 _ \u2022 _) .(_ \u2022 _) .(_ \u2022 _) (dvv v\u2022 d\u03c1\u03c1) =\n    cong\u2082 _\u2022_ (fromto\u2192\u2295 _ _ _ dvv) (efromto\u2192\u2295 _ _ _ d\u03c1\u03c1)\n\n  e\u229d-fromto : \u2200 {\u0393} \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 \u03c12 e\u229d \u03c11 from \u03c11 to \u03c12\n  e\u229d-fromto \u2205 \u2205 = v\u2205\n  e\u229d-fromto (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) = \u229d-fromto v1 v2 v\u2022 e\u229d-fromto \u03c11 \u03c12\n\n  _e\u229a[_]_ : \u2200 {\u0393} \u2192 Ch\u0393 \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0393 \u0393 \u2192 Ch\u0393 \u0393\n  _e\u229a[_]_ {\u2205} \u2205 \u2205 \u2205 = \u2205\n  _e\u229a[_]_ {\u03c4 \u2022 \u0393} (dv1 \u2022 _ \u2022 d\u03c11) (v1 \u2022 \u03c11) (dv2 \u2022 _ \u2022 d\u03c12) = (dv1 \u229a[ v1 ] dv2) \u2022 v1 \u2022 (d\u03c11 e\u229a[ \u03c11 ] d\u03c12)\n\n  e\u229a-fromto : \u2200 \u0393 \u2192 (\u03c11 \u03c12 \u03c13 : \u27e6 \u0393 \u27e7Context) (d\u03c11 d\u03c12 : Ch\u0393 \u0393) \u2192\n    [ \u0393 ]\u0393 d\u03c11 from \u03c11 to \u03c12 \u2192\n    [ \u0393 ]\u0393 d\u03c12 from \u03c12 to \u03c13 \u2192 [ \u0393 ]\u0393 (d\u03c11 e\u229a[ \u03c11 ] d\u03c12) from \u03c11 to \u03c13\n  e\u229a-fromto \u2205 \u2205 \u2205 \u2205 \u2205 \u2205 v\u2205 v\u2205 = v\u2205\n  e\u229a-fromto (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (v3 \u2022 \u03c13)\n    (dv1 \u2022 (.v1 \u2022 d\u03c11)) (dv2 \u2022 (.v2 \u2022 d\u03c12))\n    (dvv1 v\u2022 d\u03c1\u03c11) (dvv2 v\u2022 d\u03c1\u03c12) =\n        \u229a-fromto v1 v2 v3 dv1 dv2 dvv1 dvv2\n      v\u2022\n        e\u229a-fromto \u0393 \u03c11 \u03c12 \u03c13 d\u03c11 d\u03c12 d\u03c1\u03c11 d\u03c1\u03c12\n\n  isEnvCompCS \u0393 = record\n    { isChangeStructure = record\n      { _\u2295_ = _e\u2295_\n      ; fromto\u2192\u2295 = efromto\u2192\u2295\n      ; _\u229d_ = _e\u229d_\n      ; \u229d-fromto = e\u229d-fromto\n      }\n    ; _\u229a[_]_ = _e\u229a[_]_\n    ; \u229a-fromto = e\u229a-fromto \u0393\n    }\n\nchangeStructure\u0393 : \u2200 \u0393 \u2192 ChangeStructure \u27e6 \u0393 \u27e7Context\nchangeStructure\u0393 \u0393 = record { isCompChangeStructure = isEnvCompCS \u0393 }\n\ninstance\n  ichangeStructure\u0393 : \u2200 {\u0393} \u2192 ChangeStructure \u27e6 \u0393 \u27e7Context\n  ichangeStructure\u0393 {\u0393} = changeStructure\u0393 \u0393\n\nchangeStructure\u0393\u03c4 : \u2200 \u0393 \u03c4 \u2192 ChangeStructure (\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type)\nchangeStructure\u0393\u03c4 \u0393 \u03c4 = funCS\n\ninstance\n  ichangeStructure\u0393\u03c4 : \u2200 {\u0393 \u03c4} \u2192 ChangeStructure (\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type)\n  ichangeStructure\u0393\u03c4 {\u0393} {\u03c4} = changeStructure\u0393\u03c4 \u0393 \u03c4\n","old_contents":"module Thesis.LangChanges where\n\nopen import Thesis.Changes\nopen import Thesis.IntChanges\nopen import Thesis.Lang\nopen import Relation.Binary.PropositionalEquality\n\nCh\u03c4 : (\u03c4 : Type) \u2192 Set\nCh\u03c4 \u03c4 = \u27e6 \u0394t \u03c4 \u27e7Type\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Ch\u03c4 \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\nisCompChangeStructure\u03c4 : \u2200 \u03c4 \u2192 IsCompChangeStructure \u27e6 \u03c4 \u27e7Type \u27e6 \u0394t \u03c4 \u27e7Type [ \u03c4 ]\u03c4_from_to_\n\nchangeStructure\u03c4 : \u2200 \u03c4 \u2192 ChangeStructure \u27e6 \u03c4 \u27e7Type\nchangeStructure\u03c4 \u03c4 = record { isCompChangeStructure = isCompChangeStructure\u03c4 \u03c4 }\n\ninstance\n  ichangeStructure\u03c4 : \u2200 {\u03c4} \u2192 ChangeStructure \u27e6 \u03c4 \u27e7Type\n  ichangeStructure\u03c4 {\u03c4} = changeStructure\u03c4 \u03c4\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Ch\u03c4 \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v1 + dv \u2261 v2\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sch_from_to_ dv v1 v2\n\nisCompChangeStructure\u03c4 (\u03c3 \u21d2 \u03c4) = ChangeStructure.isCompChangeStructure (funCS {{changeStructure\u03c4 \u03c3}} {{changeStructure\u03c4 \u03c4}})\nisCompChangeStructure\u03c4 int = ChangeStructure.isCompChangeStructure intCS\nisCompChangeStructure\u03c4 (pair \u03c3 \u03c4) = ChangeStructure.isCompChangeStructure (pairCS {{changeStructure\u03c4 \u03c3}} {{changeStructure\u03c4 \u03c4}})\nisCompChangeStructure\u03c4 (sum \u03c3 \u03c4) = ChangeStructure.isCompChangeStructure ((sumCS {{changeStructure\u03c4 \u03c3}} {{changeStructure\u03c4 \u03c4}}))\n\nopen import Data.Unit\n-- We can define an alternative semantics for contexts, that defines environments as nested products:\n\u27e6_\u27e7Context-prod : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context-prod = \u22a4\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context-prod = \u27e6 \u03c4 \u27e7Type \u00d7 \u27e6 \u0393 \u27e7Context-prod\n\n-- And define a change structure for such environments, reusing the change\n-- structure constructor for pairs. However, this change structure does not\n-- duplicate base values in the environment. We probably *could* define a\n-- different change structure for pairs that gives the right result.\nchangeStructure\u0393-old : \u2200 \u0393 \u2192 ChangeStructure \u27e6 \u0393 \u27e7Context-prod\nchangeStructure\u0393-old \u2205 = unitCS\nchangeStructure\u0393-old (\u03c4 \u2022 \u0393) = pairCS {{changeStructure\u03c4 \u03c4}}  {{changeStructure\u0393-old \u0393}}\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nmodule _ where\n  _e\u2295_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0393 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n  _e\u2295_ \u2205 \u2205 = \u2205\n  _e\u2295_ (_ \u2022 \u03c1) (dv \u2022 v \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c1 e\u2295 d\u03c1\n  _e\u229d_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0393 \u0393\n  _e\u229d_ \u2205 \u2205 = \u2205\n  _e\u229d_ (v\u2082 \u2022 \u03c1\u2082) (v\u2081 \u2022 \u03c1\u2081) = v\u2082 \u229d v\u2081 \u2022 v\u2081 \u2022 \u03c1\u2082 e\u229d \u03c1\u2081\n\n  isEnvCompCS : \u2200 \u0393 \u2192 IsCompChangeStructure \u27e6 \u0393 \u27e7Context (Ch\u0393 \u0393) [ \u0393 ]\u0393_from_to_\n\n  efromto\u2192\u2295 : \u2200 {\u0393} (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192\n      [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192 (\u03c11 e\u2295 d\u03c1) \u2261 \u03c12\n  efromto\u2192\u2295 .\u2205 .\u2205 .\u2205 v\u2205 = refl\n  efromto\u2192\u2295 .(_ \u2022 _ \u2022 _) .(_ \u2022 _) .(_ \u2022 _) (dvv v\u2022 d\u03c1\u03c1) =\n    cong\u2082 _\u2022_ (fromto\u2192\u2295 _ _ _ dvv) (efromto\u2192\u2295 _ _ _ d\u03c1\u03c1)\n\n  e\u229d-fromto : \u2200 {\u0393} \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 [ \u0393 ]\u0393 \u03c12 e\u229d \u03c11 from \u03c11 to \u03c12\n  e\u229d-fromto \u2205 \u2205 = v\u2205\n  e\u229d-fromto (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) = \u229d-fromto v1 v2 v\u2022 e\u229d-fromto \u03c11 \u03c12\n\n  _e\u229a[_]_ : \u2200 {\u0393} \u2192 Ch\u0393 \u0393 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0393 \u0393 \u2192 Ch\u0393 \u0393\n  _e\u229a[_]_ {\u2205} \u2205 \u2205 \u2205 = \u2205\n  _e\u229a[_]_ {\u03c4 \u2022 \u0393} (dv1 \u2022 _ \u2022 d\u03c11) (v1 \u2022 \u03c11) (dv2 \u2022 _ \u2022 d\u03c12) = (dv1 \u229a[ v1 ] dv2) \u2022 v1 \u2022 (d\u03c11 e\u229a[ \u03c11 ] d\u03c12)\n\n  e\u229a-fromto : \u2200 \u0393 \u2192 (\u03c11 \u03c12 \u03c13 : \u27e6 \u0393 \u27e7Context) (d\u03c11 d\u03c12 : Ch\u0393 \u0393) \u2192\n    [ \u0393 ]\u0393 d\u03c11 from \u03c11 to \u03c12 \u2192\n    [ \u0393 ]\u0393 d\u03c12 from \u03c12 to \u03c13 \u2192 [ \u0393 ]\u0393 (d\u03c11 e\u229a[ \u03c11 ] d\u03c12) from \u03c11 to \u03c13\n  e\u229a-fromto \u2205 \u2205 \u2205 \u2205 \u2205 \u2205 v\u2205 v\u2205 = v\u2205\n  e\u229a-fromto (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (v3 \u2022 \u03c13)\n    (dv1 \u2022 (.v1 \u2022 d\u03c11)) (dv2 \u2022 (.v2 \u2022 d\u03c12))\n    (dvv1 v\u2022 d\u03c1\u03c11) (dvv2 v\u2022 d\u03c1\u03c12) =\n        \u229a-fromto v1 v2 v3 dv1 dv2 dvv1 dvv2\n      v\u2022\n        e\u229a-fromto \u0393 \u03c11 \u03c12 \u03c13 d\u03c11 d\u03c12 d\u03c1\u03c11 d\u03c1\u03c12\n\n  isEnvCompCS \u0393 = record\n    { isChangeStructure = record\n      { _\u2295_ = _e\u2295_\n      ; fromto\u2192\u2295 = efromto\u2192\u2295\n      ; _\u229d_ = _e\u229d_\n      ; \u229d-fromto = e\u229d-fromto\n      }\n    ; _\u229a[_]_ = _e\u229a[_]_\n    ; \u229a-fromto = e\u229a-fromto \u0393\n    }\n\nchangeStructure\u0393 : \u2200 \u0393 \u2192 ChangeStructure \u27e6 \u0393 \u27e7Context\nchangeStructure\u0393 \u0393 = record { isCompChangeStructure = isEnvCompCS \u0393 }\n\ninstance\n  ichangeStructure\u0393 : \u2200 {\u0393} \u2192 ChangeStructure \u27e6 \u0393 \u27e7Context\n  ichangeStructure\u0393 {\u0393} = changeStructure\u0393 \u0393\n\nchangeStructure\u0393\u03c4 : \u2200 \u0393 \u03c4 \u2192 ChangeStructure (\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type)\nchangeStructure\u0393\u03c4 \u0393 \u03c4 = funCS\n\ninstance\n  ichangeStructure\u0393\u03c4 : \u2200 {\u0393 \u03c4} \u2192 ChangeStructure (\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type)\n  ichangeStructure\u0393\u03c4 {\u0393} {\u03c4} = changeStructure\u0393\u03c4 \u0393 \u03c4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1e32a2c333bf5790343d317dd505b4f1e5753453","subject":"Start to formalize changes (WIP).","message":"Start to formalize changes (WIP).\n\nThis is intended to be the approach Paolo describes in the last\nsubsection of alternative-defs-proofs.md.\n\nOld-commit-hash: 68ae9d3ef37f16aa12f3cc6d85367de6aeb73ca7\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","old_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"91350be8effc67083cb6387219634ced37b99c8a","subject":"Desc model: generic proof that map id = id.","message":"Desc model: generic proof that map id = id.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A : Set}{B : A -> Set}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","old_contents":"\n {-# OPTIONS --type-in-type #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : (D : Desc)(X : Set)(P : X -> Set) -> [| D |] X -> Set\nAll id          X P x        = P x\nAll (const Z)   X P x        = Z\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = z\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\nmodule Elim (D : Desc)\n            (P : Mu D -> Set)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = z\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Suc : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsuc : Nat -> Nat\nsuc n = con (Suc , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n--********************************************\n-- Finite sets\n--********************************************\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = suc e\n\n{-\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n-}\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (P' : EnumT e -> Set) -> Set) xs -> \n           (P' : EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Suc , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P = induction NatD (\\e -> (P : EnumT e -> Set) -> Set) casesSpi e  P\n\n{-\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n-}\n\n\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set)(b' : spi e P')(x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set)(b' : spi (con xs) P')(x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Suc , n) hs P' b' EZe = fst b'\ncasesSwitch (Suc , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch e P b x = induction NatD (\\e -> (P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x) casesSwitch e P b x\n\n{-\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n--********************************************\n-- Tagged description\n--********************************************\n\nTagDesc : Set\nTagDesc = Sigma EnumU (\\e -> spi e (\\_ -> Desc))\n\ntoDesc : TagDesc -> Desc\ntoDesc (B , F) = sigma (EnumT B) (\\e -> switch B (\\_ -> Desc) F e)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ba8ac9fe26eb0d3dafab4c11ec603e11b285f4e9","subject":"Vec: map2*-++","message":"Vec: map2*-++\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (Type)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd; map to \u00d7-map)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; _\u2219_; !_) renaming (refl to idp)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : Type a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 Type a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : Type a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Type a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Type a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id= : \u2200 {a n} {A : Type a}{f : A \u2192 A}\n             \u2192 f \u2257 id \u2192 map f \u2257 id {A = Vec A n}\n    map-id= f= []       = idp\n    map-id= f= (x \u2237 xs) = ap-\u2237 (f= x) (map-id= f= xs)\n\n    map-id : \u2200 {a n} {A : Type a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id = map-id= (\u03bb _ \u2192 idp)\n\n    map-\u2218= : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n               {f : B \u2192 C}{g : A \u2192 B}{h : A \u2192 C} \u2192\n               f \u2218 g \u2257 h \u2192\n               _\u2257_ {A = Vec A n} (map f \u2218 map g) (map h)\n    map-\u2218= fg= []       = idp\n    map-\u2218= fg= (x \u2237 xs) = ap-\u2237 (fg= x) (map-\u2218= fg= xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n              (f : B \u2192 C) (g : A \u2192 B) \u2192\n              _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g v = ! map-\u2218= (\u03bb _ \u2192 idp) v\n\n    map-ext : \u2200 {a b} {A : Type a} {B : Type b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : Type a}\n    (_\u2248_ : A \u2192 A \u2192 Type \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Type \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 Type _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : Type a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= idp idp = idp\n\nmodule With\u2261 {a}{A : Type a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                           ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Type a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Type a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Type a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\ncount\u1da0 : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : Type a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : Type a} where\n  count-++ : \u2200 {m n} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m b} {B : Type b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , idp = idp\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , idp = idp\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  map2*-++ : \u2200 {m n}(f : Vec A m \u2192 Vec A n) \u2192 map2* m n f \u2218 uncurry _++_ \u2257 uncurry _++_ \u2218 \u00d7-map f f\n  map2*-++ {m} f (x , y) rewrite take-++ m x y | drop-++ m x y = idp\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\n  vec= : \u2200 {n}{xs ys : Vec A n} \u2192 (\u2200 i \u2192 (xs \u203c i) \u2261 (ys \u203c i)) \u2192 xs \u2261 ys\n  vec= {xs = []}     {[]}     f= = idp\n  vec= {xs = x \u2237 xs} {y \u2237 ys} f= = ap-\u2237 (f= zero) (vec= (f= \u2218 suc))\n\n  concat-group : \u2200 m n (xs : Vec A (m * n)) \u2192 concat (group m n xs) \u2261 xs\n  concat-group zero    n [] = idp\n  concat-group (suc m) n xs rewrite concat-group m n (drop n xs) = take-drop-lem n xs\n\n  group-concat : \u2200 {m n}(xss : Vec (Vec A n) m) \u2192 group m n (concat xss) \u2261 xss\n  group-concat [] = idp\n  group-concat (xs \u2237 xss)\n    rewrite take-++ _ xs (concat xss)\n          | drop-++ _ xs (concat xss)\n          | group-concat xss\n          = idp\n\n  -- These are also in the stdlib as Data.Vec.Properties.lookup-morphism\n  \u203c-replicate : \u2200 {n}(i : Fin n)(x : A) \u2192 (replicate x \u203c i) \u2261 x\n  \u203c-replicate zero    = \u03bb _ \u2192 idp\n  \u203c-replicate (suc i) = \u203c-replicate i\n\nmodule _ {a b}{A : Type a}{B : Type b} where\n  \u203c-\u229b= : \u2200 {n} i {fs : Vec (A \u2192 B) n}{xs : Vec A n}{f x}\n           (fs= : (fs \u203c i) \u2261 f)\n           (xs= : (xs \u203c i) \u2261 x)\n         \u2192 (fs \u229b xs \u203c i) \u2261 f x\n  \u203c-\u229b= zero    {f \u2237 fs} {x \u2237 xs} f= xs= = ap\u2082 _$_ f= xs=\n  \u203c-\u229b= (suc i) {f \u2237 fs} {x \u2237 xs} f= xs= = \u203c-\u229b= i f= xs=\n\n  \u203c-\u229b : \u2200 {n} i (fs : Vec (A \u2192 B) n) (xs : Vec A n) \u2192\n          (fs \u229b xs \u203c i) \u2261 (fs \u203c i) (xs \u203c i)\n  \u203c-\u229b i fs xs = \u203c-\u229b= i idp idp\n\n  \u203c-map= : \u2200 {n} i (f : A \u2192 B) {xs : Vec A n}{x : A}\n             (xs= : (xs \u203c i) \u2261 x)\n           \u2192 (map f xs \u203c i) \u2261 f x\n  \u203c-map= i f = \u203c-\u229b= i (\u203c-replicate i f)\n\n  \u203c-map : \u2200 {n} i (f : A \u2192 B) (xs : Vec A n) \u2192\n            (map f xs \u203c i) \u2261 f (xs \u203c i)\n  \u203c-map i f xs = \u203c-map= i f idp\n\n  vuncurry : \u2200 {n}(f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\n  vuncurry f (x \u2237 xs) = f x xs\n\n  count-\u2218 : \u2200 {n}(f : A \u2192 B)(pred : B \u2192 Bool) \u2192\n              count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\n  count-\u2218 f pred [] = idp\n  count-\u2218 f pred (x \u2237 xs) with pred (f x)\n  ... | 1b rewrite count-\u2218 f pred xs = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                 | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                 | count-\u2218 f pred xs = idp\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (Type)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; 2*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_; concat)\nimport Data.Fin.Properties as F\nopen import Data.Bit\nopen import Data.Bool using (Bool; if_then_else_)\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; ap\u2082; _\u2219_; !_) renaming (refl to idp)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : Type a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 Type a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : Type a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : Type a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Type a} {B : Type b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : Type a} {B : Type b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : Type a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id= : \u2200 {a n} {A : Type a}{f : A \u2192 A}\n             \u2192 f \u2257 id \u2192 map f \u2257 id {A = Vec A n}\n    map-id= f= []       = idp\n    map-id= f= (x \u2237 xs) = ap-\u2237 (f= x) (map-id= f= xs)\n\n    map-id : \u2200 {a n} {A : Type a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id = map-id= (\u03bb _ \u2192 idp)\n\n    map-\u2218= : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n               {f : B \u2192 C}{g : A \u2192 B}{h : A \u2192 C} \u2192\n               f \u2218 g \u2257 h \u2192\n               _\u2257_ {A = Vec A n} (map f \u2218 map g) (map h)\n    map-\u2218= fg= []       = idp\n    map-\u2218= fg= (x \u2237 xs) = ap-\u2237 (fg= x) (map-\u2218= fg= xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : Type a} {B : Type b} {C : Type c}\n              (f : B \u2192 C) (g : A \u2192 B) \u2192\n              _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g v = ! map-\u2218= (\u03bb _ \u2192 idp) v\n\n    map-ext : \u2200 {a b} {A : Type a} {B : Type b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule With\u2248\n    {a \u2113 \u2113'}{A : Type a}\n    (_\u2248_ : A \u2192 A \u2192 Type \u2113)\n    {_\u2248\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Type \u2113'}\n    ([]-cong  : [] \u2248\u1d5b [])\n    (_\u2237-cong_ : \u2200 {n} {x\u00b9} {xs\u00b9 : Vec A n} {x\u00b2} {xs\u00b2 : Vec A n}\n                  (x\u00b9\u2248x\u00b2 : x\u00b9 \u2248 x\u00b2) (xs\u00b9\u2248xs\u00b2 : xs\u00b9 \u2248\u1d5b xs\u00b2) \u2192\n                  (x\u00b9 \u2237 xs\u00b9) \u2248\u1d5b (x\u00b2 \u2237 xs\u00b2))\n  where\n  open Algebra.FunctionProperties\n\n  module _ {_\u2219_ _\u2219\u2219_ : A \u2192 A \u2192 A}\n           (assoc : \u2200 x y z \u2192 ((x \u2219 y) \u2219\u2219 z) \u2248 (x \u2219 (y \u2219\u2219 z))) where\n    private\n      _\u2219\u1d5b_ _\u2219\u2219\u1d5b_ : \u2200 {n}(xs ys : Vec A n) \u2192 Vec A n\n      _\u2219\u1d5b_  = zipWith _\u2219_\n      _\u2219\u2219\u1d5b_ = zipWith _\u2219\u2219_\n\n    zipWith-assoc : \u2200 {n} (xs ys zs : Vec A n) \u2192 ((xs \u2219\u1d5b ys) \u2219\u2219\u1d5b zs) \u2248\u1d5b (xs \u2219\u1d5b (ys \u2219\u2219\u1d5b zs))\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = assoc x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} where\n    module _ (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n      zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n      zipWith-comm [] [] = []-cong\n      zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n    module _ {\u03b5 : A} where\n      module _ (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-left [] = []-cong\n        zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n      module _ (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n        zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n        zipWith-id-right [] = []-cong\n        zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-left-inverse [] = []-cong\n        zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n      module _ {_\u207b\u00b9 : A \u2192 A}\n               (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n        zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n        zipWith-right-inverse [] = []-cong\n        zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 Type _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  open With\u2248 _\u2248_ {_\u2248\u1d5b_} []-cong (\u03bb x y \u2192 x \u2237-cong y) public\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n\u2237= : \u2200 {a}{A : Type a}{n x} {xs : Vec A n} {y} {ys : Vec A n}\n       (p : x \u2261 y) (q : xs \u2261 ys) \u2192\n        x \u2237 xs \u2261 y \u2237 ys\n\u2237= idp idp = idp\n\nmodule With\u2261 {a}{A : Type a} where\n  open With\u2248 (_\u2261_ {A = A}) {_\u2261_} idp (\u03bb x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2 \u2192 \u2237= x\u00b9\u2248x\u00b2 xs\u00b9\u2248xs\u00b2) public\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                           ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\nopen Algebra.FunctionProperties.Eq.Implicits\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : Type a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : Type a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : Type a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\ncount\u1da0 : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Type a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nmodule _ {a} {A : Type a} where\n  -- Exchange elements at positions 0 and 1 of a given vector\n  -- (this only apply if the vector is long enough).\n  0\u21941 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n  0\u21941 xs = xs\n\nmodule _ {a} {A : Type a} where\n  count-++ : \u2200 {m n} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n              \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\n  count-++ pred [] ys = idp\n  count-++ pred (x \u2237 xs) ys with pred x\n  ... | 1b rewrite count-++ pred xs ys = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                    | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\n  ext-count\u1da0 : \u2200 {n} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\n  ext-count\u1da0 f\u2257g [] = idp\n  ext-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\n  filter : \u2200 {n} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\n  filter pred [] = []\n  filter pred (x \u2237 xs) with pred x\n  ... | 1b = x \u2237 filter pred xs\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n  transpose : \u2200 {m n} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\n  transpose [] = replicate []\n  transpose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\n  vap : \u2200 {m b} {B : Type b} (f : Vec A m \u2192 B)\n          \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\n  vap f = map f \u2218 transpose\n\n  infixl 2 _\u203c_\n  _\u203c_ : \u2200 {n} \u2192 Vec A n \u2192 Fin n \u2192 A\n  _\u203c_ = flip lookup\n\n  \u03b7 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7 = tabulate \u2218 _\u203c_\n\n  \u03b7\u2032 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  \u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n  \u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n  shallow-\u03b7 : \u2200 {n} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\n  shallow-\u03b7 (x \u2237 xs) = idp\n\n  uncons : \u2200 {n} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\n  \u2237-uncons : \u2200 {n} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n  \u2237-uncons (x \u2237 xs) = idp\n\n  splitAt\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\n  splitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  group : \u2200 n k \u2192 Vec A (n * k) \u2192 Vec (Vec A k) n\n  group zero    k v = []\n  group (suc n) k v = take k v \u2237 group n k (drop k v)\n\n  map2* : \u2200 m n (f : Vec A m \u2192 Vec A n) \u2192 Vec A (2* m) \u2192 Vec A (2* n)\n  map2* m _ f xs = f (take m xs) ++ f (drop m xs)\n\n  map* : \u2200 k {m n}(f : Vec A m \u2192 Vec A n) \u2192 Vec A (k * m) \u2192 Vec A (k * n)\n  map* k f xs = concat (map f (group k _ xs))\n\n  take2* : \u2200 n \u2192 Bit \u2192 Vec A (2* n) \u2192 Vec A n\n  take2* n 0b v = take n v\n  take2* n 1b v = drop n v\n\n  ++-decomp : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n  ++-decomp {zero} {xs = []} {[]} p = idp , p\n  ++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n  ... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n  ++-inj\u2081 : \u2200 {m n} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n  ++-inj\u2081 eq = fst (++-decomp eq)\n\n  ++-inj\u2082 : \u2200 {m n} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n  ++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\n  take-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  drop-\u2237 : \u2200 {m} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , idp = idp\n\n  take-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\n  take-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  take-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2081 eq)\n\n  drop-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\n  drop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n  ... | zs | eq with splitAt m zs\n  drop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , idp = !(++-inj\u2082 xs\u2081 xs eq)\n\n  take-drop-lem : \u2200 m {n} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\n  take-drop-lem m xs with splitAt m xs\n  take-drop-lem m .(ys ++ zs) | ys , zs , idp = idp\n\n  take-them-all : \u2200 n (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\n  take-them-all n xs with splitAt n xs\n  take-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\n  drop\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A n\n  drop\u2032 zero    = id\n  drop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\n  drop\u2032-spec : \u2200 m {n} \u2192 drop\u2032 m {n} \u2257 drop m {n}\n  drop\u2032-spec zero    _        = idp\n  drop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\n  take\u2032 : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A m\n  take\u2032 zero    _  = []\n  take\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\n  take\u2032-spec : \u2200 m {n} \u2192 take\u2032 m {n} \u2257 take m {n}\n  take\u2032-spec zero    xs       = idp\n  take\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\n  swap : \u2200 m {n} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\n  swap m xs = drop m xs ++ take m xs\n\n  swap-++ : \u2200 m {n} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\n  swap-++ m xs ys = ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (_\u203c_ v \u2218 f)\n\n  take-drop= : \u2200 m {n} (xs ys : Vec A (m + n))\n                 \u2192 take m xs \u2261 take m ys\n                 \u2192 drop m xs \u2261 drop m ys\n                 \u2192 xs \u2261 ys\n  take-drop= m xs ys take= drop= =\n    ! take-drop-lem m xs \u2219 ap\u2082 _++_ take= drop= \u2219 take-drop-lem m ys\n\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = map (_\u203c_ v) tbl\n\n  on\u1d62 : \u2200 (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\n  on\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\n  on\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n  \u229b-dist-0\u21941 : \u2200 {n}(fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n  \u229b-dist-0\u21941 _           []          = idp\n  \u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n  \u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\n  map-tail : \u2200 {m n} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\n  map-tail f (x \u2237 xs) = x \u2237 f xs\n\n  map-tail-id : \u2200 {n} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\n  map-tail-id (x \u2237 xs) = idp\n\n  map-tail\u2218map-tail : \u2200 {m n o}\n                        (f : Vec A o \u2192 Vec A m)\n                        (g : Vec A n \u2192 Vec A o)\n                      \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\n  map-tail\u2218map-tail f g (x \u2237 xs) = idp\n\n  map-tail-\u2257 : \u2200 {m n}(f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\n  map-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\n  dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n  dup xs = xs ++ xs\n\n  dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n  dup-inj = ++-inj\u2081\n\n  rot\u2081 : \u2200 {n} \u2192 Vec A n \u2192 Vec A n\n  rot\u2081 []       = []\n  rot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\n  rot : \u2200 {n} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\n  rot zero    xs = xs\n  rot (suc n) xs = rot n (rot\u2081 xs)\n\n  map-\u2237\u02b3 : \u2200 {n}(f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\n  map-\u2237\u02b3 f x []       = idp\n  map-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\n  sum-map-rot\u2081 : \u2200 {n}(f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\n  sum-map-rot\u2081 f [] = idp\n  sum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                          \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                          \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\n  vec= : \u2200 {n}{xs ys : Vec A n} \u2192 (\u2200 i \u2192 (xs \u203c i) \u2261 (ys \u203c i)) \u2192 xs \u2261 ys\n  vec= {xs = []}     {[]}     f= = idp\n  vec= {xs = x \u2237 xs} {y \u2237 ys} f= = ap-\u2237 (f= zero) (vec= (f= \u2218 suc))\n\n  concat-group : \u2200 m n (xs : Vec A (m * n)) \u2192 concat (group m n xs) \u2261 xs\n  concat-group zero    n [] = idp\n  concat-group (suc m) n xs rewrite concat-group m n (drop n xs) = take-drop-lem n xs\n\n  group-concat : \u2200 {m n}(xss : Vec (Vec A n) m) \u2192 group m n (concat xss) \u2261 xss\n  group-concat [] = idp\n  group-concat (xs \u2237 xss)\n    rewrite take-++ _ xs (concat xss)\n          | drop-++ _ xs (concat xss)\n          | group-concat xss\n          = idp\n\n  -- These are also in the stdlib as Data.Vec.Properties.lookup-morphism\n  \u203c-replicate : \u2200 {n}(i : Fin n)(x : A) \u2192 (replicate x \u203c i) \u2261 x\n  \u203c-replicate zero    = \u03bb _ \u2192 idp\n  \u203c-replicate (suc i) = \u203c-replicate i\n\nmodule _ {a b}{A : Type a}{B : Type b} where\n  \u203c-\u229b= : \u2200 {n} i {fs : Vec (A \u2192 B) n}{xs : Vec A n}{f x}\n           (fs= : (fs \u203c i) \u2261 f)\n           (xs= : (xs \u203c i) \u2261 x)\n         \u2192 (fs \u229b xs \u203c i) \u2261 f x\n  \u203c-\u229b= zero    {f \u2237 fs} {x \u2237 xs} f= xs= = ap\u2082 _$_ f= xs=\n  \u203c-\u229b= (suc i) {f \u2237 fs} {x \u2237 xs} f= xs= = \u203c-\u229b= i f= xs=\n\n  \u203c-\u229b : \u2200 {n} i (fs : Vec (A \u2192 B) n) (xs : Vec A n) \u2192\n          (fs \u229b xs \u203c i) \u2261 (fs \u203c i) (xs \u203c i)\n  \u203c-\u229b i fs xs = \u203c-\u229b= i idp idp\n\n  \u203c-map= : \u2200 {n} i (f : A \u2192 B) {xs : Vec A n}{x : A}\n             (xs= : (xs \u203c i) \u2261 x)\n           \u2192 (map f xs \u203c i) \u2261 f x\n  \u203c-map= i f = \u203c-\u229b= i (\u203c-replicate i f)\n\n  \u203c-map : \u2200 {n} i (f : A \u2192 B) (xs : Vec A n) \u2192\n            (map f xs \u203c i) \u2261 f (xs \u203c i)\n  \u203c-map i f xs = \u203c-map= i f idp\n\n  vuncurry : \u2200 {n}(f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\n  vuncurry f (x \u2237 xs) = f x xs\n\n  count-\u2218 : \u2200 {n}(f : A \u2192 B)(pred : B \u2192 Bool) \u2192\n              count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\n  count-\u2218 f pred [] = idp\n  count-\u2218 f pred (x \u2237 xs) with pred (f x)\n  ... | 1b rewrite count-\u2218 f pred xs = idp\n  ... | 0b rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                 | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                 | count-\u2218 f pred xs = idp\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative; group)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"49b349b2f6df97c82e223c7ba03c3d80d14fdd82","subject":"Removed FOTC dependency in issue #8.","message":"Removed FOTC dependency in issue #8.\n","repos":"asr\/apia,asr\/apia","old_file":"issues\/Issue8.agda","new_file":"issues\/Issue8.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issue8 where\n\npostulate D : Set\n\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\npostulate foo : (A : D \u2192 Set) \u2192 (\u2203 \u03bb x \u2192 A x) \u2192 (\u2203 \u03bb x \u2192 A x)\n{-# ATP prove foo #-}\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issue8 where\n\nopen import Common.FOL.FOL\n\npostulate foo : (A : D \u2192 Set) \u2192 (\u2203 \u03bb x \u2192 A x) \u2192 (\u2203 \u03bb x \u2192 A x)\n{-# ATP prove foo #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7ee4c1bb72bd8f965845534b0af5c05d5163e3ac","subject":"every case except for holes, which require thinking about substitition things to cough up the type and i dont know that my encoding of that is correct yet","message":"every case except for holes, which require thinking about substitition things to cough up the type and i dont know that my encoding of that is correct yet\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"type-assignment-unicity.agda","new_file":"type-assignment-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule type-assignment-unicity where\n  type-assignment-unicity : {\u0393 : tctx} {d : dhexp} {\u03c4' \u03c4 : htyp} {\u0394 : hctx} \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                              \u03c4 == \u03c4'\n  type-assignment-unicity TAConst TAConst = refl\n  type-assignment-unicity {\u0393 = \u0393} (TAVar x\u2081) (TAVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082\n  type-assignment-unicity (TALam d1) (TALam d2)\n    with type-assignment-unicity d1 d2\n  ... | refl = refl\n  type-assignment-unicity (TAAp d3 MAHole d4) (TAAp d5 MAHole d6) = refl\n  type-assignment-unicity (TAAp d3 MAHole d4) (TAAp d5 MAArr d6)\n      with type-assignment-unicity d3 d5\n  ... | ()\n  type-assignment-unicity (TAAp d3 MAArr d4) (TAAp d5 MAHole d6)\n    with type-assignment-unicity d3 d5\n  ... | ()\n  type-assignment-unicity (TAAp d3 MAArr d4) (TAAp d5 MAArr d6)\n    with type-assignment-unicity d3 d5 | type-assignment-unicity d4 d6\n  ... | refl | refl = refl\n  type-assignment-unicity (TAEHole x) d2 = {!!}\n  type-assignment-unicity (TANEHole d1 x) d2 = {!!}\n  type-assignment-unicity (TACast d1 x) (TACast d2 x\u2081)\n    with type-assignment-unicity d1 d2\n  ... | refl = refl\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule type-assignment-unicity where\n  type-assignment-unicity : (\u0393 : tctx) (d : dhexp) (\u03c4' \u03c4 : htyp) (\u0394 : hctx) \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n                              \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                              \u03c4 == \u03c4'\n  type-assignment-unicity = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dd40e9080437a9a681f55282bbd3b79a51c7b3d5","subject":"+-interchange is now with implicit arguments...","message":"+-interchange is now with implicit arguments...\n","repos":"crypto-agda\/crypto-agda","old_file":"Neglible.agda","new_file":"Neglible.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {fN k} {gD k} {f'D k} {g'D k} \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange {gN k} {fD k} {g'D k} {f'D k})\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange {f'N k} {fD k} {g'D k} {gD k})\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange {g'N k} {gD k} {f'D k} {fD k}\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Algebra\n\nopen import Function\n\nopen import Data.Nat.NP\nopen import Data.Nat.Distance\nopen import Data.Nat.Properties\nopen import Data.Two\nopen import Data.Zero\nopen import Data.Product\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import HoTT\nopen Equivalences\n\nopen import Explore.Core\nopen import Explore.Universe.Type {\ud835\udfd8}\nopen import Explore.Universe.Base\n\nmodule Neglible where\n\nmodule prop = CommutativeSemiring commutativeSemiring\nmodule OR = Poset (DecTotalOrder.poset decTotalOrder)\n\n\u2264-*-cancel : \u2200 {x m n} \u2192 1 \u2264 x \u2192  x * m \u2264 x * n \u2192 m \u2264 n\n\u2264-*-cancel {suc x} {m} {n} (s\u2264s le) mn\n  rewrite prop.*-comm (suc x) m | prop.*-comm (suc x) n = cancel-*-right-\u2264 _ _ _ mn\n\nrecord \u2115\u2192\u211a : Set where\n  constructor _\/_[_]\n  field\n    \u03b5N : (n : \u2115) \u2192 \u2115\n    \u03b5D : (n : \u2115) \u2192 \u2115\n    \u03b5D-pos : \u2200 n \u2192 \u03b5D n > 0\n\nrecord Is-Neg (\u03b5 : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a \u03b5\n  field\n    c\u2099 : (c : \u2115) \u2192 \u2115\n    prf : \u2200(c n : \u2115) \u2192 n > c\u2099 n \u2192 n ^ c * \u03b5N n \u2264 \u03b5D n\nopen Is-Neg\n\n0\u2115\u211a : \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N 0\u2115\u211a _ = 0\n\u2115\u2192\u211a.\u03b5D 0\u2115\u211a _ = 1\n\u2115\u2192\u211a.\u03b5D-pos 0\u2115\u211a _ = s\u2264s z\u2264n\n\n0\u2115\u211a-neg : Is-Neg 0\u2115\u211a\nc\u2099 0\u2115\u211a-neg _ = 0\nprf 0\u2115\u211a-neg c n x = OR.trans (OR.reflexive (proj\u2082 prop.zero (n ^ c))) z\u2264n\n\n_+\u2115\u211a_ : \u2115\u2192\u211a \u2192 \u2115\u2192\u211a \u2192 \u2115\u2192\u211a\n\u2115\u2192\u211a.\u03b5N ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n\n\u2115\u2192\u211a.\u03b5D ((\u03b5N \/ \u03b5D [ _ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ _ ])) n = \u03b5D n * \u03bcD n\n\u2115\u2192\u211a.\u03b5D-pos ((\u03b5N \/ \u03b5D [ \u03b5D+ ]) +\u2115\u211a (\u03bcN \/ \u03bcD [ \u03bcD+ ])) n = \u03b5D+ n *-mono \u03bcD+ n\n\n\n+\u2115\u211a-neg : {\u03b5 \u03bc : \u2115\u2192\u211a} \u2192 Is-Neg \u03b5 \u2192 Is-Neg \u03bc \u2192 Is-Neg (\u03b5 +\u2115\u211a \u03bc)\nc\u2099 (+\u2115\u211a-neg \u03b5 \u03bc) n = 1 + c\u2099 \u03b5 n + c\u2099 \u03bc n\nprf (+\u2115\u211a-neg {\u03b5M} {\u03bcM} \u03b5 \u03bc) c n n>nc = \u2264-*-cancel {x = n} (OR.trans (s\u2264s z\u2264n) n>nc) lemma\n  where\n\n  open \u2264-Reasoning\n  open \u2115\u2192\u211a \u03b5M\n  open \u2115\u2192\u211a \u03bcM renaming (\u03b5N to \u03bcN; \u03b5D to \u03bcD; \u03b5D-pos to \u03bcD-pos)\n\n  lemma =  n * (n ^ c * (\u03b5N n * \u03bcD n + \u03bcN n * \u03b5D n))\n        \u2261\u27e8 ! prop.*-assoc n (n ^ c) _\n         \u2219 proj\u2081 prop.distrib (n ^ (1 + c)) (\u03b5N n * \u03bcD n) (\u03bcN n * \u03b5D n)\n         \u2219 ap\u2082 _+_ (! prop.*-assoc (n ^ (1 + c)) (\u03b5N n) (\u03bcD n))\n                   (! (prop.*-assoc (n ^ (1 + c)) (\u03bcN n) (\u03b5D n))) \u27e9\n           n ^ (1 + c) * \u03b5N n * \u03bcD n + n ^ (1 + c) * \u03bcN n * \u03b5D n\n        \u2264\u27e8     (prf \u03b5 (1 + c) n (OR.trans (s\u2264s (\u2264-step (m\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03bcD n \u220e))\n        +-mono (prf \u03bc (1 + c) n (OR.trans (s\u2264s (\u2264-step (n\u2264m+n (c\u2099 \u03b5 n) (c\u2099 \u03bc n)))) n>nc) *-mono (\u03b5D n \u220e)) \u27e9\n           \u03b5D n * \u03bcD n + \u03bcD n * \u03b5D n\n        \u2261\u27e8 ap\u2082 _+_ (refl {x = \u03b5D n * \u03bcD n}) (prop.*-comm (\u03bcD n) (\u03b5D n) \u2219 ! proj\u2082 prop.+-identity (\u03b5D n * \u03bcD n)) \u27e9\n           2 * (\u03b5D n * \u03bcD n)\n        \u2264\u27e8 OR.trans (s\u2264s (s\u2264s z\u2264n)) n>nc *-mono (\u03b5D n * \u03bcD n \u220e) \u27e9\n           n * (\u03b5D n * \u03bcD n)\n        \u220e\n\ninfix 4 _\u2264\u2192_\nrecord _\u2264\u2192_ (f g : \u2115\u2192\u211a) : Set where\n  constructor mk\n  open \u2115\u2192\u211a f renaming (\u03b5N to fN; \u03b5D to fD)\n  open \u2115\u2192\u211a g renaming (\u03b5N to gN; \u03b5D to gD)\n  field\n    -- fN k \/ fD k \u2264 gN k \/ gD k\n    \u2264\u2192 : \u2200 k \u2192 fN k * gD k \u2264 gN k * fD k\n\n\u2264\u2192-trans : \u2200 {f g h} \u2192 f \u2264\u2192 g \u2192 g \u2264\u2192 h \u2192 f \u2264\u2192 h\n_\u2264\u2192_.\u2264\u2192 (\u2264\u2192-trans {fN \/ fD [ fD-pos ]} {gN \/ gD [ gD-pos ]} {hN \/ hD [ hD-pos ]} (mk fg) (mk gh)) k\n  = \u2264-*-cancel (gD-pos k) lemma\n  where\n    open \u2264-Reasoning\n    lemma : gD k * (fN k * hD k) \u2264 gD k * (hN k * fD k)\n    lemma = gD k * (fN k * hD k)\n          \u2261\u27e8 ! prop.*-assoc (gD k) (fN k) (hD k)\n             \u2219 ap (flip _*_ (hD k)) (prop.*-comm (gD k) (fN k))\n           \u27e9\n            (fN k * gD k) * hD k\n          \u2264\u27e8 fg k *-mono OR.refl \u27e9\n            (gN k * fD k) * hD k\n          \u2261\u27e8 prop.*-assoc (gN k) (fD k) (hD k)\n             \u2219 ap (_*_ (gN k)) (prop.*-comm (fD k) (hD k))\n             \u2219 ! prop.*-assoc (gN k) (hD k) (fD k)\n           \u27e9\n            (gN k * hD k) * fD k\n          \u2264\u27e8 gh k *-mono OR.refl \u27e9\n            (hN k * gD k) * fD k\n          \u2261\u27e8 ap (flip _*_ (fD k)) (prop.*-comm (hN k) (gD k))\n             \u2219 prop.*-assoc (gD k) (hN k) (fD k)\n           \u27e9\n            gD k * (hN k * fD k)\n          \u220e\n\n+\u2115\u211a-mono : \u2200 {f f' g g'} \u2192 f \u2264\u2192 f' \u2192 g \u2264\u2192 g' \u2192 f +\u2115\u211a g \u2264\u2192 f' +\u2115\u211a g'\n_\u2264\u2192_.\u2264\u2192 (+\u2115\u211a-mono {fN \/ fD [ _ ]} {f'N \/ f'D [ _ ]} {gN \/ gD [ _ ]} {g'N \/ g'D [ _ ]} (mk ff) (mk gg)) k\n  = (fN k * gD k + gN k * fD k) * (f'D k * g'D k)\n  \u2261\u27e8 proj\u2082 prop.distrib (f'D k * g'D k) (fN k * gD k) (gN k * fD k)  \u27e9\n    fN k * gD k * (f'D k * g'D k) + gN k * fD k * (f'D k * g'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (fN k) (gD k) (f'D k) (g'D k) \u2219 ap (_*_ (fN k * f'D k)) (prop.*-comm (gD k) (g'D k)))\n             (ap (_*_ (gN k * fD k)) (prop.*-comm (f'D k) (g'D k)) \u2219 *-interchange (gN k) (fD k) (g'D k) (f'D k))\n   \u27e9\n    fN k * f'D k * (g'D k * gD k) + gN k * g'D k * (fD k * f'D k)\n  \u2264\u27e8 (ff k *-mono OR.refl) +-mono (gg k *-mono OR.refl) \u27e9\n    f'N k * fD k * (g'D k * gD k) + g'N k * gD k * (fD k * f'D k)\n  \u2261\u27e8 ap\u2082 _+_ (*-interchange (f'N k) (fD k) (g'D k) (gD k))\n             (ap (_*_ (g'N k * gD k)) (prop.*-comm (fD k) (f'D k))\n             \u2219 *-interchange (g'N k) (gD k) (f'D k) (fD k)\n             \u2219 ap (_*_ (g'N k * f'D k)) (prop.*-comm (gD k) (fD k)))\n   \u27e9\n    f'N k * g'D k * (fD k * gD k) + g'N k * f'D k * (fD k * gD k)\n  \u2261\u27e8 ! proj\u2082 prop.distrib (fD k * gD k) (f'N k * g'D k) (g'N k * f'D k) \u27e9\n    (f'N k * g'D k + g'N k * f'D k) * (fD k * gD k)\n  \u220e\n  where\n    open \u2264-Reasoning\n\nrecord NegBounded (f : \u2115\u2192\u211a) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : f \u2264\u2192 \u03b5\n\nmodule _ where\n  open NegBounded\n  \u2264-NB : {f g : \u2115\u2192\u211a} \u2192 f \u2264\u2192 g \u2192 NegBounded g \u2192 NegBounded f\n  \u03b5 (\u2264-NB le nb) = \u03b5 nb\n  \u03b5-neg (\u2264-NB le nb) = \u03b5-neg nb\n  bounded (\u2264-NB le nb) = \u2264\u2192-trans le (bounded nb)\n\n  _+NB_ : {f g : \u2115\u2192\u211a} \u2192 NegBounded f \u2192 NegBounded g \u2192 NegBounded (f +\u2115\u211a g)\n  \u03b5 (fNB +NB gNB) = \u03b5 fNB +\u2115\u211a \u03b5 gNB\n  \u03b5-neg (fNB +NB gNB) = +\u2115\u211a-neg (\u03b5-neg fNB) (\u03b5-neg gNB)\n  bounded (fNB +NB gNB) = +\u2115\u211a-mono (bounded fNB) (bounded gNB)\n\nmodule ~-NegBounded (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n))(inh : \u2200 x \u2192 0 < Card (R\u1d41 x)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  ~dist : (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) \u2192 \u2115\u2192\u211a\n  \u2115\u2192\u211a.\u03b5N (~dist f g) n = dist (# (f n)) (# (g n))\n  \u2115\u2192\u211a.\u03b5D (~dist f g) n = Card (R\u1d41 n)\n  \u2115\u2192\u211a.\u03b5D-pos (~dist f g) n = inh n\n\n  ~dist-sum : \u2200 f g h \u2192 ~dist f h \u2264\u2192 ~dist f g +\u2115\u211a ~dist g h\n  _\u2264\u2192_.\u2264\u2192 (~dist-sum f g h) k\n      = #fh * (|R| * |R|)\n      \u2264\u27e8 dist-sum #f #g #h *-mono OR.refl \u27e9\n        (#fg + #gh) * (|R| * |R|)\n      \u2261\u27e8 ! prop.*-assoc (#fg + #gh) |R| |R| \u2219 ap (flip _*_ |R|) (proj\u2082 prop.distrib |R| #fg #gh) \u27e9\n        (#fg * |R| + #gh * |R|) * |R|\n      \u220e\n    where\n      open \u2264-Reasoning\n      |R| = Card (R\u1d41 k)\n      #f = # (f k)\n      #g = # (g k)\n      #h = # (h k)\n      #fh = dist #f #h\n      #fg = dist #f #g\n      #gh = dist #g #h\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      ~ : NegBounded (~dist f g)\n\n  ~-trans : Transitive _~_\n  _~_.~ (~-trans {f}{g}{h} (mk fg) (mk gh)) = \u2264-NB (~dist-sum f g h) (fg +NB gh)\n\nmodule ~-Inlined (R\u1d41 : \u2115 \u2192 U)(let R = \u03bb n \u2192 El (R\u1d41 n)) where\n\n  # : \u2200 {n} \u2192 Count (R n)\n  # {n} = count (R\u1d41 n)\n\n  record _~_ (f g : (x : \u2115) \u2192 R x \u2192 \ud835\udfda) : Set where\n    constructor mk\n    field\n      \u03b5 : \u2115\u2192\u211a\n    open \u2115\u2192\u211a \u03b5\n    field\n      \u03b5-neg : Is-Neg \u03b5\n      bounded : \u2200 k \u2192 \u03b5D k * dist (# (f k)) (# (g k)) \u2264 Card (R\u1d41 k) * \u03b5N k\n\n\n  ~-trans : Transitive _~_\n  _~_.\u03b5 (~-trans x x\u2081) = _\n  _~_.\u03b5-neg (~-trans x x\u2081) = +\u2115\u211a-neg (_~_.\u03b5-neg x) (_~_.\u03b5-neg x\u2081)\n  _~_.bounded (~-trans {f}{g}{h}(mk \u03b5\u2080 \u03b5\u2080-neg fg) (mk \u03b5\u2081 \u03b5\u2081-neg gh)) k\n      = (b * d) * dist #f #h\n      \u2264\u27e8 (b * d \u220e) *-mono dist-sum #f #g #h \u27e9\n        (b * d) * (dist #f #g + dist #g #h)\n      \u2261\u27e8 proj\u2081 prop.distrib (b * d) (dist #f #g) (dist #g #h)\n         \u2219 ap\u2082 _+_ (ap\u2082 _*_ (prop.*-comm b d) refl\n         \u2219 prop.*-assoc d b (dist #f #g)) (prop.*-assoc b d (dist #g #h))\n       \u27e9\n        d * (b * dist #f #g) + b * (d * dist #g #h)\n      \u2264\u27e8 ((d \u220e) *-mono fg k) +-mono ((b \u220e) *-mono gh k) \u27e9\n        d * (|R| * a) + b * (|R| * c)\n      \u2261\u27e8 ap\u2082 _+_ (rot d |R| a) (rot b |R| c) \u2219 ! proj\u2081 prop.distrib |R| (a * d) (c * b) \u27e9\n        |R| * \u2115\u2192\u211a.\u03b5N (\u03b5\u2080 +\u2115\u211a \u03b5\u2081) k\n      \u220e\n   where\n     open \u2264-Reasoning\n     rot : \u2200 x y z \u2192 x * (y * z) \u2261 y * (z * x)\n     rot x y z = prop.*-comm x (y * z) \u2219 prop.*-assoc y z x\n     |R| = Card (R\u1d41 k)\n     #f = # (f k)\n     #g = # (g k)\n     #h = # (h k)\n     a = \u2115\u2192\u211a.\u03b5N \u03b5\u2080 k\n     b = \u2115\u2192\u211a.\u03b5D \u03b5\u2080 k\n     c = \u2115\u2192\u211a.\u03b5N \u03b5\u2081 k\n     d = \u2115\u2192\u211a.\u03b5D \u03b5\u2081 k\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"05fcd17d4ae402deee20157ede6bafacd7a00059","subject":"#3 scraps","message":"#3 scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = {!!} -- I (IAp {!!} x (FIndet x\u2081)) -- todo: check that it's not that form, otherwise the cast can progress maybe\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) --\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\n\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = V (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- todo: check that it's not that form, otherwise the cast can progress maybe\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) --\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FIndet i)) --cyrus, as below\n  progress (TAAp wt1 wt2) | V v | V v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | Inl (x , d' , refl , qq) = S (_ , Step FHOuter ITLam FHOuter)\n  ... | Inr (d' , \u03c41' , \u03c42' , refl , neq , qq) = I (IAp {!!} (ICastArr neq {!!}) (FBoxed v\u2082)) --cyrus\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"cdb8794dfd31832cb3ed5466a45a7fcffae1941c","subject":"fussing with whitespace and name spaces","message":"fussing with whitespace and name spaces\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S  : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I  : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- similar if the left is indetermiante but the right is a boxed val\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes are indeterminate\n  progress (TAEHole _ _ ) = I IEHole\n\n    -- nonempty holes step if the innards step, indet otherwise\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I  x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n    -- step if the innards step\n  progress (TACast wt con) | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n    -- if indet, inspect how the types in the cast are realted by consistency:\n    -- if they're the same, step by ID\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n    -- if first type is hole\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n    -- if second type is hole\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHEHole (_ , _ , _ , refl , f)           = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHNEHole (_ , _ , _ , _ , _ , refl , _ ) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHAp (_ , _ , _ , refl , _ )             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHCast (_ , \u03c4 , refl , _)\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , .b , refl , \u03c02 , _) | Inl refl = S (_ , Step FHOuter (ITCastSucceed \u03c02) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , _ , refl , \u03c02 , _)  | Inr x =    S (_ , Step FHOuter (ITCastFail \u03c02 GBase x) FHOuter)\n  progress (TACast wt (TCHole2 {\u2987\u2988}))| I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHEHole (_ , _ , _ , refl , _)          = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHNEHole (_ , _ , _ , _ , _ , refl , _) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHAp (_ , _ , _ , refl , _ )            = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , GBase , _)      = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , GHole , _)      = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n    -- if both are arrows\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n    -- boxed value cases, inspect how the casts are realted by consistency\n    -- step by ID if the casts are the same\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n    -- if left is hole\n  progress (TACast wt (TCHole1 {\u03c4 = \u03c4})) | BV x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole1) | BV x\u2081 | Inl g = BV (BVHoleCast g x\u2081)\n  progress (TACast wt (TCHole1 {b})) | BV x\u2081 | Inr x = abort (x GBase)\n  progress (TACast wt (TCHole1 {\u2987\u2988})) | BV x\u2081 | Inr x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2081 | Inr x\n    with (htype-dec  (\u03c41 ==> \u03c42) (\u2987\u2988 ==> \u2987\u2988))\n  progress (TACast wt (TCHole1 {.\u2987\u2988 ==> .\u2987\u2988})) | BV x\u2082 | Inr x\u2081 | Inl refl = BV (BVHoleCast GHole x\u2082)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2082 | Inr x\u2081 | Inr x = S (_ , Step FHOuter (ITGround (MGArr x)) FHOuter)\n    -- if right is hole\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4 \u03c4'\n  progress (TACast wt TCHole2) | BV x\u2081 | d' , \u03c4 , refl , gnd , wt' | Inl refl = S (_  , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | _ , _ , refl , _ , _ | Inr _\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole2) | BV x\u2082 | _ , _ , refl , gnd , _ | Inr x\u2081 | Inl x = S(_ , Step FHOuter (ITCastFail gnd x (flip x\u2081)) FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2082 | _ , _ , refl , _ , _ | Inr x\u2081 | Inr x\n    with notground x\n  progress (TACast wt TCHole2) | BV x\u2083 | _ , _ , refl , _ , _ | Inr _ | Inr _ | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2083 | _ , _ , refl , _ , _ | Inr _ | Inr x | Inr (_ , _ , refl) = S(_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter )\n    -- if both arrows\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | BV x\n    with htype-dec (\u03c41 ==> \u03c42) (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inr x = BV (BVArrCast x x\u2081)\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S (_ , Step (FHFailedCast x) a (FHFailedCast q))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxed x) y z w)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import lemmas-ground\n\nopen import progress-checks\n\nopen import canonical-boxed-forms\nopen import canonical-value-forms\nopen import canonical-indeterminate-forms\n\nopen import ground-decidable\nopen import htype-decidable\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    BV : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n    -- constants\n  progress TAConst = BV (BVVal VConst)\n\n    -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n    -- lambdas\n  progress (TALam wt) = BV (BVVal VLam)\n\n    -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is indeterminate, step the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 x) y (FHAp2  z))\n    -- if they're both indeterminate, step when the cast steps and indet otherwise\n  progress (TAAp wt1 wt2) | I x | I x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | I y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | I y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n  progress (TAAp wt1 wt2) | I x | I y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FIndet y))\n    -- if the left is indetermiante but the right is a value\n  progress (TAAp wt1 wt2) | I x | BV x\u2081\n    with canonical-indeterminate-forms-arr wt1 x\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACast (_ , _ , _ , _ , _ , refl , _ , _ ) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAEHole (_ , _ , _ , refl , _)             = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFANEHole (_ , _ , _ , _ , _ , refl , _)    = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAAp (_ , _ , _ , _ , _ , refl , _)        = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFACastHole (_ , refl , refl , refl , _ )   = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n  progress (TAAp wt1 wt2) | I x | BV y | CIFAFailedCast (_ , _ , refl , _ )           = I (IAp (\u03bb _ _ _ _ _ ()) x (FBoxed y))\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | BV v | S (_ , Step x y z) = S (_ , Step (FHAp2  x) y (FHAp2  z))\n  progress (TAAp wt1 wt2) | BV v | I i\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n  progress (TAAp wt1 wt2) | BV v | BV v\u2082\n    with canonical-boxed-forms-arr wt1 v\n  ... | CBFLam (_ , _ , refl , _)             = S (_ , Step FHOuter ITLam FHOuter)\n  ... | CBFCastArr (_ , _ , _ , refl , _ , _) = S (_ , Step FHOuter ITApCast FHOuter)\n\n    -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n    -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | I x = I (INEHole (FIndet x))\n  ... | BV x = I (INEHole (FBoxed x))\n\n    -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast {\u03c41 = \u03c41} wt TCHole1) | I x\n    with \u03c41\n  progress (TACast wt TCHole1) | I x | b = I (ICastGroundHole GBase x)\n  progress (TACast wt TCHole1) | I x | \u2987\u2988 = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole1) | I x | \u03c411 ==> \u03c412\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt TCHole1) | I x\u2081 | .\u2987\u2988 ==> .\u2987\u2988 | Inl GHole = I (ICastGroundHole GHole x\u2081)\n  progress (TACast wt TCHole1) | I x\u2081 | \u03c411 ==> \u03c412 | Inr x =  S (_ , Step FHOuter (ITGround (MGArr (ground-arr-not-hole x))) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\n    with canonical-indeterminate-forms-hole wt x\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHEHole (_ , _ , _ , refl , f)           = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHNEHole (_ , _ , _ , _ , _ , refl , _ ) = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHAp (_ , _ , _ , refl , _ )             = I (ICastHoleGround (\u03bb _ _ ()) x GBase)\n  progress (TACast wt (TCHole2 {b})) | I x | CIFHCast (_ , \u03c4 , refl , _)\n    with htype-dec \u03c4 b\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , .b , refl , \u03c02 , _) | Inl refl = S (_ , Step FHOuter (ITCastSucceed \u03c02) FHOuter)\n  progress (TACast wt (TCHole2 {b})) | I x\u2081 | CIFHCast (_ , _ , refl , \u03c02 , _)  | Inr x =    S (_ , Step FHOuter (ITCastFail \u03c02 GBase x) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u2987\u2988})) | I x = S (_ , Step FHOuter ITCastID FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\n    with ground-decidable (\u03c411 ==> \u03c412)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x\u2081 | Inl GHole\n    with canonical-indeterminate-forms-hole wt x\u2081\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHEHole (_ , _ , _ , refl , _)          = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHNEHole (_ , _ , _ , _ , _ , refl , _) = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHAp (_ , _ , _ , refl , _ )            = I (ICastHoleGround (\u03bb _ _ ()) x GHole)\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , GBase , _)      = S (_ , Step FHOuter (ITCastFail GBase GHole (\u03bb ())) FHOuter )\n  progress (TACast wt (TCHole2 {.\u2987\u2988 ==> .\u2987\u2988})) | I x | Inl GHole | CIFHCast (_ , ._ , refl , GHole , _)      = S (_ , Step FHOuter (ITCastSucceed GHole) FHOuter)\n\n  progress (TACast wt (TCHole2 {\u03c411 ==> \u03c412})) | I x\u2081 | Inr x = S (_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter)\n\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | I x\n    with htype-dec (\u03c41 ==> \u03c42)  (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | I x\u2081 | Inr x = I (ICastArr x x\u2081)\n\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | BV x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c4 = \u03c4})) | BV x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole1) | BV x\u2081 | Inl g = BV (BVHoleCast g x\u2081)\n  progress (TACast wt (TCHole1 {b})) | BV x\u2081 | Inr x = abort (x GBase)\n  progress (TACast wt (TCHole1 {\u2987\u2988})) | BV x\u2081 | Inr x = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2081 | Inr x\n    with (htype-dec  (\u03c41 ==> \u03c42) (\u2987\u2988 ==> \u2987\u2988))\n  progress (TACast wt (TCHole1 {.\u2987\u2988 ==> .\u2987\u2988})) | BV x\u2082 | Inr x\u2081 | Inl refl = BV (BVHoleCast GHole x\u2082)\n  progress (TACast wt (TCHole1 {\u03c41 ==> \u03c42})) | BV x\u2082 | Inr x\u2081 | Inr x = S (_ , Step FHOuter (ITGround (MGArr x)) FHOuter)\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\n    with canonical-boxed-forms-hole wt x\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x | d' , \u03c4' , refl , gnd , wt'\n    with htype-dec \u03c4 \u03c4'\n  progress (TACast wt TCHole2) | BV x\u2081 | d' , \u03c4 , refl , gnd , wt' | Inl refl = S (_  , Step FHOuter (ITCastSucceed gnd) FHOuter)\n  progress {\u03c4 = \u03c4} (TACast wt TCHole2) | BV x\u2081 | d' , \u03c4' , refl , gnd , wt' | Inr x\n    with ground-decidable \u03c4\n  progress (TACast wt TCHole2) | BV x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x\u2081 | Inl x = S(_ , Step FHOuter (ITCastFail gnd x (flip x\u2081)) FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2082 | d' , \u03c4' , refl , gnd , wt' | Inr x\u2081 | Inr x\n    with notground x\n  progress (TACast wt TCHole2) | BV x\u2083 | d' , \u03c4' , refl , gnd , wt' | Inr x\u2082 | Inr x | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt TCHole2) | BV x\u2083 | d' , \u03c4' , refl , gnd , wt' | Inr x\u2082 | Inr x | Inr (\u03c41 , \u03c42 , refl) = S(_ , Step FHOuter (ITExpand (MGArr (ground-arr-not-hole x))) FHOuter )\n\n  progress (TACast wt (TCArr {\u03c41} {\u03c42} {\u03c41'} {\u03c42'} c1 c2)) | BV x\n    with htype-dec (\u03c41 ==> \u03c42) (\u03c41' ==> \u03c42')\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inl refl = S (_ , Step FHOuter ITCastID FHOuter)\n  progress (TACast wt (TCArr c1 c2)) | BV x\u2081 | Inr x = BV (BVArrCast x x\u2081)\n\n   -- failed casts\n  progress (TAFailedCast wt y z w)\n    with progress wt\n  progress (TAFailedCast wt y z w) | S (d' , Step x a q) = S (_ , Step (FHFailedCast x) a (FHFailedCast q))\n  progress (TAFailedCast wt y z w) | I x = I (IFailedCast (FIndet x) y z w)\n  progress (TAFailedCast wt y z w) | BV x = I (IFailedCast (FBoxed x) y z w)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b9efdadb157196f21a07ba50086562a24e651679","subject":"Working in issue #8.","message":"Working in issue #8.\n","repos":"asr\/apia,asr\/apia","old_file":"issues\/Issue8.agda","new_file":"issues\/Issue8.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issue8 where\n\npostulate\n  D : Set\n  \u2203 : (A : D \u2192 Set) \u2192 Set\n\npostulate foo : (A : D \u2192 Set) \u2192 \u2203 A \u2192 \u2203 A\n{-# ATP prove foo #-}\n\n-- 25 February 2013: The issue is due to the \u03b7-expansion.\n-- $ agda2atp -v 20 Issue8.agda\n--- ...\n--- The type and the eta-expanded type: equals\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Issue8 where\n\npostulate D : Set\n\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\npostulate foo : (A : D \u2192 Set) \u2192 (\u2203 \u03bb x \u2192 A x) \u2192 (\u2203 \u03bb x \u2192 A x)\n{-# ATP prove foo #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"810ef11b0eb413a664163bd26597ff14ca4aedc9","subject":"Typo.","message":"Typo.\n\nIgnore-this: 203292a98a6e474e921983623c84ada7\n\ndarcs-hash:20110216162043-3bd4e-98383cda71486452f52845f4a8f3bac848b29de9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/WellFoundedInductionATP.agda","new_file":"Draft\/McCarthy91\/WellFoundedInductionATP.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"072ea672a30c599ee9ae7a3fbfefbf8ca4249465","subject":"Derive elim specialized to types in examples.","message":"Derive elim specialized to types in examples.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_file":"formalization\/agda\/Spire\/DarkwingDuck\/Examples.agda","new_contents":"{-# OPTIONS --type-in-type --no-pattern-matching #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\nNatN : String\nNatN = \"Nat\"\n\nNatP : Tel\nNatP = Emp\n\nNatI : Scope NatP \u2192 Tel\nNatI _ = Emp\n\nNatE : Enum\nNatE = \"zero\" \u2237 \"suc\" \u2237 []\n\nNatB : (p : Scope NatP) \u2192 BranchesD NatE (NatI p)\nNatB _ = End tt , Rec tt (End tt) , tt\n\nNat : Set\nNat = Form NatN NatP NatI NatE NatB\n\ninjNat : CurriedScope NatP \u03bb p\n  \u2192 CurriedFunc (Scope (NatI p))\n      (sumD NatE (NatI p) (NatB p))\n      (FormUncurried NatN NatP NatI NatE NatB p)\ninjNat = inj NatN NatP NatI NatE NatB\n\nelimNat : CurriedScope NatP \u03bb p\n  \u2192 (M : CurriedScope (NatI p) (\u03bb i \u2192 FormUncurried NatN NatP NatI NatE NatB p i \u2192 Set))\n  \u2192 let unM = uncurryScope (NatI p) (\u03bb i \u2192 FormUncurried NatN NatP NatI NatE NatB p i \u2192 Set) M\n  in CurriedBranches NatE\n     (SumCurriedHyps NatN NatP NatI NatE NatB p M)\n     (CurriedScope (NatI p) (\u03bb i \u2192 (x : FormUncurried NatN NatP NatI NatE NatB p i) \u2192 unM i x))\nelimNat = elim NatN NatP NatI NatE NatB\n\nzero : Nat\nzero = injNat here\n\nsuc : Nat \u2192 Nat\nsuc = injNat (there here)\n\n----------------------------------------------------------------------\n\nVecN : String\nVecN = \"Vec\"\n\nVecP : Tel\nVecP = Ext Set \u03bb _ \u2192 Emp\n\nVecI : Scope VecP \u2192 Tel\nVecI _ = Ext Nat \u03bb _ \u2192 Emp\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\nVecB : (p : Scope VecP) \u2192 BranchesD VecE (VecI p)\nVecB = uncurryScope VecP (\u03bb p \u2192 BranchesD VecE (VecI p)) \u03bb A\n  \u2192 End (zero , tt)\n  , Arg Nat (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n  , tt\n\nVec : (A : Set) \u2192 Nat \u2192 Set\nVec = Form VecN VecP VecI VecE VecB\n\ninjVec : CurriedScope VecP \u03bb p\n  \u2192 CurriedFunc (Scope (VecI p))\n      (sumD VecE (VecI p) (VecB p))\n      (FormUncurried VecN VecP VecI VecE VecB p)\ninjVec = inj VecN VecP VecI VecE VecB\n\nelimVec : CurriedScope VecP \u03bb p\n  \u2192 (M : CurriedScope (VecI p) (\u03bb i \u2192 FormUncurried VecN VecP VecI VecE VecB p i \u2192 Set))\n  \u2192 let unM = uncurryScope (VecI p) (\u03bb i \u2192 FormUncurried VecN VecP VecI VecE VecB p i \u2192 Set) M\n  in CurriedBranches VecE\n     (SumCurriedHyps VecN VecP VecI VecE VecB p M)\n     (CurriedScope (VecI p) (\u03bb i \u2192 (x : FormUncurried VecN VecP VecI VecE VecB p i) \u2192 unM i x))\nelimVec = elim VecN VecP VecI VecE VecB\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = injVec A here\n\ncons : (A : Set) (n : Nat) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A = injVec A (there here)\n\n----------------------------------------------------------------------\n\nadd : Nat \u2192 Nat \u2192 Nat\nadd = elimNat\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : Nat \u2192 Nat \u2192 Nat\nmult = elimNat\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : (A : Set) (m n : Nat) (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend A m n = elimVec A\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb m' x xs ih ys \u2192 cons A (add m' n) x (ih ys))\n  m\n\nconcat : (A : Set) (m n : Nat) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m n = elimVec (Vec A m)\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  (nil A)\n  (\u03bb m' xs xss ih \u2192 append A m (mult m' m) xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : Nat\none = suc zero\n\ntwo : Nat\ntwo = suc one\n\nthree : Nat\nthree = suc two\n\n[1,2] : Vec Nat two\n[1,2] = cons Nat one one (cons Nat zero two (nil Nat))\n\n[3] : Vec Nat one\n[3] = cons Nat zero three (nil Nat)\n\n[1,2,3] : Vec Nat three\n[1,2,3] = cons Nat two one (cons Nat one two (cons Nat zero three (nil Nat)))\n\ntest-append : [1,2,3] \u2261 append Nat two one [1,2] [3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type --no-pattern-matching #-}\nopen import Spire.DarkwingDuck.Primitive\nopen import Spire.DarkwingDuck.Derived\nmodule Spire.DarkwingDuck.Examples where\n\n----------------------------------------------------------------------\n\nNatN : String\nNatN = \"Nat\"\n\nNatP : Tel\nNatP = Emp\n\nNatI : Scope NatP \u2192 Tel\nNatI _ = Emp\n\nNatE : Enum\nNatE = \"zero\" \u2237 \"suc\" \u2237 []\n\nNatB : (p : Scope NatP) \u2192 BranchesD NatE (NatI p)\nNatB _ = End tt , Rec tt (End tt) , tt\n\nNat : Set\nNat = Form NatN NatP NatI NatE NatB\n\nzero : Nat\nzero = inj NatN NatP NatI NatE NatB here\n\nsuc : Nat \u2192 Nat\nsuc = inj NatN NatP NatI NatE NatB (there here)\n\n----------------------------------------------------------------------\n\nVecN : String\nVecN = \"Vec\"\n\nVecP : Tel\nVecP = Ext Set \u03bb _ \u2192 Emp\n\nVecI : Scope VecP \u2192 Tel\nVecI _ = Ext Nat \u03bb _ \u2192 Emp\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\nVecB : (p : Scope VecP) \u2192 BranchesD VecE (VecI p)\nVecB = uncurryScope VecP (\u03bb p \u2192 BranchesD VecE (VecI p)) \u03bb A\n  \u2192 End (zero , tt)\n  , Arg Nat (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n  , tt\n\nVec : (A : Set) \u2192 Nat \u2192 Set\nVec = Form VecN VecP VecI VecE VecB\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = inj VecN VecP VecI VecE VecB A here\n\ncons : (A : Set) (n : Nat) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A = inj VecN VecP VecI VecE VecB A (there here)\n\n----------------------------------------------------------------------\n\nadd : Nat \u2192 Nat \u2192 Nat\nadd = elim NatN NatP NatI NatE NatB\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n\nmult : Nat \u2192 Nat \u2192 Nat\nmult = elim NatN NatP NatI NatE NatB\n  (\u03bb n \u2192 Nat \u2192 Nat)\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n\nappend : (A : Set) (m n : Nat) (xs : Vec A m) (ys : Vec A n) \u2192 Vec A (add m n)\nappend A m n = elim VecN VecP VecI VecE VecB A\n  (\u03bb m xs \u2192 (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb ys \u2192 ys)\n  (\u03bb m' x xs ih ys \u2192 cons A (add m' n) x (ih ys))\n  m\n\nconcat : (A : Set) (m n : Nat) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m n = elim VecN VecP VecI VecE VecB (Vec A m)\n  (\u03bb n xss \u2192 Vec A (mult n m))\n  (nil A)\n  (\u03bb m' xs xss ih \u2192 append A m (mult m' m) xs ih)\n  n\n\n----------------------------------------------------------------------\n\none : Nat\none = suc zero\n\ntwo : Nat\ntwo = suc one\n\nthree : Nat\nthree = suc two\n\n[1,2] : Vec Nat two\n[1,2] = cons Nat one one (cons Nat zero two (nil Nat))\n\n[3] : Vec Nat one\n[3] = cons Nat zero three (nil Nat)\n\n[1,2,3] : Vec Nat three\n[1,2,3] = cons Nat two one (cons Nat one two (cons Nat zero three (nil Nat)))\n\ntest-append : [1,2,3] \u2261 append Nat two one [1,2] [3]\ntest-append = refl\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4c8990f1323496415af84ce8c49e276f0c366745","subject":"Improve commentary in Base.Syntax.Context.","message":"Improve commentary in Base.Syntax.Context.\n\nOld-commit-hash: 4886ab0c5f6e2cdc47142a8ae0c717890f449f93\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Syntax\/Context.agda","new_file":"Base\/Syntax\/Context.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- Typing Contexts\n-- ===============\n\nimport Data.List as List\nopen List public\n  using ()\n  renaming\n    ( [] to \u2205 ; _\u2237_ to _\u2022_\n    ; map to mapContext\n    )\n\nContext : Set\nContext = List.List Type\n\n-- Variables\n-- =========\n--\n-- Here it is clear that we are using de Bruijn indices,\n-- encoded as natural numbers, more or less.\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- Weakening\n-- =========\n--\n-- We provide two approaches for weakening: context prefixes and\n-- and subcontext relationship. In this formalization, we mostly\n-- (maybe exclusively?) use the latter.\n\n-- Context Prefixes\n-- ----------------\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Subcontexts\n-- -----------\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Variables and contexts\n--\n-- This module defines the syntax of contexts, prefixes of\n-- contexts and variables and properties of these notions.\n--\n-- This module is parametric in the syntax of types, so it\n-- can be reused for different calculi.\n--\n------------------------------------------------------------------------\n\nmodule Base.Syntax.Context\n    (Type : Set)\n  where\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\nimport Data.List as List\nopen List public\n  using ()\n  renaming\n    ( [] to \u2205 ; _\u2237_ to _\u2022_\n    ; map to mapContext\n    )\n\nContext : Set\nContext = List.List Type\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c3 \u2022 \u0393) \u03c4\n\n-- WEAKENING\n\n-- CONTEXT PREFIXES\n--\n-- Useful for making lemmas strong enough to prove by induction.\n--\n-- Consider using the Subcontexts module instead.\n\nmodule Prefixes where\n\n-- Prefix of a context\n\n  data Prefix : Context \u2192 Set where\n    \u2205 : \u2200 {\u0393} \u2192 Prefix \u0393\n    _\u2022_ : \u2200 {\u0393} \u2192 (\u03c4 : Type) \u2192 Prefix \u0393 \u2192 Prefix (\u03c4 \u2022 \u0393)\n\n-- take only the prefix of a context\n\n  take : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  take \u0393 \u2205 = \u2205\n  take (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 take \u0393 \u0393\u2032\n\n-- drop the prefix of a context\n\n  drop : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n  drop \u0393 \u2205 = \u0393\n  drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) = drop \u0393 \u0393\u2032\n\n-- Extend a context to a super context\n\n  infixr 10 _\u22ce_\n\n  _\u22ce_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Context\n  \u2205 \u22ce \u0393\u2082 = \u0393\u2082\n  (\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 (\u0393\u2081 \u22ce \u0393\u2082)\n\n  take-drop : \u2200 \u0393 \u0393\u2032 \u2192 take \u0393 \u0393\u2032 \u22ce drop \u0393 \u0393\u2032 \u2261 \u0393\n  take-drop \u2205 \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) \u2205 = refl\n  take-drop (\u03c4 \u2022 \u0393) (.\u03c4 \u2022 \u0393\u2032) rewrite take-drop \u0393 \u0393\u2032 = refl\n\n  or-unit : \u2200 \u0393 \u2192 \u0393 \u22ce \u2205 \u2261 \u0393\n  or-unit \u2205 = refl\n  or-unit (\u03c4 \u2022 \u0393) rewrite or-unit \u0393 = refl\n\n  move-prefix : \u2200 \u0393 \u03c4 \u0393\u2032 \u2192\n    \u0393 \u22ce (\u03c4 \u2022 \u0393\u2032) \u2261 (\u0393 \u22ce (\u03c4 \u2022 \u2205)) \u22ce \u0393\u2032\n  move-prefix \u2205 \u03c4 \u0393\u2032 = refl\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 \u2205 = sym (or-unit (\u03c4 \u2022 \u0393 \u22ce (\u03c4\u2081 \u2022 \u2205)))\n  move-prefix (\u03c4 \u2022 \u0393) \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) rewrite move-prefix \u0393 \u03c4\u2081 (\u03c4\u2082 \u2022 \u0393\u2032) = refl\n\n-- Lift a variable to a super context\n\n  lift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\n  lift {\u2205} {\u2205} x = x\n  lift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\n  lift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- SUBCONTEXTS\n--\n-- Useful as a reified weakening operation,\n-- and for making theorems strong enough to prove by induction.\n--\n-- The contents of this module are currently exported at the end\n-- of this file.\n\nmodule Subcontexts where\n  infix 4 _\u227c_\n\n  data _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n    \u2205 : \u2205 \u227c \u2205\n    keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n    drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192\n      (\u03c4 : Type) \u2192\n      \u0393\u2081 \u227c \u0393\u2082 \u2192\n      \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n  -- Properties\n\n  \u2205\u227c\u0393 : \u2200 {\u0393} \u2192 \u2205 \u227c \u0393\n  \u2205\u227c\u0393 {\u2205} = \u2205\n  \u2205\u227c\u0393 {\u03c4 \u2022 \u0393} = drop \u03c4 \u2022 \u2205\u227c\u0393\n\n  \u227c-refl : Reflexive _\u227c_\n  \u227c-refl {\u2205} = \u2205\n  \u227c-refl {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u227c-refl\n\n  \u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n  \u227c-reflexive refl = \u227c-refl\n\n  \u227c-trans : Transitive _\u227c_\n  \u227c-trans \u227c\u2081 \u2205 = \u227c\u2081\n  \u227c-trans (keep .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = keep \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans (drop .\u03c4 \u2022 \u227c\u2081) (keep \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n  \u227c-trans \u227c\u2081 (drop \u03c4 \u2022 \u227c\u2082) = drop \u03c4 \u2022 \u227c-trans \u227c\u2081 \u227c\u2082\n\n  \u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n  \u227c-isPreorder = record\n    { isEquivalence = isEquivalence\n    ; reflexive = \u227c-reflexive\n    ; trans = \u227c-trans\n    }\n\n  \u227c-preorder : Preorder _ _ _\n  \u227c-preorder = record\n    { Carrier = Context\n    ; _\u2248_ = _\u2261_\n    ; _\u223c_ = _\u227c_\n    ; isPreorder = \u227c-isPreorder\n    }\n\n  module \u227c-Reasoning where\n    open import Relation.Binary.PreorderReasoning \u227c-preorder public\n      renaming\n        ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n        ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n        ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n        )\n\n  -- Lift a variable to a super context\n\n  weaken-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) this = this\n  weaken-var (keep \u03c4 \u2022 \u227c\u2081) (that x) = that (weaken-var \u227c\u2081 x)\n  weaken-var (drop \u03c4 \u2022 \u227c\u2081) x = that (weaken-var \u227c\u2081 x)\n\n-- Currently, we export the subcontext relation as well as the\n-- definition of _\u22ce_.\n\nopen Subcontexts public\nopen Prefixes public using (_\u22ce_)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7bcd3c1ee41ec3b612852f7ffc3aa6dc27cd79d3","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: ab1d6ae5d4e9f4c3b6baf58aabc8da1e\n\ndarcs-hash:20120216160130-3bd4e-2d7ccd4434c4cf23da1ca7682675d1af29b4251c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Data\/Product.agda","new_file":"src\/Common\/Data\/Product.agda","new_contents":"------------------------------------------------------------------------------\n-- Products\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Common.Data.Product where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : \u2200 {A B} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , _) = x\n\n\u2227-proj\u2082 : \u2200 {A B} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (_ , y) = y\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n-- Sugar syntax for the existential.\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n\u2203-proj\u2081 : \u2200 {P} \u2192 \u2203 P \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {P} \u2192 (p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n\u2203-proj\u2082 (_ , Px) = Px\n","old_contents":"------------------------------------------------------------------------------\n-- Products\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Common.Data.Product where\n\nopen import Common.Universe using ( D )\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n-- The conjunction data type.\n\n-- It is not necessary to add the data constructor _,_ as an\n-- axiom because the ATPs implement it.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n-- It is not strictly necessary define the projections \u2227-proj\u2081 and\n-- \u2227-proj\u2082 because the ATPs implement them. For the same reason, it is\n-- not necessary to add them as (general\/local) hints.\n\u2227-proj\u2081 : \u2200 {A B} \u2192 A \u2227 B \u2192 A\n\u2227-proj\u2081 (x , _) = x\n\n\u2227-proj\u2082 : \u2200 {A B} \u2192 A \u2227 B \u2192 B\n\u2227-proj\u2082 (_ , y) = y\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n\u2203-proj\u2081 : \u2200 {P} \u2192 \u2203 P \u2192 D\n\u2203-proj\u2081 (x , _) = x\n\n\u2203-proj\u2082 : \u2200 {P} \u2192 (p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n\u2203-proj\u2082 (_ , Px) = Px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a7637ec5e00efd81a55621bf0fd9dfc7dfac44ef","subject":"Bits: add a SearchUnit module","message":"Bits: add a SearchUnit module\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    infixr 1 _`\u204f_\n    data Op : Set where\n      `id `0\u21941 `not : Op\n      `tl : Op \u2192 Op\n      `if0 : Op \u2192 Op\n      _`\u204f_ : Op \u2192 Op \u2192 Op\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n\n    infixr 9 _\u2219_\n    _\u2219_ : Op \u2192 \u2200 {n} \u2192 Endo (Bits n)\n    `id   \u2219 xs       = xs\n    `0\u21941  \u2219 xs       = 0\u21941 xs\n    `not  \u2219 []       = []\n    `not  \u2219 (x \u2237 xs) = not x \u2237 xs\n    `tl f \u2219 []       = []\n    `tl f \u2219 (x \u2237 xs) = x \u2237 f \u2219 xs\n    `if0 f \u2219 []           = []\n    `if0 f \u2219 (false \u2237 xs) = false \u2237 f \u2219 xs\n    `if0 f \u2219 (true  \u2237 xs) = true  \u2237 xs\n    (f `\u204f g) \u2219 xs = g \u2219 f \u2219 xs\n\n    `if1 : Op \u2192 Op\n    `if1 f = `not `\u204f `if0 f `\u204f `not\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : Op\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : Op \u2192 Op\n    on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    on-firsts : Op \u2192 Op\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Op\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    open PermutationSyntax using (Perm; `id; `0\u21941; `tl; _`\u204f_)\n    module P = PermutationSemantics\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Op\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Op\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\n    `xor-head : Bit \u2192 Op\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Op\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 []       (`tl \u03c0)   [] = refl\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 {!!} \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    search-const\u03b5\u2261\u03b5 : \u2200 \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n    search-const\u03b5\u2261\u03b5 \u03b5 \u03b5\u2219\u03b5 = go\n      where\n        go : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        go zero = refl\n        go (suc n) rewrite go n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e\n                        where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Op (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax renaming (_\u2219_ to op)\n        search-op : \u2200 {n} (f : Bits n \u2192 A) (g : Op) \u2192 search {n} (f \u2218 op g) \u2261 search {n} f\n        search-op f `id = refl\n        search-op f `0\u21941 = search-0\u21941 f\n        search-op {zero} f `not = refl\n        search-op {suc n} f `not = \u2219-comm _ _\n        search-op {zero} f (`tl g) = refl\n        search-op {suc n} f (`tl g) rewrite search-op (f \u2218 0\u2237_) g | search-op (f \u2218 1\u2237_) g = refl\n        search-op {zero} f (`if0 g) = refl\n        search-op {suc n} f (`if0 g) rewrite search-op (f \u2218 0\u2237_) g = refl\n        search-op f (g `\u204f h) rewrite search-op (f \u2218 op h) g = search-op f h\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; search-const\u03b5\u2261\u03b5; module Op)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Op \u2115\u00b0.+-comm +-interchange public renaming (search-op to sum-op)\n\n    sum-const0\u22610 : \u2200 n \u2192 sum {n = n} (const 0) \u2261 0\n    sum-const0\u22610 n = search-const\u03b5\u2261\u03b5 0 refl n\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax renaming (_\u2219_ to op)\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-op : \u2200 {n} (f : Bits n \u2192 Bit) (g : Op) \u2192 #\u27e8 f \u2218 op g \u27e9 \u2261 #\u27e8 f \u27e9\n    #-op f = sum-op (Bool.to\u2115 \u2218 f)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"24a1a302286c0d935b44176f1e9c2d1e9d79086a","subject":"asked some questions about syntactic versions of compose\/apply\/diff","message":"asked some questions about syntactic versions of compose\/apply\/diff\n\nOld-commit-hash: 946e948c40de1d7fdcfb2b08e6a350f9f4c7d259\n","repos":"inc-lc\/ilc-agda","old_file":"derivation.agda","new_file":"derivation.agda","new_contents":"module derivation where\n\nopen import lambda\n\n-- KO: Is it correct that this file is supposed to contain the syntactic \n-- variants of the operations in Changes.agda?\n-- If so, why does it have a different (and rather strange) \u0394-Type definition?\n-- Why are the base cases (bool) all missing? The inductive cases look\n-- boring since they don't actually do anything (which explains why compose and apply are the same)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","old_contents":"module derivation where\n\nopen import lambda\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type bool = {!!}\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\napply {bool} = {!!}\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app (compose {\u03c4\u2082}) (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\ncompose {bool} = {!!}\n\nnil : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4)\nnil {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (nil {\u03c4\u2082})\n -- \u03bbx. nil\nnil {bool} = {!!}\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0394-Context \u0393\n\n-- CHANGE VARIABLES\n\n-- changes 'x' to 'dx'\n\nrename : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nrename this = that this\nrename (that x) = that (that (rename x))\n\n-- Weakening of a term needed during derivation - change x to x in a context which also includes dx.\n-- The actual specification is more complicated, study the type signature.\nadaptVar : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) this = this\nadaptVar \u2205 (\u03c4\u2082 \u2022 \u0393\u2082) (that x) = that (that (adaptVar \u2205 \u0393\u2082 x))\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 this = this\nadaptVar (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 (that x) = that (adaptVar \u0393\u2081 \u0393\u2082 x)\n\nadapt : \u2200 {\u03c4} \u0393\u2081 \u0393\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) \u03c4\nadapt {\u03c4\u2081 \u21d2 \u03c4\u2082} \u0393\u2081 \u0393\u2082 (abs t) = abs (adapt (\u03c4\u2081 \u2022 \u0393\u2081) \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 (app t\u2081 t\u2082) = app (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\nadapt \u0393\u2081 \u0393\u2082 (var t) = var (adaptVar \u0393\u2081 \u0393\u2082 t)\nadapt \u0393\u2081 \u0393\u2082 true = true\nadapt \u0393\u2081 \u0393\u2082 false = false\nadapt \u0393\u2081 \u0393\u2082 (cond tc t\u2081 t\u2082) = cond (adapt \u0393\u2081 \u0393\u2082 tc) (adapt \u0393\u2081 \u0393\u2082 t\u2081) (adapt \u0393\u2081 \u0393\u2082 t\u2082)\n\n-- changes 'x' to 'nil' (if internally bound) or 'dx' (if externally bound)\n\n\u0394-var : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-var {\u2205} x = var (rename x)\n\u0394-var {\u03c4 \u2022 \u0393} this = nil {\u03c4}\n\u0394-var {\u03c4 \u2022 \u0393} (that x) = \u0394-var {\u0393} x\n\n-- Note: this should be the derivative with respect to the first variable.\nnabla : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2080 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082)\nnabla = {!!}\n\n-- CHANGE TERMS\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\n-- To recheck: which is the order in which to apply the changes? When I did my live proof to Klaus, I came up with this order:\n-- (\u0394-term t\u2081) (t\u2082 \u2295 (\u0394-term t\u2082)) \u2218 (\u2207 t\u2081) t\u2082 (\u0394-term t\u2082)\n-- corresponding to:\n\u0394-term {\u0393\u2081} {\u0393\u2082} {\u03c4} (app t\u2081 t\u2082) = app (app (compose {\u03c4}) (app (\u0394-term {\u0393\u2081} t\u2081) (app (app apply (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082)) (adapt \u0393\u2081 \u0393\u2082 t\u2082))))\n  (app (adapt \u0393\u2081 \u0393\u2082 (app (nabla {\u0393\u2081 \u22ce \u0393\u2082} t\u2081) t\u2082)) (\u0394-term {\u0393\u2081} {\u0393\u2082} t\u2082))\n\n\u0394-term {\u0393} (var x) = weaken {\u2205} {\u0393} (\u0394-var {\u0393} x)\n\u0394-term {\u0393} true = {!!}\n\u0394-term {\u0393} false = {!!}\n\u0394-term {\u0393} (cond t\u2081 t\u2082 t\u2083) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"af1dfc7832f2f9e9a35b87e4e9ac755968627859","subject":"Change Desc model to make Mu take a Desc rather than a \"tagged desc\".","message":"Change Desc model to make Mu take a Desc rather than a \"tagged desc\".\n\nI have decided to favor treating the canonical theory as lower level.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) : ISet I where\n  init : UncurriedEl D (\u03bc D)\n\nData : (I : Set) \u2192 Set\nData I = \u03a3 Enum (BranchesD I)\n\nTagR : \u2200{I} \u2192 Data I \u2192 Set\nTagR (E , xs) = Tag E\n\nBranchesR : \u2200{I} (R : Data I) (P : TagR R \u2192 Set) \u2192 Set\nBranchesR (E , xs) = Branches E\n\nCurriedBranchesR : \u2200{I} (R : Data I) (P : TagR R \u2192 Set) (X : Set) \u2192 Set\nCurriedBranchesR (E , xs) = CurriedBranches E\n\nDescR : \u2200{I} \u2192 Data I \u2192 Desc I\nDescR (E , xs) = Arg (Tag E) (caseD xs)\n\ncaseR : \u2200{I} (R : Data I) \u2192 TagR R \u2192 Desc I\ncaseR (E , xs) = case (\u03bb _ \u2192 Desc _) xs\n\n----------------------------------------------------------------------\n\ninj : {I : Set} (R : Data I)\n  \u2192 let D = DescR R\n  in CurriedEl D (\u03bc D)\ninj R = let D = DescR R\n  in curryEl D (\u03bc D) init\n\n----------------------------------------------------------------------\n\nprove : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nprove (End j) X P \u03b1 i q = tt\nprove (Rec j D) X P \u03b1 i (x , xs) = \u03b1 j x , prove D X P \u03b1 i xs\nprove (Arg A B) X P \u03b1 i (a , xs) = prove (B a) X P \u03b1 i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : UncurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nind D P \u03b1 i (init xs) = \u03b1 i xs $\n  prove D (\u03bc D) P (ind D P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nindCurried D P f i x = ind D P (uncurryHyps D (\u03bc D) P init f) i x\n\nSumCurriedHyps : {I : Set} (R : Data I)\n  \u2192 let D = DescR R in\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  \u2192 TagR R \u2192 Set\nSumCurriedHyps R P t =\n  CurriedHyps (caseR R t) (\u03bc (DescR R)) P (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : {I : Set} (R : Data I)\n  \u2192 let D = DescR R in\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  \u2192 BranchesR R (SumCurriedHyps R P)\n  \u2192 (i : I) (x : \u03bc D i) \u2192 P i x\nelimUncurried R P cs i x =\n  indCurried (DescR R) P\n    (case (SumCurriedHyps R P) cs)\n    i x\n\nelim : {I : Set} (R : Data I)\n  \u2192 let D = DescR R in\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  \u2192 CurriedBranchesR R\n      (SumCurriedHyps R P)\n      ((i : I) (x : \u03bc D i) \u2192 P i x)\nelim R P = curryBranches (elimUncurried R P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115R : Data \u22a4\n\u2115R = \u2115E\n  , End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = DescR \u2115R\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115D\n\nzero : \u2115 tt\nzero = init (zeroT , refl)\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init (sucT , n , refl)\n\nVecR : (A : Set) \u2192 Data (\u2115 tt)\nVecR A = VecE\n  , End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = DescR (VecR A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc (VecD A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init (nilT , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init (consT , n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj (VecR A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj (VecR A) consT\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115R _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115R _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim (VecR A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim (VecR (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (Ds : BranchesD I E) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) (\u03bc E Ds)\n\ninj : {I : Set} (E : Enum) (Ds : BranchesD I E) (t : Tag E) \u2192 CurriedEl (caseD Ds t) (\u03bc E Ds)\ninj E Ds t = curryEl (caseD Ds t) (\u03bc E Ds) (init t)\n\n----------------------------------------------------------------------\n\nprove : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nprove (End j) X P \u03b1 i q = tt\nprove (Rec j D) X P \u03b1 i (x , xs) = \u03b1 j x , prove D X P \u03b1 i xs\nprove (Arg A B) X P \u03b1 i (a , xs) = prove (B a) X P \u03b1 i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nind E Ds P \u03b1 i (init t xs) = \u03b1 t i xs $\n  prove (caseD Ds t) (\u03bc E Ds) P (ind E Ds P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nindCurried E Ds P f i x = ind E Ds P (\u03bb t \u2192 uncurryHyps (caseD Ds t) (\u03bc E Ds) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E Ds X cn P t = CurriedHyps (caseD Ds t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E Ds P t = Summer E Ds (\u03bc E Ds) init P t\n\nelimUncurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E Ds P)\n  \u2192 (i : I) (x : \u03bc E Ds i) \u2192 P i x\nelimUncurried E Ds P cs i x =\n  indCurried E Ds P\n    (case (SumCurriedHyps E Ds P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E Ds P)\n      ((i : I) (x : \u03bc E Ds i) \u2192 P i x)\nelim E Ds P = curryBranches (elimUncurried E Ds P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115Ds : BranchesD \u22a4 \u2115E\n\u2115Ds =\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T (caseD \u2115Ds)\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115Ds\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecDs : (A : Set) \u2192 BranchesD (\u2115 tt) VecE\nVecDs A =\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (caseD (VecDs A))\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecDs A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecDs A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecDs A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecDs A) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs A) t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecDs (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs (Vec A m)) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecDs A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecDs (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0ad36a15b665b3c1ef964bd90e030a683fa87b5b","subject":"Start to replace pattern matching with elims in model.","message":"Start to replace pattern matching with elims in model.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P c,cs here = proj\u2081 c,cs\ncase P c,cs (there t) = case (\u03bb t \u2192 P (there t)) (proj\u2082 c,cs) t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i x,xs = P j (proj\u2081 x,xs) \u00d7 Hyps D X P i (proj\u2082 x,xs)\nHyps (Arg A B) X P i a,b = Hyps (B (proj\u2081 a,b)) X P i (proj\u2082 a,b)\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (C t) (\u03bc E C)\n\ninj : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) (t : Tag E) \u2192 CurriedEl (C t) (\u03bc E C)\ninj E C t = curryEl (C t) (\u03bc E C) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i x,xs = \u03b1 j (proj\u2081 x,xs) , hyps D X P \u03b1 i (proj\u2082 x,xs)\nhyps (Arg A B) X P \u03b1 i a,xs = hyps (B (proj\u2081 a,xs)) X P \u03b1 i (proj\u2082 a,xs)\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nind E C P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (C t) (\u03bc E C) P (ind E C P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nindCurried E C P f i x = ind E C P (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E C X cn P t = CurriedHyps (C t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E C P t = Summer E C (\u03bc E C) init P t\n\nelimUncurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E C P)\n  \u2192 (i : I) (x : \u03bc E C i) \u2192 P i x\nelimUncurried E C P cs i x =\n  indCurried E C P\n    (case (SumCurriedHyps E C P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E C P)\n      ((i : I) (x : \u03bc E C i) \u2192 P i x)\nelim E C P = curryBranches (elimUncurried E C P)\n\n----------------------------------------------------------------------\n\nSoundness : Set\u2081\nSoundness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (cs : Branches E (SumCurriedHyps E C P))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb \u03b1\n  \u2192 elimUncurried E C P cs i x \u2261 ind E C P \u03b1 i x\n\nsound : Soundness\nsound E C P cs i xs =\n  let D = Arg (Tag E) C in\n  (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) cs t)) , refl\n\nCompleteness : Set\u2081\nCompleteness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb cs\n  \u2192 ind E C P \u03b1 i x \u2261 elimUncurried E C P cs i x\n\nuncurryHypsIdent : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  (\u03b1 : UncurriedHyps D X P cn)\n  (i : I) (xs : El D X i) (ihs : Hyps D X P i xs)\n  \u2192 \u03b1 i xs ihs \u2261 uncurryHyps D X P cn (curryHyps D X P cn \u03b1) i xs ihs\nuncurryHypsIdent (End .i) X P cn \u03b1 i refl tt = refl\nuncurryHypsIdent (Rec j D) X P cn \u03b1 i (x , xs) (p , ps) =\n  uncurryHypsIdent D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb k ys rs \u2192 \u03b1 k (x , ys) (p , rs)) i xs ps\nuncurryHypsIdent (Arg A B) X P cn \u03b1 i (a , xs) ps =\n  uncurryHypsIdent (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb j ys \u2192 \u03b1 j (a , ys)) i xs ps\n\npostulate\n  ext4 : {A : Set} {B : A \u2192 Set} {C : (a : A) \u2192 B a \u2192 Set}\n    {D : (a : A) (b : B a) \u2192 C a b \u2192 Set}\n    {Z : (a : A) (b : B a) (c : C a b) \u2192 D a b c \u2192 Set}\n    (f g : (a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 Z a b c d)\n    \u2192 ((a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 f a b c d \u2261 g a b c d)\n    \u2192 f \u2261 g\n\ntoBranches : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  \u2192 Branches E (Summer E C X cn P)\ntoBranches [] C X cn P \u03b1 = tt\ntoBranches (l \u2237 E) C X cn P \u03b1 =\n    curryHyps (C here) X P (\u03bb xs \u2192 cn here xs) (\u03bb i xs \u2192 \u03b1 here i xs)\n  , toBranches E (\u03bb t \u2192 C (there t)) X\n     (\u03bb t \u2192 cn (there t))\n     P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih)\n\nToBranches : {I : Set} {E : Enum} (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  (t : Tag E)\n  \u2192 let \u03b2 = toBranches E C X cn P \u03b1 in\n  case (Summer E C X cn P) \u03b2 t \u2261 curryHyps (C t) X P (cn t) (\u03b1 t)\nToBranches C X cn P \u03b1 here = refl\nToBranches C X cn P \u03b1 (there t)\n  with ToBranches (\u03bb t \u2192 C (there t)) X\n    (\u03bb t xs \u2192 cn (there t) xs)\n    P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih) t\n... | ih rewrite ih = refl\n\ncomplete\u03b1 : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (t : Tag E) (i : I) (xs : El (C t) (\u03bc E C) i) (ihs : Hyps (C t) (\u03bc E C) P i xs)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  \u03b1 t i xs ihs \u2261 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t) i xs ihs\ncomplete\u03b1 E C P \u03b1 t i xs ihs\n  with ToBranches C (\u03bc E C) init P \u03b1 t\n... | q rewrite q = uncurryHypsIdent (C t) (\u03bc E C) P (init t) (\u03b1 t) i xs ihs\n\ncomplete' : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  ind E C P \u03b1 i x \u2261 elimUncurried E C P \u03b2 i x\ncomplete' E C P \u03b1 i (init t xs) = cong\n  (\u03bb f \u2192 ind E C P f i (init t xs))\n  (ext4 \u03b1\n    (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t))\n    (complete\u03b1 E C P \u03b1)\n  )\n  where \u03b2 = toBranches E C (\u03bc E C) init P \u03b1\n\ncomplete : Completeness\ncomplete E C P \u03b1 i x =\n  let D = Arg (Tag E) C in\n    toBranches E C (\u03bc E C) init P \u03b1\n  , complete' E C P \u03b1 i x\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115C : \u2115T \u2192 Desc \u22a4\n\u2115C = caseD $\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T \u2115C\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115C\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecC : (A : Set) \u2192 VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD $\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (VecC A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecC A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecC A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecC A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecC A) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC A t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecC (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC (Vec A m) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115C _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115C _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecC A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecC (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (C t) (\u03bc E C)\n\ninj : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) (t : Tag E) \u2192 CurriedEl (C t) (\u03bc E C)\ninj E C t = curryEl (C t) (\u03bc E C) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i (x , xs) = \u03b1 j x , hyps D X P \u03b1 i xs\nhyps (Arg A B) X P \u03b1 i (a , xs) = hyps (B a) X P \u03b1 i xs\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nind E C P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (C t) (\u03bc E C) P (ind  E C P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nindCurried E C P f i x = ind E C P (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E C X cn P t = CurriedHyps (C t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E C P t = Summer E C (\u03bc E C) init P t\n\nelimUncurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E C P)\n  \u2192 (i : I) (x : \u03bc E C i) \u2192 P i x\nelimUncurried E C P cs i x =\n  indCurried E C P\n    (case (SumCurriedHyps E C P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E C P)\n      ((i : I) (x : \u03bc E C i) \u2192 P i x)\nelim E C P = curryBranches (elimUncurried E C P)\n\n----------------------------------------------------------------------\n\nSoundness : Set\u2081\nSoundness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (cs : Branches E (SumCurriedHyps E C P))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb \u03b1\n  \u2192 elimUncurried E C P cs i x \u2261 ind E C P \u03b1 i x\n\nsound : Soundness\nsound E C P cs i xs =\n  let D = Arg (Tag E) C in\n  (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) cs t)) , refl\n\nCompleteness : Set\u2081\nCompleteness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb cs\n  \u2192 ind E C P \u03b1 i x \u2261 elimUncurried E C P cs i x\n\nuncurryHypsIdent : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  (\u03b1 : UncurriedHyps D X P cn)\n  (i : I) (xs : El D X i) (ihs : Hyps D X P i xs)\n  \u2192 \u03b1 i xs ihs \u2261 uncurryHyps D X P cn (curryHyps D X P cn \u03b1) i xs ihs\nuncurryHypsIdent (End .i) X P cn \u03b1 i refl tt = refl\nuncurryHypsIdent (Rec j D) X P cn \u03b1 i (x , xs) (p , ps) =\n  uncurryHypsIdent D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb k ys rs \u2192 \u03b1 k (x , ys) (p , rs)) i xs ps\nuncurryHypsIdent (Arg A B) X P cn \u03b1 i (a , xs) ps =\n  uncurryHypsIdent (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb j ys \u2192 \u03b1 j (a , ys)) i xs ps\n\npostulate\n  ext4 : {A : Set} {B : A \u2192 Set} {C : (a : A) \u2192 B a \u2192 Set}\n    {D : (a : A) (b : B a) \u2192 C a b \u2192 Set}\n    {Z : (a : A) (b : B a) (c : C a b) \u2192 D a b c \u2192 Set}\n    (f g : (a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 Z a b c d)\n    \u2192 ((a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 f a b c d \u2261 g a b c d)\n    \u2192 f \u2261 g\n\ntoBranches : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  \u2192 Branches E (Summer E C X cn P)\ntoBranches [] C X cn P \u03b1 = tt\ntoBranches (l \u2237 E) C X cn P \u03b1 =\n    curryHyps (C here) X P (\u03bb xs \u2192 cn here xs) (\u03bb i xs \u2192 \u03b1 here i xs)\n  , toBranches E (\u03bb t \u2192 C (there t)) X\n     (\u03bb t \u2192 cn (there t))\n     P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih)\n\nToBranches : {I : Set} {E : Enum} (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  (t : Tag E)\n  \u2192 let \u03b2 = toBranches E C X cn P \u03b1 in\n  case (Summer E C X cn P) \u03b2 t \u2261 curryHyps (C t) X P (cn t) (\u03b1 t)\nToBranches C X cn P \u03b1 here = refl\nToBranches C X cn P \u03b1 (there t)\n  with ToBranches (\u03bb t \u2192 C (there t)) X\n    (\u03bb t xs \u2192 cn (there t) xs)\n    P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih) t\n... | ih rewrite ih = refl\n\ncomplete\u03b1 : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (t : Tag E) (i : I) (xs : El (C t) (\u03bc E C) i) (ihs : Hyps (C t) (\u03bc E C) P i xs)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  \u03b1 t i xs ihs \u2261 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t) i xs ihs\ncomplete\u03b1 E C P \u03b1 t i xs ihs\n  with ToBranches C (\u03bc E C) init P \u03b1 t\n... | q rewrite q = uncurryHypsIdent (C t) (\u03bc E C) P (init t) (\u03b1 t) i xs ihs\n\ncomplete' : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  ind E C P \u03b1 i x \u2261 elimUncurried E C P \u03b2 i x\ncomplete' E C P \u03b1 i (init t xs) = cong\n  (\u03bb f \u2192 ind E C P f i (init t xs))\n  (ext4 \u03b1\n    (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t))\n    (complete\u03b1 E C P \u03b1)\n  )\n  where \u03b2 = toBranches E C (\u03bc E C) init P \u03b1\n\ncomplete : Completeness\ncomplete E C P \u03b1 i x =\n  let D = Arg (Tag E) C in\n    toBranches E C (\u03bc E C) init P \u03b1\n  , complete' E C P \u03b1 i x\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115C : \u2115T \u2192 Desc \u22a4\n\u2115C = caseD $\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T \u2115C\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115C\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecC : (A : Set) \u2192 VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD $\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (VecC A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecC A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecC A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecC A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecC A) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC A t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecC (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC (Vec A m) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115C _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115C _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecC A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecC (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d9d0b04a3aba0824a0ea7b63c10b06c8ab3692e3","subject":"0,1: Comment on _+I_","message":"0,1: Comment on _+I_\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  1-I_ : [0,1] \u2192 [0,1]\n  1-I 0I    = 1I\n  1-I 1I    = 0I\n  1-I (x I) = (1-E x) I\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I  +I x    = x\n  x I +I 0I   = x I\n  1I  +I _    = 1I -- troublesome\n  x I +I 1I   = 1I -- troublesome\n  x I +I x\u2081 I = (x +E x\u2081) I -- faithful\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field\n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n    1-E_ : ]0,1[ \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248\n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n\n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n\n  -- 1 - ((1 + x) \/ (2 + x + y))\n  --  \u2261 (2 + x + y - (1 + x)) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + x + y)\n  --  \u2261 (1 + y) \/ (2 + y + x)\n  1-E_ : ]0,1[ \u2192 ]0,1[\n  1-E (1+ x \/2+x+ y) = 1+ y \/2+x+ x\n\n  -- ((1 + x) \/ (2 + x + y)) \u00b7 ((1 + x') \/ (2 + x' + y'))\n  --  \u2261 ((1 + x) \u00b7 (1 + x')) \/ ((2 + x + y) \u00b7 (2 + x' + y'))\n  --  \u2261 (1 + x + x' + x \u00b7 x') \/ (4 + 2x + 2y + 2x' + 2y' + xx' + xy' + yx' + yy')\n  --  \u2261 (1 + (x + x' + xx')) \/ (2 + (x + x' + xx') + (2 + x + 2y + x' + xy' + yx' + yy'))\n  _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n  (1+ x \/2+x+ y) \u00b7E (1+ x' \/2+x+ y') = 1+ x'' \/2+x+ y''\n    where x'' = x + x' + x * x'\n          y'' = 2 + x + 2 * y + x' + x * y' + y * x' + y * y'\n\n  -- ((1 + x) \/ (2 + x + y)) \/ ((1 + x') \/ (2 + x' + y'))\n  --   \u2261 ((1 + x) \u00b7 (2 + x' + y')) \/ ((1 + x') \u00b7 (2 + x + y))\n  --   \u2261 (2 + x' + y' + 2x + xx' + xy') \/ (2 + x + y + 2x' + xx' + x'y)\n  --   \u2261 (1 + (1 + x' + y' + 2x + xx' + xy')) \/ (2 + (1 + x' + y' + 2x + xx' + xy') + y + x' + x'y - (1 + y' + x + xy'))\n  --   ok provided that:\n  --   ((1 + x) \/ (2 + x + y)) < ((1 + x') \/ (2 + x' + y'))\n  --   implies\n  --   (1 + y' + x + xy') \u2264 (y + x' + x'y)\n  --   which remains to be checked\n  _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n  (1+ x \/2+x+ y) \/E (1+ x' \/2+x+ y') < _ > = 1+ x'' \/2+x+ y''\n    where x'' = 1 + x' + y' + 2 * x + x * x' + x * y'\n          y'' = y + x' + x' * y \u2238 1 \u2238 y' \u2238 x \u2238 x * y'\n\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> 1-E_ <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[\n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  1-I_ : [0,1] \u2192 [0,1]\n  1-I 0I    = 1I\n  1-I 1I    = 0I\n  1-I (x I) = (1-E x) I\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n\n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  >\n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115)\n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where\n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP pmf []       = 0I\n  sumP pmf (x \u2237 xs) = (pmf x) +I (sumP pmf xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      -- Probability mass function: http:\/\/en.wikipedia.org\/wiki\/Probability_mass_function\n      pmf    : U \u2192 [0,1]\n      sum\u22611 : sumP pmf allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n\n    Event : Set\n    Event = U \u2192 Bool\n\n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then pmf x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]\n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x\n      ... | true  | true  rewrite +I-assoc (pmf x) (sA\u2081 xs) (pmf x +I sA\u2082 xs)\n                                | +I-sym (pmf x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (pmf x))\n                                = \u2264I-pres (pmf x) (\u2264I-mono (pmf x) (go xs))\n      ... | true  | false rewrite +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (pmf x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (pmf x)\n                                | +I-assoc (pmf x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (pmf x) (go xs)\n      ... | false | false = go xs\n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v\n\n       _\u2261r_ : RV \u2192 V \u2192 Event\n       RV \u2261r v = RV ^-1 v\n\n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n\n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bd16dd91fd35f2b3e6242e75ab012864bb9d4051","subject":"dapply-sound: reindent for clarity","message":"dapply-sound: reindent for clarity\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192\n    eval (app (app ds t) dt) \u03c1 (suc (suc n))\n  \u2261\n    dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with\n  eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Failed variants, the step-indexes are wrong:\n-- dapply-equiv-2 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc n)) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-2 (suc n) t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 (suc zero) (const c) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (var x) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (app t t\u2081) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (abs t) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc (suc n)) t d\u03c1 \u03c11 da a1 = refl\n\n-- Worse:\n-- dapply-equiv-0 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-0 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc zero) t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc (suc n)) t d\u03c1 \u03c11 da a1 = {!!}\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192 eval (app (app ds t) dt) \u03c1 (suc (suc n)) \u2261 dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Failed variants, the step-indexes are wrong:\n-- dapply-equiv-2 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc n)) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-2 (suc n) t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 (suc zero) (const c) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (var x) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (app t t\u2081) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (abs t) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc (suc n)) t d\u03c1 \u03c11 da a1 = refl\n\n-- Worse:\n-- dapply-equiv-0 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-0 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc zero) t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc (suc n)) t d\u03c1 \u03c11 da a1 = {!!}\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3b3cc5127e4531c6f78580d798411d084309b94c","subject":"Insert empty lines to improve diffs of future commits.","message":"Insert empty lines to improve diffs of future commits.\n\nI'm planning to merge Atlas-term and Atlas-lookup into a single\ndefinition, and these empty lines will make the diffs somewhat\nreadable.\n\nOld-commit-hash: cce63fc3e11f5a6cfc12adbdb3fdb895dda24142\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n\n  false : Atlas-const\n\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\nAtlas-lookup (lookup {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9\n\n-- Model of zip = Haskell Data.List.zipWith\n--\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n--\n-- Behavioral difference: all key-value pairs present\n-- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n-- be iterated over. Neutral element of type `a` or `b`\n-- will be supplied if the key is missing in the\n-- corresponding map.\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\n-- Model of fold = Haskell Data.Map.foldWithKey\n--\n-- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n--\nAtlas-lookup (fold {\u03ba} {a} {b}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n  base b \u21d2 base (Map \u03ba a) \u21d2 base b\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t (const empty)\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app\u2082 (const xor) \u0394x \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  lookup : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n  fold   : \u2200 {\u03ba a b : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\nAtlas-lookup (lookup {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base (Map \u03ba \u03b9) \u21d2 base \u03b9\n\n-- Model of zip = Haskell Data.List.zipWith\n--\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\n--\n-- Behavioral difference: all key-value pairs present\n-- in *either* (m\u2081 : Map \u03ba a) *or* (m\u2082 : Map \u03ba b) will\n-- be iterated over. Neutral element of type `a` or `b`\n-- will be supplied if the key is missing in the\n-- corresponding map.\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\n-- Model of fold = Haskell Data.Map.foldWithKey\n--\n-- foldWithKey :: (k \u2192 a \u2192 b \u2192 b) \u2192 b \u2192 Map k a \u2192 b\n--\nAtlas-lookup (fold {\u03ba} {a} {b}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u21d2\n  base b \u21d2 base (Map \u03ba a) \u21d2 base b\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 _ Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\nupdate! = app\u2083 (const update)\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\nlookup! = app\u2082 (const lookup)\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\nzip! = app\u2083 (const zip)\n\nfold! : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base b) \u2192\n  Atlas-term \u0393 (base b) \u2192 Atlas-term \u0393 (base (Map \u03ba a)) \u2192\n  Atlas-term \u0393 (base b)\nfold! = app\u2083 (const fold)\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Nonfunctional products can be encoded.\n-- The incremental behavior of products thus encoded is weird:\n-- \u0394(\u03b1 \u00d7 \u03b2) = \u03b1 \u00d7 \u0394\u03b2\nPair : Atlas-type \u2192 Atlas-type \u2192 Atlas-type\nPair \u03b1 \u03b2 = Map \u03b1 \u03b2\n\npair : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1) \u2192 Atlas-term \u0393 (base \u03b2) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2))\npair s t = update! s t (const empty)\n\npair-term : \u2200 {\u03b1 \u03b2 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base (Pair \u03b1 \u03b2))\npair-term = abs (abs (pair (var (that this)) (var this)))\n\nuncurry : \u2200 {\u03b1 \u03b2 \u03b3 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b1 \u21d2 base \u03b2 \u21d2 base \u03b3) \u2192\n  Atlas-term \u0393 (base (Pair \u03b1 \u03b2)) \u2192\n  Atlas-term \u0393 (base \u03b3)\nuncurry f p =\n  let\n    a = var (that (that this))\n    b = var (that this)\n    g = abs (abs (abs (app\u2082 (weaken\u2083 f) a b)))\n  in\n    fold! g neutral-term p\n\nzip-pair : \u2200 {\u03ba a b \u0393} \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba (Pair a b)))\nzip-pair = zip! (abs pair-term)\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff = app\u2082 (lift-diff Atlas-diff Atlas-apply)\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply = app\u2082 (lift-apply Atlas-diff Atlas-apply)\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\n-- zip\u2084 f m\u2081 m\u2082 m\u2083 m\u2084 =\n-- zip (\u03bb k p\u2081\u2082 p\u2083\u2084 \u2192 uncurry (\u03bb v\u2081 v\u2082 \u2192 uncurry (\u03bb v\u2083 v\u2084 \u2192\n--       f k v\u2081 v\u2082 v\u2083 v\u2084)\n--       p\u2083\u2084) p\u2081\u2082)\n--     (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    f\u2032 = weaken\u2083 (weaken\u2083 (weaken\u2081 f))\n    k = var (that (that (that (that (that (that this))))))\n    p\u2081\u2082 = var (that this)\n    p\u2083\u2084 = var (that (that this))\n    v\u2081 = var (that (that (that this)))\n    v\u2082 = var (that (that this))\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n        (app\u2085 f\u2032 k v\u2081 v\u2082 v\u2083 v\u2084)))\n        p\u2083\u2084))) p\u2081\u2082)))\n  in\n    zip! g (zip-pair m\u2081 m\u2082) (zip-pair m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app\u2082 (const xor) \u0394x \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394lookup k \u0394k m \u0394m | true? (k \u2295 \u0394k \u2261 k)\n-- ... | true  = lookup k \u0394m\n-- ... | false =\n--   (lookup (k \u2295 \u0394k) m \u2295 lookup (k \u2295 \u0394k) \u0394m)\n--     \u229d lookup k m\n--\n-- Only the false-branch is implemented.\nAtlas-\u0394const lookup =\n  let\n    k  = var (that (that (that this)))\n    \u0394k = var (that (that this))\n    m  = var (that this)\n    \u0394m = var this\n    k\u2032 = apply \u0394k k\n  in\n    abs (abs (abs (abs\n      (diff (apply (lookup! k\u2032 \u0394m) (lookup! k\u2032 m))\n            (lookup! k m)))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app\u2082 (weaken\u2081 \u0394f) (var this) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\n-- \u0394fold f \u0394f z \u0394z m \u0394m = proj\u2082\n--   (fold (\u03bb k [a,\u0394a] [b,\u0394b] \u2192\n--           uncurry (\u03bb a \u0394a \u2192 uncurry (\u03bb b \u0394b \u2192\n--             pair (f k a b) (\u0394f k nil a \u0394a b \u0394b))\n--             [b,\u0394b]) [a,\u0394a])\n--        (pair z \u0394z)\n--        (zip-pair m \u0394m))\n--\n-- \u0394fold is efficient only if evaluation is lazy and \u0394f is\n-- self-maintainable: it doesn't look at the argument\n-- (b = fold f k a b\u2080) at all.\nAtlas-\u0394const (fold {\u03ba} {\u03b1} {\u03b2}) =\n    let -- TODO (tedius): write weaken\u2087\n      f  = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that (that this))))))))\n      \u0394f = weaken\u2083 (weaken\u2083 (weaken\u2081\n        (var (that (that (that (that this)))))))\n      z  = var (that (that (that this)))\n      \u0394z = var (that (that this))\n      m  = var (that this)\n      \u0394m = var this\n      k = weaken\u2083 (weaken\u2081 (var (that (that this))))\n      [a,\u0394a] = var (that this)\n      [b,\u0394b] = var this\n      a  = var (that (that (that this)))\n      \u0394a = var (that (that this))\n      b  = var (that this)\n      \u0394b = var this\n      g = abs (abs (abs (uncurry (abs (abs (uncurry (abs (abs\n            (pair (app\u2083 f k a b)\n                  (app\u2086 \u0394f k nil-term a \u0394a b \u0394b))))\n            (weaken\u2082 [b,\u0394b])))) [a,\u0394a])))\n      proj\u2082 = uncurry (abs (abs (var this)))\n    in\n      abs (abs (abs (abs (abs (abs\n        (proj\u2082 (fold! g (pair z \u0394z) (zip-pair m \u0394m))))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7b518550cfa9e3ad9e59a04506f9846fd6df806b","subject":"Nat: lower bound: \u2200 n \u2192 2^ n \u2265 1 + n","message":"Nat: lower bound: \u2200 n \u2192 2^ n \u2265 1 + n\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Empty using (\u22a5-elim)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\nassoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\nassoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite assoc-comm x 1 x\n                              | assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite assoc-comm x 1 k | q | \u2261.sym (assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite assoc-comm a 1 a\n                                  | assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  sound : \u2200 m n \u2192 T (m <= n) \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 T (m <= n)\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a1b64bd6f0c7ee64a3981d6c27a1f95b45e4fc44","subject":"Updated a note on the existential.","message":"Updated a note on the existential.\n\nIgnore-this: 7293968fd045d35df9d66c1cb5beae62\n\ndarcs-hash:20120227220311-3bd4e-50e10ddd55956eaede0dbea4f19196795efaebb1.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Existential.agda","new_file":"notes\/Existential.agda","new_contents":"-- Tested with the development version of Agda on 27 February 2012.\n\n-- Thm: (\u2203x)A(x), (\u2200x)(A(x) \u21d2 B(x)) \u22a2 (\u2203x)B(x)\n\n-- From: Elliott Mendelson. Introduction to mathematical logic. Chapman &\n-- Hall, 4th edition, 1997, p. 155.\n\n-- Rule A4: (\u2200x)A(x) \u21d2 B(t) (with some conditions)\n-- Rule E4: A(t) \u21d2 (\u2203x)A(x) (with some conditions)\n-- Rule C: From (\u2203x)A(x) to A(t) (with some conditions)\n\n-- 1. (\u2203x)A(x)          Hyp\n-- 2. (\u2200x)(A(x) \u21d2 B(x)) Hyp\n-- 3. A(b)              1, rule C\n-- 4. A(b) \u21d2 B(b)       2, rule A4\n-- 5. B(b)              4,3 MP\n-- 6. (\u2203x)B(x)          5, rule E4\n\nmodule Existential where\n\npostulate\n  D   : Set\n  A B : D \u2192 Set\n\nmodule Witness where\n\n  infixr 7 _,_\n\n  data \u2203 (A : D \u2192 Set) : Set where\n    _,_ : \u2200 x \u2192 A x \u2192 \u2203 A\n\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 x \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (x , Ax) h = h x Ax\n\n  -- A proof using the existential elimination.\n  prf\u2081 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2081 h\u2081 h\u2082 = \u2203-elim h\u2081 (\u03bb x Ax \u2192 x , (h\u2082 Ax))\n\n  -- A proof using pattern matching.\n  prf\u2082 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2082 (x , Ax) h = x , h Ax\n\nmodule NonWitness\u2081 where\n\n  data \u2203 (A : D \u2192 Set) : Set where\n    \u2203-intro : \u2200 {x} \u2192 A x \u2192 \u2203 A\n\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (\u2203-intro Ax) h = h Ax\n\n  -- A proof using the existential elimination.\n  prf\u2081 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2081 h\u2081 h\u2082 = \u2203-elim h\u2081 (\u03bb Ax \u2192 \u2203-intro (h\u2082 Ax))\n\n  -- A proof using pattern matching.\n  prf\u2082 : \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf\u2082 (\u2203-intro Ax) h = \u2203-intro (h Ax)\n\nmodule NonWitness\u2082 where\n\n  -- We add 3 to the fixities of the standard library.\n  infix 6 \u00ac_\n\n  -- The empty type.\n  data \u22a5 : Set where\n\n  \u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n  \u22a5-elim ()\n\n  \u00ac_ : Set \u2192 Set\n  \u00ac A = A \u2192 \u22a5\n\n  \u2203 : (D \u2192 Set) \u2192 Set\n  \u2203 A = \u00ac (\u2200 x \u2192 \u00ac (A x))\n\n  prf : \u2203 A \u2192 (\u2200 x \u2192 A x \u2192 B x) \u2192 \u2203 B\n  prf h\u2081 h\u2082 h\u2083 = h\u2081 (\u03bb x Ax \u2192 h\u2083 x (h\u2082 x Ax))\n","old_contents":"-- Tested with the development version of Agda on 26 February 2012.\n\nmodule Existential where\n\ninfixr 7 _,_\n\npostulate\n  D   : Set\n  P Q : D \u2192 Set\n  P\u2192Q : \u2200 {x} \u2192 P x \u2192 Q x\n\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 (\u2200 x \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n\n-- A proof using the existential elimination.\nprf\u2081 : \u2203 P \u2192 \u2203 Q\nprf\u2081 h = \u2203-elim h (\u03bb x Px \u2192 x , P\u2192Q Px)\n\n-- A proof using the existential elimination with a helper function.\nprf\u2082 : \u2203 P \u2192 \u2203 Q\nprf\u2082 h = \u2203-elim h helper\n  where\n  helper : \u2200 x \u2192 P x \u2192 \u2203 Q\n  helper x Px = x , P\u2192Q Px\n\n-- A proof using pattern matching.\nprf\u2083 : \u2203 P \u2192 \u2203 Q\nprf\u2083 (x , Px) = x , P\u2192Q Px\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ab581eebcdf1294dc84fbc332fba270cb1f5002f","subject":"minor","message":"minor\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 x \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192\n                    let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ind  : SearchInd search\n\n  search-ind' : SearchInd' search\n  search-ind' P P\u2219 {f} Pf = search-ind (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\n  search-ext : SearchExt search\n  search-ext sg {f} {g} f\u2248\u00b0g = search-ind (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  search-mono : SearchMono search\n  search-mono _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = search-ind (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\n  search-swap : SearchSwap search\n  search-swap sg f {s\u1d2e} pf = search-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2248 s\u1d2e (s \u2218 flip f)) (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _))) (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\n  searchMon : SearchMon A\n  searchMon m = search _\u2219_\n    where open Mon m\n\n  search-\u03b5 : Search\u03b5 searchMon\n  search-\u03b5 m = search-ind' Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n    where open Mon m\n\n  search-hom : SearchMonHom searchMon\n  search-hom cm f g = search-ind (\u03bb s \u2192 s (f \u2219\u00b0 g) \u2248 s f \u2219 s g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  sum : Sum A\n  sum = search _+_\n\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ind\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    {- derived from ind\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n    -}\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\n    {- derived from ind\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n    -}\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk _ (search-ind \u03bcA +SearchInd search-ind \u03bcB)\n    {- derived from ind\n  where\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n    -}\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) = mk _ (search-ind \u03bcA \u00d7SearchInd search-ind \u03bcB)\n{- derived from SearchInd\n   where\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n             -}\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n    {- derived from ind\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n    -}\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x))\n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192\n                    let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n                  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n                  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ind  : SearchInd search\n\n  search-ind' : SearchInd' search\n  search-ind' P P\u2219 {f} Pf = search-ind (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\n  search-ext : SearchExt search\n  search-ext sg {f} {g} f\u2248\u00b0g = search-ind (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n    where open Sgrp sg\n\n  search-mono : SearchMono search\n  search-mono _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = search-ind (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\n  search-swap : SearchSwap search\n  search-swap sg f {s\u1d2e} pf = search-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2248 s\u1d2e (s \u2218 flip f)) (\u03bb p q \u2192 trans (\u2219-cong p q) (sym (pf _ _))) (\u03bb _ \u2192 refl)\n    where open Sgrp sg\n\n  searchMon : SearchMon A\n  searchMon m = search _\u2219_\n    where open Mon m\n\n  search-\u03b5 : Search\u03b5 searchMon\n  search-\u03b5 m = search-ind' Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n    where open Mon m\n\n  search-hom : SearchMonHom searchMon\n  search-hom cm f g = search-ind (\u03bb s \u2192 s (f \u2219\u00b0 g) \u2248 s f \u2219 s g)\n                                 (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _)) (\u03bb _ \u2192 refl)\n    where open CMon cm\n\n  sum : Sum A\n  sum = search _+_\n\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ind\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    {- derived from ind\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n    -}\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\n    {- derived from ind\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n    -}\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk _ (search-ind \u03bcA +SearchInd search-ind \u03bcB)\n    {- derived from ind\n  where\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n    -}\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) = mk _ (search-ind \u03bcA \u00d7SearchInd search-ind \u03bcB)\n{- derived from SearchInd\n   where\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n             -}\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n    {- derived from ind\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n    -}\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x))\n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"08dbc7d60b15a02ae4039444bf0e6d8a7ac2216a","subject":"sum: private sA","message":"sum: private sA\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen L using (Lift)\nopen import Type hiding (\u2605)\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to SumProp)\nopen import Search.Searchable.Product\n\nmodule sum where\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bcLift\u22a4 : SumProp (Lift \u22a4)\n\u03bcLift\u22a4 = _ , ind\n  where\n    srch : Search (Lift \u22a4)\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search \u22a4\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nvsgsum : \u2200 sg {n} \u2192 let open Sgrp sg in\n                    Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  private\n    sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf = \n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) \n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen L using (Lift)\nopen import Type hiding (\u2605)\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nimport Function.Related as FR\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Search.Type\nopen import Search.Searchable renaming (Searchable to SumProp)\nopen import Search.Searchable.Product\n\nmodule sum where\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605 _\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n_\u2248Search_ : \u2200 {A} \u2192 (s\u2080 s\u2081 : Search A) \u2192 \u2605\u2081\ns\u2080 \u2248Search s\u2081 = \u2200 {B} (op : Op\u2082 B) f \u2192 s\u2080 op f \u2261 s\u2081 op f\n\n\u03bcLift\u22a4 : SumProp (Lift \u22a4)\n\u03bcLift\u22a4 = _ , ind\n  where\n    srch : Search (Lift \u22a4)\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = srch , ind\n  where\n    srch : Search \u22a4\n    srch _ f = f _\n\n    ind : SearchInd srch\n    ind _ _ Pf = Pf _\n\n\u03bcBit : SumProp Bit\n\u03bcBit = searchBit , ind\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    ind : SearchInd searchBit\n    ind _ _P\u2219_ Pf = Pf 0b P\u2219 Pf 1b\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e B)\n\u03bcA +\u03bc \u03bcB = _ , search-ind \u03bcA +SearchInd search-ind \u03bcB\n\n\n_\u228e'_ : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\nA \u228e' B = \u03a3 Bool (cond A B)\n\n_\u03bc\u228e'_ : \u2200 {A B} \u2192 SumProp A \u2192 SumProp B \u2192 SumProp (A \u228e' B)\n\u03bcA \u03bc\u228e' \u03bcB = \u03bc\u03a3 \u03bcBit (\u03bb { {true} \u2192 \u03bcA ; {false} \u2192 \u03bcB })\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = search\u1d2e , ind\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n\n    ind : SearchInd search\u1d2e\n    ind P P\u2219 Pf = search-ind \u03bcA (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 A\u2192B))) P\u2219 (Pf \u2218 A\u2192B)\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec.NP as Vec using (Vec; tabulate; _++_) renaming (map to vmap; sum to vsum; foldr to vfoldr; foldr\u2081 to vfoldr\u2081)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nvsgsum : \u2200 sg {n} \u2192 let open Sgrp sg in\n                    Vec C (suc n) \u2192 C\nvsgsum sg = vfoldr\u2081 _\u2219_\n  where open Sgrp sg\n\n-- let's recall that: tabulate f \u2257 vmap f (allFin n)\n\n-- searchMonFin : \u2200 n \u2192 SearchMon (Fin n)\n-- searchMonFin n m f = vmsum m (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr\u2081 _\u2219_ (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFinSuc : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSuc n = searchFinSuc n , ind n\n  where ind : \u2200 n \u2192 SearchInd (searchFinSuc n)\n        ind zero    P P\u2219 Pf = Pf zero\n        ind (suc n) P P\u2219 Pf = P\u2219 (Pf zero) (ind n (\u03bb s \u2192 P (\u03bb op f \u2192 s op (f \u2218 suc))) P\u2219 (Pf \u2218 suc))\n\n\u03bcFinSucIso : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFinSucIso n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n\n\u03bc^ : \u2200 {A} (\u03bcA : SumProp A) n \u2192 SumProp (A ^ n)\n\u03bc^ \u03bcA zero    = \u03bcLift\u22a4\n\u03bc^ \u03bcA (suc n) = \u03bcA \u00d7\u03bc \u03bc^ \u03bcA n\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA n = \u03bc-iso (^\u2194Vec n) (\u03bc^ \u03bcA n)\n\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c op f = f []\nsearchVec (suc n) search\u1d2c op f = search\u1d2c op (\u03bb x \u2192 searchVec n search\u1d2c op (f \u2218 _\u2237_ x))\n\nsearchVec-spec : \u2200 {A} (\u03bcA : SumProp A) n \u2192 searchVec n (search \u03bcA) \u2248Search search (\u03bcVec \u03bcA n)\nsearchVec-spec \u03bcA zero    op f = \u2261.refl\nsearchVec-spec \u03bcA (suc n) op f = search-ext \u03bcA op (\u03bb x \u2192 searchVec-spec \u03bcA n op (f \u2218 _\u2237_ x))\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192\n                   search (\u03bcVec \u03bcA m) _\u2219_ (\u03bb ys \u2192\n                     f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-sg-ext \u03bcA sg (\u03bb x \u2192\n                                 searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-iso swap-iso\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 swap)\nswapS-preserve = \u03bc-iso-preserve swap-iso\n\nmodule _ {A : Set}(\u03bcA : SumProp A) where\n\n  sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  abs : Fin 0 \u2192 A\n  abs ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search (Fin n \u2192 A)\n  sFun zero    op f = f abs\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = Pf abs\n  Ind (suc n) P P\u2219 Pf = \n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) \n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 SumProp (Fin n \u2192 A)\n  \u03bcFun = sFun _ , Ind _\n\n{-\n  -- If we want to force non-empty domain\n\n  sFun : \u2200 n \u2192 Search (Fin (suc n) \u2192 A)\n  sFun zero    op f = sA op (f \u2218 const)\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  Ind : \u2200 n \u2192 SearchInd (sFun n)\n  Ind zero    P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (f \u2218 const))) P\u2219 (Pf \u2218 const)\n  Ind (suc n) P P\u2219 Pf = search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x)))) \n      P\u2219 \n      (\u03bb x \u2192 Ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x))) P\u2219 (Pf \u2218 extend x))\n\n-}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"eec887d44cbcc922a09205e4324a31177d49b804","subject":"bintree: function sorting","message":"bintree: function sorting\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bits hiding (replicate; _<=_)\nimport Data.Bits.Search as Search\nopen Search.SimpleSearch\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Data.Sum using (_\u228e_; inj\u2081; inj\u2082)\nopen Alternative-Reverse\nopen import Relation.Binary\nimport Level as L\nopen import Data.Bool\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\ndata _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n  here  : x \u2208 leaf x\n  left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n  right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x)   = leaf (f x)\nmap f (fork t u) = fork (map f t) (map f u)\n\nprivate\n  module Dummy {a} {A : Set a} where\n    replicate : \u2200 n \u2192 A \u2192 Tree A n\n    replicate zero    x = leaf x\n    replicate (suc n) x = fork t t where t = replicate n x\n\n    fromFun : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Tree A n\n    fromFun {zero}  f = leaf (f [])\n    fromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\n    toFun : \u2200 {n} \u2192 Tree A n \u2192 Bits n \u2192 A\n    toFun (leaf x)   _        = x\n    toFun (fork t u) (b \u2237 bs) = toFun (if b then u else t) bs\n\n    toFun\u2218fromFun : \u2200 {n} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\n    toFun\u2218fromFun {zero}  f [] = \u2261.refl\n    toFun\u2218fromFun {suc n} f (false \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\n    toFun\u2218fromFun {suc n} f (true \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\n    fromFun\u2218toFun : \u2200 {n} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\n    fromFun\u2218toFun (leaf x) = \u2261.refl\n    fromFun\u2218toFun (fork t\u2080 t\u2081)\n      rewrite fromFun\u2218toFun t\u2080\n            | fromFun\u2218toFun t\u2081 = \u2261.refl\n\n    toFun\u2192fromFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\n    toFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\n    toFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n      rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n            | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\n    fromFun\u2192toFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\n    fromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\n    fromFun-\u2257 : \u2200 {n} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\n    fromFun-\u2257 {zero}  f\u2257g\n      rewrite f\u2257g [] = \u2261.refl\n    fromFun-\u2257 {suc n} f\u2257g\n      rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n            | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n            = \u2261.refl\n\n    lookup : \u2200 {n} \u2192 Bits n \u2192 Tree A n \u2192 A\n    lookup = flip toFun\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\n\n    first : \u2200 {n} \u2192 Tree A n \u2192 A\n    first (leaf x)   = x\n    first (fork t _) = first t\n\n    last : \u2200 {n} \u2192 Tree A n \u2192 A\n    last (leaf x)   = x\n    last (fork _ t) = last t\n\n    -- Returns the flat vector of leaves underlying the perfect binary tree.\n    toVec : \u2200 {n} \u2192 Tree A n \u2192 Vec A (2^ n)\n    toVec (leaf x)     = x \u2237 []\n    toVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\n    lookup' : \u2200 {m n} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\n    lookup' [] t = t\n    lookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n    update' : \u2200 {m n} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\n    update' []        val t = val\n    update' (b \u2237 key) val (fork t u) = if b then fork t (update' key val u)\n                                            else fork (update' key val t) u\n\n    \u2208-toBits : \u2200 {x n} {t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n    \u2208-toBits here = []\n    \u2208-toBits (left key) = 0b \u2237 \u2208-toBits key\n    \u2208-toBits (right key) = 1b \u2237 \u2208-toBits key\n\n    \u2208-lookup : \u2200 {x n} {t : Tree A n} (path : x \u2208 t) \u2192 lookup (\u2208-toBits path) t \u2261 x\n    \u2208-lookup here = \u2261.refl\n    \u2208-lookup (left path) = \u2208-lookup path\n    \u2208-lookup (right path) = \u2208-lookup path\n\n    lookup-\u2208 : \u2200 {n} key (t : Tree A n) \u2192 lookup key t \u2208 t\n    lookup-\u2208 [] (leaf x) = here\n    lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n    lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\nopen Dummy public\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n = n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                              (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = \u2208-toBits p \u2264\u1d2e \u2208-toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                                (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting-\u2293-\u2294 {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x , y) = (x \u2293\u1d2c y , x \u2294\u1d2c y)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc _} = merge \u2218 map-outer sort sort\n\nmodule \u2293-\u2294\u1d2c {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n    _\u2293\u1d2c_ : A \u2192 A \u2192 A\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n    _\u2294\u1d2c_ : A \u2192 A \u2192 A\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\nmodule Sorting-<= {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n    open \u2293-\u2294\u1d2c _<=\u1d2c_\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_ public\n\nmodule EvalTree {a} {A : Set a} where\n    open import Data.Bits.OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {n} \u2192 Bij n \u2192 Endo (Tree A n)\n    evalTree `id          = id\n    evalTree (f `\u204f g)      = evalTree g \u2218 evalTree f\n    evalTree (`id   `\u2237 f) = map-outer (evalTree (f 0b)) (evalTree (f 1b))\n    evalTree (`not\u1d2e `\u2237 f) = map-outer (evalTree (f 1b)) (evalTree (f 0b)) \u2218 swap\n    evalTree `0\u21941         = interchange\n\n    evalTree-eval : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id  _ _ = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval (f `\u204f g) t xs\n      rewrite evalTree-eval f t xs\n            | evalTree-eval g (evalTree f t) (eval f xs)\n            = \u2261.refl\n    evalTree-eval (`id   `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id   `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule BijSpec {a} {A : Set a} where\n    open EvalTree\n    open import Data.Bits.OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n\n    record Bij[_\u21a6_] {n} (t u : Tree A n) : Set a where\n      constructor _,_\n      field\n        bij   : Bij n\n        proof : evalTree bij t \u2261 u\n    open Bij[_\u21a6_] public\n\n    Bij[\u2257_] : \u2200 {n} (f : Endo (Tree A n)) \u2192 Set a\n    Bij[\u2257 f ] = \u2200 t \u2192 Bij[ t \u21a6 f t ]\n\n    evalB : \u2200 {n} {f : Endo (Tree A n)} (b : Bij[\u2257 f ]) \u2192 Endo (Tree A n)\n    evalB b t = evalTree (bij (b t)) t\n\n    bij-evalB-spec : \u2200 {n} {f : Endo (Tree A n)} (b : Bij[\u2257 f ]) \u2192 evalB b \u2257 f\n    bij-evalB-spec b = proof \u2218 b\n\n    id-bij : \u2200 {n} \u2192 Bij[\u2257 id {A = Tree A n} ]\n    id-bij _ = `id , \u2261.refl\n\n    infixr 9 _\u2218-bij_\n    _\u2218-bij_ : \u2200 {n} {f g : Endo (Tree A n)} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 f \u2218 g ]\n    _\u2218-bij_ {f = f} {g} `f `g t\n      = `bij , helper\n      where `bij = bij (`g t) `\u204f bij (`f (g t))\n            helper : evalTree `bij t \u2261 f (g t)\n            helper rewrite proof (`g t) = proof (`f (g t))\n\n    swap-bij : \u2200 {n} \u2192 Bij[\u2257 swap {n = n} ]\n    swap-bij (fork _ _) = `not , \u2261.refl\n\n    map-outer-bij : \u2200 {n} {f g : Endo (Tree A n)}\n                      \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 map-outer f g ]\n    map-outer-bij `f `g (fork t u)\n      = `map-outer (bij (`f t)) (bij (`g u))\n      , \u2261.cong\u2082 fork (proof (`f t)) (proof (`g u))\n\n    map-inner-bij : \u2200 {n} {f : Endo (Tree A (1 + n))} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 map-inner f ]\n    map-inner-bij {f = f} `f t = map-inner-perm , helper\n       where map-inner-perm = `map-inner (bij (`f (inner t)))\n             helper : evalTree map-inner-perm t \u2261 map-inner f t\n             helper with t\n             ... | fork (fork a b) (fork c d) with f (fork b c) | proof (`f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\n    interchange-bij : \u2200 {n} \u2192 Bij[\u2257 interchange {n = n} ]\n    interchange-bij = map-inner-bij swap-bij\n\nmodule SortingBijSpec {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool)\n                      (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    open Sorting-<= _<=\u1d2c_\n    open EvalTree\n    open import Data.Bits.OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open BijSpec\n\n    `sort\u2081 : Tree A 1 \u2192 Bij 1\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    sort\u2081-bij : Bij[\u2257 sort\u2081 ]\n    sort\u2081-bij t = `sort\u2081 t , helper t\n      where helper : \u2200 t \u2192 evalTree (`sort\u2081 t) t \u2261 sort\u2081 t\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-bij : \u2200 {n} \u2192 Bij[\u2257 merge {n} ]\n    merge-bij {zero}  = sort\u2081-bij\n    merge-bij {suc _} = map-inner-bij merge-bij\n                  \u2218-bij map-outer-bij merge-bij merge-bij\n                  \u2218-bij interchange-bij\n\n    sort-bij : \u2200 {n} \u2192 Bij[\u2257 sort {n} ]\n    sort-bij {zero}  = id-bij\n    sort-bij {suc _} = merge-bij \u2218-bij map-outer-bij sort-bij sort-bij\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule FunctionSorting {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n  _\u2264\u1d2c_ = \u03bb x y \u2192 T (x <=\u1d2c y)\n  open \u2293-\u2294\u1d2c _<=\u1d2c_\n  module S = Sorting-<= _<=\u1d2c_\n  open BijSpec\n\n  sort : \u2200 {n} \u2192 Endo (Bits n \u2192 A)\n  sort = toFun \u2218 S.sort \u2218 fromFun\n\n  module Properties (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n                    (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                    (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                    (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                    (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                    (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                    (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                    (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                    (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z) where\n\n    module B = SortingBijSpec _<=\u1d2c_  isTotalOrder\n    open IsTotalOrder isTotalOrder\n    open SortedMonotonicFunctions _\u2264\u1d2c_ isPreorder\n    module SP = SortingProperties _\u2264\u1d2c_ _\u2293\u1d2c_ _\u2294\u1d2c_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open import Data.Bits.OperationSyntax using (eval)\n    open EvalTree\n\n    sort-is-sorting : \u2200 {n} (f : Bits n \u2192 A) \u2192 Monotone (sort f)\n    sort-is-sorting = DataSorted\u2192Sorted \u2218 SP.sort-spec \u2218 fromFun\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting-\u2293-\u2294 _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    module FS = FunctionSorting _<=_\n    module FSP = FS.Properties isTotalOrder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    open SortingBijSpec _<=_ isTotalOrder public\n    open EvalTree public using (evalTree)\n    open BijSpec public\n    open OperationSyntax public\n    open FS public using () renaming (sort to sortFun)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bits hiding (replicate; _<=_)\nimport Data.Bits.Search as Search\nopen Search.SimpleSearch\nopen import Function.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Data.Sum using (_\u228e_; inj\u2081; inj\u2082)\nopen Alternative-Reverse\nopen import Relation.Binary\nimport Level as L\nopen import Data.Bool\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\ndata _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n  here  : x \u2208 leaf x\n  left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n  right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x)   = leaf (f x)\nmap f (fork t u) = fork (map f t) (map f u)\n\nprivate\n  module Dummy {a} {A : Set a} where\n    replicate : \u2200 n \u2192 A \u2192 Tree A n\n    replicate zero    x = leaf x\n    replicate (suc n) x = fork t t where t = replicate n x\n\n    fromFun : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Tree A n\n    fromFun {zero}  f = leaf (f [])\n    fromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\n    toFun : \u2200 {n} \u2192 Tree A n \u2192 Bits n \u2192 A\n    toFun (leaf x)   _        = x\n    toFun (fork t u) (b \u2237 bs) = toFun (if b then u else t) bs\n\n    toFun\u2218fromFun : \u2200 {n} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\n    toFun\u2218fromFun {zero}  f [] = \u2261.refl\n    toFun\u2218fromFun {suc n} f (false \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\n    toFun\u2218fromFun {suc n} f (true \u2237 xs)\n      rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\n    fromFun\u2218toFun : \u2200 {n} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\n    fromFun\u2218toFun (leaf x) = \u2261.refl\n    fromFun\u2218toFun (fork t\u2080 t\u2081)\n      rewrite fromFun\u2218toFun t\u2080\n            | fromFun\u2218toFun t\u2081 = \u2261.refl\n\n    toFun\u2192fromFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\n    toFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\n    toFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n      rewrite toFun\u2192fromFun t\u2080 _ (t\u2257f \u2218 0\u2237_)\n            | toFun\u2192fromFun t\u2081 _ (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\n    fromFun\u2192toFun : \u2200 {n} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\n    fromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\n    fromFun-\u2257 : \u2200 {n} {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 fromFun f \u2261 fromFun g\n    fromFun-\u2257 {zero}  f\u2257g\n      rewrite f\u2257g [] = \u2261.refl\n    fromFun-\u2257 {suc n} f\u2257g\n      rewrite fromFun-\u2257 (f\u2257g \u2218 0\u2237_)\n            | fromFun-\u2257 (f\u2257g \u2218 1\u2237_)\n            = \u2261.refl\n\n    lookup : \u2200 {n} \u2192 Bits n \u2192 Tree A n \u2192 A\n    lookup = flip toFun\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\n\n    first : \u2200 {n} \u2192 Tree A n \u2192 A\n    first (leaf x)   = x\n    first (fork t _) = first t\n\n    last : \u2200 {n} \u2192 Tree A n \u2192 A\n    last (leaf x)   = x\n    last (fork _ t) = last t\n\n    -- Returns the flat vector of leaves underlying the perfect binary tree.\n    toVec : \u2200 {n} \u2192 Tree A n \u2192 Vec A (2^ n)\n    toVec (leaf x)     = x \u2237 []\n    toVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\n    lookup' : \u2200 {m n} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\n    lookup' [] t = t\n    lookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n    update' : \u2200 {m n} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\n    update' []        val t = val\n    update' (b \u2237 key) val (fork t u) = if b then fork t (update' key val u)\n                                            else fork (update' key val t) u\n\n    \u2208-toBits : \u2200 {x n} {t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n    \u2208-toBits here = []\n    \u2208-toBits (left key) = 0b \u2237 \u2208-toBits key\n    \u2208-toBits (right key) = 1b \u2237 \u2208-toBits key\n\n    \u2208-lookup : \u2200 {x n} {t : Tree A n} (path : x \u2208 t) \u2192 lookup (\u2208-toBits path) t \u2261 x\n    \u2208-lookup here = \u2261.refl\n    \u2208-lookup (left path) = \u2208-lookup path\n    \u2208-lookup (right path) = \u2208-lookup path\n\n    lookup-\u2208 : \u2200 {n} key (t : Tree A n) \u2192 lookup key t \u2208 t\n    lookup-\u2208 [] (leaf x) = here\n    lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n    lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\nopen Dummy public\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n = n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 first t \u2264\u1d2c last t\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 first t \u2264\u1d2c lookup p t\n    Sorted\u2192lb leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork st su h\u2264l) (true  \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st) h\u2264l) (Sorted\u2192lb su p)\n    Sorted\u2192lb (fork st _  _)   (false \u2237 p) = Sorted\u2192lb st p\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 \u2200 (p : Bits n) \u2192 lookup p t \u2264\u1d2c last t\n    Sorted\u2192ub leaf             []          = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _  su _)   (true  \u2237 p) = Sorted\u2192ub su p\n    Sorted\u2192ub (fork st su h\u2264l) (false \u2237 p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (\u2264\u1d2c-bounds su)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set \u2113\n    Bounded {n} t l h = \u2200 (p : Bits n) \u2192 (l \u2264\u1d2c lookup p t) \u00d7 (lookup p t \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} \u2192 Sorted t \u2192 Bounded t (first t) (last t)\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\nmodule SortedMembershipProofs {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                              (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = \u2208-toBits p \u2264\u1d2e \u2208-toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedMonotonicFunctions {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                                (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n\n    Monotone : \u2200 {n} \u2192 (Bits n \u2192 A) \u2192 Set _\n    Monotone {n} f = \u2200 {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2c f q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted = Monotone \u2218 toFun\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    open SortedData _\u2264\u1d2c_ isPreorder renaming (Sorted to DataSorted)\n    DataSorted\u2192Sorted : \u2200 {n} {t : Tree A n} \u2192 DataSorted t \u2192 Sorted t\n    DataSorted\u2192Sorted leaf             []                = \u2264\u1d2c.refl\n    DataSorted\u2192Sorted (fork _  su _)   (there true  p\u2264q) = DataSorted\u2192Sorted su p\u2264q\n    DataSorted\u2192Sorted (fork st _  _)   (there false p\u2264q) = DataSorted\u2192Sorted st p\u2264q\n    DataSorted\u2192Sorted (fork st su h\u2264l) (0-1 p q)         = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub st p) h\u2264l) (Sorted\u2192lb su q)\n\nmodule Sorting-\u2293-\u2294 {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x , y) = (x \u2293\u1d2c y , x \u2294\u1d2c y)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc _} = merge \u2218 map-outer sort sort\n\nmodule \u2293-\u2294\u1d2c {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n    _\u2293\u1d2c_ : A \u2192 A \u2192 A\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n    _\u2294\u1d2c_ : A \u2192 A \u2192 A\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\nmodule Sorting-<= {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n    open \u2293-\u2294\u1d2c _<=\u1d2c_\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_ public\n\nmodule EvalTree {a} {A : Set a} where\n    open import Data.Bits.OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    evalTree : \u2200 {n} \u2192 Bij n \u2192 Endo (Tree A n)\n    evalTree `id          = id\n    evalTree (f `\u204f g)      = evalTree g \u2218 evalTree f\n    evalTree (`id   `\u2237 f) = map-outer (evalTree (f 0b)) (evalTree (f 1b))\n    evalTree (`not\u1d2e `\u2237 f) = map-outer (evalTree (f 1b)) (evalTree (f 0b)) \u2218 swap\n    evalTree `0\u21941         = interchange\n\n    evalTree-eval : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun t \u2257 toFun (evalTree f t) \u2218 eval f\n    evalTree-eval `id  _ _ = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 true  \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (true  \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval `0\u21941 (fork (fork _ _) (fork _ _)) (false \u2237 false \u2237 _) = \u2261.refl\n    evalTree-eval (f `\u204f g) t xs\n      rewrite evalTree-eval f t xs\n            | evalTree-eval g (evalTree f t) (eval f xs)\n            = \u2261.refl\n    evalTree-eval (`id   `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`id   `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (true  \u2237 xs) = evalTree-eval (f 1b) u xs\n    evalTree-eval (`not\u1d2e `\u2237 f) (fork t u) (false \u2237 xs) = evalTree-eval (f 0b) t xs\n\n    evalTree-eval\u2032 : \u2200 {n} (f : Bij n) (t : Tree A n) \u2192 toFun (evalTree f t) \u2257 toFun t \u2218 eval (f \u207b\u00b9)\n    evalTree-eval\u2032 f t x = toFun (evalTree f t) x\n                         \u2261\u27e8 \u2261.cong (toFun (evalTree f t)) (\u2261.sym (VecBijKit._\u207b\u00b9-inverse\u2032 _ f x)) \u27e9\n                           toFun (evalTree f t) (eval f (eval (f \u207b\u00b9) x))\n                         \u2261\u27e8 \u2261.sym (evalTree-eval f t (eval (f \u207b\u00b9) x)) \u27e9\n                           toFun t (eval (f \u207b\u00b9) x)\n                         \u220e where open \u2261-Reasoning\n\nmodule BijSpec {a} {A : Set a} where\n    open EvalTree\n    open import Data.Bits.OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n\n    record Bij[_\u21a6_] {n} (t u : Tree A n) : Set a where\n      constructor _,_\n      field\n        bij   : Bij n\n        proof : evalTree bij t \u2261 u\n    open Bij[_\u21a6_] public\n\n    Bij[\u2257_] : \u2200 {n} (f : Endo (Tree A n)) \u2192 Set a\n    Bij[\u2257 f ] = \u2200 t \u2192 Bij[ t \u21a6 f t ]\n\n    evalB : \u2200 {n} {f : Endo (Tree A n)} (b : Bij[\u2257 f ]) \u2192 Endo (Tree A n)\n    evalB b t = evalTree (bij (b t)) t\n\n    bij-evalB-spec : \u2200 {n} {f : Endo (Tree A n)} (b : Bij[\u2257 f ]) \u2192 evalB b \u2257 f\n    bij-evalB-spec b = proof \u2218 b\n\n    id-bij : \u2200 {n} \u2192 Bij[\u2257 id {A = Tree A n} ]\n    id-bij _ = `id , \u2261.refl\n\n    infixr 9 _\u2218-bij_\n    _\u2218-bij_ : \u2200 {n} {f g : Endo (Tree A n)} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 f \u2218 g ]\n    _\u2218-bij_ {f = f} {g} `f `g t\n      = `bij , helper\n      where `bij = bij (`g t) `\u204f bij (`f (g t))\n            helper : evalTree `bij t \u2261 f (g t)\n            helper rewrite proof (`g t) = proof (`f (g t))\n\n    swap-bij : \u2200 {n} \u2192 Bij[\u2257 swap {n = n} ]\n    swap-bij (fork _ _) = `not , \u2261.refl\n\n    map-outer-bij : \u2200 {n} {f g : Endo (Tree A n)}\n                      \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 g ] \u2192 Bij[\u2257 map-outer f g ]\n    map-outer-bij `f `g (fork t u)\n      = `map-outer (bij (`f t)) (bij (`g u))\n      , \u2261.cong\u2082 fork (proof (`f t)) (proof (`g u))\n\n    map-inner-bij : \u2200 {n} {f : Endo (Tree A (1 + n))} \u2192 Bij[\u2257 f ] \u2192 Bij[\u2257 map-inner f ]\n    map-inner-bij {f = f} `f t = map-inner-perm , helper\n       where map-inner-perm = `map-inner (bij (`f (inner t)))\n             helper : evalTree map-inner-perm t \u2261 map-inner f t\n             helper with t\n             ... | fork (fork a b) (fork c d) with f (fork b c) | proof (`f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\n    interchange-bij : \u2200 {n} \u2192 Bij[\u2257 interchange {n = n} ]\n    interchange-bij = map-inner-bij swap-bij\n\nmodule SortingBijSpec {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool)\n                      (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    open Sorting-<= _<=\u1d2c_\n    open EvalTree\n    open import Data.Bits.OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open BijSpec\n\n    `sort\u2081 : Tree A 1 \u2192 Bij 1\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    sort\u2081-bij : Bij[\u2257 sort\u2081 ]\n    sort\u2081-bij t = `sort\u2081 t , helper t\n      where helper : \u2200 t \u2192 evalTree (`sort\u2081 t) t \u2261 sort\u2081 t\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-bij : \u2200 {n} \u2192 Bij[\u2257 merge {n} ]\n    merge-bij {zero}  = sort\u2081-bij\n    merge-bij {suc _} = map-inner-bij merge-bij\n                  \u2218-bij map-outer-bij merge-bij merge-bij\n                  \u2218-bij interchange-bij\n\n    sort-bij : \u2200 {n} \u2192 Bij[\u2257 sort {n} ]\n    sort-bij {zero}  = id-bij\n    sort-bij {suc _} = merge-bij \u2218-bij map-outer-bij sort-bij sort-bij\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting-\u2293-\u2294 _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule FunctionSorting {a} {A : Set a} (_<=\u1d2c_ : A \u2192 A \u2192 Bool) where\n  _\u2264\u1d2c_ = \u03bb x y \u2192 T (x <=\u1d2c y)\n  open \u2293-\u2294\u1d2c _<=\u1d2c_\n  module S = Sorting-<= _<=\u1d2c_\n  open BijSpec\n\n  sort : \u2200 {n} \u2192 Endo (Bits n \u2192 A)\n  sort = toFun \u2218 S.sort \u2218 fromFun\n\n  module Properties (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n                    (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                    (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                    (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                    (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                    (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                    (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                    (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                    (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z) where\n\n    module B = SortingBijSpec _<=\u1d2c_  isTotalOrder\n    open IsTotalOrder isTotalOrder\n    open SortedMonotonicFunctions _\u2264\u1d2c_ isPreorder\n    module SP = SortingProperties _\u2264\u1d2c_ _\u2293\u1d2c_ _\u2294\u1d2c_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open import Data.Bits.OperationSyntax using (eval)\n    open EvalTree\n\n    sort-is-sorting : \u2200 {n} (f : Bits n \u2192 A) \u2192 Monotone (sort f)\n    sort-is-sorting = DataSorted\u2192Sorted \u2218 SP.sort-spec \u2218 fromFun\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y) public\n\n    module S = Sorting-\u2293-\u2294 _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_ isPreorder public\n    open SortingBijSpec _<=_ isTotalOrder public\n    open EvalTree public using (evalTree)\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a46a63ee380b76b2baeb6a5a49427ff64ca28f16","subject":"scraps for #14","message":"scraps for #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' val\n  lem-valfill VConst FHOuter = VConst\n  lem-valfill VLam FHOuter = VLam\n\n  -- values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFinal x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFinal x\u2081 x\u2082 () x\u2084)\n  ve (BVVal x) (CECong x\u2081 er) = ve (BVVal (lem-valfill x x\u2081)) er\n  ve (BVArrCast x bv) (CECong FHOuter (CECong x\u2081 er)) = {!!} -- ve bv {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d' , Step x\u2081 x\u2082 x\u2083) = {!x\u2082!}\n  vs (BVHoleCast x bv) s = {!!}\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole err = {!err!}\n    ie (INEHole x) err = {!!}\n    ie (IAp x indet x\u2081) err = {!!}\n    ie (ICastArr x indet) err = {!!}\n    ie (ICastGroundHole x indet) err = {!!}\n    ie (ICastHoleGround x indet x\u2081) err = {!!}\n    -- ie IEHole ()\n    -- ie (INEHole x) (ENEHole e) = fe x e\n    -- ie (IAp i x) (EAp1 e) = ie i e\n    -- ie (IAp i x) (EAp2 y e) = fe x e\n    -- ie (ICast i) (ECastError x x\u2081) = {!!} -- todo: this is evidence that casts are busted\n    -- ie (ICast i) (ECastProp x) = ie i x\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = {!!} -- cyrus\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = {!!} -- cyrus\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5 -- boxed?\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) st = {!!}\n  lem3 (BVHoleCast x bv) st = {!!}\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem1' : \u2200{d d'} \u2192 d final \u2192 d \u21a6 d' \u2192 \u22a5\n  lem1' fin = {!!}\n\n  -- lem1 (FVal x) st = lem3 x st\n  -- lem1 (FIndet x) st = lem2 x st\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 = {!!}\n  -- lem4 f (FHFinal x) = x\n  -- lem4 (FVal ()) (FHAp1 x\u2082 sub)\n  -- lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp1 x\u2083 sub) = lem4 x\u2082 sub\n  -- lem4 (FVal ()) (FHAp2 sub)\n  -- lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp2 sub) = lem4 (FIndet x\u2081) sub\n  -- lem4 f FHEHole = f\n  -- lem4 f (FHNEHoleFinal x) = f\n  -- lem4 (FVal ()) (FHCast sub)\n  -- lem4 (FIndet (ICast i)) (FHCast sub) = lem4 (FIndet i) sub\n  -- lem4 f (FHCastFinal x) = f\n  -- lem4 (FVal ()) (FHNEHole y)\n  -- lem4 (FIndet (INEHole x\u2081)) (FHNEHole y) = lem4 x\u2081 y\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  is IEHole (d' , Step FHOuter () FHOuter)\n  is (INEHole x) (d' , Step FHOuter () FHOuter)\n  is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n  is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n  is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = {!!}\n  is (IAp x ind x\u2081) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n  is (IAp x ind x\u2081) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = lem1 x\u2081 {!\u03c02 (lem6 (Step x\u2083 x\u2084 x\u2086))!}\n  is (ICastArr x ind) stp = {!!}\n  is (ICastGroundHole x ind) stp = {!!}\n  is (ICastHoleGround x ind g) stp = {!!}\n  -- is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  -- is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  -- is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  -- is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n  -- is (ICast a) (d' , Step (FHFinal x) q x\u2084) = lem1 x q\n  -- is (ICast a) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = is a (_ , Step x x\u2081 x\u2082)\n  -- is (ICast a) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = lem5 (FIndet (ICast a)) (FHCastFinal x) x\u2081 -- todo: this feels odd\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!e!}\n  -- -- cast error cases\n  -- es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  -- es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!} -- todo: this is evidence that casts are busted\n  -- es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n  --   with type-assignment-unicity x x\u2084\n  -- ... | refl = ~apart x\u2081 x\u2085\n\n  -- -- ap1 cases\n  -- es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  -- es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- -- ap2 cases\n  -- es (EAp2 a er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (EAp2 a er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  -- es (EAp2 a er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = lem5 a x x\u2081\n\n  -- -- nehole cases\n  -- es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (ENEHole er) (_ , Step (FHNEHole a) x (FHNEHole x\u2082)) = es er (_ , Step a x x\u2082)\n  -- es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- -- castprop cases\n  -- es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  -- es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' val\n  lem-valfill VConst FHOuter = VConst\n  lem-valfill VLam FHOuter = VLam\n\n  -- values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFinal x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFinal x\u2081 x\u2082 () x\u2084)\n  ve (BVVal x) (CECong x\u2081 er) = ve (BVVal (lem-valfill x x\u2081)) er\n  ve (BVArrCast x bv) (CECong FHOuter (CECong x\u2081 er)) = {!!} -- ve bv {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d' , Step x\u2081 x\u2082 x\u2083) = {!x\u2082!}\n  vs (BVHoleCast x bv) s = {!!}\n\n  mutual\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole err = {!err!}\n    ie (INEHole x) err = {!!}\n    ie (IAp x indet x\u2081) err = {!!}\n    ie (ICastArr x indet) err = {!!}\n    ie (ICastGroundHole x indet) err = {!!}\n    ie (ICastHoleGround x indet x\u2081) err = {!!}\n    -- ie IEHole ()\n    -- ie (INEHole x) (ENEHole e) = fe x e\n    -- ie (IAp i x) (EAp1 e) = ie i e\n    -- ie (IAp i x) (EAp2 y e) = fe x e\n    -- ie (ICast i) (ECastError x x\u2081) = {!!} -- todo: this is evidence that casts are busted\n    -- ie (ICast i) (ECastProp x) = ie i x\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 = {!!}\n  -- lem2 IEHole ()\n  -- lem2 (INEHole f) ()\n  -- lem2 (IAp () _) (ITLam _)\n  -- lem2 (ICast x) (ITCast (FVal x\u2081) x\u2082 x\u2083) = vi x\u2081 x\n  -- lem2 (ICast x) (ITCast (FIndet x\u2081) x\u2082 x\u2083) = {!!} -- todo: this is evidence that casts are busted\n\n  lem3 : \u2200{d d'} \u2192 d val \u2192 d \u2192> d' \u2192 \u22a5 -- boxed?\n  lem3 = {!!}\n  -- lem3 VConst ()\n  -- lem3 VLam ()\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 = {!!}\n  -- lem1 (FVal x) st = lem3 x st\n  -- lem1 (FIndet x) st = lem2 x st\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 = {!!}\n  -- lem4 f (FHFinal x) = x\n  -- lem4 (FVal ()) (FHAp1 x\u2082 sub)\n  -- lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp1 x\u2083 sub) = lem4 x\u2082 sub\n  -- lem4 (FVal ()) (FHAp2 sub)\n  -- lem4 (FIndet (IAp x\u2081 x\u2082)) (FHAp2 sub) = lem4 (FIndet x\u2081) sub\n  -- lem4 f FHEHole = f\n  -- lem4 f (FHNEHoleFinal x) = f\n  -- lem4 (FVal ()) (FHCast sub)\n  -- lem4 (FIndet (ICast i)) (FHCast sub) = lem4 (FIndet i) sub\n  -- lem4 f (FHCastFinal x) = f\n  -- lem4 (FVal ()) (FHNEHole y)\n  -- lem4 (FIndet (INEHole x\u2081)) (FHNEHole y) = lem4 x\u2081 y\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  is = {!!}\n  -- is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  -- is IEHole (_ , Step FHEHole () _)\n  -- is (INEHole _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  -- is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n  -- is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () FHEHole)\n  -- is (INEHole x) (d , Step (FHNEHoleFinal x\u2081) () (FHFinal x\u2083))\n  -- is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  -- is (INEHole x) (_ , Step (FHNEHoleFinal x\u2081) () (FHCastFinal x\u2083))\n  -- is (IAp _ _) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  -- is (IAp _ (FVal x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = vs x (_ , Step p q r)\n  -- is (IAp _ (FIndet x)) (_ , Step (FHAp1 _ p) q (FHAp1 _ r)) = is x (_ , Step p q r)\n  -- is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = is i (_ , (Step p q r))\n  -- is (ICast a) (d' , Step (FHFinal x) q x\u2084) = lem1 x q\n  -- is (ICast a) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = is a (_ , Step x x\u2081 x\u2082)\n  -- is (ICast a) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = lem5 (FIndet (ICast a)) (FHCastFinal x) x\u2081 -- todo: this feels odd\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es = {!!}\n  -- -- cast error cases\n  -- es (ECastError x x\u2081) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = lem1 x\u2082 x\u2083\n  -- es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!!} -- todo: this is evidence that casts are busted\n  -- es (ECastError x x\u2081) (d' , Step (FHCastFinal x\u2082) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n  --   with type-assignment-unicity x x\u2084\n  -- ... | refl = ~apart x\u2081 x\u2085\n\n  -- -- ap1 cases\n  -- es (EAp1 er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (EAp1 er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = fe x er\n  -- es (EAp1 er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n\n  -- -- ap2 cases\n  -- es (EAp2 a er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (EAp2 a er) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = es er (_ , Step x\u2081 x\u2082 x\u2084)\n  -- es (EAp2 a er) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = lem5 a x x\u2081\n\n  -- -- nehole cases\n  -- es (ENEHole er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (ENEHole er) (_ , Step (FHNEHole a) x (FHNEHole x\u2082)) = es er (_ , Step a x x\u2082)\n  -- es (ENEHole er) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = fe x er\n\n  -- -- castprop cases\n  -- es (ECastProp er) (d' , Step (FHFinal x) x\u2081 x\u2082) = lem1 x x\u2081\n  -- es (ECastProp er) (_ , Step (FHCast x) x\u2081 (FHCast x\u2082)) = es er (_ , Step x x\u2081 x\u2082)\n  -- es (ECastProp er) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = fe x er\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d57efc3d211b594ad1f24803df8ff1199af2f8bb","subject":"reduction definition","message":"reduction definition\n","repos":"ademinn\/RefactoringVerification","old_file":"src\/prover\/Main.agda","new_file":"src\/prover\/Main.agda","new_contents":"module Main where\nopen import IO\nopen import Data.String\nopen import Data.List\nopen import Data.Bool\n\n{-# IMPORT AST #-}\n\n{-# COMPILED_TYPE String AST.Type #-}\n{-# COMPILED_TYPE String AST.Identifier #-}\n\ndata Variable : Set where\n  V : String \u2192 String \u2192 Variable\n\nvarName : Variable \u2192 String\nvarName (V _ name) = name\n\n{-# COMPILED_DATA Variable AST.Variable AST.Variable #-}\n\ndata Expression : Set where\n  New : String \u2192 List Expression \u2192 Expression\n  FieldAccess : String \u2192 Expression \u2192 String \u2192 Expression\n  MethodCall : String \u2192 Expression \u2192 String \u2192 List Expression \u2192 Expression\n  Var : String \u2192 String \u2192 Expression\n\nexprType : Expression \u2192 String\nexprType (New t _) = t\nexprType (FieldAccess t _ _) = t\nexprType (MethodCall t _ _ _) = t\nexprType (Var t _) = t\n\n{-# COMPILED_DATA Expression AST.Expression AST.New AST.FieldAccess AST.MethodCall AST.Var #-}\n\ndata Method : Set where\n  Mth : String \u2192 List Variable \u2192 Expression \u2192 Method\n\n{-# COMPILED_DATA Method AST.Method AST.Method #-}\n\ndata Constructor : Set where\n  Cons : List Variable \u2192 Constructor\n\n{-# COMPILED_DATA Constructor AST.Constructor AST.Constructor #-}\n\ndata Pair (A B : Set) : Set where\n  P : A \u2192 B \u2192 Pair A B\n\n{-# COMPILED_DATA Pair (,) (,) #-}\n\ndata SimpleMap (A B : Set) : Set where\n  SM : List (Pair A B) \u2192 SimpleMap A B\n\n{-# COMPILED_DATA SimpleMap AST.SimpleMap AST.SimpleMap #-}\n\ndata Class : Set where\n  Cls : String \u2192 SimpleMap String String \u2192 Constructor \u2192 SimpleMap String Method \u2192 Class\n\n{-# COMPILED_DATA Class AST.Class AST.Class #-}\n\ndata Program : Set where\n  Prog : SimpleMap String Class \u2192 Program\n\n{-# COMPILED_DATA Program AST.Program AST.Program #-}\n\npostulate\n  parseUnsafe : String \u2192 Program\n  showProgram : Program \u2192 String\n  getValue : {A B : Set} \u2192 A \u2192 SimpleMap A B \u2192 B\n\n{-# COMPILED parseUnsafe AST.parseUnsafe #-}\n{-# COMPILED showProgram AST.showProgram #-}\n\ninfixr 5 _++l_\n\n_++l_ : \u2200 {a} {A : Set a} \u2192 List A \u2192 List A \u2192 List A\nxs ++l ys = Data.List._++_ xs ys\n\n{-# NO_TERMINATION_CHECK #-}\nsubst : Pair String Expression \u2192 Expression \u2192 Expression\nsubst substExpr (New t ps) = New t (map (subst substExpr) ps)\nsubst substExpr (FieldAccess t e f) = FieldAccess t (subst substExpr e) f\nsubst substExpr (MethodCall t e m ps) = MethodCall t (subst substExpr e) m (map (subst substExpr) ps)\nsubst (P name expr) (Var t vName) = if name == vName then expr else (Var t vName)\n\ngetMethod : Program \u2192 String \u2192 String \u2192 Method\ngetMethod (Prog classes) clsName mthName with getValue clsName classes\n... | Cls _ _ _ methods = getValue mthName methods\n\nreduceInvk : Program \u2192 Expression \u2192 String \u2192 List Expression \u2192 Expression\nreduceInvk p e mthName params with getMethod p (exprType e) mthName\n... | Mth _ vars body = foldr subst body ((P \"this\" e) \u2237 (zipWith P (map varName vars) params))\n\ndata Reduce {Pr : Program} : Expression \u2192 Expression \u2192 Set where\n  RInvk : \u2200 {t consParams mthType mthName mthParams} \u2192 Reduce {Pr} (MethodCall mthType (New t consParams) mthName mthParams) (reduceInvk Pr (New t consParams) mthName mthParams)\n  RCField : \u2200 {e\u2081 e\u2082 t f} \u2192 Reduce {Pr} e\u2081 e\u2082 \u2192 Reduce {Pr} (FieldAccess t e\u2081 f) (FieldAccess t e\u2082 f)\n  RCInvk : \u2200 {e\u2081 e\u2082 t m p} \u2192 Reduce {Pr} e\u2081 e\u2082 \u2192 Reduce {Pr} (MethodCall t e\u2081 m p) (MethodCall t e\u2082 m p)\n  RCInvkArg : \u2200 {e e\u2081 e\u2082 t m p\u2081 p\u2082} \u2192 Reduce {Pr} e\u2081 e\u2082 \u2192 Reduce {Pr} (MethodCall t e m (p\u2081 ++l [ e\u2081 ] ++l p\u2082)) (MethodCall t e m (p\u2081 ++l [ e\u2082 ] ++l p\u2082))\n  RCNewArg : \u2200 {e\u2081 e\u2082 t p\u2081 p\u2082} \u2192 Reduce {Pr} e\u2081 e\u2082 \u2192 Reduce {Pr} (New t (p\u2081 ++l [ e\u2081 ] ++l p\u2082)) (New t (p\u2081 ++l [ e\u2082 ] ++l p\u2082))\n\nmain = run (putStrLn (showProgram (parseUnsafe \"class A extends Object { A() {super();} }\")))\n","old_contents":"module Main where\nopen import IO\nopen import Data.String\nopen import Data.List\n\n{-# IMPORT AST #-}\n\n{-# COMPILED_TYPE String AST.Type #-}\n{-# COMPILED_TYPE String AST.Identifier #-}\n\ndata Variable : Set where\n  V : String \u2192 String \u2192 Variable\n\n{-# COMPILED_DATA Variable AST.Variable AST.Variable #-}\n\ndata Expression : Set where\n  New : String \u2192 List Expression \u2192 Expression\n  FieldAccess : String \u2192 Expression \u2192 String \u2192 Expression\n  MethodCall : String \u2192 Expression \u2192 String \u2192 List Expression \u2192 Expression\n  Var : String \u2192 String \u2192 Expression\n\n{-# COMPILED_DATA Expression AST.Expression AST.New AST.FieldAccess AST.MethodCall AST.Var #-}\n\ndata Method : Set where\n  Mth : String \u2192 List Variable \u2192 Expression \u2192 Method\n\n{-# COMPILED_DATA Method AST.Method AST.Method #-}\n\ndata Constructor : Set where\n  Cons : List Variable \u2192 Constructor\n\n{-# COMPILED_DATA Constructor AST.Constructor AST.Constructor #-}\n\ndata Pair (A B : Set) : Set where\n  P : A \u2192 B \u2192 Pair A B\n\n{-# COMPILED_DATA Pair (,) (,) #-}\n\ndata SimpleMap (A B : Set) : Set where\n  SM : List (Pair A B) \u2192 SimpleMap A B\n\n{-# COMPILED_DATA SimpleMap AST.SimpleMap AST.SimpleMap #-}\n\ndata Class : Set where\n  Cls : String \u2192 SimpleMap String String \u2192 Constructor \u2192 SimpleMap String Method \u2192 Class\n\n{-# COMPILED_DATA Class AST.Class AST.Class #-}\n\ndata Program : Set where\n  Prog : SimpleMap String Class \u2192 Program\n\n{-# COMPILED_DATA Program AST.Program AST.Program #-}\n\npostulate\n  parseUnsafe : String \u2192 Program\n  showProgram : Program \u2192 String\n\n{-# COMPILED parseUnsafe AST.parseUnsafe #-}\n{-# COMPILED showProgram AST.showProgram #-}\n\nmain = run (putStrLn (showProgram (parseUnsafe \"class A extends Object { A() {super();} }\")))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ce27d979622c23720f115dbd933df4e18f579e51","subject":"IDesc model: finer grain universe control","message":"IDesc model: finer grain universe control\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc zero I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc zero I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc zero I)(P : I -> Set) -> desc I D P -> IDesc zero (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc zero I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc zero I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc zero I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set1 where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc (suc zero) Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc (suc zero) Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0583257a3d833ee683a9bf109f7a72b3f9ce4132","subject":"Desc stratified model: from embedding to host","message":"Desc stratified model: from embedding to host","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\n-- desc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\n-- descD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\n-- IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l\n\n-- cases : (l : Level)\n--         (I : Set l)\n--         (xs : desc (suc l) (lift I) (descD l I) (IMu (suc l) Unit (\\ _ -> descD l I)))\n--         (hs : desc l I (box l I (descD l I) (IMu l Unit (\u03bb _ -> descD l I)) xs) (\u03bb _ -> IDesc (suc l) I)) ->\n--         IDesc (suc l) I\n-- cases x I ( lvar , i ) hs =  var i\n-- cases x I ( lconst , X ) hs =  const X\n-- cases x I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\n-- cases x I ( lpi , ( S , T ) ) hs =  pi S hs\n-- cases x I ( lsigma , ( S , T ) ) hs = sigma S hs\n\n-- phi : {I : Set} -> IDescl I -> IDesc I\n-- phi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bc199d61e20de7f0af5df969e0b713d1e13e40cc","subject":"Algebra: missing constructor","message":"Algebra: missing constructor\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Raw.agda","new_file":"lib\/Algebra\/Raw.agda","new_contents":"open import Type using (Type_)\nopen import Function.NP\nopen import Data.Nat.Base using (\u2115) renaming (suc to 1+_)\nopen import Data.Integer using (\u2124; +_; -[1+_])\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\n\nmodule Algebra.Raw where\n\nrecord Magma {\u2113}(A : Type \u2113) : Type \u2113 where\n  constructor \u27e8_\u27e9\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n\n  infix 8 _\u00b2 _\u00b3 _\u2074\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x \u2219 x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 \u2219 x\n\n  _\u2074 : A \u2192 A\n  x \u2074 = x \u00b3 \u2219 x\n\n  infix 8 _^\u00b9\u207a_\n  _^\u00b9\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u00b9\u207a n = nest n (_\u2219_ x) x\n\n  module _ {x x' y y'}(p : x \u2261 x')(q : y \u2261 y') where\n    \u2219= : x \u2219 y \u2261 x' \u2219 y'\n    \u2219= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\nrecord Monoid-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  constructor _,_\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n    \u03b5   : A\n\n  \u2219-magma : Magma A\n  \u2219-magma = \u27e8 _\u2219_ \u27e9\n\n  open Magma \u2219-magma public hiding (_\u2219_)\n\n  infix 8 _^\u207a_\n  _^\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u207a n = nest n (_\u2219_ x) \u03b5\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Monoid-Ops {\u2113}{M : Set \u2113} (mon : Monoid-Ops M) where\n  private\n   module M = Monoid-Ops mon\n    using    ()\n    renaming ( _\u2219_ to _+_\n             ; \u03b5 to 0#\n             ; _\u00b2 to 2\u2297_\n             ; _\u00b3 to 3\u2297_\n             ; _\u2074 to 4\u2297_\n             ; _^\u00b9\u207a_ to _\u2297\u00b9\u207a_\n             ; _^\u207a_ to _\u2297\u207a_\n             ; \u2219= to +=\n             )\n  open M public using (0#; +=)\n  infixl 6 _+_\n  infixl 7 _\u2297\u00b9\u207a_ _\u2297\u207a_ 2\u2297_ 3\u2297_ 4\u2297_\n  _+_   = M._+_\n  _\u2297\u00b9\u207a_ = M._\u2297\u00b9\u207a_\n  _\u2297\u207a_  = M._\u2297\u207a_\n  2\u2297_   = M.2\u2297_\n  3\u2297_   = M.3\u2297_\n  4\u2297_   = M.4\u2297_\n\n-- A renaming of Monoid-Ops with multiplicative notation\nmodule Multiplicative-Monoid-Ops {\u2113}{M : Type \u2113} (mon-ops : Monoid-Ops M)\n  = Monoid-Ops mon-ops\n    renaming ( _\u2219_ to _*_\n             ; \u03b5 to 1#\n             ; \u2219= to *=\n             )\n\nrecord Group-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  constructor _,_\n\n  field\n    mon-ops : Monoid-Ops A\n    _\u207b\u00b9     : A \u2192 A\n\n  open Monoid-Ops mon-ops public\n\n  \u207b\u00b9= : \u2200 {x y} \u2192 x \u2261 y \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9\n  \u207b\u00b9= = ap _\u207b\u00b9\n\n  infixl 7 _\/_\n\n  _\/_ : A \u2192 A \u2192 A\n  x \/ y = x \u2219 y \u207b\u00b9\n\n  \/-magma : Magma A\n  \/-magma = \u27e8 _\/_ \u27e9\n\n  open Magma \/-magma public using () renaming (\u2219= to \/=)\n\n  _^\u207b_ _^\u207b\u2032_ : A \u2192 \u2115 \u2192 A\n  x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n  x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n  _^_ : A \u2192 \u2124 \u2192 A\n  x ^ -[1+ n ] = x ^\u207b(1+ n)\n  x ^ (+ n)    = x ^\u207a n\n\n-- A renaming of Group-Ops with additive notation\nmodule Additive-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) where\n  private\n   module M = Group-Ops grp\n    using    ()\n    renaming ( _\u207b\u00b9 to 0\u2212_\n             ; \u207b\u00b9= to 0\u2212=\n             ; _\/_ to _\u2212_\n             ; _^\u207b_ to _\u2297\u207b_\n             ; _^_ to _\u2297_\n             ; mon-ops to +-mon-ops\n             ; \/= to \u2212=)\n  open M public using (0\u2212_; +-mon-ops; \u2212=; 0\u2212=)\n  open Additive-Monoid-Ops +-mon-ops public\n  infixl 6 _\u2212_\n  infixl 7 _\u2297\u207b_ _\u2297_\n  _\u2212_   = M._\u2212_\n  _\u2297\u207b_  = M._\u2297\u207b_\n  _\u2297_   = M._\u2297_\n\n-- A renaming of Group-Ops with multiplicative notation\nmodule Multiplicative-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) = Group-Ops grp\n    using    ( _^\u207a_; _\u00b2; _\u00b3; _\u2074; _\u207b\u00b9; _\/_; \/=; _^\u207b_; _^_; \/-magma; \u207b\u00b9= )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1#; mon-ops to *-mon-ops; \u2219= to *= )\n\nrecord Ring-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  constructor _,_\n  infixl 6 2+_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-mon-ops : Monoid-Ops A\n\n  open Additive-Group-Ops        +-grp-ops public\n    renaming (2\u2297_ to 2*_)\n  open Multiplicative-Monoid-Ops *-mon-ops public\n\n  suc : A \u2192 A\n  suc = _+_ 1#\n\n  pred : A \u2192 A\n  pred x = x \u2212 1#\n\n  2# 3# -0# -1# : A\n  2#  = suc 1#\n  3#  = suc 2#\n  -0# = 0\u2212 0#\n  -1# = 0\u2212 1#\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0    ] = 0#\n  \u2115[ 1+ n ] = nest n suc 1#\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ + x      ] = \u2115[ x ]\n  \u2124[ -[1+ x ] ] = 0\u2212 \u2115[ 1+ x ]\n\n  -- TODO use _^\u207a_ and _^_ from group\/monoid\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0      = 1#\n  b ^\u2115 (1+ n) = nest n (_*_ b) b\n\n  2+_ : A \u2192 A\n  2+ x = 2# + x\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  constructor _,_\n\n  field\n    +-grp-ops : Group-Ops A\n    *-grp-ops : Group-Ops A\n\n  open Multiplicative-Group-Ops *-grp-ops public\n    using ( _\u207b\u00b9; \u207b\u00b9=; _\/_; \/-magma; \/=; _^\u207b_; _^_; *-mon-ops )\n\n  rng-ops : Ring-Ops A\n  rng-ops = +-grp-ops , *-mon-ops\n\n  open Ring-Ops rng-ops public\n    hiding (+-grp-ops; *-mon-ops)\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (+ n)    = b ^\u2115 n\n  b ^\u2124 -[1+ n ] = (b ^\u2115 (1+ n))\u207b\u00b9\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type using (Type_)\nopen import Function.NP\nopen import Data.Nat using (\u2115) renaming (suc to 1+_)\nopen import Data.Integer using (\u2124; +_; -[1+_])\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\n\nmodule Algebra.Raw where\n\nrecord Magma {\u2113}(A : Type \u2113) : Type \u2113 where\n  constructor \u27e8_\u27e9\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n\n  infix 8 _\u00b2 _\u00b3 _\u2074\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x \u2219 x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 \u2219 x\n\n  _\u2074 : A \u2192 A\n  x \u2074 = x \u00b3 \u2219 x\n\n  infix 8 _^\u00b9\u207a_\n  _^\u00b9\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u00b9\u207a n = nest n (_\u2219_ x) x\n\n  module _ {x x' y y'}(p : x \u2261 x')(q : y \u2261 y') where\n    \u2219= : x \u2219 y \u2261 x' \u2219 y'\n    \u2219= = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\nrecord Monoid-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  constructor _,_\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : A \u2192 A \u2192 A\n    \u03b5   : A\n\n  \u2219-magma : Magma A\n  \u2219-magma = \u27e8 _\u2219_ \u27e9\n\n  open Magma \u2219-magma public hiding (_\u2219_)\n\n  infix 8 _^\u207a_\n  _^\u207a_ : A \u2192 \u2115 \u2192 A\n  x ^\u207a n = nest n (_\u2219_ x) \u03b5\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Monoid-Ops {\u2113}{M : Set \u2113} (mon : Monoid-Ops M) where\n  private\n   module M = Monoid-Ops mon\n    using    ()\n    renaming ( _\u2219_ to _+_\n             ; \u03b5 to 0#\n             ; _\u00b2 to 2\u2297_\n             ; _\u00b3 to 3\u2297_\n             ; _\u2074 to 4\u2297_\n             ; _^\u00b9\u207a_ to _\u2297\u00b9\u207a_\n             ; _^\u207a_ to _\u2297\u207a_\n             ; \u2219= to +=\n             )\n  open M public using (0#; +=)\n  infixl 6 _+_\n  infixl 7 _\u2297\u00b9\u207a_ _\u2297\u207a_ 2\u2297_ 3\u2297_ 4\u2297_\n  _+_   = M._+_\n  _\u2297\u00b9\u207a_ = M._\u2297\u00b9\u207a_\n  _\u2297\u207a_  = M._\u2297\u207a_\n  2\u2297_   = M.2\u2297_\n  3\u2297_   = M.3\u2297_\n  4\u2297_   = M.4\u2297_\n\n-- A renaming of Monoid-Ops with multiplicative notation\nmodule Multiplicative-Monoid-Ops {\u2113}{M : Type \u2113} (mon-ops : Monoid-Ops M)\n  = Monoid-Ops mon-ops\n    renaming ( _\u2219_ to _*_\n             ; \u03b5 to 1#\n             ; \u2219= to *=\n             )\n\nrecord Group-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  constructor _,_\n\n  field\n    mon-ops : Monoid-Ops A\n    _\u207b\u00b9     : A \u2192 A\n\n  open Monoid-Ops mon-ops public\n\n  \u207b\u00b9= : \u2200 {x y} \u2192 x \u2261 y \u2192 x \u207b\u00b9 \u2261 y \u207b\u00b9\n  \u207b\u00b9= = ap _\u207b\u00b9\n\n  infixl 7 _\/_\n\n  _\/_ : A \u2192 A \u2192 A\n  x \/ y = x \u2219 y \u207b\u00b9\n\n  \/-magma : Magma A\n  \/-magma = \u27e8 _\/_ \u27e9\n\n  open Magma \/-magma public using () renaming (\u2219= to \/=)\n\n  _^\u207b_ _^\u207b\u2032_ : A \u2192 \u2115 \u2192 A\n  x ^\u207b n = (x ^\u207a n)\u207b\u00b9\n  x ^\u207b\u2032 n = (x \u207b\u00b9)^\u207a n\n\n  _^_ : A \u2192 \u2124 \u2192 A\n  x ^ -[1+ n ] = x ^\u207b(1+ n)\n  x ^ (+ n)    = x ^\u207a n\n\n-- A renaming of Group-Ops with additive notation\nmodule Additive-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) where\n  private\n   module M = Group-Ops grp\n    using    ()\n    renaming ( _\u207b\u00b9 to 0\u2212_\n             ; \u207b\u00b9= to 0\u2212=\n             ; _\/_ to _\u2212_\n             ; _^\u207b_ to _\u2297\u207b_\n             ; _^_ to _\u2297_\n             ; mon-ops to +-mon-ops\n             ; \/= to \u2212=)\n  open M public using (0\u2212_; +-mon-ops; \u2212=; 0\u2212=)\n  open Additive-Monoid-Ops +-mon-ops public\n  infixl 6 _\u2212_\n  infixl 7 _\u2297\u207b_ _\u2297_\n  _\u2212_   = M._\u2212_\n  _\u2297\u207b_  = M._\u2297\u207b_\n  _\u2297_   = M._\u2297_\n\n-- A renaming of Group-Ops with multiplicative notation\nmodule Multiplicative-Group-Ops {\u2113}{G : Type \u2113} (grp : Group-Ops G) = Group-Ops grp\n    using    ( _^\u207a_; _\u00b2; _\u00b3; _\u2074; _\u207b\u00b9; _\/_; \/=; _^\u207b_; _^_; \/-magma; \u207b\u00b9= )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1#; mon-ops to *-mon-ops; \u2219= to *= )\n\nrecord Ring-Ops {\u2113} (A : Type \u2113) : Type \u2113 where\n  constructor _,_\n  infixl 6 2+_\n\n  field\n    +-grp-ops  : Group-Ops A\n    *-mon-ops : Monoid-Ops A\n\n  open Additive-Group-Ops        +-grp-ops public\n    renaming (2\u2297_ to 2*_)\n  open Multiplicative-Monoid-Ops *-mon-ops public\n\n  suc : A \u2192 A\n  suc = _+_ 1#\n\n  pred : A \u2192 A\n  pred x = x \u2212 1#\n\n  2# 3# -0# -1# : A\n  2#  = suc 1#\n  3#  = suc 2#\n  -0# = 0\u2212 0#\n  -1# = 0\u2212 1#\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0    ] = 0#\n  \u2115[ 1+ n ] = nest n suc 1#\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ + x      ] = \u2115[ x ]\n  \u2124[ -[1+ x ] ] = 0\u2212 \u2115[ 1+ x ]\n\n  -- TODO use _^\u207a_ and _^_ from group\/monoid\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0      = 1#\n  b ^\u2115 (1+ n) = nest n (_*_ b) b\n\n  2+_ : A \u2192 A\n  2+ x = 2# + x\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n\n  field\n    +-grp-ops : Group-Ops A\n    *-grp-ops : Group-Ops A\n\n  open Multiplicative-Group-Ops *-grp-ops public\n    using ( _\u207b\u00b9; \u207b\u00b9=; _\/_; \/-magma; \/=; _^\u207b_; _^_; *-mon-ops )\n\n  rng-ops : Ring-Ops A\n  rng-ops = +-grp-ops , *-mon-ops\n\n  open Ring-Ops rng-ops public\n    hiding (+-grp-ops; *-mon-ops)\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (+ n)    = b ^\u2115 n\n  b ^\u2124 -[1+ n ] = (b ^\u2115 (1+ n))\u207b\u00b9\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9deee4a415c1667ab53cb0f43fabb9607859b5ca","subject":"Data.Nat","message":"Data.Nat\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.Eq\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromAssocComm _+_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.+-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.+-comm x y)\n  renaming ( assoc-comm to +-assoc-comm\n           ; interchange to +-interchange\n           ; !assoc-comm to +-!assoc-comm\n           ; comm= to +-comm=\n           ; assoc= to +-assoc=\n           ; !assoc= to +-!assoc=\n           ; inner= to +-inner=\n           ; outer= to +-outer=\n           )\n  public\n\nopen FromAssocComm _*_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.*-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.*-comm x y)\n  renaming ( assoc-comm to *-assoc-comm\n           ; interchange to *-interchange\n           ; !assoc-comm to *-!assoc-comm\n           ; comm= to *-comm=\n           ; inner= to *-inner=\n           ; outer= to *-outer=\n           ; assoc= to *-assoc=\n           ; !assoc= to *-!assoc=\n           )\n  public\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm {a} {1} {b}\n                                  | +-assoc-comm {a \u2294 b} {1} {a \u2293 b}\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm {1} {a} {b}))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm {1} {n} {n} = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange {x} {x} {y} {y}\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm {a} {1} {a}\n                                  | +-assoc-comm {b} {1} {b} = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = snd \u2115\u00b0.*-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = ! snd \u2115\u00b0.+-identity (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! \u2115\u00b0.*-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.Eq\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\nopen FromAssocComm _+_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.+-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.+-comm x y)\n  renaming ( assoc-comm to +-assoc-comm\n           ; interchange to +-interchange\n           )\n\nopen FromAssocComm _*_ (\u03bb {x}{y}{z} \u2192 \u2115\u00b0.*-assoc x y z) (\u03bb {x}{y} \u2192 \u2115\u00b0.*-comm x y)\n  renaming ( assoc-comm to *-assoc-comm\n           ; interchange to *-interchange\n           )\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm {a} {1} {b}\n                                  | +-assoc-comm {a \u2294 b} {1} {a \u2293 b}\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm {1} {a} {b}))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm {1} {n} {n} = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange {x} {x} {y} {y}\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm {a} {1} {a}\n                                  | +-assoc-comm {b} {1} {b} = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1-id : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1-id = snd \u2115\u00b0.*-identity\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b 0 n = ! snd \u2115\u00b0.+-identity (b ^ n)\n  ^-+ b (1+ m) n rewrite ^-+ b m n = ! \u2115\u00b0.*-assoc b (b ^ m) (b ^ n)\n\n  ^-* : \u2200 b m n \u2192 b ^(m * n) \u2261 (b ^ n) ^ m\n  ^-* b 0   n = idp\n  ^-* b (1+ m) n\n    = b ^ (n + m * n)      \u2261\u27e8 ^-+ b n (m * n) \u27e9\n      b ^ n * b ^(m * n)   \u2261\u27e8 ap (_*_ (b ^ n)) (^-* b m n) \u27e9\n      b ^ n * (b ^ n) ^ m  \u220e\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1a9213ba64792163a7e8c6747dd129dd9b91534c","subject":"Typo.","message":"Typo.\n\nIgnore-this: f7a1648b47497cbe77ab2c3533bc1860\n\ndarcs-hash:20100701205739-3bd4e-a68be27acd723ef23490c840a10a894694e127a7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/Division\/IsDIV-PCF.agda","new_file":"Examples\/Division\/IsDIV-PCF.agda","new_contents":"------------------------------------------------------------------------------\n-- The division satisfies the specification\n------------------------------------------------------------------------------\n\n-- This module formalizes the division algorithm following [1].\n-- [1] Peter Dybjer. Program verification in a logical theory of\n-- constructions. In Jean-Pierre Jouannaud, editor. Functional\n-- Programming Languages and Computer Architecture, volume 201 of\n-- LNCS, 1985, pages 334-349. Appears in revised form as Programming\n-- Methodology Group Report 26, June 1986.\n\nmodule Examples.Division.IsDIV-PCF where\n\nopen import LTC.Minimal\n\nopen import Examples.DivisionPCF\nopen import Examples.Division.EquationsPCF\nopen import Examples.Division.IsCorrectPCF\nopen import Examples.Division.IsN-PCF\nopen import Examples.Division.SpecificationPCF\n\nopen import LTC.Data.N\nopen import LTC.Data.N.Induction.WellFoundedPCF\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF\nopen import LTC.Relation.InequalitiesPCF\nopen import LTC.Relation.Inequalities.PropertiesPCF\n\n------------------------------------------------------------------------------\n\n-- The division result satifies the specification DIV\n-- when the dividend is less than the divisor\ndiv-x<y-DIV : {i j : D} \u2192 N i \u2192 N j -> LT i j \u2192 DIV i j (div i j)\ndiv-x<y-DIV Ni Nj i<j = div-x<y-N i<j , div-x<y-correct Ni Nj i<j\n\n-- The division result satisfies the specification DIV when the\n-- dividend is greater or equal than the divisor.\ndiv-x\u2265y-DIV : {i j : D} \u2192 N i -> N j \u2192\n              ({i' : D} \u2192 N i' \u2192 LT i' i \u2192 DIV i' j (div i' j)) \u2192\n              GT j zero \u2192\n              GE i j \u2192\n              DIV i j (div i j)\ndiv-x\u2265y-DIV {i} {j} Ni Nj accH j>0 i\u2265j =\n  (div-x\u2265y-N ih i\u2265j) , div-x\u2265y-correct Ni Nj ih i\u2265j\n\n    where\n    -- The inductive hypothesis on 'i - j'.\n    ih : DIV (i - j) j (div (i - j) j)\n    ih = accH {i - j}\n              (minus-N Ni Nj )\n              (x\u2265y\u2192y>0\u2192x-y<x Ni Nj i\u2265j j>0)\n\n------------------------------------------------------------------------------\n-- The division satisfies the specification\n\n-- We do the well-founded induction on 'i' and we keep 'j' fixed.\ndiv-DIV : {i j : D} \u2192 N i \u2192 N j \u2192 GT j zero \u2192 DIV i j (div i j)\ndiv-DIV {j = j} Ni Nj j>0 = wfIndN-LT P iStep Ni\n\n   where\n     P : D \u2192 Set\n     P d = DIV d j (div d j)\n\n     -- The inductive step doesn't use the variable 'i'\n     -- (nor 'Ni'). To make this\n     -- clear we write down the inductive step using the variables\n     -- 'm' and 'n'.\n     iStep : {n : D} \u2192 N n \u2192\n             (accH : {m : D} \u2192 N m \u2192 LT m n \u2192 P m) \u2192\n             P n\n     iStep {n} Nn accH =\n       [ div-x<y-DIV Nn Nj , div-x\u2265y-DIV Nn Nj accH j>0 ] (x<y\u2228x\u2265y Nn Nj)\n","old_contents":"------------------------------------------------------------------------------\n-- The division satisfies the specification\n------------------------------------------------------------------------------\n\n-- This module formalize the division algorithm following [1].\n-- [1] Peter Dybjer. Program verification in a logical theory of\n-- constructions. In Jean-Pierre Jouannaud, editor. Functional\n-- Programming Languages and Computer Architecture, volume 201 of\n-- LNCS, 1985, pages 334-349. Appears in revised form as Programming\n-- Methodology Group Report 26, June 1986.\n\nmodule Examples.Division.IsDIV-PCF where\n\nopen import LTC.Minimal\n\nopen import Examples.DivisionPCF\nopen import Examples.Division.EquationsPCF\nopen import Examples.Division.IsCorrectPCF\nopen import Examples.Division.IsN-PCF\nopen import Examples.Division.SpecificationPCF\n\nopen import LTC.Data.N\nopen import LTC.Data.N.Induction.WellFoundedPCF\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF\nopen import LTC.Relation.InequalitiesPCF\nopen import LTC.Relation.Inequalities.PropertiesPCF\n\n------------------------------------------------------------------------------\n\n-- The division result satifies the specification DIV\n-- when the dividend is less than the divisor\ndiv-x<y-DIV : {i j : D} \u2192 N i \u2192 N j -> LT i j \u2192 DIV i j (div i j)\ndiv-x<y-DIV Ni Nj i<j = div-x<y-N i<j , div-x<y-correct Ni Nj i<j\n\n-- The division result satisfies the specification DIV when the\n-- dividend is greater or equal than the divisor.\ndiv-x\u2265y-DIV : {i j : D} \u2192 N i -> N j \u2192\n              ({i' : D} \u2192 N i' \u2192 LT i' i \u2192 DIV i' j (div i' j)) \u2192\n              GT j zero \u2192\n              GE i j \u2192\n              DIV i j (div i j)\ndiv-x\u2265y-DIV {i} {j} Ni Nj accH j>0 i\u2265j =\n  (div-x\u2265y-N ih i\u2265j) , div-x\u2265y-correct Ni Nj ih i\u2265j\n\n    where\n    -- The inductive hypothesis on 'i - j'.\n    ih : DIV (i - j) j (div (i - j) j)\n    ih = accH {i - j}\n              (minus-N Ni Nj )\n              (x\u2265y\u2192y>0\u2192x-y<x Ni Nj i\u2265j j>0)\n\n------------------------------------------------------------------------------\n-- The division satisfies the specification\n\n-- We do the well-founded induction on 'i' and we keep 'j' fixed.\ndiv-DIV : {i j : D} \u2192 N i \u2192 N j \u2192 GT j zero \u2192 DIV i j (div i j)\ndiv-DIV {j = j} Ni Nj j>0 = wfIndN-LT P iStep Ni\n\n   where\n     P : D \u2192 Set\n     P d = DIV d j (div d j)\n\n     -- The inductive step doesn't use the variable 'i'\n     -- (nor 'Ni'). To make this\n     -- clear we write down the inductive step using the variables\n     -- 'm' and 'n'.\n     iStep : {n : D} \u2192 N n \u2192\n             (accH : {m : D} \u2192 N m \u2192 LT m n \u2192 P m) \u2192\n             P n\n     iStep {n} Nn accH =\n       [ div-x<y-DIV Nn Nj , div-x\u2265y-DIV Nn Nj accH j>0 ] (x<y\u2228x\u2265y Nn Nj)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a3b1db76fbdbf97e5638eca08c0475817ff1afca","subject":"Monoid: add the Pointwise construction","message":"Monoid: add the Pointwise construction\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Monoid.agda","new_file":"lib\/Algebra\/Monoid.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type_)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; idp; ap; ap\u2082)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\n\nmodule Algebra.Monoid where\n\nrecord Monoid-Ops {\u2113} (M : Type \u2113) : Type \u2113 where\n  constructor _,_\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : M \u2192 M \u2192 M\n    \u03b5   : M\n\n  open From-Op\u2082 _\u2219_ public renaming (op= to \u2219=)\n  open From-Monoid-Ops \u03b5 _\u2219_ public\n\nrecord Monoid-Struct {\u2113} {M : Type \u2113} (mon-ops : Monoid-Ops M) : Type \u2113 where\n  constructor _,_\n  open Monoid-Ops mon-ops\n\n  -- laws\n  field\n    assocs   : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    identity : Identity  \u03b5 _\u2219_\n\n  \u03b5\u2219-identity : LeftIdentity \u03b5 _\u2219_\n  \u03b5\u2219-identity = fst identity\n\n  \u2219\u03b5-identity : RightIdentity \u03b5 _\u2219_\n  \u2219\u03b5-identity = snd identity\n\n  assoc  = fst assocs\n  !assoc = snd assocs\n\n  open From-Assoc assoc public hiding (assocs)\n  open From-Assoc-LeftIdentity  assoc \u03b5\u2219-identity public\n  open From-Assoc-RightIdentity assoc \u2219\u03b5-identity public\n\nrecord Monoid {\u2113}(M : Type \u2113) : Type \u2113 where\n  constructor _,_\n  field\n    mon-ops    : Monoid-Ops M\n    mon-struct : Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Monoid-Struct mon-struct public\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Monoid-Ops {\u2113}{M : Set \u2113} (mon : Monoid-Ops M) where\n  private\n   module M = Monoid-Ops mon\n    using    ()\n    renaming ( _\u2219_ to _+_\n             ; \u03b5 to 0#\n             ; _\u00b2 to 2\u2297_\n             ; _\u00b3 to 3\u2297_\n             ; _\u2074 to 4\u2297_\n             ; _^\u00b9\u207a_ to _\u2297\u00b9\u207a_\n             ; _^\u207a_ to _\u2297\u207a_\n             ; \u2219= to +=\n             )\n  open M public using (0#; +=)\n  infixl 6 _+_\n  infixl 7 _\u2297\u00b9\u207a_ _\u2297\u207a_ 2\u2297_ 3\u2297_ 4\u2297_\n  _+_   = M._+_\n  _\u2297\u00b9\u207a_ = M._\u2297\u00b9\u207a_\n  _\u2297\u207a_  = M._\u2297\u207a_\n  2\u2297_   = M.2\u2297_\n  3\u2297_   = M.3\u2297_\n  4\u2297_   = M.4\u2297_\n\n-- A renaming of Monoid-Struct with additive notation\nmodule Additive-Monoid-Struct {\u2113}{M : Type \u2113}{mon-ops : Monoid-Ops M}\n                              (mon-struct : Monoid-Struct mon-ops)\n  = Monoid-Struct mon-struct\n    renaming ( identity to +-identity\n             ; \u03b5\u2219-identity to 0+-identity\n             ; \u2219\u03b5-identity to +0-identity\n             ; assocs to +-assocs\n             ; assoc to +-assoc\n             ; !assoc to +-!assoc\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             )\n\n-- A renaming of Monoid with additive notation\nmodule Additive-Monoid {\u2113}{M : Type \u2113} (mon : Monoid M) where\n  open Additive-Monoid-Ops    (Monoid.mon-ops    mon) public\n  open Additive-Monoid-Struct (Monoid.mon-struct mon) public\n\n-- A renaming of Monoid-Ops with multiplicative notation\nmodule Multiplicative-Monoid-Ops {\u2113}{M : Type \u2113} (mon-ops : Monoid-Ops M)\n  = Monoid-Ops mon-ops\n    renaming ( _\u2219_ to _*_\n             ; \u03b5 to 1#\n             ; \u2219= to *=\n             )\n\n-- A renaming of Monoid-Struct with multiplicative notation\nmodule Multiplicative-Monoid-Struct {\u2113}{M : Type \u2113}{mon-ops : Monoid-Ops M}\n                                    (mon-struct : Monoid-Struct mon-ops)\n  = Monoid-Struct mon-struct\n    renaming ( identity to *-identity\n             ; \u03b5\u2219-identity to 1*-identity\n             ; \u2219\u03b5-identity to *1-identity\n             ; assocs to *-assocs\n             ; assoc to *-assoc\n             ; !assoc to *-!assoc\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             )\n\n-- A renaming of Monoid with multiplicative notation\nmodule Multiplicative-Monoid {\u2113}{M : Type \u2113} (mon : Monoid M) where\n  open Multiplicative-Monoid-Ops    (Monoid.mon-ops    mon) public\n  open Multiplicative-Monoid-Struct (Monoid.mon-struct mon) public\n\nmodule Monoid\u1d52\u1d56 {\u2113}{M : Type \u2113} where\n  _\u1d52\u1d56-ops : Monoid-Ops M \u2192 Monoid-Ops M\n  (_\u2219_ , \u03b5) \u1d52\u1d56-ops = flip _\u2219_ , \u03b5\n\n  _\u1d52\u1d56-struct : {mon : Monoid-Ops M} \u2192 Monoid-Struct mon \u2192 Monoid-Struct (mon \u1d52\u1d56-ops)\n  (assocs , identities) \u1d52\u1d56-struct = (swap assocs , swap identities)\n\n  _\u1d52\u1d56 : Monoid M \u2192 Monoid M\n  (ops , struct)\u1d52\u1d56 = _ , struct \u1d52\u1d56-struct\n\n  \u1d52\u1d56\u2218\u1d52\u1d56-id : \u2200 {mon} \u2192 (mon \u1d52\u1d56) \u1d52\u1d56 \u2261 mon\n  \u1d52\u1d56\u2218\u1d52\u1d56-id = idp\n\nmodule _ {a}{A : Type a}(_\u2219_ : Op\u2082 A)(assoc : Associative _\u2219_) where\n  from-assoc = From-Op\u2082.From-Assoc.assocs _\u2219_ assoc\n\n--import Data.Vec.NP as V\n{-\nmodule _ {A B : Type} where\n  (f : Vec  : Vec M n) \u2192 f \u229b x (\u2200 i \u2192 )\n\nmodule VecMonoid {M : Type} (mon : Monoid M) where\n  open V\n  open Monoid mon\n\n  module _ n where\n    \u00d7-mon-ops : Monoid-Ops (Vec M n)\n    \u00d7-mon-ops = zipWith _\u2219_ , replicate \u03b5\n\n    \u00d7-mon-struct : Monoid-Struct \u00d7-mon-ops\n    \u00d7-mon-struct = (\u03bb {x}{y}{z} \u2192 {!replicate ? \u229b ?!}) , {!!} , {!!}\n-}\n\n-- The monoidal structure of endomorphisms\nmodule _ {a}(A : Type a) where\n  \u2218-mon-ops : Monoid-Ops (Endo A)\n  \u2218-mon-ops = _\u2218\u2032_ , id\n\n  \u2218-mon-struct : Monoid-Struct \u2218-mon-ops\n  \u2218-mon-struct = (idp , idp) , (idp , idp)\n\n  \u2218-mon : Monoid (Endo A)\n  \u2218-mon = \u2218-mon-ops , \u2218-mon-struct\n\nmodule Product {a}{A : Type a}{b}{B : Type b}\n               (\ud835\udd38 : Monoid A)(\ud835\udd39 : Monoid B) where\n  open Additive-Monoid \ud835\udd38\n  open Multiplicative-Monoid \ud835\udd39\n\n  \u00d7-mon-ops    : Monoid-Ops (A \u00d7 B)\n  \u00d7-mon-ops    = zip _+_ _*_ , 0# , 1#\n\n  \u00d7-mon-struct : Monoid-Struct \u00d7-mon-ops\n  \u00d7-mon-struct = (ap\u2082 _,_ +-assoc *-assoc , ap\u2082 _,_ +-!assoc *-!assoc)\n               , ap\u2082 _,_ (fst +-identity) (fst *-identity)\n               , ap\u2082 _,_ (snd +-identity) (snd *-identity)\n\n  \u00d7-mon : Monoid (A \u00d7 B)\n  \u00d7-mon = \u00d7-mon-ops , \u00d7-mon-struct\n\n  open Monoid \u00d7-mon public\n\n{-\nThis module shows how properties of a monoid \ud835\udd44 are carried on\nfunctions from any type A to \ud835\udd44.\nHowever since the function type can be dependent, this also generalises\nthe product of monoids (since \u03a0 \ud835\udfda [ A , B ] \u2243 A \u00d7 B).\n-}\nmodule Pointwise {{_ : FunExt}}{a}(A : Type a){m}{M : A \u2192 Type m}\n                 (\ud835\udd44 : (x : A) \u2192 Monoid (M x)) where\n  private\n    module \ud835\udd44 {x} = Monoid (\ud835\udd44 x)\n  open \ud835\udd44 hiding (mon-ops; mon-struct)\n\n  \u27e8\u03b5\u27e9 : \u03a0 A M\n  \u27e8\u03b5\u27e9 = \u03bb _ \u2192 \u03b5\n\n  _\u27e8\u2219\u27e9_ : Op\u2082 (\u03a0 A M)\n  (f \u27e8\u2219\u27e9 g) x = f x \u2219 g x\n\n  mon-ops : Monoid-Ops (\u03a0 A M)\n  mon-ops = _\u27e8\u2219\u27e9_ , \u27e8\u03b5\u27e9\n\n  mon-struct : Monoid-Struct mon-ops\n  mon-struct = (\u03bb=\u2071 assoc , \u03bb=\u2071 !assoc) , \u03bb=\u2071 \u03b5\u2219-identity , \u03bb=\u2071 \u2219\u03b5-identity\n\n  mon : Monoid (\u03a0 A M)\n  mon = mon-ops , mon-struct\n\n  open Monoid mon public hiding (mon-ops; mon-struct)\n\n-- Non-dependent version of Pointwise\u2032\nmodule Pointwise\u2032 {{_ : FunExt}}{a}(A : Type a){m}{M : Type m}(\ud835\udd44 : Monoid M) =\n  Pointwise A (\u03bb _ \u2192 \ud835\udd44)\n  {- OR\n  open Monoid \ud835\udd44 hiding (mon-ops; mon-struct)\n\n  \u27e8\u03b5\u27e9 : A \u2192 M\n  \u27e8\u03b5\u27e9 = \u03bb _ \u2192 \u03b5\n\n  _\u27e8\u2219\u27e9_ : Op\u2082 (A \u2192 M)\n  (f \u27e8\u2219\u27e9 g) x = f x \u2219 g x\n\n  mon-ops : Monoid-Ops (A \u2192 M)\n  mon-ops = _\u27e8\u2219\u27e9_ , \u27e8\u03b5\u27e9\n\n  mon-struct : Monoid-Struct mon-ops\n  mon-struct = (\u03bb=\u2071 assoc , \u03bb=\u2071 !assoc) , \u03bb=\u2071 \u03b5\u2219-identity , \u03bb=\u2071 \u2219\u03b5-identity\n\n  mon : Monoid (A \u2192 M)\n  mon = mon-ops , mon-struct\n\n  open Monoid mon public hiding (mon-ops; mon-struct)\n  -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using () renaming (Type_ to Type)\nopen import Level.NP\nopen import Function.NP\nopen import Data.Product.NP\nopen import Data.Nat\n  using    (\u2115; zero)\n  renaming (_+_ to _+\u2115_; _*_ to _*\u2115_; suc to 1+_)\nimport Data.Nat.Properties.Simple as \u2115\u00b0\nopen import Data.Integer.NP\n  using    (\u2124; +_; -[1+_]; _\u2296_; -_; module \u2124\u00b0)\n  renaming ( _+_ to _+\u2124_; _-_ to _\u2212\u2124_; _*_ to _*\u2124_\n           ; suc to suc\u2124; pred to pred\u2124\n           )\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen \u2261-Reasoning\n\nmodule Algebra.Monoid where\n\nrecord Monoid-Ops {\u2113} (M : Type \u2113) : Type \u2113 where\n  constructor _,_\n  infixl 7 _\u2219_\n\n  field\n    _\u2219_ : M \u2192 M \u2192 M\n    \u03b5   : M\n\n  open From-Op\u2082 _\u2219_ public renaming (op= to \u2219=)\n  open From-Monoid-Ops \u03b5 _\u2219_ public\n\nrecord Monoid-Struct {\u2113} {M : Type \u2113} (mon-ops : Monoid-Ops M) : Type \u2113 where\n  constructor _,_\n  open Monoid-Ops mon-ops\n\n  -- laws\n  field\n    assocs   : Associative _\u2219_ \u00d7 Associative (flip _\u2219_)\n    identity : Identity  \u03b5 _\u2219_\n\n  \u03b5\u2219-identity : LeftIdentity \u03b5 _\u2219_\n  \u03b5\u2219-identity = fst identity\n\n  \u2219\u03b5-identity : RightIdentity \u03b5 _\u2219_\n  \u2219\u03b5-identity = snd identity\n\n  assoc  = fst assocs\n  !assoc = snd assocs\n\n  open From-Assoc assoc public hiding (assocs)\n  open From-Assoc-LeftIdentity  assoc \u03b5\u2219-identity public\n  open From-Assoc-RightIdentity assoc \u2219\u03b5-identity public\n\nrecord Monoid {\u2113}(M : Type \u2113) : Type \u2113 where\n  constructor _,_\n  field\n    mon-ops    : Monoid-Ops M\n    mon-struct : Monoid-Struct mon-ops\n  open Monoid-Ops    mon-ops    public\n  open Monoid-Struct mon-struct public\n\n-- A renaming of Monoid-Ops with additive notation\nmodule Additive-Monoid-Ops {\u2113}{M : Set \u2113} (mon : Monoid-Ops M) where\n  private\n   module M = Monoid-Ops mon\n    using    ()\n    renaming ( _\u2219_ to _+_\n             ; \u03b5 to 0#\n             ; _\u00b2 to 2\u2297_\n             ; _\u00b3 to 3\u2297_\n             ; _\u2074 to 4\u2297_\n             ; _^\u00b9\u207a_ to _\u2297\u00b9\u207a_\n             ; _^\u207a_ to _\u2297\u207a_\n             ; \u2219= to +=\n             )\n  open M public using (0#; +=)\n  infixl 6 _+_\n  infixl 7 _\u2297\u00b9\u207a_ _\u2297\u207a_ 2\u2297_ 3\u2297_ 4\u2297_\n  _+_   = M._+_\n  _\u2297\u00b9\u207a_ = M._\u2297\u00b9\u207a_\n  _\u2297\u207a_  = M._\u2297\u207a_\n  2\u2297_   = M.2\u2297_\n  3\u2297_   = M.3\u2297_\n  4\u2297_   = M.4\u2297_\n\n-- A renaming of Monoid-Struct with additive notation\nmodule Additive-Monoid-Struct {\u2113}{M : Type \u2113}{mon-ops : Monoid-Ops M}\n                              (mon-struct : Monoid-Struct mon-ops)\n  = Monoid-Struct mon-struct\n    renaming ( identity to +-identity\n             ; \u03b5\u2219-identity to 0+-identity\n             ; \u2219\u03b5-identity to +0-identity\n             ; assocs to +-assocs\n             ; assoc to +-assoc\n             ; !assoc to +-!assoc\n             ; assoc= to +-assoc=\n             ; !assoc= to +-!assoc=\n             ; inner= to +-inner=\n             )\n\n-- A renaming of Monoid with additive notation\nmodule Additive-Monoid {\u2113}{M : Type \u2113} (mon : Monoid M) where\n  open Additive-Monoid-Ops    (Monoid.mon-ops    mon) public\n  open Additive-Monoid-Struct (Monoid.mon-struct mon) public\n\n-- A renaming of Monoid-Ops with multiplicative notation\nmodule Multiplicative-Monoid-Ops {\u2113}{M : Type \u2113} (mon-ops : Monoid-Ops M)\n  = Monoid-Ops mon-ops\n    renaming ( _\u2219_ to _*_\n             ; \u03b5 to 1#\n             ; \u2219= to *=\n             )\n\n-- A renaming of Monoid-Struct with multiplicative notation\nmodule Multiplicative-Monoid-Struct {\u2113}{M : Type \u2113}{mon-ops : Monoid-Ops M}\n                                    (mon-struct : Monoid-Struct mon-ops)\n  = Monoid-Struct mon-struct\n    renaming ( identity to *-identity\n             ; \u03b5\u2219-identity to 1*-identity\n             ; \u2219\u03b5-identity to *1-identity\n             ; assocs to *-assocs\n             ; assoc to *-assoc\n             ; !assoc to *-!assoc\n             ; assoc= to *-assoc=\n             ; !assoc= to *-!assoc=\n             ; inner= to *-inner=\n             )\n\n-- A renaming of Monoid with multiplicative notation\nmodule Multiplicative-Monoid {\u2113}{M : Type \u2113} (mon : Monoid M) where\n  open Multiplicative-Monoid-Ops    (Monoid.mon-ops    mon) public\n  open Multiplicative-Monoid-Struct (Monoid.mon-struct mon) public\n\nmodule Monoid\u1d52\u1d56 {\u2113}{M : Type \u2113} where\n  _\u1d52\u1d56-ops : Monoid-Ops M \u2192 Monoid-Ops M\n  (_\u2219_ , \u03b5) \u1d52\u1d56-ops = flip _\u2219_ , \u03b5\n\n  _\u1d52\u1d56-struct : {mon : Monoid-Ops M} \u2192 Monoid-Struct mon \u2192 Monoid-Struct (mon \u1d52\u1d56-ops)\n  (assocs , identities) \u1d52\u1d56-struct = (swap assocs , swap identities)\n\n  _\u1d52\u1d56 : Monoid M \u2192 Monoid M\n  (ops , struct)\u1d52\u1d56 = _ , struct \u1d52\u1d56-struct\n\n  \u1d52\u1d56\u2218\u1d52\u1d56-id : \u2200 {mon} \u2192 (mon \u1d52\u1d56) \u1d52\u1d56 \u2261 mon\n  \u1d52\u1d56\u2218\u1d52\u1d56-id = idp\n\nmodule _ {a}{A : Type a}(_\u2219_ : Op\u2082 A)(assoc : Associative _\u2219_) where\n  from-assoc = From-Op\u2082.From-Assoc.assocs _\u2219_ assoc\n\n--import Data.Vec.NP as V\n{-\nmodule _ {A B : Type} where\n  (f : Vec  : Vec M n) \u2192 f \u229b x (\u2200 i \u2192 )\n\nmodule VecMonoid {M : Type} (mon : Monoid M) where\n  open V\n  open Monoid mon\n\n  module _ n where\n    \u00d7-mon-ops : Monoid-Ops (Vec M n)\n    \u00d7-mon-ops = zipWith _\u2219_ , replicate \u03b5\n\n    \u00d7-mon-struct : Monoid-Struct \u00d7-mon-ops\n    \u00d7-mon-struct = (\u03bb {x}{y}{z} \u2192 {!replicate ? \u229b ?!}) , {!!} , {!!}\n-}\n\nmodule MonoidProduct {a}{A : Type a}{b}{B : Type b}\n                     (monA0+ : Monoid A)(monB1* : Monoid B) where\n  open Additive-Monoid monA0+\n  open Multiplicative-Monoid monB1*\n\n  \u00d7-mon-ops    : Monoid-Ops (A \u00d7 B)\n  \u00d7-mon-ops    = zip _+_ _*_ , 0# , 1#\n\n  \u00d7-mon-struct : Monoid-Struct \u00d7-mon-ops\n  \u00d7-mon-struct = (ap\u2082 _,_ +-assoc *-assoc , ap\u2082 _,_ +-!assoc *-!assoc)\n               , ap\u2082 _,_ (fst +-identity) (fst *-identity)\n               , ap\u2082 _,_ (snd +-identity) (snd *-identity)\n\n  \u00d7-mon : Monoid (A \u00d7 B)\n  \u00d7-mon = \u00d7-mon-ops , \u00d7-mon-struct\n\n  open Monoid \u00d7-mon public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5ef26165a819d689fd846742e6b8c2660e0494c3","subject":"first cut at making rule names and variable choices match up with the TeX","message":"first cut at making rule names and variable choices match up with the TeX\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- substitution\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 = \u2987\u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] (d1 \u2218 d2) =  ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 evalctx \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2081 \u03b5) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2082 d) evalctx\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 evalctx \u2192\n             (< \u03c4 > \u03b5 ) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FDot : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d' \u03b5 x \u03c4 } \u2192\n           d == \u03b5 \u27e6 d' \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d' \u27e7\n    FAp1 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d2' \u27e7\n    FAp2 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d1' \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d' \u03b5 \u03c4 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (< \u03c4 > d) == < \u03c4 > \u03b5 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  -- used to mark which dhexp holes have been evaluated\n  data mark : Set where\n    \u2713 : mark\n    \u2717 : mark\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c41 \u03c42 \u03942} \u2192\n                -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: bneed to think about disjointness and context rep\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4 \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 , \u2717 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4 m} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 m} \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , m \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  -- substitution\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 , m \u27e9 = \u2987\u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3 , m \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 , m \u27e9\n  [ d \/ y ] (d1 \u2218 d2) =  ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (< \u03c4 > d') = < \u03c4 > ([ d \/ y ] d')\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n  -- error\n  data _\u22a2_err : (\u0394 : hctx) (d : dhexp) \u2192 Set where\n    ECastError : \u2200{ \u0394 d \u03c41 \u03c42 } \u2192\n                 \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n                 \u03c41 ~\u0338 \u03c42 \u2192\n                 \u0394 \u22a2 (< \u03c41 > d) err\n    EAp1 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d1 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    EAp2 : \u2200{ \u0394 d1 d2} \u2192\n           \u0394 \u22a2 d2 err \u2192\n           \u0394 \u22a2 (d1 \u2218 d2) err\n    ENEHole : \u2200{ \u0394 d u \u03c3 m} \u2192\n           \u0394 \u22a2 d err \u2192\n           \u0394 \u22a2 (\u2987 d \u2988\u27e8 (u , \u03c3 , m)\u27e9) err\n    ECastProp : \u2200{ \u0394 d \u03c4} \u2192\n                \u0394 \u22a2 d err \u2192\n                \u0394 \u22a2 (< \u03c4 > d) err\n    EConst : \u2200{ \u0394 \u03c4 } \u2192 \u0394 \u22a2 (< \u03c4 > c) err\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 ectx \u2192 ectx\n    _\u2218\u2081_ : dhexp \u2192 ectx \u2192 ectx\n    _\u2218\u2082_ : ectx \u2192 dhexp \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst \u00d7 mark) \u2192 ectx\n    <_>_   : htyp \u2192 ectx \u2192 ectx\n\n  -- instruction transition judgement\n  data _\u22a2_\u2192>_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ \u0394 x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            \u0394 \u22a2 ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCast : \u2200{d \u0394 \u03c41 \u03c42 } \u2192\n             d final \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c42 \u2192\n             \u03c41 ~ \u03c42 \u2192 -- maybe?\n             \u0394 \u22a2 < \u03c41 > d \u2192> d\n    ITEHole : \u2200{ \u0394 u \u03c3} \u2192\n              \u0394 \u22a2 \u2987\u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987\u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n    ITNEHole : \u2200{ \u0394 u \u03c3 d } \u2192\n               d final \u2192\n               \u0394 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 , \u2717 \u27e9 \u2192> \u2987 d \u2988\u27e8 u , \u03c3 , \u2713 \u27e9\n\n\n  data _ectxt : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 ectxt\n    ECLam : \u2200{\u03b5 x \u03c4} \u2192\n            \u03b5 ectxt \u2192\n            (\u00b7\u03bb x [ \u03c4 ] \u03b5) ectxt\n    ECAp1 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 ectxt \u2192\n            (d \u2218\u2081 \u03b5) ectxt\n    ECAp2 : \u2200{d \u03b5} \u2192\n            \u03b5 ectxt \u2192\n            (\u03b5 \u2218\u2082 d) ectxt\n    ECNEHole : \u2200{\u03b5 m u \u03c3} \u2192\n               \u03b5 ectxt \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 , m \u27e9 ectxt\n    ECCast : \u2200{ \u03b5 \u03c4 } \u2192\n             \u03b5 ectxt \u2192\n             (< \u03c4 > \u03b5 ) ectxt\n\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FRefl : \u2200{d : dhexp} \u2192\n            d == \u2299 \u27e6 d \u27e7\n    FLam : \u2200{ d d1 \u03b5 x \u03c4 } \u2192\n           d1 == \u03b5 \u27e6 d \u27e7 \u2192\n           (\u00b7\u03bb x [ \u03c4 ] d1) == (\u00b7\u03bb x [ \u03c4 ] \u03b5) \u27e6 d \u27e7\n    FAp1 : \u2200{ d1 d2 d \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2081 \u03b5) \u27e6 d \u27e7\n    FAp2 : \u2200{ d1 d2 d \u03b5} \u2192\n           d2 == \u03b5 \u27e6 d \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2082 d2) \u27e6 d \u27e7\n    FNEHole : \u2200{ d d1 \u03b5 u \u03c3 m} \u2192\n              d1 == \u03b5 \u27e6 d \u27e7 \u2192\n              \u2987 d1 \u2988\u27e8 (u , \u03c3 , m) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 , m) \u27e9 \u27e6 d \u27e7\n    FCast : \u2200{ d d1 \u03b5 \u03c4 } \u2192\n            d1 == \u03b5 \u27e6 d \u27e7 \u2192\n            (< \u03c4 > d1) == < \u03c4 > \u03b5 \u27e6 d \u27e7\n\n  data _\u22a2_\u21a6_ : (\u0394 : hctx) (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u0394 \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           \u0394 \u22a2 d0 \u2192> d0' \u2192 -- should this \u0394 be \u2205?\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192 -- why is this the same \u03b5\n           \u0394 \u22a2 d \u21a6 d\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e5aaa499b712814badda7b7a410eea2ce7ee003b","subject":"Another way to pick functions at random","message":"Another way to pick functions at random\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function hiding (_\u27e8_\u27e9_)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Bool\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head; tail)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrandomFunExt : \u2200 {n k a} {A : Set a} \u2192 M k (Bits n \u2192 A) \u2192 M (k + k) (Bits (suc n) \u2192 A)\nrandomFunExt f = \u27ea comb \u00b7 f \u00b7 f \u27eb where comb = \u03bb g\u2081 g\u2082 xs \u2192 (if head xs then g\u2081 else g\u2082) (tail xs)\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2^_ : \u2115 \u2192 \u2115\n2^ 0       = 1\n2^ (suc n) = 2* (2^ n)\n\ncostRndFun : \u2115 \u2192 \u2115 \u2192 \u2115\ncostRndFun zero n = n\ncostRndFun (suc m) n = 2* (costRndFun m n)\n\nlem : \u2200 m n \u2192 costRndFun m n \u2261 2 ^ m * n\nlem zero n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\nlem (suc m) n rewrite lem m n | \u2115\u00b0.*-assoc 2 (2 ^ m) n | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\nrandomFun\u2032 : \u2200 {m n} \u2192 M (costRndFun m n) (Bits m \u2192 Bits n)\nrandomFun\u2032 {zero}  = \u27ea const \u00b7 random \u27eb\nrandomFun\u2032 {suc m} = randomFunExt (randomFun\u2032 {m})\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","old_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop; head)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\ntoss\u2032 : M 1 Bit\ntoss\u2032 = mk head\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 = mk \u2218 const\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 M m A \u2192 M n A\nweaken\u2264 p = comap (take\u2264 p)\n\ncoerce : \u2200 {m n a} {A : Set a} \u2192 m \u2261 n \u2192 M m A \u2192 M n A\ncoerce \u2261.refl = id\n\ntoss : \u2200 {n} \u2192 M (1 + n) Bit\ntoss = weaken\u2032 toss\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\n\u27ea_\u27eb : \u2200 {n} {a} {A : Set a} \u2192 A \u2192 M n A\n\u27ea_\u27eb = pure\n\n\u27ea_\u27eb\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\n\u27ea_\u27eb\u2032 = pure\u2032\n\n\u27ea_\u00b7_\u27eb : \u2200 {a b} {A : Set a} {B : Set b} {n} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\n\u27ea f \u00b7 x \u27eb = map f x\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n            (A \u2192 B \u2192 C) \u2192 M m A \u2192 M n B \u2192 M (m + n) C\n\u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} {m n o} \u2192\n              (A \u2192 B \u2192 C \u2192 D) \u2192 M m A \u2192 M n B \u2192 M o C \u2192 M (m + n + o) D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n_\u27e8,\u27e9_ : \u2200 {a b} {A : Set a} {B : Set b} {m n} \u2192 M m A \u2192 M n B \u2192 M (m + n) (A \u00d7 B)\nx \u27e8,\u27e9 y = \u27ea _,_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8xor\u27e9_ : \u2200 {n\u2081 n\u2082} \u2192 M n\u2081 Bit \u2192 M n\u2082 Bit \u2192 M (n\u2081 + n\u2082) Bit\nx \u27e8xor\u27e9 y = \u27ea _xor_ \u00b7 x \u00b7 y \u27eb\n\n_\u27e8\u2295\u27e9_ : \u2200 {n\u2081 n\u2082 m} \u2192 M n\u2081 (Bits m) \u2192 M n\u2082 (Bits m) \u2192 M (n\u2081 + n\u2082) (Bits m)\nx \u27e8\u2295\u27e9 y = \u27ea _\u2295_ \u00b7 x \u00b7 y \u27eb\n\nreplicateM : \u2200 {n m} {a} {A : Set a} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {zero}  _ = \u27ea [] \u27eb\nreplicateM {suc _} x = \u27ea _\u2237_ \u00b7 x \u00b7 replicateM x \u27eb\n\nrandom : \u2200 {n} \u2192 M n (Bits n)\n-- random = coerce ? (replicateM toss) -- specialized version for now to avoid coerce\nrandom {zero}  = \u27ea [] \u27eb\nrandom {suc _} = \u27ea _\u2237_ \u00b7 toss\u2032 \u00b7 random \u27eb\n\nrandomTbl : \u2200 m n \u2192 M (2 ^ m * n) (Vec (Bits n) (2 ^ m))\nrandomTbl m n = replicateM random\n\nrandomFun : \u2200 m n \u2192 M (2 ^ m * n) (Bits m \u2192 Bits n)\nrandomFun m n = \u27ea funFromTbl \u00b7 randomTbl m n \u27eb\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_ _\u224b_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n    refl  : \u2200 {n A} \u2192 Reflexive {A = M n A} _\u2248_\n    sym   : \u2200 {n A} \u2192 Symmetric {A = M n A} _\u2248_\n\n    -- not strictly transitive\n\n  reflexive : \u2200 {n A} \u2192 _\u2261_ \u21d2 _\u2248_ {n} {A}\n  reflexive \u2261.refl = refl\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\n  -- Another name for _\u224b_\n  _looks_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _looks_ = _\u224b_\n\nmodule WithEquiv (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n\n  SecPRG : \u2200 {k n} (prg : (key : Bits k) \u2192 Bits n) \u2192 Set\n  SecPRG prg = this looks random where this = \u27ea prg \u00b7 random \u27eb\n\n  record PRG k n : Set where\n    constructor _,_\n    field\n      prg : Bits k \u2192 Bits n\n      sec : SecPRG prg\n\n  OneTimeSecPRF : \u2200 {k m n} (prf : (key : Bits k) (msg : Bits m) \u2192 Bits n) \u2192 Set\n  OneTimeSecPRF prf = \u2200 {xs} \u2192 let this = \u27ea prf \u00b7 random \u00b7 \u27ea xs \u27eb\u2032 \u27eb in\n                                this looks random\n\n  record PRF k m n : Set where\n    constructor _,_\n    field\n      prf : Bits k \u2192 Bits m \u2192 Bits n\n      sec : OneTimeSecPRF prf\n\nOTP : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\nOTP key msg = key \u2295 msg\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = \u27ea f \u00b7 random \u27eb\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  open WithEquiv progEq\n\n  left-unit-law = \u2200 {A B : Set} {n} {x : A} {f : A \u2192 M n B} \u2192 return\u2032 x >>= f \u2248 f x\n\n  right-unit-law = \u2200 {A : Set} {n} {x : M n A} \u2192 x >>=\u2032 return\u2032 \u2248 x\n\n  assoc-law = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u224b x >>= (\u03bb x \u2192 f x >>= g)\n\n  assoc-law\u2032 = \u2200 {A B C : Set} {n\u2081 n\u2082 n\u2083} {x : M n\u2081 A} {f : A \u2192 M n\u2082 B} {g : B \u2192 M n\u2083 C}\n              \u2192 (x >>= f) >>= g \u2248 coerce (\u2261.sym (\u2115\u00b0.+-assoc n\u2081 n\u2082 n\u2083)) (x >>= (\u03bb x \u2192 f x >>= g))\n\n  ex\u2081 = \u2200 {x} \u2192 toss\u2032 \u27e8xor\u27e9 \u27ea x \u27eb\u2032 \u2248 \u27ea x \u27eb\n\n  ex\u2082 = p \u2248 map swap p where p = toss\u2032 \u27e8,\u27e9 toss\u2032\n\n  ex\u2083 = \u2200 {n} \u2192 OneTimeSecPRF {n} OTP\n\n  ex\u2084 = \u2200 {k n} (prg : PRG k n) \u2192 OneTimeSecPRF (\u03bb key xs \u2192 xs \u2295 PRG.prg prg key)\n\n  ex\u2085 = \u2200 {k n} \u2192 PRG k n \u2192 PRF k n n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d62231b3076ff653a44813df738d411c3332ba73","subject":"Used as an example of a possible agda error.","message":"Used as an example of a possible agda error.\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/LinFun.agda","new_file":"agda\/LinFun.agda","new_contents":"{-# OPTIONS --exact-split #-}\n--{-# OPTIONS --show-implicit #-}\n\nmodule LinFun where\n\nopen import Common\nopen import LinLogic\nopen import LinDepT\nopen import LinT \nopen import SetLL\nopen import SetLLProp\n\nopen import Size\nopen import Function\nopen import Data.Bool\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Empty\nopen import Data.Unit hiding (_\u2264_ ; _\u2264?_)\nopen import Relation.Nullary\n\n\n\n-- We send to the receiver both the values the type depends and the value of the type. This way, we achieve locality in terms of finding the type of the incoming value.\n-- We need to point that the receiver and the sender are the same node.\n\nmodule _ where\n\n  mutual\n    data LFun {u} : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 Set (lsuc u) where\n     I   : {i : Size} \u2192 \u2200{rll} \u2192 LFun {u} {i} {_} {rll} {rll}\n     _\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{pll ll ell rll} \u2192 {ind : IndexLL {i} pll ll} \u2192 (elf : LFun {_} {i} {j} {ell} {pll})\n           \u2192 {{prf : usesInput elf}} \u2192 LFun {_} {j} {k} {rll} {(replLL ll ind ell)}\n           \u2192 LFun {_} {i} {k} {rll} {ll}\n     --Do we need to set ltr as an instance variable?\n     tr  : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{ll orll rll} \u2192 {{ltr : LLTr orll ll}} \u2192 LFun {_} {i} {j} {rll} {orll} \u2192 LFun {_} {i} {j} {rll} {ll}\n     obs : {i : Size} \u2192 {j : Size< i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{\u221ell rll} \u2192 LFun {_} {j} {k} {rll} {(step \u221ell)} \u2192 LFun {_} {i} {k} {rll} {call \u221ell}\n     com : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{rll ll} \u2192 {frll : LinLogic i}\n           \u2192 \u2983 prfi : onlyOneNilOrNoNilFinite ll \u2261 true \u2984 \u2192 \u2983 prfo : onlyOneNilOrNoNilFinite frll \u2261 true \u2984\n           \u2192 (df : (ldt : LinDepT ll) \u2192 LinT ldt \u2192 LinDepT frll) \u2192 LFun {rll = rll} {ll = frll}\n           \u2192 LFun {_} {i} {j} {rll = rll} {ll = ll}\n     call : {i : Size} \u2192 {j : Size< i} \u2192 \u2200{ll \u221erll rll} \u2192 \u221eLFun {i} {_} {\u221erll} {ll} \u2192 LFun {_} {i} {j} {rll} {call \u221erll} \u2192 LFun {_} {i} {j} {rll} {ll}\n  \n  \n    record \u221eLFun {i : Size} {u} {\u221erll : \u221eLinLogic i {u}} {ll : LinLogic i {u}} : Set (lsuc u) where\n      coinductive\n      field\n        step : {j : Size< i} \u2192 LFun {_} {i} {j} {(step \u221erll)} {ll}\n\n\n    usesInput : {i : Size} \u2192 {j : Size< \u2191 i } \u2192 \u2200{u rll ll} \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set\n    usesInput x = usesInput` x \u2205 where\n      usesInput` : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u} \u2192 {rll : LinLogic j} \u2192 {ll : LinLogic i} \u2192 LFun {u} {_} {_} {rll} {ll} \u2192 MSetLL ll \u2192 Set\n      usesInput` I s = \u22a5\n      usesInput` (_\u2282_ {j = j} {k = k} {ell = ell} {ind = ind} elf lf) \u2205 = usesInput` lf (\u00ac\u2205 (\u2205-add ind ell))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2193} elf lf) (\u00ac\u2205 s) with (contruct $ add {i = i} {j = j} s \u2193 ell)\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2193} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2227 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2227))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2227 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | \u2227\u2192 ns =  usesInput` lf (\u00ac\u2205 (\u2227\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2227 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227\u2192 ns\u2081 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2227\u2192 ns\u2081))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2228 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2228))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2228 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | \u2228\u2192 ns =  usesInput` lf (\u00ac\u2205 (\u2228\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2228 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228\u2192 ns\u2081 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2228\u2192 ns\u2081))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_\u2202_ _ _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2202))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_\u2202_ _ _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | \u2202\u2192 ns =  usesInput` lf (\u00ac\u2205 (\u2202\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_\u2202_ _ _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202\u2192 ns\u2081 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2202\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = ind \u2190\u2227} elf lf) (\u00ac\u2205 s) with (contruct $ add s (ind \u2190\u2227) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | \u2227\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2227\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2227\u2192 ind} elf lf) (\u00ac\u2205 s) with (contruct $ add s (\u2227\u2192 ind) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2227\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2227\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = ind \u2190\u2228} elf lf) (\u00ac\u2205 s) with (contruct $ add s (ind \u2190\u2228) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | \u2228\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2228\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2228\u2192 ind} elf lf) (\u00ac\u2205 s) with (contruct $ add s (\u2228\u2192 ind) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2228\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2228\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = ind \u2190\u2202} elf lf) (\u00ac\u2205 s) with (contruct $ add s (ind \u2190\u2202) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | \u2202\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2202\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2202\u2192 ind} elf lf) (\u00ac\u2205 s) with (contruct $ add s (\u2202\u2192 ind) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2202\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2202\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202\u2192 ns\u2081))\n      usesInput` (tr lf) \u2205 = usesInput` lf \u2205\n      usesInput` (tr {{ltr = ltr}} lf) (\u00ac\u2205 s) = usesInput` lf $ \u00ac\u2205 (SetLL.tran s ltr)\n      usesInput` (obs {j = lj} {k = lk} {\u221ell = \u221ell} x) s = usesInput` {i = lj} {j = lk} x (\u2205 {ll = (step \u221ell)})\n      usesInput` (com df lf) s = \u22a4\n      usesInput` (call \u221ex x) s = \u22a5\n\nopen \u221eLFun public\n\n\n\nmodule _ where\n\n  open import Data.List\n  open import Data.Nat\n\n\n  module _ where\n\n    _\u2264un_ : {a : \u2115} \u2192 {b : \u2115} \u2192 (c : (a \u2264 b)) \u2192 (d : (a \u2264 b)) \u2192 c \u2261 d  \n    z\u2264n \u2264un z\u2264n = refl\n    s\u2264s c \u2264un s\u2264s d with ( c \u2264un d )\n    s\u2264s c \u2264un s\u2264s .c | refl = refl\n\n    \u2264rsuc : {a : \u2115} \u2192 {b : \u2115} \u2192 (a \u2264 b) \u2192 a \u2264 suc b\n    \u2264rsuc z\u2264n = z\u2264n\n    \u2264rsuc (s\u2264s x) = s\u2264s $ \u2264rsuc x\n\n\n\n\n  -- Here we assume that cut removes call and obs as soon as the call is possible to be executed,\n  -- thus if we reach at a call, we do not continue, it means that this specific subtree will not contain a com\n  -- to execute this communication pattern.\n\n  fstSp : \u2200 {i u ll rll} \u2192 (s : SetLL {i} {u} ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 \u2983 prf : (zero < (length (sptran s ltr))) \u2984 \n          \u2192 SetLL {i} {u} rll\n  fstSp s ltr {{prf = prf}} with (sptran s ltr)\n  fstSp s ltr {{()}} | []\n  fstSp s ltr {{prf}} | x \u2237 r = x\n\n  sndSp : \u2200 {i u ll rll} \u2192 (s : SetLL {i} {u} ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 \u2983 prf : ((suc zero) < (length (sptran s ltr))) \u2984 \n          \u2192 SetLL {i} {u} rll\n  sndSp s ltr {{prf = prf}} with (sptran s ltr)\n  sndSp s ltr {{()}} | []\n  sndSp s ltr {{s\u2264s ()}} | x \u2237 []\n  sndSp s ltr {{prf}} | x \u2237 x\u2081 \u2237 r = x\u2081\n\n  data cuttable {u} : \u2200{i} \u2192 {j : Size< \u2191 i} \u2192 \u2200{rll ll} \u2192 SetLL ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set (lsuc u) where\n    cuttable-s-com : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{rll ll s frll prfi prfo  df lf}\n                     \u2192 \u2983 prf : contruct s \u2261 \u2193 \u2984\n                     \u2192 cuttable {u = u} {i = i} {j = j} s (com {rll = rll} {ll = ll} {frll = frll}  \u2983 prfi = prfi \u2984 \u2983 prfo = prfo \u2984 df lf)\n    cuttable-s\u2282-ex : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{ll rll ell pll s ind lf prf lf\u2081}\n                     \u2192 \u2983 ex : onlyInside s ind \u2984\n                     \u2192 cuttable {rll = ell} {ll = pll} (truncOISetLL {pll = pll} s ind {{prf = ex}}) lf\n                     \u2192 cuttable {i = i} {j = k} {rll = rll} {ll = ll} s (_\u2282_ {i = i} {j = j} {k = k} {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf \u2983 prf = prf \u2984 lf\u2081)\n    cuttable-s\u2282-ho : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{ll rll pll ell s ind lf prf lf\u2081}\n                     \u2192 \u2983 \u00acho : \u00ac (hitsAtLeastOnce {ll = ll} {rll = pll} s ind) \u2984\n                     \u2192 cuttable {rll = rll} (replSetLL s ind {{prf = \u00acho }} ell) lf\u2081\n                     \u2192 cuttable {i = i} {j = k} {rll = rll} {ll = ll} s (_\u2282_ {i = i} {j = j} {k = k} {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf \u2983 prf = prf \u2984 lf\u2081)\n    cuttable-s-tr-fst : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{ll orll rll lf s ltr prftr}\n                        \u2192 cuttable {i = i} {j = j} {rll = rll} {ll = orll} (fstSp s ltr \u2983 prf = prftr \u2984) lf\n                        \u2192 cuttable {i = i} {j = j} s (tr {ll = ll} {orll = orll} {rll = rll} \u2983 ltr = ltr \u2984 lf)\n    cuttable-s-tr-snd : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{ll orll rll lf s ltr prftr}\n                        \u2192 cuttable {i = i} {j = j} {rll = rll} {ll = orll} (sndSp s ltr \u2983 prf = prftr \u2984) lf\n                        \u2192 cuttable {i = i} {j = j} s (tr {ll = ll} {orll = orll} {rll = rll} \u2983 ltr = ltr \u2984 lf)\n\n\n  canItBeCut : \u2200{i} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 (s : SetLL ll) \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 Dec (cuttable {i = i} {j = j} {rll = rll} {ll = ll} s lf)\n  canItBeCut s I = no (\u03bb ())\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) with (isOnlyInside {ll = ll} {rll = pll} s ind)\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) | yes ex with (canItBeCut {rll = ell} {ll = pll} (truncOISetLL {pll = pll} s ind {{prf = ex}}) lf)\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) | yes ex | (yes p) = yes (cuttable-s\u2282-ex {i = i} {j = j} {k = k} \u2983 ex = ex \u2984 p)\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) | yes ex | (no \u00acp) = no (\u03bb x \u2192 helpFunEx x ex \u00acp) where\n    helpFunEx : cuttable {u = u} s (_\u2282_ {i = i} {j = j} {k = k} {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf lf\u2081)\n                \u2192 (ex : onlyInside {i} s ind)\n                \u2192 \u00ac (cuttable {u} {i} {j} {ell} {pll} (truncOISetLL {pll = pll} s ind {{prf = ex}})) lf\n                \u2192 \u22a5\n    helpFunEx (cuttable-s\u2282-ex {{ex = ex}} ct) ex\u2081 \u00acp with (onlyInsideUnique s ind ex ex\u2081)\n    helpFunEx (cuttable-s\u2282-ex {{ex = .ex\u2081}} ct) ex\u2081 \u00acp | refl = \u00acp ct\n    helpFunEx (cuttable-s\u2282-ho {s = s} {ind = ind} {{\u00acho = \u00acho}} ct) ex \u00acp = onlyInside\u00achitsAtLeastOnce\u2192\u22a5 s ind ex \u00acho\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex with (doesItHitAtLeastOnce s ind)\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex | (yes ho) = no (\u03bb { (cuttable-s\u2282-ex {{ex = ex}}   ct) \u2192 \u00acex ex\n                                                                                                      ; (cuttable-s\u2282-ho {{\u00acho = \u00acho}} ct) \u2192 \u00acho ho     })\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {ell} {_} {ind} lf lf\u2081)  | no \u00acex | (no \u00acho) with (canItBeCut (replSetLL s ind {{prf = \u00acho }} ell) lf\u2081)\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex | (no \u00acho) | (yes p) = yes (cuttable-s\u2282-ho \u2983 \u00acho = \u00acho \u2984 p)\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex | (no \u00acho) | (no \u00acp) = no (\u03bb x \u2192 helpFunho x) where\n    helpFunho : cuttable s (_\u2282_ lf lf\u2081)\n                \u2192 \u22a5\n    helpFunho (cuttable-s\u2282-ex {{ex = ex}} x) = \u00acex ex\n    helpFunho (cuttable-s\u2282-ho {{\u00acho = \u00acho\u2081}} x) = \u00acp x\n  canItBeCut s (tr {{ltr = ltr}} lf) with (( suc zero) \u2264? length (sptran s ltr))\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p with (canItBeCut (fstSp s ltr \u2983 prf = p \u2984) lf)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p | (yes p\u2081) = yes (cuttable-s-tr-fst {prftr = p} p\u2081)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p | (no \u00acp) with (( suc $ suc zero) \u2264? length (sptran s ltr))\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p\u2081 | (no \u00acp) | (yes p) with (canItBeCut (sndSp s ltr \u2983 prf = p \u2984) lf)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p\u2082 | (no \u00acp) | (yes p) | (yes p\u2081) = yes (cuttable-s-tr-snd p\u2081)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p\u2081 | (no \u00acp\u2081) | (yes p) | (no \u00acp) = no hf where\n    hf : cuttable s (tr {{ltr = ltr}} lf) \u2192 \u22a5\n    hf (cuttable-s-tr-fst x) = \u00acp\u2081 x\n    hf (cuttable-s-tr-snd {prftr = prftr} x) with (prftr \u2264un p)\n    hf (cuttable-s-tr-snd {_} {_} {_} {_} {_} {_} {_} {_} {_} x) | refl = \u00acp x\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p | (no \u00acp\u2081) | (no \u00acp) = no hf where\n    hf : cuttable s (tr {{ltr = ltr}} lf) \u2192 \u22a5\n    hf (cuttable-s-tr-fst x) = \u00acp\u2081 x\n    hf (cuttable-s-tr-snd {prftr = prftr} x) = \u00acp prftr\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | no \u00acp = no hf where\n    hf : cuttable s (tr {{ltr = ltr}} lf) \u2192 \u22a5\n    hf (cuttable-s-tr-fst {prftr = prftr} x) = \u00acp prftr\n    hf (cuttable-s-tr-snd {prftr = prftr} x) = \u00acp ( \u2264-pred $ \u2264rsuc prftr)\n  canItBeCut s (obs lf) = no (\u03bb ())\n  canItBeCut s (com df lf) with (isEq (contruct s) \u2193)\n  canItBeCut s (com df lf) | yes p = yes (cuttable-s-com {s = s} {{ prf = p }})\n  canItBeCut s (com df lf) | no \u00acp = no hf where\n    hf : cuttable s (com df lf) \u2192 \u22a5\n    hf (cuttable-s-com {{prf = prf}}) = \u00acp prf\n  canItBeCut s (call x lf) = no (\u03bb ())\n\n\n\nmodule _ where\n\n-- cll is the linear logic that is introduced after the last Com.\n-- The index points us to the last transformation of the LinLogic, the last place we received data.\n-- We need to preserve the \u2228(or) choices of the previous inputs.\n  mutual\n    data Spec :  {i : Size} \u2192 {j : Size< i} \u2192 \u2200{u ll rll} \u2192 LinDepT ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set where  \n\n\n--  canBeCut : \u2200{i} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 SetLL ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Bool \u00d7 LinLogic j {u}\n    data Input {u} : {i : Size} {j : Size< \u2191 i} \u2192 \u2200{rll ll} \u2192  LinDepT ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set (lsuc u) where\n--      I    : {i : Size} {j : Size< \u2191 i} \u2192 \u2200{rll ll ldt lf} \u2192 \u2983 prf : nextLFun {i} {j} {u} {rll} {ll} lf \u2261 I \u2984 \u2192 Input {rll = rll} ldt lf\n--      next : {i : Size} {j : Size< \u2191 i} \u2192 \u2200{rll ll ldt lf} \u2192 (s : SetLL ll) \u2192 let cbc = canBeCut s lf in LinT (proj\u2082 cbc) \u2192 \u2983 prf : nextLFun {i} {j} {u} {rll} {ll} lf \u2261 com \u2984 \u2192 Input {u} {i} {j} {rll} {ll} ldt lf\n--      next : in \u2192 Input \u2192 Input\n--      call : \u221einput \u2192 Input \u2192 Input\n\n--    record \u221eInput {i u ll \u221erll} ( ldt : LinDepT ll) ( \u221elf : \u221eLFun {i} {u} {\u221erll} {ll}) : Set (suc u) where\n--      coinductive\n--      field\n--        step : {j : Size< i} \u2192 Input {u} {i = i} {j = j} {rll = step \u221erll} {ll} ldt ((step \u221elf) {j = j})\n--\n\n\n-- As soon as all the inputs of a specific LFun has been received and the resulst is \u2205 for all except an embedding, we replace the linear function with that embedding.\n","old_contents":"{-# OPTIONS --exact-split #-}\n--{-# OPTIONS --show-implicit #-}\n\nmodule LinFun where\n\nopen import Common\nopen import LinLogic\nopen import LinDepT\nopen import LinT \nopen import SetLL\nopen import SetLLProp\n\nopen import Size\nopen import Function\nopen import Data.Bool\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Empty\nopen import Data.Unit hiding (_\u2264_ ; _\u2264?_)\nopen import Relation.Nullary\n\n\n\n-- We send to the receiver both the values the type depends and the value of the type. This way, we achieve locality in terms of finding the type of the incoming value.\n-- We need to point that the receiver and the sender are the same node.\n\nmodule _ where\n\n  mutual\n    data LFun {u} : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {rll : LinLogic j {u}} \u2192 {ll : LinLogic i {u}} \u2192 Set (lsuc u) where\n     I   : {i : Size} \u2192 \u2200{rll} \u2192 LFun {u} {i} {_} {rll} {rll}\n     _\u2282_ : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{pll ll ell rll} \u2192 {ind : IndexLL {i} pll ll} \u2192 (elf : LFun {_} {i} {j} {ell} {pll})\n           \u2192 {{prf : usesInput elf}} \u2192 LFun {_} {j} {k} {rll} {(replLL ll ind ell)}\n           \u2192 LFun {_} {i} {k} {rll} {ll}\n     --Do we need to set ltr as an instance variable?\n     tr  : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{ll orll rll} \u2192 {{ltr : LLTr orll ll}} \u2192 LFun {_} {i} {j} {rll} {orll} \u2192 LFun {_} {i} {j} {rll} {ll}\n     obs : {i : Size} \u2192 {j : Size< i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{\u221ell rll} \u2192 LFun {_} {j} {k} {rll} {(step \u221ell)} \u2192 LFun {_} {i} {k} {rll} {call \u221ell}\n     com : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{rll ll} \u2192 {frll : LinLogic i}\n           \u2192 \u2983 prfi : onlyOneNilOrNoNilFinite ll \u2261 true \u2984 \u2192 \u2983 prfo : onlyOneNilOrNoNilFinite frll \u2261 true \u2984\n           \u2192 (df : (ldt : LinDepT ll) \u2192 LinT ldt \u2192 LinDepT frll) \u2192 LFun {rll = rll} {ll = frll}\n           \u2192 LFun {_} {i} {j} {rll = rll} {ll = ll}\n     call : {i : Size} \u2192 {j : Size< i} \u2192 \u2200{ll \u221erll rll} \u2192 \u221eLFun {i} {_} {\u221erll} {ll} \u2192 LFun {_} {i} {j} {rll} {call \u221erll} \u2192 LFun {_} {i} {j} {rll} {ll}\n  \n  \n    record \u221eLFun {i : Size} {u} {\u221erll : \u221eLinLogic i {u}} {ll : LinLogic i {u}} : Set (lsuc u) where\n      coinductive\n      field\n        step : {j : Size< i} \u2192 LFun {_} {i} {j} {(step \u221erll)} {ll}\n\n\n    usesInput : {i : Size} \u2192 {j : Size< \u2191 i } \u2192 \u2200{u rll ll} \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set\n    usesInput x = usesInput` x \u2205 where\n      usesInput` : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u} \u2192 {rll : LinLogic j} \u2192 {ll : LinLogic i} \u2192 LFun {u} {_} {_} {rll} {ll} \u2192 MSetLL ll \u2192 Set\n      usesInput` I s = \u22a5\n      usesInput` (_\u2282_ {j = j} {k = k} {ell = ell} {ind = ind} elf lf) \u2205 = usesInput` {i = j} {j = k} lf (\u00ac\u2205 (\u2205-add ind ell))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2193} elf lf) (\u00ac\u2205 s) with (contruct $ add {i = i} {j = j} s \u2193 ell)\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2193} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2227 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2227))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2227 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | \u2227\u2192 ns =  usesInput` lf (\u00ac\u2205 (\u2227\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2227 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227\u2192 ns\u2081 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2227\u2192 ns\u2081))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2228 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2228))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2228 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | \u2228\u2192 ns =  usesInput` lf (\u00ac\u2205 (\u2228\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_ LinLogic.\u2228 _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228\u2192 ns\u2081 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2228\u2192 ns\u2081))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_\u2202_ _ _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2202))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_\u2202_ _ _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | \u2202\u2192 ns =  usesInput` lf (\u00ac\u2205 (\u2202\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {.(_\u2202_ _ _)} {_} {\u2193} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202\u2192 ns\u2081 =  usesInput` lf (\u00ac\u2205 (ns \u2190\u2202\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = ind \u2190\u2227} elf lf) (\u00ac\u2205 s) with (contruct $ add s (ind \u2190\u2227) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | \u2227\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2227\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2227} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2227\u2192 ind} elf lf) (\u00ac\u2205 s) with (contruct $ add s (\u2227\u2192 ind) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2227\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2227\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2227\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2227\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2227\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = ind \u2190\u2228} elf lf) (\u00ac\u2205 s) with (contruct $ add s (ind \u2190\u2228) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | \u2228\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2228\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2228} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2228\u2192 ind} elf lf) (\u00ac\u2205 s) with (contruct $ add s (\u2228\u2192 ind) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2228\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2228\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2228\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2228\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2228\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = ind \u2190\u2202} elf lf) (\u00ac\u2205 s) with (contruct $ add s (ind \u2190\u2202) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | \u2202\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2202\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {ind \u2190\u2202} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202\u2192 ns\u2081))\n      usesInput` {i = i} {j = .k} (_\u2282_ {i = .i} {j = j} {k = k} {ell = ell} {rll = rll} {ind = \u2202\u2192 ind} elf lf) (\u00ac\u2205 s) with (contruct $ add s (\u2202\u2192 ind) ell)\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2193 = \u22a4\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | \u2202\u2192 ns = usesInput` lf (\u00ac\u2205 (\u2202\u2192 ns))\n      usesInput` {i} {.k} (_\u2282_ {.i} {j} {k} {_} {_} {ell} {_} {\u2202\u2192 ind} elf lf) (\u00ac\u2205 s) | ns \u2190\u2202\u2192 ns\u2081 = usesInput` lf (\u00ac\u2205 (ns \u2190\u2202\u2192 ns\u2081))\n      usesInput` (tr lf) \u2205 = usesInput` lf \u2205\n      usesInput` (tr {{ltr = ltr}} lf) (\u00ac\u2205 s) = usesInput` lf $ \u00ac\u2205 (SetLL.tran s ltr)\n      usesInput` (obs {j = lj} {k = lk} {\u221ell = \u221ell} x) s = usesInput` {i = lj} {j = lk} x (\u2205 {ll = (step \u221ell)})\n      usesInput` (com df lf) s = \u22a4\n      usesInput` (call \u221ex x) s = \u22a5\n\nopen \u221eLFun public\n\n\n\nmodule _ where\n\n  open import Data.List\n  open import Data.Nat\n\n\n  module _ where\n\n    _\u2264un_ : {a : \u2115} \u2192 {b : \u2115} \u2192 (c : (a \u2264 b)) \u2192 (d : (a \u2264 b)) \u2192 c \u2261 d  \n    z\u2264n \u2264un z\u2264n = refl\n    s\u2264s c \u2264un s\u2264s d with ( c \u2264un d )\n    s\u2264s c \u2264un s\u2264s .c | refl = refl\n\n    \u2264rsuc : {a : \u2115} \u2192 {b : \u2115} \u2192 (a \u2264 b) \u2192 a \u2264 suc b\n    \u2264rsuc z\u2264n = z\u2264n\n    \u2264rsuc (s\u2264s x) = s\u2264s $ \u2264rsuc x\n\n\n\n\n  -- Here we assume that cut removes call and obs as soon as the call is possible to be executed,\n  -- thus if we reach at a call, we do not continue, it means that this specific subtree will not contain a com\n  -- to execute this communication pattern.\n\n  fstSp : \u2200 {i u ll rll} \u2192 (s : SetLL {i} {u} ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 \u2983 prf : (zero < (length (sptran s ltr))) \u2984 \n          \u2192 SetLL {i} {u} rll\n  fstSp s ltr {{prf = prf}} with (sptran s ltr)\n  fstSp s ltr {{()}} | []\n  fstSp s ltr {{prf}} | x \u2237 r = x\n\n  sndSp : \u2200 {i u ll rll} \u2192 (s : SetLL {i} {u} ll) \u2192 (ltr : LLTr {i} {u} rll ll) \u2192 \u2983 prf : ((suc zero) < (length (sptran s ltr))) \u2984 \n          \u2192 SetLL {i} {u} rll\n  sndSp s ltr {{prf = prf}} with (sptran s ltr)\n  sndSp s ltr {{()}} | []\n  sndSp s ltr {{s\u2264s ()}} | x \u2237 []\n  sndSp s ltr {{prf}} | x \u2237 x\u2081 \u2237 r = x\u2081\n\n  data cuttable {u} : \u2200{i} \u2192 {j : Size< \u2191 i} \u2192 \u2200{rll ll} \u2192 SetLL ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set (lsuc u) where\n    cuttable-s-com : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{rll ll s frll prfi prfo  df lf}\n                     \u2192 \u2983 prf : contruct s \u2261 \u2193 \u2984\n                     \u2192 cuttable {u = u} {i = i} {j = j} s (com {rll = rll} {ll = ll} {frll = frll}  \u2983 prfi = prfi \u2984 \u2983 prfo = prfo \u2984 df lf)\n    cuttable-s\u2282-ex : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{ll rll ell pll s ind lf prf lf\u2081}\n                     \u2192 \u2983 ex : onlyInside s ind \u2984\n                     \u2192 cuttable {rll = ell} {ll = pll} (truncOISetLL {pll = pll} s ind {{prf = ex}}) lf\n                     \u2192 cuttable {i = i} {j = k} {rll = rll} {ll = ll} s (_\u2282_ {i = i} {j = j} {k = k} {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf \u2983 prf = prf \u2984 lf\u2081)\n    cuttable-s\u2282-ho : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 {k : Size< \u2191 j} \u2192 \u2200{ll rll pll ell s ind lf prf lf\u2081}\n                     \u2192 \u2983 \u00acho : \u00ac (hitsAtLeastOnce {ll = ll} {rll = pll} s ind) \u2984\n                     \u2192 cuttable {rll = rll} (replSetLL s ind {{prf = \u00acho }} ell) lf\u2081\n                     \u2192 cuttable {i = i} {j = k} {rll = rll} {ll = ll} s (_\u2282_ {i = i} {j = j} {k = k} {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf \u2983 prf = prf \u2984 lf\u2081)\n    cuttable-s-tr-fst : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{ll orll rll lf s ltr prftr}\n                        \u2192 cuttable {i = i} {j = j} {rll = rll} {ll = orll} (fstSp s ltr \u2983 prf = prftr \u2984) lf\n                        \u2192 cuttable {i = i} {j = j} s (tr {ll = ll} {orll = orll} {rll = rll} \u2983 ltr = ltr \u2984 lf)\n    cuttable-s-tr-snd : {i : Size} \u2192 {j : Size< \u2191 i} \u2192 \u2200{ll orll rll lf s ltr prftr}\n                        \u2192 cuttable {i = i} {j = j} {rll = rll} {ll = orll} (sndSp s ltr \u2983 prf = prftr \u2984) lf\n                        \u2192 cuttable {i = i} {j = j} s (tr {ll = ll} {orll = orll} {rll = rll} \u2983 ltr = ltr \u2984 lf)\n\n\n  canItBeCut : \u2200{i} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 (s : SetLL ll) \u2192 (lf : LFun {u} {i} {j} {rll} {ll}) \u2192 Dec (cuttable {i = i} {j = j} {rll = rll} {ll = ll} s lf)\n  canItBeCut s I = no (\u03bb ())\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) with (isOnlyInside {ll = ll} {rll = pll} s ind)\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) | yes ex with (canItBeCut {rll = ell} {ll = pll} (truncOISetLL {pll = pll} s ind {{prf = ex}}) lf)\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) | yes ex | (yes p) = yes (cuttable-s\u2282-ex {i = i} {j = j} {k = k} \u2983 ex = ex \u2984 p)\n  canItBeCut {.i} {.k} {u} {_} {_} s (_\u2282_ {i} {j} {k} {pll} {ll} {ell} {_} {ind} lf lf\u2081) | yes ex | (no \u00acp) = no (\u03bb x \u2192 helpFunEx x ex \u00acp) where\n    helpFunEx : cuttable {u = u} s (_\u2282_ {i = i} {j = j} {k = k} {pll = pll} {ll = ll} {ell = ell} {ind = ind} lf lf\u2081)\n                \u2192 (ex : onlyInside {i} s ind)\n                \u2192 \u00ac (cuttable {u} {i} {j} {ell} {pll} (truncOISetLL {pll = pll} s ind {{prf = ex}})) lf\n                \u2192 \u22a5\n    helpFunEx (cuttable-s\u2282-ex {{ex = ex}} ct) ex\u2081 \u00acp with (onlyInsideUnique s ind ex ex\u2081)\n    helpFunEx (cuttable-s\u2282-ex {{ex = .ex\u2081}} ct) ex\u2081 \u00acp | refl = \u00acp ct\n    helpFunEx (cuttable-s\u2282-ho {s = s} {ind = ind} {{\u00acho = \u00acho}} ct) ex \u00acp = onlyInside\u00achitsAtLeastOnce\u2192\u22a5 s ind ex \u00acho\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex with (doesItHitAtLeastOnce s ind)\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex | (yes ho) = no (\u03bb { (cuttable-s\u2282-ex {{ex = ex}}   ct) \u2192 \u00acex ex\n                                                                                                      ; (cuttable-s\u2282-ho {{\u00acho = \u00acho}} ct) \u2192 \u00acho ho     })\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {ell} {_} {ind} lf lf\u2081)  | no \u00acex | (no \u00acho) with (canItBeCut (replSetLL s ind {{prf = \u00acho }} ell) lf\u2081)\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex | (no \u00acho) | (yes p) = yes (cuttable-s\u2282-ho \u2983 \u00acho = \u00acho \u2984 p)\n  canItBeCut {_} {.k} s (_\u2282_ {_} {_} {k} {_} {_} {_} {_} {ind} lf lf\u2081)    | no \u00acex | (no \u00acho) | (no \u00acp) = no (\u03bb x \u2192 helpFunho x) where\n    helpFunho : cuttable s (_\u2282_ lf lf\u2081)\n                \u2192 \u22a5\n    helpFunho (cuttable-s\u2282-ex {{ex = ex}} x) = \u00acex ex\n    helpFunho (cuttable-s\u2282-ho {{\u00acho = \u00acho\u2081}} x) = \u00acp x\n  canItBeCut s (tr {{ltr = ltr}} lf) with (( suc zero) \u2264? length (sptran s ltr))\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p with (canItBeCut (fstSp s ltr \u2983 prf = p \u2984) lf)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p | (yes p\u2081) = yes (cuttable-s-tr-fst {prftr = p} p\u2081)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p | (no \u00acp) with (( suc $ suc zero) \u2264? length (sptran s ltr))\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p\u2081 | (no \u00acp) | (yes p) with (canItBeCut (sndSp s ltr \u2983 prf = p \u2984) lf)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p\u2082 | (no \u00acp) | (yes p) | (yes p\u2081) = yes (cuttable-s-tr-snd p\u2081)\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p\u2081 | (no \u00acp\u2081) | (yes p) | (no \u00acp) = no hf where\n    hf : cuttable s (tr {{ltr = ltr}} lf) \u2192 \u22a5\n    hf (cuttable-s-tr-fst x) = \u00acp\u2081 x\n    hf (cuttable-s-tr-snd {prftr = prftr} x) with (prftr \u2264un p)\n    hf (cuttable-s-tr-snd {_} {_} {_} {_} {_} {_} {_} {_} {_} x) | refl = \u00acp x\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | yes p | (no \u00acp\u2081) | (no \u00acp) = no hf where\n    hf : cuttable s (tr {{ltr = ltr}} lf) \u2192 \u22a5\n    hf (cuttable-s-tr-fst x) = \u00acp\u2081 x\n    hf (cuttable-s-tr-snd {prftr = prftr} x) = \u00acp prftr\n  canItBeCut s (tr {_} {_} {_} {_} {_} {{ltr}} lf) | no \u00acp = no hf where\n    hf : cuttable s (tr {{ltr = ltr}} lf) \u2192 \u22a5\n    hf (cuttable-s-tr-fst {prftr = prftr} x) = \u00acp prftr\n    hf (cuttable-s-tr-snd {prftr = prftr} x) = \u00acp ( \u2264-pred $ \u2264rsuc prftr)\n  canItBeCut s (obs lf) = no (\u03bb ())\n  canItBeCut s (com df lf) with (isEq (contruct s) \u2193)\n  canItBeCut s (com df lf) | yes p = yes (cuttable-s-com {s = s} {{ prf = p }})\n  canItBeCut s (com df lf) | no \u00acp = no hf where\n    hf : cuttable s (com df lf) \u2192 \u22a5\n    hf (cuttable-s-com {{prf = prf}}) = \u00acp prf\n  canItBeCut s (call x lf) = no (\u03bb ())\n\n\n\nmodule _ where\n\n-- cll is the linear logic that is introduced after the last Com.\n-- The index points us to the last transformation of the LinLogic, the last place we received data.\n-- We need to preserve the \u2228(or) choices of the previous inputs.\n  mutual\n    data Spec :  {i : Size} \u2192 {j : Size< i} \u2192 \u2200{u ll rll} \u2192 LinDepT ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set where  \n\n\n--  canBeCut : \u2200{i} \u2192 {j : Size< \u2191 i} \u2192 \u2200{u rll ll} \u2192 SetLL ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Bool \u00d7 LinLogic j {u}\n    data Input {u} : {i : Size} {j : Size< \u2191 i} \u2192 \u2200{rll ll} \u2192  LinDepT ll \u2192 LFun {u} {i} {j} {rll} {ll} \u2192 Set (lsuc u) where\n--      I    : {i : Size} {j : Size< \u2191 i} \u2192 \u2200{rll ll ldt lf} \u2192 \u2983 prf : nextLFun {i} {j} {u} {rll} {ll} lf \u2261 I \u2984 \u2192 Input {rll = rll} ldt lf\n--      next : {i : Size} {j : Size< \u2191 i} \u2192 \u2200{rll ll ldt lf} \u2192 (s : SetLL ll) \u2192 let cbc = canBeCut s lf in LinT (proj\u2082 cbc) \u2192 \u2983 prf : nextLFun {i} {j} {u} {rll} {ll} lf \u2261 com \u2984 \u2192 Input {u} {i} {j} {rll} {ll} ldt lf\n--      next : in \u2192 Input \u2192 Input\n--      call : \u221einput \u2192 Input \u2192 Input\n\n--    record \u221eInput {i u ll \u221erll} ( ldt : LinDepT ll) ( \u221elf : \u221eLFun {i} {u} {\u221erll} {ll}) : Set (suc u) where\n--      coinductive\n--      field\n--        step : {j : Size< i} \u2192 Input {u} {i = i} {j = j} {rll = step \u221erll} {ll} ldt ((step \u221elf) {j = j})\n--\n\n\n-- As soon as all the inputs of a specific LFun has been received and the resulst is \u2205 for all except an embedding, we replace the linear function with that embedding.\n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"a04c9ff1b87354586f7b0daad0d9b8c64f7e39d5","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: 6d947224b6fb2cf53d7584c2a11adf06\n\ndarcs-hash:20110428154435-3bd4e-a8ed66e5cfbf21adde8d36eb9a5773087048f78d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/List.agda","new_file":"src\/FOTC\/Data\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.List where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixr 8 _++_\n\n------------------------------------------------------------------------------\n-- The FOTC list type.\n-- open import FOTC.Data.List.Type public\n\n------------------------------------------------------------------------------\n-- Basic functions\n\npostulate\n  length    : D \u2192 D\n  length-[] :          length []       \u2261 zero\n  length-\u2237  : \u2200 d ds \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : \u2200 ds \u2192      []       ++ ds \u2261 ds\n  ++-\u2237  : \u2200 d ds es \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  -- NB. The function map is not a higher-order function.\n  map    : D \u2192 D \u2192 D\n  map-[] : \u2200 f \u2192      map f []       \u2261 []\n  map-\u2237  : \u2200 f d ds \u2192 map f (d \u2237 ds) \u2261 f \u00b7 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  rev    : D \u2192 D \u2192 D\n  rev-[] : \u2200 es \u2192      rev []       es \u2261 es\n  rev-\u2237  : \u2200 d ds es \u2192 rev (d \u2237 ds) es \u2261 rev ds (d \u2237 es)\n{-# ATP axiom rev-[] #-}\n{-# ATP axiom rev-\u2237 #-}\n\nreverse : D \u2192 D\nreverse ds = rev ds []\n{-# ATP definition reverse #-}\n\npostulate\n  replicate   : D \u2192 D \u2192 D\n  replicate-0 : \u2200 d \u2192   replicate zero     d \u2261 []\n  replicate-S : \u2200 d e \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n-- Reducing lists\n\npostulate\n  -- NB. The function foldr is not a higher-order function.\n  foldr    : D \u2192 D \u2192 D \u2192 D\n  foldr-[] : \u2200 f n  \u2192     foldr f n []       \u2261 n\n  foldr-\u2237  : \u2200 f n d ds \u2192 foldr f n (d \u2237 ds) \u2261 f \u00b7 d \u00b7 (foldr f n ds)\n{-# ATP axiom foldr-[] #-}\n{-# ATP axiom foldr-\u2237 #-}\n\n-- Building lists\n\npostulate\n  -- NB. The function iterate is not a higher-order function.\n  iterate    : D \u2192 D \u2192 D\n  iterate-eq : \u2200 f x \u2192 iterate f x \u2261 x \u2237 iterate f (f \u00b7 x)\n{-# ATP axiom iterate-eq #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Lists\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.List where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixr 8 _++_\n\n------------------------------------------------------------------------------\n-- The FOTC list type.\n-- open import FOTC.Data.List.Type public\n\n------------------------------------------------------------------------------\n-- Basic functions\n\npostulate\n  length    : D \u2192 D\n  length-[] :          length []       \u2261 zero\n  length-\u2237  : \u2200 d ds \u2192 length (d \u2237 ds) \u2261 succ (length ds)\n{-# ATP axiom length-[] #-}\n{-# ATP axiom length-\u2237 #-}\n\npostulate\n  _++_  : D \u2192 D \u2192 D\n  ++-[] : \u2200 ds \u2192      []       ++ ds \u2261 ds\n  ++-\u2237  : \u2200 d ds es \u2192 (d \u2237 ds) ++ es \u2261 d \u2237 (ds ++ es)\n{-# ATP axiom ++-[] #-}\n{-# ATP axiom ++-\u2237 #-}\n\n-- List transformations\n\npostulate\n  map    : D \u2192 D \u2192 D\n  map-[] : (f : D) \u2192        map f []       \u2261 []\n  map-\u2237  : \u2200 (f : D) d ds \u2192 map f (d \u2237 ds) \u2261 f \u00b7 d \u2237 map f ds\n{-# ATP axiom map-[] #-}\n{-# ATP axiom map-\u2237 #-}\n\npostulate\n  rev    : D \u2192 D \u2192 D\n  rev-[] : \u2200 es \u2192      rev []       es \u2261 es\n  rev-\u2237  : \u2200 d ds es \u2192 rev (d \u2237 ds) es \u2261 rev ds (d \u2237 es)\n{-# ATP axiom rev-[] #-}\n{-# ATP axiom rev-\u2237 #-}\n\nreverse : D \u2192 D\nreverse ds = rev ds []\n{-# ATP definition reverse #-}\n\npostulate\n  replicate   : D \u2192 D \u2192 D\n  replicate-0 : \u2200 d \u2192   replicate zero     d \u2261 []\n  replicate-S : \u2200 d e \u2192 replicate (succ e) d \u2261 d \u2237 replicate e d\n{-# ATP axiom replicate-0 #-}\n{-# ATP axiom replicate-S #-}\n\n-- Reducing lists\n\npostulate\n  foldr    : D \u2192 D \u2192 D \u2192 D\n  foldr-[] : \u2200 (f : D) n  \u2192     foldr f n []       \u2261 n\n  foldr-\u2237  : \u2200 (f : D) n d ds \u2192 foldr f n (d \u2237 ds) \u2261 f \u00b7 d \u00b7 (foldr f n ds)\n{-# ATP axiom foldr-[] #-}\n{-# ATP axiom foldr-\u2237 #-}\n\n-- Building lists\n\npostulate\n  iterate    : D \u2192 D \u2192 D\n  iterate-eq : \u2200 (f : D) x \u2192 iterate f x \u2261 x \u2237 iterate f (f \u00b7 x)\n{-# ATP axiom iterate-eq #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"24af905e5de0171ac9c9a281c67a3f0a3b912e09","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\n\nmodule typed-expansion where\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 x x\u2081 x\u2082 x\u2083) = TAAp {!!} MAHole {!!}\n    typed-expansion-synth (ESAp2 x ex x\u2081 x\u2082) = TAAp {!!} MAArr {!!}\n    typed-expansion-synth (ESAp3 x ex x\u2081)    = TAAp {!!} MAArr {!!}\n    typed-expansion-synth ESEHole = TAEHole (\u03bb x d x\u2081 \u2192 \u2987\u2988 , {!!} , {!!})\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) (\u03bb x d x\u2081 \u2192 {!!})\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = {!!}\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex) = {!!}\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    typed-expansion-ana EAEHole = TCRefl , TAEHole {!!}\n    typed-expansion-ana (EANEHole x x\u2081) = TCRefl , TANEHole (typed-expansion-synth x) {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\n\nmodule typed-expansion where\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 x x\u2081 x\u2082 x\u2083) = TAAp {!!} MAHole {!!}\n    typed-expansion-synth (ESAp2 x ex x\u2081 x\u2082) = TAAp {!!} MAArr {!!}\n    typed-expansion-synth (ESAp3 x ex x\u2081)    = TAAp {!!} MAArr {!!}\n    typed-expansion-synth ESEHole = TAEHole (\u03bb x d x\u2081 \u2192 \u2987\u2988 , {!!} , {!!})\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) (\u03bb x d x\u2081 \u2192 {!!})\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x) with typed-expansion-ana x\n    ... | con , ih = {!ih!}\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex) = {!!}\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    typed-expansion-ana EAEHole = TCRefl , TAEHole {!!}\n    typed-expansion-ana (EANEHole x x\u2081) = TCRefl , TANEHole (typed-expansion-synth x) {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db85c51bd067f64e27e0e4029dab987c8894e106","subject":"agda: Comments on sorting","message":"agda: Comments on sorting\n\nOld-commit-hash: 399565c71bf148667ea798bd7bda7fd00af371d4\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/Sorting.agda","new_file":"experimental\/Sorting.agda","new_contents":"module Sorting where\n\nimport Level as L\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary\nopen import Relation.Nullary\n\n-- Here I mostly follow the insertion sort developed in:\n-- http:\/\/mazzo.li\/posts\/AgdaSort.html\n--\n-- The main difference, apart from a few identifiers, is that I use the standard\n-- library.\n\nmodule Sort {c \u2113\u2081 \u2113\u2082}  {X : Set c}\n                       {_\u2248_ : Rel X \u2113\u2081} {_\u2264_ : Rel X \u2113\u2082}\n                       (ord : IsDecTotalOrder _\u2248_ _\u2264_)\n                       where\n  open IsDecTotalOrder ord\n\n  data \u22a5X\u22a4 : Set c where\n    \u27e6\u22a5\u27e7 \u27e6\u22a4\u27e7 : \u22a5X\u22a4\n    \u27e6_\u27e7 : X \u2192 \u22a5X\u22a4\n\n  data _\u2272_ : Rel \u22a5X\u22a4 (c L.\u2294 \u2113\u2082) where\n    \u22a5\u2272_ : \u2200 {x} \u2192 \u27e6\u22a5\u27e7 \u2272 x\n    _\u2272\u22a4 : \u2200 {x} \u2192 x \u2272 \u27e6\u22a4\u27e7\n    \u2264-lift : \u2200 {x y} \u2192 x \u2264 y \u2192 \u27e6 x \u27e7 \u2272 \u27e6 y \u27e7\n\n  data OList (l u : \u22a5X\u22a4) : Set (c L.\u2294 \u2113\u2082) where\n    nil : l \u2272 u \u2192 OList l u\n    cons : \u2200 x (xs : OList \u27e6 x \u27e7 u) \u2192 l \u2272 \u27e6 x \u27e7 \u2192 OList l u\n\n  toList : \u2200 {l u} \u2192 OList l u \u2192 List X\n  toList (nil _) = []\n  toList (cons x xs _) = x \u2237 toList xs\n\n  insert : \u2200 {l u} y \u2192 OList l u \u2192 l \u2272 \u27e6 y \u27e7 \u2192 \u27e6 y \u27e7 \u2272 u \u2192 OList l u\n  insert y (nil _) l\u2272y y\u2272u = cons y (nil y\u2272u) l\u2272y\n  insert y (cons x xs l\u2272x) l\u2272y y\u2272u with y \u2264? x\n  insert y (cons x xs l\u2272x) l\u2272y y\u2272u | yes p = cons y (cons x xs (\u2264-lift p)) l\u2272y\n  insert y (cons x xs l\u2272x) l\u2272y y\u2272u | no y>x = cons x (insert y xs x\u2272y y\u2272u) l\u2272x\n    where\n      x\u2272y : \u27e6 x \u27e7 \u2272 \u27e6 y \u27e7\n      -- Ambiguous\n      --x\u2272y = [ \u2264-lift , \u22a5-elim \u2218 y>x ] (total x y)\n      --x\u2272y = (\u03bb r \u2192 [ \u2264-lift , \u22a5-elim \u2218 y>x ] r) (total x y)\n\n      -- This is instead not ambiguous, somehow.\n      x\u2272y with total x y \n      ... | r = [ \u2264-lift , \u22a5-elim \u2218 y>x ] r\n\n  UList = OList \u27e6\u22a5\u27e7 \u27e6\u22a4\u27e7\n  unil : UList\n  unil = nil \u22a5\u2272_\n\n  ucons : X \u2192 UList \u2192 UList\n  ucons x xs = insert x xs \u22a5\u2272_ _\u2272\u22a4\n\n  uunion-asym : UList \u2192 List X \u2192 UList\n  uunion-asym = foldr (\u03bb x xs \u2192 insert {\u27e6\u22a5\u27e7} {\u27e6\u22a4\u27e7} x xs \u22a5\u2272_ _\u2272\u22a4)\n\n  uunion : UList \u2192 UList \u2192 UList\n  uunion b\u2081 b\u2082 = uunion-asym b\u2081 (toList b\u2082)\n\n  uToList : UList \u2192 List X\n  uToList = toList\n\n  isort\u2032 : List X \u2192 UList\n  isort\u2032 = uunion-asym (nil \u22a5\u2272_)\n\n  isort = toList \u2218 isort\u2032\n\n--open Sort using (isort) public\n","old_contents":"module Sorting where\n\nimport Level as L\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Sum\nopen import Function\nopen import Relation.Binary\nopen import Relation.Nullary\n\nmodule Sort {c \u2113\u2081 \u2113\u2082}  {X : Set c}\n                       {_\u2248_ : Rel X \u2113\u2081} {_\u2264_ : Rel X \u2113\u2082}\n                       (ord : IsDecTotalOrder _\u2248_ _\u2264_)\n                       where\n  open IsDecTotalOrder ord\n\n  --open IsTotalOrder isTotalOrder using (total)\n  --open IsEquivalence isEquivalence using (refl)\n\n  data \u22a5X\u22a4 : Set c where\n    \u27e6\u22a5\u27e7 \u27e6\u22a4\u27e7 : \u22a5X\u22a4\n    \u27e6_\u27e7 : X \u2192 \u22a5X\u22a4\n\n  data _\u2272_ : Rel \u22a5X\u22a4 (c L.\u2294 \u2113\u2082) where\n    \u22a5\u2272_ : \u2200 {x} \u2192 \u27e6\u22a5\u27e7 \u2272 x\n    _\u2272\u22a4 : \u2200 {x} \u2192 x \u2272 \u27e6\u22a4\u27e7\n    \u2264-lift : \u2200 {x y} \u2192 x \u2264 y \u2192 \u27e6 x \u27e7 \u2272 \u27e6 y \u27e7\n\n  data OList (l u : \u22a5X\u22a4) : Set (c L.\u2294 \u2113\u2082) where\n    nil : l \u2272 u \u2192 OList l u\n    cons : \u2200 x (xs : OList \u27e6 x \u27e7 u) \u2192 l \u2272 \u27e6 x \u27e7 \u2192 OList l u\n\n  toList : \u2200 {l u} \u2192 OList l u \u2192 List X\n  toList (nil _) = []\n  toList (cons x xs _) = x \u2237 toList xs\n\n  insert : \u2200 {l u} y \u2192 OList l u \u2192 l \u2272 \u27e6 y \u27e7 \u2192 \u27e6 y \u27e7 \u2272 u \u2192 OList l u\n  insert y (nil _) l\u2272y y\u2272u = cons y (nil y\u2272u) l\u2272y\n  insert y (cons x xs l\u2272x) l\u2272y y\u2272u with y \u2264? x\n  insert y (cons x xs l\u2272x) l\u2272y y\u2272u | yes p = cons y (cons x xs (\u2264-lift p)) l\u2272y\n  insert y (cons x xs l\u2272x) l\u2272y y\u2272u | no y>x = cons x (insert y xs x\u2272y y\u2272u) l\u2272x\n    where\n      x\u2272y : \u27e6 x \u27e7 \u2272 \u27e6 y \u27e7\n      -- Ambiguous\n      --x\u2272y = [ \u2264-lift , \u22a5-elim \u2218 y>x ] (total x y)\n      x\u2272y with total x y \n      ... | r = [ \u2264-lift , \u22a5-elim \u2218 y>x ] r\n{-\n      x\u2272y | inj\u2081 x\u2264y = \u2264-lift x\u2264y\n      x\u2272y | inj\u2082 y\u2264x = \u22a5-elim \u2218 y>x $ y\u2264x\n-}\n  UList = OList \u27e6\u22a5\u27e7 \u27e6\u22a4\u27e7\n  unil : UList\n  unil = nil \u22a5\u2272_\n\n  ucons : X \u2192 UList \u2192 UList\n  ucons x xs = insert x xs \u22a5\u2272_ _\u2272\u22a4\n\n  uunion-asym : UList \u2192 List X \u2192 UList\n  uunion-asym = foldr (\u03bb x xs \u2192 insert {\u27e6\u22a5\u27e7} {\u27e6\u22a4\u27e7} x xs \u22a5\u2272_ _\u2272\u22a4)\n\n  uunion : UList \u2192 UList \u2192 UList\n  uunion b\u2081 b\u2082 = uunion-asym b\u2081 (toList b\u2082)\n\n  uToList : UList \u2192 List X\n  uToList = toList\n\n  isort\u2032 : List X \u2192 UList\n  isort\u2032 = uunion-asym (nil \u22a5\u2272_)\n\n  isort = toList \u2218 isort\u2032\n\n--open Sort using (isort) public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7813ae9f077f7718b236662de80693c3ca0038a5","subject":"Added x\u2223S\u2192x\u2264S-ah\u2081 (PCF version).","message":"Added x\u2223S\u2192x\u2264S-ah\u2081 (PCF version).\n\nIgnore-this: 692a6f2824cf447189a28eb84d3b84e8\n\ndarcs-hash:20100603031930-3bd4e-87846c88094f80c6d4bffd0d52bea3b9ed351bc5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Divisibility\/PropertiesPCF.agda","new_file":"LTC\/Relation\/Divisibility\/PropertiesPCF.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the divisibility relation\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Divisibility.PropertiesPCF where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF\nopen import LTC.Relation.DivisibilityPCF\nopen import LTC.Relation.Equalities.Properties\nopen import LTC.Relation.InequalitiesPCF\nopen import LTC.Relation.Inequalities.PropertiesPCF\n\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n-- Any positive number divides 0.\n\npostulate S\u22230 : {n : D} \u2192 N n \u2192  succ n \u2223 zero\n{-# ATP prove S\u22230 zN *-0x #-}\n\n------------------------------------------------------------------------------\n-- The divisibility relation is reflexive for positive numbers.\n\n-- For the proof using the ATP we added the auxiliar hypothesis\n-- N (succ zero).\npostulate \u2223-refl-S-ah : {n : D} \u2192 N n \u2192 N (succ zero) \u2192 succ n \u2223 succ n\n{-# ATP prove \u2223-refl-S-ah sN *-leftIdentity #-}\n\n\u2223-refl-S : {n : D} \u2192 N n \u2192 succ n \u2223 succ n\n\u2223-refl-S Nn = \u2223-refl-S-ah Nn (sN zN)\n\n------------------------------------------------------------------------------\n-- 0 doesn't divide any number.\n0\u2224x : {d : D} \u2192 \u00ac (zero \u2223 d)\n0\u2224x ( 0\u22600 , _ ) = \u22a5-elim $ 0\u22600 refl\n\n------------------------------------------------------------------------------\n-- If 'x' divides 'y' and 'z' then 'x' divides 'y - z'.\npostulate\n  x\u2223y\u2192x\u2223z\u2192x\u2223y-z-ah : {m n p k\u2081 k\u2082 : D} \u2192 N m \u2192 N n \u2192 N k\u2081 \u2192 N k\u2082 \u2192\n                      n \u2261 k\u2081 * succ m \u2192\n                      p \u2261 k\u2082 * succ m \u2192\n                      n - p \u2261 (k\u2081 - k\u2082) * succ m\n{-# ATP prove x\u2223y\u2192x\u2223z\u2192x\u2223y-z-ah [x-y]z\u2261xz*yz sN #-}\n\nx\u2223y\u2192x\u2223z\u2192x\u2223y-z : {m n p : D} \u2192 N m \u2192 N n \u2192 N p \u2192 m \u2223 n \u2192 m \u2223 p \u2192 m \u2223 n - p\nx\u2223y\u2192x\u2223z\u2192x\u2223y-z zN _ _ ( 0\u22600 , _) m\u2223p = \u22a5-elim (0\u22600 refl)\nx\u2223y\u2192x\u2223z\u2192x\u2223y-z (sN Nm) Nn Np\n              ( 0\u22600 , ( k\u2081 , Nk\u2081 , n\u2261k\u2081Sm ))\n              ( _   , ( k\u2082 , Nk\u2082 , p\u2261k\u2082Sm )) =\n  (\u03bb S\u22610 \u2192 \u22a5-elim (\u00acS\u22610 S\u22610)) ,\n  (k\u2081 - k\u2082) ,\n  minus-N Nk\u2081 Nk\u2082 ,\n  x\u2223y\u2192x\u2223z\u2192x\u2223y-z-ah Nm Nn Nk\u2081 Nk\u2082 n\u2261k\u2081Sm p\u2261k\u2082Sm\n\n------------------------------------------------------------------------------\n-- If 'x' divides 'y' and 'z' then 'x' divides 'y + z'.\npostulate\n  x\u2223y\u2192x\u2223z\u2192x\u2223y+z-ah : {m n p k\u2081 k\u2082 : D} \u2192 N m \u2192 N n \u2192 N k\u2081 \u2192 N k\u2082 \u2192\n                      n \u2261 k\u2081 * succ m \u2192\n                      p \u2261 k\u2082 * succ m \u2192\n                      n + p \u2261 (k\u2081 + k\u2082) * succ m\n{-# ATP prove x\u2223y\u2192x\u2223z\u2192x\u2223y+z-ah [x+y]z\u2261xz*yz sN #-}\n\nx\u2223y\u2192x\u2223z\u2192x\u2223y+z : {m n p : D} \u2192 N m \u2192 N n \u2192 N p \u2192 m \u2223 n \u2192 m \u2223 p \u2192 m \u2223 n + p\nx\u2223y\u2192x\u2223z\u2192x\u2223y+z zN      _  _ ( 0\u22600 , _) m\u2223p = \u22a5-elim (0\u22600 refl)\nx\u2223y\u2192x\u2223z\u2192x\u2223y+z (sN Nm) Nn Np\n              ( 0\u22600 , ( k\u2081 , Nk\u2081 , n\u2261k\u2081Sm ))\n              ( _   , ( k\u2082 , Nk\u2082 , p\u2261k\u2082Sm )) =\n  (\u03bb S\u22610 \u2192 \u22a5-elim (\u00acS\u22610 S\u22610)) ,\n  (k\u2081 + k\u2082) ,\n  +-N Nk\u2081 Nk\u2082 ,\n  x\u2223y\u2192x\u2223z\u2192x\u2223y+z-ah Nm Nn Nk\u2081 Nk\u2082 n\u2261k\u2081Sm p\u2261k\u2082Sm\n\n------------------------------------------------------------------------------\n-- If x divides y, and y is positive, then x \u2264 y.\n\npostulate\n  x\u2223S\u2192x\u2264S-ah\u2081 : {m n : D} \u2192 succ n \u2261 zero * succ m \u2192 \u22a5\n{-# ATP prove x\u2223S\u2192x\u2264S-ah\u2081 *-leftZero *-0x sN #-}\n\n-- Nice proof by the ATP.\npostulate\n  x\u2223S\u2192x\u2264S-ah\u2082 : {m n k : D} \u2192 N m \u2192 N n \u2192 N k \u2192\n                succ n \u2261 succ k * succ m \u2192\n                LE (succ m) (succ n)\n{-# ATP prove x\u2223S\u2192x\u2264S-ah\u2082 x\u2264x+y *-N sN *-Sx #-}\n\nx\u2223S\u2192x\u2264S : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2223 (succ n) \u2192 LE m (succ n)\nx\u2223S\u2192x\u2264S  zN     Nn ( 0\u22600 , _)                  = \u22a5-elim (0\u22600 refl)\nx\u2223S\u2192x\u2264S (sN Nm) Nn ( _ , .zero , zN , Sn\u22610*Sm) = \u22a5-elim (x\u2223S\u2192x\u2264S-ah\u2081 Sn\u22610*Sm)\nx\u2223S\u2192x\u2264S (sN {m} Nm) Nn ( _ , .(succ k) , sN {k} Nk , Sn\u2261Sk*Sm) =\n  x\u2223S\u2192x\u2264S-ah\u2082 Nm Nn Nk Sn\u2261Sk*Sm\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the divisibility relation\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Divisibility.PropertiesPCF where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.ArithmeticPCF\nopen import LTC.Function.Arithmetic.PropertiesPCF\nopen import LTC.Relation.DivisibilityPCF\nopen import LTC.Relation.Equalities.Properties\nopen import LTC.Relation.InequalitiesPCF\nopen import LTC.Relation.Inequalities.PropertiesPCF\n\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n-- Any positive number divides 0.\n\npostulate S\u22230 : {n : D} \u2192 N n \u2192  succ n \u2223 zero\n{-# ATP prove S\u22230 zN *-0x #-}\n\n------------------------------------------------------------------------------\n-- The divisibility relation is reflexive for positive numbers.\n\n-- For the proof using the ATP we added the auxiliar hypothesis\n-- N (succ zero).\npostulate \u2223-refl-S-ah : {n : D} \u2192 N n \u2192 N (succ zero) \u2192 succ n \u2223 succ n\n{-# ATP prove \u2223-refl-S-ah sN *-leftIdentity #-}\n\n\u2223-refl-S : {n : D} \u2192 N n \u2192 succ n \u2223 succ n\n\u2223-refl-S Nn = \u2223-refl-S-ah Nn (sN zN)\n\n------------------------------------------------------------------------------\n-- 0 doesn't divide any number.\n0\u2224x : {d : D} \u2192 \u00ac (zero \u2223 d)\n0\u2224x ( 0\u22600 , _ ) = \u22a5-elim $ 0\u22600 refl\n\n------------------------------------------------------------------------------\n-- If 'x' divides 'y' and 'z' then 'x' divides 'y - z'.\npostulate\n  x\u2223y\u2192x\u2223z\u2192x\u2223y-z-ah : {m n p k\u2081 k\u2082 : D} \u2192 N m \u2192 N n \u2192 N k\u2081 \u2192 N k\u2082 \u2192\n                      n \u2261 k\u2081 * succ m \u2192\n                      p \u2261 k\u2082 * succ m \u2192\n                      n - p \u2261 (k\u2081 - k\u2082) * succ m\n{-# ATP prove x\u2223y\u2192x\u2223z\u2192x\u2223y-z-ah [x-y]z\u2261xz*yz sN #-}\n\nx\u2223y\u2192x\u2223z\u2192x\u2223y-z : {m n p : D} \u2192 N m \u2192 N n \u2192 N p \u2192 m \u2223 n \u2192 m \u2223 p \u2192 m \u2223 n - p\nx\u2223y\u2192x\u2223z\u2192x\u2223y-z zN _ _ ( 0\u22600 , _) m\u2223p = \u22a5-elim (0\u22600 refl)\nx\u2223y\u2192x\u2223z\u2192x\u2223y-z (sN Nm) Nn Np\n              ( 0\u22600 , ( k\u2081 , Nk\u2081 , n\u2261k\u2081Sm ))\n              ( _   , ( k\u2082 , Nk\u2082 , p\u2261k\u2082Sm )) =\n  (\u03bb S\u22610 \u2192 \u22a5-elim (\u00acS\u22610 S\u22610)) ,\n  (k\u2081 - k\u2082) ,\n  minus-N Nk\u2081 Nk\u2082 ,\n  x\u2223y\u2192x\u2223z\u2192x\u2223y-z-ah Nm Nn Nk\u2081 Nk\u2082 n\u2261k\u2081Sm p\u2261k\u2082Sm\n\n------------------------------------------------------------------------------\n-- If 'x' divides 'y' and 'z' then 'x' divides 'y + z'.\npostulate\n  x\u2223y\u2192x\u2223z\u2192x\u2223y+z-ah : {m n p k\u2081 k\u2082 : D} \u2192 N m \u2192 N n \u2192 N k\u2081 \u2192 N k\u2082 \u2192\n                      n \u2261 k\u2081 * succ m \u2192\n                      p \u2261 k\u2082 * succ m \u2192\n                      n + p \u2261 (k\u2081 + k\u2082) * succ m\n{-# ATP prove x\u2223y\u2192x\u2223z\u2192x\u2223y+z-ah [x+y]z\u2261xz*yz sN #-}\n\nx\u2223y\u2192x\u2223z\u2192x\u2223y+z : {m n p : D} \u2192 N m \u2192 N n \u2192 N p \u2192 m \u2223 n \u2192 m \u2223 p \u2192 m \u2223 n + p\nx\u2223y\u2192x\u2223z\u2192x\u2223y+z zN      _  _ ( 0\u22600 , _) m\u2223p = \u22a5-elim (0\u22600 refl)\nx\u2223y\u2192x\u2223z\u2192x\u2223y+z (sN Nm) Nn Np\n              ( 0\u22600 , ( k\u2081 , Nk\u2081 , n\u2261k\u2081Sm ))\n              ( _   , ( k\u2082 , Nk\u2082 , p\u2261k\u2082Sm )) =\n  (\u03bb S\u22610 \u2192 \u22a5-elim (\u00acS\u22610 S\u22610)) ,\n  (k\u2081 + k\u2082) ,\n  +-N Nk\u2081 Nk\u2082 ,\n  x\u2223y\u2192x\u2223z\u2192x\u2223y+z-ah Nm Nn Nk\u2081 Nk\u2082 n\u2261k\u2081Sm p\u2261k\u2082Sm\n\n------------------------------------------------------------------------------\n-- If x divides y, and y is positive, then x \u2264 y.\n\npostulate\n  x\u2223S\u2192x\u2264S-ah\u2081 : {m n : D} \u2192 succ n \u2261 zero * succ m \u2192 \u22a5\n-- {-# ATP prove x\u2223S\u2192x\u2264S-ah\u2081 *-leftZero sN #-}\n\n-- Nice proof by the ATP.\npostulate\n  x\u2223S\u2192x\u2264S-ah\u2082 : {m n k : D} \u2192 N m \u2192 N n \u2192 N k \u2192\n                succ n \u2261 succ k * succ m \u2192\n                LE (succ m) (succ n)\n{-# ATP prove x\u2223S\u2192x\u2264S-ah\u2082 x\u2264x+y *-N sN *-Sx #-}\n\nx\u2223S\u2192x\u2264S : {m n : D} \u2192 N m \u2192 N n \u2192 m \u2223 (succ n) \u2192 LE m (succ n)\nx\u2223S\u2192x\u2264S  zN     Nn ( 0\u22600 , _)                  = \u22a5-elim (0\u22600 refl)\nx\u2223S\u2192x\u2264S (sN Nm) Nn ( _ , .zero , zN , Sn\u22610*Sm) = \u22a5-elim (x\u2223S\u2192x\u2264S-ah\u2081 Sn\u22610*Sm)\nx\u2223S\u2192x\u2264S (sN {m} Nm) Nn ( _ , .(succ k) , sN {k} Nk , Sn\u2261Sk*Sm) =\n  x\u2223S\u2192x\u2264S-ah\u2082 Nm Nn Nk Sn\u2261Sk*Sm\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"68245d3724575719e4155604645cd403a368fe6b","subject":"Inline absNBase away and normalize resulting code","message":"Inline absNBase away and normalize resulting code\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Term.agda","new_file":"Parametric\/Syntax\/Term.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_])\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    drop-\u227c-l : \u2200 {\u0393 \u0393\u2032 \u03c4} \u2192 (\u03c4 \u2022 \u0393 \u227c \u0393\u2032) \u2192 \u0393 \u227c \u0393\u2032\n    drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081 = \u227c-trans (drop _ \u2022 \u227c-refl) \u0393\u2032\u227c\u0393\u2032\u2081\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN []        f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN (\u03c4\u2081 \u2237 \u03c4s) f =\n      abs (absN \u03c4s\n        (\u03bb {_} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f\n            {\u0393\u227c\u0393\u2032 = drop-\u227c-l \u0393\u2032\u227c\u0393\u2032\u2081}\n            (weaken \u0393\u2032\u227c\u0393\u2032\u2081 (var this))))\n\n    -- Using a similar trick, we can declare absV which takes the N implicit\n    -- type arguments individually, collects them and passes them on to absN.\n    -- This is inspired by what's shown in the Agda 2.4.0 release notes, and\n    -- relies critically on support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType 0       \u03c4s = absNType \u03c4s\n    absVType (suc n) \u03c4s = {\u03c4\u1d62 : Type} \u2192 absVType n (\u03c4\u1d62 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN \u03c4s\n    absVAux \u03c4s (suc n) = \u03bb {\u03c4\u1d62} \u2192 absVAux (\u03c4\u1d62 \u2237 \u03c4s) n\n\n    absV = absVAux []\n\n  open AbsNHelpers using (absV) public\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n\n  abs\u2081 = absV 1\n  abs\u2082 = absV 2\n  abs\u2083 = absV 3\n  abs\u2084 = absV 4\n  abs\u2085 = absV 5\n  abs\u2086 = absV 6\n\n  {-\n  Example types:\n  abs\u2081:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n        ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n        Term \u0393 (\u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n  abs\u2082:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n        ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n         Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n        Term \u0393 (\u03c4\u2082 Type.Structure.\u21d2 \u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n  -}\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of terms (Fig. 1a and 1b).\n------------------------------------------------------------------------\n\n-- The syntax of terms depends on the syntax of simple types\n-- (because terms are indexed by types in order to rule out\n-- ill-typed terms). But we are in the Parametric.* hierarchy, so\n-- we don't know the full syntax of types, only how to lift the\n-- syntax of base types into the syntax of simple types. This\n-- means that we have to be parametric in the syntax of base\n-- types, too.\n--\n-- In such parametric modules that depend on other parametric\n-- modules, we first import our dependencies under a more\n-- convenient name.\n\nimport Parametric.Syntax.Type as Type\n\n-- Then we start the module proper, with parameters for all\n-- extension points of our dependencies. Note that here, the\n-- \"Structure\" naming convenion makes some sense, because we can\n-- say that we need some \"Type.Structure\" in order to define the\n-- \"Term.Structure\".\n\nmodule Parametric.Syntax.Term\n    (Base : Type.Structure)\n  where\n\n-- Now inside the module, we can open our dependencies with the\n-- parameters for their extension points. Again, here the name\n-- \"Structure\" makes some sense, because we can say that we want\n-- to access the \"Type.Structure\" that is induced by Base.\n\nopen Type.Structure Base\n\n-- At this point, we have dealt with the extension points of our\n-- dependencies, and we have all the definitions about simple\n-- types, contexts, variables, and variable sets in scope that we\n-- provided in Parametric.Syntax.Type. Now we can proceed to\n-- define our own extension point, following the pattern\n-- explained in Parametric.Syntax.Type.\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\nopen import Function using (_\u2218_)\nopen import Data.Unit\nopen import Data.Sum\n\n-- Our extension point is a set of primitives, indexed by the\n-- types of their arguments and their return type. In general, if\n-- you're confused about what an extension point means, you might\n-- want to open the corresponding module in the Nehemiah\n-- hierarchy to see how it is implemented in the example\n-- plugin. In this case, that would be the Nehemiah.Syntax.Term\n-- module.\n\nStructure : Set\u2081\nStructure = Context \u2192 Type \u2192 Set\n\nmodule Structure (Const : Structure) where\n  import Base.Data.DependentList as DependentList\n  open DependentList public using (\u2205 ; _\u2022_)\n  open DependentList\n\n  -- Declarations of Term and Terms to enable mutual recursion.\n  --\n  -- Note that terms are indexed by contexts and types. In the\n  -- paper, we define the abstract syntax of terms in Fig 1a and\n  -- then define a type system in Fig 1b. All lemmas and theorems\n  -- then explicitly specify that they only hold for well-typed\n  -- terms. Here, we use the indices to define a type that can\n  -- only hold well-typed terms in the first place.\n  data Term\n    (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set\n\n  -- (Terms \u0393 \u03a3) represents a list of terms with types from \u03a3\n  -- with free variables bound in \u0393.\n  Terms : Context \u2192 Context \u2192 Set\n  Terms \u0393 = DependentList (Term \u0393)\n\n  -- (Term \u0393 \u03c4) represents a term of type \u03c4\n  -- with free variables bound in \u0393.\n  data Term \u0393 where\n    -- constants aka. primitives can only occur fully applied.\n    const : \u2200 {\u03a3 \u03c4} \u2192\n      (c : Const \u03a3 \u03c4) \u2192\n      (args : Terms \u0393 \u03a3) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Free variables\n  FV : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Vars \u0393\n  FV-terms : \u2200 {\u03a3 \u0393} \u2192 Terms \u0393 \u03a3 \u2192 Vars \u0393\n\n  FV (const \u03b9 ts) = FV-terms ts\n  FV (var x) = singleton x\n  FV (abs t) = tail (FV t)\n  FV (app s t) = FV s \u222a FV t\n\n  FV-terms \u2205 = none\n  FV-terms (t \u2022 ts) = FV t \u222a FV-terms ts\n\n  closed? : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) \u2192 (FV t \u2261 none) \u228e \u22a4\n  closed? t = empty? (FV t)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n\n  weaken-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Terms \u0393\u2081 \u03a3 \u2192\n    Terms \u0393\u2082 \u03a3\n\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c ts) = const c (weaken-terms \u0393\u2081\u227c\u0393\u2082 ts)\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 \u2205 = \u2205\n  weaken-terms \u0393\u2081\u227c\u0393\u2082 (t \u2022 ts) = weaken \u0393\u2081\u227c\u0393\u2082 t \u2022 weaken-terms \u0393\u2081\u227c\u0393\u2082 ts\n\n  -- Specialized weakening\n  weaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\n  weaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\n  weaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\n  weaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  weaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n    Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\n  weaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\n  UncurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  UncurriedTermConstructor \u0393 \u03a3 \u03c4 = Terms \u0393 \u03a3 \u2192 Term \u0393 \u03c4\n\n  uncurriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4\n  uncurriedConst constant = const constant\n\n  CurriedTermConstructor : (\u0393 \u03a3 : Context) (\u03c4 : Type) \u2192 Set\n  CurriedTermConstructor \u0393 \u2205 \u03c4\u2032 = Term \u0393 \u03c4\u2032\n  CurriedTermConstructor \u0393 (\u03c4 \u2022 \u03a3) \u03c4\u2032 = Term \u0393 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\u2032\n\n  curryTermConstructor : \u2200 {\u03a3 \u0393 \u03c4} \u2192 UncurriedTermConstructor \u0393 \u03a3 \u03c4 \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curryTermConstructor {\u2205} k = k \u2205\n  curryTermConstructor {\u03c4 \u2022 \u03a3} k = \u03bb t \u2192 curryTermConstructor (\u03bb ts \u2192 k (t \u2022 ts))\n\n  curriedConst : \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u2200 {\u0393} \u2192 CurriedTermConstructor \u0393 \u03a3 \u03c4\n  curriedConst constant = curryTermConstructor (uncurriedConst constant)\n\n\n  -- HOAS-like smart constructors for lambdas, for different arities.\n\n  -- We could also write this:\n  module NamespaceForBadAbs\u2081 where\n    abs\u2081\u2032 :\n      \u2200 {\u0393 \u03c4\u2081 \u03c4} \u2192\n        (Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2081 \u2192 Term (\u03c4\u2081 \u2022 \u0393) \u03c4) \u2192\n        (Term \u0393 (\u03c4\u2081 \u21d2 \u03c4))\n    abs\u2081\u2032 {\u0393} {\u03c4\u2081} = \u03bb f \u2192 abs (f (var this))\n\n  -- However, this is less general, and it is harder to reuse. In particular,\n  -- this cannot be used inside abs\u2082, ..., abs\u2086.\n\n  -- Now, let's write other variants with a loop!\n  open import Data.Vec using (_\u2237_; []; Vec; foldr; [_])\n  open import Data.Nat\n  module AbsNHelpers where\n    open import Function\n    hoasArgType : \u2200 {n} \u2192 Context \u2192 Type \u2192 Vec Type n \u2192 Set\n    hoasArgType \u0393 \u03c4 = foldr _ (\u03bb a b \u2192 a \u2192 b) (Term \u0393 \u03c4) \u2218 Data.Vec.map (Term \u0393)\n    -- That is,\n    --hoasArgType \u0393 \u03c4 [] = Term \u0393 \u03c4\n    --hoasArgType \u0393 \u03c4 (\u03c4\u2080 \u2237 \u03c4s) = Term \u0393 \u03c4\u2080 \u2192 hoasArgType \u0393 \u03c4 \u03c4s\n\n    hoasResType : \u2200 {n} \u2192 Type \u2192 Vec Type n \u2192 Type\n    hoasResType \u03c4 = foldr _ _\u21d2_ \u03c4\n\n    absNType : {n : \u2115} \u2192 Vec _ n \u2192 Set\n    absNType \u03c4s = \u2200 {\u0393 \u03c4} \u2192\n      (f : \u2200 {\u0393\u2032} \u2192 {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 hoasArgType \u0393\u2032 \u03c4 \u03c4s) \u2192\n      Term \u0393 (hoasResType \u03c4 \u03c4s)\n\n    -- A better type for absN but a mess to use due to the proofs (which aren't synthesized, even though maybe they should be?)\n    -- absN : (n : \u2115) \u2192 {_ : n > 0} \u2192 absNType n\n    -- XXX See \"how to keep your neighbours in order\" for tricks.\n\n    -- Please the termination checker by keeping this case separate.\n    absNBase : \u2200 {\u03c4\u2081} \u2192 absNType [ \u03c4\u2081 ]\n    absNBase {\u03c4\u2081} f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    -- Otherwise, the recursive step of absN would invoke absN twice, and the\n    -- termination checker does not figure out that the calls are in fact\n    -- terminating.\n\n    -- What I'd like to write, avoiding the need for absNBase, but can't because of the termination checker.\n    {-\n    absN {zero}  (\u03c4\u2081 \u2237 []) f = abs (f {\u0393\u227c\u0393\u2032 = drop \u03c4\u2081 \u2022 \u227c-refl} (var this))\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4s) f =\n      absN (\u03c4\u2081 \u2237 []) (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4\u2082 \u2237 \u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n    -}\n    --What I have to write instead:\n\n    absN : {n : \u2115} \u2192 (\u03c4s : Vec _ n) \u2192 absNType \u03c4s\n    absN {zero}  [] f = f {\u0393\u227c\u0393\u2032 = \u227c-refl}\n    absN {suc n} (\u03c4\u2081 \u2237 \u03c4s) f =\n      absNBase (\u03bb {_} {\u0393\u227c\u0393\u2032} x\u2081 \u2192\n        absN {n} (\u03c4s) (\u03bb {\u0393\u2032\u2081} {\u0393\u2032\u227c\u0393\u2032\u2081} \u2192\n          f {\u0393\u227c\u0393\u2032 = \u227c-trans \u0393\u227c\u0393\u2032 \u0393\u2032\u227c\u0393\u2032\u2081} (weaken \u0393\u2032\u227c\u0393\u2032\u2081 x\u2081)))\n\n    -- Using a similar trick, we can declare absV which takes the N implicit\n    -- type arguments individually, collects them and passes them on to absN.\n    -- This is inspired by what's shown in the Agda 2.4.0 release notes, and\n    -- relies critically on support for varying arity. To collect them, we need\n    -- to use an accumulator argument.\n\n    absVType : \u2200 n {m} (\u03c4s : Vec Type m) \u2192 Set\n    absVType 0       \u03c4s = absNType \u03c4s\n    absVType (suc n) \u03c4s = {\u03c4\u1d62 : Type} \u2192 absVType n (\u03c4\u1d62 \u2237 \u03c4s)\n\n    absVAux : \u2200 {m} \u2192 (\u03c4s : Vec Type m) \u2192 \u2200 n \u2192 absVType n \u03c4s\n    absVAux \u03c4s zero    = absN \u03c4s\n    absVAux \u03c4s (suc n) = \u03bb {\u03c4\u1d62} \u2192 absVAux (\u03c4\u1d62 \u2237 \u03c4s) n\n\n    absV = absVAux []\n\n  open AbsNHelpers using (absV) public\n\n  -- Declare abs\u2081 .. abs\u2086 wrappers for more convenient use, allowing implicit\n  -- type arguments to be synthesized. Somehow, Agda does not manage to\n  -- synthesize \u03c4s by unification.\n  -- Implicit arguments are reversed when assembling the list, but that's no real problem.\n\n  abs\u2081 = absV 1\n  abs\u2082 = absV 2\n  abs\u2083 = absV 3\n  abs\u2084 = absV 4\n  abs\u2085 = absV 5\n  abs\u2086 = absV 6\n\n  {-\n  Example types:\n  abs\u2081:\n    {\u03c4\u2081 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n        ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n        Term \u0393 (\u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n  abs\u2082:\n    {\u03c4\u2081 : Type} {\u03c4\u2081 = \u03c4\u2082 : Type} {\u0393 : Context} {\u03c4 : Type} \u2192\n        ({\u0393\u2032 : Context} {\u0393\u227c\u0393\u2032 : \u0393 \u227c \u0393\u2032} \u2192\n         Term \u0393\u2032 \u03c4\u2082 \u2192 Term \u0393\u2032 \u03c4\u2081 \u2192 Term \u0393\u2032 \u03c4) \u2192\n        Term \u0393 (\u03c4\u2082 Type.Structure.\u21d2 \u03c4\u2081 Type.Structure.\u21d2 \u03c4)\n\n  -}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"10e9de946bdf22fa97db8506d27df3087e8a5152","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val n \u27e7     =  return n\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\nl ->\n                   \u27e6 er \u27e7 >>= \\nr ->\n                   nl \u00f7 nr\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val _)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val _) _ = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Znr = magic (ernz er\u21d3Znr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3nl           | er\u21d3Snr = \u21d3Step el\u21d3nl er\u21d3Snr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val _) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {n : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure n\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n : (e : Expr) (n : Nat)\n               -> \u27e6 e \u27e7 \u2261 Pure n\n               -> e \u21d3 n\n  \u27e6e\u27e7\u2261Puren\u2192e\u21d3n e _ eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = \u27e6e\u27e7\u2261Puren\u2192e\u21d3n er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val _) _ = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3vl           | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val x) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {v : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure v\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = aux el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = aux er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"d9a198218dd67187b3ede89ab3d63ac082c4ee41","subject":"Param: Take advantage of the latest dev version of Agda works better","message":"Param: Take advantage of the latest dev version of Agda works better\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Param.agda","new_file":"lib\/Reflection\/Param.agda","new_contents":"open import Function\nopen import Opaque\nopen import Type\nopen import Data.Zero\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Two.Logical\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.Fin using (Fin; zero; suc; to\u2115)\nopen import Data.Vec using (Vec; []; _\u2237_; replicate; tabulate; allFin; reverse; _\u229b_; toList) renaming (map to vmap)\nopen import Data.Nat.Logical hiding (_\u225f_) renaming (zero to \u27e6zero\u27e7; suc to \u27e6suc\u27e7)\nopen import Data.List hiding (replicate; reverse)\nopen import Data.Sum.NP\nopen import Reflection.NP\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen import Relation.Nullary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Param where\n\nmodule Revelator (tyH : Type) where\n    tm : Term \u2192 \u2115 \u2192 Args \u2192 Term\n    tm (pi (arg (arg-info v _) t\u2081) (el _ t\u2082)) y as = lam visible (tm t\u2082 (suc y) (mapVarArgs suc as ++ arg\u02b3 v (var 0 []) \u2237 []))\n    tm (var x args) = var\n    tm (def f args) = var\n    tm (sort s)     = var\n    tm unknown _ _ = unknown\n    tm (con c args) _ _ = opaque \"revealator\/tm\/con: impossible\" unknown\n    tm (lam v ty) _ _ = opaque \"revealator\/tm\/lam: impossible\" unknown\n    tm (lit l) _ _ = opaque \"revealator\/tm\/lit: impossible\" unknown\n    tm (pat-lam cs args) _ _ = opaque \"revealator\/tm\/pat-lam: TODO\" unknown\n    t\u00ffpe : Type\n    t\u00ffpe = tyH `\u2192\u1d5b\u02b3 Reveal-args.t\u00ffpe tyH\n    term : Term\n    term = lam visible (tm (unEl tyH) 0 [])\n    fun : FunctionDef\n    fun = fun-def t\u00ffpe (clause (arg\u1d5b\u02b3 var \u2237 []) (tm (unEl tyH) 0 []) \u2237 [])\n\ndata VarKind (n : \u2115) : Set where\n  K\u1d62 : Fin n \u2192 VarKind n\n  K\u1d63 : VarKind n\n\nrecord Env (n : \u2115)(A B : Set) : Set where\n  field\n    pVar : VarKind n \u2192 A \u2192 B\n    pCon : Name \u2192 Name\n    pDef : Name \u2192 Name\nopen Env\n\nEnv' = \u03bb n \u2192 Env n \u2115 \u2115\n\n\u03b5 : \u2200 n \u2192 Env' n\n\u03b5 n = record { pVar = \u03bb _ _ \u2192 opaque \"\u03b5.pVar\" 0\n             ; pCon = const (opaque \"\u03b5.pCon\" (quote \u03b5))\n             ; pDef = const (opaque \"\u03b5.pDef\" (quote \u03b5)) }\n\nextDefEnv : ((Name \u2192 Name) \u2192 (Name \u2192 Name)) \u2192 \u2200 {n A B} \u2192 Env n A B \u2192 Env n A B\nextDefEnv ext \u0393 = record \u0393 { pDef = ext (pDef \u0393) }\n\next1DefEnv : (old new : Name) \u2192 \u2200 {n A B} \u2192 Env n A B \u2192 Env n A B\next1DefEnv old new = extDefEnv (\u03bb f x \u2192 [yes: (\u03bb _ \u2192 new) no: (\u03bb _ \u2192 f x) ]\u2032 (x \u225f-Name old))\n\n\u2191pVar : \u2115 \u2192 (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u2191pVar zero = id\n\u2191pVar (suc n) = \u2191pVar n \u2218 mapVar\u2191\n\non-pVar : \u2200 {n A B C D} \u2192 (VarKind n \u2192 (A \u2192 B) \u2192 (C \u2192 D)) \u2192 Env n A B \u2192 Env n C D\non-pVar f \u0393 = record\n  { pVar = \u03bb k \u2192 f k (pVar \u0393 k)\n  ; pCon = pCon \u0393\n  ; pDef = pDef \u0393 }\n\n_+\u2191 : \u2200 {n} \u2192 Env' n \u2192 Env' n\n_+\u2191 {n} = on-pVar goK\n  where\n    goK : VarKind n \u2192 (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n    goK (K\u1d62 x) f = _+_ (n \u2238 to\u2115 x) \u2218 \u2191pVar 1 (_+_ (to\u2115 x) \u2218 f)\n    goK K\u1d63     f = \u2191pVar 1 (_+_ n \u2218 f)\n\n_+'_ : \u2200 {w} \u2192 Env' w \u2192 \u2115 \u2192 Env' w\n\u0393 +' n = record { pVar = \u03bb k \u2192 _+_ n \u2218 pVar \u0393 k ; pCon = pCon \u0393 ; pDef = pDef \u0393 }\n\n_+1 : \u2200 {w} \u2192 Env' w \u2192 Env' w\n\u0393 +1 = \u0393 +' 1\n\npattern `\u27e6zero\u27e7  = con (quote \u27e6\u2115\u27e7.zero) []\npattern `\u27e6suc\u27e7 t = con (quote \u27e6\u2115\u27e7.suc) (arg\u1d5b\u02b3 t \u2237 [])\npNat : \u2115 \u2192 Term\npNat zero    = `\u27e6zero\u27e7\npNat (suc n) = `\u27e6suc\u27e7 (pNat n)\n\nlam^ : \u2115 \u2192 Visibility \u2192 Term \u2192 Term\nlam^ zero    v x = x\nlam^ (suc n) v x = lam v (lam^ n v x)\n\nlam\u2208 : \u2200 {n} \u2192 Env' n \u2192 (f : Env' n \u2192 Vec Term n \u2192 Term) \u2192 Term\nlam\u2208 {n} \u0393 f = lam^ n visible (f (\u0393 +' n) (reverse (tabulate (\u03bb x \u2192 var (to\u2115 x) []))))\n\napp-tabulate : \u2200 {n a} {A : Set a} \u2192 (Fin n \u2192 A) \u2192 List A \u2192 List A\napp-tabulate {zero}  f xs = xs\napp-tabulate {suc n} f xs = f zero \u2237 app-tabulate (f \u2218 suc) xs\n\np^args : \u2200 {n A} \u2192 Relevance \u2192 (Fin n \u2192 A) \u2192 A \u2192 List (Arg A) \u2192 List (Arg A)\np^args r f x args = app-tabulate (\u03bb k \u2192 arg\u02b0 r (f k)) (arg\u1d5b\u02b3 x \u2237 args)\n\np^lam : \u2115 \u2192 Term \u2192 Term\np^lam n t = lam^ n hidden (lam visible t)\n\n{-\npi^ : \u2115 \u2192 Arg Type \u2192 Type \u2192 Type\npi^ zero    i t = t\npi^ (suc n) i t = `\u03a0 i (pi^ n i t)\n-}\n\nallVarsFrom : \u2200 n \u2192 \u2115 \u2192 Vec Term n\nallVarsFrom zero    k = []\nallVarsFrom (suc n) k = var (k + n) [] \u2237 allVarsFrom n k\n\np3pi\u2208 : \u2200 {n} (\u0393 : Env' n) (r : Relevance)\n          (s : Type)\n          (t : Env' n \u2192 Vec Term n \u2192 Type)\n          (u : Env' n \u2192 Vec Term n \u2192 Type) \u2192 Term\np3pi\u2208 {n} \u0393 r s t u = unEl (go \u0393 (allFin n))\n  where\n  go : \u2200 {m} \u2192 Env' n \u2192 Vec (Fin n) m \u2192 Type\n  go \u0394 [] = \n      `\u03a0\u1d5b\u02b3 (t \u0394 (allVarsFrom n 0)) (u (\u0393 +\u2191) (allVarsFrom n 1))\n  go \u0394 (x \u2237 xs) =\n      `\u03a0 (arg\u02b0 r (mapVarType (pVar \u0394 (K\u1d62 x)) s))\n         (go (\u0394 +1) xs)\n\npSort\u2208 : \u2200 {n} \u2192 Sort \u2192 Vec Term n \u2192 Type\npSort\u2208 s = go 0\n  where\n    go : \u2115 \u2192 \u2200 {n} \u2192 Vec Term n \u2192 Type\n    go k []       = el (suc-sort s) (sort s)\n    go k (t \u2237 ts) = `\u03a0\u1d5b\u02b3 (mapVarType (_+_ k) (el s t)) (go (suc k) ts)\n\npEnvPats : List (Arg Pattern) \u2192 \u2200 {n} \u2192 Env' n \u2192 Env' n\npEnvPat : Pattern \u2192 \u2200 {n} \u2192 Env' n \u2192 Env' n\n\npEnvPat (con _ pats) = pEnvPats pats\npEnvPat dot = id\npEnvPat var = _+\u2191\npEnvPat (lit _) = id\npEnvPat (proj _) = opaque \"pEnvPats\/proj\" id\npEnvPat absurd = id\n\npEnvPats [] = id\npEnvPats (arg i p \u2237 ps) = pEnvPat p \u2218 pEnvPats ps\n\npPats : \u2200 {n} \u2192 Env' n \u2192 List (Arg Pattern) \u2192 List (Arg Pattern)\npPat  : \u2200 {n} \u2192 Env' n \u2192 Arg Pattern \u2192 List (Arg Pattern)\n\npPats \u0393 [] = []\npPats \u0393 (pat \u2237 ps) = pPat \u0393 pat ++ pPats \u0393 ps\n\n--nodot = var\n\npPat {n} \u0393 (arg (arg-info _ r) (con c pats))\n  = p^args {n} r (const dot) (con (pCon \u0393 c) (pPats \u0393 pats)) []\npPat {n} \u0393 (arg (arg-info _ r) dot)\n  = p^args {n} r (const dot) dot []\npPat {n} \u0393 (arg (arg-info _ r) var)\n  = p^args {n} r (const var) var []\npPat {n} \u0393 (arg i (lit (nat l))) = opaque \"pPat\/lit\/nat\" []\npPat {n} \u0393 (arg (arg-info _ r) (lit l))\n  = p^args {n} r go\n                 (con (quote refl) []) []\n      where\n        go : \u2200 {n} \u2192 Fin n \u2192 Pattern\n        go zero = lit l\n        go (suc _) = dot\npPat {n} \u0393 (arg (arg-info _ r) absurd) = p^args {n} r (const var) absurd []\npPat {n} \u0393 (arg i (proj p)) = opaque \"pPat\/proj\" []\n\npLit : Literal \u2192 Term\npLit (nat n) = pNat n\npLit _       = con (quote refl) []\n\npTerm  : \u2200 {n} \u2192 Env' n \u2192 Term \u2192 Term\npArgs  : \u2200 {n} \u2192 Env' n \u2192 Args \u2192 Args\npType\u2208 : \u2200 {n} \u2192 Env' n \u2192 Type             \u2192 Vec Term n \u2192 Type\npTerm\u2208 : \u2200 {n} \u2192 Env' n \u2192 Term             \u2192 Vec Term n \u2192 Term\npPi\u2208   : \u2200 {n} \u2192 Env' n \u2192 Arg Type \u2192 Type \u2192 Vec Term n \u2192 Term\n\npType\u2208 \u0393 (el s t) as = el s (pTerm\u2208 \u0393 t as)\n\npTerm {n} \u0393 (lam _ t) = p^lam n (pTerm (\u0393 +\u2191) t)\npTerm \u0393 (var v args)  = var (pVar \u0393 K\u1d63 v) (pArgs \u0393 args)\npTerm \u0393 (con c args)  = con (pCon \u0393 c) (pArgs \u0393 args)\npTerm \u0393 (def d args)  = def (pDef \u0393 d) (pArgs \u0393 args)\npTerm \u0393 (lit l)       = pLit l\npTerm \u0393 (pat-lam _ _) = opaque \"pTerm\/pat-lam\" unknown\npTerm \u0393 unknown       = unknown\n\npTerm \u0393 (sort s) = lam\u2208 \u0393 \u03bb _ as \u2192 unEl (pSort\u2208 s as)\npTerm \u0393 (pi s t) = lam\u2208 \u0393 \u03bb \u0394 \u2192 pPi\u2208 \u0394 s t\n\npPi\u2208 {n} \u0393 (arg (arg-info _ r) s) t as =\n  p3pi\u2208 \u0393 r s (\u03bb \u0394 \u2192 pType\u2208 \u0394 s) (\u03bb \u0394 vs \u2192 pType\u2208 \u0394 t\n    (vmap (\u03bb a v \u2192 app (mapVar (_+_ (suc n)) a) (arg\u1d5b\u02b3 v \u2237 [])) as \u229b vs))\n\npTerm\u2208 \u0393 (sort s) as = unEl (pSort\u2208 s as)\npTerm\u2208 \u0393 (pi s t) = pPi\u2208 \u0393 s t\npTerm\u2208 \u0393 t as = app (pTerm \u0393 t) (toList (vmap arg\u1d5b\u02b3 as))\n\npArgs \u0393 [] = []\npArgs \u0393 (arg (arg-info _ r) t \u2237 as)\n  = p^args r (\u03bb k \u2192 mapVar (pVar (\u0393 +\u2191) (K\u1d62 k)) t) (pTerm \u0393 t) (pArgs \u0393 as)\n\npClause  : \u2200 {n} \u2192 Env' n \u2192 Clause \u2192 Clause\npClauses : \u2200 {n} \u2192 Env' n \u2192 Clauses \u2192 Clauses\n\npClause \u0393 (clause pats body) = clause (pPats \u0393 pats) (pTerm (pEnvPats pats \u0393) body)\npClause \u0393 (absurd-clause pats) = absurd-clause (pPats \u0393 pats)\n\npClauses \u0393 [] = []\npClauses \u0393 (c \u2237 cs) = pClause \u0393 c \u2237 pClauses \u0393 cs\n\npFunDef : \u2200 {n} \u2192 Env' n \u2192 Name \u2192 FunctionDef \u2192 FunctionDef\npFunDef \u0393 d (fun-def t cs)\n  = fun-def (pType\u2208 \u0393 t (replicate (def d []))) (pClauses \u0393 cs)\n\npostulate\n  IMPOSSIBLE : FunctionDef\n\npFunName : \u2200 {n} \u2192 Env' n \u2192 Name \u2192 FunctionDef\npFunName \u0393 x with definition x\n... | function d = pFunDef \u0393 x d\n... | _ = IMPOSSIBLE -- opaque \"pFunName\" (fun-def {!!} {!!})\n\npFunNameRec : \u2200 {n} \u2192 Env' n \u2192 (x x\u209a : Name) \u2192 FunctionDef\npFunNameRec \u0393 x x\u209a = pFunName (ext1DefEnv x x\u209a \u0393) x\n\n{-\npDefinition : Env' 2 \u2192 Name \u2192 Definition \u2192 Definition\npDefinition \u0393 n (function x) = function {!pFunDef!}\npDefinition \u0393 n (data-type x) = {!!}\npDefinition \u0393 n (record\u2032 x) = {!!}\npDefinition \u0393 n constructor\u2032 = {!!}\npDefinition \u0393 n axiom = {!!}\npDefinition \u0393 n primitive\u2032 = {!!}\n\npName : Env' 2 \u2192 Name \u2192 Definition\npName \u0393 n = pDefinition \u0393 n (definition n)\n-}\n\nopen import Data.String.Core using (String)\nopen import Data.Float       using (Float)\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n\np[Set\u2080] = pFunName (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl (Get-type.from-fun-def p[Set\u2080]))\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261 Get-term.from-fun-def p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nunquoteDecl u-p[Set\u2080] = p[Set\u2080]\n\n{-\np-[\u2192]-type = unEl (Get-type.from-fun-def p-[\u2192])\np-[\u2192]' : \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n                {A\u209a : A \u2192 Set\u2080} (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 Set\u2080)\n                {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n                {B\u209a : B \u2192 Set\u2080} (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 Set\u2080)\n                {f : A \u2192 B}    (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n              \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f \u2192\n              {C : Set\u2080} \u2192 (C \u2192 Set\u2080) \u2192 {t : {!!}} \u2192 (A \u2192 B) \u2192 Set\u2080 -- _[\u2080\u2192\u2080]_ {A} {-A\u2080\u209a} {A\u209a-} {-A\u2081\u209a-} {!!} {B} {!!} {-B\u2080\u209a} B\u2081\u209a-} f -- f\u209a\np-[\u2192]' = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n-}\n\n[String] : [\u2605\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [\u2605\u2080] Float\n[Float] _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6\u2605\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6\u2605\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\nunquoteDecl revealed-[\u2192]' = un-function revealed-[\u2192]\n\nunquoteDecl revelator-[\u2192] = Revelator.fun (type (quote _[\u2080\u2192\u2080]_))\n\n{-\np-[\u2192] = pFunName (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]-type = unEl (Get-type.from-fun-def p-[\u2192])\n\n{-\np-[\u2192]' : (x0 : Set) \u2192 (x1 : (x1 : x0) \u2192 Set) \u2192 (x2 : (x2 : x0) \u2192 Set) \u2192 (x3 : (x3 : x0) \u2192 (x4 : x1 x3) \u2192 (x5 : x2 x3) \u2192 Set) \u2192 (x4 : Set) \u2192 (x5 : (x5 : x4) \u2192 Set) \u2192 (x6 : (x6 : x4) \u2192 Set) \u2192 (x7 : (x7 : x4) \u2192 (x8 : x5 x7) \u2192 (x9 : x6 x7) \u2192 Set) \u2192 (x8 : (x8 : x0) \u2192 x4) \u2192 (x9 : (x9 : x0) \u2192 (x10 : x1 x9) \u2192 x5 (x8 x9)) \u2192 (x10 : _[\u2080\u2192\u2080]_ {x0} x2 {x4} x6 x8) \u2192 Set\n{-\np-[\u2192]' : \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n                {A\u209a : A \u2192 Set\u2080} (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 Set\u2080)\n                {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n                {B\u209a : B \u2192 Set\u2080} (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 Set\u2080)\n                {f : A \u2192 B}    (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n              \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f \u2192\n              {C : Set\u2080} \u2192 (C \u2192 Set\u2080) \u2192 {t : {!!}} \u2192 (A \u2192 B) \u2192 Set\u2080 -- _[\u2080\u2192\u2080]_ {A} {-A\u2080\u209a} {A\u209a-} {-A\u2081\u209a-} {!!} {B} {!!} {-B\u2080\u209a} B\u2081\u209a-} f -- f\u209a\n-}\np-[\u2192]' = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n\ntest : {!showTerm p-[\u2192]-type!}\ntest = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n-}\n-- unquoteDecl _[[\u2192]]_ = p-[\u2192]\n\n{-\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : ([[\u2115]] [[\u2192]] [[\u2115]]) [suc] [suc]\n-}\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda) = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115) = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float) = quote [Float]\ndefDefEnv1 (quote \u2605\u2080) = quote [\u2605\u2080]\ndefDefEnv1 (quote \u2605\u2081) = quote [\u2605\u2081]\ndefDefEnv1 n         = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082) = quote [0\u2082]\ndefConEnv1 (quote 1\u2082) = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero) = quote [zero]\ndefConEnv1 (quote \u2115.suc)  = quote [suc]\ndefConEnv1 n              = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda) = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115) = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float) = quote \u27e6Float\u27e7\ndefDefEnv2 (quote \u2605\u2080) = quote \u27e6\u2605\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081) = quote \u27e6\u2605\u2081\u27e7\ndefDefEnv2 n         = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082) = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082) = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero) = quote \u27e6\u2115\u27e7.zero\ndefConEnv2 (quote \u2115.suc)  = quote \u27e6\u2115\u27e7.suc\ndefConEnv2 n              = opaque \"defConEnv2\" n\n\ndefEnv0 : Env' 0\ndefEnv0 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv1.pVar\" 0\n                 ; pCon = id\n                 ; pDef = id }\n\ndefEnv1 : Env' 1\ndefEnv1 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv1.pVar\" 0\n                 ; pCon = defConEnv1\n                 ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv2.pVar\" 0\n                 ; pCon = defConEnv2\n                 ; pDef = defDefEnv2 }\n\n{-\n_\u27e6+\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) _+_ _+_\nzero  \u27e6+\u27e7 n = n\nsuc m \u27e6+\u27e7 n = suc (m \u27e6+\u27e7 n)\n-}\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7         \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261 quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7-type = unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7)))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : \u27e6\u27e6Set\u2080\u27e7\u27e7-type \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = pFunName defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261 quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261 quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261 Get-term.from-fun-def p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\nunquoteDecl \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' : _\u2261_ {A = \u27e6\u27e6Set\u2080\u27e7\u27e7-type} \u27e6\u27e6Set\u2080\u27e7\u27e7'' \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7'' = refl\n\ntest-p0-\u27e6Set\u2080\u27e7 : pTerm defEnv0 (quoteTerm \u27e6Set\u2080\u27e7) \u2261 quoteTerm \u27e6Set\u2080\u27e7\ntest-p0-\u27e6Set\u2080\u27e7 = refl\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\n_\u2261clauses_ = \u03bb x y \u2192 Get-clauses.from-def x \u2261 Get-clauses.from-def y\n_\u2261type_ = \u03bb x y \u2192 Get-type.from-def x \u2261 Get-type.from-def y\n  \nmodule Test where\n  p1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n  [\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\n  test-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\n  test-p1\u2115\u2192\u2115 = refl\n\n  p2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n  \u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n  test-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\n  test-p2\u2115\u2192\u2115 = refl\n\n  p\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n  \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n  test-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261 quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\n  test-p\u2115\u2192\u2115\u2192\u2115 = refl\n  ZERO : Set\u2081\n  ZERO = (A : Set\u2080) \u2192 A\n  \u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n  \u27e6ZERO\u27e7 f\u2080 f\u2081 =\n    {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n    \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\n  pZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\n  q\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\n  test-pZERO : pZERO \u2261 q\u27e6ZERO\u27e7\n  test-pZERO = refl\n  ID : Set\u2081\n  ID = (A : Set\u2080) \u2192 A \u2192 A\n  \u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n  \u27e6ID\u27e7 f\u2080 f\u2081 =\n    {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n    {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n    \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\n  pID = pTerm (\u03b5 2) (quoteTerm ID)\n  q\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\n  test-param : absTerm q\u27e6ID\u27e7 \u2261 absTerm pID\n  test-param = refl\n  test-ID : q\u27e6ID\u27e7 \u2261 pID\n  test-ID = refl\n\n  p-not = pFunName defEnv2 (quote not)\n  unquoteDecl P-not = p-not\n  test-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 P-not x\u1d63\n  test-not \u27e60\u2082\u27e7 = refl\n  test-not \u27e61\u2082\u27e7 = refl\n\n  p1-pred = pFunName defEnv1 (quote pred)\n  q-[pred] = quoteTerm [pred]\n  unquoteDecl [pred]' = p1-pred\n  test-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\n  test-p1-pred [zero] = refl\n  test-p1-pred ([suc] n\u209a) = refl\n\n  p2-pred = pFunName defEnv2 (quote pred)\n  q-\u27e6pred\u27e7 = definition (quote \u27e6pred\u27e7)\n  unquoteDecl \u27e6pred\u27e7' = p2-pred\n  test-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\n  test-p2-pred \u27e6zero\u27e7 = refl\n  test-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\n  p\/2 = pFunNameRec defEnv2 (quote _\/2)\n  q\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\n  unquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\n  test-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261 q\u27e6\/2\u27e7\n  test-\/2 = refl\n  test-\/2' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\n  test-\/2' \u27e6zero\u27e7 = refl\n  test-\/2' (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\n  test-\/2' (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) = ap \u27e6suc\u27e7 (test-\/2' n\u1d63)\n\n  p+ = pFunNameRec defEnv2 (quote _+\u2115_)\n  q\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\n  unquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\n  test-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261 q\u27e6+\u27e7\n  test-+ = refl\n  test-+' : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\n  test-+' \u27e6zero\u27e7    n'\u1d63 = refl\n  test-+' (\u27e6suc\u27e7 n\u1d63) n'\u1d63 = ap \u27e6suc\u27e7 (test-+' n\u1d63 n'\u1d63)\n\n  {-\n  is-good : String \u2192 \ud835\udfda\n  is-good \"good\" = 1\u2082\n  is-good _      = 0\u2082\n\n  \u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n  \u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n  \u27e6is-good\u27e7 {_}      refl = {!!}\n\n  p-is-good = pFunName defEnv2 (quote is-good)\n  unquoteDecl \u27e6is-good\u27e7' = p-is-good\n  test-is-good = {!!}\n  -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Function\nopen import Opaque\nopen import Type\nopen import Data.Zero\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Two.Logical\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.Fin using (Fin; zero; suc; to\u2115)\nopen import Data.Vec using (Vec; []; _\u2237_; replicate; tabulate; allFin; reverse; _\u229b_; toList) renaming (map to vmap)\nopen import Data.Nat.Logical hiding (_\u225f_) renaming (zero to \u27e6zero\u27e7; suc to \u27e6suc\u27e7)\nopen import Data.List hiding (replicate; reverse)\nopen import Data.Sum.NP\nopen import Reflection.NP\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen import Relation.Nullary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Param where\n\nmodule Revelator (tyH : Type) where\n    tm : Term \u2192 \u2115 \u2192 Args \u2192 Term\n    tm (pi (arg (arg-info v _) t\u2081) (el _ t\u2082)) y as = lam visible (tm t\u2082 (suc y) (mapVarArgs suc as ++ arg\u02b3 v (var 0 []) \u2237 []))\n    tm (var x args) = var\n    tm (def f args) = var\n    tm (sort s)     = var\n    tm unknown _ _ = unknown\n    tm (con c args) _ _ = opaque \"revealator\/tm\/con: impossible\" unknown\n    tm (lam v ty) _ _ = opaque \"revealator\/tm\/lam: impossible\" unknown\n    tm (lit l) _ _ = opaque \"revealator\/tm\/lit: impossible\" unknown\n    tm (pat-lam cs args) _ _ = opaque \"revealator\/tm\/pat-lam: TODO\" unknown\n    t\u00ffpe : Type\n    t\u00ffpe = tyH `\u2192\u1d5b\u02b3 Reveal-args.t\u00ffpe tyH\n    term : Term\n    term = lam visible (tm (unEl tyH) 0 [])\n    fun : FunctionDef\n    fun = fun-def t\u00ffpe (clause (arg\u1d5b\u02b3 var \u2237 []) (tm (unEl tyH) 0 []) \u2237 [])\n\ndata VarKind (n : \u2115) : Set where\n  K\u1d62 : Fin n \u2192 VarKind n\n  K\u1d63 : VarKind n\n\nrecord Env (n : \u2115)(A B : Set) : Set where\n  field\n    pVar : VarKind n \u2192 A \u2192 B\n    pCon : Name \u2192 Name\n    pDef : Name \u2192 Name\nopen Env\n\nEnv' = \u03bb n \u2192 Env n \u2115 \u2115\n\n\u03b5 : \u2200 n \u2192 Env' n\n\u03b5 n = record { pVar = \u03bb _ _ \u2192 opaque \"\u03b5.pVar\" 0\n             ; pCon = const (opaque \"\u03b5.pCon\" (quote \u03b5))\n             ; pDef = const (opaque \"\u03b5.pDef\" (quote \u03b5)) }\n\nextDefEnv : ((Name \u2192 Name) \u2192 (Name \u2192 Name)) \u2192 \u2200 {n A B} \u2192 Env n A B \u2192 Env n A B\nextDefEnv ext \u0393 = record \u0393 { pDef = ext (pDef \u0393) }\n\next1DefEnv : (old new : Name) \u2192 \u2200 {n A B} \u2192 Env n A B \u2192 Env n A B\next1DefEnv old new = extDefEnv (\u03bb f x \u2192 [yes: (\u03bb _ \u2192 new) no: (\u03bb _ \u2192 f x) ]\u2032 (x \u225f-Name old))\n\n\u2191pVar : \u2115 \u2192 (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u2191pVar zero = id\n\u2191pVar (suc n) = \u2191pVar n \u2218 mapVar\u2191\n\non-pVar : \u2200 {n A B C D} \u2192 (VarKind n \u2192 (A \u2192 B) \u2192 (C \u2192 D)) \u2192 Env n A B \u2192 Env n C D\non-pVar f \u0393 = record\n  { pVar = \u03bb k \u2192 f k (pVar \u0393 k)\n  ; pCon = pCon \u0393\n  ; pDef = pDef \u0393 }\n\n_+\u2191 : \u2200 {n} \u2192 Env' n \u2192 Env' n\n_+\u2191 {n} = on-pVar goK\n  where\n    goK : VarKind n \u2192 (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n    goK (K\u1d62 x) f = _+_ (n \u2238 to\u2115 x) \u2218 \u2191pVar 1 (_+_ (to\u2115 x) \u2218 f)\n    goK K\u1d63     f = \u2191pVar 1 (_+_ n \u2218 f)\n\n_+'_ : \u2200 {w} \u2192 Env' w \u2192 \u2115 \u2192 Env' w\n\u0393 +' n = record { pVar = \u03bb k \u2192 _+_ n \u2218 pVar \u0393 k ; pCon = pCon \u0393 ; pDef = pDef \u0393 }\n\n_+1 : \u2200 {w} \u2192 Env' w \u2192 Env' w\n\u0393 +1 = \u0393 +' 1\n\npattern `\u27e6zero\u27e7  = con (quote \u27e6\u2115\u27e7.zero) []\npattern `\u27e6suc\u27e7 t = con (quote \u27e6\u2115\u27e7.suc) (arg\u1d5b\u02b3 t \u2237 [])\npNat : \u2115 \u2192 Term\npNat zero    = `\u27e6zero\u27e7\npNat (suc n) = `\u27e6suc\u27e7 (pNat n)\n\nlam^ : \u2115 \u2192 Visibility \u2192 Term \u2192 Term\nlam^ zero    v x = x\nlam^ (suc n) v x = lam v (lam^ n v x)\n\nlam\u2208 : \u2200 {n} \u2192 Env' n \u2192 (f : Env' n \u2192 Vec Term n \u2192 Term) \u2192 Term\nlam\u2208 {n} \u0393 f = lam^ n visible (f (\u0393 +' n) (reverse (tabulate (\u03bb x \u2192 var (to\u2115 x) []))))\n\napp-tabulate : \u2200 {n a} {A : Set a} \u2192 (Fin n \u2192 A) \u2192 List A \u2192 List A\napp-tabulate {zero}  f xs = xs\napp-tabulate {suc n} f xs = f zero \u2237 app-tabulate (f \u2218 suc) xs\n\np^args : \u2200 {n A} \u2192 Relevance \u2192 (Fin n \u2192 A) \u2192 A \u2192 List (Arg A) \u2192 List (Arg A)\np^args r f x args = app-tabulate (\u03bb k \u2192 arg\u02b0 r (f k)) (arg\u1d5b\u02b3 x \u2237 args)\n\np^lam : \u2115 \u2192 Term \u2192 Term\np^lam n t = lam^ n hidden (lam visible t)\n\n{-\npi^ : \u2115 \u2192 Arg Type \u2192 Type \u2192 Type\npi^ zero    i t = t\npi^ (suc n) i t = `\u03a0 i (pi^ n i t)\n-}\n\nallVarsFrom : \u2200 n \u2192 \u2115 \u2192 Vec Term n\nallVarsFrom zero    k = []\nallVarsFrom (suc n) k = var (k + n) [] \u2237 allVarsFrom n k\n\np3pi\u2208 : \u2200 {n} (\u0393 : Env' n) (r : Relevance)\n          (s : Type)\n          (t : Env' n \u2192 Vec Term n \u2192 Type)\n          (u : Env' n \u2192 Vec Term n \u2192 Type) \u2192 Term\np3pi\u2208 {n} \u0393 r s t u = unEl (go \u0393 (allFin n))\n  where\n  go : \u2200 {m} \u2192 Env' n \u2192 Vec (Fin n) m \u2192 Type\n  go \u0394 [] = \n      `\u03a0\u1d5b\u02b3 (t \u0394 (allVarsFrom n 0)) (u (\u0393 +\u2191) (allVarsFrom n 1))\n  go \u0394 (x \u2237 xs) =\n      `\u03a0 (arg\u02b0 r (mapVarType (pVar \u0394 (K\u1d62 x)) s))\n         (go (\u0394 +1) xs)\n\npSort\u2208 : \u2200 {n} \u2192 Sort \u2192 Vec Term n \u2192 Type\npSort\u2208 s = go 0\n  where\n    go : \u2115 \u2192 \u2200 {n} \u2192 Vec Term n \u2192 Type\n    go k []       = el (suc-sort s) (sort s)\n    go k (t \u2237 ts) = `\u03a0\u1d5b\u02b3 (mapVarType (_+_ k) (el s t)) (go (suc k) ts)\n\npEnvPats : List (Arg Pattern) \u2192 \u2200 {n} \u2192 Env' n \u2192 Env' n\npEnvPat : Pattern \u2192 \u2200 {n} \u2192 Env' n \u2192 Env' n\n\npEnvPat (con _ pats) = pEnvPats pats\npEnvPat dot = id\npEnvPat var = _+\u2191\npEnvPat (lit _) = id\npEnvPat (proj _) = opaque \"pEnvPats\/proj\" id\npEnvPat absurd = id\n\npEnvPats [] = id\npEnvPats (arg i p \u2237 ps) = pEnvPat p \u2218 pEnvPats ps\n\npPats : \u2200 {n} \u2192 Env' n \u2192 List (Arg Pattern) \u2192 List (Arg Pattern)\npPat  : \u2200 {n} \u2192 Env' n \u2192 Arg Pattern \u2192 List (Arg Pattern)\n\npPats \u0393 [] = []\npPats \u0393 (pat \u2237 ps) = pPat \u0393 pat ++ pPats \u0393 ps\n\nnodot = var\n\npPat {n} \u0393 (arg (arg-info _ r) (con c pats))\n  = p^args {n} r (const nodot) (con (pCon \u0393 c) (pPats \u0393 pats)) []\npPat {n} \u0393 (arg (arg-info _ r) dot)\n  = p^args {n} r (const dot) dot []\npPat {n} \u0393 (arg (arg-info _ r) var)\n  = p^args {n} r (const var) var []\npPat {n} \u0393 (arg i (lit (nat l))) = opaque \"pPat\/lit\/nat\" []\npPat {n} \u0393 (arg (arg-info _ r) (lit l))\n  = p^args {n} r go\n                 (con (quote refl) []) []\n      where\n        go : \u2200 {n} \u2192 Fin n \u2192 Pattern\n        go zero = lit l\n        go (suc _) = nodot\npPat {n} \u0393 (arg (arg-info _ r) absurd) = p^args {n} r (const var) absurd []\npPat {n} \u0393 (arg i (proj p)) = opaque \"pPat\/proj\" []\n\npLit : Literal \u2192 Term\npLit (nat n) = pNat n\npLit _       = con (quote refl) []\n\npTerm  : \u2200 {n} \u2192 Env' n \u2192 Term \u2192 Term\npArgs  : \u2200 {n} \u2192 Env' n \u2192 Args \u2192 Args\npType\u2208 : \u2200 {n} \u2192 Env' n \u2192 Type             \u2192 Vec Term n \u2192 Type\npTerm\u2208 : \u2200 {n} \u2192 Env' n \u2192 Term             \u2192 Vec Term n \u2192 Term\npPi\u2208   : \u2200 {n} \u2192 Env' n \u2192 Arg Type \u2192 Type \u2192 Vec Term n \u2192 Term\n\npType\u2208 \u0393 (el s t) as = el s (pTerm\u2208 \u0393 t as)\n\npTerm {n} \u0393 (lam _ t) = p^lam n (pTerm (\u0393 +\u2191) t)\npTerm \u0393 (var v args)  = var (pVar \u0393 K\u1d63 v) (pArgs \u0393 args)\npTerm \u0393 (con c args)  = con (pCon \u0393 c) (pArgs \u0393 args)\npTerm \u0393 (def d args)  = def (pDef \u0393 d) (pArgs \u0393 args)\npTerm \u0393 (lit l)       = pLit l\npTerm \u0393 (pat-lam _ _) = opaque \"pTerm\/pat-lam\" unknown\npTerm \u0393 unknown       = unknown\n\npTerm \u0393 (sort s) = lam\u2208 \u0393 \u03bb _ as \u2192 unEl (pSort\u2208 s as)\npTerm \u0393 (pi s t) = lam\u2208 \u0393 \u03bb \u0394 \u2192 pPi\u2208 \u0394 s t\n\npPi\u2208 {n} \u0393 (arg (arg-info _ r) s) t as =\n  p3pi\u2208 \u0393 r s (\u03bb \u0394 \u2192 pType\u2208 \u0394 s) (\u03bb \u0394 vs \u2192 pType\u2208 \u0394 t\n    (vmap (\u03bb a v \u2192 app (mapVar (_+_ (suc n)) a) (arg\u1d5b\u02b3 v \u2237 [])) as \u229b vs))\n\npTerm\u2208 \u0393 (sort s) as = unEl (pSort\u2208 s as)\npTerm\u2208 \u0393 (pi s t) = pPi\u2208 \u0393 s t\npTerm\u2208 \u0393 t as = app (pTerm \u0393 t) (toList (vmap arg\u1d5b\u02b3 as))\n\npArgs \u0393 [] = []\npArgs \u0393 (arg (arg-info _ r) t \u2237 as)\n  = p^args r (\u03bb k \u2192 mapVar (pVar (\u0393 +\u2191) (K\u1d62 k)) t) (pTerm \u0393 t) (pArgs \u0393 as)\n\npClause  : \u2200 {n} \u2192 Env' n \u2192 Clause \u2192 Clause\npClauses : \u2200 {n} \u2192 Env' n \u2192 Clauses \u2192 Clauses\n\npClause \u0393 (clause pats body) = clause (pPats \u0393 pats) (pTerm (pEnvPats pats \u0393) body)\npClause \u0393 (absurd-clause pats) = absurd-clause (pPats \u0393 pats)\n\npClauses \u0393 [] = []\npClauses \u0393 (c \u2237 cs) = pClause \u0393 c \u2237 pClauses \u0393 cs\n\npFunDef : \u2200 {n} \u2192 Env' n \u2192 Name \u2192 FunctionDef \u2192 FunctionDef\npFunDef \u0393 d (fun-def t cs)\n  = fun-def (pType\u2208 \u0393 t (replicate (def d []))) (pClauses \u0393 cs)\n\npostulate\n  IMPOSSIBLE : FunctionDef\n\npFunName : \u2200 {n} \u2192 Env' n \u2192 Name \u2192 FunctionDef\npFunName \u0393 x with definition x\n... | function d = pFunDef \u0393 x d\n... | _ = IMPOSSIBLE -- opaque \"pFunName\" (fun-def {!!} {!!})\n\npFunNameRec : \u2200 {n} \u2192 Env' n \u2192 (x x\u209a : Name) \u2192 FunctionDef\npFunNameRec \u0393 x x\u209a = pFunName (ext1DefEnv x x\u209a \u0393) x\n\n{-\npDefinition : Env' 2 \u2192 Name \u2192 Definition \u2192 Definition\npDefinition \u0393 n (function x) = function {!pFunDef!}\npDefinition \u0393 n (data-type x) = {!!}\npDefinition \u0393 n (record\u2032 x) = {!!}\npDefinition \u0393 n constructor\u2032 = {!!}\npDefinition \u0393 n axiom = {!!}\npDefinition \u0393 n primitive\u2032 = {!!}\n\npName : Env' 2 \u2192 Name \u2192 Definition\npName \u0393 n = pDefinition \u0393 n (definition n)\n-}\n\nopen import Data.String.Core using (String)\nopen import Data.Float       using (Float)\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n\np[Set\u2080] = pFunName (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl (Get-type.from-fun-def p[Set\u2080]))\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261 Get-term.from-fun-def p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nunquoteDecl u-p[Set\u2080] = p[Set\u2080]\n\n{-\np-[\u2192]-type = unEl (Get-type.from-fun-def p-[\u2192])\np-[\u2192]' : \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n                {A\u209a : A \u2192 Set\u2080} (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 Set\u2080)\n                {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n                {B\u209a : B \u2192 Set\u2080} (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 Set\u2080)\n                {f : A \u2192 B}    (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n              \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f \u2192\n              {C : Set\u2080} \u2192 (C \u2192 Set\u2080) \u2192 {t : {!!}} \u2192 (A \u2192 B) \u2192 Set\u2080 -- _[\u2080\u2192\u2080]_ {A} {-A\u2080\u209a} {A\u209a-} {-A\u2081\u209a-} {!!} {B} {!!} {-B\u2080\u209a} B\u2081\u209a-} f -- f\u209a\np-[\u2192]' = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n-}\n\n[String] : [\u2605\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [\u2605\u2080] Float\n[Float] _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6\u2605\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6\u2605\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\nunquoteDecl revealed-[\u2192]' = un-function revealed-[\u2192]\n\nunquoteDecl revelator-[\u2192] = Revelator.fun (type (quote _[\u2080\u2192\u2080]_))\n\n{-\np-[\u2192] = pFunName (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]-type = unEl (Get-type.from-fun-def p-[\u2192])\n\np-[\u2192]' : (x0 : Set) \u2192 (x1 : (x1 : x0) \u2192 Set) \u2192 (x2 : (x2 : x0) \u2192 Set) \u2192 (x3 : (x3 : x0) \u2192 (x4 : x1 x3) \u2192 (x5 : x2 x3) \u2192 Set) \u2192 (x4 : Set) \u2192 (x5 : (x5 : x4) \u2192 Set) \u2192 (x6 : (x6 : x4) \u2192 Set) \u2192 (x7 : (x7 : x4) \u2192 (x8 : x5 x7) \u2192 (x9 : x6 x7) \u2192 Set) \u2192 (x8 : (x8 : x0) \u2192 x4) \u2192 (x9 : (x9 : x0) \u2192 (x10 : x1 x9) \u2192 x5 (x8 x9)) \u2192 (x10 : _[\u2080\u2192\u2080]_ {x0} x2 {x4} x6 x8) \u2192 Set\n{-\np-[\u2192]' : \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n                {A\u209a : A \u2192 Set\u2080} (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 Set\u2080)\n                {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n                {B\u209a : B \u2192 Set\u2080} (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 Set\u2080)\n                {f : A \u2192 B}    (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n              \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f \u2192\n              {C : Set\u2080} \u2192 (C \u2192 Set\u2080) \u2192 {t : {!!}} \u2192 (A \u2192 B) \u2192 Set\u2080 -- _[\u2080\u2192\u2080]_ {A} {-A\u2080\u209a} {A\u209a-} {-A\u2081\u209a-} {!!} {B} {!!} {-B\u2080\u209a} B\u2081\u209a-} f -- f\u209a\n-}\np-[\u2192]' = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n\ntest : {!showTerm p-[\u2192]-type!}\ntest = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n{-\n-- unquoteDecl _[[\u2192]]_ = p-[\u2192]\n\n{-\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : ([[\u2115]] [[\u2192]] [[\u2115]]) [suc] [suc]\n-}\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda) = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115) = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float) = quote [Float]\ndefDefEnv1 (quote \u2605\u2080) = quote [\u2605\u2080]\ndefDefEnv1 (quote \u2605\u2081) = quote [\u2605\u2081]\ndefDefEnv1 n         = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082) = quote [0\u2082]\ndefConEnv1 (quote 1\u2082) = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero) = quote [zero]\ndefConEnv1 (quote \u2115.suc)  = quote [suc]\ndefConEnv1 n              = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda) = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115) = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float) = quote \u27e6Float\u27e7\ndefDefEnv2 (quote \u2605\u2080) = quote \u27e6\u2605\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081) = quote \u27e6\u2605\u2081\u27e7\ndefDefEnv2 n         = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082) = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082) = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero) = quote \u27e6\u2115\u27e7.zero\ndefConEnv2 (quote \u2115.suc)  = quote \u27e6\u2115\u27e7.suc\ndefConEnv2 n              = opaque \"defConEnv2\" n\n\ndefEnv1 : Env' 1\ndefEnv1 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv1.pVar\" 0\n                 ; pCon = defConEnv1\n                 ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv2.pVar\" 0\n                 ; pCon = defConEnv2\n                 ; pDef = defDefEnv2 }\n\n{-\n_\u27e6+\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) _+_ _+_\nzero  \u27e6+\u27e7 n = n\nsuc m \u27e6+\u27e7 n = suc (m \u27e6+\u27e7 n)\n-}\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7         \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261 quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7))) \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = pFunName defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261 quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261 quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261 Get-term.from-fun-def p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\n-- unquoteDecl \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\n_\u2261clauses_ = \u03bb x y \u2192 Get-clauses.from-def x \u2261 Get-clauses.from-def y\n_\u2261type_ = \u03bb x y \u2192 Get-type.from-def x \u2261 Get-type.from-def y\n  \nmodule Test where\n  p1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n  [\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\n  test-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\n  test-p1\u2115\u2192\u2115 = refl\n\n  p2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n  \u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n  test-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\n  test-p2\u2115\u2192\u2115 = refl\n\n  p\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n  \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n  test-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261 quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\n  test-p\u2115\u2192\u2115\u2192\u2115 = refl\n  ZERO : Set\u2081\n  ZERO = (A : Set\u2080) \u2192 A\n  \u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n  \u27e6ZERO\u27e7 f\u2080 f\u2081 =\n    {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n    \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\n  pZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\n  q\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\n  test-pZERO : pZERO \u2261 q\u27e6ZERO\u27e7\n  test-pZERO = refl\n  ID : Set\u2081\n  ID = (A : Set\u2080) \u2192 A \u2192 A\n  \u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n  \u27e6ID\u27e7 f\u2080 f\u2081 =\n    {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n    {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n    \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\n  pID = pTerm (\u03b5 2) (quoteTerm ID)\n  q\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\n  test-param : absTerm q\u27e6ID\u27e7 \u2261 absTerm pID\n  test-param = refl\n  test-ID : q\u27e6ID\u27e7 \u2261 pID\n  test-ID = refl\n\n  p-not = pFunName defEnv2 (quote not)\n  unquoteDecl P-not = p-not\n  test-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 P-not x\u1d63\n  test-not \u27e60\u2082\u27e7 = refl\n  test-not \u27e61\u2082\u27e7 = refl\n\n  p1-pred = pFunName defEnv1 (quote pred)\n  q-[pred] = quoteTerm [pred]\n  unquoteDecl [pred]' = p1-pred\n  test-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\n  test-p1-pred [zero] = refl\n  test-p1-pred ([suc] n\u209a) = refl\n\n  p2-pred = pFunName defEnv2 (quote pred)\n  q-\u27e6pred\u27e7 = definition (quote \u27e6pred\u27e7)\n  unquoteDecl \u27e6pred\u27e7' = p2-pred\n  test-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\n  test-p2-pred \u27e6zero\u27e7 = refl\n  test-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\n  p\/2 = pFunNameRec defEnv2 (quote _\/2)\n  q\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\n  unquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\n  {-\n  test-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261 q\u27e6\/2\u27e7\n  test-\/2 = refl\n  -}\n  test-\/2 : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\n  test-\/2 \u27e6zero\u27e7 = refl\n  test-\/2 (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\n  test-\/2 (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) = ap \u27e6suc\u27e7 (test-\/2 n\u1d63)\n\n  p+ = pFunNameRec defEnv2 (quote _+\u2115_)\n  q\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\n  unquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\n  -- test-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261 q\u27e6+\u27e7\n  -- test-+ = refl\n  test-+ : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\n  test-+ \u27e6zero\u27e7    n'\u1d63 = refl\n  test-+ (\u27e6suc\u27e7 n\u1d63) n'\u1d63 = ap \u27e6suc\u27e7 (test-+ n\u1d63 n'\u1d63)\n\n  {-\n  is-good : String \u2192 \ud835\udfda\n  is-good \"good\" = 1\u2082\n  is-good _      = 0\u2082\n\n  \u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n  \u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n  \u27e6is-good\u27e7 {_}      refl = {!!}\n\n  p-is-good = pFunName defEnv2 (quote is-good)\n  unquoteDecl \u27e6is-good\u27e7' = p-is-good\n  test-is-good = {!!}\n  -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b6915db12b7c450d87def9967c9830d36ca626b4","subject":"Simplify change semantics","message":"Simplify change semantics\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Specification.agda","new_file":"Parametric\/Change\/Specification.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e (\u27e6 c \u27e7Const)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = record\n    { apply = \u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    ; correct = \u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e\n    } where open \u2261-Reasoning\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e (\u27e6 const c \u27e7Term \u03c1)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = record\n    { apply = \u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    ; correct = \u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e\n    } where open \u2261-Reasoning\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5c76b2f48299a010ec3ccf23921c0a130d17d754","subject":"Drop redundant import","message":"Drop redundant import\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 v1 v2 dv \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb v1 v2 dv x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) (da , db) (daa , dbb) = (fromto\u2192valid  _ _ _ daa) , (fromto\u2192valid _ _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} f1 f2 df dff = \u03bb a da ada \u2192\n  fromto\u2192valid _ _ _ (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto _ _ ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid _ _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 _ _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Types\nopen import New.Derive\n\n[_]_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n[ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ] da from a1 to a2 \u2192 [ \u03c4 ] df a1 da from f1 a1 to f2 a2\n[ int ] dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ] (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ] da from a1 to a2 \u00d7 [ \u03c4 ] db from b1 to b2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ] dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ] \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ] (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\nopen import Postulate.Extensionality\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ] dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 _ _ _ daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ] nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ] da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ] da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ] df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ] da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da a1 a2 daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 v1 v2 dv \u2192 [ \u03c4 ] dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ] dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb v1 v2 dv x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) (da , db) (daa , dbb) = (fromto\u2192valid  _ _ _ daa) , (fromto\u2192valid _ _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} f1 f2 df dff = \u03bb a da ada \u2192\n  fromto\u2192valid _ _ _ (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto _ _ ada)\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid _ _ _ daa)\n    body : [ \u03c4 ] df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 _ _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c41c41315f6dad2bfb20aeb507061183c7a3be9d","subject":"Enable missing ChangeAlgebra instance for sums","message":"Enable missing ChangeAlgebra instance for sums\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da (ada : valid a da) \u2192 WellDefinedFunChangePoint f df a da\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  instance\n    sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","old_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public hiding ([_])\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da (ada : valid a da) \u2192 WellDefinedFunChangePoint f df a da\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"06eb823d67f432686eb5e6a95ab0b097794c2e33","subject":"Simplified a proof (from the Agsy proof).","message":"Simplified a proof (from the Agsy proof).\n\nIgnore-this: 7afe6f629a94a3225b9cdedb7794794e\n\ndarcs-hash:20110121131044-3bd4e-7fa7af65e37fd604644549d4eaab9685218abfa3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/AxiomaticPA\/Relation\/Binary\/PropositionalEqualityI.agda","new_file":"src\/AxiomaticPA\/Relation\/Binary\/PropositionalEqualityI.agda","new_contents":"------------------------------------------------------------------------------\n-- PA propositional equality\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Relation.Binary.PropositionalEqualityI where\n\nopen import AxiomaticPA.Base\n\n------------------------------------------------------------------------------\n-- Identity properties\n\nrefl : \u2200 {n} \u2192 n \u2263 n\nrefl {n} = S\u2081 (S\u2085 n) (S\u2085 n)\n{-# ATP hint refl #-}\n\nsym : \u2200 {m n} \u2192 m \u2263 n \u2192 n \u2263 m\nsym m\u2263n = S\u2081 m\u2263n refl\n{-# ATP hint sym #-}\n\ntrans : \u2200 {m n o} \u2192 m \u2263 n \u2192 n \u2263 o \u2192 m \u2263 o\ntrans m\u2263n = S\u2081 (sym m\u2263n)\n{-# ATP hint trans #-}\n","old_contents":"------------------------------------------------------------------------------\n-- PA propositional equality\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Relation.Binary.PropositionalEqualityI where\n\nopen import AxiomaticPA.Base\n\n------------------------------------------------------------------------------\n-- Identity properties\n\nrefl : \u2200 {n} \u2192 n \u2263 n\nrefl {n} = S\u2081 (S\u2085 n) (S\u2085 n)\n{-# ATP hint refl #-}\n\nsym : \u2200 {m n} \u2192 m \u2263 n \u2192 n \u2263 m\nsym {m} = aux S\u2081 refl\n  where\n    aux : \u2200 {r t} \u2192 (t \u2263 r \u2192 t \u2263 t \u2192 r \u2263 t) \u2192 t \u2263 t \u2192 t \u2263 r \u2192 r \u2263 t\n    aux hyp t\u2263t t\u2263r = hyp t\u2263r t\u2263t\n{-# ATP hint sym #-}\n\ntrans : \u2200 {m n o} \u2192 m \u2263 n \u2192 n \u2263 o \u2192 m \u2263 o\ntrans m\u2263n = S\u2081 (sym m\u2263n)\n{-# ATP hint trans #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a36f7936402f8d9e95bbaf6c573766c06eacbabc","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: ed9b9992b5c344edb014fc3c88be6946\n\ndarcs-hash:20120214135729-3bd4e-1a5e26411543607ffa3ae825731118608932af7d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/Base.agda","new_file":"src\/GroupTheory\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- Group theory base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfix  11 _\u207b\u00b9\ninfixl 10 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n-- The group universe\nopen import Common.Universe public renaming ( D to G )\n\n-- The equality\n-- The equality is the propositional identity on the group universe.\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.Inductive public\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Group theory axioms\npostulate\n  \u03b5   : G          -- The identity element.\n  _\u00b7_ : G \u2192 G \u2192 G  -- The binary operation.\n  _\u207b\u00b9 : G \u2192 G      -- The inverse function.\n\n  assoc         : \u2200 a b c \u2192 a \u00b7 b \u00b7 c \u2261 a \u00b7 (b \u00b7 c)\n  leftIdentity  : \u2200 a \u2192 \u03b5 \u00b7 a         \u2261 a\n  rightIdentity : \u2200 a \u2192 a \u00b7 \u03b5         \u2261 a\n  leftInverse   : \u2200 a \u2192 a \u207b\u00b9 \u00b7 a      \u2261 \u03b5\n  rightInverse  : \u2200 a \u2192 a \u00b7 a \u207b\u00b9      \u2261 \u03b5\n{-# ATP axiom assoc leftIdentity rightIdentity leftInverse rightInverse #-}\n\n-- Congruence property.\n\u00b7-cong : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2081 \u00b7 x\u2082 \u2261 y\u2081 \u00b7 y\u2082\n\u00b7-cong = cong\u2082 _\u00b7_\n","old_contents":"------------------------------------------------------------------------------\n-- Group theory base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Base where\n\n-- We add 3 to the fixities of the standard library.\ninfix  11 _\u207b\u00b9\ninfixl 10 _\u00b7_  -- The symbol is '\\cdot'.\n\n------------------------------------------------------------------------------\n-- The group universe\nopen import Common.Universe public renaming ( D to G )\n\n-- The equality\n-- The equality is the propositional identity on the group universe.\nimport Common.Relation.Binary.PropositionalEquality\nopen module Eq =\n  Common.Relation.Binary.PropositionalEquality.Inductive public\n\n-- Logical constants\nopen import Common.LogicalConstants public\n\n-- Group theory axioms\npostulate\n  \u03b5   : G          -- The identity element.\n  _\u00b7_ : G \u2192 G \u2192 G  -- The binary operation.\n  _\u207b\u00b9 : G \u2192 G      -- The inverse function.\n\n  assoc         : \u2200 a b c \u2192 a \u00b7 b \u00b7 c    \u2261 a \u00b7 (b \u00b7 c)\n  leftIdentity  : \u2200 a     \u2192     \u03b5 \u00b7 a    \u2261 a\n  rightIdentity : \u2200 a     \u2192     a \u00b7 \u03b5    \u2261 a\n  leftInverse   : \u2200 a     \u2192  a \u207b\u00b9 \u00b7 a    \u2261 \u03b5\n  rightInverse  : \u2200 a     \u2192  a    \u00b7 a \u207b\u00b9 \u2261 \u03b5\n{-# ATP axiom assoc leftIdentity rightIdentity leftInverse rightInverse #-}\n\n-- Congruence property.\n\u00b7-cong : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2261 y\u2081 \u2192 x\u2082 \u2261 y\u2082 \u2192 x\u2081 \u00b7 x\u2082 \u2261 y\u2081 \u00b7 y\u2082\n\u00b7-cong = cong\u2082 _\u00b7_\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0c369642a333d5a9786f2bbbc209b72bb219da9a","subject":"Hutton's razor: shave it with Conor(tm)","message":"Hutton's razor: shave it with Conor(tm)","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/RevisedHutton.agda","new_file":"models\/RevisedHutton.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- A context is a snoc-list of types\n-- put otherwise, a context is a type telescope\ndata Context : Set where\n  [] : Context\n  _,_ : Context -> Type -> Context\n\n-- The environment realizes the context, having a value for each type\nEnv : Context -> Set\nEnv []      = Unit\nEnv (G , S)  = Env G * Val S\n\n-- A typed-variable indexes into the context, obtaining a proof that\n-- what we get is what you want (WWGIWYW)\nVar : Context -> Type -> Set\nVar []      T = Zero\nVar (G , S) T = Var G T + (S == T)\n\n-- The lookup gets into the context to extract the value\nlookup : (G : Context) -> Env G -> (T : Type) -> Var G T -> Val T\nlookup []        _       T ()\nlookup (G , .T)  (g , t) T (r refl) = t\nlookup (G , S)   (g , t) T (l x)    = lookup G g T x \n\n-- Open term: holes are either values or variables in a context\nopenTerm : Context -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup\ndischarge : (context : Context) ->\n            Env context ->\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ctxt env ty (l value) = con (EZe , value)\ndischarge ctxt env ty (r variable) = con (EZe , lookup ctxt env ty variable ) \n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {ty : Type}\n            (context : Context)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {ty : Type}(context : Context) ->\n           Env context ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen ctxt env tm = eval (substExpr ctxt (discharge ctxt env) tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- Test context:\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context\ntestContext = (([] , bool) , nat) , pair bool nat\ntestEnv : Env testContext\ntestEnv = ((Void , true ) , su (su ze)) , (false , su ze) \n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( l (r refl) ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext testEnv test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (l (r refl)) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                       (discharge testContext testEnv)\n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext testEnv test2\n\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (l (l (r refl)))) ,\n                       (con (EZe , r (l (r refl))) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext testEnv test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( r refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       (discharge testContext testEnv)\n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext testEnv test4","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\n-- Open term: holes are either values or variables in a context\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = \n    subst (cong (IMu closeTerm) (snd variable)) \n          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0a1a8cb15553b264233f9409dd7af49f55d1b0eb","subject":"lib\/Relation\/Binary\/Logical\/LEM.agda","message":"lib\/Relation\/Binary\/Logical\/LEM.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Logical\/LEM.agda","new_file":"lib\/Relation\/Binary\/Logical\/LEM.agda","new_contents":"-- Why I don't want LEM (another way to see that parametricity is\n-- anti-classical).\n\nopen import Type\nopen import Function\nopen import Data.Two\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Relation.Binary.Logical.LEM\n  (LEM : (P : \u2605\u2081) \u2192 Dec P) where\n\nbroken-id : \u2200 (A : \u2605) \u2192 A \u2192 A\nbroken-id A with LEM (A \u2261 \ud835\udfda)\n... | yes p = coe! p \u2218 not \u2218 coe p\n... | no \u00acp = id\n","old_contents":"-- Why I don't want LEM\n\nopen import Type\nopen import Function\nopen import Data.Two\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Relation.Binary.Logical.LEM\n  (LEM : (P : \u2605\u2081) \u2192 Dec P) where\n\nbroken-id : \u2200 (A : \u2605) \u2192 A \u2192 A\nbroken-id A with LEM (A \u2261 \ud835\udfda)\n... | yes p = coe! p \u2218 not \u2218 coe p\n... | no \u00acp = id\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"de69e08550a7e7d86b91c9248f4a41a3c4f1bf7c","subject":"too many changes...not much working","message":"too many changes...not much working\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping.agda","new_file":"Agda\/ITyping.agda","new_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\ndata IType : Set\ndata IType\u1d62 : Set\n\ndata IType where\n  o : IType\n  _~>_ : IType\u1d62 -> IType\u1d62 -> IType\n\ndata IType\u1d62 where\n  \u2229 : List IType -> IType\u1d62\n\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\u1d62\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\n_\u225fTI\u1d62_ : Decidable {A = IType\u1d62} _\u2261_\n\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI\u1d62 \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI\u1d62 \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n\n\u2229 [] \u225fTI\u1d62 \u2229 [] = yes refl\n\u2229 [] \u225fTI\u1d62 \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI\u1d62 (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI\u1d62 \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 IType\u1d62)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\ndata _\u2237'_ : IType -> Type -> Set\ndata _\u2237'\u1d62_ : IType\u1d62 -> Type -> Set\n\ndata _\u2237'_ where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237'\u1d62 A -> \u03c4 \u2237'\u1d62 B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n\ndata _\u2237'\u1d62_ where\n  nil : \u2200 {A} ->  \u03c9 \u2237'\u1d62 A\n  cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237'\u1d62 A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237'\u1d62 A\n\n\ndata _\u2286[_]_ : IType -> Type -> IType -> Set\ndata _\u2286\u1d62[_]_ : IType\u1d62 -> Type -> IType\u1d62 -> Set\n\ndata _\u2286[_]_ where\n  base : o \u2286[ \u03c3 ] o\n  arr : \u2200 {A B \u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} ->\n    \u03c4\u2082\u2081 \u2286\u1d62[ A ] \u03c4\u2081\u2081 -> \u03c4\u2081\u2082 \u2286\u1d62[ B ] \u03c4\u2082\u2082 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2237' (A \u27f6 B) -> (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) \u2237' (A \u27f6 B) ->\n    -------------------------------------------------------------------------------------------\n                            (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2286[ A \u27f6 B ] (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  -- \u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  --   \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  --   -------------------------------\n  --             \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\ndata _\u2286\u1d62[_]_ where\n  nil : \u2200 {A \u03c4} ->\n    \u03c4 \u2237'\u1d62 A ->\n    -----------\n    \u03c9 \u2286\u1d62[ A ] \u03c4\n  -- \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229' \u03c4) \u2286[ A ] (\u2229 \u03c4\u1d62)\n  cons : \u2200 {A \u03c4\u1d62 \u03c4' \u03c4'\u1d62} ->\n    \u2203(\u03bb \u03c4 -> (\u03c4 \u2208 \u03c4\u1d62) \u00d7 (\u03c4' \u2286[ A ] \u03c4)) -> (\u2229 \u03c4'\u1d62) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62) ->\n    -------------------------------------------------------------\n                  (\u2229 (\u03c4' \u2237 \u03c4'\u1d62)) \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62)\n\n-- data _\u2264\u2229_ : IType -> IType -> Set where\n--   base : o \u2264\u2229 o\n--   arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n--   \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n--   \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n--   \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\u2237'\u1d62-\u2208 : \u2200 {A \u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A -> \u03c4 \u2237' A\n\u2237'\u1d62-\u2208 {\u03c4\u1d62 = []} () _\n\u2237'\u1d62-\u2208 {\u03c4 = \u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 \u03c4'\u03c4\u1d62\u2237A with \u03c4' \u225fTI \u03c4\n\u2237'\u1d62-\u2208 {A} {\u03c4} {.\u03c4 \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4\u2237A \u03c4'\u03c4\u1d62\u2237A) | yes refl = \u03c4\u2237A\n\u2237'\u1d62-\u2208 {A} {\u03c4} {\u03c4' \u2237 \u03c4\u1d62} \u03c4\u2208\u03c4'\u03c4\u1d62 (cons \u03c4'\u2237A \u03c4'\u03c4\u1d62\u2237A) | no \u03c4'\u2260\u03c4 = \u2237'\u1d62-\u2208 (\u2208-\u2237-elim \u03c4 \u03c4\u1d62 \u03c4'\u2260\u03c4 \u03c4\u2208\u03c4'\u03c4\u1d62) \u03c4'\u03c4\u1d62\u2237A\n\n\n-- \u2237'\u1d62-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A\n-- \u2237'\u1d62-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237'A = nil\n-- \u2237'\u1d62-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} {\u03c4\u2c7c} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237'A =\n--   cons (\u2237'\u1d62-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237'A) (\u2237'\u1d62-\u2286 {\u03c4\u1d62 = \u03c4\u1d62} {\u03c4\u2c7c} (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237'A)\n\n\n\n\u2286-refl : \u2200 {A \u03c4} -> \u03c4 \u2237' A -> \u03c4 \u2286[ A ] \u03c4\n\u2286\u1d62-refl : \u2200 {A \u03c4} -> \u03c4 \u2237'\u1d62 A -> \u03c4 \u2286\u1d62[ A ] \u03c4\n\u2286\u1d62-\u2286 : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> \u2229 \u03c4\u1d62 \u2286\u1d62[ A ] \u2229 \u03c4\u2c7c\n\n\u2286-refl {\u03c4 = o} base = base\n\u2286-refl {\u03c4 = \u03c4 ~> \u03c4'} (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) =\n  arr (\u2286\u1d62-refl \u03c4\u2237\u1d62A) (\u2286\u1d62-refl \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B) (arr \u03c4\u2237\u1d62A \u03c4'\u2237\u1d62B)\n\n\u2286\u1d62-refl {\u03c4 = \u2229 []} nil = nil nil\n\u2286\u1d62-refl {A} {\u2229 (\u03c4 \u2237 \u03c4\u1d62)} \u03c4\u03c4\u1d62\u2237A = \u2286\u1d62-\u2286 (\u03bb {x} z \u2192 z) \u03c4\u03c4\u1d62\u2237A\n\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A = nil \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2286 {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2237A =\n  cons (\u03c4 , (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) , \u2286-refl (\u2237'\u1d62-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl)) \u03c4\u2c7c\u2237A)) (\u2286\u1d62-\u2286 (\u03bb {x} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)) \u03c4\u2c7c\u2237A)\n\n--\n-- list-trans : \u2200 {a} {A : Set a} {\u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096 : List A} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u03c4\u2c7c \u2286 \u03c4\u2096 -> \u03c4\u1d62 \u2286 \u03c4\u2096\n-- list-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = \u03bb {x} z \u2192 \u03c4\u2c7c\u2286\u03c4\u2096 (\u03c4\u1d62\u2286\u03c4\u2c7c z)\n--\n-- list-\u2208-\u2286-nil : \u2200 {a} {A : Set a} {\u03c4 : A} {\u03c4\u2c7c : List A} -> \u03c4 \u2208 \u03c4\u2c7c -> \u03c4 \u2237 [] \u2286 \u03c4\u2c7c\n-- list-\u2208-\u2286-nil \u03c4\u2208\u03c4\u2c7c (here refl) = \u03c4\u2208\u03c4\u2c7c\n-- list-\u2208-\u2286-nil \u03c4\u2208\u03c4\u2c7c (there ())\n--\n-- list-\u2208-\u2286-cons : \u2200 {\u03c4 : IType} {\u03c4\u1d62 \u03c4\u2c7c : List IType} -> \u03c4 \u2208 \u03c4\u2c7c -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u03c4 \u2237 \u03c4\u1d62 \u2286 \u03c4\u2c7c\n-- list-\u2208-\u2286-cons {\u03c4} \u03c4\u2208\u03c4\u2c7c \u03c4\u1d62\u2286\u03c4\u2c7c {x = x} x\u2208\u03c4\u03c4\u1d62 with x \u225fTI \u03c4\n-- list-\u2208-\u2286-cons \u03c4\u2208\u03c4\u2c7c \u03c4\u1d62\u2286\u03c4\u2c7c x\u2208\u03c4\u03c4\u1d62 | yes refl = \u03c4\u2208\u03c4\u2c7c\n-- list-\u2208-\u2286-cons \u03c4\u2208\u03c4\u2c7c \u03c4\u1d62\u2286\u03c4\u2c7c (here x\u2261\u03c4) | no x\u2260\u03c4 = \u22a5-elim (x\u2260\u03c4 x\u2261\u03c4)\n-- list-\u2208-\u2286-cons \u03c4\u2208\u03c4\u2c7c \u03c4\u1d62\u2286\u03c4\u2c7c (there x\u2208\u03c4\u1d62) | no x\u2260\u03c4 = \u03c4\u1d62\u2286\u03c4\u2c7c x\u2208\u03c4\u1d62\n\n-- \u2286\u1d62-\u2286\u2082-l : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 \u03c4\u1d62 \u2286\u1d62[ A ] \u2229 \u03c4\u2c7c -> \u03c4\u1d62 \u2286 \u03c4\u2c7c\n-- \u2286\u1d62-\u2286\u2082-l {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c = \u03bb {x} \u2192 \u03bb ()\n-- \u2286\u1d62-\u2286\u2082-l {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = {!   !}\n--\n--\n-- -- \u2286\u1d62-\u2286\u2082-l {\u03c4\u1d62 = []} \u03c4\u1d62\u2286\u03c4\u2c7c = \u03bb {x} \u2192 \u03bb ()\n-- -- \u2286\u1d62-\u2286\u2082-l {\u03c4\u1d62 = \u03c4 \u2237 \u03c4\u1d62} (cons \u03c4\u2208\u03c4\u2c7c _ \u03c4\u1d62\u2286\u03c4\u2c7c) = list-\u2208-\u2286-cons \u03c4\u2208\u03c4\u2c7c (\u2286\u1d62-\u2286\u2082-l \u03c4\u1d62\u2286\u03c4\u2c7c)\n\n\u2286\u1d62-\u2208-\u2203 : \u2200 {A \u03c4 \u03c4\u2081 \u03c4\u2082} -> \u2229 \u03c4\u2081 \u2286\u1d62[ A ] \u2229 \u03c4\u2082 -> \u03c4 \u2208 \u03c4\u2081 -> \u2203(\u03bb \u03c4' -> (\u03c4' \u2208 \u03c4\u2082) \u00d7 (\u03c4 \u2286[ A ] \u03c4'))\n\u2286\u1d62-\u2208-\u2203 (cons \u2203\u03c4 \u03c4\u2081\u2286\u03c4\u2082) (here refl) = \u2203\u03c4\n\u2286\u1d62-\u2208-\u2203 (cons _ \u03c4\u2081\u2286\u03c4\u2082) (there \u03c4\u2208\u03c4\u2081) = \u2286\u1d62-\u2208-\u2203 \u03c4\u2081\u2286\u03c4\u2082 \u03c4\u2208\u03c4\u2081\n\n\u2286\u1d62-\u03c9-\u22a5 : \u2200 {A \u03c4 \u03c4\u1d62} -> (\u2229 (\u03c4 \u2237 \u03c4\u1d62)) \u2286\u1d62[ A ] \u03c9 -> \u22a5\n\u2286\u1d62-\u03c9-\u22a5 (cons (_ , () , _) _)\n\n\u2286\u1d62-\u2237'\u1d62-r : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2237'\u1d62 A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 []} (nil \u03c4\u2c7c\u2237A) = \u03c4\u2c7c\u2237A\n\u2286\u1d62-\u2237'\u1d62-r {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} (cons x \u03c4\u1d62\u2286\u03c4\u2c7c) = \u2286\u1d62-\u2237'\u1d62-r \u03c4\u1d62\u2286\u03c4\u2c7c\n\n\u2286\u1d62-trans : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c \u03c4\u2096} ->\n  \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2c7c -> \u03c4\u2c7c \u2286\u1d62[ A ] \u03c4\u2096 ->\n  ---------------------------------\n            \u03c4\u1d62 \u2286\u1d62[ A ] \u03c4\u2096\n\n\u2286-trans : \u2200 {A \u03c4\u2081 \u03c4\u2082 \u03c4\u2083} ->\n  \u03c4\u2081 \u2286[ A ] \u03c4\u2082 -> \u03c4\u2082 \u2286[ A ] \u03c4\u2083 ->\n  -------------------------------\n            \u03c4\u2081 \u2286[ A ] \u03c4\u2083\n\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 []} \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096 = nil (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2c7c\u2286\u03c4\u2096)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} \u03c4\u1d62\u2286\u03c4\u2c7c (nil x) = \u22a5-elim (\u2286\u1d62-\u03c9-\u22a5 \u03c4\u1d62\u2286\u03c4\u2c7c)\n\u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4 \u2237 \u03c4\u1d62)} {\u2229 \u03c4\u2c7c} {\u2229 \u03c4\u2096} (cons (\u03c4' , \u03c4'\u2208\u03c4\u2c7c , \u03c4\u2286\u03c4') \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 =\n  cons (\u03c4'' , (\u03c4''\u2208\u03c4\u2096 , (\u2286-trans \u03c4\u2286\u03c4' \u03c4'\u2286\u03c4''))) (\u2286\u1d62-trans \u03c4\u1d62\u2286\u03c4\u2c7c \u03c4\u2c7c\u2286\u03c4\u2096)\n    where\n    \u03c4'' = proj\u2081 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c)\n    \u03c4''\u2208\u03c4\u2096 = proj\u2081 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n    \u03c4'\u2286\u03c4'' = proj\u2082 (proj\u2082 (\u2286\u1d62-\u2208-\u2203 \u03c4\u2c7c\u2286\u03c4\u2096 \u03c4'\u2208\u03c4\u2c7c))\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} {\u2229 (_ \u2237 \u03c4\u2c7c)} {\u2229 \u03c4\u2096} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) (cons x \u03c4\u2c7c\u2286\u03c4\u2096) = cons (\u03c4 , ({!   !} , {!   !})) {!   !}\n\n\n-- \u2286\u1d62-trans {\u03c4\u1d62 = \u2229 (\u03c4' \u2237 \u03c4\u1d62)} (cons (\u03c4 , \u03c4\u2208\u03c4\u2c7c , \u03c4'\u2286\u03c4) \u03c4\u1d62\u2286\u03c4\u2c7c) \u03c4\u2c7c\u2286\u03c4\u2096 = {!   !}\n\n\u2286-trans base base = base\n\u2286-trans (arr \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081 \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B _) (arr \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084 \u03c4\u2082\u2081~>\u03c4\u2082\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B) =\n  arr (\u2286\u1d62-trans \u03c4\u2082\u2083\u2286\u03c4\u2082\u2081 \u03c4\u2082\u2081\u2286\u03c4\u2081\u2081) (\u2286\u1d62-trans \u03c4\u2081\u2082\u2286\u03c4\u2082\u2082 \u03c4\u2082\u2082\u2286\u03c4\u2082\u2084) \u03c4\u2081\u2081~>\u03c4\u2081\u2082\u2237A\u27f6B \u03c4\u2082\u2083~>\u03c4\u2082\u2084\u2237A\u27f6B\n\n\n\u2286-\u2237'-r : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> \u03c4' \u2237' A\n\u2286-\u2237'-r base = base\n\u2286-\u2237'-r (arr _ _ _ x) = x\n\n\u2286->\u2286\u1d62 : \u2200 {A \u03c4 \u03c4'} -> \u03c4 \u2286[ A ] \u03c4' -> (\u2229' \u03c4) \u2286\u1d62[ A ] (\u2229' \u03c4')\n\u2286->\u2286\u1d62 {\u03c4 = \u03c4} {\u03c4'} \u03c4\u2286\u03c4' = cons (\u03c4' , (here refl , \u03c4\u2286\u03c4')) (nil (cons (\u2286-\u2237'-r \u03c4\u2286\u03c4') nil))\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app s t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y _) = Y \u03c4\n\n-- \u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n-- \u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n-- -- ers : \u2200 {t} -> \u039b t -> PTerm\n-- -- ers (bv i) = bv i\n-- -- ers (fv x) = fv x\n-- -- ers (lam A \u039bt) = lam (ers \u039bt)\n-- -- ers (app \u039bs \u039bt) = app (ers \u039bs) (ers \u039bt)\n-- -- ers (Y t) = Y t\n\n\n-- data _\u209b : IType -> Set where\n--   o : o \u209b\n--   arr : \u2200 {\u03c4 \u03c4'} -> \u03c4 \u209b -> \u03c4' \u209b -> (\u03c4 ~> \u03c4') \u209b\n\n\n-- data _\u209b\u209b : IType -> Set where\n--   o : o \u209b\u209b\n--   arr :  \u2200 {\u03c4 \u03c4'} -> \u03c4 \u209b\u209b -> \u03c4' \u209b\u209b -> (\u03c4 ~> \u03c4') \u209b\u209b\n--   \u2229-nil : \u03c9 \u209b\u209b\n--   \u2229-cons : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u209b -> (\u2229 \u03c4\u1d62) \u209b\u209b -> (\u2229 (\u03c4 \u2237 \u03c4\u1d62)) \u209b\u209b\n\n\n-- \u03c4\u209b->\u03c4\u209b\u209b : \u2200 {\u03c4} -> \u03c4 \u209b -> \u03c4 \u209b\u209b\n-- \u03c4\u209b->\u03c4\u209b\u209b o = o\n-- \u03c4\u209b->\u03c4\u209b\u209b (arr \u03c4\u209b \u03c4\u209b\u2081) = arr (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b) (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b\u2081)\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\n-- data Y-shape : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n--   intro\u2081 : \u2200 {A m} -> Y-shape (app (Y A) m)\n--   intro\u2082 : \u2200 {A m} -> Y-shape (app m (app (Y A) m))\n--\n\n~>' : IType\u1d62 \u00d7 IType\u1d62 -> IType\n~>' (a , b) = a ~> b\n\n\u2237-++-\u2261 : \u2200 {a} {A : Set a} {x : A} {xs ys : List A} -> (x \u2237 xs) ++ ys \u2261 x \u2237 (xs ++ ys)\n\u2237-++-\u2261 {a} {A} {x} {xs} {ys} = refl\n\n\n\u2237\u1d62-++ : \u2200 {A \u03c4\u1d62 \u03c4\u2c7c} -> (\u2229 \u03c4\u1d62) \u2237'\u1d62 A -> (\u2229 \u03c4\u2c7c) \u2237'\u1d62 A -> \u2229 (\u03c4\u1d62 ++ \u03c4\u2c7c) \u2237'\u1d62 A\n\u2237\u1d62-++ nil \u03c4\u2c7c\u2237A = \u03c4\u2c7c\u2237A\n\u2237\u1d62-++ (cons x \u03c4\u1d62\u2237A) \u03c4\u2c7c\u2237A = cons x (\u2237\u1d62-++ \u03c4\u1d62\u2237A \u03c4\u2c7c\u2237A)\n\n\u2286\u1d62-++ : \u2200 {A \u03c4 \u03c4\u1d62 \u03c4\u2c7c} -> \u03c4 \u2286\u1d62[ A ] (\u2229 \u03c4\u1d62) -> \u03c4 \u2286\u1d62[ A ] (\u2229 \u03c4\u2c7c) -> \u03c4 \u2286\u1d62[ A ] \u2229 (\u03c4\u1d62 ++ \u03c4\u2c7c)\n\u2286\u1d62-++ {\u03c4 = \u2229 []} \u03c4\u2286\u03c4\u1d62 \u03c4\u2286\u03c4\u2c7c = nil (\u2237\u1d62-++ (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2286\u03c4\u1d62) (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2286\u03c4\u2c7c))\n\u2286\u1d62-++ {\u03c4 = \u2229 (x \u2237 x\u1d62)} (cons \u2203\u03c4 x\u2081\u2286\u03c4\u1d62) (cons \u2203\u03c4\u2082 x\u2081\u2286\u03c4\u2c7c) =\n  cons ? (\u2286\u1d62-++ {\u03c4 = \u2229 x\u1d62} x\u2081\u2286\u03c4\u1d62 x\u2081\u2286\u03c4\u2c7c)\n\n-- \u2286\u1d62-++ {\u03c4\u1d62 = \u03c4' \u2237 \u03c4\u1d62} (nil x) \u03c4\u2286\u03c4\u2c7c = nil (\u2237\u1d62-++ x (\u2286\u1d62-\u2237'\u1d62-r \u03c4\u2286\u03c4\u2c7c))\n-- \u2286\u1d62-++ {\u03c4\u1d62 = \u03c4' \u2237 \u03c4\u1d62} (cons (\u03c4 , \u03c4\u2208\u03c4\u1d62 , \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62) \u03c4\u2286\u03c4\u2c7c = cons (\u03c4 , ({!   !} , \u03c4'\u2286\u03c4)) {!   !}\n\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set\ndata _\u22a9\u1d62_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType\u1d62 -> Set\n\n\ndata _\u22a9_\u2236_ where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : IType\u1d62} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , \u03c4\u1d62) \u2208 \u0393) -> (\u03c4\u2286\u03c4\u1d62 : (\u2229' \u03c4) \u2286\u1d62[ A ] \u03c4\u1d62) ->\n    -----------------------------------------------------------------------------\n                                    \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4 \u03c4\u2081 \u03c4\u2082} ->\n    \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9\u1d62 t \u2236 \u03c4\u2081 -> (\u2229' \u03c4) \u2286\u1d62[ B ] \u03c4\u2082 -> \u03c4\u2081 \u2237'\u1d62 A ->\n    ---------------------------------------------------------------------\n                        \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\n  abs : \u2200 {A B \u0393 \u03c4 \u03c4'} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u03c4) \u2237 \u0393) \u22a9\u1d62 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4' ) -> \u03c4 \u2237'\u1d62 A -> \u03c4' \u2237'\u1d62 B ->\n    -------------------------------------------------------------------------------------------------------\n                                        \u0393 \u22a9 lam A t \u2236 (\u03c4 ~> \u03c4')\n  Y : \u2200 {\u0393 A \u03c4\u2082} {\u03c4\u00d7\u03c4\u2081 : List (IType\u1d62 \u00d7 IType\u1d62)} ->\n    Wf-ICtxt \u0393 -> -- \u03c4\u2082 \u2286\u1d62[ A ] (\u2229 (Data.List.map proj\u2082 \u03c4\u00d7\u03c4\u2081)) -> (\u2229 (Data.List.map proj\u2082 \u03c4\u00d7\u03c4\u2081)) \u2286\u1d62[ A ] (\u2229 (Data.List.map proj\u2081 \u03c4\u00d7\u03c4\u2081)) ->\n    ----------------------------------------------\n            \u0393 \u22a9 Y A \u2236 ((\u2229 (Data.List.map ~>' \u03c4\u00d7\u03c4\u2081)) ~> \u03c4\u2082)\n\n  -- Y : \u2200 {\u0393 A \u03c4} {\u03c4\u2081\u00d7\u03c4\u2082 : List (IType\u1d62 \u00d7 IType\u1d62)} ->\n  --   Wf-ICtxt \u0393 -> \u03c4\u2082 \u2286\u1d62[ A ] \u03c4\u2081 -> \u03c4\u2081 \u2286\u1d62[ A ] \u03c4 ->\n  --   ----------------------------------------------\n  --           \u0393 \u22a9 Y A \u2236 ((\u2229 Data.List.map ~>' \u03c4\u2081\u00d7\u03c4\u2082) ~> \u03c4\u2082)\n\n\n\ndata _\u22a9\u1d62_\u2236_ where\n  nil : \u2200 {A \u0393} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) ->\n    ----------------------\n          \u0393 \u22a9\u1d62 m\u2005 \u2236 \u03c9\n\n  cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} ->\n    (wf-\u0393 : Wf-ICtxt \u0393) -> \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 \u03c4\u1d62) ->\n    -------------------------------------------------------\n                    \u0393 \u22a9\u1d62 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n\n\n\n\n\n-- -- data _\u22a9_\u2236_ : \u2200 {A \u03c4} -> ICtxt -> \u039b A -> \u03c4 \u209b\u209b -> Set where\n-- --   var : \u2200 {A \u0393 x \u03c4} {\u03c4\u209b : \u03c4 \u209b} {\u03c4\u1d62 : List IType} ->\n-- --     (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u2229 \u03c4\u1d62)) \u2208 \u0393) -> (\u03c4\u1d62\u2264\u2229\u03c4 : \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4) ->\n-- --     --------------------------------------------------------------------------\n-- --                     \u03c4 \u2237' A -> \u0393 \u22a9 fv {A} x \u2236 (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b)\n-- --   app : \u2200 {A B \u0393 s t \u03c4\u2081 \u03c4\u2082} {\u03c4\u2081\u209b\u209b : \u03c4\u2081 \u209b\u209b} {\u03c4\u2082\u209b\u209b : \u03c4\u2082 \u209b\u209b} ->\n-- --     \u0393 \u22a9 s \u2236 (arr \u03c4\u2081\u209b\u209b \u03c4\u2082\u209b\u209b) -> \u0393 \u22a9 t \u2236 \u03c4\u2081\u209b\u209b -> (\u03c4\u2081 ~> \u03c4\u2082) \u2237' (A \u27f6 B) ->\n-- --     -------------------------------------------------------------------\n-- --                       \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\u2082\u209b\u209b\n-- --   \u2229-nil : \u2200 {A \u0393} {m : \u039b A} ->\n-- --     (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) ->\n-- --     --------------------------------------------------\n-- --                       \u0393 \u22a9 m\u2005 \u2236 \u2229-nil\n-- --   \u2229-cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {\u03c4\u209b : \u03c4 \u209b} {\u03c4\u1d62\u209b\u209b : (\u2229 \u03c4\u1d62) \u209b\u209b} {m : \u039b A} ->\n-- --     (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) -> \u0393 \u22a9 m\u2005 \u2236 (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b) -> \u0393 \u22a9 m\u2005 \u2236 \u03c4\u1d62\u209b\u209b ->\n-- --     -------------------------------------------------------------------------------------------\n-- --                                   \u0393 \u22a9 m\u2005 \u2236 (\u2229-cons \u03c4\u209b \u03c4\u1d62\u209b\u209b)\n-- --   abs : \u2200 {A B \u0393 \u03c4\u1d62 \u03c4} {\u03c4\u209b\u209b : \u03c4 \u209b\u209b} {\u03c4\u1d62\u209b\u209b : (\u2229 \u03c4\u1d62) \u209b\u209b} (L : FVars) -> \u2200 {t : \u039b B} ->\n-- --     ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4\u209b\u209b ) -> \u2229 \u03c4\u1d62 \u2237' A -> \u03c4 \u2237' B ->\n-- --     ----------------------------------------------------------------------------------------------------------\n-- --                                         \u0393 \u22a9 lam A t \u2236 (arr \u03c4\u1d62\u209b\u209b \u03c4\u209b\u209b)\n-- --   Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} {\u03c4\u209b\u209b : \u03c4 \u209b\u209b} {\u03c4\u2081\u209b\u209b : \u03c4\u2081 \u209b\u209b} {\u03c4\u2082\u209b\u209b : \u03c4\u2082 \u209b\u209b} ->\n-- --     Wf-ICtxt \u0393 -> ((\u03c4 ~> \u03c4\u2081) ~> \u03c4\u2082) \u2237' ((A \u27f6 A) \u27f6 A) -> \u03c4 \u2264\u2229 \u03c4\u2081 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2081 ->\n-- --     --------------------------------------------------------------------------\n-- --                         \u0393 \u22a9 Y A \u2236 (arr (arr \u03c4\u209b\u209b \u03c4\u2081\u209b\u209b) \u03c4\u2082\u209b\u209b)\n\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n-- \u22a2->\u039b-\u039bTerm : \u2200 {\u0393 t \u03c4} -> {\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4} -> (\u039bTerm (\u22a2->\u039b \u0393\u22a2t))\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = var x\u2081 x\u2082} = var\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = app \u0393\u22a2t \u0393\u22a2t\u2081} = app \u22a2->\u039b-\u039bTerm \u22a2->\u039b-\u039bTerm\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = abs L cf} = {!   !}\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = Y x} = Y\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n-- sub : \u2200 {A \u0393} {m : \u039b A} {\u03c4 \u03c4'} -> \u0393 \u22a9 m \u2236 \u03c4 -> \u03c4' \u2286[ A ] \u03c4 -> \u0393 \u22a9 m \u2236 \u03c4'\n-- sub (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u2286\u03c4\u1d62) \u03c4'\u2286\u03c4 = var wf-\u0393 \u03c4\u1d62\u2208\u0393 (\u2286\u1d62-trans (\u2286->\u2286\u1d62 \u03c4'\u2286\u03c4) \u03c4\u2286\u03c4\u1d62)\n-- sub (app \u0393\u22a9m\u2236\u03c4 x x\u2081 x\u2082) \u03c4'\u2286\u03c4 = {!   !}\n-- sub (abs L cf x x\u2081) \u03c4'\u2286\u03c4 = {!   !}\n-- sub (Y {\u03c4 = \u03c4} {\u03c4\u2081} {\u03c4\u2082} wf-\u0393 \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u1d62\u03c4) (arr x x\u2081 x\u2082 x\u2083) = {!   !}\n\n-- -- weakening-\u03c4 {A} {\u0393} {_} {\u03c4} {\u03c4'} {_} {\u03c4'\u209b\u209b} (var wf-\u0393 \u03c4\u1d62\u2208\u0393 \u03c4\u1d62\u2264\u2229\u03c4 \u03c4\u2237A) \u03c4\u2264\u03c4' \u03c4'\u2237A = var {A} {\u0393} {_} {\u03c4'} {\u03c4'\u209b\u209b} wf-\u0393 \u03c4\u1d62\u2208\u0393 (\u2229-trans \u03c4\u1d62\u2264\u2229\u03c4 \u03c4\u2264\u03c4') \u03c4'\u2237A\n-- -- weakening-\u03c4 (app \u0393\u22a9m\u2236\u03c4\u209b\u209b \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081 x) \u03c4\u2264\u03c4' \u03c4'\u2237A =\n-- --   app (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b (arr \u03c4\u2264\u03c4' \u2264\u2229-refl) (arr {!   !} \u03c4'\u2237A)) (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081 \u2264\u2229-refl {!   !}) (arr {!   !} \u03c4'\u2237A)\n-- --   -- (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b (arr \u03c4\u2264\u03c4' \u2264\u2229-refl) ?) (weakening-\u03c4 \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081 \u2264\u2229-refl ?) ?)\n-- -- weakening-\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) \u03c4\u2264\u03c4' \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m\u2236\u03c4\u209b\u209b \u0393\u22a9m\u2236\u03c4\u209b\u209b\u2081) \u03c4\u2264\u03c4' \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (abs L cf x x\u2081) \u03c4\u2264\u03c4' \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) (arr \u03c4\u2264\u03c4' \u03c4\u2264\u03c4'') \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) \u2229-nil \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) (\u2229-cons \u03c4\u2264\u03c4' \u03c4\u2264\u03c4'') \u03c4'\u2237A = {!   !}\n-- -- weakening-\u03c4 (Y x (arr (arr x\u2081 x\u2082) x\u2083) x\u2084 x\u2085) (\u2229-trans \u03c4\u2264\u03c4' \u03c4\u2264\u03c4'') \u03c4'\u2237A = {!   !}\n\n\n\n-- \u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c9~>\u03c4\u2082 (nil wf-\u0393) \u03c4\u2286\u03c4\u2082 nil) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2083\u03c4\u1d62~>\u03c4\u2085 (cons _ (app (Y _ \u03c4\u2082\u2286\u03c4\u2081 \u03c4\u2081\u2286\u03c4) (cons _ \u0393\u22a9m\u2236\u03c4~>\u03c4\u2081 (nil wf-\u0393)) \u03c4\u2083\u2286\u03c4\u2082 (cons \u03c4~>\u03c4\u2081\u2237A\u27f6A _)) \u0393\u22a9\u1d62Ym\u2236\u2229\u03c4\u1d62) \u03c4\u2084\u2286\u03c4\u2085 \u03c4\u2083\u03c4\u1d62\u2237A) (Y trm-m) = {!   !}\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- -- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 (arr (arr ::' ::'') ::''') \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m'\u2236\u03c4\u2082 x\u2084) x\u2085) (Y x\u2086) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app {\u03c4\u2082 = \u03c4\u2081} (Y {\u03c4 = \u03c4\u2082} {\u03c4\u2083} wf-\u0393 (arr (arr \u03c4\u2082\u2237A \u03c4\u2083\u2237A) _) \u03c4\u2264\u03c4\u2081 \u03c4\u2082\u2264\u03c4\u2081) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 _) (arr {A = A} \u03c4\u2081\u2237A \u03c4\u2237A)) (Y x\u2086) =\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2082 \u2237 \u03c4\u2081 \u2237 [])} {\u2229 (\u03c4\u2083 \u2237 \u03c4 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--       {!   !}\n--       (\u2229-\u2208 (there (here refl))))\n--     {!   !}\n--     (arr (arr (\u2229-cons \u03c4\u2082\u2237A (\u2229-cons \u03c4\u2081\u2237A \u2229-nil)) (\u2229-cons \u03c4\u2083\u2237A (\u2229-cons \u03c4\u2237A \u2229-nil))) \u03c4\u2237A)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082) x) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {A B \u0393} {m m' : \u039b (A \u27f6 B)} {n : \u039b A} {\u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081) (redL x\u2082 m->\u039b\u03b2m') =\n--     app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u039b\u03b2m') \u0393\u22a9m'n\u2236\u03c4\u2081 x x\u2081\n--   \u22a9->\u03b2-redL (\u2229-nil \u00acY-shape wf-\u0393) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (redR x m->\u039b\u03b2m') = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (abs L x) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (beta x x\u2081) = {!   !}\n--   \u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 (Y x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app {s = m} {\u03c4\u2082 = \u03c4} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y {_} {_} {\u03c4\u2082} {\u03c4\u2083} {\u03c4\u2081} wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A \u03c4\u2082\u2264\u2229\u03c4\u2083 \u03c4\u2081\u2264\u2229\u03c4\u2083) \u0393\u22a9m\u2236\u03c4\u2082~>\u03c4\u2083 x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n--   -- app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A {!   !} {!   !}) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n--   app {A = A \u27f6 A}\n--     (Y {_} {A} {\u2229 (\u03c4\u2081 \u2237 \u03c4\u2082 \u2237 \u03c4\u2083 \u2237 [])} {\u2229 (\u03c4 \u2237 \u03c4\u2083 \u2237 [])} {\u03c4}\n--       wf-\u0393\n--       {!   !}\n--       (\u2229-cons \u03c4\u2237A (\u2229-cons \u03c4\u2083\u2237A \u2229-nil))\n--       \u03c4\u2237A\n--       {!   !}\n--       {!   !})\n--     {!   !}\n--     (arr {!   !} {!   !})\n--     {!   !}\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n","old_contents":"module ITyping where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\n-- open import Data.Maybe\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.Core\n\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\nopen import Reduction\n\n\ndata IType : Set where\n  o : IType\n  _~>_ : IType -> IType -> IType\n  \u2229 : List IType -> IType\n\n\u03c9 = \u2229 []\n\n\u2229' : IType -> IType\n\u2229' x = \u2229 (x \u2237 [])\n\n~>-inj-l : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2081 \u2261 \u03c4\u2082\u2081\n~>-inj-l refl = refl\n\n~>-inj-r : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2261 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) -> \u03c4\u2081\u2082 \u2261 \u03c4\u2082\u2082\n~>-inj-r refl = refl\n\n\u2229-inj-cons : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> \u2229 \u03c4\u1d62 \u2261 \u2229 \u03c4\u2c7c\n\u2229-inj-cons refl = refl\n\n\u2229-inj : \u2200 {x y \u03c4\u1d62 \u03c4\u2c7c} -> \u2229 (x \u2237 \u03c4\u1d62) \u2261 \u2229 (y \u2237 \u03c4\u2c7c) -> x \u2261 y\n\u2229-inj refl = refl\n\n\n_\u225fTI_ : Decidable {A = IType} _\u2261_\no \u225fTI o = yes refl\no \u225fTI (_ ~> _) = no (\u03bb ())\no \u225fTI (\u2229 _) = no (\u03bb ())\n\n(_ ~> _) \u225fTI o = no (\u03bb ())\n(_ ~> _) \u225fTI (\u2229 _) = no (\u03bb ())\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) with \u03c4\u2081\u2081 \u225fTI \u03c4\u2082\u2081 | \u03c4\u2081\u2082 \u225fTI \u03c4\u2082\u2082\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> .\u03c4\u2081\u2082) | yes refl | yes refl = yes refl\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (.\u03c4\u2081\u2081 ~> \u03c4\u2082\u2082) | yes refl | no \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 = no (\u03bb eq \u2192 \u03c4\u2081\u2082\u2260\u03c4\u2082\u2082 (~>-inj-r eq))\n(\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u225fTI (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082) | no \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 | _ = no (\u03bb eq \u2192 \u03c4\u2081\u2081\u2260\u03c4\u2082\u2081 (~>-inj-l eq))\n\n(\u2229 _) \u225fTI o = no (\u03bb ())\n(\u2229 _) \u225fTI (_ ~> _) = no (\u03bb ())\n\u2229 [] \u225fTI \u2229 [] = yes refl\n\u2229 [] \u225fTI \u2229 (x \u2237 \u03c4\u2c7c) = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 [] = no (\u03bb ())\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) with x \u225fTI y | (\u2229 \u03c4\u1d62) \u225fTI (\u2229 \u03c4\u2c7c)\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 .\u03c4\u1d62) | yes refl | yes refl = yes refl\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (.x \u2237 \u03c4\u2c7c) | yes refl | no \u03c4\u1d62\u2260\u03c4\u2c7c = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c \u2192 \u03c4\u1d62\u2260\u03c4\u2c7c (\u2229-inj-cons \u2229x\u2237\u03c4\u1d62\u2261\u2229x\u2237\u03c4\u2c7c))\n\u2229 (x \u2237 \u03c4\u1d62) \u225fTI \u2229 (y \u2237 \u03c4\u2c7c) | no x\u2260y | _ = no (\u03bb \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c \u2192 x\u2260y (\u2229-inj \u2229x\u2237\u03c4\u1d62\u2261\u2229y\u2237\u03c4\u2c7c))\n\n\n\nICtxt = List (Atom \u00d7 IType)\n\n\ndata Wf-ICtxt : ICtxt -> Set where\n  nil : Wf-ICtxt []\n  cons : \u2200 {\u0393 x \u03c4} -> (x\u2209 : x \u2209 dom \u0393) -> Wf-ICtxt \u0393 ->\n    Wf-ICtxt ((x , \u03c4) \u2237 \u0393)\n\n\ndata _\u2237'_ : IType -> Type -> Set where\n  base : o \u2237' \u03c3\n  arr : \u2200 {\u03b4 \u03c4 A B} -> \u03b4 \u2237' A -> \u03c4 \u2237' B -> (\u03b4 ~> \u03c4) \u2237' (A \u27f6 B)\n  \u2229-nil : \u2200 {A} ->  \u03c9 \u2237' A\n  \u2229-cons : \u2200 {\u03c4\u1d62 \u03c4 A} ->  \u03c4 \u2237' A -> \u2229 \u03c4\u1d62 \u2237' A -> \u2229 (\u03c4 \u2237 \u03c4\u1d62) \u2237' A\n\n\ndata _\u2264\u2229_ : IType -> IType -> Set where\n  base : o \u2264\u2229 o\n  arr : \u2200 {\u03c4\u2081\u2081 \u03c4\u2081\u2082 \u03c4\u2082\u2081 \u03c4\u2082\u2082} -> \u03c4\u2081\u2082 \u2264\u2229 \u03c4\u2082\u2082 -> \u03c4\u2082\u2081 \u2264\u2229 \u03c4\u2081\u2081 -> (\u03c4\u2081\u2081 ~> \u03c4\u2081\u2082) \u2264\u2229 (\u03c4\u2082\u2081 ~> \u03c4\u2082\u2082)\n  \u2229-\u2208 : \u2200 {\u03c4 \u03c4\u1d62} -> \u03c4 \u2208 \u03c4\u1d62 -> \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4\n  \u2229-nil : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c9\n  \u2229-cons : \u2200 {\u03c4 \u03c4' \u03c4\u1d62} -> \u03c4 \u2264\u2229 \u03c4' -> \u03c4 \u2264\u2229 \u2229 \u03c4\u1d62 -> \u03c4 \u2264\u2229 \u2229 (\u03c4' \u2237 \u03c4\u1d62)\n  -- \u2229-trans : \u2200 {\u03c4\u2081 \u03c4\u2082 \u03c4\u2083} -> \u03c4\u2081 \u2264\u2229 \u03c4\u2082 -> \u03c4\u2082 \u2264\u2229 \u03c4\u2083 -> \u03c4\u2081 \u2264\u2229 \u03c4\u2083\n\n\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 : \u2200 {\u03c4\u1d62 \u03c4\u2c7c} -> \u03c4\u1d62 \u2286 \u03c4\u2c7c -> \u2229 \u03c4\u2c7c \u2264\u2229 \u2229 \u03c4\u1d62\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {[]} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-nil\n\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 {x \u2237 \u03c4\u1d62} \u03c4\u1d62\u2286\u03c4\u2c7c = \u2229-cons (\u2229-\u2208 (\u03c4\u1d62\u2286\u03c4\u2c7c (here refl))) (\u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2081} z \u2192 \u03c4\u1d62\u2286\u03c4\u2c7c (there z)))\n\n\u2264\u2229-refl : \u2200 {\u03c4} -> \u03c4 \u2264\u2229 \u03c4\n\u2264\u2229-refl {o} = base\n\u2264\u2229-refl {\u03c4 ~> \u03c4\u2081} = arr \u2264\u2229-refl \u2264\u2229-refl\n\u2264\u2229-refl {\u2229 []} = \u2229-nil\n\u2264\u2229-refl {\u2229 (x \u2237 x\u2081)} = \u2229\u03c4\u2c7c\u2264\u2229\u03c4\u1d62 (\u03bb {x\u2082} z \u2192 z)\n\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\n-- _\u2208?_ : \u2200 {a} {A : Set a} -> Atom -> List (Atom \u00d7 A) -> Maybe A\n-- a \u2208? [] = nothing\n-- a \u2208? (l \u2237 ist) with a \u225f proj\u2081 l\n-- ... | yes _ = just (proj\u2082 l)\n-- a \u2208? (l \u2237 ist) | no _ = a \u2208? ist\n\n-- \u22a2->\u039b : \u2200 {\u0393 m t} -> (List (Atom \u00d7 \u2115)) -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- \u22a2->\u039b {m = bv i} _ ()\n-- \u22a2->\u039b {m = fv x} bound \u0393\u22a2m\u2236t with x \u2208? bound\n-- \u22a2->\u039b {m = fv x} {t} bound \u0393\u22a2m\u2236t | just i = bv {t} i\n-- \u22a2->\u039b {m = fv x} bound \u0393\u22a2m\u2236t | nothing = fv x\n-- \u22a2->\u039b {m = lam m} bound (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) = lam \u03c4\u2081 (\u22a2->\u039b ((x , 0) \u2237 bound') (cf (\u2209-cons-l _ _ x\u2209)))\n--   where\n--   x = \u2203fresh (L ++ FV m)\n--   x\u2209 : x \u2209 (L ++ FV m)\n--   x\u2209 = \u2203fresh-spec (L ++ FV m)\n--\n--   bound' : List (Atom \u00d7 \u2115)\n--   bound' = Data.List.map (\u03bb a,i \u2192 (proj\u2081 a,i) , suc (proj\u2082 a,i)) bound\n--\n-- \u22a2->\u039b {m = app t1 t} bound (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b bound \u0393\u22a2s) (\u22a2->\u039b bound \u0393\u22a2t)\n-- \u22a2->\u039b {m = Y \u03c4} bound (Y x) = Y \u03c4\n--\n-- \u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n-- \u22a2->\u039b = \u22a2->\u039b []\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app t1 t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y x) = Y \u03c4\n\n\u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n\u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n-- ers : \u2200 {t} -> \u039b t -> PTerm\n-- ers (bv i) = bv i\n-- ers (fv x) = fv x\n-- ers (lam A \u039bt) = lam (ers \u039bt)\n-- ers (app \u039bs \u039bt) = app (ers \u039bs) (ers \u039bt)\n-- ers (Y t) = Y t\n\n\n-- data IType\u209b : IType -> Set where\n--   o : IType\u209b o\n--   arr : \u2200 {\u03c4 \u03c4'} -> IType\u209b \u03c4 -> IType\u209b \u03c4' -> IType\u209b (\u03c4 ~> \u03c4')\n--\n-- data IType\u209b\u209b : IType -> Set where\n--   o : IType\u209b\u209b o\n--   arr :  \u2200 {\u03c4 \u03c4'} -> IType\u209b\u209b \u03c4 -> IType\u209b\u209b \u03c4' -> IType\u209b\u209b (\u03c4 ~> \u03c4')\n--   \u2229-nil : IType\u209b\u209b \u03c9\n--   \u2229-cons : \u2200 {\u03c4 \u03c4\u1d62} -> IType\u209b \u03c4 ->  IType\u209b\u209b (\u2229 \u03c4\u1d62) -> IType\u209b\u209b (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n--\n-- \u03c4\u209b->\u03c4\u209b\u209b : \u2200 {\u03c4} -> IType\u209b \u03c4 -> IType\u209b\u209b \u03c4\n-- \u03c4\u209b->\u03c4\u209b\u209b o = o\n-- \u03c4\u209b->\u03c4\u209b\u209b (arr \u03c4\u209b \u03c4\u209b\u2081) = arr (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b) (\u03c4\u209b->\u03c4\u209b\u209b \u03c4\u209b\u2081)\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n\n\ndata Y-shape : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  intro\u2081 : \u2200 {A m} -> Y-shape (app (Y A) m)\n  intro\u2082 : \u2200 {A m} -> Y-shape (app m (app (Y A) m))\n\ndata _\u22a9_\u2236_ : \u2200 {A} -> ICtxt -> \u039b A -> IType -> Set where\n  var : \u2200 {A \u0393 x \u03c4} {\u03c4\u1d62 : List IType} -> (wf-\u0393 : Wf-ICtxt \u0393) -> (\u03c4\u1d62\u2208\u0393 : (x , (\u2229 \u03c4\u1d62)) \u2208 \u0393) -> (\u03c4\u1d62\u2264\u2229\u03c4 : \u2229 \u03c4\u1d62 \u2264\u2229 \u03c4) ->\n    \u03c4 \u2237' A -> \u0393 \u22a9 fv {A} x \u2236 \u03c4\n  app : \u2200 {A B \u0393 s t \u03c4\u2081 \u03c4\u2082} -> \u0393 \u22a9 s \u2236 (\u03c4\u2081 ~> \u03c4\u2082) -> \u0393 \u22a9 t \u2236 \u03c4\u2081 -> (\u03c4\u2081 ~> \u03c4\u2082) \u2237' (A \u27f6 B) -> \u03c4\u2081 \u2237' A ->\n    \u0393 \u22a9 (app {A} {B} s t) \u2236 \u03c4\u2082\n  \u2229-nil : \u2200 {A \u0393} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) -> \u0393 \u22a9 m\u2005 \u2236 \u03c9\n  \u2229-cons : \u2200 {A \u0393 \u03c4 \u03c4\u1d62} {m : \u039b A} -> (\u00acY-shape : \u00ac Y-shape m) -> (wf-\u0393 : Wf-ICtxt \u0393) ->\n    \u0393 \u22a9 m\u2005 \u2236 \u03c4 -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 \u03c4\u1d62) -> \u0393 \u22a9 m\u2005 \u2236 (\u2229 (\u03c4 \u2237 \u03c4\u1d62))\n  abs : \u2200 {A B \u0393 \u03c4\u1d62 \u03c4} (L : FVars) -> \u2200 {t : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> ((x , \u2229 \u03c4\u1d62) \u2237 \u0393) \u22a9 \u039b[ 0 >> fv {A} x ] t \u2236 \u03c4 ) -> \u2229 \u03c4\u1d62 \u2237' A -> \u03c4 \u2237' B ->\n    \u0393 \u22a9 lam A t \u2236 (\u2229 \u03c4\u1d62 ~> \u03c4)\n  Y : \u2200 {\u0393 A \u03c4 \u03c4\u2081 \u03c4\u2082} -> Wf-ICtxt \u0393 -> \u03c4 \u2237' A -> \u03c4\u2081 \u2237' A -> \u03c4\u2082 \u2237' A ->\n    \u0393 \u22a9 Y A \u2236 ((\u03c4 ~> \u03c4\u2081) ~> \u03c4\u2082)\n\ndata \u039bTerm : \u2200 {\u03c4} -> \u039b \u03c4 -> Set where\n  var : \u2200 {A x} -> \u039bTerm (fv {A} x)\n  lam : \u2200 {A B} (L : FVars) -> \u2200 {e : \u039b B} ->\n    (cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> \u039bTerm (\u039b[ 0 >> fv {A} x ] e)) -> \u039bTerm (lam A e)\n  app : \u2200 {A B} {e\u2081 : \u039b (A \u27f6 B)} {e\u2082 : \u039b A} -> \u039bTerm e\u2081 -> \u039bTerm e\u2082 -> \u039bTerm (app e\u2081 e\u2082)\n  Y : \u2200 {t} -> \u039bTerm (Y t)\n\n\n-- \u22a2->\u039b-\u039bTerm : \u2200 {\u0393 t \u03c4} -> {\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4} -> (\u039bTerm (\u22a2->\u039b \u0393\u22a2t))\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = var x\u2081 x\u2082} = var\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = app \u0393\u22a2t \u0393\u22a2t\u2081} = app \u22a2->\u039b-\u039bTerm \u22a2->\u039b-\u039bTerm\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = abs L cf} = {!   !}\n-- \u22a2->\u039b-\u039bTerm {\u0393\u22a2t = Y x} = Y\n\n\ndata _->\u039b\u03b2_ : \u2200 {\u03c4} -> \u039b \u03c4 \u219d \u039b \u03c4 where\n  redL : \u2200 {A B} {n : \u039b A} {m m' : \u039b (A \u27f6 B)} -> \u039bTerm n -> m ->\u039b\u03b2 m' -> app m n ->\u039b\u03b2 app m' n\n  redR : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n n' : \u039b A} -> \u039bTerm m -> n ->\u039b\u03b2 n' -> app m n ->\u039b\u03b2 app m n'\n  abs : \u2200 L {A B} {m m' : \u039b B} -> ( \u2200 {x} -> x \u2209 L -> \u039b[ 0 >> fv {A} x ] m ->\u039b\u03b2 \u039b[ 0 >> fv {A} x ] m' ) ->\n    lam A m ->\u039b\u03b2 lam A m'\n  beta : \u2200 {A B} {m : \u039b (A \u27f6 B)} {n : \u039b A} -> \u039bTerm (lam A m) -> \u039bTerm n -> app (lam A m) n ->\u039b\u03b2 (\u039b[ 0 >> n ] m)\n  Y : \u2200 {A} {m : \u039b (A \u27f6 A)} -> \u039bTerm m -> app (Y A) m ->\u039b\u03b2 app m (app (Y A) m)\n\n\n\n\u22a9->\u03b2 : \u2200 {A \u0393} {m m' : \u039b A} {\u03c4} -> \u0393 \u22a9 m' \u2236 \u03c4 -> m ->\u039b\u03b2 m' -> \u0393 \u22a9 m \u2236 \u03c4\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL trm-n m->\u03b2m') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR trm-m n->\u03b2n') = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n\u22a9->\u03b2 (app {s = m} \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (app (Y wf-\u0393 \u03c4\u2082\u2237A \u03c4\u2083\u2237A \u03c4\u2081\u2237A) \u0393\u22a9m x \u03c4\u2082~>\u03c4\u2083\u2237A) (arr {A = A} _ \u03c4\u2237A) _) (Y trm-m) =\n  app {A = A \u27f6 A} (Y wf-\u0393 \u03c4\u2081\u2237A \u03c4\u2237A \u03c4\u2237A) \u0393\u22a9m\u2236\u03c4\u2081~>\u03c4 (arr (arr \u03c4\u2081\u2237A \u03c4\u2237A) \u03c4\u2237A) (arr \u03c4\u2081\u2237A \u03c4\u2237A)\n\u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-nil \u00acY-shape wf-\u0393) x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n\u22a9->\u03b2 (app \u0393\u22a9m\u2236\u03c4~>\u03c4' (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'') x \u03c4\u2237A) (Y trm-m) = \u22a5-elim (\u00acY-shape intro\u2081)\n\u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redR x m->\u03b2m') = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (abs L x) = {!   !}\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 {\u03c4 = \u03c4} (app \u0393\u22a9m\u2236\u03c4'~>\u03c4 (app (Y {\u03c4 = \u03c4'} wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 \u03c4\u2082\u2237) \u0393\u22a9m\u2236\u03c4'~>\u03c4')) (Y trm-m) =\n--     app (Y wf-\u0393 \u03c4\u2237 \u03c4\u2081\u2237 {!   !}) \u0393\u22a9m\u2236\u03c4'~>\u03c4'\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-nil \u00acY-shape wf-\u0393 trm-m)) (Y x) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4\u2081 \u0393\u22a9m'\u2236\u03c4\u2082)) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2081)\n-- \u22a9->\u03b2 (\u2229-nil \u00acY-shape wf-\u0393 trm-m) (Y x) = \u22a5-elim (\u00acY-shape intro\u2082)\n-- \u22a9->\u03b2 (\u2229-cons \u00acY-shape wf-\u0393 trm-m \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (Y x\u2081) = \u22a5-elim (\u00acY-shape intro\u2082)\n--\n--\n\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (redL x m->\u03b2m') = \u22a9->\u03b2-redL \u0393\u22a9m'\u2236\u03c4 m->\u03b2m'\n--   where\n--   \u22a9->\u03b2-redL : \u2200 {\u0393 m m' n \u03c4} -> \u0393 \u22a9 app m' n \u2236 \u03c4 -> m ->\u03b2 m' -> \u0393 \u22a9 app m n \u2236 \u03c4\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redL x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2-redL \u0393\u22a9m'n\u2236\u03c4 m->\u03b2m'') \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-n\u2082)) (redL trm-n\u2081 m->\u03b2m'') =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2-redL (\u0393\u22a9m''n\u2081n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 m->\u03b2m'')) wf-\u0393 (app (app (->\u03b2-Term-l m->\u03b2m'') trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (redR x\u2081 m->\u03b2m'') = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (redR x\u2081 m->\u03b2m'')) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-e\u2082) trm-n\u2082)) (redR trm-m\u2081 n\u2081->e\u2082) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081e\u2082n\u2082\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (redR trm-m\u2081 n\u2081->e\u2082))) wf-\u0393 (app (app trm-m\u2081 (->\u03b2-Term-l n\u2081->e\u2082)) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (abs L cf) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (abs L cf)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app trm-lam-m'' trm-n\u2081)) (abs L cf) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m''n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (abs L cf))) wf-\u0393 (app (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (cf x\u2209L))) trm-n\u2081)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (beta trm-lam-m\u2081 trm-n\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (beta trm-lam-m\u2081 trm-n\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 wf-\u0393 (app trm-m\u2081^n\u2081 trm-n\u2082)) (beta trm-lam-m\u2081 trm-n\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081^n\u2081n\u2082\u2236\u03c4\u2081 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2082 (beta trm-lam-m\u2081 trm-n\u2081))) wf-\u0393 (app (app trm-lam-m\u2081 trm-n\u2081) trm-n\u2082)\n--   \u22a9->\u03b2-redL (app \u0393\u22a9m'n\u2236\u03c4 \u0393\u22a9m'n\u2236\u03c4\u2081) (Y trm-m\u2081) = app (\u22a9->\u03b2 \u0393\u22a9m'n\u2236\u03c4 (Y trm-m\u2081)) \u0393\u22a9m'n\u2236\u03c4\u2081\n--   \u22a9->\u03b2-redL (\u2229-intro \u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 wf-\u0393 (app (app _ trm-Ym\u2081) trm-n\u2081)) (Y trm-m\u2081) =\n--     \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9m\u2081Ym\u2081n\u2081\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redL trm-n\u2081 (Y trm-m\u2081))) wf-\u0393 (app trm-Ym\u2081 trm-n\u2081)\n--\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4 \u0393\u22a9m'\u2236\u03c4\u2081) (redR trm-n m->\u03b2m') = app \u0393\u22a9m'\u2236\u03c4 (\u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4\u2081 m->\u03b2m')\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mn'\u2236\u03c4\u1d62 wf-\u0393 trm-mn') (redR trm-m n->\u03b2n') =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mn'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (redR trm-m n->\u03b2n')) wf-\u0393 (app trm-m (->\u03b2-Term-l n->\u03b2n'))\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9lam-m'\u2236\u03c4\u1d62 wf-\u0393 trm-lam-x) (abs L x\u2082) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9lam-m'\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (abs L x\u2082)) wf-\u0393 (lam L (\u03bb x\u2209L \u2192 ->\u03b2-Term-l (x\u2082 x\u2209L)))\n-- \u22a9->\u03b2 (abs L cf) (abs L\u2081 x) = abs (L ++ L\u2081) (\u03bb x\u2209L \u2192 \u22a9->\u03b2 (cf (\u2209-cons-l _ _ x\u2209L)) (x (\u2209-cons-r L _ x\u2209L)))\n-- \u22a9->\u03b2 \u0393\u22a9m'\u2236\u03c4 (beta x x\u2081) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (app \u0393\u22a9Ym\u2236\u03c4' \u0393\u22a9Ym\u2236\u03c4'')) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 (app \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = []} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n-- \u22a9->\u03b2 {\u0393} (app {s = m} \u0393\u22a9m'\u2236\u03c4'~>\u03c4 (\u2229-intro {\u03c4\u1d62 = x \u2237 \u03c4\u1d62} \u0393\u22a9Ym\u2236\u03c4\u1d62 wf-\u0393 trm-Ym)) (Y trm-m) = {!   !}\n--   where\n--   \u0393\u22a9m\u2236x~>x : \u0393 \u22a9 m \u2236 (x ~> x)\n--   \u0393\u22a9m\u2236x~>x = {!   !}\n-- \u22a9->\u03b2 (\u2229-intro \u0393\u22a9mYm\u2236\u03c4\u1d62 wf-\u0393 (app _ trm-Ym)) (Y trm-m) =\n--   \u2229-intro (\u03bb \u03c4\u2208\u03c4\u1d62 \u2192 \u22a9->\u03b2 (\u0393\u22a9mYm\u2236\u03c4\u1d62 \u03c4\u2208\u03c4\u1d62) (Y trm-m)) wf-\u0393 trm-Ym\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1bece65efc64700949a5946cdeb3d2ed8a5f670f","subject":"NomPa: Update use of NP library","message":"NomPa: Update use of NP library\n","repos":"np\/NomPa","old_file":"lib\/NomPa\/Implem\/LogicalRelation\/Internals.agda","new_file":"lib\/NomPa\/Implem\/LogicalRelation\/Internals.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nimport Level as L\nopen import Data.Bool.NP\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Sum.NP\nopen import Data.Sum.Logical\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Nullary.Negation using (contraposition)\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP as Bin\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Equivalence as \u21d4 using (_\u21d4_; equivalence; module Equivalence)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nimport Data.Nat.NP as \u2115\nopen \u2115 renaming (_==_ to _==\u2115_)\nimport Data.Nat.Properties as \u2115\nopen import NomPa.Record using (module NomPa)\nopen import NomPa.Implem\nimport NaPa\nopen NaPa using (\u2208-uniq)\nopen import NomPa.Worlds\nimport NomPa.Derived\n\nmodule NomPa.Implem.LogicalRelation.Internals where\n\nopen \u2261 using (_\u2261_; _\u2262_; \u27e6\u2261\u27e7)\nopen \u2261.\u2261-Reasoning\nopen NomPa.Implem.Internals\nopen NomPa NomPa.Implem.nomPa using (worldSym; Name\u00f8-elim; suc\u1d3a; suc\u1d3a\u2191)\n\n-- move to record\npred\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 +1) \u2192 Name \u03b1\npred\u1d3a = subtract\u1d3a 1\n\n\u2262\u2192\u2713-not-==\u2115 : \u2200 {x y} \u2192 x \u2262 y \u2192 \u2713 (not (x ==\u2115 y))\n\u2262\u2192\u2713-not-==\u2115 \u00acp = \u2713-\u00ac-not (\u00acp \u2218 \u2115.==.sound _ _)\n\n#\u21d2\u2209 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 b \u2209 \u03b1\n#\u21d2\u2209 (_ #\u00f8)      = id\n#\u21d2\u2209 (suc#\u2237 b#\u03b1) = #\u21d2\u2209 b#\u03b1\n#\u21d2\u2209 (0# _)      = id\n\n#\u21d2\u2262 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 (x : Name \u03b1) \u2192 binder\u1d3a x \u2262 b\n#\u21d2\u2262 b# (b , b\u2208) \u2261.refl = #\u21d2\u2209 b# b\u2208\n\nbinder\u1d3a-injective : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u2192 x \u2261 y\nbinder\u1d3a-injective {\u03b1} {_ , p\u2081} {a , p\u2082} eq rewrite eq = \u2261.cong (_,_ a) (\u2208-uniq \u03b1 p\u2081 p\u2082)\n\npf-irr\u2032 : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x x\u2032 : Name \u03b1} {y y\u2032 : Name \u03b2}\n         \u2192 binder\u1d3a x \u2261 binder\u1d3a x\u2032 \u2192 binder\u1d3a y \u2261 binder\u1d3a y\u2032 \u2192 \u211b x y \u2192 \u211b x\u2032 y\u2032\npf-irr\u2032 {\u03b1} {\u03b2} \u211b {x , x\u2208} {x\u2032 , x\u2032\u2208} {y , y\u2208} {y\u2032 , y\u2032\u2208} eq\u2081 eq\u2082 x\u1d63y\n  rewrite eq\u2081 | eq\u2082 | \u2208-uniq \u03b1 x\u2208 x\u2032\u2208 | \u2208-uniq \u03b2 y\u2208 y\u2032\u2208 = x\u1d63y\n\npf-irr : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x y x\u2208 x\u2208\u2032 y\u2208 y\u2208\u2032}\n         \u2192 \u211b (x , x\u2208) (y , y\u2208) \u2192 \u211b (x , x\u2208\u2032) (y , y\u2208\u2032)\npf-irr \u211b = pf-irr\u2032 \u211b \u2261.refl \u2261.refl\n\nPreserve-\u2248 : \u2200 {a b \u2113 \u2113a \u2113b} {A : Set a} {B : Set b} \u2192 Rel A \u2113a \u2192 Rel (B) \u2113b \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2248 _\u2248a_ _\u2248b_ _\u223c_ = \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u223c x\u2082 \u2192 y\u2081 \u223c y\u2082 \u2192 (x\u2081 \u2248a y\u2081) \u21d4 (x\u2082 \u2248b y\u2082)\n\n==\u2115\u21d4\u2261 : \u2200 {x y} \u2192 \u2713 (x ==\u2115 y) \u21d4 x \u2261 y\n==\u2115\u21d4\u2261 {x} {y} = equivalence (\u2115.==.sound _ _) \u2115.==.reflexive\n\n\u2261-on-name\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u21d4 x \u2261 y\n\u2261-on-name\u21d4\u2261 {\u03b1} {x} {y} = equivalence binder\u1d3a-injective (\u2261.cong binder\u1d3a)\n\nT\u21d4T\u2192\u2261 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u21d4 \u2713 b\u2082 \u2192 b\u2081 \u2261 b\u2082\nT\u21d4T\u2192\u2261 {1\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\nT\u21d4T\u2192\u2261 {1\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.to b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.from b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\n\n==\u1d3a\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 \u2713 (x ==\u1d3a y) \u21d4 x \u2261 y\n==\u1d3a\u21d4\u2261 = \u2261-on-name\u21d4\u2261 \u27e8\u2218\u27e9 ==\u2115\u21d4\u2261 where open \u21d4 renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n-- Preserving the equalities also mean that the relation is functional and injective\nPreserve-\u2261 : \u2200 {a b \u2113} {A : Set a} {B : Set b} \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2261 _\u223c_ = Preserve-\u2248 _\u2261_ _\u2261_ _\u223c_\n\nexport\u1d3a?-nothing : \u2200 {b \u03b1} (x : Name (b \u25c5 \u03b1)) \u2192 \u2713 (binder\u1d3a x ==\u2115 b) \u2192 export\u1d3a? {b} x \u2261 nothing\nexport\u1d3a?-nothing {b} {\u03b1} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u2261.refl\n... | 0\u2082 | _ = \u22a5-elim p\n\nexport\u1d3a?-just : \u2200 {\u03b1 b} (x : Name (b \u25c5 \u03b1)) {x\u2208} \u2192 \u2713 (not (binder\u1d3a x ==\u2115 b)) \u2192 export\u1d3a? {b} x \u2261 just (binder\u1d3a x , x\u2208)\nexport\u1d3a?-just {\u03b1} {b} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u22a5-elim p\n... | 0\u2082 | _ = \u2261.cong just (binder\u1d3a-injective \u2261.refl)\n\nrecord \u27e6World\u27e7 (\u03b1\u2081 \u03b1\u2082 : World) : Set\u2081 where\n  constructor _,_\n  field\n    _\u223c_ : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set\n    \u223c-pres-\u2261 : Preserve-\u2261 _\u223c_\n\n  \u223c-inj : \u2200 {x y z} \u2192 x \u223c z \u2192 y \u223c z \u2192 x \u2261 y\n  \u223c-inj p q = Equivalence.from (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n  \u223c-fun : \u2200 {x y z} \u2192 x \u223c y \u2192 x \u223c z \u2192 y \u2261 z\n  \u223c-fun p q = Equivalence.to (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n\n  \u223c-\u2208-uniq : \u2200 {x\u2081 x\u2082} {x\u2081\u2208\u2032 x\u2082\u2208\u2032} \u2192\n        x\u2081 \u223c x\u2082 \u2261 (binder\u1d3a x\u2081 , x\u2081\u2208\u2032) \u223c (binder\u1d3a x\u2082 , x\u2082\u2208\u2032)\n  \u223c-\u2208-uniq {_ , x\u2081\u2208} {_ , x\u2082\u2208} {x\u2081\u2208\u2032} {x\u2082\u2208\u2032} = \u2261.ap\u2082 (\u03bb x\u2081\u2208 x\u2082\u2208 \u2192 (_ , x\u2081\u2208) \u223c (_ , x\u2082\u2208)) (\u2208-uniq \u03b1\u2081 x\u2081\u2208 x\u2081\u2208\u2032) (\u2208-uniq \u03b1\u2082 x\u2082\u2208 x\u2082\u2208\u2032)\n\n\u27e6sym\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 \u27e6World\u27e7 \u03b1\u2082 \u03b1\u2081\n\u27e6sym\u27e7 (\u211b , \u211b-pres-\u2261) = \u211b-sym , \u211b-sym-pres-\u2261\n  where \u211b-sym = flip \u211b\n        \u211b-sym-pres-\u2261 : Preserve-\u2261 \u211b-sym\n        \u211b-sym-pres-\u2261 p q = \u21d4.sym (\u211b-pres-\u2261 p q)\n\nrecord \u27e6Binder\u27e7 (b\u2081 b\u2082 : Binder) : Set where\n\nmodule \u27e6\u25c5\u27e7 (b\u2081 b\u2082 : Binder) {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  data _\u211b_ (x : Name (b\u2081 \u25c5 \u03b1\u2081)) (y : Name (b\u2082 \u25c5 \u03b1\u2082)) : Set where\n    here  : (x\u2261b\u2081 : binder\u1d3a x \u2261 b\u2081) (y\u2261b\u2082 : binder\u1d3a y \u2261 b\u2082) \u2192 x \u211b y\n    there : \u2200 (x\u2262b\u2081 : binder\u1d3a x \u2262 b\u2081) (y\u2262b\u2082 : binder\u1d3a y \u2262 b\u2082) {x\u2208 y\u2208} (x\u223cy : (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a y , y\u2208)) \u2192 x \u211b y\n\n  \u211b-inj : \u2200 {x y z} \u2192 x \u211b z \u2192 y \u211b z \u2192 x \u2261 y\n  \u211b-inj (here x\u2261b\u2081 _)    (here y\u2261b\u2081 _)     = binder\u1d3a-injective (\u2261.trans x\u2261b\u2081 (\u2261.sym y\u2261b\u2081))\n  \u211b-inj (here _ z\u2261b\u2082)    (there _ z\u2262b\u2082 _)  = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj (there _ z\u2262b\u2082 _) (here _ z\u2261b\u2082)     = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj {x} {y} {z} (there x\u2262b\u2081 z\u2262b\u2082 {x\u2208} {z\u2208} x\u223cz) (there y\u2262b\u2081 _ {y\u2208} {z\u2208\u2032} y\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-inj x\u223cz (\u2261.tr (\u03bb z\u2208 \u2192 (binder\u1d3a y , y\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2082 z\u2208\u2032 z\u2208) y\u223cz )))\n\n  \u211b-fun : \u2200 {x y z} \u2192 x \u211b y \u2192 x \u211b z \u2192 y \u2261 z\n  \u211b-fun (here _ y\u2261b\u2082)  (here _ z\u2261b\u2082)    = binder\u1d3a-injective (\u2261.trans y\u2261b\u2082 (\u2261.sym z\u2261b\u2082))\n  \u211b-fun (here p _)     (there \u00acp _ _)    = \u22a5-elim (\u00acp p)\n  \u211b-fun (there \u00acp _ _) (here p _)        = \u22a5-elim (\u00acp p)\n  \u211b-fun {x} {y} {z} (there x\u2262b\u2081 y\u2262b\u2082 {x\u2208} {y\u2208} x\u223cy) (there p q {x\u2208\u2032} {z\u2208} x\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-fun x\u223cy (\u2261.tr (\u03bb x\u2208 \u2192 (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2081 x\u2208\u2032 x\u2208) x\u223cz )))\n\n  \u211b-pres-\u2261 : Preserve-\u2261 _\u211b_\n  \u211b-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 x\u2081 \u211b x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b-inj x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 y\u2081 \u211b y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n  \u27e6world\u27e7 : \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n  \u27e6world\u27e7 = _\u211b_ , \u211b-pres-\u2261\n\nhere\u2032 : \u2200 {\u03b1\u2081 \u03b1\u2082 b\u2081 b\u2082 pf\u2081 pf\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} \u2192 \u27e6\u25c5\u27e7._\u211b_ b\u2081 b\u2082 \u03b1\u1d63 (b\u2081 , pf\u2081) (b\u2082 , pf\u2082)\nhere\u2032 = \u27e6\u25c5\u27e7.here \u2261.refl \u2261.refl\n\n_\u27e6\u25c5\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u25c5_ _\u25c5_\n_\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63 = \u27e6\u25c5\u27e7.\u27e6world\u27e7 b\u2081 b\u2082 \u03b1\u1d63\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 (\u211b , _) x\u2081 x\u2082 = \u211b x\u2081 x\u2082\n\n\u27e6zero\u1d2e\u27e7 : \u27e6Binder\u27e7 zero\u1d2e zero\u1d2e\n\u27e6zero\u1d2e\u27e7 = _\n\n\u27e6suc\u1d2e\u27e7 : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) suc\u1d2e suc\u1d2e\n\u27e6suc\u1d2e\u27e7 = _\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = (\u03bb _ _ \u2192 \u22a5) , (\u03bb())\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 x\u1d63 y\u1d63 =\n   \u27e6\ud835\udfda\u27e7-Props.reflexive (T\u21d4T\u2192\u2261 (sym ==\u1d3a\u21d4\u2261 \u27e8\u2218\u27e9 \u223c-pres-\u2261 x\u1d63 y\u1d63 \u27e8\u2218\u27e9 ==\u1d3a\u21d4\u2261)) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  open \u21d4 using (sym) renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n\u27e6name\u1d2e\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) name\u1d2e name\u1d2e\n\u27e6name\u1d2e\u27e7 _ _ = here\u2032\n\n\u27e6binder\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) binder\u1d3a binder\u1d3a\n\u27e6binder\u1d3a\u27e7 _ _ = _\n\n\u27e6nothing\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082}\n            \u2192 x \u2261 nothing \u2192 y \u2261 nothing \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6nothing\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6nothing\u27e7\n\n\u27e6just\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082} {x\u2032 y\u2032}\n            \u2192 x \u2261 just x\u2032 \u2192 y \u2261 just y\u2032 \u2192 A\u1d63 x\u2032 y\u2032 \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6just\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6just\u27e7\n\n\u27e6export\u1d3a?\u27e7 : (\u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) (\u03bb {b} \u2192 export\u1d3a? {b}) (\u03bb {b} \u2192 export\u1d3a? {b})\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.here e\u2081 e\u2082)\n  = \u27e6nothing\u27e7\u2032 (export\u1d3a?-nothing x\u2081 (\u2115.==.reflexive e\u2081)) (export\u1d3a?-nothing x\u2082 (\u2115.==.reflexive e\u2082))\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.there x\u2081\u2262b\u2081 x\u2082\u2262b\u2082 x\u2081\u223cx\u2082)\n  = \u27e6just\u27e7\u2032 (export\u1d3a?-just x\u2081 (\u2262\u2192\u2713-not-==\u2115 x\u2081\u2262b\u2081))\n           (export\u1d3a?-just x\u2082 (\u2262\u2192\u2713-not-==\u2115 x\u2082\u2262b\u2082)) x\u2081\u223cx\u2082\n\n_\u27e6\u2286\u27e7b_ : \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a x) (coerce\u1d3a y)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x y} \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) x y \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7b \u03b2\u1d63) x y\nun\u27e6\u2286\u27e7 (mk x) {a} {b} c = x {a} {b} c\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {mk f\u2081} {mk f\u2082} f {mk g\u2081} {mk g\u2082} g\n  = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {coerce\u1d3a (mk f\u2081) x\u2081} {coerce\u1d3a (mk f\u2082) x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 ()\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk \u03bb { {_ , ()} }\n\n_\u27e6#\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) _#_ _#_\n_\u27e6#\u27e7_ _ _ _ _ = \u22a4\n\n_\u27e6#\u00f8\u27e7 : (\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u27e6\u00f8\u27e7) _#\u00f8 _#\u00f8\n_\u27e6#\u00f8\u27e7 _ = _\n\n\u27e6suc#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 (\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6#\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) suc# suc#\n\u27e6suc#\u27e7 _ _ _ = _\n\n\u27e6\u2286-#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                  (b\u1d63 \u27e6#\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) \u2286-# \u2286-#\n\u27e6\u2286-#\u27e7 _ {b\u2081} {b\u2082} _ {b\u2081#\u03b1\u2081} {b\u2082#\u03b1\u2082} _ = mk (\u03bb {x\u2081} {x\u2082} \u2192 \u27e6\u25c5\u27e7.there (#\u21d2\u2262 b\u2081#\u03b1\u2081 x\u2081) (#\u21d2\u2262 b\u2082#\u03b1\u2082 x\u2082))\n\n\u27e6\u2286-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b2\u1d63)) \u2286-\u25c5 \u2286-\u25c5\n\u27e6\u2286-\u25c5\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {b\u2081} {b\u2082} _ {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} (mk f) = mk g where\n  open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n  g : \u2200 {x\u2081 x\u2082} \u2192 _\u211b_ \u03b1\u1d63 x\u2081 x\u2082 \u2192 _\u211b_ \u03b2\u1d63 (coerce\u1d3a (\u2286-\u25c5 b\u2081 \u03b1\u2286\u03b2\u2081) x\u2081) (coerce\u1d3a (\u2286-\u25c5 b\u2082 \u03b1\u2286\u03b2\u2082) x\u2082)\n  g (here x\u2261b\u2081 y\u2261b\u2082)       = here x\u2261b\u2081 y\u2261b\u2082\n  g (there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = there x\u2262b\u2081 y\u2262b\u2082 (f x\u223cy)\n\ncong\u2115 : \u2200 {\u03b1 \u03b2 : World} {x y : Name \u03b1} (f : \u2115 \u2192 \u2115) {fx\u2208 fy\u2208} \u2192 x \u2261 y \u2192 _\u2261_ {A = Name \u03b2} (f (binder\u1d3a x) , fx\u2208) (f (binder\u1d3a y) , fy\u2208)\ncong\u2115 f = binder\u1d3a-injective \u2218 \u2261.cong (f \u2218 binder\u1d3a)\n\nmodule \u27e6+1\u27e7 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63 renaming (_\u223c_ to \u211b; \u223c-inj to \u211b-inj; \u223c-fun to \u211b-fun)\n  \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set\n  \u211b+1 x y = \u27e6Name\u27e7 \u03b1\u1d63 (pred\u1d3a x) (pred\u1d3a y)\n{-\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x x\u2208 y y\u2208} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (suc x , ) (sucN y)\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x y} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (sucN x) (sucN y)\n-}\n\n  \u211b+1-inj : \u2200 {x y z} \u2192 \u211b+1 x z \u2192  \u211b+1 y z \u2192 x \u2261 y\n  -- \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} (suc a) (suc b) = {!!}\n  \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-inj a b)\n  \u211b+1-inj {zero , ()} _ _\n  \u211b+1-inj {_} {zero , ()} _ _\n  \u211b+1-inj {_} {_} {zero , ()} _ _\n  \u211b+1-fun : \u2200 {x y z} \u2192 \u211b+1 x y \u2192  \u211b+1 x z \u2192 y \u2261 z\n  \u211b+1-fun {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-fun a b)\n  \u211b+1-fun {zero , ()} _ _\n  \u211b+1-fun {_} {zero , ()} _ _\n  \u211b+1-fun {_} {_} {zero , ()} _ _\n\n  \u211b+1-pres-\u2261 : Preserve-\u2261 \u211b+1\n  -- \u211b+1-pres-\u2261 x y = equivalence {!\u211b+1-inj!} {!!}\n  \u211b+1-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     -- factor this move\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b+1-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 \u211b+1 x\u2081 x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b+1-inj {x\u2081} {y\u2081} {x\u2082} x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 \u211b+1 y\u2081 y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 \u03b1\u1d63 = \u211b+1 , \u211b+1-pres-\u2261 where open \u27e6+1\u27e7 \u03b1\u1d63\n\nopen WorldOps worldSym\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 \u03b1\u1d63 = \u27e6zero\u1d2e\u27e7 \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)\n\n_\u27e6\u2191\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6\u2191\u27e7 k) \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082)\n                 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 _  (\u27e6\u25c5\u27e7.here () _)\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b1\u1d63 (\u27e6\u25c5\u27e7.there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63) x\u223cy\n\n\u27e6\u00f8+1\u2286\u00f8\u27e7 : (\u27e6\u00f8\u27e7 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u00f8+1\u2286\u00f8 \u00f8+1\u2286\u00f8\n\u27e6\u00f8+1\u2286\u00f8\u27e7 = mk (\u03bb {x} _ \u2192 Name\u00f8-elim (pred\u1d3a x))\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 _ \u03b2\u1d63 \u03b1\u2286\u03b2\u1d63 = mk (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) \u2218 un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk helper where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper (here x\u2261b\u2081 y\u2261b\u2082) = here x\u2261b\u2081 y\u2261b\u2082\n  helper {x\u2081 , _} {x\u2082 , _} (there x\u2262b\u2081 y\u2262b\u2082 {p\u2081} {p\u2082} x\u223cy)\n     = there x\u2262b\u2081 y\u2262b\u2082 {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081) x\u2081 p\u2081}\n                        {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082) x\u2082 p\u2082}\n                        (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 x\u223cy))\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {add\u1d3a 1 x\u2081} {add\u1d3a 1 x\u2082}) \n\n\u22620 : \u2200 {\u03b1} (x : Name (\u03b1 +1)) \u2192 binder\u1d3a x \u2262 0\n\u22620 (0 , ()) \u2261.refl\n\u22620 (suc _ , _) ()\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 there (\u22620 x\u2081) (\u22620 x\u2082)) where open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ \u03b2\u1d63 \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b2\u1d63 (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (there (\u03bb()) (\u03bb()) x\u1d63))) where\n  open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ \u03b2\u1d63 \u03b1+\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} x\u1d63)) where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b2\u1d63 x\u2081 x\u2082\n  helper (here () _)\n  helper (there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) x\u223cy\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 \u03b1\u1d63 x\u1d63 = \u27e6coerce\u1d3a\u27e7 _ _ (\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63) (\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 x\u1d63)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _  zero x\u1d63 = x\u1d63\n\u27e6add\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) x\u1d63 = pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = pf-irr\u2032 (\u27e6Name\u27e7 \u03b1\u1d63) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2081) 1 k\u2081) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2082) 1 k\u2082) (\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {pf\u2081 : 0 \u2208 \u03b1\u2081 \u21911} {n} {pf\u2082 : suc n \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (0 , pf\u2081) (suc n , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.there p _ _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.here _ ())\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {n} {pf\u2081 : suc n \u2208 \u03b1\u2081 \u21911} {pf\u2082 : 0 \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc n , pf\u2081) (0 , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.there _ p _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.here () _)\n\n-- cmp\u1d3a-bool : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 \ud835\udfda\n-- cmp\u1d3a-bool \u2113 x = suc (binder\u1d3a x) <= \u2113\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) NaPa.cmp\u1d3a-bool NaPa.cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 _ zero _ = \u27e60\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 _ (suc _) {zero , _} {zero , _} _ = \u27e61\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) NaPa.easy-cmp\u1d3a NaPa.easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 _ zero x\u1d63 = \u27e6inr\u27e7 x\u1d63\n\u27e6easy-cmp\u1d3a\u27e7 _ (suc _) {zero , _} {zero , _} x\u1d63 = \u27e6inl\u27e7 here\u2032\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6map\u27e7 _ _ _ _ (\u27e6suc\u1d3a\u2191\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63)) (pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63))\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (NaPa.easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6\u2286-dist-+1-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) \u27e6\u2286\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))))\n                 \u2286-dist-+1-\u25c5 \u2286-dist-+1-\u25c5\n\u27e6\u2286-dist-+1-\u25c5\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083) where\n  -- open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n\n  hper :   \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (suc\u1d3a x\u2081) (suc\u1d3a x\u2082)\n  hper (\u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082) = \u27e6\u25c5\u27e7.here (\u2261.cong suc x\u2261b\u2081) (\u2261.cong suc y\u2261b\u2082)\n  hper (\u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong pred) (y\u2262b\u2082 \u2218 \u2261.cong pred) x\u223cy\n\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 = hper x\u1d63\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n\u27e6\u2286-dist-\u25c5-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) \u27e6\u2286\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)))\n                 \u2286-dist-\u25c5-+1 \u2286-dist-\u25c5-+1\n\u27e6\u2286-dist-\u25c5-+1\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk helper where\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 with x\u1d63\n  ... | \u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082 = \u27e6\u25c5\u27e7.here (\u2261.cong (\u03bb x \u2192 x \u2238 1) x\u2261b\u2081) (\u2261.cong (\u03bb x \u2192 x \u2238 1) y\u2261b\u2082)\n  ... | \u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong suc) (y\u2262b\u2082 \u2218 \u2261.cong suc) x\u223cy\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n{-\nBinder\u2261\u2115 : Binder \u2261 \u2115\nBinder\u2261\u2115 = \u2261.refl\n\npostulate\n  \u27e6Binder\u27e7\u2262\u27e6\u2115\u27e7 : \u27e6Binder\u27e7 \u2262 \u27e6\u2115\u27e7\n\n\u27e6\u2261\u27e7-lem : \u2200 {a\u2081 a\u2082 a\u1d63 A\u2081 A\u2082} {A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082} {x\u2081 x\u2082} {x\u1d63 : A\u1d63 x\u2081 x\u2082} {y\u2081 y\u2082} {y\u1d63 : A\u1d63 y\u2081 y\u2082} {pf\u2081 pf\u2082} \u2192 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 y\u1d63 pf\u2081 pf\u2082 \u2192 \u2200 (P : \u2200 {z\u2081 z\u2082} \u2192 A\u1d63 z\u2081 z\u2082 \u2192 Set) \u2192 P x\u1d63 \u2192 P y\u1d63\n\u27e6\u2261\u27e7-lem = {!!}\n\n\u00ac\u27e6Binder\u2261\u2115\u27e7 : \u00ac((\u27e6\u2261\u27e7 \u27e6Set\u2080\u27e7 \u27e6Binder\u27e7 \u27e6\u2115\u27e7) Binder\u2261\u2115 Binder\u2261\u2115)\n\u00ac\u27e6Binder\u2261\u2115\u27e7 = \u03bb pf \u2192 \u27e6\u2261\u27e7-lem pf (\u03bb x \u2192 \u22a5) {!!}\n-}\n\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 : (\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                   \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                    \u27e6\u2261\u27e7 \u27e6Binder\u27e7 (\u27e6binder\u1d3a\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) (\u27e6name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63)) b\u1d63) binder\u1d3a\u2218name\u1d2e binder\u1d3a\u2218name\u1d2e\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63 = \u2261.\u27e6refl\u27e7\n\n\u27e8_,_\u27e9\u27e6\u25c5\u27e7_ : (b\u2081 b\u2082 : Binder) \u2192 \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) \u2192 \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n\u27e8 b\u2081 , b\u2082 \u27e9\u27e6\u25c5\u27e7 \u03b1\u1d63 = _\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63\n\nmodule Perm (m n : \u2115) (m\u2262n : m \u2262 n) where\n  mn\u1d42 : World\n  mn\u1d42 = m \u25c5 n \u25c5 \u00f8\n  nm\u1d42 : World\n  nm\u1d42 = n \u25c5 m \u25c5 \u00f8\n  perm-m-n : \u27e6World\u27e7 mn\u1d42 nm\u1d42\n  perm-m-n = \u27e8 m , n \u27e9\u27e6\u25c5\u27e7 (\u27e8 n , m \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7)\n  m-n : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (m , m\u2208) (n , n\u2208)\n  m-n = here\u2032\n  n-m : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (n , n\u2208) (m , m\u2208)\n  n-m = \u27e6\u25c5\u27e7.there (m\u2262n \u2218 \u2261.sym) m\u2262n {b\u2208b\u25c5 n \u00f8} {b\u2208b\u25c5 m \u00f8} here\u2032\n\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 : \u00ac(\u27e6\ud835\udfda\u27e7 1\u2082 0\u2082)\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 ()\n\nbinder-irrelevance : \u2200 (f : Binder \u2192 \ud835\udfda)\n                     \u2192 (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f\n                     \u2192 \u2200 {b\u2081 b\u2082} \u2192 f b\u2081 \u2261 f b\u2082\nbinder-irrelevance _ f\u1d63 = \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 _)\n\ncontrab : \u2200 (f : Binder \u2192 \ud835\udfda) {b\u2081 b\u2082}\n          \u2192 f b\u2081 \u2262 f b\u2082\n          \u2192 \u00ac((\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\ncontrab f = contraposition (\u03bb f\u1d63 \u2192 binder-irrelevance f (\u03bb x\u1d63 \u2192 f\u1d63 x\u1d63))\n\nmodule Single \u03b1\u2081 \u03b1\u2082 x\u2081 x\u2082 where\n  data \u211b : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set where\n    refl : \u211b x\u2081 x\u2082\n  \u211b-pres-\u2261 : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 \u211b x\u2081 x\u2082 \u2192 \u211b y\u2081 y\u2082 \u2192 (x\u2081 \u2261 y\u2081) \u21d4 (x\u2082 \u2261 y\u2082)\n  \u211b-pres-\u2261 refl refl = equivalence (const \u2261.refl) (const \u2261.refl)\n  \u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082\n  \u03b1\u1d63 = \u211b , \u211b-pres-\u2261\n\npoly-name-uniq : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1} (x : Name \u03b1) \u2192 f x \u2261 f {\u00f8 \u21911} (0 , _)\npoly-name-uniq f f\u1d63 {\u03b1} x =\n  \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 {\u03b1} {\u00f8 \u21911} \u03b1\u1d63 {x} {0 , _} refl)\n  where open Single \u03b1 (\u00f8 \u21911) x (0 , _)\n\npoly-name-irrelevance : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1\u2081 \u03b1\u2082} (x\u2081 : Name \u03b1\u2081) (x\u2082 : Name \u03b1\u2082)\n                        \u2192 f x\u2081 \u2261 f x\u2082\npoly-name-irrelevance f f\u1d63 x\u2081 x\u2082 =\n  \u2261.trans (poly-name-uniq f f\u1d63 x\u2081) (\u2261.sym (poly-name-uniq f f\u1d63 x\u2082))\n\nmodule Broken where\n  _<=\u1d3a_ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name \u03b1 \u2192 \ud835\udfda\n  (m , _) <=\u1d3a (n , _) = \u2115._<=_ m n\n\n  \u00ac\u27e6<=\u1d3a\u27e7 : \u00ac((\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _<=\u1d3a_ _<=\u1d3a_)\n  \u00ac\u27e6<=\u1d3a\u27e7 \u27e6<=\u27e7 = \u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 (\u27e6<=\u27e7 perm-0-1 {0 , _} {1 , _} 0-1 {1 , _} {0 , _} 1-0)\n     where open Perm 0 1 (\u03bb()) renaming (perm-m-n to perm-0-1; m-n to 0-1; n-m to 1-0)\n\n  \u2286-broken : \u2200 \u03b1 b \u2192 \u03b1 \u2286 (b \u25c5 \u03b1)\n  \u2286-broken \u03b1 b = mk (\u03bb b\u2032 b\u2032\u2208\u03b1 \u2192 \u2261.tr id (\u2261.sym (\u25c5-sem \u03b1 b\u2032 b)) (If\u2032 \u2115._==_ b\u2032 b then _ else b\u2032\u2208\u03b1))\n\n  \u00ac\u27e6\u2286-broken\u27e7 : \u00ac((\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63))\n                  \u2286-broken \u2286-broken)\n  \u00ac\u27e6\u2286-broken\u27e7 \u27e6\u2286-broken\u27e7 = bot\n     where 0\u21940 : \u27e6Name\u27e7 (\u27e8 0 , 1 \u27e9\u27e6\u25c5\u27e7 \u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) (0 , _) (0 , _)\n           0\u21940 = un\u27e6\u2286\u27e7 (\u27e6\u2286-broken\u27e7 (\u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) _) {0 , _} {0 , _} here\u2032\n\n           bot : \u22a5\n           bot with 0\u21940\n           bot | \u27e6\u25c5\u27e7.here _ ()\n           bot | \u27e6\u25c5\u27e7.there 0\u22620 _ _ = 0\u22620 \u2261.refl\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nimport Level as L\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Unit\nopen import Data.Empty\nopen import Data.Sum.NP\nopen import Data.Sum.Logical\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Nullary.Negation using (contraposition)\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP as Bin\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Equivalence as \u21d4 using (_\u21d4_; equivalence; module Equivalence)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nimport Data.Nat.NP as \u2115\nopen \u2115 renaming (_==_ to _==\u2115_)\nimport Data.Nat.Properties as \u2115\nopen import NomPa.Record using (module NomPa)\nopen import NomPa.Implem\nimport NaPa\nopen NaPa using (\u2208-uniq)\nopen import NomPa.Worlds\nimport NomPa.Derived\n\nmodule NomPa.Implem.LogicalRelation.Internals where\n\nopen \u2261 using (_\u2261_; _\u2262_; \u27e6\u2261\u27e7)\nopen \u2261.\u2261-Reasoning\nopen NomPa.Implem.Internals\nopen NomPa NomPa.Implem.nomPa using (worldSym; Name\u00f8-elim; suc\u1d3a; suc\u1d3a\u2191)\n\n-- move to record\npred\u1d3a : \u2200 {\u03b1} \u2192 Name (\u03b1 +1) \u2192 Name \u03b1\npred\u1d3a = subtract\u1d3a 1\n\n\u2262\u2192\u2713-not-==\u2115 : \u2200 {x y} \u2192 x \u2262 y \u2192 \u2713 (not (x ==\u2115 y))\n\u2262\u2192\u2713-not-==\u2115 \u00acp = \u2713-\u00ac-not (\u00acp \u2218 \u2115.==.sound _ _)\n\n#\u21d2\u2209 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 b \u2209 \u03b1\n#\u21d2\u2209 (_ #\u00f8)      = id\n#\u21d2\u2209 (suc#\u2237 b#\u03b1) = #\u21d2\u2209 b#\u03b1\n#\u21d2\u2209 (0# _)      = id\n\n#\u21d2\u2262 : \u2200 {\u03b1 b} \u2192 b # \u03b1 \u2192 (x : Name \u03b1) \u2192 binder\u1d3a x \u2262 b\n#\u21d2\u2262 b# (b , b\u2208) \u2261.refl = #\u21d2\u2209 b# b\u2208\n\nbinder\u1d3a-injective : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u2192 x \u2261 y\nbinder\u1d3a-injective {\u03b1} {_ , p\u2081} {a , p\u2082} eq rewrite eq = \u2261.cong (_,_ a) (\u2208-uniq \u03b1 p\u2081 p\u2082)\n\npf-irr\u2032 : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x x\u2032 : Name \u03b1} {y y\u2032 : Name \u03b2}\n         \u2192 binder\u1d3a x \u2261 binder\u1d3a x\u2032 \u2192 binder\u1d3a y \u2261 binder\u1d3a y\u2032 \u2192 \u211b x y \u2192 \u211b x\u2032 y\u2032\npf-irr\u2032 {\u03b1} {\u03b2} \u211b {x , x\u2208} {x\u2032 , x\u2032\u2208} {y , y\u2208} {y\u2032 , y\u2032\u2208} eq\u2081 eq\u2082 x\u1d63y\n  rewrite eq\u2081 | eq\u2082 | \u2208-uniq \u03b1 x\u2208 x\u2032\u2208 | \u2208-uniq \u03b2 y\u2208 y\u2032\u2208 = x\u1d63y\n\npf-irr : \u2200 {\u03b1 \u03b2} (\u211b : Name \u03b1 \u2192 Name \u03b2 \u2192 Set) {x y x\u2208 x\u2208\u2032 y\u2208 y\u2208\u2032}\n         \u2192 \u211b (x , x\u2208) (y , y\u2208) \u2192 \u211b (x , x\u2208\u2032) (y , y\u2208\u2032)\npf-irr \u211b = pf-irr\u2032 \u211b \u2261.refl \u2261.refl\n\nPreserve-\u2248 : \u2200 {a b \u2113 \u2113a \u2113b} {A : Set a} {B : Set b} \u2192 Rel A \u2113a \u2192 Rel (B) \u2113b \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2248 _\u2248a_ _\u2248b_ _\u223c_ = \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u223c x\u2082 \u2192 y\u2081 \u223c y\u2082 \u2192 (x\u2081 \u2248a y\u2081) \u21d4 (x\u2082 \u2248b y\u2082)\n\n==\u2115\u21d4\u2261 : \u2200 {x y} \u2192 \u2713 (x ==\u2115 y) \u21d4 x \u2261 y\n==\u2115\u21d4\u2261 {x} {y} = equivalence (\u2115.==.sound _ _) \u2115.==.reflexive\n\n\u2261-on-name\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 binder\u1d3a x \u2261 binder\u1d3a y \u21d4 x \u2261 y\n\u2261-on-name\u21d4\u2261 {\u03b1} {x} {y} = equivalence binder\u1d3a-injective (\u2261.cong binder\u1d3a)\n\nT\u21d4T\u2192\u2261 : \u2200 {b\u2081 b\u2082} \u2192 \u2713 b\u2081 \u21d4 \u2713 b\u2082 \u2192 b\u2081 \u2261 b\u2082\nT\u21d4T\u2192\u2261 {1\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\nT\u21d4T\u2192\u2261 {1\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.to b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {1\u2082} b\u2081\u21d4b\u2082 = \u22a5-elim (Equivalence.from b\u2081\u21d4b\u2082 \u27e8$\u27e9 _)\nT\u21d4T\u2192\u2261 {0\u2082} {0\u2082} b\u2081\u21d4b\u2082 = \u2261.refl\n\n==\u1d3a\u21d4\u2261 : \u2200 {\u03b1} {x y : Name \u03b1} \u2192 \u2713 (x ==\u1d3a y) \u21d4 x \u2261 y\n==\u1d3a\u21d4\u2261 = \u2261-on-name\u21d4\u2261 \u27e8\u2218\u27e9 ==\u2115\u21d4\u2261 where open \u21d4 renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n-- Preserving the equalities also mean that the relation is functional and injective\nPreserve-\u2261 : \u2200 {a b \u2113} {A : Set a} {B : Set b} \u2192 REL A (B) \u2113 \u2192 Set _\nPreserve-\u2261 _\u223c_ = Preserve-\u2248 _\u2261_ _\u2261_ _\u223c_\n\nexport\u1d3a?-nothing : \u2200 {b \u03b1} (x : Name (b \u25c5 \u03b1)) \u2192 \u2713 (binder\u1d3a x ==\u2115 b) \u2192 export\u1d3a? {b} x \u2261 nothing\nexport\u1d3a?-nothing {b} {\u03b1} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u2261.refl\n... | 0\u2082 | _ = \u22a5-elim p\n\nexport\u1d3a?-just : \u2200 {\u03b1 b} (x : Name (b \u25c5 \u03b1)) {x\u2208} \u2192 \u2713 (not (binder\u1d3a x ==\u2115 b)) \u2192 export\u1d3a? {b} x \u2261 just (binder\u1d3a x , x\u2208)\nexport\u1d3a?-just {\u03b1} {b} (x , x\u2208) p with x ==\u2115 b | export\u2208 \u03b1 x b\n... | 1\u2082 | _ = \u22a5-elim p\n... | 0\u2082 | _ = \u2261.cong just (binder\u1d3a-injective \u2261.refl)\n\nrecord \u27e6World\u27e7 (\u03b1\u2081 \u03b1\u2082 : World) : Set\u2081 where\n  constructor _,_\n  field\n    _\u223c_ : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set\n    \u223c-pres-\u2261 : Preserve-\u2261 _\u223c_\n\n  \u223c-inj : \u2200 {x y z} \u2192 x \u223c z \u2192 y \u223c z \u2192 x \u2261 y\n  \u223c-inj p q = Equivalence.from (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n  \u223c-fun : \u2200 {x y z} \u2192 x \u223c y \u2192 x \u223c z \u2192 y \u2261 z\n  \u223c-fun p q = Equivalence.to (\u223c-pres-\u2261 p q) \u27e8$\u27e9 \u2261.refl\n\n  \u223c-\u2208-uniq : \u2200 {x\u2081 x\u2082} {x\u2081\u2208\u2032 x\u2082\u2208\u2032} \u2192\n        x\u2081 \u223c x\u2082 \u2261 (binder\u1d3a x\u2081 , x\u2081\u2208\u2032) \u223c (binder\u1d3a x\u2082 , x\u2082\u2208\u2032)\n  \u223c-\u2208-uniq {_ , x\u2081\u2208} {_ , x\u2082\u2208} {x\u2081\u2208\u2032} {x\u2082\u2208\u2032} = \u2261.ap\u2082 (\u03bb x\u2081\u2208 x\u2082\u2208 \u2192 (_ , x\u2081\u2208) \u223c (_ , x\u2082\u2208)) (\u2208-uniq \u03b1\u2081 x\u2081\u2208 x\u2081\u2208\u2032) (\u2208-uniq \u03b1\u2082 x\u2082\u2208 x\u2082\u2208\u2032)\n\n\u27e6sym\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} \u2192 \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082 \u2192 \u27e6World\u27e7 \u03b1\u2082 \u03b1\u2081\n\u27e6sym\u27e7 (\u211b , \u211b-pres-\u2261) = \u211b-sym , \u211b-sym-pres-\u2261\n  where \u211b-sym = flip \u211b\n        \u211b-sym-pres-\u2261 : Preserve-\u2261 \u211b-sym\n        \u211b-sym-pres-\u2261 p q = \u21d4.sym (\u211b-pres-\u2261 p q)\n\nrecord \u27e6Binder\u27e7 (b\u2081 b\u2082 : Binder) : Set where\n\nmodule \u27e6\u25c5\u27e7 (b\u2081 b\u2082 : Binder) {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  data _\u211b_ (x : Name (b\u2081 \u25c5 \u03b1\u2081)) (y : Name (b\u2082 \u25c5 \u03b1\u2082)) : Set where\n    here  : (x\u2261b\u2081 : binder\u1d3a x \u2261 b\u2081) (y\u2261b\u2082 : binder\u1d3a y \u2261 b\u2082) \u2192 x \u211b y\n    there : \u2200 (x\u2262b\u2081 : binder\u1d3a x \u2262 b\u2081) (y\u2262b\u2082 : binder\u1d3a y \u2262 b\u2082) {x\u2208 y\u2208} (x\u223cy : (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a y , y\u2208)) \u2192 x \u211b y\n\n  \u211b-inj : \u2200 {x y z} \u2192 x \u211b z \u2192 y \u211b z \u2192 x \u2261 y\n  \u211b-inj (here x\u2261b\u2081 _)    (here y\u2261b\u2081 _)     = binder\u1d3a-injective (\u2261.trans x\u2261b\u2081 (\u2261.sym y\u2261b\u2081))\n  \u211b-inj (here _ z\u2261b\u2082)    (there _ z\u2262b\u2082 _)  = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj (there _ z\u2262b\u2082 _) (here _ z\u2261b\u2082)     = \u22a5-elim (z\u2262b\u2082 z\u2261b\u2082)\n  \u211b-inj {x} {y} {z} (there x\u2262b\u2081 z\u2262b\u2082 {x\u2208} {z\u2208} x\u223cz) (there y\u2262b\u2081 _ {y\u2208} {z\u2208\u2032} y\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-inj x\u223cz (\u2261.tr (\u03bb z\u2208 \u2192 (binder\u1d3a y , y\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2082 z\u2208\u2032 z\u2208) y\u223cz )))\n\n  \u211b-fun : \u2200 {x y z} \u2192 x \u211b y \u2192 x \u211b z \u2192 y \u2261 z\n  \u211b-fun (here _ y\u2261b\u2082)  (here _ z\u2261b\u2082)    = binder\u1d3a-injective (\u2261.trans y\u2261b\u2082 (\u2261.sym z\u2261b\u2082))\n  \u211b-fun (here p _)     (there \u00acp _ _)    = \u22a5-elim (\u00acp p)\n  \u211b-fun (there \u00acp _ _) (here p _)        = \u22a5-elim (\u00acp p)\n  \u211b-fun {x} {y} {z} (there x\u2262b\u2081 y\u2262b\u2082 {x\u2208} {y\u2208} x\u223cy) (there p q {x\u2208\u2032} {z\u2208} x\u223cz) =\n    binder\u1d3a-injective (\u2261.cong binder\u1d3a (\u223c-fun x\u223cy (\u2261.tr (\u03bb x\u2208 \u2192 (binder\u1d3a x , x\u2208) \u223c (binder\u1d3a z , z\u2208)) (\u2208-uniq \u03b1\u2081 x\u2208\u2032 x\u2208) x\u223cz )))\n\n  \u211b-pres-\u2261 : Preserve-\u2261 _\u211b_\n  \u211b-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 x\u2081 \u211b x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b-inj x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 y\u2081 \u211b y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n  \u27e6world\u27e7 : \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n  \u27e6world\u27e7 = _\u211b_ , \u211b-pres-\u2261\n\nhere\u2032 : \u2200 {\u03b1\u2081 \u03b1\u2082 b\u2081 b\u2082 pf\u2081 pf\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} \u2192 \u27e6\u25c5\u27e7._\u211b_ b\u2081 b\u2082 \u03b1\u1d63 (b\u2081 , pf\u2081) (b\u2082 , pf\u2082)\nhere\u2032 = \u27e6\u25c5\u27e7.here \u2261.refl \u2261.refl\n\n_\u27e6\u25c5\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u25c5_ _\u25c5_\n_\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63 = \u27e6\u25c5\u27e7.\u27e6world\u27e7 b\u2081 b\u2082 \u03b1\u1d63\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 (\u211b , _) x\u2081 x\u2082 = \u211b x\u2081 x\u2082\n\n\u27e6zero\u1d2e\u27e7 : \u27e6Binder\u27e7 zero\u1d2e zero\u1d2e\n\u27e6zero\u1d2e\u27e7 = _\n\n\u27e6suc\u1d2e\u27e7 : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) suc\u1d2e suc\u1d2e\n\u27e6suc\u1d2e\u27e7 = _\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = (\u03bb _ _ \u2192 \u22a5) , (\u03bb())\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 x\u1d63 y\u1d63 =\n   \u27e6\ud835\udfda\u27e7-Props.reflexive (T\u21d4T\u2192\u2261 (sym ==\u1d3a\u21d4\u2261 \u27e8\u2218\u27e9 \u223c-pres-\u2261 x\u1d63 y\u1d63 \u27e8\u2218\u27e9 ==\u1d3a\u21d4\u2261)) where\n  open \u27e6World\u27e7 \u03b1\u1d63\n  open \u21d4 using (sym) renaming (_\u2218_ to _\u27e8\u2218\u27e9_)\n\n\u27e6name\u1d2e\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) name\u1d2e name\u1d2e\n\u27e6name\u1d2e\u27e7 _ _ = here\u2032\n\n\u27e6binder\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Binder\u27e7) binder\u1d3a binder\u1d3a\n\u27e6binder\u1d3a\u27e7 _ _ = _\n\n\u27e6nothing\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082}\n            \u2192 x \u2261 nothing \u2192 y \u2261 nothing \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6nothing\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6nothing\u27e7\n\n\u27e6just\u27e7\u2032 : \u2200 {a} {A\u2081 A\u2082 : Set a} {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set} {x : Maybe A\u2081} {y : Maybe A\u2082} {x\u2032 y\u2032}\n            \u2192 x \u2261 just x\u2032 \u2192 y \u2261 just y\u2032 \u2192 A\u1d63 x\u2032 y\u2032 \u2192 \u27e6Maybe\u27e7 A\u1d63 x y\n\u27e6just\u27e7\u2032 \u2261.refl \u2261.refl = \u27e6just\u27e7\n\n\u27e6export\u1d3a?\u27e7 : (\u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) (\u03bb {b} \u2192 export\u1d3a? {b}) (\u03bb {b} \u2192 export\u1d3a? {b})\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.here e\u2081 e\u2082)\n  = \u27e6nothing\u27e7\u2032 (export\u1d3a?-nothing x\u2081 (\u2115.==.reflexive e\u2081)) (export\u1d3a?-nothing x\u2082 (\u2115.==.reflexive e\u2082))\n\u27e6export\u1d3a?\u27e7 _ \u03b1\u1d63 {x\u2081} {x\u2082} (\u27e6\u25c5\u27e7.there x\u2081\u2262b\u2081 x\u2082\u2262b\u2082 x\u2081\u223cx\u2082)\n  = \u27e6just\u27e7\u2032 (export\u1d3a?-just x\u2081 (\u2262\u2192\u2713-not-==\u2115 x\u2081\u2262b\u2081))\n           (export\u1d3a?-just x\u2082 (\u2262\u2192\u2713-not-==\u2115 x\u2082\u2262b\u2082)) x\u2081\u223cx\u2082\n\n_\u27e6\u2286\u27e7b_ : \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a x) (coerce\u1d3a y)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x y} \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) x y \u2192 (\u03b1\u1d63 \u27e6\u2286\u27e7b \u03b2\u1d63) x y\nun\u27e6\u2286\u27e7 (mk x) {a} {b} c = x {a} {b} c\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {mk f\u2081} {mk f\u2082} f {mk g\u2081} {mk g\u2082} g\n  = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {coerce\u1d3a (mk f\u2081) x\u2081} {coerce\u1d3a (mk f\u2082) x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 ()\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk \u03bb { {_ , ()} }\n\n_\u27e6#\u27e7_ : (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6Set\u2080\u27e7) _#_ _#_\n_\u27e6#\u27e7_ _ _ _ _ = \u22a4\n\n_\u27e6#\u00f8\u27e7 : (\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u27e6\u00f8\u27e7) _#\u00f8 _#\u00f8\n_\u27e6#\u00f8\u27e7 _ = _\n\n\u27e6suc#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 b\u1d63 \u27e6#\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 (\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6#\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) suc# suc#\n\u27e6suc#\u27e7 _ _ _ = _\n\n\u27e6\u2286-#\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                  (b\u1d63 \u27e6#\u27e7 \u03b1\u1d63) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63)) \u2286-# \u2286-#\n\u27e6\u2286-#\u27e7 _ {b\u2081} {b\u2082} _ {b\u2081#\u03b1\u2081} {b\u2082#\u03b1\u2082} _ = mk (\u03bb {x\u2081} {x\u2082} \u2192 \u27e6\u25c5\u27e7.there (#\u21d2\u2262 b\u2081#\u03b1\u2081 x\u2081) (#\u21d2\u2262 b\u2082#\u03b1\u2082 x\u2082))\n\n\u27e6\u2286-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b2\u1d63)) \u2286-\u25c5 \u2286-\u25c5\n\u27e6\u2286-\u25c5\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {b\u2081} {b\u2082} _ {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} (mk f) = mk g where\n  open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n  g : \u2200 {x\u2081 x\u2082} \u2192 _\u211b_ \u03b1\u1d63 x\u2081 x\u2082 \u2192 _\u211b_ \u03b2\u1d63 (coerce\u1d3a (\u2286-\u25c5 b\u2081 \u03b1\u2286\u03b2\u2081) x\u2081) (coerce\u1d3a (\u2286-\u25c5 b\u2082 \u03b1\u2286\u03b2\u2082) x\u2082)\n  g (here x\u2261b\u2081 y\u2261b\u2082)       = here x\u2261b\u2081 y\u2261b\u2082\n  g (there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = there x\u2262b\u2081 y\u2262b\u2082 (f x\u223cy)\n\ncong\u2115 : \u2200 {\u03b1 \u03b2 : World} {x y : Name \u03b1} (f : \u2115 \u2192 \u2115) {fx\u2208 fy\u2208} \u2192 x \u2261 y \u2192 _\u2261_ {A = Name \u03b2} (f (binder\u1d3a x) , fx\u2208) (f (binder\u1d3a y) , fy\u2208)\ncong\u2115 f = binder\u1d3a-injective \u2218 \u2261.cong (f \u2218 binder\u1d3a)\n\nmodule \u27e6+1\u27e7 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) where\n  open \u27e6World\u27e7 \u03b1\u1d63 renaming (_\u223c_ to \u211b; \u223c-inj to \u211b-inj; \u223c-fun to \u211b-fun)\n  \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set\n  \u211b+1 x y = \u27e6Name\u27e7 \u03b1\u1d63 (pred\u1d3a x) (pred\u1d3a y)\n{-\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x x\u2208 y y\u2208} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (suc x , ) (sucN y)\n  data \u211b+1 : Name (\u03b1\u2081 +1) \u2192 Name (\u03b1\u2082 +1) \u2192 Set where\n    suc : \u2200 {x y} \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x y \u2192 \u211b+1 (sucN x) (sucN y)\n-}\n\n  \u211b+1-inj : \u2200 {x y z} \u2192 \u211b+1 x z \u2192  \u211b+1 y z \u2192 x \u2261 y\n  -- \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} (suc a) (suc b) = {!!}\n  \u211b+1-inj {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-inj a b)\n  \u211b+1-inj {zero , ()} _ _\n  \u211b+1-inj {_} {zero , ()} _ _\n  \u211b+1-inj {_} {_} {zero , ()} _ _\n  \u211b+1-fun : \u2200 {x y z} \u2192 \u211b+1 x y \u2192  \u211b+1 x z \u2192 y \u2261 z\n  \u211b+1-fun {suc x , _} {suc y , _} {suc z , _} a b = cong\u2115 suc (\u211b-fun a b)\n  \u211b+1-fun {zero , ()} _ _\n  \u211b+1-fun {_} {zero , ()} _ _\n  \u211b+1-fun {_} {_} {zero , ()} _ _\n\n  \u211b+1-pres-\u2261 : Preserve-\u2261 \u211b+1\n  -- \u211b+1-pres-\u2261 x y = equivalence {!\u211b+1-inj!} {!!}\n  \u211b+1-pres-\u2261 {x\u2081} {x\u2082} {y\u2081} {y\u2082} x\u2081\u223cx\u2082 y\u2081\u223cy\u2082 =\n     -- factor this move\n     equivalence (\u03bb x\u2081=y\u2081 \u2192 \u211b+1-fun {y\u2081} (\u2261.tr (\u03bb x\u2081 \u2192 \u211b+1 x\u2081 x\u2082) x\u2081=y\u2081 x\u2081\u223cx\u2082) y\u2081\u223cy\u2082)\n                 (\u03bb x\u2082=y\u2082 \u2192 \u211b+1-inj {x\u2081} {y\u2081} {x\u2082} x\u2081\u223cx\u2082 (\u2261.tr (\u03bb y\u2082 \u2192 \u211b+1 y\u2081 y\u2082) (\u2261.sym x\u2082=y\u2082) y\u2081\u223cy\u2082))\n\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 \u03b1\u1d63 = \u211b+1 , \u211b+1-pres-\u2261 where open \u27e6+1\u27e7 \u03b1\u1d63\n\nopen WorldOps worldSym\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 \u03b1\u1d63 = \u27e6zero\u1d2e\u27e7 \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)\n\n_\u27e6\u2191\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6\u2191\u27e7 k) \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082)\n                 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 _  (\u27e6\u25c5\u27e7.here () _)\n\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b1\u1d63 (\u27e6\u25c5\u27e7.there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63) x\u223cy\n\n\u27e6\u00f8+1\u2286\u00f8\u27e7 : (\u27e6\u00f8\u27e7 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u00f8+1\u2286\u00f8 \u00f8+1\u2286\u00f8\n\u27e6\u00f8+1\u2286\u00f8\u27e7 = mk (\u03bb {x} _ \u2192 Name\u00f8-elim (pred\u1d3a x))\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 _ \u03b2\u1d63 \u03b1\u2286\u03b2\u1d63 = mk (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) \u2218 un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk helper where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper (here x\u2261b\u2081 y\u2261b\u2082) = here x\u2261b\u2081 y\u2261b\u2082\n  helper {x\u2081 , _} {x\u2082 , _} (there x\u2262b\u2081 y\u2262b\u2082 {p\u2081} {p\u2082} x\u223cy)\n     = there x\u2262b\u2081 y\u2262b\u2082 {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081) x\u2081 p\u2081}\n                        {coe (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082) x\u2082 p\u2082}\n                        (pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 x\u223cy))\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {add\u1d3a 1 x\u2081} {add\u1d3a 1 x\u2082}) \n\n\u22620 : \u2200 {\u03b1} (x : Name (\u03b1 +1)) \u2192 binder\u1d3a x \u2262 0\n\u22620 (0 , ()) \u2261.refl\n\u22620 (suc _ , _) ()\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk (\u03bb {x\u2081} {x\u2082} \u2192 there (\u22620 x\u2081) (\u22620 x\u2082)) where open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ \u03b2\u1d63 \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 \u03b2\u1d63 (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (there (\u03bb()) (\u03bb()) x\u1d63))) where\n  open \u27e6\u25c5\u27e7 0 0\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ \u03b2\u1d63 \u03b1+\u2286\u03b2\u2191\u1d63 = mk (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} x\u1d63)) where\n  open \u27e6\u25c5\u27e7 0 0\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) (suc\u1d3a\u2191 x\u2081) (suc\u1d3a\u2191 x\u2082)) \u2192 \u27e6Name\u27e7 \u03b2\u1d63 x\u2081 x\u2082\n  helper (here () _)\n  helper (there _ _ x\u223cy) = pf-irr (\u27e6Name\u27e7 \u03b2\u1d63) x\u223cy\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 \u03b1\u1d63 = pf-irr (\u27e6Name\u27e7 \u03b1\u1d63)\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 \u03b1\u1d63 x\u1d63 = \u27e6coerce\u1d3a\u27e7 _ _ (\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63) (\u27e6suc\u1d3a\u27e7 \u03b1\u1d63 x\u1d63)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _  zero x\u1d63 = x\u1d63\n\u27e6add\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) x\u1d63 = pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = pf-irr\u2032 (\u27e6Name\u27e7 \u03b1\u1d63) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2081) 1 k\u2081) (\u2115.\u2238-+-assoc (binder\u1d3a x\u2082) 1 k\u2082) (\u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 x\u1d63)\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {pf\u2081 : 0 \u2208 \u03b1\u2081 \u21911} {n} {pf\u2082 : suc n \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (0 , pf\u2081) (suc n , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.there p _ _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc _ (\u27e6\u25c5\u27e7.here _ ())\n\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {n} {pf\u2081 : suc n \u2208 \u03b1\u2081 \u21911} {pf\u2082 : 0 \u2208 \u03b1\u2082 \u21911} \u2192 \u00ac(\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) (suc n , pf\u2081) (0 , pf\u2082))\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.there _ p _) = p \u2261.refl\n\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 _ (\u27e6\u25c5\u27e7.here () _)\n\n-- cmp\u1d3a-bool : \u2200 {\u03b1} \u2113 \u2192 Name (\u03b1 \u2191 \u2113) \u2192 \ud835\udfda\n-- cmp\u1d3a-bool \u2113 x = suc (binder\u1d3a x) <= \u2113\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) NaPa.cmp\u1d3a-bool NaPa.cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 _ zero _ = \u27e60\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 _ (suc _) {zero , _} {zero , _} _ = \u27e61\u2082\u27e7\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) NaPa.easy-cmp\u1d3a NaPa.easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 _ zero x\u1d63 = \u27e6inr\u27e7 x\u1d63\n\u27e6easy-cmp\u1d3a\u27e7 _ (suc _) {zero , _} {zero , _} x\u1d63 = \u27e6inl\u27e7 here\u2032\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {suc _ , _} x\u1d63 = \u27e6map\u27e7 _ _ _ _ (\u27e6suc\u1d3a\u2191\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63)) (pf-irr (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 (\u27e6pred\u1d3a\u27e7-suc\u1d3a\u2191 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63))\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {zero , _} {suc _ , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-0-suc (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 (suc k\u1d63) {suc _ , _} {zero , _} x\u1d63 = \u22a5-elim (\u00ac\u27e6Name\u27e7-\u03b1\u1d63-\u27e6\u21911\u27e7-suc-0 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) x\u1d63)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.tr (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191\u27e7 k\u1d63) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (NaPa.easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (NaPa.easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6\u2286-dist-+1-\u25c5\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) \u27e6\u2286\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))))\n                 \u2286-dist-+1-\u25c5 \u2286-dist-+1-\u25c5\n\u27e6\u2286-dist-+1-\u25c5\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7\u2083) where\n  -- open \u27e6\u25c5\u27e7 b\u2081 b\u2082\n\n  hper :   \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (suc\u1d3a x\u2081) (suc\u1d3a x\u2082)\n  hper (\u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082) = \u27e6\u25c5\u27e7.here (\u2261.cong suc x\u2261b\u2081) (\u2261.cong suc y\u2261b\u2082)\n  hper (\u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy) = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong pred) (y\u2262b\u2082 \u2218 \u2261.cong pred) x\u223cy\n\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7))\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-+1-\u25c5 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 = hper x\u1d63\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n\u27e6\u2286-dist-\u25c5-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) \u27e6\u2286\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)))\n                 \u2286-dist-\u25c5-+1 \u2286-dist-\u25c5-+1\n\u27e6\u2286-dist-\u25c5-+1\u27e7 \u03b1\u1d63 {b\u2081} {b\u2082} b\u1d63 = mk helper where\n  helper : \u2200 {x\u2081 x\u2082} (x\u1d63 : \u27e6Name\u27e7 ((\u27e6suc\u1d2e\u27e7 b\u1d63) \u27e6\u25c5\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) x\u2081 x\u2082)\n           \u2192 \u27e6Name\u27e7 ((b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) \u27e6+1\u27e7)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2081) x\u2081)\n                     (coerce\u1d3a (\u2286-dist-\u25c5-+1 b\u2082) x\u2082)\n  helper {suc x\u2081 , pf\u2081} {suc x\u2082 , pf\u2082} x\u1d63 with x\u1d63\n  ... | \u27e6\u25c5\u27e7.here x\u2261b\u2081 y\u2261b\u2082 = \u27e6\u25c5\u27e7.here (\u2261.cong (\u03bb x \u2192 x \u2238 1) x\u2261b\u2081) (\u2261.cong (\u03bb x \u2192 x \u2238 1) y\u2261b\u2082)\n  ... | \u27e6\u25c5\u27e7.there x\u2262b\u2081 y\u2262b\u2082 x\u223cy = \u27e6\u25c5\u27e7.there (x\u2262b\u2081 \u2218 \u2261.cong suc) (y\u2262b\u2082 \u2218 \u2261.cong suc) x\u223cy\n  helper {zero , ()} {_ , _} _\n  helper {_ , _} {zero , ()} _\n\n{-\nBinder\u2261\u2115 : Binder \u2261 \u2115\nBinder\u2261\u2115 = \u2261.refl\n\npostulate\n  \u27e6Binder\u27e7\u2262\u27e6\u2115\u27e7 : \u27e6Binder\u27e7 \u2262 \u27e6\u2115\u27e7\n\n\u27e6\u2261\u27e7-lem : \u2200 {a\u2081 a\u2082 a\u1d63 A\u2081 A\u2082} {A\u1d63 : \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 A\u2081 A\u2082} {x\u2081 x\u2082} {x\u1d63 : A\u1d63 x\u2081 x\u2082} {y\u2081 y\u2082} {y\u1d63 : A\u1d63 y\u2081 y\u2082} {pf\u2081 pf\u2082} \u2192 \u27e6\u2261\u27e7 A\u1d63 x\u1d63 y\u1d63 pf\u2081 pf\u2082 \u2192 \u2200 (P : \u2200 {z\u2081 z\u2082} \u2192 A\u1d63 z\u2081 z\u2082 \u2192 Set) \u2192 P x\u1d63 \u2192 P y\u1d63\n\u27e6\u2261\u27e7-lem = {!!}\n\n\u00ac\u27e6Binder\u2261\u2115\u27e7 : \u00ac((\u27e6\u2261\u27e7 \u27e6Set\u2080\u27e7 \u27e6Binder\u27e7 \u27e6\u2115\u27e7) Binder\u2261\u2115 Binder\u2261\u2115)\n\u00ac\u27e6Binder\u2261\u2115\u27e7 = \u03bb pf \u2192 \u27e6\u2261\u27e7-lem pf (\u03bb x \u2192 \u22a5) {!!}\n-}\n\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 : (\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                   \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7\n                    \u27e6\u2261\u27e7 \u27e6Binder\u27e7 (\u27e6binder\u1d3a\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63) (\u27e6name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63)) b\u1d63) binder\u1d3a\u2218name\u1d2e binder\u1d3a\u2218name\u1d2e\n\u27e6binder\u1d3a\u2218name\u1d2e\u27e7 \u03b1\u1d63 b\u1d63 = \u2261.\u27e6refl\u27e7\n\n\u27e8_,_\u27e9\u27e6\u25c5\u27e7_ : (b\u2081 b\u2082 : Binder) \u2192 \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) \u2192 \u27e6World\u27e7 (b\u2081 \u25c5 \u03b1\u2081) (b\u2082 \u25c5 \u03b1\u2082)\n\u27e8 b\u2081 , b\u2082 \u27e9\u27e6\u25c5\u27e7 \u03b1\u1d63 = _\u27e6\u25c5\u27e7_ {b\u2081} {b\u2082} _ \u03b1\u1d63\n\nmodule Perm (m n : \u2115) (m\u2262n : m \u2262 n) where\n  mn\u1d42 : World\n  mn\u1d42 = m \u25c5 n \u25c5 \u00f8\n  nm\u1d42 : World\n  nm\u1d42 = n \u25c5 m \u25c5 \u00f8\n  perm-m-n : \u27e6World\u27e7 mn\u1d42 nm\u1d42\n  perm-m-n = \u27e8 m , n \u27e9\u27e6\u25c5\u27e7 (\u27e8 n , m \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7)\n  m-n : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (m , m\u2208) (n , n\u2208)\n  m-n = here\u2032\n  n-m : \u2200 {m\u2208 n\u2208} \u2192 \u27e6Name\u27e7 perm-m-n (n , n\u2208) (m , m\u2208)\n  n-m = \u27e6\u25c5\u27e7.there (m\u2262n \u2218 \u2261.sym) m\u2262n {b\u2208b\u25c5 n \u00f8} {b\u2208b\u25c5 m \u00f8} here\u2032\n\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 : \u00ac(\u27e6\ud835\udfda\u27e7 1\u2082 0\u2082)\n\u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 ()\n\nbinder-irrelevance : \u2200 (f : Binder \u2192 \ud835\udfda)\n                     \u2192 (\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f\n                     \u2192 \u2200 {b\u2081 b\u2082} \u2192 f b\u2081 \u2261 f b\u2082\nbinder-irrelevance _ f\u1d63 = \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 _)\n\ncontrab : \u2200 (f : Binder \u2192 \ud835\udfda) {b\u2081 b\u2082}\n          \u2192 f b\u2081 \u2262 f b\u2082\n          \u2192 \u00ac((\u27e6Binder\u27e7 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\ncontrab f = contraposition (\u03bb f\u1d63 \u2192 binder-irrelevance f (\u03bb x\u1d63 \u2192 f\u1d63 x\u1d63))\n\nmodule Single \u03b1\u2081 \u03b1\u2082 x\u2081 x\u2082 where\n  data \u211b : Name \u03b1\u2081 \u2192 Name \u03b1\u2082 \u2192 Set where\n    refl : \u211b x\u2081 x\u2082\n  \u211b-pres-\u2261 : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 \u211b x\u2081 x\u2082 \u2192 \u211b y\u2081 y\u2082 \u2192 (x\u2081 \u2261 y\u2081) \u21d4 (x\u2082 \u2261 y\u2082)\n  \u211b-pres-\u2261 refl refl = equivalence (const \u2261.refl) (const \u2261.refl)\n  \u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082\n  \u03b1\u1d63 = \u211b , \u211b-pres-\u2261\n\npoly-name-uniq : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1} (x : Name \u03b1) \u2192 f x \u2261 f {\u00f8 \u21911} (0 , _)\npoly-name-uniq f f\u1d63 {\u03b1} x =\n  \u27e6\ud835\udfda\u27e7-Props.to-propositional (f\u1d63 {\u03b1} {\u00f8 \u21911} \u03b1\u1d63 {x} {0 , _} refl)\n  where open Single \u03b1 (\u00f8 \u21911) x (0 , _)\n\npoly-name-irrelevance : \u2200 (f : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 \ud835\udfda)\n                          (f\u1d63 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) f f)\n                          {\u03b1\u2081 \u03b1\u2082} (x\u2081 : Name \u03b1\u2081) (x\u2082 : Name \u03b1\u2082)\n                        \u2192 f x\u2081 \u2261 f x\u2082\npoly-name-irrelevance f f\u1d63 x\u2081 x\u2082 =\n  \u2261.trans (poly-name-uniq f f\u1d63 x\u2081) (\u2261.sym (poly-name-uniq f f\u1d63 x\u2082))\n\nmodule Broken where\n  _<=\u1d3a_ : \u2200 {\u03b1} \u2192 Name \u03b1 \u2192 Name \u03b1 \u2192 \ud835\udfda\n  (m , _) <=\u1d3a (n , _) = \u2115._<=_ m n\n\n  \u00ac\u27e6<=\u1d3a\u27e7 : \u00ac((\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _<=\u1d3a_ _<=\u1d3a_)\n  \u00ac\u27e6<=\u1d3a\u27e7 \u27e6<=\u27e7 = \u00ac\u27e6\ud835\udfda\u27e7-1\u2082-0\u2082 (\u27e6<=\u27e7 perm-0-1 {0 , _} {1 , _} 0-1 {1 , _} {0 , _} 1-0)\n     where open Perm 0 1 (\u03bb()) renaming (perm-m-n to perm-0-1; m-n to 0-1; n-m to 1-0)\n\n  \u2286-broken : \u2200 \u03b1 b \u2192 \u03b1 \u2286 (b \u25c5 \u03b1)\n  \u2286-broken \u03b1 b = mk (\u03bb b\u2032 b\u2032\u2208\u03b1 \u2192 \u2261.tr id (\u2261.sym (\u25c5-sem \u03b1 b\u2032 b)) (If\u2032 \u2115._==_ b\u2032 b then _ else b\u2032\u2208\u03b1))\n\n  \u00ac\u27e6\u2286-broken\u27e7 : \u00ac((\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 b\u1d63 \u2236 \u27e6Binder\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 (b\u1d63 \u27e6\u25c5\u27e7 \u03b1\u1d63))\n                  \u2286-broken \u2286-broken)\n  \u00ac\u27e6\u2286-broken\u27e7 \u27e6\u2286-broken\u27e7 = bot\n     where 0\u21940 : \u27e6Name\u27e7 (\u27e8 0 , 1 \u27e9\u27e6\u25c5\u27e7 \u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) (0 , _) (0 , _)\n           0\u21940 = un\u27e6\u2286\u27e7 (\u27e6\u2286-broken\u27e7 (\u27e8 0 , 0 \u27e9\u27e6\u25c5\u27e7 \u27e6\u00f8\u27e7) _) {0 , _} {0 , _} here\u2032\n\n           bot : \u22a5\n           bot with 0\u21940\n           bot | \u27e6\u25c5\u27e7.here _ ()\n           bot | \u27e6\u25c5\u27e7.there 0\u22620 _ _ = 0\u22620 \u2261.refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0f9ba331639f6a5ed11f09341e5e55897688ce1d","subject":"Function: +lift-op\u2082","message":"Function: +lift-op\u2082\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/NP.agda","new_file":"lib\/Function\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.NP where\n\nopen import Level\n  using (_\u2294_; Lift; lift; lower)\nopen import Type\n  hiding (\u2605)\nopen import Algebra\n  using (module Monoid; Monoid)\nopen import Algebra.Structures\n  using (IsSemigroup)\nopen import Data.Nat.Base\n  using (\u2115; zero; suc)\nopen import Data.Bool.Base\n  renaming (Bool to \ud835\udfda)\nopen import Data.Product\n  using (_\u00d7_; \u03a3; _,_)\nopen import Data.Vec.N-ary\n  using (N-ary; N-ary-level)\nopen import Category.Monad\n  using () renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Monad.Identity\n  using (IdentityMonad)\nopen import Category.Applicative\n  renaming (module RawApplicative to Applicative;\n            RawApplicative to Applicative)\nopen import Relation.Binary\n  using (IsEquivalence; module IsEquivalence; _Preserves\u2082_\u27f6_\u27f6_;\n         module Setoid)\nopen import Relation.Binary.PropositionalEquality.NP\n  using (_\u2261_; _\u2257_; refl; ap; ap\u2082; module \u2261-Reasoning; _\u2192-setoid_; _\u2219_)\n  renaming (isEquivalence to \u2261-isEquivalence)\n\nopen import Function public\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = Monad.rawIApplicative IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : \u2605 a) (B : \u2605 b) \u2192 \u2605 (a \u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : \u2605 a) (n : \u2115) (B : \u2605 b) \u2192 \u2605 (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\n_\u2194_ : \u2200 {a b} (A : \u2605 a) (B : \u2605 b) \u2192 \u2605 (a \u2294 b)\nA \u2194 B = (A \u2192 B) \u00d7 (B \u2192 A)\n\nEndo : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nCmp A = A \u2192 A \u2192 \ud835\udfda\n\nOp\u2081 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nOp\u2081 A = A \u2192 A\n\nOp\u2082 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nOp\u2082 A = A \u2192 A \u2192 A\n\nlift-op\u2082 : \u2200 {a}{A : \u2605_ a}(op : Op\u2082 A){\u2113} \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A\nlift-op\u2082 op x y = lift (op (lower x) (lower y))\n\n-- More properties about nest\/fold are in Data.Nat.NP\nnest : \u2200 {a} {A : \u2605 a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero    f x = x\nnest (suc n) f x = f (nest n f x)\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : \u2605 a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\nit : \u2200 {a} {A : \u2605 a} \u2983 x : A \u2984 \u2192 A\nit \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : \u2605 i} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : (x : A) \u2192 B x \u2192 \u2605 c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : \u2605 a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 \u2605 \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : \u2605 a) = EndoMonoid-\u2248 {A = A} \u2261-isEquivalence (ap\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : \u2605 a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 p (h x) \u2219 ap g (q x))\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\n\u03a0\u2071 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0\u2071 A B = {x : A} \u2192 B x\n\nmodule From\u03a0 (\u03a0 : \u2200 {a b}(A : \u2605 a)(B : A \u2192 \u2605 b) \u2192 \u2605(a \u2294 b)) where\n\n  module _ {a b}{A : \u2605 a}(B : A \u2192 \u2605 b) where\n    \u2200\u2081 : \u2605(a \u2294 b)\n    \u2200\u2081 = \u03a0 A B\n\n  module _ {a b c}{A : \u2605 a}{B : A \u2192 \u2605 b}(C : (x : A)(y : B x) \u2192 \u2605 c) where\n    \u2200\u2082 : \u2605(a \u2294 b \u2294 c)\n    \u2200\u2082 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) (C _)\n\n  module _ {a b c d}{A : \u2605 a}{B : A \u2192 \u2605 b}{C : {x : A}(y : B x) \u2192 \u2605 c}\n           (D : (x : A)(y : B x)(z : C y) \u2192 \u2605 d) where\n    \u2200\u2083 : \u2605(a \u2294 b \u2294 c \u2294 d)\n    \u2200\u2083 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C y) (D _ _)\n\n  module _ {a b c d e}{A : \u2605 a}{B : A \u2192 \u2605 b}{C : {x : A}(y : B x) \u2192 \u2605 c}\n           {D : {x : A}{y : B x}(z : C y) \u2192 \u2605 d}\n           (E : (x : A)(y : B x)(z : C y)(t : D z) \u2192 \u2605 e)\n           where\n    \u2200\u2084 : \u2605(a \u2294 b \u2294 c \u2294 d \u2294 e)\n    \u2200\u2084 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C y) \u03bb z \u2192 \u03a0 (D z) (E _ _ _)\n\nmodule From\u03a3 (\u03a3 : \u2200 {a b}(A : \u2605 a)(B : A \u2192 \u2605 b) \u2192 \u2605(a \u2294 b)) =\n  From\u03a0 \u03a3 renaming (\u2200\u2081 to \u2203\u2081; \u2200\u2082 to \u2203\u2082; \u2200\u2083 to \u2203\u2083; \u2200\u2084 to \u2203\u2084)\n\nmodule From\u03a0\u03a3 (\u03a0 \u03a3 : \u2200 {a b}(A : \u2605 a)(B : A \u2192 \u2605 b) \u2192 \u2605(a \u2294 b))\n              (_,_ : \u2200 {a b}{A : \u2605 a}{B : A \u2192 \u2605 b}(x : A)(y : B x) \u2192 \u03a3 A B) where\n  module _ {a b c}(A : \u2605 a)(B : A \u2192 \u2605 b)(C : \u03a3 A B \u2192 \u2605 c) where\n    \u03a0\u03a0 \u03a0\u03a3 \u03a3\u03a0 : \u2605 _\n    \u03a0\u03a0 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n    \u03a0\u03a3 = \u03a0 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 C (x , y)\n    \u03a3\u03a0 = \u03a3 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n    \u03a3\u03a3 = \u03a3 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 C (x , y)\n\n  module _ {a b c d}(A : \u2605 a)(B : A \u2192 \u2605 b)(C : \u03a3 A B \u2192 \u2605 c)\n           (D : \u03a3 (\u03a3 A B) C \u2192 \u2605 d) where\n    \u03a0\u03a0\u03a0 \u03a0\u03a3\u03a0 \u03a0\u03a3\u03a3 \u03a3\u03a0\u03a3 \u03a3\u03a0\u03a0 : \u2605 _\n    \u03a0\u03a0\u03a0 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a0\u03a3\u03a0 = \u03a0 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a0\u03a3\u03a3 = \u03a0 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 \u03a3 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a3\u03a0\u03a3 = \u03a3 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a3 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a3\u03a0\u03a0 = \u03a3 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n\nmodule Implicits where\n  open From\u03a0  \u03a0\u2071       public\n  open From\u03a0\u03a3 \u03a0\u2071 \u03a3 _,_ public\n\nmodule Explicits where\n  open From\u03a0  \u03a0       public\n  open From\u03a3  \u03a3       public\n  open From\u03a0\u03a3 \u03a0 \u03a3 _,_ public\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Function.NP where\n\nopen import Level\n  using (_\u2294_)\nopen import Type\n  hiding (\u2605)\nopen import Algebra\n  using (module Monoid; Monoid)\nopen import Algebra.Structures\n  using (IsSemigroup)\nopen import Data.Nat.Base\n  using (\u2115; zero; suc)\nopen import Data.Bool.Base\n  renaming (Bool to \ud835\udfda)\nopen import Data.Product\n  using (_\u00d7_; \u03a3; _,_)\nopen import Data.Vec.N-ary\n  using (N-ary; N-ary-level)\nopen import Category.Monad\n  using () renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Monad.Identity\n  using (IdentityMonad)\nopen import Category.Applicative\n  renaming (module RawApplicative to Applicative;\n            RawApplicative to Applicative)\nopen import Relation.Binary\n  using (IsEquivalence; module IsEquivalence; _Preserves\u2082_\u27f6_\u27f6_;\n         module Setoid)\nopen import Relation.Binary.PropositionalEquality.NP\n  using (_\u2261_; _\u2257_; refl; ap; ap\u2082; module \u2261-Reasoning; _\u2192-setoid_; _\u2219_)\n  renaming (isEquivalence to \u2261-isEquivalence)\n\nopen import Function public\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = Monad.rawIApplicative IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : \u2605 a) (B : \u2605 b) \u2192 \u2605 (a \u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : \u2605 a) (n : \u2115) (B : \u2605 b) \u2192 \u2605 (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\n_\u2194_ : \u2200 {a b} (A : \u2605 a) (B : \u2605 b) \u2192 \u2605 (a \u2294 b)\nA \u2194 B = (A \u2192 B) \u00d7 (B \u2192 A)\n\nEndo : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nCmp A = A \u2192 A \u2192 \ud835\udfda\n\nOp\u2081 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nOp\u2081 A = A \u2192 A\n\nOp\u2082 : \u2200 {a} \u2192 \u2605 a \u2192 \u2605 a\nOp\u2082 A = A \u2192 A \u2192 A\n\n-- More properties about nest\/fold are in Data.Nat.NP\nnest : \u2200 {a} {A : \u2605 a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero    f x = x\nnest (suc n) f x = f (nest n f x)\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : \u2605 a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\nit : \u2200 {a} {A : \u2605 a} \u2983 x : A \u2984 \u2192 A\nit \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : \u2605 i} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : (x : A) \u2192 B x \u2192 \u2605 c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : \u2605 a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 \u2605 \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : \u2605 a) = EndoMonoid-\u2248 {A = A} \u2261-isEquivalence (ap\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : \u2605 a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 p (h x) \u2219 ap g (q x))\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\n\u03a0\u2071 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0\u2071 A B = {x : A} \u2192 B x\n\nmodule From\u03a0 (\u03a0 : \u2200 {a b}(A : \u2605 a)(B : A \u2192 \u2605 b) \u2192 \u2605(a \u2294 b)) where\n\n  module _ {a b}{A : \u2605 a}(B : A \u2192 \u2605 b) where\n    \u2200\u2081 : \u2605(a \u2294 b)\n    \u2200\u2081 = \u03a0 A B\n\n  module _ {a b c}{A : \u2605 a}{B : A \u2192 \u2605 b}(C : (x : A)(y : B x) \u2192 \u2605 c) where\n    \u2200\u2082 : \u2605(a \u2294 b \u2294 c)\n    \u2200\u2082 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) (C _)\n\n  module _ {a b c d}{A : \u2605 a}{B : A \u2192 \u2605 b}{C : {x : A}(y : B x) \u2192 \u2605 c}\n           (D : (x : A)(y : B x)(z : C y) \u2192 \u2605 d) where\n    \u2200\u2083 : \u2605(a \u2294 b \u2294 c \u2294 d)\n    \u2200\u2083 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C y) (D _ _)\n\n  module _ {a b c d e}{A : \u2605 a}{B : A \u2192 \u2605 b}{C : {x : A}(y : B x) \u2192 \u2605 c}\n           {D : {x : A}{y : B x}(z : C y) \u2192 \u2605 d}\n           (E : (x : A)(y : B x)(z : C y)(t : D z) \u2192 \u2605 e)\n           where\n    \u2200\u2084 : \u2605(a \u2294 b \u2294 c \u2294 d \u2294 e)\n    \u2200\u2084 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C y) \u03bb z \u2192 \u03a0 (D z) (E _ _ _)\n\nmodule From\u03a3 (\u03a3 : \u2200 {a b}(A : \u2605 a)(B : A \u2192 \u2605 b) \u2192 \u2605(a \u2294 b)) =\n  From\u03a0 \u03a3 renaming (\u2200\u2081 to \u2203\u2081; \u2200\u2082 to \u2203\u2082; \u2200\u2083 to \u2203\u2083; \u2200\u2084 to \u2203\u2084)\n\nmodule From\u03a0\u03a3 (\u03a0 \u03a3 : \u2200 {a b}(A : \u2605 a)(B : A \u2192 \u2605 b) \u2192 \u2605(a \u2294 b))\n              (_,_ : \u2200 {a b}{A : \u2605 a}{B : A \u2192 \u2605 b}(x : A)(y : B x) \u2192 \u03a3 A B) where\n  module _ {a b c}(A : \u2605 a)(B : A \u2192 \u2605 b)(C : \u03a3 A B \u2192 \u2605 c) where\n    \u03a0\u03a0 \u03a0\u03a3 \u03a3\u03a0 : \u2605 _\n    \u03a0\u03a0 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n    \u03a0\u03a3 = \u03a0 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 C (x , y)\n    \u03a3\u03a0 = \u03a3 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n    \u03a3\u03a3 = \u03a3 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 C (x , y)\n\n  module _ {a b c d}(A : \u2605 a)(B : A \u2192 \u2605 b)(C : \u03a3 A B \u2192 \u2605 c)\n           (D : \u03a3 (\u03a3 A B) C \u2192 \u2605 d) where\n    \u03a0\u03a0\u03a0 \u03a0\u03a3\u03a0 \u03a0\u03a3\u03a3 \u03a3\u03a0\u03a3 \u03a3\u03a0\u03a0 : \u2605 _\n    \u03a0\u03a0\u03a0 = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a0\u03a3\u03a0 = \u03a0 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a0\u03a3\u03a3 = \u03a0 A \u03bb x \u2192 \u03a3 (B x) \u03bb y \u2192 \u03a3 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a3\u03a0\u03a3 = \u03a3 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a3 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n    \u03a3\u03a0\u03a0 = \u03a3 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n\nmodule Implicits where\n  open From\u03a0  \u03a0\u2071       public\n  open From\u03a0\u03a3 \u03a0\u2071 \u03a3 _,_ public\n\nmodule Explicits where\n  open From\u03a0  \u03a0       public\n  open From\u03a3  \u03a3       public\n  open From\u03a0\u03a3 \u03a0 \u03a3 _,_ public\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1245d22fa20cf416f1b898ed29ddfa531ba8ed80","subject":"Implement booleans.","message":"Implement booleans.\n\nThis was harder than I expected. I had to adapt some of the\nolder proofs. Apparently I used the fact that Type had only\none constructor somehow.\n\nMissing features:\n\n - deriveTerm (if ...)\n - congruence of equivalence relation (ok, ok, it isn't easy).\n\nOld-commit-hash: 8b6ea9150951d6c6e3c0cf05d2a788f4a59c9deb\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n  bool : Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\ndata Bool : Set where\n  true false : Bool\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 bool \u27e7Type = Bool\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n  bool : \u2200 {b : Bool} \u2192\n    b \u2261 b\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\u2261-refl {bool} = bool\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\u2261-sym {bool} bool = bool\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\u2261-trans {bool} bool bool = bool\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n\u2261-cong {bool} f \u2261 = {!!}\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\u2261-cong\u2082 {bool} f \u2261\u2081 \u2261\u2082 = {!!}\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\u0394-Type (bool) = bool -- true means negate, false means nil change\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\ndiff {bool} true true = false\ndiff {bool} true false = true\ndiff {bool} false true = true\ndiff {bool} false false = false\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\nderive {bool} b = false\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\napply {bool} true true = false\napply {bool} true false = true\napply {bool} false true = true\napply {bool} false false = false\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\ncompose {bool} true true = false\ncompose {bool} true false = true\ncompose {bool} false true = true\ncompose {bool} false false = false\n\n-- CONGRUENCE rules for change operations\n\n\u2261-diff : \u2200 {\u03c4 : Type} {v\u2081 v\u2082 v\u2083 v\u2084 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 diff v\u2081 v\u2083 \u2261 diff v\u2082 v\u2084\n\u2261-diff = \u2261-cong\u2082 diff\n\n\u2261-apply : \u2200 {\u03c4 : Type} {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  dv\u2081 \u2261 dv\u2082 \u2192 v\u2081 \u2261 v\u2082 \u2192 apply dv\u2081 v\u2081 \u2261 apply dv\u2082 v\u2082\n\u2261-apply = \u2261-cong\u2082 apply\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\ndiff-derive {bool} true = bool\ndiff-derive {bool} false = bool\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\ndiff-apply {bool} true true = bool\ndiff-apply {bool} true false = bool\ndiff-apply {bool} false true = bool\ndiff-apply {bool} false false = bool\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\n\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192\n  begin\n    apply (diff f\u2082 f\u2081) f\u2081 v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f\u2082 (apply (derive v) v)) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 \u2261-apply (\u2261-diff (\u2261-cong f\u2082 (apply-derive v)) \u2261-refl) \u2261-refl \u27e9\n    apply (diff (f\u2082 v) (f\u2081 v)) (f\u2081 v)\n  \u2261\u27e8 apply-diff (f\u2081 v) (f\u2082 v) \u27e9\n    f\u2082 v\n  \u220e) where open \u2261-Reasoning\napply-diff {bool} true true = bool\napply-diff {bool} true false = bool\napply-diff {bool} false true = bool\napply-diff {bool} false false = bool\n\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\napply-derive {bool} true = bool\napply-derive {bool} false = bool\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2081 v (derive v)) (df\u2082 v (derive v)))\napply-compose {bool} true true true = bool\napply-compose {bool} true true false = bool\napply-compose {bool} true false true = bool\napply-compose {bool} true false false = bool\napply-compose {bool} false true true = bool\napply-compose {bool} false true false = bool\napply-compose {bool} false false true = bool\napply-compose {bool} false false false = bool\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\ncompose-assoc {bool} true true true = bool\ncompose-assoc {bool} true true false = bool\ncompose-assoc {bool} true false true = bool\ncompose-assoc {bool} true false false = bool\ncompose-assoc {bool} false true true = bool\ncompose-assoc {bool} false true false = bool\ncompose-assoc {bool} false false true = bool\ncompose-assoc {bool} false false false = bool\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192 apply-diff (f\u2081 v) (f\u2082 v))\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2082 v (derive v)) (df\u2082 v (derive v)))\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n\u0394-Context\u2032 : (\u0393 : Context) \u2192 Prefix \u0393 \u2192 Context\n\u0394-Context\u2032 \u0393 \u2205 = \u0394-Context \u0393\n\u0394-Context\u2032 (.\u03c4 \u2022 \u0393) (\u03c4 \u2022 \u0393\u2032) = \u03c4 \u2022 \u0394-Context\u2032 \u0393 \u0393\u2032\n\nupdate\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate\u2032 \u2205 \u03c1 = update \u03c1\nupdate\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 update\u2032 \u0393\u2032 \u03c1\n\nignore\u2032 : \u2200 {\u0393} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore\u2032 \u2205 \u03c1 = ignore \u03c1\nignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) = v \u2022 ignore\u2032 \u0393\u2032 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\nfv : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Context\nfv {\u0393} _ = \u0393\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2248) = ext (\u03bb \u03c1 \u2192 ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 diff (x v) (y v)) (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv = meaning \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal = meaning \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"68f1a4ca7868d7b75f634edb2a779509d01e7ceb","subject":"auxiliary functions to define levels compactly; example level","message":"auxiliary functions to define levels compactly; example level\n","repos":"toonn\/popartt","old_file":"koopa.agda","new_file":"koopa.agda","new_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin renaming (_+_ to _F+_)\n  open import Data.List\n  open import Data.Vec renaming (map to vmap; lookup to vlookup)\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n  open Matrix\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa Color : Set where\n    _KT : Color \u2192 KoopaTroopa Color\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n\n  data _follows_ : Position \u2192 Position \u2192 Set where\n    stay  : \u2200 {p} \u2192 p follows p\n    next  : \u2200 {x y mat} \u2192 pos (suc x) y mat follows pos x y mat\n    back  : \u2200 {x y mat} \u2192 pos x y mat follows pos (suc x) y mat\n    -- jump  : \u2200 {x y mat} \u2192 pos x (suc y) mat follows pos x y mat\n    fall  : \u2200 {x y mat} \u2192 pos x y mat follows pos x (suc y) mat\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  ex_path : Path (Red KT) (pos 0 0 solid) (pos 0 0 solid)\n  ex_path = pos 0 0 solid \u21a0\u27e8 next \u27e9\n            pos 1 0 solid \u21a0\u27e8 back \u27e9\n            pos 0 0 solid \u21a0\u27e8 stay \u27e9 []\n\n  matToPosVec : {n : \u2115} \u2192 Vec Material n \u2192 \u2115 \u2192 \u2115 \u2192 Vec Position n\n  matToPosVec []           _ _ = []\n  matToPosVec (mat \u2237 mats) x y = pos x y mat \u2237 matToPosVec mats (x + 1) y\n\n  matToPosVecs : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Vec (Vec Position w) h\n  matToPosVecs []                   = []\n  matToPosVecs (_\u2237_ {y} mats matss) = matToPosVec mats 0 y \u2237 matToPosVecs matss\n\n  matToMat : {w h : \u2115} \u2192 Vec (Vec Material w) h \u2192 Matrix Position w h\n  matToMat matss = Mat (matToPosVecs matss)\n\n  \u25a1 : Material\n  \u25a1 = gas\n  \u25a0 : Material\n  \u25a0 = solid\n  example_level : Matrix Position 10 7\n  example_level = Mat (matToPosVecs (\n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 []) \u2237 \n    (\u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a1 \u2237 \u25a1 \u2237 \u25a0 \u2237 \u25a0 \u2237 \u25a0 \u2237 []) \u2237 []))\n\n","old_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\n  open import Data.Nat\n  open import Data.Fin\n  open import Data.List\n  open import Data.Vec renaming (map to vmap; lookup to vlookup)\n\n  module Matrix where\n    data Matrix (A : Set) : \u2115 \u2192 \u2115 \u2192 Set where\n      Mat : {w h : \u2115} \u2192 Vec (Vec A w) h \u2192 Matrix A w h\n\n    lookup : \u2200 {w h} {A : Set} \u2192 Fin h \u2192 Fin w \u2192 Matrix A w h \u2192 A\n    lookup row column (Mat rows) = vlookup column (vlookup row rows)\n\n  data Color : Set where\n    Green : Color\n    Red : Color\n\n  data KoopaTroopa Color : Set where\n    _KT : Color \u2192 KoopaTroopa Color\n\n  data Material : Set where\n    gas    : Material\n    -- liquid : Material\n    solid  : Material\n\n  record Position : Set where\n    constructor pos\n    field\n      x   : \u2115\n      y   : \u2115\n      mat : Material\n\n  data _follows_ : Position \u2192 Position \u2192 Set where\n    stay  : \u2200 {p} \u2192 p follows p\n    next  : \u2200 {x y mat} \u2192 pos (suc x) y mat follows pos x y mat\n    back  : \u2200 {x y mat} \u2192 pos x y mat follows pos (suc x) y mat\n    -- jump  : \u2200 {x y mat} \u2192 pos x (suc y) mat follows pos x y mat\n    fall  : \u2200 {x y mat} \u2192 pos x y mat follows pos x (suc y) mat\n\n\n  infixr 5 _\u21a0\u27e8_\u27e9_\n  data Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n    []  : \u2200 {p} \u2192 Path Koopa p p\n    _\u21a0\u27e8_\u27e9_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n                \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\n  example_level : Matrix Position \n  example_level = []\n\n  ex_path : Path (Red KT) (pos 0 0 solid) (pos 0 0 solid)\n  ex_path = pos 0 0 solid \u21a0\u27e8 next \u27e9\n            pos 1 0 solid \u21a0\u27e8 back \u27e9\n            pos 0 0 solid \u21a0\u27e8 stay \u27e9 []\n","returncode":0,"stderr":"","license":"bsd-2-clause","lang":"Agda"}
{"commit":"4fa1bdd8b3ebc60c9e0e35d29c64fab78c71de4a","subject":"Added draft about the use of the existential elimination.","message":"Added draft about the use of the existential elimination.\n\nIgnore-this: 3eaa2ea7fe0572858239df18e1843c2c\n\ndarcs-hash:20120219124310-3bd4e-aad6b1f41dbc1f59201ed878ba50fec74bbc21ae.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/Existential.agda","new_file":"Draft\/FOTC\/Data\/Stream\/Existential.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"22461d80f96884e08432e0e7c6886e34bed883f6","subject":"Get rid of \u209c","message":"Get rid of \u209c\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n  -}\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n{-\n    data _\u2264\u1d3e_ : \u2200 {x y t} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set where\n      here-here   : here \u2264\u1d3e here\n      left-left   : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 left p \u2264\u1d3e left q\n      right-right : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 right p \u2264\u1d3e right q\n      left-right  : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 left p \u2264\u1d3e right q\n      -}\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\nmodule Sorting {a} {A : Set a} -- (sort\u1d2c : A \u00d7 A \u2192 A \u00d7 A)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    {-\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\n    {-\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted' _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    postulate dec-\u2264 : \u2200 x y \u2192 Dec (x \u2264\u1d2c y)\n    postulate dec-\u2294 : \u2200 x y \u2192 x \u2294\u1d2c y \u2261 x \u228e x \u2294\u1d2c y \u2261 y\n    postulate dec-\u2293 : \u2200 x y \u2192 x \u2293\u1d2c y \u2261 x \u228e x \u2293\u1d2c y \u2261 y\n\n    merge-spec : \u2200 {n} (t u : Tree A n) \u2192\n                 Sorted t \u2192 Sorted u \u2192 Sorted (merge t u)\n    merge-spec (leaf x) (leaf y) sx sy (left here) (left here) pf = \u2264\u1d2c.refl\n    merge-spec (leaf x) (leaf y) sx sy (left here) (right here) (0-1 ._ ._) = \u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)\n    merge-spec (leaf x) (leaf y) sx sy (right p) (left q) ()\n    merge-spec (leaf x) (leaf y) sx sy (right here) (right here) pf = \u2264\u1d2c.refl\n    merge-spec (fork t\u2080 t\u2081) (fork u\u2080 u\u2081) st su p q p\u2264q\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 -- | merge-spec t\u2080 u\u2080 ? ? | merge-spec t\u2081 u\u2081 ? ?\n    ... | fork l m\u2080    | fork m\u2081 h\n      with merge m\u2080 m\u2081 -- | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 -- | fork {high_t = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with p | q | p\u2264q\n    ... | left  pp | left  qq | there ._ pp\u2264qq = {!merge-spec t\u2080 u\u2080 !}\n    ... | left  pp | right qq | 0-1 ._ ._ = {!!}\n    ... | right pp | left qq  | ()\n    ... | right pp | right qq | there ._ pp\u2264qq = {!!}\n    -}\n    -}\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf)\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = refl\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = refl\n\n             {-\n    \u2203Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    \u2203Sorted t = \u2203 \u03bb l \u2192 \u2203 \u03bb h \u2192 Sorted t l h\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = \u2203Sorted t \u00d7 \u2203Sorted u\n    -}\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = SD.Sorted t \u00d7 SD.Sorted u\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork t t\u2081) = refl\n\n    {-\n    record MergeInnerHyp {n} (t : Tree A (2 + n)) : Set (a L.\u2294 \u2113) where\n      constructor mk\n      field\n        st\u2080 : SD.Sorted (lft t)\n        st\u2081 : SD.Sorted (rght t)\n      u = interchange t\n      field\n        su\u2080 : SD.Sorted (lft u)\n        su\u2081 : SD.Sorted (rght u)\n\n    merge-inner : \u2200 {n} {t : Tree A (2 + n)} \u2192\n                 MergeInnerHyp t \u2192 SD.Sorted (map-inner merge t)\n    merge-inner {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n                (mk (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                    (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n                    (fork sv\u2080 sv\u2081 hv\u2080\u2264lv\u2081)\n                    (fork sw\u2080 sw\u2081 lw\u2080\u2264hw\u2081)) = {!!}\n                    -}\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n                -- last  (merge t u) \u2261 last  t \u2293\u1d2c last  u\n    merge-spec : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192 SD.Sorted (merge (fork t u))\n    merge-spec (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec st\u2080 su\u2080 | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec st\u2081 su\u2081\n    ... | fork v\u2080 w\u2080 | fork sv\u2080 sw\u2080 p1 | fpf1\n        | fork v\u2081 w\u2081 | fork sv\u2081 sw\u2081 p2\n      with merge (fork w\u2080 v\u2081) | merge-spec sw\u2080 sv\u2081\n    ... | fork a b | fork sa sb p3\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3\n         where\n             postulate\n                pf3 : last v\u2080 \u2264\u1d2c first t\u2081\n                pf4 : last v\u2080 \u2264\u1d2c first u\u2081\n                -- Sorted (merge t u) \u2192\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 = {!first-merge !}\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 = {!!}\n    {-\n      -}\n\n\n--      map-outer merge id \u2218 interchange\n\n      {-\n\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high_t = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high_t = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high_t = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      {-with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl-} =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n  -}\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low\u209c high\u209c low\u1d64 high\u1d64} \u2192\n             Sorted t low\u209c high\u209c \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high\u209c \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low\u209c high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n{-\n    data _\u2264\u1d3e_ : \u2200 {x y t} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set where\n      here-here   : here \u2264\u1d3e here\n      left-left   : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 left p \u2264\u1d3e left q\n      right-right : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 right p \u2264\u1d3e right q\n      left-right  : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 left p \u2264\u1d3e right q\n      -}\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\nmodule Sorting {a} {A : Set a} -- (sort\u1d2c : A \u00d7 A \u2192 A \u00d7 A)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    {-\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\n    {-\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted' _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    postulate dec-\u2264 : \u2200 x y \u2192 Dec (x \u2264\u1d2c y)\n    postulate dec-\u2294 : \u2200 x y \u2192 x \u2294\u1d2c y \u2261 x \u228e x \u2294\u1d2c y \u2261 y\n    postulate dec-\u2293 : \u2200 x y \u2192 x \u2293\u1d2c y \u2261 x \u228e x \u2293\u1d2c y \u2261 y\n\n    merge-spec : \u2200 {n} (t u : Tree A n) \u2192\n                 Sorted t \u2192 Sorted u \u2192 Sorted (merge t u)\n    merge-spec (leaf x) (leaf y) sx sy (left here) (left here) pf = \u2264\u1d2c.refl\n    merge-spec (leaf x) (leaf y) sx sy (left here) (right here) (0-1 ._ ._) = \u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)\n    merge-spec (leaf x) (leaf y) sx sy (right p) (left q) ()\n    merge-spec (leaf x) (leaf y) sx sy (right here) (right here) pf = \u2264\u1d2c.refl\n    merge-spec (fork t\u2080 t\u2081) (fork u\u2080 u\u2081) st su p q p\u2264q\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 -- | merge-spec t\u2080 u\u2080 ? ? | merge-spec t\u2081 u\u2081 ? ?\n    ... | fork l m\u2080    | fork m\u2081 h\n      with merge m\u2080 m\u2081 -- | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 -- | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with p | q | p\u2264q\n    ... | left  pp | left  qq | there ._ pp\u2264qq = {!merge-spec t\u2080 u\u2080 !}\n    ... | left  pp | right qq | 0-1 ._ ._ = {!!}\n    ... | right pp | left qq  | ()\n    ... | right pp | right qq | there ._ pp\u2264qq = {!!}\n    -}\n    -}\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf)\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = refl\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = refl\n\n             {-\n    \u2203Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    \u2203Sorted t = \u2203 \u03bb l \u2192 \u2203 \u03bb h \u2192 Sorted t l h\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = \u2203Sorted t \u00d7 \u2203Sorted u\n    -}\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = SD.Sorted t \u00d7 SD.Sorted u\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork t t\u2081) = refl\n\n    {-\n    record MergeInnerHyp {n} (t : Tree A (2 + n)) : Set (a L.\u2294 \u2113) where\n      constructor mk\n      field\n        st\u2080 : SD.Sorted (lft t)\n        st\u2081 : SD.Sorted (rght t)\n      u = interchange t\n      field\n        su\u2080 : SD.Sorted (lft u)\n        su\u2081 : SD.Sorted (rght u)\n\n    merge-inner : \u2200 {n} {t : Tree A (2 + n)} \u2192\n                 MergeInnerHyp t \u2192 SD.Sorted (map-inner merge t)\n    merge-inner {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n                (mk (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                    (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n                    (fork sv\u2080 sv\u2081 hv\u2080\u2264lv\u2081)\n                    (fork sw\u2080 sw\u2081 lw\u2080\u2264hw\u2081)) = {!!}\n                    -}\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n                -- last  (merge t u) \u2261 last  t \u2293\u1d2c last  u\n    merge-spec : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192 SD.Sorted (merge (fork t u))\n    merge-spec (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec st\u2080 su\u2080 | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec st\u2081 su\u2081\n    ... | fork v\u2080 w\u2080 | fork sv\u2080 sw\u2080 p1 | fpf1\n        | fork v\u2081 w\u2081 | fork sv\u2081 sw\u2081 p2\n      with merge (fork w\u2080 v\u2081) | merge-spec sw\u2080 sv\u2081\n    ... | fork a b | fork sa sb p3\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3\n         where\n             postulate\n                pf3 : last v\u2080 \u2264\u1d2c first t\u2081\n                pf4 : last v\u2080 \u2264\u1d2c first u\u2081\n                -- Sorted (merge t u) \u2192\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 = {!first-merge !}\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 = {!!}\n    {-\n      -}\n\n\n--      map-outer merge id \u2218 interchange\n\n      {-\n\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high\u209c = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high\u209c = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      {-with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl-} =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"03a5fae73d699a763ae6c39c316350902dfa651e","subject":"close #16","message":"close #16\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d1' d2 \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)  \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d d' \u03c4 \u03c4' } \u2192\n                        d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFinal : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of u?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of u?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts? should this\n  -- be judgemental rather than functional?\n\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              -- \u03941 ## \u03942 \u2192 -- todo: need to think about disjointness and context rep\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x e \u03c4 d \u0394 } \u2192\n              (x # \u0393) \u2192\n              (\u0393 ,, (x , \u2987\u2988)) \u22a2 e \u21d0 \u2987\u2988 ~> d :: \u03c4 \u22a3 \u0394  \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ \u2987\u2988 ] d :: \u2987\u2988 ==> \u03c4 \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X y\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d1' d2 \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)  \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d d' \u03c4 \u03c4' } \u2192\n                        d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9   : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            d final \u2192\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7 -- this used to a have a premise of being final for some reason\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           d1 final \u2192\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            d2 final \u2192\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    d final \u2192\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               d1 final \u2192\n               d2 final \u2192\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c41 \u03c42 } \u2192\n               d final \u2192\n               \u03c41 ==> \u03c42 \u2260 \u2987\u2988 ==> \u2987\u2988 \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u2987\u2988 ==> \u2987\u2988 \u21d2 \u03c41 ==> \u03c42 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _casterr : (d : dhexp) \u2192 Set where\n    CECastFinal : \u2200 {d \u03c41 \u03c42} \u2192\n                  d final \u2192\n                  \u03c41 ground \u2192\n                  \u03c42 ground \u2192\n                  \u03c41 \u2260 \u03c42 \u2192\n                  d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9 casterr\n    CECong : \u2200{ d \u03b5 d0} \u2192\n             d == \u03b5 \u27e6 d0 \u27e7 \u2192\n             d0 casterr \u2192\n             d casterr\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e6b366592e727138ab845e5190ed6c35fa3d28d4","subject":"make \u00d7 universe polymorphic","message":"make \u00d7 universe polymorphic\n","repos":"crypto-agda\/protocols","old_file":"js-experiment\/prelude.agda","new_file":"js-experiment\/prelude.agda","new_contents":"module _ where\n\nopen import Agda.Primitive public\n  using    (Level; _\u2294_)\n  renaming (lzero to \u2080; lsuc to \u209b)\n\n-- open import Function.NP\nid : \u2200 {a} {A : Set a} (x : A) \u2192 A\nid x = x\n\ninfixr 9 _\u2218_\n_\u2218_ : \u2200 {a b c}\n        {A : Set a} {B : A \u2192 Set b} {C : {x : A} \u2192 B x \u2192 Set c} \u2192\n        (\u2200 {x} (y : B x) \u2192 C y) \u2192 (g : (x : A) \u2192 B x) \u2192\n        ((x : A) \u2192 C (g x))\nf \u2218 g = \u03bb x \u2192 f (g x)\n\ninfixr 0 _$_\n_$_ : \u2200 {a b} {A : Set a} {B : A \u2192 Set b} \u2192\n      ((x : A) \u2192 B x) \u2192 ((x : A) \u2192 B x)\nf $ x = f x\n\n_$\u2032_ : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n      (A \u2192 B) \u2192 (A \u2192 B)\nf $\u2032 x = f x\n\ndata \ud835\udfd8 : Set where\n\n-- open import Data.One\nrecord \ud835\udfd9 : Set\u2080 where\n  constructor <>\nopen \ud835\udfd9\n\ndata Bool : Set where\n  true false : Bool\n\n[true:_,false:_] : \u2200 {c} {C : Bool \u2192 Set c} \u2192\n        C true \u2192 C false \u2192\n        ((x : Bool) \u2192 C x)\n[true: f ,false: g ] true  = f\n[true: f ,false: g ] false = g\n\ndata LR : Set where L R : LR\n\n[L:_,R:_] : \u2200 {c}{C : LR \u2192 Set c} \u2192\n  C L \u2192 C R \u2192 (x : LR) \u2192 C x\n[L: f ,R: g ] L = f\n[L: f ,R: g ] R = g\n\n-- open import Data.Product.NP\nrecord \u03a3 {a b}(A : Set a)(B : A \u2192 Set b) : Set(a \u2294 b) where\n  constructor _,_\n  field\n    fst : A\n    snd : B fst\nopen \u03a3 public\n\n_\u00d7_ : \u2200{a b}(A : Set a)(B : Set b) \u2192 Set _\nA \u00d7 B = \u03a3 A (\u03bb _ \u2192 B)\n\n-- open import Data.Sum.NP\ndata _\u228e_ (A B : Set) : Set where\n  inl : A \u2192 A \u228e B\n  inr : B \u2192 A \u228e B\n\n[inl:_,inr:_] : \u2200 {c} {A B : Set} {C : A \u228e B \u2192 Set c} \u2192\n        ((x : A) \u2192 C (inl x)) \u2192 ((x : B) \u2192 C (inr x)) \u2192\n        ((x : A \u228e B) \u2192 C x)\n[inl: f ,inr: g ] (inl x) = f x\n[inl: f ,inr: g ] (inr y) = g y\n\n-- open import Data.List using (List; []; _\u2237_)\ninfixr 5 _\u2237_\n\ndata List {a} (A : Set a) : Set a where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\n[_] : \u2200 {a}{A : Set a} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n{-# BUILTIN LIST List #-}\n{-# BUILTIN NIL  []   #-}\n{-# BUILTIN CONS _\u2237_  #-}\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse {A} = go []\n  where\n    go : List A \u2192 List A \u2192 List A\n    go acc []       = acc\n    go acc (x \u2237 xs) = go (x \u2237 acc) xs\n\nmodule _ {A : Set} (_\u2264_ : A \u2192 A \u2192 Bool) where\n\n    merge-sort-list : (l\u2080 l\u2081 : List A) \u2192 List A\n    merge-sort-list [] l\u2081 = l\u2081\n    merge-sort-list l\u2080 [] = l\u2080\n    merge-sort-list (x\u2080 \u2237 l\u2080) (x\u2081 \u2237 l\u2081) with x\u2080 \u2264 x\u2081\n    ... | true  = x\u2080 \u2237 merge-sort-list l\u2080 (x\u2081 \u2237 l\u2081)\n    ... | false = x\u2081 \u2237 merge-sort-list (x\u2080 \u2237 l\u2080) l\u2081\n\n-- open import Relation.Binary.PropositionalEquality\ninfix 0 _\u2261_\ndata _\u2261_ {a}{A : Set a} (x : A) : A \u2192 Set a where\n  refl : x \u2261 x\n\n{-# BUILTIN EQUALITY _\u2261_ #-}\n{-# BUILTIN REFL refl #-}\n\n_\u2262_ : \u2200 {a}{A : Set a}(x y : A) \u2192 Set a\nx \u2262 y = x \u2261 y \u2192 \ud835\udfd8\n\nap : \u2200{a b}{A : Set a}{B : Set b} (f : A \u2192 B) {x y : A} (p : x \u2261 y) \u2192 f x \u2261 f y\nap f refl = refl\n\nsym : \u2200{a}{A : Set a}{x y : A} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n! : \u2200{a}{A : Set a}{x y : A} \u2192 x \u2261 y \u2192 y \u2261 x\n! refl = refl\n\nsubst : \u2200{a p}{A : Set a}(P : A \u2192 Set p){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n\ntr = subst\n\nap\u2082 : \u2200 {a b c}{A : Set a}{B : Set b}{C : Set c}(f : A \u2192 B \u2192 C){x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y'\n    \u2192 f x y \u2261 f x' y'\nap\u2082 f refl refl = refl\n\ninfixr 6 _\u2219_\n_\u2219_ : \u2200 {a}{A : Set a}{x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\nrefl \u2219 p = p\n\npostulate\n  String : Set\n  Char   : Set\n\n{-# BUILTIN STRING String #-}\n{-# BUILTIN CHAR   Char #-}\n\ndata Error (A : Set) : Set where\n  succeed : A \u2192 Error A\n  fail    : (err : String) \u2192 Error A\n\n[succeed:_,fail:_] : \u2200 {c} {A : Set} {C : Error A \u2192 Set c} \u2192\n        ((x : A) \u2192 C (succeed x)) \u2192 ((x : String) \u2192 C (fail x)) \u2192\n        ((x : Error A) \u2192 C x)\n[succeed: f ,fail: g ] (succeed x) = f x\n[succeed: f ,fail: g ] (fail y) = g y\n\nmapE : {A B : Set} \u2192 (A \u2192 B) \u2192 Error A \u2192 Error B\nmapE f (fail err) = fail err\nmapE f (succeed x) = succeed (f x)\n\ndata All {A : Set} (P : A \u2192 Set) : Error A \u2192 Set where\n  fail    : \u2200 s \u2192 All P (fail s)\n  succeed : \u2200 {x} \u2192 P x \u2192 All P (succeed x)\n\nrecord _\u2243?_ (A B : Set) : Set where\n  field\n    serialize : A \u2192 B\n    parse     : B \u2192 Error A\n    linv      : \u2200 x \u2192 All (_\u2261_ x \u2218 serialize) (parse x)\n    rinv      : \u2200 x \u2192 parse (serialize x) \u2261 succeed x\nopen _\u2243?_ {{...}} public\n\n{-\nopen import Category.Profunctor\n\nPrism : (S T A B : Set) \u2192 Set\u2081\nPrism S T A B = \u2200 {_\u219d_}{{_ : Choice {\u2080} _\u219d_}} \u2192 (A \u219d B) \u2192 (S \u219d T)\n\nPrism' : (S A : Set) \u2192 Set\u2081\nPrism' S A = Prism S S A A\n\nmodule _ {S T A B : Set} where\n    prism : (B \u2192 T) \u2192 (S \u2192 T \u228e A) \u2192 Prism S T A B\n    prism bt sta = dimap sta [inl: id ,inr: bt ] \u2218 right\n      where open Choice {{...}}\n-}\n\nPrism : (S T A B : Set) \u2192 Set\nPrism S T A B = (B \u2192 T) \u00d7 (S \u2192 T \u228e A)\n\nPrism' : (S A : Set) \u2192 Set\nPrism' S A = Prism S S A A\n\nmodule _ {S T A B : Set} where\n    prism : (B \u2192 T) \u2192 (S \u2192 T \u228e A) \u2192 Prism S T A B\n    prism = _,_\n\n    -- This is particular case of lens' _#_\n    _#_ : Prism S T A B \u2192 B \u2192 T\n    _#_ = fst\n\n    -- This is particular case of lens' review\n    review = _#_\n\n    -- _^._ : Prism S T A B \u2192 S \u2192 ...\n\nmodule _ {S A : Set} where\n    prism' : (A \u2192 S) \u2192 (S \u2192 S \u228e A) \u2192 Prism' S A\n    prism' = prism\n\nmodule _ {a b}{A : Set a}{B : A \u2192 Set b} where\n  case_of_ : (x : A) \u2192 ((y : A) \u2192 B y) \u2192 B x\n  case x of f = f x\n\nmodule _ {a b}{A : Set a}{B : Set b} where\n  case_of\u2032_ : A \u2192 (A \u2192 B) \u2192 B\n  case_of\u2032_ = case_of_\n\n\u2026 : {A : Set}{{x : A}} \u2192 A\n\u2026 {{x}} = x\n","old_contents":"module _ where\n\nopen import Agda.Primitive public\n  using    (Level; _\u2294_)\n  renaming (lzero to \u2080; lsuc to \u209b)\n\n-- open import Function.NP\nid : \u2200 {a} {A : Set a} (x : A) \u2192 A\nid x = x\n\ninfixr 9 _\u2218_\n_\u2218_ : \u2200 {a b c}\n        {A : Set a} {B : A \u2192 Set b} {C : {x : A} \u2192 B x \u2192 Set c} \u2192\n        (\u2200 {x} (y : B x) \u2192 C y) \u2192 (g : (x : A) \u2192 B x) \u2192\n        ((x : A) \u2192 C (g x))\nf \u2218 g = \u03bb x \u2192 f (g x)\n\ninfixr 0 _$_\n_$_ : \u2200 {a b} {A : Set a} {B : A \u2192 Set b} \u2192\n      ((x : A) \u2192 B x) \u2192 ((x : A) \u2192 B x)\nf $ x = f x\n\n_$\u2032_ : \u2200 {a b} {A : Set a} {B : Set b} \u2192\n      (A \u2192 B) \u2192 (A \u2192 B)\nf $\u2032 x = f x\n\ndata \ud835\udfd8 : Set where\n\n-- open import Data.One\nrecord \ud835\udfd9 : Set\u2080 where\n  constructor <>\nopen \ud835\udfd9\n\ndata Bool : Set where\n  true false : Bool\n\n[true:_,false:_] : \u2200 {c} {C : Bool \u2192 Set c} \u2192\n        C true \u2192 C false \u2192\n        ((x : Bool) \u2192 C x)\n[true: f ,false: g ] true  = f\n[true: f ,false: g ] false = g\n\ndata LR : Set where L R : LR\n\n[L:_,R:_] : \u2200 {c}{C : LR \u2192 Set c} \u2192\n  C L \u2192 C R \u2192 (x : LR) \u2192 C x\n[L: f ,R: g ] L = f\n[L: f ,R: g ] R = g\n\n-- open import Data.Product.NP\nrecord \u03a3 {a b}(A : Set a)(B : A \u2192 Set b) : Set(a \u2294 b) where\n  constructor _,_\n  field\n    fst : A\n    snd : B fst\nopen \u03a3 public\n\n_\u00d7_ : (A B : Set) \u2192 Set\nA \u00d7 B = \u03a3 A (\u03bb _ \u2192 B)\n\n-- open import Data.Sum.NP\ndata _\u228e_ (A B : Set) : Set where\n  inl : A \u2192 A \u228e B\n  inr : B \u2192 A \u228e B\n\n[inl:_,inr:_] : \u2200 {c} {A B : Set} {C : A \u228e B \u2192 Set c} \u2192\n        ((x : A) \u2192 C (inl x)) \u2192 ((x : B) \u2192 C (inr x)) \u2192\n        ((x : A \u228e B) \u2192 C x)\n[inl: f ,inr: g ] (inl x) = f x\n[inl: f ,inr: g ] (inr y) = g y\n\n-- open import Data.List using (List; []; _\u2237_)\ninfixr 5 _\u2237_\n\ndata List {a} (A : Set a) : Set a where\n  []  : List A\n  _\u2237_ : (x : A) (xs : List A) \u2192 List A\n\n[_] : \u2200 {a}{A : Set a} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n{-# BUILTIN LIST List #-}\n{-# BUILTIN NIL  []   #-}\n{-# BUILTIN CONS _\u2237_  #-}\n\nreverse : {A : Set} \u2192 List A \u2192 List A\nreverse {A} = go []\n  where\n    go : List A \u2192 List A \u2192 List A\n    go acc []       = acc\n    go acc (x \u2237 xs) = go (x \u2237 acc) xs\n\nmodule _ {A : Set} (_\u2264_ : A \u2192 A \u2192 Bool) where\n\n    merge-sort-list : (l\u2080 l\u2081 : List A) \u2192 List A\n    merge-sort-list [] l\u2081 = l\u2081\n    merge-sort-list l\u2080 [] = l\u2080\n    merge-sort-list (x\u2080 \u2237 l\u2080) (x\u2081 \u2237 l\u2081) with x\u2080 \u2264 x\u2081\n    ... | true  = x\u2080 \u2237 merge-sort-list l\u2080 (x\u2081 \u2237 l\u2081)\n    ... | false = x\u2081 \u2237 merge-sort-list (x\u2080 \u2237 l\u2080) l\u2081\n\n-- open import Relation.Binary.PropositionalEquality\ninfix 0 _\u2261_\ndata _\u2261_ {a}{A : Set a} (x : A) : A \u2192 Set a where\n  refl : x \u2261 x\n\n{-# BUILTIN EQUALITY _\u2261_ #-}\n{-# BUILTIN REFL refl #-}\n\n_\u2262_ : \u2200 {a}{A : Set a}(x y : A) \u2192 Set a\nx \u2262 y = x \u2261 y \u2192 \ud835\udfd8\n\nap : \u2200{a b}{A : Set a}{B : Set b} (f : A \u2192 B) {x y : A} (p : x \u2261 y) \u2192 f x \u2261 f y\nap f refl = refl\n\nsym : \u2200{a}{A : Set a}{x y : A} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n! : \u2200{a}{A : Set a}{x y : A} \u2192 x \u2261 y \u2192 y \u2261 x\n! refl = refl\n\nsubst : \u2200{a p}{A : Set a}(P : A \u2192 Set p){x y : A} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n\ntr = subst\n\nap\u2082 : \u2200 {a b c}{A : Set a}{B : Set b}{C : Set c}(f : A \u2192 B \u2192 C){x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y'\n    \u2192 f x y \u2261 f x' y'\nap\u2082 f refl refl = refl\n\ninfixr 6 _\u2219_\n_\u2219_ : \u2200 {a}{A : Set a}{x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\nrefl \u2219 p = p\n\npostulate\n  String : Set\n  Char   : Set\n\n{-# BUILTIN STRING String #-}\n{-# BUILTIN CHAR   Char #-}\n\ndata Error (A : Set) : Set where\n  succeed : A \u2192 Error A\n  fail    : (err : String) \u2192 Error A\n\n[succeed:_,fail:_] : \u2200 {c} {A : Set} {C : Error A \u2192 Set c} \u2192\n        ((x : A) \u2192 C (succeed x)) \u2192 ((x : String) \u2192 C (fail x)) \u2192\n        ((x : Error A) \u2192 C x)\n[succeed: f ,fail: g ] (succeed x) = f x\n[succeed: f ,fail: g ] (fail y) = g y\n\nmapE : {A B : Set} \u2192 (A \u2192 B) \u2192 Error A \u2192 Error B\nmapE f (fail err) = fail err\nmapE f (succeed x) = succeed (f x)\n\ndata All {A : Set} (P : A \u2192 Set) : Error A \u2192 Set where\n  fail    : \u2200 s \u2192 All P (fail s)\n  succeed : \u2200 {x} \u2192 P x \u2192 All P (succeed x)\n\nrecord _\u2243?_ (A B : Set) : Set where\n  field\n    serialize : A \u2192 B\n    parse     : B \u2192 Error A\n    linv      : \u2200 x \u2192 All (_\u2261_ x \u2218 serialize) (parse x)\n    rinv      : \u2200 x \u2192 parse (serialize x) \u2261 succeed x\nopen _\u2243?_ {{...}} public\n\n{-\nopen import Category.Profunctor\n\nPrism : (S T A B : Set) \u2192 Set\u2081\nPrism S T A B = \u2200 {_\u219d_}{{_ : Choice {\u2080} _\u219d_}} \u2192 (A \u219d B) \u2192 (S \u219d T)\n\nPrism' : (S A : Set) \u2192 Set\u2081\nPrism' S A = Prism S S A A\n\nmodule _ {S T A B : Set} where\n    prism : (B \u2192 T) \u2192 (S \u2192 T \u228e A) \u2192 Prism S T A B\n    prism bt sta = dimap sta [inl: id ,inr: bt ] \u2218 right\n      where open Choice {{...}}\n-}\n\nPrism : (S T A B : Set) \u2192 Set\nPrism S T A B = (B \u2192 T) \u00d7 (S \u2192 T \u228e A)\n\nPrism' : (S A : Set) \u2192 Set\nPrism' S A = Prism S S A A\n\nmodule _ {S T A B : Set} where\n    prism : (B \u2192 T) \u2192 (S \u2192 T \u228e A) \u2192 Prism S T A B\n    prism = _,_\n\n    -- This is particular case of lens' _#_\n    _#_ : Prism S T A B \u2192 B \u2192 T\n    _#_ = fst\n\n    -- This is particular case of lens' review\n    review = _#_\n\n    -- _^._ : Prism S T A B \u2192 S \u2192 ...\n\nmodule _ {S A : Set} where\n    prism' : (A \u2192 S) \u2192 (S \u2192 S \u228e A) \u2192 Prism' S A\n    prism' = prism\n\nmodule _ {a b}{A : Set a}{B : A \u2192 Set b} where\n  case_of_ : (x : A) \u2192 ((y : A) \u2192 B y) \u2192 B x\n  case x of f = f x\n\nmodule _ {a b}{A : Set a}{B : Set b} where\n  case_of\u2032_ : A \u2192 (A \u2192 B) \u2192 B\n  case_of\u2032_ = case_of_\n\n\u2026 : {A : Set}{{x : A}} \u2192 A\n\u2026 {{x}} = x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"87ccae64e17e632f7df9b19625d246429531dca7","subject":"Control\/Protocol\/Extend.agda","message":"Control\/Protocol\/Extend.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Extend.agda","new_file":"Control\/Protocol\/Extend.agda","new_contents":"open import Type\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Function\nopen import Function.Extensionality\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Extend where\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n\n    extend-send : Proto \u2192 Proto\n    extend-send end      = end\n    extend-send (send P) = send [inl: (\u03bb m \u2192 extend-send (P m)) ,inr: A\u1d3e ]\n    extend-send (recv P) = recv \u03bb m \u2192 extend-send (P m)\n\n    extend-recv : Proto \u2192 Proto\n    extend-recv end      = end\n    extend-recv (recv P) = recv [inl: (\u03bb m \u2192 extend-recv (P m)) ,inr: A\u1d3e ]\n    extend-recv (send P) = send \u03bb m \u2192 extend-recv (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto){{_ : FunExt}} where\n\n    dual-extend-send : \u2200 P \u2192 dual (extend-send A\u1d3e P) \u2261 extend-recv (dual \u2218 A\u1d3e) (dual P)\n    dual-extend-send end      = refl\n    dual-extend-send (recv P) = send=\u2032 \u03bb m \u2192 dual-extend-send (P m)\n    dual-extend-send (send P) = recv=\u2032 [inl: (\u03bb m \u2192 dual-extend-send (P m))\n                                       ,inr: (\u03bb x \u2192 refl) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = extend-send Abort\u1d3e\n","old_contents":"open import Type\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Function\nopen import Function.Extensionality\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Extend where\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n\n    extend-send : Proto \u2192 Proto\n    extend-send end      = end\n    extend-send (send P) = send [inl: (\u03bb m \u2192 extend-send (P m)) ,inr: A\u1d3e ]\n    extend-send (recv P) = recv \u03bb m \u2192 extend-send (P m)\n\n    extend-recv : Proto \u2192 Proto\n    extend-recv end      = end\n    extend-recv (recv P) = recv [inl: (\u03bb m \u2192 extend-recv (P m)) ,inr: A\u1d3e ]\n    extend-recv (send P) = send \u03bb m \u2192 extend-recv (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto){{_ : FunExt}} where\n\n    dual-extend-send : \u2200 P \u2192 dual (extend-send A\u1d3e P) \u2261 extend-recv (dual \u2218 A\u1d3e) (dual P)\n    dual-extend-send end      = refl\n    dual-extend-send (recv P) = send=\u2032 \u03bb m \u2192 dual-extend-send (P m)\n    dual-extend-send (send P) = recv=\u2032 [inl: (\u03bb m \u2192 dual-extend-send (P m))\n                                       ,inr: (\u03bb x \u2192 refl) ]\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1f1412f9c2a5249f583f6606fb2b7a588ee2651e","subject":"New.Changes: Enable reusing more","message":"New.Changes: Enable reusing more\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChangePoint : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 \u2200 a da (v : valid a da) \u2192 Set \u2113\u2082\n  WellDefinedFunChangePoint f df a da ada = (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n\n  WellDefinedFunChange : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  WellDefinedFunChange f df = \u2200 a da ada \u2192 WellDefinedFunChangePoint f df a da ada\n\n  private\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (ada : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        WellDefinedFunChangePoint f df a da ada\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","old_contents":"module New.Changes where\n\nopen import Data.Product public hiding (map)\nopen import Data.Sum public hiding (map)\nopen import Data.Unit.Base public\n\nopen import Relation.Binary.PropositionalEquality public\nopen import Postulate.Extensionality public\nopen import Level\n\nopen import Theorem.Groups-Nehemiah public\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    _\u229a[_]_ : Ch \u2192 A \u2192 Ch \u2192 Ch\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n    \u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 da1 \u229a[ a ] da2 \u2261 a \u2295 da1 \u2295 da2\n    \u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (da1 \u229a[ a ] da2)\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\n  default-\u229a : Ch \u2192 A \u2192 Ch \u2192 Ch\n  default-\u229a da1 a da2 = a \u2295 da1 \u2295 da2 \u229d a\n  default-\u229a-valid : (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192 valid a (default-\u229a da1 a da2)\n  default-\u229a-valid a da1 ada1 da2 ada2 = \u229d-valid a (a \u2295 da1 \u2295 da2)\n  default-\u229a-correct : \u2200 (a : A) \u2192 (da1 : Ch) \u2192 valid a da1 \u2192 (da2 : Ch) \u2192 valid (a \u2295 da1) da2 \u2192\n      a \u2295 default-\u229a da1 a da2 \u2261 a \u2295 da1 \u2295 da2\n  default-\u229a-correct a da1 ada1 da2 ada2 = update-diff (a \u2295 da1 \u2295 da2) a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n    fvalid : (A \u2192 B) \u2192 fCh \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n    fvalid =  \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n    f\u229d-valid : \u2200 (f g : A \u2192 B) \u2192 fvalid f (g f\u229d f)\n    f\u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n\n    -- Two alternatives. Proofs on f\u229a are a bit easier.\n    -- f\u229a is optimized relying on the incrementalization property for df2.\n    f\u229a f\u229a2 : fCh \u2192 (A \u2192 B) \u2192 fCh \u2192 fCh\n    f\u229a df1 f df2 = \u03bb a da \u2192 (df1 a (nil a)) \u229a[ f a ] (df2 a da)\n    f\u229a-valid : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a df1 f df2)\n    f\u229a-valid f df1 fdf1 df2 fdf2 a da ada\n      rewrite \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      | \u229a-correct\n        (f (a \u2295 da))\n        (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n        (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n      =\n        \u229a-valid (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a da) (proj\u2081 (fdf2 a da ada))\n      , proj\u2082 (fdf2 a da ada)\n\n    -- f\u229a2 is optimized relying on the incrementalization property for df1, so we need to use that property to rewrite some goals.\n    f\u229a2 df1 f df2 = \u03bb a da \u2192 df1 a da \u229a[ f a ] df2 (a \u2295 da) (nil (a \u2295 da))\n    f\u229a-valid2 : (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 fvalid f (f\u229a2 df1 f df2)\n    f\u229a-valid2 f df1 fdf1 df2 fdf2 = lemma\n      where\n        -- Prove that (df2 (a \u2295 da) (nil (a \u2295 da))) is valid for (f a \u2295 df1 a da).\n        -- Using the incrementalization property for df1, that becomes (f \u2295 df1) (a \u2295 da).\n        --\n        -- Since df2 is valid for f \u2295 df1, it follows that (df2 (a \u2295 da) (nil (a \u2295 da)))\n        -- is indeed valid for (f \u2295 df1) (a \u2295 da).\n        valid-df2-a2-nila2 : \u2200 a da \u2192 (valid a da) \u2192 valid (f a \u2295 df1 a da) (df2 (a \u2295 da) (nil (a \u2295 da)))\n        valid-df2-a2-nila2 a da ada rewrite sym (proj\u2082 (fdf1 a da ada)) = proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da)))\n\n        lemma : fvalid f (f\u229a2 df1 f df2)\n        lemma a da ada\n          rewrite \u229a-correct\n            (f a)\n            (df1 a da) (proj\u2081 (fdf1 a da ada))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          | update-nil (a \u2295 da)\n          | \u229a-correct\n            (f (a \u2295 da))\n            (df1 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf1 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n            (df2 (a \u2295 da) (nil (a \u2295 da))) (proj\u2081 (fdf2 (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))))\n          | proj\u2082 (fdf1 a da ada)\n          = \u229a-valid (f a) (df1 a da) (proj\u2081 (fdf1 a da ada)) (df2 (a \u2295 da) (nil (a \u2295 da))) (valid-df2-a2-nila2 a da ada)\n          , refl\n\n    f\u229a-correct : \u2200 (f : A \u2192 B) \u2192 (df1 : fCh) \u2192 fvalid f df1 \u2192 (df2 : fCh) \u2192 fvalid (f f\u2295 df1) df2 \u2192 (a : A) \u2192\n      (f f\u2295 f\u229a df1 f df2) a \u2261 ((f f\u2295 df1) f\u2295 df2) a\n    f\u229a-correct f df1 fdf1 df2 fdf2 a = \u229a-correct (f a) (df1 a (nil a)) (proj\u2081 (fdf1 a (nil a) (nil-valid a))) (df2 a (nil a)) (proj\u2081 (fdf2 a (nil a) (nil-valid a)))\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = fvalid\n      ; _\u229a[_]_ = f\u229a\n      ; \u229d-valid = f\u229d-valid\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      ; \u229a-correct = \u03bb f df1 fdf1 df2 fdf2 \u2192 ext (f\u229a-correct f df1 fdf1 df2 fdf2)\n      ; \u229a-valid = f\u229a-valid\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    pCh = Ch A \u00d7 Ch B\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 pCh\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 pCh \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n    _p\u229a[_]_ : pCh \u2192 A \u00d7 B \u2192 pCh \u2192 pCh\n    (da1 , db1) p\u229a[ a , b ] (da2 , db2) = da1 \u229a[ a ] da2 , db1 \u229a[ b ] db2\n    p\u229a-valid : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 pvalid p (dp1 p\u229a[ p ] dp2)\n    p\u229a-valid (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) = \u229a-valid a da1 ada1 da2 ada2 , \u229a-valid b db1 bdb1 db2 bdb2\n    p\u229a-correct : (p : A \u00d7 B) (dp1 : pCh) \u2192\n      pvalid p dp1 \u2192\n      (dp2 : pCh) \u2192 pvalid (p p\u2295 dp1) dp2 \u2192 p p\u2295 (dp1 p\u229a[ p ] dp2) \u2261 (p p\u2295 dp1) p\u2295 dp2\n    p\u229a-correct (a , b) (da1 , db1) (ada1 , bdb1) (da2 , db2) (ada2 , bdb2) rewrite \u229a-correct a da1 ada1 da2 ada2 | \u229a-correct b db1 bdb1 db2 bdb2 = refl\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = pCh\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    ; _\u229a[_]_ = _p\u229a[_]_\n    ; \u229a-valid = p\u229a-valid\n    ; \u229a-correct = p\u229a-correct\n    } }\n\n  private\n    SumChange = (Ch A \u228e Ch B) \u228e (A \u228e B)\n\n  data SumChange2 : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    ch\u2081 : (da : Ch A) \u2192 SumChange2\n    ch\u2082 : (db : Ch B) \u2192 SumChange2\n    rp : (s : A \u228e B) \u2192 SumChange2\n\n  convert : SumChange \u2192 SumChange2\n  convert (inj\u2081 (inj\u2081 da)) = ch\u2081 da\n  convert (inj\u2081 (inj\u2082 db)) = ch\u2082 db\n  convert (inj\u2082 s) = rp s\n  convert\u2081 : SumChange2 \u2192 SumChange\n  convert\u2081 (ch\u2081 da) = inj\u2081 (inj\u2081 da)\n  convert\u2081 (ch\u2082 db) = inj\u2081 (inj\u2082 db)\n  convert\u2081 (rp s) = inj\u2082 s\n\n  data SValid : A \u228e B \u2192 SumChange \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    sv\u2081 : \u2200 (a : A) (da : Ch A) (ada : valid a da) \u2192 SValid (inj\u2081 a) (convert\u2081 (ch\u2081 da))\n    sv\u2082 : \u2200 (b : B) (db : Ch B) (bdb : valid b db) \u2192 SValid (inj\u2082 b) (convert\u2081 (ch\u2082 db))\n    svrp\u2081 : \u2200 a1 b2 \u2192 SValid (inj\u2081 a1) (convert\u2081 (rp (inj\u2082 b2)))\n    svrp\u2082 : \u2200 b1 a2 \u2192 SValid (inj\u2082 b1) (convert\u2081 (rp (inj\u2081 a2)))\n\n  inv1 : \u2200 ds \u2192 convert\u2081 (convert ds) \u2261 ds\n  inv1 (inj\u2081 (inj\u2081 da)) = refl\n  inv1 (inj\u2081 (inj\u2082 db)) = refl\n  inv1 (inj\u2082 s) = refl\n  inv2 : \u2200 ds \u2192 convert (convert\u2081 ds) \u2261 ds\n  inv2 (ch\u2081 da) = refl\n  inv2 (ch\u2082 db) = refl\n  inv2 (rp s) = refl\n\n  private\n    s\u22952 : A \u228e B \u2192 SumChange2 \u2192 A \u228e B\n    s\u22952 (inj\u2081 a) (ch\u2081 da) = inj\u2081 (a \u2295 da)\n    s\u22952 (inj\u2082 b) (ch\u2082 db) = inj\u2082 (b \u2295 db)\n    s\u22952 (inj\u2082 b) (ch\u2081 da) = inj\u2082 b -- invalid\n    s\u22952 (inj\u2081 a) (ch\u2082 db) = inj\u2081 a -- invalid\n    s\u22952 s (rp s\u2081) = s\u2081\n\n    s\u2295 : A \u228e B \u2192 SumChange \u2192 A \u228e B\n    s\u2295 s ds = s\u22952 s (convert ds)\n\n    s\u229d2 : A \u228e B \u2192 A \u228e B \u2192 SumChange2\n    s\u229d2 (inj\u2081 x2) (inj\u2081 x1) = ch\u2081 (x2 \u229d x1)\n    s\u229d2 (inj\u2082 y2) (inj\u2082 y1) = ch\u2082 (y2 \u229d y1)\n    s\u229d2 s2 s1 = rp s2\n\n    s\u229d : A \u228e B \u2192 A \u228e B \u2192 SumChange\n    s\u229d s2 s1 = convert\u2081 (s\u229d2 s2 s1)\n\n    s\u229d-valid : (a b : A \u228e B) \u2192 SValid a (s\u229d b a)\n    s\u229d-valid (inj\u2081 x1) (inj\u2081 x2) = sv\u2081 x1 (x2 \u229d x1) (\u229d-valid x1 x2)\n    s\u229d-valid (inj\u2082 y1) (inj\u2082 y2) = sv\u2082 y1 (y2 \u229d y1) (\u229d-valid y1 y2)\n    s\u229d-valid s1@(inj\u2081 x) s2@(inj\u2082 y) = svrp\u2081 x y\n    s\u229d-valid s1@(inj\u2082 y) s2@(inj\u2081 x) = svrp\u2082 y x\n\n    s\u2295-\u229d : (b a : A \u228e B) \u2192 s\u2295 a (s\u229d b a) \u2261 b\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2081 x1) rewrite \u2295-\u229d x2 x1 = refl\n    s\u2295-\u229d (inj\u2081 x2) (inj\u2082 y1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2081 x1) = refl\n    s\u2295-\u229d (inj\u2082 y2) (inj\u2082 y1) rewrite \u2295-\u229d y2 y1 = refl\n    {-# TERMINATING #-}\n    isSumCA : IsChAlg (A \u228e B) SumChange\n    s\u229a-valid : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      SValid a (IsChAlg.default-\u229a isSumCA da1 a da2)\n    s\u229a-correct : (a : A \u228e B) (da1 : SumChange) \u2192\n      SValid a da1 \u2192\n      (da2 : SumChange) \u2192\n      SValid (s\u2295 a da1) da2 \u2192\n      s\u2295 a (IsChAlg.default-\u229a isSumCA da1 a da2) \u2261\n      s\u2295 (s\u2295 a da1) da2\n\n  sumCA : ChAlg (A \u228e B)\n  isSumCA = record\n    { _\u2295_ = s\u2295\n    ; _\u229d_ = s\u229d\n    ; valid = SValid\n    ; \u229d-valid = s\u229d-valid\n    ; \u2295-\u229d = s\u2295-\u229d\n    ; _\u229a[_]_ = IsChAlg.default-\u229a isSumCA\n    ; \u229a-valid = s\u229a-valid\n    ; \u229a-correct = s\u229a-correct\n    }\n\n  s\u229a-valid = IsChAlg.default-\u229a-valid isSumCA\n  s\u229a-correct = IsChAlg.default-\u229a-correct isSumCA\n\n  sumCA = record\n    { Ch = SumChange\n    ; isChAlg = isSumCA }\n\nopen import Data.Integer\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b}\n  ; \u229a-valid = \u03bb a da1 _ da2 _ \u2192 tt\n  ; \u229a-correct = \u03bb a da1 _ da2 _ \u2192 sym (associative-int a da1 da2)\n  } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4f80914ab5c624cdbcdfd54a84b137331638ccdb","subject":"agda\/...: avoid closing over parameters in theorem","message":"agda\/...: avoid closing over parameters in theorem\n\nIMHO local lemmas should still be closed terms, so that their\nsignature is adequately parametric.\n\nOld-commit-hash: dceb2333ba4708054afb02c05b19f9df97e9c76a\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore \u0393\u2032 t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore {\u0393} \u0393\u2032 t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import IlcModel\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 {.(\u0394-Context \u0393)} {.(\u0394-Type \u03c4)} \u0393\u2032 (\u0394 {\u0393} {\u03c4} t) = weakenMore t\n  where\n    open import Relation.Binary.PropositionalEquality using (sym)\n\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) (\u0394-Type \u03c4) \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) (\u0394-Type \u03c4)\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u2205 (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u2205 (\u0394-Type \u03c4\u2082) (doWeakenMore (\u03c4\u2082 \u2022 \u2205) \u0393rest t\u2081)\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) {\u03c4} t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\n    prefix : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    prefix \u0393 \u0393\u2032 = take (\u0394-Context \u0393) \u0393\u2032\n\n    rest   : \u2200 \u0393 \u2192 (\u0393\u2032 : Prefix (\u0394-Context \u0393)) \u2192 Context\n    rest   \u0393 \u0393\u2032 = drop (\u0394-Context \u0393) \u0393\u2032\n\n    weakenMore2 : \u2200 \u0393 \u0393\u2032 {\u03c4} \u2192\n      Term \u0393 \u03c4 \u2192\n      Term (prefix \u0393 \u0393\u2032 \u22ce \u0394-Context (rest \u0393 \u0393\u2032)) (\u0394-Type \u03c4)\n    weakenMore2 \u0393 \u0393\u2032 t =\n      doWeakenMore (prefix \u0393 \u0393\u2032) (rest \u0393 \u0393\u2032) (\n      substTerm (sym (take-drop (\u0394-Context \u0393) \u0393\u2032)) (\u0394 t))\n\n    weakenMore : --\u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n      Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032) (\u0394-Type \u03c4)\n    weakenMore t =\n      substTerm\n        (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 (\u0394-Context \u0393) \u0393\u2032))\n        (weakenMore2 \u0393 \u0393\u2032 t)\n\nlift-term\u2032 {._} {_} _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | bool = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\ndiff-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term = {!!}\n\napply-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term = {!!}\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = {!!}\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = {!!}\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6489171b7a2e26f15efa74f4e435b3d3c0977542","subject":"README: flat-funs-prod is broken","message":"README: flat-funs-prod is broken\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import flat-funs-implem\n-- open import flat-funs-prod\n\nopen import program-distance\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","old_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import flat-funs-implem\nopen import flat-funs-prod\n\nopen import program-distance\n\nopen import flipbased\nopen import flipbased-implem\nopen import flipbased-tree\n\nopen import bit-guessing-game\n\nopen import circuit\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b79b6386af309325661e541c718f713bd15e8617","subject":"proved typed expansion, modulo two lemmas about weakening Delta that I need to combine IHs into the right form. i believe these should be true, but actually proving them may require figuring out the barendrecht-style premises i need to add to rules about deltas being disjoint.","message":"proved typed expansion, modulo two lemmas about weakening Delta that I need\nto combine IHs into the right form. i believe these should be true, but\nactually proving them may require figuring out the barendrecht-style\npremises i need to add to rules about deltas being disjoint.\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!} -- this hole should be refl from pattern matching but it isn't\n  lem-idsub x d xin       | None   = abort (somenotnone (! xin))\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D x D\u2081) = TAAp (lem-weaken\u03941 D) x (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole x) = {!!}\n  lem-weaken\u03941 (TANEHole D x) = {!!}\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 = {!!}\n  -- lem-weaken\u03942 TAConst = TAConst\n  -- lem-weaken\u03942 (TAVar x\u2081) = TAVar x\u2081\n  -- lem-weaken\u03942 (TALam D) = TALam (lem-weaken\u03942 D)\n  -- lem-weaken\u03942 (TAAp D x D\u2081) = TAAp (lem-weaken\u03942 D) x (lem-weaken\u03942 D\u2081)\n  -- lem-weaken\u03942 (TAEHole x) = {!!}\n  -- lem-weaken\u03942 (TANEHole D x) = {!!}\n  -- lem-weaken\u03942 (TACast D x) = TACast (lem-weaken\u03942 D) x\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp1 {\u03941 = \u03941} x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp (TACast (lem-weaken\u03941 ih1) con1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth (ESAp2 {\u03941 = \u03941} ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (TACast (lem-weaken\u03942 {\u03941 = \u03941} ih2) con2)\n    typed-expansion-synth (ESAp3 {\u03941 = \u03941} ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp (lem-weaken\u03941 ih1) MAArr (lem-weaken\u03942 {\u03941 = \u03941} ih2)\n    typed-expansion-synth ESEHole = TAEHole lem-idsub\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = ih\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana EAEHole = TCRefl , TAEHole lem-idsub\n    typed-expansion-ana (EANEHole x) = TCRefl , TANEHole (typed-expansion-synth x) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0393 = \u0393} x d xin with \u0393 x\n  lem-idsub x .(X x) refl | Some \u03c4 = \u03c4 , refl , TAVar {!!} -- this hole should be refl from pattern matching but it isn't\n  lem-idsub x d xin       | None   = abort (somenotnone (! xin))\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    -- need a lemma for these two cases that says that if ## and d1 works,\n    -- then the union works. it'll use funext\n    typed-expansion-synth (ESAp1 x x\u2081 x\u2082 x\u2083)\n      with typed-expansion-ana x\u2082 | typed-expansion-ana x\u2083\n    ... | con1 , ih1 | con2 , ih2 = TAAp {!!} MAHole {!!}\n    typed-expansion-synth (ESAp2 x ex x\u2081 x\u2082)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp {!!} MAArr {!!}\n    typed-expansion-synth (ESAp3 x ex x\u2081)\n      with typed-expansion-synth ex | typed-expansion-ana x\u2081\n    ... | ih1 | con2 , ih2 = TAAp {!ih1!} MAArr {!!}\n    typed-expansion-synth ESEHole = TAEHole lem-idsub\n    typed-expansion-synth (ESNEHole ex) = TANEHole (typed-expansion-synth ex) lem-idsub\n    typed-expansion-synth (ESAsc1 x x\u2081)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n    typed-expansion-synth (ESAsc2 x)\n      with typed-expansion-ana x\n    ... | con , ih = {!!}\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4 ~ \u03c4') \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 ex)\n      with typed-expansion-ana ex\n    ... | con , D = (TCArr TCRefl con) , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana EAEHole = TCRefl , TAEHole lem-idsub\n    typed-expansion-ana (EANEHole x) = TCRefl , TANEHole (typed-expansion-synth x) lem-idsub\n    typed-expansion-ana (EALamHole x y)\n      with typed-expansion-ana y\n    ... | _ , ih = TCHole2 , TALam ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"324a7673109b75c3400506f3a0cbd4b09226bfff","subject":"Desc stratified model: I hate yellow. Almost done with the proof, but stuck on yellow.","message":"Desc stratified model: I hate yellow. Almost done with the proof, but stuck on yellow.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n\nIDescl0 : {l : Level}(I : Set l) -> Unit -> Set (suc l)\nIDescl0 I = IMu (\\_ -> descD I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set l} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\nproof-psi-phi : {l : Level}(I : Set l) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      lvar' : DescDConst {l = suc x}\n                      lvar' = lvar\n                      lpi' : DescDConst {l = suc x}\n                      lpi' = lpi\n                      lsigma' : DescDConst {l = suc x}\n                      lsigma' = lsigma\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = cong (\\t -> con (lvar' , t)) \n                                                                    (proof-lift-unlift-eq i)\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\t ->  con ((lpi' , ((S , \\ s -> t s))))) \n                                                                          (reflFun (\\s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = {!!} {- cong (\\t -> con (lsigma' , ( S , \\s -> t (unlift s) ))) \n                                                                              (reflFun (\\s -> psi (phi (T (lifter s)))) \n                                                                                       (\\s -> T (lifter s)) \n                                                                                       (\\s -> hs (lifter s))) -}","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"37237300729bb823df237c752fa35464d2bf58a6","subject":"IDesc stratified model: GOT IT! The proof type-checks with no fancy color: the embedded IDescl is isomorph to the host IDesc. Fully stratified, on any universe you want my Lord.","message":"IDesc stratified model: GOT IT! The proof type-checks with no fancy color: the embedded IDescl is isomorph to the host IDesc. Fully stratified, on any universe you want my Lord.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n\nIDescl0 : {l : Level}(I : Set l) -> Unit -> Set (suc l)\nIDescl0 I = IMu (\\_ -> descD I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set l} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\nproof-psi-phi : {l : Level}(I : Set l) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = cong (\\t -> con (lvar' , t)) \n                                                                    (proof-lift-unlift-eq i)\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                                           (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                           (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                           (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                                       (proof-lift-unlift-eq s)))\n                                                                                  (reflFun (\\s -> psi (phi (T s))) \n                                                                                           T \n                                                                                           hs)) \n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                              (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                              (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                              (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                                          (proof-lift-unlift-eq s)))\n                                                                                     (reflFun (\\s -> psi (phi (T s))) \n                                                                                              T \n                                                                                              hs))","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n\nIDescl0 : {l : Level}(I : Set l) -> Unit -> Set (suc l)\nIDescl0 I = IMu (\\_ -> descD I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set l} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\nproof-psi-phi : {l : Level}(I : Set l) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      lvar' : DescDConst {l = suc x}\n                      lvar' = lvar\n                      lpi' : DescDConst {l = suc x}\n                      lpi' = lpi\n                      lsigma' : DescDConst {l = suc x}\n                      lsigma' = lsigma\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = cong (\\t -> con (lvar' , t)) \n                                                                    (proof-lift-unlift-eq i)\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\t ->  con ((lpi' , ((S , \\ s -> t s))))) \n                                                                          (reflFun (\\s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = {!!} {- cong (\\t -> con (lsigma' , ( S , \\s -> t (unlift s) ))) \n                                                                              (reflFun (\\s -> psi (phi (T (lifter s)))) \n                                                                                       (\\s -> T (lifter s)) \n                                                                                       (\\s -> hs (lifter s))) -}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"473a3e3258868f0f98b57781ec716a38a7aca0fa","subject":"Fixed wrong url.","message":"Fixed wrong url.\n\nIgnore-this: 1efade942b0f368001984228e770b4fe\n\ndarcs-hash:20111225145504-3bd4e-972ef6455283a5402d7c404e4be1e74d685fec7c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fossacs-2012\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"46974ae6ecf312cad6975e1e06cc1c115eaefac1","subject":"cleaning up some scraps and easy holes from last night","message":"cleaning up some scraps and easy holes from last night\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"528d4f13dd6bffb83c13506bc7eae9439036b237","subject":"Alternate definition of relT","message":"Alternate definition of relT\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 with svv n \u2264-refl tv1 tv2 {! tvv !} v1 {! eqv1 !}\n... | v2 , eqv2 , final-v = v2 , {! eqv2 !} , {! final-v !}\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","old_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl)\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 with svv n \u2264-refl tv1 tv2 {! tvv !} v1 {! eqv1 !}\n... | v2 , eqv2 , final-v = v2 , {! eqv2 !} , {! final-v !}\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 n)\n  (tvv : relV \u03c3 tv1 tv2 n)\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n--... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- TODO: match sv2 before matching on fundamental s.\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n--   where\n--     v2 : Val \u03c4\n--     v2 = {!!}\n--     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n--     -- t\u03c12\u2193v2 = ?\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2e69b8b84c1ccab5550bbd4dce9125180003733f","subject":"Data.LR: +[L:_R:_]\u2032 +isL?-is-equiv","message":"Data.LR: +[L:_R:_]\u2032 +isL?-is-equiv\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/LR.agda","new_file":"lib\/Data\/LR.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.LR where\n\nopen import Type\nopen import Function.NP\nopen import HoTT\nopen import Relation.Binary.PropositionalEquality.NP using () renaming (refl to idp)\nopen import Data.Two\nopen Equivalences\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {\u2113}{C : LR \u2192 \u2605_ \u2113}(l : C `L)(r : C `R)(lr : LR) \u2192 C lr\n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n[L:_R:_]\u2032 : \u2200 {\u2113}{C : \u2605_ \u2113}(l : C)(r : C)(lr : LR) \u2192 C\n[L:_R:_]\u2032 = [L:_R:_]\n\nLR! : LR \u2192 LR\nLR! `L = `R\nLR! `R = `L\n\nLR!-equiv : LR \u2243 LR\nLR!-equiv = equiv LR! LR! [L: idp R: idp ] [L: idp R: idp ]\n\nisR? : LR \u2192 \ud835\udfda\nisR? `L = 0\u2082\nisR? `R = 1\u2082\n\nisL? : LR \u2192 \ud835\udfda\nisL? = not \u2218 isR?\n\nisR?-is-equiv : Is-equiv isR?\nisR?-is-equiv = is-equiv [0: `L 1: `R ] [0: idp 1: idp ] [L: idp R: idp ]\n\nisL?-is-equiv : Is-equiv isL?\nisL?-is-equiv = is-equiv [0: `R 1: `L ] [0: idp 1: idp ] [L: idp R: idp ]\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.LR where\n\nopen import Type\nopen import Function.NP\nopen import HoTT\nopen import Relation.Binary.PropositionalEquality.NP using () renaming (refl to idp)\nopen import Data.Two\nopen Equivalences\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {\u2113}{C : LR \u2192 \u2605_ \u2113}(l : C `L)(r : C `R)(lr : LR) \u2192 C lr\n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\nLR! : LR \u2192 LR\nLR! `L = `R\nLR! `R = `L\n\nLR!-equiv : LR \u2243 LR\nLR!-equiv = equiv LR! LR! [L: idp R: idp ] [L: idp R: idp ]\n\nisR? : LR \u2192 \ud835\udfda\nisR? `L = 0\u2082\nisR? `R = 1\u2082\n\nisL? : LR \u2192 \ud835\udfda\nisL? = not \u2218 isR?\n\nisR?-is-equiv : Is-equiv isR?\nisR?-is-equiv = is-equiv [0: `L 1: `R ] [0: idp 1: idp ] [L: idp R: idp ]\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cdbb5c65eb9c6c7a691550f7fbe6da379c62d53b","subject":"Show what would happen with fixpoints","message":"Show what would happen with fixpoints\n\nI looked on paper at what would happen with fixpoints for typed lambda calculus,\nand we'd immediately need to use well-founded recursion to prove the fundamental\nproperty. I conjecture that would go through, and Ahmed does use similar tricks.\nAt least, that might be less ugly than defining the relation via\nwell-founded induction, which still does not appear necessary.\n\nHere I mark the corresponding location in the formal proof.\n\nI haven't redone in full on paper the appplication case, but looking at\ndefinitions and at the formalized proof, it appears I would incur no further\nobligation: if we have related functions `(k, f1, df, f2) \\in RV[sigma -> tau]`,\napplying them now puts f1, df and f2 in the environment when evaluating their\nbody, but in the definition I sketched it does not require reproving that\n`(j, f1, df, f2) \\in RV[sigma -> tau]` for some `j < k`, while it requires for\nthe argument that `(j, v1, dv, v2) \\in RV[sigma]`.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/FundamentalProperty.agda","new_file":"Thesis\/SIRelBigStep\/FundamentalProperty.agda","new_contents":"module Thesis.SIRelBigStep.FundamentalProperty where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb j j<k v1 dv v2 vv \u2192\n    -- If we replace abs t by rec t, here we would need to invoke (by\n    -- well-founded recursion) rfundamental3svv on (abs t) at a smaller j, as follows:\n    --   rfundamental3svv j (abs t) \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n    -- As you'd expect, Agda's termination checker does not accept that call directly.\n      rfundamental3 j t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrelV3\u2192\u2295 vtv1 dtv vtv2 dtvv) =\n        bang v2 , bangapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"module Thesis.SIRelBigStep.FundamentalProperty where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\n\nrfundamentalV3v : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\nrfundamentalV3v x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamental3constV : \u2200 {\u03c4} k (c : Const \u03c4) \u2192\n  rrelV3 \u03c4 (eval-const c) (deval (derive-const c) \u2205 \u2205) (eval-const c) k\nrfundamental3constV k (lit n) = refl\n\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\nrfundamental3svv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelV3 \u03c4 (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\nrfundamental3svv k (var x) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3v x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (cons sv1 sv2) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamental3svv k sv1 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , rfundamental3svv k sv2 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3svv k (const c) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 rewrite deval-derive-const-inv c \u03c11 d\u03c1 = rfundamental3constV k c\nrfundamental3svv k (abs t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb j j<k v1 dv v2 vv \u2192\n      rfundamental3 j t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono j k (lt1 j<k) _ _ _ _ \u03c1\u03c1)\n\nrfundamental3sv : \u2200 {\u03c4 \u0393} k (sv : SVal \u0393 \u03c4) \u2192\n  \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192 rrelT3 (val sv) (dval (derive-dsval sv)) (val sv) \u03c11 d\u03c1 \u03c12 k\nrfundamental3sv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(eval sv \u03c11) .(eval sv \u03c12) .0 j<k (val .sv) (val .sv) = deval (derive-dsval sv) \u03c11 d\u03c1 , dval (derive-dsval sv) , rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nopen import Theorem.Groups-Nehemiah\nrfundamental3primv : \u2200 {\u03c3 \u03c4} k p \u2192\n  \u2200 v1 dv v2 \u2192 (vv : rrelV3 \u03c3 v1 dv v2 k) \u2192\n  rrelV3 \u03c4 (eval-primitive p v1) (deval-primitive p v1 dv) (eval-primitive p v2) k\nrfundamental3primv k succ (natV n\u2081) (bang .(natV n)) (natV n) refl = refl\nrfundamental3primv k succ (natV n) (dnatV dn) (natV .(dn + n)) refl = +-suc dn n\nrfundamental3primv k add (pairV (natV a1) (natV b1))\n  (dpairV (dnatV da) (dnatV db))\n  (pairV (natV .(da + a1)) (natV .(db + b1)))\n  (refl , refl) = \u2115-mn\u00b7pq=mp\u00b7nq {da} {db} {a1} {b1}\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (dnatV da) (bang b2))\n  (pairV a2 .b2) (aa , refl) rewrite rrelV3\u2192\u2295 a1 (dnatV da) a2 aa = refl\nrfundamental3primv k add (pairV a1 b1)\n  (dpairV (bang a2) db)\n  (pairV .a2 b2) (refl , bb) rewrite rrelV3\u2192\u2295 b1 db b2 bb = refl\nrfundamental3primv k add (pairV a1 b1) (bang p2) .p2 refl = refl\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 k (val sv) = rfundamental3sv k sv\nrfundamental3 (suc k) (primapp p sv) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n  .(eval-primitive p (eval sv \u03c11)) .(eval-primitive p (eval sv \u03c12)) .1 (s\u2264s j<k) (primapp .p .sv) (primapp .p .sv) =\n   deval-primitive p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) , dprimapp p sv (derive-dsval sv) ,\n     rfundamental3primv k p (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12)\n       (rfundamental3svv k sv \u03c11 d\u03c1 \u03c12 (rrel\u03c13-mono k (suc k) (\u2264-step \u2264-refl) _ \u03c11 d\u03c1 \u03c12 \u03c1\u03c1))\n-- (eval sv \u03c11) (deval (derive-dsval sv) \u03c11 d\u03c1) (eval sv \u03c12) k\n\nrfundamental3 (suc (suc k)) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s (s\u2264s j<k))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc k)) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc k)) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrelV3\u2192\u2295 vtv1 dtv vtv2 dtvv) =\n        bang v2 , bangapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) \u2264-refl vtv1 dtv vtv2\n           (rrelV3-mono (suc k) (suc (suc k)) (s\u2264s (\u2264-step \u2264-refl)) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\n\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5760ecf94ca99d0cf810abe16aba48c31d177ad8","subject":"Simplify code","message":"Simplify code\n","repos":"inc-lc\/ilc-agda","old_file":"Structure\/Bag\/Nehemiah.agda","new_file":"Structure\/Bag\/Nehemiah.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Bags of integers, for Nehemiah plugin.\n--\n-- This module imports postulates about bags of integers\n-- with negative multiplicities as a group under additive union.\n------------------------------------------------------------------------\n\nmodule Structure.Bag.Nehemiah where\n\nopen import Postulate.Bag-Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra using (CommutativeRing)\nopen import Algebra.Structures\nopen import Data.Integer\nopen import Data.Integer.Properties\n  using ()\n  renaming (commutativeRing to \u2124-is-commutativeRing)\nopen import Data.Product\n\ninfixl 9 _\\\\_ -- same as Data.Map.(\\\\)\n_\\\\_ : Bag \u2192 Bag \u2192 Bag\nd \\\\ b = d ++ (negateBag b)\n\n-- Useful properties of abelian groups\ncommutative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (m n : A) \u2192 f m n \u2261 f n m\ncommutative = IsAbelianGroup.comm\n\nassociative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (k m n : A) \u2192 f (f k m) n \u2261 f k (f m n)\nassociative abelian = IsAbelianGroup.assoc abelian\n\nleft-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f (neg n) n \u2261 z\nleft-inverse abelian = proj\u2081 (IsAbelianGroup.inverse abelian)\nright-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n (neg n) \u2261 z\nright-inverse abelian = proj\u2082 (IsAbelianGroup.inverse abelian)\n\nleft-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f z n \u2261 n\nleft-identity abelian = proj\u2081 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\nright-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n z \u2261 n\nright-identity abelian = proj\u2082 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\n\ninstance\n  abelian-int : IsAbelianGroup _\u2261_ _+_ (+ 0) (-_)\n  abelian-int =\n    CommutativeRing.+-isAbelianGroup \u2124-is-commutativeRing\n\ncommutative-int : (m n : \u2124) \u2192 m + n \u2261 n + m\ncommutative-int = commutative abelian-int\nassociative-int : (k m n : \u2124) \u2192 (k + m) + n \u2261 k + (m + n)\nassociative-int = associative abelian-int\nright-inv-int : (n : \u2124) \u2192 n - n \u2261 + 0\nright-inv-int = right-inverse abelian-int\nleft-id-int : (n : \u2124) \u2192 (+ 0) + n \u2261 n\nleft-id-int = left-identity abelian-int\nright-id-int : (n : \u2124) \u2192 n + (+ 0) \u2261 n\nright-id-int = right-identity abelian-int\n\ncommutative-bag : (a b : Bag) \u2192 a ++ b \u2261 b ++ a\ncommutative-bag = commutative abelian-bag\nassociative-bag : (a b c : Bag) \u2192 (a ++ b) ++ c \u2261 a ++ (b ++ c)\nassociative-bag = associative abelian-bag\nright-inv-bag : (b : Bag) \u2192 b \\\\ b \u2261 emptyBag\nright-inv-bag = right-inverse abelian-bag\nleft-id-bag : (b : Bag) \u2192 emptyBag ++ b \u2261 b\nleft-id-bag = left-identity abelian-bag\nright-id-bag : (b : Bag) \u2192 b ++ emptyBag \u2261 b\nright-id-bag = right-identity abelian-bag\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Bags of integers, for Nehemiah plugin.\n--\n-- This module imports postulates about bags of integers\n-- with negative multiplicities as a group under additive union.\n------------------------------------------------------------------------\n\nmodule Structure.Bag.Nehemiah where\n\nopen import Postulate.Bag-Nehemiah public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra using (CommutativeRing)\nopen import Algebra.Structures\nopen import Data.Integer\nopen import Data.Integer.Properties\n  using ()\n  renaming (commutativeRing to \u2124-is-commutativeRing)\nopen import Data.Product\n\ninfixl 9 _\\\\_ -- same as Data.Map.(\\\\)\n_\\\\_ : Bag \u2192 Bag \u2192 Bag\nd \\\\ b = d ++ (negateBag b)\n\n-- Useful properties of abelian groups\ncommutative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (m n : A) \u2192 f m n \u2261 f n m\ncommutative = IsAbelianGroup.comm\n\nassociative : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (k m n : A) \u2192 f (f k m) n \u2261 f k (f m n)\nassociative abelian = IsAbelianGroup.assoc abelian\n\nleft-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f (neg n) n \u2261 z\nleft-inverse abelian = proj\u2081 (IsAbelianGroup.inverse abelian)\nright-inverse : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n (neg n) \u2261 z\nright-inverse abelian = proj\u2082 (IsAbelianGroup.inverse abelian)\n\nleft-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f z n \u2261 n\nleft-identity abelian = proj\u2081 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\nright-identity : \u2200 {A : Set} {f : A \u2192 A \u2192 A} {z neg} \u2192\n  IsAbelianGroup _\u2261_ f z neg \u2192 (n : A) \u2192 f n z \u2261 n\nright-identity abelian = proj\u2082 (IsMonoid.identity\n  (IsGroup.isMonoid (IsAbelianGroup.isGroup abelian)))\n\ninstance\n  abelian-int : IsAbelianGroup _\u2261_ _+_ (+ 0) (-_)\n  abelian-int =\n    IsRing.+-isAbelianGroup\n    (IsCommutativeRing.isRing\n    (CommutativeRing.isCommutativeRing\n    \u2124-is-commutativeRing))\ncommutative-int : (m n : \u2124) \u2192 m + n \u2261 n + m\ncommutative-int = commutative abelian-int\nassociative-int : (k m n : \u2124) \u2192 (k + m) + n \u2261 k + (m + n)\nassociative-int = associative abelian-int\nright-inv-int : (n : \u2124) \u2192 n - n \u2261 + 0\nright-inv-int = right-inverse abelian-int\nleft-id-int : (n : \u2124) \u2192 (+ 0) + n \u2261 n\nleft-id-int = left-identity abelian-int\nright-id-int : (n : \u2124) \u2192 n + (+ 0) \u2261 n\nright-id-int = right-identity abelian-int\n\ncommutative-bag : (a b : Bag) \u2192 a ++ b \u2261 b ++ a\ncommutative-bag = commutative abelian-bag\nassociative-bag : (a b c : Bag) \u2192 (a ++ b) ++ c \u2261 a ++ (b ++ c)\nassociative-bag = associative abelian-bag\nright-inv-bag : (b : Bag) \u2192 b \\\\ b \u2261 emptyBag\nright-inv-bag = right-inverse abelian-bag\nleft-id-bag : (b : Bag) \u2192 emptyBag ++ b \u2261 b\nleft-id-bag = left-identity abelian-bag\nright-id-bag : (b : Bag) \u2192 b ++ emptyBag \u2261 b\nright-id-bag = right-identity abelian-bag\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"541e0cdd6b049f37f4dff7587a88559f69b1db17","subject":"Added doc.","message":"Added doc.\n\nIgnore-this: c1455481229df75e5be9f89856ffb5de\n\ndarcs-hash:20110301054603-3bd4e-4ac1de2906d09058197378b316fe83b04bd5c72c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/Induction\/Acc\/WellFoundedInductionATP.agda","new_file":"src\/FOTC\/Program\/McCarthy91\/MCR\/Induction\/Acc\/WellFoundedInductionATP.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n----------------------------------------------------------------------------\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\n\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.PropertiesATP\n\nimport FOTC.Data.Nat.Induction.Acc.WellFoundedInduction\nopen module WF-LT =\n  FOTC.Data.Nat.Induction.Acc.WellFoundedInduction.WF-LT\u2082\n  x<0\u2192\u22a5 x\u2264y\u2192x<y\u2228x\u2261y x<Sy\u2192x\u2264y\n\nopen module II =\n  FOTC.Data.Nat.Induction.Acc.WellFounded.InverseImage {LT} MCR-f MCR-f-N\n\n----------------------------------------------------------------------------\n\n-- The relation MCR is well-founded.\nwf-MCR : WellFounded MCR\nwf-MCR Nn = wellFounded wf-LT Nn\n\n-- Well-founded induction on the relation MCR.\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 MCR m n \u2192 P m) \u2192 P n) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P = WellFoundedInduction wf-MCR\n","old_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation MCR\n----------------------------------------------------------------------------\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.MCR.Induction.Acc.WellFoundedInductionATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFounded\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\n\nopen import FOTC.Program.McCarthy91.MCR\nopen import FOTC.Program.McCarthy91.MCR.PropertiesATP\n\nimport FOTC.Data.Nat.Induction.Acc.WellFoundedInduction\nopen module WF-LT =\n  FOTC.Data.Nat.Induction.Acc.WellFoundedInduction.WF-LT\u2082\n  x<0\u2192\u22a5 x\u2264y\u2192x<y\u2228x\u2261y x<Sy\u2192x\u2264y\n\nopen module II =\n  FOTC.Data.Nat.Induction.Acc.WellFounded.InverseImage {LT} MCR-f MCR-f-N\n\n----------------------------------------------------------------------------\n\nwf-MCR : WellFounded MCR\nwf-MCR Nn = wellFounded wf-LT Nn\n\nwfInd-MCR : (P : D \u2192 Set) \u2192\n            (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 MCR m n \u2192 P m) \u2192 P n) \u2192\n            \u2200 {n} \u2192 N n \u2192 P n\nwfInd-MCR P = WellFoundedInduction wf-MCR\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d69eb0df6d726f4eaeb86e149dd6d8b9716c3cce","subject":"FLABloM: error in fromNat?","message":"FLABloM: error in fromNat?\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/Shape.agda","new_file":"FLABloM\/Shape.agda","new_contents":"module Shape where\nopen import Data.Product\nopen import Data.Maybe\n\n-- shape of matrix\ndata Shape : Set where\n  -- 1\n  L : Shape\n  -- s\u2081 + s\u2082\n  B : (s\u2081 s\u2082 : Shape) \u2192 Shape\n\nopen import Data.Nat\n\ntoNat : Shape \u2192 \u2115\ntoNat L         = 1\ntoNat (B s s\u2081)  = toNat s + toNat s\u2081\n\n-- | Divide in two (almost) equal parts\nsplit : \u2115  ->  \u2115 \u00d7 \u2115\nsplit zero       = (zero , zero)\nsplit (suc zero) = (suc zero , zero)\nsplit (suc (suc n)) with split n\n... | (n1 , n2)  = (suc n1 , suc n2)\n\n\n{-# NO_TERMINATION_CHECK #-}\n-- Compute a balanced shape\nfromNat : \u2115 \u2192 Maybe Shape\nfromNat zero     = nothing\nfromNat (suc n)  with split n\n... | (n1 , n2) with fromNat n1 | fromNat n2\n...   | nothing  | nothing  = nothing\n...   | just s1  | nothing  = just s1\n...   | nothing  | just s2  = just s2\n...   | just s1  | just s2  = just (B s1 s2)\n\n-- TODO: perhaps add an empty shape (but probably in a separate\n-- experiment file because many things change in the matrix\n-- representation).\n\n-- TODO: `\u2200 n \u2192 fromNat n \u2261 nothing` seems like its true\n","old_contents":"module Shape where\nopen import Data.Product\nopen import Data.Maybe\n\n-- shape of matrix\ndata Shape : Set where\n  -- 1\n  L : Shape\n  -- s\u2081 + s\u2082\n  B : (s\u2081 s\u2082 : Shape) \u2192 Shape\n\nopen import Data.Nat\n\ntoNat : Shape \u2192 \u2115\ntoNat L         = 1\ntoNat (B s s\u2081)  = toNat s + toNat s\u2081\n\n-- | Divide in two (almost) equal parts\nsplit : \u2115  ->  \u2115 \u00d7 \u2115\nsplit zero       = (zero , zero)\nsplit (suc zero) = (suc zero , zero)\nsplit (suc (suc n)) with split n\n... | (n1 , n2)  = (suc n1 , suc n2)\n\n\n{-# NO_TERMINATION_CHECK #-}\n-- Compute a balanced shape\nfromNat : \u2115 \u2192 Maybe Shape\nfromNat zero     = nothing\nfromNat (suc n)  with split n\n... | (n1 , n2) with fromNat n1 | fromNat n2\n...   | nothing  | nothing  = nothing\n...   | just s1  | nothing  = just s1\n...   | nothing  | just s2  = just s2\n...   | just s1  | just s2  = just (B s1 s2)\n\n-- TODO: perhaps add an empty shape (but probably in a separate\n-- experiment file because many things change in the matrix\n-- representation).\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"28f4b329a68e6e4fbca342bbfa79199437a7d8f4","subject":"Another more specialized type example in the tactics file.","message":"Another more specialized type example in the tactics file.\n","repos":"spire\/spire","old_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin : Tactic\nrm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin a (fin i) = proj\u2082\n  (rm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n","old_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 b) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) b)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c8ddee2eedb4010e9db81bf9251ec542b3c28e5e","subject":"refactoring a bit #4","message":"refactoring a bit #4\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n\n  -- todo: these are bad names\n  lem2 : \u2200{d \u0394 d'} \u2192 d indet \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp () x\u2081) (ITLam x\u2082)\n\n  lem3 : \u2200{d \u0394 d'} \u2192 d val \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem3 VConst ()\n  lem3 VLam ()\n\n  lem1 : \u2200{d \u0394 d'} \u2192 d final \u2192 \u0394 \u22a2 d \u2192> d' \u2192 \u22a5\n  lem1 (FVal x) st = lem3 x st\n  lem1 (FIndet x) st = lem2 x st\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (_ , Step (FHFinal x) q _) = lem1 x q\n  is IEHole (_ , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (_ , Step (FHFinal x\u2081) q _) = lem1 x\u2081 q\n  is (IAp i x) (_ , Step (FHAp1 x\u2081 p) q (FHAp1 x\u2082 r)) = {!!}\n  is (IAp i x) (_ , Step (FHAp2 p) q (FHAp2 r)) = {!!}\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n--  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n\n  --es (ENEHole e) x = {!e!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step FHNEHoleEvaled x\u2082 x\u2083) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleInside x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  -- es (ENEHole e) (d' , Step (FHFinal (FVal ())) x\u2081 x\u2082)\n  -- es (ENEHole e) (d' , Step (FHFinal (FIndet (INEHole x))) () x\u2082)\n  -- es (ENEHole e) (d' , Step FHNEHoleEvaled () x\u2082)\n  -- es (ENEHole e) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FVal ())))\n  -- es (ENEHole (ECastError x x\u2081)) (_ , Step (FHNEHoleFinal x\u2082) (ITNEHole x\u2083) (FHFinal (FIndet (INEHole x\u2084)))) = {!x\u2081!}\n  -- es (ENEHole (EAp1 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (EAp2 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ENEHole e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ECastProp e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) FHNEHoleEvaled) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (d0' , Step (FHFinal x) () (FHFinal x\u2081))\n  is IEHole (d0' , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step (FHFinal x) () FHEHole)\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step (FHFinal x) () (FHCastFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (d , Step (FHFinal x\u2081) () (FHFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHEHole)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2082))\n  is (INEHole x) (d , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (d' , Step p q r) = {!!}\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n--  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2082!}\n\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n\n  --es (ENEHole e) x = {!e!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step FHNEHoleEvaled x\u2082 x\u2083) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleInside x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (ECastError x x\u2081)) (d' , Step (FHNEHoleFinal x\u2082) x\u2083 x\u2084) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp1 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (EAp2 e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ENEHole e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHFinal x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step FHNEHoleEvaled x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleInside x) x\u2081 x\u2082) = {!!}\n  es (ENEHole (ECastProp e)) (d' , Step (FHNEHoleFinal x) x\u2081 x\u2082) = {!!}\n  -- es (ENEHole e) (d' , Step (FHFinal (FVal ())) x\u2081 x\u2082)\n  -- es (ENEHole e) (d' , Step (FHFinal (FIndet (INEHole x))) () x\u2082)\n  -- es (ENEHole e) (d' , Step FHNEHoleEvaled () x\u2082)\n  -- es (ENEHole e) (_ , Step (FHNEHoleInside x) x\u2081 (FHNEHoleInside x\u2082)) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FVal ())))\n  -- es (ENEHole (ECastError x x\u2081)) (_ , Step (FHNEHoleFinal x\u2082) (ITNEHole x\u2083) (FHFinal (FIndet (INEHole x\u2084)))) = {!x\u2081!}\n  -- es (ENEHole (EAp1 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (EAp2 e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ENEHole e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole (ECastProp e)) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) (FHFinal (FIndet (INEHole x\u2082)))) = {!!}\n  -- es (ENEHole e) (_ , Step (FHNEHoleFinal x) (ITNEHole x\u2081) FHNEHoleEvaled) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bc5c7c12f19c3ee807b23dc5f6207a5405125dc0","subject":"Prove congruence for diff-term and apply-term.","message":"Prove congruence for diff-term and apply-term.\n\nOld-commit-hash: 1c6b015a57f1d5bf9fb8e40745a8350e352fc478\n","repos":"inc-lc\/ilc-agda","old_file":"Syntactic\/Changes.agda","new_file":"Syntactic\/Changes.agda","new_contents":"module Syntactic.Changes where\n\n-- TERMS that operate on CHANGES\n--\n-- This module defines terms that correspond to the basic change\n-- operations, as well as some other helper terms.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Changes\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\n-- Other problem: in fact, \u0394 t is not the nil change of t, in this context. That's a problem.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs (abs (var this xor-term var (that this)))\n\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} =\n-- \u03bbdf. \u03bbf.  \u03bbx.            apply          (          df                       x          (x \u2296 x))                             (     f                x        )\n   abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) (diff-term (var this) (var this)))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb df f \u2192 app (app apply-pure-term df) f\napply-term {bool} = _xor-term_\n\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} =\n  \u03bb f\u2081 f\u2082 \u2192\n  -- The following can be written as:\n  -- * Using Unicode symbols for change operations:\n  -- \u03bb x dx. (f1 (dx \u2295 x)) \u229d (f2 x)\n  -- * In lambda calculus with names:\n  --\u03bbx.  \u03bbdx. diff           (     f\u2081                   (apply      dx         x))                 (f\u2082                        x)\n  -- * As a deBrujin-encoded term in Agda:\n    abs (abs (diff-term {\u03c4\u2081} (app (weaken \u227c-in-body f\u2081) (apply-term (var this) (var (that this)))) (app (weaken \u227c-in-body f\u2082) (var (that this)))))\n  where\n    \u227c-in-body = drop (\u0394-Type \u03c4) \u2022 (drop \u03c4 \u2022 \u227c-refl)\n\ndiff-term {bool} = _xor-term_\n\n-- Derived CONGRUENCE RULES\nmodule _ where\n  open import Denotational.Equivalence\n\n  _\u2248-and_ : \u2200 {\u0393} {t\u2081 t\u2082 t\u2083 t\u2084 : Term \u0393 bool} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 (t\u2081 and t\u2083) \u2248 (t\u2082 and t\u2084)\n  _\u2248-and_ \u2248\u2081 \u2248\u2082 = \u2248-if \u2248\u2081 \u2248\u2082 \u2248-false\n\n  \u2248-!_ : \u2200 {\u0393} {t\u2081 t\u2082 : Term \u0393 bool} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 ! t\u2081 \u2248 ! t\u2082\n  \u2248-!_ \u2248\u2081 = \u2248-if \u2248\u2081 \u2248-false \u2248-true\n\n  _\u2248-xor-term_ : \u2200 {\u0393} {t\u2081 t\u2082 t\u2083 t\u2084 : Term \u0393 bool} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 (t\u2081 xor-term t\u2083) \u2248 (t\u2082 xor-term t\u2084)\n  _\u2248-xor-term_ \u2248\u2081 \u2248\u2082 = \u2248-if \u2248\u2081 (\u2248-! \u2248\u2082) \u2248\u2082\n\n  \u2248-diff-term : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 t\u2083 t\u2084 : Term \u0393 \u03c4} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 diff-term t\u2081 t\u2083 \u2248 diff-term t\u2082 t\u2084\n  \u2248-diff-term {\u03c4\u2081 \u21d2 \u03c4\u2082} \u2248\u2081 \u2248\u2082 =\n    \u2248-abs (\u2248-abs\n      (\u2248-diff-term\n        (\u2248-app (\u2248-weaken \u0393\u2033 \u2248\u2081) \u2248-refl)\n        (\u2248-app (\u2248-weaken \u0393\u2033 \u2248\u2082) \u2248-refl)))\n    where\n      \u0393\u2033 = drop (\u0394-Type \u03c4\u2081) \u2022 (drop \u03c4\u2081 \u2022 \u227c-refl)\n  \u2248-diff-term {bool} \u2248\u2081 \u2248\u2082 = \u2248\u2081 \u2248-xor-term \u2248\u2082\n\n  \u2248-apply-term : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 (\u0394-Type \u03c4)} {t\u2083 t\u2084 : Term \u0393 \u03c4} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 apply-term t\u2081 t\u2083 \u2248 apply-term t\u2082 t\u2084\n  \u2248-apply-term {\u03c4\u2081 \u21d2 \u03c4\u2082} \u2248\u2081 \u2248\u2082 = \u2248-app (\u2248-app \u2248-refl \u2248\u2081) \u2248\u2082\n  \u2248-apply-term {bool} \u2248\u2081 \u2248\u2082 = \u2248\u2081 \u2248-xor-term \u2248\u2082\n","old_contents":"module Syntactic.Changes where\n\n-- TERMS that operate on CHANGES\n--\n-- This module defines terms that correspond to the basic change\n-- operations, as well as some other helper terms.\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Changes\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\n-- Other problem: in fact, \u0394 t is not the nil change of t, in this context. That's a problem.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs (abs (var this xor-term var (that this)))\n\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} =\n-- \u03bbdf. \u03bbf.  \u03bbx.            apply          (          df                       x          (x \u2296 x))                             (     f                x        )\n   abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) (diff-term (var this) (var this)))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb df f \u2192 app (app apply-pure-term df) f\napply-term {bool} = _xor-term_\n\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} =\n  \u03bb f\u2081 f\u2082 \u2192\n  -- The following can be written as:\n  -- * Using Unicode symbols for change operations:\n  -- \u03bb x dx. (f1 (dx \u2295 x)) \u229d (f2 x)\n  -- * In lambda calculus with names:\n  --\u03bbx.  \u03bbdx. diff           (     f\u2081                   (apply      dx         x))                 (f\u2082                        x)\n  -- * As a deBrujin-encoded term in Agda:\n    abs (abs (diff-term {\u03c4\u2081} (app (weaken \u227c-in-body f\u2081) (apply-term (var this) (var (that this)))) (app (weaken \u227c-in-body f\u2082) (var (that this)))))\n  where\n    \u227c-in-body = drop (\u0394-Type \u03c4) \u2022 (drop \u03c4 \u2022 \u227c-refl)\n\ndiff-term {bool} = _xor-term_\n\n-- Derived CONGRUENCE RULES\nmodule _ where\n  open import Denotational.Equivalence\n\n  _\u2248-and_ : \u2200 {\u0393} {t\u2081 t\u2082 t\u2083 t\u2084 : Term \u0393 bool} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 (t\u2081 and t\u2083) \u2248 (t\u2082 and t\u2084)\n  _\u2248-and_ \u2248\u2081 \u2248\u2082 = \u2248-if \u2248\u2081 \u2248\u2082 \u2248-false\n\n  \u2248-!_ : \u2200 {\u0393} {t\u2081 t\u2082 : Term \u0393 bool} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 ! t\u2081 \u2248 ! t\u2082\n  \u2248-!_ \u2248\u2081 = \u2248-if \u2248\u2081 \u2248-false \u2248-true\n\n  _\u2248-xor-term_ : \u2200 {\u0393} {t\u2081 t\u2082 t\u2083 t\u2084 : Term \u0393 bool} \u2192\n    t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 (t\u2081 xor-term t\u2083) \u2248 (t\u2082 xor-term t\u2084)\n  _\u2248-xor-term_ \u2248\u2081 \u2248\u2082 = \u2248-if \u2248\u2081 (\u2248-! \u2248\u2082) \u2248\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"870795b84662dcd477173ba74a5e5788e75211b7","subject":"ZK\/JSChecker: avoid what seems to be a bug in JS extraction (open public)","message":"ZK\/JSChecker: avoid what seems to be a bug in JS extraction (open public)\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker.agda","new_file":"ZK\/JSChecker.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\n  hiding (check; trace)\n  renaming (_*_ to _*Number_)\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FiniteField.JS as \ud835\udd3d\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\ntrace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\ntrace _ inp f = f inp\n\nbignum : Number \u2192 BigI\nbignum n = bigI (Number\u25b9String n) \"10\"\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\nrecord ZK-chaum-pedersen-pok-elgamal-rnd (\u2124q \u2124p\u2605 : Set) : Set where\n  field\n    m c s : \u2124q\n    g p q y \u03b1 \u03b2 A B : \u2124p\u2605\n\n-- TODO dynamise me\nt : Number\nt = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\ncheck : (title  : String)\n        (pred   : Bool)\n        (errmsg : \ud835\udfd9 \u2192 String)\n     \u2192 JS!\ncheck title true  errmsg = Console.log (title ++ \": PASS\")\ncheck title false errmsg = Console.log (title ++ \": FAIL [\" ++ errmsg _ ++ \"]\")\n\ncheck-size : Number \u2192 String \u2192 BigI \u2192 JS!\ncheck-size min-bits name value =\n  check (\"check size of \" ++ name)\n        (len \u2265Number min-bits)\n        (\u03bb _ \u2192 name ++ \" is not a necessarily a safe prime: \"\n             ++ BigI.toString value    ++ \" has \"\n             ++ Number\u25b9String len      ++ \" bits which is less than \"\n             ++ Number\u25b9String min-bits ++ \" bits\")\n  module Check-size where\n    len = BigI.byteLength value *Number 8N\n\ncheck-pq-relation : (p q : BigI) \u2192 JS!\ncheck-pq-relation p q =\n  check (\"check p and q relation p-1\/q=\" ++ BigI.toString s)\n        (equals r 0I)\n        (\u03bb _ \u2192 \"Not necessarily a safe group: (p-1) mod q != 0\\np=\"\n             ++ BigI.toString p\n             ++ \", q=\" ++ BigI.toString q)\n  module Check-pq-relation where\n    open BigI\n    p-1 = subtract p 1I\n    r   = mod    p-1 q\n    s   = divide p-1 q\n\ncheck-primality : String \u2192 BigI \u2192 JS!\ncheck-primality name value =\n  check (\"check primality of \" ++ name)\n        (BigI.isProbablePrime value t)\n        (\u03bb _ \u2192 \"Not a prime number: \" ++ BigI.toString value)\n\ncheck-generator-group-order : (g q p : BigI) \u2192 JS!\ncheck-generator-group-order g q p =\n  check \"check generator & group order\"\n        (BigI.equals (BigI.modPow g q p) BigI.1I)\n        (\u03bb _ \u2192 \"Not a generator of a group of order q: modPow \"\n             ++ BigI.toString g ++ \" \" ++ BigI.toString q ++ \" \"\n             ++ BigI.toString p)\n\nmodule [\u2124q]\u2124p\u2605 (qI pI gI : BigI) where\n\n  checks : JS!\n  checks =\n    check-pq-relation      pI qI >>\n    check-size min-bits-q \"q\" qI >>\n    check-size min-bits-p \"p\" pI >>\n    check-primality       \"q\" qI >>\n    check-primality       \"p\" pI >>\n    check-generator-group-order gI qI pI\n\n  module \u2124q = \ud835\udd3d qI\n    using (0#; 1#; _+_; _\u2212_; _*_; _\/_)\n    renaming (\ud835\udd3d to \u2124q; fromBigI to BigI\u25b9\u2124q; repr to \u2124q-repr)\n  module \u2124p\u2605 = \ud835\udd3d pI\n    using (_==_)\n    renaming ( fromBigI to BigI\u25b9\u2124p\u2605; \ud835\udd3d to \u2124p\u2605; _*_ to _\u00b7_\n             ; repr to \u2124p\u2605-repr; _\/_ to _\u00b7\/_)\n\n  open \u2124q  -- public -- <- BUG\n  open \u2124p\u2605 public\n\n  g : \u2124p\u2605\n  g = BigI\u25b9\u2124p\u2605 gI\n\n  _^_ : \u2124p\u2605 \u2192 \u2124q \u2192 \u2124p\u2605\n  b ^ e = BigI\u25b9\u2124p\u2605 (BigI.modPow (\u2124p\u2605-repr b) (\u2124q-repr e) pI)\n\nzk-check-chaum-pedersen-pok-elgamal-rnd : ZK-chaum-pedersen-pok-elgamal-rnd BigI BigI \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks\n      >> check \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module Zk-check-chaume-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    open [\u2124q]\u2124p\u2605 I.q I.p I.g\n    open \u2124q\n--  open \u2124p\u2605 -- <- BUG\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check : JSValue \u2192 JS!\nzk-check arg =\n    check \"type of statement\" (typ === fromString \"chaum-pedersen-pok-elgamal-rnd\")\n                              (\u03bb _ \u2192 \"\")\n >> zk-check-chaum-pedersen-pok-elgamal-rnd pok\n module Zk-check where\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check (JSON-parse (castString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Function         using (id; _\u2218\u2032_; case_of_)\nopen import Data.Bool.Base   using (Bool; true; false; _\u2227_)\nopen import Data.List.Base   using (List; []; _\u2237_; and; foldr)\nopen import Data.String.Base using (String)\n\nopen import FFI.JS\n  hiding (check; trace)\n  renaming (_*_ to _*Number_)\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FiniteField.JS as \ud835\udd3d\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\ntrace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\ntrace _ inp f = f inp\n\nbignum : Number \u2192 BigI\nbignum n = bigI (Number\u25b9String n) \"10\"\n\n-- TODO check with undefined\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\nrecord ZK-chaum-pedersen-pok-elgamal-rnd (\u2124q \u2124p\u2605 : Set) : Set where\n  field\n    m c s : \u2124q\n    g p q y \u03b1 \u03b2 A B : \u2124p\u2605\n\n-- TODO dynamise me\nt : Number\nt = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\ncheck : (title  : String)\n        (pred   : Bool)\n        (errmsg : \ud835\udfd9 \u2192 String)\n     \u2192 JS!\ncheck title true  errmsg = Console.log (title ++ \": PASS\")\ncheck title false errmsg = Console.log (title ++ \": FAIL [\" ++ errmsg _ ++ \"]\")\n\ncheck-size : Number \u2192 String \u2192 BigI \u2192 JS!\ncheck-size min-bits name value =\n  check (\"check size of \" ++ name)\n        (len \u2265Number min-bits)\n        (\u03bb _ \u2192 name ++ \" is not a necessarily a safe prime: \"\n             ++ BigI.toString value    ++ \" has \"\n             ++ Number\u25b9String len      ++ \" bits which is less than \"\n             ++ Number\u25b9String min-bits ++ \" bits\")\n  module Check-size where\n    len = BigI.byteLength value *Number 8N\n\ncheck-pq-relation : (p q : BigI) \u2192 JS!\ncheck-pq-relation p q =\n  check (\"check p and q relation p-1\/q=\" ++ BigI.toString s)\n        (equals r 0I)\n        (\u03bb _ \u2192 \"Not necessarily a safe group: (p-1) mod q != 0\\np=\"\n             ++ BigI.toString p\n             ++ \", q=\" ++ BigI.toString q)\n  module Check-pq-relation where\n    open BigI\n    p-1 = subtract p 1I\n    r   = mod    p-1 q\n    s   = divide p-1 q\n\ncheck-primality : String \u2192 BigI \u2192 JS!\ncheck-primality name value =\n  check (\"check primality of \" ++ name)\n        (BigI.isProbablePrime value t)\n        (\u03bb _ \u2192 \"Not a prime number: \" ++ BigI.toString value)\n\ncheck-generator-group-order : (g q p : BigI) \u2192 JS!\ncheck-generator-group-order g q p =\n  check \"check generator & group order\"\n        (BigI.equals (BigI.modPow g q p) BigI.1I)\n        (\u03bb _ \u2192 \"Not a generator of a group of order q: modPow \"\n             ++ BigI.toString g ++ \" \" ++ BigI.toString q ++ \" \"\n             ++ BigI.toString p)\n\nmodule [\u2124q]\u2124p\u2605 (qI pI gI : BigI) where\n\n  checks : JS!\n  checks =\n    check-pq-relation      pI qI >>\n    check-size min-bits-q \"q\" qI >>\n    check-size min-bits-p \"p\" pI >>\n    check-primality       \"q\" qI >>\n    check-primality       \"p\" pI >>\n    check-generator-group-order gI qI pI\n\n  open \ud835\udd3d qI\n    public\n    using (0#; 1#; _+_; _\u2212_; _*_; _\/_)\n    renaming (\ud835\udd3d to \u2124q; fromBigI to BigI\u25b9\u2124q; repr to \u2124q-repr)\n\n  open \ud835\udd3d pI\n    public\n    using (_==_)\n    renaming ( fromBigI to BigI\u25b9\u2124p\u2605; \ud835\udd3d to \u2124p\u2605; _*_ to _\u00b7_\n             ; repr to \u2124p\u2605-repr; _\/_ to _\u00b7\/_)\n\n  g : \u2124p\u2605\n  g = BigI\u25b9\u2124p\u2605 gI\n\n  _^_ : \u2124p\u2605 \u2192 \u2124q \u2192 \u2124p\u2605\n  b ^ e = BigI\u25b9\u2124p\u2605 (BigI.modPow (\u2124p\u2605-repr b) (\u2124q-repr e) pI)\n\nzk-check-chaum-pedersen-pok-elgamal-rnd : ZK-chaum-pedersen-pok-elgamal-rnd BigI BigI \u2192 JS!\nzk-check-chaum-pedersen-pok-elgamal-rnd pf\n      = trace \"g=\" g \u03bb _ \u2192\n        trace \"p=\" I.p \u03bb _ \u2192\n        trace \"q=\" I.q \u03bb _ \u2192\n        trace \"y=\" y \u03bb _ \u2192\n        trace \"\u03b1=\" \u03b1 \u03bb _ \u2192\n        trace \"\u03b2=\" \u03b2 \u03bb _ \u2192\n        trace \"m=\" m \u03bb _ \u2192\n        trace \"M=\" M \u03bb _ \u2192\n        trace \"A=\" A \u03bb _ \u2192\n        trace \"B=\" B \u03bb _ \u2192\n        trace \"c=\" c \u03bb _ \u2192\n        trace \"s=\" s \u03bb _ \u2192\n         checks\n      >> check \"g^s==A\u00b7\u03b1^c\"     ((g ^ s) == (A \u00b7 (\u03b1 ^ c)))        (\u03bb _ \u2192 \"\")\n      >> check \"y^s==B\u00b7(\u03b2\/M)^c\" ((y ^ s) == (B \u00b7 ((\u03b2 \u00b7\/ M) ^ c))) (\u03bb _ \u2192 \"\")\n  module Zk-check-chaume-pedersen-pok-elgamal-rnd where\n    module I = ZK-chaum-pedersen-pok-elgamal-rnd pf\n    open [\u2124q]\u2124p\u2605 I.q I.p I.g\n    A = BigI\u25b9\u2124p\u2605 I.A\n    B = BigI\u25b9\u2124p\u2605 I.B\n    \u03b1 = BigI\u25b9\u2124p\u2605 I.\u03b1\n    \u03b2 = BigI\u25b9\u2124p\u2605 I.\u03b2\n    y = BigI\u25b9\u2124p\u2605 I.y\n    s = BigI\u25b9\u2124q  I.s\n    c = BigI\u25b9\u2124q  I.c\n    m = BigI\u25b9\u2124q  I.m\n    M = g ^ m\n\nzk-check : JSValue \u2192 JS!\nzk-check arg =\n    check \"type of statement\" (typ === fromString \"chaum-pedersen-pok-elgamal-rnd\")\n                              (\u03bb _ \u2192 \"\")\n >> zk-check-chaum-pedersen-pok-elgamal-rnd pok\n module Zk-check where\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    g   = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    p   = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    q   = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    y   = bigdec (dat \u00b7\u00ab \"y\" \u00bb)\n    m   = bigdec (dat \u00b7\u00ab \"plain\" \u00bb)\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    \u03b1   = bigdec (enc \u00b7\u00ab \"alpha\" \u00bb)\n    \u03b2   = bigdec (enc \u00b7\u00ab \"beta\"  \u00bb)\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    A   = bigdec (com \u00b7\u00ab \"A\" \u00bb)\n    B   = bigdec (com \u00b7\u00ab \"B\" \u00bb)\n    c   = bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)\n    s   = bigdec (prf \u00b7\u00ab \"response\" \u00bb)\n    pok = record { g = g; p = p; q = q; y = y; \u03b1 = \u03b1; \u03b2 = \u03b2; A = A; B = B; c = c; s = s; m = m }\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv !\u2081 \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv !\u2081 \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              FS.readFile arg nullJS !\u2082 \u03bb err dat \u2192\n                check \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check (JSON-parse (castString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2d357d2da573fa557939a00d1973548ed2ad132c","subject":"Added Even\u2192N.","message":"Added Even\u2192N.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Even.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Even.agda","new_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n\ndata Even : D \u2192 Set where\n  ezero :                  Even zero\n  enext : \u2200 {n} \u2192 Even n \u2192 Even (succ\u2081 (succ\u2081 n))\n\nEven-ind : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 (succ\u2081 n))) \u2192\n           \u2200 {n} \u2192 Even n \u2192 A n\nEven-ind A A0 h ezero      = A0\nEven-ind A A0 h (enext En) = h (Even-ind A A0 h En)\n\nEven\u2192N : \u2200 {n} \u2192 Even n \u2192 N n\nEven\u2192N ezero      = nzero\nEven\u2192N (enext En) = nsucc (nsucc (Even\u2192N En))\n","old_contents":"------------------------------------------------------------------------------\n-- Even predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Even where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata Even : D \u2192 Set where\n  ezero :                  Even zero\n  enext : \u2200 {n} \u2192 Even n \u2192 Even (succ\u2081 (succ\u2081 n))\n\nEven-ind : (A : D \u2192 Set) \u2192\n           A zero \u2192\n           (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 (succ\u2081 n))) \u2192\n           \u2200 {n} \u2192 Even n \u2192 A n\nEven-ind A A0 h ezero      = A0\nEven-ind A A0 h (enext En) = h (Even-ind A A0 h En)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d7232a93452dede4f8fe5c82202ff26225bda56e","subject":"Homogenous again","message":"Homogenous again\n","repos":"crypto-agda\/crypto-agda","old_file":"program-distance.agda","new_file":"program-distance.agda","new_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nmodule HomImplem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {n} (f g : \u2141 n) \u2192 Set\n  _]-[_ {n} f g = dist (count\u21ba f) (count\u21ba g) \u2265 2^(n \u2238 k)\n\n  ]-[-sym : \u2200 {n} {f g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {n} {f} {g} f]-[g rewrite dist-sym (count\u21ba f) (count\u21ba g) = f]-[g\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {n} {f g h : \u2141 n} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {n} {f} {g} f\u2248g g]-[h rewrite \u2248\u2141\u21d2\u2248\u2141\u2032 {n} {f} {g} f\u2248g = g]-[h\n  -- dist #g #h \u2265 2^(n \u2238 k)\n  -- dist #f #h \u2265 2^(n \u2238 k)\n\nmodule Implem k where\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) \u2265 2^((m + n) \u2238 k)\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 x y z t \u2192 x + ((y + z) \u2238 t) \u2261 y + ((x + z) \u2238 t)\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n  ... | q rewrite sym (dist-2^* m (\u27e82^ o \u27e9* (count\u21ba g)) (\u27e82^ n \u27e9* (count\u21ba h)))\n                | 2^-comm m o (count\u21ba g)\n                | sym f\u2248g\n                | 2^-comm o n (count\u21ba f)\n                | 2^-comm m n (count\u21ba h)\n                | dist-2^* n (\u27e82^ o \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba h))\n                | 2^-+ m (n + o \u2238 k) 1\n                | helper m n o k\n                | sym (2^-+ n (m + o \u2238 k) 1)\n                = 2^*-mono\u2032 n q\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_ (\u03bb {m n f g} x \u2192 ]-[-sym {m} {n} {f} {g} x) (\u03bb {m n o f g h} x y \u2192 ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} x y)\n","old_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count; count\u1da0)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n    ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n    ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n\n  ]-[-cong-right-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f ]-[ g \u2192 g \u2248\u2141 h \u2192 f ]-[ h\n  ]-[-cong-right-\u2248\u21ba pf pf' = ]-[-sym (]-[-cong-left-\u2248\u21ba (sym pf') (]-[-sym pf))\n\n  ]-[-cong-\u2257\u21ba : \u2200 {c c'} {f g : \u2141 c} {f' g' : \u2141 c'} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong-\u2257\u21ba {c} {c'} {f} {g} {f'} {g'} f\u2257g f'\u2257g' pf\n     = ]-[-cong-left-\u2248\u21ba {f = g} {g = f} {h = g'}\n         ((\u2257\u21d2\u2248\u2141 (\u03bb x \u2192 sym (f\u2257g x)))) (]-[-cong-right-\u2248\u21ba pf (\u2257\u21d2\u2248\u2141 f'\u2257g'))\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u2141 c) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u2141 c\u2080) (\u2141\u2081 : Bit \u2192 \u2141 c\u2081) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u2141 c}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong-\u2257\u21ba (same-games 0b) (same-games 1b)\n\nmodule Implem k where\n  --  | Pr[ f \u2261 1 ] - Pr[ g \u2261 1 ] | \u2265 \u03b5            [ on reals ]\n  --  dist Pr[ f \u2261 1 ] Pr[ g \u2261 1 ] \u2265 \u03b5             [ on reals ]\n  --  dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) \u2265 \u03b5          [ on reals ]\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c where \u03b5 = 2^ -k [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(-k) * 2^ c            [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c - k)                [ on rationals ]\n  --  dist (#1 f) (#1 g) \u2265 2^(c \u2238 k)               [ on natural ]\n  _]-[_ : \u2200 {m n} \u2192 \u2141 m \u2192 \u2141 n \u2192 Set\n  _]-[_ {m} {n} f g = dist (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) \u2265 2^((m + n) \u2238 k)\n\n  ]-[-sym : \u2200 {m n} {f : \u2141 m} {g : \u2141 n} \u2192 f ]-[ g \u2192 g ]-[ f\n  ]-[-sym {m} {n} {f} {g} f]-[g rewrite dist-sym (\u27e82^ n \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba g)) | \u2115\u00b0.+-comm m n = f]-[g\n\n  postulate\n      helper : \u2200 x y z t \u2192 x + ((y + z) \u2238 t) \u2261 y + ((x + z) \u2238 t)\n\n  ]-[-cong-left-\u2248\u21ba : \u2200 {m n o} {f : \u2141 m} {g : \u2141 n} {h : \u2141 o} \u2192 f \u2248\u2141 g \u2192 g ]-[ h \u2192 f ]-[ h\n  ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} f\u2248g g]-[h\n      with 2^*-mono m g]-[h\n  ... | q rewrite sym (dist-2^* m (\u27e82^ o \u27e9* (count\u21ba g)) (\u27e82^ n \u27e9* (count\u21ba h)))\n                | 2^-comm m o (count\u21ba g)\n                | sym f\u2248g\n                | 2^-comm o n (count\u21ba f)\n                | 2^-comm m n (count\u21ba h)\n                | dist-2^* n (\u27e82^ o \u27e9* (count\u21ba f)) (\u27e82^ m \u27e9* (count\u21ba h))\n                | 2^-+ m (n + o \u2238 k) 1\n                | helper m n o k\n                | sym (2^-+ n (m + o \u2238 k) 1)\n                = 2^*-mono\u2032 n q\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_ (\u03bb {m n f g} x \u2192 ]-[-sym {m} {n} {f} {g} x) (\u03bb {m n o f g h} x y \u2192 ]-[-cong-left-\u2248\u21ba {m} {n} {o} {f} {g} {h} x y)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"98913dff03c23ec41390da83fdc97a418746a9f2","subject":"Avoid using congruence to prove flatmap correct","message":"Avoid using congruence to prove flatmap correct\n","repos":"inc-lc\/ilc-agda","old_file":"Nehemiah\/Change\/Correctness.agda","new_file":"Nehemiah\/Change\/Correctness.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Correctness of differentiation with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Change.Correctness where\n\n-- The denotational properties of the `derive` transformation\n-- for Calculus Nehemiah. In particular, the main theorem\n-- about it producing the correct incremental behavior.\n\nopen import Nehemiah.Syntax.Type\nopen import Nehemiah.Syntax.Term\nopen import Nehemiah.Denotation.Value\nopen import Nehemiah.Denotation.Evaluation\nopen import Nehemiah.Change.Type\nopen import Nehemiah.Change.Term\nopen import Nehemiah.Change.Value\nopen import Nehemiah.Change.Evaluation\nopen import Nehemiah.Change.Validity\nopen import Nehemiah.Change.Derive\nopen import Nehemiah.Change.Implementation\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\nopen import Postulate.Extensionality\n\nimport Parametric.Change.Correctness\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base nil-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 \u27e6nil-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base meaning-onil-base\n  deriveConst implementation-structure as Correctness\n\nopen import Algebra.Structures\n\nprivate\n  open import Level using () renaming (zero to lzero)\n  open FunctionChanges {c = lzero} {d = lzero} \u2124 Bag using (FunctionChange; changeAlgebra)\n\n  flatmap-funarg-equal : \u2200 (f : \u2124 \u2192 Bag) (\u0394f : \u0394 f) \u0394f\u2032 (\u0394f\u2248\u0394f\u2032 : \u0394f \u2248\u208d int \u21d2 bag \u208e \u0394f\u2032) \u2192\n    (f \u229e \u0394f) \u2261 (f \u27e6\u2295\u208d int \u21d2 bag \u208e\u27e7 \u0394f\u2032)\n  flatmap-funarg-equal f \u0394f \u0394f\u2032 \u0394f\u2248\u0394f\u2032 = ext lemma\n    where\n    lemma : \u2200 v \u2192 f v ++ FunctionChange.apply \u0394f v (+ 0) \u2261 f v ++ \u0394f\u2032 v (+ 0)\n    lemma v rewrite \u0394f\u2248\u0394f\u2032 v (+ 0) (+ 0) refl = refl\n\nderive-const-correct : Correctness.Structure\nderive-const-correct (intlit-const n) = refl\nderive-const-correct add-const w \u0394w .\u0394w refl w\u2081 \u0394w\u2081 .\u0394w\u2081 refl\n  rewrite mn\u00b7pq=mp\u00b7nq {w} {\u0394w} {w\u2081} {\u0394w\u2081}\n  | associative-int (w + w\u2081) (\u0394w + \u0394w\u2081) (- (w + w\u2081))\n  = n+[m-n]=m {w + w\u2081} {\u0394w + \u0394w\u2081}\nderive-const-correct minus-const w \u0394w .\u0394w refl\n  rewrite sym (-m\u00b7-n=-mn {w} {\u0394w})\n  | associative-int (- w) (- \u0394w) (- (- w)) = n+[m-n]=m { - w} { - \u0394w}\nderive-const-correct empty-const = refl\nderive-const-correct insert-const w \u0394w .\u0394w refl w\u2081 \u0394w\u2081 .\u0394w\u2081 refl = refl\nderive-const-correct union-const w \u0394w .\u0394w refl w\u2081 \u0394w\u2081 .\u0394w\u2081 refl\n  rewrite ab\u00b7cd=ac\u00b7bd {w} {\u0394w} {w\u2081} {\u0394w\u2081}\n  | associative-bag (w ++ w\u2081) (\u0394w ++ \u0394w\u2081) (negateBag (w ++ w\u2081))\n  = a++[b\\\\a]=b {w ++ w\u2081} {\u0394w ++ \u0394w\u2081}\nderive-const-correct negate-const w \u0394w .\u0394w refl\n  rewrite sym (-a\u00b7-b=-ab {w} {\u0394w})\n  | associative-bag (negateBag w) (negateBag \u0394w) (negateBag (negateBag w))\n  = a++[b\\\\a]=b {negateBag w} {negateBag \u0394w}\nderive-const-correct flatmap-const w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032 w\u2081 \u0394w\u2081 .\u0394w\u2081 refl\n  rewrite flatmap-funarg-equal w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032 = refl\nderive-const-correct sum-const w \u0394w .\u0394w refl\n  rewrite homo-sum {w} {\u0394w}\n  | associative-int (sumBag w) (sumBag \u0394w) (- sumBag w)\n  = n+[m-n]=m {sumBag w} {sumBag \u0394w}\n\nopen Correctness.Structure derive-const-correct public\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Correctness of differentiation with the Nehemiah plugin.\n------------------------------------------------------------------------\n\nmodule Nehemiah.Change.Correctness where\n\n-- The denotational properties of the `derive` transformation\n-- for Calculus Nehemiah. In particular, the main theorem\n-- about it producing the correct incremental behavior.\n\nopen import Nehemiah.Syntax.Type\nopen import Nehemiah.Syntax.Term\nopen import Nehemiah.Denotation.Value\nopen import Nehemiah.Denotation.Evaluation\nopen import Nehemiah.Change.Type\nopen import Nehemiah.Change.Term\nopen import Nehemiah.Change.Value\nopen import Nehemiah.Change.Evaluation\nopen import Nehemiah.Change.Validity\nopen import Nehemiah.Change.Derive\nopen import Nehemiah.Change.Implementation\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\nopen import Postulate.Extensionality\n\nimport Parametric.Change.Correctness\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base nil-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 \u27e6nil-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base meaning-onil-base\n  deriveConst implementation-structure as Correctness\n\nopen import Algebra.Structures\n\nderive-const-correct : Correctness.Structure\nderive-const-correct (intlit-const n) = refl\nderive-const-correct add-const w \u0394w .\u0394w refl w\u2081 \u0394w\u2081 .\u0394w\u2081 refl\n  rewrite mn\u00b7pq=mp\u00b7nq {w} {\u0394w} {w\u2081} {\u0394w\u2081}\n  | associative-int (w + w\u2081) (\u0394w + \u0394w\u2081) (- (w + w\u2081))\n  = n+[m-n]=m {w + w\u2081} {\u0394w + \u0394w\u2081}\nderive-const-correct minus-const w \u0394w .\u0394w refl\n  rewrite sym (-m\u00b7-n=-mn {w} {\u0394w})\n  | associative-int (- w) (- \u0394w) (- (- w)) = n+[m-n]=m { - w} { - \u0394w}\nderive-const-correct empty-const = refl\nderive-const-correct insert-const w \u0394w .\u0394w refl w\u2081 \u0394w\u2081 .\u0394w\u2081 refl = refl\nderive-const-correct union-const w \u0394w .\u0394w refl w\u2081 \u0394w\u2081 .\u0394w\u2081 refl\n  rewrite ab\u00b7cd=ac\u00b7bd {w} {\u0394w} {w\u2081} {\u0394w\u2081}\n  | associative-bag (w ++ w\u2081) (\u0394w ++ \u0394w\u2081) (negateBag (w ++ w\u2081))\n  = a++[b\\\\a]=b {w ++ w\u2081} {\u0394w ++ \u0394w\u2081}\nderive-const-correct negate-const w \u0394w .\u0394w refl\n  rewrite sym (-a\u00b7-b=-ab {w} {\u0394w})\n  | associative-bag (negateBag w) (negateBag \u0394w) (negateBag (negateBag w))\n  = a++[b\\\\a]=b {negateBag w} {negateBag \u0394w}\nderive-const-correct flatmap-const w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032 w\u2081 \u0394w\u2081 .\u0394w\u2081 refl =\n  cong (\u03bb \u25a1 \u2192 flatmapBag \u25a1 (w\u2081 ++ \u0394w\u2081) ++ negateBag (flatmapBag w w\u2081))\n  (ext (\u03bb n \u2192 cong (_++_ (w n)) (\u0394w\u2248\u0394w\u2032 n (+ 0) (+ 0) refl)))\nderive-const-correct sum-const w \u0394w .\u0394w refl\n  rewrite homo-sum {w} {\u0394w}\n  | associative-int (sumBag w) (sumBag \u0394w) (- sumBag w)\n  = n+[m-n]=m {sumBag w} {sumBag \u0394w}\n\nopen Correctness.Structure derive-const-correct public\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"497767b3608a37de2b6c94d8a0d61ed37a7c0e00","subject":"Adding stub for expanding lambda getting checked against hole","message":"Adding stub for expanding lambda getting checked against hole\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- not a hole\n\n  data _not-hole : dhexp \u2192 Set where\n    CNotHole : c not-hole\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EALamHole : \u2200{\u0393 x  e d \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              {!!} \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u2987\u2988 ~> \u00b7\u03bb x [ {!!}  ] d :: {!!} \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set -- todo: no idea if this is right; mutual thing is weird\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      <_>_     : htyp \u2192 dhexp \u2192 dhexp\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 tctx \u00d7 htyp)\n  u ::[ \u0393 ] \u03c4 = u , \u0393 , \u03c4\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988     -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => \u03c4 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- todo: do we care about completeness of hexp or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : htyp \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (\u03c41 ==> \u03c42) = tcomplete \u03c41 \u00d7 tcomplete \u03c42\n\n  -- those expressions without holes anywhere\n  ecomplete : hexp \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: \u03c4)  = ecomplete e1 \u00d7 tcomplete \u03c4\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ \u03c4 ] e) = tcomplete \u03c4 \u00d7 ecomplete e\n\n  -- not a hole\n\n  data _not-hole : dhexp \u2192 Set where\n    CNotHole : c not-hole\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp1   : \u2200{\u0393 e1 e2 d2 d1 \u03941 \u03c42 \u03c41 \u03942} \u2192\n                \u03941 ## \u03942 \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u2987\u2988) ~> d1 :: \u03c41 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< \u03c42 ==> \u2987\u2988 > d1) \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 \u03c42 \u03c4 d1 d2 \u03941 \u03942 \u03c42' e2} \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              (\u03c42 == \u03c42' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 (< \u03c42 > d2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 \u03c4 d1 \u03941 e2 \u03c42 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 \u21d2 (\u03c42 ==> \u03c4) ~> d1 \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> d1 \u2218 d2 \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 (\u03c4 == \u03c4' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> (< \u03c4 > d) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e \u03c4 d \u03c4' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192 -- todo: i added this\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c41 ==> \u03c42 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 (e == \u2987\u2988[ u ] \u2192 \u22a5)) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 (e == \u2987 e' \u2988[ u ] \u2192 \u22a5)) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393 \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) \u2192\n          (x , d) \u2208 \u03c3 \u2192\n          \u03a3[ \u03c4 \u2208 htyp ] (\u0393' x == Some \u03c4 \u00d7 \u0394 , \u0393 \u22a2 d :: \u03c4)\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c42 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n             \u03c41 \u25b8arr (\u03c42 ==> \u03c4) \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 (\u0394 ,, u ::[ \u0393' ] \u03c4) , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c4 \u03c4'} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n             \u03c4 ~ \u03c4' \u2192\n             \u0394 , \u0393 \u22a2 < \u03c4 > d :: \u03c4\n\n  postulate -- todo: write this later\n    [_]_ : subst \u2192 dhexp \u2192 dhexp\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 d1 indet \u2192 d2 final \u2192 (d1 \u2218 d2) indet\n      ICast : \u2200{d \u03c4} \u2192 d indet \u2192 (< \u03c4 > d) indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FVal : \u2200{d} \u2192 d val \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n------------- these two judgements are still being figured out; form\n------------- changing, etc. double check everything here once it settles\n------------- before doing anything with it\n  -- error\n  data _err[_] : (d : dhexp) \u2192 (\u0394 : hctx) \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : (d d' : dhexp)  \u2192 Set where\n    STHole : \u2200{ d d' u \u03c3 } \u2192\n             d \u21a6 d' \u2192\n             \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 \u21a6  \u2987 d' \u2988\u27e8 u , \u03c3 \u27e9\n    -- STCast\n    STAp1 : \u2200{ d1 d2 d1' } \u2192\n            d1 \u21a6 d1' \u2192\n            (d1 \u2218 d2) \u21a6 (d1' \u2218 d2)\n    STAp2 : \u2200{ d1 d2 d2' } \u2192\n            d1 final \u2192\n            d2 \u21a6 d2' \u2192\n            (d1 \u2218 d2) \u21a6 (d1 \u2218 d2')\n    -- STAp\u03b2 : \u2200{ d1 d2 \u03c4 x } \u2192\n    --         d2 final \u2192\n    --         ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] d2)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5013dd25be51aea4374492d685bcc066ae526e82","subject":"Agda: Define setoid for d.o.e. (not very succesfully)","message":"Agda: Define setoid for d.o.e. (not very succesfully)\n\nThis setoid can be defined, but using it for equational reasoning seems\nnot so convenient. I guess a reason is that unifying against _\u2259_ might\nbe harder (see comments at the end).\n\nThe other, bigger problem is that most reasonings here need to rewrite\ndx\u2081 \u2259 dx\u2082 into x \u229e dx\u2081 \u2261 x \u229e dx\u2082 to perform other transformations, so\nthis setoid is not powerful enough.\n\nMaybe, however, other files can make more use of this setoid.\n\nOld-commit-hash: a37b3b7baeb5c99c881dcf6dc7dced39f9311ed3\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-symm\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid \u2113 a\n  \u2259-setoid = record\n    { Carrier       = \u0394 x\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\n  ------------------------------------------------------------------------\n  -- Convenient syntax for equational reasoning\n\n  import Relation.Binary.EqReasoning as EqR\n\n  -- Relation.Binary.EqReasoning is more convenient to use with _\u2261_ if\n  -- the combinators take the type argument (a) as a hidden argument,\n  -- instead of being locked to a fixed type at module instantiation\n  -- time.\n\n  module \u2259-Reasoning where\n    open EqR \u2259-setoid public\n      renaming (_\u2248\u27e8_\u27e9_ to _\u2259\u27e8_\u27e9_)\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      df\n    \u2259\u27e8 derivative-is-nil df fdf \u27e9\n      nil f\n    \u2259\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      dg\n    \u220e\n    where\n      open \u2259-Reasoning\n{-\n      lemma : nil f \u2259 dg\n      -- Goes through\n      --lemma = sym (derivative-is-nil dg fdg)\n      -- Generates tons of crazy yellow ambiguities of equality type. Apparently, unifying against \u2259 does not work so well.\n      lemma = \u2259-symm {{changeAlgebra}} {f} {dg} {nil f} (derivative-is-nil dg fdg)\n-}\n\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nmodule Base.Change.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\nmodule _ {a \u2113} {A : Set a} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n  _\u2259_ : \u2200 dx dy \u2192 Set a\n  dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = refl\n\n  \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-symm \u2259 = sym \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n  -- By update-nil, if dx = nil x, then x \u229e dx \u2261 x.\n  -- As a consequence, if dx \u2259 nil x, then x \u229e dx \u2261 x\n  nil-is-\u229e-unit : \u2200 dx \u2192 dx \u2259 nil x \u2192 x \u229e dx \u2261 x\n  nil-is-\u229e-unit dx dx\u2259nil-x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 dx\u2259nil-x \u27e9\n      x \u229e (nil x)\n    \u2261\u27e8 update-nil x \u27e9\n      x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- Here we prove the inverse:\n  \u229e-unit-is-nil : \u2200 dx \u2192 x \u229e dx \u2261 x \u2192 dx \u2259 nil x\n  \u229e-unit-is-nil dx x\u229edx\u2261x =\n    begin\n      x \u229e dx\n    \u2261\u27e8 x\u229edx\u2261x \u27e9\n      x\n    \u2261\u27e8 sym (update-nil x) \u27e9\n      x \u229e nil x\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- TODO: we want to show that all functions of interest respect\n  -- delta-observational equivalence, so that two d.o.e. changes can be\n  -- substituted for each other freely.\n  --\n  -- * That should be be true for\n  --   functions using changes parametrically.\n  --\n  -- * Moreover, d.o.e. should be respected by contexts [ ] x dx and df x [ ];\n  --   this is proved below on both contexts at once by fun-change-respects.\n  --\n  -- * Finally, change algebra operations should respect d.o.e. But \u229e respects\n  --   it by definition, and \u229f doesn't take change arguments - we will only\n  --   need a proof for compose, when we define it.\n  --\n  -- Stating the general result, though, seems hard, we should\n  -- rather have lemmas proving that certain classes of functions respect this\n  -- equivalence.\n\nmodule _ {a} {b} {c} {d} {A : Set a} {B : Set b}\n  {{CA : ChangeAlgebra c A}} {{CB : ChangeAlgebra d B}} where\n\n  module FC = FunctionChanges A B {{CA}} {{CB}}\n  open FC using (changeAlgebra; incrementalization)\n  open FC.FunctionChange\n\n  fun-change-respects : \u2200 {x : A} {dx\u2081 dx\u2082 : \u0394 x} {f : A \u2192 B} {df\u2081 df\u2082} \u2192\n    df\u2081 \u2259 df\u2082 \u2192 dx\u2081 \u2259 dx\u2082 \u2192 apply df\u2081 x dx\u2081 \u2259 apply df\u2082 x dx\u2082\n  fun-change-respects {x} {dx\u2081} {dx\u2082} {f} {df\u2081} {df\u2082} df\u2081\u2259df\u2082 dx\u2081\u2259dx\u2082 = lemma\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma : f x \u229e apply df\u2081 x dx\u2081 \u2261 f x \u229e apply df\u2082 x dx\u2082\n      -- Informally: use incrementalization on both sides and then apply\n      -- congruence.\n      lemma =\n        begin\n          f x \u229e apply df\u2081 x dx\u2081\n        \u2261\u27e8 sym (incrementalization f df\u2081 x dx\u2081) \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2081)\n        \u2261\u27e8 cong (f \u229e df\u2081) dx\u2081\u2259dx\u2082 \u27e9\n          (f \u229e df\u2081) (x \u229e dx\u2082)\n        \u2261\u27e8 cong (\u03bb f \u2192 f (x \u229e dx\u2082)) df\u2081\u2259df\u2082 \u27e9\n          (f \u229e df\u2082) (x \u229e dx\u2082)\n        \u2261\u27e8 incrementalization f df\u2082 x dx\u2082 \u27e9\n          f x \u229e apply df\u2082 x dx\u2082\n        \u220e\n\n  open import Postulate.Extensionality\n\n  -- An extensionality principle for delta-observational equivalence: if\n  -- applying two function changes to the same base value and input change gives\n  -- a d.o.e. result, then the two function changes are d.o.e. themselves.\n\n  delta-ext : \u2200 {f : A \u2192 B} \u2192 \u2200 {df dg : \u0394 f} \u2192 (\u2200 x dx \u2192 apply df x dx \u2259 apply dg x dx) \u2192 df \u2259 dg\n  delta-ext {f} {df} {dg} df-x-dx\u2259dg-x-dx = lemma\u2082\n    where\n      open \u2261-Reasoning\n      -- This type signature just expands the goal manually a bit.\n      lemma\u2081 : \u2200 x dx \u2192 f x \u229e apply df x dx \u2261 f x \u229e apply dg x dx\n      lemma\u2081 = df-x-dx\u2259dg-x-dx\n      lemma\u2082 : f \u229e df \u2261 f \u229e dg\n      lemma\u2082 = ext (\u03bb x \u2192 lemma\u2081 x (nil x))\n\n  -- We know that Derivative f (apply (nil f)) (by nil-is-derivative).\n  -- That is, df = nil f -> Derivative f (apply df).\n  -- Now, we try to prove that if Derivative f (apply df) -> df = nil f.\n  -- But first, we prove that f \u229e df = f.\n  derivative-is-\u229e-unit : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 f \u229e df \u2261 f\n  derivative-is-\u229e-unit {f} df fdf =\n    begin\n      f \u229e df\n    \u2261\u27e8\u27e9\n      (\u03bb x \u2192 f x \u229e apply df x (nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 fdf x (nil x)) \u27e9\n      (\u03bb x \u2192 f (x \u229e nil x))\n    \u2261\u27e8 ext (\u03bb x \u2192 cong f (update-nil x)) \u27e9\n      (\u03bb x \u2192 f x)\n    \u2261\u27e8\u27e9\n      f\n    \u220e\n    where\n      open \u2261-Reasoning\n\n  -- We can restate the above as \"df is a nil change\".\n\n  derivative-is-nil : \u2200 {f : A \u2192 B} df \u2192\n    Derivative f (apply df) \u2192 df \u2259 nil f\n  derivative-is-nil df fdf = \u229e-unit-is-nil df (derivative-is-\u229e-unit df fdf)\n\n  -- If we have two derivatives, they're both nil, hence they're equal.\n  derivative-unique : \u2200 {f : A \u2192 B} {df dg : \u0394 f} \u2192 Derivative f (apply df) \u2192 Derivative f (apply dg) \u2192 df \u2259 dg\n  derivative-unique {f} {df} {dg} fdf fdg =\n    begin\n      f \u229e df\n    \u2261\u27e8 derivative-is-nil df fdf \u27e9\n      f \u229e nil f\n    \u2261\u27e8 sym (derivative-is-nil dg fdg) \u27e9\n      f \u229e dg\n    \u220e\n    where\n      open \u2261-Reasoning\n  -- We could also use derivative-is-\u229e-unit, but the proof above matches better\n  -- with the text above.\n  --\n  -- Small TODO: It could be even more elegant to define a setoid for \u2259 and\n  -- use equational reasoning on it directly.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"669a6c291093468469c67d58acdbcb96f0c1b221","subject":"circuit: fixes","message":"circuit: fixes\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits hiding (rewire; rewireTbl)\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp;\n                tabulate-\u2218)\n         renaming (map to vmap)\nimport Data.Vec.Properties as VecProps\nopen VecProps using (tabulate\u2218lookup)\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Vec.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Vec.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Vec.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Vec.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Vec.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = Vec.rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Vec.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Vec.rewire r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Vec.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Vec.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","old_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits hiding (rewire; rewireTbl)\nopen import Data.Bits.Bits2\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin; concat; ++-decomp;\n                tabulate-\u2218)\n         renaming (map to vmap)\nimport Data.Vec.Properties as VecProps\nopen VecProps using (tabulate\u2218lookup)\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nRunCircuit : CircuitType \u2192 Set\nRunCircuit C = \u2200 {i o} \u2192 C i o \u2192 Bits i \u2192 Bits o\n\nRewireFun : CircuitType\nRewireFun i o = Fin o \u2192 Fin i\n\nrecord RewiringBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n\n  field\n    isComposable  : Composable C\n    isVComposable : VComposable _+_ C\n  open Composable isComposable\n  open VComposable isVComposable\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idCDefault : \u2200 {i} \u2192 C i i\n  idCDefault = rewire id\n\n  field\n    idC : \u2200 {i} \u2192 C i i\n\n  field\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) is \u2192 is =[ rewire r ]= Vec.rewire r is\n\n{-\n    _>>>-spec_ : \u2200 {i m o} {c\u2080 : C i m} {c\u2081 : C m o} {is ms os} \u2192\n                 is =[ c\u2080 ]= ms \u2192 ms =[ c\u2081 ]= os \u2192 is =[ c\u2080 >>> c\u2081 ]= os\n\n    _***-spec_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {c\u2080 : C i\u2080 o\u2080} {c\u2081 : C i\u2081 o\u2081} {is\u2080 is\u2081 os\u2080 os\u2081} \u2192\n                 is\u2080 =[ c\u2080 ]= os\u2080 \u2192 is\u2081 =[ c\u2081 ]= os\u2081 \u2192 (is\u2080 ++ is\u2081) =[ c\u2080 *** c\u2081 ]= (os\u2080 ++ os\u2081)\n\n-}\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idC ]= bs\n{-\n  rewireWithTbl-spec : \u2200 {i o} (t : RewireTbl i o) is\n                       \u2192 is =[ rewireWithTbl t ]= Vec.rewireTbl t is\n  rewireWithTbl-spec t is = {!rewire-spec ? ?!}\n-}\n\n  idCDefault-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ idCDefault ]= bs\n  idCDefault-spec bs\n    = subst (\u03bb bs' \u2192 bs =[ idCDefault ]= bs') (tabulate\u2218lookup bs) (rewire-spec id bs)\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n-- sink-spec : bs =[ sink i ]= []\n\n  dup\u2081 : \u2200 o \u2192 C 1 o\n  dup\u2081 _ = rewire (const zero)\n-- dup\u2081-spec : (b \u2237 []) =[ dup o ]= replicate o\n\n  dup\u2081\u00b2 : C 1 2\n  dup\u2081\u00b2 = dup\u2081 2\n-- dup\u2081\u00b2-spec : (b \u2237 []) =[ dup\u2081\u00b2 ]= (b \u2237 b \u2237 [])\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i o} \u2192 i \u2261 o \u2192 C i o\n  coerce refl = idC\n\n{-\n  coerce-spec : \u2200 {i o} {i\u2261o : i \u2261 o} {is} \u2192\n                  is =[ coerce i\u2261o ]= subst Bits i\u2261o is\n  coerce-spec = {!!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,1,2,...,n,0,1,2,...,n,0,1,2,...,n,...\n  -}\n  dup\u207f : \u2200 {i} k \u2192 C i (k * i)\n  dup\u207f {i} k = rewireWithTbl (concat (replicate {n = k} (allFin i)))\n\n{-\n  dup\u207f-spec : \u2200 {i} {is : Bits i} k \u2192 is =[ dup\u207f k ]= concat (replicate {n = k} is)\n  dup\u207f-spec {i} {is} k = {!rewireWithTbl-spec (concat (replicate {n = k} (allFin _))) ?!}\n-}\n\n  {-\n    Sends inputs 0,1,2,...,n to outputs 0,0,0,...,1,1,1,...,2,2,2,...,n,n,n,...\n  -}\n  dup\u207f\u2032 : \u2200 {i} k \u2192 C i (i * k)\n  dup\u207f\u2032 {i} k = rewireWithTbl (concat (vmap replicate (allFin i)))\n{-\n  dup\u207f\u2032 {i} k = coerce (proj\u2082 \u2115\u00b0.*-identity i) >>>\n                      (vcat {n = i} (replicate (dup\u2081 _)))\n-}\n\n  dup\u00b2 : \u2200 {i} \u2192 C i (i + i)\n  dup\u00b2 {i} = dup\u207f 2 >>> coerce (cong (_+_ i) (sym (\u2115\u00b0.+-comm 0 i)))\n\n{-\n  dup\u00b2-spec : \u2200 {n} {is : Bits n} \u2192 is =[ dup\u00b2 ]= (is ++ is)\n  dup\u00b2-spec = coerce-spec (rewireWithTbl-spec {!!} {!!})\n-}\n\n  _&&&_ : \u2200 {i o\u2080 o\u2081} \u2192 C i o\u2080 \u2192 C i o\u2081 \u2192 C i (o\u2080 + o\u2081)\n  c\u2080 &&& c\u2081 = dup\u00b2 >>> c\u2080 *** c\u2081\n\n{-\n  _&&&-spec_ : \u2200 {i o\u2080 o\u2081} {c\u2080 : C i o\u2080} {c\u2081 : C i o\u2081} {is os\u2080 os\u2081}\n               \u2192 is =[ c\u2080 ]= os\u2080 \u2192 is =[ c\u2081 ]= os\u2081 \u2192 is =[ c\u2080 &&& c\u2081 ]= (os\u2080 ++ os\u2081)\n  pf\u2080 &&&-spec pf\u2081 = dup\u00b2-spec >>>-spec (pf\u2080 ***-spec pf\u2081)\n-}\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  commC : \u2200 m n \u2192 C (m + n) (n + m)\n  commC m n = rewireWithTbl (vmap (raise m) (allFin n) ++ vmap (inject+ n) (allFin m))\n\n  dropC : \u2200 {i} k \u2192 C (k + i) i\n  dropC k = sink k *** idC\n\n  takeC : \u2200 {i} k \u2192 C (k + i) k\n  takeC {i} k = commC k _ >>> dropC i\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\n  headC : \u2200 {i} \u2192 C (1 + i) 1\n  headC = takeC 1\n\n  tailC : \u2200 {i} \u2192 C (1 + i) i\n  tailC = dropC 1\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nrecord CircuitBuilder (C : CircuitType) : Set\u2081 where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    leafC : \u2200 {o} \u2192 Bits o \u2192 C 0 o\n    isForkable : Forkable suc C\n\n  open RewiringBuilder isRewiringBuilder\n  open Forkable isForkable renaming (fork to forkC)\n\n{-\n  field\n    leafC-spec : \u2200 {o} (os : Bits o) \u2192 [] =[ leafC os ]= os\n    forkC-left-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                      \u2192 is =[ c\u2080 ]= os \u2192 (0\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n    forkC-right-spec : \u2200 {i o} {c\u2080 c\u2081 : C i o} {is os}\n                       \u2192 is =[ c\u2081 ]= os \u2192 (1\u2237 is) =[ forkC c\u2080 c\u2081 ]= os\n-}\n  bit : Bit \u2192 C 0 1\n  bit b = leafC (b \u2237 [])\n\n{-\n  bit-spec : \u2200 b \u2192 [] =[ bit b ]= (b \u2237 [])\n  bit-spec b = leafC-spec (b \u2237 [])\n-}\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  0\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  0\u02b7\u207f = leafC 0\u207f\n\n{-\n  0\u02b7-spec : [] =[ 0\u02b7 ]= 0\u2237 []\n  0\u02b7-spec = bit-spec 0b\n-}\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  1\u02b7\u207f : \u2200 {o} \u2192 C 0 o\n  1\u02b7\u207f = leafC 1\u207f\n\n{-\n  1\u02b7-spec : [] =[ 1\u02b7 ]= 1\u2237 []\n  1\u02b7-spec = bit-spec 1b\n-}\n\n  padL : \u2200 {i} k \u2192 C i (k + i)\n  padL k = 0\u02b7\u207f {k} *** idC\n\n  padR : \u2200 {i} k \u2192 C i (i + k)\n  padR k = padL k >>> commC k _\n\n  arr' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n  arr' {zero}  f = leafC (f [])\n  arr' {suc i} f = forkC (arr' {i} (f \u2218 0\u2237_)) (arr' (f \u2218 1\u2237_))\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  terOp : (Bit \u2192 Bit \u2192 Bit \u2192 Bit) \u2192 C 3 1\n  terOp op = arr (\u03bb { (x \u2237 y \u2237 z \u2237 []) \u2192 (op x y z) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n  if\u27e8head=0\u27e9then_else_ : \u2200 {i o} (c\u2080 c\u2081 : C i o) \u2192 C (1 + i) o\n  if\u27e8head=0\u27e9then_else_ = forkC\n\n  -- Base addition with carry:\n  --   * Any input can be used as the carry\n  --   * First output is the carry\n  --\n  -- This can also be seen as the addition of\n  -- three bits which result is between 0 and 3\n  -- and thus fits in two bit word in binary\n  -- representation.\n  --\n  -- Alternatively you can see it as counting the\n  -- number of 1s in a three bits vector.\n  add\u2082 : C 3 2\n  add\u2082 = carry &&& result\n    where carry  : C 3 1\n          carry  = terOp (\u03bb x y z \u2192 x xor (y \u2227 z))\n          result : C 3 1\n          result = terOp (\u03bb x y z \u2192 x xor (y xor z))\n\n  open RewiringBuilder isRewiringBuilder public\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk (opComp (ixFunComp Fin)) finFunOpVComp id id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1da0_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = Vec.rewire f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec = tabulate\u2218lookup\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk (mk _>>>_) (mk _***_) tabulate (allFin _) _=[_]=_ rewire-spec idC-spec\n  where\n    C = RewireTbl\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tbl ]= output = Vec.rewireTbl tbl input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ tabulate r ]= Vec.rewire r bs\n    rewire-spec r bs = sym (tabulate-\u2218 (flip lookup bs) r)\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ allFin _ ]= bs\n    -- idC-spec = map-lookup-allFin\n    idC-spec bs rewrite rewire-spec id bs = tabulate\u2218lookup bs\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = Vec.rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk bitsFunComp bitsFunVComp Vec.rewire id _=[_]=_ rewire-spec idC-spec\n  where\n    C = _\u2192\u1d47_\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ f ]= output = f input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ Vec.rewire r ]= Vec.rewire r bs\n    rewire-spec r bs = refl\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ id ]= bs\n    idC-spec bs = refl\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id (\u03bb { bs [] \u2192 bs }) bitsFunFork\n\nmodule BitsFunExtras where\n  open CircuitBuilder bitsFunCircuitBuilder\n  C = _\u2192\u1d47_\n  >>>-spec : \u2200 {i m o} (c\u2080 : C i m) (c\u2081 : C m o) xs \u2192 (c\u2080 >>> c\u2081) xs \u2261 c\u2081 (c\u2080 xs)\n  >>>-spec _ _ _ = refl\n  ***-spec : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} (c\u2080 : C i\u2080 o\u2080) (c\u2081 : C i\u2081 o\u2081) xs {ys}\n             \u2192 (c\u2080 *** c\u2081) (xs ++ ys) \u2261 c\u2080 xs ++ c\u2081 ys\n  ***-spec {i\u2080} c\u2080 c\u2081 xs {ys} with splitAt i\u2080 (xs ++ ys)\n  ... | pre , post , eq with ++-decomp {xs = xs} {pre} {ys} {post} eq\n  ... | eq1 , eq2 rewrite eq1 | eq2 = refl\n\nopen import bintree hiding (_>>>_; _***_)\nopen import flipbased-tree -- hiding (_***_)\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk bintree.composable bintree.vcomposable rewire (rewire id) _=[_]=_ rewire-spec idC-spec\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Vec.rewire f)\n\n    _=[_]=_ : \u2200 {i o} \u2192 Bits i \u2192 C i o \u2192 Bits o \u2192 Set\n    input =[ tree ]= output = toFun tree input \u2261 output\n\n    rewire-spec : \u2200 {i o} (r : RewireFun i o) bs \u2192 bs =[ rewire r ]= Vec.rewire r bs\n    rewire-spec r = toFun\u2218fromFun (tabulate \u2218 flip (Vec.lookup \u2218 r))\n\n    idC-spec : \u2200 {i} (bs : Bits i) \u2192 bs =[ rewire id ]= bs\n    idC-spec bs rewrite toFun\u2218fromFun (tabulate \u2218 flip Vec.lookup) bs | tabulate\u2218lookup bs = refl\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder fromFun leaf bintree.forkable\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2081\u00b2 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  -- open RewiringWith2^Outputs\n  -- test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  -- test\u2085 = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8aac5b938c8838cfc7e84259d3497fd1f6160af9","subject":"Cleanups","message":"Cleanups\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import Function hiding (const)\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\nopen import New.FunctionLemmas\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\n\nmodule _ {t1 t2 t3 : Type}\n  (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (df : Cht (t1 \u21d2 t3))\n  (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (dg : Cht (t2 \u21d2 t3))\n  where\n  private\n    \u0393 = sum t1 t2 \u2022\n      (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n      (t2 \u21d2 t3) \u2022\n      (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n      (t1 \u21d2 t3) \u2022\n      sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n      sum t1 t2 \u2022 \u2205\n  module _ where\n    private\n      \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n    changeMatchSem-lem1 :\n      \u2200 a1 b2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n      \u2261\n        g b2 \u2295 dg b2 (nil b2) \u229d f a1\n    changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n  module _ where\n    private\n      \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n    changeMatchSem-lem2 :\n      \u2200 b1 a2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n      \u2261\n        f a2 \u2295 df a2 (nil a2) \u229d g b1\n    changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\ncorrectDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\ncorrectDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\ncorrectDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n  where\n    lemma : \u2200 s f g \u2192\n      \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n      (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n    lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n    lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\nvalidDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\nvalidDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\nvalidDeriveConst (match {t1} {t2} {t3}) =\n  ternary-valid dmatch-valid dmatch-eq\n  where\n    open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n    dmatch-valid : ternary-valid-preserve-hp\n    dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n    dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n    dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n    dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n    dmatch-eq : ternary-valid-eq-hp\n    dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n      rewrite update-nil (a \u2295 da)\n      | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n    dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n      rewrite update-nil (b \u2295 db)\n      | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n    dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      | update-nil b2\n      | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n      | update-nil (g b2 \u2295 dg b2 (nil b2))\n      = refl\n    dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      | update-nil a2\n      | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n      | update-nil (f a2 \u2295 df a2 (nil a2))\n      = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext $ \u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term \u03c12 \u2295 \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","old_contents":"module New.Correctness where\n\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\nopen import New.FunctionLemmas\n\n-- Lemmas\nalternate : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 eCh \u0393 \u2192 \u27e6 \u0394\u0393 \u0393 \u27e7Context\nalternate {\u2205} \u2205 \u2205 = \u2205\nalternate {\u03c4 \u2022 \u0393} (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv \u2022 v \u2022 alternate \u03c1 d\u03c1\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c (alternate \u03c1 d\u03c1)\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term (alternate \u03c1 d\u03c1)\nfit-sound t \u03c1 d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1))\n  (sym (weaken-sound t _))\n\n-- The change semantics is just the semantics composed with derivation!\n\u27e6_\u27e7\u0394Var : \u2200 {\u0393 \u03c4} \u2192 (t : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 = \u27e6 deriveVar x \u27e7Var (alternate \u03c1 d\u03c1)\n\n\u27e6_\u27e7\u0394Term : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 Cht \u03c4\n\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (vdv , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\u27e6_\u27e7\u0394Const : \u2200 {\u03c4} (c : Const \u03c4) \u2192 Cht \u03c4\n\u27e6 c \u27e7\u0394Const = \u27e6 deriveConst c \u27e7Term \u2205\n\n\u27e6_\u27e7\u0394Const-rewrite : \u2200 {\u03c4 \u0393} (c : Const \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7Context) d\u03c1 \u2192 \u27e6_\u27e7\u0394Term (const c) \u03c1 d\u03c1 \u2261 \u27e6 c \u27e7\u0394Const\n\u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 rewrite weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u2205\u227c\u0393} (deriveConst c) (alternate \u03c1 d\u03c1) | \u27e6\u2205\u227c\u0393\u27e7-\u2205 (alternate \u03c1 d\u03c1) = refl\n\n\nmodule _ {t1 t2 t3 : Type}\n  (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (df : Cht (t1 \u21d2 t3))\n  (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n  (dg : Cht (t2 \u21d2 t3))\n  where\n  private\n    \u0393 = sum t1 t2 \u2022\n      (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n      (t2 \u21d2 t3) \u2022\n      (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n      (t1 \u21d2 t3) \u2022\n      sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n      sum t1 t2 \u2022 \u2205\n  module _ where\n    private\n      \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n    changeMatchSem-lem1 :\n      \u2200 a1 b2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n      \u2261\n        g b2 \u2295 dg b2 (nil b2) \u229d f a1\n    changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n  module _ where\n    private\n      \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n      \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n    changeMatchSem-lem2 :\n      \u2200 b1 a2 \u2192\n        \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n      \u2261\n        f a2 \u2295 df a2 (nil a2) \u229d g b1\n    changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\ncorrectDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\ncorrectDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\ncorrectDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n  where\n    lemma : \u2200 s f g \u2192\n      \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n      (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n    lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n    lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\nvalidDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\nvalidDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\nvalidDeriveConst (match {t1} {t2} {t3}) =\n  ternary-valid dmatch-valid dmatch-eq\n  where\n    open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n    dmatch-valid : ternary-valid-preserve-hp\n    dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n    dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n    dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n    dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n    dmatch-eq : ternary-valid-eq-hp\n    dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n      rewrite update-nil (a \u2295 da)\n      | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n    dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n      rewrite update-nil (b \u2295 db)\n      | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n    dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem1 f df g dg a1 b2\n      | update-nil b2\n      | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n      | update-nil (g b2 \u2295 dg b2 (nil b2))\n      = refl\n    dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n      rewrite changeMatchSem-lem2 f df g dg b1 a2\n      | update-nil a2\n      | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n      | update-nil (f a2 \u2295 df a2 (nil a2))\n      = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext (\u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid a , \u03c1d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e)\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term (alternate \u03c1 d\u03c1)\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term (alternate \u03c1 d\u03c1)\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 d\u03c1\n    \u03c11d\u03c11 : valid \u03c11 d\u03c11\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , \u03c1d\u03c1\n    \u03c12d\u03c12 : valid \u03c12 d\u03c12\n    \u03c12d\u03c12 = ada , \u03c1d\u03c1\n    fv = \u27e6 t \u27e7Term \u03c12\n    dfvdv = \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     (begin\n       \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n      \u2261\u27e8 sym ( correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11) \u27e9\n       \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n      \u2261\u27e8\u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2295 (nil (a \u2295 da)) \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)) \u27e9\n       \u27e6 t \u27e7Term (a \u2295 da \u2022 \u03c1 \u2295 d\u03c1)\n      \u2261\u27e8 correctDerive t \u03c12 d\u03c12 \u03c12d\u03c12 \u27e9\n        \u27e6 t \u27e7Term \u03c12 \u2295 \u27e6 t \u27e7\u0394Term \u03c12 d\u03c12\n      \u220e)\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8a87967ac025b29e18245cd7c7a49fffa8f6bad9","subject":"Cleaning.","message":"Cleaning.\n\nIgnore-this: 90aec585ae75dbc0572456e97b313939\n\ndarcs-hash:20110730142949-3bd4e-06262695fac88710e572435dc3aa8138b50b97d7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/BadProofTermErasure.agda","new_file":"Issues\/BadProofTermErasure.agda","new_contents":"","old_contents":"module Issues.BadProofTermErasure where\n\npostulate\n  D : Set\n\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (witness : D) \u2192 P witness \u2192 \u2203 P\n\npostulate\n  something    : D \u2192 D \u2192 Set\n  somethingAll : \u2200 a \u2192 something a a\n\nfoo : \u2200 d \u2192 d \u2261 d\nfoo d = bar d (d , (somethingAll d))\n  where\n  unnecesaryR : D \u2192 Set\n  unnecesaryR e = \u2203 \u03bb e' \u2192 something e e'\n\n  bar : \u2200 f \u2192 unnecesaryR f \u2192 f \u2261 f\n  bar f h = barfoo f\n    where\n    postulate barfoo : \u2200 g \u2192 g \u2261 g\n    -- We don't know how to erase the proof term h, therefore the\n    -- agda2atp should abort. However, the current version does the\n    -- translation.\n    {-# ATP prove barfoo #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6e89a5e8f4748ae387e6d11c8580e17693a5339f","subject":"Desc model: varl, constl, prodl, sigmal, pil implicit.","message":"Desc model: varl, constl, prodl, sigmal, pil implicit.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl i\niso2 I (const X) = constl X\niso2 I (prod D D') = prodl (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : (I : Set) \n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases I ( lvar , i ) hs =  var i\ncases I ( lconst , X ) hs =  const X\ncases I ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases I ( lpi , ( S , T ) ) hs =  pi S hs\ncases I ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases I) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : (I : Set) -> IDesc I -> IDescl I\niso2 I (var i) = varl I i\niso2 I (const X) = constl I X\niso2 I (prod D D') = prodl I (iso2 I D) (iso2 I D')\niso2 I (pi S T) = pil I S (\\s -> iso2 I (T s))\niso2 I (sigma S T) = sigmal I S (\\s -> iso2 I (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 I D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 I (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 I (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 I (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 (prodl I) p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil I S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal I S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 I (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c4bb998146cea67e573ed0cc357cd9fb8cbc04dd","subject":"\\or commutes","message":"\\or commutes\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/IPL.agda","new_file":"learyouanagda\/IPL.agda","new_contents":"module IPL where\n\n\ndata _\u2227_ (P : Set) (Q : Set) : Set where\n  \u2227-intro : P \u2192 Q \u2192 (P \u2227 Q)\n\n\nproof\u2081 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 P\nproof\u2081 (\u2227-intro p q) = p\n\nproof\u2082 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 Q\nproof\u2082 (\u2227-intro p q) = q\n\n\n_\u21d4_ : (P : Set) \u2192 (Q : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\n\u2227-comm\u2032 : (P Q : Set) \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm\u2032 _ _ (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm : (P Q : Set) \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm P Q = \u2227-intro (\u2227-comm\u2032 P Q) (\u2227-comm\u2032 Q P)\n\n\n\u2227-comm1\u2032 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm1\u2032 (\u2227-intro p q) = \u2227-intro q p\n\n\n\ndata _\u2228_ (P Q : Set) : Set where\n   \u2228-intro\u2081 : P \u2192 P \u2228 Q\n   \u2228-intro\u2082 : Q \u2192 P \u2228 Q\n\n\n\u2228-elim : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 (A \u2228 B) \u2192 C\n\u2228-elim ac bc (\u2228-intro\u2081 a) = ac a\n\u2228-elim ac bc (\u2228-intro\u2082 b) = bc b\n\n\n\u2228-comm\u2032 : {A B : Set} \u2192 (A \u2228 B) \u2192 (B \u2228 A)\n\u2228-comm\u2032 (\u2228-intro\u2081 a) = \u2228-intro\u2082 a\n\u2228-comm\u2032 (\u2228-intro\u2082 b) = \u2228-intro\u2081 b\n\n\u2228-comm : {A B : Set} \u2192 (A \u2228 B) \u21d4 (B \u2228 A)\n\u2228-comm = \u2227-intro \u2228-comm\u2032 \u2228-comm\u2032\n\n","old_contents":"module IPL where\n\n\ndata _\u2227_ (P : Set) (Q : Set) : Set where\n  \u2227-intro : P \u2192 Q \u2192 (P \u2227 Q)\n\n\nproof\u2081 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 P\nproof\u2081 (\u2227-intro p q) = p\n\nproof\u2082 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 Q\nproof\u2082 (\u2227-intro p q) = q\n\n\n_\u21d4_ : (P : Set) \u2192 (Q : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\n\u2227-comm\u2032 : (P Q : Set) \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm\u2032 _ _ (\u2227-intro p q) = \u2227-intro q p\n\n\u2227-comm : (P Q : Set) \u2192 (P \u2227 Q) \u21d4 (Q \u2227 P)\n\u2227-comm P Q = \u2227-intro (\u2227-comm\u2032 P Q) (\u2227-comm\u2032 Q P)\n\n\n\u2227-comm1\u2032 : {P Q : Set} \u2192 (P \u2227 Q) \u2192 (Q \u2227 P)\n\u2227-comm1\u2032 (\u2227-intro p q) = \u2227-intro q p\n\n\n\ndata _\u2228_ (P Q : Set) : Set where\n   \u2228-intro\u2081 : P \u2192 P \u2228 Q\n   \u2228-intro\u2082 : Q \u2192 P \u2228 Q\n\n\n\u2228-elim : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 (A \u2228 B) \u2192 C\n\u2228-elim ac bc (\u2228-intro\u2081 a) = ac a\n\u2228-elim ac bc (\u2228-intro\u2082 b) = bc b\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57da8ab233fdca3b992d585fa703f2c305a61295","subject":"Fix sha1...","message":"Fix sha1...\n","repos":"crypto-agda\/crypto-agda","old_file":"sha1.agda","new_file":"sha1.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _\u2238_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Function.NP using (Endo; _\u2218_)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nopen import Solver.Linear\n\nmodule sha1 where\n\nmodule FunSHA1\n  {t}\n  {T : Set t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    Word : T\n    Word = `Bits 32\n\n    map\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\n    map\u00b2\u02b7 = zipWith\n\n    map\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\n    map\u02b7 = map\n\n    lift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\n    lift f g = g \u204f f\n\n    lift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n    lift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n    `not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    `not = lift (map\u02b7 not)\n\n    infixr 3 _`\u2295_\n    _`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\n    infixr 3 _`\u2227_\n    _`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\n    infixr 2 _`\u2228_\n    _`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                   \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                   \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    <\u229e> adder : Word `\u00d7 Word `\u2192 Word\n    adder = <tt\u204f <0\u2082> , id > \u204f iter full-adder\n    <\u229e> = adder\n\n    infixl 4 _`\u229e_\n    _`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n    _`\u229e_ = lift\u2082 <\u229e>\n\n    <_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n    < f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n    <_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                                  \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                                  \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n    < f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n    <<<\u2085 : `Endo Word\n    <<<\u2085 = rot-left 5\n\n    <<<\u2083\u2080 : `Endo Word\n    <<<\u2083\u2080 = rot-left 30\n\n    Word\u00b2 = Word `\u00d7 Word\n    Word\u00b3 = Word `\u00d7 Word\u00b2\n    Word\u2074 = Word `\u00d7 Word\u00b3\n    Word\u2075 = Word `\u00d7 Word\u2074\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 =  constBits (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0\u2082\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0\u2082\n            F\u25b9\ud835\udfda (suc _) = 1\u2082\n\n            {-\n    [_-_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m - n mod p ] = {!!}\n\n    [_+_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m + n mod p ] = {!!}\n    -}\n\n    #\u02b7 = bits 32\n\n    <\u229e\u2075> : Word\u2075 `\u00d7 Word\u2075 `\u2192 Word\u2075\n    <\u229e\u2075> = < <\u229e> `zip` < <\u229e> `zip` < <\u229e> `zip` < <\u229e> `zip` <\u229e> > > > >\n\n    iterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\n    iterate\u207f zero    f = id\n    iterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n    _\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n    _\u00b2\u2070 = iterate\u207f 20\n\n    module _ where\n\n      K\u2080 = #\u02b7 0x5A827999\n      K\u2082 = #\u02b7 0x6ED9EBA1\n      K\u2084 = #\u02b7 0x8F1BBCDC\n      K\u2086 = #\u02b7 0xCA62C1D6\n\n      H0 = #\u02b7 0x67452301\n      H1 = #\u02b7 0xEFCDAB89\n      H2 = #\u02b7 0x98BADCFE\n      H3 = #\u02b7 0x10325476\n      H4 = #\u02b7 0xC3D2E1F0\n\n      A B C D E : Word\u2075 `\u2192 Word\n      A = fst\n      B = snd \u204f fst\n      C = snd \u204f snd \u204f fst\n      D = snd \u204f snd \u204f snd \u204f fst\n      E = snd \u204f snd \u204f snd \u204f snd\n\n      F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n      F\u2082 = B `\u2295 C `\u2295 D\n      F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n      F\u2086 = F\u2082\n\n      module _ (F : Word\u2075 `\u2192 Word)\n               (K : `\ud835\udfd9  `\u2192 Word)\n               (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n          where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n      module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n        W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n        W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n        W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n        W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n        W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n        Iteration\u2078\u2070 : `Endo Word\u2075\n        Iteration\u2078\u2070 =\n          (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n          (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n          (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n          (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n        pad0s : \u2115 \u2192 \u2115\n        pad0s zero = 512 \u2238 65\n        pad0s (suc _) = STUCK where postulate STUCK : \u2115\n        -- pad0s n = [ 512 - [ n + 65 mod 512 ] mod 512 ]\n\n        paddedLength : \u2115 \u2192 \u2115\n        paddedLength n = n + (1 + pad0s n + 64)\n\n        padding : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (paddedLength n)\n        padding {n} = < id ,tt\u204f <1\u2237 < <0\u207f> {pad0s n} ++ bits 64 n > > > \u204f append\n\n        ite : Endo (`Endo Word\u2075)\n        ite f = dup \u204f first f \u204f <\u229e\u2075>\n\n        hash-block : `Endo Word\u2075\n        hash-block = ite Iteration\u2078\u2070\n\n      ite' : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word)\n             \u2192 ((Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `Endo Word\u2075) \u2192 `Endo Word\u2075\n      ite' zero    W f = id\n      ite' (suc n) W f = f (W zero) \u204f ite' n (W \u2218 suc) f\n\n      SHA1 : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1 n W = < H0 , H1 , H2 , H3 , H4 >\n               \u204f ite' n W hash-block\n\n      SHA1-on-0s : `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1-on-0s = SHA1 1 (\u03bb _ _ \u2192 <0\u207f>)\n\nmodule AgdaSHA1 where\n  open import FunUniverse.Agda\n  open FunSHA1 agdaFunOps\n  open import Data.Two\n\nopen import IO\nimport IO.Primitive\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Coinduction\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1\u2082 = putStr \"1\"\nputBit 0\u2082 = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\nput\u00d7 : \u2200 {A B : Set} \u2192 (A \u2192 IO \ud835\udfd9) \u2192 (B \u2192 IO \ud835\udfd9) \u2192 (A \u00d7 B) \u2192 IO \ud835\udfd9\nput\u00d7 pA pB (x , y) = \u266f pA x >> \u266f pB y\n{-\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits\n          putBits))) AgdaSHA1.SHA1-on-0s)\n-}\nfirstBit : \u2200 {A : Set} \u2192 (V.Vec \ud835\udfda 32 \u00d7 A) \u2192 \ud835\udfda\nfirstBit ((b \u2237 _) , _) = b\nimport FunUniverse.Cost as Cost\nopen import Data.Nat.Show\nsha1-cost : \u2115\nsha1-cost = FunSHA1.SHA1-on-0s Cost.timeOps\nmain : IO.Primitive.IO \ud835\udfd9\n--main = IO.run (putBit (firstBit (AgdaSHA1.SHA1-on-0s _)))\nmain = IO.run (putStrLn (show sha1-cost))\n\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1 where\n\nmodule FunSHA1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n    Word : T\n    Word = `Bits 32\n\n    map\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\n    map\u00b2\u02b7 = zipWith\n\n    map\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\n    map\u02b7 = map\n\n    lift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\n    lift f g = g \u204f f\n\n    lift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n    lift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n    `not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    `not = lift (map\u02b7 not)\n\n    infixr 3 _`\u2295_\n    _`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\n    infixr 3 _`\u2227_\n    _`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\n    infixr 2 _`\u2228_\n    _`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n    _`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\n    open import Solver.Linear\n\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                   \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                   \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    <\u229e> adder : Word `\u00d7 Word `\u2192 Word\n    adder = <tt\u204f <0\u2082> , id > \u204f iter full-adder\n    <\u229e> = adder\n\n    infixl 4 _`\u229e_\n    _`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n    _`\u229e_ = lift\u2082 <\u229e>\n\n    <_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n    < f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n    <_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                                  \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                                  \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n    < f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n    <<<\u2085 : `Endo Word\n    <<<\u2085 = rot-left 5\n\n    <<<\u2083\u2080 : `Endo Word\n    <<<\u2083\u2080 = rot-left 30\n\n    Word\u00b2 = Word `\u00d7 Word\n    Word\u00b3 = Word `\u00d7 Word\u00b2\n    Word\u2074 = Word `\u00d7 Word\u00b3\n    Word\u2075 = Word `\u00d7 Word\u2074\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 =  constBits (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0\u2082\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0\u2082\n            F\u25b9\ud835\udfda (suc _) = 1\u2082\n\n            {-\n    [_-_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m - n mod p ] = {!!}\n\n    [_+_mod_] : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    [ m + n mod p ] = {!!}\n    -}\n\n    #\u02b7 = bits 32\n\n    <\u229e\u2075> : Word\u2075 `\u00d7 Word\u2075 `\u2192 Word\u2075\n    <\u229e\u2075> = < <\u229e> `zip` < <\u229e> `zip` < <\u229e> `zip` < <\u229e> `zip` <\u229e> > > > >\n\n    iterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 `Endo A) \u2192 `Endo A\n    iterate\u207f zero    f = id\n    iterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n    _\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 `Endo A) \u2192 `Endo A\n    _\u00b2\u2070 = iterate\u207f 20\n\n    module _ where\n\n      K\u2080 = #\u02b7 0x5A827999\n      K\u2082 = #\u02b7 0x6ED9EBA1\n      K\u2084 = #\u02b7 0x8F1BBCDC\n      K\u2086 = #\u02b7 0xCA62C1D6\n\n      H0 = #\u02b7 0x67452301\n      H1 = #\u02b7 0xEFCDAB89\n      H2 = #\u02b7 0x98BADCFE\n      H3 = #\u02b7 0x10325476\n      H4 = #\u02b7 0xC3D2E1F0\n\n      A B C D E : Word\u2075 `\u2192 Word\n      A = fst\n      B = snd \u204f fst\n      C = snd \u204f snd \u204f fst\n      D = snd \u204f snd \u204f snd \u204f fst\n      E = snd \u204f snd \u204f snd \u204f snd\n\n      F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n      F\u2082 = B `\u2295 C `\u2295 D\n      F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n      F\u2086 = F\u2082\n\n      module _ (F : Word\u2075 `\u2192 Word)\n               (K : `\ud835\udfd9  `\u2192 Word)\n               (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n          where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n      module _ (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n        W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n        W\u2080 = W \u2218 inject+ 60 \u2218 raise  0\n        W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n        W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n        W\u2086 = W \u2218 inject+  0 \u2218 raise 60\n\n        Iteration\u2078\u2070 : `Endo Word\u2075\n        Iteration\u2078\u2070 =\n          (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n          (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n          (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n          (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n\n        pad0s : \u2115 \u2192 \u2115\n        pad0s zero = 512 \u2238 65\n        pad0s (suc _) = STUCK where postulate STUCK : \u2115\n        -- pad0s n = [ 512 - [ n + 65 mod 512 ] mod 512 ]\n\n        paddedLength : \u2115 \u2192 \u2115\n        paddedLength n = n + (1 + pad0s n + 64)\n\n        padding : \u2200 {n} \u2192 `Bits n `\u2192 `Bits (paddedLength n)\n        padding {n} = < id ,tt\u204f <1\u2237 < <0\u207f> {pad0s n} ++ bits 64 n > > > \u204f append\n\n        ite : Endo (`Endo Word\u2075)\n        ite f = dup \u204f first f \u204f <\u229e\u2075>\n\n        hash-block : `Endo Word\u2075\n        hash-block = ite Iteration\u2078\u2070\n\n      ite' : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word)\n             \u2192 ((Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `Endo Word\u2075) \u2192 `Endo Word\u2075\n      ite' zero    W f = id\n      ite' (suc n) W f = f (W zero) \u204f ite' n (W \u2218 suc) f\n\n      SHA1 : \u2200 n (W : Fin n \u2192 Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1 n W = < H0 , H1 , H2 , H3 , H4 >\n               \u204f ite' n W hash-block\n\n      SHA1-on-0s : `\ud835\udfd9 `\u2192 Word\u2075\n      SHA1-on-0s = SHA1 1 (\u03bb _ _ \u2192 <0\u207f>)\n\nmodule AgdaSHA1 where\n  open import FunUniverse.Agda\n  open FunSHA1 agdaFunOps\n  open import Data.Two\n\nopen import IO\nimport IO.Primitive\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Coinduction\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1\u2082 = putStr \"1\"\nputBit 0\u2082 = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\nput\u00d7 : \u2200 {A B : Set} \u2192 (A \u2192 IO \ud835\udfd9) \u2192 (B \u2192 IO \ud835\udfd9) \u2192 (A \u00d7 B) \u2192 IO \ud835\udfd9\nput\u00d7 pA pB (x , y) = \u266f pA x >> \u266f pB y\n{-\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits (put\u00d7 putBits\n          putBits))) AgdaSHA1.SHA1-on-0s)\n-}\nfirstBit : \u2200 {A : Set} \u2192 (V.Vec \ud835\udfda 32 \u00d7 A) \u2192 \ud835\udfda\nfirstBit ((b \u2237 _) , _) = b\nimport FunUniverse.Cost as Cost\nopen import Data.Nat.Show\nsha1-cost : \u2115\nsha1-cost = FunSHA1.SHA1-on-0s Cost.timeOps\nmain : IO.Primitive.IO \ud835\udfd9\n--main = IO.run (putBit (firstBit (AgdaSHA1.SHA1-on-0s _)))\nmain = IO.run (putStrLn (show sha1-cost))\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bfcd2eb5d4fab7bcb698719c3f02fc2aea768cf3","subject":"Add Any predicate for binary tree","message":"Add Any predicate for binary tree\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Tree\/Binary.agda","new_file":"lib\/Data\/Tree\/Binary.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type hiding (\u2605)\n\nopen import Level\nopen import Data.Zero\nopen import Data.Sum\n\nmodule Data.Tree.Binary where\n\ndata BinTree {a} (A : \u2605 a) : \u2605 a where\n  empty : BinTree A\n  leaf  : A \u2192 BinTree A\n  fork  : (\u2113 r : BinTree A) \u2192 BinTree A\n\nAny : \u2200 {a p}{A : \u2605 a}(P : A \u2192 \u2605 p) \u2192 BinTree A \u2192 \u2605 p\nAny P empty = Lift \ud835\udfd8\nAny P (leaf x) = P x\nAny P (fork ts ts\u2081) = Any P ts \u228e Any P ts\u2081\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type hiding (\u2605)\n\nmodule Data.Tree.Binary where\n\ndata BinTree {a} (A : \u2605 a) : \u2605 a where\n  empty : BinTree A\n  leaf  : A \u2192 BinTree A\n  fork  : (\u2113 r : BinTree A) \u2192 BinTree A\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ec00642d7b0541cc7fa6b43dfdafbb6203bd3dfe","subject":"IDesc model: 0wn3d. No type-in-type but a serious Otis lift. Got up to making Desc constructors in Desc.","message":"IDesc model: 0wn3d. No type-in-type but a serious Otis lift. Got up to making Desc constructors in Desc.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\nvarl : (l : Level)(I : Set l)(i : I) -> IDescl l I\nvarl x I i = con (lv , lifter i) \n     where lv : DescDConst (suc x)\n           lv = lvar\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lc , X)\n       where lc : DescDConst (suc x)\n             lc = lconst\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lp , (D , D'))\n      where lp : DescDConst (suc x)\n            lp = lprod\n\n\npil : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\npil x I S T = con (lp , ( S , Tl))\n    where lp : DescDConst (suc x)\n          lp = lpi\n          Tl : Lifted S -> IDescl x I\n          Tl (lifter s) = T s\n\nsigmal : (l : Level)(I : Set l)(S : Set l)(T : S -> IDescl l I) -> IDescl l I\nsigmal x I S T = con (ls , ( S , Tl))\n       where ls : DescDConst (suc x)\n             ls = lsigma\n             Tl : Lifted S -> IDescl x I\n             Tl (lifter s) = T s\n             ","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (l : Level)(I : Set l) -> IDesc l I -> (I -> Set l) -> Set l\ndesc _ I (var i) P = P i\ndesc _ I (const X) P = X\ndesc x I (prod D D') P = desc x I D P * desc x I D' P\ndesc x I (sigma S T) P = Sigma S (\\s -> desc x I (T s) P)\ndesc x I (pi S T) P = (s : S) -> desc x I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (l : Level)(I : Set l)(R : I -> IDesc l I)(i : I) : Set l where\n  con : desc l I (R i) (\\j -> IMu l I R j) -> IMu l I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (l : Level)(I : Set l)(D : IDesc l I)(P : I -> Set l) -> desc l I D P -> IDesc l (Sigma I P)\nbox _ I (var i)     P x        = var (i , x)\nbox _ I (const X)   P x        = const X\nbox x I (prod D D') P (d , d') = prod (box x I D P d) (box x I D' P d')\nbox x I (sigma S T) P (a , b)  = box x I (T a) P b\nbox x I (pi S T)    P f        = pi S (\\s -> box x I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu l I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc l I) -> \n           (xs : desc l I D (IMu l I R)) -> \n           desc l (Sigma I (IMu l I R)) (box l I D (IMu l I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (l : Level)\n            (I : Set l)\n            (R : I -> IDesc l I)\n            (P : Sigma I (IMu l I R) -> Set l)\n            (m : (i : I)\n                 (xs : desc l I (R i) (IMu l I R))\n                 (hs : desc l (Sigma I (IMu l I R)) (box l I (R i) (IMu l I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu l I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst (l : Level) : Set l where\n  lvar   : DescDConst l\n  lconst : DescDConst l\n  lprod  : DescDConst l\n  lpi    : DescDConst l\n  lsigma : DescDConst l\n\ndescDChoice : (l : Level) -> Set l -> DescDConst (suc l) -> IDesc (suc l) Unit\ndescDChoice _ I lvar = const (lift I)\ndescDChoice x _ lconst = const (Set x)\ndescDChoice _ _ lprod = prod (var Void) (var Void)\ndescDChoice x _ lpi = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice x _ lsigma = sigma (Set x) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (l : Level)(I : Set l) -> IDesc (suc l) Unit\ndescD x I = sigma (DescDConst (suc x)) (descDChoice x I)\n\nIDescl : (l : Level)(I : Set l) -> Set (suc l)\nIDescl x I = IMu (suc x) Unit (\\_ -> descD x I) Void\n\n{-\nvarl : (l : Level)(I : Set l) -> IDescl l I\nvarl x I = con (lvar , {!!})\n-}\n\nconstl : (l : Level)(I : Set l)(X : Set l) -> IDescl l I\nconstl x I X = con (lconst , X)\n\nprodl : (l : Level)(I : Set l)(D D' : IDescl l I) -> IDescl l I\nprodl x I D D' = con (lprod , (D , D'))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4fafa6d205a9f1b41fca2dd7b4ccb0eea14d011f","subject":"Finished proving ! is a comonad.","message":"Finished proving ! is a comonad.\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/DC2Sets.agda","new_file":"dialectica-cats\/DC2Sets.agda","new_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = let h = \u03bb r \u2192 fst (f (fst r)) (snd r)\n        H = \u03bb r z \u2192 F (fst r) (snd r , z) , snd (f (fst r)) (snd r) z\n     in h , (H , cond)\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Z} \u2192\n      (\u03b1 \u2297\u1d63 \u03b2) u (F (fst u) (snd u , y) , snd (f (fst u)) (snd u) y) \u2192\n      \u03b3 (fst (f (fst u)) (snd u)) y\n  cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u} {v , z}\n  ... | p\u2084 with f u\n  ... | i , I = p\u2084 p\u2082 p\u2083\n  \ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   aux\u2082 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 (\u03bb z \u2192 fst (F (fst a , snd a) z) , snd (F (fst a , snd a) z)) \u2261 F a\n   aux\u2082 {u , v} = ext-set aux\u2083\n    where\n      aux\u2083 : {a : Z} \u2192 (fst (F (u , v) a) , snd (F (u , v) a)) \u2261 F (u , v) a\n      aux\u2083 {z} with F (u , v) z\n      ... | x , y = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = (ext-set aux) , ext-set (ext-set aux')\n where\n  aux : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb v z \u2192 snd (g a) v z)) \u2261 g a\n  aux {u} with g u\n  ... | i , I = refl\n  aux' : {u : U}{r : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G u (fst r , snd r) \u2261 G u r\n  aux' {u}{v , z} = refl\n\n\n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set} \u2192 (\u03b1 : U \u2192 X \u2192 Set) \u2192 U \u2192 \ud835\udd43 X \u2192 Set  \n!\u2092-cond {U}{X} \u03b1 u [] = \u22a4\n!\u2092-cond {U}{X} \u03b1 u (x :: xs) = (\u03b1 u x) \u00d7 (!\u2092-cond \u03b1 u xs)\n\n!\u2092-cond-++ : \u2200{U X : Set}{\u03b1 : U \u2192 X \u2192 Set}{u : U}{l\u2081 l\u2082 : \ud835\udd43 X}\n  \u2192 !\u2092-cond \u03b1 u (l\u2081 ++ l\u2082) \u2261 ((!\u2092-cond \u03b1 u l\u2081) \u00d7 (!\u2092-cond \u03b1 u l\u2082))\n!\u2092-cond-++ {U}{X}{\u03b1}{u}{[]}{l\u2082} = \u2227-unit\n!\u2092-cond-++ {U}{X}{\u03b1}{u}{x :: xs}{l\u2082} rewrite !\u2092-cond-++ {U}{X}{\u03b1}{u}{xs}{l\u2082} = \u2227-assoc\n\n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U ,  X * , !\u2092-cond \u03b1\n\n!\u2090-s : \u2200{U Y X : Set}\n  \u2192 (U \u2192 Y \u2192 X)\n  \u2192 (U \u2192 Y * \u2192 X *)\n!\u2090-s f u l = map (f u) l\n\n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , (!\u2090-s F , aux)\n where\n   aux : {u : U} {y : \ud835\udd43 Y} \u2192 !\u2092-cond \u03b1 u (!\u2090-s F u y) \u2192 !\u2092-cond \u03b2 (f u) y\n   aux {u}{[]} p\u2081 = triv\n   aux {u}{y :: ys} (p\u2081 , p\u2082) = p p\u2081 , aux p\u2082\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb u x \u2192 [ x ]) , fst\n\n\u03b4-s : {U X : Set} \u2192 U \u2192 \ud835\udd43 (\ud835\udd43 X) \u2192 \ud835\udd43 X\n\u03b4-s u xs = foldr _++_ [] xs\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-s , cond\n where\n   cond : {u : U} {y : \ud835\udd43 (\ud835\udd43 X)} \u2192 !\u2092-cond \u03b1 u (foldr _++_ [] y) \u2192 !\u2092-cond (!\u2092-cond \u03b1) u y\n   cond {u}{[]} p = triv\n   cond {u}{l :: ls} p with !\u2092-cond-++ {U}{X}{\u03b1}{u}{l}{foldr _++_ [] ls}\n   ... | p' rewrite p' with p\n   ... | p\u2082 , p\u2083 = p\u2082 , cond {u}{ls} p\u2083    \n\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} = refl , ext-set (\u03bb {x} \u2192 ext-set (\u03bb {l} \u2192 aux {x} {l}))\n where\n  aux : \u2200{x : U}{l : \ud835\udd43 (\ud835\udd43 (\ud835\udd43 X))}\n    \u2192 foldr _++_ [] (!\u2090-s (\u03bb u xs\n    \u2192 foldr _++_ [] xs) x l) \u2261 foldr _++_ [] (foldr _++_ [] l)\n  aux {u}{[]} = refl\n  aux {u}{x :: xs} rewrite aux {u}{xs} = foldr-append {_}{_}{X}{X}{x}{foldr _++_ [] xs}\n\n\nfoldr-map : \u2200{\u2113}{A : Set \u2113}{l : \ud835\udd43 A} \u2192 l \u2261 foldr _++_ [] (map (\u03bb x\u2081 \u2192 x\u2081 :: []) l)\nfoldr-map {_}{_}{[]} = refl\nfoldr-map {\u2113}{A}{x :: xs} rewrite sym (foldr-map {\u2113}{A}{xs}) = refl\n\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {u} \u2192 ext-set (\u03bb {l} \u2192 aux {l}))\n  where\n    aux : \u2200{a : \ud835\udd43 X} \u2192 a ++ [] \u2261 foldr _++_ [] (map (\u03bb x \u2192 x :: []) a)\n    aux {[]} = refl\n    aux {x :: xs} rewrite (++[] xs) | sym (foldr-map {_}{X}{xs}) = refl    \n","old_contents":"-----------------------------------------------------------------------\n-- This file defines DC\u2082(Sets) and its SMC structure.              --\n-----------------------------------------------------------------------\nmodule DC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (U \u2192 X \u2192 Set))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , \u03b1) (V , Y , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)} (f , F , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , \u03b1)}{(V , Y , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , \u03b1}{V , Y , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , \u03b1}{V , Y , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , \u03b1}\n         {V , Y , \u03b2}\n         {W , Z , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{S , T , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _}{V , Y , _}{f , F , _} = refl , refl\n\n\n-----------------------------------------------------------------------\n-- DC\u2082(Sets) is a SMC                                              --\n-----------------------------------------------------------------------\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , \u03b1) \u2297\u2092 (V , Y , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\n-- The unit for tensor:\n\u03b9 : \u22a4 \u2192 \u22a4 \u2192 Set\n\u03b9 triv triv = \u22a4\n\nI : Obj\nI = (\u22a4 , \u22a4 , \u03b9)\n\nJ : Obj\nJ = (\u22a4 , \u22a4 , (\u03bb x y \u2192 \u22a5))\n\n\n-- The left-unitor:\n\u03bb\u2297-p : \u2200{U X \u03b1}{u : \u03a3 \u22a4 (\u03bb x \u2192 U)} {y : X} \u2192 (\u03b9 \u2297\u1d63 \u03b1) u (triv , y) \u2192 \u03b1 (snd u) y\n\u03bb\u2297-p {U}{X}{\u03b1}{(triv , u)}{x} = snd\n   \n\u03bb\u2297 : \u2200{A : Obj} \u2192 Hom (I \u2297\u2092 A) A\n\u03bb\u2297 {(U , X , \u03b1)} = snd , (\u03bb _ x \u2192 triv , x) , \u03bb\u2297-p\n\n\u03bb\u2297-inv : \u2200{A : Obj} \u2192 Hom A (I \u2297\u2092 A)\n\u03bb\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 triv , u) , (\u03bb _ r \u2192 snd r) , \u03bb\u2297-inv-p\n where\n  \u03bb\u2297-inv-p : \u2200{U X \u03b1}{u : U} {y : \u03a3 \u22a4 (\u03bb x \u2192 X)} \u2192 \u03b1 u (snd y) \u2192 (\u03b9 \u2297\u1d63 \u03b1) (triv , u) y\n  \u03bb\u2297-inv-p {U}{X}{\u03b1}{u}{triv , x} p = triv , p\n\n-- The right-unitor:\n\u03c1\u2297 : \u2200{A : Obj} \u2192 Hom (A \u2297\u2092 I) A\n\u03c1\u2297 {(U , X , \u03b1)} = fst , (\u03bb r x \u2192 x , triv) , \u03c1\u2297-p\n where\n  \u03c1\u2297-p : \u2200{U X \u03b1}{u : \u03a3 U (\u03bb x \u2192 \u22a4)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b9) u (y , triv) \u2192 \u03b1 (fst u) y\n  \u03c1\u2297-p {U}{X}{\u03b1}{(u , _)}{x} (p , _) = p\n\n\n\u03c1\u2297-inv : \u2200{A : Obj} \u2192 Hom A (A \u2297\u2092 I)\n\u03c1\u2297-inv {(U , X , \u03b1)} = (\u03bb u \u2192 u , triv) , (\u03bb u r \u2192 fst r) , \u03c1\u2297-p-inv\n where\n  \u03c1\u2297-p-inv : \u2200{U X \u03b1}{u : U} {y : \u03a3 X (\u03bb x \u2192 \u22a4)} \u2192 \u03b1 u (fst y) \u2192 (\u03b1 \u2297\u1d63 \u03b9) (u , triv) y\n  \u03c1\u2297-p-inv {U}{X}{\u03b1}{u}{x , triv} p = p , triv\n\n-- Symmetry:\n\u03b2\u2297 : \u2200{A B : Obj} \u2192 Hom (A \u2297\u2092 B) (B \u2297\u2092 A)\n\u03b2\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)} = twist-\u00d7 , (\u03bb r\u2081 r\u2082 \u2192 twist-\u00d7 r\u2082) , \u03b2\u2297-p\n where\n   \u03b2\u2297-p : \u2200{U V Y X \u03b1 \u03b2}{u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Y (\u03bb x \u2192 X)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (twist-\u00d7 y) \u2192 (\u03b2 \u2297\u1d63 \u03b1) (twist-\u00d7 u) y\n   \u03b2\u2297-p {U}{V}{Y}{X}{\u03b1}{\u03b2}{u , v}{y , x} = twist-\u00d7\n\n\n-- The associator:\nF\u03b1-inv : \u2200{\u2113}{U V W X Y Z : Set \u2113} \u2192 (U \u00d7 (V \u00d7 W)) \u2192 ((X \u00d7 Y) \u00d7 Z) \u2192 (X \u00d7 (Y \u00d7 Z))\nF\u03b1-inv (u , (v , w)) ((x , y) , z) = x , y , z\n   \n\u03b1\u2297-inv : \u2200{A B C : Obj} \u2192 Hom (A \u2297\u2092 (B \u2297\u2092 C)) ((A \u2297\u2092 B) \u2297\u2092 C)\n\u03b1\u2297-inv {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = rl-assoc-\u00d7 , F\u03b1-inv , \u03b1-inv-cond\n where\n   \u03b1-inv-cond : {u : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))}{y : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)}\n     \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) u (F\u03b1-inv u y)\n     \u2192 ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) (rl-assoc-\u00d7 u) y\n   \u03b1-inv-cond {u , (v , w)}{(x , y) , z} (p\u2081 , (p\u2082 , p\u2083)) = (p\u2081 , p\u2082) , p\u2083\n\n\nF\u03b1 : \u2200{V W X Y U Z : Set} \u2192 ((U \u00d7 V) \u00d7 W) \u2192 (X \u00d7 (Y \u00d7 Z)) \u2192 ((X \u00d7 Y) \u00d7 Z)\nF\u03b1 {V}{W}{X}{Y}{U}{Z} ((u , v) , w) (x , (y , z)) = (x , y) , z\n\n\u03b1\u2297 : \u2200{A B C : Obj} \u2192 Hom ((A \u2297\u2092 B) \u2297\u2092 C) (A \u2297\u2092 (B \u2297\u2092 C)) \n\u03b1\u2297 {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)} = (lr-assoc-\u00d7 , F\u03b1 , \u03b1-cond)\n where\n  \u03b1-cond : {u : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)}\n      {y : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192\n      ((\u03b1 \u2297\u1d63 \u03b2) \u2297\u1d63 \u03b3) u (F\u03b1 u y) \u2192 (\u03b1 \u2297\u1d63 (\u03b2 \u2297\u1d63 \u03b3)) (lr-assoc-\u00d7 u) y\n  \u03b1-cond {(u , v) , w}{x , (y , z)} ((p\u2081 , p\u2082) , p\u2083) = p\u2081 , p\u2082 , p\u2083\n\n\u03b1\u2297-id\u2081 : \u2200{A B C} \u2192 (\u03b1\u2297 {A}{B}{C}) \u25cb \u03b1\u2297-inv \u2261h id\n\u03b1\u2297-id\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 rl-assoc-\u00d7 (lr-assoc-\u00d7 a) \u2261 a\n   aux {(u , v) , w} = refl\n\n   aux' : {a : \u03a3 (\u03a3 U (\u03bb x \u2192 V)) (\u03bb x \u2192 W)} \u2192 (\u03bb z \u2192 F\u03b1 {V}{W}{X}{Y}{U}{Z} a (F\u03b1-inv (lr-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {(u , v), w} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 (\u03a3 X (\u03bb x \u2192 Y)) (\u03bb x \u2192 Z)} \u2192 F\u03b1 ((u , v) , w) (F\u03b1-inv (u , v , w) a) \u2261 a\n      aux'' {(x , y) , z} = refl\n\n\u03b1\u2297-id\u2082 : \u2200{A B C} \u2192 (\u03b1\u2297-inv {A}{B}{C}) \u25cb \u03b1\u2297 \u2261h id\n\u03b1\u2297-id\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} = ext-set aux , ext-set aux'\n where\n   aux : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 lr-assoc-\u00d7 (rl-assoc-\u00d7 a) \u2261 a\n   aux {u , (v , w)} = refl\n   aux' : {a : \u03a3 U (\u03bb x \u2192 \u03a3 V (\u03bb x\u2081 \u2192 W))} \u2192 (\u03bb z \u2192 F\u03b1-inv {_}{U}{V}{W}{X}{Y}{Z} a (F\u03b1 (rl-assoc-\u00d7 a) z)) \u2261 (\u03bb y \u2192 y)\n   aux' {u , (v , w)} = ext-set aux''\n    where\n      aux'' : {a : \u03a3 X (\u03bb x \u2192 \u03a3 Y (\u03bb x\u2081 \u2192 Z))} \u2192 F\u03b1-inv (u , v , w) (F\u03b1 ((u , v) , w) a) \u2261 a\n      aux'' {x , (y , z)} = refl\n       \n-- Internal hom:\n\u22b8-cond : \u2200{U V X Y : Set}{\u03b1 : U \u2192 X \u2192 Set}{\u03b2 : V \u2192 Y \u2192 Set}\n  \u2192 \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)\n  \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  \u2192 Set\n\u22b8-cond {\u03b1 = \u03b1}{\u03b2} (f , F) (u , y) = \u03b1 u (F u y) \u2192 \u03b2 (f u) y\n\n_\u22b8\u2092_ : Obj \u2192 Obj \u2192 Obj\n(U , X , \u03b1) \u22b8\u2092 (V , Y , \u03b2) = ((U \u2192 V) \u00d7 (U \u2192 Y \u2192 X)) , ((U \u00d7 Y) , \u22b8-cond {\u03b1 = \u03b1}{\u03b2})\n\n\n_\u22b8\u2090_ : {A B C D : Obj} \u2192 Hom C A \u2192 Hom B D \u2192 Hom (A \u22b8\u2092 B) (C \u22b8\u2092 D)\n_\u22b8\u2090_ {(U , X , \u03b1)}{(V , Y , \u03b2)}{(W , Z , \u03b3)}{(S , T , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) =\n  h , H , cond\n where\n  h : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 (W \u2192 S) (\u03bb x \u2192 W \u2192 T \u2192 Z)\n  h (i , I) = (\u03bb w \u2192 g (i (f w))) , (\u03bb w t \u2192 F w (I (f w) (G (i (f w)) t)))\n  H : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X) \u2192 \u03a3 W (\u03bb x \u2192 T) \u2192 \u03a3 U (\u03bb x \u2192 Y)\n  H (i , I) (w , t) = f w , G (i (f w)) t\n  cond : {u : \u03a3 (U \u2192 V) (\u03bb x \u2192 U \u2192 Y \u2192 X)} {y : \u03a3 W (\u03bb x \u2192 T)} \u2192 \u22b8-cond {\u03b1 = \u03b1}{\u03b2} u (H u y) \u2192 \u22b8-cond {\u03b1 = \u03b3}{\u03b4} (h u) y\n  cond {i , I}{w , y} p\u2083 p\u2084 = p\u2082 (p\u2083 (p\u2081 p\u2084))\n\ncur : {A B C : Obj}\n  \u2192 Hom (A \u2297\u2092 B) C\n  \u2192 Hom A (B \u22b8\u2092 C)\ncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = (\u03bb u \u2192 (\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) , (\u03bb u r \u2192 fst (F (u , (fst r)) (snd r))) , cond \n where\n   cond : {u : U} {y : \u03a3 V (\u03bb x \u2192 Z)}\n     \u2192 \u03b1 u (fst (F (u , fst y) (snd y)))\n     \u2192 \u22b8-cond {\u03b1 = \u03b2}{\u03b3} ((\u03bb v \u2192 f (u , v)) , (\u03bb v z \u2192 snd (F (u , v) z))) y   \n   cond {u}{v , z} p\u2082 p\u2083 with (p\u2081 {u , v}{z})\n   ... | p\u2084 with F (u , v) z\n   ... | (x , y) = p\u2084 (p\u2082 , p\u2083)\n   \n\ncur-\u2261h : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-\u2261h {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}\n       {f\u2081 , F\u2081 , p\u2081}{f\u2082 , F\u2082 , p\u2082} (p\u2083 , p\u2084)\n       rewrite p\u2083 | p\u2084 = refl , refl\n\ncur-cong : \u2200{A B C}{f\u2081 f\u2082 : Hom (A \u2297\u2092 B) C} \u2192 f\u2081 \u2261h f\u2082 \u2192 cur f\u2081 \u2261h cur f\u2082\ncur-cong {(U , X , \u03b1)} {(V , Y , \u03b2)} {(W , Z , \u03b3)}{f\u2081 , F\u2081 , _}{f\u2082 , F\u2082 , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\nuncur : {A B C : Obj}\n  \u2192 Hom A (B \u22b8\u2092 C)\n  \u2192 Hom (A \u2297\u2092 B) C\nuncur {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3} (f , F , p\u2081)\n  = let h = \u03bb r \u2192 fst (f (fst r)) (snd r)\n        H = \u03bb r z \u2192 F (fst r) (snd r , z) , snd (f (fst r)) (snd r) z\n     in h , (H , cond)\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Z} \u2192\n      (\u03b1 \u2297\u1d63 \u03b2) u (F (fst u) (snd u , y) , snd (f (fst u)) (snd u) y) \u2192\n      \u03b3 (fst (f (fst u)) (snd u)) y\n  cond {u , v}{z} (p\u2082 , p\u2083) with p\u2081 {u} {v , z}\n  ... | p\u2084 with f u\n  ... | i , I = p\u2084 p\u2082 p\u2083\n  \ncur-uncur-bij\u2081 : \u2200{A B C}{f : Hom (A \u2297\u2092 B) C}\n  \u2192 uncur (cur f) \u2261h f\ncur-uncur-bij\u2081 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{f , F , p\u2081} = ext-set aux\u2081 , ext-set aux\u2082\n where\n   aux\u2081 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 f (fst a , snd a) \u2261 f a\n   aux\u2081 {u , v} = refl\n   aux\u2082 : {a : \u03a3 U (\u03bb x \u2192 V)} \u2192 (\u03bb z \u2192 fst (F (fst a , snd a) z) , snd (F (fst a , snd a) z)) \u2261 F a\n   aux\u2082 {u , v} = ext-set aux\u2083\n    where\n      aux\u2083 : {a : Z} \u2192 (fst (F (u , v) a) , snd (F (u , v) a)) \u2261 F (u , v) a\n      aux\u2083 {z} with F (u , v) z\n      ... | x , y = refl\n\ncur-uncur-bij\u2082 : \u2200{A B C}{g : Hom A (B \u22b8\u2092 C)}\n  \u2192 cur (uncur g) \u2261h g\ncur-uncur-bij\u2082 {U , X , \u03b1}{V , Y , \u03b2}{W , Z , \u03b3}{g , G , p\u2081} = (ext-set aux) , ext-set (ext-set aux')\n where\n  aux : {a : U} \u2192 ((\u03bb v \u2192 fst (g a) v) , (\u03bb v z \u2192 snd (g a) v z)) \u2261 g a\n  aux {u} with g u\n  ... | i , I = refl\n  aux' : {u : U}{r : \u03a3 V (\u03bb x \u2192 Z)} \u2192 G u (fst r , snd r) \u2261 G u r\n  aux' {u}{v , z} = refl\n\n\n-- The of-course exponential:\n!\u2092-cond : \u2200{U X : Set} \u2192 (\u03b1 : U \u2192 X \u2192 Set) \u2192 U \u2192 \ud835\udd43 X \u2192 Set  \n!\u2092-cond {U}{X} \u03b1 u [] = \u22a4\n!\u2092-cond {U}{X} \u03b1 u (x :: xs) = (\u03b1 u x) \u00d7 (!\u2092-cond \u03b1 u xs)\n\n!\u2092-cond-++ : \u2200{U X : Set}{\u03b1 : U \u2192 X \u2192 Set}{u : U}{l\u2081 l\u2082 : \ud835\udd43 X}\n  \u2192 !\u2092-cond \u03b1 u (l\u2081 ++ l\u2082) \u2261 ((!\u2092-cond \u03b1 u l\u2081) \u00d7 (!\u2092-cond \u03b1 u l\u2082))\n!\u2092-cond-++ {U}{X}{\u03b1}{u}{[]}{l\u2082} = \u2227-unit\n!\u2092-cond-++ {U}{X}{\u03b1}{u}{x :: xs}{l\u2082} rewrite !\u2092-cond-++ {U}{X}{\u03b1}{u}{xs}{l\u2082} = \u2227-assoc\n\n!\u2092 : Obj \u2192 Obj\n!\u2092 (U , X , \u03b1) = U ,  X * , !\u2092-cond \u03b1\n\n!\u2090-s : \u2200{U Y X : Set}\n  \u2192 (U \u2192 Y \u2192 X)\n  \u2192 (U \u2192 Y * \u2192 X *)\n!\u2090-s f u [] = []\n!\u2090-s f u (y :: ys) = f u y :: !\u2090-s f u ys       \n\n!\u2090 : {A B : Obj} \u2192 Hom A B \u2192 Hom (!\u2092 A) (!\u2092 B)\n!\u2090 {U , X , \u03b1}{V , Y , \u03b2} (f , F , p) = f , (!\u2090-s F , aux)\n where\n   aux : {u : U} {y : \ud835\udd43 Y} \u2192 !\u2092-cond \u03b1 u (!\u2090-s F u y) \u2192 !\u2092-cond \u03b2 (f u) y\n   aux {u}{[]} p\u2081 = triv\n   aux {u}{y :: ys} (p\u2081 , p\u2082) = p p\u2081 , aux p\u2082\n\n-- Of-course is a comonad:\n\u03b5 : \u2200{A} \u2192 Hom (!\u2092 A) A\n\u03b5 {U , X , \u03b1} = id-set , (\u03bb u x \u2192 [ x ]) , fst\n\n\u03b4-s : {U X : Set} \u2192 U \u2192 \ud835\udd43 (\ud835\udd43 X) \u2192 \ud835\udd43 X\n\u03b4-s u xs = foldr _++_ [] xs\n  \n\u03b4 : \u2200{A} \u2192 Hom (!\u2092 A) (!\u2092 (!\u2092 A))\n\u03b4 {U , X , \u03b1} = id-set , \u03b4-s , cond\n where\n   cond : {u : U} {y : \ud835\udd43 (\ud835\udd43 X)} \u2192 !\u2092-cond \u03b1 u (foldr _++_ [] y) \u2192 !\u2092-cond (!\u2092-cond \u03b1) u y\n   cond {u}{[]} p = triv\n   cond {u}{l :: ls} p with !\u2092-cond-++ {U}{X}{\u03b1}{u}{l}{foldr _++_ [] ls}\n   ... | p' rewrite p' with p\n   ... | p\u2082 , p\u2083 = p\u2082 , cond {u}{ls} p\u2083    \n\n\n-- These diagrams can be found on page 22 of the report.\ncomonand-diag\u2081 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (!\u2090 (\u03b4 {A})) \u2261h (\u03b4 {A}) \u25cb (\u03b4 { !\u2092 A})\ncomonand-diag\u2081 {U , X , \u03b1} = refl , ext-set (\u03bb {x} \u2192 ext-set (\u03bb {l} \u2192 aux {x} {l}))\n where\n  aux : \u2200{x : U}{l : \ud835\udd43 (\ud835\udd43 (\ud835\udd43 X))}\n    \u2192 foldr _++_ [] (!\u2090-s (\u03bb u xs\n    \u2192 foldr _++_ [] xs) x l) \u2261 foldr _++_ [] (foldr _++_ [] l)\n  aux {u}{[]} = refl\n  aux {u}{x :: xs} rewrite aux {u}{xs} = foldr-append {_}{_}{X}{X}{x}{foldr _++_ [] xs}\n\n{-\ncomonand-diag\u2082 : \u2200{A}\n  \u2192 (\u03b4 {A}) \u25cb (\u03b5 { !\u2092 A}) \u2261h (\u03b4 {A}) \u25cb (!\u2090 (\u03b5 {A}))\ncomonand-diag\u2082 {U , X , \u03b1} =\n  refl , ext-set (\u03bb {f} \u2192 ext-set (\u03bb {a} \u2192 aux {a}{f a}))\n where\n  aux : \u2200{a : U}{l : X *}\n    \u2192 l ++ [] \u2261 foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x y \u2192 x :: []) l)\n  aux {a}{[]} = refl\n  aux {a}{x :: l} with aux {a}{l}\n  ... | IH rewrite ++[] l =\n    cong2 {a = x} {x} {l}\n          {foldr (\u03bb f\u2081 \u2192 _++_ (f\u2081 a)) [] (map (\u03bb x\u2081 y \u2192 x\u2081 :: []) l)} _::_ refl\n          IH\n          \nmodule Cartesian where\n  \u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (U , X , \u03b1))\n  \u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2} = fst , (\u03bb f \u2192 (\u03bb v u \u2192 f u) , (\u03bb u v \u2192 [])) , \u03c0\u2081-cond\n    where\n      \u03c0\u2081-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : U \u2192 \ud835\udd43 X} \u2192\n        ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n        (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n        u ((\u03bb v u\u2081 \u2192 y u\u2081) , (\u03bb u\u2081 v \u2192 [])) \u2192\n        all-pred (\u03b1 (fst u)) (y (fst u))\n      \u03c0\u2081-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2081\n\n  \u03c0\u2082 : {U X V Y : Set}\n      \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n      \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n      \u2192 Hom ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2))) (!\u2092 (V , Y , \u03b2))\n  \u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2} = snd , (\u03bb f \u2192 (\u03bb v u \u2192 []) , (\u03bb u v \u2192 f v)) , \u03c0\u2082-cond\n      where\n        \u03c0\u2082-cond : \u2200{u : \u03a3 U (\u03bb x \u2192 V)} {y : V \u2192 \ud835\udd43 Y} \u2192\n          ((\u03bb u\u2081 f \u2192 all-pred (\u03b1 u\u2081) (f u\u2081)) \u2297\u1d63\n            (\u03bb u\u2081 f \u2192 all-pred (\u03b2 u\u2081) (f u\u2081)))\n              u ((\u03bb v u\u2081 \u2192 []) , (\u03bb u\u2081 v \u2192 y v)) \u2192\n            all-pred (\u03b2 (snd u)) (y (snd u))\n        \u03c0\u2082-cond {u , v}{f} (p\u2081 , p\u2082) = p\u2082\n\n  cart-ar-crt : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y) \u2192 W \u2192 \ud835\udd43 Z\n  cart-ar-crt  (f , F , p\u2081) (g , G , p\u2082) (j\u2081 , j\u2082) w = F (j\u2081 (g w)) w ++ G (j\u2082 (f w)) w\n\n  cart-ar : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (U , X , \u03b1))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) (!\u2092 (V , Y , \u03b2))\n    \u2192 Hom (!\u2092 (W , Z , \u03b3)) ((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))\n  cart-ar {U}{X}{V}{Y}{W}{Z}{\u03b1}{\u03b2}{\u03b3} (f , F , p\u2081) (g , G , p\u2082)\n    = (\u03bb w \u2192 f w , g w) , cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) , cart-ar-cond\n      where\n        cart-ar-cond : \u2200{u : W} {y : \u03a3 (V \u2192 U \u2192 \ud835\udd43 X) (\u03bb x \u2192 U \u2192 V \u2192 \ud835\udd43 Y)} \u2192\n          all-pred (\u03b3 u) (cart-ar-crt {\u03b1 = \u03b1}{\u03b2} (f , F , p\u2081) (g , G , p\u2082) y u) \u2192\n          ((\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b1 u\u2081) (f\u2081 u\u2081)) \u2297\u1d63\n          (\u03bb u\u2081 f\u2081 \u2192 all-pred (\u03b2 u\u2081) (f\u2081 u\u2081)))\n          (f u , g u) y\n        cart-ar-cond {w}{j\u2081 , j\u2082} p\n          rewrite\n            all-pred-append {f = \u03b3 w}{F (j\u2081 (g w)) w}{G (j\u2082 (f w)) w} \u2227-unit \u2227-assoc with p\n        ... | (a , b) = p\u2081 a , p\u2082 b\n\n  cart-diag\u2081 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (U , X , \u03b1)}\n      (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (U , X , \u03b1)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2081)\n  cart-diag\u2081 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 sym (++[] (map F (j (f w))))))\n\n  cart-diag\u2082 : {U X V Y W Z : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {\u03b3 : W \u2192 Z \u2192 Set}\n    \u2192 {f : Hom (W , Z , \u03b3) (U , X , \u03b1)}\n    \u2192 {g : Hom (W , Z , \u03b3) (V , Y , \u03b2)}\n    \u2192 _\u2261h_ { !\u2092 (W , Z , \u03b3)}{ !\u2092 (V , Y , \u03b2)}\n      (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g)\n      (comp { !\u2092 (W , Z , \u03b3)}\n            {((!\u2092 (U , X , \u03b1)) \u2297\u2092 (!\u2092 (V , Y , \u03b2)))}\n            { !\u2092 (V , Y , \u03b2)}\n            (cart-ar {\u03b1 = \u03b1}{\u03b2}{\u03b3} (!\u2090 {W , Z , \u03b3}{U , X , \u03b1} f) (!\u2090 {W , Z , \u03b3}{V , Y , \u03b2} g))\n            \u03c0\u2082)\n  cart-diag\u2082 {f = f , F , p\u2081}{g , G , p\u2082}\n    = refl , ext-set (\u03bb {j} \u2192 ext-set (\u03bb {w} \u2192 refl))\n-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9d21ea480115dbdcfc5e286820b16afee440f6bd","subject":"agda: finish comment","message":"agda: finish comment\n\nOld-commit-hash: c7a38d021c3ee43ffa291ebff0475f6fe60291ec\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Evaluation\/Total.agda","new_file":"Denotational\/Evaluation\/Total.agda","new_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\n{-\n\nHere is an example to understand the semantics of \u0394. I will use a\nnamed variable representation for the task.\n\nConsider the typing judgment:\n\n   x: T |- x: T\n\nThus, we have that:\n\n   dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nThanks to weakening, we also have:\n\n   y : S, dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nIn the formalization, we need a proof \u0393\u2032 that the context \u0393\u2081 = dx : \u0394\nT, x: T is a subcontext of \u0393\u2082 = y : S, dx : \u0394 T, x: T. Thus, \u0393\u2032 has\ntype \u0393\u2081 \u227c \u0393\u2082.\n\nNow take the environment:\n\n   \u03c1 = y \u21a6 w, dx \u21a6 dv, x \u21a6 v\n\nSince the semantics of \u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082 is a function from environments\nfor \u0393\u2082 to environments for \u0393\u2081, we have that:\n\n   \u27e6 \u0393\u2032 \u27e7 \u03c1 = dx \u21a6 dv, x \u21a6 v\n\nFrom the definitions of update and ignore, it follows that:\n\n   update (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 dv \u2295 v\n   ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 v\n\nHence, finally, we have that:\n\n   diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nis simply diff (dv \u2295 v) v (or (dv \u2295 v) \u229d v). If dv is a valid change,\nthat's just dv, that is \u27e6 dx \u27e7 \u03c1. In other words, if dv is a valid\nchange then \u27e6 \u0394 {{\u0393\u2032}} x \u27e7 \u03c1 \u2261 \u27e6 dx \u27e7 \u03c1 \u2261 \u27e6 derive-term x \u27e7 \u03c1. This\nfact, generalized for arbitrary terms, is proven formally by\nderive-term-correct.\n\n-}\n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n","old_contents":"module Denotational.Evaluation.Total where\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\n\nopen import Changes\nopen import ChangeContexts\n\n-- TERMS\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 = if \u27e6 t\u2081 \u27e7Term \u03c1 then \u27e6 t\u2082 \u27e7Term \u03c1 else \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 {{\u0393\u2032}} t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\n{-\n\nHere is an example to understand the semantics of \u0394. I will use a\nnamed variable representation for the task.\n\nConsider the typing judgment:\n\n   x: T |- x: T\n\nThus, we have that:\n\n   dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nThanks to weakening, we also have:\n\n   y : S, dx : \u0394 T, x: T |- \u0394 x : \u0394 T\n\nIn the formalization, we need a proof \u0393\u2032 that the context \u0393\u2081 = dx : \u0394\nT, x: T is a subcontext of \u0393\u2082 = y : S, dx : \u0394 T, x: T. Thus, \u0393\u2032 has\ntype \u0393\u2081 \u227c \u0393\u2082.\n\nNow take the environment:\n\n   \u03c1 = y \u21a6 w, dx \u21a6 dv, x \u21a6 v\n\nSince the semantics of \u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082 is a function from environments\nfor \u0393\u2082 to environments for \u0393\u2081, we have that:\n\n   \u27e6 \u0393\u2032 \u27e7 \u03c1 = dx \u21a6 dv, x \u21a6 v\n\nFrom the definitions of update and ignore, it follows that:\n\n   update (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 dv \u2295 v\n   ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1) =  x \u21a6 v\n\nHence, finally, we have that:\n\n   diff (\u27e6 t \u27e7Term (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u27e6 t \u27e7Term (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n\nis simply diff (dv \u2295 v) v (or (dv \u2295 v) \u229d v). If dv is a valid change,\nthat's just dv, that is \u27e6 dx \u27e7 \u03c1. In other words\n\n-}\n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- PROPERTIES of WEAKENING\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (t : Term \u0393\u2081 \u03c4) {\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound (var x) \u03c1 = lift-sound _ x \u03c1\nweaken-sound true \u03c1 = refl\nweaken-sound false \u03c1 = refl\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 with weaken-sound t\u2081 {\u0393\u2032} \u03c1\n... | H with \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1 | \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u2032 \u27e7 \u03c1)\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | true | true = weaken-sound t\u2082 {\u0393\u2032} \u03c1\nweaken-sound (if t\u2081 t\u2082 t\u2083) {\u0393\u2032} \u03c1 | refl | false | false = weaken-sound t\u2083 {\u0393\u2032} \u03c1\nweaken-sound (\u0394 {{\u0393\u2032}} t) {\u0393\u2033} \u03c1 =\n  cong (\u03bb x \u2192 diff (\u27e6 t \u27e7 (update x)) (\u27e6 t \u27e7 (ignore x))) (\u27e6\u27e7-\u227c-trans \u0393\u2032 \u0393\u2033 \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"31ad64d7c2727cc85e7671d5191c6a079a186a79","subject":"minor","message":"minor\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {zero}  _   f = f []\nsearch {suc n} _\u00b7_ f = search _\u00b7_ (f \u2218 0\u2237_) \u00b7 search _\u00b7_ (f \u2218 1\u2237_)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = search _+_ (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Nat.DivMod\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec.NP hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nopen import Data.Bool.NP public using (_xor_)\nopen import Data.Vec.NP public using ([]; _\u2237_; head; tail; replicate)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot\u2257id xs)\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nprivate\n module M {a} {A : Set a} {M : Set a \u2192 Set a} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n\n  where open RawMonad L.monad\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nsearch : \u2200 {n a} {A : Set a} \u2192 (A \u2192 A \u2192 A) \u2192 (Bits n \u2192 A) \u2192 A\nsearch {zero}  _   f = f []\nsearch {suc n} _\u00b7_ f = search {n} _\u00b7_ (f \u2218 0\u2237_) \u00b7 search {n} _\u00b7_ (f \u2218 1\u2237_)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\n#\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n#\u27e8 pred \u27e9 = search _+_ (\u03bb x \u2192 if pred x then 1 else 0)\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Maybe (Bits n)\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2 ^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2 ^ n) (inject+ 0 (toFin xs))\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2 ^ n + to\u2115 xs\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2 ^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2 ^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (take (2 ^ n) (drop (2 ^ n) tbl))\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2 ^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2 ^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_) ++ []\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ []) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2 ^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_) ++ [])\n        | take-++ (2 ^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2 ^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (take (2 ^ n) (drop (2 ^ n) tbl))\n        | take-them-all (2 ^ n) (drop (2 ^ n) tbl)\n        | take-drop-lem (2 ^ n) tbl\n   = refl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2 ^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- sucB-lem : \u2200 {n} x \u2192 (sucB (from\u2115 x)) [mod 2 ^ n ] \u2261 from\u2115 ((suc x) [mod 2 ^ n ])\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 to\u2115 (from\u2115 {n} x) \u2261 x\nto\u2115\u2218from\u2115 zero = {!!}\nto\u2115\u2218from\u2115 (suc x) = {!to\u2115\u2218from\u2115 x!}\n\nto\u2115\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 to\u2115 (fromFin x) \u2261 Fin.to\u2115 x\nto\u2115\u2218fromFin x = {!!}\n\ntoFin\u2218fromFin : \u2200 {n} (x : Fin (2 ^ n)) \u2192 toFin (fromFin x) \u2261 x\ntoFin\u2218fromFin x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b67e66a9d83b416aaf2d8f49be76094429a596f7","subject":"Desc model: rename iso1 to phi.","message":"Desc model: rename iso1 to phi.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> phi (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (phi D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (phi D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : {I : Set} -> IDescl I -> IDesc I\niso1 {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"72a66b24c052e5fa86b6fccb5493ca5a3bd7379d","subject":"Using pure FOTC for the definition of addition.","message":"Using pure FOTC for the definition of addition.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/FOTC\/AddConstant.agda","new_file":"notes\/thesis\/FOTC\/AddConstant.agda","new_contents":"------------------------------------------------------------------------------\n-- Addition as a function constant\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.AddConstant where\n\n------------------------------------------------------------------------------\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_\ninfix  7 _\u2261_\n\npostulate\n  D                      : Set\n  _\u00b7_                    : D \u2192 D \u2192 D\n  true false if          : D\n  zero succ pred iszero  : D\n  _\u2261_                    : D \u2192 D \u2192 Set\n\npostulate\n  if-true  : \u2200 d\u2081 {d\u2082} \u2192 if \u00b7 true  \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2081\n  if-false : \u2200 {d\u2081} d\u2082 \u2192 if \u00b7 false \u00b7 d\u2081 \u00b7 d\u2082 \u2261 d\u2082\n  pred-0   : pred \u00b7 zero \u2261 zero\n  pred-S   : \u2200 n \u2192 pred \u00b7 (succ \u00b7 n) \u2261 n\n  iszero-0 : iszero \u00b7 zero \u2261 true\n  iszero-S : \u2200 n \u2192 iszero \u00b7 (succ \u00b7 n) \u2261 false\n{-# ATP axiom if-true if-false pred-0 pred-S iszero-0 iszero-S #-}\n\npostulate\n  +    : D\n  +-eq : \u2200 m n \u2192 + \u00b7 m \u00b7 n \u2261\n         if \u00b7 (iszero \u00b7 m) \u00b7 n \u00b7 (succ \u00b7 (+ \u00b7 (pred \u00b7 m) \u00b7 n))\n{-# ATP axiom +-eq #-}\n\npostulate\n  +-0x : \u2200 n \u2192 + \u00b7 zero \u00b7 n \u2261 n\n  +-Sx : \u2200 m n \u2192 + \u00b7 (succ \u00b7 m) \u00b7 n \u2261 succ \u00b7 (+ \u00b7 m \u00b7 n)\n{-# ATP prove +-0x #-}\n{-# ATP prove +-Sx #-}\n\n-- $ cd fotc\/notes\/thesis\n-- $ agda -i. -i ~\/fot FOTC\/AddConstant.agda\n-- $ agda2atp -i. -i ~\/fot FOTC\/AddConstant.agda\n-- Proving the conjecture in \/tmp\/FOTC\/AddConstant\/37-43-0x.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture\n-- Proving the conjecture in \/tmp\/FOTC\/AddConstant\/38-43-Sx.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture\n","old_contents":"------------------------------------------------------------------------------\n-- Addition as a function constant\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.AddConstant where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  +    : D\n  +-eq : \u2200 m n \u2192 + \u00b7 m \u00b7 n \u2261 if (iszero\u2081 m) then n else succ\u2081 (+ \u00b7 (pred\u2081 m) \u00b7 n)\n{-# ATP axiom +-eq #-}\n\npostulate\n  +-0x : \u2200 n \u2192 + \u00b7 zero \u00b7 n \u2261 n\n  +-Sx : \u2200 m n \u2192 + \u00b7 (succ\u2081 m) \u00b7 n \u2261 succ\u2081 (+ \u00b7 m \u00b7 n)\n{-# ATP prove +-0x #-}\n{-# ATP prove +-Sx #-}\n\n-- $ cd fotc\/notes\/thesis\n-- $ agda -i. -i ~\/fot FOTC\/AddConstant.agda\n-- $ agda2atp -i. -i ~\/fot FOTC\/AddConstant.agda\n-- Proving the conjecture in \/tmp\/FOTC\/AddConstant\/21-43-Sx.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture\n-- Proving the conjecture in \/tmp\/FOTC\/AddConstant\/20-43-0x.tptp ...\n-- E 1.7 Jun Chiabari proved the conjecture\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5c1ef9d73912d2e59b329ff1866bf20c6dddefc2","subject":"Remove a useless check: the point of \u2124p\u2605 is to not have 0","message":"Remove a useless check: the point of \u2124p\u2605 is to not have 0\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/JS\/BigI\/CyclicGroup.agda","new_file":"Crypto\/JS\/BigI\/CyclicGroup.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import FFI.JS using (Bool; trace-call; _++_)\nopen import FFI.JS.Check\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Raw\nopen import Algebra.Group\n\n-- TODO carry on a primality proof of p\nmodule Crypto.JS.BigI.CyclicGroup (p : BigI) where\n\nabstract\n  \u2124[_]\u2605 : Set\n  \u2124[_]\u2605 = BigI\n\n  private\n    \u2124p\u2605 : Set\n    \u2124p\u2605 = BigI\n\n    mod-p : BigI \u2192 \u2124p\u2605\n    mod-p x = mod x p\n\n  -- There are two ways to go from BigI to \u2124p\u2605: check and mod-p\n  -- Use check for untrusted input data and mod-p for internal\n  -- computation.\n  BigI\u25b9\u2124[_]\u2605 : BigI \u2192 \u2124p\u2605\n  BigI\u25b9\u2124[_]\u2605 = -- trace-call \"BigI\u25b9\u2124[_]\u2605 \"\n    \u03bb x \u2192\n      (check? (x <I p)\n         (\u03bb _ \u2192 \"Not below the modulus: p:\" ++ toString p ++ \" is less than x:\" ++ toString x)\n         (check? (x >I 0I)\n            (\u03bb _ \u2192 \"Should be strictly positive: \" ++ toString x ++ \" <= 0\") x))\n\n  repr : \u2124p\u2605 \u2192 BigI\n  repr x = x\n\n  1# : \u2124p\u2605\n  1# = 1I\n\n  1\/_ : \u2124p\u2605 \u2192 \u2124p\u2605\n  1\/ x = modInv x p\n\n  _^_ : \u2124p\u2605 \u2192 BigI \u2192 \u2124p\u2605\n  x ^ y = modPow x y p\n\n_*_ _\/_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \u2124p\u2605\n\nx * y = mod-p (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\ninstance\n  \u2124[_]\u2605-Eq? : Eq? \u2124p\u2605\n  \u2124[_]\u2605-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\nprod : List \u2124p\u2605 \u2192 \u2124p\u2605\nprod = foldr _*_ 1#\n\nmon-ops : Monoid-Ops \u2124p\u2605\nmon-ops = _*_ , 1#\n\ngrp-ops : Group-Ops \u2124p\u2605\ngrp-ops = mon-ops , 1\/_\n\npostulate\n  grp-struct : Group-Struct grp-ops\n\ngrp : Group \u2124p\u2605\ngrp = grp-ops , grp-struct\n\nmodule grp = Group grp\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Eq\nopen import FFI.JS using (Bool; trace-call; _++_)\nopen import FFI.JS.Check\n  renaming (check      to check?)\n--renaming (warn-check to check?)\n\nopen import FFI.JS.BigI\nopen import Data.List.Base using (List; foldr)\nopen import Data.Two hiding (_==_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Raw\nopen import Algebra.Group\n\n-- TODO carry on a primality proof of p\nmodule Crypto.JS.BigI.CyclicGroup (p : BigI) where\n\nabstract\n  \u2124[_]\u2605 : Set\n  \u2124[_]\u2605 = BigI\n\n  private\n    \u2124p\u2605 : Set\n    \u2124p\u2605 = BigI\n\n    mod-p : BigI \u2192 \u2124p\u2605\n    mod-p x = mod x p\n\n  -- There are two ways to go from BigI to \u2124p\u2605: check and mod-p\n  -- Use check for untrusted input data and mod-p for internal\n  -- computation.\n  BigI\u25b9\u2124[_]\u2605 : BigI \u2192 \u2124p\u2605\n  BigI\u25b9\u2124[_]\u2605 = -- trace-call \"BigI\u25b9\u2124[_]\u2605 \"\n    \u03bb x \u2192\n      (check? (x <I p)\n         (\u03bb _ \u2192 \"Not below the modulus: p:\" ++ toString p ++ \" is less than x:\" ++ toString x)\n         (check? (x >I 0I)\n            (\u03bb _ \u2192 \"Should be strictly positive: \" ++ toString x ++ \" <= 0\") x))\n\n  check-non-zero : \u2124p\u2605 \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check? (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \u2124p\u2605 \u2192 BigI\n  repr x = x\n\n  1# : \u2124p\u2605\n  1# = 1I\n\n  1\/_ : \u2124p\u2605 \u2192 \u2124p\u2605\n  1\/ x = modInv (check-non-zero x) p\n\n  _^_ : \u2124p\u2605 \u2192 BigI \u2192 \u2124p\u2605\n  x ^ y = modPow x y p\n\n_*_ _\/_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \u2124p\u2605\n\nx * y = mod-p (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\ninstance\n  \u2124[_]\u2605-Eq? : Eq? \u2124p\u2605\n  \u2124[_]\u2605-Eq? = record\n    { _==_ = _=='_\n    ; \u2261\u21d2== = \u2261\u21d2=='\n    ; ==\u21d2\u2261 = ==\u21d2\u2261' }\n    where\n      _=='_ : \u2124p\u2605 \u2192 \u2124p\u2605 \u2192 \ud835\udfda\n      x ==' y = equals (repr x) (repr y)\n      postulate\n        \u2261\u21d2==' : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x ==' y)\n        ==\u21d2\u2261' : \u2200 {x y} \u2192 \u2713 (x ==' y) \u2192 x \u2261 y\n\nprod : List \u2124p\u2605 \u2192 \u2124p\u2605\nprod = foldr _*_ 1#\n\nmon-ops : Monoid-Ops \u2124p\u2605\nmon-ops = _*_ , 1#\n\ngrp-ops : Group-Ops \u2124p\u2605\ngrp-ops = mon-ops , 1\/_\n\npostulate\n  grp-struct : Group-Struct grp-ops\n\ngrp : Group \u2124p\u2605\ngrp = grp-ops , grp-struct\n\nmodule grp = Group grp\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e6dc4b68e44d8d340a5227f0ad0a4240a5f086a9","subject":"Attack.Compression: work around the instance argument becoming weaker...","message":"Attack.Compression: work around the instance argument becoming weaker...\n","repos":"crypto-agda\/crypto-agda","old_file":"Attack\/Compression.agda","new_file":"Attack\/Compression.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- Compression can be used an an Oracle to defeat encryption.\n-- Here we show how compressing before encrypting lead to a\n-- NOT semantically secure construction (IND-CPA).\nmodule Attack.Compression where\n\nopen import Type using (\u2605)\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Product\nopen import Data.Zero\nopen import Relation.Binary.PropositionalEquality.NP\n\nimport Game.IND-CPA\n\nrecord Sized (A : \u2605) : \u2605 where\n  field\n    size  : A \u2192 \u2115\n\nopen Sized {{...}}\n\nmodule EqSized {A B : \u2605} {{_ : Sized A}} {{_ : Sized B}} where\n    -- Same size\n    _==\u02e2_ : A \u2192 B \u2192 \ud835\udfda\n    x ==\u02e2 y = size x == size y\n\n    -- Same size\n    _\u2261\u02e2_ : A \u2192 B \u2192 \u2605\n    x \u2261\u02e2 y = size x \u2261 size y\n\n    \u2261\u02e2\u2192==\u02e2 : \u2200 {x y} \u2192 x \u2261\u02e2 y \u2192 (x ==\u02e2 y) \u2261 1\u2082\n    \u2261\u02e2\u2192==\u02e2 {x} {y} x\u2261\u02e2y rewrite x\u2261\u02e2y = \u2713\u2192\u2261 (==.refl {size y})\n\n    ==\u02e2\u2192\u2261\u02e2 : \u2200 {x y} \u2192 (x ==\u02e2 y) \u2261 1\u2082 \u2192 x \u2261\u02e2 y\n    ==\u02e2\u2192\u2261\u02e2 p = ==.sound _ _ (\u2261\u2192\u2713 p)\n\nmodule EncSized\n         {PubKey Message CipherText R\u2091 : \u2605}\n         (enc  : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n         {{_ : Sized Message}}\n         {{_ : Sized CipherText}}\n  where\n    open EqSized\n\n    -- Encryption size is independant of the randomness\n    EncSizeRndInd =\n      \u2200 {pk m r\u2080 r\u2081} \u2192 enc pk m r\u2080 \u2261\u02e2 enc pk m r\u2081\n\n    -- Encrypted ciphertexts of the same size, will lead to messages of the same size\n    EncLeakSize =\n      \u2200 {pk m\u2080 m\u2081 r\u2080 r\u2081} \u2192 enc pk m\u2080 r\u2080 \u2261\u02e2 enc pk m\u2081 r\u2081 \u2192 m\u2080 \u2261\u02e2 m\u2081\n\nmodule M\n  {Message CompressedMessage : \u2605}\n  {{_ : Sized CompressedMessage}}\n\n  (compress : Message \u2192 CompressedMessage)\n\n  -- 2 messages which have different size after compression\n  (m\u2080 m\u2081 : Message)\n  (different-compression\n     : size (compress m\u2080) \u2262 size (compress m\u2081))\n\n  (PubKey     : \u2605)\n  (SecKey     : \u2605)\n  (CipherText : \u2605)\n  {{_ : Sized CipherText}}\n  (R\u2091 R\u2096 R\u2093 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (enc : PubKey \u2192 CompressedMessage \u2192 R\u2091 \u2192 CipherText)\n  (open EncSized enc)\n  (encSizeRndInd : EncSizeRndInd)\n  (encLeakSize : EncLeakSize)\n  where\n\n  -- Our adversary runs one encryption\n  R\u2090 = R\u2091\n\n  CEnc : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\n  CEnc pk m r\u2091 = enc pk (compress m) r\u2091\n\n  module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText\n                                R\u2091 R\u2096 R\u2090 R\u2093 KeyGen CEnc\n  open IND-CPA.Adversary\n  open EqSized {CipherText}{CipherText} {{it}} {{it}}\n\n  A : IND-CPA.Adversary\n  m  A = \u03bb _ _ \u2192 [0: m\u2080 1: m\u2081 ]\n  b\u2032 A = \u03bb r\u2091 pk c \u2192 c ==\u02e2 CEnc pk m\u2081 r\u2091\n\n  -- The adversary A is always winning.\n  A-always-wins : \u2200 b r \u2192 IND-CPA.EXP b A r \u2261 b\n  A-always-wins 0\u2082 _ = \u22621\u2192\u22610 (different-compression \u2218\u2032 encLeakSize \u2218\u2032 ==\u02e2\u2192\u2261\u02e2)\n  A-always-wins 1\u2082 _ = \u2261\u02e2\u2192==\u02e2 encSizeRndInd\n\n  -- One should be able to derive this one from A-always-wins and the game-flipping general lemma in the exploration lib\n  {-\n  A-always-wins' : \u2200 r \u2192 IND-CPA.game A r \u2261 1\u2082\n  A-always-wins' (0\u2082 , r) = {!lem (not (IND-CPA.EXP 0\u2082 {!A!} r)) (IND-CPA.EXP 1\u2082 A r) (A-always-wins 0\u2082 r)!}\n    where\n    lem : \u2200 x y \u2192 (x ==\u1d47 y) \u2261 0\u2082 \u2192 not (x ==\u1d47 y) \u2261 1\u2082\n    lem 1\u2082 1\u2082 = \u03bb ()\n    lem 1\u2082 0\u2082 = \u03bb _ \u2192 refl\n    lem 0\u2082 1\u2082 = \u03bb _ \u2192 refl\n    lem 0\u2082 0\u2082 = \u03bb ()\n  A-always-wins' (1\u2082 , r) = A-always-wins 1\u2082 r\n  -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- Compression can be used an an Oracle to defeat encryption.\n-- Here we show how compressing before encrypting lead to a\n-- NOT semantically secure construction (IND-CPA).\nmodule Attack.Compression where\n\nopen import Type using (\u2605)\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Two renaming (_==_ to _==\u1d47_)\nopen import Data.Product\nopen import Data.Zero\nopen import Relation.Binary.PropositionalEquality.NP\n\nimport Game.IND-CPA\n\nrecord Sized (A : \u2605) : \u2605 where\n  field\n    size  : A \u2192 \u2115\n\nopen Sized {{...}}\n\nmodule _ {A B} {{_ : Sized A}} {{_ : Sized B}} where\n    -- Same size\n    _==\u02e2_ : A \u2192 B \u2192 \ud835\udfda\n    x ==\u02e2 y = size x == size y\n\n    -- Same size\n    _\u2261\u02e2_ : A \u2192 B \u2192 \u2605\n    x \u2261\u02e2 y = size x \u2261 size y\n\n    \u2261\u02e2\u2192==\u02e2 : \u2200 {x y} \u2192 x \u2261\u02e2 y \u2192 (x ==\u02e2 y) \u2261 1\u2082\n    \u2261\u02e2\u2192==\u02e2 {x} {y} x\u2261\u02e2y rewrite x\u2261\u02e2y = \u2713\u2192\u2261 (==.refl {size y})\n\n    ==\u02e2\u2192\u2261\u02e2 : \u2200 {x y} \u2192 (x ==\u02e2 y) \u2261 1\u2082 \u2192 x \u2261\u02e2 y\n    ==\u02e2\u2192\u2261\u02e2 p = ==.sound _ _ (\u2261\u2192\u2713 p)\n\nmodule _ {PubKey Message CipherText R\u2091 : \u2605}\n         (Enc  : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n         {{_ : Sized Message}}\n         {{_ : Sized CipherText}}\n  where\n    -- Encryption size is independant of the randomness\n    EncSizeRndInd =\n      \u2200 {pk m r\u2080 r\u2081} \u2192 Enc pk m r\u2080 \u2261\u02e2 Enc pk m r\u2081\n\n    -- Encrypted ciphertexts of the same size, will lead to messages of the same size\n    EncLeakSize =\n      \u2200 {pk m\u2080 m\u2081 r\u2080 r\u2081} \u2192 Enc pk m\u2080 r\u2080 \u2261\u02e2 Enc pk m\u2081 r\u2081 \u2192 m\u2080 \u2261\u02e2 m\u2081\n\nmodule M\n  {Message CompressedMessage : \u2605}\n  {{_ : Sized CompressedMessage}}\n\n  (compress : Message \u2192 CompressedMessage)\n\n  -- 2 messages which have different size after compression\n  (m\u2080 m\u2081 : Message)\n  (different-compression\n     : size (compress m\u2080) \u2262 size (compress m\u2081))\n\n  (PubKey     : \u2605)\n  (SecKey     : \u2605)\n  (CipherText : \u2605)\n  {{_ : Sized CipherText}}\n  (R\u2091 R\u2096 R\u2093 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc : PubKey \u2192 CompressedMessage \u2192 R\u2091 \u2192 CipherText)\n  (EncSizeRndInd : EncSizeRndInd Enc)\n  (EncLeakSize : EncLeakSize Enc)\n  where\n\n  -- Our adversary runs one encryption\n  R\u2090 = R\u2091\n\n  CEnc : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\n  CEnc pk m r\u2091 = Enc pk (compress m) r\u2091\n\n  module IND-CPA = Game.IND-CPA PubKey SecKey Message CipherText\n                                R\u2091 R\u2096 R\u2090 R\u2093 KeyGen CEnc\n  open IND-CPA.Adversary\n\n  A : IND-CPA.Adversary\n  m  A = \u03bb _ _ \u2192 [0: m\u2080 1: m\u2081 ]\n  b\u2032 A = \u03bb r\u2091 pk c \u2192 c ==\u02e2 CEnc pk m\u2081 r\u2091\n\n  -- The adversary A is always winning.\n  A-always-wins : \u2200 b r \u2192 IND-CPA.EXP b A r \u2261 b\n  A-always-wins 0\u2082 _ = \u22621\u2192\u22610 (different-compression \u2218 EncLeakSize \u2218 ==\u02e2\u2192\u2261\u02e2)\n  A-always-wins 1\u2082 _ = \u2261\u02e2\u2192==\u02e2 EncSizeRndInd\n\n  lem : \u2200 x y \u2192 (x ==\u1d47 y) \u2261 0\u2082 \u2192 not (x ==\u1d47 y) \u2261 1\u2082\n  lem 1\u2082 1\u2082 = \u03bb ()\n  lem 1\u2082 0\u2082 = \u03bb _ \u2192 refl\n  lem 0\u2082 1\u2082 = \u03bb _ \u2192 refl\n  lem 0\u2082 0\u2082 = \u03bb ()\n\n  {-\n  A-always-wins' : \u2200 r \u2192 IND-CPA.game A r \u2261 1\u2082\n  A-always-wins' (0\u2082 , r) = {!lem (not (IND-CPA.EXP 0\u2082 {!A!} r)) (IND-CPA.EXP 1\u2082 A r) (A-always-wins 0\u2082 r)!}\n  A-always-wins' (1\u2082 , r) = A-always-wins 1\u2082 r\n  -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fa147f718cf49d0e4919b4e7229ea963a4335825","subject":"s\/\u2203 (\u03bb x \u2192 ...\/\u2203[ x ] ...","message":"s\/\u2203 (\u03bb x \u2192 ...\/\u2203[ x ] ...\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Nat\/PropertiesByInductionI.agda","new_file":"src\/fot\/FOTC\/Data\/Nat\/PropertiesByInductionI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the FOTC natural numbers type)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the FOTC natural numbers. The following\n-- examples show some proofs using it.\n\nmodule FOTC.Data.Nat.PropertiesByInductionI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a + c \u2261 b + c\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a + b \u2261 a + c\n+-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\nN\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN\u21920\u2228S = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n  A0 : A zero\n  A0 = inj\u2081 refl\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = case prf\u2081 prf\u2082 Ai\n    where\n    prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n    prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n           succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\nSx\u2262x : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2262 n\nSx\u2262x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = succ\u2081 i \u2262 i\n\n  A0 : A zero\n  A0 = S\u22620\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = \u2192-trans succInjective Ai\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N {n} Nn = case h\u2081 h\u2082 (N\u21920\u2228S Nn)\n  where\n  h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n  h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n  h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n  h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  A0 : A zero\n  A0 = subst N (sym (+-leftIdentity n)) Nn\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n+-assoc : \u2200 {m} \u2192 N m \u2192 \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc Nm n o = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n + o \u2261 i + (n + o)\n\n  A0 : A zero\n  A0 = zero + n + o   \u2261\u27e8 +-leftCong (+-leftIdentity n) \u27e9\n       n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n       zero + (n + o) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n + o     \u2261\u27e8 +-leftCong (+-Sx i n) \u27e9\n              succ\u2081 (i + n) + o   \u2261\u27e8 +-Sx (i + n) o \u27e9\n              succ\u2081 (i + n + o)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (i + (n + o)) \u2261\u27e8 sym (+-Sx i (n + o)) \u27e9\n              succ\u2081 i + (n + o)   \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i   \u220e\n\n0\u2238x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n0\u2238x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = zero \u2238 i \u2261 zero\n\n  A0 : A zero\n  A0 = \u2238-x0 zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = zero \u2238 succ\u2081 i   \u2261\u27e8 \u2238-xS zero i \u27e9\n              pred\u2081 (zero \u2238 i) \u2261\u27e8 predCong Ai \u27e9\n              pred\u2081 zero       \u2261\u27e8 pred-0 \u27e9\n              zero \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the FOTC natural numbers type)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the FOTC natural numbers. The following\n-- examples show some proofs using it.\n\nmodule FOTC.Data.Nat.PropertiesByInductionI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a + c \u2261 b + c\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a + b \u2261 a + c\n+-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\nN\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN\u21920\u2228S = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n  A0 : A zero\n  A0 = inj\u2081 refl\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = case prf\u2081 prf\u2082 Ai\n    where\n    prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 \u2203 (\u03bb i' \u2192 succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n    prf\u2082 : \u2203 (\u03bb i' \u2192 i \u2261 succ\u2081 i' \u2227 N i') \u2192\n           succ\u2081 i \u2261 zero \u2228 \u2203 (\u03bb i' \u2192 succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\nSx\u2262x : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2262 n\nSx\u2262x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = succ\u2081 i \u2262 i\n\n  A0 : A zero\n  A0 = S\u22620\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = \u2192-trans succInjective Ai\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N {n} Nn = case h\u2081 h\u2082 (N\u21920\u2228S Nn)\n  where\n  h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n  h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n  h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n  h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  A0 : A zero\n  A0 = subst N (sym (+-leftIdentity n)) Nn\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n+-assoc : \u2200 {m} \u2192 N m \u2192 \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc Nm n o = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n + o \u2261 i + (n + o)\n\n  A0 : A zero\n  A0 = zero + n + o   \u2261\u27e8 +-leftCong (+-leftIdentity n) \u27e9\n       n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n       zero + (n + o) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n + o     \u2261\u27e8 +-leftCong (+-Sx i n) \u27e9\n              succ\u2081 (i + n) + o   \u2261\u27e8 +-Sx (i + n) o \u27e9\n              succ\u2081 (i + n + o)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (i + (n + o)) \u2261\u27e8 sym (+-Sx i (n + o)) \u27e9\n              succ\u2081 i + (n + o)   \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i   \u220e\n\n0\u2238x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n0\u2238x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = zero \u2238 i \u2261 zero\n\n  A0 : A zero\n  A0 = \u2238-x0 zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = zero \u2238 succ\u2081 i   \u2261\u27e8 \u2238-xS zero i \u27e9\n              pred\u2081 (zero \u2238 i) \u2261\u27e8 predCong Ai \u27e9\n              pred\u2081 zero       \u2261\u27e8 pred-0 \u27e9\n              zero \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7de7425fbeec715875f0e5d451e5f79823970c89","subject":"Added inductive version of S\u2238S.","message":"Added inductive version of S\u2238S.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Nat\/PropertiesByInductionI.agda","new_file":"src\/fot\/FOTC\/Data\/Nat\/PropertiesByInductionI.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the FOTC natural numbers type)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the FOTC natural numbers. The following\n-- examples show some proofs using it.\n\nmodule FOTC.Data.Nat.PropertiesByInductionI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a + c \u2261 b + c\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a + b \u2261 a + c\n+-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\nN\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN\u21920\u2228S = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n  A0 : A zero\n  A0 = inj\u2081 refl\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = case prf\u2081 prf\u2082 Ai\n    where\n    prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n    prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n           succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\nSx\u2262x : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2262 n\nSx\u2262x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = succ\u2081 i \u2262 i\n\n  A0 : A zero\n  A0 = S\u22620\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = \u2192-trans succInjective Ai\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N {n} Nn = case h\u2081 h\u2082 (N\u21920\u2228S Nn)\n  where\n  h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n  h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n  h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n  h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  A0 : A zero\n  A0 = subst N (sym (+-leftIdentity n)) Nn\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = N (m \u2238 i)\n\n  A0 : A zero\n  A0 = subst N (sym (\u2238-x0 m)) Nm\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = subst N (sym (\u2238-xS m i)) (pred-N Ai)\n\n+-assoc : \u2200 {m} \u2192 N m \u2192 \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc Nm n o = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n + o \u2261 i + (n + o)\n\n  A0 : A zero\n  A0 = zero + n + o   \u2261\u27e8 +-leftCong (+-leftIdentity n) \u27e9\n       n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n       zero + (n + o) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n + o     \u2261\u27e8 +-leftCong (+-Sx i n) \u27e9\n              succ\u2081 (i + n) + o   \u2261\u27e8 +-Sx (i + n) o \u27e9\n              succ\u2081 (i + n + o)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (i + (n + o)) \u2261\u27e8 sym (+-Sx i (n + o)) \u27e9\n              succ\u2081 i + (n + o)   \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i   \u220e\n\n0\u2238x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n0\u2238x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = zero \u2238 i \u2261 zero\n\n  A0 : A zero\n  A0 = \u2238-x0 zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = zero \u2238 succ\u2081 i   \u2261\u27e8 \u2238-xS zero i \u27e9\n              pred\u2081 (zero \u2238 i) \u2261\u27e8 predCong Ai \u27e9\n              pred\u2081 zero       \u2261\u27e8 pred-0 \u27e9\n              zero \u220e\n\nS\u2238S : \u2200 {m n} \u2192 N m \u2192 N n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\nS\u2238S {m} Nm = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = succ\u2081 m \u2238 succ\u2081 i \u2261 m \u2238 i\n\n  A0 : A zero\n  A0 = succ\u2081 m \u2238 succ\u2081 zero   \u2261\u27e8 \u2238-xS (succ\u2081 m) zero \u27e9\n       pred\u2081 (succ\u2081 m \u2238 zero) \u2261\u27e8 predCong (\u2238-x0 (succ\u2081 m)) \u27e9\n       pred\u2081 (succ\u2081 m)        \u2261\u27e8 pred-S m \u27e9\n       m                      \u2261\u27e8 sym (\u2238-x0 m) \u27e9\n       m \u2238 zero               \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 m \u2238 succ\u2081 (succ\u2081 i) \u2261\u27e8 \u2238-xS (succ\u2081 m) (succ\u2081 i) \u27e9\n              pred\u2081 (succ\u2081 m \u2238 succ\u2081 i) \u2261\u27e8 predCong Ai \u27e9\n              pred\u2081 (m \u2238 i)             \u2261\u27e8 sym (\u2238-xS m i) \u27e9\n              m \u2238 succ\u2081 i               \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties (using induction on the FOTC natural numbers type)\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Usually our proofs use pattern matching instead of the induction\n-- principle associated with the FOTC natural numbers. The following\n-- examples show some proofs using it.\n\nmodule FOTC.Data.Nat.PropertiesByInductionI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- Congruence properties\n\n+-leftCong : \u2200 {a b c} \u2192 a \u2261 b \u2192 a + c \u2261 b + c\n+-leftCong refl = refl\n\n+-rightCong : \u2200 {a b c} \u2192 b \u2261 c \u2192 a + b \u2261 a + c\n+-rightCong refl = refl\n\n------------------------------------------------------------------------------\n\nN\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN\u21920\u2228S = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n  A0 : A zero\n  A0 = inj\u2081 refl\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = case prf\u2081 prf\u2082 Ai\n    where\n    prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n    prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n           succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\nSx\u2262x : \u2200 {n} \u2192 N n \u2192 succ\u2081 n \u2262 n\nSx\u2262x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = succ\u2081 i \u2262 i\n\n  A0 : A zero\n  A0 = S\u22620\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = \u2192-trans succInjective Ai\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N {n} Nn = case h\u2081 h\u2082 (N\u21920\u2228S Nn)\n  where\n  h\u2081 : n \u2261 zero \u2192 N (pred\u2081 n)\n  h\u2081 n\u22610 = subst N (sym (trans (predCong n\u22610) pred-0)) nzero\n\n  h\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N (pred\u2081 n)\n  h\u2082 (n' , prf , Nn') = subst N (sym (trans (predCong prf) (pred-S n'))) Nn'\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  A0 : A zero\n  A0 = subst N (sym (+-leftIdentity n)) Nn\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = subst N (sym (+-Sx i n)) (nsucc Ai)\n\n\u2238-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m \u2238 n)\n\u2238-N {m} Nm Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = N (m \u2238 i)\n\n  A0 : A zero\n  A0 = subst N (sym (\u2238-x0 m)) Nm\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = subst N (sym (\u2238-xS m i)) (pred-N Ai)\n\n+-assoc : \u2200 {m} \u2192 N m \u2192 \u2200 n o \u2192 m + n + o \u2261 m + (n + o)\n+-assoc Nm n o = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n + o \u2261 i + (n + o)\n\n  A0 : A zero\n  A0 = zero + n + o   \u2261\u27e8 +-leftCong (+-leftIdentity n) \u27e9\n       n + o          \u2261\u27e8 sym (+-leftIdentity (n + o)) \u27e9\n       zero + (n + o) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n + o     \u2261\u27e8 +-leftCong (+-Sx i n) \u27e9\n              succ\u2081 (i + n) + o   \u2261\u27e8 +-Sx (i + n) o \u27e9\n              succ\u2081 (i + n + o)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (i + (n + o)) \u2261\u27e8 sym (+-Sx i (n + o)) \u27e9\n              succ\u2081 i + (n + o)   \u220e\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i   \u220e\n\n0\u2238x : \u2200 {n} \u2192 N n \u2192 zero \u2238 n \u2261 zero\n0\u2238x = N-ind A A0 is\n  where\n  A : D \u2192 Set\n  A i = zero \u2238 i \u2261 zero\n\n  A0 : A zero\n  A0 = \u2238-x0 zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = zero \u2238 succ\u2081 i   \u2261\u27e8 \u2238-xS zero i \u27e9\n              pred\u2081 (zero \u2238 i) \u2261\u27e8 predCong Ai \u27e9\n              pred\u2081 zero       \u2261\u27e8 pred-0 \u27e9\n              zero \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5ccd226a36aa6e81aed71d85e28985068b764437","subject":"Changed doc.","message":"Changed doc.\n\nIgnore-this: da862b89e8d3385df314a5f8bdc72028\n\ndarcs-hash:20100916164741-3bd4e-aa26a260bc8c3fb80ffff828ec485ed5fd3062a6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Consistency\/Readme.agda","new_file":"Test\/Consistency\/Readme.agda","new_contents":"------------------------------------------------------------------------------\n-- Test the consistency of the LTC axioms\n------------------------------------------------------------------------------\n\n-- In some modules, we declare Agda postulates as FOL axioms. We\n-- test in these modules if it is possible to prove an unprovable\n-- theorem from these axioms.\n\nmodule Test.Consistency.Readme where\n\nopen import Test.Consistency.LTC.Minimal\nopen import Test.Consistency.LTC.Data.Bool\nopen import Test.Consistency.LTC.Data.List\nopen import Test.Consistency.LTC.Data.List.Bisimulation\nopen import Test.Consistency.LTC.Data.Nat\nopen import Test.Consistency.LTC.Data.Nat.Inequalities\n","old_contents":"------------------------------------------------------------------------------\n-- Test the consistency of the LTC axioms\n------------------------------------------------------------------------------\n\n-- In some modules, we declare some Agda postulates as FOL axioms. We\n-- test in these modules if it is possible to prove an unprovable\n-- theorem.\n\nmodule Test.Consistency.Readme where\n\nopen import Test.Consistency.LTC.Minimal\nopen import Test.Consistency.LTC.Data.Bool\nopen import Test.Consistency.LTC.Data.List\nopen import Test.Consistency.LTC.Data.List.Bisimulation\nopen import Test.Consistency.LTC.Data.Nat\nopen import Test.Consistency.LTC.Data.Nat.Inequalities\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f0f55406f62939473b4ed44fefd97a529c8407ac","subject":"Modified an Agsy proof (it is more similar to the PA proof).","message":"Modified an Agsy proof (it is more similar to the PA proof).\n\nIgnore-this: 59410d7030fbcca0732245f0667d8f86\n\ndarcs-hash:20110111151946-3bd4e-c072880a875ff594f23c4155ce8d6107a8e70812.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Agsy\/Nat.agda","new_file":"src\/Agsy\/Nat.agda","new_contents":"------------------------------------------------------------------------------\n-- Natural numbers properties using Agsy\n------------------------------------------------------------------------------\n\nmodule Agsy.Nat where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n  -- via Agsy {-c}\n+-rightIdentity zero    = refl\n+-rightIdentity (suc n) = cong suc (+-rightIdentity n)\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero    n o = refl\n+-assoc (suc m) n o = cong suc (+-assoc m n o)\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + suc n \u2261 suc (m + n)  -- via Agsy {-c}\nx+Sy\u2261S[x+y] zero    n = refl\nx+Sy\u2261S[x+y] (suc m) n = cong suc (x+Sy\u2261S[x+y] m n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = sym (+-rightIdentity n)\n+-comm (suc m) n =\n  begin\n    suc (m + n) \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n    n + suc m\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Natural numbers properties using Agsy\n------------------------------------------------------------------------------\n\nmodule Agsy.Nat where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n+-assoc : \u2200 m n o \u2192 m + n + o \u2261 m + (n + o)  -- via Agsy {-c}\n+-assoc zero    n o = refl\n+-assoc (suc m) n o = cong suc (+-assoc m n o)\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + suc n \u2261 suc (m + n)  -- via Agsy {-c}\nx+Sy\u2261S[x+y] zero    n = refl\nx+Sy\u2261S[x+y] (suc m) n = cong suc (x+Sy\u2261S[x+y] m n)\n\n0+x\u2261x+0 : \u2200 x \u2192 0 + x \u2261 x + 0  -- via Agsy {-c}\n0+x\u2261x+0 zero    = refl\n0+x\u2261x+0 (suc n) = cong suc (0+x\u2261x+0 n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m  -- via Agsy {-c -m}\n+-comm zero    n = 0+x\u2261x+0 n\n+-comm (suc m) n =\n  begin\n    suc (m + n) \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n    n + suc m\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"87029ae914bfd1455bc9578f282daf471448ad01","subject":"Minimal -> Base","message":"Minimal -> Base\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Schnorr.agda","new_file":"ZK\/Schnorr.agda","new_contents":"open import Type using (Type)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; idp; ap; ap\u2082; !_; module \u2261-Reasoning)\nopen import ZK.Types using (Cyclic-group; module Cyclic-group\n                           ; Cyclic-group-properties; module Cyclic-group-properties)\nimport ZK.SigmaProtocol\n\nmodule ZK.Schnorr\n    {G \u2124q : Type}\n    (cg : Cyclic-group G \u2124q)\n    where\n  open Cyclic-group cg\n\n  Commitment = G\n  Challenge  = \u2124q\n  Response   = \u2124q\n\n  open ZK.SigmaProtocol Commitment Challenge Response\n\n  module _ (x a : \u2124q) where\n    prover-commitment : Commitment\n    prover-commitment = g ^ a\n\n    prover-response : Challenge \u2192 Response\n    prover-response c = a + (x * c)\n\n    prover : Prover\n    prover = prover-commitment , prover-response\n\n  module _ (y : G) where\n    verifier : Verifier\n    verifier (mk A c s) = (g ^ s) == (A \u00b7 (y ^ c))\n\n    -- This simulator shows why it is so important for the\n    -- challenge to be picked once the commitment is known.\n\n    -- To fake a transcript, the challenge and response can\n    -- be arbitrarily chosen. However to be indistinguishable\n    -- from a valid proof it they need to be picked at random.\n    simulator : Simulator\n    simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (g ^ s) \/ (y ^ c)\n\n    witness-extractor : Transcript\u00b2 verifier \u2192 \u2124q\n    witness-extractor t\u00b2 = x\n      module Witness-extractor where\n        open Transcript\u00b2 t\u00b2\n        fd = get-f\u2080 - get-f\u2081\n        cd = get-c\u2080 - get-c\u2081\n        x  = fd * modinv cd\n\n    extractor : \u2200 a \u2192 Extractor verifier\n    extractor a t\u00b2 = prover (witness-extractor t\u00b2) a\n\n  module Proofs (cg-props : Cyclic-group-properties cg) where\n    open Cyclic-group-properties cg-props\n\n    correct : \u2200 x a \u2192 Correct (prover x a) (let y = g ^ x in verifier y)\n    correct x a c\n      = \u2713-== (g ^(a + (x * c))\n           \u2261\u27e8 ^-+ \u27e9\n              g\u02b7 \u00b7 (g ^(x * c))\n           \u2261\u27e8 ap (\u03bb z \u2192 g\u02b7 \u00b7 z) ^-* \u27e9\n              g\u02b7 \u00b7 (y ^ c)\n           \u220e)\n      where\n        open \u2261-Reasoning\n        g\u02b7 = g ^ a\n        y  = g ^ x\n\n    module _ (y : G) where\n      shvzk : Special-Honest-Verifier-Zero-Knowledge (verifier y)\n      shvzk = record { simulator = simulator y\n                     ; correct-simulator = \u03bb _ _ \u2192 \u2713-== \/-\u00b7 }\n\n    module _ (x : \u2124q) (t\u00b2 : Transcript\u00b2 (verifier (g ^ x))) where\n      private\n        y = g ^ x\n        x' = witness-extractor y t\u00b2\n\n      open Transcript\u00b2 t\u00b2 renaming (get-A to A; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                                   ;get-f\u2080 to f\u2080; get-f\u2081 to f\u2081)\n      open Witness-extractor y t\u00b2 hiding (x)\n      open \u2261-Reasoning\n\n      g^xcd\u2261g^fd = g ^(x * cd)\n                 \u2261\u27e8 ^-* \u27e9\n                   y ^ (c\u2080 - c\u2081)\n                 \u2261\u27e8 ^-- \u27e9\n                   (y ^ c\u2080) \/ (y ^ c\u2081)\n                 \u2261\u27e8 ! cancels-\/ \u27e9\n                   (A \u00b7 (y ^ c\u2080)) \/ (A \u00b7 (y ^ c\u2081))\n                 \u2261\u27e8 ap\u2082 _\/_ (! ==-\u2713 verify\u2080) (! ==-\u2713 verify\u2081) \u27e9\n                   (g ^ f\u2080) \/ (g ^ f\u2081)\n                 \u2261\u27e8 ! ^-- \u27e9\n                   g ^ fd\n                 \u220e\n\n      -- The extracted x is correct\n      x\u2261x' : x \u2261 x'\n      x\u2261x' = left-*-to-right-\/ (^-inj g^xcd\u2261g^fd)\n\n      extractor-exact : \u2200 a \u2192 EqProver (extractor y a t\u00b2) (prover x a)\n      extractor-exact a = idp , (\u03bb c \u2192 ap (\u03bb z \u2192 _+_ a (_*_ z c)) (! x\u2261x'))\n\n      special-soundness : Special-Soundness (verifier y)\n      special-soundness = record { extractor = extractor y a\n                                 ; extractor-correct = extractor-correct }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type using (Type)\nopen import Data.Bool.Minimal using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; idp; ap; ap\u2082; !_; module \u2261-Reasoning)\nopen import ZK.Types using (Cyclic-group; module Cyclic-group\n                           ; Cyclic-group-properties; module Cyclic-group-properties)\nimport ZK.SigmaProtocol\n\nmodule ZK.Schnorr\n    {G \u2124q : Type}\n    (cg : Cyclic-group G \u2124q)\n    where\n  open Cyclic-group cg\n\n  Commitment = G\n  Challenge  = \u2124q\n  Response   = \u2124q\n\n  open ZK.SigmaProtocol Commitment Challenge Response\n\n  module _ (x a : \u2124q) where\n    prover-commitment : Commitment\n    prover-commitment = g ^ a\n\n    prover-response : Challenge \u2192 Response\n    prover-response c = a + (x * c)\n\n    prover : Prover\n    prover = prover-commitment , prover-response\n\n  module _ (y : G) where\n    verifier : Verifier\n    verifier (mk A c s) = (g ^ s) == (A \u00b7 (y ^ c))\n\n    -- This simulator shows why it is so important for the\n    -- challenge to be picked once the commitment is known.\n\n    -- To fake a transcript, the challenge and response can\n    -- be arbitrarily chosen. However to be indistinguishable\n    -- from a valid proof it they need to be picked at random.\n    simulator : Simulator\n    simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (g ^ s) \/ (y ^ c)\n\n    witness-extractor : Transcript\u00b2 verifier \u2192 \u2124q\n    witness-extractor t\u00b2 = x\n      module Witness-extractor where\n        open Transcript\u00b2 t\u00b2\n        fd = get-f\u2080 - get-f\u2081\n        cd = get-c\u2080 - get-c\u2081\n        x  = fd * modinv cd\n\n    extractor : \u2200 a \u2192 Extractor verifier\n    extractor a t\u00b2 = prover (witness-extractor t\u00b2) a\n\n  module Proofs (cg-props : Cyclic-group-properties cg) where\n    open Cyclic-group-properties cg-props\n\n    correct : \u2200 x a \u2192 Correct (prover x a) (let y = g ^ x in verifier y)\n    correct x a c\n      = \u2713-== (g ^(a + (x * c))\n           \u2261\u27e8 ^-+ \u27e9\n              g\u02b7 \u00b7 (g ^(x * c))\n           \u2261\u27e8 ap (\u03bb z \u2192 g\u02b7 \u00b7 z) ^-* \u27e9\n              g\u02b7 \u00b7 (y ^ c)\n           \u220e)\n      where\n        open \u2261-Reasoning\n        g\u02b7 = g ^ a\n        y  = g ^ x\n\n    module _ (y : G) where\n      shvzk : Special-Honest-Verifier-Zero-Knowledge (verifier y)\n      shvzk = record { simulator = simulator y\n                     ; correct-simulator = \u03bb _ _ \u2192 \u2713-== \/-\u00b7 }\n\n    module _ (x : \u2124q) (t\u00b2 : Transcript\u00b2 (verifier (g ^ x))) where\n      private\n        y = g ^ x\n        x' = witness-extractor y t\u00b2\n\n      open Transcript\u00b2 t\u00b2 renaming (get-A to A; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                                   ;get-f\u2080 to f\u2080; get-f\u2081 to f\u2081)\n      open Witness-extractor y t\u00b2 hiding (x)\n      open \u2261-Reasoning\n\n      g^xcd\u2261g^fd = g ^(x * cd)\n                 \u2261\u27e8 ^-* \u27e9\n                   y ^ (c\u2080 - c\u2081)\n                 \u2261\u27e8 ^-- \u27e9\n                   (y ^ c\u2080) \/ (y ^ c\u2081)\n                 \u2261\u27e8 ! cancels-\/ \u27e9\n                   (A \u00b7 (y ^ c\u2080)) \/ (A \u00b7 (y ^ c\u2081))\n                 \u2261\u27e8 ap\u2082 _\/_ (! ==-\u2713 verify\u2080) (! ==-\u2713 verify\u2081) \u27e9\n                   (g ^ f\u2080) \/ (g ^ f\u2081)\n                 \u2261\u27e8 ! ^-- \u27e9\n                   g ^ fd\n                 \u220e\n\n      -- The extracted x is correct\n      x\u2261x' : x \u2261 x'\n      x\u2261x' = left-*-to-right-\/ (^-inj g^xcd\u2261g^fd)\n\n      extractor-exact : \u2200 a \u2192 EqProver (extractor y a t\u00b2) (prover x a)\n      extractor-exact a = idp , (\u03bb c \u2192 ap (\u03bb z \u2192 _+_ a (_*_ z c)) (! x\u2261x'))\n\n      special-soundness : Special-Soundness (verifier y)\n      special-soundness = record { extractor = extractor y a\n                                 ; extractor-correct = extractor-correct }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"eb332321cb55523e6068bacced76b6f0bc2f6f7b","subject":"Simplify reexports.","message":"Simplify reexports.\n\nMerge open declarations, reexport everything.\n\nOld-commit-hash: 69a4b0415ae43f3db65282495914ddf7a6fe06fd\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra as CA\n  using (ChangeAlgebraFamily)\nopen import Level\n\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  -- change algebras\n\n  open CA public renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  -- change algebra for every type\n\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  -- function changes\n\n  open FunctionChanges public using (cons)\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  -- environment changes\n\n  open ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Base.Change.Algebra as CA using\n  ( ChangeAlgebra\n  ; ChangeAlgebraFamily\n  ; change-algebra\u208d_\u208e\n  ; family\n  ; \u0394\u208d_\u208e\n  )\nopen import Level\n\nStructure : Set\u2081\nStructure = ChangeAlgebraFamily zero \u27e6_\u27e7Base\n\nmodule Structure (change-algebra-base : Structure) where\n  change-algebra : \u2200 \u03c4 \u2192 ChangeAlgebra zero \u27e6 \u03c4 \u27e7Type\n  change-algebra (base \u03b9) = change-algebra\u208d \u03b9 \u208e\n  change-algebra (\u03c4\u2081 \u21d2 \u03c4\u2082) = CA.FunctionChanges.changeAlgebra _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}}\n\n  change-algebra-family : ChangeAlgebraFamily zero \u27e6_\u27e7Type\n  change-algebra-family = family change-algebra\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  open CA public\n    using\n      ( \u0394\u208d_\u208e\n      ; update\u2032\n      ; diff\u2032\n      ; nil\u208d_\u208e\n      ; update-diff\u208d_\u208e\n      ; update-nil\u208d_\u208e\n      )\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  module _ {\u03c4\u2081 \u03c4\u2082 : Type} where\n    open CA.FunctionChanges.FunctionChange _ _ {{change-algebra \u03c4\u2081}} {{change-algebra \u03c4\u2082}} public\n      using\n        (\n        )\n      renaming\n        ( correct to is-valid\n        ; apply to call-change\n        )\n\n  open CA public using\n    ( before\n    ; after\n    ; before\u208d_\u208e\n    ; after\u208d_\u208e\n    )\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open CA.FunctionChanges public using (cons)\n  open CA public using () renaming\n    ( [] to \u2205\n    ; _\u2237_ to _\u2022_\n    )\n\n  open CA.ListChanges \u27e6_\u27e7Type {{change-algebra-family}} public using () renaming\n    ( changeAlgebra to environment-changes\n    )\n\n  after-env : \u2200 {\u0393 : Context} \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d  \u0393 \u208e \u03c1) \u2192 \u27e6 \u0393 \u27e7\n  after-env {\u0393} = after\u208d \u0393 \u208e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1d2b458091ad0372771b9ca150d19c3cfb458b08","subject":"Explore.Product: typo in a name","message":"Explore.Product: typo in a name\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Product.agda","new_file":"lib\/Explore\/Product.agda","new_contents":"{-# OPTIONS --without-K #-}\n{- The main definitions are the following:\n\n  * explore\u03a3\n  * explore\u03a3-ind\n  * adequate-sum\u03a3\n-}\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Function.NP\nimport Function.Related as FR\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Bool using (true ; false ; _\u2227_ ; if_then_else_)\nopen import Data.Product\nopen import Data.Fin\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_ ; module \u2261-Reasoning)\nopen import Category.Monad.Continuation.Alias\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\nopen import Explore.Explorable\n\nmodule Explore.Product where\n\nmodule _ {m A} {B : A \u2192 \u2605\u2080} where\n\n    explore\u03a3 : Explore m A \u2192 (\u2200 {x} \u2192 Explore m (B x)) \u2192 Explore m (\u03a3 A B)\n    explore\u03a3 explore\u1d2c explore\u1d2e z op = explore\u1d2c z op \u27e8,\u27e9 explore\u1d2e z op\n\n    module _ {e\u1d2c : Explore m A} {e\u1d2e : \u2200 {x} \u2192 Explore m (B x)} where\n\n        explore\u03a3-ind : \u2200 {p} \u2192 ExploreInd p e\u1d2c \u2192 (\u2200 {x} \u2192 ExploreInd p (e\u1d2e {x})) \u2192 ExploreInd p (explore\u03a3 e\u1d2c e\u1d2e)\n        explore\u03a3-ind Pe\u1d2c Pe\u1d2e P Pz P\u2219 Pf =\n          Pe\u1d2c (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 (\u03bb x \u2192 Pe\u1d2e {x} (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 (curry Pf x))\n\nmodule _ {A}{B : A \u2192 \u2605\u2080}{e\u1d2c : Explore \u2080 A}{e\u1d2e : \u2200 {x} \u2192 Explore \u2080 (B x)} where\n   explore\u03a3-adq : AdequateExplore e\u1d2c \u2192 (\u2200 {x} \u2192 AdequateExplore (e\u1d2e {x})) \u2192 AdequateExplore (explore\u03a3 e\u1d2c e\u1d2e)\n   explore\u03a3-adq a\u1d2c a\u1d2e F \u03b5 _\u2295_ f F-proof\n     = F (big\u2295\u1d2c (\u03bb a \u2192 big\u2295\u1d2e (\u03bb b \u2192 f (a , b))))\n     \u2194\u27e8 a\u1d2c F _ _ _ F-proof \u27e9\n         \u03a3 A (F \u2218 (\u03bb a \u2192 big\u2295\u1d2e (\u03bb b \u2192 f (a , b))))\n     \u2194\u27e8 second-iso (\u03bb _ \u2192 a\u1d2e F _ _ _ F-proof) \u27e9\n       \u03a3 A (\u03bb a \u2192 \u03a3 (B a) (\u03bb b \u2192 F (f (a , b))))\n     \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n       \u03a3 (\u03a3 A B) (F \u2218 f)\n     \u220e\n     where open FR.EquationalReasoning\n           big\u2295\u1d2c = e\u1d2c \u03b5 _\u2295_\n           big\u2295\u1d2e = \u03bb {x} \u2192 e\u1d2e {x} \u03b5 _\u2295_\n\nmodule _ {A} {B : A \u2192 \u2605\u2080} {sum\u1d2c : Sum A} {sum\u1d2e : \u2200 {x} \u2192 Sum (B x)} where\n\n    private\n        sum\u1d2c\u1d2e = sum\u1d2c \u27e8,\u27e9 (\u03bb {x} \u2192 sum\u1d2e {x})\n\n    adequate-sum\u03a3 : AdequateSum sum\u1d2c \u2192 (\u2200 {x} \u2192 AdequateSum (sum\u1d2e {x})) \u2192 AdequateSum sum\u1d2c\u1d2e\n    adequate-sum\u03a3 asum\u1d2c asum\u1d2e f = Fin (sum\u1d2c\u1d2e f)\n                                \u2194\u27e8 FI.id \u27e9\n                                  Fin (sum\u1d2c (\u03bb a \u2192 sum\u1d2e (\u03bb b \u2192 f (a , b))))\n                                \u2194\u27e8 asum\u1d2c _ \u27e9\n                                  \u03a3 A (\u03bb a \u2192 Fin (sum\u1d2e (\u03bb b \u2192 f (a , b))))\n                                \u2194\u27e8 second-iso (\u03bb _ \u2192 asum\u1d2e _) \u27e9\n                                  \u03a3 A (\u03bb a \u2192 \u03a3 (B a) (\u03bb b \u2192 Fin (f (a , b))))\n                                \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n                                  \u03a3 (\u03a3 A B) (Fin \u2218 f)\n                                \u220e\n      where open FR.EquationalReasoning\n\n-- From now on, these are derived definitions for convenience and pedagogical reasons\n\nexplore\u00d7 : \u2200 {m A B} \u2192 Explore m A \u2192 Explore m B \u2192 Explore m (A \u00d7 B)\nexplore\u00d7 explore\u1d2c explore\u1d2e = explore\u03a3 explore\u1d2c explore\u1d2e\n\nexplore\u00d7-ind : \u2200 {m p A B} {e\u1d2c : Explore m A} {e\u1d2e : Explore m B}\n               \u2192 ExploreInd p e\u1d2c \u2192 ExploreInd p e\u1d2e\n               \u2192 ExploreInd p (explore\u00d7 e\u1d2c e\u1d2e)\nexplore\u00d7-ind Pe\u1d2c Pe\u1d2e = explore\u03a3-ind Pe\u1d2c Pe\u1d2e\n\nsum\u03a3 : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 {x} \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\nsum\u03a3 = _\u27e8,\u27e9_\n\nsum\u00d7 : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsum\u00d7 = _\u27e8,\u27e9\u2032_\n\n{-\n\u03bc\u03a3 : \u2200 {A} {B : A \u2192 \u2605 _} \u2192 Explorable A \u2192 (\u2200 {x} \u2192 Explorable (B x)) \u2192 Explorable (\u03a3 A B)\n\u03bc\u03a3 \u03bcA \u03bcB = mk _ (explore\u03a3-ind (explore-ind \u03bcA) (explore-ind \u03bcB))\n                (adequate-sum\u03a3 (adequate-sum \u03bcA) (adequate-sum \u03bcB))\n\ninfixr 4 _\u00d7-\u03bc_\n_\u00d7-\u03bc_ : \u2200 {A B} \u2192 Explorable A \u2192 Explorable B \u2192 Explorable (A \u00d7 B)\n\u03bcA \u00d7-\u03bc \u03bcB = \u03bc\u03a3 \u03bcA \u03bcB\n\n_\u00d7-cmp_ : \u2200 {A B : \u2605\u2080 } \u2192 Cmp A \u2192 Cmp B \u2192 Cmp (A \u00d7 B)\n(ca \u00d7-cmp cb) (a , b) (a' , b') = ca a a' \u2227 cb b b'\n\n\u00d7-unique : \u2200 {A B}(\u03bcA : Explorable A)(\u03bcB : Explorable B)(cA : Cmp A)(cB : Cmp B)\n           \u2192 Unique cA (count \u03bcA) \u2192 Unique cB (count \u03bcB) \u2192 Unique (cA \u00d7-cmp cB) (count (\u03bcA \u00d7-\u03bc \u03bcB))\n\u00d7-unique \u03bcA \u03bcB cA cB uA uB (x , y) = count (\u03bcA \u00d7-\u03bc \u03bcB) ((cA \u00d7-cmp cB) (x , y)) \u2261\u27e8 explore-ext \u03bcA _ (\u03bb x' \u2192 help (cA x x')) \u27e9\n                                     count \u03bcA (cA x) \u2261\u27e8 uA x \u27e9\n                                     1 \u220e\n  where\n    open \u2261-Reasoning\n    help : \u2200 b \u2192 count \u03bcB (\u03bb y' \u2192 b \u2227 cB y y') \u2261 (if b then 1 else 0)\n    help true = uB y\n    help false = sum-zero \u03bcB\n-}\n\nmodule _ {\u2113} {A} {B : A \u2192 _} {e\u1d2c : Explore (\u209b \u2113) A} {e\u1d2e : \u2200 {x} \u2192 Explore (\u209b \u2113) (B x)} where\n  focus\u03a3 : Focus e\u1d2c \u2192 (\u2200 {x} \u2192 Focus (e\u1d2e {x})) \u2192 Focus (explore\u03a3 e\u1d2c (\u03bb {x} \u2192 e\u1d2e {x}))\n  focus\u03a3 f\u1d2c f\u1d2e ((x , y) , z) = f\u1d2c (x , f\u1d2e (y , z))\n\n  lookup\u03a3 : Lookup e\u1d2c \u2192 (\u2200 {x} \u2192 Lookup (e\u1d2e {x})) \u2192 Lookup (explore\u03a3 e\u1d2c (\u03bb {x} \u2192 e\u1d2e {x}))\n  lookup\u03a3 lookup\u1d2c lookup\u1d2e d = uncurry (lookup\u1d2e \u2218 lookup\u1d2c d)\n\n  -- can also be derived from explore-ind\n  reify\u03a3 : Reify e\u1d2c \u2192 (\u2200 {x} \u2192 Reify (e\u1d2e {x})) \u2192 Reify (explore\u03a3 e\u1d2c (\u03bb {x} \u2192 e\u1d2e {x}))\n  reify\u03a3 reify\u1d2c reify\u1d2e f = reify\u1d2c (reify\u1d2e \u2218 curry f)\n\nmodule _ {\u2113} {A B} {e\u1d2c : Explore (\u209b \u2113) A} {e\u1d2e : Explore (\u209b \u2113) B} where\n  focus\u00d7 : Focus e\u1d2c \u2192 Focus e\u1d2e \u2192 Focus (explore\u00d7 e\u1d2c e\u1d2e)\n  focus\u00d7 f\u1d2c f\u1d2e = focus\u03a3 {e\u1d2c = e\u1d2c} {e\u1d2e = e\u1d2e} f\u1d2c f\u1d2e\n  lookup\u00d7 : Lookup e\u1d2c \u2192 Lookup e\u1d2e \u2192 Lookup (explore\u00d7 e\u1d2c e\u1d2e)\n  lookup\u00d7 f\u1d2c f\u1d2e = lookup\u03a3 {e\u1d2c = e\u1d2c} {e\u1d2e = e\u1d2e} f\u1d2c f\u1d2e\n\nmodule Operators where\n    infixr 4 _\u00d7\u1d49_ _\u00d7\u2071_ _\u00d7\u02e2_\n    _\u00d7\u1d49_ = explore\u00d7\n    _\u00d7\u2071_ = explore\u00d7-ind\n    _\u00d7\u1d43_ = adequate-sum\u03a3\n    _\u00d7\u02e2_ = sum\u00d7\n    _\u00d7\u1da0_ = focus\u00d7\n    _\u00d7\u02e1_ = lookup\u00d7\n\n-- Those are here only for pedagogical use\nprivate\n    proj\u2081-explore : \u2200 {m A} {B : A \u2192 \u2605 _} \u2192 Explore m (\u03a3 A B) \u2192 Explore m A\n    proj\u2081-explore = EM.map _ proj\u2081\n\n    proj\u2082-explore : \u2200 {m A B} \u2192 Explore m (A \u00d7 B) \u2192 Explore m B\n    proj\u2082-explore = EM.map _ proj\u2082\n\n    sum'\u03a3 : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 x \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\n    sum'\u03a3 sum\u1d2c sum\u1d2e f = sum\u1d2c (\u03bb x\u2080 \u2192\n                          sum\u1d2e x\u2080 (\u03bb x\u2081 \u2192\n                            f (x\u2080 , x\u2081)))\n\n    explore\u00d7' : \u2200 {A B} \u2192 Explore\u2080 A \u2192 Explore _ B \u2192 Explore _ (A \u00d7 B)\n    explore\u00d7' explore\u1d2c explore\u1d2e z op f = explore\u1d2c z op (\u03bb x \u2192 explore\u1d2e z op (curry f x))\n\n    explore\u00d7-ind' : \u2200 {A B} {e\u1d2c : Explore _ A} {e\u1d2e : Explore _ B}\n                    \u2192 ExploreInd\u2080 e\u1d2c \u2192 ExploreInd\u2080 e\u1d2e \u2192 ExploreInd\u2080 (explore\u00d7' e\u1d2c e\u1d2e)\n    explore\u00d7-ind' Pe\u1d2c Pe\u1d2e P Pz P\u2219 Pf =\n      Pe\u1d2c (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 (Pe\u1d2e (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 \u2218 curry Pf)\n\n    -- liftM2 _,_ in the continuation monad\n    sum\u00d7' : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n    sum\u00d7' sum\u1d2c sum\u1d2e f = sum\u1d2c \u03bb x\u2080 \u2192\n                          sum\u1d2e \u03bb x\u2081 \u2192\n                            f (x\u2080 , x\u2081)\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n{- The main definitions are the following:\n\n  * explore\u03a3\n  * explore\u03a3-ind\n  * adequate-sum\u03a3\n-}\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Function.NP\nimport Function.Related as FR\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Bool using (true ; false ; _\u2227_ ; if_then_else_)\nopen import Data.Product\nopen import Data.Fin\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_ ; module \u2261-Reasoning)\nopen import Category.Monad.Continuation.Alias\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\nopen import Explore.Explorable\n\nmodule Explore.Product where\n\nmodule _ {m A} {B : A \u2192 \u2605\u2080} where\n\n    explore\u03a3 : Explore m A \u2192 (\u2200 {x} \u2192 Explore m (B x)) \u2192 Explore m (\u03a3 A B)\n    explore\u03a3 explore\u1d2c explore\u1d2e z op = explore\u1d2c z op \u27e8,\u27e9 explore\u1d2e z op\n\n    module _ {e\u1d2c : Explore m A} {e\u1d2e : \u2200 {x} \u2192 Explore m (B x)} where\n\n        explore\u03a3-ind : \u2200 {p} \u2192 ExploreInd p e\u1d2c \u2192 (\u2200 {x} \u2192 ExploreInd p (e\u1d2e {x})) \u2192 ExploreInd p (explore\u03a3 e\u1d2c e\u1d2e)\n        explore\u03a3-ind Pe\u1d2c Pe\u1d2e P Pz P\u2219 Pf =\n          Pe\u1d2c (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 (\u03bb x \u2192 Pe\u1d2e {x} (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 (curry Pf x))\n\nmodule _ {A}{B : A \u2192 \u2605\u2080}{e\u1d2c : Explore \u2080 A}{e\u1d2e : \u2200 {x} \u2192 Explore \u2080 (B x)} where\n   explore\u228e-adq : AdequateExplore e\u1d2c \u2192 (\u2200 {x} \u2192 AdequateExplore (e\u1d2e {x})) \u2192 AdequateExplore (explore\u03a3 e\u1d2c e\u1d2e)\n   explore\u228e-adq a\u1d2c a\u1d2e F \u03b5 _\u2295_ f F-proof\n     = F (big\u2295\u1d2c (\u03bb a \u2192 big\u2295\u1d2e (\u03bb b \u2192 f (a , b))))\n     \u2194\u27e8 a\u1d2c F _ _ _ F-proof \u27e9\n         \u03a3 A (F \u2218 (\u03bb a \u2192 big\u2295\u1d2e (\u03bb b \u2192 f (a , b))))\n     \u2194\u27e8 second-iso (\u03bb _ \u2192 a\u1d2e F _ _ _ F-proof) \u27e9\n       \u03a3 A (\u03bb a \u2192 \u03a3 (B a) (\u03bb b \u2192 F (f (a , b))))\n     \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n       \u03a3 (\u03a3 A B) (F \u2218 f)\n     \u220e\n     where open FR.EquationalReasoning\n           big\u2295\u1d2c = e\u1d2c \u03b5 _\u2295_\n           big\u2295\u1d2e = \u03bb {x} \u2192 e\u1d2e {x} \u03b5 _\u2295_\n\nmodule _ {A} {B : A \u2192 \u2605\u2080} {sum\u1d2c : Sum A} {sum\u1d2e : \u2200 {x} \u2192 Sum (B x)} where\n\n    private\n        sum\u1d2c\u1d2e = sum\u1d2c \u27e8,\u27e9 (\u03bb {x} \u2192 sum\u1d2e {x})\n\n    adequate-sum\u03a3 : AdequateSum sum\u1d2c \u2192 (\u2200 {x} \u2192 AdequateSum (sum\u1d2e {x})) \u2192 AdequateSum sum\u1d2c\u1d2e\n    adequate-sum\u03a3 asum\u1d2c asum\u1d2e f = Fin (sum\u1d2c\u1d2e f)\n                                \u2194\u27e8 FI.id \u27e9\n                                  Fin (sum\u1d2c (\u03bb a \u2192 sum\u1d2e (\u03bb b \u2192 f (a , b))))\n                                \u2194\u27e8 asum\u1d2c _ \u27e9\n                                  \u03a3 A (\u03bb a \u2192 Fin (sum\u1d2e (\u03bb b \u2192 f (a , b))))\n                                \u2194\u27e8 second-iso (\u03bb _ \u2192 asum\u1d2e _) \u27e9\n                                  \u03a3 A (\u03bb a \u2192 \u03a3 (B a) (\u03bb b \u2192 Fin (f (a , b))))\n                                \u2194\u27e8 FI.sym \u03a3-assoc \u27e9\n                                  \u03a3 (\u03a3 A B) (Fin \u2218 f)\n                                \u220e\n      where open FR.EquationalReasoning\n\n-- From now on, these are derived definitions for convenience and pedagogical reasons\n\nexplore\u00d7 : \u2200 {m A B} \u2192 Explore m A \u2192 Explore m B \u2192 Explore m (A \u00d7 B)\nexplore\u00d7 explore\u1d2c explore\u1d2e = explore\u03a3 explore\u1d2c explore\u1d2e\n\nexplore\u00d7-ind : \u2200 {m p A B} {e\u1d2c : Explore m A} {e\u1d2e : Explore m B}\n               \u2192 ExploreInd p e\u1d2c \u2192 ExploreInd p e\u1d2e\n               \u2192 ExploreInd p (explore\u00d7 e\u1d2c e\u1d2e)\nexplore\u00d7-ind Pe\u1d2c Pe\u1d2e = explore\u03a3-ind Pe\u1d2c Pe\u1d2e\n\nsum\u03a3 : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 {x} \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\nsum\u03a3 = _\u27e8,\u27e9_\n\nsum\u00d7 : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsum\u00d7 = _\u27e8,\u27e9\u2032_\n\n{-\n\u03bc\u03a3 : \u2200 {A} {B : A \u2192 \u2605 _} \u2192 Explorable A \u2192 (\u2200 {x} \u2192 Explorable (B x)) \u2192 Explorable (\u03a3 A B)\n\u03bc\u03a3 \u03bcA \u03bcB = mk _ (explore\u03a3-ind (explore-ind \u03bcA) (explore-ind \u03bcB))\n                (adequate-sum\u03a3 (adequate-sum \u03bcA) (adequate-sum \u03bcB))\n\ninfixr 4 _\u00d7-\u03bc_\n_\u00d7-\u03bc_ : \u2200 {A B} \u2192 Explorable A \u2192 Explorable B \u2192 Explorable (A \u00d7 B)\n\u03bcA \u00d7-\u03bc \u03bcB = \u03bc\u03a3 \u03bcA \u03bcB\n\n_\u00d7-cmp_ : \u2200 {A B : \u2605\u2080 } \u2192 Cmp A \u2192 Cmp B \u2192 Cmp (A \u00d7 B)\n(ca \u00d7-cmp cb) (a , b) (a' , b') = ca a a' \u2227 cb b b'\n\n\u00d7-unique : \u2200 {A B}(\u03bcA : Explorable A)(\u03bcB : Explorable B)(cA : Cmp A)(cB : Cmp B)\n           \u2192 Unique cA (count \u03bcA) \u2192 Unique cB (count \u03bcB) \u2192 Unique (cA \u00d7-cmp cB) (count (\u03bcA \u00d7-\u03bc \u03bcB))\n\u00d7-unique \u03bcA \u03bcB cA cB uA uB (x , y) = count (\u03bcA \u00d7-\u03bc \u03bcB) ((cA \u00d7-cmp cB) (x , y)) \u2261\u27e8 explore-ext \u03bcA _ (\u03bb x' \u2192 help (cA x x')) \u27e9\n                                     count \u03bcA (cA x) \u2261\u27e8 uA x \u27e9\n                                     1 \u220e\n  where\n    open \u2261-Reasoning\n    help : \u2200 b \u2192 count \u03bcB (\u03bb y' \u2192 b \u2227 cB y y') \u2261 (if b then 1 else 0)\n    help true = uB y\n    help false = sum-zero \u03bcB\n-}\n\nmodule _ {\u2113} {A} {B : A \u2192 _} {e\u1d2c : Explore (\u209b \u2113) A} {e\u1d2e : \u2200 {x} \u2192 Explore (\u209b \u2113) (B x)} where\n  focus\u03a3 : Focus e\u1d2c \u2192 (\u2200 {x} \u2192 Focus (e\u1d2e {x})) \u2192 Focus (explore\u03a3 e\u1d2c (\u03bb {x} \u2192 e\u1d2e {x}))\n  focus\u03a3 f\u1d2c f\u1d2e ((x , y) , z) = f\u1d2c (x , f\u1d2e (y , z))\n\n  lookup\u03a3 : Lookup e\u1d2c \u2192 (\u2200 {x} \u2192 Lookup (e\u1d2e {x})) \u2192 Lookup (explore\u03a3 e\u1d2c (\u03bb {x} \u2192 e\u1d2e {x}))\n  lookup\u03a3 lookup\u1d2c lookup\u1d2e d = uncurry (lookup\u1d2e \u2218 lookup\u1d2c d)\n\n  -- can also be derived from explore-ind\n  reify\u03a3 : Reify e\u1d2c \u2192 (\u2200 {x} \u2192 Reify (e\u1d2e {x})) \u2192 Reify (explore\u03a3 e\u1d2c (\u03bb {x} \u2192 e\u1d2e {x}))\n  reify\u03a3 reify\u1d2c reify\u1d2e f = reify\u1d2c (reify\u1d2e \u2218 curry f)\n\nmodule _ {\u2113} {A B} {e\u1d2c : Explore (\u209b \u2113) A} {e\u1d2e : Explore (\u209b \u2113) B} where\n  focus\u00d7 : Focus e\u1d2c \u2192 Focus e\u1d2e \u2192 Focus (explore\u00d7 e\u1d2c e\u1d2e)\n  focus\u00d7 f\u1d2c f\u1d2e = focus\u03a3 {e\u1d2c = e\u1d2c} {e\u1d2e = e\u1d2e} f\u1d2c f\u1d2e\n  lookup\u00d7 : Lookup e\u1d2c \u2192 Lookup e\u1d2e \u2192 Lookup (explore\u00d7 e\u1d2c e\u1d2e)\n  lookup\u00d7 f\u1d2c f\u1d2e = lookup\u03a3 {e\u1d2c = e\u1d2c} {e\u1d2e = e\u1d2e} f\u1d2c f\u1d2e\n\nmodule Operators where\n    infixr 4 _\u00d7\u1d49_ _\u00d7\u2071_ _\u00d7\u02e2_\n    _\u00d7\u1d49_ = explore\u00d7\n    _\u00d7\u2071_ = explore\u00d7-ind\n    _\u00d7\u1d43_ = adequate-sum\u03a3\n    _\u00d7\u02e2_ = sum\u00d7\n    _\u00d7\u1da0_ = focus\u00d7\n    _\u00d7\u02e1_ = lookup\u00d7\n\n-- Those are here only for pedagogical use\nprivate\n    proj\u2081-explore : \u2200 {m A} {B : A \u2192 \u2605 _} \u2192 Explore m (\u03a3 A B) \u2192 Explore m A\n    proj\u2081-explore = EM.map _ proj\u2081\n\n    proj\u2082-explore : \u2200 {m A B} \u2192 Explore m (A \u00d7 B) \u2192 Explore m B\n    proj\u2082-explore = EM.map _ proj\u2082\n\n    sum'\u03a3 : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 x \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\n    sum'\u03a3 sum\u1d2c sum\u1d2e f = sum\u1d2c (\u03bb x\u2080 \u2192\n                          sum\u1d2e x\u2080 (\u03bb x\u2081 \u2192\n                            f (x\u2080 , x\u2081)))\n\n    explore\u00d7' : \u2200 {A B} \u2192 Explore\u2080 A \u2192 Explore _ B \u2192 Explore _ (A \u00d7 B)\n    explore\u00d7' explore\u1d2c explore\u1d2e z op f = explore\u1d2c z op (\u03bb x \u2192 explore\u1d2e z op (curry f x))\n\n    explore\u00d7-ind' : \u2200 {A B} {e\u1d2c : Explore _ A} {e\u1d2e : Explore _ B}\n                    \u2192 ExploreInd\u2080 e\u1d2c \u2192 ExploreInd\u2080 e\u1d2e \u2192 ExploreInd\u2080 (explore\u00d7' e\u1d2c e\u1d2e)\n    explore\u00d7-ind' Pe\u1d2c Pe\u1d2e P Pz P\u2219 Pf =\n      Pe\u1d2c (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 (Pe\u1d2e (\u03bb e \u2192 P (\u03bb _ _ _ \u2192 e _ _ _)) Pz P\u2219 \u2218 curry Pf)\n\n    -- liftM2 _,_ in the continuation monad\n    sum\u00d7' : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n    sum\u00d7' sum\u1d2c sum\u1d2e f = sum\u1d2c \u03bb x\u2080 \u2192\n                          sum\u1d2e \u03bb x\u2081 \u2192\n                            f (x\u2080 , x\u2081)\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3060bb57fffdab72775a55fb36a23285794efc75","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: 6da64a122b3e8c86cd742cb4cbe5f7d3\n\ndarcs-hash:20110220153706-3bd4e-46ea9e1fdf44b254cfbd61801079a9df91fe3438.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/PropertiesI.agda","new_file":"Draft\/Mirror\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.PropertiesI where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesI\nopen import Draft.Mirror.ListTree.Closures\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesI using ( reverse-[x]\u2261[x] )\n\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove the postulate\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\n-- mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n-- mirror-Tree (treeT d nilLT) =\n--   subst Tree (sym (mirror-eq d [])) (treeT d helper\u2082)\n--     where\n--       helper\u2081 : rev (map mirror []) [] \u2261 []\n--       helper\u2081 =\n--         begin\n--           rev (map mirror []) []\n--           \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n--                    (map-[] mirror)\n--                    refl\n--           \u27e9\n--           rev [] []\n--             \u2261\u27e8 rev-[] [] \u27e9\n--           []\n--         \u220e\n\n--       helper\u2082 : ListTree (rev (map mirror []) [])\n--       helper\u2082 = subst ListTree (sym helper\u2081) nilLT\n\n-- mirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n--   subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d helper)\n\n--   where\n--     helper : ListTree (reverse (map mirror (t \u2237 ts)))\n--     helper = rev-ListTree (map-ListTree mirror {!!} (consLT Tt LTts)) nilLT\n\nmutual\n  mirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\n  mirror\u00b2 (treeT d nilLT) =\n    begin\n      mirror \u00b7 (mirror \u00b7 (node d []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 (node d [])) \u2261 mirror \u00b7 x )\n                 (mirror-eq d [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror [])) \u2261\n                        mirror \u00b7 node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse []) \u2261 mirror \u00b7 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d []\n        \u2261\u27e8 mirror-eq d [] \u27e9\n      node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror [])) \u2261\n                        node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse []) \u2261 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      node d []\n    \u220e\n\n  mirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n    begin\n      mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 mirror \u00b7 x)\n                 (mirror-eq d (t \u2237 ts))\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror (t \u2237 ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror (t \u2237 ts))) \u2261\n                        x)\n                 (mirror-eq d (reverse (map mirror (t \u2237 ts))))\n                 refl\n        \u27e9\n      node d (reverse (map mirror (reverse (map mirror (t \u2237 ts)))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (reverse (map mirror (t \u2237 ts))))) \u2261\n                      node d x)\n               (helper (consLT Tt LTts))\n               refl\n      \u27e9\n      node d (t \u2237 ts)\n    \u220e\n\n  helper : \u2200 {ts} \u2192 ListTree ts \u2192\n           reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  helper nilLT =\n    begin\n      reverse (map mirror (reverse (map mirror [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror []))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse []))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse [])) \u2261\n                        reverse (map mirror x))\n                 (rev-[] [])\n                 refl\n        \u27e9\n      reverse (map mirror [])\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror []) \u2261\n                        reverse x)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse []\n        \u2261\u27e8 rev-[] [] \u27e9\n      []\n    \u220e\n\n  helper (consLT {t} {ts} Tt LTts) =\n    begin\n      reverse (map mirror (reverse (map mirror (t \u2237 ts))))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror (t \u2237 ts)))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-\u2237 mirror t ts)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts))) \u2261\n                        reverse (map mirror x))\n                 (reverse-\u2237 (mirror-Tree Tt) (map-ListTree mirror mirror-Tree LTts))\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror \u00b7 t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror\n                   mirror-Tree\n                   (rev-ListTree (map-ListTree mirror mirror-Tree LTts) nilLT)\n                   (consLT (mirror-Tree Tt) nilLT))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror \u00b7 t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++-commute (map-ListTree mirror\n                                           mirror-Tree\n                                           (rev-ListTree (map-ListTree mirror\n                                                                       mirror-Tree\n                                                                       LTts)\n                                                         nilLT))\n                             (map-ListTree mirror\n                                           mirror-Tree\n                                           (consLT (mirror-Tree Tt) nilLT)))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (mirror \u00b7 t \u2237 [])) ++ n\u2081 \u2261\n                        reverse x ++ n\u2081)\n                 (map-\u2237 mirror (mirror \u00b7 t) [])\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081 \u2261\n                        reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 x) ++ n\u2081)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081 \u2261\n                        x ++ n\u2081)\n                 (reverse-[x]\u2261[x] (mirror \u00b7 (mirror \u00b7 t)))\n                 refl\n        \u27e9\n      (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 ++-\u2237 (mirror \u00b7 (mirror \u00b7 t)) [] n\u2081 \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192  mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081 \u2261\n                         mirror \u00b7 (mirror \u00b7 t) \u2237 x)\n                 (++-leftIdentity LTn\u2081)\n                 refl\n        \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror \u00b7 (mirror \u00b7 t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 Tt)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (helper LTts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n      where\n        n\u2081 : D\n        n\u2081 = reverse (map mirror (reverse (map mirror ts)))\n\n        LTn\u2081 : ListTree n\u2081\n        LTn\u2081 = rev-ListTree (map-ListTree mirror\n                                          mirror-Tree\n                                          (rev-ListTree (map-ListTree mirror\n                                                                      mirror-Tree\n                                                                      LTts)\n                                                        nilLT))\n                            nilLT\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the mirror function\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.PropertiesI where\n\nopen import LTC.Base\n\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.ListTree.PropertiesI\nopen import Draft.Mirror.ListTree.Closures\n\nopen import LTC.Data.List\nopen import LTC.Data.List.PropertiesI using ( reverse-[x]\u2261[x] )\n\n\nopen import LTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- TODO: To remove the postulate\npostulate\n  mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n\n-- mirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\n-- mirror-Tree (treeT d nilLT) =\n--   subst Tree (sym (mirror-eq d [])) (treeT d aux\u2082)\n--     where\n--       aux\u2081 : rev (map mirror []) [] \u2261 []\n--       aux\u2081 =\n--         begin\n--           rev (map mirror []) []\n--           \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n--                    (map-[] mirror)\n--                    refl\n--           \u27e9\n--           rev [] []\n--             \u2261\u27e8 rev-[] [] \u27e9\n--           []\n--         \u220e\n\n--       aux\u2082 : ListTree (rev (map mirror []) [])\n--       aux\u2082 = subst ListTree (sym aux\u2081) nilLT\n\n-- mirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n--   subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d aux)\n\n--   where\n--     aux : ListTree (reverse (map mirror (t \u2237 ts)))\n--     aux = rev-ListTree (map-ListTree mirror {!!} (consLT Tt LTts)) nilLT\n\nmutual\n  mirror\u00b2 : \u2200 {t} \u2192 Tree t \u2192 mirror \u00b7 (mirror \u00b7 t) \u2261 t\n  mirror\u00b2 (treeT d nilLT) =\n    begin\n      mirror \u00b7 (mirror \u00b7 (node d []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 (node d [])) \u2261 mirror \u00b7 x )\n                 (mirror-eq d [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror [])) \u2261\n                        mirror \u00b7 node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse []) \u2261 mirror \u00b7 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      mirror \u00b7 node d []\n        \u2261\u27e8 mirror-eq d [] \u27e9\n      node d (reverse (map mirror []))\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror [])) \u2261\n                        node d (reverse x))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      node d (reverse [])\n        \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse []) \u2261 node d x)\n                 (rev-[] [])\n                 refl\n        \u27e9\n      node d []\n    \u220e\n\n  mirror\u00b2 (treeT d (consLT {t} {ts} Tt LTts)) =\n    begin\n      mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 (mirror \u00b7 node d (t \u2237 ts)) \u2261 mirror \u00b7 x)\n                 (mirror-eq d (t \u2237 ts))\n                 refl\n        \u27e9\n      mirror \u00b7 node d (reverse (map mirror (t \u2237 ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 mirror \u00b7 node d (reverse (map mirror (t \u2237 ts))) \u2261\n                        x)\n                 (mirror-eq d (reverse (map mirror (t \u2237 ts))))\n                 refl\n        \u27e9\n      node d (reverse (map mirror (reverse (map mirror (t \u2237 ts)))))\n      \u2261\u27e8 subst (\u03bb x \u2192 node d (reverse (map mirror (reverse (map mirror (t \u2237 ts))))) \u2261\n                      node d x)\n               (helper (consLT Tt LTts))\n               refl\n      \u27e9\n      node d (t \u2237 ts)\n    \u220e\n\n  helper : \u2200 {ts} \u2192 ListTree ts \u2192\n           reverse (map mirror (reverse (map mirror ts))) \u2261 ts\n  helper nilLT =\n    begin\n      reverse (map mirror (reverse (map mirror [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror []))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse []))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse [])) \u2261\n                        reverse (map mirror x))\n                 (rev-[] [])\n                 refl\n        \u27e9\n      reverse (map mirror [])\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror []) \u2261\n                        reverse x)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse []\n        \u2261\u27e8 rev-[] [] \u27e9\n      []\n    \u220e\n\n  helper (consLT {t} {ts} Tt LTts) =\n    begin\n      reverse (map mirror (reverse (map mirror (t \u2237 ts))))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (map mirror (t \u2237 ts)))) \u2261\n                        reverse (map mirror (reverse x)))\n                 (map-\u2237 mirror t ts)\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (reverse (mirror \u00b7 t \u2237 map mirror ts))) \u2261\n                        reverse (map mirror x))\n                 (reverse-\u2237 (mirror-Tree Tt) (map-ListTree mirror mirror-Tree LTts))\n                 refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts) ++ (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts) ++\n                                              mirror \u00b7 t \u2237 []))) \u2261\n                        reverse x)\n           (map-++-commute mirror\n                   mirror-Tree\n                   (rev-ListTree (map-ListTree mirror mirror-Tree LTts) nilLT)\n                   (consLT (mirror-Tree Tt) nilLT))\n           refl\n        \u27e9\n      reverse (map mirror (reverse (map mirror ts)) ++\n              (map mirror (mirror \u00b7 t \u2237 [])))\n        \u2261\u27e8 subst (\u03bb x \u2192 (reverse (map mirror (reverse (map mirror ts)) ++\n                                             (map mirror (mirror \u00b7 t \u2237 [])))) \u2261\n                        x)\n                 (reverse-++-commute (map-ListTree mirror\n                                           mirror-Tree\n                                           (rev-ListTree (map-ListTree mirror\n                                                                       mirror-Tree\n                                                                       LTts)\n                                                         nilLT))\n                             (map-ListTree mirror\n                                           mirror-Tree\n                                           (consLT (mirror-Tree Tt) nilLT)))\n                 refl\n        \u27e9\n      reverse (map mirror (mirror \u00b7 t \u2237 [])) ++\n      reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (map mirror (mirror \u00b7 t \u2237 [])) ++ n\u2081 \u2261\n                        reverse x ++ n\u2081)\n                 (map-\u2237 mirror (mirror \u00b7 t) [])\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 map mirror []) ++ n\u2081 \u2261\n                        reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 x) ++ n\u2081)\n                 (map-[] mirror)\n                 refl\n        \u27e9\n      reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192 reverse (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081 \u2261\n                        x ++ n\u2081)\n                 (reverse-[x]\u2261[x] (mirror \u00b7 (mirror \u00b7 t)))\n                 refl\n        \u27e9\n      (mirror \u00b7 (mirror \u00b7 t) \u2237 []) ++ n\u2081\n        \u2261\u27e8 ++-\u2237 (mirror \u00b7 (mirror \u00b7 t)) [] n\u2081 \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081\n        \u2261\u27e8 subst (\u03bb x \u2192  mirror \u00b7 (mirror \u00b7 t) \u2237 [] ++ n\u2081 \u2261\n                         mirror \u00b7 (mirror \u00b7 t) \u2237 x)\n                 (++-leftIdentity LTn\u2081)\n                 refl\n        \u27e9\n      mirror \u00b7 (mirror \u00b7 t) \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 (mirror \u00b7 (mirror \u00b7 t) \u2237\n                                reverse (map mirror (reverse (map mirror ts)))) \u2261\n                        (x \u2237 reverse (map mirror (reverse (map mirror ts)))))\n                 (mirror\u00b2 Tt)  -- IH.\n                 refl\n        \u27e9\n      t \u2237 reverse (map mirror (reverse (map mirror ts)))\n        \u2261\u27e8 subst (\u03bb x \u2192 t \u2237 reverse (map mirror (reverse (map mirror ts))) \u2261\n                        t \u2237 x)\n                 (helper LTts)\n                 refl\n        \u27e9\n      t \u2237 ts\n    \u220e\n      where\n        n\u2081 : D\n        n\u2081 = reverse (map mirror (reverse (map mirror ts)))\n\n        LTn\u2081 : ListTree n\u2081\n        LTn\u2081 = rev-ListTree (map-ListTree mirror\n                                          mirror-Tree\n                                          (rev-ListTree (map-ListTree mirror\n                                                                      mirror-Tree\n                                                                      LTts)\n                                                        nilLT))\n                            nilLT\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b11afb649d30d0390d376d773a727268686b5e24","subject":"Simplified a proof.","message":"Simplified a proof.\n\nIgnore-this: 61c45ca949212bfeb3a46a5b703dcc1d\n\ndarcs-hash:20120219133102-3bd4e-d3939af7e4c1da1b94a3e84a18f59c8e2e66db74.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/GroupTheory\/Commutator\/PropertiesI.agda","new_file":"src\/GroupTheory\/Commutator\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with the group commutator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Commutator.PropertiesI where\n\nopen import GroupTheory.Base\nopen import GroupTheory.Commutator\nopen import GroupTheory.PropertiesI\nopen import GroupTheory.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- From: A. G. Kurosh. The Theory of Groups, vol. 1. Chelsea Publising\n-- Company, 2nd edition, 1960. p. 99.\ncommutatorInverse : \u2200 a b \u2192 \u27e6 a , b \u27e7 \u00b7 \u27e6 b , a \u27e7 \u2261 \u03b5\ncommutatorInverse a b =\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a) b (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 t))\n             (assoc (b \u207b\u00b9 \u00b7 a \u207b\u00b9) b a)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 t))\n             (assoc (b \u207b\u00b9) (a \u207b\u00b9) (b \u00b7 a))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 t)\n             (sym (assoc b (b \u207b\u00b9) (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (t \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n             (rightInverse b)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 t)\n             (leftIdentity (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9) a (a \u207b\u00b9 \u00b7 (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 t)\n             (sym (assoc a (a \u207b\u00b9) (b \u00b7 a)))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (rightInverse a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 \u00b7-rightCong (leftIdentity (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9) (b \u207b\u00b9) (b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 (b \u207b\u00b9 \u00b7 (b \u00b7 a))\n     \u2261\u27e8 \u00b7-rightCong (sym (assoc (b \u207b\u00b9) b a)) \u27e9\n  a \u207b\u00b9 \u00b7 ((b \u207b\u00b9 \u00b7 b) \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (leftInverse b)) \u27e9\n  a \u207b\u00b9 \u00b7 (\u03b5 \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (leftIdentity a) \u27e9\n  a \u207b\u00b9 \u00b7 a\n     \u2261\u27e8 leftInverse a \u27e9\n  \u03b5 \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with the group commutator\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GroupTheory.Commutator.PropertiesI where\n\nopen import GroupTheory.Base\nopen import GroupTheory.Commutator\nopen import GroupTheory.PropertiesI\nopen import GroupTheory.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\n-- From: A. G. Kurosh. The Theory of Groups, vol. 1. Chelsea Publising\n-- Company, 2nd edition, 1960. p. 99.\ncommutatorInverse : \u2200 a b \u2192 \u27e6 a , b \u27e7 \u00b7 \u27e6 b , a \u27e7 \u2261 \u03b5\ncommutatorInverse a b =\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a) b (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 b \u00b7 a)) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 t))\n             (assoc (b \u207b\u00b9 \u00b7 a \u207b\u00b9) b a)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 t))\n             (assoc (b \u207b\u00b9) (a \u207b\u00b9) (b \u00b7 a))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 (b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 t)\n             (sym (assoc b (b \u207b\u00b9) (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (b \u00b7 b \u207b\u00b9 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (t \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))))\n             (rightInverse b)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (\u03b5 \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 t)\n             (leftIdentity (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 assoc (a \u207b\u00b9 \u00b7 b \u207b\u00b9) a (a \u207b\u00b9 \u00b7 (b \u00b7 a)) \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a)))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 (a \u207b\u00b9 \u00b7 (b \u00b7 a))) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 t)\n             (sym (assoc a (a \u207b\u00b9) (b \u00b7 a)))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (a \u00b7 a \u207b\u00b9 \u00b7 (b \u00b7 a)) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (t \u00b7 (b \u00b7 a)))\n             (rightInverse a)\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 (b \u00b7 a))\n    \u2261\u27e8 subst (\u03bb t \u2192 a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (\u03b5 \u00b7 (b \u00b7 a)) \u2261\n                    a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 t)\n             (leftIdentity (b \u00b7 a))\n             refl\n    \u27e9\n  a \u207b\u00b9 \u00b7 b \u207b\u00b9 \u00b7 (b \u00b7 a)\n    \u2261\u27e8 assoc (a \u207b\u00b9) (b \u207b\u00b9) (b \u00b7 a) \u27e9\n  a \u207b\u00b9 \u00b7 (b \u207b\u00b9 \u00b7 (b \u00b7 a))\n     \u2261\u27e8 \u00b7-rightCong (sym (assoc (b \u207b\u00b9) b a)) \u27e9\n  a \u207b\u00b9 \u00b7 ((b \u207b\u00b9 \u00b7 b) \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (\u00b7-leftCong (leftInverse b)) \u27e9\n  a \u207b\u00b9 \u00b7 (\u03b5 \u00b7 a)\n     \u2261\u27e8 \u00b7-rightCong (leftIdentity a) \u27e9\n  a \u207b\u00b9 \u00b7 a\n     \u2261\u27e8 leftInverse a \u27e9\n  \u03b5 \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4039cedda59826535107a513ae1a70735d45621c","subject":"Multiplicative: adapt to the termination checker who became less smart","message":"Multiplicative: adapt to the termination checker who became less smart\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Multiplicative.agda","new_file":"Control\/Protocol\/Multiplicative.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol.Core\nopen import Control.Protocol.End\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\ninfixr 4 _\u22b8_\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                             ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            [inl: (\u03bb m \u2192 \u2297-assoc (P m) (recv Q) (recv R))\n                                            ,inr: [inl: (\u03bb m \u2192 \u2297-assoc (recv P) (Q m) (recv R))\n                                                  ,inr: (\u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)) ] ]\n                                            {- This use to work (and should) with older versions of Agda (11-Dec 2013)\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n                                            -}\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                             ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            [inl: (\u03bb m \u2192 \u214b-assoc (P m) (send Q) (send R))\n                                            ,inr: [inl: (\u03bb m \u2192 \u214b-assoc (send P) (Q m) (send R))\n                                                  ,inr: (\u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)) ] ]\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 (source-of P >>= \u03bb log \u2192 (Q log \u2297 R))\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 (sink-of P >>= \u03bb log \u2192 (Q log \u214b R))\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  dual-\u214b : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  dual-\u214b end Q = refl\n  dual-\u214b (recv P) Q = com=\u2032 _ \u03bb m \u2192 dual-\u214b (P m) _\n  dual-\u214b (send P) end = refl\n  dual-\u214b (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 dual-\u214b (send P) (Q n)\n  dual-\u214b (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 dual-\u214b (P m) (send Q))\n    ,inr: (\u03bb n \u2192 dual-\u214b (send P) (Q n)) ]\n\n  dual-\u2297 : \u2200 P Q \u2192 dual (P \u2297 Q) \u2261 dual P \u214b dual Q\n  dual-\u2297 end Q = refl\n  dual-\u2297 (send P) Q = com=\u2032 _ \u03bb m \u2192 dual-\u2297 (P m) _\n  dual-\u2297 (recv P) end = refl\n  dual-\u2297 (recv P) (send Q) = com=\u2032 _ \u03bb n \u2192 dual-\u2297 (recv P) (Q n)\n  dual-\u2297 (recv P) (recv Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 dual-\u2297 (P m) (recv Q))\n    ,inr: (\u03bb n \u2192 dual-\u2297 (recv P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  fwd : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  fwd end      = end\n  fwd (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (fwd (P x))\n  fwd (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (fwd (P x))\n\n  \u214b-id = fwd\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\ndata View-\u2218-proto : (P Q R : Proto) \u2192 \u2605\u2081 where\n  sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R\n           \u2192 View-\u2218-proto (send P) (send Q) R\n  recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n           \u2192 View-\u2218-proto (recv P) Q R\n  recvR-sendR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) (send R)\n  recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n           \u2192 View-\u2218-proto (send P) (recv Q) (recv R)\n  endL : \u2200 Q R \u2192 View-\u2218-proto end Q R\n  sendLM : \u2200 {MP}(P : MP \u2192 Proto)R \u2192 View-\u2218-proto (send P) end R\n  recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) end\n\nview-\u2218-proto : \u2200 P Q R \u2192 View-\u2218-proto P Q R\nview-\u2218-proto end      Q        R        = endL Q R\nview-\u2218-proto (recv P) Q        R        = recvLL P Q R\nview-\u2218-proto (send P) end      R        = sendLM P R\nview-\u2218-proto (send P) (recv Q) end      = recvL-sendR P Q\nview-\u2218-proto (send P) (recv Q) (recv R) = recvRR P Q R\nview-\u2218-proto (send P) (recv Q) (send R) = recvR-sendR P Q R\nview-\u2218-proto (send P) (send Q) R        = sendLL P Q R\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    \u214b-\u2218 : \u2200 {P Q R} \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218 {P} {Q} {R} pq qr with view-\u2218-proto P Q R\n    \u214b-\u2218 (inl m , pq) qr | sendLL P Q R = \u214b-sendL R m (\u214b-\u2218 {P m} {send Q} {R} pq qr)\n    \u214b-\u2218 (inr m , pq) qr | sendLL P Q R = \u214b-\u2218 {send P} {Q m} pq (qr m)\n    \u214b-\u2218 pq qr           | recvLL P Q M = \u03bb m \u2192 \u214b-\u2218 {P m} (pq m) qr\n    \u214b-\u2218 pq (inl m , qr) | recvR-sendR P Q R = \u214b-\u2218 {send P} {Q m} {send R} (coe (\u214b-recvR (send P) Q) pq m) qr\n    \u214b-\u2218 pq (inr m , qr) | recvR-sendR P Q R = \u214b-sendR (send P) m (\u214b-\u2218 {send P} {recv Q} {R m} pq qr)\n    \u214b-\u2218 pq qr           | recvRR P Q R = \u03bb m \u2192 \u214b-\u2218 {send P} {recv Q} {R m} pq (qr m)\n    \u214b-\u2218 pq qr           | endL Q R = \u22b8-apply {Q} {R} qr pq\n    \u214b-\u2218 (m , pq) qr     | sendLM P R = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218 pq (m , qr)     | recvL-sendR P Q = \u214b-\u2218 {send P} {Q m} {end} (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol.Core\nopen import Control.Protocol.End\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                              ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                              ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                              ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                              ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  dual-\u214b : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  dual-\u214b end Q = refl\n  dual-\u214b (recv P) Q = com=\u2032 _ \u03bb m \u2192 dual-\u214b (P m) _\n  dual-\u214b (send P) end = refl\n  dual-\u214b (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 dual-\u214b (send P) (Q n)\n  dual-\u214b (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 dual-\u214b (P m) (send Q))\n    ,inr: (\u03bb n \u2192 dual-\u214b (send P) (Q n)) ]\n\n  dual-\u2297 : \u2200 P Q \u2192 dual (P \u2297 Q) \u2261 dual P \u214b dual Q\n  dual-\u2297 end Q = refl\n  dual-\u2297 (send P) Q = com=\u2032 _ \u03bb m \u2192 dual-\u2297 (P m) _\n  dual-\u2297 (recv P) end = refl\n  dual-\u2297 (recv P) (send Q) = com=\u2032 _ \u03bb n \u2192 dual-\u2297 (recv P) (Q n)\n  dual-\u2297 (recv P) (recv Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 dual-\u2297 (P m) (recv Q))\n    ,inr: (\u03bb n \u2192 dual-\u2297 (recv P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  fwd : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  fwd end      = end\n  fwd (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (fwd (P x))\n  fwd (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (fwd (P x))\n\n  \u214b-id = fwd\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\ndata View-\u2218-proto : (P Q R : Proto) \u2192 \u2605\u2081 where\n  sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R\n           \u2192 View-\u2218-proto (send P) (send Q) R\n  recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n           \u2192 View-\u2218-proto (recv P) Q R\n  recvR-sendR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) (send R)\n  recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n           \u2192 View-\u2218-proto (send P) (recv Q) (recv R)\n  endL : \u2200 Q R \u2192 View-\u2218-proto end Q R\n  sendLM : \u2200 {MP}(P : MP \u2192 Proto)R \u2192 View-\u2218-proto (send P) end R\n  recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) end\n\nview-\u2218-proto : \u2200 P Q R \u2192 View-\u2218-proto P Q R\nview-\u2218-proto end      Q        R        = endL Q R\nview-\u2218-proto (recv P) Q        R        = recvLL P Q R\nview-\u2218-proto (send P) end      R        = sendLM P R\nview-\u2218-proto (send P) (recv Q) end      = recvL-sendR P Q\nview-\u2218-proto (send P) (recv Q) (recv R) = recvRR P Q R\nview-\u2218-proto (send P) (recv Q) (send R) = recvR-sendR P Q R\nview-\u2218-proto (send P) (send Q) R        = sendLL P Q R\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    \u214b-\u2218 : \u2200 {P Q R} \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218 {P} {Q} {R} pq qr with view-\u2218-proto P Q R\n    \u214b-\u2218 (inl m , pq) qr | sendLL P Q R = \u214b-sendL R m (\u214b-\u2218 {P m} {send Q} {R} pq qr)\n    \u214b-\u2218 (inr m , pq) qr | sendLL P Q R = \u214b-\u2218 {send P} {Q m} pq (qr m)\n    \u214b-\u2218 pq qr           | recvLL P Q M = \u03bb m \u2192 \u214b-\u2218 {P m} (pq m) qr\n    \u214b-\u2218 pq (inl m , qr) | recvR-sendR P Q R = \u214b-\u2218 {send P} {Q m} {send R} (coe (\u214b-recvR (send P) Q) pq m) qr\n    \u214b-\u2218 pq (inr m , qr) | recvR-sendR P Q R = \u214b-sendR (send P) m (\u214b-\u2218 {send P} {recv Q} {R m} pq qr)\n    \u214b-\u2218 pq qr           | recvRR P Q R = \u03bb m \u2192 \u214b-\u2218 {send P} {recv Q} {R m} pq (qr m)\n    \u214b-\u2218 pq qr           | endL Q R = \u22b8-apply {Q} {R} qr pq\n    \u214b-\u2218 (m , pq) qr     | sendLM P R = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218 pq (m , qr)     | recvL-sendR P Q = \u214b-\u2218 {send P} {Q m} {end} (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9e4e42995f871b15f24591bd95cac30bfecb1bda","subject":"fix copattern error in CPAd-CPA","message":"fix copattern error in CPAd-CPA\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/Transformation\/CPAd-CPA.agda","new_file":"Game\/Transformation\/CPAd-CPA.agda","new_contents":"\n{-# OPTIONS --without-K --copatterns #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.One\nopen import Control.Strategy renaming (run to runStrategy; map to mapStrategy)\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Game.IND-CPA-dagger\nimport Game.IND-CPA\n\nmodule Game.Transformation.CPAd-CPA\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n  \nwhere\n\nmodule CPA\u2020 = Game.IND-CPA-dagger PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 \ud835\udfd9 KeyGen Enc\nmodule CPA  = Game.IND-CPA        PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 \ud835\udfd9 KeyGen Enc\n\nR-transform : CPA\u2020.R \u2192 CPA.R\nR-transform (r\u2090 , r\u2096 , r\u2091 , _ , _) = r\u2090 , r\u2096 , r\u2091 , _\n\nmodule _ (A : CPA.Adversary) where\n  open CPA\u2020.Adversary\n  module A = CPA.Adversary A\n\n  A\u2020 : CPA\u2020.Adversary\n  m  A\u2020 = A.m\n  b\u2032 A\u2020 r\u2090 pk c\u2080 c\u2081 = A.b\u2032 r\u2090 pk c\u2080\n\n  lemma : \u2200 b t r\n          \u2192 CPA.EXP  b   A  (R-transform r)\n          \u2261 CPA\u2020.EXP b t A\u2020 r\n  lemma _ _ _ = refl\n\n  -- If we are able to do the transformation, then we get the same advantage\n  correct : \u2200 b r\n            \u2192 CPA.EXP  b A          (R-transform r)\n            \u2261 CPA\u2020.EXP b (not b) A\u2020 r\n  correct _ _ = refl\n\n-- -}\n","old_contents":"\n{-# OPTIONS --without-K --copatterns #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.One\nopen import Control.Strategy renaming (run to runStrategy; map to mapStrategy)\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Game.IND-CPA-dagger\nimport Game.IND-CPA\n\nmodule Game.Transformation.CPAd-CPA\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n  \nwhere\n\nmodule CPA\u2020 = Game.IND-CPA-dagger PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 \ud835\udfd9 KeyGen Enc\nmodule CPA  = Game.IND-CPA        PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 \ud835\udfd9 KeyGen Enc\n\n{-\nf : (Message \u00d7 Message) \u00d7 (CipherText \u2192 DecRound Bit)\n  \u2192 (Message \u00d7 Message) \u00d7 (CipherText \u2192 CipherText \u2192 DecRound Bit)\nf (m , g) = m , \u03bb c _ \u2192 g c\n-}\n\nR-transform : CPA\u2020.R \u2192 CPA.R\nR-transform (r\u2090 , r\u2096 , r\u2091 , _ , _) = r\u2090 , r\u2096 , r\u2091 , _\n\nmodule _ (A : CPA.Adversary) where\n  open CPA.Adversary\n\n  A\u2020 : CPA\u2020.Adversary\n  m  A\u2020 = m A\n  b\u2032 A\u2020 r\u2090 pk c\u2080 c\u2081 = b\u2032 A r\u2090 pk c\u2080\n\n  lemma : \u2200 b t r\n          \u2192 CPA.EXP  b   A  (R-transform r)\n          \u2261 CPA\u2020.EXP b t A\u2020 r\n  lemma _ _ _ = refl\n\n  -- If we are able to do the transformation, then we get the same advantage\n  correct : \u2200 b r\n            \u2192 CPA.EXP  b A          (R-transform r)\n            \u2261 CPA\u2020.EXP b (not b) A\u2020 r\n  correct _ _ = refl\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"495402798a0e92c17e10cea3ec8e3fcfa29748c0","subject":"first proof is made!","message":"first proof is made!\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/peano.agda","new_file":"learyouanagda\/peano.agda","new_contents":"module peano where\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\nzero + zero  = zero\nzero + n     = n\n(suc n) + n\u2032 = suc (n + n\u2032)\n\n\ndata _even : \u2115 \u2192 Set where\n  ZERO : zero even\n  STEP : \u2200 x \u2192 x even \u2192 (suc (suc x)) even\n\n\n-- To prove four is even\nproof\u2081 : suc (suc (suc (suc zero))) even\nproof\u2081 = STEP (suc (suc zero)) (STEP zero ZERO)\n","old_contents":"module peano where\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\nzero + zero  = zero\nzero + n     = n\n(suc n) + n\u2032 = suc (n + n\u2032)\n\n\ndata _even : \u2115 \u2192 Set where\n  ZERO : zero even\n  STEP : \u2200 x \u2192 x even \u2192 (suc (suc x)) even\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7505985ea6246300b76e4ab4eb7ee4750d566e2a","subject":"Nicer notation for Data.Indexed","message":"Nicer notation for Data.Indexed\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Indexed.agda","new_file":"lib\/Data\/Indexed.agda","new_contents":"open import Level           using (_\u2294_; suc; Level)\nopen import Function.NP     using (id; const; _\u2218_; _\u2218\u2032_; _\u02e2_; _\u27e8_\u27e9\u00b0_)\nopen import Data.Sum        using (_\u228e_; inj\u2081; inj\u2082)\nopen import Data.Bool       using (Bool; true; false)\nopen import Data.Unit       using (\u22a4)\nopen import Data.List       using (List; []; _\u2237_)\nopen import Data.Maybe      using (Maybe; nothing; just)\nopen import Data.Empty      using (\u22a5)\nopen import Data.Product.NP using (\u2203; _,_; _\u00d7_)\n\nmodule Data.Indexed {i} {Ix : Set i} where\n\n\u2605\u00b0 : \u2200 \u2113 \u2192 Set _\n\u2605\u00b0 \u2113 = Ix \u2192 Set \u2113\n\n\u2605\u2080\u00b0 : Set _\n\u2605\u2080\u00b0 = Ix \u2192 Set\n\n\u2605\u2081\u00b0 : Set _\n\u2605\u2081\u00b0 = Ix \u2192 Set\u2082\n\n\u2200\u00b0 : \u2200 {f} (F : \u2605\u00b0 f) \u2192 Set _\n\u2200\u00b0 F = \u2200 {x} \u2192 F x\n\n\u2203\u00b0 : \u2200 {f} (F : \u2605\u00b0 f) \u2192 Set _\n\u2203\u00b0 F = \u2203[ x ] F x\n\npure\u00b0 : \u2200 {a} {A : Set a} \u2192 A \u2192 (Ix \u2192 A)\npure\u00b0 = const\n\n-- an alias for pure\u00b0\nK\u00b0 : \u2200 {a} (A : Set a) \u2192 \u2605\u00b0 _\nK\u00b0 = pure\u00b0\n\nCmp\u00b0 : (F : \u2605\u00b0 _) (i j : Ix) \u2192 Set\nCmp\u00b0 F i j = F i \u2192 F j \u2192 Bool\n\n\u03a0\u00b0 : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2200 {i} \u2192 F i \u2192 \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n\u03a0\u00b0 F G i = (x : F i) \u2192 G x i\n\n-- this version used to work (type-checking its uses) better than that\ninfixr 0 _\u2192\u00b0_\n_\u2192\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 (f \u2294 g)\nF \u2192\u00b0 G = \u03a0\u00b0 F (const G)\n-- expanded: (F \u2192\u00b0 G) i = F i \u2192 G i\n\ninfixr 0 _\u21a6\u00b0_\n_\u21a6\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 Set _\n(F \u21a6\u00b0 G) = \u2200\u00b0(F \u2192\u00b0 G)\n\nid\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 F \u21a6\u00b0 F\nid\u00b0 = id\n\n_\u2218\u00b0_ : \u2200 {f g h} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} {H : \u2605\u00b0 h} \u2192 G \u21a6\u00b0 H \u2192 F \u21a6\u00b0 G \u2192 F \u21a6\u00b0 H\nf \u2218\u00b0 g = f \u2218 g\n\ninfixr 0 _$\u00b0_ _$\u00b0\u2032_\n\n_$\u00b0_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2200 {i} \u2192 F i \u2192 \u2605\u00b0 g}\n        \u2192 \u03a0\u00b0 F G \u21a6\u00b0 \u03a0\u00b0 F G\n_$\u00b0_ = id\n\n_$\u00b0\u2032_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g}\n        \u2192 \u2200\u00b0 ((F \u2192\u00b0 G) \u2192\u00b0 F \u2192\u00b0 G)\n_$\u00b0\u2032_ = id\n\ninfixl 4 _\u229b\u00b0_\n\n_\u229b\u00b0_ : \u2200 {a b} {A : Set a} {B : Set b}\n         \u2192 (Ix \u2192 A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\n_\u229b\u00b0_ = _\u02e2_\n\nliftA\u00b0 : \u2200 {a b} {A : Set a} {B : Set b}\n          \u2192 (A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\nliftA\u00b0 = _\u2218\u2032_\n-- liftA\u00b0 f x = pure\u00b0 f \u229b\u00b0 x\n\nliftA\u2082\u00b0 : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n           \u2192 (A \u2192 B \u2192 C)\n           \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B) \u2192 (Ix \u2192 C))\nliftA\u2082\u00b0 f x y = x \u27e8 f \u27e9\u00b0 y\n-- liftA\u2082\u00b0 f x y = pure\u00b0 f \u229b\u00b0 x \u229b\u00b0 y\n\nList\u00b0 : \u2200 {a} \u2192 \u2605\u00b0 a \u2192 \u2605\u00b0 a\nList\u00b0 = liftA\u00b0 List\n\n[]\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(List\u00b0 F)\n[]\u00b0 = []\n\n_\u2237\u00b0_ : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(F \u2192\u00b0 List\u00b0 F \u2192\u00b0 List\u00b0 F)\n_\u2237\u00b0_ = _\u2237_\n\nMaybe\u00b0 : \u2200 {a} \u2192 \u2605\u00b0 a \u2192 \u2605\u00b0 a\nMaybe\u00b0 = liftA\u00b0 Maybe\n\nnothing\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(Maybe\u00b0 F)\nnothing\u00b0 = nothing\n\njust\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 \u2200\u00b0(F \u2192\u00b0 Maybe\u00b0 F)\njust\u00b0 = just\n\ninfixr 2 _\u00d7\u00b0_\n_\u00d7\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n_\u00d7\u00b0_ = liftA\u2082\u00b0 _\u00d7_\n\n_,\u00b0_ : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 \u2200\u00b0 (F \u2192\u00b0 G \u2192\u00b0 F \u00d7\u00b0 G)\n_,\u00b0_ = _,_\n\npack\u00b0 : \u2200 {f} {F : \u2605\u00b0 f} \u2192 (x : Ix) \u2192 F x \u2192 \u2203\u00b0 F\npack\u00b0 = _,_\n\ninfixr 1 _\u228e\u00b0_\n\n_\u228e\u00b0_ : \u2200 {f g} (F : \u2605\u00b0 f) (G : \u2605\u00b0 g) \u2192 \u2605\u00b0 _\n_\u228e\u00b0_ = liftA\u2082\u00b0 _\u228e_\n\ninj\u2081\u00b0 : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 F \u21a6\u00b0 F \u228e\u00b0 G\ninj\u2081\u00b0 = inj\u2081\n\ninj\u2082\u00b0 : \u2200 {f g} {F : \u2605\u00b0 f} {G : \u2605\u00b0 g} \u2192 G \u21a6\u00b0 F \u228e\u00b0 G\ninj\u2082\u00b0 = inj\u2082\n\n\u22a4\u00b0 : \u2605\u00b0 _\n\u22a4\u00b0 = pure\u00b0 \u22a4\n\n\u22a5\u00b0 : \u2605\u00b0 _\n\u22a5\u00b0 = pure\u00b0 \u22a5\n\n\u00ac\u00b0 : \u2200 {\u2113} \u2192 \u2605\u00b0 \u2113 \u2192 \u2605\u00b0 _\n\u00ac\u00b0 F = F \u2192\u00b0 \u22a5\u00b0\n\n-- This is the type of |map| functions, the fmap function on Ix-indexed\n-- functors.\nMap\u00b0 : \u2200 {a a' b b'} (F : \u2605\u00b0 a \u2192 \u2605\u00b0 a')\n                     (G : \u2605\u00b0 b \u2192 \u2605\u00b0 b') \u2192 Set _\nMap\u00b0 F G = \u2200 {A B} \u2192 (A \u21a6\u00b0 B) \u2192 F A \u21a6\u00b0 G B\n\nmap\u00b0-K\u00b0 : \u2200 {a a' b b'}\n            {F : Set a \u2192 Set a'}\n            {G : Set b \u2192 Set b'}\n            (fmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 F A \u2192 G B)\n            {A B} \u2192 (K\u00b0 A \u21a6\u00b0 K\u00b0 B) \u2192 K\u00b0 (F A) \u21a6\u00b0 K\u00b0 (G B)\nmap\u00b0-K\u00b0 fmap f {i} = fmap (f {i})\n","old_contents":"open import Level           using (_\u2294_; suc; Level)\nopen import Function.NP     using (id; const; _\u2218_; _\u2218\u2032_; _\u02e2_; _\u27e8_\u27e9\u00b0_)\nopen import Data.Sum        using (_\u228e_; inj\u2081; inj\u2082)\nopen import Data.Bool       using (Bool; true; false)\nopen import Data.Unit       using (\u22a4)\nopen import Data.List       using (List; []; _\u2237_)\nopen import Data.Maybe      using (Maybe; nothing; just)\nopen import Data.Empty      using (\u22a5)\nopen import Data.Product.NP using (\u2203; _,_; _\u00d7_)\n\nmodule Data.Indexed {i} {Ix : Set i} where\n\n|Set| : \u2200 \u2113 \u2192 Set _\n|Set| \u2113 = Ix \u2192 Set \u2113\n\n|Set\u2080| : Set _\n|Set\u2080| = Ix \u2192 Set\n\n|Set\u2081| : Set _\n|Set\u2081| = Ix \u2192 Set\u2082\n\n|\u2200| : \u2200 {f} (F : |Set| f) \u2192 Set _\n|\u2200| F = \u2200 {x} \u2192 F x\n\n|\u2203| : \u2200 {f} (F : |Set| f) \u2192 Set _\n|\u2203| F = \u2203[ x ] F x\n\n|pure| : \u2200 {a} {A : Set a} \u2192 A \u2192 (Ix \u2192 A)\n|pure| = const\n\n-- an alias for |pure|\n|K| : \u2200 {a} (A : Set a) \u2192 |Set| _\n|K| = |pure|\n\n|Cmp| : (F : |Set| _) (i j : Ix) \u2192 Set\n|Cmp| F i j = F i \u2192 F j \u2192 Bool\n\n|\u03a0| : \u2200 {f g} (F : |Set| f) (G : \u2200 {i} \u2192 F i \u2192 |Set| g) \u2192 |Set| _\n|\u03a0| F G i = (x : F i) \u2192 G x i\n\n-- this version used to work (type-checking its uses) better than that\ninfixr 0 _|\u2192|_\n_|\u2192|_ : \u2200 {f g} (F : |Set| f) (G : |Set| g) \u2192 |Set| (f \u2294 g)\nF |\u2192| G = |\u03a0| F (const G)\n-- expanded: (F |\u2192| G) i = F i \u2192 G i\n\ninfixr 0 _|\u21a6|_\n_|\u21a6|_ : \u2200 {f g} (F : |Set| f) (G : |Set| g) \u2192 Set _\n(F |\u21a6| G) = |\u2200|(F |\u2192| G)\n\n|id| : \u2200 {f} {F : |Set| f} \u2192 F |\u21a6| F\n|id| = id\n\n_|\u2218|_ : \u2200 {f g h} {F : |Set| f} {G : |Set| g} {H : |Set| h} \u2192 G |\u21a6| H \u2192 F |\u21a6| G \u2192 F |\u21a6| H\nf |\u2218| g = f \u2218 g\n\ninfixr 0 _|$|_ _|$\u2032|_\n\n_|$|_ : \u2200 {f g} {F : |Set| f} {G : \u2200 {i} \u2192 F i \u2192 |Set| g}\n        \u2192 |\u03a0| F G |\u21a6| |\u03a0| F G\n_|$|_ = id\n\n_|$\u2032|_ : \u2200 {f g} {F : |Set| f} {G : |Set| g}\n        \u2192 |\u2200| ((F |\u2192| G) |\u2192| F |\u2192| G)\n_|$\u2032|_ = id\n\ninfixl 4 _|\u229b|_\n\n_|\u229b|_ : \u2200 {a b} {A : Set a} {B : Set b}\n         \u2192 (Ix \u2192 A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\n_|\u229b|_ = _\u02e2_\n\n|liftA| : \u2200 {a b} {A : Set a} {B : Set b}\n          \u2192 (A \u2192 B) \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B))\n-- |liftA| f x = |pure| f |\u229b| x\n|liftA| = _\u2218\u2032_\n\n|liftA2| : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c}\n           \u2192 (A \u2192 B \u2192 C)\n           \u2192 ((Ix \u2192 A) \u2192 (Ix \u2192 B) \u2192 (Ix \u2192 C))\n|liftA2| f x y = x \u27e8 f \u27e9\u00b0 y\n-- |liftA2| f x y = |pure| f |\u229b| x |\u229b| y\n\n|List| : \u2200 {a} \u2192 |Set| a \u2192 |Set| a\n|List| = |liftA| List\n\n|[]| : \u2200 {f} {F : |Set| f} \u2192 |\u2200|(|List| {f} F) -- Agda could not infer this {f}\n|[]| = []\n\n_|\u2237|_ : \u2200 {f} {F : |Set| f}\n         \u2192 |\u2200|(F |\u2192| |List| F |\u2192| |List| F)\n_|\u2237|_ = _\u2237_\n\n|Maybe| : \u2200 {a} \u2192 |Set| a \u2192 |Set| a\n|Maybe| = |liftA| Maybe\n\n|nothing| : \u2200 {f} {F : |Set| f} \u2192 |\u2200|(|Maybe| {f} F) -- Agda could not infer this {f}\n|nothing| = nothing\n\n|just| : \u2200 {f} {F : |Set| f} \u2192 |\u2200|(F |\u2192| |Maybe| F)\n|just| = just\n\ninfixr 2 _|\u00d7|_\n_|\u00d7|_ : \u2200 {f g} (F : |Set| f) (G : |Set| g) \u2192 |Set| _\n_|\u00d7|_ = |liftA2| _\u00d7_\n-- or alternatively:\n-- (F |\u00d7| G) i = F i \u00d7 G i\n\n_|,|_ : \u2200 {f g} {F : |Set| f} {G : |Set| g} \u2192 |\u2200| (F |\u2192| G |\u2192| F |\u00d7| G)\n_|,|_ = _,_\n\npack : \u2200 {f} {F : |Set| f} \u2192 (x : Ix) \u2192 F x \u2192 |\u2203| F\npack = _,_\n\ninfixr 1 _|\u228e|_\n\n_|\u228e|_ : \u2200 {f g} (F : |Set| f) (G : |Set| g) \u2192 |Set| _\n_|\u228e|_ = |liftA2| _\u228e_\n\n|inj\u2081| : \u2200 {f g} {F : |Set| f} {G : |Set| g} \u2192 F |\u21a6| F |\u228e| G\n|inj\u2081| = inj\u2081\n\n|inj\u2082| : \u2200 {f g} {F : |Set| f} {G : |Set| g} \u2192 G |\u21a6| F |\u228e| G\n|inj\u2082| = inj\u2082\n\n|\u22a4| : |Set| _\n|\u22a4| = |pure| \u22a4\n\n|\u22a5| : |Set| _\n|\u22a5| = |pure| \u22a5\n\n|\u00ac| : \u2200 {\u2113} \u2192 |Set| \u2113 \u2192 |Set| _\n|\u00ac| F = F |\u2192| |\u22a5|\n\n-- This is the type of |map| functions, the fmap function on Ix-indexed\n-- functors.\n|Map| : \u2200 {a a' b b'} (F : |Set| a \u2192 |Set| a')\n                      (G : |Set| b \u2192 |Set| b') \u2192 Set _\n|Map| F G = \u2200 {A B} \u2192 (A |\u21a6| B) \u2192 F A |\u21a6| G B\n\n|map|-|K| : \u2200 {a a' b b'}\n              {F : Set a \u2192 Set a'}\n              {G : Set b \u2192 Set b'}\n              (fmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 F A \u2192 G B)\n              {A B} \u2192 (|K| A |\u21a6| |K| B) \u2192 |K| (F A) |\u21a6| |K| (G B)\n|map|-|K| fmap f {i} x = fmap (f {i}) x\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4a65823a1ac5b3718d390a669991f8927d94f765","subject":"Swierstra_A_Predicate_Transformer_for_Effects","message":"Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3vl           | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val x) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {v : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure v\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} _    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = aux el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = aux er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (ernz , (sdel , sder))\n   with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  ... | Pure _       | Pure Zero     | _               | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  ... | Pure _       | Pure (Succ _) | el\u21d3vl           | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  ... | Pure _       | Step Abort _  | _               | ()\n  ... | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val x) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {v : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure v\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} h    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = aux el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = aux er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"68b3da18797900003afeb84c885a9af66378b6f7","subject":"collatz function","message":"collatz function\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/Evens.agda","new_file":"livecode\/Evens.agda","new_contents":"open import Base\n\nopen import Nat\n\nmodule Evens where\n\ndata Even : \u2115 \u2192 Type where\n  0even : Even 0\n  _+2even : {n : \u2115} \u2192 Even n \u2192 Even (succ (succ n))\n\n4even : Even 4\n4even = (0even +2even) +2even\n\nopen import Logic\n\n1odd : Even 1 \u2192 False\n1odd ()\n\nmodule n|n+1even where\n  P = \u03bb n \u2192 (Even n) \u2295 (Even (succ n))\n\n  step : {n : \u2115} \u2192 P n \u2192 P (succ n)\n  step (\u03b9\u2081 p) = \u03b9\u2082(p +2even)\n  step (\u03b9\u2082 p) = \u03b9\u2081(p)\n\n  thm : (n : \u2115) \u2192 P n\n  thm 0 = \u03b9\u2081 0even\n  thm (succ n) = step (thm n)\n\nvalue : {n : \u2115} \u2192 Even n \u2192 \u2115\nvalue 0even = 0\nvalue (p +2even) = succ (succ (value p))\n\nevenValue : {n : \u2115} \u2192 (pf : Even n) \u2192 Even (value pf)\nevenValue 0even = 0even\nevenValue (pf +2even) = evenValue pf +2even\n\nhalf : (n : \u2115) \u2192 Even n \u2192 \u2115\nhalf .0 0even = 0\nhalf (succ (succ n)) (p +2even) = succ (half n p) \n\n-- wrong = half 2 0even\n\ndouble : \u2115 \u2192 \u03a3 \u2115 Even\ndouble 0 = [ 0 , 0even ]\ndouble (succ n) = [ succ (succ (proj\u2081(double n))) , (proj\u2082 (double n) +2even) ]\n\nhalfdouble : \u2115 \u2192 \u2115\nhalfdouble n = half (proj\u2081 (double n)) (proj\u2082 (double n))\n\nmodule Collatz where\n  open n|n+1even\n\n  fn : {n : \u2115} \u2192 P n \u2192 \u2115\n  fn (\u03b9\u2081 x) = half (value x) (evenValue x)\n  fn (\u03b9\u2082 (x +2even)) = succ (half (value x) (evenValue x) + (value x))\n","old_contents":"open import Base\n\nopen import Nat\n\nmodule Evens where\n\ndata Even : \u2115 \u2192 Type where\n  0even : Even 0\n  _+2even : {n : \u2115} \u2192 Even n \u2192 Even (succ (succ n))\n\n4even : Even 4\n4even = (0even +2even) +2even\n\nopen import Logic\n\n1odd : Even 1 \u2192 False\n1odd ()\n\nmodule n|n+1even where\n  P = \u03bb n \u2192 (Even n) \u2295 (Even (succ n))\n\n  step : {n : \u2115} \u2192 P n \u2192 P (succ n)\n  step (\u03b9\u2081 p) = \u03b9\u2082(p +2even)\n  step (\u03b9\u2082 p) = \u03b9\u2081(p)\n\n  thm : (n : \u2115) \u2192 P n\n  thm 0 = \u03b9\u2081 0even\n  thm (succ n) = step (thm n)\n\nvalue : {n : \u2115} \u2192 Even n \u2192 \u2115\nvalue 0even = 0\nvalue (p +2even) = succ (succ (value p))\n\nhalf : (n : \u2115) \u2192 Even n \u2192 \u2115\nhalf .0 0even = 0\nhalf (succ (succ n)) (p +2even) = succ (half n p) \n\n-- wrong = half 2 0even\n\ndouble : \u2115 \u2192 \u03a3 \u2115 Even\ndouble 0 = [ 0 , 0even ]\ndouble (succ n) = [ succ (succ (proj\u2081(double n))) , (proj\u2082 (double n) +2even) ]\n\nhalfdouble : \u2115 \u2192 \u2115\nhalfdouble n = half (proj\u2081 (double n)) (proj\u2082 (double n))\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"58aaeec7e0952f04d046ac8cc9bc0f573af3e7f2","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"all.agda","new_file":"all.agda","new_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import lemmas-gcomplete\n\nopen import lemmas-disjointness\n--open import disjointness -- working here\n\nopen import structural-assumptions\n\nopen import focus-formation\nopen import ground-decidable\nopen import matched-ground-invariant\nopen import finality\n\n--open import lemmas-subst-ta -- working here\nopen import lemmas-ground\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\n--open import expansion-generality -- depend on disjointness\n--open import expandability -- ditto\nopen import expansion-unicity\nopen import type-assignment-unicity\nopen import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import lemmas-progress-checks\nopen import progress-checks\nopen import progress\nopen import preservation -- depends on subst-ta\n\nopen import lemmas-complete\nopen import cast-inert\nopen import complete-preservation\nopen import complete-progress\nopen import complete-expansion\n\nopen import weakening\nopen import exchange\nopen import lemmas-freshness\nopen import contraction\n","old_contents":"open import List\nopen import Nat\nopen import Prelude\n\nopen import contexts\nopen import core\n\nopen import lemmas-gcomplete\n\nopen import lemmas-disjointness\n--open import disjointness -- working here\n\nopen import structural-assumptions\n\nopen import focus-formation\nopen import ground-decidable\nopen import matched-ground-invariant\nopen import finality\n\n--open import lemmas-subst-ta -- working here\nopen import lemmas-ground\nopen import lemmas-consistency\nopen import lemmas-matching\nopen import synth-unicity\nopen import htype-decidable\n\n--open import expansion-generality -- depend on disjointness\n--open import expandability -- ditto\nopen import expansion-unicity\nopen import type-assignment-unicity\nopen import typed-expansion\n\nopen import canonical-value-forms\nopen import canonical-boxed-forms\nopen import canonical-indeterminate-forms\n\nopen import lemmas-progress-checks\nopen import progress-checks\nopen import progress\nopen import preservation\n\nopen import lemmas-complete\nopen import cast-inert\nopen import complete-preservation\nopen import complete-progress\nopen import complete-expansion\n\nopen import weakening\nopen import exchange\nopen import lemmas-freshness\nopen import contraction\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d16564e962657ecd40c7c4053e46c889a592d1f3","subject":"Control\/Protocol\/CoreBind.agda","message":"Control\/Protocol\/CoreBind.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/CoreBind.agda","new_file":"Control\/Protocol\/CoreBind.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\nopen import Data.Two\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Control.Protocol.CoreBind where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n{-\ndata Inf : \u2605 where\n  mu nu : Inf\n-}\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (A : \u2605)(B : A \u2192 Proto) \u2192 Proto\n  -- mu nu : \u221e Proto \u2192 Proto\n\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\n-- Tele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n-- later i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n-- ++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  -- right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  -- assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\n{-\ndualInf : Inf \u2192 Inf\ndualInf mu = nu\ndualInf nu = mu\n-}\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' A B) = \u03a3' A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' A B) = \u03a0' A (\u03bb x \u2192 dual (B x))\n-- dual (mu P) = nu (\u266f (dual (\u266d P)))\n-- dual (nu P) = mu (\u266f (dual (\u266d P)))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n{-\ndata Process (A : \u2605): Proto \u2192 \u2605\u2081 where\n  send : \u2200 {M P} (m : M) \u2192 Process A (P m) \u2192 Process A (\u03a3' M P)\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 Process A (P m)) \u2192 Process A (\u03a0' M P)\n  end  : A \u2192 Process A end\n  -}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  -- mu   : \u2200 {P} \u2192 this (\u266d P) \u2192 ProcessF this (mu P)\n  -- nu   : \u2200 {P} \u2192 \u221e (this (\u266d P)) \u2192 ProcessF this (nu P)\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\ndata Proc : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF Proc P \u2192 Proc P\n  end : Proc end\n\nproc : \u2200 {A P} \u2192 Process A P \u2192 Proc (P >> (A \u00d7' end))\nproc (do (send m x)) = do (send m (proc x))\nproc (do (recv x)) = do (recv (\u03bb m \u2192 proc (x m)))\nproc (end x) = do (send x end)\n\nprocess : \u2200 {A P} \u2192 Proc (P >> (A \u00d7' end)) \u2192 Process A P\nprocess {P = end} (do (send m _)) = end m\nprocess {P = \u03a0' A\u2081 B} (do (recv x)) = do (recv (\u03bb m \u2192 process (x m)))\nprocess {P = \u03a3' A\u2081 B} (do (send m p)) = do (send m (process p))\n\ndata Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  left  : \u2200 {P Q} \u2192 ProcessF (flip Sim Q) P \u2192 Sim P Q\n  right : \u2200 {P Q} \u2192 ProcessF (Sim P) Q \u2192 Sim P Q\n  end   : Sim end end\n\nSimL : Proto \u2192 Proto \u2192 \u2605\u2081\nSimL P Q = ProcessF (flip Sim Q) P\n\nSimR : Proto \u2192 Proto \u2192 \u2605\u2081\nSimR P Q = ProcessF (Sim P) Q\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3' M Q)\nsendR m = right \u2218 send m\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3' M P) Q\nsendL m = left \u2218 send m\n\nrecvR : \u2200 {M P Q} \u2192 ((m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0' M Q)\nrecvR = right \u2218 recv\n\nrecvL : \u2200 {M P Q} \u2192 ((m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0' M P) Q\nrecvL = left \u2218 recv\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = recvR (\u03bb m \u2192 sendL m (sim-id (B m)))\nsim-id (\u03a3' A B) = recvL (\u03bb m \u2192 sendR m (sim-id (B m)))\n--sim-id (mu P) = right (mu (left (nu (\u266f sim-id (\u266d P)))))\n--sim-id (nu P) = left (mu (right (nu (\u266f (sim-id (\u266d P))))))\n\nsim-id! : \u2200 P \u2192 Sim P (dual P)\nsim-id! end = end\nsim-id! (\u03a0' A B) = recvL (\u03bb m \u2192 sendR m (sim-id! (B m)))\nsim-id! (\u03a3' A B) = recvR (\u03bb m \u2192 sendL m (sim-id! (B m)))\n\nsim-comp : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q R} \u2192 SimL P Q \u2192 Sim (dual Q) R \u2192 SimL P R\nsim-compR : \u2200 {P Q R} \u2192 Sim P Q \u2192 SimR (dual Q) R \u2192 SimR P R\n-- sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual Q) R \u2192 Sim P R\n\nsim-comp (right (send m PQ)) (left (recv QR)) = sim-comp PQ (QR m)\nsim-comp (right (recv PQ))   (left (send m QR)) = sim-comp (PQ m) QR\n-- sim-comp (right (mu x)) (left (nu x\u2081)) = sim-comp x (\u266d x\u2081)\n-- sim-comp (right (nu x)) (left (mu x\u2081)) = sim-comp (\u266d x) x\u2081\nsim-comp end QR = QR\nsim-comp (left PQ)  QR = left (sim-compL PQ QR)\nsim-comp (right PQ) (right QR) = right (sim-compR (right PQ) QR)\n\n{-\nsim-compRL (send m PQ) (recv QR)   = sim-comp PQ (QR m)\nsim-compRL (recv PQ)   (send m QR) = sim-comp (PQ m) QR\n-}\n\nsim-compL (send m PQ) QR = send m (sim-comp PQ QR)\nsim-compL (recv PQ)   QR = recv (\u03bb m \u2192 sim-comp (PQ m) QR)\n-- sim-compL (mu PQ) QR = mu (sim-comp PQ QR)\n-- sim-compL (nu PQ) QR = nu (\u266f sim-comp (\u266d PQ) QR)\n\nsim-compR PQ (send m QR) = send m (sim-comp PQ QR)\nsim-compR PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp PQ (QR m))\n-- sim-comp PQ (mu QR) = mu (sim-comp PQ QR)\n-- sim-comp PQ (nu QR) = nu (\u266f sim-comp PQ (\u266d QR))\n\n-- sim-comp (sendR m ?) (sendR m' ?)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp\n\n\u2666-end-L : \u2200 {P Q} \u2192 Sim P end \u2192 Sim end Q \u2192 Sim P Q\n\u2666-end-L (left (send m PE)) EQ = sendL m (\u2666-end-L PE EQ)\n\u2666-end-L (left (recv   PE)) EQ = recvL (\u03bb m \u2192 \u2666-end-L (PE m) EQ)\n\u2666-end-L end                EQ = EQ\n\u2666-end-L (right ()) _\n\n\u2666-end-R : \u2200 {P Q} \u2192 Sim P end \u2192 Sim end Q \u2192 Sim P Q\n\u2666-end-R PE (right (send m EQ)) = sendR m (\u2666-end-R PE EQ)\n\u2666-end-R PE (right (recv   EQ)) = recvR \u03bb m \u2192 \u2666-end-R PE (EQ m)\n\u2666-end-R PE end                 = PE\n\u2666-end-R _  (left ())\n\ntestL : \u2666-end-L {\ud835\udfda \u00d7' end} {\ud835\udfda \u00d7' end} (sendL 0\u2082 end) (sendR 1\u2082 end)\n      \u2262 \u2666-end-R {\ud835\udfda \u00d7' end} {\ud835\udfda \u00d7' end} (sendL 0\u2082 end) (sendR 1\u2082 end)\ntestL ()\n\n{-\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\n!\u02e2 (left (send m x)) = sendR m (!\u02e2 x)\n!\u02e2 (left (recv x)) = recvR (\u03bb m \u2192 !\u02e2 (x m))\n!\u02e2 (right (send m x)) = sendL m (!\u02e2 x)\n!\u02e2 (right (recv x)) = recvL (\u03bb m \u2192 !\u02e2 (x m))\n!\u02e2 end = end\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n    !!\u02e2 :\n      \u2200 {P Q}\n        (PQ : Sim P Q)\n      \u2192 !\u02e2 (!\u02e2 PQ) \u2261 PQ\n    !!\u02e2 (left (send m x)) = cong (sendL m) (!!\u02e2 x)\n    !!\u02e2 (left (recv x)) = cong recvL (funExt \u03bb m \u2192 !!\u02e2 (x m))\n    !!\u02e2 (right (send m x)) = cong (sendR m) (!!\u02e2 x)\n    !!\u02e2 (right (recv x)) = cong recvR (funExt \u03bb m \u2192 !!\u02e2 (x m))\n    !!\u02e2 end = refl\n\n    sim-comp-end : \u2200 {P} (PE : Sim P end) \u2192 sim-comp PE end \u2261 PE\n    sim-comp-end (left (send m PE)) = cong (sendL m) (sim-comp-end PE)\n    sim-comp-end (left (recv PE)) = cong recvL (funExt (\u03bb x \u2192 sim-comp-end (PE x)))\n    sim-comp-end (right ())\n    sim-comp-end end = refl\n\n    sim-comp-end' : \u2200 {P Q} (PE : Sim P end) (EQ : Sim end Q) \u2192 sim-comp PE EQ \u2261 {!!}\n    sim-comp-end' PE EQ = {!!}\n\n    sim-comp-left :\n      \u2200 {P Q R}\n        (PQ : SimL P Q)\n        (QR : Sim (dual Q) R)\n      \u2192 sim-comp (left PQ) QR \u2261 left (sim-compL PQ QR)\n    sim-comp-left PQ QR = refl\n\n    sim-comp-right :\n      \u2200 {P Q R}\n        (PQ : SimR P Q)\n        (QR : SimR (dual Q) R)\n      \u2192 sim-comp (right PQ) (right QR) \u2261 right (sim-compR (right PQ) QR)\n    sim-comp-right (send m x) QR = refl\n    sim-comp-right (recv x) QR = refl\n\n    sim-comp-assoc :\n      \u2200 {P Q R S}\n        (PQ : Sim P Q)\n        (QR : Sim (dual Q) R)\n        (RS : Sim (dual R) S)\n      \u2192 sim-comp (sim-comp PQ QR) RS \u2261 sim-comp PQ (sim-comp QR RS)\n    sim-comp-assoc (left (send m PQ)) QR RS = cong (sendL m) (sim-comp-assoc PQ QR RS)\n    sim-comp-assoc (left (recv PQ)) QR RS = cong recvL (funExt (\u03bb m \u2192 sim-comp-assoc (PQ m) QR RS))\n    sim-comp-assoc (right (send m PQ)) (left (recv QR)) RS = sim-comp-assoc PQ (QR m) RS\n    sim-comp-assoc (right (send m PQ)) (right (send m\u2081 QR)) RS = {!!}\n    sim-comp-assoc (right (send m PQ)) (right (recv QR)) (left (send m\u2081 RS)) with QR m\u2081\n    sim-comp-assoc (right (send m PQ)) (right (recv _ )) (left (send m\u2081 RS)) | left (recv QR) = sim-comp-assoc PQ (QR m) RS\n    sim-comp-assoc (right (send m PQ)) (right (recv _ )) (left (send m\u2081 RS)) | right QR = {!!}\n    sim-comp-assoc (right (send m PQ)) (right (recv QR)) (right (send m\u2081 RS))\n      = {!sim-comp-assoc (sendR m PQ) (QR ?) ?!}\n    sim-comp-assoc (right (send m PQ)) (right (recv x)) (right (recv x\u2081)) = {!!}\n    sim-comp-assoc (right (recv PQ)) QR RS = {!!}\n    sim-comp-assoc end QR RS = refl\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = sim-comp\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (send m x)) = do (send m (sim-unit x))\nsim-unit (right (recv x)) = do (recv (\u03bb m \u2192 sim-unit (x m)))\nsim-unit end = end 0\u2081\n\nsim\u2192proc : \u2200 {P} \u2192 Sim end P \u2192 Proc P\nsim\u2192proc (left ())\nsim\u2192proc (right (send m x)) = do (send m (sim\u2192proc x))\nsim\u2192proc (right (recv x)) = do (recv (\u03bb m \u2192 sim\u2192proc (x m)))\nsim\u2192proc end = end\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\nproc\u2192sim : \u2200 {P} \u2192 Proc P \u2192 Sim end P\nproc\u2192sim (do (send m x)) = right (send m (proc\u2192sim x))\nproc\u2192sim (do (recv x)) = right (recv (\u03bb m \u2192 proc\u2192sim (x m)))\nproc\u2192sim end = end\n\n\u27f9-dual : \u2200 {P Q} \u2192 P \u27f9 dual Q \u2192 Q \u27f9 dual P\n\u27f9-dual = !\u02e2\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim0 : \u2200 {P Q} \u2192 P \u27f9 Q \u2192 Proc P \u2192 Proc Q\napply-sim0 PQ P = sim\u2192proc (sim-comp (proc\u2192sim P) PQ)\n\napply-sim1 : \u2200 {P Q} \u2192 Sim P Q \u2192 Proc (dual Q) \u2192 Proc P\napply-sim1 PQ Q = sim\u2192proc (!\u02e2 (sim-comp PQ (!\u02e2 (proc\u2192sim Q))))\n\napply-sim2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Proc (dual P) \u2192 Proc Q\napply-sim2 PQ P = apply-sim1 (!\u02e2 PQ) P\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (!\u02e2 PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n{-\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n  -}\n\n  apply-comp : \u2200 {P Q R}(EP : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n               \u2192 sim-comp EP (sim-comp PQ QR) \u2261 sim-comp (sim-comp EP PQ) QR\n  apply-comp (left ()) PQ QR\n  apply-comp (right (send m EP)) (left (recv PQ)) QR = apply-comp EP (PQ m) QR\n  apply-comp (right (send m EP)) (right (send m\u2081 x)) QR = {!apply-comp!}\n  apply-comp (right (send m EP)) (right (recv x)) QR = {!!}\n  apply-comp (right (recv x)) PQ QR = {!!}\n  apply-comp end PQ QR = refl\n\n               {-\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Control.Protocol.CoreBind where\nopen import Control.Strategy renaming (Strategy to Client) public\n\ndata Inf : \u2605 where\n  mu nu : Inf\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (A : \u2605)(B : A \u2192 Proto) \u2192 Proto\n  later : Inf \u2192 \u221e Proto \u2192 Proto\n\n{-\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndualInf : Inf \u2192 Inf\ndualInf mu = nu\ndualInf nu = mu\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' A B) = \u03a3' A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' A B) = \u03a0' A (\u03bb x \u2192 dual (B x))\ndual (later i P) = later (dualInf i) (\u266f (dual (\u266d P)))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n{-\ndata Process (A : \u2605): Proto \u2192 \u2605\u2081 where\n  send : \u2200 {M P} (m : M) \u2192 Process A (P m) \u2192 Process A (\u03a3' M P)\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 Process A (P m)) \u2192 Process A (\u03a0' M P)\n  end  : A \u2192 Process A end\n  -}\nelInf : Inf \u2192 \u2605\u2081 \u2192 \u2605\u2081\nelInf mu x = x\nelInf nu x = \u221e x\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  latr : \u2200 {P} i \u2192 elInf i (this (\u266d P)) \u2192 ProcessF this (later i P)\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\ndata Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  left  : \u2200 {P Q} \u2192 ProcessF (flip Sim Q) P \u2192 Sim P Q\n  right : \u2200 {P Q} \u2192 ProcessF (Sim P) Q \u2192 Sim P Q\n  end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb m \u2192 left (send m (sim-id (B m)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb m \u2192 right (send m (sim-id (B m)))))\nsim-id (later mu P) = right (latr mu (left (latr nu (\u266f sim-id (\u266d P)))))\nsim-id (later nu P) = left (latr mu (right (latr nu (\u266f (sim-id (\u266d P))))))\n\nsim-comp : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\nsim-comp (left (send m x)) QR = left (send m (sim-comp x QR))\nsim-comp (left (recv x)) QR = left (recv (\u03bb m \u2192 sim-comp (x m) QR))\nsim-comp (left (latr mu x)) QR = left (latr mu (sim-comp x QR))\nsim-comp (left (latr nu x)) QR = left (latr nu (\u266f sim-comp (\u266d x) QR))\nsim-comp (right (send m x)) (left (recv x\u2081)) = sim-comp x (x\u2081 m)\nsim-comp (right (recv x)) (left (send m x\u2081)) = sim-comp (x m) x\u2081\nsim-comp (right (latr mu x)) (left (latr .nu x\u2081)) = sim-comp x (\u266d x\u2081)\nsim-comp (right (latr nu x)) (left (latr .mu x\u2081)) = sim-comp (\u266d x) x\u2081\nsim-comp PQ (right (send m x)) = right (send m (sim-comp PQ x))\nsim-comp PQ (right (recv x)) = right (recv (\u03bb m \u2192 sim-comp PQ (x m)))\nsim-comp PQ (right (latr mu x)) = right (latr mu (sim-comp PQ x))\nsim-comp PQ (right (latr nu x)) = right (latr nu (\u266f sim-comp PQ (\u266d x)))\nsim-comp end QR = QR\n{-\nsim-comp end QR = QR\nsim-comp (right (send m x)) (left (recv x\u2081)) = sim-comp x (x\u2081 m)\nsim-comp (right (recv x)) (left (send m x\u2081)) = sim-comp (x m) x\u2081\nsim-comp (left (send m x)) QR = left (send m (sim-comp x QR))\nsim-comp (left (recv x)) QR = left (recv (\u03bb m \u2192 sim-comp (x m) QR))\nsim-comp PQ (right (send m\u2081 x\u2081)) = right (send m\u2081 (sim-comp PQ x\u2081))\nsim-comp PQ (right (recv x\u2081)) = right (recv (\u03bb m\u2081 \u2192 sim-comp PQ (x\u2081 m\u2081)))\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = sim-comp\n\nsim-sym : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-sym (left (send m x)) = right (send m (sim-sym x))\nsim-sym (left (recv x)) = right (recv (\u03bb m \u2192 sim-sym (x m)))\nsim-sym (right (send m x)) = left (send m (sim-sym x))\nsim-sym (right (recv x)) = left (recv (\u03bb m \u2192 sim-sym (x m)))\nsim-sym end = end\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (send m x)) = do (send m (sim-unit x))\nsim-unit (right (recv x)) = do (recv (\u03bb m \u2192 sim-unit (x m)))\nsim-unit end = end 0\u2081\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n{-\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = {!!}\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a9f01bec9a9da85813999f9078b5054609f3e184","subject":"The GHC ticket about gcd 0 0 = 0 was reopened.","message":"The GHC ticket about gcd 0 0 = 0 was reopened.\n\nIgnore-this: 20d5a440fe472435bde36f7c1588ad4e\n\ndarcs-hash:20110510154802-3bd4e-2a151ee77bd96f87a6ab52b4f1c0300c0c880e53.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Divisibility.agda","new_file":"src\/FOTC\/Data\/Nat\/Divisibility.agda","new_contents":"------------------------------------------------------------------------------\n-- The relation of divisibility on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Divisibility where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat using ( _*_ ; N )\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2223_\n\n------------------------------------------------------------------------------\n-- The relation of divisibility.\n-- The symbol is '\\mid' not '|'.\n-- It seems there is not agreement about if 0\u22230, e.g.\n-- Coq 8.3: 0\u22230\n-- Agda standard library, version 0.5: 0|0\n-- Hardy and Wright. An introduction to the theory of numbers. 1975. 4ed: 0\u22240\n--\n-- In our definition 0\u22240, therefore gcd 0 0 is undefined as it is in\n-- GHC (v. 7.0.3) (but see http:\/\/hackage.haskell.org\/trac\/ghc\/ticket\/3304).\n_\u2223_ : D \u2192 D \u2192 Set\nd \u2223 e = \u00ac (d \u2261 zero) \u2227 \u2203 \u03bb k \u2192 N k \u2227 e \u2261 k * d\n{-# ATP definition _\u2223_ #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The relation of divisibility on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Divisibility where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat using ( _*_ ; N )\n\n-- We add 3 to the fixities of the standard library.\ninfix 7 _\u2223_\n\n------------------------------------------------------------------------------\n-- The relation of divisibility.\n-- The symbol is '\\mid' not '|'.\n-- It seems there is not agreement about if 0\u22230, e.g.\n-- Coq 8.3: 0\u22230\n-- Agda standard library, version 0.5: 0|0\n-- Hardy and Wright. An introduction to the theory of numbers. 1975. 4ed: 0\u22240\n--\n-- In our definition 0\u22240, therefore gcd 0 0 is undefined as it is in\n-- GHC (v. 7.0.3) (but see the discussion about it in\n-- http:\/\/www.haskell.org\/pipermail\/haskell-cafe\/2009-May\/060788.html).\n_\u2223_ : D \u2192 D \u2192 Set\nd \u2223 e = \u00ac (d \u2261 zero) \u2227 \u2203 \u03bb k \u2192 N k \u2227 e \u2261 k * d\n{-# ATP definition _\u2223_ #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0b848c7f1bc49d5ae50193b8abe25a13694ce804","subject":"make agda spec of `coerce` terminating for contractive types","message":"make agda spec of `coerce` terminating for contractive types\n","repos":"yfcai\/CREG","old_file":"experiments\/Coerce.agda","new_file":"experiments\/Coerce.agda","new_contents":"{-# OPTIONS --guardedness-preserving-type-constructors #-}\n\n-- Agda specification of `coerce`\n--\n-- Incomplete: too much work just to make type system happy.\n-- Conversions between morally identical types obscure\n-- the operational content.\n\nopen import Function\nopen import Data.Product\nopen import Data.Maybe\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Sum\nopen import Data.Nat\nimport Data.Fin as Fin\nopen Fin using (Fin ; zero ; suc)\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Coinduction renaming (Rec to Fix ; fold to roll ; unfold to unroll)\n\n-- Things fail sometimes\npostulate fail : \u2200 {\u2113} {A : Set \u2113} \u2192 A\n\n-- I know what I'm doing\npostulate unsafeCast : \u2200 {A B : Set} \u2192 A \u2192 B\n\ndata Data : Set where\n  base : \u2115 \u2192 Data\n  pair : Data \u2192 Data \u2192 Data\n  plus : Data \u2192 Data \u2192 Data\n  var  : \u2115 \u2192 Data\n  fix  : Data \u2192 Data\n\n-- infinite environments\nEnv : \u2200 {\u2113} \u2192 Set \u2113 \u2192 Set \u2113\nEnv A = \u2115 \u2192 A\n\ninfixr 0 decide_then_else_\n\ndecide_then_else_ : \u2200 {p \u2113} {P : Set p} {A : Set \u2113} \u2192 Dec P \u2192 A \u2192 A \u2192 A\ndecide yes p then x else y = x\ndecide no \u00acp then x else y = y\n\n--update : \u2200 {\u2113} {A : Set \u2113} \u2192 \u2115 \u2192 A \u2192 Env A \u2192 Env A\n--update n v env = \u03bb m \u2192 decide n \u225f m then v else env m\n\nprepend : \u2200 {\u2113} {A : Set \u2113} \u2192 A \u2192 Env A \u2192 Env A\nprepend v env zero = v\nprepend v env (suc m) = env m\n\n-- universe construction; always terminating.\n{-# NO_TERMINATION_CHECK #-}\nuc : Data \u2192 Env Set \u2192 Set\nuc (base x) \u03c1 = Fin x\nuc (pair S T) \u03c1 = uc S \u03c1 \u00d7 uc T \u03c1\nuc (plus S T) \u03c1 = uc S \u03c1 \u228e uc T \u03c1\nuc (var x) \u03c1 = \u03c1 x\nuc (fix D) \u03c1 = set where set = Fix (\u266f (uc D (prepend set \u03c1)))\n\n\u2205 : \u2200 {\u2113} {A : Set \u2113} \u2192 Env A\n\u2205 = fail\n\nmy-nat-data : Data\nmy-nat-data = fix (plus (base 1) (var zero))\n\nMy-nat : Set\nMy-nat = uc my-nat-data \u2205\n\nmy-zero : My-nat\nmy-zero = roll (inj\u2081 zero)\n\nmy-one : My-nat\nmy-one = roll (inj\u2082 (roll (inj\u2081 zero)))\n\n-- nonterminating stuff\nbotData : Data\nbotData = fix (var zero)\n\nbotType : Set\nbotType = uc botData \u2205\n\n{-# NON_TERMINATING #-}\n\u22ef : botType\n\u22ef = roll (roll (roll (roll (roll (roll \u22ef)))))\n\nshift : \u2115 \u2192 \u2115 \u2192 Data \u2192 Data\nshift c d (base x) = base x\nshift c d (pair S T) = pair (shift c d S) (shift c d T)\nshift c d (plus S T) = plus (shift c d S) (shift c d T)\nshift c d (var k) = decide suc k \u2264? c then var k else var (k + d)\nshift c d (fix D) = fix (shift c d D)\n\n_[_\u21a6_] : Data \u2192 \u2115 \u2192 Data \u2192 Data\nbase x [ n \u21a6 S ] = base x\npair T T\u2081 [ n \u21a6 S ] = pair (T [ n \u21a6 S ]) (T\u2081 [ n \u21a6 S ])\nplus T T\u2081 [ n \u21a6 S ] = plus (T [ n \u21a6 S ]) (T\u2081 [ n \u21a6 S ])\nvar k [ n \u21a6 S ] = decide k \u225f n then S else var k\nfix T [ n \u21a6 S ] = fix (T [ n + 1 \u21a6 shift 0 1 S ])\n\n\nimport Level\nopen import Category.Monad\nopen RawMonad {Level.zero} monad\n\n\n-- terminating specification of `coerce`\n-- it is terminating because:\n--\n-- 1. if `fix T` is contractive (i. e., fix T != \u03bcX. X),\n--    then the `fix T` case will trigger one of the other cases.\n--\n-- 2. in every other case, the input data `s` is stripped of\n--    a constructor.\n--\n-- This `coerce` is an interpreter.\n-- The Scala `coerce` is the corresponding compiler.\n--\n{-# NON_TERMINATING #-}\ncoerce : (S : Data) \u2192 (T : Data) \u2192 uc S \u2205 \u2192 Maybe (uc T \u2205)\n\ncoerce (base m) (base n) s with m \u2264? n\n... | yes m\u2264n = just (Fin.inject\u2264 s m\u2264n)\n... | no  m>n = nothing\n\ncoerce (pair S\u2081 S\u2082) (pair T\u2081 T\u2082) (s\u2081 , s\u2082) =\n  coerce S\u2081 T\u2081 s\u2081 >>= (\u03bb t\u2081 \u2192\n  coerce S\u2082 T\u2082 s\u2082 >>= (\u03bb t\u2082 \u2192\n  return (t\u2081 , t\u2082)))\n\ncoerce (plus S\u2081 S\u2082) (plus T\u2081 T\u2082) (inj\u2081 s\u2081) =\n  coerce S\u2081 T\u2081 s\u2081 >>= (\u03bb t\u2081 \u2192 just (inj\u2081 t\u2081))\n\ncoerce (plus S\u2081 S\u2082) (plus T\u2081 T\u2082) (inj\u2082 s\u2082) =\n  coerce S\u2082 T\u2082 s\u2082 >>= (\u03bb t\u2082 \u2192 just (inj\u2082 t\u2082))\n\ncoerce (var j) T s =\n  nothing -- should not happen\n\ncoerce (fix S) T (roll x) =\n  let\n    S\u2032 = S [ 0 \u21a6 fix S ]\n  in\n    coerce S\u2032 T (unsafeCast x)\n\ncoerce S (fix T) s =\n  let\n    T\u2032 = T [ 0 \u21a6 fix T ]\n  in\n    coerce S T\u2032 s >>= (\u03bb t \u2192 just (roll (unsafeCast t)))\n\ncoerce _ _ _ = nothing\n","old_contents":"{-# OPTIONS --guardedness-preserving-type-constructors #-}\n\n-- Agda specification of `coerce`\n--\n-- Incomplete: too much work just to make type system happy.\n-- Conversions between morally identical types obscure\n-- the operational content.\n\nopen import Function\nopen import Data.Product\nopen import Data.Maybe\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Sum\nopen import Data.Nat\nimport Data.Fin as Fin\nopen Fin using (Fin ; zero ; suc)\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Coinduction renaming (Rec to Fix ; fold to roll ; unfold to unroll)\n\n-- Things fail sometimes\npostulate fail : \u2200 {\u2113} {A : Set \u2113} \u2192 A\n\n-- I know what I'm doing\npostulate unsafeCast : \u2200 {A B : Set} \u2192 A \u2192 B\n\ndata Data : Set where\n  base : \u2115 \u2192 Data\n  pair : Data \u2192 Data \u2192 Data\n  plus : Data \u2192 Data \u2192 Data\n  var  : \u2115 \u2192 Data\n  fix  : Data \u2192 Data\n\n-- infinite environments\nEnv : \u2200 {\u2113} \u2192 Set \u2113 \u2192 Set \u2113\nEnv A = \u2115 \u2192 A\n\ninfixr 0 decide_then_else_\n\ndecide_then_else_ : \u2200 {p \u2113} {P : Set p} {A : Set \u2113} \u2192 Dec P \u2192 A \u2192 A \u2192 A\ndecide yes p then x else y = x\ndecide no \u00acp then x else y = y\n\n--update : \u2200 {\u2113} {A : Set \u2113} \u2192 \u2115 \u2192 A \u2192 Env A \u2192 Env A\n--update n v env = \u03bb m \u2192 decide n \u225f m then v else env m\n\nprepend : \u2200 {\u2113} {A : Set \u2113} \u2192 A \u2192 Env A \u2192 Env A\nprepend v env zero = v\nprepend v env (suc m) = env m\n\n-- universe construction; always terminating.\n{-# NO_TERMINATION_CHECK #-}\nuc : Data \u2192 Env Set \u2192 Set\nuc (base x) \u03c1 = Fin x\nuc (pair S T) \u03c1 = uc S \u03c1 \u00d7 uc T \u03c1\nuc (plus S T) \u03c1 = uc S \u03c1 \u228e uc T \u03c1\nuc (var x) \u03c1 = \u03c1 x\nuc (fix D) \u03c1 = set where set = Fix (\u266f (uc D (prepend set \u03c1)))\n\n\u2205 : \u2200 {\u2113} {A : Set \u2113} \u2192 Env A\n\u2205 = fail\n\nmy-nat-data : Data\nmy-nat-data = fix (plus (base 1) (var zero))\n\nMy-nat : Set\nMy-nat = uc my-nat-data \u2205\n\nmy-zero : My-nat\nmy-zero = roll (inj\u2081 zero)\n\nmy-one : My-nat\nmy-one = roll (inj\u2082 (roll (inj\u2081 zero)))\n\n-- nonterminating stuff\nbotData : Data\nbotData = fix (var zero)\n\nbotType : Set\nbotType = uc botData \u2205\n\n{-# NON_TERMINATING #-}\n\u22ef : botType\n\u22ef = roll (roll (roll (roll (roll (roll \u22ef)))))\n\nshift : \u2115 \u2192 \u2115 \u2192 Data \u2192 Data\nshift c d (base x) = base x\nshift c d (pair S T) = pair (shift c d S) (shift c d T)\nshift c d (plus S T) = plus (shift c d S) (shift c d T)\nshift c d (var k) = decide suc k \u2264? c then var k else var (k + d)\nshift c d (fix D) = fix (shift c d D)\n\n_[_\u21a6_] : Data \u2192 \u2115 \u2192 Data \u2192 Data\nbase x [ n \u21a6 S ] = base x\npair T T\u2081 [ n \u21a6 S ] = pair (T [ n \u21a6 S ]) (T\u2081 [ n \u21a6 S ])\nplus T T\u2081 [ n \u21a6 S ] = plus (T [ n \u21a6 S ]) (T\u2081 [ n \u21a6 S ])\nvar k [ n \u21a6 S ] = decide k \u225f n then S else var k\nfix T [ n \u21a6 S ] = fix (T [ n + 1 \u21a6 shift 0 1 S ])\n\n\nimport Level\nopen import Category.Monad\nopen RawMonad {Level.zero} monad\n\n{-# NON_TERMINATING #-}\ncoerce : (S : Data) \u2192 (T : Data) \u2192 Maybe (uc S \u2205 \u2192 (uc T \u2205))\n\ncoerce (base m) (base n) with m \u2264? n\n... | yes m\u2264n = just (\u03bb s \u2192 Fin.inject\u2264 s m\u2264n)\n... | no  m>n = nothing\n\ncoerce (pair S\u2081 S\u2082) (pair T\u2081 T\u2082) =\n  coerce S\u2081 T\u2081 >>= (\u03bb f\u2081 \u2192\n  coerce S\u2082 T\u2082 >>= (\u03bb f\u2082 \u2192\n  return (\u03bb s \u2192 f\u2081 (proj\u2081 s) , f\u2082 (proj\u2082 s))))\n\ncoerce (plus S\u2081 S\u2082) (plus T\u2081 T\u2082) =\n  coerce S\u2081 T\u2081 >>= (\u03bb f\u2081 \u2192\n  coerce S\u2082 T\u2082 >>= (\u03bb f\u2082 \u2192\n  return [ inj\u2081 \u2218 f\u2081 , inj\u2082 \u2218 f\u2082 ]))\n\ncoerce (var j) T =\n  nothing -- should not happen\n\ncoerce (fix S) T =\n  let\n    S\u2032 = S [ 0 \u21a6 fix S ]\n  in\n    coerce S\u2032 T >>= (\u03bb f \u2192\n    return (\u03bb { (roll x) \u2192 f (unsafeCast x) }))\n\ncoerce S (fix T) =\n  let\n    T\u2032 = T [ 0 \u21a6 fix T ]\n  in\n    coerce S T\u2032 >>= (\u03bb f \u2192\n    return (\u03bb x \u2192 (roll (unsafeCast (f x)))))\n\ncoerce _ _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0240be272dccd81d3faee61f0b1d047305947952","subject":"dot pattern","message":"dot pattern\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n\n\ndata Vec (A : Set) : \u2115 -> Set where\n  nil  : Vec A O\n  cons : (n : \u2115) -> A -> Vec A n -> Vec A (S n)\n\nhead : {A : Set} {n : \u2115} -> Vec A (S n) -> A\nhead (cons n v vs) = v\n\n\nvmap : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap .O f nil = nil\nvmap .(S n) f (cons n x xs) = cons n (f x) (vmap n f xs)\n\n\nvmap\u2032 : {A B : Set} (n : \u2115) -> (A -> B) -> Vec A n -> Vec B n\nvmap\u2032 O f nil = nil\nvmap\u2032 (S n) f (cons .n x xs) = cons n (f x) (vmap n f xs)\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\n\ndata \u2115 : Set where\n  O : \u2115\n  S : \u2115 \u2192 \u2115\n\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   + a = a\nS a + b = S (a + b)\n\n_*_ : \u2115 \u2192 \u2115 \u2192 \u2115\nO   * a = O\nS a * b = a + (a * b)\n\n_or_ : Bool \u2192 Bool \u2192 Bool\ntrue  or _ = true\nfalse or b = b\n\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then x else y = x\nif false then x else y = y\n\n\ninfixl 60 _*_\ninfixl 40 _+_\ninfixr 20 _or_\ninfix  5 if_then_else_\n\n\ninfixr 40 _::_\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A -> List A -> List A\n\n\n_\u2218_ : {A : Set} -> {B : A -> Set} -> {C : (x : A) -> B x -> Set} ->\n      (f : {x : A} -> (y : B x) -> C x y) -> (g : (x : A) -> B x) ->\n      (x : A) -> C x (g x)\n_\u2218_ f g a = f (g a)\n\n\nplus-two = S \u2218 S\n\nmap : {A B : Set} -> (A -> B) -> List A -> List B\nmap f []        = []\nmap f (x :: xs) = f x :: map f xs\n\n_++_ : {A : Set} -> List A -> List A -> List A\n[]      ++ ys = ys\nx :: xs ++ ys = x :: (xs ++ ys)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"28d3562728511f354e8e57a54957853d46c788a0","subject":"Group.Homomorphism: the easy direction of stability preservation","message":"Group.Homomorphism: the easy direction of stability preservation\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_file":"lib\/Algebra\/Group\/Homomorphism.agda","new_contents":"module Algebra.Group.Homomorphism where\n\nopen import Type using (Type_)\nopen import Function.NP using (Op\u2082; _\u2218_)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Type a}{b}{B : Type b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Type (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Type \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6*   : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  where\n\n  +-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)\n  +-stable {k} =\n    F \u03c6                \u2261\u27e8 F\u03c6* \u27e9\n    F (_*_ (\u03c6 k) \u2218 \u03c6)  \u2261\u27e8 F= (\u03bb x \u2192 ! \u03c6-+-*) \u27e9\n    F (\u03c6 \u2218 _+_ k)      \u220e\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6 : A \u2192 B)\n  (\u03c6-hom : GroupHomomorphism G+ G* \u03c6)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupHomomorphism \u03c6-hom\n\n  open Stability-Minimal \u03c6 _+_ _*_ hom public\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Algebra.Group.Homomorphism where\n\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Raw\nopen import Algebra.Group\nopen import Algebra.Group.Constructions\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat.NP     using (1+_)\nopen import Data.Integer.NP using (\u2124; -[1+_]; +_; -_; module \u2124\u00b0)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nrecord GroupHomomorphism {a}{A : Set a}{b}{B : Set b}\n                         (grpA0+ : Group A)(grpB1* : Group B)\n                         (f : A \u2192 B) : Set (a \u2294 b) where\n  constructor mk\n\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  field\n    hom : Homomorphic\u2082 f _+_ _*_\n\n  pres-unit : f 0# \u2261 1#\n  pres-unit = unique-1-left part\n    where part = f 0# * f 0#  \u2261\u27e8 ! hom \u27e9\n                 f (0# + 0#)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                 f 0#         \u220e\n\n  mon-hom : MonoidHomomorphism +-mon *-mon f\n  mon-hom = pres-unit , hom\n\n  open MonoidHomomorphism mon-hom public\n\n  pres-inv : \u2200 {x} \u2192 f (0\u2212 x) \u2261 (f x)\u207b\u00b9\n  pres-inv {x} = unique-\u207b\u00b9 part\n    where part = f (0\u2212 x) * f x  \u2261\u27e8 ! hom \u27e9\n                 f (0\u2212 x + x)    \u2261\u27e8 ap f (fst 0\u2212-inverse) \u27e9\n                 f 0#            \u2261\u27e8 pres-unit \u27e9\n                 1#              \u220e\n\n  0\u2212-\u207b\u00b9 = pres-inv\n\n  \u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n  \u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 hom \u27e9\n                f x * f (0\u2212 y)  \u2261\u27e8 ap (_*_ (f x)) pres-inv \u27e9\n                f x \/ f y       \u220e\n\n  hom-iterated\u207b : \u2200 {x} n \u2192 f (x \u2297\u207b n) \u2261 f x ^\u207b n\n  hom-iterated\u207b {x} n =\n    f (x \u2297\u207b n)      \u2261\u27e8by-definition\u27e9\n    f (0\u2212(x \u2297\u207a n))  \u2261\u27e8 pres-inv \u27e9\n    f(x \u2297\u207a n)\u207b\u00b9     \u2261\u27e8 ap _\u207b\u00b9 (hom-iterated\u207a n) \u27e9\n    (f x ^\u207a n)\u207b\u00b9    \u2261\u27e8by-definition\u27e9\n    f x ^\u207b n        \u220e\n\n  hom-iterated : \u2200 {x} i \u2192 f (x \u2297 i) \u2261 f x ^ i\n  hom-iterated -[1+ n ] = hom-iterated\u207b (1+ n)\n  hom-iterated (+ n)    = hom-iterated\u207a n\n\n\u2124+-grp-ops : Group-Ops \u2124\n\u2124+-grp-ops = \u2124+-mon-ops , -_\n\n\u2124+-grp-struct : Group-Struct \u2124+-grp-ops\n\u2124+-grp-struct = \u2124+-mon-struct\n              , (\u03bb{x} \u2192 fst \u2124\u00b0.-\u203finverse x)\n              , (\u03bb{x} \u2192 snd \u2124\u00b0.-\u203finverse x)\n\n\u2124+-grp : Group \u2124\n\u2124+-grp = _ , \u2124+-grp-struct\n\nmodule \u2124+ = Additive-Group \u2124+-grp\n\nmodule _ {\u2113}{G : Set \u2113}(\ud835\udd3e : Group G) where\n  open Group\u1d52\u1d56\n  open Group \ud835\udd3e\n\n  module \u207b\u00b9-Hom where\n    -- The proper type for \u207b\u00b9-hom\u2032\n    \u207b\u00b9-hom' : GroupHomomorphism \ud835\udd3e (\ud835\udd3e \u1d52\u1d56) _\u207b\u00b9\n    \u207b\u00b9-hom' = mk \u207b\u00b9-hom\u2032\n    open GroupHomomorphism \u207b\u00b9-hom' public\n\n  module \u2124+-^-Hom {b} where\n    ^-+-hom : GroupHomomorphism \u2124+-grp \ud835\udd3e (_^_ b)\n    ^-+-hom = mk (\u03bb {i} {j} \u2192 ^-+ i j)\n\n    open GroupHomomorphism ^-+-hom public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b30a78ae45d4773a715f5d749b8b5378d0bcbbd7","subject":"Renaming without so it doesnt clash with substitution","message":"Renaming without so it doesnt clash with substitution\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"contexts.agda","new_file":"contexts.agda","new_contents":"open import Prelude\nopen import List\nopen import Nat\n\nmodule contexts where\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  _ctx : Set \u2192 Set\n  A ctx = Nat \u2192 Maybe A\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : {A : Set} \u2192 A ctx\n  \u2205 _ = None\n\n  infixr 100 \u25a0_\n\n  -- membership, or presence, in a context\n  _\u2208_ : {A : Set} (p : Nat \u00d7 A) \u2192 (\u0393 : A ctx) \u2192 Set\n  (x , y) \u2208 \u0393 = (\u0393 x) == Some y\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : {A : Set} (n : Nat) \u2192 (\u0393 : A ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- disjointness for contexts\n  _##_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 Set\n  _##_ {A} \u0393 \u0393'  = ((n : Nat) (a : A) \u2192 \u0393 n == Some a \u2192 \u0393' n == None) \u00d7\n                   ((n : Nat) (a : A) \u2192 \u0393' n == Some a \u2192 \u0393 n == None)\n\n  -- without: remove a variable from a context\n  _\/\/_ : {A : Set} \u2192 A ctx \u2192 Nat \u2192 A ctx\n  (\u0393 \/\/ x) y with natEQ x y\n  (\u0393 \/\/ x) .x | Inl refl = None\n  (\u0393 \/\/ x) y  | Inr neq  = \u0393 y\n\n  -- a variable is apart from any context from which it is removed\n  aar : {A : Set} \u2192 (\u0393 : A ctx) (x : Nat) \u2192 x # (\u0393 \/\/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {A : Set} \u2192 {\u0393 : A ctx} {n : Nat} {t t' : A} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- warning: this is union but it assumes WITHOUT CHECKING that the\n  -- domains are disjoint.\n  _\u222a_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 A ctx\n  (C1 \u222a C2) x with C1 x\n  (C1 \u222a C2) x | Some x\u2081 = Some x\u2081\n  (C1 \u222a C2) x | None = C2 x\n\n  -- the singleton context\n  \u25a0_ : {A : Set} \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u25a0 (x , a)) y with natEQ x y\n  (\u25a0 (x , a)) .x | Inl refl = Some a\n  ... | Inr _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : {A : Set} \u2192 A ctx \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u0393 ,, (x , t)) = \u0393 \u222a (\u25a0 (x , t))\n\n  infixl 10 _,,_\n\n  x\u2208\u222a1 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  x\u2208\u222a1 \u0393 \u0393' n x xin with \u0393 n\n  x\u2208\u222a1 \u0393 \u0393' n x\u2081 xin | Some x = xin\n  x\u2208\u222a1 \u0393 \u0393' n x ()   | None\n\n\n  x\u2208\u25a0 : {A : Set} (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u25a0 (n , a))\n  x\u2208\u25a0 n a with natEQ n n\n  x\u2208\u25a0 n a | Inl refl = refl\n  x\u2208\u25a0 n a | Inr x = abort (x refl)\n\n  postulate -- TODO\n    x\u2208sing : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\n    \u222acomm : {A : Set} \u2192 (C1 C2 : A ctx) \u2192 (C1 \u222a C2) == (C2 \u222a C1)\n\n-- x\u2208sing \u0393 n x with \u0393 n\n  -- x\u2208sing \u0393 n x  | Some y with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inl refl = {!!}\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inr x = abort (x refl)\n  -- x\u2208sing \u0393 n x  | None with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | None | Inl refl = refl\n  -- x\u2208sing \u0393 n x\u2081 | None | Inr x = abort (x refl)\n\n  -- x\u2208\u222a2 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  -- x\u2208\u222a2 \u0393 \u0393' n x xin with \u0393' n | \u0393 n\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2082 xin | Some x | Some x\u2081 = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x refl | Some .x | None = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2081 xin | None   | _ = abort (somenotnone (! xin))\n\n-- x\u2208\u222a\u25a0 : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 \u222a (\u25a0 (n , a)))\n  -- x\u2208\u222a\u25a0 \u0393 n a with natEQ n n\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inl refl = {!!} -- it might be in \u0393 because i don't know that they're disjoint. this is where that premise gets you\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inr x = {!!}\n\n  -- x\u2208sing \u0393 n a  = {!!} -- x\u2208\u222a2 \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a)\n","old_contents":"open import Prelude\nopen import List\nopen import Nat\n\nmodule contexts where\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  _ctx : Set \u2192 Set\n  A ctx = Nat \u2192 Maybe A\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : {A : Set} \u2192 A ctx\n  \u2205 _ = None\n\n  infixr 100 \u25a0_\n\n  -- membership, or presence, in a context\n  _\u2208_ : {A : Set} (p : Nat \u00d7 A) \u2192 (\u0393 : A ctx) \u2192 Set\n  (x , y) \u2208 \u0393 = (\u0393 x) == Some y\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : {A : Set} (n : Nat) \u2192 (\u0393 : A ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- disjointness for contexts\n  _##_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 Set\n  _##_ {A} \u0393 \u0393'  = ((n : Nat) (a : A) \u2192 \u0393 n == Some a \u2192 \u0393' n == None) \u00d7\n                   ((n : Nat) (a : A) \u2192 \u0393' n == Some a \u2192 \u0393 n == None)\n\n  -- without: remove a variable from a context\n  _\/_ : {A : Set} \u2192 A ctx \u2192 Nat \u2192 A ctx\n  (\u0393 \/ x) y with natEQ x y\n  (\u0393 \/ x) .x | Inl refl = None\n  (\u0393 \/ x) y  | Inr neq  = \u0393 y\n\n  -- a variable is apart from any context from which it is removed\n  aar : {A : Set} \u2192 (\u0393 : A ctx) (x : Nat) \u2192 x # (\u0393 \/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {A : Set} \u2192 {\u0393 : A ctx} {n : Nat} {t t' : A} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- warning: this is union but it assumes WITHOUT CHECKING that the\n  -- domains are disjoint.\n  _\u222a_ : {A : Set} \u2192 A ctx \u2192 A ctx \u2192 A ctx\n  (C1 \u222a C2) x with C1 x\n  (C1 \u222a C2) x | Some x\u2081 = Some x\u2081\n  (C1 \u222a C2) x | None = C2 x\n\n  -- the singleton context\n  \u25a0_ : {A : Set} \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u25a0 (x , a)) y with natEQ x y\n  (\u25a0 (x , a)) .x | Inl refl = Some a\n  ... | Inr _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : {A : Set} \u2192 A ctx \u2192 (Nat \u00d7 A) \u2192 A ctx\n  (\u0393 ,, (x , t)) = \u0393 \u222a (\u25a0 (x , t))\n\n  infixl 10 _,,_\n\n  x\u2208\u222a1 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393 \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  x\u2208\u222a1 \u0393 \u0393' n x xin with \u0393 n\n  x\u2208\u222a1 \u0393 \u0393' n x\u2081 xin | Some x = xin\n  x\u2208\u222a1 \u0393 \u0393' n x ()   | None\n\n\n  x\u2208\u25a0 : {A : Set} (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u25a0 (n , a))\n  x\u2208\u25a0 n a with natEQ n n\n  x\u2208\u25a0 n a | Inl refl = refl\n  x\u2208\u25a0 n a | Inr x = abort (x refl)\n\n  postulate -- TODO\n    x\u2208sing : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 ,, (n , a))\n    \u222acomm : {A : Set} \u2192 (C1 C2 : A ctx) \u2192 (C1 \u222a C2) == (C2 \u222a C1)\n\n-- x\u2208sing \u0393 n x with \u0393 n\n  -- x\u2208sing \u0393 n x  | Some y with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inl refl = {!!}\n  -- x\u2208sing \u0393 n x\u2081 | Some y | Inr x = abort (x refl)\n  -- x\u2208sing \u0393 n x  | None with natEQ n n\n  -- x\u2208sing \u0393 n x\u2081 | None | Inl refl = refl\n  -- x\u2208sing \u0393 n x\u2081 | None | Inr x = abort (x refl)\n\n  -- x\u2208\u222a2 : {A : Set} \u2192 (\u0393 \u0393' : A ctx) (n : Nat) (x : A) \u2192 (n , x) \u2208 \u0393' \u2192 (n , x) \u2208 (\u0393 \u222a \u0393')\n  -- x\u2208\u222a2 \u0393 \u0393' n x xin with \u0393' n | \u0393 n\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2082 xin | Some x | Some x\u2081 = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x refl | Some .x | None = {!!}\n  -- x\u2208\u222a2 \u0393 \u0393' n x\u2081 xin | None   | _ = abort (somenotnone (! xin))\n\n-- x\u2208\u222a\u25a0 : {A : Set} \u2192 (\u0393 : A ctx) (n : Nat) (a : A) \u2192 (n , a) \u2208 (\u0393 \u222a (\u25a0 (n , a)))\n  -- x\u2208\u222a\u25a0 \u0393 n a with natEQ n n\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inl refl = {!!} -- it might be in \u0393 because i don't know that they're disjoint. this is where that premise gets you\n  -- x\u2208\u222a\u25a0 \u0393 n a | Inr x = {!!}\n\n  -- x\u2208sing \u0393 n a  = {!!} -- x\u2208\u222a2 \u0393 (\u25a0 (n , a)) n a (x\u2208\u25a0 n a)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"09e20d9c2273b80ffe7b71077db2ed3c9f7ed71b","subject":"vcomp: typo","message":"vcomp: typo\n","repos":"crypto-agda\/crypto-agda","old_file":"vcomp.agda","new_file":"vcomp.agda","new_contents":"module vcomp where\n\nopen import Level\nopen import Function\nopen import Data.Unit using (\u22a4)\nopen import composable using (ConstArr)\n\nVComposition : \u2200 {i j} {I : Set i} (_\u00d7_ : I \u2192 I \u2192 I) (_\u219d_ : I \u2192 I \u2192 Set j) \u2192 Set (i \u2294 j)\nVComposition _\u00d7_ _\u219d_ = \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u219d o\u2080 \u2192 i\u2081 \u219d o\u2081 \u2192 (i\u2080 \u00d7 i\u2081) \u219d (o\u2080 \u00d7 o\u2081)\n\nVCIComposition : \u2200 {i t a} {I : Set i} {T : Set t}\n                   (_\u00d7\u1d62_ : I \u2192 I \u2192 I)\n                   (_\u00d7_ : T \u2192 T \u2192 T)\n                   (\u27e8_\u27e9_\u219d_ : I \u2192 T \u2192 T \u2192 Set a) \u2192 Set (a \u2294 t \u2294 i)\nVCIComposition _\u00d7\u1d62_ _\u00d7_ \u27e8_\u27e9_\u219d_ =\n  \u2200 {j\u2080 j\u2081 i\u2080 i\u2081 o\u2080 o\u2081} \u2192 \u27e8 j\u2080 \u27e9 i\u2080 \u219d o\u2080 \u2192 \u27e8 j\u2081 \u27e9 i\u2081 \u219d o\u2081 \u2192 \u27e8 j\u2080 \u00d7\u1d62 j\u2081 \u27e9 (i\u2080 \u00d7 i\u2081) \u219d (o\u2080 \u00d7 o\u2081)\n\nrecord VIComposable {i j t a} {I : Set i} {_\u00d7\u1d62_ : I \u2192 I \u2192 I} {_\u219d\u1d62_ : I \u2192 I \u2192 Set j} {T : I \u2192 Set t}\n                    (_\u00b7_ : VComposition _\u00d7\u1d62_ _\u219d\u1d62_)\n                    (_\u00d7_ : \u2200 {i\u2080 i\u2081} \u2192 T i\u2080 \u2192 T i\u2081 \u2192 T (i\u2080 \u00d7\u1d62 i\u2081))\n                    (\u27e8_\u27e9_\u219d_ : \u2200 {i\u2080 i\u2081} \u2192 i\u2080 \u219d\u1d62 i\u2081 \u2192 T i\u2080 \u2192 T i\u2081 \u2192 Set a) : Set (a \u2294 t \u2294 i \u2294 j) where\n  constructor mk\n  infixr 3 _***_\n  field\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {j\u2080 : i\u2080 \u219d\u1d62 o\u2080} {j\u2081 : i\u2081 \u219d\u1d62 o\u2081} {A B C D}\n            \u2192 \u27e8 j\u2080 \u27e9 A \u219d C \u2192 \u27e8 j\u2081 \u27e9 B \u219d D\n            \u2192 \u27e8 j\u2080 \u00b7 j\u2081 \u27e9 (A \u00d7 B) \u219d (C \u00d7 D)\n\nvComposition : \u2200 {t a} {T : Set t} {_\u00d7_ : T \u2192 T \u2192 T} {_\u219d_ : T \u2192 T \u2192 Set a}\n               \u2192 VIComposable {I = \u22a4} {_\u219d\u1d62_ = \u03bb _ _ \u2192 \u22a4} {T = const T} _ _\u00d7_ (const _\u219d_)\n               \u2192 VComposition _\u00d7_ _\u219d_\nvComposition (mk _***_) = _***_\n\nvCIComposition : \u2200 {i t a} {T : Set t} {_\u00d7_ : T \u2192 T \u2192 T}\n                 {I : Set i} {_\u00b7_ : I \u2192 I \u2192 I}\n                 {\u27e8_\u27e9_\u219d_ : I \u2192 T \u2192 T \u2192 Set a}\n               \u2192 VIComposable {I = \u22a4} _\u00b7_ _\u00d7_ \u27e8_\u27e9_\u219d_\n               \u2192 VCIComposition _\u00b7_ _\u00d7_ \u27e8_\u27e9_\u219d_\nvCIComposition (mk _***_) = _***_\n\nVComposable : \u2200 {t a} {T : Set t} (_\u00d7_ : T \u2192 T \u2192 T) (_\u219d_ : T \u2192 T \u2192 Set a) \u2192 Set (t \u2294 a)\nVComposable _\u00d7_ _\u219d_ = VIComposable {i = zero} {_\u219d\u1d62_ = ConstArr \u22a4} _ _\u00d7_ (const _\u219d_)\n\n{- VComposable unfolds to:\nrecord VComposable {t a} {T : Set t} (_\u00b7_ : T \u2192 T \u2192 T) (_\u219d_ : T \u2192 T \u2192 Set a) : Set (t \u2294 a) where\n  constructor mk\n  infixr 3 _***_\n  field\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u219d o\u2080 \u2192 i\u2081 \u219d o\u2081 \u2192 (i\u2080 \u00b7 i\u2081) \u219d (o\u2080 \u00b7 o\u2081)\n-}\n\nmodule VComposable = VIComposable\n\nopen import Data.Nat\nopen import Data.Fin.NP as Fin using (Fin; inject+; raise; bound; free)\n\nopVComp : \u2200 {t a} {T : Set t} {_\u00d7_ : T \u2192 T \u2192 T} {_\u219d_ : T \u2192 T \u2192 Set a}\n          \u2192 VComposable _\u00d7_ _\u219d_ \u2192 VComposable (flip _\u00d7_) (flip _\u219d_)\nopVComp (mk _***_) = mk (flip _***_)\n\nopen import Data.Product\n\nfunVComp : \u2200 {t} \u2192 VComposable _\u00d7_ (\u03bb (A B : Set t) \u2192 A \u2192 B)\nfunVComp {t} = mk _***_ where\n  _***_ : \u2200 {A B C D : Set t} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n\nfinFunVComp : VComposable _+_ (\u03bb i o \u2192 Fin i \u2192 Fin o)\nfinFunVComp = mk _***_ where\n  C = \u03bb i o \u2192 Fin i \u2192 Fin o\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  _***_ {i}      f g x with Fin.cmp i _ x\n  _***_          f _ ._ | Fin.bound x = inject+ _ (f x)\n  _***_ {o\u2080 = o} _ g ._ | Fin.free  x = raise o (g x)\n\n-- finFunOpVComp = opVComp finFunVComp [modulo commutativity of _+_]\nfinFunOpVComp : VComposable _+_ (\u03bb i o \u2192 Fin o \u2192 Fin i)\nfinFunOpVComp = mk _***_ where\n  C = \u03bb i o \u2192 Fin o \u2192 Fin i\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  _***_ {o\u2080 = o} f g x with Fin.cmp o _ x\n  _***_          f _ ._ | Fin.bound x = inject+ _ (f x)\n  _***_ {i}      _ g ._ | Fin.free  x = raise i (g x)\n\nopen import Data.Vec\n\nvecFunVComp : \u2200 {a} (A : Set a) \u2192 VComposable _+_ (\u03bb i o \u2192 Vec A i \u2192 Vec A o)\nvecFunVComp A = mk _***_ where\n  C = \u03bb i o \u2192 Vec A i \u2192 Vec A o\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  (f *** g) xs with splitAt _ xs\n  ... | ys , zs , _ = f ys ++ g zs\n\nopen import Data.Unit using (\u22a4)\n\nconstVComp : \u2200 {a} {A : Set a} (_***_ : A \u2192 A \u2192 A) \u2192 VComposable _ (ConstArr A)\nconstVComp _***_ = mk _***_\n","old_contents":"module vcomp where\n\nopen import Level\nopen import Function\nopen import Data.Unit using (\u22a4)\nopen import composable using (ConstArr)\n\nVComposition : \u2200 {i j} {I : Set i} (_\u00d7_ : I \u2192 I \u2192 I) (_\u219d_ : I \u2192 I \u2192 Set j) \u2192 Set (i \u2294 j)\nVComposition _\u00d7_ _\u219d_ = \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u219d o\u2080 \u2192 i\u2081 \u219d o\u2081 \u2192 (i\u2080 \u00d7 i\u2081) \u219d (o\u2080 \u00d7 o\u2081)\n\nVCIComposition : \u2200 {i t a} {I : Set i} {T : Set t}\n                   (_\u00d7\u1d62_ : I \u2192 I \u2192 I)\n                   (_\u00d7_ : T \u2192 T \u2192 T)\n                   (\u27e8_\u27e9_\u219d_ : I \u2192 T \u2192 T \u2192 Set a) \u2192 Set (a \u2294 t \u2294 i)\nVCIComposition _\u00d7\u1d62_ _\u00d7_ \u27e8_\u27e9_\u219d_ =\n  \u2200 {j\u2080 j\u2081 i\u2080 i\u2081 o\u2080 o\u2081} \u2192 \u27e8 j\u2080 \u27e9 i\u2080 \u219d o\u2080 \u2192 \u27e8 j\u2081 \u27e9 i\u2081 \u219d o\u2081 \u2192 \u27e8 j\u2080 \u00d7\u1d62 j\u2081 \u27e9 (i\u2080 \u00d7 i\u2081) \u219d (o\u2080 \u00d7 o\u2081)\n\nrecord VIComposable {i j t a} {I : Set i} {_\u00d7\u1d62_ : I \u2192 I \u2192 I} {_\u219d\u1d62_ : I \u2192 I \u2192 Set j} {T : I \u2192 Set t}\n                    (_\u00b7_ : VComposition _\u00d7\u1d62_ _\u219d\u1d62_)\n                    (_\u00d7_ : \u2200 {i\u2080 i\u2081} \u2192 T i\u2080 \u2192 T i\u2081 \u2192 T (i\u2080 \u00d7\u1d62 i\u2081))\n                    (\u27e8_\u27e9_\u219d_ : \u2200 {i\u2080 i\u2081} \u2192 i\u2080 \u219d\u1d62 i\u2081 \u2192 T i\u2080 \u2192 T i\u2081 \u2192 Set a) : Set (a \u2294 t \u2294 i \u2294 j) where\n  constructor mk\n  infixr 3 _***_\n  field\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} {j\u2080 : i\u2080 \u219d\u1d62 o\u2080} {j\u2081 : i\u2081 \u219d\u1d62 o\u2081} {A B C D}\n            \u2192 \u27e8 j\u2080 \u27e9 A \u219d C \u2192 \u27e8 j\u2081 \u27e9 B \u219d D\n            \u2192 \u27e8 j\u2080 \u00b7 j\u2081 \u27e9 (A \u00d7 B) \u219d (C \u00d7 D)\n\nvComposition : \u2200 {t a} {T : Set t} {_\u00d7_ : T \u2192 T \u2192 T} {_\u219d_ : T \u2192 T \u2192 Set a}\n               \u2192 VIComposable {I = \u22a4} {_\u219d\u1d62_ = \u03bb _ _ \u2192 \u22a4} {T = const T} _ _\u00d7_ (const _\u219d_)\n               \u2192 VComposition _\u00d7_ _\u219d_\nvComposition (mk _***_) = _***_\n\nvCIComposition : \u2200 {i t a} {T : Set t} {_\u00d7_ : T \u2192 T \u2192 T}\n                 {I : Set i} {_\u00b7_ : I \u2192 I \u2192 I}\n                 {\u27e8_\u27e9_\u219d_ : I \u2192 T \u2192 T \u2192 Set a}\n               \u2192 VIComposable {I = \u22a4} _\u00b7_ _\u00d7_ \u27e8_\u27e9_\u219d_\n               \u2192 VCIComposition _\u00b7_ _\u00d7_ \u27e8_\u27e9_\u219d_\nvCIComposition (mk _***_) = _***_\n\nVComposable : \u2200 {t a} {T : Set t} (_\u00d7_ : T \u2192 T \u2192 T) (_\u219d_ : T \u2192 T \u2192 Set a) \u2192 Set (t \u2294 a)\nVComposable _\u00d7_ _\u219d_ = VIComposable {i = zero} {_\u219d\u1d62_ = ConstArr \u22a4} _ _\u00d7_ (const _\u219d_)\n\n{- VComposable unfolds to:\nrecord VComposable {t a} {T : Set t} (_\u00b7_ : T \u2192 T \u2192 T) (_\u219d_ : T \u2192 T \u2192 Set a) : Set (t \u2294 a) where\n  constructor mk\n  infixr 3 _***_\n  field\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u219d o\u2080 \u2192 i\u2081 \u219d o\u2081 \u2192 (i\u2080 \u00b7 i\u2081) \u219d (o\u2080 \u00b7 o\u2081)\n-}\n\nmodule VComposable = VIComposable\n\nopen import Data.Nat\nopen import Data.Fin.NP as Fin using (Fin; inject+; raise; bound; free)\n\nopVComp : \u2200 {t a} {T : Set t} {_\u00d7_ : T \u2192 T \u2192 T} {_\u219d_ : T \u2192 T \u2192 Set a}\n          \u2192 VComposable _\u00d7_ _\u219d_ \u2192 VComposable (flip _\u00d7_) (flip _\u219d_)\nopVComp (mk _***_) = mk (flip _***_)\n\nopen import Data.Product\n\nfunComp : \u2200 {t} \u2192 VComposable _\u00d7_ (\u03bb (A B : Set t) \u2192 A \u2192 B)\nfunComp {t} = mk _***_ where\n  _***_ : \u2200 {A B C D : Set t} \u2192 (A \u2192 B) \u2192 (C \u2192 D) \u2192 A \u00d7 C \u2192 B \u00d7 D\n  (f *** g) (x , y) = (f x , g y)\n\nfinFunVComp : VComposable _+_ (\u03bb i o \u2192 Fin i \u2192 Fin o)\nfinFunVComp = mk _***_ where\n  C = \u03bb i o \u2192 Fin i \u2192 Fin o\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  _***_ {i}      f g x with Fin.cmp i _ x\n  _***_          f _ ._ | Fin.bound x = inject+ _ (f x)\n  _***_ {o\u2080 = o} _ g ._ | Fin.free  x = raise o (g x)\n\n-- finFunOpVComp = opVComp finFunVComp [modulo commutativity of _+_]\nfinFunOpVComp : VComposable _+_ (\u03bb i o \u2192 Fin o \u2192 Fin i)\nfinFunOpVComp = mk _***_ where\n  C = \u03bb i o \u2192 Fin o \u2192 Fin i\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  _***_ {o\u2080 = o} f g x with Fin.cmp o _ x\n  _***_          f _ ._ | Fin.bound x = inject+ _ (f x)\n  _***_ {i}      _ g ._ | Fin.free  x = raise i (g x)\n\nopen import Data.Vec\n\nvecFunVComp : \u2200 {a} (A : Set a) \u2192 VComposable _+_ (\u03bb i o \u2192 Vec A i \u2192 Vec A o)\nvecFunVComp A = mk _***_ where\n  C = \u03bb i o \u2192 Vec A i \u2192 Vec A o\n  _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n  (f *** g) xs with splitAt _ xs\n  ... | ys , zs , _ = f ys ++ g zs\n\nopen import Data.Unit using (\u22a4)\n\nconstVComp : \u2200 {a} {A : Set a} (_***_ : A \u2192 A \u2192 A) \u2192 VComposable _ (ConstArr A)\nconstVComp _***_ = mk _***_\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c58a075ccee125982373089129d1a491d18d1652","subject":"bintree: -AC","message":"bintree: -AC\n","repos":"crypto-agda\/crypto-agda","old_file":"bintree.agda","new_file":"bintree.agda","new_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; _\u2264_; s\u2264s; _+_; module \u2115\u2264; module \u2115\u00b0)\nopen import Data.Nat.Properties\nopen import Data.Bool\nopen import Data.Vec using (_++_)\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero}  f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (false {-0b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (true  {-1b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\nleaf\u207f : \u2200 {n a} {A : Set a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : Set a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x rewrite fromConst\u2261leaf\u207f {n} x = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x) = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 Tree A m \u2192 Tree A n\nweaken\u2264 _       (leaf x)          = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 n {m a} {A : Set a} \u2192 Tree A m \u2192 Tree A (n + m)\nweaken+ n = weaken\u2264 (m\u2264n+m _ n)\n\njoin : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 Tree (Tree A c\u2082) c\u2081 \u2192 Tree A (c\u2081 + c\u2082)\njoin {c} (leaf x)          = weaken+ c x\njoin     (fork left right) = fork (join left) (join right)\n\n_>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\nf >>> g = map (flip lookup g) f\n\n_\u2192\u1d57_ : (i o : \u2115) \u2192 Set\ni \u2192\u1d57 o = Tree (Bits o) i\n\ncomposable : Composable _\u2192\u1d57_\ncomposable = mk _>>>_\n\n_***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2192\u1d57 o\u2080 \u2192 i\u2081 \u2192\u1d57 o\u2081 \u2192 (i\u2080 + i\u2081) \u2192\u1d57 (o\u2080 + o\u2081)\n(f *** g) = join (map (\u03bb xs \u2192 map (_++_ xs) g) f)\n\nvcomposable : VComposable _+_ _\u2192\u1d57_\nvcomposable = mk _***_\n\nforkable : Forkable suc _\u2192\u1d57_\nforkable = mk fork\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","old_contents":"module bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; _\u2264_; s\u2264s; _+_; module \u2115\u2264; module \u2115\u00b0)\nopen import Data.Nat.Properties\nopen import Data.Bool\nopen import Data.Vec using (_++_)\nopen import Data.Bits\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import composable\nopen import vcomp\nopen import forkable\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : \u2200 {n} \u2192 A \u2192 Tree A n\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero}  f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f []        = refl\ntoFun\u2218fromFun {suc n} f (false {-0b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 0\u2237_) bs\ntoFun\u2218fromFun {suc n} f (true  {-1b-} \u2237 bs) = toFun\u2218fromFun {n} (f \u2218 1\u2237_) bs\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\nleaf\u207f : \u2200 {n a} {A : Set a} \u2192 A \u2192 Tree A n\nleaf\u207f {zero}  x = leaf x\nleaf\u207f {suc n} x = fork t t where t = leaf\u207f x\n\nexpand : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Tree A n\nexpand (leaf x) = leaf\u207f x\nexpand (fork t\u2080 t\u2081) = fork (expand t\u2080) (expand t\u2081)\n\nfromConst\u2261leaf\u207f : \u2200 {n a} {A : Set a} (x : A) \u2192 fromFun (const x) \u2261 leaf\u207f {n} x\nfromConst\u2261leaf\u207f {zero}  _ = refl\nfromConst\u2261leaf\u207f {suc n} x rewrite fromConst\u2261leaf\u207f {n} x = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 expand t\nfromFun\u2218toFun (leaf x) = fromConst\u2261leaf\u207f x\nfromFun\u2218toFun (fork t\u2080 t\u2081) = cong\u2082 fork (fromFun\u2218toFun t\u2080) (fromFun\u2218toFun t\u2081)\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 m \u2264 n \u2192 Tree A m \u2192 Tree A n\nweaken\u2264 _       (leaf x)          = leaf x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nm\u2264n+m : \u2200 m n \u2192 m \u2264 n + m\nm\u2264n+m m n = \u2115\u2264.trans (m\u2264m+n m n) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m n))\n\nweaken+ : \u2200 n {m a} {A : Set a} \u2192 Tree A m \u2192 Tree A (n + m)\nweaken+ n = weaken\u2264 (m\u2264n+m _ n)\n\njoin : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 Tree (Tree A c\u2082) c\u2081 \u2192 Tree A (c\u2081 + c\u2082)\njoin {c} (leaf x)          = weaken+ c x\njoin     (fork left right) = fork (join left) (join right)\n\n_>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\nf >>> g = map (flip lookup g) f\n\n_\u2192\u1d57_ : (i o : \u2115) \u2192 Set\ni \u2192\u1d57 o = Tree (Bits o) i\n\ncomposable : Composable _\u2192\u1d57_\ncomposable = mk _>>>_\n\n_***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2192\u1d57 o\u2080 \u2192 i\u2081 \u2192\u1d57 o\u2081 \u2192 (i\u2080 + i\u2081) \u2192\u1d57 (o\u2080 + o\u2081)\n(f *** g) = join (map (\u03bb xs \u2192 map (_++_ xs) g) f)\n\nvcomposable : VComposable _+_ _\u2192\u1d57_\nvcomposable = mk _***_\n\nforkable : Forkable suc _\u2192\u1d57_\nforkable = mk fork\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 {n} x \u2192 Rot {n = n} (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata AC {a} {A : Set a} : \u2200 {m n} (left : Tree A m) (right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 AC t t\n\n  _\u204f_ : \u2200 {m n o} {t : Tree A m} {u : Tree A n} {v : Tree A o} \u2192 AC t u \u2192 AC u v \u2192 AC t v\n\n  first : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         AC left\u2080 left\u2081 \u2192\n         AC (fork left\u2080 right) (fork left\u2081 right)\n\n  swp : \u2200 {m n} {left : Tree A m} {right : Tree A n} \u2192\n         AC (fork left right) (fork right left)\n\n  assoc : \u2200 {n} {t\u2080 t\u2081 : Tree A n} {t\u2082 : Tree A (suc n)} \u2192\n          AC (fork (fork t\u2080 t\u2081) t\u2082) (fork t\u2080 (fork t\u2081 t\u2082))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ff23fcc7d38198cdc5601d297c43bca536910433","subject":"documented the thought process of otp-lem","message":"documented the thought process of otp-lem\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-lem.agda","new_file":"otp-lem.agda","new_contents":"module otp-lem where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\n\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Search.Type\nopen import Search.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)(f : A \u2192 B)(searchA : Search zero A)(f-homo : GroupHomomorphism GA GB f)([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n              (sui : \u2200 k \u2192 StableUnder searchA (flip (Group._\u2219_ GA) k))(search-ext : SearchExt searchA) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  {- How all this is related to elgamal\n\n  the Group GA is \u2124q with modular addition as operation\n  the Group GB is the cyclic group with order q\n\n  f is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the search (for type A) function (should work with only summation)\n  is Stable under addition of GA (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the search function respects extensionality\n  -}\n  \n  {-\n   I had some problems with using the standard library definiton of Groups\n   so I rolled my own, therefor I need some boring proofs first\n\n  -}\n  \n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  {-\n    While this proof looks complicated it basically just adds inverse of m\u2080 and then adds m\u2081 (from image of f)\n    we need the homomorphic property to pull out the values.\n\n    Should be possible to simplify a little, or atleast name some parts betweeen calls of sui..\n  -}\n\n               \n  -- this proof isn't actually any hard..\n  thm : \u2200 {X}(op : X \u2192 X \u2192 X)(O : B \u2192 X) m\u2080 m\u2081 \u2192 searchA op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 searchA op (\u03bb x \u2192 O (f x * m\u2081))\n  thm op O m\u2080 m\u2081 = searchA op (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) op (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong (\u03bb p \u2192 O (p * m\u2080)) (f-homo x (- [f] m\u2080))) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * f(- [f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O ((f x * p) * m\u2080))) (f-pres-inv ([f] m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * 1\/ f([f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (Group.assoc GB (f x) (1\/ f ([f] m\u2080)) m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ f([f] m\u2080)  * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * (1\/ p * m\u2080)))) (f-sur m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ m\u2080 * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (proj\u2081 (Group.inverse GB) m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * 1g))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (proj\u2082 (Group.identity GB) (f x))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) op (\u03bb x \u2192 O (f x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (f-homo x ([f] m\u2081))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * f ([f] m\u2081)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (f-sur m\u2081) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n    where\n      open \u2261-Reasoning\n      \n","old_contents":"module otp-lem where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\n\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Search.Type\nopen import Search.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)(f : A \u2192 B)(searchA : Search zero A)(f-homo : GroupHomomorphism GA GB f)([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n              (sui : \u2200 k \u2192 StableUnder searchA (flip (Group._\u2219_ GA) k))(search-ext : SearchExt searchA) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  \n  -- this proof isn't actually any hard..\n  thm : \u2200 {X}(op : X \u2192 X \u2192 X)(O : B \u2192 X) m\u2080 m\u2081 \u2192 searchA op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 searchA op (\u03bb x \u2192 O (f x * m\u2081))\n  thm op O m\u2080 m\u2081 = searchA op (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) op (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong (\u03bb p \u2192 O (p * m\u2080)) (f-homo x (- [f] m\u2080))) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * f(- [f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O ((f x * p) * m\u2080))) (f-pres-inv ([f] m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * 1\/ f([f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (Group.assoc GB (f x) (1\/ f ([f] m\u2080)) m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ f([f] m\u2080)  * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * (1\/ p * m\u2080)))) (f-sur m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ m\u2080 * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (proj\u2081 (Group.inverse GB) m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * 1g))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (proj\u2082 (Group.identity GB) (f x))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) op (\u03bb x \u2192 O (f x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (f-homo x ([f] m\u2081))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * f ([f] m\u2081)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (f-sur m\u2081) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n    where\n      open \u2261-Reasoning\n      \n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"634f458ffffacc3e5bc977bed4de2be936703233","subject":"Add helpers","message":"Add helpers\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Lang.agda","new_file":"New\/Lang.agda","new_contents":"module New.Lang where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Sum\n\nopen import New.Types public\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\nopen import Base.Data.DependentList public\n\nmodule _ where\n  data Const : (\u03c4 : Type) \u2192 Set where\n  data Term (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set where\n    -- constants aka. primitives\n    const : \u2200 {\u03c4} \u2192\n      (c : Const \u03c4) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n  -- Shorthands for nested applications\n  app\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\n  app\u2082 f x = app (app f x)\n\n  app\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\n  app\u2083 f x = app\u2082 (app f x)\n\n  app\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\n  app\u2084 f x = app\u2083 (app f x)\n\n  app\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\n  app\u2085 f x = app\u2084 (app f x)\n\n  app\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\n  app\u2086 f x = app\u2085 (app f x)\n\nopen import Base.Denotation.Environment Type \u27e6_\u27e7Type public\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Const ()\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1)\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7Context) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7Term \u03c1 \u2261 \u27e6 t \u27e7Term (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7\u227c \u03c1)\nweaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\nweaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (const c) \u03c1 = refl\n","old_contents":"module New.Lang where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Sum\n\nopen import New.Types public\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\nopen import Base.Data.DependentList public\n\nmodule _ where\n  data Const : (\u03c4 : Type) \u2192 Set where\n  data Term (\u0393 : Context) :\n    (\u03c4 : Type) \u2192 Set where\n    -- constants aka. primitives\n    const : \u2200 {\u03c4} \u2192\n      (c : Const \u03c4) \u2192\n      Term \u0393 \u03c4\n    var : \u2200 {\u03c4} \u2192\n      (x : Var \u0393 \u03c4) \u2192\n      Term \u0393 \u03c4\n    app : \u2200 {\u03c3 \u03c4}\n      (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n      (t : Term \u0393 \u03c3) \u2192\n      Term \u0393 \u03c4\n    -- we use de Bruijn indicies, so we don't need binding occurrences.\n    abs : \u2200 {\u03c3 \u03c4}\n      (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n  -- Weakening\n\n  weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n    (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n    Term \u0393\u2081 \u03c4 \u2192\n    Term \u0393\u2082 \u03c4\n  weaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\n  weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\n  weaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\n  weaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\nopen import Base.Denotation.Environment Type \u27e6_\u27e7Type public\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Const ()\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7Var \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1)\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7Context) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7Term \u03c1 \u2261 \u27e6 t \u27e7Term (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7\u227c \u03c1)\nweaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\nweaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (const c) \u03c1 = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"872c56d277bdcd62f4a8191fac36bdfb5017a65c","subject":"updating the proof for automation","message":"updating the proof for automation\n","repos":"crypto-agda\/crypto-agda","old_file":"tactic.agda","new_file":"tactic.agda","new_contents":"module tactic where\n\nopen import Type\nopen import Search.Type\nopen import Search.Searchable\nopen import Search.Searchable.Product\n\nopen import Data.Unit\nopen import Data.Product\nopen import Function\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule _ {R S} (sum-R : Sum R)(sum-R-ext : SumExt sum-R)(sum-S : Sum S) where\n  su-\u00d7 : \u2200 {f : R \u2192 R}{g : S \u2192 S} \u2192 SumStableUnder sum-R f \u2192 SumStableUnder sum-S g\n       \u2192 SumStableUnder (sum-R \u00d7-sum sum-S) (map f g)\n  su-\u00d7 {f}{g} su-f su-g F\n    = (sum-R \u00d7-sum sum-S) F\n    \u2261\u27e8 sum-R-ext (\u03bb x \u2192 su-g (\u03bb y \u2192 F ( x , y))) \u27e9\n      (sum-R \u00d7-sum sum-S) (F \u2218 map id g)\n    \u2261\u27e8 su-f (\u03bb x \u2192 sum-S (\u03bb y \u2192 F (map id g (x , y)))) \u27e9\n      (sum-R \u00d7-sum sum-S) (F \u2218 map f g)\n    \u220e where open \u2261-Reasoning\n\nmodule _ {R}(sum-R : Sum R) where\n  su-id : SumStableUnder sum-R id\n  su-id f = refl\n\n  su-\u2218 : \u2200 {f g} \u2192 SumStableUnder sum-R f \u2192 SumStableUnder sum-R g \u2192 SumStableUnder sum-R (f \u2218 g)\n  su-\u2218 {f} su-f su-g F = trans (su-f F) (su-g (F \u2218 f))\n\nmodule with-sum {R R' S'}(sum-R : Sum R)(sum-R' : Sum R')(sum-S' : Sum S')(sum-R'-ext : SumExt sum-R')\n           (h : R' \u2192 R')(su-h : SumStableUnder sum-R' h)\n           (to : R' \u2192 S' \u2192 R)(sum-same : \u2200 f \u2192 sum-R f \u2261 sum-R' (\u03bb r' \u2192 sum-S' (f \u2218 to r'))) where\n\n  principle : \u2200 {f g} \u2192 (\u2200 r' \u2192 sum-S' (f \u2218 to r') \u2261 sum-S' (g \u2218 to (h r'))) \u2192 sum-R f \u2261 sum-R g\n  principle {f}{g} prf\n    = sum-R f\n    \u2261\u27e8 sum-same f \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 f (to r' s')))\n    \u2261\u27e8 sum-R'-ext (\u03bb r' \u2192 prf r') \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to (h r') s')))\n    \u2261\u27e8 sym (su-h (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to r' s')))) \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to r' s')))\n    \u2261\u27e8 sym (sum-same g) \u27e9\n      sum-R g\n    \u220e\n    where open \u2261-Reasoning\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_) renaming (_$\u2081_ to to ; _$\u2082_ to from)\n    \nmodule with-iso {R R' S' : Set}(iso : R \u2194 (R' \u00d7 S'))(h : R' \u2194 R') where\n\n  open import Function.Related\n  open import Data.Fin using (Fin)\n\n  open import Function.Related.TypeIsomorphisms.NP\n\n  principle : \u2200 {f g} \u2192 (\u2200 r' \u2192 \u03a3 S' (Fin \u2218 f \u2218 curry (from iso) r')\n                              \u2194 \u03a3 S' (Fin \u2218 g \u2218 curry (from iso) (to h r')))\n            \u2192 \u03a3 R (Fin \u2218 f) \u2194 \u03a3 R (Fin \u2218 g)\n  principle {f} {g} prf\n    = \u03a3 R (Fin \u2218 f)\n    \u2194\u27e8 first-iso iso \u27e9\n      \u03a3 (R' \u00d7 S') (Fin \u2218 f \u2218 from iso)\n    \u2194\u27e8 \u03a3-assoc \u27e9\n      \u03a3 R' (\u03bb r' \u2192 \u03a3 S' (\u03bb s' \u2192 Fin (f (from iso (r' , s')))))\n    \u2194\u27e8 second-iso prf \u27e9\n      \u03a3 R' (\u03bb r' \u2192 \u03a3 S' (\u03bb s' \u2192 Fin (g (from iso (to h r' , s')))))\n    \u2194\u27e8 Inv.sym (first-iso (Inv.sym h)) \u27e9\n      \u03a3 R' (\u03bb r' \u2192 \u03a3 S' (\u03bb s' \u2192 Fin (g (from iso (r' , s')))))\n    \u2194\u27e8 Inv.sym \u03a3-assoc \u27e9\n      \u03a3 (R' \u00d7 S') (Fin \u2218 g \u2218 from iso)\n    \u2194\u27e8 Inv.sym (first-iso iso) \u27e9\n      \u03a3 R (Fin \u2218 g)\n    \u220e where open EquationalReasoning \n\n  module _ (sum-R : Sum R)(sum-S' : Sum S')(sum-R-adq : AdequateSum sum-R)(sum-S'-adq : AdequateSum sum-S') where\n    cool : \u2200 {f g} \u2192 (\u2200 r' \u2192 sum-S' (f \u2218 curry (from iso) r')\n                           \u2261 sum-S' (g \u2218 curry (from iso) (to h r')))\n         \u2192 sum-R f \u2261 sum-R g\n    cool {f} {g} prf = Fin-injective (Inv.sym (sum-R-adq g)\n                               Inv.\u2218 principle lemma\n                               Inv.\u2218 sum-R-adq f)\n\n     where\n        open EquationalReasoning\n        lemma : (_ : _) \u2192 _\n        lemma r' = \u03a3 S' (Fin \u2218 f \u2218 curry (from iso) r')\n                 \u2194\u27e8 Inv.sym (sum-S'-adq (f \u2218 curry (from iso) r')) \u27e9\n                   Fin (sum-S' (f \u2218 curry (from iso) r'))\n                 \u2194\u27e8 Fin-cong (prf r') \u27e9\n                   Fin (sum-S' (g \u2218 curry (from iso) (to h r')))\n                 \u2194\u27e8 sum-S'-adq (g \u2218 curry (from iso) (to h r')) \u27e9\n                   \u03a3 S' (Fin \u2218 g \u2218 curry (from iso) (to h r'))\n                 \u220e\n        \n","old_contents":"module tactic where\n\nopen import Type\nopen import Search.Type\nopen import Search.Searchable\nopen import Search.Searchable.Product\n\nopen import Data.Unit\nopen import Data.Product\nopen import Function\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule _ {R S} (sum-R : Sum R)(sum-R-ext : SumExt sum-R)(sum-S : Sum S) where\n  su-\u00d7 : \u2200 {f : R \u2192 R}{g : S \u2192 S} \u2192 SumStableUnder sum-R f \u2192 SumStableUnder sum-S g\n       \u2192 SumStableUnder (sum-R \u00d7-sum sum-S) (map f g)\n  su-\u00d7 {f}{g} su-f su-g F\n    = (sum-R \u00d7-sum sum-S) F\n    \u2261\u27e8 sum-R-ext (\u03bb x \u2192 su-g (\u03bb y \u2192 F ( x , y))) \u27e9\n      (sum-R \u00d7-sum sum-S) (F \u2218 map id g)\n    \u2261\u27e8 su-f (\u03bb x \u2192 sum-S (\u03bb y \u2192 F (map id g (x , y)))) \u27e9\n      (sum-R \u00d7-sum sum-S) (F \u2218 map f g)\n    \u220e where open \u2261-Reasoning\n\nmodule _ {R}(sum-R : Sum R) where\n  su-id : SumStableUnder sum-R id\n  su-id f = refl\n\n  su-\u2218 : \u2200 {f g} \u2192 SumStableUnder sum-R f \u2192 SumStableUnder sum-R g \u2192 SumStableUnder sum-R (f \u2218 g)\n  su-\u2218 {f} su-f su-g F = trans (su-f F) (su-g (F \u2218 f))\n\nmodule _ {R R' S'}(sum-R : Sum R)(sum-R' : Sum R')(sum-S' : Sum S')(sum-R'-ext : SumExt sum-R')\n           (h : R' \u2192 R')(su-h : SumStableUnder sum-R' h)\n           (to : R' \u2192 S' \u2192 R)(sum-same : \u2200 f \u2192 sum-R f \u2261 sum-R' (\u03bb r' \u2192 sum-S' (f \u2218 to r'))) where\n\n  principle : \u2200 {f g} \u2192 (\u2200 r' \u2192 sum-S' (f \u2218 to r') \u2261 sum-S' (g \u2218 to (h r'))) \u2192 sum-R f \u2261 sum-R g\n  principle {f}{g} prf\n    = sum-R f\n    \u2261\u27e8 sum-same f \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 f (to r' s')))\n    \u2261\u27e8 sum-R'-ext (\u03bb r' \u2192 prf r') \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to (h r') s')))\n    \u2261\u27e8 sym (su-h (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to r' s')))) \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to r' s')))\n    \u2261\u27e8 sym (sum-same g) \u27e9\n      sum-R g\n    \u220e\n    where open \u2261-Reasoning\n  \n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4c166f1235965703622eb1d357a0f1e078b60f8f","subject":"Group.Isomorphism: the interesting part of stability preservation","message":"Group.Isomorphism: the interesting part of stability preservation\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Isomorphism.agda","new_file":"lib\/Algebra\/Group\/Isomorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type_)\nopen import Level.NP\nopen import Function.NP\nopen import Data.Product using (_,_)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nmodule Algebra.Group.Isomorphism where\n\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n  hiding (module Stability; module Stability-Minimal)\n\nrecord GroupIsomorphism\n         {a}{A : Type a}\n         {b}{B : Type b}\n         (G+   : Group A)\n         (G*   : Group B) : Set(a \u2294 b) where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n\n  field\n    \u03c6 : A \u2192 B -- TODO\n    \u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y\n    \u03c6\u207b\u00b9   : B \u2192 A\n    \u03c6\u207b\u00b9-\u03c6 : \u2200 {x} \u2192 \u03c6\u207b\u00b9 (\u03c6 x) \u2261 x\n    \u03c6-\u03c6\u207b\u00b9 : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x\n\n  \u03c6-hom : GroupHomomorphism G+ G* \u03c6\n  \u03c6-hom = mk \u03c6-+-*\n\n  open GroupHomomorphism \u03c6-hom public\n\n{- How this proof can be used for crypto, in particular ElGamal to DDH\n\n  the Group A is \u2124q with modular addition as operation\n  the Group B is the cyclic group with order q\n\n  \u03c6 is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  F is the summation operator and is required to be stable\n  under addition of A.\n\n  F should respects extensionality\n\n  This proof adds \u03c6\u207b\u00b9 k, because adding a constant is stable under\n  the big op F, this addition can then be pulled homomorphically\n  through f, to become a, multiplication by k.\n-}\nmodule Stability-Minimal\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (\u03c6     : A \u2192 B)\n  (\u03c6\u207b\u00b9   : B \u2192 A)\n  (\u03c6-\u03c6\u207b\u00b9 : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  {c}{C  : Type c}\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  where\n\n  module _ (F\u03c6+ : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)) where\n\n    *-stable : \u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6)\n    *-stable {k} =\n      F \u03c6                  \u2261\u27e8 F\u03c6+ \u27e9\n      F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 k))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 k + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                         \u03c6 (\u03c6\u207b\u00b9 k) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-\u03c6\u207b\u00b9 idp \u27e9\n                                         k * \u03c6 x         \u220e) \u27e9\n      F (_*_ k \u2218 \u03c6)        \u220e\n\n  {- The reverse direction comes from the homomorphism -}\n  open Algebra.Group.Homomorphism.Stability-Minimal\n    \u03c6 _+_ _*_ \u03c6-+-* F F=\n\n  stability : (\u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k)) \u2194 (\u2200 {k} \u2192 F \u03c6 \u2261 F (_*_ k \u2218 \u03c6))\n  stability = *-stable , +-stable\n\nmodule Stability\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6-iso : GroupIsomorphism G+ G*)\n  where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n  open GroupIsomorphism \u03c6-iso\n\n  open Stability-Minimal \u03c6 \u03c6\u207b\u00b9 \u03c6-\u03c6\u207b\u00b9 _+_ _*_ \u03c6-+-* public\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{- How this proof can be used for crypto, in particular ElGamal to DDH\n\n  the Group A is \u2124q with modular addition as operation\n  the Group B is the cyclic group with order q\n\n  \u03c6 is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of A (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n\n  This proof adds \u03c6\u207b\u00b9 m, because adding a constant is stable under the\n  big op F, this addition can then be pulled homomorphically through\n  f, to become a, multiplication by m.\n-}\nopen import Type using (Type_)\nopen import Level.NP\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nmodule Algebra.Group.Isomorphism where\n\nmodule M\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  {c}{C  : Type c}\n  (\u03c6     : A \u2192 B)\n  (\u03c6\u207b\u00b9   : B \u2192 A)\n  (\u03c6-sur : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6+   : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k))\n  {m     : B}\n  where\n\n  *-stable : F \u03c6 \u2261 F (_*_ m \u2218 \u03c6)\n  *-stable =\n    F \u03c6                  \u2261\u27e8 F\u03c6+ \u27e9\n    F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 m))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 m + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                       \u03c6 (\u03c6\u207b\u00b9 m) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-sur idp \u27e9\n                                       m * \u03c6 x         \u220e) \u27e9\n    F (_*_ m \u2218 \u03c6)        \u220e\n\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n\nrecord GroupIsomorphism \n         {a}{A  : Type a}\n         {b}{B  : Type b}\n         (G+ : Group A)\n         (G* : Group B)\n          : Set(a \u2294 b) where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n\n  field\n    \u03c6 : A \u2192 B -- TODO\n    -- hom : GroupHomomorphism G+ G* \u03c6\n    \u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y\n    \u03c6\u207b\u00b9   : B \u2192 A\n    \u03c6-sur : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x\n    -- \u03c6-inj\n\nmodule N\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  {c}{C  : Type c}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6-iso : GroupIsomorphism G+ G*)\n  (open Additive-Group G+)\n  (open Multiplicative-Group G*)\n  (open GroupIsomorphism \u03c6-iso)\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F+    : \u2200 {k \u03c6} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k))\n  {m     : B}\n  where\n\n  *-stable : F \u03c6 \u2261 F (_*_ m \u2218 \u03c6)\n  *-stable =\n    F \u03c6                  \u2261\u27e8 F+ \u27e9\n    F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 m))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 m + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                       \u03c6 (\u03c6\u207b\u00b9 m) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-sur idp \u27e9\n                                       m * \u03c6 x         \u220e) \u27e9\n    F (_*_ m \u2218 \u03c6)        \u220e\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dc8fa267d4bde5baedcd2cd8d18114e56dd46a90","subject":"working on #3, lots of new questions","message":"working on #3, lots of new questions\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-boxed-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus\n    where\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n    lem (TACast wt TCRefl) (ICastArr x\u2083 ind) \u03c41 \u03c42 .\u03c41 .\u03c42 d refl = x\u2083 refl\n    lem (TACast wt (TCArr x\u2082 x\u2083)) (ICastArr x\u2084 ind) \u03c41 \u03c42 \u03c43 \u03c4' d refl = {!!}\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i = I (IAp {!!} {!!} (FIndet i)) -- cyrus (issue from above, and also i think missing a rule for indet applications)\n  progress (TAAp wt1 wt2) | V v | V v\u2082 = {!!} -- {!canonical-boxed-forms-arr wt1 x !}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  -- indet cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | I x = S (_ , Step FHOuter (ITCastID (FIndet x)) FHOuter)\n  progress (TACast wt TCHole1) | I x = I (ICastGroundHole {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | I x = I (ICastHoleGround {!!} x {!!}) -- cyrus\n  progress (TACast wt (TCArr c1 c2)) | I x = I (ICastArr {!!} x) -- cyrus\n  -- boxed value cases, inspect how the casts are realted by consistency\n  progress (TACast wt TCRefl)  | V x = S (_ , Step FHOuter (ITCastID (FBoxed x)) FHOuter)\n  progress (TACast wt TCHole1) | V x = V (BVHoleCast {!!} x) -- cyrus\n  progress (TACast wt TCHole2) | V x = V {!!} -- cyrus: missing rule for boxed values?\n  progress (TACast wt (TCArr c1 c2)) | V x = V (BVArrCast {!!} x) -- cyrus\n\n\n  -- this would fill two above, but it's false: \u2987\u2988 indet but its type, \u2987\u2988, is not ground\n  -- lem-groundindet : \u2200{ \u0394 d \u03c4} \u2192 \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192 d indet \u2192 \u03c4 ground\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-boxed-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n    -- if the left steps, the whole thing steps\n  progress (TAAp wt1 wt2) | S (_ , Step x y z) | _ = S (_ , Step (FHAp1 x) y (FHAp1 z))\n    -- if the left is an error, the whole thing is an error\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n    -- if the left is indeterminate, inspect the right\n  progress (TAAp wt1 wt2) | I i | S (_ , Step x y z) = S (_ , Step (FHAp2 (FIndet i) x) y (FHAp2 (FIndet i) z))\n  progress (TAAp wt1 wt2) | I x | E x\u2081 = E (CECong (FHAp2 (FIndet x) FHOuter) x\u2081)\n  progress (TAAp wt1 wt2) | I x | I x\u2081 = I (IAp {!!} x (FIndet x\u2081)) -- cyrus\n    where\n    lem : \u2200{ \u0394 d1 \u03c4 \u03c4'} \u2192 \u0394 , \u2205 \u22a2 d1 :: (\u03c4 ==> \u03c4') \u2192\n                          d1 indet \u2192\n                          ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192 d1 \u2260 (d1' \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9))\n    lem (TACast wt TCRefl) (ICastArr x\u2083 ind) \u03c41 \u03c42 .\u03c41 .\u03c42 d refl = x\u2083 refl\n    lem (TACast wt (TCArr x\u2082 x\u2083)) (ICastArr x\u2084 ind) \u03c41 \u03c42 \u03c43 \u03c4' d refl = {!!}\n  progress (TAAp wt1 wt2) | I x | V x\u2081 = I (IAp {!!} x (FBoxed x\u2081)) -- cyrus\n    -- if the left is a boxed value, inspect the right\n  progress (TAAp wt1 wt2) | V v | S (_ , Step x y z) = S (_ , Step (FHAp2 (FBoxed v) x) y (FHAp2 (FBoxed v) z))\n  progress (TAAp wt1 wt2) | V v | E e = E (CECong (FHAp2 (FBoxed v) FHOuter) e)\n  progress (TAAp wt1 wt2) | V v | I i = I (IAp {!!} {!!} (FIndet i)) -- cyrus (issue from above, and also i think missing a rule for indet applications)\n  progress (TAAp wt1 wt2) | V v | V v\u2082 = {!!} -- {!canonical-boxed-forms-arr wt1 x !}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHNEHole x) y (FHNEHole z))\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt con)\n    with progress wt\n  ... | S (_ , Step x y z) = S (_ , Step (FHCast x) y (FHCast z))\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  ... | I x = I {!!}\n  ... | V x = V {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"16e4674089315f65f2fe2b2160b09f2da165c9ab","subject":"Add a simple record for probability distributions","message":"Add a simple record for probability distributions\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field     \n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248 \n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n  \n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[ \n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[ \n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n \n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  > \n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  record Distr : Set where\n    constructor _,_\n    field\n      P : U \u2192 [0,1]\n      sumP\u22611 : sumP P allU \u2261 1I\n\n  module Prob (d : Distr) where\n    open Distr d\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)","old_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Sum\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1[-ops (]0,1[ : Set) (_<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set) : Set where\n  constructor mk\n\n  field     \n    -- 1E : ]0,1]\n    1+_\/1+x_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n   -- proofs <E\n  field\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  field\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[){y : ]0,1[} \u2192 (x \u00b7E y) <E y\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  -- proofs \u2261 maybe should be some Equivalence \u2248 \n  field\n    +E-sym : (x y : ]0,1[) \u2192 x +E y \u2261 y +E x\n    +E-assoc : (x y z : ]0,1[) \u2192 (x +E y) +E z \u2261 x +E (y +E z)\n    \u00b7\/E-identity : (x : ]0,1[){y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\nmodule ]0,1[-\u211a where\n  data ]0,1[ : Set where\n    1+_\/2+x+_ : \u2115 \u2192 \u2115 \u2192 ]0,1[\n  -- to\u211a : ]0,1] \u2192 \u211a\n  -- to\u211a (1+ x \/1+x+ y) = (1 + x) \/\u211a (1 + x + y)\n  \n  postulate\n    _<E_ : ]0,1[ \u2192 ]0,1[ \u2192 Set\n  -- (1+ x \/1+x+ y) \u2264E (1+ x' \/1+x+ y') = ?\n  -- (1 + x' + y') * (1 + x) \u2264E (1 + x + y) * (1 + x') = ?\n  postulate\n    _+E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[\n    _\u00b7E_ : ]0,1[ \u2192 ]0,1[ \u2192 ]0,1[ \n    _\/E_<_> : (x y : ]0,1[) \u2192 x <E y \u2192 ]0,1[\n\n  postulate\n    <E-trans : \u2200 {x y z} \u2192 x <E y \u2192 y <E z \u2192 x <E z\n\n  postulate\n    +E-mono\u2081 : \u2200 x {y} \u2192 y <E (y +E x)\n    +E-mono\u2082 : \u2200 x {y z} \u2192 y <E z \u2192 (x +E y) <E (x +E z)\n    \u00b7E-anti\u2081 : (x : ]0,1[) {x\u2081 : ]0,1[} \u2192 (x \u00b7E x\u2081) <E x\u2081\n    \u00b7E-anti\u2082 : (x : ]0,1[){y z : ]0,1[} \u2192 y <E z \u2192 (x \u00b7E y) <E z\n    \u00b7\/E-assoc : (x y z : ]0,1[)(pf : y <E z) \u2192 x \u00b7E (y \/E z < pf >) \u2261 (x \u00b7E y) \/E z < \u00b7E-anti\u2082 x pf >\n\n  postulate\n    +E-sym : (x x\u2081 : ]0,1[) \u2192 x +E x\u2081 \u2261 x\u2081 +E x\n    +E-assoc : (x x\u2081 x\u2082 : ]0,1[) \u2192 (x +E x\u2081) +E x\u2082 \u2261 x +E (x\u2081 +E x\u2082)\n    \u00b7\/E-identity : (x : ]0,1[) {y : ]0,1[} \u2192 x \u2261 (x \u00b7E y) \/E y < \u00b7E-anti\u2081 x >\n\n  ops : ]0,1[-ops ]0,1[ _<E_\n  ops = mk 1+_\/2+x+_ _+E_ _\u00b7E_ _\/E_<_> <E-trans +E-mono\u2081 +E-mono\u2082 \u00b7E-anti\u2081 \u00b7E-anti\u2082 \u00b7\/E-assoc +E-sym +E-assoc \u00b7\/E-identity\n\nmodule [0,1] {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_) where\n\n  open ]0,1[-ops ]0,1[R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   1I : [0,1]\n   _I : ]0,1[ \u2192 [0,1]\n\n  data _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set where\n    z\u2264n : \u2200 {n} \u2192 0I \u2264I n\n    n\u22641 : \u2200 {n} \u2192 n \u2264I 1I\n    E\u2264E : \u2200 {x y} \u2192 x <E y \u2192 (x I) \u2264I (y I)\n    E\u2261E : \u2200 {x} \u2192 x I \u2264I x I\n\n\n  \u2264I-refl : \u2200 {x} \u2192 x \u2264I x\n  \u2264I-refl {0I} = z\u2264n\n  \u2264I-refl {1I} = n\u22641\n  \u2264I-refl {x I} = E\u2261E\n\n  Pos : [0,1] \u2192 Set\n  Pos 0I    = \u22a5\n  Pos 1I    = \u22a4\n  Pos (_ I) = \u22a4\n\n  Inc : [0,1] \u2192 Set\n  Inc 0I = \u22a5\n  Inc 1I = \u22a5\n  Inc (x I) = \u22a4\n\n  _<_> : (x : [0,1]) \u2192 Inc x \u2192 ]0,1[ \n  0I < () >\n  1I < () >\n  (x I) < pos > = x\n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  1I +I _ = 1I\n  x I +I 0I = x I\n  x I +I 1I = 1I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  1I \u00b7I y = y\n  (x I) \u00b7I 0I     = 0I\n  (x I) \u00b7I 1I     = x I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  -- 1\/I1+_ : \u2115 \u2192 [0,1]\n\n \n  _\/I_<_,_> : (x y : [0,1]) \u2192 x \u2264I y \u2192 Pos y \u2192 [0,1]\n  x \/I 0I    < _        , () >\n  x \/I 1I    < _        , _  > = x\n  0I \/I _    < _        , _  > = 0I\n  1I \/I _ I  < ()       , _  > \n  (x I) \/I y I < E\u2264E pf , _  > = (x \/E y < pf >) I\n  (x I) \/I .x I < E\u2261E   , _  > = 1I\n\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  *-anti : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x \u00b7I y \u2264I z\n  *-anti 0I le = z\u2264n\n  *-anti 1I le = le\n  *-anti (x I) z\u2264n = z\u2264n\n  *-anti (x I) n\u22641 = n\u22641\n  *-anti (x I) (E\u2264E pf) = E\u2264E (\u00b7E-anti\u2082 x pf)\n  *-anti (x I) E\u2261E = E\u2264E (\u00b7E-anti\u2081 x)\n\n  *\/-assoc : (x y z : [0,1])(pr : y \u2264I z)(pos : Pos z) \u2192 (x \u00b7I (y \/I z < pr , pos >)) \u2261 (x \u00b7I y) \/I z < *-anti x pr , pos >\n  *\/-assoc x y 0I pr ()\n  *\/-assoc x y 1I pr pos = refl\n  *\/-assoc 0I y (z I) pr pos = refl\n  *\/-assoc 1I y (z I) pr pos = refl\n  *\/-assoc (x I) 0I (z I) pr pos = refl\n  *\/-assoc (x I) 1I (z I) () pos\n  *\/-assoc (x I) (y I) (z I) (E\u2264E pf) pos = cong _I (\u00b7\/E-assoc x y z pf)\n  *\/-assoc (x I) (y I) (.y I) E\u2261E pos = cong _I (\u00b7\/E-identity x)\n\n  +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n  +I-sym 0I 0I = refl\n  +I-sym 0I 1I = refl\n  +I-sym 0I (x I) = refl\n  +I-sym 1I 0I = refl\n  +I-sym 1I 1I = refl\n  +I-sym 1I (x I) = refl\n  +I-sym (x I) 0I = refl\n  +I-sym (x I) 1I = refl\n  +I-sym (x I) (y I) = cong _I (+E-sym x y)\n\n  +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n  +I-assoc 0I y z = refl\n  +I-assoc 1I y z = refl\n  +I-assoc (x I) 0I z = refl\n  +I-assoc (x I) 1I z = refl\n  +I-assoc (x I) (y I) 0I = refl\n  +I-assoc (x I) (y I) 1I = refl\n  +I-assoc (x I) (y I) (z I) = cong _I (+E-assoc x y z)\n\n  \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n  \u2264I-trans z\u2264n le2 = z\u2264n\n  \u2264I-trans n\u22641 n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) n\u22641 = n\u22641\n  \u2264I-trans (E\u2264E x\u2081) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2081 x\u2082)\n  \u2264I-trans (E\u2264E x\u2081) E\u2261E = E\u2264E x\u2081\n  \u2264I-trans E\u2261E le2 = le2\n\n  \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n  \u2264I-mono 0I {z = z} le rewrite +I-sym z 0I = le\n  \u2264I-mono 1I {z = z} le rewrite +I-sym z 1I = n\u22641\n  \u2264I-mono (x I) z\u2264n = z\u2264n\n  \u2264I-mono (x I) n\u22641 = n\u22641\n  \u2264I-mono (x I) (E\u2264E x\u2082) = E\u2264E (<E-trans x\u2082 (+E-mono\u2081 x))\n  \u2264I-mono (x I) E\u2261E = E\u2264E (+E-mono\u2081 x)\n\n  \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n  \u2264I-pres 0I le = le\n  \u2264I-pres 1I le = n\u22641\n  \u2264I-pres (x I) {.0I} {0I} z\u2264n = E\u2261E\n  \u2264I-pres (x I) {.0I} {1I} z\u2264n = n\u22641\n  \u2264I-pres (x I) {.0I} {x\u2081 I} z\u2264n = E\u2264E (+E-mono\u2081 x\u2081)\n  \u2264I-pres (x I) n\u22641 = n\u22641\n  \u2264I-pres (x I) (E\u2264E x\u2082) = E\u2264E (+E-mono\u2082 x x\u2082)\n  \u2264I-pres (x I) E\u2261E = E\u2261E\n\n\nmodule Univ {]0,1[ _<E_} (]0,1[R : ]0,1[-ops ]0,1[ _<E_)\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1[R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  module Prob (P : U \u2192 [0,1])\n              (sumP\u22611 : sumP P allU \u2261 1I) where\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    _\u2286_ : Event \u2192 Event \u2192 Set\n    A \u2286 B = \u2200 x \u2192 T(A x) \u2192 T(B x)\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    postulate\n      Pr-mono : \u2200 {A B} \u2192 A \u2286 B \u2192 Pr[ A ] \u2264I Pr[ B ]\n\n    \u222a-lem : \u2200 {A} B \u2192 A \u2286 (A \u222a B)\n    \u222a-lem {A} _ x with A x\n    ... | true = id\n    ... | false = \u03bb()\n\n    \u2229-lem : \u2200 A {B} \u2192 (A \u2229 B) \u2286 B\n    \u2229-lem A x with A x\n    ... | true  = id\n    ... | false = \u03bb()\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < Pr-mono (\u2229-lem A) , pr >\n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b)","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d1098ca013c11b6341878675dd6ba7bf35982cfe","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOL\/Base.agda","new_file":"src\/fot\/FOL\/Base.agda","new_contents":"------------------------------------------------------------------------------\n-- First-order logic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOL.Base where\n\ninfix  2 \u22c0\ninfixr 1 _\u21d2_\ninfixr 0 _\u21d4_\n\n------------------------------------------------------------------------------\n-- First-order logic (without equality).\nopen import Common.FOL.FOL public\n\n------------------------------------------------------------------------------\n-- We added extra symbols for the implication, the biconditional and\n-- the universal quantification (see module Common.FOL.FOL).\n\n-- The implication data type.\ndata _\u21d2_ (A B : Set) : Set where\n  fun : (A \u2192 B) \u2192 A \u21d2 B\n\napp : {A B : Set} \u2192 A \u21d2 B \u2192 A \u2192 B\napp (fun f) a = f a\n\n-- Biconditional.\n_\u21d4_ : Set \u2192 Set \u2192 Set\nA \u21d4 B = (A \u21d2 B) \u2227 (B \u21d2 A)\n\n-- The universal quantifier type on D.\ndata  \u22c0 (A : D \u2192 Set) : Set where\n  dfun : ((t : D) \u2192 A t) \u2192 \u22c0 A\n\n-- Sugar syntax for the universal quantifier.\nsyntax \u22c0 (\u03bb x \u2192 e) = \u22c0[ x ] e\n\ndapp : {A : D \u2192 Set}(t : D) \u2192 \u22c0 A \u2192 A t\ndapp t (dfun f) = f t\n\n------------------------------------------------------------------------------\n-- In first-order logic it is assumed that the universe of discourse\n-- is non-empty.\npostulate D\u2262\u2205 : D\n\n------------------------------------------------------------------------------\n-- The ATPs work in classical logic, therefore we add the principle of\n-- the exclude middle for prove some non-intuitionistic theorems. Note\n-- that we do not need to add the postulate as an ATP axiom.\n\n-- The principle of the excluded middle.\npostulate pem : \u2200 {A} \u2192 A \u2228 \u00ac A\n","old_contents":"------------------------------------------------------------------------------\n-- First-order logic base\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOL.Base where\n\ninfix  2 \u22c0\ninfixr 1 _\u21d2_\ninfixr 0 _\u21d4_\n\n------------------------------------------------------------------------------\n-- First-order logic (without equality).\nopen import Common.FOL.FOL public\n\n------------------------------------------------------------------------------\n-- We added extra symbols for the implication, the biconditional and\n-- the universal quantification (see module Common.FOL.FOL).\n\n-- The implication data type.\ndata _\u21d2_ (A B : Set) : Set where\n  fun : (A \u2192 B) \u2192 A \u21d2 B\n\napp : {A B : Set} \u2192 A \u21d2 B \u2192 A \u2192 B\napp (fun f) a = f a\n\n-- Biconditional.\n_\u21d4_ : Set \u2192 Set \u2192 Set\nA \u21d4 B = (A \u21d2 B) \u2227 (B \u21d2 A)\n\n-- The universal quantifier type on D.\ndata  \u22c0 (A : D \u2192 Set) : Set where\n  dfun : ((t : D) \u2192 A t) \u2192 \u22c0 A\n\n-- Sugar syntax for the universal quantifier.\nsyntax \u22c0 (\u03bb x \u2192 e) = \u22c0[ x ] e\n\ndapp : {A : D \u2192 Set}(t : D) \u2192 \u22c0 A \u2192 A t\ndapp t (dfun f) = f t\n\n------------------------------------------------------------------------------\n-- In first-order logic it is assumed that the universe of discourse\n-- is nonempty.\npostulate D\u2262\u2205 : D\n\n------------------------------------------------------------------------------\n-- The ATPs work in classical logic, therefore we add the principle of\n-- the exclude middle for prove some non-intuitionistic theorems. Note\n-- that we do not need to add the postulate as an ATP axiom.\n\n-- The principle of the excluded middle.\npostulate pem : \u2200 {A} \u2192 A \u2228 \u00ac A\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9932e76b5b54e88e54dc74ba3485c58f35c63313","subject":"Fixed doc.","message":"Fixed doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"examples\/README.agda","new_file":"examples\/README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Paper, prerequisites, tested versions of the ATPS, and use\n\n-- See https:\/\/github.com\/asr\/agda2atp\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see examples\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper Combining Interactive and Automatic\n-- Reasoning in First Order Theories of Functional Programs by Ana\n-- Bove, Peter Dybjer, and Andr\u00e9s Sicard-Ram\u00edrez (FoSSaCS 2012).\n\n-- The code presented here does not match the paper exactly.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Paper, prerequisites, tested versions of the ATPS, and use\n\n-- See https:\/\/github.com\/asr\/agda2atp\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- In the modules with the formalization of the first-order theories,\n-- if the module's name ends in 'I' the module contains interactive\n-- proofs, if it ends in 'ATP' the module contains combined proofs,\n-- otherwise the module contains definitions and\/or interactive proofs\n-- that are used by the interactive and combined proofs.\n\n------------------------------------------------------------------------------\n-- First-order theories\n------------------------------------------------------------------------------\n\n-- \u2022 First-order logic with equality\n\n-- First-order logic (FOL)\nopen import FOL.README\n\n-- Propositional equality\nopen import Common.FOL.Relation.Binary.PropositionalEquality\n\n-- Equality reasoning\nopen import Common.FOL.Relation.Binary.EqReasoning\n\n-- \u2022 Group theory\n\nopen import GroupTheory.README\n\n-- \u2022 Distributive laws on a binary operation (Stanovsk\u00fd example)\n\nopen import DistributiveLaws.README\n\n-- \u2022 First-order Peano arithmetic (PA)\n\nopen import PA.README\n\n-- \u2022 First-Order Theory of Combinators (FOTC)\n\nopen import FOTC.README\n\n-- \u2022 Logical Theory of Constructions for PCF (LTC-PCF)\n\nopen import LTC-PCF.README\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"45a6182cb5916d691b87fe23987dfff2fce70412","subject":"woops missed one commentsign","message":"woops missed one commentsign\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nsearchInd' : \u2200 {A} {s\u1d2c : Search A} \u2192 SearchInd s\u1d2c \u2192 SearchInd' s\u1d2c\nsearchInd' Ps\u1d2c P P\u2219 {f} Pf = Ps\u1d2c (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nsearch-mono' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchMono s\u1d2c\nsearch-mono' s\u1d2c Ps\u1d2c _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nsearch-\u03b5' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 Search\u03b5 (searchMonFromSearch s\u1d2c)\nsearch-\u03b5' s\u1d2c Ps\u1d2c m = searchInd' Ps\u1d2c Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n  where open Mon m\n\nsearch-ext' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchExt s\u1d2c\nsearch-ext' s\u1d2c Ps\u1d2c sg {f} {g} f\u2248\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n  where open Sgrp sg\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n-- _+\u03bc_ {A} {B} \u03bcA \u03bcB = {!!}\n-- {-\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk srch ext mono swp eps hom\n  where\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e {C} P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) -- = {!!}\n-- {-\n   = mk srch ext mono swp eps hom\n   where\n     srch : Search _ -- (A \u00d7 B)\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nsearchInd' : \u2200 {A} {s\u1d2c : Search A} \u2192 SearchInd s\u1d2c \u2192 SearchInd' s\u1d2c\nsearchInd' Ps\u1d2c P P\u2219 {f} Pf = Ps\u1d2c (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nsearch-mono' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchMono s\u1d2c\nsearch-mono' s\u1d2c Ps\u1d2c _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nsearch-\u03b5' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 Search\u03b5 (searchMonFromSearch s\u1d2c)\nsearch-\u03b5' s\u1d2c Ps\u1d2c m = searchInd' Ps\u1d2c Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n  where open Mon m\n\nsearch-ext' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchExt s\u1d2c\nsearch-ext' s\u1d2c Ps\u1d2c sg {f} {g} f\u2248\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n  where open Sgrp sg\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n-- _+\u03bc_ {A} {B} \u03bcA \u03bcB = {!!}\n-- {-\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk srch ext mono swp eps hom\n  where\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e {C} P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) -- = {!!}\n-- {-\n   = mk srch ext mono swp eps hom\n   where\n     srch : Search _ -- (A \u00d7 B)\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n-}\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"65d48b8fa56a336f1ed2fb0595f0b7218102a693","subject":"RF: the proof is getting closer","message":"RF: the proof is getting closer\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/Transformation\/ReceiptFreeness-CCA2d.agda","new_file":"Game\/Transformation\/ReceiptFreeness-CCA2d.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.One\nopen import Data.Two\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Nat\nopen import Data.Vec hiding (_>>=_)\nopen import Data.List\nopen import Data.Fin as Fin using (Fin)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy renaming (map to mapS)\nimport Game.ReceiptFreeness\nimport Game.IND-CCA2-dagger\nimport Game.IND-CPA-utils\n\nmodule Game.Transformation.ReceiptFreeness-CCA2d\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (#q : \u2115) (max#q : Fin #q)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CipherText \u2192 Message)\n  (Check    : CipherText \u2192 \ud835\udfda)\n  (CheckEnc : \u2200 pk m r\u2091 \u2192 Check (Enc pk m r\u2091) \u2261 1\u2082)\n  where\n\n_\u00b2' : \u2605 \u2192 \u2605\nX \u00b2' = X \u00d7 X\n\nmodule _ where --StrategyUtils where\n  record Proto : \u2605\u2081 where\n    constructor P[_,_]\n    field\n      Q : \u2605\n      R : Q \u2192 \u2605\n\n  Client : Proto \u2192 \u2605 \u2192 \u2605\n  Client P[ Q , R ] A = Strategy Q R A\n\n  {-\n  data Bisim {P P' A A'} (RA : A \u2192 A' \u2192 \u2605) : Client P A \u2192 Client P' A' \u2192 \u2605 where\n    ask-nop : \u2200 {q? cont clt} r\n              \u2192 Bisim RA (cont r) clt\n              \u2192 Bisim RA (ask q? cont) clt\n    ask-ask : \u2200 q\u2080 q\u2081 cont\u2080 cont\u2081 r\u2080 r\u2081\n              \u2192 ({!!} \u2192 Bisim RA (cont\u2080 r\u2080) (cont\u2081 r\u2081))\n              \u2192 Bisim RA (ask q\u2080 cont\u2080) (ask q\u2081 cont\u2081)\n-}\n  module Unused where\n    mapS' : \u2200 {A B Q Q' R} (f : A \u2192 B) (g : Q \u2192 Q') \u2192 Strategy Q (R \u2218 g) A \u2192 Strategy Q' R B\n    mapS' f g (ask q cont) = ask (g q) (\u03bb r \u2192 mapS' f g (cont r))\n    mapS' f g (done x)     = done (f x)\n\n  [_,_]=<<_ : \u2200 {A B Q Q' R} (f : A \u2192 Strategy Q' R B) (g : Q \u2192 Q') \u2192 Strategy Q (R \u2218 g) A \u2192 Strategy Q' R B\n  [ f , g ]=<< ask q? cont = ask (g q?) (\u03bb r \u2192 [ f , g ]=<< cont r)\n  [ f , g ]=<< done x      = f x\n\n  module Rec {A B Q : \u2605} {R : Q \u2192 \u2605} {M : \u2605 \u2192 \u2605}\n             (runAsk : \u2200 {A} \u2192 (q : Q) \u2192 (R q \u2192 M A) \u2192 M A)\n             (runDone : A \u2192 M B) where\n    runMM : Strategy Q R A \u2192 M B\n    runMM (ask q? cont) = runAsk q? (\u03bb r \u2192 runMM (cont r))\n    runMM (done x)      = runDone x\n\n  module MM {A B Q : \u2605} {R : Q \u2192 \u2605} {M : \u2605 \u2192 \u2605}\n           (_>>=M_ : \u2200 {A B : \u2605} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B)\n           (runAsk : (q : Q) \u2192 M (R q))\n           (runDone : A \u2192 M B) where\n    runMM : Strategy Q R A \u2192 M B\n    runMM (ask q? cont) = runAsk q? >>=M (\u03bb r \u2192 runMM (cont r))\n    runMM (done x)      = runDone x\n\n  module _ {A B Q Q' : \u2605} {R : Q \u2192 \u2605} {R'}\n           (f : (q : Q) \u2192 Strategy Q' R' (R q))\n           (g : A \u2192 Strategy Q' R' B) where\n    [_,_]=<<'_ : Strategy Q R A \u2192 Strategy Q' R' B\n    [_,_]=<<'_ (ask q? cont) = f q? >>= \u03bb r \u2192 [_,_]=<<'_ (cont r)\n    [_,_]=<<'_ (done x)      = g x\n\n  record ServerResp (P : Proto) (q : Proto.Q P) (A : \u2605\u2080) : \u2605\u2080 where\n    coinductive\n    open Proto P\n    field\n        srv-resp : R q\n        srv-cont : \u2200 q \u2192 ServerResp P q A\n  open ServerResp\n\n  Server : Proto \u2192 \u2605\u2080 \u2192 \u2605\u2080\n  Server P A = \u2200 q \u2192 ServerResp P q A\n\n  OracleServer : Proto \u2192 \u2605\u2080\n  OracleServer P[ Q , R ] = (q : Q) \u2192 R q\n \n  module _ {P : Proto} {A : \u2605} where\n    open Proto P\n    com : Server P A \u2192 Client P A \u2192 A\n    com srv (ask q \u03bac) = com (srv-cont r) (\u03bac (srv-resp r)) where r = srv q\n    com srv (done x)   = x\n\n  module _ {P P' : Proto} {A A' : \u2605} where\n    module P = Proto P\n    module P' = Proto P'\n    record MITM : \u2605 where\n      coinductive\n      field\n        hack-query : (q' : P'.Q) \u2192 Client P (P'.R q' \u00d7 MITM)\n        hack-result : A' \u2192 Client P A\n    open MITM\n\n    module WithOutBind where\n        hacked-com-client : Server P A \u2192 MITM \u2192 Client P' A' \u2192 A\n        hacked-com-mitm : \u2200 {q'} \u2192 Server P A \u2192 Client P (P'.R q' \u00d7 MITM) \u2192 (P'.R q' \u2192 Client P' A') \u2192 A\n        hacked-com-srv-resp : \u2200 {q q'} \u2192 ServerResp P q A \u2192 (P.R q \u2192 Client P (P'.R q' \u00d7 MITM)) \u2192 (P'.R q' \u2192 Client P' A') \u2192 A\n\n        hacked-com-srv-resp r mitm clt = hacked-com-mitm (srv-cont r) (mitm (srv-resp r)) clt\n\n        hacked-com-mitm srv (ask q? mitm) clt = hacked-com-srv-resp (srv q?) mitm clt\n        hacked-com-mitm srv (done (r' , mitm)) clt = hacked-com-client srv mitm (clt r')\n\n        hacked-com-client srv mitm (ask q' \u03bac) = hacked-com-mitm srv (hack-query mitm q') \u03bac\n        hacked-com-client srv mitm (done x) = com srv (hack-result mitm x)\n\n    mitm-to-client-trans : MITM \u2192 Client P' A' \u2192 Client P A\n    mitm-to-client-trans mitm (ask q? cont) = hack-query mitm q? >>= \u03bb { (r' , mitm') \u2192 mitm-to-client-trans mitm' (cont r') }\n    mitm-to-client-trans mitm (done x)      = hack-result mitm x\n\n    hacked-com : Server P A \u2192 MITM \u2192 Client P' A' \u2192 A\n    hacked-com srv mitm clt = com srv (mitm-to-client-trans mitm clt)\n\n  module _ (P : Proto) (A : \u2605) where\n    open Proto P\n    open MITM\n    honest : MITM {P} {P} {A} {A}\n    hack-query  honest q = ask q \u03bb r \u2192 done (r , honest)\n    hack-result honest a = done a\n\n  module _ (P : Proto) (A : \u2605) (Oracle : OracleServer P) where\n    oracle-server : Server P A\n    srv-resp (oracle-server q) = Oracle q\n    srv-cont (oracle-server q) = oracle-server\n\nMessage = \ud835\udfda\nopen Game.IND-CPA-utils Message CipherText\nmodule RF    = Game.ReceiptFreeness PubKey SecKey         CipherText SerialNumber R\u2091 R\u2096 R\u2090  #q max#q KeyGen Enc Dec Check CheckEnc\nopen RF renaming (Phase to RFPhase; Q to RFQ; Resp to RFResp)\n\nR\u2090\u2020 : \u2605\nR\u2090\u2020 = R\u2090 \u00d7 {-SerialNumber \u00b2 \u00d7-} (Vec Rgb #q)\u00b2\nmodule CCA2\u2020 = Game.IND-CCA2-dagger PubKey SecKey Message CipherText              R\u2091 R\u2096 R\u2090\u2020          KeyGen Enc Dec\n\nCPAChallenger : (Next : \u2605) \u2192 \u2605\nCPAChallenger Next = Message \u00b2 \u2192 CipherText \u00b2 \u00d7 Next\n\nCCAProto : Proto\nCCAProto = P[ CipherText , const Message ]\n\nRFProto : Proto\nRFProto = P[ RFQ , RFResp ]\n\nDecRoundChallenger : (Next : \u2605) \u2192 \u2605\nDecRoundChallenger = Server CCAProto\n\nMITMState : \u2605 \u2192 \u2605\nMITMState X = X \u00d7 BB \u00d7 Tally\n\nmodule Receipts (m : \ud835\udfda) (sn : SerialNumber \u00b2) (ct : CipherText \u00b2) where\n  receipts : Candidate \u2192 Receipt\n  receipts c = marked m , sn c , ct c\n\n  trBB : BB \u2192 BB\n  trBB bb = receipts 0\u2082 \u2237 receipts 1\u2082 \u2237 bb\n\nmodule Simulator (m : \ud835\udfda {-which mark to put on the two receipts-})\n                 (t : \ud835\udfda {-which message to ask for in the challenge -})\n                 (RFA : RF.Adversary) where\n  module SecondLayer (rgb : (Vec Rgb #q)\u00b2) (pk : PubKey) where\n    open MITM\n\n    {-\n    askDecBB : BB \u2192 DecRound ClearBB\n    askDecBB [] = done []\n    askDecBB ((m? , sn , enc-co) \u2237 bb) = ask enc-co (\u03bb co \u2192 askDecBB bb >>= \u03bb dec-bb \u2192 done ((co , m?) \u2237 dec-bb))\n    -}\n\n    ballot : RF.PhaseNumber \u2192 Fin #q \u2192 Ballot\n    ballot p# n = RF.genBallot pk (lookup n (rgb p#))\n\n    MITM-phase : (p# : RF.PhaseNumber) \u2192 Fin #q \u2192 BB \u2192 Tally \u2192 \u2200 {X} \u2192 MITM {CCAProto} {RFProto} {MITMState X} {X}\n    hack-query (MITM-phase p# n bb ta) REB = done (ballot p# n , MITM-phase p# (Fin.pred n) bb ta)\n    hack-query (MITM-phase p# n bb ta) RBB = done (bb , MITM-phase p# (Fin.pred n) bb ta)\n    hack-query (MITM-phase p# n bb ta) RTally = done (ta , MITM-phase p# (Fin.pred n) bb ta)\n    hack-query (MITM-phase p# n bb ta) (RCO (m? , sn , enc-co)) =\n       -- if receipt in DB then ...\n       ask enc-co \u03bb co \u2192 done (co , MITM-phase p# (Fin.pred n) bb ta)\n    hack-query (MITM-phase p# n bb ta) (Vote (m? , sn , enc-co))\n       -- lots of cases\n       = ask enc-co \u03bb co \u2192\n           let (res , cbb' , bb') =\n                  case Check enc-co\n                    0: (RF.reject ,\u2032 ta , bb)\n                    1: (RF.accept , tallyMarkedReceipt? co m? +,+ ta , (m? , sn , enc-co) \u2237 bb) in\n           done (res , MITM-phase p# (Fin.pred n) bb' cbb')\n    hack-result (MITM-phase p# n bb ta) r = done (r , bb , ta)\n\n    MITM-RFChallenge : \u2200 {X} \u2192 MITM {_} {_} {MITMState X} {X}\n    MITM-RFChallenge = MITM-phase 0\u2082 max#q [] (0 , 0)\n\n    hack-challenge : \u2200 {X} \u2192 RFChallenge X \u2192 CPA\u2020Adversary (X \u00d7 (BB \u2192 BB))\n    get-m (hack-challenge _) = t , not t\n    put-c (hack-challenge rfc) c\u2080 c\u2081 = proj\u2082 rfc receipts , trBB\n      where\n        ct = proj (c\u2080 , c\u2081)\n        open Receipts m (proj\u2081 rfc) ct\n\n    module _ (bb : BB) (ta : Tally) (Aphase[II] : RFPhase Candidate) where\n\n      decRoundAdv2 : DecRound (MITMState Candidate)\n      decRoundAdv2 = mitm-to-client-trans (MITM-phase 1\u2082 max#q bb ta) Aphase[II]\n\n    mapCPAAdv = TransformAdversaryResponse.A*\n\n    A\u20203 : BB \u2192 Tally \u2192 (CipherText \u2192 RFPhase Candidate \u00d7 (BB \u2192 BB)) \u2192 CipherText \u2192 DecRound Candidate\n    A\u20203 bb ta = \u03bb f c \u2192 mapS proj\u2081 (decRoundAdv2 (proj\u2082 (f c) bb) ((1 , 1) +,+ ta) (proj\u2081 (f c)))\n\n    A\u20202 : MITMState (RFChallenge (RFPhase Candidate)) \u2192 CPA\u2020Adversary (DecRound Candidate)\n    A\u20202 (rfc , bb , ta) = mapCPAAdv (A\u20203 bb ta) (hack-challenge rfc)\n\n  module AdversaryParts (rgb : (Vec Rgb #q)\u00b2) (pk : PubKey) (r\u2090 : R\u2090) where\n    open SecondLayer rgb pk public\n    A\u20201 : Client CCAProto (MITMState (RFChallenge (RFPhase Candidate)))\n    A\u20201 = mitm-to-client-trans MITM-RFChallenge (RFA r\u2090 pk)\n\n  A\u2020 : CCA2\u2020.Adversary\n  A\u2020 (r\u2090 , rgb) pk = mapS A\u20202 A\u20201\n     where open AdversaryParts rgb pk r\u2090\n\nopen StatefulRun\nmodule Pfff1\n  (m : \ud835\udfda) (RFA : RF.Adversary) (pk : PubKey) (sk : SecKey)\n  (DecEnc : \u2200 r\u2091 m \u2192 Dec sk (Enc pk m r\u2091) \u2261 m)\n  (r\u2090 : R\u2090) (rgb : Candidate \u2192 Rgb)\n  (rgbs : PhaseNumber \u2192 Vec Rgb #q) (sn : Candidate \u2192 SerialNumber)\n  (ext\ud835\udfda : \u2200 {A : \u2605} {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n\n module PB (b : \ud835\udfda) where\n\n  -- When t = b then the simulator is behaving the same as an RF challenger\n  t = b\n\n  module Sim = Simulator m t RFA\n  module Tr = Sim.SecondLayer rgbs pk\n  open Tr using (ballot; hack-challenge; MITM-phase)\n  open Sim.AdversaryParts rgbs pk r\u2090 using (A\u20201; A\u20202; A\u20203)\n  A\u2020 = Sim.A\u2020\n\n  r\u2091 = proj\u2082 \u2218 proj\u2082 \u2218 rgb\n\n  module RFEXP = RF.EXP b RFA pk sk r\u2090 rgbs (Enc pk \u02e2 r\u2091) (const (marked m))\n  module EXP\u2020  = CCA2\u2020.EXP b A\u2020 (r\u2090 , rgbs) pk sk (r\u2091 0\u2082) (r\u2091 1\u2082)\n\n  module _ {X} (p# : PhaseNumber) where\n    RX : X \u00d7 RFEXP.S \u2192 MITMState X \u2192 \u2605\n    RX (x , bb , _) (x' , bb' , ta) = bb \u2261 bb' \u00d7 x \u2261 x' \u00d7 ta \u2261 tally sk bb'\n\n    Bisim' : (n : Fin #q) (bb : BB) \u2192 Client RFProto X \u2192 Client CCAProto (MITMState X) \u2192 \u2605\n    Bisim' n bb clt0 clt1 = RX (runS (RFEXP.OracleS p#) clt0 (bb , n)) (run (Dec sk) clt1)\n\n    pf-phase : (n : Fin #q) (bb : BB) (clt : Client _ _)\n                   \u2192 Bisim' n bb clt (mitm-to-client-trans (Tr.MITM-phase p# n bb (tally sk bb)) clt)\n    pf-phase n bb (ask REB cont) = pf-phase (Fin.pred n) bb (cont (Tr.ballot p# n))\n    pf-phase n bb (ask RBB cont) = pf-phase (Fin.pred n) bb (cont bb)\n    pf-phase n bb (ask RTally cont) = pf-phase (Fin.pred n) bb (cont (tally sk bb))\n    pf-phase n bb (ask (RCO (m? , sn , enc-co)) cont) = pf-phase (Fin.pred n) bb (cont (Dec sk enc-co))\n    pf-phase n bb (ask (Vote r) cont) with Check (enc-co r)\n    ... | 0\u2082 = pf-phase (Fin.pred n) bb (cont reject)\n    ... | 1\u2082 = pf-phase (Fin.pred n) (r \u2237 bb) (cont accept)\n    pf-phase n bb (done x) = refl , refl , refl\n\n  pf-phase[I] : Bisim' 0\u2082 max#q [] RFEXP.Aphase[I] A\u20201\n  pf-phase[I] = pf-phase 0\u2082 max#q [] RFEXP.Aphase[I]\n\n  MITM[I] = run (Dec sk) A\u20201\n  MITM-S[I] = proj\u2082 MITM[I]\n  MITM-BB[I] = proj\u2081 MITM-S[I]\n  MITM-tally[I] = proj\u2082 MITM-S[I]\n\n  tally[I] = tally sk RFEXP.BBphase[I]\n\n  BBphase[I]-pf : RFEXP.BBphase[I] \u2261 MITM-BB[I]\n  BBphase[I]-pf = proj\u2081 pf-phase[I]\n\n  tally[I]-pf : tally[I] \u2261 MITM-tally[I]\n  tally[I]-pf rewrite BBphase[I]-pf = sym (proj\u2082 (proj\u2082 pf-phase[I]))\n\n  CPA\u2020Challenge : CPA\u2020Adversary (RFPhase Candidate \u00d7 (BB \u2192 BB))\n  CPA\u2020Challenge = Tr.hack-challenge RFEXP.AdversaryRFChallenge\n\n  tally-pf : tally sk RFEXP.BBrfc \u2261 (1 , 1) +,+ tally[I]\n  tally-pf rewrite\n             DecEnc (proj\u2082 (proj\u2082 (rgb 0\u2082))) 0\u2082\n           | DecEnc (proj\u2082 (proj\u2082 (rgb 1\u2082))) 1\u2082\n           with m\n  ... | 0\u2082 = refl\n  ... | 1\u2082 = refl\n\n  pf-phase[II] : Bisim' 1\u2082 max#q RFEXP.BBrfc RFEXP.Aphase[II] (Tr.decRoundAdv2 RFEXP.BBrfc ((1 , 1) +,+ tally[I]) RFEXP.Aphase[II])\n  pf-phase[II] rewrite sym tally-pf = pf-phase 1\u2082 max#q RFEXP.BBrfc RFEXP.Aphase[II]\n\n  pf-phase[II]' : Bisim' 1\u2082 max#q RFEXP.BBrfc RFEXP.Aphase[II] (Tr.decRoundAdv2 RFEXP.BBrfc ((1 , 1) +,+ MITM-tally[I]) RFEXP.Aphase[II])\n  pf-phase[II]' rewrite sym tally[I]-pf = pf-phase[II]\n\n  pf-A\u2020 : run (Dec sk) (A\u2020 (r\u2090 , rgbs) pk) \u2261 A\u20202 (run (Dec sk) A\u20201)\n  pf-A\u2020 = run-map (Dec sk) A\u20202 A\u20201\n\n  open \u2261-Reasoning\n\n  put-c\u2080c\u2081 = put-c (Tr.hack-challenge (proj\u2081 (run (Dec sk) A\u20201))) EXP\u2020.c\u2080 EXP\u2020.c\u2081\n\n  rfc = proj\u2081 (run (Dec sk) A\u20201)\n  rfc' = proj\u2081 (runS (RFEXP.OracleS 0\u2082) RFEXP.Aphase[I] ([] , max#q))\n\n  proj\u2081-put-c\u2080c\u2081 : (rfc : RFChallenge (RFPhase Candidate)) \u2192 _\n  proj\u2081-put-c\u2080c\u2081 rfc = proj\u2082 rfc receipts\n      where open Receipts m (proj\u2081 rfc) (proj (EXP\u2020.c\u2080 , EXP\u2020.c\u2081))\n\n  proj\u2081-put-c\u2080c\u2081' : (ct : Candidate \u2192 CipherText) \u2192 _\n  proj\u2081-put-c\u2080c\u2081' ct = proj\u2082 rfc' receipts\n      where open Receipts m (proj\u2081 rfc') ct\n\n  proj\u2082-put-c\u2080c\u2081 : (rfc : RFChallenge (RFPhase Candidate)) \u2192 BB \u2192 BB\n  proj\u2082-put-c\u2080c\u2081 rfc = trBB\n      where open Receipts m (proj\u2081 rfc) EXP\u2020.ct\n\n  mb-pf : \u2200 c \u2192 EXP\u2020.mb (c xor b) \u2261 c\n  mb-pf 0\u2082 rewrite pf-A\u2020 with b\n  ... | 0\u2082 = refl\n  ... | 1\u2082 = refl\n  mb-pf 1\u2082 rewrite pf-A\u2020 with b\n  ... | 0\u2082 = refl\n  ... | 1\u2082 = refl\n\n  ct-pf : \u2200 c \u2192 EXP\u2020.ct c \u2261 (Enc pk \u02e2 r\u2091) c\n  ct-pf 0\u2082 = cong (\u03bb x \u2192 Enc pk x (proj\u2082 (proj\u2082 (rgb 0\u2082)))) (mb-pf 0\u2082)\n  ct-pf 1\u2082 = cong (\u03bb x \u2192 Enc pk x (proj\u2082 (proj\u2082 (rgb 1\u2082)))) (mb-pf 1\u2082)\n\n  BBrfc-pf' : RFEXP.BBrfc \u2261 proj\u2082-put-c\u2080c\u2081 rfc' MITM-BB[I]\n  BBrfc-pf' rewrite ct-pf 0\u2082 | ct-pf 1\u2082 = cong\u2082 _\u2237_ refl (cong\u2082 _\u2237_ refl (proj\u2081 pf-phase[I]))\n\n  BBrfc-pf : RFEXP.BBrfc \u2261 proj\u2082 put-c\u2080c\u2081 MITM-BB[I]\n  BBrfc-pf = trans BBrfc-pf' (cong (\u03bb x \u2192 proj\u2082-put-c\u2080c\u2081 x MITM-BB[I]) (proj\u2081 (proj\u2082 pf-phase[I])))\n\n  Aphase[II]-pf : RFEXP.Aphase[II] \u2261 proj\u2081 put-c\u2080c\u2081\n  Aphase[II]-pf = trans (ap proj\u2081-put-c\u2080c\u2081' (ext\ud835\udfda (sym \u2218 ct-pf))) (ap proj\u2081-put-c\u2080c\u2081 (proj\u2081 (proj\u2082 pf-phase[I])))\n\n  {-\n  BiSim3 : (p q : RFPhase Candidate) \u2192 \u2605\n  BiSim3 p q = \u2200 s \u2192 runS (RFEXP.OracleS 1\u2082) p s \u2261 runS (RFEXP.OracleS 1\u2082) q s\n  -}\n\n  pf-b\u2032 : RFEXP.b\u2032 \u2261 EXP\u2020.b\u2032\n  pf-b\u2032 = RFEXP.b\u2032\n        \u2261\u27e8 refl \u27e9\n          proj\u2081 (runS (RFEXP.OracleS 1\u2082) RFEXP.Aphase[II] (RFEXP.BBrfc , max#q))\n        \u2261\u27e8 proj\u2081 (proj\u2082 pf-phase[II]') \u27e9\n          proj\u2081 (run (Dec sk) (Tr.decRoundAdv2 RFEXP.BBrfc ((1 , 1) +,+ MITM-tally[I]) RFEXP.Aphase[II]))\n        \u2261\u27e8 ap (\u03bb x \u2192 proj\u2081 (run (Dec sk) (Tr.decRoundAdv2 x ((1 , 1) +,+ MITM-tally[I]) RFEXP.Aphase[II]))) BBrfc-pf \u27e9\n          proj\u2081 (run (Dec sk) (Tr.decRoundAdv2 (proj\u2082 put-c\u2080c\u2081 MITM-BB[I]) ((1 , 1) +,+ MITM-tally[I]) RFEXP.Aphase[II]))\n        \u2261\u27e8 ap (\u03bb x \u2192 proj\u2081 (run (Dec sk) (Tr.decRoundAdv2 (proj\u2082 put-c\u2080c\u2081 MITM-BB[I]) ((1 , 1) +,+ MITM-tally[I]) x))) Aphase[II]-pf \u27e9\n          proj\u2081 (run (Dec sk) (Tr.decRoundAdv2 (proj\u2082 put-c\u2080c\u2081 MITM-BB[I]) ((1 , 1) +,+ MITM-tally[I]) (proj\u2081 put-c\u2080c\u2081)))\n        \u2261\u27e8 sym (run-map (Dec sk) proj\u2081 (Tr.decRoundAdv2 (proj\u2082 put-c\u2080c\u2081 MITM-BB[I]) ((1 , 1) +,+ MITM-tally[I]) (proj\u2081 put-c\u2080c\u2081))) \u27e9\n          run (Dec sk) (mapS proj\u2081 (Tr.decRoundAdv2 (proj\u2082 put-c\u2080c\u2081 MITM-BB[I]) ((1 , 1) +,+ MITM-tally[I]) (proj\u2081 put-c\u2080c\u2081)))\n        \u2261\u27e8 refl \u27e9\n          run (Dec sk) ((A\u20203 MITM-BB[I] MITM-tally[I] (put-c (Tr.hack-challenge (proj\u2081 (run (Dec sk) A\u20201))) EXP\u2020.c\u2080)) EXP\u2020.c\u2081)\n        \u2261\u27e8 refl \u27e9\n          run (Dec sk) (put-c (A\u20202 (run (Dec sk) A\u20201)) EXP\u2020.c\u2080 EXP\u2020.c\u2081)\n        \u2261\u27e8 ap (\u03bb x \u2192 run (Dec sk) (put-c x EXP\u2020.c\u2080 EXP\u2020.c\u2081)) (sym pf-A\u2020) \u27e9\n          run (Dec sk) (put-c (run (Dec sk) (A\u2020 (r\u2090 , rgbs) pk)) EXP\u2020.c\u2080 EXP\u2020.c\u2081)\n        \u2261\u27e8 refl \u27e9\n          EXP\u2020.b\u2032\n        \u220e\n\n open PB\n foo : EXP\u2020.b\u2032 0\u2082 \u2261 EXP\u2020.b\u2032 1\u2082\n foo = {!refl!}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.One\nopen import Data.Two\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Nat\nopen import Data.Vec hiding (_>>=_)\nopen import Data.List\nopen import Data.Fin as Fin using (Fin)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy --renaming (map to mapS)\nimport Game.ReceiptFreeness\nimport Game.IND-CCA2-dagger\nimport Game.IND-CPA-utils\n\nmodule Game.Transformation.ReceiptFreeness-CCA2d\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (#q : \u2115) (max#q : Fin #q)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CipherText \u2192 Message)\n  (Check    : CipherText \u2192 \ud835\udfda)\n  (CheckEnc : \u2200 pk m r\u2091 \u2192 Check (Enc pk m r\u2091) \u2261 1\u2082)\n  where\n\nmodule _ where --StrategyUtils where\n  record Proto : \u2605\u2081 where\n    constructor P[_,_]\n    field\n      Q : \u2605\n      R : Q \u2192 \u2605\n\n  Client : Proto \u2192 \u2605 \u2192 \u2605\n  Client P[ Q , R ] A = Strategy Q R A\n\n  {-\n  data Bisim {P P' A A'} (RA : A \u2192 A' \u2192 \u2605) : Client P A \u2192 Client P' A' \u2192 \u2605 where\n    ask-nop : \u2200 {q? cont clt} r\n              \u2192 Bisim RA (cont r) clt\n              \u2192 Bisim RA (ask q? cont) clt\n    ask-ask : \u2200 q\u2080 q\u2081 cont\u2080 cont\u2081 r\u2080 r\u2081\n              \u2192 ({!!} \u2192 Bisim RA (cont\u2080 r\u2080) (cont\u2081 r\u2081))\n              \u2192 Bisim RA (ask q\u2080 cont\u2080) (ask q\u2081 cont\u2081)\n-}\n  mapS : \u2200 {A B Q Q' R} (f : A \u2192 B) (g : Q \u2192 Q') \u2192 Strategy Q (R \u2218 g) A \u2192 Strategy Q' R B\n  mapS f g (ask q cont) = ask (g q) (\u03bb r \u2192 mapS f g (cont r))\n  mapS f g (done x)     = done (f x)\n\n  [_,_]=<<_ : \u2200 {A B Q Q' R} (f : A \u2192 Strategy Q' R B) (g : Q \u2192 Q') \u2192 Strategy Q (R \u2218 g) A \u2192 Strategy Q' R B\n  [ f , g ]=<< ask q? cont = ask (g q?) (\u03bb r \u2192 [ f , g ]=<< cont r)\n  [ f , g ]=<< done x      = f x\n\n  module Rec {A B Q : \u2605} {R : Q \u2192 \u2605} {M : \u2605 \u2192 \u2605}\n             (runAsk : \u2200 {A} \u2192 (q : Q) \u2192 (R q \u2192 M A) \u2192 M A)\n             (runDone : A \u2192 M B) where\n    runMM : Strategy Q R A \u2192 M B\n    runMM (ask q? cont) = runAsk q? (\u03bb r \u2192 runMM (cont r))\n    runMM (done x)      = runDone x\n\n  module MM {A B Q : \u2605} {R : Q \u2192 \u2605} {M : \u2605 \u2192 \u2605}\n           (_>>=M_ : \u2200 {A B : \u2605} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B)\n           (runAsk : (q : Q) \u2192 M (R q))\n           (runDone : A \u2192 M B) where\n    runMM : Strategy Q R A \u2192 M B\n    runMM (ask q? cont) = runAsk q? >>=M (\u03bb r \u2192 runMM (cont r))\n    runMM (done x)      = runDone x\n\n  module _ {A B Q Q' : \u2605} {R : Q \u2192 \u2605} {R'}\n           (f : (q : Q) \u2192 Strategy Q' R' (R q))\n           (g : A \u2192 Strategy Q' R' B) where\n    [_,_]=<<'_ : Strategy Q R A \u2192 Strategy Q' R' B\n    [_,_]=<<'_ (ask q? cont) = f q? >>= \u03bb r \u2192 [_,_]=<<'_ (cont r)\n    [_,_]=<<'_ (done x)      = g x\n\n  record ServerResp (P : Proto) (q : Proto.Q P) (A : \u2605\u2080) : \u2605\u2080 where\n    coinductive\n    open Proto P\n    field\n        srv-resp : R q\n        srv-cont : \u2200 q \u2192 ServerResp P q A\n  open ServerResp\n\n  Server : Proto \u2192 \u2605\u2080 \u2192 \u2605\u2080\n  Server P A = \u2200 q \u2192 ServerResp P q A\n\n  OracleServer : Proto \u2192 \u2605\u2080\n  OracleServer P[ Q , R ] = (q : Q) \u2192 R q\n \n  module _ {P : Proto} {A : \u2605} where\n    open Proto P\n    com : Server P A \u2192 Client P A \u2192 A\n    com srv (ask q \u03bac) = com (srv-cont r) (\u03bac (srv-resp r)) where r = srv q\n    com srv (done x)   = x\n\n  module _ (P P' : Proto) (A A' : \u2605) where\n    module P = Proto P\n    module P' = Proto P'\n    record MITM : \u2605 where\n      coinductive\n      field\n        hack-query : (q' : P'.Q) \u2192 Client P (P'.R q' \u00d7 MITM)\n        hack-result : A' \u2192 Client P A\n    open MITM\n\n    module WithOutBind where\n        hacked-com-client : Server P A \u2192 MITM \u2192 Client P' A' \u2192 A\n        hacked-com-mitm : \u2200 {q'} \u2192 Server P A \u2192 Client P (P'.R q' \u00d7 MITM) \u2192 (P'.R q' \u2192 Client P' A') \u2192 A\n        hacked-com-srv-resp : \u2200 {q q'} \u2192 ServerResp P q A \u2192 (P.R q \u2192 Client P (P'.R q' \u00d7 MITM)) \u2192 (P'.R q' \u2192 Client P' A') \u2192 A\n\n        hacked-com-srv-resp r mitm clt = hacked-com-mitm (srv-cont r) (mitm (srv-resp r)) clt\n\n        hacked-com-mitm srv (ask q? mitm) clt = hacked-com-srv-resp (srv q?) mitm clt\n        hacked-com-mitm srv (done (r' , mitm)) clt = hacked-com-client srv mitm (clt r')\n\n        hacked-com-client srv mitm (ask q' \u03bac) = hacked-com-mitm srv (hack-query mitm q') \u03bac\n        hacked-com-client srv mitm (done x) = com srv (hack-result mitm x)\n\n    mitm-to-client-trans : MITM \u2192 Client P' A' \u2192 Client P A\n    mitm-to-client-trans mitm (ask q? cont) = hack-query mitm q? >>= \u03bb { (r' , mitm') \u2192 mitm-to-client-trans mitm' (cont r') }\n    mitm-to-client-trans mitm (done x)      = hack-result mitm x\n\n    hacked-com : Server P A \u2192 MITM \u2192 Client P' A' \u2192 A\n    hacked-com srv mitm clt = com srv (mitm-to-client-trans mitm clt)\n\n  module _ (P : Proto) (A : \u2605) where\n    open Proto P\n    open MITM\n    honest : MITM P P A A\n    hack-query  honest q = ask q \u03bb r \u2192 done (r , honest)\n    hack-result honest a = done a\n\n  module _ (P : Proto) (A : \u2605) (Oracle : OracleServer P) where\n    oracle-server : Server P A\n    srv-resp (oracle-server q) = Oracle q\n    srv-cont (oracle-server q) = oracle-server\n\nMessage = \ud835\udfda\nopen Game.IND-CPA-utils Message CipherText\nmodule RF    = Game.ReceiptFreeness PubKey SecKey         CipherText SerialNumber R\u2091 R\u2096 R\u2090  #q max#q KeyGen Enc Dec Check CheckEnc\nopen RF renaming (Phase to RFPhase; Q to RFQ; Resp to RFResp)\n\nR\u2090\u2020 : \u2605\nR\u2090\u2020 = R\u2090 \u00d7 {-SerialNumber \u00b2 \u00d7-} (Vec Rgb #q)\u00b2\nmodule CCA2\u2020 = Game.IND-CCA2-dagger PubKey SecKey Message CipherText              R\u2091 R\u2096 R\u2090\u2020          KeyGen Enc Dec\n\nCPAChallenger : (Next : \u2605) \u2192 \u2605\nCPAChallenger Next = Message \u00b2 \u2192 CipherText \u00b2 \u00d7 Next\n\nCCAProto : Proto\nCCAProto = P[ CipherText , const Message ]\n\nRFProto : Proto\nRFProto = P[ RFQ , RFResp ]\n\nDecRoundChallenger : (Next : \u2605) \u2192 \u2605\nDecRoundChallenger = Server CCAProto\n\nmodule AdversaryTrans (m : \ud835\udfda {-which mark to put on the two receipts-})\n                      (RFA : RF.Adversary) (r\u2090 : R\u2090) --(sn : SerialNumber \u00b2)\n                      (rgb : (Vec Rgb #q)\u00b2) (pk : PubKey) where\n  open MITM\n\n  {-\n  askDecBB : BB \u2192 DecRound ClearBB\n  askDecBB [] = done []\n  askDecBB ((m? , sn , enc-co) \u2237 bb) = ask enc-co (\u03bb co \u2192 askDecBB bb >>= \u03bb dec-bb \u2192 done ((co , m?) \u2237 dec-bb))\n  -}\n\n  MITMState : \u2605 \u2192 \u2605\n  MITMState X = X \u00d7 BB \u00d7 Tally\n\n  ballot : RF.PhaseNumber \u2192 Fin #q \u2192 Ballot\n  ballot p# n = RF.genBallot pk (lookup n (rgb p#))\n\n  mitmPhase : (p# : RF.PhaseNumber) \u2192 Fin #q \u2192 BB \u2192 Tally \u2192 \u2200 {X} \u2192 MITM CCAProto RFProto (MITMState X) X\n  hack-query (mitmPhase p# n bb ta) REB = done (ballot p# n , mitmPhase p# (Fin.pred n) bb ta)\n  hack-query (mitmPhase p# n bb ta) RBB = done (bb , mitmPhase p# (Fin.pred n) bb ta)\n  hack-query (mitmPhase p# n bb ta) RTally = done (ta , mitmPhase p# (Fin.pred n) bb ta)\n  hack-query (mitmPhase p# n bb ta) (RCO (m? , sn , enc-co)) =\n     -- if receipt in DB then ...\n     ask enc-co \u03bb co \u2192 done (co , mitmPhase p# (Fin.pred n) bb ta)\n  hack-query (mitmPhase p# n bb ta) (Vote (m? , sn , enc-co))\n     -- lots of cases\n     = ask enc-co \u03bb co \u2192\n         let (res , cbb' , bb') =\n                case Check enc-co\n                  0: (RF.reject ,\u2032 ta , bb)\n                  1: (RF.accept , tallyMarkedReceipt? co m? +,+ ta , (m? , sn , enc-co) \u2237 bb) in\n         done (res , mitmPhase p# (Fin.pred n) bb' cbb')\n  hack-result (mitmPhase p# n bb ta) r = done (r , bb , ta)\n\n  MITM-RFChallenge : \u2200 {X} \u2192 MITM _ _ (MITMState X) X\n  MITM-RFChallenge = mitmPhase 0\u2082 max#q [] (0 , 0)\n\n  decRoundAdvTr : \u2200 {X} \u2192 Client RFProto X \u2192 Client CCAProto (MITMState X)\n  decRoundAdvTr = mitm-to-client-trans _ _ _ _ MITM-RFChallenge\n\n  decRoundAdv1 : Client CCAProto (MITMState (RFChallenge (RFPhase Candidate)))\n  decRoundAdv1 = decRoundAdvTr (RFA r\u2090 pk)\n\n  module Receipts (sn : SerialNumber \u00b2) (ct : CipherText \u00b2) where\n      receipts : Candidate \u2192 Receipt\n      receipts c = marked m , sn c , ct c\n\n\n{-\n      trCBB : ClearBB \u2192 ClearBB\n      trCBB cbb = ? \u2237 ? \u2237\n      -}\n\n      trBB : BB \u2192 BB\n      trBB bb = receipts 0\u2082 \u2237 receipts 1\u2082 \u2237 bb\n\n  hack-challenge : \u2200 {X} \u2192 RFChallenge X \u2192 CPA\u2020Adversary (X \u00d7 (BB \u2192 BB))\n  get-m (hack-challenge _) = alice-then-bob , bob-then-alice\n  put-c (hack-challenge rfc) c\u2080 c\u2081 = proj\u2082 rfc receipts , trBB\n    where\n      ct = proj (c\u2080 , c\u2081)\n      open Receipts (proj\u2081 rfc) ct\n\n  module _ (bb : BB) (ta : Tally) (Aphase[II] : RFPhase Candidate) where\n\n    decRoundAdv2 : DecRound (MITMState Candidate)\n    decRoundAdv2 = mitm-to-client-trans _ _ _ _ (mitmPhase 1\u2082 max#q bb ta) Aphase[II]\n\n  mapCPAAdv = TransformAdversaryResponse.A*\n\n  A\u2020 : DecRound (CPA\u2020Adversary (DecRound \ud835\udfda))\n  A\u2020 = decRoundAdv1 >>= \u03bb { (rfc , bb , ta) \u2192\n         done (mapCPAAdv (\u03bb f c \u2192 decRoundAdv2 (proj\u2082 (f c) bb) ((1 , 1) +,+ ta) (proj\u2081 (f c)) >>= (done \u2218 proj\u2081))\n                  (hack-challenge rfc)) }\n\nA\u2020-trans : \u2200 m \u2192 RF.Adversary \u2192 CCA2\u2020.Adversary\nA\u2020-trans m RFA (r\u2090 , rgb) pk = AdversaryTrans.A\u2020 m RFA r\u2090 rgb pk\n\nopen StatefulRun\nmodule Pfff\n  {-(b : \ud835\udfda)-} (RFA : RF.Adversary) (pk : PubKey) (sk : SecKey)\n  (DecEnc : \u2200 r\u2091 m \u2192 Dec sk (Enc pk m r\u2091) \u2261 m)\n  (r\u2090 : R\u2090) (rgb : Candidate \u2192 Rgb) -- (rco : Candidate \u2192 CO) (r\u2091 : Candidate \u2192 R\u2091)\n  (rgbs : PhaseNumber \u2192 Vec Rgb #q) (sn : Candidate \u2192 SerialNumber) where\n\n  postulate b m : \ud835\udfda\n  {-\n  b = 0\u2082\n  t = 0\u2082\n  m = 0\u2082\n  -}\n  module Tr = AdversaryTrans m RFA r\u2090 {-sn-} rgbs pk\n\n  r\u2091 = proj\u2082 \u2218 proj\u2082 \u2218 rgb\n\n  module RFEXP = RF.EXP b RFA pk sk r\u2090 rgbs (Enc pk \u02e2 r\u2091) (const (marked 0\u2082))\n  module EXP\u2020  = CCA2\u2020.EXP b (A\u2020-trans m RFA) (r\u2090 , {-sn ,-} rgbs) pk sk (r\u2091 0\u2082) (r\u2091 1\u2082)\n\n  module _ {X} (p# : PhaseNumber) where\n    RX : X \u00d7 RFEXP.S \u2192 Tr.MITMState X \u2192 \u2605\n    RX (x , bb , _) (x' , bb' , _) = bb \u2261 bb' \u00d7 x \u2261 x'\n\n    Bisim' : (n : Fin #q) (bb : BB) \u2192 Client RFProto X \u2192 Client CCAProto (Tr.MITMState X) \u2192 \u2605\n    Bisim' n bb clt0 clt1 = RX (runS (RFEXP.OracleS p#) clt0 (bb , n)) (run (Dec sk) clt1)\n\n    pf-phase : (n : Fin #q) (bb : BB) (clt : Client _ _)\n                   \u2192 Bisim' n bb clt (mitm-to-client-trans _ _ _ _ (Tr.mitmPhase p# n bb (tally sk bb)) clt)\n    pf-phase n bb (ask REB cont) = pf-phase (Fin.pred n) bb (cont (Tr.ballot p# n))\n    pf-phase n bb (ask RBB cont) = pf-phase (Fin.pred n) bb (cont bb)\n    pf-phase n bb (ask RTally cont) = pf-phase (Fin.pred n) bb (cont (tally sk bb))\n    pf-phase n bb (ask (RCO (m? , sn , enc-co)) cont) = pf-phase (Fin.pred n) bb (cont (Dec sk enc-co))\n    pf-phase n bb (ask (Vote r) cont) with Check (enc-co r)\n    ... | 0\u2082 = pf-phase (Fin.pred n) bb (cont reject)\n    ... | 1\u2082 = pf-phase (Fin.pred n) (r \u2237 bb) (cont accept)\n    pf-phase n bb (done x) = refl , refl\n\n  pf-phase[I] : Bisim' 0\u2082 max#q [] RFEXP.Aphase[I] Tr.decRoundAdv1\n  pf-phase[I] = pf-phase 0\u2082 max#q [] RFEXP.Aphase[I]\n\n  tally[I] = tally sk RFEXP.BBphase[I]\n\n  CPA\u2020Challenge : CPA\u2020Adversary (RFPhase Candidate \u00d7 (BB \u2192 BB))\n  CPA\u2020Challenge = Tr.hack-challenge RFEXP.AdversaryRFChallenge\n\n  tally-pf : tally sk RFEXP.BBrfc \u2261 (1 , 1) +,+ tally[I]\n  tally-pf rewrite\n             DecEnc (proj\u2082 (proj\u2082 (rgb 0\u2082))) 0\u2082\n           | DecEnc (proj\u2082 (proj\u2082 (rgb 1\u2082))) 1\u2082\n           = refl\n\n  pf-phase[II] : Bisim' 1\u2082 max#q RFEXP.BBrfc RFEXP.Aphase[II] (Tr.decRoundAdv2 RFEXP.BBrfc (tally sk RFEXP.BBrfc) RFEXP.Aphase[II])\n  pf-phase[II] = pf-phase 1\u2082 max#q RFEXP.BBrfc RFEXP.Aphase[II]\n\n  pf-phase[II]' : Bisim' 1\u2082 max#q RFEXP.BBrfc RFEXP.Aphase[II] (Tr.decRoundAdv2 RFEXP.BBrfc ((1 , 1) +,+ tally[I]) RFEXP.Aphase[II])\n  pf-phase[II]' rewrite sym tally-pf = pf-phase[II]\n\n  {-\n  module DecRoundAdv2 = Tr.DecRoundAdv2 RFEXP.BBphase[I] CBB[I] RFEXP.AdversaryRFChallenge\n\n--  msg = DecRoundAdv2.get-m\n\n  ct = enc-co \u2218 RFEXP.receipts\n\n  {-\n\n  pf-b\u2032 : RFEXP.b\u2032 \u2261 EXP\u2020.b\u2032\n  pf-b\u2032 = {!!}\n\n  {-\n  pf-chal : \u2200 c \u2192 RFEXP.randomly-swapped-receipts c \u2261 DecRoundAdv2.ContA.randomly-swapped-receipts ([0: {!!} 1: Enc pk m (r\u2091 (b xor 1\u2082)) ]) c\n  pf-chal 1\u2082 = {!!}\n  pf-chal 0\u2082 = {!!}\n\n{-\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e3439a3dd5452c70afecd2905843eae3ad4b1125","subject":"Added [x+y]-[x+z]\u2261y-z (ER version).","message":"Added [x+y]-[x+z]\u2261y-z (ER version).\n\nIgnore-this: d28cfa3428610bb9e0092c8efc3525e3\n\ndarcs-hash:20100429193538-3bd4e-8b78f8e261843db4875e4be1be22297657da3d27.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\n  using ( +-leftIdentity ; *-leftZero ; *-comm ; minus-0x )\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN          = subst (\u03bb t \u2192 N t) (sym cP\u2081) zN\npred-N (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (cP\u2082 n)) Nn\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN                   = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN (sN {n} Nn)          = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst ((\u03bb t \u2192 N t)) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n------------------------------------------------------------------------------\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                 succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                        (minus-x0 m)\n                        refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                              t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Postulates using ( [x+y]-[x+z]\u2261y-z )\nopen import LTC.Function.Arithmetic.Properties\n  using ( +-leftIdentity ; *-leftZero ; *-comm ; minus-0x )\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN          = subst (\u03bb t \u2192 N t) (sym cP\u2081) zN\npred-N (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (cP\u2082 n)) Nn\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN                   = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN (sN {n} Nn)          = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst ((\u03bb t \u2192 N t)) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n------------------------------------------------------------------------------\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                        (minus-x0 m)\n                        refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                              t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c557ae57d1c35294b25134a5cb726f2dca2b6c77","subject":"Data.Nat: missing ()","message":"Data.Nat: missing ()\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"-- NOTE with-K so far\n-- TODO {-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nimport Data.Two.Equality as \ud835\udfda==\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; idp; !; _\u2219_; ap)\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = \u2261.refl\na\u2261a\u2293b+a\u2238b zero (suc b) = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) zero = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite \u2261.sym (a\u2261a\u2293b+a\u2238b a b) = \u2261.refl\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z = !(\u2115\u00b0.+-assoc x y z) \u2219 ap (flip _+_ z) (\u2115\u00b0.+-comm x y) \u2219 \u2115\u00b0.+-assoc y x z\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = \u2261.refl\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = \u2261.refl\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = \u2261.refl\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = \u2261.refl\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = \u2261.cong (suc \u2218\u2032 suc)\n                         (\u2261.sym (proj\u2082 \u2115\u00b0.*-identity m))\n*\u2032-spec (suc (suc m)) (suc (suc n)) = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 \u2605\u2080\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : \u2605\u2080} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 \u2605\u2080) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 \u2605\u2080\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n","old_contents":"-- NOTE with-K so far\n-- TODO {-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Two hiding (_==_)\nimport Data.Two.Equality as \ud835\udfda==\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; idp; !; _\u2219_; ap)\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: \u03bb {n} \u2192 s n ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = \u2261.refl\na\u2261a\u2293b+a\u2238b zero (suc b) = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) zero = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite \u2261.sym (a\u2261a\u2293b+a\u2238b a b) = \u2261.refl\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z = !(\u2115\u00b0.+-assoc x y z) \u2219 ap (flip _+_ z) (\u2115\u00b0.+-comm x y) \u2219 \u2115\u00b0.+-assoc y x z\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = \u2261.refl\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = \u2261.refl\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = \u2261.refl\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = \u2261.refl\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = \u2261.cong (suc \u2218\u2032 suc)\n                         (\u2261.sym (proj\u2082 \u2115\u00b0.*-identity m))\n*\u2032-spec (suc (suc m)) (suc (suc n)) = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 \u2605\u2080\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : \u2605\u2080} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 \u2605\u2080) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 \u2605\u2080\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b5a7bd579e94a8a269bef69c1e8cd6305f77f472","subject":"Some numbers (powers of 2 up to 4096)","message":"Some numbers (powers of 2 up to 4096)\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS.agda","new_file":"lib\/FFI\/JS.agda","new_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate 4N : Number\n{-# COMPILED_JS 4N 4 #-}\n\npostulate 8N : Number\n{-# COMPILED_JS 8N 8 #-}\n\npostulate 16N : Number\n{-# COMPILED_JS 16N 16 #-}\n\npostulate 32N : Number\n{-# COMPILED_JS 32N 32 #-}\n\npostulate 64N : Number\n{-# COMPILED_JS 64N 64 #-}\n\npostulate 128N : Number\n{-# COMPILED_JS 128N 128 #-}\n\npostulate 256N : Number\n{-# COMPILED_JS 256N 256 #-}\n\npostulate 512N : Number\n{-# COMPILED_JS 512N 512 #-}\n\npostulate 1024N : Number\n{-# COMPILED_JS 1024N 1024 #-}\n\npostulate 2048N : Number\n{-# COMPILED_JS 2048N 2048 #-}\n\npostulate 4096N : Number\n{-# COMPILED_JS 4096N 4096 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber Number #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString String #-}\n\n-- TODO dyn check of length 1?\npostulate castChar : JSValue \u2192 Char\n{-# COMPILED_JS castChar String #-}\n\n-- TODO dyn check of length 1?\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char String #-}\n\n-- TODO dyn check?\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray function(x) { return x; } #-}\n\n-- TODO dyn check?\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject function(x) { return x; } #-}\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\npostulate onJSArray : {A : Set} (f : JSArray JSValue \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onJSArray require(\"libagda\").onJSArray #-}\n\npostulate onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onString require(\"libagda\").onString #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = castString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback0 : Set\nCallback0 = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 Callback0\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ncheck : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\ncheck true  errmsg x = x\ncheck false errmsg x = throw (errmsg _) x\n\nwarn-check : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\nwarn-check true  errmsg x = x\nwarn-check false errmsg x = trace (\"Warning: \" ++ errmsg _) x id\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\ndata JS! : Set\u2081 where\n  end  : JS!\n  _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n  _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n\n_>>_ : Callback0 \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module FFI.JS where\n\nopen import Data.Empty       public renaming (\u22a5 to \ud835\udfd8)\nopen import Data.Unit.Base   public renaming (\u22a4 to \ud835\udfd9)\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List; []; _\u2237_)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function         using (id; _\u2218_)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\npostulate\n  Number   : Set\n  JSArray  : Set \u2192 Set\n  JSObject : Set\n  JSValue  : Set\n\npostulate readNumber : String \u2192 Number\n{-# COMPILED_JS readNumber Number #-}\n\npostulate 0N : Number\n{-# COMPILED_JS 0N 0 #-}\n\npostulate 1N : Number\n{-# COMPILED_JS 1N 1 #-}\n\npostulate 2N : Number\n{-# COMPILED_JS 2N 2 #-}\n\npostulate _+_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _+_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2212_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\u2212_ function(x) { return function(y) { return x - y; }; } #-}\n\npostulate _*_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _*_ function(x) { return function(y) { return x * y; }; } #-}\n\npostulate _\/_ : Number \u2192 Number \u2192 Number\n{-# COMPILED_JS _\/_ function(x) { return function(y) { return x \/ y; }; } #-}\n\ninfixr 5  _++_\npostulate _++_ : String \u2192 String \u2192 String\n{-# COMPILED_JS _++_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _+JS_ : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _+JS_ function(x) { return function(y) { return x + y; }; } #-}\n\npostulate _\u2264JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2264JS_ function(x) { return function(y) { return x <= y; }; } #-}\n\npostulate _<JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _<JS_ function(x) { return function(y) { return x < y; }; } #-}\n\npostulate _>JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _>JS_ function(x) { return function(y) { return x > y; }; } #-}\n\npostulate _\u2265JS_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _\u2265JS_ function(x) { return function(y) { return x >= y; }; } #-}\n\npostulate _===_ : JSValue \u2192 JSValue \u2192 Bool\n{-# COMPILED_JS _===_ function(x) { return function(y) { return x === y; }; } #-}\n\npostulate reverse : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS reverse function(ty) { return function(x) { return x.reverse(); }; } #-}\n\npostulate sort : {A : Set} \u2192 JSArray A \u2192 JSArray A\n{-# COMPILED_JS sort function(ty) { return function(x) { return x.sort(); }; } #-}\n\npostulate split : (sep target : String) \u2192 JSArray String\n{-# COMPILED_JS split function(sep) { return function(target) { return target.split(sep); }; } #-}\n\npostulate join : (sep : String)(target : JSArray String) \u2192 String\n{-# COMPILED_JS join function(sep) { return function(target) { return target.join(sep); }; } #-}\n\npostulate fromList : {A B : Set}(xs : List A)(fromElt : A \u2192 B) \u2192 JSArray B\n{-# COMPILED_JS fromList require(\"libagda\").fromList #-}\n\npostulate length : String \u2192 Number\n{-# COMPILED_JS length function(s) { return s.length; } #-}\n\npostulate JSON-stringify : JSValue \u2192 String\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n\npostulate JSON-parse : String \u2192 JSValue\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n\npostulate toString : JSValue \u2192 String\n{-# COMPILED_JS toString function(x) { return x.toString(); } #-}\n\npostulate fromBool : Bool \u2192 JSValue\n{-# COMPILED_JS fromBool function(x) { return x; } #-}\n\npostulate fromString : String \u2192 JSValue\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n\npostulate fromChar : Char \u2192 JSValue\n{-# COMPILED_JS fromChar String #-}\n\npostulate Char\u25b9String : Char \u2192 String\n{-# COMPILED_JS Char\u25b9String String #-}\n\npostulate fromNumber : Number \u2192 JSValue\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n\npostulate fromJSArray : {A : Set} \u2192 JSArray A \u2192 JSValue\n{-# COMPILED_JS fromJSArray function(ty) { return function(x) { return x; }; } #-}\n\npostulate fromJSObject : JSObject \u2192 JSValue\n{-# COMPILED_JS fromJSObject function(x) { return x; } #-}\n\npostulate objectFromList : {A : Set}(xs : List A)(fromKey : A \u2192 String)(fromVal : A \u2192 JSValue) \u2192 JSObject\n{-# COMPILED_JS objectFromList require(\"libagda\").objectFromList #-}\n\npostulate decodeJSArray : {A B : Set}(arr : JSArray A)(fromElt : Number \u2192 A \u2192 B) \u2192 List B\n{-# COMPILED_JS decodeJSArray require(\"libagda\").decodeJSArray #-}\n\npostulate castNumber : JSValue \u2192 Number\n{-# COMPILED_JS castNumber Number #-}\n\npostulate castString : JSValue \u2192 String\n{-# COMPILED_JS castString String #-}\n\n-- TODO dyn check of length 1?\npostulate castChar : JSValue \u2192 Char\n{-# COMPILED_JS castChar String #-}\n\n-- TODO dyn check of length 1?\npostulate String\u25b9Char : String \u2192 Char\n{-# COMPILED_JS String\u25b9Char String #-}\n\n-- TODO dyn check?\npostulate castJSArray : JSValue \u2192 JSArray JSValue\n{-# COMPILED_JS castJSArray function(x) { return x; } #-}\n\n-- TODO dyn check?\npostulate castJSObject : JSValue \u2192 JSObject\n{-# COMPILED_JS castJSObject function(x) { return x; } #-}\n\npostulate nullJS : JSValue\n{-# COMPILED_JS nullJS null #-}\n\npostulate _\u00b7[_] : JSValue \u2192 JSValue \u2192 JSValue\n{-# COMPILED_JS _\u00b7[_] require(\"libagda\").readProp #-}\n\npostulate _Array[_] : {A : Set} \u2192 JSArray A \u2192 Number \u2192 A\n{-# COMPILED_JS _Array[_] function(ty) { return require(\"libagda\").readProp; } #-}\n\npostulate onJSArray : {A : Set} (f : JSArray JSValue \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onJSArray require(\"libagda\").onJSArray #-}\n\npostulate onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n{-# COMPILED_JS onString require(\"libagda\").onString #-}\n\n-- Writes 'msg' and 'inp' to the console and then returns `f inp`\npostulate trace : {A B : Set}(msg : String)(inp : A)(f : A \u2192 B) \u2192 B\n{-# COMPILED_JS trace require(\"libagda\").trace #-}\n\npostulate throw : {A : Set} \u2192 String \u2192 A \u2192 A\n{-# COMPILED_JS throw require(\"libagda\").throw #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number \u2192 Value\n  bool   : Bool   \u2192 Value\n  null   : Value\n\nObject = List (String \u00d7 JSValue)\n\npostulate fromValue : Value \u2192 JSValue\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\n-- TODO we could make it a COMPILED type and remove the encoding by using JSValue as the internal repr.\ndata ValueView : Set\u2080 where\n  array  : JSArray JSValue \u2192 ValueView\n  object : JSObject        \u2192 ValueView\n  string : String          \u2192 ValueView\n  number : Number          \u2192 ValueView\n  bool   : Bool            \u2192 ValueView\n  null   : ValueView\n\n-- TODO not yet tested\npostulate viewJSValue : JSValue \u2192 ValueView\n{-# COMPILED_JS viewJSValue require(\"libagda\").viewJSValue #-}\n\nBool\u25b9String : Bool \u2192 String\nBool\u25b9String true  = \"true\"\nBool\u25b9String false = \"false\"\n\nList\u25b9String : List Char \u2192 String\nList\u25b9String xs = join \"\" (fromList xs Char\u25b9String)\n\nString\u25b9List : String \u2192 List Char\nString\u25b9List s = decodeJSArray (split \"\" s) (\u03bb _ \u2192 String\u25b9Char)\n\nNumber\u25b9String : Number \u2192 String\nNumber\u25b9String = castString \u2218 fromNumber\n\nJSArray\u25b9ListString : {A : Set} \u2192 JSArray A \u2192 List A\nJSArray\u25b9ListString a = decodeJSArray a (\u03bb _ \u2192 id)\n\nfromObject : Object \u2192 JSObject\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_<Char_ : Char \u2192 Char \u2192 Bool\nx <Char y = fromChar x <JS fromChar y\n\n_>Char_ : Char \u2192 Char \u2192 Bool\nx >Char y = fromChar x >JS fromChar y\n\n_\u2265Char_ : Char \u2192 Char \u2192 Bool\nx \u2265Char y = fromChar x \u2265JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_<String_ : String \u2192 String \u2192 Bool\nx <String y = fromString x <JS fromString y\n\n_>String_ : String \u2192 String \u2192 Bool\nx >String y = fromString x >JS fromString y\n\n_\u2265String_ : String \u2192 String \u2192 Bool\nx \u2265String y = fromString x \u2265JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_<Number_ : Number \u2192 Number \u2192 Bool\nx <Number y = fromNumber x <JS fromNumber y\n\n_>Number_ : Number \u2192 Number \u2192 Bool\nx >Number y = fromNumber x >JS fromNumber y\n\n_\u2265Number_ : Number \u2192 Number \u2192 Bool\nx \u2265Number y = fromNumber x \u2265JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\n_\u00b7\u00ab_\u00bbA : JSValue \u2192 String \u2192 JSArray JSValue\nv \u00b7\u00ab s \u00bbA = castJSArray (v \u00b7\u00ab s \u00bb)\n\ntrace-call : {A B : Set} \u2192 String \u2192 (A \u2192 B) \u2192 A \u2192 B\ntrace-call s f x = trace s (f x) id\n\npostulate JSCmd : Set \u2192 Set\n\nCallback1 : Set \u2192 Set\nCallback1 A = JSCmd ((A \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\nCallback0 : Set\nCallback0 = Callback1 \ud835\udfd9\n\nCallback2 : Set \u2192 Set \u2192 Set\nCallback2 A B = JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8)\n\npostulate assert : Bool \u2192 Callback0\n{-# COMPILED_JS assert require(\"libagda\").assert #-}\n\ncheck : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\ncheck true  errmsg x = x\ncheck false errmsg x = throw (errmsg _) x\n\nwarn-check : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\nwarn-check true  errmsg x = x\nwarn-check false errmsg x = trace (\"Warning: \" ++ errmsg _) x id\n\ninfixr 0  _>>_ _!\u2081_ _!\u2082_\ndata JS! : Set\u2081 where\n  end  : JS!\n  _!\u2081_ : {A : Set}(cmd : Callback1 A)(cb : A \u2192 JS!) \u2192 JS!\n  _!\u2082_ : {A B : Set}(cmd : JSCmd ((A \u2192 B \u2192 \ud835\udfd8) \u2192 \ud835\udfd8))(cb : A \u2192 B \u2192 JS!) \u2192 JS!\n\n_>>_ : Callback0 \u2192 JS! \u2192 JS!\ncmd >> cont = cmd !\u2081 \u03bb _ \u2192 cont\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"780ea6e9760f1ac4e150779c01d07a3da9552e78","subject":"sorting: WIP","message":"sorting: WIP\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n  -}\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low\u209c high\u209c low\u1d64 high\u1d64} \u2192\n             Sorted t low\u209c high\u209c \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high\u209c \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low\u209c high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n{-\n    data _\u2264\u1d3e_ : \u2200 {x y t} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set where\n      here-here   : here \u2264\u1d3e here\n      left-left   : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 left p \u2264\u1d3e left q\n      right-right : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 right p \u2264\u1d3e right q\n      left-right  : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 left p \u2264\u1d3e right q\n      -}\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\nmodule Sorting {a} {A : Set a} -- (sort\u1d2c : A \u00d7 A \u2192 A \u00d7 A)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    {-\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\n    {-\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted' _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    postulate dec-\u2264 : \u2200 x y \u2192 Dec (x \u2264\u1d2c y)\n    postulate dec-\u2294 : \u2200 x y \u2192 x \u2294\u1d2c y \u2261 x \u228e x \u2294\u1d2c y \u2261 y\n    postulate dec-\u2293 : \u2200 x y \u2192 x \u2293\u1d2c y \u2261 x \u228e x \u2293\u1d2c y \u2261 y\n\n    merge-spec : \u2200 {n} (t u : Tree A n) \u2192\n                 Sorted t \u2192 Sorted u \u2192 Sorted (merge t u)\n    merge-spec (leaf x) (leaf y) sx sy (left here) (left here) pf = \u2264\u1d2c.refl\n    merge-spec (leaf x) (leaf y) sx sy (left here) (right here) (0-1 ._ ._) = \u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)\n    merge-spec (leaf x) (leaf y) sx sy (right p) (left q) ()\n    merge-spec (leaf x) (leaf y) sx sy (right here) (right here) pf = \u2264\u1d2c.refl\n    merge-spec (fork t\u2080 t\u2081) (fork u\u2080 u\u2081) st su p q p\u2264q\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 -- | merge-spec t\u2080 u\u2080 ? ? | merge-spec t\u2081 u\u2081 ? ?\n    ... | fork l m\u2080    | fork m\u2081 h\n      with merge m\u2080 m\u2081 -- | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 -- | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with p | q | p\u2264q\n    ... | left  pp | left  qq | there ._ pp\u2264qq = {!merge-spec t\u2080 u\u2080 !}\n    ... | left  pp | right qq | 0-1 ._ ._ = {!!}\n    ... | right pp | left qq  | ()\n    ... | right pp | right qq | there ._ pp\u2264qq = {!!}\n    -}\n    -}\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_ isPreorder\n    open SD using (fork; leaf)\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = refl\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = refl\n\n             {-\n    \u2203Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    \u2203Sorted t = \u2203 \u03bb l \u2192 \u2203 \u03bb h \u2192 Sorted t l h\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = \u2203Sorted t \u00d7 \u2203Sorted u\n    -}\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted (fork t u) = SD.Sorted t \u00d7 SD.Sorted u\n\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork t t\u2081) = refl\n\n    {-\n    record MergeInnerHyp {n} (t : Tree A (2 + n)) : Set (a L.\u2294 \u2113) where\n      constructor mk\n      field\n        st\u2080 : SD.Sorted (lft t)\n        st\u2081 : SD.Sorted (rght t)\n      u = interchange t\n      field\n        su\u2080 : SD.Sorted (lft u)\n        su\u2081 : SD.Sorted (rght u)\n\n    merge-inner : \u2200 {n} {t : Tree A (2 + n)} \u2192\n                 MergeInnerHyp t \u2192 SD.Sorted (map-inner merge t)\n    merge-inner {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n                (mk (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                    (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n                    (fork sv\u2080 sv\u2081 hv\u2080\u2264lv\u2081)\n                    (fork sw\u2080 sw\u2081 lw\u2080\u2264hw\u2081)) = {!!}\n                    -}\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n                -- last  (merge t u) \u2261 last  t \u2293\u1d2c last  u\n    merge-spec : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192 SD.Sorted (merge (fork t u))\n    merge-spec (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec st\u2080 su\u2080 | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec st\u2081 su\u2081\n    ... | fork v\u2080 w\u2080 | fork sv\u2080 sw\u2080 p1 | fpf1\n        | fork v\u2081 w\u2081 | fork sv\u2081 sw\u2081 p2\n      with merge (fork w\u2080 v\u2081) | merge-spec sw\u2080 sv\u2081\n    ... | fork a b | fork sa sb p3\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3\n         where\n             postulate\n                pf3 : last v\u2080 \u2264\u1d2c first t\u2081\n                pf4 : last v\u2080 \u2264\u1d2c first u\u2081\n                -- Sorted (merge t u) \u2192\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 = {!first-merge !}\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 = {!!}\n    {-\n      -}\n\n\n--      map-outer merge id \u2218 interchange\n\n      {-\n\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high\u209c = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high\u209c = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      {-with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl-} =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n  -}\n\nmodule All {a} (A : Set a) where\n\n  All : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  All f (leaf x)     = f [] x\n  All f (fork t\u2080 t\u2081) = All (f \u2218 0\u2237_) t\u2080 \u00d7 All (f \u2218 1\u2237_) t\u2081\n\n  Any : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\n  Any f (leaf x)     = f [] x\n  Any f (fork t\u2080 t\u2081) = Any (f \u2218 0\u2237_) t\u2080 \u228e Any (f \u2218 1\u2237_) t\u2081\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  module M {m} = All (Bits m)\n  open M\n\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n{-\n    rev-app : \u2200 {a} {A : Set a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n              -}\n\nbar : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\nbar f here      = refl\nbar f (left p)  = bar (f \u2218 0\u2237_) p\nbar f (right p) = bar (f \u2218 1\u2237_) p\n\nfoo : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\nfoo _ = bar \u2218 rev-app\n\nmodule Sorted {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : (x : A) \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low\u209c high\u209c low\u1d64 high\u1d64} \u2192\n             Sorted t low\u209c high\u209c \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             high\u209c \u2264\u1d2c low\u1d64 \u2192\n             Sorted (fork t u) low\u209c high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds (leaf ._)       = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub (leaf ._)       here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    {-\n    bounded\u2192sorted : \u2200 {n} {t : Tree A n} {l h} \u2192 Bounded t l h \u2192 Sorted t l h\n    bounded\u2192sorted {t = leaf x} b = {!b here!}\n    bounded\u2192sorted {t = fork t\u2080 t\u2081} b = fork (bounded\u2192sorted {t = t\u2080} {!!}) (bounded\u2192sorted {t = t\u2081} {!!}) {!!}\n    -}\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n{-\n    data _\u2264\u1d3e_ : \u2200 {x y t} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set where\n      here-here   : here \u2264\u1d3e here\n      left-left   : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 left p \u2264\u1d3e left q\n      right-right : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 p \u2264\u1d3e q \u2192 right p \u2264\u1d3e right q\n      left-right  : \u2200 {x y t} {p : x \u2208 t} {q : y \u2208 t} \u2192 left p \u2264\u1d3e right q\n      -}\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = Sorted _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' (leaf ._)       here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)    (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)    (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)    (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 (t u : Tree A n) \u2192 Tree A (1 + n)\n    merge (leaf x\u2080)    (leaf x\u2081)    =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081\n    ...  | fork l m\u2080   | fork m\u2081 h   with merge m\u2080 m\u2081\n    ...                                 | fork m\u2080\u2032 m\u2081\u2032 = fork (fork l m\u2080\u2032) (fork m\u2081\u2032 h)\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort (leaf x)     = leaf x\n    sort (fork t\u2080 t\u2081) = merge (sort t\u2080) (sort t\u2081)\n\n    open new-approach\n    InjTree : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    InjTree t = \u2200 x \u2192 (p q : x \u2208 t) \u2192 p \u2261 q\n\n    InjTree-\u00d7 : \u2200 {n} (t u : Tree A n) \u2192 InjTree (fork t u) \u2192 InjTree t \u00d7 InjTree u\n    InjTree-\u00d7 t u pf = pf\u2080 , pf\u2081\n      where pf\u2080 : InjTree t\n            pf\u2080 x p q with pf x (left p) (left q)\n            pf\u2080 x p .p | refl = refl\n            pf\u2081 : InjTree u\n            pf\u2081 x p q with pf x (right p) (right q)\n            pf\u2081 x p .p | refl = refl\n\n    _\u2257T_ : \u2200 {n} (t u : Tree A n) \u2192 Set _\n    t \u2257T u = toFun t \u2257 toFun u\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted' _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    postulate dec-\u2264 : \u2200 x y \u2192 Dec (x \u2264\u1d2c y)\n    postulate dec-\u2294 : \u2200 x y \u2192 x \u2294\u1d2c y \u2261 x \u228e x \u2294\u1d2c y \u2261 y\n    postulate dec-\u2293 : \u2200 x y \u2192 x \u2293\u1d2c y \u2261 x \u228e x \u2293\u1d2c y \u2261 y\n\n    merge-spec : \u2200 {n} (t u : Tree A n) \u2192\n                 Sorted t \u2192 Sorted u \u2192 Sorted (merge t u)\n    merge-spec (leaf x) (leaf y) sx sy (left here) (left here) pf = \u2264\u1d2c.refl\n    merge-spec (leaf x) (leaf y) sx sy (left here) (right here) (0-1 ._ ._) = \u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)\n    merge-spec (leaf x) (leaf y) sx sy (right p) (left q) ()\n    merge-spec (leaf x) (leaf y) sx sy (right here) (right here) pf = \u2264\u1d2c.refl\n    merge-spec (fork t\u2080 t\u2081) (fork u\u2080 u\u2081) st su p q p\u2264q\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 -- | merge-spec t\u2080 u\u2080 ? ? | merge-spec t\u2081 u\u2081 ? ?\n    ... | fork l m\u2080    | fork m\u2081 h\n      with merge m\u2080 m\u2081 -- | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 -- | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with p | q | p\u2264q\n    ... | left  pp | left  qq | there ._ pp\u2264qq = {!merge-spec t\u2080 u\u2080 !}\n    ... | left  pp | right qq | 0-1 ._ ._ = {!!}\n    ... | right pp | left qq  | ()\n    ... | right pp | right qq | there ._ pp\u2264qq = {!!}\n\n    {-\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-\u2293 : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2294-\u2264 : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorted _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-spec : \u2200 {n lt ht lu hu} {t u : Tree A n} \u2192\n                 Sorted t lt ht \u2192 Sorted u lu hu \u2192 Sorted (merge t u) (lt \u2293\u1d2c lu) (ht \u2294\u1d2c hu)\n    merge-spec (leaf x) (leaf y) = fork (leaf _) (leaf _) (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x))\n    merge-spec {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081} (fork {low\u209c = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                                                 (fork {low\u209c = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264hu\u2081)\n      with merge t\u2080 u\u2080 | merge t\u2081 u\u2081 | merge-spec st\u2080 su\u2080 | merge-spec st\u2081 su\u2081\n    ... | fork l m\u2080    | fork m\u2081 h   | fork {high\u209c = hl} {lm\u2080} sl sm\u2080 pf1\n                                     | fork {high\u209c = hm\u2081} {lh} sm\u2081 sh pf2\n      with merge m\u2080 m\u2081 | merge-spec sm\u2080 sm\u2081\n    ... | fork m\u2080\u2032 m\u2081\u2032 | fork {high\u209c = hm\u2080} {lm\u2081} sm\u2080\u2032 sm\u2081\u2032 pf3\n      with \u2264\u1d2c-bounds st\u2080  | \u2264\u1d2c-bounds st\u2081\n         | \u2264\u1d2c-bounds su\u2080  | \u2264\u1d2c-bounds su\u2081\n         | \u2264\u1d2c-bounds sm\u2080  | \u2264\u1d2c-bounds sm\u2081\n         | \u2264\u1d2c-bounds sm\u2080\u2032 | \u2264\u1d2c-bounds sm\u2081\u2032\n         | \u2264\u1d2c-bounds sh   | \u2264\u1d2c-bounds sl\n    ...  | lt\u2080\u2264ht\u2080        | lt\u2081\u2264ht\u2081\n         | lu\u2080\u2264hu\u2081        | lu\u2081\u2264hu\u2081\n         | lm\u2080\u2264\u2605          | \u2605\u2264hm\u2081\n         | \u2605\u2264hm\u2080          | lm\u2081\u2264\u2605\n         | lh\u2264\u2605           | \u2605\u2264hl =\n        fork\n          (fork sl sm\u2080\u2032 (proj\u2081 pf))\n          (fork sm\u2081\u2032 sh (proj\u2082 pf)) pf3\n          module M where\n                hl\u2264lt\u2081 : hl  \u2264\u1d2c lt\u2081\n                hl\u2264lt\u2081 = {!!}\n                hl\u2264lu\u2081 : hl  \u2264\u1d2c lu\u2081\n                hl\u2264lu\u2081 = {!!}\n                ht\u2080\u2264lh : ht\u2080 \u2264\u1d2c lh\n                ht\u2080\u2264lh = {!!}\n                hu\u2080\u2264lh : hu\u2080 \u2264\u1d2c lh\n                hu\u2080\u2264lh = {!!}\n\n                pf : (hl \u2264\u1d2c (lm\u2080 \u2293\u1d2c (lt\u2081 \u2293\u1d2c lu\u2081))) \u00d7 (((ht\u2080 \u2294\u1d2c hu\u2080) \u2294\u1d2c hm\u2081) \u2264\u1d2c lh)\n                pf = \u2264-\u2293 pf1 (\u2264-\u2293 hl\u2264lt\u2081 hl\u2264lu\u2081) , \u2294-\u2264 (\u2294-\u2264 ht\u2080\u2264lh hu\u2080\u2264lh) pf2\n\npostulate\n    _\u2294_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : \u2200 {n} \u2192 Bits n \u2192 Bits n \u2192 Bits n\n\nmodule BitsSorting {m} where\n\n    module S = Sorting (_\u2293_ {m}) (_\u2294_ {m})\n    open S public using (InjTree; InjTree-\u00d7)\n\n    merge : \u2200 {n} \u2192 (t u : Tree (Bits m) n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dadf2a8cb9c04418e5a508ec2962efe25e3a08cf","subject":"Desc model: replace All with Pi. It's all Set now.","message":"Desc model: replace All with Pi. It's all Set now.\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  ((y : H) -> p (proj\u2081 v y))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v = (\\y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> Set) ->\n         ((x : descOp d (Mu d)) -> (boxOp d (Mu d) bp x) -> bp (Con x)) ->\n         (v : Mu d) -> bp v\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","old_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n-- Bringing in an hand-made Prop universe\n\ndata PROP : Set where\n  All : (S : Set) -> (S -> PROP) -> PROP\n  And : PROP -> PROP -> PROP\n  Trivial : PROP\n  Absurd : PROP\n\nPrf : PROP -> Set\nPrf Absurd = \u22a5\nPrf Trivial = \u22a4\nPrf (All S P) = (x : S) -> Prf (P x)\nPrf (And A B) = \u03a3 (Prf A)  (\\ _ -> Prf B)\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  Prf ( All H (\\y -> p (proj\u2081 v y)))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) ->\n           (Prf (All D (\\y -> bp y))) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n-- On the Ind case, the Pig code is saying something along those lines:\n-- mapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (p?! (proj\u2082 v))\n\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> PROP) ->\n         (Prf (All (descOp d (Mu d)) (\\ x -> (All (boxOp d (Mu d) bp x) (\\ _ -> bp (Con x)))))) ->\n         (v : Mu d) -> Prf (bp v)\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4823ee78a6c7d4b61d093904537f68db94708dc0","subject":"Make-over of the Desc model, including natural numbers and doc.","message":"Make-over of the Desc model, including natural numbers and doc.","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\nrecord One : Set where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\n-- Inductive types are implemented as a Universe. Hence, in this\n-- section, we implement their code.\n\n-- We can read this code as follow (see Conor's \"Ornamental\n-- algebras\"): a description in |Desc| is a program to read one node\n-- of the described tree.\n\ndata Desc : Set where\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  -- Read a field in |X|; continue, given its value\n\n  -- Often, |X| is an |EnumT|, hence allowing to choose a constructor\n  -- among a finite set of constructors\n\n  Ind : (H : Set) -> Desc -> Desc\n  -- Read a field in H; read a recursive subnode given the field,\n  -- continue regarless of the subnode\n\n  -- Often, |H| is |1|, hence |Ind| simplifies to |Inf : Desc -> Desc|,\n  -- meaning: read a recursive subnode and continue regardless\n\n  Done : Desc\n  -- Stop reading\n\n\n--********************************************\n-- Desc decoder\n--********************************************\n\n-- Provided the type of the recursive subnodes |R|, we decode a\n-- description as a record describing the node.\n\n[|_|]_ : Desc -> Set -> Set\n[| Arg A D |] R = Sigma A (\\ a -> [| D a |] R)\n[| Ind H D |] R = (H -> R) * [| D |] R\n[| Done |] R = One\n\n\n--********************************************\n-- Functions on codes\n--********************************************\n\n-- Saying that a \"predicate\" |p| holds everywhere in |v| amounts to\n-- write something of the following type:\n\nEverywhere : (d : Desc) (D : Set) (bp : D -> Set) (V : [| d |] D) -> Set\nEverywhere (Arg A f) d p v =  Everywhere (f (fst v)) d p (snd v)\n-- It must hold for this constructor\n\nEverywhere (Ind H x) d p v =  ((y : H) -> p (fst v y)) * Everywhere x d p (snd v)\n-- It must hold for the subtrees \n\nEverywhere Done _ _ _ =  _\n-- It trivially holds at endpoints\n\n-- Then, we can build terms that inhabits this type. That is, a\n-- function that takes a \"predicate\" |bp| and makes it hold everywhere\n-- in the data-structure. It is the equivalent of a \"map\", but in a\n-- dependently-typed setting.\n\neverywhere : (d : Desc) (D : Set) (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : [| d |] D) ->\n           Everywhere d D bp v\neverywhere (Arg a f) d bp p v =  everywhere (f (fst v)) d bp p (snd v)\n-- It holds everywhere on this constructor\n\neverywhere (Ind H x) d bp p v = (\\y -> p (fst v y)) ,  everywhere x d bp p (snd v)\n-- It holds here, and down in the recursive subtrees\n\neverywhere Done _ _ _ _ =  One\n-- Nothing needs to be done on endpoints\n\n\n-- Looking at the decoder, a natural thing to do is to define its\n-- fixpoint |Mu D|, hence instantiating |R| with |Mu D| itself.\n\ndata Mu (D : Desc) : Set where\n  Con : [| D |] (Mu D) -> Mu D\n\n-- Using the \"map\" defined by |everywhere|, we can implement a \"fold\"\n-- over the |Mu| fixpoint:\n\nfoldDesc : (D : Desc) (bp : Mu D -> Set) ->\n         ((x : [| D |] (Mu D)) -> Everywhere D (Mu D) bp x -> bp (Con x)) ->\n         (v : Mu D) ->\n         bp v\nfoldDesc D bp p (Con v) =  p v (everywhere D (Mu D) bp (\\x -> foldDesc D bp p x) v) \n\n\n--********************************************\n-- Nat\n--********************************************\n\ndata NatConst : Set where\n  ZE : NatConst\n  SU : NatConst\n\nnatc : NatConst -> Desc\nnatc ZE = Done\nnatc SU = Ind One Done\n\nnatd : Desc\nnatd = Arg NatConst natc\n\nnat : Set\nnat = Mu natd\n\nzero : nat\nzero = Con ( ZE ,  _ )\n\nsuc : nat -> nat\nsuc n = Con ( SU , ( (\\_ -> n) ,  _ ) )\n\ntwo : nat\ntwo = suc (suc zero)\n\nfour : nat\nfour = suc (suc (suc (suc zero)))\n\nsum : nat -> \n      ((x : Sigma NatConst (\\ a -> [| natc a |] Mu (Arg NatConst natc))) ->\n      Everywhere (natc (fst x)) (Mu (Arg NatConst natc)) (\\ _ -> Mu (Arg NatConst natc)) (snd x) ->\n      Mu (Arg NatConst natc))\nsum n2 (ZE , _) p =  n2\nsum n2 (SU , f) p =  suc ( fst p _) \n\n\nplus : nat -> nat -> nat\nplus n1 n2 = foldDesc natd (\\_ -> nat) (sum n2) n1 \n\nx : nat\nx = plus two two\n\n\n\n","old_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  ((y : H) -> p (proj\u2081 v y))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> Set) ->\n           ((y : D) -> bp y) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v = (\\y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> Set) ->\n         ((x : descOp d (Mu d)) -> (boxOp d (Mu d) bp x) -> bp (Con x)) ->\n         (v : Mu d) -> bp v\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"be5ee7eec0dd2371c546256f7625b36c21943d03","subject":"Remove unnecessary imports.","message":"Remove unnecessary imports.\n\nOld-commit-hash: 01c6cb718906e9a41b8156076fe9e837257c7a1f\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Change\/Specification.agda","new_file":"Popl14\/Change\/Specification.agda","new_contents":"module Popl14.Change.Specification where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Integer\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 Change \u03c4 (\u27e6 t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1)\n  \u2192 after {\u03c4} (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394Env \u0393 \u03c1 \u2192 Change \u03c4 (\u27e6 x \u27e7Var \u03c1)\n\u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) \u03c1 d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) \u03c1 d\u03c1 = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1 + \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) \u03c1 d\u03c1 = - \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 empty \u03c1 d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) \u03c1 d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) \u03c1 d\u03c1 = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1 ++ \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) \u03c1 d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) \u03c1 d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 \u03c1\n    b = \u27e6 t \u27e7 \u03c1\n    \u0394f : Change (int \u21d2 bag) f\n    \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) \u03c1 d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n     call-change (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = cons\n  (\u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1))\n  (\u03bb v dv \u2192\n    begin\n      \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n      \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n    \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n      \u27e6 t \u27e7 (update ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n    \u2261\u27e8\u27e9\n      \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))) \u2022 update d\u03c1)\n    \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3})  \u27e9\n      \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 update d\u03c1)\n    \u2261\u27e8\u27e9\n      \u27e6 t \u27e7 (update (dv \u2022 d\u03c1))\n    \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n      \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n  \u27e6 x \u27e7 \u03c1 \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\ncorrectVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\ncorrectness (intlit n) \u03c1 d\u03c1 = right-id-int n\ncorrectness (add s t) \u03c1 d\u03c1 = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 \u03c1} {\u27e6 t \u27e7 \u03c1} {\u27e6 s \u27e7\u0394 \u03c1 d\u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong\u2082 _+_ (correctness s \u03c1 d\u03c1) (correctness t \u03c1 d\u03c1))\ncorrectness (minus t) \u03c1 d\u03c1 = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 \u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong -_ (correctness t \u03c1 d\u03c1))\n\ncorrectness empty \u03c1 d\u03c1 = right-id-bag emptyBag\ncorrectness (insert s t) \u03c1 d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 \u03c1\n    b = \u27e6 t \u27e7 \u03c1\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness s \u03c1 d\u03c1))\n         (correctness t \u03c1 d\u03c1) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness (union s t) \u03c1 d\u03c1 = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 \u03c1} {\u27e6 t \u27e7 \u03c1} {\u27e6 s \u27e7\u0394 \u03c1 d\u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong\u2082 _++_ (correctness s \u03c1 d\u03c1) (correctness t \u03c1 d\u03c1))\ncorrectness (negate t) \u03c1 d\u03c1 = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 \u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong negateBag (correctness t \u03c1 d\u03c1))\n\ncorrectness (flatmap s t) \u03c1 d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 \u03c1\n    b = \u27e6 t \u27e7 \u03c1\n    \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness s \u03c1 d\u03c1) (correctness t \u03c1 d\u03c1))\ncorrectness (sum t) \u03c1 d\u03c1 =\n  let\n    b = \u27e6 t \u27e7 \u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness t \u03c1 d\u03c1))\n\ncorrectness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\ncorrectness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 \u03c1\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 \u03c1\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e call-change \u0394f u \u0394u\n    \u2261\u27e8  sym (is-valid \u0394f u \u0394u) \u27e9\n       (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n  let\n    -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n    d\u03c1\u2032 = nil-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n    (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n      (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n    \u229e\u208d \u03c4 \u208e\n    (call-change (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\ncorollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n  is-valid (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n  (v : \u27e6 \u03c3 \u27e7) (dv : Change \u03c3 v)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} dv)\n  \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\ncorollary-closed {\u03c3} {\u03c4} t v dv =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n  in\n    begin\n      f (after {\u03c3} dv)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} dv))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} dv)\n    \u2261\u27e8 is-valid (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n      f (before {\u03c3} dv) \u229e\u208d \u03c4 \u208e call-change \u0394f v dv\n    \u220e where open \u2261-Reasoning\n","old_contents":"module Popl14.Change.Specification where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192 Change \u03c4 (\u27e6 t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1)\n  \u2192 after {\u03c4} (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394Env \u0393 \u03c1 \u2192 Change \u03c4 (\u27e6 x \u27e7Var \u03c1)\n\u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) \u03c1 d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) \u03c1 d\u03c1 = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1 + \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) \u03c1 d\u03c1 = - \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 empty \u03c1 d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) \u03c1 d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) \u03c1 d\u03c1 = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1 ++ \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) \u03c1 d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) \u03c1 d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 \u03c1\n    b = \u27e6 t \u27e7 \u03c1\n    \u0394f : Change (int \u21d2 bag) f\n    \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) \u03c1 d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n     call-change (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = cons\n  (\u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1))\n  (\u03bb v dv \u2192\n    begin\n      \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n      \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n    \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n      \u27e6 t \u27e7 (update ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n    \u2261\u27e8\u27e9\n      \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e ((v \u229e\u208d \u03c3 \u208e dv) \u229f\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))) \u2022 update d\u03c1)\n    \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3})  \u27e9\n      \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 update d\u03c1)\n    \u2261\u27e8\u27e9\n      \u27e6 t \u27e7 (update (dv \u2022 d\u03c1))\n    \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n      \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n  \u27e6 x \u27e7 \u03c1 \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\ncorrectVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\ncorrectness (intlit n) \u03c1 d\u03c1 = right-id-int n\ncorrectness (add s t) \u03c1 d\u03c1 = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 \u03c1} {\u27e6 t \u27e7 \u03c1} {\u27e6 s \u27e7\u0394 \u03c1 d\u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong\u2082 _+_ (correctness s \u03c1 d\u03c1) (correctness t \u03c1 d\u03c1))\ncorrectness (minus t) \u03c1 d\u03c1 = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 \u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong -_ (correctness t \u03c1 d\u03c1))\n\ncorrectness empty \u03c1 d\u03c1 = right-id-bag emptyBag\ncorrectness (insert s t) \u03c1 d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 \u03c1\n    b = \u27e6 t \u27e7 \u03c1\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness s \u03c1 d\u03c1))\n         (correctness t \u03c1 d\u03c1) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness (union s t) \u03c1 d\u03c1 = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 \u03c1} {\u27e6 t \u27e7 \u03c1} {\u27e6 s \u27e7\u0394 \u03c1 d\u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong\u2082 _++_ (correctness s \u03c1 d\u03c1) (correctness t \u03c1 d\u03c1))\ncorrectness (negate t) \u03c1 d\u03c1 = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 \u03c1} {\u27e6 t \u27e7\u0394 \u03c1 d\u03c1})\n  (cong negateBag (correctness t \u03c1 d\u03c1))\n\ncorrectness (flatmap s t) \u03c1 d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 \u03c1\n    b = \u27e6 t \u27e7 \u03c1\n    \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness s \u03c1 d\u03c1) (correctness t \u03c1 d\u03c1))\ncorrectness (sum t) \u03c1 d\u03c1 =\n  let\n    b = \u27e6 t \u27e7 \u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness t \u03c1 d\u03c1))\n\ncorrectness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\ncorrectness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 \u03c1\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 \u03c1\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e call-change \u0394f u \u0394u\n    \u2261\u27e8  sym (is-valid \u0394f u \u0394u) \u27e9\n       (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n  let\n    -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n    d\u03c1\u2032 = nil-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394Env \u0393 \u03c1) \u2192\n    (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n      (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n    \u229e\u208d \u03c4 \u208e\n    (call-change (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\ncorollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n  is-valid (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n  (v : \u27e6 \u03c3 \u27e7) (dv : Change \u03c3 v)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} dv)\n  \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\ncorollary-closed {\u03c3} {\u03c4} t v dv =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n  in\n    begin\n      f (after {\u03c3} dv)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} dv))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} dv)\n    \u2261\u27e8 is-valid (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n      f (before {\u03c3} dv) \u229e\u208d \u03c4 \u208e call-change \u0394f v dv\n    \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d90b16ea45533f37292736b66abbf94a6f7a4a08","subject":"Atlas: finish deriving the primitives `zip` and `update`","message":"Atlas: finish deriving the primitives `zip` and `update`\n\nPrimitives to add: lookup, fold\n\nOld-commit-hash: 7de6fdc8217beb1c74bd1b71c9e60ef1132c3ec9\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Atlas.agda","new_file":"Syntax\/Language\/Atlas.agda","new_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394base : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394base Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394base (Map key val) = (Map key (Atlas-\u0394base val))\n\nAtlas-\u0394type : Type Atlas-type \u2192 Type Atlas-type\nAtlas-\u0394type = lift-\u0394type\u2080 Atlas-\u0394base\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nneutral-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base \u03b9)\nneutral-term {Bool}   = const (neutral {Bool})\nneutral-term {Map \u03ba \u03b9} = const (neutral {Map \u03ba \u03b9})\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394base \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394base \u03b9))\nnil-term {Bool}   = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Shorthands of constants\n--\n-- There's probably a uniform way to lift constants\n-- into term constructors.\n--\n-- TODO: write this and call it Syntax.Term.Plotkin.lift-term\n\nzip! : \u2200 {\u03ba a b c \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u2192\n  Atlas-term \u0393 (base (Map \u03ba a)) \u2192 Atlas-term \u0393 (base (Map \u03ba b)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba c))\n\nzip! f m\u2081 m\u2082 = app (app (app (const zip) f) m\u2081) m\u2082\n\nlookup! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base \u03b9)\n\nlookup! = {!!} -- TODO: add constant `lookup`\n\n-- diff-term and apply-term\n\n-- b\u2080 \u229d b\u2081 = b\u2080 xor b\u2081\n-- m\u2080 \u229d m\u2081 = zip _\u229d_ m\u2080 m\u2081\n\nAtlas-diff : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 Atlas-\u0394type (base \u03b9))\nAtlas-diff {Bool} = const xor\nAtlas-diff {Map \u03ba \u03b9} = app (const zip) (abs Atlas-diff)\n\n-- b \u2295 \u0394b = b xor \u0394b\n-- m \u2295 \u0394m = zip _\u2295_ m \u0394m\n\nAtlas-apply : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)\nAtlas-apply {Bool} = const xor\nAtlas-apply {Map \u03ba \u03b9} = app (const zip) (abs Atlas-apply)\n\n-- Shorthands for working with diff-term and apply-term\n\ndiff : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 \u03c4 \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4)\ndiff s t = app (app (lift-diff Atlas-diff Atlas-apply) s) t\n\napply : \u2200 {\u03c4 \u0393} \u2192\n  Atlas-term \u0393 (Atlas-\u0394type \u03c4) \u2192 Atlas-term \u0393 \u03c4 \u2192\n  Atlas-term \u0393 \u03c4\napply s t = app (app (lift-apply Atlas-diff Atlas-apply) s) t\n\n-- Shorthands for creating changes corresponding to\n-- insertion\/deletion.\n\nupdate! : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9)) \u2192\n  Atlas-term \u0393 (base (Map \u03ba \u03b9))\n\nupdate! k v m = app (app (app (const update) k) v) m\n\ninsert : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  -- last argument is the change accumulator\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ndelete : \u2200 {\u03ba \u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03ba) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9))) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (base (Map \u03ba \u03b9)))\n\ninsert k v acc = update! k (diff v neutral-term) acc\ndelete k v acc = update! k (diff neutral-term v) acc\n\n-- The binary operator with which all base-type values\n-- form a group\n\nunion : \u2200 {\u03b9 \u0393} \u2192\n  Atlas-term \u0393 (base \u03b9) \u2192 Atlas-term \u0393 (base \u03b9) \u2192\n  Atlas-term \u0393 (base \u03b9)\nunion {Bool} s t = app (app (const xor) s) t\nunion {Map \u03ba \u03b9} s t =\n  let\n    union-term = abs (abs (union (var (that this)) (var this)))\n  in\n    zip! (abs union-term) s t\n\n-- Shorthand for 4-way zip\nzip4! : \u2200 {\u03ba a b c d e \u0393} \u2192\n  let\n    t:_ = \u03bb \u03b9 \u2192 Atlas-term \u0393 (base \u03b9)\n  in\n    Atlas-term \u0393\n      (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c \u21d2 base d \u21d2 base e) \u2192\n    t: Map \u03ba a \u2192 t: Map \u03ba b \u2192 t: Map \u03ba c \u2192 t: Map \u03ba d \u2192 t: Map \u03ba e\n\nzip4! f m\u2081 m\u2082 m\u2083 m\u2084 =\n  let\n    v\u2081 = var (that this)\n    v\u2082 = var this\n    v\u2083 = var (that this)\n    v\u2084 = var this\n    k\u2081\u2082 = var (that (that this))\n    k\u2083\u2084 = var (that (that this))\n    f\u2081\u2082 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2081\u2082) v\u2081) v\u2082)\n      (lookup! k\u2081\u2082 (weaken\u2083 m\u2083))) (lookup! k\u2081\u2082 (weaken\u2083 m\u2084)))))\n    f\u2083\u2084 = abs (abs (abs (app (app (app (app (app\n      (weaken\u2083 f) k\u2083\u2084)\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2081)))\n      (lookup! k\u2083\u2084 (weaken\u2083 m\u2082))) v\u2083) v\u2084)))\n  in\n    -- A correct but inefficient implementation.\n    -- May want to speed it up after constants are finalized.\n    union (zip! f\u2081\u2082 m\u2081 m\u2082) (zip! f\u2083\u2084 m\u2083 m\u2084)\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n  Atlas-term \u0393 (Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- else it is a deletion followed by insertion:\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- We implement the else-branch only for the moment.\nAtlas-\u0394const update =\n  let\n    k  = var (that (that (that (that (that this)))))\n    \u0394k = var (that (that (that (that this))))\n    v  = var (that (that (that this)))\n    \u0394v = var (that (that this))\n    -- m = var (that this) -- unused parameter\n    \u0394m = var this\n  in\n    abs (abs (abs (abs (abs (abs\n      (insert (apply \u0394k k) (apply \u0394v v)\n        (delete k v \u0394m)))))))\n\n-- \u0394zip f \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082 | true? (f \u2295 \u0394f \u2261 f)\n--\n-- ... | true =\n--   zip (\u03bb k \u0394v\u2081 \u0394v\u2082 \u2192 \u0394f (lookup k m\u2081) \u0394v\u2081 (lookup k m\u2082) \u0394v\u2082)\n--       \u0394m\u2081 \u0394m\u2082\n--\n-- ... | false = zip\u2084 \u0394f m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082\n--\n-- we implement the false-branch for the moment.\nAtlas-\u0394const zip =\n  let\n    \u0394f = var (that (that (that (that this))))\n    m\u2081  = var (that (that (that this)))\n    \u0394m\u2081 = var (that (that this))\n    m\u2082  = var (that this)\n    \u0394m\u2082 = var this\n    g = abs (app (app (weaken\u2081 \u0394f) (var this)) nil-term)\n  in\n    abs (abs (abs (abs (abs (abs (zip4! g m\u2081 \u0394m\u2081 m\u2082 \u0394m\u2082))))))\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  Atlas-\u0394type\n  Atlas-\u0394const\n","old_contents":"module Syntax.Language.Atlas where\n\n-- Base types of the calculus Atlas\n-- to be used with Plotkin-style language description\n--\n-- Atlas supports maps with neutral elements. The change type to\n-- `Map \u03ba \u03b9` is `Map \u03ba \u0394\u03b9`, where \u0394\u03b9 is the change type of the\n-- ground type \u03b9. Such a change to maps can support insertions\n-- and deletions as well: Inserting `k -> v` means mapping `k` to\n-- the change from the neutral element to `v`, and deleting\n-- `k -> v` means mapping `k` to the change from `v` to the\n-- neutral element.\n\nopen import Syntax.Language.Calculus\n\ndata Atlas-type : Set where\n  Bool : Atlas-type\n  Map : (\u03ba : Atlas-type) (\u03b9 : Atlas-type) \u2192 Atlas-type\n\ndata Atlas-const : Set where\n\n  true  : Atlas-const\n  false : Atlas-const\n  xor   : Atlas-const\n\n  empty  : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  update : \u2200 {\u03ba \u03b9 : Atlas-type} \u2192 Atlas-const\n  zip    : \u2200 {\u03ba a b c : Atlas-type} \u2192 Atlas-const\n\nAtlas-lookup : Atlas-const \u2192 Type Atlas-type\n\nAtlas-lookup true  = base Bool\nAtlas-lookup false = base Bool\nAtlas-lookup xor   = base Bool \u21d2 base Bool \u21d2 base Bool\n\nAtlas-lookup (empty {\u03ba} {\u03b9}) = base (Map \u03ba \u03b9)\n\n-- `update key val my-map` would\n-- - insert if `key` is not present in `my-map`\n-- - delete if `val` is the neutral element\n-- - make an update otherwise\nAtlas-lookup (update {\u03ba} {\u03b9}) =\n  base \u03ba \u21d2 base \u03b9 \u21d2 base (Map \u03ba \u03b9) \u21d2 base (Map \u03ba \u03b9)\n\n-- Model of zip = Haskell Data.List.zipWith\n-- zipWith :: (a \u2192 b \u2192 c) \u2192 [a] \u2192 [b] \u2192 [c]\nAtlas-lookup (zip {\u03ba} {a} {b} {c}) =\n  (base \u03ba \u21d2 base a \u21d2 base b \u21d2 base c) \u21d2\n  base (Map \u03ba a) \u21d2 base (Map \u03ba b) \u21d2 base (Map \u03ba c)\n\nAtlas-\u0394type : Atlas-type \u2192 Atlas-type\n-- change to a boolean is a xor-rand\nAtlas-\u0394type Bool = Bool\n-- change to a map is change to its values\nAtlas-\u0394type (Map key val) = (Map key (Atlas-\u0394type val))\n\nAtlas-context : Set\nAtlas-context = Context {Type Atlas-type}\n\nAtlas-term : Atlas-context \u2192 Type Atlas-type \u2192 Set\nAtlas-term = Term {Atlas-type} {Atlas-const} {Atlas-lookup}\n\n-- Every base type has a known nil-change.\n-- The nil-change of \u03b9 is also the neutral element of Map \u03ba \u0394\u03b9.\n\nneutral : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nneutral {Bool} = false\nneutral {Map \u03ba \u03b9} = empty {\u03ba} {\u03b9}\n\nnil-const : \u2200 {\u03b9 : Atlas-type} \u2192 Atlas-const\nnil-const {\u03b9} = neutral {Atlas-\u0394type \u03b9}\n\nnil-term : \u2200 {\u03b9 \u0393} \u2192 Atlas-term \u0393 (base (Atlas-\u0394type \u03b9))\nnil-term {Bool} = const (nil-const {Bool})\nnil-term {Map \u03ba \u03b9} = const (nil-const {Map \u03ba \u03b9})\n\n-- Type signature of Atlas-\u0394const is boilerplate.\nAtlas-\u0394const : \u2200 {\u0393} \u2192 (c : Atlas-const) \u2192\n      Atlas-term \u0393 (lift-\u0394type\u2080 Atlas-\u0394type (Atlas-lookup c))\n\nAtlas-\u0394const true  = const false\nAtlas-\u0394const false = const false\n\n-- \u0394xor = \u03bb x \u0394x y \u0394y \u2192 \u0394x xor \u0394y\nAtlas-\u0394const xor =\n  let\n    \u0394x = var (that (that this))\n    \u0394y = var this\n  in abs (abs (abs (abs (app (app (const xor) \u0394x) \u0394y))))\n\nAtlas-\u0394const empty = const empty\n\n-- If k \u2295 \u0394k \u2261 k, then\n--   \u0394update k \u0394k v \u0394v m \u0394m = update k \u0394v \u0394m\n-- otherwise it is a deletion followed by insertion.\n--   \u0394update k \u0394k v \u0394v m \u0394m =\n--     insert (k \u2295 \u0394k) (v \u2295 \u0394v) (delete k v \u0394m)\n--\n-- TODO: diff-term, apply-term\nAtlas-\u0394const update = {!!}\n\nAtlas-\u0394const zip = {!!}\n\nAtlas = calculus-with\n  Atlas-type\n  Atlas-const\n  Atlas-lookup\n  (lift-\u0394type\u2080 Atlas-\u0394type)\n  Atlas-\u0394const\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"239fae927ec7cfd1d245611caec9648ff523c126","subject":"Updated thesis' examples.","message":"Updated thesis' examples.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/AgdaIntroduction.agda","new_file":"notes\/thesis\/AgdaIntroduction.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n{-# OPTIONS --no-coverage-check #-}\n\nmodule AgdaIntroduction where\n\ninfixr 5 _\u2237_ _++_\ninfix  4 _\u2261_\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115     #-}\n{-# BUILTIN ZERO    zero  #-}\n{-# BUILTIN SUC     succ  #-}\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A 0\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0      + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\nf : \u2115 \u2192 \u2115\nf 0 = 0\nf _ = 1\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin 0 \u2192 A\nmagic ()\n\n-- The with constructor\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven 0        = true\neven (succ n) = odd n\n\nodd 0        = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Normalisation\n\ndata _\u2261_ {A : Set}: A \u2192 A \u2192 Set where\n  refl : {a : A} \u2192 a \u2261 a\n\nlength : {A : Set} \u2192 List A \u2192 \u2115\nlength []       = 0\nlength (x \u2237 xs) = 1 + length xs\n\n_++_ : {A : Set} \u2192 List A \u2192 List A \u2192 List A\n[]       ++ ys = ys\n(x \u2237 xs) ++ ys = x \u2237 xs ++ ys\n\nsuccCong : {m n : \u2115} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsuccCong refl = refl\n\nthm : {A : Set}(xs ys : List A) \u2192 length (xs ++ ys) \u2261 length xs + length ys\nthm [] ys       = refl\nthm (x \u2237 xs) ys = succCong (thm xs ys)\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack 0        n        = succ n\nack (succ m) 0        = ack m 1\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n-- Combinators for equational reasoning\n\npostulate\n  sym   : {A : Set}{a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\n  trans : {A : Set}{a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\n  subst : {A : Set}(P : A \u2192 Set){a b : A} \u2192 a \u2261 b \u2192 P a \u2192 P b\n\npostulate\n  _*_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  *-comm          : \u2200 m n \u2192 m * n \u2261 n * m\n  *-rightIdentity : \u2200 n \u2192 n * 1 \u2261 n\n\n*-leftIdentity : \u2200 n \u2192 1 * n \u2261 n\n*-leftIdentity n =\n  trans {\u2115} {1 * n} {n * 1} {n} (*-comm 1 n) (*-rightIdentity n)\n\nmodule ER\n  {A     : Set}\n  (_\u223c_   : A \u2192 A \u2192 Set)\n  (\u223c-refl  : \u2200 {x} \u2192 x \u223c x)\n  (\u223c-trans : \u2200 {x y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z)\n  where\n\n  infixr 5 _\u223c\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  _\u223c\u27e8_\u27e9_ : \u2200 x {y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z\n  _ \u223c\u27e8 x\u223cy \u27e9 y\u223cz = \u223c-trans x\u223cy y\u223cz\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _\u220e _ = \u223c-refl\n\nopen module \u2261-Reasoning = ER _\u2261_ (refl {\u2115}) (trans {\u2115})\n  renaming ( _\u223c\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_ )\n\n*-leftIdentity' : \u2200 n \u2192 1 * n \u2261 n\n*-leftIdentity' n =\n  1 * n \u2261\u27e8 *-comm 1 n \u27e9\n  n * 1 \u2261\u27e8 *-rightIdentity n \u27e9\n  n     \u220e\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n{-# OPTIONS --no-coverage-check #-}\n\nmodule AgdaIntroduction where\n\ninfixr 5 _\u2237_ _++_\ninfix  4 _\u2261_\n\n-- Dependent function types\nid : (A : Set) \u2192 A \u2192 A\nid A x = x\n\n-- \u03bb-notation\nid\u2082 : (A : Set) \u2192 A \u2192 A\nid\u2082 = \u03bb A \u2192 \u03bb x \u2192 x\n\nid\u2083 : (A : Set) \u2192 A \u2192 A\nid\u2083 = \u03bb A x \u2192 x\n\n-- Implicit arguments\nid\u2084 : {A : Set} \u2192 A \u2192 A\nid\u2084 x = x\n\nid\u2085 : {A : Set} \u2192 A \u2192 A\nid\u2085 = \u03bb x \u2192 x\n\n-- Inductively defined sets and families\ndata Bool : Set where\n  false true : Bool\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115     #-}\n{-# BUILTIN ZERO    zero  #-}\n{-# BUILTIN SUC     succ  #-}\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Vec (A : Set) : \u2115 \u2192 Set where\n  []  : Vec A 0\n  _\u2237_ : {n : \u2115} \u2192 A \u2192 Vec A n \u2192 Vec A (succ n)\n\ndata Fin : \u2115 \u2192 Set where\n  fzero : {n : \u2115} \u2192 Fin (succ n)\n  fsucc : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n\n-- Structurally recursive functions and pattern matching\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0      + n = n\nsucc m + n = succ (m + n)\n\nmap : {A B : Set} \u2192 (A \u2192 B) \u2192 List A \u2192 List B\nmap f []       = []\nmap f (x \u2237 xs) = f x \u2237 map f xs\n\n-- The absurd pattern\nmagic : {A : Set} \u2192 Fin 0 \u2192 A\nmagic ()\n\n-- The with constructor\nfilter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter p [] = []\nfilter p (x \u2237 xs) with p x\n... | true  = x \u2237 filter p xs\n... | false = filter p xs\n\nfilter' : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\nfilter' p [] = []\nfilter' p (x \u2237 xs) with p x\nfilter' p (x \u2237 xs) | true  = x \u2237 filter' p xs\nfilter' p (x \u2237 xs) | false = filter' p xs\n\n-- Mutual definitions\neven : \u2115 \u2192 Bool\nodd  : \u2115 \u2192 Bool\n\neven 0        = true\neven (succ n) = odd n\n\nodd 0        = false\nodd (succ n) = even n\n\ndata EvenList : Set\ndata OddList  : Set\n\ndata EvenList where\n  []  : EvenList\n  _\u2237_ : \u2115 \u2192 OddList \u2192 EvenList\n\ndata OddList where\n  _\u2237_ : \u2115 \u2192 EvenList \u2192 OddList\n\n-- Normalisation\n\ndata _\u2261_ {A : Set}: A \u2192 A \u2192 Set where\n  refl : {a : A} \u2192 a \u2261 a\n\nlength : {A : Set} \u2192 List A \u2192 \u2115\nlength []       = 0\nlength (x \u2237 xs) = 1 + length xs\n\n_++_ : {A : Set} \u2192 List A \u2192 List A \u2192 List A\n[]       ++ ys = ys\n(x \u2237 xs) ++ ys = x \u2237 xs ++ ys\n\nsuccCong : {m n : \u2115} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsuccCong refl = refl\n\nthm : {A : Set}(xs ys : List A) \u2192 length (xs ++ ys) \u2261 length xs + length ys\nthm [] ys       = refl\nthm (x \u2237 xs) ys = succCong (thm xs ys)\n\n-- Coverage and termination checkers\nhead : {A : Set} \u2192 List A \u2192 A\nhead (x \u2237 xs) = x\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack 0        n        = succ n\nack (succ m) 0        = ack m 1\nack (succ m) (succ n) = ack m (ack (succ m) n)\n\n{-# NO_TERMINATION_CHECK #-}\nf : \u2115 \u2192 \u2115\nf n = f n\n\n{-# NO_TERMINATION_CHECK #-}\nnest : \u2115 \u2192 \u2115\nnest 0        = 0\nnest (succ n) = nest (nest n)\n\n-- Combinators for equational reasoning\n\npostulate\n  sym   : {A : Set}{a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\n  trans : {A : Set}{a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\n  subst : {A : Set}(P : A \u2192 Set){a b : A} \u2192 a \u2261 b \u2192 P a \u2192 P b\n\npostulate\n  _*_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  *-comm          : \u2200 m n \u2192 m * n \u2261 n * m\n  *-rightIdentity : \u2200 n \u2192 n * 1 \u2261 n\n\n*-leftIdentity : \u2200 n \u2192 1 * n \u2261 n\n*-leftIdentity n =\n  trans {\u2115} {1 * n} {n * 1} {n} (*-comm 1 n) (*-rightIdentity n)\n\nmodule ER\n  {A     : Set}\n  (_\u223c_   : A \u2192 A \u2192 Set)\n  (refl  : \u2200 {x} \u2192 x \u223c x)\n  (trans : \u2200 {x y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z)\n  where\n\n  infixr 5 _\u223c\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  _\u223c\u27e8_\u27e9_ : \u2200 x {y z} \u2192 x \u223c y \u2192 y \u223c z \u2192 x \u223c z\n  _ \u223c\u27e8 x\u223cy \u27e9 y\u223cz = trans x\u223cy y\u223cz\n\n  _\u220e : \u2200 x \u2192 x \u223c x\n  _\u220e _ = refl\n\nopen module \u2261-Reasoning = ER _\u2261_ (refl {\u2115}) (trans {\u2115})\n  renaming ( _\u223c\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_ )\n\n*-leftIdentity' : \u2200 n \u2192 1 * n \u2261 n\n*-leftIdentity' n =\n  1 * n \u2261\u27e8 *-comm 1 n \u27e9\n  n * 1 \u2261\u27e8 *-rightIdentity n \u27e9\n  n     \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57ffb7b8e13a6bdd56fd4e960bdec24b159a26fe","subject":"Added missing local hint.","message":"Added missing local hint.\n\nIgnore-this: 38c71d36e14d0bf8e8fe831d2f2baa98\n\ndarcs-hash:20110219041623-3bd4e-8e0d484d261b3040cbb34f3b7a133387fbedfae5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_file":"src\/LTC\/Program\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule LTC.Program.McCarthy91.Properties.AuxiliaryATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\nopen import LTC.Program.McCarthy91.ArithmeticATP\nopen import LTC.Program.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 N10 \u2238-N #-}\n{-# ATP prove x<mc91x+11>100 +-N \u2238-N N10 N11 x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11\n              +-comm #-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n              Nmc91>100 N111 N101\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 N91 #-}\n{-# ATP prove mc91<mc91+11 mc91-res-100 102\u226191+11 100<102 #-}\n\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","old_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule LTC.Program.McCarthy91.Properties.AuxiliaryATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\nopen import LTC.Program.McCarthy91.ArithmeticATP\nopen import LTC.Program.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 N10 \u2238-N #-}\n{-# ATP prove x<mc91x+11>100 +-N \u2238-N N10 N11 x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11\n              +-comm #-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n              Nmc91>100 N111 N101\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 N91 #-}\n-- TODO\n-- {-# ATP prove mc91<mc91+11 mc91\u226191 n102 100<102 #-}\n\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"584a95def9b81558cd9bc660db3fc2eb687dd915","subject":"correct list of theorems","message":"correct list of theorems\n\nOld-commit-hash: c915c2bba4a8fd18c2b4d5af429fdedf6042ca83\n","repos":"inc-lc\/ilc-agda","old_file":"PLDI14-List-of-Theorems.agda","new_file":"PLDI14-List-of-Theorems.agda","new_contents":"module PLDI14-List-of-Theorems where\n\nopen import Function\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important name. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\n-- (d) ChangeAlgebra.diff\n-- (e) IsChangeAlgebra.update-diff\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- For each plugin requirement, we include its definition and\n-- a concrete instantiation called \"Popl14\" with integers and\n-- bags of integers as base types.\n\n-- Plugin Requirement 3.1 (Domains of base types)\nopen import Parametric.Denotation.Value using (Structure)\nopen import     Popl14.Denotation.Value using (\u27e6_\u27e7Base)\n\n-- Definition 3.2 (Domains)\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Plugin Requirement 3.3 (Evaluation of constants)\nopen import Parametric.Denotation.Evaluation using (Structure)\nopen import     Popl14.Denotation.Evaluation using (\u27e6_\u27e7Const)\n\n-- Definition 3.4 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.5 (Evaluation)\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Plugin Requirement 3.6 (Changes on base types)\nopen import Parametric.Change.Validity using (Structure)\nopen import     Popl14.Change.Validity using (change-algebra-base-family)\n\n-- Definition 3.7 (Changes)\nopen Parametric.Change.Validity.Structure using (change-algebra)\n\n-- Definition 3.8 (Change environments)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Plugin Requirement 3.9 (Change semantics for constants)\nopen import Parametric.Change.Specification using (Structure)\nopen import     Popl14.Change.Specification using (specification-structure)\n\n-- Definition 3.10 (Change semantics)\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\n\n-- Lemma 3.11 (Change semantics is the derivative of semantics)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.12 (Erasure)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen import Popl14.Change.Implementation using (implements-base)\n\n-- Lemma 3.13 (The erased version of a change is almost the same)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.14 (\u27e6 t \u27e7\u0394 erases to Derive(t))\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (derive-correct-closed)\n\n-- Theorem 3.15 (Correctness of differentiation)\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","old_contents":"module PLDI14-List-of-Theorems where\n\nopen import Function\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important name. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\n-- (d) ChangeAlgebra.diff\n-- (e) IsChangeAlgebra.update-diff\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- For each plugin requirement, we include its definition and\n-- a concrete instantiation called \"Popl14\" with integers and\n-- bags of integers as base types.\n\n-- Plugin Requirement 3.1 (Domains of base types)\nopen import Parametric.Denotation.Value using (Structure)\nopen import     Popl14.Denotation.Value using (\u27e6_\u27e7Base)\n\n-- Definition 3.2 (Domains)\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Plugin Requirement 3.3 (Evaluation of constants)\nopen import Parametric.Denotation.Evaluation using (Structure)\nopen import     Popl14.Denotation.Evaluation using (\u27e6_\u27e7Const)\n\n-- Definition 3.4 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.5 (Evaluation)\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Plugin Requirement 3.6 (Changes on base types)\nopen import Parametric.Change.Validity using (Structure)\nopen import     Popl14.Change.Validity using (change-algebra-base-family)\n\n-- Definition 3.7 (Changes)\nopen Parametric.Change.Validity.Structure using (change-algebra)\n\n-- Definition 3.8 (Change environments)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Plugin Requirement 3.9 (Change semantics for constants)\nopen import Parametric.Change.Specification using (Structure)\nopen import     Popl14.Change.Specification using (specification-structure)\n\n-- Definition 3.10 (Change semantics)\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\n\n-- Lemma 3.11 (Change semantics is the derivative of semantics)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.12 (Erasure)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen import Popl14.Change.Implementation using (implements-base)\n\n-- Lemma 3.13 (The erased version of a change is almost the same)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.14 (\u27e6 t \u27e7\u0394 erases to Derive(t))\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a7110647de4a66dda0726a937b1b2090851508bb","subject":"Maybe: monad things","message":"Maybe: monad things\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Maybe\/NP.agda","new_file":"lib\/Data\/Maybe\/NP.agda","new_contents":"module Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Relation.Unary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\ninfixr 0 _\u2192?_\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\ninfixl 1 _>>=?_\n\n-- More universe-polymorphic than M?._>>=_\n_>>=?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 Maybe A \u2192 (A \u2192 Maybe B) \u2192 Maybe B\nmx >>=? f = maybe f nothing mx\n\ninfixr 1 _=<<?_\n\n-- More universe-polymorphic than M?._=<<_\n_=<<?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 Maybe B) \u2192 Maybe A \u2192 Maybe B\nf =<<? mx = mx >>=? f\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f mx = mx >>=? (just \u2218 f)\n\n_<$?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2192 Maybe B \u2192 Maybe A\n_<$?_ x = map? (const x)\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = _=<<?_ id\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\njust?\u2192IsJust : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata [Maybe] {a p} {A : \u2605 a} (A\u209a : A \u2192 \u2605 p) : Maybe A \u2192 \u2605 (a \u2294 p) where\n  nothing : [Maybe] A\u209a nothing\n  just    : (A\u209a [\u2192] [Maybe] A\u209a) just\n\n[maybe] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] (\u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] ((A\u209a [\u2192] B\u209a) [\u2192] (B\u209a [\u2192] ([Maybe] A\u209a [\u2192] B\u209a)))))\n                     (maybe {a} {b})\n[maybe] _ _ just\u209a nothing\u209a (just x\u209a) = just\u209a x\u209a\n[maybe] _ _ just\u209a nothing\u209a nothing   = nothing\u209a\n\n[map?] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] \u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] (A\u209a [\u2192] B\u209a) [\u2192] [Maybe] A\u209a [\u2192] [Maybe] B\u209a) (map? {a} {b})\n[map?] _ _ f\u209a (just x\u209a) = just (f\u209a x\u209a)\n[map?] _ _ f\u209a nothing   = nothing\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  just    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  nothing : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (just x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 nothing   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (just x\u1d63) = just (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 nothing   = nothing\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (just x\u1d63) = just x\u1d63\n\u27e6map?-id\u27e7 _ nothing   = nothing\n\nAny-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\n(f \u2218? g) x = g x >>=? f\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_[\u2192?]_ : \u2200 {a b pa pb} \u2192 ([\u2605] {a} pa [\u2192] [\u2605] {b} pb [\u2192] [\u2605] _) _\u2192?_\nA\u209a [\u2192?] B\u209a = A\u209a [\u2192] [Maybe] B\u209a\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = just (refl-A x)\n  refl refl-A nothing  = nothing\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (just x\u223c\u2081y) = just (sym-A x\u223c\u2081y)\n  sym sym-A nothing     = nothing\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (just x\u223cy) (just y\u223cz) = just (trans' x\u223cy y\u223cz)\n  trans trans' nothing    nothing    = nothing\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (just x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f nothing    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \u22a5-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n-- -}\n","old_contents":"module Data.Maybe.NP where\n\nopen import Type hiding (\u2605)\nopen import Function\nimport Level as L\nopen L using (_\u2294_; lift; Lift)\nopen import Data.Maybe public\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Category.Applicative\nimport      Category.Monad as Cat\nopen import Relation.Binary.PropositionalEquality as \u2261 using (_\u2261_;_\u2257_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.Logical\nopen import Relation.Unary.Logical\nopen import Function using (_$_;flip;id)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\n\n\u03a0? : \u2200 {a b} (A : \u2605 a) (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0? A B = (x : A) \u2192 Maybe (B x)\n\ninfixr 0 _\u2192?_\n_\u2192?_ : \u2200 {a b} \u2192 \u2605 a \u2192 \u2605 b \u2192 \u2605 _\nA \u2192? B = A \u2192 Maybe B\n\nmodule M? \u2113 where\n  open Cat.RawMonadPlus (monadPlus {\u2113}) public\n  applicative = rawIApplicative\n\nopen M? public using (applicative)\n\ninfixl 4 _\u229b?_\n\n-- More universe-polymorphic than M?._\u229b_\n_\u229b?_ : \u2200 {a b}{A : \u2605 a}{B : \u2605 b} \u2192 Maybe (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\njust f  \u229b? just x = just (f x)\n_       \u229b? _      = nothing\n\n-- More universe-polymorphic than M?._<$>_\nmap? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\nmap? f = maybe (just \u2218\u2032 f) nothing\n\n\u27ea_\u00b7_\u27eb? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u2192 B) \u2192 Maybe A \u2192 Maybe B\n\u27ea f \u00b7 x \u27eb? = map? f x\n\n\u27ea_\u00b7_\u00b7_\u27eb? : \u2200 {a b c}\n             {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n             (A \u2192 B \u2192 C) \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C\n\u27ea f \u00b7 x \u00b7 y \u27eb? = map? f x \u229b? y\n\n\u27ea_\u00b7_\u00b7_\u00b7_\u27eb? : \u2200 {a b c d}\n               {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             \u2192 (A \u2192 B \u2192 C \u2192 D)\n             \u2192 Maybe A \u2192 Maybe B \u2192 Maybe C \u2192 Maybe D\n\u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb? = map? f x \u229b? y \u229b? z\n\njoin? : \u2200 {a} {A : \u2605 a} \u2192 Maybe (Maybe A) \u2192 Maybe A\njoin? = M?.join _\n\nMaybe^ : \u2200 {a} \u2192 \u2115 \u2192 \u2605 a \u2192 \u2605 a\nMaybe^ zero    = id\nMaybe^ (suc n) = Maybe \u2218 Maybe^ n\n\njust^ : \u2200 {a} {A : \u2605 a} n \u2192 A \u2192 Maybe^ n A\njust^ zero x = x\njust^ (suc n) x = just (just^ n x)\n\nMaybe^-\u2218-+ : \u2200 {a} m n (A : \u2605 a) \u2192 Maybe^ m (Maybe^ n A) \u2261 Maybe^ (m + n) A\nMaybe^-\u2218-+ zero    _ _ = \u2261.refl\nMaybe^-\u2218-+ (suc m) _ _ = \u2261.cong Maybe (Maybe^-\u2218-+ m _ _)\n\njust-injective : \u2200 {a} {A : \u2605 a} {x y : A}\n                 \u2192 Maybe.just x \u2261 just y \u2192 x \u2261 y\njust-injective \u2261.refl = \u2261.refl\n\nmaybe-just-nothing : \u2200 {a} {A : \u2605 a} \u2192 maybe {A = A} just nothing \u2257 id\nmaybe-just-nothing (just _)  = \u2261.refl\nmaybe-just-nothing nothing   = \u2261.refl\n\n_\u2261JAll_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAll y = All (\u03bb y' \u2192 All (_\u2261_ y') y) x\n\n_\u2261JAny_ : \u2200 {a} {A : \u2605 a} (x y : Maybe A) \u2192 \u2605 a\nx \u2261JAny y = Any (\u03bb y' \u2192 Any (_\u2261_ y') y) x\n\n\u2261JAll-refl : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261JAll x\n\u2261JAll-refl {x = just x}  = just (just \u2261.refl)\n\u2261JAll-refl {x = nothing} = nothing\n\njust? : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2192 \u2605\u2080\njust? nothing  = \u22a5\njust? (just _) = \u22a4\n\njust?\u2192IsJust : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 just? x \u2192 IsJust x\njust?\u2192IsJust {x = just _}  p = just _\njust?\u2192IsJust {x = nothing} ()\n\nAny\u2192just? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 just? x\nAny\u2192just? (just _) = _\n\ndata [Maybe] {a p} {A : \u2605 a} (A\u209a : A \u2192 \u2605 p) : Maybe A \u2192 \u2605 (a \u2294 p) where\n  nothing : [Maybe] A\u209a nothing\n  just    : (A\u209a [\u2192] [Maybe] A\u209a) just\n\n[maybe] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] (\u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] ((A\u209a [\u2192] B\u209a) [\u2192] (B\u209a [\u2192] ([Maybe] A\u209a [\u2192] B\u209a)))))\n                     (maybe {a} {b})\n[maybe] _ _ just\u209a nothing\u209a (just x\u209a) = just\u209a x\u209a\n[maybe] _ _ just\u209a nothing\u209a nothing   = nothing\u209a\n\n[map?] : \u2200 {a b} \u2192 (\u2200\u27e8 A\u209a \u2236 [\u2605] a \u27e9[\u2192] \u2200\u27e8 B\u209a \u2236 [\u2605] b \u27e9[\u2192] (A\u209a [\u2192] B\u209a) [\u2192] [Maybe] A\u209a [\u2192] [Maybe] B\u209a) (map? {a} {b})\n[map?] _ _ f\u209a (just x\u209a) = just (f\u209a x\u209a)\n[map?] _ _ f\u209a nothing   = nothing\n\ndata \u27e6Maybe\u27e7 {a b r} {A : \u2605 a} {B : \u2605 b} (_\u223c_ : A \u2192 B \u2192 \u2605 r) : Maybe A \u2192 Maybe B \u2192 \u2605 (a \u2294 b \u2294 r) where\n  just    : \u2200 {x\u2081 x\u2082} \u2192 (x\u1d63 : x\u2081 \u223c x\u2082) \u2192 \u27e6Maybe\u27e7 _\u223c_ (just x\u2081) (just x\u2082)\n  nothing : \u27e6Maybe\u27e7 _\u223c_ nothing nothing\n\n\u27e6maybe\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 (\u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 ((A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 (B\u1d63 \u27e6\u2192\u27e7 (\u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63)))))\n                     (maybe {a} {b}) (maybe {a} {b})\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 (just x\u1d63) = just\u1d63 x\u1d63\n\u27e6maybe\u27e7 _ _ just\u1d63 nothing\u1d63 nothing   = nothing\u1d63\n\n\u27e6map?\u27e7 : \u2200 {a b} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 a \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6\u2605\u27e7 b \u27e9\u27e6\u2192\u27e7 (A\u1d63 \u27e6\u2192\u27e7 B\u1d63) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63) (map? {a} {b}) (map? {a} {b})\n\u27e6map?\u27e7 _ _ f\u1d63 (just x\u1d63) = just (f\u1d63 x\u1d63)\n\u27e6map?\u27e7 _ _ f\u1d63 nothing   = nothing\n\n\u27e6map?-id\u27e7 : \u2200 {a} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6\u2605\u27e7 {a} {a} a \u27e9\u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 A\u1d63) (map? id) id\n\u27e6map?-id\u27e7 _ (just x\u1d63) = just x\u1d63\n\u27e6map?-id\u27e7 _ nothing   = nothing\n\nAny-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any (Any P) x \u2192 Any P (join? x)\nAny-join? (just p) = p\n\nAll-join? : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All (All P) x \u2192 All P (join? x)\nAll-join? (just p) = p\nAll-join? nothing  = nothing\n\nAny-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P (join? x) \u2192 Any (Any P) x\nAny-join?\u2032 {x = just x}  p = just p\nAny-join?\u2032 {x = nothing} ()\n\nAll-join?\u2032 : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 All P (join? x) \u2192 All (All P) x\nAll-join?\u2032 {x = just x}  p       = just p\nAll-join?\u2032 {x = nothing} nothing = nothing\n\nAny\u2192IsJust : \u2200 {a} {A : \u2605 a} {P : A \u2192 \u2605 a} {x} \u2192 Any P x \u2192 IsJust x\nAny\u2192IsJust (just _) = just _\n\njust\u2261\u2192just? : \u2200 {a} {A : \u2605 a} {x} {y : A} \u2192 x \u2261 just y \u2192 just? x\njust\u2261\u2192just? \u2261.refl = _\n\njust?-join? : \u2200 {a} {A : \u2605 a} {x : Maybe (Maybe A)} \u2192 just? (join? x) \u2192 just? x\njust?-join? = Any\u2192just? \u2218 Any-join?\u2032 \u2218 just?\u2192IsJust\n\nAny-just?-join? : \u2200 {A : \u2605\u2080} (x : Maybe (Maybe A)) \u2192 just? (join? x) \u2192 Any just? x\nAny-just?-join? (just (just _)) _ = just _\nAny-just?-join? (just nothing)  ()\nAny-just?-join? nothing         ()\n\njust?-map? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B)\n               (x : Maybe A) \u2192 just? (map? f x) \u2192 just? x\njust?-map? f (just x) pf = _\njust?-map? f nothing  ()\n\ninfix 4 _\u2257?_\n\n_\u2257?_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n         (f g : A \u2192? B) \u2192 \u2605 _\n(f \u2257? g) = \u2200 x \u2192 f x \u2261JAll g x\n\n_\u2218?_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n       \u2192 B \u2192? C \u2192 A \u2192? B \u2192 A \u2192? C\nf \u2218? g = join? \u2218 map? f \u2218 g\n\n\u2218?-just : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 f \u2218? just \u2257? f\n\u2218?-just f x = \u2261JAll-refl\n\njust-\u2218? : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192\n            (f : A \u2192? B) \u2192 just \u2218? f \u2257? f\njust-\u2218? f x with f x\njust-\u2218? f x | just _  = just (just \u2261.refl)\njust-\u2218? f x | nothing = nothing\n\n\u2218?-assoc : \u2200 {a b c d} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} {D : \u2605 d}\n             (f : C \u2192? D) (g : B \u2192? C) (h : A \u2192? B)\n             \u2192 (f \u2218? g) \u2218? h \u2257 f \u2218? (g \u2218? h)\n\u2218?-assoc f g h x with h x\n\u2218?-assoc f g h x | just _  = \u2261.refl\n\u2218?-assoc f g h x | nothing = \u2261.refl\n\nT[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT[_] {A = A} {B} f? = (x : A) \u2192 .{pf : just? (f? x)} \u2192 B\n\nF[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T[ f? ]\nF[ f? ] x {pf} with f? x\nF[ f? ] x      | just r  = r\nF[ f? ] x {()} | nothing\n\nT'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 \u2605 (a L.\u2294 b)\nT'[_] {A = A} {B} f? = (x : A) \u2192 IsJust (f? x) \u2192 B\n\nF'[_] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} (f? : A \u2192? B) \u2192 T'[ f? ]\nF'[ f? ] x pf with f? x\nF'[ f? ] x (just {y} _) | .(just y) = y\n\n-- F[ f? ] \u27f6 F'[ f? ]\n\nmodule F[] where\n    _[\u2257]_ : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n              {f? g? : A \u2192? B}\n              (f : T[ f? ]) (g : T[ g? ]) \u2192 \u2605 _\n    f [\u2257] g = \u2200 x {pf1} {pf2} \u2192 f x {pf1} \u2261 g x {pf2}\n\n    [id] : \u2200 {a} {A : \u2605 a} \u2192 T[ just {A = A} ]\n    [id] = F[ just ]\n\n    {- might actually be wrong\n    []-just-\u2261 : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f? : A \u2192? B} (f : T[ f? ]) {x} (pf : just? (f? x)) \u2192 just (f x {pf}) \u2261 f? x\n    []-just-\u2261 {f? = f?} f {x} pf = {!!}\n    -}\n\n    _[\u2218]_ : \u2200 {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n              {f? : B \u2192? C} {g? : A \u2192? B}\n            \u2192 T[ f? ] \u2192 T[ g? ] \u2192 T[ f? \u2218? g? ]\n    _[\u2218]_ {f? = f?} {g?} f g x {pf} with g? x\n    _[\u2218]_ f g x {pf} | just y  = f y {pf}\n    _[\u2218]_ f g x {()} | nothing\n\n    {-\n    [id]-[\u2218] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (F[ just ] [\u2218] f) [\u2257] f\n    [id]-[\u2218] {f? = f?} f x {pf1} {pf2} = just-injective {!(\u2261.sym (\u2261.trans ([]-just-\u2261 f pf2) ?))!}\n    -}\n\n    [\u2218]-[id] : \u2200 {a b} {A : \u2605 a} {B : \u2605 b}\n                 {f? : A \u2192? B} (f : T[ f? ]) \u2192 (f [\u2218] [id]) [\u2257] f\n    [\u2218]-[id] {f? = f?} f x {pf1} {pf2} = \u2261.refl\n\n_[\u2192?]_ : \u2200 {a b pa pb} \u2192 ([\u2605] {a} pa [\u2192] [\u2605] {b} pb [\u2192] [\u2605] _) _\u2192?_\nA\u209a [\u2192?] B\u209a = A\u209a [\u2192] [Maybe] B\u209a\n\n_\u27e6\u2192?\u27e7_ : \u2200 {a0 a1 ar b0 b1 br} \u2192 (\u27e6\u2605\u27e7 {a0} {a1} ar \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 {b0} {b1} br \u27e6\u2192\u27e7 \u27e6\u2605\u27e7 _) _\u2192?_ _\u2192?_\nA\u1d63 \u27e6\u2192?\u27e7 B\u1d63 = A\u1d63 \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 B\u1d63\n\nmodule \u27e6Maybe\u27e7-Properties where\n\n  refl : \u2200 {a p} {A : \u2605 a} {_\u223c_ : A \u2192 A \u2192 \u2605 p} (refl-A : \u2200 x \u2192 x \u223c x) (mx : Maybe A) \u2192 \u27e6Maybe\u27e7 _\u223c_ mx mx\n  refl refl-A (just x) = just (refl-A x)\n  refl refl-A nothing  = nothing\n\n  sym : \u2200 {a b r\u2081 r\u2082} {A : \u2605 a} {B : \u2605 b} {_\u223c\u2081_ : A \u2192 B \u2192 \u2605 r\u2081} {_\u223c\u2082_ : B \u2192 A \u2192 \u2605 r\u2082}\n          (sym-AB : \u2200 {x y} \u2192 x \u223c\u2081 y \u2192 y \u223c\u2082 x) {mx : Maybe A} {my : Maybe B}\n        \u2192 \u27e6Maybe\u27e7 _\u223c\u2081_ mx my \u2192 \u27e6Maybe\u27e7 _\u223c\u2082_ my mx\n  sym sym-A (just x\u223c\u2081y) = just (sym-A x\u223c\u2081y)\n  sym sym-A nothing     = nothing\n\n  trans : \u2200 {a b c r\u2081 r\u2082 r\u2083} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            {_\u27e6AB\u27e7_ : A \u2192 B \u2192 \u2605 r\u2081}\n            {_\u27e6BC\u27e7_ : B \u2192 C \u2192 \u2605 r\u2082}\n            {_\u27e6AC\u27e7_ : A \u2192 C \u2192 \u2605 r\u2083}\n            (trans : \u2200 {x y z} \u2192 x \u27e6AB\u27e7 y \u2192 y \u27e6BC\u27e7 z \u2192 x \u27e6AC\u27e7 z)\n            {mx : Maybe A} {my : Maybe B} {mz : Maybe C}\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AB\u27e7_ mx my \u2192 \u27e6Maybe\u27e7 _\u27e6BC\u27e7_ my mz\n          \u2192 \u27e6Maybe\u27e7 _\u27e6AC\u27e7_ mx mz\n  trans trans' (just x\u223cy) (just y\u223cz) = just (trans' x\u223cy y\u223cz)\n  trans trans' nothing    nothing    = nothing\n\n  subst-\u27e6AB\u27e7 : \u2200 {a b p q r} {A : \u2605 a} {B : \u2605 b}\n                 (P : Maybe A \u2192 \u2605 p)\n                 (Q : Maybe B \u2192 \u2605 q)\n                 (\u27e6AB\u27e7 : A \u2192 B \u2192 \u2605 r)\n                 (subst-\u27e6AB\u27e7-just : \u2200 {x y} \u2192 \u27e6AB\u27e7 x y \u2192 P (just x) \u2192 Q (just y))\n                 (Pnothing\u2192Qnothing : P nothing \u2192 Q nothing)\n                 {mx : Maybe A} {my : Maybe B}\n               \u2192 (\u27e6Maybe\u27e7 \u27e6AB\u27e7 mx my) \u2192 P mx \u2192 Q my\n  subst-\u27e6AB\u27e7 _ _ _ subst-\u27e6AB\u27e7-just _ (just x\u223cy) Pmx = subst-\u27e6AB\u27e7-just x\u223cy Pmx\n  subst-\u27e6AB\u27e7 _ _ _ _               f nothing    Pnothing = f Pnothing\n\n  subst : \u2200 {a p r} {A : \u2605 a}\n            (P : Maybe A \u2192 \u2605 p)\n            (A\u1d63 : A \u2192 A \u2192 \u2605 r)\n            (subst-A\u1d63 : \u2200 {x y} \u2192 A\u1d63 x y \u2192 P (just x) \u2192 P (just y))\n            {mx my}\n          \u2192 (\u27e6Maybe\u27e7 A\u1d63 mx my) \u2192 P mx \u2192 P my\n  subst P A\u1d63 subst-A\u1d63 = subst-\u27e6AB\u27e7 P P A\u1d63 subst-A\u1d63 id\n\nIsNothing'\u2261nothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 IsNothing x \u2192 x \u2261 nothing\nIsNothing'\u2261nothing nothing = \u2261.refl\nIsNothing'\u2261nothing (just (lift ()))\n\n\u2261nothing'IsNothing : \u2200 {a} {A : \u2605 a} {x : Maybe A} \u2192 x \u2261 nothing \u2192 IsNothing x\n\u2261nothing'IsNothing \u2261.refl = nothing\n\nmodule FunctorLemmas {a} where\n  open M? a\n\n  <$>-injective\u2081 : \u2200 {A B}\n                     {f : A \u2192 B} {x y : Maybe A}\n                     (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y)\n                   \u2192 f <$> x \u2261 f <$> y \u2192 x \u2261 y\n  <$>-injective\u2081 {x = just _}  {just _}  f-inj eq = \u2261.cong just (f-inj (just-injective eq))\n  <$>-injective\u2081 {x = nothing} {nothing} _     _  = \u2261.refl\n  <$>-injective\u2081 {x = just _}  {nothing} _     ()\n  <$>-injective\u2081 {x = nothing} {just _}  _     ()\n\n  <$>-assoc : \u2200 {A B C} {f : A \u2192 B} {g : C \u2192 A} (x : Maybe C) \u2192 f \u2218 g <$> x \u2261 f <$> (g <$> x)\n  <$>-assoc (just _) = \u2261.refl\n  <$>-assoc nothing  = \u2261.refl\n\nmodule MonadLemmas {a} where\n\n  open M? a public\n --  open RawApplicative applicative public\n\n  cong-Maybe : \u2200 {A B}\n                 (f : A \u2192 B) {x y} \u2192 x \u2261 pure y \u2192 f <$> x \u2261 pure (f y)\n  cong-Maybe f \u2261.refl = \u2261.refl\n\n  cong\u2082-Maybe : \u2200 {A B C}\n                  (f : A \u2192 B \u2192 C) {x y u v} \u2192 x \u2261 pure y \u2192 u \u2261 pure v \u2192 pure f \u229b x \u229b u \u2261 pure (f y v)\n  cong\u2082-Maybe f \u2261.refl \u2261.refl = \u2261.refl\n\n  Maybe-comm-monad :\n    \u2200 {A B C} {x y} {f : A \u2192 B \u2192 Maybe C} \u2192\n      (x >>= \u03bb x' \u2192 y >>= \u03bb y' \u2192 f x' y')\n    \u2261 (y >>= \u03bb y' \u2192 x >>= \u03bb x' \u2192 f x' y')\n  Maybe-comm-monad {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-monad {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-monad {x = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl : \u2200 {A B} {f : Maybe (A \u2192 B)} {x} \u2192 f \u229b x \u2261 (flip _$_) <$> x \u229b f\n  Maybe-comm-appl {f = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl {f = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl {f = just _}  {just _}   = \u2261.refl\n\n  Maybe-comm-appl\u2082 : \u2200 {A B C} {f : A \u2192 B \u2192 C} {x y} \u2192 f <$> x \u229b y \u2261 flip f <$> y \u229b x\n  Maybe-comm-appl\u2082 {x = nothing} {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = nothing} {just _}   = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {nothing}  = \u2261.refl\n  Maybe-comm-appl\u2082 {x = just _}  {just _}   = \u2261.refl\n\nmodule MonoidFromSemigroup {c \u2113} (sg   : Semigroup c \u2113)\n                                 {_\u2248?_ : let open Semigroup sg\n                                         in Maybe Carrier \u2192 Maybe Carrier \u2192 \u2605 \u2113}\n                                 (isEquivalence : IsEquivalence _\u2248?_)\n                                 (just-cong : let open Semigroup sg in\n                                              just Preserves _\u2248_ \u27f6 _\u2248?_)\n                                 (just-inj  : let open Semigroup sg in\n                                              (_\u2248?_ on just) \u21d2 _\u2248_)\n                                 (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248? nothing)) where\n  private\n    module SG = Semigroup sg\n    open SG using (_\u2248_) renaming (Carrier to A; _\u2219_ to op)\n    module \u2248  = IsEquivalence SG.isEquivalence\n    module \u2248? = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    just x  \u2219 just y  = just (op x y)\n    just x  \u2219 nothing = just x\n    nothing \u2219 y?      = y?\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248?_ _\u2219_\n    assoc (just x) (just y) (just z) = just-cong (SG.assoc x y z)\n    assoc (just _) (just _) nothing  = \u2248?.refl\n    assoc (just _) nothing  _        = \u2248?.refl\n    assoc nothing  _        _        = \u2248?.refl\n\n    right-identity : RightIdentity _\u2248?_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248?.refl\n    right-identity nothing  = \u2248?.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248?_ \u27f6 _\u2248?_ \u27f6 _\u2248?_\n    \u2219-cong {just _}{just _}{just _}{just _}   p q\n      = just-cong (SG.\u2219-cong (just-inj p) (just-inj q))\n    \u2219-cong {just _}{just _}{just _}{nothing}  p q = \u22a5-elim (just\u2249nothing q)\n    \u2219-cong {just _}{just _}{nothing}{just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym q))\n    \u2219-cong {just _}{just _}{nothing}{nothing} p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _}  {nothing} p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248?.sym p))\n\n  monoid : Monoid c \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248?_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248?.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule Monoid-\u2261 {a} {A : \u2605 a} {op : Op\u2082 A} (isSg : IsSemigroup _\u2261_ op)\n  = MonoidFromSemigroup (record { isSemigroup = isSg })\n                        \u2261.isEquivalence (\u2261.cong just) just-injective\n\nmodule First-\u2248 {a \u2113} {A : \u2605 a} {_\u2248_ : Maybe A \u2192 Maybe A \u2192 \u2605 \u2113}\n               (isEquivalence : IsEquivalence _\u2248_)\n               (just\u2249nothing : \u2200 {x} \u2192 \u00ac(just x \u2248 nothing)) where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    _\u2219_ : Op\u2082 (Maybe A)\n    x \u2219 y = maybe just y x\n\n    \u03b5 : Maybe A\n    \u03b5 = nothing\n\n    assoc : Associative _\u2248_ _\u2219_\n    assoc (just _) _ _ = \u2248.refl\n    assoc nothing  _ _ = \u2248.refl\n\n    right-identity : RightIdentity _\u2248_ \u03b5 _\u2219_\n    right-identity (just _) = \u2248.refl\n    right-identity nothing  = \u2248.refl\n\n    \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n    \u2219-cong {just _} {just _}   p q = p\n    \u2219-cong {nothing} {nothing} p q = q\n    \u2219-cong {just _} {nothing}  p q = \u22a5-elim (just\u2249nothing p)\n    \u2219-cong {nothing} {just _}  p q = \u22a5-elim (just\u2249nothing (\u2248.sym p))\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Maybe A\n                  ; _\u2248_ = _\u2248_\n                  ; _\u2219_ = _\u2219_\n                  ; \u03b5 = \u03b5\n                  ; isMonoid\n                    = record { isSemigroup\n                               = record { isEquivalence = isEquivalence\n                                        ; assoc = assoc; \u2219-cong = \u2219-cong }\n                             ; identity = (\u03bb _ \u2192 \u2248.refl) , right-identity } }\n\n  open Monoid monoid public\n\nmodule First {a} (A : \u2605 a) = First-\u2248 {A = A} \u2261.isEquivalence (\u03bb())\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a6c837cc001c04e01bceef68c282b551a001dcbd","subject":"Added ListN-ind.","message":"Added ListN-ind.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Nat\/List\/Type.agda","new_file":"src\/fot\/FOTC\/Data\/Nat\/List\/Type.agda","new_contents":"----------------------------------------------------------------------------\n-- The FOTC lists of natural numbers type\n----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Nat.List.Type where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The FOTC lists of natural numbers type (inductive predicate for\n-- total lists of natural numbers).\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n{-# ATP axiom lnnil lncons #-}\n\n-- Induction principle.\nListN-ind : (A : D \u2192 Set) \u2192\n            A [] \u2192\n            (\u2200 {n ns} \u2192 N n \u2192 A ns \u2192 A (n \u2237 ns)) \u2192\n            \u2200 {ns} \u2192 ListN ns \u2192 A ns\nListN-ind A Anil h lnnil           = Anil\nListN-ind A Anil h (lncons Nn Lns) = h Nn (ListN-ind A Anil h Lns)\n","old_contents":"----------------------------------------------------------------------------\n-- The FOTC lists of natural numbers type\n----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Nat.List.Type where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- The FOTC lists of natural numbers type (inductive predicate for\n-- total lists of natural numbers).\ndata ListN : D \u2192 Set where\n  lnnil  :                             ListN []\n  lncons : \u2200 {n ns} \u2192 N n \u2192 ListN ns \u2192 ListN (n \u2237 ns)\n{-# ATP axiom lnnil lncons #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e49234d8d4b71ee5271d777c9a5e694bb6c6ee66","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Induction principle generated by Coq when we define the data type N\n-- in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n-- N-ind\u2082 from N-ind\u2081.\nN-ind\u2082' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n-- We cannot prove N-ind\u2081 from N-ind\u2082.\nN-ind\u2081' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081' A A0 h = N-ind\u2082 A A0 {!!}\n","old_contents":"------------------------------------------------------------------------------\n-- Induction principle for N.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Induction principle generated by Coq when we define the data type N\n-- in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n-- N-ind\u2082 from N-ind\u2081.\nN-ind\u2082' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n-- We cannot prove N-ind\u2081 from N-ind\u2082.\nN-ind\u2081' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081' A A0 h = N-ind\u2082 A A0 {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"80325601bb7c894ee0237d8a6f7f3c3aa3044b73","subject":"Nat: +2^*-spec","message":"Nat: +2^*-spec\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 (suc n \u2236 \u2115) \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : Set a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n{-\npost--ulate\n  dist-sum   : \u2200 x y z \u2192 dist x y + dist y z \u2264 dist x z\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 Set\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : Set} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 Set\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 Set) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 Set\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  postulate\n    \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n    -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n    mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n  -- mon\u21913+' b = {!!}\n\n  mon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\n  mon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  open \u2191-Props\n  lem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\n  lem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n               2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n               2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n               2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n               2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n    where open \u2264-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n  lem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\n  lem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n  lem>=3'' (suc (suc m)) n = lem>=3 m n\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  ack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\n  ack-\u2191 zero n = \u2261.refl\n  ack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                       2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                       2 \u2191\u27e8 suc m \u27e9 3 \u220e\n    where open \u2261-Reasoning\n  ack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m (ack (suc m) n)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                           3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                        \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                        \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                           2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n    where open \u2261-Reasoning\n\n  postulate\n    1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n  --  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  -- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\n  fold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\n  fold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n  \u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n  \u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : Set a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 Set\n  m \u2248 n = T (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 T (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 Set where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 Set\n  \u211b x y = T (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cb78a702818ecaff8f87e4a2eb1e4a14b0693470","subject":"Added doc.","message":"Added doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/ConversionRules.agda","new_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/ConversionRules.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion rules for inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for lt it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (lt-s\u2081,\n  -- lt-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q, e.g.\n  --\n  -- proof\u2081\u2192proof\u2082 : \u2200 m n \u2192 lt-s\u2082 m n \u2192 lt-s\u2083 m n.\n\n  -- The terms lt-00, lt-0S, lt-S0, and lt-SS show the use of the\n  -- states lt-s\u2081, lt-s\u2082, ..., and the proofs associated with the\n  -- execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt.\n\n  -- Initially, the conversion rule fix-eq is applied.\n  lt-s\u2081 : D \u2192 D \u2192 D\n  lt-s\u2081 m n = lth (fix lth) \u00b7 m \u00b7 n\n\n  -- First argument application.\n  lt-s\u2082 : D \u2192 D\n  lt-s\u2082 m = lam (\u03bb n \u2192\n               if (iszero\u2081 n)\n                  then false\n                  else (if (iszero\u2081 m)\n                           then true\n                           else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)))\n\n\n  -- Second argument application.\n  lt-s\u2083 : D \u2192 D \u2192 D\n  lt-s\u2083 m n = if (iszero\u2081 n)\n                 then false\n                 else (if (iszero\u2081 m)\n                          then true\n                          else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  lt-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2084 m n b = if b\n                   then false\n                   else (if (iszero\u2081 m)\n                            then true\n                            else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 true.\n  -- It should be\n  -- lt-s\u2085 : D \u2192 D \u2192 D\n  -- lt-s\u2085 m n = false\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  lt-s\u2085 : D \u2192 D \u2192 D\n  lt-s\u2085 m n = if (iszero\u2081 m)\n                 then true\n                 else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b.\n  lt-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2086 m n b = if b\n                   then true\n                   else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n  -- Reduction iszero\u2081 m \u2261 true.\n  -- It should be\n  -- lt-s\u2087 : D \u2192 D \u2192 D\n  -- lt-s\u2087 m n = true\n  -- but we do not give a name to this step.\n\n   -- Reduction iszero\u2081 m \u2261 false.\n  lt-s\u2087 : D \u2192 D \u2192 D\n  lt-s\u2087 m n = fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ m) \u2261 m.\n  lt-s\u2088 : D \u2192 D \u2192 D\n  lt-s\u2088 m n = fix lth \u00b7 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  lt-s\u2089 : D \u2192 D \u2192 D\n  lt-s\u2089 m n = fix lth \u00b7 m \u00b7 n\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n\n    To prove the execution steps, e.g.\n\n    proof\u2083\u2192proof\u2084 : \u2200 m n \u2192 lt-s\u2083 m n \u2192 lt-s\u2084 m n)\n\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n\n    We prove (1) using\n\n    subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\n    where\n      \u2022 P is given by \u03bb t \u2192 C [m] \u2261 C [t],\n      \u2022 x \u2261 y is given m \u2261 n, and\n      \u2022 P x is given by C [m] \u2261 C [m] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 m n \u2192 fix lth \u00b7 m \u00b7 n  \u2261 lt-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u00b7 m \u00b7 n  \u2261 lth (fix lth) \u00b7 m \u00b7 n )\n                       (sym (fix-eq lth ))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 m n \u2192 lt-s\u2081 m n \u2261 lt-s\u2082 m \u00b7 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u00b7 n \u2261 lt-s\u2082 m \u00b7 n)\n                       (sym (beta lt-s\u2082 m))\n                       refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 m n \u2192 lt-s\u2082 m \u00b7 n \u2261 lt-s\u2083 m n\n  proof\u2082\u208b\u2083 m n = beta (lt-s\u2083 m) n\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n  --\n  -- Reduction iszero n \u2261 b using that proof.\n  proof\u2083\u208b\u2084 : \u2200 m n b \u2192 iszero\u2081 n \u2261 b \u2192 lt-s\u2083 m n \u2261 lt-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b h = subst (\u03bb x \u2192 lt-s\u2084 m n x \u2261 lt-s\u2084 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 n \u2261 true using the conversion rule if-true.\n  proof\u2084\u208a : \u2200 m n \u2192 lt-s\u2084 m n true \u2261 false\n  proof\u2084\u208a m n = if-true false\n\n  -- Reduction of iszero\u2081 n \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2084\u208b\u2085 : \u2200 m n \u2192 lt-s\u2084 m n false \u2261 lt-s\u2085 m n\n  proof\u2084\u208b\u2085 m n = if-false (lt-s\u2085 m n)\n\n  -- 01 June 2013. This proof could use pattern matching on _\u2261_. See\n  -- Agda issue 865.\n  --\n  -- Reduction iszero\u2081 m \u2261 b using that proof.\n  proof\u2085\u208b\u2086 : \u2200 m n b \u2192 iszero\u2081 m \u2261 b \u2192 lt-s\u2085 m n \u2261 lt-s\u2086 m n b\n  proof\u2085\u208b\u2086 m n b h = subst (\u03bb x \u2192 lt-s\u2086 m n x \u2261 lt-s\u2086 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 m \u2261 true using the conversion rule if-true.\n  proof\u2086\u208a : \u2200 m n \u2192 lt-s\u2086 m n true \u2261 true\n  proof\u2086\u208a m n = if-true true\n\n  -- Reduction of iszero\u2081 m \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2086\u208b\u2087 : \u2200 m n \u2192 lt-s\u2086 m n false \u2261 lt-s\u2087 m n\n  proof\u2086\u208b\u2087 m n = if-false (lt-s\u2087 m n)\n\n  -- Reduction pred (succ m) \u2261 m using the conversion rule pred-S.\n  proof\u2087\u208b\u2088 : \u2200 m n \u2192 lt-s\u2087 (succ\u2081 m) n  \u2261 lt-s\u2088 m n\n  proof\u2087\u208b\u2088 m n = subst (\u03bb x \u2192 lt-s\u2088 x n \u2261 lt-s\u2088 m n)\n                       (sym (pred-S m ))\n                       refl\n\n  -- Reduction pred (succ n) \u2261 n using the conversion rule pred-S.\n  proof\u2088\u208b\u2089 : \u2200 m n \u2192 lt-s\u2088 m (succ\u2081 n)  \u2261 lt-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = subst (\u03bb x \u2192 lt-s\u2089 m x \u2261 lt-s\u2089 m n)\n                       (sym (pred-S n ))\n                       refl\n\n------------------------------------------------------------------------------\n\nprivate\n  X\u226e0 : \u2200 n \u2192 n \u226e zero\n  X\u226e0 n = fix lth \u00b7 n \u00b7 zero  \u2261\u27e8 proof\u2080\u208b\u2081 n zero \u27e9\n          lt-s\u2081 n zero        \u2261\u27e8 proof\u2081\u208b\u2082 n zero \u27e9\n          lt-s\u2082 n \u00b7 zero      \u2261\u27e8 proof\u2082\u208b\u2083 n zero \u27e9\n          lt-s\u2083 n zero        \u2261\u27e8 proof\u2083\u208b\u2084 n zero true iszero-0 \u27e9\n          lt-s\u2084 n zero true   \u2261\u27e8 proof\u2084\u208a n zero \u27e9\n          false              \u220e\n\nlt-00 : zero \u226e zero\nlt-00 = X\u226e0 zero\n\nlt-0S : \u2200 n \u2192 zero < succ\u2081 n\nlt-0S n =\n  fix lth \u00b7 zero \u00b7 (succ\u2081 n)    \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ\u2081 n) \u27e9\n  lt-s\u2081 zero (succ\u2081 n)          \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ\u2081 n) \u27e9\n  lt-s\u2082 zero \u00b7 (succ\u2081 n)        \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ\u2081 n) \u27e9\n  lt-s\u2083 zero (succ\u2081 n)          \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 zero (succ\u2081 n) false    \u2261\u27e8 proof\u2084\u208b\u2085 zero (succ\u2081 n) \u27e9\n  lt-s\u2085 zero (succ\u2081 n)          \u2261\u27e8 proof\u2085\u208b\u2086 zero (succ\u2081 n) true iszero-0 \u27e9\n  lt-s\u2086 zero (succ\u2081 n) true     \u2261\u27e8 proof\u2086\u208a  zero (succ\u2081 n) \u27e9\n  true                         \u220e\n\nlt-S0 : \u2200 n \u2192 succ\u2081 n \u226e zero\nlt-S0 n = X\u226e0 (succ\u2081 n)\n\nlt-SS : \u2200 m n \u2192 lt (succ\u2081 m) (succ\u2081 n) \u2261 lt m n\nlt-SS m n =\n  fix lth \u00b7 (succ\u2081 m) \u00b7 (succ\u2081 n)  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2081 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2082 (succ\u2081 m) \u00b7 (succ\u2081 n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2083 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 m) (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2085 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 m) (succ\u2081 n) false (iszero-S m) \u27e9\n  lt-s\u2086 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2086\u208b\u2087 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2087 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2087\u208b\u2088 m (succ\u2081 n) \u27e9\n  lt-s\u2088 m (succ\u2081 n)                \u2261\u27e8 proof\u2088\u208b\u2089 m n \u27e9\n  lt-s\u2089 m n                        \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Conversion rules for inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for lt it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (lt-s\u2081,\n  -- lt-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q, e.g.\n  --\n  -- proof\u2081\u2192proof\u2082 : \u2200 m n \u2192 lt-s\u2082 m n \u2192 lt-s\u2083 m n.\n\n  -- The terms lt-00, lt-0S, lt-S0, and lt-SS show the use of the\n  -- states lt-s\u2081, lt-s\u2082, ..., and the proofs associated with the\n  -- execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt.\n\n  -- Initially, the conversion rule fix-eq is applied.\n  lt-s\u2081 : D \u2192 D \u2192 D\n  lt-s\u2081 m n = lth (fix lth) \u00b7 m \u00b7 n\n\n  -- First argument application.\n  lt-s\u2082 : D \u2192 D\n  lt-s\u2082 m = lam (\u03bb n \u2192\n               if (iszero\u2081 n)\n                  then false\n                  else (if (iszero\u2081 m)\n                           then true\n                           else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)))\n\n\n  -- Second argument application.\n  lt-s\u2083 : D \u2192 D \u2192 D\n  lt-s\u2083 m n = if (iszero\u2081 n)\n                 then false\n                 else (if (iszero\u2081 m)\n                          then true\n                          else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  lt-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2084 m n b = if b\n                   then false\n                   else (if (iszero\u2081 m)\n                            then true\n                            else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 true.\n  -- It should be\n  -- lt-s\u2085 : D \u2192 D \u2192 D\n  -- lt-s\u2085 m n = false\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  lt-s\u2085 : D \u2192 D \u2192 D\n  lt-s\u2085 m n = if (iszero\u2081 m)\n                 then true\n                 else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b.\n  lt-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2086 m n b = if b\n                   then true\n                   else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n  -- Reduction iszero\u2081 m \u2261 true.\n  -- It should be\n  -- lt-s\u2087 : D \u2192 D \u2192 D\n  -- lt-s\u2087 m n = true\n  -- but we do not give a name to this step.\n\n   -- Reduction iszero\u2081 m \u2261 false.\n  lt-s\u2087 : D \u2192 D \u2192 D\n  lt-s\u2087 m n = fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ m) \u2261 m.\n  lt-s\u2088 : D \u2192 D \u2192 D\n  lt-s\u2088 m n = fix lth \u00b7 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  lt-s\u2089 : D \u2192 D \u2192 D\n  lt-s\u2089 m n = fix lth \u00b7 m \u00b7 n\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n\n    To prove the execution steps, e.g.\n\n    proof\u2083\u2192proof\u2084 : \u2200 m n \u2192 lt-s\u2083 m n \u2192 lt-s\u2084 m n)\n\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n\n    We prove (1) using\n\n    subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\n    where\n      \u2022 P is given by \u03bb t \u2192 C [m] \u2261 C [t],\n      \u2022 x \u2261 y is given m \u2261 n, and\n      \u2022 P x is given by C [m] \u2261 C [m] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 m n \u2192 fix lth \u00b7 m \u00b7 n  \u2261 lt-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u00b7 m \u00b7 n  \u2261 lth (fix lth) \u00b7 m \u00b7 n )\n                       (sym (fix-eq lth ))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 m n \u2192 lt-s\u2081 m n \u2261 lt-s\u2082 m \u00b7 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u00b7 n \u2261 lt-s\u2082 m \u00b7 n)\n                       (sym (beta lt-s\u2082 m))\n                       refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 m n \u2192 lt-s\u2082 m \u00b7 n \u2261 lt-s\u2083 m n\n  proof\u2082\u208b\u2083 m n = beta (lt-s\u2083 m) n\n\n\n  -- Reduction iszero n \u2261 b using that proof.\n  proof\u2083\u208b\u2084 : \u2200 m n b \u2192 iszero\u2081 n \u2261 b \u2192 lt-s\u2083 m n \u2261 lt-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b h = subst (\u03bb x \u2192 lt-s\u2084 m n x \u2261 lt-s\u2084 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 n \u2261 true using the conversion rule if-true.\n  proof\u2084\u208a : \u2200 m n \u2192 lt-s\u2084 m n true \u2261 false\n  proof\u2084\u208a m n = if-true false\n\n  -- Reduction of iszero\u2081 n \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2084\u208b\u2085 : \u2200 m n \u2192 lt-s\u2084 m n false \u2261 lt-s\u2085 m n\n  proof\u2084\u208b\u2085 m n = if-false (lt-s\u2085 m n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b using that proof.\n  proof\u2085\u208b\u2086 : \u2200 m n b \u2192 iszero\u2081 m \u2261 b \u2192 lt-s\u2085 m n \u2261 lt-s\u2086 m n b\n  proof\u2085\u208b\u2086 m n b h = subst (\u03bb x \u2192 lt-s\u2086 m n x \u2261 lt-s\u2086 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 m \u2261 true using the conversion rule if-true.\n  proof\u2086\u208a : \u2200 m n \u2192 lt-s\u2086 m n true \u2261 true\n  proof\u2086\u208a m n = if-true true\n\n  -- Reduction of iszero\u2081 m \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2086\u208b\u2087 : \u2200 m n \u2192 lt-s\u2086 m n false \u2261 lt-s\u2087 m n\n  proof\u2086\u208b\u2087 m n = if-false (lt-s\u2087 m n)\n\n  -- Reduction pred (succ m) \u2261 m using the conversion rule pred-S.\n  proof\u2087\u208b\u2088 : \u2200 m n \u2192 lt-s\u2087 (succ\u2081 m) n  \u2261 lt-s\u2088 m n\n  proof\u2087\u208b\u2088 m n = subst (\u03bb x \u2192 lt-s\u2088 x n \u2261 lt-s\u2088 m n)\n                       (sym (pred-S m ))\n                       refl\n\n  -- Reduction pred (succ n) \u2261 n using the conversion rule pred-S.\n  proof\u2088\u208b\u2089 : \u2200 m n \u2192 lt-s\u2088 m (succ\u2081 n)  \u2261 lt-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = subst (\u03bb x \u2192 lt-s\u2089 m x \u2261 lt-s\u2089 m n)\n                       (sym (pred-S n ))\n                       refl\n\n------------------------------------------------------------------------------\n\nprivate\n  X\u226e0 : \u2200 n \u2192 n \u226e zero\n  X\u226e0 n = fix lth \u00b7 n \u00b7 zero  \u2261\u27e8 proof\u2080\u208b\u2081 n zero \u27e9\n          lt-s\u2081 n zero        \u2261\u27e8 proof\u2081\u208b\u2082 n zero \u27e9\n          lt-s\u2082 n \u00b7 zero      \u2261\u27e8 proof\u2082\u208b\u2083 n zero \u27e9\n          lt-s\u2083 n zero        \u2261\u27e8 proof\u2083\u208b\u2084 n zero true iszero-0 \u27e9\n          lt-s\u2084 n zero true   \u2261\u27e8 proof\u2084\u208a n zero \u27e9\n          false              \u220e\n\nlt-00 : zero \u226e zero\nlt-00 = X\u226e0 zero\n\nlt-0S : \u2200 n \u2192 zero < succ\u2081 n\nlt-0S n =\n  fix lth \u00b7 zero \u00b7 (succ\u2081 n)    \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ\u2081 n) \u27e9\n  lt-s\u2081 zero (succ\u2081 n)          \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ\u2081 n) \u27e9\n  lt-s\u2082 zero \u00b7 (succ\u2081 n)        \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ\u2081 n) \u27e9\n  lt-s\u2083 zero (succ\u2081 n)          \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 zero (succ\u2081 n) false    \u2261\u27e8 proof\u2084\u208b\u2085 zero (succ\u2081 n) \u27e9\n  lt-s\u2085 zero (succ\u2081 n)          \u2261\u27e8 proof\u2085\u208b\u2086 zero (succ\u2081 n) true iszero-0 \u27e9\n  lt-s\u2086 zero (succ\u2081 n) true     \u2261\u27e8 proof\u2086\u208a  zero (succ\u2081 n) \u27e9\n  true                         \u220e\n\nlt-S0 : \u2200 n \u2192 succ\u2081 n \u226e zero\nlt-S0 n = X\u226e0 (succ\u2081 n)\n\nlt-SS : \u2200 m n \u2192 lt (succ\u2081 m) (succ\u2081 n) \u2261 lt m n\nlt-SS m n =\n  fix lth \u00b7 (succ\u2081 m) \u00b7 (succ\u2081 n)  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2081 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2082 (succ\u2081 m) \u00b7 (succ\u2081 n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2083 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 m) (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2085 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 m) (succ\u2081 n) false (iszero-S m) \u27e9\n  lt-s\u2086 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2086\u208b\u2087 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2087 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2087\u208b\u2088 m (succ\u2081 n) \u27e9\n  lt-s\u2088 m (succ\u2081 n)                \u2261\u27e8 proof\u2088\u208b\u2089 m n \u27e9\n  lt-s\u2089 m n                        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"abdb76b33ea917b52a63cb820db97de2d5439cdb","subject":"Added +-comm using an instance of the induction principle.","message":"Added +-comm using an instance of the induction principle.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddComm.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddComm.agda","new_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition of total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Auxiliary properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n+-comm-ind-instance :\n  \u2200 n \u2192\n  zero + n \u2261 n + zero \u2192\n  (\u2200 {m} \u2192 m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m) \u2192\n  \u2200 {m} \u2192 N m \u2192 m + n \u2261 n + m\n+-comm-ind-instance n = N-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n------------------------------------------------------------------------------\n-- Approach 1: Interactive proof using pattern matching\n\nmodule M\u2081 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} nzero Nn =\n    zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n    n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n    n + zero \u220e\n\n  +-comm {n = n} (nsucc {m} Nm) Nn =\n    succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n    succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm Nm Nn) \u27e9\n    succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n    n + succ\u2081 m   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 2: Combined proof using pattern matching\n\nmodule M\u2082 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} nzero Nn = prf\n    where postulate prf : zero + n \u2261 n + zero\n          {-# ATP prove prf +-rightIdentity #-}\n  +-comm {n = n} (nsucc {m} Nm) Nn = prf (+-comm Nm Nn)\n    where postulate prf : m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m\n          {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 3: Interactive proof using the induction principle\n\nmodule M\u2083 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = i + n \u2261 n + i\n\n    A0 : A zero\n    A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n         n + zero \u220e\n\n    is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n                succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n                succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n                n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 4: Combined proof using the induction principle\n\nmodule M\u2084 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = N-ind A A0 is Nm\n    where\n    A : D \u2192 Set\n    A i = i + n \u2261 n + i\n    {-# ATP definition A #-}\n\n    postulate A0 : A zero\n    {-# ATP prove A0 +-rightIdentity #-}\n\n    postulate is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n    {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 5: Interactive proof using an instance of the induction\n-- principle\n\nmodule M\u2085 where\n\n  +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  +-comm {n = n} Nm Nn = +-comm-ind-instance n A0 is Nm\n    where\n    A0 : zero + n \u2261 n + zero\n    A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n         n + zero \u220e\n\n    is : \u2200 {i} \u2192 i + n \u2261 n + i \u2192 succ\u2081 i + n \u2261 n + succ\u2081 i\n    is {i} ih = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n                succ\u2081 (i + n) \u2261\u27e8 succCong ih \u27e9\n                succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n                n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 6: Combined proof using an instance of the induction\n-- principle\n\nmodule M\u2086 where\n\n  -- TODO (25 October 2012): Why is it not necessary the hypothesis\n  -- +-rightIdentity?\n  postulate +-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n  {-# ATP prove +-comm +-comm-ind-instance x+Sy\u2261S[x+y] #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition of total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n-- Auxiliary properties\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ\u2081 m \u2261 succ\u2081 n\nsuccCong refl = refl\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity n = +-0x n\n\n+-rightIdentity : \u2200 {n} \u2192 N n \u2192 n + zero \u2261 n\n+-rightIdentity Nn = N-ind A A0 is Nn\n  where\n  A : D \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = trans (+-Sx i zero) (succCong Ai)\n\nx+Sy\u2261S[x+y] : \u2200 {m} \u2192 N m \u2192 \u2200 n \u2192 m + succ\u2081 n \u2261 succ\u2081 (m + n)\nx+Sy\u2261S[x+y] Nm n = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + succ\u2081 n \u2261 succ\u2081 (i + n)\n\n  A0 : A zero\n  A0 = zero + succ\u2081 n   \u2261\u27e8 +-leftIdentity (succ\u2081 n) \u27e9\n       succ\u2081 n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ\u2081 (zero + n) \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + succ\u2081 n     \u2261\u27e8 +-Sx i (succ\u2081 n) \u27e9\n              succ\u2081 (i + succ\u2081 n)   \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (succ\u2081 (i + n)) \u2261\u27e8 succCong (sym (+-Sx i n)) \u27e9\n              succ\u2081 (succ\u2081 i + n)   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 1: Interactive proof using pattern matching\n\n+-comm\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2081 {n = n} nzero Nn =\n  zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n  n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n  n + zero \u220e\n\n+-comm\u2081 {n = n} (nsucc {m} Nm) Nn =\n  succ\u2081 m + n   \u2261\u27e8 +-Sx m n \u27e9\n  succ\u2081 (m + n) \u2261\u27e8 succCong (+-comm\u2081 Nm Nn) \u27e9\n  succ\u2081 (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn m) \u27e9\n  n + succ\u2081 m   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 2: Interactive proof using the induction principle\n\n+-comm\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2082 {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity Nn) \u27e9\n       n + zero \u220e\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = succ\u2081 i + n   \u2261\u27e8 +-Sx i n \u27e9\n              succ\u2081 (i + n) \u2261\u27e8 succCong Ai \u27e9\n              succ\u2081 (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] Nn i) \u27e9\n              n + succ\u2081 i   \u220e\n\n------------------------------------------------------------------------------\n-- Approach 3: Combined proof using pattern matching\n\n+-comm\u2083 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2083 {n = n} nzero Nn = prf\n  where postulate prf : zero + n \u2261 n + zero\n        {-# ATP prove prf +-rightIdentity #-}\n+-comm\u2083 {n = n} (nsucc {m} Nm) Nn = prf (+-comm\u2083 Nm Nn)\n  where postulate prf : m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m\n        {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 4: Combined proof using the induction principle\n\n+-comm\u2084 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n+-comm\u2084 {n = n} Nm Nn = N-ind A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = i + n \u2261 n + i\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-rightIdentity #-}\n\n  postulate is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Approach 5: Combined proof using an instance of the induction\n-- principle\n\n+-comm-ind-instance :\n  \u2200 n \u2192\n  zero + n \u2261 n + zero \u2192\n  (\u2200 {m} \u2192 m + n \u2261 n + m \u2192 succ\u2081 m + n \u2261 n + succ\u2081 m) \u2192\n  \u2200 {m} \u2192 N m \u2192 m + n \u2261 n + m\n+-comm-ind-instance n = N-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n-- TODO (25 October 2012): Why is it not necessary the hypothesis\n-- +-rightIdentity?\npostulate +-comm\u2085 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 m + n \u2261 n + m\n{-# ATP prove +-comm\u2085 +-comm-ind-instance x+Sy\u2261S[x+y] #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b90f5a73530c7644e6ace151d23563b257356d03","subject":"Experimental way for SEE","message":"Experimental way for SEE\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Strong-Fiat-Shamir.agda","new_file":"ZK\/Strong-Fiat-Shamir.agda","new_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Nat\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.List.Any using (module Membership-\u2261 ; Any ; here ; there)\nopen Membership-\u2261 using (_\u2208_)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; module TranscriptConstRun; module RepeatIndex ; done; ask ; run)\n  renaming (map to mapS)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp E-State : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (state : E-State)                    {- The current query -}\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy E-State (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-initial-run : \ud835\udfda\n        K-winning-initial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h e-s \u2192 runT (\u03bb t q \u2192 Kf e-s (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\n-- This module changes the game from the paper to be simpler to understand\n-- this is done purely for educational reasons, we make no claim about the security implications\nmodule Simulation-Sound-Extractability-[EXPERIMENTAL]\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf hiding (Prfs)\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    Res = \u03a3 \u039b Prf\n\n    ExtractorServerPart : \u2605\n    ExtractorServerPart =\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Res \u2192 (init-transcript : Transcript) \u2192 ExtractorServerPart \u00d7 (Res \u00d7 Transcript \u2192 W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Res)\n        (\u03c9 : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv \u03c9) rs\n\n        initial-prf : Res\n        initial-prf = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prf initial-transcript initial-prf\n\n        -- Second run\n\n        K = K' initial-prf initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        open TranscriptRun\n        w : W\n        w = Ks (runT Kf (Adv \u03c9) [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witness? w (fst initial-prf)\n\nmodule Lift-to-list\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  where\n\n  E-State : \u2605\n  E-State = \u03a3 \u039b Prf\n\n  open Game-Types Q Resp Prf\n  module Normal = Simulation-Sound-Extractability-[EXPERIMENTAL] PF Prf? Q Resp\n  module Lifted = Simulation-Sound-Extractability                PF Prf? Q Resp E-State\n  open RepeatIndex\n\n  lookup : {A : Set} \u2192 \u2115 \u2192 List A \u2192 Maybe A\n  lookup n [] = nothing\n  lookup zero (x \u2237 xs) = just x\n  lookup (suc n) (x \u2237 xs) = lookup n xs\n\n  trans-server : Normal.Extractor \u2192 Transcript \u2192 Lifted.ExtractorServerPart\n  trans-server K sim-tran \u03a3\u03c0 _ this-tran q = fst (K \u03a3\u03c0 sim-tran) this-tran q\n\n  trans-strat : Normal.Extractor \u2192 List Normal.Res \u2192 Transcript \u2192 Strategy E-State (const (Prfs \u00d7 Transcript)) (List W)\n  trans-strat K xs tran = map-list (\u03bb { i \u03a3\u03c0 (p , t) \u2192 snd (K \u03a3\u03c0 tran) (maybe\u2032 id \u03a3\u03c0 (lookup i p) , t)}) xs\n  {-\n  trans-strat K []         tran = done []\n  trans-strat K (\u03a3\u03c0 \u2237 res) tran = ask \u03a3\u03c0 (\u03bb { (p , t) \u2192 let w = snd (K \u03a3\u03c0 tran) ({!p!} , t) in mapS (_\u2237_ w) (trans-strat K res tran) } )\n  -}\n\n  transformation : Normal.Extractor \u2192 Lifted.Extractor\n  transformation K res tran = trans-server K tran , trans-strat K res tran\n\n  module Game\n      (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n      (sim : Simulator Q Resp PF)\n      (open Is-Zero-Knowledge L-to-Prf PF sim)\n      {RA : \u2605}\n\n      {- The malicious prover -}\n      (Adv : RA \u2192 Adversary Prfs)\n      (\u03c9 : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Normal.Extractor) where\n\n      module LGame = Lifted.Game L-to-Prf sim Adv \u03c9 rs ro (transformation K')\n      open TranscriptRun\n      open \u2261-Reasoning\n\n\n      Oracle : E-State \u2192 Prfs \u00d7 Transcript\n      Oracle e-s = runT (LGame.Kf e-s []) (Adv \u03c9) []\n\n      ws' : List W\n      ws' = run Oracle LGame.Ks\n\n      ws'-correct : LGame.ws \u2261 {!!}\n      ws'-correct =\n         LGame.ws\n        \u2261\u27e8 TranscriptConstRun.runT-run _ LGame.Ks \u27e9\n          run Oracle LGame.Ks\n        \u2261\u27e8 run-map-list Oracle LGame.initial-prfs _ \u27e9\n          mapIndex\n            (\u03bb i q \u2192\n               let (p , t) = Oracle q\n                in snd (K' q LGame.initial-transcript)\n                       (maybe\u2032 id q (lookup i p) , t))\n            LGame.initial-prfs\n        \u2261\u27e8 {!!} \u27e9\n          {!!}\n        \u220e\n\n      thm : LGame.K-winning-initial-run \u2261 0\u2082 \u2192 LGame.K-winning-second-run \u2261 1\u2082\n      thm eq = {!\u03bb e \u2192 LGame.Kf e []!}\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  v \u21c4 p = \u03a3-Protocol.\u03a3-game (v , p)\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct p = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Protocol p\n    in \u03a3-game r Y c \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 \u03a3-proto Y : \u2605 where\n    constructor mk\n    open \u03a3-Protocol \u03a3-proto using (\u03a3-verifier)\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n      -- The \u03a3-transcript verify\n      verify\u2080 : \u03a3-verifier Y (mk get-A get-c\u2080 get-f\u2080) \u2261 1\u2082\n      verify\u2081 : \u03a3-verifier Y (mk get-A get-c\u2081 get-f\u2081) \u2261 1\u2082\n\n  record Special-Soundness \u03a3-proto : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 \u03a3-proto Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 \u03a3-proto Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              -- For the transformation we technically only need Simulate, no proofs...\n              -- but we take the proofs as well\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 FS-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness \u03a3-proto)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      E-State : \u2605\n      E-State = \ud835\udfd9\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp E-State\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        K0 : Prfs \u2192 Transcript \u2192 ExtractorServerPart\n        K0 prfs init-transcript past-history on-going-transcript q = {!!}\n\n        K1 : Prfs \u2192 Transcript \u2192 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n        K1 [] init-transcript = done {!!}\n        K1 ((Y , \u03c0) \u2237 prfs) init-transcript = ask _ (\u03bb pt \u2192 {!!})\n\n        K : Extractor\n        K prfs init-transcript = K0 prfs init-transcript , K1 prfs init-transcript\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type\nopen import Function.NP\nopen import Data.Maybe\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum using (_\u228e_)\nopen import Data.List.NP renaming (map to map\u1d38)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Control.Strategy using (Strategy; module TranscriptRun; done; ask)\n\nmodule ZK.Strong-Fiat-Shamir\n  {W \u039b : \u2605}{L : W \u2192 \u039b \u2192 \u2605}\n  (any-W : W)\n  {RS : \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))\n  (\u039b? : Decidable (_\u2261_ {A = \u039b}))\n  (Eps  : \u2605)\n  (\u03b5[_] : Eps \u2192 \u2605)\n  (\u03b50 : Eps)\n  (\u03b5[0] : \u03b5[ \u03b50 ] \u2261 \ud835\udfd8)\n  where\n\nmodule _ {R : \u2605} where\n    _\u224b_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u224b g = (\u03a3 R (\u2713 \u2218 f)) \u2261 (\u03a3 R (\u2713 \u2218 g))\n\n    _\u2248_ : (f g : R \u2192 \ud835\udfda) \u2192 \u2605\u2081\n    f \u2248 g = \u2203\u2082 \u03bb \u03b5\u2080 \u03b5\u2081 \u2192\n            (\u03a3 R (\u2713 \u2218 f) \u228e \u03b5[ \u03b5\u2080 ]) \u2261 (\u03a3 R (\u2713 \u2218 g) \u228e \u03b5[ \u03b5\u2081 ])\n\n    \u224b\u2192\u2248 : \u2200 {f g} \u2192 f \u224b g \u2192 f \u2248 g\n    \u224b\u2192\u2248 f\u224bg = \u03b50 , \u03b50 , ap (flip _\u228e_ \u03b5[ \u03b50 ]) f\u224bg\n\n{-\nRandom-Oracle-List : \u2605\nRandom-Oracle-List = List (Q \u00d7 Resp)\n-}\n\nmodule Game-Types (Q Resp : \u2605)(Prf : \u039b \u2192 \u2605) where\n  Random-Oracle : \u2605\n  Random-Oracle = Q \u2192 Resp\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : \u039b) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605 \u2192 \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp\n\n  Transcript = List (\u03a3 Adversary-Query Challenger-Resp)\n\n  Prfs : \u2605\n  Prfs = List (\u03a3 \u039b Prf)\n\nrecord Proof-System (RP : \u2605)(Prf : \u039b \u2192 \u2605) : \u2605 where\n  field\n    Prove  : RP \u2192 (w : W)(Y : \u039b) \u2192 Prf Y\n    Verify : (Y : \u039b)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\n  Complete : \u2605\n  Complete = \u2200 rp {w Y} \u2192 L w Y \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082\n\n  -- Not in the paper but...\n  Sound : \u2605\n  Sound = \u2200 rp {w Y} \u2192 Verify Y (Prove rp w Y) \u2261 1\u2082 \u2192 L w Y\n\nrecord Simulator (Q : \u2605)(Resp : \u2605){Prf RP}(PF : Proof-System RP Prf) : \u2605 where\n  open Proof-System PF\n  open Game-Types Q Resp Prf\n\n  field\n    -- Computing the compound patch to the random oracle\n    H-patch  : RS \u2192 Transcript \u2192 Random-Oracle \u2192 Random-Oracle\n\n    -- Simulate does not patch itself but H-patch does\n    Simulate : RS \u2192 Transcript \u2192 (Y : \u039b) \u2192 Prf Y\n    verify-sim-spec : \u2200 rs t Y \u2192 Verify Y (Simulate rs t Y) \u2261 1\u2082\n\n  open Proof-System PF public\n  open Game-Types Q Resp Prf public\n\nmodule Is-Zero-Knowledge\n  {RP Prf Q Resp}\n  (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  (PF : Proof-System RP Prf)\n  (open Proof-System PF)\n  (sim : Simulator Q Resp PF)\n  (open Simulator sim)\n  (ro : Random-Oracle)\n  where\n\n    simulated-Challenger : RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    simulated-Challenger rs t (query-H q) = H-patch rs t ro q\n    simulated-Challenger rs t (query-create-proof w Y) with L? w Y\n    ... | yes p = just (Simulate rs t Y)\n    ... | no \u00acp = nothing\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    challenger : \ud835\udfda \u2192 RS \u2192 Transcript \u2192 \u2200 q \u2192 Challenger-Resp q\n    challenger 0\u2082 _  _ = real-Challenger\n    challenger 1\u2082 rs t = simulated-Challenger rs t\n\n    open TranscriptRun\n    -- \u03b2 = 0\u2082 is real\n    -- \u03b2 = 1\u2082 is simulated\n    Experiment : \u2200 {Output} \u2192 \ud835\udfda \u2192 Adversary Output \u2192 RS \u2192 Output \u00d7 Transcript\n    Experiment \u03b2 adv rs = runT (challenger \u03b2 rs) adv []\n\n    EXP\u2080 EXP\u2081 : Adversary \ud835\udfda \u2192 RS \u2192 \ud835\udfda\n    EXP\u2080 Adv = fst \u2218 Experiment 0\u2082 Adv\n    EXP\u2081 Adv = fst \u2218 Experiment 1\u2082 Adv\n\n    -- Too strong\n    IsZK = \u2200 Adv \u2192 EXP\u2080 Adv \u2248 EXP\u2081 Adv\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Simulation-Sound-Extractability\n           {RP}{Prf : \u039b \u2192 \u2605}\n           (PF : Proof-System RP Prf)\n           (Prf? : \u2200 {Y Y'} \u2192 Prf Y \u2192 Prf Y' \u2192 \ud835\udfda)\n           (Q Resp : \u2605)\n  -- (Prf? : \u2200 Y \u2192 Decidable (_\u2261_ {A = Prf Y}))\n           where\n    open Proof-System PF\n    open Game-Types Q Resp Prf\n\n    Prf-in-Q : \u2200 {Y} \u2192 Prf Y \u2192 \u03a3 Adversary-Query Challenger-Resp \u2192 \ud835\udfda\n    Prf-in-Q \u03c0 (query-create-proof _ _ , just \u03c0') = Prf? \u03c0 \u03c0'\n    Prf-in-Q \u03c0 _                                  = 0\u2082\n\n    HistoryForExtractor = List (Prfs \u00d7 Transcript)\n\n    ExtractorServerPart =\n         (past-history : HistoryForExtractor) {- previous invocations of Adv -}\n         (on-going-transcript : Transcript)   {- about the current invocation of Adv -}\n       \u2192 \u03a0 Adversary-Query Challenger-Resp\n\n    Extractor : \u2605\n    Extractor = Prfs \u2192 (init-transcript : Transcript)\n                     \u2192 ExtractorServerPart\n                     \u00d7 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n\n    valid-witness? : W \u2192 \u039b \u2192 \ud835\udfda\n    valid-witness? w Y = \u230a L? w Y \u230b\n\n    valid-witnesses? : List W \u2192 List \u039b \u2192 \ud835\udfda\n    valid-witnesses? [] [] = 1\u2082\n    valid-witnesses? (w \u2237 ws) (prf \u2237 prfs) = valid-witness? w prf \u2227 valid-witnesses? ws prfs\n    valid-witnesses? _ _ = 0\u2082\n\n    open TranscriptRun\n\n    module _ (t : Transcript) where\n        Prf-in-Transcript : \u2200 {Y} \u2192 Prf Y \u2192 \ud835\udfda\n        Prf-in-Transcript \u03c0 = any (Prf-in-Q \u03c0) t\n\n        K-winning-prf : \u03a3 \u039b Prf \u2192 \ud835\udfda\n        K-winning-prf (Y , \u03c0) = not (Verify Y \u03c0)\n                              \u2228 Prf-in-Transcript \u03c0\n\n        K-winning-prfs : Prfs \u2192 \ud835\udfda\n        K-winning-prfs []   = 1\u2082\n        K-winning-prfs prfs = any K-winning-prf prfs\n\n    module Game\n        (L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n        (sim : Simulator Q Resp PF)\n        (open Is-Zero-Knowledge L-to-Prf PF sim)\n        {RA : \u2605}\n\n        {- The malicious prover -}\n        (Adv : RA \u2192 Adversary Prfs)\n        (w : RA)(rs : RS)(ro : Q \u2192 Resp)(K' : Extractor) where\n\n        initial-result = Experiment ro 1\u2082 (Adv w) rs\n\n        initial-prfs : Prfs\n        initial-prfs = fst initial-result\n\n        initial-transcript : Transcript\n        initial-transcript = snd initial-result\n\n        K-winning-intial-run : \ud835\udfda\n        K-winning-intial-run = K-winning-prfs initial-transcript initial-prfs\n\n        -- Second run\n\n        K = K' initial-prfs initial-transcript\n\n        Kf = fst K\n        Ks = snd K\n\n        ws = fst (runT (\u03bb h _ \u2192 runT (\u03bb t q \u2192 Kf (map\u1d38 snd h) t q) (Adv w) []) Ks [])\n\n        K-winning-second-run : \ud835\udfda\n        K-winning-second-run = valid-witnesses? ws (map\u1d38 fst initial-prfs)\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : \u039b \u2192 \u2605)\n  {R\u03a3P : \u2605}\n  (any-R\u03a3P : R\u03a3P)\n  where\n\n  record \u03a3-Prover : \u2605 where\n    field\n      get-A : R\u03a3P \u2192 (Y : \u039b) \u2192 Commitment\n      get-f : R\u03a3P \u2192 (Y : \u039b) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  record \u03a3-Transcript (Y : \u039b) : \u2605 where\n    constructor mk\n    field\n      get-A : Commitment\n      get-c : Challenge\n      get-f : \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : \u039b)(t : \u03a3-Transcript Y) \u2192 \ud835\udfda\n\n  record \u03a3-Protocol : \u2605 where\n    constructor _,_\n    field\n      \u03a3-verifier : \u03a3-Verifier\n      \u03a3-prover   : \u03a3-Prover\n    open \u03a3-Prover \u03a3-prover public\n\n    \u03a3-game : (r : R\u03a3P)(Y : \u039b)(c : Challenge) \u2192 \ud835\udfda\n    \u03a3-game r Y c = \u03a3-verifier Y (mk A c f)\n      where\n        A = get-A r Y\n        f = get-f r Y c\n\n  _\u21c4_ : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 R\u03a3P \u2192 \u039b \u2192 Challenge \u2192 \ud835\udfda\n  v \u21c4 p = \u03a3-Protocol.\u03a3-game (v , p)\n\n  Correct : \u03a3-Protocol \u2192 \u2605\n  Correct p = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Protocol p\n    in \u03a3-game r Y c \u2261 1\u2082\n\n  record Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : \u2605 where\n    open \u03a3-Protocol \u03a3-proto\n    field\n      Simulate : (Y : \u039b)(c : Challenge)(f : \u03a3-Prf Y) \u2192 Commitment\n      Simulate-ok : \u2200 Y c f \u2192 \u03a3-verifier Y (mk (Simulate Y c f) c f) \u2261 1\u2082\n      -- If (c,f) uniformly distributed then (Simulate Y c f , c , f) is\n      -- distributed equaly to ((\u03a3-verifier \u21c4 \u03a3-prover) r Y c)\n\n  -- A pair of \"\u03a3-Transcript\"s such that the commitment is shared\n  -- and the challenges are different.\n  record \u03a3-Transcript\u00b2 \u03a3-proto Y : \u2605 where\n    constructor mk\n    open \u03a3-Protocol \u03a3-proto using (\u03a3-verifier)\n    field\n      -- The commitment is shared\n      get-A         : Commitment\n\n      -- The challenges...\n      get-c\u2080 get-c\u2081 : Challenge\n\n      -- ...are different\n      c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n      -- The proofs are arbitrary\n      get-f\u2080 get-f\u2081 : \u03a3-Prf Y\n\n      -- The \u03a3-transcript verify\n      verify\u2080 : \u03a3-verifier Y (mk get-A get-c\u2080 get-f\u2080) \u2261 1\u2082\n      verify\u2081 : \u03a3-verifier Y (mk get-A get-c\u2081 get-f\u2081) \u2261 1\u2082\n\n  record Special-Soundness \u03a3-proto : \u2605 where\n    field\n      Extract    : \u2200 {Y}(t : \u03a3-Transcript\u00b2 \u03a3-proto Y) \u2192 W\n      Extract-ok : \u2200 {Y}(t : \u03a3-Transcript\u00b2 \u03a3-proto Y) \u2192 L (Extract t) Y\n\n  module Fiat-Shamir-Transformation\n              (\u03a3-proto : \u03a3-Protocol)\n              -- For the transformation we technically only need Simulate, no proofs...\n              -- but we take the proofs as well\n              (shvzk : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              where\n\n      open \u03a3-Protocol \u03a3-proto\n      open Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      FS-Prf : \u039b \u2192 \u2605\n      FS-Prf Y = Challenge \u00d7 \u03a3-Prf Y\n\n      sFS : (H : (\u039b \u00d7 Commitment) \u2192 Challenge) \u2192 Proof-System R\u03a3P FS-Prf\n      sFS H = record { Prove = sFS-Prove ; Verify = sFS-Verify }\n        where\n          sFS-Prove : R\u03a3P \u2192 W \u2192 (Y : \u039b) \u2192 FS-Prf Y\n          sFS-Prove r w Y = let c = H (Y , get-A r Y) in c , get-f r Y c\n          sFS-Verify : \u2200 Y \u2192 FS-Prf Y \u2192 \ud835\udfda\n          sFS-Verify Y (c , \u03c0) = \u03a3-verifier Y (mk (Simulate Y c \u03c0) c \u03c0)\n\n      -- The weak fiat-shamir is like the strong one but the H function do not get to see\n      -- the statement, hence the 'snd'\n      wFS : (H : Commitment \u2192 Challenge) \u2192 Proof-System R\u03a3P (\u03bb Y \u2192 Challenge \u00d7 \u03a3-Prf Y)\n      wFS H = sFS (H \u2218 snd)\n\n  module Theorem1\n              (\u03a3-proto : \u03a3-Protocol)\n              -- (\u03a3-correct : Correct \u03a3-proto)\n              (shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-proto)\n              (ssound  : Special-Soundness \u03a3-proto)\n              (open \u03a3-Protocol \u03a3-proto)\n              (H : (\u039b \u00d7 Commitment) \u2192 Challenge)\n              where\n      module SHVZK = Special-Honest-Verifier-Zero-Knowledge shvzk\n\n      module FST = Fiat-Shamir-Transformation \u03a3-proto shvzk\n      open FST using (FS-Prf)\n\n      Q = \u039b \u00d7 Commitment\n      Resp = Challenge\n\n      sFS : Proof-System R\u03a3P FS-Prf\n      sFS = FST.sFS H\n      module SFS = Proof-System sFS\n\n      FS-Prf? : {Y Y' : \u039b} \u2192 FS-Prf Y \u2192 FS-Prf Y' \u2192 \ud835\udfda\n      FS-Prf? \u03c0 \u03c0' = {!!}\n\n      open Simulation-Sound-Extractability sFS FS-Prf? Q Resp\n\n      L-to-FS-Prf : \u2200 {w Y} \u2192 L w Y \u2192 FS-Prf Y\n      L-to-FS-Prf {w} w\u2208Y = SFS.Prove any-R\u03a3P w _\n\n      FS-Random-Oracle = (\u039b \u00d7 Commitment) \u2192 Challenge\n      FS-Patch = FS-Random-Oracle \u2192 FS-Random-Oracle\n\n      open Special-Soundness ssound\n      open Game-Types Q Resp FS-Prf\n\n      module _ (rnd-c : \u039b \u2192 Challenge)(rnd-\u03a3-Prf : \u2200 Y \u2192 \u03a3-Prf Y) where\n\n        module _ (rs : RS) where\n        {-\n          \u03a3-t\u00b2 : \u2200{Y} \u2192 SFS.Transcript \u2192 Maybe (\u03a3-Transcript\u00b2 Y)\n          \u03a3-t\u00b2 t = {!!}\n          -}\n\n\n    {-\nmodule H-def (ro : Random-Oracle)(t : Random-Oracle-List)(q : Q) where\n  H : Resp\n  H with find (\u03bb { (query-H q' , r) \u2192 \u230a Q? q' q \u230b\n                 ; (query-create-proof _ _ , _) \u2192 0\u2082 }) t\n  ... | just (query-H q\u2081 , r) = r\n  ... | _ = ro q\n  -}\n          H-patch-1 : (q : Adversary-Query) \u2192 Challenger-Resp q \u2192 FS-Random-Oracle \u2192 FS-Random-Oracle\n          H-patch-1 (query-H q) r = id\n          H-patch-1 (query-create-proof w Y) r ro (Y' , c')\n            = ro (Y' , case \u230a \u039b? Y Y' \u230b 0: c' 1: SHVZK.Simulate Y (rnd-c Y) (rnd-\u03a3-Prf Y))\n\n          H-patch : Transcript \u2192 FS-Patch\n          H-patch []            = id\n          H-patch ((q , r) \u2237 t) = H-patch-1 q r \u2218 H-patch t\n\n          module _ (t : Transcript)(Y : \u039b) where\n              Simulate : FS-Prf Y\n              Simulate = rnd-c Y , rnd-\u03a3-Prf Y\n              -- SFS.Prove any-R\u03a3P (maybe\u2032 (Extract {Y}) any-W (\u03a3-t\u00b2 t)) Y\n\n              c : Challenge\n              c = fst Simulate\n\n              \u03a3-prf : \u03a3-Prf Y\n              \u03a3-prf = snd Simulate\n\n              verify-sim-spec : SFS.Verify Y Simulate \u2261 1\u2082\n              verify-sim-spec = SHVZK.Simulate-ok Y c \u03a3-prf\n\n        S : Simulator (\u039b \u00d7 Commitment) Challenge sFS\n        S = record { H-patch = H-patch\n                   ; Simulate = Simulate\n                   ; verify-sim-spec = verify-sim-spec }\n\n        open Is-Zero-Knowledge L-to-FS-Prf sFS S {!!}\n\n        K0 : Prfs \u2192 Transcript \u2192 ExtractorServerPart\n        K0 prfs init-transcript past-history on-going-transcript q = {!!}\n\n        K1 : Prfs \u2192 Transcript \u2192 Strategy \ud835\udfd9 (const (Prfs \u00d7 Transcript)) (List W)\n        K1 [] init-transcript = done {!!}\n        K1 ((Y , \u03c0) \u2237 prfs) init-transcript = ask _ (\u03bb pt \u2192 {!!})\n\n        K : Extractor\n        K prfs init-transcript = K0 prfs init-transcript , K1 prfs init-transcript\n\n        module _ Adv where\n          is-zk' : EXP\u2080 Adv \u224b EXP\u2081 Adv\n          is-zk' = {!!}\n\n        is-zk : IsZK\n        is-zk Adv = \u224b\u2192\u2248 (is-zk' Adv)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"725e1c1e530aa4f2fe79f25b115ceabc6d9d12dd","subject":"Boring.","message":"Boring.\n\nIgnore-this: 38f9ae3275b2e0cdd3fefe714cfe33aa\n\ndarcs-hash:20100723040455-3bd4e-0c819da0901b462860c435e4e2c4eb7f4fd9c3ed.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LogicalConstants.agda","new_file":"Test\/Succeed\/LogicalConstants.agda","new_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  5 \u00ac_\ninfixr 4 _\u2227_\ninfixr 3 _\u2228_\ninfixr 2 _\u21d2_\ninfixr 1 _\u2194_\n\n------------------------------------------------------------------------------\n-- Propositional logic\n\n-- The logical connectives\n\n-- The logical connectives are hard-coded in our implementation,\n-- i.e. the following symbols must be used.\npostulate\n  \u22a5 \u22a4     : Set -- N.B. the name of the tautology symbol is \"\\top\", not T.\n  \u00ac_      : Set \u2192 Set -- N.B. the right hole.\n  _\u2227_ _\u2228_ : Set \u2192 Set \u2192 Set\n  _\u21d2_ _\u2194_ : Set \u2192 Set \u2192 Set\n\n-- We postulate some propositional variables (which are translated as\n-- 0-ary predicates).\npostulate\n  P Q R : Set\n\n-- The introduction and elimination rules for the propositional\n-- connectives are theorems.\npostulate\n  \u2192I  : (P \u2192 Q) \u2192 P \u21d2 Q\n  \u2192E  : (P \u21d2 Q) \u2192 P \u2192 Q\n  \u2227I  : P \u2192 Q \u2192 P \u2227 Q\n  \u2227E\u2081 : P \u2227 Q \u2192 P\n  \u2227E\u2082 : P \u2227 Q \u2192 Q\n  \u2228I\u2081 : P \u2192 P \u2228 Q\n  \u2228I\u2082 : Q \u2192 P \u2228 Q\n  \u2228E  : (P \u21d2 R) \u2192 (Q \u21d2 R) \u2192 P \u2228 Q \u2192 R\n  \u22a5E  : \u22a5 \u2192 P\n  \u00acE : (\u00ac P \u2192 \u22a5 ) \u2192 P\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u00acE #-}\n\n------------------------------------------------------------------------------\n-- Predicate logic\n\n-- The universe of discourse.\npostulate\n  D : Set\n\n-- The quantifiers are hard-coded in our implementation, i.e. the\n-- following symbols must be used.\npostulate\n  \u2203D : (P : D \u2192 Set) \u2192 Set\n  \u2200D : (P : D \u2192 Set) \u2192 Set\n\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 \u2200D P\u00b9\n  \u2200E : \u2200D P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- This elimination rule cannot prove in Coq because in Coq we can\n  -- have empty domains. We do not have this problem because the ATPs\n  -- assume a non-empty domain.\n  \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203D P\u00b9\n  -- TODO: \u2203E : (x : D) \u2192 \u2203D P\u00b9 \u2192 (P\u00b9 x \u2192 Q\u00b9 x) \u2192 Q\u00b9 x\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203I #-}\n-- {-# ATP prove \u2203E #-}\n","old_contents":"module Test.Succeed.LogicalConstants where\n\ninfix  5 \u00ac_\ninfixr 4 _\u2227_\ninfixr 3 _\u2228_\ninfixr 2 _\u21d2_\ninfixr 1 _\u2194_\n\n------------------------------------------------------------------------------\n-- Propositional logic\n\n-- The logical connectives\n\n-- The logical connectives are hard-coded in our implementation,\n-- i.e. the following symbols must be used.\npostulate\n  \u22a5 \u22a4     : Set -- N.B. the name of the tautology symbol is \"\\top\", not T.\n  \u00ac_      : Set \u2192 Set -- N.B. the right hole.\n  _\u2227_ _\u2228_ : Set \u2192 Set \u2192 Set\n  _\u21d2_ _\u2194_ : Set \u2192 Set \u2192 Set\n\n-- We postulate some propositional variables (which are translated as\n-- 0-ary predicates).\npostulate\n  P Q R : Set\n\n-- The introduction and elimination rules for the propositional\n-- connectives are theorems.\npostulate\n  \u2192I  : (P \u2192 Q) \u2192 P \u21d2 Q\n  \u2192E  : (P \u21d2 Q) \u2192 P \u2192 Q\n  \u2227I  : P \u2192 Q \u2192 P \u2227 Q\n  \u2227E\u2081 : P \u2227 Q \u2192 P\n  \u2227E\u2082 : P \u2227 Q \u2192 Q\n  \u2228I\u2081 : P \u2192 P \u2228 Q\n  \u2228I\u2082 : Q \u2192 P \u2228 Q\n  \u2228E  : (P \u21d2 R) \u2192 (Q \u21d2 R) \u2192 P \u2228 Q \u2192 R\n  \u22a5E  : \u22a5 \u2192 P\n  \u00acE : (\u00ac P \u2192 \u22a5 ) \u2192 P\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u00acE #-}\n\n------------------------------------------------------------------------------\n-- Predicate logic\n\n-- The universe of discourse.\npostulate\n  D : Set\n\n-- The quantifiers are hard-coded in our implementation, i.e. the\n-- following symbols must be used.\npostulate\n  \u2203D : (P : D \u2192 Set) \u2192 Set\n  \u2200D : (P : D \u2192 Set) \u2192 Set\n\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 \u2200D P\u00b9\n  \u2200E : \u2200D P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- This elimination rule cannot prove in Coq because in Coq we can\n  -- have empty domains. We do not have this problem because the ATPs\n  -- assume a non-empty domain.\n  \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203D P\u00b9\n--  TODO \u2203E : \u2203D (\u03bb x \u2192 P\u00b9 x) \u2192 ((y : D) \u2192 P\u00b9 y \u2192 Q\u00b9 y) \u2192 (z : D) \u2192 Q\u00b9 z\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203I #-}\n-- {-# ATP prove \u2203E #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a516c442f633d2f94ab05bed08db54457aadc08a","subject":"flat-funs: cleanup the function combinators","message":"flat-funs: cleanup the function combinators\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; 2^_; _\u2294_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_; _++_)\nimport Level as L\nimport Function as F\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nimport Data.Product as \u00d7\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bits using (Bits; _\u2192\u1d47_)\n\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    El      : T \u2192 Set\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    -- Functions\n    id : \u2200 {A} \u2192 A `\u2192 A\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n    const : \u2200 {_A B} \u2192 El B \u2192 _A `\u2192 B\n\n    -- Products\n    <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    -- Unit\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    -- Vectors\n    nil : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  f &&& g = < f , g >\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = < fst >>> f , snd >>> g >\n\n  _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  f *** g = < f \u00d7 g >\n\n  swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = f *** id\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = id *** f\n\n  assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc = < fst >>> fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd >>> snd >\n\n  `\u00d7-zip : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  `\u00d7-zip f g = < < fst >>> fst , snd >>> fst > >>> f ,\n                 < fst >>> snd , snd >>> snd > >>> g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons >>> fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons >>> snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons >>> < f \u00d7 g > >>> cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > >>> cons\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                >>> assoc\u2032\n                >>> first f\n                >>> foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons >>> foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                >>> assoc\n                >>> second (foldr f)\n                >>> f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons >>> swap >>> foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map {zero}  f = nil\n  map {suc n} f = < f \u2237 map f >\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                  >>> `\u00d7-zip f (zipWith f)\n                  >>> cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith id\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < id , nil > >>> cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd >>> singleton\n  snoc {suc n} = first uncons >>> assoc >>> second snoc >>> cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons >>> swap >>> first reverse >>> snoc\n\n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n               >>> assoc\n               >>> second append\n               >>> cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , id >\n  splitAt (suc m) = uncons\n                >>> second (splitAt m)\n                >>> assoc\u2032\n                >>> first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < id \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = id\n  drop (suc m) = tail >>> drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) >>> < folda n f \u00d7 folda n f > >>> f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < id \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail >>> last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons >>> second concat >>> append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k >>> second (group n k) >>> cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f >>> concat\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = uncons >>> < f \u00d7 \u229b fs > >>> cons\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < id , replicate > >>> cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz >>> f , tabulate (fs >>> f) > >>> cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst >>> elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail >>> lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate id\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U F.id -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U Bits _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U Fin _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk F.id (\u03bb f g x \u2192 g (f x)) F.const \u00d7.<_,_> proj\u2081 proj\u2082\n             _ (F.const []) (uncurry _\u2237_) uncons\n  where\n  uncons : \u2200 {n A} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id _>>>_ const <_,_> (\u03bb {A} \u2192 V.take A)\n                    (\u03bb {A} \u2192 V.drop A) (const []) (const []) id id\n  where\n  open FlatFuns bitsFun\u266dFuns\n  open FlatFunsOps fun\u266dOps using (id; _>>>_; const)\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > x = f x ++ g x\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) \u2192 S.El A\u2080 \u00d7 T.El A\u2081 })\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe (\u03bb A \u2192 S.El A \u00d7 T.El _)\n                            (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T\n  = mk (S.id , T.id) (\u00d7.zip S._>>>_ T._>>>_) < S.const \u00d7 T.const >\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.cons) (S.uncons , T.uncons)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4\n  = mk (S.id , T.id) (\u00d7.zip S._>>>_ T._>>>_) < S.const \u00d7 T.const >\n       (\u00d7.zip S.<_,_> T.<_,_>) (S.fst , T.fst) (S.snd , T.snd)\n       (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.id) (S.uncons , T.id)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ \u2192 A) (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ (F.const 0) _\u2294_ 0 0 0 0 0 0\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ const _+_ 0 0 0 0 0 0\n  where open FlatFuns (constFuns \u2115)\n        const : \u2200 {_A B} \u2192 El B \u2192 _A `\u2192 B\n        const {_} {_} x = x\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Nat.NP\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Data.Bits using (Bits)\nimport Level as L\nimport Function as F\nopen import composable\nopen import vcomp\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n\n    _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    nil : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n    constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\n  swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = f *** idO\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = idO *** f\n\n  assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc = < fst >>> fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd >>> snd >\n\n  `\u00d7-zip : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  `\u00d7-zip f g = < < fst >>> fst , snd >>> fst > >>> f ,\n                 < fst >>> snd , snd >>> snd > >>> g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons >>> fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons >>> snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons >>> < f \u00d7 g > >>> cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > >>> cons\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                >>> assoc\u2032\n                >>> first f\n                >>> foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons >>> foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                >>> assoc\n                >>> second (foldr f)\n                >>> f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons >>> swap >>> foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map {zero}  f = nil\n  map {suc n} f = < f \u2237 map f >\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                  >>> `\u00d7-zip f (zipWith f)\n                  >>> cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith idO\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < idO , nil > >>> cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd >>> singleton\n  snoc {suc n} = first uncons >>> assoc >>> second snoc >>> cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons >>> swap >>> first reverse >>> snoc\n  \n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n               >>> assoc\n               >>> second append\n               >>> cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , idO >\n  splitAt (suc m) = uncons\n                >>> second (splitAt m)\n                >>> assoc\u2032\n                >>> first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < idO \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = idO\n  drop (suc m) = tail >>> drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) >>> < folda n f \u00d7 folda n f > >>> f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < idO \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail >>> last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons >>> second concat >>> append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k >>> second (group n k) >>> cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f >>> concat\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = uncons >>> < f \u00d7 \u229b fs > >>> cons\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < idO , replicate > >>> cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz >>> f , tabulate (fs >>> f) > >>> cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst >>> elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail >>> lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate idO\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Product renaming (zip to \u00d7-zip; map to \u00d7-map)\nopen import Function\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id (\u03bb f g x \u2192 g (f x)) (\u03bb f g \u2192 \u00d7-map f g) _&&&_ proj\u2081 proj\u2082 _ (const []) (uncurry _\u2237_) uncons const\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n  uncons : \u2200 {n A} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id (\u03bb f g \u2192 g \u2218 f) (VComposable._***_ bitsFunVComp) _&&&_ (\u03bb {A} \u2192 take A)\n                    (\u03bb {A} \u2192 drop A) (const []) (const []) id id (\u03bb xs _ \u2192 concat (map [_] xs) {- ugly? -})\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe\n                           (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                        (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                        (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.cons)\n                        (S.uncons , T.uncons) (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4 = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                         (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                         (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.idO)\n                         (S.uncons , T.idO) (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ _\u2294_ _\u2294_ 0 0 0 0 0 0 (const 0)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ _+_ _+_ 0 0 0 0 0 0 (\u03bb {n} _ \u2192 n)\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"55c7aa343a6340cc1b91aa2086699e3df587a653","subject":"sum typechecks again","message":"sum typechecks again\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nsearchInd' : \u2200 {A} {s\u1d2c : Search A} \u2192 SearchInd s\u1d2c \u2192 SearchInd' s\u1d2c\nsearchInd' Ps\u1d2c P P\u2219 {f} Pf = Ps\u1d2c (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nsearch-mono' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchMono s\u1d2c\nsearch-mono' s\u1d2c Ps\u1d2c _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nsearch-\u03b5' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 Search\u03b5 (searchMonFromSearch s\u1d2c)\nsearch-\u03b5' s\u1d2c Ps\u1d2c m = searchInd' Ps\u1d2c Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n  where open Mon m\n\nsearch-ext' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchExt s\u1d2c\nsearch-ext' s\u1d2c Ps\u1d2c sg {f} {g} f\u2248\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n  where open Sgrp sg\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.cong\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n-- _+\u03bc_ {A} {B} \u03bcA \u03bcB = {!!}\n-- {-\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk srch ext mono swp eps hom\n  where\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e {C} P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) -- = {!!}\n-- {-\n   = mk srch ext mono swp eps hom\n   where\n     srch : Search _ -- (A \u00d7 B)\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n-}\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"import Level as L\nopen import Type\nopen import Function\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Maybe.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Function.Equality using (_\u27e8$\u27e9_)\nimport Function.Inverse as FI\nopen FI using (_\u2194_; module Inverse)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nmodule sum where\n\n_\u2264\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 \u2115) \u2192 \u2605\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSemigroup\u2080 = Semigroup L.zero L.zero\nMonoid\u2080 = Monoid L.zero L.zero\nCommutativeMonoid\u2080 = CommutativeMonoid L.zero L.zero\nPreorder\u2080 = Preorder L.zero L.zero L.zero\n\nmodule PO (po : Preorder\u2080) where\n  open Preorder po public renaming (_\u223c_ to _\u2286_)\n  _\u2286\u00b0_ : \u2200 {A : \u2605}(f g : A \u2192 Carrier) \u2192 \u2605\n  f \u2286\u00b0 g = \u2200 x \u2192 f x \u2286 g x\n\nmodule SgrpExtra (sg : Semigroup\u2080) where\n  open Semigroup sg\n  open Setoid-Reasoning (Semigroup.setoid sg) public\n  C : \u2605\n  C = Carrier\n  _\u2248\u00b0_ : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 \u2605\n  f \u2248\u00b0 g = \u2200 x \u2192 f x \u2248 g x\n  _\u2219\u00b0_   : \u2200 {A : \u2605} (f g : A \u2192 C) \u2192 A \u2192 C\n  (f \u2219\u00b0 g) x = f x \u2219 g x\n  infixl 7 _-\u2219-_\n  _-\u2219-_ : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n  _-\u2219-_ = \u2219-cong\n\nmodule Sgrp (sg : Semigroup\u2080) where\n  open Semigroup sg public\n  open SgrpExtra sg public\n\nmodule Mon (m : Monoid\u2080) where\n  open Monoid m public\n  sg = semigroup\n  open SgrpExtra semigroup public\n  Is-\u03b5 : Carrier \u2192 \u2605\n  Is-\u03b5 x = x \u2248 \u03b5\n\nmodule CMon (cm : CommutativeMonoid\u2080) where\n  open CommutativeMonoid cm public\n  sg = semigroup\n  m = monoid\n  open SgrpExtra sg public\n\n  \u2219-interchange : Interchange _\u2248_ _\u2219_ _\u2219_\n  \u2219-interchange = InterchangeFromAssocCommCong.\u2219-interchange\n                    _\u2248_ isEquivalence\n                    _\u2219_ assoc comm (\u03bb _ \u2192 flip \u2219-cong refl)\n\nSearch : \u2605 \u2192 \u2605\u2081\nSearch A = \u2200 {B} \u2192 (_\u2219_ : B \u2192 B \u2192 B) \u2192 (A \u2192 B) \u2192 B\n-- Search A = \u2200 {I : \u2605} {F : I \u2192 \u2605} \u2192 (_\u2219_ : \u2200 {i} \u2192 F i \u2192 F i \u2192 F i) \u2192 \u2200 {i} \u2192 (A \u2192 F i) \u2192 F i\n\nSearchMon : \u2605 \u2192 \u2605\u2081\nSearchMon A = (m : Monoid\u2080) \u2192 let open Mon m in\n                              (A \u2192 C) \u2192 C\n\nsearchMonFromSearch : \u2200 {A} \u2192 Search A \u2192 SearchMon A\nsearchMonFromSearch s m = s _\u2219_ where open Mon m\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSearchInd : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd {A} srch = \u2200 {C}\n                       (P   : ((A \u2192 C) \u2192 C) \u2192 \u2605)\n                       {_\u2219_ : Op\u2082 C}\n                       (P\u2219  : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f \u2219 s\u2081 f))\n                       (Pf  : \u2200 (x : A) \u2192 P (\u03bb f \u2192 f x))\n                     \u2192 P (srch _\u2219_)\n\nSearchInd' : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchInd' {A} srch = \u2200 {C}\n                        (P   : C \u2192 \u2605)\n                        {_\u2219_ : C \u2192 C \u2192 C}\n                        (P\u2219  : \u2200 {x y} \u2192 P x \u2192 P y \u2192 P (x \u2219 y))\n                        {f   : A \u2192 C}\n                        (Pf  : \u2200 x \u2192 P (f x))\n                      \u2192 P (srch _\u2219_ f)\n\nsearchInd' : \u2200 {A} {s\u1d2c : Search A} \u2192 SearchInd s\u1d2c \u2192 SearchInd' s\u1d2c\nsearchInd' Ps\u1d2c P P\u2219 {f} Pf = Ps\u1d2c (\u03bb s \u2192 P (s f)) P\u2219 Pf\n\nSearchMono : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchMono s\u1d2c = \u2200 {C : \u2605} (_\u2286_ : C \u2192 C \u2192 \u2605) \u2192 -- let open PO {C} _\u2286_ in\n                \u2200 {_\u2219_} (\u2219-mono : _\u2219_ Preserves\u2082 _\u2286_ \u27f6 _\u2286_ \u27f6 _\u2286_)\n                  {f g} \u2192\n                  (\u2200 x \u2192 f x \u2286 g x) \u2192 s\u1d2c _\u2219_ f \u2286 s\u1d2c _\u2219_ g\n\nsearch-mono' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchMono s\u1d2c\nsearch-mono' s\u1d2c Ps\u1d2c _\u2286_ {_\u2219_} _\u2219-mono_ {f} {g} f\u2286\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2286 s g) _\u2219-mono_ f\u2286\u00b0g\n\nSearchExt : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchExt s\u1d2c = \u2200 sg {f g} \u2192 let open Sgrp sg in\n                            f \u2248\u00b0 g \u2192 s\u1d2c _\u2219_ f \u2248 s\u1d2c _\u2219_ g\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSearch\u03b5 : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearch\u03b5 s\u1d2c = \u2200 m \u2192 let open Mon m in\n                   s\u1d2c m (const \u03b5) \u2248 \u03b5\n\nsearch-\u03b5' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 Search\u03b5 (searchMonFromSearch s\u1d2c)\nsearch-\u03b5' s\u1d2c Ps\u1d2c m = searchInd' Ps\u1d2c Is-\u03b5 (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (proj\u2081 identity \u03b5)) (\u03bb _ \u2192 refl)\n  where open Mon m\n\nsearch-ext' : \u2200 {A} (s\u1d2c : Search A) \u2192 SearchInd s\u1d2c \u2192 SearchExt s\u1d2c\nsearch-ext' s\u1d2c Ps\u1d2c sg {f} {g} f\u2248\u00b0g = Ps\u1d2c (\u03bb s \u2192 s f \u2248 s g) \u2219-cong f\u2248\u00b0g\n  where open Sgrp sg\n\nSumZero : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumZero sum\u1d2c = sum\u1d2c (\u03bb _ \u2192 0) \u2261 0\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSearchHom : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchHom s\u1d2c = \u2200 sg f g \u2192 let open Sgrp sg in\n                          s\u1d2c _\u2219_ (f \u2219\u00b0 g) \u2248 s\u1d2c _\u2219_ f \u2219 s\u1d2c _\u2219_ g\n\nSearchMonHom : \u2200 {A} \u2192 SearchMon A \u2192 \u2605\u2081\nSearchMonHom s\u1d2c = \u2200 (cm : CommutativeMonoid\u2080) f g \u2192\n                         let open CMon cm in\n                         s\u1d2c m (f \u2219\u00b0 g) \u2248 s\u1d2c m f \u2219 s\u1d2c m g\n\n{-search-hom\u2032 :\n  \u2200 {a b}\n    {A : Set a} {B : Set b} {R}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : R \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom\u2032 = ?\n{-\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n-}\n-}\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMono : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMono sum\u1d2c = \u2200 {f g} \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nSearchSwap : \u2200 {A} \u2192 Search A \u2192 \u2605\u2081\nSearchSwap {A} s\u1d2c = \u2200 {B} sg f \u2192 let open Sgrp sg in \u2200 {s\u1d2e : (B \u2192 C) \u2192 C}\n\n  \u2192 (hom : \u2200 f g \u2192 s\u1d2e (f \u2219\u00b0 g) \u2248 s\u1d2e f \u2219 s\u1d2e g)\n  \u2192 s\u1d2c _\u2219_ (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c _\u2219_ \u2218 flip f)\n\nSumLin\u21d2SumZero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nSumLin\u21d2SumZero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nmkSumExt : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nmkSumExt sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nmkSumMon : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nmkSumMon {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nrecord SumProp A : \u2605\u2081 where\n  constructor mk\n  field\n    search      : Search A\n    search-ext  : SearchExt search\n    search-mono : SearchMono search\n    search-swap : SearchSwap search\n  searchMon : SearchMon A\n  searchMon m = let open Mon m in search _\u2219_\n  field\n    search-\u03b5   : Search\u03b5 searchMon\n    search-hom : SearchMonHom searchMon\n  sum : Sum A\n  sum = search _+_\n  sum-ext : SumExt sum\n  sum-ext = search-ext \u2115+.semigroup\n  sum-zero : SumZero sum\n  sum-zero = search-\u03b5 \u2115+.monoid\n  sum-hom : SumHom sum\n  sum-hom = search-hom \u2115\u00b0.+-commutativeMonoid\n  sum-mono : SumMono sum\n  sum-mono = search-mono _\u2264_ _+-mono_\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = {!!}\n  {-\n    rewrite sum-hom f (\u03bb x \u2192 k * f x)\n          | sum-lin f k = {!\u2261.refl!}\n          -}\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : SumProp A) (\u03bcB : SumProp B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk search\u22a4 ext mono search\u22a4-swap eps hom\n  where\n    search\u22a4 : Search \u22a4\n    search\u22a4 _ f = f _\n\n    searchMon\u22a4 : SearchMon \u22a4\n    searchMon\u22a4 _ f = f _\n\n    ind : SearchInd search\u22a4\n    ind _ _ Pf = Pf _\n\n    ext : SearchExt search\u22a4\n    ext _ f\u2257g = f\u2257g _\n\n    mono : SearchMono search\u22a4\n    mono _ _ f\u2264\u00b0g = f\u2264\u00b0g _\n\n    eps : Search\u03b5 searchMon\u22a4\n    eps m = Monoid.refl m\n\n    search\u22a4-swap : SearchSwap search\u22a4\n    search\u22a4-swap sg f hom = refl\n      where open Sgrp sg\n\n    hom : SearchMonHom searchMon\u22a4\n    hom m f g = CommutativeMonoid.refl m\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk searchBit ext mono swp eps hom\n  where\n    searchBit : Search Bit\n    searchBit _\u2219_ f = f 0b \u2219 f 1b\n\n    searchMonBit : SearchMon Bit\n    searchMonBit m f = f 0b \u2219 f 1b\n      where open Mon m\n\n    ext : SearchExt searchBit\n    ext sg f\u2257g = f\u2257g 0b -\u2219- f\u2257g 1b\n      where open Sgrp sg\n\n    mono : SearchMono searchBit\n    mono po _\u2219-mono_ f\u2286\u00b0g = f\u2286\u00b0g 0b \u2219-mono f\u2286\u00b0g 1b\n\n    eps : Search\u03b5 searchMonBit\n    eps m = proj\u2081 identity \u03b5\n      where open Monoid m\n\n    hom : SearchMonHom searchMonBit\n    hom cm f g = \u2219-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n      where open CMon cm\n\n    sumBit : Sum Bit\n    sumBit f = f 0b + f 1b\n\n    swp : SearchSwap searchBit\n    swp sg f hom\u1d2e = sym (hom\u1d2e (f 0b) (f 1b))\n      where open Sgrp sg\n\ninfixr 4 _+Search_\n\n_+Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u228e B)\n(search\u1d2c +Search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_+SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd' s\u1d2c \u2192 SearchInd' s\u1d2e \u2192 SearchInd' (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd' Ps\u1d2e) P P\u2219 Pf = P\u2219 (Ps\u1d2c P P\u2219 (Pf \u2218 inj\u2081)) (Ps\u1d2e P P\u2219 (Pf \u2218 inj\u2082))\n\n_+SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n                       SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c +Search s\u1d2e)\n(Ps\u1d2c +SearchInd Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = ?\n{-\n_+\u03bc_ {A} {B} \u03bcA \u03bcB = mk srch ext mono swp eps hom\n  where\n    s\u1d2c : Search A\n    s\u1d2c = search \u03bcA\n    s\u1d2e : Search B\n    s\u1d2e = search \u03bcB\n    srch : Search (A \u228e B)\n    srch = s\u1d2c +Search s\u1d2e\n    sMon\u1d2c = searchMon \u03bcA\n    sMon\u1d2e = searchMon \u03bcB\n    srchMon = searchMonFromSearch srch\n    ext : SearchExt srch\n    ext sg f\u2257g = search-ext \u03bcA sg (f\u2257g \u2218 inj\u2081) -\u2219- search-ext \u03bcB sg (f\u2257g \u2218 inj\u2082)\n      where open Sgrp sg\n\n    eps : Search\u03b5 srchMon\n    eps m = srchMon m (const \u03b5)\n          \u2248\u27e8 search-\u03b5 \u03bcA m -\u2219- search-\u03b5 \u03bcB m \u27e9\n            \u03b5 \u2219 \u03b5\n          \u2248\u27e8 proj\u2081 identity \u03b5 \u27e9\n            \u03b5\n          \u220e\n      where open Mon m\n\n    mono : SearchMono srch\n    mono po _\u2219-mono_ f\u2286\u00b0g = mono\u1d2c (f\u2286\u00b0g \u2218 inj\u2081) \u2219-mono mono\u1d2e (f\u2286\u00b0g \u2218 inj\u2082)\n      where\n      mono\u1d2c = search-mono \u03bcA po _\u2219-mono_\n      mono\u1d2e = search-mono \u03bcB po _\u2219-mono_\n\n    hom : SearchMonHom srchMon\n    hom cm f g\n      = srchMon m (f \u2219\u00b0 g)\n      \u2248\u27e8 search-hom \u03bcA cm (f \u2218 inj\u2081) (g \u2218 inj\u2081) -\u2219-\n         search-hom \u03bcB cm (f \u2218 inj\u2082) (g \u2218 inj\u2082) \u27e9\n        (sMon\u1d2c m (f \u2218 inj\u2081) \u2219 sMon\u1d2c m (g \u2218 inj\u2081)) \u2219\n        (sMon\u1d2e m (f \u2218 inj\u2082) \u2219 sMon\u1d2e m (g \u2218 inj\u2082))\n      \u2248\u27e8 \u2219-interchange (sMon\u1d2c m (f \u2218 inj\u2081)) (sMon\u1d2c m (g \u2218 inj\u2081))\n                       (sMon\u1d2e m (f \u2218 inj\u2082)) (sMon\u1d2e m (g \u2218 inj\u2082)) \u27e9\n        srchMon m f \u2219 srchMon m g\n      \u220e\n      where open CMon cm\n\n    swp : SearchSwap srch\n    swp sg f hom = trans (\u2219-cong (search-swap \u03bcA sg (f \u2218 inj\u2081) hom) (search-swap \u03bcB sg (f \u2218 inj\u2082) hom)) (sym (hom g h))\n      where open Sgrp sg\n            g = \u03bb x \u2192 search \u03bcA _\u2219_ (\u03bb y \u2192 f (inj\u2081 y) x)\n            h = \u03bb x \u2192 search \u03bcB _\u2219_ (\u03bb y \u2192 f (inj\u2082 y) x)\n-}\ninfixr 4 _\u00d7Search_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n(search\u1d2c \u00d7Search search\u1d2e) m f = search\u1d2c m (\u03bb x\u2080 \u2192\n                                  search\u1d2e m (\u03bb x\u2081 \u2192\n                                    f (x\u2080 , x\u2081)))\n\n_\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd' s\u1d2c)\n                       (Ps\u1d2e : SearchInd' s\u1d2e) \u2192 SearchInd' (s\u1d2c \u00d7Search s\u1d2e)\n(Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf = Ps\u1d2c P P\u2219 (\u03bb x \u2192 Ps\u1d2e P P\u2219 (curry Pf x))\n\n_\u00d7SearchInd_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                       (Ps\u1d2c : SearchInd s\u1d2c)\n                       (Ps\u1d2e : SearchInd s\u1d2e) \u2192 SearchInd (s\u1d2c \u00d7Search s\u1d2e)\n_\u00d7SearchInd_ {s\u1d2c = s\u1d2c} {s\u1d2e} Ps\u1d2c Ps\u1d2e {C} P {_\u2219_} P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb f \u2192 s (\u03bb x \u2192 s\u1d2e _\u2219_ (curry f x)))) P\u2219 (\u03bb x \u2192 Ps\u1d2e (\u03bb s \u2192 P (\u03bb f \u2192 s (curry f x))) P\u2219 (curry Pf x))\n\n_\u00d7SearchExt_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B} \u2192\n              SearchExt s\u1d2c \u2192 SearchExt s\u1d2e \u2192 SearchExt (s\u1d2c \u00d7Search s\u1d2e)\n(s\u1d2c-ext \u00d7SearchExt s\u1d2e-ext) sg f\u2257g = s\u1d2c-ext sg (s\u1d2e-ext sg \u2218 curry f\u2257g)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB) = ?\n{-\n   = mk srch ext mono swp eps hom\n   where\n     srch : Search _ -- (A \u00d7 B)\n     srch = search \u03bcA \u00d7Search search \u03bcB\n     srchMon = searchMonFromSearch srch\n\n     ext : SearchExt srch\n     ext sg f\u2257g = search-ext \u03bcA sg (search-ext \u03bcB sg \u2218 curry f\u2257g)\n\n     eps : Search\u03b5 srchMon\n     eps m = srchMon m (const \u03b5)\n           \u2248\u27e8 search-ext \u03bcA sg (const (search-\u03b5 \u03bcB m)) \u27e9\n             searchMon \u03bcA m (const \u03b5)\n           \u2248\u27e8 search-\u03b5 \u03bcA m \u27e9\n             \u03b5\n           \u220e\n       where open Mon m\n\n     mono : SearchMono srch\n     mono po \u2219mono f\u2264\u00b0g = mono\u1d2c (mono\u1d2e \u2218 curry f\u2264\u00b0g)\n      where\n      mono\u1d2c = search-mono \u03bcA po \u2219mono\n      mono\u1d2e = search-mono \u03bcB po \u2219mono\n\n     hom : SearchMonHom srchMon\n     hom cm f g\n       = srch _\u2219_ (f \u2219\u00b0 g)\n       \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-hom \u03bcB cm (curry f x) (curry g x)) \u27e9\n         search \u03bcA _\u2219_ (\u03bb x \u2192 search \u03bcB _\u2219_ (curry f x) \u2219 search \u03bcB _\u2219_ (curry g x))\n       \u2248\u27e8 search-hom \u03bcA cm (search \u03bcB _\u2219_ \u2218 curry f) (search \u03bcB _\u2219_ \u2218 curry g) \u27e9\n         srch _\u2219_ f \u2219 srch _\u2219_ g\n       \u220e where open CMon cm\n\n     swp : SearchSwap srch\n     swp sg f {g} hom = s\u1d2c\u1d2e (g \u2218 f)\n                      \u2248\u27e8 search-ext \u03bcA sg (\u03bb x \u2192 search-swap \u03bcB sg (curry f x) hom) \u27e9\n                        s\u1d2c (\u03bb z \u2192 g (\u03bb x \u2192 s\u1d2e (curry (flip f x) z)))\n                      \u2248\u27e8 search-swap \u03bcA sg (\u03bb z x \u2192 s\u1d2e (curry (flip f x) z)) hom \u27e9\n                        g (s\u1d2c\u1d2e \u2218 flip f)\n                      \u220e\n       where open Sgrp sg\n             s\u1d2c = search \u03bcA _\u2219_\n             s\u1d2e = search \u03bcB _\u2219_\n             s\u1d2c\u1d2e = srch _\u2219_\n-}\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k\nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | \u2261.sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = \u2261.refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = \u2261.refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\n\u03bc-view : \u2200 {A B} \u2192 (A \u2192 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-view {A}{B} A\u2192B \u03bcA = mk search\u1d2e ext mono swp eps hom\n  where\n    search\u1d2e : Search B\n    search\u1d2e m f = search \u03bcA m (f \u2218 A\u2192B)\n    searchMon\u1d2e = searchMonFromSearch search\u1d2e\n\n    ext : SearchExt search\u1d2e\n    ext m f\u2257g = search-ext \u03bcA m (f\u2257g \u2218 A\u2192B)\n\n    eps : Search\u03b5 searchMon\u1d2e\n    eps = search-\u03b5 \u03bcA\n\n    mono : SearchMono search\u1d2e\n    mono po \u2219mono f\u2264\u00b0g = search-mono \u03bcA po \u2219mono (f\u2264\u00b0g \u2218 A\u2192B)\n\n    hom : SearchMonHom searchMon\u1d2e\n    hom m f g = search-hom \u03bcA m (f \u2218 A\u2192B) (g \u2218 A\u2192B)\n\n    swp : SearchSwap search\u1d2e\n    swp sg f hom = search-swap \u03bcA sg (f \u2218 A\u2192B) hom\n\n\u03bc-iso : \u2200 {A B} \u2192 (A \u2194 B) \u2192 SumProp A \u2192 SumProp B\n\u03bc-iso A\u2194B = \u03bc-view (_\u27e8$\u27e9_ (Inverse.to A\u2194B))\n\n\u03bc-view-preserve : \u2200 {A B} (A\u2192B : A \u2192 B)(B\u2192A : B \u2192 A)(A\u2248B : id \u2257 B\u2192A \u2218 A\u2192B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-view A\u2192B \u03bcA) (f \u2218 B\u2192A)\n\u03bc-view-preserve A\u2192B B\u2192A A\u2248B f \u03bcA = sum-ext \u03bcA (\u2261.cong f \u2218 A\u2248B)\n\n\u03bc-iso-preserve : \u2200 {A B} (A\u2194B : A \u2194 B) f (\u03bcA : SumProp A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2194B \u03bcA) (f \u2218 _\u27e8$\u27e9_ (Inverse.from A\u2194B))\n\u03bc-iso-preserve A\u2194B f \u03bcA = \u03bc-view-preserve (_\u27e8$\u27e9_ (Inverse.to A\u2194B)) (_\u27e8$\u27e9_ (Inverse.from A\u2194B))\n                            (\u2261.sym \u2218 Inverse.left-inverse-of A\u2194B) f \u03bcA\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum; foldr to vfoldr)\n\nvmsum : \u2200 m {n} \u2192 let open Mon m in\n                  Vec C n \u2192 C\nvmsum m = vfoldr _ _\u2219_ \u03b5\n  where open Monoid m\n\nsearchMonFin : \u2200 n \u2192 SearchMon (Fin n)\nsearchMonFin n m f = vmsum m (vmap f (allFin n)) -- or vsum (tabulate f)\n\nsearchFinSuc : \u2200 n \u2192 Search (Fin (suc n))\nsearchFinSuc n _\u2219_ f = vfoldr _ _\u2219_ (f zero) (vmap (f \u2218 suc) (allFin n))\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n)) -- or vsum (tabulate f)\n\n\u03bcMaybe : \u2200 {A} \u2192 SumProp A \u2192 SumProp (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (FI.sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 +\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 SumProp A \u2192 SumProp (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin (suc n))\n\u03bcFin n = \u03bc-iso (Maybe^\u22a4\u2194Fin1+ n) (\u03bcMaybe^ n \u03bc\u22a4)\n{-\n\u03bcFin n = mk (searchFin n) ext eps hom sumFin-mon sumFin-swap\n  module SumFin where\n    ext : SearchMonExt (searchFin n)\n    ext m f\u2257g = {!map-ext f\u2257g (allFin n)!}\n      where open Mon m\n\n    eps : Search\u03b5 (searchFin n)\n    eps = {!!}\n\n    hom : SearchMonHom (searchFin n)\n    hom m f g = {!sum-linear f g (allFin n)!}\n\n    sumFin-mon : SumMono (sumFin n)\n    sumFin-mon f\u2264\u00b0g = sum-mono f\u2264\u00b0g (allFin n)\n\n    sumFin-swap : SumSwap (sumFin n)\n    sumFin-swap f {sum\u02e3} sum\u02e3-linear = inner (allFin n) where\n      inner : \u2200 {m}(xs : Vec (Fin n) m) \u2192 vsum (vmap (sum\u02e3 \u2218 f) xs) \u2261 sum\u02e3 (\u03bb x \u2192 vsum (vmap (flip f x) xs))\n      inner [] = \u2261.sym (SumLinear.sum-lin sum\u02e3-linear (const 1337) 0)\n      inner (x \u2237 xs) rewrite inner xs = \u2261.sym\n                                          (SumLinear.sum-hom sum\u02e3-linear (f x)\n                                                (\u03bb y \u2192 vsum (vmap (flip f y) xs)))\n-}\n\n\u03bcVec : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec \u03bcA zero    = \u03bc-view (const []) \u03bc\u22a4\n\u03bcVec \u03bcA (suc n) = \u03bc-view (uncurry _\u2237_) (\u03bcA \u00d7\u03bc \u03bcVec \u03bcA n)\n\n{-\nsearchVec : \u2200 {A} n \u2192 Search A \u2192 Search (Vec A n)\nsearchVec zero    search\u1d2c m f = f []\nsearchVec (suc n) search\u1d2c m f = search\u1d2c m (\u03bb x \u2192 searchVec n search\u1d2c m (f \u2218 _\u2237_ x))\n\nsumVec : \u2200 {A} n \u2192 Sum A \u2192 Sum (Vec A n)\nsumVec n sum\u1d2c = searchVec n (\u03bb m \u2192 {!sum\u1d2c!}) \u2115+.monoid\n\nsumVec zero    sum\u1d2c f = f []\nsumVec (suc n) sum\u1d2c f = (sum\u1d2c \u00d7Sum sumVec n sum\u1d2c) (\u03bb { (x , xs) \u2192 f (x \u2237 xs) })\n\n\u03bcVec' : \u2200 {A} (\u03bcA : SumProp A) n  \u2192 SumProp (Vec A n)\n\u03bcVec' {A} \u03bcA n = mk (searchVec n (search \u03bcA))  (mk (sumVec-lin n) (sumVec-hom n)) (sumVec-ext n) (sumVec-mon n) (sumVec-swap n)\n  where\n    sV = flip sumVec (sum \u03bcA)\n\n    sumVec-ext : \u2200 n \u2192 SumExt (sV n)\n    sumVec-ext zero    f\u2257g = f\u2257g []\n    sumVec-ext (suc n) f\u2257g = sum-ext \u03bcA (\u03bb x \u2192 sumVec-ext n (f\u2257g \u2218 _\u2237_ x))\n\n    sumVec-lin : \u2200 n \u2192 SumLin (sV n)\n    sumVec-lin zero    f k = \u2261.refl\n    sumVec-lin (suc n) f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-lin n (f \u2218 _\u2237_ x) k)\n      = sum-lin \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) k\n\n    sumVec-hom : \u2200 n \u2192 SumHom (sV n)\n    sumVec-hom zero    f g = \u2261.refl\n    sumVec-hom (suc n) f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-hom n (f \u2218 _\u2237_ x) (g \u2218 _\u2237_ x)) \n      = sum-hom \u03bcA (\u03bb x \u2192 sV n (f \u2218 _\u2237_ x)) (\u03bb x \u2192 sV n (g \u2218 _\u2237_ x))\n\n    sumVec-mon : \u2200 n \u2192 SumMono (sV n)\n    sumVec-mon zero    f\u2264\u00b0g = f\u2264\u00b0g []\n    sumVec-mon (suc n) f\u2264\u00b0g = sum-mono \u03bcA (\u03bb x \u2192 sumVec-mon n (f\u2264\u00b0g \u2218 _\u2237_ x))\n\n    sumVec-swap : \u2200 n \u2192 SumSwap (sV n)\n    sumVec-swap zero    f        \u03bc\u02e3-linear = \u2261.refl\n    sumVec-swap (suc n) f {sum\u02e3} \u03bc\u02e3-linear rewrite sum-ext \u03bcA (\u03bb x \u2192 sumVec-swap n (f \u2218 _\u2237_ x) \u03bc\u02e3-linear)\n      = sum-swap \u03bcA (\u03bb z x \u2192 sumVec n (sum \u03bcA) (\u03bb y \u2192 f (z \u2237 y) x)) \u03bc\u02e3-linear\n-}\n\nsearchVec-++ : \u2200 {A} n {m} (\u03bcA : SumProp A) sg\n               \u2192 let open Sgrp sg in\n                 (f : Vec A (n + m) \u2192 C)\n               \u2192 search (\u03bcVec \u03bcA (n + m)) _\u2219_ f\n               \u2248 search (\u03bcVec \u03bcA n) _\u2219_ (\u03bb xs \u2192 search (\u03bcVec \u03bcA m) _\u2219_\n                  (\u03bb ys \u2192 f (xs ++ ys)))\nsearchVec-++ zero    \u03bcA sg f = Sgrp.refl sg\nsearchVec-++ (suc n) \u03bcA sg f = search-ext \u03bcA sg (\u03bb x \u2192\n                                  searchVec-++ n \u03bcA sg (f \u2218 _\u2237_ x))\n\nsumVec-swap : \u2200 {A} n {m} (\u03bcA : SumProp A)(f : Vec A (n + m) \u2192 \u2115)\n            \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 f (xs ++ ys)))\n            \u2261 sum (\u03bcVec \u03bcA m) (\u03bb ys \u2192 sum (\u03bcVec \u03bcA n) (\u03bb xs \u2192 f (xs ++ ys)))\nsumVec-swap n {m} \u03bcA f = sum-swap (\u03bcVec \u03bcA n) (\u03bcVec \u03bcA m) (\u03bb xs ys \u2192 f (xs ++ ys))\n\nswapS : \u2200 {A B} \u2192 SumProp (A \u00d7 B) \u2192 SumProp (B \u00d7 A)\nswapS = \u03bc-view Data.Product.swap\n  -- \u03bc-iso \u00d7-comm\n\nswapS-preserve : \u2200 {A B} f (\u03bcA\u00d7B : SumProp (A \u00d7 B)) \u2192 sum \u03bcA\u00d7B f \u2261 sum (swapS \u03bcA\u00d7B) (f \u2218 Data.Product.swap)\nswapS-preserve f \u03bcA\u00d7B =  \u03bc-view-preserve Data.Product.swap Data.Product.swap (\u03bb x \u2192 \u2261.refl) f \u03bcA\u00d7B {- or \u2261.refl -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"abdefc0bf9ff9a9f162a9ece25086662e1309247","subject":"NewNew: Simplify proof","message":"NewNew: Simplify proof\n","repos":"inc-lc\/ilc-agda","old_file":"New\/NewNew.agda","new_file":"New\/NewNew.agda","new_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ]\u03c4 da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ]\u03c4 db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ]\u03c4 \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = sym (right-id-int n)\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ]\u03c4 da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8 cong (\u03bb \u25a1 \u2192 \u25a1 (a \u2295 da)) (fromto\u2192\u2295 df f1 f2 dff)\u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ]\u03c4 da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ]\u03c4 da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ]\u03c4 dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ]\u03c4 df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","old_contents":"module New.NewNew where\n\nopen import New.Changes\nopen import New.LangChanges\nopen import New.Lang\nopen import New.Derive\nopen import Data.Empty\n\n[_]\u03c4_from_to_ : \u2200 (\u03c4 : Type) \u2192 (dv : Cht \u03c4) \u2192 (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 Set\n\n-- This can't be a datatype, since it wouldn't be strictly positive as it\n-- appears on the left of an arrow in the function case, which then can be\n-- contained in nested \"fromto-validity\" proofs.\nsumfromto : \u2200 (\u03c3 \u03c4 : Type) \u2192 (dv : SumChange2 {A = \u27e6 \u03c3 \u27e7Type} {B = \u27e6 \u03c4 \u27e7Type}) \u2192 (v1 v2 : \u27e6 sum \u03c3 \u03c4 \u27e7Type) \u2192 Set\nsumfromto \u03c3 \u03c4 (ch\u2081 da) (inj\u2081 a1) (inj\u2081 a2) = [ \u03c3 ]\u03c4 da from a1 to a2\n-- These fallback equations unfortunately don't hold definitionally, they're\n-- split on multiple cases, so the pattern match is a mess.\n--\n-- To \"case split\" on validity, I typically copy-paste the pattern match from\n-- fromto\u2192\u2295 and change the function name :-(.\nsumfromto \u03c3 \u03c4 (ch\u2081 da) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (ch\u2082 db) (inj\u2082 b1) (inj\u2082 b2) = [ \u03c4 ]\u03c4 db from b1 to b2\nsumfromto \u03c3 \u03c4 (ch\u2082 db) _ _ = \u22a5\nsumfromto \u03c3 \u03c4 (rp (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 b2') = b2 \u2261 b2'\nsumfromto \u03c3 \u03c4 (rp (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 a2') = a2 \u2261 a2'\nsumfromto \u03c3 \u03c4 (rp s) _ _ = \u22a5\n\n[ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n  \u2200 (da : Cht \u03c3) (a1 a2 : \u27e6 \u03c3 \u27e7Type) \u2192\n  [ \u03c3 ]\u03c4 da from a1 to a2 \u2192 [ \u03c4 ]\u03c4 df a1 da from f1 a1 to f2 a2\n[ int ]\u03c4 dv from v1 to v2 = v2 \u2261 v1 + dv\n[ pair \u03c3 \u03c4 ]\u03c4 (da , db) from (a1 , b1) to (a2 , b2) = [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 [ \u03c4 ]\u03c4 db from b1 to b2\n[ sum \u03c3 \u03c4 ]\u03c4 dv from v1 to v2 = sumfromto \u03c3 \u03c4 (convert dv) v1 v2\n\ndata [_]\u0393_from_to_ : \u2200 \u0393 \u2192 eCh \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {\u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393 \u03c11 \u03c12 d\u03c1} \u2192 (d\u03c1\u03c1 : [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n  \u03c11 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 v\u2205 = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (dvv v\u2022 d\u03c1\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n  \u27e6 t \u27e7Term \u03c11 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t d\u03c1\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 d\u03c1\u03c1))\n  (sym (weaken-sound t _))\n\n-- Now relate this validity with \u2295. To know that nil and so on are valid, also\n-- relate it to the other definition.\n\nfromto\u2192\u2295 : \u2200 {\u03c4} dv v1 v2 \u2192\n  [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  v1 \u2295 dv \u2261 v2\n\n\u229d-fromto : \u2200 {\u03c4} (v1 v2 : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 v2 \u229d v1 from v1 to v2\n\u229d-fromto {\u03c3 \u21d2 \u03c4} f1 f2 da a1 a2 daa rewrite sym (fromto\u2192\u2295 da a1 a2 daa) = \u229d-fromto (f1 a1) (f2 (a1 \u2295 da))\n\u229d-fromto {int} v1 v2 = sym (update-diff v2 v1)\n\u229d-fromto {pair \u03c3 \u03c4} (a1 , b1) (a2 , b2) = \u229d-fromto a1 a2 , \u229d-fromto b1 b2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2081 a2) = \u229d-fromto a1 a2\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2081 a1) (inj\u2082 b2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2081 a2) = refl\n\u229d-fromto {sum \u03c3 \u03c4} (inj\u2082 b1) (inj\u2082 b2) = \u229d-fromto b1 b2\n\nnil-fromto : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7Type) \u2192 [ \u03c4 ]\u03c4 nil v from v to v\nnil-fromto v = \u229d-fromto v v\n\nfromto\u2192\u2295 {\u03c3 \u21d2 \u03c4} df f1 f2 dff =\n  ext (\u03bb v \u2192 fromto\u2192\u2295 {\u03c4} (df v (nil v)) (f1 v) (f2 v) (dff (nil v) v v (nil-fromto v)))\nfromto\u2192\u2295 {int} dn n1 n2 refl = refl\nfromto\u2192\u2295 {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) =\n  cong\u2082 _,_ (fromto\u2192\u2295 _ _ _ daa) (fromto\u2192\u2295 _ _ _ dbb)\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa rewrite fromto\u2192\u2295 da a1 a2 daa = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb rewrite fromto\u2192\u2295 db b1 b2 dbb = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = refl\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192\u2295 {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveConst : \u2200 {\u03c4} c \u2192\n  [ \u03c4 ]\u03c4 \u27e6 deriveConst c \u27e7Term \u2205 from \u27e6 c \u27e7Const to \u27e6 c \u27e7Const\nfromtoDeriveConst (lit n) = sym (right-id-int n)\nfromtoDeriveConst plus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb = mn\u00b7pq=mp\u00b7nq {a1} {da} {b1} {db}\nfromtoDeriveConst minus da a1 a2 daa db b1 b2 dbb rewrite daa | dbb | sym (-m\u00b7-n=-mn {b1} {db}) = mn\u00b7pq=mp\u00b7nq {a1} {da} { - b1} { - db}\nfromtoDeriveConst cons da a1 a2 daa db b1 b2 dbb = daa , dbb\nfromtoDeriveConst fst (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = daa\nfromtoDeriveConst snd (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = dbb\nfromtoDeriveConst linj da a1 a2 daa = daa\nfromtoDeriveConst rinj db b1 b2 dbb = dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa df f1 f2 dff dg g1 g2 dgg = dff da a1 a2 daa\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb df f1 f2 dff dg g1 g2 dgg = dgg db b1 b2 dbb\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem2 f1 df g1 dg b1 a2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (g1 b1) ((f1 \u2295 df) a2)\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl df f1 f2 dff dg g1 g2 dgg rewrite changeMatchSem-lem1 f1 df g1 dg a1 b2 | sym (fromto\u2192\u2295 df _ _ dff) | sym (fromto\u2192\u2295 dg _ _ dgg) = \u229d-fromto (f1 a1) ((g1 \u2295 dg) b2)\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromtoDeriveConst match (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nfromtoDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {d\u03c1 \u03c11 \u03c12}  \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 x \u27e7\u0394Var \u03c11 d\u03c1) from (\u27e6 x \u27e7Var \u03c11) to (\u27e6 x \u27e7Var \u03c12)\nfromtoDeriveVar this (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar (that x) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar x d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 \u2192 (t : Term \u0393 \u03c4) \u2192\n  {d\u03c1 : eCh \u0393} {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192 [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    [ \u03c4 ]\u03c4 (\u27e6 t \u27e7\u0394Term \u03c11 d\u03c1) from (\u27e6 t \u27e7Term \u03c11) to (\u27e6 t \u27e7Term \u03c12)\nfromtoDerive \u03c4 (const c) {d\u03c1} {\u03c11} d\u03c1\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c11 d\u03c1 = fromtoDeriveConst c\nfromtoDerive \u03c4 (var x) d\u03c1\u03c1 = fromtoDeriveVar x d\u03c1\u03c1\nfromtoDerive \u03c4 (app {\u03c3} s t) d\u03c1\u03c1 rewrite sym (fit-sound t d\u03c1\u03c1) =\n  let fromToF = fromtoDerive (\u03c3 \u21d2 \u03c4) s d\u03c1\u03c1\n  in let fromToB = fromtoDerive \u03c3 t d\u03c1\u03c1 in fromToF _ _ _ fromToB\nfromtoDerive (\u03c3 \u21d2 \u03c4) (abs t) d\u03c1\u03c1 = \u03bb dv v1 v2 dvv \u2192\n  fromtoDerive \u03c4 t (dvv v\u2022 d\u03c1\u03c1)\n\nopen import Postulate.Extensionality\n\nopen \u2261-Reasoning\n\n-- If df is valid, prove (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da.\n\n-- This statement uses a \u2295 da instead of a2, which is not the style of this formalization but fits better with the other one.\n-- Instead, WellDefinedFunChangeFromTo (without prime) fits this formalization.\nWellDefinedFunChangeFromTo\u2032 : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo\u2032 f1 df = \u2200 da a \u2192 [ _ ]\u03c4 da from a to (a \u2295 da) \u2192 WellDefinedFunChangePoint f1 df a da\n\nfromto\u2192WellDefined\u2032 : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo\u2032 f1 df\nfromto\u2192WellDefined\u2032 {f1 = f1} {f2} {df} dff da a daa =\n  begin\n    (f1 \u2295 df) (a \u2295 da)\n  \u2261\u27e8\u27e9\n    f1 (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))\n  \u2261\u27e8 (fromto\u2192\u2295\n     (df (a \u2295 da) (nil (a \u2295 da))) _ _\n     (dff (nil (a \u2295 da)) _ _ (nil-fromto (a \u2295 da))))\n  \u27e9\n    f2 (a \u2295 da)\n  \u2261\u27e8 sym (fromto\u2192\u2295 _ _ _ (dff da _ _ daa)) \u27e9\n    f1 a \u2295 df a da\n  \u220e\n\nWellDefinedFunChangeFromTo : \u2200 {\u03c3 \u03c4} (f1 : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 (df : Cht (\u03c3 \u21d2 \u03c4)) \u2192 Set\nWellDefinedFunChangeFromTo f1 df = \u2200 da a1 a2 \u2192 [ _ ]\u03c4 da from a1 to a2 \u2192 WellDefinedFunChangePoint f1 df a1 da\n\nfromto\u2192WellDefined : \u2200 {\u03c3 \u03c4 f1 f2 df} \u2192 [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 \u2192\n  WellDefinedFunChangeFromTo f1 df\nfromto\u2192WellDefined {f1 = f1} {f2} {df} dff da a1 a2 daa =\n  fromto\u2192WellDefined\u2032 dff da a1 daa\u2032\n  where\n    daa\u2032 : [ _ ]\u03c4 da from a1 to (a1 \u2295 da)\n    daa\u2032 rewrite fromto\u2192\u2295 da _ _ daa = daa\n\n-- Recursive isomorphism between the two validities.\n--\n-- Among other things, valid\u2192fromto proves that a validity-preserving function,\n-- with validity defined via (f1 \u2295 df) (a \u2295 da) \u2261 f1 a \u2295 df a da, is also valid\n-- in the \"fromto\" sense.\n--\n-- We can't hope for a better statement, since we need the equation to be\n-- satisfied also by returned or argument functions.\n\nfromto\u2192valid : \u2200 {\u03c4} \u2192\n  \u2200 dv v1 v2 \u2192 [ \u03c4 ]\u03c4 dv from v1 to v2 \u2192\n  valid v1 dv\nvalid\u2192fromto : \u2200 {\u03c4} v (dv : Cht \u03c4) \u2192 valid v dv \u2192 [ \u03c4 ]\u03c4 dv from v to (v \u2295 dv)\n\nfromto\u2192valid {int} = \u03bb dv v1 v2 x \u2192 tt\nfromto\u2192valid {pair \u03c3 \u03c4} (da , db) (a1 , b1) (a2 , b2) (daa , dbb) = (fromto\u2192valid da _ _ daa) , (fromto\u2192valid db _ _ dbb)\nfromto\u2192valid {\u03c3 \u21d2 \u03c4} df f1 f2 dff = \u03bb a da ada \u2192\n  fromto\u2192valid (df a da) (f1 a) (f2 (a \u2295 da)) (dff da a (a \u2295 da) (valid\u2192fromto a da ada)) ,\n  fromto\u2192WellDefined\u2032 dff da a (valid\u2192fromto a da ada)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 a1) (inj\u2081 a2) daa = sv\u2081 a1 da (fromto\u2192valid da a1 a2 daa)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2081 da)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 b1) (inj\u2082 b2) dbb = sv\u2082 b1 db (fromto\u2192valid db b1 b2 dbb)\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2081 (inj\u2082 db)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 b1) (inj\u2081 .a2) refl = svrp\u2082 b1 a2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2081 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2081 a2)) (inj\u2082 _) (inj\u2082 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 a1) (inj\u2082 .b2) refl = svrp\u2081 a1 b2\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2081 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2081 _) ()\nfromto\u2192valid {sum \u03c3 \u03c4} (inj\u2082 (inj\u2082 b2)) (inj\u2082 _) (inj\u2082 _) ()\n\nvalid\u2192fromto {int} v dv tt = refl\nvalid\u2192fromto {pair \u03c3 \u03c4} (a , b) (da , db) (ada , bdb) = valid\u2192fromto a da ada , valid\u2192fromto b db bdb\nvalid\u2192fromto {\u03c3 \u21d2 \u03c4} f df fdf da a1 a2 daa = body\n  where\n    fa1da-valid :\n      valid (f a1) (df a1 da) \u00d7\n      WellDefinedFunChangePoint f df a1 da\n    fa1da-valid = fdf a1 da (fromto\u2192valid da _ _ daa)\n    body : [ \u03c4 ]\u03c4 df a1 da from f a1 to (f \u2295 df) a2\n    body rewrite sym (fromto\u2192\u2295 da _ _ daa) | proj\u2082 fa1da-valid = valid\u2192fromto (f a1) (df a1 da) (proj\u2081 fa1da-valid)\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) = valid\u2192fromto a da ada\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) = valid\u2192fromto b db bdb\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) = refl\nvalid\u2192fromto {sum \u03c3 \u03c4} .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) = refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"683afb522e345315858025284786e3506e21d0ea","subject":"working on lemmas","message":"working on lemmas\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"disjointness.agda","new_file":"disjointness.agda","new_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192 dom-eq \u03941 H1 \u2192 dom-eq \u03942 H2 \u2192 dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union (\u03c01 , \u03c02) (\u03c03 , \u03c04) = (\u03bb n x \u2192 {!!}) ,\n                                  (\u03bb n x \u2192 {!!})\n\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) with holes-delta-ana h x\u2084 | holes-delta-ana h\u2081 x\u2085\n    ... | ih1 | ih2 = dom-union ih1 ih2\n\n  lem-apart-new : \u2200{e H u} \u2192 holes e H \u2192 hole-name-new e u \u2192 u # H\n  lem-apart-new HConst HNConst = refl\n  lem-apart-new (HAsc h) (HNAsc hn) = lem-apart-new h hn\n  lem-apart-new HVar HNVar = refl\n  lem-apart-new (HLam1 h) (HNLam1 hn) = lem-apart-new h hn\n  lem-apart-new (HLam2 h) (HNLam2 hn) = lem-apart-new h hn\n  lem-apart-new HEHole (HNHole x) = apart-singleton (flip x)\n  lem-apart-new (HNEHole {u = u'} {H = H} h) (HNNEHole  {u = u}  x hn) = apart-parts H (\u25a0 (u' , <>)) u (lem-apart-new h hn) (apart-singleton (flip x))\n  lem-apart-new (HAp {H1 = H1} {H2 = H2} h h\u2081) (HNAp hn hn\u2081) = apart-parts H1 H2 _ (lem-apart-new h hn) (lem-apart-new h\u2081 hn\u2081)\n\n  lem-apart-disjoint : {A : Set} {H : A ctx} {u : Nat} {x : A} \u2192 u # H \u2192 (\u25a0 (u , x)) ## H\n  lem-apart-disjoint {H = H} apt = (\u03bb n x \u2192 tr (\u03bb qq \u2192 qq # H) (singleton-eq (\u03c02 x)) apt) ,\n                                   (\u03bb n x \u2192 {!!})\n\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint HConst he2 HDConst = empty-disj _\n  holes-disjoint-disjoint (HAsc he1) he2 (HDAsc hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HVar he2 HDVar = empty-disj _\n  holes-disjoint-disjoint (HLam1 he1) he2 (HDLam1 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint (HLam2 he1) he2 (HDLam2 hd) = holes-disjoint-disjoint he1 he2 hd\n  holes-disjoint-disjoint HEHole he2 (HDHole x) = lem-apart-disjoint (lem-apart-new he2 x)\n  holes-disjoint-disjoint (HNEHole he1) he2 (HDNEHole x hd) = disjoint-parts (holes-disjoint-disjoint he1 he2 hd) (lem-apart-disjoint (lem-apart-new he2 x))\n  holes-disjoint-disjoint (HAp he1 he2) he3 (HDAp hd hd\u2081) = disjoint-parts (holes-disjoint-disjoint he1 he3 hd) (holes-disjoint-disjoint he2 he3 hd\u2081)\n\n  mutual\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint = {!!}\n\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","old_contents":"open import Prelude\nopen import Nat\nopen import core\nopen import contexts\nopen import lemmas-disjointness\nopen import exchange\nopen import lemmas-freshness\nopen import weakening\n\nmodule disjointness where\n  mutual\n    expand-new-disjoint-synth : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4'} \u2192\n                          hole-name-new e u \u2192\n                          \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                          \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-synth HNConst ESConst = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNAsc hn) (ESAsc x) = expand-new-disjoint-ana hn x\n    expand-new-disjoint-synth HNVar (ESVar x\u2081) = empty-disj (\u25a0 (_ , _ , _))\n    expand-new-disjoint-synth (HNLam1 hn) ()\n    expand-new-disjoint-synth (HNLam2 hn) (ESLam x\u2081 exp) = expand-new-disjoint-synth hn exp\n    expand-new-disjoint-synth (HNHole x) ESEHole = disjoint-singles x\n    expand-new-disjoint-synth (HNNEHole x hn) (ESNEHole x\u2081 exp) = disjoint-parts (expand-new-disjoint-synth hn exp) (disjoint-singles x)\n    expand-new-disjoint-synth (HNAp hn hn\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) =\n                                            disjoint-parts (expand-new-disjoint-ana hn x\u2084)\n                                                  (expand-new-disjoint-ana hn\u2081 x\u2085)\n\n    expand-new-disjoint-ana : \u2200 { e u \u03c4 d \u0394 \u0393 \u0393' \u03c4' \u03c42} \u2192\n                              hole-name-new e u \u2192\n                              \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                              \u0394 ## (\u25a0 (u , \u0393' , \u03c4'))\n    expand-new-disjoint-ana hn (EASubsume x x\u2081 x\u2082 x\u2083) = expand-new-disjoint-synth hn x\u2082\n    expand-new-disjoint-ana (HNLam1 hn) (EALam x\u2081 x\u2082 ex) = expand-new-disjoint-ana hn ex\n    expand-new-disjoint-ana (HNHole x) EAEHole = disjoint-singles x\n    expand-new-disjoint-ana (HNNEHole x hn) (EANEHole x\u2081 x\u2082) = disjoint-parts (expand-new-disjoint-synth hn x\u2082) (disjoint-singles x)\n\n  mutual\n    expand-disjoint-new-synth : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-synth ESConst disj = HNConst\n    expand-disjoint-new-synth (ESVar x\u2081) disj = HNVar\n    expand-disjoint-new-synth (ESLam x\u2081 ex) disj = HNLam2 (expand-disjoint-new-synth ex disj)\n    expand-disjoint-new-synth (ESAp {\u03941 = \u03941} x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) disj\n      with expand-disjoint-new-ana x\u2084 (disjoint-union1 disj) | expand-disjoint-new-ana x\u2085 (disjoint-union2 {\u03931 = \u03941} disj)\n    ... | ih1 | ih2 = HNAp ih1 ih2\n    expand-disjoint-new-synth {\u0393 = \u0393} ESEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-synth (ESNEHole {\u0394 = \u0394} x ex) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                      (expand-disjoint-new-synth ex (disjoint-union1 disj))\n    expand-disjoint-new-synth (ESAsc x) disj = HNAsc (expand-disjoint-new-ana x disj)\n\n    expand-disjoint-new-ana : \u2200{ e \u03c4 d \u0394 u \u0393 \u0393' \u03c42 \u03c4'} \u2192\n                                \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c42 \u22a3 \u0394 \u2192\n                                \u0394 ## (\u25a0 (u , \u0393' , \u03c4')) \u2192\n                                hole-name-new e u\n    expand-disjoint-new-ana (EALam x\u2081 x\u2082 ex) disj = HNLam1 (expand-disjoint-new-ana ex disj)\n    expand-disjoint-new-ana (EASubsume x x\u2081 x\u2082 x\u2083) disj = expand-disjoint-new-synth x\u2082 disj\n    expand-disjoint-new-ana EAEHole disj = HNHole (singles-notequal disj)\n    expand-disjoint-new-ana (EANEHole {\u0394 = \u0394} x x\u2081) disj = HNNEHole (singles-notequal (disjoint-union2 {\u03931 = \u0394} disj))\n                                                                    (expand-disjoint-new-synth x\u2081 (disjoint-union1 disj))\n\n  data holes : (e : hexp) (H : \u22a4 ctx) \u2192 Set where\n    HConst : holes c \u2205\n    HAsc   : \u2200{e \u03c4 H} \u2192 holes e H \u2192 holes (e \u00b7: \u03c4) H\n    HVar   : \u2200{x} \u2192 holes (X x) \u2205\n    HLam1  : \u2200{x e H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x e) H\n    HLam2  : \u2200{x e \u03c4 H} \u2192 holes e H \u2192 holes (\u00b7\u03bb x [ \u03c4 ] e) H\n    HEHole : \u2200{u} \u2192 holes (\u2987\u2988[ u ]) (\u25a0 (u , <>))\n    HNEHole : \u2200{e u H} \u2192 holes e H \u2192 holes (\u2987 e \u2988[ u ]) (H ,, (u , <>))\n    HAp : \u2200{e1 e2 H1 H2} \u2192 holes e1 H1 \u2192 holes e2 H2 \u2192 holes (e1 \u2218 e2) (H1 \u222a H2)\n\n  dom-eq : {A B : Set} \u2192 A ctx \u2192 B ctx \u2192 Set\n  dom-eq {A} {B} C1 C2 = ((n : Nat) \u2192 \u03a3[ x \u2208 A ]( C1 n == Some x) \u2192 (\u03a3[ y \u2208 B ](C2 n == Some y)))\u00d7\n                         ((n : Nat) \u2192 \u03a3[ y \u2208 B ]( C2 n == Some y) \u2192 (\u03a3[ x \u2208 A ](C1 n == Some x)))\n\n  dom-\u2205 : {A B : Set} \u2192 dom-eq (\u03bb _ \u2192 None {A}) (\u03bb _ \u2192 None {B})\n  dom-\u2205 {A} {B} = (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x)))) , (\u03bb n x \u2192 abort (somenotnone (! (\u03c02 x))))\n\n  -- todo: this seems like i would have proven it already\n  singleton-eq : {A : Set} {a : A} \u2192 \u2200{x n y} \u2192 (\u25a0 (x , a)) n == Some y \u2192 x == n\n  singleton-eq {A} {a} {x} {n} {y} eq with natEQ x n\n  singleton-eq eq | Inl x\u2081 = x\u2081\n  singleton-eq eq | Inr x\u2081 = abort (somenotnone (! eq))\n\n  singleton-lookup-refl : {A : Set} {n : Nat} {\u03b2 : A} \u2192 (\u25a0 (n , \u03b2)) n == Some \u03b2\n  singleton-lookup-refl {n = n} with natEQ n n\n  singleton-lookup-refl | Inl refl = \u03bb {\u03b2} \u2192 refl\n  singleton-lookup-refl | Inr x = abort (x refl)\n\n  dom-single : {A B : Set} (x : Nat) (a : A) (b : B) \u2192 dom-eq (\u25a0 (x , a)) (\u25a0 (x , b))\n  dom-single {A} {B} x \u03b1 \u03b2 = (\u03bb n x\u2081 \u2192 \u03b2 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b2)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl)) ,\n                             (\u03bb n x\u2081 \u2192 \u03b1 , (ap1 (\u03bb qq \u2192 (\u25a0 (qq , \u03b1)) n) (singleton-eq (\u03c02 x\u2081)) \u00b7 singleton-lookup-refl))\n\n  dom-union : {A B : Set} {\u03941 \u03942 : A ctx} {H1 H2 : B ctx} \u2192 dom-eq \u03941 H1 \u2192 dom-eq \u03942 H2 \u2192 dom-eq (\u03941 \u222a \u03942) (H1 \u222a H2)\n  dom-union (\u03c01 , \u03c02) (\u03c03 , \u03c04) = (\u03bb n x \u2192 {!!}) ,\n                                  {!!}\n\n  mutual\n    holes-delta-ana : \u2200{\u0393 H e \u03c4 d \u03c4' \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-ana (HLam1 h) (EALam x\u2081 x\u2082 exp) = holes-delta-ana h exp\n    holes-delta-ana h (EASubsume x x\u2081 x\u2082 x\u2083) = holes-delta-synth h x\u2082\n    holes-delta-ana (HEHole {u = u}) EAEHole = dom-single u _ _\n    holes-delta-ana (HNEHole {u = u} h) (EANEHole x x\u2081) = dom-union (holes-delta-synth h x\u2081) (dom-single u _ _ )\n\n    holes-delta-synth : \u2200{\u0393 H e \u03c4 d \u0394} \u2192\n                    holes e H \u2192\n                    \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                    dom-eq \u0394 H\n    holes-delta-synth HConst ESConst = dom-\u2205\n    holes-delta-synth (HAsc h) (ESAsc x) = holes-delta-ana h x\n    holes-delta-synth HVar (ESVar x\u2081) = dom-\u2205\n    holes-delta-synth (HLam2 h) (ESLam x\u2081 exp) = holes-delta-synth h exp\n    holes-delta-synth (HEHole {u = u}) ESEHole = dom-single u _ _\n    holes-delta-synth (HNEHole {u = u} h) (ESNEHole x exp) = dom-union (holes-delta-synth h exp) (dom-single u _ _)\n    holes-delta-synth (HAp h h\u2081) (ESAp x x\u2081 x\u2082 x\u2083 x\u2084 x\u2085) with holes-delta-ana h x\u2084 | holes-delta-ana h\u2081 x\u2085\n    ... | ih1 | ih2 = dom-union ih1 ih2\n\n  holes-disjoint-disjoint : \u2200{ e1 e2 H1 H2} \u2192\n                    holes e1 H1 \u2192\n                    holes e2 H2 \u2192\n                    holes-disjoint e1 e2 \u2192\n                    H1 ## H2\n  holes-disjoint-disjoint = {!!}\n\n  mutual\n    expand-ana-disjoint : \u2200{ e1 e2 \u03c41 \u03c42 e1' e2' \u03c41' \u03c42' \u0393 \u03941 \u03942 } \u2192\n          holes-disjoint e1 e2 \u2192\n          \u0393 \u22a2 e1 \u21d0 \u03c41 ~> e1' :: \u03c41' \u22a3 \u03941 \u2192\n          \u0393 \u22a2 e2 \u21d0 \u03c42 ~> e2' :: \u03c42' \u22a3 \u03942 \u2192\n          \u03941 ## \u03942\n    expand-ana-disjoint = {!!}\n\n\n  -- these lemmas are all structurally recursive and quite\n  -- mechanical. morally, they establish the properties about reduction\n  -- that would be obvious \/ baked into Agda if holes-disjoint was defined\n  -- as a function rather than a judgement (datatype), or if we had defined\n  -- all the O(n^2) cases rather than relying on a little indirection to\n  -- only have O(n) cases. that work has to go somewhwere, and we prefer\n  -- that it goes here.\n  ds-lem-asc : \u2200{e1 e2 \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (e1 \u00b7: \u03c4)\n  ds-lem-asc HDConst = HDConst\n  ds-lem-asc (HDAsc hd) = HDAsc (ds-lem-asc hd)\n  ds-lem-asc HDVar = HDVar\n  ds-lem-asc (HDLam1 hd) = HDLam1 (ds-lem-asc hd)\n  ds-lem-asc (HDLam2 hd) = HDLam2 (ds-lem-asc hd)\n  ds-lem-asc (HDHole x) = HDHole (HNAsc x)\n  ds-lem-asc (HDNEHole x hd) = HDNEHole (HNAsc x) (ds-lem-asc hd)\n  ds-lem-asc (HDAp hd hd\u2081) = HDAp (ds-lem-asc hd) (ds-lem-asc hd\u2081)\n\n  ds-lem-lam1 : \u2200{e1 e2 x} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x e1)\n  ds-lem-lam1 HDConst = HDConst\n  ds-lem-lam1 (HDAsc hd) = HDAsc (ds-lem-lam1 hd)\n  ds-lem-lam1 HDVar = HDVar\n  ds-lem-lam1 (HDLam1 hd) = HDLam1 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDLam2 hd) = HDLam2 (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDHole x\u2081) = HDHole (HNLam1 x\u2081)\n  ds-lem-lam1 (HDNEHole x\u2081 hd) = HDNEHole (HNLam1 x\u2081) (ds-lem-lam1 hd)\n  ds-lem-lam1 (HDAp hd hd\u2081) = HDAp (ds-lem-lam1 hd) (ds-lem-lam1 hd\u2081)\n\n  ds-lem-lam2 : \u2200{e1 e2 x \u03c4} \u2192 holes-disjoint e2 e1 \u2192 holes-disjoint e2 (\u00b7\u03bb x [ \u03c4 ] e1)\n  ds-lem-lam2 HDConst = HDConst\n  ds-lem-lam2 (HDAsc hd) = HDAsc (ds-lem-lam2 hd)\n  ds-lem-lam2 HDVar = HDVar\n  ds-lem-lam2 (HDLam1 hd) = HDLam1 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDLam2 hd) = HDLam2 (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDHole x\u2081) = HDHole (HNLam2 x\u2081)\n  ds-lem-lam2 (HDNEHole x\u2081 hd) = HDNEHole (HNLam2 x\u2081) (ds-lem-lam2 hd)\n  ds-lem-lam2 (HDAp hd hd\u2081) = HDAp (ds-lem-lam2 hd) (ds-lem-lam2 hd\u2081)\n\n  ds-lem-nehole : \u2200{e e1 u} \u2192 holes-disjoint e e1 \u2192 hole-name-new e u \u2192 holes-disjoint e \u2987 e1 \u2988[ u ]\n  ds-lem-nehole HDConst \u03bd = HDConst\n  ds-lem-nehole (HDAsc hd) (HNAsc \u03bd) = HDAsc (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole HDVar \u03bd = HDVar\n  ds-lem-nehole (HDLam1 hd) (HNLam1 \u03bd) = HDLam1 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDLam2 hd) (HNLam2 \u03bd) = HDLam2 (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDHole x) (HNHole x\u2081) = HDHole (HNNEHole (flip x\u2081) x)\n  ds-lem-nehole (HDNEHole x hd) (HNNEHole x\u2081 \u03bd) = HDNEHole (HNNEHole (flip x\u2081) x) (ds-lem-nehole hd \u03bd)\n  ds-lem-nehole (HDAp hd hd\u2081) (HNAp \u03bd \u03bd\u2081) = HDAp (ds-lem-nehole hd \u03bd) (ds-lem-nehole hd\u2081 \u03bd\u2081)\n\n  ds-lem-ap : \u2200{e1 e2 e3} \u2192 holes-disjoint e3 e1 \u2192 holes-disjoint e3 e2 \u2192 holes-disjoint e3 (e1 \u2218 e2)\n  ds-lem-ap HDConst hd2 = HDConst\n  ds-lem-ap (HDAsc hd1) (HDAsc hd2) = HDAsc (ds-lem-ap hd1 hd2)\n  ds-lem-ap HDVar hd2 = HDVar\n  ds-lem-ap (HDLam1 hd1) (HDLam1 hd2) = HDLam1 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDLam2 hd1) (HDLam2 hd2) = HDLam2 (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDHole x) (HDHole x\u2081) = HDHole (HNAp x x\u2081)\n  ds-lem-ap (HDNEHole x hd1) (HDNEHole x\u2081 hd2) = HDNEHole (HNAp x x\u2081) (ds-lem-ap hd1 hd2)\n  ds-lem-ap (HDAp hd1 hd2) (HDAp hd3 hd4) = HDAp (ds-lem-ap hd1 hd3) (ds-lem-ap hd2 hd4)\n\n  -- holes-disjoint is symmetric\n  disjoint-sym : (e1 e2 : hexp) \u2192 holes-disjoint e1 e2 \u2192 holes-disjoint e2 e1\n  disjoint-sym .c c HDConst = HDConst\n  disjoint-sym .c (e2 \u00b7: x) HDConst = HDAsc (disjoint-sym _ _ HDConst)\n  disjoint-sym .c (X x) HDConst = HDVar\n  disjoint-sym .c (\u00b7\u03bb x e2) HDConst = HDLam1 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (\u00b7\u03bb x [ x\u2081 ] e2) HDConst = HDLam2 (disjoint-sym c e2 HDConst)\n  disjoint-sym .c \u2987\u2988[ x ] HDConst = HDHole HNConst\n  disjoint-sym .c \u2987 e2 \u2988[ x ] HDConst = HDNEHole HNConst (disjoint-sym c e2 HDConst)\n  disjoint-sym .c (e2 \u2218 e3) HDConst = HDAp (disjoint-sym c e2 HDConst) (disjoint-sym c e3 HDConst)\n\n  disjoint-sym _ c (HDAsc hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x) (HDAsc hd) | HDAsc ih = HDAsc (ds-lem-asc ih)\n  disjoint-sym _ (X x) (HDAsc hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDAsc hd) | HDLam1 ih = HDLam1 (ds-lem-asc ih)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDAsc hd) | HDLam2 ih = HDLam2 (ds-lem-asc ih)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDAsc hd) | HDHole x\u2081 = HDHole (HNAsc x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDAsc hd) | HDNEHole x\u2081 ih = HDNEHole (HNAsc x\u2081) (ds-lem-asc ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDAsc hd) | HDAp ih ih\u2081 = HDAp (ds-lem-asc ih) (ds-lem-asc ih\u2081)\n\n  disjoint-sym _ c HDVar = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) HDVar = HDAsc (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (X x\u2081) HDVar = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) HDVar = HDLam1 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) HDVar = HDLam2 (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] HDVar = HDHole HNVar\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] HDVar = HDNEHole HNVar (disjoint-sym _ e2 HDVar)\n  disjoint-sym _ (e2 \u2218 e3) HDVar = HDAp (disjoint-sym _ e2 HDVar) (disjoint-sym _ e3 HDVar)\n\n  disjoint-sym _ c (HDLam1 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam1 hd) | HDAsc ih = HDAsc (ds-lem-lam1 ih)\n  disjoint-sym _ (X x\u2081) (HDLam1 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam1 hd) | HDLam1 ih = HDLam1 (ds-lem-lam1 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam1 hd) | HDLam2 ih = HDLam2 (ds-lem-lam1 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam1 hd) | HDHole x = HDHole (HNLam1 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam1 hd) | HDNEHole x ih = HDNEHole (HNLam1 x) (ds-lem-lam1 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam1 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam1 ih) (ds-lem-lam1 ih\u2081)\n\n  disjoint-sym _ c (HDLam2 hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u00b7: x\u2081) (HDLam2 hd) | HDAsc ih = HDAsc (ds-lem-lam2 ih)\n  disjoint-sym _ (X x\u2081) (HDLam2 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 e2) (HDLam2 hd) | HDLam1 ih = HDLam1 (ds-lem-lam2 ih)\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x\u2081 [ x\u2082 ] e2) (HDLam2 hd) | HDLam2 ih = HDLam2 (ds-lem-lam2 ih)\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x\u2081 ] (HDLam2 hd) | HDHole x = HDHole (HNLam2 x)\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x\u2081 ] (HDLam2 hd) | HDNEHole x ih = HDNEHole (HNLam2 x) (ds-lem-lam2 ih)\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e2 \u2218 e3) (HDLam2 hd) | HDAp ih ih\u2081 = HDAp (ds-lem-lam2 ih) (ds-lem-lam2 ih\u2081)\n\n  disjoint-sym _ c (HDHole x) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDHole (HNAsc x\u2081)) = HDAsc (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (X x) (HDHole x\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDHole (HNLam1 x\u2081)) = HDLam1 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDHole (HNLam2 x\u2082)) = HDLam2 (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2082))\n  disjoint-sym _ \u2987\u2988[ x ] (HDHole (HNHole x\u2081)) =  HDHole (HNHole (flip x\u2081))\n  disjoint-sym _ \u2987 e2 \u2988[ u' ] (HDHole (HNNEHole x x\u2081)) = HDNEHole (HNHole (flip x)) (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x\u2081))\n  disjoint-sym _ (e2 \u2218 e3) (HDHole (HNAp x x\u2081)) = HDAp (disjoint-sym \u2987\u2988[ _ ] e2 (HDHole x))\n                                                    (disjoint-sym \u2987\u2988[ _ ] e3 (HDHole x\u2081))\n\n  disjoint-sym _ c (HDNEHole x hd) = HDConst\n  disjoint-sym _ (e2 \u00b7: x) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e \u00b7: x) (HDNEHole (HNAsc x\u2081) hd) | HDAsc ih = HDAsc (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (X x) (HDNEHole x\u2081 hd) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x e2) (HDNEHole (HNLam1 x\u2081) hd) | HDLam1 ih = HDLam1 (ds-lem-nehole ih x\u2081)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole x\u2082 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e2) (HDNEHole (HNLam2 x\u2082) hd) | HDLam2 ih = HDLam2 (ds-lem-nehole ih x\u2082)\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987\u2988[ x ] (HDNEHole (HNHole x\u2082) hd) | HDHole x\u2081 = HDHole (HNNEHole (flip x\u2082) x\u2081)\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole x\u2081 hd) with disjoint-sym _ _ hd\n  disjoint-sym _ \u2987 e2 \u2988[ x ] (HDNEHole (HNNEHole x\u2082 x\u2083) hd) | HDNEHole x\u2081 ih = HDNEHole (HNNEHole (flip x\u2082) x\u2081) (ds-lem-nehole ih x\u2083)\n  disjoint-sym _ (e2 \u2218 e3) (HDNEHole x hd) with disjoint-sym _ _ hd\n  disjoint-sym _ (e1 \u2218 e3) (HDNEHole (HNAp x x\u2081) hd) | HDAp ih ih\u2081 = HDAp (ds-lem-nehole ih x) (ds-lem-nehole ih\u2081 x\u2081)\n\n  disjoint-sym _ c (HDAp hd hd\u2081) = HDConst\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u00b7: x) (HDAp hd hd\u2081) | HDAsc ih | HDAsc ih1 = HDAsc (ds-lem-ap ih ih1)\n  disjoint-sym _ (X x) (HDAp hd hd\u2081) = HDVar\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x e3) (HDAp hd hd\u2081) | HDLam1 ih | HDLam1 ih1 = HDLam1 (ds-lem-ap ih ih1)\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (\u00b7\u03bb x [ x\u2081 ] e3) (HDAp hd hd\u2081) | HDLam2 ih | HDLam2 ih1 = HDLam2 (ds-lem-ap ih ih1)\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987\u2988[ x ] (HDAp hd hd\u2081) | HDHole x\u2081 | HDHole x\u2082 = HDHole (HNAp x\u2081 x\u2082)\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ \u2987 e3 \u2988[ x ] (HDAp hd hd\u2081) | HDNEHole x\u2081 ih | HDNEHole x\u2082 ih1 = HDNEHole (HNAp x\u2081 x\u2082) (ds-lem-ap ih ih1)\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) with disjoint-sym _ _ hd | disjoint-sym _ _ hd\u2081\n  disjoint-sym _ (e3 \u2218 e4) (HDAp hd hd\u2081) | HDAp ih ih\u2081 | HDAp ih1 ih2 = HDAp (ds-lem-ap ih ih1) (ds-lem-ap ih\u2081 ih2)\n\n\n  -- note that this is false, so holes-disjoint isn't transitive\n  -- disjoint-new : \u2200{e1 e2 u} \u2192 holes-disjoint e1 e2 \u2192 hole-name-new e1 u \u2192 hole-name-new e2 u\n\n  -- it's also not reflexive, because \u2987\u2988[ u ] isn't hole-disjoint with\n  -- itself since refl : u == u; it's also not anti-reflexive, because the\n  -- expression c *is* hole-disjoint with itself (albeit vacuously)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3434a00acb3e4442b6a72989801219cdbbbd6775","subject":"Added subst\u2083.","message":"Added subst\u2083.\n\nIgnore-this: 560fe8716715efaaf8a849450468e60c\n\ndarcs-hash:20110812224721-3bd4e-046a8a3e342834685fa2a84e22246837452f119e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Common\/Relation\/Binary\/PropositionalEquality.agda","new_file":"src\/Common\/Relation\/Binary\/PropositionalEquality.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"2b84cac6bf59cadb238898ce1205107283b7f5aa","subject":"--without-K","message":"--without-K\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\nimport ZK.SigmaProtocol.Types\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n  -- TODO Challenge should exp large wrt the security param\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- See the module ZK.Types for details about the types:\n--   * Prover-Interaction\n--   * Verifier\n--   * Transcript\n--   * Transcript\u00b2\nopen ZK.SigmaProtocol.Types Commitment Challenge Response public\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n_\u21c6_ : Verifier \u2192 Prover \u2192 (r : Randomness)(w : Witness)(c : Challenge) \u2192 Bool\n(verifier \u21c6 prover) r w c = verifier (run (prover r w) c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = \u2713 (verifier t)\n\n  prover-commitment : (r : Randomness)(w : Witness) \u2192 Commitment\n  prover-commitment r w = Prover-Interaction.get-A (prover r w)\n\n  prover-response : (r : Randomness)(w : Witness)(c : Challenge) \u2192 Response\n  prover-response r w = Prover-Interaction.get-f (prover r w)\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 w r c (w? : ValidWitness w) \u2192\n                              \u2713 ((verifier \u21c6 prover) r w c)\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713 (verifier (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator          : Simulator\n    .correct-simulator : Correct-simulator verifier simulator\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = (t\u00b2 : Transcript\u00b2 verifier) \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\n-- Also called 2-extractable\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor              : Extractor verifier\n    .extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    .correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"open import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\nimport ZK.SigmaProtocol.Types\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n  -- TODO Challenge should exp large wrt the security param\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- See the module ZK.Types for details about the types:\n--   * Prover-Interaction\n--   * Verifier\n--   * Transcript\n--   * Transcript\u00b2\nopen ZK.SigmaProtocol.Types Commitment Challenge Response public\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n_\u21c6_ : Verifier \u2192 Prover \u2192 (r : Randomness)(w : Witness)(c : Challenge) \u2192 Bool\n(verifier \u21c6 prover) r w c = verifier (run (prover r w) c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = \u2713 (verifier t)\n\n  prover-commitment : (r : Randomness)(w : Witness) \u2192 Commitment\n  prover-commitment r w = Prover-Interaction.get-A (prover r w)\n\n  prover-response : (r : Randomness)(w : Witness)(c : Challenge) \u2192 Response\n  prover-response r w = Prover-Interaction.get-f (prover r w)\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 w r c (w? : ValidWitness w) \u2192\n                              \u2713 ((verifier \u21c6 prover) r w c)\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            \u2713 (verifier (mk A c s))\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator          : Simulator\n    .correct-simulator : Correct-simulator verifier simulator\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = (t\u00b2 : Transcript\u00b2 verifier) \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\n-- Also called 2-extractable\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    extractor              : Extractor verifier\n    .extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    .correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a7b78ee63370dd5827f735bf58cc67702eed57f0","subject":"Fixed indentation.","message":"Fixed indentation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/co-data\/Conat\/PropertiesSL.agda","new_file":"notes\/co-data\/Conat\/PropertiesSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Conat properties using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Conat.PropertiesSL where\n\nopen import Coinduction\nopen import Data.Conat\nopen import Data.Nat\n\n------------------------------------------------------------------------------\n\n\u2115\u2192Co\u2115 : \u2115 \u2192 Co\u2115\n\u2115\u2192Co\u2115 zero    = zero\n\u2115\u2192Co\u2115 (suc n) = suc (\u266f (\u2115\u2192Co\u2115 n))\n\n{-# NO_TERMINATION_CHECK #-}\nCo\u2115\u2192\u2115 : Co\u2115 \u2192 \u2115\nCo\u2115\u2192\u2115 zero    = zero\nCo\u2115\u2192\u2115 (suc n) = suc (Co\u2115\u2192\u2115 (\u266d n))\n","old_contents":"------------------------------------------------------------------------------\n-- Conat properties using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Conat.PropertiesSL where\n\nopen import Coinduction\nopen import Data.Conat\nopen import Data.Nat\n\n------------------------------------------------------------------------------\n\n\u2115\u2192Co\u2115 : \u2115 \u2192 Co\u2115\n\u2115\u2192Co\u2115 zero    = zero\n\u2115\u2192Co\u2115 (suc n) = suc (\u266f (\u2115\u2192Co\u2115 n))\n\n{-# NO_TERMINATION_CHECK #-}\nCo\u2115\u2192\u2115 : Co\u2115 \u2192 \u2115\nCo\u2115\u2192\u2115 zero     = zero\nCo\u2115\u2192\u2115 (suc n) = suc (Co\u2115\u2192\u2115 (\u266d n))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ee85f4b963d3956908da8af2a06f7a0b6c007c83","subject":"Desc stratified model: stratify phi and psi.","message":"Desc stratified model: stratify phi and psi.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l : Level}{A B : Set l}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l : Level}{A B C : Set l}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {l : Level}{A B : Set l}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e82479ac436b69e0e07f2b197c9936aea319f6b1","subject":"Only indentation.","message":"Only indentation.\n\nIgnore-this: 7cb5bbe5f2e6e2cf5be423aec0798b44\n\ndarcs-hash:20110204214625-3bd4e-eb00618e3c96c3ac71cfe41e5b7817040200a53c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Program\/SortList\/Properties\/Closures\/OrdList\/FlattenATP.agda","new_file":"src\/LTC\/Program\/SortList\/Properties\/Closures\/OrdList\/FlattenATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to OrdList (flatten-OrdList-aux)\n------------------------------------------------------------------------------\n\nmodule LTC.Program.SortList.Properties.Closures.OrdList.FlattenATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Type\n\nopen import LTC.Program.SortList.Properties.Closures.BoolATP\nopen import LTC.Program.SortList.Properties.Closures.ListN-ATP\nopen import LTC.Program.SortList.Properties.Closures.OrdTreeATP\nopen import LTC.Program.SortList.Properties.MiscellaneousATP\nopen import LTC.Program.SortList.SortList\n\n------------------------------------------------------------------------------\n\nflatten-OrdList-aux : \u2200 {t\u2081 i t\u2082} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192\n                      OrdTree (node t\u2081 i t\u2082) \u2192\n                      LE-Lists (flatten t\u2081) (flatten t\u2082)\n\nflatten-OrdList-aux {t\u2082 = t\u2082} nilT Ni Tt\u2082 OTt =\n  subst (\u03bb t \u2192 LE-Lists t (flatten t\u2082))\n        (sym (flatten-nilTree))\n        (\u2264-Lists-[] (flatten t\u2082))\n\nflatten-OrdList-aux (tipT {i\u2081} Ni\u2081) _ nilT OTt = prf\n  where\n    postulate prf : LE-Lists (flatten (tip i\u2081)) (flatten nilTree)\n    -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    -- Metis 2.3 : Non-tested.\n    -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf #-}\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni (tipT {i\u2082} Ni\u2082) OTt = prf\n  where\n    postulate lemma : LE i\u2081 i\u2082\n    -- E 1.2: Non-tested.\n    -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    -- Metis 2.3 : Non-tested.\n    {-# ATP prove lemma \u2264-ItemTree-Bool \u2264-TreeItem-Bool ordTree-Bool\n                  \u2264-trans &&\u2083-proj\u2083 &&\u2083-proj\u2084\n    #-}\n\n    postulate prf : LE-Lists (flatten (tip i\u2081)) (flatten (tip i\u2082))\n    -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    -- Metis 2.3 : Non-tested.\n    -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf lemma #-}\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt = prf\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (tipT Ni\u2081)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (tipT Ni\u2081) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (tip i\u2081) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the fourth conjunct of OTt\n    aux\u2086 = \u2264-ItemTree-Bool Ni Tt\u2082\u2081\n    aux\u2087 = \u2264-ItemTree-Bool Ni Tt\u2082\u2082\n    aux\u2088 = trans (sym (\u2264-ItemTree-node i t\u2082\u2081 i\u2082 t\u2082\u2082))\n                 (&&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    -- The ATPs could not figure out them.\n    OrdTree-tip-i\u2081 : OrdTree (tip i\u2081)\n    OrdTree-tip-i\u2081 = &&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-TreeItem-tip-i\u2081-i : LE-TreeItem (tip i\u2081) i\n    LE-TreeItem-tip-i\u2081-i = &&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2081)\n    lemma\u2081 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2081 OT  -- IH.\n      where\n        -- The ATPs could not figure these terms.\n        OrdTree-t\u2082\u2081 : OrdTree t\u2082\u2081\n        OrdTree-t\u2082\u2081 = leftSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                          (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2081 : LE-ItemTree i t\u2082\u2081\n        LE-ItemTree-i-t\u2082\u2081 = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        postulate OT : OrdTree (node (tip i\u2081) i t\u2082\u2081)\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT OrdTree-tip-i\u2081\n                         OrdTree-t\u2082\u2081\n                         LE-TreeItem-tip-i\u2081-i\n                         LE-ItemTree-i-t\u2082\u2081\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2082)\n    lemma\u2082 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2082 OT  -- IH.\n      where\n        -- The ATPs could not figure these terms.\n        OrdTree-t\u2082\u2082 : OrdTree t\u2082\u2082\n        OrdTree-t\u2082\u2082 = rightSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                           (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2082 : LE-ItemTree i t\u2082\u2082\n        LE-ItemTree-i-t\u2082\u2082 = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        postulate OT : OrdTree (node (tip i\u2081) i t\u2082\u2082)\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT OrdTree-tip-i\u2081\n                         OrdTree-t\u2082\u2082\n                         LE-TreeItem-tip-i\u2081-i\n                         LE-ItemTree-i-t\u2082\u2082\n        #-}\n\n    postulate prf : LE-Lists (flatten (tip i\u2081)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    -- E 1.2: Non-tested.\n    -- Metis 2.3 : Non-tested.\n    -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf xs\u2264ys\u2192xs\u2264zs\u2192xs\u2264ys++zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n                    Ni nilT OTt = prf\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool nilT\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni nilT\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i nilTree )) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten nilTree)\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni nilT OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2081 i nilTree)\n        -- E 1.2: Non-tested.\n        -- Equinox 5.0alpha (2010-06-29): Non-tested.\n        -- Metis 2.3 : Non-tested.\n        {-# ATP prove OT leftSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&-proj\u2081 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten nilTree)\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni nilT OT\n      where\n        postulate OT : OrdTree (node t\u2081\u2082 i nilTree)\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT rightSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    postulate prf : LE-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten nilTree)\n      -- E 1.2: Non-tested.\n      -- Equinox 5.0alpha (2010-06-29): Non-tested.\n      -- Metis 2.3 : Non-tested.\n    {-# ATP prove prf xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (tipT {i\u2082} Ni\u2082) OTt = prf\n  where\n        -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (tipT Ni\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (tipT Ni\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (tip i\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (tip i\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2081 i (tip i\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT leftSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&-proj\u2081 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (tip i\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2082 i (tip i\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT rightSubTree-OrdTree\n                         &&-proj\u2082 &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088 #-}\n\n    postulate prf : LE-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (tip i\u2082))\n      -- E 1.2: Non-tested.\n      -- Metis 2.3 : Non-tested.\n      -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt = prf\n\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT leftSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&-proj\u2081 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT rightSubTree-OrdTree\n                         &&-proj\u2082 &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088 #-}\n\n    postulate prf : LE-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082))\n                    (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      -- E 1.2: Non-tested.\n      -- Metis 2.3 : Non-tested.\n      -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    {-# ATP prove prf xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Closures properties respect to OrdList (flatten-OrdList-aux)\n------------------------------------------------------------------------------\n\nmodule LTC.Program.SortList.Properties.Closures.OrdList.FlattenATP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Bool.PropertiesATP\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Type\n\nopen import LTC.Program.SortList.Properties.Closures.BoolATP\nopen import LTC.Program.SortList.Properties.Closures.ListN-ATP\nopen import LTC.Program.SortList.Properties.Closures.OrdTreeATP\nopen import LTC.Program.SortList.Properties.MiscellaneousATP\nopen import LTC.Program.SortList.SortList\n\n------------------------------------------------------------------------------\n\nflatten-OrdList-aux : \u2200 {t\u2081 i t\u2082} \u2192 Tree t\u2081 \u2192 N i \u2192 Tree t\u2082 \u2192\n                      OrdTree (node t\u2081 i t\u2082) \u2192\n                      LE-Lists (flatten t\u2081) (flatten t\u2082)\n\nflatten-OrdList-aux {t\u2082 = t\u2082} nilT Ni Tt\u2082 OTt =\n  subst (\u03bb t \u2192 LE-Lists t (flatten t\u2082))\n        (sym (flatten-nilTree))\n        (\u2264-Lists-[] (flatten t\u2082))\n\nflatten-OrdList-aux (tipT {i\u2081} Ni\u2081) _ nilT OTt = prf\n  where\n    postulate prf : LE-Lists (flatten (tip i\u2081)) (flatten nilTree)\n    -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    -- Metis 2.3 : Non-tested.\n    -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf #-}\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni (tipT {i\u2082} Ni\u2082) OTt = prf\n  where\n    postulate lemma : LE i\u2081 i\u2082\n    -- E 1.2: Non-tested.\n    -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    -- Metis 2.3 : Non-tested.\n    {-# ATP prove lemma \u2264-ItemTree-Bool \u2264-TreeItem-Bool ordTree-Bool\n                  \u2264-trans &&\u2083-proj\u2083 &&\u2083-proj\u2084\n    #-}\n\n    postulate prf : LE-Lists (flatten (tip i\u2081)) (flatten (tip i\u2082))\n    -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    -- Metis 2.3 : Non-tested.\n    -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf lemma #-}\n\nflatten-OrdList-aux {i = i} (tipT {i\u2081} Ni\u2081) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt = prf\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (tipT Ni\u2081)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (tipT Ni\u2081) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (tip i\u2081) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the fourth conjunct of OTt\n    aux\u2086 = \u2264-ItemTree-Bool Ni Tt\u2082\u2081\n    aux\u2087 = \u2264-ItemTree-Bool Ni Tt\u2082\u2082\n    aux\u2088 = trans (sym (\u2264-ItemTree-node i t\u2082\u2081 i\u2082 t\u2082\u2082))\n                 (&&\u2083-proj\u2084 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    -- Common terms for the lemma\u2081 and lemma\u2082.\n    -- The ATPs could not figure out them.\n    OrdTree-tip-i\u2081 : OrdTree (tip i\u2081)\n    OrdTree-tip-i\u2081 = &&\u2083-proj\u2081 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    LE-TreeItem-tip-i\u2081-i : LE-TreeItem (tip i\u2081) i\n    LE-TreeItem-tip-i\u2081-i = &&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085\n\n    lemma\u2081 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2081)\n    lemma\u2081 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2081 OT  -- IH.\n      where\n        -- The ATPs could not figure these terms.\n        OrdTree-t\u2082\u2081 : OrdTree t\u2082\u2081\n        OrdTree-t\u2082\u2081 = leftSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                          (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2081 : LE-ItemTree i t\u2082\u2081\n        LE-ItemTree-i-t\u2082\u2081 = &&-proj\u2081 aux\u2086 aux\u2087 aux\u2088\n\n        postulate OT : OrdTree (node (tip i\u2081) i t\u2082\u2081)\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT OrdTree-tip-i\u2081\n                         OrdTree-t\u2082\u2081\n                         LE-TreeItem-tip-i\u2081-i\n                         LE-ItemTree-i-t\u2082\u2081\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten (tip i\u2081)) (flatten t\u2082\u2082)\n    lemma\u2082 = flatten-OrdList-aux (tipT Ni\u2081) Ni Tt\u2082\u2082 OT  -- IH.\n      where\n        -- The ATPs could not figure these terms.\n        OrdTree-t\u2082\u2082 : OrdTree t\u2082\u2082\n        OrdTree-t\u2082\u2082 = rightSubTree-OrdTree Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082\n                                           (&&\u2083-proj\u2082 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n        LE-ItemTree-i-t\u2082\u2082 : LE-ItemTree i t\u2082\u2082\n        LE-ItemTree-i-t\u2082\u2082 = &&-proj\u2082 aux\u2086 aux\u2087 aux\u2088\n\n        postulate OT : OrdTree (node (tip i\u2081) i t\u2082\u2082)\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT OrdTree-tip-i\u2081\n                         OrdTree-t\u2082\u2082\n                         LE-TreeItem-tip-i\u2081-i\n                         LE-ItemTree-i-t\u2082\u2082\n        #-}\n\n    postulate prf : LE-Lists (flatten (tip i\u2081)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    -- E 1.2: Non-tested.\n    -- Metis 2.3 : Non-tested.\n    -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf xs\u2264ys\u2192xs\u2264zs\u2192xs\u2264ys++zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n                    Ni nilT OTt = prf\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool nilT\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni nilT\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i nilTree )) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten nilTree)\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni nilT OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2081 i nilTree)\n        -- E 1.2: Non-tested.\n        -- Equinox 5.0alpha (2010-06-29): Non-tested.\n        -- Metis 2.3 : Non-tested.\n        {-# ATP prove OT leftSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&-proj\u2081 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten nilTree)\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni nilT OT\n      where\n        postulate OT : OrdTree (node t\u2081\u2082 i nilTree)\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT rightSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    postulate prf : LE-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten nilTree)\n      -- E 1.2: Non-tested.\n      -- Equinox 5.0alpha (2010-06-29): Non-tested.\n      -- Metis 2.3 : Non-tested.\n    {-# ATP prove prf xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (tipT {i\u2082} Ni\u2082) OTt = prf\n  where\n        -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (tipT Ni\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (tipT Ni\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (tip i\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (tip i\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2081 i (tip i\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT leftSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&-proj\u2081 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (tip i\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (tipT Ni\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2082 i (tip i\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT rightSubTree-OrdTree\n                         &&-proj\u2082 &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088 #-}\n\n    postulate prf : LE-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (tip i\u2082))\n      -- E 1.2: Non-tested.\n      -- Metis 2.3 : Non-tested.\n      -- Vampire 0.6 (revision 903): Non-tested.\n    {-# ATP prove prf xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n\nflatten-OrdList-aux {i = i} (nodeT {t\u2081\u2081} {i\u2081} {t\u2081\u2082} Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n                    (nodeT {t\u2082\u2081} {i\u2082} {t\u2082\u2082} Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OTt = prf\n\n  where\n    -- Auxiliary terms to get the conjuncts from OTt.\n    aux\u2081 = ordTree-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082)\n    aux\u2082 = ordTree-Bool (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2083 = \u2264-TreeItem-Bool (nodeT Tt\u2081\u2081 Ni\u2081 Tt\u2081\u2082) Ni\n    aux\u2084 = \u2264-ItemTree-Bool Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082)\n    aux\u2085 = trans (sym (ordTree-node (node t\u2081\u2081 i\u2081 t\u2081\u2082) i (node t\u2082\u2081 i\u2082 t\u2082\u2082))) OTt\n\n    -- Auxiliary terms to get the conjuncts from the third conjunct of OTt.\n    aux\u2086 = \u2264-TreeItem-Bool Tt\u2081\u2081 Ni\n    aux\u2087 = \u2264-TreeItem-Bool Tt\u2081\u2082 Ni\n    aux\u2088 = trans (sym (\u2264-TreeItem-node t\u2081\u2081 i\u2081 t\u2081\u2082 i))\n                 (&&\u2083-proj\u2083 aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085)\n\n    lemma\u2081 : LE-Lists (flatten t\u2081\u2081) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2081 = flatten-OrdList-aux Tt\u2081\u2081 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2081 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT leftSubTree-OrdTree\n                         &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&-proj\u2081 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088\n        #-}\n\n    lemma\u2082 : LE-Lists (flatten t\u2081\u2082) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n    lemma\u2082 = flatten-OrdList-aux Tt\u2081\u2082 Ni (nodeT Tt\u2082\u2081 Ni\u2082 Tt\u2082\u2082) OT  -- IH.\n      where\n        postulate OT : OrdTree (node t\u2081\u2082 i (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n        -- E 1.2: Non-tested.\n        -- Metis 2.3 : Non-tested.\n        -- Vampire 0.6 (revision 903): Non-tested.\n        {-# ATP prove OT rightSubTree-OrdTree\n                         &&-proj\u2082 &&\u2083-proj\u2081 &&\u2083-proj\u2082 &&\u2083-proj\u2084\n                         aux\u2081 aux\u2082 aux\u2083 aux\u2084 aux\u2085 aux\u2086 aux\u2087 aux\u2088 #-}\n\n    postulate prf : LE-Lists (flatten (node t\u2081\u2081 i\u2081 t\u2081\u2082)) (flatten (node t\u2082\u2081 i\u2082 t\u2082\u2082))\n      -- E 1.2: Non-tested.\n      -- Metis 2.3 : Non-tested.\n      -- Equinox 5.0alpha (2010-06-29): Non-tested.\n    {-# ATP prove prf xs\u2264zs\u2192ys\u2264zs\u2192xs++ys\u2264zs flatten-ListN lemma\u2081 lemma\u2082 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ea9cebaf77fba9aaf6d9808003b2ca94984a151d","subject":"Embed62: correctness of embed (fix #62)","message":"Embed62: correctness of embed (fix #62)\n\nOld-commit-hash: d609ac1436ac9262b4332f8ffefc7020d487a547\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/Embed62.agda","new_file":"experimental\/Embed62.agda","new_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import ExplicitNil using (ext\u00b3)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Data.Unit using (\u22a4 ; tt)\nopen import Data.Nat\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd! : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd! _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) s)) (weaken (drop _ \u2022 \u0393\u227c\u0393) t))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = abs (app (app (var this) (nat old)) (nat new))\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = abs (app (app (var this)\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) fst)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) fst)))\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) snd)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) snd)))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\n_\u2248_ : \u2200 {\u03c4} \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\n_\u2248_ {nats} (old , new) v = old \u2261 v \u27e6fst\u27e7 \u00d7 new \u2261 v \u27e6snd\u27e7\n_\u2248_ {bags} u v = u \u2261 v\n_\u2248_ {\u03c3 \u21d2 \u03c4} u v = \n  (w : \u27e6 \u03c3 \u27e7) (dw : \u0394Val \u03c3) (dw\u2032 : \u27e6 \u0394-Type \u03c3 \u27e7)\n  (R[w,dw] : valid w dw) (eq : dw \u2248 dw\u2032) \u2192\n  u w dw R[w,dw] \u2248 v w dw\u2032\n\ninfix 4 _\u2248_\n\ncompatible : \u2200 {\u0393} \u2192 \u0394Env \u0393 \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\ncompatible {\u2205} \u2205 \u2205 = EmptySet\ncompatible {\u03c4 \u2022 \u0393} (cons v dv _ \u03c1) (dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032) =\n  triple (v \u2261 v\u2032) (dv \u2248 dv\u2032) (compatible \u03c1 \u03c1\u2032)\n\nderiveVar-correct : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 x \u27e7\u0394Var \u03c1 \u2248 \u27e6 deriveVar x \u27e7 \u03c1\u2032\n\nderiveVar-correct {x = this} -- pattern-matching on \u03c1, \u03c1\u2032 NOT optional\n  {cons _ _ _ _} {_ \u2022 _ \u2022 _} {cons _ dv\u2248dv\u2032 _ _} = dv\u2248dv\u2032\nderiveVar-correct {x = that y}\n  {cons _ _ _ \u03c1} {_ \u2022 _ \u2022 \u03c1\u2032} {cons _ _ C _} =\n  deriveVar-correct {x = y} {\u03c1} {\u03c1\u2032} {C}\n\nweak-id : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nweak-id {\u2205} {\u2205} = refl\nweak-id {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} = cong\u2082 _\u2022_ {x = v} refl (weak-id {\u0393} {\u03c1})\n\nweak-eq : \u2200 {\u0393} {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7}\n  (C : compatible \u03c1 \u03c1\u2032) \u2192 ignore \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1\u2032\n\nweak-eq {\u2205} {\u2205} {\u2205} _ = refl\nweak-eq {\u03c4 \u2022 \u0393} {cons v dv _ \u03c1} {dv\u2032 \u2022 v\u2032 \u2022 \u03c1\u2032} (cons v\u2261v\u2032 _ C _) =\n  cong\u2082 _\u2022_ v\u2261v\u2032 (weak-eq C)\n\nweak-eq-term : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  {\u03c1 : \u0394Env \u0393} {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} (C : compatible \u03c1 \u03c1\u2032) \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2261 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1\u2032\n\nweak-eq-term t {\u03c1} {\u03c1\u2032} C =\n  trans (cong \u27e6 t \u27e7 (weak-eq C)) (sym (weaken-sound t \u03c1\u2032))\n\nweaken-once : \u2200 {\u03c4 \u0393 \u03c3} {t : Term \u0393 \u03c4} {v : \u27e6 \u03c3 \u27e7} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192\n  \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 weaken (drop \u03c3 \u2022 \u0393\u227c\u0393) t \u27e7 (v \u2022 \u03c1)\n\nweaken-once {t = t} {v} {\u03c1} = trans\n  (cong \u27e6 t \u27e7 (sym weak-id))\n  (sym (weaken-sound t (v \u2022 \u03c1)))\n\nembed-correct : \u2200 {\u03c4 \u0393}\n  {dt : \u0394Term \u0393 \u03c4}\n  {\u03c1 : \u0394Env \u0393} {V : dt is-valid-for \u03c1}\n  {\u03c1\u2032 : \u27e6 \u0394-Context \u0393 \u27e7} {C : compatible \u03c1 \u03c1\u2032} \u2192\n  \u27e6 dt \u27e7\u0394 \u03c1 V \u2248 \u27e6 embed dt \u27e7 \u03c1\u2032\n\nembed-correct {dt = \u0394nat old new} = refl , refl\n\nembed-correct {dt = \u0394bag db} = refl\n\nembed-correct {dt = \u0394var x} {\u03c1} {\u03c1\u2032 = \u03c1\u2032} {C} =\n  deriveVar-correct {x = x} {\u03c1} {\u03c1\u2032} {C}\n\nembed-correct {dt = \u0394add ds dt} {\u03c1} {V} {\u03c1\u2032} {C} =\n  let\n    s00 , s01 = \u27e6 ds \u27e7\u0394 \u03c1 (car V)\n    t00 , t01 = \u27e6 dt \u27e7\u0394 \u03c1 (cadr V)\n    s10 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    s11 = \u27e6 embed ds \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    t10 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6fst\u27e7\n    t11 = \u27e6 embed dt \u27e7 \u03c1\u2032 \u27e6snd\u27e7\n    rec-s = embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n    rec-t = embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C}\n  in\n    \n    (begin\n      s00 + t00\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2081 rec-s) (proj\u2081 rec-t) \u27e9\n      s10 + t10\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6fst\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6fst\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6fst\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6fst\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed ds) (\u27e6fst\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6fst\u27e7) (sym (weaken-sound (embed dt) (\u27e6fst\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6fst\u27e7 \u2022 \u03c1\u2032) \u27e6fst\u27e7\n    \u220e)\n    ,\n    (begin\n      s01 + t01\n    \u2261\u27e8 cong\u2082 _+_ (proj\u2082 rec-s) (proj\u2082 rec-t) \u27e9\n      s11 + t11\n    \u2261\u27e8 cong\u2082 _+_\n        (cong (\u03bb x \u2192 \u27e6 embed ds \u27e7 x \u27e6snd\u27e7) (sym weak-id))\n        (cong (\u03bb x \u2192 \u27e6 embed dt \u27e7 x \u27e6snd\u27e7) (sym weak-id)) \u27e9\n      \u27e6 embed ds \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7 +\n      \u27e6 embed dt \u27e7 (\u27e6 drop _ \u2022 \u0393\u227c\u0393 \u27e7 (\u27e6snd\u27e7 {\u2115} \u2022 \u03c1\u2032)) \u27e6snd\u27e7\n    \u2261\u27e8 cong\u2082 _+_\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed ds) (\u27e6snd\u27e7 \u2022 \u03c1\u2032))))\n      (cong (\u03bb f \u2192 f \u27e6snd\u27e7) (sym (weaken-sound (embed dt) (\u27e6snd\u27e7 \u2022 \u03c1\u2032)))) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7 +\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt) \u27e7 (\u27e6snd\u27e7 \u2022 \u03c1\u2032) \u27e6snd\u27e7\n    \u220e)\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394union ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _++_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394diff ds dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (embed-correct {dt = ds} {\u03c1} {car  V} {\u03c1\u2032} {C})\n  (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2081 s dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 mapBag\n  (weak-eq-term s C)\n  (embed-correct {dt = dt} {\u03c1} {car V} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394map\u2080 s ds t dt} {\u03c1} {V} {\u03c1\u2032} {C} = cong\u2082 _\\\\_\n  (cong\u2082 mapBag\n    (extensionality (\u03bb v \u2192\n    begin\n      proj\u2082 (\u27e6 ds \u27e7\u0394 \u03c1 (car V) v (v , v) refl)\n    \u2261\u27e8 proj\u2082 (embed-correct {dt = ds} {\u03c1} {car V} {\u03c1\u2032} {C}\n        v (v , v) (\u03bb f \u2192 f v v) refl (refl , refl)) \u27e9\n      \u27e6 embed ds \u27e7 \u03c1\u2032 v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole v (\u03bb f \u2192 f v v) \u27e6snd\u27e7)\n        (weaken-once {t = embed ds} {v} {\u03c1\u2032}) \u27e9\n      \u27e6 weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds) \u27e7 (v \u2022 \u03c1\u2032) v (\u03bb f \u2192 f v v) \u27e6snd\u27e7\n    \u220e))\n    (cong\u2082 _++_\n      (weak-eq-term t C)\n      (embed-correct {dt = dt} {\u03c1} {cadr V} {\u03c1\u2032} {C})))\n  (cong\u2082 mapBag (weak-eq-term s C) (weak-eq-term t C))\n  where open \u2261-Reasoning\n\nembed-correct {dt = \u0394app ds t dt} {\u03c1} {cons vds vdt R[t,dt] _} {\u03c1\u2032} {C}\n  rewrite sym (weak-eq-term t C) =\n  embed-correct {dt = ds} {\u03c1} {vds} {\u03c1\u2032} {C}\n    (\u27e6 t \u27e7Term (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 vdt) (\u27e6 embed dt \u27e7Term \u03c1\u2032)\n    R[t,dt] (embed-correct {dt = dt} {\u03c1} {vdt} {\u03c1\u2032} {C})\n\nembed-correct {dt = \u0394abs dt} {\u03c1} {V} {\u03c1\u2032} {C} = \u03bb w dw dw\u2032 R[w,dw] dw\u2248dw\u2032 \u2192\n  embed-correct {dt = dt}\n    {cons w dw R[w,dw] \u03c1} {V w dw R[w,dw]}\n    {dw\u2032 \u2022 w \u2022 \u03c1\u2032} {cons refl dw\u2248dw\u2032 C tt}\n","old_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\ntranslate : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ntranslate {nats} dn = dn \u27e6fst\u27e7 , dn \u27e6snd\u27e7\ntranslate {bags} db = db\ntranslate {\u03c3 \u21d2 \u03c4} df = \u03bb v dv R[v,dv] \u2192 translate (df v {!dv!})\n-- TODO figure out what to do to dv to put it in the hole\n\nconsistent : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\nconsistent {\u2205} \u2205 = EmptySet\nconsistent {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = couple (valid v (translate dv)) (consistent \u03c1)\n\nrecall : \u2200 {\u0393} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (\u03b3 : consistent \u03c1)  \u2192 \u0394Env \u0393\nrecall {\u2205} \u2205 \u2205 = \u2205\nrecall {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) (cons R[v,dv] \u03b3 _ _) = {!!}\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) s)) (weaken (drop _ \u2022 \u0393\u227c\u0393) t))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = abs (app (app (var this) (nat old)) (nat new))\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = abs (app (app (var this)\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) fst)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) fst)))\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) snd)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) snd)))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n-- embed-correct : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env}\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"67753164bfe339da23cfc88edd35d97fb22c6c8a","subject":"Desc model: rename iso2 to psi.","message":"Desc model: rename iso2 to psi.","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : (I : Set) -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi I (var i) = refl\nproof-phi-psi I (const x) = refl\nproof-phi-psi I (prod D D') with proof-phi-psi I D | proof-phi-psi I D'\n...                             | p | q = cong2 prod p q\nproof-phi-psi I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi I (T s)))\nproof-phi-psi I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (psi (T s)))\n                                              T\n                                              (\\s -> proof-phi-psi I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> phi (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 phi (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (phi D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (phi D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"146a094b8a78494503293fa50f87d1c201c2ed03","subject":"FunUniverse.Bits: update","message":"FunUniverse.Bits: update\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Bits.agda","new_file":"FunUniverse\/Bits.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.Bits where\n\nopen import Data.Nat.NP using (\u2115; _*_; module \u2115\u00b0)\nopen import Data.Bool using (if_then_else_)\nopen import Data.Fin using (Fin)\nopen import Function using (const; _\u2218\u2032_; id)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; _++_; [_]; take; drop; rewire; rewireTbl)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bit using (0b; 1b)\nopen import Data.Bits using (_\u2192\u1d47_; RewireTbl)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Core\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\n\nbitsFunU : FunUniverse \u2115\nbitsFunU = Bits-U , _\u2192\u1d47_\n\nmodule BitsFunUniverse = FunUniverse bitsFunU\nopen BitsFunUniverse\nopen Cong-*1 _`\u2192_\nopen Defaults bitsFunU\n\nbitsFunOps : FunOps bitsFunU\nbitsFunOps = mk rewiring (cond , fork\u1d47) (const [ 0b ]) (const [ 1b ])\n  where\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = take A\n  snd\u1d47 : \u2200 {B} A \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 A = drop A\n  <_,_>\u1d47 : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <_,_>\u1d47 f g x = f x ++ g x\n  fork\u1d47 : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u1d47 f g (b \u2237 xs) = (if b then f else g) xs\n\n  open DefaultsGroup2 id _\u2218\u2032_ (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n  open DefaultCondFromFork fork\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n\n  cat : Category _`\u2192_\n  cat = id , _\u2218\u2032_\n\n  lin : LinRewiring bitsFunU\n  lin = mk cat first (\u03bb {x} \u2192 swap {x}) (\u03bb {x} \u2192 assoc {x}) id id\n           <_\u00d7_> (\u03bb {x} \u2192 second {x}) id id id id\n\n  rewiring : Rewiring bitsFunU\n  rewiring = mk lin (const []) dup (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n                (cong-*1 \u2218\u2032 rewire) (cong-*1 \u2218\u2032 rewireTbl)\n\nbitsFunRewiring : Rewiring bitsFunU\nbitsFunRewiring = FunOps.rewiring bitsFunOps\n\nbitsFunLin : LinRewiring bitsFunU\nbitsFunLin = Rewiring.linRewiring bitsFunRewiring\n\nmodule BitsFunOps = FunOps bitsFunOps\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule FunUniverse.Bits where\n\nopen import Data.Nat.NP using (\u2115; _*_; module \u2115\u00b0)\nopen import Data.Bool using (if_then_else_)\nopen import Data.Fin using (Fin)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using (Vec; []; _\u2237_; _++_; [_])\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bit using (0b; 1b)\nopen import Data.Bits using (_\u2192\u1d47_; RewireTbl)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Core\n\nbitsFunU : FunUniverse \u2115\nbitsFunU = Bits-U , _\u2192\u1d47_\n\nmodule BitsFunUniverse = FunUniverse bitsFunU\n\nbitsFunOps : FunOps bitsFunU\nbitsFunOps = mk rewiring (cond , fork\u1d47) (const [ 0b ]) (const [ 1b ])\n  where\n  open BitsFunUniverse\n  open F using (id; _\u2218\u2032_)\n  fst\u1d47 : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  fst\u1d47 {A} = V.take A\n  snd\u1d47 : \u2200 {B} A \u2192 A `\u00d7 B `\u2192 B\n  snd\u1d47 A = V.drop A\n  <_,_>\u1d47 : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  <_,_>\u1d47 f g x = f x ++ g x\n  fork\u1d47 : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  fork\u1d47 f g (b \u2237 xs) = (if b then f else g) xs\n\n  cong-*1 : \u2200 {i o} \u2192 i `\u2192 o \u2192 (i * 1) `\u2192 (o * 1)\n  cong-*1 {i} {o} rewrite proj\u2082 \u2115\u00b0.*-identity i\n                        | proj\u2082 \u2115\u00b0.*-identity o = id\n\n  open Defaults bitsFunU\n  open DefaultsGroup2 id _\u2218\u2032_ (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n  open DefaultCondFromFork fork\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n\n  lin : LinRewiring bitsFunU\n  lin = mk id _\u2218\u2032_ first (\u03bb {x} \u2192 swap {x}) (\u03bb {x} \u2192 assoc {x}) id id\n           <_\u00d7_> (\u03bb {x} \u2192 second {x}) id id id id\n\n  rewiring : Rewiring bitsFunU\n  rewiring = mk lin (const []) dup (const []) <_,_>\u1d47 fst\u1d47 (\u03bb {x} \u2192 snd\u1d47 x)\n                (cong-*1 \u2218\u2032 V.rewire) (cong-*1 \u2218\u2032 V.rewireTbl)\n\nbitsFunRewiring : Rewiring bitsFunU\nbitsFunRewiring = FunOps.rewiring bitsFunOps\n\nbitsFunLin : LinRewiring bitsFunU\nbitsFunLin = Rewiring.linRewiring bitsFunRewiring\n\nmodule BitsFunOps = FunOps bitsFunOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5bf06aa027244da83061d4926c27a5c35a87af1b","subject":"layout","message":"layout\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Product.agda","new_file":"Search\/Searchable\/Product.agda","new_contents":"open import Type hiding (\u2605)\nopen import Function.NP\nopen import Data.Product\nopen import Search.Type\nopen import Search.Searchable\n\nmodule Search.Searchable.Product where\n\nprivate\n    Cont : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080\n    Cont M A = (A \u2192 M) \u2192 M\n\n    -- liftM2 _,_ in the continuation monad\n    _\u00d7F_ : \u2200 {A B M} \u2192 Cont M A \u2192 Cont M B \u2192 Cont M (A \u00d7 B)\n    (fA \u00d7F fB) f = fA (fB \u2218 curry f)\n    -- (fA \u00d7F fB) f = fA \u03bb x \u2192 fB (f (x , y)\n\n    \u03a3F : \u2200 {A B M} \u2192 Cont M A \u2192 (\u2200 {x} \u2192 Cont M (B x)) \u2192 Cont M (\u03a3 A B)\n    \u03a3F fA fB f = fA (fB \u2218 curry f)\n\n\u03a3Search : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Search A \u2192 (\u2200 {x} \u2192 Search (B x)) \u2192 Search (\u03a3 A B)\n\u03a3Search search\u1d2c search\u1d2e op = \u03a3F (search\u1d2c op) (search\u1d2e op)\n\n\u03a3SearchInd : \u2200 {A} {B : A \u2192 \u2605\u2080}\n               {s\u1d2c : Search A} {s\u1d2e : \u2200 {x} \u2192 Search (B x)}\n             \u2192 SearchInd s\u1d2c \u2192 (\u2200 {x} \u2192 SearchInd (s\u1d2e {x})) \u2192 SearchInd (\u03a3Search s\u1d2c s\u1d2e)\n\u03a3SearchInd Ps\u1d2c Ps\u1d2e P P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (\u03bb x \u2192 Ps\u1d2e {x} (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (curry Pf x))\n\n\u03bc\u03a3 : \u2200 {A} {B : A \u2192 \u2605 _} \u2192 Searchable A \u2192 (\u2200 {x} \u2192 Searchable (B x)) \u2192 Searchable (\u03a3 A B)\n\u03bc\u03a3 \u03bcA \u03bcB = _ , \u03a3SearchInd (search-ind \u03bcA) (search-ind \u03bcB)\n\n-- From now on, these are derived definitions for convenience and pedagogical reasons\n\n\u03a3Sum : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 {x} \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\n\u03a3Sum = \u03a3F\n\ninfixr 4 _\u00d7Search_\n\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\nsearch\u1d2c \u00d7Search search\u1d2e = \u03a3Search search\u1d2c search\u1d2e\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u00d7 B)\n\u03bcA \u00d7\u03bc \u03bcB = \u03bc\u03a3 \u03bcA \u03bcB\n\ninfixr 4 _\u00d7Sum_\n\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsum\u1d2c \u00d7Sum sum\u1d2e = \u03a3Sum sum\u1d2c sum\u1d2e\n\n-- Those are here only for pedagogical use\nprivate\n    \u03a3Sum' : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 x \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\n    \u03a3Sum' sum\u1d2c sum\u1d2e f = sum\u1d2c (\u03bb x\u2080 \u2192\n                          sum\u1d2e x\u2080 (\u03bb x\u2081 \u2192\n                            f (x\u2080 , x\u2081)))\n\n    _\u00d7Search'_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n    (search\u1d2c \u00d7Search' search\u1d2e) op f = search\u1d2c op (\u03bb x \u2192 search\u1d2e op (curry f x))\n\n    _\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                    \u2192 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c \u00d7Search' s\u1d2e)\n    (Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf =\n      Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 \u2218 curry Pf)\n\n    -- liftM2 _,_ in the continuation monad\n    _\u00d7Sum'_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n    (sum\u1d2c \u00d7Sum' sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                            sum\u1d2e (\u03bb x\u2081 \u2192\n                              f (x\u2080 , x\u2081)))\n","old_contents":"open import Type hiding (\u2605)\nopen import Function.NP\nopen import Data.Product\nopen import Search.Type\nopen import Search.Searchable\n\nmodule Search.Searchable.Product where\n\nprivate\n    Cont : \u2605\u2080 \u2192 \u2605\u2080 \u2192 \u2605\u2080  \n    Cont M A = (A \u2192 M) \u2192 M\n\n    -- liftM2 _,_ in the continuation monad\n    _\u00d7F_ : \u2200 {A B M} \u2192 Cont M A \u2192 Cont M B \u2192 Cont M (A \u00d7 B)\n    (fA \u00d7F fB) f = fA (fB \u2218 curry f)\n    -- (fA \u00d7F fB) f = fA \u03bb x \u2192 fB (f (x , y)\n\n    \u03a3F : \u2200 {A B M} \u2192 Cont M A \u2192 (\u2200 {x} \u2192 Cont M (B x)) \u2192 Cont M (\u03a3 A B)\n    \u03a3F fA fB f = fA (fB \u2218 curry f)\n\n\u03a3Search : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Search A \u2192 (\u2200 {x} \u2192 Search (B x)) \u2192 Search (\u03a3 A B)\n\u03a3Search search\u1d2c search\u1d2e op = \u03a3F (search\u1d2c op) (search\u1d2e op)\n\n\u03a3SearchInd : \u2200 {A} {B : A \u2192 \u2605\u2080}\n               {s\u1d2c : Search A} {s\u1d2e : \u2200 {x} \u2192 Search (B x)}\n             \u2192 SearchInd s\u1d2c \u2192 (\u2200 {x} \u2192 SearchInd (s\u1d2e {x})) \u2192 SearchInd (\u03a3Search s\u1d2c s\u1d2e)\n\u03a3SearchInd Ps\u1d2c Ps\u1d2e P P\u2219 Pf =\n  Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (\u03bb x \u2192 Ps\u1d2e {x} (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (curry Pf x))\n\n\u03bc\u03a3 : \u2200 {A} {B : A \u2192 \u2605 _} \u2192 Searchable A \u2192 (\u2200 {x} \u2192 Searchable (B x)) \u2192 Searchable (\u03a3 A B)\n\u03bc\u03a3 \u03bcA \u03bcB = _ , \u03a3SearchInd (search-ind \u03bcA) (search-ind \u03bcB)\n\n-- From now on, these are derived definitions for convenience and pedagogical reasons\n\n\u03a3Sum : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 {x} \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\n\u03a3Sum = \u03a3F\n\ninfixr 4 _\u00d7Search_\n\n_\u00d7Search_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\nsearch\u1d2c \u00d7Search search\u1d2e = \u03a3Search search\u1d2c search\u1d2e\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u00d7 B)\n\u03bcA \u00d7\u03bc \u03bcB = \u03bc\u03a3 \u03bcA \u03bcB\n\ninfixr 4 _\u00d7Sum_\n\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\nsum\u1d2c \u00d7Sum sum\u1d2e = \u03a3Sum sum\u1d2c sum\u1d2e\n\n-- Those are here only for pedagogical use\nprivate\n    \u03a3Sum' : \u2200 {A} {B : A \u2192 \u2605\u2080} \u2192 Sum A \u2192 (\u2200 x \u2192 Sum (B x)) \u2192 Sum (\u03a3 A B)\n    \u03a3Sum' sum\u1d2c sum\u1d2e f = sum\u1d2c (\u03bb x\u2080 \u2192\n                          sum\u1d2e x\u2080 (\u03bb x\u2081 \u2192\n                            f (x\u2080 , x\u2081)))\n\n    _\u00d7Search'_ : \u2200 {A B} \u2192 Search A \u2192 Search B \u2192 Search (A \u00d7 B)\n    (search\u1d2c \u00d7Search' search\u1d2e) op f = search\u1d2c op (\u03bb x \u2192 search\u1d2e op (curry f x))\n\n    _\u00d7SearchInd'_ : \u2200 {A B} {s\u1d2c : Search A} {s\u1d2e : Search B}\n                    \u2192 SearchInd s\u1d2c \u2192 SearchInd s\u1d2e \u2192 SearchInd (s\u1d2c \u00d7Search' s\u1d2e)\n    (Ps\u1d2c \u00d7SearchInd' Ps\u1d2e) P P\u2219 Pf =\n      Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ _ \u2192 s _ _)) P\u2219 \u2218 curry Pf)\n\n    -- liftM2 _,_ in the continuation monad\n    _\u00d7Sum'_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n    (sum\u1d2c \u00d7Sum' sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                            sum\u1d2e (\u03bb x\u2081 \u2192\n                              f (x\u2080 , x\u2081)))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"baf9d9e76a34316fce810a93541b10309ab193d0","subject":"flat-funs: <_,_> and <_\u00d7_> shortcuts","message":"flat-funs: <_,_> and <_\u00d7_> shortcuts\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk \u22a4 Bit _\u00d7_ Vec (\u03bb A B \u2192 A \u2192 B)\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk 0 1 _+_ _*_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_ proj\u2081 proj\u2082\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id bitsFunComp bitsFunVComp _&&&_ (\u03bb {A} \u2192 take A) (\u03bb {A} \u2192 drop A)\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nimport Level as L\nopen import composable\nopen import vcomp\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n    _`\u2192_  : T \u2192 T \u2192 Set\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n  infix 0 _`\u2192_\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n    isComposable  : Composable  _`\u2192_\n    isVComposable : VComposable _`\u00d7_ _`\u2192_\n\n    -- Fanout\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n  open FlatFuns \u266dFuns public\n  open Composable isComposable public\n  open VComposable isVComposable public\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Product\nopen import Function\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk \u22a4 Bit _\u00d7_ Vec (\u03bb A B \u2192 A \u2192 B)\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk 0 1 _+_ _*_ (\u03bb i o \u2192 Bits i \u2192 Bits o)\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id funComp funVComp _&&&_ proj\u2081 proj\u2082\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id bitsFunComp bitsFunVComp _&&&_ (\u03bb {A} \u2192 take A) (\u03bb {A} \u2192 drop A)\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0825f4444e8c74376e24c3c0414e978d9006feff","subject":"Reorder","message":"Reorder\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Changes.agda","new_file":"New\/Changes.agda","new_contents":"module New.Changes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Unit\nopen import Data.Product\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  infixl 6 _\u2295_ _\u229d_\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n      ; \u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 Ch A \u00d7 Ch B\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = Ch A \u00d7 Ch B\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    } }\n\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b} } }\n","old_contents":"module New.Changes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\nopen import Data.Unit\nopen import Data.Product\n\nrecord IsChAlg {\u2113 : Level} (A : Set \u2113) (Ch : Set \u2113) : Set (suc \u2113) where\n  field\n    _\u2295_ : A \u2192 Ch \u2192 A\n    _\u229d_ : A \u2192 A \u2192 Ch\n    valid : A \u2192 Ch \u2192 Set \u2113\n    \u229d-valid : \u2200 (a b : A) \u2192 valid a (b \u229d a)\n    \u2295-\u229d : (b a : A) \u2192 a \u2295 (b \u229d a) \u2261 b\n  infixl 6 _\u2295_ _\u229d_\n\n  nil : A \u2192 Ch\n  nil a = a \u229d a\n  nil-valid : (a : A) \u2192 valid a (nil a)\n  nil-valid a = \u229d-valid a a\n\n  \u0394 : A \u2192 Set \u2113\n  \u0394 a = \u03a3[ da \u2208 Ch ] (valid a da)\n\n  update-diff = \u2295-\u229d\n  update-nil : (a : A) \u2192 a \u2295 nil a \u2261 a\n  update-nil a = update-diff a a\n\nrecord ChAlg {\u2113 : Level} (A : Set \u2113) : Set (suc \u2113) where\n  field\n    Ch : Set \u2113\n    isChAlg : IsChAlg A Ch\n  open IsChAlg isChAlg public\n\nopen ChAlg {{...}} public hiding (Ch)\nCh : \u2200 {\u2113} (A : Set \u2113) \u2192 {{CA : ChAlg A}} \u2192 Set \u2113\nCh A {{CA}} = ChAlg.Ch CA\n\n-- Huge hack, but it does gives the output you want to see in all cases I've seen.\n\n{-# DISPLAY IsChAlg.valid x = valid #-}\n{-# DISPLAY ChAlg.valid x = valid #-}\n{-# DISPLAY IsChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY ChAlg._\u2295_ x = _\u2295_ #-}\n{-# DISPLAY IsChAlg.nil x = nil #-}\n{-# DISPLAY ChAlg.nil x = nil #-}\n{-# DISPLAY IsChAlg._\u229d_ x = _\u229d_ #-}\n{-# DISPLAY ChAlg._\u229d_ x = _\u229d_ #-}\n\nmodule _ {\u2113\u2081} {\u2113\u2082}\n  {A : Set \u2113\u2081} {B : Set \u2113\u2082} {{CA : ChAlg A}} {{CB : ChAlg B}} where\n  private\n    fCh = A \u2192 Ch A \u2192 Ch B\n    _f\u2295_ : (A \u2192 B) \u2192 fCh \u2192 A \u2192 B\n    _f\u2295_ = \u03bb f df a \u2192 f a \u2295 df a (nil a)\n    _f\u229d_ : (g f : A \u2192 B) \u2192 fCh\n    _f\u229d_ = \u03bb g f a da \u2192 g (a \u2295 da) \u229d f a\n  open \u2261-Reasoning\n  open import Postulate.Extensionality\n\n  IsDerivative : \u2200 (f : A \u2192 B) \u2192 (df : fCh) \u2192 Set (\u2113\u2081 \u2294 \u2113\u2082)\n  IsDerivative f df = \u2200 a da (v : valid a da) \u2192 f (a \u2295 da) \u2261 f a \u2295 df a da\n\n  instance\n    funCA : ChAlg (A \u2192 B)\n  private\n    funUpdateDiff : \u2200 g f a \u2192 (f f\u2295 (g f\u229d f)) a \u2261 g a\n  funUpdateDiff g f a rewrite update-nil a = update-diff (g a) (f a)\n  funCA = record\n    { Ch = A \u2192 Ch A \u2192 Ch B\n    ; isChAlg = record\n      { _\u2295_ = _f\u2295_\n      ; _\u229d_ = _f\u229d_\n      ; valid = \u03bb f df \u2192 \u2200 a da (v : valid a da) \u2192\n        valid (f a) (df a da) \u00d7\n        (f f\u2295 df) (a \u2295 da) \u2261 f a \u2295 df a da\n      ; \u229d-valid = \u03bb f g a da (v : valid a da) \u2192\n         \u229d-valid (f a) (g (a \u2295 da))\n        , ( begin\n          f (a \u2295 da) \u2295 (g (a \u2295 da \u2295 nil (a \u2295 da)) \u229d f (a \u2295 da))\n        \u2261\u27e8 cong (\u03bb \u25a1 \u2192 f (a \u2295 da) \u2295 (g \u25a1 \u229d f (a \u2295 da)))\n             (update-nil (a \u2295 da)) \u27e9\n          f (a \u2295 da) \u2295 (g (a \u2295 da) \u229d f (a \u2295 da))\n        \u2261\u27e8 update-diff (g (a \u2295 da)) (f (a \u2295 da)) \u27e9\n          g (a \u2295 da)\n        \u2261\u27e8 sym (update-diff (g (a \u2295 da)) (f a)) \u27e9\n          f a \u2295 (g (a \u2295 da) \u229d f a)\n        \u220e)\n      ; \u2295-\u229d = \u03bb g f \u2192 ext (funUpdateDiff g f)\n      } }\n\n  nil-is-derivative : \u2200 (f : A \u2192 B) \u2192 IsDerivative f (nil f)\n  nil-is-derivative f a da v =\n    begin\n      f (a \u2295 da)\n    \u2261\u27e8 sym (cong (\u03bb \u25a1 \u2192 \u25a1 (_\u2295_ a da)) (update-nil f)) \u27e9\n      (f \u2295 nil f) (a \u2295 da)\n    \u2261\u27e8 proj\u2082 (nil-valid f a da v) \u27e9\n      f a \u2295 (nil f a da)\n    \u220e\n\n  private\n    _p\u2295_ : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 A \u00d7 B\n    _p\u2295_ (a , b) (da , db) = a \u2295 da , b \u2295 db\n    _p\u229d_ : A \u00d7 B \u2192 A \u00d7 B \u2192 Ch A \u00d7 Ch B\n    _p\u229d_ (a2 , b2) (a1 , b1) = a2 \u229d a1 , b2 \u229d b1\n    pvalid : A \u00d7 B \u2192 Ch A \u00d7 Ch B \u2192 Set (\u2113\u2082 \u2294 \u2113\u2081)\n    pvalid (a , b) (da , db) = valid a da \u00d7 valid b db\n    p\u229d-valid : (p1 p2 : A \u00d7 B) \u2192 pvalid p1 (p2 p\u229d p1)\n    p\u229d-valid (a1 , b1) (a2 , b2) = \u229d-valid a1 a2 , \u229d-valid b1 b2\n    p\u2295-\u229d : (p2 p1 : A \u00d7 B) \u2192 p1 p\u2295 (p2 p\u229d p1) \u2261 p2\n    p\u2295-\u229d (a2 , b2) (a1 , b1) = cong\u2082 _,_ (\u2295-\u229d a2 a1) (\u2295-\u229d b2 b1)\n  instance\n    pairCA : ChAlg (A \u00d7 B)\n  pairCA = record\n    { Ch = Ch A \u00d7 Ch B\n    ; isChAlg = record\n    { _\u2295_ = _p\u2295_\n    ; _\u229d_ = _p\u229d_\n    ; valid = pvalid\n    ; \u229d-valid = p\u229d-valid\n    ; \u2295-\u229d = p\u2295-\u229d\n    } }\n\nopen import Data.Integer\nopen import Theorem.Groups-Nehemiah\n\ninstance\n  intCA : ChAlg \u2124\nintCA = record\n  { Ch = \u2124\n  ; isChAlg = record\n  { _\u2295_ = _+_\n  ; _\u229d_ = _-_\n  ; valid = \u03bb a b \u2192 \u22a4\n  ; \u229d-valid = \u03bb a b \u2192 tt\n  ; \u2295-\u229d = \u03bb b a \u2192 n+[m-n]=m {a} {b} } }\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"435d14447683e13e31f39fb02f751a10dec2396f","subject":"Added nest-\u2264.","message":"Added nest-\u2264.\n\nIgnore-this: 66798d074a18d6463535b689c1646140\n\ndarcs-hash:20110405141851-3bd4e-7fe2b36202a77d7104679148f24241e6b1336d6c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-N : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-N dom0           = subst N (sym nest-0) zN\nnest-N (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-N h\u2082)\n\nnest-\u2264 : \u2200 {n} \u2192 N n \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 Nz dom0           =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 NSn (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (domS n h\u2081 h\u2082)) (nest-N h\u2081) NSn prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 (dom-N (nest n) h\u2082) h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N h\u2081) Nn NSn (nest-\u2264 Nn h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N dn-x) (x\u2264y\u2192x<Sy (nest-N dn-x) Nx (nest-\u2264 Nx dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\n-- The nest function is total (non-terminating version).\n-- nest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\n-- nest-N zN          = subst N (sym nest-0) zN\n-- nest-N (sN {n} Nn) = subst N (sym (nest-S n)) (nest-N (nest-N Nn))\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-N : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-N dom0           = subst N (sym nest-0) zN\nnest-N (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-N h\u2082)\n\npostulate\n  nest-\u2264 : \u2200 {n} \u2192 N n \u2192 LE (nest n) n\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h = domS x dn-x (h (nest-N dn-x)\n                                      (x\u2264y\u2192x<Sy (nest-N dn-x) Nx (nest-\u2264 Nx)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"16b461ffe856060f636c01a2b50dbb9c0018189e","subject":"close #18","message":"close #18\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress.agda","new_file":"progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d boxedval + d indet + d casterr[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n  progress (TAAp wt1 wt2) | S x | ih2 = {!!}\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n  progress (TAAp wt1 wt2) | I x | ih2 = {!!}\n  progress (TAAp wt1 wt2) | V x | ih2 = {!!}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (d' , Step x y z) = S (_ , Step {!!} {!!} {!!})\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt y)\n    with progress wt\n  ... | S x = {!!}\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  ... | I x = {!x!}\n  ... | V x = {!!}\n\n\n  -- -- applications\n  -- progress (TAAp D1 x D2)\n  --   with progress D1 | progress D2\n  --   -- left applicand value\n  -- progress (TAAp TAConst () D2) | V VConst | _\n  -- progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  -- progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n  --   -- errors propagate\n  -- progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = E (EAp2 (FVal x) x\u2081)\n  -- progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = E (EAp2 (FIndet x) x\u2081)\n  -- progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = E (EAp1 x) -- NB: could have picked the other one here, too; this is a source of non-determinism\n  -- progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = S {!!}\n  -- progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n  --   -- indeterminates\n  -- progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  -- progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  --   -- either applicand steps\n  -- progress (TAAp D1 x\u2083 D2) | S (d1' , Step x x\u2081 x\u2082)  | _ = S (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082))\n  -- progress (TAAp D1 x\u2084 D2) | _ | S (d2' , Step x\u2081 x\u2082 x\u2083) = S (_ , Step (FHAp1 {!!} x\u2081) x\u2082 (FHAp1 {!!} x\u2083))\n\n\n  -- -- non-empty holes\n  -- progress (TANEHole x D x\u2081)\n  --   with progress D\n  -- progress (TANEHole x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  -- progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  -- progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  -- progress (TANEHole x\u2083 D x\u2084) | S (_ , Step x x\u2081 x\u2082) = S (_ , Step (FHNEHole x) x\u2081 (FHNEHole x\u2082))\n\n  -- -- casts\n  -- progress (TACast D x)\n  --   with progress D\n  -- progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  -- progress (TACast D m)  | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  -- progress (TACast D x\u2081) | I x = I (ICast x)\n  -- progress (TACast D x\u2081) | E x = E (ECastProp x)\n  -- progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\n\nmodule progress where\n  -- this is a little bit of syntactic sugar to avoid many layer nested Inl\n  -- and Inrs that you would get from the more literal transcription of the\n  -- consequent of progress as:\n  --\n  -- d val + d indet + d err[ \u0394 ] + \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 ok d \u0394\n    E : \u2200{d \u0394} \u2192 d casterr \u2192 ok d \u0394\n    I : \u2200{d \u0394} \u2192 d indet \u2192 ok d \u0394\n    V : \u2200{d \u0394} \u2192 d boxedval \u2192 ok d \u0394\n\n\n  progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             ok d \u0394\n  -- constants\n  progress TAConst = V (BVVal VConst)\n\n  -- variables\n  progress (TAVar x\u2081) = abort (somenotnone (! x\u2081))\n\n  -- lambdas\n  progress (TALam wt) = V (BVVal VLam)\n\n  -- applications\n  progress (TAAp wt1 wt2)\n    with progress wt1 | progress wt2\n  progress (TAAp wt1 wt2) | S x | ih2 = {!!}\n  progress (TAAp wt1 wt2) | E x | _ = E (CECong (FHAp1 FHOuter) x)\n  progress (TAAp wt1 wt2) | I x | ih2 = {!!}\n  progress (TAAp wt1 wt2) | V x | ih2 = {!!}\n\n  -- empty holes\n  progress (TAEHole x x\u2081) = I IEHole\n\n  -- nonempty holes\n  progress (TANEHole xin wt x\u2081)\n    with progress wt\n  ... | S (d' , Step x y z) = S (_ , Step {!!} {!!} {!!})\n  ... | E x = E (CECong (FHNEHole FHOuter) x)\n  ... | I x = I (INEHole (FIndet x))\n  ... | V x = I (INEHole (FBoxed x))\n\n  -- casts\n  progress (TACast wt y)\n    with progress wt\n  ... | S x = {!!}\n  ... | E x = E (CECong (FHCast FHOuter) x)\n  ... | I x = {!x!}\n  ... | V x = {!!}\n\n\n  -- -- applications\n  -- progress (TAAp D1 x D2)\n  --   with progress D1 | progress D2\n  --   -- left applicand value\n  -- progress (TAAp TAConst () D2) | V VConst | _\n  -- progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | V x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FVal x\u2081)) {!!} )\n  -- progress (TAAp {d2 = d2} D1 MAArr D2) | V (VLam {x = x} {d = d}) | I x\u2081 = S ([ d2 \/ x ] d , Step (FHAp1 (FVal VLam) {!!}) (ITLam (FIndet x\u2081)) {!!} )\n  --   -- errors propagate\n  -- progress (TAAp D1 x\u2082 D2) | V x | E x\u2081 = E (EAp2 (FVal x) x\u2081)\n  -- progress (TAAp D1 x\u2082 D2) | I x | E x\u2081 = E (EAp2 (FIndet x) x\u2081)\n  -- progress (TAAp D1 x\u2082 D2) | E x | E x\u2081 = E (EAp1 x) -- NB: could have picked the other one here, too; this is a source of non-determinism\n  -- progress (TAAp D1 x\u2082 D2) | S x | E x\u2081 = S {!!}\n  -- progress (TAAp D1 x\u2082 D2) | E x | _    = E (EAp1 x)\n  --   -- indeterminates\n  -- progress (TAAp D1 x\u2082 D2) | I i | V v = I (IAp i  (FVal v))\n  -- progress (TAAp D1 x\u2082 D2) | I x | I x\u2081 = I (IAp x (FIndet x\u2081))\n  --   -- either applicand steps\n  -- progress (TAAp D1 x\u2083 D2) | S (d1' , Step x x\u2081 x\u2082)  | _ = S (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082))\n  -- progress (TAAp D1 x\u2084 D2) | _ | S (d2' , Step x\u2081 x\u2082 x\u2083) = S (_ , Step (FHAp1 {!!} x\u2081) x\u2082 (FHAp1 {!!} x\u2083))\n\n\n  -- -- non-empty holes\n  -- progress (TANEHole x D x\u2081)\n  --   with progress D\n  -- progress (TANEHole x\u2081 D x\u2082) | V v = I (INEHole (FVal v))\n  -- progress (TANEHole x\u2081 D x\u2082) | I x = I (INEHole (FIndet x))\n  -- progress (TANEHole x\u2081 D x\u2082) | E x = E (ENEHole x)\n  -- progress (TANEHole x\u2083 D x\u2084) | S (_ , Step x x\u2081 x\u2082) = S (_ , Step (FHNEHole x) x\u2081 (FHNEHole x\u2082))\n\n  -- -- casts\n  -- progress (TACast D x)\n  --   with progress D\n  -- progress (TACast TAConst con) | V VConst = S (c , Step (FHCastFinal (FVal VConst)) (ITCast (FVal VConst) TAConst con) (FHFinal (FVal VConst)))\n  -- progress (TACast D m)  | V VLam = S (_ , Step (FHCastFinal (FVal VLam)) (ITCast (FVal VLam) D m) (FHFinal (FVal VLam)))\n  -- progress (TACast D x\u2081) | I x = I (ICast x)\n  -- progress (TACast D x\u2081) | E x = E (ECastProp x)\n  -- progress (TACast D x\u2083) | S (d , Step x x\u2081 x\u2082) = S (_ , Step (FHCast x) x\u2081 (FHCast x\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a52361daab67d1986ef5f3449d90d72bccb0c16a","subject":"ZK.GroupHom: can swap transcripts to refine their order, also make ^-\u2212\u1d9c ^-^-1\/ more precise","message":"ZK.GroupHom: can swap transcripts to refine their order, also make ^-\u2212\u1d9c ^-^-1\/ more precise\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom.agda","new_file":"ZK\/GroupHom.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function using (flip; case_of_)\nopen import Data.Sum.NP\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nimport ZK.SigmaProtocol.KnownStatement\n\nmodule ZK.GroupHom\n    {G+ G* : Type} (grp-G+ : Group G+) (grp-G* : Group G*)\n    (_==_ : G* \u2192 G* \u2192 Bool)\n    (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n    (==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y)\n\n    (open Additive-Group grp-G+ hiding (_\u2297_))\n    (open Multiplicative-Group grp-G* hiding (_^_))\n\n    {C : Type} -- e.g. C \u2286 \u2115\n    (_>_ : C \u2192 C \u2192 Type)\n    (swap? : {c\u2080 c\u2081 : C} \u2192 c\u2080 \u2262 c\u2081 \u2192 (c\u2080 > c\u2081) \u228e (c\u2081 > c\u2080))\n    (_\u2297_ : G+ \u2192 C \u2192 G+)\n    (_^_ : G* \u2192 C \u2192 G*)\n    (_\u2212\u1d9c_ : C \u2192 C \u2192 C)\n    (1\/ : C \u2192 C)\n\n    (\u03c6 : G+ \u2192 G*) -- For instance \u03c6(x) = g^x\n    (\u03c6-hom : GroupHomomorphism grp-G+ grp-G* \u03c6)\n    (\u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297 c) \u2261 \u03c6 x ^ c)\n\n    (Y : G*)\n\n    (^-\u2212\u1d9c  : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 Y ^(c\u2080 \u2212\u1d9c c\u2081) \u2261 (Y ^ c\u2080) \/ (Y ^ c\u2081))\n    (^-^-1\/ : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 (Y ^(c\u2080 \u2212\u1d9c c\u2081))^(1\/(c\u2080 \u2212\u1d9c c\u2081)) \u2261 Y)\n    where\n\n  Commitment = G*\n  Challenge  = C\n  Response   = G+\n  Witness    = G+\n  Randomness = G+\n  _\u2208y : Witness \u2192 Type\n  x \u2208y = \u03c6 x \u2261 Y\n\n  open ZK.SigmaProtocol.KnownStatement Commitment Challenge Response Randomness Witness _\u2208y public\n\n  prover : Prover\n  prover a x = \u03c6 a , response\n     where response : Challenge \u2192 Response\n           response c = (x \u2297 c) + a\n\n  verifier : Verifier\n  verifier (mk A c s) = \u03c6 s == ((Y ^ c) * A)\n\n  -- We still call it Schnorr since essentially not much changed.\n  -- The scheme abstracts away g^ as an homomorphism called \u03c6\n  -- and one get to distinguish between the types C vs G+ and\n  -- their operations _\u2297_ vs _*_.\n  Schnorr : \u03a3-Protocol\n  Schnorr = (prover , verifier)\n\n  -- The simulator shows why it is so important for the\n  -- challenge to be picked once the commitment is known.\n  -- To fake a transcript, the challenge and response can\n  -- be arbitrarily chosen. However to be indistinguishable\n  -- from a valid proof they need to be picked at random.\n  simulator : Simulator\n  simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (Y ^ c)\u207b\u00b9 * \u03c6 s\n\n  swap-t\u00b2? : Transcript\u00b2[ _\u2262_ ] verifier \u2192 Transcript\u00b2[ _>_ ] verifier\n  swap-t\u00b2? t\u00b2 = t\u00b2'\n      module Swap? where\n        open Transcript\u00b2 t\u00b2 public\n          using (verify\u2080; verify\u2081; c\u2080\u2262c\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   )\n        t\u00b2' = case swap? c\u2080\u2262c\u2081 of \u03bb\n              { (inl c\u2080>c\u2081) \u2192 mk A c\u2080 c\u2081 c\u2080>c\u2081 r\u2080 r\u2081 verify\u2080 verify\u2081\n              ; (inr c\u2080<c\u2081) \u2192 mk A c\u2081 c\u2080 c\u2080<c\u2081 r\u2081 r\u2080 verify\u2081 verify\u2080\n              }\n\n  -- The extractor shows the importance of never reusing a\n  -- commitment. If the prover answers to two different challenges\n  -- comming from the same commitment then the knowledge of the\n  -- prover (the witness) can be extracted.\n  extractor : Extractor verifier\n  extractor t\u00b2 = x'\n      module Witness-extractor where\n        open Transcript\u00b2 (swap-t\u00b2? t\u00b2) public\n          using (verify\u2080; verify\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   ; c\u2080\u2262c\u2081 to c\u2080>c\u2081\n                   )\n        rd   = r\u2080 \u2212 r\u2081\n        cd   = c\u2080 \u2212\u1d9c c\u2081\n        1\/cd = 1\/ cd\n        x'   = rd \u2297 1\/cd\n\n  open \u2261-Reasoning\n  open GroupHomomorphism \u03c6-hom\n\n  correct : Correct Schnorr\n  correct x a c w\n    = \u2713-== (\u03c6((x \u2297 c) + a)  \u2261\u27e8 hom \u27e9\n            \u03c6(x \u2297 c)  * A   \u2261\u27e8 *= \u03c6-hom-iterated idp \u27e9\n            (\u03c6 x ^ c) * A   \u2261\u27e8 ap (\u03bb z \u2192 (z ^ c) * A) w \u27e9\n            (Y ^ c)   * A   \u220e)\n    where\n      A = \u03c6 a\n\n  shvzk : Special-Honest-Verifier-Zero-Knowledge Schnorr\n  shvzk = record { simulator = simulator\n                 ; correct-simulator =\n                    \u03bb _ _ \u2192 \u2713-== (! elim-*-!assoc= (snd \u207b\u00b9-inverse)) }\n\n  module _ (t\u00b2 : Transcript\u00b2 verifier) where\n    open Witness-extractor t\u00b2\n\n    \u03c6rd\u2261ycd : _\n    \u03c6rd\u2261ycd\n      = \u03c6 rd                        \u2261\u27e8by-definition\u27e9\n        \u03c6 (r\u2080 \u2212 r\u2081)                 \u2261\u27e8 \u2212-\/ \u27e9\n        \u03c6 r\u2080 \/ \u03c6 r\u2081                 \u2261\u27e8 \/= (==-\u2713 verify\u2080) (==-\u2713 verify\u2081) \u27e9\n        (Y ^ c\u2080 * A) \/ (Y ^ c\u2081 * A) \u2261\u27e8 elim-*-right-\/ \u27e9\n        Y ^ c\u2080 \/ Y ^ c\u2081             \u2261\u27e8 ! ^-\u2212\u1d9c c\u2080>c\u2081 \u27e9\n        Y ^(c\u2080 \u2212\u1d9c c\u2081)               \u2261\u27e8by-definition\u27e9\n        Y ^ cd                      \u220e\n\n    -- The extracted x is correct\n    extractor-ok : \u03c6 x' \u2261 Y\n    extractor-ok\n      = \u03c6(rd \u2297 1\/cd)    \u2261\u27e8 \u03c6-hom-iterated \u27e9\n        \u03c6 rd ^ 1\/cd     \u2261\u27e8 ap\u2082 _^_ \u03c6rd\u2261ycd idp \u27e9\n        (Y ^ cd)^ 1\/cd  \u2261\u27e8 ^-^-1\/ c\u2080>c\u2081 \u27e9\n        Y               \u220e\n\n  special-soundness : Special-Soundness Schnorr\n  special-soundness = record { extractor = extractor\n                             ; extract-valid-witness = extractor-ok }\n\n  special-\u03a3-protocol : Special-\u03a3-Protocol\n  special-\u03a3-protocol = record { \u03a3-protocol = Schnorr ; correct = correct ; shvzk = shvzk ; ssound = special-soundness }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function using (flip)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nimport ZK.SigmaProtocol.KnownStatement\n\nmodule ZK.GroupHom\n    {G+ G* : Type} (grp-G+ : Group G+) (grp-G* : Group G*)\n    (_==_ : G* \u2192 G* \u2192 Bool)\n    (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n    (==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y)\n\n    (open Additive-Group grp-G+ hiding (_\u2297_))\n    (open Multiplicative-Group grp-G* hiding (_^_))\n\n    {C D : Type} -- e.g. C \u2286 \u2115, D = C\/0\n    (_\u2297_ : G+ \u2192 C \u2192 G+)\n    (_^_ : G* \u2192 C \u2192 G*)\n    (_\u2212\u1d9c_ : C \u2192 C \u2192 D)\n    (1* 1\/ : D \u2192 C) -- Two ways to go from D to C, 1*d embeds D in C, 1\/d does the inverse operation\n    (^-\u2212\u1d9c  : \u2200 {b x y}(x\u2262y : x \u2262 y) \u2192 b ^ 1*(x \u2212\u1d9c y) \u2261 (b ^ x) \/ (b ^ y))\n    (^-^-1\/ : \u2200 {b x y}(x\u2262y : x \u2262 y) \u2192 (b ^ 1*(x \u2212\u1d9c y))^(1\/(x \u2212\u1d9c y)) \u2261 b)\n\n    (\u03c6 : G+ \u2192 G*) -- For instance \u03c6(x) = g^x\n    (\u03c6-hom : GroupHomomorphism grp-G+ grp-G* \u03c6)\n    (\u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297 c) \u2261 \u03c6 x ^ c)\n\n    (y : G*)\n    where\n\n  Commitment = G*\n  Challenge  = C\n  Response   = G+\n  Witness    = G+\n  Randomness = G+\n  _\u2208y : Witness \u2192 Type\n  x \u2208y = \u03c6 x \u2261 y\n\n  open ZK.SigmaProtocol.KnownStatement Commitment Challenge Response Randomness Witness _\u2208y public\n\n  prover : Prover\n  prover a x = \u03c6 a , response\n     where response : Challenge \u2192 Response\n           response c = (x \u2297 c) + a\n\n  verifier : Verifier\n  verifier (mk A c s) = \u03c6 s == ((y ^ c) * A)\n\n  -- We still call it Schnorr since essentially not much changed.\n  -- The scheme abstracts away g^ as an homomorphism called \u03c6\n  -- and one get to distinguish between the types C vs G+ and\n  -- their operations _\u2297_ vs _*_.\n  Schnorr : \u03a3-Protocol\n  Schnorr = (prover , verifier)\n\n  -- The simulator shows why it is so important for the\n  -- challenge to be picked once the commitment is known.\n  -- To fake a transcript, the challenge and response can\n  -- be arbitrarily chosen. However to be indistinguishable\n  -- from a valid proof they need to be picked at random.\n  simulator : Simulator\n  simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (y ^ c)\u207b\u00b9 * \u03c6 s\n\n  -- The extractor shows the importance of never reusing a\n  -- commitment. If the prover answers to two different challenges\n  -- comming from the same commitment then the knowledge of the\n  -- prover (the witness) can be extracted.\n  extractor : Extractor verifier\n  extractor t\u00b2 = x'\n      module Witness-extractor where\n        open Transcript\u00b2 t\u00b2 public\n          using (verify\u2080; verify\u2081; c\u2080\u2262c\u2081)\n          renaming ( get-A to A\n                   ; get-f\u2080 to r\u2080; get-f\u2081 to r\u2081\n                   ; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                   )\n        rd   = r\u2080 \u2212 r\u2081\n        cd   = c\u2080 \u2212\u1d9c c\u2081\n        1\/cd = 1\/ cd\n        x'   = rd \u2297 1\/cd\n\n  open \u2261-Reasoning\n  open GroupHomomorphism \u03c6-hom\n\n  correct : Correct Schnorr\n  correct x a c w\n    = \u2713-== (\u03c6((x \u2297 c) + a)  \u2261\u27e8 hom \u27e9\n            \u03c6(x \u2297 c)  * A   \u2261\u27e8 *= \u03c6-hom-iterated idp \u27e9\n            (\u03c6 x ^ c) * A   \u2261\u27e8 ap (\u03bb z \u2192 (z ^ c) * A) w \u27e9\n            (y ^ c)   * A   \u220e)\n    where\n      A = \u03c6 a\n\n  shvzk : Special-Honest-Verifier-Zero-Knowledge Schnorr\n  shvzk = record { simulator = simulator\n                 ; correct-simulator =\n                    \u03bb _ _ \u2192 \u2713-== (! elim-*-!assoc= (snd \u207b\u00b9-inverse)) }\n\n  module _ (t\u00b2 : Transcript\u00b2 verifier) where\n    open Witness-extractor t\u00b2\n\n    \u03c6rd\u2261ycd : _\n    \u03c6rd\u2261ycd\n      = \u03c6 rd                        \u2261\u27e8by-definition\u27e9\n        \u03c6 (r\u2080 \u2212 r\u2081)                 \u2261\u27e8 \u2212-\/ \u27e9\n        \u03c6 r\u2080 \/ \u03c6 r\u2081                 \u2261\u27e8 \/= (==-\u2713 verify\u2080) (==-\u2713 verify\u2081) \u27e9\n        (y ^ c\u2080 * A) \/ (y ^ c\u2081 * A) \u2261\u27e8 elim-*-right-\/ \u27e9\n        y ^ c\u2080 \/ y ^ c\u2081             \u2261\u27e8 ! ^-\u2212\u1d9c c\u2080\u2262c\u2081 \u27e9\n        y ^ 1*(c\u2080 \u2212\u1d9c c\u2081)            \u2261\u27e8by-definition\u27e9\n        y ^ 1* cd                   \u220e\n\n    -- The extracted x is correct\n    extractor-ok : \u03c6 x' \u2261 y\n    extractor-ok\n      = \u03c6(rd \u2297 1\/cd)       \u2261\u27e8 \u03c6-hom-iterated \u27e9\n        \u03c6 rd ^ 1\/cd        \u2261\u27e8 ap\u2082 _^_ \u03c6rd\u2261ycd idp \u27e9\n        (y ^ 1* cd)^ 1\/cd  \u2261\u27e8 ^-^-1\/ c\u2080\u2262c\u2081 \u27e9\n        y                  \u220e\n\n  special-soundness : Special-Soundness Schnorr\n  special-soundness = record { extractor = extractor\n                             ; extract-valid-witness = extractor-ok }\n\n  special-\u03a3-protocol : Special-\u03a3-Protocol\n  special-\u03a3-protocol = record { \u03a3-protocol = Schnorr ; correct = correct ; shvzk = shvzk ; ssound = special-soundness }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a4fc3d6f4972c9d415904434ecf9b9490c536205","subject":"Removed note on induction principles for N.","message":"Removed note on induction principles for N.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Induction\/AdditionalHypothesis\/PropertiesByInductionI.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Induction\/AdditionalHypothesis\/PropertiesByInductionI.agda","new_contents":"","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties: Testing the induction principle\n------------------------------------------------------------------------------\n\nmodule FOT.FOTC.Data.Nat.Induction.AdditionalHypothesis.PropertiesByInductionI\n  where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat hiding ( N-ind )\n\n------------------------------------------------------------------------------\n-- Induction principle with the additional hypothesis.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- Induction principle without the additional hypothesis.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n        A zero \u2192\n        (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n        \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n------------------------------------------------------------------------------\n\npred-N\u2081 : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N\u2081 = N-ind\u2081 A A0 is\n  where\n  A : D \u2192 Set\n  A i = N (pred\u2081 i)\n\n  A0 : A zero\n  A0 = subst N (sym pred-0) nzero\n\n  is : \u2200 {i} \u2192 N i \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ni ih = subst N (sym (pred-S i)) Ni\n\n-- It is not possible to prove pred-N using N-ind\u2082.\n-- pred-N\u2082 : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n-- pred-N\u2082 = N-ind\u2082 A A0 is\n--   where\n--   A : D \u2192 Set\n--   A i = N (pred\u2081 i)\n\n--   A0 : A zero\n--   A0 = subst N (sym pred-0) nzero\n\n--   -- We need the hypothesis N i.\n--   is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n--   is {i} ih = subst N (sym (pred-S i)) {!!}\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 {n = n} Nm Nn = N-ind\u2081 A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  A0 : A zero\n  A0 = subst N (sym (+-0x n)) Nn\n\n  is : \u2200 {i} \u2192 N i \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} _ ih = subst N (sym (+-Sx i n)) (nsucc ih)\n\n+-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2082 {n = n} Nm Nn = N-ind\u2082 A A0 is Nm\n  where\n  A : D \u2192 Set\n  A i = N (i + n)\n\n  A0 : A zero\n  A0 = subst N (sym (+-0x n)) Nn\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} ih = subst N (sym (+-Sx i n)) (nsucc ih)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"64e72cbed8cdc8987dc660f59aaf1d361cf907dd","subject":"ClientServer: some renamings","message":"ClientServer: some renamings\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/ClientServer.agda","new_file":"Control\/Protocol\/ClientServer.agda","new_contents":"open import Type\nopen import Function.NP\nopen import Data.Nat\nopen import Data.Product.NP using (_,_)\n\nopen import Control.Protocol.Core\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.ClientServer (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = send \u03bb (q : Query) \u2192 recv \u03bb (r : Resp q) \u2192 end\n    ServerRound = dual ClientRound\n\n    ClientRounds ServerRounds : \u2115 \u2192 Proto\n    ClientRounds n = replicate\u1d3e n ClientRound\n    ServerRounds = dual \u2218 ClientRounds\n\n    Server Client : Proto\n    Client = send \u03bb n \u2192\n             ClientRounds n\n    Server = recv \u03bb n \u2192\n             ServerRounds n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u27e6 Server \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n","old_contents":"open import Type\nopen import Function.NP\nopen import Data.Nat\nopen import Data.Product.NP using (_,_)\n\nopen import Control.Protocol.Core\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.ClientServer (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = send \u03bb (q : Query) \u2192 recv \u03bb (r : Resp q) \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = recv \u03bb n \u2192\n                    Server n\n    StaticServer  = send \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3c679447abcf2db5d97d1de201980955ca090c8a","subject":"Added theorem 0^0\u22611.","message":"Added theorem 0^0\u22611.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Pow.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Pow.agda","new_contents":"------------------------------------------------------------------------------\n-- Some proofs related to the power function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.Nat.Pow where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\ninfixr 11 _^_\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192 n ^ zero      \u2261 1'\n  ^-S : \u2200 m n \u2192 m ^ succ\u2081 n \u2261 m * m ^ n\n{-# ATP axioms ^-0 ^-S #-}\n\npostulate 0^0\u22611 : 0' ^ 0' \u2261 1'\n{-# ATP prove 0^0\u22611 #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ 5' \u2264 5' ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn)\n  where\n  postulate prf : (5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n) \u2192 (succ\u2081 n) ^ 5' \u2264 5' ^ (succ\u2081 n)\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261 2' ^ (n + 1') \u2238 1'\nthm\u2082 nzero = prf\n  where\n  postulate prf : ((2' ^ zero) \u2238 1') + 1' + ((2' ^ zero) \u2238 1') \u2261\n                  2' ^ (zero + 1') \u2238 1'\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261\n                  2' ^ (n + 1') \u2238 1' \u2192\n                  ((2' ^ succ\u2081 n) \u2238 1') + 1' + ((2' ^ succ\u2081 n) \u2238 1') \u2261\n                  2' ^ (succ\u2081 n + 1') \u2238 1'\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Some proofs related to the power function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.Nat.Pow where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n------------------------------------------------------------------------------\n\ninfixr 11 _^_\n\npostulate\n  _^_ : D \u2192 D \u2192 D\n  ^-0 : \u2200 n \u2192 n ^ zero      \u2261 1'\n  ^-S : \u2200 m n \u2192 m ^ succ\u2081 n \u2261 m * m ^ n\n{-# ATP axioms ^-0 ^-S #-}\n\nthm\u2081 : \u2200 {n} \u2192 N n \u2192 5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n\nthm\u2081 nzero h = prf\n  where postulate prf : zero ^ 5' \u2264 5' ^ zero\n        {-# ATP prove prf #-}\nthm\u2081 (nsucc {n} Nn) h = prf (thm\u2081 Nn)\n  where\n  postulate prf : (5' \u2264 n \u2192 n ^ 5' \u2264 5' ^ n) \u2192 (succ\u2081 n) ^ 5' \u2264 5' ^ (succ\u2081 n)\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf 5-N #-}\n\nthm\u2082 : \u2200 {n} \u2192 N n \u2192\n       ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261 2' ^ (n + 1') \u2238 1'\nthm\u2082 nzero = prf\n  where\n  postulate prf : ((2' ^ zero) \u2238 1') + 1' + ((2' ^ zero) \u2238 1') \u2261\n                  2' ^ (zero + 1') \u2238 1'\n  {-# ATP prove prf #-}\nthm\u2082 (nsucc {n} Nn) = prf (thm\u2082 Nn)\n  where\n  postulate prf : ((2' ^ n) \u2238 1') + 1' + ((2' ^ n) \u2238 1') \u2261\n                  2' ^ (n + 1') \u2238 1' \u2192\n                  ((2' ^ succ\u2081 n) \u2238 1') + 1' + ((2' ^ succ\u2081 n) \u2238 1') \u2261\n                  2' ^ (succ\u2081 n + 1') \u2238 1'\n  -- 06 January 2016: The ATPs could not prove the theorem (240 sec).\n  -- {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5697a9e64ef64b963304fa3802a2e7ee1855dad6","subject":"pushing whitespace etc.","message":"pushing whitespace etc.\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081) with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {e = c} (ASubsume D x\u2081)                    | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = e \u00b7: x} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = X x} (ASubsume D x\u2081)                  | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x e} (ASubsume D x\u2081)               | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {e = \u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {e = e1 \u2218 e2} (ASubsume D x\u2081)              | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {e = \u2987\u2988[ x ]} (ASubsume _ _ )              | _ , _ , _  = _ , _ , _ , EAEHole\n    expandability-ana {e = \u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _  = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import htype-decidable\n\nmodule expandability where\n  mutual\n    expandability-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                          \u0393 \u22a2 e => \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ]\n                            (\u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394)\n    expandability-synth SConst = c , \u2205 , ESConst\n    expandability-synth (SAsc {\u03c4 = \u03c4} wt)\n      with expandability-ana wt\n    ... | d' , \u0394' , \u03c4' , D with htype-dec \u03c4 \u03c4'\n    ... | Inl _ = _ , _ , ESAsc2 D\n    ... | Inr x = _ , _ , ESAsc1 D x\n    expandability-synth (SVar {n = n} x) = _ , _ , ESVar x\n    expandability-synth (SAp wt1 MAHole wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42 , D2  = _ , _ , ESAp1 {!!} wt1 {!!} D2\n    expandability-synth (SAp wt1 (MAArr {\u03c42 = \u03c42}) wt2)\n      with expandability-synth wt1 | expandability-ana wt2\n    ... | d1 , \u03941 , D1\n        | d2 , \u03942 , \u03c42' , D2\n      with htype-dec \u03c42 \u03c42'\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42' , D2 | Inr neq  = _ , _ , ESAp2 {!!} {!D1!} {!!} neq\n    expandability-synth (SAp wt1 MAArr wt2) | d1 , \u03941 , D1 | d2 , \u03942 , \u03c42 , D2  | Inl refl = _ , _ , ESAp3 {!!} {!D1!} {!!}\n    expandability-synth SEHole = _ , _ , ESEHole\n    expandability-synth (SNEHole wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESNEHole wt'\n    expandability-synth (SLam x\u2081 wt) with expandability-synth wt\n    ... | d' , \u0394' , wt' = _ , _ , ESLam x\u2081 wt'\n\n    expandability-ana : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                         \u0393 \u22a2 e <= \u03c4 \u2192\n                          \u03a3[ d \u2208 dhexp ] \u03a3[ \u0394 \u2208 hctx ] \u03a3[ \u03c4' \u2208 htyp ]\n                            (\u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394)\n    expandability-ana {e = e} (ASubsume D x\u2081) with expandability-synth D\n    -- these cases just pass through, but we need to pattern match so we can prove things aren't holes\n    expandability-ana {\u0393} {c} (ASubsume D x\u2081)             | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {e \u00b7: x} (ASubsume D x\u2081)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {X x} (ASubsume D x\u2081)           | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {\u00b7\u03bb x e} (ASubsume D x\u2081)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    expandability-ana {\u0393} {\u00b7\u03bb x [ x\u2081 ] e} (ASubsume D x\u2082) | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2082\n    expandability-ana {\u0393} {e \u2218 e\u2081} (ASubsume D x\u2081)        | _ , _ , D' = _ , _ , _ , EASubsume (\u03bb _ ()) (\u03bb _ _ ()) D' x\u2081\n    -- the two holes are special-cased\n    expandability-ana {\u0393} {\u2987\u2988[ x ]} (ASubsume _ _ )       | _ , _ , _ = _ , _ , _ , EAEHole\n    expandability-ana {\u0393} {\u2987 e \u2988[ x ]} (ASubsume (SNEHole wt) _) | _ , _ , _ = _ , _ , _ , EANEHole (\u03c02( \u03c02 (expandability-synth wt)))\n    -- the lambda cases\n    expandability-ana (ALam x\u2081 MAHole wt) with expandability-ana wt\n    ... | _ , _ , _ , D' = _ , _ , _ , EALamHole x\u2081 D'\n    expandability-ana (ALam x\u2081 MAArr wt) with expandability-ana wt\n    ... | _ , _\u202f, _ , D' = _ , _ , _ , EALam x\u2081 D'\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7158eae7b05fe9e31d75d927f6857ba32e2a8f3d","subject":"knocking out some easy cases of expansion unicity, hoping to figure out more about whats wrong with the ap rules","message":"knocking out some easy cases of expansion unicity, hoping to figure out more about whats wrong with the ap rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expansion-unicity.agda","new_file":"expansion-unicity.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import correspondence\nopen import synth-unicity\n\nmodule expansion-unicity where\n  \u2987\u2988\u2260arr : \u2200{t1 t2} \u2192 t1 ==> t2 == \u2987\u2988 \u2192 \u22a5\n  \u2987\u2988\u2260arr ()\n\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , ih3\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp1 x\u2084 x\u2085 x\u2086 x\u2087)\n      with expansion-unicity-ana x\u2083 x\u2087 | expansion-unicity-ana x\u2082 x\u2086\n    ... | refl , refl , refl , refl | refl , refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp2 x\u2084 d5 x\u2085 x\u2086) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2081))\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp3 x\u2084 d5 x\u2085) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2081))\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp1 x\u2083 x\u2084 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp2 x\u2083 d6 x\u2084 x\u2085)\n      with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2084 x\u2081\n    ... | refl , refl , refl | refl , refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp3 x\u2083 d6 x\u2084)\n      with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2084 x\u2081\n    ...| refl , refl , refl | refl , refl , refl , refl  = abort (x\u2082 refl)\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp1 x\u2082 x\u2083 x\u2084 x\u2085) = abort (\u2987\u2988\u2260arr (synthunicity (correspondence-synth d5) x\u2083))\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp2 x\u2082 d6 x\u2083 x\u2084)\n      with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2083 x\u2081\n    ...| refl , refl , refl | refl , refl , refl , refl  = abort (x\u2084 refl)\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp3 x\u2082 d6 x\u2083)\n      with expansion-unicity-synth d5 d6 | expansion-unicity-ana x\u2083 x\u2081\n    ... | refl , refl , refl | refl , refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083)\n      with expansion-unicity-ana x x\u2082\n    ... | refl , refl , refl , refl = refl , refl , refl\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc2 x\u2082)\n      with expansion-unicity-ana x x\u2082\n    ... | refl , refl , refl , refl = refl , {!!} , refl -- should fail, can't show it though\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , refl , refl , refl = refl , {!!} , refl  -- ditto\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081)\n      with expansion-unicity-ana x x\u2081\n    ... | refl , refl , refl , refl = refl , refl , refl\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c41' \u03c42 \u03c42' : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n                          \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03c41' == \u03c42' \u00d7 \u03941 == \u03942\n    expansion-unicity-ana (EALam apt1 d1) (EALam apt2 d2) = {!expansion-unicity-ana d1 d2!} -- doesn't go because the contexts are different\n    expansion-unicity-ana (EALam apt1 d1) (EASubsume x\u2081 x\u2082 () x\u2084)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALam apt2 d2)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087)\n      with expansion-unicity-synth x\u2086 x\u2082\n    ... | refl , refl , refl = {!!} , refl , refl , refl\n    expansion-unicity-ana (EASubsume x x\u2081 ESEHole x\u2083) EAEHole = abort (x _ refl)\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084) = abort (x\u2081 _ _ refl)\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = abort (x _ refl)\n    expansion-unicity-ana EAEHole EAEHole = {!!} , refl , {!!} , {!!}\n    expansion-unicity-ana (EANEHole x) (EASubsume x\u2082 x\u2083 x\u2084 x\u2085) = abort (x\u2083 _ _ refl)\n    expansion-unicity-ana (EANEHole x) (EANEHole x\u2081) =  {!!}\n    expansion-unicity-ana (EALam x\u2081 x\u2082) (EALamHole x\u2083 y) = {!!} -- should be an abort\n    expansion-unicity-ana (EALamHole x\u2081 x\u2082) (EALam x\u2083 y) = {!!} -- should be same abort\n    expansion-unicity-ana (EALamHole x\u2081 x\u2082) (EALamHole x\u2083 y)\n      with expansion-unicity-ana x\u2082 y\n    ... | refl , refl , refl , refl = refl , refl , refl , refl\n    expansion-unicity-ana (EALamHole x\u2081 x\u2082) (EASubsume x\u2083 x\u2084 () x\u2086)\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 () x\u2084) (EALamHole x\u2085 y)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule expansion-unicity where\n  mutual\n    expansion-unicity-synth : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c41 ~> d1 \u22a3 \u03941 \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c42 ~> d2 \u22a3 \u03942 \u2192\n                            \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03941 == \u03942\n    expansion-unicity-synth ESConst ESConst = refl , refl , refl\n    expansion-unicity-synth (ESVar {\u0393 = \u0393} x\u2081) (ESVar x\u2082) = ctxunicity {\u0393 = \u0393} x\u2081 x\u2082 , refl , refl\n    expansion-unicity-synth (ESLam apt1 d1) (ESLam apt2 d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = ap1 _ ih1  , ap1 _ ih2 , {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp1 x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp2 x\u2084 d5 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp1 x x\u2081 x\u2082 x\u2083) (ESAp3 x\u2084 d5 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp1 x\u2083 x\u2084 x\u2085 x\u2086) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp2 x\u2083 d6 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp2 x d5 x\u2081 x\u2082) (ESAp3 x\u2083 d6 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp1 x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp2 x\u2082 d6 x\u2083 x\u2084) = {!!}\n    expansion-unicity-synth (ESAp3 x d5 x\u2081) (ESAp3 x\u2082 d6 x\u2083) = {!!}\n    expansion-unicity-synth ESEHole ESEHole = refl , refl , refl\n    expansion-unicity-synth (ESNEHole d1) (ESNEHole d2) with expansion-unicity-synth d1 d2\n    ... | ih1 , ih2 , ih3 = refl , ap1 _ ih2 , ap1 _ ih3\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc1 x\u2082 x\u2083) = {!!}\n    expansion-unicity-synth (ESAsc1 x x\u2081) (ESAsc2 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc1 x\u2081 x\u2082) = {!!}\n    expansion-unicity-synth (ESAsc2 x) (ESAsc2 x\u2081) = {!!}\n\n    expansion-unicity-ana : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c41' \u03c42 \u03c42' : htyp} {d1 d2 : dhexp} {\u03941 \u03942 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c41 ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n                          \u03c41 == \u03c42 \u00d7 d1 == d2 \u00d7 \u03c41' == \u03c42' \u00d7 \u03941 == \u03942\n    expansion-unicity-ana (EALam apt1 d1) (EALam apt2 d2) = {!!}\n    expansion-unicity-ana (EALam apt1 d1) (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) = {!!}\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) (EALam apt2 d2) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EASubsume x\u2084 x\u2085 x\u2086 x\u2087) = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) EAEHole = {!!}\n    expansion-unicity-ana (EASubsume x x\u2081 x\u2082 x\u2083) (EANEHole x\u2084) = {!!}\n    expansion-unicity-ana EAEHole (EASubsume x x\u2081 x\u2082 x\u2083) = {!!}\n    expansion-unicity-ana EAEHole EAEHole = {!!}\n    expansion-unicity-ana (EANEHole x) (EASubsume x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n    expansion-unicity-ana (EANEHole x) (EANEHole x\u2081) = {!!}\n    expansion-unicity-ana (EALam x\u2081 x\u2082) (EALamHole x\u2083 y) = {!!}\n    expansion-unicity-ana (EALamHole x\u2081 x\u2082) (EALam x\u2083 y) = {!!}\n    expansion-unicity-ana (EALamHole x\u2081 x\u2082) (EALamHole x\u2083 y) = {!!}\n    expansion-unicity-ana (EALamHole x\u2081 x\u2082) (EASubsume x\u2083 x\u2084 x\u2085 x\u2086) = {!!}\n    expansion-unicity-ana (EASubsume x\u2081 x\u2082 x\u2083 x\u2084) (EALamHole x\u2085 y) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ed945eff06441d7a203120cae9b1f383d552bf5d","subject":"Give examples of using the new generic Desc eliminator.","message":"Give examples of using the new generic Desc eliminator.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n\n  module GenericEliminator where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind2 \u22a4 \u2115D _\n      (case \u2115T _\n        ( (\u03bb n \u2192 n)\n        , (\u03bb m ih n \u2192 suc (ih n))\n        , tt\n        )\n      )\n      tt\n\n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind2 \u22a4 \u2115D _\n      (case \u2115T _\n        ( (\u03bb n \u2192 zero)\n        , (\u03bb m ih n \u2192 add n (ih n))\n        , tt\n        )\n      )\n      tt\n\n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = ind2 (\u2115 tt) (VecD A) _\n      (case VecT _\n        ( (\u03bb n ys \u2192 ys)\n        , (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n        , tt\n        )\n      )\n\n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind2 (\u2115 tt) (VecD (Vec A m)) _\n      (case VecT _\n        ( (nil A)\n        , (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n        , tt\n        )\n      )\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.PropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : (A : Set) (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq A .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j) X i = j \u2261 i\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El I D X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`End j) X P i q = \u22a4\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`Arg A B) X P i (a , b) = All I (B a) X P i b\nAll I (`RecFun A B D) X P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All I D X P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nUncurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl I D X = {i : I} \u2192 El I D X i \u2192 X i\n\nCurriedEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl I (`End i) X = X i\nCurriedEl I (`Rec j D) X = (x : X j) \u2192 CurriedEl I D X\nCurriedEl I (`Arg A B) X = (a : A) \u2192 CurriedEl I (B a) X\nCurriedEl I (`RecFun A B D) X = ((a : A) \u2192 X (B a)) \u2192 CurriedEl I D X\n\ncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : UncurriedEl I D X) \u2192 CurriedEl I D X\ncurryEl I (`End i) X cn = cn refl\ncurryEl I (`Rec i D) X cn = \u03bb x \u2192 curryEl I D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl I (`Arg A B) X cn = \u03bb a \u2192 curryEl I (B a) X (\u03bb xs \u2192 cn (a , xs))\ncurryEl I (`RecFun A B D) X cn = \u03bb f \u2192 curryEl I D X (\u03bb xs \u2192 cn (f , xs))\n\nuncurryEl : (I : Set) (D : Desc I) (X : ISet I)\n  (cn : CurriedEl I D X) \u2192 UncurriedEl I D X\nuncurryEl I (`End i) X cn refl = cn\nuncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs\nuncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs\nuncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs\n\ncon2 : (I : Set) (D : Desc I) \u2192 CurriedEl I D (\u03bc I D)\ncon2 I D = curryEl I D (\u03bc I D) con\n\n----------------------------------------------------------------------\n\nUncurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : {i : I} \u2192 El I D X i \u2192 X i)\n  \u2192 Set\nUncurriedAll I D X P cn =\n  (i : I) (xs : El I D X i) \u2192 All I D X P i xs \u2192 P i (cn xs)\n\nCurriedAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  \u2192 Set\nCurriedAll I (`End i) X P cn =\n  P i (cn refl)\nCurriedAll I (`Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedAll I (`Arg A B) X P cn =\n  (a : A) \u2192 CurriedAll I (B a) X P (\u03bb xs \u2192 cn (a , xs))\nCurriedAll I (`RecFun A B D) X P cn =\n  (f : (a : A) \u2192 X (B a)) (ihf : (a : A) \u2192 P (B a) (f a)) \u2192 CurriedAll I D X P (\u03bb xs \u2192 cn (f , xs))\n\ncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : UncurriedAll I D X P cn)\n  \u2192 CurriedAll I D X P cn\ncurryAll I (`End i) X P cn pf =\n  pf i refl tt\ncurryAll I (`Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryAll I D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryAll I (`Arg A B) X P cn pf =\n  \u03bb a \u2192 curryAll I (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\ncurryAll I (`RecFun A B D) X P cn pf =\n  \u03bb f ihf \u2192 curryAll I D X P (\u03bb xs \u2192 cn (f , xs)) (\u03bb i xs ihs \u2192 pf i (f , xs) (ihf , ihs))\n\nuncurryAll : (I : Set) (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl I D X)\n  (pf : CurriedAll I D X P cn)\n  \u2192 UncurriedAll I D X P cn\nuncurryAll I (`End .i) X P cn pf i refl tt =\n  pf\nuncurryAll I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryAll I (`Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryAll I (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\nuncurryAll I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =\n  uncurryAll I D X P (\u03bb ys \u2192 cn (f , ys)) (pf f ihf) i xs ihs\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : UncurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : UncurriedAll I D\u2081 (\u03bc I D\u2081) P con)\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`RecFun A B E) i (f , xs) = (\u03bb a \u2192 ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs\n\n----------------------------------------------------------------------\n\nind2 :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : CurriedAll I D (\u03bc I D) P con)\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\nind2 I D P pcon i x = ind I D P (uncurryAll I D (\u03bc I D) P con pcon) i x\n\n----------------------------------------------------------------------\n\nmodule Sugared where\n\n  data \u2115T : Set where `zero `suc : \u2115T\n  data VecT : Set where `nil `cons : VecT\n\n  \u2115D : Desc \u22a4\n  \u2115D = `Arg \u2115T \u03bb\n    { `zero \u2192 `End tt\n    ; `suc \u2192 `Rec tt (`End tt)\n    }\n\n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n\n  zero : \u2115 tt\n  zero = con (`zero , refl)\n\n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (`suc , n , refl)\n\n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg VecT \u03bb\n    { `nil  \u2192 `End zero\n    ; `cons \u2192 `Arg (\u2115 tt) \u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n))\n    }\n\n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n\n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (`nil , refl)\n\n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (`cons , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 n\n      ; tt (`suc , m , q) (ih , tt) n \u2192 suc (ih n)\n      }\n    )\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n    (\u03bb\n      { tt (`zero , q) tt n \u2192 zero\n      ; tt (`suc , m , q) (ih , tt) n \u2192 add n (ih n)\n      }\n    )\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) ih n ys \u2192 ys\n      ; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys \u2192 cons A (add m n) x (ih n ys)\n      }\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (\u03bb\n      { .(con (`zero , refl)) (`nil , refl) tt \u2192 nil A\n      ; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) \u2192 append A m xs (mult n m) ih\n      }\n    )\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u22a4 \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  zero2 : \u2115 tt\n  zero2 = con2 \u22a4 \u2115D here\n\n  suc2 : \u2115 tt \u2192 \u2115 tt\n  suc2 = con2 \u22a4 \u2115D (there here)\n\n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (\u2115 tt) (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n  nil2 : (A : Set) \u2192 Vec A zero\n  nil2 A = con2 (\u2115 tt) (VecD A) here\n\n  cons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons2 A = con2 (\u2115 tt) (VecD A) (there here)\n\n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) m)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (\u2115 tt) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u22a4 \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case \u2115T\n        (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u)\n               (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq \u22a4 tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n      tt\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons = ind (\u2115 tt) (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case VecT\n        (\u03bb t \u2192 (c : El (\u2115 tt) (VecC A t) (Vec A) n)\n               (ih : All (\u2115 tt) (VecD A) (Vec A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq (\u2115 tt) zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (\u2115 tt) (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d0bb4f7bbbfa2acbfe952a24f1f347e8b7bac989","subject":"Add more derived Field properties","message":"Add more derived Field properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  -0\u1da0 -1\u1da0 : A\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[_] = fold 0\u1da0 suc\u1da0\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 e = fold 1\u1da0 (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-= : \u2200 {x x'} \u2192 x \u2261 x' \u2192 0- x \u2261 0- x'\n  0-= = ap 0-_\n\n  -- Correct naming scheme?\n  +-*-distr : _*_ DistributesOver\u02b3 _+_\n  +-*-distr = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  *0-zero : RightZero 0\u1da0 _*_\n  *0-zero = +-right-cancel  (+= refl (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= refl 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero 0\u1da0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  -- name ?\n  *-0- : \u2200 {x y} \u2192 0- (x * y) \u2261 (0- x) * y\n  *-0- = +-right-cancel (0--inverse \u2219 ! 0*-zero \u2219 *= (! 0--inverse) refl \u2219 +-*-distr)\n\n  -1*-neg : \u2200 {x} \u2192 -1\u1da0 * x \u2261 0- x\n  -1*-neg = ! *-0- \u2219 0-= 1*-identity\n\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  noZeroDivisor : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 x * y \u2262 0\u1da0\n  noZeroDivisor nx ny x*y\u22610\u1da0 = ny (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse nx)\n                                   \u2219 *= refl *-comm \u2219 ! *-assoc\n                                   \u2219 *= (*-comm \u2219 x*y\u22610\u1da0) refl \u2219 0*-zero\n                                   )\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : Field A) where\n  open Field F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : Field-Ops B)(default : Decode-Term B) where\n    module G = Field-Ops G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_\ndata Tm (A : Set) : Set where\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\npattern  0' = lit (\u2124.+ 0)\npattern  1' = lit (\u2124.+ 1)\npattern  2' = lit (\u2124.+ 1)\npattern  3' = lit (\u2124.+ 1)\npattern -1' = lit \u2124.-[1+ 0 ]\npattern -2' = lit \u2124.-[1+ 1 ]\npattern -3' = lit \u2124.-[1+ 2 ]\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : Field F) where\n    open Field F-field renaming (0-_ to -_)\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = \u2124[ x ]\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  -0\u1da0 -1\u1da0 : A\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[_] = fold 0\u1da0 suc\u1da0\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 e = fold 1\u1da0 (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : Field A) where\n  open Field F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : Field-Ops B)(default : Decode-Term B) where\n    module G = Field-Ops G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_\ndata Tm (A : Set) : Set where\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\npattern  0' = lit (\u2124.+ 0)\npattern  1' = lit (\u2124.+ 1)\npattern  2' = lit (\u2124.+ 1)\npattern  3' = lit (\u2124.+ 1)\npattern -1' = lit \u2124.-[1+ 0 ]\npattern -2' = lit \u2124.-[1+ 1 ]\npattern -3' = lit \u2124.-[1+ 2 ]\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : Field F) where\n    open Field F-field renaming (0-_ to -_)\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = \u2124[ x ]\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1481e800b1453a7850b116c29fc35dc57c8aacca","subject":"Bits: style","message":"Bits: style\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : \u2200 {n} \u2192 Bij (1 + n)\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if0 f = `if `id f\n\n    `if1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : \u2200 {n} \u2192 Bij (2 + n)\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","old_contents":"module Data.Bits where\n\n-- cleanup\nimport Level\nopen import Category.Applicative.NP\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_==_) renaming (_<=_ to _\u2115<=_)\nopen import Data.Nat.Properties\nopen import Data.Nat.DivMod\nimport Data.Bool.NP as Bool\nopen Bool hiding (_==_; to\u2115)\nopen import Data.Bool.Properties using (not-involutive)\nopen import Data.Maybe.NP\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc; #_; inject\u2081; inject+; raise) renaming (_+_ to _+\u1da0_)\nimport Data.Vec.NP as V\nopen V hiding (rewire; rewireTbl; sum) renaming (map to vmap; swap to vswap)\nopen import Data.Vec.N-ary.NP\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; uncurry; proj\u2081; proj\u2082)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Nullary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\nopen import Algebra.FunctionProperties.NP\nimport Data.List.NP as L\nopen import Function.Bijection.SyntaxKit\n\nopen import Data.Bool.NP public using (_xor_; not; true; false; if_then_else_)\nopen V public using ([]; _\u2237_; head; tail; replicate; RewireTbl)\n\nBit : Set\nBit = Bool\n\nmodule Defs where\n  0b = false\n  1b = true\nmodule Patterns where\n  pattern 0b = false\n  pattern 1b = true\nopen Patterns\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n_\u2192\u1d47_ : \u2115 \u2192 \u2115 \u2192 Set\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n0\u207f : \u2200 {n} \u2192 Bits n\n0\u207f = replicate 0b\n\n-- Warning: 0\u207f {0} \u2261 1\u207f {0}\n1\u207f : \u2200 {n} \u2192 Bits n\n1\u207f = replicate 1b\n\n0\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n0\u2237 xs = 0b \u2237 xs\n\n-- can't we make these pattern aliases?\n1\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bits (suc n)\n1\u2237 xs = 1b \u2237 xs\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\n==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n==-comm [] [] = refl\n==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_<=_ : \u2200 {n} (xs ys : Bits n) \u2192 Bool\n[]        <= []        = 1b\n(1b \u2237 xs) <= (0b \u2237 ys) = 0b\n(0b \u2237 xs) <= (1b \u2237 ys) = 1b\n(_  \u2237 xs) <= (_  \u2237 ys) = xs <= ys\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\n-- Negate all bits, i.e. \"xor\"ing them by one.\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u207f\n\nvnot\u2218vnot\u2257id : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\nvnot\u2218vnot\u2257id [] = refl\nvnot\u2218vnot\u2257id (x \u2237 xs) rewrite not-involutive x | vnot\u2218vnot\u2257id xs = refl\n\n-- Negate the i-th bit.\nnot\u1d62 : \u2200 {n} (i : Fin n) \u2192 Bits n \u2192 Bits n\nnot\u1d62 = on\u1d62 not\n\n\u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n\u2295-assoc [] [] [] = refl\n\u2295-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs) rewrite \u2295-assoc xs ys zs | Xor\u00b0.+-assoc x y z = refl\n\n\u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n\u2295-comm [] [] = refl\n\u2295-comm (x \u2237 xs) (y \u2237 ys) rewrite \u2295-comm xs ys | Xor\u00b0.+-comm x y = refl\n\n\u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-left-identity [] = refl\n\u2295-left-identity (x \u2237 xs) rewrite \u2295-left-identity xs = refl\n\n\u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ 0\u207f (_\u2295_ {n})\n\u2295-right-identity [] = refl\n\u2295-right-identity (x \u2237 xs) rewrite \u2295-right-identity xs | proj\u2082 Xor\u00b0.+-identity x = refl\n\n\u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 0\u207f\n\u2295-\u2261 [] = refl\n\u2295-\u2261 (x \u2237 xs) rewrite \u2295-\u2261 xs | proj\u2082 Xor\u00b0.-\u203finverse x = refl\n\n\u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 1\u207f\n\u2295-\u2262 x = x \u2295 vnot x   \u2261\u27e8 refl \u27e9\n         x \u2295 (1\u207f \u2295 x) \u2261\u27e8 cong (_\u2295_ x) (\u2295-comm 1\u207f x) \u27e9\n         x \u2295 (x \u2295 1\u207f) \u2261\u27e8 sym (\u2295-assoc x x 1\u207f) \u27e9\n         (x \u2295 x) \u2295 1\u207f \u2261\u27e8 cong (flip _\u2295_ 1\u207f) (\u2295-\u2261 x) \u27e9\n         0\u207f \u2295 1\u207f       \u2261\u27e8 \u2295-left-identity 1\u207f \u27e9\n         1\u207f \u220e where open \u2261-Reasoning\n\n-- \"Xor\"ing the i-th bit with `b' is the same thing as \"xor\"ing with a vector of zeros\n-- except at the i-th position.\n-- Such a vector can be obtained by \"xor\"ing the i-th bit of a vector of zeros.\non\u1d62-xor-\u2295 : \u2200 b {n} (i : Fin n) \u2192 on\u1d62 (_xor_ b) i \u2257 _\u2295_ (on\u1d62 (_xor_ b) i 0\u207f)\non\u1d62-xor-\u2295 b zero    (x \u2237 xs) rewrite proj\u2082 Xor\u00b0.+-identity b | \u2295-left-identity xs = refl\non\u1d62-xor-\u2295 b (suc i) (x \u2237 xs) rewrite on\u1d62-xor-\u2295 b i xs = refl\n\nmsb : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k\nmsb = take\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\nallBitsL : \u2200 n \u2192 L.List (Bits n)\nallBitsL _ = replicateM (toList (0b \u2237 1b \u2237 []))\n  where open L.Monad\n\nallBits : \u2200 n \u2192 Vec (Bits n) (2^ n)\nallBits zero    = [] \u2237 []\nallBits (suc n) = vmap 0\u2237_ bs ++ vmap 1\u2237_ bs\n  where bs = allBits n\n\nalways : \u2200 n \u2192 Bits n \u2192 Bit\nalways _ _ = 1b\nnever  : \u2200 n \u2192 Bits n \u2192 Bit\nnever _ _ = 0b\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule Search {i} {I : Set i} (`1 : I) (`2*_ : I \u2192 I)\n              {a} {A : I \u2192 Set a} (_\u2219_ : \u2200 {m} \u2192 A m \u2192 A m \u2192 A (`2* m)) where\n\n  `2^_ : \u2115 \u2192 I\n  `2^_ = fold `1 `2*_\n\n  search : \u2200 {n} \u2192 (Bits n \u2192 A `1) \u2192 A (`2^ n)\n  search {zero}  f = f []\n  search {suc n} f = search (f \u2218 0\u2237_) \u2219 search (f \u2218 1\u2237_)\n\n  searchBit : (Bit \u2192 A `1) \u2192 A (`2* `1)\n  searchBit f = f 0b \u2219 f 1b\n\n  -- search-ext\n  search-\u2257 : \u2200 {n} (f g : Bits n \u2192 A `1) \u2192 f \u2257 g \u2192 search f \u2261 search g\n  search-\u2257 {zero}  f g f\u2257g = f\u2257g []\n  search-\u2257 {suc n} f g f\u2257g\n    rewrite search-\u2257 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u2257g \u2218 0\u2237_)\n          | search-\u2257 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u2257g \u2218 1\u2237_) = refl\n\n  module Comm (\u2219-comm : \u2200 {m} (x y : A m) \u2192 x \u2219 y \u2261 y \u2219 x) where\n\n    {- This pad bit vector allows to specify which bit do we negate in the vector. -}\n    search-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 A `1) \u2192 search f \u2261 search (f \u2218 _\u2295_ pad)\n    search-comm {zero} pad f = refl\n    search-comm {suc n} (b \u2237 pad) f\n      rewrite search-comm pad (f \u2218 0\u2237_)\n            | search-comm pad (f \u2218 1\u2237_)\n      with b\n    ... | true  = \u2219-comm (search (f \u2218 0\u2237_ \u2218 _\u2295_ pad)) _\n    ... | false = refl\n  open Comm public\n\nopen Search 1 2*_ public using () renaming (search to search\u2032; search-\u2257 to search\u2032-\u2257; search-comm to search\u2032-comm)\n\nmodule OperationSyntax where\n    module BitBij = BoolBijection\n    open BitBij public using (`id) renaming (BoolBij to BitBij; bool-bijKit to bitBijKit; `not to `not\u1d2e)\n    open BijectionSyntax Bit BitBij public\n    open BijectionSemantics bitBijKit public\n\n    {-\n       id   \u03b5          id\n       0\u21941  swp-inners interchange\n       not  swap       comm\n       if0  first      ...\n       if1  second     ...\n     -}\n    `not : \u2200 {n} \u2192 Bij (1 + n)\n    `not = BitBij.`not `\u2237 const `id\n\n    `xor : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor b = BitBij.`xor b `\u2237 const `id\n\n    `if : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    `if f g = BitBij.`id `\u2237 cond f g\n\n    `if0 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if0 f = `if `id f\n\n    `if1 : \u2200 {n} \u2192 Bij n \u2192 Bij (1 + n)\n    `if1 f = `if f `id\n\n    -- law: `if0 f `\u204f `if1 g \u2261 `if1 g `; `if0 f\n\n    on-firsts : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    on-firsts f = `0\u21941 `\u204f `if0 f `\u204f `0\u21941\n\n    --   (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap\n    --   (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange\n    --   (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 right swap\n    --   (a \u2219 d) \u2219 (c \u2219 b)\n    swp-seconds : \u2200 {n} \u2192 Bij (2 + n)\n    swp-seconds = `if1 `not `\u204f `0\u21941 `\u204f `if1 `not\n\n    on-extremes : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    -- on-extremes f = swp-seconds `\u204f `if0 f `\u204f swp-seconds\n\n    -- (a \u2219 b) \u2219 (c \u2219 d)\n    -- \u2261 right swap \u2261 if1 not\n    -- (a \u2219 b) \u2219 (d \u2219 c)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (a \u2219 d) \u2219 (b \u2219 c)\n    -- \u2261 left f \u2261 if0 f\n    -- (A \u2219 D) \u2219 (b \u2219 c)\n    --     where A \u2219 D = f (a \u2219 d)\n    -- \u2261 interchange \u2261 0\u21941\n    -- (A \u2219 b) \u2219 (D \u2219 c)\n    -- \u2261 right swap \u2261 if1 not\n    -- (A \u2219 b) \u2219 (c \u2219 D)\n    on-extremes f = `if1 `not `\u204f `0\u21941 `\u204f `if0 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-inner : \u2200 {n} \u2192 Bij (1 + n) \u2192 Bij (2 + n)\n    map-inner f = `if1 `not `\u204f `0\u21941 `\u204f `if1 f `\u204f `0\u21941 `\u204f `if1 `not\n\n    map-outer : \u2200 {n} \u2192 Bij n \u2192 Bij n \u2192 Bij (1 + n)\n    map-outer f g = `if g f\n\n    0\u21941\u2237_ : \u2200 {n} \u2192 Bits n \u2192 Bij (1 + n)\n    0\u21941\u2237 [] = `not\n    0\u21941\u2237 (true {-1-} \u2237 p) = on-extremes (0\u21941\u2237 p)\n    0\u21941\u2237 (false{-0-} \u2237 p) = on-firsts   (0\u21941\u2237 p)\n\n    0\u2194_ : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    0\u2194 [] = `id\n    0\u2194 (false{-0-} \u2237 p) = `if0 (0\u2194 p)\n    0\u2194 (true{-1-}  \u2237 p) = 0\u21941\u2237 p\n\n    \u27e80\u2194_\u27e9-sem : \u2200 {n} (p : Bits n) \u2192 Bits n \u2192 Bits n\n    \u27e80\u2194 p \u27e9-sem xs = if 0\u207f == xs then p else if p == xs then 0\u207f else xs\n\n    if\u2237 : \u2200 {n} a x (xs ys : Bits n) \u2192 (if a then (x \u2237 xs) else (x \u2237 ys)) \u2261 x \u2237 (if a then xs else ys)\n    if\u2237 true x xs ys = refl\n    if\u2237 false x xs ys = refl\n\n    if-not\u2237 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (false \u2237 xs) else (true \u2237 ys)) \u2261 (not a) \u2237 (if a then xs else ys)\n    if-not\u2237 true xs ys = refl\n    if-not\u2237 false xs ys = refl\n\n    if\u2237\u2032 : \u2200 {n} a (xs ys : Bits n) \u2192 (if a then (true \u2237 xs) else (false \u2237 ys)) \u2261 a \u2237 (if a then xs else ys)\n    if\u2237\u2032 true xs ys = refl\n    if\u2237\u2032 false xs ys = refl\n\n    \u27e80\u21941\u2237_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u21941\u2237 p \u2219 xs \u2261 \u27e80\u2194 (1\u2237 p) \u27e9-sem xs\n    \u27e80\u21941\u2237_\u27e9-spec [] (true \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec [] (false \u2237 []) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 true \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (true \u2237 false \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (true \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (true \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (1\u2237 xs)\n         with ps == xs\n    ... | true = refl\n    ... | false = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 true \u2237 xs) = refl\n    \u27e80\u21941\u2237_\u27e9-spec (false \u2237 ps) (false \u2237 false \u2237 xs)\n       rewrite \u27e80\u21941\u2237_\u27e9-spec ps (0\u2237 xs)\n         with 0\u207f == xs\n    ... | true = refl\n    ... | false = refl\n\n    \u27e80\u2194_\u27e9-spec : \u2200 {n} (p : Bits n) xs \u2192 0\u2194 p \u2219 xs \u2261 \u27e80\u2194 p \u27e9-sem xs\n    \u27e80\u2194_\u27e9-spec [] [] = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (true \u2237 xs) = refl\n    \u27e80\u2194_\u27e9-spec (false \u2237 ps) (false \u2237 xs)\n       rewrite \u27e80\u2194 ps \u27e9-spec xs\n             | if\u2237 (ps == xs) 0b 0\u207f xs\n             | if\u2237 (0\u207f == xs) 0b ps (if ps == xs then 0\u207f else xs)\n        = refl\n    \u27e80\u2194_\u27e9-spec (true \u2237 p) xs = \u27e80\u21941\u2237 p \u27e9-spec xs\n\n    private\n        module P where\n           open PermutationSyntax public\n           open PermutationSemantics public\n    open P using (Perm; `id; `0\u21941; _`\u204f_)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u21941+ i \u27e9 \u2219 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 _ \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Bij n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) xs \u2192 `\u27e80\u2194 i \u27e9 \u2219 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    {-\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Bij n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) xs \u2192 `\u27e8 i \u2194 j \u27e9 \u2219 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs) rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n    -}\n\n    `xor-head : \u2200 {n} \u2192 Bit \u2192 Bij (1 + n)\n    `xor-head b = if b then `not else `id\n\n    `xor-head-spec : \u2200 b {n} x (xs : Bits n) \u2192 `xor-head b \u2219 (x \u2237 xs) \u2261 (b xor x) \u2237 xs\n    `xor-head-spec true x xs  = refl\n    `xor-head-spec false x xs = refl\n\n    `\u27e8_\u2295\u27e9 : \u2200 {n} \u2192 Bits n \u2192 Bij n\n    `\u27e8 []     \u2295\u27e9 = `id\n    `\u27e8 b \u2237 xs \u2295\u27e9 = `xor-head b `\u204f `tl `\u27e8 xs \u2295\u27e9\n\n    `\u27e8_\u2295\u27e9-spec : \u2200 {n} (xs ys : Bits n) \u2192 `\u27e8 xs \u2295\u27e9 \u2219 ys \u2261 xs \u2295 ys\n    `\u27e8 []         \u2295\u27e9-spec []       = refl\n    `\u27e8 true  \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n    `\u27e8 false \u2237 xs \u2295\u27e9-spec (y \u2237 ys) rewrite `\u27e8 xs \u2295\u27e9-spec ys = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) x \u2192 0\u21941 pad \u2295 0\u21941 x \u2261 0\u21941 (pad \u2295 x)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\nmodule PermutationSyntax-Props where\n    open PermutationSyntax\n    open PermutationSemantics\n    -- open PermutationProperties\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u2219\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 fs      `id        xs = refl\n    \u2295-dist-\u2219 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u2219 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u2219 fs \u03c0 xs = refl\n    \u2295-dist-\u2219 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u2219 (\u03c0\u2080 \u2219 fs) \u03c0\u2081 (\u03c0\u2080 \u2219 xs)\n                                         | \u2295-dist-\u2219 fs \u03c0\u2080 xs = refl\n    {-\n -- \u229b-dist-\u2219 : \u2200 {n a} {A : Set a} (fs : Vec (A \u2192 A) n) \u03c0 xs \u2192 \u03c0 \u2219 fs \u229b \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (fs \u229b xs)\n    \u2295-dist-\u2219 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs \u2261 \u03c0 \u2219 (pad \u2295 xs)\n    \u2295-dist-\u2219 pad \u03c0 xs = \u03c0 \u2219 pad \u2295 \u03c0 \u2219 xs\n                      \u2261\u27e8 refl \u27e9\n                        vmap _xor_ (\u03c0 \u2219 pad) \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 TODO \u27e9\n                        \u03c0 \u2219 vmap _xor_ pad \u229b \u03c0 \u2219 xs\n                      \u2261\u27e8 \u229b-dist-\u2219 _ (vmap _xor_ pad) \u03c0 xs \u27e9\n                        \u03c0 \u2219 (vmap _xor_ pad \u229b xs)\n                      \u2261\u27e8 refl \u27e9\n                        \u03c0 \u2219 (pad \u2295 xs)\n                      \u220e where open \u2261-Reasoning\n     -- rans {!\u229b-dist-\u2219 (vmap _xor_ (op \u2219 pad)) op xs!} (\u229b-dist-\u2219 (vmap _xor_ pad) op xs)\n-}\nmodule SimpleSearch {a} {A : Set a} (_\u2219_ : A \u2192 A \u2192 A) where\n\n    open Search 1 2*_ {A = const A} _\u2219_ public\n\n    module SearchUnit \u03b5 (\u03b5\u2219\u03b5 : \u03b5 \u2219 \u03b5 \u2261 \u03b5) where\n        search-const\u03b5\u2261\u03b5 : \u2200 n \u2192 search {n = n} (const \u03b5) \u2261 \u03b5\n        search-const\u03b5\u2261\u03b5 zero = refl\n        search-const\u03b5\u2261\u03b5 (suc n) rewrite search-const\u03b5\u2261\u03b5 n = \u03b5\u2219\u03b5\n\n    searchBit-search : \u2200 n (f : Bits (suc n) \u2192 A) \u2192 searchBit (\u03bb b \u2192 search (f \u2218 _\u2237_ b)) \u2261 search f\n    searchBit-search n f = refl\n\n    search-\u2257\u2082 : \u2200 {m n} (f g : Bits m \u2192 Bits n \u2192 A) \u2192 f \u2257\u2082 g\n                  \u2192 search (search \u2218 f) \u2261 search (search \u2218 g)\n    search-\u2257\u2082 f g f\u2257g = search-\u2257 (search \u2218 f) (search \u2218 g) (\u03bb xs \u2192\n                          search-\u2257 (f xs) (g xs) (\u03bb ys \u2192\n                            f\u2257g xs ys))\n\n    search-+ : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                 search {m + n} f\n               \u2261 search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n    search-+ {zero} f = refl\n    search-+ {suc m} f rewrite search-+ {m} (f \u2218 0\u2237_)\n                             | search-+ {m} (f \u2218 1\u2237_) = refl\n\n    module SearchInterchange (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n\n        search-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 A) \u2192 search (\u03bb x \u2192 f\u2080 x \u2219 f\u2081 x) \u2261 search f\u2080 \u2219 search f\u2081\n        search-dist {zero}  _ _ = refl\n        search-dist {suc n} f\u2080 f\u2081\n          rewrite search-dist (f\u2080 \u2218 0\u2237_) (f\u2081 \u2218 0\u2237_)\n                | search-dist (f\u2080 \u2218 1\u2237_) (f\u2081 \u2218 1\u2237_)\n                = \u2219-interchange _ _ _ _\n\n        search-searchBit : \u2200 {n} (f : Bits (suc n) \u2192 A) \u2192\n                             search (\u03bb xs \u2192 searchBit (\u03bb b \u2192 f (b \u2237 xs))) \u2261 search f\n        search-searchBit f = search-dist (f \u2218 0\u2237_) (f \u2218 1\u2237_)\n\n        search-search : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192\n                          search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                        \u2261 search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n        search-search {zero} f = refl\n        search-search {suc m} {n} f\n          rewrite search-search {m} {n} (f \u2218 0\u2237_)\n                | search-search {m} {n} (f \u2218 1\u2237_)\n                | search-searchBit {n} (\u03bb { (b \u2237 ys) \u2192 search {m} (\u03bb xs \u2192 f (b \u2237 xs ++ ys)) })\n                = refl\n        {- -- It might also be done by using search-dist twice and commutativity of addition.\n           -- However, this also affect 'f' and makes this proof actually longer.\n           search-search {m} {n} f =\n                             search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {m + n} f\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n + m} (f \u2218 vswap n)\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                           \u2261\u27e8 {!!} \u27e9\n                             search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                           \u220e\n                           where open \u2261-Reasoning\n         -}\n\n        search-swap : \u2200 {m n} (f : Bits (m + n) \u2192 A) \u2192 search {n + m} (f \u2218 vswap n) \u2261 search {m + n} f\n        search-swap {m} {n} f =\n                     search {n + m} (f \u2218 vswap n)\n                   \u2261\u27e8 search-+ {n} {m} (f \u2218 vswap n) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (vswap n (ys ++ xs))))\n                   \u2261\u27e8 search-\u2257\u2082 {n} {m}\n                                (\u03bb ys \u2192 f \u2218 vswap n \u2218 _++_ ys)\n                                (\u03bb ys \u2192 f \u2218 flip _++_ ys)\n                                (\u03bb ys xs \u2192 cong f (swap-++ n ys xs)) \u27e9\n                     search {n} (\u03bb ys \u2192 search {m} (\u03bb xs \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-search {m} {n} f) \u27e9\n                     search {m} (\u03bb xs \u2192 search {n} (\u03bb ys \u2192 f (xs ++ ys)))\n                   \u2261\u27e8 sym (search-+ {m} {n} f) \u27e9\n                     search {m + n} f\n                   \u220e where open \u2261-Reasoning\n\n        search-0\u21941 : \u2200 {n} (f : Bits n \u2192 A) \u2192 search {n} (f \u2218 0\u21941) \u2261 search {n} f\n        search-0\u21941 {zero}        _ = refl\n        search-0\u21941 {suc zero}    _ = refl\n        search-0\u21941 {suc (suc n)} _ = \u2219-interchange _ _ _ _\n\n    module Bij (\u2219-comm : Commutative _\u2261_ _\u2219_)\n              (\u2219-interchange : Interchange _\u2261_ _\u2219_ _\u2219_) where\n        open SearchInterchange \u2219-interchange using (search-0\u21941)\n        open OperationSyntax hiding (_\u2219_)\n        search-bij : \u2200 {n} f (g : Bits n \u2192 A) \u2192 search (g \u2218 eval f) \u2261 search g\n        search-bij `id     _ = refl\n        search-bij `0\u21941    f = search-0\u21941 f\n        search-bij (f `\u204f g) h\n          rewrite search-bij f (h \u2218 eval g)\n                | search-bij g h\n                = refl\n        search-bij {suc n} (`id   `\u2237 f) g\n          rewrite search-bij (f 0b) (g \u2218 0\u2237_)\n                | search-bij (f 1b) (g \u2218 1\u2237_)\n                = refl\n        search-bij {suc n} (`not\u1d2e `\u2237 f) g\n          rewrite search-bij (f 1b) (g \u2218 0\u2237_)\n                | search-bij (f 0b) (g \u2218 1\u2237_)\n                = \u2219-comm _ _\n\nmodule Sum where\n    open SimpleSearch _+_ using (module Comm; module SearchInterchange; module SearchUnit; module Bij)\n    open SimpleSearch _+_ public using () renaming (search to sum; search-\u2257 to sum-\u2257; searchBit to sumBit;\n                                                    search-\u2257\u2082 to sum-\u2257\u2082;\n                                                    search-+ to sum-+)\n    open Comm \u2115\u00b0.+-comm public renaming (search-comm to sum-comm)\n    open SearchUnit 0 refl public renaming\n       (search-const\u03b5\u2261\u03b5 to sum-const0\u22610)\n    open SearchInterchange +-interchange public renaming (\n        search-dist to sum-dist;\n        search-searchBit to sum-sumBit;\n        search-search to sum-sum;\n        search-swap to sum-swap)\n    open Bij \u2115\u00b0.+-comm +-interchange public renaming (search-bij to sum-bij)\n\n    sum-const : \u2200 n x \u2192 sum {n} (const x) \u2261 \u27e82^ n * x \u27e9\n    sum-const zero    _ = refl\n    sum-const (suc n) x = cong\u2082 _+_ (sum-const n x) (sum-const n x)\n\n#\u27e8_\u27e9\u1da0 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 Fin (suc (2^ n))\n#\u27e8 pred \u27e9\u1da0 = count\u1da0 pred (allBits _)\n\nmodule Count where\n    open Sum\n    open OperationSyntax\n\n    #\u27e8_\u27e9 : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192 \u2115\n    #\u27e8 pred \u27e9 = sum (Bool.to\u2115 \u2218 pred)\n\n    -- #-ext\n    #-\u2257 : \u2200 {n} (f g : Bits n \u2192 Bool) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-\u2257 f g f\u2257g = sum-\u2257 (Bool.to\u2115 \u2218 f) (Bool.to\u2115 \u2218 g) (\u03bb x \u2192 \u2261.cong Bool.to\u2115 (f\u2257g x))\n\n    #-comm : \u2200 {n} (pad : Bits n) (f : Bits n \u2192 Bool) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ pad \u27e9\n    #-comm pad f = sum-comm pad (Bool.to\u2115 \u2218 f)\n\n    #-bij : \u2200 {n} f (g : Bits n \u2192 Bit) \u2192 #\u27e8 g \u2218 eval f \u27e9 \u2261 #\u27e8 g \u27e9\n    #-bij f g = sum-bij f (Bool.to\u2115 \u2218 g)\n\n    #-\u2295 : \u2200 {c} (bs : Bits c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 _\u2295_ bs \u27e9\n    #-\u2295 = #-comm\n\n    #-const : \u2200 n b \u2192 #\u27e8 (\u03bb (_ : Bits n) \u2192 b) \u27e9 \u2261 \u27e82^ n * Bool.to\u2115 b \u27e9\n    #-const n b = sum-const n (Bool.to\u2115 b)\n\n    #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n    #never\u22610 = sum-const0\u22610\n\n    #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n    #always\u22612^ n = sum-const n 1\n\n    #-dist : \u2200 {n} (f\u2080 f\u2081 : Bits n \u2192 Bit) \u2192 sum (\u03bb x \u2192 Bool.to\u2115 (f\u2080 x) + Bool.to\u2115 (f\u2081 x)) \u2261 #\u27e8 f\u2080 \u27e9 + #\u27e8 f\u2081 \u27e9\n    #-dist f\u2080 f\u2081 = sum-dist (Bool.to\u2115 \u2218 f\u2080) (Bool.to\u2115 \u2218 f\u2081)\n\n    #-+ : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                     #\u27e8 f \u27e9 \u2261 sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9 )\n    #-+ {m} {n} f = sum-+ {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-# : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192\n                          sum {m} (\u03bb xs \u2192 #\u27e8 (\u03bb ys \u2192 f (xs ++ ys)) \u27e9)\n                        \u2261 sum {n} (\u03bb ys \u2192 #\u27e8 (\u03bb (xs : Bits m) \u2192 f (xs ++ ys)) \u27e9)\n    #-# {m} {n} f = sum-sum {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #-swap : \u2200 {m n} (f : Bits (m + n) \u2192 Bit) \u2192 #\u27e8 f \u2218 vswap n {m} \u27e9 \u2261 #\u27e8 f \u27e9\n    #-swap {m} {n} f = sum-swap {m} {n} (Bool.to\u2115 \u2218 f)\n\n    #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n    #\u27e8== [] \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n    #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n    \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n    \u2257-cong-# f g f\u2257g = sum-\u2257 _ _ (cong Bool.to\u2115 \u2218 f\u2257g)\n\n    -- #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n    -- #-+ f f0 f1 rewrite f0 | f1 = refl\n\n    #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-take-drop zero n f g with f []\n    ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n    ... | false = #never\u22610 n\n    #-take-drop (suc m) n f g\n      rewrite \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                       ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x))\n            | #-take-drop m n (f \u2218 0\u2237_) g\n            | \u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                       ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                       (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x))\n            | #-take-drop m n (f \u2218 1\u2237_) g\n            = sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9)\n\n    #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                    \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n    #-drop-take m n f g =\n               #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n             \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n               #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n             \u2261\u27e8 #-take-drop m n g f \u27e9\n               #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n             \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n               #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n             \u220e\n      where open \u2261-Reasoning\n\n    #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n                 \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n    #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n                 \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                   #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n                 \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                   2^ n * #\u27e8 f \u27e9\n                 \u220e\n      where open \u2261-Reasoning\n\n    #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n    #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n    #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n    #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n    ... | true  | true  | _ = s\u2264s z\u2264n\n    ... | true  | false | p = \u22a5-elim (p _)\n    ... | false | _     | _ = z\u2264n\n    #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                    +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n    #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 Bool.to\u2115 (x \u2227 y) + Bool.to\u2115 (x \u2228 y) \u2261 Bool.to\u2115 x + Bool.to\u2115 y\n    #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (Bool.to\u2115 y) 1 = refl\n    #-\u2227-\u2228\u1d47 false y = refl\n\n    #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n    #-\u2227-\u2228 {suc n} f g =\n      trans\n        (trans\n           (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                   #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                   #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n           (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                      (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n    #\u2228' {zero} f g with f []\n    ... | true  = s\u2264s z\u2264n\n    ... | false = \u2115\u2264.refl\n    #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                                 #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                        (\u2115\u2264.reflexive\n                          (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n        where open SemiringSolver\n              helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n              helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n    #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n    #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n    #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n    #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n    #-bound : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2264 2^ c\n    #-bound zero    f = Bool.to\u2115\u22641 (f [])\n    #-bound (suc c) f = #-bound c (f \u2218 0\u2237_) +-mono #-bound c (f \u2218 1\u2237_)\n\n    #-\u2218vnot : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 vnot \u27e9\n    #-\u2218vnot _ f = #-\u2295 1\u207f f\n\n    #-\u2218xor\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) b \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n    #-\u2218xor\u1d62 i f b = pf\n      where pad = on\u1d62 (_xor_ b) i 0\u207f\n            pf : #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 on\u1d62 (_xor_ b) i \u27e9\n            pf rewrite #-\u2295 pad f = \u2257-cong-# (f \u2218 _\u2295_ pad) (f \u2218 on\u1d62 (_xor_ b) i) (cong (_$_ f) \u2218 sym \u2218 on\u1d62-xor-\u2295 b i)\n\n    #-\u2218not\u1d62 : \u2200 {c} (i : Fin c) (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 f \u2218 not\u1d62 i \u27e9\n    #-\u2218not\u1d62 i f = #-\u2218xor\u1d62 i f true\n\n    #-not\u2218 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 f \u27e9 \u2261 2^ c \u2238 #\u27e8 not \u2218 f \u27e9\n    #-not\u2218 zero f with f []\n    ... | true  = \u2261.refl\n    ... | false = \u2261.refl\n    #-not\u2218 (suc c) f\n      rewrite #-not\u2218 c (f \u2218 0\u2237_)\n            | #-not\u2218 c (f \u2218 1\u2237_) = factor-+-\u2238 (#-bound c (not \u2218 f \u2218 0\u2237_)) (#-bound c (not \u2218 f \u2218 1\u2237_))\n\n    #-not\u2218\u2032 : \u2200 c (f : Bits c \u2192 Bit) \u2192 #\u27e8 not \u2218 f \u27e9 \u2261 2^ c \u2238 #\u27e8 f \u27e9\n    #-not\u2218\u2032 c f = #\u27e8 not \u2218 f \u27e9\n               \u2261\u27e8 #-not\u2218 c (not \u2218 f) \u27e9\n                 2^ c \u2238 #\u27e8 not \u2218 not \u2218 f \u27e9\n               \u2261\u27e8 \u2261.cong (\u03bb g \u2192 2^ c \u2238 g) (\u2257-cong-# (not \u2218 not \u2218 f) f (not-involutive \u2218 f)) \u27e9\n                 2^ c \u2238 #\u27e8 f \u27e9\n               \u220e\n      where open \u2261-Reasoning\n\nopen SimpleSearch public\nopen Sum public\nopen Count public\n\n#\u27e8\u27e9-spec : \u2200 {n} (pred : Bits n \u2192 Bool) \u2192 #\u27e8 pred \u27e9 \u2261 Fin.to\u2115 #\u27e8 pred \u27e9\u1da0\n#\u27e8\u27e9-spec {zero}  pred with pred []\n... | true = refl\n... | false = refl\n#\u27e8\u27e9-spec {suc n} pred rewrite count-++ pred (vmap 0\u2237_ (allBits n)) (vmap 1\u2237_ (allBits n))\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 0\u2237_)\n                            | #\u27e8\u27e9-spec {n} (pred \u2218 1\u2237_)\n                            | count-\u2218 0\u2237_ pred (allBits n)\n                            | count-\u2218 1\u2237_ pred (allBits n) = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9\u1da0 \u2261 #\u27e8 g \u27e9\u1da0\next-# f\u2257g = ext-count\u1da0 f\u2257g (allBits _)\n\nfind? : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192? A) \u2192? A\nfind? = search (M?._\u2223_ _)\n\nfindB : \u2200 {n} \u2192 (Bits n \u2192 Bool) \u2192? Bits n\nfindB pred = find? (\u03bb x \u2192 if pred x then just x else nothing)\n\n|de-morgan| : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n|de-morgan| f g x with f x\n... | true = refl\n... | false = sym (not-involutive _)\n\nsearch-de-morgan : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192\n                   search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\nsearch-de-morgan op f g = search-\u2257 op _ _ (|de-morgan| f g)\n\nsearch-hom :\n  \u2200 {n a b}\n    {A : Set a} {B : Set b}\n    (_+_ : A \u2192 A \u2192 A)\n    (_*_ : B \u2192 B \u2192 B)\n    (f : A \u2192 B)\n    (p : Bits n \u2192 A)\n    (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n    \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\nsearch-hom {zero}  _   _   _ _ _   = refl\nsearch-hom {suc n} _+_ _*_ f p hom =\n   trans (hom _ _)\n         (cong\u2082 _*_ (search-hom _+_ _*_ f (p \u2218 0\u2237_) hom)\n                    (search-hom _+_ _*_ f (p \u2218 1\u2237_) hom))\n\nsucBCarry : \u2200 {n} \u2192 Bits n \u2192 Bits (1 + n)\nsucBCarry [] = 0b \u2237 []\nsucBCarry (0b \u2237 xs) = 0b \u2237 sucBCarry xs\nsucBCarry (1b \u2237 xs) with sucBCarry xs\n... | 0b \u2237 bs = 0b \u2237 1b \u2237 bs\n... | 1b \u2237 bs = 1b \u2237 0b \u2237 bs\n\nsucB : \u2200 {n} \u2192 Bits n \u2192 Bits n\nsucB = tail \u2218 sucBCarry\n\n_[mod_] : \u2115 \u2192 \u2115 \u2192 Set\na [mod b ] = DivMod' a b\n\nproj : \u2200 {a} {A : Set a} \u2192 A \u00d7 A \u2192 Bit \u2192 A\nproj (x\u2080 , x\u2081) 0b = x\u2080\nproj (x\u2080 , x\u2081) 1b = x\u2081\n\nrewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2192\u1d47 o\nrewire = V.rewire\n\nrewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 i \u2192\u1d47 o\nrewireTbl = V.rewireTbl\n\nmodule ReversedBits where\n  sucRB : \u2200 {n} \u2192 Bits n \u2192 Bits n\n  sucRB [] = []\n  sucRB (0b \u2237 xs) = 1b \u2237 xs\n  sucRB (1b \u2237 xs) = 0b \u2237 sucRB xs\n\ntoFin : \u2200 {n} \u2192 Bits n \u2192 Fin (2^ n)\ntoFin         []        = zero\ntoFin         (0b \u2237 xs) = inject+ _ (toFin xs)\ntoFin {suc n} (1b \u2237 xs) = raise (2^ n) (toFin xs)\n\n{-\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115 = Fin.to\u2115 \u2218 toFin\n-}\n\nto\u2115 : \u2200 {n} \u2192 Bits n \u2192 \u2115\nto\u2115         []        = zero\nto\u2115         (0b \u2237 xs) = to\u2115 xs\nto\u2115 {suc n} (1b \u2237 xs) = 2^ n + to\u2115 xs\n\nto\u2115-bound : \u2200 {n} (xs : Bits n) \u2192 to\u2115 xs < 2^ n \nto\u2115-bound         [] = s\u2264s z\u2264n\nto\u2115-bound {suc n} (1b \u2237 xs) rewrite +-assoc-comm 1 (2^ n) (to\u2115 xs) = \u2115\u2264.refl {2^ n} +-mono to\u2115-bound xs\nto\u2115-bound {suc n} (0b \u2237 xs) = \u2264-steps (2^ n) (to\u2115-bound xs)\n\nto\u2115\u22642\u207f+ : \u2200 {n} (x : Bits n) {y} \u2192 to\u2115 {n} x \u2264 2^ n + y\nto\u2115\u22642\u207f+ {n} x {y} = \u2115\u2264.trans (\u2264-steps y (\u2264-pred (\u2264-steps 1 (to\u2115-bound x))))\n                             (\u2115\u2264.reflexive (\u2115\u00b0.+-comm y (2^ n)))\n\n2\u207f+\u2270to\u2115 : \u2200 {n x} (y : Bits n) \u2192 2^ n + x \u2270 to\u2115 {n} y\n2\u207f+\u2270to\u2115 {n} {x} y p = \u00acn+\u2264y<n (2^ n) p (to\u2115-bound y)\n\nto\u2115-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2261 to\u2115 y \u2192 x \u2261 y\nto\u2115-inj         []        []        _ = refl\nto\u2115-inj         (0b \u2237 xs) (0b \u2237 ys) p = cong 0\u2237_ (to\u2115-inj xs ys p)\nto\u2115-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = cong 1\u2237_ (to\u2115-inj xs ys (cancel-+-left (2^ n) p))\nto\u2115-inj {suc n} (0b \u2237 xs) (1b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 xs (\u2115\u2264.reflexive (\u2261.sym p)))\nto\u2115-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115\u2264.reflexive p))\n\ndata _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n  []    : [] \u2264\u1d2e []\n  there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n  0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n\u2264\u1d2e\u2192<= : \u2200 {n} {p q : Bits n} \u2192 p \u2264\u1d2e q \u2192 T (p <= q)\n\u2264\u1d2e\u2192<= [] = _\n\u2264\u1d2e\u2192<= (there 0b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (there 1b pf) = \u2264\u1d2e\u2192<= pf\n\u2264\u1d2e\u2192<= (0-1 p q) = _\n\n<=\u2192\u2264\u1d2e : \u2200 {n} (p q : Bits n) \u2192 T (p <= q) \u2192 p \u2264\u1d2e q\n<=\u2192\u2264\u1d2e [] [] _ = []\n<=\u2192\u2264\u1d2e (1b \u2237 p) (0b \u2237 q) ()\n<=\u2192\u2264\u1d2e (0b \u2237 p) (1b \u2237 q) _  = 0-1 p q\n<=\u2192\u2264\u1d2e (0b \u2237 p) (0b \u2237 q) pf = there 0b (<=\u2192\u2264\u1d2e p q pf)\n<=\u2192\u2264\u1d2e (1b \u2237 p) (1b \u2237 q) pf = there 1b (<=\u2192\u2264\u1d2e p q pf)\n\nto\u2115-\u2264-inj : \u2200 {n} (x y : Bits n) \u2192 to\u2115 x \u2264 to\u2115 y \u2192 x \u2264\u1d2e y\nto\u2115-\u2264-inj     [] [] p = []\nto\u2115-\u2264-inj         (0b \u2237 xs) (0b \u2237 ys) p = there 0b (to\u2115-\u2264-inj xs ys p)\nto\u2115-\u2264-inj         (0b \u2237 xs) (1b \u2237 ys) p = 0-1 _ _\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (0b \u2237 ys) p = \u22a5-elim (2\u207f+\u2270to\u2115 ys p)\nto\u2115-\u2264-inj {suc n} (1b \u2237 xs) (1b \u2237 ys) p = there 1b (to\u2115-\u2264-inj xs ys (+-\u2264-inj (2^ n) p))\n\nfrom\u2115 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115 {zero}  _ = []\nfrom\u2115 {suc n} x = if 2^ n \u2115<= x then 1\u2237 from\u2115 (x \u2238 2^ n) else 0\u2237 from\u2115 x\n\nfrom\u2115\u2032 : \u2200 {n} \u2192 \u2115 \u2192 Bits n\nfrom\u2115\u2032 = fold 0\u207f sucB\n\nfromFin : \u2200 {n} \u2192 Fin (2^ n) \u2192 Bits n\nfromFin = from\u2115 \u2218 Fin.to\u2115\n\nlookupTbl : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Vec A (2^ n) \u2192 A\nlookupTbl         []         (x \u2237 []) = x\nlookupTbl         (0b \u2237 key) tbl      = lookupTbl key (take _ tbl)\nlookupTbl {suc n} (1b \u2237 key) tbl      = lookupTbl key (drop (2^ n) tbl)\n\nfunFromTbl : \u2200 {n a} {A : Set a} \u2192 Vec A (2^ n) \u2192 (Bits n \u2192 A)\nfunFromTbl = flip lookupTbl\n\ntblFromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Vec A (2^ n)\n-- tblFromFun f = tabulate (f \u2218 fromFin)\ntblFromFun {zero} f = f [] \u2237 []\ntblFromFun {suc n} f = tblFromFun {n} (f \u2218 0\u2237_) ++ tblFromFun {n} (f \u2218 1\u2237_)\n\nfunFromTbl\u2218tblFromFun : \u2200 {n a} {A : Set a} (fun : Bits n \u2192 A) \u2192 funFromTbl (tblFromFun fun) \u2257 fun\nfunFromTbl\u2218tblFromFun {zero} f [] = refl\nfunFromTbl\u2218tblFromFun {suc n} f (0b \u2237 xs)\n  rewrite take-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_)) =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 0\u2237_) xs\nfunFromTbl\u2218tblFromFun {suc n} f (1b \u2237 xs)\n  rewrite drop-++ (2^ n) (tblFromFun {n} (f \u2218 0\u2237_)) (tblFromFun {n} (f \u2218 1\u2237_))\n        | take-++ (2^ n) (tblFromFun {n} (f \u2218 1\u2237_)) [] =\n    funFromTbl\u2218tblFromFun {n} (f \u2218 1\u2237_) xs\n\ntblFromFun\u2218funFromTbl : \u2200 {n a} {A : Set a} (tbl : Vec A (2^ n)) \u2192 tblFromFun {n} (funFromTbl tbl) \u2261 tbl\ntblFromFun\u2218funFromTbl {zero} (x \u2237 []) = refl\ntblFromFun\u2218funFromTbl {suc n} tbl\n  rewrite tblFromFun\u2218funFromTbl {n} (take _ tbl)\n        | tblFromFun\u2218funFromTbl {n} (drop (2^ n) tbl)\n        = take-drop-lem (2^ n) tbl\n\n{-\nsucB-lem : \u2200 {n} x \u2192 to\u2115 {2^ n} (sucB x) [mod 2 ^ n ] \u2261 (suc (to\u2115 x)) [mod 2 ^ n ]\nsucB-lem x = {!!}\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n-}\n\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | \u2261.[ p ] = 2\u207f\u2270to\u2115 xs (<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |     _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x < 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | \u2261.[ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | \u2261.[ p ] = to\u2115\u2218from\u2115 {n} x (<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x < 2^ n \u2192 y < 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y\nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nopen Defs public\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"02f76eb9e7cbae765526c4deee1763be6c0c0f59","subject":"Minor changes to perfect-bin-tree.agda","message":"Minor changes to perfect-bin-tree.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = FI.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081) \n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary using (Dec ; yes ; no)\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  open import Function.Inverse\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082) \n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = Function.Inverse.id\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}(t\u2081 t\u2082 : Tree A n) \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 t\u2081 t\u2082 = \u03bb x \u2192 record \n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv \n                          ; right-inverse-of = swap-inv } \n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record \n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  } \n    } where\n        \n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n        \n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 Function.Inverse.sym (t1\u2248s1 x)) (\u03bb x \u2192 Function.Inverse.sym (t2\u2248s2 x)))\n\n       \n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        y = Function.Inverse.id\n  Rot\u27f6\u2248 (fork rot rot\u2081) y = (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) y = \n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 r' l' y \u27e9 \n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081) \n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  (x \u2237\u2262 neg) eq = neg (cong (_\u2237_ x) eq)\n\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl \n  swap-perm\u2081 (left path)  = swap-perm\u2081 path \u27e8fork\u27e9 \u2248-refl\n  swap-perm\u2081 (right path) = \u2248-refl \u27e8fork\u27e9 swap-perm\u2081 path\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t) \n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left \n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"42eba4cf2cdf0edfe17778400b79ee9dc0061d79","subject":"Improved a proof.","message":"Improved a proof.\n\nIgnore-this: 1589ea4960e3c9cc42eca4ce73439d16\n\ndarcs-hash:20100524233058-3bd4e-33aa79cfe3c992350bd9067acbd6e8a1d2f8dbbf.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Equalities\/Properties.agda","new_file":"LTC\/Relation\/Equalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Functions on equalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Equalities.Properties where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\npostulate\n  \u00acS\u22610 : {d : D} \u2192 \u00ac (succ d \u2261 zero)\n{-# ATP prove \u00acS\u22610 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Functions on equalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Equalities.Properties where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\n\u00acS\u22610 : {d : D} \u2192 \u00ac (succ d \u2261 zero)\n\u00acS\u22610 S\u22610 = \u22a5-elim prf\n  where\n    -- The proof uses the axiom 0\u2260S\n    postulate prf : \u22a5\n    {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"40ac98e28ba5915262fe24eb18e5cd83f6bf8cc3","subject":"Stratified Desc model: checkpoint Set0\/Set1","message":"Stratified Desc model: checkpoint Set0\/Set1\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : A -> Set m}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc : Set1 where\n  id : Desc\n  const : Set -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set) -> (S -> Desc) -> Desc\n  pi : (S : Set) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : Desc -> Set -> Set\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu (D : Desc) : Set where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {l : Level}(D : Desc)(X : Set)(P : X -> Set l) -> [| D |] X -> Set l\nAll id          X P x        = P x\nAll (const Z)   X P x        = Unit\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\n\nall : (D : Desc)(X : Set)(P : X -> Set)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = Void\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : (D : Desc)(X Y : Set)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : (D : Desc)(X : Set)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\nproof-map-compos : (D : Desc)(X Y Z : Set)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\ninduction : (D : Desc) \n            (P : Mu D -> Set) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\n-- But the termination checker is unhappy.\n-- So we write the following:\n\nmodule Elim {l : Level}\n            (D : Desc)\n            (P : Mu D -> Set l)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = Void\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> Desc\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : Desc\nNatD = sigma NatConst  natCases\n\nNat : Set\nNat = Mu NatD\n\nze : Nat\nze = con (Ze , Void)\n\nsu : Nat -> Nat\nsu n = con (Su , n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n\n--********************************************\n-- Finite sets\n--********************************************\n\n-- If we weren't such bug fans of levitating things, we would\n-- implement finite sets with:\n\n{-\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n-- But no, we make it fly in Desc:\n\nEnumU : Set\nEnumU = Nat\n\nnilE : EnumU\nnilE = ze\n\nconsE : EnumU -> EnumU\nconsE e = su e \n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n-- All : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l) -> [| D |] X -> Set l\n\n{-\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD (Nat {l = zero}) (\\e -> (EnumT e -> Set1) -> Set1) xs -> \n           (EnumT (con xs) -> Set1) -> Set1\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n-}\n\n{-\ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\n-}\n\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD Nat (\\e -> (EnumT e -> Set) -> Set) xs -> \n           (EnumT (con xs) -> Set) -> Set\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi e P =  induction NatD (\\E -> (EnumT E -> Set) -> Set) casesSpi e P\n\n{-\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set1)\n                                  (b' : spi e P')\n                                  (x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set1)\n              (b' : spi (con xs) P')\n              (x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Su , n) hs P' b' EZe = fst b'\ncasesSwitch (Su , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\n\nswitch : (e : EnumU)\n         (P : EnumT e -> Set1)\n         (b : spi e P)\n         (x : EnumT e) -> P x\nswitch e P b xs =  induction NatD\n                            (\\e -> (P : EnumT e -> Set1)\n                                   (b : spi e P)\n                                   (xs : EnumT e) -> P xs) \n                            casesSwitch e P b xs \n\n\n--********************************************\n-- Tagged description\n--********************************************\n\n-- data Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j)\n-- spi : {l : Level}(e : EnumU)(P : EnumT e -> Set l) -> Set l\n\nTagDesc : Set2\nTagDesc = Sigma (EnumU {l = suc (suc zero)}) (\\e -> spi e (\\_ -> Desc {l = zero}))\n\ntoDesc : {l : Level} -> TagDesc -> Desc {l = zero}\ntoDesc {x} (B , F) = {!!} -- sigma (EnumT B) (\\e -> ?) --switch B (\\_ -> Desc {l = x}) F e)\n\ntest : (E : EnumU {l = suc (suc zero)}) -> spi E (\\_ -> Desc {l = zero}) -> EnumT E -> Desc {l = zero}\ntest B F e = {!switch B (\\_ -> Desc {l = zero}) F e!}\n\n-}\n--********************************************\n-- Catamorphism\n--********************************************\n\n{-\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n-}","old_contents":"\n {-# OPTIONS --universe-polymorphism #-}\n\nmodule Desc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\npair : {i j : Level}{A : Set i}{B : A -> Set j} -> \n       (x : A) (y : B x) -> Sigma {i = i}{j = j} A B\npair x y = x , y\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : A -> Set m}(f : (a : A) -> B a)(g : (a : A) -> B a)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata Desc {l : Level} : Set (suc l) where\n  id : Desc\n  const : Set l -> Desc\n  prod : Desc -> Desc -> Desc\n  sigma : (S : Set l) -> (S -> Desc) -> Desc\n  pi : (S : Set l) -> (S -> Desc) -> Desc\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|]_ : {l : Level} -> Desc -> Set l -> Set l\n[| id |] Z        = Z\n[| const X |] Z   = X\n[| prod D D' |] Z = [| D |] Z * [| D' |] Z\n[| sigma S T |] Z = Sigma S (\\s -> [| T s |] Z)\n[| pi S T |] Z    = (s : S) -> [| T s |] Z\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata Mu {l : Level}(D : Desc {l = l}) : Set l where\n  con : [| D |] (Mu D) -> Mu D\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l) -> [| D |] X -> Set l\nAll id          X P x        = P x\nAll (const Z)   X P x        = Unit\nAll (prod D D') X P (d , d') = (All D X P d) * (All D' X P d')\nAll (sigma S T) X P (a , b)  = All (T a) X P b\nAll (pi S T)    X P f        = (s : S) -> All (T s) X P (f s)\n\nall : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l)(R : (x : X) -> P x)(x : [| D |] X) -> All D X P x\nall id X P R x = R x\nall (const Z) X P R z = Void\nall (prod D D') X P R (d , d') = all D X P R d , all D' X P R d'\nall (sigma S T) X P R (a , b) = all (T a) X P R b\nall (pi S T) X P R f = \\ s -> all (T s) X P R (f s)\n\n--********************************************\n-- Map\n--********************************************\n\nmap : {l : Level}(D : Desc)(X Y : Set l)(f : X -> Y)(v : [| D |] X) -> [| D |] Y\nmap id X Y sig x = sig x\nmap (const Z) X Y sig z = z\nmap (prod D D') X Y sig (d , d') = map D X Y sig d , map D' X Y sig d'\nmap (sigma S T) X Y sig (a , b) = (a , map (T a) X Y sig b)\nmap (pi S T) X Y sig f = \\x -> map (T x) X Y sig (f x)\n\n\nproof-map-id : {l : Level}(D : Desc)(X : Set l)(v : [| D |] X) -> map D X X (\\x -> x) v == v\nproof-map-id id X v = refl\nproof-map-id (const Z) X v = refl\nproof-map-id (prod D D') X (v , v') = cong2 (\\x y -> (x , y)) (proof-map-id D X v) (proof-map-id D' X v')\nproof-map-id (sigma S T) X (a , b) = cong (\\x -> (a , x)) (proof-map-id (T a) X b) \nproof-map-id (pi S T) X f = reflFun (\\a -> map (T a) X X (\\x -> x) (f a)) f (\\a -> proof-map-id (T a) X (f a))\n\n\nproof-map-compos : {l : Level}(D : Desc)(X Y Z : Set l)\n                   (f : X -> Y)(g : Y -> Z)\n                   (v : [| D |] X) -> \n                   map D X Z (\\x -> g (f x)) v == map D Y Z g (map D X Y f v)\nproof-map-compos id X Y Z f g v = refl\nproof-map-compos (const K) X Y Z f g v = refl\nproof-map-compos (prod D D') X Y Z f g (v , v') = cong2 (\\x y -> (x , y)) \n                                                        (proof-map-compos D X Y Z f g v)\n                                                        (proof-map-compos D' X Y Z f g v')\nproof-map-compos (sigma S T) X Y Z f g (a , b) = cong (\\x -> (a , x)) (proof-map-compos (T a) X Y Z f g b)\nproof-map-compos (pi S T) X Y Z f g fc = reflFun (\\a -> map (T a) X Z (\\x -> g (f x)) (fc a))\n                                                 (\\a -> map (T a) Y Z g (map (T a) X Y f (fc a)))\n                                                 (\\a -> proof-map-compos (T a) X Y Z f g (fc a))\n\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\n-- One would like to write the following:\n\n{-\ninduction : {l : Set}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms (con xs) = ms xs (all D (Mu D) P (\\x -> induction D P ms x) xs)\n-}\n\n-- But the termination checker is unhappy.\n-- So we write the following:\n\nmodule Elim {l : Level}\n            (D : Desc)\n            (P : Mu D -> Set l)\n            (ms : (x : [| D |] (Mu D)) -> \n                  All D (Mu D) P x -> P (con x))\n       where\n\n    mutual\n        induction : (x : Mu D) -> P x\n        induction (con xs) =  ms xs (hyps D xs)\n    \n        hyps : (D' : Desc)\n               (xs : [| D' |] (Mu D)) ->\n               All D' (Mu D) P xs\n        hyps id x = induction x\n        hyps (const Z) z = Void\n        hyps (prod D D') (d , d') = hyps D d , hyps D' d'\n        hyps (sigma S T) (a , b) = hyps (T a) b\n        hyps (pi S T) f = \\s -> hyps (T s) (f s)\n            \ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\ninduction D P ms x = Elim.induction D P ms x\n\n--********************************************\n-- Examples\n--********************************************\n\n--****************\n-- Nat\n--****************\n\ndata NatConst {l : Level} : Set l where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : {l : Level} -> NatConst {l = l} -> Desc {l = l}\nnatCases Ze = const Unit\nnatCases Suc = id\n\nNatD : {l : Level} -> Desc {l = l}\nNatD {x} = sigma {l = x} NatConst  natCases\n\nNat : {l : Level} -> Set l\nNat = Mu NatD\n\nze : {l : Level} -> Nat {l = l}\nze {x} = con (pair {i = x} {j = x} Ze Void)\n\nsu : {l : Level} -> Nat {l = l} -> Nat {l = l}\nsu {x} n = con (pair {i = x} {j = x} Su n)\n\n--****************\n-- List\n--****************\n\ndata ListConst : Set where\n  Nil : ListConst\n  Cons : ListConst\n\nlistCases : Set -> ListConst -> Desc\nlistCases X Nil = const Unit\nlistCases X Cons = sigma X (\\_ -> id)\n\nListD : Set -> Desc\nListD X = sigma ListConst (listCases X)\n\nList : Set -> Set\nList X = Mu (ListD X)\n\nnil : {X : Set} -> List X\nnil = con ( Nil , Void )\n\ncons : {X : Set} -> X -> List X -> List X\ncons x t = con ( Cons , ( x , t ))\n\n--****************\n-- Tree\n--****************\n\ndata TreeConst : Set where\n  Leaf : TreeConst\n  Node : TreeConst\n\ntreeCases : Set -> TreeConst -> Desc\ntreeCases X Leaf = const Unit\ntreeCases X Node = sigma X (\\_ -> prod id id)\n\nTreeD : Set -> Desc\nTreeD X = sigma TreeConst (treeCases X)\n\nTree : Set -> Set\nTree X = Mu (TreeD X)\n\nleaf : {X : Set} -> Tree X\nleaf = con (Leaf , Void)\n\nnode : {X : Set} -> X -> Tree X -> Tree X -> Tree X\nnode x le ri = con (Node , (x , (le , ri)))\n\n\n--********************************************\n-- Finite sets\n--********************************************\n\n-- If we weren't such bug fans of levitating things, we would\n-- implement finite sets with:\n\n{-\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n-}\n\n-- But no, we make it fly in Desc:\n\nEnumU : {l : Level} -> Set l\nEnumU = Nat\n\nnilE : {l : Level} -> EnumU {l = l}\nnilE {x} = ze {l = x}\n\nconsE : {l : Level} -> EnumU -> EnumU {l = l}\nconsE {x} e = su {l = x} e \n\ndata EnumT {l : Level} : (e : EnumU {l = l}) -> Set l where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\n-- All : {l : Level}(D : Desc)(X : Set l)(P : X -> Set l) -> [| D |] X -> Set l\n\n{-\ncasesSpi : (xs : [| NatD |] Nat) -> \n           All NatD (Nat {l = zero}) (\\e -> (EnumT e -> Set1) -> Set1) xs -> \n           (EnumT (con xs) -> Set1) -> Set1\ncasesSpi (Ze , Void) hs P' = Unit\ncasesSpi (Su , n) hs P' = P' EZe * hs (\\e -> P' (ESu e))\n-}\n\n{-\ninduction : {l : Level}\n            (D : Desc) \n            (P : Mu D -> Set l) ->\n            ( (x : [| D |] (Mu D)) -> \n              All D (Mu D) P x -> P (con x)) ->\n            (v : Mu D) ->\n            P v\n-}\n\nspi : (e : EnumU {l = zero})(P : EnumT {l = zero} e -> Set) -> Set\nspi e P =  induction NatD (\\E -> {!(EnumT {l = zero} e -> Set) -> Set!}) {!!} e  ?\n\n{-\ncasesSwitch : (xs : [| NatD |] Nat) ->\n              All NatD Nat (\\e -> (P' : EnumT e -> Set1)\n                                  (b' : spi e P')\n                                  (x' : EnumT e) -> P' x') xs ->\n              (P' : EnumT (con xs) -> Set1)\n              (b' : spi (con xs) P')\n              (x' : EnumT (con xs)) -> P' x'\ncasesSwitch (Ze , Void) hs P' b' ()\ncasesSwitch (Su , n) hs P' b' EZe = fst b'\ncasesSwitch (Su , n) hs P' b' (ESu e') = hs (\\e -> P' (ESu e)) (snd b') e'\n\n\nswitch : (e : EnumU)\n         (P : EnumT e -> Set1)\n         (b : spi e P)\n         (x : EnumT e) -> P x\nswitch e P b xs =  induction NatD\n                            (\\e -> (P : EnumT e -> Set1)\n                                   (b : spi e P)\n                                   (xs : EnumT e) -> P xs) \n                            casesSwitch e P b xs \n\n\n--********************************************\n-- Tagged description\n--********************************************\n\n-- data Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j)\n-- spi : {l : Level}(e : EnumU)(P : EnumT e -> Set l) -> Set l\n\nTagDesc : Set2\nTagDesc = Sigma (EnumU {l = suc (suc zero)}) (\\e -> spi e (\\_ -> Desc {l = zero}))\n\ntoDesc : {l : Level} -> TagDesc -> Desc {l = zero}\ntoDesc {x} (B , F) = {!!} -- sigma (EnumT B) (\\e -> ?) --switch B (\\_ -> Desc {l = x}) F e)\n\ntest : (E : EnumU {l = suc (suc zero)}) -> spi E (\\_ -> Desc {l = zero}) -> EnumT E -> Desc {l = zero}\ntest B F e = {!switch B (\\_ -> Desc {l = zero}) F e!}\n\n-}\n--********************************************\n-- Catamorphism\n--********************************************\n\n{-\n\ncata : (D : Desc)\n       (T : Set) ->\n       ([| D |] T -> T) ->\n       (Mu D) -> T\ncata D T phi x = induction D (\\_ -> T) (\\x ms -> phi (replace D T x ms)) x\n  where replace : (D' : Desc)(T : Set)(xs : [| D' |] (Mu D))(ms : All D' (Mu D) (\\_ -> T) xs) -> [| D' |] T\n        replace id T x y = y\n        replace (const Z) T z z' = z'\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : TagDesc -> (X : Set) -> TagDesc\n(e , D) ** X = consE e , (const X , D)\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : (D : TagDesc)(X Y : Set) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        [| toDesc (D ** X) |] (Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\napply (E , B) X Y sig (EZe , x) = sig x\napply (E , B) X Y sig (ESu n , t) = con (ESu n , t)\n\nsubst : (D : TagDesc)(X Y : Set) ->\n        Mu (toDesc (D ** X)) ->\n        (X -> Mu (toDesc (D ** Y))) ->\n        Mu (toDesc (D ** Y))\nsubst D X Y x sig = cata (toDesc (D ** X)) (Mu (toDesc (D ** Y))) (apply D X Y sig) x\n-}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e295b6ccc86d810b0a1b3567f66adcdaf0bb065f","subject":"Add tests","message":"Add tests\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\ntrue \u2228 _  = true\nfalse \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nfalse \u2227 _ = false\ntrue \u2227 b  = b\n\nif_then_else_ : {A : Set} \u2192 Bool \u2192 A \u2192 A \u2192 A\nif true  then a else _ = a\nif false then _ else b = b\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[_] : {A : Set} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x == y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q == s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p == r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs ([ n ]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 []\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 [])) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (5 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (6 \u2237 3 \u2237 [])) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (6 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 []) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 []) (3 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 [])) \u2237\n  (model (10 \u2237 9 \u2237 []) (1 \u2237 [])) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 []) (4 \u2237 11 \u2237 8 \u2237 [])) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 []) (11 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 []) (11 \u2237 3 \u2237 [])) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 []) ([])) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 []) (12 \u2237 [])) \u2237 []\n\ntestp : Relation\ntestp = consider proofs cms (5 \u21d2 (\u21d2 10))\n\ntestd : Relation\ntestd = consider proofs cms (5 \u21d2 (\u21d2 4))\n\ntests : Relation\ntests = consider proofs cms (5 \u21d2 (\u21d2 3))\n\ntestu : Relation\ntestu = consider proofs cms (6 \u21d2 (\u21d2 1))\n\ntestcl : List \u2115\ntestcl = closure proofs (5 \u2237 [])\n","old_contents":"data Bool : Set where\n  true  : Bool\n  false : Bool\n\n\n_\u2228_ : Bool \u2192 Bool \u2192 Bool\ntrue \u2228 _  = true\nfalse \u2228 b = b\n\n\n_\u2227_ : Bool \u2192 Bool \u2192 Bool\nfalse \u2227 _ = false\ntrue \u2227 b  = b\n\n\n----------------------------------------\n\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n{-# BUILTIN NATURAL \u2115 #-}\n\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = n == m\n_ == _         = false\n\n\n\n----------------------------------------\n\n\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\ninfixr 5 _\u2237_\n\n\n[_] : {A : Set} \u2192 A \u2192 List A\n[ x ] = x \u2237 []\n\n\nany : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Bool\nany _ []       = false\nany f (x \u2237 xs) = (f x) \u2228 (any f xs)\n\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []       = false\nx \u2208 (y \u2237 ys) with x == y\n...             | true  = true\n...             | false = x \u2208 ys\n\n\n_\u220b_ : List \u2115 \u2192 \u2115 \u2192 Bool\nxs \u220b y = y \u2208 xs\n\n\n\n----------------------------------------\n\n\n\ndata Arrow : Set where\n  \u21d2_  : \u2115 \u2192 Arrow\n  _\u21d2_ : \u2115 \u2192 Arrow \u2192 Arrow\n\n\n_\u2261\u2261_ : Arrow \u2192 Arrow \u2192 Bool\n(\u21d2 q) \u2261\u2261 (\u21d2 s)     = q == s\n(p \u21d2 q) \u2261\u2261 (r \u21d2 s) = (p == r) \u2227 (q \u2261\u2261 s)\n_ \u2261\u2261 _             = false\n\n\n_\u2208\u2208_ : Arrow \u2192 List Arrow \u2192 Bool\nx \u2208\u2208 []       = false\nx \u2208\u2208 (y \u2237 ys) with x \u2261\u2261 y\n...              | true  = true\n...              | false = x \u2208\u2208 ys\n\n\nclosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nclosure [] found               = found\nclosure ((\u21d2 n) \u2237 rest) found   = n \u2237 (closure rest (n \u2237 found))\nclosure ((n \u21d2 q) \u2237 rest) found with (n \u2208 found) \u2228 (n \u2208 (closure rest found))\n...                               | true  = closure (q \u2237 rest) found\n...                               | false = closure rest found\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\ncs , ps \u22a2 q = q \u2208 (closure cs ps)\n\n\n_\u22a2_ : List Arrow \u2192 Arrow \u2192 Bool\ncs \u22a2 (\u21d2 q)   = q \u2208 (closure cs [])\ncs \u22a2 (p \u21d2 q) = ((\u21d2 p) \u2237 cs) \u22a2 q\n\n\n\n----------------------------------------\n\n\n\ndata Separation : Set where\n  model : List \u2115 \u2192 List \u2115 \u2192 Separation\n\n\nmodelsupports : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodelsupports (model holds _) cs n = cs , holds \u22a2 n\n\n\nmodeldenies : Separation \u2192 List Arrow \u2192 \u2115 \u2192 Bool\nmodeldenies (model _ fails) cs n = any (_\u220b_ (closure cs ([ n ]))) fails\n\n\n_\u27ea!_\u27eb_ : List Arrow \u2192 Separation \u2192 Arrow \u2192 Bool\ncs \u27ea! m \u27eb (\u21d2 q) = modeldenies m cs q\ncs \u27ea! m \u27eb (p \u21d2 q) = (modelsupports m cs p) \u2227 (cs \u27ea! m \u27eb q)\n\n\n_\u27ea_\u27eb_ : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Bool\ncs \u27ea [] \u27eb arr     = false\ncs \u27ea m \u2237 ms \u27eb arr = (cs \u27ea! m \u27eb arr) \u2228 (cs \u27ea ms \u27eb arr)\n\n\n\n----------------------------------------\n\n\n\ndata Relation : Set where\n  Proved    : Relation\n  Derivable : Relation\n  Separated : Relation\n  Unknown   : Relation\n\n\nconsider : List Arrow \u2192 List Separation \u2192 Arrow \u2192 Relation\nconsider cs ms arr with (arr \u2208\u2208 cs)\n...                   | true  = Proved\n...                   | false with (cs \u22a2 arr)\n...                              | true  = Derivable\n...                              | false with (cs \u27ea ms \u27eb arr)\n...                                         | true  = Separated\n...                                         | false = Unknown\n\n\nproofs : List Arrow\nproofs =\n   (3 \u21d2 (\u21d2 4)) \u2237\n--   (5 \u21d2 (\u21d2 4)) \u2237\n   (6 \u21d2 (\u21d2 4)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (3 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (9 \u21d2 (\u21d2 3)) \u2237\n   (5 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (6 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 8)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (5 \u21d2 (\u21d2 10)) \u2237\n   (10 \u21d2 (\u21d2 4)) \u2237\n   (5 \u21d2 (\u21d2 11)) \u2237\n   (6 \u21d2 (\u21d2 11)) \u2237\n   (11 \u21d2 (\u21d2 4)) \u2237\n   (10 \u21d2 (\u21d2 7)) \u2237\n   (8 \u21d2 (\u21d2 4)) \u2237\n   (9 \u21d2 (\u21d2 7)) \u2237\n   (9 \u21d2 (\u21d2 8)) \u2237\n   (3 \u21d2 (\u21d2 8)) \u2237\n   (5 \u21d2 (3 \u21d2 (\u21d2 9))) \u2237\n   (6 \u21d2 (7 \u21d2 (\u21d2 10))) \u2237\n   (6 \u21d2 (3 \u21d2 (\u21d2 3))) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (\u21d2 7)) \u2237\n   (7 \u21d2 (8 \u21d2 (\u21d2 10))) \u2237\n   (3 \u21d2 (10 \u21d2 (\u21d2 9))) \u2237\n   (5 \u21d2 (\u21d2 1)) \u2237\n   (3 \u21d2 (1 \u21d2 (\u21d2 9))) \u2237\n   (1 \u21d2 (\u21d2 2)) \u2237\n   (10 \u21d2 (\u21d2 2)) \u2237 []\n\ncms : List Separation\ncms =\n  (model (12 \u2237 6 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (5 \u2237 3 \u2237 7 \u2237 7 \u2237 [])) \u2237\n  (model (6 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (5 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 11 \u2237 4 \u2237 1 \u2237 []) (6 \u2237 3 \u2237 [])) \u2237\n  (model (5 \u2237 3 \u2237 11 \u2237 4 \u2237 7 \u2237 8 \u2237 3 \u2237 9 \u2237 10 \u2237 1 \u2237 []) (6 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 []) (5 \u2237 6 \u2237 3 \u2237 8 \u2237 9 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 5 \u2237 6 \u2237 4 \u2237 11 \u2237 1 \u2237 []) (3 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 11 \u2237 7 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 8 \u2237 1 \u2237 [])) \u2237\n  (model (10 \u2237 9 \u2237 []) (1 \u2237 [])) \u2237\n  (model (3 \u2237 4 \u2237 11 \u2237 []) (9 \u2237 5 \u2237 6 \u2237 10 \u2237 7 \u2237 1 \u2237 [])) \u2237\n  (model (12 \u2237 7 \u2237 1 \u2237 []) (4 \u2237 11 \u2237 8 \u2237 [])) \u2237\n  (model (9 \u2237 3 \u2237 10 \u2237 8 \u2237 1 \u2237 []) (11 \u2237 [])) \u2237\n  (model (12 \u2237 4 \u2237 10 \u2237 1 \u2237 []) (11 \u2237 3 \u2237 [])) \u2237\n  (model (3 \u2237 6 \u2237 5 \u2237 []) ([])) \u2237\n  (model (1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 6 \u2237 7 \u2237 8 \u2237 9 \u2237 10 \u2237 11 \u2237 []) (12 \u2237 [])) \u2237 []\n\ntestp : Arrow\ntestp = (5 \u21d2 (\u21d2 10))\n\ntestd : Arrow\ntestd = (5 \u21d2 (\u21d2 4))\n\ntests : Arrow\ntests = (5 \u21d2 (\u21d2 3))\n\ntestu : Arrow\ntestu = (6 \u21d2 (\u21d2 1))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"85e0d8138be0cef992a8fca3430d92f3602b85f0","subject":"scraps","message":"scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\nopen import lemmas-consistency\n\nmodule typed-expansion where\n  lem : (\u0393 : tctx) (x : Nat) \u2192 (((id \u0393) x) == Some (X x) \u00d7 (\u03a3[ \u03c4 \u2208 htyp ]((\u0393 x) == Some \u03c4)))\n                             + (((id \u0393) x) == None)\n  lem \u0393 x with \u0393 x\n  lem \u0393 x | Some x\u2081 = Inl (refl , x\u2081 , refl)\n  lem \u0393 x | None = Inr refl\n\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0394} {\u0393} x d xin with lem \u0393 x\n  lem-idsub {\u0394} {\u0393} x d xin | Inl (w , y , z) with (! xin) \u00b7 w\n  lem-idsub x .(X x) xin | Inl (w , y , z) | refl = y , z , TAVar z\n  lem-idsub x d xin | Inr x\u2081 = abort (somenotnone ((! xin) \u00b7 x\u2081))\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D D\u2081) = TAAp (lem-weaken\u03941 D) (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222al \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222al \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n  lem-weaken\u03941 (TAFailedCast x y z w) = TAFailedCast (lem-weaken\u03941 x) y z w\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4} D = tr (\u03bb q \u2192 q , \u0393 \u22a2 d :: \u03c4) (\u222acomm \u03942 \u03941) (lem-weaken\u03941 D)\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp {\u03941 = \u03941} x\u2081 x\u2082 x\u2083 x\u2084)\n      with typed-expansion-ana x\u2083 | typed-expansion-ana x\u2084\n    ... | con1 , ih1 | con2 , ih2  = TAAp (TACast (lem-weaken\u03941 ih1) con1) (TACast (lem-weaken\u03942 {\u03941 = \u03941 }ih2) con2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-synth (ESAsc x)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 MAHole ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCHole1 , TALam D\n    typed-expansion-ana (EALam x\u2081 MAArr ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCArr TCRefl con , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ~sym x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u03941 ih1) lem-idsub\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\nopen import lemmas-consistency\n\nmodule typed-expansion where\n  lem : (\u0393 : tctx) (x : Nat) \u2192 (((id \u0393) x) == Some (X x) \u00d7 (\u03a3[ \u03c4 \u2208 htyp ]((\u0393 x) == Some \u03c4)))\n                             + (((id \u0393) x) == None)\n  lem \u0393 x with \u0393 x\n  lem \u0393 x | Some x\u2081 = Inl (refl , x\u2081 , refl)\n  lem \u0393 x | None = Inr refl\n\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0394} {\u0393} x d xin with lem \u0393 x\n  lem-idsub {\u0394} {\u0393} x d xin | Inl (w , y , z) with (! xin) \u00b7 w\n  lem-idsub x .(X x) xin | Inl (w , y , z) | refl = y , z , TAVar z\n  lem-idsub x d xin | Inr x\u2081 = abort (somenotnone ((! xin) \u00b7 x\u2081))\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D D\u2081) = TAAp (lem-weaken\u03941 D) (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a1 \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a1 \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n  lem-weaken\u03941 (TAFailedCast x y z w) = TAFailedCast (lem-weaken\u03941 x) y z w\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4} D = tr (\u03bb q \u2192 q , \u0393 \u22a2 d :: \u03c4) (\u222acomm \u03942 \u03941) (lem-weaken\u03941 D)\n\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp {\u03941 = \u03941} x\u2081 x\u2082 x\u2083 x\u2084)\n      with typed-expansion-ana x\u2083 | typed-expansion-ana x\u2084\n    ... | con1 , ih1 | con2 , ih2  = TAAp (TACast (lem-weaken\u03941 ih1) con1) (TACast (lem-weaken\u03942 {\u03941 = \u03941 }ih2) con2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-synth (ESAsc x)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 MAHole ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCHole1 , TALam D\n    typed-expansion-ana (EALam x\u2081 MAArr ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCArr TCRefl con , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ~sym x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u03941 ih1) lem-idsub\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d41c050cc2930313a7dc71510e7f38e52ec1344c","subject":"FunUniverse.Agda: fork\/bijFork were broken","message":"FunUniverse.Agda: fork\/bijFork were broken\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Agda.agda","new_file":"FunUniverse\/Agda.agda","new_contents":"module FunUniverse.Agda where\n\nopen import Type\nopen import Data.Two using (proj\u2032)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using ([]; _\u2237_)\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bit using (0b; 1b)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Core\n\n-\u2192- : \u2605 \u2192 \u2605 \u2192 \u2605\n-\u2192- A B = A \u2192 B\n\nagdaFunU : FunUniverse \u2605\nagdaFunU = \u2605-U , -\u2192-\n\nmodule AgdaFunUniverse = FunUniverse agdaFunU\n\nfunCat : Category -\u2192-\nfunCat = F.id , _\u2218\u2032_\n\nfunLin : LinRewiring agdaFunU\nfunLin = mk funCat\n            (\u03bb f \u2192 \u00d7.map f F.id)\n            \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n            (\u03bb f g \u2192 \u00d7.map f g) (\u03bb f \u2192 \u00d7.map F.id f)\n            (F.const []) _ (uncurry _\u2237_) V.uncons\n\nfunRewiring : Rewiring agdaFunU\nfunRewiring = mk funLin _ (\u03bb x \u2192 x , x) (F.const []) \u00d7.<_,_> proj\u2081 proj\u2082\n                 V.rewire V.rewireTbl\n\nfunFork : HasFork agdaFunU\nfunFork = (\u03bb { (b , xy)    \u2192 proj\u2032 xy b })\n        , (\u03bb { f g (b , x) \u2192 proj\u2032 (f , g) b x })\n\nagdaFunOps : FunOps agdaFunU\nagdaFunOps = mk funRewiring funFork (F.const 0b) (F.const 1b)\n\nmodule AgdaFunOps = FunOps agdaFunOps\n","old_contents":"module FunUniverse.Agda where\n\nopen import Type\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec.NP as V\nimport Function as F\nimport Data.Product as \u00d7\nopen F using (const; _\u2218\u2032_)\nopen V using ([]; _\u2237_)\nopen \u00d7 using (_\u00d7_; _,_; proj\u2081; proj\u2082; uncurry)\n\nopen import Data.Bit using (0b; 1b)\n\nopen import FunUniverse.Data\nopen import FunUniverse.Category\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Core\n\n-\u2192- : \u2605 \u2192 \u2605 \u2192 \u2605\n-\u2192- A B = A \u2192 B\n\nagdaFunU : FunUniverse \u2605\nagdaFunU = \u2605-U , -\u2192-\n\nmodule AgdaFunUniverse = FunUniverse agdaFunU\n\nfunCat : Category -\u2192-\nfunCat = F.id , _\u2218\u2032_\n\nfunLin : LinRewiring agdaFunU\nfunLin = mk funCat\n            (\u03bb f \u2192 \u00d7.map f F.id)\n            \u00d7.swap (\u03bb {((x , y) , z) \u2192 x , (y , z) }) (\u03bb x \u2192 _ , x) proj\u2082\n            (\u03bb f g \u2192 \u00d7.map f g) (\u03bb f \u2192 \u00d7.map F.id f)\n            (F.const []) _ (uncurry _\u2237_) V.uncons\n\nfunRewiring : Rewiring agdaFunU\nfunRewiring = mk funLin _ (\u03bb x \u2192 x , x) (F.const []) \u00d7.<_,_> proj\u2081 proj\u2082\n                 V.rewire V.rewireTbl\n\nfunFork : HasFork agdaFunU\nfunFork = (\u03bb { (b , x , y) \u2192 if b then x else y })\n        , (\u03bb { f g (b , x) \u2192 (if b then f else g) x })\n\nagdaFunOps : FunOps agdaFunU\nagdaFunOps = mk funRewiring funFork (F.const 0b) (F.const 1b)\n\nmodule AgdaFunOps = FunOps agdaFunOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a5faa0bd005291a857cd94232a94053d38db1d58","subject":"Added x\u2238Sy\u2264x\u2238y.","message":"Added x\u2238Sy\u2264x\u2238y.\n\nIgnore-this: f709e5a9a8388e91fa15b954f50333e2\n\ndarcs-hash:20110216151638-3bd4e-19d717234f84e2d15423b7f2bd7b3faeada5867e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/LTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x\u2238y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x\u2238y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x\u2238y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import LTC.Base\nopen import LTC.Base.Properties using ( x\u2261y\u2192Sx\u2261Sy )\n\nopen import Common.Function using ( _$_ )\n\nopen import LTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The LTC natural numbers type\n        )\nopen import LTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import LTC.Data.Nat.Inequalities.Properties public\nopen import LTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<x\u2238y Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<x\u2238y zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<x\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<x\u2238y Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}  -- Use the hint sN.\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}  -- Use the hint sN.\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (x\u2261y\u2192Sx\u2261Sy m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}  -- Use the hint sN.\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}  -- Use the hint sN.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7256ab4f7bb4043b92bddbbceae0cc0f07ffca14","subject":"game: adapt to flipbased","message":"game: adapt to flipbased\n","repos":"crypto-agda\/crypto-agda","old_file":"game.agda","new_file":"game.agda","new_contents":"module game where\n\nopen import Level using (Lift) renaming (_\u2294_ to _L\u2294_)\nopen import Function.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.List.Any\nopen import Data.Bool\nopen import Relation.Binary.PropositionalEquality\nopen Membership-\u2261\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\n{-\n- cost\n- non-det\n- queries\n- partial (like decryption with a mac)\n-}\n\nopen import Data.Bits\nopen import flipbased-implem\n\nBits\u27e8_\u27e9\u2192_ : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a\nBits\u27e8 n \u27e9\u2192 A = Bits n \u2192 A\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : \u2200 k \u2192 Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R 0\n  ask : \u2200 {k} {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R k) \u2192 Strategy \u2610_ R k\n  rnd : \u2200 {n k} {A : Set} \u2192 \u21ba n A \u2192 (A \u2192 Strategy \u2610_ R k) \u2192 Strategy \u2610_ R (n + k)\n\nrun : \u2200 {k R} \u2192 Strategy id R k \u2192 \u21ba k R\nrun (say r)    = \u27ea r \u27eb\nrun (ask a f)  = run (f a)\nrun (rnd nd f) = nd >>= (\u03bb xs \u2192 run (f xs))\n\npostulate\n      Proba : Set\n      Pr[_\u22611] : \u2200 {k} (EXP : \u21ba k Bit) \u2192 Proba\n      _\u22611\/2^_ : Proba \u2192 \u2115 \u2192 Set\n      Pr[toss\u22611] : Pr[ toss \u22611] \u22611\/2^ 1\n      Pr[random\u2261_] : \u2200 {n} (x : Bits n) \u2192 Pr[ \u27ea _==_ x \u00b7 random \u27eb \u22611] \u22611\/2^ n\n      negligible-proba : Proba\n      negligible-advantage-proba : Proba \u2192 Proba \u2192 Set\n\n_is-negligible : Proba \u2192 Set\np is-negligible = negligible-advantage-proba p negligible-proba\n\nnegligible-advantage : \u2200 {k} (EXP : Bit \u2192 \u21ba k Bit) \u2192 Set\nnegligible-advantage EXP = negligible-advantage-proba Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\npostulate\n  negligible-advantage-xor : \u2200 {k} (EXP\u2081 EXP\u2082 : \u21ba k Bit) \u2192 EXP\u2081 \u2261 \u27ea _xor_ 1b \u00b7 EXP\u2082 \u27eb \u2192 negligible-advantage-proba Pr[ EXP\u2081 \u22611] Pr[ EXP\u2082 \u22611]\n\n  -- whatch out\n  negligible-advantage-trans : \u2200 {p\u2081 p\u2082 p\u2083} \u2192 negligible-advantage-proba p\u2081 p\u2082 \u2192 negligible-advantage-proba p\u2082 p\u2083 \u2192 negligible-advantage-proba p\u2081 p\u2083\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nmodule CryptoGames where\n\n  module CipherGames (|K| |M| |C| : \u2115) where\n    K = Bits |K|\n    M = Bits |M|\n    C = Bits |C|\n\n    record SemSecAdv k : Set where\n      constructor mk\n      field\n        {k\u2080 k\u2081}   : \u2115\n        k\u2261k\u2080+k\u2081  : k \u2261 k\u2080 + k\u2081\n        gen-m\u2080m\u2081  : \u21ba k\u2080 (M \u00d7 M)\n        stat-test : C \u2192 \u21ba k\u2081 Bit\n\n    mk\u2032 : \u2200 {k\u2080 k\u2081} \u2192 \u21ba k\u2080 (M \u00d7 M) \u2192 (C \u2192 \u21ba k\u2081 Bit) \u2192 SemSecAdv (k\u2080 + k\u2081)\n    mk\u2032 = mk refl\n\n    module DeterministicSemanticSecurity where\n      runDetSemSec : \u2200 {k} (enc : M \u2192 C) \u2192 SemSecAdv k \u2192 Bit \u2192 \u21ba k Bit\n      runDetSemSec enc (mk refl gen-m\u2080m\u2081 f) b = gen-m\u2080m\u2081 >>= \u03bb m\u2080m\u2081 \u2192 f (EXP b enc m\u2080m\u2081)\n\n      DetSemSecWinner : \u2200 (enc : M \u2192 C) b {k} (Adv : SemSecAdv k) \u2192 \u21ba k Set\n      DetSemSecWinner enc b Adv = T\u21ba (runDetSemSec enc Adv b)\n\n      DetSemSec : \u2200 k (enc : M \u2192 C) \u2192 Set\n      DetSemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runDetSemSec enc Adv)\n\n      luckySemSec : SemSecAdv 1\n      luckySemSec = mk\u2032 \u27ea 0\u207f , 0\u207f \u27eb\u1d30 (\u03bb _ \u2192 toss)\n\n      -- see OTPDetSemSec\n\n    module DeterministicCPASecurity where\n      DetCPAAdv : \u2200 k \u2192 Set\u2081\n      DetCPAAdv k = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit k\n\n      data DetCPAWinner (enc : M \u00d7 M \u2192 C) b (ms : List (M \u00d7 M)) : \u2200 {k} (q : \u2115) \u2192 Strategy id Bit k \u2192 Set\u2081 where\n        win : \u2200 {q} \u2192 DetCPAWinner enc b ms q (say b)\n        ask : \u2200 {k q m\u2080m\u2081 f} \u2192 m\u2080m\u2081 \u2209 ms \u2192 DetCPAWinner enc b (m\u2080m\u2081 \u2237 ms) q (f (enc m\u2080m\u2081)) \u2192\n              DetCPAWinner enc b ms {k} (suc q) (ask (enc m\u2080m\u2081) f)\n        rnd : \u2200 {A : Set} {n k q bs} {r : \u21ba n A} {f : A \u2192 _}\n              \u2192 DetCPAWinner enc b ms {k} q (f bs) \u2192 DetCPAWinner enc b ms q (rnd r f)\n\n      runDetCPA : \u2200 {k} (enc : M \u2192 C) (Adv : DetCPAAdv k) \u2192 Bit \u2192 \u21ba k Bit\n      runDetCPA enc Adv b = run (Adv (EXP b enc))\n\n      -- no use of CPAWinner\n      DetCPASec : \u2200 k (enc : M \u2192 C) \u2192 Set\u2081\n      DetCPASec k enc = \u2200 (Adv : DetCPAAdv k) \u2192 negligible-advantage (runDetCPA enc Adv)\n\n    CPAAdv : \u2200 k \u2192 Set\u2081\n    CPAAdv k = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 (\u21ba |K| C)) \u2192 Strategy \u2610_ Bit k\n\n    sem2CPAAdv : \u2200 {k} \u2192 SemSecAdv k \u2192 CPAAdv (|K| + k)\n    sem2CPAAdv (mk {k\u2080} {k\u2081} refl gen-m\u2080m\u2081 f) {\u2610_} enc =\n      subst (Strategy \u2610_ Bit) helper\n        (rnd gen-m\u2080m\u2081 (\u03bb m\u2080m\u2081 \u2192\n          ask (enc m\u2080m\u2081) (\u03bb c \u2192\n            rnd c (\u03bb c' \u2192 rnd (f c') say))))\n      where open import Data.Nat.Properties\n            open SemiringSolver\n            helper : k\u2080 + (|K| + (k\u2081 + 0)) \u2261 |K| + (k\u2080 + k\u2081)\n            helper = solve 3 (\u03bb k\u2080 |K| k\u2081 \u2192 k\u2080 :+ (|K| :+ (k\u2081 :+ con 0)) := |K| :+ (k\u2080 :+ k\u2081))\n                             refl k\u2080 |K| k\u2081\n\n    data CPAWinner (enc : M \u00d7 M \u2192 \u21ba |K| C) b : \u2200 {k} q \u2192 Strategy id Bit k \u2192 Set\u2081 where\n      win : \u2200 {q} \u2192 CPAWinner enc b q (say b)\n      ask : \u2200 {k q m\u2080m\u2081 f} \u2192 CPAWinner enc b {k} q (f (enc m\u2080m\u2081)) \u2192\n            CPAWinner enc b (suc q) (ask (enc m\u2080m\u2081) f)\n      rnd : \u2200 {A : Set} {n k q bs} {r : \u21ba n A} {f : A \u2192 _}\n            \u2192 CPAWinner enc b {k} q (f bs) \u2192 CPAWinner enc b q (rnd r f)\n\n    det : \u2200 {k a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 A \u2192 \u21ba k B\n    det f x = \u27ea f x \u27eb\n\n    runCPA : \u2200 {k} (enc : M \u2192 \u21ba |K| C) (Adv : CPAAdv k) \u2192 Bit \u2192 \u21ba k Bit\n    runCPA enc Adv b = run (Adv (EXP b enc))\n\n    -- no use of CPAWinner\n    CPASec : \u2200 k (enc : M \u2192 \u21ba |K| C) \u2192 Set\u2081\n    CPASec k enc = \u2200 (Adv : CPAAdv k) \u2192 negligible-advantage (runCPA enc Adv)\n\n    SemSecWinner : \u2200 (enc : M \u00d7 M \u2192 \u21ba |K| C) b k \u2192 SemSecAdv k \u2192 Set\u2081\n    SemSecWinner enc b k Adv = CPAWinner enc b 1 (sem2CPAAdv Adv enc)\n\n    runSemSec : \u2200 {k} (enc : M \u2192 \u21ba |K| C) \u2192 SemSecAdv k \u2192 Bit \u2192 \u21ba (|K| + k) Bit\n    runSemSec enc Adv = runCPA enc (sem2CPAAdv Adv)\n\n    -- no use of SemSecWinner\n    SemSec : \u2200 k (enc : M \u2192 \u21ba |K| C) \u2192 Set\n    SemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runSemSec enc Adv)\n\n    weakenWinner : \u2200 {enc b k q\u2080 q\u2081} {s : Strategy id Bit k} \u2192 CPAWinner enc b q\u2080 s \u2192 CPAWinner enc b (q\u2080 + q\u2081) s\n    weakenWinner win     = win\n    weakenWinner (ask w) = ask (weakenWinner w)\n    weakenWinner (rnd {n = n} {k} {bs = bs} w) = rnd {n = n} {k} {bs = bs} (weakenWinner w)\n\n    CPA2Sem-correct : \u2200 {enc b k q} {semAdv : SemSecAdv k}\n                      \u2192 SemSecWinner enc b k semAdv \u2192 CPAWinner enc b (suc q) (sem2CPAAdv semAdv enc)\n    CPA2Sem-correct = weakenWinner\n\n  record Forgery {a} {A : Set a} (P : A \u2192 Set) (queries : List A) : Set a where\n    field\n      forgery         : A\n      fresh-forgery   : forgery \u2209 queries\n      correct-forgery : P forgery\n  open Forgery public\n\n  module MACGames (|M| |Tag| : \u2115) where\n    M = Bits |M|\n    Tag = Bits |Tag|\n\n    -- One might also want that the adversary sends the trials to the challenger.\n    -- The challenger might then either end the game or maybe return the info to\n    -- the adversary.\n    MACAdv : Set\u2081\n    MACAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 Tag) \u2192 Strategy \u2610_ (M \u00d7 Tag) 0\n\n    -- or\n    -- MACAdv = \u2200 {\u2610_ Done} (mac : M \u2192 \u2610 Tag) (submit : M \u00d7 Tag \u2192 \u2610 Done) \u2192 Strategy \u2610_ Done\n\n    -- or\n    -- MACAdv = \u2200 {\u2610_ Done} (mac : M \u2192 \u2610 Tag)\n    --                       (try : M \u00d7 Tag \u2192 \u2610 Bool)\n    --                       (submit : M \u00d7 Tag \u2192 \u2610 Done) \u2192 Strategy \u2610_ Done\n\n    module MACGames' (mac : M \u2192 Tag) where\n      CorrectMAC : M \u00d7 Tag \u2192 Set\n      CorrectMAC (m , t) = mac m \u2261 t\n\n      correctMAC? : M \u00d7 Tag \u2192 Bool\n      correctMAC? (m , t) = mac m == t\n\n      data MACWinner (mts : List (M \u00d7 Tag)) : (q : \u2115) \u2192 Strategy id (M \u00d7 Tag) 0 \u2192 Set\u2081 where\n        win : \u2200 {q} (f : Forgery CorrectMAC mts) \u2192 MACWinner mts q (say (forgery f))\n        ask : \u2200 {m f q} \u2192 MACWinner ((m , mac m) \u2237 mts) q (f (mac m))\n                        \u2192 MACWinner mts (suc q) (ask (mac m) f)\n\n      SecMAC : Set\u2081\n      SecMAC = \u2200 (A : MACAdv) \u2192 Pr[ \u27ea correctMAC? \u00b7 run (A mac) \u27eb \u22611] \u22611\/2^ |Tag| -- check that bound\n\n  module CiphertextIntegrity (M C : Set) where\n    CIAdv : Set\u2081\n    CIAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ C 0\n\n    module CiphertextIntegrity' (enc : M \u2192 C) (dec : C \u2192? M) where\n      CorrectCiphertext : C \u2192 Set\n      CorrectCiphertext c = IsJust (dec c)\n\n      data CIWinner (cs : List C) : (q : \u2115) \u2192 Strategy id C 0 \u2192 Set\u2081 where\n        win : \u2200 {q} (f : Forgery CorrectCiphertext cs) \u2192 CIWinner cs q (say (forgery f))\n        ask : \u2200 {m f q} \u2192 CIWinner (enc m \u2237 cs) q (f (enc m))\n                        \u2192 CIWinner cs (suc q) (ask (enc m) f)\n\n  module AuthenticatedEncryption (|K| |M| |C| : \u2115) where\n    open CipherGames |K| |M| |C|\n    open CiphertextIntegrity M C\n\n    data AERes : Set where\n      distinguish   : Bit \u2192 AERes\n      found-forgery : C \u2192 AERes\n\n    AEAdv : Set\u2081\n    AEAdv = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) \u2192 Strategy \u2610_ AERes 0\n\n    CCAAdv : Set\u2081\n    CCAAdv = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) (dec : C \u2192 \u2610 (Maybe M)) \u2192 Strategy \u2610_ Bit 0\n\n    module AE' (enc : M \u00d7 M \u2192 C) (dec : C \u2192? M) where\n      open CiphertextIntegrity' (\u03bb m \u2192 enc (m , m)) dec\n\n      data AEWinner (cs : List C) b : (q : \u2115) \u2192 Strategy id AERes 0 \u2192 Set\u2081 where\n        win-distinguish : \u2200 {q} \u2192 AEWinner cs b q (say (distinguish b))\n        win-forgery : \u2200 {q} (f : Forgery CorrectCiphertext cs)\n                       \u2192 AEWinner cs b q (say (found-forgery (forgery f)))\n        ask : \u2200 {m\u2080m\u2081 f q} \u2192 AEWinner cs b q (f (enc m\u2080m\u2081)) \u2192\n              AEWinner cs b (suc q) (ask (enc m\u2080m\u2081) f)\n\n      data CCAWinner b : (q : \u2115) \u2192 Strategy id Bit 0 \u2192 Set\u2081 where\n        win : \u2200 {q} \u2192 CCAWinner b q (say b)\n        ask-enc : \u2200 {m\u2080m\u2081 f q} \u2192 CCAWinner b q (f (enc m\u2080m\u2081)) \u2192\n                    CCAWinner b (suc q) (ask (enc m\u2080m\u2081) f)\n        ask-dec : \u2200 {c f q} \u2192 CCAWinner b q (f (dec c)) \u2192\n                    CCAWinner b (suc q) (ask (dec c) f)\n\n  -- CBC with rand IV is not CCA secure\n\n      -- AE \u21d2 CCA Secure\n\n  module OTPDetSemSec {n} (key : Bits n) where\n    open CipherGames n n n\n    open DeterministicSemanticSecurity\n\n    postulate\n      OTPDetSemSec : \u2200 {k} \u2192 DetSemSec k (OTP key)\n{-\n    OTPDetSemSec (mk refl gen-m\u2080m\u2081 stat-test) = {!!}\n    -- DetSemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runDetSemSec enc Adv)\n    -- runDetSemSec enc (mk refl gen-m\u2080m\u2081 f) b = gen-m\u2080m\u2081 >>= \u03bb m\u2080m\u2081 \u2192 f (EXP b enc m\u2080m\u2081)\n\nPr[ runDetSemSec enc Adv b \u2261 b ]\n-}\n\n  Enc : \u2200 |K| |M| |C| \u2192 Set\n  Enc |K| |M| |C| = Bits |K| \u2192 Bits |M| \u2192 Bits |C|\n\n  module DerivedBlockCiphers\n    {|K| |M| |C|} {E : Enc |K| |M| |C|}\n    (E-sec : \u2200 {k} key \u2192 CipherGames.DeterministicSemanticSecurity.DetSemSec |K| |M| |C| k (E key)) where\n\n    open module CipherGames' {|M| |C|} = CipherGames |K| |M| |C|\n    open DeterministicSemanticSecurity\n\n    E\u2080 : Enc |K| |M| |C|\n    E\u2080 k m = 1\u207f \u2295 E k m\n\n    postulate\n      lem' : \u2200 {k} (Adv : SemSecAdv k) b (f : Bits k \u2192 Bits |C|) \u2192 runDetSemSec {k} (_\u2295_ 1\u207f \u2218 f) Adv b \u2261 \u27ea _xor_ 1b \u00b7 runDetSemSec f Adv b \u27eb\n    -- lem' Adv b f = {!!}\n\n      lem'' : \u2200 {k} (Adv : SemSecAdv k) b (f : Bits k \u2192 Bits |C|) \u2192 \u27ea _xor_ 1b \u00b7 runDetSemSec {k} (_\u2295_ 1\u207f \u2218 f) Adv b \u27eb \u2261 runDetSemSec f Adv b\n    -- lem'' Adv b f = {!!}\n\n    -- postulate\n    E\u2080-sec : \u2200 key \u2192 DetSemSec |M| (E\u2080 key)\n    E\u2080-sec key Adv = negligible-advantage-trans (negligible-advantage-xor (runDetSemSec (E\u2080 key) Adv 0b) (runDetSemSec (E key) Adv 0b) (lem' Adv 0b (E key))) (negligible-advantage-trans (E-sec key Adv) (negligible-advantage-xor (runDetSemSec (E key) Adv 1b) (runDetSemSec (E\u2080 key) Adv 1b) (sym (lem'' Adv 1b (E key)))))\n\n    E\u2081 : Enc |K| |M| (1 + |C|)\n    E\u2081 k m = 0\u2237 E k m\n\n    -- E\u2081-adv : \u2200 {k} key \u2192 SemSecAdv k (E key) \u2192 SemSecAdv k (E\u2081 key)\n    -- E\u2081-adv key Adv = {!E-sec!}\n\n    postulate\n      E\u2081-sec : \u2200 {k} key \u2192 DetSemSec k (E\u2081 key)\n    -- E\u2081-sec key Adv = {!E-sec key ?!}\n\n    -- runDetSemSec enc adv b \u2248 runDetSemSec enc' adv b\n","old_contents":"module game where\n\nopen import Level using (Lift) renaming (_\u2294_ to _L\u2294_)\nopen import Function.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.List.Any\nopen import Data.Bool\nopen import Relation.Binary.PropositionalEquality\nopen Membership-\u2261\n\n{-\n-- Game\ndata \u2141 ...\n-}\n\n{-\n- cost\n- non-det\n- queries\n- partial (like decryption with a mac)\n-}\n\nopen import Data.Bits\nopen import flipbased\n\nBits\u27e8_\u27e9\u2192_ : \u2200 {a} \u2192 \u2115 \u2192 Set a \u2192 Set a\nBits\u27e8 n \u27e9\u2192 A = Bits n \u2192 A\n\ndata Strategy (\u2610_ : Set \u2192 Set) (R : Set) : \u2200 k \u2192 Set\u2081 where\n  say : R \u2192 Strategy \u2610_ R 0\n  ask : \u2200 {k} {A : Set} \u2192 \u2610 A \u2192 (A \u2192 Strategy \u2610_ R k) \u2192 Strategy \u2610_ R k\n  rnd : \u2200 {n k} {A : Set} \u2192 \u21ba n A \u2192 (A \u2192 Strategy \u2610_ R k) \u2192 Strategy \u2610_ R (n + k)\n\nrun : \u2200 {k R} \u2192 Strategy id R k \u2192 \u21ba k R\nrun (say r)    = \u27ea r \u27eb\nrun (ask a f)  = run (f a)\nrun (rnd nd f) = nd >>= (\u03bb xs \u2192 run (f xs))\n\npostulate\n      Proba : Set\n      Pr[_\u22611] : \u2200 {k} (EXP : \u21ba k Bit) \u2192 Proba\n      _\u22611\/2^_ : Proba \u2192 \u2115 \u2192 Set\n      Pr[toss\u22611] : Pr[ toss \u22611] \u22611\/2^ 1\n      Pr[random\u2261_] : \u2200 {n} (x : Bits n) \u2192 Pr[ \u27ea _==_ x \u00b7 random \u27eb \u22611] \u22611\/2^ n\n      negligible-proba : Proba\n      negligible-advantage-proba : Proba \u2192 Proba \u2192 Set\n\n_is-negligible : Proba \u2192 Set\np is-negligible = negligible-advantage-proba p negligible-proba\n\nnegligible-advantage : \u2200 {k} (EXP : Bit \u2192 \u21ba k Bit) \u2192 Set\nnegligible-advantage EXP = negligible-advantage-proba Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\npostulate\n  negligible-advantage-xor : \u2200 {k} (EXP\u2081 EXP\u2082 : \u21ba k Bit) \u2192 EXP\u2081 \u2261 \u27ea _xor_ 1b \u00b7 EXP\u2082 \u27eb \u2192 negligible-advantage-proba Pr[ EXP\u2081 \u22611] Pr[ EXP\u2082 \u22611]\n\n  -- whatch out\n  negligible-advantage-trans : \u2200 {p\u2081 p\u2082 p\u2083} \u2192 negligible-advantage-proba p\u2081 p\u2082 \u2192 negligible-advantage-proba p\u2082 p\u2083 \u2192 negligible-advantage-proba p\u2081 p\u2083\n\npattern looks-random = 0b\npattern not-random   = 1b\n\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nmodule CryptoGames where\n\n  module CipherGames (|K| |M| |C| : \u2115) where\n    K = Bits |K|\n    M = Bits |M|\n    C = Bits |C|\n\n    record SemSecAdv k : Set where\n      constructor mk\n      field\n        {k\u2080 k\u2081}   : \u2115\n        k\u2261k\u2080+k\u2081  : k \u2261 k\u2080 + k\u2081\n        gen-m\u2080m\u2081  : \u21ba k\u2080 (M \u00d7 M)\n        stat-test : C \u2192 \u21ba k\u2081 Bit\n\n    mk\u2032 : \u2200 {k\u2080 k\u2081} \u2192 \u21ba k\u2080 (M \u00d7 M) \u2192 (C \u2192 \u21ba k\u2081 Bit) \u2192 SemSecAdv (k\u2080 + k\u2081)\n    mk\u2032 = mk refl\n\n    module DeterministicSemanticSecurity where\n      runDetSemSec : \u2200 {k} (enc : M \u2192 C) \u2192 SemSecAdv k \u2192 Bit \u2192 \u21ba k Bit\n      runDetSemSec enc (mk refl gen-m\u2080m\u2081 f) b = gen-m\u2080m\u2081 >>= \u03bb m\u2080m\u2081 \u2192 f (EXP b enc m\u2080m\u2081)\n\n      DetSemSecWinner : \u2200 (enc : M \u2192 C) b {k} (Adv : SemSecAdv k) \u2192 \u21ba k Set\n      DetSemSecWinner enc b Adv = \u21baT (runDetSemSec enc Adv b)\n\n      DetSemSec : \u2200 k (enc : M \u2192 C) \u2192 Set\n      DetSemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runDetSemSec enc Adv)\n\n      luckySemSec : SemSecAdv 1\n      luckySemSec = mk\u2032 \u27ea 0\u207f , 0\u207f \u27eb\u1d30 (\u03bb _ \u2192 toss)\n\n      -- see OTPDetSemSec\n\n    module DeterministicCPASecurity where\n      DetCPAAdv : \u2200 k \u2192 Set\u2081\n      DetCPAAdv k = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) \u2192 Strategy \u2610_ Bit k\n\n      data DetCPAWinner (enc : M \u00d7 M \u2192 C) b (ms : List (M \u00d7 M)) : \u2200 {k} (q : \u2115) \u2192 Strategy id Bit k \u2192 Set\u2081 where\n        win : \u2200 {q} \u2192 DetCPAWinner enc b ms q (say b)\n        ask : \u2200 {k q m\u2080m\u2081 f} \u2192 m\u2080m\u2081 \u2209 ms \u2192 DetCPAWinner enc b (m\u2080m\u2081 \u2237 ms) q (f (enc m\u2080m\u2081)) \u2192\n              DetCPAWinner enc b ms {k} (suc q) (ask (enc m\u2080m\u2081) f)\n        rnd : \u2200 {A : Set} {n k q bs} {r : \u21ba n A} {f : A \u2192 _}\n              \u2192 DetCPAWinner enc b ms {k} q (f bs) \u2192 DetCPAWinner enc b ms q (rnd r f)\n\n      runDetCPA : \u2200 {k} (enc : M \u2192 C) (Adv : DetCPAAdv k) \u2192 Bit \u2192 \u21ba k Bit\n      runDetCPA enc Adv b = run (Adv (EXP b enc))\n\n      -- no use of CPAWinner\n      DetCPASec : \u2200 k (enc : M \u2192 C) \u2192 Set\u2081\n      DetCPASec k enc = \u2200 (Adv : DetCPAAdv k) \u2192 negligible-advantage (runDetCPA enc Adv)\n\n    CPAAdv : \u2200 k \u2192 Set\u2081\n    CPAAdv k = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 (\u21ba |K| C)) \u2192 Strategy \u2610_ Bit k\n\n    sem2CPAAdv : \u2200 {k} \u2192 SemSecAdv k \u2192 CPAAdv (|K| + k)\n    sem2CPAAdv (mk {k\u2080} {k\u2081} refl gen-m\u2080m\u2081 f) {\u2610_} enc =\n      subst (Strategy \u2610_ Bit) helper\n        (rnd gen-m\u2080m\u2081 (\u03bb m\u2080m\u2081 \u2192\n          ask (enc m\u2080m\u2081) (\u03bb c \u2192\n            rnd c (\u03bb c' \u2192 rnd (f c') say))))\n      where open import Data.Nat.Properties\n            open SemiringSolver\n            helper : k\u2080 + (|K| + (k\u2081 + 0)) \u2261 |K| + (k\u2080 + k\u2081)\n            helper = solve 3 (\u03bb k\u2080 |K| k\u2081 \u2192 k\u2080 :+ (|K| :+ (k\u2081 :+ con 0)) := |K| :+ (k\u2080 :+ k\u2081))\n                             refl k\u2080 |K| k\u2081\n\n    data CPAWinner (enc : M \u00d7 M \u2192 \u21ba |K| C) b : \u2200 {k} q \u2192 Strategy id Bit k \u2192 Set\u2081 where\n      win : \u2200 {q} \u2192 CPAWinner enc b q (say b)\n      ask : \u2200 {k q m\u2080m\u2081 f} \u2192 CPAWinner enc b {k} q (f (enc m\u2080m\u2081)) \u2192\n            CPAWinner enc b (suc q) (ask (enc m\u2080m\u2081) f)\n      rnd : \u2200 {A : Set} {n k q bs} {r : \u21ba n A} {f : A \u2192 _}\n            \u2192 CPAWinner enc b {k} q (f bs) \u2192 CPAWinner enc b q (rnd r f)\n\n    det : \u2200 {k a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 A \u2192 \u21ba k B\n    det f x = \u27ea f x \u27eb\n\n    runCPA : \u2200 {k} (enc : M \u2192 \u21ba |K| C) (Adv : CPAAdv k) \u2192 Bit \u2192 \u21ba k Bit\n    runCPA enc Adv b = run (Adv (EXP b enc))\n\n    -- no use of CPAWinner\n    CPASec : \u2200 k (enc : M \u2192 \u21ba |K| C) \u2192 Set\u2081\n    CPASec k enc = \u2200 (Adv : CPAAdv k) \u2192 negligible-advantage (runCPA enc Adv)\n\n    SemSecWinner : \u2200 (enc : M \u00d7 M \u2192 \u21ba |K| C) b k \u2192 SemSecAdv k \u2192 Set\u2081\n    SemSecWinner enc b k Adv = CPAWinner enc b 1 (sem2CPAAdv Adv enc)\n\n    runSemSec : \u2200 {k} (enc : M \u2192 \u21ba |K| C) \u2192 SemSecAdv k \u2192 Bit \u2192 \u21ba (|K| + k) Bit\n    runSemSec enc Adv = runCPA enc (sem2CPAAdv Adv)\n\n    -- no use of SemSecWinner\n    SemSec : \u2200 k (enc : M \u2192 \u21ba |K| C) \u2192 Set\n    SemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runSemSec enc Adv)\n\n    weakenWinner : \u2200 {enc b k q\u2080 q\u2081} {s : Strategy id Bit k} \u2192 CPAWinner enc b q\u2080 s \u2192 CPAWinner enc b (q\u2080 + q\u2081) s\n    weakenWinner win     = win\n    weakenWinner (ask w) = ask (weakenWinner w)\n    weakenWinner (rnd {n = n} {k} {bs = bs} w) = rnd {n = n} {k} {bs = bs} (weakenWinner w)\n\n    CPA2Sem-correct : \u2200 {enc b k q} {semAdv : SemSecAdv k}\n                      \u2192 SemSecWinner enc b k semAdv \u2192 CPAWinner enc b (suc q) (sem2CPAAdv semAdv enc)\n    CPA2Sem-correct = weakenWinner\n\n  record Forgery {a} {A : Set a} (P : A \u2192 Set) (queries : List A) : Set a where\n    field\n      forgery         : A\n      fresh-forgery   : forgery \u2209 queries\n      correct-forgery : P forgery\n  open Forgery public\n\n  module MACGames (|M| |Tag| : \u2115) where\n    M = Bits |M|\n    Tag = Bits |Tag|\n\n    -- One might also want that the adversary sends the trials to the challenger.\n    -- The challenger might then either end the game or maybe return the info to\n    -- the adversary.\n    MACAdv : Set\u2081\n    MACAdv = \u2200 {\u2610_} (mac : M \u2192 \u2610 Tag) \u2192 Strategy \u2610_ (M \u00d7 Tag) 0\n\n    -- or\n    -- MACAdv = \u2200 {\u2610_ Done} (mac : M \u2192 \u2610 Tag) (submit : M \u00d7 Tag \u2192 \u2610 Done) \u2192 Strategy \u2610_ Done\n\n    -- or\n    -- MACAdv = \u2200 {\u2610_ Done} (mac : M \u2192 \u2610 Tag)\n    --                       (try : M \u00d7 Tag \u2192 \u2610 Bool)\n    --                       (submit : M \u00d7 Tag \u2192 \u2610 Done) \u2192 Strategy \u2610_ Done\n\n    module MACGames' (mac : M \u2192 Tag) where\n      CorrectMAC : M \u00d7 Tag \u2192 Set\n      CorrectMAC (m , t) = mac m \u2261 t\n\n      correctMAC? : M \u00d7 Tag \u2192 Bool\n      correctMAC? (m , t) = mac m == t\n\n      data MACWinner (mts : List (M \u00d7 Tag)) : (q : \u2115) \u2192 Strategy id (M \u00d7 Tag) 0 \u2192 Set\u2081 where\n        win : \u2200 {q} (f : Forgery CorrectMAC mts) \u2192 MACWinner mts q (say (forgery f))\n        ask : \u2200 {m f q} \u2192 MACWinner ((m , mac m) \u2237 mts) q (f (mac m))\n                        \u2192 MACWinner mts (suc q) (ask (mac m) f)\n\n      SecMAC : Set\u2081\n      SecMAC = \u2200 (A : MACAdv) \u2192 Pr[ \u27ea correctMAC? \u00b7 run (A mac) \u27eb \u22611] \u22611\/2^ |Tag| -- check that bound\n\n  module CiphertextIntegrity (M C : Set) where\n    CIAdv : Set\u2081\n    CIAdv = \u2200 {\u2610_} (enc : M \u2192 \u2610 C) \u2192 Strategy \u2610_ C 0\n\n    module CiphertextIntegrity' (enc : M \u2192 C) (dec : C \u2192? M) where\n      CorrectCiphertext : C \u2192 Set\n      CorrectCiphertext c = IsJust (dec c)\n\n      data CIWinner (cs : List C) : (q : \u2115) \u2192 Strategy id C 0 \u2192 Set\u2081 where\n        win : \u2200 {q} (f : Forgery CorrectCiphertext cs) \u2192 CIWinner cs q (say (forgery f))\n        ask : \u2200 {m f q} \u2192 CIWinner (enc m \u2237 cs) q (f (enc m))\n                        \u2192 CIWinner cs (suc q) (ask (enc m) f)\n\n  module AuthenticatedEncryption (|K| |M| |C| : \u2115) where\n    open CipherGames |K| |M| |C|\n    open CiphertextIntegrity M C\n\n    data AERes : Set where\n      distinguish   : Bit \u2192 AERes\n      found-forgery : C \u2192 AERes\n\n    AEAdv : Set\u2081\n    AEAdv = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) \u2192 Strategy \u2610_ AERes 0\n\n    CCAAdv : Set\u2081\n    CCAAdv = \u2200 {\u2610_} (enc : M \u00d7 M \u2192 \u2610 C) (dec : C \u2192 \u2610 (Maybe M)) \u2192 Strategy \u2610_ Bit 0\n\n    module AE' (enc : M \u00d7 M \u2192 C) (dec : C \u2192? M) where\n      open CiphertextIntegrity' (\u03bb m \u2192 enc (m , m)) dec\n\n      data AEWinner (cs : List C) b : (q : \u2115) \u2192 Strategy id AERes 0 \u2192 Set\u2081 where\n        win-distinguish : \u2200 {q} \u2192 AEWinner cs b q (say (distinguish b))\n        win-forgery : \u2200 {q} (f : Forgery CorrectCiphertext cs)\n                       \u2192 AEWinner cs b q (say (found-forgery (forgery f)))\n        ask : \u2200 {m\u2080m\u2081 f q} \u2192 AEWinner cs b q (f (enc m\u2080m\u2081)) \u2192\n              AEWinner cs b (suc q) (ask (enc m\u2080m\u2081) f)\n\n      data CCAWinner b : (q : \u2115) \u2192 Strategy id Bit 0 \u2192 Set\u2081 where\n        win : \u2200 {q} \u2192 CCAWinner b q (say b)\n        ask-enc : \u2200 {m\u2080m\u2081 f q} \u2192 CCAWinner b q (f (enc m\u2080m\u2081)) \u2192\n                    CCAWinner b (suc q) (ask (enc m\u2080m\u2081) f)\n        ask-dec : \u2200 {c f q} \u2192 CCAWinner b q (f (dec c)) \u2192\n                    CCAWinner b (suc q) (ask (dec c) f)\n\n  -- CBC with rand IV is not CCA secure\n\n      -- AE \u21d2 CCA Secure\n\n  module OTPDetSemSec {n} (key : Bits n) where\n    open CipherGames n n n\n    open DeterministicSemanticSecurity\n\n    postulate\n      OTPDetSemSec : \u2200 {k} \u2192 DetSemSec k (OTP key)\n{-\n    OTPDetSemSec (mk refl gen-m\u2080m\u2081 stat-test) = {!!}\n    -- DetSemSec k enc = \u2200 (Adv : SemSecAdv k) \u2192 negligible-advantage (runDetSemSec enc Adv)\n    -- runDetSemSec enc (mk refl gen-m\u2080m\u2081 f) b = gen-m\u2080m\u2081 >>= \u03bb m\u2080m\u2081 \u2192 f (EXP b enc m\u2080m\u2081)\n\nPr[ runDetSemSec enc Adv b \u2261 b ]\n-}\n\n  Enc : \u2200 |K| |M| |C| \u2192 Set\n  Enc |K| |M| |C| = Bits |K| \u2192 Bits |M| \u2192 Bits |C|\n\n  module DerivedBlockCiphers\n    {|K| |M| |C|} {E : Enc |K| |M| |C|}\n    (E-sec : \u2200 {k} key \u2192 CipherGames.DeterministicSemanticSecurity.DetSemSec |K| |M| |C| k (E key)) where\n\n    open module CipherGames' {|M| |C|} = CipherGames |K| |M| |C|\n    open DeterministicSemanticSecurity\n\n    E\u2080 : Enc |K| |M| |C|\n    E\u2080 k m = 1\u207f \u2295 E k m\n\n    postulate\n      lem' : \u2200 {k} (Adv : SemSecAdv k) b (f : Bits k \u2192 Bits |C|) \u2192 runDetSemSec {k} (_\u2295_ 1\u207f \u2218 f) Adv b \u2261 \u27ea _xor_ 1b \u00b7 runDetSemSec f Adv b \u27eb\n    -- lem' Adv b f = {!!}\n\n      lem'' : \u2200 {k} (Adv : SemSecAdv k) b (f : Bits k \u2192 Bits |C|) \u2192 \u27ea _xor_ 1b \u00b7 runDetSemSec {k} (_\u2295_ 1\u207f \u2218 f) Adv b \u27eb \u2261 runDetSemSec f Adv b\n    -- lem'' Adv b f = {!!}\n\n    -- postulate\n    E\u2080-sec : \u2200 key \u2192 DetSemSec |M| (E\u2080 key)\n    E\u2080-sec key Adv = negligible-advantage-trans (negligible-advantage-xor (runDetSemSec (E\u2080 key) Adv 0b) (runDetSemSec (E key) Adv 0b) (lem' Adv 0b (E key))) (negligible-advantage-trans (E-sec key Adv) (negligible-advantage-xor (runDetSemSec (E key) Adv 1b) (runDetSemSec (E\u2080 key) Adv 1b) (sym (lem'' Adv 1b (E key)))))\n\n    E\u2081 : Enc |K| |M| (1 + |C|)\n    E\u2081 k m = 0\u2237 E k m\n\n    -- E\u2081-adv : \u2200 {k} key \u2192 SemSecAdv k (E key) \u2192 SemSecAdv k (E\u2081 key)\n    -- E\u2081-adv key Adv = {!E-sec!}\n\n    postulate\n      E\u2081-sec : \u2200 {k} key \u2192 DetSemSec k (E\u2081 key)\n    -- E\u2081-sec key Adv = {!E-sec key ?!}\n\n    -- runDetSemSec enc adv b \u2248 runDetSemSec enc' adv b\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1df741121f11612b4871c4ea4c9a7b5659b99a96","subject":"flat-funs: vector combinators!","message":"flat-funs: vector combinators!\n","repos":"crypto-agda\/crypto-agda","old_file":"flat-funs.agda","new_file":"flat-funs.agda","new_contents":"module flat-funs where\n\nopen import Data.Nat.NP\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Data.Bits using (Bits)\nimport Level as L\nimport Function as F\nopen import composable\nopen import vcomp\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n\n    _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    tt : \u2200 {_A} \u2192 _A `\u2192 `\u22a4\n\n    nil : \u2200 {_A B} \u2192 _A `\u2192 `Vec B 0\n    cons : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n\n    uncons : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n\n    constBits : \u2200 {n _A} \u2192 Bits n \u2192 _A `\u2192 `Bits n\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\n  swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  swap = < snd , fst >\n\n  first : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  first f = f *** idO\n\n  second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  second f = idO *** f\n\n  assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  assoc = < fst >>> fst , first snd >\n\n  assoc\u2032 : \u2200 {A B C} \u2192 (A `\u00d7 (B `\u00d7 C)) `\u2192 ((A `\u00d7 B) `\u00d7 C)\n  assoc\u2032 = < second fst , snd >>> snd >\n\n  `\u00d7-zip : \u2200 {A B C D E F} \u2192 ((A `\u00d7 B) `\u2192 C)\n                           \u2192 ((D `\u00d7 E) `\u2192 F)\n                           \u2192 ((A `\u00d7 D) `\u00d7 (B `\u00d7 E)) `\u2192 (C `\u00d7 F)\n  `\u00d7-zip f g = < < fst >>> fst , snd >>> fst > >>> f ,\n                 < fst >>> snd , snd >>> snd > >>> g >\n\n  head : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  head = uncons >>> fst\n\n  tail : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  tail = uncons >>> snd\n\n  <_\u2237_> : \u2200 {m n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A m `\u2192 `Vec B n)\n                  \u2192 `Vec A (1 + m) `\u2192 `Vec B (1 + n)\n  < f \u2237 g > = uncons >>> < f \u00d7 g > >>> cons\n\n  <_\u2237\u2032_> : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (A `\u2192 `Vec B n)\n                     \u2192 A `\u2192 `Vec B (1 + n)\n  < f \u2237\u2032 g > = < f , g > >>> cons\n\n  foldl : \u2200 {n A B} \u2192 (B `\u00d7 A `\u2192 B) \u2192 (B `\u00d7 `Vec A n) `\u2192 B\n  foldl {zero}  f = fst\n  foldl {suc n} f = second uncons\n                >>> assoc\u2032\n                >>> first f\n                >>> foldl f\n\n  foldl\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldl\u2081 f = uncons >>> foldl f\n\n  foldr : \u2200 {n A B} \u2192 (A `\u00d7 B `\u2192 B) \u2192 (`Vec A n `\u00d7 B) `\u2192 B\n  foldr {zero}  f = snd\n  foldr {suc n} f = first uncons\n                >>> assoc\n                >>> second (foldr f)\n                >>> f\n\n  foldr\u2081 : \u2200 {n A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (1 + n) `\u2192 A\n  foldr\u2081 f = uncons >>> swap >>> foldr f\n\n  map : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Vec A n `\u2192 `Vec B n)\n  map {zero}  f = nil\n  map {suc n} f = < f \u2237 map f >\n\n  zipWith : \u2200 {n A B C} \u2192 ((A `\u00d7 B) `\u2192 C)\n                        \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec C n\n  zipWith {zero}  f = nil\n  zipWith {suc n} f = < uncons \u00d7 uncons >\n                  >>> `\u00d7-zip f (zipWith f)\n                  >>> cons\n\n  zip : \u2200 {n A B} \u2192 (`Vec A n `\u00d7 `Vec B n) `\u2192 `Vec (A `\u00d7 B) n\n  zip = zipWith idO\n\n  singleton : \u2200 {A} \u2192 A `\u2192 `Vec A 1\n  singleton = < idO , nil > >>> cons\n\n  snoc : \u2200 {n A} \u2192 (`Vec A n `\u00d7 A) `\u2192 `Vec A (1 + n)\n  snoc {zero}  = snd >>> singleton\n  snoc {suc n} = first uncons >>> assoc >>> second snoc >>> cons\n\n  reverse : \u2200 {n A} \u2192 `Vec A n `\u2192 `Vec A n\n  reverse {zero}  = nil\n  reverse {suc n} = uncons >>> swap >>> first reverse >>> snoc\n  \n  append : \u2200 {m n A} \u2192 (`Vec A m `\u00d7 `Vec A n) `\u2192 `Vec A (m + n)\n  append {zero}  = snd\n  append {suc m} = first uncons\n               >>> assoc\n               >>> second append\n               >>> cons\n\n  splitAt : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 (`Vec A m `\u00d7 `Vec A n)\n  splitAt zero    = < nil , idO >\n  splitAt (suc m) = uncons\n                >>> second (splitAt m)\n                >>> assoc\u2032\n                >>> first cons\n\n  take : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A m\n  take zero    = nil\n  take (suc m) = < idO \u2237 take m >\n\n  drop : \u2200 m {n A} \u2192 `Vec A (m + n) `\u2192 `Vec A n\n  drop zero    = idO\n  drop (suc m) = tail >>> drop m\n\n  folda : \u2200 n {A} \u2192 (A `\u00d7 A `\u2192 A) \u2192 `Vec A (2^ n) `\u2192 A\n  folda zero    f = head\n  folda (suc n) f = splitAt (2^ n) >>> < folda n f \u00d7 folda n f > >>> f\n\n  init : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 `Vec A n\n  init {zero}  = nil\n  init {suc n} = < idO \u2237 init >\n\n  last : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 A\n  last {zero}  = head\n  last {suc n} = tail >>> last\n\n  concat : \u2200 {m n A} \u2192 `Vec (`Vec A m) n `\u2192 `Vec A (n * m)\n  concat {n = zero}  = nil\n  concat {n = suc n} = uncons >>> second concat >>> append\n\n  group : \u2200 {A} n k \u2192 `Vec A (n * k) `\u2192 `Vec (`Vec A k) n\n  group zero    k = nil\n  group (suc n) k = splitAt k >>> second (group n k) >>> cons\n\n  bind : \u2200 {m n A B} \u2192 (A `\u2192 `Vec B m) \u2192 `Vec A n `\u2192 `Vec B (n * m)\n  bind f = map f >>> concat\n\n  \u229b : \u2200 {n A B} \u2192 Vec (A `\u2192 B) n \u2192 `Vec A n `\u2192 `Vec B n\n  \u229b []       = nil\n  \u229b (f \u2237 fs) = uncons >>> < f \u00d7 \u229b fs > >>> cons\n\n  replicate : \u2200 {n A} \u2192 A `\u2192 `Vec A n\n  replicate {zero}  = nil\n  replicate {suc n} = < idO , replicate > >>> cons\n\n  module WithFin\n    (`Fin : \u2115 \u2192 T)\n    (fz : \u2200 {n _A} \u2192 _A `\u2192 `Fin (suc n))\n    (fs : \u2200 {n} \u2192 `Fin n `\u2192 `Fin (suc n))\n    (elim-Fin0 : \u2200 {A} \u2192 `Fin 0 `\u2192 A)\n    (elim-Fin1+ : \u2200 {n A B} \u2192 (A `\u2192 B) \u2192 (`Fin n `\u00d7 A `\u2192 B) \u2192 `Fin (suc n) `\u00d7 A `\u2192 B) where\n\n    tabulate : \u2200 {n A _B} \u2192 (`Fin n `\u2192 A) \u2192 _B `\u2192 `Vec A n\n    tabulate {zero}  f = nil\n    tabulate {suc n} f = < fz >>> f , tabulate (fs >>> f) > >>> cons\n\n    lookup : \u2200 {n A} \u2192 `Fin n `\u00d7 `Vec A n `\u2192 A\n    lookup {zero}  = fst >>> elim-Fin0\n    lookup {suc n} = elim-Fin1+ head (second tail >>> lookup)\n\n    allFin : \u2200 {n _A} \u2192 _A `\u2192 `Vec (`Fin n) n\n    allFin = tabulate idO\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Product renaming (zip to \u00d7-zip; map to \u00d7-map)\nopen import Function\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id (\u03bb f g x \u2192 g (f x)) (\u03bb f g \u2192 \u00d7-map f g) _&&&_ proj\u2081 proj\u2082 _ (const []) (uncurry _\u2237_) uncons const\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n  uncons : \u2200 {n A} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\n  uncons (x \u2237 xs) = x , xs\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id (\u03bb f g \u2192 g \u2218 f) (VComposable._***_ bitsFunVComp) _&&&_ (\u03bb {A} \u2192 take A)\n                    (\u03bb {A} \u2192 drop A) (const []) (const []) id id (\u03bb xs _ \u2192 concat (map [_] xs) {- ugly? -})\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe\n                           (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                        (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                        (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.cons)\n                        (S.uncons , T.uncons) (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4 = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                         (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                         (S.tt , T.tt) (S.nil , T.nil) (S.cons , T.idO)\n                         (S.uncons , T.idO) (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ _\u2294_ _\u2294_ 0 0 0 0 0 0 (const 0)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ _+_ _+_ 0 0 0 0 0 0 (\u03bb {n} _ \u2192 n)\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","old_contents":"module flat-funs where\n\nopen import Data.Nat\nopen import Data.Bits using (Bits)\nimport Level as L\nimport Function as F\nopen import composable\nopen import vcomp\nopen import universe\n\nrecord FlatFuns {t} (T : Set t) : Set (L.suc t) where\n  constructor mk\n  field\n    universe : Universe T\n    _`\u2192_    : T \u2192 T \u2192 Set\n  infix 0 _`\u2192_\n  open Universe universe public\n\nrecord FlatFunsOps {t} {T : Set t} (\u266dFuns : FlatFuns T) : Set t where\n  constructor mk\n  open FlatFuns \u266dFuns\n  infixr 1 _>>>_\n  infixr 3 _***_\n  infixr 3 _&&&_\n  field\n    idO : \u2200 {A} \u2192 A `\u2192 A\n\n    _>>>_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (B `\u2192 C) \u2192 (A `\u2192 C)\n\n    _***_ : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n\n    _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n\n    fst : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n    snd : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n\n    constBits : \u2200 {n} \u2192 Bits n \u2192 `\u22a4 `\u2192 `Bits n\n\n  <_,_> : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  < f , g > = f &&& g\n\n  <_\u00d7_> : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n  < f \u00d7 g > = f *** g\n\nopen import Data.Unit using (\u22a4)\nopen import Data.Vec\nopen import Data.Bits\nopen import Data.Fin using (Fin) renaming (_+_ to _+\u1da0_)\nopen import Data.Product renaming (zip to \u00d7-zip; map to \u00d7-map)\nopen import Function\n\n-\u2192- : Set \u2192 Set \u2192 Set\n-\u2192- A B = A \u2192 B\n\n_\u2192\u1da0_ : \u2115 \u2192 \u2115 \u2192 Set\n_\u2192\u1da0_ i o = Fin i \u2192 Fin o\n\nmapArr : \u2200 {s t} {S : Set s} {T : Set t} (F G : T \u2192 S)\n          \u2192 (S \u2192 S \u2192 Set) \u2192 (T \u2192 T \u2192 Set)\nmapArr F G _`\u2192_ A B = F A `\u2192 G B\n\nfun\u266dFuns : FlatFuns Set\nfun\u266dFuns = mk Set-U -\u2192-\n\nbitsFun\u266dFuns : FlatFuns \u2115\nbitsFun\u266dFuns = mk Bits-U _\u2192\u1d47_\n\nfinFun\u266dFuns : FlatFuns \u2115\nfinFun\u266dFuns = mk Fin-U _\u2192\u1da0_\n\nfun\u266dOps : FlatFunsOps fun\u266dFuns\nfun\u266dOps = mk id (\u03bb f g x \u2192 g (f x)) (\u03bb f g \u2192 \u00d7-map f g) _&&&_ proj\u2081 proj\u2082 const\n  where\n  _&&&_ : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u2192 C) \u2192 A \u2192 B \u00d7 C\n  (f &&& g) x = (f x , g x)\n\nbitsFun\u266dOps : FlatFunsOps bitsFun\u266dFuns\nbitsFun\u266dOps = mk id (\u03bb f g \u2192 g \u2218 f) (VComposable._***_ bitsFunVComp) _&&&_ (\u03bb {A} \u2192 take A)\n                    (\u03bb {A} \u2192 drop A) (\u03bb xs _ \u2192 xs ++ [] {- ugly? -})\n  where\n  open FlatFuns bitsFun\u266dFuns\n  _&&&_ : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  (f &&& g) x = (f x ++ g x)\n\n\u00d7-\u266dFuns : \u2200 {s t} {S : Set s} {T : Set t} \u2192 FlatFuns S \u2192 FlatFuns T \u2192 FlatFuns (S \u00d7 T)\n\u00d7-\u266dFuns funs-S funs-T = mk (\u00d7-U S.universe T.universe)\n                           (\u03bb { (A\u2080 , A\u2081) (B\u2080 , B\u2081) \u2192 (A\u2080 S.`\u2192 B\u2080) \u00d7 (A\u2081 T.`\u2192 B\u2081) })\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7\u22a4-\u266dFuns : \u2200 {s} {S : Set s} \u2192 FlatFuns S \u2192 FlatFuns \u22a4 \u2192 FlatFuns S\n\u00d7\u22a4-\u266dFuns funs-S funs-T = mk S.universe\n                           (\u03bb A B \u2192 (A S.`\u2192 B) \u00d7 (_ T.`\u2192 _))\n  where module S = FlatFuns funs-S\n        module T = FlatFuns funs-T\n\n\u00d7-\u266dOps : \u2200 {s t} {S : Set s} {T : Set t} {funs-S : FlatFuns S} {funs-T : FlatFuns T}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-T\n         \u2192 FlatFunsOps (\u00d7-\u266dFuns funs-S funs-T)\n\u00d7-\u266dOps ops-S ops-T = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                        (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                        (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-T\n        open FlatFunsOps fun\u266dOps\n\n\u00d7\u22a4-\u266dOps : \u2200 {s} {S : Set s} {funs-S : FlatFuns S} {funs-\u22a4 : FlatFuns \u22a4}\n         \u2192 FlatFunsOps funs-S \u2192 FlatFunsOps funs-\u22a4\n         \u2192 FlatFunsOps (\u00d7\u22a4-\u266dFuns funs-S funs-\u22a4)\n\u00d7\u22a4-\u266dOps ops-S ops-\u22a4 = mk (S.idO , T.idO) (\u00d7-zip S._>>>_ T._>>>_) (\u00d7-zip S._***_ T._***_)\n                         (\u00d7-zip S._&&&_ T._&&&_) (S.fst , T.fst) (S.snd , T.snd)\n                         (S.constBits &&& T.constBits)\n  where module S = FlatFunsOps ops-S\n        module T = FlatFunsOps ops-\u22a4\n        open FlatFunsOps fun\u266dOps\n\n\nconstFuns : Set \u2192 FlatFuns \u22a4\nconstFuns A = mk \u22a4-U (\u03bb _ _ \u2192 A)\n\ntimeOps : FlatFunsOps (constFuns \u2115)\ntimeOps = mk 0 _+_ _\u2294_ _\u2294_ 0 0 (const 0)\n\nspaceOps : FlatFunsOps (constFuns \u2115)\nspaceOps = mk 0 _+_ _+_ _+_ 0 0 (\u03bb {n} _ \u2192 n)\n\ntime\u00d7spaceOps : FlatFunsOps (constFuns (\u2115 \u00d7 \u2115))\ntime\u00d7spaceOps = \u00d7\u22a4-\u266dOps timeOps spaceOps\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"feebbe1b65dd5ca3a92cd9412a134a1e001236de","subject":"pisearch: P\u2190Ind","message":"pisearch: P\u2190Ind\n","repos":"crypto-agda\/crypto-agda","old_file":"pisearch.agda","new_file":"pisearch.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\n\u03a0' : \u2200 {A} \u2192 Search A \u2192 (A \u2192 \u2605) \u2192 \u2605\n\u03a0' sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605\u2080) \u2192 (B : A \u2192 \u2605\u2080) \u2192 \u2605\u2080\n\u03a0 A B = (x : A) \u2192 B x\n\nP\u2192 : \u2200 {A} (sA : Search A) \u2192 \u2605\u2080\nP\u2192 {A} sA = \u2200 {B} \u2192 \u03a0' sA B \u2192 \u03a0 A B\n\nP\u2190 : \u2200 {A} (sA : Search A) \u2192 \u2605\u2080\nP\u2190 {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 \u03a0' sA B\n\nP\u2192\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n     \u2192 P\u2192 sA\n     \u2192 (\u2200 {x} \u2192 P\u2192 (sB {x}))\n     \u2192 P\u2192 (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nP\u2192\u03a3 _ _ PA PB t = uncurry (PB \u2218 PA t)\n\nP\u2190\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x))\n      \u2192 P\u2190 sA\n      \u2192 (\u2200 {x} \u2192 P\u2190 (sB {x}))\n      \u2192 P\u2190 (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nP\u2190\u03a3 _ _ PA PB f = PA (PB \u2218 curry f)\n\nP\u2192\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2192 sA \u2192 P\u2192 sB \u2192 P\u2192 (sA \u00d7Search sB)\nP\u2192\u00d7 sA sB PA PB = P\u2192\u03a3 sA sB PA PB\n\nP\u2190\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2190 sA \u2192 P\u2190 sB \u2192 P\u2190 (sA \u00d7Search sB)\nP\u2190\u00d7 sA sB PA PB = P\u2190\u03a3 sA sB PA PB\n\nP\u2192Bit : P\u2192 (search \u03bcBit)\nP\u2192Bit (f , t) false = f\nP\u2192Bit (f , t) true  = t\n\nP\u2190Bit : P\u2190 (search \u03bcBit)\nP\u2190Bit f = f false , f true\n\nP\u2192\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2192 sA \u2192 P\u2192 sB \u2192 P\u2192 (sA +Search sB)\nP\u2192\u228e sA sB PA PB (t , u) (inj\u2081 x) = PA t x\nP\u2192\u228e sA sB PA PB (t , u) (inj\u2082 x) = PB u x\n\nP\u2190\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2190 sA \u2192 P\u2190 sB \u2192 P\u2190 (sA +Search sB)\nP\u2190\u228e sA sB PA PB f = PA (f \u2218 inj\u2081) , PB (f \u2218 inj\u2082)\n\n-- P\u2192Ind : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 P\u2192 sA\n-- P\u2192Ind {A} {sA} indA {B} t = ?\n\nP\u2190Ind : \u2200 {A} {sA : Search A} \u2192 SearchInd sA \u2192 P\u2190 sA\nP\u2190Ind indA = indA (\u03bb s \u2192 \u03a0' s _) _,_\n","old_contents":"{-# OPTIONS --type-in-type #-}\nmodule pisearch where\nopen import Type hiding (\u2605_)\nopen import Function\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Bool.NP\nopen import Search.Type\nopen import Search.Searchable.Product\nopen import Search.Searchable\nopen import sum\n\n\u03a0' : \u2200 {A} \u2192 Search A \u2192 (B : A \u2192 \u2605) \u2192 \u2605\n\u03a0' sA B = sA _\u00d7_ B\n\n\u03a0 : (A : \u2605\u2080) \u2192 (B : A \u2192 \u2605\u2080) \u2192 \u2605\u2080\n\u03a0 A B = (x : A) \u2192 B x\n\nP\u2192 : \u2200 {A} (sA : Search A) \u2192 \u2605\u2080\nP\u2192 {A} sA = \u2200 {B} \u2192 \u03a0' sA B \u2192 \u03a0 A B\n\nP\u2190 : \u2200 {A} (sA : Search A) \u2192 \u2605\u2080\nP\u2190 {A} sA = \u2200 {B} \u2192 \u03a0 A B \u2192 \u03a0' sA B\n\nP\u2192\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2192 sA \u2192 P\u2192 sB \u2192 P\u2192 (sA \u00d7Search sB)\nP\u2192\u00d7 sA sB PA PB t (x , y) = PB (PA t x) y\n\nP\u2190\u00d7 : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2190 sA \u2192 P\u2190 sB \u2192 P\u2190 (sA \u00d7Search sB)\nP\u2190\u00d7 sA sB PA PB f = PA (\u03bb x \u2192 PB (curry f x))\n\nP\u2192\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x)) \u2192 P\u2192 sA \u2192 (\u2200 {x} \u2192 P\u2192 (sB {x})) \u2192 P\u2192 (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nP\u2192\u03a3 sA sB PA PB t (x , y) = PB (PA t x) y\n\nP\u2190\u03a3 : \u2200 {A} {B : A \u2192 \u2605} (sA : Search A) (sB : \u2200 {x} \u2192 Search (B x)) \u2192 P\u2190 sA \u2192 (\u2200 {x} \u2192 P\u2190 (sB {x})) \u2192 P\u2190 (\u03a3Search sA (\u03bb {x} \u2192 sB {x}))\nP\u2190\u03a3 sA sB PA PB f = PA (\u03bb x \u2192 PB (curry f x))\n\nP\u2192Bit : P\u2192 (search \u03bcBit)\nP\u2192Bit (f , t) false = f\nP\u2192Bit (f , t) true  = t\n\nP\u2190Bit : P\u2190 (search \u03bcBit)\nP\u2190Bit f = f false , f true\n\nP\u2192\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2192 sA \u2192 P\u2192 sB \u2192 P\u2192 (sA +Search sB)\nP\u2192\u228e sA sB PA PB (t , u) (inj\u2081 x) = PA t x\nP\u2192\u228e sA sB PA PB (t , u) (inj\u2082 x) = PB u x\n\nP\u2190\u228e : \u2200 {A B} (sA : Search A) (sB : Search B) \u2192 P\u2190 sA \u2192 P\u2190 sB \u2192 P\u2190 (sA +Search sB)\nP\u2190\u228e sA sB PA PB f = PA (f \u2218 inj\u2081) , PB (f \u2218 inj\u2082)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"25c5602cdd3e8458291abfd1227a1cc311e20a21","subject":"Fixed typo.","message":"Fixed typo.\n\nIgnore-this: 4ea02904e3ef7c53e5920ebc03362216\n\ndarcs-hash:20110513150633-3bd4e-18969dcf53bcc26c1aa0adf3fbb24637513eb69d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function using ( _$_ )\n\nopen import FOTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The FOTC natural numbers type\n        )\nopen import FOTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import FOTC.Data.Nat.Inequalities.Properties public\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 GE n n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\nx<Sy\u2192x\u2264y zN Nn 0<Sn       = 0\u2264x Nn\nx<Sy\u2192x\u2264y (sN Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m (succ n)\nx\u2264y\u2192x<Sy {n = n} zN      Nn 0\u2264n  = <-0S n\nx\u2264y\u2192x<Sy         (sN Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 NGT m n\nx\u2264y\u2192x\u226fy zN          Nn          0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (sN Nm)     zN          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (<-SS m (succ n))) Sm\u2264Sn))\n  where\n    postulate prf : NGT m n \u2192  -- IH.\n                    NGT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx\u226fy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 NGT m n \u2192 LE m n\nx\u226fy\u2192x\u2264y zN          Nn          _     = 0\u2264x Nn\nx\u226fy\u2192x\u2264y (sN {m} Nm) zN          Sm\u226f0  = \u22a5-elim (true\u2260false\n                                                 (trans (sym (<-0S m)) Sm\u226f0))\nx\u226fy\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm\u226fSn =\n  prf (x\u226fy\u2192x\u2264y Nm Nn (trans (sym (<-SS n m)) Sm\u226fSn))\n  where\n    postulate prf : LE m n \u2192  -- IH.\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 NGT m n\nx>y\u2228x\u226fy Nm Nn = [ (\u03bb m>n \u2192 inj\u2081 m>n) ,\n                  (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192x\u226fy Nm Nn m\u2264n))\n                ] (x>y\u2228x\u2264y Nm Nn)\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<y\u2238x Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (cong succ m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u226fSy\u2192x\u226fy\u2228x\u2261Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 NGT m (succ n) \u2192 NGT m n \u2228 m \u2261 succ n\nx\u226fSy\u2192x\u226fy\u2228x\u2261Sy {m} {n} Nm Nn m\u226fSn =\n  [ (\u03bb m<Sn \u2192 inj\u2081 (x\u2264y\u2192x\u226fy Nm Nn (x<Sy\u2192x\u2264y Nm Nn m<Sn)))\n  , (\u03bb m\u2261Sn \u2192 inj\u2082 m\u2261Sn)\n  ]\n  (x<Sy\u2192x<y\u2228x\u2261y Nm (sN Nn) (x\u2264y\u2192x<Sy Nm (sN Nn) (x\u226fy\u2192x\u2264y Nm (sN Nn) m\u226fSn)))\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function using ( _$_ )\n\nopen import FOTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The FOTC natural numbers type\n        )\nopen import FOTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import FOTC.Data.Nat.Inequalities.Properties public\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 GE n n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\nx<Sy\u2192x\u2264y zN Nn 0<Sn       = 0\u2264x Nn\nx<Sy\u2192x\u2264y (sN Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m (succ n)\nx\u2264y\u2192x<Sy {n = n} zN      Nn 0\u2264n  = <-0S n\nx\u2264y\u2192x<Sy         (sN Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 NGT m n\nx\u2264y\u2192x\u226fy zN          Nn          0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (sN Nm)     zN          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (<-SS m (succ n))) Sm\u2264Sn))\n  where\n    postulate prf : NGT m n \u2192  -- IH.\n                    NGT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx\u226fy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 NGT m n \u2192 LE m n\nx\u226fy\u2192x\u2264y zN          Nn          _     = 0\u2264x Nn\nx\u226fy\u2192x\u2264y (sN {m} Nm) zN          Sm\u226f0  = \u22a5-elim (true\u2260false\n                                                 (trans (sym (<-0S m)) Sm\u226f0))\nx\u226fy\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm\u226fSn =\n  prf (x\u226fy\u2192x\u2264y Nm Nn (trans (sym (<-SS n m)) Sm\u226fSn))\n  where\n    postulate prf : LE m n \u2192  -- IH.\n                    LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 NGT m n\nx>y\u2228x\u226fy Nm Nn = [ (\u03bb m>n \u2192 inj\u2081 m>n) ,\n                  (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192x\u226fy Nm Nn m\u2264n))\n                ] (x>y\u2228x\u2264y Nm Nn)\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<y\u2238x Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (cong succ m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u226fSy\u2192x\u226fy\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 NGT m (succ n) \u2192 NGT m n \u2228 m \u2261 succ n\nx\u226fSy\u2192x\u226fy\u2228x\u2261y {m} {n} Nm Nn m\u226fSn =\n  [ (\u03bb m<Sn \u2192 inj\u2081 (x\u2264y\u2192x\u226fy Nm Nn (x<Sy\u2192x\u2264y Nm Nn m<Sn)))\n  , (\u03bb m\u2261Sn \u2192 inj\u2082 m\u2261Sn)\n  ]\n  (x<Sy\u2192x<y\u2228x\u2261y Nm (sN Nn) (x\u2264y\u2192x<Sy Nm (sN Nn) (x\u226fy\u2192x\u2264y Nm (sN Nn) m\u226fSn)))\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6f7ab8c3c08753b21a02281f03b869e332d90756","subject":"Lift right zero","message":"Lift right zero\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SquareMatrix.agda","new_file":"FLABloM\/SquareMatrix.agda","new_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  setoidS : (r c : Shape) \u2192 Setoid _ _\n  setoidS r c =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = \u2243S r c\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S' 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning (setoidS a L)\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 L)\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS a c)\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a c\u2081)\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c)\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c\u2081)\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n  zeroS\u02b3 : (a b c : Shape) (x : M s a b) \u2192\n    (x \u2219S 0S b c) \u2243S' 0S a c\n  zeroS\u02b3 L L L (One x) = zero\u02b3 x\n  zeroS\u02b3 L L (B c c\u2081) (One x) =\n    (zeroS\u02b3 L L c (One x)) , (zeroS\u02b3 L L c\u2081 (One x))\n  zeroS\u02b3 L (B b b\u2081) L (Row x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02b3 L b L x\n      ih\u2081 = zeroS\u02b3 L b\u2081 L x\u2081\n    in begin\n      Row x x\u2081 \u2219S Col (0S b L) (0S b\u2081 L)\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (x \u2219S 0S b L) +S (x\u2081 \u2219S 0S b\u2081 L)\n    \u2248\u27e8 <+S> L L {x \u2219S 0S b L} {0S L L} {x\u2081 \u2219S 0S b\u2081 L} {0S L L} ih ih\u2081 \u27e9\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02b3 L (B b b\u2081) (B c c\u2081) (Row x x\u2081) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02b3 L b c x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n      0S L c +S 0S L c\n    \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n      0S L c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02b3 L b c\u2081 x\n      ih\u2081 = zeroS\u02b3 L b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 (0S L c\u2081) \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) L L (Col x x\u2081) = zeroS\u02b3 a L L x , zeroS\u02b3 a\u2081 L L x\u2081\n  zeroS\u02b3 (B a a\u2081) L (B c c\u2081) (Col x x\u2081) =\n    zeroS\u02b3 a L c x ,\n    zeroS\u02b3 a L c\u2081 x ,\n    zeroS\u02b3 a\u2081 L c x\u2081 ,\n    zeroS\u02b3 a\u2081 L c\u2081 x\u2081\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) L (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b L x\n      ih\u2081 = zeroS\u02b3 a b\u2081 L x\u2081\n    in begin\n      x \u2219S 0S b L +S x\u2081 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L (0S _ _) \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b L x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 L x\u2083\n    in begin\n      x\u2082 \u2219S 0S b L +S x\u2083 \u2219S 0S b\u2081 L\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 L (0S _ _) \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02b3 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c x\u2081\n    in begin\n      x \u2219S 0S b c +S x\u2081 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c (0S _ _) \u27e9\n      0S _ _\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a b c\u2081 x\n      ih\u2081 = zeroS\u02b3 a b\u2081 c\u2081 x\u2081\n    in begin\n      x \u2219S 0S b c\u2081 +S x\u2081 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a c\u2081 (0S _ _) \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c +S x\u2083 \u2219S 0S b\u2081 c\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c (0S _ _) \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS _ _)\n      ih = zeroS\u02b3 a\u2081 b c\u2081 x\u2082\n      ih\u2081 = zeroS\u02b3 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      x\u2082 \u2219S 0S b c\u2081 +S x\u2083 \u2219S 0S b\u2081 c\u2081\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S _ _ +S 0S _ _\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 (0S _ _) \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = zeroS\u02b3 shape shape shape\n      ; _<\u2219>_ = {!!}\n      }\n","old_contents":"-- Matrices indexed by shape\nmodule SquareMatrix where\n\nopen import Shape\nopen import Matrix\n\n-- Square matrices have the same shape in both dimensions\nSq : Set \u2192 Shape \u2192 Set\nSq a shape = M a shape shape\n\nopen import SemiNearRingRecord\nopen import Algebra.Structures\nopen import Relation.Binary\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality hiding (trans; sym) renaming (refl to refl-\u2261)\nimport Relation.Binary.EqReasoning as EqReasoning\n\n-- Lifting a SNR to a to a Square matrix of some shape\nSquare : SemiNearRing \u2192 Shape \u2192 SemiNearRing\nSquare snr shape = SNR\n  where\n  open SemiNearRing snr\n\n  S = Sq s shape\n\n  open Operations s _\u2219\u209b_ _+\u209b_\n    renaming (_+_ to _+S_; _*_ to _\u2219S_)\n\n  0S : (r c : Shape) \u2192 M s r c\n  0S L L = One 0\u209b\n  0S L (B s s\u2081) = Row (0S L s) (0S L s\u2081)\n  0S (B r r\u2081) L = Col (0S r L) (0S r\u2081 L)\n  0S (B r r\u2081) (B s s\u2081) =\n      Q (0S r s) (0S r s\u2081)\n        (0S r\u2081 s) (0S r\u2081 s\u2081)\n\n  \u2243S : (r c : Shape) \u2192\n        M s r c \u2192 M s r c \u2192 Set\n  \u2243S L L (One x) (One x\u2081) = x \u2243\u209b x\u2081\n  \u2243S L (B c\u2081 c\u2082) (Row m m\u2081) (Row n n\u2081) = \u2243S L c\u2081 m n \u00d7 \u2243S L c\u2082 m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) L (Col m m\u2081) (Col n n\u2081) = \u2243S r\u2081 L m n \u00d7 \u2243S r\u2082 L m\u2081 n\u2081\n  \u2243S (B r\u2081 r\u2082) (B c\u2081 c\u2082) (Q m00 m01 m10 m11) (Q n00 n01 n10 n11) =\n    \u2243S r\u2081 c\u2081 m00 n00 \u00d7 \u2243S r\u2081 c\u2082 m01 n01 \u00d7\n    \u2243S r\u2082 c\u2081 m10 n10 \u00d7 \u2243S r\u2082 c\u2082 m11 n11\n\n\n  _\u2243S'_ : {r c : Shape} \u2192 M s r c \u2192 M s r c \u2192 Set\n  _\u2243S'_ {r} {c} m n = \u2243S r c m n\n\n\n  reflS : (r c : Shape) \u2192\n    {X : M s r c} \u2192 X \u2243S' X\n  reflS L L {X = One x} = refl\u209b {x}\n  reflS L (B c\u2081 c\u2082) {X = Row X Y} = reflS L c\u2081 , reflS L c\u2082\n  reflS (B r\u2081 r\u2082) L {X = Col X Y} = reflS r\u2081 L , reflS r\u2082 L\n  reflS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {X = Q X Y Z W} =\n    reflS r\u2081 c\u2081 , reflS r\u2081 c\u2082 ,\n    reflS r\u2082 c\u2081 , reflS r\u2082 c\u2082\n\n  symS : (r c : Shape) \u2192\n    {i j : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' i\n  symS L L {One x} {One x\u2081} p = sym\u209b p\n  symS L (B c\u2081 c\u2082) {Row i\u2081 i\u2082} {Row j\u2081 j\u2082} (p , q) = symS L c\u2081 p , symS L c\u2082 q\n  symS (B r\u2081 r\u2082) L {Col i\u2081 i} {Col j j\u2081} (p , q) = symS r\u2081 L p , symS r\u2082 L q\n  symS (B r r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j j\u2081 j\u2082 j\u2083} (p , q , x , y) =\n    symS r c\u2081 p , symS r c\u2082 q ,\n    symS r\u2082 c\u2081 x , symS r\u2082 c\u2082 y\n\n  transS : (r c : Shape) \u2192\n    {i j k : M s r c} \u2192 i \u2243S' j \u2192 j \u2243S' k \u2192 i \u2243S' k\n  transS L L {One x} {One x\u2081} {One x\u2082} p q = trans p q\n  transS L (B c\u2081 c\u2082) {Row i i\u2081} {Row j j\u2081} {Row k k\u2081} (p , q) (p' , q') =\n    transS L c\u2081 p p' , transS L c\u2082 q q'\n  transS (B r\u2081 r\u2082) L {Col i i\u2081} {Col j j\u2081} {Col k k\u2081} (p , q) (p' , q') =\n    transS r\u2081 L p p' , transS r\u2082 L q q'\n  transS (B r\u2081 r\u2082) (B c\u2081 c\u2082) {Q i\u2082 i i\u2083 i\u2081} {Q j\u2082 j j\u2083 j\u2081} {Q k k\u2081 k\u2082 k\u2083}\n    (p , q , x , y) (p' , q' , x' , y') =\n    transS r\u2081 c\u2081 p p' , transS r\u2081 c\u2082 q q' ,\n    transS r\u2082 c\u2081 x x' , transS r\u2082 c\u2082 y y'\n\n  isEquivS =\n    record\n      { refl = reflS shape shape\n      ; sym = symS shape shape\n      ; trans = transS shape shape }\n\n  liftAssoc : (r c : Shape) (x y z : M s r c) \u2192 ((x +S y) +S z) \u2243S' (x +S (y +S z))\n  liftAssoc L L (One x) (One y) (One z) = assoc\u209b x y z\n  liftAssoc L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) (Row z z\u2081) =\n    liftAssoc L c x y z  , liftAssoc L c\u2081 x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) (Col z z\u2081) =\n    liftAssoc r L x y z , liftAssoc r\u2081 L x\u2081 y\u2081 z\u2081\n  liftAssoc (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) (Q z z\u2081 z\u2082 z\u2083) =\n    (liftAssoc r c x y z) , (liftAssoc r c\u2081 x\u2081 y\u2081 z\u2081) ,\n    (liftAssoc r\u2081 c x\u2082 y\u2082 z\u2082) , (liftAssoc r\u2081 c\u2081 x\u2083 y\u2083 z\u2083)\n\n  <+S> : (r c : Shape) {x y u v : M s r c} \u2192\n    x \u2243S' y \u2192 u \u2243S' v \u2192 (x +S u) \u2243S' (y +S v)\n  <+S> L L {One x} {One x\u2081} {One x\u2082} {One x\u2083} p q = p <+> q\n  <+S> L (B c c\u2081) {Row x x\u2081} {Row y y\u2081} {Row u u\u2081} {Row v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> L c p q , <+S> L c\u2081 p\u2081 q\u2081\n  <+S> (B r r\u2081) L {Col x x\u2081} {Col y y\u2081} {Col u u\u2081} {Col v v\u2081} (p , p\u2081) (q , q\u2081) =\n    <+S> r L p q , <+S> r\u2081 L p\u2081 q\u2081\n  <+S> (B r r\u2081) (B c c\u2081) {Q x x\u2081 x\u2082 x\u2083} {Q y y\u2081 y\u2082 y\u2083} {Q u u\u2081 u\u2082 u\u2083} {Q v v\u2081 v\u2082 v\u2083}\n    (p , p\u2081 , p\u2082 , p\u2083) (q , q\u2081 , q\u2082 , q\u2083) =\n    <+S> r c p q , <+S> r c\u2081 p\u2081 q\u2081 ,\n    <+S> r\u2081 c p\u2082 q\u2082 , <+S> r\u2081 c\u2081 p\u2083 q\u2083\n\n  isSemgroupS =\n    record\n      { isEquivalence = isEquivS\n      ; assoc = liftAssoc shape shape\n      ; \u2219-cong = <+S> shape shape }\n\n\n  identS\u02e1 : (r c : Shape) (x : M s r c) \u2192\n     0S r c +S x \u2243S' x\n  identS\u02e1 L L (One x) = identity\u02e1\u209b x\n  identS\u02e1 L (B c c\u2081) (Row x x\u2081) = identS\u02e1 L c x , identS\u02e1 L c\u2081 x\u2081\n  identS\u02e1 (B r r\u2081) L (Col x x\u2081) = identS\u02e1 r L x , identS\u02e1 r\u2081 L x\u2081\n  identS\u02e1 (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    identS\u02e1 r c x , identS\u02e1 r c\u2081 x\u2081 ,\n    identS\u02e1 r\u2081 c x\u2082 , identS\u02e1 r\u2081 c\u2081 x\u2083\n\n  liftComm : (r c : Shape) \u2192 (x y : M s r c) \u2192\n    (x +S y) \u2243S' (y +S x)\n  liftComm L L (One x) (One x\u2081) = comm\u209b x x\u2081\n  liftComm L (B c c\u2081) (Row x x\u2081) (Row y y\u2081) = (liftComm L c x y) , (liftComm L c\u2081 x\u2081 y\u2081)\n  liftComm (B r r\u2081) L (Col x x\u2081) (Col y y\u2081) = (liftComm r L x y) , (liftComm r\u2081 L x\u2081 y\u2081)\n  liftComm (B r r\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) (Q y y\u2081 y\u2082 y\u2083) =\n    liftComm r c x y , liftComm r c\u2081 x\u2081 y\u2081 ,\n    liftComm r\u2081 c x\u2082 y\u2082 , liftComm r\u2081 c\u2081 x\u2083 y\u2083\n\n  isCommMonS =\n    record\n      { isSemigroup = isSemgroupS\n      ; identity\u02e1 = identS\u02e1 shape shape\n      ; comm = liftComm shape shape }\n\n\n  setoidS : (r c : Shape) \u2192 Setoid _ _\n  setoidS r c =\n    record\n      { Carrier = M s r c\n      ; _\u2248_ = \u2243S r c\n      ; isEquivalence =\n        record\n          { refl = reflS r c ; sym = symS r c ; trans = transS r c } }\n\n\n  zeroS\u02e1 : (a b c : Shape) (x : M s b c) \u2192\n    (0S a b \u2219S x) \u2243S' 0S a c\n  zeroS\u02e1 L L L (One x) = zero\u02e1 x\n  zeroS\u02e1 L L (B c c\u2081) (Row x x\u2081) = (zeroS\u02e1 L L c x) , (zeroS\u02e1 L L c\u2081 x\u2081)\n  zeroS\u02e1 L (B b b\u2081) L (Col x x\u2081) =\n    let\n      open EqReasoning (setoidS L L)\n      ih = zeroS\u02e1 L b L x\n      ih\u2081 = zeroS\u02e1 L b\u2081 L x\u2081\n    in begin\n      (Row (0S L b) (0S L b\u2081) \u2219S (Col x x\u2081))\n    \u2261\u27e8 refl-\u2261 \u27e9\n      (0S L b) \u2219S x +S (0S L b\u2081) \u2219S x\u2081\n    \u2248\u27e8 <+S> L L {0S L b \u2219S x} {0S L L} {0S L b\u2081 \u2219S x\u2081} {0S L L} ih ih\u2081 \u27e9 -- TODO: make nicer?\n      0S L L +S 0S L L\n    \u2248\u27e8 identS\u02e1 L L (0S L L) \u27e9\n      0S L L\n    \u220e\n  zeroS\u02e1 L (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS L c)\n      ih = zeroS\u02e1 L b c x\n      ih\u2081 = zeroS\u02e1 L b\u2081 c x\u2082\n     in begin\n       0S L b \u2219S x +S 0S L b\u2081 \u2219S x\u2082\n     \u2248\u27e8 <+S> L c ih ih\u2081 \u27e9\n       0S L c +S 0S L c\n     \u2248\u27e8 identS\u02e1 L c (0S L c) \u27e9\n       0S L c\n     \u220e) ,\n    (let\n      open EqReasoning (setoidS L c\u2081)\n      ih = zeroS\u02e1 L b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 L b\u2081 c\u2081 x\u2083\n    in begin\n      0S L b \u2219S x\u2081 +S 0S L b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> L c\u2081 ih ih\u2081 \u27e9\n      0S L c\u2081 +S 0S L c\u2081\n    \u2248\u27e8 identS\u02e1 L c\u2081 _ \u27e9\n      0S L c\u2081\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) L L (One x) = zeroS\u02e1 a L L (One x) , zeroS\u02e1 a\u2081 L L (One x)\n  zeroS\u02e1 (B a a\u2081) L (B c c\u2081) (Row x x\u2081) =\n    zeroS\u02e1 a L c x , zeroS\u02e1 a L c\u2081 x\u2081 ,\n    zeroS\u02e1 a\u2081 L c x , zeroS\u02e1 a\u2081 L c\u2081 x\u2081\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) L (Col x x\u2081) =\n    (let\n      open EqReasoning (setoidS a L)\n      ih = zeroS\u02e1 a b L x\n      ih\u2081 = zeroS\u02e1 a b\u2081 L x\u2081\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a L ih ih\u2081 \u27e9\n      0S a L +S 0S a L\n    \u2248\u27e8 identS\u02e1 a L _ \u27e9\n      0S a L\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 L)\n      ih = zeroS\u02e1 a\u2081 b L x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 L x\u2081\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2081\n    \u2248\u27e8 <+S> a\u2081 L ih ih\u2081 \u27e9\n      0S a\u2081 L +S 0S a\u2081 L\n    \u2248\u27e8 identS\u02e1 a\u2081 L _ \u27e9\n      0S a\u2081 L\n    \u220e)\n  zeroS\u02e1 (B a a\u2081) (B b b\u2081) (B c c\u2081) (Q x x\u2081 x\u2082 x\u2083) =\n    (let\n      open EqReasoning (setoidS a c)\n      ih = zeroS\u02e1 a b c x\n      ih\u2081 = zeroS\u02e1 a b\u2081 c x\u2082\n    in begin\n      0S a b \u2219S x +S 0S a b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a c ih ih\u2081 \u27e9\n      0S a c +S 0S a c\n    \u2248\u27e8 identS\u02e1 a c _ \u27e9\n      0S a c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a c\u2081)\n      ih = zeroS\u02e1 a b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a b\u2081 c\u2081 x\u2083\n    in begin\n      0S a b \u2219S x\u2081 +S 0S a b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a c\u2081 ih ih\u2081 \u27e9\n      0S a c\u2081 +S 0S a c\u2081\n    \u2248\u27e8 identS\u02e1 a c\u2081 _ \u27e9\n      0S a c\u2081\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c)\n      ih = zeroS\u02e1 a\u2081 b c x\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c x\u2082\n    in begin\n      0S a\u2081 b \u2219S x +S 0S a\u2081 b\u2081 \u2219S x\u2082\n    \u2248\u27e8 <+S> a\u2081 c ih ih\u2081 \u27e9\n      0S a\u2081 c +S 0S a\u2081 c\n    \u2248\u27e8 identS\u02e1 a\u2081 c _ \u27e9\n      0S a\u2081 c\n    \u220e) ,\n    (let\n      open EqReasoning (setoidS a\u2081 c\u2081)\n      ih = zeroS\u02e1 a\u2081 b c\u2081 x\u2081\n      ih\u2081 = zeroS\u02e1 a\u2081 b\u2081 c\u2081 x\u2083\n    in begin\n      0S a\u2081 b \u2219S x\u2081 +S 0S a\u2081 b\u2081 \u2219S x\u2083\n    \u2248\u27e8 <+S> a\u2081 c\u2081 ih ih\u2081 \u27e9\n      0S a\u2081 c\u2081 +S 0S a\u2081 c\u2081\n    \u2248\u27e8 identS\u02e1 a\u2081 c\u2081 _ \u27e9\n      0S a\u2081 c\u2081\n    \u220e)\n\n\n  SNR : SemiNearRing\n  SNR =\n    record\n      { s = S\n      ; _\u2243\u209b_ = \u2243S shape shape\n      ; 0\u209b = 0S shape shape\n      ; _+\u209b_ = _+S_\n      ; _\u2219\u209b_ = _\u2219S_\n      ; isCommMon = isCommMonS\n      ; zero\u02e1 = zeroS\u02e1 shape shape shape\n      ; zero\u02b3 = {!!}\n      ; _<\u2219>_ = {!!}\n      }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f584e5bc9c34e6764e8a6ec1a21ee5f2326d3304","subject":"Remove instance arguments from Agda code.","message":"Remove instance arguments from Agda code.\n\nInstance argument behavior was changed in a recent release. Remove use of instance arguments to avoid the hassle of changing them.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 v vowel \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 v vowel \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 v vowel \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 v vowel \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 v vowel \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 v vowel \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2113 with-rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 v vowel \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2113 vowel \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2113 vowel \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Set where\n  unmarked : (\u2113 : Letter) \u2192 (c : Case) \u2192 Token\n  with-accent : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 Token\n  with-breathing : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token\n  with-accent-breathing : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token\n  with-accent-breathing-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token\n  with-diaeresis : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 diaeresis \u2192 Token\n  with-accent-diaeresis : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token\n  with-accent-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token\n  with-breathing-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token\n  with-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 iota-subscript \u2192 Token\n  final : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token\n\n-- Constructions\n\n-- \u0391 \u03b1\n\u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short x (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u03b1-vowel \u00ac\u03b1-always-short\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel (is-vowel here)\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (is-vowel here) (\u03bb ())\n\n\u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel (is-vowel here)\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript (is-vowel here) here\n\n-- \u0395 \u03b5\n\u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short (is-vowel (there here)) here\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel (is-vowel (there here))\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (is-vowel (there here)) (\u03bb ())\n\n\u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel (is-vowel (there here))\n\n-- \u0397 \u03b7\n\u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short x (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel (is-vowel (there (there here))) \u00ac\u03b7-always-short\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel (is-vowel (there (there here)))\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (is-vowel (there (there here))) (\u03bb ())\n\n\u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel (is-vowel (there (there here)))\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (is-vowel (there (there here))) (there here)\n\n-- \u0399 \u03b9\n\n\u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short x (there (there ())))\n\n\u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel (is-vowel (there (there (there here)))) \u00ac\u03b9-always-short\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel (is-vowel (there (there (there here))))\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5\n             (is-vowel (there (there (there here)))) (\u03bb ())\n\n\u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel (is-vowel (there (there (there here))))\n\n\u03b9-diaeresis : \u03b9\u2032 diaeresis\n\u03b9-diaeresis = add-diaeresis (is-vowel (there (there (there here)))) here\n\n-- \u039f \u03bf\n\u03bf-vowel : \u03bf\u2032 vowel\n\u03bf-vowel = is-vowel (there (there (there (there here))))\n\n\u03bf-always-short : \u03bf\u2032 always-short\n\u03bf-always-short = is-always-short (is-vowel (there (there (there (there here)))))\n                   (there here)\n\n\u03bf-smooth : \u03bf\u2032 \u27e6 lower \u27e7-smooth\n\u03bf-smooth = add-smooth-lower-vowel\n             (is-vowel (there (there (there (there here)))))\n\n\u039f-smooth : \u03bf\u2032 \u27e6 upper \u27e7-smooth\n\u039f-smooth = add-smooth-upper-vowel-not-\u03a5\n             (is-vowel (there (there (there (there here))))) (\u03bb ())\n\n\u03bf-rough : \u03bf\u2032 with-rough\n\u03bf-rough = add-rough-vowel (is-vowel (there (there (there (there here)))))\n\n-- \u03a1 \u03c1\n\u03c1-smooth : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n\u03c1-smooth = add-smooth-\u03c1\n\n\u03c1-rough : \u03c1\u2032 with-rough\n\u03c1-rough = add-rough-\u03c1\n\n-- \u03a3 \u03c3\n\u03c3-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\u03c3-final = make-final\n\n-- \u03a5 \u03c5\n\u03c5-vowel : \u03c5\u2032 vowel\n\u03c5-vowel = is-vowel (there (there (there (there (there here)))))\n\n\u00ac\u03c5-always-short : \u00ac \u03c5\u2032 always-short\n\u00ac\u03c5-always-short (is-always-short x (there (there ())))\n\n\u03c5-long-vowel : \u03c5\u2032 long-vowel\n\u03c5-long-vowel = make-long-vowel \u03c5-vowel \u00ac\u03c5-always-short\n\n\u03c5-smooth : \u03c5\u2032 \u27e6 lower \u27e7-smooth\n\u03c5-smooth = add-smooth-lower-vowel\n             (is-vowel (there (there (there (there (there here))))))\n\n\u03c5-rough : \u03c5\u2032 with-rough\n\u03c5-rough = add-rough-vowel\n            (is-vowel (there (there (there (there (there here))))))\n\n\u03c5-diaeresis : \u03c5\u2032 diaeresis\n\u03c5-diaeresis = add-diaeresis\n                (is-vowel (there (there (there (there (there here))))))\n                (there here)\n\n-- \u03a9 \u03c9\n\u03c9-vowel : \u03c9\u2032 vowel\n\u03c9-vowel = is-vowel (there (there (there (there (there (there here))))))\n\n\u00ac\u03c9-always-short : \u00ac \u03c9\u2032 always-short\n\u00ac\u03c9-always-short (is-always-short x (there (there ())))\n\n\u03c9-long-vowel : \u03c9\u2032 long-vowel\n\u03c9-long-vowel = make-long-vowel \u03c9-vowel \u00ac\u03c9-always-short\n\n\u03c9-smooth : \u03c9\u2032 \u27e6 lower \u27e7-smooth\n\u03c9-smooth = add-smooth-lower-vowel\n             (is-vowel (there (there (there (there (there (there here)))))))\n\n\u03a9-smooth : \u03c9\u2032 \u27e6 upper \u27e7-smooth\n\u03a9-smooth = add-smooth-upper-vowel-not-\u03a5\n             (is-vowel (there (there (there (there (there (there here)))))))\n             (\u03bb ())\n\n\u03c9-rough : \u03c9\u2032 with-rough\n\u03c9-rough = add-rough-vowel\n            (is-vowel (there (there (there (there (there (there here)))))))\n\n\u03c9-iota-subscript : \u03c9\u2032 iota-subscript\n\u03c9-iota-subscript = add-iota-subscript\n                     (is-vowel (there (there (there (there (there (there here)))))))\n                     (there (there here))\n\n\n-- Mapping\n\nget-letter : Token \u2192 Letter\nget-letter (unmarked \u2113 _) = \u2113\nget-letter (with-accent \u2113 _ _) = \u2113\nget-letter (with-breathing \u2113 _ _) = \u2113\nget-letter (with-accent-breathing \u2113 _ _ _) = \u2113\nget-letter (with-accent-breathing-iota \u2113 _ _ _ _) = \u2113\nget-letter (with-diaeresis \u2113 _ _) = \u2113\nget-letter (with-accent-diaeresis \u2113 _ _ _) = \u2113\nget-letter (with-accent-iota \u2113 _ _ _) = \u2113\nget-letter (with-breathing-iota \u2113 _ _ _) = \u2113\nget-letter (with-iota \u2113 _ _) = \u2113\nget-letter (final \u2113 _ _) = \u2113\n\nget-case : Token \u2192 Case\nget-case (unmarked _ c) = c\nget-case (with-accent _ c _) = c\nget-case (with-breathing _ c _) = c\nget-case (with-accent-breathing _ c _ _) = c\nget-case (with-accent-breathing-iota _ c _ _ _) = c\nget-case (with-diaeresis _ c _) = c\nget-case (with-accent-diaeresis _ c _ _) = c\nget-case (with-accent-iota _ c _ _) = c\nget-case (with-breathing-iota _ c _ _) = c\nget-case (with-iota _ c _) = c\nget-case (final _ c _) = c\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent (acute x) = acute-mark\nletter-to-accent (grave x) = grave-mark\nletter-to-accent (circumflex x) = circumflex-mark\n\nget-accent : Token \u2192 Maybe Accent\nget-accent (with-accent _ _ a) = just (letter-to-accent a)\nget-accent (with-accent-breathing _ _ a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota _ _ a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis _ _ a _) = just (letter-to-accent a)\nget-accent (with-accent-iota _ _ a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (smooth x) = smooth-mark\nletter-to-breathing (rough x) = rough-mark\n\nget-breathing : Token \u2192 Maybe Breathing\nget-breathing (with-breathing _ _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ _ _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ _ _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota _ _ x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : Token \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _ _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : Token \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _ _ _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _ _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : Token \u2192 Maybe FinalForm\nget-final-form (final _ _ _) = just final-form\nget-final-form _ = nothing\n","old_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _with-rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v with-rough\n  add-rough-\u03c1 : \u03c1\u2032 with-rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  smooth : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-smooth \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n  rough : \u2200 {\u2113 c} \u2192 \u2983 p : \u2113 with-rough \u2984 \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 \u2983 p : v vowel \u2984 \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  acute : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  grave : \u2200 {\u2113} \u2192 \u2983 p : \u2113 vowel \u2984 \u2192 \u2113 accent\n  circumflex : \u2200 {\u2113} \u2192 \u2983 p : \u2113 long-vowel \u2984 \u2192 \u2113 accent\n\ndata Token : Set where\n  unmarked : (\u2113 : Letter) \u2192 (c : Case) \u2192 Token\n  with-accent : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 Token\n  with-breathing : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token\n  with-accent-breathing : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token\n  with-accent-breathing-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token\n  with-diaeresis : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token\n  with-accent-diaeresis : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2983 p : \u2113 diaeresis \u2984 \u2192 Token\n  with-accent-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 accent \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token\n  with-breathing-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token\n  with-iota : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2983 p : \u2113 iota-subscript \u2984 \u2192 Token\n  final : (\u2113 : Letter) \u2192 (c : Case) \u2192 \u2983 p : \u2113 \u27e6 c \u27e7-final \u2984 \u2192 Token\n\n-- Constructions\n\n-- \u0391 \u03b1\ninstance \u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short (there (there ())))\n\ninstance \u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u00ac\u03b1-always-short\n\ninstance \u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-smooth\n\u03b1-smooth = add-smooth-lower-vowel\n\ninstance \u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-smooth\n\u0391-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b1-rough : \u03b1\u2032 with-rough\n\u03b1-rough = add-rough-vowel\n\ninstance \u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript here\n\n-- \u0395 \u03b5\ninstance \u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short here\n\ninstance \u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-smooth\n\u03b5-smooth = add-smooth-lower-vowel\n\ninstance \u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-smooth\n\u0395-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b5-rough : \u03b5\u2032 with-rough\n\u03b5-rough = add-rough-vowel\n\n-- \u0397 \u03b7\ninstance \u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short (there (there ())))\n\ninstance \u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u00ac\u03b7-always-short\n\ninstance \u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-smooth\n\u03b7-smooth = add-smooth-lower-vowel\n\ninstance \u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-smooth\n\u0397-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b7-rough : \u03b7\u2032 with-rough\n\u03b7-rough = add-rough-vowel\n\ninstance \u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript (there here)\n\n-- \u0399 \u03b9\n\ninstance \u03b9-vowel : \u03b9\u2032 vowel\n\u03b9-vowel = is-vowel (there (there (there here)))\n\n\u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n\u00ac\u03b9-always-short (is-always-short (there (there ())))\n\ninstance \u03b9-long-vowel : \u03b9\u2032 long-vowel\n\u03b9-long-vowel = make-long-vowel \u00ac\u03b9-always-short\n\ninstance \u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-smooth\n\u03b9-smooth = add-smooth-lower-vowel\n\ninstance \u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-smooth\n\u0399-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03b9-rough : \u03b9\u2032 with-rough\n\u03b9-rough = add-rough-vowel\n\ninstance \u03b9-diaeresis : \u03b9\u2032 diaeresis\n\u03b9-diaeresis = add-diaeresis here\n\n-- \u039f \u03bf\ninstance \u03bf-vowel : \u03bf\u2032 vowel\n\u03bf-vowel = is-vowel (there (there (there (there here))))\n\n\u03bf-always-short : \u03bf\u2032 always-short\n\u03bf-always-short = is-always-short (there here)\n\ninstance \u03bf-smooth : \u03bf\u2032 \u27e6 lower \u27e7-smooth\n\u03bf-smooth = add-smooth-lower-vowel\n\ninstance \u039f-smooth : \u03bf\u2032 \u27e6 upper \u27e7-smooth\n\u039f-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03bf-rough : \u03bf\u2032 with-rough\n\u03bf-rough = add-rough-vowel\n\n-- \u03a1 \u03c1\ninstance \u03c1-smooth : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n\u03c1-smooth = add-smooth-\u03c1\n\ninstance \u03c1-rough : \u03c1\u2032 with-rough\n\u03c1-rough = add-rough-\u03c1\n\n-- \u03a3 \u03c3\ninstance \u03c3-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\u03c3-final = make-final\n\n-- \u03a5 \u03c5\ninstance \u03c5-vowel : \u03c5\u2032 vowel\n\u03c5-vowel = is-vowel (there (there (there (there (there here)))))\n\n\u00ac\u03c5-always-short : \u00ac \u03c5\u2032 always-short\n\u00ac\u03c5-always-short (is-always-short (there (there ())))\n\ninstance \u03c5-long-vowel : \u03c5\u2032 long-vowel\n\u03c5-long-vowel = make-long-vowel \u00ac\u03c5-always-short\n\ninstance \u03c5-smooth : \u03c5\u2032 \u27e6 lower \u27e7-smooth\n\u03c5-smooth = add-smooth-lower-vowel\n\ninstance \u03c5-rough : \u03c5\u2032 with-rough\n\u03c5-rough = add-rough-vowel\n\ninstance \u03c5-diaeresis : \u03c5\u2032 diaeresis\n\u03c5-diaeresis = add-diaeresis (there here)\n\n-- \u03a9 \u03c9\ninstance \u03c9-vowel : \u03c9\u2032 vowel\n\u03c9-vowel = is-vowel (there (there (there (there (there (there here))))))\n\n\u00ac\u03c9-always-short : \u00ac \u03c9\u2032 always-short\n\u00ac\u03c9-always-short (is-always-short (there (there ())))\n\ninstance \u03c9-long-vowel : \u03c9\u2032 long-vowel\n\u03c9-long-vowel = make-long-vowel \u00ac\u03c9-always-short\n\ninstance \u03c9-smooth : \u03c9\u2032 \u27e6 lower \u27e7-smooth\n\u03c9-smooth = add-smooth-lower-vowel\n\ninstance \u03a9-smooth : \u03c9\u2032 \u27e6 upper \u27e7-smooth\n\u03a9-smooth = add-smooth-upper-vowel-not-\u03a5 (\u03bb ())\n\ninstance \u03c9-rough : \u03c9\u2032 with-rough\n\u03c9-rough = add-rough-vowel\n\ninstance \u03c9-iota-subscript : \u03c9\u2032 iota-subscript\n\u03c9-iota-subscript = add-iota-subscript (there (there here))\n\n\n-- Mapping\n\nget-letter : Token \u2192 Letter\nget-letter (unmarked \u2113 _) = \u2113\nget-letter (with-accent \u2113 _ _) = \u2113\nget-letter (with-breathing \u2113 _ _) = \u2113\nget-letter (with-accent-breathing \u2113 _ _ _) = \u2113\nget-letter (with-accent-breathing-iota \u2113 _ _ _) = \u2113\nget-letter (with-diaeresis \u2113 _) = \u2113\nget-letter (with-accent-diaeresis \u2113 _ _) = \u2113\nget-letter (with-accent-iota \u2113 _ _) = \u2113\nget-letter (with-breathing-iota \u2113 _ _) = \u2113\nget-letter (with-iota \u2113 _) = \u2113\nget-letter (final \u2113 _) = \u2113\n\nget-case : Token \u2192 Case\nget-case (unmarked _ c) = c\nget-case (with-accent _ c _) = c\nget-case (with-breathing _ c _) = c\nget-case (with-accent-breathing _ c _ _) = c\nget-case (with-accent-breathing-iota _ c _ _) = c\nget-case (with-diaeresis _ c) = c\nget-case (with-accent-diaeresis _ c _) = c\nget-case (with-accent-iota _ c _) = c\nget-case (with-breathing-iota _ c _) = c\nget-case (with-iota _ c) = c\nget-case (final _ c) = c\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent acute = acute-mark\nletter-to-accent grave = grave-mark\nletter-to-accent circumflex = circumflex-mark\n\nget-accent : Token \u2192 Maybe Accent\nget-accent (with-accent _ _ a) = just (letter-to-accent a)\nget-accent (with-accent-breathing _ _ a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota _ _ a _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis _ _ a) = just (letter-to-accent a)\nget-accent (with-accent-iota _ _ a) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing smooth = smooth-mark\nletter-to-breathing rough = rough-mark\n\nget-breathing : Token \u2192 Maybe Breathing\nget-breathing (with-breathing _ _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ _ _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ _ _ x) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota _ _ x) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : Token \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : Token \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _ _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : Token \u2192 Maybe FinalForm\nget-final-form (final _ _) = just final-form\nget-final-form _ = nothing\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c6294111d477e969e1351c3c2235cc0eb1064318","subject":"IDesc in IDesc, stratified, with a 'lift'ing axiom","message":"IDesc in IDesc, stratified, with a 'lift'ing axiom\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift = {!!}\n\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (l : Level)(I : Set l) : Set (suc l) where\n  var : I -> IDesc l I\n  const : Set l -> IDesc l I\n  prod : IDesc l I -> IDesc l I -> IDesc l I\n  sigma : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n  pi : (S : Set l) -> (S -> IDesc l I) -> IDesc l I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc zero I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc zero I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc zero I)(P : I -> Set) -> desc I D P -> IDesc zero (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc zero I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc zero I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc zero I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set1 where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc (suc zero) Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc (suc zero) Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set1 where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\n{-\ndata IMu (I : Set)(R : I -> IDesc I) : IDesc I -> Set1 where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)\n-}\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)(xs : desc I (R i) (IMu I R))(hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) -> P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> (xs : desc I D (IMu I R)) -> desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)(R : I -> IDesc I)(P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7182b69f26a2e04641f656468e7362e72e77cc10","subject":"Renamed a proof.","message":"Renamed a proof.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Program\/McCarthy91\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n-- For all n > 100, then mc91 n = n - 10.\npostulate mc91-res>100 : \u2200 n \u2192 n > [100] \u2192 mc91 n \u2261 n \u2238 [10]\n{-# ATP prove mc91-res>100 #-}\n\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f [100] \u2192 mc91 n \u2261 [91]\nmc91-res\u226f100 = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d \u226f [100] \u2192 mc91 d \u2261 [91]\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = \u03bb m\u226f100 \u2192\n                     \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nm 100-N m>100 (x\u226fy\u2192x\u2264y Nm 100-N m\u226f100))\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = \u03bb _ \u2192 mc91-res-aux mc91-res-100 m\u2261100\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = \u03bb _ \u2192 mc91-res-aux mc91-res-99 m\u226199\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = \u03bb _ \u2192 mc91-res-aux mc91-res-98 m\u226198\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = \u03bb _ \u2192 mc91-res-aux mc91-res-97 m\u226197\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = \u03bb _ \u2192 mc91-res-aux mc91-res-96 m\u226196\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = \u03bb _ \u2192 mc91-res-aux mc91-res-95 m\u226195\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = \u03bb _ \u2192 mc91-res-aux mc91-res-94 m\u226194\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = \u03bb _ \u2192 mc91-res-aux mc91-res-93 m\u226193\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = \u03bb _ \u2192 mc91-res-aux mc91-res-92 m\u226192\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = \u03bb _ \u2192 mc91-res-aux mc91-res-91 m\u226191\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = \u03bb _ \u2192 mc91-res-aux mc91-res-90 m\u226190\n  ... | inj\u2081 m\u226f89 = \u03bb _ \u2192 mc91-res-m\u226f89\n    where\n    m\u226489 : m \u2264 [89]\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : m + [11] \u226f [100] \u2192 mc91 (m + [11]) \u2261 [91]\n    mc91-res-m+11 = f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n    mc91-res-m\u226f89 : mc91 m \u2261 [91]\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m [91] m\u226f100\n                                  (mc91-res-m+11 (x\u226489\u2192x+11\u226f100 Nm m\u226489))\n                                  mc91-res-91\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N {n} Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (mc91-res>100 n n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226e100 = subst N (sym (mc91-res\u226f100 Nn n\u226e100)) 91-N\n\n-- For all n, n < mc91 n + 11.\nmc91-ineq : \u2200 {n} \u2192 N n \u2192 n < mc91 n + [11]\nmc91-ineq = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < mc91 d + [11]\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x<mc91x+11>100 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let mc91-m+11-N : N (mc91 (m + [11]))\n        mc91-m+11-N = mc91-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + [11])\n        h\u2081 = f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<mc91m+11 : m < mc91 (m + [11])\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm mc91-m+11-N 11-N h\u2081\n\n        h\u2082 : A (mc91 (m + [11]))\n        h\u2082 = f mc91-m+11-N (<\u2192\u226a mc91-m+11-N Nm m\u226f100 m<mc91m+11)\n\n    in (<-trans Nm mc91-m+11-N (x+11-N (mc91-N Nm))\n                m<mc91m+11\n                (mc91x+11<mc91x+11 m m\u226f100 h\u2082))\n","old_contents":"------------------------------------------------------------------------------\n-- Main properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- The main properties proved of the McCarthy 91 function (called\n-- mc91) are\n\n-- 1. The function always terminates.\n-- 2. For all n, n < mc91 n + 11.\n-- 3. For all n > 100, then mc91 n = n - 10.\n-- 4. For all n <= 100, then mc91 n = 91.\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\nopen import FOTC.Program.McCarthy91.ArithmeticATP\nopen import FOTC.Program.McCarthy91.AuxiliaryPropertiesATP\nopen import FOTC.Program.McCarthy91.McCarthy91\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.LT2WF-RelationATP\nopen import FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP\n\n------------------------------------------------------------------------------\n-- For all n > 100, then mc91 n = n - 10.\npostulate mc91-res>100 : \u2200 n \u2192 n > [100] \u2192 mc91 n \u2261 n \u2238 [10]\n{-# ATP prove mc91-res>100 #-}\n\n-- For all n <= 100, then mc91 n = 91.\nmc91-res\u226f100 : \u2200 {n} \u2192 N n \u2192 n \u226f [100] \u2192 mc91 n \u2261 [91]\nmc91-res\u226f100 = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d \u226f [100] \u2192 mc91 d \u2261 [91]\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = \u03bb m\u226f100 \u2192\n                     \u22a5-elim (x>y\u2192x\u2264y\u2192\u22a5 Nm 100-N m>100 (x\u226fy\u2192x\u2264y Nm 100-N m\u226f100))\n  ... | inj\u2082 m\u226f100 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 99-N m\u226f100\n  ... | inj\u2082 m\u2261100 = \u03bb _ \u2192 mc91-res-aux mc91-res-100 m\u2261100\n  ... | inj\u2081 m\u226f99 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 98-N m\u226f99\n  ... | inj\u2082 m\u226199 = \u03bb _ \u2192 mc91-res-aux mc91-res-99 m\u226199\n  ... | inj\u2081 m\u226f98 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 97-N m\u226f98\n  ... | inj\u2082 m\u226198 = \u03bb _ \u2192 mc91-res-aux mc91-res-98 m\u226198\n  ... | inj\u2081 m\u226f97 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 96-N m\u226f97\n  ... | inj\u2082 m\u226197 = \u03bb _ \u2192 mc91-res-aux mc91-res-97 m\u226197\n  ... | inj\u2081 m\u226f96 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 95-N m\u226f96\n  ... | inj\u2082 m\u226196 = \u03bb _ \u2192 mc91-res-aux mc91-res-96 m\u226196\n  ... | inj\u2081 m\u226f95 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 94-N m\u226f95\n  ... | inj\u2082 m\u226195 = \u03bb _ \u2192 mc91-res-aux mc91-res-95 m\u226195\n  ... | inj\u2081 m\u226f94 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 93-N m\u226f94\n  ... | inj\u2082 m\u226194 = \u03bb _ \u2192 mc91-res-aux mc91-res-94 m\u226194\n  ... | inj\u2081 m\u226f93 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 92-N m\u226f93\n  ... | inj\u2082 m\u226193 = \u03bb _ \u2192 mc91-res-aux mc91-res-93 m\u226193\n  ... | inj\u2081 m\u226f92 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 91-N m\u226f92\n  ... | inj\u2082 m\u226192 = \u03bb _ \u2192 mc91-res-aux mc91-res-92 m\u226192\n  ... | inj\u2081 m\u226f91 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 90-N m\u226f91\n  ... | inj\u2082 m\u226191 = \u03bb _ \u2192 mc91-res-aux mc91-res-91 m\u226191\n  ... | inj\u2081 m\u226f90 with x\u226fSy\u2192x\u226fy\u2228x\u2261Sy Nm 89-N m\u226f90\n  ... | inj\u2082 m\u226190 = \u03bb _ \u2192 mc91-res-aux mc91-res-90 m\u226190\n  ... | inj\u2081 m\u226f89 = \u03bb _ \u2192 mc91-res-m\u226f89\n    where\n    m\u226489 : m \u2264 [89]\n    m\u226489 = x\u226fy\u2192x\u2264y Nm 89-N m\u226f89\n\n    mc91-res-m+11 : m + [11] \u226f [100] \u2192 mc91 (m + [11]) \u2261 [91]\n    mc91-res-m+11 = f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n    mc91-res-m\u226f89 : mc91 m \u2261 [91]\n    mc91-res-m\u226f89 = mc91x-res\u226f100 m [91] m\u226f100\n                                  (mc91-res-m+11 (x\u226489\u2192x+11\u226f100 Nm m\u226489))\n                                  mc91-res-91\n-- The function always terminates.\nmc91-N : \u2200 {n} \u2192 N n \u2192 N (mc91 n)\nmc91-N {n} Nn with x>y\u2228x\u226fy Nn 100-N\n... | inj\u2081 n>100 = subst N (sym (mc91-res>100 n n>100)) (\u2238-N Nn 10-N)\n... | inj\u2082 n\u226e100 = subst N (sym (mc91-res\u226f100 Nn n\u226e100)) 91-N\n\n-- For all n, n < mc91 n + 11.\nmc91-N-ineq : \u2200 {n} \u2192 N n \u2192 n < mc91 n + [11]\nmc91-N-ineq = \u226a-wfind A h\n  where\n  A : D \u2192 Set\n  A d = d < mc91 d + [11]\n\n  h : \u2200 {m} \u2192 N m \u2192 (\u2200 {k} \u2192 N k \u2192 k \u226a m \u2192 A k) \u2192 A m\n  h {m} Nm f with x>y\u2228x\u226fy Nm 100-N\n  ... | inj\u2081 m>100 = x<mc91x+11>100 Nm m>100\n  ... | inj\u2082 m\u226f100 =\n    let mc91-m+11-N : N (mc91 (m + [11]))\n        mc91-m+11-N = mc91-N (+-N Nm 11-N)\n\n        h\u2081 : A (m + [11])\n        h\u2081 = f (x+11-N Nm) (<\u2192\u226a (x+11-N Nm) Nm m\u226f100 (x<x+11 Nm))\n\n        m<mc91m+11 : m < mc91 (m + [11])\n        m<mc91m+11 = x+k<y+k\u2192x<y Nm mc91-m+11-N 11-N h\u2081\n\n        h\u2082 : A (mc91 (m + [11]))\n        h\u2082 = f mc91-m+11-N (<\u2192\u226a mc91-m+11-N Nm m\u226f100 m<mc91m+11)\n\n    in (<-trans Nm mc91-m+11-N (x+11-N (mc91-N Nm))\n                m<mc91m+11\n                (mc91x+11<mc91x+11 m m\u226f100 h\u2082))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"89a0dbbd05a306ce23cdd621c8864440bd9383af","subject":"adding some rule names and small step rules","message":"adding some rule names and small step rules\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 \u00eb\n    \u2987_\u2988[_]   : \u00eb \u2192 (Nat \u00d7 subst) \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< e2' > t2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u27e6 u ::[ \u0393 ] \u2987\u2988 \u27e7\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< e' > t) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x t1 t2 e e' t2' \u0394 } \u2192\n              (\u0393 ,, (x , t1)) \u22a2 e \u21d0 t2 ~> e' :: t2' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' :: (t1 ==> t2') \u22a3 \u0394\n      EASubsume : \u2200{e u m \u0393 t' e' \u0394 t} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 m \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                  t ~ t' \u2192\n                  \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u t  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 t ~> \u2987\u2988[ u , id \u0393 ] :: t \u22a3 \u27e6 u ::[ \u0393 ] t \u27e7\n      EANEHole : \u2200{ \u0393 e u t e' t' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 t ~> \u2987 e' \u2988[ u , id \u0393 ] :: t \u22a3 (\u0394 ,, u ::[ \u0393 ] t)\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n    TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n    TAVar : \u2200{\u0394 \u0393 x t} \u2192 \u0394 , (\u0393 ,, (x , t)) \u22a2 X x :: t\n    TALam : \u2200{ \u0394 \u0393 x t1 e t2} \u2192\n            \u0394 , (\u0393 ,, (x , t1)) \u22a2 e :: t2 \u2192\n            \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e :: (t1 ==> t2)\n    TAAp : \u2200{ \u0394 \u0393 e1 e2 t1 t2 t} \u2192\n           \u0394 , \u0393 \u22a2 e1 :: t1 \u2192\n           t1 \u25b8arr (t2 ==> t) \u2192\n           \u0394 , \u0393 \u22a2 e2 :: t2 \u2192\n           \u0394 , \u0393 \u22a2 e1 \u2218 e2 :: t\n    -- TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' } \u2192 -- todo: overloaded form in the paper here\n    -- TANEHole : \u2200 { \u0394 \u0393 e t' \u0393' u \u03c3 t }\n    TACast : \u2200{ \u0394 \u0393 e t t'} \u2192\n           \u0394 , \u0393 \u22a2 e :: t' \u2192\n           t ~ t' \u2192\n           \u0394 , \u0393 \u22a2 < e > t :: t\n\n  -- todo: substition goes here\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where\n    -- ERNEHole\n    -- ERCastError\n    -- ERLam\n    -- ERAp1\n    -- ERAp2\n    -- ERCast\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n    STHole : \u2200{ e e' u \u03c3 } \u2192\n             e \u21a6 e' \u2192\n             \u2987 e \u2988[ u , \u03c3 ] \u21a6 \u2987 e' \u2988[ u , \u03c3 ]\n    -- STCast\n    STAp1 : \u2200{ e1 e2 e1' } \u2192\n            e1 \u21a6 e1' \u2192\n            (e1 \u2218 e2) \u21a6 (e1' \u2218 e2)\n    STAp2 : \u2200{ e1 e2 e2' } \u2192\n            e1 final \u2192\n            e2 \u21a6 e2' \u2192\n            (e1 \u2218 e2) \u21a6 (e1 \u2218 e2')\n    STAp\u03b2 : \u2200{ e1 e2 t x } \u2192\n            e2 final \u2192\n            ((\u00b7\u03bb x [ t ] e1) \u2218 e2) \u21a6 ([ \u27e6 x , {!!} \u27e7 ] e2)\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    b     : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data e\u0307 : Set where\n    c       : e\u0307\n    _\u00b7:_    : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X       : Nat \u2192 e\u0307\n    \u00b7\u03bb      : Nat \u2192 e\u0307 \u2192 e\u0307\n    \u00b7\u03bb_[_]_ : Nat \u2192 \u03c4\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]   : Nat \u2192 e\u0307\n    \u2987_\u2988[_]  : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_     : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n\n  subst : Set -- todo: no idea if this is right\n  subst = e\u0307 ctx\n\n  -- expressions without ascriptions but with casts\n  data \u00eb : Set where\n    c        : \u00eb\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb_[_]_  : Nat \u2192 \u03c4\u0307 \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_]    : (Nat \u00d7 subst) \u2192 \u00eb\n    \u2987_\u2988[_]   : \u00eb \u2192 (Nat \u00d7 subst) \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               t1 ==> t2 ~ t1' ==> t2'\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICBaseArr1 : {t1 t2 : \u03c4\u0307} \u2192 b ~\u0338 t1 ==> t2\n    ICBaseArr2 : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 ~\u0338 b\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               t1 ==> t2 ~\u0338 t3 ==> t4\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 t1 ==> t2 \u25b8arr t1 ==> t2\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = \u03c4\u0307 ctx\n\n  hctx : Set\n  hctx = (\u03c4\u0307 ctx \u00d7 \u03c4\u0307) ctx\n\n  postulate -- todo: write this stuff later\n    id : {A : Set} \u2192 A ctx \u2192 subst\n    [_]_ : subst \u2192 \u00eb \u2192 \u00eb\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 \u03c4\u0307 \u2192 (Nat \u00d7 tctx \u00d7 \u03c4\u0307)\n  u ::[ \u0393 ] t = u , \u0393 , t\n\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : tctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : tctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr t2 ==> t' \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : tctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : e\u0307} {t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192 -- todo\n                 (\u0393 ,, (x , t1)) \u22a2 e => t2 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e => t1 ==> t2\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u03c4\u0307 ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : tctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr t1 ==> t2 \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n\n  -- todo: do we care about completeness of e\u0307 or e-umlauts?\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete b         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n\n  -- those expressions without holes anywhere\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete c = \u22a4\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (\u00b7\u03bb x [ t ] e) = tcomplete t \u00d7 ecomplete e\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x t} \u2192 (\u0393 ,, (x , t)) \u22a2 X x \u21d2 t ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x t1 t2 e e' \u0394 } \u2192\n                     (\u0393 ,, (x , t1)) \u22a2 e \u21d2 t2 ~> e' \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e \u21d2 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' \u22a3 \u2205\n\n      -- todo: really ought to check disjointness of domains here ..\n      ESAp1   : \u2200{\u0393 e1 e2 e2' e1' \u03941 t2 t1 \u03942} \u2192\n                \u0393 \u22a2 e1 => \u2987\u2988 \u2192\n                \u0393 \u22a2 e2 \u21d0 \u2987\u2988 ~> e2' :: t2 \u22a3 \u03942 \u2192\n                \u0393 \u22a2 e1 \u21d0 (t2 ==> \u2987\u2988) ~> e1' :: t1 \u22a3 \u03941 \u2192\n                \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u2987\u2988 ~> (< e1' > t2) \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESAp2 : \u2200{\u0393 e1 t2 t e1' e2' \u03941 \u03942 t2' e2} \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2' \u22a3 \u03942 \u2192\n              (t2 == t2' \u2192 \u22a5) \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 (< e2' > t2) \u22a3 (\u03941 \u222a \u03942)\n      ESAp3 : \u2200{\u0393 e1 t e1' \u03941 e2 t2 e2' \u03942 } \u2192\n              \u0393 \u22a2 e1 \u21d2 (t2 ==> t) ~> e1' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 t2 ~> e2' :: t2 \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 t ~> e1' \u2218 e2' \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988[ u , id \u0393 ] \u22a3  \u27e6 u ::[ \u0393 ] \u2987\u2988 \u27e7\n      ESNEHole : \u2200{ \u0393 e t e' u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 e' \u2988[ u , id \u0393 ] \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc1 : \u2200 {\u0393 e t e' t' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n                 (t == t' \u2192 \u22a5) \u2192\n                 \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> (< e' > t) \u22a3 \u0394\n      ESAsc2 : \u2200{\u0393 e t e' t' \u0394 } \u2192\n               \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394 \u2192\n               \u0393 \u22a2 (e \u00b7: t) \u21d2 t ~> e' \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : e\u0307) (t : \u03c4\u0307) (e' : \u00eb) (t' : \u03c4\u0307)(\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x t1 t2 e e' t2' \u0394 } \u2192\n              (\u0393 ,, (x , t1)) \u22a2 e \u21d0 t2 ~> e' :: t2' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 (t1 ==> t2) ~> \u00b7\u03bb x [ t1 ] e' :: (t1 ==> t2') \u22a3 \u0394\n      EASubsume : \u2200{e u m \u0393 t' e' \u0394 t} \u2192\n                  (e == \u2987\u2988[ u ] \u2192 \u22a5) \u2192\n                  (e == \u2987 m \u2988[ u ] \u2192 \u22a5) \u2192\n                  \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                  t ~ t' \u2192\n                  \u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u t  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 t ~> \u2987\u2988[ u , id \u0393 ] :: t \u22a3 \u27e6 u ::[ \u0393 ] t \u27e7\n      EANEHole : \u2200{ \u0393 e u t e' t' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 t' ~> e' \u22a3 \u0394 \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 t ~> \u2987 e' \u2988[ u , id \u0393 ] :: t \u22a3 (\u0394 ,, u ::[ \u0393 ] t)\n\n  -- type assignment\n  data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (e' : \u00eb) (t : \u03c4\u0307) \u2192 Set where\n    TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n    TAVar : \u2200{\u0394 \u0393 x t} \u2192 \u0394 , (\u0393 ,, (x , t)) \u22a2 X x :: t\n    TALam : \u2200{ \u0394 \u0393 x t1 e t2} \u2192\n            \u0394 , (\u0393 ,, (x , t1)) \u22a2 e :: t2 \u2192\n            \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ t1 ] e :: (t1 ==> t2)\n    TAAp : \u2200{ \u0394 \u0393 e1 e2 t1 t2 t} \u2192\n           \u0394 , \u0393 \u22a2 e1 :: t1 \u2192\n           t1 \u25b8arr (t2 ==> t) \u2192\n           \u0394 , \u0393 \u22a2 e2 :: t2 \u2192\n           \u0394 , \u0393 \u22a2 e1 \u2218 e2 :: t\n    -- TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' } \u2192 -- todo: overloaded form in the paper here\n    -- TANEHole : \u2200 { \u0394 \u0393 e t' \u0393' u \u03c3 t }\n    TACast : \u2200{ \u0394 \u0393 e t t'} \u2192\n           \u0394 , \u0393 \u22a2 e :: t' \u2192\n           t ~ t' \u2192\n           \u0394 , \u0393 \u22a2 < e > t :: t\n\n  -- todo: substition goes here\n\n  -- value\n  data _val : \u00eb \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x t e} \u2192 (\u00b7\u03bb x [ t ] e) val\n\n  -- error\n  data _err[_] : \u00eb \u2192 hctx \u2192 Set where\n\n  mutual\n    -- indeterminate\n    data _indet : \u00eb \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988[ u , \u03c3 ] indet\n      INEHole : \u2200{e u \u03c3} \u2192 e final \u2192 \u2987 e \u2988[ u , \u03c3 ] indet\n      IAp : \u2200{e1 e2} \u2192 e1 indet \u2192 e2 final \u2192 (e1 \u2218 e2) indet\n      ICast : \u2200{e t} \u2192 e indet \u2192 (< e > t) indet\n\n    -- final\n    data _final : \u00eb \u2192 Set where\n      FVal : \u2200{e} \u2192 e val \u2192 e final\n      FIndet : \u2200{e} \u2192 e indet \u2192 e final\n\n  -- small step semantics\n  data _\u21a6_ : \u00eb \u2192 \u00eb \u2192 Set where\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4a0796e6660e467230df977598ccd6f198a90651","subject":"Removed Agda with constructor from the proof \u2237-Stream.","message":"Removed Agda with constructor from the proof \u2237-Stream.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality.Type\nopen import FOTC.Data.Colist\nopen import FOTC.Data.Colist.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\n\n-----------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functional\n-- StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream (see FOTC.Data.Stream.Type).\nStream-in : \u2200 {xs} \u2192\n            \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n            Stream xs\nStream-in h = Stream-coind A h' h\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h' (x' , xs' , prf , Sxs') = x' , xs' , prf , Stream-out Sxs'\n\nfoo : \u2200 {xs} \u2192 Stream xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\nfoo Sxs = Stream-out Sxs\n\n\u2237-Stream : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream {x} {xs} h = \u2237-Stream-helper (Stream-out h)\n  where\n  \u2237-Stream-helper : \u2203[ x' ] \u2203[ xs' ] x \u2237 xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n                    Stream xs\n  \u2237-Stream-helper (x' , xs' , prf , Sxs') =\n    subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\n-- Version using Agda with constructor.\n\u2237-Stream' : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream' h with Stream-out h\n... | x' , xs' , prf , Sxs' =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\nStream\u2192Colist : \u2200 {xs} \u2192 Stream xs \u2192 Colist xs\nStream\u2192Colist {xs} Sxs = Colist-coind A h\u2081 h\u2082\n  where\n  A : D \u2192 Set\n  A ys = Stream ys\n\n  h\u2081 : \u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n  h\u2081 Axs with Stream-out Axs\n  ... | x' , xs' , prf , Sxs' = inj\u2082 (x' , xs' , prf , Sxs')\n\n  h\u2082 : A xs\n  h\u2082 = Sxs\n\n-- Adapted from (Sander 1992, p. 59).\nstreamLength : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength {xs} Sxs = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R m n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 R m n \u2192\n       m \u2261 zero \u2227 n \u2261 zero\n         \u2228 (\u2203[ m' ] \u2203[ n' ] m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n' \u2227 R m' n')\n  h\u2081 {m} (xs , Sxs , m=lxs , n\u2261\u221e) with Stream-out Sxs\n  ... | x' , xs' , xs\u2261x'\u2237xs' , Sxs' =\n    inj\u2082 (length xs' , \u221e , helper , trans n\u2261\u221e \u221e-eq , (xs' , Sxs' , refl , refl))\n    where\n    helper : m \u2261 succ\u2081 (length xs')\n    helper = trans m=lxs (trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs'))\n\n  h\u2082 : R (length xs) \u221e\n  h\u2082 = xs , Sxs , refl , refl\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Sander, Herbert P. (1992). A Logic of Functional Programs with an\n-- Application to Concurrency. PhD thesis. Department of Computer\n-- Sciences: Chalmers University of Technology and University of\n-- Gothenburg.\n","old_contents":"------------------------------------------------------------------------------\n-- Streams properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality.Type\nopen import FOTC.Data.Colist\nopen import FOTC.Data.Colist.PropertiesI\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream.Type\n\n-----------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Stream predicate is also a pre-fixed point of the functional\n-- StreamF, i.e.\n--\n-- StreamF Stream \u2264 Stream (see FOTC.Data.Stream.Type).\nStream-in : \u2200 {xs} \u2192\n            \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs' \u2192\n            Stream xs\nStream-in h = Stream-coind A h' h\n  where\n  A : D \u2192 Set\n  A xs = \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 Stream xs'\n\n  h' : \u2200 {xs} \u2192 A xs \u2192 \u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs'\n  h' (x' , xs' , prf , Sxs') = x' , xs' , prf , Stream-out Sxs'\n\n\u2237-Stream : \u2200 {x xs} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\u2237-Stream h with Stream-out h\n... | x' , xs' , prf , Sxs' =\n  subst Stream (sym (\u2227-proj\u2082 (\u2237-injective prf))) Sxs'\n\nStream\u2192Colist : \u2200 {xs} \u2192 Stream xs \u2192 Colist xs\nStream\u2192Colist {xs} Sxs = Colist-coind A h\u2081 h\u2082\n  where\n  A : D \u2192 Set\n  A ys = Stream ys\n\n  h\u2081 : \u2200 {xs} \u2192 A xs \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] xs \u2261 x' \u2237 xs' \u2227 A xs')\n  h\u2081 Axs with Stream-out Axs\n  ... | x' , xs' , prf , Sxs' = inj\u2082 (x' , xs' , prf , Sxs')\n\n  h\u2082 : A xs\n  h\u2082 = Sxs\n\n-- Adapted from (Sander 1992, p. 59).\nstreamLength : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\nstreamLength {xs} Sxs = \u2248N-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R m n = \u2203[ xs ] Stream xs \u2227 m \u2261 length xs \u2227 n \u2261 \u221e\n\n  h\u2081 : \u2200 {m n} \u2192 R m n \u2192\n       m \u2261 zero \u2227 n \u2261 zero\n         \u2228 (\u2203[ m' ] \u2203[ n' ] m \u2261 succ\u2081 m' \u2227 n \u2261 succ\u2081 n' \u2227 R m' n')\n  h\u2081 {m} (xs , Sxs , m=lxs , n\u2261\u221e) with Stream-out Sxs\n  ... | x' , xs' , xs\u2261x'\u2237xs' , Sxs' =\n    inj\u2082 (length xs' , \u221e , helper , trans n\u2261\u221e \u221e-eq , (xs' , Sxs' , refl , refl))\n    where\n    helper : m \u2261 succ\u2081 (length xs')\n    helper = trans m=lxs (trans (lengthCong xs\u2261x'\u2237xs') (length-\u2237 x' xs'))\n\n  h\u2082 : R (length xs) \u221e\n  h\u2082 = xs , Sxs , refl , refl\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Sander, Herbert P. (1992). A Logic of Functional Programs with an\n-- Application to Concurrency. PhD thesis. Department of Computer\n-- Sciences: Chalmers University of Technology and University of\n-- Gothenburg.\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"38caf1c378f3acd54cdebe3e3fb1714898336a43","subject":"Describe the purpose of TotalTerms.agda (see #28).","message":"Describe the purpose of TotalTerms.agda (see #28).\n\nThere are three comments because I intend to split this file.\n\nOld-commit-hash: 9fddcf21b9f59d47ad8a67a88f1304cc9098741a\n","repos":"inc-lc\/ilc-agda","old_file":"TotalTerms.agda","new_file":"TotalTerms.agda","new_contents":"module TotalTerms where\n\n-- TERMS with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the syntax of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\n-- EVALUATION with a primitive for TOTAL DERIVATIVES\n--\n-- This module defines the semantics of terms that support a\n-- primitive (\u0394 e) for computing the total derivative according\n-- to all free variables in e and all future arguments of e if e\n-- is a function.\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Binary.PropositionalEquality as P\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n  weakenOne : \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext-t :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n\nsubstTerm : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 P.\u2261 \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nsubstTerm {\u03c4} {\u0393\u2081} {\u0393\u2082} \u2261\u2081 t = subst (\u03bb \u0393 \u2192 Term \u0393 \u03c4) \u2261\u2081 t\n","old_contents":"module TotalTerms where\n\nopen import Relation.Binary.PropositionalEquality as P\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  true false : \u2200 {\u0393} \u2192 Term \u0393 bool\n  if : \u2200 {\u0393 \u03c4} \u2192 (t\u2081 : Term \u0393 bool) (t\u2082 t\u2083 : Term \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n  weakenOne : \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u03c4\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 true \u27e7Term \u03c1 = true\n\u27e6 false \u27e7Term \u03c1 = false\n\u27e6 if t\u2081 t\u2082 t\u2083 \u27e7Term \u03c1 with \u27e6 t\u2081 \u27e7Term \u03c1\n... | true = \u27e6 t\u2082 \u27e7Term \u03c1\n... | false = \u27e6 t\u2083 \u27e7Term \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7Term \u03c1 = \u27e6 t \u27e7Term (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext-t :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n\nsubstTerm : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 P.\u2261 \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nsubstTerm {\u03c4} {\u0393\u2081} {\u0393\u2082} \u2261\u2081 t = subst (\u03bb \u0393 \u2192 Term \u0393 \u03c4) \u2261\u2081 t\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9b9f837762792c3a0616d4a16e8420a7aff225db","subject":"less asinine names + moving lemmas around #14","message":"less asinine names + moving lemmas around #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-progress-checks.agda","new_file":"lemmas-progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule lemmas-progress-checks where\n  indet-not-trans : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  indet-not-trans IEHole ()\n  indet-not-trans (INEHole x) ()\n  indet-not-trans (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  indet-not-trans (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  indet-not-trans (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = x refl\n  indet-not-trans (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  indet-not-trans (ICastGroundHole () ind) (ITCastID x\u2081)\n  indet-not-trans (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  indet-not-trans (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  indet-not-trans (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  indet-not-trans (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  indet-not-trans (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  boxedval-not-trans : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  boxedval-not-trans (BVVal VConst) ()\n  boxedval-not-trans (BVVal VLam) ()\n  boxedval-not-trans (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  boxedval-not-trans (BVHoleCast () bv) (ITCastID x\u2081)\n  boxedval-not-trans (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  boxedval-not-trans (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  final-not-trans : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  final-not-trans (FBoxed x) = boxedval-not-trans x\n  final-not-trans (FIndet x) = indet-not-trans x\n\n  final-sub-final : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  final-sub-final x FHOuter = x\n  final-sub-final (FBoxed (BVVal ())) (FHAp1 eps)\n  final-sub-final (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  final-sub-final (FBoxed (BVVal ())) (FHNEHole eps)\n  final-sub-final (FBoxed (BVVal ())) (FHCast eps)\n  final-sub-final (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = final-sub-final (FBoxed x\u2082) eps\n  final-sub-final (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = final-sub-final (FBoxed x\u2082) eps\n  final-sub-final (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = final-sub-final (FIndet x\u2082) eps\n  final-sub-final (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = final-sub-final x\u2083 eps\n  final-sub-final (FIndet (INEHole x\u2081)) (FHNEHole eps) = final-sub-final x\u2081 eps\n  final-sub-final (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = final-sub-final (FIndet x\u2082) eps\n  final-sub-final (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = final-sub-final (FIndet x\u2082) eps\n  final-sub-final (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = final-sub-final (FIndet x\u2082) eps\n\n  final-sub-not-trans : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  final-sub-not-trans f sub step = final-not-trans (final-sub-final f sub) step\n\n  ce-castarr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n  ce-castarr (CECong FHOuter ce) = ce-castarr ce\n  ce-castarr (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n\n  ce-castth : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n  ce-castth (CECastFail x\u2081 x\u2082 () x\u2084)\n  ce-castth (CECong FHOuter ce) = ce-castth ce\n  ce-castth (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n\n  ce-nehole : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n  ce-nehole (CECong FHOuter ce) = ce-nehole ce\n  ce-nehole (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n\n  ce-ap : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n  ce-ap (CECong FHOuter ce) = ce-ap ce\n  ce-ap (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n  ce-ap (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n\n  ce-castht : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n  ce-castht (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n  ce-castht (CECong FHOuter ce) = ce-castht ce\n  ce-castht (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n\n  -- ce-cast : \u2200{d \u03c41 \u03c42} \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) casterr \u2192 d casterr\n  -- ce-cast (CECastFail x x\u2081 x\u2082 x\u2083) = {!!}\n  -- ce-cast (CECong x ce) = {!!}\n\n  ce-out-cast : \u2200{d d' \u03b5} \u2192\n                d casterr \u2192\n                d' == \u03b5 \u27e6 d \u27e7 \u2192\n                d' casterr\n  ce-out-cast (CECong x ce) FHOuter = ce-out-cast ce x\n  ce-out-cast (CECong x ce) y = CECong y (ce-out-cast ce x)\n  ce-out-cast z FHOuter = z\n  ce-out-cast z y = CECong y z\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule lemmas-progress-checks where\n  ce-castarr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n  ce-castarr (CECong FHOuter ce) = ce-castarr ce\n  ce-castarr (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n\n  ce-castth : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n  ce-castth (CECastFail x\u2081 x\u2082 () x\u2084)\n  ce-castth (CECong FHOuter ce) = ce-castth ce\n  ce-castth (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n\n  ce-nehole : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n  ce-nehole (CECong FHOuter ce) = ce-nehole ce\n  ce-nehole (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n\n  ce-ap : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n  ce-ap (CECong FHOuter ce) = ce-ap ce\n  ce-ap (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n  ce-ap (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n\n  ce-castht : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n  ce-castht (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n  ce-castht (CECong FHOuter ce) = ce-castht ce\n  ce-castht (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n\n  -- ce-cast : \u2200{d \u03c41 \u03c42} \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) casterr \u2192 d casterr\n  -- ce-cast (CECastFail x x\u2081 x\u2082 x\u2083) = {!!}\n  -- ce-cast (CECong x ce) = {!!}\n\n  ce-out-cast : \u2200{d d' \u03b5} \u2192\n                d casterr \u2192\n                d' == \u03b5 \u27e6 d \u27e7 \u2192\n                d' casterr\n  ce-out-cast (CECong x ce) FHOuter = ce-out-cast ce x\n  ce-out-cast (CECong x ce) y = CECong y (ce-out-cast ce x)\n  ce-out-cast z FHOuter = z\n  ce-out-cast z y = CECong y z\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"77d4cdbd1a3c9495c6cd2a70a8b60d40bacb202f","subject":"List monad module","message":"List monad module\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/List\/NP.agda","new_file":"lib\/Data\/List\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.List.NP where\n\nopen import Category.Monad\nopen import Category.Applicative.NP\nopen import Data.List  public\nopen import Data.Bool  using (Bool; not; if_then_else_)\nopen import Data.Nat   using (\u2115; zero; suc)\nopen import Data.Maybe using (Maybe; just; nothing)\nopen import Function   using (_\u2218_)\n\nmodule Monad {\u2113} where\n  open RawMonad (monad {\u2113}) public\n  open RawApplicative rawIApplicative public using (replicateM)\n\n-- Move this\nEq : \u2200 {a} \u2192 Set a \u2192 Set a\nEq A = (x\u2081 x\u2082 : A) \u2192 Bool\n\nuniqBy : \u2200 {A : Set} \u2192 Eq A \u2192 List A \u2192 List A\nuniqBy _==_ []       = []\nuniqBy _==_ (x \u2237 xs) = x \u2237 filter (not \u2218 _==_ x) (uniqBy _==_ xs)\n                    -- x \u2237 uniqBy _==_ (filter (not \u2218 _==_ x) xs)\n\nlistToMaybe : \u2200 {a} {A : Set a} \u2192 List A \u2192 Maybe A\nlistToMaybe []      = nothing\nlistToMaybe (x \u2237 _) = just x\n\nfindIndices : \u2200 {a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List \u2115\n-- findIndices p xs = [ i | (x,i) <- zip xs [0..], p x ]\nfindIndices {A = A} p = go 0 where\n  go : \u2115 \u2192 List A \u2192 List \u2115\n  go _ []       = []\n  go i (x \u2237 xs) = if p x then i \u2237 go (suc i) xs else go (suc i) xs\n\nfindIndex : \u2200 {a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Maybe \u2115\nfindIndex p = listToMaybe \u2218 findIndices p\n\nindex : \u2200 {a} {A : Set a} \u2192 Eq A \u2192 A \u2192 List A \u2192 Maybe \u2115\nindex _==_ x = findIndex (_==_ x)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.List.NP where\n\nopen import Data.List  public\nopen import Data.Bool  using (Bool; not; if_then_else_)\nopen import Data.Nat   using (\u2115; zero; suc)\nopen import Data.Maybe using (Maybe; just; nothing)\nopen import Function   using (_\u2218_)\n\n-- Move this\nEq : \u2200 {a} \u2192 Set a \u2192 Set a\nEq A = (x\u2081 x\u2082 : A) \u2192 Bool\n\nuniqBy : \u2200 {A : Set} \u2192 Eq A \u2192 List A \u2192 List A\nuniqBy _==_ []       = []\nuniqBy _==_ (x \u2237 xs) = x \u2237 filter (not \u2218 _==_ x) (uniqBy _==_ xs)\n                    -- x \u2237 uniqBy _==_ (filter (not \u2218 _==_ x) xs)\n\nlistToMaybe : \u2200 {a} {A : Set a} \u2192 List A \u2192 Maybe A\nlistToMaybe []      = nothing\nlistToMaybe (x \u2237 _) = just x\n\nfindIndices : \u2200 {a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List \u2115\n-- findIndices p xs = [ i | (x,i) <- zip xs [0..], p x ]\nfindIndices {A = A} p = go 0 where\n  go : \u2115 \u2192 List A \u2192 List \u2115\n  go _ []       = []\n  go i (x \u2237 xs) = if p x then i \u2237 go (suc i) xs else go (suc i) xs\n\nfindIndex : \u2200 {a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 List A \u2192 Maybe \u2115\nfindIndex p = listToMaybe \u2218 findIndices p\n\nindex : \u2200 {a} {A : Set a} \u2192 Eq A \u2192 A \u2192 List A \u2192 Maybe \u2115\nindex _==_ x = findIndex (_==_ x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6deb837cb8897cb6ce4dc9c8734658bf52f149f6","subject":"Isos: cleaner Lift\u2194id","message":"Isos: cleaner Lift\u2194id\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nopen L using (Lift; lower; lift)\nopen import Type\nimport Function as F\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat using (\u2115; zero; suc)\nopen import Data.Maybe.NP\nopen import Data.Product\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms public\nopen import Function.Inverse using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = record { to = \u2192-to-\u27f6 lower\n                 ; from = \u2192-to-\u27f6 lift\n                 ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                       ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = (proj\u2082 CMon.identity \u22a4 \u2218 id \u228e-cong (sym (Lift\u2194id {A = \u22a5}))) \u2218 Maybe\u2194\u22a4\u228e\n  where module CMon = CommutativeMonoid (\u228e-CommutativeMonoid bijection L.zero)\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n","old_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nopen L using (Lift; lower; lift)\nopen import Type\nimport Function as F\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat using (\u2115; zero; suc)\nopen import Data.Maybe.NP\nopen import Data.Product\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms public\nopen import Function.Inverse using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = \u03bb {a} {A} \u2192 record { to = \u2192-to-\u27f6 lower\n                             ; from = \u2192-to-\u27f6 lift\n                             ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                                   ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = (proj\u2082 CMon.identity \u22a4 \u2218 id \u228e-cong (sym (Lift\u2194id {A = \u22a5}))) \u2218 Maybe\u2194\u22a4\u228e\n  where module CMon = CommutativeMonoid (\u228e-CommutativeMonoid bijection L.zero)\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"16f3e21650411f9cf452b30bdfdfe4f2bee1804d","subject":"Desc model: iso1 implicit.","message":"Desc model: iso1 implicit.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : {I : Set} -> IDescl I -> IDesc I\niso1 {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\niso1 : (I : Set) -> IDescl I -> IDesc I\niso1 I d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\niso2 : {I : Set} -> IDesc I -> IDescl I\niso2 (var i) = varl i\niso2 (const X) = constl X\niso2 (prod D D') = prodl (iso2 D) (iso2 D')\niso2 (pi S T) = pil S (\\s -> iso2 (T s))\niso2 (sigma S T) = sigmal S (\\s -> iso2 (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-iso1-iso2 : (I : Set) -> (D : IDesc I) -> iso1 I (iso2 D) == D\nproof-iso1-iso2 I (var i) = refl\nproof-iso1-iso2 I (const x) = refl\nproof-iso1-iso2 I (prod D D') with proof-iso1-iso2 I D | proof-iso1-iso2 I D'\n...                             | p | q = cong2 prod p q\nproof-iso1-iso2 I (pi S T) = cong (pi S)  \n                                  (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                           T\n                                           (\\s -> proof-iso1-iso2 I (T s)))\nproof-iso1-iso2 I (sigma S T) = cong (sigma S) \n                                     (reflFun (\u03bb s \u2192 iso1 I (iso2 (T s)))\n                                              T\n                                              (\\s -> proof-iso1-iso2 I (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = iso2 (iso1 I D) == D\n\nproof-iso2-iso1 : (I : Set) -> (D : IDescl I) -> iso2 (iso1 I D) == D\nproof-iso2-iso1 I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-iso2-iso1-casesW I) \n                                 Void\n                                 D\n                where proof-iso2-iso1-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-iso2-iso1-cases I (lvar , i) hs = refl\n                      proof-iso2-iso1-cases I (lconst , x) hs = refl\n                      proof-iso2-iso1-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-iso2-iso1-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-iso2-iso1-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 iso2 (iso1 I (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-iso2-iso1-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-iso2-iso1-casesW I Void = proof-iso2-iso1-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1ef99259bd4c08172831359f47bda1541f187b41","subject":"Desc introduction in operational semantics.","message":"Desc introduction in operational semantics.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Operational.agda","new_file":"formalization\/agda\/Spire\/Operational.agda","new_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Desc `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Desc `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Value \u0393 (`Type \u2113))\n    (B : Value (\u0393 , \u27e6 A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Value \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elim\u22a5 : \u2200{A} \u2192 Neutral \u0393 `\u22a5 \u2192 Neutral \u0393 A\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `Desc {\u2113} \u27e7 = `Desc \u2113\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nelim\u22a5 : \u2200{\u0393 A} \u2192 Value \u0393 `\u22a5 \u2192 Value \u0393 A\nelim\u22a5 (`neut bot) = `neut (`elim\u22a5 bot)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Term \u0393 (`Type \u2113))\n    (D : Term (\u0393 , \u27e6 eval A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Term \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B}\n    (b : Term (\u0393 , A) B)\n    \u2192 Term \u0393 (`\u03a0 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elim\u22a5 : \u2200{A} \u2192 Term \u0393 `\u22a5 \u2192 Term \u0393 A\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Desc introduction -}\neval `\u22a4\u1d48 = `\u22a4\u1d48\neval `X\u1d48 = `X\u1d48\neval (`\u03a0\u1d48 A D) = `\u03a0\u1d48 (eval A) (eval D)\neval (`\u03a3\u1d48 A D) = `\u03a3\u1d48 (eval A) (eval D)\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\neval (`\u03bb b) = `\u03bb (eval b)\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elim\u22a5 bot) = elim\u22a5 (eval bot)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\n\n----------------------------------------------------------------------\n\n","old_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elim\u22a5 : \u2200{A} \u2192 Neutral \u0393 `\u22a5 \u2192 Neutral \u0393 A\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\nelim\u22a5 : \u2200{\u0393 A} \u2192 Value \u0393 `\u22a5 \u2192 Value \u0393 A\nelim\u22a5 (`neut bot) = `neut (`elim\u22a5 bot)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B}\n    (b : Term (\u0393 , A) B)\n    \u2192 Term \u0393 (`\u03a0 A B)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elim\u22a5 : \u2200{A} \u2192 Term \u0393 `\u22a5 \u2192 Term \u0393 A\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\neval (`\u03bb b) = `\u03bb (eval b)\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elim\u22a5 bot) = elim\u22a5 (eval bot)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f37d4bbfe846ae9bfa96642964754e442e960cf2","subject":"Fixed typo.","message":"Fixed typo.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/LogicalFramework\/Existential.agda","new_file":"notes\/thesis\/LogicalFramework\/Existential.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.Existential where\n\nmodule LF where\n  postulate\n    D : Set\n\n    -- Disjunction.\n    _\u2228_  : Set \u2192 Set \u2192 Set\n    inj\u2081 : {A B : Set} \u2192 A \u2192 A \u2228 B\n    inj\u2082 : {A B : Set} \u2192 B \u2192 A \u2228 B\n    case : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n\n    -- The existential quantifier type on D.\n    \u2203       : (A : D \u2192 Set) \u2192 Set\n    _,_     : {A : D \u2192 Set}(t : D) \u2192 A t \u2192 \u2203 A\n    \u2203-proj\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    \u2203-proj\u2082 : {A : D \u2192 Set}(h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n    \u2203-elim  : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = case (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n                 (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n                 (\u2203-proj\u2082 h)\n\n    -- Using the elimination.\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 case (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                      (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                      ah)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\nmodule Inductive where\n\n  open import Common.FOL.FOL\n\n  -- The existential proyections.\n  \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n  \u2203-proj\u2082 (_ , Ax) = Ax\n\n  -- The existential elimination.\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (_ , Ax) h = h Ax\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = case (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n                 (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n                 (\u2203-proj\u2082 h)\n\n    -- Using the elimination.\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 case (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                      (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                      ah)\n\n    -- Using pattern matching.\n    \u2203\u2200\u2083 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2083 (x , Ax) y = x , Ax y\n\n    \u2203\u2228\u2083 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2083 (x , inj\u2081 Ax) = inj\u2081 (x , Ax)\n    \u2203\u2228\u2083 (x , inj\u2082 Bx) = inj\u2082 (x , Bx)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\n    -- Using the pattern matching.\n    non-FOL\u2083 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2083 (x , _) = x\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.Existential where\n\nmodule LF where\n  postulate\n    D : Set\n\n    -- Disjunction.\n    _\u2228_  : Set \u2192 Set \u2192 Set\n    inj\u2081 : {A B : Set} \u2192 A \u2192 A \u2228 B\n    inj\u2082 : {A B : Set} \u2192 B \u2192 A \u2228 B\n    case : {A B C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n\n    -- The existential quantifier type on D.\n    \u2203       : (A : D \u2192 Set) \u2192 Set\n    _,_     : {A : D \u2192 Set}(t : D) \u2192 A t \u2192 \u2203 A\n    \u2203-proj\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    \u2203-proj\u2082 : {A : D \u2192 Set}(h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n    \u2203-elim  : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n\n  syntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = case (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n                 (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n                 (\u2203-proj\u2082 h)\n\n    -- Using the elimination\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 case (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                      (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                      ah)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\nmodule Inductive where\n\n  open import Common.FOL.FOL\n\n  -- The existential proyections.\n  \u2203-proj\u2081 : \u2200 {A} \u2192 \u2203 A \u2192 D\n  \u2203-proj\u2081 (x , _) = x\n\n  \u2203-proj\u2082 : \u2200 {A} \u2192 (h : \u2203 A) \u2192 A (\u2203-proj\u2081 h)\n  \u2203-proj\u2082 (_ , Ax) = Ax\n\n  -- The existential elimination.\n  \u2203-elim : {A : D \u2192 Set}{B : Set} \u2192 \u2203 A \u2192 (\u2200 {x} \u2192 A x \u2192 B) \u2192 B\n  \u2203-elim (_ , Ax) h = h Ax\n\n  module FOL-Examples where\n    -- Using the projections.\n    \u2203\u2200\u2081 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2081 h y = \u2203-proj\u2081 h , (\u2203-proj\u2082 h) y\n\n    \u2203\u2228\u2081 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2081 h = case (\u03bb Ax \u2192 inj\u2081 (\u2203-proj\u2081 h , Ax))\n                 (\u03bb Bx \u2192 inj\u2082 (\u2203-proj\u2081 h , Bx))\n                 (\u2203-proj\u2082 h)\n\n    -- Using the elimination.\n    \u2203\u2200\u2082 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2082 h y = \u2203-elim h (\u03bb {x} ah \u2192 x , ah y)\n\n    \u2203\u2228\u2082 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2082 h = \u2203-elim h (\u03bb {x} ah \u2192 case (\u03bb Ax \u2192 inj\u2081 (x , Ax))\n                                      (\u03bb Bx \u2192 inj\u2082 (x , Bx))\n                                      ah)\n\n    -- Using pattern matching.\n    \u2203\u2200\u2083 : {A : D \u2192 D \u2192 Set} \u2192 \u2203[ x ](\u2200 y \u2192 A x y) \u2192 \u2200 y \u2192 \u2203[ x ] A x y\n    \u2203\u2200\u2083 (x , Ax) y = x , Ax y\n\n    \u2203\u2228\u2083 : {A B : D \u2192 Set} \u2192 \u2203[ x ](A x \u2228 B x) \u2192 (\u2203[ x ] A x) \u2228 (\u2203[ x ] B x)\n    \u2203\u2228\u2083 (x , inj\u2081 Ax) = inj\u2081 (x , Ax)\n    \u2203\u2228\u2083 (x , inj\u2082 Bx) = inj\u2082 (x , Bx)\n\n  module NonFOL-Examples where\n\n    -- Using the projections.\n    non-FOL\u2081 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2081 h = \u2203-proj\u2081 h\n\n    -- Using the elimination.\n    non-FOL\u2082 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2082 h = \u2203-elim h (\u03bb {x} _ \u2192 x)\n\n    -- Using the pattern matching.\n    non-FOL\u2083 : {A : D \u2192 Set} \u2192 \u2203 A \u2192 D\n    non-FOL\u2083 (x , _) = x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1ab9564a3e920bcc55846015f5ad6db766a9b16d","subject":"Bool: more logical rels","message":"Bool: more logical rels\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bool\/NP.agda","new_file":"lib\/Data\/Bool\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nopen import Algebra.FunctionProperties using (Op\u2081; Op\u2082)\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\nopen import Function.NP\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6not\u27e7 : (\u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7) not not\n\u27e6not\u27e7 \u27e6true\u27e7  = \u27e6false\u27e7\n\u27e6not\u27e7 \u27e6false\u27e7 = \u27e6true\u27e7\n\n_\u27e6\u2227\u27e7_ : (\u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7) _\u2227_ _\u2227_\n\u27e6true\u27e7  \u27e6\u2227\u27e7 x = x\n\u27e6false\u27e7 \u27e6\u2227\u27e7 _ = \u27e6false\u27e7\n\n_\u27e6\u2228\u27e7_ : (\u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7) _\u2228_ _\u2228_\n\u27e6true\u27e7  \u27e6\u2228\u27e7 _ = \u27e6true\u27e7\n\u27e6false\u27e7 \u27e6\u2228\u27e7 x = x\n\n_\u27e6xor\u27e7_ : (\u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 \u27e6Bool\u27e7) _xor_ _xor_\n\u27e6true\u27e7  \u27e6xor\u27e7 x = \u27e6not\u27e7 x\n\u27e6false\u27e7 \u27e6xor\u27e7 x = x\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\n\u2261\u2192T : \u2200 {b} \u2192 b \u2261 true \u2192 T b\n\u2261\u2192T \u2261.refl = _\n\n\u2261\u2192Tnot : \u2200 {b} \u2192 b \u2261 false \u2192 T (not b)\n\u2261\u2192Tnot \u2261.refl = _\n\nT\u2192\u2261 : \u2200 {b} \u2192 T b \u2192 b \u2261 true\nT\u2192\u2261 {true}  _  = \u2261.refl\nT\u2192\u2261 {false} ()\n\nTnot\u2192\u2261 : \u2200 {b} \u2192 T (not b) \u2192 b \u2261 false\nTnot\u2192\u2261 {false} _  = \u2261.refl\nTnot\u2192\u2261 {true}  ()\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n\nmodule Indexed {a} {A : Set a} where\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 Bool)\n    not\u00b0 f = not \u2218 f\n\ndata So : Bool \u2192 Set where\n  oh! : So true\n\nSo\u2192T : \u2200 {b} \u2192 So b \u2192 T b\nSo\u2192T oh! = _\n\nT\u2192So : \u2200 {b} \u2192 T b \u2192 So b\nT\u2192So {true}  _  = oh!\nT\u2192So {false} ()\n\nSo\u2192\u2261 : \u2200 {b} \u2192 So b \u2192 b \u2261 true\nSo\u2192\u2261 oh! = \u2261.refl\n\n\u2261\u2192So : \u2200 {b} \u2192 b \u2261 true \u2192 So b\n\u2261\u2192So \u2261.refl = oh!\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Bool.NP where\n\nopen import Data.Bool using (Bool; true; false; T; if_then_else_; not)\nimport Algebra\nopen import Algebra.FunctionProperties using (Op\u2081; Op\u2082)\nimport Data.Bool.Properties as B\nopen import Data.Unit using (\u22a4)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Nat using (\u2115; _\u2264_; z\u2264n; s\u2264s; _\u2293_; _\u2294_; _\u2238_)\nopen import Function.NP\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nimport Relation.Binary.PropositionalEquality as \u2261\nimport Function.Equivalence as E\nopen E.Equivalence using (to; from)\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen \u2261 using (_\u2261_)\n\ncond : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\ncond x y b = if b then x else y\n\nmodule Xor\u00b0 = Algebra.CommutativeRing B.commutativeRing-xor-\u2227\nmodule Bool\u00b0 = Algebra.CommutativeSemiring B.commutativeSemiring-\u2227-\u2228\n\ncheck : \u2200 b \u2192 {pf : T b} \u2192 \u22a4\ncheck = _\n\nIf_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 (T b \u2192 A) \u2192 (T (not b) \u2192 B) \u2192 if b then A else B\nIf_then_else_ true  x _ = x _\nIf_then_else_ false _ x = x _\n\nIf\u2032_then_else_ : \u2200 {\u2113} {A B : Set \u2113} b \u2192 A \u2192 B \u2192 if b then A else B\nIf\u2032_then_else_ true  x _ = x\nIf\u2032_then_else_ false _ x = x\n\nIf-map : \u2200 {A B C D : Set} b (f : T b \u2192 A \u2192 C) (g : T (not b) \u2192 B \u2192 D) \u2192\n           if b then A else B \u2192 if b then C else D\nIf-map true  f _ = f _\nIf-map false _ f = f _\n\nIf-elim : \u2200 {A B : Set} {P : Bool \u2192 Set}\n            b (f : T b \u2192 A \u2192 P true) (g : T (not b) \u2192 B \u2192 P false) \u2192 if b then A else B \u2192 P b\nIf-elim true  f _ = f _\nIf-elim false _ f = f _\n\nIf-true : \u2200 {A B : Set} {b} \u2192 T b \u2192 (if b then A else B) \u2261 A\nIf-true {b = true}  _ = \u2261.refl\nIf-true {b = false} ()\n\nIf-false : \u2200 {A B : Set} {b} \u2192 T (not b) \u2192 (if b then A else B) \u2261 B\nIf-false {b = true}  ()\nIf-false {b = false} _ = \u2261.refl\n\ncong-if : \u2200 {A B : Set} b {t\u2080 t\u2081} (f : A \u2192 B) \u2192 (if b then f t\u2080 else f t\u2081) \u2261 f (if b then t\u2080 else t\u2081)\ncong-if true  _ = \u2261.refl\ncong-if false _ = \u2261.refl\n\nif-not : \u2200 {a} {A : Set a} b {t\u2080 t\u2081 : A} \u2192 (if b then t\u2080 else t\u2081) \u2261 (if not b then t\u2081 else t\u2080)\nif-not true  = \u2261.refl\nif-not false = \u2261.refl\n\ndata \u27e6Bool\u27e7 : (b\u2081 b\u2082 : Bool) \u2192 Set where\n  \u27e6true\u27e7   : \u27e6Bool\u27e7 true true\n  \u27e6false\u27e7  : \u27e6Bool\u27e7 false false\n\nprivate\n module \u27e6Bool\u27e7-Internals where\n  refl : Reflexive \u27e6Bool\u27e7\n  refl {true}   = \u27e6true\u27e7\n  refl {false}  = \u27e6false\u27e7\n\n  sym : Symmetric \u27e6Bool\u27e7\n  sym \u27e6true\u27e7  = \u27e6true\u27e7\n  sym \u27e6false\u27e7 = \u27e6false\u27e7\n\n  trans : Transitive \u27e6Bool\u27e7\n  trans \u27e6true\u27e7   = id\n  trans \u27e6false\u27e7  = id\n\n  subst : \u2200 {\u2113} \u2192 Substitutive \u27e6Bool\u27e7 \u2113\n  subst _ \u27e6true\u27e7   = id\n  subst _ \u27e6false\u27e7  = id\n\n  _\u225f_ : Decidable \u27e6Bool\u27e7\n  true   \u225f  true   = yes \u27e6true\u27e7\n  false  \u225f  false  = yes \u27e6false\u27e7\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence \u27e6Bool\u27e7\n  isEquivalence = record { refl = refl; sym = sym; trans = trans }\n\n  isDecEquivalence : IsDecEquivalence \u27e6Bool\u27e7\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = \u27e6Bool\u27e7; isDecEquivalence = isDecEquivalence }\n\n  equality : Equality \u27e6Bool\u27e7\n  equality = record { isEquivalence = isEquivalence; subst = subst }\n\nmodule \u27e6Bool\u27e7-Props where\n  open \u27e6Bool\u27e7-Internals public using (subst; decSetoid; equality)\n  open DecSetoid decSetoid public\n  open Equality equality public hiding (subst; isEquivalence; refl; reflexive; sym; trans)\n\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 : \u2200 {a\u2081 a\u2082 a\u1d63} \u2192 (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {a\u2081} {a\u2082} a\u1d63 \u27e9\u27e6\u2192\u27e7 \u27e6Bool\u27e7 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 A\u1d63) if_then_else_ if_then_else_\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6if\u27e8_\u27e9_then_else_\u27e7 _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 : \u2200 {\u2113\u2081 \u2113\u2082 \u2113\u1d63} \u2192\n                       (\u2200\u27e8 A\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 B\u1d63 \u2236 \u27e6Set\u27e7 {\u2113\u2081} {\u2113\u2082} \u2113\u1d63 \u27e9\u27e6\u2192\u27e7\n                           \u27e8 b\u1d63 \u2236 \u27e6Bool\u27e7 \u27e9\u27e6\u2192\u27e7 A\u1d63 \u27e6\u2192\u27e7 B\u1d63 \u27e6\u2192\u27e7 \u27e6if\u27e8 \u27e6Set\u27e7 _ \u27e9 b\u1d63 then A\u1d63 else B\u1d63 \u27e7)\n                       If\u2032_then_else_ If\u2032_then_else_\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6true\u27e7  x\u1d63 _ = x\u1d63\n\u27e6If\u2032\u27e8_,_\u27e9_then_else_\u27e7 _ _ \u27e6false\u27e7 _ x\u1d63 = x\u1d63\n\n_==_ : (x y : Bool) \u2192 Bool\ntrue == true = true\ntrue == false = false\nfalse == true = false\nfalse == false = true\n\nmodule == where\n  _\u2248_ : (x y : Bool) \u2192 Set\n  x \u2248 y = T (x == y)\n\n  refl : Reflexive _\u2248_\n  refl {true}  = _\n  refl {false} = _\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {true}   {true}   _ = id\n  subst _ {false}  {false}  _ = id\n  subst _ {true}   {false}  ()\n  subst _ {false}  {true}   ()\n\n  sym : Symmetric _\u2248_\n  sym {x} {y} eq = subst (\u03bb y \u2192 y \u2248 x) {x} {y} eq (refl {x})\n\n  trans : Transitive _\u2248_\n  trans {x} {y} {z} x\u2248y y\u2248z = subst (_\u2248_ x) {y} {z} y\u2248z x\u2248y\n\n  _\u225f_ : Decidable _\u2248_\n  true   \u225f  true   = yes _\n  false  \u225f  false  = yes _\n  true   \u225f  false  = no (\u03bb())\n  false  \u225f  true   = no (\u03bb())\n\n  isEquivalence : IsEquivalence _\u2248_\n  isEquivalence = record { refl = \u03bb {x} \u2192 refl {x}\n                         ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                         ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n\n  isDecEquivalence : IsDecEquivalence _\u2248_\n  isDecEquivalence = record { isEquivalence = isEquivalence; _\u225f_ = _\u225f_ }\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = Bool; _\u2248_ = _\u2248_ ; isEquivalence = isEquivalence }\n\n  decSetoid : DecSetoid _ _\n  decSetoid = record { Carrier = Bool; _\u2248_ = _\u2248_; isDecEquivalence = isDecEquivalence }\n\nmodule \u27e6Bool\u27e7-Reasoning = Setoid-Reasoning \u27e6Bool\u27e7-Props.setoid\n\nopen Data.Bool public\n\n\u27e6true\u27e7\u2032 : \u2200 {b} \u2192 T b \u2192 \u27e6Bool\u27e7 true b\n\u27e6true\u27e7\u2032 {true}  _ = \u27e6true\u27e7\n\u27e6true\u27e7\u2032 {false} ()\n\n\u27e6false\u27e7\u2032 : \u2200 {b} \u2192 T (not b) \u2192 \u27e6Bool\u27e7 false b\n\u27e6false\u27e7\u2032 {true} ()\n\u27e6false\u27e7\u2032 {false} _ = \u27e6false\u27e7\n\n\u2261\u2192T : \u2200 {b} \u2192 b \u2261 true \u2192 T b\n\u2261\u2192T \u2261.refl = _\n\n\u2261\u2192Tnot : \u2200 {b} \u2192 b \u2261 false \u2192 T (not b)\n\u2261\u2192Tnot \u2261.refl = _\n\nT\u2192\u2261 : \u2200 {b} \u2192 T b \u2192 b \u2261 true\nT\u2192\u2261 {true}  _  = \u2261.refl\nT\u2192\u2261 {false} ()\n\nTnot\u2192\u2261 : \u2200 {b} \u2192 T (not b) \u2192 b \u2261 false\nTnot\u2192\u2261 {false} _  = \u2261.refl\nTnot\u2192\u2261 {true}  ()\n\nT\u2227 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T b\u2082 \u2192 T (b\u2081 \u2227 b\u2082)\nT\u2227 p q = _\u27e8$\u27e9_ (from B.T-\u2227) (p , q)\n\nT\u2227\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2081\nT\u2227\u2081 = proj\u2081 \u2218 _\u27e8$\u27e9_ (to B.T-\u2227)\n\nT\u2227\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2227 b\u2082) \u2192 T b\u2082\nT\u2227\u2082 {b\u2081} = proj\u2082 \u2218 _\u27e8$\u27e9_ (to (B.T-\u2227 {b\u2081}))\n\nT\u2228'\u228e : \u2200 {b\u2081 b\u2082} \u2192 T (b\u2081 \u2228 b\u2082) \u2192 T b\u2081 \u228e T b\u2082\nT\u2228'\u228e {b\u2081} = _\u27e8$\u27e9_ (to (B.T-\u2228 {b\u2081}))\n\nT\u2228\u2081 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2081 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2081 = _\u27e8$\u27e9_ (from B.T-\u2228) \u2218 inj\u2081\n\nT\u2228\u2082 : \u2200 {b\u2081 b\u2082} \u2192 T b\u2082 \u2192 T (b\u2081 \u2228 b\u2082)\nT\u2228\u2082 {b\u2081} = _\u27e8$\u27e9_ (from (B.T-\u2228 {b\u2081})) \u2218 inj\u2082\n\nT'not'\u00ac : \u2200 {b} \u2192 T (not b) \u2192 \u00ac (T b)\nT'not'\u00ac {false} _ = \u03bb()\nT'not'\u00ac {true} ()\n\nT'\u00ac'not : \u2200 {b} \u2192 \u00ac (T b) \u2192 T (not b)\nT'\u00ac'not {true}  f = f _\nT'\u00ac'not {false} _ = _\n\n\u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u2228 y)\n\u2227\u21d2\u2228 true y = _\n\u2227\u21d2\u2228 false y = \u03bb ()\n\nTdec : \u2200 b \u2192 Dec (T b)\nTdec true = yes _\nTdec false = no \u03bb()\n\nde-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\nde-morgan true  _ = \u2261.refl\nde-morgan false _ = \u2261.refl\n\n-- false is 0 and true is 1\nto\u2115 : Bool \u2192 \u2115\nto\u2115 b = if b then 1 else 0\n\nto\u2115\u22641 : \u2200 b \u2192 to\u2115 b \u2264 1\nto\u2115\u22641 true  = s\u2264s z\u2264n\nto\u2115\u22641 false = z\u2264n\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\nxor-not-not : \u2200 x y \u2192 (not x) xor (not y) \u2261 x xor y\nxor-not-not true  y = \u2261.refl\nxor-not-not false y = B.not-involutive y\n\nnot-inj : \u2200 {x y} \u2192 not x \u2261 not y \u2192 x \u2261 y\nnot-inj {true} {true} _ = \u2261.refl\nnot-inj {true} {false} ()\nnot-inj {false} {true} ()\nnot-inj {false} {false} _ = \u2261.refl\n\nxor-inj\u2081 : \u2200 x {y z} \u2192 x xor y \u2261 x xor z \u2192 y \u2261 z\nxor-inj\u2081 true  = not-inj\nxor-inj\u2081 false = id\n\nxor-inj\u2082 : \u2200 x {y z} \u2192 y xor x \u2261 z xor x \u2192 y \u2261 z\nxor-inj\u2082 x {y} {z} rewrite Xor\u00b0.+-comm y x | Xor\u00b0.+-comm z x = xor-inj\u2081 x\n\nmodule Indexed {a} {A : Set a} where\n    _\u2227\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x \u2227\u00b0 y = x \u27e8 _\u2227_ \u27e9\u00b0 y\n\n    _\u2228\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x \u2228\u00b0 y = x \u27e8 _\u2228_ \u27e9\u00b0 y\n\n    _xor\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x xor\u00b0 y = x \u27e8 _xor_ \u27e9\u00b0 y\n\n    _==\u00b0_ : Op\u2082 (A \u2192 Bool)\n    x ==\u00b0 y = x \u27e8 _==_ \u27e9\u00b0 y\n\n    not\u00b0 : Op\u2081 (A \u2192 Bool)\n    not\u00b0 f = not \u2218 f\n\ndata So : Bool \u2192 Set where\n  oh! : So true\n\nSo\u2192T : \u2200 {b} \u2192 So b \u2192 T b\nSo\u2192T oh! = _\n\nT\u2192So : \u2200 {b} \u2192 T b \u2192 So b\nT\u2192So {true}  _  = oh!\nT\u2192So {false} ()\n\nSo\u2192\u2261 : \u2200 {b} \u2192 So b \u2192 b \u2261 true\nSo\u2192\u2261 oh! = \u2261.refl\n\n\u2261\u2192So : \u2200 {b} \u2192 b \u2261 true \u2192 So b\n\u2261\u2192So \u2261.refl = oh!\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fab1df4c820031858a8111d3a016a96fcf1bd2e4","subject":"Clarify comment","message":"Clarify comment\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but instead of producing object language terms, we\n  -- produce host language terms to take advantage of the richer type system of\n  -- the host language (in particular, here we need the unit type, product types\n  -- and *existentials*).\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache2 : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache2 (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache2 (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache2 (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache2 \u03c1) (\u27e6 t \u27e7TermCache2 \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache2 (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache2 (x \u2022 \u03c1) , tt)\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache2, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4355e3c0b951411fb9c7e909706a939236eeded9","subject":"Instantiate the whole Context module.","message":"Instantiate the whole Context module.\n\nSince Type never changes in this module, we can as well instantiate\nthe definitions inside the Context module once and for all.\n\nOld-commit-hash: 0c1357babc76a15515aedf1b933256d42cbfb588\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Term\/Plotkin.agda","new_file":"Syntax\/Term\/Plotkin.agda","new_contents":"import Syntax.Type.Plotkin as Type\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen Type B\nopen import Syntax.Context {Type}\n\ndata Term\n  (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set\n  where\n  const : \u2200 {\u03c4} \u2192 (c : C \u03c4) \u2192 Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","old_contents":"import Syntax.Type.Plotkin as Type\n\nmodule Syntax.Term.Plotkin\n    {B : Set {- of base types -}}\n    {C : Type.Type B \u2192 Set {- of constants -}}\n  where\n\n-- Terms of languages described in Plotkin style\n\nopen import Function using (_\u2218_)\nopen import Data.Product\nopen Type B\nopen import Syntax.Context\n\ndata Term\n  (\u0393 : Context {Type}) :\n  (\u03c4 : Type) \u2192 Set\n  where\n  const : \u2200 {\u03c4} \u2192 (c : C \u03c4) \u2192 Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4))\n    (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n\n-- g \u229d f  = \u03bb x . \u03bb \u0394x . g (x \u2295 \u0394x) \u229d f x\n-- f \u2295 \u0394f = \u03bb x . f x \u2295 \u0394f x (x \u229d x)\n\nlift-diff-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192\n    term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4) \u00d7 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-diff-apply diff apply {base \u03b9} = diff , apply\nlift-diff-apply diff apply {\u03c3 \u21d2 \u03c4} =\n  let\n    -- for diff\n    g  = var (that (that (that this)))\n    f  = var (that (that this))\n    x  = var (that this)\n    \u0394x = var this\n    -- for apply\n    \u0394h = var (that (that this))\n    h  = var (that this)\n    y  = var this\n    -- syntactic sugars\n    diff\u03c3  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    diff\u03c4  = \u03bb {\u0393} \u2192 proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    apply\u03c3 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c3} {\u0393})\n    apply\u03c4 = \u03bb {\u0393} \u2192 proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n    _\u229d\u03c3_ = \u03bb s t  \u2192 app (app diff\u03c3 s) t\n    _\u229d\u03c4_ = \u03bb s t  \u2192 app (app diff\u03c4 s) t\n    _\u2295\u03c3_ = \u03bb t \u0394t \u2192 app (app apply\u03c3 \u0394t) t\n    _\u2295\u03c4_ = \u03bb t \u0394t \u2192 app (app apply\u03c4 \u0394t) t\n  in\n    abs (abs (abs (abs (app f (x \u2295\u03c3 \u0394x) \u229d\u03c4 app g x))))\n    ,\n    abs (abs (abs (app h y \u2295\u03c4 app (app \u0394h y) (y \u229d\u03c3 y))))\n\nlift-diff : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u0394type \u03c4)\n\nlift-diff diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2081 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\nlift-apply : \u2200 {\u0394base : B \u2192 Type} \u2192\n  let\n    \u0394type = lift-\u0394type \u0394base\n    term = Term\n  in\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (base \u03b9 \u21d2 base \u03b9 \u21d2 \u0394type (base \u03b9))) \u2192\n  (\u2200 {\u03b9 \u0393} \u2192 term \u0393 (\u0394type (base \u03b9) \u21d2 base \u03b9 \u21d2 base \u03b9)) \u2192\n  \u2200 {\u03c4 \u0393} \u2192 term \u0393 (\u0394type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\n\nlift-apply diff apply = \u03bb {\u03c4 \u0393} \u2192\n  proj\u2082 (lift-diff-apply diff apply {\u03c4} {\u0393})\n\n-- Weakening\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\n-- Specialized weakening\nweaken\u2081 : \u2200 {\u0393 \u03c3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4\nweaken\u2081 t = weaken (drop _ \u2022 \u227c-refl) t\n\nweaken\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u0393) \u03c4\nweaken\u2082 t = weaken (drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\nweaken\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03c4} \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u03b1 \u2022 \u03b2 \u2022 \u03b3 \u2022 \u0393) \u03c4\nweaken\u2083 t = weaken (drop _ \u2022 drop _ \u2022 drop _ \u2022 \u227c-refl) t\n\n-- Shorthands for nested applications\napp\u2082 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3\napp\u2082 f x = app (app f x)\n\napp\u2083 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4\napp\u2083 f x = app\u2082 (app f x)\n\napp\u2084 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5\napp\u2084 f x = app\u2083 (app f x)\n\napp\u2085 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6\napp\u2085 f x = app\u2084 (app f x)\n\napp\u2086 : \u2200 {\u0393 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7} \u2192\n    Term \u0393 (\u03b1 \u21d2 \u03b2 \u21d2 \u03b3 \u21d2 \u03b4 \u21d2 \u03b5 \u21d2 \u03b6 \u21d2 \u03b7) \u2192\n    Term \u0393 \u03b1 \u2192 Term \u0393 \u03b2 \u2192 Term \u0393 \u03b3 \u2192 Term \u0393 \u03b4 \u2192\n    Term \u0393 \u03b5 \u2192 Term \u0393 \u03b6 \u2192 Term \u0393 \u03b7\napp\u2086 f x = app\u2085 (app f x)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"52948d9a0752c59d2dd56584505f277a6d5b125a","subject":"removing a few of the more pessimistic todos","message":"removing a few of the more pessimistic todos\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192 -- need a premise that the hole names are disjoint and something to relate that to the dom(\u0394)s in expansion\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192   -- premise here that u is disjoint from hole names of e\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes anywhere\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those expressions without holes anywhere\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only know about complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1)\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- application of a substution to a term\n  -- todo: this is not well-founded with the current representation of \u03c3s\n  postulate\n    apply : subst \u2192 dhexp \u2192 dhexp\n  -- apply \u03c3 c = c\n  -- apply \u03c3 (X x) with \u03c3 x\n  -- apply \u03c3 (X x) | Some d' = d'\n  -- apply \u03c3 (X x) | None = X x\n  -- apply \u03c3 (\u00b7\u03bb x [ \u03c4 ] d) = (\u00b7\u03bb x [ \u03c4 ] (apply \u03c3 d))\n  -- apply \u03c3 \u2987\u2988\u27e8 u , \u03c3' \u27e9 =  \u2987\u2988\u27e8 u , ((\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))) \u27e9 -- (\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))\n  -- apply \u03c3 \u2987 d \u2988\u27e8 u , \u03c3' \u27e9 = \u2987 apply \u03c3 d \u2988\u27e8 u ,{!!} \u27e9\n  -- apply \u03c3 (d1 \u2218 d2) = ((apply \u03c3 d1) \u2218 (apply \u03c3 d2))\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  --hole instantiation; todo: judgemental or functional?\n  \u27e6_\/_\u27e7_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  \u27e6 d \/ u \u27e7 c = c\n  \u27e6 d \/ u \u27e7 X x = X x\n  \u27e6 d \/ u \u27e7 (\u00b7\u03bb x [ \u03c4 ] d') = \u00b7\u03bb x [ \u03c4 ] (\u27e6 d \/ u \u27e7 d')\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9  | Inr x = \u2987\u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9 | Inr x = \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 (d1 \u2218 d2) = (\u27e6 d \/ u \u27e7 d1) \u2218 (\u27e6 d \/ u \u27e7 d2)\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n","old_contents":"open import Nat\nopen import Prelude\nopen import contexts\n\nmodule core where\n  -- types\n  data htyp : Set where\n    b     : htyp\n    \u2987\u2988    : htyp\n    _==>_ : htyp \u2192 htyp \u2192 htyp\n\n  -- arrow type constructors bind very tightly\n  infixr 25  _==>_\n\n  -- expressions\n  data hexp : Set where\n    c       : hexp\n    _\u00b7:_    : hexp \u2192 htyp \u2192 hexp\n    X       : Nat \u2192 hexp\n    \u00b7\u03bb      : Nat \u2192 hexp \u2192 hexp\n    \u00b7\u03bb_[_]_ : Nat \u2192 htyp \u2192 hexp \u2192 hexp\n    \u2987\u2988[_]   : Nat \u2192 hexp\n    \u2987_\u2988[_]  : hexp \u2192 Nat \u2192 hexp\n    _\u2218_     : hexp \u2192 hexp \u2192 hexp\n\n  mutual\n    subst : Set\n    subst = dhexp ctx\n\n    -- expressions without ascriptions but with casts\n    data dhexp : Set where\n      c        : dhexp\n      X        : Nat \u2192 dhexp\n      \u00b7\u03bb_[_]_  : Nat \u2192 htyp \u2192 dhexp \u2192 dhexp\n      \u2987\u2988\u27e8_\u27e9    : (Nat \u00d7 subst) \u2192 dhexp\n      \u2987_\u2988\u27e8_\u27e9   : dhexp \u2192 (Nat \u00d7 subst) \u2192 dhexp\n      _\u2218_      : dhexp \u2192 dhexp \u2192 dhexp\n      _\u27e8_\u21d2_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n      _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9   : dhexp \u2192 htyp \u2192 htyp \u2192 dhexp\n\n  -- notation for chaining together agreeable casts\n  _\u27e8_\u21d2_\u21d2_\u27e9 : dhexp \u2192 htyp \u2192 htyp \u2192 htyp \u2192 dhexp\n  d \u27e8 t1 \u21d2 t2 \u21d2 t3 \u27e9 = d \u27e8 t1 \u21d2 t2 \u27e9 \u27e8 t2 \u21d2 t3 \u27e9\n\n  -- type consistency\n  data _~_ : (t1 t2 : htyp) \u2192 Set where\n    TCRefl  : {\u03c4 : htyp} \u2192 \u03c4 ~ \u03c4\n    TCHole1 : {\u03c4 : htyp} \u2192 \u03c4 ~ \u2987\u2988\n    TCHole2 : {\u03c4 : htyp} \u2192 \u2987\u2988 ~ \u03c4\n    TCArr   : {\u03c41 \u03c42 \u03c41' \u03c42' : htyp} \u2192\n               \u03c41 ~ \u03c41' \u2192\n               \u03c42 ~ \u03c42' \u2192\n               \u03c41 ==> \u03c42 ~ \u03c41' ==> \u03c42'\n\n  -- type inconsistency\n  data _~\u0338_ : (\u03c41 \u03c42 : htyp) \u2192 Set where\n    ICBaseArr1 : {\u03c41 \u03c42 : htyp} \u2192 b ~\u0338 \u03c41 ==> \u03c42\n    ICBaseArr2 : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 ~\u0338 b\n    ICArr1 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c41 ~\u0338 \u03c43 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n    ICArr2 : {\u03c41 \u03c42 \u03c43 \u03c44 : htyp} \u2192\n               \u03c42 ~\u0338 \u03c44 \u2192\n               \u03c41 ==> \u03c42 ~\u0338 \u03c43 ==> \u03c44\n\n  --- matching for arrows\n  data _\u25b8arr_ : htyp \u2192 htyp \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr \u2987\u2988 ==> \u2987\u2988\n    MAArr  : {\u03c41 \u03c42 : htyp} \u2192 \u03c41 ==> \u03c42 \u25b8arr \u03c41 ==> \u03c42\n\n  -- aliases for type and hole contexts\n  tctx : Set\n  tctx = htyp ctx\n\n  hctx : Set\n  hctx = (htyp ctx \u00d7 htyp) ctx\n\n  -- todo: this probably belongs in contexts, but need to abstract it.\n  id : tctx \u2192 subst\n  id ctx x with ctx x\n  id ctx x | Some \u03c4 = Some (X x)\n  id ctx x | None   = None\n\n  -- this is just fancy notation to match the paper\n  _::[_]_ : Nat \u2192 tctx \u2192 htyp \u2192 (Nat \u00d7 (tctx \u00d7 htyp))\n  u ::[ \u0393 ] \u03c4 = u , (\u0393 , \u03c4)\n\n  -- bidirectional type checking judgements for hexp\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      SConst  : {\u0393 : tctx} \u2192 \u0393 \u22a2 c => b\n      SAsc    : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} \u2192\n                 \u0393 \u22a2 e <= \u03c4 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) => \u03c4\n      SVar    : {\u0393 : tctx} {\u03c4 : htyp} {n : Nat} \u2192\n                 (n , \u03c4) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => \u03c4\n      SAp     : {\u0393 : tctx} {e1 e2 : hexp} {\u03c4 \u03c41 \u03c42 : htyp} \u2192\n                 \u0393 \u22a2 e1 => \u03c41 \u2192 -- need a premise that the hole names are disjoint and something to relate that to the dom(\u0394)s in expansion\n                 \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n                 \u0393 \u22a2 e2 <= \u03c42 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => \u03c4\n      SEHole  : {\u0393 : tctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988\n      SNEHole : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {u : Nat} \u2192\n                 \u0393 \u22a2 e => \u03c4 \u2192   -- premise here that u is disjoint from hole names of e\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n      SLam    : {\u0393 : tctx} {e : hexp} {\u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e => \u03c42 \u2192\n                 \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e => \u03c41 ==> \u03c42\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : htyp ctx) (e : hexp) (\u03c4 : htyp) \u2192 Set where\n      ASubsume : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} \u2192\n                 \u0393 \u22a2 e => \u03c4' \u2192\n                 \u03c4 ~ \u03c4' \u2192\n                 \u0393 \u22a2 e <= \u03c4\n      ALam : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c41 \u03c42 : htyp} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n                 (\u0393 ,, (x , \u03c41)) \u22a2 e <= \u03c42 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= \u03c4\n\n  -- those types without holes anywhere\n  data _tcomplete : htyp \u2192 Set where\n    TCBase : b tcomplete\n    TCArr : \u2200{\u03c41 \u03c42} \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (\u03c41 ==> \u03c42) tcomplete\n\n  -- those expressions without holes anywhere\n  data _ecomplete : hexp \u2192 Set where\n    ECConst : c ecomplete\n    ECAsc : \u2200{\u03c4 e} \u2192 \u03c4 tcomplete \u2192 e ecomplete \u2192 (e \u00b7: \u03c4) ecomplete\n    ECVar : \u2200{x} \u2192 (X x) ecomplete\n    ECLam1 : \u2200{x e} \u2192 e ecomplete \u2192 (\u00b7\u03bb x e) ecomplete\n    ECLam2 : \u2200{x e \u03c4} \u2192 e ecomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] e) ecomplete\n    ECAp : \u2200{e1 e2} \u2192 e1 ecomplete \u2192 e2 ecomplete \u2192 (e1 \u2218 e2) ecomplete\n\n  data _dcomplete : dhexp \u2192 Set where\n    DCVar : \u2200{x} \u2192 (X x) dcomplete\n    DCConst : c dcomplete\n    DCLam : \u2200{x \u03c4 d} \u2192 d dcomplete \u2192 \u03c4 tcomplete \u2192 (\u00b7\u03bb x [ \u03c4 ] d) dcomplete\n    DCAp : \u2200{d1 d2} \u2192 d1 dcomplete \u2192 d2 dcomplete \u2192 (d1 \u2218 d2) dcomplete\n    DCCast : \u2200{d \u03c41 \u03c42} \u2192 d dcomplete \u2192 \u03c41 tcomplete \u2192 \u03c42 tcomplete \u2192 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) dcomplete\n\n  -- contexts that only know about complete types\n  _gcomplete : tctx \u2192 Set\n  \u0393 gcomplete = (x : Nat) (\u03c4 : htyp) \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u03c4 tcomplete\n\n\n  -- expansion\n  mutual\n    data _\u22a2_\u21d2_~>_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u0394 : hctx) \u2192 Set where\n      ESConst : \u2200{\u0393} \u2192 \u0393 \u22a2 c \u21d2 b ~> c \u22a3 \u2205\n      ESVar   : \u2200{\u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192\n                         \u0393 \u22a2 X x \u21d2 \u03c4 ~> X x \u22a3 \u2205\n      ESLam   : \u2200{\u0393 x \u03c41 \u03c42 e d \u0394 } \u2192\n                     (x # \u0393) \u2192 -- todo: i added this\n                     (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d2 \u03c42 ~> d \u22a3 \u0394 \u2192\n                      \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] e \u21d2 (\u03c41 ==> \u03c42) ~> \u00b7\u03bb x [ \u03c41 ] d \u22a3 \u0394\n      ESAp : \u2200{\u0393 e1 \u03c4 \u03c41 \u03c41' \u03c42 \u03c42' d1 \u03941 e2 d2 \u03942 } \u2192\n              \u03941 ## \u03942 \u2192\n              \u0393 \u22a2 e1 => \u03c41 \u2192\n              \u03c41 \u25b8arr \u03c42 ==> \u03c4 \u2192\n              \u0393 \u22a2 e1 \u21d0 (\u03c42 ==> \u03c4) ~> d1 :: \u03c41' \u22a3 \u03941 \u2192\n              \u0393 \u22a2 e2 \u21d0 \u03c42 ~> d2 :: \u03c42' \u22a3 \u03942 \u2192\n              \u0393 \u22a2 e1 \u2218 e2 \u21d2 \u03c4 ~> (d1 \u27e8 \u03c41' \u21d2 \u03c42 ==> \u03c4 \u27e9) \u2218 (d2 \u27e8 \u03c42' \u21d2 \u03c42 \u27e9) \u22a3 (\u03941 \u222a \u03942)\n      ESEHole : \u2200{ \u0393 u } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987\u2988\u27e8 u , id \u0393 \u27e9 \u22a3  \u25a0 (u ::[ \u0393 ] \u2987\u2988)\n      ESNEHole : \u2200{ \u0393 e \u03c4 d u \u0394 } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d2 \u2987\u2988 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u2987\u2988)\n      ESAsc : \u2200 {\u0393 e \u03c4 d \u03c4' \u0394} \u2192\n                 \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 (e \u00b7: \u03c4) \u21d2 \u03c4 ~> d \u27e8 \u03c4' \u21d2 \u03c4 \u27e9 \u22a3 \u0394\n\n    data _\u22a2_\u21d0_~>_::_\u22a3_ : (\u0393 : tctx) (e : hexp) (\u03c4 : htyp) (d : dhexp) (\u03c4' : htyp) (\u0394 : hctx) \u2192 Set where\n      EALam : \u2200{\u0393 x \u03c4 \u03c41 \u03c42 e d \u03c42' \u0394 } \u2192\n              (x # \u0393) \u2192\n              \u03c4 \u25b8arr \u03c41 ==> \u03c42 \u2192\n              (\u0393 ,, (x , \u03c41)) \u22a2 e \u21d0 \u03c42 ~> d :: \u03c42' \u22a3 \u0394 \u2192\n              \u0393 \u22a2 \u00b7\u03bb x e \u21d0 \u03c4 ~> \u00b7\u03bb x [ \u03c41 ] d :: \u03c41 ==> \u03c42' \u22a3 \u0394\n      EASubsume : \u2200{e \u0393 \u03c4' d \u0394 \u03c4} \u2192\n                  ((u : Nat) \u2192 e \u2260 \u2987\u2988[ u ]) \u2192\n                  ((e' : hexp) (u : Nat) \u2192 e \u2260 \u2987 e' \u2988[ u ]) \u2192\n                  \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                  \u03c4 ~ \u03c4' \u2192\n                  \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394\n      EAEHole : \u2200{ \u0393 u \u03c4  } \u2192\n                \u0393 \u22a2 \u2987\u2988[ u ] \u21d0 \u03c4 ~> \u2987\u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 \u25a0 (u ::[ \u0393 ] \u03c4)\n      EANEHole : \u2200{ \u0393 e u \u03c4 d \u03c4' \u0394  } \u2192\n                 \u0393 \u22a2 e \u21d2 \u03c4' ~> d \u22a3 \u0394 \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] \u21d0 \u03c4 ~> \u2987 d \u2988\u27e8 u , id \u0393  \u27e9 :: \u03c4 \u22a3 (\u0394 ,, u ::[ \u0393 ] \u03c4)\n\n  -- ground\n  data _ground : (\u03c4 : htyp) \u2192 Set where\n    GBase : b ground\n    GHole : \u2987\u2988 ==> \u2987\u2988 ground\n\n  mutual\n    -- substitition type assignment\n    _,_\u22a2_:s:_ : hctx \u2192 tctx \u2192 subst \u2192 tctx \u2192 Set\n    \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' =\n        (x : Nat) (d : dhexp) (xd\u2208\u03c3 : (x , d) \u2208 \u03c3) \u2192\n            \u03a3[ \u03c4 \u2208 htyp ] ((\u0393' x == Some \u03c4 \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4)))\n\n    -- type assignment\n    data _,_\u22a2_::_ : (\u0394 : hctx) (\u0393 : tctx) (d : dhexp) (\u03c4 : htyp) \u2192 Set where\n      TAConst : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 c :: b\n      TAVar : \u2200{\u0394 \u0393 x \u03c4} \u2192 (x , \u03c4) \u2208 \u0393 \u2192 \u0394 , \u0393 \u22a2 X x :: \u03c4\n      TALam : \u2200{ \u0394 \u0393 x \u03c41 d \u03c42} \u2192\n              \u0394 , (\u0393 ,, (x , \u03c41)) \u22a2 d :: \u03c42 \u2192\n              \u0394 , \u0393 \u22a2 \u00b7\u03bb x [ \u03c41 ] d :: (\u03c41 ==> \u03c42)\n      TAAp : \u2200{ \u0394 \u0393 d1 d2 \u03c41 \u03c4} \u2192\n             \u0394 , \u0393 \u22a2 d1 :: \u03c41 ==> \u03c4 \u2192\n             \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n             \u0394 , \u0393 \u22a2 d1 \u2218 d2 :: \u03c4\n      TAEHole : \u2200{ \u0394 \u0393 \u03c3 u \u0393' \u03c4} \u2192\n                (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                \u0394 , \u0393 \u22a2 \u2987\u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TANEHole : \u2200 { \u0394 \u0393 d \u03c4' \u0393' u \u03c3 \u03c4 } \u2192\n                 (u , (\u0393' , \u03c4)) \u2208 \u0394 \u2192\n                 \u0394 , \u0393 \u22a2 d :: \u03c4' \u2192\n                 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192\n                 \u0394 , \u0393 \u22a2 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 :: \u03c4\n      TACast : \u2200{ \u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ~ \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 :: \u03c42\n      TAFailedCast : \u2200{\u0394 \u0393 d \u03c41 \u03c42} \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c41 \u2192\n             \u03c41 ground \u2192\n             \u03c42 ground \u2192\n             \u03c41 \u2260 \u03c42 \u2192\n             \u0394 , \u0393 \u22a2 d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 :: \u03c42\n\n  -- substitution;; todo: maybe get a premise that it's final; analagous to \"value substitution\"\n  [_\/_]_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  [ d \/ y ] c = c\n  [ d \/ y ] X x\n    with natEQ x y\n  [ d \/ y ] X .y | Inl refl = d\n  [ d \/ y ] X x  | Inr neq = X x\n  [ d \/ y ] (\u00b7\u03bb x [ x\u2081 ] d') = \u00b7\u03bb x [ x\u2081 ] ( [ d \/ y ] d') -- TODO: i *think* barendrecht's saves us here, or at least i want it to. may need to reformulat this as a relation --> set\n  [ d \/ y ] \u2987\u2988\u27e8 u , \u03c3 \u27e9 = \u2987\u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] \u2987 d' \u2988\u27e8 u , \u03c3  \u27e9 =  \u2987 [ d \/ y ] d' \u2988\u27e8 u , \u03c3 \u27e9\n  [ d \/ y ] (d1 \u2218 d2) = ([ d \/ y ] d1) \u2218 ([ d \/ y ] d2)\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  [ d \/ y ] (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 ) = ([ d \/ y ] d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n\n  -- value\n  data _val : (d : dhexp) \u2192 Set where\n    VConst : c val\n    VLam   : \u2200{x \u03c4 d} \u2192 (\u00b7\u03bb x [ \u03c4 ] d) val\n\n  data _boxedval : (d : dhexp) \u2192 Set where\n    BVVal : \u2200{d} \u2192 d val \u2192 d boxedval\n    BVArrCast : \u2200{ d \u03c41 \u03c42 \u03c43 \u03c44 } \u2192\n                \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                d boxedval \u2192\n                d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 boxedval\n    BVHoleCast : \u2200{ \u03c4 d } \u2192 \u03c4 ground \u2192 d boxedval \u2192 d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9 boxedval\n\n  mutual\n    -- indeterminate\n    data _indet : (d : dhexp) \u2192 Set where\n      IEHole : \u2200{u \u03c3} \u2192 \u2987\u2988\u27e8 u , \u03c3 \u27e9 indet\n      INEHole : \u2200{d u \u03c3} \u2192 d final \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 indet\n      IAp : \u2200{d1 d2} \u2192 ((\u03c41 \u03c42 \u03c43 \u03c44 : htyp) (d1' : dhexp) \u2192\n                       d1 \u2260 (d1' \u27e8(\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44)\u27e9)) \u2192\n                       d1 indet \u2192\n                       d2 final \u2192\n                       (d1 \u2218 d2) indet -- todo: should there be two ap rules?\n      ICastArr : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192\n                 \u03c41 ==> \u03c42 \u2260 \u03c43 ==> \u03c44 \u2192\n                 d indet \u2192\n                 d \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c43 ==> \u03c44) \u27e9 indet\n      ICastGroundHole : \u2200{ \u03c4 d } \u2192\n                        \u03c4 ground \u2192\n                        d indet \u2192\n                        d \u27e8 \u03c4 \u21d2  \u2987\u2988 \u27e9 indet\n      ICastHoleGround : \u2200 { d \u03c4 } \u2192\n                        ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192\n                        d indet \u2192\n                        \u03c4 ground \u2192\n                        d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9 indet\n      IFailedCast : \u2200{ d \u03c41 \u03c42 } \u2192\n                    d final \u2192\n                    \u03c41 ground \u2192\n                    \u03c42 ground \u2192\n                    \u03c41 \u2260 \u03c42 \u2192\n                    d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 indet\n\n    -- final\n    data _final : (d : dhexp) \u2192 Set where\n      FBoxed : \u2200{d} \u2192 d boxedval \u2192 d final\n      FIndet : \u2200{d} \u2192 d indet \u2192 d final\n\n\n --  -- contextual dynamics\n\n  -- evaluation contexts\n  data ectx : Set where\n    \u2299 : ectx\n    _\u2218\u2081_ : ectx \u2192 dhexp \u2192 ectx\n    _\u2218\u2082_ : dhexp \u2192 ectx \u2192 ectx\n    \u2987_\u2988\u27e8_\u27e9 : ectx \u2192 (Nat \u00d7 subst ) \u2192 ectx\n    _\u27e8_\u21d2_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n    _\u27e8_\u21d2\u2987\u2988\u21d2\u0338_\u27e9 : ectx \u2192 htyp \u2192 htyp \u2192 ectx\n\n -- todo\/ note: this judgement is redundant now; in the absence of the\n -- premises in the red brackets in the notes PDF, all syntactically well\n -- formed ectxs are valid; with finality premises, that's not true. so it\n -- might make sense to remove this judgement entirely, but need to make\n -- sure to describe why we don't have the redbox things and how we'd patch\n -- it up if we wanted to force a particular evaluation order in some\n -- document somewhere (probably a README for this repo, or a sentence in\n -- the paper text or both)\n\n --\u03b5 is an evaluation context\n  data _evalctx : (\u03b5 : ectx) \u2192 Set where\n    ECDot : \u2299 evalctx\n    ECAp1 : \u2200{d \u03b5} \u2192\n            \u03b5 evalctx \u2192\n            (\u03b5 \u2218\u2081 d) evalctx\n    ECAp2 : \u2200{d \u03b5} \u2192\n            -- d final \u2192 -- red box\n            \u03b5 evalctx \u2192\n            (d \u2218\u2082 \u03b5) evalctx\n    ECNEHole : \u2200{\u03b5 u \u03c3} \u2192\n               \u03b5 evalctx \u2192\n               \u2987 \u03b5 \u2988\u27e8 u , \u03c3 \u27e9 evalctx\n    ECCast : \u2200{ \u03b5 \u03c41 \u03c42} \u2192\n             \u03b5 evalctx \u2192\n             (\u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) evalctx\n    ECFailedCast : \u2200{ \u03b5 \u03c41 \u03c42 } \u2192\n                   \u03b5 evalctx \u2192\n                   \u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9 evalctx\n\n  -- d is the result of filling the hole in \u03b5 with d'\n  data _==_\u27e6_\u27e7 : (d : dhexp) (\u03b5 : ectx) (d' : dhexp) \u2192 Set where\n    FHOuter : \u2200{d} \u2192 d == \u2299 \u27e6 d \u27e7\n    FHAp1 : \u2200{d1 d1' d2 \u03b5} \u2192\n           d1 == \u03b5 \u27e6 d1' \u27e7 \u2192\n           (d1 \u2218 d2) == (\u03b5 \u2218\u2081 d2) \u27e6 d1' \u27e7\n    FHAp2 : \u2200{d1 d2 d2' \u03b5} \u2192\n           -- d1 final \u2192 -- red box\n           d2 == \u03b5 \u27e6 d2' \u27e7 \u2192\n           (d1 \u2218 d2) == (d1 \u2218\u2082 \u03b5) \u27e6 d2' \u27e7\n    FHNEHole : \u2200{ d d' \u03b5 u \u03c3} \u2192\n              d == \u03b5 \u27e6 d' \u27e7 \u2192\n              \u2987 d \u2988\u27e8 (u , \u03c3 ) \u27e9 ==  \u2987 \u03b5 \u2988\u27e8 (u , \u03c3 ) \u27e9 \u27e6 d' \u27e7\n    FHCast : \u2200{ d d' \u03b5 \u03c41 \u03c42 } \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 == \u03b5 \u27e8 \u03c41 \u21d2 \u03c42 \u27e9 \u27e6 d' \u27e7\n    FHFailedCast : \u2200{ d d' \u03b5 \u03c41 \u03c42} \u2192\n            d == \u03b5 \u27e6 d' \u27e7 \u2192\n            (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) == (\u03b5 \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) \u27e6 d' \u27e7\n\n  data _\u25b8gnd_ : htyp \u2192 htyp \u2192 Set where\n    MGArr : \u2200{\u03c41 \u03c42} \u2192\n            (\u03c41 ==> \u03c42) \u2260 (\u2987\u2988 ==> \u2987\u2988) \u2192\n            (\u03c41 ==> \u03c42) \u25b8gnd (\u2987\u2988 ==> \u2987\u2988)\n\n  -- instruction transition judgement\n  data _\u2192>_ : (d d' : dhexp) \u2192 Set where\n    ITLam : \u2200{ x \u03c4 d1 d2 } \u2192\n            -- d2 final \u2192 -- red box\n            ((\u00b7\u03bb x [ \u03c4 ] d1) \u2218 d2) \u2192> ([ d2 \/ x ] d1) -- todo: this is very unlikely to work long term\n    ITCastID : \u2200{d \u03c4 } \u2192\n               -- d final \u2192 -- red box\n               (d \u27e8 \u03c4 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastSucceed : \u2200{d \u03c4 } \u2192\n                    -- d final \u2192 -- red box\n                    \u03c4 ground \u2192\n                    (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> d\n    ITCastFail : \u2200{ d \u03c41 \u03c42} \u2192\n                 -- d final \u2192 -- red box\n                 \u03c41 ground \u2192\n                 \u03c42 ground \u2192\n                 \u03c41 \u2260 \u03c42 \u2192\n                 (d \u27e8 \u03c41 \u21d2 \u2987\u2988 \u21d2 \u03c42 \u27e9) \u2192> (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n    ITApCast : \u2200{d1 d2 \u03c41 \u03c42 \u03c41' \u03c42' } \u2192\n               -- d1 final \u2192 -- red box\n               -- d2 final \u2192 -- red box\n               ((d1 \u27e8 (\u03c41 ==> \u03c42) \u21d2 (\u03c41' ==> \u03c42')\u27e9) \u2218 d2) \u2192> ((d1 \u2218 (d2 \u27e8 \u03c41' \u21d2 \u03c41 \u27e9)) \u27e8 \u03c42 \u21d2 \u03c42' \u27e9)\n    ITGround : \u2200{ d \u03c4 \u03c4'} \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) \u2192> (d \u27e8 \u03c4 \u21d2 \u03c4' \u21d2 \u2987\u2988 \u27e9)\n    ITExpand : \u2200{d \u03c4 \u03c4' } \u2192\n               -- d final \u2192 -- red box\n               \u03c4 \u25b8gnd \u03c4' \u2192\n               (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) \u2192> (d \u27e8 \u2987\u2988 \u21d2 \u03c4' \u21d2 \u03c4 \u27e9)\n\n  data _\u21a6_ : (d d' : dhexp) \u2192 Set where\n    Step : \u2200{ d d0 d' d0' \u03b5} \u2192\n           d == \u03b5 \u27e6 d0 \u27e7 \u2192\n           d0 \u2192> d0' \u2192\n           d' == \u03b5 \u27e6 d0' \u27e7 \u2192\n           d \u21a6 d'\n\n  data _\u21a6*_ : (d d' : dhexp) \u2192 Set where\n    MSRefl : \u2200{d} \u2192 d \u21a6* d\n    MSStep : \u2200{d d' d''} \u2192\n                 d \u21a6 d' \u2192\n                 d' \u21a6* d'' \u2192\n                 d  \u21a6* d''\n\n  -- application of a substution to a term\n  -- todo: this is currently not well-founded! fun!\n  postulate\n    apply : subst \u2192 dhexp \u2192 dhexp\n  -- apply \u03c3 c = c\n  -- apply \u03c3 (X x) with \u03c3 x\n  -- apply \u03c3 (X x) | Some d' = d'\n  -- apply \u03c3 (X x) | None = X x\n  -- apply \u03c3 (\u00b7\u03bb x [ \u03c4 ] d) = (\u00b7\u03bb x [ \u03c4 ] (apply \u03c3 d))\n  -- apply \u03c3 \u2987\u2988\u27e8 u , \u03c3' \u27e9 =  \u2987\u2988\u27e8 u , ((\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))) \u27e9 -- (\u03bb x \u2192 (lift (apply \u03c3)) (\u03c3' x))\n  -- apply \u03c3 \u2987 d \u2988\u27e8 u , \u03c3' \u27e9 = \u2987 apply \u03c3 d \u2988\u27e8 u ,{!!} \u27e9\n  -- apply \u03c3 (d1 \u2218 d2) = ((apply \u03c3 d1) \u2218 (apply \u03c3 d2))\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2 \u03c42 \u27e9)\n  -- apply \u03c3 (d \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = ((apply \u03c3 d) \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9)\n\n  --hole instantiation; todo: judgemental or functional?\n  \u27e6_\/_\u27e7_ : dhexp \u2192 Nat \u2192 dhexp \u2192 dhexp\n  \u27e6 d \/ u \u27e7 c = c\n  \u27e6 d \/ u \u27e7 X x = X x\n  \u27e6 d \/ u \u27e7 (\u00b7\u03bb x [ \u03c4 ] d') = \u00b7\u03bb x [ \u03c4 ] (\u27e6 d \/ u \u27e7 d')\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987\u2988\u27e8 n , \u03c3 \u27e9  | Inr x = \u2987\u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n    with natEQ n u\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 .u , \u03c3 \u27e9 | Inl refl = apply \u03c3 d\n  \u27e6 d \/ u \u27e7 \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9 | Inr x = \u2987 d' \u2988\u27e8 n , \u03c3 \u27e9\n  \u27e6 d \/ u \u27e7 (d1 \u2218 d2) = (\u27e6 d \/ u \u27e7 d1) \u2218 (\u27e6 d \/ u \u27e7 d2)\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2 \u03c42 \u27e9\n  \u27e6 d \/ u \u27e7 (d' \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9) = (\u27e6 d \/ u \u27e7 d') \u27e8 \u03c41 \u21d2\u2987\u2988\u21d2\u0338 \u03c42 \u27e9\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c1c8bbd8530b7745e884bb791424c3d5dd7eb1f","subject":"Vec: dup","message":"Vec: dup\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nimport Algebra.FunctionProperties\nimport Algebra.FunctionProperties.Eq\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = \u2261.ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open Algebra.FunctionProperties\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f g : A \u2192 A \u2192 A}\n                 (f-g : \u2200 x y z \u2192 f (g x y) z \u2248 f x (g y z)) where\n    zipWith-assoc :\n      \u2200 {n} (xs ys zs : Vec A n) \u2192\n      zipWith f (zipWith g xs ys) zs \u2248\u1d5b zipWith f xs (zipWith g ys zs)\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = f-g x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-left [] = []-cong\n    zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-right [] = []-cong\n    zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n    zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n    zipWith-comm [] [] = []-cong\n    zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-left-inverse [] = []-cong\n     zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-right-inverse [] = []-cong\n     zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = idp\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = idp\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\n\nmodule Here {a} {A : \u2605 a} where\n  open Data.Vec.Equality.Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} {xs ys : Vec A m} {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 eq = fst (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {zs = ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = \u2261.ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = idp\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\nopen Algebra.FunctionProperties.Eq\n\nmodule _ {a} {A : \u2605 a} where\n    dup : \u2200 {n} \u2192 Vec A n \u2192 Vec A (n + n)\n    dup xs = xs ++ xs\n\n    dup-inj : \u2200 {n} \u2192 Injective (dup {n})\n    dup-inj = ++-inj\u2081\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --copatterns #-}\n-- NOTE with-K\nmodule Data.Vec.NP where\n\nopen import Algebra\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Type hiding (\u2605)\nimport Level as L\nopen import Category.Applicative\nopen import Data.Nat.NP using (\u2115; suc; zero; _+_; _*_; module \u2115\u00b0 ; +-interchange ; _\u2264_)\nopen import Data.Nat.Properties using (_+-mono_)\nopen import Data.Fin hiding (_\u2264_) renaming (_+_ to _+\u1da0_)\nopen import Data.Vec using (Vec; []; _\u2237_; head; tail; replicate; tabulate; foldr; _++_; lookup; splitAt; take; drop; sum; _\u2237\u02b3_)\nimport Data.Fin.Props as F\nopen import Data.Bool\nopen import Data.Product hiding (map; zip; swap) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Function.NP\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; ap; idp; _\u2219_; !_)\nimport Data.Vec.Equality\n\nmodule FunVec {a} {A : \u2605 a} where\n    _\u2192\u1d5b_ : \u2115 \u2192 \u2115 \u2192 \u2605 a\n    i \u2192\u1d5b o = Vec A i \u2192 Vec A o\n\nap-\u2237 : \u2200 {a} {A : \u2605 a} {n}\n         {x y : A} {xs ys : Vec A n} \u2192 x \u2261 y \u2192 xs \u2261 ys \u2192 x \u2237 xs \u2261 y \u2237 ys\nap-\u2237 = \u2261.ap\u2082 _\u2237_\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    applicative : \u2200 {a n} \u2192 RawApplicative (\u03bb (A : \u2605 a) \u2192 Vec A n)\n    applicative = record\n      { pure = replicate\n      ; _\u229b_  = _\u229b_\n      }\n\n    map : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : \u2605 a} {B : \u2605 b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\n    tabulate-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b}\n                 (f : A \u2192 B) (g : Fin n \u2192 A) \u2192\n                 tabulate (f \u2218 g) \u2261 map f (tabulate g)\n    tabulate-\u2218 {zero}  f g = idp\n    tabulate-\u2218 {suc n} f g = ap (_\u2237_ _) (tabulate-\u2218 f (g \u2218 suc))\n\n    tabulate-ext : \u2200 {n a}{A : \u2605 a}{f g : Fin n \u2192 A} \u2192 f \u2257 g \u2192 tabulate f \u2261 tabulate g\n    tabulate-ext {zero}  f\u2257g = idp\n    tabulate-ext {suc n} f\u2257g = ap-\u2237 (f\u2257g zero) (tabulate-ext (f\u2257g \u2218 suc))\n\n    -- map is functorial.\n\n    map-id : \u2200 {a n} {A : \u2605 a} \u2192 map id \u2257 id {A = Vec A n}\n    map-id []       = idp\n    map-id (x \u2237 xs) = ap (_\u2237_ x) (map-id xs)\n\n    map-\u2218 : \u2200 {a b c n} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c}\n            (f : B \u2192 C) (g : A \u2192 B) \u2192\n            _\u2257_ {A = Vec A n} (map (f \u2218 g)) (map f \u2218 map g)\n    map-\u2218 f g []       = idp\n    map-\u2218 f g (x \u2237 xs) = ap (_\u2237_ (f (g x))) (map-\u2218 f g xs)\n\n    map-ext : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} {f g : A \u2192 B} {n} \u2192 f \u2257 g \u2192 map f \u2257 map {n = n} g\n    map-ext f\u2257g []       = idp\n    map-ext f\u2257g (x \u2237 xs) = ap-\u2237 (f\u2257g x) (map-ext f\u2257g xs)\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nmodule WithSetoid {c \u2113} (S : Setoid c \u2113) where\n  A = Setoid.Carrier S\n  open Setoid S\n  V = Vec A\n  module V\u2248 = Data.Vec.Equality.Equality S\n  open V\u2248 hiding (_\u2248_)\n  _\u2248\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 \u2605 _\n  xs \u2248\u1d5b ys = V\u2248._\u2248_ xs ys\n\n  module _ {f : A \u2192 A} (f-cong : f Preserves _\u2248_ \u27f6 _\u2248_) where\n    map-cong : \u2200 {n} \u2192 map {n = n} f Preserves _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    map-cong []-cong = []-cong\n    map-cong (x\u2248y \u2237-cong xs\u2248ys) = f-cong x\u2248y \u2237-cong map-cong xs\u2248ys\n\n  module _ {f g : A \u2192 A \u2192 A}\n                 (f-g : \u2200 x y z \u2192 f (g x y) z \u2248 f x (g y z)) where\n    zipWith-assoc :\n      \u2200 {n} (xs ys zs : Vec A n) \u2192\n      zipWith f (zipWith g xs ys) zs \u2248\u1d5b zipWith f xs (zipWith g ys zs)\n    zipWith-assoc [] [] [] = []-cong\n    zipWith-assoc (x \u2237 xs) (y \u2237 ys) (z \u2237 zs)\n       = f-g x y z \u2237-cong (zipWith-assoc xs ys zs)\n\n  module _ {f : A \u2192 A \u2192 A} (f-cong : f Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n    zipWith-cong : \u2200 {n} \u2192 zipWith {n = n} f Preserves\u2082 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_ \u27f6 _\u2248\u1d5b_\n    zipWith-cong []-cong []-cong = []-cong\n    zipWith-cong (x\u2248y \u2237-cong xs\u2248ys) (z\u2248t \u2237-cong zs\u2248ts) = f-cong x\u2248y z\u2248t \u2237-cong zipWith-cong xs\u2248ys zs\u2248ts\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-left : LeftIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-left : \u2200 {n} \u2192 LeftIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-left [] = []-cong\n    zipWith-id-left (x \u2237 xs) = \u2219-id-left x \u2237-cong zipWith-id-left xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} {\u03b5 : A} (\u2219-id-right : RightIdentity _\u2248_ \u03b5 _\u2219_) where\n    zipWith-id-right : \u2200 {n} \u2192 RightIdentity _\u2248\u1d5b_ (replicate \u03b5) (zipWith {n = n} _\u2219_)\n    zipWith-id-right [] = []-cong\n    zipWith-id-right (x \u2237 xs) = \u2219-id-right x \u2237-cong zipWith-id-right xs\n\n  module _ {_\u2219_ : A \u2192 A \u2192 A} (\u2219-comm : Commutative _\u2248_ _\u2219_) where\n    zipWith-comm : \u2200 {n} \u2192 Commutative _\u2248\u1d5b_ (zipWith {n = n} _\u2219_)\n    zipWith-comm [] [] = []-cong\n    zipWith-comm (x \u2237 xs) (y \u2237 ys) = \u2219-comm x y \u2237-cong zipWith-comm xs ys\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : LeftInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-left-inverse : \u2200 {n} \u2192 LeftInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-left-inverse [] = []-cong\n     zipWith-left-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-left-inverse xs\n\n  module _ {\u03b5 : A} {_\u207b\u00b9 : A \u2192 A} {_\u2219_ : A \u2192 A \u2192 A}\n                 (inverse : RightInverse _\u2248_ \u03b5 _\u207b\u00b9 _\u2219_) where\n     zipWith-right-inverse : \u2200 {n} \u2192 RightInverse _\u2248\u1d5b_ (replicate \u03b5) (map _\u207b\u00b9) (zipWith {n = n} _\u2219_)\n     zipWith-right-inverse [] = []-cong\n     zipWith-right-inverse (x \u2237 xs) = inverse x \u2237-cong zipWith-right-inverse xs\n\nmodule LiftSemigroup {c \u2113} (Sg : Semigroup c \u2113) where\n    module Sg = Semigroup Sg\n    open WithSetoid Sg.setoid public\n\n    _\u2219\u1d5b_ : \u2200 {n} \u2192 V n \u2192 V n \u2192 V n\n    _\u2219\u1d5b_ = zipWith Sg._\u2219_\n\n    -- this should be in Data.Vec.Equality\n    isEquivalence : \u2200 {n} \u2192 IsEquivalence (_\u2248\u1d5b_ {n})\n    isEquivalence = record { refl = \u03bb {xs} \u2192 V\u2248.refl xs\n                                         ; sym = V\u2248.sym ; trans = V\u2248.trans }\n\n    isSemigroup : \u2200 {n} \u2192 IsSemigroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_\n    isSemigroup = record { isEquivalence = isEquivalence\n                                       ; assoc = zipWith-assoc Sg.assoc\n                                       ; \u2219-cong = zipWith-cong Sg.\u2219-cong }\n\n    semigroup : \u2200 n \u2192 Semigroup c (\u2113 L.\u2294 c)\n    semigroup n = record { isSemigroup = isSemigroup {n} }\n\nmodule LiftMonoid {c \u2113} (M : Monoid c \u2113) where\n    module M = Monoid M\n    open LiftSemigroup M.semigroup public\n\n    \u03b5\u1d5b : \u2200 {n} \u2192 V n\n    \u03b5\u1d5b = replicate M.\u03b5\n\n    isMonoid : \u2200 {n} \u2192 IsMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isMonoid = record { isSemigroup = isSemigroup\n                      ; identity = zipWith-id-left (fst M.identity)\n                                 , zipWith-id-right (snd M.identity) }\n\n    monoid : \u2115 \u2192 Monoid c (\u2113 L.\u2294 c)\n    monoid n = record { isMonoid = isMonoid {n} }\n\nmodule LiftCommutativeMonoid {c \u2113} (CM : CommutativeMonoid c \u2113) where\n    module CM = CommutativeMonoid CM\n    open LiftMonoid CM.monoid public\n\n    isCommutativeMonoid : \u2200 {n} \u2192 IsCommutativeMonoid (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b\n    isCommutativeMonoid =\n       record { isSemigroup = isSemigroup\n                  ; identity\u02e1 = zipWith-id-left  CM.identity\u02e1\n                  ; comm = zipWith-comm CM.comm }\n\n    commutative-monoid : \u2115 \u2192 CommutativeMonoid c (\u2113 L.\u2294 c)\n    commutative-monoid n = record { isCommutativeMonoid = isCommutativeMonoid {n} }\n\nmodule LiftGroup {c \u2113} (G : Group c \u2113) where\n    module G = Group G\n    open LiftMonoid G.monoid public\n\n    _\u207b\u00b9\u1d5b : \u2200 {n} \u2192 V n \u2192 V n\n    _\u207b\u00b9\u1d5b = map G._\u207b\u00b9\n\n    isGroup : \u2200 {n} \u2192 IsGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isGroup = record { isMonoid = isMonoid\n                                ; inverse = (zipWith-left-inverse (fst G.inverse))\n                                                , (zipWith-right-inverse (snd G.inverse))\n                                ; \u207b\u00b9-cong = map-cong G.\u207b\u00b9-cong }\n\n    group : \u2115 \u2192 Group c _\n    group n = record { isGroup = isGroup {n} }\n\nmodule LiftAbelianGroup {c \u2113} (AG : AbelianGroup c \u2113) where\n    module AG = AbelianGroup AG\n    open LiftGroup AG.group public\n\n    isAbelianGroup : \u2200 {n} \u2192 IsAbelianGroup (_\u2248\u1d5b_ {n}) _\u2219\u1d5b_ \u03b5\u1d5b _\u207b\u00b9\u1d5b\n    isAbelianGroup = record { isGroup = isGroup ; comm = zipWith-comm AG.comm }\n\n    abelianGroup : \u2115 \u2192 AbelianGroup c _\n    abelianGroup n = record { isAbelianGroup = isAbelianGroup {n} }\n\n-- Trying to get rid of the foldl in the definition of reverse and\n-- without using equations on natural numbers.\n-- In the end that's not very convincing.\nmodule Alternative-Reverse where\n    rev-+ : \u2115 \u2192 \u2115 \u2192 \u2115\n    rev-+ zero    = id\n    rev-+ (suc x) = rev-+ x \u2218 suc\n\n    rev-app : \u2200 {a} {A : \u2605 a} {m n} \u2192\n              Vec A n \u2192 Vec A m \u2192 Vec A (rev-+ n m)\n    rev-app []       = id\n    rev-app (x \u2237 xs) = rev-app xs \u2218 _\u2237_ x\n\n    rev-aux : \u2200 {a} {A : \u2605 a} {m} n \u2192\n              Vec A (rev-+ n zero) \u2192\n              (\u2200 {m} \u2192 A \u2192 Vec A (rev-+ n m) \u2192 Vec A (rev-+ n (suc m))) \u2192\n              Vec A m \u2192 Vec A (rev-+ n m)\n    rev-aux m acc op []       = acc\n    rev-aux m acc op (x \u2237 xs) = rev-aux (suc m) (op x acc) op xs\n\n    alt-reverse : \u2200 {a n} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    alt-reverse = rev-aux 0 [] _\u2237_\n\nvuncurry : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : \u2605 a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : \u2605 a} {B : \u2605 b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = idp\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | F.inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = idp\n\ncount-++ : \u2200 {m n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = idp\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = idp\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | F.inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = idp\n\next-count\u1da0 : \u2200 {n a} {A : \u2605 a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = idp\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = idp\n\nfilter : \u2200 {n a} {A : \u2605 a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite F.inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\ntranspose : \u2200 {m n a} {A : \u2605 a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap : \u2200 {m a b} {A : \u2605 a} {B : \u2605 b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap f = map f \u2218 transpose\n\n\u03b7 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nshallow-\u03b7 : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 xs \u2261 head xs \u2237 tail xs\nshallow-\u03b7 (x \u2237 xs) = idp\n\nuncons : \u2200 {n a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\n\u2237-uncons : \u2200 {n a} {A : \u2605 a} (xs : Vec A (1 + n)) \u2192 uncurry _\u2237_ (uncons xs) \u2261 xs\n\u2237-uncons (x \u2237 xs) = idp\n\nsplitAt\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : \u2605 a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = idp , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (ap tail eq)\n... | (q\u2081 , q\u2082) = (ap-\u2237 (ap head eq) q\u2081) , q\u2082\n\n{-\n\nmodule Here {a} {A : \u2605 a} where\n  open Data.Vec.Equality.Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = fst (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : \u2605 a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = snd (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ndrop-\u2237 : \u2200 {m a} {A : \u2605 a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , \u2261.refl = \u2261.refl\n\ntake-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | \u2261.inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | \u2261.[ eq ] | xs , ys , \u2261.refl = !(++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , \u2261.refl = \u2261.refl\n\ntake-them-all : \u2200 n {a} {A : \u2605 a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , \u2261.refl = \u2261.refl\n\ndrop\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero    _        = idp\ndrop\u2032-spec (suc m) (x \u2237 xs) = drop\u2032-spec m xs \u2219 !(drop-\u2237 m x xs)\n\ntake\u2032 : \u2200 {a} {A : \u2605 a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : \u2605 a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero    xs       = idp\ntake\u2032-spec (suc m) (x \u2237 xs) = ap (_\u2237_ x) (take\u2032-spec m xs) \u2219 !(take-\u2237 m x xs)\n\nswap : \u2200 m {n} {a} {A : \u2605 a} \u2192 Vec A (m + n) \u2192 Vec A (n + m)\nswap m xs = drop m xs ++ take m xs\n\nswap-++ : \u2200 m {n} {a} {A : \u2605 a} (xs : Vec A m) (ys : Vec A n) \u2192 swap m (xs ++ ys) \u2261 ys ++ xs\nswap-++ m xs ys = \u2261.ap\u2082 _++_ (drop-++ m xs ys) (take-++ m xs ys)\n\nrewire : \u2200 {a i o} {A : \u2605 a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 \u2605\u2080\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : \u2605 a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : \u2605 a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\n\u229b-dist-0\u21941 : \u2200 {n a} {A : \u2605 a} (fs : Vec (Endo A) n) xs \u2192 0\u21941 fs \u229b 0\u21941 xs \u2261 0\u21941 (fs \u229b xs)\n\u229b-dist-0\u21941 _           []          = idp\n\u229b-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = idp\n\u229b-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = idp\n\nmap-tail : \u2200 {m n a} {A : \u2605 a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : \u2605 a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = idp\n\nmap-tail\u2218map-tail : \u2200 {m n o a} {A : \u2605 a}\n                      (f : Vec A o \u2192 Vec A m)\n                      (g : Vec A n \u2192 Vec A o)\n                    \u2192 map-tail f \u2218 map-tail g \u2257 map-tail (f \u2218 g)\nmap-tail\u2218map-tail f g (x \u2237 xs) = idp\n\nmap-tail-\u2257 : \u2200 {m n a} {A : \u2605 a} (f g : Vec A m \u2192 Vec A n) \u2192 f \u2257 g \u2192 map-tail f \u2257 map-tail g\nmap-tail-\u2257 f g f\u2257g (x \u2237 xs) = ap (_\u2237_ x) (f\u2257g xs)\n\nsum-\u2237\u02b3 : \u2200 {n} x (xs : Vec \u2115 n) \u2192 sum (xs \u2237\u02b3 x) \u2261 sum xs + x\nsum-\u2237\u02b3 x []        = \u2115\u00b0.+-comm x 0\nsum-\u2237\u02b3 x (x\u2081 \u2237 xs) = ap (_+_ x\u2081) (sum-\u2237\u02b3 x xs) \u2219 !(\u2115\u00b0.+-assoc x\u2081 (sum xs) x)\n\nrot\u2081 : \u2200 {n a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\nrot\u2081 []       = []\nrot\u2081 (x \u2237 xs) = xs \u2237\u02b3 x\n\nrot : \u2200 {n a} {A : \u2605 a} \u2192 \u2115 \u2192 Vec A n \u2192 Vec A n\nrot zero    xs = xs\nrot (suc n) xs = rot n (rot\u2081 xs)\n\nsum-distrib\u02e1 : \u2200 {A : \u2605\u2080} {n} f k (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 k * f x) xs) \u2261 k * sum (map f xs)\nsum-distrib\u02e1 f k [] = \u2115\u00b0.*-comm 0 k\nsum-distrib\u02e1 f k (x \u2237 xs) rewrite sum-distrib\u02e1 f k xs = !(fst \u2115\u00b0.distrib k _ _)\n\nsum-linear : \u2200 {A : \u2605\u2080} {n} f g (xs : Vec A n) \u2192 sum (map (\u03bb x \u2192 f x + g x) xs) \u2261 sum (map f xs) + sum (map g xs)\nsum-linear f g [] = idp\nsum-linear f g (x \u2237 xs) rewrite sum-linear f g xs = +-interchange (f x) (g x) (sum (map f xs)) (sum (map g xs))\n\nsum-mono : \u2200 {A : \u2605\u2080} {n f g} (mono : \u2200 x \u2192 f x \u2264 g x)(xs : Vec A n) \u2192 sum (map f xs) \u2264 sum (map g xs)\nsum-mono f\u2264\u00b0g [] = Data.Nat.NP.z\u2264n\nsum-mono f\u2264\u00b0g (x \u2237 xs) = f\u2264\u00b0g x +-mono sum-mono f\u2264\u00b0g xs\n\nsum-rot\u2081 : \u2200 {n} (xs : Vec \u2115 n) \u2192 sum xs \u2261 sum (rot\u2081 xs)\nsum-rot\u2081 []       = idp\nsum-rot\u2081 (x \u2237 xs) = \u2115\u00b0.+-comm x (sum xs) \u2219 !(sum-\u2237\u02b3 x xs)\n\nmap-\u2237\u02b3 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) x (xs : Vec A n) \u2192 map f (xs \u2237\u02b3 x) \u2261 map f xs \u2237\u02b3 f x\nmap-\u2237\u02b3 f x []       = idp\nmap-\u2237\u02b3 f x (_ \u2237 xs) = ap (_\u2237_ _) (map-\u2237\u02b3 f x xs)\n\nsum-map-rot\u2081 : \u2200 {n a} {A : \u2605 a} (f : A \u2192 \u2115) (xs : Vec A n) \u2192 sum (map f (rot\u2081 xs)) \u2261 sum (map f xs)\nsum-map-rot\u2081 f [] = idp\nsum-map-rot\u2081 f (x \u2237 xs) = ap sum (map-\u2237\u02b3 f x xs)\n                        \u2219 sum-\u2237\u02b3 (f x) (map f xs)\n                        \u2219 \u2115\u00b0.+-comm (sum (map f xs)) (f x)\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map; applicative)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"02723433a15894a6b11154b1e82769b86bde0182","subject":"Updated a thesis' note.","message":"Updated a thesis' note.\n\nIgnore-this: e71c7f0b9d8d7149cdd2cc19706a51c0\n\ndarcs-hash:20120517141804-3bd4e-a1a13d5f9d673b164d85791c8e17983027798e96.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/AdditionComm.agda","new_file":"notes\/thesis\/logical-framework\/AdditionComm.agda","new_contents":"------------------------------------------------------------------------------\n-- A Peano arithmetic proof without using a where clause\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 17 May 2012.\n\nmodule AdditionComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import PA.Axiomatic.Standard.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  x+Sy\u2261S[x+y]     : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\n  succ-cong       : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n\nA : M \u2192 Set\nA m = \u2200 n \u2192 m + n \u2261 n + m\n\nA0 : A zero\nA0 n = zero + n   \u2261\u27e8 PA\u2083 n \u27e9\n       n          \u2261\u27e8 sym (+-rightIdentity n) \u27e9\n       n + zero   \u220e\n\nis : \u2200 m \u2192 A m \u2192 A (succ m)\nis m ih n = succ m + n   \u2261\u27e8 PA\u2084 m n \u27e9\n            succ (m + n) \u2261\u27e8 succ-cong (ih n) \u27e9\n            succ (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n            n + succ m   \u220e\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm = PA-ind A A0 is\n","old_contents":"------------------------------------------------------------------------------\n-- A Peano arithmetic proof without using a where clause\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT on 02 March 2012.\n\nmodule AdditionComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import PA.Axiomatic.Standard.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  x+Sy\u2261S[x+y]     : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\n  succ-cong       : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n\nA : M \u2192 Set\nA m = \u2200 n \u2192 m + n \u2261 n + m\n\nA0 : A zero\nA0 n = zero + n   \u2261\u27e8 A\u2083 n \u27e9\n       n          \u2261\u27e8 sym (+-rightIdentity n) \u27e9\n       n + zero \u220e\n\nis : \u2200 m \u2192 A m \u2192 A (succ m)\nis m ih n = succ m + n   \u2261\u27e8 A\u2084 m n \u27e9\n            succ (m + n) \u2261\u27e8 succ-cong (ih n) \u27e9\n            succ (n + m) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n m) \u27e9\n            n + succ m \u220e\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm = PA-ind A A0 is\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6ff590167cc799d215983f4e4413c3070c81ad03","subject":"otp-sem-sec.agda","message":"otp-sem-sec.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec\nopen import Data.Product\nopen import circuit\nopen import Relation.Binary.PropositionalEquality\n\n-- dup from Game\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nCoins = \u2115\nPorts = \u2115\n\nmodule F\n  (Size Time : Set)\n  (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  (Cp : Ports \u2192 Ports \u2192 Set)\n  (builder : CircuitBuilder Cp)\n  (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n  (runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o)\n\n  (|M| |C| : \u2115)\n\n  (Proba : Set)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (dist : Proba \u2192 Proba \u2192 Proba)\n  (negligible : Proba \u2192 Set)\n\n  where\n\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n      size  : Size\n      time  : Time\n  open Power public\n\n  M = Bits |M|\n  C = Bits |C|\n\n  \u21ba : Coins \u2192 Set \u2192 Set\n  \u21ba c A = Bits c \u2192 A\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n\n  runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n  runSemSec E {mk c s t} A b R\n      = case SemSecAdv.beh A R\u2080 of \u03bb\n          { (m0m1 , kont) \u2192 {- b == -} kont (EXP b E m0m1) R\u2081 }\n    where R\u2080 = take (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n          R\u2081 = drop (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n\n  -- runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n\n  negligible-advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  negligible-advantage EXP = negligible (dist Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611])\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible-advantage (runSemSec E A)\n\n  |K| = |M|\n  K = Bits |K|\n  -- OTP : K \u2192 M \u2192 C\n  -- OTP = _\u2295_\n\n  -- OTP-SemSec : SemSec (OTP \n\n  Enc = M \u2192 C\n\n  Tr = Enc \u2192 Enc\n\n  -- In general the power might change\n  SemSecTr : Tr \u2192 Set\n  SemSecTr tr = \u2200 {E p} \u2192 SemSec (tr E) p \u2192 SemSec E p\n\n  neg-pres-sem-sec : SemSecTr (_\u2218_ vnot)\n  neg-pres-sem-sec {E} {p} E'-sec A = A-breaks-E\n     where E' : Enc\n           E' = vnot \u2218 E\n           open SemSecAdv A using (c\u2261c\u2080+c\u2081 ; c\u2080 ; c\u2081) renaming (beh to A-beh)\n           A'-beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n           A'-beh = {!!}\n           A' : SemSecAdv p\n           A' = {!!}\n           A'-breaks-E' : negligible-advantage (runSemSec E' A')\n           A'-breaks-E' = E'-sec A'\n           A-breaks-E : negligible-advantage (runSemSec E A)\n           A-breaks-E = ?\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecTr (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec mask {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = (_\u2295_ mask) \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n-}\n","old_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec\nopen import Data.Product\nopen import circuit\nopen import Relation.Binary.PropositionalEquality\n\n-- dup from Game\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nCoins = \u2115\nPorts = \u2115\n\nmodule F\n  (Size Time : Set)\n  (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  (Cp : Ports \u2192 Ports \u2192 Set)\n  (builder : CircuitBuilder Cp)\n  (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n  (runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o)\n\n  (|M| |C| : Ports)\n\n  (Proba : Set)\n  -- (Pr[_\u22611] : \u2200 {c} (EXP : \u21ba c Bit) \u2192 Proba)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (negligible-advantage-proba : Proba \u2192 Proba \u2192 Set)\n\n  where\n\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n      size  : Size\n      time  : Time\n  open Power public\n\n  M = Bits |M|\n  C = Bits |C|\n\n  \u21ba : Coins \u2192 Set \u2192 Set\n  \u21ba c A = Bits c \u2192 A\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n\n  runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n  runSemSec E {mk c s t} A b R\n      = case SemSecAdv.beh A R\u2080 of \u03bb\n          { (m0m1 , kont) \u2192 {- b == -} kont (EXP b E m0m1) R\u2081 }\n    where R\u2080 = take (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n          R\u2081 = drop (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n\n  negligible-advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  negligible-advantage EXP = negligible-advantage-proba Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible-advantage (runSemSec E A)\n\n  |K| = |M|\n  K = Bits |K|\n  -- OTP : K \u2192 M \u2192 C\n  -- OTP = _\u2295_\n\n  -- OTP-SemSec : SemSec (OTP \n\n  Enc = M \u2192 C\n\n  Tr = Enc \u2192 Enc\n\n  -- In general the power might change\n  SemSecTr : Tr \u2192 Set\n  SemSecTr tr = \u2200 {E p} \u2192 SemSec E p \u2192 SemSec (tr E) p\n\n  neg-pres-sem-sec : SemSecTr (_\u2218_ vnot)\n  neg-pres-sem-sec {E} {p} E-sec A' = A'-breaks-E'\n     where E' : Enc\n           E' = vnot \u2218 E\n           open SemSecAdv A' using (c\u2261c\u2080+c\u2081 ; c\u2080 ; c\u2081)\n           A'-beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n           A'-beh = {!Sec!}\n           A-beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n           A-beh = {!!}\n           A : SemSecAdv p\n           A = {!!}\n           A-breaks-E : negligible-advantage (runSemSec E A)\n           A-breaks-E = E-sec A\n           A'-breaks-E' : negligible-advantage (runSemSec E' A')\n           A'-breaks-E' = {!!}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecTr (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec mask {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = (_\u2295_ mask) \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n-}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a8eeec60c56417a12567e8bff6998fd46d5e8836","subject":"bintree: sorting a tree now can produce a syntactic bijection doing the same thing","message":"bintree: sorting a tree now can produce a syntactic bijection doing the same thing\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    from-\u00d7 : A \u00d7 A \u2192 Tree A 1\n    from-\u00d7 (x , y) = fork (leaf x) (leaf y)\n\n    to-\u00d7 : Tree A 1 \u2192 A \u00d7 A\n    to-\u00d7 (fork (leaf x) (leaf y)) = x , y\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap t = fork (rght t) (lft t)\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\n\n    inner : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (1 + n)\n    inner t = fork (rght (lft t)) (lft (rght t))\nopen Dummy public\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    sort-\u00d7 : Endo (A \u00d7 A)\n    sort-\u00d7 (x\u2080 , x\u2081) = (x\u2080 \u2293\u1d2c x\u2081 , x\u2080 \u2294\u1d2c x\u2081)\n\n    sort\u2081 : Endo (Tree A 1)\n    sort\u2081 = from-\u00d7 \u2218 sort-\u00d7 \u2218 to-\u00d7\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero}  = sort\u2081\n    merge {suc _} = map-inner merge \u2218 map-outer merge merge \u2218 interchange\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule Sorting-Perm-Properties {OT : Set} (_<=\u1d2c_ : OT \u2192 OT \u2192 Bool)\n    (isTotalOrder : IsTotalOrder _\u2261_ (\u03bb x y \u2192 T (x <=\u1d2c y)))\n    where\n    open IsTotalOrder isTotalOrder\n\n    _\u2293\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2293\u1d2c y = if x <=\u1d2c y then x else y\n\n    _\u2294\u1d2c_ : OT \u2192 OT \u2192 OT\n    x \u2294\u1d2c y = if x <=\u1d2c y then y else x\n\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    open OperationSyntax renaming (map-inner to `map-inner; map-outer to `map-outer)\n    open import Function.Bijection.SyntaxKit\n\n    evalTree : \u2200 {a} {A : Set a} \u2192 Bij \u2192 \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    evalTree `id         = id\n    evalTree (op\u2080 `\u204f op\u2081) = evalTree op\u2081 \u2218 evalTree op\u2080\n    evalTree op {zero} = id\n    evalTree (`id   `\u2237 g) {suc n} = map-outer (evalTree (g 0b)) (evalTree (g 1b))\n    evalTree (`not\u1d2e `\u2237 g) {suc n} = map-outer (evalTree (g 0b)) (evalTree (g 1b)) \u2218 swap\n    evalTree `0\u21941 {suc zero} = id\n    evalTree `0\u21941 {suc (suc n)} = interchange\n\n\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Bij\n        proof : (t : Tree A n) \u2192 t \u2261 evalTree (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {f = f}{g} (mk `f pf) (mk `g pg)\n      = mk (\u03bb t \u2192 `f (g t) `\u204f `g t)\n           (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (evalTree (`g t)) (pf (g t))))\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = mk (\u03bb _ \u2192 `not) \u03b7fork\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof `f `g = mk (\u03bb t \u2192 `map-outer (perm `f (lft t)) (perm `g (rght t)))\n                               (\u03bb { (fork t u) \u2192 \u2261.cong\u2082 fork (proof `f t) (proof `g u) })\n       where open Perm\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof {A} {f = f} `f = mk map-inner-perm helper\n       where open Perm\n             map-inner-perm = `map-inner \u2218 perm `f \u2218 inner\n             helper : \u2200 t \u2192 t \u2261 evalTree (map-inner-perm t) (map-inner f t)\n             helper (fork (fork a b) (fork c d)) with f (fork b c) | \u2261.sym (proof `f (fork b c))\n             ... | fork B C | p rewrite p = \u2261.refl\n\n    `sort\u2081 : Tree OT 1 \u2192 Bij\n    `sort\u2081 = `xor \u2218 uncurry _<=\u1d2c_ \u2218 swap-\u00d7 \u2218 to-\u00d7\n\n    -- \u2200 x  T (y <=\u1d2c x) \u2192 fork (leaf x) (leaf y) \u2261 fork (leaf (x \u2293 y)) (leaf (x \u2294 y))\n    sort\u2081-proof : Perm {OT} 1 sort\u2081\n    sort\u2081-proof = mk `sort\u2081 helper\n      where helper : \u2200 t \u2192 t \u2261 evalTree (`sort\u2081 t) (sort\u2081 t)\n            helper (fork (leaf x) (leaf y)) with y <=\u1d2c x | x <=\u1d2c y | antisym {x} {y} | total x y\n            ... | true  | true  | p | _ rewrite p _ _ = \u2261.refl\n            ... | false | true  | _ | _ = \u2261.refl\n            ... | true  | false | _ | _ = \u2261.refl\n            ... | false | false | _ | inj\u2081 ()\n            ... | false | false | _ | inj\u2082 ()\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = sort\u2081-proof\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\nmodule BitsSorting {m} where\n    open ToNat {A = Bits m} to\u2115 (\u03bb {x} {y} \u2192 to\u2115-inj x y)\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder (\u03bb {x} {y} z \u2192 \u2294-spec {x} {y} z)\n                   (\u03bb {x} {y} \u2192 \u2293-spec {x} {y}) \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder\n                   \u2264-\u2294 \u2293-\u2264\n                   (\u03bb {x} {y} {z} \u2192 \u2264-<_,_> {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-[_,_] {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2293\u2081 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2080 {x} {y} {z})\n                   (\u03bb {x} {y} {z} \u2192 \u2264-\u2294\u2081 {x} {y} {z})\n    open SortedData _\u2264_\n    module SPP = Sorting-Perm-Properties _<=_ isTotalOrder\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open OperationSyntax\n    mkBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij\n    mkBij f = SPP.Perm.perm SPP.sort-proof (fromFun f)\n\n    {-\n    thm : \u2200 {n} (f : Endo (Bits n)) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) \u2192\n            eval (mkBij f) \u2257 f\n    thm f f-inj = \u03bb xs \u2192 {!\u2261.cong toFun (SPP.Perm.proof SPP.sort-proof (fromFun f))!}\n    -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree where\n\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\ntoFun\u2218fromFun : \u2200 {n a} {A : Set a} (f : Bits n \u2192 A) \u2192 toFun (fromFun f) \u2257 f\ntoFun\u2218fromFun {zero}  f [] = \u2261.refl\ntoFun\u2218fromFun {suc n} f (false \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 0\u2237_) xs = \u2261.refl\ntoFun\u2218fromFun {suc n} f (true \u2237 xs)\n  rewrite toFun\u2218fromFun (f \u2218 1\u2237_) xs = \u2261.refl\n\nfromFun\u2218toFun : \u2200 {n a} {A : Set a} (t : Tree A n) \u2192 fromFun (toFun t) \u2261 t\nfromFun\u2218toFun (leaf x) = \u2261.refl\nfromFun\u2218toFun (fork t\u2080 t\u2081)\n  rewrite fromFun\u2218toFun t\u2080\n        | fromFun\u2218toFun t\u2081 = \u2261.refl\n\ntoFun\u2192fromFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 toFun t \u2257 f \u2192 t \u2261 fromFun f\ntoFun\u2192fromFun (leaf x) f t\u2257f = \u2261.cong leaf (t\u2257f [])\ntoFun\u2192fromFun (fork t\u2080 t\u2081) f t\u2257f\n  rewrite toFun\u2192fromFun t\u2080 (f \u2218 0\u2237_) (t\u2257f \u2218 0\u2237_)\n        | toFun\u2192fromFun t\u2081 (f \u2218 1\u2237_) (t\u2257f \u2218 1\u2237_) = \u2261.refl\n\nfromFun\u2192toFun : \u2200 {n a} {A : Set a} (t : Tree A n) (f : Bits n \u2192 A) \u2192 t \u2261 fromFun f \u2192 toFun t \u2257 f\nfromFun\u2192toFun ._ _ \u2261.refl = toFun\u2218fromFun _\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\n{-\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = \u2261.refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = \u2261.refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = \u2261.refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = \u2261.refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = \u2261.refl\nswpOp-sym-involutive swp = \u2261.refl\nswpOp-sym-involutive swp-seconds = \u2261.refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = \u2261.refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = \u2261.refl\nswpOp-sym-sound swp (fork _ _) = \u2261.refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = \u2261.refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n-}\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = \u2261.refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  {-\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = \u2261.cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = \u2261.cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = \u2261.cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = \u2261.cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ \u2261.cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = \u2261.cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 \u2261.cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 \u2261.sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 \u2261.cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n-}\n\nmodule FoldProp {a \u2113} {A : Set a} (_\u24cd_ : Set \u2113 \u2192 Set \u2113 \u2192 Set \u2113) where\n    Fold : \u2200 {n} \u2192 (Bits n \u2192 A \u2192 Set \u2113) \u2192 Tree A n \u2192 Set \u2113\n    Fold f (leaf x)     = f [] x\n    Fold f (fork t\u2080 t\u2081) = Fold (f \u2218 0\u2237_) t\u2080 \u24cd Fold (f \u2218 1\u2237_) t\u2081\n\nAll : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAll = FoldProp.Fold _\u00d7_\n\nAny : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A \u2192 Set) \u2192 Tree A n \u2192 Set\nAny = FoldProp.Fold _\u228e_\n\nopen Alternative-Reverse\n\nmodule AllBits where\n  _IsRevPrefixOf_ : \u2200 {m n} \u2192 Bits m \u2192 Bits (rev-+ m n) \u2192 Set\n  _IsRevPrefixOf_ {m} {n} p xs = \u2203 \u03bb (ys : Bits n) \u2192 rev-app p ys \u2261 xs\n\n  RevPrefix : \u2200 {m n o} (p : Bits m) \u2192 Tree (Bits (rev-+ m n)) o \u2192 Set\n  RevPrefix p = All (\u03bb _ \u2192 _IsRevPrefixOf_ p)\n\n  RevPrefix-[]-\u22a4 : \u2200 {m n} (t : Tree (Bits m) n) \u2192 RevPrefix [] t\n  RevPrefix-[]-\u22a4 (leaf x) = x , \u2261.refl\n  RevPrefix-[]-\u22a4 (fork t u) = RevPrefix-[]-\u22a4 t , RevPrefix-[]-\u22a4 u\n\n  All-fromFun : \u2200 {m} n (p : Bits m) \u2192 All (_\u2261_ \u2218 rev-app p) (fromFun {n} (rev-app p))\n  All-fromFun zero    p = \u2261.refl\n  All-fromFun (suc n) p = All-fromFun n (0\u2237 p) , All-fromFun n (1\u2237 p)\n\n  All-id : \u2200 n \u2192 All {n} _\u2261_ (fromFun id)\n  All-id n = All-fromFun n []\n\nopen new-approach\n\n\u2208-fromFun : \u2200 {m n x} (f : Bits m \u2192 Bits n) (p : x \u2208 fromFun f) \u2192 f (toBits p) \u2261 x\n\u2208-fromFun f here      = \u2261.refl\n\u2208-fromFun f (left p)  = \u2208-fromFun (f \u2218 0\u2237_) p\n\u2208-fromFun f (right p) = \u2208-fromFun (f \u2218 1\u2237_) p\n\n\u2208-rev-app : \u2200 {m} n {x : Bits (rev-+ m n)} (q : Bits m) (p : x \u2208 fromFun (rev-app q)) \u2192 rev-app q (toBits p) \u2261 x\n\u2208-rev-app _ = \u2208-fromFun \u2218 rev-app\n\nfirst : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nfirst (leaf x)   = x\nfirst (fork t _) = first t\n\nlast : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 A\nlast (leaf x)   = x\nlast (fork _ t) = last t\n\nmodule SortedDataIx {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x) x x\n      fork : \u2200 {n} {t u : Tree A n} {low_t high_t low\u1d64 high\u1d64} \u2192\n             Sorted t low_t high_t \u2192\n             Sorted u low\u1d64 high\u1d64 \u2192\n             (h\u2264l : high_t \u2264\u1d2c low\u1d64) \u2192\n             Sorted (fork t u) low_t high\u1d64\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    \u2264\u1d2c-bounds : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 l \u2264\u1d2c h\n    \u2264\u1d2c-bounds leaf            = \u2264\u1d2c.refl\n    \u2264\u1d2c-bounds (fork s\u2080 s\u2081 pf) = \u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) (\u2264\u1d2c.trans pf (\u2264\u1d2c-bounds s\u2081))\n\n    Sorted\u2192lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 l \u2264\u1d2c x\n    Sorted\u2192lb leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192lb (fork s _ _)    (left  p) = Sorted\u2192lb s p\n    Sorted\u2192lb (fork s\u2080 s\u2081 pf) (right p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (\u2264\u1d2c-bounds s\u2080) pf) (Sorted\u2192lb s\u2081 p)\n\n    Sorted\u2192ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 \u2200 {x} \u2192 x \u2208 t \u2192 x \u2264\u1d2c h\n    Sorted\u2192ub leaf            here      = \u2264\u1d2c.refl\n    Sorted\u2192ub (fork _ s _)    (right p) = Sorted\u2192ub s p\n    Sorted\u2192ub (fork s\u2080 s\u2081 pf) (left  p) = \u2264\u1d2c.trans (\u2264\u1d2c.trans (Sorted\u2192ub s\u2080 p) pf) (\u2264\u1d2c-bounds s\u2081)\n\n    Bounded : \u2200 {n} \u2192 Tree A n \u2192 A \u2192 A \u2192 Set (a L.\u2294 \u2113)\n    Bounded t l h = \u2200 {x} \u2192 x \u2208 t \u2192 (l \u2264\u1d2c x) \u00d7 (x \u2264\u1d2c h)\n\n    Sorted\u2192Bounded : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 Bounded t l h\n    Sorted\u2192Bounded s x = Sorted\u2192lb s x , Sorted\u2192ub s x\n\n    first-lb : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 first t \u2261 l\n    first-lb leaf          = \u2261.refl\n    first-lb (fork st _ _) = first-lb st\n\n    last-ub : \u2200 {n} {t : Tree A n} {l h} \u2192 Sorted t l h \u2192 last t \u2261 h\n    last-ub leaf          = \u2261.refl\n    last-ub (fork _ st _) = last-ub st\n\n    uniq-lb : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 l\u2080 \u2261 l\u2081\n    uniq-lb leaf leaf = \u2261.refl\n    uniq-lb (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-lb p q\n\n    uniq-ub : \u2200 {n} {t : Tree A n} {l\u2080 h\u2080 l\u2081 h\u2081}\n                  \u2192 Sorted t l\u2080 h\u2080 \u2192 Sorted t l\u2081 h\u2081 \u2192 h\u2080 \u2261 h\u2081\n    uniq-ub leaf leaf = \u2261.refl\n    uniq-ub (fork p p\u2081 h\u2264l) (fork q q\u2081 h\u2264l\u2081) = uniq-ub p\u2081 q\u2081\n\n    Sorted-trans : \u2200 {n} {t u v : Tree A n} {lt hu lu hv}\n                   \u2192 Sorted (fork t u) lt hu \u2192 Sorted (fork u v) lu hv \u2192 Sorted (fork t v) lt hv\n    Sorted-trans {lt = lt} {hu} {lu} {hv} (fork tu tu\u2081 h\u2264l) (fork uv uv\u2081 h\u2264l\u2081)\n       rewrite uniq-lb uv tu\u2081\n             | uniq-ub uv tu\u2081\n         = fork tu uv\u2081 (\u2264\u1d2c.trans h\u2264l (\u2264\u1d2c.trans (\u2264\u1d2c-bounds tu\u2081) h\u2264l\u2081))\n\nmodule Sorted' {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_) where\n    data _\u2264\u1d2e_ : \u2200 {n} (p q : Bits n) \u2192 Set where\n      []    : [] \u2264\u1d2e []\n      there : \u2200 {n} {p q : Bits n} b \u2192 p \u2264\u1d2e q \u2192 (b \u2237 p) \u2264\u1d2e (b \u2237 q)\n      0-1   : \u2200 {n} (p q : Bits n) \u2192 0\u2237 p \u2264\u1d2e 1\u2237 q\n\n    _\u2264\u1d3e_ : \u2200 {n x y} {t : Tree A n} \u2192 x \u2208 t \u2192 y \u2208 t \u2192 Set\n    p \u2264\u1d3e q = toBits p \u2264\u1d2e toBits q\n\n    Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set _\n    Sorted t = \u2200 {x} (p : x \u2208 t) {y} (q : y \u2208 t) \u2192 p \u2264\u1d3e q \u2192 x \u2264\u1d2c y\n\n    private\n        module \u2264\u1d2c = IsPreorder isPreorder\n\n    module S = SortedDataIx _\u2264\u1d2c_ isPreorder\n    open S using (leaf; fork)\n    Sorted\u2192Sorted' : \u2200 {n l h} {t : Tree A n} \u2192 S.Sorted t l h \u2192 Sorted t\n    Sorted\u2192Sorted' leaf             here     here       p\u2264q = \u2264\u1d2c.refl\n    Sorted\u2192Sorted' (fork s _ _)     (left p) (left q)   (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n    Sorted\u2192Sorted' (fork s\u2080 s\u2081 l\u2264h) (left p) (right q)  p\u2264q = \u2264\u1d2c.trans (S.Sorted\u2192ub s\u2080 p) (\u2264\u1d2c.trans l\u2264h (S.Sorted\u2192lb s\u2081 q))\n    Sorted\u2192Sorted' (fork _ _ _)     (right _) (left _)  ()\n    Sorted\u2192Sorted' (fork _ s _)     (right p) (right q) (there ._ p\u2264q) = Sorted\u2192Sorted' s p q p\u2264q\n\nprivate\n  module Dummy {a} {A : Set a} where\n    lft : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    lft (fork t _) = t\n\n    rght : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A n\n    rght (fork _ t) = t\n\n    \u03b7fork : \u2200 {n} (t : Tree A (1 + n)) \u2192 t \u2261 fork (lft t) (rght t)\n    \u03b7fork (fork _ _) = \u2261.refl\n\n    swap : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Tree A (1 + n)\n    swap (fork t u) = fork u t\n\n    map-inner : \u2200 {n} \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n)) \u2192 (Tree A (2 + n) \u2192 Tree A (2 + n))\n    map-inner f (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) =\n      case f (fork t\u2081 t\u2082) of \u03bb { (fork t\u2084 t\u2085) \u2192 fork (fork t\u2080 t\u2084) (fork t\u2085 t\u2083) }\n\n    map-outer : \u2200 {n} \u2192 (f g : Tree A n \u2192 Tree A n) \u2192 (Tree A (1 + n) \u2192 Tree A (1 + n))\n    map-outer f g (fork t u) = fork (f t) (g u)\n\n    interchange : \u2200 {n} \u2192 Tree A (2 + n) \u2192 Tree A (2 + n)\n    interchange = map-inner swap\nopen Dummy public\n\nmodule Sorting {a} {A : Set a} (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A) where\n\n    merge : \u2200 {n} \u2192 Endo (Tree A (1 + n))\n    merge {zero} (fork (leaf x\u2080) (leaf x\u2081)) =\n      fork (leaf (x\u2080 \u2293\u1d2c x\u2081)) (leaf (x\u2080 \u2294\u1d2c x\u2081))\n    merge {suc _} t\n      = (map-inner merge \u2218 map-outer merge merge \u2218 interchange) t\n\n    sort : \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    sort {zero}  = id\n    sort {suc n} = merge \u2218 map-outer sort sort\n\nmodule Sorting-Perm-Properties {OT : Set}\n  (_\u2293\u1d2c_ _\u2294\u1d2c_ : (xs ys : OT) \u2192 OT) where\n    \n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    import Data.Bits as B\n    open B.OperationSyntax renaming (_\u2219_ to _\u2022_) -- \n\n\n    infixr 1 _`\u2218_\n    _`\u2218_ = \u03bb x y \u2192 y `\u204f x\n\n    eval : {A : Set} \u2192 Op \u2192 \u2200 {n} \u2192 Tree A n \u2192 Tree A n\n    eval `id t        = t\n    eval `0\u21941 t       = {!!}\n    eval `not t       = {!!}\n    eval (`tl op) t   = {!!}\n    eval (`if0 op) t  = {!!}\n    eval (op `\u204f op\u2081) t = eval op\u2081 (eval op t)\n\n    record Perm {A : Set} n (f : Endo (Tree A n)) : Set where\n      constructor mk\n      field\n        perm  :  Tree A n \u2192 Op\n        proof : (t : Tree A n) \u2192 t \u2261  eval (perm t) (f t)\n\n    id-proof : \u2200 {A : Set}{n} \u2192 Perm {A} n id\n    id-proof = mk (\u03bb _ \u2192 `id) (\u03bb t \u2192 \u2261.refl)\n\n    _\u2218-proof_ : \u2200{A : Set}{n }{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm n (f \u2218 g)\n    _\u2218-proof_ {A}{n}{f}{g} (mk \u03c0f pf) (mk \u03c0g pg) = mk (\u03bb t \u2192 \u03c0g t `\u2218 \u03c0f (g t)) \n      (\u03bb t \u2192 \u2261.trans (pg t) (\u2261.cong (eval (\u03c0g t)) (pf (g t))) )\n\n    swap-proof : \u2200 {A : Set}{n} \u2192 Perm {A} (suc n) swap\n    swap-proof = {!!}\n\n    map-outer-proof : \u2200 {A : Set}{n}{f g : Endo (Tree A n)} \u2192 Perm n f \u2192 Perm n g \u2192 Perm {A} (suc n) (map-outer f g)\n    map-outer-proof = {!!}\n\n    map-inner-proof : \u2200 {A : Set}{n}{f : Endo (Tree A (1 + n))} \u2192 Perm (1 + n) f \u2192 Perm (2 + n) (map-inner f)\n    map-inner-proof = {!!}\n\n    merge-proof : \u2200 {n} \u2192 Perm {OT} (suc n) merge\n    merge-proof {zero}  = mk {!!} (\u03bb { (fork (leaf x) (leaf y))  \u2192 {!!} })\n    merge-proof {suc n} = map-inner-proof merge-proof \u2218-proof\n                            (map-outer-proof merge-proof merge-proof \u2218-proof\n                             map-inner-proof swap-proof)\n\n    sort-proof : \u2200 {n} \u2192 Perm {OT} n sort\n    sort-proof {zero}  = id-proof\n    sort-proof {suc n} = merge-proof \u2218-proof map-outer-proof sort-proof sort-proof\n\nmodule SortedData {a \u2113} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113) where\n    data Sorted : \u2200 {n} \u2192 Tree A n \u2192 Set (a L.\u2294 \u2113) where\n      leaf : {x : A} \u2192 Sorted (leaf x)\n      fork : \u2200 {n} {t u : Tree A n} \u2192\n             Sorted t \u2192\n             Sorted u \u2192\n             (h\u2264l : last t \u2264\u1d2c first u) \u2192\n             Sorted (fork t u)\n\n    PreSorted : \u2200 {n} \u2192 Tree A (1 + n) \u2192 Set _\n    PreSorted t = Sorted (lft t) \u00d7 Sorted (rght t)\n\nmodule MergeSwap {a} {A : Set a}\n                 (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                 (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                 (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_) where\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    merge-swap : \u2200 {n} (t : Tree A (1 + n)) \u2192 merge t \u2261 merge (swap t)\n    merge-swap (fork (leaf x) (leaf y)) rewrite \u2294-comm x y | \u2293-comm y x = \u2261.refl\n    merge-swap (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      rewrite merge-swap (fork t\u2080 u\u2080)\n            | merge-swap (fork t\u2081 u\u2081) = \u2261.refl\n\nmodule SortingDataIxProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2294-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2294\u1d2c y \u2261 y)\n                               (\u2293-spec : \u2200 {x y} \u2192 x \u2264\u1d2c y \u2192 x \u2293\u1d2c y \u2261 x)\n                               (\u2293-comm : Commutative _\u2261_ _\u2293\u1d2c_)\n                               (\u2294-comm : Commutative _\u2261_ _\u2294\u1d2c_)\n                               where\n    open MergeSwap _\u2293\u1d2c_ _\u2294\u1d2c_ \u2293-comm \u2294-comm\n    module \u2264\u1d2c = IsPreorder isPreorder\n    open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n\n    merge-pres : \u2200 {n} {t : Tree A (1 + n)} {l h} \u2192 Sorted t l h \u2192 merge t \u2261 t\n    merge-pres (fork leaf leaf x) = \u2261.cong\u2082 (fork on leaf) (\u2293-spec x) (\u2294-spec x)\n    merge-pres {t = fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081)}\n               (fork (fork {low_t = lt\u2080} {ht\u2080} {lt\u2081} {ht\u2081} st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n                     (fork {low_t = lu\u2080} {hu\u2080} {lu\u2081} {hu\u2081} su\u2080 su\u2081 hu\u2080\u2264lu\u2081) ht\u2081\u2264lu\u2080)\n       rewrite merge-pres (fork st\u2080 su\u2080 (\u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds st\u2081) ht\u2081\u2264lu\u2080)))\n             | merge-pres (fork st\u2081 su\u2081 (\u2264\u1d2c.trans ht\u2081\u2264lu\u2080 (\u2264\u1d2c.trans (\u2264\u1d2c-bounds su\u2080) hu\u2080\u2264lu\u2081)))\n             | merge-swap (fork u\u2080 t\u2081)\n             | merge-pres (fork st\u2081 su\u2080 ht\u2081\u2264lu\u2080) = \u2261.refl\n\nmodule SortingProperties {\u2113 a} {A : Set a} (_\u2264\u1d2c_ : A \u2192 A \u2192 Set \u2113)\n                               (_\u2293\u1d2c_ _\u2294\u1d2c_ : A \u2192 A \u2192 A)\n                               (isPreorder : IsPreorder _\u2261_ _\u2264\u1d2c_)\n                               (\u2264-\u2294 : \u2200 x y \u2192 x \u2264\u1d2c (y \u2294\u1d2c x))\n                               (\u2293-\u2264 : \u2200 x y \u2192 (x \u2293\u1d2c y) \u2264\u1d2c y)\n                               (\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264\u1d2c y \u2192 x \u2264\u1d2c z \u2192 x \u2264\u1d2c (y \u2293\u1d2c z))\n                               (\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264\u1d2c z \u2192 y \u2264\u1d2c z \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z)\n                               (\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c y)\n                               (\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264\u1d2c (y \u2293\u1d2c z) \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 x \u2264\u1d2c z)\n                               (\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294\u1d2c y) \u2264\u1d2c z \u2192 y \u2264\u1d2c z)\n                               where\n    module \u2264\u1d2c = IsPreorder isPreorder\n    -- open SortedDataIx _\u2264\u1d2c_ isPreorder\n    open Sorting _\u2293\u1d2c_ _\u2294\u1d2c_\n    module SD = SortedData _\u2264\u1d2c_\n    open SD using (fork; leaf; PreSorted)\n\n    first-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                first (merge t) \u2261 first (lft t) \u2293\u1d2c first (rght t)\n    first-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    first-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080) | first-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | pf\n        | fork v\u2081 w\u2081\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    last-merge : \u2200 {n} (t : Tree A (1 + n)) \u2192\n                last (merge t) \u2261 last (lft t) \u2294\u1d2c last (rght t)\n    last-merge (fork (leaf x) (leaf y)) = \u2261.refl\n    last-merge (fork (fork t\u2080 t\u2081) (fork u\u2080 u\u2081))\n      with merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | last-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080\n        | fork v\u2081 w\u2081 | pf\n      with merge (fork w\u2080 v\u2081)\n    ... | fork a b\n      = pf\n\n    merge-spec\u2032 : \u2200 {n} {t u : Tree A n} \u2192\n                 SD.Sorted t \u2192 SD.Sorted u \u2192\n                 let tu' = merge (fork t u) in\n                 SD.Sorted tu'\n                 \u00d7 last (lft tu') \u2264\u1d2c (last t \u2293\u1d2c last u)\n                 \u00d7 (first t \u2294\u1d2c first u) \u2264\u1d2c first (rght tu')\n    merge-spec\u2032 (leaf {x}) (leaf {y}) = fork leaf leaf (\u2264\u1d2c.trans (\u2293-\u2264 x y) (\u2264-\u2294 y x)) , \u2264\u1d2c.refl , \u2264\u1d2c.refl\n    merge-spec\u2032 {t = fork t\u2080 t\u2081} {u = fork u\u2080 u\u2081}\n               (fork st\u2080 st\u2081 ht\u2080\u2264lt\u2081)\n               (fork su\u2080 su\u2081 lu\u2080\u2264hu\u2081)\n      with merge (fork t\u2080 u\u2080) | merge-spec\u2032 st\u2080 su\u2080 | last-merge (fork t\u2080 u\u2080)\n         | merge (fork t\u2081 u\u2081) | merge-spec\u2032 st\u2081 su\u2081 | first-merge (fork t\u2081 u\u2081)\n    ... | fork v\u2080 w\u2080 | (fork sv\u2080 sw\u2080 p1 , lpf1 , rpf1) | lastw\u2080\n        | fork v\u2081 w\u2081 | (fork sv\u2081 sw\u2081 p2 , lpf2 , rpf2) | firstv\u2081\n      with merge (fork w\u2080 v\u2081) | merge-spec\u2032 sw\u2080 sv\u2081 | first-merge (fork w\u2080 v\u2081) | last-merge (fork w\u2080 v\u2081)\n    ... | fork a b | (fork sa sb p3 , lpf3 , rpf3) | firsta | lastb\n      = fork (fork sv\u2080 sa pf1) (fork sb sw\u2081 pf2) p3 , lpf4 , rpf4\n         where\n             pf1 : last v\u2080 \u2264\u1d2c first a\n             pf1 rewrite firsta | firstv\u2081 = \u2264-< p1 , \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2080 lpf1) ht\u2080\u2264lt\u2081 , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf1) lu\u2080\u2264hu\u2081 > >\n             pf2 : last b \u2264\u1d2c first w\u2081\n             pf2 rewrite lastb | lastw\u2080 = \u2264-[ \u2264-[ \u2264\u1d2c.trans ht\u2080\u2264lt\u2081 (\u2264-\u2294\u2080 rpf2) , \u2264\u1d2c.trans lu\u2080\u2264hu\u2081 (\u2264-\u2294\u2081 rpf2) ] , p2 ]\n             lpf4 = \u2264-< \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2080 lpf2) , \u2264\u1d2c.trans (\u2264-\u2293\u2081 lpf3) (\u2264-\u2293\u2081 lpf2) >\n             rpf4 = \u2264-[ \u2264\u1d2c.trans (\u2264-\u2294\u2080 rpf1) (\u2264-\u2294\u2080 rpf3) , \u2264\u1d2c.trans (\u2264-\u2294\u2081 rpf1) (\u2264-\u2294\u2080 rpf3) ]\n\n    merge-spec : \u2200 {n} {t : Tree A (1 + n)} \u2192 PreSorted t \u2192 SD.Sorted (merge t)\n    merge-spec {t = fork t u} (st , su) = proj\u2081 (merge-spec\u2032 st su)\n\n    sort-spec : \u2200 {n} (t : Tree A n) \u2192 SD.Sorted (sort t)\n    sort-spec (leaf _)   = leaf\n    sort-spec (fork t u) = merge-spec (sort-spec t , sort-spec u)\n\n    {-\nmodule M {n} where\n  postulate\n    _\u2264_ : {-\u2200 {n} \u2192-} Bits n \u2192 Bits n \u2192 Set\n    _\u2294_ : Bits n \u2192 Bits n \u2192 Bits n\n    _\u2293_ : Bits n \u2192 Bits n \u2192 Bits n\n    isPreorder : IsPreorder _\u2261_ _\u2264_\n    \u2264-\u2294 : \u2200 x y \u2192 x \u2264 (y \u2294 x)\n    \u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n    \u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n    \u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n    \u2293-comm : Commutative _\u2261_ _\u2293_\n    \u2294-comm : Commutative _\u2261_ _\u2294_\n    \u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n    \u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n    \u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n    \u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n    \u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n    \u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\nmodule BitsSorting {m} where\n    open M {m}\n\n    module S = Sorting _\u2293_ _\u2294_\n    module SDP = SortingDataIxProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2294-spec \u2293-spec \u2293-comm \u2294-comm\n    module SP = SortingProperties _\u2264_ _\u2293_ _\u2294_ isPreorder \u2264-\u2294 \u2293-\u2264 \u2264-<_,_> \u2264-[_,_] \u2264-\u2293\u2080 \u2264-\u2293\u2081 \u2264-\u2294\u2080 \u2264-\u2294\u2081\n    open SortedData _\u2264_\n\n    merge : \u2200 {n} \u2192 Tree (Bits m) (1 + n) \u2192 Tree (Bits m) (1 + n)\n    merge = S.merge\n\n    sort : \u2200 {n} \u2192 Tree (Bits m) n \u2192 Tree (Bits m) n\n    sort = S.sort\n\n    merge-spec : \u2200 {n} {t : Tree (Bits m) (1 + n)} \u2192 PreSorted t \u2192 Sorted (merge t)\n    merge-spec = SP.merge-spec\n\n    sort-spec : \u2200 {n} (t : Tree (Bits m) n) \u2192 Sorted (sort t)\n    sort-spec = SP.sort-spec\n\nmodule BitsSorting\u2032 where\n    open BitsSorting\n    open AllBits\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"07cae51cc16edefb7e8225f94559a896323c1dd2","subject":"Remove comment.","message":"Remove comment.\n\nOld-commit-hash: aabb08e2144e33b92062ac6a93d3722869d7ba2f\n","repos":"inc-lc\/ilc-agda","old_file":"Popl14\/Syntax\/Term.agda","new_file":"Popl14\/Syntax\/Term.agda","new_contents":"module Popl14.Syntax.Term where\n\nopen import Popl14.Syntax.Type\n\nopen import Data.Integer\n\nimport Parametric.Syntax.Term Base as Term\n\ndata Const : Term.Structure where\n  intlit-const : (n : \u2124) \u2192 Const \u2205 int\n  add-const : Const (int \u2022 int \u2022 \u2205) int\n  minus-const : Const (int \u2022 \u2205) (int)\n\n  empty-const : Const \u2205 (bag)\n  insert-const : Const (int \u2022 bag \u2022 \u2205) (bag)\n  union-const : Const (bag \u2022 bag \u2022 \u2205) (bag)\n  negate-const : Const (bag \u2022 \u2205) (bag)\n\n  flatmap-const : Const ((int \u21d2 bag) \u2022 bag \u2022 \u2205) (bag)\n  sum-const : Const (bag \u2022 \u2205) (int)\n\nopen Term.Structure Const public\n\n-- Shorthands of constants\n\npattern intlit n = const (intlit-const n) \u2205\npattern add s t = const add-const (s \u2022 t \u2022 \u2205)\npattern minus t = const minus-const (t \u2022 \u2205)\npattern empty = const empty-const \u2205\npattern insert s t = const insert-const (s \u2022 t \u2022 \u2205)\npattern union s t = const union-const (s \u2022 t \u2022 \u2205)\npattern negate t = const negate-const (t \u2022 \u2205)\npattern flatmap s t = const flatmap-const (s \u2022 t \u2022 \u2205)\npattern sum t = const sum-const (t \u2022 \u2205)\n","old_contents":"module Popl14.Syntax.Term where\n\nopen import Popl14.Syntax.Type\n\nopen import Data.Integer\n\nimport Parametric.Syntax.Term Base as Term\n\n-- Popl14-Const ? No.\ndata Const : Term.Structure where\n  intlit-const : (n : \u2124) \u2192 Const \u2205 int\n  add-const : Const (int \u2022 int \u2022 \u2205) int\n  minus-const : Const (int \u2022 \u2205) (int)\n\n  empty-const : Const \u2205 (bag)\n  insert-const : Const (int \u2022 bag \u2022 \u2205) (bag)\n  union-const : Const (bag \u2022 bag \u2022 \u2205) (bag)\n  negate-const : Const (bag \u2022 \u2205) (bag)\n\n  flatmap-const : Const ((int \u21d2 bag) \u2022 bag \u2022 \u2205) (bag)\n  sum-const : Const (bag \u2022 \u2205) (int)\n\nopen Term.Structure Const public\n\n-- Shorthands of constants\n\npattern intlit n = const (intlit-const n) \u2205\npattern add s t = const add-const (s \u2022 t \u2022 \u2205)\npattern minus t = const minus-const (t \u2022 \u2205)\npattern empty = const empty-const \u2205\npattern insert s t = const insert-const (s \u2022 t \u2022 \u2205)\npattern union s t = const union-const (s \u2022 t \u2022 \u2205)\npattern negate t = const negate-const (t \u2022 \u2205)\npattern flatmap s t = const flatmap-const (s \u2022 t \u2022 \u2205)\npattern sum t = const sum-const (t \u2022 \u2205)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ae5fcf25be1ffe0b1df8686c79b685676574d274","subject":"Fixed an draft example.","message":"Fixed an draft example.\n\nIgnore-this: 6f00f72910ff4493fe8221384b6bd1ee\n\ndarcs-hash:20110923165036-3bd4e-e601f964ecbe784ed6a51ce23eba0d19847f7bd6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/LambdaLifting.agda","new_file":"Draft\/LambdaLifting.agda","new_contents":"------------------------------------------------------------------------------\n-- Example of lambda-lifting\n------------------------------------------------------------------------------\n\nmodule Draft.LambdaLifting where\n\nopen import LTC-PCF.Base\n\nopen import LTC-PCF.Data.Nat using ( _*_ )\n\n------------------------------------------------------------------------------\n\n-- The original fach\n-- fach : D \u2192 D\n-- fach f = lam (\u03bb n \u2192 if (isZero n) then (succ zero) else n * (f \u00b7 (pred n)))\n\n-- Lambda-lifting via super-combinators (Hughes. Super-combinators. 1982).\n\n\u03b1 : D \u2192 D \u2192 D\n\u03b1 f n = if (isZero n) then (succ zero) else n * (f \u00b7 (pred n))\n{-# ATP definition \u03b1 #-}\n\nfach : D \u2192 D\nfach f = lam (\u03b1 f)\n{-# ATP definition fach #-}\n\nfac : D \u2192 D\nfac n = fix fach \u00b7 n\n{-# ATP definition fac #-}\n\npostulate\n  fac0 : fac zero \u2261 succ zero\n{-# ATP prove fac0 #-}\n\npostulate\n  fac1 : fac (succ zero) \u2261 succ zero\n{-# ATP prove fac1 #-}\n\npostulate\n  fac2 : fac (succ (succ zero)) \u2261 succ (succ zero)\n{-# ATP prove fac2 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Example of lambda-lifting\n------------------------------------------------------------------------------\n\nmodule Draft.LambdaLifting where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat using ( _*_ )\n\n------------------------------------------------------------------------------\n\n-- The original fach\n-- fach : D \u2192 D\n-- fach f = lam (\u03bb n \u2192 if (isZero n) then (succ zero) else n * (f \u00b7 (pred n)))\n\n-- Lambda-lifting via super-combinators (Hughes. Super-combinators. 1982).\n\n\u03b1 : D \u2192 D \u2192 D\n\u03b1 f n = if (isZero n) then (succ zero) else n * (f \u00b7 (pred n))\n{-# ATP definition \u03b1 #-}\n\nfach : D \u2192 D\nfach f = lam (\u03b1 f)\n{-# ATP definition fach #-}\n\nfac : D \u2192 D\nfac n = fix fach \u00b7 n\n{-# ATP definition fac #-}\n\npostulate\n  fac0 : fac zero \u2261 succ zero\n{-# ATP prove fac0 #-}\n\npostulate\n  fac1 : fac (succ zero) \u2261 succ zero\n{-# ATP prove fac1 #-}\n\npostulate\n  fac2 : fac (succ (succ zero)) \u2261 succ (succ zero)\n{-# ATP prove fac2 #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c1b1e677974f6ebc81e83e5c1674955e4e300fb9","subject":"[ doc ] Pattern matching and with in the ABP's lemma1.","message":"[ doc ] Pattern matching and with in the ABP's lemma1.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1NoHelperATP.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1NoHelperATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1NoHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- 30 November 2013. If we don't have the following definitions\n-- outside, the ATPs cannot prove the theorems.\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\n-- 26 January 2014. Pattern matching after a @with@ it is not accepted\n-- by the termination checker. See Agda issue 59, comment 18.\n\n{-# NO_TERMINATION_CHECK #-}\nlemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n         Bit b \u2192\n         Fair os\u2081 \u2192\n         Fair os\u2082 \u2192\n         S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n         \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n           Fair os\u2081'\n           \u2227 Fair os\u2082'\n           \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n           \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n            Fair os\u2081'\n            \u2227 Fair os\u2082'\n            \u2227 (as' \u2261 await b i' is' ds'\n              \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n              \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n              \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n              \u2227 js' \u2261 out (not b) \u00b7 bs')\n            \u2227 js \u2261 i' \u2237 js'\n  {-# ATP prove prf #-}\n\n... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n  lemma\u2081 b i' is'\n         (ft\u2081^ ++ os\u2081')\n         (tail\u2081 os\u2082)\n         (await b i' is' ds)\n         (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n         (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n         (corrupt (tail\u2081 os\u2082) \u00b7\n           (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n         js Bb (Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')) (tail-Fair Fos\u2082) ihS\n     where\n     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n     {-# ATP prove os\u2081-eq-helper #-}\n\n     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n     {-# ATP prove as-eq #-}\n\n     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove cs-eq bs-eq #-}\n\n     postulate\n       ds-eq-helper\u2081 :\n         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n     postulate\n       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2082)\n\n     postulate\n       as^-eq-helper\u2081 :\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n     postulate\n       as^-eq-helper\u2082 :\n         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2082 #-}\n\n     as^-eq : as^ b i' is' ds \u2261\n              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove js-eq bs-eq #-}\n\n     ihS : S b\n             (i' \u2237 is')\n             (os\u2081^ os\u2081' ft\u2081^)\n             (os\u2082^ os\u2082)\n             (as^ b i' is' ds)\n             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n             js\n     ihS = as^-eq , refl , refl , refl , js-eq\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1NoHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- 30 November 2013. If we don't have the following definitions\n-- outside, the ATPs cannot prove the theorems.\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\nlemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n         Bit b \u2192\n         Fair os\u2081 \u2192\n         Fair os\u2082 \u2192\n         S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n         \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n           Fair os\u2081'\n           \u2227 Fair os\u2082'\n           \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n           \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n            Fair os\u2081'\n            \u2227 Fair os\u2082'\n            \u2227 (as' \u2261 await b i' is' ds'\n              \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n              \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n              \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n              \u2227 js' \u2261 out (not b) \u00b7 bs')\n            \u2227 js \u2261 i' \u2237 js'\n  {-# ATP prove prf #-}\n\n... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n  lemma\u2081 b i' is'\n         (ft\u2081^ ++ os\u2081')\n         (tail\u2081 os\u2082)\n         (await b i' is' ds)\n         (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n         (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n         (corrupt (tail\u2081 os\u2082) \u00b7\n           (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n         js Bb (Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')) (tail-Fair Fos\u2082) ihS\n     where\n     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n     {-# ATP prove os\u2081-eq-helper #-}\n\n     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n     {-# ATP prove as-eq #-}\n\n     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove cs-eq bs-eq #-}\n\n     postulate\n       ds-eq-helper\u2081 :\n         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n     postulate\n       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2082)\n\n     postulate\n       as^-eq-helper\u2081 :\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n     postulate\n       as^-eq-helper\u2082 :\n         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2082 #-}\n\n     as^-eq : as^ b i' is' ds \u2261\n              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove js-eq bs-eq #-}\n\n     ihS : S b\n             (i' \u2237 is')\n             (os\u2081^ os\u2081' ft\u2081^)\n             (os\u2082^ os\u2082)\n             (as^ b i' is' ds)\n             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n             js\n     ihS = as^-eq , refl , refl , refl , js-eq\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"daef8583632895e241161e26df3f99b280657513","subject":"#33 scraps","message":"#33 scraps\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"preservation.agda","new_file":"preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = {!!}\n  pres-lem (FHAp2 eps) D1 D2 D3 (FHAp2 D4) = {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n  pres-lem (FHFailedCast eps) D1 D2 D3 (FHFailedCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n  pres-lem2 (TAFailedCast x y z w) eps = {!!}\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp ta ta\u2081) (ITLam) = {!!}\n  pres-lem3 (TAAp ta ta\u2081) (ITApCast) = {!!}\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) () -- todo: this is a little surprising; nehole doesn't take a transition but it defieately can still step\n  pres-lem3 (TACast ta x) (ITCastID) = {!!}\n  pres-lem3 (TACast ta x) (ITCastSucceed x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITGround y) = {!!}\n  pres-lem3 (TACast ta x) (ITExpand x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITCastFail w y z) = {!!}\n  pres-lem3 (TAFailedCast x y z q) xx  = {!!}\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nopen import type-assignment-unicity\n\nmodule preservation where\n  -- todo: rename\n  pres-lem : \u2200{ \u0394 \u0393 d \u03b5 d1 d2 d3 \u03c4 \u03c41 } \u2192\n            d == \u03b5 \u27e6 d1 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            \u0394 , \u0393 \u22a2 d1 :: \u03c41 \u2192\n            \u0394 , \u0393 \u22a2 d2 :: \u03c41 \u2192\n            d3 == \u03b5 \u27e6 d2 \u27e7 \u2192\n            \u0394 , \u0393 \u22a2 d3 :: \u03c4\n  pres-lem FHOuter D1 D2 D3 FHOuter\n    with type-assignment-unicity D1 D2\n  ... | refl = D3\n  pres-lem (FHAp1 eps) D1 D2 D3 (FHAp1 D4) = {!!}\n  pres-lem (FHAp2 eps) D1 D2 D3 (FHAp2 D4) = {!!}\n  pres-lem (FHNEHole eps) D1 D2 D3 (FHNEHole D4) = {!!}\n  pres-lem (FHCast eps) D1 D2 D3 (FHCast D4) = {!!}\n\n  -- todo: rename\n  pres-lem2 : \u2200{ \u03b5 \u0394 \u0393 d \u03c4 d' } \u2192\n             \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n             d == \u03b5 \u27e6 d' \u27e7 \u2192\n             \u03a3[ \u03c4' \u2208 htyp ] (\u0394 , \u0393 \u22a2 d' :: \u03c4')\n  pres-lem2 TAConst FHOuter = _ , TAConst\n  pres-lem2 (TAVar x\u2081) FHOuter = _ , TAVar x\u2081\n  pres-lem2 (TALam ta) FHOuter = _ , TALam ta\n\n  pres-lem2 (TAAp ta ta\u2081) FHOuter = _ , TAAp ta ta\u2081\n  pres-lem2 (TAAp ta ta\u2081) (FHAp1 eps) = pres-lem2 ta eps\n  pres-lem2 (TAAp ta ta\u2081) (FHAp2 eps) = pres-lem2 ta\u2081 eps\n\n  pres-lem2 (TAEHole x x\u2081) FHOuter = _ , TAEHole x x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) FHOuter = _ , TANEHole x ta x\u2081\n  pres-lem2 (TANEHole x ta x\u2081) (FHNEHole eps) = pres-lem2 ta eps\n  pres-lem2 (TACast ta x) FHOuter = _ , TACast ta x\n  pres-lem2 (TACast ta x) (FHCast eps) = pres-lem2 ta eps\n\n  -- todo: rename\n  pres-lem3 : \u2200{ \u0394 \u0393 d \u03c4 d' } \u2192\n            \u0394 , \u0393 \u22a2 d :: \u03c4 \u2192\n            d \u2192> d' \u2192\n            \u0394 , \u0393 \u22a2 d' :: \u03c4\n  pres-lem3 TAConst ()\n  pres-lem3 (TAVar x\u2081) ()\n  pres-lem3 (TALam ta) ()\n  pres-lem3 (TAAp ta ta\u2081) (ITLam) = {!!}\n  pres-lem3 (TAAp ta ta\u2081) (ITApCast) = {!!}\n  pres-lem3 (TAEHole x x\u2081) ()\n  pres-lem3 (TANEHole x ta x\u2081) () -- todo: this is a little surprising; nehole doesn't take a transition but it defieately can still step\n  pres-lem3 (TACast ta x) (ITCastID) = {!!}\n  pres-lem3 (TACast ta x) (ITCastSucceed x\u2082) = {!!}\n  pres-lem3 (TACast ta x) (ITGround y) = {!!}\n  pres-lem3 (TACast ta x) (ITExpand x\u2082) = {!!}\n\n  preservation : {\u0394 : hctx} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             \u0394 , \u2205 \u22a2 d :: \u03c4 \u2192\n             d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  preservation D (Step x x\u2081 x\u2082)\n    with pres-lem2 D x\n  ... | (_ , wt) = pres-lem x D wt (pres-lem3 wt x\u2081) x\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ce0860eb54642cc4fd246103f18313790c9d29f4","subject":"rename \u2261-splitAt into ++-decomp","message":"rename \u2261-splitAt into ++-decomp\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin hiding (_+_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount pred [] = zero\ncount pred (x \u2237 xs) = (if pred x then suc else inject\u2081) (count pred xs)\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (to\u2115 (count pred xs))\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin hiding (_+_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount pred [] = zero\ncount pred (x \u2237 xs) = (if pred x then suc else inject\u2081) (count pred xs)\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (to\u2115 (count pred xs))\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\n\u2261-splitAt : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n\u2261-splitAt {zero} {xs = []} {[]} p = refl , p\n\u2261-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with \u2261-splitAt {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (\u2261-splitAt eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (\u2261-splitAt {xs = xs} {ys} eq)\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7c9c84e96c4fed7033d313b4f2c7cd9d8dd39754","subject":"[Agda] BUILTIN NATURAL now binds ZERO and SUC.","message":"[Agda] BUILTIN NATURAL now binds ZERO and SUC.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/NumericLiterals.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/NumericLiterals.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the use of numeric literals\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.NumericLiterals where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\ndata \u2115 : Set where\n  z : \u2115\n  s : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ntoD : \u2115 \u2192 D\ntoD z     = zero\ntoD (s n) = succ\u2081 (toD n)\n\nten : D\nten = toD 10\n","old_contents":"------------------------------------------------------------------------------\n-- Testing the use of numeric literals\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.NumericLiterals where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\ndata \u2115 : Set where\n  z : \u2115\n  s : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115 #-}\n{-# BUILTIN ZERO    z #-}\n{-# BUILTIN SUC     s #-}\n\ntoD : \u2115 \u2192 D\ntoD z     = zero\ntoD (s n) = succ\u2081 (toD n)\n\nten : D\nten = toD 10\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9a173b770f197c6b30d41ce679f7a2b1864026a1","subject":"Update ZK.JSChecker to monadic JS operations","message":"Update ZK.JSChecker to monadic JS operations\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker.agda","new_file":"ZK\/JSChecker.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Type.Eq\nopen import Function         using (case_of_)\nopen import Data.Two.Base    using (_\u2227_)\nopen import Data.List.Base   using ([]; _\u2237_)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.FiniteField as Zq\nimport Crypto.JS.BigI.CyclicGroup as Zp\nopen import Crypto.JS.BigI.Params using (Params; module Params)\nopen import Crypto.JS.BigI.Checks using (check-params!)\nopen import ZK.Types\n\nverify-PoK-CP-EG-rnd :\n  (p q : BigI)\n  (pok : PoK-CP-EG-rnd Zq.\u2124[ q ] Zp.\u2124[ p ]\u2605) \u2192 Bool\nverify-PoK-CP-EG-rnd p q pok = g\u02e2==\u03b1\u1d9c\u00b7A \u2227 y\u02e2==\u27e8\u03b2\/M\u27e9\u1d9c\u00b7B\n  module verify-PoK-CP-EG-rnd where\n    open module \u2124q  = Zq q\n    open module \u2124p\u2605 = Zp p\n    open module ^-h = ^-hom {q}\n    open module pok = PoK-CP-EG-rnd pok\n    M = g ^ m\n    g\u02e2==\u03b1\u1d9c\u00b7A     = g ^ s == (\u03b1 ^ c) ** A\n    y\u02e2==\u27e8\u03b2\/M\u27e9\u1d9c\u00b7B = y ^ s == ((\u03b2 \/\/ M) ^ c) ** B\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\nbigdec : JSValue \u2192 JS[ BigI ]\nbigdec v = bigI <$> castString v <*> return \"10\"\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n  arg \u00b7\u00ab \"statement\" \u00bb  >>= \u03bb stm \u2192\n  stm \u00b7\u00ab \"type\"  \u00bb      >>= \u03bb typ \u2192\n  let\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n  in\n  check! \"type of statement\" (typ === fromString cpt)\n         (\u03bb _ \u2192 \"Expected type of statement: \" ++\n                cpt ++ \" not \" ++ toString typ) >>\n  stm \u00b7\u00ab \"data\"  \u00bb      >>= \u03bb dat \u2192\n  dat \u00b7\u00ab \"enc\"   \u00bb      >>= \u03bb enc \u2192\n  arg \u00b7\u00ab \"proof\" \u00bb      >>= \u03bb prf \u2192\n  prf \u00b7\u00ab \"commitment\" \u00bb >>= \u03bb com \u2192\n  (bigdec =<< dat \u00b7\u00ab \"g\" \u00bb) >>= \u03bb gI \u2192\n  (bigdec =<< dat \u00b7\u00ab \"p\" \u00bb) >>= \u03bb pI \u2192\n  (bigdec =<< dat \u00b7\u00ab \"q\" \u00bb) >>= \u03bb qI \u2192\n  let\n    open module \u2124q  = Zq qI using (BigI\u25b9\u2124q)\n    open module \u2124p\u2605 = Zp pI using (BigI\u25b9\u2124p\u2605)\n    gpq = record\n            { primality-test-probability-bound = primality-test-probability-bound\n            ; min-bits-q = min-bits-q\n            ; min-bits-p = min-bits-p\n            ; qI = qI\n            ; pI = pI\n            ; gI = gI\n            }\n  in\n  BigI\u25b9\u2124p\u2605 gI >>= \u03bb g \u2192\n  (BigI\u25b9\u2124p\u2605 =<< bigdec =<< dat \u00b7\u00ab \"y\"         \u00bb) >>= \u03bb y \u2192\n  (BigI\u25b9\u2124p\u2605 =<< bigdec =<< enc \u00b7\u00ab \"alpha\"     \u00bb) >>= \u03bb \u03b1 \u2192\n  (BigI\u25b9\u2124p\u2605 =<< bigdec =<< enc \u00b7\u00ab \"beta\"      \u00bb) >>= \u03bb \u03b2 \u2192\n  (BigI\u25b9\u2124p\u2605 =<< bigdec =<< com \u00b7\u00ab \"A\"         \u00bb) >>= \u03bb A \u2192\n  (BigI\u25b9\u2124p\u2605 =<< bigdec =<< com \u00b7\u00ab \"B\"         \u00bb) >>= \u03bb B \u2192\n  (BigI\u25b9\u2124q  =<< bigdec =<< prf \u00b7\u00ab \"challenge\" \u00bb) >>= \u03bb c \u2192\n  (BigI\u25b9\u2124q  =<< bigdec =<< prf \u00b7\u00ab \"response\"  \u00bb) >>= \u03bb s \u2192\n  (BigI\u25b9\u2124q  =<< bigdec =<< dat \u00b7\u00ab \"plain\"     \u00bb) >>= \u03bb m \u2192\n  check-params! gpq >>\n  -- Console.log (\"pok=\" ++ JSON-stringify (fromAny pok)) >>\n  check! \"PoK-CP-EG-rnd\" (verify-PoK-CP-EG-rnd pI qI\n    (record\n       { g = g\n       ; y = y\n       ; \u03b1 = \u03b1\n       ; \u03b2 = \u03b2\n       ; A = A\n       ; B = B\n       ; c = c\n       ; s = s\n       ; m = m\n       }))\n    (\u03bb _ \u2192 \"Invalid transcript\")\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv >>= \u03bb args \u2192\n  case JSArray\u25b9List args of \u03bb\n  { (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb\n      { [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv >>= \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb\n          { [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS >>== \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule ZK.JSChecker where\n\nopen import Type.Eq\nopen import Function         using (case_of_)\nopen import Data.Two.Base    using (_\u2227_)\nopen import Data.List.Base   using ([]; _\u2237_)\n\nopen import FFI.JS\nopen import FFI.JS.Check\n-- open import FFI.JS.Proc using (URI; JSProc; showURI; server)\n-- open import Control.Process.Type\nimport FFI.JS.Console as Console\nimport FFI.JS.Process as Process\nimport FFI.JS.FS as FS\n\nimport FFI.JS.BigI as BigI\nopen BigI using (BigI; bigI)\n\nimport Crypto.JS.BigI.FiniteField as Zq\nimport Crypto.JS.BigI.CyclicGroup as Zp\nopen import Crypto.JS.BigI.Params using (Params; module Params)\nopen import Crypto.JS.BigI.Checks using (check-params!)\n\n-- TODO dynamise me\nprimality-test-probability-bound : Number\nprimality-test-probability-bound = readNumber \"10\"\n\n-- TODO: check if this is large enough\nmin-bits-q : Number\nmin-bits-q = 256N\n\nmin-bits-p : Number\nmin-bits-p = 2048N\n\nbigdec : JSValue \u2192 BigI\nbigdec v = bigI (castString v) \"10\"\n\n-- PoK: (Zero-Knowledge) Proof of Knowledge\n-- CP: Chaum-Pedersen\n-- EG: ElGamal\n-- rnd: Knowledge of the secret, random exponent used in ElGamal encryption\nrecord PoK-CP-EG-rnd (\u2124q \u2124p\u2605 : Set) : Set where\n  inductive -- NO_ETA\n  field\n    g y \u03b1 \u03b2 A B : \u2124p\u2605\n    m c s : \u2124q\n\nverify-PoK-CP-EG-rnd :\n  (p q : BigI)\n  (pok : PoK-CP-EG-rnd Zq.\u2124[ q ] Zp.\u2124[ p ]\u2605) \u2192 Bool\nverify-PoK-CP-EG-rnd p q pok = g\u02e2==\u03b1\u1d9c\u00b7A \u2227 y\u02e2==\u27e8\u03b2\/M\u27e9\u1d9c\u00b7B\n  module verify-PoK-CP-EG-rnd where\n    open module \u2124q  = Zq q\n    open module \u2124p\u2605 = Zp p\n    open module ^-h = ^-hom {q}\n    open module pok = PoK-CP-EG-rnd pok\n    M = g ^ m\n    g\u02e2==\u03b1\u1d9c\u00b7A     = g ^ s == (\u03b1 ^ c) ** A\n    y\u02e2==\u27e8\u03b2\/M\u27e9\u1d9c\u00b7B = y ^ s == ((\u03b2 \/\/ M) ^ c) ** B\n\nzk-check! : JSValue \u2192 JS!\nzk-check! arg =\n  check! \"type of statement\" (typ === fromString cpt)\n         (\u03bb _ \u2192 \"Expected type of statement: \" ++\n                cpt ++ \" not \" ++ toString typ) >>\n  BigI\u25b9\u2124p\u2605 gI >>= \u03bb g \u2192\n  BigI\u25b9\u2124p\u2605 (bigdec (dat \u00b7\u00ab \"y\"         \u00bb)) >>= \u03bb y \u2192\n  BigI\u25b9\u2124p\u2605 (bigdec (enc \u00b7\u00ab \"alpha\"     \u00bb)) >>= \u03bb \u03b1 \u2192\n  BigI\u25b9\u2124p\u2605 (bigdec (enc \u00b7\u00ab \"beta\"      \u00bb)) >>= \u03bb \u03b2 \u2192\n  BigI\u25b9\u2124p\u2605 (bigdec (com \u00b7\u00ab \"A\"         \u00bb)) >>= \u03bb A \u2192\n  BigI\u25b9\u2124p\u2605 (bigdec (com \u00b7\u00ab \"B\"         \u00bb)) >>= \u03bb B \u2192\n  BigI\u25b9\u2124q  (bigdec (prf \u00b7\u00ab \"challenge\" \u00bb)) >>= \u03bb c \u2192\n  BigI\u25b9\u2124q  (bigdec (prf \u00b7\u00ab \"response\"  \u00bb)) >>= \u03bb s \u2192\n  BigI\u25b9\u2124q  (bigdec (dat \u00b7\u00ab \"plain\"     \u00bb)) >>= \u03bb m \u2192\n  check-params! gpq >>\n  -- Console.log (\"pok=\" ++ JSON-stringify (fromAny pok)) >>\n  check! \"PoK-CP-EG-rnd\" (verify-PoK-CP-EG-rnd pI qI\n    (record\n       { g = g\n       ; y = y\n       ; \u03b1 = \u03b1\n       ; \u03b2 = \u03b2\n       ; A = A\n       ; B = B\n       ; c = c\n       ; s = s\n       ; m = m\n       }))\n    (\u03bb _ \u2192 \"Invalid transcript\")\n module zk-check where\n    cpt = \"chaum-pedersen-pok-elgamal-rnd\"\n    stm = arg \u00b7\u00ab \"statement\" \u00bb\n    typ = stm \u00b7\u00ab \"type\" \u00bb\n    dat = stm \u00b7\u00ab \"data\" \u00bb\n    enc = dat \u00b7\u00ab \"enc\" \u00bb\n    prf = arg \u00b7\u00ab \"proof\" \u00bb\n    com = prf \u00b7\u00ab \"commitment\" \u00bb\n    gI  = bigdec (dat \u00b7\u00ab \"g\" \u00bb)\n    pI  = bigdec (dat \u00b7\u00ab \"p\" \u00bb)\n    qI  = bigdec (dat \u00b7\u00ab \"q\" \u00bb)\n    gpq = record\n            { primality-test-probability-bound = primality-test-probability-bound\n            ; min-bits-q = min-bits-q\n            ; min-bits-p = min-bits-p\n            ; qI = qI\n            ; pI = pI\n            ; gI = gI\n            }\n    open module \u2124q  = Zq qI using (BigI\u25b9\u2124q)\n    open module \u2124p\u2605 = Zp pI using (BigI\u25b9\u2124p\u2605)\n    -- open module [\u2124q]\u2124p\u2605 = ZqZp gpq\n\n{-\nsrv : URI \u2192 JSProc\nsrv d =\n  recv d \u03bb q \u2192\n  send d (fromBool (zk-check q))\n  end\n-}\n\n-- Working around Agda.Primitive.lsuc being undefined\n-- case_of_ : {A : Set} {B : Set} \u2192 A \u2192 (A \u2192 B) \u2192 B\n-- case x of f = f x\n\nmain : JS!\nmain =\n  Process.argv >>= \u03bb args \u2192\n  case JSArray\u25b9ListString args of \u03bb {\n    (_node \u2237 _run \u2237 _test \u2237 args') \u2192\n      case args' of \u03bb {\n        [] \u2192\n        Console.log \"usage: No arguments\"\n        {- server \"127.0.0.1\" \"1337\" srv >>= \u03bb uri \u2192\n           Console.log (showURI uri)\n         -}\n      ; (arg \u2237 args'') \u2192\n          case args'' of \u03bb {\n            [] \u2192\n              Console.log (\"Reading input file: \" ++ arg) >>\n              FS.readFile arg nullJS >>== \u03bb err dat \u2192\n                check! \"reading input file\" (is-null err)\n                       (\u03bb _ \u2192 \"readFile error: \" ++ toString err) >>\n                zk-check! (JSON-parse (toString dat))\n          ; _ \u2192\n              Console.log \"usage: Too many arguments\"\n          }\n      }\n  ; _ \u2192\n      Console.log \"usage\"\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6bb176b4f67427ad261d7b1e2ae39d3610508c16","subject":"Profunctors: levels","message":"Profunctors: levels\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Profunctor.agda","new_file":"lib\/Category\/Profunctor.agda","new_contents":"import Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor {i} (_\u219d_ : \u2605 i \u2192 \u2605 i \u2192 \u2605 \u2113) : \u2605 (\u2113 L.\u2294 L.suc i) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n\u2192Profunctor : Profunctor {\u2113} -\u2192-\n\u2192Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nupStarRawFunctor : \u2200 {F A} \u2192 RawFunctor F \u2192 RawFunctor (UpStar F A)\nupStarRawFunctor fun = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n  where open RawFunctor fun\n\nupStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (UpStar F)\nupStarProfunctor fun = flip _\u2218\u2032_\n                     , RawFunctor._<$>_ (upStarRawFunctor fun)\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStarProfunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\ndownStarRawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStarRawFunctor = record { _<$>_ = _\u2218\u2032_ }\n","old_contents":"import Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor (_\u219d_ : \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113) : \u2605 (L.suc \u2113) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n\u2192Profunctor : Profunctor -\u2192-\n\u2192Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nupStarRawFunctor : \u2200 {F A} \u2192 RawFunctor F \u2192 RawFunctor (UpStar F A)\nupStarRawFunctor fun = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n  where open RawFunctor fun\n\nupStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (UpStar F)\nupStarProfunctor fun = flip _\u2218\u2032_\n                     , RawFunctor._<$>_ (upStarRawFunctor fun)\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStarProfunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\ndownStarRawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStarRawFunctor = record { _<$>_ = _\u2218\u2032_ }\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8e3669ff6f054abcca50532a7e3a59e70325cc20","subject":"Added Conat-stronger-coind\u2082.","message":"Added Conat-stronger-coind\u2082.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_file":"notes\/fixed-points\/GreatestFixedPoints\/Conat.agda","new_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-out-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e.\n  --\n  -- \u2200 A. A \u2264 NatF A \u21d2 A \u2264 Conat.\n  Conat-coind :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial predicate A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification\/proof for\n  -- this principle.\n  Conat-stronger-coind\u2081 :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n  -- Other stronger co-induction principle\n  --\n  -- Adapted from (Paulson, 1997. p. 16).\n  Conat-stronger-coind\u2082 :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 A n \u2192 (n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2228 Conat n) \u2192\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-out and Conat-out-ho are equivalents\n\nConat-out-fo : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out-fo = Conat-out-ho\n\nConat-out-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-out-ho' = Conat-out\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind-fo :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-fo = Conat-coind-ho\n\nConat-coind-ho' :\n  (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e.\n--\n-- NatF Conat \u2264 Conat.\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in h = Conat-coind A h' h\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  h' : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h' (inj\u2081 n\u22610)              = inj\u2081 n\u22610\n  h' (inj\u2082 (n' , prf , Cn')) = inj\u2082 (n' , prf , Conat-out Cn')\n\n-- The higher-order version.\nConat-in-ho : \u2200 {n} \u2192 NatF Conat n \u2192 Conat n\nConat-in-ho = Conat-in\n\n-- A different definition.\nConat-in' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n            \u2200 {n} \u2192 Conat n\nConat-in' h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\nConat-in-ho' : (\u2200 {n} \u2192 NatF Conat n) \u2192 \u2200 {n} \u2192 Conat n\nConat-in-ho' = Conat-in'\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-stronger-coind\u2081 to Conat-stronger-coind\u2081\/Conat-coind\n\nConat-coind'' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-stronger-coind\u2081 A h An\n\n-- 22 December 2013: We couln't prove Conat-stronger-coind\u2081 using\n-- Conat-coind.\nConat-stronger-coind\u2081' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind\u2081' A {n} h An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n\n------------------------------------------------------------------------------\n-- From Conat-stronger-coind\u2082 to Conat-stronger-coind\u2081\n\n-- 13 January 2014: We couln't prove Conat-stronger-coind\u2081 using\n-- Conat-stronger-coind\u2082.\nConat-stronger-coind\u2081'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind\u2081'' A h An = Conat-stronger-coind\u2082 A {!!} An\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in\n-- Higher-order Logic. In: Journal of Logic and Computation 7.2,\n-- pp. 175\u2013204.\n","old_contents":"------------------------------------------------------------------------------\n-- Co-inductive natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule GreatestFixedPoints.Conat where\n\nopen import FOTC.Base\nopen import FOTC.Base.PropertiesI\n\n------------------------------------------------------------------------------\n-- Conat is a greatest fixed-point of a functor\n\n-- The functor.\nNatF : (D \u2192 Set) \u2192 D \u2192 Set\nNatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n-- The co-natural numbers are the greatest fixed-point of NatF.\npostulate\n  Conat : D \u2192 Set\n\n  -- Conat is a post-fixed point of NatF, i.e.\n  --\n  -- Conat \u2264 NatF Conat.\n  Conat-out : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  -- The higher-order version.\n  Conat-out-ho : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\n\n  -- Conat is the greatest post-fixed point of NatF, i.e.\n  --\n  -- \u2200 A. A \u2264 NatF A \u21d2 A \u2264 Conat.\n  Conat-coind :\n    (A : D \u2192 Set) \u2192\n    -- A is post-fixed point of ConatF.\n    (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    -- Conat is greater than A.\n    \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- The higher-order version.\n  Conat-coind-ho :\n    (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\n\n  -- 22 December 2013. This is a stronger induction principle. If we\n  -- use it, we can use the trivial predicate A = \u03bb x \u2192 x \u2261 x in the\n  -- proofs. Unfortunately, we don't have a justification\/proof for\n  -- this principle.\n  Conat-stronger-coind :\n    \u2200 (A : D \u2192 Set) {n} \u2192\n    (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n    A n \u2192 Conat n\n\n------------------------------------------------------------------------------\n-- Conat-out and Conat-out-ho are equivalents\n\nConat-out-fo : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\nConat-out-fo = Conat-out-ho\n\nConat-out-ho' : \u2200 {n} \u2192 Conat n \u2192 NatF Conat n\nConat-out-ho' = Conat-out\n\n------------------------------------------------------------------------------\n-- Conat-coind and Conat-coind-ho are equivalents\n\nConat-coind-fo :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-fo = Conat-coind-ho\n\nConat-coind-ho' :\n  (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 A n \u2192 NatF A n) \u2192 \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind-ho' = Conat-coind\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, then the\n-- Conat predicate is also a pre-fixed point of the functional NatF,\n-- i.e.\n--\n-- NatF Conat \u2264 Conat.\nConat-in : \u2200 {n} \u2192\n           n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n') \u2192\n           Conat n\nConat-in h = Conat-coind A h' h\n  where\n  A : D \u2192 Set\n  A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n\n  h' : \u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n  h' (inj\u2081 n\u22610)              = inj\u2081 n\u22610\n  h' (inj\u2082 (n' , prf , Cn')) = inj\u2082 (n' , prf , Conat-out Cn')\n\n-- The higher-order version.\nConat-in-ho : \u2200 {n} \u2192 NatF Conat n \u2192 Conat n\nConat-in-ho = Conat-in\n\n-- A different definition.\nConat-in' : (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')) \u2192\n            \u2200 {n} \u2192 Conat n\nConat-in' h = Conat-coind (\u03bb m \u2192 m \u2261 m) h' refl\n  where\n  h' : \u2200 {m} \u2192 m \u2261 m \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 m' \u2261 m')\n  h' _ with h\n  ... | inj\u2081 m\u22610            = inj\u2081 m\u22610\n  ... | inj\u2082 (m' , prf , _) = inj\u2082 (m' , prf , refl)\n\nConat-in-ho' : (\u2200 {n} \u2192 NatF Conat n) \u2192 \u2200 {n} \u2192 Conat n\nConat-in-ho' = Conat-in'\n\n------------------------------------------------------------------------------\n-- From Conat-coind\/Conat-stronger-coind to Conat-stronger-coind\/Conat-coind\n\nConat-coind'' :\n  (A : D \u2192 Set) \u2192\n  (\u2200 {n} \u2192 A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  \u2200 {n} \u2192 A n \u2192 Conat n\nConat-coind'' A h An = Conat-stronger-coind A h An\n\n-- 22 December 2013: We cannot prove Conat-stronger-coind using\n-- Conat-coind.\nConat-stronger-coind'' :\n  \u2200 (A : D \u2192 Set) {n} \u2192\n  (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')) \u2192\n  A n \u2192 Conat n\nConat-stronger-coind'' A {n} h An = Conat-in (case prf\u2081 prf\u2082 (h An))\n  where\n  prf\u2081 : n \u2261 zero \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2081 n\u22610 = inj\u2081 n\u22610\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n' \u2192\n         n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 Conat n')\n  prf\u2082 (n' , prf , An') = inj\u2082 (n' , prf , {!!})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e230556386f4963e640326db2df538946f91b83c","subject":"agda: move doWeakenMore to the top-level to ease proofs (see #22)","message":"agda: move doWeakenMore to the top-level to ease proofs (see #22)\n\nThis will be required in other to state and prove its properties, on\nthe way to completing the proof of lift-term-ignore.\n\nOld-commit-hash: 7304eb30848c2dd2babc32f1446423bae138578a\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n  where\n    doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n      Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n      Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\n    doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\n    doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n      weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n        (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n          (doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n            (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6fab60c63de82d5040c6a6adc0f29eafeda11379","subject":"agda: show that diff-is-valid is the right thing to prove (#33)","message":"agda: show that diff-is-valid is the right thing to prove (#33)\n\nderive-is-valid needs to be proved using induction, but as induction\nhypothesis it's not strong enough and a hole is left in the proof.\n\nHowever, another conjecture is the validity of the result of diff,\ndiff-is-valid: it is a slight generalization of derive-is-valid, and\nas shown in this commit, would be sufficient as induction hypothesis.\n\nHence, in subsequent commits I will try to prove diff-is-valid\ninstead (and I will succeed).\n\nOld-commit-hash: 470e7716f46766a6ca912382551d466ed5a8068e\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid = {!!}\n\n-- This proof could be finished using diff-is-valid:\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid {bool} v = tt\nderive-is-valid {\u03c4\u2081 \u21d2 \u03c4\u2082} v =\n  \u03bb s ds valid-\u0394-s-ds \u2192 diff-is-valid (v (apply ds s)) (v s) , (\n    begin\n      (apply (derive v) v) (apply ds s)\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 x (apply ds s)) (apply-derive v) \u27e9\n      v (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v (apply ds s))) \u27e9\n      apply (derive v s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\n-- This is a postulate elsewhere, but here I want to start developing a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (derive-is-valid (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid {bool} v = tt\nderive-is-valid {\u03c4\u2081 \u21d2 \u03c4\u2082} v =\n  \u03bb s ds valid-\u0394-s-ds \u2192 {!!} , (\n    begin\n      (apply (derive v) v) (apply ds s)\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 x (apply ds s)) (apply-derive v) \u27e9\n      v (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v (apply ds s))) \u27e9\n      apply (derive v s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\n-- This is a postulate elsewhere, but here I want to start developing a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (derive-is-valid (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"de67371e77fa5141377217c251faae2f423cd3b3","subject":"agda: state correctness lemmas for lift-term-ignore (#22)","message":"agda: state correctness lemmas for lift-term-ignore (#22)\n\nProofs are still missing, and auxiliary definitions belong elsewhere.\n\nOld-commit-hash: 6d433a1b91ca988719faf1ffd49d2f6ae453e0d7\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nweakenOne-ignore :\n \u2200 \u0393\u2081 \u03c4\u2082 {\u0393\u2083 \u03c4} \u2192 (t : Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4) \u2192\n   {\u03c1 : \u27e6 \u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083) \u27e7} \u2192\n   \u27e6 weakenOne \u0393\u2081 \u03c4\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (weakenEnv \u0393\u2081 \u03c4\u2082 \u03c1)\nweakenOne-ignore \u0393\u2081 \u03c4\u2082 t {\u03c1} = \u2261-refl\n\n_\u222a_ : \u2200 {\u0393 \u0393\u2032} \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 (\u03c1\u2032 : \u27e6 \u0393\u2032 \u27e7) \u2192 \u27e6 \u0393 \u22ce \u0393\u2032 \u27e7\n_\u222a_ {\u2205} {\u0393\u2032} \u2205 \u03c1\u2032 = \u03c1\u2032\n_\u222a_ {\u03c4 \u2022 \u0393} {\u0393\u2032} (v \u2022 \u03c1) \u03c1\u2032 = v \u2022 \u03c1 \u222a \u03c1\u2032\n\nlift-term-doWeakenOne-ignore\u2032 : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  {\u03c1\u2081 : \u27e6 \u0393prefix \u27e7} {\u03c1\u2082 : \u27e6 \u0394-Context \u0393rest \u27e7}\n  (t : Term (\u0393prefix \u22ce \u0393rest) \u03c4) \u2192\n  \u27e6 lift-term\u2032-doWeakenMore \u0393prefix \u0393rest t \u27e7 (\u03c1\u2081 \u222a \u03c1\u2082) \u2261 \u27e6 t \u27e7 (\u03c1\u2081 \u222a ignore \u03c1\u2082)\nlift-term-doWeakenOne-ignore\u2032 = {!!}\n\nlift-term-weakenOne-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032-weakenOne \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-weakenOne-ignore\u2032 {\u0393} {\u03c4} \u0393\u2032 {\u03c1} t =\n  begin\n    \u27e6\n      substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n        (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n          (substTerm (sym (take-drop \u0393 \u0393\u2032))\n            t))\n      \u27e7 \u03c1\n  --\u2261\u27e8 {!\u2261-cong!} \u27e9\n    --{!!}\n  \u2261\u27e8 {!!} \u27e9\n    \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n  \u220e\n  where\n    open \u2261-Reasoning\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import meaning\nopen import Model\nopen import Changes\nopen import ChangeContexts\nopen import binding Type \u27e6_\u27e7Type\nopen import TotalTerms\n\nopen import Relation.Binary.PropositionalEquality\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-var\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-var\u2032 \u2205 x = lift-var x\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) this = this\nlift-var\u2032 (\u03c4 \u2022 \u0393\u2032) (that x) = that (lift-var\u2032 \u0393\u2032 x)\n\n-- This version of lift-term\u2032 uses just weakenOne to accomplish its\n-- job.\n\nlift-term\u2032-doWeakenMore : \u2200 \u0393prefix \u0393rest {\u03c4} \u2192\n  Term (\u0393prefix \u22ce \u0393rest) \u03c4 \u2192\n  Term (\u0393prefix \u22ce \u0394-Context \u0393rest) \u03c4\n\nlift-term\u2032-doWeakenMore \u0393prefix \u2205 t\u2081 = t\u2081\nlift-term\u2032-doWeakenMore \u0393prefix (\u03c4\u2082 \u2022 \u0393rest) t\u2081 =\n  weakenOne \u0393prefix (\u0394-Type \u03c4\u2082)\n    (substTerm (sym (move-prefix \u0393prefix \u03c4\u2082 (\u0394-Context \u0393rest)))\n      (lift-term\u2032-doWeakenMore (\u0393prefix \u22ce (\u03c4\u2082 \u2022 \u2205)) \u0393rest\n        (substTerm (move-prefix \u0393prefix \u03c4\u2082 \u0393rest) t\u2081)))\n\nlift-term\u2032-weakenOne : \u2200 {\u0393 \u03c4} \u0393\u2032 \u2192\n  Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032-weakenOne {\u0393} \u0393\u2032 t =\n  substTerm (sym (take-\u22ce-\u0394-Context-drop-\u0394-Context\u2032 \u0393 \u0393\u2032))\n    (lift-term\u2032-doWeakenMore (take \u0393 \u0393\u2032) (drop \u0393 \u0393\u2032)\n      (substTerm (sym (take-drop \u0393 \u0393\u2032))\n        t))\n\nlift-term\u2032 : \u2200 {\u0393 \u03c4} \u2192 (\u0393\u2032 : Prefix \u0393) \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context\u2032 \u0393 \u0393\u2032) \u03c4\nlift-term\u2032 \u0393\u2032 (abs t) = abs (lift-term\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term\u2032 \u0393\u2032 (app t\u2081 t\u2082) = app (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082)\nlift-term\u2032 \u0393\u2032 (var x) = var (lift-var\u2032 \u0393\u2032 x)\nlift-term\u2032 \u0393\u2032 true = true\nlift-term\u2032 \u0393\u2032 false = false\nlift-term\u2032 \u0393\u2032 (if t\u2081 t\u2082 t\u2083) = if (lift-term\u2032 \u0393\u2032 t\u2081) (lift-term\u2032 \u0393\u2032 t\u2082) (lift-term\u2032 \u0393\u2032 t\u2083)\nlift-term\u2032 \u0393\u2032 (\u0394 t) = lift-term\u2032-weakenOne \u0393\u2032 (\u0394 t)\nlift-term\u2032 {\u03c4 = \u03c4} \u0393\u2032 (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) = lift-weakened-term \u0393\u2081 \u03c4\u2082 \u0393\u2032 t\n  where\n    double-weakenOne :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083))) \u03c4\n    double-weakenOne \u0393\u2081 {\u0393\u2083} \u03c4\u2082 t =\n      substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n        (weakenOne (\u0394-Context \u0393\u2081) _\n          (weakenOne (\u0394-Context \u0393\u2081) _\n            (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083) (lift-term\u2032 \u2205 t))))\n\n    lift-weakened-term :\n      \u2200 \u0393\u2081 {\u0393\u2083 \u03c4} \u03c4\u2082 \u0393\u2032 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 \n      Term (\u0394-Context\u2032 (\u0393\u2081 \u22ce (\u03c4\u2082 \u2022 \u0393\u2083)) \u0393\u2032) \u03c4\n    lift-weakened-term \u2205 \u03c4\u2082 \u2205 t = double-weakenOne \u2205 \u03c4\u2082 t\n    lift-weakened-term \u0393\u2081 \u03c4\u2082 \u2205 t = double-weakenOne \u0393\u2081 \u03c4\u2082 t\n    lift-weakened-term \u2205 \u03c4\u2083 (.\u03c4\u2083 \u2022 \u0393\u2032\u2081) t\u2081 = weakenOne \u2205 \u03c4\u2083 (lift-term\u2032 \u0393\u2032\u2081 t\u2081)\n    lift-weakened-term (\u03c4\u2081 \u2022 \u0393\u2081) {\u0393\u2083} \u03c4\u2082 (.\u03c4\u2081 \u2022 \u0393\u2032\u2081) t\u2081 =\n      lift-term\u2032-weakenOne (\u03c4\u2081 \u2022 \u0393\u2032\u2081) (weakenOne (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082 t\u2081)\n    -- In the last case, I was not able to finish without using the expensive\n    -- lift-term\u2032-weakenOne\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term = lift-term\u2032 \u2205\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-var-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) (\u03c1 : \u27e6 \u0394-Context\u2032 \u0393 \u0393\u2032 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var\u2032 \u0393\u2032 x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-var-ignore\u2032 \u2205 \u03c1 x = lift-var-ignore \u03c1 x\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore\u2032 (\u03c4 \u2022 \u0393\u2032) (v \u2022 \u03c1) (that x) = lift-var-ignore\u2032 \u0393\u2032 \u03c1 x\n\nlift-term-ignore\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  (\u0393\u2032 : Prefix \u0393) {\u03c1 : \u27e6 \u0394-Context\u2032  \u0393 \u0393\u2032 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term\u2032 \u0393\u2032 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\nlift-term-ignore\u2032 \u0393\u2032 (abs t) =\n  ext (\u03bb v \u2192 lift-term-ignore\u2032 (_ \u2022 \u0393\u2032) t)\nlift-term-ignore\u2032 \u0393\u2032 (app t\u2081 t\u2082) =\n  \u2261-app (lift-term-ignore\u2032 \u0393\u2032 t\u2081) (lift-term-ignore\u2032 \u0393\u2032 t\u2082)\nlift-term-ignore\u2032 \u0393\u2032 (var x) = lift-var-ignore\u2032 \u0393\u2032 _ x\nlift-term-ignore\u2032 \u0393\u2032 true = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 false = \u2261-refl\nlift-term-ignore\u2032 \u0393\u2032 {\u03c1} (if t\u2081 t\u2082 t\u2083)\n  with \u27e6 lift-term\u2032 \u0393\u2032 t\u2081 \u27e7 \u03c1\n     | \u27e6 t\u2081 \u27e7 (ignore\u2032 \u0393\u2032 \u03c1)\n     | lift-term-ignore\u2032 \u0393\u2032 {\u03c1} t\u2081\n... | true | true | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2082\n... | false | false | refl = lift-term-ignore\u2032 \u0393\u2032 t\u2083\nlift-term-ignore\u2032 \u0393\u2032 (\u0394 t) = {!!}\nlift-term-ignore\u2032 _ (weakenOne _ _ {_} {._} _) = {!!}\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore = lift-term-ignore\u2032 \u2205\n\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\n_and_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na and b = if a b false\n\n!_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool\n! x = if x false true\n\n-- XXX: to finish this, we just need to call lift-term, like in\n-- derive-term. Should be easy, just a bit boring.\napply-pure-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply-pure-term {bool} = abs {!!}\napply-pure-term {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} = abs (abs (abs (app (app apply-pure-term (app (app (var (that (that this))) (var this)) ({!\u0394 {\u0393} (var this)!}))) (app (var (that this)) (var this)))))\n--abs (abs (abs (app (app apply-compose-term (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x       (\u0394x))  (     f                 x        )\n\n-- Term xor\n_xor-term_ : \u2200 {\u0393} \u2192 Term \u0393 bool \u2192 Term \u0393 bool \u2192 Term \u0393 bool\na xor-term b = if a (! b) b\n\ndiff-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\ndiff-term {\u03c4 \u21d2 \u03c4\u2081} = \u03bb f\u2081 f\u2082 \u2192 {!diff-term (f\u2081 x) (f\u2082 x)!}\ndiff-term {bool} = _xor-term_\n\napply-term : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\napply-term {\u03c4 \u21d2 \u03c4\u2081} = {!!}\napply-term {bool} = _xor-term_\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term true = false\nderive-term false = false\nderive-term (if c t e) =\n  if ((derive-term c) and (lift-term c))\n    (diff-term (apply-term (derive-term e) (lift-term e)) (lift-term t))\n    (if ((derive-term c) and (lift-term (! c)))\n      (diff-term (apply-term (derive-term t) (lift-term t)) (lift-term e))\n      (if (lift-term c)\n        (derive-term t)\n        (derive-term e)))\n\nderive-term (\u0394 t) = \u0394 (derive-term t)\nderive-term (weakenOne \u0393\u2081 \u03c4\u2082 {\u0393\u2083} t) =\n  substTerm (\u0394-Context-\u22ce-expanded \u0393\u2081 \u03c4\u2082 \u0393\u2083)\n    (weakenOne (\u0394-Context \u0393\u2081) (\u0394-Type \u03c4\u2082)\n      (weakenOne (\u0394-Context \u0393\u2081) \u03c4\u2082\n        (substTerm (\u0394-Context-\u22ce \u0393\u2081 \u0393\u2083)\n          (derive-term t))))\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext-t (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\nderive-term-correct (weakenOne _ _ t) = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e7431ce4a65422d78dcf90124d87c8258353580c","subject":"agda: we don't need a proof that the input change is valid (#33)","message":"agda: we don't need a proof that the input change is valid (#33)\n\ndiff is defined on the semantic domain, so it produces a valid change\non functions even when the input is not a valid change (maybe this\nextends to the syntactic domain, but I won't claim that).\n\nHowever, the proof of correctness of symbolic derivation uses\ndiff-apply, which is still only valid on valid changes, as visible in\nthe type of diff-apply-proof.\n\nCurrently, diff-is-valid and derive-is-valid are unused. However, when\nthe proof of derive-term will use diff-apply-proof, derive-term will\nonly be valid on valid changes; so diff-is-valid and derive-is-valid\nwill be required to prove that we can produce such valid changes.\n\nAdditionally, remark that now validity proofs form a strictly positive\ndatatype, so they could be encoded with a standard datatype.\n\nMoreover, validity proofs are not any more a logical relation.\n\nOld-commit-hash: 1a3d91d1304701dead9c75f987d7e19b4fb45401\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : valid-\u0394 s ds) -} \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","old_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Note that \u0394 is *not* the same as the \u2202 operator in\n-- definition\/intro.tex. See discussion at:\n--\n--   https:\/\/github.com\/ps-mr\/ilc\/pull\/34#discussion_r4290325\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nopen import Data.Product\nopen import Data.Unit\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\nopen import Syntactic.Changes\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\nopen import Denotational.Equivalence\n\nopen import Changes\nopen import ChangeContexts\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds (valid : Valid\u0394 s ds) \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n-- What I had to write:\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {bool} v dv = \u22a4\nvalid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds (valid-w : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds valid-\u0394-s-ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I want to start developing a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v)) (derive-is-valid (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv {!!})) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv {!!})) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nlift-term {\u0393\u2081} {\u0393\u2082} {{\u0393\u2032}} = weaken (\u227c-trans \u227c-\u0394-Context \u0393\u2032)\n\n-- PROPERTIES of lift-term\n\nlift-term-ignore : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {\u03c1 : \u27e6 \u0393\u2082 \u27e7} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\nlift-term-ignore {{\u0393\u2032}} {\u03c1} t = let \u0393\u2033 = \u227c-trans \u227c-\u0394-Context \u0393\u2032 in\n  begin\n    \u27e6 lift-term {{\u0393\u2032}} t \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    \u27e6 weaken \u0393\u2033 t \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t \u03c1 \u27e9\n    \u27e6 t \u27e7 (\u27e6 \u227c-trans \u227c-\u0394-Context \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb x \u2192 \u27e6 t \u27e7 x) (\u27e6\u27e7-\u227c-trans \u227c-\u0394-Context \u0393\u2032 \u03c1) \u27e9\n    \u27e6 t \u27e7Term (\u27e6 \u227c-\u0394-Context \u27e7\u227c (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1))\n  \u2261\u27e8\u27e9\n    \u27e6 t \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))\n  \u220e where open \u2261-Reasoning\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t : Term (\u03c4\u2081 \u2022 \u0393\u2081) \u03c4\u2082) \u2192\n  let \u0393\u2033 = keep \u0394-Type \u03c4\u2081 \u2022 keep \u03c4\u2081 \u2022 \u0393\u2032 in\n  \u0394 {{\u0393\u2032}} (abs t) \u2248 abs (abs (\u0394 {\u03c4\u2081 \u2022 \u0393\u2081} t))\n\u0394-abs t = ext-t (\u03bb \u03c1 \u2192 refl)\n\n\u0394-app : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4\u2081 \u03c4\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} (t\u2081 : Term \u0393\u2081 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393\u2081 \u03c4\u2081) \u2192\n  \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u2248 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n\u0394-app {{\u0393\u2032}} t\u2081 t\u2082 = \u2248-sym (ext-t (\u03bb \u03c1\u2032 \u2192 let \u03c1 = \u27e6 \u0393\u2032 \u27e7 \u03c1\u2032 in\n  begin\n    \u27e6 app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082) \u27e7 \u03c1\u2032\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term {{\u0393\u2032}} t\u2082 \u27e7 \u03c1\u2032))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {{\u0393\u2032}} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8  \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 t\u2082 \u27e7 (update \u03c1)))  \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8\u27e9\n     \u27e6 \u0394 {{\u0393\u2032}} (app t\u2081 t\u2082) \u27e7 \u03c1\u2032\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var {\u03c4 \u2022 \u0393} this = this\nderive-var {\u03c4 \u2022 \u0393} (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 (\u0394-Type \u03c4)\nderive-term {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) = abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} {{\u0393\u2033}} t))\n  where \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term {{\u0393\u2032}} (app t\u2081 t\u2082) = app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\nderive-term {{\u0393\u2032}} (var x) = var (lift \u0393\u2032 (derive-var x))\nderive-term {{\u0393\u2032}} true = false\nderive-term {{\u0393\u2032}} false = false\nderive-term {{\u0393\u2032}} (if c t e) =\n  if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} c))\n    (diff-term (apply-term (derive-term {{\u0393\u2032}} e) (lift-term {{\u0393\u2032}} e)) (lift-term {{\u0393\u2032}} t))\n    (if ((derive-term {{\u0393\u2032}} c) and (lift-term {{\u0393\u2032}} (! c)))\n      (diff-term (apply-term (derive-term {{\u0393\u2032}} t) (lift-term {{\u0393\u2032}} t)) (lift-term {{\u0393\u2032}} e))\n      (if (lift-term {{\u0393\u2032}} c)\n        (derive-term {{\u0393\u2032}} t)\n        (derive-term {{\u0393\u2032}} e)))\nderive-term {{\u0393\u2032}} (\u0394 {{\u0393\u2033}} t) = \u0394 {{\u0393\u2032}} (derive-term {{\u0393\u2033}} t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} \u2192 (t : Term \u0393\u2081 \u03c4) \u2192\n  \u0394 {{\u0393\u2032}} t \u2248 derive-term {{\u0393\u2032}} t\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (abs {\u03c4} t) =\n  begin\n     \u0394 (abs t)\n  \u2248\u27e8  \u0394-abs t  \u27e9\n     abs (abs (\u0394 {\u03c4 \u2022 \u0393\u2081} t))\n  \u2248\u27e8  \u2248-abs (\u2248-abs (derive-term-correct {\u03c4 \u2022 \u0393\u2081} t))  \u27e9\n     abs (abs (derive-term {\u03c4 \u2022 \u0393\u2081} t))\n  \u2261\u27e8\u27e9\n     derive-term (abs t)\n  \u220e where\n      open \u2248-Reasoning\n      \u0393\u2033 = keep \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u2032\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8  \u0394-app t\u2081 t\u2082  \u27e9\n     app (app (\u0394 {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (\u0394 {{\u0393\u2032}} t\u2082)\n  \u2248\u27e8  \u2248-app (\u2248-app (derive-term-correct {{\u0393\u2032}} t\u2081) \u2248-refl) (derive-term-correct {{\u0393\u2032}} t\u2082)  \u27e9\n     app (app (derive-term {{\u0393\u2032}} t\u2081) (lift-term {{\u0393\u2032}} t\u2082)) (derive-term {{\u0393\u2032}} t\u2082)\n  \u2261\u27e8\u27e9\n    derive-term {{\u0393\u2032}} (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct {\u0393\u2081} {{\u0393\u2032}} (var x) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 \u0394 {{\u0393\u2032}} (var x) \u27e7 \u03c1\n  \u2261\u27e8\u27e9\n    diff\n      (\u27e6 x \u27e7 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n      (\u27e6 x \u27e7 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1)))\n  \u2261\u27e8  derive-var-correct {\u0393\u2081} (\u27e6 \u0393\u2032 \u27e7 \u03c1) x  \u27e9\n    \u27e6 derive-var x \u27e7Var (\u27e6 \u0393\u2032 \u27e7 \u03c1)\n  \u2261\u27e8 sym (lift-sound \u0393\u2032 (derive-var x) \u03c1) \u27e9\n    \u27e6 lift \u0393\u2032 (derive-var x) \u27e7Var \u03c1\n  \u220e) where open \u2261-Reasoning\nderive-term-correct true = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct false = ext-t (\u03bb \u03c1 \u2192 \u2261-refl)\nderive-term-correct (if t\u2081 t\u2082 t\u2083) = {!!}\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4abb43884e83ac40ba0a379ce9eb0606e824adaf","subject":"bintree: adapt sorting proofs","message":"bintree: adapt sorting proofs\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree-sorting.agda","new_file":"prefect-bintree-sorting.agda","new_contents":"module prefect-bintree-sorting where\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; _\u2238_; module \u2115\u00b0; module \u2115\u2264; +-\u2264-inj; \u00acn+\u2264y<n; sucx\u2270x; \u00acn\u2264x<n; module \u2115cmp) renaming (module <= to \u2115<=; _<=_ to _\u2115<=_)\nopen import Data.Bool.NP hiding (to\u2115)\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse; shallow-\u03b7)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning; [_])\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen import prefect-bintree\nopen import Data.Nat.Properties\nopen Nat using (z\u2264n; s\u2264s; \u2264-pred; 2*_; 2*\u2032_; 2*-spec; 2*\u2032-spec; suc-injective) renaming (_\u2264_ to _\u2115\u2264_; _<_ to _\u2115<_)\n\nmodule MM where\n open Nat using (_\u2264_; _<_)\n split-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\n split-\u2264 {zero} {zero} p = inj\u2081 \u2261.refl\n split-\u2264 {zero} {suc y} p = inj\u2082 (s\u2264s z\u2264n)\n split-\u2264 {suc x} {zero} ()\n split-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n ... | inj\u2081 q rewrite q = inj\u2081 \u2261.refl\n ... | inj\u2082 q = inj\u2082 (s\u2264s q)\n\n <\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n <\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\n Monotone : \u2115 \u2192 Endo \u2115 \u2192 Set\n Monotone ub f = \u2200 {x y} \u2192 x \u2264 y \u2192 y < ub \u2192 f x \u2264 f y\n\n IsInj : \u2115 \u2192 Endo \u2115 \u2192 Set\n IsInj ub f = \u2200 {x y} \u2192 x < ub \u2192 y < ub \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n Bounded : \u2115 \u2192 Endo \u2115 \u2192 Set\n Bounded ub f = \u2200 x \u2192 x < ub \u2192 f x < ub\n\n module M (f : \u2115 \u2192 \u2115) {ub}\n          (f-mono : Monotone ub f)\n          (f-inj : IsInj ub f)\n          (f-bounded : Bounded ub f) where\n\n  f-mono-< : \u2200 {x y} \u2192 x < y \u2192 y < ub \u2192 f x < f y\n  f-mono-< {x} {y} p y<ub with split-\u2264 (f-mono {x} {y} (<\u2192\u2264 p) y<ub)\n  ... | inj\u2081 q = \u22a5-elim (sucx\u2270x y (\u2261.subst (\u03bb z \u2192 suc z \u2115\u2264 y) (f-inj {x} {y} (<-trans p y<ub) y<ub q) p))\n  ... | inj\u2082 q = q\n\n  -- f-mono-<\u2032 : \u2200 {x} \u2192 suc x < ub \u2192 f x < f (suc x)\n\n  le : \u2200 n \u2192 suc n < ub \u2192 f (suc n) \u2264 n \u2192 f n < n\n  le n 1+n<ub p = \u2115\u2264.trans (f-mono-< {n} {suc n} \u2115\u2264.refl 1+n<ub) p\n\n  bo : \u2200 b \u2192 b < ub \u2192 \u00ac(f b < b)\n  bo zero _ ()\n  bo (suc b) b<ub (s\u2264s p) = bo b (\u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) b<ub) (le b b<ub p)\n\n  fp : \u2200 b \u2192 b < ub \u2192 Bounded (suc b) f \u2192 f b \u2261 b\n  fp b b<ub bub with split-\u2264 (bub b \u2115\u2264.refl)\n  ... | inj\u2081 p = suc-injective p\n  ... | inj\u2082 p = \u22a5-elim (bo b b<ub (\u2264-pred p))\n\n  ob : \u2200 b \u2192 b \u2264 ub \u2192 Bounded b f \u2192 \u2200 x \u2192 x < b \u2192 f x \u2261 x\n  ob zero b\u2264ub bub _ ()\n  ob (suc b) b\u2264ub bub x pf with split-\u2264 pf\n  ... | inj\u2081 p rewrite suc-injective p = fp b b\u2264ub bub\n  ... | inj\u2082 (s\u2264s p) = ob b (<\u2192\u2264 b\u2264ub) ((\u03bb y y<b \u2192 \u2115\u2264.trans (f-mono-< y<b b\u2264ub) (\u2115\u2264.reflexive (fp b b\u2264ub bub)))) x p\n\n  is-id : \u2200 x \u2192 x < ub \u2192 f x \u2261 x\n  is-id = ob ub \u2115\u2264.refl f-bounded\n\nopen BitsSorting\nopen OperationSyntax\nopen BijSpec\nopen EvalTree\nopen Alternative-Reverse\nopen SPP\n\nmkBijT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nmkBijT = bij \u2218 sort-bij\n\nmkBijT\u207b\u00b9 : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nmkBijT\u207b\u00b9 f = mkBijT f \u207b\u00b9\n\nmkBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij n\nmkBij f = mkBijT (fromFun f) \u207b\u00b9\n\nmkBij-p : \u2200 {n} (t : Tree (Bits n) n) \u2192 t \u2261 evalTree (mkBijT t) (sort t)\nmkBij-p = proof \u2218 sort-bij\n\nIsInj : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set\nIsInj f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nidT : \u2200 {n} \u2192 Tree (Bits n) n\nidT = fromFun id\n\ntoFun-idT : \u2200 {n} \u2192 toFun idT \u2257 id {A = Bits n}\ntoFun-idT x = toFun\u2218fromFun id x\n\nInjT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Set\nInjT = IsInj \u2218 toFun\n\n_\u207b\u00b9-inverseTree : \u2200 {n} f \u2192 evalTree (f `\u204f f \u207b\u00b9) \u2257 id {A = Tree (Bits n) n}\n(f \u207b\u00b9-inverseTree) t = evalTree (f `\u204f f \u207b\u00b9) t\n                     \u2261\u27e8 \u2261.sym (fromFun\u2218toFun (evalTree (f `\u204f f \u207b\u00b9) t)) \u27e9\n                       fromFun (toFun (evalTree (f `\u204f f \u207b\u00b9) t))\n                     \u2261\u27e8 fromFun-\u2257 (evalTree-eval\u2032 (f `\u204f f \u207b\u00b9) t) \u27e9\n                       fromFun (toFun t \u2218 eval (f \u207b\u00b9 \u207b\u00b9 `\u204f f \u207b\u00b9))\n                     \u2261\u27e8 fromFun-\u2257 (\u03bb x \u2192 \u2261.cong (toFun t) (VecBijKit._\u207b\u00b9-inverse\u2032 _ (f \u207b\u00b9) x)) \u27e9\n                       fromFun (toFun t)\n                     \u2261\u27e8 fromFun\u2218toFun t \u27e9\n                       t\n                     \u220e where open \u2261-Reasoning\n\nmkBij-p\u207b\u00b9 : \u2200 {n} (t : Tree (Bits n) n) \u2192 evalTree (mkBijT\u207b\u00b9 t) t \u2261 sort t\nmkBij-p\u207b\u00b9 t = \u2261.trans (\u2261.cong (evalTree (mkBijT\u207b\u00b9 t)) (mkBij-p t)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t))\n\nfoo : \u2200 {n} (t : Tree (Bits n) n) \u2192 toFun t \u2257 toFun (sort t) \u2218 eval (mkBijT\u207b\u00b9 t)\nfoo t xs = toFun t xs\n         \u2261\u27e8 \u2261.cong (\u03bb t \u2192 toFun t xs) (mkBij-p t) \u27e9\n           toFun (evalTree (mkBijT t) (sort t)) xs\n         \u2261\u27e8 evalTree-eval (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t)) xs \u27e9\n           toFun (evalTree (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t))) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.refl \u27e9\n           toFun (evalTree (mkBijT t `\u204f mkBijT\u207b\u00b9 t) (sort t)) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.cong (\u03bb u \u2192 toFun u (eval (mkBijT\u207b\u00b9 t) xs)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t)) \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u220e where open \u2261-Reasoning\n\n\nmodule SortedMonotonicFunctionsBits {n} where\n    open SortedMonotonicFunctions (_\u2264_ {n}) isPreorder public\nopen SortedMonotonicFunctionsBits hiding (Sorted)\n\n-- _\u2218-inj_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {f : B \u2192 C} {g : A \u2192 B} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\n_\u2218-inj_ : \u2200 {a b c} {f : Bits b \u2192 Bits c} {g : Bits a \u2192 Bits b} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\nf-inj \u2218-inj g-inj = g-inj \u2218 f-inj\n\n_\u2237-inj : \u2200 {n} b \u2192 IsInj (_\u2237_ {n = n} b)\n(_ \u2237-inj) \u2261.refl = \u2261.refl\n\nlemvec : \u2200 {n} (xs ys : Bits (suc n)) \u2192 head xs \u2261 head ys \u2192 tail xs \u2261 tail ys \u2192 xs \u2261 ys\nlemvec (x \u2237 xs) (.x \u2237 .xs) \u2261.refl \u2261.refl = \u2261.refl\n\n\u2264\u2192\u2264\u1d2e : \u2200 {n} {x y : Bits n} \u2192 x \u2264 y \u2192 x \u2264\u1d2e y\n\u2264\u2192\u2264\u1d2e {x = []}         {[]}         p = []\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {true \u2237 ys}  p = there 1b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} (\u2115<=.complete {to\u2115 xs} {to\u2115 ys} (+-\u2264-inj (2^ n) (\u2115<=.sound _ _ p))))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {false \u2237 ys} p = there 0b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} p)\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {false \u2237 ys} p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115<=.sound (2^ n + to\u2115 xs) (to\u2115 ys) p))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {true \u2237 ys}  p = 0-1 xs ys\n\n\u2264\u1d2e\u2192\u2264 : \u2200 {n} {x y : Bits n} \u2192 x \u2264\u1d2e y \u2192 x \u2264 y\n\u2264\u1d2e\u2192\u2264 [] = _\n\u2264\u1d2e\u2192\u2264 {suc n} (there true p) = \u2115<=.complete (\u2115\u2264.refl {2^ n} +-mono (\u2115<=.sound _ _ (\u2264\u1d2e\u2192\u2264 p)))\n\u2264\u1d2e\u2192\u2264 (there false p) = \u2264\u1d2e\u2192\u2264 p\n\u2264\u1d2e\u2192\u2264 (0-1 p q) = \u2115<=.complete (to\u2115\u22642\u207f+ p {to\u2115 q})\n\nMonotone' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set _\nMonotone' {i} {o} f = \u2200 {p q : Bits i} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2e f q\n\ntoEndo\u2115 : \u2200 {n} \u2192 Endo (Bits n) \u2192 Endo \u2115\ntoEndo\u2115 f = to\u2115 \u2218 f \u2218 from\u2115\n\nopen Nat using (_\u2270_)\n\nmodule ToEndo\u2115 {n} (f : Endo (Bits n)) where\n    f\u2115 = toEndo\u2115 f\n\n    from\u2115n = from\u2115 {n}\n\n    mono : Monotone f \u2192 MM.Monotone (2^ n) f\u2115\n    mono f-mono {x} {y} x\u2264y y<2\u207f\n       = \u2115<=.sound _ _\n          (f-mono\n            (\u2264\u2192\u2264\u1d2e (\u2115<=.complete {to\u2115 (from\u2115n x)} {to\u2115 (from\u2115n y)}\n                     (\u2115\u2264.trans (\u2115\u2264.reflexive (to\u2115\u2218from\u2115 {n} x (\u2115\u2264.trans (s\u2264s x\u2264y) y<2\u207f)))\n                        (\u2115\u2264.trans x\u2264y (\u2115\u2264.reflexive (\u2261.sym (to\u2115\u2218from\u2115 {n} y y<2\u207f))))))))\n    \n    inj : IsInj f \u2192 MM.IsInj (2^ n) f\u2115\n    inj f-inj {x} {y} x< y< = from\u2115-inj x< y< \u2218 f-inj \u2218 to\u2115-inj _ _\n\n    bounded : MM.Bounded (2^ n) f\u2115\n    bounded x x\u22642\u207f = to\u2115-bound (f (from\u2115 x))\n\nlem : \u2200 {n} (f : Endo (Bits n)) \u2192 Monotone f \u2192 IsInj f \u2192 f \u2257 id\nlem {n} f f-mono f-inj x = to\u2115-inj _ _ (\u2261.trans (\u2261.cong (to\u2115 \u2218 f) (\u2261.sym (from\u2115\u2218to\u2115 x))) (MM.M.is-id (toEndo\u2115 f) {2^ n} (mono f-mono) (inj f-inj) bounded (to\u2115 x) (to\u2115-bound x)))\n  where open ToEndo\u2115 f\n\nIsInj-toFunFromFun : \u2200 {n} (f : Endo (Bits n)) \u2192 IsInj f \u2192 IsInj (toFun (fromFun f))\nIsInj-toFunFromFun f f-inj {x} {y} eq\n  rewrite toFun\u2218fromFun f x\n        | toFun\u2218fromFun f y\n        = f-inj eq\n\nsort-spec\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 Sorted (evalTree (mkBijT\u207b\u00b9 t) t)\nsort-spec\u2032 t rewrite mkBij-p\u207b\u00b9 t = sort-spec t\n\nsort-id\u2032\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted t \u2192 toFun t \u2257 id\nsort-id\u2032\u2032 t Pt st x = lem (toFun t) (DataSorted\u2192Sorted st) Pt x\n\ntoFun-eval-inj : \u2200 {n} (f : Bij n) \u2192 IsInj (eval f)\ntoFun-eval-inj f {x} {y} eq = x\n                            \u2261\u27e8 \u2261.sym ((f \u207b\u00b9-inverse) x) \u27e9\n                              eval (f \u207b\u00b9) (eval f x)\n                            \u2261\u27e8 \u2261.cong (eval (f \u207b\u00b9)) eq \u27e9\n                              eval (f \u207b\u00b9) (eval f y)\n                            \u2261\u27e8 (f \u207b\u00b9-inverse) y \u27e9\n                              y\n                            \u220e where open \u2261-Reasoning\n\ntoFun-evalTree-inj : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 InjT (evalTree f t)\ntoFun-evalTree-inj f t Pt {x} {y} eq\n  rewrite evalTree-eval\u2032 f t x\n        | evalTree-eval\u2032 f t y = toFun-eval-inj (f \u207b\u00b9) (Pt eq)\n\nsort-id\u2032 : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted (evalTree f t) \u2192 toFun (evalTree f t) \u2257 id\nsort-id\u2032 f t Pt = sort-id\u2032\u2032 (evalTree f t) (\u03bb {x} {y} eq \u2192 toFun-evalTree-inj f t Pt eq)\n\nsort-id : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun (sort t) \u2257 id\nsort-id t Pt rewrite \u2261.sym (mkBij-p\u207b\u00b9 t) = sort-id\u2032 (mkBijT\u207b\u00b9 t) t Pt (sort-spec\u2032 t)\n\nbar : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun t \u2257 eval (mkBijT\u207b\u00b9 t)\nbar t Pt xs = toFun t xs\n         \u2261\u27e8 foo t xs \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 sort-id t Pt (eval (mkBijT\u207b\u00b9 t) xs) \u27e9\n           eval (mkBijT\u207b\u00b9 t) xs\n         \u220e where open \u2261-Reasoning\n\nthm : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) \u2192\n        eval (mkBij f) \u2257 f\nthm f f-inj xs = eval (mkBij f) xs\n               \u2261\u27e8 \u2261.sym (bar (fromFun f) (IsInj-toFunFromFun f f-inj) xs) \u27e9\n                 toFun (fromFun f) xs\n               \u2261\u27e8 toFun\u2218fromFun f xs \u27e9\n                 f xs\n               \u220e where open \u2261-Reasoning\n\nthm# : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) (g : Bits n \u2192 Bit) \u2192\n       #\u27e8 g \u2218 f \u27e9 \u2261 #\u27e8 g \u27e9\nthm# f f-inj g = #\u27e8 g \u2218 f \u27e9\n               \u2261\u27e8 #-\u2257 (g \u2218 f) (g \u2218 eval (mkBij f)) (\u03bb x \u2192 \u2261.cong g (\u2261.sym (thm f f-inj x))) \u27e9\n                 #\u27e8 g \u2218 eval (mkBij f) \u27e9\n               \u2261\u27e8 #-bij (mkBij f) g \u27e9\n                 #\u27e8 g \u27e9\n               \u220e where open \u2261-Reasoning\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module prefect-bintree-sorting where\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; _\u2238_; module \u2115\u00b0; module \u2115\u2264; +-\u2264-inj; \u00acn+\u2264y<n; sucx\u2270x; \u00acn\u2264x<n; module \u2115cmp) renaming (module <= to \u2115<=; _<=_ to _\u2115<=_)\nopen import Data.Bool.NP hiding (to\u2115)\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse; shallow-\u03b7)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning; [_])\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen import prefect-bintree\nopen import Data.Nat.Properties\nopen Nat using (z\u2264n; s\u2264s; \u2264-pred; 2*_; 2*\u2032_; 2*-spec; 2*\u2032-spec; suc-injective) renaming (_\u2264_ to _\u2115\u2264_; _<_ to _\u2115<_)\n\nmodule MM where\n open Nat using (_\u2264_; _<_)\n split-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\n split-\u2264 {zero} {zero} p = inj\u2081 \u2261.refl\n split-\u2264 {zero} {suc y} p = inj\u2082 (s\u2264s z\u2264n)\n split-\u2264 {suc x} {zero} ()\n split-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n ... | inj\u2081 q rewrite q = inj\u2081 \u2261.refl\n ... | inj\u2082 q = inj\u2082 (s\u2264s q)\n\n <\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n <\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\n Monotone : \u2115 \u2192 Endo \u2115 \u2192 Set\n Monotone ub f = \u2200 {x y} \u2192 x \u2264 y \u2192 y < ub \u2192 f x \u2264 f y\n\n IsInj : \u2115 \u2192 Endo \u2115 \u2192 Set\n IsInj ub f = \u2200 {x y} \u2192 x < ub \u2192 y < ub \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n Bounded : \u2115 \u2192 Endo \u2115 \u2192 Set\n Bounded ub f = \u2200 x \u2192 x < ub \u2192 f x < ub\n\n module M (f : \u2115 \u2192 \u2115) {ub}\n          (f-mono : Monotone ub f)\n          (f-inj : IsInj ub f)\n          (f-bounded : Bounded ub f) where\n\n  f-mono-< : \u2200 {x y} \u2192 x < y \u2192 y < ub \u2192 f x < f y\n  f-mono-< {x} {y} p y<ub with split-\u2264 (f-mono {x} {y} (<\u2192\u2264 p) y<ub)\n  ... | inj\u2081 q = \u22a5-elim (sucx\u2270x y (\u2261.subst (\u03bb z \u2192 suc z \u2115\u2264 y) (f-inj {x} {y} (<-trans p y<ub) y<ub q) p))\n  ... | inj\u2082 q = q\n\n  -- f-mono-<\u2032 : \u2200 {x} \u2192 suc x < ub \u2192 f x < f (suc x)\n\n  le : \u2200 n \u2192 suc n < ub \u2192 f (suc n) \u2264 n \u2192 f n < n\n  le n 1+n<ub p = \u2115\u2264.trans (f-mono-< {n} {suc n} \u2115\u2264.refl 1+n<ub) p\n\n  bo : \u2200 b \u2192 b < ub \u2192 \u00ac(f b < b)\n  bo zero _ ()\n  bo (suc b) b<ub (s\u2264s p) = bo b (\u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) b<ub) (le b b<ub p)\n\n  fp : \u2200 b \u2192 b < ub \u2192 Bounded (suc b) f \u2192 f b \u2261 b\n  fp b b<ub bub with split-\u2264 (bub b \u2115\u2264.refl)\n  ... | inj\u2081 p = suc-injective p\n  ... | inj\u2082 p = \u22a5-elim (bo b b<ub (\u2264-pred p))\n\n  ob : \u2200 b \u2192 b \u2264 ub \u2192 Bounded b f \u2192 \u2200 x \u2192 x < b \u2192 f x \u2261 x\n  ob zero b\u2264ub bub _ ()\n  ob (suc b) b\u2264ub bub x pf with split-\u2264 pf\n  ... | inj\u2081 p rewrite suc-injective p = fp b b\u2264ub bub\n  ... | inj\u2082 (s\u2264s p) = ob b (<\u2192\u2264 b\u2264ub) ((\u03bb y y<b \u2192 \u2115\u2264.trans (f-mono-< y<b b\u2264ub) (\u2115\u2264.reflexive (fp b b\u2264ub bub)))) x p\n\n  is-id : \u2200 x \u2192 x < ub \u2192 f x \u2261 x\n  is-id = ob ub \u2115\u2264.refl f-bounded\n\nopen BitsSorting\nopen OperationSyntax\nopen PermTreeProof\nopen EvalTree\nopen Alternative-Reverse\n\nmkBijT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nmkBijT = Perm.perm SPP.sort-proof\n\nmkBijT\u207b\u00b9 : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij n\nmkBijT\u207b\u00b9 f = mkBijT f \u207b\u00b9\n\nmkBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij n\nmkBij f = mkBijT (fromFun f) \u207b\u00b9\n\nmkBij-p : \u2200 {n} (t : Tree (Bits n) n) \u2192 t \u2261 evalTree (mkBijT t) (sort t)\nmkBij-p t = Perm.proof SPP.sort-proof t\n\nIsInj : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set\nIsInj f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nidT : \u2200 {n} \u2192 Tree (Bits n) n\nidT = fromFun id\n\ntoFun-idT : \u2200 {n} \u2192 toFun idT \u2257 id {A = Bits n}\ntoFun-idT x = toFun\u2218fromFun id x\n\nInjT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Set\nInjT = IsInj \u2218 toFun\n\n_\u207b\u00b9-inverseTree : \u2200 {n} f \u2192 evalTree (f `\u204f f \u207b\u00b9) \u2257 id {A = Tree (Bits n) n}\n(f \u207b\u00b9-inverseTree) t = evalTree (f `\u204f f \u207b\u00b9) t\n                     \u2261\u27e8 \u2261.sym (fromFun\u2218toFun (evalTree (f `\u204f f \u207b\u00b9) t)) \u27e9\n                       fromFun (toFun (evalTree (f `\u204f f \u207b\u00b9) t))\n                     \u2261\u27e8 fromFun-\u2257 (evalTree-eval\u2032 (f `\u204f f \u207b\u00b9) t) \u27e9\n                       fromFun (toFun t \u2218 eval (f \u207b\u00b9 \u207b\u00b9 `\u204f f \u207b\u00b9))\n                     \u2261\u27e8 fromFun-\u2257 (\u03bb x \u2192 \u2261.cong (toFun t) (VecBijKit._\u207b\u00b9-inverse\u2032 _ (f \u207b\u00b9) x)) \u27e9\n                       fromFun (toFun t)\n                     \u2261\u27e8 fromFun\u2218toFun t \u27e9\n                       t\n                     \u220e where open \u2261-Reasoning\n\nmkBij-p\u207b\u00b9 : \u2200 {n} (t : Tree (Bits n) n) \u2192 evalTree (mkBijT\u207b\u00b9 t) t \u2261 sort t\nmkBij-p\u207b\u00b9 t = \u2261.trans (\u2261.cong (evalTree (mkBijT\u207b\u00b9 t)) (mkBij-p t)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t))\n\nfoo : \u2200 {n} (t : Tree (Bits n) n) \u2192 toFun t \u2257 toFun (sort t) \u2218 eval (mkBijT\u207b\u00b9 t)\nfoo t xs = toFun t xs\n         \u2261\u27e8 \u2261.cong (\u03bb t \u2192 toFun t xs) (Perm.proof SPP.sort-proof t) \u27e9\n           toFun (evalTree (mkBijT t) (sort t)) xs\n         \u2261\u27e8 evalTree-eval (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t)) xs \u27e9\n           toFun (evalTree (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t))) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.refl \u27e9\n           toFun (evalTree (mkBijT t `\u204f mkBijT\u207b\u00b9 t) (sort t)) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.cong (\u03bb u \u2192 toFun u (eval (mkBijT\u207b\u00b9 t) xs)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t)) \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u220e where open \u2261-Reasoning\n\n\nmodule SortedMonotonicFunctionsBits {n} where\n    open SortedMonotonicFunctions (_\u2264_ {n}) isPreorder public\nopen SortedMonotonicFunctionsBits hiding (Sorted)\n\n-- _\u2218-inj_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {f : B \u2192 C} {g : A \u2192 B} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\n_\u2218-inj_ : \u2200 {a b c} {f : Bits b \u2192 Bits c} {g : Bits a \u2192 Bits b} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\nf-inj \u2218-inj g-inj = g-inj \u2218 f-inj\n\n_\u2237-inj : \u2200 {n} b \u2192 IsInj (_\u2237_ {n = n} b)\n(_ \u2237-inj) \u2261.refl = \u2261.refl\n\nlemvec : \u2200 {n} (xs ys : Bits (suc n)) \u2192 head xs \u2261 head ys \u2192 tail xs \u2261 tail ys \u2192 xs \u2261 ys\nlemvec (x \u2237 xs) (.x \u2237 .xs) \u2261.refl \u2261.refl = \u2261.refl\n\n\u2264\u2192\u2264\u1d2e : \u2200 {n} {x y : Bits n} \u2192 x \u2264 y \u2192 x \u2264\u1d2e y\n\u2264\u2192\u2264\u1d2e {x = []}         {[]}         p = []\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {true \u2237 ys}  p = there 1b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} (\u2115<=.complete {to\u2115 xs} {to\u2115 ys} (+-\u2264-inj (2^ n) (\u2115<=.sound _ _ p))))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {false \u2237 ys} p = there 0b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} p)\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {false \u2237 ys} p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115<=.sound (2^ n + to\u2115 xs) (to\u2115 ys) p))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {true \u2237 ys}  p = 0-1 xs ys\n\n\u2264\u1d2e\u2192\u2264 : \u2200 {n} {x y : Bits n} \u2192 x \u2264\u1d2e y \u2192 x \u2264 y\n\u2264\u1d2e\u2192\u2264 [] = _\n\u2264\u1d2e\u2192\u2264 {suc n} (there true p) = \u2115<=.complete (\u2115\u2264.refl {2^ n} +-mono (\u2115<=.sound _ _ (\u2264\u1d2e\u2192\u2264 p)))\n\u2264\u1d2e\u2192\u2264 (there false p) = \u2264\u1d2e\u2192\u2264 p\n\u2264\u1d2e\u2192\u2264 (0-1 p q) = \u2115<=.complete (to\u2115\u22642\u207f+ p {to\u2115 q})\n\nMonotone' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set _\nMonotone' {i} {o} f = \u2200 {p q : Bits i} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2e f q\n\ntoEndo\u2115 : \u2200 {n} \u2192 Endo (Bits n) \u2192 Endo \u2115\ntoEndo\u2115 f = to\u2115 \u2218 f \u2218 from\u2115\n\nopen Nat using (_\u2270_)\n\nmodule ToEndo\u2115 {n} (f : Endo (Bits n)) where\n    f\u2115 = toEndo\u2115 f\n\n    from\u2115n = from\u2115 {n}\n\n    mono : Monotone f \u2192 MM.Monotone (2^ n) f\u2115\n    mono f-mono {x} {y} x\u2264y y<2\u207f\n       = \u2115<=.sound _ _\n          (f-mono\n            (\u2264\u2192\u2264\u1d2e (\u2115<=.complete {to\u2115 (from\u2115n x)} {to\u2115 (from\u2115n y)}\n                     (\u2115\u2264.trans (\u2115\u2264.reflexive (to\u2115\u2218from\u2115 {n} x (\u2115\u2264.trans (s\u2264s x\u2264y) y<2\u207f)))\n                        (\u2115\u2264.trans x\u2264y (\u2115\u2264.reflexive (\u2261.sym (to\u2115\u2218from\u2115 {n} y y<2\u207f))))))))\n    \n    inj : IsInj f \u2192 MM.IsInj (2^ n) f\u2115\n    inj f-inj {x} {y} x< y< = from\u2115-inj x< y< \u2218 f-inj \u2218 to\u2115-inj _ _\n\n    bounded : MM.Bounded (2^ n) f\u2115\n    bounded x x\u22642\u207f = to\u2115-bound (f (from\u2115 x))\n\nlem : \u2200 {n} (f : Endo (Bits n)) \u2192 Monotone f \u2192 IsInj f \u2192 f \u2257 id\nlem {n} f f-mono f-inj x = to\u2115-inj _ _ (\u2261.trans (\u2261.cong (to\u2115 \u2218 f) (\u2261.sym (from\u2115\u2218to\u2115 x))) (MM.M.is-id (toEndo\u2115 f) {2^ n} (mono f-mono) (inj f-inj) bounded (to\u2115 x) (to\u2115-bound x)))\n  where open ToEndo\u2115 f\n\nIsInj-toFunFromFun : \u2200 {n} (f : Endo (Bits n)) \u2192 IsInj f \u2192 IsInj (toFun (fromFun f))\nIsInj-toFunFromFun f f-inj {x} {y} eq\n  rewrite toFun\u2218fromFun f x\n        | toFun\u2218fromFun f y\n        = f-inj eq\n\nsort-spec\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 Sorted (evalTree (mkBijT\u207b\u00b9 t) t)\nsort-spec\u2032 t rewrite mkBij-p\u207b\u00b9 t = sort-spec t\n\nsort-id\u2032\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted t \u2192 toFun t \u2257 id\nsort-id\u2032\u2032 t Pt st x = lem (toFun t) (DataSorted\u2192Sorted st) Pt x\n\ntoFun-eval-inj : \u2200 {n} (f : Bij n) \u2192 IsInj (eval f)\ntoFun-eval-inj f {x} {y} eq = x\n                            \u2261\u27e8 \u2261.sym ((f \u207b\u00b9-inverse) x) \u27e9\n                              eval (f \u207b\u00b9) (eval f x)\n                            \u2261\u27e8 \u2261.cong (eval (f \u207b\u00b9)) eq \u27e9\n                              eval (f \u207b\u00b9) (eval f y)\n                            \u2261\u27e8 (f \u207b\u00b9-inverse) y \u27e9\n                              y\n                            \u220e where open \u2261-Reasoning\n\ntoFun-evalTree-inj : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 InjT (evalTree f t)\ntoFun-evalTree-inj f t Pt {x} {y} eq\n  rewrite evalTree-eval\u2032 f t x\n        | evalTree-eval\u2032 f t y = toFun-eval-inj (f \u207b\u00b9) (Pt eq)\n\nsort-id\u2032 : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted (evalTree f t) \u2192 toFun (evalTree f t) \u2257 id\nsort-id\u2032 f t Pt = sort-id\u2032\u2032 (evalTree f t) (\u03bb {x} {y} eq \u2192 toFun-evalTree-inj f t Pt eq)\n\nsort-id : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun (sort t) \u2257 id\nsort-id t Pt rewrite \u2261.sym (mkBij-p\u207b\u00b9 t) = sort-id\u2032 (mkBijT\u207b\u00b9 t) t Pt (sort-spec\u2032 t)\n\nbar : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun t \u2257 eval (mkBijT\u207b\u00b9 t)\nbar t Pt xs = toFun t xs\n         \u2261\u27e8 foo t xs \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 sort-id t Pt (eval (mkBijT\u207b\u00b9 t) xs) \u27e9\n           eval (mkBijT\u207b\u00b9 t) xs\n         \u220e where open \u2261-Reasoning\n\nthm : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) \u2192\n        eval (mkBij f) \u2257 f\nthm f f-inj xs = eval (mkBij f) xs\n               \u2261\u27e8 \u2261.sym (bar (fromFun f) (IsInj-toFunFromFun f f-inj) xs) \u27e9\n                 toFun (fromFun f) xs\n               \u2261\u27e8 toFun\u2218fromFun f xs \u27e9\n                 f xs\n               \u220e where open \u2261-Reasoning\n\nthm# : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) (g : Bits n \u2192 Bit) \u2192\n       #\u27e8 g \u2218 f \u27e9 \u2261 #\u27e8 g \u27e9\nthm# f f-inj g = #\u27e8 g \u2218 f \u27e9\n               \u2261\u27e8 #-\u2257 (g \u2218 f) (g \u2218 eval (mkBij f)) (\u03bb x \u2192 \u2261.cong g (\u2261.sym (thm f f-inj x))) \u27e9\n                 #\u27e8 g \u2218 eval (mkBij f) \u27e9\n               \u2261\u27e8 #-bij (mkBij f) g \u27e9\n                 #\u27e8 g \u27e9\n               \u220e where open \u2261-Reasoning\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"55aea3833c696bde6e43897f5335a51272c43b4f","subject":"inline type function Soundness-of-weakening at its 1 call site","message":"inline type function Soundness-of-weakening at its 1 call site\n\nOld-commit-hash: bfe5b5057816935a44cd3465f3669d62cd210dca\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/WeakenSound.agda","new_file":"Parametric\/Denotation\/WeakenSound.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.WeakenSound\n  {Base : Type.Structure}\n  (Const : Term.Structure Base)\n  {\u27e6_\u27e7Base : Value.Structure Base}\n  (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n  -- importing modules couldn't resolve constraints\n  -- unless Const is explicit.\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\nopen import Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 lift subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound = lift-sound\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\nweaken-sound-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n  \u27e6 weakenAll \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\nweaken-sound-terms \u2205 \u03c1 = refl\nweaken-sound-terms (t \u2022 terms) \u03c1 =\n  cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-sound-terms terms \u03c1)\n\nweaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weakenVar-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\nweaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n  cong \u27e6 c \u27e7Const (weaken-sound-terms args \u03c1)\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.WeakenSound\n  {Base : Type.Structure}\n  (Const : Term.Structure Base)\n  {\u27e6_\u27e7Base : Value.Structure Base}\n  (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n  -- importing modules couldn't resolve constraints\n  -- unless Const is explicit.\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Denotation.Notation\nopen import Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 lift subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound = lift-sound\n\nSoundness-of-weakening : \u2200 {\u0393\u2081 \u0393\u2082 : Context} {\u03c4}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192 Set\nSoundness-of-weakening t \u03c1 \u0393\u2081\u227c\u0393\u2082 =\n  \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 Soundness-of-weakening t \u03c1 \u0393\u2081\u227c\u0393\u2082\n\nweaken-sound-terms : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n  (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n  \u27e6 weakenAll \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\nweaken-sound-terms \u2205 \u03c1 = refl\nweaken-sound-terms (t \u2022 terms) \u03c1 =\n  cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-sound-terms terms \u03c1)\n\nweaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weakenVar-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\nweaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\nweaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n  cong \u27e6 c \u27e7Const (weaken-sound-terms args \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b2234b277277edade3894a57df3c1340b30ce026","subject":"Simplified two proofs.","message":"Simplified two proofs.\n\nIgnore-this: d8fd5428625a1fa253d896b6e6b7f1a0\n\ndarcs-hash:20110805172720-3bd4e-b3562ca0bdde5411ed84b7d8effec24c86d5a044.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Base\/PropertiesI.agda","new_file":"src\/FOTC\/Base\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC terms properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Base.PropertiesI where\n\nopen import FOTC.Base\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nsuccInjective : \u2200 {d e} \u2192 succ d \u2261 succ e \u2192 d \u2261 e\nsuccInjective {d} {e} h =\n  begin\n    d              \u2261\u27e8 sym (pred-S d) \u27e9\n    pred (succ d)  \u2261\u27e8 cong pred h \u27e9\n    pred (succ e)  \u2261\u27e8 pred-S e \u27e9\n    e\n  \u220e\n\n\u2237-injective : \u2200 {x y xs ys} \u2192 x \u2237 xs \u2261 y \u2237 ys \u2192 x \u2261 y \u2227 xs \u2261 ys\n\u2237-injective {x} {y} {xs} {ys} h = x\u2261y , xs\u2261ys\n  where\n  x\u2261y : x \u2261 y\n  x\u2261y =\n    begin\n      x              \u2261\u27e8 sym (head-\u2237 x xs) \u27e9\n      head (x \u2237 xs)  \u2261\u27e8 cong head h \u27e9\n      head (y \u2237 ys)  \u2261\u27e8 head-\u2237 y ys \u27e9\n      y\n    \u220e\n\n  xs\u2261ys : xs \u2261 ys\n  xs\u2261ys =\n    begin\n      xs             \u2261\u27e8 sym (tail-\u2237 x xs) \u27e9\n      tail (x \u2237 xs)  \u2261\u27e8 cong tail h \u27e9\n      tail (y \u2237 ys)  \u2261\u27e8 tail-\u2237 y ys \u27e9\n      ys\n    \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC terms properties\n------------------------------------------------------------------------------\n\nmodule FOTC.Base.PropertiesI where\n\nopen import FOTC.Base\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nsuccInjective : \u2200 {d e} \u2192 succ d \u2261 succ e \u2192 d \u2261 e\nsuccInjective {d} {e} Sd\u2261Se =\n  begin\n    d              \u2261\u27e8 sym (pred-S d) \u27e9\n    pred (succ d)  \u2261\u27e8 subst (\u03bb t \u2192 pred (succ d) \u2261 pred t)\n                            Sd\u2261Se\n                            refl\n                   \u27e9\n    pred (succ e)  \u2261\u27e8 pred-S e \u27e9\n    e\n  \u220e\n\n\u2237-injective : \u2200 {x y xs ys} \u2192 x \u2237 xs \u2261 y \u2237 ys \u2192 x \u2261 y \u2227 xs \u2261 ys\n\u2237-injective {x} {y} {xs} {ys} x\u2237xs\u2261y\u2237ys = x\u2261y , xs\u2261ys\n  where\n  x\u2261y : x \u2261 y\n  x\u2261y =\n    begin\n      x              \u2261\u27e8 sym (head-\u2237 x xs) \u27e9\n      head (x \u2237 xs)  \u2261\u27e8 subst (\u03bb t \u2192 head (x \u2237 xs) \u2261 head t)\n                              x\u2237xs\u2261y\u2237ys\n                              refl\n                     \u27e9\n      head (y \u2237 ys)  \u2261\u27e8 head-\u2237 y ys \u27e9\n      y\n    \u220e\n\n  xs\u2261ys : xs \u2261 ys\n  xs\u2261ys =\n    begin\n      xs             \u2261\u27e8 sym (tail-\u2237 x xs) \u27e9\n      tail (x \u2237 xs)  \u2261\u27e8 subst (\u03bb t \u2192 tail (x \u2237 xs) \u2261 tail t)\n                              x\u2237xs\u2261y\u2237ys\n                              refl\n                     \u27e9\n      tail (y \u2237 ys)  \u2261\u27e8 tail-\u2237 y ys \u27e9\n      ys\n    \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"13ef6fc79c73003cf82b179fbf73127b0e3772b8","subject":"Drop non-CBPV caching semantics & update comments","message":"Drop non-CBPV caching semantics & update comments\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_file":"Parametric\/Denotation\/CachingEvaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Defining a caching semantics for Term proves to be hard, requiring to\n  -- insert and remove caches where we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Caching evaluation\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\n\nmodule Parametric.Denotation.CachingEvaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Structure where\n  {-\n  -- Prototype here the type-correctness of a simple non-standard semantics.\n  -- This describes a simplified version of the transformation by Liu and\n  -- Tanenbaum, PEPM 1995 - but for now, instead of producing object language\n  -- terms, we produce host language terms to take advantage of the richer type\n  -- system of the host language (in particular, here we need the unit type,\n  -- product types and *existentials*).\n  --\n  -- As usual, we'll later switch to a real term transformation.\n  -}\n  open import Data.Product hiding (map)\n  open import Data.Sum hiding (map)\n  open import Data.Unit\n  open import Level\n\n  -- Type semantics for this scenario.\n  \u27e6_\u27e7TypeHidCache : (\u03c4 : Type) \u2192 Set\u2081\n  \u27e6 base \u03b9 \u27e7TypeHidCache = Lift \u27e6 base \u03b9 \u27e7\n  \u27e6 \u03c3 \u21d2 \u03c4 \u27e7TypeHidCache = \u27e6 \u03c3 \u27e7TypeHidCache \u2192 (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7TypeHidCache \u00d7 \u03c4\u2081 )\n\n\n  -- Now, let us define a caching semantics for terms.\n\n  -- This proves to be hard, because we need to insert and remove caches where\n  -- we apply constants.\n\n  -- Indeed, our plugin interface is not satisfactory for adding caching. CBPV can help us.\n\n  --\n  -- Inserting and removing caches --\n  --\n\n  -- Implementation note: The mutual recursion looks like a fold on exponentials, where you need to define the function and the inverse at the same time.\n  -- Indeed, both functions seem structurally recursive on \u03c4.\n  dropCache : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7TypeHidCache \u2192 \u27e6 \u03c4 \u27e7\n  extend : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n\n  dropCache {base \u03b9} v = lower v\n  dropCache {\u03c3 \u21d2 \u03c4} v x = dropCache (proj\u2081 (proj\u2082 (v (extend x))))\n\n  extend {base \u03b9} v = lift v\n  extend {\u03c3 \u21d2 \u03c4} v = \u03bb x \u2192 , (extend (v (dropCache x)) , tt)\n\n  \u27e6_\u27e7CtxHidCache : (\u0393 : Context) \u2192 Set\u2081\n  \u27e6_\u27e7CtxHidCache = DependentList \u27e6_\u27e7TypeHidCache\n\n  -- It's questionable that this is not polymorphic enough.\n  \u27e6_\u27e7VarHidCache : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6 this \u27e7VarHidCache (v \u2022 \u03c1) = v\n  \u27e6 that x \u27e7VarHidCache (v \u2022 \u03c1) = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- OK, this version is syntax-directed, luckily enough, except on primitives\n  -- (as expected). This reveals a bug of ours on higher-order primitives.\n  --\n  -- Moreover, we can somewhat safely assume that each call to extend and to\n  -- dropCache is bad: then we see that the handling of constants is bad. That's\n  -- correct, because constants will not return intermediate results in this\n  -- schema :-(.\n  \u27e6_\u27e7TermCache : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7CtxHidCache \u2192 \u27e6 \u03c4 \u27e7TypeHidCache\n  \u27e6_\u27e7TermCache (const c args) \u03c1 = extend (\u27e6 const c args \u27e7 (map dropCache \u03c1))\n  \u27e6_\u27e7TermCache (var x) \u03c1 = \u27e6 x \u27e7VarHidCache \u03c1\n\n  -- It seems odd (a probable bug?) that the result of t needn't be stripped of\n  -- its cache.\n  \u27e6_\u27e7TermCache (app s t) \u03c1 = proj\u2081 (proj\u2082 ((\u27e6 s \u27e7TermCache \u03c1) (\u27e6 t \u27e7TermCache \u03c1)))\n\n  -- Provide an empty cache!\n  \u27e6_\u27e7TermCache (abs t) \u03c1 x = , (\u27e6 t \u27e7TermCache (x \u2022 \u03c1) , tt)\n\n  open import Parametric.Syntax.MType as MType\n  open MType.Structure Base\n  open import Parametric.Syntax.MTerm as MTerm\n  open MTerm.Structure Const\n\n  open import Function hiding (const)\n\n  \u27e6_\u27e7ValType : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompType : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValType = \u27e6 c \u27e7CompType\n  \u27e6 B \u03b9 \u27e7ValType = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValType = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u00d7 \u27e6 \u03c4\u2082 \u27e7ValType\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValType = \u27e6 \u03c4\u2081 \u27e7ValType \u228e \u27e6 \u03c4\u2082 \u27e7ValType\n\n  \u27e6 F \u03c4 \u27e7CompType = \u27e6 \u03c4 \u27e7ValType\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompType = \u27e6 \u03c3 \u27e7ValType \u2192 \u27e6 \u03c4 \u27e7CompType\n\n  -- XXX: below we need to override just a few cases. Inheritance would handle\n  -- this precisely; without inheritance, we might want to use one of the\n  -- standard encodings of related features (delegation?).\n\n  \u27e6_\u27e7ValTypeHidCacheWrong : (\u03c4 : ValType) \u2192 Set\u2081\n  \u27e6_\u27e7CompTypeHidCacheWrong : (\u03c4 : CompType) \u2192 Set\u2081\n\n  -- This line is the only change, up to now, for the caching semantics starting from CBPV.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCacheWrong = (\u03a3[ \u03c4\u2081 \u2208 Set ] \u27e6 \u03c4 \u27e7ValTypeHidCacheWrong \u00d7 \u03c4\u2081 )\n  -- Delegation upward.\n  \u27e6 \u03c4 \u27e7CompTypeHidCacheWrong = Lift \u27e6 \u03c4 \u27e7CompType\n  \u27e6_\u27e7ValTypeHidCacheWrong = Lift \u2218 \u27e6_\u27e7ValType\n  -- The above does not override what happens in recursive occurrences.\n\n  {-# NO_TERMINATION_CHECK #-}\n  \u27e6_\u27e7ValTypeHidCache : (\u03c4 : ValType) \u2192 Set\n  \u27e6_\u27e7CompTypeHidCache : (\u03c4 : CompType) \u2192 Set\n\n  \u27e6 U c \u27e7ValTypeHidCache = \u27e6 c \u27e7CompTypeHidCache\n  \u27e6 B \u03b9 \u27e7ValTypeHidCache = \u27e6 base \u03b9 \u27e7\n  \u27e6 vUnit \u27e7ValTypeHidCache = \u22a4\n  \u27e6 \u03c4\u2081 v\u00d7 \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n  \u27e6 \u03c4\u2081 v+ \u03c4\u2082 \u27e7ValTypeHidCache = \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache \u228e \u27e6 \u03c4\u2082 \u27e7ValTypeHidCache\n\n  -- This line is the only change, up to now, for the caching semantics.\n  --\n  -- XXX The termination checker isn't happy with it, and it may be right \u2500 if\n  -- you keep substituting \u03c4\u2081 = U (F \u03c4), you can make the cache arbitrarily big.\n  -- I think we don't do that unless we are caching a non-terminating\n  -- computation to begin with, but I'm not entirely sure.\n  --\n  -- However, the termination checker can't prove that the function is\n  -- terminating because it's not structurally recursive, but one call of the\n  -- function will produce another call of the function stuck on a neutral term:\n  -- So the computation will have terminated, just in an unusual way!\n  --\n  -- Anyway, I need not mechanize this part of the proof for my goals.\n  \u27e6 F \u03c4 \u27e7CompTypeHidCache = (\u03a3[ \u03c4\u2081 \u2208 ValType ] \u27e6 \u03c4 \u27e7ValTypeHidCache \u00d7 \u27e6 \u03c4\u2081 \u27e7ValTypeHidCache )\n  \u27e6 \u03c3 \u21db \u03c4 \u27e7CompTypeHidCache = \u27e6 \u03c3 \u27e7ValTypeHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n\n  \u27e6_\u27e7ValCtxHidCache : (\u0393 : ValContext) \u2192 Set\n  \u27e6_\u27e7ValCtxHidCache = DependentList \u27e6_\u27e7ValTypeHidCache\n\n  {-\n  \u27e6_\u27e7CompCtxHidCache : (\u0393 : CompContext) \u2192 Set\u2081\n  \u27e6_\u27e7CompCtxHidCache = DependentList \u27e6_\u27e7CompTypeHidCache\n  -}\n\n  -- The solution is to distinguish among different kinds of constants. Some are\n  -- value constructors (and thus do not return caches), while others are\n  -- computation constructors (and thus should return caches). For products, I\n  -- believe we will only use the products which are values, not computations\n  -- (XXX check CBPV paper for the name).\n  \u27e6_\u27e7CompTermCache : \u2200 {\u03c4 \u0393} \u2192 Comp \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7CompTypeHidCache\n  \u27e6_\u27e7ValTermCache : \u2200 {\u03c4 \u0393} \u2192 Val \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7ValCtxHidCache \u2192 \u27e6 \u03c4 \u27e7ValTypeHidCache\n\n  open import Base.Denotation.Environment ValType \u27e6_\u27e7ValTypeHidCache public\n    using ()\n    renaming (\u27e6_\u27e7Var to \u27e6_\u27e7ValVar)\n\n  -- This says that the environment does not contain caches... sounds wrong!\n  -- Either we add extra variables for the caches, or we store computations in\n  -- the environment (but that does not make sense), or we store caches in\n  -- values, by acting not on F but on something else (U?).\n\n  -- I suspect the plan was to use extra variables; that's annoying to model in\n  -- Agda but easier in implementations.\n\n  \u27e6 vVar   x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7ValVar \u03c1\n  \u27e6 vThunk x \u27e7ValTermCache \u03c1 = \u27e6 x \u27e7CompTermCache \u03c1\n\n  -- The real deal, finally.\n  open import UNDEFINED\n  -- XXX constants are still a slight mess because I'm abusing CBPV...\n  -- (Actually, I just forgot the difference, and believe I had too little clue\n  -- when I wrote these constructors... but some of them did make sense).\n  \u27e6_\u27e7CompTermCache (cConst c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV c args) \u03c1 = reveal UNDEFINED\n  \u27e6_\u27e7CompTermCache (cConstV2 c args) \u03c1 = reveal UNDEFINED\n\n  -- Also, where are introduction forms for pairs and sums among values? With\n  -- them, we should see that we can interpret them without adding a cache.\n\n  -- Thunks keep seeming noops.\n  \u27e6_\u27e7CompTermCache (cForce x) \u03c1 = \u27e6 x \u27e7ValTermCache \u03c1\n\n  -- Here, in an actual implementation, we would return the actual cache with\n  -- all variables.\n  --\n  -- The effect of F is a writer monad of cached values, where the monoid is\n  -- (isomorphic to) the free monoid over (\u2203 \u03c4 . \u03c4), but we push the\n  -- existentials up when pairing things!\n\n  -- That's what we're interpreting computations in. XXX maybe consider using\n  -- monads directly. But that doesn't deal with arity.\n  \u27e6_\u27e7CompTermCache (cReturn v) \u03c1 = vUnit , (\u27e6 v \u27e7ValTermCache \u03c1 , tt)\n\n  -- For this to be enough, a lambda needs to return a produce, not to forward\n  -- the underlying one (unless there are no intermediate results). The correct\n  -- requirements can maybe be enforced through a linear typing discipline.\n\n  {-\n  -- Here we'd have a problem with the original into from CBPV, because it does\n  -- not require converting expressions to the \"CBPV A-normal form\".\n\n  \u27e6_\u27e7CompTermCache (v\u2081 into v\u2082) \u03c1 =\n  -- Sequence commands and combine their caches.\n  {-\n    let (_ , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (_ , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in , (r\u2082 , (c\u2081 ,\u2032 c\u2082))\n  -}\n\n  -- The code above does not work, because we only guarantee that v\u2082 is\n  -- a computation, not that it's an F-computation - v\u2082 could also be function\n  -- type or a computation product.\n\n  -- We need a restricted CBPV, where these two possibilities are forbidden. If\n  -- you want to save something between different lambdas, you need to add an F\n  -- U to reflect that. (Double-check the papers which show how to encode arity\n  -- using CBPV, it seems that they should find the same problem --- XXX they\n  -- don't).\n\n  -- Instead, just drop the first cache (XXX WRONG).\n    \u27e6 v\u2082 \u27e7CompTermCache (proj\u2081 (proj\u2082 (\u27e6 v\u2081 \u27e7CompTermCache \u03c1)) \u2022 \u03c1)\n  -}\n\n  -- But if we alter _into_ as described above, composing the caches works!\n  -- However, we should not forget we also need to save the new intermediate\n  -- result, that is the one produced by the first part of the let.\n  \u27e6_\u27e7CompTermCache (_into_ {\u03c3} {\u03c4} v\u2081 v\u2082) \u03c1 =\n    let (\u03c4\u2081 , (r\u2081 , c\u2081)) = \u27e6 v\u2081 \u27e7CompTermCache \u03c1\n        (\u03c4\u2082 , (r\u2082 , c\u2082)) = \u27e6 v\u2082 \u27e7CompTermCache (r\u2081 \u2022 \u03c1)\n    in  (\u03c3 v\u00d7 \u03c4\u2081 v\u00d7 \u03c4\u2082) , (r\u2082 ,\u2032 ( r\u2081  ,\u2032 c\u2081 ,\u2032 c\u2082))\n\n  -- Note the compositionality and luck: we don't need to do anything at the\n  -- cReturn time, we just need the nested into to do their job, because as I\n  -- said intermediate results are a writer monad.\n  --\n  -- Q: But then, do we still need to do all the other stuff? IOW, do we still\n  -- need to forbid (\u03bb x . y <- f args; g args') and replace it with (\u03bb x . y <-\n  -- f args; z <- g args'; z)?\n  --\n  -- A: One thing we still need is using the monadic version of into for the\n  -- same reasons - which makes sense, since it has the type of monadic bind.\n  --\n  -- Maybe not: if we use a monad, which respects left and right identity, the\n  -- two above forms are equivalent. But what about associativity? We don't have\n  -- associativity with nested tuples in the middle. That's why the monad uses\n  -- lists! We can also use nested tuple, as long as in the into case we don't\n  -- do (a, b) but append a b (ahem, where?), which decomposes the first list\n  -- and prepends it to the second. To this end, we need to know the type of the\n  -- first element, or to ensure it's always a pair. Maybe we just want to reuse\n  -- an HList.\n\n  -- In abstractions, we should start collecting all variables...\n\n  -- Here, unlike in \u27e6_\u27e7TermCache, we don't need to invent an empty cache,\n  -- that's moved into the handling of cReturn. This makes *the* difference for\n  -- nested lambdas, where we don't need to create caches multiple times!\n\n  \u27e6_\u27e7CompTermCache (cAbs v) \u03c1 x = \u27e6 v \u27e7CompTermCache (x \u2022 \u03c1)\n\n  -- Here we see that we are in a sort of A-normal form, because the argument is\n  -- a value (not quite ANF though, since values can be thunks - that is,\n  -- computations which haven't been run yet, I guess. Do we have an use for\n  -- that? That allows passing lambdas as arguments directly - which is fine,\n  -- because producing a closure indeed does not have intermediate results!).\n  \u27e6_\u27e7CompTermCache (cApp t v) \u03c1 = \u27e6 t \u27e7CompTermCache \u03c1 (\u27e6 v \u27e7ValTermCache \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9c686e047573740e63520196a076309570ddfaea","subject":"Cosmetic change.","message":"Cosmetic change.\n\nIgnore-this: c0ac3d6b1eb5edebf6be7139a698a541\n\ndarcs-hash:20110919015938-3bd4e-1c475cd3e1a7d2fdb71d05033e334b1a57666e58.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddTotality.agda","new_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : \u2200 x \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","old_contents":"module Draft.FOTC.Data.Nat.AddTotality where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n-- Interactive proof using the induction principle for natural numbers.\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = indN P P0 ih Nm\n  where\n    P : D \u2192 Set\n    P = \u03bb i \u2192 N (i + n)\n\n    P0 : P zero\n    P0 = subst N (sym (+-0x n)) Nn\n\n    ih : \u2200 {i} \u2192 N i \u2192 P i \u2192 P (succ i)\n    ih {i} Ni Pi = subst N (sym (+-Sx i n)) (sN Pi)\n\n-- Combined proof using an instance of the induction principle.\nindN-instance : (x : D) \u2192\n                N (zero + x) \u2192\n                (\u2200 {n} \u2192 N n \u2192 N (n + x) \u2192 N (succ n + x)) \u2192\n                \u2200 {n} \u2192 N n \u2192 N (n + x)\nindN-instance x = indN (\u03bb i \u2192 N (i + x))\n\npostulate\n  +-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n{-# ATP prove +-N\u2081 indN-instance #-}\n\n-- Combined proof using the induction principle.\n\n-- The translation is\n-- \u2200 p. app\u2081(p,zero) \u2192\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x) \u2192 app\u2081(p,appFn(succ,x))) \u2192   -- indN\n--      (\u2200 x. app\u2081(n,x) \u2192 app\u2081(p,x))\n----------------------------------------------------------------\n-- \u2200 x y. app\u2081(n,x) \u2192 app\u2081(n,y) \u2192 app\u2081(n,appFn(appFn(+,x),y))    -- +-N\u2082\n\npostulate\n  +-N\u2082 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n-- The ATPs could not prove this postulate.\n{-# ATP prove +-N\u2082 indN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"8f9df8c8f4f2317fb947fdad28a8f3efa1133937","subject":"IDesc stratified model: GOT IT! The proof type-checks with no fancy color: the embedded IDescl is isomorph to the host IDesc. Fully stratified, on any universe you want my Lord.","message":"IDesc stratified model: GOT IT! The proof type-checks with no fancy color: the embedded IDescl is isomorph to the host IDesc. Fully stratified, on any universe you want my Lord.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\ntrans : {l : Level}{A : Set l}{x y z : A} -> x == y -> y == z -> x == z\ntrans refl refl = refl\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n\nIDescl0 : {l : Level}(I : Set l) -> Unit -> Set (suc l)\nIDescl0 I = IMu (\\_ -> descD I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set l} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\nproof-psi-phi : {l : Level}(I : Set l) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = cong (\\t -> con (lvar' , t)) \n                                                                    (proof-lift-unlift-eq i)\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\T -> con (lpi {l = suc x} , ( S , T ) )) \n                                                                           (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                           (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                           (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                                       (proof-lift-unlift-eq s)))\n                                                                                  (reflFun (\\s -> psi (phi (T s))) \n                                                                                           T \n                                                                                           hs)) \n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = cong (\\T -> con (lsigma {l = suc x} , ( S , T ) )) \n                                                                              (trans (reflFun (\u03bb s \u2192 psi (phi (T (lifter (unlift s)))))\n                                                                                              (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                              (\\s -> cong (\u03bb s \u2192 psi (phi (T (s))))\n                                                                                                          (proof-lift-unlift-eq s)))\n                                                                                     (reflFun (\\s -> psi (phi (T s))) \n                                                                                              T \n                                                                                              hs))","old_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\ndata Lifted {l : Level} (A : Set l) : Set (suc l) where\n       lifter : A \u2192 Lifted A\n\nlift : {i : Level} -> Set i -> Set (suc i) \nlift x =  Lifted x\n\nunlift : {l : Level}{A : Set l} -> Lifted A -> A\nunlift (lifter a) = a\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i j : Level}(A : Set i)(B : Set j) -> Set (max i j)\nA * B = Sigma A \\_ -> B\n\nfst : {i j : Level}{A : Set i}{B : A -> Set j} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i j : Level}{A : Set i}{B : A -> Set j} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\ndata Unit  {i : Level} : Set i where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i j : Level}(A : Set i)(B : Set j) : Set (max i j) where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {l : Level}{A : Set l}(x : A) : A -> Set l where\n  refl : x == x\n\ncong : {l m : Level}{A : Set l}{B : Set m}\n       (f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {l m n : Level}{A : Set l}{B : Set m}{C : Set n}\n        (f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n-- Intensionally extensional\npostulate \n  reflFun : {l m : Level}{A : Set l}{B : Set m}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {l : Level}(I : Set l) : Set (suc l) where\n  var : I -> IDesc I\n  const : Set l -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set l) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set l) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {l : Level}{I : Set l} -> IDesc I -> (I -> Set l) -> Set l\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {l : Level}{I : Set l}(R : I -> IDesc I)(i : I) : Set l where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {l : Level}{I : Set l}(D : IDesc I)(P : I -> Set l) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {l : Level}\n            {I : Set l}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set l)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst {l : Level} : Set l where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : {l : Level} -> Set l -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const (lift I)\ndescDChoice _ lconst = const (Set _)\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\ndescDChoice _ lsigma = sigma (Set _) (\\S -> pi (lift S) (\\s -> var Void))\n\ndescD : {l : Level}(I : Set l) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\n\nIDescl0 : {l : Level}(I : Set l) -> Unit -> Set (suc l)\nIDescl0 I = IMu (\\_ -> descD I)\n\nIDescl : {l : Level}(I : Set l) -> Set (suc l)\nIDescl I = IDescl0 I Void\n\nvarl : {l : Level}{I : Set l}(i : I) -> IDescl I\nvarl {x} i = con (lv , lifter i) \n     where lv : DescDConst {l = suc x}\n           lv = lvar\n\nconstl : {l : Level}{I : Set l}(X : Set l) -> IDescl I\nconstl {x} X = con (lc , X)\n       where lc : DescDConst {l = suc x}\n             lc = lconst\n\nprodl : {l : Level}{I : Set l}(D D' : IDescl I) -> IDescl I\nprodl {x} D D' = con (lp , (D , D'))\n      where lp : DescDConst {l = suc x}\n            lp = lprod\n\n\npil : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\npil {x} S T = con (lp , ( S , Tl))\n    where lp : DescDConst {l = suc x}\n          lp = lpi\n          Tl : Lifted S -> IDescl _\n          Tl (lifter s) = T s\n\nsigmal : {l : Level}{I : Set l}(S : Set l)(T : S -> IDescl I) -> IDescl I\nsigmal {x} S T = con (ls , ( S , Tl))\n       where ls : DescDConst {l = suc x}\n             ls = lsigma\n             Tl : Lifted S -> IDescl _\n             Tl (lifter s) = T s\n             \n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {l : Level}\n        {I : Set l}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var (unlift i)\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S (\\s -> hs (lifter s))\ncases ( lsigma , ( S , T ) ) hs = sigma S (\\s -> hs (lifter s))\n\nphi : {l : Level}{I : Set l} -> IDescl I -> IDesc I\nphi {x} {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n--********************************************\n-- From the host to the embedding\n--********************************************\n\npsi : {l : Level}{I : Set l} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {l : Level}{I : Set l} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi {x} (pi S T) = cong (pi S) \n                                  (reflFun (\\s -> phi (psi (T s)))\n                                           T\n                                           (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S)\n                                 (reflFun (\\ s -> phi (psi (T s)))\n                                          T\n                                          (\\s -> proof-phi-psi (T s)))\n\n-- From embedding to embedding\n\nproof-lift-unlift-eq : {l : Level}{A : Set l}(x : Lifted A) -> lifter (unlift x) == x\nproof-lift-unlift-eq (lifter a) = refl\n\nproof-psi-phi : {l : Level}(I : Set l) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi {x} I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-cases\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set (suc x)\n                      P ( Void , D ) = psi (phi D) == D\n                      lvar' : DescDConst {l = suc x}\n                      lvar' = lvar\n                      lpi' : DescDConst {l = suc x}\n                      lpi' = lpi\n                      lsigma' : DescDConst {l = suc x}\n                      lsigma' = lsigma\n                      proof-psi-phi-cases : (i : Unit)\n                                            (xs : desc (descD I) (IDescl0 I))\n                                            (hs : desc (box (descD I) (IDescl0 I) xs) P)\n                                            -> P (i , con xs)\n                      proof-psi-phi-cases Void (lvar , i) hs = cong (\\t -> con (lvar' , t)) \n                                                                    (proof-lift-unlift-eq i)\n                      proof-psi-phi-cases Void (lconst , x) hs = refl\n                      proof-psi-phi-cases Void (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases Void (lpi , ( S , T )) hs = cong (\\t ->  con ((lpi' , ((S , \\ s -> t s))))) \n                                                                          (reflFun (\\s -> psi (phi (T s)))\n                                                                                    T\n                                                                                    hs)\n                      proof-psi-phi-cases Void (lsigma , ( S , T )) hs = {!!} {- cong (\\t -> con (lsigma' , ( S , \\s -> t (unlift s) ))) \n                                                                              (reflFun (\\s -> psi (phi (T (lifter s)))) \n                                                                                       (\\s -> T (lifter s)) \n                                                                                       (\\s -> hs (lifter s))) -}","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"dfb4cbdbf57896ed01ec968466c352b639cabbc6","subject":"Mult: re-add composition","message":"Mult: re-add composition\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Multiplicative.agda","new_file":"Control\/Protocol\/Multiplicative.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                              ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                              ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                              ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                              ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  \u2297\u214b-dual end Q = refl\n  \u2297\u214b-dual (recv P) Q = com=\u2032 _ \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (send P) end = refl\n  \u2297\u214b-dual (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)\n  \u2297\u214b-dual (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (send Q))\n    ,inr: (\u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  fwd : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  fwd end      = end\n  fwd (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (fwd (P x))\n  fwd (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (fwd (P x))\n\n  \u214b-id = fwd\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\ndata View-\u2218-proto : (P Q R : Proto) \u2192 \u2605\u2081 where\n  sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R\n           \u2192 View-\u2218-proto (send P) (send Q) R\n  recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n           \u2192 View-\u2218-proto (recv P) Q R\n  recvR-sendR : \u2200 {MP MQ MR}ioP(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                \u2192 View-\u2218-proto (com ioP P) (recv Q) (send R)\n  recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n           \u2192 View-\u2218-proto (send P) (recv Q) (recv R)\n  endL : \u2200 Q R \u2192 View-\u2218-proto end Q R\n  sendLM : \u2200 {MP}(P : MP \u2192 Proto)R \u2192 View-\u2218-proto (send P) end R\n  recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                \u2192 View-\u2218-proto (send P) (recv Q) end\n\nview-\u2218-proto : \u2200 P Q R \u2192 View-\u2218-proto P Q R\nview-\u2218-proto end      Q        R        = endL Q R\nview-\u2218-proto (recv P) Q        R        = recvLL P Q R\nview-\u2218-proto (send P) end      R        = sendLM P R\nview-\u2218-proto (send P) (recv Q) end      = recvL-sendR P Q\nview-\u2218-proto (send P) (recv Q) (recv R) = recvRR P Q R\nview-\u2218-proto (send P) (recv Q) (send R) = recvR-sendR Out P Q R\nview-\u2218-proto (send P) (send Q) R        = sendLL P Q R\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    \u214b-\u2218 : \u2200 {P Q R} \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218 {P} {Q} {R} pq qr with view-\u2218-proto P Q R\n    \u214b-\u2218 (inl m , pq) qr | sendLL P Q R = \u214b-sendL R m (\u214b-\u2218 {P m} {send Q} {R} pq qr)\n    \u214b-\u2218 (inr m , pq) qr | sendLL P Q R = \u214b-\u2218 {send P} {Q m} pq (qr m)\n    \u214b-\u2218 pq qr           | recvLL P Q M = \u03bb m \u2192 \u214b-\u2218 {P m} (pq m) qr\n    \u214b-\u2218 pq (inl m , qr) | recvR-sendR ioP P Q R = \u214b-\u2218 {com ioP P} {Q m} {send R} (coe (\u214b-recvR (com ioP P) Q) pq m) qr\n    \u214b-\u2218 pq (inr m , qr) | recvR-sendR ioP P Q R = \u214b-sendR (com ioP P) m (\u214b-\u2218 {com ioP P} {recv Q} {R m} pq qr)\n    \u214b-\u2218 pq qr           | recvRR P Q R = \u03bb m \u2192 \u214b-\u2218 {send P} {recv Q} {R m} pq (qr m)\n    \u214b-\u2218 pq qr           | endL Q R = \u22b8-apply {Q} {R} qr pq\n    \u214b-\u2218 (m , pq) qr     | sendLM P R = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218 pq (m , qr)     | recvL-sendR P Q = \u214b-\u2218 {send P} {Q m} {end} (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                              ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                              ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                              ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                              ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  \u2297\u214b-dual end Q = refl\n  \u2297\u214b-dual (recv P) Q = com=\u2032 _ \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (send P) end = refl\n  \u2297\u214b-dual (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)\n  \u2297\u214b-dual (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (send Q))\n    ,inr: (\u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  \u214b-id : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  \u214b-id end      = end\n  \u214b-id (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (\u214b-id (P x))\n  \u214b-id (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (\u214b-id (P x))\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e4bd4f4bbdd84e651d60329e0fe6f6b692bbd1ae","subject":"t-app case of correctness-of-derive","message":"t-app case of correctness-of-derive\n\nOld-commit-hash: d99aab61c6cd13aa46660a4df8f701ce4b4fc4ec\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/A.agda","new_file":"nats\/A.agda","new_contents":"\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 (app t t\u2081) = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\n","old_contents":"-- TODO: Include proof of\n-- \"not all correctly-typed values are valid changes\"\n\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- Question: Would it not have been better if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`?\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 (app t t\u2081) = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 {!!} \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e\n  ) where open \u2261-Reasoning\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9626c21abbb031bbc774b55e0a20adc3e2f56c9a","subject":"agda : Swierstra_A_Predicate_Transformer_for_Effects","message":"agda : Swierstra_A_Predicate_Transformer_for_Effects\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0Test where\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  module \u21d3Test where\n    exv       : Expr\n    exv       = Val 3\n    exd       : Expr\n    exd       = Div (Val 3) (Val 0)\n\n    exb0      : Val 0 \u21d3 0\n    exb0      = \u21d3Base\n\n    exb3      : Val 3 \u21d3 3\n    exb3      = \u21d3Base\n\n    exs331    : Div (Val 3) (Val 3) \u21d3 1\n    exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs931    : Div (Val 9) (Val 3) \u21d3 3\n    exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n    exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n    exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n    exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n    exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  module InterpTest where\n    evv   : Free C R Nat\n    evv   = \u27e6 Val 3 \u27e7\n    evv'  : evv \u2261 Pure 3\n    evv'  = refl\n\n    evd   : Free C R Nat\n    evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n    evd'  : evd \u2261 Pure 1\n    evd'  = refl\n\n    evd0  : Free C R Nat\n    evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n    evd0' : evd0 \u2261 Step Abort (\u03bb ())\n    evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MustPTTest where\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WpPartialTest where\n    exwpp   : Expr -> Set\n    exwpp   = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp'  : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp'  = refl\n\n    wppv1   : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1   = \u21d3Base\n\n    wppd11  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd11  = \u21d3Step \u21d3Base \u21d3Base\n    wppd11' : Set\n    wppd11' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n    {- this type cannot be constructed\n    wppd10  : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n    wppd10  = {!!}\n    -}\n    wppd10' : Set\n    wppd10' = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 0))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given Expr, satisfaction of predicate means\n  - Expr evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n    which will NOT agree with \u21d3 big-step semantics since \u21d3 does not allow div-by-0\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define SafeDiv : safety criteria that we believe in\n  - prove every expression that satisfies SafeDiv satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SafeDivTest where\n    exsdv3  : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv3  = refl\n    exsdv0  : SafeDiv (Val 0) \u2261 \u22a4\n    exsdv0  = refl\n    exsdv0' : Set\n    exsdv0' = SafeDiv (Val 0)\n\n    exsd33  : SafeDiv (Div (Val 3) (Val 3))\n            \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd33  = refl\n    exsd33' : Set\n    exsd33' = SafeDiv (Div (Val 3) (Val 3))\n\n    exsd30  : SafeDiv (Div (Val 3) (Val 0))\n            \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsd30  = refl\n    exsd30' : Set\n    exsd30' = SafeDiv (Div (Val 3) (Val 0))\n\n  {-\n  Expect : any expr e for which SafeDiv e holds can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (   _ , (sdel , sder))\n                                          with \u27e6 el \u27e7       | \u27e6 er \u27e7         | correct el sdel | correct er sder\n  correct (Div  _  _) (ernz , (   _ ,    _)) | Pure _       | Pure Zero     | _               | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Pure (Succ _) | el\u21d3vl           | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Step Abort _  | _               | ()\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Step Abort _ | _             | ()              | _\n\n  {-\n  Generalize above.\n  Instead of manually defining SafeDiv,\n  define more general predicate\n  characterising the domain of a partial function using wpPartial\n  -}\n\n  -- HC: compare to 'mustPT'\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a) -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  --can show that the two semantics agree precisely on the domain of the interpreter:\n\n  -- everything in domain of interpreter\n  -- gets evaluated in agreement with big step semantics\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  -- only things interpreted in agreement with big step semantics\n  -- are those that are in domain of interpreter\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on argument expression\n\n  sound (Val x) _ = \u21d3Base\n  sound (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7          | sound el    | sound er\n  sound (Div  _  _) ()   | Pure _       | Pure Zero      | _            | _\n  sound (Div  _  _) _    | Pure nl      | Pure (Succ nr) | \u22a4'zero\u2192el\u21d3nl | \u22a4'zero\u2192er\u21d3Succnr\n    = \u21d3Step (\u22a4'zero\u2192el\u21d3nl tt) (\u22a4'zero\u2192er\u21d3Succnr tt)\n  sound (Div  _  _) ()   | Pure _       | Step Abort _   | _            | _\n  sound (Div  _  _) ()   | Step Abort _ | _              | _            | _\n\n  inDom : {v : Nat}\n       -> (e : Expr)\n       -> \u27e6 e \u27e7 == Pure v\n       -> dom \u27e6_\u27e7 e\n  inDom (Val _) _ = tt\n  inDom (Div el er) _ with \u27e6 el \u27e7       | \u27e6 er \u27e7\n  inDom (Div  _  _) ()   | Pure _       | Pure Zero\n  inDom (Div  _  _) _    | Pure _       | Pure (Succ _) = tt\n  inDom (Div  _  _) ()   | Pure _       | Step Abort _\n  inDom (Div  _  _) ()   | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_      el\n  wpPartial1 {el} {er} Diveler\u21d3_nldivSuccn\n                         with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Pure Zero\n  wpPartial1 {el} { _} h    | Pure nl      | [[[ eq ]]]    | Pure (Succ _) = aux el nl eq\n  wpPartial1 { _} { _} ()   | Pure _       | _             | Step Abort _\n  wpPartial1 { _} { _} ()   | Step Abort _ | _             | _\n\n  wpPartial2 : {el er : Expr}\n            -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er)\n            -> wpPartial \u27e6_\u27e7 _\u21d3_         er\n  wpPartial2 {el} {er} _ with \u27e6 el \u27e7       | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7       | inspect \u27e6_\u27e7 er\n  wpPartial2 { _} {er} _    | Pure _       | _             | Pure nr      | [[[ eqnr ]]] = aux er nr eqnr\n  wpPartial2 { _} { _} ()   | Pure _       | _             | Step Abort _ | _\n  wpPartial2 { _} { _} ()   | Step Abort _ | _             | _            | _\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with           \u27e6 el \u27e7\n       | inspect \u27e6_\u27e7 el\n       |           \u27e6 er \u27e7\n       | inspect \u27e6_\u27e7 er\n       | complete    el (wpPartial1 {el} {er} h)\n       | complete    er (wpPartial2 {el} {er} h)\n  complete (Div _ _) h   | Pure nl      | [[[ \u27e6el\u27e7\u2261Purenl ]]] | Pure Zero     | [[[ \u27e6er\u27e7\u2261Pure0 ]]] | _  | _\n                    -- Goal\n                    -- mustPT  (\u03bb _ _ \u2192 \u22a4' zero)  (Div el er)  (Pure nl >>= (\u03bb v1 \u2192 Pure Zero >>= _\u00f7_ v1))\n                    -- \u22a5\n                    -- Context\n                    -- h : mustPT _\u21d3_ (Div el er) (\u27e6 el \u27e7 >>= (\u03bb v1 \u2192 \u27e6 er \u27e7 >>= _\u00f7_ v1))\n    rewrite\n        \u27e6el\u27e7\u2261Purenl -- h : mustPT _\u21d3_ (Div el er)                    (\u27e6 er \u27e7 >>= _\u00f7_ nl)\n      | \u27e6er\u27e7\u2261Pure0  -- h : \u22a5\n    = magic h\n  complete (Div _ _)   _ | Pure _       | _            | Pure (Succ _) | _            | _  | _ = tt\n  complete (Div _ _)   _ | Pure _       | _            | Step Abort _  | _            | _  | ()\n  complete (Div _ _)   _ | Step Abort _ | _            | _             | _            | () | _\n\n  module CompleteTest where\n    {-\n    xxx  : mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n    xxx  = {!!}\n    -}\n    xxx' : Set\n    xxx' = mustPT _\u21d3_ (Div (Val 1) (Val 0)) (\u27e6 Val 0 \u27e7 >>= _\u00f7_ 1)\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level   hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\n------------------------------------------------------------------------------\nABSTRACT\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\nrefinement type https:\/\/en.wikipedia.org\/wiki\/Refinement_(computing)\n- type with a predicate which must hold for any element of the refined type\n- can express\n  - preconditions  when used as function args\n  - postconditions when used as return types\n\nthe problem statement is centered around \/compositionality\/\n- pure functions enable compositional reasoning\n- effectful programs require reasoning about context\n- paper provides mechanism for reasoning about effectful programs (and for program calculation)\n\nmain idea\n- represent effectful computation as a free monad\n- write an \"interpreter\" that computes a predicate transformer\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to use this refinement relation to\n    - show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- Free       : represents syntax of an effectful computation\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- - R : C -> Set :  is type family of responses for commands\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure a)      = Pure (f a)\n  map f (Step c Rc\u2192F) = Step c (\\Rc -> map f (Rc\u2192F Rc))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure a      >>= a\u2192F = a\u2192F a\n  Step c Rc\u2192F >>= a\u2192F = Step c (\\Rc -> Rc\u2192F Rc >>= a\u2192F)\n  infixr 20  _>>=_\n\n  _>>_  : forall {l l' C R} {a : Set l} {b : Set l'}\n       -> Free C R a ->       Free C R b\n       -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition (WP) semantics\n\n  associating WP semantics with imperative programs dates to\n  Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  values of type a -> Set are called \"predicate on type a\"\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\" input to output\n  -- input\n  -- - pure function 'a\u2192b : a -> b'\n  -- - postcondition 'b -> Set' (on output of a\u2192b)\n  -- output\n  -- - weakest precondition 'a -> Set' on input to a\u2192b that ensures postcondition satisfied\n  --\n  -- non-dependent version\n  -- note: definition IS reverse function composition\n  wp0 : forall {a : Set} {b : Set}\n     -> (a -> b)\n     ->      (b -> Set)\n     -> (a ->      Set)\n  wp0 a\u2192b b\u2192Set = \\a -> b\u2192Set (a\u2192b a)\n\n  module WP0 where\n\n    open import Data.Nat using (\u2115; zero; suc; _+_)\n    data isEven : \u2115 \u2192 Set where\n      even-z : isEven Nat.zero\n      even-s : {n : \u2115} \u2192 isEven n \u2192 isEven (Nat.suc (Nat.suc n))\n\n    isEvenPT : \u2115 -> Set\n    isEvenPT = wp0 (_+ 2) isEven\n\n    isEven5  : Set\n    isEven5  = isEvenPT 5\n\n    isEven5' : isEvenPT 5 \u2261 isEven 7\n    isEven5' = refl\n\n  {-\n  wp0 semantics\n  - no way to specify that output is related to input\n  - enable via making f dependent:\n  -}\n\n  -- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} {b : a -> Set l'}\n    -> ((x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (     a  ->        Set l'')\n  wp a\u2192ba a\u2192ba\u2192Set = \\a -> a\u2192ba\u2192Set a (a\u2192ba a)\n\n  -- shorthand for working with predicates and predicates transformers\n  -- every element a that satisfies P is contained in the set of things that satifies Q\n  --\n  -- implication between predicates\n  --\n  --  P a says a \u2208 P,\n  --   if the extents of P, Q : A \u2192 Set is taken to be\n  --   the subset of A satisfying predicates P, Q\n  --\n  -- the extent of P : A -> Set is the subset of terms of type A satisfying P.\n  -- \"extent\" indicates a kind of black-box view of things,\n  -- e.g.,  the \"extent\" of a function is the set of its input-output pairs.\n  --\n  -- https:\/\/en.wikipedia.org\/wiki\/Extension_%28semantics%29\n  -- the extension of a concept, idea, or sign consists of the things to which it applies\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 a -> P a -> Q a\n\n  -- refinement relation between PTs : pt1 refined by pt2\n  --\n  -- (same thing as \u2286, but one level up)\n  -- for every post condition, pt2 weakens the precondition\n  -- \"weakest\" is the right side : pt2 P\n  --\n  -- pt1 , pt2 : preconditions for satisfying a program\n  -- refinement : pt1 maybe be a stronger requirement than pt2\n  --\n  -- pt2 is a refinement because it weakens what is required by pt1\n  -- e.g., pt1      pt2\n  --      >= 20    >= 10\n  --\n  -- weakening the input assumptions gives a stronger result\n  --\n  -- contract says establish a pre and I will give you a post, i.e.,\n  -- \"stronger\" post condition\n  -- or \"weaker\" pre condition\n  --\n  -- refinement of predicate transformers\n  --\n  -- use case: relate different implementations of the \"same program\"\n  -- pt1 \u2291 pt2 : pt1 is refined by pt2\n  -- - if pt1 is understood as giving preconditions which satisfy\n  --   postcondition P for some given program f1\n  --   then his says the precondition can be \/weakened\/ and still guarantee the postcondition\n  _\u2291_ : {a : Set} {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} {b : a -> Set} {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans P\u2291Q Q\u2291R a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = Q\u2291R a\u2192ba\u2192Set a (P\u2291Q a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a)\n\n  \u2291-refl  : {a : Set} {b : a -> Set} {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl a\u2192ba\u2192Set a P_a\u2192ba\u2192Set_a = P_a\u2192ba\u2192Set_a\n\n  \u2291-eq    : {a b : Set}\n         ->   (f      g : a -> b)\n         -> wp f \u2291 wp g\n         ->     (x : a)\n         ->    f x == g x\n  \u2291-eq f _ wpf\u2291wpg a = wpf\u2291wpg (\\_ b -> f a == b) a refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b)\n         -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq a\u2192b\u2192Set a a\u2192b\u2192Set_a_b with f a | g a | eq a\n  ... | _ | _ | refl = a\u2192b\u2192Set_a_b\n\n  {-\n  use refinement relation to\n  - relate PT semantics between programs and specifications\n  - show a program satisfies its specification\n  - show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  - corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  note : for =Free C R b= programs means generating the same AST\n\n  instead, define predicate transformers that give \/interpretations\/\n  to a particular choice of command type C and response family R\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat            public using (_+_; _>_; _*_)\n                                  renaming (\u2115 to Nat; zero to Zero; suc to Succ)\n  open import Data.Nat.DivMod     using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  -- one command\n  data C : Set where\n    Abort : C -- no continuation\n\n  -- response code : the program does not execute further\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  -- responses to Abort command are empty\n  -- use emptiness to discharge continuation in 2nd arg of Step\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type 'Partial a' will either\n  - return a value of type 'a' or\n  - fail, issuing abort command\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- BIG-STEP SEMANTICS specified using inductively defined RELATION:\n  -- requires divisor to evaluate to non-zero : rules out bad results\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {n}\n         -> Val n \u21d3 n\n    \u21d3Step : forall {el er nl nr}\n         ->     el    \u21d3  nl\n         ->        er \u21d3         (Succ nr) -- divisor is non-zero\n         -> Div el er \u21d3 (nl div (Succ nr))\n\n  exb0      : Val 0 \u21d3 0\n  exb0      = \u21d3Base\n\n  exb3      : Val 3 \u21d3 3\n  exb3      = \u21d3Base\n\n  exs331    : Div (Val 3) (Val 3) \u21d3 1\n  exs331    = \u21d3Step \u21d3Base \u21d3Base\n\n  exs931    : Div (Val 9) (Val 3) \u21d3 3\n  exs931    = \u21d3Step \u21d3Base \u21d3Base\n\n  exs9d31   : Div (Val 9) (Div (Val 3) (Val 1)) \u21d3 3\n  exs9d31   = \u21d3Step \u21d3Base (\u21d3Step \u21d3Base \u21d3Base)\n\n  exsd91d31 : Div (Div (Val 9) (Val 1)) (Div (Val 3) (Val 1)) \u21d3 3\n  exsd91d31 = \u21d3Step (\u21d3Step \u21d3Base \u21d3Base) (\u21d3Step \u21d3Base \u21d3Base)\n\n  -- semantics can also be specified by an INTERPRETER\n  -- monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- div operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div el er \u27e7 =  \u27e6 el \u27e7 >>= \\v1 ->\n                   \u27e6 er \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  to relate big step semantics to monadic interpreter\n  - stdlib 'div' requires implicit proof that divisor is non-zero\n    - _\u21d3_ relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n\n  to call 'wp', must show how to transform\n  -      predicate                    P  :         b -> Set\n  - to a predicate on partial results P' : Partial b -> Set\n  done via 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by definition of 'mustPT'\n  here: rule out failure entirely\n  - so Abort case returns empty type\n  -}\n\n  -- convert post-condition for a pure function  'f : (x : a) ->          b x'\n  -- to      post-condition on partial functions 'f : (x : A) -> Partial (b x)'\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> ((x : a) -> b x -> Set)\n        ->  (x : a)\n        ->    Partial (b x)\n        ->                   Set\n  mustPT a\u2192ba\u2192Set a (Pure ba)      = a\u2192ba\u2192Set a ba -- P holds if operation produces a value\n  mustPT _        _ (Step Abort _) = \u22a5 -- if operation fails, there is no result to apply P to\n                                       -- note: could give \u22a4 here, but want to rule out failure\n                                       -- (total correctness)\n\n  module MPT where\n\n    bs\u21d3     : Expr -> Nat -> Set\n    bs\u21d3     = _\u21d3_\n\n    bs\u21d3'    : Set\n    bs\u21d3'    = Val 1 \u21d3 1\n\n    bsp\u21d3    : Expr -> Partial Nat -> Set\n    bsp\u21d3    = mustPT _\u21d3_\n\n    bsp\u21d3'   : Set\n    bsp\u21d3'   = mustPT _\u21d3_ (Val 1) (Pure 1)\n\n    bsp\u21d3'x  : mustPT _\u21d3_ (Val 1) (Pure 1)\n    bsp\u21d3'x  = \u21d3Base\n\n    bsp\u21d3''  : Set\n    bsp\u21d3''  = mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n\n    {- nothing can be constructed with this type\n    bsp\u21d3''x : mustPT _\u21d3_ (Val 1) (Step Abort (\u03bb ()))\n    bsp\u21d3''x = {!!}\n    -}\n    xxx     : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v -> Pure 3 >>= _\u00f7 v))\n    xxx     = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx'    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n    xxx'    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n    xxx''   : Set\n    xxx''   = mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> ((x : a) -> Partial (b x))\n           -> ((x : a) ->          b x -> Set)\n           -> (     a  ->                 Set)\n  wpPartial a\u2192partialBa a\u2192ba\u2192Set = wp a\u2192partialBa (mustPT a\u2192ba\u2192Set)\n\n  {-\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter via passing\n  - interpreter           : \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  module WPP where\n\n    exwpp  : Expr -> Set\n    exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n    exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n    exwpp' = refl\n\n    wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n    wppv1  = \u21d3Base\n    wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n    wppd   = \u21d3Step \u21d3Base \u21d3Base\n    wppd'  : Set\n    wppd'  = wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n\n  {-\n  wpPartial \u27e6_\u27e7 _\u21d3_ : a predicate on expressions\n  - a way to express SUCCESSFUL evaluation of an Expr with intended semantics\n\n  \u2200 exprs satisfying this predicate\n  - monadic interpreter and\n  - relational big-step specification\n  must agree on result of evaluation\n\n  for a given e : Expr, satisfaction of predicate means\n  - 'e' evaluates to a value by big-step operational semantics\n  - if interpreter returns a value at all\n    then that value agrees with big step semantics\n  - if interpreter aborts\n    then have a proof of \u22a5 (so, it does not abort)\n\n  What does this say about correctness of interpreter?\n\n  does not specify how many Expr that satisfy that characterization also satisfy wpPartial \u27e6_\u27e7 _\u21d3_\n\n  to understand wpPartial \u27e6_\u27e7 _\u21d3_ better\n  - define safety criteria that we believe in\n  - prove every expression that satisfies that criteria satisfies wpPartial \u27e6_\u27e7 _\u21d3_\n    - so SafeDiv refines wpPartial \u27e6_\u27e7 _\u21d3_\n  - if we gave \u22a4 above, then 'correct' would not say that SafeDiv rules out undefined behavior\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div el er) = (er \u21d3 Zero -> \u22a5) \u2227 SafeDiv el \u2227 SafeDiv er\n\n  module SD where\n\n    exsdv : SafeDiv (Val 3) \u2261 \u22a4\n    exsdv = refl\n    exsdd : SafeDiv (Div (Val 3) (Val 3))\n          \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsdd = refl\n\n    exsdx : SafeDiv (Div (Val 3) (Val 0))\n          \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n    exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div el er) (   _ , (sdel , sder)) with \u27e6 el \u27e7 | \u27e6 er \u27e7 | correct el sdel | correct er sder\n  correct (Div  _  _) (ernz , (   _ ,    _)) | Pure _       | Pure Zero     |     _ | er\u21d3Zvr = magic (ernz er\u21d3Zvr)\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Pure (Succ _) | el\u21d3vl | er\u21d3Svr = \u21d3Step el\u21d3vl er\u21d3Svr\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Pure _       | Step Abort _  |     _ | ()\n  correct (Div  _  _) (   _ , (   _ ,    _)) | Step Abort _ | _             | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7 | sound el | sound er\n  sound (Div el er) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div el er) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div el er) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div el er) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div el er) h with \u27e6 el \u27e7 | \u27e6 er \u27e7\n  inDom (Div el er) () | Pure v1      | Pure Zero\n  inDom (Div el er) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div el er) () | Pure _       | Step Abort _\n  inDom (Div el er) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ el\n  wpPartial1 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7\n  wpPartial1 {el} {er} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {el} {er} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux el x eq\n  wpPartial1 {el} {er} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {el} {er} () | Step Abort x | eq         | ver\n\n  wpPartial2 : {el er : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div el er) -> wpPartial \u27e6_\u27e7 _\u21d3_ er\n  wpPartial2 {el} {er} h with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n  wpPartial2 {el} {er} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux er y eqy\n  wpPartial2 {el} {er} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | ser           | eq2\n\n  complete (Val x) h = tt\n  complete (Div el er) h\n    with \u27e6 el \u27e7 | inspect \u27e6_\u27e7 el | \u27e6 er \u27e7 | inspect \u27e6_\u27e7 er\n      | complete el (wpPartial1 {el} {er} h)\n      | complete er (wpPartial2 {el} {er} h)\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div el er) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div el er) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div el er) h | Step Abort x | eq1         | ser           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and post- condition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"27a4ba88511dda3572cdb1d98a77ac8d6af2e8e1","subject":"Improve commentary on Base.Change.Context.","message":"Improve commentary on Base.Change.Context.\n\nOld-commit-hash: ad5f3d68627e46f68376b1c994a9ce3787870847\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Context.agda","new_file":"Base\/Change\/Context.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change contexts\n--\n-- This module specifies how a context of values is merged\n-- together with the corresponding context of changes.\n--\n-- In the PLDI paper, instead of merging the contexts together, we just\n-- concatenate them. For example, in the \"typing rule\" for Derive\n-- in Sec. 3.2 of the paper, we write in the conclusion of the rule:\n-- \"\u0393, \u0394\u0393 \u22a2 Derive(t) : \u0394\u03c4\". Simple concatenation is possible because\n-- the paper uses named variables and assumes that no user variables\n-- start with \"d\". In this formalization, we use de Bruijn indices, so\n-- it is easier to alternate values and their changes in the context.\n------------------------------------------------------------------------\n\nmodule Base.Change.Context\n    {Type : Set}\n    (\u0394Type : Type \u2192 Type) where\n\nopen import Base.Syntax.Context Type\n\n-- Transform a context of values into a context of values and\n-- changes.\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- like \u0394Context, but \u0394Type \u03c4 and \u03c4 are swapped\n\u0394Context\u2032 : Context \u2192 Context\n\u0394Context\u2032 \u2205 = \u2205\n\u0394Context\u2032 (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394Type \u03c4 \u2022 \u0394Context\u2032 \u0393\n\n-- This sub-context relationship explains how to go back from\n-- \u0394Context \u0393 to \u0393: You have to drop every other binding.\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change contexts\n------------------------------------------------------------------------\n\nmodule Base.Change.Context\n    {Type : Set}\n    (\u0394Type : Type \u2192 Type) where\n\nopen import Base.Syntax.Context Type\n\n-- Transform a context of values into a context of values and\n-- changes.\n\n\u0394Context : Context \u2192 Context\n\u0394Context \u2205 = \u2205\n\u0394Context (\u03c4 \u2022 \u0393) = \u0394Type \u03c4 \u2022 \u03c4 \u2022 \u0394Context \u0393\n\n-- like \u0394Context, but \u0394Type \u03c4 and \u03c4 are swapped\n\u0394Context\u2032 : Context \u2192 Context\n\u0394Context\u2032 \u2205 = \u2205\n\u0394Context\u2032 (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394Type \u03c4 \u2022 \u0394Context\u2032 \u0393\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"670c3d1576725334e48ff0039610f28d8afc13f2","subject":"Document P.C.Correctness.","message":"Document P.C.Correctness.\n\nOld-commit-hash: 688c41e30d805aa8b9ac8494f9ec53ddf0b1eb5a\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Correctness.agda","new_file":"Parametric\/Change\/Correctness.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Correctness of differentiation (Lemma 3.10 and Theorem 3.11).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\n-- Extension point: A proof that change evaluation for a fully\n-- applied primitive is related to the value of incrementalizing\n-- this primitive, if their arguments are so related.\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- We provide: A variant of Lemma 3.10 for arbitrary contexts.\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  derive-correct-closed : \u2200 {\u03c4} (t : Term \u2205 \u03c4) \u2192\n    \u27e6 t \u27e7\u0394 \u2205 \u2205 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 \u2205\n\n  derive-correct-closed t = derive-correct t \u2205 \u2205 \u2205 \u2205\n\n  -- And we proof Theorem 3.11, finally.\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {s : Term \u2205 \u03c3} {ds : Term \u2205 (\u0394Type \u03c3)} \u2192\n    {dv : \u0394\u208d \u03c3 \u208e (\u27e6 s \u27e7 \u2205)} {erasure : dv \u2248\u208d \u03c3 \u208e (\u27e6 ds \u27e7 \u2205)} \u2192\n\n    \u27e6 app f (s \u2295\u208d \u03c3 \u208e ds) \u27e7 \u2261 \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {s} {ds} {dv} {erasure} =\n    let\n      g  = \u27e6 f \u27e7 \u2205\n      \u0394g = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394g\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 s \u27e7 \u2205\n      dv\u2032 = \u27e6 ds \u27e7 \u2205\n      u  = \u27e6 s \u2295\u208d \u03c3 \u208e ds \u27e7 \u2205\n      -- \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        g u\n      \u2261\u27e8 cong g (sym (meaning-\u2295 {t = s} {\u0394t = ds})) \u27e9\n        g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 dv\u2032)\n      \u2261\u27e8 cong g (sym (carry-over {\u03c3} dv erasure)) \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v dv \u27e9\n        g v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394g v dv\n      \u2261\u27e8 carry-over {\u03c4} (call-change {\u03c3} {\u03c4} \u0394g v dv)\n           (derive-correct-closed f v dv dv\u2032 erasure) \u27e9\n        g v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394g\u2032 v dv\u2032\n      \u2261\u27e8 meaning-\u2295 {t = app f s} {\u0394t = app (app (derive f) s) ds} \u27e9\n        \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Term as ChangeTerm\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Evaluation as ChangeEvaluation\nimport Parametric.Change.Derive as Derive\nimport Parametric.Change.Implementation as Implementation\n\nmodule Parametric.Change.Correctness\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (diff-base : ChangeTerm.DiffStructure Const \u0394Base)\n    (apply-base : ChangeTerm.ApplyStructure Const \u0394Base)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (meaning-\u2295-base : ChangeEvaluation.ApplyStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (meaning-\u229d-base : ChangeEvaluation.DiffStructure\n      \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base apply-base diff-base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (derive-const : Derive.Structure Const \u0394Base)\n    (implementation-structure : Implementation.Structure\n      Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n      validity-structure specification-structure\n      \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeTerm.Structure Const \u0394Base diff-base apply-base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen ChangeEvaluation.Structure\n  \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base\n  apply-base diff-base\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\n  meaning-\u2295-base meaning-\u229d-base\nopen Derive.Structure Const \u0394Base derive-const\nopen Implementation.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const \u0394Base validity-structure specification-structure\n  \u27e6apply-base\u27e7 \u27e6diff-base\u27e7 derive-const implementation-structure\n\n-- The denotational properties of the `derive` transformation.\n-- In particular, the main theorem about it producing the correct\n-- incremental behavior.\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nStructure : Set\nStructure = \u2200 {\u03a3 \u0393 \u03c4} (c : Const \u03a3 \u03c4) (ts : Terms \u0393 \u03a3)\n  (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n  (ts-correct : implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))) \u2192\n  \u27e6 c \u27e7\u0394Const (\u27e6 ts \u27e7Terms \u03c1) (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) \u2248\u208d \u03c4 \u208e \u27e6 derive-const c (fit-terms ts) (derive-terms ts) \u27e7 (alternate \u03c1 \u03c1\u2032)\n\nmodule Structure (derive-const-correct : Structure) where\n  deriveVar-correct : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 deriveVar x \u27e7 (alternate \u03c1 \u03c1\u2032)\n  deriveVar-correct this (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = dv\u2248dv\u2032\n  deriveVar-correct (that x) (v \u2022 \u03c1) (dv \u2022 d\u03c1) (dv\u2032 \u2022 d\u03c1\u2032) (dv\u2248dv\u2032 \u2022 d\u03c1\u2248d\u03c1\u2032) = deriveVar-correct x \u03c1 d\u03c1 d\u03c1\u2032 d\u03c1\u2248d\u03c1\u2032\n\n  -- That `derive t` implements \u27e6 t \u27e7\u0394\n  derive-correct : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    \u27e6 t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)\n\n  derive-terms-correct : \u2200 {\u03a3 \u0393} (ts : Terms \u0393 \u03a3)\n    (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) (\u03c1\u2032 : \u27e6 mapContext \u0394Type \u0393 \u27e7) (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 d\u03c1 \u03c1\u2032) \u2192\n    implements-env \u03a3 (\u27e6 ts \u27e7\u0394Terms \u03c1 d\u03c1) (\u27e6 derive-terms ts \u27e7Terms (alternate \u03c1 \u03c1\u2032))\n\n  derive-terms-correct \u2205 \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 = \u2205\n  derive-terms-correct (t \u2022 ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-correct t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 \u2022 derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n\n  derive-correct (const c ts) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    derive-const-correct c ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n      (derive-terms-correct ts \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032)\n  derive-correct (var x) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    deriveVar-correct x \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n  derive-correct (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n   = subst (\u03bb \u27e6t\u27e7 \u2192 \u27e6 app s t \u27e7\u0394 \u03c1 d\u03c1 \u2248\u208d \u03c4 \u208e (\u27e6 derive s \u27e7Term (alternate \u03c1 \u03c1\u2032)) \u27e6t\u27e7 (\u27e6 derive t \u27e7Term (alternate \u03c1 \u03c1\u2032))) (\u27e6fit\u27e7 t \u03c1 \u03c1\u2032)\n       (derive-correct {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032\n          (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 derive t \u27e7 (alternate \u03c1 \u03c1\u2032)) (derive-correct {\u03c3} t \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032))\n\n  derive-correct (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 \u03c1\u2032 d\u03c1\u2248\u03c1\u2032 =\n    \u03bb w dw w\u2032 dw\u2248w\u2032 \u2192\n      derive-correct t (w \u2022 \u03c1) (dw \u2022 d\u03c1) (w\u2032 \u2022 \u03c1\u2032) (dw\u2248w\u2032 \u2022 d\u03c1\u2248\u03c1\u2032)\n\n  derive-correct-closed : \u2200 {\u03c4} (t : Term \u2205 \u03c4) \u2192\n    \u27e6 t \u27e7\u0394 \u2205 \u2205 \u2248\u208d \u03c4 \u208e \u27e6 derive t \u27e7 \u2205\n\n  derive-correct-closed t = derive-correct t \u2205 \u2205 \u2205 \u2205\n\n  main-theorem : \u2200 {\u03c3 \u03c4}\n    {f : Term \u2205 (\u03c3 \u21d2 \u03c4)} {s : Term \u2205 \u03c3} {ds : Term \u2205 (\u0394Type \u03c3)} \u2192\n    {dv : \u0394\u208d \u03c3 \u208e (\u27e6 s \u27e7 \u2205)} {erasure : dv \u2248\u208d \u03c3 \u208e (\u27e6 ds \u27e7 \u2205)} \u2192\n\n    \u27e6 app f (s \u2295\u208d \u03c3 \u208e ds) \u27e7 \u2261 \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7\n\n  main-theorem {\u03c3} {\u03c4} {f} {s} {ds} {dv} {erasure} =\n    let\n      g  = \u27e6 f \u27e7 \u2205\n      \u0394g = \u27e6 f \u27e7\u0394 \u2205 \u2205\n      \u0394g\u2032 = \u27e6 derive f \u27e7 \u2205\n      v  = \u27e6 s \u27e7 \u2205\n      dv\u2032 = \u27e6 ds \u27e7 \u2205\n      u  = \u27e6 s \u2295\u208d \u03c3 \u208e ds \u27e7 \u2205\n      -- \u0394output-term = app (app (derive f) x) (y \u229d x)\n    in\n      ext {A = \u27e6 \u2205 \u27e7Context} (\u03bb { \u2205 \u2192\n      begin\n        g u\n      \u2261\u27e8 cong g (sym (meaning-\u2295 {t = s} {\u0394t = ds})) \u27e9\n        g (v \u27e6\u2295\u208d \u03c3 \u208e\u27e7 dv\u2032)\n      \u2261\u27e8 cong g (sym (carry-over {\u03c3} dv erasure)) \u27e9\n        g (v \u229e\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 corollary-closed {\u03c3} {\u03c4} f v dv \u27e9\n        g v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394g v dv\n      \u2261\u27e8 carry-over {\u03c4} (call-change {\u03c3} {\u03c4} \u0394g v dv)\n           (derive-correct-closed f v dv dv\u2032 erasure) \u27e9\n        g v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394g\u2032 v dv\u2032\n      \u2261\u27e8 meaning-\u2295 {t = app f s} {\u0394t = app (app (derive f) s) ds} \u27e9\n        \u27e6 app f s \u2295\u208d \u03c4 \u208e app (app (derive f) s) ds \u27e7 \u2205\n      \u220e}) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"57dd15c14074878a5698f4ec39e14cf45e2b99aa","subject":"circuit","message":"circuit\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Circuit.agda","new_file":"FunUniverse\/Circuit.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Nat  using (\u2115; _+_)\nopen import Data.Bits using (Bits)\nopen import Data.Fin  using (Fin)\n\nopen import FunUniverse.Data\n\nmodule FunUniverse.Circuit where\n\ninfix 0 _\u2325_\ndata _\u2325_ : \u2115 \u2192 \u2115 \u2192 Type where\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2325 o\n    -- cost: 0 time, o space\n  _\u2218_    : \u2200 {m n o} \u2192 (n \u2325 o) \u2192 (m \u2325 n) \u2192 (m \u2325 o)\n    -- cost: sum time and space\n  <_\u00d7_>  : \u2200 {m n o p} \u2192 (m \u2325 o) \u2192 (n \u2325 p) \u2192 (m + n) \u2325 (o + p)\n    -- cost: max time and sum space\n  cond   : \u2200 {n} \u2192 (1 + (n + n)) \u2325 n\n    -- cost: 1 time, 1 space\n  bits   : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 \u2325 n\n    -- cost: 0 time, n space\n  xor    : \u2200 {n} \u2192 Bits n \u2192 n \u2325 n\n    -- cost: 1 time, n space\n\n  {- derived rewiring\n  id        : \u2200 {A} \u2192 A `\u2192 A\n  dup       : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n  fst       : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  snd       : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n  swap      : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  assoc     : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  tt        : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\ud835\udfd9\n  <[]>      : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n  <\u2237>       : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n  uncons    : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n  <tt,id>   : \u2200 {A} \u2192 A `\u2192 `\ud835\udfd9 `\u00d7 A\n  snd<tt,>  : \u2200 {A} \u2192 `\ud835\udfd9 `\u00d7 A `\u2192 A\n  tt\u2192[]     : \u2200 {A} \u2192 `\ud835\udfd9 `\u2192 `Vec A 0\n  []\u2192tt     : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\ud835\udfd9\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o   \u2192 `Bits i `\u2192 `Bits o\n  -}\n  {- derived from <_\u00d7_> and id:\n  first     : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  second    : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  -}\n  {- derived from <_\u00d7_> and dup:\n  <_,_>     : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  -}\n  {- derived from cond:\n  bijFork   : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  -}\n  {- derived from bits\n  <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n  -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Nat  using (\u2115; _+_)\nopen import Data.Bits using (Bits)\nopen import Data.Fin  using (Fin)\n\nopen import FunUniverse.Data\n\nmodule FunUniverse.Circuit where\n\ninfix 0 _\u2325_\ndata _\u2325_ : \u2115 \u2192 \u2115 \u2192 \u2605 where\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2325 o\n    -- cost: 0 time, o space\n  _\u2218_    : \u2200 {m n o} \u2192 (n \u2325 o) \u2192 (m \u2325 n) \u2192 (m \u2325 o)\n    -- cost: sum time and space\n  <_\u00d7_>  : \u2200 {m n o p} \u2192 (m \u2325 o) \u2192 (n \u2325 p) \u2192 (m + n) \u2325 (o + p)\n    -- cost: max time and sum space\n  cond   : \u2200 {n} \u2192 (1 + (n + n)) \u2325 n\n    -- cost: 1 time, 1 space\n  bits   : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 \u2325 n\n    -- cost: 0 time, n space\n  xor    : \u2200 {n} \u2192 Bits n \u2192 n \u2325 n\n    -- cost: 1 time, n space\n\n  {- derived rewiring\n  id        : \u2200 {A} \u2192 A `\u2192 A\n  dup       : \u2200 {A} \u2192 A `\u2192 A `\u00d7 A\n  fst       : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  snd       : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n  swap      : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  assoc     : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  tt        : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\ud835\udfd9\n  <[]>      : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n  <\u2237>       : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n  uncons    : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n  <tt,id>   : \u2200 {A} \u2192 A `\u2192 `\ud835\udfd9 `\u00d7 A\n  snd<tt,>  : \u2200 {A} \u2192 `\ud835\udfd9 `\u00d7 A `\u2192 A\n  tt\u2192[]     : \u2200 {A} \u2192 `\ud835\udfd9 `\u2192 `Vec A 0\n  []\u2192tt     : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\ud835\udfd9\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o   \u2192 `Bits i `\u2192 `Bits o\n  -}\n  {- derived from <_\u00d7_> and id:\n  first     : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  second    : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  -}\n  {- derived from <_\u00d7_> and dup:\n  <_,_>     : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  -}\n  {- derived from cond:\n  bijFork   : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  -}\n  {- derived from bits\n  <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n  -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"50a6d3336783aba6e726257ab367eeb8618342ff","subject":"agda:new proof of assoc","message":"agda:new proof of assoc\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat1.agda","new_file":"agda\/Nat1.agda","new_contents":"module Nat1 where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\nopen import Equality\none   = succ zero\ntwo   = succ one\nthree = succ two\n\n0-is-id : \u2200 (n : \u2115) \u2192 (n + zero) \u2261 n\n0-is-id zero     =\n  begin\n    (zero + zero) \u2248 zero by definition\n  \u220e\n0-is-id (succ y) =\n  begin\n    (succ y + zero) \u2248 succ (y + zero) by definition\n                    \u2248 succ y          by cong succ (0-is-id y)\n  \u220e\n\n+-assoc : \u2200 (a b c : \u2115) \u2192 (a + b) + c \u2261 a + (b + c)\n+-assoc zero     b c = definition\n+-assoc (succ a) b c =\n  begin ((succ a + b) + c)\n               \u2248 succ (a + b) + c by definition\n               \u2248 succ ((a + b) + c) by definition\n               \u2248 succ (a + (b + c)) by cong succ (+-assoc a b c)\n               \u2248 succ a + (b + c) by definition\n  \u220e\n{-\n+-assoc zero     b c     = definition\n+-assoc (succ a) b c     =\n  begin ((succ a + b) + c)\n               \u2248 succ (a + b)  + c   by definition\n               \u2248 succ ((a + b) + c)  by definition\n               \u2248 succ (a  + (b + c)) by cong succ (+-assoc a b c)\n               \u2248 succ a   + (b + c)  by definition\n  \u220e\n-}\n","old_contents":"module Nat1 where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\nopen import Equality\none   = succ zero\ntwo   = succ one\nthree = succ two\n\n0-is-id : \u2200 (n : \u2115) \u2192 (n + zero) \u2261 n\n0-is-id zero     =\n  begin\n    (zero + zero) \u2248 zero by definition\n  \u220e\n0-is-id (succ y) =\n  begin\n    (succ y + zero) \u2248 succ (y + zero) by definition\n                    \u2248 succ y          by cong succ (0-is-id y)\n  \u220e\n\n+-assoc : \u2200 (a b c : \u2115) \u2192 (a + b) + c \u2261 a + (b + c)\n+-assoc zero     b c     = definition\n+-assoc (succ a) b c     =\n  begin ((succ a + b) + c)\n               \u2248 succ (a + b)  + c   by definition\n               \u2248 succ ((a + b) + c)  by definition\n               \u2248 succ (a  + (b + c)) by cong succ (+-assoc a b c)\n               \u2248 succ a   + (b + c)  by definition\n  \u220e\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"2fbf9654c1cd556317dc4eaee7c81283b276d720","subject":"Fun...Fun: use the Regular version","message":"Fun...Fun: use the Regular version\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Function\/Fin.agda","new_file":"lib\/Explore\/Function\/Fin.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Function\nopen import Data.Fin using (Fin)\nopen import Level.NP\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat\nopen import Data.Product\nopen import Relation.Binary.NP\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Explorable\nimport Explore.Fin\nopen Explore.Fin.Regular\n\nmodule Explore.Function.Fin where\n\npostulate\n  Postulate-FinFun\u02e2-ok : \u2605\n\nmodule _ {A : \u2605}(\u03bcA : Explorable A) where\n\n  e\u1d2c = explore \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  FinFun\u1d49 : \u2200 n \u2192 Explore _ (Fin n \u2192 A)\n  FinFun\u1d49 zero    z op f = f \u00acFin0\n  FinFun\u1d49 (suc n) z op f = e\u1d2c z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))\n\n  FinFun\u2071 : \u2200 n \u2192 ExploreInd _ (FinFun\u1d49 n)\n  FinFun\u2071 zero    P Pz P\u2219 Pf = Pf _\n  FinFun\u2071 (suc n) P Pz P\u2219 Pf =\n    explore-ind \u03bcA (\u03bb sa \u2192 P (\u03bb z op f \u2192 sa z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))))\n      Pz P\u2219\n      (\u03bb x \u2192 FinFun\u2071 n (\u03bb sf \u2192 P (\u03bb z op f \u2192 sf z op (f \u2218 extend x)))\n        Pz P\u2219 (Pf \u2218 extend x))\n\n  FinFun\u02e2 : \u2200 n \u2192 Sum (Fin n \u2192 A)\n  FinFun\u02e2 n = FinFun\u1d49 n 0 _+_\n\n  postulate\n    FinFun\u02e2-ok : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} n \u2192 AdequateSum (FinFun\u02e2 n)\n\n  \u03bcFinFun : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} {n} \u2192 Explorable (Fin n \u2192 A)\n  \u03bcFinFun = mk _ (FinFun\u2071 _) (FinFun\u02e2-ok _)\n\nmodule BigDistr\n  {{_ : Postulate-FinFun\u02e2-ok}}\n  {A : \u2605}(\u03bcA : Explorable A)\n  (cm           : CommutativeMonoid \u2080 \u2080)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_          : let open CMon cm in C \u2192 C \u2192 C)\n  (\u03b5-\u25ce          : let open CMon cm in Zero _\u2248_ \u03b5 _\u25ce_)\n  (distrib      : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_     : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm renaming (sym to \u2248-sym)\n\n  \u03bcF\u2192A = \u03bcFinFun \u03bcA\n\n  -- Sum over A\n  \u03a3\u1d2c = explore \u03bcA \u03b5 _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin I \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = explore \u03bcF\u2192A \u03b5 _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 explore (\u03bcFin I) \u03b5 _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))\n    \u2248\u27e8 \u2248-sym (explore-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))) (proj\u2081 \u03b5-\u25ce _) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)))\n    \u2248\u27e8 explore-mon-ext \u03bcA monoid (\u03bb j \u2192 \u2248-sym (explore-lin\u02e1 \u03bcF\u2192A monoid _\u25ce_ (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)) (F zero j) (proj\u2082 \u03b5-\u25ce _) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I ((F \u2218 suc) \u02e2 f))\n    \u220e\n\nmodule _\n  {{_ : Postulate-FinFun\u02e2-ok}} where\n\n  FinDist : \u2200 {n} \u2192 DistFun (\u03bcFin n) (\u03bb \u03bcX \u2192 \u03bcFinFun \u03bcX)\n  FinDist \u03bcB c \u25ce distrib \u25ce-cong f = BigDistr.bigDistr \u03bcB c \u25ce distrib \u25ce-cong f _\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Function\nopen import Data.Fin using (Fin)\nopen import Level.NP\nopen import Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat\nopen import Data.Product\nopen import Relation.Binary.NP\n\nopen import Explore.Core\nopen import Explore.Properties\nopen import Explore.Explorable\nopen import Explore.Fin\n\nmodule Explore.Function.Fin where\n\npostulate\n  Postulate-FinFun\u02e2-ok : \u2605\n\nmodule _ {A : \u2605}(\u03bcA : Explorable A) where\n\n  e\u1d2c = explore \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  FinFun\u1d49 : \u2200 n \u2192 Explore _ (Fin n \u2192 A)\n  FinFun\u1d49 zero    z op f = f \u00acFin0\n  FinFun\u1d49 (suc n) z op f = e\u1d2c z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))\n\n  FinFun\u2071 : \u2200 n \u2192 ExploreInd _ (FinFun\u1d49 n)\n  FinFun\u2071 zero    P Pz P\u2219 Pf = Pf _\n  FinFun\u2071 (suc n) P Pz P\u2219 Pf =\n    explore-ind \u03bcA (\u03bb sa \u2192 P (\u03bb z op f \u2192 sa z op (\u03bb x \u2192 FinFun\u1d49 n z op (f \u2218 extend x))))\n      Pz P\u2219\n      (\u03bb x \u2192 FinFun\u2071 n (\u03bb sf \u2192 P (\u03bb z op f \u2192 sf z op (f \u2218 extend x)))\n        Pz P\u2219 (Pf \u2218 extend x))\n\n  FinFun\u02e2 : \u2200 n \u2192 Sum (Fin n \u2192 A)\n  FinFun\u02e2 n = FinFun\u1d49 n 0 _+_\n\n  postulate\n    FinFun\u02e2-ok : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} n \u2192 AdequateSum (FinFun\u02e2 n)\n\n  \u03bcFinFun : \u2200 {{_ : Postulate-FinFun\u02e2-ok}} {n} \u2192 Explorable (Fin n \u2192 A)\n  \u03bcFinFun = mk _ (FinFun\u2071 _) (FinFun\u02e2-ok _)\n\nmodule BigDistr\n  {{_ : Postulate-FinFun\u02e2-ok}}\n  {A : \u2605}(\u03bcA : Explorable A)\n  (cm           : CommutativeMonoid \u2080 \u2080)\n  -- we want (open CMon cm) !!!\n  (_\u25ce_          : let open CMon cm in C \u2192 C \u2192 C)\n  (\u03b5-\u25ce          : let open CMon cm in Zero _\u2248_ \u03b5 _\u25ce_)\n  (distrib      : let open CMon cm in _DistributesOver_ _\u2248_ _\u25ce_ _\u2219_)\n  (_\u25ce-cong_     : let open CMon cm in _\u25ce_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_) where\n\n  open CMon cm renaming (sym to \u2248-sym)\n\n  \u03bcF\u2192A = \u03bcFinFun \u03bcA\n\n  -- Sum over A\n  \u03a3\u1d2c = explore \u03bcA \u03b5 _\u2219_\n\n  -- Sum over (Fin(1+I)\u2192A) functions\n  \u03a3' : \u2200 {I} \u2192 ((Fin I \u2192 A) \u2192 C) \u2192 C\n  \u03a3' = explore \u03bcF\u2192A \u03b5 _\u2219_\n\n  -- Product over Fin(1+I) values\n  \u03a0' = \u03bb I \u2192 explore (\u03bcFin I) \u03b5 _\u25ce_\n\n  bigDistr : \u2200 I F \u2192 \u03a0' I (\u03a3\u1d2c \u2218 F) \u2248 \u03a3' (\u03a0' I \u2218 _\u02e2_ F)\n  bigDistr zero    _ = refl\n  bigDistr (suc I) F\n    = \u03a3\u1d2c (F zero) \u25ce \u03a0' I (\u03a3\u1d2c \u2218 F \u2218 suc)\n    \u2248\u27e8 refl \u25ce-cong bigDistr I (F \u2218 suc) \u27e9\n      \u03a3\u1d2c (F zero) \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))\n    \u2248\u27e8 \u2248-sym (explore-lin\u02b3 \u03bcA monoid _\u25ce_ (F zero) (\u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc))) (proj\u2081 \u03b5-\u25ce _) (proj\u2082 distrib)) \u27e9\n      \u03a3\u1d2c (\u03bb j \u2192 F zero j \u25ce \u03a3' (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)))\n    \u2248\u27e8 explore-mon-ext \u03bcA monoid (\u03bb j \u2192 \u2248-sym (explore-lin\u02e1 \u03bcF\u2192A monoid _\u25ce_ (\u03a0' I \u2218 _\u02e2_ (F \u2218 suc)) (F zero j) (proj\u2082 \u03b5-\u25ce _) (proj\u2081 distrib))) \u27e9\n      (\u03a3\u1d2c \u03bb j \u2192 \u03a3' \u03bb f \u2192 F zero j \u25ce \u03a0' I ((F \u2218 suc) \u02e2 f))\n    \u220e\n\nmodule _\n  {{_ : Postulate-FinFun\u02e2-ok}} where\n\n  FinDist : \u2200 {n} \u2192 DistFun (\u03bcFin n) (\u03bb \u03bcX \u2192 \u03bcFinFun \u03bcX)\n  FinDist \u03bcB c \u25ce distrib \u25ce-cong f = BigDistr.bigDistr \u03bcB c \u25ce distrib \u25ce-cong f _\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"18cdc5c3e0989baa42c6669ff2552c5f95ed8867","subject":"Document Postulate.Bag-Nehemiah (for #275).","message":"Document Postulate.Bag-Nehemiah (for #275).\n\nOld-commit-hash: f22f7ab7d307f61d4bb165003c6eb917869d49a5\n","repos":"inc-lc\/ilc-agda","old_file":"Postulate\/Bag-Nehemiah.agda","new_file":"Postulate\/Bag-Nehemiah.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Bags with negative multiplicities, for Nehemiah.\n--\n-- Instead of implementing bags (with negative multiplicities,\n-- like in the paper) in Agda, we postulate that a group of such\n-- bags exist. Note that integer bags with integer multiplicities\n-- are actually the free group given a singleton operation\n-- `Integer -> Bag`, so this should be easy to formalize in\n-- principle.\n------------------------------------------------------------------------\nmodule Postulate.Bag-Nehemiah where\n\n-- Postulates about bags of integers, version Nehemiah\n--\n-- This module postulates bags of integers with negative\n-- multiplicities as a group under additive union.\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Structures\nopen import Data.Integer\n\n-- postulate Bag as an abelion group\npostulate Bag : Set\n-- singleton\npostulate singletonBag : \u2124 \u2192 Bag\n-- union\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\n-- negate = mapMultiplicities (\u03bb z \u2192 - z)\npostulate negateBag : Bag \u2192 Bag\npostulate emptyBag : Bag\npostulate abelian-bag : IsAbelianGroup _\u2261_ _++_ emptyBag negateBag\n\n-- Naming convention follows Algebra.Morphism\n-- Homomorphic\u2081 : morphism preserves negation\nHomomorphic\u2081 :\n  {A B : Set} (f : A \u2192 B) (negA : A \u2192 A) (negB : B \u2192 B) \u2192 Set\nHomomorphic\u2081 {A} {B} f negA negB = \u2200 {x} \u2192 f (negA x) \u2261 negB (f x)\n\n-- Homomorphic\u2082 : morphism preserves binary operation.\nHomomorphic\u2082 :\n  {A B : Set} (f : A \u2192 B) (_+_ : A \u2192 A \u2192 A) (_*_ : B \u2192 B \u2192 B) \u2192 Set\nHomomorphic\u2082 {A} {B} f _+_ _*_ = \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n\n-- postulate map, flatmap and sum to be homomorphisms\npostulate mapBag : (f : \u2124 \u2192 \u2124) (b : Bag) \u2192 Bag\npostulate flatmapBag : (f : \u2124 \u2192 Bag) (b : Bag) \u2192 Bag\npostulate sumBag : Bag \u2192 \u2124\npostulate homo-map : \u2200 {f} \u2192 Homomorphic\u2082 (mapBag f) _++_ _++_\npostulate homo-flatmap : \u2200 {f} \u2192 Homomorphic\u2082 (flatmapBag f) _++_ _++_\npostulate homo-sum : Homomorphic\u2082 sumBag _++_ _+_\npostulate neg-map : \u2200 {f} \u2192 Homomorphic\u2081 (mapBag f) negateBag negateBag\npostulate neg-flatmap : \u2200 {f} \u2192 Homomorphic\u2081 (flatmapBag f) negateBag negateBag\n","old_contents":"module Postulate.Bag-Nehemiah where\n\n-- Postulates about bags of integers, version Nehemiah\n--\n-- This module postulates bags of integers with negative\n-- multiplicities as a group under additive union.\n\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.Structures\nopen import Data.Integer\n\n-- postulate Bag as an abelion group\n-- [Argue that no inconsistency can be introduced]\npostulate Bag : Set\n-- singleton\npostulate singletonBag : \u2124 \u2192 Bag\n-- union\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\n-- negate = mapMultiplicities (\u03bb z \u2192 - z)\npostulate negateBag : Bag \u2192 Bag\npostulate emptyBag : Bag\npostulate abelian-bag : IsAbelianGroup _\u2261_ _++_ emptyBag negateBag\n\n-- Naming convention follows Algebra.Morphism\n-- Homomorphic\u2081 : morphism preserves negation\nHomomorphic\u2081 :\n  {A B : Set} (f : A \u2192 B) (negA : A \u2192 A) (negB : B \u2192 B) \u2192 Set\nHomomorphic\u2081 {A} {B} f negA negB = \u2200 {x} \u2192 f (negA x) \u2261 negB (f x)\n\n-- Homomorphic\u2082 : morphism preserves binary operation.\nHomomorphic\u2082 :\n  {A B : Set} (f : A \u2192 B) (_+_ : A \u2192 A \u2192 A) (_*_ : B \u2192 B \u2192 B) \u2192 Set\nHomomorphic\u2082 {A} {B} f _+_ _*_ = \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n\n-- postulate map, flatmap and sum to be homomorphisms\npostulate mapBag : (f : \u2124 \u2192 \u2124) (b : Bag) \u2192 Bag\npostulate flatmapBag : (f : \u2124 \u2192 Bag) (b : Bag) \u2192 Bag\npostulate sumBag : Bag \u2192 \u2124\npostulate homo-map : \u2200 {f} \u2192 Homomorphic\u2082 (mapBag f) _++_ _++_\npostulate homo-flatmap : \u2200 {f} \u2192 Homomorphic\u2082 (flatmapBag f) _++_ _++_\npostulate homo-sum : Homomorphic\u2082 sumBag _++_ _+_\npostulate neg-map : \u2200 {f} \u2192 Homomorphic\u2081 (mapBag f) negateBag negateBag\npostulate neg-flatmap : \u2200 {f} \u2192 Homomorphic\u2081 (flatmapBag f) negateBag negateBag\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6ececbf2f8abde287e790622ae7faf0775a223d2","subject":"a better go with clearer holes","message":"a better go with clearer holes","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/Data1.agda","new_file":"models\/Data1.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check --no-positivity-check\n #-}\n\nmodule Data1 where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Cx : Set where\n  E : Cx\n  _\/_ : Cx -> Set -> Cx\n\ndata _:>_ : Cx -> Set -> Set where\n  ze : {G : Cx}{S : Set} -> (G \/ S) :> S\n  su : {G : Cx}{S T : Set} -> G :> T -> (G \/ S) :> T\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (G : Cx) : Set where\n  ?? : {S : Set} -> G :> S -> S -> CON G\n  PrfC : PROP -> CON G\n  _*C_ : CON G -> CON G -> CON G\n  PiC : (S : Set) -> (S -> CON G) -> CON G\n  SiC : (S : Set) -> (S -> CON G) -> CON G\n  MuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n  NuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n\nBox : (G : Cx) -> ((S : Set) -> (G :> S) -> Set) -> Set\nBox E F = One\nBox (G \/ S) F = Box G (\\ S n -> F S (su n)) * F S ze\n\npr : {G : Cx}{S : Set}{F : (S : Set) -> (G :> S) -> Set} -> Box G F -> (n : G :> S) -> F S n\npr fsf ze = snd fsf\npr fsf (su n) = pr (fst fsf) n\n\nPay : Cx -> Set\nPay G = Box G (\\ I _ -> I -> Set)\n\nArr : (G : Cx) -> Pay G -> Pay G -> Set\nArr G X Y = Box G (\\ S n -> (s : S) -> pr X n s -> pr Y n s)\n\n\n\nmutual\n  [|_|] : {G : Cx} -> CON G -> Pay G -> Set\n  [| ?? n i |]     X = pr X n i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Mu C X ) -> Mu C X o\n  codata Nu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Nu C X ) -> Nu C X o\n\nnoc :  {G : Cx}{O : Set}{C : O -> CON (G \/ O)}{X : Pay G}{o : O} ->\n       Nu C X o -> [| C o |] ( X , Nu C X )\nnoc (con xs) = xs\n\nmutual\n  map : {G : Cx}{X Y : Pay G} -> (C : CON G) ->\n        Arr G X Y ->\n        [| C |] X -> [| C |] Y\n  map (?? n i)     f = pr f n i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = foldMap C (Mu C _) f (\\ o -> con) o\n  map (NuC O C o)  f = unfoldMap C (Nu C _) f (\\ o -> noc) o\n\n  foldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n          Arr G X Y ->\n          ((o : O) -> [| C o |] ( Y , Z ) -> Z o) ->\n          (o : O) -> Mu C X o -> Z o\n  foldMap C Z f alg o (con xs) = alg o (map (C o) (f , foldMap C Z f alg) xs)\n\n  unfoldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n            Arr G X Y ->\n            ((o : O) -> Z o -> [| C o |] ( X , Z )) ->\n            (o : O) -> Z o -> Nu C Y o\n  unfoldMap C Z f coalg o y = con (map (C o) (f , unfoldMap C Z f coalg) (coalg o y))\n\nida : (G : Cx) -> (X : Pay G) -> Arr G X X\nida E = id\nida (G \/ S) = sig \\ Xs X -> ida G Xs , \\ s x -> x\n\ncompa : (G : Cx) -> (X Y Z : Pay G) -> Arr G Y Z -> Arr G X Y -> Arr G X Z\ncompa E = \\ _ _ _ _ _ -> _\ncompa (G \/ S) = sig \\ Xs X -> sig \\ Ys Y -> sig \\ Zs Z -> sig \\ fs f -> sig \\ gs g ->\n  compa G Xs Ys Zs fs gs , \\ s x -> f s (g s x)\n\ndata _==_ {X : Set}(x : X) : {Y : Set} -> Y -> Set where\n  refl : x == x\n\nsubst : {X : Set}{P : X -> Set}{x y : X} -> x == y -> P x -> P y\nsubst refl p = p\n\n_=,=_ : {S : Set}{T : S -> Set}\n        {s0 : S}{s1 : S} -> s0 == s1 ->\n        {t0 : T s0}{t1 : T s1} -> t0 == t1 ->\n        _==_ {Sig S T} (s0 , t0) {Sig S T} (s1 , t1)\nrefl =,= refl = refl\n        \n\nmylem :  (G : Cx) -> (X : Pay G) -> compa G X X X (ida G X) (ida G X) == ida G X\nmylem E X = refl\nmylem (G \/ S) X = mylem G (fst X) =,= refl\n\nclaim : (G : Cx){O : Set}(C : O -> CON (G \/ O))(X : Pay G)\n        (P : (o : O) -> Mu C X o -> Set) -> (p : (o : O)(y : Mu C X o) -> P o y)\n        (o : O) ->\n        compa (G \/ O) (X , Mu C X) (X , (\\ o -> Sig (Mu C X o) (P o))) (X , Mu C X)\n         (ida G X , konst fst) (ida G X , (\\ o y -> (y , p o y)))\n        == ida (G \/ O) (X , Mu C X)\nclaim G C X P p o = mylem G X =,= refl\n\nidLem : {G : Cx}{S : Set}{X : Pay G}(n : G :> S) ->\n        pr (ida G X) n == (\\ (s : S)(x : pr X n s) -> x)\nidLem ze = refl\nidLem (su y) = idLem y\n\n_=$_ : {S : Set}{T : S -> Set}{f g : (s : S) -> T s} ->\n        f == g -> (s : S) -> f s == g s\nrefl =$ s = refl\n\nmutual  -- and too intensional\n  mapId : {G : Cx}{X : Pay G} -> (C : CON G) -> (xs : [| C |] X) ->\n          map C (ida G X) xs == xs\n  mapId (?? y y') xs = (idLem y =$ y') =$ xs\n  mapId (PrfC y) xs = refl\n  mapId (y *C y') xs = mapId y (fst xs) =,= mapId y' (snd xs)\n  mapId (PiC S y) xs = {!!} -- no chance\n  mapId (SiC S y) xs = refl =,= mapId (y (fst xs)) (snd xs)\n  mapId (MuC O y y') xs = foldMapId y y' xs\n  mapId (NuC O y y') xs = {!!}\n\n  foldMapId : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n              (o : O)(z : Mu C X o) ->\n              foldMap C (Mu C X) (ida G X) (\\ o -> con) o z == z\n  foldMapId C o (con xzs) = {!!} -- close but no banana\n\nmapComp :  {G : Cx}{X Y Z : Pay G} (f : Arr G Y Z)(g : Arr G X Y)\n           (C : CON G) -> (xs : [| C |] X) ->\n           map C f (map C g xs) == map C (compa G X Y Z f g) xs\nmapComp = {!!}\n\n\n_-=-_ : {X : Set}{x y z : X} -> x == y -> y == z -> x == z\nrefl -=- refl = refl\n\ninduction : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] (X , (\\ o -> Sig (Mu C X o) (P o)))) ->\n             P o (con (map (C o) (ida G X , konst fst) xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction {G} C {X} P p o (con xys) =\n  subst {P = \\ xys -> P o (con xys)}{y = xys}\n    ((mapComp \n     (ida G X , konst fst) (ida G X , (\\ o y -> (y , induction C P p o y))) (C o) xys\n     -=- {!!}) \n      -=- mapId (C o) xys)\n    (p o (map (C o) (ida G X , (\\ o y -> (y , induction C P p o y))) xys))\n","old_contents":"{-# OPTIONS --type-in-type --no-termination-check --no-positivity-check\n #-}\n\nmodule Data1 where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Cx : Set where\n  E : Cx\n  _\/_ : Cx -> Set -> Cx\n\ndata _:>_ : Cx -> Set -> Set where\n  ze : {G : Cx}{S : Set} -> (G \/ S) :> S\n  su : {G : Cx}{S T : Set} -> G :> T -> (G \/ S) :> T\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (G : Cx) : Set where\n  ?? : {S : Set} -> G :> S -> S -> CON G\n  PrfC : PROP -> CON G\n  _*C_ : CON G -> CON G -> CON G\n  PiC : (S : Set) -> (S -> CON G) -> CON G\n  SiC : (S : Set) -> (S -> CON G) -> CON G\n  MuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n  NuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n\nBox : (G : Cx) -> ((S : Set) -> (G :> S) -> Set) -> Set\nBox E F = One\nBox (G \/ S) F = Box G (\\ S n -> F S (su n)) * F S ze\n\npr : {G : Cx}{S : Set}{F : (S : Set) -> (G :> S) -> Set} -> Box G F -> (n : G :> S) -> F S n\npr fsf ze = snd fsf\npr fsf (su n) = pr (fst fsf) n\n\nPay : Cx -> Set\nPay G = Box G (\\ I _ -> I -> Set)\n\nArr : (G : Cx) -> Pay G -> Pay G -> Set\nArr G X Y = Box G (\\ S n -> (s : S) -> pr X n s -> pr Y n s)\n\n\n\nmutual\n  [|_|] : {G : Cx} -> CON G -> Pay G -> Set\n  [| ?? n i |]     X = pr X n i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Mu C X ) -> Mu C X o\n  codata Nu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Nu C X ) -> Nu C X o\n\nnoc :  {G : Cx}{O : Set}{C : O -> CON (G \/ O)}{X : Pay G}{o : O} ->\n       Nu C X o -> [| C o |] ( X , Nu C X )\nnoc (con xs) = xs\n\nmutual\n  map : {G : Cx}{X Y : Pay G} -> (C : CON G) ->\n        Arr G X Y ->\n        [| C |] X -> [| C |] Y\n  map (?? n i)     f = pr f n i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = foldMap C (Mu C _) f (\\ o -> con) o\n  map (NuC O C o)  f = unfoldMap C (Nu C _) f (\\ o -> noc) o\n\n  foldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n          Arr G X Y ->\n          ((o : O) -> [| C o |] ( Y , Z ) -> Z o) ->\n          (o : O) -> Mu C X o -> Z o\n  foldMap C Z f alg o (con xs) = alg o (map (C o) (f , foldMap C Z f alg) xs)\n\n  unfoldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n            Arr G X Y ->\n            ((o : O) -> Z o -> [| C o |] ( X , Z )) ->\n            (o : O) -> Z o -> Nu C Y o\n  unfoldMap C Z f coalg o y = con (map (C o) (f , unfoldMap C Z f coalg) (coalg o y))\n\nida : (G : Cx) -> (X : Pay G) -> Arr G X X\nida E = id\nida (G \/ S) = sig \\ Xs X -> ida G Xs , \\ s x -> x\n\ncompa : (G : Cx) -> (X Y Z : Pay G) -> Arr G Y Z -> Arr G X Y -> Arr G X Z\ncompa E = \\ _ _ _ _ _ -> _\ncompa (G \/ S) = sig \\ Xs X -> sig \\ Ys Y -> sig \\ Zs Z -> sig \\ fs f -> sig \\ gs g ->\n  compa G Xs Ys Zs fs gs , \\ s x -> f s (g s x)\n\ninduction : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] (X , (\\ o -> Sig (Mu C X o) (P o)))) ->\n             P o (con (map (C o) (ida G X , konst fst) xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction {G} C {X} P p o (con xys) =\n  p o (map (C o) (ida G X , (\\ o y -> (y , induction C P p o y))) xys)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e047957caacbba3c00568183dc07c1230b362d8e","subject":"Make some arguments of correctVar explicit.","message":"Make some arguments of correctVar explicit.\n\nOld-commit-hash: 2596b6835434e84c2b902574a029b3b1644baa1b\n","repos":"inc-lc\/ilc-agda","old_file":"Denotation\/Specification\/Canon-Popl14.agda","new_file":"Denotation\/Specification\/Canon-Popl14.agda","new_contents":"module Denotation.Specification.Canon-Popl14 where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192 Change \u03c4\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to future arguments\nvalidity : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  valid {\u03c4} (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to free variables\ncorrectness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393)\n  \u2192 \u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1 \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Change \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 + \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) d\u03c1 = - \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 empty d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 ++ \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) d\u03c1 = \u27e6 x \u27e7\u0394Var d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) d\u03c1 =\n     (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n  (validity {\u03c3} t d\u03c1))\n\u27e6_\u27e7\u0394 (abs t) d\u03c1 = \u03bb v \u2192\n  \u27e6 t \u27e7\u0394 (v \u2022 d\u03c1)\n\nvalidVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : \u0394Env \u0393) \u2192 valid {\u03c4} (\u27e6 x \u27e7 (ignore d\u03c1)) (\u27e6 x \u27e7\u0394Var d\u03c1)\nvalidVar this (cons v \u0394v R[v,\u0394v] \u2022 _) = R[v,\u0394v]\nvalidVar {\u03c4} (that x) (cons _ _ _ \u2022 d\u03c1) = validVar {\u03c4} x d\u03c1\n\nvalidity (intlit n)    d\u03c1 = tt\nvalidity (add s t)     d\u03c1 = tt\nvalidity (minus t)     d\u03c1 = tt\nvalidity (empty)       d\u03c1 = tt\nvalidity (insert s t)  d\u03c1 = tt\nvalidity (union s t)   d\u03c1 = tt\nvalidity (negate t)    d\u03c1 = tt\nvalidity (flatmap s t) d\u03c1 = tt\nvalidity (sum t)       d\u03c1 = tt\n\nvalidity {\u03c4} (var x) d\u03c1 = validVar {\u03c4} x d\u03c1\n\nvalidity (app s t) d\u03c1 =\n  let\n    v = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394v = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    proj\u2081 (validity s d\u03c1 (cons v \u0394v (validity t d\u03c1)))\n\nvalidity {\u03c3 \u21d2 \u03c4} (abs t) d\u03c1 = \u03bb v \u2192\n  let\n    v\u2032 = after {\u03c3} v\n    \u0394v\u2032 = v\u2032 \u229f\u208d \u03c3 \u208e v\u2032\n    d\u03c1\u2081 = v \u2022 d\u03c1\n    d\u03c1\u2082 = nil-valid-change \u03c3 v\u2032 \u2022 d\u03c1\n  in\n    validity t d\u03c1\u2081\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2082) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2082\n    \u2261\u27e8 correctness t d\u03c1\u2082 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2081)\n    \u2261\u27e8 sym (correctness t d\u03c1\u2081) \u27e9\n      \u27e6 t \u27e7 (ignore d\u03c1\u2081) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  \u27e6 x \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar (this) (cons v dv R[v,dv] \u2022 d\u03c1) = refl\ncorrectVar (that y) (cons v dv R[v,dv] \u2022 d\u03c1) = correctVar y d\u03c1\n\ncorrectness (intlit n) d\u03c1 = right-id-int n\ncorrectness (add s t) d\u03c1 = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _+_ (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (minus t) d\u03c1 = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong -_ (correctness t d\u03c1))\n\ncorrectness empty d\u03c1 = right-id-bag emptyBag\ncorrectness (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness s d\u03c1))\n         (correctness t d\u03c1) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness (union s t) d\u03c1 = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _++_ (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (negate t) d\u03c1 = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong negateBag (correctness t d\u03c1))\n\ncorrectness (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (sum t) d\u03c1 =\n  let\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness t d\u03c1))\n\ncorrectness {\u03c4} (var x) d\u03c1 = correctVar {\u03c4} x d\u03c1\ncorrectness (app {\u03c3} {\u03c4} s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 (ignore d\u03c1)\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e \u0394f (cons u \u0394u (validity {\u03c3} t d\u03c1))\n    \u2261\u27e8 sym (proj\u2082 (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons u \u0394u (validity {\u03c3} t d\u03c1)))) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t d\u03c1 \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) d\u03c1 = ext (\u03bb v \u2192\n  let\n    d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393)\n    d\u03c1\u2032 = nil-valid-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} t d\u03c1\u2032 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {d\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 d\u03c1)\n      (\u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7 (ignore d\u03c1))\n    \u229e\u208d \u03c4 \u208e\n    (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1))\n\ncorollary {\u03c3} {\u03c4} s t {d\u03c1} = proj\u2082\n  (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons (\u27e6 t \u27e7 (ignore d\u03c1))\n     (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1)))\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  (v : ValidChange \u03c3)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} v)\n  \u2261 \u27e6 t \u27e7 \u2205 (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 \u2205 v\n\ncorollary-closed {\u03c3} {\u03c4} {t = t} v =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205\n  in\n    begin\n      f (after {\u03c3} v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} v))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205)) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} v)\n    \u2261\u27e8 proj\u2082 (validity {\u03c3 \u21d2 \u03c4} t \u2205 v) \u27e9\n      f (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u0394f v\n    \u220e where open \u2261-Reasoning\n","old_contents":"module Denotation.Specification.Canon-Popl14 where\n\n-- Denotation-as-specification for Calculus Popl14\n--\n-- Contents\n-- - \u27e6_\u27e7\u0394 : binding-time-shifted derive\n-- - Validity and correctness lemmas for \u27e6_\u27e7\u0394\n-- - `corollary`: \u27e6_\u27e7\u0394 reacts to both environment and arguments\n-- - `corollary-closed`: binding-time-shifted main theorem\n\nopen import Popl14.Syntax.Term\nopen import Popl14.Denotation.Value\nopen import Popl14.Change.Validity\nopen import Popl14.Denotation.Evaluation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Unit\nopen import Data.Product\nopen import Data.Integer\nopen import Structure.Bag.Popl14\nopen import Theorem.Groups-Popl14\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192 Change \u03c4\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to future arguments\nvalidity : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393) \u2192\n  valid {\u03c4} (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n\n-- better name is: \u27e6_\u27e7\u0394 reacts to free variables\ncorrectness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (d\u03c1 : \u0394Env \u0393)\n  \u2192 \u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1 \u2261 \u27e6 t \u27e7 (update d\u03c1)\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Change \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var d\u03c1\n\n\u27e6_\u27e7\u0394 (intlit n) d\u03c1 = + 0\n\u27e6_\u27e7\u0394 (add s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 + \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (minus t) d\u03c1 = - \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 empty d\u03c1 = emptyBag\n\u27e6_\u27e7\u0394 (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    (singletonBag (n \u229e\u208d int \u208e \u0394n) ++ (b \u229e\u208d bag \u208e \u0394b)) \u229f\u208d bag \u208e (singletonBag n ++ b)\n\u27e6_\u27e7\u0394 (union s t) d\u03c1 = \u27e6 s \u27e7\u0394 d\u03c1 ++ \u27e6 t \u27e7\u0394 d\u03c1\n\u27e6_\u27e7\u0394 (negate t) d\u03c1 = negateBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    flatmapBag (f \u229e\u208d int \u21d2 bag \u208e \u0394f) (b \u229e\u208d bag \u208e \u0394b) \u229f\u208d bag \u208e flatmapBag f b\n\u27e6_\u27e7\u0394 (sum t) d\u03c1 = sumBag (\u27e6 t \u27e7\u0394 d\u03c1)\n\u27e6_\u27e7\u0394 (var x) d\u03c1 = \u27e6 x \u27e7\u0394Var d\u03c1\n\u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) d\u03c1 =\n     (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1)\n  (validity {\u03c3} t d\u03c1))\n\u27e6_\u27e7\u0394 (abs t) d\u03c1 = \u03bb v \u2192\n  \u27e6 t \u27e7\u0394 (v \u2022 d\u03c1)\n\nvalidVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : \u0394Env \u0393) \u2192 valid {\u03c4} (\u27e6 x \u27e7 (ignore d\u03c1)) (\u27e6 x \u27e7\u0394Var d\u03c1)\nvalidVar this (cons v \u0394v R[v,\u0394v] \u2022 _) = R[v,\u0394v]\nvalidVar {\u03c4} (that x) (cons _ _ _ \u2022 d\u03c1) = validVar {\u03c4} x d\u03c1\n\nvalidity (intlit n)    d\u03c1 = tt\nvalidity (add s t)     d\u03c1 = tt\nvalidity (minus t)     d\u03c1 = tt\nvalidity (empty)       d\u03c1 = tt\nvalidity (insert s t)  d\u03c1 = tt\nvalidity (union s t)   d\u03c1 = tt\nvalidity (negate t)    d\u03c1 = tt\nvalidity (flatmap s t) d\u03c1 = tt\nvalidity (sum t)       d\u03c1 = tt\n\nvalidity {\u03c4} (var x) d\u03c1 = validVar {\u03c4} x d\u03c1\n\nvalidity (app s t) d\u03c1 =\n  let\n    v = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394v = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    proj\u2081 (validity s d\u03c1 (cons v \u0394v (validity t d\u03c1)))\n\nvalidity {\u03c3 \u21d2 \u03c4} (abs t) d\u03c1 = \u03bb v \u2192\n  let\n    v\u2032 = after {\u03c3} v\n    \u0394v\u2032 = v\u2032 \u229f\u208d \u03c3 \u208e v\u2032\n    d\u03c1\u2081 = v \u2022 d\u03c1\n    d\u03c1\u2082 = nil-valid-change \u03c3 v\u2032 \u2022 d\u03c1\n  in\n    validity t d\u03c1\u2081\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2082) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2082\n    \u2261\u27e8 correctness t d\u03c1\u2082 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2081)\n    \u2261\u27e8 sym (correctness t d\u03c1\u2081) \u27e9\n      \u27e6 t \u27e7 (ignore d\u03c1\u2081) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {d\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore d\u03c1) \u229e\u208d \u03c4 \u208e \u27e6 x \u27e7\u0394Var d\u03c1 \u2261 \u27e6 x \u27e7 (update d\u03c1)\ncorrectVar {x = this  } {cons v dv R[v,dv] \u2022 d\u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u2022 d\u03c1} = correctVar {x = y} {d\u03c1}\n\ncorrectness (intlit n) d\u03c1 = right-id-int n\ncorrectness (add s t) d\u03c1 = trans\n  (mn\u00b7pq=mp\u00b7nq\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _+_ (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (minus t) d\u03c1 = trans\n  (-m\u00b7-n=-mn {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong -_ (correctness t d\u03c1))\n\ncorrectness empty d\u03c1 = right-id-bag emptyBag\ncorrectness (insert s t) d\u03c1 =\n  let\n    n = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    n\u2032 = \u27e6 s \u27e7 (update d\u03c1)\n    b\u2032 = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394n = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      (singletonBag n ++ b) ++\n         (singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n           (b \u229e\u208d base base-bag \u208e \u0394b)) \\\\ (singletonBag n ++ b)\n    \u2261\u27e8 a++[b\\\\a]=b \u27e9\n      singletonBag (n \u229e\u208d base base-int \u208e \u0394n) ++\n        (b \u229e\u208d base base-bag \u208e \u0394b)\n    \u2261\u27e8 cong\u2082 _++_\n         (cong singletonBag (correctness s d\u03c1))\n         (correctness t d\u03c1) \u27e9\n      singletonBag n\u2032 ++ b\u2032\n    \u220e where open \u2261-Reasoning\ncorrectness (union s t) d\u03c1 = trans\n  (ab\u00b7cd=ac\u00b7bd\n    {\u27e6 s \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 s \u27e7\u0394 d\u03c1} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong\u2082 _++_ (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (negate t) d\u03c1 = trans\n  (-a\u00b7-b=-ab {\u27e6 t \u27e7 (ignore d\u03c1)} {\u27e6 t \u27e7\u0394 d\u03c1})\n  (cong negateBag (correctness t d\u03c1))\n\ncorrectness (flatmap s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in trans\n      (a++[b\\\\a]=b {flatmapBag f b}\n        {flatmapBag (f \u229e\u208d base base-int \u21d2 base base-bag \u208e \u0394f)\n                    (b \u229e\u208d base base-bag \u208e \u0394b)})\n      (cong\u2082 flatmapBag (correctness s d\u03c1) (correctness t d\u03c1))\ncorrectness (sum t) d\u03c1 =\n  let\n    b = \u27e6 t \u27e7 (ignore d\u03c1)\n    \u0394b = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    trans (sym homo-sum) (cong sumBag (correctness t d\u03c1))\n\ncorrectness {\u03c4} (var x) d\u03c1 = correctVar {\u03c4} {x = x}\ncorrectness (app {\u03c3} {\u03c4} s t) d\u03c1 =\n  let\n    f = \u27e6 s \u27e7 (ignore d\u03c1)\n    g = \u27e6 s \u27e7 (update d\u03c1)\n    u = \u27e6 t \u27e7 (ignore d\u03c1)\n    v = \u27e6 t \u27e7 (update d\u03c1)\n    \u0394f = \u27e6 s \u27e7\u0394 d\u03c1\n    \u0394u = \u27e6 t \u27e7\u0394 d\u03c1\n  in\n    begin\n      f u \u229e\u208d \u03c4 \u208e \u0394f (cons u \u0394u (validity {\u03c3} t d\u03c1))\n    \u2261\u27e8 sym (proj\u2082 (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons u \u0394u (validity {\u03c3} t d\u03c1)))) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u \u229e\u208d \u03c3 \u208e \u0394u)\n    \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t d\u03c1 \u27e9\n      g v\n    \u220e where open \u2261-Reasoning\ncorrectness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) d\u03c1 = ext (\u03bb v \u2192\n  let\n    d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393)\n    d\u03c1\u2032 = nil-valid-change \u03c3 v \u2022 d\u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore d\u03c1\u2032) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 d\u03c1\u2032\n    \u2261\u27e8 correctness {\u03c4} t d\u03c1\u2032 \u27e9\n      \u27e6 t \u27e7 (update d\u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update d\u03c1)) (v+[u-v]=u {\u03c3}) \u27e9\n      \u27e6 t \u27e7 (v \u2022 update d\u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n\n-- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\ncorollary : \u2200 {\u03c3 \u03c4 \u0393}\n  (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) {d\u03c1 : \u0394Env \u0393} \u2192\n    (\u27e6 s \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 d\u03c1)\n      (\u27e6 t \u27e7 (ignore d\u03c1) \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 d\u03c1)\n  \u2261 (\u27e6 s \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7 (ignore d\u03c1))\n    \u229e\u208d \u03c4 \u208e\n    (\u27e6 s \u27e7\u0394 d\u03c1) (cons (\u27e6 t \u27e7 (ignore d\u03c1)) (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1))\n\ncorollary {\u03c3} {\u03c4} s t {d\u03c1} = proj\u2082\n  (validity {\u03c3 \u21d2 \u03c4} s d\u03c1 (cons (\u27e6 t \u27e7 (ignore d\u03c1))\n     (\u27e6 t \u27e7\u0394 d\u03c1) (validity {\u03c3} t d\u03c1)))\n\ncorollary-closed : \u2200 {\u03c3 \u03c4} {t : Term \u2205 (\u03c3 \u21d2 \u03c4)}\n  (v : ValidChange \u03c3)\n  \u2192 \u27e6 t \u27e7 \u2205 (after {\u03c3} v)\n  \u2261 \u27e6 t \u27e7 \u2205 (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 \u2205 v\n\ncorollary-closed {\u03c3} {\u03c4} {t = t} v =\n  let\n    f  = \u27e6 t \u27e7 \u2205\n    \u0394f = \u27e6 t \u27e7\u0394 \u2205\n  in\n    begin\n      f (after {\u03c3} v)\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole (after {\u03c3} v))\n            (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205)) \u27e9\n      (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after {\u03c3} v)\n    \u2261\u27e8 proj\u2082 (validity {\u03c3 \u21d2 \u03c4} t \u2205 v) \u27e9\n      f (before {\u03c3} v) \u229e\u208d \u03c4 \u208e \u0394f v\n    \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7efcd0778cf6cb1e18356e1f869fbf2f6fe65840","subject":"Reenable code after Agda bugfix!","message":"Reenable code after Agda bugfix!\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Denotation\/Evaluation.agda","new_file":"Parametric\/Denotation\/Evaluation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n\n  instance\n    meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n    meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound \u2205 \u03c1 = refl\n  weaken-terms-sound (t \u2022 terms) \u03c1 =\n    cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Standard evaluation (Def. 3.3 and Fig. 4i)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Denotation.Evaluation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\n-- Extension Point: Evaluation of fully applied constants.\nStructure : Set\nStructure = \u2200 {\u03a3 \u03c4} \u2192 Const \u03a3 \u03c4 \u2192 \u27e6 \u03a3 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\nmodule Structure (\u27e6_\u27e7Const : Structure) where\n  \u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\n  \u27e6_\u27e7Terms : \u2200 {\u0393 \u03a3} \u2192 Terms \u0393 \u03a3 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03a3 \u27e7\n\n  -- We provide: Evaluation of arbitrary terms.\n  \u27e6 const c args \u27e7Term \u03c1 = \u27e6 c \u27e7Const (\u27e6 args \u27e7Terms \u03c1)\n  \u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n  \u27e6 app s t \u27e7Term \u03c1 = (\u27e6 s \u27e7Term \u03c1) (\u27e6 t \u27e7Term \u03c1)\n  \u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\n  -- this is what we'd like to write.\n  -- unfortunately termination checker complains.\n  --\n  --   \u27e6 terms \u27e7Terms \u03c1 = map (\u03bb t \u2192 \u27e6 t \u27e7Term \u03c1) terms\n  --\n  -- so we do explicit pattern matching instead.\n  \u27e6 \u2205 \u27e7Terms \u03c1 = \u2205\n  \u27e6 s \u2022 terms \u27e7Terms \u03c1 = \u27e6 s \u27e7Term \u03c1 \u2022 \u27e6 terms \u27e7Terms \u03c1\n\n  instance\n    meaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\n    meaningOfTerm = meaning \u27e6_\u27e7Term\n\n  weaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (t : Term \u0393\u2081 \u03c4) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken \u0393\u2081\u227c\u0393\u2082 t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03a3} {\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082}\n    (terms : Terms \u0393\u2081 \u03a3) (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192\n    \u27e6 weaken-terms \u0393\u2081\u227c\u0393\u2082 terms \u27e7Terms \u03c1 \u2261 \u27e6 terms \u27e7Terms (\u27e6 \u0393\u2081\u227c\u0393\u2082 \u27e7 \u03c1)\n\n  weaken-terms-sound = {!!}\n  -- This is irrelevant for the crash\n  -- weaken-terms-sound \u2205 \u03c1 = refl\n  -- weaken-terms-sound (t \u2022 terms) \u03c1 =\n  --   cong\u2082 _\u2022_ (weaken-sound t \u03c1) (weaken-terms-sound terms \u03c1)\n\n  -- Agda crashes here:\n  weaken-sound {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (var x) \u03c1 = weaken-var-sound \u0393\u2081\u227c\u0393\u2082 x \u03c1\n  weaken-sound (app s t) \u03c1 = weaken-sound s \u03c1 \u27e8$\u27e9 weaken-sound t \u03c1\n  weaken-sound (abs t) \u03c1 = ext (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\n  weaken-sound {\u0393\u2081} {\u0393\u2082} {\u0393\u2081\u227c\u0393\u2082 = \u0393\u2081\u227c\u0393\u2082} (const {\u03a3} {\u03c4} c args) \u03c1 =\n    cong \u27e6 c \u27e7Const (weaken-terms-sound args \u03c1)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bd5b798e4b344f70b4950ea02e6f29a2d4c07a5d","subject":"Data.Vec.N-ary: lifting properties to vectors","message":"Data.Vec.N-ary: lifting properties to vectors\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/N-ary\/NP.agda","new_file":"lib\/Data\/Vec\/N-ary\/NP.agda","new_contents":"module Data.Vec.N-ary.NP where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Vec.N-ary public renaming (curry\u207f to curry\u207f\u2032; _$\u207f_ to _$\u207f\u2032_)\nopen import Relation.Binary.PropositionalEquality\n\ncurry\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192\n         ((xs : Vec A n) \u2192 B $\u207f\u2032 xs) \u2192 \u2200\u207f n B\ncurry\u207f {zero}  f = f []\ncurry\u207f {suc n} f = \u03bb x \u2192 curry\u207f (f \u2218 _\u2237_ x)\n\n_$\u207f_ : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192 \u2200\u207f n B \u2192 ((xs : Vec A n) \u2192 B $\u207f\u2032 xs)\nf $\u207f []       = f\nf $\u207f (x \u2237 xs) = f x $\u207f xs\n\ncurry-$\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} (f : (xs : Vec A n) \u2192 B $\u207f\u2032 xs) (xs : Vec A n)\n           \u2192 curry\u207f f $\u207f xs \u2261 f xs\ncurry-$\u207f f []       = refl\ncurry-$\u207f f (x \u2237 xs) = curry-$\u207f (f \u2218 _\u2237_ x) xs\n\ncurry-$\u207f\u2032 : \u2200 {n a b} {A : Set a} {B : Set b} (f : Vec A n \u2192 B) (xs : Vec A n)\n           \u2192 curry\u207f\u2032 f $\u207f\u2032 xs \u2261 f xs\ncurry-$\u207f\u2032 f []       = refl\ncurry-$\u207f\u2032 f (x \u2237 xs) = curry-$\u207f\u2032 (f \u2218 _\u2237_ x) xs\n\nconst\u207f : \u2200 n {a b} {A : Set a} {B : Set b} \u2192 B \u2192 N-ary n A B\nconst\u207f zero x = x\nconst\u207f (suc n) x = const (const\u207f n x)\n\nlift\u2081 : \u2200 {a b} {A : Set a} {B : Set b} (f g : A \u2192 B) \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 \u2200 {n} (xs : Vec A n) \u2192 map f xs \u2261 map g xs\nlift\u2081 f g pf [] = refl\nlift\u2081 f g pf (x \u2237 xs) rewrite lift\u2081 f g pf xs | pf x = refl\n\n-- move it\ntranspose : \u2200 {m n a} {A : Set a} \u2192 Vec (Vec A m) n \u2192 Vec (Vec A n) m\ntranspose [] = replicate []\ntranspose (xs \u2237 xss) = zipWith _\u2237_ xs (transpose xss)\n\nvap' : \u2200 {m a b} {A : Set a} {B : Set b} (f : Vec A m \u2192 B)\n       \u2192 \u2200 {n} \u2192 Vec (Vec A m) n \u2192 Vec B n\nvap' f [] = []\nvap' f (xs \u2237 xs\u2081) = f xs \u2237 vap' f xs\u2081\n\nvap'' : \u2200 {m a b} {A : Set a} {B : Set b} (f : Vec A m \u2192 B)\n        \u2192 \u2200 {n} \u2192 Vec (Vec A n) m \u2192 Vec B n\nvap'' f = vap' f \u2218 transpose\n\nvap : \u2200 {m a b} {A : Set a} {B : Set b} (f : N-ary m A B)\n       \u2192 \u2200 {n} \u2192 N-ary m (Vec A n) (Vec B n)\nvap {m} f = curry\u207f\u2032 {m} (\u03bb xs \u2192 vap'' (_$\u207f\u2032_ f) xs)\n\nlift' : \u2200 {m a b} {A : Set a} {B : Set b} (f g : Vec A m \u2192 B)\n       \u2192 (\u2200 (xs : Vec A m) \u2192 f xs \u2261 g xs)\n       \u2192 \u2200 {n : \u2115} (xs : Vec (Vec A m) n) \u2192 vap' f xs \u2261 vap' g xs\nlift' f g pf [] = refl\nlift' f g pf (x \u2237 xs) rewrite lift' f g pf xs | pf x = refl\n\nlift : \u2200 {m a b} {A : Set a} {B : Set b} (f g : N-ary m A B)\n       \u2192 (\u2200\u207f m (curry\u207f\u2032 (\u03bb (xs : Vec A m) \u2192 f $\u207f\u2032 xs \u2261 g $\u207f\u2032 xs)))\n       \u2192 \u2200 {n} \u2192 \u2200\u207f m (curry\u207f\u2032 (\u03bb (xs : Vec (Vec A n) m) \u2192 vap {m} f $\u207f\u2032 xs \u2261 vap {m} g $\u207f\u2032 xs))\nlift {m} f g pf = curry\u207f helper\n  where f' = vap'' {m} (_$\u207f\u2032_ f)\n        g' = vap'' {m} (_$\u207f\u2032_ g)\n        h  = \u03bb (xs : Vec _ m) \u2192 curry\u207f\u2032 f' $\u207f\u2032 xs \u2261 curry\u207f\u2032 g' $\u207f\u2032 xs\n        hh = \u03bb (xs : Vec _ m) \u2192 f $\u207f\u2032 xs \u2261 g $\u207f\u2032 xs\n        pf' : \u2200 xs \u2192 hh xs\n        pf' xs rewrite sym (curry-$\u207f\u2032 hh xs) = pf $\u207f xs\n        helper : \u2200 xs \u2192 curry\u207f\u2032 h $\u207f\u2032 xs\n        helper xs rewrite curry-$\u207f\u2032 h xs | curry-$\u207f\u2032 f' xs | curry-$\u207f\u2032 g' xs\n          = lift' (_$\u207f\u2032_ f) (_$\u207f\u2032_ g) pf' (transpose xs)\n","old_contents":"module Data.Vec.N-ary.NP where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Vec.N-ary public renaming (curry\u207f to curry\u207f\u2032; _$\u207f_ to _$\u207f\u2032_)\nopen import Relation.Binary.PropositionalEquality\n\ncurry\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192\n         ((xs : Vec A n) \u2192 B $\u207f\u2032 xs) \u2192 \u2200\u207f n B\ncurry\u207f {zero}  f = f []\ncurry\u207f {suc n} f = \u03bb x \u2192 curry\u207f (f \u2218 _\u2237_ x)\n\n_$\u207f_ : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192 \u2200\u207f n B \u2192 ((xs : Vec A n) \u2192 B $\u207f\u2032 xs)\nf $\u207f []       = f\nf $\u207f (x \u2237 xs) = f x $\u207f xs\n\ncurry-$\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} (f : (xs : Vec A n) \u2192 B $\u207f\u2032 xs) (xs : Vec A n)\n           \u2192 curry\u207f f $\u207f xs \u2261 f xs\ncurry-$\u207f f []       = refl\ncurry-$\u207f f (x \u2237 xs) = curry-$\u207f (f \u2218 _\u2237_ x) xs\n\ncurry-$\u207f\u2032 : \u2200 {n a b} {A : Set a} {B : Set b} (f : Vec A n \u2192 B) (xs : Vec A n)\n           \u2192 curry\u207f\u2032 f $\u207f\u2032 xs \u2261 f xs\ncurry-$\u207f\u2032 f []       = refl\ncurry-$\u207f\u2032 f (x \u2237 xs) = curry-$\u207f\u2032 (f \u2218 _\u2237_ x) xs\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2b24d361d123e140416254d555065fbf73f695d0","subject":"Reordered the README.agda","message":"Reordered the README.agda\n\nIgnore-this: a48e1f702643206de51d433845945acb\n\ndarcs-hash:20111207205530-3bd4e-d3a9bc66680aa4b76ab971c1be5cbd8fd40ee301.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"README.agda","new_file":"README.agda","new_contents":"------------------------------------------------------------------------------\n-- FOT (First-Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Reasoning about Functional Programs\" by Ana Bove, Peter Dybjer, and\n-- Andr\u00e9s Sicard-Ram\u00edrez.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first-order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first-order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first-order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- First order theories\n------------------------------------------------------------------------------\n\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- LTC-PCF (Almost a first-order theory)\n\n-- Formalization of (a version of) Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 7.1. The axioms\nopen import LTC-PCF.Base\n\n-- 7.2. Natural numberes\n\n-- 7.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 7.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 7.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 7.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 7.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 7.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 7.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 7.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 7.3 Programs\n\n-- 7.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 7.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n","old_contents":"------------------------------------------------------------------------------\n-- FOT (First Order Theories)\n------------------------------------------------------------------------------\n\n-- Code accompanying the paper \"Combining Interactive and Automatic\n-- Reasoning about Functional Programs\" by Ana Bove, Peter Dybjer, and\n-- Andr\u00e9s Sicard-Ram\u00edrez.\n\nmodule README where\n\n------------------------------------------------------------------------------\n-- Description\n\n-- Examples of the formalization of first order theories showing the\n-- combination of interactive proofs with automatics proofs carried\n-- out by first order automatic theorem provers (ATPs).\n\n------------------------------------------------------------------------------\n-- Prerequisites and use\n\n-- See http:\/\/www1.eafit.edu.co\/asicard\/code\/fotc\/.\n\n------------------------------------------------------------------------------\n-- Conventions\n\n-- The following modules show the formalization of some first order\n-- theories. If the module's name ends in 'I' the module contains\n-- interactive proofs, if it ends in 'ATP' the module contains\n-- combined proofs, otherwise the module contains definions and\/or\n-- interactive proofs that are used by the interactive and combined\n-- proofs.\n\n------------------------------------------------------------------------------\n-- 1. Predicate logic\n\n-- 1.1 Definition of the connectives and quantifiers\nopen import Common.LogicalConstants\nopen import PredicateLogic.Constants\n\n-- 1.2 Propositional logic theorems\nopen import PredicateLogic.Propositional.TheoremsATP\nopen import PredicateLogic.Propositional.TheoremsI\n\n-- 1.3 Predicate logic theorems\nopen import PredicateLogic.TheoremsATP\nopen import PredicateLogic.TheoremsI\n\n-- 1.4 Logical schemas\nopen import PredicateLogic.SchemasATP\n\n-- 1.5 Non-empty domains\nopen import PredicateLogic.NonEmptyDomain.TheoremsATP\nopen import PredicateLogic.NonEmptyDomain.TheoremsI\n\n-- 1.6 Classical predicate logic theorems\nopen import PredicateLogic.ClassicalATP\nopen import PredicateLogic.ClassicalI\n\n------------------------------------------------------------------------------\n-- 2. Equality\n\n-- 2.1 Definition of equality and some properties about it\nopen import Common.Relation.Binary.PropositionalEquality\n\n-- 2.2 The equality reasoning\nopen import Common.Relation.Binary.PreorderReasoning\n\n------------------------------------------------------------------------------\n-- 3. Group theory\n\n-- 3.1 The axioms\nopen import GroupTheory.Base\n\n-- 3.2 Basic properties\nopen import GroupTheory.PropertiesATP\nopen import GroupTheory.PropertiesI\n\n-- 3.3 Commutator properties\nopen import GroupTheory.Commutator.PropertiesATP\nopen import GroupTheory.Commutator.PropertiesI\n\n-- 3.4 Abelian groups\nopen import GroupTheory.AbelianGroup.PropertiesATP\n\n------------------------------------------------------------------------------\n-- 4. Stanovsk\u00fd example (distributive laws on a binary operation)\n\n-- We prove the proposition 2 (task B) of [1]. Let _\u00b7_ be a\n-- left-associative binary operation, the task B consist in given the\n-- left and right distributive axioms\n\n--   \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n--   \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- to prove the theorem\n\n--   \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n--               (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n--               x \u2219 z \u2219 (y \u2219 u)\n\n-- [1] David Stanovsk\u00fd. Distributive groupoids are symmetrical-by-medial:\n--     An elementary proof. Commentations Mathematicae Universitatis\n--     Carolinae, 49(4):541\u2013546, 2008.\n\n-- 4.1 The ATPs could not prove the theorem\nopen import DistributiveLaws.TaskB-ATP\n\n-- 4.2 The interactive and combined proofs\nopen import DistributiveLaws.TaskB-HalvedStepsATP\nopen import DistributiveLaws.TaskB-I\nopen import DistributiveLaws.TaskB-TopDownATP\n\n------------------------------------------------------------------------------\n-- 5. Peano arithmetic (PA)\n\n-- We write two formalizations of PA. In the axiomatic formalization\n-- we use Agda postulates for Peano's axioms. In the inductive\n-- formalization, we use Agda data types and primitive recursive\n-- functions.\n\n-- 5.1 Axiomatic PA\n-- 5.1.1 The axioms\nopen import PA.Axiomatic.Base\n\n-- 5.1.2 Some properties\nopen import PA.Axiomatic.PropertiesATP\nopen import PA.Axiomatic.PropertiesI\n\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityATP\nopen import PA.Axiomatic.Relation.Binary.PropositionalEqualityI\n\n-- 5.2. Inductive PA\n-- 5.2.1 Some properties\nopen import PA.Inductive.Properties\nopen import PA.Inductive.PropertiesATP\nopen import PA.Inductive.PropertiesI\n\nopen import PA.Inductive.PropertiesByInduction\nopen import PA.Inductive.PropertiesByInductionATP\nopen import PA.Inductive.PropertiesByInductionI\n\n------------------------------------------------------------------------------\n-- 6. FOTC\n\n-- Formalization of (a version of) Azcel's First Order Theory of Combinators.\n\n-- 6.1 The axioms\nopen import FOTC.Base\n\n-- 6.2 Booleans\n\n-- 6.2.1 The axioms\nopen import FOTC.Data.Bool\n\n-- 6.2.2 The inductive predicate\nopen import FOTC.Data.Bool.Type\n\n-- 6.2.3 Properties\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.Bool.PropertiesI\n\n-- 6.3. Natural numbers\n\n-- 6.3.1 The axioms\nopen import FOTC.Data.Nat\n\n-- 6.3.2 The inductive predicate\nopen import FOTC.Data.Nat.Type\n\n-- 6.3.3 Properties\nopen import FOTC.Data.Nat.PropertiesATP\nopen import FOTC.Data.Nat.PropertiesI\n\nopen import FOTC.Data.Nat.PropertiesByInductionATP\nopen import FOTC.Data.Nat.PropertiesByInductionI\n\n-- 6.3.4 Divisibility relation\nopen import FOTC.Data.Nat.Divisibility.By0.Properties\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.By0.PropertiesI\nopen import FOTC.Data.Nat.Divisibility.NotBy0.Properties\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import FOTC.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 6.3.5 Induction\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\n\n-- 6.3.6 Inequalites\nopen import FOTC.Data.Nat.Inequalities.Properties\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n-- 6.3.7 Unary numbers\nopen import FOTC.Data.Nat.UnaryNumbers.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers.TotalityATP\n\n-- 6.4 Lists\n\n-- 6.4.1 The axioms\nopen import FOTC.Data.List\n\n-- 6.4.2 The inductive predicate\nopen import FOTC.Data.List.Type\n\n-- 6.4.3 Properties\nopen import FOTC.Data.List.PropertiesATP\nopen import FOTC.Data.List.PropertiesI\n\n-- 6.4.4 Well-founded induction\nopen import FOTC.Data.List.LT-Cons.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Cons.PropertiesI\nopen import FOTC.Data.List.LT-Length.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.List.LT-Length.PropertiesI\n\n-- 6.4.5 Lists of natural numbers\n\n-- 6.4.5.1 The inductive predicate\nopen import FOTC.Data.Nat.List.Type\n\n-- 6.4.5.2 Properties\nopen import FOTC.Data.Nat.List.PropertiesATP\nopen import FOTC.Data.Nat.List.PropertiesI\n\n-- 6.5 Coinductive natural numbers\n\n-- 6.5.1 The coinductive predicate\nopen import FOTC.Data.Conat\n\n-- 6.5.2 Equality on coinductive natural numbers\nopen import FOTC.Data.Conat.Equality\n\n-- 6.5.3 Properties\nopen import FOTC.Data.Conat.PropertiesI\n\n-- 6.6 Streams\n\n-- 6.6.1 The coinductive predicate (axioms)\nopen import FOTC.Data.Stream\n\n-- 6.6.2 Equality on streams\nopen import FOTC.Data.Stream.Equality\nopen import FOTC.Data.Stream.Equality.PropertiesATP\nopen import FOTC.Data.Stream.Equality.PropertiesI\n\n-- 6.6.3 Properties\nopen import FOTC.Data.Stream.PropertiesATP\nopen import FOTC.Data.Stream.PropertiesI\n\n-- 6.7 Programs\n\n-- 6.7.1 The Collatz function: A function without a termination proof\nopen import FOTC.Program.Collatz.PropertiesATP\nopen import FOTC.Program.Collatz.PropertiesI\n\n-- 6.7.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.GCD.Partial.ProofSpecificationATP\nopen import FOTC.Program.GCD.Partial.ProofSpecificationI\nopen import FOTC.Program.GCD.Total.ProofSpecificationATP\nopen import FOTC.Program.GCD.Total.ProofSpecificationI\n\n-- 6.7.3 The McCarthy 91 function: A function with nested recursion\nopen import FOTC.Program.McCarthy91.Properties.MainATP\n\n-- 6.7.4 The mirror function: A function with higher-order recursion\nopen import FOTC.Program.Mirror.PropertiesATP\nopen import FOTC.Program.Mirror.PropertiesI\n\n-- 6.7.5 Burstall's sort list algorithm: A structurally recursive algorithm\nopen import FOTC.Program.SortList.ProofSpecificationATP\nopen import FOTC.Program.SortList.ProofSpecificationI\n\n-- 6.7.6 The division algorithm: A non-structurally recursive algorithm\nopen import FOTC.Program.Division.ProofSpecificationATP\nopen import FOTC.Program.Division.ProofSpecificationI\n\n-- 6.7.7 The map-iterate property: A property using coinduction\nopen import FOTC.Program.MapIterate.MapIterateATP\n\n-- 6.7.8 The alternating bit protocol: A program using co-inductive types\nopen import FOTC.Program.ABP.ProofSpecificationATP\nopen import FOTC.Program.ABP.ProofSpecificationI\n\n-- 6.8 Other modules\n-- (These modules are not imported from any module).\nopen import FOTC.Program.Division.EquationsATP\nopen import FOTC.Program.GCD.Partial.EquationsATP\nopen import FOTC.Program.GCD.Total.EquationsATP\n\n------------------------------------------------------------------------------\n-- ATPs failures\n------------------------------------------------------------------------------\n\n-- The ATPs (E, Equinox, Metis, SPASS and Vampire) could not prove some\n-- theorems.\n\nopen import DistributiveLaws.TaskB-ATP\nopen import FOTC.Program.ABP.MayorPremiseATP\nopen import PA.Axiomatic.PropertiesATP\n\n------------------------------------------------------------------------------\n-- Agsy examples\n------------------------------------------------------------------------------\n\n-- We cannot import the Agsy examples because some modules contain\n-- unsolved metas, therefore see src\/Agsy\/README.txt\n\n------------------------------------------------------------------------------\n-- Other theories\n------------------------------------------------------------------------------\n\n-- 1. LTC-PCF\n-- Formalization of a version of Azcel's Logical Theory of Constructions.\n-- N.B. This was the theory shown in our PLPV'09 paper.\n\n-- 1.1. The axioms\nopen import LTC-PCF.Base\n\n-- 1.2 Natural numberes\n\n-- 1.2.1 The axioms\nopen import LTC-PCF.Data.Nat\n\n-- 1.2.2 The inductive predicate\nopen import LTC-PCF.Data.Nat.Type\n\n-- 1.2.3 Properties\nopen import LTC-PCF.Data.Nat.PropertiesATP\nopen import LTC-PCF.Data.Nat.PropertiesI\n\n-- 1.2.4 Divisibility relation\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.Properties\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesATP\nopen import LTC-PCF.Data.Nat.Divisibility.NotBy0.PropertiesI\n\n-- 1.2.5 Induction\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.LexicographicI\nopen import LTC-PCF.Data.Nat.Induction.NonAcc.WellFoundedInductionI\n\n-- 1.2.6 Inequalites\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesATP\nopen import LTC-PCF.Data.Nat.Inequalities.PropertiesI\n\n-- 1.2.7 The recursive operator\nopen import LTC-PCF.Data.Nat.Rec.EquationsATP\nopen import LTC-PCF.Data.Nat.Rec.EquationsI\n\n-- 1.2.8 A looping combinator\nopen import LTC-PCF.Loop\n\n-- 1.3 Programs\n\n-- 1.3.1 The division algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.Division.ProofSpecificationATP\nopen import LTC-PCF.Program.Division.ProofSpecificationI\n\n-- 1.3.2 The GCD algorithm: A non-structurally recursive algorithm\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationATP\nopen import LTC-PCF.Program.GCD.Partial.ProofSpecificationI\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"45c054d3fb6bfe73b01d5c2337702cfd329484df","subject":"Removed unnecessary local hint.","message":"Removed unnecessary local hint.\n\nIgnore-this: 7aec4faaaf599dcbca5817e82e272cd5\n\ndarcs-hash:20110218171250-3bd4e-98d47b1b3922adddec0d910bc589adbed2d23f6d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_file":"Draft\/McCarthy91\/Properties\/AuxiliaryATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Properties.AuxiliaryATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.ArithmeticATP\nopen import Draft.McCarthy91.McCarthy91\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 x\u223810-N #-}\n{-# ATP prove x<mc91x+11>100 x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 1+x-N 11+x-N\n              x\u223810-N +-comm #-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n              Nmc91>100 N111 N101\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 N91 #-}\n-- {-# ATP prove mc91<mc91+11 mc91\u226191 n102 100<102 #-}\n\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","old_contents":"------------------------------------------------------------------------------\n-- Auxiliary properties of the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.Properties.AuxiliaryATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.ArithmeticATP\nopen import Draft.McCarthy91.McCarthy91\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.PropertiesATP\nopen import LTC.Data.Nat.Unary.IsN-ATP\nopen import LTC.Data.Nat.Unary.Numbers\nopen import LTC.Data.Nat.Unary.Inequalities.PropertiesATP\n\n------------------------------------------------------------------------------\n\n--- Auxiliary properties\n\n---- Case n > 100\npostulate\n  Nmc91>100      : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192 N (mc91 n)\n  x<mc91x+11>100 : \u2200 {n} \u2192 N n \u2192 GT n one-hundred \u2192\n                           LT n (mc91 n + eleven)\n{-# ATP prove Nmc91>100 x\u223810-N #-}\n{-# ATP prove x<mc91x+11>100 x<y\u2192y\u2264z\u2192x<z x<x+1 x+1\u2264x\u223810+11 1+x-N 11+x-N\n              x\u223810-N +-comm #-}\n\n-- Most of them not needed\n-- Case n \u2261 100 can be proved automatically\npostulate\n  Nmc91\u2261100     : N (mc91 one-hundred)\n  mc91-res-100  : mc91 one-hundred \u2261 ninety-one\n  mc91-res-100' : \u2200 {n} \u2192 n \u2261 one-hundred \u2192 mc91 n \u2261 ninety-one\n  mc91<mc91+11  : LT  one-hundred (mc91 one-hundred + eleven)\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810\n              Nmc91>100 N111 N101\n#-}\n{-# ATP prove mc91-res-100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 #-}\n{-# ATP prove mc91-res-100' mc91-res-100 #-}\n{-# ATP prove Nmc91\u2261100 111>100 101>100 101\u2261100+11-10 91\u2261[100+11\u223810]\u223810 N91 #-}\n-- {-# ATP prove mc91<mc91+11 mc91\u226191 n102 100<102 #-}\n\n\n---- Case n \u2264 100\n\npostulate\n  Nmc91\u2264100'        : \u2200 n \u2192 LE n one-hundred \u2192 N (mc91 (mc91 (n + eleven))) \u2192\n                      N (mc91 n)\n  x<mc91x+11\u2264100'   : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT n (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT n (mc91 n + eleven)\n  mc91x+11<mc91x+11 : \u2200 n \u2192 LE n one-hundred \u2192\n                      LT (mc91 (n + eleven)) (mc91 (mc91 (n + eleven)) + eleven) \u2192\n                      LT (mc91 (n + eleven)) (mc91 n + eleven)\n  mc91x-res\u2264100     : \u2200 m n \u2192 LE m one-hundred \u2192\n                      mc91 (m + eleven) \u2261 n \u2192 mc91 n \u2261 ninety-one \u2192\n                      mc91 m \u2261 ninety-one\n{-# ATP prove Nmc91\u2264100' #-}\n{-# ATP prove x<mc91x+11\u2264100' #-}\n{-# ATP prove mc91x+11<mc91x+11 #-}\n{-# ATP prove mc91x-res\u2264100 #-}\n\npostulate\n  mc91-res-110 : mc91 (ninety-nine + eleven)  \u2261 one-hundred\n  mc91-res-109 : mc91 (ninety-eight + eleven) \u2261 ninety-nine\n  mc91-res-108 : mc91 (ninety-seven + eleven) \u2261 ninety-eight\n  mc91-res-107 : mc91 (ninety-six + eleven)   \u2261 ninety-seven\n  mc91-res-106 : mc91 (ninety-five + eleven)  \u2261 ninety-six\n  mc91-res-105 : mc91 (ninety-four + eleven)  \u2261 ninety-five\n  mc91-res-104 : mc91 (ninety-three + eleven) \u2261 ninety-four\n  mc91-res-103 : mc91 (ninety-two + eleven)   \u2261 ninety-three\n  mc91-res-102 : mc91 (ninety-one + eleven)   \u2261 ninety-two\n  mc91-res-101 : mc91 (ninety + eleven)       \u2261 ninety-one\n{-# ATP prove mc91-res-110 mc91-eq\u2081 99+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-109 mc91-eq\u2081 98+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-108 mc91-eq\u2081 97+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-107 mc91-eq\u2081 96+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-106 mc91-eq\u2081 95+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-105 mc91-eq\u2081 94+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-104 mc91-eq\u2081 93+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-103 mc91-eq\u2081 92+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-102 mc91-eq\u2081 91+11>100 x+11\u223810\u2261Sx #-}\n{-# ATP prove mc91-res-101 mc91-eq\u2081 90+11>100 x+11\u223810\u2261Sx #-}\n\npostulate\n  mc91-res-99 : mc91 ninety-nine  \u2261 ninety-one\n  mc91-res-98 : mc91 ninety-eight \u2261 ninety-one\n  mc91-res-97 : mc91 ninety-seven \u2261 ninety-one\n  mc91-res-96 : mc91 ninety-six   \u2261 ninety-one\n  mc91-res-95 : mc91 ninety-five  \u2261 ninety-one\n  mc91-res-94 : mc91 ninety-four  \u2261 ninety-one\n  mc91-res-93 : mc91 ninety-three \u2261 ninety-one\n  mc91-res-92 : mc91 ninety-two   \u2261 ninety-one\n  mc91-res-91 : mc91 ninety-one   \u2261 ninety-one\n  mc91-res-90 : mc91 ninety       \u2261 ninety-one\n{-# ATP prove mc91-res-99 mc91x-res\u2264100 mc91-res-110 mc91-res-100 #-}\n{-# ATP prove mc91-res-98 mc91x-res\u2264100 mc91-res-109 mc91-res-99 #-}\n{-# ATP prove mc91-res-97 mc91x-res\u2264100 mc91-res-108 mc91-res-98 #-}\n{-# ATP prove mc91-res-96 mc91x-res\u2264100 mc91-res-107 mc91-res-97 #-}\n{-# ATP prove mc91-res-95 mc91x-res\u2264100 mc91-res-106 mc91-res-96 #-}\n{-# ATP prove mc91-res-94 mc91x-res\u2264100 mc91-res-105 mc91-res-95 #-}\n{-# ATP prove mc91-res-93 mc91x-res\u2264100 mc91-res-104 mc91-res-94 #-}\n{-# ATP prove mc91-res-92 mc91x-res\u2264100 mc91-res-103 mc91-res-93 #-}\n{-# ATP prove mc91-res-91 mc91x-res\u2264100 mc91-res-102 mc91-res-92 #-}\n{-# ATP prove mc91-res-90 mc91x-res\u2264100 mc91-res-101 mc91-res-91 #-}\n\nmc91-res-99' : \u2200 {n} \u2192 n \u2261 ninety-nine \u2192 mc91 n \u2261 ninety-one\nmc91-res-99' refl = mc91-res-99\n\nmc91-res-98' : \u2200 {n} \u2192 n \u2261 ninety-eight \u2192 mc91 n \u2261 ninety-one\nmc91-res-98' refl = mc91-res-98\n\nmc91-res-97' : \u2200 {n} \u2192 n \u2261 ninety-seven \u2192 mc91 n \u2261 ninety-one\nmc91-res-97' refl = mc91-res-97\n\nmc91-res-96' : \u2200 {n} \u2192 n \u2261 ninety-six \u2192 mc91 n \u2261 ninety-one\nmc91-res-96' refl = mc91-res-96\n\nmc91-res-95' : \u2200 {n} \u2192 n \u2261 ninety-five \u2192 mc91 n \u2261 ninety-one\nmc91-res-95' refl = mc91-res-95\n\nmc91-res-94' : \u2200 {n} \u2192 n \u2261 ninety-four \u2192 mc91 n \u2261 ninety-one\nmc91-res-94' refl = mc91-res-94\n\nmc91-res-93' : \u2200 {n} \u2192 n \u2261 ninety-three \u2192 mc91 n \u2261 ninety-one\nmc91-res-93' refl = mc91-res-93\n\nmc91-res-92' : \u2200 {n} \u2192 n \u2261 ninety-two \u2192 mc91 n \u2261 ninety-one\nmc91-res-92' refl = mc91-res-92\n\nmc91-res-91' : \u2200 {n} \u2192 n \u2261 ninety-one \u2192 mc91 n \u2261 ninety-one\nmc91-res-91' refl = mc91-res-91\n\nmc91-res-90' : \u2200 {n} \u2192 n \u2261 ninety \u2192 mc91 n \u2261 ninety-one\nmc91-res-90' refl = mc91-res-90\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"da6da46c3fedc6e0063b151bb1910c5af498a3fb","subject":"Fixed whitespace issue","message":"Fixed whitespace issue\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/List\/Induction\/Instances\/PropertiesATP.agda","new_file":"notes\/FOT\/FOTC\/Data\/List\/Induction\/Instances\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists using instances of the induction principle\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.List.Induction.Instances.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\nlengthList-N-ind-instance :\n  N (length []) \u2192\n  (\u2200 x {xs} \u2192 N (length xs) \u2192 N (length (x \u2237 xs))) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 N (length xs)\nlengthList-N-ind-instance = List-ind (\u03bb as \u2192 N (length as))\n\npostulate lengthList-N : \u2200 {xs} \u2192 List xs \u2192 N (length xs)\n{-# ATP prove lengthList-N lengthList-N-ind-instance #-}\n\n++-List-ind-instance :\n  \u2200 {ys} \u2192\n  List ([] ++ ys) \u2192\n  (\u2200 x {xs} \u2192 List (xs ++ ys) \u2192 List ((x \u2237 xs) ++ ys)) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 List (xs ++ ys)\n++-List-ind-instance {ys} = List-ind (\u03bb as \u2192 List (as ++ ys))\n\npostulate ++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n{-# ATP prove ++-List ++-List-ind-instance #-}\n\nmap-List-ind-instance :\n  \u2200 {f} \u2192\n  List (map f []) \u2192\n  (\u2200 x {xs} \u2192 List (map f xs) \u2192 List (map f (x \u2237 xs))) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List-ind-instance {f} = List-ind (\u03bb as \u2192 List (map f as))\n\npostulate map-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\n{-# ATP prove map-List map-List-ind-instance #-}\n\n------------------------------------------------------------------------------\n\n++-assoc-ind-instance :\n  \u2200 {ys zs} \u2192\n  ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs \u2192\n  (\u2200 x {xs} \u2192\n    (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192\n    ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs\n++-assoc-ind-instance {ys} {zs} =\n  List-ind (\u03bb as \u2192 (as ++ ys) ++ zs \u2261 as ++ ys ++ zs )\n\npostulate\n  ++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs\n{-# ATP prove ++-assoc ++-assoc-ind-instance #-}\n\nmap-++-ind-instance :\n  \u2200 {f} {ys} \u2192\n  map f ([] ++ ys) \u2261 map f [] ++ map f ys \u2192\n  (\u2200 x {xs} \u2192 map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n    map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 map f (xs ++ ys) \u2261 map f xs ++ map f ys\nmap-++-ind-instance {f} {ys} =\n  List-ind (\u03bb as \u2192 map f (as ++ ys) \u2261 map f as ++ map f ys)\n\npostulate\n  map-++ : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n           map f (xs ++ ys) \u2261 map f xs ++ map f ys\n{-# ATP prove map-++ map-++-ind-instance #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties related with lists using instances of the induction principle\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule FOT.FOTC.Data.List.Induction.Instances.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Totality properties\n\nlengthList-N-ind-instance :\n  N (length []) \u2192\n  (\u2200 x {xs} \u2192 N (length xs) \u2192 N (length (x \u2237 xs))) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 N (length xs)\nlengthList-N-ind-instance = List-ind (\u03bb as \u2192 N (length as))\n\npostulate lengthList-N : \u2200 {xs} \u2192 List xs \u2192 N (length xs)\n{-# ATP prove lengthList-N lengthList-N-ind-instance #-}\n\n++-List-ind-instance :\n  \u2200 {ys} \u2192\n  List ([] ++ ys) \u2192\n  (\u2200 x {xs} \u2192 List (xs ++ ys) \u2192 List ((x \u2237 xs) ++ ys)) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 List (xs ++ ys)\n++-List-ind-instance {ys} = List-ind (\u03bb as \u2192 List (as ++ ys))\n\npostulate ++-List : \u2200 {xs ys} \u2192 List xs \u2192 List ys \u2192 List (xs ++ ys)\n{-# ATP prove ++-List ++-List-ind-instance #-}\n\nmap-List-ind-instance :\n  \u2200 {f} \u2192\n  List (map f []) \u2192\n  (\u2200 x {xs} \u2192 List (map f xs) \u2192 List (map f (x \u2237 xs))) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 List (map f xs)\nmap-List-ind-instance {f} = List-ind (\u03bb as \u2192 List (map f as))\n\npostulate map-List : \u2200 f {xs} \u2192 List xs \u2192 List (map f xs)\n{-# ATP prove map-List map-List-ind-instance #-}\n\n------------------------------------------------------------------------------\n\n++-assoc-ind-instance :\n  \u2200 {ys zs} \u2192\n  ([] ++ ys) ++ zs \u2261 [] ++ ys ++ zs \u2192\n  (\u2200 x {xs} \u2192\n    (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs \u2192\n    ((x \u2237 xs) ++ ys) ++ zs \u2261 (x \u2237 xs) ++ ys ++ zs) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs\n++-assoc-ind-instance {ys} {zs} =\n  List-ind (\u03bb as \u2192 (as ++ ys) ++ zs \u2261 as ++ ys ++ zs )\n\npostulate\n  ++-assoc : \u2200 {xs} \u2192 List xs \u2192 \u2200 ys zs \u2192 (xs ++ ys) ++ zs \u2261 xs ++ ys ++ zs\n{-# ATP prove ++-assoc ++-assoc-ind-instance #-}\n\nmap-++-ind-instance :\n  \u2200 {f} {ys} \u2192\n  map f ([] ++ ys) \u2261 map f [] ++ map f ys \u2192\n  (\u2200 x {xs} \u2192 map f (xs ++ ys) \u2261 map f xs ++ map f ys \u2192\n    map f ((x \u2237 xs) ++ ys) \u2261 map f (x \u2237 xs) ++ map f ys) \u2192\n  \u2200 {xs} \u2192 List xs \u2192 map f (xs ++ ys) \u200c\u2261 map f xs ++ map f ys\nmap-++-ind-instance {f} {ys} =\n  List-ind (\u03bb as \u2192 map f (as ++ ys) \u2261 map f as ++ map f ys)\n\npostulate\n  map-++ : \u2200 f {xs} \u2192 List xs \u2192 \u2200 ys \u2192\n           map f (xs ++ ys) \u2261 map f xs ++ map f ys\n{-# ATP prove map-++ map-++-ind-instance #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b3310d52d2be45305707db0f9151c40f8d395a1d","subject":"hiding _\u2194_","message":"hiding _\u2194_\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/Permutation.agda","new_file":"lib\/Data\/Vec\/Permutation.agda","new_contents":"-- NOTE with-K\nopen import Type hiding (\u2605)\nopen import Level.NP\nopen import Function.NP hiding (_\u2194_)\nopen import Data.Nat.NP hiding (\u27e8_\u2194+1\u27e9; \u27e80\u21941+_\u27e9)\nopen import Data.Nat.Properties\nopen import Data.Vec.NP\nopen import Data.Bits\nopen import Data.Product hiding (map)\nopen import Data.Fin using (Fin; zero; suc; inject\u2081)\nopen import Relation.Binary.PropositionalEquality\n\nmodule Data.Vec.Permutation where\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : \u2605 a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = refl\n\nprivate\n    _\u00b2 : \u2200 {a} {A : \u2605 a} \u2192 Endo (Endo A)\n    f \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : \u2605 a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = refl\n    comm zero    (suc _) _ = refl\n    comm (suc _) zero    _ = refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : \u2605 a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 \u2605\u2080 where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : \u2605 a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u00b7_\n    _\u00b7_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u00b7_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u00b7 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u00b7 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 \u2605 _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u00b7 xs \u2261 g \u00b7 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u00b7 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u00b7 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u00b7 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u00b7 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u00b7 : \u2200 {n} {A : \u2605 a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u00b7 fs \u229b f \u00b7 xs) \u2261 f \u00b7 (fs \u229b xs)\n    \u229b-dist-\u00b7 _       `id        _  = refl\n    \u229b-dist-\u00b7 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u00b7 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u00b7 fs f xs = refl\n    \u229b-dist-\u00b7 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u00b7 (f \u00b7 fs) g (f \u00b7 xs)\n                                         | \u229b-dist-\u00b7 fs f xs = refl\n\n    \u00b7-replicate : \u2200 {n} {A : \u2605 a} (x : A) (f : Perm n) \u2192 f \u00b7 replicate x \u2261 replicate x\n    \u00b7-replicate x `id = refl\n    \u00b7-replicate x `0\u21941 = refl\n    \u00b7-replicate x (`tl f) rewrite \u00b7-replicate x f = refl\n    \u00b7-replicate x (f `\u204f g) rewrite \u00b7-replicate x f | \u00b7-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : \u2605 a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u00b7 xs \u2261 f \u00b7 (f \u207b\u00b9 \u00b7 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u00b7 (f \u207b\u00b9 \u00b7 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u00b7-map : \u2200 {n} {A : \u2605 a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u00b7 xs) \u2261 g \u00b7 map f xs\n    \u00b7-map f g xs rewrite sym (\u229b-dist-\u00b7 (replicate f) g xs) | \u00b7-replicate f g = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u00b7\n    \u2295-dist-\u00b7 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u00b7 pad \u2295 \u03c0 \u00b7 xs \u2261 \u03c0 \u00b7 (pad \u2295 xs)\n    \u2295-dist-\u00b7 fs      `id        xs = refl\n    \u2295-dist-\u00b7 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u00b7 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u00b7 fs \u03c0 xs = refl\n    \u2295-dist-\u00b7 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u00b7 (\u03c0\u2080 \u00b7 fs) \u03c0\u2081 (\u03c0\u2080 \u00b7 xs)\n                                         | \u2295-dist-\u00b7 fs \u03c0\u2080 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = refl\n","old_contents":"-- NOTE with-K\nopen import Type hiding (\u2605)\nopen import Level.NP\nopen import Function.NP\nopen import Data.Nat.NP hiding (\u27e8_\u2194+1\u27e9; \u27e80\u21941+_\u27e9)\nopen import Data.Nat.Properties\nopen import Data.Vec.NP\nopen import Data.Bits\nopen import Data.Product hiding (map)\nopen import Data.Fin using (Fin; zero; suc; inject\u2081)\nopen import Relation.Binary.PropositionalEquality\n\nmodule Data.Vec.Permutation where\n\n-- \u27e80\u21941+ i \u27e9: Exchange elements at position 0 and 1+i.\n\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A (1 + n) \u2192 Vec A (1 + n)\n\u27e80\u21941+ zero  \u27e9 = 0\u21941\n\u27e80\u21941+ suc i \u27e9 = 0\u21941 \u2218 (map-tail \u27e80\u21941+ i \u27e9) \u2218 0\u21941\n  {- 0   1   2 3 ... i 1+i ... n\n     1   0   2 3 ... i 1+i ... n\n     1   1+i 2 3 ... i 0   ... n\n\n     1+i 1   2 3 ... i 0   ... n\n   -}\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = \u27e80\u21941+ i \u27e9\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : \u2605 a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = refl\n\nprivate\n    _\u00b2 : \u2200 {a} {A : \u2605 a} \u2192 Endo (Endo A)\n    f \u00b2 = f \u2218 f\n\nmodule \u27e8\u2194\u27e9 {a} (A : \u2605 a) where\n\n    \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194 j     \u27e9 = \u27e80\u2194 j \u27e9\n    \u27e8 i     \u2194 zero  \u27e9 = \u27e80\u2194 i \u27e9\n    \u27e8 suc i \u2194 suc j \u27e9 = map-tail \u27e8 i \u2194 j \u27e9\n-- \u27e8 # 0 \u2194 # 1 \u27e9\n\n    comm : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u2257 \u27e8 j \u2194 i \u27e9\n    comm zero    zero    _ = refl\n    comm zero    (suc _) _ = refl\n    comm (suc _) zero    _ = refl\n    comm (suc i) (suc j) (x \u2237 xs) rewrite comm i j xs = refl\n\n    0\u21941\u00b2-cancel : \u2200 {n} \u2192 0\u21941 \u00b2 \u2257 id {A = Vec A n}\n    0\u21941\u00b2-cancel [] = refl\n    0\u21941\u00b2-cancel (_ \u2237 []) = refl\n    0\u21941\u00b2-cancel (x \u2237 x\u2081 \u2237 xs) = refl\n\n    \u27e80\u21941+_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u21941+ i \u27e9 \u00b2 \u2257 id {A = Vec A (1 + n)}\n    \u27e80\u21941+ zero  \u27e9\u00b2-cancel xs = 0\u21941\u00b2-cancel xs\n    \u27e80\u21941+ suc i \u27e9\u00b2-cancel xs\n      rewrite 0\u21941\u00b2-cancel (map-tail \u27e80\u21941+ i \u27e9 (0\u21941 xs))\n            | map-tail\u2218map-tail \u27e80\u21941+ i \u27e9 \u27e80\u21941+ i \u27e9 (0\u21941 xs)\n            | map-tail-\u2257 _ _ \u27e80\u21941+ i \u27e9\u00b2-cancel (0\u21941 xs)\n            | map-tail-id (0\u21941 xs)\n            | 0\u21941\u00b2-cancel xs = refl\n\n    \u27e80\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i : Fin n) \u2192 \u27e80\u2194 i \u27e9 \u00b2 \u2257 id {A = Vec A n}\n    \u27e80\u2194 zero  \u27e9\u00b2-cancel _  = refl\n    \u27e80\u2194 suc i \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n\n    \u27e8_\u2194_\u27e9\u00b2-cancel : \u2200 {n} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9 \u00b2 \u2257 id\n    \u27e8 zero  \u2194 j     \u27e9\u00b2-cancel xs = \u27e80\u2194 j   \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 zero  \u27e9\u00b2-cancel xs = \u27e80\u21941+ i \u27e9\u00b2-cancel xs\n    \u27e8 suc i \u2194 suc j \u27e9\u00b2-cancel xs\n      rewrite map-tail\u2218map-tail \u27e8 i \u2194 j \u27e9 \u27e8 i \u2194 j \u27e9 xs\n            | map-tail-\u2257 _ _ \u27e8 i \u2194 j \u27e9\u00b2-cancel xs\n            | map-tail-id xs = refl\n\n    lem01maptail2 : \u2200 {m n a} {A : \u2605 a} (f : Vec A m \u2192 Vec A n) \u2192\n                      0\u21941 \u2218 map-tail (map-tail f) \u2218 0\u21941 \u2257 map-tail (map-tail f)\n    lem01maptail2 _ (_ \u2237 _ \u2237 _) = refl\n\n    \u2194-refl : \u2200 {n} (i : Fin n) \u2192 \u27e8 i \u2194 i \u27e9 \u2257 id\n    \u2194-refl zero    _  = refl\n    \u2194-refl (suc i) xs rewrite map-tail-\u2257 _ _ (\u2194-refl i) xs = map-tail-id xs\n\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n\u27e8_\u2194_\u27e9 i j = \u27e8\u2194\u27e9.\u27e8_\u2194_\u27e9 _ i j\n\nmodule PermutationSyntax where\n    infixr 1 _`\u204f_\n    data Perm : \u2115 \u2192 \u2605\u2080 where\n      `id : \u2200 {n} \u2192 Perm n\n      `0\u21941 : \u2200 {n} \u2192 Perm (2 + n)\n      _`\u204f_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 Perm n\n      `tl : \u2200 {n} \u2192 Perm n \u2192 Perm (1 + n)\n\n    _\u207b\u00b9 : \u2200 {n} \u2192 Endo (Perm n)\n    `id \u207b\u00b9 = `id\n    (f\u2080 `\u204f f\u2081) \u207b\u00b9 = f\u2081 \u207b\u00b9 `\u204f f\u2080 \u207b\u00b9\n    `0\u21941 \u207b\u00b9 = `0\u21941\n    (`tl f) \u207b\u00b9 = `tl (f \u207b\u00b9)\n\n    `\u27e80\u21941+_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm (1 + n)\n    `\u27e80\u21941+ zero  \u27e9 = `0\u21941\n    `\u27e80\u21941+ suc i \u27e9 = `0\u21941 `\u204f `tl `\u27e80\u21941+ i \u27e9 `\u204f `0\u21941\n\n    `\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) \u2192 Perm n\n    `\u27e80\u2194 zero  \u27e9 = `id\n    `\u27e80\u2194 suc i \u27e9 = `\u27e80\u21941+ i \u27e9\n\n    `\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) \u2192 Perm n\n    `\u27e8 zero  \u2194 j     \u27e9 = `\u27e80\u2194 j \u27e9\n    `\u27e8 i     \u2194 zero  \u27e9 = `\u27e80\u2194 i \u27e9\n    `\u27e8 suc i \u2194 suc j \u27e9 = `tl `\u27e8 i \u2194 j \u27e9\n\nmodule PermutationSemantics {a} {A : \u2605 a} where\n    open PermutationSyntax\n\n    eval : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    eval `id      = id\n    eval (f `\u204f g)  = eval g \u2218 eval f\n    eval `0\u21941     = 0\u21941\n    eval (`tl f)  = \u03bb xs \u2192 head xs \u2237 eval f (tail xs)\n\n    infixr 9 _\u00b7_\n    _\u00b7_ : \u2200 {n} \u2192 Perm n \u2192 Endo (Vec A n)\n    _\u00b7_ = eval\n\n    `\u27e80\u21941+_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A (suc n)) \u2192 `\u27e80\u21941+ i \u27e9 \u00b7 xs \u2261 \u27e80\u21941+ i \u27e9 xs\n    `\u27e80\u21941+ zero  \u27e9-spec xs = refl\n    `\u27e80\u21941+ suc i \u27e9-spec (x \u2237 y \u2237 xs) rewrite `\u27e80\u21941+ i \u27e9-spec (x \u2237 xs) = refl\n\n    `\u27e80\u2194_\u27e9-spec : \u2200 {n} (i : Fin n) (xs : Vec A n) \u2192 `\u27e80\u2194 i \u27e9 \u00b7 xs \u2261 \u27e80\u2194 i \u27e9 xs\n    `\u27e80\u2194 zero  \u27e9-spec xs = refl\n    `\u27e80\u2194 suc i \u27e9-spec xs = `\u27e80\u21941+ i \u27e9-spec xs\n\n    _\u2257\u2032_ : \u2200 {n} \u2192 Perm n \u2192 Perm n \u2192 \u2605 _\n    f \u2257\u2032 g = \u2200 xs \u2192 f \u00b7 xs \u2261 g \u00b7 xs\n\n    open \u27e8\u2194\u27e9 A hiding (\u27e8_\u2194_\u27e9)\n\n    _\u207b\u00b9-inverse : \u2200 {n} (f : Perm n) \u2192 (f `\u204f f \u207b\u00b9) \u2257\u2032 `id\n    (`id \u207b\u00b9-inverse) xs = refl\n    ((f `\u204f g) \u207b\u00b9-inverse) xs\n      rewrite (g \u207b\u00b9-inverse) (f \u00b7 xs)\n            | (f \u207b\u00b9-inverse) xs = refl\n    (`0\u21941 \u207b\u00b9-inverse) xs = 0\u21941\u00b2-cancel xs\n    ((`tl f) \u207b\u00b9-inverse) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-inverse) xs = refl\n\n    _\u207b\u00b9-involutive : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257\u2032 f\n    (`id \u207b\u00b9-involutive) _ = refl\n    ((f `\u204f g) \u207b\u00b9-involutive) x\n      rewrite (f \u207b\u00b9-involutive) x\n            | (g \u207b\u00b9-involutive) (f \u00b7 x) = refl\n    (`0\u21941 \u207b\u00b9-involutive) _ = refl\n    ((`tl f) \u207b\u00b9-involutive) (x \u2237 xs)\n      rewrite (f \u207b\u00b9-involutive) xs\n            = refl\n\n    _\u207b\u00b9-inverse\u2032 : \u2200 {n} (f : Perm n) \u2192 (f \u207b\u00b9 `\u204f f) \u2257\u2032 `id\n    (f \u207b\u00b9-inverse\u2032) xs with ((f \u207b\u00b9) \u207b\u00b9-inverse) xs\n    ... | p rewrite (f \u207b\u00b9-involutive) (f \u207b\u00b9 \u00b7 xs) = p\n\n    `\u27e8_\u2194_\u27e9-spec : \u2200 {n} (i j : Fin n) (xs : Vec A n) \u2192 `\u27e8 i \u2194 j \u27e9 \u00b7 xs \u2261 \u27e8 i \u2194 j \u27e9 xs\n    `\u27e8_\u2194_\u27e9-spec zero    j       xs = `\u27e80\u2194   j \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) zero    xs = `\u27e80\u21941+ i \u27e9-spec xs\n    `\u27e8_\u2194_\u27e9-spec (suc i) (suc j) (x \u2237 xs)\n       rewrite `\u27e8 i \u2194 j \u27e9-spec xs = refl\n\nmodule PermutationProperties {a : Level} where\n    open PermutationSyntax\n    open PermutationSemantics\n\n    \u229b-dist-\u00b7 : \u2200 {n} {A : \u2605 a} (fs : Vec (Endo A) n) (f : Perm n) xs \u2192 (f \u00b7 fs \u229b f \u00b7 xs) \u2261 f \u00b7 (fs \u229b xs)\n    \u229b-dist-\u00b7 _       `id        _  = refl\n    \u229b-dist-\u00b7 fs      `0\u21941       xs = \u229b-dist-0\u21941 fs xs\n    \u229b-dist-\u00b7 (_ \u2237 fs) (`tl f)   (_ \u2237 xs) rewrite \u229b-dist-\u00b7 fs f xs = refl\n    \u229b-dist-\u00b7 fs       (f `\u204f g)   xs rewrite \u229b-dist-\u00b7 (f \u00b7 fs) g (f \u00b7 xs)\n                                         | \u229b-dist-\u00b7 fs f xs = refl\n\n    \u00b7-replicate : \u2200 {n} {A : \u2605 a} (x : A) (f : Perm n) \u2192 f \u00b7 replicate x \u2261 replicate x\n    \u00b7-replicate x `id = refl\n    \u00b7-replicate x `0\u21941 = refl\n    \u00b7-replicate x (`tl f) rewrite \u00b7-replicate x f = refl\n    \u00b7-replicate x (f `\u204f g) rewrite \u00b7-replicate x f | \u00b7-replicate x g = refl\n\n    private\n        lem : \u2200 {n} {A : \u2605 a} (fs : Vec (Endo A) n) f xs\n              \u2192 fs \u229b f \u00b7 xs \u2261 f \u00b7 (f \u207b\u00b9 \u00b7 fs \u229b xs)\n        lem fs f xs rewrite sym (\u229b-dist-\u00b7 (f \u207b\u00b9 \u00b7 fs) f xs) | (f \u207b\u00b9-inverse\u2032) fs = refl\n\n    \u00b7-map : \u2200 {n} {A : \u2605 a} (f : Endo A) g (xs : Vec A n) \u2192 map f (g \u00b7 xs) \u2261 g \u00b7 map f xs\n    \u00b7-map f g xs rewrite sym (\u229b-dist-\u00b7 (replicate f) g xs) | \u00b7-replicate f g = refl\n\n    \u2295-dist-0\u21941 : \u2200 {n} (pad : Bits n) xs \u2192 0\u21941 pad \u2295 0\u21941 xs \u2261 0\u21941 (pad \u2295 xs)\n    \u2295-dist-0\u21941 _           []          = refl\n    \u2295-dist-0\u21941 (_ \u2237 [])    (_ \u2237 [])    = refl\n    \u2295-dist-0\u21941 (_ \u2237 _ \u2237 _) (_ \u2237 _ \u2237 _) = refl\n\n    -- TODO make use of \u229b-dist-\u00b7\n    \u2295-dist-\u00b7 : \u2200 {n} (pad : Bits n) \u03c0 xs \u2192 \u03c0 \u00b7 pad \u2295 \u03c0 \u00b7 xs \u2261 \u03c0 \u00b7 (pad \u2295 xs)\n    \u2295-dist-\u00b7 fs      `id        xs = refl\n    \u2295-dist-\u00b7 fs      `0\u21941       xs = \u2295-dist-0\u21941 fs xs\n    \u2295-dist-\u00b7 (f \u2237 fs) (`tl \u03c0)   (x \u2237 xs) rewrite \u2295-dist-\u00b7 fs \u03c0 xs = refl\n    \u2295-dist-\u00b7 fs       (\u03c0\u2080 `\u204f \u03c0\u2081) xs rewrite \u2295-dist-\u00b7 (\u03c0\u2080 \u00b7 fs) \u03c0\u2081 (\u03c0\u2080 \u00b7 xs)\n                                         | \u2295-dist-\u00b7 fs \u03c0\u2080 xs = refl\n\nprivate\n  module Unused where\n    module Foo where\n    {-\n       WRONG\n        \u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n        \u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n        \u27e80\u2194\u27e9-spec : \u2200 {n a} {A : \u2605 a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n        \u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n        -}\n\n    \u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    \u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n    \u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n    -- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n    \u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    \u27e8 zero  \u2194+1\u27e9 = 0\u21941\n    \u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n    \u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9 {A = A}\n    \u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = refl\n    \u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = refl\n\n    -- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n    --           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\n    rot-up-to : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    rot-up-to zero    = id\n    rot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n    -- Inverse of rot-up-to\n    rot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 Vec A n \u2192 Vec A n\n    rot\u207b\u00b9-up-to zero    = id\n    rot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\n    rot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : \u2605 a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\n    rot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = refl\n    rot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fc24f3ed8e691a865e43d3713264e63cb2ea19e7","subject":"NaPa: Remove broken import","message":"NaPa: Remove broken import\n","repos":"np\/NomPa","old_file":"lib\/NaPa\/LogicalRelation.agda","new_file":"lib\/NaPa\/LogicalRelation.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule NaPa.LogicalRelation where\n\nimport Level as L\nopen import Irrelevance\nopen import Data.Nat.NP hiding (_\u2294_) renaming (_+_ to _+\u2115_; _\u2264_ to _\u2264\u2115_; _\u2264?_ to _\u2264?\u2115_; _<_ to _<\u2115_\n                                              ;module == to ==\u2115; _==_ to _==\u2115_)\nopen import Data.Nat.Logical renaming (_\u225f_ to _\u225f\u2115_; _\u27e6+\u27e7_ to _\u27e6+\u2115\u27e7_)\nopen import Data.Nat.Properties as Nat\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Maybe.NP\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Sum.NP\nopen import Data.Product.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen \u2261.\u2261-Reasoning\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality as \u27f6\u2261 using (_\u27f6_; _\u27e8$\u27e9_)\nopen import Function.Injection.NP using (Injection; Injective; module\n    Injection; _\u2208_)\nopen import NaPa hiding (_\u2208_)\n\nprivate\n  module Inj = Injection\n  module \u2115s = Setoid \u27e6\u2115\u27e7-setoid\n  module ==\u2115s = Setoid ==\u2115.setoid\n  module \u2115e = Equality \u27e6\u2115\u27e7-equality\n\n_\u27e8$\u27e9\u1d62_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Injection From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\n_\u27e8$\u27e9\u1d62_ = \u27f6\u2261._\u27e8$\u27e9_ \u2218 Inj.to -- inj x = Inj.to inj \u27e8$\u27e9 x\n\n==\u2115-dec : \u2200 x y \u2192 \u27e6\ud835\udfda\u27e7 \u230a x \u225f\u2115 y \u230b (x ==\u2115 y)\n==\u2115-dec x y with x \u225f\u2115 y\n... | yes p = \u27e61\u2082\u27e7\u2032 (==\u2115s.reflexive (\u2115e.to-propositional p))\n... | no \u00acp = \u27e60\u2082\u27e7\u2032 (T'\u00ac'not (\u00acp \u2218 \u2115s.reflexive \u2218 ==\u2115.sound _ _))\n\n\u27e6dec\u27e7 : \u2200 {P Q : Set} (decP : Dec P) (decQ : Dec Q) \u2192 (P \u2192 Q) \u2192 (Q \u2192 P) \u2192 \u27e6\ud835\udfda\u27e7 \u230a decP \u230b \u230a decQ \u230b\n\u27e6dec\u27e7 (yes _) (yes _) _ _ = \u27e61\u2082\u27e7\n\u27e6dec\u27e7 (no  _) (no  _) _ _ = \u27e60\u2082\u27e7\n\u27e6dec\u27e7 (yes p) (no \u00acq) f _ = \u22a5-elim (\u00acq (f p))\n\u27e6dec\u27e7 (no \u00acp) (yes q) _ g = \u22a5-elim (\u00acp (g q))\n\n\u27e6World\u27e7 : (\u03b1\u2081 \u03b1\u2082 : World) \u2192 Set\n\u27e6World\u27e7 = Injection on Nm\n\n-- dup with NotSoFresh.LogicalRelation\n\u27e6WorldPred\u27e7 : (P\u2081 P\u2082 : Pred L.zero World) \u2192 Set\u2081\n\u27e6WorldPred\u27e7 = \u27e6Pred\u27e7 _ \u27e6World\u27e7\n\n\u27e6WorldRel\u27e7 : (R\u2081 R\u2082 : Rel World L.zero) \u2192 Set\u2081\n\u27e6WorldRel\u27e7 = \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = to-inj } where\n  to : Nm (\u03b1\u2081 \u21911) \u27f6 Nm (\u03b1\u2082 \u21911)\n  to = Name\u2192-to-Nm\u27f6 (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63))\n  to-inj : Injective to\n  to-inj {zero  , _} {zero  , _} zero    = zero\n  to-inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  to-inj {suc _ , _} {zero  , _} ()\n  to-inj {zero  , _} {suc _ , _} ()\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = inj } where\n  to : Nm (\u03b1\u2081 +1) \u27f6 Nm (\u03b1\u2082 +1)\n  to = Name\u2192-to-Nm\u27f6 (\u03bb x \u2192 suc\u1d3a (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (pred\u1d3a x)))\n  inj : Injective to\n  inj {zero  , _} {zero  , _} _       = zero\n  inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  inj {suc _ , _} {zero  , ()} _\n  inj {zero  , ()} {suc _ , _} _\n\n_\u27e6\u2191_\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 (suc k) = \u03b1\u1d63 \u27e6\u2191 k \u27e7 \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082 = ( x\u2081 , x\u2082 ) \u2208 \u03b1\u1d63\n\n\u27e6Name\u27e7\u2261 : \u27e6WorldPred\u27e7 Name Name\n\u27e6Name\u27e7\u2261 \u03b1\u1d63 x\u2081 x\u2082 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\u2081 \u2261 x\u2082\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = record { to = Name\u2192-to-Nm\u27f6 Name\u00f8-elim\n             ; injective = \u03bb{x} \u2192 Name\u00f8-elim x }\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 {_} {x\u2082} x\u1d63 {_} {y\u2082} y\u1d63 = helper (\u2261\u1d3a\u21d2\u2261 {_} {_} {x\u2082} x\u1d63) (\u2261\u1d3a\u21d2\u2261 {_} {_} {y\u2082} y\u1d63) where\n  helper : (\u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n  helper {x} \u2261.refl {y} \u2261.refl =\n       x ==\u1d3a y       \u2248\u27e8 sym (==\u2115-dec _ _) \u27e9\n       \u230a x \u225f\u1d3a y \u230b   \u2248\u27e8 \u27e6dec\u27e7 (x \u225f\u1d3a y) (x\u2032 \u225f\u1d3a y\u2032) (\u27f6\u2261.cong (Inj.to \u03b1\u1d63)) (Inj.injective \u03b1\u1d63) \u27e9\n       \u230a x\u2032 \u225f\u1d3a y\u2032 \u230b \u2248\u27e8 ==\u2115-dec _ _ \u27e9\n       x\u2032 ==\u1d3a y\u2032 \u220e\n    where\n      x\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\n      y\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 y\n      open \u27e6\ud835\udfda\u27e7-Props using (sym)\n      open \u27e6\ud835\udfda\u27e7-Reasoning\n\n\u27e6zero\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) zero\u1d3a zero\u1d3a\n\u27e6zero\u1d3a\u27e7 _ = zero\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 _ = suc\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 _ = suc\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 _ = \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _ zero x = x\n\u27e6add\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\n   = \u2115.suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-p-})))\n   \u2248\u27e8 \u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 x))) \u2115e.refl \u27e9\n     suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-q-})))\n   \u2248\u27e8 suc (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x) \u27e9\n     suc (k\u2082 +\u2115 name x\u2082)\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 suc k\u2081 , _{-p-}))\n   \u2248\u27e8 \u2261-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) (name-injective (\u2261.sym (\u2238-+-assoc (name x\u2081) 1 k\u2081))) \u27e9\n      name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 1 \u2238 k\u2081 , _{-r-}))\n   \u2248\u27e8 \u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {pred\u1d3a x\u2081} {pred\u1d3a x\u2082} (\u27e6pred\u1d3a\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) {x\u2081} {x\u2082} x\u1d63)  \u27e9\n      (name x\u2082 \u2238 1) \u2238 k\u2082\n   \u2248\u27e8 \u2115e.reflexive (\u2238-+-assoc (name x\u2082) 1 k\u2082) \u27e9\n      name x\u2082 \u2238 suc k\u2082\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n  helper zero                   \u2261.refl  = \u27e60\u2082\u27e7\n  helper (suc k\u1d63) {zero  , _ }  \u2261.refl  = \u27e61\u2082\u27e7\n  helper (suc k\u1d63) {suc x , pf\u2081} \u2261.refl  = helper k\u1d63 {x , pf\u2081} \u2261.refl\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) easy-cmp\u1d3a easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) easy-cmp\u1d3a easy-cmp\u1d3a\n  helper zero                             \u2261.refl = inj\u2082 (\u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) \u2115e.refl)\n  helper (suc k\u1d63)           {zero  , _ }  \u2261.refl = inj\u2081 zero\n  helper (suc {k\u2081} {k\u2082} k\u1d63) {suc x , pf\u2081} \u2261.refl\n    = \u27e6map\u27e7 _ _ _ _ suc suc (helper k\u1d63 {x , pf\u2081} \u2261.refl)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6protect\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n           (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) \u27e6\u2192\u27e7 (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7))) protect\u21911 protect\u21911\n\u27e6protect\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {f\u2081} {f\u2082} f\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n   helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (protect\u21911 f\u2081) (protect\u21911 f\u2082)\n   helper {zero ,  _}  \u2261.refl = zero\n   helper {suc x , pf} \u2261.refl = suc (f\u1d63 \u2115e.refl)\n\n_\u27e6\u2286\u27e7b_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk\u27e6\u2286\u27e7 : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x : \u03b1\u2081 \u2286 \u03b2\u2081} {y : \u03b1\u2082 \u2286 \u03b2\u2082} \u2192\n  _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y\nun\u27e6\u2286\u27e7 (mk\u27e6\u2286\u27e7 x) {a} {b} c = x {a} {b} c\n\n-- this is the start of an experimentation to see if the issue with \u27e6dropName\u27e7 is due\n-- to implicit lambdas.\n_\u27e6\u2286\u27e7e_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7e_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7e \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk\u27e6\u2286\u27e7 (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {_} {f\u2082} f g = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {_} {coerce\u1d3a f\u2082 x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 {_ , ()}\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , _} \u2261.refl = zero\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a \u2286-+1\u21911) (coerce\u1d3a \u2286-+1\u21911)\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc \u2115e.refl\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7) where\n   helper : (\u27e6\u00f8\u27e7 \u27e6\u2286\u27e7b \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n   helper {_ , ()}\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ _ \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ _ \u03b1+\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8 \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 _ {\u03b1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1\u2286\u00f8 (pred\u1d3a x))\n\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8 \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 _ {\u03b1+1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1+1\u2286\u00f8 (suc\u1d3a x))\n\n-- not primitive\n\n\u27e6pred\u1d3a?\u27e7-core : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) x\n                    \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x) (pred\u1d3a? (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x))\n\u27e6pred\u1d3a?\u27e7-core _  (zero  , _) = \u27e6nothing\u27e7\n\u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 (suc _ , _) = \u27e6just\u27e7 (\u27f6\u2261.cong (Inj.to \u03b1\u1d63) \u2115e.refl)\n\n\u27e6pred\u1d3a?\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) pred\u1d3a? pred\u1d3a?\n\u27e6pred\u1d3a?\u27e7 {_} {\u03b1\u2082} \u03b1\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)\n    -- BUG: the 'with' notation failed here\n  where helper : \u2200 {x\u2082} (eq : protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x\u2081 \u2261 x\u2082) \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x\u2081) (pred\u1d3a? x\u2082)\n        helper \u2261.refl = \u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 x\u2081\n\n\u27e6\u2286-ctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-ctx-+1\u21911 \u2286-ctx-+1\u21911\n\u27e6\u2286-ctx-+1\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u00acName\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u03b1\u1d63))) \u00acName \u00acName\n\u27e6\u00acName\u27e7 _ {\u03b1\u2286\u00f8\u2081} _ {x\u2081} _ = Name-elim \u03b1\u2286\u00f8\u2081 x\u2081\n\n\u27e6\u2286-cong-+\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)))) \u2286-cong-+ \u2286-cong-+\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 zero    = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 (suc k\u1d63) = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 k\u1d63)\n\n\u27e6\u2286-+-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+-inj \u2286-+-inj\n\u27e6\u2286-+-inj\u27e7 _ _ zero    \u03b1\u2286\u03b2\u1d63 = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-+-inj\u27e7 _ _ (suc k\u1d63) \u03b1\u2286\u03b2\u1d63 = \u27e6\u2286-+-inj\u27e7 _ _ k\u1d63 (\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-assoc-+\u2032\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u2081\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e8 k\u2082\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 (k\u2082\u1d63 \u27e6+\u2115\u27e7 k\u2081\u1d63)) \u27e6\u2286\u27e7 ((\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u2081\u1d63) \u27e6+\u1d42\u27e7 k\u2082\u1d63)))) \u2286-assoc-+\u2032 \u2286-assoc-+\u2032\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 zero      = \u27e6\u2286-cong-+\u27e7 _ _ base k\u2081\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 (suc k\u2082)  = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 k\u2082)\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule NaPa.LogicalRelation where\n\nimport Level as L\nopen import Abstractor\nopen import Irrelevance\nopen import Data.Nat.NP hiding (_\u2294_) renaming (_+_ to _+\u2115_; _\u2264_ to _\u2264\u2115_; _\u2264?_ to _\u2264?\u2115_; _<_ to _<\u2115_\n                                              ;module == to ==\u2115; _==_ to _==\u2115_)\nopen import Data.Nat.Logical renaming (_\u225f_ to _\u225f\u2115_; _\u27e6+\u27e7_ to _\u27e6+\u2115\u27e7_)\nopen import Data.Nat.Properties as Nat\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Maybe.NP\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Two.Logical\nopen import Data.Sum.NP\nopen import Data.Product.NP\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen \u2261.\u2261-Reasoning\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.Logical\nopen import Function\nopen import Function.Equality as \u27f6\u2261 using (_\u27f6_; _\u27e8$\u27e9_)\nopen import Function.Injection.NP using (Injection; Injective; module\n    Injection; _\u2208_)\nopen import NaPa hiding (_\u2208_)\n\nprivate\n  module Inj = Injection\n  module \u2115s = Setoid \u27e6\u2115\u27e7-setoid\n  module ==\u2115s = Setoid ==\u2115.setoid\n  module \u2115e = Equality \u27e6\u2115\u27e7-equality\n\n_\u27e8$\u27e9\u1d62_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Injection From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\n_\u27e8$\u27e9\u1d62_ = \u27f6\u2261._\u27e8$\u27e9_ \u2218 Inj.to -- inj x = Inj.to inj \u27e8$\u27e9 x\n\n==\u2115-dec : \u2200 x y \u2192 \u27e6\ud835\udfda\u27e7 \u230a x \u225f\u2115 y \u230b (x ==\u2115 y)\n==\u2115-dec x y with x \u225f\u2115 y\n... | yes p = \u27e61\u2082\u27e7\u2032 (==\u2115s.reflexive (\u2115e.to-propositional p))\n... | no \u00acp = \u27e60\u2082\u27e7\u2032 (T'\u00ac'not (\u00acp \u2218 \u2115s.reflexive \u2218 ==\u2115.sound _ _))\n\n\u27e6dec\u27e7 : \u2200 {P Q : Set} (decP : Dec P) (decQ : Dec Q) \u2192 (P \u2192 Q) \u2192 (Q \u2192 P) \u2192 \u27e6\ud835\udfda\u27e7 \u230a decP \u230b \u230a decQ \u230b\n\u27e6dec\u27e7 (yes _) (yes _) _ _ = \u27e61\u2082\u27e7\n\u27e6dec\u27e7 (no  _) (no  _) _ _ = \u27e60\u2082\u27e7\n\u27e6dec\u27e7 (yes p) (no \u00acq) f _ = \u22a5-elim (\u00acq (f p))\n\u27e6dec\u27e7 (no \u00acp) (yes q) _ g = \u22a5-elim (\u00acp (g q))\n\n\u27e6World\u27e7 : (\u03b1\u2081 \u03b1\u2082 : World) \u2192 Set\n\u27e6World\u27e7 = Injection on Nm\n\n-- dup with NotSoFresh.LogicalRelation\n\u27e6WorldPred\u27e7 : (P\u2081 P\u2082 : Pred L.zero World) \u2192 Set\u2081\n\u27e6WorldPred\u27e7 = \u27e6Pred\u27e7 _ \u27e6World\u27e7\n\n\u27e6WorldRel\u27e7 : (R\u2081 R\u2082 : Rel World L.zero) \u2192 Set\u2081\n\u27e6WorldRel\u27e7 = \u27e6Rel\u27e7 \u27e6World\u27e7 L.zero\n\n_\u27e6\u21911\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u21911 _\u21911\n_\u27e6\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = to-inj } where\n  to : Nm (\u03b1\u2081 \u21911) \u27f6 Nm (\u03b1\u2082 \u21911)\n  to = Name\u2192-to-Nm\u27f6 (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63))\n  to-inj : Injective to\n  to-inj {zero  , _} {zero  , _} zero    = zero\n  to-inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  to-inj {suc _ , _} {zero  , _} ()\n  to-inj {zero  , _} {suc _ , _} ()\n\n_\u27e6+1\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+1 _+1\n_\u27e6+1\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 = record { to = to; injective = inj } where\n  to : Nm (\u03b1\u2081 +1) \u27f6 Nm (\u03b1\u2082 +1)\n  to = Name\u2192-to-Nm\u27f6 (\u03bb x \u2192 suc\u1d3a (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (pred\u1d3a x)))\n  inj : Injective to\n  inj {zero  , _} {zero  , _} _       = zero\n  inj {suc _ , _} {suc _ , _} (suc e) = suc (Inj.injective \u03b1\u1d63 e)\n  inj {suc _ , _} {zero  , ()} _\n  inj {zero  , ()} {suc _ , _} _\n\n_\u27e6\u2191_\u27e7 : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _\u2191_ _\u2191_\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6\u2191_\u27e7 \u03b1\u1d63 (suc k) = \u03b1\u1d63 \u27e6\u2191 k \u27e7 \u27e6\u21911\u27e7\n\n_\u27e6+\u1d42\u27e7_ : (\u27e6World\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6World\u27e7) _+\u1d42_ _+\u1d42_\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 zero    = \u03b1\u1d63\n_\u27e6+\u1d42\u27e7_ \u03b1\u1d63 (suc k) = (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k) \u27e6+1\u27e7\n\n\u27e6Name\u27e7 : \u27e6Pred\u27e7 L.zero \u27e6World\u27e7 Name Name\n\u27e6Name\u27e7 \u03b1\u1d63 x\u2081 x\u2082 = ( x\u2081 , x\u2082 ) \u2208 \u03b1\u1d63\n\n\u27e6Name\u27e7\u2261 : \u27e6WorldPred\u27e7 Name Name\n\u27e6Name\u27e7\u2261 \u03b1\u1d63 x\u2081 x\u2082 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\u2081 \u2261 x\u2082\n\n\u27e6\u00f8\u27e7 : \u27e6World\u27e7 \u00f8 \u00f8\n\u27e6\u00f8\u27e7 = record { to = Name\u2192-to-Nm\u27f6 Name\u00f8-elim\n             ; injective = \u03bb{x} \u2192 Name\u00f8-elim x }\n\n_\u27e6==\u1d3a\u27e7_ : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n_\u27e6==\u1d3a\u27e7_ \u03b1\u1d63 {_} {x\u2082} x\u1d63 {_} {y\u2082} y\u1d63 = helper (\u2261\u1d3a\u21d2\u2261 {_} {_} {x\u2082} x\u1d63) (\u2261\u1d3a\u21d2\u2261 {_} {_} {y\u2082} y\u1d63) where\n  helper : (\u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) _==\u1d3a_ _==\u1d3a_\n  helper {x} \u2261.refl {y} \u2261.refl =\n       x ==\u1d3a y       \u2248\u27e8 sym (==\u2115-dec _ _) \u27e9\n       \u230a x \u225f\u1d3a y \u230b   \u2248\u27e8 \u27e6dec\u27e7 (x \u225f\u1d3a y) (x\u2032 \u225f\u1d3a y\u2032) (\u27f6\u2261.cong (Inj.to \u03b1\u1d63)) (Inj.injective \u03b1\u1d63) \u27e9\n       \u230a x\u2032 \u225f\u1d3a y\u2032 \u230b \u2248\u27e8 ==\u2115-dec _ _ \u27e9\n       x\u2032 ==\u1d3a y\u2032 \u220e\n    where\n      x\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 x\n      y\u2032 = \u03b1\u1d63 \u27e8$\u27e9\u1d62 y\n      open \u27e6\ud835\udfda\u27e7-Props using (sym)\n      open \u27e6\ud835\udfda\u27e7-Reasoning\n\n\u27e6zero\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) zero\u1d3a zero\u1d3a\n\u27e6zero\u1d3a\u27e7 _ = zero\n\n\u27e6suc\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7)) suc\u1d3a suc\u1d3a\n\u27e6suc\u1d3a\u27e7 _ = suc\n\n\u27e6suc\u1d3a\u2191\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) suc\u1d3a\u2191 suc\u1d3a\u2191\n\u27e6suc\u1d3a\u2191\u27e7 _ = suc\n\n\u27e6pred\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) pred\u1d3a pred\u1d3a\n\u27e6pred\u1d3a\u27e7 _ = \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1)\n\n\u27e6add\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63))) add\u1d3a add\u1d3a\n\u27e6add\u1d3a\u27e7 _ zero x = x\n\u27e6add\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\n   = \u2115.suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-p-})))\n   \u2248\u27e8 \u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 x))) \u2115e.refl \u27e9\n     suc (name ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e8$\u27e9\u1d62 (k\u2081 +\u2115 name x\u2081 , _{-q-})))\n   \u2248\u27e8 suc (\u27e6add\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x) \u27e9\n     suc (k\u2082 +\u2115 name x\u2082)\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6subtract\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b1\u1d63) subtract\u1d3a subtract\u1d3a\n\u27e6subtract\u1d3a\u27e7 _ zero x = x\n\u27e6subtract\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 (suc {k\u2081} {k\u2082} k\u1d63) {x\u2081} {x\u2082} x\u1d63\n   = name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 suc k\u2081 , _{-p-}))\n   \u2248\u27e8 \u2261-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) (name-injective (\u2261.sym (\u2238-+-assoc (name x\u2081) 1 k\u2081))) \u27e9\n      name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 (name x\u2081 \u2238 1 \u2238 k\u2081 , _{-r-}))\n   \u2248\u27e8 \u27e6subtract\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {pred\u1d3a x\u2081} {pred\u1d3a x\u2082} (\u27e6pred\u1d3a\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) {x\u2081} {x\u2082} x\u1d63)  \u27e9\n      (name x\u2082 \u2238 1) \u2238 k\u2082\n   \u2248\u27e8 \u2115e.reflexive (\u2238-+-assoc (name x\u2082) 1 k\u2082) \u27e9\n      name x\u2082 \u2238 suc k\u2082\n   \u220e where open \u27e6\u2115\u27e7-Reasoning\n\n\u27e6cmp\u1d3a-bool\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n\u27e6cmp\u1d3a-bool\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6\ud835\udfda\u27e7) cmp\u1d3a-bool cmp\u1d3a-bool\n  helper zero                   \u2261.refl  = \u27e60\u2082\u27e7\n  helper (suc k\u1d63) {zero  , _ }  \u2261.refl  = \u27e61\u2082\u27e7\n  helper (suc k\u1d63) {suc x , pf\u2081} \u2261.refl  = helper k\u1d63 {x , pf\u2081} \u2261.refl\n\n\u27e6easy-cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n             ) easy-cmp\u1d3a easy-cmp\u1d3a\n\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper k\u1d63 {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n  helper : (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) easy-cmp\u1d3a easy-cmp\u1d3a\n  helper zero                             \u2261.refl = inj\u2082 (\u2261\u1d3a-\u27e6\u2115\u27e7-cong (\u03bb x \u2192 name (\u03b1\u1d63 \u27e8$\u27e9\u1d62 x)) \u2115e.refl)\n  helper (suc k\u1d63)           {zero  , _ }  \u2261.refl = inj\u2081 zero\n  helper (suc {k\u2081} {k\u2082} k\u1d63) {suc x , pf\u2081} \u2261.refl\n    = \u27e6map\u27e7 _ _ _ _ suc suc (helper k\u1d63 {x , pf\u2081} \u2261.refl)\n\n\u27e6cmp\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)\n         ) cmp\u1d3a cmp\u1d3a\n\u27e6cmp\u1d3a\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {k\u2081} {k\u2082} k\u1d63 {x\u2081} {x\u2082} x\u1d63 =\n   \u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) (cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) x)\n           (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082)\n           (\u2261.subst (\u03bb x \u2192 (\u27e6Name\u27e7 (\u27e6\u00f8\u27e7 \u27e6\u2191 k\u1d63 \u27e7) \u27e6\u228e\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) x\n                    (easy-cmp\u1d3a {\u03b1\u2082} k\u2082 x\u2082))\n                    (easy-cmp\u1d3a\u2257cmp\u1d3a {\u03b1\u2081} k\u2081 x\u2081) (\u27e6easy-cmp\u1d3a\u27e7 \u03b1\u1d63 k\u1d63 {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6protect\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n           (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) \u27e6\u2192\u27e7 (\u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7))) protect\u21911 protect\u21911\n\u27e6protect\u21911\u27e7 {\u03b1\u2081} {\u03b1\u2082} \u03b1\u1d63 {\u03b2\u2081} {\u03b2\u2082} \u03b2\u1d63 {f\u2081} {f\u2082} f\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63) where\n   helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (protect\u21911 f\u2081) (protect\u21911 f\u2082)\n   helper {zero ,  _}  \u2261.refl = zero\n   helper {suc x , pf} \u2261.refl = suc (f\u1d63 \u2115e.refl)\n\n_\u27e6\u2286\u27e7b_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\ndata _\u27e6\u2286\u27e7_ {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) {\u03b2\u2081 \u03b2\u2082} (\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082) (x : \u03b1\u2081 \u2286 \u03b2\u2081) (y : \u03b1\u2082 \u2286 \u03b2\u2082) : Set where\n  mk\u27e6\u2286\u27e7 : _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y\n\nun\u27e6\u2286\u27e7 : \u2200 {\u03b1\u2081 \u03b1\u2082} {\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082} {\u03b2\u2081 \u03b2\u2082} {\u03b2\u1d63 : \u27e6World\u27e7 \u03b2\u2081 \u03b2\u2082} {x : \u03b1\u2081 \u2286 \u03b2\u2081} {y : \u03b1\u2082 \u2286 \u03b2\u2082} \u2192\n  _\u27e6\u2286\u27e7_ \u03b1\u1d63 \u03b2\u1d63 x y \u2192 _\u27e6\u2286\u27e7b_ \u03b1\u1d63 \u03b2\u1d63 x y\nun\u27e6\u2286\u27e7 (mk\u27e6\u2286\u27e7 x) {a} {b} c = x {a} {b} c\n\n-- this is the start of an experimentation to see if the issue with \u27e6dropName\u27e7 is due\n-- to implicit lambdas.\n_\u27e6\u2286\u27e7e_ : \u27e6WorldRel\u27e7 _\u2286_ _\u2286_\n_\u27e6\u2286\u27e7e_ \u03b1\u1d63 \u03b2\u1d63 \u03b1\u2286\u03b2\u2081 \u03b1\u2286\u03b2\u2082\n  = (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7e \u27e6Name\u27e7 \u03b2\u1d63) (coerce\u1d3a \u03b1\u2286\u03b2\u2081) (coerce\u1d3a \u03b1\u2286\u03b2\u2082)\n\n\u27e6\u2286-refl\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-refl \u2286-refl\n\u27e6\u2286-refl\u27e7 _ = mk\u27e6\u2286\u27e7 (\u03bb x\u1d63 \u2192 x\u1d63)\n\n\u27e6\u2286-trans\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b3\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7\n                  \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b2\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u2286\u27e7 \u03b3\u1d63))) \u2286-trans \u2286-trans\n\u27e6\u2286-trans\u27e7 _ _ _ {_} {f\u2082} f g = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 un\u27e6\u2286\u27e7 g {_} {coerce\u1d3a f\u2082 x\u2082} (un\u27e6\u2286\u27e7 f {x\u2081} {x\u2082} x\u1d63))\n\n\u27e6coerce\u1d3a\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63 \u27e6\u2192\u27e7 \u27e6Name\u27e7 \u03b2\u1d63)) coerce\u1d3a coerce\u1d3a\n\u27e6coerce\u1d3a\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63 = un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x\u2081} {x\u2082} x\u1d63\n\n\u27e6\u00acName\u00f8\u27e7 : (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u27e6\u00f8\u27e7)) \u00acName\u00f8 \u00acName\u00f8\n\u27e6\u00acName\u00f8\u27e7 {_ , ()}\n\n\u27e6\u2286-cong-\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-cong-\u21911 \u2286-cong-\u21911\n\u27e6\u2286-cong-\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , _} \u2261.refl = zero\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-cong-+1\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) \u2286-cong-+1 \u2286-cong-+1\n\u27e6\u2286-cong-+1\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-cong-+1 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u2286-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) \u2286-+1\u21911 \u2286-+1\u21911\n\u27e6\u2286-+1\u21911\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a \u2286-+1\u21911) (coerce\u1d3a \u2286-+1\u21911)\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc \u2115e.refl\n\n\u27e6\u2286-\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2286\u27e7 \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n\u27e6\u2286-\u00f8\u27e7 \u03b1\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {\u03b7\u2081} {\u03b7\u2082} \u03b7 \u2192 helper {\u03b7\u2081} {\u03b7\u2082} \u03b7) where\n   helper : (\u27e6\u00f8\u27e7 \u27e6\u2286\u27e7b \u03b1\u1d63) \u2286-\u00f8 \u2286-\u00f8\n   helper {_ , ()}\n\n\u27e6\u2286-+1-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+1-inj \u2286-+1-inj\n\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1+\u2286\u03b2+\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2+\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-\u21911-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-\u21911-inj \u2286-\u21911-inj\n\u27e6\u2286-\u21911-inj\u27e7 _ _ \u03b1\u2191\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1\u2191\u2286\u03b2\u2191\u1d63 {suc\u1d3a\u2191 x\u2081} {suc\u1d3a\u2191 x\u2082} (suc x\u1d63)))\n\n\u27e6\u2286-unctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-unctx-+1\u21911 \u2286-unctx-+1\u21911\n\u27e6\u2286-unctx-+1\u21911\u27e7 _ _ \u03b1+\u2286\u03b2\u2191\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 \u27e6\u2115\u27e7-cong (\u03bb x \u2192 x \u2238 1) (un\u27e6\u2286\u27e7 \u03b1+\u2286\u03b2\u2191\u1d63 {suc\u1d3a x\u2081} {suc\u1d3a x\u2082} (suc x\u1d63)))\n\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8 \u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\n\u27e6\u03b1\u2286\u00f8\u2192\u03b1+1\u2286\u00f8\u27e7 _ {\u03b1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1\u2286\u00f8 (pred\u1d3a x))\n\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6+1\u27e7 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7) \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8 \u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\n\u27e6\u03b1+1\u2286\u00f8\u2192\u03b1\u2286\u00f8\u27e7 _ {\u03b1+1\u2286\u00f8} _ = mk\u27e6\u2286\u27e7 (\u03bb {x} _ \u2192 Name-elim \u03b1+1\u2286\u00f8 (suc\u1d3a x))\n\n-- not primitive\n\n\u27e6pred\u1d3a?\u27e7-core : \u2200 {\u03b1\u2081 \u03b1\u2082} (\u03b1\u1d63 : \u27e6World\u27e7 \u03b1\u2081 \u03b1\u2082) x\n                    \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x) (pred\u1d3a? (protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x))\n\u27e6pred\u1d3a?\u27e7-core _  (zero  , _) = \u27e6nothing\u27e7\n\u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 (suc _ , _) = \u27e6just\u27e7 (\u27f6\u2261.cong (Inj.to \u03b1\u1d63) \u2115e.refl)\n\n\u27e6pred\u1d3a?\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b1\u1d63 \u27e6\u21911\u27e7) \u27e6\u2192\u27e7 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63)) pred\u1d3a? pred\u1d3a?\n\u27e6pred\u1d3a?\u27e7 {_} {\u03b1\u2082} \u03b1\u1d63 {x\u2081} {x\u2082} x\u1d63 = helper {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)\n    -- BUG: the 'with' notation failed here\n  where helper : \u2200 {x\u2082} (eq : protect\u21911 (_\u27e8$\u27e9\u1d62_ \u03b1\u1d63) x\u2081 \u2261 x\u2082) \u2192 \u27e6Maybe\u27e7 (\u27e6Name\u27e7 \u03b1\u1d63) (pred\u1d3a? x\u2081) (pred\u1d3a? x\u2082)\n        helper \u2261.refl = \u27e6pred\u1d3a?\u27e7-core \u03b1\u1d63 x\u2081\n\n\u27e6\u2286-ctx-+1\u21911\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) \u2286-ctx-+1\u21911 \u2286-ctx-+1\u21911\n\u27e6\u2286-ctx-+1\u21911\u27e7 \u03b1\u1d63 \u03b2\u1d63 {\u03b1\u2286\u03b2\u2081} {\u03b1\u2286\u03b2\u2082} \u03b1\u2286\u03b2\u1d63 = mk\u27e6\u2286\u27e7 (\u03bb {x\u2081} {x\u2082} x\u1d63 \u2192 helper {x\u2081} {x\u2082} (\u2261\u1d3a\u21d2\u2261 x\u1d63)) where\n  helper : (\u27e6Name\u27e7\u2261 (\u03b1\u1d63 \u27e6+1\u27e7) \u27e6\u2192\u27e7 \u27e6Name\u27e7 (\u03b2\u1d63 \u27e6\u21911\u27e7)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2081)) (coerce\u1d3a (\u2286-ctx-+1\u21911 \u03b1\u2286\u03b2\u2082))\n  helper {zero  , ()} _\n  helper {suc x , p} \u2261.refl = suc (un\u27e6\u2286\u27e7 \u03b1\u2286\u03b2\u1d63 {x , p} {\u03b1\u1d63 \u27e8$\u27e9\u1d62 (x , p)} \u2115e.refl)\n\n\u27e6\u00acName\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u27e6\u00f8\u27e7 \u27e6\u2192\u27e7 (\u27e6\u00ac\u27e7(\u27e6Name\u27e7 \u03b1\u1d63))) \u00acName \u00acName\n\u27e6\u00acName\u27e7 _ {\u03b1\u2286\u00f8\u2081} _ {x\u2081} _ = Name-elim \u03b1\u2286\u00f8\u2081 x\u2081\n\n\u27e6\u2286-cong-+\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)))) \u2286-cong-+ \u2286-cong-+\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 zero    = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 (suc k\u1d63) = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-cong-+\u27e7 _ _ \u03b1\u2286\u03b2\u1d63 k\u1d63)\n\n\u27e6\u2286-+-inj\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u27e8 k\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 ((\u03b1\u1d63 \u27e6+\u1d42\u27e7 k\u1d63) \u27e6\u2286\u27e7 (\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u1d63)) \u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63) \u2286-+-inj \u2286-+-inj\n\u27e6\u2286-+-inj\u27e7 _ _ zero    \u03b1\u2286\u03b2\u1d63 = \u03b1\u2286\u03b2\u1d63\n\u27e6\u2286-+-inj\u27e7 _ _ (suc k\u1d63) \u03b1\u2286\u03b2\u1d63 = \u27e6\u2286-+-inj\u27e7 _ _ k\u1d63 (\u27e6\u2286-+1-inj\u27e7 _ _ \u03b1\u2286\u03b2\u1d63)\n\n\u27e6\u2286-assoc-+\u2032\u27e7 : (\u2200\u27e8 \u03b1\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u2200\u27e8 \u03b2\u1d63 \u2236 \u27e6World\u27e7 \u27e9\u27e6\u2192\u27e7 \u03b1\u1d63 \u27e6\u2286\u27e7 \u03b2\u1d63 \u27e6\u2192\u27e7 (\u27e8 k\u2081\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7 (\u27e8 k\u2082\u1d63 \u2236 \u27e6\u2115\u27e7 \u27e9\u27e6\u2192\u27e7\n                 (\u03b1\u1d63 \u27e6+\u1d42\u27e7 (k\u2082\u1d63 \u27e6+\u2115\u27e7 k\u2081\u1d63)) \u27e6\u2286\u27e7 ((\u03b2\u1d63 \u27e6+\u1d42\u27e7 k\u2081\u1d63) \u27e6+\u1d42\u27e7 k\u2082\u1d63)))) \u2286-assoc-+\u2032 \u2286-assoc-+\u2032\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 zero      = \u27e6\u2286-cong-+\u27e7 _ _ base k\u2081\n\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 (suc k\u2082)  = \u27e6\u2286-cong-+1\u27e7 _ _ (\u27e6\u2286-assoc-+\u2032\u27e7 _ _ base k\u2081 k\u2082)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c9c4841ba2fc0056950103df3a6997c29cd38a1f","subject":"Agda: \u2259-symm \u21a6 \u2259-sym, for consistency with sym for \u2261","message":"Agda: \u2259-symm \u21a6 \u2259-sym, for consistency with sym for \u2261\n\nOld-commit-hash: 590f75f6eea150a3f02693ff2ac470479b6a7cfe\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Equivalence.agda","new_file":"Base\/Change\/Equivalence.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"8cadba32beaf296786f42a5f8d8070424a35b379","subject":"Added automatic proofs.","message":"Added automatic proofs.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Bool\/PropertiesATP.agda","new_file":"src\/fot\/FOTC\/Data\/Bool\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- The Booleans properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Bool.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Basic properties\n\npostulate t&&x\u2261x : \u2200 b \u2192 true && b \u2261 b\n{-# ATP prove t&&x\u2261x #-}\n\npostulate f&&x\u2261f : \u2200 b \u2192 false && b \u2261 false\n{-# ATP prove f&&x\u2261f #-}\n\n&&-Bool : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 Bool (a && b)\n&&-Bool {b = b} btrue Bb = prf\n  where postulate prf : Bool (true && b)\n        {-# ATP prove prf #-}\n&&-Bool {b = b} bfalse Bb = prf\n  where postulate prf : Bool (false && b)\n        {-# ATP prove prf #-}\n\nnot-Bool : \u2200 {b} \u2192 Bool b \u2192 Bool (not b)\nnot-Bool btrue = prf\n  where postulate prf : Bool (not true)\n        {-# ATP prove prf #-}\nnot-Bool bfalse = prf\n  where postulate prf : Bool (not false)\n        {-# ATP prove prf #-}\n\n&&-list\u2082-t : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 a && b \u2261 true \u2192 a \u2261 true \u2227 b \u2261 true\n&&-list\u2082-t btrue btrue   h = refl , refl\n&&-list\u2082-t btrue bfalse  h = \u22a5-elim (t\u2262f (trans (sym h) (t&&x\u2261x false)))\n&&-list\u2082-t bfalse btrue  h = \u22a5-elim (t\u2262f (trans (sym h) (f&&x\u2261f true)))\n&&-list\u2082-t bfalse bfalse h = \u22a5-elim (t\u2262f (trans (sym h) (f&&x\u2261f false)))\n\n&&-list\u2082-t\u2081 : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 a && b \u2261 true \u2192 a \u2261 true\n&&-list\u2082-t\u2081 Ba Bb h = \u2227-proj\u2081 (&&-list\u2082-t Ba Bb h)\n\n&&-list\u2082-t\u2082 : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 a && b \u2261 true \u2192 b \u2261 true\n&&-list\u2082-t\u2082 Ba Bb h = \u2227-proj\u2082 (&&-list\u2082-t Ba Bb h)\n\n&&-list\u2084-some-f : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n                  a \u2261 false \u2228 b \u2261 false \u2228 c \u2261 false \u2228 d \u2261 false \u2192\n                  a && b && c && d \u2261 false\n&&-list\u2084-some-f btrue Bb Bc Bd (inj\u2081 h) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue Bc Bd (inj\u2082 (inj\u2081 h)) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue btrue Bd (inj\u2082 (inj\u2082 (inj\u2081 h))) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue btrue btrue (inj\u2082 (inj\u2082 (inj\u2082 h))) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue btrue bfalse (inj\u2082 (inj\u2082 (inj\u2082 h))) =\n  trans (t&&x\u2261x (true && true && false))\n        (trans (t&&x\u2261x (true && false)) (t&&x\u2261x false))\n&&-list\u2084-some-f btrue btrue bfalse btrue (inj\u2082 (inj\u2082 (inj\u2081 h))) =\n  trans (t&&x\u2261x (true && false && true))\n        (trans (t&&x\u2261x (false && true)) (f&&x\u2261f true))\n&&-list\u2084-some-f btrue btrue bfalse btrue (inj\u2082 (inj\u2082 (inj\u2082 h))) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue bfalse bfalse (inj\u2082 (inj\u2082 h)) =\n  trans (t&&x\u2261x (true && false && false))\n        (trans (t&&x\u2261x (false && false)) (f&&x\u2261f false))\n&&-list\u2084-some-f {c = c} {d} btrue bfalse Bc Bd (inj\u2082 h) =\n  trans (t&&x\u2261x (false && c && d)) (f&&x\u2261f (c && d))\n&&-list\u2084-some-f {b = b} {c} {d} bfalse Bb Bc Bd _ = f&&x\u2261f (b && c && d)\n\n&&-list\u2084-t : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n             a && b && c && d \u2261 true \u2192\n             a \u2261 true \u2227 b \u2261 true \u2227 c \u2261 true \u2227 d \u2261 true\n&&-list\u2084-t btrue btrue btrue btrue h = refl , refl , refl , refl\n&&-list\u2084-t btrue btrue btrue bfalse h =\n  \u22a5-elim (t\u2262f\n           (trans (sym h)\n                  (&&-list\u2084-some-f btrue btrue btrue bfalse (inj\u2082 (inj\u2082 (inj\u2082 refl))))))\n&&-list\u2084-t btrue btrue bfalse Bd h =\n  \u22a5-elim (t\u2262f\n           (trans (sym h)\n                  (&&-list\u2084-some-f btrue btrue bfalse Bd (inj\u2082 (inj\u2082 (inj\u2081 refl))))))\n&&-list\u2084-t btrue bfalse Bc Bd h =\n  \u22a5-elim (t\u2262f\n           (trans (sym h) (&&-list\u2084-some-f btrue bfalse Bc Bd (inj\u2082 (inj\u2081 refl)))))\n&&-list\u2084-t bfalse Bb Bc Bd h =\n  \u22a5-elim (t\u2262f (trans (sym h) (&&-list\u2084-some-f bfalse Bb Bc Bd (inj\u2081 refl))))\n\n&&-list\u2084-t\u2081 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 a \u2261 true\n&&-list\u2084-t\u2081 {a} Ba Bb Bc Bd h = prf\n  where postulate prf : a \u2261 true\n        {-# ATP prove prf &&-list\u2084-t #-}\n\n&&-list\u2084-t\u2082 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 b \u2261 true\n&&-list\u2084-t\u2082 {b = b} Ba Bb Bc Bd h = prf\n  where postulate prf : b \u2261 true\n        {-# ATP prove prf &&-list\u2084-t #-}\n\n&&-list\u2084-t\u2083 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 c \u2261 true\n&&-list\u2084-t\u2083 {c = c} Ba Bb Bc Bd h = prf\n  where postulate prf : c \u2261 true\n        {-# ATP prove prf &&-list\u2084-t #-}\n\n&&-list\u2084-t\u2084 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 d \u2261 true\n&&-list\u2084-t\u2084 {d = d} Ba Bb Bc Bd h = prf\n  where postulate prf : d \u2261 true\n        {-# ATP prove prf &&-list\u2084-t #-}\n\nx\u2262not-x : \u2200 {b} \u2192 Bool b \u2192 b \u2262 not b\nx\u2262not-x btrue = prf\n  where postulate prf : true \u2261 not true \u2192 \u22a5\n        {-# ATP prove prf #-}\nx\u2262not-x bfalse = prf\n  where postulate prf : false \u2261 not false \u2192 \u22a5\n        {-# ATP prove prf #-}\n\nnot-x\u2262x : \u2200 {b} \u2192 Bool b \u2192 not b \u2262 b\nnot-x\u2262x Bb h = x\u2262not-x Bb (sym h)\n\nnot-involutive : \u2200 {b} \u2192 Bool b \u2192 not (not b) \u2261 b\nnot-involutive btrue = prf\n  where postulate prf : not (not true) \u2261 true\n        {-# ATP prove prf #-}\nnot-involutive bfalse = prf\n  where postulate prf : not (not false) \u2261 false\n        {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties with inequalities\n\nlt-Bool : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Bool (lt m n)\nlt-Bool nzero nzero = prf\n  where postulate prf : Bool (lt zero zero)\n        {-# ATP prove prf #-}\nlt-Bool nzero (nsucc {n} Nn) = prf\n  where postulate prf : Bool (lt zero (succ\u2081 n))\n        {-# ATP prove prf #-}\nlt-Bool (nsucc {m} Nm) nzero = prf\n  where postulate prf : Bool (lt (succ\u2081 m)  zero)\n        {-# ATP prove prf #-}\nlt-Bool (nsucc {m} Nm) (nsucc {n} Nn) = prf (lt-Bool Nm Nn)\n  where postulate prf : Bool (lt m n) \u2192 Bool (lt (succ\u2081 m) (succ\u2081 n))\n        {-# ATP prove prf #-}\n\nle-Bool : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Bool (le m n)\nle-Bool {n = n} nzero Nn = prf\n  where postulate prf : Bool (le zero n)\n        {-# ATP prove prf #-}\nle-Bool (nsucc {m} Nm) nzero = prf\n  where postulate prf : Bool (le (succ\u2081 m) zero)\n        {-# ATP prove prf Sx\u22700 #-}\nle-Bool (nsucc {m} Nm) (nsucc {n} Nn) = prf (le-Bool Nm Nn)\n  where postulate prf : Bool (le m n) \u2192 Bool (le (succ\u2081 m) (succ\u2081 n))\n        {-# ATP prove prf #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The Booleans properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Bool.PropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Basic properties\n\npostulate t&&x\u2261x : \u2200 b \u2192 true && b \u2261 b\n{-# ATP prove t&&x\u2261x #-}\n\npostulate f&&x\u2261f : \u2200 b \u2192 false && b \u2261 false\n{-# ATP prove f&&x\u2261f #-}\n\n&&-Bool : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 Bool (a && b)\n&&-Bool {b = b} btrue Bb = prf\n  where postulate prf : Bool (true && b)\n        {-# ATP prove prf #-}\n&&-Bool {b = b} bfalse Bb = prf\n  where postulate prf : Bool (false && b)\n        {-# ATP prove prf #-}\n\nnot-Bool : \u2200 {b} \u2192 Bool b \u2192 Bool (not b)\nnot-Bool btrue = prf\n  where postulate prf : Bool (not true)\n        {-# ATP prove prf #-}\nnot-Bool bfalse = prf\n  where postulate prf : Bool (not false)\n        {-# ATP prove prf #-}\n\n&&-list\u2082-t : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 a && b \u2261 true \u2192 a \u2261 true \u2227 b \u2261 true\n&&-list\u2082-t btrue btrue   h = refl , refl\n&&-list\u2082-t btrue bfalse  h = \u22a5-elim (t\u2262f (trans (sym h) (t&&x\u2261x false)))\n&&-list\u2082-t bfalse btrue  h = \u22a5-elim (t\u2262f (trans (sym h) (f&&x\u2261f true)))\n&&-list\u2082-t bfalse bfalse h = \u22a5-elim (t\u2262f (trans (sym h) (f&&x\u2261f false)))\n\n&&-list\u2082-t\u2081 : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 a && b \u2261 true \u2192 a \u2261 true\n&&-list\u2082-t\u2081 Ba Bb h = \u2227-proj\u2081 (&&-list\u2082-t Ba Bb h)\n\n&&-list\u2082-t\u2082 : \u2200 {a b} \u2192 Bool a \u2192 Bool b \u2192 a && b \u2261 true \u2192 b \u2261 true\n&&-list\u2082-t\u2082 Ba Bb h = \u2227-proj\u2082 (&&-list\u2082-t Ba Bb h)\n\n&&-list\u2084-some-f : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n                  a \u2261 false \u2228 b \u2261 false \u2228 c \u2261 false \u2228 d \u2261 false \u2192\n                  a && b && c && d \u2261 false\n&&-list\u2084-some-f btrue Bb Bc Bd (inj\u2081 h) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue Bc Bd (inj\u2082 (inj\u2081 h)) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue btrue Bd (inj\u2082 (inj\u2082 (inj\u2081 h))) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue btrue btrue (inj\u2082 (inj\u2082 (inj\u2082 h))) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue btrue bfalse (inj\u2082 (inj\u2082 (inj\u2082 h))) =\n  trans (t&&x\u2261x (true && true && false))\n        (trans (t&&x\u2261x (true && false)) (t&&x\u2261x false))\n&&-list\u2084-some-f btrue btrue bfalse btrue (inj\u2082 (inj\u2082 (inj\u2081 h))) =\n  trans (t&&x\u2261x (true && false && true))\n        (trans (t&&x\u2261x (false && true)) (f&&x\u2261f true))\n&&-list\u2084-some-f btrue btrue bfalse btrue (inj\u2082 (inj\u2082 (inj\u2082 h))) = \u22a5-elim (t\u2262f h)\n&&-list\u2084-some-f btrue btrue bfalse bfalse (inj\u2082 (inj\u2082 h)) =\n  trans (t&&x\u2261x (true && false && false))\n        (trans (t&&x\u2261x (false && false)) (f&&x\u2261f false))\n&&-list\u2084-some-f {c = c} {d} btrue bfalse Bc Bd (inj\u2082 h) =\n  trans (t&&x\u2261x (false && c && d)) (f&&x\u2261f (c && d))\n&&-list\u2084-some-f {b = b} {c} {d} bfalse Bb Bc Bd _ = f&&x\u2261f (b && c && d)\n\n&&-list\u2084-t : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n             a && b && c && d \u2261 true \u2192\n             a \u2261 true \u2227 b \u2261 true \u2227 c \u2261 true \u2227 d \u2261 true\n&&-list\u2084-t btrue btrue btrue btrue h = refl , refl , refl , refl\n&&-list\u2084-t btrue btrue btrue bfalse h =\n  \u22a5-elim (t\u2262f\n           (trans (sym h)\n                  (&&-list\u2084-some-f btrue btrue btrue bfalse (inj\u2082 (inj\u2082 (inj\u2082 refl))))))\n&&-list\u2084-t btrue btrue bfalse Bd h =\n  \u22a5-elim (t\u2262f\n           (trans (sym h)\n                  (&&-list\u2084-some-f btrue btrue bfalse Bd (inj\u2082 (inj\u2082 (inj\u2081 refl))))))\n&&-list\u2084-t btrue bfalse Bc Bd h =\n  \u22a5-elim (t\u2262f\n           (trans (sym h) (&&-list\u2084-some-f btrue bfalse Bc Bd (inj\u2082 (inj\u2081 refl)))))\n&&-list\u2084-t bfalse Bb Bc Bd h =\n  \u22a5-elim (t\u2262f (trans (sym h) (&&-list\u2084-some-f bfalse Bb Bc Bd (inj\u2081 refl))))\n\n&&-list\u2084-t\u2081 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 a \u2261 true\n&&-list\u2084-t\u2081 Ba Bb Bc Bd h = \u2227-proj\u2081 (&&-list\u2084-t Ba Bb Bc Bd h)\n\n&&-list\u2084-t\u2082 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 b \u2261 true\n&&-list\u2084-t\u2082 Ba Bb Bc Bd h = \u2227-proj\u2081 (\u2227-proj\u2082 (&&-list\u2084-t Ba Bb Bc Bd h))\n\n&&-list\u2084-t\u2083 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 c \u2261 true\n&&-list\u2084-t\u2083 Ba Bb Bc Bd h =\n  \u2227-proj\u2081 (\u2227-proj\u2082 (\u2227-proj\u2082 (&&-list\u2084-t Ba Bb Bc Bd h)))\n\n&&-list\u2084-t\u2084 : \u2200 {a b c d} \u2192 Bool a \u2192 Bool b \u2192 Bool c \u2192 Bool d \u2192\n              a && b && c && d \u2261 true \u2192 d \u2261 true\n&&-list\u2084-t\u2084 Ba Bb Bc Bd h =\n  \u2227-proj\u2082 (\u2227-proj\u2082 (\u2227-proj\u2082 (&&-list\u2084-t Ba Bb Bc Bd h)))\n\nx\u2262not-x : \u2200 {b} \u2192 Bool b \u2192 b \u2262 not b\nx\u2262not-x btrue = prf\n  where postulate prf : true \u2261 not true \u2192 \u22a5\n        {-# ATP prove prf #-}\nx\u2262not-x bfalse = prf\n  where postulate prf : false \u2261 not false \u2192 \u22a5\n        {-# ATP prove prf #-}\n\nnot-x\u2262x : \u2200 {b} \u2192 Bool b \u2192 not b \u2262 b\nnot-x\u2262x Bb h = x\u2262not-x Bb (sym h)\n\nnot-involutive : \u2200 {b} \u2192 Bool b \u2192 not (not b) \u2261 b\nnot-involutive btrue = prf\n  where postulate prf : not (not true) \u2261 true\n        {-# ATP prove prf #-}\nnot-involutive bfalse = prf\n  where postulate prf : not (not false) \u2261 false\n        {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties with inequalities\n\nlt-Bool : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Bool (lt m n)\nlt-Bool nzero nzero = prf\n  where postulate prf : Bool (lt zero zero)\n        {-# ATP prove prf #-}\nlt-Bool nzero (nsucc {n} Nn) = prf\n  where postulate prf : Bool (lt zero (succ\u2081 n))\n        {-# ATP prove prf #-}\nlt-Bool (nsucc {m} Nm) nzero = prf\n  where postulate prf : Bool (lt (succ\u2081 m)  zero)\n        {-# ATP prove prf #-}\nlt-Bool (nsucc {m} Nm) (nsucc {n} Nn) = prf (lt-Bool Nm Nn)\n  where postulate prf : Bool (lt m n) \u2192 Bool (lt (succ\u2081 m) (succ\u2081 n))\n        {-# ATP prove prf #-}\n\nle-Bool : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Bool (le m n)\nle-Bool {n = n} nzero Nn = prf\n  where postulate prf : Bool (le zero n)\n        {-# ATP prove prf #-}\nle-Bool (nsucc {m} Nm) nzero = prf\n  where postulate prf : Bool (le (succ\u2081 m) zero)\n        {-# ATP prove prf Sx\u22700 #-}\nle-Bool (nsucc {m} Nm) (nsucc {n} Nn) = prf (le-Bool Nm Nn)\n  where postulate prf : Bool (le m n) \u2192 Bool (le (succ\u2081 m) (succ\u2081 n))\n        {-# ATP prove prf #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"733710e4aaf27ddfb2695c88abba783af283161e","subject":"Decouple syntax of semantics of proof-embedded changes ... by embedding the proof in the interpretation as opposed to embedding them in the \u0394-terms. Invalid programs are well-formed terms now, but they may only denote the empty function.","message":"Decouple syntax of semantics of proof-embedded changes\n... by embedding the proof in the interpretation as opposed to\nembedding them in the \u0394-terms. Invalid programs are well-formed\nterms now, but they may only denote the empty function.\n\nOld-commit-hash: 4d82a603c0824b2e4c070a22ddb7ca78562be994\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n---------------------------------------------------------\n-- Postulates: Extensionality and bag properties (#55) --\n---------------------------------------------------------\n\npostulate extensionality : Extensionality Level.zero Level.zero\n-- Instead of:\n-- open import Data.NatBag renaming\n---  (map to mapBag ; empty to emptyBag ; update to updateBag)\n-- open import Data.NatBag.Properties\npostulate Bag : Set\npostulate emptyBag : Bag\npostulate mapBag : (\u2115 \u2192 \u2115) \u2192 Bag \u2192 Bag\npostulate _++_ : Bag \u2192 Bag \u2192 Bag\npostulate _\\\\_ : Bag \u2192 Bag \u2192 Bag\ninfixr 5 _++_\ninfixl 9 _\\\\_\npostulate b\\\\b=\u2205 : \u2200 {b : Bag} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {b : Bag} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {b : Bag} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n----------------------------\n-- Useful data structures --\n----------------------------\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\ncouple : Set \u2192 Set \u2192 Set\ncouple A B = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 \u22a4) (\u03bb _ _ _ \u2192 \u22a4)\n\ntriple : Set \u2192 Set \u2192 Set \u2192 Set\ntriple A B C = Quadruple A (\u03bb _ \u2192 B) (\u03bb _ _ \u2192 C) (\u03bb _ _ _ \u2192 \u22a4)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (x : Var \u0393 \u03c3) \u2192 Var (\u03c4 \u2022 \u0393) \u03c3\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 Term \u0393 (\u03c3 \u21d2 \u03c4)\n  app : \u2200 {\u03c3 \u03c4 \u0393} \u2192 (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) \u2192 Term \u0393 \u03c4\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192\n    Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n-----------------------\n-- Syntax of changes --\n-----------------------\n\n-- Validity proofs are not literally embedded in terms.\n-- They are introduced and checked at interpretation time.\n-- Invalid programs are well-formed terms,\n-- they just denote the empty function.\n--\n-- Thus do we avoid the horrible mutual recursions between\n-- the syntax and semantics of changes and between the\n-- program transformation and its correctness, which drives\n-- the type checker to thrashing.\n\ndata \u0394Term : Context \u2192 Type \u2192 Set where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394Term \u0393 nats\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394Term \u0393 bags\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394Term \u0393 \u03c4\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (dt : \u0394Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n         \u0394Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c3 \u03c4 \u0393}\n    (ds : \u0394Term \u0393 (\u03c3 \u21d2 \u03c4))\n    ( t :  Term \u0393 \u03c3)\n    (dt : \u0394Term \u0393 \u03c3) \u2192\n    \u0394Term \u0393 \u03c4\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393}\n    (ds : \u0394Term \u0393 nats)\n    (dt : \u0394Term \u0393 nats) \u2192\n    \u0394Term \u0393 nats\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (ds : \u0394Term \u0393 (nats \u21d2 nats))\n    ( t :   Term \u0393 bags)\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n  \u0394map\u2081 : \u2200 {\u0393}\n    ( s :   Term \u0393 (nats \u21d2 nats))\n    (dt : \u0394Term \u0393 bags) \u2192\n    \u0394Term \u0393 bags\n\n---------------------------------\n-- Semantic domains of changes --\n---------------------------------\n\n\u0394Val : Type \u2192 Set\n\u0394Env : Context \u2192 Set\nvalid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\n\n-- \u0394Val : Type \u2192 Set\n\u0394Val nats = \u2115 \u00d7 \u2115\n\u0394Val bags = Bag\n\u0394Val (\u03c3 \u21d2 \u03c4) = (v : \u27e6 \u03c3 \u27e7) \u2192 (dv : \u0394Val \u03c3) \u2192 valid v dv \u2192 \u0394Val \u03c4\n\n-- \u0394Env : Context \u2192 Set\n\u0394Env \u2205 = EmptySet\n\u0394Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb _ \u2192 \u0394Val \u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb _ _ _ \u2192 \u0394Env \u0393)\n\n_\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\n-- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c3 \u21d2 \u03c4} f df =\n  \u2200 (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\n\n-- _\u2295_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n-- _\u229d_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u0394Val \u03c4\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nv\u2295[u\u229dv]=u {nats} {u} {v} = refl\nv\u2295[u\u229dv]=u {bags} {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c3 \u21d2 \u03c4} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\n-- R[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv]\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n--------------------------\n-- Semantics of changes --\n--------------------------\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 \u0394Val \u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n_is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192\n  (t : \u0394Term \u0393 \u03c4) \u2192 (\u03c1 : \u0394Env \u0393) \u2192 t is-valid-wrt \u03c1 \u2192\n  \u0394Val \u03c4\n\n-- _is-valid-wrt_ : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 \u0394Env \u0393 \u2192 Set\n\u0394nat old new is-valid-wrt \u03c1 = \u22a4\n\u0394bag db is-valid-wrt \u03c1 = \u22a4\n\u0394var x is-valid-wrt \u03c1 = \u22a4\n\n_is-valid-wrt_ {\u03c3 \u21d2 \u03c4} (\u0394abs dt) \u03c1 =\n  (v : \u27e6 \u03c3 \u27e7) (dv : \u0394Val \u03c3) (R[v,dv] : valid v dv) \u2192\n  _is-valid-wrt_ dt (cons v dv R[v,dv] \u03c1)\n\n\u0394app ds t dt is-valid-wrt \u03c1 = Quadruple\n  (ds is-valid-wrt \u03c1)\n  (\u03bb _ \u2192 dt is-valid-wrt \u03c1)\n  (\u03bb _ v-dt \u2192 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt))\n  (\u03bb _ _ _ \u2192 \u22a4)\n\n\u0394add ds dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n\n\u0394map\u2080 s ds t dt is-valid-wrt \u03c1 = couple\n  (ds is-valid-wrt \u03c1)\n  (dt is-valid-wrt \u03c1)\n_is-valid-wrt_ (\u0394map\u2081 s db) \u03c1 = db is-valid-wrt \u03c1\n\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 tt = old , new\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 tt = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 tt = \u27e6 x \u27e7\u0394Var \u03c1\n\n\u27e6 \u0394abs dt \u27e7\u0394 \u03c1 make-valid = \u03bb v dv R[v,dv] \u2192\n  \u27e6 dt \u27e7\u0394 (cons v dv R[v,dv] \u03c1) (make-valid v dv R[v,dv])\n\n\u27e6 \u0394app ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt R[ds,dt] _) =\n  \u27e6 ds \u27e7\u0394 \u03c1 v-ds (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt) R[ds,dt]\n\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    (old-s + old-t , new-s + new-t)\n\n\u27e6 \u0394map\u2080 s ds t dt \u27e7\u0394 \u03c1 (cons v-ds v-dt _ _) =\n  let\n    f  = \u27e6 s \u27e7 (ignore \u03c1)\n    v  = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 ds \u27e7\u0394 \u03c1 v-ds\n    dv = \u27e6 dt \u27e7\u0394 \u03c1 v-dt\n  in\n    mapBag (f \u2295 df) (v \u2295 dv) \u229d mapBag f v\n\n\u27e6 \u0394map\u2081 s dt \u27e7\u0394 \u03c1 v-dt = mapBag (\u27e6 s \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1 v-dt)\n\n-- Minor issue about concrete syntax\n--\n-- Because \u27e6_\u27e7\u0394 have dependently typed arguments,\n-- we can't make it an instance of the Meaning\n-- type class and can't use \u27e6_\u27e7 on \u0394Terms.\n--\n-- Error message:\n-- Cannot instantiate the metavariable _872 to solution ((\u03c1 : \u0394Env .\u0393)\n-- \u2192 t is-valid-wrt \u03c1 \u2192 \u0394Val .\u03c4) since it contains the variable t\n-- which is not in scope of the metavariable or irrelevant in the\n-- metavariable but relevant in the solution\n-- when checking that the expression \u27e6_\u27e7\u0394 has type\n-- \u0394Term .\u0393 .\u03c4 \u2192 _Semantics_872\n--\n--     meaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394Term \u0393 \u03c4)\n--     meaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n----------------------------\n-- Program transformation --\n----------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394Term \u0393 \u03c4\n\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t)\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\n-----------------\n-- Correctness --\n-----------------\n\nunrestricted : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  derive t is-valid-wrt \u03c1\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t}))\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env \u0393}\n  \u2192 \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  \u2261 \u27e6 t \u27e7 (update \u03c1)\n\nunrestricted {t = nat n} = tt\nunrestricted {t = bag b} = tt\nunrestricted {t = var x} {\u03c1} = tt\nunrestricted {t = abs t} {\u03c1} = (\u03bb _ _ _ \u2192 unrestricted {t = t})\nunrestricted {t = app s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t})\n  (validity {t = t}) tt\nunrestricted {t = add s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\nunrestricted {t = map s t} {\u03c1} = cons\n  (unrestricted {t = s})\n  (unrestricted {t = t}) tt tt\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2082 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2081 (unrestricted {t = t})\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7\u0394 \u03c1 (unrestricted {t = s})\n    db = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7\u0394 \u03c1 (unrestricted {t = t})\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7\u0394 \u03c1\u2032 (unrestricted {t = t})\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","old_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\ndata \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms\n-- (because it embeds validity proofs),\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\ndata \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                          valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {u = u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {u = u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car   : A\n    cadr  : B car\n    caddr : C car cadr\n    cdddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n\u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n\u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n\u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n\u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n  \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n\u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n  let\n    (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n    (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n  in\n    (old-s + old-t , new-s + new-t)\n\u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n  let\n    v  = \u27e6 b \u27e7 (ignore \u03c1)\n    h  = \u27e6 f \u27e7 (ignore \u03c1)\n    dv = \u27e6 db \u27e7\u0394 \u03c1\n    dh = \u27e6 df \u27e7\u0394 \u03c1\n  in\n    mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n\u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\ncorrectness : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 t \u27e7 (ignore \u03c1) \u2295 \u27e6 derive t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (update \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag emptyBag\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {t = nat n} = refl\nvalidity {t = bag b} = tt\nvalidity {t = var x} = validity-var x\nvalidity {t = map f b} = tt\nvalidity {t = add s t} = cong\u2082 _+_ (validity {t = s}) (validity {t = t})\n\nvalidity {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in\n    proj\u2081 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))\n\nvalidity {t = abs t} {\u03c1} = \u03bb v dv R[v,dv] \u2192\n  let\n    v\u2032 = v \u2295 dv\n    dv\u2032 = v\u2032 \u229d v\u2032\n    \u03c1\u2081 = cons v dv R[v,dv] \u03c1\n    \u03c1\u2082 = cons v\u2032 dv\u2032 R[v,u\u229dv] \u03c1\n  in\n    validity {t = t} {\u03c1\u2081}\n    ,\n    (begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2082) \u2295 \u27e6 derive t \u27e7 \u03c1\u2082\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2082} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2082)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2081)\n    \u2261\u27e8 sym (correctness {t = t} {\u03c1\u2081}) \u27e9\n      \u27e6 t \u27e7 (ignore \u03c1\u2081) \u2295 \u27e6 derive t \u27e7 \u03c1\u2081\n    \u220e) where open \u2261-Reasoning\n\ncorrectVar : \u2200 {\u03c4 \u0393} {x : Var \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  \u27e6 x \u27e7 (ignore \u03c1) \u2295 \u27e6 x \u27e7\u0394Var \u03c1 \u2261 \u27e6 x \u27e7 (update \u03c1)\n\ncorrectVar {x = this  } {cons v dv R[v,dv] \u03c1} = refl\ncorrectVar {x = that y} {cons v dv R[v,dv] \u03c1} = correctVar {x = y} {\u03c1}\n\ncorrectness {t = nat n} = refl\ncorrectness {t = bag b} = b++\u2205=b\ncorrectness {t = var x} = correctVar {x = x}\n\ncorrectness {t = add s t} =\n  cong\u2082 _+_ (correctness {t = s}) (correctness {t = t})\n\ncorrectness {t = map s t} {\u03c1} =\n  trans (b++[d\\\\b]=d {mapBag f b} {mapBag (f \u2295 df) (b \u2295 db)})\n        (cong\u2082 mapBag (correctness {t = s}) (correctness {t = t}))\n  where\n    f = \u27e6 s \u27e7 (ignore \u03c1)\n    b = \u27e6 t \u27e7 (ignore \u03c1)\n    df = \u27e6 derive s \u27e7 \u03c1\n    db = \u27e6 derive t \u27e7 \u03c1\n\ncorrectness {t = app s t} {\u03c1} =\n  let\n    v = \u27e6 t \u27e7 (ignore \u03c1)\n    dv = \u27e6 derive t \u27e7 \u03c1\n  in trans\n     (sym (proj\u2082 (validity {t = s} {\u03c1} v dv (validity {t = t} {\u03c1}))))\n     (correctness {t = s} \u27e8$\u27e9 correctness {t = t})\n\ncorrectness {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} {\u03c1} = extensionality (\u03bb v \u2192\n  let\n    \u03c1\u2032 : \u0394-Env (\u03c4\u2081 \u2022 \u0393)\n    \u03c1\u2032 = cons v (v \u229d v) R[v,u\u229dv] \u03c1\n  in\n    begin\n      \u27e6 t \u27e7 (ignore \u03c1\u2032) \u2295 \u27e6 derive t \u27e7 \u03c1\u2032\n    \u2261\u27e8 correctness {t = t} {\u03c1\u2032} \u27e9\n      \u27e6 t \u27e7 (update \u03c1\u2032)\n    \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 update \u03c1)) v\u2295[u\u229dv]=u \u27e9\n      \u27e6 t \u27e7 (v \u2022 update \u03c1)\n    \u220e\n  ) where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9b6a3b9baa5a58efdb2547966b013d71b25bc882","subject":"Function: +\u03a0\u03a0 and +\u03a0\u03a0\u03a0","message":"Function: +\u03a0\u03a0 and +\u03a0\u03a0\u03a0\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/NP.agda","new_file":"lib\/Function\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Algebra\nopen import Algebra.Structures\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Product\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\n\u03a0\u03a0 : \u2200 {a b c} (A : \u2605 a) (B : A \u2192 \u2605 b) (C : \u03a3 A B \u2192 \u2605 c) \u2192 \u2605 _\n\u03a0\u03a0 A B C = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 C (x , y)\n\n\u03a0\u03a0\u03a0 : \u2200 {a b c d} (A : \u2605 a) (B : A \u2192 \u2605 b)\n                  (C : \u03a3 A B \u2192 \u2605 c) (D : \u03a3 (\u03a3 A B) C \u2192 \u2605 d) \u2192 \u2605 _\n\u03a0\u03a0\u03a0 A B C D = \u03a0 A \u03bb x \u2192 \u03a0 (B x) \u03bb y \u2192 \u03a0 (C (x , y)) \u03bb z \u2192 D ((x , y) , z)\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : Set a) (B : Set b) \u2192 Set (a L.\u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = \u2261.refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = \u2261.refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = \u2261.refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = \u2261.refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = \u2261.refl\n  nest-+ (suc m) n = \u2261.cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = \u2261.cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = \u2261.refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 \u2261.refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 \u2261.cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 \u2261.refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 \u2261.refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261.\u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\n\u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : Set i} {A : Set a} {B : A \u2192 Set b} {C : (x : A) \u2192 B x \u2192 Set c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : Set a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 Set \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : Set a) = EndoMonoid-\u2248 {A = A} \u2261.isEquivalence (\u2261.cong\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : Set a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2261.\u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 \u2261.trans (p (h x)) (\u2261.cong g (q x)))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Function.NP where\n\nimport Level as L\nopen import Type hiding (\u2605)\nopen import Algebra\nopen import Algebra.Structures\nopen import Function       public\nopen import Data.Nat       using (\u2115; zero; suc; _+_; _*_; fold)\nopen import Data.Bool      using (Bool)\nopen import Data.Product\nopen import Data.Vec.N-ary using (N-ary; N-ary-level)\nimport Category.Monad.Identity as Id\nopen import Category.Monad renaming (module RawMonad to Monad; RawMonad to Monad)\nopen import Category.Applicative renaming (module RawApplicative to Applicative; RawApplicative to Applicative)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\n\u03a0 : \u2200 {a b} (A : \u2605 a) \u2192 (B : A \u2192 \u2605 b) \u2192 \u2605 _\n\u03a0 A B = (x : A) \u2192 B x\n\nid-app : \u2200 {f} \u2192 Applicative {f} id\nid-app = rawIApplicative\n  where open Monad Id.IdentityMonad\n\n-\u2192- : \u2200 {a b} (A : Set a) (B : Set b) \u2192 Set (a L.\u2294 b)\n-\u2192- A B = A \u2192 B\n\n_\u2192\u27e8_\u27e9_ : \u2200 {a b} (A : Set a) (n : \u2115) (B : Set b) \u2192 Set (N-ary-level a b n)\nA \u2192\u27e8 n \u27e9 B = N-ary n A B\n\n_\u2192\u27e8_\u27e9\u2080_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2080) \u2192 \u2605\u2080\nA \u2192\u27e8 zero  \u27e9\u2080 B = B\nA \u2192\u27e8 suc n \u27e9\u2080 B = A \u2192 A \u2192\u27e8 n \u27e9\u2080 B\n\n_\u2192\u27e8_\u27e9\u2081_ : \u2200 (A : \u2605\u2080) (n : \u2115) (B : \u2605\u2081) \u2192 \u2605\u2081\nA \u2192\u27e8 zero  \u27e9\u2081 B = B\nA \u2192\u27e8 suc n \u27e9\u2081 B = A \u2192 A \u2192\u27e8 n \u27e9\u2081 B\n\nEndo : \u2200 {a} \u2192 Set a \u2192 Set a\nEndo A = A \u2192 A\n\nCmp : \u2200 {a} \u2192 Set a \u2192 Set a\nCmp A = A \u2192 A \u2192 Bool\n\n-- More properties about fold are in Data.Nat.NP\nnest : \u2200 {a} {A : Set a} \u2192 \u2115 \u2192 Endo (Endo A)\n-- TMP nest n f x = fold x f n\nnest zero f x = x\nnest (suc n) f x = f (nest n f x)\n\nmodule nest-Properties {a} {A : Set a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = \u2261.refl\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = \u2261.refl\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = \u2261.refl\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = \u2261.refl\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = \u2261.refl\n  nest-+ (suc m) n = \u2261.cong (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = \u2261.cong (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = \u2261.refl\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 \u2261.refl \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 \u2261.cong (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 \u2261.refl \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 \u2261.refl \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261.\u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : Set a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n_$\u27e8_\u27e9_ : \u2200 {a} {A : Set a} \u2192 Endo A \u2192 \u2115 \u2192 Endo A\n_$\u27e8_\u27e9_ f n = nest n f\n\n-- If you run a version of Agda without the support of instance\n-- arguments, simply comment this definition, very little code rely on it.\n\u2026 : \u2200 {a} {A : Set a} \u2983 x : A \u2984 \u2192 A\n\u2026 \u2983 x \u2984 = x\n\n_\u27e8_\u27e9\u00b0_ : \u2200 {i a b c} {Ix : Set i} {A : Set a} {B : A \u2192 Set b} {C : (x : A) \u2192 B x \u2192 Set c}\n         \u2192 (f  : Ix \u2192 A)\n         \u2192 (op : (x : A) (y : B x) \u2192 C x y)\n         \u2192 (g  : (i : Ix) \u2192 B (f i))\n         \u2192 (i : Ix) \u2192 C (f i) (g i)\n(f \u27e8 _\u2219_ \u27e9\u00b0 g) x = f x \u2219 g x\n\nmodule Combinators where\n    S : \u2200 {A B C : \u2605\u2080} \u2192\n          (A \u2192 B \u2192 C) \u2192\n          (A \u2192 B) \u2192\n          (A \u2192 C)\n    S = _\u02e2_\n\n    K : \u2200 {A B : \u2605\u2080} \u2192 A \u2192 B \u2192 A\n    K = const\n\n    -- B \u2257 _\u2218_\n    B : \u2200 {A B C : \u2605\u2080} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n    B = S (K S) K\n\n    -- C \u2257 flip\n    C : \u2200 {A B C : \u2605\u2080} \u2192 (A \u2192 B \u2192 C) \u2192 B \u2192 A \u2192 C\n    C = S (S (K (S (K S) K)) S) (K K)\n\nmodule EndoMonoid-\u2248 {a \u2113} {A : Set a}\n                    {_\u2248_ : Endo A \u2192 Endo A \u2192 Set \u2113}\n                    (isEquivalence : IsEquivalence _\u2248_)\n                    (\u2218-cong : _\u2218\u2032_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n                   where\n  private\n    module \u2248 = IsEquivalence isEquivalence\n    isSemigroup : IsSemigroup _\u2248_ _\u2218\u2032_\n    isSemigroup = record { isEquivalence = isEquivalence; assoc = \u03bb _ _ _ \u2192 \u2248.refl; \u2219-cong = \u2218-cong }\n\n  monoid : Monoid a \u2113\n  monoid = record { Carrier = Endo A; _\u2248_ = _\u2248_; _\u2219_ = _\u2218\u2032_; \u03b5 = id\n                  ; isMonoid = record { isSemigroup = isSemigroup\n                                      ; identity = (\u03bb _ \u2192 \u2248.refl) , (\u03bb _ \u2192 \u2248.refl) } }\n\n  open Monoid monoid public\n\nmodule EndoMonoid-\u2261 {a} (A : Set a) = EndoMonoid-\u2248 {A = A} \u2261.isEquivalence (\u2261.cong\u2082 _\u2218\u2032_)\n\nmodule EndoMonoid-\u2257 {a} (A : Set a) = EndoMonoid-\u2248 (Setoid.isEquivalence (A \u2261.\u2192-setoid A))\n                                                   (\u03bb {f} {g} {h} {i} p q x \u2192 \u2261.trans (p (h x)) (\u2261.cong g (q x)))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"95fabbfe85838015d0aaa9d7c8dd55cac4ca54db","subject":"bintree: generalized fold","message":"bintree: generalized fold\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nmodule Fold {a b i} {I : Set i} (ze : I) (su : I \u2192 I)\n            {A : Set a} {B : I \u2192 Set b}\n            (f : A \u2192 B ze) (_\u00b7_ : \u2200 {n} \u2192 B n \u2192 B n \u2192 B (su n)) where\n\n  `_ : \u2115 \u2192 I\n  `_ = Nat.fold ze su\n\n  fold : \u2200 {n} \u2192 Tree A n \u2192 B(` n)\n  fold (leaf x)    = f x\n  fold (fork t\u2080 t\u2081) = fold t\u2080 \u00b7 fold t\u2081\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold {A = A} op = Fold.fold 0 suc {B = const A} id op\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","old_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat.NP using (\u2115; zero; suc; 2^_; _+_; module \u2115\u00b0)\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Vec using (Vec; _++_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Algebra.FunctionProperties\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nfold : \u2200 {n a} {A : Set a} (op : A \u2192 A \u2192 A) \u2192 Tree A n \u2192 A\nfold _   (leaf x) = x\nfold _\u00b7_ (fork t\u2080 t\u2081) = fold _\u00b7_ t\u2080 \u00b7 fold _\u00b7_ t\u2081\n\nsearch\u2261fold\u2218fromFun : \u2200 {n a} {A : Set a} op (f : Bits n \u2192 A) \u2192 search op f \u2261 fold op (fromFun f)\nsearch\u2261fold\u2218fromFun {zero}  op f = refl\nsearch\u2261fold\u2218fromFun {suc n} op f\n  rewrite search\u2261fold\u2218fromFun op (f \u2218 0\u2237_)\n        | search\u2261fold\u2218fromFun op (f \u2218 1\u2237_) = refl\n\n-- Returns the flat vector of leaves underlying the perfect binary tree.\ntoVec : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Vec A (2^ n)\ntoVec (leaf x)     = x \u2237 []\ntoVec (fork t\u2080 t\u2081) = toVec t\u2080 ++ toVec t\u2081\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' [] val tree = val\nupdate' (b \u2237 key) val (fork tree tree\u2081) = if b then fork tree (update' key val tree\u2081)\n                                               else fork (update' key val tree) tree\u2081\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star using (Star; \u03b5; _\u25c5_)\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  left : \u2200 {n} {left\u2080 left\u2081 right : Tree A n} \u2192\n         Swp left\u2080 left\u2081 \u2192\n         Swp (fork left\u2080 right) (fork left\u2081 right)\n  right : \u2200 {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n         Swp right\u2080 right\u2081 \u2192\n         Swp (fork left right\u2080) (fork left right\u2081)\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {n a} {A : Set a} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym (left s)  = left (Swp-sym s)\nSwp-sym (right s) = right (Swp-sym s)\nSwp-sym swp\u2081      = swp\u2081\nSwp-sym swp\u2082      = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork _ _} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n\ndata SwpOp : \u2115 \u2192 Set where\n  \u03b5 : \u2200 {n} \u2192 SwpOp n\n\n  _\u204f_ : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n \u2192 SwpOp n\n\n  first : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\n\n  swp : \u2200 {n} \u2192 SwpOp (suc n)\n\n  swp-seconds : \u2200 {n} \u2192 SwpOp (2 + n)\n\ndata Perm {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm t t\n\n  _\u204f_ : \u2200 {n} {t u v : Tree A n} \u2192 Perm t u \u2192 Perm u v \u2192 Perm t v\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm tA tB \u2192\n         Perm (fork tA tC) (fork tB tC)\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm (fork tA tB) (fork tB tA)\n\n  swp-seconds : \u2200 {n} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tA tD) (fork tC tB))\n\ndata Perm0\u2194 {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  \u03b5 : \u2200 {n} {t : Tree A n} \u2192 Perm0\u2194 t t\n\n  swp : \u2200 {n} {tA tB : Tree A n} \u2192\n         Perm0\u2194 (fork tA tB) (fork tB tA)\n\n  first : \u2200 {n} {tA tB tC : Tree A n} \u2192\n         Perm0\u2194 tA tB \u2192\n         Perm0\u2194 (fork tA tC) (fork tB tC)\n\n  firsts : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tC) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tF tD))\n\n  extremes : \u2200 {n} {tA tB tC tD tE tF : Tree A n} \u2192\n                 Perm0\u2194 (fork tA tD) (fork tE tF) \u2192\n                 Perm0\u2194 (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tE tB) (fork tC tF))\n\n-- Star Perm0\u2194 can then model any permutation\n\ninfixr 1 _\u204f_\n\nsecond-perm : \u2200 {a} {A : Set a} {n} {left right\u2080 right\u2081 : Tree A n} \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left right\u2080) (fork left right\u2081)\nsecond-perm f = swp \u204f first f \u204f swp\n\nsecond-swpop : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp (suc n)\nsecond-swpop f = swp \u204f first f \u204f swp\n\n<_\u00d7_>-perm : \u2200 {a} {A : Set a} {n} {left\u2080 right\u2080 left\u2081 right\u2081 : Tree A n} \u2192\n           Perm left\u2080 left\u2081 \u2192\n           Perm right\u2080 right\u2081 \u2192\n           Perm (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n< f \u00d7 g >-perm = first f \u204f second-perm g\n\nswp\u2082-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n          Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\nswp\u2082-perm = first swp \u204f swp-seconds \u204f first swp\n\nswp\u2083-perm : \u2200 {a n} {A : Set a} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Perm (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2080\u2080 t\u2081\u2080) (fork t\u2080\u2081 t\u2081\u2081))\nswp\u2083-perm = second-perm swp \u204f swp-seconds \u204f second-perm swp\n\nswp-firsts-perm : \u2200 {n a} {A : Set a} {tA tB tC tD : Tree A n} \u2192\n                 Perm (fork (fork tA tB) (fork tC tD))\n                          (fork (fork tC tB) (fork tA tD))\nswp-firsts-perm = < swp \u00d7 swp >-perm \u204f swp-seconds \u204f < swp \u00d7 swp >-perm\n\nSwp\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp {a} {A} {n} \u21d2 Perm {n = n}\nSwp\u21d2Perm (left pf) = first (Swp\u21d2Perm pf)\nSwp\u21d2Perm (right pf) = second-perm (Swp\u21d2Perm pf)\nSwp\u21d2Perm swp\u2081 = swp\nSwp\u21d2Perm swp\u2082 = swp\u2082-perm\n\nSwp\u2605\u21d2Perm : \u2200 {n a} {A : Set a} \u2192 Swp\u2605 {n} {a} {A} \u21d2 Perm {n = n}\nSwp\u2605\u21d2Perm \u03b5         = \u03b5\nSwp\u2605\u21d2Perm (x \u25c5 xs) = Swp\u21d2Perm x \u204f Swp\u2605\u21d2Perm xs\n\nswp-inners : \u2200 {n} \u2192 SwpOp (2 + n)\nswp-inners = second-swpop swp \u204f swp-seconds \u204f second-swpop swp\n\non-extremes : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-extremes f = swp-seconds \u204f first f \u204f swp-seconds\n\non-firsts : \u2200 {n} \u2192 SwpOp (1 + n) \u2192 SwpOp (2 + n)\non-firsts f = swp-inners \u204f first f \u204f swp-inners\n\n0\u2194_ : \u2200 {m n} \u2192 Bits m \u2192 SwpOp (m + n)\n0\u2194 [] = \u03b5\n0\u2194 (false{-0-} \u2237 p) = first (0\u2194 p)\n0\u2194 (true{-1-}  \u2237 []) = swp\n0\u2194 (true{-1-}  \u2237 true {-1-} \u2237 p) = on-extremes (0\u2194 (1b \u2237 p))\n0\u2194 (true{-1-}  \u2237 false{-0-} \u2237 p) = on-firsts   (0\u2194 (1b \u2237 p))\n\ncommSwpOp : \u2200 m n \u2192 SwpOp (m + n) \u2192 SwpOp (n + m)\ncommSwpOp m n x rewrite \u2115\u00b0.+-comm m n = x\n\n[_\u2194_] : \u2200 {m n} (p q : Bits m) \u2192 SwpOp (m + n)\n[ p \u2194 q ] = 0\u2194 p \u204f 0\u2194 q \u204f 0\u2194 p\n\n[_\u2194\u2032_] : \u2200 {n} (p q : Bits n) \u2192 SwpOp n\n[ p \u2194\u2032 q ] = commSwpOp _ 0 [ p \u2194 q ]\n\n_$swp_ : \u2200 {n a} {A : Set a} \u2192 SwpOp n \u2192 Tree A n \u2192 Tree A n\n\u03b5           $swp t = t\n(f \u204f g)     $swp t = g $swp (f $swp t)\n(first f)   $swp (fork t\u2080 t\u2081) = fork (f $swp t\u2080) t\u2081\nswp         $swp (fork t\u2080 t\u2081) = fork t\u2081 t\u2080\nswp-seconds $swp (fork (fork t\u2080 t\u2081) (fork t\u2082 t\u2083)) = fork (fork t\u2080 t\u2083) (fork t\u2082 t\u2081)\n\nswpRel : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 Perm t (f $swp t)\nswpRel \u03b5           _          = \u03b5\nswpRel (f \u204f g)     _          = swpRel f _ \u204f swpRel g _\nswpRel (first f)   (fork _ _) = first (swpRel f _)\nswpRel swp         (fork _ _) = swp\nswpRel swp-seconds\n (fork (fork _ _) (fork _ _)) = swp-seconds\n\n[0\u2194_]-Rel : \u2200 {m n a} {A : Set a} (p : Bits m) (t : Tree A (m + n)) \u2192 Perm t ((0\u2194 p) $swp t)\n[0\u2194 p ]-Rel = swpRel (0\u2194 p)\n\nswpOp' : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm0\u2194 t u \u2192 SwpOp n\nswpOp' \u03b5 = \u03b5\nswpOp' (first f) = first (swpOp' f)\nswpOp' swp = swp\nswpOp' (firsts f) = on-firsts (swpOp' f)\nswpOp' (extremes f) = on-extremes (swpOp' f)\n\nswpOp : \u2200 {n a} {A : Set a} {t u : Tree A n} \u2192 Perm t u \u2192 SwpOp n\nswpOp \u03b5 = \u03b5\nswpOp (f \u204f g) = swpOp f \u204f  swpOp g\nswpOp (first f) = first (swpOp f)\nswpOp swp = swp\nswpOp swp-seconds = swp-seconds\n\nswpOp-sym : \u2200 {n} \u2192 SwpOp n \u2192 SwpOp n\nswpOp-sym \u03b5 = \u03b5\nswpOp-sym (f \u204f g) = swpOp-sym g \u204f swpOp-sym f\nswpOp-sym (first f) = first (swpOp-sym f)\nswpOp-sym swp = swp\nswpOp-sym swp-seconds = swp-seconds\n\nswpOp-sym-involutive : \u2200 {n} (f : SwpOp n) \u2192 swpOp-sym (swpOp-sym f) \u2261 f\nswpOp-sym-involutive \u03b5 = refl\nswpOp-sym-involutive (f \u204f g) rewrite swpOp-sym-involutive f | swpOp-sym-involutive g = refl\nswpOp-sym-involutive (first f) rewrite swpOp-sym-involutive f = refl\nswpOp-sym-involutive swp = refl\nswpOp-sym-involutive swp-seconds = refl\n\nswpOp-sym-sound : \u2200 {n a} {A : Set a} (f : SwpOp n) (t : Tree A n) \u2192 swpOp-sym f $swp (f $swp t) \u2261 t\nswpOp-sym-sound \u03b5 t = refl\nswpOp-sym-sound (f \u204f g) t rewrite swpOp-sym-sound g (f $swp t) | swpOp-sym-sound f t = refl\nswpOp-sym-sound (first f) (fork t _) rewrite swpOp-sym-sound f t = refl\nswpOp-sym-sound swp (fork _ _) = refl\nswpOp-sym-sound swp-seconds (fork (fork _ _) (fork _ _)) = refl\n\nmodule \u00acswp-comm where\n  data X : Set where\n    A B C D E F G H : X\n  n : \u2115\n  n = 3\n  t : Tree X n\n  t = fork (fork (fork (leaf A) (leaf B))(fork (leaf C) (leaf D))) (fork (fork (leaf E) (leaf F))(fork (leaf G) (leaf H)))\n  f : SwpOp n\n  f = swp\n  g : SwpOp n\n  g = first swp\n  pf : f $swp (g $swp t) \u2262 g $swp (f $swp t)\n  pf ()\n\nswp-leaf : \u2200 {a} {A : Set a} (f : SwpOp 0) (x : A) \u2192 f $swp (leaf x) \u2261 leaf x\nswp-leaf \u03b5 x = refl\nswp-leaf (f \u204f g) x rewrite swp-leaf f x | swp-leaf g x = refl\n\nswpOp-sound : \u2200 {n a} {A : Set a} {t u : Tree A n} (perm : Perm t u) \u2192 (swpOp perm $swp t \u2261 u)\nswpOp-sound \u03b5 = refl\nswpOp-sound (f \u204f f\u2081) rewrite swpOp-sound f | swpOp-sound f\u2081 = refl\nswpOp-sound (first f) rewrite swpOp-sound f = refl\nswpOp-sound swp = refl\nswpOp-sound swp-seconds = refl\n\nopen import Relation.Nullary using (Dec ; yes ; no)\nopen import Relation.Nullary.Negation\n\nmodule new-approach where\n\n  open import Data.Empty\n\n  import Function.Inverse as FI\n  open FI using (_\u2194_; module Inverse; _InverseOf_)\n  open import Function.Related\n  import Function.Equality\n  import Relation.Binary.PropositionalEquality as P\n\n  data _\u2208_ {a}{A : Set a}(x : A) : {n : \u2115} \u2192 Tree A n \u2192 Set a where\n    here  : x \u2208 leaf x\n    left  : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2081 \u2192 x \u2208 fork t\u2081 t\u2082\n    right : {n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 t\u2082 \u2192 x \u2208 fork t\u2081 t\u2082\n\n  toBits : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n} \u2192 x \u2208 t \u2192 Bits n\n  toBits here = []\n  toBits (left key) = 0b \u2237 toBits key\n  toBits (right key) = 1b \u2237 toBits key\n\n  \u2208-lookup : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t : Tree A n}(path : x \u2208 t) \u2192 lookup (toBits path) t \u2261 x\n  \u2208-lookup here = refl\n  \u2208-lookup (left path) = \u2208-lookup path\n  \u2208-lookup (right path) = \u2208-lookup path\n\n  lookup-\u2208 : \u2200 {a}{A : Set a}{n : \u2115}(key : Bits n)(t : Tree A n) \u2192 lookup key t \u2208 t\n  lookup-\u2208 [] (leaf x) = here\n  lookup-\u2208 (true \u2237 key) (fork tree tree\u2081) = right (lookup-\u2208 key tree\u2081)\n  lookup-\u2208 (false \u2237 key) (fork tree tree\u2081) = left (lookup-\u2208 key tree)\n\n  _\u2248_ : \u2200 {a}{A : Set a}{n : \u2115} \u2192 Tree A n \u2192 Tree A n \u2192 Set _\n  t\u2081 \u2248 t\u2082 = \u2200 x \u2192 (x \u2208 t\u2081) \u2194 (x \u2208 t\u2082)\n\n  \u2248-refl : {a : _}{A : Set a}{n : \u2115}{t : Tree A n} \u2192 t \u2248 t\n  \u2248-refl _ = FI.id\n\n  \u2248-trans : {a : _}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 u \u2248 v \u2192 t \u2248 v\n  \u2248-trans f g x = g x FI.\u2218 f x\n\n  move : \u2200 {a}{A : Set a}{n : \u2115}{t s : Tree A n}{x : A} \u2192 t \u2248 s \u2192 x \u2208 t \u2192 x \u2208 s\n  move t\u2248s x\u2208t = Inverse.to (t\u2248s _) Function.Equality.\u27e8$\u27e9 x\u2208t\n\n  swap\u2080 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 fork t\u2081 t\u2082 \u2248 fork t\u2082 t\u2081\n  swap\u2080 _ = record\n    { to         = \u2192-to-\u27f6 swap\n    ; from       = \u2192-to-\u27f6 swap\n    ; inverse-of = record { left-inverse-of  = swap-inv\n                          ; right-inverse-of = swap-inv }\n    } where\n       swap : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork t\u2082 t\u2081\n       swap (left path)  = right path\n       swap (right path) = left path\n\n       swap-inv : \u2200 {a}{A : Set a}{x : A}{n : \u2115}{t\u2081 t\u2082 : Tree A n}(p : x \u2208 fork t\u2081 t\u2082) \u2192 swap (swap p) \u2261 p\n       swap-inv (left p)  = refl\n       swap-inv (right p) = refl\n\n  swap\u2082 : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tD) (fork tC tB)\n  swap\u2082 _ = record\n    { to         = \u2192-to-\u27f6 fun\n    ; from       = \u2192-to-\u27f6 fun\n    ; inverse-of = record { left-inverse-of  = inv\n                          ; right-inverse-of = inv }\n    } where\n       fun : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 x \u2208 fork (fork tA tB) (fork tC tD) \u2192 x \u2208 fork (fork tA tD) (fork tC tB)\n       fun (left (left path))  = left (left path)\n       fun (left (right path)) = right (right path)\n       fun (right (left path)) = right (left path)\n       fun (right (right path)) = left (right path)\n\n       inv : \u2200 {a}{A : Set a}{x n}{tA tB tC tD : Tree A n}\n             \u2192 (p : x \u2208 fork (fork tA tB) (fork tC tD)) \u2192 fun (fun p) \u2261 p\n       inv (left (left p)) = refl\n       inv (left (right p)) = refl\n       inv (right (left p)) = refl\n       inv (right (right p)) = refl\n\n  _\u27e8fork\u27e9_ : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 fork t\u2081 t\u2082 \u2248 fork s\u2081 s\u2082\n  (t1\u2248s1 \u27e8fork\u27e9 t2\u2248s2) y = record\n    { to         = to\n    ; from       = from\n    ; inverse-of = record { left-inverse-of  = frk-linv\n                          ; right-inverse-of = frk-rinv  }\n    } where\n\n        frk : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 s\u2081 s\u2082 : Tree A n}{x : A} \u2192 t\u2081 \u2248 s\u2081 \u2192 t\u2082 \u2248 s\u2082 \u2192 x \u2208 fork t\u2081 t\u2082 \u2192 x \u2208 fork s\u2081 s\u2082\n        frk t1\u2248s1 t2\u2248s2 (left x\u2208t1) = left (move t1\u2248s1 x\u2208t1)\n        frk t1\u2248s1 t2\u2248s2 (right x\u2208t2) = right (move t2\u2248s2 x\u2208t2)\n\n        to = \u2192-to-\u27f6 (frk t1\u2248s1 t2\u2248s2)\n        from = \u2192-to-\u27f6 (frk (\u03bb x \u2192 FI.sym (t1\u2248s1 x)) (\u03bb x \u2192 FI.sym (t2\u2248s2 x)))\n\n\n        open Function.Equality using (_\u27e8$\u27e9_)\n        open import Function.LeftInverse\n\n        frk-linv : from LeftInverseOf to\n        frk-linv (left x) = cong left (_InverseOf_.left-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-linv (right x) = cong right (_InverseOf_.left-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n        frk-rinv : from RightInverseOf to -- \u2200 x \u2192 to \u27e8$\u27e9 (from \u27e8$\u27e9 x) \u2261 x\n        frk-rinv (left x) = cong left (_InverseOf_.right-inverse-of (Inverse.inverse-of (t1\u2248s1 y)) x)\n        frk-rinv (right x) = cong right (_InverseOf_.right-inverse-of (Inverse.inverse-of (t2\u2248s2 y)) x)\n\n  \u2248-first : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork t v \u2248 fork u v\n  \u2248-first f = f \u27e8fork\u27e9 \u2248-refl\n\n  \u2248-second : \u2200 {a}{A : Set a}{n : \u2115}{t u v : Tree A n} \u2192 t \u2248 u \u2192 fork v t \u2248 fork v u\n  \u2248-second f = \u2248-refl \u27e8fork\u27e9 f\n\n  swap-inner : \u2200 {a}{A : Set a}{n : \u2115}{tA tB tC tD : Tree A n}\n          \u2192 fork (fork tA tB) (fork tC tD) \u2248 fork (fork tA tC) (fork tB tD)\n  swap-inner = \u2248-trans (\u2248-second swap\u2080) (\u2248-trans swap\u2082 (\u2248-second swap\u2080))\n\n  Rot\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Rot t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Rot\u27f6\u2248 (leaf x)        = \u2248-refl\n  Rot\u27f6\u2248 (fork rot rot\u2081) = Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081\n  Rot\u27f6\u2248 (krof {_} {l} {l'} {r} {r'} rot rot\u2081) = \u03bb y \u2192\n        y \u2208 fork l r \u2194\u27e8 (Rot\u27f6\u2248 rot \u27e8fork\u27e9 Rot\u27f6\u2248 rot\u2081) y \u27e9\n        y \u2208 fork r' l' \u2194\u27e8 swap\u2080 y \u27e9\n        y \u2208 fork l' r' \u220e\n    where open EquationalReasoning\n\n  Perm\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm\u27f6\u2248 (f \u204f g) = \u2248-trans (Perm\u27f6\u2248 f) (Perm\u27f6\u2248 g)\n  Perm\u27f6\u2248 (first f) = \u2248-first (Perm\u27f6\u2248 f)\n  Perm\u27f6\u2248 swp = swap\u2080\n  Perm\u27f6\u2248 swp-seconds = swap\u2082\n\n  Perm0\u2194\u27f6\u2248 : \u2200 {a}{A : Set a}{n : \u2115}{t\u2081 t\u2082 : Tree A n} \u2192 Perm0\u2194 t\u2081 t\u2082 \u2192 t\u2081 \u2248 t\u2082\n  Perm0\u2194\u27f6\u2248 \u03b5 = \u2248-refl\n  Perm0\u2194\u27f6\u2248 swp = swap\u2080\n  Perm0\u2194\u27f6\u2248 (first t) = \u2248-first (Perm0\u2194\u27f6\u2248 t)\n  Perm0\u2194\u27f6\u2248 (firsts t) = \u2248-trans swap-inner (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap-inner)\n  Perm0\u2194\u27f6\u2248 (extremes t) = \u2248-trans swap\u2082 (\u2248-trans (\u2248-first (Perm0\u2194\u27f6\u2248 t)) swap\u2082)\n\n  put : {a : _}{A : Set a}{n : \u2115} \u2192 Bits n \u2192 A \u2192 Tree A n \u2192 Tree A n\n  put [] val tree = leaf val\n  put (x \u2237 key) val (fork tree tree\u2081) = if x then fork tree (put key val tree\u2081)\n                                             else fork (put key val tree) tree\u2081\n\n  -- move-me\n  _\u2237\u2262_ : {n : \u2115}{xs ys : Bits n}(x : Bit) \u2192 x \u2237 xs \u2262 x \u2237 ys \u2192 xs \u2262 ys\n  _\u2237\u2262_ x = contraposition $ cong $ _\u2237_ x\n\n  \u2208-put : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x : A}(t : Tree A n) \u2192 x \u2208 put p x t\n  \u2208-put [] t = here\n  \u2208-put (true \u2237 p) (fork t t\u2081) = right (\u2208-put p t\u2081)\n  \u2208-put (false \u2237 p) (fork t t\u2081) = left (\u2208-put p t)\n\n  \u2208-put-\u2262  : {a : _}{A : Set a}{n : \u2115}(p : Bits n){x y : A}{t : Tree A n}(path : x \u2208 t)\n          \u2192 p \u2262 toBits path \u2192 x \u2208 put p y t\n  \u2208-put-\u2262 [] here neg = \u22a5-elim (neg refl)\n  \u2208-put-\u2262 (true \u2237 p) (left path) neg   = left path\n  \u2208-put-\u2262 (false \u2237 p) (left path) neg  = left (\u2208-put-\u2262 p path (false \u2237\u2262 neg))\n  \u2208-put-\u2262 (true \u2237 p) (right path) neg  = right (\u2208-put-\u2262 p path (true \u2237\u2262 neg))\n  \u2208-put-\u2262 (false \u2237 p) (right path) neg = right path\n\n  swap : {a : _}{A : Set a}{n : \u2115} \u2192 (p\u2081 p\u2082 : Bits n) \u2192 Tree A n \u2192 Tree A n\n  swap p\u2081 p\u2082 t = put p\u2081 a\u2082 (put p\u2082 a\u2081 t)\n    where\n      a\u2081 = lookup p\u2081 t\n      a\u2082 = lookup p\u2082 t\n\n  swap-perm\u2081 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p : x \u2208 t) \u2192 t \u2248 swap (toBits p) (toBits p) t\n  swap-perm\u2081 here         = \u2248-refl\n  swap-perm\u2081 (left path)  = \u2248-first (swap-perm\u2081 path)\n  swap-perm\u2081 (right path) = \u2248-second (swap-perm\u2081 path)\n\n  swap-comm : {a : _}{A : Set a}{n : \u2115} (p\u2081 p\u2082 : Bits n)(t : Tree A n) \u2192 swap p\u2082 p\u2081 t \u2261 swap p\u2081 p\u2082 t\n  swap-comm [] [] (leaf x) = refl\n  swap-comm (true \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = cong (fork t) (swap-comm p\u2081 p\u2082 t\u2081)\n  swap-comm (true \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (true \u2237 p\u2082) (fork t t\u2081) = refl\n  swap-comm (false \u2237 p\u2081) (false \u2237 p\u2082) (fork t t\u2081) = cong (flip fork t\u2081) (swap-comm p\u2081 p\u2082 t)\n\n  swap-perm\u2082 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p' : Bits n)(p : x \u2208 t)\n             \u2192 x \u2208 swap (toBits p) p' t\n  swap-perm\u2082 _ here = here\n  swap-perm\u2082 (true \u2237 p) (left path) rewrite \u2208-lookup path = right (\u2208-put p _)\n  swap-perm\u2082 (false \u2237 p) (left path) = left (swap-perm\u2082 p path)\n  swap-perm\u2082 (true \u2237 p) (right path) = right (swap-perm\u2082 p path)\n  swap-perm\u2082 (false \u2237 p) (right path) rewrite \u2208-lookup path = left (\u2208-put p _)\n\n  swap-perm\u2083 : {a : _}{A : Set a}{n : \u2115}{t : Tree A n}{x : A}(p\u2081 p\u2082 : Bits n)(p : x \u2208 t)\n              \u2192 p\u2081 \u2262 toBits p \u2192 p\u2082 \u2262 toBits p \u2192 x \u2208 swap p\u2081 p\u2082 t\n  swap-perm\u2083 [] [] here neg\u2081 neg\u2082 = here\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082   = left path\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (left path) neg\u2081 neg\u2082  = left (\u2208-put-\u2262 _ path (false \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (left path) neg\u2081 neg\u2082 = left\n             (swap-perm\u2083 p\u2081 p\u2082 path (false \u2237\u2262 neg\u2081) (false \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082   = right\n             (swap-perm\u2083 p\u2081 p\u2082 path (true \u2237\u2262 neg\u2081) (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (true \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2081))\n  swap-perm\u2083 (false \u2237 p\u2081) (true \u2237 p\u2082) (right path) neg\u2081 neg\u2082  = right (\u2208-put-\u2262 _ path (true \u2237\u2262 neg\u2082))\n  swap-perm\u2083 (false \u2237 p\u2081) (false \u2237 p\u2082) (right path) neg\u2081 neg\u2082 = right path\n\n  \u2208-swp : \u2200 {n a} {A : Set a} (f : SwpOp n) {x : A} {t : Tree A n} \u2192 x \u2208 t \u2192 x \u2208 (f $swp t)\n  \u2208-swp \u03b5 pf = pf\n  \u2208-swp (f \u204f g) pf = \u2208-swp g (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (left pf) = left (\u2208-swp f pf)\n  \u2208-swp (first f) {t = fork _ _} (right pf) = right pf\n  \u2208-swp swp {t = fork t u} (left pf) = right pf\n  \u2208-swp swp {t = fork t u} (right pf) = left pf\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (left pf)) = left (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (left (right pf)) = right (right pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (left pf)) = right (left pf)\n  \u2208-swp swp-seconds {t = fork (fork _ _) (fork _ _)} (right (right pf)) = left (right pf)\n\nmodule FoldProp {a} {A : Set a} (_\u00b7_ : Op\u2082 A) (op-comm : Commutative _\u2261_ _\u00b7_) (op-assoc : Associative _\u2261_ _\u00b7_) where\n\n  \u27ea_\u27eb : \u2200 {n} \u2192 Tree A n \u2192 A\n  \u27ea_\u27eb = fold _\u00b7_\n\n  _=[fold]\u21d2\u2032_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {m n} \u2192 REL (Tree A m) (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  -- _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {m n} \u2192 _\u223c\u2080_ {m} {n} =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n  _\u223c\u2080_ =[fold]\u21d2\u2032 _\u223c\u2081_ = \u2200 {m n} {t : Tree A m} {u : Tree A n} \u2192 t \u223c\u2080 u \u2192 \u27ea t \u27eb \u223c\u2081 \u27ea u \u27eb\n\n  _=[fold]\u21d2_ : \u2200 {\u2113\u2081 \u2113\u2082} \u2192 (\u2200 {n} \u2192 Rel (Tree A n) \u2113\u2081) \u2192 Rel A \u2113\u2082 \u2192 Set _\n  _\u223c\u2080_ =[fold]\u21d2 _\u223c\u2081_ = \u2200 {n} \u2192 _\u223c\u2080_ =[ fold {n} _\u00b7_ ]\u21d2 _\u223c\u2081_\n\n  fold-rot : Rot =[fold]\u21d2 _\u2261_\n  fold-rot (leaf x) = refl\n  fold-rot (fork rot rot\u2081) = cong\u2082 _\u00b7_ (fold-rot rot) (fold-rot rot\u2081)\n  fold-rot (krof rot rot\u2081) rewrite fold-rot rot | fold-rot rot\u2081 = op-comm _ _\n\n  -- t \u223c u \u2192 fork v t \u223c fork u w\n\n  lem : \u2200 x y z t \u2192 (x \u00b7 y) \u00b7 (z \u00b7 t) \u2261 (t \u00b7 y) \u00b7 (z \u00b7 x)\n  lem x y z t = (x \u00b7 y) \u00b7 (z \u00b7 t)\n              \u2261\u27e8 op-assoc x y _ \u27e9\n                x \u00b7 (y \u00b7 (z \u00b7 t))\n              \u2261\u27e8 op-comm x _ \u27e9\n                (y \u00b7 (z \u00b7 t)) \u00b7 x\n              \u2261\u27e8 op-assoc y (z \u00b7 t) _ \u27e9\n                y \u00b7 ((z \u00b7 t) \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 y \u00b7 (u \u00b7 x)) (op-comm z t) \u27e9\n                y \u00b7 ((t \u00b7 z) \u00b7 x)\n              \u2261\u27e8 cong (_\u00b7_ y) (op-assoc t z x) \u27e9\n                y \u00b7 (t \u00b7 (z \u00b7 x))\n              \u2261\u27e8 sym (op-assoc y t _) \u27e9\n                (y \u00b7 t) \u00b7 (z \u00b7 x)\n              \u2261\u27e8 cong (\u03bb u \u2192 u \u00b7 (z \u00b7 x)) (op-comm y t) \u27e9\n                (t \u00b7 y) \u00b7 (z \u00b7 x)\n              \u220e\n    where open \u2261-Reasoning\n\n  fold-swp : Swp =[fold]\u21d2 _\u2261_\n  fold-swp (left pf) rewrite fold-swp pf = refl\n  fold-swp (right pf) rewrite fold-swp pf = refl\n  fold-swp swp\u2081 = op-comm _ _\n  fold-swp (swp\u2082 {_} {t\u2080\u2080} {t\u2080\u2081} {t\u2081\u2080} {t\u2081\u2081}) = lem \u27ea t\u2080\u2080 \u27eb \u27ea t\u2080\u2081 \u27eb \u27ea t\u2081\u2080 \u27eb \u27ea t\u2081\u2081 \u27eb\n\n  fold-swp\u2605 : Swp\u2605 =[fold]\u21d2 _\u2261_\n  fold-swp\u2605 \u03b5 = refl\n  fold-swp\u2605 (x \u25c5 xs) rewrite fold-swp x | fold-swp\u2605 xs = refl\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9b2d54a0d452488646b4ad893d98b6b7c51edbc6","subject":"Add happly, and the univalence axioms","message":"Add happly, and the univalence axioms\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Extensionality.agda","new_file":"lib\/Function\/Extensionality.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP  -- renaming (subst to tr)\n\nhapply : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}\n  \u2192 f \u2261 g \u2192 (x : A) \u2192 f x \u2261 g x\nhapply p x = ap (\u03bb f \u2192 f x) p\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n           (f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n\n    happly-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 happly (\u03bb= fg) \u2261 fg\n\n    \u03bb=-happly : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n      (\u03b1 : f \u2261 g) \u2192 \u03bb= (happly \u03b1) \u2261 \u03b1\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}{{fe : FunExt}}(fg : (x : A) \u2192 f x \u2261 g x)\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n\n\n!-\u03b1-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (\u03b1 : f \u2261 g) \u2192 ! \u03b1 \u2261 \u03bb= (!_ \u2218 happly \u03b1)\n!-\u03b1-\u03bb= refl = ! \u03bb=-happly refl\n\n!-\u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f g : (x : A) \u2192 B x}{{fe : FunExt}}\n  (fg : \u2200 x \u2192 f x \u2261 g x) \u2192 ! (\u03bb= fg) \u2261 \u03bb= (!_ \u2218 fg)\n!-\u03bb= fg = !-\u03b1-\u03bb= (\u03bb= fg) \u2219 ap \u03bb= (\u03bb= (\u03bb x \u2192 ap !_ (happly (happly-\u03bb= fg) x)))\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Relation.Binary.PropositionalEquality renaming (subst to tr)\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}{{fe : FunExt}}\n           (f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x) \u2192 f\u2080 \u2261 f\u2081\n\n    -- This should be derivable if I had a proper proof of \u03bb=\n    tr-\u03bb= : \u2200 {a b p}{A : Set a}{B : A \u2192 Set b}{x}(P : B x \u2192 Set p)\n              {f g : (x : A) \u2192 B x}{{fe : FunExt}}(fg : (x : A) \u2192 f x \u2261 g x)\n            \u2192 tr (\u03bb f \u2192 P (f x)) (\u03bb= fg) \u2261 tr P (fg x)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"882236d335abb915434af70670af53e1e66e8050","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: fc158b5db71339823977877ee10bd200\n\ndarcs-hash:20101217052700-3bd4e-c6dfaddcc53b98b507900d88280a9194528935f7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/AxiomaticPA\/Equality\/Properties.agda","new_file":"src\/AxiomaticPA\/Equality\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- PA properties\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Equality.Properties where\n\nopen import AxiomaticPA.Base\n\n------------------------------------------------------------------------------\n\nrefl : \u2200 n \u2192 n \u2263 n\nrefl n = S\u2081 (S\u2085 n) (S\u2085 n)\n{-# ATP hint refl #-}\n\nsym : \u2200 {m n} \u2192 m \u2263 n \u2192 n \u2263 m\nsym {m} = aux S\u2081 (refl m)\n  where\n    aux : \u2200 {r t} \u2192 (t \u2263 r \u2192 t \u2263 t \u2192 r \u2263 t) \u2192 t \u2263 t \u2192 t \u2263 r \u2192 r \u2263 t\n    aux hyp t\u2263t t\u2263r = hyp t\u2263r t\u2263t\n{-# ATP hint sym #-}\n\ntrans : \u2200 {m n o} \u2192 m \u2263 n \u2192 n \u2263 o \u2192 m \u2263 o\ntrans m\u2263n = S\u2081 (sym m\u2263n)\n{-# ATP hint trans #-}\n","old_contents":"------------------------------------------------------------------------------\n-- PA properties\n------------------------------------------------------------------------------\n\nmodule AxiomaticPA.Equality.Properties where\n\nopen import AxiomaticPA.Base\n\n------------------------------------------------------------------------------\n\nrefl : \u2200 n \u2192 n \u2263 n\nrefl n = S\u2081 (S\u2085 n) (S\u2085 n)\n{-# ATP hint refl #-}\n\nsym : \u2200 {m n} \u2192 m \u2263 n \u2192 n \u2263 m\nsym {m} = aux S\u2081 (refl m)\n  where\n    aux : \u2200 {r t} \u2192 (t \u2263 r \u2192 t \u2263 t \u2192 r \u2263 t) \u2192 t \u2263 t \u2192 t \u2263 r \u2192 r \u2263 t\n    aux fn t\u2263t t\u2263r = fn t\u2263r t\u2263t\n{-# ATP hint sym #-}\n\ntrans : \u2200 {m n o} \u2192 m \u2263 n \u2192 n \u2263 o \u2192 m \u2263 o\ntrans m\u2263n = S\u2081 (sym m\u2263n)\n{-# ATP hint trans #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0087d65feddda54b6f2fa46d87ca1d54d8f1779a","subject":"Simplify definition","message":"Simplify definition\n\nThis variant is simpler to think about and do proofs about on paper, and\nthe two variants are definitionally equal for Agda.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Specification.agda","new_file":"Parametric\/Change\/Specification.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e (\u27e6 const c \u27e7Term \u03c1)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = record\n    { apply = \u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    ; correct = \u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e\n    } where open \u2261-Reasoning\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = record\n    { apply = \u03bb v dv \u2192 \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n    ; correct = \u03bb v dv \u2192\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e\n        \u27e6 t \u27e7\u0394 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1)\n      \u2261\u27e8  correctness t (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1) (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1) \u27e9\n        \u27e6 t \u27e7 (after-env (nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv) \u2022 d\u03c1))\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (((v \u229e\u208d \u03c3 \u208e dv) \u229e\u208d \u03c3 \u208e nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv)) \u2022 after-env d\u03c1)\n      \u2261\u27e8  cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e (v \u229e\u208d \u03c3 \u208e dv))  \u27e9\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 after-env d\u03c1)\n      \u2261\u27e8\u27e9\n        \u27e6 t \u27e7 (after-env (dv \u2022 d\u03c1))\n      \u2261\u27e8  sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1))  \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u03c1)  \u229e\u208d \u03c4 \u208e  \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e\n    } where open \u2261-Reasoning\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = ext (\u03bb v \u2192\n    let\n      -- d\u03c1\u2032 : \u0394Env (\u03c3 \u2022 \u0393) (v \u2022 \u03c1)\n      d\u03c1\u2032 = nil\u208d \u03c3 \u208e v \u2022 d\u03c1\n    in\n      begin\n        \u27e6 t \u27e7 (v \u2022 \u03c1) \u229e\u208d \u03c4 \u208e \u27e6 t \u27e7\u0394 _ d\u03c1\u2032\n      \u2261\u27e8 correctness {\u03c4} t _ d\u03c1\u2032 \u27e9\n        \u27e6 t \u27e7 (after-env d\u03c1\u2032)\n      \u2261\u27e8 cong (\u03bb hole \u2192 \u27e6 t \u27e7 (hole \u2022 after-env d\u03c1)) (update-nil\u208d \u03c3 \u208e v) \u27e9\n        \u27e6 t \u27e7 (v \u2022 after-env d\u03c1)\n      \u220e\n    ) where open \u2261-Reasoning\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"88f7224861b65dc56f3f386ef9ac0ff2bea03488","subject":"more lemmas about Fields","message":"more lemmas about Fields\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_ 2*_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  2\u1da0 3\u1da0 -0\u1da0 -1\u1da0 : A\n  2\u1da0  = suc\u1da0 1\u1da0\n  3\u1da0  = suc\u1da0 2\u1da0\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0     ] = 0\u1da0\n  \u2115[ suc n ] = fold 1\u1da0 suc\u1da0 n\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0       = 1\u1da0\n  b ^\u2115 (suc e) = fold b (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\n  2+_ : A \u2192 A\n  2+ x = 2\u1da0 + x\n\n  2*_ : A \u2192 A\n  2* x = x + x\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x * x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 * x\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  \u2212= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \u2212 y \u2261 x' \u2212 y'\n  \u2212= {x} {y' = y'} p q = ap (_\u2212_ x) q \u2219 ap (\u03bb z \u2192 z \u2212 y') p\n\n  \/= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \/ y \u2261 x' \/ y'\n  \/= {x} {y' = y'} p q = ap (_\/_ x) q \u2219 ap (\u03bb z \u2192 z \/ y') p\n\n  0-= : \u2200 {x x'} \u2192 x \u2261 x' \u2192 0- x \u2261 0- x'\n  0-= = ap 0-_\n\n\n  +-*-distr : _*_ DistributesOver\u02b3 _+_\n  +-*-distr = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  +-\/-distr : _\/_ DistributesOver\u02b3 _+_\n  +-\/-distr = +-*-distr\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  *0-zero : RightZero 0\u1da0 _*_\n  *0-zero = +-right-cancel  (+= refl (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= refl 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero 0\u1da0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  \u207b\u00b9-notZero : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 \u2262 0\u1da0\n  \u207b\u00b9-notZero x\/=0 = \u03bb p \u2192 0\u22621 (! 0*-zero \u2219 *= (! p) refl \u2219 \u207b\u00b9-inverse x\/=0)\n\n  -- name ?\n  *-0- : \u2200 {x y} \u2192 0- (x * y) \u2261 (0- x) * y\n  *-0- = +-right-cancel (0--inverse \u2219 ! 0*-zero \u2219 *= (! 0--inverse) refl \u2219 +-*-distr)\n\n  -1*-neg : \u2200 {x} \u2192 -1\u1da0 * x \u2261 0- x\n  -1*-neg = ! *-0- \u2219 0-= 1*-identity\n\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  1\u207b\u00b9\u22611 : 1\u1da0 \u207b\u00b9 \u2261 1\u1da0\n  1\u207b\u00b9\u22611 = ! *1-identity \u2219 \u207b\u00b9-inverse (\u03bb p \u2192 0\u22621 (! p))\n\n  noZeroDivisor : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 x * y \u2262 0\u1da0\n  noZeroDivisor nx ny x*y\u22610\u1da0 = ny (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse nx)\n                                   \u2219 *= refl *-comm \u2219 ! *-assoc\n                                   \u2219 *= (*-comm \u2219 x*y\u22610\u1da0) refl \u2219 0*-zero\n                                   )\n\n  2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u1da0 * n\n  2*-spec = ! += 1*-identity 1*-identity \u2219 ! +-*-distr\n\n  +-interchange : Interchange _+_ _+_\n  +-interchange = InterchangeFromAssocComm.\u00b7-interchange _+_ +-assoc +-comm\n\n  *-interchange : Interchange _*_ _*_\n  *-interchange = InterchangeFromAssocComm.\u00b7-interchange _*_ *-assoc *-comm\n\n  *-unique-inverse : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 x * y \u2261 1\u1da0 \u2192 y \u2261 x \u207b\u00b9\n  *-unique-inverse x\/=0 xy=1 = ! 1*-identity \u2219 *= (! \u207b\u00b9-inverse x\/=0) refl \u2219 *-assoc \u2219 *= refl xy=1 \u2219 *1-identity\n\n  \u207b\u00b9-involutive : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 \u207b\u00b9 \u2261 x\n  \u207b\u00b9-involutive x\/=0 = ! *-unique-inverse (\u207b\u00b9-notZero x\/=0) (\u207b\u00b9-inverse x\/=0)\n\n  \u207b\u00b9*-distr : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 (x * y)\u207b\u00b9 \u2261 x \u207b\u00b9  \/ y\n  \u207b\u00b9*-distr x\/=0 y\/=0 =\n    ! (*-unique-inverse (noZeroDivisor x\/=0 y\/=0) (*= refl *-comm \u2219 *-assoc \u2219 *= refl (! *-assoc \u2219 *= (\u207b\u00b9-right-inverse y\/=0) refl \u2219 1*-identity) \u2219 \u207b\u00b9-right-inverse x\/=0))\n\n  +-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192(a \/ b) + (a' \/ b') \u2261 (a * b' + a' * b) \/ (b * b')\n  +-quotient b b'\n    = += (\/= (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse b')) refl) (\/= (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse b)) refl)\n    \u2219 (+= (\/= (*= refl *-comm) refl) (\/= (*= refl *-comm) refl)\n    \u2219 += (*= (! *-assoc) refl \u2219 *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b' b \u2219 ap _\u207b\u00b9 *-comm)) (*-assoc \u2219 *= refl *-assoc \u2219 ! *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b b') ))\n    \u2219 ! +-\/-distr\n\n  \u2212-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192 (a \/ b) \u2212 (a' \/ b') \u2261 (a * b' \u2212 a' * b) \/ (b * b')\n  \u2212-quotient b b' = += refl *-0- \u2219 +-quotient b b' \u2219 \/= (+= refl (! *-0-)) refl\n\n  *-quotient : \u2200 {a b a' b'} \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n    \u2192 (a \/ b) * (a' \/ b') \u2261 (a * a') \/ (b * b')\n  *-quotient b\/=0 b'\/=0 = *-assoc \u2219 *= refl (! *-assoc \u2219 *= *-comm refl \u2219 *-assoc) \u2219 ! *-assoc \u2219 *= refl (! \u207b\u00b9*-distr b\/=0 b'\/=0)\n\n  \/-quotient : \u2200 {a b a' b'} \u2192 a' \u2262 0\u1da0 \u2192 b \u2262 0\u1da0 \u2192 b' \u2262 0\u1da0\n     \u2192 (a \/ b) \/ (a' \/ b') \u2261 (a * b') \/ (b * a')\n  \/-quotient a' b b' = *= refl (\u207b\u00b9*-distr a' (\u207b\u00b9-notZero b') \u2219 *= refl (\u207b\u00b9-involutive b') \u2219 *-comm)\n    \u2219 *-interchange \u2219 *= refl (! \u207b\u00b9*-distr b a')\n\n  {-\n\u2212= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \u2212 y \u2261 x' \u2212 y'\n\u2212= {x} {y' = y'} p q = ap (_\u2212_ x) q \u2219 ap (\u03bb z \u2192 z \u2212 y') p\n\n\/= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \/ y \u2261 x' \/ y'\n\/= {x} {y' = y'} p q = ap (_\/_ x) q \u2219 ap (\u03bb z \u2192 z \/ y') p\n\n0--cancel : \u2200 {x y} \u2192 0- x \u2261 0- y \u2192 x \u2261 y\n0--cancel {x} {y} p = {!!}\n\n-1*-spec : \u2200 {x} \u2192 -1\u1da0 * x \u2261 - x\n-1*-spec {x} = {!!}\n\n-*-distr : \u2200 {x y} \u2192 -(x * y) \u2261 (- x) * y\n-*-distr = ! -1*-spec \u2219 ! *-assoc \u2219 *= -1*-spec refl\n\n-*-distr' : \u2200 {x y} \u2192 -(x * y) \u2261 x * (- y)\n-*-distr' = ap -_ *-comm \u2219 -*-distr \u2219 *-comm\n\n*-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n*-\u2212-distr = *-+-distr \u2219 += refl (! -*-distr')\n\n-+-distr : \u2200 {x y} \u2192 -(x + y) \u2261 - x \u2212 y\n-+-distr =\n  +-left-cancel\n     (0--right-inverse\n      \u2219 (! 0+-identity \u2219 += (! 0--right-inverse)\n                            (! 0--right-inverse))\n                       \u2219 +-interchange)\n\n\u00b2-*-distr : \u2200 {x y} \u2192 (x * y)\u00b2 \u2261 x \u00b2 * y \u00b2\n\u00b2-*-distr = *-interchange\n\n\u00b2-\u2212-distr : \u2200 {x y} \u2192 (x \u2212 y)\u00b2 \u2261 x \u00b2 + y \u00b2 \u2212 2* x * y\n\u00b2-\u2212-distr {x} {y} = {!!}\n-- = (x - y)*(x - y) = x*(x - y) - y*(x - y) = x \u00b2 - xy - yx + y \u00b2\n\n\u00b2-+-distr : \u2200 {x y} \u2192 (x + y)\u00b2 \u2261 x \u00b2 + y \u00b2 + 2* x * y\n\u00b2-+-distr = {!!}\n\n\u00b2--distr : \u2200 {x} \u2192 (- x)\u00b2 \u2261 x \u00b2\n\u00b2--distr {x} = ! -*-distr \u2219 ap -_ (! -*-distr') \u2219 0--involutive\n-}\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : Field A) where\n  open Field F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : Field-Ops B)(default : Decode-Term B) where\n    module G = Field-Ops G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_\ndata Tm (A : Set) : Set where\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\npattern  0' = lit (\u2124.+ 0)\npattern  1' = lit (\u2124.+ 1)\npattern  2' = lit (\u2124.+ 1)\npattern  3' = lit (\u2124.+ 1)\npattern -1' = lit \u2124.-[1+ 0 ]\npattern -2' = lit \u2124.-[1+ 1 ]\npattern -3' = lit \u2124.-[1+ 2 ]\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : Field F) where\n    open Field F-field renaming (0-_ to -_)\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = \u2124[ x ]\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\nopen import Data.Nat.NP using (\u2115; zero; suc; fold)\nimport Data.Integer as \u2124\nopen \u2124 using (\u2124)\n\nrecord Field-Ops {\u2113} (A : Set \u2113) : Set \u2113 where\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_ 2*_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    0\u1da0 1\u1da0        : A\n    0-_         : A \u2192 A\n    _\u207b\u00b9         : A \u2192 A\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  suc\u1da0 : A \u2192 A\n  suc\u1da0 = _+_ 1\u1da0\n\n  pred\u1da0 : A \u2192 A\n  pred\u1da0 x = x \u2212 1\u1da0\n\n  2\u1da0 3\u1da0 -0\u1da0 -1\u1da0 : A\n  2\u1da0  = suc\u1da0 1\u1da0\n  3\u1da0  = suc\u1da0 2\u1da0\n  -0\u1da0 = 0- 0\u1da0\n  -1\u1da0 = 0- 1\u1da0\n\n  \u2115[_] : \u2115 \u2192 A\n  \u2115[ 0     ] = 0\u1da0\n  \u2115[ suc n ] = fold 1\u1da0 suc\u1da0 n\n\n  \u2124[_] : \u2124 \u2192 A\n  \u2124[ \u2124.+ x      ] = \u2115[ x ]\n  \u2124[ \u2124.-[1+ x ] ] = 0- \u2115[ suc x ]\n\n  _^\u2115_ : A \u2192 \u2115 \u2192 A\n  b ^\u2115 0       = 1\u1da0\n  b ^\u2115 (suc e) = fold b (_*_ b) e\n\n  _^\u2124_ : A \u2192 \u2124 \u2192 A\n  b ^\u2124 (\u2124.+ x)    = b ^\u2115 x\n  b ^\u2124 \u2124.-[1+ x ] = (b ^\u2115 (suc x))\u207b\u00b9\n\n  2+_ : A \u2192 A\n  2+ x = 2\u1da0 + x\n\n  2*_ : A \u2192 A\n  2* x = x + x\n\n  _\u00b2 : A \u2192 A\n  x \u00b2 = x * x\n\n  _\u00b3 : A \u2192 A\n  x \u00b3 = x \u00b2 * x\n\nrecord Field-Struct {\u2113} {A : Set \u2113} (field-ops : Field-Ops A) : Set \u2113 where\n  open FP {\u2113} {A}\n  open Field-Ops field-ops\n\n  field\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    \u207b\u00b9-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x \u207b\u00b9 * x \u2261 1\u1da0\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  \u207b\u00b9-right-inverse  : \u2200 {x} \u2192 x \u2262 0\u1da0 \u2192 x * x \u207b\u00b9 \u2261 1\u1da0\n  \u207b\u00b9-right-inverse p = *-comm \u2219 \u207b\u00b9-inverse p\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-= : \u2200 {x x'} \u2192 x \u2261 x' \u2192 0- x \u2261 0- x'\n  0-= = ap 0-_\n\n  +-*-distr : _*_ DistributesOver\u02b3 _+_\n  +-*-distr = *-comm \u2219 *-+-distr \u2219 += *-comm *-comm\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  *0-zero : RightZero 0\u1da0 _*_\n  *0-zero = +-right-cancel  (+= refl (! *1-identity) \u2219 ! *-+-distr\n                            \u2219 *= refl 0+-identity \u2219 *1-identity \u2219 ! 0+-identity)\n\n  0*-zero : LeftZero 0\u1da0 _*_\n  0*-zero = *-comm \u2219 *0-zero\n\n  -- name ?\n  *-0- : \u2200 {x y} \u2192 0- (x * y) \u2261 (0- x) * y\n  *-0- = +-right-cancel (0--inverse \u2219 ! 0*-zero \u2219 *= (! 0--inverse) refl \u2219 +-*-distr)\n\n  -1*-neg : \u2200 {x} \u2192 -1\u1da0 * x \u2261 0- x\n  -1*-neg = ! *-0- \u2219 0-= 1*-identity\n\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\n  -0\u22610 : -0\u1da0 \u2261 0\u1da0\n  -0\u22610 = +-left-cancel (0--right-inverse \u2219 ! 0+-identity)\n\n  noZeroDivisor : \u2200 {x y} \u2192 x \u2262 0\u1da0 \u2192 y \u2262 0\u1da0 \u2192 x * y \u2262 0\u1da0\n  noZeroDivisor nx ny x*y\u22610\u1da0 = ny (! *1-identity \u2219 *= refl (! \u207b\u00b9-inverse nx)\n                                   \u2219 *= refl *-comm \u2219 ! *-assoc\n                                   \u2219 *= (*-comm \u2219 x*y\u22610\u1da0) refl \u2219 0*-zero\n                                   )\n\n  2*-spec : \u2200 {n} \u2192 2* n \u2261 2\u1da0 * n\n  2*-spec = ! += 1*-identity 1*-identity \u2219 ! +-*-distr\n\n  +-interchange : Interchange _+_ _+_\n  +-interchange = InterchangeFromAssocComm.\u00b7-interchange _+_ +-assoc +-comm\n\n  *-interchange : Interchange _*_ _*_\n  *-interchange = InterchangeFromAssocComm.\u00b7-interchange _*_ *-assoc *-comm\n\n  {-\n\u2212= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \u2212 y \u2261 x' \u2212 y'\n\u2212= {x} {y' = y'} p q = ap (_\u2212_ x) q \u2219 ap (\u03bb z \u2192 z \u2212 y') p\n\n\/= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \/ y \u2261 x' \/ y'\n\/= {x} {y' = y'} p q = ap (_\/_ x) q \u2219 ap (\u03bb z \u2192 z \/ y') p\n\n0--cancel : \u2200 {x y} \u2192 0- x \u2261 0- y \u2192 x \u2261 y\n0--cancel {x} {y} p = {!!}\n\n-1*-spec : \u2200 {x} \u2192 -1\u1da0 * x \u2261 - x\n-1*-spec {x} = {!!}\n\n-*-distr : \u2200 {x y} \u2192 -(x * y) \u2261 (- x) * y\n-*-distr = ! -1*-spec \u2219 ! *-assoc \u2219 *= -1*-spec refl\n\n-*-distr' : \u2200 {x y} \u2192 -(x * y) \u2261 x * (- y)\n-*-distr' = ap -_ *-comm \u2219 -*-distr \u2219 *-comm\n\n*-\u2212-distr : \u2200 {x y z} \u2192 x * (y \u2212 z) \u2261 x * y \u2212 x * z\n*-\u2212-distr = *-+-distr \u2219 += refl (! -*-distr')\n\n-+-distr : \u2200 {x y} \u2192 -(x + y) \u2261 - x \u2212 y\n-+-distr =\n  +-left-cancel\n     (0--right-inverse\n      \u2219 (! 0+-identity \u2219 += (! 0--right-inverse)\n                            (! 0--right-inverse))\n                       \u2219 +-interchange)\n\n\u00b2-*-distr : \u2200 {x y} \u2192 (x * y)\u00b2 \u2261 x \u00b2 * y \u00b2\n\u00b2-*-distr = *-interchange\n\n\u00b2-\u2212-distr : \u2200 {x y} \u2192 (x \u2212 y)\u00b2 \u2261 x \u00b2 + y \u00b2 \u2212 2* x * y\n\u00b2-\u2212-distr {x} {y} = {!!}\n-- = (x - y)*(x - y) = x*(x - y) - y*(x - y) = x \u00b2 - xy - yx + y \u00b2\n\n\u00b2-+-distr : \u2200 {x y} \u2192 (x + y)\u00b2 \u2261 x \u00b2 + y \u00b2 + 2* x * y\n\u00b2-+-distr = {!!}\n\n\u00b2--distr : \u2200 {x} \u2192 (- x)\u00b2 \u2261 x \u00b2\n\u00b2--distr {x} = ! -*-distr \u2219 ap -_ (! -*-distr') \u2219 0--involutive\n-}\n\n  open Field-Ops field-ops public\n\nrecord Field {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    field-ops    : Field-Ops A\n    field-struct : Field-Struct field-ops\n  open Field-Struct field-struct public\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : Field A) where\n  open Field F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-RawField {b}{B : Set b}(G : Field-Ops B)(default : Decode-Term B) where\n    module G = Field-Ops G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\nopen \u2261-Reasoning\nopen import Data.List\n\ninfixl 6 _+'_\ninfixl 7 _*'_\ndata Tm (A : Set) : Set where\n  lit       : (n : \u2124) \u2192 Tm A\n  var       : (x : A) \u2192 Tm A\n  _+'_ _*'_ : Tm A \u2192 Tm A \u2192 Tm A\n\npattern  0' = lit (\u2124.+ 0)\npattern  1' = lit (\u2124.+ 1)\npattern  2' = lit (\u2124.+ 1)\npattern  3' = lit (\u2124.+ 1)\npattern -1' = lit \u2124.-[1+ 0 ]\npattern -2' = lit \u2124.-[1+ 1 ]\npattern -3' = lit \u2124.-[1+ 2 ]\n\nrecord Var (A : Set) : Set where\n  constructor _^'_\n  field\n    v : A\n    e : \u2124\n\ndata Nf' (A : Set) : Set where\n  _*'_ : (n : \u2124) (vs : List (Var A)) \u2192 Nf' A\n\ndata Nf (A : Set) : Set where\n  _+'_ : (n : \u2124) (ts' : List (Nf' A)) \u2192 Nf A\n\nmodule _ {A : Set} where\n    infixl 6 _+\u1d3a_\n\n    lit\u1d3a : \u2124 \u2192 Nf A\n    lit\u1d3a n = n +' []\n\n    var\u1d3a : (x : A) \u2192 Nf A\n    var\u1d3a x = \u2124.+ 0 +' (\u2124.+ 1 *' (x ^' (\u2124.+ 1) \u2237 []) \u2237 [])\n\n    _+\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) +\u1d3a (m +' us) = n \u2124.+ m +' (ts ++ us)\n\n    infixl 7 _**_ _***_ _*\u1d3a_\n    _**_ : \u2124 \u2192 List (Nf' A) \u2192 Nf A\n    z ** [] = z +' []\n    m ** (n *' vs \u2237 ts) with m ** ts\n    ... | z +' us = z +' ((m \u2124.* n) *' vs \u2237 us)\n\n    _***_ : List (Nf' A) \u2192 List (Nf' A) \u2192 Nf A\n    ts *** us = \u2124.+ 0 +' (ts ++ us)\n\n    _*\u1d3a_ : Nf A \u2192 Nf A \u2192 Nf A\n    (n +' ts) *\u1d3a (m +' us) =\n       n \u2124.* m +' [] +\u1d3a n ** us +\u1d3a m ** ts +\u1d3a ts *** us\n\nmodule Normalizer {F : Set}\n                  (F-field : Field F) where\n    open Field F-field renaming (0-_ to -_)\n\n    module _ {A : Set} where\n        eval : (A \u2192 F) \u2192 Tm A \u2192 F\n        eval \u03c1 (lit x) = \u2124[ x ]\n        eval \u03c1 (t +' t\u2081) = eval \u03c1 t + eval \u03c1 t\u2081\n        eval \u03c1 (t *' t\u2081) = eval \u03c1 t * eval \u03c1 t\u2081\n        eval \u03c1 (var x)  = \u03c1 x\n\n        norm\u2122 : Tm A \u2192 Nf A\n        norm\u2122 (lit x) = lit\u1d3a x\n        norm\u2122 (var x) = var\u1d3a x\n        norm\u2122 (t +' t\u2081) = norm\u2122 t +\u1d3a norm\u2122 t\u2081\n        norm\u2122 (t *' t\u2081) = norm\u2122 t *\u1d3a norm\u2122 t\u2081\n\n        _+''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) +'' u = u\n        t +'' lit (\u2124.+ 0) = t\n        t +'' u = t +' u\n\n        _*''_ : Tm A \u2192 Tm A \u2192 Tm A\n        -- lit (\u2124.+ 0) *'' u = lit (\u2124.+ 0)\n        -- lit (\u2124.-[1+_] 0) *'' u = lit (\u2124.+ 0)\n        t *'' lit (\u2124.+ 0) = lit (\u2124.+ 0)\n        t *'' lit (\u2124.-[1+_] 0) = lit (\u2124.+ 0)\n        -- lit (\u2124.+ 1) *'' u = u\n        t *'' lit (\u2124.+ 1) = t\n        t *'' u           = t *' u\n\n        reifyVar : Var A \u2192 Tm A\n        reifyVar (v ^' (\u2124.+ n))    = fold (lit (\u2124.+ 1)) (\u03bb x \u2192 var v *'' x) n\n        reifyVar (v ^' \u2124.-[1+ n ]) = fold (lit (\u2124.-[1+_] 1)) (\u03bb x \u2192 var v *'' x) n\n\n        reifyNf' : Nf' A \u2192 Tm A\n        reifyNf' (n *' vs) = foldr (\u03bb v t \u2192 reifyVar v *'' t) (lit n) vs\n\n        reifyNf : Nf A \u2192 Tm A\n        reifyNf (n +' ts') = foldr (\u03bb n t \u2192 reifyNf' n +'' t) (lit n) ts'\n\n        module _ \u03c1 where\n            _\u223c_ : (t u : Tm A) \u2192 Set\n            t \u223c u = eval \u03c1 t \u2261 eval \u03c1 u\n\n            +'\u223c+'' : \u2200 t u \u2192 (t +' u) \u223c (t +'' u)\n            +'\u223c+'' t (lit (\u2124.+ zero)) = +0-identity\n            +'\u223c+'' t (lit (\u2124.+ (suc x))) = refl\n            +'\u223c+'' t (lit \u2124.-[1+ x ]) = refl\n            +'\u223c+'' t (var x) = refl\n            +'\u223c+'' t (u +' u\u2081) = refl\n            +'\u223c+'' t (u *' u\u2081) = refl\n\n            {-\n            +-reifyNf : \u2200 t u \u2192 reifyNf (t +\u1d3a u) \u223c (reifyNf t +' reifyNf u)\n            +-reifyNf (n +' ts') (n\u2081 +' ts'') = {!!}\n\n            *-reifyNf : \u2200 t u \u2192 reifyNf (t *\u1d3a u) \u223c (reifyNf t *' reifyNf u)\n            *-reifyNf t u = {!!}\n\n            pf : \u2200 t \u2192 eval \u03c1 (reifyNf (norm\u2122 t)) \u2261 eval \u03c1 t\n            pf (lit n) = refl\n            pf (var x) = refl\n            pf (t +' t\u2081) rewrite +-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            pf (t *' t\u2081) rewrite *-reifyNf (norm\u2122 t) (norm\u2122 t\u2081) | pf t | pf t\u2081 = refl\n            -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"71e84d39b2c2f33deea86e293b42bdb49707ada3","subject":"Reflection.NP: updates and improvements","message":"Reflection.NP: updates and improvements\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/NP.agda","new_file":"lib\/Reflection\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Reflection.NP where\n\nopen import Opaque\nopen import Type\nopen import Level.NP\nopen import Data.Nat using (\u2115; zero; suc; _+_) renaming (_\u2294_ to _\u2294\u2115_)\nopen import Data.Maybe.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9; 0\u2081)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; [0:_1:_])\nopen import Data.List\nopen import Data.Vec using (Vec) -- ; []; _\u2237_)\nopen import Data.Product.NP\nopen import Function.NP\n\nopen import Reflection public\n\nArgs : \u2605\nArgs = List (Arg Term)\n\n-- lam\u1d5b : Term \u2192 Term\npattern lam\u1d5b = lam visible\n\n-- lam\u02b0 : Term \u2192 Term\npattern lam\u02b0 = lam hidden\n\n-- argi\u1d5b : Relevance \u2192 Arg-info\npattern argi\u1d5b x = arg-info visible x\n\n-- argi\u02b3 : Visibility \u2192 Arg-info\npattern argi\u02b3 x = arg-info x relevant\n\n-- argi\u1d5b\u02b3 : Arg-info\npattern argi\u1d5b\u02b3 = argi\u1d5b relevant\n\n-- argi\u02b0 : Relevance \u2192 Arg-info\npattern argi\u02b0 x = arg-info hidden x\n\n-- argi\u02b0\u02b3 : Arg-info\npattern argi\u02b0\u02b3 = argi\u02b0 relevant\n\n-- arg\u1d5b : \u2200{A} \u2192 Relevance \u2192 A \u2192 Arg A\npattern arg\u1d5b r v = arg (argi\u1d5b r) v\n\n-- arg\u1d5b\u02b3 : \u2200{A} \u2192 Visibility \u2192 A \u2192 Arg A\npattern arg\u02b3 v x = arg (argi\u02b3 v) x\n\n-- arg\u02b0 : \u2200{A} \u2192 Relevance \u2192 A \u2192 Arg A\npattern arg\u02b0 r v = arg (argi\u02b0 r) v\n\n-- arg\u1d5b\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\npattern arg\u1d5b\u02b3 v = arg argi\u1d5b\u02b3 v\n\n-- arg\u02b0\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\npattern arg\u02b0\u02b3 v = arg argi\u02b0\u02b3 v\n\npattern con\u1d5b n r t = con n (arg\u1d5b r t \u2237 [])\npattern con\u02b0 n r t = con n (arg\u02b0 r t \u2237 [])\npattern def\u1d5b n r t = def n (arg\u1d5b r t \u2237 [])\npattern def\u02b0 n r t = def n (arg\u02b0 r t \u2237 [])\n\npattern con\u1d5b\u02b3 n t = con n (arg\u1d5b\u02b3 t \u2237 [])\npattern con\u02b0\u02b3 n t = con n (arg\u02b0\u02b3 t \u2237 [])\npattern def\u1d5b\u02b3 n t = def n (arg\u1d5b\u02b3 t \u2237 [])\npattern def\u02b0\u02b3 n t = def n (arg\u02b0\u02b3 t \u2237 [])\n\nArg-infos : \u2605\nArg-infos = List Arg-info\n\napp` : (Args \u2192 Term) \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\napp` f = go [] where\n  go : Args \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\n  go args []         = f (reverse args)\n  go args (ai \u2237 ais) = \u03bb t \u2192 go (arg ai t \u2237 args) ais\n\ncon` : Name \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\ncon` x = app` (con x)\n\ndef` : Name \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\ndef` x = app` (def x)\n\nvar` : \u2115 \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\nvar` x = app` (var x)\n\ncoe : \u2200 {A : \u2605} {z : A} n \u2192 (Term \u2192\u27e8 length (replicate n z) \u27e9 Term) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncoe zero    t = t\ncoe (suc n) f = \u03bb t \u2192 coe n (f t)\n\ncon`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncon`\u207f\u02b3 x n = coe n (app` (con x) (replicate n argi\u1d5b\u02b3))\n\ndef`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ndef`\u207f\u02b3 x n = coe n (app` (def x) (replicate n argi\u1d5b\u02b3))\n\nvar`\u207f\u02b3 : \u2115 \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\nvar`\u207f\u02b3 x n = coe n (app` (var x) (replicate n argi\u1d5b\u02b3))\n\n-- sort\u2080 : Sort\npattern sort\u2080 = lit 0\n\n-- sort\u2081 : Sort\npattern sort\u2081 = lit 1\n\n-- `\u2605_ : \u2115 \u2192 Term\npattern  `\u2605_ i = sort (lit i)\n\n-- `\u2605\u2080 : Term\npattern `\u2605\u2080 = `\u2605_ 0\n\n-- el\u2080 : Term \u2192 Type\npattern el\u2080 t = el sort\u2080 t\n\n-- Builds a type variable (of type \u2605\u2080)\n``var\u2080 : \u2115 \u2192 Args \u2192 Type\n``var\u2080 n args = el\u2080 (var n args)\n\n-- ``set : \u2115 \u2192 \u2115 \u2192 Type\npattern ``set i j = el (lit (suc j)) (`\u2605_ i)\n\n``\u2605_ : \u2115 \u2192 Type\n``\u2605_ i = el (lit (suc i)) (`\u2605_ i)\n\n-- ``\u2605\u2080 : Type\npattern ``\u2605\u2080 = ``set 0 0\n\nunEl : Type \u2192 Term\nunEl (el _ tm) = tm\n\ngetSort : Type \u2192 Sort\ngetSort (el s _) = s\n\nunArg : \u2200 {A} \u2192 Arg A \u2192 A\nunArg (arg _ a) = a\n\n-- `Level : Term\npattern `Level = def (quote Level) []\n\n-- ``Level : Type\npattern ``Level = el\u2080 `Level\n\npattern `\u2080 = def (quote \u2080) []\n\n-- `\u209b_ : Term \u2192 Term\n-- `\u209b_ = def`\u207f\u02b3 (quote L.suc) 1\npattern `\u209b_ v = def (quote \u209b) (arg\u1d5b\u02b3 v \u2237 [])\n\n-- `suc-sort : Sort \u2192 Sort\npattern `suc-sort s = set (`\u209b (sort s))\n\npattern `set\u2080 = set `\u2080\npattern `set\u209b_ s = set (`\u209b s)\npattern `set_ s = set (sort s)\n\nsuc-sort : Sort \u2192 Sort\nsuc-sort (set t) = set (`\u209b t)\nsuc-sort (lit n) = lit (suc n)\nsuc-sort unknown = unknown\n\ndecode-Sort : Sort \u2192 Maybe \u2115\ndecode-Sort `set\u2080 = just zero\ndecode-Sort (`set\u209b_ s) = map? suc (decode-Sort (set s))\ndecode-Sort (`set_ s) = decode-Sort s\ndecode-Sort (set _) = nothing\ndecode-Sort (lit n) = just n\ndecode-Sort unknown = nothing\n\npattern _`\u2294_ s\u2081 s\u2082 = set (def (quote _\u2294_) (arg\u1d5b\u02b3 (sort s\u2081) \u2237 arg\u1d5b\u02b3 (sort s\u2082) \u2237 []))\n\n_`\u2294`_ : Sort \u2192 Sort \u2192 Sort\ns\u2081 `\u2294` s\u2082 with decode-Sort s\u2081 | decode-Sort s\u2082\n...          | just n\u2081        | just n\u2082        = lit (n\u2081 \u2294\u2115 n\u2082)\n...          | _              | _              = s\u2081 `\u2294 s\u2082\n\nmapVar : (\u2115 \u2192 \u2115) \u2192 Term \u2192 Term\nmapVarArgs : (\u2115 \u2192 \u2115) \u2192 Args \u2192 Args\nmapVarType : (\u2115 \u2192 \u2115) \u2192 Type \u2192 Type\nmapVarSort : (\u2115 \u2192 \u2115) \u2192 Sort \u2192 Sort\n\nmapVar\u2191 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\nmapVar\u2191 f zero    = zero\nmapVar\u2191 f (suc n) = suc (f n)\n\nmapVar f (var x args) = var (f x) (mapVarArgs f args)\nmapVar f (con c args) = con c (mapVarArgs f args)\nmapVar f (def d args) = def d (mapVarArgs f args)\nmapVar f (lam v t) = lam v (mapVar (mapVar\u2191 f) t)\nmapVar f (pat-lam cs args) = unknown\nmapVar f (pi (arg i t\u2081) t\u2082) = pi (arg i (mapVarType f t\u2081)) (mapVarType (mapVar\u2191 f) t\u2082)\nmapVar f (sort x) = sort (mapVarSort f x)\nmapVar f (lit x) = lit x\nmapVar f unknown = unknown\n\nmapVarArgs f [] = []\nmapVarArgs f (arg i x \u2237 as) = arg i (mapVar f x) \u2237 mapVarArgs f as\nmapVarType f (el s t) = el (mapVarSort f s) (mapVar f t)\nmapVarSort f (set t) = set (mapVar f t)\nmapVarSort f (lit n) = lit n\nmapVarSort f unknown = unknown\n\nliftTerm : Term \u2192 Term\nliftTerm = mapVar suc\n\nliftType : Type \u2192 Type\nliftType = mapVarType suc\n\npattern pi\u1d5b\u02b3 t u = pi (arg\u1d5b\u02b3 t) u\npattern pi\u02b0\u02b3 t u = pi (arg\u02b0\u02b3 t) u\n\n`\u03a0 : Arg Type \u2192 Type \u2192 Type\n`\u03a0 t u = el (getSort (unArg t) `\u2294` getSort u) (pi t u)\n\n`\u03a0\u1d5b\u02b3 : Type \u2192 Type \u2192 Type\n`\u03a0\u1d5b\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u1d5b\u02b3 t u)\n\n`\u03a0\u02b0\u02b3 : Type \u2192 Type \u2192 Type\n`\u03a0\u02b0\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u02b0\u02b3 t u)\n\n_`\u2192_ : Arg Type \u2192 Type \u2192 Type\nt `\u2192 u = `\u03a0 t (liftType u)\n\n_`\u2192\u02b0\u02b3_ : Type \u2192 Type \u2192 Type\nt `\u2192\u02b0\u02b3 u = `\u03a0\u02b0\u02b3 t (liftType u)\n\n_`\u2192\u1d5b\u02b3_ : Type \u2192 Type \u2192 Type\nt `\u2192\u1d5b\u02b3 u = `\u03a0\u1d5b\u02b3 t (liftType u)\n\n-- \u03b7 vs mk: performs no shifting of the result of mk.\n-- Safe values of mk are def and con for instance\n\u03b7 : List Arg-info \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7 ais\u2080 mk = go ais\u2080 where\n  vars : List Arg-info \u2192 Args\n  vars []         = []\n  vars (ai \u2237 ais) = arg ai (var (length ais) []) \u2237 vars ais\n  go : List Arg-info \u2192 Term\n  go []                   = mk (vars ais\u2080)\n  go (arg-info v _ \u2237 ais) = lam v (go ais)\n\n\u03b7\u02b0 : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u02b0 n = \u03b7 (replicate n argi\u02b0\u02b3)\n\n\u03b7\u1d5b : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u1d5b n = \u03b7 (replicate n argi\u1d5b\u02b3)\n\n\u03b7\u02b0\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u02b0\u207f n = \u03b7\u02b0 n \u2218 def\n\n\u03b7\u1d5b\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u1d5b\u207f n = \u03b7\u1d5b n \u2218 def\n\narityOfTerm : Term \u2192 List Arg-info\narityOfType : Type \u2192 List Arg-info\n\narityOfType (el _ u) = arityOfTerm u\narityOfTerm (pi (arg ai _) t) = ai \u2237 arityOfType t\n\n-- no more arguments\narityOfTerm (var _ _) = []\n\n-- TODO\narityOfTerm (con c args) = []\narityOfTerm (def f args) = []\narityOfTerm (sort s)     = []\narityOfTerm (lit _)      = []\n\n-- fail\narityOfTerm unknown = []\n\n-- absurd cases\narityOfTerm (lam _ _) = []\narityOfTerm (pat-lam _ _) = []\n\n\u03b7\u207f : Name \u2192 Term\n\u03b7\u207f nm = \u03b7 (arityOfType (type nm)) (def nm)\n\nDecode-Term : \u2200 {a} \u2192 \u2605_ a \u2192 \u2605_ a\nDecode-Term A = Term \u2192 Maybe A\n\npattern `\ud835\udfd8 = def (quote \ud835\udfd8) []\n\npattern `\ud835\udfd9  = def (quote \ud835\udfd9) []\npattern `0\u2081 = con (quote 0\u2081) []\n\ndecode-\ud835\udfd9 : Decode-Term \ud835\udfd9\ndecode-\ud835\udfd9 `0\u2081 = just 0\u2081\ndecode-\ud835\udfd9 _   = nothing\n\npattern `\ud835\udfda  = def (quote \ud835\udfda) []\npattern `0\u2082 = con (quote 0\u2082) []\npattern `1\u2082 = con (quote 1\u2082) []\n\ndecode-\ud835\udfda : Decode-Term \ud835\udfda\ndecode-\ud835\udfda `0\u2082 = just 0\u2082\ndecode-\ud835\udfda `1\u2082 = just 1\u2082\ndecode-\ud835\udfda _   = nothing\n\npattern `\u2115     = def (quote \u2115) []\npattern `zero  = con (quote zero) []\npattern `suc t = con\u1d5b\u02b3 (quote suc) t\n\ndecode-\u2115 : Decode-Term \u2115\ndecode-\u2115 (lit (nat n)) = just n\n-- these two should not happen anymore:\ndecode-\u2115 `zero         = just 0\ndecode-\u2115 (`suc t)      = map? suc (decode-\u2115 t)\ndecode-\u2115 _             = nothing\n\npattern `Maybe t = def\u1d5b\u02b3 (quote Maybe) t\npattern `nothing = con (quote Maybe.nothing) []\npattern `just  t = con\u1d5b\u02b3 (quote Maybe.just) t\n\ndecode-Maybe : \u2200 {a} {A : \u2605_ a} \u2192 Decode-Term A \u2192 Decode-Term (Maybe A)\ndecode-Maybe decode-A `nothing  = just nothing\ndecode-Maybe decode-A (`just t) = map? just (decode-A t)\ndecode-Maybe decode-A _         = nothing\n\npattern `List t = def\u1d5b\u02b3 (quote List) t\npattern `[] = con (quote List.[]) []\npattern _`\u2237_ t u = con (quote List._\u2237_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n\ndecode-List : \u2200 {a} {A : \u2605_ a} \u2192 Decode-Term A \u2192 Decode-Term (List A)\ndecode-List decode-A `[]      = just []\ndecode-List decode-A (t `\u2237 u) = \u27ea _\u2237_ \u00b7 decode-A t \u00b7 decode-List decode-A u \u27eb?\ndecode-List decode-A _        = nothing\n\npattern `Vec t u = def (quote Vec) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern `v[]      = con (quote Vec.[]) []\npattern _`v\u2237_ t u = con (quote Vec._\u2237_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n\n{-\ndecode-Vec : \u2200 {a} {A : \u2605_ a} {n} \u2192 Decode-Term A \u2192 Decode-Term (Vec A n)\ndecode-Vec decode-A `[]      = just []\ndecode-Vec decode-A (t `\u2237 u) = \u27ea _\u2237_ \u00b7 decode-A t \u00b7 decode-Vec decode-A u \u27eb?\ndecode-Vec decode-A _        = nothing\n-}\n\npattern `\u03a3 t u = def (quote \u03a3) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern _`,_ t u = con (quote _,_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern `fst t = def\u1d5b\u02b3 (quote fst) t\npattern `snd t = def\u1d5b\u02b3 (quote snd) t\n\nmodule _ {a b} {A : \u2605_ a} {B : A \u2192 \u2605_ b}\n         (decode-A : Decode-Term A)\n         (decode-B : (x : A) \u2192 Decode-Term (B x))\n         where\n    decode-\u03a3 : Decode-Term (\u03a3 A B)\n    decode-\u03a3 (t `, u) = decode-A t   >>=? \u03bb x \u2192\n                        decode-B x u >>=? \u03bb y \u2192\n                        just (x , y)\n    decode-\u03a3 _        = nothing\n\ndata Abs : Set where\n  var : \u2115 \u2192 Abs\n  []  : Abs\n  _,_ : Abs \u2192 Abs \u2192 Abs\n  abs : Abs \u2192 Abs\n\n_,,_ : Abs \u2192 Abs \u2192 Abs\n[]      ,, x  = x\nx       ,, [] = x\n(x , y) ,, z  = x ,, (y ,, z)\nx       ,, y  = x , y\n\nabs' : Abs \u2192 Abs\nabs' [] = []\nabs' x  = abs x\n\nabsTerm : Term \u2192 Abs\nabsArgs : Args \u2192 Abs\nabsType : Type \u2192 Abs\nabsSort : Sort \u2192 Abs\n\nabsTerm (var x args) = var x ,, absArgs args\nabsTerm (con c args) = absArgs args\nabsTerm (def f args) = absArgs args\nabsTerm (lam v t) = abs' (absTerm t)\nabsTerm (pat-lam cs args) = opaque \"absTm\/pat-lam\" []\nabsTerm (pi (arg _ t\u2081) t\u2082) = absType t\u2081 ,, abs' (absType t\u2082)\nabsTerm (sort x) = absSort x\nabsTerm (lit x) = []\nabsTerm unknown = []\n\nabsArgs [] = []\nabsArgs (arg i x \u2237 as) = absTerm x ,, absArgs as\nabsType (el _ t) = absTerm t\nabsSort (set t) = absTerm t\nabsSort (lit n) = []\nabsSort unknown = []\n\napp : Term \u2192 Args \u2192 Term\napp (var x args) args\u2081 = var x (args ++ args\u2081)\napp (con c args) args\u2081 = con c (args ++ args\u2081)\napp (def f args) args\u2081 = def f (args ++ args\u2081)\napp (lam v t) args = opaque \"app\/lam\" unknown\napp (pat-lam cs args) args\u2081 = opaque \"app\/pat-lam\" unknown\napp (pi t\u2081 t\u2082) args = opaque \"app\/pi\" unknown\napp (sort x) args = opaque \"app\/sort\" unknown\napp (lit x) args = opaque \"app\/lit\" unknown\napp unknown args = unknown\n\nquoteNat : \u2115 \u2192 Term\nquoteNat zero    = `zero\nquoteNat (suc n) = `suc (quoteNat n)\n\nunlit : Literal \u2192 Term\nunlit (nat x) = quoteNat x\nunlit x = lit x\n\nunknown-fun-def : FunctionDef\nunknown-fun-def = opaque \"unknown-fun-def\" (fun-def (el unknown unknown) [])\n\nunknown-definition : Definition\nunknown-definition = opaque \"unknown-definition\" (function unknown-fun-def)\n\nun-function : Definition \u2192 FunctionDef\nun-function (function x) = x\nun-function _            = unknown-fun-def\n\nmodule Map-arg-info (f : Arg-info \u2192 Arg-info) where\n\n    On : Set \u2192 Set\n    On T = T \u2192 T\n\n    pat : On Pattern\n    pats : On (List (Arg Pattern))\n    pat (con c ps) = con c (pats ps)\n    pat dot = dot\n    pat var = var\n    pat (lit l) = lit l\n    pat (proj p) = proj p\n    pat absurd = absurd\n    pats [] = []\n    pats (arg i p \u2237 ps) = arg (f i) (pat p) \u2237 pats ps\n\n    term : On Term\n    t\u00ffpe : On Type\n    \u00e5rgs : On Args\n    s\u00f8rt : On Sort\n    cl\u00e5use  : On Clause\n    cl\u00e5uses : On (List Clause)\n\n    term (var x as) = var x (\u00e5rgs as)\n    term (con c as) = con c (\u00e5rgs as)\n    term (def f as) = def f (\u00e5rgs as)\n    term (lam v t) = lam (visibility (f (arg-info v (relevant{- arbitrary choice -})))) (term t)\n    term (pat-lam cs as) = pat-lam (cl\u00e5uses cs) (\u00e5rgs as)\n    term (pi (arg i t\u2081) t\u2082) = pi (arg (f i) (t\u00ffpe t\u2081)) (t\u00ffpe t\u2082)\n    term (sort s) = sort (s\u00f8rt s)\n    term (lit l) = lit l\n    term unknown = unknown\n\n    t\u00ffpe (el s t) = el (s\u00f8rt s) (term t)\n\n    \u00e5rgs [] = []\n    \u00e5rgs (arg i t \u2237 as) = arg (f i) (term t) \u2237 \u00e5rgs as\n\n    s\u00f8rt (set t) = set (term t)\n    s\u00f8rt (lit n) = lit n\n    s\u00f8rt unknown = unknown\n\n    cl\u00e5use (clause ps body) = clause (pats ps) (term body)\n    cl\u00e5use (absurd-clause ps) = absurd-clause (pats ps)\n\n    cl\u00e5uses [] = []\n    cl\u00e5uses (x \u2237 cs) = cl\u00e5use x \u2237 cl\u00e5uses cs\n\n    f\u00fcn-def : FunctionDef \u2192 FunctionDef\n    f\u00fcn-def (fun-def t cs) = fun-def (t\u00ffpe t) (cl\u00e5uses cs)\n\n    d\u00ebf : Definition \u2192 Definition\n    d\u00ebf (function x) = function (f\u00fcn-def x)\n    d\u00ebf (data-type x) = opaque \"Map-arg-info.d\u00ebf\/data-type\" unknown-definition\n    d\u00ebf (record\u2032 x) = opaque \"Map-arg-info.d\u00ebf\/record\u2032\" unknown-definition\n    d\u00ebf constructor\u2032 = opaque \"Map-arg-info.d\u00ebf\/constructor\u2032\" unknown-definition\n    d\u00ebf axiom = opaque \"Map-arg-info.d\u00ebf\/axiom\" unknown-definition\n    d\u00ebf primitive\u2032 = opaque \"Map-arg-info.d\u00ebf\/primitive\u2032\" unknown-definition\n\n    n\u00e5me : Name \u2192 Definition\n    n\u00e5me = d\u00ebf \u2218 definition\n\nreveal-arg : Arg-info \u2192 Arg-info\nreveal-arg (arg-info v r) = arg-info visible r\n\nmodule Reveal-args = Map-arg-info reveal-arg\n\nmodule Get-clauses where\n    from-fun-def : FunctionDef \u2192 Clauses\n    from-fun-def (fun-def _ cs) = cs\n    from-def : Definition \u2192 Clauses\n    from-def (function x) = from-fun-def x\n    from-def (data-type x) = opaque \"Get-clauses.from-def\/data-type\" []\n    from-def (record\u2032 x) = opaque \"Get-clauses.from-def\/record\u2032\" []\n    from-def constructor\u2032 = opaque \"Get-clauses.from-def\/constructor\u2032\" []\n    from-def axiom = opaque \"Get-clauses.from-def\/axiom\" []\n    from-def primitive\u2032 = opaque \"Get-clauses.from-def\/primitive\u2032\" []\n    from-name : Name \u2192 Clauses\n    from-name n = from-def (definition n)\n\nmodule Get-type where\n    from-clause : Clause \u2192 Term\n    from-clause (clause [] body) = body\n    from-clause (clause (arg _ var \u2237 pats) body)\n      = lam visible (from-clause (clause pats body))\n    from-clause _ = unknown\n    from-fun-def : FunctionDef \u2192 Type\n    from-fun-def (fun-def t _) = t\n    from-def : Definition \u2192 Type\n    from-def (function x) = from-fun-def x\n    from-def (data-type x) = opaque \"Get-type.from-def\/data-type\" ``\u2605\u2080\n    from-def (record\u2032 x) = opaque \"Get-type.from-def\/record\u2032\" ``\u2605\u2080\n    from-def constructor\u2032 = opaque \"Get-type.from-def\/constructor\" ``\u2605\u2080\n    from-def axiom = opaque \"Get-type.from-def\/axiom\" ``\u2605\u2080\n    from-def primitive\u2032 = opaque \"Get-type.from-def\/primitive\u2032\" ``\u2605\u2080\n    from-name : Name \u2192 Type\n    from-name n = from-def (definition n)\n\nmodule Get-term where\n    from-clause : Clause \u2192 Term\n    from-clause (clause [] body) = body\n    from-clause (clause (arg _ var \u2237 pats) body)\n      = lam visible (from-clause (clause pats body))\n    from-clause _ = unknown\n    from-fun-def : FunctionDef \u2192 Term\n    from-fun-def (fun-def _ (c \u2237 [])) = from-clause c\n    from-fun-def _ = unknown\n    from-def : Definition \u2192 Term\n    from-def (function x) = from-fun-def x\n    from-def (data-type x) = unknown\n    from-def (record\u2032 x) = unknown\n    from-def constructor\u2032 = unknown\n    from-def axiom = unknown\n    from-def primitive\u2032 = unknown\n    from-name : Name \u2192 Term\n    from-name n = from-def (definition n)\n\nmodule Ex where\n  open import Relation.Binary.PropositionalEquality\n  postulate\n    f : \u2115 \u2192 \u2115 \u2192 \u2115\n    g : {x y : \u2115} \u2192 \u2115\n    h : {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  H : \u2605\n  H = {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  postulate\n    h\u2082 : H\n  test\u2081 : unquote (\u03b7\u1d5b\u207f 2 (quote f)) \u2261 f\n  test\u2081 = refl\n  test\u2082 : unquote (\u03b7\u02b0\u207f 2 (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2082 = refl\n  test\u2083 : unquote (\u03b7\u207f (quote f)) \u2261 f\n  test\u2083 = refl\n  test\u2084 : unquote (\u03b7\u207f (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2084 = refl\n  \u03b7h = \u03b7\u207f (quote h)\n  -- this test passes but leave an undecided instance argument\n  -- test\u2085 : unquote \u03b7h \u2261 \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  -- test\u2085 = refl\n  \u03b7h\u2082 : Term\n  \u03b7h\u2082 = \u03b7\u207f (quote h\u2082)\n  {-\n  test\u2086 : unquote \u03b7h\u2082 \u2261 {!unquote \u03b7h\u2082!} -- \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  test\u2086 = refl\n  -}\n  test\u2087 : decode-\u2115 (quoteTerm (\u2115.suc (suc zero))) \u2261 just 2\n  test\u2087 = refl\n  test\u2088 : decode-\u2115 (quoteTerm (\u2115.suc (suc 3))) \u2261 just 5\n  test\u2088 = refl\n  test\u2089 : decode-Maybe decode-\ud835\udfda (quoteTerm (Maybe.just 0\u2082)) \u2261 just (just 0\u2082)\n  test\u2089 = refl\n  test\u2081\u2080 : decode-List decode-\u2115 (quoteTerm (0 \u2237 1 \u2237 2 \u2237 [])) \u2261 just (0 \u2237 1 \u2237 2 \u2237 [])\n  test\u2081\u2080 = refl\n  test\u2081\u2081 : quoteTerm (_,\u2032_ 0\u2082 1\u2082) \u2261 `0\u2082 `, `1\u2082\n  test\u2081\u2081 = refl\n  test\u2081\u2081' : decode-List (decode-\u03a3 {A = \ud835\udfda} {B = [0: \ud835\udfda 1: \u2115 ]} decode-\ud835\udfda [0: decode-\ud835\udfda 1: decode-\u2115 ])\n                        (quoteTerm ((\u03a3._,_ {B = [0: \ud835\udfda 1: \u2115 ]} 0\u2082 1\u2082) \u2237 (1\u2082 , 4) \u2237 [])) \u2261 just ((0\u2082 , 1\u2082) \u2237 (1\u2082 , 4) \u2237 [])\n  test\u2081\u2081' = refl\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Reflection.NP where\n\nopen import Type\nopen import Level.NP\nopen import Data.Nat using (\u2115; zero; suc) renaming (_\u2294_ to _\u2294\u2115_)\nopen import Data.Maybe.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9; 0\u2081)\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; [0:_1:_])\nopen import Data.List\nopen import Data.Vec using (Vec) -- ; []; _\u2237_)\nopen import Data.Product.NP\nopen import Function.NP hiding (\u03a0)\n\nopen import Reflection public\n\nArgs : \u2605\nArgs = List (Arg Term)\n\n-- lam\u1d5b : Term \u2192 Term\npattern lam\u1d5b = lam visible\n\n-- lam\u02b0 : Term \u2192 Term\npattern lam\u02b0 = lam hidden\n\n-- argi\u1d5b\u02b3 : Arg-info\npattern argi\u1d5b\u02b3 = arg-info visible relevant\n\n-- argi\u02b0\u02b3 : Arg-info\npattern argi\u02b0\u02b3 = arg-info hidden relevant\n\n-- arg\u1d5b\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\npattern arg\u1d5b\u02b3 v = arg argi\u1d5b\u02b3 v\n\n-- arg\u02b0\u02b3 : \u2200{A} \u2192 A \u2192 Arg A\npattern arg\u02b0\u02b3 v = arg argi\u02b0\u02b3 v\n\nArg-infos : \u2605\nArg-infos = List Arg-info\n\napp` : (Args \u2192 Term) \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\napp` f = go [] where\n  go : Args \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\n  go args []         = f (reverse args)\n  go args (ai \u2237 ais) = \u03bb t \u2192 go (arg ai t \u2237 args) ais\n\npattern con\u1d5b\u02b3 n t = con n (arg\u1d5b\u02b3 t \u2237 [])\npattern con\u02b0\u02b3 n t = con n (arg\u02b0\u02b3 t \u2237 [])\npattern def\u1d5b\u02b3 n t = def n (arg\u1d5b\u02b3 t \u2237 [])\npattern def\u02b0\u02b3 n t = def n (arg\u02b0\u02b3 t \u2237 [])\n\ncon` : Name \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\ncon` x = app` (con x)\n\ndef` : Name \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\ndef` x = app` (def x)\n\nvar` : \u2115 \u2192 (ais : Arg-infos) \u2192 Term \u2192\u27e8 length ais \u27e9 Term\nvar` x = app` (var x)\n\ncoe : \u2200 {A : \u2605} {z : A} n \u2192 (Term \u2192\u27e8 length (replicate n z) \u27e9 Term) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncoe zero    t = t\ncoe (suc n) f = \u03bb t \u2192 coe n (f t)\n\ncon`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ncon`\u207f\u02b3 x n = coe n (app` (con x) (replicate n argi\u1d5b\u02b3))\n\ndef`\u207f\u02b3 : Name \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\ndef`\u207f\u02b3 x n = coe n (app` (def x) (replicate n argi\u1d5b\u02b3))\n\nvar`\u207f\u02b3 : \u2115 \u2192 (n : \u2115) \u2192 Term \u2192\u27e8 n \u27e9 Term\nvar`\u207f\u02b3 x n = coe n (app` (var x) (replicate n argi\u1d5b\u02b3))\n\n-- sort\u2080 : Sort\npattern sort\u2080 = lit 0\n\n-- sort\u2081 : Sort\npattern sort\u2081 = lit 1\n\n-- `\u2605\u2080 : Term\npattern `\u2605\u2080 = sort sort\u2080\n\n-- el\u2080 : Term \u2192 Type\npattern el\u2080 t = el sort\u2080 t\n\n-- Builds a type variable (of type \u2605\u2080)\n``var\u2080 : \u2115 \u2192 Args \u2192 Type\n``var\u2080 n args = el\u2080 (var n args)\n\n-- ``\u2605\u2080 : Type\npattern ``\u2605\u2080 = el sort\u2081 `\u2605\u2080\n\nunEl : Type \u2192 Term\nunEl (el _ tm) = tm\n\ngetSort : Type \u2192 Sort\ngetSort (el s _) = s\n\nunArg : \u2200 {A} \u2192 Arg A \u2192 A\nunArg (arg _ a) = a\n\n-- `Level : Term\npattern `Level = def (quote Level) []\n\n-- ``Level : Type\npattern ``Level = el\u2080 `Level\n\npattern `\u2080 = def (quote \u2080) []\n\n-- `\u209b_ : Term \u2192 Term\n-- `\u209b_ = def`\u207f\u02b3 (quote L.suc) 1\npattern `\u209b_ v = def (quote \u209b) (arg\u1d5b\u02b3 v \u2237 [])\n\n-- sucSort : Sort \u2192 Sort\npattern sucSort s = set (`\u209b (sort s))\n\npattern `set\u2080 = set `\u2080\npattern `set\u209b_ s = set (`\u209b s)\npattern `set_ s = set (sort s)\n\ndecode-Sort : Sort \u2192 Maybe \u2115\ndecode-Sort `set\u2080 = just zero\ndecode-Sort (`set\u209b_ s) = map? suc (decode-Sort (set s))\ndecode-Sort (`set_ s) = decode-Sort s\ndecode-Sort (set _) = nothing\ndecode-Sort (lit n) = just n\ndecode-Sort unknown = nothing\n\npattern _`\u2294_ s\u2081 s\u2082 = set (def (quote _\u2294_) (arg\u1d5b\u02b3 (sort s\u2081) \u2237 arg\u1d5b\u02b3 (sort s\u2082) \u2237 []))\n\n_`\u2294`_ : Sort \u2192 Sort \u2192 Sort\ns\u2081 `\u2294` s\u2082 with decode-Sort s\u2081 | decode-Sort s\u2082\n...          | just n\u2081        | just n\u2082        = lit (n\u2081 \u2294\u2115 n\u2082)\n...          | _              | _              = s\u2081 `\u2294 s\u2082\n\npattern pi\u1d5b\u02b3 t u = pi (arg\u1d5b\u02b3 t) u\npattern pi\u02b0\u02b3 t u = pi (arg\u02b0\u02b3 t) u\n\n\u03a0 : Arg Type \u2192 Type \u2192 Type\n\u03a0 t u = el (getSort (unArg t) `\u2294` getSort u) (pi t u)\n\n\u03a0\u1d5b\u02b3 : Type \u2192 Type \u2192 Type\n\u03a0\u1d5b\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u1d5b\u02b3 t u)\n\n\u03a0\u02b0\u02b3 : Type \u2192 Type \u2192 Type\n\u03a0\u02b0\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u02b0\u02b3 t u)\n\n-- \u03b7 vs mk: performs no shifting of the result of mk.\n-- Safe values of mk are def and con for instance\n\u03b7 : List Arg-info \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7 ais\u2080 mk = go ais\u2080 where\n  vars : List Arg-info \u2192 Args\n  vars []         = []\n  vars (ai \u2237 ais) = arg ai (var (length ais) []) \u2237 vars ais\n  go : List Arg-info \u2192 Term\n  go []                   = mk (vars ais\u2080)\n  go (arg-info v _ \u2237 ais) = lam v (go ais)\n\n\u03b7\u02b0 : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u02b0 n = \u03b7 (replicate n argi\u02b0\u02b3)\n\n\u03b7\u1d5b : \u2115 \u2192 (Args \u2192 Term) \u2192 Term\n\u03b7\u1d5b n = \u03b7 (replicate n argi\u1d5b\u02b3)\n\n\u03b7\u02b0\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u02b0\u207f n = \u03b7\u02b0 n \u2218 def\n\n\u03b7\u1d5b\u207f : \u2115 \u2192 Name \u2192 Term\n\u03b7\u1d5b\u207f n = \u03b7\u1d5b n \u2218 def\n\narityOfTerm : Term \u2192 List Arg-info\narityOfType : Type \u2192 List Arg-info\n\narityOfType (el _ u) = arityOfTerm u\narityOfTerm (pi (arg ai _) t) = ai \u2237 arityOfType t\n\n-- no more arguments\narityOfTerm (var _ _) = []\n\n-- TODO\narityOfTerm (con c args) = []\narityOfTerm (def f args) = []\narityOfTerm (sort s)     = []\n\n-- fail\narityOfTerm unknown = []\n\n-- absurd cases\narityOfTerm (lam _ _) = []\n\n\u03b7\u207f : Name \u2192 Term\n\u03b7\u207f nm = \u03b7 (arityOfType (type nm)) (def nm)\n\nDecode-Term : \u2200 {a} \u2192 \u2605_ a \u2192 \u2605_ a\nDecode-Term A = Term \u2192 Maybe A\n\npattern `\ud835\udfd8 = def (quote \ud835\udfd8) []\n\npattern `\ud835\udfd9  = def (quote \ud835\udfd9) []\npattern `0\u2081 = con (quote 0\u2081) []\n\ndecode-\ud835\udfd9 : Decode-Term \ud835\udfd9\ndecode-\ud835\udfd9 `0\u2081 = just 0\u2081\ndecode-\ud835\udfd9 _   = nothing\n\npattern `\ud835\udfda  = def (quote \ud835\udfda) []\npattern `0\u2082 = con (quote 0\u2082) []\npattern `1\u2082 = con (quote 1\u2082) []\n\ndecode-\ud835\udfda : Decode-Term \ud835\udfda\ndecode-\ud835\udfda `0\u2082 = just 0\u2082\ndecode-\ud835\udfda `1\u2082 = just 1\u2082\ndecode-\ud835\udfda _   = nothing\n\npattern `\u2115     = def (quote \u2115) []\npattern `zero  = con (quote zero) []\npattern `suc t = con\u1d5b\u02b3 (quote suc) t\n\ndecode-\u2115 : Decode-Term \u2115\ndecode-\u2115 `zero    = just 0\ndecode-\u2115 (`suc t) = map? suc (decode-\u2115 t)\ndecode-\u2115 _        = nothing\n\npattern `Maybe t = def\u1d5b\u02b3 (quote Maybe) t\npattern `nothing = con (quote Maybe.nothing) []\npattern `just  t = con\u1d5b\u02b3 (quote Maybe.just) t\n\ndecode-Maybe : \u2200 {a} {A : \u2605_ a} \u2192 Decode-Term A \u2192 Decode-Term (Maybe A)\ndecode-Maybe decode-A `nothing  = just nothing\ndecode-Maybe decode-A (`just t) = map? just (decode-A t)\ndecode-Maybe decode-A _         = nothing\n\npattern `List t = def\u1d5b\u02b3 (quote List) t\npattern `[] = con (quote List.[]) []\npattern _`\u2237_ t u = con (quote List._\u2237_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n\ndecode-List : \u2200 {a} {A : \u2605_ a} \u2192 Decode-Term A \u2192 Decode-Term (List A)\ndecode-List decode-A `[]      = just []\ndecode-List decode-A (t `\u2237 u) = \u27ea _\u2237_ \u00b7 decode-A t \u00b7 decode-List decode-A u \u27eb?\ndecode-List decode-A _        = nothing\n\npattern `Vec t u = def (quote Vec) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern `v[]      = con (quote Vec.[]) []\npattern _`v\u2237_ t u = con (quote Vec._\u2237_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n\n{-\ndecode-Vec : \u2200 {a} {A : \u2605_ a} {n} \u2192 Decode-Term A \u2192 Decode-Term (Vec A n)\ndecode-Vec decode-A `[]      = just []\ndecode-Vec decode-A (t `\u2237 u) = \u27ea _\u2237_ \u00b7 decode-A t \u00b7 decode-Vec decode-A u \u27eb?\ndecode-Vec decode-A _        = nothing\n-}\n\npattern `\u03a3 t u = def (quote \u03a3) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern _`,_ t u = con (quote _,_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\npattern `fst t = def\u1d5b\u02b3 (quote fst) t\npattern `snd t = def\u1d5b\u02b3 (quote snd) t\n\nmodule _ {a b} {A : \u2605_ a} {B : A \u2192 \u2605_ b}\n         (decode-A : Decode-Term A)\n         (decode-B : (x : A) \u2192 Decode-Term (B x))\n         where\n    decode-\u03a3 : Decode-Term (\u03a3 A B)\n    decode-\u03a3 (t `, u) = decode-A t   >>=? \u03bb x \u2192\n                        decode-B x u >>=? \u03bb y \u2192\n                        just (x , y)\n    decode-\u03a3 _        = nothing\n\nmodule Ex where\n  open import Relation.Binary.PropositionalEquality\n  postulate\n    f : \u2115 \u2192 \u2115 \u2192 \u2115\n    g : {x y : \u2115} \u2192 \u2115\n    h : {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  H : \u2605\n  H = {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 \u2115\n  postulate\n    h\u2082 : H\n  test\u2081 : unquote (\u03b7\u1d5b\u207f 2 (quote f)) \u2261 f\n  test\u2081 = refl\n  test\u2082 : unquote (\u03b7\u02b0\u207f 2 (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2082 = refl\n  test\u2083 : unquote (\u03b7\u207f (quote f)) \u2261 f\n  test\u2083 = refl\n  test\u2084 : unquote (\u03b7\u207f (quote g)) \u2261 \u03bb {x y : \u2115} \u2192 g {x} {y}\n  test\u2084 = refl\n  \u03b7h = \u03b7\u207f (quote h)\n  -- this test passes but leave an undecided instance argument\n  -- test\u2085 : unquote \u03b7h \u2261 \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  -- test\u2085 = refl\n  \u03b7h\u2082 : Term\n  \u03b7h\u2082 = \u03b7\u207f (quote h\u2082)\n  {-\n  test\u2086 : unquote \u03b7h\u2082 \u2261 {!unquote \u03b7h\u2082!} -- \u03bb {x y : \u2115} {{z : \u2115}} (t u : \u2115) {v : \u2115} \u2192 h {x} {y} {{z}} t u {v}\n  test\u2086 = refl\n  -}\n  test\u2087 : decode-\u2115 (quoteTerm (suc (suc zero))) \u2261 just 2\n  test\u2087 = refl\n  test\u2088 : decode-\u2115 (quoteTerm (suc (suc 3))) \u2261 just 5\n  test\u2088 = refl\n  test\u2089 : decode-Maybe decode-\ud835\udfda (quoteTerm (Maybe.just 0\u2082)) \u2261 just (just 0\u2082)\n  test\u2089 = refl\n  test\u2081\u2080 : decode-List decode-\u2115 (quoteTerm (0 \u2237 1 \u2237 2 \u2237 [])) \u2261 just (0 \u2237 1 \u2237 2 \u2237 [])\n  test\u2081\u2080 = refl\n  test\u2081\u2081 : quoteTerm (_,\u2032_ 0\u2082 1\u2082) \u2261 `0\u2082 `, `1\u2082\n  test\u2081\u2081 = refl\n  test\u2081\u2081' : decode-List (decode-\u03a3 {A = \ud835\udfda} {B = [0: \ud835\udfda 1: \u2115 ]} decode-\ud835\udfda [0: decode-\ud835\udfda 1: decode-\u2115 ])\n                        (quoteTerm ((_,_ {B = [0: \ud835\udfda 1: \u2115 ]} 0\u2082 1\u2082) \u2237 (1\u2082 , 4) \u2237 [])) \u2261 just ((0\u2082 , 1\u2082) \u2237 (1\u2082 , 4) \u2237 [])\n  test\u2081\u2081' = refl\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8ebd9178164cdc849d6ab9b42153822565291981","subject":"Added minus-0x (ER version).","message":"Added minus-0x (ER version).\n\nIgnore-this: 5e95c1e87f09e329be44ef0dd3f11352\n\ndarcs-hash:20100430180427-3bd4e-7ed9136ac05b1d498076d4a77bde0c33023f7429.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_file":"LTC\/Function\/Arithmetic\/PropertiesER.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\n  using ( +-leftIdentity ; *-leftZero ; *-comm )\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN          = subst (\u03bb t \u2192 N t) (sym cP\u2081) zN\npred-N (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (cP\u2082 n)) Nn\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN                   = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN (sN {n} Nn)          = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst ((\u03bb t \u2192 N t)) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n------------------------------------------------------------------------------\n\nminus-0x : {n : D} \u2192 N n \u2192 zero - n  \u2261 zero\nminus-0x zN          = minus-x0 zero\nminus-0x (sN {n} Nn) = minus-0S n\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties using equational reasoning\n------------------------------------------------------------------------------\n\nmodule LTC.Function.Arithmetic.PropertiesER where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\n  using ( +-leftIdentity ; *-leftZero ; *-comm ; minus-0x )\n\nopen import MyStdLib.Function\nimport MyStdLib.Relation.Binary.EqReasoning\nopen module APER = MyStdLib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\n------------------------------------------------------------------------------\n-- Closure properties\n\npred-N : {n : D} \u2192 N n \u2192 N (pred n)\npred-N zN          = subst (\u03bb t \u2192 N t) (sym cP\u2081) zN\npred-N (sN {n} Nn) = subst (\u03bb t \u2192 N t) (sym (cP\u2082 n)) Nn\n\nminus-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m - n)\nminus-N {m} Nm zN                   = subst (\u03bb t \u2192 N t) (sym (minus-x0 m)) Nm\nminus-N     zN (sN {n} Nn)          = subst (\u03bb t \u2192 N t) (sym (minus-0S n)) zN\nminus-N     (sN {m} Nm) (sN {n} Nn) = subst (\u03bb t \u2192 N t)\n                                            (sym (minus-SS m n))\n                                            (minus-N Nm Nn)\n\n+-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N zN Nn = subst (\u03bb t \u2192 N t) (sym (+-leftIdentity Nn)) Nn\n+-N {n = n} (sN {m} Nm ) Nn =\n  subst ((\u03bb t \u2192 N t)) (sym (+-Sx m n)) (sN (+-N Nm Nn))\n\n*-N : {m n : D} \u2192 N m \u2192 N n \u2192 N (m * n)\n*-N zN Nn = subst (\u03bb t \u2192 N t) (sym (*-leftZero Nn)) zN\n*-N {n = n} (sN {m} Nm) Nn =\n  subst (\u03bb t \u2192 N t) (sym (*-Sx m n)) (+-N Nn (*-N Nm Nn))\n\n------------------------------------------------------------------------------\n\n[x+y]-[x+z]\u2261y-z : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192\n                  (m + n) - (m + o) \u2261 n - o\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} zN Nn No =\n  begin\n    (zero + n) - (zero + o) \u2261\u27e8 subst (\u03bb t \u2192 (zero + n) - (zero + o) \u2261\n                                            t - (zero + o))\n                                      (+-0x n) refl\n                            \u27e9\n     n - (zero + o)         \u2261\u27e8 subst (\u03bb t \u2192 n - (zero + o) \u2261 n - t)\n                                     (+-0x o)\n                                     refl \u27e9\n    n - o\n  \u220e\n\n[x+y]-[x+z]\u2261y-z {n = n} {o = o} (sN {m} Nm) Nn No =\n  begin\n    (succ m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ m + n - (succ m + o) \u2261\n                                                t - (succ m + o))\n                                         (+-Sx m n)\n                                         refl\n                                \u27e9\n    succ (m + n) - (succ m + o) \u2261\u27e8 subst (\u03bb t \u2192 succ (m + n) - (succ m + o) \u2261\n                                                succ (m + n) - t)\n                                         (+-Sx m o)\n                                         refl \u27e9\n    succ (m + n) - succ (m + o) \u2261\u27e8 minus-SS (m + n) (m + o) \u27e9\n    (m + n) - (m + o) \u2261\u27e8 [x+y]-[x+z]\u2261y-z Nm Nn No \u27e9\n    n - o\n  \u220e\n\n[x-y]z\u2261xz*yz : {m n o : D} \u2192 N m \u2192 N n \u2192 N o \u2192 (m - n) * o \u2261 m * o - n * o\n[x-y]z\u2261xz*yz {m} .{zero} {o} Nm zN No =\n  begin\n    (m - zero) * o   \u2261\u27e8 subst (\u03bb t \u2192 (m - zero) * o \u2261 t * o)\n                              (minus-x0 m)\n                              refl\n                      \u27e9\n    m * o            \u2261\u27e8 sym (minus-x0 (m * o)) \u27e9\n    m * o - zero     \u2261\u27e8 subst (\u03bb t \u2192 m * o - zero \u2261 m * o - t)\n                              (sym (*-0x o))\n                              refl\n                     \u27e9\n    m * o - zero * o\n  \u220e\n\n[x-y]z\u2261xz*yz {o = o} zN (sN {n} Nn) No =\n  begin\n    (zero - succ n) * o  \u2261\u27e8 subst (\u03bb t \u2192 (zero - succ n) * o \u2261 t * o)\n                                  (minus-0S n)\n                                  refl\n                         \u27e9\n    zero * o             \u2261\u27e8 *-0x o \u27e9\n    zero                 \u2261\u27e8 sym (minus-0x (*-N (sN Nn) No)) \u27e9\n    zero - succ n * o    \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * o \u2261 t - succ n * o)\n                                   (sym (*-0x o))\n                                   refl\n                         \u27e9\n    zero * o - succ n * o\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) zN =\n  begin\n    (succ m - succ n) * zero      \u2261\u27e8 *-comm (minus-N (sN Nm) (sN Nn)) zN \u27e9\n    zero * (succ m - succ n)      \u2261\u27e8 *-0x (succ m - succ n) \u27e9\n    zero                          \u2261\u27e8 sym (minus-0x (*-N (sN Nn) zN)) \u27e9\n    zero - succ n * zero          \u2261\u27e8 subst (\u03bb t \u2192 zero - succ n * zero \u2261\n                                                  t - succ n * zero)\n                                           (sym (*-0x (succ m)))\n                                           refl\n                                  \u27e9\n    zero * succ m - succ n * zero \u2261\u27e8 subst\n                                       (\u03bb t \u2192 zero * succ m - succ n * zero \u2261\n                                              t - succ n * zero)\n                                       (*-comm zN (sN Nm))\n                                       refl\n                                  \u27e9\n    succ m * zero - succ n * zero\n  \u220e\n\n[x-y]z\u2261xz*yz (sN {m} Nm) (sN {n} Nn) (sN {o} No) =\n  begin\n    (succ m - succ n) * succ o  \u2261\u27e8 subst (\u03bb t \u2192 (succ m - succ n) * succ o \u2261\n                                                t * succ o)\n                                         (minus-SS m n)\n                                         refl\n                                \u27e9\n    (m - n) * succ o            \u2261\u27e8 [x-y]z\u2261xz*yz Nm Nn (sN No) \u27e9\n    m * succ o - n * succ o     \u2261\u27e8 sym ([x+y]-[x+z]\u2261y-z (sN No)\n                                                        (*-N Nm (sN No))\n                                                        (*-N Nn (sN No)))\n                                \u27e9\n    (succ o + m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ o + m * succ o) - (succ o + n * succ o) \u2261\n                      t - (succ o + n * succ o))\n               (sym (*-Sx m (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ o + n * succ o)\n      \u2261\u27e8 subst (\u03bb t \u2192 (succ m * succ o) - (succ o + n * succ o) \u2261\n                      (succ m * succ o) - t)\n               (sym (*-Sx n (succ o)))\n               refl\n      \u27e9\n    (succ m * succ o) - (succ n * succ o)\n  \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"0b10c159c05b326958e33f1f6aeb21ba6d7e5637","subject":"Only indentation.","message":"Only indentation.\n\nIgnore-this: 78623b802d70c19e5a8e22e87d8d0439\n\ndarcs-hash:20100505231828-3bd4e-f52abd1371ed97700ba20e864a1ee177d0708b75.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_file":"LTC\/Relation\/Inequalities\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate\n      prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sx\u22640 = \u22a5-elim prf\n  where\n    postulate\n      -- The proof uses the axiom true\u2260false\n      prf : \u22a5\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n  postulate prf : \u00ac (LT m m) \u2192 \u22a5\n  {-# ATP prove prf #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n  postulate prf : lt (m + n) m \u2261 false \u2192\n                  lt (succ m + n) (succ m) \u2261 false\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n  postulate prf : (succ m - zero) + zero \u2261 succ m\n  {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n  postulate m>n : GT m n\n  {-# ATP prove m>n #-}\n\n  postulate prf : (m - n) + n \u2261 m \u2192\n                  (succ m - succ n) + succ n \u2261 succ m\n  {-# ATP prove prf +-comm minus-N sN #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Inequalities.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.N\nopen import LTC.Function.Arithmetic\nopen import LTC.Function.Arithmetic.Properties\nopen import LTC.Relation.Inequalities\n\nopen import MyStdLib.Data.Sum\nopen import MyStdLib.Function\n\n------------------------------------------------------------------------------\n\n-- TODO: Why the ATP couldn't prove it?\n-- postulate\n--   x\u22650 : (n : D) \u2192 N n \u2192 GE n zero\n-- {-# ATP prove x\u22650 zN sN #-}\n\nx\u22650 : {n : D} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = lt-00\nx\u22650 (sN {n} Nn) = lt-S0 n\n-- {-# ATP hint x\u22650 #-}\n\n\u00ac0>x : {n : D} \u2192 N n \u2192 \u00ac (GT zero n)\n\u00ac0>x Nn 0>n = \u22a5-elim prf\n  where\n    postulate\n      prf : \u22a5\n    {-# ATP prove prf x\u22650 #-}\n\n\u00acS\u22640 : {d : D} \u2192 \u00ac (LE (succ d) zero)\n\u00acS\u22640 Sx\u22640 = \u22a5-elim prf\n  where\n    postulate\n      -- The proof uses the axiom true\u2260false\n      prf : \u22a5\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u2264y : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ lt-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) = prf $ x>y\u2228x\u2264y Nm Nn\n  where\n    postulate\n      prf : (GT m n) \u2228 (LE m n) \u2192\n            GT (succ m) (succ n) \u2228 LE (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n\u00acx<x : {m : D} \u2192 N m \u2192 \u00ac (LT m m)\n\u00acx<x zN _ = \u22a5-elim prf\n  where\n  postulate prf : \u22a5\n  {-# ATP prove prf #-}\n\u00acx<x (sN {m} Nm) _ = \u22a5-elim (prf (\u00acx<x Nm))\n  where\n  postulate prf : \u00ac (LT m m) \u2192 \u22a5\n  {-# ATP prove prf #-}\n\nx\u2264x+y : {m n : D} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf (x\u2264x+y Nm Nn)\n  where\n  postulate prf : lt (m + n) m \u2261 false \u2192\n                  lt (succ m + n) (succ m) \u2261 false\n\nx>y\u2192x-y+y\u2261x : {m n : D} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m - n) + n \u2261 m\nx>y\u2192x-y+y\u2261x zN Nn 0>n = \u22a5-elim (\u00ac0>x Nn 0>n)\nx>y\u2192x-y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n  postulate prf : (succ m - zero) + zero \u2261 succ m\n  {-# ATP prove prf +-rightIdentity minus-N #-}\n\nx>y\u2192x-y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  prf (x>y\u2192x-y+y\u2261x Nm Nn m>n )\n  where\n  postulate m>n : GT m n\n  {-# ATP prove m>n #-}\n\n  postulate prf : (m - n) + n \u2261 m \u2192\n                  (succ m - succ n) + succ n \u2261 succ m\n  {-# ATP prove prf +-comm minus-N sN #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"aca2e0c6e7ec210618f1274cfe5b86bd37305585","subject":"Compile fix","message":"Compile fix\n\nNot sure how I missed this.\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Specification.agda","new_file":"Parametric\/Change\/Specification.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base {{validity-structure}}\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n  open import Base.Change.Equivalence\n  open import Base.Change.Equivalence.Realizers\n\n  module _ {\u03c3 \u03c4 : Type} {\u0393 : Context} (t : Term (\u03c3 \u2022 \u0393) \u03c4) where\n    instance\n      c\u0393 : ChangeAlgebra \u27e6 \u0393 \u27e7Context\n      c\u0393 = change-algebra\u208d \u0393 \u208e\n      c\u03c3 : ChangeAlgebra \u27e6 \u03c3 \u27e7Type\n      c\u03c3 = change-algebra \u03c3\n      c\u03c4 : ChangeAlgebra \u27e6 \u03c4 \u27e7Type\n      c\u03c4 = change-algebra \u03c4\n\n    private\n      \u27e6\u27e7\u0394\u03bb = nil {{changeAlgebraFun {{c\u0393}} {{change-algebra\u208d \u03c3 \u21d2 \u03c4 \u208e}}}} \u27e6 abs t \u27e7\n\n    \u27e6\u27e7\u0394\u03bb-realizer : \u2200\n      (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1)\n      (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v) \u2192\n      \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 (v \u2022 \u03c1))\n    \u27e6\u27e7\u0394\u03bb-realizer \u03c1 d\u03c1 v dv = \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n\n    \u27e6\u27e7\u0394\u03bb-realizer-correct :\n      equiv-hp-binary {{c\u0393}} {{c\u03c3}} {{c\u03c4}} (\u27e6 abs t \u27e7) \u27e6\u27e7\u0394\u03bb \u27e6\u27e7\u0394\u03bb-realizer\n    \u27e6\u27e7\u0394\u03bb-realizer-correct \u03c1 d\u03c1 v dv =\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1 \u229e\u208d \u0393 \u208e d\u03c1) \u229f\u208d \u03c4 \u208e \u27e6 t \u27e7 (v \u2022 \u03c1)\n      \u2259\u27e8 equiv-cancel-2 {{change-algebra \u03c4}}\n           (\u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1 \u229e\u208d \u0393 \u208e d\u03c1))\n           (\u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1))\n           (sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1)))\n      \u27e9\n         \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e where open \u2259-Reasoning\n\n    \u27e6_\u27e7\u0394\u03bb\u2032-faster = proj\u2081 (equiv-raw-change-to-change-binary {{c\u0393}} {{c\u03c3}} {{c\u03c4}} \u27e6 abs t \u27e7 \u27e6\u27e7\u0394\u03bb \u27e6\u27e7\u0394\u03bb-realizer \u27e6\u27e7\u0394\u03bb-realizer-correct)\n      where\n        open import Data.Product\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e (\u27e6 c \u27e7Const)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = apply \u27e6 t \u27e7\u0394\u03bb\u2032-faster \u03c1 d\u03c1\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = equiv-raw-deriv-to-deriv-binary {{change-algebra\u208d \u0393 \u208e}} {{change-algebra \u03c3}} \u27e6 abs t \u27e7Term (\u27e6\u27e7\u0394\u03bb-realizer t) (\u27e6\u27e7\u0394\u03bb-realizer-correct t) \u03c1 d\u03c1\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Change evaluation (Def 3.6 and Fig. 4h).\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\n\nmodule Parametric.Change.Specification\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    {{validity-structure : Validity.Structure \u27e6_\u27e7Base}}\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation\nopen import Relation.Binary.PropositionalEquality\nopen import Theorem.CongApp\nopen import Postulate.Extensionality\n\nmodule Structure where\n  ---------------\n  -- Interface --\n  ---------------\n\n  -- We provide: Derivatives of terms.\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 \u03c1)\n\n  -- And we provide correctness proofs about the derivatives.\n  correctness : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 t \u27e7 \u27e6 t \u27e7\u0394\n\n  --------------------\n  -- Implementation --\n  --------------------\n  open import Base.Change.Equivalence\n  open import Base.Change.Equivalence.Realizers\n\n  module _ {\u03c3 \u03c4 : Type} {\u0393 : Context} (t : Term (\u03c3 \u2022 \u0393) \u03c4) where\n    instance\n      c\u0393 : ChangeAlgebra \u27e6 \u0393 \u27e7Context\n      c\u0393 = change-algebra\u208d \u0393 \u208e\n      c\u03c3 : ChangeAlgebra \u27e6 \u03c3 \u27e7Type\n      c\u03c3 = change-algebra \u03c3\n      c\u03c4 : ChangeAlgebra \u27e6 \u03c4 \u27e7Type\n      c\u03c4 = change-algebra \u03c4\n\n    private\n      \u27e6\u27e7\u0394\u03bb = nil {{changeAlgebraFun {{c\u0393}} {{change-algebra\u208d \u03c3 \u21d2 \u03c4 \u208e}}}} \u27e6 abs t \u27e7\n\n    \u27e6\u27e7\u0394\u03bb-realizer : \u2200\n      (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1)\n      (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v) \u2192\n      \u0394\u208d \u03c4 \u208e (\u27e6 t \u27e7 (v \u2022 \u03c1))\n    \u27e6\u27e7\u0394\u03bb-realizer \u03c1 d\u03c1 v dv = \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n\n    \u27e6\u27e7\u0394\u03bb-realizer-correct :\n      equiv-hp-binary {{c\u0393}} {{c\u03c3}} {{c\u03c4}} (\u27e6 abs t \u27e7) \u27e6\u27e7\u0394\u03bb \u27e6\u27e7\u0394\u03bb-realizer\n    \u27e6\u27e7\u0394\u03bb-realizer-correct \u03c1 d\u03c1 v dv =\n      begin\n        \u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1 \u229e\u208d \u0393 \u208e d\u03c1) \u229f\u208d \u03c4 \u208e \u27e6 t \u27e7 (v \u2022 \u03c1)\n      \u2259\u27e8 equiv-cancel-2 {{change-algebra \u03c4}}\n           (\u27e6 t \u27e7 (v \u229e\u208d \u03c3 \u208e dv \u2022 \u03c1 \u229e\u208d \u0393 \u208e d\u03c1))\n           (\u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1))\n           (sym (correctness t (v \u2022 \u03c1) (dv \u2022 d\u03c1)))\n      \u27e9\n         \u27e6 t \u27e7\u0394 (v \u2022 \u03c1) (dv \u2022 d\u03c1)\n      \u220e where open \u2259-Reasoning\n\n    \u27e6_\u27e7\u0394\u03bb\u2032-faster = proj\u2081 (equiv-raw-change-to-change-binary {{c\u0393}} {{c\u03c3}} {{c\u03c4}} \u27e6 abs t \u27e7 \u27e6\u27e7\u0394\u03bb \u27e6\u27e7\u0394\u03bb-realizer \u27e6\u27e7\u0394\u03bb-realizer-correct)\n      where\n        open import Data.Product\n\n  \u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7) \u2192 \u0394\u208d \u0393 \u208e \u03c1 \u2192 \u0394\u208d \u03c4 \u208e (\u27e6 x \u27e7Var \u03c1)\n  \u27e6 this   \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = dv\n  \u27e6 that x \u27e7\u0394Var (v \u2022 \u03c1) (dv \u2022 d\u03c1) = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n\n  \u27e6_\u27e7\u0394 (const {\u03c4} c) \u03c1 d\u03c1 = nil\u208d \u03c4 \u208e (\u27e6 c \u27e7Const)\n  \u27e6_\u27e7\u0394 (var x) \u03c1 d\u03c1 = \u27e6 x \u27e7\u0394Var \u03c1 d\u03c1\n  \u27e6_\u27e7\u0394 (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n       call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n  \u27e6_\u27e7\u0394 (abs {\u03c3} {\u03c4} t) \u03c1 d\u03c1 = apply \u27e6 t \u27e7\u0394\u03bb\u2032-faster \u03c1 d\u03c1\n\n  correctVar : \u2200 {\u03c4 \u0393} (x : Var \u0393 \u03c4) \u2192\n    IsDerivative\u208d \u0393 , \u03c4 \u208e \u27e6 x \u27e7 \u27e6 x \u27e7\u0394Var\n  correctVar (this) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = refl\n  correctVar (that y) (v \u2022 \u03c1) (dv \u2022 d\u03c1) = correctVar y \u03c1 d\u03c1\n\n  correctness (const {\u03c4} c) \u03c1 d\u03c1 = update-nil\u208d \u03c4 \u208e \u27e6 c \u27e7Const\n  correctness {\u03c4} (var x) \u03c1 d\u03c1 = correctVar {\u03c4} x \u03c1 d\u03c1\n  correctness (app {\u03c3} {\u03c4} s t) \u03c1 d\u03c1 =\n    let\n      f\u2081 = \u27e6 s \u27e7 \u03c1\n      f\u2082 = \u27e6 s \u27e7 (after-env d\u03c1)\n      \u0394f = \u27e6 s \u27e7\u0394 \u03c1 d\u03c1\n      u\u2081 = \u27e6 t \u27e7 \u03c1\n      u\u2082 = \u27e6 t \u27e7 (after-env d\u03c1)\n      \u0394u = \u27e6 t \u27e7\u0394 \u03c1 d\u03c1\n    in\n      begin\n        f\u2081 u\u2081 \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u\n      \u2261\u27e8  sym (is-valid {\u03c3} {\u03c4} \u0394f u\u2081 \u0394u) \u27e9\n        (f\u2081 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (u\u2081 \u229e\u208d \u03c3 \u208e \u0394u)\n      \u2261\u27e8 correctness {\u03c3 \u21d2 \u03c4} s \u03c1 d\u03c1 \u27e8$\u27e9 correctness {\u03c3} t \u03c1 d\u03c1 \u27e9\n        f\u2082 u\u2082\n      \u220e where open \u2261-Reasoning\n  correctness {\u03c3 \u21d2 \u03c4} {\u0393} (abs t) \u03c1 d\u03c1 = equiv-raw-deriv-to-deriv-binary {{change-algebra\u208d \u0393 \u208e}} {{change-algebra \u03c3}} \u27e6 abs t \u27e7Term (\u27e6\u27e7\u0394\u03bb-realizer t) (\u27e6\u27e7\u0394\u03bb-realizer-correct t) \u03c1 d\u03c1\n\n  -- Corollary: (f \u229e df) (v \u229e dv) = f v \u229e df v dv\n\n  corollary : \u2200 {\u03c3 \u03c4 \u0393}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) (t : Term \u0393 \u03c3) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192\n      (\u27e6 s \u27e7 \u03c1 \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u27e6 s \u27e7\u0394 \u03c1 d\u03c1)\n        (\u27e6 t \u27e7 \u03c1 \u229e\u208d \u03c3 \u208e \u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n    \u2261 (\u27e6 s \u27e7 \u03c1) (\u27e6 t \u27e7 \u03c1)\n      \u229e\u208d \u03c4 \u208e\n      (call-change {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1)) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary {\u03c3} {\u03c4} s t \u03c1 d\u03c1 =\n    is-valid {\u03c3} {\u03c4} (\u27e6 s \u27e7\u0394 \u03c1 d\u03c1) (\u27e6 t \u27e7 \u03c1) (\u27e6 t \u27e7\u0394 \u03c1 d\u03c1)\n\n  corollary-closed : \u2200 {\u03c3 \u03c4} (t : Term \u2205 (\u03c3 \u21d2 \u03c4))\n    (v : \u27e6 \u03c3 \u27e7) (dv : \u0394\u208d \u03c3 \u208e v)\n    \u2192 \u27e6 t \u27e7 \u2205 (after\u208d \u03c3 \u208e dv)\n    \u2261 \u27e6 t \u27e7 \u2205 v \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv\n\n  corollary-closed {\u03c3} {\u03c4} t v dv =\n    let\n      f  = \u27e6 t \u27e7 \u2205\n      \u0394f = \u27e6 t \u27e7\u0394 \u2205 \u2205\n    in\n      begin\n        f (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole (after\u208d \u03c3 \u208e dv))\n              (sym (correctness {\u03c3 \u21d2 \u03c4} t \u2205 \u2205)) \u27e9\n         (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after\u208d \u03c3 \u208e dv)\n      \u2261\u27e8 is-valid {\u03c3} {\u03c4} (\u27e6 t \u27e7\u0394 \u2205 \u2205) v dv \u27e9\n        f (before\u208d \u03c3 \u208e dv) \u229e\u208d \u03c4 \u208e call-change {\u03c3} {\u03c4} \u0394f v dv\n      \u220e where open \u2261-Reasoning\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ca22d87fd854053fa2453c37e2f745872289b5a0","subject":"Removed unnecessary definitions.","message":"Removed unnecessary definitions.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, the\n-- bisimilarity relation _\u2248_ on unbounded lists is also a pre-fixed\n-- point of the bisimulation functional (see\n-- FOTC.Relation.Binary.Bisimulation).\n\u2248-pre-fixed : \u2200 {xs ys} \u2192\n              (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n                xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 xs' \u2248 ys') \u2192\n              xs \u2248 ys\n\u2248-pre-fixed {xs} {ys} h = \u2248-coind (\u03bb zs _ \u2192 zs \u2261 zs) h' refl\n  where\n  h' : xs \u2261 xs \u2192\n       \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 xs' \u2261 xs'\n  h' _ with h\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , _ = x' , xs' , ys' , prf\u2081 , prf\u2082 , refl\n\n\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\n\u2248-refl {xs} Sxs = \u2248-coind (\u03bb ys _ \u2192 ys \u2261 ys) h refl\n  where\n  h : xs \u2261 xs \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 xs \u2261 x' \u2237 ys' \u2227 xs' \u2261 xs'\n  h _ with Stream-unf Sxs\n  ... | x' , xs' , prf , _ = x' , xs' , xs' , prf , prf , refl\n\n\u2248-sym : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 ys \u2248 xs\n\u2248-sym {xs} {ys} xs\u2248ys = \u2248-coind (\u03bb zs _ \u2192 zs \u2261 zs) h refl\n  where\n  h : ys \u2261 ys \u2192\n      \u2203[ y' ] \u2203[ ys' ] \u2203[ xs' ] ys \u2261 y' \u2237 ys' \u2227 xs \u2261 y' \u2237 xs' \u2227 ys' \u2261 ys'\n  h _ with \u2248-unf xs\u2248ys\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , _ = x' , ys' , xs' , prf\u2082 , prf\u2081 , refl\n\n\n\u2248-trans : \u2200 {xs ys zs} \u2192 xs \u2248 ys \u2192 ys \u2248 zs \u2192 xs \u2248 zs\n\u2248-trans {xs} {ys} {zs} xs\u2248ys ys\u2248zs = \u2248-coind (\u03bb ws _ \u2192 ws \u2261 ws) h refl\n  where\n  h : xs \u2261 xs \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ zs' ] xs \u2261 x' \u2237 xs' \u2227 zs \u2261 x' \u2237 zs' \u2227 xs' \u2261 xs'\n  h _ with \u2248-unf xs\u2248ys\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , _ with \u2248-unf ys\u2248zs\n  ... | y' , ys'' , zs' , prf\u2083 , prf\u2084 , _ =\n    x'\n    , xs'\n    , zs'\n    , prf\u2081\n    , subst (\u03bb t \u2192 zs \u2261 t \u2237 zs') y'\u2261x' prf\u2084\n    , refl\n    where\n    y'\u2261x' : y' \u2261 x'\n    y'\u2261x' = \u2227-proj\u2081 (\u2237-injective (trans (sym prf\u2083) prf\u2082))\n\n\u2237-injective\u2248 : \u2200 {x xs ys} \u2192 x \u2237 xs \u2248 x \u2237 ys \u2192 xs \u2248 ys\n\u2237-injective\u2248 {x} {xs} {ys} h with \u2248-unf h\n... | x' , xs' , ys' , prf\u2081 , prf\u2082 , prf\u2083 = xs\u2248ys\n  where\n  xs\u2261xs' : xs \u2261 xs'\n  xs\u2261xs' = \u2227-proj\u2082 (\u2237-injective prf\u2081)\n\n  ys\u2261ys' : ys \u2261 ys'\n  ys\u2261ys' = \u2227-proj\u2082 (\u2237-injective prf\u2082)\n\n  xs\u2248ys : xs \u2248 ys\n  xs\u2248ys = subst (\u03bb t \u2192 t \u2248 ys)\n                (sym xs\u2261xs')\n                (subst (_\u2248_ xs') (sym ys\u2261ys') prf\u2083)\n\n\u2237-rightCong\u2248 : \u2200 {x xs ys} \u2192 xs \u2248 ys \u2192 x \u2237 xs \u2248 x \u2237 ys\n\u2237-rightCong\u2248 {x} {xs} {ys} h = \u2248-pre-fixed (x , xs , ys , refl , refl , h)\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for the bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Relation.Binary.Bisimilarity.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Because a greatest post-fixed point is a fixed-point, the\n-- bisimilarity relation _\u2248_ on unbounded lists is also a pre-fixed\n-- point of the bisimulation functional (see\n-- FOTC.Relation.Binary.Bisimulation).\n\u2248-pre-fixed : \u2200 {xs ys} \u2192\n              (\u2203[ x' ]  \u2203[ xs' ] \u2203[ ys' ]\n                xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 xs' \u2248 ys') \u2192\n              xs \u2248 ys\n\u2248-pre-fixed {xs} {ys} h = \u2248-coind (\u03bb zs _ \u2192 zs \u2261 zs) h' refl\n  where\n  h' : xs \u2261 xs \u2192\n       \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys' \u2227 xs' \u2261 xs'\n  h' _ with h\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , _ = x' , xs' , ys' , prf\u2081 , prf\u2082 , refl\n\n\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\n\u2248-refl {xs} Sxs = \u2248-coind (\u03bb ys _ \u2192 ys \u2261 ys) h refl\n  where\n  h : xs \u2261 xs \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 xs \u2261 x' \u2237 ys' \u2227 xs' \u2261 xs'\n  h _ with Stream-unf Sxs\n  ... | x' , xs' , prf , _ = x' , xs' , xs' , prf , prf , refl\n\n\u2248-sym : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 ys \u2248 xs\n\u2248-sym {xs} {ys} xs\u2248ys = \u2248-coind (\u03bb zs _ \u2192 zs \u2261 zs) h refl\n  where\n  B : D \u2192 D \u2192 Set\n  B xs ys = xs \u2261 xs\n\n  h : ys \u2261 ys \u2192\n      \u2203[ y' ] \u2203[ ys' ] \u2203[ xs' ] ys \u2261 y' \u2237 ys' \u2227 xs \u2261 y' \u2237 xs' \u2227 ys' \u2261 ys'\n  h _ with \u2248-unf xs\u2248ys\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , _ = x' , ys' , xs' , prf\u2082 , prf\u2081 , refl\n\n\n\u2248-trans : \u2200 {xs ys zs} \u2192 xs \u2248 ys \u2192 ys \u2248 zs \u2192 xs \u2248 zs\n\u2248-trans {xs} {ys} {zs} xs\u2248ys ys\u2248zs = \u2248-coind (\u03bb ws _ \u2192 ws \u2261 ws) h refl\n  where\n  B : D \u2192 D \u2192 Set\n  B xs zs = xs \u2261 xs\n\n  h : xs \u2261 xs \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ zs' ] xs \u2261 x' \u2237 xs' \u2227 zs \u2261 x' \u2237 zs' \u2227 xs' \u2261 xs'\n  h _ with \u2248-unf xs\u2248ys\n  ... | x' , xs' , ys' , prf\u2081 , prf\u2082 , _ with \u2248-unf ys\u2248zs\n  ... | y' , ys'' , zs' , prf\u2083 , prf\u2084 , _ =\n    x'\n    , xs'\n    , zs'\n    , prf\u2081\n    , subst (\u03bb t \u2192 zs \u2261 t \u2237 zs') y'\u2261x' prf\u2084\n    , refl\n\n    where\n    y'\u2261x' : y' \u2261 x'\n    y'\u2261x' = \u2227-proj\u2081 (\u2237-injective (trans (sym prf\u2083) prf\u2082))\n\n\u2237-injective\u2248 : \u2200 {x xs ys} \u2192 x \u2237 xs \u2248 x \u2237 ys \u2192 xs \u2248 ys\n\u2237-injective\u2248 {x} {xs} {ys} h with \u2248-unf h\n... | x' , xs' , ys' , prf\u2081 , prf\u2082 , prf\u2083 = xs\u2248ys\n  where\n  xs\u2261xs' : xs \u2261 xs'\n  xs\u2261xs' = \u2227-proj\u2082 (\u2237-injective prf\u2081)\n\n  ys\u2261ys' : ys \u2261 ys'\n  ys\u2261ys' = \u2227-proj\u2082 (\u2237-injective prf\u2082)\n\n  xs\u2248ys : xs \u2248 ys\n  xs\u2248ys = subst (\u03bb t \u2192 t \u2248 ys)\n                (sym xs\u2261xs')\n                (subst (_\u2248_ xs') (sym ys\u2261ys') prf\u2083)\n\n\u2237-rightCong\u2248 : \u2200 {x xs ys} \u2192 xs \u2248 ys \u2192 x \u2237 xs \u2248 x \u2237 ys\n\u2237-rightCong\u2248 {x} {xs} {ys} h = \u2248-pre-fixed (x , xs , ys , refl , refl , h)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7279d087b0be289a27ba52ea14a82f3c9882848c","subject":"Revert \"Added thesis' example.\"","message":"Revert \"Added thesis' example.\"\n\nThis reverts commit 1000130700500fc9f0c74f3feded244488fba784.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/report\/CombiningProofs\/ConatPropertiesATP.agda","new_file":"notes\/thesis\/report\/CombiningProofs\/ConatPropertiesATP.agda","new_contents":"","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule CombiningProofs.ConatPropertiesATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\n\u221e-Conat : Conat \u221e\n\u221e-Conat = Conat-coind (\u03bb n \u2192 n \u2261 \u221e) h refl\n  where\n  postulate h : \u2200 {n} \u2192 n \u2261 \u221e \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 n' \u2261 \u221e)\n  {-# ATP prove h #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"df3e9873d4d0c9dac1df76aa623fc13494cecc2d","subject":"NomPa: Disable broken import","message":"NomPa: Disable broken import\n","repos":"np\/NomPa","old_file":"lib\/NomPa\/All.agda","new_file":"lib\/NomPa\/All.agda","new_contents":"module NomPa.All where\nimport NomPa.Record\nimport NomPa.Derived\nimport NomPa.Derived.NaPa\nimport NomPa.Traverse\nimport NomPa.Implem\nimport NomPa.Examples.LC.DataTypes\nimport NomPa.Examples.LC.Contexts\nimport NomPa.Examples.LC.Equiv\n-- import NomPa.Examples.LC.Monadic\n-- import NomPa.Examples.LC.Monadic.Tests\nimport NomPa.Examples.LC.DbLevels\nimport NomPa.Examples.NaPa.LC\nimport NomPa.Examples.LC.Combined\nimport NomPa.Examples.LC\nimport NomPa.Examples.Records\n-- import NomPa.Examples.Records.TypedVec\n-- import NomPa.Examples.Records.Typed\nimport NomPa.Laws\nimport NomPa.Link\nimport NomPa.NomT\nimport NomPa.NomT.Maybe\n\n-- Normalization By Evaluation\nimport NomPa.Examples.NBE\nimport NomPa.Examples.NBE.Can\nimport NomPa.Examples.NBE.Env\nimport NomPa.Examples.NBE.Neu\nimport NomPa.Examples.NBE.Tests\n\n-- bare, raw, non-fancy\u2026\nimport NomPa.Examples.BareDB\nimport NomPa.Examples.Raw\nimport NomPa.Examples.Raw.Parser\nimport NomPa.Examples.Raw.Printer\nimport NomPa.Examples.LC.WellScopedJudgment\n\n-- Encodings: Emebedding of nominal types\nimport NomPa.Encodings.NominalTypes\nimport NomPa.Encodings.NominalTypes.Generic\nimport NomPa.Encodings.NominalTypes.Generic.Subst\nimport NomPa.Encodings.NominalTypes.Generic.Combined\nimport NomPa.Encodings.NominalTypes.MultiSorts\nimport NomPa.Encodings.NominalTypes.MultiSorts.Test\nimport NomPa.Encodings.AlphaCaml\nimport NomPa.Encodings.AlphaCaml.Test\nimport NomPa.Encodings.BindersUnbound\nimport NomPa.Encodings.BindersUnbound.Generic\n\n-- Algorithms\nimport NomPa.Examples.LC.CC\nimport NomPa.Examples.LC.Monadic\nimport NomPa.Examples.LC.Monadic.Tests\n\n-- hybrid\nimport NomPa.Examples.LN\nimport NomPa.Examples.LL\nimport NomPa.Examples.LocallyNamed\n\n-- More examples\nimport NomPa.Examples.LC.Maybe\n--import NomPa.Examples.LC.SymanticsTy\nimport NomPa.Examples.LC.Tests\nimport NomPa.Examples.LocallyClosed\nimport NomPa.Examples.PTS\nimport NomPa.Examples.PTS.F\nimport NomPa.Examples.Reg\n\n-- Tests\nimport NomPa.Examples.Tests\n\n-- Logical relations\nimport NomPa.Record.LogicalRelation\nimport NomPa.Implem.LogicalRelation\nimport NomPa.Examples.LC.DataTypes.Logical\n\n-- Universal types\nimport NomPa.Universal.RegularTree\nimport NomPa.Universal.Sexp\n\nimport NomPa.Derived.WorldInclusion\n","old_contents":"module NomPa.All where\nimport NomPa.Record\nimport NomPa.Derived\nimport NomPa.Derived.NaPa\nimport NomPa.Traverse\nimport NomPa.Implem\nimport NomPa.Examples.LC.DataTypes\nimport NomPa.Examples.LC.Contexts\nimport NomPa.Examples.LC.Equiv\n-- import NomPa.Examples.LC.Monadic\n-- import NomPa.Examples.LC.Monadic.Tests\nimport NomPa.Examples.LC.DbLevels\nimport NomPa.Examples.NaPa.LC\nimport NomPa.Examples.LC.Combined\nimport NomPa.Examples.LC\nimport NomPa.Examples.Records\n-- import NomPa.Examples.Records.TypedVec\n-- import NomPa.Examples.Records.Typed\nimport NomPa.Laws\nimport NomPa.Link\nimport NomPa.NomT\nimport NomPa.NomT.Maybe\n\n-- Normalization By Evaluation\nimport NomPa.Examples.NBE\nimport NomPa.Examples.NBE.Can\nimport NomPa.Examples.NBE.Env\nimport NomPa.Examples.NBE.Neu\nimport NomPa.Examples.NBE.Tests\n\n-- bare, raw, non-fancy\u2026\nimport NomPa.Examples.BareDB\nimport NomPa.Examples.Raw\nimport NomPa.Examples.Raw.Parser\nimport NomPa.Examples.Raw.Printer\nimport NomPa.Examples.LC.WellScopedJudgment\n\n-- Encodings: Emebedding of nominal types\nimport NomPa.Encodings.NominalTypes\nimport NomPa.Encodings.NominalTypes.Generic\nimport NomPa.Encodings.NominalTypes.Generic.Subst\nimport NomPa.Encodings.NominalTypes.Generic.Combined\nimport NomPa.Encodings.NominalTypes.MultiSorts\nimport NomPa.Encodings.NominalTypes.MultiSorts.Test\nimport NomPa.Encodings.AlphaCaml\nimport NomPa.Encodings.AlphaCaml.Test\nimport NomPa.Encodings.BindersUnbound\nimport NomPa.Encodings.BindersUnbound.Generic\n\n-- Algorithms\nimport NomPa.Examples.LC.CC\nimport NomPa.Examples.LC.Monadic\nimport NomPa.Examples.LC.Monadic.Tests\n\n-- hybrid\nimport NomPa.Examples.LN\nimport NomPa.Examples.LL\nimport NomPa.Examples.LocallyNamed\n\n-- More examples\nimport NomPa.Examples.LC.Maybe\nimport NomPa.Examples.LC.SymanticsTy\nimport NomPa.Examples.LC.Tests\nimport NomPa.Examples.LocallyClosed\nimport NomPa.Examples.PTS\nimport NomPa.Examples.PTS.F\nimport NomPa.Examples.Reg\n\n-- Tests\nimport NomPa.Examples.Tests\n\n-- Logical relations\nimport NomPa.Record.LogicalRelation\nimport NomPa.Implem.LogicalRelation\nimport NomPa.Examples.LC.DataTypes.Logical\n\n-- Universal types\nimport NomPa.Universal.RegularTree\nimport NomPa.Universal.Sexp\n\nimport NomPa.Derived.WorldInclusion\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"247680fb5c13d8db678e248830c7b9b70a009441","subject":"Finished natural semantics ifthenelse proof thanks to Tillmanns pattern matching magic","message":"Finished natural semantics ifthenelse proof thanks to Tillmanns pattern matching magic\n\nOld-commit-hash: 269afa06d5278fb47afaf3509104feafd3c6d67a\n","repos":"inc-lc\/ilc-agda","old_file":"Natural\/Soundness.agda","new_file":"Natural\/Soundness.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"29f70aa411847fa9a6d573e62d3fe12a9a4ee779","subject":"proved a -> a, and therefore \\bn -> \\bn","message":"proved a -> a, and therefore \\bn -> \\bn\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/peano.agda","new_file":"learyouanagda\/peano.agda","new_contents":"module peano where\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\nzero + n     = n\n(suc n) + n\u2032 = suc (n + n\u2032)\n\n\ndata _even : \u2115 \u2192 Set where\n  ZERO : zero even\n  STEP : \u2200 x \u2192 x even \u2192 (suc (suc x)) even\n\n\n-- To prove four is even\nproof\u2081 : suc (suc (suc (suc zero))) even\nproof\u2081 = STEP (suc (suc zero)) (STEP zero ZERO)\n\nproof\u2082\u2032 : (A : Set) \u2192 A \u2192 A\nproof\u2082\u2032 _ a = a\n\nproof\u2082 : \u2115 \u2192 \u2115\nproof\u2082 = proof\u2082\u2032 \u2115\n","old_contents":"module peano where\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\nzero + zero  = zero\nzero + n     = n\n(suc n) + n\u2032 = suc (n + n\u2032)\n\n\ndata _even : \u2115 \u2192 Set where\n  ZERO : zero even\n  STEP : \u2200 x \u2192 x even \u2192 (suc (suc x)) even\n\n\n-- To prove four is even\nproof\u2081 : suc (suc (suc (suc zero))) even\nproof\u2081 = STEP (suc (suc zero)) (STEP zero ZERO)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"70177fd53d9a8f771d08255ffe684a6233a8b874","subject":"Adapt Base.Change.Algebra to Agda master","message":"Adapt Base.Change.Algebra to Agda master\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"1edc24705001056aa608cdde782b47b8a88273a5","subject":"list functions","message":"list functions\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"agda-tutorial\/basics.agda","new_file":"agda-tutorial\/basics.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"a192431287c18e439b3105dda3553a9797b3d9e5","subject":"Missing changes.","message":"Missing changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/LogicalFramework\/AdequacyTheorems.agda","new_file":"notes\/thesis\/LogicalFramework\/AdequacyTheorems.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.AdequacyTheorems where\n\nmodule Example5 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081    : A \u2192 C\n    f\u2082    : B \u2192 C\n\n  g : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  g f\u2081 f\u2082 (inj\u2081 a) = f\u2081 a\n  g f\u2081 f\u2082 (inj\u2082 b) = f\u2082 b\n\n  g' : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  g' f\u2081 f\u2082 x = case f\u2081 f\u2082 x\n\nmodule Example7 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    C  : D \u2192 D \u2192 Set\n    d  : \u2200 {a} \u2192 C a a\n\n  g : \u2200 {a b} \u2192 a \u2261 b \u2192 C a b\n  g refl = d\n\n  g' : \u2200 {a b} \u2192 a \u2261 b \u2192 C a b\n  g' {a} h = subst (\u03bb x \u2192 C a x) h d\n\nmodule Example10 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081 : A \u2192 C\n    f\u2082 : B \u2192 C\n\n  f : A \u2228 B \u2192 C\n  f (inj\u2081 a) = f\u2081 a\n  f (inj\u2082 b) = f\u2082 b\n\n  f' : A \u2228 B \u2192 C\n  f' = case f\u2081 f\u2082\n\nmodule Example20 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  f : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f {A} {t} {.t} refl At = d At\n    where\n    postulate d : A t \u2192 A t\n\n  f' : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f' {A} h At = subst A h At\n\nmodule Example30 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C E : Set\n    f\u2081 : A \u2192 E\n    f\u2082 : B \u2192 E\n    f\u2083 : C \u2192 E\n\n  g : (A \u2228 B) \u2228 C \u2192 E\n  g (inj\u2081 (inj\u2081 a)) = f\u2081 a\n  g (inj\u2081 (inj\u2082 b)) = f\u2082 b\n  g (inj\u2082 c)        = f\u2083 c\n\n  g' : (A \u2228 B) \u2228 C \u2192 E\n  g' = case (case f\u2081 f\u2082) f\u2083\n\nmodule Example40 where\n\n  infixl 9  _+_ _+'_\n  infix  7  _\u2261_\n\n  data \u2115 : Set where\n    zero :     \u2115\n    succ : \u2115 \u2192 \u2115\n\n  PA-ind : (A : \u2115 \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n  PA-ind A A0 h zero     = A0\n  PA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n  data _\u2261_ (x : \u2115) : \u2115 \u2192 Set where\n    refl : x \u2261 x\n\n  subst : (A : \u2115 \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A refl Ax = Ax\n\n  _+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  zero   + n = n\n  succ m + n = succ (m + n)\n\n  _+'_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  m +' n = PA-ind (\u03bb _ \u2192 \u2115) n (\u03bb x y \u2192 succ y) m\n\n  -- Properties using pattern matching.\n  succCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong refl = refl\n\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  +-rightIdentity zero     = refl\n  +-rightIdentity (succ n) = succCong (+-rightIdentity n)\n\n  -- Properties using the basic inductive constants.\n  succCong' : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong' {m} h = subst (\u03bb x \u2192 succ m \u2261 succ x) h refl\n\n  +'-leftIdentity : \u2200 n \u2192 zero +' n \u2261 n\n  +'-leftIdentity n = refl\n\n  +'-rightIdentity : \u2200 n \u2192 n +' zero \u2261 n\n  +'-rightIdentity = PA-ind A A0 is\n    where\n    A : \u2115 \u2192 Set\n    A n = n +' zero \u2261 n\n\n    A0 : A zero\n    A0 = refl\n\n    is : \u2200 n \u2192 A n \u2192 A (succ n)\n    is n ih = succCong' ih\n\nmodule Example50 where\n\n  infixl 10 _*_\n  infixl 9  _+_\n  infix  7  _\u2261_\n\n  data \u2115 : Set where\n    zero :     \u2115\n    succ : \u2115 \u2192 \u2115\n\n  PA-ind : (A : \u2115 \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n  PA-ind A A0 h zero     = A0\n  PA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n  data _\u2261_ (x : \u2115) : \u2115 \u2192 Set where\n    refl : x \u2261 x\n\n  rec\u2115 : {A : Set} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n  rec\u2115 {A} = PA-ind (\u03bb _ \u2192 A)\n\n  _+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  m + n = rec\u2115 n (\u03bb _ x \u2192 succ x) m\n\n  +-0x : \u2200 n \u2192 zero + n \u2261 n\n  +-0x n = refl\n\n  +-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n  +-Sx m n = refl\n\n  _*_ : \u2115 \u2192 \u2115 \u2192 \u2115\n  m * n = rec\u2115 zero (\u03bb _ x \u2192 n + x) m\n\n  *-0x : \u2200 n \u2192 zero * n \u2261 zero\n  *-0x n = refl\n\n  *-Sx : \u2200 m n \u2192 succ m * n \u2261 n + m * n\n  *-Sx m n = refl\n","old_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LogicalFramework.AdequacyTheorems where\n\nmodule Example5 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081    : A \u2192 C\n    f\u2082    : B \u2192 C\n\n  g : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  g f\u2081 f\u2082 (inj\u2081 a) = f\u2081 a\n  g f\u2081 f\u2082 (inj\u2082 b) = f\u2082 b\n\n  g' : (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\n  g' f\u2081 f\u2082 x = case f\u2081 f\u2082 x\n\nmodule Example7 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    C  : D \u2192 D \u2192 Set\n    d  : \u2200 {a} \u2192 C a a\n\n  g : \u2200 {a b} \u2192 a \u2261 b \u2192 C a b\n  g refl = d\n\n  g' : \u2200 {a b} \u2192 a \u2261 b \u2192 C a b\n  g' {a} h = subst (\u03bb x \u2192 C a x) h d\n\nmodule Example10 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C : Set\n    f\u2081 : A \u2192 C\n    f\u2082 : B \u2192 C\n\n  f : A \u2228 B \u2192 C\n  f (inj\u2081 a) = f\u2081 a\n  f (inj\u2082 b) = f\u2082 b\n\n  f' : A \u2228 B \u2192 C\n  f' = case f\u2081 f\u2082\n\nmodule Example20 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  f : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f {A} {t} {.t} refl At = d At\n    where\n    postulate d : A t \u2192 A t\n\n  f' : {A : D \u2192 Set}{t t' : D} \u2192 t \u2261 t' \u2192 A t \u2192 A t'\n  f' {A} h At = subst A h At\n\nmodule Example30 where\n\n  -- First-order logic with equality.\n  open import Common.FOL.FOL-Eq public\n\n  postulate\n    A B C E : Set\n    f\u2081 : A \u2192 E\n    f\u2082 : B \u2192 E\n    f\u2083 : C \u2192 E\n\n  g : (A \u2228 B) \u2228 C \u2192 E\n  g (inj\u2081 (inj\u2081 a)) = f\u2081 a\n  g (inj\u2081 (inj\u2082 b)) = f\u2082 b\n  g (inj\u2082 c)        = f\u2083 c\n\n  g' : (A \u2228 B) \u2228 C \u2192 E\n  g' = case (case f\u2081 f\u2082) f\u2083\n\nmodule Example40 where\n\n  infixl 9  _+_ _+'_\n  infix  7  _\u2261_\n\n  data M : Set where\n    zero :     M\n    succ : M \u2192 M\n\n  PA-ind : (A : M \u2192 Set) \u2192 A zero \u2192 (\u2200 n \u2192 A n \u2192 A (succ n)) \u2192 \u2200 n \u2192 A n\n  PA-ind A A0 h zero     = A0\n  PA-ind A A0 h (succ n) = h n (PA-ind A A0 h n)\n\n  data _\u2261_ (x : M) : M \u2192 Set where\n    refl : x \u2261 x\n\n  subst : (A : M \u2192 Set) \u2192 \u2200 {x y} \u2192 x \u2261 y \u2192 A x \u2192 A y\n  subst A refl Ax = Ax\n\n  _+_ : M \u2192 M \u2192 M\n  zero   + n = n\n  succ m + n = succ (m + n)\n\n  _+'_ : M \u2192 M \u2192 M\n  m +' n = PA-ind (\u03bb _ \u2192 M) n (\u03bb x y \u2192 succ y) m\n\n  -- Properties using pattern matching.\n  succCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong refl = refl\n\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  +-rightIdentity zero     = refl\n  +-rightIdentity (succ n) = succCong (+-rightIdentity n)\n\n  -- Properties using the basic inductive constants.\n  succCong' : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\n  succCong' {m} h = subst (\u03bb x \u2192 succ m \u2261 succ x) h refl\n\n  +'-leftIdentity : \u2200 n \u2192 zero +' n \u2261 n\n  +'-leftIdentity n = refl\n\n  +'-rightIdentity : \u2200 n \u2192 n +' zero \u2261 n\n  +'-rightIdentity = PA-ind A A0 is\n    where\n    A : M \u2192 Set\n    A n = n +' zero \u2261 n\n\n    A0 : A zero\n    A0 = refl\n\n    is : \u2200 n \u2192 A n \u2192 A (succ n)\n    is n ih = succCong' ih\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"85d2c1c465b079730a47287ec5817987f35aab41","subject":"proved one of the two remaining stupid lemmas #8","message":"proved one of the two remaining stupid lemmas #8\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"typed-expansion.agda","new_file":"typed-expansion.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\nopen import lemmas-consistency\n\nmodule typed-expansion where\n  lem : (\u0393 : tctx) (x : Nat) \u2192 (((id \u0393) x) == Some (X x) \u00d7 (\u03a3[ \u03c4 \u2208 htyp ]((\u0393 x) == Some \u03c4)))\n                             + (((id \u0393) x) == None)\n  lem \u0393 x with \u0393 x\n  lem \u0393 x | Some x\u2081 = Inl (refl , x\u2081 , refl)\n  lem \u0393 x | None = Inr refl\n\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0394} {\u0393} x d xin with lem \u0393 x\n  lem-idsub {\u0394} {\u0393} x d xin | Inl (w , y , z) with (! xin) \u00b7 w\n  lem-idsub x .(X x) xin | Inl (w , y , z) | refl = y , z , TAVar z\n  lem-idsub x d xin | Inr x\u2081 = abort (somenotnone ((! xin) \u00b7 x\u2081))\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D D\u2081) = TAAp (lem-weaken\u03941 D) (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a1 \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a1 \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n  lem-weaken\u03941 (TAFailedCast x y z w) = TAFailedCast (lem-weaken\u03941 x) y z w\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4} D = tr (\u03bb q \u2192 q , \u0393 \u22a2 d :: \u03c4) (\u222acomm \u03942 \u03941) (lem-weaken\u03941 D)\n\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp {\u03941 = \u03941} x\u2081 x\u2082 x\u2083 x\u2084)\n      with typed-expansion-ana x\u2083 | typed-expansion-ana x\u2084\n    ... | con1 , ih1 | con2 , ih2  = TAAp (TACast (lem-weaken\u03941 ih1) con1) (TACast (lem-weaken\u03942 {\u03941 = \u03941 }ih2) con2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-synth (ESAsc x)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 MAHole ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCHole1 , TALam D\n    typed-expansion-ana (EALam x\u2081 MAArr ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCArr TCRefl con , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ~sym x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u03941 ih1) lem-idsub\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import correspondence\nopen import contexts\nopen import lemmas-consistency\n\nmodule typed-expansion where\n  lem-idsub : \u2200{\u0394 \u0393} \u2192 \u0394 , \u0393 \u22a2 id \u0393 :s: \u0393\n  lem-idsub {\u0394} {\u0393} x d xd\u2208\u03c3 with (id \u0393) x\n  lem-idsub x y refl | Some .y = {!!}\n  lem-idsub x d xd\u2208\u03c3 | None = abort (somenotnone (! xd\u2208\u03c3))\n\n  lem-subweak : \u2200{\u0394 \u0393 \u0393' \u0394' \u03c3} \u2192 \u0394 , \u0393 \u22a2 \u03c3 :s: \u0393' \u2192 (\u0394 \u222a \u0394') , \u0393 \u22a2 \u03c3 :s: \u0393'\n  lem-subweak sub x d xd\u2208\u03c3 with sub x d xd\u2208\u03c3\n  ... | \u03c4 , some , sub' = \u03c4 , some , {!!}\n\n  lem-weaken\u03941 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03941 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03941 TAConst = TAConst\n  lem-weaken\u03941 (TAVar x\u2081) = TAVar x\u2081\n  lem-weaken\u03941 (TALam D) = TALam (lem-weaken\u03941 D)\n  lem-weaken\u03941 (TAAp D D\u2081) = TAAp (lem-weaken\u03941 D) (lem-weaken\u03941 D\u2081)\n  lem-weaken\u03941 (TAEHole {\u0394 = \u0394} x y) = TAEHole (x\u2208\u222a1 \u0394 _ _ _ x) (lem-subweak y)\n  lem-weaken\u03941 (TANEHole {\u0394 = \u0394} D x y) = TANEHole (x\u2208\u222a1 \u0394 _ _ _ D) (lem-weaken\u03941 x) (lem-subweak y)\n  lem-weaken\u03941 (TACast D x) = TACast (lem-weaken\u03941 D) x\n  lem-weaken\u03941 (TAFailedCast x y z w) = TAFailedCast (lem-weaken\u03941 x) y z w\n\n  lem-weaken\u03942 : \u2200{\u03941 \u03942 \u0393 d \u03c4} \u2192 \u03942 , \u0393 \u22a2 d :: \u03c4 \u2192 (\u03941 \u222a \u03942) , \u0393 \u22a2 d :: \u03c4\n  lem-weaken\u03942 {\u03941} {\u03942} {\u0393} {d} {\u03c4} D = tr (\u03bb q \u2192 q , \u0393 \u22a2 d :: \u03c4) (\u222acomm \u03942 \u03941) (lem-weaken\u03941 D)\n\n\n  mutual\n    typed-expansion-synth : {\u0393 : tctx} {e : hexp} {\u03c4 : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                            \u0393 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                            \u0394 , \u0393 \u22a2 d :: \u03c4\n    typed-expansion-synth ESConst = TAConst\n    typed-expansion-synth (ESVar x\u2081) = TAVar x\u2081\n    typed-expansion-synth (ESLam x\u2081 ex) = TALam (typed-expansion-synth ex)\n    typed-expansion-synth (ESAp {\u03941 = \u03941} x\u2081 x\u2082 x\u2083 x\u2084)\n      with typed-expansion-ana x\u2083 | typed-expansion-ana x\u2084\n    ... | con1 , ih1 | con2 , ih2  = TAAp (TACast (lem-weaken\u03941 ih1) con1) (TACast (lem-weaken\u03942 {\u03941 = \u03941 }ih2) con2)\n    typed-expansion-synth (ESEHole {\u0393 = \u0393} {u = u})  = TAEHole (x\u2208sing \u2205 u (\u0393 , \u2987\u2988)) lem-idsub\n    typed-expansion-synth (ESNEHole {\u0393 = \u0393} {\u03c4 = \u03c4} {u = u} {\u0394 = \u0394} ex)\n      with typed-expansion-synth ex\n    ... | ih1 = TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u2987\u2988)} (x\u2208sing \u0394 u (\u0393 , \u2987\u2988)) (lem-weaken\u03941 ih1) lem-idsub\n    typed-expansion-synth (ESAsc x)\n      with typed-expansion-ana x\n    ... | con , ih = TACast ih con\n\n    typed-expansion-ana : {\u0393 : tctx} {e : hexp} {\u03c4 \u03c4' : htyp} {d : dhexp} {\u0394 : hctx} \u2192\n                          \u0393 \u22a2 e \u21d0 \u03c4 ~> d :: \u03c4' \u22a3 \u0394 \u2192\n                          (\u03c4' ~ \u03c4) \u00d7 (\u0394 , \u0393 \u22a2 d :: \u03c4')\n    typed-expansion-ana (EALam x\u2081 MAHole ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCHole1 , TALam D\n    typed-expansion-ana (EALam x\u2081 MAArr ex)\n      with typed-expansion-ana ex\n    ... | con , D = TCArr TCRefl con , TALam D\n    typed-expansion-ana (EASubsume x x\u2081 x\u2082 x\u2083) = ~sym x\u2083 , typed-expansion-synth x\u2082\n    typed-expansion-ana (EAEHole {\u0393 = \u0393} {u = u}) = TCRefl , TAEHole (x\u2208sing \u2205 u (\u0393 , _)) lem-idsub\n    typed-expansion-ana (EANEHole {\u0393 = \u0393} {u = u} {\u03c4 = \u03c4} {\u0394 = \u0394}  x)\n      with typed-expansion-synth x\n    ... | ih1 = TCRefl , TANEHole {\u0394 = \u0394 ,, (u , \u0393 , \u03c4)} (x\u2208sing \u0394 u (\u0393 , \u03c4)) (lem-weaken\u03941 ih1) lem-idsub\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"7c370be85bd6cb58479b11facd3b2873d2178920","subject":"Simplified combined proof for ABP's lemma2.","message":"Simplified combined proof for ABP's lemma2.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2ATP.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2ATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\n  using ( not-x\u2262x\n        ; not-involutive\n        )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- 30 November 2013. If we don't have the following definitions\n-- outside the where clause, the ATPs cannot prove the theorems.\n\nds^ : D \u2192 D \u2192 D\nds^ cs' os\u2082^ = corrupt os\u2082^ \u00b7 cs'\n{-# ATP definition ds^ #-}\n\nas^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nas^ b i' is' cs' os\u2082^ = await b i' is' (ds^ cs' os\u2082^)\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' cs' os\u2081^ os\u2082^ = corrupt os\u2081^ \u00b7 as^ b i' is' cs' os\u2082^\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' cs' os\u2081^ os\u2082^ = ack (not b) \u00b7 bs^ b i' is' cs' os\u2081^ os\u2082^\n{-# ATP definition cs^ #-}\n\nos\u2081^ : D \u2192 D\nos\u2081^ os\u2081' = tail\u2081 os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D \u2192 D\nos\u2082^ ft\u2082 os\u2082'' = ft\u2082 ++ os\u2082''\n{-# ATP definition os\u2082^ #-}\n\n-- Helper function for the lemma 2.\nhelper : \u2200 {b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2081' \u2192\n         S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' \u2192\n         \u2200 ft\u2082 os\u2082'' \u2192 F*T ft\u2082 \u2192 Fair os\u2082'' \u2192 os\u2082' \u2261 ft\u2082 ++ os\u2082'' \u2192\n         \u2203[ os\u2081'' ] \u2203[ os\u2082'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2081''\n           \u2227 Fair os\u2082''\n           \u2227 S (not b) is' os\u2081'' os\u2082'' as'' bs'' cs'' ds'' js'\nhelper {b} {i'} {is'} {js' = js'} Bb Fos\u2081' s'\n       .(T \u2237 []) os\u2082'' f*tnil Fos\u2082'' os\u2082'-eq = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081'' ] \u2203[ os\u2082'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n          Fair os\u2081''\n          \u2227 Fair os\u2082''\n          \u2227 as'' \u2261 send (not b) \u00b7 is' \u00b7 ds''\n          \u2227 bs'' \u2261 corrupt os\u2081'' \u00b7 as''\n          \u2227 cs'' \u2261 ack (not b) \u00b7 bs''\n          \u2227 ds'' \u2261 corrupt os\u2082'' \u00b7 cs''\n          \u2227 js'  \u2261 out (not b) \u00b7 bs''\n  {-# ATP prove prf #-}\n\nhelper {b} {i'} {is'} {os\u2081'} {os\u2082'} {as'} {bs'} {cs'} {ds'} {js'}\n       Bb Fos\u2081' s'\n       .(F \u2237 ft\u2082) os\u2082'' (f*tcons {ft\u2082} FTft\u2082) Fos\u2082'' os\u2082'-eq =\n       helper Bb (tail-Fair Fos\u2081') ihS' ft\u2082 os\u2082'' FTft\u2082 Fos\u2082'' refl\n  where\n  postulate os\u2082'-eq-helper : os\u2082' \u2261 F \u2237 os\u2082^ ft\u2082 os\u2082''\n  {-# ATP prove os\u2082'-eq-helper #-}\n\n  postulate ds'-eq : ds' \u2261 error \u2237 ds^ cs' (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove ds'-eq os\u2082'-eq-helper #-}\n\n  postulate as'-eq : as' \u2261 < i' , b > \u2237 as^ b i' is' cs' (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove as'-eq #-}\n\n  postulate\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n               \u2228 bs' \u2261 error \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove bs'-eq as'-eq head-tail-Fair #-}\n\n  postulate\n    cs'-eq-helper\u2081 :\n      bs' \u2261 ok < i' , b > \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'') \u2192\n      cs' \u2261 b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove cs'-eq-helper\u2081 not-x\u2262x not-involutive #-}\n\n  postulate\n    cs'-eq-helper\u2082 :\n      bs' \u2261 error \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'') \u2192\n      cs' \u2261 b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove cs'-eq-helper\u2082 not-involutive #-}\n\n  cs'-eq : cs' \u2261 b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n  postulate\n    js'-eq : js' \u2261 out (not b) \u00b7 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove js'-eq not-x\u2262x bs'-eq #-}\n\n  postulate\n    ds^-eq : ds^ cs' (os\u2082^ ft\u2082 os\u2082'') \u2261\n             corrupt (os\u2082^ ft\u2082 os\u2082'') \u00b7\n               (b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082''))\n\n  ihS' : S' b i' is'\n            (os\u2081^ os\u2081')\n            (os\u2082^ ft\u2082 os\u2082'')\n            (as^ b i' is' cs' (os\u2082^ ft\u2082 os\u2082''))\n            (bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082''))\n            (cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082''))\n            (ds^ cs' (os\u2082^ ft\u2082 os\u2082''))\n            js'\n  ihS' = refl , refl , refl , ds^-eq , js'-eq\n\n-- From Dybjer and Sander's paper: From the assumption that\n-- os\u2082'\u00a0\u2208\u00a0Fair and hence by unfolding Fair, we conclude that there are\n-- ft\u2082\u00a0:\u00a0F*T and os\u2082''\u00a0:\u00a0Fair, such that os\u2082'\u00a0=\u00a0ft\u2082\u00a0++\u00a0os\u2082''.\n--\n-- We proceed by induction on ft\u2082\u00a0:\u00a0F*T using helper.\n\nlemma\u2082 : \u2200 {b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2081' \u2192\n         Fair os\u2082' \u2192\n         S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2081'' ] \u2203[ os\u2082'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2081''\n           \u2227 Fair os\u2082''\n           \u2227 S (not b) is' os\u2081'' os\u2082'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2081' Fos\u2082' s' with Fair-out Fos\u2082'\n... | ft , os\u2081'' , FTft , prf , Fos\u2081'' =\n  helper Bb Fos\u2081' s' ft os\u2081'' FTft Fos\u2081'' prf\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2ATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\n  using ( not-x\u2262x\n        ; not-involutive\n        )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- 30 November 2013. If we don't have the following definitions\n-- outside the where clause, the ATPs cannot prove the theorems.\n\nds^ : D \u2192 D \u2192 D\nds^ cs' os\u2082^ = corrupt os\u2082^ \u00b7 cs'\n{-# ATP definition ds^ #-}\n\nas^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nas^ b i' is' cs' os\u2082^ = await b i' is' (ds^ cs' os\u2082^)\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' cs' os\u2081^ os\u2082^ = corrupt os\u2081^ \u00b7 as^ b i' is' cs' os\u2082^\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' cs' os\u2081^ os\u2082^ = ack (not b) \u00b7 bs^ b i' is' cs' os\u2081^ os\u2082^\n{-# ATP definition cs^ #-}\n\nos\u2081^ : D \u2192 D\nos\u2081^ os\u2081' = tail\u2081 os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D \u2192 D\nos\u2082^ ft\u2082 os\u2082'' = ft\u2082 ++ os\u2082''\n{-# ATP definition os\u2082^ #-}\n\n-- Helper function for the lemma 2.\nhelper : \u2200 {b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2081' \u2192\n         S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' \u2192\n         \u2200 ft\u2082 os\u2082'' \u2192 F*T ft\u2082 \u2192 Fair os\u2082'' \u2192 os\u2082' \u2261 ft\u2082 ++ os\u2082'' \u2192\n         \u2203[ os\u2081'' ] \u2203[ os\u2082'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2081''\n           \u2227 Fair os\u2082''\n           \u2227 S (not b) is' os\u2081'' os\u2082'' as'' bs'' cs'' ds'' js'\nhelper {b} {i'} {is'} {js' = js'} Bb Fos\u2081' s'\n       .(T \u2237 []) os\u2082'' f*tnil Fos\u2082'' os\u2082'-eq = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081'' ] \u2203[ os\u2082'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n          Fair os\u2081''\n          \u2227 Fair os\u2082''\n          \u2227 as'' \u2261 send (not b) \u00b7 is' \u00b7 ds''\n          \u2227 bs'' \u2261 corrupt os\u2081'' \u00b7 as''\n          \u2227 cs'' \u2261 ack (not b) \u00b7 bs''\n          \u2227 ds'' \u2261 corrupt os\u2082'' \u00b7 cs''\n          \u2227 js'  \u2261 out (not b) \u00b7 bs''\n  {-# ATP prove prf #-}\n\nhelper {b} {i'} {is'} {os\u2081'} {os\u2082'} {as'} {bs'} {cs'} {ds'} {js'}\n       Bb Fos\u2081' s'\n       .(F \u2237 ft\u2082) os\u2082'' (f*tcons {ft\u2082} FTft\u2082) Fos\u2082'' os\u2082'-eq =\n       helper Bb (tail-Fair Fos\u2081') ihS' ft\u2082 os\u2082'' FTft\u2082 Fos\u2082'' refl\n  where\n  postulate os\u2082'-eq-helper : os\u2082' \u2261 F \u2237 os\u2082^ ft\u2082 os\u2082''\n  {-# ATP prove os\u2082'-eq-helper #-}\n\n  postulate ds'-eq : ds' \u2261 error \u2237 ds^ cs' (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove ds'-eq os\u2082'-eq-helper #-}\n\n  postulate as'-eq : as' \u2261 < i' , b > \u2237 as^ b i' is' cs' (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove as'-eq #-}\n\n  postulate\n    bs'-eq-helper\u2081 :\n      os\u2081' \u2261 T \u2237 tail\u2081 os\u2081' \u2192\n      bs' \u2261 ok < i' , b > \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove bs'-eq-helper\u2081 as'-eq #-}\n\n  postulate\n    bs'-eq-helper\u2082 :\n      os\u2081' \u2261 F \u2237 tail\u2081 os\u2081' \u2192\n      bs' \u2261 error \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove bs'-eq-helper\u2082 as'-eq #-}\n\n  bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n             \u2228 bs' \u2261 error \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                (head-tail-Fair Fos\u2081')\n\n  postulate\n    cs'-eq-helper\u2081 :\n      bs' \u2261 ok < i' , b > \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'') \u2192\n      cs' \u2261 b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove cs'-eq-helper\u2081 not-x\u2262x not-involutive #-}\n\n  postulate\n    cs'-eq-helper\u2082 :\n      bs' \u2261 error \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'') \u2192\n      cs' \u2261 b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove cs'-eq-helper\u2082 not-involutive #-}\n\n  cs'-eq : cs' \u2261 b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n  postulate\n    js'-eq-helper\u2081 :\n      bs' \u2261 ok < i' , b > \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'') \u2192\n      js' \u2261 out (not b) \u00b7 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove js'-eq-helper\u2081 not-x\u2262x #-}\n\n  postulate\n    js'-eq-helper\u2082 :\n      bs' \u2261 error \u2237 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'') \u2192\n      js' \u2261 out (not b) \u00b7 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  {-# ATP prove js'-eq-helper\u2082 #-}\n\n  js'-eq : js' \u2261 out (not b) \u00b7 bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082'')\n  js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n  postulate\n    ds^-eq : ds^ cs' (os\u2082^ ft\u2082 os\u2082'') \u2261\n             corrupt (os\u2082^ ft\u2082 os\u2082'') \u00b7\n               (b \u2237 cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082''))\n\n  ihS' : S' b i' is'\n            (os\u2081^ os\u2081')\n            (os\u2082^ ft\u2082 os\u2082'')\n            (as^ b i' is' cs' (os\u2082^ ft\u2082 os\u2082''))\n            (bs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082''))\n            (cs^ b i' is' cs' (os\u2081^ os\u2081') (os\u2082^ ft\u2082 os\u2082''))\n            (ds^ cs' (os\u2082^ ft\u2082 os\u2082''))\n            js'\n  ihS' = refl , refl , refl , ds^-eq , js'-eq\n\n-- From Dybjer and Sander's paper: From the assumption that\n-- os\u2082'\u00a0\u2208\u00a0Fair and hence by unfolding Fair, we conclude that there are\n-- ft\u2082\u00a0:\u00a0F*T and os\u2082''\u00a0:\u00a0Fair, such that os\u2082'\u00a0=\u00a0ft\u2082\u00a0++\u00a0os\u2082''.\n--\n-- We proceed by induction on ft\u2082\u00a0:\u00a0F*T using helper.\n\nlemma\u2082 : \u2200 {b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2081' \u2192\n         Fair os\u2082' \u2192\n         S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2081'' ] \u2203[ os\u2082'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2081''\n           \u2227 Fair os\u2082''\n           \u2227 S (not b) is' os\u2081'' os\u2082'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2081' Fos\u2082' s' with Fair-out Fos\u2082'\n... | ft , os\u2081'' , FTft , prf , Fos\u2081'' =\n  helper Bb Fos\u2081' s' ft os\u2081'' FTft Fos\u2081'' prf\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"95e2bfa97f5f84f48cf5fe9cd6a123b8e8ddb4bb","subject":"Add missing projection for Receipts","message":"Add missing projection for Receipts\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/ReceiptFreeness\/Definitions.agda","new_file":"Game\/ReceiptFreeness\/Definitions.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function\nopen import Type\nopen import Data.Product renaming (zip to zip-\u00d7)\nopen import Data.Two\nopen import Data.List as L\nopen import Data.Nat.NP hiding (_==_)\n\nmodule Game.ReceiptFreeness.Definitions\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2090 : \u2605)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CipherText \u2192 Message)\n  where\n\nCandidate : \u2605\nCandidate = \ud835\udfda -- as in the paper: \"for simplicity\"\n\nalice bob : Candidate\nalice = 0\u2082\nbob   = 1\u2082\n\n-- candidate order\n-- also known as LHS or Message\n-- represented by who is the first candidate\nCO = \ud835\udfda\n\nalice-then-bob bob-then-alice : CO\nalice-then-bob = alice\nbob-then-alice = bob\n\ndata CO-spec : CO \u2192 Candidate \u2192 Candidate \u2192 \u2605 where\n  alice-then-bob-spec : CO-spec alice-then-bob alice bob\n  bob-then-alice-spec : CO-spec bob-then-alice bob alice\n\nMarkedReceipt : \u2605\nMarkedReceipt = \ud835\udfda\n\nmarked-on-first-cell marked-on-second-cell : MarkedReceipt\nmarked-on-first-cell  = 0\u2082\nmarked-on-second-cell = 1\u2082\n\ndata MarkedReceipt-spec : CO \u2192 MarkedReceipt \u2192 Candidate \u2192 \u2605 where\n  m1 : MarkedReceipt-spec alice-then-bob marked-on-first-cell  alice\n  m2 : MarkedReceipt-spec alice-then-bob marked-on-second-cell bob\n  m3 : MarkedReceipt-spec bob-then-alice marked-on-first-cell  bob\n  m4 : MarkedReceipt-spec bob-then-alice marked-on-second-cell alice\n\ndata MarkedReceipt? : \u2605 where\n  not-marked : MarkedReceipt?\n  marked     : MarkedReceipt \u2192 MarkedReceipt?\n\n-- Receipt or also called RHS\n-- Made of a potential mark, a serial number, and an encrypted candidate order\nReceipt : \u2605\nReceipt = MarkedReceipt? \u00d7 SerialNumber \u00d7 CipherText\n\nmarkedReceipt? : Receipt \u2192 MarkedReceipt?\nmarkedReceipt? = proj\u2081\n\n-- Marked when there is a 1\nmarked? : MarkedReceipt? \u2192 \ud835\udfda\nmarked? not-marked = 0\u2082\nmarked? (marked _) = 1\u2082\n\nmarked-on-first-cell? : MarkedReceipt? \u2192 \ud835\udfda\nmarked-on-first-cell? not-marked = 0\u2082\nmarked-on-first-cell? (marked x) = x == 0\u2082\n\nmarked-on-second-cell? : MarkedReceipt? \u2192 \ud835\udfda\nmarked-on-second-cell? not-marked = 0\u2082\nmarked-on-second-cell? (marked x) = x == 1\u2082\n\nenc-co : Receipt \u2192 CipherText\nenc-co = proj\u2082 \u2218 proj\u2082\n\nm? : Receipt \u2192 MarkedReceipt?\nm? = proj\u2081\n\nBallot : \u2605\nBallot = CO \u00d7 Receipt\n\n-- co or also called LHS\nco : Ballot \u2192 CO\nco = proj\u2081\n\n-- receipt or also called RHS\nreceipt : Ballot \u2192 Receipt\nreceipt = proj\u2082\n\n-- randomness for genBallot\nRgb : \u2605\nRgb = CO \u00d7 SerialNumber \u00d7 R\u2091\n\ngenBallot : PubKey \u2192 Rgb \u2192 Ballot\ngenBallot pk (r-co , sn , r\u2091) = r-co , not-marked , sn , Enc pk r-co r\u2091\n\nmark : CO \u2192 Candidate \u2192 MarkedReceipt\nmark co c = co xor c\n\nmark-ok : \u2200 co c \u2192 MarkedReceipt-spec co (mark co c) c\nmark-ok 1\u2082 1\u2082 = m3\nmark-ok 1\u2082 0\u2082 = m4\nmark-ok 0\u2082 1\u2082 = m2\nmark-ok 0\u2082 0\u2082 = m1\n\nfillBallot : Candidate \u2192 Ballot \u2192 Ballot\nfillBallot c (co , _ , sn , enc-co) = co , marked (mark co c) , sn , enc-co\n\n-- TODO Ballot-spec c (fillBallot b)\n\nClearReceipt : \u2605\nClearReceipt = CO \u00d7 MarkedReceipt?\n\nClearBB : \u2605\nClearBB = List ClearReceipt\n\nTally : \u2605\nTally = \u2115 \u00d7 \u2115\n\nalice-score : Tally \u2192 \u2115\nalice-score = proj\u2081\n\nbob-score : Tally \u2192 \u2115\nbob-score = proj\u2082\n\n1:alice-0:bob 0:alice-1:bob : Tally\n1:alice-0:bob = 1 , 0\n0:alice-1:bob = 0 , 1\n\ndata TallyMarkedReceipt-spec : CO \u2192 MarkedReceipt \u2192 Tally \u2192 \u2605 where\n  c1 : TallyMarkedReceipt-spec alice-then-bob marked-on-first-cell  1:alice-0:bob\n  c2 : TallyMarkedReceipt-spec alice-then-bob marked-on-second-cell 0:alice-1:bob\n  c3 : TallyMarkedReceipt-spec bob-then-alice marked-on-first-cell  0:alice-1:bob\n  c4 : TallyMarkedReceipt-spec bob-then-alice marked-on-second-cell 1:alice-0:bob\n\nmarked-for-alice? : CO \u2192 MarkedReceipt \u2192 \ud835\udfda\nmarked-for-alice? co m = co == m\n\nmarked-for-bob? : CO \u2192 MarkedReceipt \u2192 \ud835\udfda\nmarked-for-bob? co m = not (marked-for-alice? co m)\n\ntallyMarkedReceipt : CO \u2192 MarkedReceipt \u2192 Tally\ntallyMarkedReceipt co m = \ud835\udfda\u25b9\u2115 for-alice , \ud835\udfda\u25b9\u2115 (not for-alice)\n  where for-alice = marked-for-alice? co m\n\ntallyMarkedReceipt-ok : \u2200 co m \u2192 TallyMarkedReceipt-spec co m (tallyMarkedReceipt co m)\ntallyMarkedReceipt-ok 1\u2082 1\u2082 = c4\ntallyMarkedReceipt-ok 1\u2082 0\u2082 = c3\ntallyMarkedReceipt-ok 0\u2082 1\u2082 = c2\ntallyMarkedReceipt-ok 0\u2082 0\u2082 = c1\n\ntallyMarkedReceipt? : CO \u2192 MarkedReceipt? \u2192 Tally\ntallyMarkedReceipt? co not-marked    = 0 , 0\ntallyMarkedReceipt? co (marked mark) = tallyMarkedReceipt co mark\n\n_+,+_ : Tally \u2192 Tally \u2192 Tally\n_+,+_ = zip-\u00d7 _+_ _+_\n\n-- Not taking advantage of any homomorphic encryption\ntallyClearBB : ClearBB \u2192 Tally\ntallyClearBB = L.foldr _+,+_ (0 , 0) \u2218 L.map (uncurry tallyMarkedReceipt?)\n\nBB : \u2605\nBB = List Receipt\n\nDecReceipt : SecKey \u2192 Receipt \u2192 CO \u00d7 MarkedReceipt?\nDecReceipt sk (m? , sn , enc-co) = Dec sk enc-co , m?\n\nDecBB : SecKey \u2192 BB \u2192 ClearBB\nDecBB = L.map \u2218 DecReceipt\n\n-- Not taking advantage of any homomorphic encryption\ntally : SecKey \u2192 BB \u2192 Tally\ntally sk bb = tallyClearBB (DecBB sk bb)\n\nopen import Game.ReceiptFreeness.Adversary PubKey (SerialNumber \u00b2) R\u2090 Receipt Ballot Tally CO BB public\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nopen import Function\nopen import Type\nopen import Data.Product renaming (zip to zip-\u00d7)\nopen import Data.Two\nopen import Data.List as L\nopen import Data.Nat.NP hiding (_==_)\n\nmodule Game.ReceiptFreeness.Definitions\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2090 : \u2605)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CipherText \u2192 Message)\n  where\n\nCandidate : \u2605\nCandidate = \ud835\udfda -- as in the paper: \"for simplicity\"\n\nalice bob : Candidate\nalice = 0\u2082\nbob   = 1\u2082\n\n-- candidate order\n-- also known as LHS or Message\n-- represented by who is the first candidate\nCO = \ud835\udfda\n\nalice-then-bob bob-then-alice : CO\nalice-then-bob = alice\nbob-then-alice = bob\n\ndata CO-spec : CO \u2192 Candidate \u2192 Candidate \u2192 \u2605 where\n  alice-then-bob-spec : CO-spec alice-then-bob alice bob\n  bob-then-alice-spec : CO-spec bob-then-alice bob alice\n\nMarkedReceipt : \u2605\nMarkedReceipt = \ud835\udfda\n\nmarked-on-first-cell marked-on-second-cell : MarkedReceipt\nmarked-on-first-cell  = 0\u2082\nmarked-on-second-cell = 1\u2082\n\ndata MarkedReceipt-spec : CO \u2192 MarkedReceipt \u2192 Candidate \u2192 \u2605 where\n  m1 : MarkedReceipt-spec alice-then-bob marked-on-first-cell  alice\n  m2 : MarkedReceipt-spec alice-then-bob marked-on-second-cell bob\n  m3 : MarkedReceipt-spec bob-then-alice marked-on-first-cell  bob\n  m4 : MarkedReceipt-spec bob-then-alice marked-on-second-cell alice\n\ndata MarkedReceipt? : \u2605 where\n  not-marked : MarkedReceipt?\n  marked     : MarkedReceipt \u2192 MarkedReceipt?\n\n-- Receipt or also called RHS\n-- Made of a potential mark, a serial number, and an encrypted candidate order\nReceipt : \u2605\nReceipt = MarkedReceipt? \u00d7 SerialNumber \u00d7 CipherText\n\nmarkedReceipt? : Receipt \u2192 MarkedReceipt?\nmarkedReceipt? = proj\u2081\n\n-- Marked when there is a 1\nmarked? : MarkedReceipt? \u2192 \ud835\udfda\nmarked? not-marked = 0\u2082\nmarked? (marked _) = 1\u2082\n\nmarked-on-first-cell? : MarkedReceipt? \u2192 \ud835\udfda\nmarked-on-first-cell? not-marked = 0\u2082\nmarked-on-first-cell? (marked x) = x == 0\u2082\n\nmarked-on-second-cell? : MarkedReceipt? \u2192 \ud835\udfda\nmarked-on-second-cell? not-marked = 0\u2082\nmarked-on-second-cell? (marked x) = x == 1\u2082\n\nenc-co : Receipt \u2192 CipherText\nenc-co = proj\u2082 \u2218 proj\u2082\n\nBallot : \u2605\nBallot = CO \u00d7 Receipt\n\n-- co or also called LHS\nco : Ballot \u2192 CO\nco = proj\u2081\n\n-- receipt or also called RHS\nreceipt : Ballot \u2192 Receipt\nreceipt = proj\u2082\n\n-- randomness for genBallot\nRgb : \u2605\nRgb = CO \u00d7 SerialNumber \u00d7 R\u2091\n\ngenBallot : PubKey \u2192 Rgb \u2192 Ballot\ngenBallot pk (r-co , sn , r\u2091) = r-co , not-marked , sn , Enc pk r-co r\u2091\n\nmark : CO \u2192 Candidate \u2192 MarkedReceipt\nmark co c = co xor c\n\nmark-ok : \u2200 co c \u2192 MarkedReceipt-spec co (mark co c) c\nmark-ok 1\u2082 1\u2082 = m3\nmark-ok 1\u2082 0\u2082 = m4\nmark-ok 0\u2082 1\u2082 = m2\nmark-ok 0\u2082 0\u2082 = m1\n\nfillBallot : Candidate \u2192 Ballot \u2192 Ballot\nfillBallot c (co , _ , sn , enc-co) = co , marked (mark co c) , sn , enc-co\n\n-- TODO Ballot-spec c (fillBallot b)\n\nClearReceipt : \u2605\nClearReceipt = CO \u00d7 MarkedReceipt?\n\nClearBB : \u2605\nClearBB = List ClearReceipt\n\nTally : \u2605\nTally = \u2115 \u00d7 \u2115\n\nalice-score : Tally \u2192 \u2115\nalice-score = proj\u2081\n\nbob-score : Tally \u2192 \u2115\nbob-score = proj\u2082\n\n1:alice-0:bob 0:alice-1:bob : Tally\n1:alice-0:bob = 1 , 0\n0:alice-1:bob = 0 , 1\n\ndata TallyMarkedReceipt-spec : CO \u2192 MarkedReceipt \u2192 Tally \u2192 \u2605 where\n  c1 : TallyMarkedReceipt-spec alice-then-bob marked-on-first-cell  1:alice-0:bob\n  c2 : TallyMarkedReceipt-spec alice-then-bob marked-on-second-cell 0:alice-1:bob\n  c3 : TallyMarkedReceipt-spec bob-then-alice marked-on-first-cell  0:alice-1:bob\n  c4 : TallyMarkedReceipt-spec bob-then-alice marked-on-second-cell 1:alice-0:bob\n\nmarked-for-alice? : CO \u2192 MarkedReceipt \u2192 \ud835\udfda\nmarked-for-alice? co m = co == m\n\nmarked-for-bob? : CO \u2192 MarkedReceipt \u2192 \ud835\udfda\nmarked-for-bob? co m = not (marked-for-alice? co m)\n\ntallyMarkedReceipt : CO \u2192 MarkedReceipt \u2192 Tally\ntallyMarkedReceipt co m = \ud835\udfda\u25b9\u2115 for-alice , \ud835\udfda\u25b9\u2115 (not for-alice)\n  where for-alice = marked-for-alice? co m\n\ntallyMarkedReceipt-ok : \u2200 co m \u2192 TallyMarkedReceipt-spec co m (tallyMarkedReceipt co m)\ntallyMarkedReceipt-ok 1\u2082 1\u2082 = c4\ntallyMarkedReceipt-ok 1\u2082 0\u2082 = c3\ntallyMarkedReceipt-ok 0\u2082 1\u2082 = c2\ntallyMarkedReceipt-ok 0\u2082 0\u2082 = c1\n\ntallyMarkedReceipt? : CO \u2192 MarkedReceipt? \u2192 Tally\ntallyMarkedReceipt? co not-marked    = 0 , 0\ntallyMarkedReceipt? co (marked mark) = tallyMarkedReceipt co mark\n\n_+,+_ : Tally \u2192 Tally \u2192 Tally\n_+,+_ = zip-\u00d7 _+_ _+_\n\n-- Not taking advantage of any homomorphic encryption\ntallyClearBB : ClearBB \u2192 Tally\ntallyClearBB = L.foldr _+,+_ (0 , 0) \u2218 L.map (uncurry tallyMarkedReceipt?)\n\nBB : \u2605\nBB = List Receipt\n\nDecReceipt : SecKey \u2192 Receipt \u2192 CO \u00d7 MarkedReceipt?\nDecReceipt sk (m? , sn , enc-co) = Dec sk enc-co , m?\n\nDecBB : SecKey \u2192 BB \u2192 ClearBB\nDecBB = L.map \u2218 DecReceipt\n\n-- Not taking advantage of any homomorphic encryption\ntally : SecKey \u2192 BB \u2192 Tally\ntally sk bb = tallyClearBB (DecBB sk bb)\n\nopen import Game.ReceiptFreeness.Adversary PubKey (SerialNumber \u00b2) R\u2090 Receipt Ballot Tally CO BB public\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a74d5f4d0654808362f2583716c758bb12b35288","subject":"Document P.C.Implementation.","message":"Document P.C.Implementation.\n\nOld-commit-hash: 62c4109c756f345ba501b4ab6a99669cec1f3a21\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Implementation.agda","new_file":"Parametric\/Change\/Implementation.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Logical relation for erasure (Def. 3.8 and Lemma 3.9)\n------------------------------------------------------------------------\n\nimport Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Derive as Derive\n\nmodule Parametric.Change.Implementation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n       Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (derive-const : Derive.Structure Const \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen Derive.Structure Const \u0394Base derive-const\n\nopen import Base.Denotation.Notation\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nrecord Structure : Set\u2081 where\n\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    -- Extension point 1: Logical relation on base types.\n    --\n    -- In the paper, we assume that the logical relation is equality on base types\n    -- (see Def. 3.8a). Here, we only require that plugins define what the logical\n    -- relation is on base types, and provide proofs for the two extension points\n    -- below.\n    implements-base : \u2200 \u03b9 {v : \u27e6 \u03b9 \u27e7Base} \u2192 \u0394\u208d \u03b9 \u208e v \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 Set\n\n    -- Extension point 2: Differences on base types are logically related.\n    u\u229fv\u2248u\u229dv-base : \u2200 \u03b9 {u v : \u27e6 \u03b9 \u27e7Base} \u2192\n      implements-base \u03b9 (u \u229f\u208d \u03b9 \u208e v) (\u27e6diff-base\u27e7 \u03b9 u v)\n\n    -- Extension point 3: Lemma 3.1 for base types.\n    carry-over-base : \u2200 {\u03b9}\n      {v : \u27e6 \u03b9 \u27e7Base}\n      (\u0394v : \u0394\u208d \u03b9 \u208e v)\n      {\u0394v\u2032 : \u27e6 \u0394Base \u03b9 \u27e7Base} (\u0394v\u2248\u0394v\u2032 : implements-base \u03b9 \u0394v \u0394v\u2032) \u2192\n        v \u229e\u208d base \u03b9 \u208e \u0394v \u2261 v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v\u2032\n\n  ------------------------\n  -- Logical relation \u2248 --\n  ------------------------\n\n  infix 4 implements\n  syntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\n  implements : \u2200 \u03c4 {v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\n  implements (base \u03b9) \u0394f \u0394f\u2032 = implements-base \u03b9 \u0394f \u0394f\u2032\n  implements (\u03c3 \u21d2 \u03c4) {f} \u0394f \u0394f\u2032 =\n    (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w)\n    (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 {w} \u0394w \u0394w\u2032) \u2192\n    implements \u03c4 {f w} (call-change {\u03c3} {\u03c4} \u0394f w \u0394w) (\u0394f\u2032 w \u0394w\u2032)\n\n  infix 4 _\u2248_\n  _\u2248_ : \u2200 {\u03c4 v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2248_ {\u03c4} {v} = implements \u03c4 {v}\n\n  data implements-env : \u2200 \u0393 \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 Set where\n    \u2205 : implements-env \u2205 {\u2205} \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1 dv d\u03c1 v\u2032 \u03c1\u2032} \u2192\n      (dv\u2248v\u2032 : implements \u03c4 {v} dv v\u2032) \u2192\n      (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 {\u03c1} d\u03c1 \u03c1\u2032) \u2192\n      implements-env (\u03c4 \u2022 \u0393) {v \u2022 \u03c1} (dv \u2022 d\u03c1) (v\u2032 \u2022 \u03c1\u2032)\n\n  ----------------\n  -- carry-over --\n  ----------------\n\n  -- This is lemma 3.10.\n  carry-over : \u2200 {\u03c4}\n    {v : \u27e6 \u03c4 \u27e7}\n    (\u0394v : \u0394\u208d \u03c4 \u208e v)\n    {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7} (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) \u2192\n      after\u208d \u03c4 \u208e \u0394v \u2261 before\u208d \u03c4 \u208e \u0394v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394v\u2032\n\n  u\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    u \u229f\u208d \u03c4 \u208e v \u2248\u208d \u03c4 \u208e u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  u\u229fv\u2248u\u229dv {base \u03b9} {u} {v} = u\u229fv\u2248u\u229dv-base \u03b9 {u} {v}\n  u\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n    result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w) \u2192\n      (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192 \u0394w \u2248\u208d \u03c3 \u208e \u0394w\u2032 \u2192\n        (g (after\u208d \u03c3 \u208e \u0394w) \u229f\u208d \u03c4 \u208e f (before\u208d \u03c3 \u208e \u0394w)) \u2248\u208d \u03c4 \u208e g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 f (before\u208d \u03c3 \u208e \u0394w)\n    result w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032\n      rewrite carry-over {\u03c3} \u0394w \u0394w\u2248\u0394w\u2032 =\n      u\u229fv\u2248u\u229dv {\u03c4} {g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032)} {f (before\u208d \u03c3 \u208e \u0394w)}\n\n  carry-over {base \u03b9} \u0394v \u0394v\u2248\u0394v\u2032 = carry-over-base \u0394v \u0394v\u2248\u0394v\u2032\n  carry-over {\u03c3 \u21d2 \u03c4} {f} \u0394f {\u0394f\u2032} \u0394f\u2248\u0394f\u2032 =\n    ext (\u03bb v \u2192\n      carry-over {\u03c4} {f v} (call-change {\u03c3} {\u03c4} \u0394f v (nil\u208d \u03c3 \u208e v))\n        {\u0394f\u2032 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)}\n        (\u0394f\u2248\u0394f\u2032 v (nil\u208d \u03c3 \u208e v) (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) ( u\u229fv\u2248u\u229dv {\u03c3} {v} {v})))\n\n  -- A property relating `alternate` and the subcontext relation \u0393\u227c\u0394\u0393\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\n  -- A specialization of the soundness of weakening\n  \u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 fit t \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6fit\u27e7 t \u03c1 d\u03c1 =\n    trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)) (sym (weaken-sound t _))\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Syntax.Term as Term\nimport Parametric.Denotation.Value as Value\nimport Parametric.Denotation.Evaluation as Evaluation\nimport Parametric.Change.Validity as Validity\nimport Parametric.Change.Specification as Specification\nimport Parametric.Change.Type as ChangeType\nimport Parametric.Change.Value as ChangeValue\nimport Parametric.Change.Derive as Derive\n\nmodule Parametric.Change.Implementation\n    {Base : Type.Structure}\n    (Const : Term.Structure Base)\n    (\u27e6_\u27e7Base : Value.Structure Base)\n    (\u27e6_\u27e7Const : Evaluation.Structure Const \u27e6_\u27e7Base)\n    (\u0394Base : ChangeType.Structure Base)\n    (validity-structure : Validity.Structure \u27e6_\u27e7Base)\n    (specification-structure : Specification.Structure\n       Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure)\n    (\u27e6apply-base\u27e7 : ChangeValue.ApplyStructure Const \u27e6_\u27e7Base \u0394Base)\n    (\u27e6diff-base\u27e7 : ChangeValue.DiffStructure Const \u27e6_\u27e7Base \u0394Base)\n    (derive-const : Derive.Structure Const \u0394Base)\n  where\n\nopen Type.Structure Base\nopen Term.Structure Base Const\nopen Value.Structure Base \u27e6_\u27e7Base\nopen Evaluation.Structure Const \u27e6_\u27e7Base \u27e6_\u27e7Const\nopen Validity.Structure \u27e6_\u27e7Base validity-structure\nopen Specification.Structure\n  Const \u27e6_\u27e7Base \u27e6_\u27e7Const validity-structure specification-structure\nopen ChangeType.Structure Base \u0394Base\nopen ChangeValue.Structure Const \u27e6_\u27e7Base \u0394Base \u27e6apply-base\u27e7 \u27e6diff-base\u27e7\nopen Derive.Structure Const \u0394Base derive-const\n\nopen import Base.Denotation.Notation\n\n-- Notions of programs being implementations of specifications\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\n\nrecord Structure : Set\u2081 where\n\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    implements-base : \u2200 \u03b9 {v : \u27e6 \u03b9 \u27e7Base} \u2192 \u0394\u208d \u03b9 \u208e v \u2192 \u27e6 \u0394Base \u03b9 \u27e7Base \u2192 Set\n    u\u229fv\u2248u\u229dv-base : \u2200 \u03b9 {u v : \u27e6 \u03b9 \u27e7Base} \u2192\n      implements-base \u03b9 (u \u229f\u208d \u03b9 \u208e v) (\u27e6diff-base\u27e7 \u03b9 u v)\n    carry-over-base : \u2200 {\u03b9}\n      {v : \u27e6 \u03b9 \u27e7Base}\n      (\u0394v : \u0394\u208d \u03b9 \u208e v)\n      {\u0394v\u2032 : \u27e6 \u0394Base \u03b9 \u27e7Base} (\u0394v\u2248\u0394v\u2032 : implements-base \u03b9 \u0394v \u0394v\u2032) \u2192\n        v \u229e\u208d base \u03b9 \u208e \u0394v \u2261 v \u27e6\u2295\u208d base \u03b9 \u208e\u27e7 \u0394v\u2032\n\n  ------------------------\n  -- Logical relation \u2248 --\n  ------------------------\n\n  infix 4 implements\n  syntax implements \u03c4 u v = u \u2248\u208d \u03c4 \u208e v\n  implements : \u2200 \u03c4 {v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n\n  implements (base \u03b9) \u0394f \u0394f\u2032 = implements-base \u03b9 \u0394f \u0394f\u2032\n  implements (\u03c3 \u21d2 \u03c4) {f} \u0394f \u0394f\u2032 =\n    (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w)\n    (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) (\u0394w\u2248\u0394w\u2032 : implements \u03c3 {w} \u0394w \u0394w\u2032) \u2192\n    implements \u03c4 {f w} (call-change {\u03c3} {\u03c4} \u0394f w \u0394w) (\u0394f\u2032 w \u0394w\u2032)\n\n  infix 4 _\u2248_\n  _\u2248_ : \u2200 {\u03c4 v} \u2192 \u0394\u208d \u03c4 \u208e v \u2192 \u27e6 \u0394Type \u03c4 \u27e7 \u2192 Set\n  _\u2248_ {\u03c4} {v} = implements \u03c4 {v}\n\n  data implements-env : \u2200 \u0393 \u2192 {\u03c1 : \u27e6 \u0393 \u27e7} (d\u03c1 : \u0394\u208d \u0393 \u208e \u03c1) \u2192 \u27e6 mapContext \u0394Type \u0393 \u27e7 \u2192 Set where\n    \u2205 : implements-env \u2205 {\u2205} \u2205 \u2205\n    _\u2022_ : \u2200 {\u03c4 \u0393 v \u03c1 dv d\u03c1 v\u2032 \u03c1\u2032} \u2192\n      (dv\u2248v\u2032 : implements \u03c4 {v} dv v\u2032) \u2192\n      (d\u03c1\u2248\u03c1\u2032 : implements-env \u0393 {\u03c1} d\u03c1 \u03c1\u2032) \u2192\n      implements-env (\u03c4 \u2022 \u0393) {v \u2022 \u03c1} (dv \u2022 d\u03c1) (v\u2032 \u2022 \u03c1\u2032)\n\n  ----------------\n  -- carry-over --\n  ----------------\n\n  -- If a program implements a specification, then certain things\n  -- proven about the specification carry over to the programs.\n  carry-over : \u2200 {\u03c4}\n    {v : \u27e6 \u03c4 \u27e7}\n    (\u0394v : \u0394\u208d \u03c4 \u208e v)\n    {\u0394v\u2032 : \u27e6 \u0394Type \u03c4 \u27e7} (\u0394v\u2248\u0394v\u2032 : \u0394v \u2248\u208d \u03c4 \u208e \u0394v\u2032) \u2192\n      after\u208d \u03c4 \u208e \u0394v \u2261 before\u208d \u03c4 \u208e \u0394v \u27e6\u2295\u208d \u03c4 \u208e\u27e7 \u0394v\u2032\n\n  u\u229fv\u2248u\u229dv : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    u \u229f\u208d \u03c4 \u208e v \u2248\u208d \u03c4 \u208e u \u27e6\u229d\u208d \u03c4 \u208e\u27e7 v\n\n  u\u229fv\u2248u\u229dv {base \u03b9} {u} {v} = u\u229fv\u2248u\u229dv-base \u03b9 {u} {v}\n  u\u229fv\u2248u\u229dv {\u03c3 \u21d2 \u03c4} {g} {f} = result where\n    result : (w : \u27e6 \u03c3 \u27e7) (\u0394w : \u0394\u208d \u03c3 \u208e w) \u2192\n      (\u0394w\u2032 : \u27e6 \u0394Type \u03c3 \u27e7) \u2192 \u0394w \u2248\u208d \u03c3 \u208e \u0394w\u2032 \u2192\n        (g (after\u208d \u03c3 \u208e \u0394w) \u229f\u208d \u03c4 \u208e f (before\u208d \u03c3 \u208e \u0394w)) \u2248\u208d \u03c4 \u208e g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032) \u27e6\u229d\u208d \u03c4 \u208e\u27e7 f (before\u208d \u03c3 \u208e \u0394w)\n    result w \u0394w \u0394w\u2032 \u0394w\u2248\u0394w\u2032\n      rewrite carry-over {\u03c3} \u0394w \u0394w\u2248\u0394w\u2032 =\n      u\u229fv\u2248u\u229dv {\u03c4} {g (before\u208d \u03c3 \u208e \u0394w \u27e6\u2295\u208d \u03c3 \u208e\u27e7 \u0394w\u2032)} {f (before\u208d \u03c3 \u208e \u0394w)}\n\n  carry-over {base \u03b9} \u0394v \u0394v\u2248\u0394v\u2032 = carry-over-base \u0394v \u0394v\u2248\u0394v\u2032\n  carry-over {\u03c3 \u21d2 \u03c4} {f} \u0394f {\u0394f\u2032} \u0394f\u2248\u0394f\u2032 =\n    ext (\u03bb v \u2192\n      carry-over {\u03c4} {f v} (call-change {\u03c3} {\u03c4} \u0394f v (nil\u208d \u03c3 \u208e v))\n        {\u0394f\u2032 v (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v)}\n        (\u0394f\u2248\u0394f\u2032 v (nil\u208d \u03c3 \u208e v) (v \u27e6\u229d\u208d \u03c3 \u208e\u27e7 v) ( u\u229fv\u2248u\u229dv {\u03c3} {v} {v})))\n\n  -- A property relating `alternate` and the subcontext relation \u0393\u227c\u0394\u0393\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 = refl\n  \u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)\n\n  -- A specialization of the soundness of weakening\n  \u27e6fit\u27e7 : \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4) (\u03c1 : \u27e6 \u0393 \u27e7) (d\u03c1 : \u27e6 mapContext \u0394Type \u0393 \u27e7) \u2192\n    \u27e6 t \u27e7 \u03c1 \u2261 \u27e6 fit t \u27e7 (alternate \u03c1 d\u03c1)\n  \u27e6fit\u27e7 t \u03c1 d\u03c1 =\n    trans (cong \u27e6 t \u27e7 (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1)) (sym (weaken-sound t _))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"14cfac67791f75bdf456a7774da62935891a6640","subject":"Add some more ways of computing Strategy","message":"Add some more ways of computing Strategy\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Strategy.agda","new_file":"Control\/Strategy.agda","new_contents":"module Control.Strategy where\n\nopen import Data.Nat\nopen import Function.NP\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Data.List as List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f)\n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  module Repeat {I O : \u2605} where\n\n    ind : (I \u2192 M O) \u2192 List I \u2192 M (List O)\n    ind g [] = done []\n    ind g (x \u2237 xs) = g x >>= (\u03bb o \u2192 map (_\u2237_ o) (ind g xs))\n\n    module _ (Oracle : (q : Q) \u2192 R q)(g : I \u2192 M O) where\n      run-ind : \u2200 xs \u2192 run Oracle (ind g xs) \u2261 List.map (run Oracle \u2218 g) xs\n      run-ind [] = refl\n      run-ind (x \u2237 xs) = run->>= Oracle (\u03bb o \u2192 map (_\u2237_ o) (ind g xs)) (g x)\n                       \u2219 run-map Oracle (_\u2237_ (run Oracle (g x))) (ind g xs)\n                       \u2219 ap (_\u2237_ (run Oracle (g x))) (run-ind xs)\n\n  module RepeatIndex {O : \u2605} where\n\n    map-list : (\u2115 \u2192 (q : Q) \u2192 R q \u2192 O) \u2192 List Q \u2192 M (List O)\n    map-list _ [] = done []\n    map-list f (x \u2237 xs) = ask x \u03bb r \u2192 map (_\u2237_ (f 0 x r)) (map-list (f \u2218 suc) xs)\n\n    mapIndex : {A B : Set} \u2192 (\u2115 \u2192 A \u2192 B) \u2192 List A \u2192 List B\n    mapIndex f []       = []\n    mapIndex f (x \u2237 xs) = f 0 x \u2237 mapIndex (f \u2218 suc) xs\n\n    module _ (Oracle : (q : Q) \u2192 R q) where\n      run-map-list : \u2200 xs (g : \u2115 \u2192 (q : Q) \u2192 R q \u2192 O) \u2192 run Oracle (map-list g xs) \u2261 mapIndex (\u03bb i q \u2192 g i q (Oracle q)) xs\n      run-map-list [] _       = refl\n      run-map-list (x \u2237 xs) g = run-map Oracle (_\u2237_ (g 0 x (Oracle x))) (map-list (g \u2218 suc) xs) \u2219 ap (_\u2237_ (g 0 x (Oracle x))) (run-map-list xs (g \u2218 suc))\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n\n  private\n    Transcript = List (\u03a3 Q R)\n  module TranscriptRun (Oracle : Transcript \u2192 \u03a0 Q R){A} where\n    runT : M A \u2192 Transcript \u2192 A \u00d7 Transcript\n    runT (ask q? cont) t = let r = Oracle t q? in runT (cont r) ((q? , r) \u2237 t)\n    runT (done x)      t = x , t\n\n    open StatefulRun\n    -- runT is runS where the state is the transcript and the Oracle cannot change it\n    runT-runS : \u2200 (str : M A) t \u2192 runT str t \u2261 runS (\u03bb q s \u2192 let r = Oracle s q in r , (q , r) \u2237 s) str t\n    runT-runS (ask q? cont) t = let r = Oracle t q? in runT-runS (cont r) ((q? , r) \u2237 t)\n    runT-runS (done x)      t = refl\n\n  module TranscriptConstRun (Oracle : \u03a0 Q R){A} where\n    open TranscriptRun\n    runT-run : \u2200 (str : M A) {t} \u2192 proj\u2081 (runT (const Oracle) str t) \u2261 run Oracle str\n    runT-run (ask q? cont) = let r = Oracle q? in runT-run (cont r)\n    runT-run (done x) = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"module Control.Strategy where\n\nopen import Function.NP\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Data.Product hiding (map)\nopen import Data.List using (List; []; _\u2237_)\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 (q? : Q) {k\u2080 k\u2081} (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : (q : Q) \u2192 E (R q)) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : (q : Q) \u2192 R q) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f)\n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n\n  private\n    State : (S A : \u2605) \u2192 \u2605\n    State S A = S \u2192 A \u00d7 S\n\n  -- redundant since we have EffectfulRun, but...\n  module StatefulRun {S : \u2605} (Oracle : (q : Q) \u2192 State S (R q)) where\n    runS : \u2200 {A} \u2192 M A \u2192 State S A\n    runS (ask q? cont) s = case Oracle q? s of \u03bb { (x , s') \u2192 runS (cont x) s' }\n    runS (done x)      s = x , s\n\n    evalS : \u2200 {A} \u2192 M A \u2192 S \u2192 A\n    evalS x s = proj\u2081 (runS x s)\n\n    execS : \u2200 {A} \u2192 M A \u2192 S \u2192 S\n    execS x s = proj\u2082 (runS x s)\n\n  private\n    Transcript = List (\u03a3 Q R)\n  module TranscriptRun (Oracle : Transcript \u2192 \u03a0 Q R){A} where\n    runT : M A \u2192 Transcript \u2192 A \u00d7 Transcript\n    runT (ask q? cont) t = let r = Oracle t q? in runT (cont r) ((q? , r) \u2237 t)\n    runT (done x)      t = x , t\n\n    open StatefulRun\n    -- runT is runS where the state is the transcript and the Oracle cannot change it\n    runT-runS : \u2200 (str : M A) t \u2192 runT str t \u2261 runS (\u03bb q s \u2192 let r = Oracle s q in r , (q , r) \u2237 s) str t\n    runT-runS (ask q? cont) t = let r = Oracle t q? in runT-runS (cont r) ((q? , r) \u2237 t)\n    runT-runS (done x)      t = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b5fbfc78f94b04b11521f7dd729d9876d9ab0764","subject":"Added some more properties about Finable","message":"Added some more properties about Finable\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-properties.agda","new_file":"sum-properties.agda","new_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Empty using (\u22a5)\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Data.Product renaming (map to pmap)\nopen import Data.Sum\nopen import Data.Unit using (\u22a4)\n\nopen import Function.NP\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\ninfix 4 _\u2248_\n\nrecord _\u2248_ {A B} (\u03bcA : SumProp A)(\u03bcB : SumProp B) : Set where\n  constructor mk\n  field\n    iso : A \u2194 B\n  from : B \u2192 A\n  from x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  to : A \u2192 B\n  to x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  from-inj : Injective from\n  from-inj = Inv.Inverse.injective (Inv.sym iso)\n\n  to-inj : Injective to\n  to-inj = Inv.Inverse.injective iso\n\n  field\n    sums-ok : \u2200 f \u2192 sum \u03bcA f \u2261 sum \u03bcB (f \u2218 from)\n\n  sums-ok' : \u2200 f \u2192 sum \u03bcB f \u2261 sum \u03bcA (f \u2218 to)\n  sums-ok' f = sum \u03bcB f\n             \u2261\u27e8 sum-ext \u03bcB (\u2261.cong f \u2218 \u2261.sym \u2218 Inv.Inverse.right-inverse-of iso) \u27e9\n               sum \u03bcB (f \u2218 to \u2218 from)\n             \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 to)) \u27e9\n               sum \u03bcA (f \u2218 to)\n             \u220e\n    where open \u2261.\u2261-Reasoning\n\n\u2248-refl : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u03bcA \u2248 \u03bcA\n\u2248-refl \u03bcA = mk Inv.id (\u03bb f \u2192 \u2261.refl)\n\n\u2248-id : \u2200 {A} {\u03bcA : SumProp A} \u2192 \u03bcA \u2248 \u03bcA\n\u2248-id = \u2248-refl _\n\n\u2248-sym : \u2200 {A B}{\u03bcA : SumProp A}{\u03bcB : SumProp B} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcA\n\u2248-sym A\u2248B = mk (Inv.sym iso) sums-ok'\n  where open _\u2248_ A\u2248B\n\n\u2248-trans : \u2200 {A B C}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcC \u2192 \u03bcA \u2248 \u03bcC\n\u2248-trans A\u2248B B\u2248C = mk (iso B\u2248C Inv.\u2218 iso A\u2248B) (\u03bb f \u2192 \u2261.trans (sums-ok A\u2248B f) (sums-ok B\u2248C (f \u2218 from A\u2248B)))\n  where open _\u2248_\n\ninfix 2 _\u2248\u220e\ninfixr 2 _\u2248\u27e8_\u27e9_\n\n_\u2248\u220e : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u03bcA \u2248 \u03bcA\n_\u2248\u220e = \u2248-refl\n\n_\u2248\u27e8_\u27e9_ : \u2200 {A B C} (\u03bcA : SumProp A){\u03bcB : SumProp B}{\u03bcC : SumProp C} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcC \u2192 \u03bcA \u2248 \u03bcC\n_ \u2248\u27e8 A\u2248B \u27e9 B\u2248C = \u2248-trans A\u2248B B\u2248C\n\nFin0\u2248\u22a4 : \u03bcFinSuc zero \u2248 \u03bc\u22a4\nFin0\u2248\u22a4 = mk iso sums-ok where\n  open import Relation.Binary.Sum\n  iso : _\n  iso = (A\u228e\u22a5\u2194A Inv.\u2218 Inv.id \u228e-cong Fin0\u2194\u22a5) Inv.\u2218 Fin\u2218suc\u2194\u22a4\u228eFin\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u22a4+Fin : \u2200 {n} \u2192 \u03bc\u22a4 +\u03bc \u03bcFinSuc n \u2248 \u03bcFinSuc (suc n)\n\u22a4+Fin {n} = mk iso sums-ok where\n  iso : _\n  iso = Inv.sym Fin\u2218suc\u2194\u22a4\u228eFin\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u22a4\u00d7A\u2248A : \u2200 {A}{\u03bcA : SumProp A} \u2192 \u03bc\u22a4 \u00d7\u03bc \u03bcA \u2248 \u03bcA\n\u22a4\u00d7A\u2248A = mk iso sums-ok where\n  iso : _\n  iso = \u22a4\u00d7A\u2194A\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u03bcFinPres : \u2200 {m n} \u2192 m \u2261 n \u2192 \u03bcFinSuc m \u2248 \u03bcFinSuc n\n\u03bcFinPres eq rewrite eq = \u2248-refl _\n\n_+\u03bc-cong_ : \u2200 {A B C D}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C}{\u03bcD : SumProp D}\n          \u2192 \u03bcA \u2248 \u03bcC \u2192 \u03bcB \u2248 \u03bcD \u2192 \u03bcA +\u03bc \u03bcB \u2248 \u03bcC +\u03bc \u03bcD\nA\u2248C +\u03bc-cong B\u2248D = mk iso sums-ok where\n  open import Relation.Binary.Sum\n  iso : _\n  iso = (_\u2248_.iso A\u2248C) \u228e-cong (_\u2248_.iso B\u2248D)\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.cong\u2082 _+_ (_\u2248_.sums-ok A\u2248C (f \u2218 inj\u2081)) (_\u2248_.sums-ok B\u2248D (f \u2218 inj\u2082))\n\n+\u03bc-assoc : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n         \u2192 (\u03bcA +\u03bc \u03bcB) +\u03bc \u03bcC \u2248 \u03bcA +\u03bc (\u03bcB +\u03bc \u03bcC)\n+\u03bc-assoc \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u228e-CMon.assoc _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2115\u00b0.+-assoc (sum \u03bcA (f \u2218 inj\u2081 \u2218 inj\u2081)) (sum \u03bcB (f \u2218 inj\u2081 \u2218 inj\u2082)) (sum \u03bcC (f \u2218 inj\u2082))\n\n+\u03bc-comm : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B)\n        \u2192 \u03bcA +\u03bc \u03bcB \u2248 \u03bcB +\u03bc \u03bcA\n+\u03bc-comm \u03bcA \u03bcB = mk iso sums-ok where\n  iso : _\n  iso = \u228e-CMon.comm _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2115\u00b0.+-comm (sum \u03bcA (f \u2218 inj\u2081)) (sum \u03bcB (f \u2218 inj\u2082))\n\n_\u00d7\u03bc-cong_ :  \u2200 {A B C D}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C}{\u03bcD : SumProp D}\n          \u2192 \u03bcA \u2248 \u03bcC \u2192 \u03bcB \u2248 \u03bcD \u2192 \u03bcA \u00d7\u03bc \u03bcB \u2248 \u03bcC \u00d7\u03bc \u03bcD\n_\u00d7\u03bc-cong_ {\u03bcA = \u03bcA}{\u03bcD = \u03bcD} A\u2248C B\u2248D = mk iso sums-ok where\n  open import Relation.Binary.Product.Pointwise\n  iso : _\n  iso = _\u2248_.iso A\u2248C \u00d7-cong _\u2248_.iso B\u2248D\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.trans (sum-ext \u03bcA (_\u2248_.sums-ok B\u2248D \u2218 curry f))\n                      (_\u2248_.sums-ok A\u2248C (\u03bb a \u2192 sum \u03bcD (curry f a \u2218 (_\u2248_.from B\u2248D))))\n\n\u00d7\u03bc-assoc : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n         \u2192 (\u03bcA \u00d7\u03bc \u03bcB) \u00d7\u03bc \u03bcC \u2248 \u03bcA \u00d7\u03bc (\u03bcB \u00d7\u03bc \u03bcC)\n\u00d7\u03bc-assoc \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u00d7-CMon.assoc _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u00d7\u03bc-comm : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B)\n        \u2192 \u03bcA \u00d7\u03bc \u03bcB \u2248 \u03bcB \u00d7\u03bc \u03bcA\n\u00d7\u03bc-comm \u03bcA \u03bcB = mk iso sums-ok where\n  iso : _\n  iso = \u00d7-CMon.comm _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = sum-swap \u03bcA \u03bcB (curry f)\n\n\u00d7+-distrib : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n           \u2192 (\u03bcA +\u03bc \u03bcB) \u00d7\u03bc \u03bcC \u2248 (\u03bcA \u00d7\u03bc \u03bcC) +\u03bc (\u03bcB \u00d7\u03bc \u03bcC)\n\u00d7+-distrib \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u00d7\u228e\u00b0.distrib\u02b3 _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n+-\u2248 : \u2200 m n \u2192 (\u03bcFinSuc m) +\u03bc (\u03bcFinSuc n) \u2248 \u03bcFinSuc (m + suc n)\n+-\u2248 zero n    = \u03bcFinSuc zero +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 Fin0\u2248\u22a4 +\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                \u03bc\u22a4 +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u22a4+Fin \u27e9\n                \u03bcFinSuc (suc n)\n              \u2248\u220e\n+-\u2248 (suc m) n = \u03bcFinSuc (suc m) +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u2248-sym \u22a4+Fin +\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 +\u03bc \u03bcFinSuc m) +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 +\u03bc-assoc \u03bc\u22a4 (\u03bcFinSuc m) (\u03bcFinSuc n) \u27e9\n                \u03bc\u22a4 +\u03bc (\u03bcFinSuc m +\u03bc \u03bcFinSuc n)\n              \u2248\u27e8 \u2248-refl \u03bc\u22a4 +\u03bc-cong +-\u2248 m n \u27e9\n                \u03bc\u22a4 +\u03bc \u03bcFinSuc (m + suc n)\n              \u2248\u27e8 \u22a4+Fin \u27e9\n                \u03bcFinSuc (suc m + suc n)\n              \u2248\u220e\n\n\u00d7-\u2248 : \u2200 m n \u2192 \u03bcFinSuc m \u00d7\u03bc \u03bcFinSuc n \u2248 \u03bcFinSuc (n + m * suc n)\n\u00d7-\u2248 zero n    = \u03bcFinSuc 0 \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 Fin0\u2248\u22a4 \u00d7\u03bc-cong (\u2248-refl (\u03bcFinSuc n)) \u27e9\n                \u03bc\u22a4 \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u22a4\u00d7A\u2248A \u27e9\n                \u03bcFinSuc n\n              \u2248\u27e8 \u03bcFinPres (\u2115\u00b0.+-comm 0 n) \u27e9\n                \u03bcFinSuc (n + 0)\n              \u2248\u220e\n\u00d7-\u2248 (suc m) n = \u03bcFinSuc (suc m) \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u2248-sym \u22a4+Fin \u00d7\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 +\u03bc \u03bcFinSuc m) \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u00d7+-distrib \u03bc\u22a4 (\u03bcFinSuc m) (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 \u00d7\u03bc \u03bcFinSuc n) +\u03bc (\u03bcFinSuc m \u00d7\u03bc \u03bcFinSuc n)\n              \u2248\u27e8 \u22a4\u00d7A\u2248A {\u03bcA = \u03bcFinSuc n} +\u03bc-cong \u00d7-\u2248 m n \u27e9\n                \u03bcFinSuc n +\u03bc \u03bcFinSuc (n + m * suc n)\n              \u2248\u27e8 +-\u2248 n (n + m * suc n) \u27e9\n                \u03bcFinSuc (n + suc m * suc n)\n              \u2248\u220e\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nFinable : \u2200 {A} \u2192 SumProp A \u2192 Set\nFinable \u03bcA = \u03a3 \u2115 \u03bb FinCard \u2192 \u03bcA \u2248 \u03bcFinSuc FinCard\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = 0 , \u2248-sym Fin0\u2248\u22a4\n\nFin-Finable : \u2200 {n} \u2192 Finable (\u03bcFinSuc n)\nFin-Finable {n} = n , \u2248-refl (\u03bcFinSuc n)\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable \u03bcA \u03bcB (|A| , A\u2248) (|B| , B\u2248) = (|A| + suc |B|) ,\n  ( \u03bcA +\u03bc \u03bcB\n  \u2248\u27e8 A\u2248 +\u03bc-cong B\u2248 \u27e9\n    \u03bcFinSuc |A| +\u03bc \u03bcFinSuc |B|\n  \u2248\u27e8 +-\u2248 |A| |B| \u27e9\n    \u03bcFinSuc (|A| + suc |B|) \u2248\u220e)\n\n\u00d7-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA \u00d7\u03bc \u03bcB)\n\u00d7-Finable \u03bcA \u03bcB (|A| , A\u2248) (|B| , B\u2248) = (|B| + |A| * suc |B|) ,\n  ( \u03bcA \u00d7\u03bc \u03bcB\n  \u2248\u27e8 A\u2248 \u00d7\u03bc-cong B\u2248 \u27e9\n    \u03bcFinSuc |A| \u00d7\u03bc \u03bcFinSuc |B|\n  \u2248\u27e8 \u00d7-\u2248 |A| |B| \u27e9\n    \u03bcFinSuc (|B| + |A| * suc |B|)\n  \u2248\u220e)\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\nStableUnder\u2248 : \u2200 {A B} (\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 \u03bcA \u2248 \u03bcB\n             \u2192 StableUnderInjection \u03bcA \u2192 StableUnderInjection \u03bcB\nStableUnder\u2248 \u03bcA \u03bcB \u03bcA\u2248\u03bcB \u03bcA-SUI f p p-inj\n  = sum \u03bcB f\n  \u2261\u27e8 sums-ok' f \u27e9\n    sum \u03bcA (f \u2218 to)\n  \u2261\u27e8 \u03bcA-SUI (f \u2218 to) (from \u2218 p \u2218 to) (to-inj \u2218 p-inj \u2218 from-inj) \u27e9\n    sum \u03bcA (f \u2218 to \u2218 from \u2218 p \u2218 to)\n  \u2261\u27e8 \u2261.sym (sums-ok' (f \u2218 to \u2218 from \u2218 p)) \u27e9\n    sum \u03bcB (f \u2218 to \u2218 from \u2218 p)\n  \u2261\u27e8 sum-ext \u03bcB (\u2261.cong f \u2218 Inv.Inverse.right-inverse-of iso \u2218 p) \u27e9\n    sum \u03bcB (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open _\u2248_ \u03bcA\u2248\u03bcB\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA (FinCard , A\u2248Fin)\n  = StableUnder\u2248 (\u03bcFinSuc FinCard) \u03bcA (\u2248-sym A\u2248Fin) \u03bcFinSUI\n\n-- -}\n","old_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Empty using (\u22a5)\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Data.Product renaming (map to pmap)\nopen import Data.Sum\nopen import Data.Unit using (\u22a4)\n\nopen import Function.NP\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\ninfix 4 _\u2248_\n\nrecord _\u2248_ {A B} (\u03bcA : SumProp A)(\u03bcB : SumProp B) : Set where\n  constructor mk\n  field\n    iso : A \u2194 B\n  from : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 B \u2192 A\n  from iso x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  to : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 A \u2192 B\n  to iso x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  from-inj : \u2200 {A B : Set} (eq : A \u2194 B) \u2192 Injective (from eq)\n  from-inj eq = Inv.Inverse.injective (Inv.sym eq)\n\n  to-inj : \u2200 {A B : Set} (eq : A \u2194 B) \u2192 Injective (to eq)\n  to-inj eq = Inv.Inverse.injective eq\n\n  field\n    sums-ok : \u2200 f \u2192 sum \u03bcA f \u2261 sum \u03bcB (f \u2218 from iso)\n\n  sums-ok' : \u2200 f \u2192 sum \u03bcB f \u2261 sum \u03bcA (f \u2218 to iso)\n  sums-ok' f = sum \u03bcB f\n             \u2261\u27e8 sum-ext \u03bcB (\u2261.cong f \u2218 \u2261.sym \u2218 Inv.Inverse.right-inverse-of iso) \u27e9\n               sum \u03bcB (f \u2218 to iso \u2218 from iso)\n             \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 to iso)) \u27e9\n               sum \u03bcA (f \u2218 to iso)\n             \u220e\n    where open \u2261.\u2261-Reasoning\n\n\u2248-refl : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u03bcA \u2248 \u03bcA\n\u2248-refl \u03bcA = mk Inv.id (\u03bb f \u2192 \u2261.refl)\n\n\u2248-id : \u2200 {A} {\u03bcA : SumProp A} \u2192 \u03bcA \u2248 \u03bcA\n\u2248-id = \u2248-refl _\n\n\u2248-sym : \u2200 {A B}{\u03bcA : SumProp A}{\u03bcB : SumProp B} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcA\n\u2248-sym A\u2248B = mk (Inv.sym iso) sums-ok'\n  where open _\u2248_ A\u2248B\n\n\u2248-trans : \u2200 {A B C}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcC \u2192 \u03bcA \u2248 \u03bcC\n\u2248-trans A\u2248B B\u2248C = mk (iso B\u2248C Inv.\u2218 iso A\u2248B) (\u03bb f \u2192 \u2261.trans (sums-ok A\u2248B f) (sums-ok B\u2248C (f \u2218 from A\u2248B (iso A\u2248B))))\n  where open _\u2248_\n\ninfix 2 _\u2248\u220e\ninfixr 2 _\u2248\u27e8_\u27e9_\n\n_\u2248\u220e : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u03bcA \u2248 \u03bcA\n_\u2248\u220e = \u2248-refl\n\n_\u2248\u27e8_\u27e9_ : \u2200 {A B C} (\u03bcA : SumProp A){\u03bcB : SumProp B}{\u03bcC : SumProp C} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcC \u2192 \u03bcA \u2248 \u03bcC\n_ \u2248\u27e8 A\u2248B \u27e9 B\u2248C = \u2248-trans A\u2248B B\u2248C\n\nFin0\u2248\u22a4 : \u03bcFinSuc zero \u2248 \u03bc\u22a4\nFin0\u2248\u22a4 = mk iso sums-ok where\n  open import Relation.Binary.Sum\n  iso : _\n  iso = (A\u228e\u22a5\u2194A Inv.\u2218 Inv.id \u228e-cong Fin0\u2194\u22a5) Inv.\u2218 Fin\u2218suc\u2194\u22a4\u228eFin\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok = \u03bb x \u2192 \u2261.refl\n\n\u22a4+Fin : \u2200 {n} \u2192 \u03bc\u22a4 +\u03bc \u03bcFinSuc n \u2248 \u03bcFinSuc (suc n)\n\u22a4+Fin {n} = mk iso sums-ok where\n  iso : _\n  iso = Inv.sym Fin\u2218suc\u2194\u22a4\u228eFin\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u22a4\u00d7A\u2248A : \u2200 {A}{\u03bcA : SumProp A} \u2192 \u03bc\u22a4 \u00d7\u03bc \u03bcA \u2248 \u03bcA\n\u22a4\u00d7A\u2248A = mk iso sums-ok where\n  iso : _\n  iso = \u22a4\u00d7A\u2194A\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u03bcFinPres : \u2200 {m n} \u2192 m \u2261 n \u2192 \u03bcFinSuc m \u2248 \u03bcFinSuc n\n\u03bcFinPres eq rewrite eq = \u2248-refl _\n\n_+\u03bc-cong_ : \u2200 {A B C D}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C}{\u03bcD : SumProp D}\n          \u2192 \u03bcA \u2248 \u03bcC \u2192 \u03bcB \u2248 \u03bcD \u2192 \u03bcA +\u03bc \u03bcB \u2248 \u03bcC +\u03bc \u03bcD\nA\u2248C +\u03bc-cong B\u2248D = mk iso sums-ok where\n  open import Relation.Binary.Sum\n  iso : _\n  iso = (_\u2248_.iso A\u2248C) \u228e-cong (_\u2248_.iso B\u2248D)\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.cong\u2082 _+_ (_\u2248_.sums-ok A\u2248C (f \u2218 inj\u2081)) (_\u2248_.sums-ok B\u2248D (f \u2218 inj\u2082))\n\n+\u03bc-assoc : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n         \u2192 (\u03bcA +\u03bc \u03bcB) +\u03bc \u03bcC \u2248 \u03bcA +\u03bc (\u03bcB +\u03bc \u03bcC)\n+\u03bc-assoc \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u228e-CMon.assoc _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2115\u00b0.+-assoc (sum \u03bcA (f \u2218 inj\u2081 \u2218 inj\u2081)) (sum \u03bcB (f \u2218 inj\u2081 \u2218 inj\u2082)) (sum \u03bcC (f \u2218 inj\u2082))\n\n_\u00d7\u03bc-cong_ :  \u2200 {A B C D}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C}{\u03bcD : SumProp D}\n          \u2192 \u03bcA \u2248 \u03bcC \u2192 \u03bcB \u2248 \u03bcD \u2192 \u03bcA \u00d7\u03bc \u03bcB \u2248 \u03bcC \u00d7\u03bc \u03bcD\n_\u00d7\u03bc-cong_ {\u03bcA = \u03bcA}{\u03bcD = \u03bcD} A\u2248C B\u2248D = mk iso sums-ok where\n  open import Relation.Binary.Product.Pointwise\n  iso : _\n  iso = _\u2248_.iso A\u2248C \u00d7-cong _\u2248_.iso B\u2248D\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.trans (sum-ext \u03bcA (_\u2248_.sums-ok B\u2248D \u2218 curry f))\n                      (_\u2248_.sums-ok A\u2248C (\u03bb a \u2192\n            sum \u03bcD (\u03bb d \u2192 f (a , Inv.Inverse.from (_\u2248_.iso B\u2248D) \u27e8$\u27e9 d))))\n\n\u00d7+-distrib : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n           \u2192 (\u03bcA +\u03bc \u03bcB) \u00d7\u03bc \u03bcC \u2248 (\u03bcA \u00d7\u03bc \u03bcC) +\u03bc (\u03bcB \u00d7\u03bc \u03bcC)\n\u00d7+-distrib \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u00d7\u228e\u00b0.distrib\u02b3 _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n+-\u2248 : \u2200 m n \u2192 (\u03bcFinSuc m) +\u03bc (\u03bcFinSuc n) \u2248 \u03bcFinSuc (m + suc n)\n+-\u2248 zero n = \u03bcFinSuc zero +\u03bc \u03bcFinSuc n\n           \u2248\u27e8 Fin0\u2248\u22a4 +\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n             \u03bc\u22a4 +\u03bc \u03bcFinSuc n\n           \u2248\u27e8 \u22a4+Fin \u27e9\n             \u03bcFinSuc (suc n)\n           \u2248\u220e\n+-\u2248 (suc m) n = \u03bcFinSuc (suc m) +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u2248-sym \u22a4+Fin +\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 +\u03bc \u03bcFinSuc m) +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 +\u03bc-assoc \u03bc\u22a4 (\u03bcFinSuc m) (\u03bcFinSuc n) \u27e9\n                \u03bc\u22a4 +\u03bc (\u03bcFinSuc m +\u03bc \u03bcFinSuc n)\n              \u2248\u27e8 \u2248-refl \u03bc\u22a4 +\u03bc-cong +-\u2248 m n \u27e9\n                \u03bc\u22a4 +\u03bc \u03bcFinSuc (m + suc n)\n              \u2248\u27e8 \u22a4+Fin \u27e9\n                \u03bcFinSuc (suc m + suc n)\n              \u2248\u220e\n\n\u00d7-\u2248 : \u2200 m n \u2192 \u03bcFinSuc m \u00d7\u03bc \u03bcFinSuc n \u2248 \u03bcFinSuc (n + m * suc n)\n\u00d7-\u2248 zero n    = \u03bcFinSuc 0 \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 Fin0\u2248\u22a4 \u00d7\u03bc-cong (\u2248-refl (\u03bcFinSuc n)) \u27e9\n                \u03bc\u22a4 \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u22a4\u00d7A\u2248A \u27e9\n                \u03bcFinSuc n\n              \u2248\u27e8 \u03bcFinPres (\u2115\u00b0.+-comm 0 n) \u27e9\n                \u03bcFinSuc (n + 0)\n              \u2248\u220e\n\u00d7-\u2248 (suc m) n = \u03bcFinSuc (suc m) \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u2248-sym \u22a4+Fin \u00d7\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 +\u03bc \u03bcFinSuc m) \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u00d7+-distrib \u03bc\u22a4 (\u03bcFinSuc m) (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 \u00d7\u03bc \u03bcFinSuc n) +\u03bc (\u03bcFinSuc m \u00d7\u03bc \u03bcFinSuc n)\n              \u2248\u27e8 \u22a4\u00d7A\u2248A {\u03bcA = \u03bcFinSuc n} +\u03bc-cong \u00d7-\u2248 m n \u27e9\n                \u03bcFinSuc n +\u03bc \u03bcFinSuc (n + m * suc n)\n              \u2248\u27e8 +-\u2248 n (n + m * suc n) \u27e9\n                \u03bcFinSuc (n + suc m * suc n)\n              \u2248\u220e\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nFinable : \u2200 {A} \u2192 SumProp A \u2192 Set\nFinable \u03bcA = \u03a3 \u2115 \u03bb FinCard \u2192 \u03bcA \u2248 \u03bcFinSuc FinCard\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = 0 , \u2248-sym Fin0\u2248\u22a4\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable \u03bcA \u03bcB (|A| , A\u2248) (|B| , B\u2248) = (|A| + suc |B|) ,\n  ( \u03bcA +\u03bc \u03bcB\n  \u2248\u27e8 A\u2248 +\u03bc-cong B\u2248 \u27e9\n    \u03bcFinSuc |A| +\u03bc \u03bcFinSuc |B|\n  \u2248\u27e8 +-\u2248 |A| |B| \u27e9\n    \u03bcFinSuc (|A| + suc |B|) \u2248\u220e)\n\n\u00d7-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA \u00d7\u03bc \u03bcB)\n\u00d7-Finable \u03bcA \u03bcB (|A| , A\u2248) (|B| , B\u2248) = (|B| + |A| * suc |B|) ,\n  ( \u03bcA \u00d7\u03bc \u03bcB\n  \u2248\u27e8 A\u2248 \u00d7\u03bc-cong B\u2248 \u27e9\n    \u03bcFinSuc |A| \u00d7\u03bc \u03bcFinSuc |B|\n  \u2248\u27e8 \u00d7-\u2248 |A| |B| \u27e9\n    \u03bcFinSuc (|B| + |A| * suc |B|)\n  \u2248\u220e)\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\nStableUnder\u2248 : \u2200 {A B} (\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 \u03bcA \u2248 \u03bcB\n             \u2192 StableUnderInjection \u03bcA \u2192 StableUnderInjection \u03bcB\nStableUnder\u2248 \u03bcA \u03bcB \u03bcA\u2248\u03bcB \u03bcA-SUI f p p-inj\n  = sum \u03bcB f\n  \u2261\u27e8 sums-ok' f \u27e9\n    sum \u03bcA (f \u2218 to iso)\n  \u2261\u27e8 \u03bcA-SUI (f \u2218 to iso) (from iso \u2218 p \u2218 to iso) (to-inj iso \u2218 p-inj \u2218 from-inj iso) \u27e9\n    sum \u03bcA (f \u2218 to iso \u2218 from iso \u2218 p \u2218 to iso)\n  \u2261\u27e8 \u2261.sym (sums-ok' (f \u2218 to iso \u2218 from iso \u2218 p)) \u27e9\n    sum \u03bcB (f \u2218 to iso \u2218 from iso \u2218 p)\n  \u2261\u27e8 sum-ext \u03bcB (\u2261.cong f \u2218 Inv.Inverse.right-inverse-of iso \u2218 p) \u27e9\n    sum \u03bcB (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open _\u2248_ \u03bcA\u2248\u03bcB\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA (FinCard , A\u2248Fin)\n  = StableUnder\u2248 (\u03bcFinSuc FinCard) \u03bcA (\u2248-sym A\u2248Fin) \u03bcFinSUI\n\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"67fd8619f57fb7d6c2645164b204da631eb44e57","subject":"New approach to finable","message":"New approach to finable\n","repos":"crypto-agda\/crypto-agda","old_file":"sum-properties.agda","new_file":"sum-properties.agda","new_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Empty using (\u22a5)\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Data.Product renaming (map to pmap)\nopen import Data.Sum\nopen import Data.Unit using (\u22a4)\n\nopen import Function.NP\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\ninfix 4 _\u2248_\n\nrecord _\u2248_ {A B} (\u03bcA : SumProp A)(\u03bcB : SumProp B) : Set where\n  constructor mk\n  field\n    iso : A \u2194 B\n  from : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 B \u2192 A\n  from iso x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  to : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 A \u2192 B\n  to iso x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  from-inj : \u2200 {A B : Set} (eq : A \u2194 B) \u2192 Injective (from eq)\n  from-inj eq = Inv.Inverse.injective (Inv.sym eq)\n\n  to-inj : \u2200 {A B : Set} (eq : A \u2194 B) \u2192 Injective (to eq)\n  to-inj eq = Inv.Inverse.injective eq\n\n  field\n    sums-ok : \u2200 f \u2192 sum \u03bcA f \u2261 sum \u03bcB (f \u2218 from iso)\n\n  sums-ok' : \u2200 f \u2192 sum \u03bcB f \u2261 sum \u03bcA (f \u2218 to iso)\n  sums-ok' f = sum \u03bcB f\n             \u2261\u27e8 sum-ext \u03bcB (\u2261.cong f \u2218 \u2261.sym \u2218 Inv.Inverse.right-inverse-of iso) \u27e9\n               sum \u03bcB (f \u2218 to iso \u2218 from iso)\n             \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 to iso)) \u27e9\n               sum \u03bcA (f \u2218 to iso)\n             \u220e\n    where open \u2261.\u2261-Reasoning\n\n\u2248-refl : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u03bcA \u2248 \u03bcA\n\u2248-refl \u03bcA = mk Inv.id (\u03bb f \u2192 \u2261.refl)\n\n\u2248-id : \u2200 {A} {\u03bcA : SumProp A} \u2192 \u03bcA \u2248 \u03bcA\n\u2248-id = \u2248-refl _\n\n\u2248-sym : \u2200 {A B}{\u03bcA : SumProp A}{\u03bcB : SumProp B} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcA\n\u2248-sym A\u2248B = mk (Inv.sym iso) sums-ok'\n  where open _\u2248_ A\u2248B\n\n\u2248-trans : \u2200 {A B C}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcC \u2192 \u03bcA \u2248 \u03bcC\n\u2248-trans A\u2248B B\u2248C = mk (iso B\u2248C Inv.\u2218 iso A\u2248B) (\u03bb f \u2192 \u2261.trans (sums-ok A\u2248B f) (sums-ok B\u2248C (f \u2218 from A\u2248B (iso A\u2248B))))\n  where open _\u2248_\n\ninfix 2 _\u2248\u220e\ninfixr 2 _\u2248\u27e8_\u27e9_\n\n_\u2248\u220e : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u03bcA \u2248 \u03bcA\n_\u2248\u220e = \u2248-refl\n\n_\u2248\u27e8_\u27e9_ : \u2200 {A B C} (\u03bcA : SumProp A){\u03bcB : SumProp B}{\u03bcC : SumProp C} \u2192 \u03bcA \u2248 \u03bcB \u2192 \u03bcB \u2248 \u03bcC \u2192 \u03bcA \u2248 \u03bcC\n_ \u2248\u27e8 A\u2248B \u27e9 B\u2248C = \u2248-trans A\u2248B B\u2248C\n\nFin0\u2248\u22a4 : \u03bcFinSuc zero \u2248 \u03bc\u22a4\nFin0\u2248\u22a4 = mk iso sums-ok where\n  open import Relation.Binary.Sum\n  iso : _\n  iso = (A\u228e\u22a5\u2194A Inv.\u2218 Inv.id \u228e-cong Fin0\u2194\u22a5) Inv.\u2218 Fin\u2218suc\u2194\u22a4\u228eFin\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok = \u03bb x \u2192 \u2261.refl\n\n\u22a4+Fin : \u2200 {n} \u2192 \u03bc\u22a4 +\u03bc \u03bcFinSuc n \u2248 \u03bcFinSuc (suc n)\n\u22a4+Fin {n} = mk iso sums-ok where\n  iso : _\n  iso = Inv.sym Fin\u2218suc\u2194\u22a4\u228eFin\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u22a4\u00d7A\u2248A : \u2200 {A}{\u03bcA : SumProp A} \u2192 \u03bc\u22a4 \u00d7\u03bc \u03bcA \u2248 \u03bcA\n\u22a4\u00d7A\u2248A = mk iso sums-ok where\n  iso : _\n  iso = \u22a4\u00d7A\u2194A\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n\u03bcFinPres : \u2200 {m n} \u2192 m \u2261 n \u2192 \u03bcFinSuc m \u2248 \u03bcFinSuc n\n\u03bcFinPres eq rewrite eq = \u2248-refl _\n\n_+\u03bc-cong_ : \u2200 {A B C D}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C}{\u03bcD : SumProp D}\n          \u2192 \u03bcA \u2248 \u03bcC \u2192 \u03bcB \u2248 \u03bcD \u2192 \u03bcA +\u03bc \u03bcB \u2248 \u03bcC +\u03bc \u03bcD\nA\u2248C +\u03bc-cong B\u2248D = mk iso sums-ok where\n  open import Relation.Binary.Sum\n  iso : _\n  iso = (_\u2248_.iso A\u2248C) \u228e-cong (_\u2248_.iso B\u2248D)\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.cong\u2082 _+_ (_\u2248_.sums-ok A\u2248C (f \u2218 inj\u2081)) (_\u2248_.sums-ok B\u2248D (f \u2218 inj\u2082))\n\n+\u03bc-assoc : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n         \u2192 (\u03bcA +\u03bc \u03bcB) +\u03bc \u03bcC \u2248 \u03bcA +\u03bc (\u03bcB +\u03bc \u03bcC)\n+\u03bc-assoc \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u228e-CMon.assoc _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2115\u00b0.+-assoc (sum \u03bcA (f \u2218 inj\u2081 \u2218 inj\u2081)) (sum \u03bcB (f \u2218 inj\u2081 \u2218 inj\u2082)) (sum \u03bcC (f \u2218 inj\u2082))\n\n_\u00d7\u03bc-cong_ :  \u2200 {A B C D}{\u03bcA : SumProp A}{\u03bcB : SumProp B}{\u03bcC : SumProp C}{\u03bcD : SumProp D}\n          \u2192 \u03bcA \u2248 \u03bcC \u2192 \u03bcB \u2248 \u03bcD \u2192 \u03bcA \u00d7\u03bc \u03bcB \u2248 \u03bcC \u00d7\u03bc \u03bcD\n_\u00d7\u03bc-cong_ {\u03bcA = \u03bcA}{\u03bcD = \u03bcD} A\u2248C B\u2248D = mk iso sums-ok where\n  open import Relation.Binary.Product.Pointwise\n  iso : _\n  iso = _\u2248_.iso A\u2248C \u00d7-cong _\u2248_.iso B\u2248D\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.trans (sum-ext \u03bcA (_\u2248_.sums-ok B\u2248D \u2218 curry f))\n                      (_\u2248_.sums-ok A\u2248C (\u03bb a \u2192\n            sum \u03bcD (\u03bb d \u2192 f (a , Inv.Inverse.from (_\u2248_.iso B\u2248D) \u27e8$\u27e9 d))))\n\n\u00d7+-distrib : \u2200 {A B C}(\u03bcA : SumProp A)(\u03bcB : SumProp B)(\u03bcC : SumProp C)\n           \u2192 (\u03bcA +\u03bc \u03bcB) \u00d7\u03bc \u03bcC \u2248 (\u03bcA \u00d7\u03bc \u03bcC) +\u03bc (\u03bcB \u00d7\u03bc \u03bcC)\n\u00d7+-distrib \u03bcA \u03bcB \u03bcC = mk iso sums-ok where\n  iso : _\n  iso = \u00d7\u228e\u00b0.distrib\u02b3 _ _ _\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n+-\u2248 : \u2200 m n \u2192 (\u03bcFinSuc m) +\u03bc (\u03bcFinSuc n) \u2248 \u03bcFinSuc (m + suc n)\n+-\u2248 zero n = \u03bcFinSuc zero +\u03bc \u03bcFinSuc n\n           \u2248\u27e8 Fin0\u2248\u22a4 +\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n             \u03bc\u22a4 +\u03bc \u03bcFinSuc n\n           \u2248\u27e8 \u22a4+Fin \u27e9\n             \u03bcFinSuc (suc n)\n           \u2248\u220e\n+-\u2248 (suc m) n = \u03bcFinSuc (suc m) +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u2248-sym \u22a4+Fin +\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 +\u03bc \u03bcFinSuc m) +\u03bc \u03bcFinSuc n\n              \u2248\u27e8 +\u03bc-assoc \u03bc\u22a4 (\u03bcFinSuc m) (\u03bcFinSuc n) \u27e9\n                \u03bc\u22a4 +\u03bc (\u03bcFinSuc m +\u03bc \u03bcFinSuc n)\n              \u2248\u27e8 \u2248-refl \u03bc\u22a4 +\u03bc-cong +-\u2248 m n \u27e9\n                \u03bc\u22a4 +\u03bc \u03bcFinSuc (m + suc n)\n              \u2248\u27e8 \u22a4+Fin \u27e9\n                \u03bcFinSuc (suc m + suc n)\n              \u2248\u220e\n\n\u00d7-\u2248 : \u2200 m n \u2192 \u03bcFinSuc m \u00d7\u03bc \u03bcFinSuc n \u2248 \u03bcFinSuc (n + m * suc n)\n\u00d7-\u2248 zero n    = \u03bcFinSuc 0 \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 Fin0\u2248\u22a4 \u00d7\u03bc-cong (\u2248-refl (\u03bcFinSuc n)) \u27e9\n                \u03bc\u22a4 \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u22a4\u00d7A\u2248A \u27e9\n                \u03bcFinSuc n\n              \u2248\u27e8 \u03bcFinPres (\u2115\u00b0.+-comm 0 n) \u27e9\n                \u03bcFinSuc (n + 0)\n              \u2248\u220e\n\u00d7-\u2248 (suc m) n = \u03bcFinSuc (suc m) \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u2248-sym \u22a4+Fin \u00d7\u03bc-cong \u2248-refl (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 +\u03bc \u03bcFinSuc m) \u00d7\u03bc \u03bcFinSuc n\n              \u2248\u27e8 \u00d7+-distrib \u03bc\u22a4 (\u03bcFinSuc m) (\u03bcFinSuc n) \u27e9\n                (\u03bc\u22a4 \u00d7\u03bc \u03bcFinSuc n) +\u03bc (\u03bcFinSuc m \u00d7\u03bc \u03bcFinSuc n)\n              \u2248\u27e8 \u22a4\u00d7A\u2248A {\u03bcA = \u03bcFinSuc n} +\u03bc-cong \u00d7-\u2248 m n \u27e9\n                \u03bcFinSuc n +\u03bc \u03bcFinSuc (n + m * suc n)\n              \u2248\u27e8 +-\u2248 n (n + m * suc n) \u27e9\n                \u03bcFinSuc (n + suc m * suc n)\n              \u2248\u220e\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nFinable : \u2200 {A} \u2192 SumProp A \u2192 Set\nFinable \u03bcA = \u03a3 \u2115 \u03bb FinCard \u2192 \u03bcA \u2248 \u03bcFinSuc FinCard\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = 0 , \u2248-sym Fin0\u2248\u22a4\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable \u03bcA \u03bcB (|A| , A\u2248) (|B| , B\u2248) = (|A| + suc |B|) ,\n  ( \u03bcA +\u03bc \u03bcB\n  \u2248\u27e8 A\u2248 +\u03bc-cong B\u2248 \u27e9\n    \u03bcFinSuc |A| +\u03bc \u03bcFinSuc |B|\n  \u2248\u27e8 +-\u2248 |A| |B| \u27e9\n    \u03bcFinSuc (|A| + suc |B|) \u2248\u220e)\n\n\u00d7-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA \u00d7\u03bc \u03bcB)\n\u00d7-Finable \u03bcA \u03bcB (|A| , A\u2248) (|B| , B\u2248) = (|B| + |A| * suc |B|) ,\n  ( \u03bcA \u00d7\u03bc \u03bcB\n  \u2248\u27e8 A\u2248 \u00d7\u03bc-cong B\u2248 \u27e9\n    \u03bcFinSuc |A| \u00d7\u03bc \u03bcFinSuc |B|\n  \u2248\u27e8 \u00d7-\u2248 |A| |B| \u27e9\n    \u03bcFinSuc (|B| + |A| * suc |B|)\n  \u2248\u220e)\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\nStableUnder\u2248 : \u2200 {A B} (\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 \u03bcA \u2248 \u03bcB\n             \u2192 StableUnderInjection \u03bcA \u2192 StableUnderInjection \u03bcB\nStableUnder\u2248 \u03bcA \u03bcB \u03bcA\u2248\u03bcB \u03bcA-SUI f p p-inj\n  = sum \u03bcB f\n  \u2261\u27e8 sums-ok' f \u27e9\n    sum \u03bcA (f \u2218 to iso)\n  \u2261\u27e8 \u03bcA-SUI (f \u2218 to iso) (from iso \u2218 p \u2218 to iso) (to-inj iso \u2218 p-inj \u2218 from-inj iso) \u27e9\n    sum \u03bcA (f \u2218 to iso \u2218 from iso \u2218 p \u2218 to iso)\n  \u2261\u27e8 \u2261.sym (sums-ok' (f \u2218 to iso \u2218 from iso \u2218 p)) \u27e9\n    sum \u03bcB (f \u2218 to iso \u2218 from iso \u2218 p)\n  \u2261\u27e8 sum-ext \u03bcB (\u2261.cong f \u2218 Inv.Inverse.right-inverse-of iso \u2218 p) \u27e9\n    sum \u03bcB (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open _\u2248_ \u03bcA\u2248\u03bcB\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA (FinCard , A\u2248Fin)\n  = StableUnder\u2248 (\u03bcFinSuc FinCard) \u03bcA (\u2248-sym A\u2248Fin) \u03bcFinSUI\n\n-- -}\n","old_contents":"module sum-properties where\n\nopen import Type\n\nimport Level as L\n\nopen import Data.Empty using (\u22a5)\n\nopen import Data.Bool.NP\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit using (\u22a4)\n\nopen import Function.NP\nimport Function.Inverse as Inv\nopen Inv using (_\u2194_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Function.Equality using (_\u27e8$\u27e9_)\n\nopen import sum\n\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2257_ ; _\u2257\u2082_)\n\nsum-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2238 g x)\nsum-lem \u03bcA f g = \u2261.trans (sum-ext \u03bcA f\u2257f\u2293g+f\u2238g) (sum-hom \u03bcA (\u03bb x \u2192 f x \u2293 g x) (\u03bb x \u2192 f x \u2238 g x))\n  where\n    f\u2257f\u2293g+f\u2238g : f \u2257 (\u03bb x \u2192 f x \u2293 g x + (f x \u2238 g x))\n    f\u2257f\u2293g+f\u2238g x = a\u2261a\u2293b+a\u2238b (f x) (g x)\n\nsum-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA f + sum \u03bcA g \u2261 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x)\nsum-lem\u2082 \u03bcA f g =\n         sum \u03bcA f + sum \u03bcA g \u2261\u27e8 \u2261.sym (sum-hom \u03bcA f g) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x + g x) \u2261\u27e8 sum-ext \u03bcA (\u03bb x \u2192 lemma (f x) (g x)) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x + f x \u2293 g x) \u2261\u27e8 sum-hom \u03bcA (\u03bb x \u2192 f x \u2294 g x) (\u03bb x \u2192 f x \u2293 g x) \u27e9\n         sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) + sum \u03bcA (\u03bb x \u2192 f x \u2293 g x) \u220e\n  where\n    open \u2261.\u2261-Reasoning\n    lemma : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\n    lemma zero b rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\n    lemma (suc a) zero = \u2261.refl\n    lemma (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                | +-assoc-comm (a \u2294 b) 1 (a \u2293 b) = \u2261.cong (suc \u2218 suc) (lemma a b)\n\nto\u2115-\u2293 : \u2200 a b \u2192 to\u2115 a \u2293 to\u2115 b \u2261 to\u2115 (a \u2227 b)\nto\u2115-\u2293 true true = \u2261.refl\nto\u2115-\u2293 true false = \u2261.refl\nto\u2115-\u2293 false b = \u2261.refl\n\nto\u2115-\u2294 : \u2200 a b \u2192 to\u2115 a \u2294 to\u2115 b \u2261 to\u2115 (a \u2228 b)\nto\u2115-\u2294 true true = \u2261.refl\nto\u2115-\u2294 true false = \u2261.refl\nto\u2115-\u2294 false b = \u2261.refl\n\nto\u2115-\u2238 : \u2200 a b \u2192 to\u2115 a \u2238 to\u2115 b \u2261 to\u2115 (a \u2227 not b)\nto\u2115-\u2238 true true = \u2261.refl\nto\u2115-\u2238 true false = \u2261.refl\nto\u2115-\u2238 false true = \u2261.refl\nto\u2115-\u2238 false false = \u2261.refl\n\ncount-lem : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool)\n          \u2192 count \u03bcA f \u2261 count \u03bcA (\u03bb x \u2192 f x \u2227 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 not (g x))\ncount-lem \u03bcA f g rewrite sum-lem \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) \n                       | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2238 (f x) (g x)) = \u2261.refl\n\ncount-lem\u2082 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA f + count \u03bcA g \u2261 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) + count \u03bcA (\u03bb x \u2192 f x \u2227 g x)\ncount-lem\u2082 \u03bcA f g rewrite sum-lem\u2082 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g)\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2294 (f x) (g x))\n  | sum-ext \u03bcA (\u03bb x \u2192 to\u2115-\u2293 (f x) (g x)) = \u2261.refl\n\n\nsum-\u2294 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 \u2115) \u2192 sum \u03bcA (\u03bb x \u2192 f x \u2294 g x) \u2264 sum \u03bcA f + sum \u03bcA g\nsum-\u2294 \u03bcA f g = \u2115\u2264.trans\n                 (sum-mono \u03bcA (\u03bb x \u2192 \u2294\u2264+ (f x) (g x)))\n                 (\u2115\u2264.reflexive (sum-hom \u03bcA f g))\n  where\n    \u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n    \u2294\u2264+ zero b = \u2115\u2264.refl\n    \u2294\u2264+ (suc a) zero = \u2115\u2264.reflexive (\u2261.cong suc (\u2115\u00b0.+-comm 0 a))\n    \u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\ncount-\u2228 : \u2200 {A : \u2605}(\u03bcA : SumProp A)(f g : A \u2192 Bool) \u2192 count \u03bcA (\u03bb x \u2192 f x \u2228 g x) \u2264 count \u03bcA f + count \u03bcA g\ncount-\u2228 \u03bcA f g = \u2115\u2264.trans (\u2115\u2264.reflexive (sum-ext \u03bcA (\u03bb x \u2192 \u2261.sym (to\u2115-\u2294 (f x) (g x))))) \n                          (sum-\u2294 \u03bcA (to\u2115 \u2218 f) (to\u2115 \u2218 g))\n\n\nsum-ext\u2082 : \u2200 {A B}{f g : A \u2192 B \u2192 \u2115}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 f \u2257\u2082 g \u2192 sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcA (sum \u03bcB \u2218 g)\nsum-ext\u2082 \u03bcA \u03bcB f\u2257g = sum-ext \u03bcA (sum-ext \u03bcB \u2218 f\u2257g)\n\nInjective : \u2200 {a b}{A : Set a}{B : Set b}(f : A \u2192 B) \u2192 Set (a L.\u2294 b)\nInjective f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nStableUnderInjection : \u2200 {A} \u2192 SumProp A \u2192 Set\nStableUnderInjection \u03bc = \u2200 f p \u2192 Injective p \u2192 sum \u03bc f \u2261 sum \u03bc (f \u2218 p)\n\n#-StableUnderInjection : \u2200 {A}{\u03bc : SumProp A} \u2192 StableUnderInjection \u03bc\n    \u2192 \u2200 f p \u2192 Injective p \u2192 count \u03bc f \u2261 count \u03bc (f \u2218 p)\n#-StableUnderInjection sui f p p-inj = sui (to\u2115 \u2218 f) p p-inj\n\nopen import Data.Fin using (Fin ; zero ; suc)\n\nrecord Finable {A : Set}(\u03bcA : SumProp A) : Set where\n  constructor mk\n  field\n    FinCard  : \u2115\n\n  FIN = Fin (suc FinCard)\n\n  field\n    iso      : A \u2194 Fin (suc FinCard)\n    \n  toFin : A \u2192 FIN\n  toFin x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  fromFin : FIN \u2192 A\n  fromFin x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  field\n    sums-ok  : \u2200 f \u2192 sum \u03bcA f \u2261 sum (\u03bcFinSuc FinCard) (f \u2218 fromFin)\n\n  from-inj : Injective fromFin\n  from-inj = Inv.Inverse.injective (Inv.sym iso)\n\n  to-inj : Injective toFin\n  to-inj = Inv.Inverse.injective iso\n\n  iso' : fromFin \u2218 toFin \u2257 id\n  iso' = Inv.Inverse.left-inverse-of iso\n\n\u22a4-Finable : Finable \u03bc\u22a4\n\u22a4-Finable = mk 0 iso sums-ok where\n\n  iso : _\n  iso = \u22a4 \u2194\u27e8 sym A\u228e\u22a5\u2194A \u27e9\n        (\u22a4 \u228e \u22a5) \u2194\u27e8 Inv.id \u228e-cong sym Fin0\u2194\u22a5 \u27e9\n        (\u22a4 \u228e Fin 0) \u2194\u27e8 sym Fin\u2218suc\u2194\u22a4\u228eFin \u27e9\n        Fin (suc 0) \u220e\n    where open EquationalReasoning\n          open import Relation.Binary.Sum\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = \u2261.refl\n\n+-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA +\u03bc \u03bcB)\n+-Finable {A}{B} \u03bcA \u03bcB finA finB = mk FinCard iso sums-ok where\n\n  |A| |B| : \u2115\n  |A| = suc (Finable.FinCard finA)\n  |B| = suc (Finable.FinCard finB)\n\n  \u03bcFinA = \u03bcFinSuc (Finable.FinCard finA)\n  \u03bcFinB = \u03bcFinSuc (Finable.FinCard finB)\n\n  FinCard : _\n  FinCard = Finable.FinCard finA + |B|\n\n  iso : _\n  iso = (A \u228e B) \n      \u2194\u27e8 (Finable.iso finA) \u228e-cong (Finable.iso finB) \u27e9 \n        (Fin |A| \u228e Fin |B|)\n      \u2194\u27e8 Fin-\u228e-+ |A| |B| \u27e9\n        Fin (|A| + |B|)\n      \u220e\n    where \n      open import Relation.Binary.Sum\n      open EquationalReasoning\n\n  from : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 B \u2192 A\n  from iso x = Inv.Inverse.from iso \u27e8$\u27e9 x\n\n  to : \u2200 {A B : Set} \u2192 A \u2194 B \u2192 A \u2192 B\n  to iso x = Inv.Inverse.to iso \u27e8$\u27e9 x\n\n  fromFin : Fin (|A| + |B|) \u2192 A \u228e B\n  fromFin x = from iso x\n  \n  open import Data.Vec.NP using (Vec ; foldr\u2081 ; tabulate-ext)\n  vsum : \u2200 {n} \u2192 Vec \u2115 (suc n) \u2192 \u2115\n  vsum = foldr\u2081 _+_\n\n  fin-proof : \u2200 n m (f : Fin (suc n) \u228e Fin (suc m) \u2192 \u2115) \u2192 \n                      sum (\u03bcFinSuc n) (f \u2218 inj\u2081) \n                    + sum (\u03bcFinSuc m) (f \u2218 inj\u2082) \n                    \u2261 sum (\u03bcFinSuc (n + suc m)) (f \u2218 from  (Fin-\u228e-+ (suc n) (suc m)))\n  fin-proof zero m f    rewrite tabulate-ext (\u2261.cong (f \u2218 inj\u2082 \u2218 suc) \u2218 \u2261.sym \u2218 Inv.Inverse.right-inverse-of (Maybe^\u22a5\u2194Fin m)) = \u2261.refl\n  fin-proof (suc n) m f = sum (\u03bcFinSuc (suc n)) (f \u2218 inj\u2081) + sum (\u03bcFinSuc m) (f \u2218 inj\u2082) \n                        \u2261\u27e8 \u2115\u00b0.+-assoc (f (inj\u2081 zero)) (sum (\u03bcFinSuc n) (f \u2218 inj\u2081 \u2218 suc)) (sum (\u03bcFinSuc m) (f \u2218 inj\u2082)) \u27e9 \n                          f (inj\u2081 zero) + (sum (\u03bcFinSuc n) (f \u2218 inj\u2081 \u2218 suc) + sum (\u03bcFinSuc m) (f \u2218 inj\u2082))\n                        \u2261\u27e8 \u2261.cong (_+_ (f (inj\u2081 zero))) (fin-proof n m (F f)) \u27e9\n                          f (inj\u2081 zero) + sum (\u03bcFinSuc (n + suc m)) (F f \u2218 from (Fin-\u228e-+ (suc n) (suc m)))\n                        \u2261\u27e8 \u2261.cong (_+_ (f (inj\u2081 zero))) (sum-ext (\u03bcFinSuc (n + suc m)) (F-proof (suc n) (suc m) f)) \u27e9\n                          f (inj\u2081 zero) + sum (\u03bcFinSuc (n + suc m)) (f \u2218 from (Fin-\u228e-+ (suc (suc n)) (suc m)) \u2218 suc)\n                        \u2261\u27e8 \u2261.refl \u27e9\n                          sum (\u03bcFinSuc (suc n + suc m))\n                                     (f \u2218 from (Fin-\u228e-+ (suc (suc n)) (suc m))) \n                        \u220e\n    where \n      open \u2261.\u2261-Reasoning\n      F : \u2200 {n m} \u2192 (Fin (suc n) \u228e Fin m \u2192 \u2115) \u2192 Fin n \u228e Fin m \u2192 \u2115\n      F f = [ (f \u2218 inj\u2081 \u2218 suc) , (f \u2218 inj\u2082) ]\n\n\n      F-proof : \u2200 n m f \u2192 F f \u2218 from (Fin-\u228e-+ n m) \u2257 f \u2218 from (Fin-\u228e-+ (suc n) m) \u2218 suc\n      F-proof n m f x with (Inv.Inverse.from (Maybe^-\u228e-+ n m) \u27e8$\u27e9 (Inv.Inverse.from (Maybe^\u22a5\u2194Fin (n + m)) \u27e8$\u27e9 x))\n      ... | inj\u2081 y = \u2261.refl\n      ... | inj\u2082 y = \u2261.refl\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = sum (\u03bcA +\u03bc \u03bcB) f\n            \u2261\u27e8 \u2261.cong\u2082 _+_ (Finable.sums-ok finA (f \u2218 inj\u2081)) (Finable.sums-ok finB (f \u2218 inj\u2082)) \u27e9\n              sum \u03bcFinA (f \u2218 inj\u2081 \u2218 from (Finable.iso finA))\n            + sum \u03bcFinB (f \u2218 inj\u2082 \u2218 from (Finable.iso finB))\n            \u2261\u27e8 \u2261.cong\u2082 _+_ (sum-ext \u03bcFinA {f = f \u2218 inj\u2081 \u2218 from (Finable.iso finA)} (\u03bb x \u2192 \u2261.refl)) \n                           (sum-ext \u03bcFinB {f = f \u2218 inj\u2082 \u2218 from (Finable.iso finB)} (\u03bb _ \u2192 \u2261.refl)) \u27e9 \n              sum \u03bcFinA (f \u2218 from F \u2218 inj\u2081)\n            + sum \u03bcFinB (f \u2218 from F \u2218 inj\u2082)\n            \u2261\u27e8 fin-proof _ _ (f \u2218 from F) \u27e9\n              sum (\u03bcFinSuc FinCard) (f \u2218 from F \u2218 from (Fin-\u228e-+ |A| |B|))\n            \u2261\u27e8 \u2261.refl \u27e9\n              sum (\u03bcFinSuc FinCard) (f \u2218 from iso) \n            \u220e\n    where\n      open \u2261.\u2261-Reasoning\n      open import Relation.Binary.Sum\n      F :  (A \u228e B) \u2194 (Fin |A| \u228e Fin |B|)\n      F = Finable.iso finA \u228e-cong Finable.iso finB\n\n\u00d7-Finable : \u2200 {A B}(\u03bcA : SumProp A)(\u03bcB : SumProp B) \u2192 Finable \u03bcA \u2192 Finable \u03bcB \u2192 Finable (\u03bcA \u00d7\u03bc \u03bcB)\n\u00d7-Finable {A} {B} \u03bcA \u03bcB finA finB = mk FinCard iso sums-ok where\n  |A| |B| : \u2115\n  |A| = suc (Finable.FinCard finA)\n  |B| = suc (Finable.FinCard finB)\n\n  \u03bcFinA = \u03bcFinSuc (Finable.FinCard finA)\n  \u03bcFinB = \u03bcFinSuc (Finable.FinCard finB)\n\n  FinCard = Finable.FinCard finB + Finable.FinCard finA * |B|\n\n  prf : suc FinCard \u2261 |A| * |B|\n  prf = \u2261.refl\n\n  iso : _\n  iso = (A \u00d7 B) \u2194\u27e8 (Finable.iso finA) \u00d7-cong (Finable.iso finB) \u27e9 \n        (Fin |A| \u00d7 Fin |B|) \u2194\u27e8 {!!} \u27e9\n        Fin (|A| * |B|) \u220e\n    where open EquationalReasoning\n          open import Relation.Binary.Product.Pointwise\n\n  sums-ok : (_ : _) \u2192 _\n  sums-ok f = {!!}\n\nmodule _ where\n  open import bijection-fin\n  open import Data.Fin using (Fin; zero; suc)\n  open import Data.Vec.NP renaming (sum to vsum)\n\n  sumFin : \u2200 n \u2192 Sum (Fin n)\n  sumFin n f = vsum (tabulate f)\n\n  sumFin-spec : \u2200 n \u2192 sumFin (suc n) \u2257 sum (\u03bcFinSuc n)\n  sumFin-spec zero    f = \u2115\u00b0.+-comm (f zero) 0\n  sumFin-spec (suc n) f = \u2261.cong (_+_ (f zero)) (sumFin-spec n (f \u2218 suc))\n\n  sumFinSUI : \u2200 n f p \u2192 Injective p \u2192 sumFin n f \u2261 sumFin n (f \u2218 p)\n  sumFinSUI n f p p-inj = count-perm f p (\u03bb x y \u2192 p-inj)\n\n  \u03bcFinSUI : \u2200 {n} \u2192 StableUnderInjection (\u03bcFinSuc n)\n  \u03bcFinSUI {n} f p p-inj rewrite \u2261.sym (sumFin-spec n f)\n                              | \u2261.sym (sumFin-spec n (f \u2218 p))\n                              = sumFinSUI (suc n) f p p-inj\n\nStableIfFinable : \u2200 {A} (\u03bcA : SumProp A) \u2192 Finable \u03bcA \u2192 StableUnderInjection \u03bcA\nStableIfFinable \u03bcA fin f p p-inj\n  = sum \u03bcA f\n  \u2261\u27e8 sums-ok f \u27e9\n    sum (\u03bcFinSuc FinCard) (f \u2218 fromFin)\n  \u2261\u27e8 \u03bcFinSUI (f \u2218 fromFin) (toFin \u2218 p \u2218 fromFin) (from-inj \u2218 p-inj \u2218 to-inj) \u27e9\n    sum (\u03bcFinSuc FinCard) (f \u2218 fromFin \u2218 toFin \u2218 p \u2218 fromFin)\n  \u2261\u27e8 sum-ext (\u03bcFinSuc FinCard) (\u03bb x \u2192 \u2261.cong f (iso' (p (fromFin x)))) \u27e9\n    sum (\u03bcFinSuc FinCard) (f \u2218 p \u2218 fromFin)\n  \u2261\u27e8 \u2261.sym (sums-ok (f \u2218 p)) \u27e9\n    sum \u03bcA (f \u2218 p)\n  \u220e\n  where open \u2261.\u2261-Reasoning\n        open Finable fin\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"feb3041008cf201bd497fc7ebc6189be35cc621a","subject":"Removed TODO.","message":"Removed TODO.\n\nIgnore-this: 7bcc7a75642a63fa785e40d4e79d2972\n\ndarcs-hash:20100722035909-3bd4e-3e4b505c2c9f2f3db16a5f4f70c4e2305a87db41.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/Nullary.agda","new_file":"LTC\/Relation\/Nullary.agda","new_contents":"------------------------------------------------------------------------------\n-- Operations on nullary relations\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Nullary where\n\nopen import LTC.Data.Empty\n\ninfix 3 \u00ac_\n\n------------------------------------------------------------------------------\n-- Negation.\n-- The underscore allows to write for example '\u00ac \u00ac A' instead of '\u00ac (\u00ac A)'.\n\u00ac_ : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n","old_contents":"------------------------------------------------------------------------------\n-- Operations on nullary relations\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.Nullary where\n\nopen import LTC.Data.Empty\n\ninfix 3 \u00ac\n\n------------------------------------------------------------------------------\n-- Negation.\n\n-- TODO: It is necessary an underscore (i.e. \u00ac_ ) (the standard\n-- library uses it)?\n\u00ac : Set \u2192 Set\n\u00ac A = A \u2192 \u22a5\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db0c09075cb7958ca194b147cf01aa5270dbf747","subject":"Make Mu take Branches parameter.","message":"Make Mu take Branches parameter.\n\nThis makes it possible to support levitation later.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (cs : BranchesD I E) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (caseD cs t) (\u03bc E cs)\n\ninj : {I : Set} (E : Enum) (cs : BranchesD I E) (t : Tag E) \u2192 CurriedEl (caseD cs t) (\u03bc E cs)\ninj E cs t = curryEl (caseD cs t) (\u03bc E cs) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i x,xs = \u03b1 j (proj\u2081 x,xs) , hyps D X P \u03b1 i (proj\u2082 x,xs)\nhyps (Arg A B) X P \u03b1 i a,xs = hyps (B (proj\u2081 a,xs)) X P \u03b1 i (proj\u2082 a,xs)\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nind E Ds P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (caseD Ds t) (\u03bc E Ds) P (ind E Ds P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (caseD Ds t) (\u03bc E Ds) P (init t))\n  (i : I)\n  (x : \u03bc E Ds i)\n  \u2192 P i x\nindCurried E Ds P f i x = ind E Ds P (\u03bb t \u2192 uncurryHyps (caseD Ds t) (\u03bc E Ds) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (caseD Ds t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E Ds X cn P t = CurriedHyps (caseD Ds t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E Ds P t = Summer E Ds (\u03bc E Ds) init P t\n\nelimUncurried : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E Ds P)\n  \u2192 (i : I) (x : \u03bc E Ds i) \u2192 P i x\nelimUncurried E Ds P cs i x =\n  indCurried E Ds P\n    (case (SumCurriedHyps E Ds P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (Ds : BranchesD I E)\n  (P : (i : I) \u2192 \u03bc E Ds i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E Ds P)\n      ((i : I) (x : \u03bc E Ds i) \u2192 P i x)\nelim E Ds P = curryBranches (elimUncurried E Ds P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115Ds : BranchesD \u22a4 \u2115E\n\u2115Ds =\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T (caseD \u2115Ds)\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115Ds\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecDs : (A : Set) \u2192 BranchesD (\u2115 tt) VecE\nVecDs A =\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (caseD (VecDs A))\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecDs A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecDs A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecDs A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115Ds _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD \u2115Ds t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecDs A) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs A) t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecDs (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (caseD (VecDs (Vec A m)) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115Ds _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecDs A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecDs (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (C t) (\u03bc E C)\n\ninj : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) (t : Tag E) \u2192 CurriedEl (C t) (\u03bc E C)\ninj E C t = curryEl (C t) (\u03bc E C) (init t)\n\n----------------------------------------------------------------------\n\nhyps : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nhyps (End j) X P \u03b1 i xs = tt\nhyps (Rec j D) X P \u03b1 i x,xs = \u03b1 j (proj\u2081 x,xs) , hyps D X P \u03b1 i (proj\u2082 x,xs)\nhyps (Arg A B) X P \u03b1 i a,xs = hyps (B (proj\u2081 a,xs)) X P \u03b1 i (proj\u2082 a,xs)\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nind E C P \u03b1 i (init t xs) = \u03b1 t i xs $\n  hyps (C t) (\u03bc E C) P (ind E C P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nindCurried E C P f i x = ind E C P (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E C X cn P t = CurriedHyps (C t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E C P t = Summer E C (\u03bc E C) init P t\n\nelimUncurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E C P)\n  \u2192 (i : I) (x : \u03bc E C i) \u2192 P i x\nelimUncurried E C P cs i x =\n  indCurried E C P\n    (case (SumCurriedHyps E C P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E C P)\n      ((i : I) (x : \u03bc E C i) \u2192 P i x)\nelim E C P = curryBranches (elimUncurried E C P)\n\n----------------------------------------------------------------------\n\nSoundness : Set\u2081\nSoundness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (cs : Branches E (SumCurriedHyps E C P))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb \u03b1\n  \u2192 elimUncurried E C P cs i x \u2261 ind E C P \u03b1 i x\n\nsound : Soundness\nsound E C P cs i xs =\n  let D = Arg (Tag E) C in\n  (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) cs t)) , refl\n\nCompleteness : Set\u2081\nCompleteness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb cs\n  \u2192 ind E C P \u03b1 i x \u2261 elimUncurried E C P cs i x\n\nuncurryHypsIdent : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  (\u03b1 : UncurriedHyps D X P cn)\n  (i : I) (xs : El D X i) (ihs : Hyps D X P i xs)\n  \u2192 \u03b1 i xs ihs \u2261 uncurryHyps D X P cn (curryHyps D X P cn \u03b1) i xs ihs\nuncurryHypsIdent (End .i) X P cn \u03b1 i refl tt = refl\nuncurryHypsIdent (Rec j D) X P cn \u03b1 i (x , xs) (p , ps) =\n  uncurryHypsIdent D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb k ys rs \u2192 \u03b1 k (x , ys) (p , rs)) i xs ps\nuncurryHypsIdent (Arg A B) X P cn \u03b1 i (a , xs) ps =\n  uncurryHypsIdent (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb j ys \u2192 \u03b1 j (a , ys)) i xs ps\n\npostulate\n  ext4 : {A : Set} {B : A \u2192 Set} {C : (a : A) \u2192 B a \u2192 Set}\n    {D : (a : A) (b : B a) \u2192 C a b \u2192 Set}\n    {Z : (a : A) (b : B a) (c : C a b) \u2192 D a b c \u2192 Set}\n    (f g : (a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 Z a b c d)\n    \u2192 ((a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 f a b c d \u2261 g a b c d)\n    \u2192 f \u2261 g\n\ntoBranches : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  \u2192 Branches E (Summer E C X cn P)\ntoBranches [] C X cn P \u03b1 = tt\ntoBranches (l \u2237 E) C X cn P \u03b1 =\n    curryHyps (C here) X P (\u03bb xs \u2192 cn here xs) (\u03bb i xs \u2192 \u03b1 here i xs)\n  , toBranches E (\u03bb t \u2192 C (there t)) X\n     (\u03bb t \u2192 cn (there t))\n     P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih)\n\nToBranches : {I : Set} {E : Enum} (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  (t : Tag E)\n  \u2192 let \u03b2 = toBranches E C X cn P \u03b1 in\n  case (Summer E C X cn P) \u03b2 t \u2261 curryHyps (C t) X P (cn t) (\u03b1 t)\nToBranches C X cn P \u03b1 here = refl\nToBranches C X cn P \u03b1 (there t)\n  with ToBranches (\u03bb t \u2192 C (there t)) X\n    (\u03bb t xs \u2192 cn (there t) xs)\n    P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih) t\n... | ih rewrite ih = refl\n\ncomplete\u03b1 : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (t : Tag E) (i : I) (xs : El (C t) (\u03bc E C) i) (ihs : Hyps (C t) (\u03bc E C) P i xs)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  \u03b1 t i xs ihs \u2261 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t) i xs ihs\ncomplete\u03b1 E C P \u03b1 t i xs ihs\n  with ToBranches C (\u03bc E C) init P \u03b1 t\n... | q rewrite q = uncurryHypsIdent (C t) (\u03bc E C) P (init t) (\u03b1 t) i xs ihs\n\ncomplete' : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  ind E C P \u03b1 i x \u2261 elimUncurried E C P \u03b2 i x\ncomplete' E C P \u03b1 i (init t xs) = cong\n  (\u03bb f \u2192 ind E C P f i (init t xs))\n  (ext4 \u03b1\n    (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t))\n    (complete\u03b1 E C P \u03b1)\n  )\n  where \u03b2 = toBranches E C (\u03bc E C) init P \u03b1\n\ncomplete : Completeness\ncomplete E C P \u03b1 i x =\n  let D = Arg (Tag E) C in\n    toBranches E C (\u03bc E C) init P \u03b1\n  , complete' E C P \u03b1 i x\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115C : \u2115T \u2192 Desc \u22a4\n\u2115C = caseD $\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T \u2115C\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115C\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecC : (A : Set) \u2192 VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD $\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (VecC A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecC A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecC A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecC A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecC A) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC A t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecC (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC (Vec A m) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115C _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115C _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecC A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecC (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"00bee87e47150d5062c2e99e9711f35f367885f2","subject":"Add Mu eliminator to operational semantics.","message":"Add Mu eliminator to operational semantics.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Operational.agda","new_file":"formalization\/agda\/Spire\/Operational.agda","new_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Desc `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u03bc : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Neutral \u0393 (`Desc \u2113) \u2192 Type \u0393 \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\u27e6_\u27e7\u1d48 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Type \u0393 \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Desc `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u03bc : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type \u2113)\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Value \u0393 (`Type \u2113))\n    (B : Value (\u0393 , \u27e6 A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Value \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `con : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Value \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D)) \u2192 Value \u0393 (`\u03bc D)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elim\u22a5 : \u2200{A} \u2192 Neutral \u0393 `\u22a5 \u2192 Neutral \u0393 A\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n  `des : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Neutral \u0393 (`\u03bc D) \u2192 Neutral \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D))\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `\u03bc D \u27e7 = `\u03bc D\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `Desc {\u2113} \u27e7 = `Desc \u2113\n\u27e6 `\u27e6 D \u27e7\u1d48 X \u27e7 = \u27e6 D \u27e7\u1d48 \u27e6 X \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\n\u27e6 `\u22a4\u1d48 \u27e7\u1d48 X = `\u22a4\n\u27e6 `X\u1d48 \u27e7\u1d48 X = X\n\u27e6 `\u03a0\u1d48 A D \u27e7\u1d48 X = `\u03a0 \u27e6 A \u27e7 (\u27e6 D \u27e7\u1d48 (wknT X))\n\u27e6 `\u03a3\u1d48 A D \u27e7\u1d48 X = `\u03a3 \u27e6 A \u27e7 (\u27e6 D \u27e7\u1d48 (wknT X))\n\u27e6 `neut D \u27e7\u1d48 X = `\u27e6 D \u27e7\u1d48 X\n\n----------------------------------------------------------------------\n\nelim\u22a5 : \u2200{\u0393 A} \u2192 Value \u0393 `\u22a5 \u2192 Value \u0393 A\nelim\u22a5 (`neut bot) = `neut (`elim\u22a5 bot)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\ndes : \u2200{\u0393 \u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Value \u0393 (`\u03bc D) \u2192 Value \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D))\ndes (`con x) = x\ndes (`neut x) = `neut (`des x)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u03bc : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113) \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type \u2113)\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Term \u0393 (`Type \u2113))\n    (D : Term (\u0393 , \u27e6 eval A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Term \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B}\n    (b : Term (\u0393 , A) B)\n    \u2192 Term \u0393 (`\u03a0 A B)\n  `con : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Term \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D)) \u2192 Term \u0393 (`\u03bc D)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elim\u22a5 : \u2200{A} \u2192 Term \u0393 `\u22a5 \u2192 Term \u0393 A\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n  `des : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Term \u0393 (`\u03bc D) \u2192 Term \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval (`\u03bc D) = `\u03bc (eval D)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\neval (`\u27e6 D \u27e7\u1d48 X) = `\u27e6 eval D \u27e7\u1d48 (eval X)\n\n{- Desc introduction -}\neval `\u22a4\u1d48 = `\u22a4\u1d48\neval `X\u1d48 = `X\u1d48\neval (`\u03a0\u1d48 A D) = `\u03a0\u1d48 (eval A) (eval D)\neval (`\u03a3\u1d48 A D) = `\u03a3\u1d48 (eval A) (eval D)\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\neval (`\u03bb b) = `\u03bb (eval b)\neval (`con x) = `con (eval x)\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elim\u22a5 bot) = elim\u22a5 (eval bot)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\neval (`des x) = des (eval x)\n\n----------------------------------------------------------------------\n\n","old_contents":"module Spire.Operational where\n\n----------------------------------------------------------------------\n\ndata Level : Set where\n  zero : Level\n  suc : Level \u2192 Level\n\n----------------------------------------------------------------------\n\ndata Context : Set\ndata Type (\u0393 : Context) : Set\ndata Value (\u0393 : Context) : Type \u0393 \u2192 Set\ndata Neutral (\u0393 : Context) : Type \u0393 \u2192 Set\n\n----------------------------------------------------------------------\n\ndata Context where\n  \u2205 : Context\n  _,_ : (\u0393 : Context) \u2192 Type \u0393 \u2192 Context\n\ndata Type \u0393 where\n  `\u22a5 `\u22a4 `Bool : Type \u0393\n  `Desc `Type : (\u2113 : Level) \u2192 Type \u0393\n  `\u03a0 `\u03a3 : (A : Type \u0393) (B : Type (\u0393 , A)) \u2192 Type \u0393\n  `\u03bc : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Type \u0393\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Neutral \u0393 (`Type \u2113) \u2192 Type \u0393\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Neutral \u0393 (`Desc \u2113) \u2192 Type \u0393 \u2192 Type \u0393\n\n----------------------------------------------------------------------\n\n\u27e6_\u27e7 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Type \u2113) \u2192  Type \u0393\n\u27e6_\u27e7\u1d48 : \u2200{\u0393 \u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Type \u0393 \u2192  Type \u0393\n\npostulate\n  wknT : \u2200{\u0393 A} \u2192 Type \u0393 \u2192 Type (\u0393 , A)\n  subT : \u2200{\u0393 A} \u2192 Type (\u0393 , A) \u2192 Value \u0393 A \u2192 Type \u0393\n  subV : \u2200{\u0393 A B} \u2192 Value (\u0393 , A) B \u2192 (x : Value \u0393 A) \u2192 Value \u0393 (subT B x)\n\ndata Var : (\u0393 : Context) (A : Type \u0393) \u2192 Set where\n  here : \u2200{\u0393 A} \u2192 Var (\u0393 , A) (wknT A)\n  there : \u2200{\u0393 A B} \u2192 Var \u0393 A \u2192 Var (\u0393 , B) (wknT A)\n\n----------------------------------------------------------------------\n\ndata Value \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Desc `Type : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Value \u0393 (`Type \u2113)) (B : Value (\u0393 , \u27e6 A \u27e7) (`Type \u2113)) \u2192 Value \u0393 (`Type \u2113)\n  `\u03bc : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Value \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type (suc \u2113))\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113) \u2192 Value \u0393 (`Type \u2113) \u2192 Value \u0393 (`Type \u2113)\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Value \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Value \u0393 (`Type \u2113))\n    (B : Value (\u0393 , \u27e6 A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Value \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Value \u0393 `\u22a4\n  `true `false : Value \u0393 `Bool\n  _`,_ : \u2200{A B} (a : Value \u0393 A) (b : Value \u0393 (subT B a)) \u2192 Value \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B} \u2192 Value (\u0393 , A) B \u2192 Value \u0393 (`\u03a0 A B)\n  `con : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Value \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D)) \u2192 Value \u0393 (`\u03bc D)\n  `neut : \u2200{A} \u2192 Neutral \u0393 A \u2192 Value \u0393 A\n\n----------------------------------------------------------------------\n\ndata Neutral \u0393 where\n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Neutral \u0393 A\n  `if_`then_`else_ : \u2200{C} (b : Neutral \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Neutral \u0393 C\n  `elim\u22a5 : \u2200{A} \u2192 Neutral \u0393 `\u22a5 \u2192 Neutral \u0393 A\n  `elimBool : \u2200{\u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n    (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n    (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n    (b : Neutral \u0393 `Bool) \u2192 Neutral \u0393 (subT \u27e6 P \u27e7 (`neut b))\n  `proj\u2081 : \u2200{A B} \u2192 Neutral \u0393 (`\u03a3 A B) \u2192 Neutral \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Neutral \u0393 (`\u03a3 A B)) \u2192 Neutral \u0393 (subT B (`neut (`proj\u2081 ab)))\n  _`$_ : \u2200{A B} (f : Neutral \u0393 (`\u03a0 A B)) (a : Value \u0393 A) \u2192 Neutral \u0393 (subT B a)\n\n----------------------------------------------------------------------\n\n\u27e6 `\u03a0 A B \u27e7 = `\u03a0 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u03a3 A B \u27e7 = `\u03a3 \u27e6 A \u27e7 \u27e6 B \u27e7\n\u27e6 `\u22a5 \u27e7 = `\u22a5\n\u27e6 `\u22a4 \u27e7 = `\u22a4\n\u27e6 `Bool \u27e7 = `Bool\n\u27e6 `\u03bc D \u27e7 = `\u03bc D\n\u27e6 `Type {zero} \u27e7 = `\u22a5\n\u27e6 `Type {suc \u2113} \u27e7 = `Type \u2113\n\u27e6 `\u27e6 A \u27e7 \u27e7 = \u27e6 A \u27e7\n\u27e6 `Desc {\u2113} \u27e7 = `Desc \u2113\n\u27e6 `\u27e6 D \u27e7\u1d48 X \u27e7 = \u27e6 D \u27e7\u1d48 \u27e6 X \u27e7\n\u27e6 `neut A \u27e7 = `\u27e6 A \u27e7\n\n----------------------------------------------------------------------\n\n\u27e6 `\u22a4\u1d48 \u27e7\u1d48 X = `\u22a4\n\u27e6 `X\u1d48 \u27e7\u1d48 X = X\n\u27e6 `\u03a0\u1d48 A D \u27e7\u1d48 X = `\u03a0 \u27e6 A \u27e7 (\u27e6 D \u27e7\u1d48 (wknT X))\n\u27e6 `\u03a3\u1d48 A D \u27e7\u1d48 X = `\u03a3 \u27e6 A \u27e7 (\u27e6 D \u27e7\u1d48 (wknT X))\n\u27e6 `neut D \u27e7\u1d48 X = `\u27e6 D \u27e7\u1d48 X\n\n----------------------------------------------------------------------\n\nelim\u22a5 : \u2200{\u0393 A} \u2192 Value \u0393 `\u22a5 \u2192 Value \u0393 A\nelim\u22a5 (`neut bot) = `neut (`elim\u22a5 bot)\n\n----------------------------------------------------------------------\n\nif_then_else_ : \u2200{\u0393 C} (b : Value \u0393 `Bool) (c\u2081 c\u2082 : Value \u0393 C) \u2192 Value \u0393 C\nif `true then c\u2081 else c\u2082 = c\u2081\nif `false then c\u2081 else c\u2082 = c\u2082\nif `neut b then c\u2081 else c\u2082 = `neut (`if b `then c\u2081 `else c\u2082)\n\nelimBool : \u2200{\u0393 \u2113} (P : Value (\u0393 , `Bool) (`Type \u2113))\n  (pt : Value \u0393 (subT \u27e6 P \u27e7 `true))\n  (pf : Value \u0393 (subT \u27e6 P \u27e7 `false))\n  (b : Value \u0393 `Bool)\n  \u2192 Value \u0393 (subT \u27e6 P \u27e7 b)\nelimBool P pt pf `true = pt\nelimBool P pt pf `false = pf\nelimBool P pt pf (`neut b) = `neut (`elimBool P pt pf b)\n\n----------------------------------------------------------------------\n\nproj\u2081 : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a3 A B) \u2192 Value \u0393 A\nproj\u2081 (a `, b) = a\nproj\u2081 (`neut ab) = `neut (`proj\u2081 ab)\n\nproj\u2082 : \u2200{\u0393 A B} (ab : Value \u0393 (`\u03a3 A B)) \u2192 Value \u0393 (subT B (proj\u2081 ab))\nproj\u2082 (a `, b) = b\nproj\u2082 (`neut ab) = `neut (`proj\u2082 ab)\n\n----------------------------------------------------------------------\n\n_$_ : \u2200{\u0393 A B} \u2192 Value \u0393 (`\u03a0 A B) \u2192 (a : Value \u0393 A) \u2192 Value \u0393 (subT B a)\n`\u03bb b $ a = subV b a\n`neut f $ a = `neut (f `$ a)\n\n----------------------------------------------------------------------\n\ndata Term (\u0393 : Context) : Type \u0393 \u2192 Set\neval : \u2200{\u0393 A} \u2192 Term \u0393 A \u2192 Value \u0393 A\n\ndata Term \u0393 where\n  {- Type introduction -}\n  `\u22a5 `\u22a4 `Bool `Type : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113)\n  `\u03a0 `\u03a3 : \u2200{\u2113} (A : Term \u0393 (`Type \u2113)) (B : Term (\u0393 , \u27e6 eval A \u27e7) (`Type \u2113)) \u2192 Term \u0393 (`Type \u2113)\n  `\u03bc : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113) \u2192 Term \u0393 (`Type \u2113)\n  `\u27e6_\u27e7 : \u2200{\u2113} \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type (suc \u2113))\n  `\u27e6_\u27e7\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113) \u2192 Term \u0393 (`Type \u2113) \u2192 Term \u0393 (`Type \u2113)\n\n  {- Desc introduction -}\n  `\u22a4\u1d48 `X\u1d48 : \u2200{\u2113} \u2192 Term \u0393 (`Desc \u2113)\n  `\u03a0\u1d48 `\u03a3\u1d48 : \u2200{\u2113}\n    (A : Term \u0393 (`Type \u2113))\n    (D : Term (\u0393 , \u27e6 eval A \u27e7) (`Desc (suc \u2113)))\n    \u2192 Term \u0393 (`Desc (suc \u2113))\n\n  {- Value introduction -}\n  `tt : Term \u0393 `\u22a4\n  `true `false : Term \u0393 `Bool\n  _`,_ : \u2200{A B}\n    (a : Term \u0393 A) (b : Term \u0393 (subT B (eval a)))\n    \u2192 Term \u0393 (`\u03a3 A B)\n  `\u03bb : \u2200{A B}\n    (b : Term (\u0393 , A) B)\n    \u2192 Term \u0393 (`\u03a0 A B)\n  `con : \u2200{\u2113} {D : Value \u0393 (`Desc \u2113)} \u2192 Term \u0393 (\u27e6 D \u27e7\u1d48 (`\u03bc D)) \u2192 Term \u0393 (`\u03bc D)\n  \n  {- Value elimination -}\n  `var : \u2200{A} \u2192 Var \u0393 A \u2192 Term \u0393 A\n  `if_`then_`else_ : \u2200{C}\n    (b : Term \u0393 `Bool)\n    (c\u2081 c\u2082 : Term \u0393 C)\n    \u2192 Term \u0393 C\n  _`$_ : \u2200{A B} (f : Term \u0393 (`\u03a0 A B)) (a : Term \u0393 A) \u2192 Term \u0393 (subT B (eval a))\n  `proj\u2081 : \u2200{A B} \u2192 Term \u0393 (`\u03a3 A B) \u2192 Term \u0393 A\n  `proj\u2082 : \u2200{A B} (ab : Term \u0393 (`\u03a3 A B)) \u2192 Term \u0393 (subT B (proj\u2081 (eval ab)))\n  `elim\u22a5 : \u2200{A} \u2192 Term \u0393 `\u22a5 \u2192 Term \u0393 A\n  `elimBool : \u2200{\u2113} (P : Term (\u0393 , `Bool) (`Type \u2113))\n    (pt : Term \u0393 (subT \u27e6 eval P \u27e7 `true))\n    (pf : Term \u0393 (subT \u27e6 eval P \u27e7 `false))\n    (b : Term \u0393 `Bool)\n    \u2192 Term \u0393 (subT \u27e6 eval P \u27e7 (eval b))\n\n----------------------------------------------------------------------\n\n{- Type introduction -}\neval `\u22a5 = `\u22a5\neval `\u22a4 = `\u22a4\neval `Bool = `Bool\neval `Type = `Type\neval (`\u03a0 A B) = `\u03a0 (eval A) (eval B)\neval (`\u03a3 A B) = `\u03a3 (eval A) (eval B)\neval (`\u03bc D) = `\u03bc (eval D)\neval `\u27e6 A \u27e7 = `\u27e6 eval A \u27e7\neval (`\u27e6 D \u27e7\u1d48 X) = `\u27e6 eval D \u27e7\u1d48 (eval X)\n\n{- Desc introduction -}\neval `\u22a4\u1d48 = `\u22a4\u1d48\neval `X\u1d48 = `X\u1d48\neval (`\u03a0\u1d48 A D) = `\u03a0\u1d48 (eval A) (eval D)\neval (`\u03a3\u1d48 A D) = `\u03a3\u1d48 (eval A) (eval D)\n\n{- Value introduction -}\neval `tt = `tt\neval `true = `true\neval `false = `false\neval (a `, b) = eval a `, eval b\neval (`\u03bb b) = `\u03bb (eval b)\neval (`con x) = `con (eval x)\n\n{- Value elimination -}\neval (`var i) = `neut (`var i)\neval (`if b `then c\u2081 `else c\u2082) = if eval b then eval c\u2081 else eval c\u2082\neval (f `$ a) = eval f $ eval a\neval (`proj\u2081 ab) = proj\u2081 (eval ab)\neval (`proj\u2082 ab) = proj\u2082 (eval ab)\neval (`elim\u22a5 bot) = elim\u22a5 (eval bot)\neval (`elimBool P pt pf b) = elimBool (eval P) (eval pt) (eval pf) (eval b)\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7006d0f34be4053a1ba3b97108bf3f15dbf3fecb","subject":"lecture rough","message":"lecture rough\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"code\/GoaTest.agda","new_file":"code\/GoaTest.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"ace6b1740d999d059d3a52854436d8e38133db73","subject":"Added missing ATP pragma.","message":"Added missing ATP pragma.\n\nIgnore-this: 35750d7ab3f49bf9ac17e50181c7427f\n\ndarcs-hash:20120430141705-3bd4e-97325bb46e704d3caa042733380b7157713d800a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOL\/NonIntuitionistic\/TheoremsATP.agda","new_file":"src\/FOL\/NonIntuitionistic\/TheoremsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Non-intuitionistic logic theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOL.NonIntuitionistic.TheoremsATP where\n\n-- The ATPs work in classical logic, therefore we also can prove some\n-- non-intuitionistic logic theorems.\n\nopen import FOL.Base hiding ( pem )\n\n------------------------------------------------------------------------------\n-- The principle of the excluded middle.\npostulate pem : \u2200 {A} \u2192 A \u2228 \u00ac A\n{-# ATP prove pem #-}\n\n-- The principle of indirect proof (proof by contradiction).\npostulate \u00ac-elim : \u2200 {A} \u2192 (\u00ac A \u2192 \u22a5) \u2192 A\n{-# ATP prove \u00ac-elim #-}\n\n-- Double negation elimination.\npostulate \u00ac\u00ac-elim : \u2200 {A} \u2192 \u00ac \u00ac A \u2192 A\n{-# ATP prove \u00ac\u00ac-elim #-}\n\n-- The reductio ab absurdum rule. (Some authors uses this name for the\n-- principle of indirect proof).\npostulate raa : \u2200 {A} \u2192 (\u00ac A \u2192 A) \u2192 A\n{-# ATP prove raa #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate \u00ac\u2203\u00ac\u2192\u2200 : {A : D \u2192 Set} \u2192 \u00ac (\u2203[ x ] \u00ac A x) \u2192 \u2200 {x} \u2192 A x\n{-# ATP prove \u00ac\u2203\u00ac\u2192\u2200 #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Non-intuitionistic logic theorems\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOL.NonIntuitionistic.TheoremsATP where\n\n-- The ATPs work in classical logic, therefore we also can prove some\n-- non-intuitionistic logic theorems.\n\nopen import FOL.Base hiding ( pem )\n\n------------------------------------------------------------------------------\n-- The principle of the excluded middle.\npostulate pem : \u2200 {A} \u2192 A \u2228 \u00ac A\n{-# ATP prove pem #-}\n\n-- The principle of indirect proof (proof by contradiction).\npostulate \u00ac-elim : \u2200 {A} \u2192 (\u00ac A \u2192 \u22a5) \u2192 A\n{-# ATP prove \u00ac-elim #-}\n\n-- Double negation elimination.\npostulate \u00ac\u00ac-elim : \u2200 {A} \u2192 \u00ac \u00ac A \u2192 A\n{-# ATP prove \u00ac\u00ac-elim #-}\n\n-- The reductio ab absurdum rule. (Some authors uses this name for the\n-- principle of indirect proof).\npostulate raa : \u2200 {A} \u2192 (\u00ac A \u2192 A) \u2192 A\n{-# ATP prove raa #-}\n\n-- \u2203 in terms of \u2200 and \u00ac.\npostulate \u00ac\u2203\u00ac\u2192\u2200 : {A : D \u2192 Set} \u2192 \u00ac (\u2203[ x ] \u00ac A x) \u2192 \u2200 {x} \u2192 A x\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"bf48517ecef7f91ec0d1628cc7636fd8139d5206","subject":"Typos.","message":"Typos.\n\nIgnore-this: 7034a047e46c9c8ca3be2cfedf871eda\n\ndarcs-hash:20110719140945-3bd4e-31a13a583b7c766e5b96868fcd29d5d4f47d7234.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Program\/MapIterate\/MapIterateATP.agda","new_file":"src\/FOTC\/Program\/MapIterate\/MapIterateATP.agda","new_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using coinduction\n------------------------------------------------------------------------------\n\n-- The map-iterate property [1]:\n-- map f (iterate f x) = iterate f (f \u00b7 x)\n\n-- [1] Jeremy Gibbons and Graham Hutton. Proof methods for corecursive\n-- programs. Fundamenta Informaticae, XX:1\u201314, 2005.\n\nmodule FOTC.Program.MapIterate.MapIterateATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- The map-iterate property.\n\n\u2248-map-iterate : (f x : D) \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-gfp\u2082 R (\u03bb {xs} {ys} \u2192 helper xs ys) (x , refl , refl)\n  where\n  -- The relation R was based on the relation used by Eduardo Gim\u00e9nez\n  -- and Pierre Cast\u00e9ran. A Tutorial on [Co-]Inductive Types in\n  -- Coq. May 1998 -- August 17, 2007.\n  R : D \u2192 D \u2192 Set\n  R xs ys = \u2203 (\u03bb y \u2192 xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y))\n\n  helper : \u2200 xs ys \u2192 R xs ys \u2192 \u2203 (\u03bb x' \u2192 \u2203 (\u03bb xs' \u2192 \u2203 (\u03bb ys' \u2192\n             R xs' ys' \u2227\n             xs \u2261 x' \u2237 xs' \u2227\n             ys \u2261 x' \u2237 ys')))\n  helper xs ys h = (f \u00b7 y)\n                   , (map f (iterate f (f \u00b7 y)))\n                   , (iterate f (f \u00b7 (f \u00b7 y)))\n                   , ((f \u00b7 y) , refl , refl)\n                   , (trans xs\u2261map unfoldMap)\n                   , trans ys\u2261iterate unfoldIterate\n    where\n    y : D\n    y = \u2203-proj\u2081 h\n\n    xs\u2261map : xs \u2261 map f (iterate f y)\n    xs\u2261map = \u2227-proj\u2081 (\u2203-proj\u2082 h)\n\n    ys\u2261iterate : ys \u2261 iterate f (f \u00b7 y)\n    ys\u2261iterate = \u2227-proj\u2082 (\u2203-proj\u2082 h)\n\n    postulate\n      unfoldMap : map f (iterate f y) \u2261 f \u00b7 y \u2237 map f (iterate f (f \u00b7 y))\n      unfoldIterate : iterate f (f \u00b7 y) \u2261 f \u00b7 y \u2237 iterate f (f \u00b7 (f \u00b7 y))\n    {-# ATP prove unfoldMap #-}\n    {-# ATP prove unfoldIterate #-}\n","old_contents":"------------------------------------------------------------------------------\n-- The map-iterate property: A property using coinduction\n------------------------------------------------------------------------------\n\n-- The map-iterate property [1]:\n-- map f (iterate f x) \u2261 iterate f (f \u00b7 x)\n\n-- [1] Jeremy Gibbons and Graham Hutton. Proof methods for corecursive\n-- programs. Fundamenta Informaticae, XX:1\u201314, 2005.\n\nmodule FOTC.Program.MapIterate.MapIterateATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- The map-iterate property.\n\n\u2248-map-iterate : (f x : D) \u2192 map f (iterate f x) \u2248 iterate f (f \u00b7 x)\n\u2248-map-iterate f x = \u2248-gfp\u2082 R (\u03bb {xs} {ys} \u2192 helper xs ys) (x , refl , refl)\n  where\n  -- The relation R was based on the relation used by Eduardo Gim\u00e9nez\n  -- Pierre Cast\u00e9ran. A Tutorial on [Co-]Inductive Types in Coq May\n  -- 1998 -- August 17, 2007.\n  R : D \u2192 D \u2192 Set\n  R xs ys = \u2203 (\u03bb y \u2192 xs \u2261 map f (iterate f y) \u2227 ys \u2261 iterate f (f \u00b7 y))\n\n  helper : \u2200 xs ys \u2192 R xs ys \u2192 \u2203 (\u03bb x' \u2192 \u2203 (\u03bb xs' \u2192 \u2203 (\u03bb ys' \u2192\n             R xs' ys' \u2227\n             xs \u2261 x' \u2237 xs' \u2227\n             ys \u2261 x' \u2237 ys')))\n  helper xs ys h = (f \u00b7 y)\n                   , (map f (iterate f (f \u00b7 y)))\n                   , (iterate f (f \u00b7 (f \u00b7 y)))\n                   , ((f \u00b7 y) , refl , refl)\n                   , (trans xs\u2261map unfoldMap)\n                   , trans ys\u2261iterate unfoldIterate\n    where\n    y : D\n    y = \u2203-proj\u2081 h\n\n    xs\u2261map : xs \u2261 map f (iterate f y)\n    xs\u2261map = \u2227-proj\u2081 (\u2203-proj\u2082 h)\n\n    ys\u2261iterate : ys \u2261 iterate f (f \u00b7 y)\n    ys\u2261iterate = \u2227-proj\u2082 (\u2203-proj\u2082 h)\n\n    postulate\n      unfoldMap : map f (iterate f y) \u2261 f \u00b7 y \u2237 map f (iterate f (f \u00b7 y))\n      unfoldIterate : iterate f (f \u00b7 y) \u2261 f \u00b7 y \u2237 iterate f (f \u00b7 (f \u00b7 y))\n    {-# ATP prove unfoldMap #-}\n    {-# ATP prove unfoldIterate #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d8eea7ea8f46fd7381f148e288d82fee056d9110","subject":"Learn we can't prove change composition correct in this setting","message":"Learn we can't prove change composition correct in this setting\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/SIRelBigStep\/DeriveCorrect.agda","new_file":"Thesis\/SIRelBigStep\/DeriveCorrect.agda","new_contents":"--\n-- In this module we conclude our proof: we have shown that t, derive-dterm t\n-- and produce related values (in the sense of the fundamental property,\n-- rfundamental3), and for related values v1, dv and v2 we have v1 \u2295 dv \u2261\n-- v2 (because \u2295 agrees with validity, rrelV3\u2192\u2295).\n--\n-- We now put these facts together via derive-correct-si and derive-correct.\n-- This is immediate, even though I spend so much code on it:\n-- Indeed, all theorems are immediate corollaries of what we established (as\n-- explained above); only the statements are longer, especially because I bother\n-- expanding them.\n\nmodule Thesis.SIRelBigStep.DeriveCorrect where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\nopen import Thesis.SIRelBigStep.FundamentalProperty\n\n-- Theorem statement. This theorem still mentions step-indexes explicitly.\nderive-correct-si-type =\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4) \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3-skeleton (\u03bb v1 dv v2 _ \u2192 v1 \u2295 dv \u2261 v2) t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\n-- A verified expansion of the theorem statement.\nderive-correct-si-type-means :\n  derive-correct-si-type \u2261\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j (j<k : j < k) \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct-si-type-means = refl\n\nderive-correct-si : derive-correct-si-type\nderive-correct-si t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rrelT3\u2192\u2295 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 (rfundamental3 _ t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1)\n\n-- Infinitely related environments:\nrrel\u03c13-inf : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12 = \u2200 k \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k\n\n-- A theorem statement for infinitely-related environments. On paper, if we informally\n-- omit the arbitrary derivation step-index as usual, we get a statement without\n-- step-indexes.\nderive-correct :\n  \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j = derive-correct-si t \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 (suc j)) v1 v2 j \u2264-refl\n\nopen import Data.Unit\n\nnil\u03c1 : \u2200 {\u0393 n} (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c1 d\u03c1 \u03c1 n\nnilV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\nnilV (closure t \u03c1) = let d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dclosure (derive-dterm t) \u03c1 d\u03c1 , rfundamental3svv _ (abs t) \u03c1 d\u03c1 \u03c1 \u03c1\u03c1\nnilV (natV n\u2081) = dnatV zero , refl\nnilV (pairV a b) = let 0a , aa = nilV a; 0b , bb = nilV b in dpairV 0a 0b , aa , bb\n\nnil\u03c1 \u2205 = \u2205 , tt\nnil\u03c1 (v \u2022 \u03c1) = let dv , vv = nilV v ; d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n\n-- Try to prove to define validity-preserving change composition.\n-- Sadly, if we don't store proofs that environment changes in closures are valid, we can't finish the proof for the closure case :-(.\n-- Moreover, the proof is annoying to do unless we build a datatype to invert\n-- validity proofs (as suggested by Ezyang in\n-- http:\/\/blog.ezyang.com\/2013\/09\/induction-and-logical-relations\/).\n\n-- ocompose\u03c1 : \u2200 {\u0393} n \u03c11 d\u03c11 \u03c12 d\u03c12 \u03c13 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c11 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c12 d\u03c12 \u03c13 n \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c13 n\n\n-- ocomposev : \u2200 {\u03c4} n v1 dv1 v2 dv2 v3 \u2192 rrelV3 \u03c4 v1 dv1 v2 n \u2192 rrelV3 \u03c4 v2 dv2 v3 n \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v1 dv v3 n\n-- ocomposev n v1 dv1 v2 (bang v3) .v3 vv1 refl = bang v3 , refl\n-- ocomposev n v1 (bang x) v2 (dclosure dt \u03c1 d\u03c1) v3 vv1 vv2 = {!!}\n-- ocomposev n (closure t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c11) (closure .t .(\u03c1 \u2295\u03c1 d\u03c11)) (dclosure .(derive-dterm t) .(\u03c1 \u2295\u03c1 d\u03c11) d\u03c12) (closure .t .((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12)) ((refl , refl) , refl , refl , refl , refl , p5) ((refl , refl) , refl , refl , refl , refl , p5q) =\n--   let d\u03c1 , \u03c1\u03c1 = ocompose\u03c1 n \u03c1 d\u03c11 (\u03c1 \u2295\u03c1 d\u03c11) d\u03c12 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12) {!!} {!!}\n--   in dclosure (derive-dterm t) \u03c1 d\u03c1 , (refl , refl) , refl , {!!} , refl , refl ,\n--     \u03bb { k (s\u2264s k<n) v1 dv v2 vv \u2192\n--       rfundamental3 k t (v1 \u2022 \u03c1) (dv \u2022 d\u03c1) (v2 \u2022 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12)) (vv , rrel\u03c13-mono k n (\u2264-step k<n) _ \u03c1 d\u03c1 ((\u03c1 \u2295\u03c1 d\u03c11) \u2295\u03c1 d\u03c12) \u03c1\u03c1)}\n-- -- p1 , p2 , p3 , p4 , p5\n-- ocomposev n v1 dv1 v2 (dnatV n\u2081) v3 vv1 vv2 = {!!}\n-- ocomposev n v1 dv1 v2 (dpairV dv2 dv3) v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (bang v2) .v2 dv2 v3 refl vv2 = bang (v2 \u2295 dv2) , rrelV3\u2192\u2295 v2 dv2 v3 vv2\n-- -- ocomposev n v1 (dclosure dt \u03c1 d\u03c1) v2 dv2 v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (dnatV n\u2081) v2 dv2 v3 vv1 vv2 = {!!}\n-- -- ocomposev n v1 (dpairV dv1 dv2) v2 dv3 v3 vv1 vv2 = {!!}\n-- -- ocomposeV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\n\n-- ocompose\u03c1 {\u2205} n \u2205 d\u03c11 \u03c12 d\u03c12 \u2205 \u03c1\u03c11 \u03c1\u03c12 = \u2205 , tt\n-- ocompose\u03c1 {\u03c4 \u2022 \u0393} n (v1 \u2022 \u03c11) (dv1 \u2022 d\u03c11) (v2 \u2022 \u03c12) (dv2 \u2022 d\u03c12) (v3 \u2022 \u03c13) (vv1 , \u03c1\u03c11) (vv2 , \u03c1\u03c12) =\n--   let dv , vv = ocomposev n v1 dv1 v2 dv2 v3 vv1 vv2\n--       d\u03c1 , \u03c1\u03c1 = ocompose\u03c1 n \u03c11 d\u03c11 \u03c12 d\u03c12 \u03c13 \u03c1\u03c11 \u03c1\u03c12\n--   in  dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n\n-- But we sort of know how to store environments validity proofs in closures, you just need to use well-founded inductions, even if you're using types. Sad, yes, that's insanely tedious. Let's leave that to Coq.\n-- What is also interesting: if that were fixed, could we prove correct\n-- composition for change *terms* and rrelT3? And for *open expressions*? The\n-- last is the main problem Ahmed run into.\n","old_contents":"--\n-- In this module we conclude our proof: we have shown that t, derive-dterm t\n-- and produce related values (in the sense of the fundamental property,\n-- rfundamental3), and for related values v1, dv and v2 we have v1 \u2295 dv \u2261\n-- v2 (because \u2295 agrees with validity, rrelV3\u2192\u2295).\n--\n-- We now put these facts together via derive-correct-si and derive-correct.\n-- This is immediate, even though I spend so much code on it:\n-- Indeed, all theorems are immediate corollaries of what we established (as\n-- explained above); only the statements are longer, especially because I bother\n-- expanding them.\n\nmodule Thesis.SIRelBigStep.DeriveCorrect where\n\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nopen import Thesis.SIRelBigStep.IlcSILR\nopen import Thesis.SIRelBigStep.FundamentalProperty\n\n-- Theorem statement. This theorem still mentions step-indexes explicitly.\nderive-correct-si-type =\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4) \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3-skeleton (\u03bb v1 dv v2 _ \u2192 v1 \u2295 dv \u2261 v2) t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\n\n-- A verified expansion of the theorem statement.\nderive-correct-si-type-means :\n  derive-correct-si-type \u2261\n  \u2200 {\u03c4 \u0393 k} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j (j<k : j < k) \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct-si-type-means = refl\n\nderive-correct-si : derive-correct-si-type\nderive-correct-si t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rrelT3\u2192\u2295 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 (rfundamental3 _ t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1)\n\n-- Infinitely related environments:\nrrel\u03c13-inf : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set\nrrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12 = \u2200 k \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k\n\n-- A theorem statement for infinitely-related environments. On paper, if we informally\n-- omit the arbitrary derivation step-index as usual, we get a statement without\n-- step-indexes.\nderive-correct :\n  \u2200 {\u03c4 \u0393} (t : Term \u0393 \u03c4)\n  \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13-inf \u0393 \u03c11 d\u03c1 \u03c12) \u2192\n  (v1 v2 : Val \u03c4) \u2192\n  \u2200 j \u2192\n  (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t \u2193[ i' j ] v1) \u2192\n  (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t \u2193[ no ] v2) \u2192\n  \u03a3[ dv \u2208 DVal \u03c4 ]\n  \u03c11 D d\u03c1 \u22a2 derive-dterm t \u2193 dv \u00d7\n  v1 \u2295 dv \u2261 v2\nderive-correct t \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 j = derive-correct-si t \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 (suc j)) v1 v2 j \u2264-refl\n\nopen import Data.Unit\n\nnil\u03c1 : \u2200 {\u0393 n} (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u03a3[ d\u03c1 \u2208 Ch\u0394 \u0393 ] rrel\u03c13 \u0393 \u03c1 d\u03c1 \u03c1 n\nnilV : \u2200 {\u03c4 n} (v : Val \u03c4) \u2192 \u03a3[ dv \u2208 DVal \u03c4 ] rrelV3 \u03c4 v dv v n\nnilV (closure t \u03c1) = let d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dclosure (derive-dterm t) \u03c1 d\u03c1 , rfundamental3svv _ (abs t) \u03c1 d\u03c1 \u03c1 \u03c1\u03c1\nnilV (natV n\u2081) = dnatV zero , refl\nnilV (pairV a b) = let 0a , aa = nilV a; 0b , bb = nilV b in dpairV 0a 0b , aa , bb\n\nnil\u03c1 \u2205 = \u2205 , tt\nnil\u03c1 (v \u2022 \u03c1) = let dv , vv = nilV v ; d\u03c1 , \u03c1\u03c1 = nil\u03c1 \u03c1 in dv \u2022 d\u03c1 , vv , \u03c1\u03c1\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b01133115d09b68bf5e1f95bbfba9dcfa2a593c3","subject":"proved that indets arent errors #14","message":"proved that indets arent errors #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err x)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 ((d' : dhexp) (\u03c4' : htyp) \u2192 d \u2260 (d' \u27e8 \u03c4' \u21d2 \u2987\u2988 \u27e9)) \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) f = abort (f _ _ refl)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2082) ce) _ = CECong x\u2082 ce\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  vi (BVVal VConst) ()\n  vi (BVVal VLam) ()\n  vi (BVArrCast x bv) (ICastArr x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = vi bv ind\n  vi (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = vi bv ind\n\n  -- boxed values and errors are disjoint\n  ve : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  ve (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  ve (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  ve (BVArrCast x bv) (CECong FHOuter (CECong eps er)) = {!!}\n  ve (BVArrCast x bv) (CECong (FHCast x\u2081) er) = ve bv (CECong x\u2081 er)\n  ve (BVHoleCast x bv) (CECong x\u2081 er) = {!!}\n  ve (BVVal x) (CECong x\u2081 er) = ve (lem-valfill x x\u2081) er\n    where\n    lem-valfill : \u2200{\u03b5 d d'} \u2192 d val \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' boxedval\n    lem-valfill VConst FHOuter = BVVal VConst\n    lem-valfill VLam FHOuter = BVVal VLam\n\n  -- boxed values and expressions that step are disjoint\n  vs : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  vs (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  vs (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  vs (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  vs (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n  vs (BVHoleCast x bv) (d' , Step FHOuter x\u2082 FHOuter) = {!x\u2082!} -- cyrus\n  vs (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = vs bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- todo: there's something going on with these lemmas here that's\n    -- repetative and i don't quite understand how to make it more compact\n\n    -- indeterminates and errors are disjoint\n    ie : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    ie IEHole (CECong FHOuter err) = ie IEHole err\n    ie (INEHole x) (CECong FHOuter err) = fe x (lem err)\n      where\n      lem : \u2200{d u \u03c3} \u2192 \u2987 d \u2988\u27e8 u , \u03c3 \u27e9 casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHNEHole x\u2081) ce) = CECong x\u2081 ce\n    ie (INEHole x) (CECong (FHNEHole x\u2081) err) = fe x (CECong x\u2081 err)\n    ie (IAp x indet x\u2081) (CECong FHOuter err)\n      with lem err\n      where\n      lem : \u2200{d1 d2} \u2192 (d1 \u2218 d2) casterr \u2192 (d1 casterr + d2 casterr)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHAp1 x\u2082) ce) = Inl (CECong x\u2082 ce)\n      lem (CECong (FHAp2 x\u2082 x\u2083) ce) = Inr (CECong x\u2083 ce)\n    ... | Inl d1err = ie indet d1err\n    ... | Inr d2err = fe x\u2081 d2err\n      where\n    ie (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = ie indet (CECong x\u2082 err)\n    ie (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = fe x\u2081 (CECong x\u2083 err)\n    ie (ICastArr x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c41 \u03c42 \u03c43 \u03c44} \u2192 (d \u27e8 \u03c41 ==> \u03c42 \u21d2 \u03c43 ==> \u03c44 \u27e9) casterr \u2192 d casterr\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastArr x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    ie (ICastGroundHole x indet) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u03c4 \u21d2 \u2987\u2988 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2081 x\u2082 () x\u2084)\n      lem (CECong FHOuter ce) = lem ce\n      lem (CECong (FHCast x\u2081) ce) = CECong x\u2081 ce\n    ie (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = ie indet (CECong x\u2081 err)\n    ie (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    ie (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = ie indet (lem err)\n      where\n      lem : \u2200{d \u03c4} \u2192 (d \u27e8 \u2987\u2988 \u21d2 \u03c4 \u27e9) casterr \u2192 d casterr\n      lem (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = {!!}\n      lem (CECong x\u2082 ce) = {!!}\n    ie (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = ie indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    fe : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    fe (FBoxed x) err = ve x err\n    fe (FIndet x) err = ie x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = vi x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole x ind) stp = {!!}\n  lem2 (ICastHoleGround x ind x\u2081) stp = {!!}\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081) = {!!} -- cyrus\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    is : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    is IEHole (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (d' , Step FHOuter () FHOuter)\n    is (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    is (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    is (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    is (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = is ind (_ , Step x\u2082 x\u2083 x\u2084)\n    is (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = fs f (_ , Step x\u2083 x\u2084 x\u2086)\n    is (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    is (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FBoxed x)) FHOuter) = vi x ind\n    is (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround (FIndet x)) FHOuter) = {!!} -- cyrus\n    is (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n    is (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    is (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    is (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    is (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = is ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    fs : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    fs (FBoxed x) stp = vs x stp\n    fs (FIndet x) stp = is x stp\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  es er stp = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"88aae6869cedaa8d81844a18827528db6dc5d98e","subject":"Added indDom.","message":"Added indDom.\n\nIgnore-this: 4caeff32954b707404a01a96a0c95c86\n\ndarcs-hash:20110405194626-3bd4e-f1bde760aba5296a166fe70f48370dd068760c5f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_file":"Draft\/FOTC\/Program\/Nest\/DomainPredicate.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- Inductive principle associated to the domain predicate.\nindDom : (P : D \u2192 Set) \u2192\n         P zero \u2192\n         (\u2200 {d} \u2192 Dom d \u2192 P d \u2192 Dom (nest d) \u2192 P (nest d) \u2192 P (succ d)) \u2192\n         \u2200 {d} \u2192 Dom d \u2192 P d\nindDom P P0 ih dom0           = P0\nindDom P P0 ih (domS d h\u2081 h\u2082) = ih h\u2081 (indDom P P0 ih h\u2081) h\u2082 (indDom P P0 ih h\u2082)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 dom0 =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) (sN Nn) prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn (sN Nn) (nest-\u2264 h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC version of the domain predicate of a nested recursive function\n-- given by the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. Vol. 2152 of LNCS. 2001\n\nmodule Draft.FOTC.Program.Nest.DomainPredicate where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.Acc.WellFoundedInductionI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\npostulate\n  nest   : D \u2192 D\n  nest-0 :       nest zero     \u2261 zero\n  nest-S : \u2200 d \u2192 nest (succ d) \u2261 nest (nest d)\n\ndata Dom : D \u2192 Set where\n  dom0 :                              Dom zero\n  domS : \u2200 d \u2192 Dom d \u2192 Dom (nest d) \u2192 Dom (succ d)\n\n-- The domain predicate is total.\ndom-N : \u2200 d \u2192 Dom d \u2192 N d\ndom-N .zero     dom0           = zN\ndom-N .(succ d) (domS d h\u2081 h\u2082) = sN (dom-N d h\u2081)\n\nnest-x\u22610 : \u2200 {n} \u2192 N n \u2192 nest n \u2261 zero\nnest-x\u22610 zN      = nest-0\nnest-x\u22610 (sN {n} Nn) =\n  begin\n    nest (succ n)\n      \u2261\u27e8 nest-S n \u27e9\n    nest (nest n)\n      \u2261\u27e8 subst (\u03bb t \u2192 nest (nest n) \u2261 nest t)\n               (nest-x\u22610 Nn)  -- IH.\n               refl\n      \u27e9\n    nest zero\n      \u2261\u27e8 nest-0 \u27e9\n    zero\n  \u220e\n\n-- The nest function is total in its domain (via structural recursion\n-- in the domain predicate).\nnest-DN : \u2200 {d} \u2192 Dom d \u2192 N (nest d)\nnest-DN dom0           = subst N (sym nest-0) zN\nnest-DN (domS d h\u2081 h\u2082) = subst N (sym (nest-S d)) (nest-DN h\u2082)\n\n-- The nest function is total.\nnest-N : \u2200 {n} \u2192 N n \u2192 N (nest n)\nnest-N Nn = subst N (sym (nest-x\u22610 Nn)) zN\n\nnest-\u2264 : \u2200 {n} \u2192 Dom n \u2192 LE (nest n) n\nnest-\u2264 dom0 =\n  begin\n    nest zero \u2264 zero\n      \u2261\u27e8 subst (\u03bb t \u2192 nest zero \u2264 zero \u2261 t \u2264 zero)\n               nest-0\n               refl\n      \u27e9\n    zero \u2264 zero\n      \u2261\u27e8 x\u2264x zN \u27e9\n    true\n  \u220e\n\nnest-\u2264 (domS n h\u2081 h\u2082) =\n  \u2264-trans (nest-N (sN (dom-N n h\u2081))) (nest-N (dom-N n h\u2081)) (sN Nn) prf\u2081 prf\u2082\n    where\n      Nn : N n\n      Nn = dom-N n h\u2081\n\n      prf\u2081 : LE (nest (succ n)) (nest n)\n      prf\u2081 =\n        begin\n          nest (succ n) \u2264 nest n\n            \u2261\u27e8 subst (\u03bb t \u2192 nest (succ n) \u2264 nest n \u2261 t \u2264 nest n)\n                     (nest-S n)\n                     refl\n            \u27e9\n          nest (nest n) \u2264 nest n\n            \u2261\u27e8 nest-\u2264 h\u2082 \u27e9\n          true\n          \u220e\n\n      prf\u2082 : LE (nest n) (succ n)\n      prf\u2082 = \u2264-trans (nest-N (dom-N n h\u2081)) Nn (sN Nn) (nest-\u2264 h\u2081) (x\u2264Sx Nn)\n\nN\u2192Dom : \u2200 {n} \u2192 N n \u2192 Dom n\nN\u2192Dom = wfInd-LT P ih\n  where\n    P : D \u2192 Set\n    P = Dom\n\n    ih : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 LT y x \u2192 P y) \u2192 P x\n    ih zN          h = dom0\n    ih (sN {x} Nx) h =\n      domS x dn-x (h (nest-N Nx ) (x\u2264y\u2192x<Sy (nest-N Nx) Nx (nest-\u2264 dn-x)))\n      where\n        dn-x : Dom x\n        dn-x = h Nx (x<Sx Nx)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"01f70c455ce9f9f3502ca6db9606a739e7b2837c","subject":"Fixed issue #5 (in Agda).","message":"Fixed issue #5 (in Agda).\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2ATP.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/Lemma2ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2ATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- Helper function for the ABP lemma 2\n\nmodule Helper where\n  -- We have these definitions outside the where clause to keep them\n  -- simple for the ATPs.\n\n  ds\u2075 : D \u2192 D \u2192 D\n  ds\u2075 cs' os\u2081\u2075 = corrupt \u00b7 os\u2081\u2075 \u00b7 cs'\n  {-# ATP definition ds\u2075 #-}\n\n  as\u2075 : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  as\u2075 b i' is' cs' os\u2081\u2075 = await b i' is' (ds\u2075 cs' os\u2081\u2075)\n  {-# ATP definition as\u2075 #-}\n\n  bs\u2075 : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  bs\u2075 b i' is' cs' os\u2080\u2075 os\u2081\u2075 = corrupt \u00b7 os\u2080\u2075 \u00b7 as\u2075 b i' is' cs' os\u2081\u2075\n  {-# ATP definition bs\u2075 #-}\n\n  cs\u2075 : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  cs\u2075 b i' is' cs' os\u2080\u2075 os\u2081\u2075 = ack \u00b7 not b \u00b7 bs\u2075 b i' is' cs' os\u2080\u2075 os\u2081\u2075\n  {-# ATP definition cs\u2075 #-}\n\n  os\u2080\u2075 : D \u2192 D\n  os\u2080\u2075 os\u2080' = tail\u2081 os\u2080'\n  {-# ATP definition os\u2080\u2075 #-}\n\n  os\u2081\u2075 : D \u2192 D \u2192 D\n  os\u2081\u2075 ft\u2081 os\u2081'' = ft\u2081 ++ os\u2081''\n  {-# ATP definition os\u2081\u2075 #-}\n\n  helper : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n           Bit b \u2192\n           Fair os\u2080' \u2192\n           ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n           \u2203[ ft\u2081 ] \u2203[ os\u2081'' ] F*T ft\u2081 \u2227 Fair os\u2081'' \u2227 os\u2081' \u2261 ft\u2081 ++ os\u2081'' \u2192\n           \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2080''\n           \u2227 Fair os\u2081''\n           \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n  helper {b} {i'} {is'} {js' = js'} Bb Fos\u2080' abp'\n         (.(T \u2237 []) , os\u2081'' , f*tnil , Fos\u2081'' , os\u2081'-eq) = prf\n    where\n    postulate\n      prf : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n            Fair os\u2080''\n            \u2227 Fair os\u2081''\n            \u2227 as'' \u2261 send \u00b7 not b \u00b7 is' \u00b7 ds''\n            \u2227 bs'' \u2261 corrupt \u00b7 os\u2080'' \u00b7 as''\n            \u2227 cs'' \u2261 ack \u00b7 not b \u00b7 bs''\n            \u2227 ds'' \u2261 corrupt \u00b7 os\u2081'' \u00b7 cs''\n            \u2227 js'  \u2261 out \u00b7 not b \u00b7 bs''\n    {-# ATP prove prf #-}\n\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' abp'\n         (.(F \u2237 ft\u2081) , os\u2081'' , f*tcons {ft\u2081} FTft\u2081 , Fos\u2081'' , os\u2081'-eq)\n         = helper Bb (tail-Fair Fos\u2080') ABP'IH (ft\u2081 , os\u2081'' , FTft\u2081 , Fos\u2081'' , refl)\n    where\n    postulate os\u2081'-eq-helper : os\u2081' \u2261 F \u2237 os\u2081\u2075 ft\u2081 os\u2081''\n    {-# ATP prove os\u2081'-eq-helper #-}\n\n    postulate ds'-eq : ds' \u2261 error \u2237 ds\u2075 cs' (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove ds'-eq os\u2081'-eq-helper #-}\n\n    postulate as'-eq : as' \u2261 < i' , b > \u2237 as\u2075 b i' is' cs' (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove as'-eq #-}\n\n    postulate\n      bs'-eq-helper\u2081 : os\u2080' \u2261 T \u2237 tail\u2081 os\u2080' \u2192\n                       bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove bs'-eq-helper\u2081 as'-eq #-}\n\n    postulate\n      bs'-eq-helper\u2082 : os\u2080' \u2261 F \u2237 tail\u2081 os\u2080' \u2192\n                       bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove bs'-eq-helper\u2082 as'-eq #-}\n\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n             \u2228 bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2080')\n\n    postulate\n      cs'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       cs' \u2261 b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove cs'-eq-helper\u2081 not-x\u2262x not-involutive #-}\n\n    postulate\n      cs'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       cs' \u2261 b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove cs'-eq-helper\u2082 not-involutive #-}\n\n    cs'-eq : cs' \u2261 b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n    postulate\n      js'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       js' \u2261 out \u00b7 not b\n                                 \u00b7 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove js'-eq-helper\u2081 not-x\u2262x #-}\n\n    postulate\n      js'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       js' \u2261 out \u00b7 not b\n                                 \u00b7 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove js'-eq-helper\u2082 #-}\n\n    js'-eq : js' \u2261 out \u00b7 not b \u00b7 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n    postulate ds\u2075-eq : ds\u2075 cs' (os\u2081\u2075 ft\u2081 os\u2081'') \u2261\n                       corrupt \u00b7 (os\u2081\u2075 ft\u2081 os\u2081'')\n                               \u00b7 (b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081''))\n\n    ABP'IH : ABP' b i' is'\n                  (os\u2080\u2075 os\u2080')\n                  (os\u2081\u2075 ft\u2081 os\u2081'')\n                  (as\u2075 b i' is' cs' (os\u2081\u2075 ft\u2081 os\u2081''))\n                  (bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081''))\n                  (cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081''))\n                  (ds\u2075 cs' (os\u2081\u2075 ft\u2081 os\u2081''))\n                  js'\n    ABP'IH = ds\u2075-eq , refl , refl , refl , js'-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081 \u2208\n-- Fair, and hence by unfolding Fair we conclude that there are ft\u2081 :\n-- F*T and os\u2081'' : Fair, such that os\u2081' = ft\u2081 ++ os\u2081''.\n--\n-- We proceed by induction on ft\u2081 : F*T using helper.\n\nopen Helper\nlemma\u2082 : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2080' \u2192\n         Fair os\u2081' \u2192\n         ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n         Fair os\u2080''\n         \u2227 Fair os\u2081''\n         \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2080' Fos\u2081' abp' = helper Bb Fos\u2080' abp' (Fair-unf Fos\u2081')\n","old_contents":"------------------------------------------------------------------------------\n-- ABP lemma 2\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The second lemma states that given\n-- a state of the latter kind (see lemma 1) we will arrive at a new\n-- start state, which is identical to the old start state except that\n-- the bit has alternated and the first item in the input stream has\n-- been removed.\n\nmodule FOTC.Program.ABP.Lemma2ATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n-- Helper function for the ABP lemma 2\n\nmodule Helper where\n  -- We have these definitions outside the where clause to keep them\n  -- simple for the ATPs.\n\n  ds\u2075 : D \u2192 D \u2192 D\n  ds\u2075 cs' os\u2081\u2075 = corrupt \u00b7 os\u2081\u2075 \u00b7 cs'\n  {-# ATP definition ds\u2075 #-}\n\n  as\u2075 : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  as\u2075 b i' is' cs' os\u2081\u2075 = await b i' is' (ds\u2075 cs' os\u2081\u2075)\n  {-# ATP definition as\u2075 #-}\n\n  bs\u2075 : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  bs\u2075 b i' is' cs' os\u2080\u2075 os\u2081\u2075 = corrupt \u00b7 os\u2080\u2075 \u00b7 as\u2075 b i' is' cs' os\u2081\u2075\n  {-# ATP definition bs\u2075 #-}\n\n  cs\u2075 : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\n  cs\u2075 b i' is' cs' os\u2080\u2075 os\u2081\u2075 = ack \u00b7 not b \u00b7 bs\u2075 b i' is' cs' os\u2080\u2075 os\u2081\u2075\n  {-# ATP definition cs\u2075 #-}\n\n  os\u2080\u2075 : D \u2192 D\n  os\u2080\u2075 os\u2080' = tail\u2081 os\u2080'\n  {-# ATP definition os\u2080\u2075 #-}\n\n  os\u2081\u2075 : D \u2192 D \u2192 D\n  os\u2081\u2075 ft\u2081 os\u2081'' = ft\u2081 ++ os\u2081''\n  {-# ATP definition os\u2081\u2075 #-}\n\n  helper : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n           Bit b \u2192\n           Fair os\u2080' \u2192\n           ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n           \u2203[ ft\u2081 ] \u2203[ os\u2081'' ] F*T ft\u2081 \u2227 Fair os\u2081'' \u2227 os\u2081' \u2261 ft\u2081 ++ os\u2081'' \u2192\n           \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n           Fair os\u2080''\n           \u2227 Fair os\u2081''\n           \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n  helper {b} {i'} {is'} {js' = js'} Bb Fos\u2080' abp'\n         (.(T \u2237 []) , os\u2081'' , f*tnil , Fos\u2081'' , os\u2081'-eq) = prf\n    where\n    postulate\n      prf : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n            Fair os\u2080''\n            \u2227 Fair os\u2081''\n            \u2227 as'' \u2261 send \u00b7 not b \u00b7 is' \u00b7 ds''\n            \u2227 bs'' \u2261 corrupt \u00b7 os\u2080'' \u00b7 as''\n            \u2227 cs'' \u2261 ack \u00b7 not b \u00b7 bs''\n            \u2227 ds'' \u2261 corrupt \u00b7 os\u2081'' \u00b7 cs''\n            \u2227 js'  \u2261 out \u00b7 not b \u00b7 bs''\n    {-# ATP prove prf #-}\n\n  helper {b} {i'} {is'} {os\u2080'} {os\u2081'} {as'} {bs'} {cs'} {ds'} {js'}\n         Bb Fos\u2080' abp'\n         (.(F \u2237 ft\u2081) , os\u2081'' , f*tcons {ft\u2081} FTft\u2081 , Fos\u2081'' , os\u2081'-eq)\n         = helper Bb (tail-Fair Fos\u2080') ABP'IH (ft\u2081 , os\u2081'' , FTft\u2081 , Fos\u2081'' , refl)\n    where\n    postulate os\u2081'-eq-helper : os\u2081' \u2261 F \u2237 os\u2081\u2075 ft\u2081 os\u2081''\n    {-# ATP prove os\u2081'-eq-helper #-}\n\n    postulate ds'-eq : ds' \u2261 error \u2237 ds\u2075 cs' (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove ds'-eq os\u2081'-eq-helper #-}\n\n    postulate as'-eq : as' \u2261 < i' , b > \u2237 as\u2075 b i' is' cs' (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove as'-eq #-}\n\n    postulate\n      bs'-eq-helper\u2081 : os\u2080' \u2261 T \u2237 tail\u2081 os\u2080' \u2192\n                       bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove bs'-eq-helper\u2081 as'-eq #-}\n\n    postulate\n      bs'-eq-helper\u2082 : os\u2080' \u2261 F \u2237 tail\u2081 os\u2080' \u2192\n                       bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove bs'-eq-helper\u2081 as'-eq #-}\n\n    bs'-eq : bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n             \u2228 bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    bs'-eq = case (\u03bb h \u2192 inj\u2081 (bs'-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (bs'-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2080')\n\n    postulate\n      cs'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       cs' \u2261 b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove cs'-eq-helper\u2081 not-x\u2262x not-involutive #-}\n\n    postulate\n      cs'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       cs' \u2261 b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove cs'-eq-helper\u2082 not-involutive #-}\n\n    cs'-eq : cs' \u2261 b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    cs'-eq = case cs'-eq-helper\u2081 cs'-eq-helper\u2082 bs'-eq\n\n    postulate\n      js'-eq-helper\u2081 : bs' \u2261 ok < i' , b > \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       js' \u2261 out \u00b7 not b\n                                 \u00b7 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove js'-eq-helper\u2081 not-x\u2262x #-}\n\n    postulate\n      js'-eq-helper\u2082 : bs' \u2261 error \u2237 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'') \u2192\n                       js' \u2261 out \u00b7 not b\n                                 \u00b7 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    {-# ATP prove js'-eq-helper\u2082 #-}\n\n    js'-eq : js' \u2261 out \u00b7 not b \u00b7 bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081'')\n    js'-eq = case js'-eq-helper\u2081 js'-eq-helper\u2082 bs'-eq\n\n    postulate ds\u2075-eq : ds\u2075 cs' (os\u2081\u2075 ft\u2081 os\u2081'') \u2261\n                       corrupt \u00b7 (os\u2081\u2075 ft\u2081 os\u2081'')\n                               \u00b7 (b \u2237 cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081''))\n\n    ABP'IH : ABP' b i' is'\n                  (os\u2080\u2075 os\u2080')\n                  (os\u2081\u2075 ft\u2081 os\u2081'')\n                  (as\u2075 b i' is' cs' (os\u2081\u2075 ft\u2081 os\u2081''))\n                  (bs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081''))\n                  (cs\u2075 b i' is' cs' (os\u2080\u2075 os\u2080') (os\u2081\u2075 ft\u2081 os\u2081''))\n                  (ds\u2075 cs' (os\u2081\u2075 ft\u2081 os\u2081''))\n                  js'\n    ABP'IH = ds\u2075-eq , refl , refl , refl , js'-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081 \u2208\n-- Fair, and hence by unfolding Fair we conclude that there are ft\u2081 :\n-- F*T and os\u2081'' : Fair, such that os\u2081' = ft\u2081 ++ os\u2081''.\n--\n-- We proceed by induction on ft\u2081 : F*T using helper.\n\nopen Helper\nlemma\u2082 : \u2200 {b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'} \u2192\n         Bit b \u2192\n         Fair os\u2080' \u2192\n         Fair os\u2081' \u2192\n         ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js' \u2192\n         \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n         Fair os\u2080''\n         \u2227 Fair os\u2081''\n         \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\nlemma\u2082 Bb Fos\u2080' Fos\u2081' abp' = helper Bb Fos\u2080' abp' (Fair-unf Fos\u2081')\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3e950703614d0acb4c03e368cb393ba6f1922fe0","subject":"Fix hole in SILR for validity","message":"Fix hole in SILR for validity\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR2.agda","new_file":"Thesis\/FunBigStepSILR2.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192\n  (\u0393\u2081\u227c\u0393\u2082 : \u0393\u2081 \u227c \u0393\u2082) \u2192\n  Term \u0393\u2081 \u03c4 \u2192\n  Term \u0393\u2082 \u03c4\nweaken \u0393\u2081\u227c\u0393\u2082 (const c) = const c\nweaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (weaken-var \u0393\u2081\u227c\u0393\u2082 x)\nweaken \u0393\u2081\u227c\u0393\u2082 (app s t) = app (weaken \u0393\u2081\u227c\u0393\u2082 s) (weaken \u0393\u2081\u227c\u0393\u2082 t)\nweaken \u0393\u2081\u227c\u0393\u2082 (abs {\u03c3} t) = abs (weaken (keep \u03c3 \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3[ \u2261\u0393 \u2208 \u03931 \u2261 \u03932 ]\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT\n        t1\n        (subst (\u03bb \u0393 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4) (sym \u2261\u0393) t2)\n        (v1 \u2022 \u03c11)\n        (subst (\u03bb \u0393 \u2192 \u27e6 \u03c3 \u2022 \u0393 \u27e7Context) (sym \u2261\u0393) (v2 \u2022 \u03c12))\n        k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus. Now relT demands\n  -- environments with matching contexts, we could do the same here.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  -- Weakening in this definition is going to be annoying to use. And having to\n  -- construct terms is ugly.\n\n  -- Weakening could be avoided if we use a separate language of change terms\n  -- with two environments, and with a dclosure binding two variables at once,\n  -- and so on.\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 (app (weaken (drop (\u0394\u03c4 \u03c3) \u2022 \u227c-refl) dt) (var this)) t2 (v1 \u2022 \u03c11) (dv \u2022 v1 \u2022 d\u03c1) (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- But to betray the eventual goal, I can also relate integer values with a\n-- change in the relation witness. That was a completely local change. But that\n-- might also be because we only have few primitives.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, in preference to Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but I deviate\n-- somewhere, especially to try following \"Functional Big-Step Semantics\"),\n-- though I deviate somewhere.\n\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- interchangeable.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- CHEATS:\n-- \"Fuctional big-step semantics\" requires an external termination proof for the\n-- semantics. There it is also mechanized, here it isn't. Worse, the same\n-- termination problem affects some lemmas about the semantics.\n\nmodule Thesis.FunBigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value.\n\n-- WARNING: ISAC's big-step semantics produces a step count as \"output\". But\n-- that would not help Agda establish termination. That's only a problem for a\n-- functional big-step semantics, not for a relational semantics.\n--\n-- So, instead, I tried to use a sort of writer monad: the interpreter gets fuel\n-- and returns the remaining fuel. That's the same trick as in \"functional\n-- big-step semantics\". That *makes* the function terminating, even though Agda\n-- cannot see this because it does not know that the returned fuel is no bigger.\n\n-- Since we focus for now on STLC, unlike that\n-- paper, we could avoid error values because we keep types.\n--\n-- One could drop types and add error values instead.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 (n1 : \u2115) \u2192 ErrVal \u03c4\n  Error : ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = (n : \u2115) \u2192 ErrVal \u03c4\n\n_>>=_ : \u2200 {\u03c3 \u03c4} \u2192 Res \u03c3 \u2192 (Val \u03c3 \u2192 Res \u03c4) \u2192 Res \u03c4\n(s >>= t) n0 with s n0\n... | Done v n1 = t v n1\n... | Error = Error\n... | TimeOut = TimeOut\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v) n\n\n{-# TERMINATING #-}\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval (var x) \u03c1 n = Done (\u27e6 x \u27e7Var \u03c1) n\neval (abs t) \u03c1 n = Done (closure t \u03c1) n\neval (const c) \u03c1 n = evalConst c n\neval _ \u03c1 zero = TimeOut\neval (app s t) \u03c1 (suc n) = (eval s \u03c1 >>= (\u03bb sv \u2192 eval t \u03c1 >>= \u03bb tv \u2192 apply sv tv)) n\n\neval-const-dec : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192  n1 \u2264 n0\neval-const-dec (lit v) n0 .n0 refl = \u2264-refl\n\n{-# TERMINATING #-}\neval-dec : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 n1 \u2264 n0\neval-dec (const c) \u03c1 v n0 n1 eq = eval-const-dec c n0 n1 eq\neval-dec (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = \u2264-refl\neval-dec (app s t) \u03c1 v zero n1 ()\neval-dec (app s t) \u03c1 v (suc n0) n3 eq  with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv sn1 | [ seq ] with eval t \u03c1 sn1 | inspect (eval t \u03c1) sn1\neval-dec (app s t) \u03c1 v (suc n0) n3 eq | Done sv@(closure st s\u03c1) sn1 | [ seq ] | (Done tv tn2) | [ teq ] = \u2264-step (\u2264-trans (\u2264-trans (eval-dec st _ _ _ _ eq) (eval-dec t _ _ _ _ teq)) (eval-dec s _ _ _ _ seq))\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | Error | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Done sv sn1 | [ seq ] | TimeOut | [ teq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | Error | [ seq ]\neval-dec (app s t) \u03c1 v (suc n0) n3 () | TimeOut | [ seq ]\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c n0 \u2261 Done v n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1)\neval-const-mono (lit v) n0 .n0 refl = refl\n\n-- ARGH\n{-# TERMINATING #-}\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1)\neval-mono (const c) \u03c1 v n0 n1 eq = eval-const-mono c n0 n1 eq\neval-mono (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\neval-mono (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\neval-mono (app s t) \u03c1 v zero n1 ()\neval-mono (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\neval-mono (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\neval-mono (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-mono t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n2 () | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\neval-mono (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n\neval-adjust-plus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 n0 \u2261 Done v n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1)\neval-adjust-plus zero t \u03c1 v n0 n1 eq = eq\neval-adjust-plus (suc d) t \u03c1 v n0 n1 eq = eval-mono t \u03c1 v (d + n0) (d + n1) (eval-adjust-plus d t \u03c1 v n0 n1 eq)\n\neval-const-strengthen : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n0 n1 \u2192 evalConst c (suc n0) \u2261 Done v (suc n1) \u2192 evalConst c n0 \u2261 Done v n1\neval-const-strengthen (lit v) n0 .n0 refl = refl\n\n-- I started trying to prove eval-strengthen, which I appeal to informally\n-- below, but I gave up. I still guess the lemma is true but proving it looks\n-- too painful to bother.\n\n-- Without this lemma, I can't fully prove that this logical relation is\n-- equivalent to the original one.\n-- But this one works (well, at least up to the fundamental theorem, haven't\n-- attempted other lemmas), so it should be good enough.\n\n-- eval-mono-err : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 eval t \u03c1 n \u2261 Error \u2192 eval t \u03c1 (suc n) \u2261 Error\n-- eval-mono-err (const (lit x)) \u03c1 zero eq = {!!}\n-- eval-mono-err (const (lit x)) \u03c1 (suc n) eq = {!!}\n-- eval-mono-err (var x) \u03c1 n eq = {!!}\n-- eval-mono-err (app t t\u2081) \u03c1 n eq = {!!}\n-- eval-mono-err (abs t) \u03c1 n eq = {!!}\n\n-- -- eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n\n-- eval-aux : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 n \u2192 (\u03a3[ res0 \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 res0) \u00d7 (\u03a3[ resS \u2208 ErrVal \u03c4 ] eval t \u03c1 n \u2261 resS)\n-- eval-aux t \u03c1 n with\n--   eval t \u03c1 n | inspect (eval t \u03c1) n |\n--   eval t \u03c1 (suc n) | inspect (eval t \u03c1) (suc n)\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | (Done v1 n1) | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | res0 | [ eq0 ] | Error | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Done v n1 | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | Error | [ eq0 ] | TimeOut | [ eq1 ] = {!!}\n-- eval-aux t \u03c1 n | TimeOut | [ eq0 ] | TimeOut | [ eq1 ] = (TimeOut , refl) , (TimeOut , refl)\n\n-- {-# TERMINATING #-}\n-- eval-strengthen : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v n0 n1 \u2192 eval t \u03c1 (suc n0) \u2261 Done v (suc n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-strengthen (const c) \u03c1 v n0 n1 eq = eval-const-strengthen c n0 n1 eq\n-- eval-strengthen (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (abs t) \u03c1 .(closure t \u03c1) n0 .n0 refl = refl\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq with eval s \u03c1 0 | inspect (eval s \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv sn1 | [ seq ] with eval-dec s \u03c1 sv 0 sn1 seq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv .0 | [ seq ] | z\u2264n with eval t \u03c1 0 | inspect (eval t \u03c1) 0\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done sv _ | [ seq ] | z\u2264n | Done tv tn1 | [ teq ]  with eval-dec t \u03c1 tv 0 tn1 teq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv .0) | [ teq ] | z\u2264n with eval-dec st _ v 0 (suc n1) eq\n-- eval-strengthen (app s t) \u03c1 v zero n1 eq | Done (closure st s\u03c1) _ | [ seq ] | z\u2264n | (Done tv _) | [ teq ] | z\u2264n | ()\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | Error | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Done sv _ | [ seq ] | z\u2264n | TimeOut | [ teq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | Error | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v zero n1 () | TimeOut | [ seq ]\n-- -- eval-dec s \u03c1\n-- --  {!eval-dec s \u03c1 ? (suc zero) (suc n1) !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 (suc n0) | inspect (eval s \u03c1) (suc n0)\n-- -- eval-strengthen (app s t) \u03c1 v\u2081 (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 n0 = {!eval-strengthen s \u03c1 v n0 n1 seq !}\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | Error | [ seq ]\n-- -- eval-strengthen (app s t) \u03c1 v (suc n0) n1 () | TimeOut | [ seq ]\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq with eval s \u03c1 n0 | inspect (eval s \u03c1) n0\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] with eval s \u03c1 (suc n0) | eval-mono s \u03c1 sv n0 n1 seq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Done tv n2 | [ teq ] with eval t \u03c1 (suc n1) | eval-mono t \u03c1 tv n1 n2 teq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n3 eq | Done (closure t\u2081 \u03c1\u2081) n1 | [ seq ] | .(Done (closure {\u0393 = _} {\u03c3 = _} {\u03c4 = _} t\u2081 \u03c1\u2081) (suc n1)) | refl | (Done tv n2) | [ teq ] | .(Done tv (suc n2)) | refl = eval-strengthen t\u2081 (tv \u2022 \u03c1\u2081) v n2 n3 eq\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | Error | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n2 eq | Done sv n1 | [ seq ] | .(Done sv (suc n1)) | refl | TimeOut | [ teq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | Error | [ seq ] = {!!}\n-- eval-strengthen (app s t) \u03c1 v (suc n0) n1 eq | TimeOut | [ seq ] = {!!}\n\n-- eval-adjust-minus : \u2200 d {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 {\u03c1 v} n0 n1 \u2192 eval t \u03c1 (d + n0) \u2261 Done v (d + n1) \u2192 eval t \u03c1 n0 \u2261 Done v n1\n-- eval-adjust-minus zero t n0 n1 eq = eq\n-- eval-adjust-minus (suc d) t n0 n1 eq = eval-adjust-minus d t n0 n1 (eval-strengthen t _ _ (d + n0) (d + n1) eq)\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  -- Warning: compared to Ahmed's papers, this definition for relT also requires\n  -- t1 to be well-typed, not just t2.\n  --\n  -- This difference might affect the status of some proofs in Ahmed's papers,\n  -- but that's not a problem here.\n\n  -- Also: can't confirm this in any of the papers I'm using, but I'd guess that\n  -- all papers using environments allow to relate closures with different\n  -- implementations and different hidden environments.\n  --\n  -- To check if the proof goes through with equal context, I changed the proof.\n  -- Now a proof that two closures are equivalent contains a proof that their\n  -- typing contexts are equivalent. The changes were limited softawre\n  -- engineering, the same proofs go through.\n\n  -- This is not the same definition of relT, but it is equivalent.\n  relT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- This equation is a lemma in the original definition.\n  relT t1 t2 \u03c11 \u03c12 zero = \u22a4\n  -- To compare this definition, note that the original k is suc n here.\n  relT {\u03c4} t1 t2 \u03c11 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7 relV \u03c4 v1 v2 (suc n-j)\n    -- Here, computing t2 is allowed to take an unbounded number of steps. Having to write a number at all is annoying.\n\n  relV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Show the proof still goes through if we relate clearly different values by\n  -- inserting changes in the relation.\n  -- There's no syntax to produce such changes, but you can add changes to the\n  -- environment.\n  relV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  relV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3[ \u2261\u0393 \u2208 \u03931 \u2261 \u03932 ]\n      \u2200 (k : \u2115) (k\u2264n : k < n) v1 v2 \u2192\n      relV \u03c3 v1 v2 k \u2192\n      relT\n        t1\n        (subst (\u03bb \u0393 \u2192 Term (\u03c3 \u2022 \u0393) \u03c4) (sym \u2261\u0393) t2)\n        (v1 \u2022 \u03c11)\n        (subst (\u03bb \u0393 \u2192 \u27e6 \u03c3 \u2022 \u0393 \u27e7Context) (sym \u2261\u0393) (v2 \u2022 \u03c12))\n        k\n  -- Above, in the conclusion, I'm not relating app (closure t1 \u03c11) v1 with app\n  -- (closure t2 \u03c12) v2 (or some encoding of that that actually works), but the\n  -- result of taking a step from that configuration. That is important, because\n  -- both Pitts' \"Step-Indexed Biorthogonality: a Tutorial Example\" and\n  -- \"Imperative Self-Adjusting Computation\" do the same thing (and point out it's\n  -- important).\n\n  \u0394\u03c4 : Type \u2192 Type\n  \u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 (\u0394\u03c4 \u03c3) \u21d2 \u0394\u03c4 \u03c4\n  \u0394\u03c4 nat = nat\n\n  -- Since the original relation allows unrelated environments, we do that here\n  -- too. However, while that is fine as a logical relation, it's not OK if we\n  -- want to prove that validity agrees with oplus.\n  relT3 : \u2200 {\u03c4 \u03931 \u03932 \u0394\u0393} (t1 : Term \u03931 \u03c4) (dt : Term \u0394\u0393 (\u0394\u03c4 \u03c4)) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (d\u03c1 : \u27e6 \u0394\u0393 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\n  relT3 t1 dt t2 \u03c11 d\u03c1 \u03c12 zero = \u22a4\n  relT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 (suc n) =\n    (v1 : Val \u03c4) \u2192\n    \u2200 n-j (n-j\u2264n : n-j \u2264 n) \u2192\n    (eq : eval t1 \u03c11 n \u2261 Done v1 n-j) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] eval t2 \u03c12 n2 \u2261 Done v2 0 \u00d7\n    \u03a3[ dv \u2208 Val (\u0394\u03c4 \u03c4) ] \u03a3[ dn \u2208 \u2115 ] eval dt d\u03c1 dn \u2261 Done dv 0 \u00d7\n    relV3 \u03c4 v1 dv v2 (suc n-j)\n\n  relV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : Val (\u0394\u03c4 \u03c4)) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  relV3 nat (intV v1) (intV dv) (intV v2) n = dv + v1 \u2261 v2\n  relV3 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure dt d\u03c1) (closure t2 \u03c12) n =\n    \u2200 (k : \u2115) (k\u2264n : k < n) v1 dv v2 \u2192\n    relV3 \u03c3 v1 dv v2 k \u2192\n    relT3 t1 {!dt!} t2 (v1 \u2022 \u03c11) d\u03c1 (v2 \u2022 \u03c12) k\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 1 at any step count.\n  example1 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (const (lit 1)) \u2205) n\n  example1 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n v1 v2 x .(intV 0) .k n-j\u2264n refl \u2192 intV 1 , 0 , refl , (I.+ 1 , refl)\n      }\n\n  -- Relate \u03bb x \u2192 0 and \u03bb x \u2192 x at any step count.\n  example2 : \u2200 n \u2192 relV (nat \u21d2 nat) (closure (const (lit 0)) \u2205) (closure (var this) \u2205) n\n  example2 n = refl ,\n    \u03bb { zero k\u2264n v1 v2 x \u2192 tt\n      ; (suc k) k\u2264n (intV v1) (intV v2) x .(intV 0) .k n-j\u2264n refl \u2192 intV v2 , 0 , refl , (I.+ v2 , cong I.+_ (+-right-identity v2))\n      }\n\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 .n n-j\u2264n refl = v2 , zero , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 = tt\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\nfundamental (const (lit v)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV v) .n n-j\u2264n refl = intV v , zero , refl , I.+ zero , refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .n n-j\u2264n refl =  closure t \u03c12 , zero , refl , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k (suc n) (lt1 k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc zero) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n ()\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq with eval s \u03c11 n | inspect (eval s \u03c11) n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done sv1 n1 | [ s1eq ] with eval-dec s _ _ n n1 s1eq | eval t \u03c11 n1 | inspect (eval t \u03c11) n1\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n eq | Done (closure st1 s\u03c11) n1 | [ s1eq ] | n1\u2264n | Done tv1 n2 | [ t1eq ] with eval-dec t _ _ n1 n2 t1eq\n... | n2\u2264n1 with fundamental s (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 (closure st1 s\u03c11) (suc n1) (s\u2264s n1\u2264n) (eval-mono s \u03c11 (closure st1 s\u03c11) n n1 s1eq)\n  | fundamental t (suc (suc n1)) \u03c11 \u03c12 (rel\u03c1-mono (suc (suc n1)) (suc (suc n)) (s\u2264s (s\u2264s n1\u2264n)) _ _ _ \u03c1\u03c1) tv1 (suc n2) (s\u2264s n2\u2264n1) (eval-mono t \u03c11 tv1 n1 n2 t1eq)\n... | sv2@(closure st2 s\u03c12) , sn3 , s2eq , refl , svv | tv2 , tn3 , t2eq , tvv with svv (suc n2) (s\u2264s (s\u2264s n2\u2264n1)) tv1 tv2 (relV-mono (suc n2) (suc (suc n2)) (s\u2264s (n\u22641+n n2)) _ tv1 tv2 tvv) v1 n-j (eval-dec st1 _ _ _ _ eq) eq\n... | v2 , n3 , eq2 , vv = v2 , suc (sn3 + (tn3 + n3)) , comp , vv\n  where\n    s2eq-adj : eval s \u03c12 (sn3 + (tn3 + n3)) \u2261 Done (closure st2 s\u03c12) (tn3 + n3)\n    s2eq-adj rewrite +-comm sn3 (tn3 + n3)| cong (Done (closure st2 s\u03c12)) (sym (+-right-identity (tn3 + n3))) = eval-adjust-plus (tn3 + n3) s _ sv2 _ _ s2eq\n    t2eq-adj : eval t \u03c12 (tn3 + n3) \u2261 Done tv2 n3\n    t2eq-adj rewrite +-comm tn3 n3 | cong (Done tv2) (sym (+-right-identity n3)) = eval-adjust-plus n3 t _ tv2 _ _ t2eq\n\n    comp : (eval s \u03c12 >>= (\u03bb sv \u2192 eval t \u03c12 >>= apply sv))\n      (sn3 + (tn3 + n3))\n      \u2261 Done v2 0\n    comp rewrite s2eq-adj | t2eq-adj = eq2\n\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | Error | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Done sv1 n1 | [ s1eq ] | n1\u2264n | TimeOut | [ t1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | Error | [ s1eq ]\nfundamental (app s t) (suc (suc n)) \u03c11 \u03c12 \u03c1\u03c1 v1 n-j n-j\u2264n () | TimeOut | [ s1eq ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c9a0e322742287bc9c2a9d3e057a06d23971c82c","subject":"Refactor Syntax\/Language\/Calculus.agda","message":"Refactor Syntax\/Language\/Calculus.agda\n\nMove helpers inside record, so that if we open the record module with a\nrecord value:\n\n  open Calculus some-specific-calculus\n\nWe get all of the helper functions specialized to\n`some-specific-calculus`, like the fields.\n\nOld-commit-hash: dafe67297ca04cb5ab0369e8d1d7a8e2109a6c6d\n","repos":"inc-lc\/ilc-agda","old_file":"Syntax\/Language\/Calculus.agda","new_file":"Syntax\/Language\/Calculus.agda","new_contents":"module Syntax.Language.Calculus where\n\n-- Full description of a calculus in Plotkin style\n--\n-- Users should supply\n-- - base types\n-- - constants\n-- - \u0394type of base types\n-- - derivatives of constants\n\nopen import Syntax.Type.Plotkin public\nopen import Syntax.Term.Plotkin public\nopen import Syntax.Context public\nopen import Syntax.Context.Plotkin public\nopen import Syntax.DeltaContext public\n\nrecord Calculus : Set\u2081 where\n  constructor\n    calculus-with\n  field\n    basetype : Set\n    constant : Context {Type basetype} \u2192 Type basetype \u2192 Set\n    \u0394type : Type basetype \u2192 Type basetype\n    \u0394const : \u2200 {\u0393 \u03a3 \u03c4} \u2192 (c : constant \u03a3 \u03c4) \u2192\n      Term {basetype} {constant} \u0393\n        (internalizeContext basetype (\u0394Context\u2032 (Type basetype) \u0394type \u03a3) (\u0394type \u03c4))\n\n  type : Set\n  type = Type basetype\n\n  context : Set\n  context = Context {type}\n\n  term : context \u2192 type \u2192 Set\n  term = Term {basetype} {constant}\n\nopen Calculus public\n","old_contents":"module Syntax.Language.Calculus where\n\n-- Full description of a calculus in Plotkin style\n--\n-- Users should supply\n-- - base types\n-- - constants\n-- - \u0394type of base types\n-- - derivatives of constants\n\nopen import Syntax.Type.Plotkin public\nopen import Syntax.Term.Plotkin public\nopen import Syntax.Context public\nopen import Syntax.Context.Plotkin public\nopen import Syntax.DeltaContext public\n\nrecord Calculus : Set\u2081 where\n  constructor\n    calculus-with\n  field\n    basetype : Set\n    constant : Context {Type basetype} \u2192 Type basetype \u2192 Set\n    \u0394type : Type basetype \u2192 Type basetype\n    \u0394const : \u2200 {\u0393 \u03a3 \u03c4} \u2192 (c : constant \u03a3 \u03c4) \u2192\n      Term {basetype} {constant} \u0393\n        (internalizeContext basetype (\u0394Context\u2032 (Type basetype) \u0394type \u03a3) (\u0394type \u03c4))\n\nopen Calculus public\n\ntype : Calculus \u2192 Set\ntype L = Type (basetype L)\n\ncontext : Calculus \u2192 Set\ncontext L = Context {type L}\n\nterm : (L : Calculus) \u2192 context L \u2192 type L \u2192 Set\nterm L = Term {basetype L} {constant L}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4774069900d749795b6c1f69a5a137c07bce6b5e","subject":"TypeIso: update","message":"TypeIso: update\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nopen import Level.NP\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen import Type hiding (\u2605)\n-- open import Type.Identities\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum.NP renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive; [0:_1:_]; \ud835\udfda-is-set)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise public using (_\u00d7-cong_)\nopen import Relation.Binary.Sum public using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\nopen import HoTT\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 fst (f x)) , (\u03bb x \u2192 snd (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid \u2080 \u2080\n\n-- requires extensionality\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c}\n         (extB : {f g : \u03a0 A B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         (extC : {f g : \u03a0 A C} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a0 A B \u2192 \u03a0 A C\n    \u21d2 g x = to (f x) (g x)\n    \u21d0 : \u03a0 A C \u2192 \u03a0 A B\n    \u21d0 g x = from (f x) (g x)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 g = extB \u03bb x \u2192 left-f x (g x)\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 g = extC \u03bb x \u2192 right-f x (g x)\n\n  fiber-iso : \u03a0 A B \u2194 \u03a0 A C\n  fiber-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.tr C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (fst p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.tr (C F.\u2218 from f) (right-f (fst p)) (coe (fst (\u21d0 p)) (snd (\u21d0 p))) \u2261 snd p\n                helper p with to f (from f (fst p)) | right-f (fst p) | left-f (from f (fst p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inl) \u228e \u03a3 B (C F.\u2218 inr)\n    \u21d2 : S \u2192 T\n    \u21d2 (inl x , y) = inl (x , y)\n    \u21d2 (inr x , y) = inr (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inl (x , y)) = inl x , y\n    \u21d0 (inr (x , y)) = inr x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inl _ , _) = \u2261.refl\n    \u21d0\u21d2 (inr _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inl _) = \u2261.refl\n    \u21d2\u21d0 (inr _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inl) \u228e \u03a3 B (C F.\u2218 inr))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inl y) = inl (x , y)\n    \u21d2 (x , inr y) = inr (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inl (x , y)) = x , inl y\n    \u21d0 (inr (x , y)) = x , inr y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inl _) = \u2261.refl\n    \u21d0\u21d2 (_ , inr _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inl _) = \u2261.refl\n    \u21d2\u21d0 (inr _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n-- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c}\n         (ext : {f g : \u03a0 (A \u228e B) C} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inl) \u00d7 \u03a0 B (C F.\u2218 inr)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inl , f F.\u2218 inr\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inl x) = \u2261.refl\n    \u21d0\u21d2 f (inr y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : \u03a0 (A \u228e B) C \u2194 (\u03a0 A (C F.\u2218 inl) \u00d7 \u03a0 B (C F.\u2218 inr))\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) (\u03bb f \u2192 ext (\u21d0\u21d2 f)) \u21d2\u21d0\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inr\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inl , inr F.\u2218 inl ] , inr F.\u2218 inr ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inl F.\u2218 inl , [ inl F.\u2218 inr , inr ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inr , inl ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = snd\n      ; cong = snd\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < fst F.\u2218 fst , < snd F.\u2218 fst , snd > >\n      ; cong  = < fst F.\u2218 fst , < snd F.\u2218 fst , snd > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < fst , fst F.\u2218 snd > , snd F.\u2218 snd >\n      ; cong  = < < fst , fst F.\u2218 snd > , snd F.\u2218 snd >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inl , curry inr ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inl F.id , map\u00d7 inr F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = fst\n      ; cong = fst\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (fst x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inl x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inr x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\n{- PORTED TO HoTT -}\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inr (inl _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inr (inl _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\n{- PORTED TO HoTT -}\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\n-- PORTED to HoTT\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\n-- PORTED to HoTT\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\n-- PORTED to HoTT\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\n-- PORTED to HoTT\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\n-- PORTED to HoTT\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n-- PORTED to HoTT\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = fst \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\n-- PORTED to HoTT\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = snd \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n-- PORTED to HoTT\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n-- PORTED to HoTT\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\n-- PORTED to HoTT\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n{-\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n-}\n\n-- requires extensionality\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\n-- requires extensionality\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n-- PORTED to HoTT\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = fst \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\n-- PORTED to HoTT\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = snd \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n-- PORTED to HoTT\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 fst \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\n-- PORTED to HoTT\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\n-- PORTED to HoTT\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\n-- PORTED to HoTT\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\n-- PORTED to HoTT\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\n-- PORTED to HoTT\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n-- PORTED to HoTT\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 : \u2200 {a} (F : \ud835\udfd8 \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfd8 F \u2194 \ud835\udfd8\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 F = inverses fst (\u03bb ()) (\u03bb { ((), _) }) (\u03bb ())\n\n-- PORTED to HoTT\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inl p\n    \u21d2 (1\u2082 , p) = inr p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inl x) = 0\u2082 , x\n    \u21d0 (inr y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inl _) = \u2261.refl\n    \u21d2\u21d0 (inr _) = \u2261.refl\n\n-- PORTED to HoTT\n\u228e\u21ff\u03a32 : \u2200 {\u2113} {A B : \u2605 \u2113} \u2192 (A \u228e B) \u2194 \u03a3 \ud835\udfda [0: A 1: B ]\n\u228e\u21ff\u03a32 {A = A} {B} = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : A \u228e B \u2192 _\n    \u21d2 (inl x) = 0\u2082 , x\n    \u21d2 (inr x) = 1\u2082 , x\n    \u21d0 : \u03a3 _ _ \u2192 A \u228e B\n    \u21d0 (0\u2082 , x) = inl x\n    \u21d0 (1\u2082 , x) = inr x\n    \u21d0\u21d2 : (_ : _ \u228e _) \u2192 _\n    \u21d0\u21d2 (inl x) = \u2261.refl\n    \u21d0\u21d2 (inr x) = \u2261.refl\n    \u21d2\u21d0 : (_ : \u03a3 _ _) \u2192 _\n    \u21d2\u21d0 (0\u2082 , x) = \u2261.refl\n    \u21d2\u21d0 (1\u2082 , x) = \u2261.refl\n\n-- PORTED to HoTT\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inl _ , inr _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inl _) = \u2261.refl\n  \u21d2\u21d0 (inr _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 : \u2200 n \u2192 Fin n \u2194 \ud835\udfd9\u228e^ n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 zero    = Fin0\u2194\ud835\udfd8\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 (suc n) = Inv.id \u228e-cong Fin\u2194\ud835\udfd9\u228e^\ud835\udfd8 n Inv.\u2218 Fin\u2218suc\u2194\ud835\udfd9\u228eFin\n\n-- PORTED to HoTT\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\n-- PORTED to HoTT\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"-- {-# OPTIONS --without-K #-}\nmodule Function.Related.TypeIsomorphisms.NP where\n\nopen import Level.NP\nopen import Algebra\nimport Algebra.FunctionProperties as FP\nopen import Type hiding (\u2605)\n-- open import Type.Identities\nopen import Function using (_\u02e2_; const)\n\nopen import Data.Fin using (Fin; zero; suc; pred)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; uncons; \u2237-uncons)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_) renaming (_^_ to _**_)\nopen import Data.Maybe.NP\nopen import Data.Product.NP renaming (map to map\u00d7)\n--open import Data.Product.N-ary\nopen import Data.Sum.NP renaming (map to map\u228e)\nopen import Data.One\nopen import Data.Zero\nopen import Data.Two using (\ud835\udfda; 0\u2082; 1\u2082; proj; \u2713; not; \u2261\u2192\u2713; \u2261\u2192\u2713not; \u2713\u2192\u2261; \u2713not\u2192\u2261 ; not-involutive; [0:_1:_]; \ud835\udfda-is-set)\n\nimport Function.NP as F\nopen F using (\u03a0)\nimport Function.Equality as FE\nopen FE using (_\u27e8$\u27e9_)\nopen import Function.Related as FR\nopen import Function.Related.TypeIsomorphisms public\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Relation.Binary\nimport Relation.Binary.Indexed as I\nopen import Data.Indexed using (_\u228e\u00b0_)\nopen import Relation.Binary.Product.Pointwise public using (_\u00d7-cong_)\nopen import Relation.Binary.Sum public using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_ ; _\u2262_; _\u2257_)\nopen import HoTT\n\nmodule _ {a b f} {A : Set a} {B : A \u2192 Set b}\n         (F : (x : A) \u2192 B x \u2192 Set f) where\n\n    -- Also called Axiom of dependent choice.\n    dep-choice-iso : (\u03a0 A (\u03bb x \u2192 \u03a3 (B x) (F x)))\n                   \u2194 (\u03a3 (\u03a0 A B) \u03bb f \u2192 \u03a0 A (F \u02e2 f))\n    dep-choice-iso = inverses (\u21d2) (uncurry <_,_>) (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n      where\n        \u21d2 = \u03bb f \u2192 (\u03bb x \u2192 fst (f x)) , (\u03bb x \u2192 snd (f x))\n\nMaybe-injective : \u2200 {A B : Set} \u2192 Maybe A \u2194 Maybe B \u2192 A \u2194 B\nMaybe-injective f = Iso.iso (g f) (g-empty f)\n  module Maybe-injective where\n    open Inverse using (injective; left-inverse-of; right-inverse-of)\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing)\n      (fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing) where\n\n        CancelMaybe' : A \u2194 B\n        CancelMaybe' = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : _ \u2192 _\n          \u21d2 x with to f (just x) | tof-tot x\n          \u21d2 x | just y  | _ = y\n          \u21d2 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0 : _ \u2192 _\n          \u21d0 x with from f (just x) | fof-tot x\n          \u21d0 x | just y  | p = y\n          \u21d0 x | nothing | p = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 x with to f (just x)\n                  | tof-tot x\n                  | from f (just (\u21d2 x))\n                  | fof-tot (\u21d2 x)\n                  | left-inverse-of f (just x)\n                  | right-inverse-of f (just (\u21d2 x))\n          \u21d0\u21d2 x | just x\u2081 | p | just x\u2082 | q | b | c  = just-injective (\u2261.trans (\u2261.sym (left-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (from f) c) b))\n          \u21d0\u21d2 x | just x\u2081 | p | nothing | q | _ | _  = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d0\u21d2 x | nothing | p | z       | q | _ | _  = \ud835\udfd8-elim (p \u2261.refl)\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 x with from f (just x)\n                  | fof-tot x\n                  | to f (just (\u21d0 x))\n                  | tof-tot (\u21d0 x)\n                  | right-inverse-of f (just x)\n                  | left-inverse-of f (just (\u21d0 x))\n          \u21d2\u21d0 x | just x\u2081 | p | just x\u2082 | q | b | c = just-injective (\u2261.trans (\u2261.sym (right-inverse-of f (just x\u2082))) (\u2261.trans (\u2261.cong (to f) c) b))\n          \u21d2\u21d0 x | just x\u2081 | p | nothing | q | _ | _ = \ud835\udfd8-elim (q \u2261.refl)\n          \u21d2\u21d0 x | nothing | p | z       | q | _ | _ = \ud835\udfd8-elim (p \u2261.refl)\n\n    module Iso {A B : Set}\n      (f : Maybe A \u2194 Maybe B)\n      (f-empty : to f nothing \u2261 nothing) where\n\n      tof-tot : \u2200 x \u2192 to f (just x) \u2262 nothing\n      tof-tot x eq with injective f (\u2261.trans eq (\u2261.sym f-empty))\n      ... | ()\n\n      f-empty' : from f nothing \u2261 nothing\n      f-empty' = \u2261.trans (\u2261.sym (\u2261.cong (from f) f-empty)) (left-inverse-of f nothing)\n\n      fof-tot : \u2200 x \u2192 from f (just x) \u2262 nothing\n      fof-tot x eq with injective (sym f) (\u2261.trans eq (\u2261.sym f-empty'))\n      ... | ()\n\n      iso : A \u2194 B\n      iso = CancelMaybe' f tof-tot fof-tot\n\n    module _ {A B : Set}\n      (f : Maybe A \u2194 Maybe B) where\n\n      g : Maybe A \u2194 Maybe B\n      g = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0 where\n\n          \u21d2 : Maybe A \u2192 Maybe B\n          \u21d2 (just x) with to f (just x)\n          ... | nothing = to f nothing\n          ... | just y  = just y\n          \u21d2 nothing = nothing\n\n          \u21d0 : Maybe B \u2192 Maybe A\n          \u21d0 (just x) with from f (just x)\n          ... | nothing = from f nothing\n          ... | just y  = just y\n          \u21d0 nothing = nothing\n\n          \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n          \u21d0\u21d2 (just x) with to f (just x) | left-inverse-of f (just x)\n          \u21d0\u21d2 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d0\u21d2 (just x) | nothing | p with to f nothing | left-inverse-of f nothing\n          \u21d0\u21d2 (just x) | nothing | p | just _ | q rewrite q = p\n          \u21d0\u21d2 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d0\u21d2 nothing = \u2261.refl\n\n          \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n          \u21d2\u21d0 (just x) with from f (just x) | right-inverse-of f (just x)\n          \u21d2\u21d0 (just x) | just x\u2081 | p rewrite p = \u2261.refl\n          \u21d2\u21d0 (just x) | nothing | p with from f nothing | right-inverse-of f nothing\n          \u21d2\u21d0 (just x) | nothing | p | just _  | q rewrite q = p\n          \u21d2\u21d0 (just x) | nothing | p | nothing | q = \u2261.trans (\u2261.sym q) p\n          \u21d2\u21d0 nothing = \u2261.refl\n\n      g-empty : to g nothing \u2261 nothing\n      g-empty = \u2261.refl\n\nprivate\n    Setoid\u2080 : \u2605 _\n    Setoid\u2080 = Setoid \u2080 \u2080\n\n-- requires extensionality\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c}\n         (extB : {f g : \u03a0 A B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         (extC : {f g : \u03a0 A C} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a0 A B \u2192 \u03a0 A C\n    \u21d2 g x = to (f x) (g x)\n    \u21d0 : \u03a0 A C \u2192 \u03a0 A B\n    \u21d0 g x = from (f x) (g x)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 g = extB \u03bb x \u2192 left-f x (g x)\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 g = extC \u03bb x \u2192 right-f x (g x)\n\n  fiber-iso : \u03a0 A B \u2194 \u03a0 A C\n  fiber-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n\u03a3\u2261\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} x \u2192 \u03a3 A (_\u2261_ x) \u2194 \ud835\udfd9\n\u03a3\u2261\u2194\ud835\udfd9 x = inverses (F.const _) (\u03bb _ \u2192 _ , \u2261.refl)\n                  helper (\u03bb _ \u2192 \u2261.refl)\n  where helper : (y : \u03a3 _ (_\u2261_ x)) \u2192 (x , \u2261.refl) \u2261 y\n        helper (.x , \u2261.refl) = \u2261.refl\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : \u03a3 A B \u2192 \u2605 c} where\n    curried : \u03a0 (\u03a3 A B) C \u2194 \u03a0 A \u03bb a \u2192 \u03a0 (B a) \u03bb b \u2192 C (a , b)\n    curried = inverses curry uncurry (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u2192 \u2605 c} (f : A \u2194 B) where\n  private\n    left-f = Inverse.left-inverse-of f\n    right-f = Inverse.right-inverse-of f\n    coe : \u2200 x \u2192 C x \u2192 C (from f (to f x))\n    coe x = \u2261.tr C (\u2261.sym (left-f x))\n    \u21d2 : \u03a3 A C \u2192 \u03a3 B (C F.\u2218 from f)\n    \u21d2 (x , p) = to f x , coe x p\n    \u21d0 : \u03a3 B (C F.\u2218 from f) \u2192 \u03a3 A C\n    \u21d0 = first (from f)\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , p) rewrite left-f x = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 p = mk\u03a3\u2261 (C F.\u2218 from f) (right-f (fst p)) (helper p)\n            where\n                helper : \u2200 p \u2192 \u2261.tr (C F.\u2218 from f) (right-f (fst p)) (coe (fst (\u21d0 p)) (snd (\u21d0 p))) \u2261 snd p\n                helper p with to f (from f (fst p)) | right-f (fst p) | left-f (from f (fst p))\n                helper _ | ._ | \u2261.refl | \u2261.refl = \u2261.refl\n  first-iso : \u03a3 A C \u2194 \u03a3 B (C F.\u2218 from f)\n  first-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : B \u2192 \u2605 c} (f : A \u2194 B) where\n  sym-first-iso : \u03a3 A (C F.\u2218 to f) \u2194 \u03a3 B C\n  sym-first-iso = sym (first-iso (sym f))\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} (f : \u2200 x \u2192 B x \u2194 C x) where\n  private\n    left-f = Inverse.left-inverse-of F.\u2218 f\n    right-f = Inverse.right-inverse-of F.\u2218 f\n    \u21d2 : \u03a3 A B \u2192 \u03a3 A C\n    \u21d2 = second (to (f _))\n    \u21d0 : \u03a3 A C \u2192 \u03a3 A B\n    \u21d0 = second (from (f _))\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (x , y) rewrite left-f x y = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (x , y) rewrite right-f x y = \u2261.refl\n  second-iso : \u03a3 A B \u2194 \u03a3 A C\n  second-iso = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b} {A : \u2605 a} {B : \u2605 b} (f\u2080 f\u2081 : A \u2194 B) where\n  \ud835\udfda\u00d7-second-iso : (\ud835\udfda \u00d7 A) \u2194 (\ud835\udfda \u00d7 B)\n  \ud835\udfda\u00d7-second-iso = second-iso {A = \ud835\udfda} {B = const A} {C = const B} (proj (f\u2080 , f\u2081))\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c} where\n  private\n    S = \u03a3 (A \u228e B) C\n    T = \u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 (inj\u2081 x , y) = inj\u2081 (x , y)\n    \u21d2 (inj\u2082 x , y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = inj\u2081 x , y\n    \u21d0 (inj\u2082 (x , y)) = inj\u2082 x , y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (inj\u2081 _ , _) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 _ , _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3\u228e-distrib : (\u03a3 (A \u228e B) C) \u2194 (\u03a3 A (C F.\u2218 inj\u2081) \u228e \u03a3 B (C F.\u2218 inj\u2082))\n  \u03a3\u228e-distrib = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\nmodule _ {a b c} {A : \u2605 a} {B : A \u2192 \u2605 b} {C : A \u2192 \u2605 c} where\n  private\n    S = \u03a3 A (B \u228e\u00b0 C)\n    T = \u03a3 A B \u228e \u03a3 A C\n    \u21d2 : S \u2192 T\n    \u21d2 (x , inj\u2081 y) = inj\u2081 (x , y)\n    \u21d2 (x , inj\u2082 y) = inj\u2082 (x , y)\n    \u21d0 : T \u2192 S\n    \u21d0 (inj\u2081 (x , y)) = x , inj\u2081 y\n    \u21d0 (inj\u2082 (x , y)) = x , inj\u2082 y\n    \u21d0\u21d2 : \u2200 x \u2192 \u21d0 (\u21d2 x) \u2261 x\n    \u21d0\u21d2 (_ , inj\u2081 _) = \u2261.refl\n    \u21d0\u21d2 (_ , inj\u2082 _) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n  \u03a3-\u228e-hom : \u03a3 A (B \u228e\u00b0 C) \u2194 (\u03a3 A B \u228e \u03a3 A C)\n  \u03a3-\u228e-hom = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n\n-- requires extensional equality\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u228e B \u2192 \u2605 c}\n         (ext : {f g : \u03a0 (A \u228e B) C} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n         where\n  private\n    S = \u03a0 (A \u228e B) C\n    T = \u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082)\n    \u21d2 : S \u2192 T\n    \u21d2 f = f F.\u2218 inj\u2081 , f F.\u2218 inj\u2082\n    \u21d0 : T \u2192 S\n    \u21d0 (f , g) = [ f , g ]\n    \u21d0\u21d2 : \u2200 f x \u2192 \u21d0 (\u21d2 f) x \u2261 f x\n    \u21d0\u21d2 f (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 f (inj\u2082 y) = \u2261.refl\n    \u21d2\u21d0 : \u2200 x \u2192 \u21d2 (\u21d0 x) \u2261 x\n    \u21d2\u21d0 (f , g) = \u2261.refl\n\n  \u03a0\u00d7-distrib : \u03a0 (A \u228e B) C \u2194 (\u03a0 A (C F.\u2218 inj\u2081) \u00d7 \u03a0 B (C F.\u2218 inj\u2082))\n  \u03a0\u00d7-distrib = inverses (\u21d2) (\u21d0) (\u03bb f \u2192 ext (\u21d0\u21d2 f)) \u21d2\u21d0\n\n\u228e-ICommutativeMonoid : CommutativeMonoid _ _\n\u228e-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u228e-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd8;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u228e-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd8) _\u228e-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ (\u03bb ()) , F.id ]\n      ; cong  = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = inj\u2082\n      ; cong = \u2082\u223c\u2082\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ (\u03bb ()) , (\u03bb x \u2192 \u2082\u223c\u2082 refl) ]\n      ; right-inverse-of = \u03bb x \u2192 refl\n      }\n    } where open Setoid A\n            cong-to : Setoid._\u2248_ (\u2261.setoid \ud835\udfd8 \u228e-setoid A) =[ _ ]\u21d2 _\u2248_\n            cong-to (\u2081\u223c\u2082 ())\n            cong-to (\u2081\u223c\u2081 x\u223c\u2081y) rewrite x\u223c\u2081y = refl\n            cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = x\u223c\u2082y\n\n  assoc : Associative _\u228e-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = [ [ inj\u2081 , inj\u2082 F.\u2218 inj\u2081 ] , inj\u2082 F.\u2218 inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = [ inj\u2081 F.\u2218 inj\u2081 , [ inj\u2081 F.\u2218 inj\u2082 , inj\u2082 ] ]\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of = [ [ (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2081\u223c\u2081 (refl A))) , (\u03bb _ \u2192 \u2081\u223c\u2081 (\u2082\u223c\u2082 (refl B)) ) ] , (\u03bb _ \u2192 \u2082\u223c\u2082 (refl C)) ]\n      ; right-inverse-of = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (refl A)) , [ (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2081\u223c\u2081 (refl B))) , (\u03bb _ \u2192 \u2082\u223c\u2082 (\u2082\u223c\u2082 (refl C))) ] ]\n      }\n    } where\n      open Setoid\n      cong-to : _\u2248_ ((A \u228e-setoid B) \u228e-setoid C) =[ _ ]\u21d2 _\u2248_ (A \u228e-setoid (B \u228e-setoid C))\n      cong-to (\u2081\u223c\u2082 ())\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2082 ()))\n      cong-to (\u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 x\u223c\u2081y\n      cong-to (\u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2082y)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y) = \u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)\n\n      cong-from : _\u2248_ (A \u228e-setoid (B \u228e-setoid C)) =[ _ ]\u21d2 _\u2248_ ((A \u228e-setoid B) \u228e-setoid C)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 x\u223c\u2081y) = \u2081\u223c\u2081 (\u2081\u223c\u2081 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2082 ()))\n      cong-from (\u2082\u223c\u2082 (\u2081\u223c\u2081 x\u223c\u2081y)) = \u2081\u223c\u2081 (\u2082\u223c\u2082 x\u223c\u2081y)\n      cong-from (\u2082\u223c\u2082 (\u2082\u223c\u2082 x\u223c\u2082y)) = \u2082\u223c\u2082 x\u223c\u2082y\n\n  comm : Commutative _\u228e-setoid_\n  comm A B = record\n    { to = swap'\n    ; from = swap'\n    ; inverse-of = record\n      { left-inverse-of  = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u228e-setoid B FE.\u27f6 B \u228e-setoid A\n      swap' {A} {B} = record\n        { _\u27e8$\u27e9_ = [ inj\u2082 , inj\u2081 ]\n        ; cong = cong\n        } where\n          cong : Setoid._\u2248_ (A \u228e-setoid B) =[ _ ]\u21d2 Setoid._\u2248_ (B \u228e-setoid A)\n          cong (\u2081\u223c\u2082 ())\n          cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2082\u223c\u2082 x\u223c\u2081y\n          cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2081\u223c\u2081 x\u223c\u2082y\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u228e-setoid B) (swap' FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = [ (\u03bb _ \u2192 \u2081\u223c\u2081 (Setoid.refl A)) , (\u03bb _ \u2192 \u2082\u223c\u2082 (Setoid.refl B)) ]\n\n\u00d7-ICommutativeMonoid : CommutativeMonoid _ _\n\u00d7-ICommutativeMonoid = record {\n                         _\u2248_ = Inv.Inverse;\n                         _\u2219_ = _\u00d7-setoid_;\n                         \u03b5 = \u2261.setoid \ud835\udfd9;\n                         isCommutativeMonoid = record\n                            { isSemigroup = record\n                              { isEquivalence = Inv.isEquivalence\n                              ; assoc = assoc\n                              ; \u2219-cong = _\u00d7-inverse_\n                              }\n                            ; identity\u02e1 = left-identity\n                            ; comm = comm\n                            }\n                         }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  left-identity : LeftIdentity (\u2261.setoid \ud835\udfd9) _\u00d7-setoid_\n  left-identity A = record\n    { to = record\n      { _\u27e8$\u27e9_ = snd\n      ; cong = snd\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \u03bb x \u2192 _ , x\n      ; cong = \u03bb x \u2192 \u2261.refl , x\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \u2261.refl , (Setoid.refl A)\n      ; right-inverse-of = \u03bb x \u2192 Setoid.refl A\n      }\n    }\n\n  assoc : Associative _\u00d7-setoid_\n  assoc A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = < fst F.\u2218 fst , < snd F.\u2218 fst , snd > >\n      ; cong  = < fst F.\u2218 fst , < snd F.\u2218 fst , snd > >\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = < < fst , fst F.\u2218 snd > , snd F.\u2218 snd >\n      ; cong  = < < fst , fst F.\u2218 snd > , snd F.\u2218 snd >\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb _ \u2192 Setoid.refl ((A \u00d7-setoid B) \u00d7-setoid C)\n      ; right-inverse-of = \u03bb _ \u2192 Setoid.refl (A \u00d7-setoid (B \u00d7-setoid C))\n      }\n    }\n\n  comm : Commutative _\u00d7-setoid_\n  comm A B = record\n    { to = swap' {A} {B}\n    ; from = swap' {B} {A}\n    ; inverse-of = record\n      { left-inverse-of = inv A B\n      ; right-inverse-of = inv B A\n      }\n    } where\n      swap' : \u2200 {A B} \u2192 A \u00d7-setoid B FE.\u27f6 B \u00d7-setoid A\n      swap' {A}{B} = record { _\u27e8$\u27e9_ = swap; cong = swap }\n\n      inv : \u2200 A B \u2192 \u2200 x \u2192 Setoid._\u2248_ (A \u00d7-setoid B) (swap' {B} {A} FE.\u2218 swap' {A} {B}\u27e8$\u27e9 x) x\n      inv A B = \u03bb x \u2192 Setoid.refl (A \u00d7-setoid B)\n\n\u00d7\u228e-ICommutativeSemiring : CommutativeSemiring _ _\n\u00d7\u228e-ICommutativeSemiring = record\n  { _\u2248_ = Inv.Inverse\n  ; _+_ = _\u228e-setoid_\n  ; _*_ = _\u00d7-setoid_\n  ; 0# = \u2261.setoid \ud835\udfd8\n  ; 1# = \u2261.setoid \ud835\udfd9\n  ; isCommutativeSemiring = record\n    { +-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u228e-ICommutativeMonoid\n    ; *-isCommutativeMonoid = CommutativeMonoid.isCommutativeMonoid \u00d7-ICommutativeMonoid\n    ; distrib\u02b3 = distrib\u02b3\n    ; zero\u02e1 = zero\u02e1\n    }\n  }\n  where\n  open FP (Inv.Inverse {f\u2081 = \u2080}{f\u2082 = \u2080})\n\n  distrib\u02b3 : _\u00d7-setoid_ DistributesOver\u02b3 _\u228e-setoid_\n  distrib\u02b3 A B C = record\n    { to = record\n      { _\u27e8$\u27e9_ = uncurry [ curry inj\u2081 , curry inj\u2082 ]\n      ; cong = cong-to\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = frm\n      ; cong = cong-from\n      }\n    ; inverse-of = record\n      { left-inverse-of  = uncurry [ (\u03bb x y \u2192 (\u2081\u223c\u2081 (refl B)) , (refl A)) , (\u03bb x y \u2192 (\u2082\u223c\u2082 (refl C)) , (refl A)) ]\n      ; right-inverse-of = [ uncurry (\u03bb x y \u2192 \u2081\u223c\u2081 ((refl B {x}) , (refl A {y})))\n                           , uncurry (\u03bb x y \u2192 \u2082\u223c\u2082 ((refl C {x}) , (refl A {y}))) ]\n      }\n    } where\n      open Setoid\n      A' = Carrier A\n      B' = Carrier B\n      C' = Carrier C\n\n      cong-to : _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A) =[ _ ]\u21d2 _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A))\n      cong-to (\u2081\u223c\u2082 () , _)\n      cong-to (\u2081\u223c\u2081 x\u223c\u2081y , A-rel) = \u2081\u223c\u2081 (x\u223c\u2081y , A-rel)\n      cong-to (\u2082\u223c\u2082 x\u223c\u2082y , A-rel) = \u2082\u223c\u2082 (x\u223c\u2082y , A-rel)\n\n      frm : B' \u00d7 A' \u228e C' \u00d7 A' \u2192 (B' \u228e C') \u00d7 A'\n      frm = [ map\u00d7 inj\u2081 F.id , map\u00d7 inj\u2082 F.id ]\n\n      cong-from : _\u2248_ ((B \u00d7-setoid A) \u228e-setoid (C \u00d7-setoid A)) =[ _ ]\u21d2 _\u2248_ ((B \u228e-setoid C) \u00d7-setoid A)\n      cong-from (\u2081\u223c\u2082 ())\n      cong-from (\u2081\u223c\u2081 (B-rel , A-rel)) = \u2081\u223c\u2081 B-rel , A-rel\n      cong-from (\u2082\u223c\u2082 (C-rel , A-rel)) = \u2082\u223c\u2082 C-rel , A-rel\n\n  zero\u02e1 : LeftZero (\u2261.setoid \ud835\udfd8) _\u00d7-setoid_\n  zero\u02e1 A = record\n    { to = record\n      { _\u27e8$\u27e9_ = fst\n      ; cong = fst\n      }\n    ; from = record\n      { _\u27e8$\u27e9_ = \ud835\udfd8-elim\n      ; cong = \u03bb { {()} x }\n      }\n    ; inverse-of = record\n      { left-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim (fst x)\n      ; right-inverse-of = \u03bb x \u2192 \ud835\udfd8-elim x\n      }\n    }\n\nmodule \u00d7-CMon {a} = CommutativeMonoid (\u00d7-CommutativeMonoid FR.bijection a)\nmodule \u228e-CMon {a} = CommutativeMonoid (\u228e-CommutativeMonoid FR.bijection a)\nmodule \u00d7\u228e\u00b0 {a}    = CommutativeSemiring (\u00d7\u228e-CommutativeSemiring FR.bijection a)\n\nmodule \u00d7-ICMon = CommutativeMonoid \u00d7-ICommutativeMonoid\nmodule \u228e-ICMon = CommutativeMonoid \u228e-ICommutativeMonoid\nmodule \u00d7\u228e\u00b0I    = CommutativeSemiring \u00d7\u228e-ICommutativeSemiring\n\nlift-\u228e : \u2200 {A B : Set} \u2192 Inv.Inverse ((\u2261.setoid A) \u228e-setoid (\u2261.setoid B)) (\u2261.setoid (A \u228e B))\nlift-\u228e {A}{B} = record\n  { to = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = cong\n    }\n  ; from = record\n    { _\u27e8$\u27e9_ = \u03bb x \u2192 x\n    ; cong = \u03bb x \u2192 Setoid.reflexive (\u2261.setoid A \u228e-setoid \u2261.setoid B) x\n    }\n  ; inverse-of = record\n    { left-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid A \u228e-setoid \u2261.setoid B)\n    ; right-inverse-of = \u03bb x \u2192 Setoid.refl (\u2261.setoid (A \u228e B))\n    }\n  } where\n    cong : Setoid._\u2248_ (\u2261.setoid A \u228e-setoid \u2261.setoid B)  =[ _ ]\u21d2 Setoid._\u2248_ (\u2261.setoid (A \u228e B))\n    cong (\u2081\u223c\u2082 ())\n    cong (\u2081\u223c\u2081 x\u223c\u2081y) = \u2261.cong inj\u2081 x\u223c\u2081y\n    cong (\u2082\u223c\u2082 x\u223c\u2082y) = \u2261.cong inj\u2082 x\u223c\u2082y\n\nswap-iso : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 (A \u00d7 B) \u2194 (B \u00d7 A)\nswap-iso = inverses swap swap (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : A \u00d7 B \u2192 \u2605 c} where\n  \u03a3\u00d7-swap : \u03a3 (A \u00d7 B) C \u2194 \u03a3 (B \u00d7 A) (C F.\u2218 swap)\n  \u03a3\u00d7-swap = first-iso swap-iso\n\n{- PORTED TO HoTT -}\nMaybe\u2194Lift\ud835\udfd9\u228e : \u2200 {\u2113 a} {A : \u2605 a} \u2192 Maybe A \u2194 (Lift {\u2113 = \u2113} \ud835\udfd9 \u228e A)\nMaybe\u2194Lift\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe\u2194\ud835\udfd9\u228e : \u2200 {a} {A : \u2605 a} \u2192 Maybe A \u2194 (\ud835\udfd9 \u228e A)\nMaybe\u2194\ud835\udfd9\u228e\n  = inverses (maybe inj\u2082 (inj\u2081 _))\n             [ F.const nothing , just ]\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n             [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ]\n\nMaybe-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B\nMaybe-cong A\u2194B = sym Maybe\u2194\ud835\udfd9\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\ud835\udfd9\u228e\n\n{- PORTED TO HoTT -}\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {a} {A : \u2605 a} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {a b} {A : \u2605 a} {B : \u2605 b} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194Lift\ud835\udfd9 : \u2200 {a \u2113} {A : \u2605 a} \u2192 Vec A 0 \u2194 Lift {_} {\u2113} \ud835\udfd9\nVec0\u2194Lift\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec0\u2194\ud835\udfd9 : \u2200 {a} {A : \u2605 a} \u2192 Vec A 0 \u2194 \ud835\udfd9\nVec0\u2194\ud835\udfd9 = inverses _ (F.const []) (\u03bb { [] \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : \u2605 a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = inverses uncons (uncurry _\u2237_) \u2237-uncons (\u03bb _ \u2192 \u2261.refl)\n\ninfix 8 _^_\n\n_^_ : \u2200 {a} \u2192 \u2605 a \u2192 \u2115 \u2192 \u2605 a\nA ^ 0     = Lift \ud835\udfd9\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {a} {A : \u2605 a} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194Lift\ud835\udfd9\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\n-- PORTED to HoTT\nFin0\u2194\ud835\udfd8 : Fin 0 \u2194 \ud835\udfd8\nFin0\u2194\ud835\udfd8 = inverses (\u03bb()) (\u03bb()) (\u03bb()) (\u03bb())\n\n-- PORTED to HoTT\nFin1\u2194\ud835\udfd9 : Fin 1 \u2194 \ud835\udfd9\nFin1\u2194\ud835\udfd9 = inverses _ (\u03bb _ \u2192 zero) \u21d0\u21d2 (\u03bb _ \u2192 \u2261.refl)\n  where \u21d0\u21d2 : (_ : Fin 1) \u2192 _\n        \u21d0\u21d2 zero = \u2261.refl\n        \u21d0\u21d2 (suc ())\n\n-- PORTED to HoTT\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = inverses to' (maybe suc zero)\n             (\u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl })\n             (maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl)\n  where to' : Fin (suc n) \u2192 Maybe (Fin n)\n        to' zero = nothing\n        to' (suc n) = just n\n\n-- PORTED to HoTT\nFin-injective : \u2200 {m n} \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\nFin-injective = go _ _ where\n    go : \u2200 m n \u2192 Fin m \u2194 Fin n \u2192 m \u2261 n\n    go zero    zero    iso = \u2261.refl\n    go zero    (suc n) iso with from iso zero\n    ...                       | ()\n    go (suc m) zero    iso with to iso zero\n    ...                       | ()\n    go (suc m) (suc n) iso = \u2261.cong suc (go m n (Maybe-injective (Fin\u2218suc\u2194Maybe\u2218Fin \u2218 iso \u2218 sym Fin\u2218suc\u2194Maybe\u2218Fin)))\n\n-- PORTED to HoTT\nLift\u2194id : \u2200 {a} {A : \u2605 a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = inverses lower lift (\u03bb { (lift x) \u2192 \u2261.refl }) (\u03bb _ \u2192 \u2261.refl)\n\n-- PORTED to HoTT\n\ud835\udfd9\u00d7A\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd9 \u00d7 A) \u2194 A\n\ud835\udfd9\u00d7A\u2194A = fst \u00d7-CMon.identity _ \u2218 sym Lift\u2194id \u00d7-cong id\n\n-- PORTED to HoTT\nA\u00d7\ud835\udfd9\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u00d7 \ud835\udfd9) \u2194 A\nA\u00d7\ud835\udfd9\u2194A = snd \u00d7-CMon.identity _ \u2218 id \u00d7-cong sym Lift\u2194id\n\n-- PORTED to HoTT\n\u03a0\ud835\udfd9F\u2194F : \u2200 {\u2113} {F : \ud835\udfd9 \u2192 \u2605_ \u2113} \u2192 \u03a0 \ud835\udfd9 F \u2194 F _\n\u03a0\ud835\udfd9F\u2194F = inverses (\u03bb x \u2192 x _) const (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\n-- PORTED to HoTT\n\ud835\udfd9\u2192A\u2194A : \u2200 {\u2113} {A : \u2605_ \u2113} \u2192 (\ud835\udfd9 \u2192 A) \u2194 A\n\ud835\udfd9\u2192A\u2194A = \u03a0\ud835\udfd9F\u2194F\n\n-- PORTED to HoTT\nnot-\ud835\udfda\u2194\ud835\udfda : \ud835\udfda \u2194 \ud835\udfda\nnot-\ud835\udfda\u2194\ud835\udfda = inverses not not not-involutive not-involutive\n\n{-\n\u2261-iso : \u2200 {\u2113 \u2113'}{A : \u2605_ \u2113}{B : \u2605_ \u2113'}{x y : A} \u2192 (\u03c0 : A \u2194 B) \u2192 (x \u2261 y) \u2194 (to \u03c0 x \u2261 to \u03c0 y)\n\u2261-iso {x = x}{y} \u03c0 = inverses (\u2261.cong (to \u03c0))\n                              (\u03bb p \u2192 \u2261.trans (\u2261.sym (Inverse.left-inverse-of \u03c0 x))\n                                    (\u2261.trans (\u2261.cong (from \u03c0) p)\n                                             (Inverse.left-inverse-of \u03c0 y)))\n                              (\u03bb x \u2192 \u2261.proof-irrelevance _ x) (\u03bb x \u2192 \u2261.proof-irrelevance _ x)\n-}\n\n-- requires extensionality\nmodule _ {a} {A : \u2605_ a} (ext\ud835\udfd8 : (f g : \ud835\udfd8 \u2192 A) \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 : (\ud835\udfd8 \u2192 A) \u2194 \ud835\udfd9\n    \ud835\udfd8\u2192A\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\n-- requires extensionality\nmodule _ {\u2113} {F : \ud835\udfd8 \u2192 \u2605_ \u2113} (ext\ud835\udfd8 : (f g : \u03a0 \ud835\udfd8 F) \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 : \u03a0 \ud835\udfd8 F \u2194 \ud835\udfd9\n    \u03a0\ud835\udfd8\u2194\ud835\udfd9 = inverses _ (\u03bb _ ()) (\u03bb h \u2192 ext\ud835\udfd8 _ h) (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {F : \ud835\udfda \u2192 \u2605_ \u2113} (ext\ud835\udfda : {f g : \u03a0 \ud835\udfda F} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 : \u03a0 \ud835\udfda F \u2194 (F 0\u2082 \u00d7 F 1\u2082)\n    \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 = inverses (\u03bb f \u2192 f 0\u2082 , f 1\u2082) proj\n                         (\u03bb f \u2192 ext\ud835\udfda (\u03bb { 0\u2082 \u2192 \u2261.refl ; 1\u2082 \u2192 \u2261.refl }))\n                         (\u03bb _ \u2192 \u2261.refl)\n\nmodule _ {\u2113} {A : \u2605_ \u2113} (ext\ud835\udfda : {f g : \ud835\udfda \u2192 A} \u2192 f \u2257 g \u2192 f \u2261 g) where\n-- PORTED to HoTT\n    \ud835\udfda\u2192A\u2194A\u00d7A : (\ud835\udfda \u2192 A) \u2194 (A \u00d7 A)\n    \ud835\udfda\u2192A\u2194A\u00d7A = \u03a0\ud835\udfdaF\u2194F\u2080\u00d7F\u2081 ext\ud835\udfda\n\n-- PORTED to HoTT\n\ud835\udfd8\u228eA\u2194A : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u228e A) \u2194 A\n\ud835\udfd8\u228eA\u2194A = fst \u228e-CMon.identity _ \u2218 sym Lift\u2194id \u228e-cong id\n\n-- PORTED to HoTT\nA\u228e\ud835\udfd8\u2194A : \u2200 {A : \u2605\u2080} \u2192 (A \u228e \ud835\udfd8) \u2194 A\nA\u228e\ud835\udfd8\u2194A = snd \u228e-CMon.identity _ \u2218 id \u228e-cong sym Lift\u2194id\n\n-- PORTED to HoTT\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 : \u2200 {A : \u2605\u2080} \u2192 (\ud835\udfd8 \u00d7 A) \u2194 \ud835\udfd8\n\ud835\udfd8\u00d7A\u2194\ud835\udfd8 = Lift\u2194id \u2218 fst \u00d7\u228e\u00b0.zero _ \u2218 sym (Lift\u2194id \u00d7-cong id)\n\n-- PORTED to HoTT\nMaybe\ud835\udfd8\u2194\ud835\udfd9 : Maybe \ud835\udfd8 \u2194 \ud835\udfd9\nMaybe\ud835\udfd8\u2194\ud835\udfd9 = A\u228e\ud835\udfd8\u2194A \u2218 Maybe\u2194\ud835\udfd9\u228e\n\n-- PORTED to HoTT\nMaybe^\ud835\udfd8\u2194Fin : \u2200 n \u2192 Maybe^ n \ud835\udfd8 \u2194 Fin n\nMaybe^\ud835\udfd8\u2194Fin zero    = sym Fin0\u2194\ud835\udfd8\nMaybe^\ud835\udfd8\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\ud835\udfd8\u2194Fin n)\n\n-- PORTED to HoTT\nMaybe^\ud835\udfd9\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \ud835\udfd9 \u2194 Fin (suc n)\nMaybe^\ud835\udfd9\u2194Fin1+ n = Maybe^\ud835\udfd8\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\ud835\udfd8\u2194\ud835\udfd9)\n\n-- PORTED to HoTT\nMaybe-\u228e : \u2200 {a} {A B : \u2605 a} \u2192 (Maybe A \u228e B) \u2194 Maybe (A \u228e B)\nMaybe-\u228e {a} = sym Maybe\u2194Lift\ud835\udfd9\u228e \u2218 \u228e-CMon.assoc (Lift {_} {a} \ud835\udfd9) _ _ \u2218 (Maybe\u2194Lift\ud835\udfd9\u228e \u228e-cong id)\n\n-- PORTED to HoTT\nMaybe^-\u228e-+ : \u2200 {A} m n \u2192 (Maybe^ m \ud835\udfd8 \u228e Maybe^ n A) \u2194 Maybe^ (m + n) A\nMaybe^-\u228e-+ zero    n = \ud835\udfd8\u228eA\u2194A\nMaybe^-\u228e-+ (suc m) n = Maybe-cong (Maybe^-\u228e-+ m n) \u2218 Maybe-\u228e\n\n-- PORTED to HoTT\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 : \u2200 {a} (F : \ud835\udfd8 \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfd8 F \u2194 \ud835\udfd8\n\u03a3\ud835\udfd8\u2194\ud835\udfd8 F = inverses fst (\u03bb ()) (\u03bb { ((), _) }) (\u03bb ())\n\n-- PORTED to HoTT\n\u03a3\ud835\udfda\u2194\u228e : \u2200 {a} (F : \ud835\udfda \u2192 \u2605_ a) \u2192 \u03a3 \ud835\udfda F \u2194 (F 0\u2082 \u228e F 1\u2082)\n\u03a3\ud835\udfda\u2194\u228e F = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : (x : \u03a3 _ _) \u2192 _\n    \u21d2 (0\u2082 , p) = inj\u2081 p\n    \u21d2 (1\u2082 , p) = inj\u2082 p\n    \u21d0 : (x : _ \u228e _) \u2192 _\n    \u21d0 (inj\u2081 x) = 0\u2082 , x\n    \u21d0 (inj\u2082 y) = 1\u2082 , y\n\n    \u21d0\u21d2 : (_ : \u03a3 _ _) \u2192 _\n    \u21d0\u21d2 (0\u2082 , p) = \u2261.refl\n    \u21d0\u21d2 (1\u2082 , p) = \u2261.refl\n    \u21d2\u21d0 : (_ : _ \u228e _) \u2192 _\n    \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n    \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\n-- PORTED to HoTT\n\u228e\u21ff\u03a32 : \u2200 {\u2113} {A B : \u2605 \u2113} \u2192 (A \u228e B) \u2194 \u03a3 \ud835\udfda [0: A 1: B ]\n\u228e\u21ff\u03a32 {A = A} {B} = inverses (\u21d2) (\u21d0) \u21d0\u21d2 \u21d2\u21d0\n  where\n    \u21d2 : A \u228e B \u2192 _\n    \u21d2 (inj\u2081 x) = 0\u2082 , x\n    \u21d2 (inj\u2082 x) = 1\u2082 , x\n    \u21d0 : \u03a3 _ _ \u2192 A \u228e B\n    \u21d0 (0\u2082 , x) = inj\u2081 x\n    \u21d0 (1\u2082 , x) = inj\u2082 x\n    \u21d0\u21d2 : (_ : _ \u228e _) \u2192 _\n    \u21d0\u21d2 (inj\u2081 x) = \u2261.refl\n    \u21d0\u21d2 (inj\u2082 x) = \u2261.refl\n    \u21d2\u21d0 : (_ : \u03a3 _ _) \u2192 _\n    \u21d2\u21d0 (0\u2082 , x) = \u2261.refl\n    \u21d2\u21d0 (1\u2082 , x) = \u2261.refl\n\n-- PORTED to HoTT\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 : \ud835\udfda \u2194 (\ud835\udfd9 \u228e \ud835\udfd9)\n\ud835\udfda\u2194\ud835\udfd9\u228e\ud835\udfd9 = inverses (proj (inj\u2081 _ , inj\u2082 _)) [ F.const 0\u2082 , F.const 1\u2082 ] \u21d0\u21d2 \u21d2\u21d0\n  where\n  \u21d0\u21d2 : (_ : \ud835\udfda) \u2192 _\n  \u21d0\u21d2 0\u2082 = \u2261.refl\n  \u21d0\u21d2 1\u2082 = \u2261.refl\n  \u21d2\u21d0 : (_ : \ud835\udfd9 \u228e \ud835\udfd9) \u2192 _\n  \u21d2\u21d0 (inj\u2081 _) = \u2261.refl\n  \u21d2\u21d0 (inj\u2082 _) = \u2261.refl\n\nFin-\u228e-+ : \u2200 m n \u2192 (Fin m \u228e Fin n) \u2194 Fin (m + n)\nFin-\u228e-+ m n = Maybe^\ud835\udfd8\u2194Fin (m + n)\n            \u2218 Maybe^-\u228e-+ m n\n            \u2218 sym (Maybe^\ud835\udfd8\u2194Fin m \u228e-cong Maybe^\ud835\udfd8\u2194Fin n)\n\nFin\u2218suc\u2194\ud835\udfd9\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\ud835\udfd9 \u228e Fin n)\nFin\u2218suc\u2194\ud835\udfd9\u228eFin = Maybe\u2194\ud835\udfd9\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 : \u2200 n \u2192 Fin n \u2194 \ud835\udfd9\u228e^ n\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 zero    = Fin0\u2194\ud835\udfd8\nFin\u2194\ud835\udfd9\u228e^\ud835\udfd8 (suc n) = Inv.id \u228e-cong Fin\u2194\ud835\udfd9\u228e^\ud835\udfd8 n Inv.\u2218 Fin\u2218suc\u2194\ud835\udfd9\u228eFin\n\n-- PORTED to HoTT\nFin-\u00d7-* : \u2200 m n \u2192 (Fin m \u00d7 Fin n) \u2194 Fin (m * n)\nFin-\u00d7-* zero n = (Fin 0 \u00d7 Fin n) \u2194\u27e8 Fin0\u2194\ud835\udfd8 \u00d7-cong id \u27e9\n                 (\ud835\udfd8 \u00d7 Fin n) \u2194\u27e8 \ud835\udfd8\u00d7A\u2194\ud835\udfd8 \u27e9\n                 \ud835\udfd8 \u2194\u27e8 sym Fin0\u2194\ud835\udfd8 \u27e9\n                 Fin 0 \u220e\n  where open EquationalReasoning hiding (sym)\nFin-\u00d7-* (suc m) n = (Fin (suc m) \u00d7 Fin n) \u2194\u27e8 Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u00d7-cong id \u27e9\n                    ((\ud835\udfd9 \u228e Fin m) \u00d7 Fin n) \u2194\u27e8 \u00d7\u228e\u00b0.distrib\u02b3 (Fin n) \ud835\udfd9 (Fin m) \u27e9\n                    ((\ud835\udfd9 \u00d7 Fin n) \u228e (Fin m \u00d7 Fin n)) \u2194\u27e8 \ud835\udfd9\u00d7A\u2194A \u228e-cong Fin-\u00d7-* m n \u27e9\n                    (Fin n \u228e Fin (m * n)) \u2194\u27e8 Fin-\u228e-+ n (m * n) \u27e9\n                    Fin (suc m * n) \u220e\n  where open EquationalReasoning hiding (sym)\n\n-- PORTED to HoTT\nFin\u228e-injective : \u2200 {A B : Set} n \u2192 (Fin n \u228e A) \u2194 (Fin n \u228e B) \u2192 A \u2194 B\nFin\u228e-injective zero    f = \ud835\udfd8\u228eA\u2194A \u2218 Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 f \u2218 sym Fin0\u2194\ud835\udfd8 \u228e-cong id \u2218 sym \ud835\udfd8\u228eA\u2194A\nFin\u228e-injective (suc n) f =\n  Fin\u228e-injective n\n    (Maybe-injective\n       (sym Maybe\u2194\ud835\udfd9\u228e \u2218\n        \u228e-CMon.assoc _ _ _ \u2218\n        Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        f \u2218\n        sym Fin\u2218suc\u2194\ud835\udfd9\u228eFin \u228e-cong id \u2218\n        sym (\u228e-CMon.assoc _ _ _) \u2218\n        Maybe\u2194\ud835\udfd9\u228e))\n\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"16d173cf40ecb353a197647665aaca4d88b513aa","subject":"Cosmetic change.","message":"Cosmetic change.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/List\/WF-Relation\/LT-Length\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/List\/WF-Relation\/LT-Length\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the relation LTL\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.WF-Relation.LT-Length.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI as Nat using ()\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.List.WF-Relation.LT-Length\n\n------------------------------------------------------------------------------\n\nxs<[]\u2192\u22a5 : \u2200 {xs} \u2192 List xs \u2192 \u00ac (LTL xs [])\nxs<[]\u2192\u22a5 Lxs xs<[] = lg-xs<lg-[]\u2192\u22a5 Lxs xs<[]\n\nx\u2237xs<y\u2237ys\u2192xs<ys : \u2200 {x xs y ys} \u2192 List xs \u2192 List ys \u2192\n                  LTL (x \u2237 xs) (y \u2237 ys) \u2192 LTL xs ys\nx\u2237xs<y\u2237ys\u2192xs<ys {x} {xs} {y} {ys} Lxs Lys x\u2237xs<y\u2237ys = Nat.Sx<Sy\u2192x<y helper\n  where\n  helper : succ\u2081 (length xs) < succ\u2081 (length ys)\n  helper =\n    lt (succ\u2081 (length xs)) (succ\u2081 (length ys))\n      \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 lt (succ\u2081 (length xs)) (succ\u2081 (length ys)) \u2261 lt t\u2081 t\u2082)\n                (sym (length-\u2237 x xs))\n                (sym (length-\u2237 y ys))\n                refl\n      \u27e9\n    lt (length (x \u2237 xs)) (length (y \u2237 ys))\n      \u2261\u27e8 x\u2237xs<y\u2237ys \u27e9\n    true \u220e\n\n<-trans : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n          LTL xs ys \u2192 LTL ys zs \u2192 LTL xs zs\n\n<-trans Lxs Lys Lzs xs<ys ys<zs =\n  Nat.<-trans (lengthList-N Lxs) (lengthList-N Lys) (lengthList-N Lzs) xs<ys ys<zs\n\nlg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs : \u2200 {xs ys zs} \u2192 length xs \u2261 length ys \u2192\n                          LTL ys zs \u2192 LTL xs zs\nlg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs {xs} {ys} {zs} h ys<zs =\n  lt (length xs) (length zs)\n    \u2261\u27e8 subst (\u03bb t \u2192 lt (length xs) (length zs) \u2261 lt t (length zs)) h refl \u27e9\n  lt (length ys) (length zs)\n    \u2261\u27e8 ys<zs \u27e9\n  true \u220e\n\nxs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys : \u2200 {xs y ys} \u2192 List xs \u2192 List ys \u2192\n                            LTL xs (y \u2237 ys) \u2192\n                            LTL xs ys \u2228 length xs \u2261 length ys\nxs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys {xs} {y} {ys} Lxs Lys xs<y\u2237ys =\n  Nat.x<Sy\u2192x<y\u2228x\u2261y (lengthList-N Lxs) (lengthList-N Lys) helper\n  where\n  helper : length xs < succ\u2081 (length ys)\n  helper =\n    lt (length xs) (succ\u2081 (length ys))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (length xs) (succ\u2081 (length ys)) \u2261 lt (length xs) t)\n               (sym (length-\u2237 y ys))\n               refl\n      \u27e9\n    lt (length xs) (length (y \u2237 ys))\n      \u2261\u27e8 xs<y\u2237ys \u27e9\n    true \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for the relation LTL\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.List.WF-Relation.LT-Length.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI as Nat using ()\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.List.WF-Relation.LT-Length\n\n------------------------------------------------------------------------------\n\nxs<[]\u2192\u22a5 : \u2200 {xs} \u2192 List xs \u2192 \u00ac (LTL xs [])\nxs<[]\u2192\u22a5 Lxs xs<[] = lg-xs<lg-[]\u2192\u22a5 Lxs xs<[]\n\nx\u2237xs<y\u2237ys\u2192xs<ys : \u2200 {x xs y ys} \u2192 List xs \u2192 List ys \u2192\n                  LTL (x \u2237 xs) (y \u2237 ys) \u2192 LTL xs ys\nx\u2237xs<y\u2237ys\u2192xs<ys {x} {xs} {y} {ys} Lxs Lys x\u2237xs<y\u2237ys = Nat.Sx<Sy\u2192x<y helper\n  where\n  helper : succ\u2081 (length xs) < succ\u2081 (length ys)\n  helper =\n    lt (succ\u2081 (length xs)) (succ\u2081 (length ys))\n      \u2261\u27e8 subst\u2082 (\u03bb t\u2081 t\u2082 \u2192 lt (succ\u2081 (length xs)) (succ\u2081 (length ys)) \u2261 lt t\u2081 t\u2082)\n                (sym (length-\u2237 x xs))\n                (sym (length-\u2237 y ys))\n                refl\n      \u27e9\n    lt (length (x \u2237 xs)) (length (y \u2237 ys))\n      \u2261\u27e8 x\u2237xs<y\u2237ys \u27e9\n    true \u220e\n\n<-trans : \u2200 {xs ys zs} \u2192 List xs \u2192 List ys \u2192 List zs \u2192\n          LTL xs ys \u2192 LTL ys zs \u2192 LTL xs zs\n\n<-trans Lxs Lys Lzs xs<ys ys<zs =\n  Nat.<-trans (lengthList-N Lxs) (lengthList-N Lys) (lengthList-N Lzs) xs<ys ys<zs\n\nlg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs : \u2200 {xs ys zs} \u2192 length xs \u2261 length ys \u2192\n                          LTL ys zs \u2192 LTL xs zs\nlg-xs\u2261lg-ys\u2192ys<zx\u2192xs<zs {xs} {ys} {zs} lg-xs\u2261lg-ys ys<zs =\n  lt (length xs) (length zs)\n    \u2261\u27e8 subst (\u03bb t \u2192 lt (length xs) (length zs) \u2261 lt t (length zs)) lg-xs\u2261lg-ys refl \u27e9\n  lt (length ys) (length zs)\n    \u2261\u27e8 ys<zs \u27e9\n  true \u220e\n\nxs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys : \u2200 {xs y ys} \u2192 List xs \u2192 List ys \u2192\n                            LTL xs (y \u2237 ys) \u2192\n                            LTL xs ys \u2228 length xs \u2261 length ys\nxs<y\u2237ys\u2192xs<ys\u2228lg-xs\u2261lg-ys {xs} {y} {ys} Lxs Lys xs<y\u2237ys =\n  Nat.x<Sy\u2192x<y\u2228x\u2261y (lengthList-N Lxs) (lengthList-N Lys) helper\n  where\n  helper : length xs < succ\u2081 (length ys)\n  helper =\n    lt (length xs) (succ\u2081 (length ys))\n      \u2261\u27e8 subst (\u03bb t \u2192 lt (length xs) (succ\u2081 (length ys)) \u2261 lt (length xs) t)\n               (sym (length-\u2237 y ys))\n               refl\n      \u27e9\n    lt (length xs) (length (y \u2237 ys))\n      \u2261\u27e8 xs<y\u2237ys \u27e9\n    true \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"210cda5b23481dbc720900564c7fd7d2081a739b","subject":"Updated the quicksort draft.","message":"Updated the quicksort draft.\n\nIgnore-this: 94df21f4bce6c1d1401e946d3f7536fd\n\ndarcs-hash:20110524053221-3bd4e-de21acfb8fcd445e3034d4704e4a45147a5c1e4c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/QuickSort\/QuickSortBC.agda","new_file":"Draft\/FOTC\/Program\/QuickSort\/QuickSortBC.agda","new_contents":"------------------------------------------------------------------------------\n-- Quicksort using the Bove-Capretta method\n------------------------------------------------------------------------------\n\n-- Tested with the development version of the standard library on\n-- 24 May 2011.\n\nmodule QuickSortBC where\n\nopen import Data.Bool\nopen import Data.List\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Sum\n\nopen import Induction\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nmodule InvImg =\n  Induction.WellFounded.Inverse-image {A = List \u2115} {\u2115} {_<\u2032_} length\n\n------------------------------------------------------------------------------\n\n_\u2264\u2032?_ : Decidable _\u2264\u2032_\nm \u2264\u2032? n with m \u2264? n\n... | yes p = yes (\u2264\u21d2\u2264\u2032 p)\n... | no \u00acp = no (\u03bb m\u2264\u2032n \u2192 \u00acp (\u2264\u2032\u21d2\u2264 m\u2264\u2032n))\n\n-- Non-terminating quicksort.\nqsNT : List \u2115 \u2192 List \u2115\nqsNT []       = []\nqsNT (x \u2237 xs) = qsNT (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) ++\n                x \u2237 qsNT (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)\n\n-- Domain predicate for quicksort.\ndata QSDom : List \u2115 \u2192 Set where\n  qsDom-[] : QSDom []\n  qsDom-\u2237  : \u2200 {x xs} \u2192\n             (QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             (QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs)) \u2192\n             QSDom (x \u2237 xs)\n\n-- Induction principle associated to the domain predicate of quicksort.\n-- (It was not necessary).\nindQSDom : (P : List \u2115 \u2192 Set) \u2192\n           P [] \u2192\n           (\u2200 {x xs} \u2192 QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) \u2192\n                       P (x \u2237 xs)) \u2192\n           (\u2200 {xs} \u2192 QSDom xs \u2192 P xs)\nindQSDom P P[] ih qsDom-[]        = P[]\nindQSDom P P[] ih (qsDom-\u2237 h\u2081 h\u2082) =\n  ih h\u2081 (indQSDom P P[] ih h\u2081) h\u2082 (indQSDom P P[] ih h\u2082)\n\n-- Well-founded relation on lists.\n_\u27ea\u2032_ : {A : Set} \u2192 List A \u2192 List A \u2192 Set\nxs \u27ea\u2032 ys = length xs <\u2032 length ys\n\nwf-\u27ea\u2032 : Well-founded _\u27ea\u2032_\nwf-\u27ea\u2032 = InvImg.well-founded <-well-founded\n\n-- The well-founded induction principle on _\u27ea\u2032_.\n-- postulate\n--   wfi-\u27ea\u2032 : (P : List \u2115 \u2192 Set) \u2192\n--            (\u2200 xs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 xs \u2192 P ys) \u2192 P xs) \u2192\n--            \u2200 xs \u2192 P xs\n\n-- The quicksort algorithm is total.\n\nfilter-length : \u2200 {A : Set} (p : A \u2192 Bool) xs \u2192\n                length (filter p xs) \u2264\u2032 length xs\nfilter-length p []       = \u2264\u2032-refl\nfilter-length p (x \u2237 xs) with p x\n... | true  = \u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length p xs)))\n... | false = \u2264\u2032-step (filter-length p xs)\n\nmodule AllWF = Induction.WellFounded.All wf-\u27ea\u2032\n\nallQSDom : \u2200 xs \u2192 QSDom xs\n-- If we use wfi-\u27ea\u2032 then allQSDom =  wfi-\u27ea\u2032 P ih\nallQSDom = build AllWF.wfRec-builder P ih\n  where\n    P : List \u2115 \u2192 Set\n    P = QSDom\n\n    -- If we use wfi-\u27ea\u2032 then\n    -- ih : \u2200 zs \u2192 (\u2200 ys \u2192 ys \u27ea\u2032 zs \u2192 P ys) \u2192 P zs\n    ih :  \u2200 zs \u2192 WfRec _\u27ea\u2032_ P zs \u2192 P zs\n    ih []       h = qsDom-[]\n    ih (z \u2237 zs) h = qsDom-\u2237 prf\u2081 prf\u2082\n\n       where\n         c\u2081 : \u2115 \u2192 Bool\n         c\u2081 = \u03bb y \u2192 \u230a y \u2264\u2032? z \u230b\n\n         c\u2082 : \u2115 \u2192 Bool\n         c\u2082 = \u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b\n\n         f\u2081 : List \u2115\n         f\u2081 = filter c\u2081 zs\n\n         f\u2082 : List \u2115\n         f\u2082 = filter c\u2082 zs\n\n         prf\u2081 : QSDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? z \u230b) zs)\n         prf\u2081 = h f\u2081 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2081 zs))))\n\n         prf\u2082 : QSDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? z \u230b) zs)\n         prf\u2082 = h f\u2082 (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264\u2032\u21d2\u2264 (filter-length c\u2082 zs))))\n\n-- Quicksort algorithm by structural recursion on the domain predicate.\nqsDom : \u2200 xs \u2192 QSDom xs \u2192 List \u2115\nqsDom .[]      qsDom-[]        = []\nqsDom (x \u2237 xs) (qsDom-\u2237 h\u2081 h\u2082) =\n  qsDom (filter (\u03bb y \u2192 \u230a y \u2264\u2032? x \u230b) xs) h\u2081 ++\n  x \u2237 qsDom (filter (\u03bb y \u2192 not \u230a y \u2264\u2032? x \u230b) xs) h\u2082\n\n-- The quicksort algorithm.\nqs : List \u2115 \u2192 List \u2115\nqs xs = qsDom xs (allQSDom xs)\n\n-- Testing.\nl\u2081 : List \u2115\nl\u2081 = []\nl\u2082 = 1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 []\nl\u2083 = 5 \u2237 4 \u2237 3 \u2237 2 \u2237 1 \u2237 []\nl\u2084 = 4 \u2237 1 \u2237 3 \u2237 5 \u2237 2 \u2237 []\n\nt\u2081 = qs l\u2081\nt\u2082 = qs l\u2082\nt\u2083 = qs l\u2083\nt\u2084 = qs l\u2084\n","old_contents":"------------------------------------------------------------------------------\n-- Quicksort using the Bove-Capretta method\n------------------------------------------------------------------------------\n\nmodule QuickSortBC where\n\nopen import Data.Bool\nopen import Data.List -- hiding ( filter )\nopen import Data.Nat hiding (_\u2264_ ; _<_ ; _\u2265_ ; _>_)\n\n------------------------------------------------------------------------------\n\n_\u2264_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero  \u2264 n     = true\nsuc m \u2264 zero  = false\nsuc m \u2264 suc n = m \u2264 n\n\n_<_ : \u2115 \u2192 \u2115 \u2192 Bool\nm < n = suc m \u2264 n\n\n_\u2265_ : \u2115 \u2192 \u2115 \u2192 Bool\nm \u2265 n = n \u2264 m\n\n-- filter : {A : Set} \u2192 (A \u2192 Bool) \u2192 List A \u2192 List A\n-- filter _ []    = []\n-- filter P (x \u2237 xs) with P x\n-- ... | true  = x \u2237 filter P xs\n-- ... | false = filter P xs\n\n-- Non-terminating quicksort.\nqsNT : List \u2115 \u2192 List \u2115\nqsNT []       = []\nqsNT (x \u2237 xs) = qsNT (filter (\u03bb y \u2192 y < x) xs) ++\n                x \u2237 qsNT (filter (\u03bb y \u2192 x < y) xs)\n\n-- Domain predicate for quicksort.\ndata Dqs : List \u2115 \u2192 Set where\n  dnil  : Dqs []\n  dcons : \u2200 {x xs} \u2192 Dqs (filter (\u03bb y \u2192 y < x) xs) \u2192\n                     Dqs (filter (\u03bb y \u2192 x < y) xs) \u2192\n                     Dqs (x \u2237 xs)\n\n-- Induction principle associated to the domain predicate of quicksort.\nindDqs : (P : List \u2115 \u2192 Set) \u2192\n         P [] \u2192\n         (\u2200 {x xs} \u2192 Dqs (filter (\u03bb y \u2192 y < x) xs) \u2192\n                     P (filter (\u03bb y \u2192 y < x) xs) \u2192\n                     Dqs (filter (\u03bb y \u2192 x < y) xs) \u2192\n                     P (filter (\u03bb y \u2192 x < y) xs) \u2192\n                     P (x \u2237 xs)) \u2192\n         (\u2200 {xs} \u2192 Dqs xs \u2192 P xs)\nindDqs P P[] Pcons dnil       = P[]\nindDqs P P[] ih (dcons h\u2081 h\u2082) =\n  ih h\u2081 (indDqs P P[] ih h\u2081) h\u2082 (indDqs P P[] ih h\u2082)\n\n-- Quicksort algorithm by structural recursion on the domain predicate.\nqsHelper : \u2200 xs \u2192 Dqs xs \u2192 List \u2115\nqsHelper .[]      dnil = []\nqsHelper (x \u2237 xs) (dcons h\u2081 h\u2082) =\n  qsHelper (filter (\u03bb y \u2192 y < x) xs) h\u2081 ++\n  x \u2237 qsHelper (filter (\u03bb y \u2192 x < y) xs) h\u2082\n\n-- The quicksort algorithm is total.\npostulate\n  allDqs : \u2200 xs \u2192 Dqs xs\n\n-- The quicksort algorithm.\nqs : List \u2115 \u2192 List \u2115\nqs xs = qsHelper xs (allDqs xs)\n\nl\u2081 : List \u2115\nl\u2081 = []\n\nl\u2082 = 1 \u2237 2 \u2237 3 \u2237 4 \u2237 5 \u2237 []\n\nl\u2083 = 5 \u2237 4 \u2237 3 \u2237 2 \u2237 1 \u2237 []\n\nl\u2084 = 4 \u2237 1 \u2237 3 \u2237 5 \u2237 2 \u2237 []\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"653605cc226a422c060b15ccb4f7d6c975709b10","subject":"add *-interchange for Data.Nat.NP","message":"add *-interchange for Data.Nat.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z = ! \u2115\u00b0.+-assoc x y z \u2219 ap (flip _+_ z) (\u2115\u00b0.+-comm x y) \u2219 \u2115\u00b0.+-assoc y x z\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 ap (flip _+_ z))\n\n*-assoc-comm : \u2200 x y z \u2192 x * (y * z) \u2261 y * (x * z)\n*-assoc-comm x y z = ! \u2115\u00b0.*-assoc x y z \u2219 ap (flip _*_ z) (\u2115\u00b0.*-comm x y) \u2219 \u2115\u00b0.*-assoc y x z\n\n*-interchange : Interchange _\u2261_ _*_ _*_\n*-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _*_ \u2115\u00b0.*-assoc \u2115\u00b0.*-comm (\u03bb z \u2192 ap (flip _*_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base hiding (_==_; _\u00b2)\nopen import Data.Product using (\u2203; _,_) renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Zero using (\ud835\udfd8-elim; \ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning; !_; _\u2219_; ap; ap\u2082; coe)\n       renaming (refl to idp)\nopen import HoTT\nopen Equivalences\n\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties\n\npattern 1+_ x = suc x\npattern 2+_ x = 1+ suc x\npattern 3+_ x = 2+ suc x\npattern 4+_ x = 3+ suc x\n\n\u27e80\u21941\u27e9 : \u2115 \u2192 \u2115\n\u27e80\u21941\u27e9 0 = 1\n\u27e80\u21941\u27e9 1 = 0\n\u27e80\u21941\u27e9 n = n\n\nprivate\n  _\u00b2 : \u2200 {A : \u2605\u2080} \u2192 Endo (Endo A)\n  f \u00b2 = f \u2218 f\n\n\u27e80\u21941\u27e9-involutive : \u27e80\u21941\u27e9 \u2218 \u27e80\u21941\u27e9 \u2257 id\n\u27e80\u21941\u27e9-involutive 0             = idp\n\u27e80\u21941\u27e9-involutive 1             = idp\n\u27e80\u21941\u27e9-involutive (suc (suc _)) = idp\n\n\u21d1\u27e8_\u27e9 : (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u21d1\u27e8 f \u27e9 zero    = zero\n\u21d1\u27e8 f \u27e9 (suc n) = suc (f n)\n\n\u27e80\u21941+_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e80\u21941+ 0     \u27e9 = \u27e80\u21941\u27e9\n\u27e80\u21941+ suc n \u27e9 = \u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 \u2218 \u27e80\u21941\u27e9\n\n\u27e8_\u2194+1\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e8 0     \u2194+1\u27e9 = \u27e80\u21941\u27e9\n\u27e8 suc n \u2194+1\u27e9 0       = 0\n\u27e8 suc n \u2194+1\u27e9 (suc m) = suc (\u27e8 n \u2194+1\u27e9 m)\n\n\u27e8_\u2194+1\u27e9-involutive : \u2200 n \u2192 \u27e8 n \u2194+1\u27e9 \u2218 \u27e8 n \u2194+1\u27e9 \u2257 id\n\u27e8_\u2194+1\u27e9-involutive 0 = \u27e80\u21941\u27e9-involutive\n\u27e8_\u2194+1\u27e9-involutive (suc _) 0       = idp\n\u27e8_\u2194+1\u27e9-involutive (suc n) (suc m) = ap suc (\u27e8 n \u2194+1\u27e9-involutive m)\n\n\u27e8_\u2194+1\u27e9-equiv : \u2115 \u2192 \u2115 \u2243 \u2115\n\u27e8 n \u2194+1\u27e9-equiv = self-inv-equiv \u27e8 n \u2194+1\u27e9 \u27e8 n \u2194+1\u27e9-involutive\n\n\u21d1\u27e8_\u27e9-involutive : \u2200 {f} \u2192 f \u00b2 \u2257 id \u2192 \u21d1\u27e8 f \u27e9 \u00b2 \u2257 id\n\u21d1\u27e8 f\u00b2id \u27e9-involutive zero    = idp\n\u21d1\u27e8 f\u00b2id \u27e9-involutive (suc x) = ap suc (f\u00b2id x)\n\n\u27e80\u21941+_\u27e9-involutive : \u2200 n \u2192 \u27e80\u21941+ n \u27e9 \u00b2 \u2257 id\n\u27e80\u21941+_\u27e9-involutive zero = \u27e80\u21941\u27e9-involutive\n\u27e80\u21941+_\u27e9-involutive (suc n) x = ap (\u27e80\u21941\u27e9 \u2218 \u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9) (\u27e80\u21941\u27e9-involutive (\u21d1\u27e8 \u27e80\u21941+ n \u27e9 \u27e9 (\u27e80\u21941\u27e9 x)))\n  \u2219 ap \u27e80\u21941\u27e9 (\u21d1\u27e8 \u27e80\u21941+ n \u27e9-involutive \u27e9-involutive (\u27e80\u21941\u27e9 x)) \u2219 \u27e80\u21941\u27e9-involutive x\n\nmodule _ {{_ : UA}} where\n    \u27e8_\u2194+1\u27e9-eq : \u2115 \u2192 \u2115 \u2261 \u2115\n    \u27e8_\u2194+1\u27e9-eq = ua \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 : \u2200 n m \u2192 coe \u27e8 n \u2194+1\u27e9-eq m \u2261 \u27e8 n \u2194+1\u27e9 m\n    \u27e8_\u2194+1\u27e9-eq-\u03b2 = coe-\u03b2 \u2218 \u27e8_\u2194+1\u27e9-equiv\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115cmp = StrictTotalOrder strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder decTotalOrder\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring commutativeSemiring\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\ninfixr 8 _\u2219\u2264_\n_\u2219\u2264_ = \u2115\u2264.trans\n_\u2219cmp_ = \u2115cmp.trans\n_\u2219<_ = <-trans\n\n[P:_zero:_suc:_] : \u2200 {p} (P : \u2115 \u2192 \u2605 p) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 n \u2192 P n\n[P: _ zero: z suc: _ ] zero    = z\n[P: P zero: z suc: s ] (suc n) = s ([P: P zero: z suc: s ] n)\n\n[zero:_suc:_] : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 (\u2115 \u2192 A \u2192 A) \u2192 \u2115 \u2192 A\n[zero: z suc: s ] = [P: _ zero: z suc: (\u03bb {n} \u2192 s n) ]\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 idp \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = ap pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    = id\n+-\u2264-inj (suc x) = +-\u2264-inj x \u2218 \u2264-pred\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = idp\na\u2261a\u2293b+a\u2238b zero (suc b) = idp\na\u2261a\u2293b+a\u2238b (suc a) zero = idp\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite ! a\u2261a\u2293b+a\u2238b a b = idp\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \ud835\udfd8\n\u00acn\u2264x<n n p q = sucx\u2270x _ (q \u2219\u2264 p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \ud835\udfd8\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (q \u2219\u2264 \u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n) \u2219\u2264 \u2115\u2264.refl {n} +-mono z\u2264n \u2219\u2264 p)\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\nmodule nest-Properties {a} {A : \u2605 a} (f : Endo A) where\n  nest\u2080 : nest 0 f \u2261 id\n  nest\u2080 = idp\n  nest\u2081 : nest 1 f \u2261 f\n  nest\u2081 = idp\n  nest\u2082 : nest 2 f \u2261 f \u2218 f\n  nest\u2082 = idp\n  nest\u2083 : nest 3 f \u2261 f \u2218 f \u2218 f\n  nest\u2083 = idp\n\n  nest-+ : \u2200 m n \u2192 nest (m + n) f \u2261 nest m f \u2218 nest n f\n  nest-+ zero    n = idp\n  nest-+ (suc m) n = ap (_\u2218_ f) (nest-+ m n)\n\n  nest-+' : \u2200 m n \u2192 nest (m + n) f \u2257 nest m f \u2218 nest n f\n  nest-+' m n x = ap (flip _$_ x) (nest-+ m n)\n\n  nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n  nest-* zero n x = idp\n  nest-* (suc m) n x =\n    nest (suc m * n) f x             \u2261\u27e8 idp \u27e9\n    nest (n + m * n) f x             \u2261\u27e8 nest-+' n (m * n) x \u27e9\n    (nest n f \u2218 nest (m * n) f) x    \u2261\u27e8 ap (nest n f) (nest-* m n x) \u27e9\n    (nest n f \u2218 nest m (nest n f)) x \u2261\u27e8 idp \u27e9\n    nest n f (nest m (nest n f) x)   \u2261\u27e8 idp \u27e9\n    nest (suc m) (nest n f) x \u220e\n   where open \u2261-Reasoning\n\n{- WRONG\nmodule more-nest-Properties {a} {A : \u2605 a} where\n  nest-+'' : \u2200 (f : Endo (Endo A)) g m n \u2192 nest m f g \u2218 nest n f g \u2257 nest (m + n) f g\n  nest-+'' f g zero n = {!!}\n  nest-+'' f g (suc m) n = {!!}\n-}\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = idp\n\n_==_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   == zero   = 1\u2082\nzero   == suc _  = 0\u2082\nsuc _  == zero   = 0\u2082\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z = ! \u2115\u00b0.+-assoc x y z \u2219 ap (flip _+_ z) (\u2115\u00b0.+-comm x y) \u2219 \u2115\u00b0.+-assoc y x z\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 ap (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = idp\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = idp\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = idp\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = idp\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = idp\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2294\u2264+ a b \u2219\u2264 \u2264-step \u2115\u2264.refl \u2219\u2264 \u2115\u2264.reflexive (+-assoc-comm 1 a b))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = idp\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = idp\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*\u2032-inj : \u2200 {m n} \u2192 2*\u2032 m \u2261 2*\u2032 n \u2192 m \u2261 n\n2*\u2032-inj {zero}  {zero}  _ = idp\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} p = ap suc (2*\u2032-inj (suc-injective (suc-injective p)))\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite ! 2*\u2032-spec m\n                       | ! 2*\u2032-spec n\n                       = 2*\u2032-inj p\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib x y = +-interchange x x y y\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = idp\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = ! 2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = idp\n2^-comm (suc x) y z rewrite 2^-comm x y z = ! 2^*-2*-comm y \u27e82^ x * z \u27e9\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = idp\n2^-+ (suc x) y z = ap 2*_ (2^-+ x y z)\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = idp\n2\u207f*0\u22610 (suc n) = ap\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = idp\n0\u2238 suc x \u22610 = idp\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = idp\n2*-\u2238 zero    (suc _) = idp\n2*-\u2238 (suc x) (suc y) rewrite ! 2*-\u2238 x y | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = idp\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = idp\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = ! n+k\u2238m b y\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = ! n+k\u2238m (suc b) x\u2264b\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = idp\n\n_*\u2032_ : \u2115 \u2192 \u2115 \u2192 \u2115\n0 *\u2032 n = 0\n1 *\u2032 n = n\nm *\u2032 0 = 0\nm *\u2032 1 = m\nm *\u2032 n = m * n\n\n*\u2032-spec : \u2200 m n \u2192 m *\u2032 n \u2261 m * n\n*\u2032-spec 0             n = idp\n*\u2032-spec 1             n = \u2115\u00b0.+-comm 0 n\n*\u2032-spec (suc (suc m)) 0 = \u2115\u00b0.*-comm 0 m\n*\u2032-spec (suc (suc m)) 1 = ap (suc \u2218\u2032 suc) (! snd \u2115\u00b0.*-identity m)\n*\u2032-spec (suc (suc m)) (suc (suc n)) = idp\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = idp\n\u2238-assoc-+ zero    (suc y) zero    = idp\n\u2238-assoc-+ zero    (suc y) (suc z) = idp\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = n\u2238m\u2264n y _ \u2219\u2264 t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n_^2 : \u2115 \u2192 \u2115\nn ^2 = n * n\n\n^2-spec : \u2200 n \u2192 n ^2 \u2261 n ^ 2\n^2-spec n rewrite snd \u2115\u00b0.*-identity n = idp\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = idp\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = idp\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = idp\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = idp\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = s\u2264s z\u2264n \u2219\u2264 1+\u22642^ n\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , idp\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , ap suc (snd (\u2264\u21d2\u2203 pf))\n\nis0? : \u2115 \u2192 \ud835\udfda\nis0? zero    = 1\u2082\nis0? (suc n) = 0\u2082\n\nmodule _ {{_ : UA}} where\n    open Equivalences\n    \u2203-is0?-uniq : \u2203 (\u2713 \u2218 is0?) \u2261 \ud835\udfd9\n    \u2203-is0?-uniq = ua (equiv _ (const (0 , _)) (const idp)\n                            \u03bb { (0 , _) \u2192 idp ; (suc _ , ()) })\n\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p idp\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 \ud835\udfda\nzero   <= _      = 1\u2082\nsuc _  <= zero   = 0\u2082\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} idp = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y \u2219\u2264 s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p)\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) \u2219\u2264 s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p)\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite ! 2*\u2032-spec y = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = s\u2264s (\u2264-step \u2115\u2264.refl) \u2219\u2264 a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \ud835\udfd8-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \ud835\udfd8-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (! eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713-not-\u00ac p \u2218 <=.complete))\n\neven? odd? : \u2115 \u2192 \ud835\udfda\neven? zero    = 1\u2082\neven? (suc n) = odd? n \nodd? n = not (even? n)\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e2de8ad916775e5b142d5745feb7526c5048fda5","subject":"Nat: \u2115\u02e2","message":"Nat: \u2115\u02e2\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/NP.agda","new_file":"lib\/Data\/Nat\/NP.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\n\u2115\u02e2 = \u2261.setoid \u2115\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\nmodule \u2115\u02e2   = Setoid \u2115\u02e2\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = \u2261.refl\na\u2261a\u2293b+a\u2238b zero (suc b) = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) zero = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite \u2261.sym (a\u2261a\u2293b+a\u2238b a b) = \u2261.refl\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = \u2261.refl\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = \u2261.refl\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = \u2261.refl\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 \u2605\u2080\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : \u2605\u2080} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 \u2605\u2080) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 \u2605\u2080\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713'not'\u00ac p \u2218 <=.complete))\n","old_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Data.Nat.NP where\n\nopen import Type hiding (\u2605)\nimport Algebra\nopen import Algebra.FunctionProperties.NP\nopen import Data.Nat public hiding (module GeneralisedArithmetic; module \u2264-Reasoning; fold)\nopen import Data.Nat.Properties as Props\nopen import Data.Nat.Logical\nopen import Data.Bool.NP hiding (_==_; module ==)\nopen import Data.Product using (proj\u2081; proj\u2082; \u2203; _,_)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Data.Empty using (\u22a5-elim; \u22a5)\nopen import Function.NP\nopen import Relation.Nullary\nopen import Relation.Binary.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2262_; _\u2257_; module \u2261-Reasoning)\n\nmodule \u2115\u00b0   = Algebra.CommutativeSemiring Props.commutativeSemiring\nmodule \u2115cmp = StrictTotalOrder Props.strictTotalOrder\nmodule \u2115\u2264   = DecTotalOrder    decTotalOrder\nmodule \u2115+   = Algebra.CommutativeMonoid \u2115\u00b0.+-commutativeMonoid\nmodule \u2115+\u2032  = Algebra.Monoid \u2115\u00b0.+-monoid\nmodule \u2294\u00b0   = Algebra.CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nmodule \u2264-Reasoning where\n  open Preorder-Reasoning \u2115\u2264.preorder public renaming (_\u223c\u27e8_\u27e9_ to _\u2264\u27e8_\u27e9_)\n  infixr 2 _\u2261\u27e8_\u27e9_\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x \u2261 y \u2192 y \u2264 z \u2192 x \u2264 z\n  _ \u2261\u27e8 \u2261.refl \u27e9 p = p\n  infixr 2 _<\u27e8_\u27e9_\n  _<\u27e8_\u27e9_ : \u2200 x {y z : \u2115} \u2192 x < y \u2192 y \u2264 z \u2192 x < z\n  x <\u27e8 p \u27e9 q = suc x \u2264\u27e8 p \u27e9 q\n\nsuc-injective : \u2200 {n m : \u2115} \u2192 \u2115.suc n \u2261 suc m \u2192 n \u2261 m\nsuc-injective = \u2261.cong pred\n\n+-\u2264-inj : \u2200 x {y z} \u2192 x + y \u2264 x + z \u2192 y \u2264 z\n+-\u2264-inj zero    p       = p\n+-\u2264-inj (suc x) (s\u2264s p) = +-\u2264-inj x p\n\ninfixl 6 _+\u00b0_\ninfixl 7 _*\u00b0_ _\u2293\u00b0_\ninfixl 6 _\u2238\u00b0_ _\u2294\u00b0_\n\n_+\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f +\u00b0 g) x = f x + g x\n\n_\u2238\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2238\u00b0 g) x = f x \u2238 g x\n\n_*\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f *\u00b0 g) x = f x * g x\n\n_\u2294\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2294\u00b0 g) x = f x \u2294 g x\n\n_\u2293\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 A \u2192 \u2115\n(f \u2293\u00b0 g) x = f x \u2293 g x\n\n-- this one is not completly in line with the\n-- others\n_\u2264\u00b0_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 \u2115) \u2192 \u2605 a\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nsucx\u2270x : \u2200 x \u2192 suc x \u2270 x\nsucx\u2270x zero    = \u03bb()\nsucx\u2270x (suc x) = sucx\u2270x x \u2218 \u2264-pred\n\ntotal-\u2264 : \u2200 a b \u2192 a \u2264 b \u228e b \u2264 a\ntotal-\u2264 zero b = inj\u2081 z\u2264n\ntotal-\u2264 (suc a) zero = inj\u2082 z\u2264n\ntotal-\u2264 (suc a) (suc b) with total-\u2264 a b\n... | inj\u2081 p = inj\u2081 (s\u2264s p)\n... | inj\u2082 p = inj\u2082 (s\u2264s p)\n\na\u2261a\u2293b+a\u2238b : \u2200 a b \u2192 a \u2261 a \u2293 b + (a \u2238 b)\na\u2261a\u2293b+a\u2238b zero zero = \u2261.refl\na\u2261a\u2293b+a\u2238b zero (suc b) = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) zero = \u2261.refl\na\u2261a\u2293b+a\u2238b (suc a) (suc b) rewrite \u2261.sym (a\u2261a\u2293b+a\u2238b a b) = \u2261.refl\n\n\u00acn\u2264x<n : \u2200 n {x} \u2192 n \u2264 x \u2192 x < n \u2192 \u22a5\n\u00acn\u2264x<n n p q = sucx\u2270x _ (\u2115\u2264.trans q p)\n\n\u00acn+\u2264y<n : \u2200 n {x y} \u2192 n + x \u2264 y \u2192 y < n \u2192 \u22a5\n\u00acn+\u2264y<n n p q = sucx\u2270x _ (\u2115\u2264.trans q (\u2115\u2264.trans (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 n)) ((\u2115\u2264.refl {n}) +-mono z\u2264n)) p))\n\nfold : \u2200 {a} {A : \u2605 a} \u2192 A \u2192 Endo A \u2192 \u2115 \u2192 A\nfold x f n = nest n f x\n\n+-inj-over-\u2238 : \u2200 x y z \u2192 (x + y) \u2238 (x + z) \u2261 y \u2238 z\n+-inj-over-\u2238 = [i+j]\u2238[i+k]\u2261j\u2238k \n\n2*_ : \u2115 \u2192 \u2115\n2* x = x + x\n\n2*-spec : \u2200 n \u2192 2* n \u2261 2 * n\n2*-spec n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n\n_==_ : (x y : \u2115) \u2192 Bool\nzero   == zero   = true\nzero   == suc _  = false\nsuc _  == zero   = false\nsuc m  == suc n  = m == n\n\n+-assoc-comm : \u2200 x y z \u2192 x + (y + z) \u2261 y + (x + z)\n+-assoc-comm x y z rewrite \u2261.sym (\u2115\u00b0.+-assoc x y z)\n                       | \u2115\u00b0.+-comm x y\n                       | \u2115\u00b0.+-assoc y x z = \u2261.refl\n\n+-interchange : Interchange _\u2261_ _+_ _+_\n+-interchange = InterchangeFromAssocCommCong.\u2219-interchange _\u2261_ \u2261.isEquivalence\n                                                           _+_ \u2115\u00b0.+-assoc \u2115\u00b0.+-comm (\u03bb z \u2192 \u2261.cong (flip _+_ z))\n\na+b\u2261a\u2294b+a\u2293b : \u2200 a b \u2192 a + b \u2261 a \u2294 b + a \u2293 b\na+b\u2261a\u2294b+a\u2293b zero    b       rewrite \u2115\u00b0.+-comm b 0 = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) zero    = \u2261.refl\na+b\u2261a\u2294b+a\u2293b (suc a) (suc b) rewrite +-assoc-comm a 1 b\n                                  | +-assoc-comm (a \u2294 b) 1 (a \u2293 b)\n                                  | a+b\u2261a\u2294b+a\u2293b a b\n                                  = \u2261.refl\n\na\u2293b\u2261a : \u2200 {a b} \u2192 a \u2264 b \u2192 a \u2293 b \u2261 a\na\u2293b\u2261a z\u2264n = \u2261.refl\na\u2293b\u2261a (s\u2264s a\u2264b) rewrite a\u2293b\u2261a a\u2264b = \u2261.refl\n\n\u2294\u2264+ : \u2200 a b \u2192 a \u2294 b \u2264 a + b\n\u2294\u2264+ zero b          = \u2115\u2264.refl\n\u2294\u2264+ (suc a) zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 a))\n\u2294\u2264+ (suc a) (suc b) = s\u2264s (\u2115\u2264.trans (\u2294\u2264+ a b) (\u2115\u2264.trans (\u2264-step \u2115\u2264.refl) (\u2115\u2264.reflexive (+-assoc-comm 1 a b))))\n\n2*\u2032_ : \u2115 \u2192 \u2115\n2*\u2032_ = fold 0 (suc \u2218\u2032 suc)\n\n2*\u2032-spec : \u2200 n \u2192 2*\u2032 n \u2261 2* n\n2*\u2032-spec zero = \u2261.refl\n2*\u2032-spec (suc n) rewrite 2*\u2032-spec n | +-assoc-comm 1 n n = \u2261.refl\n\ndist : \u2115 \u2192 \u2115 \u2192 \u2115\ndist zero    y       = y\ndist x       zero    = x\ndist (suc x) (suc y) = dist x y\n\ndist-refl : \u2200 x \u2192 dist x x \u2261 0\ndist-refl zero = \u2261.refl\ndist-refl (suc x) rewrite dist-refl x = \u2261.refl\n\ndist-0\u2261id : \u2200 x \u2192 dist 0 x \u2261 x\ndist-0\u2261id _ = \u2261.refl\n\ndist-x-x+y\u2261y : \u2200 x y \u2192 dist x (x + y) \u2261 y\ndist-x-x+y\u2261y zero    y = \u2261.refl\ndist-x-x+y\u2261y (suc x) y = dist-x-x+y\u2261y x y\n\ndist-sym : \u2200 x y \u2192 dist x y \u2261 dist y x\ndist-sym zero zero = \u2261.refl\ndist-sym zero (suc y) = \u2261.refl\ndist-sym (suc x) zero = \u2261.refl\ndist-sym (suc x) (suc y) = dist-sym x y\n\ndist-x+ : \u2200 x y z \u2192 dist (x + y) (x + z) \u2261 dist y z\ndist-x+ zero    y z = \u2261.refl\ndist-x+ (suc x) y z = dist-x+ x y z\n\ndist-2* : \u2200 x y \u2192 dist (2* x) (2* y) \u2261 2* dist x y\ndist-2* zero y = \u2261.refl\ndist-2* (suc x) zero = \u2261.refl\ndist-2* (suc x) (suc y) rewrite +-assoc-comm x 1 x\n                              | +-assoc-comm y 1 y = dist-2* x y\n\ndist-asym-def : \u2200 {x y} \u2192 x \u2264 y \u2192 x + dist x y \u2261 y\ndist-asym-def z\u2264n = \u2261.refl\ndist-asym-def (s\u2264s pf) = \u2261.cong suc (dist-asym-def pf)\n\ndist-sym-wlog : \u2200 (f : \u2115 \u2192 \u2115) \u2192 (\u2200 x k \u2192 dist (f x) (f (x + k)) \u2261 f k) \u2192 \u2200 x y \u2192 dist (f x) (f y) \u2261 f (dist x y)\ndist-sym-wlog f pf x y with compare x y\ndist-sym-wlog f pf x .(suc (x + k)) | less .x k with pf x (suc k)\n... | q rewrite +-assoc-comm x 1 k | q | \u2261.sym (+-assoc-comm x 1 k) | dist-x-x+y\u2261y x (suc k) = \u2261.refl\ndist-sym-wlog f pf .y y | equal .y with pf y 0\n... | q rewrite \u2115\u00b0.+-comm y 0 | dist-refl y = q\ndist-sym-wlog f pf .(suc (y + k)) y | greater .y k with pf y (suc k)\n... | q rewrite +-assoc-comm 1 y k | dist-sym (y + suc k) y | dist-x-x+y\u2261y y (suc k) | dist-sym (f (y + suc k)) (f y) = q\n\ndist-x* : \u2200 x y z \u2192 dist (x * y) (x * z) \u2261 x * dist y z\ndist-x* x = dist-sym-wlog (_*_ x) pf\n  where pf : \u2200 a k \u2192 dist (x * a) (x * (a + k)) \u2261 x * k\n        pf a k rewrite proj\u2081 \u2115\u00b0.distrib x a k = dist-x-x+y\u2261y (x * a) _\n\n2^\u27e8_\u27e9* : \u2115 \u2192 \u2115 \u2192 \u2115\n2^\u27e8 n \u27e9* x = fold x 2*_ n\n\n\u27e82^_*_\u27e9 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e82^ n * x \u27e9 = 2^\u27e8 n \u27e9* x\n\n2*-distrib : \u2200 x y \u2192 2* x + 2* y \u2261 2* (x + y) \n2*-distrib = solve 2 (\u03bb x y \u2192 2:* x :+ 2:* y := 2:* (x :+ y)) \u2261.refl\n      where open SemiringSolver\n            2:* : \u2200 {n} \u2192 Polynomial n \u2192 Polynomial n\n            2:* x = x :+ x\n\n2^*-distrib : \u2200 k x y \u2192 \u27e82^ k * (x + y)\u27e9 \u2261 \u27e82^ k * x \u27e9 + \u27e82^ k * y \u27e9\n2^*-distrib zero x y = \u2261.refl\n2^*-distrib (suc k) x y rewrite 2^*-distrib k x y = \u2261.sym (2*-distrib \u27e82^ k * x \u27e9 \u27e82^ k * y \u27e9)\n\n2^*-2*-comm : \u2200 k x \u2192 \u27e82^ k * 2* x \u27e9 \u2261 2* \u27e82^ k * x \u27e9\n2^*-2*-comm k x = 2^*-distrib k x x\n\ndist-2^* : \u2200 x y z \u2192 dist \u27e82^ x * y \u27e9 \u27e82^ x * z \u27e9 \u2261 \u27e82^ x * dist y z \u27e9\ndist-2^* x = dist-sym-wlog (2^\u27e8 x \u27e9*) pf\n  where pf : \u2200 a k \u2192 dist \u27e82^ x * a \u27e9 \u27e82^ x * (a + k) \u27e9 \u2261 \u27e82^ x * k \u27e9\n        pf a k rewrite 2^*-distrib x a k = dist-x-x+y\u2261y \u27e82^ x * a \u27e9 \u27e82^ x * k \u27e9\n\ndist-bounded : \u2200 {x y f} \u2192 x \u2264 f \u2192 y \u2264 f \u2192 dist x y \u2264 f\ndist-bounded z\u2264n       y\u2264f = y\u2264f\ndist-bounded (s\u2264s x\u2264f) z\u2264n = s\u2264s x\u2264f\ndist-bounded (s\u2264s x\u2264f) (s\u2264s y\u2264f) = \u2264-step (dist-bounded x\u2264f y\u2264f)\n\n2*-mono : \u2200 {a b} \u2192 a \u2264 b \u2192 2* a \u2264 2* b\n2*-mono pf = pf +-mono pf\n\n2^*-mono : \u2200 k {a b} \u2192 a \u2264 b \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9\n2^*-mono zero    pf = pf\n2^*-mono (suc k) pf = 2*-mono (2^*-mono k pf)\n\n2*-mono\u2032 : \u2200 {a b} \u2192 2* a \u2264 2* b \u2192 a \u2264 b\n2*-mono\u2032 {zero} pf = z\u2264n\n2*-mono\u2032 {suc a} {zero} ()\n2*-mono\u2032 {suc a} {suc b} pf rewrite +-assoc-comm a 1 a\n                                  | +-assoc-comm b 1 b = s\u2264s (2*-mono\u2032 (\u2264-pred (\u2264-pred pf)))\n\n2^*-mono\u2032 : \u2200 k {a b} \u2192 \u27e82^ k * a \u27e9 \u2264 \u27e82^ k * b \u27e9 \u2192 a \u2264 b\n2^*-mono\u2032 zero    = id\n2^*-mono\u2032 (suc k) = 2^*-mono\u2032 k \u2218 2*-mono\u2032\n\n2^-comm : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ y * \u27e82^ x * z \u27e9 \u27e9\n2^-comm zero y z = \u2261.refl\n2^-comm (suc x) y z rewrite 2^-comm x y z = \u2261.sym (2^*-2*-comm y \u27e82^ x * z \u27e9)\n\n2^-+ : \u2200 x y z \u2192 \u27e82^ x * \u27e82^ y * z \u27e9 \u27e9 \u2261 \u27e82^ (x + y) * z \u27e9\n2^-+ zero    y z = \u2261.refl\n2^-+ (suc x) y z = \u2261.cong 2*_ (2^-+ x y z)\n\n2*\u2032-inj : \u2200 {m n} \u2192 \u27e6\u2115\u27e7 (2*\u2032 m) (2*\u2032 n) \u2192 \u27e6\u2115\u27e7 m n\n2*\u2032-inj {zero}  {zero}  _ = zero\n2*\u2032-inj {zero}  {suc _} ()\n2*\u2032-inj {suc _} {zero}  ()\n2*\u2032-inj {suc m} {suc n} (suc (suc p)) = suc (2*\u2032-inj p)\n\n2*-inj : \u2200 {m n} \u2192 2* m \u2261 2* n \u2192 m \u2261 n\n2*-inj {m} {n} p rewrite \u2261.sym (2*\u2032-spec m)\n                       | \u2261.sym (2*\u2032-spec n)\n                       = \u27e6\u2115\u27e7\u21d2\u2261 (2*\u2032-inj (\u27e6\u2115\u27e7\u02e2.reflexive p))\n\n2^-inj : \u2200 k {m n} \u2192 \u27e82^ k * m \u27e9 \u2261 \u27e82^ k * n \u27e9 \u2192 m \u2261 n\n2^-inj zero    = id\n2^-inj (suc k) = 2^-inj k \u2218 2*-inj\n\ncancel-*-left : \u2200 i j {k} \u2192 suc k * i \u2261 suc k * j \u2192 i \u2261 j\ncancel-*-left i j {k}\n  rewrite \u2115\u00b0.*-comm (suc k) i\n        | \u2115\u00b0.*-comm (suc k) j = cancel-*-right i j {k}\n\n2\u207f*0\u22610 : \u2200 n \u2192 \u27e82^ n * 0 \u27e9 \u2261 0\n2\u207f*0\u22610 zero    = \u2261.refl\n2\u207f*0\u22610 (suc n) = \u2261.cong\u2082 _+_ (2\u207f*0\u22610 n) (2\u207f*0\u22610 n)\n\n0\u2238_\u22610 : \u2200 x \u2192 0 \u2238 x \u2261 0\n0\u2238 zero  \u22610 = \u2261.refl\n0\u2238 suc x \u22610 = \u2261.refl\n\n2*-\u2238 : \u2200 x y \u2192 2* x \u2238 2* y \u2261 2* (x \u2238 y)\n2*-\u2238 _       zero    = \u2261.refl\n2*-\u2238 zero    (suc _) = \u2261.refl\n2*-\u2238 (suc x) (suc y) rewrite \u2261.sym (2*-\u2238 x y) | \u2115\u00b0.+-comm x (1 + x) | \u2115\u00b0.+-comm y (1 + y) = \u2261.refl\n\nn+k\u2238m : \u2200 {m n} k \u2192 m \u2264 n \u2192 n + k \u2238 m \u2261 (n \u2238 m) + k\nn+k\u2238m k z\u2264n = \u2261.refl\nn+k\u2238m k (s\u2264s m\u2264n) = n+k\u2238m k m\u2264n\n\nfactor-+-\u2238 : \u2200 {b x y} \u2192 x \u2264 b \u2192 y \u2264 b \u2192 (b \u2238 x) + (b \u2238 y) \u2261 2* b \u2238 (x + y)\nfactor-+-\u2238 {b} {zero} {y} z\u2264n y\u2264b rewrite \u2115\u00b0.+-comm b (b \u2238 y) = \u2261.sym (n+k\u2238m b y\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) z\u2264n             rewrite \u2115\u00b0.+-comm x 0 = \u2261.sym (n+k\u2238m (suc b) x\u2264b)\nfactor-+-\u2238 (s\u2264s {x} {b} x\u2264b) (s\u2264s {y} y\u2264b)   rewrite factor-+-\u2238 x\u2264b y\u2264b\n                                              | \u2115\u00b0.+-comm b (suc b)\n                                              | \u2115\u00b0.+-comm x (suc y)\n                                              | n+k\u2238m (suc y) x\u2264b\n                                              | \u2115\u00b0.+-comm x y = \u2261.refl\n\n\u2264\u2192\u22621+ : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2262 suc y\n\u2264\u2192\u22621+ z\u2264n     ()\n\u2264\u2192\u22621+ (s\u2264s p) q = \u2264\u2192\u22621+ p (suc-injective q)\n\n<\u2192\u2262 : \u2200 {x y} \u2192 x < y \u2192 x \u2262 y\n<\u2192\u2262 (s\u2264s p) = \u2264\u2192\u22621+ p\n\n-- Triangular inequality\ndist-sum : \u2200 x y z \u2192 dist x z \u2264 dist x y + dist y z\ndist-sum zero    zero    z       = \u2115\u2264.refl\ndist-sum zero    (suc y) zero    = z\u2264n\ndist-sum zero    (suc y) (suc z) = s\u2264s (dist-sum zero y z)\ndist-sum (suc x) zero    zero    = s\u2264s (\u2115\u2264.reflexive (\u2115\u00b0.+-comm 0 x))\ndist-sum (suc x) (suc y) zero\n  rewrite \u2115\u00b0.+-comm (dist x y) (suc y)\n        | dist-sym x y = s\u2264s (dist-sum zero y x)\ndist-sum (suc x) zero    (suc z) = \u2115\u2264.trans (dist-sum x zero z)\n                                  (\u2115\u2264.trans (\u2115\u2264.reflexive (\u2261.cong\u2082 _+_ (dist-sym x 0) \u2261.refl))\n                                            (\u2264-step (\u2115\u2264.refl {x} +-mono \u2264-step \u2115\u2264.refl)))\ndist-sum (suc x) (suc y) (suc z) = dist-sum x y z\n\n\u2238-assoc-+ : \u2200 x y z \u2192 x \u2238 y \u2238 z \u2261 x \u2238 (y + z)\n\u2238-assoc-+ x       zero    z       = \u2261.refl\n\u2238-assoc-+ zero    (suc y) zero    = \u2261.refl\n\u2238-assoc-+ zero    (suc y) (suc z) = \u2261.refl\n\u2238-assoc-+ (suc x) (suc y) z       = \u2238-assoc-+ x y z\n\n_\u2238-tone_ : \u2200 {x y t u} \u2192 x \u2264 y \u2192 t \u2264 u \u2192 t \u2238 y \u2264 u \u2238 x\n_\u2238-tone_ {y = y} z\u2264n t\u2264u = \u2115\u2264.trans (n\u2238m\u2264n y _) t\u2264u\ns\u2264s x\u2264y \u2238-tone z\u2264n = z\u2264n\ns\u2264s x\u2264y \u2238-tone s\u2264s t\u2264u = x\u2264y \u2238-tone t\u2264u\n\n\u2238-mono' : \u2200 k {x y} \u2192 x \u2264 y \u2192 x \u2238 k \u2264 y \u2238 k\n\u2238-mono' k = _\u2238-tone_ (\u2115\u2264.refl {k})\n\n\u2238-anti : \u2200 k {x y} \u2192 x \u2264 y \u2192 k \u2238 y \u2264 k \u2238 x\n\u2238-anti k x\u2264y = _\u2238-tone_ x\u2264y (\u2115\u2264.refl {k})\n\n{-\npost--ulate\n  dist-\u2264     : \u2200 x y \u2192 dist x y \u2264 x\n  dist-mono\u2081 : \u2200 x y z t \u2192 x \u2264 y \u2192 dist z t \u2264 dist (x + z) (y + t)\n-}\n\n-- Haskell\n-- let dist x y = abs (x - y)\n-- quickCheck (forAll (replicateM 3 (choose (0,100))) (\\ [x,y,z] -> dist (x * y) (x * z) == x * dist y z))\n\ninfix 8 _^_\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nb ^ zero  = 1\nb ^ suc n = b * b ^ n\n\n2^_ : \u2115 \u2192 \u2115\n2^ n = \u27e82^ n * 1 \u27e9\n\n2^-spec : \u2200 n \u2192 2^ n \u2261 2 ^ n\n2^-spec zero = \u2261.refl\n2^-spec (suc n) rewrite 2^-spec n | 2*-spec (2 ^ n) = \u2261.refl\n\n2^*-spec : \u2200 m n \u2192 2^\u27e8 m \u27e9* n \u2261 2 ^ m * n\n2^*-spec zero    n rewrite \u2115\u00b0.+-comm n 0 = \u2261.refl\n2^*-spec (suc m) n rewrite 2^*-spec m n\n                         | \u2115\u00b0.*-assoc 2 (2 ^ m) n\n                         | \u2115\u00b0.+-comm (2 ^ m * n) 0 = \u2261.refl\n\n1\u22642^ : \u2200 n \u2192 2^ n \u2265 1\n1+\u22642^ : \u2200 n \u2192 2^ n \u2265 1 + n\n1+\u22642^ zero    = s\u2264s z\u2264n\n1+\u22642^ (suc n) = (1\u22642^ n) +-mono (1+\u22642^ n)\n\n1\u22642^ n  = \u2115\u2264.trans (s\u2264s z\u2264n) (1+\u22642^ n)\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x \u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Hyper_operator\n_\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\na \u2191\u27e8 suc n             \u27e9 (suc b) = a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\na \u2191\u27e8 0                 \u27e9 b       = suc b\na \u2191\u27e8 1                 \u27e9 0       = a\na \u2191\u27e8 2                 \u27e9 0       = 0\na \u2191\u27e8 suc (suc (suc n)) \u27e9 0       = 1\n\nmodule \u2191-Props where\n  \u2191-fold : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 b \u2261 fold (a \u2191\u27e8 suc n \u27e9 0) (_\u2191\u27e8_\u27e9_ a n) b\n  \u2191-fold a n zero = \u2261.refl\n  \u2191-fold a n (suc b) = \u2261.cong (_\u2191\u27e8_\u27e9_ a n) (\u2191-fold a n b)\n\n  \u2191\u2080-+ : \u2200 a b \u2192 a \u2191\u27e8 0 \u27e9 b \u2261 1 + b\n  \u2191\u2080-+ a b = \u2261.refl\n\n  \u2191\u2081-+ : \u2200 a b \u2192 a \u2191\u27e8 1 \u27e9 b \u2261 b + a\n  \u2191\u2081-+ a zero    = \u2261.refl\n  \u2191\u2081-+ a (suc b) = \u2261.cong suc (\u2191\u2081-+ a b)\n\n  \u2191\u2082-* : \u2200 a b \u2192 a \u2191\u27e8 2 \u27e9 b \u2261 b * a\n  \u2191\u2082-* a zero    = \u2261.refl\n  \u2191\u2082-* a (suc b) rewrite \u2191\u2082-* a b\n                       | \u2191\u2081-+ a (b * a)\n                       | \u2115\u00b0.+-comm (b * a) a\n                       = \u2261.refl\n\n  -- fold 1 (_*_ a) b \u2261 a ^ b\n  \u2191\u2083-^ : \u2200 a b \u2192 a \u2191\u27e8 3 \u27e9 b \u2261 a ^ b\n  \u2191\u2083-^ a zero = \u2261.refl\n  \u2191\u2083-^ a (suc b) rewrite \u2191\u2083-^ a b\n                       | \u2191\u2082-* a (a ^ b)\n                       | \u2115\u00b0.*-comm (a ^ b) a\n                       = \u2261.refl\n\n  _\u2191\u27e8_\u27e90 : \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u27e8 0                 \u27e90 = 1\n  a \u2191\u27e8 1                 \u27e90 = a\n  a \u2191\u27e8 2                 \u27e90 = 0\n  a \u2191\u27e8 suc (suc (suc n)) \u27e90 = 1\n\n  _`\u2191\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _`\u2191\u27e8_\u27e9_ a 0       = suc\n  _`\u2191\u27e8_\u27e9_ a (suc n) = fold (a \u2191\u27e8 suc n \u27e90) (_`\u2191\u27e8_\u27e9_ a n)\n\n  \u2191-ind-rule : \u2200 a n b \u2192 a \u2191\u27e8 suc n \u27e9 (suc b) \u2261 a \u2191\u27e8 n \u27e9 (a \u2191\u27e8 suc n \u27e9 b)\n  \u2191-ind-rule a n b = \u2261.refl\n\n  _\u2191\u2032\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  a \u2191\u2032\u27e8 0 \u27e9 b = suc b\n  a \u2191\u2032\u27e8 1 \u27e9 b = a + b\n  a \u2191\u2032\u27e8 2 \u27e9 b = a * b\n  a \u2191\u2032\u27e8 3 \u27e9 b = a ^ b\n  a \u2191\u2032\u27e8 suc (suc (suc (suc n))) \u27e9 b = a \u2191\u27e8 4 + n \u27e9 b\n\n  \u2191-\u2191\u2032 : \u2200 a n b \u2192 a \u2191\u27e8 n \u27e9 b \u2261 a \u2191\u2032\u27e8 n \u27e9 b\n  \u2191-\u2191\u2032 a 0 b = \u2261.refl\n  \u2191-\u2191\u2032 a 1 b = \u2261.trans (\u2191\u2081-+ a b) (\u2115\u00b0.+-comm b a)\n  \u2191-\u2191\u2032 a 2 b = \u2261.trans (\u2191\u2082-* a b) (\u2115\u00b0.*-comm b a)\n  \u2191-\u2191\u2032 a 3 b = \u2191\u2083-^ a b\n  \u2191-\u2191\u2032 a (suc (suc (suc (suc _)))) b = \u2261.refl\n\n  _\u21912+\u27e8_\u27e9_ : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  _\u21912+\u27e8_\u27e9_ a = fold (_*_ a) (fold 1)\n\nmodule InflModule where\n  -- Inflationary functions\n  Infl< : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl< f = \u2200 {x} \u2192 x < f x\n\n  Infl : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Infl f = Infl< (suc \u2218 f)\n\n  InflT< : (f : Endo (Endo \u2115)) \u2192 \u2605\u2080\n  InflT< f = \u2200 {g} \u2192 Infl< g \u2192 Infl< (f g)\n\n{-\n\u2200 {x} \u2192 x \u2264 f x\n\n\n  1 + x \u2264 1 + f x\n  x < 1 + f x\n  x < (suc \u2218 f) x\n  Infl< (suc \u2218 f)\n\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n  Infl< f = \u2200 {x} \u2192 1 + x \u2264 f x\n\nInfl< f \u2192 Infl f\nInfl< f \u2192 Infl< (suc \u2218 f)\n-}\n\n  id-infl : Infl id\n  id-infl = \u2115\u2264.refl\n\n  \u2218-infl< : \u2200 {f g} \u2192 Infl< f \u2192 Infl< g \u2192 Infl< (f \u2218 g)\n  \u2218-infl< inflf< inflg< = <-trans inflg< inflf<\n\n  suc-infl< : Infl< suc\n  suc-infl< = \u2115\u2264.refl\n\n  suc-infl : Infl suc\n  suc-infl = s\u2264s (\u2264-step \u2115\u2264.refl)\n\n  nest-infl : \u2200 f (finfl : Infl f) n \u2192 Infl (nest n f)\n  nest-infl f _ zero = \u2115\u2264.refl\n  nest-infl f finfl (suc n) = \u2115\u2264.trans (nest-infl f finfl n) finfl\n\n  nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  nest-infl< f finfl< zero = finfl<\n  nest-infl< f finfl< (suc n) = <-trans (nest-infl< f finfl< n) finfl< \n\n-- See Function.NP.nest-Properties for more properties\nmodule fold-Props where\n\n  fold-suc : \u2200 m n \u2192 fold m suc n \u2261 n + m\n  fold-suc m zero = \u2261.refl\n  fold-suc m (suc n) rewrite fold-suc m n = \u2261.refl\n\n  fold-+ : \u2200 m n \u2192 fold 0 (_+_ m) n \u2261 n * m\n  fold-+ m zero = \u2261.refl\n  fold-+ m (suc n) rewrite fold-+ m n = \u2261.refl\n\n  fold-* : \u2200 m n \u2192 fold 1 (_*_ m) n \u2261 m ^ n\n  fold-* m zero = \u2261.refl\n  fold-* m (suc n) rewrite fold-* m n = \u2261.refl\n\n{- TODO\n  fold-^ : \u2200 m n \u2192 fold 1 (flip _^_ m) n \u2261 m \u2191\u27e8 4 \u27e9 n\n  fold-^ m zero = \u2261.refl\n  fold-^ m (suc n) rewrite fold-^ m n = ?\n-}\n\n  cong-fold : \u2200 {A : \u2605\u2080} {f g : Endo A} (f\u2257g : f \u2257 g) {z} \u2192 fold z f \u2257 fold z g\n  cong-fold eq zero = \u2261.refl\n  cong-fold eq {z} (suc x) rewrite cong-fold eq {z} x = eq _\n\n  private\n   open \u2191-Props\n   module Alt\u21912 where\n    alt : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n    alt a 0       = \u03bb b \u2192 a \u2191\u27e8 2 \u27e9 b\n    alt a (suc n) = fold 1 (alt a n)\n\n    alt-ok : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 alt a n\n    alt-ok a zero = \u2261.sym \u2218 \u2191-\u2191\u2032 a 2\n    alt-ok a (suc n) = cong-fold (alt-ok a n)\n\n{- TODO\n    alt' : \u2200 a n \u2192 _\u21912+\u27e8_\u27e9_ a n \u2257 _\u2191\u27e8_\u27e9_ a (2 + n)\n    alt' a zero  b = \u2261.sym (\u2191-\u2191\u2032 a 2 _)\n    alt' a (suc n) zero = \u2261.refl\n    alt' a (suc n) (suc x) rewrite alt' a n (_\u21912+\u27e8_\u27e9_ a (suc n) x)\n       = \u2261.cong (_\u2191\u27e8_\u27e9_ a (2 + n)) { fold 1 (fold (_*_ a) (fold 1) n) x} {a \u2191\u27e8 suc (suc (suc n)) \u27e9 x} (alt' a (suc n) x)\n-}\n\n  open InflModule\n{-\n  fold1-infl : \u2200 f \u2192 Infl f \u2192 Infl (fold 1 f)\n  fold1-infl f finfl {zero} = {!z\u2264n!}\n  fold1-infl f finfl {suc x} =\n     x          <\u27e8 {!s\u2264s (fold1-infl f finfl {x})!} \u27e9\n     fold 1 f x <\u27e8 {!!} \u27e9\n     {- f (fold 1 f x) \u2264\u27e8 finfl \u27e9\n     f (fold 1 f x) \u2261\u27e8 \u2261.refl \u27e9 -}\n     fold 1 f (1 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 f \u2218 suc)\n  fold1+1-infl< f finfl< {zero} = finfl<\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + f (fold 1 f x) \u2264\u27e8 finfl< {f (fold 1 f x)} \u27e9\n     f (suc (f (fold 1 f x))) \u2264\u27e8 {!!} \u27e9\n     fold 1 f (2 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+1-infl< : \u2200 (f : \u2115 \u2192 \u2115) \u2192 Infl< (f \u2218 suc) \u2192 Infl< (fold 1 (f \u2218 suc))\n  fold1+1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1+1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 suc) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 suc) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1+-infl< : \u2200 k f \u2192 Infl< (f \u2218\u2032 _+_ k) \u2192 Infl< (fold 1 (f \u2218\u2032 _+_ k))\n  fold1+-infl< k f finfl< {zero} = s\u2264s z\u2264n\n  fold1+-infl< k f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1+-infl< k f finfl< {x}) \u27e9\n     1 + fold 1 (f \u2218 _+_ k) x \u2264\u27e8 finfl< \u27e9\n     fold 1 (f \u2218 _+_ k) (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n  \n{-\n  fold1+-infl< : \u2200 f \u2192 Infl+2< f \u2192 Infl+2< (fold 1 f)\n  fold1+-infl< f finfl< {zero} _ = \u2115\u2264.refl\n  fold1+-infl< f finfl< {suc zero} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc zero)} _ = {!!}\n  fold1+-infl< f finfl< {suc (suc (suc x))} x>1 =\n     4 + x \u2264\u27e8 s\u2264s (fold1+-infl< f finfl< {suc (suc x)} (m\u2264m+n 2 x)) \u27e9\n     1 + fold 1 f (2 + x) \u2264\u27e8 finfl< {!!} \u27e9\n     fold 1 f (3 + x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\n{-\n  fold1-infl< f finfl< {zero} = s\u2264s z\u2264n\n  fold1-infl< f finfl< {suc x} =\n     2 + x \u2264\u27e8 s\u2264s (fold1-infl< f finfl< {x}) \u27e9\n     1 + fold 1 f x \u2264\u27e8 finfl< \u27e9\n     fold 1 f (suc x)\n   \u220e\n     where\n       open Props\n       open \u2264-Reasoning\n-}\nmodule ^-Props where\n  open \u2261-Reasoning\n\n  ^1 : \u2200 n \u2192 n ^ 1 \u2261 n\n  ^1 zero = \u2261.refl\n  ^1 (suc n) = \u2261.cong suc (proj\u2082 \u2115\u00b0.*-identity n)\n\n  ^-+ : \u2200 b m n \u2192 b ^ (m + n) \u2261 b ^ m * b ^ n\n  ^-+ b zero n = \u2261.sym (proj\u2082 \u2115\u00b0.+-identity (b ^ n))\n  ^-+ b (suc m) n rewrite ^-+ b m n = \u2261.sym (\u2115\u00b0.*-assoc b (b ^ m) (b ^ n))\n\n  ^-* : \u2200 b m n \u2192 b ^ (m * n) \u2261 (b ^ n) ^ m\n  ^-* b zero    n = \u2261.refl\n  ^-* b (suc m) n = b ^ (n + m * n)\n                  \u2261\u27e8 ^-+ b n (m * n) \u27e9\n                    b ^ n * b ^ (m * n)\n                  \u2261\u27e8 \u2261.cong (_*_ (b ^ n)) (^-* b m n) \u27e9\n                    b ^ n * (b ^ n) ^ m \u220e\n\nack : \u2115 \u2192 \u2115 \u2192 \u2115\nack zero n          = suc n\nack (suc m) zero    = ack m (suc zero)\nack (suc m) (suc n) = ack m (ack (suc m) n)\n\n\u2264\u21d2\u2203 : \u2200 {m n} \u2192 m \u2264 n \u2192 \u2203 \u03bb k \u2192 m + k \u2261 n\n\u2264\u21d2\u2203 z\u2264n      = _ , \u2261.refl\n\u2264\u21d2\u2203 (s\u2264s pf) = _ , \u2261.cong suc (proj\u2082 (\u2264\u21d2\u2203 pf))\n\nmodule ack-Props where\n  lem\u2238 : \u2200 {m n} \u2192 m \u2264 n \u2192 m + (n \u2238 m) \u2261 n\n  lem\u2238 z\u2264n = \u2261.refl\n  lem\u2238 (s\u2264s {m} {n} m\u2264n) = \u2261.cong suc (lem\u2238 m\u2264n)\n\n  -- n >= m \u2192 m + (n \u2238 m) \u2261 n\n  -- n >= m \u2192 \u2203 k \u2192 n \u2261 m + k\n  -- \u2203 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 n \u2261 m + k \u2192 m + (n \u2238 m) \u2261 n\n  -- \u2200 k \u2192 m + ((m + k) \u2238 m) \u2261 m + k\n  -- \u2200 k \u2192 m + k \u2261 m + k\n\n  -- 3 \u2264 x \u2192 P x \u2192 \u2203 \u03bb k \u2192 P (3 + k)\n\n  open InflModule\n\n  fold1+-inflT< : \u2200 {z} \u2192 InflT< (fold (suc z))\n  fold1+-inflT<     _      {zero} = s\u2264s z\u2264n\n  fold1+-inflT< {z} {f} finfl< {suc x} =\n     2 + x                \u2264\u27e8 s\u2264s (fold1+-inflT< {z} {f} finfl< {x}) \u27e9\n     1 + fold (suc z) f x \u2264\u27e8 finfl< \u27e9\n     f (fold (suc z) f x) \u2261\u27e8 \u2261.refl \u27e9\n     fold (suc z) f (1 + x)\n   \u220e\n     where open \u2264-Reasoning\n\n  Mon : (f : \u2115 \u2192 \u2115) \u2192 \u2605\u2080\n  Mon f = \u2200 {x y} \u2192 x \u2264 y \u2192 f x \u2264 f y\n  open InflModule\n\n\n  \u2115-ind : \u2200 (P : \u2115 \u2192 \u2605\u2080) \u2192 P zero \u2192 (\u2200 {n} \u2192 P n \u2192 P (suc n)) \u2192 \u2200 {n} \u2192 P n\n  \u2115-ind P P0 PS {zero} = P0\n  \u2115-ind P P0 PS {suc n} = PS (\u2115-ind P P0 PS)\n\n  open fold-Props\n  fold-infl< : \u2200 {f g} \u2192 Infl< f \u2192 InflT< g \u2192 \u2200 {n} \u2192 Infl< (fold f g n)\n  fold-infl< {f} {g} inflf< inflTg< {n} = \u2115-ind (Infl< \u2218 fold f g) inflf< inflTg< {n}\n\n  fold-mon : \u2200 {f} (fmon : Mon f) (finfl< : Infl< f) {z} \u2192 Mon (fold z f)\n  fold-mon fmon finfl< {_} {y = 0} z\u2264n = \u2115\u2264.refl\n  fold-mon {f} fmon finfl< {z} {y = suc n} z\u2264n = z \u2264\u27e8 \u2264-step \u2115\u2264.refl \u27e9\n                                              suc z \u2264\u27e8 nest-infl< f finfl< n {z} \u27e9\n                                              fold z f (suc n) \u2261\u27e8 \u2261.refl \u27e9\n                                              fold z f (suc n) \u220e\n    where open \u2264-Reasoning\n  fold-mon fmon finfl< (s\u2264s m\u2264n) = fmon (fold-mon fmon finfl< m\u2264n)\n\n  MonT : Endo (Endo \u2115) \u2192 \u2605\u2080\n  MonT g = \u2200 {f} \u2192 Mon f \u2192 Infl< f \u2192 Mon (g f)\n\n  fold-mon' : \u2200 {f g} \u2192 Mon f \u2192 Infl< f \u2192 MonT g \u2192 InflT< g \u2192 \u2200 {n} \u2192 Mon (fold f g n)\n  fold-mon' {f} {g} mon-f infl-f mont-g inflTg< {n} = go n where\n    go : \u2200 n \u2192 Mon (fold f g n)\n    go zero    = mon-f\n    go (suc n) = mont-g (go n) (fold-infl< infl-f inflTg< {n})\n\n\n  --  +-fold : \u2200 a b \u2192 a + b \u2261 fold a suc b\n  --  *-fold : \u2200 a b \u2192 a * b \u2261 fold 0 (_+_ a) b\n\n  1\u22641+a^b : \u2200 a b \u2192 1 \u2264 (1 + a) ^ b\n  1\u22641+a^b a zero = s\u2264s z\u2264n\n  1\u22641+a^b a (suc b) = 1\u22641+a^b a b +-mono z\u2264n\n\n  1+a^-mon : \u2200 {a} \u2192 Mon (_^_ (1 + a))\n  1+a^-mon {a} (z\u2264n {b}) = 1\u22641+a^b a b\n  1+a^-mon {a} (s\u2264s {m} {n} m\u2264n)\n     = (1 + a) ^ (1 + m)             \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ m + a * (1 + a) ^ m \u2264\u27e8 (1+a^-mon m\u2264n) +-mono ((a \u220e) *-mono (1+a^-mon m\u2264n)) \u27e9\n       (1 + a) ^ n + a * (1 + a) ^ n \u2261\u27e8 \u2261.refl \u27e9\n       (1 + a) ^ (1 + n) \u220e\n    where open \u2264-Reasoning\n\n  lem2^3 : \u2200 n \u2192 2 ^ 3 \u2264 2 ^ (3 + n)\n  lem2^3 n = 1+a^-mon {1} {3} {3 + n} (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n  lem2221 : \u2200 m \u2192 2 \u2261 2 \u2191\u27e8 2 + m \u27e9 1\n  lem2221 zero = \u2261.refl\n  lem2221 (suc m) = lem2221 m\n\n  lem4212 : \u2200 m \u2192 4 \u2261 2 \u2191\u27e8 1 + m \u27e9 2\n  lem4212 zero = \u2261.refl\n  lem4212 (suc m) = 4                            \u2261\u27e8 lem4212 m \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 2               \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 (1 + m)) (lem2221 m) \u27e9\n                    2 \u2191\u27e8 1 + m \u27e9 (2 \u2191\u27e8 2 + m \u27e9 1) \u2261\u27e8 \u2261.refl \u27e9\n                    2 \u2191\u27e8 2 + m \u27e9 2 \u220e\n    where open \u2261-Reasoning\n\n  2* = _*_ 2\n\n  -- fold 2* n 1 \u2261 2 ^ n\n\n  -- nest-* : \u2200 m n \u2192 nest (m * n) f \u2257 nest m (nest n f)\n\n  nest-2* : \u2200 m n \u2192 nest (m * n) (fold 1) 2* \u2261 nest m (nest n (fold 1)) 2*\n  nest-2* m n = nest-Properties.nest-* (fold 1) m n 2*\n\n{-\n  lem3\u2264 : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 2\n  lem3\u2264 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264 (suc n) = \n    3 \u2264\u27e8 {!!} \u27e9\n    fold 2* f1 (1 + n) 2 \u220e\n    where open \u2264-Reasoning\n          f1 = fold 1\n\n  lem3\u2264' : \u2200 n \u2192 3 \u2264 fold 2* (fold 1) n 3\n  lem3\u2264' 0 = s\u2264s (s\u2264s (s\u2264s z\u2264n))\n    where open \u2264-Reasoning\n  lem3\u2264' (suc n) = \n    3 \u2264\u27e8 lem3\u2264 n \u27e9\n    fold 2* (fold 1) n 2 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) n (fold 2* (fold 1) 3 1) \u2261\u27e8 {!!} \u27e9\n{- \u2261\u27e8 nest-Properties.nest-* {!2*!} {!!} {!!} {!!} \u27e9\n    fold 2* (fold 1) (3 * n) 1 \u2261\u27e8 {!!} \u27e9\n-}\n    fold 1 (fold 2* (fold 1) n) 3 \u2261\u27e8 \u2261.refl \u27e9\n    fold 2* (fold 1) (1 + n) 3 \u220e\n    where open \u2264-Reasoning\n-}\n---\n\n{-\nfold 2* (fold 1) (3 * n) 1 \u2264\n\nfold 2* (fold 1) n (fold 2* (fold 1) n (fold 2* (fold 1) n 1)) \u2264\n\nfold 1 (fold 2* (fold 1) n) 3 \u2264\nfold 2* (fold 1) (1 + n) 3\n-}\n\n{-\n  open \u2191-Props\n\n  -*\u27e81+n\u27e9-infl : \u2200 n \u2192 Infl (_*_ (1 + n))\n  -*\u27e81+n\u27e9-infl n = {!!}\n\n  lem121 : \u2200 a b \u2192 b < (1 + a) \u2191\u27e8 2 \u27e9 (1 + b)\n  lem121 a b rewrite \u2191\u2082-* (1 + a) (1 + b)\n     = 1 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (1 + a) * (1 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (1 + a) (1 + b) \u27e9\n       (1 + b) * (1 + a)\n     \u220e\n     where open \u2264-Reasoning\n\n  -- \u2200 a b \u2192 b < (2 + a) * b\n  lem222 : \u2200 a b \u2192 b < (2 + a) \u2191\u27e8 2 \u27e9 (2 + b)\n  lem222 a b rewrite \u2191\u2082-* (2 + a) (2 + b)\n     = 1 + b\n              \u2264\u27e8 suc-infl \u27e9\n       2 + b\n              \u2264\u27e8 m\u2264m+n _ _ \u27e9\n       (2 + a) * (2 + b)\n              \u2261\u27e8 \u2115\u00b0.*-comm (2 + a) (2 + b) \u27e9\n       (2 + b) * (2 + a)\n     \u220e\n     where open \u2264-Reasoning\n-}\n{-\n  open fold-Props\n\n  module Foo a where\n    f = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b\n    f-lem : \u2200 n \u2192 f (suc n) \u2257 fold 1 (f n)\n    f-lem n x = \u2261.refl\n\n    f2 = \u03bb n b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n    f2-lem : \u2200 n \u2192 f2 (suc n) \u2257 fold 1 (f2 n)\n    f2-lem n x = {!!}\n\n  -- b < (1 + a) * (1 + b)\n\n -- nest-infl< : \u2200 f (finfl< : Infl< f) n \u2192 Infl< (nest (suc n) f)\n  infl3< : \u2200 a n \u2192 Infl< (\u03bb b \u2192 (1 + a) \u21912+\u27e8 n \u27e9 (1 + b))\n  infl3< a zero {b} = lem121 a b\n  infl3< a (suc n) {b}\n    = 1 + b\n --  fold1-infl< : \u2200 f \u2192 Infl< f \u2192 \u2200 x \u2192 x < fold 1 f x\n            \u2264\u27e8 {!fold1f2-infl< {_}!} \u27e9\n      fold 1 f1 b\n            \u2264\u27e8 {!!} \u27e9\n{-\n            \u2264\u27e8 fold1-infl< (f 0) (\u03bb {x} \u2192 {!infl3< a n {x}!}) {{!!}} \u27e9\n      {!!}\n            \u2264\u27e8 {!!} \u27e9\n-}\n{-\n      1 + b\n      (2 + a) \u21912+\u27e8 n \u27e9 (2 + b)\n            \u2264\u27e8 {!nest-infl< (\u03bb b \u2192 (2 + a) \u21912+\u27e8 n \u27e9 b) ? b {(2 + a) \u21912+\u27e8 n \u27e9 (2 + b)}!} \u27e9\n      fold (f 0 1) (f 0) (1 + b)\n            \u2264\u27e8 {!\u2115\u2264.reflexive (\u2261.cong (\u03bb g \u2192 g 1) (nest-Properties.nest-+ (f 0) 1 (1 + b)))!} \u27e9\n-}\n      fold 1 (f 0) (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      f 1 (1 + b)\n            \u2261\u27e8 \u2261.refl \u27e9\n      (1 + a) \u21912+\u27e8 suc n \u27e9 (1 + b)\n    \u220e\n\n     where\n       f = \u03bb k b \u2192 (1 + a) \u21912+\u27e8 k + n \u27e9 b\n       f1 = f 0 \u2218 suc\n\n       -- fold1f2-infl< : Infl< (fold 1 f2)\n       -- fold1f2-infl< = fold1-infl< f2 {!(\u03bb b \u2192 infl3< a n b ?)!}\n\n       fold1f1-infl<' : Infl< (fold 1 f1)\n       fold1f1-infl<' = {!fold1+1-infl< (f 0) {!infl3< a n!}!}\n       lem : \u2200 x \u2192 f 0 (1 + x) < f 0 (f 0 x)\n       lem x = {!!}\n       postulate\n         finfl : Infl< (f 0)\n       open \u2264-Reasoning\n-}\n\n_\u21d1\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\na \u21d1\u27e8 n , k \u27e9 b = fold k f n\n  module M\u21d1 where \n    f : \u2115 \u2192 \u2115\n    f x = a \u2191\u27e8 2 + x \u27e9 b\n\nmodule \u21d1-Props where\n  _\u21d1\u2032\u27e8_,_\u27e9_ : (a n k b : \u2115) \u2192 \u2115\n  a \u21d1\u2032\u27e8 n\u2080 , k \u27e9 b = g n\u2080\n    module M\u21d1\u2032 where\n      h : \u2115 \u2192 \u2115\n      h x = a \u2191\u27e8 2 + x \u27e9 b\n\n      g : \u2115 \u2192 \u2115\n      g 0       = k\n      g (suc n) = h (g n)\n\n      g-nest : \u2200 n \u2192 g n \u2261 h $\u27e8 n \u27e9 k\n      g-nest zero = \u2261.refl\n      g-nest (suc n) rewrite g-nest n = \u2261.refl\n\n  \u21d1\u2080 : \u2200 a k b \u2192 a \u21d1\u27e8 0 , k \u27e9 b \u2261 k\n  \u21d1\u2080 _ _ _ = \u2261.refl\n\n  \u21d1\u2081 : \u2200 a k b \u2192 a \u21d1\u27e8 1 , k \u27e9 b \u2261 a \u2191\u27e8 2 + k \u27e9 b\n  \u21d1\u2081 _ _ _ = \u2261.refl\n\n  \u21d1-\u21d1\u2032 : \u2200 a n k b \u2192 a \u21d1\u27e8 n , k \u27e9 b \u2261 a \u21d1\u2032\u27e8 n , k \u27e9 b\n  \u21d1-\u21d1\u2032 a n k b rewrite M\u21d1\u2032.g-nest a n k b n = \u2261.refl\n\n-- https:\/\/en.wikipedia.org\/wiki\/Graham%27s_number\nGraham's-number : \u2115\nGraham's-number = 3 \u21d1\u27e8 64 , 4 \u27e9 3\n  -- = (f\u2218\u2076\u2074\u2026\u2218f)4\n  -- where f x = 3 \u2191\u27e8 2 + x \u27e9 3\n\n{-\nmodule GeneralisedArithmetic {a} {A : \u2605 a} (0# : A) (1+ : A \u2192 A) where\n\n  1# : A\n  1# = 1+ 0#\n\n  open Data.Nat.GeneralisedArithmetic 0# 1+ public\n\n  exp : (* : A \u2192 A \u2192 A) \u2192 (\u2115 \u2192 A \u2192 A)\n  exp _*_ n b = fold 1# (\u03bb s \u2192 b * s) n\n-}\n  -- hyperop a n b = fold exp\n\nmodule == where\n  _\u2248_ : (m n : \u2115) \u2192 \u2605\u2080\n  m \u2248 n = \u2713 (m == n)\n\n  subst : \u2200 {\u2113} \u2192 Substitutive _\u2248_ \u2113\n  subst _ {zero}  {zero}  _  = id\n  subst P {suc _} {suc _} p  = subst (P \u2218 suc) p\n  subst _ {zero}  {suc _} ()\n  subst _ {suc _} {zero}  ()\n\n  sound : \u2200 m n \u2192 \u2713 (m == n) \u2192 m \u2261 n\n  sound m n p = subst (_\u2261_ m) p \u2261.refl\n\n  refl : Reflexive _\u2248_\n  refl {zero}  = _\n  refl {suc n} = refl {n}\n\n  sym : Symmetric _\u2248_\n  sym {m} {n} eq rewrite sound m n eq = refl {n}\n\n  trans : Transitive _\u2248_\n  trans {m} {n} {o} m\u2248n n\u2248o rewrite sound m n m\u2248n | sound n o n\u2248o = refl {o}\n\n  setoid : Setoid _ _\n  setoid = record { Carrier = \u2115; _\u2248_ = _\u2248_\n                  ; isEquivalence =\n                      record { refl = \u03bb {x} \u2192 refl {x}\n                             ; sym = \u03bb {x} {y} \u2192 sym {x} {y}\n                             ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} } }\n\n  open Setoid setoid public hiding (refl; sym; trans; _\u2248_)\n\n{-\ndata _`\u2264?`_\u219d_ : (m n : \u2115) \u2192 Dec (m \u2264 n) \u2192 \u2605\u2080 where\n  z\u2264?n     : \u2200 {n} \u2192 zero `\u2264?` n \u219d yes z\u2264n\n  s\u2264?z     : \u2200 {m} \u2192 suc m `\u2264?` zero \u219d no \u03bb()\n  s\u2264?s-yes : \u2200 {m n m\u2264n} \u2192 m `\u2264?` n \u219d yes m\u2264n \u2192 suc m `\u2264?` suc n \u219d yes (s\u2264s m\u2264n)\n  s\u2264?s-no  : \u2200 {m n m\u2270n} \u2192 m `\u2264?` n \u219d no m\u2270n \u2192 suc m `\u2264?` suc n \u219d no (m\u2270n \u2218 \u2264-pred)\n\n`\u2264?`-complete : \u2200 m n \u2192 m `\u2264?` n \u219d (m \u2264? n)\n`\u2264?`-complete zero n = z\u2264?n\n`\u2264?`-complete (suc n) zero = {!s\u2264?z!}\n`\u2264?`-complete (suc m) (suc n) with m \u2264? n | `\u2264?`-complete m n\n... | yes q | r = s\u2264?s-yes r\n... | no q  | r = s\u2264?s-no {!!}\n-}\n\n_<=_ : (x y : \u2115) \u2192 Bool\nzero   <= _      = true\nsuc _  <= zero   = false\nsuc m  <= suc n  = m <= n\n\nmodule <= where\n  \u211b : \u2115 \u2192 \u2115 \u2192 \u2605\u2080\n  \u211b x y = \u2713 (x <= y)\n\n  sound : \u2200 m n \u2192 \u211b m n \u2192 m \u2264 n\n  sound zero    _       _  = z\u2264n\n  sound (suc m) (suc n) p  = s\u2264s (sound m n p)\n  sound (suc m) zero    ()\n\n  complete : \u2200 {m n} \u2192 m \u2264 n \u2192 \u211b m n\n  complete z\u2264n       = _\n  complete (s\u2264s m\u2264n) = complete m\u2264n\n\n  isTotalOrder : IsTotalOrder _\u2261_ \u211b\n  isTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total x y) }\n   where\n    reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 \u211b i j\n    reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {i})\n    trans : Transitive \u211b\n    trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound x y p) (sound y z q))\n    isPreorder : IsPreorder _\u2261_ \u211b\n    isPreorder = record { isEquivalence = \u2261.isEquivalence\n                        ; reflexive = reflexive\n                        ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n    antisym : Antisymmetric _\u2261_ \u211b\n    antisym {x} {y} p q = \u2115\u2264.antisym (sound x y p) (sound y x q)\n    isPartialOrder : IsPartialOrder _\u2261_ \u211b\n    isPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n\n  open IsTotalOrder isTotalOrder public\n\n<=-steps\u2032 : \u2200 {x} y \u2192 \u2713 (x <= (x + y))\n<=-steps\u2032 {x} y = <=.complete (\u2264-steps\u2032 {x} y)\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2264 2*\u2032 y \u2192 x \u2238 y \u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x < 2*\u2032 y \u2192 x \u2238 y < y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x < 2* y \u2192 x \u2238 y < y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y < x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\n\u00ac\u2264 : \u2200 {m n} \u2192 \u00ac(m < n) \u2192 n \u2264 m\n\u00ac\u2264 {m} {n} p with \u2115cmp.compare m n\n... | tri< m<n _ _   = \u22a5-elim (p m<n)\n... | tri\u2248 _ eq _    = \u2115\u2264.reflexive (\u2261.sym eq)\n... | tri> _ _ 1+n\u2264m = \u2264-pred (Props.\u2264-steps 1 1+n\u2264m)\n\nnot<=\u2192< : \u2200 x y \u2192 \u2713 (not (x <= y)) \u2192 \u2713 (suc y <= x)\nnot<=\u2192< x y p = <=.complete (\u2270\u2192< x y (\u2713'not'\u00ac p \u2218 <=.complete))\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d77ee1cf2cda1faa62e8ad76b80bb0e29d8c55e6","subject":"Agda bug","message":"Agda bug\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/LinFunContructw.agda","new_file":"agda\/LinFunContructw.agda","new_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\nopen import CTT\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\n\nopen import LinFunContruct\n\n\n\nmodule _ where\n\n  private\n    module _ where\n    \n      open import PathPrelude hiding (_\u2261_)\n    \n      lem1 : \u2200{i u ll ell pll x} \u2192 (s : SetLL ll) \u2192 (lind : IndexLL {i} {u} pll ll)\n            \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 x)\n            \u2192 (\u00acho : \u00ac hitsAtLeastOnce s lind)\n            \u2192 (teq : truncSetLL s lind \u2261 \u2205)\n            \u2192 \u2200{mx}\n            \u2192 del s lind ell \u2261 \u00ac\u2205 mx\n            \u2192 \u2200{cs}\n            \u2192 complL\u209b mx \u2261 \u00ac\u2205 cs\n            \u2192 (shrink (replLL ll lind ell) cs) \u2261 replLL (shrink ll x) (\u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind (\u00acho\u21d2trunc\u2261\u2205 s lind \u00acho))) ell\n            \u2192 (shrink (replLL ll lind ell) cs) \u2261 replLL (shrink ll x) (\u00acho-shr-morph s eq lind \u00acho) ell\n      lem1 {ll = ll} {ell} {_} {x} s lind eq \u00acho _ _ {cs} _ req = PathPrelude.subst {P = \u03bb x \u2192 x} w req where\n        w = PathPrelude.cong (\u03bb z \u2192 (shrink (replLL ll lind ell) cs) \u2261 replLL (shrink ll x) (\u00acho-shr-morph s eq lind z) ell) (\u00acfun-eq (trunc\u2261\u2205\u21d2\u00acho s lind (\u00acho\u21d2trunc\u2261\u2205 s lind \u00acho)) \u00acho) \n     \n    \n    \n  open IndexLLProp\n\n  roo : \u2200{i u ll ell pll x} \u2192 (s : SetLL ll) \u2192 (lind : IndexLL {i} {u} pll ll)\n        \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 x)\n        \u2192 (\u00acho : \u00ac hitsAtLeastOnce s lind)\n        \u2192 let mx = proj\u2081 (\u00acho\u21d2del\u2261\u00ac\u2205 s lind \u00acho) in\n          \u2200{cs}\n          \u2192 complL\u209b mx \u2261 \u00ac\u2205 cs\n          \u2192 (shrink (replLL ll lind ell) cs) \u2261 replLL (shrink ll x) (\u00acho-shr-morph s eq lind \u00acho) ell\n  roo {ell = ell} s lind eq \u00acho cmeq with smx | mx | meq where\n    smx = \u00acho\u21d2del\u2261\u00ac\u2205 {fll = ell} s lind \u00acho\n    mx = proj\u2081 smx\n    meq = proj\u2082 smx\n  ... | g | mx | meq = lem1 s lind eq \u00acho ((\u00acho\u21d2trunc\u2261\u2205 s lind \u00acho)) meq cmeq r where\n    r = shrink-repl-comm s lind eq (\u00acho\u21d2trunc\u2261\u2205 s lind \u00acho) meq cmeq\n\n\n\n\n  poo : \u2200 {i u ll ell pll tll cs} \u2192 (s : SetLL ll) \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL tll ll)\n        \u2192 (\u00achob : \u00ac hitsAtLeastOnce s ind)\n        \u2192 (\u00achoh : \u00ac hitsAtLeastOnce s lind)\n        \u2192 let mx = proj\u2081 (\u00acho\u21d2del\u2261\u00ac\u2205 s lind \u00achoh) in\n        \u2200{mcs}\n        \u2192 (ceqi : complL\u209b mx \u2261 \u00ac\u2205 mcs)\n        \u2192 let hind = \u00acho-shr-morph s eq lind \u00achoh in\n        let bind = \u00acho-shr-morph s eq ind \u00achob in\n        (nord : \u00ac Ordered\u1d62 ind lind) \u2192\n        let nind = \u00acord-morph ind lind ell (flipNotOrd\u1d62 nord) in\n        let hnord = ( \u03bb z \u2192 nord (\u00acho-shr-morph-pres-\u00acord s eq ind lind \u00achob \u00achoh (flipOrd\u1d62 z))) in\n        \u00acord-morph bind hind ell hnord \u2261 subst (\u03bb z \u2192 IndexLL pll z) (roo {ell = ell} s lind eq \u00achoh ceqi) (\u00acho-shr-morph mx ceqi nind (\u00acord&\u00acho-del\u21d2\u00acho ind s \u00achob lind nord (sym (proj\u2082 (\u00acho\u21d2del\u2261\u00ac\u2205 s lind \u00achoh)))))  \n  poo \u2193 () ind lind \u00achob \u00achoh ceqi nord\n  poo (s \u2190\u2227) eq \u2193 lind \u00achob \u00achoh ceqi nord = \u22a5-elim (\u00achob hitsAtLeastOnce\u2190\u2227\u2193)\n  poo {ell = ell} (s \u2190\u2227) eq (ind \u2190\u2227) lind \u00achob \u00achoh ceqi nord with complL\u209b s | inspect complL\u209b s | nhsm where\n    nhsm = \u00acho-shr-morph (s \u2190\u2227) eq (ind \u2190\u2227) \u00achob\n--    hnord = \u03bb z \u2192 nord (\u00acho-shr-morph-pres-\u00acord (s \u2190\u2227) eq (ind \u2190\u2227) lind \u00achob \u00achoh (flipOrd\u1d62 z))\n--    omh = \u00acord-morph nhsm (\u00acho-shr-morph (s \u2190\u2227) eq lind \u00achoh) ell hnord\n  ... | g | e | nhsm = {!!}\n  poo (s \u2190\u2227) eq (\u2227\u2192 ind) lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (\u2227\u2192 s) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (s \u2190\u2227\u2192 s\u2081) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (s \u2190\u2228) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (\u2228\u2192 s) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (s \u2190\u2228\u2192 s\u2081) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (s \u2190\u2202) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (\u2202\u2192 s) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n  poo (s \u2190\u2202\u2192 s\u2081) eq ind lind \u00achob \u00achoh ceqi nord = {!!}\n\n\n\n--   private\n--     data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n--       \u2205 : M\u00acho s \u2205\n--       \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n--            \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n--     hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n--          \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n--          \u2192 LinLogic i {u}\n--     hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n--     hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n--     hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n--           \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n--           \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n--           \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n--     hf2 \u2205 mnho s lf eqs = \u2205\n--     hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (\u03bb z \u2192 tranLFMIndexLL lf (\u00ac\u2205 z)) x\n--     hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n--     hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n\n\n--   data MLFun {i u ll rll n dt df} (mind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n--              (s : SetLL ll) (m\u00acho : M\u00acho s mind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n--     \u2205 : \u2200{ts} \u2192 (\u2205 \u2261 mind) \u2192 (complL\u209b s \u2261 \u2205)\n--         \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun mind s m\u00acho lf\n--     \u00ac\u2205\u2205 : \u2200 {ind cs} \u2192 (\u00ac\u2205 ind \u2261 mind) \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n--           \u2192 (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n--           \u2192 (cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) rll\n--           \u2192 MLFun mind s m\u00acho lf \n--     \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n--            \u2192 let tind = tranLFMIndexLL lf mind in\n--            (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n--            \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 mind s m\u00acho lf eqs) cll))\n--            \u2192 MLFun mind s m\u00acho lf \n--            --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n--            -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n--            -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- -- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- -- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n--     -- s here does contain the ind.\n--   -- TODO IMPORTANT After test , we need to remove \u2282 that contain I in the first place.\n--   test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n--          \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n--   test ind s mnho lf with complL\u209b s | inspect complL\u209b s\n--   test \u2205 s \u2205 lf | \u2205 | [ e ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n--   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u2205 | [ r ] = IMPOSSIBLE\n--   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] with complL\u209b x | inspect complL\u209b x\n--   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u2205 | [ t ] = \u2205 refl e (sym r) t\n--   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n--   test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf | \u2205 | [ e ] = \u22a5-elim (\u00acho (compl\u2261\u2205\u21d2ho s e x)) \n\n--   test \u2205 s mnho I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n--   test (\u00ac\u2205 x\u2081) s (\u00ac\u2205 \u00acho) I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n--   ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n--   ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n--   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n--   ... | \u2205 indeq ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n--   ... | \u00ac\u2205\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00ac\u2205\u00ac\u2205 eq tseq ceqo ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n--     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n--     g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n--   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n--   ... | \u2205 indeq ceq eqs ceqo with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n--   ... | \u00ac\u2205 mx | r with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u2205 = IMPOSSIBLE -- IMPOSSIBLE and not provable with the current information we have.\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x with compl\u2261\u00ac\u2205\u21d2replace-compl\u2261\u00ac\u2205 s lind eq teq ceq x\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , proj\u2084 with complL\u209b (replacePartOf s to x at ind)\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 refl eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , () | .\u2205\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi1 eqs1 cll1 t1\n--   test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 {ts = ts} indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 {cs} ceqi1 eqs1 ceqo1 t1\n--     = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 ((\u03bb cll \u2192 subst (\u03bb z \u2192 LFun z (shrink rll _)) (shrink-repl\u2261\u2205 s lind eq teq ceq _ ceqo t) (t1 cll))) where\n--     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n--     t : complL\u209b (replacePartOf s to ts at lind) \u2261 \u00ac\u2205 cs\n--     t with trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n--     ... | g with replacePartOf s to ts at lind\n--     t | refl | e = ceqi1\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq refl ceqo | \u2205 | [ () ] | .(\u00ac\u2205 _)\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind\u2081} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n--   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t\n--        with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi () ceqo t | \u2205 | [ meq ] | \u2205\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ () ] | \u00ac\u2205 _\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u2205 indeq ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n--   test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl ceqo t | \u00ac\u2205 .(replacePartOf s to _ at lind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | .(\u00ac\u2205 _) with complL\u209b ts | ct where\n--     m = replacePartOf s to ts at lind\n--     mind = subst (\u03bb z \u2192 IndexLL z (replLL ll lind ell)) (replLL-\u2193 lind) ((a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))\n--     ct = compl\u2261\u2205\u21d2compltr\u2261\u2205 m ceq1 mind (sym (tr-repl\u21d2id s lind ts))\n--   test {rll = rll} {ll} {n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl () t | \u00ac\u2205 .(replacePartOf s to ts at ind) | [ refl ] | \u2205 indeq ceq1 eqs1 ceqo1 | .(\u00ac\u2205 ts) | .\u2205 | refl\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi\u2081 eqs\u2081 cll1 x\u2081\n--   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 ceqi1 eqs1 ceqo1 t1\n--       = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 (\u03bb cll \u2192 _\u2282_ {ind = hind} (t cll) (subst (\u03bb z \u2192 LFun z _) (sym (ho-shrink-repl-comm s lind eq teq ceqi ceqo w ceqi1)) (t1 cll))) where\n--     w = trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n--     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n--     hind = ho-shr-morph s eq lind (sym teq) ceqi\n--   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n--   ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n--   ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n--   ... | \u00ac\u2205 mx | [ meq ] with isLTi lind ind\n--   ... | no \u00acp with test {n = n} {dt} {df} nind mx (\u00ac\u2205 n\u00acho) lf\u2081 where\n--     nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n--     nind = \u00ac\u2205 (\u00acord-morph ind lind ell (flipNotOrd\u1d62 nord))\n--     n\u00acho = \u00acord&\u00acho-del\u21d2\u00acho ind s \u00acho lind nord (sym meq)\n--   ... | \u2205 () ceq eqs ceqo\n--   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u2205 {ind = tind} indeq ceqi eqs cll t\n--     = \u00ac\u2205\u2205 refl eq tseq cll (_\u2282_ {ind = sind} lf {!l1!}) where \n--     nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n--     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n--     hind = \u00acho-shr-morph s eq lind ((trunc\u2261\u2205\u21d2\u00acho s lind teq))\n--     bind = \u00acho-shr-morph s eq ind \u00acho\n    \n--     hord : \u00ac Ordered\u1d62 bind hind\n--     hord x = nord (\u00acho-shr-morph-pres-\u00acord s eq ind lind \u00acho (trunc\u2261\u2205\u21d2\u00acho s lind teq) x)\n    \n--     sind = \u00acord-morph hind bind cll hord \n--     l1 = replLL-\u00acordab\u2261ba bind cll hind ell (flipNotOrd\u1d62 hord)\n--     l2 = {!cong (\u03bb z \u2192 replLL z (\u00acord-morph bind hind ell ?) cll) ?!} \n\n--     find = \u00acord-morph ind lind ell (flipNotOrd\u1d62 nord)\n--     nind = \u00acord-morph lind ind cll nord\n--   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t = {!!}\n--   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | yes p = {!!}\n--   test (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] = {!!}\n--   test ind s mnho (tr ltr lf) | \u00ac\u2205 x | eq = {!!}\n--   test ind s mnho (com df\u2081 lf) | \u00ac\u2205 x | eq = {!!}\n--   test ind s mnho (call x\u2081) | \u00ac\u2205 x | eq = {!!}\n\n\n\n\n\n-- --  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | \u00acI\u00ac\u2205 ceqi () ceqo x | .(\u00ac\u2205 _)\n-- --\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n-- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n-- --\n-- --\n-- ---- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n-- ----   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n-- ----     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n-- ---- --    is = test \u2205 {!!} \u2205 lf\u2081\n-- ----   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n-- ----   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n-- ----   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n-- ----   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n-- ----     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n-- ----     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n-- ----   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n-- ----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n-- ----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n-- --  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n-- --  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n-- --  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n-- --  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n-- --  \n-- --  \n  \n  \n  \n  \n  \n","old_contents":"-- {-# OPTIONS --show-implicit #-}\n\nmodule LinFunContructw where\n\nopen import Common\nopen import LinLogic\nimport IndexLLProp \nopen import LinFun\nopen import SetLL\nopen import SetLLProp\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\n\nopen import LinFunContruct\n\n\n\n\nmodule _ where\n\n  open IndexLLProp\n\n  roo : \u2200{i u ll ell pll x} \u2192 (s : SetLL ll) \u2192 (lind : IndexLL {i} {u} pll ll)\n        \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 x)\n        \u2192 (\u00acho : \u00ac hitsAtLeastOnce s lind)\n        \u2192 let mx = proj\u2081 (\u00acho\u21d2del\u2261\u00ac\u2205 s lind \u00acho) in\n          \u2200{cs}\n          \u2192 complL\u209b mx \u2261 \u00ac\u2205 cs\n          \u2192 (shrink (replLL ll lind ell) cs) \u2261 replLL (shrink ll x) (\u00acho-shr-morph s eq lind \u00acho) ell\n  roo {ell = ell} s lind eq \u00acho cmeq with smx | mx | meq where\n    smx = \u00acho\u21d2del\u2261\u00ac\u2205 {fll = ell} s lind \u00acho\n    mx = proj\u2081 smx\n    meq = proj\u2082 smx\n  ... | g | mx | meq = {!r!} where\n    r = shrink-repl-comm s lind eq (\u00acho\u21d2trunc\u2261\u2205 s lind \u00acho) meq cmeq\n\n\n\n\n--   roo {ell = ell} (s \u2190\u2227) (lind \u2190\u2227) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ deq ] = ?\n--   roo {ell = ell} (s \u2190\u2227) (lind \u2190\u2227) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] = ?\n--   roo {ell = ell} (s \u2190\u2227) (lind \u2190\u2227) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] with complL\u209b nmx | inspect complL\u209b nmx\n--   ... | \u2205 | [ nceq ] = ?\n--   ... | \u00ac\u2205 ncs | [ nceq ] with roo s lind qeq  deq nceq \n--   roo {ll = _ \u2227 rs} {ell = ell} (s \u2190\u2227) (lind \u2190\u2227) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] | r = cong (\u03bb z \u2192 z \u2227 (shrink rs (fillAllLower rs))) r\n--   roo {ell = ell} (s \u2190\u2227) (lind \u2190\u2227) eq  {\u2227\u2192 mx} ()\u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ]\n--   roo {ell = ell} (s \u2190\u2227) (lind \u2190\u2227) eq  {mx \u2190\u2227\u2192 mx\u2081} ()\u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ]\n--   roo {ll = _ \u2227 rs} {ell} (s \u2190\u2227) (\u2227\u2192 lind) eq  \u00acho ceq with complL\u209b s | inspect complL\u209b s\n--   roo {u = _} {_ \u2227 rs} {ell} (s \u2190\u2227) (\u2227\u2192 lind) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] with shrink rs (fillAllLower rs) | shr-fAL-id rs | shrink (replLL rs lind ell) (fillAllLower (replLL rs lind ell)) | shr-fAL-id (replLL rs lind ell)\n--   roo {u = _} {_ \u2227 rs} {ell} (s \u2190\u2227) (\u2227\u2192 lind) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | .rs | refl | .(replLL rs lind ell) | refl = refl\n--   roo {ll = _ \u2227 rs} {ell} (s \u2190\u2227) (\u2227\u2192 lind) refl \u00acho refl | \u2205 | [ qeq ] with shrink rs (fillAllLower rs) | shr-fAL-id rs | shrink (replLL rs lind ell) (fillAllLower (replLL rs lind ell)) | shr-fAL-id (replLL rs lind ell)\n--   roo {u = _} {_ \u2227 rs} {ell} (s \u2190\u2227) (\u2227\u2192 lind) refl \u00acho refl | \u2205 | [ qeq ] | .rs | refl | .(replLL rs lind ell) | refl = refl\n--   roo {ll = ls \u2227 _} {ell} (\u2227\u2192 s) \u2193 eq  \u00acho ceq =  \u22a5-elim (\u00acho hitsAtLeastOnce\u2227\u2192\u2193) where\n--   roo {ll = ls \u2227 _} {ell} (\u2227\u2192 s) (lind \u2190\u2227) eq  \u00acho ceq with complL\u209b s | inspect complL\u209b s\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (lind \u2190\u2227) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] with shrink ls (fillAllLower ls) | shr-fAL-id ls | shrink (replLL ls lind ell) (fillAllLower (replLL ls lind ell)) | shr-fAL-id (replLL ls lind ell)\n--   ... | .ls | refl | .(replLL ls lind ell) | refl = refl\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (lind \u2190\u2227) refl \u00acho refl | \u2205 | [ qeq ] with shrink ls (fillAllLower ls) | shr-fAL-id ls | shrink (replLL ls lind ell) (fillAllLower (replLL ls lind ell)) | shr-fAL-id (replLL ls lind ell)\n--   ... | .ls | refl | .(replLL ls lind ell) | refl = refl\n--   roo {ll = ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq with complL\u209b s | inspect complL\u209b s | del s lind ell | inspect (del s lind) ell\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq | \u2205 | [ qeq ] | w | t = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s qeq lind)) where\n--     \u00acnho = trunc\u2261\u2205\u21d2\u00acho s lind \n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ deq ] = ? \n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] = ?\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] = ?\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] with complL\u209b nmx | inspect complL\u209b nmx\n--   ... | \u2205 | [ nceq ] = \u22a5-elim (del\u21d2\u00acho s lind (sym deq) (compl\u2261\u2205\u21d2ho nmx nceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n--   ... | \u00ac\u2205 ncs | [ nceq ] with roo s lind qeq  deq nceq\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) refl refl refl | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] | r = cong (\u03bb z \u2192 (shrink ls (fillAllLower ls)) \u2227 z) r\n--   roo {u = _} {ls \u2227 _} {ell} (\u2227\u2192 s) (\u2227\u2192 lind) eq \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 nmx | [ deq ]\n--   roo {ll = ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) \u2193 eq \u00acho ceq = \u22a5-elim (\u00acho hitsAtLeastOnce\u2190\u2227\u2192\u2193)\n--   roo {ll = ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) eq \u00acho ceq  with complL\u209b s | inspect complL\u209b s | complL\u209b s\u2081 | inspect complL\u209b s\u2081\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) eq \u00acho ceq | \u2205 | [ qeq ] | e | _ = \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s qeq lind)) where\n--     \u00acnho = trunc\u2261\u2205\u21d2\u00acho s lind \n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] with del s lind ell | inspect (del s lind) ell\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u2205 | [ deq ] =  \u22a5-elim ((\u00acho\u21d2\u00acdel\u2261\u2205 s lind (trunc\u2261\u2205\u21d2\u00acho s lind )) deq)\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] with complL\u209b nmx | inspect complL\u209b nmx\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] with complL\u209b s\u2081\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho () | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u2205\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 .x | [ refl ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u00ac\u2205 x = \u22a5-elim ((del\u21d2\u00acho s lind (sym deq)) (compl\u2261\u2205\u21d2ho nmx nceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))) \n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 x | [ nceq ] with complL\u209b s\u2081\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ () ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 x | [ nceq ] | \u2205\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 .x | [ refl ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 x\u2081 | [ nceq ] | \u00ac\u2205 x = cong (\u03bb z \u2192 z \u2227 (shrink rs x)) r where\n--     r = roo s lind qeq  deq nceq\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | eeq with del s lind ell | inspect (del s lind) ell\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u2205 | [ deq ] = \u22a5-elim ((\u00acho\u21d2\u00acdel\u2261\u2205 s lind (trunc\u2261\u2205\u21d2\u00acho s lind )) deq)\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | eeq | \u00ac\u2205 nmx | [ deq ]  with complL\u209b nmx | inspect complL\u209b nmx\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] with complL\u209b s\u2081\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] | \u2205 = roo s lind qeq  deq nceq\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ () ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] | \u00ac\u2205 x\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] with complL\u209b s\u2081\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho () | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u2205\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (lind \u2190\u2227) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ () ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u00ac\u2205 x\n--   roo {ll = ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) eq \u00acho ceq with complL\u209b s\u2081 | inspect complL\u209b s\u2081 | complL\u209b s | inspect complL\u209b s\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) eq \u00acho ceq | \u2205 | [ qeq ] | e | [ eeq ] =  \u22a5-elim (\u00acnho (compl\u2261\u2205\u21d2ho s\u2081 qeq lind)) where\n--     \u00acnho = trunc\u2261\u2205\u21d2\u00acho s\u2081 lind \n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] with del s\u2081 lind ell | inspect (del s\u2081 lind) ell\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u2205 | [ deq ] =  \u22a5-elim ((\u00acho\u21d2\u00acdel\u2261\u2205 s\u2081 lind (trunc\u2261\u2205\u21d2\u00acho s\u2081 lind )) deq)\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] with complL\u209b nmx | inspect complL\u209b nmx\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] with complL\u209b s\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho () | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u2205\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 .x | [ refl ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u00ac\u2205 x = \u22a5-elim ((del\u21d2\u00acho s\u2081 lind (sym deq)) (compl\u2261\u2205\u21d2ho nmx nceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 x | [ nceq ] with complL\u209b s\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 e | [ () ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 x | [ nceq ] | \u2205\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | \u00ac\u2205 .x | [ refl ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 x\u2081 | [ nceq ] | \u00ac\u2205 x = cong (\u03bb z \u2192 (shrink ls x) \u2227 z) r where\n--     r = roo s\u2081 lind qeq  deq nceq\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] with del s\u2081 lind ell | inspect (del s\u2081 lind) ell\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u2205 | [ deq ] = \u22a5-elim ((\u00acho\u21d2\u00acdel\u2261\u2205 s\u2081 lind (trunc\u2261\u2205\u21d2\u00acho s\u2081 lind )) deq)\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] with complL\u209b nmx | inspect complL\u209b nmx\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] with complL\u209b s\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho refl | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] | \u2205 = roo s\u2081 lind qeq  deq nceq\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ () ] | \u00ac\u2205 nmx | [ deq ] | \u00ac\u2205 ncs | [ nceq ] | \u00ac\u2205 x\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] with complL\u209b s\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho () | \u00ac\u2205 q | [ qeq ] | \u2205 | [ eeq ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u2205\n--   roo {u = _} {ls \u2227 rs} {ell} (s \u2190\u2227\u2192 s\u2081) (\u2227\u2192 lind) refl \u00acho ceq | \u00ac\u2205 q | [ qeq ] | \u2205 | [ () ] | \u00ac\u2205 nmx | [ deq ] | \u2205 | [ nceq ] | \u00ac\u2205 x\n\n\n\n-- -- --  poo : \u2200 {i u ll ell pll tll cs} \u2192 (s : SetLL ll) \u2192 (eq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (ind : IndexLL {i} {u} pll ll) \u2192 (lind : IndexLL tll ll)\n-- -- --        \u2192 (\u00achob : \u00ac hitsAtLeastOnce s ind)\n-- -- --        \u2192 (\u00achoh : \u00ac hitsAtLeastOnce s lind)\n-- -- --        \u2192 \u2200 mx ceqi meq \u00acho\n-- -- --        \u2192 let hind = \u00acho-shr-morph s eq lind \u00achoh in\n-- -- --          let bind = \u00acho-shr-morph s eq ind \u00achob in\n-- -- --          (nord : \u00ac Ordered\u1d62 ind lind) \u2192\n-- -- --          let nind = \u00acord-morph ind lind ell ? in\n-- -- --          \u00acord-morph bind hind ell ? \u2261 \u00acho-shr-morph mx ceqi nind ?\n-- -- --  poo = ?\n\n-- --   private\n-- --     data M\u00acho {i u ll n dt df} (s : SetLL ll) : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll \u2192 Set where\n-- --       \u2205 : M\u00acho s \u2205\n-- --       \u00ac\u2205 : {ind : IndexLL (\u03c4 {i} {u} {n} {dt} df) ll} \u2192 (\u00acho : \u00ac (hitsAtLeastOnce s ind))\n-- --            \u2192 M\u00acho s (\u00ac\u2205 ind)\n\n-- --     hf : \u2200{i u n dt df} \u2192 \u2200 ll \u2192 \u2200{cs} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n-- --          \u2192 (s : SetLL ll) \u2192 (ceq : complL\u209b s \u2261 \u00ac\u2205 cs) \u2192 (m\u00acho : M\u00acho s ind) \u2192 LinLogic i {u}\n-- --          \u2192 LinLogic i {u}\n-- --     hf ll {cs} \u2205 s ceq mnho cll = shrinkcms ll s cs ceq\n-- --     hf ll {cs} (\u00ac\u2205 x) s ceq (\u00ac\u2205 \u00acho) cll = replLL (shrinkcms ll s cs ceq) (\u00acho-shr-morph s ceq x \u00acho) cll\n\n-- --     hf2 : \u2200{i u ll rll n dt df ts} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n-- --           \u2192 (s : SetLL ll) \u2192 (m\u00acho : M\u00acho s ind) \u2192 (lf : LFun {i} {u} ll rll)\n-- --           \u2192 \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)\n-- --           \u2192 M\u00acho ts (tranLFMIndexLL lf ind)\n-- --     hf2 \u2205 mnho s lf eqs = \u2205\n-- --     hf2 (\u00ac\u2205 x) s mnho lf eqs with tranLFMIndexLL lf (\u00ac\u2205 x) | inspect (\u03bb z \u2192 tranLFMIndexLL lf (\u00ac\u2205 z)) x\n-- --     hf2 (\u00ac\u2205 x) s mnho lf eqs | \u2205 | [ eq ] = \u2205\n-- --     hf2 (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf eqs | \u00ac\u2205 _ | [ eq ] = \u00ac\u2205 (tranLF-preserves-\u00acho lf x s \u00acho (sym eq) eqs)\n\n\n-- --   data MLFun {i u ll rll n dt df} (mind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll)\n-- --              (s : SetLL ll) (m\u00acho : M\u00acho s mind) (lf : LFun {i} {u} ll rll) : Set (lsuc u) where\n-- --     \u2205 : \u2200{ts} \u2192 (\u2205 \u2261 mind) \u2192 (complL\u209b s \u2261 \u2205)\n-- --         \u2192 (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u2205) \u2192 MLFun mind s m\u00acho lf\n-- --     \u00ac\u2205\u2205 : \u2200 {ind cs} \u2192 (\u00ac\u2205 ind \u2261 mind) \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n-- --           \u2192 (eqs : \u2205 \u2261 tranLFMSetLL lf (\u00ac\u2205 s))\n-- --           \u2192 (cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) rll\n-- --           \u2192 MLFun mind s m\u00acho lf \n-- --     \u00ac\u2205\u00ac\u2205 : \u2200 {cs ts cts} \u2192 (ceqi : complL\u209b s \u2261 \u00ac\u2205 cs)\n-- --            \u2192 let tind = tranLFMIndexLL lf mind in\n-- --            (eqs : \u00ac\u2205 ts \u2261 tranLFMSetLL lf (\u00ac\u2205 s)) \u2192 (ceqo : complL\u209b ts \u2261 \u00ac\u2205 cts)\n-- --            \u2192 ((cll : LinLogic i {u}) \u2192 LFun (hf ll mind s ceqi m\u00acho cll) (hf rll tind ts ceqo (hf2 mind s m\u00acho lf eqs) cll))\n-- --            \u2192 MLFun mind s m\u00acho lf \n-- --            --???  We will never reach to a point where complL\u209b ts \u2261 \u2205 because\n-- --            -- the input would have the same fate. ( s becomes smaller as we go forward, thus complL\u209b increases. Here we take the case where s is not \u2205.\n-- --            -- Correction : In fact , the ordinal remains the same since all the points of the set need to to end at the same com by design. (but we might not be able to prove this now and thus need to go with the weaker argument.)\n  \n\n\n\n-- -- -- s is special, all of the input points eventually will be consumed by a single com + the point from the index. Thus if complL\u209b s \u2261 \u2205, this means that lf does not contain coms or calls.\n-- -- -- Maybe one day, we will provide a datatype that contains that information, for the time being, we use IMPOSSIBLE where necessary.\n\n-- --     -- s here does contain the ind.\n-- --   -- TODO IMPORTANT After test , we need to remove \u2282 that contain I in the first place.\n-- --   test : \u2200{i u rll ll n dt df} \u2192 (ind : MIndexLL (\u03c4 {i} {u} {n} {dt} df) ll) \u2192 (s : SetLL ll)\n-- --          \u2192 \u2200 m\u00acho \u2192 (lf : LFun ll rll) \u2192 MLFun ind s m\u00acho lf\n-- --   test ind s mnho lf with complL\u209b s | inspect complL\u209b s\n-- --   test \u2205 s \u2205 lf | \u2205 | [ e ] with tranLFMSetLL lf (\u00ac\u2205 s) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) s\n-- --   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u2205 | [ r ] = IMPOSSIBLE\n-- --   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] with complL\u209b x | inspect complL\u209b x\n-- --   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u2205 | [ t ] = \u2205 refl e (sym r) t\n-- --   test \u2205 s \u2205 lf | \u2205 | [ e ] | \u00ac\u2205 x | [ r ] | \u00ac\u2205 x\u2081 | t = IMPOSSIBLE\n-- --   test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf | \u2205 | [ e ] = \u22a5-elim (\u00acho (compl\u2261\u2205\u21d2ho s e x)) \n\n-- --   test \u2205 s mnho I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n-- --   test (\u00ac\u2205 x\u2081) s (\u00ac\u2205 \u00acho) I | \u00ac\u2205 x | [ eq ] = \u00ac\u2205\u00ac\u2205 eq refl eq (\u03bb cll \u2192 I)\n\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n-- --   ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n-- --   ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n-- --   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n-- --   ... | \u2205 indeq ceq eqs ceqo = \u22a5-elim ((del\u21d2\u00acho s lind (sym meq)) (compl\u2261\u2205\u21d2ho mx ceq (a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind))))\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n-- --   ... | \u00ac\u2205\u00ac\u2205 {ts = ts} {cts} ceqi eqs ceqo t = \u00ac\u2205\u00ac\u2205 eq tseq ceqo ((\u03bb z \u2192 _\u2282_ {ind = \u00acho-shr-morph s eq lind (trunc\u2261\u2205\u21d2\u00acho s lind teq)} lf (g (t z)))) where\n-- --     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n-- --     g = subst (\u03bb a \u2192 LFun a (shrink rll cts)) (shrink-repl-comm s lind eq teq meq ceqi)\n-- --   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n-- --   ... | \u2205 indeq ceq eqs ceqo with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n-- --   ... | \u00ac\u2205 mx | r with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u2205 = IMPOSSIBLE -- IMPOSSIBLE and not provable with the current information we have.\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x with compl\u2261\u00ac\u2205\u21d2replace-compl\u2261\u00ac\u2205 s lind eq teq ceq x\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , proj\u2084 with complL\u209b (replacePartOf s to x at ind)\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 .(replacePartOf s to x at ind) | [ refl ] | \u2205 indeq1 refl eqs1 ceqo1 | \u00ac\u2205 x | proj\u2083 , () | .\u2205\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi1 eqs1 cll1 t1\n-- --   test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 {ts = ts} indeq ceq eqs ceqo | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 {cs} ceqi1 eqs1 ceqo1 t1\n-- --     = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 ((\u03bb cll \u2192 subst (\u03bb z \u2192 LFun z (shrink rll _)) (shrink-repl\u2261\u2205 s lind eq teq ceq _ ceqo t) (t1 cll))) where\n-- --     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n-- --     t : complL\u209b (replacePartOf s to ts at lind) \u2261 \u00ac\u2205 cs\n-- --     t with trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n-- --     ... | g with replacePartOf s to ts at lind\n-- --     t | refl | e = ceqi1\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq eqs ceqo | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 indeq ceq refl ceqo | \u2205 | [ () ] | .(\u00ac\u2205 _)\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind\u2081} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u2205 () ceqi eqs cll t\n-- --   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t\n-- --        with (mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (tranLFMSetLL lf (\u00ac\u2205 trs)) at z) lind\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ meq ] with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi () ceqo t | \u2205 | [ meq ] | \u2205\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u2205 | [ () ] | \u00ac\u2205 _\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u2205 indeq ceq1 eqs1 ceqo1 with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- --   test {rll = rll} {ll = ll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl ceqo t | \u00ac\u2205 .(replacePartOf s to _ at lind) | [ refl ] | \u2205 indeq1 ceq1 eqs1 ceqo1 | .(\u00ac\u2205 _) with complL\u209b ts | ct where\n-- --     m = replacePartOf s to ts at lind\n-- --     mind = subst (\u03bb z \u2192 IndexLL z (replLL ll lind ell)) (replLL-\u2193 lind) ((a\u2264\u1d62b-morph lind lind ell (\u2264\u1d62-reflexive lind)))\n-- --     ct = compl\u2261\u2205\u21d2compltr\u2261\u2205 m ceq1 mind (sym (tr-repl\u21d2id s lind ts))\n-- --   test {rll = rll} {ll} {n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 {ts = ts} ceqi refl () t | \u00ac\u2205 .(replacePartOf s to ts at ind) | [ refl ] | \u2205 indeq ceq1 eqs1 ceqo1 | .(\u00ac\u2205 ts) | .\u2205 | refl\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u2205 () ceqi\u2081 eqs\u2081 cll1 x\u2081\n-- --   test {rll = rll} {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t | \u00ac\u2205 mx | [ meq ] | \u00ac\u2205\u00ac\u2205 ceqi1 eqs1 ceqo1 t1\n-- --       = \u00ac\u2205\u00ac\u2205 eq tseq ceqo1 (\u03bb cll \u2192 _\u2282_ {ind = hind} (t cll) (subst (\u03bb z \u2192 LFun z _) (sym (ho-shrink-repl-comm s lind eq teq ceqi ceqo w ceqi1)) (t1 cll))) where\n-- --     w = trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to z at lind) eqs) meq\n-- --     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs1))\n-- --     hind = ho-shr-morph s eq lind (sym teq) ceqi\n-- --   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] with truncSetLL s lind | inspect (truncSetLL s) lind\n-- --   ... | \u2205 | [ teq ] with (mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at lind) | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to (\u2205 {ll = ell}) at z) lind\n-- --   ... | \u2205 | [ meq ] = \u22a5-elim ((trunc\u2261\u2205\u21d2\u00acmrpls\u2261\u2205 s lind teq) meq)\n-- --   ... | \u00ac\u2205 mx | [ meq ] with isLTi lind ind\n-- --   ... | no \u00acp with test {n = n} {dt} {df} nind mx (\u00ac\u2205 n\u00acho) lf\u2081 where\n-- --     nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n-- --     nind = \u00ac\u2205 (\u00acord-morph ind lind ell (flipNotOrd\u1d62 nord))\n-- --     n\u00acho = \u00acord&\u00acho-del\u21d2\u00acho ind s \u00acho lind nord (sym meq)\n-- --   ... | \u2205 () ceq eqs ceqo\n-- --   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u2205 {ind = tind} indeq ceqi eqs cll t\n-- --     = \u00ac\u2205\u2205 refl eq tseq cll (_\u2282_ {ind = sind} lf {!l1!}) where \n-- --     nord = ind\u03c4&\u00acge\u21d2\u00acOrd ind lind \u00acp\n-- --     tseq =  sym (trans (cong (\u03bb z \u2192 tranLFMSetLL lf\u2081 z) (trans (cong (\u03bb z \u2192 mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) teq) meq)) (sym eqs))\n-- --     hind = \u00acho-shr-morph s eq lind ((trunc\u2261\u2205\u21d2\u00acho s lind teq))\n-- --     bind = \u00acho-shr-morph s eq ind \u00acho\n    \n-- --     hord : \u00ac Ordered\u1d62 bind hind\n-- --     hord x = nord (\u00acho-shr-morph-pres-\u00acord s eq ind lind \u00acho (trunc\u2261\u2205\u21d2\u00acho s lind teq) x)\n    \n-- --     sind = \u00acord-morph hind bind cll hord \n-- --     l1 = replLL-\u00acordab\u2261ba bind cll hind ell (flipNotOrd\u1d62 hord)\n-- --     l2 = {!cong (\u03bb z \u2192 replLL z (\u00acord-morph bind hind ell ?) cll) ?!} \n\n-- --     find = \u00acord-morph ind lind ell (flipNotOrd\u1d62 nord)\n-- --     nind = \u00acord-morph lind ind cll nord\n-- --   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | no \u00acp | \u00ac\u2205\u00ac\u2205 ceqi eqs ceqo t = {!!}\n-- --   test {rll = rll} {n = n} {dt} {df} (\u00ac\u2205 ind) s (\u00ac\u2205 \u00acho) (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u2205 | [ teq ] | \u00ac\u2205 mx | [ meq ] | yes p = {!!}\n-- --   test (\u00ac\u2205 ind\u2081) s (\u00ac\u2205 \u00acho) (_\u2282_ {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] = {!!}\n-- --   test ind s mnho (tr ltr lf) | \u00ac\u2205 x | eq = {!!}\n-- --   test ind s mnho (com df\u2081 lf) | \u00ac\u2205 x | eq = {!!}\n-- --   test ind s mnho (call x\u2081) | \u00ac\u2205 x | eq = {!!}\n\n\n\n\n\n-- -- --  test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] with tranLFMSetLL lf (\u00ac\u2205 trs) | inspect (\u03bb z \u2192 tranLFMSetLL lf (\u00ac\u2205 z)) trs | test {n = n} {dt} {df} \u2205 trs \u2205 lf\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | I x eqs ceqo with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | I x () ceqo | .\u2205\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ tseq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo x with tranLFMSetLL lf (\u00ac\u2205 trs)\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u2205 | [ refl ] | \u00acI\u00ac\u2205 ceqi () ceqo x | .(\u00ac\u2205 _)\n-- -- --\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | I x eqs ceqo = {!!}\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u2205 ceqi eqs x = {!!}\n-- -- --  test \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = ind} lf lf\u2081) | \u00ac\u2205 x\u2081 | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00ac\u2205 ts | tseq | \u00acI\u00ac\u2205 ceqi eqs ceqo x = {!!}\n-- -- --\n-- -- --\n-- -- ---- with test {n = n} {dt} {df} \u2205 trs \u2205 lf\n-- -- ----   ... | I ceq with mreplacePartOf (\u00ac\u2205 s) to tlf at lind | inspect (\u03bb z \u2192 mreplacePartOf (\u00ac\u2205 s) to tlf at z) lind where -- tranLFMSetLL lf\u2081 nmx | inspect (tranLFMSetLL lf\u2081) nmx | test \u2205 {!nmx!} \u2205 lf\u2081 where\n-- -- ----     tlf = tranLFMSetLL lf (\u00ac\u2205 trs)\n-- -- ---- --    is = test \u2205 {!!} \u2205 lf\u2081\n-- -- ----   ... | \u2205 | [ meq ] = IMPOSSIBLE -- Since we have I, lf cannot contain com or call, thus tlf is not \u2205.\n-- -- ----   ... | \u00ac\u2205 mx | [ meq ] with test {n = n} {dt} {df} \u2205 mx \u2205 lf\u2081\n-- -- ----   ... | I ceq1 = IMPOSSIBLE -- TODO This is impossible because eq assures us that complL\u209b s \u2261 \u00ac\u2205 x but complL\u209b of both ceq and ceq1 are \u2205.\n-- -- ----   ... | \u00acI\u2205 ceqi1 eqs1 t1 = \u00acI\u2205 eq tseq {!!} where\n-- -- ----     meeq = subst (\u03bb z \u2192 (mreplacePartOf \u00ac\u2205 s to tranLFMSetLL lf z at lind) \u2261 \u00ac\u2205 mx) (sym teq) meq\n-- -- ----     tseq = subst (\u03bb z \u2192 \u2205 \u2261 tranLFMSetLL lf\u2081 z) (sym meeq) eqs1\n-- -- ----   ... | \u00acI\u00ac\u2205 ceqi1 eqs1 ceqo1 t1 = {!!} \n-- -- ----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u2205 ceqi eqs q = {!!}\n-- -- ----   test {n = n} {dt} {df} \u2205 s \u2205 (_\u2282_ {ell = ell} {ind = lind} lf lf\u2081) | \u00ac\u2205 x | [ eq ] | \u00ac\u2205 trs | [ teq ] | \u00acI\u00ac\u2205 ceqi eqs ceqo q = {!!} where\n-- -- --  test \u2205 s \u2205 (tr ltr lf) | \u00ac\u2205 x | [ eq ] = {!!}\n-- -- --  test \u2205 s \u2205 (com df\u2081 lf) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE -- Since ind is \u2205, this is impossible.\n-- -- --  test \u2205 s \u2205 (call x\u2081) | \u00ac\u2205 x | [ eq ] = IMPOSSIBLE\n-- -- --  test (\u00ac\u2205 x) s (\u00ac\u2205 \u00acho) lf = {!!}\n-- -- --  \n-- -- --  \n  \n  \n  \n  \n  \n","returncode":0,"stderr":"","license":"mpl-2.0","lang":"Agda"}
{"commit":"a634891a919352b41720019a6bdf7b7c56ac492d","subject":"Proved N-ind\u2081 fromN-ind\u2082.","message":"Proved N-ind\u2081 fromN-ind\u2082.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Type.agda","new_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers inductive predicate\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- The induction principle generated by Coq 8.4pl3 when we define the data\n-- type N in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step (see Martin-L\u00f6f 1971, p. 190).\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n------------------------------------------------------------------------------\n-- N-ind\u2082 from N-ind\u2081.\n\nN-ind\u2082' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n------------------------------------------------------------------------------\n-- N-ind\u2081 from N-ind\u2082.\n\nN-ind\u2081' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081' A A0 h {n} Nn = \u2227-proj\u2082 (N-ind\u2082 B B0 h' Nn)\n  where\n  B : D \u2192 Set\n  B n = N n \u2227 A n\n\n  B0 : B zero\n  B0 = nzero , A0\n\n  h' : \u2200 {m} \u2192 B m \u2192 B (succ\u2081 m)\n  h' (Nm , Am) = (nsucc Nm) , (h Nm Am)\n\n------------------------------------------------------------------------------\n-- References\n--\n-- Martin-L\u00f6f, P. (1971). Hauptsatz for the Intuitionistic Theory of\n-- Iterated Inductive De\ufb01nitions. In: Proceedings of the Second\n-- Scandinavian Logic Symposium. Ed. by Fenstad,\n-- J. E. Vol. 63. Studies in Logic and the Foundations of\n-- Mathematics. North-Holland Publishing Company, pp. 179\u2013216.\n","old_contents":"------------------------------------------------------------------------------\n-- Induction principles for the total natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Type where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Induction principle generated by Coq 8.4pl3 when we define the data\n-- type N in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n------------------------------------------------------------------------------\n-- N-ind\u2082 from N-ind\u2081.\n\nN-ind\u2082' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082' A A0 h = N-ind\u2081 A A0 (\u03bb _ \u2192 h)\n\n------------------------------------------------------------------------------\n-- We cannot prove N-ind\u2081 from N-ind\u2082.\n\nN\u21920\u2228S : \u2200 {n} \u2192 N n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n')\nN\u21920\u2228S = N-ind\u2082 A A0 is\n  where\n  A : D \u2192 Set\n  A i = i \u2261 zero \u2228 (\u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i')\n\n  A0 : A zero\n  A0 = inj\u2081 refl\n\n  is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ai = case prf\u2081 prf\u2082 Ai\n    where\n    prf\u2081 : i \u2261 zero \u2192 succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2081 h' = inj\u2082 (i , refl , (subst N (sym h') nzero))\n\n    prf\u2082 : \u2203[ i' ] i \u2261 succ\u2081 i' \u2227 N i' \u2192\n           succ\u2081 i \u2261 zero \u2228 (\u2203[ i' ] succ\u2081 i \u2261 succ\u2081 i' \u2227 N i')\n    prf\u2082 (i' , prf , Ni') = inj\u2082 (i , refl , subst N (sym prf) (nsucc Ni'))\n\nN-ind\u2081' : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081' A A0 h {n} Nn = case prf\u2081 prf\u2082 (N\u21920\u2228S Nn)\n  where\n  prf\u2081 : n \u2261 zero \u2192 A n\n  prf\u2081 n\u22610 = subst A (sym n\u22610) A0\n\n  prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 A n\n  prf\u2082 (n' , prf , Nn') = subst A (sym prf) (h Nn' {!!})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"e57feb0a768c5ff75b146f04d9ade09c6c2c71e6","subject":"Refactored a proof.","message":"Refactored a proof.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationI.agda","new_file":"src\/fot\/FOTC\/Program\/ABP\/ProofSpecificationI.agda","new_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ProofSpecificationI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1I\nopen import FOTC.Program.ABP.Lemma2I\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 transfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n       \u2203[ i' ] \u2203[ is' ] \u2203[ js' ] is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  prf\u2081 {is} {js} (b , os\u2080 , os\u2081 , as , bs , cs , ds , Sis , Bb , Fos\u2080 , Fos\u2081 , h)\n     with Stream-unf Sis\n  ... | (i' , is' , is\u2261i'\u2237is , Sis') =\n    i' , is' , js' , is\u2261i'\u2237is , js\u2261i'\u2237js' , Bis'js'\n    where\n    ABP-helper : is \u2261 i' \u2237 is' \u2192\n                 ABP b is os\u2080 os\u2081 as bs cs ds js \u2192\n                 ABP b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js\n    ABP-helper h\u2081 h\u2082 = subst (\u03bb t \u2192 ABP b t os\u2080 os\u2081 as bs cs ds js) h\u2081 h\u2082\n\n    ABP'-lemma\u2081 : \u2203[ os\u2080' ] \u2203[ os\u2081' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n                  Fair os\u2080'\n                  \u2227 Fair os\u2081'\n                  \u2227 ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                  \u2227 js \u2261 i' \u2237 js'\n    ABP'-lemma\u2081 = lemma\u2081 Bb Fos\u2080 Fos\u2081 (ABP-helper is\u2261i'\u2237is h)\n\n    -- Following Martin Escardo advice (see Agda mailing list, heap\n    -- mistery) we use pattern matching instead of \u2203 eliminators to\n    -- project the elements of the existentials.\n\n    -- 2011-08-25 update: It does not seems strictly necessary because\n    -- the Agda issue 415 was fixed.\n\n    js' : D\n    js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , js' , _ = js'\n\n    js\u2261i'\u2237js' : js \u2261 i' \u2237 js'\n    js\u2261i'\u2237js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , _ , _ , _ , _ , h = h\n\n    ABP-lemma\u2082 : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n                 Fair os\u2080''\n                 \u2227 Fair os\u2081''\n                 \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n    ABP-lemma\u2082 with ABP'-lemma\u2081\n    ABP-lemma\u2082 | _ , _ , _ , _ , _ , _ , _ , Fos\u2080' , Fos\u2081' , abp' , _ =\n      lemma\u2082 Bb Fos\u2080' Fos\u2081' abp'\n\n    Bis'js' : B is' js'\n    Bis'js' with ABP-lemma\u2082\n    ... | os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds'' , Fos\u2080'' , Fos\u2081'' , abp =\n      not b , os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds''\n      , Sis' , not-Bool Bb , Fos\u2080'' , Fos\u2081'' , abp\n\n  prf\u2082 : B is (transfer b os\u2080 os\u2081 is)\n  prf\u2082 = b\n       , os\u2080\n       , os\u2081\n       , has a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , Sis\n       , Bb\n       , Fos\u2080\n       , Fos\u2081\n       , has-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hbs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hcs-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , hds-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is\n       , trans (transfer-eq b os\u2080 os\u2081 is) (genTransfer-eq a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 is)\n    where\n    a\u2081 a\u2082 a\u2083 a\u2084 a\u2085 : D\n    a\u2081 = send \u00b7 b\n    a\u2082 = ack \u00b7 b\n    a\u2083 = out \u00b7 b\n    a\u2084 = corrupt \u00b7 os\u2080\n    a\u2085 = corrupt \u00b7 os\u2081\n","old_contents":"------------------------------------------------------------------------------\n-- The alternating bit protocol (ABP) satisfies the specification\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- This module proves the correctness of the ABP following the\n-- formalization in [1].\n\n-- [1] Peter Dybjer and Herbert Sander. A functional programming\n--     approach to the specification and verification of concurrent\n--     systems. Formal Aspects of Computing, 1:303\u2013319, 1989.\n\nmodule FOTC.Program.ABP.ProofSpecificationI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesI\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Lemma1I\nopen import FOTC.Program.ABP.Lemma2I\nopen import FOTC.Program.ABP.Terms\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n-- Main theorem.\nspec : \u2200 {b is os\u2080 os\u2081} \u2192 Bit b \u2192 Stream is \u2192 Fair os\u2080 \u2192 Fair os\u2081 \u2192\n       is \u2248 transfer b os\u2080 os\u2081 is\nspec {b} {is} {os\u2080} {os\u2081} Bb Sis Fos\u2080 Fos\u2081 = \u2248-coind B prf\u2081 prf\u2082\n  where\n  prf\u2081 : \u2200 {is js} \u2192 B is js \u2192\n       \u2203[ i' ] \u2203[ is' ] \u2203[ js' ] is \u2261 i' \u2237 is' \u2227 js \u2261 i' \u2237 js' \u2227 B is' js'\n  prf\u2081 {is} {js} (b , os\u2080 , os\u2081 , as , bs , cs , ds , Sis , Bb , Fos\u2080 , Fos\u2081 , h)\n     with Stream-unf Sis\n  ... | (i' , is' , is\u2261i'\u2237is , Sis') =\n    i' , is' , js' , is\u2261i'\u2237is , js\u2261i'\u2237js' , Bis'js'\n    where\n    ABP-helper : is \u2261 i' \u2237 is' \u2192\n                 ABP b is os\u2080 os\u2081 as bs cs ds js \u2192\n                 ABP b (i' \u2237 is') os\u2080 os\u2081 as bs cs ds js\n    ABP-helper h\u2081 h\u2082 = subst (\u03bb t \u2192 ABP b t os\u2080 os\u2081 as bs cs ds js) h\u2081 h\u2082\n\n    ABP'-lemma\u2081 : \u2203[ os\u2080' ] \u2203[ os\u2081' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n                  Fair os\u2080'\n                  \u2227 Fair os\u2081'\n                  \u2227 ABP' b i' is' os\u2080' os\u2081' as' bs' cs' ds' js'\n                  \u2227 js \u2261 i' \u2237 js'\n    ABP'-lemma\u2081 = lemma\u2081 Bb Fos\u2080 Fos\u2081 (ABP-helper is\u2261i'\u2237is h)\n\n    -- Following Martin Escardo advice (see Agda mailing list, heap\n    -- mistery) we use pattern matching instead of \u2203 eliminators to\n    -- project the elements of the existentials.\n\n    -- 2011-08-25 update: It does not seems strictly necessary because\n    -- the Agda issue 415 was fixed.\n\n    js' : D\n    js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , js' , _ = js'\n\n    js\u2261i'\u2237js' : js \u2261 i' \u2237 js'\n    js\u2261i'\u2237js' with ABP'-lemma\u2081\n    ... | _ , _ , _ , _ , _ , _ , _ , _ , _ , _ , h = h\n\n    ABP-lemma\u2082 : \u2203[ os\u2080'' ] \u2203[ os\u2081'' ] \u2203[ as'' ] \u2203[ bs'' ] \u2203[ cs'' ] \u2203[ ds'' ]\n                 Fair os\u2080''\n                 \u2227 Fair os\u2081''\n                 \u2227 ABP (not b) is' os\u2080'' os\u2081'' as'' bs'' cs'' ds'' js'\n    ABP-lemma\u2082 with ABP'-lemma\u2081\n    ABP-lemma\u2082 | _ , _ , _ , _ , _ , _ , _ , Fos\u2080' , Fos\u2081' , abp' , _ =\n      lemma\u2082 Bb Fos\u2080' Fos\u2081' abp'\n\n    Bis'js' : B is' js'\n    Bis'js' with ABP-lemma\u2082\n    ... | os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds'' , Fos\u2080'' , Fos\u2081'' , abp =\n      not b , os\u2080'' , os\u2081'' , as'' , bs'' , cs'' , ds''\n      , Sis' , not-Bool Bb , Fos\u2080'' , Fos\u2081'' , abp\n\n  prf\u2082 : B is (transfer b os\u2080 os\u2081 is)\n  prf\u2082 = b\n       , os\u2080\n       , os\u2081\n       , has (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , hbs (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , hcs (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , hds (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , Sis\n       , Bb\n       , Fos\u2080\n       , Fos\u2081\n       , has-eq (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , hbs-eq (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , hcs-eq (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , hds-eq (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080) (corrupt \u00b7 os\u2081) is\n       , trans (transfer-eq b os\u2080 os\u2081 is)\n               (genTransfer-eq (send \u00b7 b) (ack \u00b7 b) (out \u00b7 b) (corrupt \u00b7 os\u2080)\n                               (corrupt \u00b7 os\u2081) is)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"fa94fd6d9613ebb3f732ab69d2963bdf1019a283","subject":"add tensor","message":"add tensor\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n  _\u2297\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297\u1d3e Q       = Q\n  \u03a3\u1d3e M P \u2297\u1d3e Q       = \u03a3\u1d3e M \u03bb m \u2192 P m \u2297\u1d3e Q\n  P      \u2297\u1d3e end     = P\n  P      \u2297\u1d3e \u03a3\u1d3e M  Q = \u03a3\u1d3e M \u03bb m \u2192 P \u2297\u1d3e Q m\n  \u03a0\u1d3e M P \u2297\u1d3e \u03a0\u1d3e M' Q = \u03a0\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u2297\u1d3e \u03a0\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a0\u1d3e M P \u2297\u1d3e Q m') ]\n\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b\u1d3e Q) \u2261\u1d3e dual P \u2297\u1d3e dual Q\n  \u2297\u214b-dual end Q = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a0\u1d3e M P) Q = com refl M (\u03bb m \u2192 \u2297\u214b-dual (P m) _)\n  \u2297\u214b-dual (\u03a3\u1d3e M P) end = \u2261\u1d3e-refl _\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) = com refl M' (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n  \u2297\u214b-dual (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) = com refl (M \u228e M')\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (\u03a3\u1d3e M' Q))\n    ,inr: (\u03bb m' \u2192 \u2297\u214b-dual (\u03a3\u1d3e M P) (Q m'))\n    ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP as \u2261\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n  {-\ndata S<_> {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  S[_] : \u2200 x \u2192 S< x >\n\nunS : \u2200 {a} {A : \u2605_ a} ..{x : A} \u2192 S< x > \u2192 A\nunS S[ y ] = y\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S<_> {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S<_> public\n\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S< x >\nS[ x ] = S[ x \u2225 refl ]\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\nmutual\n    record Com : \u2605\u2081 where\n      constructor mk\n      field\n        io : InOut\n        M  : \u2605\n        P  : M \u2192 Proto\n\n    data Proto : \u2605\u2081 where\n      end : Proto\n      com : Com \u2192 Proto\n\npattern com' q M P = com (mk q M P)\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {q\u2080 q\u2081} (q : q\u2080 \u2248\u1d35\u1d3c q\u2081) M {P Q} \u2192 (\u2200 m \u2192 P m \u2248\u1d3e Q m) \u2192 com' q\u2080 M P \u2248\u1d3e com' q\u2081 M Q\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_)\n\n{-\ninfix 0 _\u2261\u1d3e_\ndata _\u2261\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : end \u2261\u1d3e end\n  com : \u2200 q M {P Q} \u2192 (\u2200 m \u2192 P m \u2261\u1d3e Q m) \u2192 com' q M P \u2261\u1d3e com' q M Q\n  -}\n\npattern \u03a0\u1d9c M P = mk In  M P\npattern \u03a3\u1d9c M P = mk Out M P\n\npattern \u03a0\u1d3e M P = com (mk In  M P)\npattern \u03a3\u1d3e M P = com (mk Out M P)\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0\u2610\u1d3e M P = \u03a0\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3\u2610\u1d3e : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3\u2610\u1d3e M P = \u03a3\u1d3e (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\nmutual\n    Trace : Proto \u2192 Proto\n    Trace end     = end\n    Trace (com c) = com (Trace\u1d9c c)\n\n    Trace\u1d9c : Com \u2192 Com\n    Trace\u1d9c c = \u03a3\u1d9c (Com.M c) \u03bb m \u2192 Trace (Com.P c m)\n\nmutual\n    dual : Proto \u2192 Proto\n    dual end     = end\n    dual (com c) = com (dual\u1d9c c)\n\n    dual\u1d9c : Com \u2192 Com\n    dual\u1d9c c = mk (dual\u1d35\u1d3c (Com.io c)) (Com.M c) \u03bb m \u2192 dual (Com.P c m)\n\ndata IsTrace : Proto \u2192 \u2605\u2081 where\n  end : IsTrace end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsTrace (P m)) \u2192 IsTrace (\u03a3\u1d3e M P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 M {P} (PT : \u2200 m \u2192 IsSink (P m)) \u2192 IsSink (\u03a0\u1d3e M P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q M {P} (P\u2610 : \u2200 m \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com' q (\u2610 M) P)\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 (M : \u2605) (P : M \u2192 \u2605) \u2192 \u2605\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end        \u27e7 = \ud835\udfd9\n\u27e6 com' q M P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u27e6_\u27e7\u27e8_\u2248_\u27e9 : (P : Proto) (p q : \u27e6 P \u27e7) \u2192 \u2605\n\u27e6 end    \u27e7\u27e8 p \u2248 q \u27e9 = \ud835\udfd9\n\u27e6 \u03a0\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = (m : M) \u2192 \u27e6 P m \u27e7\u27e8 p m \u2248 q m \u27e9\n\u27e6 \u03a3\u1d3e M P \u27e7\u27e8 p \u2248 q \u27e9 = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u27e6 P (fst q) \u27e7\u27e8 subst (\u27e6_\u27e7 \u2218 P) e (snd p) \u2248 snd q \u27e9\n\n_\u00d7'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u00d7' P = \u03a3\u1d3e M \u03bb _ \u2192 P\n\n_\u2192'_ : \u2605 \u2192 Proto \u2192 Proto\nM \u2192' P = \u03a0\u1d3e M \u03bb _ \u2192 P\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmutual\n    data Dual\u1d9c : Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a0\u1d9c M P) (\u03a3\u1d9c M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual\u1d9c (\u03a3\u1d9c M P) (\u03a0\u1d9c M Q)\n\n    data Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : Dual end end\n      \u03a0\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a0\u1d3e M P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q x)) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e M Q)\n      {-\n      \u03a0\u2610\u00b7\u03a3 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P [ x ]) (Q x)) \u2192 Dual (\u03a0\u1d3e (\u2610 M) P) (\u03a3\u1d3e M Q)\n      \u03a3\u00b7\u03a0\u2610 : \u2200 {M P Q} \u2192 (\u2200 x \u2192 Dual (P x) (Q [ x ])) \u2192 Dual (\u03a3\u1d3e M P) (\u03a0\u1d3e (\u2610 M) Q)\n      -}\n\n      {-\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = com (case p A\n                               0: case p B\n                                    0: \u03a0\u1d9c (\u2610 M) (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: \u03a0\u1d9c M     (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: \u03a3\u1d9c M (\u03bb m \u2192 \u2102 m \/\/ p))\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com' (case p A 0: In 1: Out) M \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Trace : Choreo I \u2192 Proto\n    \u2102Trace \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Trace-IsTrace : \u2200 \u2102 \u2192 IsTrace (\u2102Trace \u2102)\n    \u2102Trace-IsTrace (A -[ M ]\u2192 B \u204f \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Trace-IsTrace (\u2102 m)\n    \u2102Trace-IsTrace end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com (\u2610 M) \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com M \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Trace \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n-}\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end              = refl\n    \u2261\u1d3e-sound (com refl M P\u2261Q) = cong (com' _ M) (funExt \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end         = end\n\u2261\u1d3e-refl (com' q M P) = com refl M \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end = end\ndual-involutive (com' q M P) = com (dual\u1d35\u1d3c-involutive q) M (\u03bb m \u2192 dual-involutive (P m))\n\nTrace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261\u1d3e Trace P\nTrace-idempotent end         = end\nTrace-idempotent (com' _ M P) = com refl M \u03bb m \u2192 Trace-idempotent (P m)\n\nTrace-dual-oblivious : \u2200 P \u2192 Trace (dual P) \u2261\u1d3e Trace P\nTrace-dual-oblivious end         = end\nTrace-dual-oblivious (com' _ M P) = com refl M \u03bb m \u2192 Trace-dual-oblivious (P m)\n\nSink : Proto \u2192 Proto\nSink = dual \u2218 Trace\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend       >>= Q = Q _\ncom' q M P >>= Q = com' q M \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n{-\n++Tele : \u2200 (P : Proto){Q : Tele P \u2192 Proto} (xs : Tele P) \u2192 Tele (Q xs) \u2192 Tele (P >>= Q)\n++Tele end         _        ys = ys\n++Tele (com' q M P) (x , xs) ys = x , ++Tele (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end         = end\n>>=-right-unit (com' q M P) = com q M \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Tele P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end         Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com' q M P) Q R = com q M \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\nmutual\n    data [_&_\u2261_]\u1d9c : Com \u2192 Com \u2192 Com \u2192 \u2605\u2081 where\n      \u03a0&   : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P m     & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c    M  P & mk q M Q \u2261 mk q M R ]\u1d9c\n      \u03a0\u2610&  : \u2200 q {M P Q R}(P& : \u2200 m \u2192 [ P [ m ] & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d9c (\u2610 M) P & mk q M Q \u2261 mk q M R ]\u1d9c\n\n    data [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n      &-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n      end : \u2200 {P} \u2192 [ end & P \u2261 P ]\n      com : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]\u1d9c \u2192 [ com P & com Q \u2261 com R ]\n\n-- pattern \u03a0&\u03a3 P = \u03a0& \u03a3' P\n\u03a0&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a0\u1d3e M Q \u2261 \u03a0\u1d3e M R ]\n\u03a0&\u03a0 P& = com (\u03a0& In P&)\n\u03a0&\u03a3 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a0\u1d3e M P & \u03a3\u1d3e M Q \u2261 \u03a3\u1d3e M R ]\n\u03a0&\u03a3 P& = com (\u03a0& Out P&)\n\u03a3&\u03a0 : \u2200 {M P Q R}(P& : \u2200 m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ \u03a3\u1d3e M P & \u03a0\u1d3e M  Q \u2261 \u03a3\u1d3e M R ]\n\u03a3&\u03a0 P& = &-comm (\u03a0&\u03a3 (\u03bb m \u2192 &-comm (P& m)))\n\n&-dual : \u2200 P \u2192 [ P & dual P \u2261 Trace P ]\n&-dual end      = end\n&-dual (\u03a3\u1d3e M P) = \u03a3&\u03a0 \u03bb m \u2192 &-dual (P m)\n&-dual (\u03a0\u1d3e M P) = \u03a0&\u03a3 \u03bb m \u2192 &-dual (P m)\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n{-\nDual-sym (\u03a0\u2610\u00b7\u03a3 f) = {!\u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))!}\nDual-sym (\u03a3\u00b7\u03a0\u2610 f) = {!\u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))!}\n-}\n-}\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0\u1d3e M P) = \u03a0\u00b7\u03a3 \u03bb m \u2192 Dual-spec (P m)\nDual-spec (\u03a3\u1d3e M P) = \u03a3\u00b7\u03a0 \u03bb m \u2192 Dual-spec (P m)\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele (dual P) \u2261 Tele P\n  dual-Tele P = cong \u27e6_\u27e7 (\u2261\u1d3e-sound funExt (Trace-dual-oblivious P))\n\nEl : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\nEl end         X = X _\nEl (com' q M P) X = \u27e6 q \u27e7\u1d35\u1d3c M \u03bb x \u2192 El (P x) (\u03bb y \u2192 X (x , y))\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  El->>= : (P : Proto){Q : Tele P \u2192 Proto}{X : Tele (P >>= Q) \u2192 \u2605} \u2192 El (P >>= Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P x y)))\n  El->>= end         = refl\n  El->>= (com' q M P) = cong (\u27e6 q \u27e7\u1d35\u1d3c M) (funExt \u03bb m \u2192 El->>= (P m))\n\ntele-com : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Tele P\ntele-com end      _       _       = _\ntele-com (\u03a0\u1d3e M P) p       (m , q) = m , tele-com (P m) (p m) q\ntele-com (\u03a3\u1d3e M P) (m , p) q       = m , tele-com (P m) p (q m)\n-}\n\n>>=-com : (P : Proto){Q : Tele P \u2192 Proto}{R : Tele P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Tele P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (\u03a3\u1d3e M P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (\u03a0\u1d3e M P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Tele P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Com \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} (s : (m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0\u1d9c M P)\n  send : \u2200 {M P} (m : M) (s : this (P m)) \u2192 ProcessF this (\u03a3\u1d9c M P)\n\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} \u2192 (..(m : M) \u2192 this (P [ m ])) \u2192 ProcessF this (\u03a0\u1d9c (\u2610 M) P)\nrecvS = recv \u2218 un\u2610\n\nsendS : \u2200 {this : Proto \u2192 \u2605\u2081}{M}{P : \u2610 M \u2192 Proto} ..(m : M) \u2192 this (P [ m ]) \u2192 ProcessF this (\u03a3\u1d9c (\u2610 M) P)\nsendS m = send [ m ]\n\ndata Process : Proto \u2192 \u2605\u2081 where\n  end : Process end\n  com : \u2200 {P} \u2192 ProcessF Process P \u2192 Process (com P)\n\nmutual\n  SimL : Com \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Com \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    comL : \u2200 {P Q} (sL : SimL P Q) \u2192 Sim (com P) Q\n    comR : \u2200 {P Q} (sR : SimR P Q) \u2192 Sim P (com Q)\n    end  : Sim end end\n\nsendL : \u2200 {M P Q} (m : M) \u2192 Sim (P m) Q \u2192 Sim (\u03a3\u1d3e M P) Q\nsendL m s = comL (send m s)\n\nsendR : \u2200 {M P Q} (m : M) \u2192 Sim P (Q m) \u2192 Sim P (\u03a3\u1d3e M Q)\nsendR m s = comR (send m s)\n\nrecvL : \u2200 {M P Q} (s : (m : M) \u2192 Sim (P m) Q) \u2192 Sim (\u03a0\u1d3e M P) Q\nrecvL s = comL (recv s)\n\nrecvR : \u2200 {M P Q} (s : (m : M) \u2192 Sim P (Q m)) \u2192 Sim P (\u03a0\u1d3e M Q)\nrecvR s = comR (recv s)\n\ndata _\u2248\u02e2_ : \u2200 {P Q} (s\u2080 s\u2081 : Sim P Q) \u2192 \u2605\u2081 where\n  \u2248-end : end \u2248\u02e2 end\n  \u2248-sendL : \u2200 {M} {P : M \u2192 Proto} {Q} (m : M) {s\u2080 s\u2081 : Sim (P m) Q}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendL {P = P} m s\u2080 \u2248\u02e2 sendL m s\u2081\n  \u2248-sendR : \u2200 {M P} {Q : M \u2192 Proto} (m : M) {s\u2080 s\u2081 : Sim P (Q m)}\n          \u2192 s\u2080 \u2248\u02e2 s\u2081\n          \u2192 sendR {Q = Q} m s\u2080 \u2248\u02e2 sendR m s\u2081\n  \u2248-recvL : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim (P m) Q}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvL {P = P} s\u2080 \u2248\u02e2 recvL s\u2081\n  \u2248-recvR : \u2200 {M P Q} {s\u2080 s\u2081 : (m : M) \u2192 Sim P (Q m)}\n          \u2192 (p : \u2200 m \u2192 s\u2080 m \u2248\u02e2 s\u2081 m)\n          \u2192 recvR {Q = Q} s\u2080 \u2248\u02e2 recvR s\u2081\n          {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080} {s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 sendR {Q = Q} r (sendL {P = P} \u2113 s\u2081)\n             \u2192 s\u2080 \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n             -}\n  {-\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2080) \u2248\u02e2 sendR r (sendL \u2113 s\u2081)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s\u2080 s\u2081 : Sim (P \u2113) (Q r)}\n             \u2192 s\u2080 \u2248\u02e2 s\u2081\n             \u2192 sendR r (sendL \u2113 s\u2080) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s\u2081)\n  -}\npostulate\n  \u2248-sendLR : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendL {P = P} \u2113 (sendR {Q = Q} r s) \u2248\u02e2 sendR r (sendL \u2113 s)\n  \u2248-sendRL : \u2200 {M\u2113 Mr P Q} (\u2113 : M\u2113) (r : Mr) {s : Sim (P \u2113) (Q r)}\n             \u2192 sendR r (sendL \u2113 s) \u2248\u02e2 sendL {P = P} \u2113 (sendR {Q = Q} r s)\n  \u2248-sendR-recvL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 sendR r (recvL s) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113))\n  \u2248-recvR-sendL : \u2200 {M\u2113 Mr P Q} (r : Mr) {s : (\u2113 : M\u2113) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 sendR {Q = Q} r (s \u2113)) \u2248\u02e2 sendR r (recvL s)\n  \u2248-recvRL : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r)) \u2248\u02e2 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113))\n  \u2248-recvLR : \u2200 {M\u2113 Mr P Q} {s : (\u2113 : M\u2113) (r : Mr) \u2192 Sim (P \u2113) (Q r)}\n             \u2192 recvL {P = P} (\u03bb \u2113 \u2192 recvR {Q = Q} (s \u2113)) \u2248\u02e2 recvR (\u03bb r \u2192 recvL (\u03bb \u2113 \u2192 s \u2113 r))\n\n\u2248\u02e2-refl : \u2200 {P Q} (s : Sim P Q) \u2192 s \u2248\u02e2 s\n\u2248\u02e2-refl (comL (recv s)) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comL (send m s)) = \u2248-sendL m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl (comR (recv s)) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-refl (s m))\n\u2248\u02e2-refl (comR (send m s)) = \u2248-sendR m (\u2248\u02e2-refl s)\n\u2248\u02e2-refl end = \u2248-end\n\n\u2248\u02e2-sym : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 s\u2081 \u2248\u02e2 s\u2080\n\u2248\u02e2-sym \u2248-end = \u2248-end\n\u2248\u02e2-sym (\u2248-sendL m p) = \u2248-sendL m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendR m p) = \u2248-sendR m (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-recvL x) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n\u2248\u02e2-sym (\u2248-recvR x) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-sym (x m))\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = {!\u2248-sendRL \u2113 r ?!}\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = {!\u2248-sendLR \u2113 r!}\n-}\n{-\n\u2248\u02e2-sym (\u2248-sendLR \u2113 r p) = \u2248-sendRL \u2113 r (\u2248\u02e2-sym p)\n\u2248\u02e2-sym (\u2248-sendRL \u2113 r p) = \u2248-sendLR \u2113 r (\u2248\u02e2-sym p)\n-}\n\n\u2248\u02e2-trans : \u2200 {P Q} \u2192 Transitive (_\u2248\u02e2_ {P} {Q})\n\u2248\u02e2-trans \u2248-end q = q\n\u2248\u02e2-trans (\u2248-sendL m x) (\u2248-sendL .m x\u2081) = \u2248-sendL m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-sendR m x) (\u2248-sendR .m x\u2081) = \u2248-sendR m (\u2248\u02e2-trans x x\u2081)\n\u2248\u02e2-trans (\u2248-recvL x) (\u2248-recvL x\u2081) = \u2248-recvL (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\u2248\u02e2-trans (\u2248-recvR x) (\u2248-recvR x\u2081) = \u2248-recvR (\u03bb m \u2192 \u2248\u02e2-trans (x m) (x\u2081 m))\n\ndata LR : \u2605 where\n  `L `R : LR\n\n[L:_R:_] : \u2200 {a} {A : \u2605_ a} (l r : A) \u2192 LR \u2192 A \n[L: l R: r ] `L = l\n[L: l R: r ] `R = r\n\n_\u2295\u1d3e_ : (l r : Proto) \u2192 Proto\nl \u2295\u1d3e r = \u03a3\u1d3e LR [L: l R: r ]\n\n_&\u1d3e_ : (l r : Proto) \u2192 Proto\nl &\u1d3e r = \u03a0\u1d3e LR [L: l R: r ]\n\n_>>\u1d9c_ : (P : Com) \u2192 (Proto \u2192 Proto) \u2192 Com\nP >>\u1d9c S = record P { P = \u03bb m \u2192 S (Com.P P m) }\n\nmodule Equivalences\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  record Equiv {A B : \u2605}(f : A \u2192 B) : \u2605 where\n    field\n      linv : B \u2192 A\n      is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n      rinv : B \u2192 A\n      is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n  id\u1d31 : \u2200 {A} \u2192 Equiv {A} id\n  id\u1d31 = record\n            { linv = id\n            ; is-linv = \u03bb _ \u2192 refl\n            ; rinv = id\n            ; is-rinv = \u03bb _ \u2192 refl\n            }\n\n  module _ {A B C}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Equiv g \u2192 Equiv f \u2192 Equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 cong F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 cong g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Equiv g\u1d31\n        module F = Equiv f\u1d31\n\n  _\u2243_ : \u2605 \u2192 \u2605 \u2192 \u2605\n  A \u2243 B = \u03a3 (A \u2192 B) Equiv\n\n  module _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    \u03a3-ext : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 subst B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    \u03a3-ext refl = cong (_,_ _)\n\ndata ViewProc : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200 M(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewProc (\u03a3\u1d3e M P) (m , p)\n  recv : \u2200 M(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewProc (\u03a0\u1d3e M P) p\n  end  : ViewProc end _\n\nview-proc : \u2200 P (p : \u27e6 P \u27e7) \u2192 ViewProc P p\nview-proc end      _       = end\nview-proc (\u03a0\u1d3e M P) p       = recv _ _ p\nview-proc (\u03a3\u1d3e M P) (m , p) = send _ _ m p\n\ndata ViewCom : \u2200 P \u2192 \u27e6 com P \u27e7 \u2192 \u2605\u2081 where\n  send : \u2200{M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 ViewCom (\u03a3\u1d9c M P) (m , p)\n  recv : \u2200{M}(P : M \u2192 Proto)(p : ((m : M) \u2192 \u27e6 P m \u27e7)) \u2192 ViewCom (\u03a0\u1d9c M P) p\n\nview-com : \u2200 P (p : \u27e6 com P \u27e7) \u2192 ViewCom P p\nview-com (\u03a0\u1d9c M P) p       = recv _ p\nview-com (\u03a3\u1d9c M P) (m , p) = send _ m p\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open Equivalences funExt\n\n  _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b\u1d3e Q       = Q\n  \u03a0\u1d3e M P \u214b\u1d3e Q       = \u03a0\u1d3e M \u03bb m \u2192 P m \u214b\u1d3e Q\n  P      \u214b\u1d3e end     = P\n  P      \u214b\u1d3e \u03a0\u1d3e M  Q = \u03a0\u1d3e M \u03bb m \u2192 P \u214b\u1d3e Q m\n  \u03a3\u1d3e M P \u214b\u1d3e \u03a3\u1d3e M' Q = \u03a3\u1d3e (M \u228e M') [inl: (\u03bb m \u2192 P m \u214b\u1d3e \u03a3\u1d3e M' Q)\n                                  ,inr: (\u03bb m' \u2192 \u03a3\u1d3e M P \u214b\u1d3e Q m') ]\n\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b\u1d3e \u03a3\u1d3e M' Q \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m  , p)\n    sendR' : \u2200 {M M'}(P : M \u2192 Proto)(Q : M' \u2192 Proto)(m' : M')(p : \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m' , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7)) \u2192 View-\u214b (\u03a0\u1d3e M P) Q p\n    recvR' : \u2200 {M M'} (P : M \u2192 Proto) (Q : M' \u2192 Proto)(p : (m' : M') \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q m' \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    endR   : \u2200 {M} (P : M \u2192 Proto) (p : \u27e6 \u03a3\u1d3e M P \u27e7) \u2192 View-\u214b (\u03a3\u1d3e M P) end p\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (\u03a0\u1d3e M P) Q p = recvL' P Q p\n  view-\u214b (\u03a3\u1d3e M P) end p = endR P p\n  view-\u214b (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = recvR' P Q p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inl x , p) = sendL' P Q x p\n  view-\u214b (com (mk Out M P)) (com (mk Out M' Q)) (inr y , p) = sendR' P Q y p\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7 \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-assoc end      Q        R         s                 = s\n  \u214b\u1d3e-assoc (\u03a0\u1d3e _ P) Q        R         s m               = \u214b\u1d3e-assoc (P m) _ _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) end      R         s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a0\u1d3e _ Q) R         s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) _ (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) end       s                 = s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a0\u1d3e M R)  s m               = \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) (s m)\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inl m , s)       = inl (inl m) , \u214b\u1d3e-assoc (P m) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inl m) , s) = inl (inr m) , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (Q m) (\u03a3\u1d3e _ R) s\n  \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (\u03a3\u1d3e Mr R) (inr (inr m) , s) = inr m       , \u214b\u1d3e-assoc (\u03a3\u1d3e _ P) (\u03a3\u1d3e _ Q) (R m) s\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end      p = p\n  \u214b\u1d3e-rend (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend (P m) (p m)\n  \u214b\u1d3e-rend (\u03a3\u1d3e _ _) p = p\n\n  \u214b\u1d3e-rend' : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b\u1d3e end \u27e7\n  \u214b\u1d3e-rend' end      p = p\n  \u214b\u1d3e-rend' (\u03a0\u1d3e _ P) p = \u03bb m \u2192 \u214b\u1d3e-rend' (P m) (p m)\n  \u214b\u1d3e-rend' (\u03a3\u1d3e _ _) p = p\n\n  {-\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n  -}\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (\u03a3\u1d3e M P) m p = inr m , p\n  \u214b\u1d3e-sendR (\u03a0\u1d3e M P) m p = \u03bb x \u2192 \u214b\u1d3e-sendR (P x) m (p x)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end      m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (\u03a0\u1d3e M Q) m p = \u03bb m' \u2192 \u214b\u1d3e-sendL (Q m') m {!p!}\n  \u214b\u1d3e-sendL         (\u03a3\u1d3e M Q) m p = {!!} , p\n\n  {-\n  \u214b\u1d3e-sendR : \u2200 {M} P Q \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7 \u2192 (m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7\n  \u214b\u1d3e-sendR P Q s m = {!!}\n  -}\n\n  \u214b\u1d3e-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e \u03a0\u1d3e M Q \u27e7\n  \u214b\u1d3e-recvR end      Q s = s\n  \u214b\u1d3e-recvR (\u03a0\u1d3e M P) Q s = \u03bb x \u2192 \u214b\u1d3e-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b\u1d3e-recvR (\u03a3\u1d3e M P) Q s = s\n\n  mutual\n    \u214b\u1d3e-! : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-! P Q p = {!!}\n\n    \u214b\u1d3e-!-view : \u2200 {P Q} {pq : \u27e6 P \u214b\u1d3e Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-!-view (sendL' P Q m p) = \u214b\u1d3e-sendR (\u03a3\u1d3e _ Q) m (\u214b\u1d3e-! (P m) (com (mk Out _ Q)) p)\n    \u214b\u1d3e-!-view (sendR' P Q m' p) = {!\u214b\u1d3e-sendL (\u03a3\u1d3e _ P) m'!}\n    \u214b\u1d3e-!-view (recvL' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (recvR' P Q pq) = {!!}\n    \u214b\u1d3e-!-view (endL Q pq) = {!!}\n    \u214b\u1d3e-!-view (endR P pq) = {!!}\n\n  {-\n  end Q p = \u214b\u1d3e-rend' Q p\n  {-\n  \u214b\u1d3e-! end Q p = \u214b\u1d3e-rend' Q p\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) end p x = \u214b\u1d3e-! (P x) end (p x)\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 {!\u214b\u1d3e-! (\u03a0 !}\n  \u214b\u1d3e-! (\u03a0\u1d3e M P) (\u03a3\u1d3e M' Q) p = \u03bb m \u2192 \u214b\u1d3e-! (P m) (com (mk Out M' Q)) (p m)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) end       p = p\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) p = \u03bb m' \u2192 \u214b\u1d3e-! (com (mk Out M P)) (Q m') (p m')\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , p) = inr m , (\u214b\u1d3e-! (P m) (com (mk Out M' Q)) p)\n  \u214b\u1d3e-! (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , p) = inl m , (\u214b\u1d3e-! (com (mk Out M P)) (Q m) p)\n\n  {-\n  \u214b\u1d3e-apply : \u2200 P Q \u2192 \u27e6 \u214b\u1d3e P Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply end      Q         s           p       = s\n  \u214b\u1d3e-apply (\u03a0\u1d3e M P) Q         s           (m , p) = \u214b\u1d3e-apply (P m) Q (s m) p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) end       s           p       = _\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a0\u1d3e M' Q) s           p m'    = \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m') (s m') p\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inl m , s) p       = \u214b\u1d3e-apply (P m) (\u03a3\u1d3e M' Q) s (p m)\n  \u214b\u1d3e-apply (\u03a3\u1d3e M P) (\u03a3\u1d3e M' Q) (inr m , s) p       = m , \u214b\u1d3e-apply (\u03a3\u1d3e M P) (Q m) s p\n  -}\n\n  {-\nmodule V4\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  mutual\n    _\u214b\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end   \u214b\u1d3e Q     = Q\n    P     \u214b\u1d3e end   = P\n    com P \u214b\u1d3e com Q = \u03a3\u1d3e LR (P \u214b\u1d9c Q)\n\n    _\u214b\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `L = P\u1d38 \u214b\u1d9cL P\u1d3f\n    (P\u1d38 \u214b\u1d9c P\u1d3f) `R = P\u1d38 \u214b\u1d9cR P\u1d3f\n\n    _\u214b\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) \u214b\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m \u214b\u1d3e com Q)\n\n    _\u214b\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P \u214b\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P \u214b\u1d3e P\u1d3f m)\n\n  mutual\n    _ox\u1d3e_ : Proto \u2192 Proto \u2192 Proto\n    end    ox\u1d3e Q      = Q\n    P      ox\u1d3e end    = P\n    com P\u1d38 ox\u1d3e com P\u1d3f = \u03a0\u1d3e LR (P\u1d38 ox\u1d9c P\u1d3f)\n\n    _ox\u1d9c_ : Com \u2192 Com \u2192 LR \u2192 Proto\n    (P\u1d38 ox\u1d9c P\u1d3f) `L = P\u1d38 ox\u1d9cL P\u1d3f\n    (P\u1d38 ox\u1d9c P\u1d3f) `R = P\u1d38 ox\u1d9cR P\u1d3f\n\n    _ox\u1d9cL_ : Com \u2192 Com \u2192 Proto\n    (mk q\u1d38 M\u1d38 P\u1d38) ox\u1d9cL Q = com' q\u1d38 M\u1d38 (\u03bb m \u2192 P\u1d38 m ox\u1d3e com Q)\n\n    _ox\u1d9cR_ : Com \u2192 Com \u2192 Proto\n    P ox\u1d9cR (mk q\u1d3f M\u1d3f P\u1d3f) = com' q\u1d3f M\u1d3f (\u03bb m \u2192 com P ox\u1d3e P\u1d3f m)\n\n  data Viewox : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u2605\u2081 where\n    com  : \u2200{P Q}(p : \u27e6 \u03a0\u1d3e LR (P ox\u1d9c Q) \u27e7) \u2192 Viewox (com P) (com Q) p\n    endL : \u2200{Q}(p : \u27e6 Q \u27e7) \u2192 Viewox end Q p\n    endR : \u2200{P}(p : \u27e6 com P \u27e7) \u2192 Viewox (com P) end p\n\n  viewox : \u2200 P Q (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 Viewox P Q p\n  viewox end Q p = endL p\n  viewox (com P) end p = endR p\n  viewox (com P) (com Q) p = com p\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M} (P : M \u2192 Proto) Q R (m : M)(p : \u27e6 P m \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a3\u1d3e M P) (com Q) R (`L , m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b\u1d3e com Q \u27e7))(q : \u27e6 dual (com Q) \u214b\u1d3e R \u27e7)\n             \u2192 View-\u2218 (\u03a0\u1d3e M P) (com Q) R (`L , p) q\n    sendRR : \u2200 {M} P Q (R : M \u2192 Proto)(m : M)(p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a3\u1d3e M R) p (`R , m , q)\n    recvRR : \u2200 {M} P Q (R : M \u2192 Proto)\n               (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : (m : M) \u2192 \u27e6 dual (com Q) \u214b\u1d3e R m \u27e7)\n             \u2192 View-\u2218 (com P) (com Q) (\u03a0\u1d3e M R) p (`R , q)\n\n    sendR-recvL : \u2200 {M} P (Q : M \u2192 Proto) R (m : M)(p : \u27e6 com P \u214b\u1d3e Q m \u27e7)(q : (m : M) \u2192 \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a3\u1d3e M Q) (com R) (`R , m , p) (`L , q)\n    recvR-sendL : \u2200 {M} P (Q : M \u2192 Proto) R (p : (m : M) \u2192 \u27e6 com P \u214b\u1d3e Q m \u27e7)(m : M)(q : \u27e6 dual (Q m) \u214b\u1d3e com R \u27e7)\n             \u2192 View-\u2218 (com P) (\u03a0\u1d3e M Q) (com R) (`R , p) (`L , m , q)\n\n    endL : \u2200 Q R\n            \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7)\n            \u2192 View-\u2218 end Q R q qr\n\n    endM  : \u2200 P R\n            \u2192 (p : \u27e6 com P \u27e7)(r : \u27e6 R \u27e7)\n            \u2192 View-\u2218 (com P) end R p r\n    endR : \u2200 P Q\n            \u2192 (p : \u27e6 com P \u214b\u1d3e com Q \u27e7)(q : \u27e6 dual (com Q) \u27e7)\n            \u2192 View-\u2218 (com P) (com Q) end p q\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7)(qr : \u27e6 dual Q \u214b\u1d3e R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvL' ._ _ _)   = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) (endR ._ _)       = endR _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = sendRR _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (endR ._ ._)      = endR _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (endR _ _)       _                 = endM _ _ _ _\n\n  \u214b\u1d3e-rend : \u2200 P \u2192 \u27e6 P \u214b\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b\u1d3e-rend end     = id\n  \u214b\u1d3e-rend (com _) = id\n\n  \u214b\u1d3e-rend-equiv : \u2200 P \u2192 Equiv (\u214b\u1d3e-rend P)\n  \u214b\u1d3e-rend-equiv end     = id\u1d31\n  \u214b\u1d3e-rend-equiv (com _) = id\u1d31\n\n  \u214b\u1d3e-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  \u214b\u1d3e-sendR end     m p = m , p\n  \u214b\u1d3e-sendR (com P) m p = `R , (m , p)\n\n  \u214b\u1d3e-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7 \u2192 \u27e6 \u03a3\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-sendL {P = P} end     m p = m , \u214b\u1d3e-rend (P m) p\n  \u214b\u1d3e-sendL         (com _) m p = `L , (m , p)\n\n  \u214b\u1d3e-recvR : \u2200 {M}P{Q : M \u2192 Proto} \u2192 ((m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7) \u2192 \u27e6 P \u214b\u1d3e com' In M Q \u27e7\n  \u214b\u1d3e-recvR end     f = f\n  \u214b\u1d3e-recvR (com _) f = `R , f\n\n  \u214b\u1d3e-recvL : \u2200 {M}{P : M \u2192 Proto}Q \u2192 ((m : M) \u2192 \u27e6 P m \u214b\u1d3e Q \u27e7) \u2192 \u27e6 \u03a0\u1d3e M P \u214b\u1d3e Q \u27e7\n  \u214b\u1d3e-recvL {P = P} end     f x = \u214b\u1d3e-rend (P x) (f x)\n  \u214b\u1d3e-recvL         (com _) f = `L , f\n\n  ox\u1d3e-rend : \u2200 P \u2192 \u27e6 P ox\u1d3e end \u27e7  \u2192 \u27e6 P \u27e7\n  ox\u1d3e-rend end     = id\n  ox\u1d3e-rend (com x) = id\n\n  ox\u1d3e-sendR : \u2200 {M P}{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b\u1d3e Q m \u27e7 \u2192 \u27e6 P \u214b\u1d3e com' Out M Q \u27e7\n  ox\u1d3e-sendR {P = end} m p = m , p\n  ox\u1d3e-sendR {P = com x} m p = `R , (m , p)\n\n  \u214b\u1d3e-id : \u2200 P \u2192 \u27e6 dual P \u214b\u1d3e P \u27e7\n  \u214b\u1d3e-id end = _\n  \u214b\u1d3e-id (com (mk In M P))  = `R , \u03bb m \u2192 \u214b\u1d3e-sendL (P m) m (\u214b\u1d3e-id (P m))\n  \u214b\u1d3e-id (com (mk Out M P)) = `L , \u03bb m \u2192 \u214b\u1d3e-sendR (dual (P m)) m (\u214b\u1d3e-id (P m))\n\n  module _ where\n    \u214b\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 Q \u214b\u1d3e P \u27e7\n    \u214b\u1d3e-comm = \u03bb P Q \u2192 to P Q , equiv P Q\n      where\n      to : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 Q \u214b\u1d3e P \u27e7\n      to end end pq = pq\n      to end (com x) pq = pq\n      to (com x) end pq = pq\n      to (com (mk In M P)) (com x\u2081) (`L , pq) = `R , (\u03bb m \u2192 to (P m) (com x\u2081) (pq m))\n      to (com (mk Out M P)) (com x\u2081) (`L , m , pq) = `R , m , to (P m) (com x\u2081) pq\n      to (com x) (com (mk In M P)) (`R , pq) = `L , (\u03bb m \u2192 to (com x) (P m) (pq m))\n      to (com x) (com (mk Out M P)) (`R , m , pq) = `L , m , to (com x) (P m) pq\n\n      toto : \u2200 P Q (x : \u27e6 P \u214b\u1d3e Q \u27e7) \u2192 to Q P (to P Q x) \u2261 x\n      toto end end x = refl\n      toto end (com (mk io M P)) x\u2081 = refl\n      toto (com (mk io M P)) end x\u2081 = refl\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' In M\u2081 P\u2081) (pq x))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`L , pq) = \u03a3-ext refl (funExt \u03bb x \u2192 toto (P x) (com' Out M\u2081 P\u2081) (pq x))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' In M\u2081 P\u2081) pq))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`L , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (P m) (com' Out M\u2081 P\u2081) pq))\n      toto (com (mk In M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' In M P) (P\u2081 x) (pq x)))\n      toto (com (mk In M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' In M P) (P\u2081 m) pq))\n      toto (com (mk Out M P)) (com (mk In M\u2081 P\u2081)) (`R , pq) = \u03a3-ext refl (funExt (\u03bb x \u2192 toto (com' Out M P) (P\u2081 x) (pq x)))\n      toto (com (mk Out M P)) (com (mk Out M\u2081 P\u2081)) (`R , m , pq) = \u03a3-ext refl (\u03a3-ext refl (toto (com' Out M P) (P\u2081 m) pq))\n\n      equiv : \u2200 P Q \u2192 Equiv (to P Q)\n      equiv P Q = record { linv = to Q P ; is-linv = toto P Q ; rinv = to Q P ; is-rinv = toto Q P }\n\n  \u214b\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) \u214b\u1d3e R \u27e7 \u2243 \u27e6 P \u214b\u1d3e (Q \u214b\u1d3e R) \u27e7\n  \u214b\u1d3e-assoc P Q R = {!!}\n\n  \u214b\u1d3e-ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2243 \u27e6 dual P ox\u1d3e dual Q \u22a5\u27e7\n  \u214b\u1d3e-ox\u1d3e P Q = {!!}\n  \n  ox\u1d3e-comm : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2243 \u27e6 Q ox\u1d3e P \u27e7\n  ox\u1d3e-comm P Q = {!!}\n\n  ox\u1d3e-assoc : \u2200 P Q R \u2192 \u27e6 (P ox\u1d3e Q) ox\u1d3e R \u27e7 \u2243 \u27e6 P ox\u1d3e (Q ox\u1d3e R) \u27e7\n  ox\u1d3e-assoc P Q R = {!!}\n\n  comma\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  comma\u1d3e end      Q        p       q = q\n  comma\u1d3e (com _)  end      p       q = p\n  comma\u1d3e (\u03a3\u1d3e M P) (com Q)  (m , p) q `L = m , comma\u1d3e (P m) (com Q) p q\n  comma\u1d3e (\u03a0\u1d3e M P) (com Q)  p       q `L = \u03bb m \u2192 comma\u1d3e (P m) (com Q) (p m) q\n  comma\u1d3e (com P)  (\u03a3\u1d3e M Q) p (m , q) `R = m , comma\u1d3e (com P) (Q m) p q\n  comma\u1d3e (com P)  (\u03a0\u1d3e M Q) p       q `R = \u03bb m \u2192 comma\u1d3e (com P) (Q m) p (q m)\n\n  \u00d7\u2192ox\u1d3e : \u2200 P Q \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P ox\u1d3e Q \u27e7\n  \u00d7\u2192ox\u1d3e P Q (p , q) = comma\u1d3e P Q p q\n\n  ox\u1d3e-fst : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7\n  ox\u1d3e-fst end      Q       pq = _\n  ox\u1d3e-fst (com _)  end     pq = pq\n  ox\u1d3e-fst (\u03a0\u1d3e M P) (com Q) pq = \u03bb m \u2192 ox\u1d3e-fst (P m) (com Q) (pq `L m)\n  ox\u1d3e-fst (\u03a3\u1d3e M P) (com Q) pq = \u00d7-map id (\u03bb {x} \u2192 ox\u1d3e-fst (P x) (com Q)) (pq `L)\n\n  ox\u1d3e-snd : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 Q \u27e7\n  ox\u1d3e-snd end     Q         pq = pq\n  ox\u1d3e-snd (com _) end       pq = _\n  ox\u1d3e-snd (com P) (\u03a0\u1d3e M' Q) pq = \u03bb m' \u2192 ox\u1d3e-snd (com P) (Q m') (pq `R m')\n  ox\u1d3e-snd (com P) (\u03a3\u1d3e M' Q) pq = \u00d7-map id (\u03bb {m'} \u2192 ox\u1d3e-snd (com P) (Q m')) (pq `R)\n\n  ox\u1d3e\u2192\u00d7 : \u2200 P Q \u2192 \u27e6 P ox\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  ox\u1d3e\u2192\u00d7 P Q p = ox\u1d3e-fst P Q p , ox\u1d3e-snd P Q p\n\n  ox\u1d3e-comma-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-fst P Q (comma\u1d3e P Q p q) \u2261 p\n  ox\u1d3e-comma-fst end      Q       p  q = refl\n  ox\u1d3e-comma-fst (com P)  end     p  q = refl\n  ox\u1d3e-comma-fst (com P)  (com Q) p  q with view-com P p\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | send P m p = cong (_,_ m) (ox\u1d3e-comma-fst (P m) (com Q) p q)\n  ox\u1d3e-comma-fst (com ._) (com Q) ._ q | recv P p   = funExt \u03bb m \u2192 ox\u1d3e-comma-fst (P m) (com Q) (p m) q\n\n  ox\u1d3e-comma-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 ox\u1d3e-snd P Q (comma\u1d3e P Q p q) \u2261 q\n  ox\u1d3e-comma-snd end     Q        p  q = refl\n  ox\u1d3e-comma-snd (com P) end      p  q = refl\n  ox\u1d3e-comma-snd (com P) (com Q)  p  q with view-com Q q\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | send Q m q = cong (_,_ m) (ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p q)\n  ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a0\u1d3e _ P) (Q m) p (q m)\n  ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (com ._) p ._ | recv Q q   = funExt \u03bb m \u2192 ox\u1d3e-comma-snd (\u03a3\u1d3e _ P) (Q m) p (q m)\n\n  {-\n  ox\u1d3e\u2192\u00d7-rinv : \u2200 {P Q} (p : \u27e6 P ox\u1d3e Q \u27e7) \u2192 \u00d7\u2192ox\u1d3e P Q (ox\u1d3e\u2192\u00d7 P Q p) \u2261 p\n  ox\u1d3e\u2192\u00d7-rinv {P} {Q} p with viewox P Q p\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk In M\u2081 P\u2081} .p = funExt \u03bb { `L \u2192 funExt \u03bb m \u2192 {!ox\u1d3e\u2192\u00d7-rinv {P m} {\u03a0\u1d3e _ P\u2081} ?!} ; `R \u2192 {!!} }\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk In M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk In M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | com {mk Out M P} {mk Out M\u2081 P\u2081} .p = {!!}\n  ox\u1d3e\u2192\u00d7-rinv p | endL .p = refl\n  ox\u1d3e\u2192\u00d7-rinv p | endR {P} .p = refl\n\n  \u00d7\u2192ox\u1d3e-equiv : \u2200 P Q \u2192 Equiv (\u00d7\u2192ox\u1d3e P Q)\n  \u00d7\u2192ox\u1d3e-equiv P Q = record { linv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-linv = \u03bb { (x , y) \u2192 cong\u2082 _,_ (ox\u1d3e-comma-fst P Q x y) (ox\u1d3e-comma-snd P Q x y) }\n                           ; rinv = ox\u1d3e\u2192\u00d7 P Q\n                           ; is-rinv = ox\u1d3e\u2192\u00d7-rinv {P} {Q} }\n  -}\n\n  -- left-biased strategy\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n  par P Q p q = par-view (view-proc P p)\n    where par-view : \u2200 {P} {p : \u27e6 P \u27e7} \u2192 ViewProc P p \u2192 \u27e6 P \u214b\u1d3e Q \u27e7\n          par-view (send M P m p) = \u214b\u1d3e-sendL Q m (par (P m) Q p q)\n          par-view (recv M P p)   = \u214b\u1d3e-recvL Q \u03bb m \u2192 par (P m) Q (p m) q\n          par-view end            = q\n\n  \u214b\u1d3e-apply : \u2200 {P Q} \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply {P} {Q} pq p with view-\u214b P Q pq\n  \u214b\u1d3e-apply ._ p       | sendL' P Q m pq = \u214b\u1d3e-apply {P m} pq (p m)\n  \u214b\u1d3e-apply ._ p       | sendR' P Q m pq = m , \u214b\u1d3e-apply {com P} {Q m} pq p\n  \u214b\u1d3e-apply ._ (m , p) | recvL' P Q pq   = \u214b\u1d3e-apply {P m} (pq m) p\n  \u214b\u1d3e-apply ._ p       | recvR' P Q pq   = \u03bb m \u2192 \u214b\u1d3e-apply {com P} {Q m} (pq m) p\n  \u214b\u1d3e-apply pq p       | endL Q .pq        = pq\n  \u214b\u1d3e-apply pq p       | endR P .pq        = _\n\n  \u214b\u1d3e-apply' : \u2200 {P Q} \u2192 \u27e6 dual P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b\u1d3e-apply' {P} {Q} pq p = \u214b\u1d3e-apply {dual P} {Q} pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound funExt (dual-involutive P))) p)\n\n  \u214b\u1d3e-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b\u1d3e Q \u27e7 \u2192 \u27e6 dual Q \u214b\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n  \u214b\u1d3e-\u2218 P Q R pq qr = \u214b\u1d3e-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b\u1d3e-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{qr : \u27e6 dual Q \u214b\u1d3e R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b\u1d3e R \u27e7\n    \u214b\u1d3e-\u2218-view (sendLL P Q R m p qr)     = \u214b\u1d3e-sendL R m (\u214b\u1d3e-\u2218 (P m) (com Q) R p qr)\n    \u214b\u1d3e-\u2218-view (recvLL P Q R p qr)       = \u214b\u1d3e-recvL R (\u03bb m \u2192 \u214b\u1d3e-\u2218 (P m) (com Q) R (p m) qr)\n    \u214b\u1d3e-\u2218-view (sendRR P Q R m pq q)     = \u214b\u1d3e-sendR (com P) m (\u214b\u1d3e-\u2218 (com P) (com Q) (R m) pq q)\n    \u214b\u1d3e-\u2218-view (recvRR P Q R pq q)       = \u214b\u1d3e-recvR (com P) (\u03bb m\u2081 \u2192 \u214b\u1d3e-\u2218 (com P) (com Q) (R m\u2081) pq (q m\u2081))\n    \u214b\u1d3e-\u2218-view (sendR-recvL P Q R m p q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) p (q m)\n    \u214b\u1d3e-\u2218-view (recvR-sendL P Q R p m q) = \u214b\u1d3e-\u2218 (com P) (Q m) (com R) (p m) q\n    \u214b\u1d3e-\u2218-view (endL Q R pq qr)          = \u214b\u1d3e-apply' {Q} {R} qr pq\n    \u214b\u1d3e-\u2218-view (endM P R pq qr)          = par (com P) R pq qr\n    \u214b\u1d3e-\u2218-view (endR P Q pq qr)          = \u214b\u1d3e-apply {com Q} {com P} (fst (\u214b\u1d3e-comm (com P) (com Q)) pq) qr\n\n  ox\u1d3e-map : \u2200 P Q R S \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P ox\u1d3e R \u27e7 \u2192 \u27e6 Q ox\u1d3e S \u27e7\n  ox\u1d3e-map P Q R S f g p = comma\u1d3e Q S (f (ox\u1d3e-fst P R p)) (g (ox\u1d3e-snd P R p))\n\n  {-\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b\u1d3e Q) ox\u1d3e R \u27e7 \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (ox\u1d3e-fst (P \u214b\u1d3e Q) R pqr) (ox\u1d3e-snd (P \u214b\u1d3e Q) R pqr)\n  -}\n\n  {-\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b\u1d3e Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n  switchL' P Q R pq r = {!switchL-view {!!}!}\n   where\n    switchL-view : \u2200 {P Q R}{pq : \u27e6 P \u214b\u1d3e Q \u27e7}{r : \u27e6 R \u27e7} \u2192 \u27e6 P \u214b\u1d3e (Q ox\u1d3e R) \u27e7\n    switchL-view {P} {Q} {R} {p\u214bq} {r} with view-\u214b P Q p\u214bq | view-proc R r\n    switchL-view | sendL' P\u2081 Q\u2081 m p | vr = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | send M\u2081 P\u2082 m\u2081 p\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | sendR' P\u2081 Q\u2081 m p | end = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvL' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | send M\u2081 P\u2082 m p\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | recv M\u2081 P\u2082 r\u2081 = {!!}\n    switchL-view | recvR' P\u2081 Q\u2081 p | end = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | send M P\u2081 m p = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | recv M P\u2081 r\u2081 = {!!}\n    switchL-view | endL Q\u2081 pq\u2081 | end = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | send M P\u2082 m p = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | recv M P\u2082 r\u2081 = {!!}\n    switchL-view | endR P\u2081 pq\u2081 | end = {!!}\n    {-\n    switchL-view {R = R}{r = r} | sendL' P Q m p = \u214b\u1d3e-sendL {!!} m (switchL' {!!} {!!} {!!} p r)\n    switchL-view {R = R}{r = r} | sendR' P Q m p with view-proc R r\n    switchL-view | sendR' {M\u2081} P\u2081 Q m\u2081 p\u2081 | send M P m p = {!!}\n    switchL-view | sendR' {M\u2081} P\u2081 Q m p | recv M P r = {!!}\n    switchL-view | sendR' {M} P Q m p | end = {!!}\n    -- {!\u214b\u1d3e-map (com P) (com P) (Q m ox\u1d3e R) (\u03a3\u1d3e M Q ox\u1d3e R) id (ox\u1d3e-map (Q m) (\u03a3\u1d3e M Q) R R (_,_ m) id) (switchL-view (com P) (Q m) R p r)!}\n    switchL-view {R = R}{r = r} | recvL' P Q p = \u214b\u1d3e-recvL (com Q ox\u1d3e R) \u03bb m \u2192 switchL' (P m) (com Q) R (p m) r\n    switchL-view {R = R}{r = r} | recvR' (\u03a0\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | recvR' (\u03a3\u1d9c M' P) Q p = {!!} -- \u214b\u1d3e-map (com P) (com P) {!!} (\u03a0\u1d3e M Q ox\u1d3e R) id {!!} {!!}\n    switchL-view {R = R}{r = r} | endL Q p\u214bq = comma\u1d3e Q R p\u214bq r\n    switchL-view {R = R}{r = r} | endR P p\u214bq = par (com P) R p\u214bq r\n  -}\n  -}\n\n  \u2295\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295\u1d3e R \u27e7 \u2192 \u27e6 Q \u2295\u1d3e S \u27e7\n  \u2295\u1d3e-map f g (`L , pr) = `L , f pr\n  \u2295\u1d3e-map f g (`R , pr) = `R , g pr\n\n  &\u1d3e-map : \u2200 {P Q R S} \u2192 (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P &\u1d3e R \u27e7 \u2192 \u27e6 Q &\u1d3e S \u27e7\n  &\u1d3e-map f g p `L = {!!}\n  &\u1d3e-map f g p `R = {!!}\n  -}\n\n  {-\n\n    {-\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = Process (dual P) \u2192 Process Q\n\n{-\nsim& : \u2200 {PA PB PAB}  \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ]\u1d9c \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& end PA PB = PB\nsim& (com P&) (comR s\u2080) (comR s\u2081) = comR (sim&R P& s\u2080 s\u2081)\nsim& (&-comm P&) PA PB = sim& P& PB PA\n\nsim&R (\u03a0&  In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 m) (s\u2081 m)\nsim&R (\u03a0&  Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 m) s\u2081)\nsim&R (\u03a0\u2610& In  P&)  (recv s\u2080) (recv s\u2081)   = recv \u03bb m \u2192 sim& (P& m) (s\u2080 [ m ]) (s\u2081 m)\nsim&R (\u03a0\u2610& Out P&)  (recv s\u2080) (send m s\u2081) = send m (sim& (P& m) (s\u2080 [ m ]) s\u2081)\n-}\n{-\nsim&-assoc : \u2200 {PA PB PC PAB PBC PABC}\n               (PAB& : [ PA & PB \u2261 PAB ])\n               (PAB&PC : [ PAB & PC \u2261 PABC ])\n               (PBC& : [ PB & PC \u2261 PBC ])\n               (PA&PBC : [ PA & PBC \u2261 PABC ])\n               (sA : Sim end PA)(sB : Sim end PB)(sC : Sim end PC)\n             \u2192 sim& PA&PBC sA (sim& PBC& sB sC) \u2261 sim& PAB&PC (sim& PAB& sA sB) sC\nsim&-assoc PAB& PAB&PC PBC& PA&PBC sA sB sC = {!!}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0\u1d3e M P) = comR (recv \u03bb m \u2192 comL  (send m (sim-id (P m))))\nsim-id (\u03a3\u1d3e M P) = comL  (recv \u03bb m \u2192 comR (send m (sim-id (P m))))\n\nid\u02e2 : \u2200 {P P'} \u2192 Dual P P' \u2192 Sim P P'\nid\u02e2 end = end\nid\u02e2 (\u03a0\u00b7\u03a3 x) = comL (recv (\u03bb m \u2192 comR (send m (id\u02e2 (x m)))))\nid\u02e2 (\u03a3\u00b7\u03a0 x) = comR (recv (\u03bb m \u2192 comL (send m (id\u02e2 (x m)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual (com Q) (com Q') \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\n--sim-compRL : \u2200 {P Q R} \u2192 SimR P Q \u2192 SimL (dual\u1d9c Q) R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (comL  PQ) QR         = comL (sim-compL Q-Q' PQ QR)\nsim-comp Q-Q' (comR PQ) (comL QR)  = sim-compRL Q-Q' PQ QR\nsim-comp Q-Q' (comR PQ) (comR QR) = comR (sim-compR Q-Q' (comR PQ) QR)\nsim-comp ()   (comR PQ) end\nsim-comp end end QR                 = QR\n\nsim-compRL (\u03a0\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ m) QR\nsim-compRL (\u03a3\u00b7\u03a0 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR m)\n{-\nsim-compRL (\u03a0\u2610\u00b7\u03a3 Q-Q') (recv PQ)   (send m QR) = sim-comp (Q-Q' m) (PQ [ m ]) QR\nsim-compRL (\u03a3\u00b7\u03a0\u2610 Q-Q') (send m PQ) (recv QR)   = sim-comp (Q-Q' m) PQ     (QR [ m ])\n-}\n\nsim-compL Q-Q' (recv PQ)   QR = recv \u03bb m \u2192 sim-comp Q-Q' (PQ m) QR\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv \u03bb m \u2192 sim-comp Q-Q' PQ (QR m)\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (comL x) = comR (sim-symL x)\n!\u02e2 (comR x) = comL (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\n!\u02e2-cong : \u2200 {P Q} {s\u2080 s\u2081 : Sim P Q} \u2192 s\u2080 \u2248\u02e2 s\u2081 \u2192 !\u02e2 s\u2080 \u2248\u02e2 !\u02e2 s\u2081\n!\u02e2-cong \u2248-end = \u2248-end\n!\u02e2-cong (\u2248-sendL m p) = \u2248-sendR m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-sendR m p) = \u2248-sendL m (!\u02e2-cong p)\n!\u02e2-cong (\u2248-recvL x) = \u2248-recvR (\u03bb m \u2192 !\u02e2-cong (x m))\n!\u02e2-cong (\u2248-recvR x) = \u2248-recvL (\u03bb m \u2192 !\u02e2-cong (x m))\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process P\nsim-unit (comR (recv P))   = com (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (comR (send m P)) = com (send m (sim-unit P))\nsim-unit end                = end\n\nmutual\n    Sim\u1d3e\u2192SimR : \u2200 {P Q} \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P)) \u2192 SimR (com P) Q\n    Sim\u1d3e\u2192SimR (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimR (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192SimL : \u2200 {P Q} \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q)) \u2192 SimL P (com Q)\n    Sim\u1d3e\u2192SimL (recv s)   = recv \u03bb m \u2192 Sim\u1d3e\u2192Sim (s m)\n    Sim\u1d3e\u2192SimL (send m s) = send m (Sim\u1d3e\u2192Sim s)\n\n    Sim\u1d3e\u2192Sim : \u2200 {P Q} \u2192 Sim end (Sim\u1d3e P Q) \u2192 Sim P Q\n    Sim\u1d3e\u2192Sim {end}           s = s\n    Sim\u1d3e\u2192Sim {com _} {end}   s = !\u02e2 s\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `L (comR s))) = comL (Sim\u1d3e\u2192SimL s)\n    Sim\u1d3e\u2192Sim {com _} {com _} (comR (send `R (comR s))) = comR (Sim\u1d3e\u2192SimR s)\n\nmutual\n    Sim\u2192Sim\u1d3eR : \u2200 {P Q} \u2192 SimR (com P) Q \u2192 ProcessF (Sim end) (Q >>\u1d9c Sim\u1d3e (com P))\n    Sim\u2192Sim\u1d3eR (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eR (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3eL : \u2200 {P Q} \u2192 SimL P (com Q) \u2192 ProcessF (Sim end) (P >>\u1d9c flip Sim\u1d3e (com Q))\n    Sim\u2192Sim\u1d3eL (recv s)   = recv \u03bb m \u2192 Sim\u2192Sim\u1d3e (s m)\n    Sim\u2192Sim\u1d3eL (send m s) = send m (Sim\u2192Sim\u1d3e s)\n\n    Sim\u2192Sim\u1d3e : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim end (Sim\u1d3e P Q)\n    Sim\u2192Sim\u1d3e {end}           s = s\n    Sim\u2192Sim\u1d3e {com _} {end}   s = !\u02e2 s\n    Sim\u2192Sim\u1d3e {com _} {com _} (comL s) = comR (send `L (comR (Sim\u2192Sim\u1d3eL s)))\n    Sim\u2192Sim\u1d3e {com _} {com _} (comR s) = comR (send `R (comR (Sim\u2192Sim\u1d3eR s)))\n\n    {-\nmutual\n    Sim\u1d3e-assocR : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cR Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocR {end}{Q}{R} s = {!!}\n    Sim\u1d3e-assocR {\u03a0\u1d3e M P}{Q}{R}(comL (recv s)) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assocR {\u03a3\u1d3e M P}{Q}{R}(comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assocR {com (mk io M P)} {Q} {mk .In M\u2082 R} (comR (recv s)) = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = com Q} {R m} (s m)))\n    Sim\u1d3e-assocR {com (mk io M P)} {mk io\u2081 M\u2081 Q} {mk .Out M\u2082 R} (comR (send m s)) = comR (send m (Sim\u1d3e-assoc {com (mk io M P)} {com (mk io\u2081 M\u2081 Q)} {R m} s))\n\n    Sim\u1d3e-assocL : \u2200 {P Q R} \u2192 Sim P (Sim\u1d9cL Q R) \u2192 Sim (Sim\u1d3e P (com Q)) (com R)\n    Sim\u1d3e-assocL s = {!!}\n\n    Sim\u1d3e-assoc : \u2200 {P Q R} \u2192 Sim P (Sim\u1d3e Q R) \u2192 Sim (Sim\u1d3e P Q) R\n    Sim\u1d3e-assoc {end}                   s = Sim\u1d3e\u2192Sim s\n    Sim\u1d3e-assoc {com _} {end}           s = s\n    Sim\u1d3e-assoc {com _} {com _} {end}   s = !\u02e2 {!s!}\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (recv s))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comL (send m s)) = comL (send `L (comL (send m (Sim\u1d3e-assoc s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `L (comL (recv s))))   = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocL (s m)))))\n    Sim\u1d3e-assoc {com ._} {com Q} {com R} (comR (send `L (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocL s))))\n    Sim\u1d3e-assoc {com P} {\u03a0\u1d3e M Q} {com R} (comR (send `L (comR (recv s))))\n      = comL (send `R (comL (recv (\u03bb m \u2192 Sim\u1d3e-assoc {Q = Q m} (s m)))))\n    Sim\u1d3e-assoc {com P} {\u03a3\u1d3e M Q} {com R} (comR (send `L (comR (send m s))))\n      = comL (send `R (comL (send m (Sim\u1d3e-assoc {Q = Q m} s))))\n    Sim\u1d3e-assoc {\u03a0\u1d3e M P} {com Q} {com R} (comR (send `R (comL (recv s)))) = comL (send `L (comL (recv (\u03bb m \u2192 Sim\u1d3e-assocR (s m)))))\n    Sim\u1d3e-assoc {\u03a3\u1d3e M P} {com Q} {com R} (comR (send `R (comL (send m s)))) = comL (send `L (comL (send m (Sim\u1d3e-assocR s))))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a0\u1d3e M R} (comR (send `R (comR (recv s))))\n      = comR (recv (\u03bb m \u2192 Sim\u1d3e-assoc {com P} {com Q} {R m} (s m)))\n    Sim\u1d3e-assoc {com P} {com Q} {\u03a3\u1d3e M R} (comR (send `R (comR (send m s))))\n      = comR (send m (Sim\u1d3e-assoc {com P} {com Q} {R m} s))\n-}\n{-\nSim\u1d3e-assoc : \u2200 {P Q R} \u2192 \u27e6 Sim\u1d3e P (Sim\u1d3e Q R) \u27e7 \u2192 \u27e6 Sim\u1d3e (Sim\u1d3e P Q) R \u27e7\nSim\u1d3e-assoc {end}                   s = s\nSim\u1d3e-assoc {com _} {end}           s = s\nSim\u1d3e-assoc {com _} {com _} {end}   s = s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`L , s) = `L , `L , Sim\u1d3e-assoc {com ?} {com Q} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `L , s) = `L , `R , Sim\u1d3e-assoc {com P} {{!com ?!}} {com R} s\nSim\u1d3e-assoc {com P} {com Q} {com R} (`R , `R , s) = {!!}\n-}\n\n{-\nmodule 3-way-trace where\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  {-\n  trace (\u03a0\u2610\u00b7\u03a3 x\u2081) Q-Q' (comR (send x x\u2082)) (comL (recv x\u2083)) Q\u00b7 = {!first (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)!}\n  trace (\u03a3\u00b7\u03a0\u2610 x\u2081) Q-Q' (comR (recv x)) (comL (send x\u2082 x\u2083)) Q\u00b7 = {!first (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)!}\n  -}\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  {-\n  trace P-P' (\u03a0\u2610\u00b7\u03a3 x\u2081) \u00b7P (comR (recv x)) (comL (send x\u2082 x\u2083)) = {!second (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)!}\n  trace P-P' (\u03a3\u00b7\u03a0\u2610 x\u2081) \u00b7P (comR (send x x\u2082)) (comL (recv x\u2083)) = {!second (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))!}\n  -}\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ where\n  trace : \u2200 {B E} \u2192 Sim (Trace B) (Trace E) \u2192 Tele B \u00d7 Tele E\n  trace {end}   {end}   end = _\n  trace {com _} {end}   (comL  (send m s)) = first  (_,_ m) (trace s)\n  trace {end}   {com _} (comR (send m s)) = second (_,_ m) (trace s)\n  trace {com _} {com c} (comL  (send m s)) = first  (_,_ m) (trace {E = com c} s)\n  trace {com c} {com _} (comR (send m s)) = second (_,_ m) (trace {com c} s)\n\n  module _ {P Q} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {B P' Q' E} \u2192 (P'-P : Dual P' P)(Q-Q' : Dual Q Q')(BP : Sim (Trace B) P')(QE : Sim Q' (Trace E))\n       \u2192 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ QE)) \u2261 trace (sim-comp P'-P BP (sim-comp Q-Q' PQ' QE))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  {-\n  Dual-sym-sym (\u03a0\u2610\u00b7\u03a3 x) = cong \u03a0\u2610\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0\u2610 x) = cong \u03a3\u00b7\u03a0\u2610 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u2261 sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' (comL (recv x)) PQ QR = cong (comL \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (x m) PQ QR))\n  sim-comp-assoc P-P' Q-Q' (comL (send m x)) PQ QR = cong (comL \u2218 send m) (sim-comp-assoc P-P' Q-Q' x PQ QR)\n  sim-comp-assoc (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (comR (recv x)) (comL (send m x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' (x m) x\u2082 QR\n  sim-comp-assoc (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (comR (send m x)) (comL (recv x\u2082)) QR = sim-comp-assoc (x\u2081 m) Q-Q' x (x\u2082 m) QR\n  sim-comp-assoc P-P' (\u03a0\u00b7\u03a3 x\u2082) (comR x) (comR (recv x\u2081)) (comL (send m x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) (x\u2081 m) x\u2083\n  sim-comp-assoc P-P' (\u03a3\u00b7\u03a0 x\u2082) (comR x) (comR (send m x\u2081)) (comL (recv x\u2083)) = sim-comp-assoc P-P' (x\u2082 m) (comR x) x\u2081 (x\u2083 m)\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (recv x\u2082)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (x\u2082 m))\n  sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) (comR (send m x\u2082)) = cong (comR \u2218 send m) (sim-comp-assoc P-P' Q-Q' (comR x) (comR x\u2081) x\u2082)\n  sim-comp-assoc end Q-Q' end PQ QR = refl\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u2261 WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n  {-\n  sim-id-comp : \u2200 {P P' Q}(P-P' : Dual P P')(s : Sim P' Q) \u2192 sim-comp P-P' (id\u02e2 (Dual-sym P-P')) s \u2261 s\n  sim-id-comp end s = refl\n  sim-id-comp (\u03a0\u00b7\u03a3 x) s = {!!}\n  sim-id-comp (\u03a3\u00b7\u03a0 x) s = {!!}\n\n  module _ (A : \u2605) where\n    Test : Proto\n    Test = A \u00d7' end\n\n    s : A \u2192 Sim Test Test\n    s m = comR (send m (comL (send m end)))\n\n    s' : Sim (dual Test) (dual Test)\n    s' = comR (recv (\u03bb m \u2192 comL (recv (\u03bb m\u2081 \u2192 end))))\n\n    prf : \u2200 x \u2192 s x \u2666 sim-id _ \u2261 s x\n    prf x = {!!}\n\n    c-prf : \u2200 x \u2192 sim-id _ \u2666 s x \u2261 s x\n    c-prf x = {!!}\n\n    c-prf' : sim-id _ \u2666 s' \u2261 s'\n    c-prf' = {!!}\n\n    prf' : s' \u2666 sim-id _ \u2261 s'\n    prf' = {!!}\n\n  sim-comp-id : \u2200 {P Q}(s : Sim P Q) \u2192 s \u2666 (sim-id Q) \u2261 s\n  sim-comp-id (comL (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comL (send m x)) = cong (comL \u2218 send m) (sim-comp-id x)\n  sim-comp-id (comR (recv x)) = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-comp-id (x m))\n  sim-comp-id (comR (send m x)) = {!cong (comR \u2218 send m) (sim-comp-id x)!}\n  sim-comp-id end = refl\n  -}\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (comL (recv x))    = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comL (send x x\u2081)) = cong (comL \u2218 send x) (sim-!! x\u2081)\n  sim-!! (comR (recv x))    = cong (comR \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (comR (send x x\u2081)) = cong (comR \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u2248\u02e2 !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (recv s))   (comR (send m s\u2081))  = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (recv s\u2081))    = TODO where postulate TODO : _\n  sim-comp-! Q-Q'       (comL (send m s)) (comR (send m\u2081 s\u2081)) = \u2248\u02e2-trans (\u2248\u02e2-trans (\u2248-sendL m\u2081 (\u2248\u02e2-trans (sim-comp-! Q-Q' (sendL m s) s\u2081) (\u2248-sendR m (\u2248\u02e2-sym (sim-comp-! Q-Q' s s\u2081))))) (\u2248-sendLR m\u2081 m)) (\u2248-sendR m (sim-comp-! Q-Q' s (comR (send m\u2081 s\u2081))))\n  sim-comp-! Q-Q'       (comL (recv s))   (comL (recv s\u2081))    = \u2248-recvR \u03bb m \u2192 sim-comp-! Q-Q' (s m) (comL (recv s\u2081))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (recv s\u2081))    = \u2248-sendR m (sim-comp-! Q-Q' s (comL (recv s\u2081)))\n  sim-comp-! Q-Q'       (comL (send m s)) (comL (send m\u2081 s\u2081)) = \u2248-sendR m (sim-comp-! Q-Q' s (comL (send m\u2081 s\u2081)))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comL (send m s\u2081))  = sim-comp-! (Q-Q' m) (s m) s\u2081\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comL (recv s))   (comL (send m s\u2081))  = \u2248-recvR \u03bb m\u2081 \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (s m\u2081) (comL (send m s\u2081))\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (recv s\u2081))    = \u2248-recvL \u03bb m \u2192 sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) (s\u2081 m)\n  sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s))   (comR (send m s\u2081))  = \u2248-sendL m (sim-comp-! (\u03a0\u00b7\u03a3 Q-Q') (comR (recv s)) s\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comL (recv s\u2081))    = sim-comp-! (Q-Q' m) s (s\u2081 m)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (recv s\u2081))    = \u2248-recvL \u03bb m\u2081 \u2192 sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (s\u2081 m\u2081)\n  sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) (comR (send m\u2081 s\u2081)) = \u2248-sendL m\u2081 (sim-comp-! (\u03a3\u00b7\u03a0 Q-Q') (comR (send m s)) s\u2081)\n  sim-comp-! end        (comL (recv s))   end                 = \u2248-recvR \u03bb m \u2192 sim-comp-! end (s m) end\n  sim-comp-! end        (comL (send m s)) end                 = \u2248-sendR m (sim-comp-! end s end)\n  sim-comp-! end        end               (comR (recv s))     = \u2248-recvL \u03bb m \u2192 sim-comp-! end end (s m)\n  sim-comp-! end        end               (comR (send m s))   = \u2248-sendL m (sim-comp-! end end s)\n  sim-comp-! end        end               end                 = \u2248-end\n\n    {-\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (comR (recv x)) (comL (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (comR (send x\u2081 x\u2082)) (comL (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (comR x) (comR (recv x\u2081))\n    = cong (comL \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (comR x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (comR x) (comR (send x\u2081 x\u2082))\n    = cong (comL \u2218 send x\u2081) (sim-comp-!-end Q-Q' (comR x) x\u2082)\n  sim-comp-!-end end end (comR (recv x)) = cong (comL \u2218 recv) (funExt \u03bb m \u2192 sim-comp-!-end end end (x m))\n  sim-comp-!-end end end (comR (send m x)) = cong (comL \u2218 send m) (sim-comp-!-end end end x)\n  sim-comp-!-end end end end = refl\n\n  open \u2261-Reasoning\n\n  module _ {P Q : Proto} where\n      infix 2 _\u223c_\n      _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n      PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n               \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc funExt Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n-}\n\n  {-\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n  -}\n  -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = \u03a3\u2610\u1d3e Secret  \u03bb s \u2192\n             \u03a0\u1d3e  Guess   \u03bb g \u2192\n             \u03a3\u1d3e  S< s >  \u03bb _ \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\n--    foo : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 Sim (dual P) Q\n--       foo can stop interacting with P as soon as Q is done\n\ndata End? : \u2605 where\n  end continue : End?\n\nEnd?\u1d3e : Proto \u2192 Proto\nEnd?\u1d3e P = \u03a3\u1d3e End? \u03bb { end \u2192 end ; continue \u2192 P }\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    add\u03a3\u1d3e : Proto \u2192 Proto\n    add\u03a3\u1d3e end      = end\n    add\u03a3\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a3\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a3\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a3\u1d3e (P m)\n\n    add\u03a0\u1d3e : Proto \u2192 Proto\n    add\u03a0\u1d3e end      = end\n    add\u03a0\u1d3e (\u03a3\u1d3e M P) = \u03a3\u1d3e (M \u228e A) [ (\u03bb m \u2192 add\u03a0\u1d3e (P m)) , A\u1d3e ]\u2032\n    add\u03a0\u1d3e (\u03a0\u1d3e M P) = \u03a0\u1d3e M \u03bb m \u2192 add\u03a0\u1d3e (P m)\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = add\u03a3\u1d3e Abort\u1d3e\n\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = \u03a3\u1d3e  G  \u03bb g\u02b3 \u2192 -- commitment\n                 \u03a0\u1d3e  \u2124q \u03bb c  \u2192 -- challenge\n                 \u03a3\u1d3e  \u2124q \u03bb s  \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover x y\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb g\u02b3 \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb s  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u03bb _ \u2192\n            end\n\n        Honest-Prover' : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Prover' x S[ y \u2225 _ ]\n          = \u03a3\u2610\u1d3e \u2124q                \u03bb r  \u2192 -- ideal commitment\n            \u03a3\u1d3e  S< g ^ r >        \u03bb { S[ g\u02b3 \u2225 _ ] \u2192 -- real  commitment\n            \u03a0\u1d3e  \u2124q                \u03bb c  \u2192 -- challenge\n            \u03a3\u1d3e  S< r + (c * x) >  \u03bb { S[ s \u2225 _ ]  \u2192 -- response\n            \u03a0\u1d3e  (Dec ((g ^ s) \u2261 (g\u02b3 \u00b7 (y ^ c)))) \u03bb _ \u2192\n            end } }\n\n        Honest-Verifier : ..(x : \u2124q) (y : S< g ^ x >) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S< g ^ x >) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S< g ^ x >) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ab63edb10a739c1368687329458a9cfe8b0449e5","subject":"Cosmetic changes.","message":"Cosmetic changes.\n\nIgnore-this: bf613f62175a0bf1746a2b56c098965f\n\ndarcs-hash:20100716151415-3bd4e-ec5694eed6a508414fc6a6872cd26d24e22c2880.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Examples\/DivisionPCF\/EquationsPCF.agda","new_file":"Examples\/DivisionPCF\/EquationsPCF.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"491e9a07a29eb9be3384cd1ad1f193bfc523c14a","subject":"Draft conversion to CBPV types","message":"Draft conversion to CBPV types\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/Type.agda","new_file":"Parametric\/Syntax\/Type.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of function types (Fig. 1a).\n------------------------------------------------------------------------\n\nmodule Parametric.Syntax.Type where\n\n-- This is a module in the Parametric.* hierarchy, so it defines\n-- an extension point for plugins. In this case, the plugin can\n-- choose the syntax for data types. We always provide a binding\n-- called \"Structure\" that defines the type of the value that has\n-- to be provided by the plugin. In this case, the plugin has to\n-- provide a type, so we say \"Structure = Set\".\n\nStructure : Set\u2081\nStructure = Set\n\n-- Here is the parametric module that defines the syntax of\n-- simply types in terms of the syntax of base types. In the\n-- Parametric.* hierarchy, we always call these parametric\n-- modules \"Structure\". Note that in Agda, there is a separate\n-- namespace for module names and for other bindings, so this\n-- doesn't clash with the \"Structure\" above.\n--\n-- The idea behind all these structures is that the parametric\n-- module lifts some structure of the plugin to the corresponding\n-- structure in the full language. In this case, we lift the\n-- structure of base types to the structure of simple types.\n-- Maybe not the best choice of names, but it seemed clever at\n-- the time.\n\nmodule Structure (Base : Structure) where\n  infixr 5 _\u21d2_\n\n  -- A simple type is either a base type or a function types.\n  -- Note that we can use our module parameter \"Base\" here just\n  -- like any other type.\n  data Type : Set where\n    base : (\u03b9 : Base) \u2192 Type\n    _\u21d2_ : (\u03c3 : Type) \u2192 (\u03c4 : Type) \u2192 Type\n\n  -- We also provide the definitions of contexts of the newly\n  -- defined simple types, variables as de Bruijn indices\n  -- pointing into such a context, and sets of bound variables.\n  open import Base.Syntax.Context Type public\n  open import Base.Syntax.Vars Type public\n\n  -- Internalize a context to a type.\n  --\n  -- Is there a better name for this function?\n  internalizeContext : (\u03a3 : Context) (\u03c4\u2032 : Type) \u2192 Type\n  internalizeContext \u2205 \u03c4\u2032 = \u03c4\u2032\n  internalizeContext (\u03c4 \u2022 \u03a3) \u03c4\u2032 = \u03c4 \u21d2 internalizeContext \u03a3 \u03c4\u2032\n\n\n  mutual\n    --  CBPV\n    data ValType : Set where\n      U : CompType \u2192 ValType\n      B : Base \u2192 ValType\n      Unit : ValType\n      _\u00d7_ : ValType \u2192 ValType \u2192 ValType\n      _+_ : ValType \u2192 ValType \u2192 ValType\n\n    data CompType : Set where\n      F : ValType \u2192 CompType\n      _\u21db_ : ValType \u2192 CompType \u2192 CompType\n      _\u03a0_ : CompType \u2192 CompType \u2192 CompType\n\n  cbnToCompType : Type \u2192 CompType\n  cbnToCompType (base \u03b9) = F (B \u03b9)\n  cbnToCompType (\u03c3 \u21d2 \u03c4) = U (cbnToCompType \u03c3) \u21db cbnToCompType \u03c4\n\n  cbvToValType : Type \u2192 ValType\n  cbvToValType (base \u03b9) = B \u03b9\n  cbvToValType (\u03c3 \u21d2 \u03c4) = U (cbvToValType \u03c3 \u21db F (cbvToValType \u03c4))\n\n  open import Base.Syntax.Context ValType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; mapContext to mapValCtx\n      ; Var to ValVar\n      ; Context to ValContext\n      ; this to vThis; that to vThat)\n\n  cbnToValType : Type \u2192 ValType\n  cbnToValType \u03c4 = U (cbnToCompType \u03c4)\n\n  cbvToCompType : Type \u2192 CompType\n  cbvToCompType \u03c4 = F (cbvToValType \u03c4)\n\n  fromCBNCtx : Context \u2192 ValContext\n  fromCBNCtx \u0393 = mapValCtx cbnToValType \u0393\n\n  fromCBVCtx : Context \u2192 ValContext\n  fromCBVCtx \u0393 = mapValCtx cbvToValType \u0393\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 ValType) \u2192 Var \u0393 \u03c4 \u2192 ValVar (mapValCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n","old_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- The syntax of function types (Fig. 1a).\n------------------------------------------------------------------------\n\nmodule Parametric.Syntax.Type where\n\n-- This is a module in the Parametric.* hierarchy, so it defines\n-- an extension point for plugins. In this case, the plugin can\n-- choose the syntax for data types. We always provide a binding\n-- called \"Structure\" that defines the type of the value that has\n-- to be provided by the plugin. In this case, the plugin has to\n-- provide a type, so we say \"Structure = Set\".\n\nStructure : Set\u2081\nStructure = Set\n\n-- Here is the parametric module that defines the syntax of\n-- simply types in terms of the syntax of base types. In the\n-- Parametric.* hierarchy, we always call these parametric\n-- modules \"Structure\". Note that in Agda, there is a separate\n-- namespace for module names and for other bindings, so this\n-- doesn't clash with the \"Structure\" above.\n--\n-- The idea behind all these structures is that the parametric\n-- module lifts some structure of the plugin to the corresponding\n-- structure in the full language. In this case, we lift the\n-- structure of base types to the structure of simple types.\n-- Maybe not the best choice of names, but it seemed clever at\n-- the time.\n\nmodule Structure (Base : Structure) where\n  infixr 5 _\u21d2_\n\n  -- A simple type is either a base type or a function types.\n  -- Note that we can use our module parameter \"Base\" here just\n  -- like any other type.\n  data Type : Set where\n    base : (\u03b9 : Base) \u2192 Type\n    _\u21d2_ : (\u03c3 : Type) \u2192 (\u03c4 : Type) \u2192 Type\n\n  -- We also provide the definitions of contexts of the newly\n  -- defined simple types, variables as de Bruijn indices\n  -- pointing into such a context, and sets of bound variables.\n  open import Base.Syntax.Context Type public\n  open import Base.Syntax.Vars Type public\n\n  -- Internalize a context to a type.\n  --\n  -- Is there a better name for this function?\n  internalizeContext : (\u03a3 : Context) (\u03c4\u2032 : Type) \u2192 Type\n  internalizeContext \u2205 \u03c4\u2032 = \u03c4\u2032\n  internalizeContext (\u03c4 \u2022 \u03a3) \u03c4\u2032 = \u03c4 \u21d2 internalizeContext \u03a3 \u03c4\u2032\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"72ce3097584e2943047ba2c533bbc23ac8caf45f","subject":"Using succ\u2081 instead of succ in an equivalence proof.","message":"Using succ\u2081 instead of succ in an equivalence proof.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/report\/FOTC\/EquivalenceInductivePredicateN.agda","new_file":"notes\/thesis\/report\/FOTC\/EquivalenceInductivePredicateN.agda","new_contents":"------------------------------------------------------------------------------\n-- Equivalent approaches for implement the inductive predicate N\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.EquivalenceInductivePredicateN where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- Using succ : D instead of succ\u2081 : D \u2192 D.\n\nmodule Constant where\n\n  module LFP where\n\n    NatF : (D \u2192 Set) \u2192 D \u2192 Set\n    NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n    postulate\n      N         : D \u2192 Set\n      N-ir-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n      N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n    N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n    N-ir = N-ir-ho\n\n    N-ind' : (A : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n             \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' = N-ind'-ho\n\n    --------------------------------------------------------------------------\n    -- The data constructors of N using data.\n\n    nzero : N zero\n    nzero = N-ir (inj\u2081 refl)\n\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n    nsucc Nn = N-ir (inj\u2082 (_ , refl , Nn))\n\n    --------------------------------------------------------------------------\n    -- The induction principle of N using data.\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h = N-ind' A h'\n      where\n      h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n      h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n      h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  module Data where\n\n    data N : D \u2192 Set where\n      nzero : N zero\n      nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h nzero      = A0\n    N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n    --------------------------------------------------------------------------\n    -- The introduction rule of N using LFP.\n    N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n    N-ir {n} h = case prf\u2081 prf\u2082 h\n      where\n      prf\u2081 : n \u2261 zero \u2192 N n\n      prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n      prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n      prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n    --------------------------------------------------------------------------\n    -- The induction principle for N using LFP.\n    N-ind' :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' A h = N-ind A h\u2081 h\u2082\n      where\n      h\u2081 :  A zero\n      h\u2081 = h (inj\u2081 refl)\n\n      h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n      h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n\n------------------------------------------------------------------------------\n-- Using succ\u2081 : D \u2192 D instead of succ : D.\n\nmodule UnaryFunction where\n\n  module LFP where\n\n    NatF : (D \u2192 Set) \u2192 D \u2192 Set\n    NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n')\n\n    postulate\n      N         : D \u2192 Set\n      N-ir-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n      N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n    N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n    N-ir = N-ir-ho\n\n    N-ind' : (A : D \u2192 Set) \u2192\n             (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n             \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' = N-ind'-ho\n\n    --------------------------------------------------------------------------\n    -- The data constructors of N using data.\n\n    nzero : N zero\n    nzero = N-ir (inj\u2081 refl)\n\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n    nsucc Nn = N-ir (inj\u2082 (_ , refl , Nn))\n\n    --------------------------------------------------------------------------\n    -- The induction principle of N using data.\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h = N-ind' A h'\n      where\n      h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ\u2081 m' \u2227 A m') \u2192 A m\n      h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n      h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n  ----------------------------------------------------------------------------\n  module Data where\n\n    data N : D \u2192 Set where\n      nzero : N zero\n      nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n    N-ind : (A : D \u2192 Set) \u2192\n            A zero \u2192\n            (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n            \u2200 {n} \u2192 N n \u2192 A n\n    N-ind A A0 h nzero      = A0\n    N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n    --------------------------------------------------------------------------\n    -- The introduction rule of N using LFP.\n    N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n') \u2192 N n\n    N-ir {n} h = case prf\u2081 prf\u2082 h\n      where\n      prf\u2081 : n \u2261 zero \u2192 N n\n      prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n      prf\u2082 : \u2203[ n' ] n \u2261 succ\u2081 n' \u2227 N n' \u2192 N n\n      prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n    --------------------------------------------------------------------------\n    -- The induction principle for N using LFP.\n    N-ind' :\n      (A : D \u2192 Set) \u2192\n      (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ\u2081 n' \u2227 A n') \u2192 A n) \u2192\n      \u2200 {n} \u2192 N n \u2192 A n\n    N-ind' A h = N-ind A h\u2081 h\u2082\n      where\n      h\u2081 :  A zero\n      h\u2081 = h (inj\u2081 refl)\n\n      h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ\u2081 m)\n      h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","old_contents":"------------------------------------------------------------------------------\n-- Equivalent approaches for implement the inductive predicate N\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.EquivalenceInductivePredicateN where\n\nopen import FOTC.Base\n\nmodule LFP where\n\n  NatF : (D \u2192 Set) \u2192 D \u2192 Set\n  NatF A n = n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n')\n\n  postulate\n    N         : D \u2192 Set\n    N-ir-ho   : \u2200 {n} \u2192 NatF N n \u2192 N n\n    N-ind'-ho : (A : D \u2192 Set) \u2192 (\u2200 {n} \u2192 NatF A n \u2192 A n) \u2192 \u2200 {n} \u2192 N n \u2192 A n\n\n  N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-ir = N-ir-ho\n\n  N-ind' : (A : D \u2192 Set) \u2192\n           (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n           \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' = N-ind'-ho\n\n  ----------------------------------------------------------------------------\n  -- The data constructors of N using data.\n\n  nzero : N zero\n  nzero = N-ir (inj\u2081 refl)\n\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n  nsucc Nn = N-ir (inj\u2082 (_ , refl , Nn))\n\n  ----------------------------------------------------------------------------\n  -- The induction principle of N using data.\n  N-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\n  N-ind A A0 h = N-ind' A h'\n    where\n    h' : \u2200 {m} \u2192 m \u2261 zero \u2228 (\u2203[ m' ] m \u2261 succ \u00b7 m' \u2227 A m') \u2192 A m\n    h' (inj\u2081 m\u22610)              = subst A (sym m\u22610) A0\n    h' (inj\u2082 (m' , prf , Am')) = subst A (sym prf) (h Am')\n\n------------------------------------------------------------------------------\nmodule Data where\n\n  data N : D \u2192 Set where\n    nzero : N zero\n    nsucc : \u2200 {n} \u2192 N n \u2192 N (succ \u00b7 n)\n\n  N-ind : (A : D \u2192 Set) \u2192\n          A zero \u2192\n          (\u2200 {n} \u2192 A n \u2192 A (succ \u00b7 n)) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\n  N-ind A A0 h nzero      = A0\n  N-ind A A0 h (nsucc Nn) = h (N-ind A A0 h Nn)\n\n  ----------------------------------------------------------------------------\n  -- The introduction rule of N using LFP.\n  N-ir : \u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n') \u2192 N n\n  N-ir {n} h = case prf\u2081 prf\u2082 h\n    where\n    prf\u2081 : n \u2261 zero \u2192 N n\n    prf\u2081 n\u22610 = subst N (sym n\u22610) nzero\n\n    prf\u2082 : \u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 N n' \u2192 N n\n    prf\u2082 (n' , prf , Nn') = subst N (sym prf) (nsucc Nn')\n\n  ----------------------------------------------------------------------------\n  -- The induction principle for N using LFP.\n  N-ind' :\n    (A : D \u2192 Set) \u2192\n    (\u2200 {n} \u2192 n \u2261 zero \u2228 (\u2203[ n' ] n \u2261 succ \u00b7 n' \u2227 A n') \u2192 A n) \u2192\n    \u2200 {n} \u2192 N n \u2192 A n\n  N-ind' A h = N-ind A h\u2081 h\u2082\n    where\n    h\u2081 :  A zero\n    h\u2081 = h (inj\u2081 refl)\n\n    h\u2082 : \u2200 {m} \u2192 A m \u2192 A (succ \u00b7 m)\n    h\u2082 {m} Am = h (inj\u2082 (m , refl , Am))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"5369218b785249dfea6dc3ddc782a92786be9d82","subject":"Desc model: proof-psi-phi implicit and slightly refactored","message":"Desc model: proof-psi-phi implicit and slightly refactored\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\\ _ -> descD I)\n                               P\n                               proof-psi-phi-casesW\n                               Void\n                               D\n                where P : Sigma Unit (IMu (\\ x -> descD I)) -> Set\n                      P ( Void , D ) = psi (phi D) == D\n                      proof-psi-phi-cases : (xs : Sigma DescDConst \n                                                  (\u03bb s \u2192 desc (descDChoice I s)\n                                                  (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                            (hs : desc \n                                                  (box (sigma DescDConst (descDChoice I))\n                                                  (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                  P)\n                                            \u2192 P (Void , con xs)\n                      proof-psi-phi-cases (lvar , i) hs = refl\n                      proof-psi-phi-cases (lconst , x) hs = refl\n                      proof-psi-phi-cases (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (i : Unit)\n                                             (xs : Sigma DescDConst\n                                                   (\u03bb s \u2192 desc (descDChoice I s)\n                                                   (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                             (hs : desc \n                                                   (box (sigma DescDConst (descDChoice I))\n                                                   (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                   P)\n                                              \u2192  P (i , con xs)\n                      proof-psi-phi-casesW Void = proof-psi-phi-cases\n                      \n\n","old_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\n\n-- Intensionally extensional\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {I : Set} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P * desc D' P\ndesc (sigma S T) P = Sigma S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : {I : Set}(D : IDesc I)(P : I -> Set) -> desc D P -> IDesc (Sigma I P)\nbox (var i)     P x        = var (i , x)\nbox (const X)   P x        = const X\nbox (prod D D') P (d , d') = prod (box D P d) (box D' P d')\nbox (sigma S T) P (a , b)  = box (T a) P b\nbox (pi S T)    P f        = pi S (\\s -> box (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc D (IMu R)) -> \n           desc (box D (IMu R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : desc (R i) (IMu R))\n                 (hs : desc (box (R i) (IMu R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu R i) -> P ( i , x )\ninduction = Elim.induction\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu (\\_ -> descD I) Void\n\nvarl : {I : Set}(i : I) -> IDescl I\nvarl i = con (lvar ,  i)\n\nconstl : {I : Set}(X : Set) -> IDescl I\nconstl X = con (lconst , X)\n\nprodl : {I : Set}(D D' : IDescl I) -> IDescl I\nprodl D D' = con (lprod , (D , D'))\n\npil : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\npil S T = con (lpi , ( S , T))\n\nsigmal : {I : Set}(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal S T = con (lsigma , ( S , T))\n\n\n\n--********************************************\n-- From the embedding to the host\n--********************************************\n\ncases : {I : Set}\n        (xs : desc (descD I) (IMu (\u03bb _ -> descD I)))\n        (hs : desc (box (descD I) (IMu (\u03bb _ -> descD I)) xs) (\u03bb _ -> IDesc I)) ->\n        IDesc I\ncases ( lvar , i ) hs =  var i\ncases ( lconst , X ) hs =  const X\ncases ( lprod , (D , D') ) ( d , d' ) =  prod d d'\ncases ( lpi , ( S , T ) ) hs =  pi S hs\ncases ( lsigma , ( S , T ) ) hs = sigma S hs\n\nphi : {I : Set} -> IDescl I -> IDesc I\nphi {I} d = induction (\\_ -> descD I) (\\_ -> IDesc I) (\\_ -> cases) Void d\n\n\n--********************************************\n-- From the host to the embedding\n--********************************************\npsi : {I : Set} -> IDesc I -> IDescl I\npsi (var i) = varl i\npsi (const X) = constl X\npsi (prod D D') = prodl (psi D) (psi D')\npsi (pi S T) = pil S (\\s -> psi (T s))\npsi (sigma S T) = sigmal S (\\s -> psi (T s))\n\n\n--********************************************\n-- Isomorphism proof\n--********************************************\n\n-- From host to host\n\nproof-phi-psi : {I : Set} -> (D : IDesc I) -> phi (psi D) == D\nproof-phi-psi (var i) = refl\nproof-phi-psi (const x) = refl\nproof-phi-psi (prod D D') with proof-phi-psi D | proof-phi-psi D'\n...  | p | q = cong2 prod p q\nproof-phi-psi (pi S T) = cong (pi S)  \n                                (reflFun (\\ s -> phi (psi (T s)))\n                                         T\n                                         (\\s -> proof-phi-psi (T s)))\nproof-phi-psi (sigma S T) = cong (sigma S) \n                                   (reflFun (\\ s -> phi (psi (T s)))\n                                            T\n                                            (\\s -> proof-phi-psi (T s)))\n\n\n-- From embedding to embedding\n\nP : (I : Set) -> Sigma Unit (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) \u2192 Set\nP I ( Void , D ) = psi (phi D) == D\n\nproof-psi-phi : (I : Set) -> (D : IDescl I) -> psi (phi D) == D\nproof-psi-phi I D =  induction (\u03bb _ \u2192 descD I)\n                                 (P I)\n                                 (proof-psi-phi-casesW I) \n                                 Void\n                                 D\n                where proof-psi-phi-cases : (I : Set)\n                                              (xs : Sigma DescDConst \n                                                    (\u03bb s \u2192 desc (descDChoice I s)\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                              (hs : desc \n                                                    (box (sigma DescDConst (descDChoice I))\n                                                    (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                    (P I))\n                                              \u2192 P I (Void , con xs)\n                      proof-psi-phi-cases I (lvar , i) hs = refl\n                      proof-psi-phi-cases I (lconst , x) hs = refl\n                      proof-psi-phi-cases I (lprod , ( D , D' )) ( p , q ) = cong2 prodl p q \n                      proof-psi-phi-cases I (lpi , ( S , T )) hs = cong (pil S) \n                                                                          (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                   T\n                                                                                   hs)\n                      proof-psi-phi-cases I (lsigma , ( S , T )) hs = cong (sigmal S) \n                                                                             (reflFun (\u03bb s \u2192 psi (phi (T s)))\n                                                                                      T\n                                                                                      hs)\n                      proof-psi-phi-casesW : (I : Set)\n                                               (i : Unit)\n                                               (xs : Sigma DescDConst\n                                                     (\u03bb s \u2192 desc (descDChoice I s)\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I)))))\n                                               (hs : desc \n                                                     (box (sigma DescDConst (descDChoice I))\n                                                     (IMu (\u03bb x \u2192 sigma DescDConst (descDChoice I))) xs)\n                                                     (P I))\n                                                \u2192  P I (i , con xs)\n                      proof-psi-phi-casesW I Void = proof-psi-phi-cases I\n                      \n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"4551345929d518f9a6978721470d5a37cfee509d","subject":"Added missing doc.","message":"Added missing doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Base\/List.agda","new_file":"src\/fot\/FOTC\/Base\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- FOTC combinators for lists, colists, streams, etc.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Base.List where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixr 8 _\u2237_\n\n------------------------------------------------------------------------------\n-- List constants.\npostulate [] cons head tail null : D  -- FOTC partial lists.\n\n-- Definitions\nabstract\n  _\u2237_  : D \u2192 D \u2192 D\n  x \u2237 xs = cons \u00b7 x \u00b7 xs\n  -- {-# ATP definition _\u2237_ #-}\n\n  head\u2081 : D \u2192 D\n  head\u2081 xs = head \u00b7 xs\n  -- {-# ATP definition head\u2081 #-}\n\n  tail\u2081 : D \u2192 D\n  tail\u2081 xs = tail \u00b7 xs\n  -- {-# ATP definition tail\u2081 #-}\n\n  null\u2081 : D \u2192 D\n  null\u2081 xs = null \u00b7 xs\n  -- {-# ATP definition null\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- Conversion rules for null.\n-- null-[] :          null \u00b7 []              \u2261 true\n-- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\npostulate\n  null-[] : null\u2081 []                \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\n-- head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\npostulate head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\n-- tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\npostulate tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\n-- postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 cons \u00b7 x \u00b7 xs\npostulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 x \u2237 xs\n","old_contents":"------------------------------------------------------------------------------\n-- FOTC combinators for lists, colists, streams, etc.\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Base.List where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixr 8 _\u2237_\n\n------------------------------------------------------------------------------\n-- List constants.\npostulate [] cons head tail null : D  -- FOTC lists.\n\n-- Definitions\nabstract\n  _\u2237_  : D \u2192 D \u2192 D\n  x \u2237 xs = cons \u00b7 x \u00b7 xs\n  -- {-# ATP definition _\u2237_ #-}\n\n  head\u2081 : D \u2192 D\n  head\u2081 xs = head \u00b7 xs\n  -- {-# ATP definition head\u2081 #-}\n\n  tail\u2081 : D \u2192 D\n  tail\u2081 xs = tail \u00b7 xs\n  -- {-# ATP definition tail\u2081 #-}\n\n  null\u2081 : D \u2192 D\n  null\u2081 xs = null \u00b7 xs\n  -- {-# ATP definition null\u2081 #-}\n\n------------------------------------------------------------------------------\n-- Conversion rules\n\n-- Conversion rules for null.\n-- null-[] :          null \u00b7 []              \u2261 true\n-- null-\u2237  : \u2200 x xs \u2192 null \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 false\npostulate\n  null-[] : null\u2081 []                \u2261 true\n  null-\u2237  : \u2200 x xs \u2192 null\u2081 (x \u2237 xs) \u2261 false\n\n-- Conversion rule for head.\n-- head-\u2237 : \u2200 x xs \u2192 head \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 x\npostulate head-\u2237 : \u2200 x xs \u2192 head\u2081 (x \u2237 xs) \u2261 x\n{-# ATP axiom head-\u2237 #-}\n\n-- Conversion rule for tail.\n-- tail-\u2237 : \u2200 x xs \u2192 tail \u00b7 (cons \u00b7 x \u00b7 xs) \u2261 xs\npostulate tail-\u2237 : \u2200 x xs \u2192 tail\u2081 (x \u2237 xs) \u2261 xs\n{-# ATP axiom tail-\u2237 #-}\n\n------------------------------------------------------------------------------\n-- Discrimination rules\n\n-- postulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 cons \u00b7 x \u00b7 xs\npostulate []\u2262cons : \u2200 {x xs} \u2192 [] \u2262 x \u2237 xs\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"b7a81abc4769054683315bc3c2b579e66699ab5b","subject":"Ahem, fix up the step-indexing properly","message":"Ahem, fix up the step-indexing properly\n\nAfter the last update, the set of big-step derivations was serious incomplete,\nsince derivations for applications and let still assumed values to evaluate in\nzero steps instead of 1.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- Adding this changes nothing without changes to the semantics.\n  succ : Const (nat \u21d2 nat)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\ndata Val : Type \u2192 Set\n\nimport Base.Denotation.Environment\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 lit n \u27e7Const = n\n\u27e6 succ \u27e7Const = suc\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Den.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7SVal : \u2200 {\u0393 \u03c4} \u2192 SVal \u0393 \u03c4 \u2192 Den.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 var x \u27e7SVal \u03c1 = Den.\u27e6 x \u27e7Var \u03c1\n\u27e6 abs t \u27e7SVal \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 val sv \u27e7Term \u03c1 = \u27e6 sv \u27e7SVal \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7SVal \u03c1 (\u27e6 t \u27e7SVal \u03c1)\n\u27e6 lett s t \u27e7Term \u03c1 = \u27e6 t \u27e7Term ((\u27e6 s \u27e7Term \u03c1) \u2022 \u03c1)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 natV n \u27e7Val = n\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 1 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 1 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc (suc (suc n))) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- Check it's fine to use i 1 in the above proofs.\n\u2193-sv-1-step : \u2200 {\u0393 \u03c4 \u03c1 v} {n} {sv : SVal \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 val sv \u2193[ i' n ] v \u2192\n  n \u2261 1\n\u2193-sv-1-step abs = refl\n\u2193-sv-1-step (var x) = refl\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v hasIdx} {n : Idx hasIdx} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193[ n ] v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\u2193-sound abs = refl\n\u2193-sound (app _ _ \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n\u2193-sound (var x) = \u21a6-sound _ x\n\u2193-sound (lit n) = refl\n\u2193-sound (lett n1 n2 vsv s t \u2193 \u2193\u2081) rewrite \u2193-sound \u2193 | \u2193-sound \u2193\u2081 = refl\n-- \u2193-sound (add \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n-- \u2193-sound (minus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\n-- No proof of completeness yet: the statement does not hold here.\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc (suc (suc (suc k)))) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (suc (suc j))) (s\u2264s (s\u2264s (s\u2264s (s\u2264s j<k))))\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app n\u2081 vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc (suc (suc (suc k)))) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ (suc zero) (s\u2264s (s\u2264s z\u2264n)) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc (suc (suc (suc k)))) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ (suc zero) (s\u2264s (s\u2264s z\u2264n)) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv (suc k) (s\u2264s (s\u2264s (\u2264-step \u2264-refl))) vtv1 dtv vtv2\n           ( (rrelV3-mono (suc k) (suc (suc (suc k))) (s\u2264s (\u2264-step (\u2264-step \u2264-refl))) _ vtv1 dtv vtv2 dtvv) )\n           v1 v2 j (s\u2264s j<k) \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- Adding this changes nothing without changes to the semantics.\n  succ : Const (nat \u21d2 nat)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\ndata Val : Type \u2192 Set\n\nimport Base.Denotation.Environment\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\nopen Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\n\u27e6_\u27e7Const : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 lit n \u27e7Const = n\n\u27e6 succ \u27e7Const = suc\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Den.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7SVal : \u2200 {\u0393 \u03c4} \u2192 SVal \u0393 \u03c4 \u2192 Den.\u27e6 \u0393 \u27e7Context \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 var x \u27e7SVal \u03c1 = Den.\u27e6 x \u27e7Var \u03c1\n\u27e6 abs t \u27e7SVal \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 const c \u27e7Term \u03c1 = \u27e6 c \u27e7Const\n\u27e6 val sv \u27e7Term \u03c1 = \u27e6 sv \u27e7SVal \u03c1\n\u27e6 app s t \u27e7Term \u03c1 = \u27e6 s \u27e7SVal \u03c1 (\u27e6 t \u27e7SVal \u03c1)\n\u27e6 lett s t \u27e7Term \u03c1 = \u27e6 t \u27e7Term ((\u27e6 s \u27e7Term \u03c1) \u2022 \u03c1)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 natV n \u27e7Val = n\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v hasIdx} {n : Idx hasIdx} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193[ n ] v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\u2193-sound abs = refl\n\u2193-sound (app _ _ \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n\u2193-sound (var x) = \u21a6-sound _ x\n\u2193-sound (lit n) = refl\n\u2193-sound (lett n1 n2 vsv s t \u2193 \u2193\u2081) rewrite \u2193-sound \u2193 | \u2193-sound \u2193\u2081 = refl\n-- \u2193-sound (add \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n-- \u2193-sound (minus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\n-- No proof of completeness yet: the statement does not hold here.\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d1f72ce229ff569e91f1fb9d7f820e91ae0cc58d","subject":"Prove congruence for \u2248-weaken.","message":"Prove congruence for \u2248-weaken.\n\nOld-commit-hash: 06cfb487c9f50ad6485a18216648fe4e22f0f24e\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/Equivalence.agda","new_file":"Denotational\/Equivalence.agda","new_contents":"module Denotational.Equivalence where\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Relation.Binary.PropositionalEquality\n\nopen import Denotational.Notation\n\n-- TERMS\n\n-- Term Equivalence\n--\n-- This module is parametric in the syntax and semantics of types\n-- and terms to make it work for different calculi.\n\nmodule _ where\n  open import Syntactic.Contexts\n  open import Denotational.Environments\n\n  module MakeEquivalence\n      (Type : Set)\n      (\u27e6_\u27e7Type : Type \u2192 Set)\n      (Term : Context Type \u2192 Type \u2192 Set)\n      (\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6_\u27e7Context Type \u27e6_\u27e7Type \u0393 \u2192 \u27e6 \u03c4 \u27e7Type) where\n    module _ {\u0393} {\u03c4} where\n      data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n        ext-t :\n          (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7Term \u03c1 \u2261 \u27e6 t\u2082 \u27e7Term \u03c1) \u2192\n          t\u2081 \u2248 t\u2082\n\n      \u2248-refl : Reflexive _\u2248_\n      \u2248-refl = ext-t (\u03bb \u03c1 \u2192 refl)\n\n      \u2248-sym : Symmetric _\u2248_\n      \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 sym (\u2248 \u03c1))\n\n      \u2248-trans : Transitive _\u2248_\n      \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n      \u2248-isEquivalence : IsEquivalence _\u2248_\n      \u2248-isEquivalence = record\n        { refl  = \u2248-refl\n        ; sym   = \u2248-sym\n        ; trans = \u2248-trans\n        }\n\n    \u2248-setoid : Context Type \u2192 Type \u2192 Setoid _ _\n    \u2248-setoid \u0393 \u03c4 = record\n      { Carrier       = Term \u0393 \u03c4\n      ; _\u2248_           = _\u2248_\n      ; isEquivalence = \u2248-isEquivalence\n      }\n\n    module \u2248-Reasoning where\n      module _ {\u0393 : Context Type} {\u03c4 : Type} where\n        open import Relation.Binary.EqReasoning (\u2248-setoid \u0393 \u03c4) public\n          hiding (_\u2261\u27e8_\u27e9_)\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\n\nopen import Changes\nopen import ChangeContexts\n\n-- Export a version of the equivalence for terms with total\n-- derivatives\n\nopen MakeEquivalence Type \u27e6_\u27e7 Term \u27e6_\u27e7 public\n\n-- Specialized congruence rules\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {t\u2081 t\u2082 : Term \u0393\u2081 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 {{\u0393\u2032}} t\u2081 \u2248 \u0394 {{\u0393\u2032}} t\u2082\n\u2248-\u0394 {{\u0393\u2032}} (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u2248 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))))\n\n\u2248-true : \u2200 {\u0393} \u2192 _\u2248_ {\u0393} true true\n\u2248-true = \u2248-refl\n\n\u2248-false : \u2200 {\u0393} \u2192 _\u2248_ {\u0393} false false\n\u2248-false = \u2248-refl\n\n\u2248-if : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 bool} {t\u2083 t\u2084 t\u2085 t\u2086 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 t\u2085 \u2248 t\u2086 \u2192 if t\u2081 t\u2083 t\u2085 \u2248 if t\u2082 t\u2084 t\u2086\n\u2248-if (ext-t \u2248\u2081) (ext-t \u2248\u2082) (ext-t \u2248\u2083) = ext-t (\u03bb \u03c1 \u2192 \u2261-if \u2248\u2081 \u03c1 then \u2248\u2082 \u03c1 else \u2248\u2083 \u03c1)\n\n\u2248-weaken : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) {t\u2081 t\u2082 : Term \u0393\u2081 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 weaken \u0393\u2032 t\u2081 \u2248 weaken \u0393\u2032 t\u2082\n\u2248-weaken \u0393\u2032 {t\u2081} {t\u2082} (ext-t \u2248\u2081) = ext-t (\u03bb \u03c1 \u2192\n  begin\n    \u27e6 weaken \u0393\u2032 t\u2081 \u27e7 \u03c1\n  \u2261\u27e8 weaken-sound t\u2081 \u03c1 \u27e9\n    \u27e6 t\u2081 \u27e7Term (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1)\n  \u2261\u27e8 \u2248\u2081 (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1) \u27e9\n    \u27e6 t\u2082 \u27e7Term (\u27e6 \u0393\u2032 \u27e7\u227c \u03c1)\n  \u2261\u27e8 \u2261-sym (weaken-sound t\u2082 \u03c1) \u27e9\n    \u27e6 weaken \u0393\u2032 t\u2082 \u27e7 \u03c1\n  \u220e) where open \u2261-Reasoning\n\n-- Consistency\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n","old_contents":"module Denotational.Equivalence where\n\n-- TERM EQUIVALENCE\n--\n-- This module defines term equivalence as the relation that\n-- identifies terms with the same meaning.\n\nopen import Relation.Nullary using (\u00ac_)\n\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Relation.Binary.PropositionalEquality\n\nopen import Denotational.Notation\n\n-- TERMS\n\n-- Term Equivalence\n--\n-- This module is parametric in the syntax and semantics of types\n-- and terms to make it work for different calculi.\n\nmodule _ where\n  open import Syntactic.Contexts\n  open import Denotational.Environments\n\n  module MakeEquivalence\n      (Type : Set)\n      (\u27e6_\u27e7Type : Type \u2192 Set)\n      (Term : Context Type \u2192 Type \u2192 Set)\n      (\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6_\u27e7Context Type \u27e6_\u27e7Type \u0393 \u2192 \u27e6 \u03c4 \u27e7Type) where\n    module _ {\u0393} {\u03c4} where\n      data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n        ext-t :\n          (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7Term \u03c1 \u2261 \u27e6 t\u2082 \u27e7Term \u03c1) \u2192\n          t\u2081 \u2248 t\u2082\n\n      \u2248-refl : Reflexive _\u2248_\n      \u2248-refl = ext-t (\u03bb \u03c1 \u2192 refl)\n\n      \u2248-sym : Symmetric _\u2248_\n      \u2248-sym (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 sym (\u2248 \u03c1))\n\n      \u2248-trans : Transitive _\u2248_\n      \u2248-trans (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192 trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n      \u2248-isEquivalence : IsEquivalence _\u2248_\n      \u2248-isEquivalence = record\n        { refl  = \u2248-refl\n        ; sym   = \u2248-sym\n        ; trans = \u2248-trans\n        }\n\n    \u2248-setoid : Context Type \u2192 Type \u2192 Setoid _ _\n    \u2248-setoid \u0393 \u03c4 = record\n      { Carrier       = Term \u0393 \u03c4\n      ; _\u2248_           = _\u2248_\n      ; isEquivalence = \u2248-isEquivalence\n      }\n\n    module \u2248-Reasoning where\n      module _ {\u0393 : Context Type} {\u03c4 : Type} where\n        open import Relation.Binary.EqReasoning (\u2248-setoid \u0393 \u03c4) public\n          hiding (_\u2261\u27e8_\u27e9_)\n\nopen import Syntactic.Types\nopen import Syntactic.Contexts Type\nopen import Syntactic.Terms.Total\n\nopen import Denotational.Values\nopen import Denotational.Environments Type \u27e6_\u27e7Type\nopen import Denotational.Evaluation.Total\n\nopen import Changes\nopen import ChangeContexts\n\n-- Export a version of the equivalence for terms with total\n-- derivatives\n\nopen MakeEquivalence Type \u27e6_\u27e7 Term \u27e6_\u27e7 public\n\n-- Specialized congruence rules\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext-t \u2248\u2081) (ext-t \u2248\u2082) = ext-t (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393\u2081 \u0393\u2082} {{\u0393\u2032 : \u0394-Context \u0393\u2081 \u227c \u0393\u2082}} {t\u2081 t\u2082 : Term \u0393\u2081 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 {{\u0393\u2032}} t\u2081 \u2248 \u0394 {{\u0393\u2032}} t\u2082\n\u2248-\u0394 {{\u0393\u2032}} (ext-t \u2248) = ext-t (\u03bb \u03c1 \u2192 \u2261-diff (\u2248 (update (\u27e6 \u0393\u2032 \u27e7 \u03c1))) (\u2248 (ignore (\u27e6 \u0393\u2032 \u27e7 \u03c1))))\n\n\u2248-true : \u2200 {\u0393} \u2192 _\u2248_ {\u0393} true true\n\u2248-true = \u2248-refl\n\n\u2248-false : \u2200 {\u0393} \u2192 _\u2248_ {\u0393} false false\n\u2248-false = \u2248-refl\n\n\u2248-if : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 bool} {t\u2083 t\u2084 t\u2085 t\u2086 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 t\u2085 \u2248 t\u2086 \u2192 if t\u2081 t\u2083 t\u2085 \u2248 if t\u2082 t\u2084 t\u2086\n\u2248-if (ext-t \u2248\u2081) (ext-t \u2248\u2082) (ext-t \u2248\u2083) = ext-t (\u03bb \u03c1 \u2192 \u2261-if \u2248\u2081 \u03c1 then \u2248\u2082 \u03c1 else \u2248\u2083 \u03c1)\n\n-- Consistency\n\n\u2248-consistent : \u00ac (\u2200 {\u0393 \u03c4} (t\u2081 t\u2082 : Term \u0393 \u03c4) \u2192 t\u2081 \u2248 t\u2082)\n\u2248-consistent H with H {\u2205} true false\n... | ext-t x with x \u2205\n... | ()\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"640b182c89c7ee086ad643793ad2687c097963c8","subject":"Added F*T\u2192List.","message":"Added F*T\u2192List.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/Fair\/Properties.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/Fair\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Fair properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.Fair.Properties where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.List\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nFair\u2192Stream : \u2200 {os} \u2192 Fair os \u2192 Stream os\nFair\u2192Stream {os} Fos = Stream-coind A h refl\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 xs\n\n  h : A os \u2192 \u2203[ o' ] \u2203[ os' ] os \u2261 o' \u2237 os' \u2227 A os'\n  h _ with head-tail-Fair Fos\n  h x\u2081 | inj\u2081 prf = T , tail\u2081 os , prf , refl\n  h x\u2081 | inj\u2082 prf = F , tail\u2081 os , prf , refl\n\nF*T\u2192List : \u2200 {xs} \u2192 F*T xs \u2192 List xs\nF*T\u2192List f*tnil              = lcons true lnil\nF*T\u2192List (f*tcons {ft} FTft) = lcons false (F*T\u2192List FTft)\n","old_contents":"------------------------------------------------------------------------------\n-- Fair properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.Fair.Properties where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nFair-Stream : \u2200 {os} \u2192 Fair os \u2192 Stream os\nFair-Stream {os} Fos = Stream-coind A h refl\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 xs\n\n  h : A os \u2192 \u2203[ o' ] \u2203[ os' ] os \u2261 o' \u2237 os' \u2227 A os'\n  h _ with head-tail-Fair Fos\n  h x\u2081 | inj\u2081 prf = T , tail\u2081 os , prf , refl\n  h x\u2081 | inj\u2082 prf = F , tail\u2081 os , prf , refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"1449ed6a218d0c91f87a8237a1066ec1fe9f5c2d","subject":"format","message":"format\n","repos":"haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc,haroldcarr\/learn-haskell-coq-ml-etc","old_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_file":"agda\/paper\/Dijkstra-monad\/2019-03-Wouter-Swierstra_A_Predicate_Transformer_for_Effects\/wouter-swierstra-predicate-transformers-6fe8f13\/PredicateTransformers.agda","new_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\nabstract\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to\n    - use this refinement relation to show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure x)   = Pure (f x)\n  map f (Step c k) = Step c (\\r -> map f (k r))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure x   >>= f = f x\n  Step c x >>= f = Step c (\\r -> x r >>= f)\n  infixr 20 _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition semantics\n\n  idea of associating weakest precondition semantics with imperative programs\n  dates to Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  call values of type a -> Set : predicate on type a\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- function f : a -> b and\n  -- desired postcondition on the function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n\n  -- non-dependent version\n  -- note: definition is just reverse function composition\n  wp0 : forall\n       {a : Set}  -> {b :     Set}\n    -> (f :      a  -> b)\n    ->            (b   ->  Set)\n    -> (a ->  Set)\n  wp0 f P = \\x -> P   (f x)\n  {-\n  above wp semantics is sometimes too restrictive\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n-- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} -> {b : a -> Set l'}\n    -> (f : (x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (a -> Set l'')\n  wp f P = \\x -> P x (f x)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 x -> P x -> Q x\n\n  -- refinement relation defined between PTs\n  _\u2291_ : {a : Set} -> {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} -> {b : a -> Set} -> {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans {a} {b} {P} {Q} {R} r1 r2 p x y = r2 p x (r1 p x y)\n\n  \u2291-refl  : {a : Set} -> {b : a -> Set} -> {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl {a} {b} {P} p x H = H\n\n  \u2291-eq    : {a b : Set}\n         -> (f g : a -> b) -> wp f \u2291 wp g -> (x : a)\n         -> f x == g x\n  \u2291-eq f g R x = R (\\_ y -> f x == y) x refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b) -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq P x H with f x | g x | eq x\n  ... | _ | _ | refl = H\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type Partial a will either\n  - return a value of type a or\n  - fail, issuing abort command\n\n  Note that responses to Abort command are empty;\n  abort smart constructor abort uses this to discharge the continuation\n  in the second argument of the Step constructor\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- def rules out erroneous results by requiring the divisor evaluates to non-zero\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {x}\n         -> Val x \u21d3 x\n    \u21d3Step : forall {l r v1 v2}\n         ->     l   \u21d3  v1\n         ->       r \u21d3         (Succ v2)\n         -> Div l r \u21d3 (v1 div (Succ v2))\n\n  exb : Val 3 \u21d3 3\n  exb = \u21d3Base\n\n  exs : Div (Val 3) (Val 3) \u21d3 1\n  exs = \u21d3Step \u21d3Base \u21d3Base\n\n  -- alternatively, semantics specified by an INTERPRETER\n  -- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- define operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  How to relate two definitions:\n  - std lib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign a weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> (P : (x : a) -> b x -> Set)\n        ->      (x : a)\n        ->                  Partial (b x)\n        -> Set\n  mustPT P _ (Pure y)       = P _ y\n  mustPT P _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> (f : (x : a) -> Partial (b x))\n           -> (P : (x : a) ->          b x -> Set)\n           -> (a -> Set)\n  wpPartial f P = wp f (mustPT P)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by def mustPT\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter: \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, consider defining this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3)) \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0)) \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1      | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1      | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x       | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2             | ih1 | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and postcondition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","old_contents":"module PredicateTransformers where\n\nopen import Prelude hiding (map; all)\nopen import Level hiding (lift)\n\n------------------------------------------------------------------------------\n\n{-\nhttps:\/\/webspace.science.uu.nl\/~swier004\/publications\/2019-icfp-submission-a.pdf\n\nA predicate transformer semantics for effects\n\nWOUTER SWIERSTRA and TIM BAANEN, Universiteit Utrecht\n\nabstract\n\nreasoning about effectful code harder than pure code\n\npredicate transformer (PT) semantics gives a refinement relation that can be used to\n- relate a program to its specification, or\n- calculate effectful programs that are correct by construction\n\n------------------------------------------------------------------------------\n1 INTRODUCTION\n\nkey techniques\n- syntax of effectful computations represented as free monads\n  - assigning meaning to these monads gives meaning to the syntactic ops each effect provides\n- paper shows how to assign PT semantics to computations arising from Kleisli arrows on free monads\n  - enables computing the weakest precondition associated with a given postcondition\n- using weakest precondition semantics\n  - define refinement on computations\n  - show how to\n    - use this refinement relation to show a program satisfies its specification, or\n    - calculate a program from its specification.\n- show how programs and specifications may be mixed,\n  enabling verified programs to be calculated from their specification one step at a time\n\n------------------------------------------------------------------------------\n2 BACKGROUND\n\n--------------------------------------------------\nFree monads\n-}\n\nmodule Free where\n  -- C          : type of commands\n  -- Free C R   : returns an 'a' or issues command c : C\n  -- For each c : C, there is a set of responses R c\n  -- 2nd arg of Step is continuation : how to proceed after receiving response R c\n  data Free {l : Level} (C : Set) (R : C -> Set) (a : Set l) : Set l where\n    Pure : a                              -> Free C R a\n    Step : (c : C) -> (R c -> Free C R a) -> Free C R a\n\n  -- show that 'Free' is a monad:\n\n  map : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n     -> (a -> b) -> Free C R a\n     -> Free C R b\n  map f (Pure x)   = Pure (f x)\n  map f (Step c k) = Step c (\\r -> map f (k r))\n\n  return : forall {l C R} -> {a : Set l}\n        -> a -> Free C R a\n  return = Pure\n\n  _>>=_ : forall {l l' C R} -> {a : Set l} -> {b : Set l'}\n       -> Free C R a -> (a -> Free C R b)\n       -> Free C R b\n  Pure x   >>= f = f x\n  Step c x >>= f = Step c (\\r -> x r >>= f)\n  infixr 20 _>>=_\n\n  _>>_ : forall {l l' C R} {a : Set l} {b : Set l'}\n      -> Free C R a -> Free C R b\n      -> Free C R b\n  c1 >> c2 = c1 >>= \\_ -> c2\n\n  {-\n  different effects choose C and R differently, depending on their ops\n\n  --------------------------------------------------\n  Weakest precondition semantics\n\n  idea of associating weakest precondition semantics with imperative programs\n  dates to Dijkstra\u2019s Guarded Command Language [1975]\n\n  ways to specify behaviour of function f : a -> b\n  - reference implementation\n  - define a relation R : a -> b -> Set\n  - write contracts and test cases\n  - PT semantics\n\n  call values of type a -> Set : predicate on type a\n\n  PTs are functions between predicates\n  e.g., weakest precondition:\n  -}\n\n  -- \"maps\"\n  -- function f : a -> b and\n  -- desired postcondition on the function\u2019s output, b -> Set\n  -- to weakest precondition a -> Set on function\u2019s input that ensures postcondition satisfied\n  --\n\n  -- non-dependent version\n  -- note: definition is just reverse function composition\n  wp0 : forall\n       {a : Set}  -> {b :     Set}\n    -> (f :      a  -> b)\n    ->            (b   ->  Set)\n    -> (a ->  Set)\n  wp0 f P = \\x -> P   (f x)\n  {-\n  above wp semantics is sometimes too restrictive\n  - no way to specify that output is related to input\n  - fix via making f dependent:\n  -}\n\n-- dependent version\n  wp : forall {l l' l''}\n    -> {a : Set l} -> {b : a -> Set l'}\n    -> (f : (x : a) -> b x)\n    -> ((x : a) -> b x -> Set l'')\n    -> (a -> Set l'')\n  wp f P = \\x -> P x (f x)\n\n  -- shorthand for working with predicates and predicates transformers\n  _\u2286_ : forall {l'} -> {a : Set}\n     -> (a -> Set l')\n     -> (a -> Set l')\n     -> Set l'\n  P \u2286 Q = \u2200 x -> P x -> Q x\n\n  -- refinement relation defined between PTs\n  _\u2291_ : {a : Set} -> {b : a -> Set}\n     -> (pt1 pt2 : ((x : a) -> b x -> Set) -> (a -> Set))\n     -> Set\u2081\n  pt1 \u2291 pt2 = forall P -> pt1 P \u2286 pt2 P\n\n  \u2291-trans : {a : Set} -> {b : a -> Set} -> {P Q R : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 Q -> Q \u2291 R\n         -> P \u2291 R\n  \u2291-trans {a} {b} {P} {Q} {R} r1 r2 p x y = r2 p x (r1 p x y)\n\n  \u2291-refl  : {a : Set} -> {b : a -> Set} -> {P : ((x : a) -> b x -> Set) -> (a -> Set)}\n         -> P \u2291 P\n  \u2291-refl {a} {b} {P} p x H = H\n\n  \u2291-eq    : {a b : Set}\n         -> (f g : a -> b) -> wp f \u2291 wp g -> (x : a)\n         -> f x == g x\n  \u2291-eq f g R x = R (\\_ y -> f x == y) x refl\n\n  eq-\u2291    : {a b : Set}\n         -> (f g : a -> b) -> ((x : a) -> f x == g x)\n         -> wp f \u2291 wp g\n  eq-\u2291 f g eq P x H with f x | g x | eq x\n  ... | _ | _ | refl = H\n\n  {-\n  use refinement relation\n  - to relate PT semantics between programs and specifications\n  - to show a program satisfies its specification; or\n  - to show that one program is somehow \u2018better\u2019 than another,\n    where \u2018better\u2019 is defined by choice of PT semantics\n\n  in pure setting, this refinement relation is not interesting:\n  the refinement relation corresponds to extensional equality between functions:\n\n  lemma follows from the \u2018Leibniz rule\u2019 for equality in intensional type theory:\n  refinement : \u2200 (f g : a -> b) -> (wp f \u2291 wp g) \u2194 (\u2200 x -> f x \u2261 g x)\n\n  this paper defines PT semantics for Kleisli arrows of form\n\n      a -> Free C R b\n\n  could use 'wp' to assign semantics to these computations directly,\n  but typically not interested in syntactic equality between free monads\n\n  rather want to study semantics of effectful programs they represent\n\n  to define a PT semantics for effects\n  define a function with form:\n\n    -- how to lift a predicate on 'a' over effectful computation returning 'a'\n    pt : (a -> Set) -> Free C R a -> Set\n\n  'pt' def depends on semantics desired for a particulr free monad\n\n  Crucially, choice of\n  - pt and\n  - weakest precondition semantics :  wp\n  together give a way to assign weakest precondition semantics\n  to Kleisli arrows representing effectful computations\n  -}\n\n------------------------------------------------------------------------------\n\nmodule Maybe where\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n  open import Data.Nat.DivMod using (_div_)\n  open import Data.Nat.Properties using (*-zero\u02b3)\n  open Free\n\n  {-\n  ------------------------------------------------------------------------------\n  3 PARTIALITY\n\n  Partial computations : i.e., 'Maybe'\n\n  make choices for commands C and responses R\n  -}\n\n  data C : Set where\n    Abort : C -- no continuation\n\n  R : C -> Set\n  R Abort = \u22a5 -- since C has no continuation, valid responses is empty\n\n  Partial : Set -> Set\n  Partial = Free C R\n\n  -- smart constructor for failure\n  abort : forall {a} -> Partial a\n  abort = Step Abort (\\())\n\n  {-\n  computation of type Partial a will either\n  - return a value of type a or\n  - fail, issuing abort command\n\n  Note that responses to Abort command are empty;\n  abort smart constructor abort uses this to discharge the continuation\n  in the second argument of the Step constructor\n\n  --------------------------------------------------\n  Example: division\n\n  expression language, closed under division and natural numbers:\n  -}\n\n  data Expr : Set where\n    Val : Nat  -> Expr\n    Div : Expr -> Expr -> Expr\n\n  exv : Expr\n  exv = Val 3\n  exd : Expr\n  exd = Div (Val 3) (Val 3)\n\n  -- semantics specified using inductively defined RELATION:\n  -- def rules out erroneous results by requiring the divisor evaluates to non-zero\n  data _\u21d3_ : Expr -> Nat -> Set where\n    \u21d3Base : forall {x}\n         -> Val x \u21d3 x\n    \u21d3Step : forall {l r v1 v2}\n         ->     l   \u21d3  v1\n         ->       r \u21d3         (Succ v2)\n         -> Div l r \u21d3 (v1 div (Succ v2))\n\n  exb : Val 3 \u21d3 3\n  exb = \u21d3Base\n\n  exs : Div (Val 3) (Val 3) \u21d3 1\n  exs = \u21d3Step \u21d3Base \u21d3Base\n\n  -- alternatively, semantics specified by an INTERPRETER\n  -- evaluate Expr via monadic INTERPRETER, using Partial to handle division-by-zero\n\n  -- define operation used by \u27e6_\u27e7 interpreter\n  _\u00f7_ : Nat -> Nat -> Partial Nat\n  n \u00f7 Zero     = abort\n  n \u00f7 (Succ k) = return (n div (Succ k))\n\n  \u27e6_\u27e7 : Expr -> Partial Nat\n  \u27e6 Val x \u27e7     =  return x\n  \u27e6 Div e1 e2 \u27e7 =  \u27e6 e1 \u27e7 >>= \\v1 ->\n                   \u27e6 e2 \u27e7 >>= \\v2 ->\n                   v1 \u00f7 v2\n\n  evv   : Free C R Nat\n  evv   = \u27e6 Val 3 \u27e7\n  evv'  : evv \u2261 Pure 3\n  evv'  = refl\n\n  evd   : Free C R Nat\n  evd   = \u27e6 Div (Val 3) (Val 3) \u27e7\n  evd'  : evd \u2261 Pure 1\n  evd'  = refl\n\n  evd0  : Free C R Nat\n  evd0  = \u27e6 Div (Val 3) (Val 0) \u27e7\n  evd0' : evd0 \u2261 Step Abort (\u03bb ())\n  evd0' = refl\n\n  {-\n  How to relate two definitions:\n  - std lib 'div' requires implicit proof that divisor is non-zero\n    - \u21d3 relation generates via pattern matching\n    - _\u00f7_ does explicit check\n  - interpreter uses _\u00f7_\n    - fails explicitly with abort when divisor is zero\n\n  Assign a weakest precondition semantics to Kleisli arrows of the form\n\n      a -> Partial b\n  -}\n\n  mustPT : forall {a : Set} -> {b : a -> Set}\n        -> (P : (x : a) -> b x -> Set)\n        ->      (x : a)\n        ->                  Partial (b x)\n        -> Set\n  mustPT P _ (Pure y)       = P _ y\n  mustPT P _ (Step Abort _) = \u22a5\n\n  wpPartial : {a : Set} -> {b : a -> Set}\n           -> (f : (x : a) -> Partial (b x))\n           -> (P : (x : a) ->          b x -> Set)\n           -> (a -> Set)\n  wpPartial f P = wp f (mustPT P)\n\n  {-\n  To call 'wp', must show how to transform\n  -      predicate                  P :         b -> Set\n  - to a predicate on partial results : Partial b -> Set\n  Done via proposition 'mustPT P c'\n  - holds when computation c of type Partial b successfully returns a 'b' that satisfies P\n\n  particular PT semantics of partial computations determined by def mustPT\n  here: rule out failure entirely\n  - so case Abort returns empty type\n\n  Given this PT semantics for Kleisli arrows in general,\n  can now study semantics of above monadic interpreter\n  via passing\n  - interpreter: \u27e6_\u27e7\n  - desired postcondition : _\u21d3_\n  as arguments to wpPartial:\n  -}\n\n  exwpp  : Expr -> Set\n  exwpp  = wpPartial \u27e6_\u27e7 _\u21d3_\n  exwpp' : wpPartial \u27e6_\u27e7 _\u21d3_ \u2261 \u03bb expr -> mustPT _\u21d3_ expr \u27e6 expr \u27e7\n  exwpp' = refl\n\n  wppv1  : wpPartial \u27e6_\u27e7 _\u21d3_ (Val 1)\n  wppv1  = \u21d3Base\n  wppd   : wpPartial \u27e6_\u27e7 _\u21d3_ (Div (Val 1) (Val 1))\n  wppd   = \u21d3Step \u21d3Base \u21d3Base\n\n  xxx    : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (Pure 3 >>= (\u03bb v1 -> Pure 3 >>= _\u00f7_ v1))\n  xxx    = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  xxx'   : mustPT _\u21d3_ (Div (Val 3) (Val 3)) (3 \u00f7 3)\n  xxx'   = \u21d3Step {Val 3} {Val 3} {3} {2} \u21d3Base \u21d3Base\n\n  {-\n  resulting in a predicate on expressions\n\n  for all expressions satisfying this predicate,\n  the monadic interpreter and the relational specification, _\u21d3_,\n  must agree on the result of evaluation\n\n  What does this say about correctness of interpreter?\n  To understand the predicate better, consider defining this predicate on expressions:\n  -}\n\n  SafeDiv : Expr -> Set\n  SafeDiv (Val x)     = \u22a4\n  SafeDiv (Div e1 e2) = (e2 \u21d3 Zero -> \u22a5) \u2227 SafeDiv e1 \u2227 SafeDiv e2\n\n  exsdv : SafeDiv (Val 3) \u2261 \u22a4\n  exsdv = refl\n  exsdd : SafeDiv (Div (Val 3) (Val 3)) \u2261 Pair ((x : Val    3 \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdd = refl\n\n  exsdx : SafeDiv (Div (Val 3) (Val 0)) \u2261 Pair ((x : Val Zero \u21d3 Zero) -> \u22a5) (Pair (\u22a4' zero) (\u22a4' zero))\n  exsdx = refl\n\n  {-\n  Expect : any expr e for which SafeDiv e holds\n  can be evaluated without division-by-zero\n\n  can prove SafeDiv is sufficient condition for two notions of evaluation to coincide:\n\n  -- lemma relates the two semantics\n  -- expressed as a relation and an evaluator\n  -- for those expressions that satisfy the SafeDiv property\n  -}\n\n  correct : SafeDiv \u2286 wpPartial \u27e6_\u27e7 _\u21d3_\n  correct (Val x) h = \u21d3Base\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | correct e1 sde1 | correct e2 sde2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure Zero         | e1\u21d3v1 | e2\u21d3Z   = magic (e2nz e2\u21d3Z)\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Pure (Succ v2)    | e1\u21d3v1 | e2\u21d3Sv2 = \u21d3Step e1\u21d3v1 e2\u21d3Sv2\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Pure v1      | Step Abort \u22a5\u2192FCRN | e1\u21d3v1 | ()\n  correct (Div e1 e2) (e2nz , (sde1 , sde2)) | Step Abort _ | _                 | ()    | _\n\n  {-\n  Instead of manually defining SafeDiv,\n  define more general predicate characterising the domain of a partial function:\n  -}\n\n  dom : {a : Set} -> {b : a -> Set}\n     -> ((x : a)\n     -> Partial (b x))\n     -> (a -> Set)\n  dom f = wpPartial f (\\_ _ -> \u22a4)\n\n  -- can show that the two semantics agree precisely on the domain of the interpreter:\n\n  sound     : dom       \u27e6_\u27e7       \u2286   wpPartial \u27e6_\u27e7 _\u21d3_\n\n  complete  : wpPartial \u27e6_\u27e7 _\u21d3_   \u2286   dom       \u27e6_\u27e7\n\n  -- both proofs proceed by induction on the argument expression\n\n  sound (Val x) h = \u21d3Base\n  sound (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7 | sound e1 | sound e2\n  sound (Div e1 e2) () | Pure v1 | Pure Zero      | ih1 | ih2\n  sound (Div e1 e2) h  | Pure v1 | Pure (Succ v2) | ih1 | ih2 = \u21d3Step (ih1 tt) (ih2 tt)\n  sound (Div e1 e2) () | Pure x  | Step Abort x\u2081  | ih1 | ih2\n  sound (Div e1 e2) () | Step Abort x | v2 | ih1  | ih2\n\n  inDom : {v : Nat} -> (e : Expr) -> \u27e6 e \u27e7 == Pure v -> dom \u27e6_\u27e7 e\n  inDom (Val x) h = tt\n  inDom (Div e1 e2) h with \u27e6 e1 \u27e7 | \u27e6 e2 \u27e7\n  inDom (Div e1 e2) () | Pure v1      | Pure Zero\n  inDom (Div e1 e2) h  | Pure v1      | Pure (Succ v2) = tt\n  inDom (Div e1 e2) () | Pure _       | Step Abort _\n  inDom (Div e1 e2) () | Step Abort _ | _\n\n  aux : (e : Expr) (v : Nat) -> \u27e6 e \u27e7 \u2261 Pure v -> e \u21d3 v\n  aux e v eq with sound e (inDom e eq)\n  ... | H rewrite eq = H\n\n  wpPartial1 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e1\n  wpPartial1 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Pure Zero\n  wpPartial1 {e1} {e2} h  | Pure x       | [[[ eq ]]] | Pure (Succ y) = aux e1 x eq\n  wpPartial1 {e1} {e2} () | Pure x       | eq         | Step Abort x\u2081\n  wpPartial1 {e1} {e2} () | Step Abort x | eq         | ve2\n\n  wpPartial2 : {e1 e2 : Expr} -> wpPartial \u27e6_\u27e7 _\u21d3_ (Div e1 e2) -> wpPartial \u27e6_\u27e7 _\u21d3_ e2\n  wpPartial2 {e1} {e2} h with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n  wpPartial2 {e1} {e2} h  | Pure x       | [[[ eqx ]]] | Pure y        | [[[ eqy ]]] = aux e2 y eqy\n  wpPartial2 {e1} {e2} () | Pure x       | [[[ eq ]]]  | Step Abort x\u2081 | eq2\n  wpPartial2 {_}  {_}  () | Step Abort x | eq1         | se2           | eq2\n\n  complete (Val x) h = tt\n  complete (Div e1 e2) h\n    with \u27e6 e1 \u27e7 | inspect \u27e6_\u27e7 e1 | \u27e6 e2 \u27e7 | inspect \u27e6_\u27e7 e2\n      | complete e1 (wpPartial1 {e1} {e2} h)\n      | complete e2 (wpPartial2 {e1} {e2} h)\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure Zero     | [[[ eqy ]]] | p1 | p2\n    rewrite eqx | eqy = magic h\n  complete (Div e1 e2) h | Pure x       | [[[ eqx ]]] | Pure (Succ y) | [[[ eqy ]]] | p1 | p2 = tt\n  complete (Div e1 e2) h | Pure x       | eq1         | Step Abort x\u2081 | eq2         | p1 | ()\n  complete (Div e1 e2) h | Step Abort x | eq1         | se2           | eq2         | () | p2\n\n  {-\n  --------------------------------------------------\n  Refinement\n\n  weakest precondition semantics on partial computations give rise\n  to a refinement relation on Kleisli arrows of the form a -> Partial b\n\n  can characterise this relation by proving:\n\n  refinement : (f g : a -> Maybe b)\n             -> (wpPartial f \u2291 wpPartial g) \u2194 (\u2200 x -> (f x \u2261 g x) \u2228 (f x \u2261 Nothing))\n\n  use refinement to relate Kleisli morphisms,\n  and to relate a program to a specification given by a pre- and postcondition\n\n  --------------------------------------------------\n  Example: Add (interpreter for stack machine)\n\n  add top two elements; can fail fail if stack has too few elements\n\n  below shows how to prove the definition meets its specification\n\n  Define specification in terms of a pre\/post condition.\n  -}\n  -- specification of a function of type (x : a) -> b x consists of:\n  record Spec {l : Level} (a : Set) (b : a -> Set) : Set (suc l) where\n    constructor [[_,_]]\n    field\n      pre   : a -> Set l              -- precondition on 'a'\n      post  : (x : a) -> b x -> Set l -- postcondition relating inputs that satisfy this precondition\n                                      -- and the corresponding outputs\n\n  {-\n  [ P , Q ] : specification consisting of precondition P and postcondition Q\n\n  for non-dependent examples (e.g., type b does not depend on x : a)\n  -}\n\n  SpecK : {l : Level} -> Set -> Set -> Set (suc l)\n  SpecK a b = Spec a (K b) -- K is constant function\n\n  -- specification for addition function : describes the desired postcondition\n  data Add : List Nat -> List Nat -> Set where\n    AddStep : {x1 x2 : Nat} -> {xs : List Nat}\n           -> Add (x1 :: x2 :: xs) ((x1 + x2) :: xs)\n\n  addSpec : SpecK (List Nat) (List Nat)\n  addSpec = [[ (\\xs -> length xs > 1) , Add ]]\n\n  {-\n  How to relate this specification to an implementation?\n  Note: 'wpPartial' assigns predicate transformer semantics to functions\n  - but do not yet have a corresponding predicate transform semantics for specifications.\n  wpSpec function does that:\n  -}\n\n  -- Given a specification, Spec a b\n  -- computes the weakest precondition that satisfies an arbitrary postcondition P\n  -- i.e., the spec\u2019s precondition should hold and its postcondition must imply P.\n  wpSpec : forall {l a} -> {b : a -> Set}\n        -> Spec {l} a b\n        -> (P : (x : a) -> b x -> Set l)\n        -> (a -> Set l)\n  wpSpec [[ pre , post ]] P = \\x -> (pre x) \u2227 (post x \u2286 P x)\n\n  -- Can now formulate program\n\n  pop : forall {a} -> List a -> Partial (a \u00d7 List a)\n  pop Nil       = abort\n  pop (x :: xs) = return (x , xs)\n\n  add : List Nat -> Partial (List Nat)\n  add xs =\n    pop xs >>= \\{(x1 , xs) ->\n    pop xs >>= \\{(x2 , xs) ->\n    return ((x1 + x2) :: xs)}}\n\n  -- that refines the specification given by addSpec:\n  correctnessAdd : wpSpec addSpec \u2291 wpPartial add\n  correctnessAdd P Nil (() , _)\n  correctnessAdd P (x :: Nil) (Data.Nat.s\u2264s () , _)\n  correctnessAdd P (x :: y :: xs) (_ , H) = H (x + y :: xs) AddStep\n\n  {-\n  This example illustrates how to use the refinement relation\n  - to relate a specification\n    - given in terms of a pre- and postcondition,\n  - to its implementation.\n\n  Compared to the refinement calculus\n  - have not yet described how to mix code and specifications (see Section 7)\n\n  --------------------------------------------------\n  Alternative semantics\n  -}\n\n  product : List Nat -> Nat\n  product = foldr _*_ 1\n\n  fastProduct : List Nat -> Partial Nat\n  fastProduct Nil          = return 1\n  fastProduct (Zero :: xs) = abort\n  fastProduct (k :: xs)    = map (_*_ k) (fastProduct xs)\n\n  defaultHandler : forall {a} -> a -> Partial a -> a\n  defaultHandler _ (Pure x)       = x\n  defaultHandler d (Step Abort _) = d\n\n  wpDefault : forall {a b : Set}\n           -> (d : b)\n           -> (f : a -> Partial b)\n           -> (P : a -> b -> Set)\n           -> (a -> Set)\n  wpDefault {a} {b} d f P = wp f defaultPT\n    where\n    defaultPT : (x : a) -> Partial b -> Set\n    defaultPT x (Pure y)       = P x y\n    defaultPT x (Step Abort _) = P x d\n\n  soundness : forall {a b}\n           -> (P : a -> b -> Set)\n           -> (d : b)\n           -> (c : a -> Partial b)\n           -> forall x\n           -> wpDefault d c P x\n           -> P x (defaultHandler d (c x))\n  soundness P d c x H with c x\n  soundness P d c x H | Pure y       = H\n  soundness P d c x H | Step Abort _ = H\n\n  correctnessProduct : wp product \u2291 wpDefault 0 fastProduct\n  correctnessProduct   P Nil H = H\n  correctnessProduct   P (Zero :: xs) H = H\n  correctnessProduct   P (Succ x :: xs) H\n    with fastProduct xs | correctnessProduct (\\xs v -> P (Succ x :: xs) _) xs H\n  correctnessProduct   P (Succ x :: xs) H | Pure v       | IH                   = IH\n  correctnessProduct   P (Succ x :: xs) H | Step Abort _ | IH rewrite *-zero\u02b3 x = IH\n\nmodule State (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  data C : Set where\n    Get : C\n    Put : s -> C\n\n  R : C -> Set\n  R Get     = s\n  R (Put _) = \u22a4\n\n  State : forall {l} -> Set l -> Set l\n  State = Free C R\n  get : State s\n  get = Step Get return\n\n  put : s -> State \u22a4\n  put s = Step (Put s) (\\_ -> return tt)\n\n  run : {a : Set} -> State a -> s -> a \u00d7 s\n  run (Pure x)         s = (x , s)\n  run (Step Get k)     s = run (k s) s\n  run (Step (Put s) k) _ = run (k tt) s\n\n  statePT : forall {l l'} -> {b : Set l} -> (b \u00d7 s -> Set l') -> State b -> (s -> Set l')\n  statePT P (Pure x)         = \\s ->         P (x , s)\n  statePT P (Step  Get    k) = \\s -> statePT P (k  s) s\n  statePT P (Step (Put s) k) = \\_ -> statePT P (k tt) s\n\n  statePT' : forall {l l'} -> {b : Set l}\n          -> (s -> b \u00d7 s -> Set l')\n          -> State b\n          -> (s -> Set l')\n  statePT' P c i = statePT (P i) c i\n\n  wpState : forall {l l' l''} -> {a : Set l} -> {b : Set l'}\n         -> (a -> State b)\n         -> (P : a \u00d7 s -> b \u00d7 s -> Set l'')\n         -> (a \u00d7 s -> Set l'')\n  wpState f P (x , i) = wp f ((const \\c -> statePT' (\\j -> P (x , j)) c i)) x\n\n  soundness : forall {a b : Set}\n           -> (P : a \u00d7 s -> b \u00d7 s -> Set)\n           -> (f : a -> State b)\n           -> forall i x\n           -> wpState f P (x , i)\n           -> P (x , i) (run (f x) i)\n  soundness {a} {b} P c i x H = lemma i (c x) H\n    where\n    lemma : (st : s) -> (statec : State b) -> (statePT (P (x , i)) statec st) -> P (x , i) (run statec st)\n    lemma i (Pure y)         H = H\n    lemma i (Step Get     k) H = lemma i (k  i) H\n    lemma i (Step (Put s) k) H = lemma s (k tt) H\n\nmodule Relabel where\n  open Free\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      )\n\n  module StateNat = State Nat\n  open StateNat\n\n  data Tree (a : Set) : Set where\n    Leaf  :      a           -> Tree a\n    Node  : Tree a -> Tree a -> Tree a\n\n  flatten : \u2200 {a} -> Tree a -> List a\n  flatten (Leaf x)   = [ x ]\n  flatten (Node l r) = flatten l ++ flatten r\n\n  size : \u2200 {a} -> Tree a -> Nat\n  size (Leaf x)   = 1\n  size (Node l r) = size l + size r\n\n  seq : Nat -> Nat -> List Nat\n  seq i Zero     = Nil\n  seq i (Succ n) = Cons i (seq (Succ i) n)\n\n  relabelSpec : forall {a} -> SpecK (Tree a \u00d7 Nat) (Tree Nat \u00d7 Nat)\n  relabelSpec = [[ K \u22a4 , relabelPost ]]\n    where\n      relabelPost : forall {a} -> Tree a \u00d7 Nat -> Tree Nat \u00d7 Nat -> Set\n      relabelPost (t , s) (t' , s') = (flatten t' == (seq (s) (size t))) \u2227 (s + size t == s')\n\n  fresh : State Nat\n  fresh =  get >>= \\n ->\n           put (Succ n) >>\n           return n\n\n  relabel : forall {a} -> Tree a -> State (Tree Nat)\n  relabel (Leaf x)   = map Leaf fresh\n  relabel (Node l r) =\n    relabel l >>= \\l' ->\n    relabel r >>= \\r' ->\n    return (Node l' r')\n\n  correctnessRelabel : forall {a : Set} -> wpSpec ( relabelSpec {a}) \u2291 wpState relabel\n\n  compositionality : forall {a b : Set} -> (c : State a) (f : a -> State b) ->\n    \u2200 i P -> statePT P (c >>= f) i == statePT (wpState f (const P)) c i\n\n  compositionality (Pure x) f i P = refl\n  compositionality (Step Get k) f i P = compositionality (k i) f i P\n  compositionality (Step (Put x) k) f i P = compositionality (k tt) f x P\n\n  correctnessRelabel = step2\n    where\n    open NaturalLemmas\n    --   Simplify proofs of refining a specification,\n    --   by first proving one side of the bind, then the second.\n    --   This is essentially the first law of consequence,\n    --   specialized to the effects of State and Spec.\n    prove-bind : \u2200 {a b} (mx : State a) (f : a -> State b) P i ->\n      statePT (wpState f \\_ -> P) mx i -> statePT P (mx >>= f) i\n    prove-bind mx f P i x = coerce {zero} (sym (compositionality mx f i P)) x\n\n    prove-bind-spec : \u2200 {a b} (mx : State a) (f : a -> State b) spec ->\n      \u2200 P i -> (\u2200 Q -> Spec.pre spec i \u00d7 (Spec.post spec i \u2286 Q) -> statePT Q mx i) ->\n      Spec.pre spec i \u00d7 (Spec.post spec i \u2286 wpState f (\\_ -> P)) ->\n      statePT P (mx >>= f) i\n    prove-bind-spec mx f spec P i Hmx Hf = prove-bind mx f P i (Hmx (wpState f (\\_ -> P)) Hf)\n\n    --   Partially apply a specification.\n    applySpec : \u2200 {a b s} -> SpecK {zero} (a \u00d7 s) (b \u00d7 s) -> a -> SpecK s (b \u00d7 s)\n    applySpec [[ pre , post ]] x = [[ (\\s -> pre (x , s)) , (\\s -> post (x , s)) ]]\n\n    --   Ingredients for proving the postcondition holds.\n    --   By abstracting over the origin of the numbers,\n    --   we can do induction on them nicely.\n    append-seq : \u2200 a b c -> seq a b ++ seq (a + b) c \u2261 seq a (b + c)\n    append-seq a Zero c = cong (\\a' -> seq a' c) (sym (plus-zero a))\n    append-seq a (Succ b) c = cong (Cons a) (trans\n      (cong (\\a+b -> seq (Succ a) b ++ seq a+b c) (+-succ a b))\n      (append-seq (Succ a) b c))\n\n    postcondition : \u2200 s s' s'' sl fl sr fr ->\n      Pair (fl \u2261 seq s sl) (s + sl \u2261 s') ->\n      Pair (fr \u2261 seq s' sr) (s' + sr \u2261 s'') ->\n      Pair (fl ++ fr \u2261 seq s (sl + sr)) (s + (sl + sr) \u2261 s'')\n    postcondition s .(s + sl) .(s + sl + sr) sl .(seq s sl) sr .(seq (s + sl) sr)\n      (refl , refl) (refl , refl) = append-seq s sl sr , +-assoc s sl sr\n\n    --   We have to rewrite the formulation of step2 slightly to make it acceptable for the termination checker.\n\n    step2' : \u2200 {a} P (t : Tree a) s -> wpSpec relabelSpec P (t , s) -> statePT (P (t , s)) (relabel t) s\n    step2' P (Leaf x) s (fst , snd) = snd (Leaf s , Succ s) (refl , plus-one s)\n    step2' P (Node l r) s (fst , snd) = prove-bind-spec (relabel l) _ (applySpec relabelSpec l) _ _\n      (\\Q -> step2' (\\_ -> Q) l s)\n      (tt , \\l',s' postL -> let l' = Pair.fst l',s' ; s' = Pair.snd l',s'\n        in prove-bind-spec (relabel r) _ (applySpec relabelSpec r) _ _\n          (\\Q -> step2' (\\_ -> Q) r s')\n          (tt , \\r',s'' postR -> let r' = Pair.fst r',s'' ; s'' = Pair.snd r',s''\n            in snd (Node l' r' , s'') (postcondition s s' s'' (size l) (flatten l') (size r) (flatten r') postL postR)))\n    step2 : wpSpec relabelSpec \u2291 wpState relabel\n    step2 P (t , s) (fst , snd) = step2' P t s (fst , snd)\n\nmodule Compositionality\n  (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set)\n  where\n  open Free\n  open Maybe using (wpSpec; [[_,_]])\n\n  postulate\n    ext : {l l' : Level} {a : Set l} {b : Set l'} -> {f g : a -> b} ->\n      ((x : a) -> f x \u2261 g x) -> f \u2261 g\n\n  pt : {a : Set} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure   x) P = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : {a : Set} {b : a -> Set} ->\n      ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  compositionality : forall {a b : Set} -> (c : Free C R a) (f : a -> Free C R b) ->\n    \u2200 P -> pt (c >>= f) P \u2261 pt c (wpCR f (const P))\n  compositionality (Pure x) f P = refl\n  compositionality (Step c x) f P =\n    cong (\\h -> ptalgebra c h) (ext (\\r -> compositionality (x r) f P))\n\n  _>=>_ : forall {l l' l''} -> {a : Set l} -> {b : Set l'} -> {c : Set l''} -> forall {C R} -> (a -> Free C R b) -> (b -> Free C R c) -> a -> Free C R c\n  f >=> g = \\x -> f x >>= g\n\n  compositionality-left : forall {a b c : Set} -> (f1 f2 : a -> Free C R b) (g : b -> Free C R c) ->\n    wpCR f1 \u2291 wpCR f2 ->\n    wpCR (f1 >=> g) \u2291 wpCR (f2 >=> g)\n  compositionality-left mx my f H P x y\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (my x) f (P x) =\n     H (\\x y -> pt (f y) (P x)) x y\n\n  compositionality-right : forall {a b c} -> (f : a -> Free C R b) (g1 g2 : b -> Free C R c) ->\n    wpCR g1 \u2291 wpCR g2 ->\n    wpCR (f >=> g1) \u2291 wpCR (f >=> g2)\n  postulate\n    monotonicity : forall {a} -> {P Q : a -> Set} -> P \u2286 Q -> (c : Free C R a) -> pt c P -> pt c Q\n  compositionality-right mx f g H P x wp1\n    rewrite compositionality (mx x) f (P x)\n    | compositionality (mx x) g (P x) = monotonicity (H _) (mx x) wp1\n\n  weakenPre          : {a : Set} -> {b : a -> Set} -> {P P' : a -> Set} -> {Q : (x : a) -> b x -> Set} -> (P \u2286 P') -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P' , Q ]])\n\n\n  strengthenPost     : {a : Set} -> {b : a -> Set} -> {P : a -> Set} -> {Q Q' : (x : a) -> b x -> Set} -> (forall (x : a) -> Q' x \u2286 Q x) -> (wpSpec [[ P , Q ]] \u2291 wpSpec [[ P , Q' ]])\n\n  weakenPre H1 p H2 (pre , post) = (H1 H2 pre , post)\n\n  strengthenPost H1 p H2 (pre , post) = (pre , \\x y -> post x (H1 _ x y))\n\nmodule Laws (s : Set) where\n  open Free\n  open Maybe using (SpecK; Spec; [[_,_]]; wpSpec)\n  module StateS = State s\n  open StateS\n\n  postulate\n    a : Set\n    k0 : State a\n    k1 : s -> State a\n    k2 : s -> s -> State a\n    x y : s\n\n  _\u2243_ : forall {b : Set} -> State b -> State b -> Set\u2081\n  t1 \u2243 t2 = (wpState' t1 \u2291 wpState' t2) \u2227 (wpState' t2 \u2291 wpState' t1)\n    where\n    wpState' : forall {b} -> State b -> (P : s -> b \u00d7 s -> Set) -> (s -> Set)\n    wpState' {b} t P s = wpState {a = \u22a4} {b} (\\_ -> t) (\\{(tt , s') y -> P s' y}) (tt , s)\n\nmodule Nondeterminism where\n\n  open Free hiding (_\u2286_)\n  open Maybe using (wpSpec; SpecK; [[_,_]])\n  data C : Set where\n    Fail : C\n    Choice : C\n\n  R : C -> Set\n  R Fail   = \u22a5\n  R Choice = Bool\n\n  ND : Set -> Set\n  ND = Free C R\n\n  fail : forall {a} -> ND a\n  fail = Step Fail (\\())\n\n  choice : forall {a} -> ND a -> ND a -> ND a\n  choice c1 c2 = Step Choice (\\b -> if b then c1 else c2)\n\n  allPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  allPT P _ (Pure x)        = P _ x\n  allPT P _ (Step Fail k)   = \u22a4\n  allPT P _ (Step Choice k) = allPT P _ (k True) \u2227 allPT P _ (k False)\n\n  wpAll : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAll f P = wp f (allPT P)\n\n  anyPT : forall {a : Set} -> {b : a -> Set} -> (P : (x : a) -> b x -> Set) -> (x : a) -> ND (b x) -> Set\n  anyPT P _ (Pure x)        = P _ x\n  anyPT P _ (Step Fail k)   = \u22a5\n  anyPT P _ (Step Choice k) = anyPT P _ (k True) \u2228 anyPT P _ (k False)\n\n  wpAny : forall {a   : Set} -> {b : a -> Set} -> ((x : a) -> ND (b x)) -> (P : (x : a) -> b x -> Set) -> (a -> Set)\n  wpAny f P = wp f (anyPT P)\n\n  run : forall {a} -> ND a -> List a\n  run (Pure x)        = [ x ]\n  run (Step Fail _)   = Nil\n  run (Step Choice k) = run (k True) ++ run (k False)\n\n  All : {a : Set} -> (a -> Set) -> List a -> Set\n  All P Nil = \u22a4\n  All P (x :: xs) = P x \u2227 All P xs\n\n  All++ : {a : Set} (P : a -> Set) (xs ys : List a) ->\n    All P xs -> All P ys -> All P (xs ++ ys)\n  All++ P Nil ys H1 H2 = H2\n  All++ P (x :: xs) ys (Px , H1) H2 = Px , All++ P xs ys H1 H2\n\n  allSoundness : {a : Set} {b : a -> Set} (P : (x : a) -> b x -> Set) (x : a) (nd : ND (b x)) ->\n    allPT P x nd -> All (P x) (run nd)\n  allSoundness P x (Pure y) H = H , tt\n  allSoundness P x (Step Fail _) H = tt\n  allSoundness P x (Step Choice k) (H1 , H2) =\n    All++ (P x) (run (k True)) (run (k False)) (allSoundness P x (k True) H1) (allSoundness P x (k False) H2)\n\n  wpAllSoundness : forall {a} -> {b : a -> Set} -> (f : (x : a) -> ND (b x)) ->\n    \u2200 P x -> wpAll f P x -> All (P x) (run (f x))\n  wpAllSoundness nd P x H = allSoundness P x (nd x) H\n  data Elem {a : Set} (x : a) : ND a -> Set where\n      Here   : Elem x (Pure x)\n      Left   : forall {k} ->  Elem x (k True)  -> Elem x (Step Choice k)\n      Right  : forall {k} ->  Elem x (k False) -> Elem x (Step Choice k)\n\n  _\u2286_ : forall {a} -> ND a -> ND a -> Set\n  nd1 \u2286 nd2 = \u2200 x -> Elem x nd1 -> Elem x nd2\n\n  _<->_ : {l l' : Level} (a : Set l) (b : Set l') -> Set (l \u2294 l')\n  a <-> b = Pair (a -> b) (b -> a)\n\n  refineAll  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAll f  \u2291 wpAll g)  <-> ((x : a) -> g x  \u2286 f x)\n\n  refineAny  : {a b : Set} {x : a} (f g : a -> ND b) -> (wpAny f  \u2291 wpAny g)  <-> ((x : a) -> f x  \u2286 g x)\n  allP : \u2200 {a b : Set} {x : a} P (S : ND b) -> allPT P x S <-> (\u2200 y -> Elem y S -> P x y)\n  Pair.fst (allP P (Pure y)) H y Here = H\n  Pair.fst (allP P (Step Choice k)) (H , _) y (Left i) = Pair.fst (allP P (k True)) H y i\n  Pair.fst (allP P (Step Choice k)) (_ , H) y (Right i) = Pair.fst (allP P (k False)) H y i\n  Pair.snd (allP P (Pure y)) H = H y Here\n  Pair.snd (allP P (Step Fail k)) H = tt\n  Pair.snd (allP P (Step Choice k)) H = (Pair.snd (allP P (k True)) \u03bb y i -> H y (Left i)) , (Pair.snd (allP P (k False)) \u03bb y i -> H y (Right i))\n\n  anyP : \u2200 {a b : Set} {x : a} P (S : ND b) -> anyPT P x S <-> Sigma b \u03bb y -> Elem y S \u2227 P x y\n  Pair.fst (anyP P (Pure y)) H = y , (Here , H)\n  Pair.fst (anyP P (Step Fail k)) ()\n  Pair.fst (anyP P (Step Choice k)) (Inl H) with Pair.fst (anyP P (k True)) H\n  Pair.fst (anyP P (Step Choice k)) (Inl H) | y , (i , IH) = y , (Left i , IH)\n  Pair.fst (anyP P (Step Choice k)) (Inr H) with Pair.fst (anyP P (k False)) H\n  Pair.fst (anyP P (Step Choice k)) (Inr H) | y , (i , IH) = y , (Right i , IH)\n  Pair.snd (anyP P (Pure y)) (.y , (Here , H)) = H\n  Pair.snd (anyP P (Step .Choice k)) (y , (Left i , H)) = Inl (Pair.snd (anyP P (k True)) (y , (i , H)))\n  Pair.snd (anyP P (Step .Choice k)) (y , (Right i , H)) = Inr (Pair.snd (anyP P (k False)) (y , (i , H)))\n\n  Pair.fst (refineAll f g) H x y i = Pair.fst (allP (\u03bb _ y' -> Elem y' (f x)) (g x)) (H _ x (Pair.snd (allP _ (f x)) (\u03bb _ -> id))) y i\n  Pair.snd (refineAll f g) r P x H = Pair.snd (allP P (g x)) \u03bb y i -> Pair.fst (allP P (f x)) H y (r x y i)\n  Pair.fst (refineAny f g) H x y i with Pair.fst (anyP (\u03bb _ y' -> y' == y) (g x)) (H _ x (Pair.snd (anyP _ (f x)) (y , (i , refl))))\n  Pair.fst (refineAny f g) H x y i | .y , (i' , refl) = i'\n  Pair.snd (refineAny f g) r P x H with Pair.fst (anyP P (f x)) H\n  Pair.snd (refineAny f g) r P x H | y , (i , IH) = Pair.snd (anyP P (g x)) (y , ((r x y i) , IH))\n\n  selectPost : forall {a} -> List a -> a \u00d7 List a -> Set\n  selectPost xs (y , ys) = Sigma (y \u2208 xs) (\\e -> delete xs e == ys)\n\n  removeSpec : forall {a} -> SpecK (List a) (a \u00d7 List a)\n  removeSpec = [[ K \u22a4 , selectPost ]]\n\n  remove : forall {a} -> List a -> ND (a \u00d7 List a)\n  remove Nil       = fail\n  remove (x :: xs) = choice  (return (x , xs)) (map (retain x) (remove xs))\n      where\n      retain : forall {a} -> a -> a \u00d7 List a -> a \u00d7 List a\n      retain x (y , ys) = (y , (x :: ys))\n\n  removeCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll remove\n  removeCorrect P Nil (tt , snd) = tt\n  removeCorrect P (x :: xs) (tt , snd) =\n    snd (x , xs) (\u2208Head , refl) ,\n    mapPT P (x :: xs) xs (remove xs) _\n      (removeCorrect _ xs (tt , (\u03bb {(x' , xs') (i , H) -> snd (x' , (x :: xs')) (\u2208Tail i , cong (x ::_) H)})))\n    where\n    mapPT : \u2200 {a b c : Set} P (x x' : a) (S : ND b) (f : b -> c) -> allPT (\u03bb _ y -> P x (f y)) x' S -> allPT P x (map f S)\n    mapPT P x x' (Pure y) f H = H\n    mapPT P x x' (Step Fail k) f H = H\n    mapPT P x x' (Step Choice k) f (fst , snd) = mapPT P x x' (k True) f fst , mapPT P x x' (k False) f snd\n\n  trivialCorrect : forall {a} -> wpSpec {a = List a} {const (a \u00d7 List a)} removeSpec \u2291 wpAll (const fail)\n  trivialCorrect = \\P xs H -> tt\n\n  completeness : forall {a} -> (y : a) (xs ys : List a) -> selectPost xs (y , ys) -> Elem (y , ys) (remove xs)\n  completeness y (y :: _) ys (\u2208Head , refl) = Left Here\n  completeness y (x :: xs) .(x :: delete xs fst) (\u2208Tail fst , refl) = Right (inMap _ (remove xs) _ (completeness y _ _ (fst , refl)))\n    where\n    inMap : \u2200 {a b : Set} (x : a) S (f : a -> b) -> Elem x S -> Elem (f x) (map f S)\n    inMap x (Pure x) f Here = Here\n    inMap x (Step Choice k) f (Left i) = Left (inMap x (k True) f i)\n    inMap x (Step Choice k) f (Right i) = Right (inMap x (k False) f i)\n\nmodule Recursion where\n  open Free\n  open import Data.Nat public\n    using\n      (_+_; _>_; _*_\n      )\n    renaming\n      ( \u2115 to Nat\n      ; zero to Zero\n      ; suc to Succ\n      ; _\u2238_ to _-_\n      )\n  open NaturalLemmas\n  open Maybe hiding (soundness)\n\n  _~~>_ : (I : Set) (O : I -> Set) -> Set\n  I ~~> O = (i : I) -> Free I O (O i)\n\n  call : forall {I O} -> (i : I) -> Free I O (O i)\n  call x = Step x Pure\n\n  f91 : Nat ~~> K Nat\n  f91 i with 100 lt i\n  f91 i | yes  _ = return (i - 10)\n  f91 i | no   _ = call (i + 11) >>= call\n\n  f91Post : Nat -> Nat -> Set\n  f91Post i o with 100 lt i\n  f91Post i o | yes _ = o == i - 10\n  f91Post i o | no _  = o == 91\n\n  f91Spec : SpecK Nat Nat\n  f91Spec = [[ K \u22a4 , f91Post ]]\n\n  invariant : forall {I} -> {O : I -> Set} -> (i : I) -> Spec I O -> Free I O (O i) -> Set\n  invariant i [[ pre , post ]] (Pure x)   =  pre i -> post i x\n  invariant i [[ pre , post ]] (Step j k) =  (pre i -> pre j)\n                                              \u2227 \u2200 o -> post j o -> invariant i [[ pre , post ]] (k o)\n  wpRec : forall {I} -> {O : I -> Set} -> Spec I O -> (f : I ~~> O) -> (P : (i : I) -> O i -> Set) -> (I -> Set)\n  wpRec spec f P i = wpSpec spec P i \u2227 invariant i spec (f i)\n\n  f91Partial-correctness : wpSpec f91Spec \u2291 wpRec f91Spec f91\n  f91Partial-correctness P i with 100 lt i\n  f91Partial-correctness P i | yes p with 100 lt i\n  f91Partial-correctness P i | yes p | yes _ = \\H -> (tt , (\\x eq -> Pair.snd H _ eq)) , (\\x -> refl)\n  f91Partial-correctness P i | yes p | no \u00acp = magic (\u00acp p)\n  f91Partial-correctness P i | no \u00acp = \\x -> (tt , (\\x\u2081 x\u2082 -> Pair.snd x x\u2081 x\u2082)) ,\n                                              ((\\_ -> tt) , (\\o x\u2081 -> (\\x\u2082 -> tt) ,\n                                              (\\o\u2081 x\u2082 x\u2083 -> lemma i o _ \u00acp x\u2081 x\u2082)))\n    where\n    open Data.Nat\n    open import Data.Nat.Properties\n\n    100-\u226e-91 : (i : Nat) -> \u00ac (i + 10 \u2264 i)\n    100-\u226e-91 Zero ()\n    100-\u226e-91 (Succ i) (s\u2264s pf) = 100-\u226e-91 i pf\n\n    plus-minus : \u2200 b c -> (b + c) - c == b\n    plus-minus b c = trans (+-\u2238-assoc b (NaturalLemmas.\u2264-refl {c})) (trans (cong (b +_) (n\u2238n\u22610 c)) (sym (plus-zero b)))\n\n    plus-plus-minus : \u2200 i -> i + 11 - 10 \u2261 Succ i\n    plus-plus-minus i = plus-minus (Succ i) 11\n\n    between : \u2200 a b -> \u00ac (a < b) -> a < Succ b -> a \u2261 b\n    between Zero Zero \u00aclt ltSucc = refl\n    between Zero (Succ b) \u00aclt ltSucc = magic (\u00aclt (s\u2264s z\u2264n))\n    between (Succ a) Zero \u00aclt (s\u2264s ())\n    between (Succ a) (Succ b) \u00aclt (s\u2264s ltSucc) = cong Succ (between a b (\u00aclt \u2218 s\u2264s) ltSucc)\n\n    lemma : \u2200 i o o' -> \u00ac (100 < i) ->\n      f91Post (i + 11) o -> f91Post o o' -> f91Post i o'\n    lemma i o o' i\u2264100 oPost o'Post with 100 lt i\n    ... | yes p = magic (i\u2264100 p)\n    ... | no \u00acp with 100 lt o\n    lemma i o .(o - 10) i\u2264100 oPost refl | no \u00acp | yes p with 100 lt (i + 11)\n    lemma i .(i + 11 - 10) .(i + 11 - 10 - 10) i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 with between 100 i i\u2264100 (subst (\\i' -> 100 < i') (plus-plus-minus i) p)\n    lemma .100 .101 .91 i\u2264100 refl refl | no \u00acp | yes p | yes p\u2081 | refl = refl\n    lemma i .91 .81 i\u2264100 refl refl | no \u00acp | yes p | no \u00acp\u2081 = magic (100-\u226e-91 91 p)\n    lemma i o o' i\u2264100 oPost o'Post | no \u00acp | no \u00acp\u2081 = o'Post\n\n  petrol : forall {I O a} -> (f : I ~~> O) -> Free I O a -> Nat -> Partial a\n  petrol f (Pure x)    n        = return x\n  petrol f (Step _ _)  Zero     = abort\n  petrol f (Step c k)  (Succ n) = petrol f (f c >>= k) n\n\n  mayPT : forall {a} -> (a -> Set) -> (Partial a -> Set)\n  mayPT P (Pure x)       = P x\n  mayPT P (Step Abort _) = \u22a4\n\n  soundness : forall {I O} -> (f : I ~~> O) (spec : Spec I O) (P : (i : I) -> O i -> Set) ->\n    (\u2200 i -> wpRec spec f P i) -> \u2200 n i -> mayPT (P i) (petrol f (f i) n)\n  soundness f spec P wpH n i = soundness' f spec P (f i) n wpH (wpH i)\n    where\n    invariant-compositionality : \u2200 {I} {O : I -> Set} {i i'} spec\n      (S : Free I O (O i)) (k : (O i) -> Free I O (O i')) ->\n      invariant i spec S -> Spec.pre spec i -> (\u2200 o -> Spec.post spec i o -> invariant i' spec (k o)) ->\n      invariant i' spec (S >>= k)\n    invariant-compositionality spec (Pure x) k SH preH kH = kH x (SH preH)\n    invariant-compositionality spec (Step c k') k (fst , snd) preH kH = (\\_ -> fst preH) , \\o postH -> invariant-compositionality spec (k' o) k (snd o postH) preH kH\n    soundness' : \u2200 {I} {O : I -> Set} {i}\n      (f : (i : I) -> Free I O (O i)) (spec : Spec I O) (P : (i : I) -> O i -> Set)\n      (S : Free I O (O i)) (n : Nat) ->\n      (\u2200 i -> wpRec spec f P i) ->\n      wpSpec spec P i \u2227 invariant i spec S ->\n      mayPT (P i) (petrol f S n)\n    soundness' f spec P (Pure x) n wpH ((preH , postH) , invH) = postH x (invH preH)\n    soundness' f spec P (Step c k) Zero wpH H = tt\n    soundness' f spec P (Step c k) (Succ n) wpH (specH , (preH , postH)) = soundness' f spec P (f c >>= k) n wpH (specH , invariant-compositionality spec (f c) k (Pair.snd (wpH c)) (preH (Pair.fst specH)) postH)\n\nmodule Mix (C : Set) (R : C -> Set) (ptalgebra : (c : C) -> (R c -> Set) -> Set) where\n  open Free hiding (_>>=_)\n  open Maybe using (SpecK; [[_,_]]; Spec; wpSpec)\n\n  SpecVal : Set -> Set\u2081\n  SpecVal a = SpecK \u22a4 a\n\n  data I (a : Set) : Set\u2081 where\n    Done  : a -> I a\n    Hole  : SpecVal a -> I a\n\n  ptI : forall {a} -> I a -> (a -> Set) -> Set\n  ptI (Done x)     P = P x\n  ptI (Hole spec)  P = wpSpec spec (const P) tt\n\n  M : Set -> Set\u2081\n  M a = Free C R (I a)\n\n  isExecutable : forall {a} -> M a -> Set\n  isExecutable (Pure (Done _)) = \u22a4\n  isExecutable (Pure (Hole _)) = \u22a5\n  isExecutable (Step c k)      = \u2200 r -> isExecutable (k r)\n\n  pt : forall {l} -> {a : Set l} -> Free C R a -> (a -> Set) -> Set\n  pt (Pure x) P   = P x\n  pt (Step c x) P = ptalgebra c (\\r -> pt (x r) P)\n\n  wpCR : forall {l l'} -> {a : Set l} -> {b : a -> Set l'} -> ((x : a) -> Free C R (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpCR f P x = pt (f x) (P x)\n\n  wpM : {a : Set} -> {b : a -> Set} -> ((x : a) -> M (b x)) -> ((x : a) -> b x -> Set) -> (a -> Set)\n  wpM f P x = wpCR f (\\x ix -> ptI ix (P x)) x\n\nmodule StateExample where\n  open Free hiding (_>>=_)\n  open Maybe using (Spec; SpecK; [[_,_]]; wpSpec)\n  open State Nat\n\n  --   We have to redo the Mix section since our specifications incorporate the state\n  SpecVal : \u2200 {l} -> Set -> Set (suc l)\n  SpecVal = SpecK Nat\n\n  data I {l : Level} (a : Set) : Set (suc l) where\n    Done  : a -> I a\n    Hole  : SpecVal {l} (a \u00d7 Nat) -> I a\n\n  M : {l : Level} -> Set -> Set (suc l)\n  M a = State (I a)\n\n  ptI : forall {l a} -> I a -> (a \u00d7 Nat -> Set l) -> Nat -> Set l\n  ptI (Done x)     P t = P (x , t)\n  ptI (Hole spec)  P t = wpSpec spec (\\_ -> P) t\n\n  wpM : forall {l l'} {a : Set l} {b : Set} -> (a -> M b) -> (a \u00d7 Nat -> b \u00d7 Nat -> Set l') -> (a \u00d7 Nat -> Set l')\n  wpM f P = wpState f (\\i o -> ptI (Pair.fst o) (P i) (Pair.snd o))\n\n  ptM : {a : Set} -> M a -> (Nat -> a \u00d7 Nat -> Set) -> (Nat -> Set)\n  ptM S post t = wpM (\u03bb _ -> S) (\u03bb _ -> (post t)) (\u22a4 , t)\n\n  liftM : \u2200 {l : Level} {a} -> M {l} a -> M {suc l} a\n  liftM (Pure (Done x)) = Pure (Done x)\n  liftM (Pure (Hole [[ pre , post ]])) = Pure (Hole [[ (\u03bb x -> Lift _ (pre x)) , (\u03bb i o -> Lift _ (post i o)) ]])\n  liftM (Step c k) = Step c \u03bb x -> liftM (k x)\n\n  _>>=_ : forall {l a b} -> (M {l} a) -> (a -> M {l} b) -> M {suc l} b\n  Pure (Done x)                >>= f = liftM (f x)\n  Pure (Hole [[ pre , post ]]) >>= f =\n    Pure (Hole [[ (\u03bb n -> Lift _ (pre n)) ,\n      (\\i ynat -> \u2200 x -> post i (x , i) -> \u2200 P -> wpM f P (x , i) -> P (x , i) ynat\n      ) ]] )\n  (Step c k) >>= f        = Step c (\\r -> k r >>= f)\n\n  _>=>_ : forall {l : Level} {a b c : Set} -> (a -> M {l} b) -> (b -> M {l} c) -> a -> M {suc l} c\n  (f >=> g) x = f x >>= g\n\n  specV : {a : Set} -> SpecVal {zero} (a \u00d7 Nat) -> M a\n  specV spec = Pure (Hole spec)\n\n  done   : forall {a} -> a -> M {zero} a\n  get'   : M Nat\n  put'   : Nat -> M \u22a4\n  done x = Pure (Done x)\n  get' = Step Get done\n  put' t = Step (Put t) done\n\n  getPost : Nat -> Nat \u00d7 Nat -> Set\n  getPost t (x , t') = (t == x) \u2227 (t == t')\n\n  putPost : Nat -> Nat -> \u22a4 \u00d7 Nat -> Set\n  putPost t _ (_ , t') = t == t'\n\n  _\u25b9_ : {a b : Set} -> (Q : a -> b -> Set) (P : a -> Set) -> b -> Set\n  _\u25b9_ {a} Q P = \\y -> Sigma a (\\x -> P x \u2227 Q x y)\n  _\u25c3_ : {a b c : Set} -> (Q : a -> b -> Set) -> (SpecK a c) -> b -> c -> Set\n  _\u25c3_ Q [[ pre , post ]] = \\y z -> \u2200 x -> pre x \u2227 Q x y -> post x z\n\n  step : forall {b} -> (c : C) (spec : SpecVal {zero} (b \u00d7 Nat)) -> SpecK {zero} (R c \u00d7 Nat) (b \u00d7 Nat)\n  step Get      [[ pre , post ]] = [[ getPost \u25b9 pre , getPost \u25c3 [[ pre , post ]] ]]\n  step (Put x)  [[ pre , post ]] = [[ (putPost x) \u25b9 pre , (putPost x) \u25c3 [[ pre , post ]] ]]\n\n  intros : forall {a b} -> SpecK {zero} (a \u00d7 Nat) (b \u00d7 Nat) -> a -> SpecVal (b \u00d7 Nat)\n  intros [[ pre , post ]] x = [[ (\\t -> pre (x , t)) , (\\t -> post (x , t)) ]]\n\n  data Derivation {b} (spec : SpecVal (b \u00d7 Nat)) : Set\u2081 where\n    Done : (x : b) -> wpSpec spec \u2291 ptM (done x) -> Derivation spec\n    Step : (c : C) -> (\u2200 (r : R c) -> Derivation (intros (step c spec) r)) -> Derivation spec\n\n  extract : forall {b} -> (spec : SpecVal (b \u00d7 Nat)) -> Derivation spec -> State b\n  extract _ (Done x _) = Pure x\n  extract _ (Step c k) = Step c (\\r -> extract _ (k r))\n\n  DerivationFun : {a b : Set} (spec : SpecK (a \u00d7 Nat) (b \u00d7 Nat)) -> Set\u2081\n  DerivationFun {a} {b} spec = (x : a) -> Derivation (intros spec x)\n\n  stepMonotone : {a : Set} (c : C) (r : R c) {spec spec' : SpecVal (a \u00d7 Nat)} ->\n    wpSpec spec \u2291 wpSpec spec' ->\n    wpSpec (intros (step c spec) r) \u2291 wpSpec (intros (step c spec') r)\n  stepMonotone {a} Get r {spec} {spec'} H P .r ((.r , (fst\u2081 , (refl , refl))) , snd) = (r , (Pair.fst (H (\\_ _ -> \u22a4) r (fst\u2081 , (\\x _ -> tt))) , (refl , refl))) , \\x x\u2081 -> snd x (postLemma r x x\u2081)\n    where\n    postLemma : \u2200 r\n      (x : Pair a Nat) ->\n      (\u2200 x\u2081 ->\n      Pair (Spec.pre spec' x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec' x\u2081 x) ->\n      \u2200 x\u2081 ->\n      Pair (Spec.pre spec x\u2081) (Pair (x\u2081 \u2261 r) (x\u2081 \u2261 r)) ->\n      Spec.post spec x\u2081 x\n    postLemma r x x\u2082 .r (fst , (refl , refl)) = Pair.snd (H (Spec.post spec) r (fst , (\\x\u2083 z -> z))) x (x\u2082 r ((Pair.fst (H (\\_ _ -> \u22a4) r (fst , (\\x\u2083 _ -> tt)))) , (refl , refl)))\n  stepMonotone {a} (Put t) r {spec} {spec'} H P .t ((fst , (fst\u2081 , refl)) , snd) = (fst , (Pair.fst (H (\\_ _ -> \u22a4) fst (fst\u2081 , (\\x _ -> tt))) , refl)) , \\x x\u2081 -> snd x (postLemma t x x\u2081)\n    where\n      postLemma : \u2200 (t : Nat)\n        (x : Pair a Nat) ->\n        (\u2200 x\u2081 -> Pair (Spec.pre spec' x\u2081) (t \u2261 t) -> Spec.post spec' x\u2081 x) ->\n        \u2200 x\u2081 ->\n        Pair (Spec.pre spec x\u2081) (t \u2261 t) -> Spec.post spec x\u2081 x\n      postLemma t x x\u2081 x\u2082 (fst , refl) = Pair.snd (H (Spec.post spec) x\u2082 (fst , (\\x\u2083 z -> z))) x (x\u2081 x\u2082 ((Pair.fst (H (\\_ _ -> \u22a4) x\u2082 (fst , (\\x\u2083 _ -> tt)))) , refl))\n\n  open import Data.Nat\n  open import Data.Nat.Properties\n  open NaturalLemmas hiding (\u2264-refl ; \u2264-trans)\n\n  data All {a : Set} (P : a -> Set) : List a -> Set where\n    AllNil : All P Nil\n    AllCons : \u2200 {x xs} -> P x -> All P xs -> All P (x :: xs)\n\n  unAllCons : \u2200 {a P x} {xs : List a} ->\n    All P (x :: xs) -> All P xs\n  unAllCons (AllCons x\u2081 x\u2082) = x\u2082\n\n  maxPre : List Nat \u00d7 Nat -> Set\n  maxPre (xs , i) = (i == 0) \u2227 (\u00ac (xs == Nil))\n\n  maxPost : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost (xs , i) (o , _) = All (o \u2265_) xs \u2227 (o \u2208 xs)\n\n  maxSpec = [[ maxPre , maxPost ]]\n\n  refineDerivation : forall {a : Set} -> {spec spec' : SpecVal (a \u00d7 Nat)} -> wpSpec spec \u2291 wpSpec spec' -> Derivation spec' -> Derivation spec\n  refineDerivation H (Done x Hx) = Done x (\u2291-trans H Hx)\n  refineDerivation H (Step c d) = Step c \\r -> refineDerivation (stepMonotone c r H) (d r)\n\n  maxPost' : List Nat \u00d7 Nat -> Nat \u00d7 Nat -> Set\n  maxPost' (xs , i) (o , _) = All (o \u2265_) (i :: xs) \u00d7 (o \u2208 (i :: xs))\n\n  maxProof : \u2200 (xs : List Nat) ->\n    wpSpec (intros [[ maxPre , maxPost ]] xs) \u2291\n    wpSpec (intros [[ K \u22a4 , maxPost' ]] xs)\n  maxProof xs P .0 ((refl , Hnil) , snd) = tt , \\o H -> snd o (unAllCons (Pair.fst H) , lemma xs Hnil (Pair.fst o) H)\n    where\n    lemma : \u2200 xs -> \u00ac (xs == Nil) ->\n      \u2200 w -> Pair (All (\\n -> n \u2264 w) (0 :: xs)) (w \u2208 (0 :: xs)) -> w \u2208 xs\n    lemma Nil Hnil w H = magic (Hnil refl)\n    lemma (.0 :: xs) _ .0 (AllCons x\u2082 (AllCons z\u2264n fst) , \u2208Head) = \u2208Head\n    lemma (x :: xs) _ w (_ , \u2208Tail snd) = snd\n\n  max'ProofNil : \u2200 i ->\n    wpSpec (intros (step Get (intros [[ K \u22a4 , maxPost' ]] Nil)) i) \u2291 ptM (done i)\n  max'ProofNil i P .i ((.i , (fst\u2081 , (refl , refl))) , snd) = snd (i , i) (lemma i)\n    where\n    lemma : \u2200 i x ->\n      Pair \u22a4 (Pair (x \u2261 i) (x \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 i) (x :: Nil)) (i \u2208 (x :: Nil))\n    lemma i .i (fst , (refl , refl)) = (AllCons \u2264-refl AllNil) , \u2208Head\n\n  max'Proof1 : \u2200 x xs i ->\n    Succ x \u2264 i ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 ->\n    Pair (Sigma \u2115 (\\x\u2082 -> Pair \u22a4 (Pair (x\u2082 \u2261 i) (x\u2082 \u2261 x\u2081))))\n    (\u2200 x\u2082 ->\n    (\u2200 x\u2083 ->\n    Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2081)) ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))) ->\n    P x\u2081 x\u2082) ->\n    Pair \u22a4\n    (\u2200 x\u2082 ->\n    Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) ->\n    P x\u2081 x\u2082)\n  max'Proof1 x xs i x<i P .i ((.i , (_ , (refl , refl))) , snd) = tt , \\x\u2082 x\u2081 -> snd x\u2082 (lemma x\u2082 x\u2081)\n    where\n    lemma : \u2200 (x\u2082 : Nat \u00d7 Nat) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (i :: xs))\n      (Pair.fst x\u2082 \u2208 (i :: xs)) ->\n      \u2200 x\u2083 -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 i)) ->\n      Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2083 :: x :: xs))\n      (Pair.fst x\u2082 \u2208 (x\u2083 :: x :: xs))\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Head) .i (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (<\u21d2\u2264 x<i) fst)) , \u2208Head\n    lemma x\u2082 (AllCons x\u2081 fst , \u2208Tail snd) _ (_ , (refl , refl)) = (AllCons x\u2081 (AllCons (\u2264-trans (<\u21d2\u2264 x<i) x\u2081) fst)) , \u2208Tail (\u2208Tail snd)\n\n  max'Proof2 : \u2200 i x xs -> (Succ x \u2264 i -> \u22a5) ->\n    \u2200 (P : Nat -> Nat \u00d7 Nat -> Set) x\u2081 -> Pair (Sigma \u2115 (\\x\u2082 -> Pair (Sigma \u2115 (\\x\u2083\n    -> Pair \u22a4 (Pair (x\u2083 \u2261 i) (x\u2083 \u2261 x\u2082)))) (x \u2261 x\u2081))) (\u2200 x\u2082 -> (\u2200 x\u2083 -> Pair (Sigma\n    \u2115 (\\x\u2084 -> Pair \u22a4 (Pair (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)))) (x \u2261 x\u2081) -> \u2200 x\u2084 -> Pair \u22a4 (Pair\n    (x\u2084 \u2261 i) (x\u2084 \u2261 x\u2083)) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2082) (x\u2084 :: x :: xs))\n    (Pair.fst x\u2082 \u2208 (x\u2084 :: x :: xs))) -> P x\u2081 x\u2082) -> Pair \u22a4 (\u2200 x\u2082 -> Pair (All (\\n\n    -> n \u2264 Pair.fst x\u2082) (x\u2081 :: xs)) (Pair.fst x\u2082 \u2208 (x\u2081 :: xs)) -> P x\u2081 x\u2082)\n  max'Proof2 i x xs x\u2265i P .x ((.i , ((.i , (fst\u2082 , (refl , refl))) , refl)) , snd) = tt , \\x\u2084 x\u2081 -> snd x\u2084 (lemma x\u2084 x\u2081)\n    where\n    lemma : \u2200 (x\u2084 : Pair Nat Nat) -> Pair (All (\\n -> n \u2264 Pair.fst x\u2084) (x :: xs))\n      (Pair.fst x\u2084 \u2208 (x :: xs)) -> \u2200 x\u2083 -> Pair (Sigma \u2115 (\\x\u2085 -> Pair \u22a4 (Pair (x\u2085\n      \u2261 i) (x\u2085 \u2261 x\u2083)))) (x \u2261 x) -> \u2200 x\u2085 -> Pair \u22a4 (Pair (x\u2085 \u2261 i) (x\u2085 \u2261 x\u2083)) -> Pair\n      (All (\\n -> n \u2264 Pair.fst x\u2084) (x\u2085 :: x :: xs)) (Pair.fst x\u2084 \u2208 (x\u2085 :: x :: xs))\n    lemma (_ , _) (AllCons x fst , \u2208Head) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u226e\u21d2\u2265 x\u2265i) (AllCons x fst)) , (\u2208Tail \u2208Head)\n    lemma x\u2084 (AllCons x\u2081 fst , \u2208Tail snd) _ ((_ , (_ , (refl , refl))) , refl) _ (_ , (refl , refl)) = (AllCons (\u2264-trans (\u226e\u21d2\u2265 x\u2265i) x\u2081) (AllCons x\u2081 fst)) , (\u2208Tail (\u2208Tail snd))\n\n  maxSpec' = [[ K \u22a4 , maxPost' ]]\n\n  max' : (xs : List Nat) -> Derivation (intros maxSpec' xs)\n  max' Nil       = Step Get \\i ->\n                     Done i (max'ProofNil i)\n  max' (x :: xs) = Step Get \\i ->\n                     if' x <? i\n                       then (\\lt  -> refineDerivation (max'Proof1 x xs i lt) (max' xs))\n                       else (\\geq -> Step (Put x) (const (refineDerivation (max'Proof2 i x xs geq) (max' xs))))\n\n  max : DerivationFun [[ maxPre , maxPost ]]\n  max xs = refineDerivation (maxProof xs) (max' xs)\n","returncode":0,"stderr":"","license":"unlicense","lang":"Agda"}
{"commit":"f92b6a6256ba4f59cf4312db4e85d4e590581f08","subject":"Typo.","message":"Typo.\n\nIgnore-this: e2f62b5566a19cf8a9d583c54f8ec894\n\ndarcs-hash:20120319115721-3bd4e-373cf4510c7d80b8fa0d5e852e64cffe382074d3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddPartialRightIdentity.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddPartialRightIdentity.agda","new_contents":"------------------------------------------------------------------------------\n-- Reasoning partially about functions\n------------------------------------------------------------------------------\n\n-- We cannot reasoning partially about partial functions intended to\n-- operate in total values.\n\n-- Tested with FOT on 19 March 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddPartialRightIdentity where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n-- How proceed?\n+-partialRightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-partialRightIdentity n = {!!}\n","old_contents":"------------------------------------------------------------------------------\n-- Reasoning partially about functions\n------------------------------------------------------------------------------\n\n-- We cannot reasoning partially about partial functions intended to\n-- operate in total values.\n\n-- Tested with FOT and agda2atp on 17 March 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddPartialRightIdentity where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n-- How proceed?\n+-partialRightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-partialRightIdentity n = {!!}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d85086d26100c1134f245889942ce080755cfa9f","subject":"Upgrade: CCA2d-CCA2","message":"Upgrade: CCA2d-CCA2\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/Transformation\/CCA2d-CCA2.agda","new_file":"Game\/Transformation\/CCA2d-CCA2.agda","new_contents":"{-# OPTIONS --without-K #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Control.Strategy renaming (run to runStrategy; map to mapStrategy)\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Game.IND-CPA-utils\nimport Game.IND-CCA2-dagger\nimport Game.IND-CCA2\nimport Game.IND-CCA\n\nmodule Game.Transformation.CCA2d-CCA2\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n  \nwhere\n\nmodule CCA2d = Game.IND-CCA2-dagger PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 KeyGen Enc Dec \nmodule CCA2  = Game.IND-CCA2        PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 KeyGen Enc Dec\nopen Game.IND-CPA-utils Message CipherText\nopen TransformAdversaryResponse {DecRound Bit} {CipherText \u2192 DecRound Bit} (\u03bb x _ \u2192 x)\n\nA-transform : (adv : CCA2.Adversary) \u2192 CCA2d.Adversary\nA-transform adv r\u2090 pk = mapStrategy A* (adv r\u2090 pk)\n\n{-\nIf we are able to do the transformation, then we get the same advantage\n-}\n\ndecRound = runStrategy \u2218 Dec\n\ncorrect : \u2200 {r\u2091 r\u2091' r\u2096 r\u2090 } b adv\n        \u2192 CCA2.EXP b adv               (r\u2090 , r\u2096 , r\u2091)\n        \u2261 CCA2d.EXP b (A-transform adv) (r\u2090 , r\u2096 , r\u2091 , r\u2091')\ncorrect {r\u2091} {r\u2091'} {r\u2096} {r\u2090} 0b m with KeyGen r\u2096\n... | pk , sk = cong (\u03bb x \u2192 decRound sk (put-c x (Enc pk (proj\u2081 (get-m x)) r\u2091)\n                                                 (Enc pk (proj\u2082 (get-m x)) r\u2091')))\n                     (sym (run-map (Dec sk) A* (m r\u2090 pk)))\ncorrect {r\u2091}{r\u2091'}{r\u2096} {r\u2090} 1b m with KeyGen r\u2096\n... | pk , sk = cong (\u03bb x \u2192 decRound sk (put-c x (Enc pk (proj\u2082 (get-m x)) r\u2091)\n                                                 (Enc pk (proj\u2081 (get-m x)) r\u2091')))\n                     (sym (run-map (Dec sk) A* (m r\u2090 pk)))\n{-\n\nNeed to show that they are valid transformation aswell:\n\n  This is not obvious that it will always work since the original adversary might\n  ask for the ciphertext from the r\u2091' supply.\n\n  The argument why this is unlikely (in the paper) is that the construction is\n  IND-CPA secure and therefor it is hard to predict a ciphertext.\n  suppose that A could predict the ciphertext, then it could use this power to\n  win the game. Since he can predict what Enc(m(not b)) is he could just check what the\n  ciphertext was and that he received and thereby know if he received b or not b.\n\n-}\n\n{-\nvalid : \u2200 adv \u2192 CCA2.Valid-Adv adv \u2192 CCA2d.Valid-Adv (A-transform adv)\nvalid adv valid = {!!} -- can't prove this strictly, but it will deviate with\n                       -- non-neg probability\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Control.Strategy renaming (run to runStrategy; map to mapStrategy)\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality\n\nimport Game.IND-CPA-utils\nimport Game.IND-CCA2-dagger\nimport Game.IND-CCA2\nimport Game.IND-CCA\n\nmodule Game.Transformation.CCA2d-CCA2\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n  \nwhere\n\nmodule CCA2d = Game.IND-CCA2-dagger PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 KeyGen Enc Dec \nmodule CCA2  = Game.IND-CCA2        PubKey SecKey Message CipherText R\u2091 R\u2096 R\u2090 KeyGen Enc Dec\nopen Game.IND-CPA-utils Message CipherText\n\nf : CPAAdversary (DecRound Bit)\n  \u2192 CPAAdversary (CipherText \u2192 DecRound Bit)\nget-m (f A) = get-m A\nput-c (\nf (m , g) = m , \u03bb c _ \u2192 g c\n\nA-transform : (adv : CCA2.Adversary) \u2192 CCA2d.Adversary\nA-transform adv r\u2090 pk = mapStrategy f (adv r\u2090 pk)\n  \n{-\n\nIf we are able to do the transformation, then we get the same advantage\n\n-}\n\ndecRound = runStrategy \u2218 Dec\n\ncorrect : \u2200 {r\u2091 r\u2091' r\u2096 r\u2090 } b adv\n        \u2192 CCA2.EXP b adv               (r\u2090 , r\u2096 , r\u2091)\n        \u2261 CCA2d.EXP b (A-transform adv) (r\u2090 , r\u2096 , r\u2091 , r\u2091')\ncorrect {r\u2091} {r\u2091'} {r\u2096} {r\u2090} 0b m with KeyGen r\u2096\n... | pk , sk = cong (\u03bb x \u2192 decRound sk (proj\u2082 x (Enc pk (proj\u2081 (proj\u2081 x)) r\u2091)\n                                             (Enc pk (proj\u2082 (proj\u2081 x)) r\u2091')))\n                     (sym (run-map (Dec sk) f (m r\u2090 pk)))\ncorrect {r\u2091}{r\u2091'}{r\u2096} {r\u2090} 1b m with KeyGen r\u2096\n... | pk , sk = cong (\u03bb x \u2192 decRound sk (proj\u2082 x (Enc pk (proj\u2082 (proj\u2081 x)) r\u2091)\n                                             (Enc pk (proj\u2081 (proj\u2081 x)) r\u2091')))\n                     (sym (run-map (Dec sk) f (m r\u2090 pk)))\n{-\n\nNeed to show that they are valid transformation aswell:\n\n  This is not obvious that it will always work since the original adversary might\n  ask for the ciphertext from the r\u2091' supply.\n\n  The argument why this is unlikely (in the paper) is that the construction is\n  IND-CPA secure and therefor it is hard to predict a ciphertext.\n  suppose that A could predict the ciphertext, then it could use this power to\n  win the game. Since he can predict what Enc(m(not b)) is he could just check what the\n  ciphertext was and that he received and thereby know if he received b or not b.\n\n-}\n\n{-\nvalid : \u2200 adv \u2192 CCA2.Valid-Adv adv \u2192 CCA2d.Valid-Adv (A-transform adv)\nvalid adv valid = {!!} -- can't prove this strictly, but it will deviate with\n                       -- non-neg probability\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"240b239b92eeb8f3e8f14e8d97b3685cf92a6a4d","subject":"Simplified a proof.","message":"Simplified a proof.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Data\/Stream\/Equality\/PropertiesI.agda","new_file":"src\/fot\/FOTC\/Data\/Stream\/Equality\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\nstream-\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\nstream-\u2248-refl {xs} Sxs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = Stream xs \u2227 xs \u2261 ys\n\n  h\u2081 : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n       R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  h\u2081 (Sxs , refl) with (Stream-unf Sxs)\n  ... | x' , xs' , Sxs' , prf = x' , xs' , xs' , (Sxs' , refl) , prf , prf\n\n  h\u2082 : R xs xs\n  h\u2082 = Sxs , refl\n\nstream-\u2261\u2192\u2248 : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2261 ys \u2192 xs \u2248 ys\nstream-\u2261\u2192\u2248 Sxs _ refl = stream-\u2248-refl Sxs\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n","old_contents":"------------------------------------------------------------------------------\n-- Properties for the equality on streams\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.Data.Stream.Equality.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\nstream-\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\nstream-\u2248-refl {xs} Sxs = \u2248-coind R h\u2081 h\u2082\n  where\n  R : D \u2192 D \u2192 Set\n  R xs ys = Stream xs \u2227 Stream xs \u2227 xs \u2261 ys\n\n  h\u2081 : \u2200 {xs ys} \u2192 R xs ys \u2192 \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ]\n       R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n  h\u2081 (Sxs , Sys , refl) with (Stream-unf Sxs)\n  ... | x' , xs' , Sxs' , prf = x' , xs' , xs' , (Sxs' , Sxs' , refl) , prf , prf\n\n  h\u2082 : R xs xs\n  h\u2082 = Sxs , Sxs , refl\n\nstream-\u2261\u2192\u2248 : \u2200 {xs ys} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2261 ys \u2192 xs \u2248 ys\nstream-\u2261\u2192\u2248 Sxs _ refl = stream-\u2248-refl Sxs\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream xs \u2227 Stream ys\n\u2248\u2192Stream {xs} {ys} h = Stream-coind P\u2081 h\u2081 (ys , h) , Stream-coind P\u2082 h\u2082 (xs , h)\n  where\n  P\u2081 : D \u2192 Set\n  P\u2081 ws = \u2203[ zs ] ws \u2248 zs\n\n  h\u2081 : \u2200 {ws} \u2192 P\u2081 ws \u2192 \u2203[ w' ] \u2203[ ws' ] P\u2081 ws' \u2227 ws \u2261 w' \u2237 ws'\n  h\u2081 {ws} (zs , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , prf\u2082 , _ = w' , ws' , (zs' , prf\u2081) , prf\u2082\n\n  P\u2082 : D \u2192 Set\n  P\u2082 zs = \u2203[ ws ] ws \u2248 zs\n\n  h\u2082 : \u2200 {zs} \u2192 P\u2082 zs \u2192 \u2203[ z' ] \u2203[ zs' ] P\u2082 zs' \u2227 zs \u2261 z' \u2237 zs'\n  h\u2082  {zs} (ws , h\u2081) with \u2248-unf h\u2081\n  ... | w' , ws' , zs' , prf\u2081 , _ , prf\u2082 = w' , zs' , (ws' , prf\u2081) , prf\u2082\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"acfdc3d2de5420e6d66443f12ddf58f5eb3d9277","subject":"Closure validity: base env in change must match original env","message":"Closure validity: base env in change must match original env\n\nThis fixes *one* weirdness.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 0 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 0 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 0 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , refl , vv) = (refl  , refl) , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .0 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x , \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .0 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv ,  rrel\u03c13-mono k\u2081 (suc k) (lt1 k<n) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .0 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 0 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 0 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 0 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  -- Note the appearance of \u03c1. I think that's odd but I'm confused.\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 \u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 \u03c1 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c1 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (v1 \u2022 \u03c1)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , vv) = (refl  , refl) , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .0 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x , \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .0 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv ,  rrel\u03c13-mono k\u2081 (suc k) (lt1 k<n) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .0 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"6b097205995c9ed56bbb14b8ede18e48a770eb70","subject":"IND-CCA2: re-export Game.CCA-Common","message":"IND-CCA2: re-export Game.CCA-Common\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/IND-CCA2.agda","new_file":"Game\/IND-CCA2.agda","new_contents":"\n{-# OPTIONS --without-K #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Unit\n\nopen import Data.Nat.NP\n--open import Rat\n\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Relation.Binary.PropositionalEquality\n\nmodule Game.IND-CCA2\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n\nwhere\nopen import Game.CCA-Common Message CipherText public\n                         \nAdv : \u2605\nAdv = R\u2090 \u2192 PubKey \u2192 Strategy ((Message \u00d7 Message) \u00d7 (CipherText \u2192 Strategy Bit))\n\n{-\nValid-Adv : Adv \u2192 Set\nValid-Adv adv = \u2200 {r\u2090 r\u2093 pk c} \u2192 Valid (\u03bb x \u2192 x \u2262 c) {!!}\n-}\n\nR : \u2605\nR = R\u2090 \u00d7 R\u2096 \u00d7 R\u2091\n\nGame : \u2605\nGame = Adv \u2192 R \u2192 Bit\n\n\n\u2141 : Bit \u2192 Game\n\u2141 b m (r\u2090 , r\u2096 , r\u2091) with KeyGen r\u2096\n... | pk , sk = b\u2032 where\n  open Eval Dec sk\n  \n  ev = eval (m r\u2090 pk)\n  mb = proj (proj\u2081 ev) b\n  d = proj\u2082 ev\n\n  c  = Enc pk mb r\u2091\n  b\u2032 = eval (d c)\n\n  \nmodule Advantage\n  (\u03bc\u2091 : Explore\u2080 R\u2091)\n  (\u03bc\u2096 : Explore\u2080 R\u2096)\n  (\u03bc\u2090 : Explore\u2080 R\u2090)\n  where\n  \u03bcR : Explore\u2080 R\n  \u03bcR = \u03bc\u2090 \u00d7\u1d49 \u03bc\u2096 \u00d7\u1d49 \u03bc\u2091\n  \n  module \u03bcR = FromExplore\u2080 \u03bcR\n  \n  run : Bit \u2192 Adv \u2192 \u2115\n  run b adv = \u03bcR.count (\u2141 b adv)\n  \n  Advantage : Adv \u2192 \u2115\n  Advantage adv = dist (run 0b adv) (run 1b adv)\n    \n  --Advantage\u211a : Adv \u2192 \u211a\n  --Advantage\u211a adv = Advantage adv \/ \u03bcR.Card\n  \n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n{-# OPTIONS --without-K #-}\n\nopen import Type\nopen import Data.Bit\nopen import Data.Maybe\nopen import Data.Product\nopen import Data.Unit\n\nopen import Data.Nat.NP\n--open import Rat\n\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Relation.Binary.PropositionalEquality\n\nmodule Game.IND-CCA2\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  (Message   : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n  (Dec    : SecKey \u2192 CipherText \u2192 Message)\n\nwhere\nopen import Game.CCA-Common Message CipherText\n                         \nAdv : \u2605\nAdv = R\u2090 \u2192 PubKey \u2192 Strategy ((Message \u00d7 Message) \u00d7 (CipherText \u2192 Strategy Bit))\n\n{-\nValid-Adv : Adv \u2192 Set\nValid-Adv adv = \u2200 {r\u2090 r\u2093 pk c} \u2192 Valid (\u03bb x \u2192 x \u2262 c) {!!}\n-}\n\nR : \u2605\nR = R\u2090 \u00d7 R\u2096 \u00d7 R\u2091\n\nGame : \u2605\nGame = Adv \u2192 R \u2192 Bit\n\n\n\u2141 : Bit \u2192 Game\n\u2141 b m (r\u2090 , r\u2096 , r\u2091) with KeyGen r\u2096\n... | pk , sk = b\u2032 where\n  open Eval Dec sk\n  \n  ev = eval (m r\u2090 pk)\n  mb = proj (proj\u2081 ev) b\n  d = proj\u2082 ev\n\n  c  = Enc pk mb r\u2091\n  b\u2032 = eval (d c)\n\n  \nmodule Advantage\n  (\u03bc\u2091 : Explore\u2080 R\u2091)\n  (\u03bc\u2096 : Explore\u2080 R\u2096)\n  (\u03bc\u2090 : Explore\u2080 R\u2090)\n  where\n  \u03bcR : Explore\u2080 R\n  \u03bcR = \u03bc\u2090 \u00d7\u1d49 \u03bc\u2096 \u00d7\u1d49 \u03bc\u2091\n  \n  module \u03bcR = FromExplore\u2080 \u03bcR\n  \n  run : Bit \u2192 Adv \u2192 \u2115\n  run b adv = \u03bcR.count (\u2141 b adv)\n  \n  Advantage : Adv \u2192 \u2115\n  Advantage adv = dist (run 0b adv) (run 1b adv)\n    \n  --Advantage\u211a : Adv \u2192 \u211a\n  --Advantage\u211a adv = Advantage adv \/ \u03bcR.Card\n  \n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ff1ad3e1a6e0d38f95bd05273f834cd1a1ff4f79","subject":"Fixed doc.","message":"Fixed doc.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/fixed-points\/LeastFixedPoints\/ListP.agda","new_file":"notes\/fixed-points\/LeastFixedPoints\/ListP.agda","new_contents":"------------------------------------------------------------------------------\n-- From ListP as the least fixed-point to ListP using data\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- We want to represent the polymorphic total lists data type\n--\n-- data ListP (A : D \u2192 Set) : D \u2192 Set where\n--   lnil  : ListP A []\n--   lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n--\n-- using the representation of ListP as the least fixed-point.\n\nmodule LeastFixedPoints.ListP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- ListP is a least fixed-point of a functor\n\n-- The functor.\n-- ListPF : (D \u2192 Set) \u2192 (D \u2192 Set) \u2192 D \u2192 Set\n-- ListPF A B xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 B xs' \u2227 xs \u2261 x' \u2237 xs')\n\n-- ListP is the least fixed-point of ListPF.\npostulate\n  ListP : (D \u2192 Set) \u2192 D \u2192 Set\n\n  -- ListP is a pre-fixed point of ListPF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  ListP-in :\n    (A : D \u2192 Set) \u2192\n    \u2200 {xs} \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 ListP A xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n    ListP A xs\n\n  -- ListP is the least pre-fixed point of ListPF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  ListP-ind :\n    (A B : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 B xs' \u2227 xs \u2261 x' \u2237 xs') \u2192 B xs) \u2192\n    \u2200 {xs} \u2192 ListP A xs \u2192 B xs\n\n------------------------------------------------------------------------------\n-- The data constructors of ListP.\nlnil  : (A : D \u2192 Set) \u2192 ListP A []\nlnil A = ListP-in A (inj\u2081 refl)\n\nlcons : (A : D \u2192 Set) \u2192 \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\nlcons A {n} {ns} An LPns = ListP-in A (inj\u2082 (n , ns , An , LPns , refl))\n\n------------------------------------------------------------------------------\n-- The induction principle for ListP.\nindListP : (A B : D \u2192 Set) \u2192\n           B [] \u2192\n           (\u2200 x {xs} \u2192 A x \u2192 B xs \u2192 B (x \u2237 xs)) \u2192\n           \u2200 {xs} \u2192 ListP A xs \u2192 B xs\nindListP A B B[] is = ListP-ind A B h\n  where\n  h : \u2200 {xs} \u2192\n      xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 B xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n      B xs\n  h (inj\u2081 xs\u2261[])                        = subst B (sym xs\u2261[]) B[]\n  h (inj\u2082 (x' , xs' , Ax' , Bxs' , h\u2081)) = subst B (sym h\u2081) (is x' Ax' Bxs')\n\n------------------------------------------------------------------------------\n-- Examples\n\n-- \"Heterogeneous\" total lists\nxs : D\nxs = [0] \u2237 true \u2237 [1] \u2237 false \u2237 []\n\nList : D \u2192 Set\nList = ListP (\u03bb d \u2192 d \u2261 d)\n\nxs-ListP : List xs\nxs-ListP =\n  lcons A refl (lcons A refl (lcons A refl (lcons A refl (lnil A))))\n  where\n  A : D \u2192 Set\n  A d = d \u2261 d\n\n-- Total lists of total natural numbers\nys : D\nys = [0] \u2237 [1] \u2237 [2] \u2237 []\n\nListN : D \u2192 Set\nListN = ListP N\n\nys-ListP : ListN ys\nys-ListP =\n  lcons N nzero (lcons N (nsucc nzero) (lcons N (nsucc (nsucc nzero)) (lnil N)))\n","old_contents":"------------------------------------------------------------------------------\n-- From ListP as the least fixed-point to ListP using data\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- We want to represent the polymorphic total lists data type\n--\n-- data ListP (A : D \u2192 Set) : D \u2192 Set\u2081 where\n--   lnil  : ListP A []\n--   lcons : \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\n--\n-- using the representation of ListP as the least fixed-point.\n\nmodule LeastFixedPoints.ListP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Nat.Type\nopen import FOTC.Data.Nat.UnaryNumbers\n\n------------------------------------------------------------------------------\n-- ListP is a least fixed-point of a functor\n\n-- The functor.\n-- ListPF : (D \u2192 Set) \u2192 (D \u2192 Set) \u2192 D \u2192 Set\n-- ListPF A B xs = xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 B xs' \u2227 xs \u2261 x' \u2237 xs')\n\n-- ListP is the least fixed-point of ListPF.\npostulate\n  ListP : (D \u2192 Set) \u2192 D \u2192 Set\n\n  -- ListP is a pre-fixed point of ListPF.\n  --\n  -- Peter: It corresponds to the introduction rules.\n  ListP-in :\n    (A : D \u2192 Set) \u2192\n    \u2200 {xs} \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 ListP A xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n    ListP A xs\n\n  -- ListP is the least pre-fixed point of ListPF.\n  --\n  -- Peter: It corresponds to the elimination rule of an inductively\n  -- defined predicate.\n  ListP-ind :\n    (A B : D \u2192 Set) \u2192\n    (\u2200 {xs} \u2192 xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 B xs' \u2227 xs \u2261 x' \u2237 xs') \u2192 B xs) \u2192\n    \u2200 {xs} \u2192 ListP A xs \u2192 B xs\n\n------------------------------------------------------------------------------\n-- The data constructors of ListP.\nlnil  : (A : D \u2192 Set) \u2192 ListP A []\nlnil A = ListP-in A (inj\u2081 refl)\n\nlcons : (A : D \u2192 Set) \u2192 \u2200 {x xs} \u2192 A x \u2192 ListP A xs \u2192 ListP A (x \u2237 xs)\nlcons A {n} {ns} An LPns = ListP-in A (inj\u2082 (n , ns , An , LPns , refl))\n\n------------------------------------------------------------------------------\n-- The induction principle for ListP.\nindListP : (A B : D \u2192 Set) \u2192\n           B [] \u2192\n           (\u2200 x {xs} \u2192 A x \u2192 B xs \u2192 B (x \u2237 xs)) \u2192\n           \u2200 {xs} \u2192 ListP A xs \u2192 B xs\nindListP A B B[] is = ListP-ind A B h\n  where\n  h : \u2200 {xs} \u2192\n      xs \u2261 [] \u2228 (\u2203[ x' ] \u2203[ xs' ] A x' \u2227 B xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n      B xs\n  h (inj\u2081 xs\u2261[])                        = subst B (sym xs\u2261[]) B[]\n  h (inj\u2082 (x' , xs' , Ax' , Bxs' , h\u2081)) = subst B (sym h\u2081) (is x' Ax' Bxs')\n\n------------------------------------------------------------------------------\n-- Examples\n\n-- \"Heterogeneous\" total lists\nxs : D\nxs = [0] \u2237 true \u2237 [1] \u2237 false \u2237 []\n\nList : D \u2192 Set\nList = ListP (\u03bb d \u2192 d \u2261 d)\n\nxs-ListP : List xs\nxs-ListP =\n  lcons A refl (lcons A refl (lcons A refl (lcons A refl (lnil A))))\n  where\n  A : D \u2192 Set\n  A d = d \u2261 d\n\n-- Total lists of total natural numbers\nys : D\nys = [0] \u2237 [1] \u2237 [2] \u2237 []\n\nListN : D \u2192 Set\nListN = ListP N\n\nys-ListP : ListN ys\nys-ListP =\n  lcons N nzero (lcons N (nsucc nzero) (lcons N (nsucc (nsucc nzero)) (lnil N)))\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"2e31b5e5a433dc178098619dd1d271fdc63cb742","subject":"Modify step counts: ensure each evaluation takes one step","message":"Modify step counts: ensure each evaluation takes one step\n\nYann suggested this should work (and it does), and this simplifies some steps of\nthe proof (it makes it easier to see that it is well-founded).\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nopen import Relation.Nullary\n\n-- Decidable equivalence for types and contexts :-|\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , refl , p , refl , refl , dvv) with \u0393 \u225fCtx \u0393\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) | yes refl = refl\n... | no \u00acq = \u22a5-elim (\u00acq refl)\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x ,  rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ _ _ _ (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 0 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 0 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 0 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032} {vtv} {dv}\n    {dvs : DSVal \u0393 (\u03c3 \u21d2 \u03c4)}\n    {vt : SVal \u0393 \u03c3} {dvt : DSVal \u0393 \u03c3}\n    {dvtv} {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nopen import Relation.Nullary\n\n-- Decidable equivalence for types and contexts :-|\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\n_\u2295_ (closure {\u0393} t \u03c1) (dclosure {\u03931} dt \u03c1\u2081 d\u03c1) with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u0393\u2081} dt \u03c1\u2081 d\u03c1 | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082) ((refl , refl) , refl , p , refl , refl , dvv) with \u0393 \u225fCtx \u0393\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) | yes refl = refl\n... | no \u00acq = \u22a5-elim (\u00acq refl)\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .0 j<n (var .x) (var .x) = (D.\u27e6 x \u27e7Var d\u03c1) , dvar x , \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .0 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv ,  rrel\u03c13-mono k\u2081 (suc k) (lt1 k<n) _ _ _ _ \u03c1\u03c1)\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .0 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n  | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n  with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2 (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ _ _ _ dtvv) v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv = dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ _ _ _ dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"db839feeb7f7f27feda06757047e5d3cbf2cddd5","subject":"Prove that invalid changes exist (see #33).","message":"Prove that invalid changes exist (see #33).\n\nOld-commit-hash: 0fc58758402d61cb6647866cd93486373152f575\n","repos":"inc-lc\/ilc-agda","old_file":"Denotational\/ValidChanges.agda","new_file":"Denotational\/ValidChanges.agda","new_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid-\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ninvalid-changes-exist : \u00ac (\u2200 {\u03c4} v dv \u2192 Valid-\u0394 {\u03c4} v dv)\ninvalid-changes-exist k with k (\u03bb x \u2192 x) (\u03bb x dx \u2192 false) false true\n... | _ , ()\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","old_contents":"module Denotational.ValidChanges where\n\nopen import Data.Product\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nopen import Syntactic.Types\n\nopen import Denotational.Notation\nopen import Denotational.Values\nopen import Denotational.Equivalence\n\nopen import Changes\n\n-- DEFINITION of valid changes via a logical relation\n\n{-\nWhat I wanted to write:\n\ndata Valid-\u0394 : {T : Type} \u2192 (v : \u27e6 T \u27e7) \u2192 (dv : \u27e6 \u0394-Type T \u27e7) \u2192 Set where\n  base : (v : \u27e6 bool \u27e7) \u2192 (dv : \u27e6 \u0394-Type bool \u27e7) \u2192 Valid\u0394 v dv\n  fun : \u2200 {S T} \u2192 (f : \u27e6 S \u21d2 T \u27e7) \u2192 (df : \u27e6 \u0394-Type (S \u21d2 T) \u27e7) \u2192\n    (\u2200 (s : \u27e6 S \u27e7) ds \u2192 (Valid\u0394 (f s) (df s ds)) \u00d7 ((apply df f) (apply ds s) \u2261 apply (df s ds) (f s))) \u2192 \n    Valid\u0394 f df\n-}\n\n-- What I had to write:\n-- Note: now I could go back to using a datatype, since the datatype is now strictly positive.\nValid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nValid-\u0394 {bool} v dv = \u22a4\nValid-\u0394 {S \u21d2 T} f df =\n  \u2200 (s : \u27e6 S \u27e7) ds {- (valid-w : Valid-\u0394 s ds) -} \u2192\n    Valid-\u0394 (f s) (df s ds) \u00d7\n    (apply df f) (apply ds s) \u2261 apply (df s ds) (f s)\n\ndiff-is-valid : \u2200 {\u03c4} (v\u2032 v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (diff v\u2032 v)\ndiff-is-valid {bool} v\u2032 v = tt\ndiff-is-valid {\u03c4 \u21d2 \u03c4\u2081} v\u2032 v =\n  \u03bb s ds \u2192\n    diff-is-valid (v\u2032 (apply ds s)) (v s) , (\n    begin\n      apply (diff v\u2032 v) v (apply ds s)\n    \u2261\u27e8 refl \u27e9\n      apply\n        (diff (v\u2032 (apply (derive (apply ds s)) (apply ds s))) (v (apply ds s)))\n        (v (apply ds s))\n    \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (v\u2032 x) (v (apply ds s))) (v (apply ds s))) (apply-derive (apply ds s)) \u27e9\n      apply (diff (v\u2032 (apply ds s)) (v (apply ds s))) (v (apply ds s))\n    \u2261\u27e8 apply-diff (v (apply ds s)) (v\u2032 (apply ds s)) \u27e9\n      v\u2032 (apply ds s)\n    \u2261\u27e8 sym (apply-diff (v s) (v\u2032 (apply ds s))) \u27e9\n      apply ((diff v\u2032 v) s ds) (v s)\n    \u220e) where open \u2261-Reasoning\n\nderive-is-valid : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192 Valid-\u0394 {\u03c4} v (derive v)\nderive-is-valid v rewrite sym (diff-derive v) = diff-is-valid v v\n\n-- This is a postulate elsewhere, but here I provide a proper proof.\n\ndiff-apply-proof : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n    (Valid-\u0394 v dv) \u2192 diff (apply dv v) v \u2261 dv\n\ndiff-apply-proof {\u03c4\u2081 \u21d2 \u03c4\u2082} df f df-valid = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  begin\n    diff (apply (df (apply dv v) (derive (apply dv v))) (f (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (sym (proj\u2082 (df-valid (apply dv v) (derive (apply dv v))))) \u27e9\n    diff ((apply df f) (apply (derive (apply dv v)) (apply dv v))) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff (apply df f x) (f v))\n       (apply-derive (apply dv v)) \u27e9\n    diff ((apply df f) (apply dv v)) (f v)\n  \u2261\u27e8 \u2261-cong\n     (\u03bb x \u2192 diff x (f v))\n       (proj\u2082 (df-valid v dv)) \u27e9\n    diff (apply (df v dv) (f v)) (f v)\n  \u2261\u27e8 diff-apply-proof {\u03c4\u2082} (df v dv) (f v) (proj\u2081 (df-valid v dv)) \u27e9\n    df v dv\n  \u220e)) where open \u2261-Reasoning\n\ndiff-apply-proof {bool} db b _ = xor-cancellative b db\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"37e44e7d8c4ae59ef7e39c510cd0f18e00322dbb","subject":"Added x>y\u2228x\u226fy.","message":"Added x>y\u2228x\u226fy.\n\nIgnore-this: 8a8a6e5b4054b358aff2a3be8264a379\n\ndarcs-hash:20110512132312-3bd4e-11ba58023babac9d38a899f342536bb84bfe8e8f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_file":"src\/FOTC\/Data\/Nat\/Inequalities\/PropertiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function using ( _$_ )\n\nopen import FOTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The FOTC natural numbers type\n        )\nopen import FOTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import FOTC.Data.Nat.Inequalities.Properties public\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 GE n n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\nx<Sy\u2192x\u2264y zN Nn 0<Sn       = 0\u2264x Nn\nx<Sy\u2192x\u2264y (sN Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m (succ n)\nx\u2264y\u2192x<Sy {n = n} zN      Nn 0\u2264n  = <-0S n\nx\u2264y\u2192x<Sy         (sN Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 NGT m n\nx\u2264y\u2192x\u226fy zN          Nn          0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (sN Nm)     zN          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (<-SS m (succ n))) Sm\u2264Sn))\n  where\n    postulate prf : NGT m n \u2192  -- IH.\n                    NGT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\nx>y\u2228x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 NGT m n\nx>y\u2228x\u226fy Nm Nn = [ (\u03bb m>n \u2192 inj\u2081 m>n) ,\n                  (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192x\u226fy Nm Nn m\u2264n))\n                ] (x>y\u2228x\u2264y Nm Nn)\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<y\u2238x Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (cong succ m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n","old_contents":"------------------------------------------------------------------------------\n-- Properties of the inequalities\n------------------------------------------------------------------------------\n\nmodule FOTC.Data.Nat.Inequalities.PropertiesATP where\n\nopen import FOTC.Base\n\nopen import Common.Function using ( _$_ )\n\nopen import FOTC.Data.Nat\n  using ( _+_ ; _\u2238_ ; \u2238-SS\n        ; N ; sN ; zN  -- The FOTC natural numbers type\n        )\nopen import FOTC.Data.Nat.Inequalities\n  using ( <-00 ; <-0S ; <-S0 ; <-SS\n        ; GE ; GT ; LE ; LT ; NGT ; NLE ; NLT\n        ; LT\u2082\n        )\nopen import FOTC.Data.Nat.Inequalities.Properties public\nopen import FOTC.Data.Nat.PropertiesATP\n  using ( +-N ; \u2238-N\n        ; +-comm\n        ; +-rightIdentity\n        ; \u2238-0x\n        )\n\n------------------------------------------------------------------------------\n\nx\u22650 : \u2200 {n} \u2192 N n \u2192 GE n zero\nx\u22650 zN          = <-0S zero\nx\u22650 (sN {n} Nn) = <-0S $ succ n\n\n0\u2264x : \u2200 {n} \u2192 N n \u2192 LE zero n\n0\u2264x Nn = x\u22650 Nn\n\n0\u226fx : \u2200 {n} \u2192 N n \u2192 NGT zero n\n0\u226fx zN          = <-00\n0\u226fx (sN {n} Nn) = <-S0 n\n\nx\u2270x : \u2200 {n} \u2192 N n \u2192 NLT n n\nx\u2270x zN          = <-00\nx\u2270x (sN {n} Nn) = trans (<-SS n n) (x\u2270x Nn)\n\nS\u22700 : \u2200 {n} \u2192 N n \u2192 NLE (succ n) zero\nS\u22700 zN          = x\u2270x (sN zN)\nS\u22700 (sN {n} Nn) = trans (<-SS (succ n) zero) (<-S0 n)\n\nx<Sx : \u2200 {n} \u2192 N n \u2192 LT n (succ n)\nx<Sx zN          = <-0S zero\nx<Sx (sN {n} Nn) = trans (<-SS n (succ n)) (x<Sx Nn)\n\npostulate\n  x<y\u2192Sx<Sy : \u2200 {m n} \u2192 LT m n \u2192 LT (succ m) (succ n)\n{-# ATP prove x<y\u2192Sx<Sy #-}\n\npostulate\n  Sx<Sy\u2192x<y : \u2200 {m n} \u2192 LT (succ m) (succ n) \u2192 LT m n\n{-# ATP prove Sx<Sy\u2192x<y #-}\n\nx\u2264x : \u2200 {n} \u2192 N n \u2192 LE n n\nx\u2264x zN          = <-0S zero\nx\u2264x (sN {n} Nn) = trans (<-SS n (succ n)) (x\u2264x Nn)\n\nx\u2265x : \u2200 {n} \u2192 N n \u2192 GE n n\nx\u2265x Nn = x\u2264x Nn\n\nx\u2264y\u2192Sx\u2264Sy : \u2200 {m n} \u2192 LE m n \u2192 LE (succ m) (succ n)\nx\u2264y\u2192Sx\u2264Sy {m} {n} m\u2264n = trans (<-SS m (succ n)) m\u2264n\n\npostulate\n  Sx\u2264Sy\u2192x\u2264y : \u2200 {m n} \u2192 LE (succ m) (succ n) \u2192 LE m n\n{-# ATP prove Sx\u2264Sy\u2192x\u2264y #-}\n\nx\u2270y\u2192Sx\u2270Sy : \u2200 m n \u2192 NLE m n \u2192 NLE (succ m) (succ n)\nx\u2270y\u2192Sx\u2270Sy m n m\u2270n = trans (<-SS m (succ n)) m\u2270n\n\nx>y\u2192y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LT n m\nx>y\u2192y<x zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y<x (sN {m} Nm) zN          _     = <-0S m\nx>y\u2192y<x (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS n m) (x>y\u2192y<x Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\nx\u2265y\u2192x\u226ey : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 NLT m n\nx\u2265y\u2192x\u226ey zN          zN          _     = x\u2270x zN\nx\u2265y\u2192x\u226ey zN          (sN Nn)     0\u2265Sn  = \u22a5-elim $ 0\u2265S\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192x\u226ey (sN {m} Nm) zN          _     = <-S0 m\nx\u2265y\u2192x\u226ey (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn =\n  trans (<-SS m n) (x\u2265y\u2192x\u226ey Nm Nn (trans (sym $ <-SS n (succ m)) Sm\u2265Sn))\n\nx>y\u2192x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NLE m n\nx>y\u2192x\u2270y zN          Nn          0>m   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>m\nx>y\u2192x\u2270y (sN Nm)     zN          _     = S\u22700 Nm\nx>y\u2192x\u2270y (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  x\u2270y\u2192Sx\u2270Sy m n (x>y\u2192x\u2270y Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x>y\u2192x\u2264y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 LE m n \u2192 \u22a5\n{-# ATP prove x>y\u2192x\u2264y\u2192\u22a5 x>y\u2192x\u2270y #-}\n\nx>y\u2228x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2228 LE m n\nx>y\u2228x\u2264y zN          Nn          = inj\u2082 $ x\u22650 Nn\nx>y\u2228x\u2264y (sN {m} Nm) zN          = inj\u2081 $ <-0S m\nx>y\u2228x\u2264y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m>n \u2192 inj\u2081 (trans (<-SS n m) m>n))\n  , (\u03bb m\u2264n \u2192 inj\u2082 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  ] (x>y\u2228x\u2264y Nm Nn)\n\nx<y\u2228x\u2265y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2228 GE m n\nx<y\u2228x\u2265y Nm Nn = x>y\u2228x\u2264y Nn Nm\n\nx\u2264y\u2228x\u2270y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2228 NLE m n\nx\u2264y\u2228x\u2270y zN Nn = inj\u2081 (0\u2264x Nn)\nx\u2264y\u2228x\u2270y (sN Nm) zN = inj\u2082 (S\u22700 Nm)\nx\u2264y\u2228x\u2270y (sN {m} Nm) (sN {n} Nn) =\n  [ (\u03bb m\u2264n \u2192 inj\u2081 (x\u2264y\u2192Sx\u2264Sy m\u2264n))\n  , (\u03bb m\u2270n \u2192 inj\u2082 (x\u2270y\u2192Sx\u2270Sy m n m\u2270n))\n  ] (x\u2264y\u2228x\u2270y Nm Nn)\n\nx\u2261y\u2192x\u2264y : \u2200 {m n} {Nm : N m} {Nn : N n} \u2192 m \u2261 n \u2192 LE m n\nx\u2261y\u2192x\u2264y {Nm = Nm} refl = x\u2264x Nm\n\nx<y\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE m n\nx<y\u2192x\u2264y Nm zN          m<0            = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x\u2264y zN (sN {n} Nn)          _     = <-0S $ succ n\nx<y\u2192x\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x\u2264y\u2192Sx\u2264Sy (x<y\u2192x\u2264y Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LE m n\nx<Sy\u2192x\u2264y zN Nn 0<Sn       = 0\u2264x Nn\nx<Sy\u2192x\u2264y (sN Nm) Nn Sm<Sn = Sm<Sn\n\nx\u2264y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m (succ n)\nx\u2264y\u2192x<Sy {n = n} zN      Nn 0\u2264n  = <-0S n\nx\u2264y\u2192x<Sy         (sN Nm) Nn Sm\u2264n = Sm\u2264n\n\nx\u2264Sx : \u2200 {m} \u2192 N m \u2192 LE m (succ m)\nx\u2264Sx Nm = x<y\u2192x\u2264y Nm (sN Nm) (x<Sx Nm)\n\nx<y\u2192Sx\u2264y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LE (succ m) n\nx<y\u2192Sx\u2264y Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192Sx\u2264y zN          (sN Nn)     0<Sn  = x\u2264y\u2192Sx\u2264Sy (0\u2264x Nn)\nx<y\u2192Sx\u2264y (sN {m} Nm) (sN {n} Nn) Sm<Sn = trans (<-SS (succ m) (succ n)) Sm<Sn\n\nSx\u2264y\u2192x<y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (succ m) n \u2192 LT m n\nSx\u2264y\u2192x<y Nm          zN          Sm\u22640   = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\nSx\u2264y\u2192x<y zN          (sN {n} Nn) _      = <-0S n\nSx\u2264y\u2192x<y (sN {m} Nm) (sN {n} Nn) SSm\u2264Sn =\n  x<y\u2192Sx<Sy (Sx\u2264y\u2192x<y Nm Nn (Sx\u2264Sy\u2192x\u2264y SSm\u2264Sn))\n\nx\u2264y\u2192x\u226fy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 NGT m n\nx\u2264y\u2192x\u226fy zN          Nn          0\u2264n   = 0\u226fx Nn\nx\u2264y\u2192x\u226fy (sN Nm)     zN          Sm\u22640  = \u22a5-elim (S\u22640\u2192\u22a5 Nm Sm\u22640)\nx\u2264y\u2192x\u226fy (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn =\n  prf (x\u2264y\u2192x\u226fy Nm Nn (trans (sym (<-SS m (succ n))) Sm\u2264Sn))\n  where\n    postulate prf : NGT m n \u2192  -- IH.\n                    NGT (succ m) (succ n)\n    {-# ATP prove prf #-}\n\n<-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LT n o \u2192 LT m o\n<-trans zN          zN          _           0<0   _     = \u22a5-elim $ 0<0\u2192\u22a5 0<0\n<-trans zN          (sN Nn)     zN          _     Sn<0  = \u22a5-elim $ S<0\u2192\u22a5 Sn<0\n<-trans zN          (sN Nn)     (sN {o} No) _     _     = <-0S o\n<-trans (sN Nm)     Nn          zN          _     n<0   = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n<-trans (sN Nm)     zN          (sN No)     Sm<0  _     = \u22a5-elim $ S<0\u2192\u22a5 Sm<0\n<-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm<Sn Sn<So =\n  x<y\u2192Sx<Sy $ <-trans Nm Nn No (Sx<Sy\u2192x<y Sm<Sn) (Sx<Sy\u2192x<y Sn<So)\n\n\u2264-trans : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LE m n \u2192 LE n o \u2192 LE m o\n\u2264-trans zN      Nn              No          _     _     = 0\u2264x No\n\u2264-trans (sN Nm) zN              No          Sm\u22640  _     = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\u2264-trans (sN Nm) (sN Nn)         zN          _     Sn\u22640  = \u22a5-elim $ S\u22640\u2192\u22a5 Nn Sn\u22640\n\u2264-trans (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2264Sn Sn\u2264So =\n  x\u2264y\u2192Sx\u2264Sy (\u2264-trans Nm Nn No (Sx\u2264Sy\u2192x\u2264y Sm\u2264Sn) (Sx\u2264Sy\u2192x\u2264y Sn\u2264So))\n\nx\u2264x+y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (m + n)\nx\u2264x+y         zN          Nn = x\u22650 (+-N zN Nn)\nx\u2264x+y {n = n} (sN {m} Nm) Nn = prf $ x\u2264x+y Nm Nn\n  where\n    postulate prf : LE m (m + n) \u2192  -- IH.\n                    LE (succ m) (succ m + n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf #-}\n\nx<x+Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (m + succ n)\nx<x+Sy {n = n} zN Nn = prf0\n  where\n    postulate prf0 : LT zero (zero + succ n)\n    {-# ATP prove prf0 #-}\nx<x+Sy {n = n} (sN {m} Nm) Nn = prfS (x<x+Sy Nm Nn)\n  where\n    postulate prfS : LT m (m + succ n) \u2192 LT (succ m) (succ m + succ n)\n    {-# ATP prove prfS #-}\n\nk+x<k+y\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LT (k + m) (k + n) \u2192 LT m n\nk+x<k+y\u2192x<y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LT m n\n    {-# ATP prove prf0 #-}\nk+x<k+y\u2192x<y {m} {n} Nm Nn (sN {k} Nk) h = k+x<k+y\u2192x<y Nm Nn Nk prfS\n  where\n    postulate prfS : LT (k + m) (k + n)\n    {-# ATP prove prfS #-}\n\npostulate x+k<y+k\u2192x<y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192\n                        LT (m + k) (n + k) \u2192 LT m n\n{-# ATP prove x+k<y+k\u2192x<y k+x<k+y\u2192x<y +-comm #-}\n\nx\u2264y\u2192k+x\u2264k+y : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192 LE (k + m) (k + n)\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn zN h = prf0\n  where\n    postulate prf0 : LE (zero + m) (zero + n)\n    {-# ATP prove prf0 #-}\nx\u2264y\u2192k+x\u2264k+y {m} {n} Nm Nn (sN {k} Nk) h = prfS (x\u2264y\u2192k+x\u2264k+y Nm Nn Nk h)\n  where\n    postulate prfS : LE (k + m) (k + n) \u2192 LE (succ k + m) (succ k + n)\n    {-# ATP prove prfS #-}\n\npostulate x\u2264y\u2192x+k\u2264y+k : \u2200 {m n k} \u2192 N m \u2192 N n \u2192 N k \u2192 LE m n \u2192\n                        LE (m + k) (n + k)\n{-# ATP prove x\u2264y\u2192x+k\u2264y+k x\u2264y\u2192k+x\u2264k+y +-comm #-}\n\nx<y\u2192Sx\u2238y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 succ m \u2238 n \u2261 zero\nx<y\u2192Sx\u2238y\u22610 Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u2192Sx\u2238y\u22610 zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : succ zero \u2238 succ n \u2261 zero\n    {-# ATP prove prf0S \u2238-0x #-}\nx<y\u2192Sx\u2238y\u22610 (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u2192Sx\u2238y\u22610 Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : succ m \u2238 n \u2261 zero \u2192 succ (succ m) \u2238 succ n \u2261 zero\n    {-# ATP prove prfSS #-}\n\npostulate x\u2264y\u2192x-y\u22610 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (m \u2238 n) \u2261 zero\n{-# ATP prove x\u2264y\u2192x-y\u22610 x<y\u2192Sx\u2238y\u22610 #-}\n\nx<y\u21920<y\u2238x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT zero (n \u2238 m)\nx<y\u21920<y\u2238x Nm zN h = \u22a5-elim (x<0\u2192\u22a5 Nm h)\nx<y\u21920<y\u2238x zN (sN {n} Nn) h = prf0S\n  where\n    postulate prf0S : LT zero (succ n \u2238 zero)\n    {-# ATP prove prf0S #-}\n\nx<y\u21920<y\u2238x (sN {m} Nm) (sN {n} Nn) h = prfSS (x<y\u21920<y\u2238x Nm Nn m<n)\n  where\n    postulate m<n : LT m n\n    {-# ATP prove m<n #-}\n\n    postulate prfSS : LT zero (n \u2238 m) \u2192 LT zero (succ n \u2238 succ m)\n    {-# ATP prove prfSS #-}\n\n0<x\u2238y\u21920<Sx\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT zero (m \u2238 n) \u2192 LT zero (succ m \u2238 n)\n0<x\u2238y\u21920<Sx\u2238y {m} Nm zN h = prfx0\n  where\n    postulate prfx0 : LT zero (succ m \u2238 zero)\n    {-# ATP prove prfx0 #-}\n\n0<x\u2238y\u21920<Sx\u2238y zN (sN {n} Nn) h = \u22a5-elim (x<0\u2192\u22a5 zN h')\n  where\n    postulate h' : LT zero zero\n    {-# ATP prove h' #-}\n\n0<x\u2238y\u21920<Sx\u2238y (sN {m} Nm) (sN {n} Nn) h = prfSS (0<x\u2238y\u21920<Sx\u2238y Nm Nn 0<m-n)\n  where\n    postulate 0<m-n : LT zero (m \u2238 n)\n    {-# ATP prove 0<m-n #-}\n\n    postulate prfSS : LT zero (succ m \u2238 n) \u2192 LT zero (succ (succ m) \u2238 succ n)\n    {-# ATP prove prfSS <-trans #-}\n\nx\u2238y<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (m \u2238 n) (succ m)\nx\u2238y<Sx {m} Nm zN = prf\n  where\n    postulate prf : LT (m \u2238 zero) (succ m)\n    {-# ATP prove prf x<Sx #-}\n\nx\u2238y<Sx zN (sN {n} Nn) = prf\n  where\n    postulate prf : LT (zero \u2238 succ n) (succ zero)\n    {-# ATP prove prf #-}\n\nx\u2238y<Sx (sN {m} Nm) (sN {n} Nn) = prf $ x\u2238y<Sx Nm Nn\n  where\n    postulate prf : LT (m \u2238 n) (succ m) \u2192  -- IH.\n                    LT (succ m \u2238 succ n) (succ (succ m))\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf <-trans \u2238-N x<Sx #-}\n\npostulate\n  Sx\u2238Sy<Sx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT (succ m \u2238 succ n) (succ m)\n{-# ATP prove Sx\u2238Sy<Sx x\u2238y<Sx #-}\n\nx<x\u2238y\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT m (m \u2238 n))\nx<x\u2238y\u2192\u22a5 {m} Nm zN m<m\u22380 = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 zN (sN Nn) 0<0\u2238Sn = prf\n where\n   postulate prf : \u22a5\n   {-# ATP prove prf x<x\u2192\u22a5 #-}\nx<x\u2238y\u2192\u22a5 (sN Nm) (sN Nn) Sm<Sm\u2238Sn = prf\n  where\n    postulate prf : \u22a5\n    {-# ATP prove prf \u2238-N x<y\u2192y<x\u2192\u22a5 x\u2238y<Sx #-}\n\nx\u2238Sy\u2264x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE (m \u2238 succ n) (m \u2238 n)\nx\u2238Sy\u2264x\u2238y {n = n} zN Nn = prf\n  where\n    postulate prf : LE (zero \u2238 succ n) (zero \u2238 n)\n    {-# ATP prove prf 0\u2264x #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) zN = prf\n  where\n    postulate prf : LE (succ m \u2238 succ zero) (succ m \u2238 zero)\n    {-# ATP prove prf x\u2264Sx #-}\n\nx\u2238Sy\u2264x\u2238y (sN {m} Nm) (sN {n} Nn) = prf (x\u2238Sy\u2264x\u2238y Nm Nn)\n  where\n    postulate prf : LE (m \u2238 succ n) (m \u2238 n) \u2192  -- IH.\n                    LE (succ m \u2238 succ (succ n)) (succ m \u2238 (succ n))\n    {-# ATP prove prf #-}\n\nx>y\u2192x\u2238y+y\u2261x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 (m \u2238 n) + n \u2261 m\nx>y\u2192x\u2238y+y\u2261x zN          Nn 0>n  = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) zN Sm>0 = prf\n  where\n    postulate prf : (succ m \u2238 zero) + zero \u2261 succ m\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx>y\u2192x\u2238y+y\u2261x (sN {m} Nm) (sN {n} Nn) Sm>Sn = prf $ x>y\u2192x\u2238y+y\u2261x Nm Nn m>n\n  where\n    postulate m>n : GT m n\n    {-# ATP prove m>n #-}\n\n    postulate prf : (m \u2238 n) + n \u2261 m \u2192  -- IH.\n                    (succ m \u2238 succ n) + succ n \u2261 succ m\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y {n = n} zN Nn 0\u2264n  = prf\n  where\n    postulate prf : (n \u2238 zero) + zero \u2261 n\n    {-# ATP prove prf +-rightIdentity \u2238-N #-}\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN Nm) zN Sm\u22640 = \u22a5-elim $ S\u22640\u2192\u22a5 Nm Sm\u22640\n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf $ x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n\n  where\n    postulate m\u2264n : LE m n\n    {-# ATP prove m\u2264n #-}\n\n    postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                    (succ n \u2238 succ m) + succ m \u2261 succ n\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    -- Vampire 0.6 (revision 903): No-success (using timeout 180 sec).\n    {-# ATP prove prf +-comm \u2238-N #-}\n\nx<y\u2192x<Sy : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m n \u2192 LT m (succ n)\nx<y\u2192x<Sy Nm          zN          m<0   = \u22a5-elim $ x<0\u2192\u22a5 Nm m<0\nx<y\u2192x<Sy zN          (sN {n} Nn) 0<Sn  = <-0S $ succ n\nx<y\u2192x<Sy (sN {m} Nm) (sN {n} Nn) Sm<Sn =\n  x<y\u2192Sx<Sy (x<y\u2192x<Sy Nm Nn (Sx<Sy\u2192x<y Sm<Sn))\n\nx<Sy\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT m (succ n) \u2192 LT m n \u2228 m \u2261 n\nx<Sy\u2192x<y\u2228x\u2261y zN zN 0<S0 = inj\u2082 refl\nx<Sy\u2192x<y\u2228x\u2261y zN (sN {n} Nn) 0<SSn = inj\u2081 (<-0S n)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) zN Sm<S0 =\n  \u22a5-elim $ x<0\u2192\u22a5 Nm (trans (sym $ <-SS m zero) Sm<S0)\nx<Sy\u2192x<y\u2228x\u2261y (sN {m} Nm) (sN {n} Nn) Sm<SSn =\n  [ (\u03bb m<n \u2192 inj\u2081 (trans (<-SS m n) m<n))\n  , (\u03bb m\u2261n \u2192 inj\u2082 (cong succ m\u2261n))\n  ]\n  m<n\u2228m\u2261n\n\n  where\n    m<n\u2228m\u2261n : LT m n \u2228 m \u2261 n\n    m<n\u2228m\u2261n = x<Sy\u2192x<y\u2228x\u2261y Nm Nn (trans (sym $ <-SS m (succ n)) Sm<SSn)\n\nx\u2264y\u2192x<y\u2228x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 LT m n \u2228 m \u2261 n\nx\u2264y\u2192x<y\u2228x\u2261y = x<Sy\u2192x<y\u2228x\u2261y\n\npostulate\n  x<y\u2192y\u2261z\u2192x<z : \u2200 {m n o} \u2192 LT m n \u2192 n \u2261 o \u2192 LT m o\n{-# ATP prove x<y\u2192y\u2261z\u2192x<z #-}\n\npostulate\n  x\u2261y\u2192y<z\u2192x<z : \u2200 {m n o} \u2192 m \u2261 n \u2192 LT n o \u2192 LT m o\n{-# ATP prove x\u2261y\u2192y<z\u2192x<z #-}\n\nx\u2265y\u2192y>0\u2192x\u2238y<x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GE m n \u2192 GT n zero \u2192 LT (m \u2238 n) m\nx\u2265y\u2192y>0\u2192x\u2238y<x Nm          zN          _     0>0  = \u22a5-elim $ x>x\u2192\u22a5 zN 0>0\nx\u2265y\u2192y>0\u2192x\u2238y<x zN          (sN Nn)     0\u2265Sn  _    = \u22a5-elim $ S\u22640\u2192\u22a5 Nn 0\u2265Sn\nx\u2265y\u2192y>0\u2192x\u2238y<x (sN {m} Nm) (sN {n} Nn) Sm\u2265Sn Sn>0 = prf\n  where\n    postulate prf : LT (succ m \u2238 succ n) (succ m)\n    {-# ATP prove prf x\u2238y<Sx #-}\n\nx<y\u2192y\u2264z\u2192x<z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192 LT m n \u2192 LE n o \u2192 LT m o\nx<y\u2192y\u2264z\u2192x<z Nm Nn No m<n n\u2264o =\n  [ (\u03bb n<o \u2192 <-trans Nm Nn No m<n n<o)\n  , (\u03bb n\u2261o \u2192 x<y\u2192y\u2261z\u2192x<z m<n n\u2261o)\n  ] (x<Sy\u2192x<y\u2228x\u2261y Nn No n\u2264o)\n\nx\u2264y+x\u2238y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m (n + (m \u2238 n))\nx\u2264y+x\u2238y {n = n} zN Nn = prf0\n  where postulate prf0 : LE zero (n + (zero \u2238 n))\n        {-# ATP prove prf0 0\u2264x +-N  #-}\nx\u2264y+x\u2238y (sN {m} Nm) zN = prfx0\n  where postulate prfx0 : LE (succ m) (zero + (succ m \u2238 zero))\n        {-# ATP prove prfx0 x<Sx #-}\nx\u2264y+x\u2238y (sN {m} Nm) (sN {n} Nn) = prfSS (x\u2264y+x\u2238y Nm Nn)\n  where postulate prfSS : LE m (n + (m \u2238 n)) \u2192  -- IH.\n                          LE (succ m) (succ n + (succ m \u2238 succ n))\n        {-# ATP prove prfSS x\u2264y\u2192Sx\u2264Sy \u2264-trans +-N \u2238-N #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z : \u2200 {m n o} \u2192 N m \u2192 N n \u2192 N o \u2192\n                    LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {n = n} {o} zN Nn No 0\u2238n<0\u2238o = prf\n  where\n    postulate prf : LT (succ zero \u2238 n) (succ zero \u2238 o)\n    {-# ATP prove prf \u2238-0x 0<0\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z {o = o} (sN {m} Nm) zN No Sm\u22380<Sm\u2238o = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 zero) (succ (succ m) \u2238 o)\n    {-# ATP prove prf x<x\u2238y\u2192\u22a5 #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) zN Sm\u2238Sn<Sm\u22380 = prf\n  where\n    postulate prf : LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 zero)\n    {-# ATP prove prf Sx\u2238Sy<Sx #-}\n\nx\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z (sN {m} Nm) (sN {n} Nn) (sN {o} No) Sm\u2238Sn<Sm\u2238So =\n  prf (x\u2238y<x\u2238z\u2192Sx\u2238y<Sx\u2238z Nm Nn No)\n  where\n    postulate\n      prf : (LT (m \u2238 n) (m \u2238 o) \u2192 LT (succ m \u2238 n) (succ m \u2238 o)) \u2192  -- IH.\n            LT (succ (succ m) \u2238 succ n) (succ (succ m) \u2238 succ o)\n    {-# ATP prove prf #-}\n\n------------------------------------------------------------------------------\n-- Properties about LT\u2082\n\npostulate\n  xy<00\u2192\u22a5 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 \u00ac (LT\u2082 m n zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  Sxy\u2081<0y\u2082\u2192\u22a5 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 \u00ac (LT\u2082 (succ m) n\u2081 zero n\u2082)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove Sxy\u2081<0y\u2082\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  0Sx<00\u2192\u22a5 : \u2200 {m} \u2192 N m \u2192 \u00ac (LT\u2082 zero (succ m) zero zero)\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove 0Sx<00\u2192\u22a5 x<0\u2192\u22a5 #-}\n\npostulate\n  x\u2081y<x\u20820\u2192x\u2081<x\u2082 : \u2200 {m\u2081 n m\u2082} \u2192 N m\u2081 \u2192 N n \u2192 N m\u2082 \u2192 LT\u2082 m\u2081 n m\u2082 zero \u2192 LT m\u2081 m\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove x\u2081y<x\u20820\u2192x\u2081<x\u2082 x<0\u2192\u22a5 #-}\n\npostulate\n  xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 : \u2200 {m n\u2081 n\u2082} \u2192 N m \u2192 N n\u2081 \u2192 N n\u2082 \u2192 LT\u2082 m n\u2081 zero n\u2082 \u2192\n                      m \u2261 zero \u2227 LT n\u2081 n\u2082\n-- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n{-# ATP prove xy\u2081<0y\u2082\u2192x\u22610\u2227y\u2081<y\u2082 x<0\u2192\u22a5 #-}\n\n[Sx\u2238Sy,Sy]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n[Sx\u2238Sy,Sy]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m \u2238 succ n) (succ n) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n\n[Sx,Sy\u2238Sx]<[Sx,Sy] : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n                     LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n[Sx,Sy\u2238Sx]<[Sx,Sy] {m} {n} Nm Nn = prf\n  where\n    postulate prf : LT\u2082 (succ m) (succ n \u2238 succ m) (succ m) (succ n)\n    -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n    {-# ATP prove prf x\u2238y<Sx #-}\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"a333e14c035eb23f5ec80043c27c9ea91a47102c","subject":"A view-based alternative to the universe-based solution to pattern match against function calls.","message":"A view-based alternative to the universe-based solution to pattern match against function calls.\n","repos":"spire\/spire","old_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\nAn alternative to the Arith codes is to create a *view* of the\nnatural numbers along with a collection of functions over them.\n-}\n\ndata ArithView : \u2115 \u2192 Set where\n  `nat : (n : \u2115) \u2192 ArithView n\n  _`+_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 + n\u2082)\n  _`*_ : (n\u2081 n\u2082 : \u2115) \u2192 ArithView (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nHere is the ArithView specialized into the Fin type.\n-}\n\ndata Fin\u2083 : \u2115 \u2192 Set where\n  fin : (n : \u2115) (i : Fin n) \u2192 Fin\u2083 n\n  fin+ : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 + n\u2082)) \u2192 Fin\u2083 (n\u2081 + n\u2082)\n  fin* : (n\u2081 n\u2082 : \u2115) (i : Fin (n\u2081 * n\u2082)) \u2192 Fin\u2083 (n\u2081 * n\u2082)\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `ArithView `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `Fin\u2083 : (n : \u2115) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `ArithView n \u27e7 = ArithView n\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `Fin\u2083 n \u27e7 = Fin\u2083 n\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin\u2082 : Tactic\nrm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin\u2082 x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin\u2082 : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin\u2082 a (fin i) = proj\u2082\n  (rm-plus0-Fin\u2082 (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic that simplifies an ArithView index.\nTo target Fin, ideally you want to match on something like:\n`\u03a3 `\u2115 (\u03bb n \u2192 `ArithView n `\u00d7 Fin n)\nBut this requires matching on a \u03bb, and a non-linear occurrence\nof n.\n\nThe example after this one accomplishes the same thing via the\nspecialized Fin\u2083 type instead.\n-}\n\nrm-plus0-ArithView : Tactic\nrm-plus0-ArithView (`ArithView .(n + 0) , (n `+ 0)) =\n  `ArithView n , `nat n \nrm-plus0-ArithView x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify to simplify the \"view\"-based Fin type Fin\u2083.\nThis is most similar to rm-plus0-Fin\u2082, except Fin\u2083 is view-based\nwhile Fin\u2082 is universe-based (where the universe is the codes of\npossible operations).\n\nOn one hand this Fin\u2083 approach is simpler than Fin\u2082, because you do\nnot need to write an eval function or tell it to reduce definitional\nequalities explicitly (like you would to get Arith to compute).\n\nOn the other hand, the Fin\u2082 approach is more powerful because it\nallows you to control when things reduce. This is more like Coq\ntactics, where you might not want something to reduce and you use\n\"simpl\" explicitly when you do. The \"simpl\" tactic is defined below\nvia the eval\u2082 function.\n-}\n\nrm-plus0-Fin\u2083 : Tactic\nrm-plus0-Fin\u2083 (`Fin\u2083 .(n + 0) , fin+ n 0 i) =\n  `Fin\u2083 n\n  ,\n  fin n (subst Fin (sym (plusrident n)) i)\nrm-plus0-Fin\u2083 x = x\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","old_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nGoing further, we could even specialize B in \u03a6Arith to Fin.\n-}\n\ndata Fin\u2082 (a : Arith) : Set where\n  fin : (i : Fin (eval a)) \u2192 Fin\u2082 a\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `Fin\u2082 : (a : Arith) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `Fin\u2082 a \u27e7 = Fin\u2082 a\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 i) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n\n{-\nFinally, here is the same tactic but using the even more \nspecialized Fin\u2082 type.\n-}\n\nrm-plus0-Fin : Tactic\nrm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i) =\n  `Fin\u2082 a\n  ,\n  fin (subst (\u03bb x \u2192 \u27e6 `Fin x \u27e7) (sym (plusrident (eval a))) i)\nrm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\neg-rm-plus0-Fin : (a : Arith)\n  \u2192 \u27e6 `Fin\u2082 (a `+ `zero) \u27e7\n  \u2192 \u27e6 `Fin\u2082 a \u27e7\neg-rm-plus0-Fin a (fin i) = proj\u2082\n  (rm-plus0-Fin (`Fin\u2082 (a `+ `zero) , fin i))\n\n----------------------------------------------------------------------\n\n{-\nA tactic to simplify by one step of `+.\n-}\n\nsimp-step-plus0-Fin : Tactic\nsimp-step-plus0-Fin (`Fin\u2082 (`zero `+ a) , fin i) =\n  `Fin\u2082 a , fin i\nsimp-step-plus0-Fin (`Fin\u2082 (`suc a\u2081 `+ a\u2082) , fin i) =\n  `Fin\u2082 (`suc (a\u2081 `+ a\u2082)) , fin i\nsimp-step-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nA semantics-preserving definitional evaluator, and a tactic\nto simplify as far as possible using definitional equality.\nThis is like the \"simp\" tactic in Coq.\n-}\n\neval\u2082 : (a\u2081 : Arith) \u2192 \u03a3 Arith (\u03bb a\u2082 \u2192 eval a\u2081 \u2261 eval a\u2082)\neval\u2082 `zero = `zero , refl\neval\u2082 (`suc a)\n  with eval\u2082 a\n... | a\u2032 , p\n  rewrite p = `suc a\u2032 , refl\neval\u2082 (`zero `+ a\u2082) = eval\u2082 a\u2082\neval\u2082 (`suc a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = `suc (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (a\u2081 `+ a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `+ a\u2082\u2032) , refl\neval\u2082 (`zero `* a\u2082) = `zero , refl\neval\u2082 (`suc a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = a\u2082\u2032 `+ (a\u2081\u2032 `* a\u2082\u2032) , refl\neval\u2082 (a\u2081 `* a\u2082)\n  with eval\u2082 a\u2081 | eval\u2082 a\u2082\n... | a\u2081\u2032 , p | a\u2082\u2032 , q\n  rewrite p | q = (a\u2081\u2032 `* a\u2082\u2032) , refl\n\nsimp-Fin : Tactic\nsimp-Fin (`Fin\u2082 a , fin i)\n  with eval\u2082 a\n... | a\u2032 , p rewrite p =\n  `Fin\u2082 a\u2032 , fin i\nsimp-Fin x = x\n\n----------------------------------------------------------------------\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2a23832b1f89001a47437db27131840307e5f806","subject":"Add replacement changes to proof","message":"Add replacement changes to proof\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_file":"Thesis\/StepIndexedRelBigStepTypedAnfIlcCorrect.agda","new_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  bang : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 DVal \u03c4\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nv1 \u2295 bang v2 = v2\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n  bangapp : \u2200 {hasIdx} {n1 n2 : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv2}\n    {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} {v2} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 bang (closure t \u03c1\u2032) \u2192\n    (\u03c1 \u2295\u03c1 d\u03c1) \u22a2 val vt \u2193[ n1 ] vtv2 \u2192\n    (vtv2 \u2022 \u03c1\u2032) \u22a2 t \u2193[ n2 ] v2 \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 bang v2\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  -- Making this case first ensures rrelV3 splits on its second argument, hence\n  -- that all its equations hold definitionally.\n  rrelV3 \u03c4 v1 (bang v2) v2' n = v2 \u2261 v2'\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 v1 (bang v2) v2' vv = vv\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n \u03c4 v1 (bang v2) v2\u2032 vv = vv\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | bang f2 , vs\u2193dsv , refl | dtv , vt\u2193dvv , dtvv\n      rewrite sym (rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1) =\n        bang v2 , bangapp vs\u2193dsv \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4 , refl\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","old_contents":"-- Step-indexed logical relations based on relational big-step semantics\n-- for ILC-based incremental computation.\n\n-- Goal: prove the fundamental lemma for a ternary logical relation (change\n-- validity) across t1, dt and t2. The fundamnetal lemma relates t, derive t and\n-- t. That is, we relate a term evaluated relative to an original environment,\n-- its derivative evaluated relative to a valid environment change, and the\n-- original term evaluated relative to an updated environment.\n--\n-- Missing goal: here \u2295 isn't defined and wouldn't yet agree with change\n-- validity.\n--\n-- This development is strongly inspired by \"Imperative self-adjusting\n-- computation\" (ISAC below), POPL'08, including the choice of using ANF syntax\n-- to simplify some step-indexing proofs.\n--\n-- In fact, this development is typed, hence some parts of the model are closer\n-- to Ahmed (ESOP 2006), \"Step-Indexed Syntactic Logical Relations for Recursive\n-- and Quantified Types\". But for many relevant aspects, the two papers are\n-- very similar. In fact, I first defined similar logical relations\n-- without types, but they require a trickier recursion scheme for well-founded\n-- recursion, and I failed to do any proof in that setting.\n--\n-- The original inspiration came from Dargaye and Leroy (2010), \"A verified\n-- framework for higher-order uncurrying optimizations\", but we ended up looking\n-- at their source.\n--\n-- The main insight from the ISAC paper missing from the other one is how to\n-- step-index a big-step semantics correctly: just ensure that the steps in the\n-- big-step semantics agree with the ones in the small-step semantics. *Then*\n-- everything just works with big-step semantics. Quite a few other details are\n-- fiddly, but those are the same in small-step semantics.\n--\n-- The crucial novelty here is that we relate two computations on different\n-- inputs. So we only conclude their results are related if both terminate; the\n-- relation for computations does not prove that if the first computation\n-- terminates, then the second terminates as well.\n--\n-- Instead, e1, de and e2 are related at k steps if, whenever e1 terminates in j\n-- < k steps and e2 terminates with any step count, then de terminates (with any\n-- step count) and their results are related at k - j steps.\n--\n-- Even when e1 terminates in j steps implies that e2 terminates, e2 gets no\n-- bound. Similarly, we do not bound in how many steps de terminates, since on\n-- big inputs it might take long.\n\nmodule Thesis.StepIndexedRelBigStepTypedAnfIlcCorrect where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\ninfixr 20 _\u21d2_\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\n-- Decidable equivalence for types and contexts. Needed later for \u2295 on closures.\n\n\u21d2-inj : \u2200 {\u03c41 \u03c42 \u03c43 \u03c44 : Type} \u2192 _\u2261_ {A = Type} (\u03c41 \u21d2 \u03c42) (\u03c43 \u21d2 \u03c44) \u2192 \u03c41 \u2261 \u03c43 \u00d7 \u03c42 \u2261 \u03c44\n\u21d2-inj refl = refl , refl\n\n_\u225fType_ : (\u03c41 \u03c42 : Type) \u2192 Dec (\u03c41 \u2261 \u03c42)\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) with \u03c41 \u225fType \u03c43 | \u03c42 \u225fType \u03c44\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 .\u03c42) | yes refl | yes refl = yes refl\n(\u03c41 \u21d2 \u03c42) \u225fType (.\u03c41 \u21d2 \u03c44) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType (\u03c43 \u21d2 \u03c44) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u21d2-inj x)))\n(\u03c41 \u21d2 \u03c42) \u225fType nat = no (\u03bb ())\nnat \u225fType (\u03c42 \u21d2 \u03c43) = no (\u03bb ())\nnat \u225fType nat = yes refl\n\n\u2022-inj : \u2200 {\u03c41 \u03c42 : Type} {\u03931 \u03932 : Context} \u2192 _\u2261_ {A = Context} (\u03c41 \u2022 \u03931) (\u03c42 \u2022 \u03932) \u2192 \u03c41 \u2261 \u03c42 \u00d7 \u03931 \u2261 \u03932\n\u2022-inj refl = refl , refl\n\n_\u225fCtx_ : (\u03931 \u03932 : Context) \u2192 Dec (\u03931 \u2261 \u03932)\n\u2205 \u225fCtx \u2205 = yes refl\n\u2205 \u225fCtx (\u03c42 \u2022 \u03932) = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx \u2205 = no (\u03bb ())\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) with \u03c41 \u225fType \u03c42 | \u03931 \u225fCtx \u03932\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 .\u03931) | yes refl | yes refl = yes refl\n(\u03c41 \u2022 \u03931) \u225fCtx (.\u03c41 \u2022 \u03932) | yes refl | no \u00acq = no (\u03bb x \u2192 \u00acq (proj\u2082 (\u2022-inj x)))\n(\u03c41 \u2022 \u03931) \u225fCtx (\u03c42 \u2022 \u03932) | no \u00acp | q = no (\u03bb x \u2192 \u00acp (proj\u2081 (\u2022-inj x)))\n\n\u225fCtx-refl : \u2200 \u0393 \u2192 \u0393 \u225fCtx \u0393 \u2261 yes refl\n\u225fCtx-refl \u0393 with \u0393 \u225fCtx \u0393\n\u225fCtx-refl \u0393 | yes refl = refl\n\u225fCtx-refl \u0393 | no \u00acp = \u22a5-elim (\u00acp refl)\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) (\u03c4 : Type) : Set\n-- Source values\ndata SVal (\u0393 : Context) : (\u03c4 : Type) \u2192 Set where\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    SVal \u0393 \u03c4\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    SVal \u0393 (\u03c3 \u21d2 \u03c4)\ndata Term (\u0393 : Context) (\u03c4 : Type) where\n  val :\n    SVal \u0393 \u03c4 \u2192\n    Term \u0393 \u03c4\n  -- constants aka. primitives\n  const :\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  app : \u2200 {\u03c3}\n    (vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  lett : \u2200 {\u03c3}\n    (s : Term \u0393 \u03c3) \u2192\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 \u03c4\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  natV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Data.Integer as I\nopen I using (\u2124)\n\n-- Yann's idea.\ndata HasIdx : Set where\n  true : HasIdx\n  false : HasIdx\ndata Idx : HasIdx \u2192 Set where\n  i' : \u2115 \u2192 Idx true\n  no : Idx false\n\ni : {hasIdx : HasIdx} \u2192 \u2115 \u2192 Idx hasIdx\ni {false} j = no\ni {true} j = i' j\n\nmodule _ {hasIdx : HasIdx} where\n  data _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Idx hasIdx \u2192 Val \u03c4 \u2192 Set where\n    abs : \u2200 {\u03c3 \u03c4} {t : Term (\u03c3 \u2022 \u0393) \u03c4} \u2192\n      \u03c1 \u22a2 val (abs t) \u2193[ i 1 ] closure t \u03c1\n    var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n      \u03c1 \u22a2 val (var x) \u2193[ i 1 ] (\u27e6 x \u27e7Var \u03c1)\n    app : \u2200 n {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032} vtv {v} {vs : SVal \u0393 (\u03c3 \u21d2 \u03c4)} {vt : SVal \u0393 \u03c3} {t : Term (\u03c3 \u2022 \u0393\u2032) \u03c4} \u2192\n      \u03c1 \u22a2 val vs \u2193[ i 0 ] closure t \u03c1\u2032 \u2192\n      \u03c1 \u22a2 val vt \u2193[ i 0 ] vtv \u2192\n      (vtv \u2022 \u03c1\u2032) \u22a2 t \u2193[ i n ] v \u2192\n      \u03c1 \u22a2 app vs vt \u2193[ i (suc n) ] v\n    lett :\n      \u2200 n1 n2 {\u03c3 \u03c4} vsv {v} (s : Term \u0393 \u03c3) (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n      \u03c1 \u22a2 s \u2193[ i n1 ] vsv \u2192\n      (vsv \u2022 \u03c1) \u22a2 t \u2193[ i n2 ] v \u2192\n      \u03c1 \u22a2 lett s t \u2193[ i (suc n1 + n2) ] v\n    lit : \u2200 n \u2192\n      \u03c1 \u22a2 const (lit n) \u2193[ i 1 ] natV n\n\n-- data DType : Set where\n--   _\u21d2_ : (\u03c3 \u03c4 : DType) \u2192 DType\n--   int : DType\nDType = Type\n\nimport Base.Syntax.Context DType as DC\n\n\u0394\u03c4 : Type \u2192 DType\n\u0394\u03c4 (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394\u03c4 \u03c3 \u21d2 \u0394\u03c4 \u03c4\n\u0394\u03c4 nat = nat\n\n\u0394\u0394 : Context \u2192 DC.Context\n\u0394\u0394 \u2205 = \u2205\n\u0394\u0394 (\u03c4 \u2022 \u0393) = \u0394\u03c4 \u03c4 \u2022 \u0394\u0394 \u0393\n--\u0394\u0394 \u0393 = \u0393\n\n-- A DTerm evaluates in normal context \u0394, change context (\u0394\u0394 \u0394), and produces\n-- a result of type (\u0394t \u03c4).\ndata DTerm (\u0394 : Context) (\u03c4 : DType) : Set\ndata DSVal (\u0394 : Context) : (\u03c4 : DType) \u2192 Set where\n  dvar : \u2200 {\u03c4} \u2192\n    (x : Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c4)) \u2192\n    DSVal \u0394 \u03c4\n  dabs : \u2200 {\u03c3 \u03c4}\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DSVal \u0394 (\u03c3 \u21d2 \u03c4)\n\ndata DTerm (\u0394 : Context) (\u03c4 : DType) where\n  dval :\n    DSVal \u0394 \u03c4 \u2192\n    DTerm \u0394 \u03c4\n  -- constants aka. primitives\n  dconst :\n    (c : Const \u03c4) \u2192\n    DTerm \u0394 \u03c4\n  dapp : \u2200 {\u03c3}\n    (dvs : DSVal \u0394 (\u03c3 \u21d2 \u03c4)) \u2192\n    (vt : SVal \u0394 \u03c3) \u2192\n    (dvt : DSVal \u0394 \u03c3) \u2192\n    DTerm \u0394 \u03c4\n  dlett : \u2200 {\u03c3}\n    (s : Term \u0394 \u03c3) \u2192\n    (ds : DTerm \u0394 \u03c3) \u2192\n    (dt : DTerm (\u03c3 \u2022 \u0394) \u03c4) \u2192\n    DTerm \u0394 \u03c4\n\nderive-dvar : \u2200 {\u0394 \u03c3} \u2192 (x : Var \u0394 \u03c3) \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3)\nderive-dvar this = this\nderive-dvar (that x) = that (derive-dvar x)\n\nderive-dterm : \u2200 {\u0394 \u03c3} \u2192 (t : Term \u0394 \u03c3) \u2192 DTerm \u0394 \u03c3\n\nderive-dsval : \u2200 {\u0394 \u03c3} \u2192 (t : SVal \u0394 \u03c3) \u2192 DSVal \u0394 \u03c3\nderive-dsval (var x) = dvar (derive-dvar x)\n\nderive-dsval (abs t) = dabs (derive-dterm t)\n\nderive-dterm (val x) = dval (derive-dsval x)\nderive-dterm (const c) = dconst c\nderive-dterm (app vs vt) = dapp (derive-dsval vs) vt (derive-dsval vt)\nderive-dterm (lett s t) = dlett s (derive-dterm s) (derive-dterm t)\n\n-- Nontrivial because of unification problems in pattern matching. I wanted to\n-- use it to define \u2295 on closures purely on terms of the closure change.\n\n-- Instead, I decided to use decidable equality on contexts: that's a lot of\n-- tedious boilerplate, but not too hard, but the proof that validity and \u2295\n-- agree becomes easier.\n-- -- Define a DVar and be done?\n-- underive-dvar :  \u2200 {\u0394 \u03c3} \u2192 Var (\u0394\u0394 \u0394) (\u0394\u03c4 \u03c3) \u2192 Var \u0394 \u03c3\n-- underive-dvar {\u2205} ()\n-- underive-dvar {\u03c4 \u2022 \u0394} x = {!!}\n\n--underive-dvar {\u03c3 \u2022 \u0394} (that x) = that (underive-dvar x)\n\ndata DVal : Type \u2192 Set\nimport Base.Denotation.Environment DType DVal as D\n\nCh\u0394 : \u2200 (\u0394 : Context) \u2192 Set\nCh\u0394 \u0394 = D.\u27e6 \u0394 \u27e7Context\n\ndata DVal where\n  dclosure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (dt : DTerm (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192  (d\u03c1 : Ch\u0394 \u0393) \u2192 DVal (\u03c3 \u21d2 \u03c4)\n  dnatV : \u2200 (n : \u2115) \u2192 DVal nat\n\n_\u2295_ : \u2200 {\u03c4} \u2192 (v1 : Val \u03c4) (dv : DVal \u03c4) \u2192 Val \u03c4\n\n_\u2295\u03c1_ : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Ch\u0394 \u0393 \u2192 \u27e6 \u0393 \u27e7Context\n\u2205 \u2295\u03c1 \u2205 = \u2205\n(v \u2022 \u03c11) \u2295\u03c1 (dv \u2022 d\u03c1) = v \u2295 dv \u2022 \u03c11 \u2295\u03c1 d\u03c1\n\nclosure {\u0393} t \u03c1 \u2295 dclosure {\u03931} dt \u03c1\u2081 d\u03c1 with \u0393 \u225fCtx \u03931\nclosure {\u0393} t \u03c1 \u2295 dclosure {.\u0393} dt \u03c1\u2081 d\u03c1 | yes refl = closure t (\u03c1 \u2295\u03c1 d\u03c1)\n... | no \u00acp = closure t \u03c1\n_\u2295_ (natV n) (dnatV dn) = natV (n + dn)\n\ndata _D_\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) : \u2200 {\u03c4} \u2192 DTerm \u0393 \u03c4 \u2192 DVal \u03c4 \u2192 Set where\n  dabs : \u2200 {\u03c3 \u03c4} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dabs dt) \u2193 dclosure dt \u03c1 d\u03c1\n  dvar : \u2200 {\u03c4} (x : DC.Var \u0393 \u03c4) \u2192\n    \u03c1 D d\u03c1 \u22a2 dval (dvar (derive-dvar x)) \u2193 D.\u27e6 x \u27e7Var d\u03c1\n  dlit : \u2200 n \u2192\n    \u03c1 D d\u03c1 \u22a2 dconst (lit n) \u2193 dnatV 0\n  dapp : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u0393\u2032 \u03c3 \u03c4 \u03c1\u2032 d\u03c1\u2032}\n    {dvs} {vt} {dvt}\n    {vtv} {dvtv}\n    {dt : DTerm (\u03c3 \u2022 \u0393\u2032) \u03c4} {dv} \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvs \u2193 dclosure dt \u03c1\u2032 d\u03c1\u2032 \u2192\n    \u03c1 \u22a2 val vt \u2193[ n ] vtv \u2192\n    \u03c1 D d\u03c1 \u22a2 dval dvt \u2193 dvtv \u2192\n    (vtv \u2022 \u03c1\u2032) D (dvtv \u2022 d\u03c1\u2032) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dapp dvs vt dvt \u2193 dv\n  dlett : \u2200 {hasIdx} {n : Idx hasIdx}\n    {\u03c3 \u03c4} {s : Term \u0393 \u03c3} {ds} {dt : DTerm (\u03c3 \u2022 \u0393) \u03c4}\n    {vsv dvsv dv} \u2192\n    \u03c1 \u22a2 s \u2193[ n ] vsv \u2192\n    \u03c1 D d\u03c1 \u22a2 ds \u2193 dvsv \u2192\n    (vsv \u2022 \u03c1) D (dvsv \u2022 d\u03c1) \u22a2 dt \u2193 dv \u2192\n    \u03c1 D d\u03c1 \u22a2 dlett s ds dt \u2193 dv\n\nmutual\n  rrelT3 : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (dt : DTerm \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  rrelT3 {\u03c4} t1 dt t2 \u03c11 d\u03c1 \u03c12 k =\n    (v1 v2 : Val \u03c4) \u2192\n    \u2200 j (j<k : j < k) \u2192\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ i' j ] v1) \u2192\n    (\u03c12\u22a2t2\u2193[n2]v2 : \u03c12 \u22a2 t2 \u2193[ no ] v2) \u2192\n    \u03a3[ dv \u2208 DVal \u03c4 ]\n    \u03c11 D d\u03c1 \u22a2 dt \u2193 dv \u00d7\n    rrelV3 \u03c4 v1 dv v2 (k \u2238 j)\n\n  rrelV3 : \u2200 \u03c4 (v1 : Val \u03c4) (dv : DVal \u03c4) (v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV3 nat (natV v1) (dnatV dv) (natV v2) n = dv + v1 \u2261 v2\n  rrelV3 (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (dclosure {\u0394\u0393} dt \u03c1 d\u03c1) (closure {\u03932} t2 \u03c12) n =\n      \u03a3 ((\u03931 \u2261 \u03932) \u00d7 (\u03931 \u2261 \u0394\u0393)) \u03bb { (refl , refl) \u2192\n        \u03c11 \u2261 \u03c1 \u00d7\n        \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12 \u00d7\n        t1 \u2261 t2 \u00d7\n        dt \u2261 derive-dterm t1 \u00d7\n        \u2200 (k : \u2115) (k<n : k < n) v1 dv v2 \u2192\n        rrelV3 \u03c3 v1 dv v2 k \u2192\n        rrelT3\n          t1\n          dt\n          t2\n          (v1 \u2022 \u03c11)\n          (dv \u2022 d\u03c1)\n          (v2 \u2022 \u03c12)\n          k\n      }\n\nrrelV3\u2192\u2295 : \u2200 {\u03c4 n} v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 v1 \u2295 dv \u2261 v2\nrrelV3\u2192\u2295 (closure {\u0393} t \u03c1) (dclosure .(derive-dterm t) .\u03c1 d\u03c1) (closure .t .(\u03c1 \u2295\u03c1 d\u03c1)) ((refl , refl) , refl , refl , refl , refl , dvv) with \u0393 \u225fCtx \u0393 | \u225fCtx-refl \u0393\n... | .(yes refl) | refl = refl\nrrelV3\u2192\u2295 (natV n) (dnatV dn) (natV .(dn + n)) refl rewrite +-comm n dn = refl\n\nrrel\u03c13 : \u2200 \u0393 (\u03c11 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0394 \u0393) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c13 \u2205 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c13 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) n = rrelV3 \u03c4 v1 dv v2 n \u00d7 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n\n\nrrel\u03c13\u2192\u2295 : \u2200 {\u0393 n} \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 \u03c11 \u2295\u03c1 d\u03c1 \u2261 \u03c12\nrrel\u03c13\u2192\u2295 \u2205 \u2205 \u2205 tt = refl\nrrel\u03c13\u2192\u2295 (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) rewrite rrelV3\u2192\u2295 v1 dv v2 vv | rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = refl\n\nrrelV3-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 dv v2 \u2192 rrelV3 \u03c4 v1 dv v2 n \u2192 rrelV3 \u03c4 v1 dv v2 m\nrrelV3-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t \u03c1) (dclosure dt \u03c1\u2081 d\u03c1) (closure t\u2081 \u03c1\u2082)\n  -- Validity\n  ((refl , refl) , refl , refl , refl , refl , vv) =\n  (refl  , refl) , refl , refl , refl , refl , \u03bb k k<m \u2192 vv k (\u2264-trans k<m m\u2264n)\nrrelV3-mono m n m\u2264n nat (natV n\u2081) (dnatV n\u2082) (natV n\u2083) vv = vv\n\nrrel\u03c13-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 d\u03c1 \u03c12 \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192 rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 m\nrrel\u03c13-mono m n m\u2264n \u2205 \u2205 \u2205 \u2205 tt = tt\nrrel\u03c13-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV3-mono m n m\u2264n _ v1 dv v2 vv , rrel\u03c13-mono m n m\u2264n \u0393 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\n\nlt1 : \u2200 {k n} \u2192 k < n \u2192 k \u2264 n\nlt1 (s\u2264s p) = \u2264-step p\n\n\u27e6_\u27e7RelVar3 : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 d\u03c1 \u03c12} \u2192\n  rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n \u2192\n  rrelV3 \u03c4 (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar3   {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar3 {v1 \u2022 \u03c11} {dv \u2022 d\u03c1} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar3 \u03c1\u03c1\n\nm\u2238n\u2264m : \u2200 m n \u2192 m \u2238 n \u2264 m\nm\u2238n\u2264m m zero = \u2264-refl\nm\u2238n\u2264m zero (suc n) = z\u2264n\nm\u2238n\u2264m (suc m) (suc n) = \u2264-step (m\u2238n\u2264m m n)\n\nrfundamentalV3 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 n) \u2192 rrelT3 (val (var x)) (dval (dvar (derive-dvar x))) (val (var x)) \u03c11 d\u03c1 \u03c12 n\nrfundamentalV3 x n \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .(\u27e6 x \u27e7Var \u03c12) .1 j<n (var .x) (var .x) =\n  D.\u27e6 x \u27e7Var d\u03c1 , dvar x , rrelV3-mono (n \u2238 1) n (m\u2238n\u2264m n 1) _ (\u27e6 x \u27e7Var \u03c11) (D.\u27e6 x \u27e7Var d\u03c1) (\u27e6 x \u27e7Var \u03c12) (\u27e6 x \u27e7RelVar3 \u03c1\u03c1)\n\nsuc\u2238 : \u2200 m n \u2192 n \u2264 m \u2192 suc (m \u2238 n) \u2261 suc m \u2238 n\nsuc\u2238 m zero z\u2264n = refl\nsuc\u2238 (suc m) (suc n) (s\u2264s n\u2264m) = suc\u2238 m n n\u2264m\n\nsub\u2238 : \u2200 m n o \u2192 m + n \u2264 o \u2192 n \u2264 o \u2238 m\nsub\u2238 m n o n+m\u2264o rewrite +-comm m n | cong (_\u2264 o \u2238 m) (sym (m+n\u2238n\u2261m n m)) = \u2238-mono n+m\u2264o (\u2264-refl {m})\n\n-- Warning: names like \u03c11\u22a2t1\u2193[j]v1 are all broken, sorry for not fixing them.\nrfundamental3 : \u2200 {\u03c4 \u0393} k (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c11 d\u03c1 \u03c12 \u2192 (\u03c1\u03c1 : rrel\u03c13 \u0393 \u03c11 d\u03c1 \u03c12 k) \u2192\n  rrelT3 t (derive-dterm t) t \u03c11 d\u03c1 \u03c12 k\nrfundamental3 k (val (var x)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 = rfundamentalV3 x k \u03c11 d\u03c1 \u03c12 \u03c1\u03c1\nrfundamental3 (suc k) (val (abs t)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .(closure t \u03c12) .1 (s\u2264s j<k) abs abs =\n  dclosure (derive-dterm t) \u03c11 d\u03c1 , dabs , (refl , refl) , refl , rrel\u03c13\u2192\u2295 \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 , refl , refl ,\n    \u03bb k\u2081 k<n v1 dv v2 vv v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2 \u2192\n    rfundamental3 k\u2081 t (v1 \u2022 \u03c11) (dv \u2022 d\u03c1) (v2 \u2022 \u03c12) (vv , rrel\u03c13-mono k\u2081 (suc k) (\u2264-step (lt1 k<n)) _ _ _ _ \u03c1\u03c1) v3 v4 j j<k\u2081 \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\nrfundamental3 k (const .(lit n)) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 (natV n) .(natV n) .1 j<k (lit .n) (lit .n) = dnatV 0 , dlit n , refl\nrfundamental3 (suc k) (app vs vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc j) (s\u2264s j<k)\n  (app j vtv1 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3)\n  (app _ vtv2 \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3 \u03c12\u22a2t2\u2193[n2]v4)\n  with rfundamental3 (suc k) (val vs) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n       | rfundamental3 (suc k) (val vt) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 _ _ zero (s\u2264s z\u2264n) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n... | dclosure dt \u03c1 d\u03c1\u2081 , vs\u2193dsv , (refl , refl) , refl , refl , refl , refl , dsvv | dtv , vt\u2193dvv , dtvv\n      with dsvv k (s\u2264s \u2264-refl) vtv1 dtv vtv2\n           (rrelV3-mono k (suc k) (\u2264-step \u2264-refl) _ vtv1 dtv vtv2 dtvv)\n           v1 v2 j j<k \u03c11\u22a2t1\u2193[j]v3 \u03c12\u22a2t2\u2193[n2]v4\n... | dv , \u2193dv , dvv =\n      dv , dapp vs\u2193dsv \u03c11\u22a2t1\u2193[j]v2 vt\u2193dvv \u2193dv , dvv\nrfundamental3 (suc (suc k)) (lett s t) \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 v1 v2 .(suc (n1 + n2)) (s\u2264s (s\u2264s n1+n2\u2264k))\n  (lett n1 n2 vs1 .s .t \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2) (lett _ _ vs2 .s .t \u03c12\u22a2t2\u2193[n2]v2 \u03c12\u22a2t2\u2193[n2]v3)\n  with rfundamental3 (suc (suc k)) s \u03c11 d\u03c1 \u03c12 \u03c1\u03c1 vs1 vs2 n1\n    (s\u2264s (\u2264-trans (\u2264-trans (m\u2264m+n n1 n2) n1+n2\u2264k) (\u2264-step \u2264-refl)))\n    \u03c11\u22a2t1\u2193[j]v1 \u03c12\u22a2t2\u2193[n2]v2\n... | dsv , \u2193dsv , vsv\n      with rfundamental3 (suc (suc k) \u2238 n1) t (vs1 \u2022 \u03c11) (dsv \u2022 d\u03c1) (vs2 \u2022 \u03c12) (vsv , rrel\u03c13-mono (suc (suc k) \u2238 n1) (suc (suc k)) (m\u2238n\u2264m (suc (suc k)) n1) _ _ _ _ \u03c1\u03c1) v1 v2 n2 (sub\u2238 n1 (suc n2) (suc (suc k)) n1+[1+n2]\u22642+k) \u03c11\u22a2t1\u2193[j]v2 \u03c12\u22a2t2\u2193[n2]v3\n      where\n        n1+[1+n2]\u22642+k : n1 + suc n2 \u2264 suc (suc k)\n        n1+[1+n2]\u22642+k rewrite +-suc n1 n2 = \u2264-step (s\u2264s n1+n2\u2264k)\n... | dv , \u2193dv , dvv = dv , dlett \u03c11\u22a2t1\u2193[j]v1 \u2193dsv \u2193dv , rrelV3-mono (suc k \u2238 (n1 + n2)) (suc (suc k) \u2238 n1 \u2238 n2) 1+k-[n1+n2]\u22642+k-n1-n2 _ v1 dv v2 dvv\n  where\n    1+k-[n1+n2]\u22642+k-n1-n2 : suc k \u2238 (n1 + n2) \u2264 suc (suc k) \u2238 n1 \u2238 n2\n    1+k-[n1+n2]\u22642+k-n1-n2 rewrite \u2238-+-assoc (suc (suc k)) n1 n2 = \u2238-mono {suc k} {suc (suc k)} {n1 + n2} {n1 + n2} (\u2264-step \u2264-refl) \u2264-refl\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"954b8dd5204255f2545829c5b2bd8cc78f8fdc50","subject":"less asinine names + moving lemmas around #14","message":"less asinine names + moving lemmas around #14\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values and errors are disjoint\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  mutual\n    -- indeterminates and expressions that step are disjoint\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = final-sub-not-trans x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    -- final expressions don't step\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors and expressions that step are disjoint\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail caserr-not-step\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!}\n\n    -- congruence caserr-not-step\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","old_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import type-assignment-unicity\nopen import lemmas-progress-checks\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that boxed values, indeterminates, cast\n-- errors, and expressions that step are pairwise disjoint. (note that as a\n-- consequence of currying and comutativity of products, this means that\n-- there are six theorems to prove)\nmodule progress-checks where\n  -- boxed values and indeterminates are disjoint\n  boxedval-not-indet : \u2200{d} \u2192 d boxedval \u2192 d indet \u2192 \u22a5\n  boxedval-not-indet (BVVal VConst) ()\n  boxedval-not-indet (BVVal VLam) ()\n  boxedval-not-indet (BVArrCast x bv) (ICastArr x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastGroundHole x\u2081 ind) = boxedval-not-indet bv ind\n  boxedval-not-indet (BVHoleCast x bv) (ICastHoleGround x\u2081 ind x\u2082) = boxedval-not-indet bv ind\n\n  -- boxed values and errors are disjoint\n  boxedval-not-err : \u2200{d} \u2192 d boxedval \u2192 d casterr \u2192 \u22a5\n  boxedval-not-err (BVVal ()) (CECastFail x\u2081 x\u2082 x\u2083 x\u2084)\n  boxedval-not-err (BVHoleCast x bv) (CECastFail x\u2081 x\u2082 () x\u2084)\n  boxedval-not-err (BVArrCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castarr er)\n  boxedval-not-err (BVArrCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVHoleCast x bv) (CECong FHOuter er) = boxedval-not-err bv (ce-castth er)\n  boxedval-not-err (BVHoleCast x bv) (CECong (FHCast x\u2081) er) = boxedval-not-err bv (CECong x\u2081 er)\n  boxedval-not-err (BVVal x) (CECong FHOuter er) = boxedval-not-err (BVVal x) er\n  boxedval-not-err (BVVal ()) (CECong (FHAp1 x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHAp2 x\u2081 x\u2082) er)\n  boxedval-not-err (BVVal ()) (CECong (FHNEHole x\u2081) er)\n  boxedval-not-err (BVVal ()) (CECong (FHCast x\u2081) er)\n\n  -- boxed values and expressions that step are disjoint\n  boxedval-not-step : \u2200{d} \u2192 d boxedval \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n  boxedval-not-step (BVVal VConst) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVVal VLam) (d' , Step FHOuter () x\u2083)\n  boxedval-not-step (BVArrCast x bv) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n  boxedval-not-step (BVArrCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n  boxedval-not-step (BVHoleCast () bv) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n  boxedval-not-step (BVHoleCast x bv) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n  boxedval-not-step (BVHoleCast GHole bv) (_ , Step FHOuter (ITGround x\u2081 x\u2082) FHOuter) = x\u2082 refl\n  boxedval-not-step (BVHoleCast x bv) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = boxedval-not-step bv (_ , Step x\u2081 x\u2082 x\u2083)\n\n  -- todo: what class of P is this true for?\n  -- lem-something : \u2200{ d \u03b5 d'} \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 P d' \u2192 P d\n\n  mutual\n    -- indeterminates and errors are disjoint\n    indet-not-err : \u2200{d} \u2192 d indet \u2192 d casterr \u2192 \u22a5\n    indet-not-err IEHole (CECong FHOuter err) = indet-not-err IEHole err\n    indet-not-err (INEHole x) (CECong FHOuter err) = final-not-err x (ce-nehole err)\n    indet-not-err (INEHole x) (CECong (FHNEHole x\u2081) err) = final-not-err x (CECong x\u2081 err)\n    indet-not-err (IAp x indet x\u2081) (CECong FHOuter err)\n      with ce-ap err\n    ... | Inl d1err = indet-not-err indet d1err\n    ... | Inr d2err = final-not-err x\u2081 d2err\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp1 x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n    indet-not-err (IAp x indet x\u2081) (CECong (FHAp2 x\u2082 x\u2083) err) = final-not-err x\u2081 (CECong x\u2083 err)\n    indet-not-err (ICastArr x indet) (CECong FHOuter err) = indet-not-err indet (ce-castarr err)\n    indet-not-err (ICastArr x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastGroundHole x indet) (CECastFail x\u2081 x\u2082 () x\u2084)\n    indet-not-err (ICastGroundHole x indet) (CECong FHOuter err) = indet-not-err indet (ce-castth err)\n    indet-not-err (ICastGroundHole x indet) (CECong (FHCast x\u2081) err) = indet-not-err indet (CECong x\u2081 err)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECastFail x\u2082 x\u2083 x\u2084 x\u2085) = x _ _ refl\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong FHOuter err) = indet-not-err indet (ce-castht err x)\n    indet-not-err (ICastHoleGround x indet x\u2081) (CECong (FHCast x\u2082) err) = indet-not-err indet (CECong x\u2082 err)\n\n    -- final expressions are not errors (not one of the 6 cases for progress, just a convenience)\n    final-not-err : \u2200{d} \u2192 d final \u2192 d casterr \u2192 \u22a5\n    final-not-err (FBoxed x) err = boxedval-not-err x err\n    final-not-err (FIndet x) err = indet-not-err x err\n\n  -- todo: these are bad names; probably some places below where i inlined\n  -- some of these lemmas before i'd come up with them\n  lem2 : \u2200{d d'} \u2192 d indet \u2192 d \u2192> d' \u2192 \u22a5\n  lem2 IEHole ()\n  lem2 (INEHole x) ()\n  lem2 (IAp x\u2081 () x\u2082) (ITLam x\u2083)\n  lem2 (IAp x (ICastArr x\u2081 ind) x\u2082) (ITApCast x\u2083 x\u2084) = x _ _ _ _ _ refl\n  lem2 (ICastArr x ind) (ITCastID (FBoxed x\u2081)) = boxedval-not-indet x\u2081 ind\n  lem2 (ICastArr x ind) (ITCastID (FIndet x\u2081)) = x refl\n  lem2 (ICastGroundHole () ind) (ITCastID x\u2081)\n  lem2 (ICastGroundHole x ind) (ITCastSucceed x\u2081 ())\n  lem2 (ICastGroundHole GHole ind) (ITGround x\u2081 x\u2082) = x\u2082 refl\n  lem2 (ICastHoleGround x ind ()) (ITCastID x\u2082)\n  lem2 (ICastHoleGround x ind x\u2081) (ITCastSucceed x\u2082 x\u2083) = x _ _ refl\n  lem2 (ICastHoleGround x ind GHole) (ITExpand x\u2082 x\u2083) = x\u2083 refl\n\n  lem3 : \u2200{d d'} \u2192 d boxedval \u2192 d \u2192> d' \u2192 \u22a5\n  lem3 (BVVal VConst) ()\n  lem3 (BVVal VLam) ()\n  lem3 (BVArrCast x bv) (ITCastID x\u2081) = x refl\n  lem3 (BVHoleCast () bv) (ITCastID x\u2081)\n  lem3 (BVHoleCast () bv) (ITCastSucceed x\u2081 x\u2082)\n  lem3 (BVHoleCast GHole bv) (ITGround x\u2081 y ) = y refl\n\n  lem1 : \u2200{d d'} \u2192 d final \u2192 d \u2192> d' \u2192 \u22a5\n  lem1 (FBoxed x) = lem3 x\n  lem1 (FIndet x) = lem2 x\n\n  lem4 : \u2200{d \u03b5 x} \u2192 d final \u2192 d == \u03b5 \u27e6 x \u27e7 \u2192 x final\n  lem4 x FHOuter = x\n  lem4 (FBoxed (BVVal ())) (FHAp1 eps)\n  lem4 (FBoxed (BVVal ())) (FHAp2 x\u2082 eps)\n  lem4 (FBoxed (BVVal ())) (FHNEHole eps)\n  lem4 (FBoxed (BVVal ())) (FHCast eps)\n  lem4 (FBoxed (BVArrCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FBoxed (BVHoleCast x\u2081 x\u2082)) (FHCast eps) = lem4 (FBoxed x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp1 eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (IAp x\u2081 x\u2082 x\u2083)) (FHAp2 x\u2084 eps) = lem4 x\u2083 eps\n  lem4 (FIndet (INEHole x\u2081)) (FHNEHole eps) = lem4 x\u2081 eps\n  lem4 (FIndet (ICastArr x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastGroundHole x\u2081 x\u2082)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n  lem4 (FIndet (ICastHoleGround x\u2081 x\u2082 x\u2083)) (FHCast eps) = lem4 (FIndet x\u2082) eps\n\n  lem5 : \u2200{d d' d'' \u03b5} \u2192  d final \u2192 d == \u03b5 \u27e6 d' \u27e7 \u2192 d' \u2192> d'' \u2192 \u22a5\n  lem5 f sub step = lem1 (lem4 f sub) step\n\n  -- indeterminates and expressions that step are disjoint\n  mutual\n    indet-not-step : \u2200{d} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    indet-not-step IEHole (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (d' , Step FHOuter () FHOuter)\n    indet-not-step (INEHole x) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = lem5 x x\u2081 x\u2082\n    indet-not-step (IAp x\u2081 () x\u2082) (_ , Step FHOuter (ITLam x\u2083) FHOuter)\n    indet-not-step (IAp x (ICastArr x\u2081 ind) x\u2082) (_ , Step FHOuter (ITApCast x\u2083 x\u2084) FHOuter) = x _ _ _ _ _  refl\n    indet-not-step (IAp x ind _) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = indet-not-step ind (_ , Step x\u2082 x\u2083 x\u2084)\n    indet-not-step (IAp x ind f) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = final-not-step f (_ , Step x\u2083 x\u2084 x\u2086)\n    indet-not-step (ICastArr x ind) (d0' , Step FHOuter (ITCastID x\u2081) FHOuter) = x refl\n    indet-not-step (ICastArr x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastGroundHole () ind) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastGroundHole x ind) (d' , Step FHOuter (ITCastSucceed x\u2081 ()) FHOuter)\n    indet-not-step (ICastGroundHole GHole ind) (_ , Step FHOuter (ITGround _ y) FHOuter) = y refl\n    indet-not-step (ICastGroundHole x ind) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n    indet-not-step (ICastHoleGround x ind ()) (d' , Step FHOuter (ITCastID x\u2081) FHOuter)\n    indet-not-step (ICastHoleGround x ind g) (d' , Step FHOuter (ITCastSucceed x\u2081 x\u2082) FHOuter) = x _ _ refl\n    indet-not-step (ICastHoleGround x ind GHole) (_ , Step FHOuter (ITExpand x\u2081 x\u2082) FHOuter) = x\u2082 refl\n    indet-not-step (ICastHoleGround x ind g) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = indet-not-step ind (_ , Step x\u2081 x\u2082 x\u2083)\n\n    final-not-step : \u2200{d} \u2192 d final \u2192 \u03a3[ d' \u2208 dhexp ] (d \u21a6 d') \u2192 \u22a5\n    final-not-step (FBoxed x) stp = boxedval-not-step x stp\n    final-not-step (FIndet x) stp = indet-not-step x stp\n\n  -- errors and expressions that step are disjoint\n  err-not-step : \u2200{d} \u2192 d casterr \u2192 (\u03a3[ d' \u2208 dhexp ] (d \u21a6 d')) \u2192 \u22a5\n    -- cast fail caserr-not-step\n  err-not-step (CECastFail x x\u2081 () x\u2083) (_ , Step FHOuter (ITCastID x\u2084) FHOuter)\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step FHOuter (ITCastSucceed x\u2084 x\u2085) FHOuter) = x\u2083 refl\n  err-not-step (CECastFail x x\u2081 GHole x\u2083) (_ , Step FHOuter (ITExpand x\u2084 x\u2085) FHOuter) = x\u2085 refl\n  err-not-step (CECastFail x x\u2081 x\u2082 x\u2083) (_ , Step (FHCast x\u2084) x\u2085 (FHCast x\u2086)) = {!!}\n\n    -- congruence caserr-not-step\n  err-not-step (CECong FHOuter ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = err-not-step ce (\u03c01 , Step FHOuter x\u2082 FHOuter)\n  err-not-step (CECong (FHAp1 FHOuter) (CECong FHOuter ce)) (_ , Step FHOuter (ITLam x\u2082) FHOuter) = boxedval-not-err (BVVal VLam) ce\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step FHOuter (ITApCast x\u2081 x\u2082) FHOuter) = {!!}  -- fe x\u2081 (CECong {!ce-out-cast ce!} ce)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!ce-out-cast ce x\u2081!}\n  err-not-step (CECong (FHNEHole x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n  err-not-step (CECong (FHCast x) ce) (\u03c01 , Step FHOuter x\u2082 FHOuter) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083))\n    with ce-ap ce\n  ... | Inl d1err = err-not-step d1err (_ , Step x\u2081 x\u2082 x\u2083)\n  ... | Inr d2err = {!Step x\u2081 x\u2082 x\u2083!} -- this is a counter example: d2 is a casterror but d1 isn't yet a value so the whole thing steps\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp1 x\u2081) x\u2082 (FHAp1 x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp1 x\u2082) x\u2083 (FHAp1 x\u2084)) = final-not-step x (_ , Step x\u2082 x\u2083 x\u2084)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085))\n    with ce-ap ce\n  ... | Inl d1err = final-not-err x\u2084 d1err\n  ... | Inr d2err = err-not-step d2err (_ , Step x\u2082 x\u2083 x\u2085)\n  err-not-step (CECong (FHAp1 x) ce) (_ , Step (FHAp2 x\u2081 x\u2082) x\u2083 (FHAp2 x\u2084 x\u2085)) = {!!}\n  err-not-step (CECong (FHAp2 x x\u2081) ce) (_ , Step (FHAp2 x\u2082 x\u2083) x\u2084 (FHAp2 x\u2085 x\u2086)) = {!!}\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-nehole ce) (_ , Step x\u2081 x\u2082 x\u2083)\n  err-not-step (CECong (FHNEHole x) ce) (_ , Step (FHNEHole x\u2081) x\u2082 (FHNEHole x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n\n  err-not-step (CECong FHOuter ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = {!!} -- err-not-step {!!} (_ , Step x\u2081 x\u2082 x\u2083) -- this might not work\n  err-not-step (CECong (FHCast x) ce) (_ , Step (FHCast x\u2081) x\u2082 (FHCast x\u2083)) = err-not-step (ce-out-cast ce x) (_ , Step x\u2081 x\u2082 x\u2083)\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"d068aa31aff8e40aa406f5f5fa70ec5fdf7ce251","subject":"Control\/Protocol\/Choreography.agda","message":"Control\/Protocol\/Choreography.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\ndata InOut : \u2605 where\n  -- \u03a0' \u03a3' : InOut\n  In Out : InOut\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  com  : (q : InOut)(M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\npattern \u03a0' M P = com In  M P\npattern \u03a3' M P = com Out M P\n{-\n\u03a0' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a0' M P = com In  M P\n\n\u03a3' : (M : \u2605)(P : M \u2192 Proto) \u2192 Proto\n\u03a3' M P = com Out M P\n-}\n\n\u03a0S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a0S' M P = \u03a0' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u03a3S' : (M : \u2605)(P : ..(_ : M) \u2192 Proto) \u2192 Proto\n\u03a3S' M P = \u03a3' (\u2610 M) (\u03bb { [ m ] \u2192 P m })\n\n\u27e6_\u27e7 : Proto \u2192 \u2605\n\u27e6 end         \u27e7 = \ud835\udfd9\n\u27e6 \u03a0' M P \u27e7 = \u03a0  M \u03bb x \u2192 \u27e6 P x \u27e7\n\u27e6 \u03a3' M P \u27e7 = \u03a3  M \u03bb x \u2192 \u27e6 P x \u27e7\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\n_-[_]\u2192\u00f8\u204f_ : \u2200 {I}(A : I) (M : \u2605)         (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\nA -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  I\u03a3D : \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  I\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n--  I\u03a0S : \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 \u03a0' (\u2610 M) \u2102C ]\n  \u2605\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200   m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ A \u2261 \u03a3' M \u2102A ]\n  \u2605\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200   m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u2605\u204f   \u2102) \/ B \u2261 \u03a0' M \u2102B ]\n  --\u00f8\u03a3D : \u2200 {A     M \u2102 \u2102A} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ A \u2261 \u03a3' S M \u2102A ]\n  --\u00f8\u03a0D : \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 ..m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192\u00f8\u204f   \u2102) \/ B \u2261 \u03a0' S M \u2102B ]\n  end : \u2200 {A} \u2192 [ end \/ A \u2261 end ]\n\nTrace : Proto \u2192 Proto\nTrace end          = end\nTrace (com _ A B) = \u03a3' A \u03bb m \u2192 Trace (B m)\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n    Trace-idempotent : \u2200 P \u2192 Trace (Trace P) \u2261 Trace P\n    Trace-idempotent end = refl\n    Trace-idempotent (com q M P) = cong (\u03a3' M) (funExt \u03bb m \u2192 Trace-idempotent (P m))\n\nTele : Proto \u2192 \u2605\nTele P = \u27e6 Trace P \u27e7\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\ncom \u03c0 A B >>\u2261 Q = com \u03c0 A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndualInOut : InOut \u2192 InOut\ndualInOut In  = Out\ndualInOut Out = In\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (com q A B) = com (dualInOut q) A (\u03bb x \u2192 dual (B x))\n\ndata DualInOut : InOut \u2192 InOut \u2192 \u2605 where\n  DInOut : DualInOut In  Out\n  DOutIn : DualInOut Out In\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u00b7\u03a3 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' A B) (\u03a3' A B')\n  \u03a3\u00b7\u03a0 : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' A B) (\u03a0' A B')\n\ndata Sing {a} {A : \u2605_ a} : ..(x : A) \u2192 \u2605_ a where\n  sing : \u2200 x \u2192 Sing x\n\n-- Commit = \u03bb M R \u2192 \u03a3' D (\u2610 M) \u03bb { [ m ] \u2192 \u03a0' D R \u03bb r \u2192 \u03a3' D (Sing m) (\u03bb _ \u2192 end) }\nCommit = \u03bb M R \u2192 \u03a3S' M \u03bb m \u2192 \u03a0' R \u03bb r \u2192 \u03a3' (Sing m) (\u03bb _ \u2192 end)\n\ncommit : \u2200 M R (m : M)  \u2192 \u27e6 Commit M R \u27e7\ncommit M R m = [ m ] , (\u03bb x \u2192 (sing m) , _)\n\ndecommit : \u2200 M R (r : R) \u2192 \u27e6 dual (Commit M R) \u27e7\ndecommit M R r = \u03bb { [ m ] \u2192 r , (\u03bb x \u2192 {!!}) }\n\ndata [_&_\u2261_]InOut : InOut \u2192 InOut \u2192 InOut \u2192 \u2605\u2081 where\n  \u03a0XX : \u2200 {X} \u2192 [ In & X  \u2261 X ]InOut\n  X\u03a0X : \u2200 {X} \u2192 [ X  & In \u2261 X ]InOut\n\n&InOut-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ]InOut \u2192 [ Q & P \u2261 R ]InOut\n&InOut-comm \u03a0XX = X\u03a0X\n&InOut-comm X\u03a0X = \u03a0XX\n\n  {-\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end& : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  &end : \u2200 {P} \u2192 [ P & end \u2261 P ]\n  XXX  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ com qP    M  P & com qQ    M  Q \u2261 com qR M R ]\n  -- SDD  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ com qP (\u2610 M) P & com qQ    M  Q \u2261 com qR M R ]\n  DSD  : \u2200 {qP qQ qR M P Q R}(q : [ qP & qQ \u2261 qR ]InOut)(P& : \u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ com qP    M  P & com qQ (\u2610 M) Q \u2261 com qR M R ]\n\n&-comm : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n&-comm end& = &end\n&-comm &end = end&\n&-comm (XXX q P&) = XXX (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (SDD q P&) = DSD (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n&-comm (DSD q P&) = SDD (&InOut-comm q) (\u03bb m \u2192 &-comm (P& m))\n-}\n\nDualInOut-sym : \u2200 {P Q} \u2192 DualInOut P Q \u2192 DualInOut Q P\nDualInOut-sym DInOut = DOutIn\nDualInOut-sym DOutIn = DInOut\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u00b7\u03a3 f) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u00b7\u03a0 f) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-sym (f x))\n\nDualInOut-spec : \u2200 P \u2192 DualInOut P (dualInOut P)\nDualInOut-spec In  = DInOut\nDualInOut-spec Out = DOutIn\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (com In M P) = \u03a0\u00b7\u03a3 (\u03bb x \u2192 Dual-spec (P x))\nDual-spec (com Out M P) = \u03a3\u00b7\u03a0 (\u03bb x \u2192 Dual-spec (P x))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n\n{-\nrecvS : \u2200 {this : Proto \u2192 \u2605\u2081}{M P} \u2192 (..(m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' (\u2610 M) P)\nrecvS = ?\nsendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' S M P))\nsendS = ?\n-}\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q M P Q} \u2192 SimL (com q M P) Q \u2192 Sim (com q M P) Q\n    right : \u2200 {P q M Q} \u2192 SimR P (com q M Q) \u2192 Sim P (com q M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recv PQA)   (send m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (send m PQB) = send m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (send m PQA) (recv PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (send x\u2081 x\u2082) (recv x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = send m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recv PQA)   (send m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recv PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recv PQB)   = ? -- recv \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (send m PQB) = ? -- send m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (send m PQA) (recv PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb x \u2192 left (send x (sim-id (B x)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb x \u2192 right (send x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u00b7\u03a3 x\u2081) (recv x) (send x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u00b7\u03a0 x)  (send x\u2081 x\u2082) (recv x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recv PQ) QR = recv (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (send m PQ) QR = send m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recv QR)   = recv (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (send m QR) = send m (sim-comp Q-Q' PQ QR)\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (send m PQ) = send m (!\u02e2 PQ)\n\nsim-symR (recv PQ)   = recv (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (send m PQ) = send m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recv P)) = do (recv (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (send m P)) = do (send m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (send x x\u2082)) (left (recv x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (\u03a3\u00b7\u03a0 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (\u03a0\u00b7\u03a3 x\u2081) \u00b7P (right (recv x)) (left (send x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (\u03a3\u00b7\u03a0 x\u2081) \u00b7P (right (send x x\u2082)) (left (recv x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (\u03a0\u00b7\u03a3 x) = cong \u03a0\u00b7\u03a3 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (\u03a3\u00b7\u03a0 x) = cong \u03a3\u00b7\u03a0 (funExt (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end (\u03a0\u00b7\u03a3 x\u2081) Q-Q' (right (recv x)) (left (send x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u00b7\u03a0 x) Q-Q' (right (send x\u2081 x\u2082)) (left (recv x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' (\u03a0\u00b7\u03a3 x\u2081) (right \u00f8P) (right (recv x)) (left (send x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u00b7\u03a0 x) (right \u00f8P) (right (send x\u2081 x\u2082)) (left (recv x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recv x\u2081))\n    = cong (right \u2218 recv) (funExt (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (send x\u2081 x\u2082))\n    = cong (right \u2218 send x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recv x))    = cong (left \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (send x x\u2081)) = cong (left \u2218 send x) (sim-!! x\u2081)\n  sim-!! (right (recv x))    = cong (right \u2218 recv) (funExt \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (send x x\u2081)) = cong (right \u2218 send x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (\u03a0\u00b7\u03a3 x\u2081) (right (recv x)) (left (send x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (\u03a3\u00b7\u03a0 x) (right (send x\u2081 x\u2082)) (left (recv x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recv x\u2081))\n    = cong (left \u2218 recv) (funExt (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (send x\u2081 x\u2082))\n    = cong (left \u2218 send x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExt R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExt Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\n{-\nrecord \u03a3\u00b7 {a b} (A : Set a) (B : ..(_ : A) \u2192 Set b) : Set (a \u2294 b) where\n  constructor _,_\n  field\n    {-..-}proj\u2081\u00b7 : A\n    proj\u2082\u00b7 : B proj\u2081\u00b7\n-}\ndata \u03a3\u00b7 {a b} (A : Set a) (B : ..(_ : A) \u2192 Set b) : Set (a \u2294 b) where\n  _,_ : ..(proj\u2081\u00b7 : A) (proj\u2082\u00b7 : B proj\u2081\u00b7) \u2192 \u03a3\u00b7 A B\n\ndata Mod : \u2605 where S D : Mod\n\n\u2192M : \u2200 {a b} \u2192 Mod \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (a \u2294 b)\n\u2192M S A B = ..(_ : A) \u2192 B\n\u2192M D A B = A \u2192 B\n\n\u03a0M : \u2200 {a b}(m : Mod) \u2192 (A : \u2605_ a) \u2192 (B : \u2192M m A (\u2605_ b)) \u2192 \u2605_ (a \u2294 b)\n\u03a0M S A B = \u03a0\u00b7 A B\n\u03a0M D A B = \u03a0 A B\n\nappM' : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2605_ b}(P : \u2192M m A B) \u2192 A \u2192 B\nappM' S P x = P x\nappM' D P x = P x\n\nappM : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2192M m A (\u2605_ b)}(P : \u03a0M m A B)(x : A) \u2192 appM' m B x\nappM S P x = P x\nappM D P x = P x\n\ndata \u03a0\u03a3 : \u2605 where\n  \u03a0' \u03a3' : \u03a0\u03a3\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  mk  : (q : \u03a0\u03a3)(d : Mod)(A : \u2605)(B : \u2192M d A Proto) \u2192 Proto\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  \u00f8         : Choreo I\n\ndata [_\/_\u2261_] {I} : Choreo I \u2192 I \u2192 Proto \u2192 \u2605\u2081 where\n  \u03a3D :  \u2200 {A B   M \u2102 \u2102A} \u2192 (\u2200 m \u2192 [ \u2102 m \/ A \u2261 \u2102A m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ A \u2261 mk \u03a3' D M \u2102A ]\n  \u03a0D :  \u2200 {A B   M \u2102 \u2102B} \u2192 (\u2200 m \u2192 [ \u2102 m \/ B \u2261 \u2102B m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ B \u2261 mk \u03a0' D M \u2102B ]\n  \u03a0S :  \u2200 {A B C M \u2102 \u2102C} \u2192 (\u2200 m \u2192 [ \u2102 m \/ C \u2261 \u2102C m ]) \u2192 [ (A -[ M ]\u2192 B \u204f \u2102) \/ C \u2261 mk \u03a0' S M \u2102C ]\n\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (mk \u03a0' D A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (mk \u03a3' D A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (mk \u03a0' S A B) = \u03a3\u00b7 A \u03bb x \u2192 Tele (B x)\nTele (mk \u03a3' S A B) = \u03a3\u00b7 A \u03bb x \u2192 Tele (B x)\n\n{-\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = mk \u03a3' D A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = mk \u03a0' D A \u03bb _ \u2192 B\n\ndual\u03a0\u03a3 : \u03a0\u03a3 \u2192 \u03a0\u03a3\ndual\u03a0\u03a3 \u03a0' = \u03a3'\ndual\u03a0\u03a3 \u03a3' = \u03a0'\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (mk q S A B) = mk (dual\u03a0\u03a3 q) S A (\u03bb x \u2192 dual (B x))\ndual (mk q D A B) = mk (dual\u03a0\u03a3 q) D A (\u03bb x \u2192 dual (B x))\n\ndata Dual\u03a0\u03a3 : \u03a0\u03a3 \u2192 \u03a0\u03a3 \u2192 \u2605 where\n  D\u03a0\u03a3 : Dual\u03a0\u03a3 \u03a0' \u03a3'\n  D\u03a3\u03a0 : Dual\u03a0\u03a3 \u03a3' \u03a0'\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  S : \u2200 {A B B' \u03c0 \u03c3} \u2192 Dual\u03a0\u03a3 \u03c0 \u03c3 \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (mk \u03c0 S A B) (mk \u03c3 S A B')\n  D : \u2200 {A B B' \u03c0 \u03c3} \u2192 Dual\u03a0\u03a3 \u03c0 \u03c3 \u2192 (\u2200   x \u2192 Dual (B x) (B' x)) \u2192 Dual (mk \u03c0 D A B) (mk \u03c3 D A B')\n\ndata [_&_\u2261_] : Proto \u2192 Proto \u2192 Proto \u2192 \u2605\u2081 where\n -- comm   : \u2200 {P Q R} \u2192 [ P & Q \u2261 R ] \u2192 [ Q & P \u2261 R ]\n  end&   : \u2200 {P} \u2192 [ end & P \u2261 P ]\n  \u03a0S-S-S : \u2200 {q M P Q R} \u2192 (\u2200 ..m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  S M Q \u2261 mk q  S M R ]\n  \u03a0S-D-D : \u2200 {q M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  D M Q \u2261 mk q  D M R ]\n  \u03a0D\u03a3D\u03a3S : \u2200   {M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' D M P & mk \u03a3' D M Q \u2261 mk \u03a3' S M R ]\n  \u03a0D\u03a0D\u03a0D : \u2200   {M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' D M P & mk \u03a0' D M Q \u2261 mk \u03a0' D M R ]\n\n  &end    : \u2200 {P} \u2192 [ P & end \u2261 P ]\n--  \u03a0S-S-S : \u2200 {q M P Q R} \u2192 (\u2200 ..m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  S M Q \u2261 mk q  S M R ]\n--  \u03a0S-D-D : \u2200 {q M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a0' S M P & mk q  D M Q \u2261 mk q  D M R ]\n \u00a0\u03a3D\u03a0D\u03a3S : \u2200   {M P Q R} \u2192 (\u2200   m \u2192 [ P m & Q m \u2261 R m ]) \u2192 [ mk \u03a3' D M P & mk \u03a0' D M Q \u2261 mk \u03a3' S M R ]\n\nDual\u03a0\u03a3-sym : \u2200 {P Q} \u2192 Dual\u03a0\u03a3 P Q \u2192 Dual\u03a0\u03a3 Q P\nDual\u03a0\u03a3-sym D\u03a0\u03a3 = D\u03a3\u03a0\nDual\u03a0\u03a3-sym D\u03a3\u03a0 = D\u03a0\u03a3\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (S d f) = S (Dual\u03a0\u03a3-sym d) (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (D d f) = D (Dual\u03a0\u03a3-sym d) (\u03bb x \u2192 Dual-sym (f x))\n\nDual\u03a0\u03a3-spec : \u2200 P \u2192 Dual\u03a0\u03a3 P (dual\u03a0\u03a3 P)\nDual\u03a0\u03a3-spec \u03a0' = D\u03a0\u03a3\nDual\u03a0\u03a3-spec \u03a3' = D\u03a3\u03a0\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (mk \u03c0 S A B) = S (Dual\u03a0\u03a3-spec \u03c0) (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (mk \u03c0 D A B) = D (Dual\u03a0\u03a3-spec \u03c0) (\u03bb x \u2192 Dual-spec (B x))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recvD : \u2200 {M P} \u2192 (\u03a0M D M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (mk \u03a0' D M P)\n  recvS : \u2200 {M P} \u2192 (\u03a0M S M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (mk \u03a0' S M P)\n  sendD : \u2200 {M P} \u2192 \u03a0M D M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (mk \u03a3' D M P))\n  sendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (mk \u03a3' S M P))\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {q d M P Q} \u2192 SimL (mk q d M P) Q \u2192 Sim (mk q d M P) Q\n    right : \u2200 {P q d M Q} \u2192 SimR P (mk q d M Q) \u2192 Sim P (mk q d M Q)\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\n{-\nsim& : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 Sim end PA \u2192 Sim end PB \u2192 Sim end PAB\nsim&R : \u2200 {PA PB PAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 SimR end PA \u2192 SimR end PB \u2192 SimR end PAB\n\nsim& P& (right \u00b7PA) (right \u00b7PB) = {! (sim&R P& \u00b7PA \u00b7PB)!}\nsim& end& PA PB = PB\nsim& &end PA PB = PA\n\nsim&R (\u03a0D\u03a3D\u03a3S P&) (recvD PQA)   (sendD m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0D\u03a0D\u03a0D P&) (recvD PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) (recvS PQA)   (sendD m PQB) = sendD m (sim& (P& m) (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) (sendD m PQA) (recvD PQB)   = sendS m (sim& (P& m) PQA (PQB m))\nsim&R &end _ ()\n-}\n\n{-\nsim&R (\u03a3D\u03a0D\u03a3S x) (sendD x\u2081 x\u2082) (recvD x\u2083) = {!!}\n-}\n\n{-\nsim& : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 Sim PA QA \u2192 Sim PB QB \u2192 Sim PAB QAB\nsim&L : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimL PA QA \u2192 SimL PB QB \u2192 SimL PAB QAB\nsim&R : \u2200 {PA PB PAB QA QB QAB} \u2192 [ PA & PB \u2261 PAB ] \u2192 [ QA & QB \u2261 QAB ] \u2192 SimR PA QA \u2192 SimR PB QB \u2192 SimR PAB QAB\n\nsim& P& Q& (left PQA) (left PQB) = left (sim&L P& Q& PQA PQB)\nsim& P& Q& (left PQA) (right PQB) = {!!}\nsim& P& Q& (left PQA) end = {!!}\nsim& P& Q& (right x) PQB = {!!}\nsim& P& Q& end PQB = {!!}\n\nsim&L (\u03a0D\u03a3D\u03a3S P&) Q& (recvD PQA)   (sendD m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0D\u03a0D\u03a0D P&) Q& (recvD PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (recvD PQB)   = recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&L (\u03a0S-D-D P&) Q& (recvS PQA)   (sendD m PQB) = sendD m (sim& (P& m) Q& (PQA m) PQB)\nsim&L (\u03a3D\u03a0D\u03a3S P&) Q& (sendD m PQA) (recvD PQB)   = sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&L &end Q& PQA ()\n\n{-\nsim&R (\u03a0D\u03a3D\u03a3S P&) Q& (recvD PQA)   (sendD m PQB) = ?\nsim&R (\u03a0D\u03a0D\u03a0D P&) Q& (recvD PQA)   (recvD PQB)   = ? -- recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (recvS PQB)   = ? -- recvS \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-S-S P&) Q& (recvS PQA)   (sendS m PQB) = ? -- sendS m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (recvD PQB)   = ? -- recvD \u03bb m \u2192 sim& (P& m) Q& (PQA m) (PQB m)\nsim&R (\u03a0S-D-D P&) Q& (recvS PQA)   (sendD m PQB) = ? -- sendD m (sim& (P& m) Q& (PQA m) PQB)\nsim&R (\u03a3D\u03a0D\u03a3S P&) Q& (sendD m PQA) (recvD PQB)   = ? -- sendS m (sim& (P& m) Q& PQA (PQB m))\nsim&R end& Q& PQA PQB = ?\n-}\n-}\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (mk \u03a0' S A B) = right (recvS (\u03bb x \u2192 left (sendS x (sim-id (B x)))))\nsim-id (mk \u03a0' D A B) = right (recvD (\u03bb x \u2192 left (sendD x (sim-id (B x)))))\nsim-id (mk \u03a3' S A B) = left (recvS (\u03bb x \u2192 right (sendS x (sim-id (B x)))))\nsim-id (mk \u03a3' D A B) = left (recvD (\u03bb x \u2192 right (sendD x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp ()   (right x) end\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (S D\u03a0\u03a3 x\u2081) (recvS x) (sendS x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (D D\u03a0\u03a3 x\u2081) (recvD x) (sendD x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (S D\u03a3\u03a0 x)  (sendS x\u2081 x\u2082) (recvS x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\nsim-compRL (D D\u03a3\u03a0 x)  (sendD x\u2081 x\u2082) (recvD x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recvD PQ) QR = recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (recvS PQ) QR = recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (sendD m PQ) QR = sendD m (sim-comp Q-Q' PQ QR)\nsim-compL Q-Q' (sendS m PQ) QR = sendS m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recvD QR)   = recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (recvS QR)   = recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x))\nsim-compR Q-Q' PQ (sendD m QR) = sendD m (sim-comp Q-Q' PQ QR)\nsim-compR Q-Q' PQ (sendS m QR) = sendS m (sim-comp Q-Q' PQ QR)\n\n{-\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-comp Q-Q' (left (recvD PQ)) QR = left (recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (recvS PQ)) QR = left (recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (sendD m PQ)) QR = left (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' (left (sendS m PQ)) QR = left (sendS m (sim-comp Q-Q' PQ QR))\nsim-comp end (right ()) (left x\u2081)\nsim-comp end end QR = QR\nsim-comp end PQ end = PQ\nsim-comp (\u03a0\u03a3'S Q-Q') (right (recvS PQ)) (left (sendS m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a0\u03a3'D Q-Q') (right (recvD PQ)) (left (sendD m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a3\u03a0'S Q-Q') (right (sendS m PQ)) (left (recvS QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp (\u03a3\u03a0'D Q-Q') (right (sendD m PQ)) (left (recvD QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp Q-Q' PQ (right (recvD QR)) = right (recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m)))\nsim-comp Q-Q' PQ (right (recvS QR)) = right (recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x)))\nsim-comp Q-Q' PQ (right (sendD m QR)) = right (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' PQ (right (sendS m QR)) = right (sendS m (sim-comp Q-Q' PQ QR))\n-}\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\n!\u02e2 : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\n!\u02e2 (left x) = right (sim-symL x)\n!\u02e2 (right x) = left (sim-symR x)\n!\u02e2 end = end\n\nsim-symL (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symL (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symL (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-symR (recvD PQ)   = recvD (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (recvS PQ)   = recvS (\u03bb m \u2192 !\u02e2 (PQ m))\nsim-symR (sendD m PQ) = sendD m (!\u02e2 PQ)\nsim-symR (sendS m PQ) = sendS m (!\u02e2 PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (right (recvD P)) = do (recvD (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (recvS P)) = do (recvS (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (sendD m P)) = do (sendD m (sim-unit P))\nsim-unit (right (sendS m P)) = do (sendS m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ where\n\n  mod\u2081 : \u2200 {A A' B : \u2605} \u2192 (A \u2192 A') \u2192 A \u00d7 B \u2192 A' \u00d7 B\n  mod\u2081 = \u03bb f \u2192 Data.Product.map f id\n\n  mod\u2082 : \u2200 {A B B' : \u2605} \u2192 (B \u2192 B') \u2192 A \u00d7 B \u2192 A \u00d7 B'\n  mod\u2082 = \u03bb f \u2192 Data.Product.map id f\n\n  trace : \u2200 {P P' Q Q'} \u2192 Dual P P' \u2192 Dual Q Q' \u2192  Sim end P' \u2192 Sim P Q \u2192 Sim Q' end\n        \u2192 Tele P \u00d7 Tele Q\n  trace (S D\u03a0\u03a3 x\u2081) Q-Q' (right (sendS x x\u2082)) (left (recvS x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (S D\u03a3\u03a0 x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace (D D\u03a0\u03a3 x\u2081) Q-Q' (right (sendD x x\u2082)) (left (recvD x\u2083)) Q\u00b7 = mod\u2081 (_,_ x) (trace (x\u2081 x) Q-Q' x\u2082 (x\u2083 x) Q\u00b7)\n  trace (D D\u03a3\u03a0 x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) Q\u00b7 = mod\u2081 (_,_ x\u2082) (trace (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 Q\u00b7)\n  trace P-P' (S D\u03a0\u03a3 x\u2081) \u00b7P (right (recvS x)) (left (sendS x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (S D\u03a3\u03a0 x\u2081) \u00b7P (right (sendS x x\u2082)) (left (recvS x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' (D D\u03a0\u03a3 x\u2081) \u00b7P (right (recvD x)) (left (sendD x\u2082 x\u2083)) = mod\u2082 (_,_ x\u2082) (trace P-P' (x\u2081 x\u2082) \u00b7P (x x\u2082) x\u2083)\n  trace P-P' (D D\u03a3\u03a0 x\u2081) \u00b7P (right (sendD x x\u2082)) (left (recvD x\u2083)) = mod\u2082 (_,_ x) (trace P-P' (x\u2081 x) \u00b7P x\u2082 (x\u2083 x))\n  trace P-P' Q-Q' \u00b7P end Q\u00b7 = _\n\n  module _ {P Q : Proto} where\n    _\u2248_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n    PQ \u2248 PQ' = \u2200 {P' Q'}(P-P' : Dual P P')(Q-Q' : Dual Q Q') \u2192 (\u00b7P : Sim end P')(Q\u00b7 : Sim Q' end)\n       \u2192 trace P-P' Q-Q' \u00b7P PQ Q\u00b7 \u2261 trace P-P' Q-Q' \u00b7P PQ' Q\u00b7\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  Dual-sym-sym : \u2200 {P Q} (P-Q : Dual P Q) \u2192 P-Q \u2261 Dual-sym (Dual-sym P-Q)\n  Dual-sym-sym end = refl\n  Dual-sym-sym (S D\u03a0\u03a3 x) = cong (S D\u03a0\u03a3) (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (D D\u03a0\u03a3 x) = cong (D D\u03a0\u03a3) (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (S D\u03a3\u03a0 x) = cong (S D\u03a3\u03a0) (funExtS (\u03bb y \u2192 Dual-sym-sym (x y)))\n  Dual-sym-sym (D D\u03a3\u03a0 x) = cong (D D\u03a3\u03a0) (funExtD (\u03bb y \u2192 Dual-sym-sym (x y)))\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end (S D\u03a0\u03a3 x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (D D\u03a0\u03a3 x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (S D\u03a3\u03a0 x) Q-Q' (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end (D D\u03a3\u03a0 x) Q-Q' (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' (S D\u03a0\u03a3 x\u2081) (right \u00f8P) (right (recvS x)) (left (sendS x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (D D\u03a0\u03a3 x\u2081) (right \u00f8P) (right (recvD x)) (left (sendD x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (S D\u03a3\u03a0 x) (right \u00f8P) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' (D D\u03a3\u03a0 x) (right \u00f8P) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvD x\u2081))\n    = cong (right \u2218 recvD) (funExtD (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvS x\u2081))\n    = cong (right \u2218 recvS) (funExtS (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendD x\u2081 x\u2082))\n    = cong (right \u2218 sendD x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendS x\u2081 x\u2082))\n    = cong (right \u2218 sendS x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\n\u223c-\u00f8 : \u2200 {P}{s s' : Sim end P} \u2192 s \u223c s' \u2192 s \u2261 s'\n\u223c-\u00f8 s\u223cs' = s\u223cs' end end\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n\n  sim-!! : \u2200 {P Q}(PQ : Sim P Q) \u2192 PQ \u2261 !\u02e2 (!\u02e2 PQ)\n  sim-!! (left (recvD x))    = cong (left \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (recvS x))    = cong (left \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (left (sendD x x\u2081)) = cong (left \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (left (sendS x x\u2081)) = cong (left \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! (right (recvD x))    = cong (right \u2218 recvD) (funExtD \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (recvS x))    = cong (right \u2218 recvS) (funExtS \u03bb m \u2192 sim-!! (x m))\n  sim-!! (right (sendD x x\u2081)) = cong (right \u2218 sendD x) (sim-!! x\u2081)\n  sim-!! (right (sendS x x\u2081)) = cong (right \u2218 sendS x) (sim-!! x\u2081)\n  sim-!! end = refl\n\n  sim-comp-!-end : \u2200 {Q Q' R}(Q-Q' : Dual Q Q')(\u00b7Q : Sim end Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 \u00b7Q) \u2261 !\u02e2 (sim-comp Q-Q' \u00b7Q Q'R)\n  sim-comp-!-end (S D\u03a0\u03a3 x\u2081) (right (recvS x)) (left (sendS x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (D D\u03a0\u03a3 x\u2081) (right (recvD x)) (left (sendD x\u2082 x\u2083)) = sim-comp-!-end (x\u2081 x\u2082) (x x\u2082) x\u2083\n  sim-comp-!-end (S D\u03a3\u03a0 x) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end (D D\u03a3\u03a0 x) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) = sim-comp-!-end (x x\u2081) x\u2082 (x\u2083 x\u2081)\n  sim-comp-!-end Q-Q' (right x) (right (recvD x\u2081))\n    = cong (left \u2218 recvD) (funExtD (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (recvS x\u2081))\n    = cong (left \u2218 recvS) (funExtS (\u03bb x\u2082 \u2192 sim-comp-!-end Q-Q' (right x) (x\u2081 x\u2082)))\n  sim-comp-!-end Q-Q' (right x) (right (sendD x\u2081 x\u2082))\n    = cong (left \u2218 sendD x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end Q-Q' (right x) (right (sendS x\u2081 x\u2082))\n    = cong (left \u2218 sendS x\u2081) (sim-comp-!-end Q-Q' (right x) x\u2082)\n  sim-comp-!-end end end QR = {!!}\n\n  open \u2261-Reasoning\n  module _ {P Q}{s s' : Sim P Q} where\n    !\u02e2-cong : s \u223c s' \u2192 !\u02e2 s \u223c !\u02e2 s'\n    !\u02e2-cong s\u223cs' Q'-Q \u00f8Q'\n      = sim-comp Q'-Q \u00f8Q' (!\u02e2 s)\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 (sim-comp (Dual-spec Q) s (sim-id _)))\n      \u2261\u27e8 {!!} \u27e9\n        sim-comp Q'-Q \u00f8Q' (!\u02e2 s')\n      \u220e\n\n  postulate\n    sim-comp-assoc-end' : \u2200 {P Q Q' R R'}(Q-Q' : Dual Q Q')(R-R' : Dual R R')\n      (PQ : Sim P Q)(QR : Sim Q' R )(R\u00f8 : Sim R' end)\n      \u2192 sim-comp R-R' (sim-comp Q-Q' PQ QR) R\u00f8\n      \u2261 sim-comp Q-Q' PQ (sim-comp R-R' QR R\u00f8)\n\n\n  sim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n    \u2192 sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ) \u223c !\u02e2 (sim-comp Q-Q' PQ Q'R)\n  sim-comp-! Q-Q' PQ Q'R {R'} R'-R \u00f8R'\n    = sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))\n    \u2261\u27e8 sim-!! (sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2( !\u02e2 ((sim-comp R'-R \u00f8R' (sim-comp (Dual-sym Q-Q') (!\u02e2 Q'R) (!\u02e2 PQ)))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 !\u02e2) (sym (sim-comp-assoc-end funExtD funExtS R'-R (Dual-sym Q-Q') \u00f8R' (!\u02e2 Q'R) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (!\u02e2\n         (sim-comp (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ)))\n    \u2261\u27e8 cong !\u02e2 (sym (sim-comp-!-end (Dual-sym Q-Q') (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)) (!\u02e2 PQ))) \u27e9\n      !\u02e2\n        (sim-comp (Dual-sym (Dual-sym Q-Q')) (!\u02e2 (!\u02e2 PQ))\n         (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong\u2082 (\u03bb X Y \u2192 !\u02e2 (sim-comp X Y (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R)))))\n         (sym (Dual-sym-sym funExtD funExtS Q-Q')) (sym (sim-!! PQ)) \u27e9\n     !\u02e2 (sim-comp Q-Q' PQ (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 Q'R))))\n    \u2261\u27e8 cong (!\u02e2 \u2218 sim-comp Q-Q' PQ) (sym (sim-comp-!-end R'-R \u00f8R' (!\u02e2 Q'R))) \u27e9\n      !\u02e2\n        (sim-comp Q-Q' PQ\n         (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 Q'R)) (!\u02e2 \u00f8R')))\n    \u2261\u27e8 cong\n         (\u03bb X \u2192 !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))))\n         (sym (sim-!! Q'R)) \u27e9\n      !\u02e2 (sim-comp Q-Q' PQ (sim-comp (Dual-sym R'-R) Q'R (!\u02e2 \u00f8R')))\n    -- \u2261\u27e8 cong !\u02e2 (sym (sim-comp-assoc-end' Q-Q' (Dual-sym R'-R) PQ Q'R (!\u02e2 \u00f8R'))) \u27e9\n    \u2261\u27e8 \u223c-\u00f8 {!!}\u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (sim-comp Q-Q' PQ Q'R) (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong (\u03bb X \u2192 !\u02e2 (sim-comp (Dual-sym R'-R) X (!\u02e2 \u00f8R'))) (sim-!! (sim-comp Q-Q' PQ Q'R)) \u27e9\n      !\u02e2 (sim-comp (Dual-sym R'-R) (!\u02e2 (!\u02e2 (sim-comp Q-Q' PQ Q'R)))\n                                   (!\u02e2 \u00f8R'))\n    \u2261\u27e8 cong !\u02e2 (sim-comp-!-end R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))) \u27e9\n      !\u02e2 (!\u02e2 (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))))\n    \u2261\u27e8 sym (sim-!! (sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R)))) \u27e9\n      sim-comp R'-R \u00f8R' (!\u02e2 (sim-comp Q-Q' PQ Q'R))\n    \u220e\n\n  \u2666-! : \u2200 {P Q R}(PQ : Sim P Q)(QR : Sim (dual Q) R)\n    \u2192 !\u02e2 (PQ \u2666 QR) \u223c (!\u02e2 QR) \u2666 (!\u02e2 {!PQ!})\n  \u2666-! = {!!}\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"433602778addc05134f8048f4a59cd0aef3ae6e0","subject":"FunUniverse.Nand: rename fork as mux","message":"FunUniverse.Nand: rename fork as mux\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Nand.agda","new_file":"FunUniverse\/Nand.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"bsd-3-clause","lang":"Agda"}
{"commit":"88f37e6cb324c7dc861f142a26635f31b915212f","subject":"Both projections are defined","message":"Both projections are defined\n","repos":"heades\/AUGL","old_file":"dialectica-cats\/WPDC2Sets.agda","new_file":"dialectica-cats\/WPDC2Sets.agda","new_contents":"module WPDC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ m \u2208 (\u22a4 \u2192 U) ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (U \u2192 X \u2192 Set))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , _ , \u03b1) (V , Y , _ , _ , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , m\u2081 , n\u2081 , \u03b1)} {(V , Y , m\u2082 , n\u2082 , \u03b2)} {(W , Z , m\u2083 , n\u2083 , \u03b3)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , m , n , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , _ , \u03b1}\n         {V , Y , _ , _ , \u03b2}\n         {W , Z , _ , _ , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{W , Z , _ , _ , \u03b3}{S , T , _ , _ , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 m\u2081 m\u2082  , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)}{(W , Z , _ , _ , \u03b3)}{(S , T , _ , _ , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (U , X , m\u2081 , n\u2081 , \u03b1)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n\n\u03c0\u2082 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (V , Y , m\u2082 , n\u2082 , \u03b2)\n\u03c0\u2082 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = snd , (\u03bb r y \u2192 n\u2081 triv , y) , cond\n where\n  cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : Y} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (n\u2081 triv , y) \u2192 \u03b2 (snd u) y\n  cond {u , v}{y} (p\u2081 , p\u2082) = p\u2082\n","old_contents":"module WPDC2Sets where\n\nopen import prelude\n\n-- The objects:\nObj : Set\u2081\nObj = \u03a3[ U \u2208 Set ] (\u03a3[ X \u2208 Set ] (\u03a3[ m \u2208 (\u22a4 \u2192 U) ] (\u03a3[ n \u2208 (\u22a4 \u2192 X) ] (U \u2192 X \u2192 Set))))\n\n-- The morphisms:\nHom : Obj \u2192 Obj \u2192 Set\nHom (U , X , _ , _ , \u03b1) (V , Y , _ , _ , \u03b2) =\n  \u03a3[ f \u2208 (U \u2192 V) ]\n    (\u03a3[ F \u2208 (U \u2192 Y \u2192 X) ] (\u2200{u : U}{y : Y} \u2192 \u03b1 u (F u y) \u2192 \u03b2 (f u) y))\n\n-- Composition:\ncomp : {A B C : Obj} \u2192 Hom A B \u2192 Hom B C \u2192 Hom A C\ncomp {(U , X , m\u2081 , n\u2081 , \u03b1)} {(V , Y , m\u2082 , n\u2082 , \u03b2)} {(W , Z , m\u2083 , n\u2083 , \u03b3)} (f , F  , p\u2081) (g , G , p\u2082) =\n  (g \u2218 f , (\u03bb u z \u2192 F u (G (f u) z)), (\u03bb {u} {y} p\u2083 \u2192 p\u2082 (p\u2081 p\u2083)))\n\ninfixl 5 _\u25cb_\n\n_\u25cb_ = comp\n\n-- The contravariant hom-functor:\nHom\u2090 :  {A' A B B' : Obj} \u2192 Hom A' A \u2192 Hom B B' \u2192 Hom A B \u2192 Hom A' B'\nHom\u2090 f h g = comp f (comp g h)\n\n-- The identity function:\nid : {A : Obj} \u2192 Hom A A \nid {(U , V , m , n , \u03b1)} = (id-set , curry snd , id-set)\n\n-- In this formalization we will only worry about proving that the\n-- data of morphisms are equivalent, and not worry about the morphism\n-- conditions.  This will make proofs shorter and faster.\n--\n-- If we have parallel morphisms (f,F) and (g,G) in which we know that\n-- f = g and F = G, then the condition for (f,F) will imply the\n-- condition of (g,G) and vice versa.  Thus, we can safely ignore it.\ninfix 4 _\u2261h_\n\n_\u2261h_ : {A B : Obj} \u2192 (f g : Hom A B) \u2192 Set\n_\u2261h_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)} (f , F , p\u2081) (g , G , p\u2082) = f \u2261 g \u00d7 F \u2261 G\n\n\u2261h-refl : {A B : Obj}{f : Hom A B} \u2192 f \u2261h f\n\u2261h-refl {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _} = refl , refl\n\n\u2261h-trans : \u2200{A B}{f g h : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h h \u2192 f \u2261h h\n\u2261h-trans {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _}{h , H , _} (p\u2081 , p\u2082) (p\u2083 , p\u2084) rewrite p\u2081 | p\u2082 | p\u2083 | p\u2084 = refl , refl\n\n\u2261h-sym : \u2200{A B}{f g : Hom A B} \u2192 f \u2261h g \u2192 g \u2261h f\n\u2261h-sym {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{f , F , _}{g , G , _} (p\u2081 , p\u2082) rewrite p\u2081 | p\u2082 = refl , refl\n\n\u2261h-subst-\u25cb : \u2200{A B C}{f\u2081 f\u2082 : Hom A B}{g\u2081 g\u2082 : Hom B C}{j : Hom A C}\n  \u2192 f\u2081 \u2261h f\u2082\n  \u2192 g\u2081 \u2261h g\u2082\n  \u2192 f\u2082 \u25cb g\u2082 \u2261h j\n  \u2192 f\u2081 \u25cb g\u2081 \u2261h j\n\u2261h-subst-\u25cb {U , X , _ , _ , \u03b1}\n         {V , Y , _ , _ , \u03b2}\n         {W , Z , _ , _ , \u03b3}\n         {f\u2081 , F\u2081 , _}\n         {f\u2082 , F\u2082 , _}\n         {g\u2081 , G\u2081 , _}\n         {g\u2082 , G\u2082 , _}\n         {j , J , _}\n         (p\u2085 , p\u2086) (p\u2087 , p\u2088) (p\u2089 , p\u2081\u2080) rewrite p\u2085 | p\u2086 | p\u2087 | p\u2088 | p\u2089 | p\u2081\u2080 = refl , refl\n\n\u25cb-assoc : \u2200{A B C D}{f : Hom A B}{g : Hom B C}{h : Hom C D}\n  \u2192 f \u25cb (g \u25cb h) \u2261h (f \u25cb g) \u25cb h\n\u25cb-assoc {U , X , _ , _ , \u03b1}{V , Y , _ , _ , \u03b2}{W , Z , _ , _ , \u03b3}{S , T , _ , _ , \u03b9}\n        {f , F , _}{g , G , _}{h , H , _} = refl , refl\n\n\u25cb-idl : \u2200{A B}{f : Hom A B} \u2192 id \u25cb f \u2261h f\n\u25cb-idl {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n\u25cb-idr : \u2200{A B}{f : Hom A B} \u2192 f \u25cb id \u2261h f\n\u25cb-idr {U , X , _ , _ , _}{V , Y , _ , _ , _}{f , F , _} = refl , refl\n\n-- The tensor functor: \u2297\n_\u2297\u1d63_ : \u2200{U X V Y : Set} \u2192 (U \u2192 X \u2192 Set) \u2192 (V \u2192 Y \u2192 Set) \u2192 ((U \u00d7 V) \u2192 (X \u00d7 Y) \u2192 Set)\n_\u2297\u1d63_ \u03b1 \u03b2 (u , v) (x , y) = (\u03b1 u x) \u00d7 (\u03b2 v y)\n\n_\u2297\u2092_ : (A B : Obj) \u2192 Obj\n(U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2) = ((U \u00d7 V) , (X \u00d7 Y) , trans-\u00d7 m\u2081 m\u2082  , trans-\u00d7 n\u2081 n\u2082 , \u03b1 \u2297\u1d63 \u03b2)\n\nF\u2297 : \u2200{Z T V X U Y : Set}{F : U \u2192 Z \u2192 X}{G : V \u2192 T \u2192 Y} \u2192 (U \u00d7 V) \u2192 (Z \u00d7 T) \u2192 (X \u00d7 Y)\nF\u2297 {F = F}{G} (u , v) (z , t) = F u z , G v t\n  \n_\u2297\u2090_ : {A B C D : Obj} \u2192 Hom A C \u2192 Hom B D \u2192 Hom (A \u2297\u2092 B) (C \u2297\u2092 D)\n_\u2297\u2090_ {(U , X , _ , _ , \u03b1)}{(V , Y , _ , _ , \u03b2)}{(W , Z , _ , _ , \u03b3)}{(S , T , _ , _ , \u03b4)} (f , F , p\u2081) (g , G , p\u2082) = \u27e8 f , g \u27e9 , F\u2297 {F = F}{G} , p\u2297\n where\n  p\u2297 : {u : \u03a3 U (\u03bb x \u2192 V)} {y : \u03a3 Z (\u03bb x \u2192 T)} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (F\u2297 {F = F}{G} u y) \u2192 (\u03b3 \u2297\u1d63 \u03b4) (\u27e8 f , g \u27e9 u) y\n  p\u2297 {u , v}{z , t} (p\u2083 , p\u2084) = p\u2081  p\u2083 , p\u2082 p\u2084\n\n\u03c0\u2081 : {U X V Y : Set}\n    \u2192 {\u03b1 : U \u2192 X \u2192 Set}\n    \u2192 {\u03b2 : V \u2192 Y \u2192 Set}\n    \u2192 {m\u2081 : \u22a4 \u2192 U}\n    \u2192 {n\u2081 : \u22a4 \u2192 X}\n    \u2192 {m\u2082 : \u22a4 \u2192 V}\n    \u2192 {n\u2082 : \u22a4 \u2192 Y}    \n    \u2192 Hom ((U , X , m\u2081 , n\u2081 , \u03b1) \u2297\u2092 (V , Y , m\u2082 , n\u2082 , \u03b2)) (U , X , m\u2081 , n\u2081 , \u03b1)\n\u03c0\u2081 {U}{X}{V}{Y}{\u03b1}{\u03b2}{m\u2081}{n\u2081}{m\u2082}{n\u2082} = fst , (\u03bb r x \u2192 x , n\u2082 triv) , cond\n where\n   cond : {u : \u03a3 U (\u03bb x \u2192 V)} {y : X} \u2192 (\u03b1 \u2297\u1d63 \u03b2) u (y , n\u2082 triv) \u2192 \u03b1 (fst u) y\n   cond {u , v}{x} (p\u2081 , p\u2082) = p\u2081\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f6cc3ac8314b4fa32d3f9b901c94b4c40b3ca594","subject":"Clarify what fails","message":"Clarify what fails\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c (suc n)\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n \u2192 evalConst c n \u2261 Done v \u2192 evalConst c (suc n) \u2261 Done v\neval-const-mono (lit v) n eq = eq\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n \u2192 eval t \u03c1 n \u2261 Done v \u2192 eval t \u03c1 (suc n) \u2261 Done v\neval-mono t \u03c1 v zero ()\neval-mono (const c) \u03c1 v (suc n) eq = eval-const-mono c (suc n) eq\neval-mono (var x) \u03c1 v (suc n) eq = eq\neval-mono (app s t) \u03c1 v (suc n) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-mono (app s t) \u03c1 v (suc n) eq | Done sv | [ seq ] | (Done tv) | [ teq ] with eval s \u03c1 (suc n) | eval-mono s \u03c1 sv n seq | eval t \u03c1 (suc n) | eval-mono t \u03c1 tv n teq\neval-mono (app s t) \u03c1 v (suc n) eq | Done (closure ct c\u03c1) | [ seq ] | (Done tv) | [ teq ] | .(Done (closure ct c\u03c1)) | refl | .(Done tv) | refl = eval-mono ct (tv \u2022 c\u03c1) _ n eq\neval-mono (app s t) \u03c1 v (suc n) () | Done _ | [ seq ] | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n) () | TimeOut | [ seq ] | tv | [ teq ]\n-- = {!eval-mono s \u03c1 (suc n)!}\neval-mono (abs t) \u03c1 v (suc n) eq = eq\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV relV2 : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 rel\u03c12 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrel\u03c12 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c12 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV2 \u03c4 v1 v2 n \u00d7 rel\u03c12 \u0393 \u03c11 \u03c12 n\n\n-- relT\/relT2 doesn't ask input environments to be related, but the fundamental lemma\n-- does do so.\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT relT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  (eq : eval t1 \u03c11 n \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  (eq : eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\n-- Copy of relV, but mentioning relT2\nrelV2 \u03c4 v1 v2 zero = \u22a4\nrelV2 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV2 \u03c3 v1 v2 k \u2192 relT2 t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV2 nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nopen import Data.Nat.Properties\n\nrelV2-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV2 \u03c4 v1 v2 n \u2192 relV2 \u03c4 v1 v2 m\nrelV2-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV2-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV2-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV2-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrel\u03c12-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 m\nrel\u03c12-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c12-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV2-mono m n m\u2264n _ v1 v2 vv , rel\u03c12-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- True but useless.\n\n-- relV-apply-go2 : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc (suc n)))\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n)))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) (\u2264-step \u2264-refl) tv1 tv2 (relV-mono _ _ (\u2264-step (\u2264-step \u2264-refl)) _ _ _ tvv) v1 eqv1\n--   | svv (suc (suc n)) \u2264-refl tv1 tv2 (relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv) v1 (eval-mono st1 (tv1 \u2022 \u03c11) v1 (suc n) eqv1)\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2' , eqv2' , final-v' | v2 , eqv2 , final-v with trans (sym (eval-mono st2 (tv2 \u2022 \u03c12) v2' (suc n) eqv2')) eqv2\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | .v2 , eqv2' , final-v' | v2 , eqv2 , final-v | refl = v2 , eqv2' , final-v\n\n\n-- relV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n--   (tvv : relV \u03c3 tv1 tv2 (suc n))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) \u2264-refl tv1 tv2 ((relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv)) v1 eqv1\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2 , eqv2 , final-v = v2 , eqv2 , {!final-v!}\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n--   (tvv : relV \u03c3 tv1 tv2 (suc n))\n--   {v1} \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n--\n\n-- -- fundamental lemma of logical relations.\n-- fundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\n-- fundamentalV x zero _ _ _ _ ()\n-- fundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- fundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\n\n-- fundamental-aux : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n--   eval t \u03c11 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 eval t \u03c12 (suc n) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = sv2 , refl , sveq , svv'\n-- --\n-- -- |\n-- fundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- -- XXX trivial case for constants.\n-- fundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\n-- fundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\n-- fundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n--   closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | eval t \u03c12 n | fundamental-aux t n \u03c11 \u03c12 \u03c1\u03c1 tv1 teq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | (.sv2 , refl , seq2' , svv') | (Done tv2) | (.tv2 , refl , teq2' , tvv') = relV-apply n svv' tvv' t\u03c11\u2193v1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- -- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = {!!}\n-- -- | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\n\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n-- --... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- -- TODO: match sv2 before matching on fundamental s.\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- -- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n-- --   where\n-- --     v2 : Val \u03c4\n-- --     v2 = {!!}\n-- --     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n-- --     -- t\u03c12\u2193v2 = ?\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n\nopen import Data.Fin.Properties\n\nltn : \u2200 n j \u2192 F.to\u2115 (n \u2115- j) \u2264 suc n\nltn n Fin.zero rewrite to-from n = \u2264-step \u2264-refl\nltn zero (Fin.suc ())\nltn (suc n) (Fin.suc j) = \u2264-step (ltn n j)\n\nltn2 : \u2200 n j \u2192 F.to\u2115 (n \u2115- Fin.suc j) \u2264 n\nltn2 zero ()\nltn2 (suc n) j = ltn n j\n\nfundamentalV2 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 (var x) (var x) \u03c11 \u03c12 n\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc ()) x\u2081\nfundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1' Fin.zero ()\nfundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = v2 , refl , relV2-mono _ _ (ltn n j) _ v1 v2 vv\nfundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = {!!}\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- --\n\n-- fundamentalV2 x zero _ _ _ _ ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nrelV-nat-refl : \u2200 nv n \u2192 relV2 nat nv nv n\nrelV-nat-refl (intV nv) zero = tt\nrelV-nat-refl (intV nv) (suc n) = refl\n\n\nrelV2-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n j\n  (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  (tvv : relV2 \u03c3 tv1 tv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  v1 \u2192\n  (eqv1 : apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n \u2115- Fin.suc j)))\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc zero) Fin.zero tt tt v1 eqv1 = {!CAN'T BE FILLED!}\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc (suc n)) Fin.zero svv tvv v1 eqv1 = {!!}\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) (Fin.suc j) svv tvv v1 eqv1 = {!!}\n\nrelV2-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n j\n  (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  (tvv : relV2 \u03c3 tv1 tv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  {v1} \u2192\n  apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n \u2115- Fin.suc j)))\nrelV2-apply n j svv tvv eqv1 = relV2-apply-go _ _ _ _ n j svv tvv _ eqv1\n\nfundamental2 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 t t \u03c11 \u03c12 n\nfundamental2 t n \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamental2 (const (lit nv)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) (Fin.suc j) refl = intV nv , refl , relV-nat-refl (intV nv) (F.to\u2115 (n \u2115- Fin.suc j))\nfundamental2 (var x) n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x n \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\nfundamental2 (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) (Fin.suc j) refl = closure t \u03c12 , refl , res-valid (F.to\u2115 (n \u2115- Fin.suc j)) (ltn2 n j)\n  -- foo (F.to\u2115 (n \u2115- Fin.suc j))\n  where\n    res-valid : \u2200 nj \u2192 nj \u2264 n \u2192 relV2 _ (closure t \u03c11) (closure t \u03c12) nj\n    res-valid zero nj\u2264n = tt\n    res-valid (suc nj) nj\u2264n k k\u2264nj v1 v2 vv v3 Fin.zero ()\n    res-valid (suc nj) nj\u2264n k k\u2264nj v1 v2 vv v3 (Fin.suc j\u2081) eq = fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c12-mono _ _ (\u2264-trans k\u2264nj (\u2264-trans (\u2264-step \u2264-refl) nj\u2264n)) _ _ _ \u03c1\u03c1) v3 (Fin.suc j\u2081) eq\n\n    -- foo : relV2 _ (closure t \u03c11) (closure t \u03c12) (F.to\u2115 (n \u2115- Fin.suc j))\n    -- foo with (F.to\u2115 (n \u2115- Fin.suc j))\n    -- foo | zero = tt\n    -- foo | (suc nj) = \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192 {!!}\n\n    -- \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192\n    -- fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264nj) _ _ _ \u03c1\u03c1)\n\n    -- foo | zero = ?\n    -- foo | (suc nj) k k\u2264nj v1 v2 vv = {!!}\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc Fin.zero) ()\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 with eval s \u03c11 (suc (F.to\u2115 j)) | inspect (eval s \u03c11) (suc (F.to\u2115 j))\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] with eval t \u03c11 (suc (F.to\u2115 j)) | inspect (eval t \u03c11) (suc (F.to\u2115 j))\nfundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | Done tv1 | [ t\u03c11\u2193tv1 ] with eval s \u03c12 (suc (F.to\u2115 j)) | fundamental2 s n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) sv1 (Fin.suc j) s\u03c11\u2193sv1\nfundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | (Done tv1) | [ t\u03c11\u2193tv1 ] | .(Done sv2) | (sv2 , refl , svv) with\n  eval t \u03c12 (suc (F.to\u2115 j)) | fundamental2 t n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) tv1 (Fin.suc j) t\u03c11\u2193tv1\n... | .(Done tv2) | (tv2 , refl , tvv) = relV2-apply n j svv tvv appst-\u03c11\u2193v1\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | Done sv1 | [ s\u03c11\u2193sv1 ] | TimeOut | [ t\u03c11\u2193tv1 ]\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | TimeOut | [ s\u03c11\u2193sv1 ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c (suc n)\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n \u2192 evalConst c n \u2261 Done v \u2192 evalConst c (suc n) \u2261 Done v\neval-const-mono (lit v) n eq = eq\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n \u2192 eval t \u03c1 n \u2261 Done v \u2192 eval t \u03c1 (suc n) \u2261 Done v\neval-mono t \u03c1 v zero ()\neval-mono (const c) \u03c1 v (suc n) eq = eval-const-mono c (suc n) eq\neval-mono (var x) \u03c1 v (suc n) eq = eq\neval-mono (app s t) \u03c1 v (suc n) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-mono (app s t) \u03c1 v (suc n) eq | Done sv | [ seq ] | (Done tv) | [ teq ] with eval s \u03c1 (suc n) | eval-mono s \u03c1 sv n seq | eval t \u03c1 (suc n) | eval-mono t \u03c1 tv n teq\neval-mono (app s t) \u03c1 v (suc n) eq | Done (closure ct c\u03c1) | [ seq ] | (Done tv) | [ teq ] | .(Done (closure ct c\u03c1)) | refl | .(Done tv) | refl = eval-mono ct (tv \u2022 c\u03c1) _ n eq\neval-mono (app s t) \u03c1 v (suc n) () | Done _ | [ seq ] | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n) () | TimeOut | [ seq ] | tv | [ teq ]\n-- = {!eval-mono s \u03c1 (suc n)!}\neval-mono (abs t) \u03c1 v (suc n) eq = eq\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV relV2 : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 rel\u03c12 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrel\u03c12 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c12 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV2 \u03c4 v1 v2 n \u00d7 rel\u03c12 \u0393 \u03c11 \u03c12 n\n\n-- relT\/relT2 doesn't ask input environments to be related, but the fundamental lemma\n-- does do so.\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT relT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  (eq : eval t1 \u03c11 n \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  (eq : eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n F.\u2115- j))\n\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\n-- Copy of relV, but mentioning relT2\nrelV2 \u03c4 v1 v2 zero = \u22a4\nrelV2 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV2 \u03c3 v1 v2 k \u2192 relT2 t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV2 nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nopen import Data.Nat.Properties\n\nrelV2-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV2 \u03c4 v1 v2 n \u2192 relV2 \u03c4 v1 v2 m\nrelV2-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV2-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV2-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV2-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrel\u03c12-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 m\nrel\u03c12-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c12-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV2-mono m n m\u2264n _ v1 v2 vv , rel\u03c12-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- True but useless.\n\n-- relV-apply-go2 : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc (suc n)))\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n)))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) (\u2264-step \u2264-refl) tv1 tv2 (relV-mono _ _ (\u2264-step (\u2264-step \u2264-refl)) _ _ _ tvv) v1 eqv1\n--   | svv (suc (suc n)) \u2264-refl tv1 tv2 (relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv) v1 (eval-mono st1 (tv1 \u2022 \u03c11) v1 (suc n) eqv1)\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2' , eqv2' , final-v' | v2 , eqv2 , final-v with trans (sym (eval-mono st2 (tv2 \u2022 \u03c12) v2' (suc n) eqv2')) eqv2\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | .v2 , eqv2' , final-v' | v2 , eqv2 , final-v | refl = v2 , eqv2' , final-v\n\n\nrelV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n  (tvv : relV \u03c3 tv1 tv2 (suc n))\n  v1 \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n  with svv (suc n) \u2264-refl tv1 tv2 ((relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv)) v1 eqv1\nrelV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2 , eqv2 , final-v = v2 , eqv2 , {!final-v!}\n\nrelV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n\n  (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n  (tvv : relV \u03c3 tv1 tv2 (suc n))\n  {v1} \u2192\n  apply sv1 tv1 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\nrelV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n--\n\n-- -- fundamental lemma of logical relations.\n-- fundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\n-- fundamentalV x zero _ _ _ _ ()\n-- fundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- fundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\n\n-- fundamental-aux : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n--   eval t \u03c11 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 eval t \u03c12 (suc n) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = sv2 , refl , sveq , svv'\n-- --\n-- -- |\n-- fundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- -- XXX trivial case for constants.\n-- fundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\n-- fundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\n-- fundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n--   closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | eval t \u03c12 n | fundamental-aux t n \u03c11 \u03c12 \u03c1\u03c1 tv1 teq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | (.sv2 , refl , seq2' , svv') | (Done tv2) | (.tv2 , refl , teq2' , tvv') = relV-apply n svv' tvv' t\u03c11\u2193v1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- -- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = {!!}\n-- -- | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\n\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n-- --... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- -- TODO: match sv2 before matching on fundamental s.\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- -- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n-- --   where\n-- --     v2 : Val \u03c4\n-- --     v2 = {!!}\n-- --     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n-- --     -- t\u03c12\u2193v2 = ?\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n\nopen import Data.Fin.Properties\n\nltn : \u2200 n j \u2192 F.to\u2115 (n \u2115- j) \u2264 suc n\nltn n Fin.zero rewrite to-from n = \u2264-step \u2264-refl\nltn zero (Fin.suc ())\nltn (suc n) (Fin.suc j) = \u2264-step (ltn n j)\n\nltn2 : \u2200 n j \u2192 F.to\u2115 (n \u2115- Fin.suc j) \u2264 n\nltn2 zero ()\nltn2 (suc n) j = ltn n j\n\nfundamentalV2 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 (var x) (var x) \u03c11 \u03c12 n\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc ()) x\u2081\nfundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1' Fin.zero ()\nfundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = v2 , refl , relV2-mono _ _ (ltn n j) _ v1 v2 vv\nfundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = {!!}\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- --\n\n-- fundamentalV2 x zero _ _ _ _ ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nrelV-nat-refl : \u2200 nv n \u2192 relV2 nat nv nv n\nrelV-nat-refl (intV nv) zero = tt\nrelV-nat-refl (intV nv) (suc n) = refl\n\n\nrelV2-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n j\n  (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  (tvv : relV2 \u03c3 tv1 tv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  v1 \u2192\n  (eqv1 : apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n \u2115- Fin.suc j)))\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc zero) Fin.zero tt tt v1 eqv1 = {!!}\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc (suc n)) Fin.zero svv tvv v1 eqv1 = {!!}\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) (Fin.suc j) svv tvv v1 eqv1 = {!!}\n\nrelV2-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n j\n  (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  (tvv : relV2 \u03c3 tv1 tv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n  {v1} \u2192\n  apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n \u2115- Fin.suc j)))\nrelV2-apply n j svv tvv eqv1 = relV2-apply-go _ _ _ _ n j svv tvv _ eqv1\n\nfundamental2 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 t t \u03c11 \u03c12 n\nfundamental2 t n \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamental2 (const (lit nv)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) (Fin.suc j) refl = intV nv , refl , relV-nat-refl (intV nv) (F.to\u2115 (n \u2115- Fin.suc j))\nfundamental2 (var x) n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x n \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\nfundamental2 (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) (Fin.suc j) refl = closure t \u03c12 , refl , res-valid (F.to\u2115 (n \u2115- Fin.suc j)) (ltn2 n j)\n  -- foo (F.to\u2115 (n \u2115- Fin.suc j))\n  where\n    res-valid : \u2200 nj \u2192 nj \u2264 n \u2192 relV2 _ (closure t \u03c11) (closure t \u03c12) nj\n    res-valid zero nj\u2264n = tt\n    res-valid (suc nj) nj\u2264n k k\u2264nj v1 v2 vv v3 Fin.zero ()\n    res-valid (suc nj) nj\u2264n k k\u2264nj v1 v2 vv v3 (Fin.suc j\u2081) eq = fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c12-mono _ _ (\u2264-trans k\u2264nj (\u2264-trans (\u2264-step \u2264-refl) nj\u2264n)) _ _ _ \u03c1\u03c1) v3 (Fin.suc j\u2081) eq\n\n    -- foo : relV2 _ (closure t \u03c11) (closure t \u03c12) (F.to\u2115 (n \u2115- Fin.suc j))\n    -- foo with (F.to\u2115 (n \u2115- Fin.suc j))\n    -- foo | zero = tt\n    -- foo | (suc nj) = \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192 {!!}\n\n    -- \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192\n    -- fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264nj) _ _ _ \u03c1\u03c1)\n\n    -- foo | zero = ?\n    -- foo | (suc nj) k k\u2264nj v1 v2 vv = {!!}\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc Fin.zero) ()\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 with eval s \u03c11 (suc (F.to\u2115 j)) | inspect (eval s \u03c11) (suc (F.to\u2115 j))\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] with eval t \u03c11 (suc (F.to\u2115 j)) | inspect (eval t \u03c11) (suc (F.to\u2115 j))\nfundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | Done tv1 | [ t\u03c11\u2193tv1 ] with eval s \u03c12 (suc (F.to\u2115 j)) | fundamental2 s n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) sv1 (Fin.suc j) s\u03c11\u2193sv1\nfundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | (Done tv1) | [ t\u03c11\u2193tv1 ] | .(Done sv2) | (sv2 , refl , svv) with\n  eval t \u03c12 (suc (F.to\u2115 j)) | fundamental2 t n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) tv1 (Fin.suc j) t\u03c11\u2193tv1\n... | .(Done tv2) | (tv2 , refl , tvv) = relV2-apply n j svv tvv appst-\u03c11\u2193v1\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | Done sv1 | [ s\u03c11\u2193sv1 ] | TimeOut | [ t\u03c11\u2193tv1 ]\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | TimeOut | [ s\u03c11\u2193sv1 ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"9cdcf85f8d4d2d3ce710c86643d4dd68dc7122d0","subject":"Reenable broken code to look at it","message":"Reenable broken code to look at it\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c (suc n)\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n \u2192 evalConst c n \u2261 Done v \u2192 evalConst c (suc n) \u2261 Done v\neval-const-mono (lit v) n eq = eq\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n \u2192 eval t \u03c1 n \u2261 Done v \u2192 eval t \u03c1 (suc n) \u2261 Done v\neval-mono t \u03c1 v zero ()\neval-mono (const c) \u03c1 v (suc n) eq = eval-const-mono c (suc n) eq\neval-mono (var x) \u03c1 v (suc n) eq = eq\neval-mono (app s t) \u03c1 v (suc n) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-mono (app s t) \u03c1 v (suc n) eq | Done sv | [ seq ] | (Done tv) | [ teq ] with eval s \u03c1 (suc n) | eval-mono s \u03c1 sv n seq | eval t \u03c1 (suc n) | eval-mono t \u03c1 tv n teq\neval-mono (app s t) \u03c1 v (suc n) eq | Done (closure ct c\u03c1) | [ seq ] | (Done tv) | [ teq ] | .(Done (closure ct c\u03c1)) | refl | .(Done tv) | refl = eval-mono ct (tv \u2022 c\u03c1) _ n eq\neval-mono (app s t) \u03c1 v (suc n) () | Done _ | [ seq ] | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n) () | TimeOut | [ seq ] | tv | [ teq ]\n-- = {!eval-mono s \u03c1 (suc n)!}\neval-mono (abs t) \u03c1 v (suc n) eq = eq\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV relV2 : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 rel\u03c12 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrel\u03c12 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c12 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV2 \u03c4 v1 v2 n \u00d7 rel\u03c12 \u0393 \u03c11 \u03c12 n\n\n-- relT\/relT2 doesn't ask input environments to be related, but the fundamental lemma\n-- does do so.\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT relT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  (eq : eval t1 \u03c11 n \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_; _\u2115-\u2115_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  (eq : eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 j)\n\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\n-- Copy of relV, but mentioning relT2\nrelV2 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV2 \u03c3 v1 v2 k \u2192 relT2 t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV2 nat v1 v2 n = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nopen import Data.Nat.Properties\n\nrelV2-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV2 \u03c4 v1 v2 n \u2192 relV2 \u03c4 v1 v2 m\nrelV2-mono m n m\u2264n (\u03c4 \u21d2 \u03c4\u2081) (closure t \u03c1) (closure t\u2081 \u03c1\u2081) vv = \u03bb k k\u2264m \u2192 vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\nrelV2-mono m n m\u2264n nat (intV n\u2081) (intV n\u2082) vv = vv\n-- relV2-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\n-- relV2-mono (suc m) zero () \u03c4 v1 v2 vv\n-- relV2-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\n-- relV2-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\n-- relV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\n-- relV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\n-- relV-mono (suc m) zero () \u03c4 v1 v2 vv\n-- relV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\n-- relV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\n-- rel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\n-- rel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\n-- rel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrel\u03c12-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 m\nrel\u03c12-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c12-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV2-mono m n m\u2264n _ v1 v2 vv , rel\u03c12-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- True but useless.\n\n-- relV-apply-go2 : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc (suc n)))\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n)))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) (\u2264-step \u2264-refl) tv1 tv2 (relV-mono _ _ (\u2264-step (\u2264-step \u2264-refl)) _ _ _ tvv) v1 eqv1\n--   | svv (suc (suc n)) \u2264-refl tv1 tv2 (relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv) v1 (eval-mono st1 (tv1 \u2022 \u03c11) v1 (suc n) eqv1)\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2' , eqv2' , final-v' | v2 , eqv2 , final-v with trans (sym (eval-mono st2 (tv2 \u2022 \u03c12) v2' (suc n) eqv2')) eqv2\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | .v2 , eqv2' , final-v' | v2 , eqv2 , final-v | refl = v2 , eqv2' , final-v\n\n\n-- relV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n--   (tvv : relV \u03c3 tv1 tv2 (suc n))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) \u2264-refl tv1 tv2 ((relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv)) v1 eqv1\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2 , eqv2 , final-v = v2 , eqv2 , {!final-v!}\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n--   (tvv : relV \u03c3 tv1 tv2 (suc n))\n--   {v1} \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n--\n\n-- -- fundamental lemma of logical relations.\n-- fundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\n-- fundamentalV x zero _ _ _ _ ()\n-- fundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- fundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\n\n-- fundamental-aux : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n--   eval t \u03c11 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 eval t \u03c12 (suc n) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = sv2 , refl , sveq , svv'\n-- --\n-- -- |\n-- fundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- -- XXX trivial case for constants.\n-- fundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\n-- fundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\n-- fundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n--   closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | eval t \u03c12 n | fundamental-aux t n \u03c11 \u03c12 \u03c1\u03c1 tv1 teq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | (.sv2 , refl , seq2' , svv') | (Done tv2) | (.tv2 , refl , teq2' , tvv') = relV-apply n svv' tvv' t\u03c11\u2193v1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- -- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = {!!}\n-- -- | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\n\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n-- --... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- -- TODO: match sv2 before matching on fundamental s.\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- -- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n-- --   where\n-- --     v2 : Val \u03c4\n-- --     v2 = {!!}\n-- --     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n-- --     -- t\u03c12\u2193v2 = ?\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n\nopen import Data.Fin.Properties\n\nltn : \u2200 n j \u2192 (n \u2115-\u2115 j) \u2264 suc n\nltn n Fin.zero = \u2264-step \u2264-refl\nltn zero (Fin.suc ())\nltn (suc n) (Fin.suc j) = \u2264-step (ltn n j)\n\nltn2 : \u2200 n j \u2192 (n \u2115-\u2115 Fin.suc j) \u2264 n\nltn2 zero ()\nltn2 (suc n) j = ltn n j\n\nfundamentalV2 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 (var x) (var x) \u03c11 \u03c12 n\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc ()) x\u2081\nfundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1' Fin.zero ()\nfundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = v2 , refl , relV2-mono _ _ (ltn n j) _ v1 v2 vv\nfundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = {!!}\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- --\n\n-- fundamentalV2 x zero _ _ _ _ ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nrelV-nat-refl : \u2200 nv n \u2192 relV2 nat nv nv n\nrelV-nat-refl (intV nv) n = refl\nmodule Bar where\n  relV2-apply-go : \u2200 {\u03c3} \u03c4 sv1 sv2 tv1 tv2\n    n j\n    (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (n \u2115-\u2115 Fin.suc j))\n    (tvv : relV2 \u03c3 tv1 tv2 (n \u2115-\u2115 Fin.suc j))\n    v1 \u2192\n    (eqv1 : apply sv1 tv1 (n \u2115-\u2115 Fin.suc j) \u2261 Done v1) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (n \u2115-\u2115 Fin.suc j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 Fin.suc j))\n  relV2-apply-go \u03c4 (closure st1 \u03c11) sv2@(closure st2 \u03c12) tv1 tv2 n j svv tvv v1 eqv1 = res --  with svv (n \u2115-\u2115 Fin.suc j) \u2264-refl tv1 tv2 tvv v1\n  -- ... | r = {!r nj eqv1'!}\n    where\n      nj0 = n \u2115-\u2115 Fin.suc j\n      nj : Fin (suc (n \u2115-\u2115 Fin.suc j))\n      nj = F.from\u2115 nj0\n      eqv1' : eval st1 (tv1 \u2022 \u03c11) (F.to\u2115 nj) \u2261 Done v1\n      eqv1' rewrite to-from nj0 = eqv1\n      res : \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (n \u2115-\u2115 Fin.suc j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 Fin.suc j))\n      res rewrite to-from nj0 = {! svv (n \u2115-\u2115 Fin.suc j) \u2264-refl tv1 tv2 tvv v1 nj eqv1'!}\n  -- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 n j svv tvv v1 eqv1 with svv _ \u2264-refl tv1 tv2 tvv v1 (n \u2115- Fin.suc j) eqv1\n  --... | v2 , eqv2 , final-v = {! !}\n\nrelV2-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n  n j\n  (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (n \u2115-\u2115 Fin.suc j))\n  (tvv : relV2 \u03c3 tv1 tv2 (n \u2115-\u2115 Fin.suc j))\n  v1 \u2192\n  (eqv1 : apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 Fin.suc j))\nrelV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 n j svv tvv v1 eqv1 with svv _ \u2264-refl tv1 tv2 tvv v1\n... | r = {!r !}\n-- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc zero) Fin.zero tt tt v1 eqv1 = {!CAN'T BE FILLED!}\n-- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc (suc n)) Fin.zero svv tvv v1 eqv1 = {!!}\n-- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) (Fin.suc j) svv tvv v1 eqv1 = {!!}\n\nrelV2-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n  n j\n  (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (n \u2115-\u2115 Fin.suc j))\n  (tvv : relV2 \u03c3 tv1 tv2 (n \u2115-\u2115 Fin.suc j))\n  {v1} \u2192\n  apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 ((n \u2115-\u2115 Fin.suc j)))\nrelV2-apply n j svv tvv eqv1 = relV2-apply-go _ _ _ _ n j svv tvv _ eqv1\n\nfundamental2 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 t t \u03c11 \u03c12 n\nfundamental2 t n \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamental2 (const (lit nv)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) (Fin.suc j) refl = intV nv , refl , relV-nat-refl (intV nv) (F.to\u2115 (n \u2115- Fin.suc j))\nfundamental2 (var x) n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x n \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\nfundamental2 (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) (Fin.suc j) refl = closure t \u03c12 , refl , res-valid (n \u2115-\u2115 Fin.suc j) (ltn2 n j)\n  -- foo (F.to\u2115 (n \u2115- Fin.suc j))\n  where\n    res-valid : \u2200 nj \u2192 nj \u2264 n \u2192 relV2 _ (closure t \u03c11) (closure t \u03c12) nj\n    res-valid nj nj\u2264n k k\u2264nj v1 v2 vv v3 Fin.zero ()\n    res-valid nj nj\u2264n k k\u2264nj v1 v2 vv v3 (Fin.suc j\u2081) eq = fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c12-mono _ _ (\u2264-trans k\u2264nj nj\u2264n) _ _ _ \u03c1\u03c1) v3 (Fin.suc j\u2081) eq\n\n    -- foo : relV2 _ (closure t \u03c11) (closure t \u03c12) (F.to\u2115 (n \u2115- Fin.suc j))\n    -- foo with (F.to\u2115 (n \u2115- Fin.suc j))\n    -- foo | zero = tt\n    -- foo | (suc nj) = \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192 {!!}\n\n    -- \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192\n    -- fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264nj) _ _ _ \u03c1\u03c1)\n\n    -- foo | zero = ?\n    -- foo | (suc nj) k k\u2264nj v1 v2 vv = {!!}\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc Fin.zero) ()\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 with eval s \u03c11 (suc (F.to\u2115 j)) | inspect (eval s \u03c11) (suc (F.to\u2115 j))\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] with eval t \u03c11 (suc (F.to\u2115 j)) | inspect (eval t \u03c11) (suc (F.to\u2115 j))\nfundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | Done tv1 | [ t\u03c11\u2193tv1 ] with eval s \u03c12 (suc (F.to\u2115 j)) | fundamental2 s n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) sv1 (Fin.suc j) s\u03c11\u2193sv1\nfundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | (Done tv1) | [ t\u03c11\u2193tv1 ] | .(Done sv2) | (sv2 , refl , svv) with\n  eval t \u03c12 (suc (F.to\u2115 j)) | fundamental2 t n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) tv1 (Fin.suc j) t\u03c11\u2193tv1\n... | .(Done tv2) | (tv2 , refl , tvv) = relV2-apply n j svv tvv appst-\u03c11\u2193v1\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | Done sv1 | [ s\u03c11\u2193sv1 ] | TimeOut | [ t\u03c11\u2193tv1 ]\nfundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | TimeOut | [ s\u03c11\u2193sv1 ]\n","old_contents":"{-# OPTIONS --allow-unsolved-metas #-}\n-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by Dargaye and Leroy (2010), \"A\n-- verified framework for higher-order uncurrying optimizations\" (and a bit by\n-- \"Functional Big-Step Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c (suc n)\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\neval-const-mono : \u2200 {\u03c4} \u2192 (c : Const \u03c4) \u2192 \u2200 {v} n \u2192 evalConst c n \u2261 Done v \u2192 evalConst c (suc n) \u2261 Done v\neval-const-mono (lit v) n eq = eq\neval-mono : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 \u2200 v n \u2192 eval t \u03c1 n \u2261 Done v \u2192 eval t \u03c1 (suc n) \u2261 Done v\neval-mono t \u03c1 v zero ()\neval-mono (const c) \u03c1 v (suc n) eq = eval-const-mono c (suc n) eq\neval-mono (var x) \u03c1 v (suc n) eq = eq\neval-mono (app s t) \u03c1 v (suc n) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-mono (app s t) \u03c1 v (suc n) eq | Done sv | [ seq ] | (Done tv) | [ teq ] with eval s \u03c1 (suc n) | eval-mono s \u03c1 sv n seq | eval t \u03c1 (suc n) | eval-mono t \u03c1 tv n teq\neval-mono (app s t) \u03c1 v (suc n) eq | Done (closure ct c\u03c1) | [ seq ] | (Done tv) | [ teq ] | .(Done (closure ct c\u03c1)) | refl | .(Done tv) | refl = eval-mono ct (tv \u2022 c\u03c1) _ n eq\neval-mono (app s t) \u03c1 v (suc n) () | Done _ | [ seq ] | TimeOut | [ teq ]\neval-mono (app s t) \u03c1 v (suc n) () | TimeOut | [ seq ] | tv | [ teq ]\n-- = {!eval-mono s \u03c1 (suc n)!}\neval-mono (abs t) \u03c1 v (suc n) eq = eq\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV relV2 : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 rel\u03c12 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\nrel\u03c12 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c12 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV2 \u03c4 v1 v2 n \u00d7 rel\u03c12 \u0393 \u03c11 \u03c12 n\n\n-- relT\/relT2 doesn't ask input environments to be related, but the fundamental lemma\n-- does do so.\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT relT2 : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  (eq : eval t1 \u03c11 n \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nimport Data.Fin as F\nopen F using (Fin; _\u2115-_; _\u2115-\u2115_)\n\n-- This is closer to what's used in Dargaye and Leroy, but not the same.\n\nrelT2 {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  \u2200 (j : Fin (suc n)) \u2192\n  (eq : eval t1 \u03c11 (F.to\u2115 j) \u2261 Done v1) \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] eval t2 \u03c12 (F.to\u2115 j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 j)\n\nrelV \u03c4 v1 v2 zero = \u22a4\n-- Seems the proof for abs would go through even if here we do not step down.\n-- However, that only works as long as we use a typed language; not stepping\n-- down here, in an untyped language, gives a non-well-founded definition.\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\n-- Copy of relV, but mentioning relT2\nrelV2 (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) n =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV2 \u03c3 v1 v2 k \u2192 relT2 t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV2 nat v1 v2 n = v1 \u2261 v2\n\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\n\nopen import Data.Nat.Properties\n\nrelV2-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV2 \u03c4 v1 v2 n \u2192 relV2 \u03c4 v1 v2 m\nrelV2-mono m n m\u2264n (\u03c4 \u21d2 \u03c4\u2081) (closure t \u03c1) (closure t\u2081 \u03c1\u2081) vv = \u03bb k k\u2264m \u2192 vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\nrelV2-mono m n m\u2264n nat (intV n\u2081) (intV n\u2082) vv = vv\n-- relV2-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\n-- relV2-mono (suc m) zero () \u03c4 v1 v2 vv\n-- relV2-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\n-- relV2-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\n-- relV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\n-- relV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\n-- relV-mono (suc m) zero () \u03c4 v1 v2 vv\n-- relV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\n-- relV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\n-- rel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\n-- rel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\n-- rel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrel\u03c12-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c12 \u0393 \u03c11 \u03c12 m\nrel\u03c12-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c12-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV2-mono m n m\u2264n _ v1 v2 vv , rel\u03c12-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- True but useless.\n\n-- relV-apply-go2 : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc (suc n)))\n--   (tvv : relV \u03c3 tv1 tv2 (suc (suc n)))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) (\u2264-step \u2264-refl) tv1 tv2 (relV-mono _ _ (\u2264-step (\u2264-step \u2264-refl)) _ _ _ tvv) v1 eqv1\n--   | svv (suc (suc n)) \u2264-refl tv1 tv2 (relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv) v1 (eval-mono st1 (tv1 \u2022 \u03c11) v1 (suc n) eqv1)\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2' , eqv2' , final-v' | v2 , eqv2 , final-v with trans (sym (eval-mono st2 (tv2 \u2022 \u03c12) v2' (suc n) eqv2')) eqv2\n-- relV-apply-go2 (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | .v2 , eqv2' , final-v' | v2 , eqv2 , final-v | refl = v2 , eqv2' , final-v\n\n\n-- relV-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n--   (tvv : relV \u03c3 tv1 tv2 (suc n))\n--   v1 \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 zero svv tvv v1 ()\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1\n--   with svv (suc n) \u2264-refl tv1 tv2 ((relV-mono _ _ (\u2264-step \u2264-refl) _ _ _ tvv)) v1 eqv1\n-- relV-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) svv tvv v1 eqv1 | v2 , eqv2 , final-v = v2 , eqv2 , {!final-v!}\n\n-- relV-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n--   n\n--   (svv : relV (\u03c3 \u21d2 \u03c4) sv1 sv2 (suc n))\n--   (tvv : relV \u03c3 tv1 tv2 (suc n))\n--   {v1} \u2192\n--   apply sv1 tv1 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- relV-apply n svv tvv eqv1 = relV-apply-go _ _ _ _ n svv tvv _ eqv1\n--\n\n-- -- fundamental lemma of logical relations.\n-- fundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\n-- fundamentalV x zero _ _ _ _ ()\n-- fundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\n-- fundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\n\n-- fundamental-aux : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 (suc n)) \u2192 (v1 : Val \u03c4) \u2192\n--   eval t \u03c11 n \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (eval t \u03c12 n \u2261 Done v2 \u00d7 eval t \u03c12 (suc n) \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 (suc n))\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = sv2 , refl , sveq , svv'\n-- --\n-- -- |\n-- fundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- -- XXX trivial case for constants.\n-- fundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\n-- fundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\n-- fundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n--   closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n | inspect (eval t \u03c11) n\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | fundamental-aux s n \u03c11 \u03c12 \u03c1\u03c1 sv1 seq1 | eval t \u03c12 n | fundamental-aux t n \u03c11 \u03c12 \u03c1\u03c1 tv1 teq1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | (.sv2 , refl , seq2' , svv') | (Done tv2) | (.tv2 , refl , teq2' , tvv') = relV-apply n svv' tvv' t\u03c11\u2193v1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1 | fundamental s (suc n) \u03c11 \u03c12 \u03c1\u03c1 sv1 (eval-mono s \u03c11 sv1 n seq1)\n-- -- ... | Done sv2 | [ seq2 ] | (.sv2 , refl , svv) | (sv2' , sveq , svv') with trans (sym (eval-mono s \u03c12 sv2 n seq2)) sveq\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | (Done tv1) | [ teq1 ] | (Done sv2) | [ seq2 ] | (.sv2 , refl , svv) | (.sv2 , sveq , svv') | refl = {!!}\n-- -- | fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 teq1\n\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] | Done sv2 | [ seq2 ] | Done tv2 | (.sv2 , refl , svv) | (.tv2 , refl , tvv) = relV-apply n svv tvv t\u03c11\u2193v1\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv1 | [ seq1 ] | TimeOut | [ teq1 ]\n-- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | [ seq1 ] | tv1 | [ teq1 ]\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ seq1 ] | Done tv1 | [ teq1 ] with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 seq1\n-- --... | sv2 | [ seq2 ] | tv2 | (sv2' , s\u03c12\u2193sv2 , svv) = ?\n\n\n-- -- TODO: match sv2 before matching on fundamental s.\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with eval s \u03c12 n | inspect (eval s \u03c12) n | eval t \u03c12 n | fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut | [ teq1 ] | (sv2' , s\u03c12\u2193sv2 , svv) with fundamental t n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) tv1 {!teq1!}\n-- -- ... | (tv2' , t\u03c12\u2193tv2 , tvv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | b | [ teq1 ] | (sv2' , () , svv)\n\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | (Done tv2) | (sv2' , s\u03c12\u2193sv2 , svv) = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | (Done sv2) | [ eq2 ] | TimeOut = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | (Done tv1) | TimeOut | [ eq2 ] | tv2 = {!!}\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq1\n-- -- fundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq1 ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n-- --   where\n-- --     v2 : Val \u03c4\n-- --     v2 = {!!}\n-- --     -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n-- --     -- t\u03c12\u2193v2 = ?\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\n-- -- fundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n\nopen import Data.Fin.Properties\n\nltn : \u2200 n j \u2192 (n \u2115-\u2115 j) \u2264 suc n\nltn n Fin.zero = \u2264-step \u2264-refl\nltn zero (Fin.suc ())\nltn (suc n) (Fin.suc j) = \u2264-step (ltn n j)\n\nltn2 : \u2200 n j \u2192 (n \u2115-\u2115 Fin.suc j) \u2264 n\nltn2 zero ()\nltn2 (suc n) j = ltn n j\n\nfundamentalV2 : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 (var x) (var x) \u03c11 \u03c12 n\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\nfundamentalV2 x zero \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc ()) x\u2081\nfundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1' Fin.zero ()\nfundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = v2 , refl , relV2-mono _ _ (ltn n j) _ v1 v2 vv\nfundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 (Fin.suc j) refl = {!!}\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) v1' Fin.zero ()\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- --\n\n-- fundamentalV2 x zero _ _ _ _ ()\n-- fundamentalV2 this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\n-- fundamentalV2 (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nrelV-nat-refl : \u2200 nv n \u2192 relV2 nat nv nv n\nrelV-nat-refl (intV nv) n = refl\nmodule Bar where\n  relV2-apply-go : \u2200 {\u03c3} \u03c4 sv1 sv2 tv1 tv2\n    n j\n    (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (n \u2115-\u2115 Fin.suc j))\n    (tvv : relV2 \u03c3 tv1 tv2 (n \u2115-\u2115 Fin.suc j))\n    v1 \u2192\n    (eqv1 : apply sv1 tv1 (n \u2115-\u2115 Fin.suc j) \u2261 Done v1) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (n \u2115-\u2115 Fin.suc j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 Fin.suc j))\n  relV2-apply-go \u03c4 (closure st1 \u03c11) sv2@(closure st2 \u03c12) tv1 tv2 n j svv tvv v1 eqv1 = res --  with svv (n \u2115-\u2115 Fin.suc j) \u2264-refl tv1 tv2 tvv v1\n  -- ... | r = {!r nj eqv1'!}\n    where\n      nj0 = n \u2115-\u2115 Fin.suc j\n      nj : Fin (suc (n \u2115-\u2115 Fin.suc j))\n      nj = F.from\u2115 nj0\n      eqv1' : eval st1 (tv1 \u2022 \u03c11) (F.to\u2115 nj) \u2261 Done v1\n      eqv1' rewrite to-from nj0 = eqv1\n      res : \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (n \u2115-\u2115 Fin.suc j) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (n \u2115-\u2115 Fin.suc j))\n      res rewrite to-from nj0 = {! svv (n \u2115-\u2115 Fin.suc j) \u2264-refl tv1 tv2 tvv v1 nj eqv1'!}\n  -- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 n j svv tvv v1 eqv1 with svv _ \u2264-refl tv1 tv2 tvv v1 (n \u2115- Fin.suc j) eqv1\n  --... | v2 , eqv2 , final-v = {! !}\n\n-- relV2-apply-go : \u2200 {\u03c3 \u03c4} sv1 sv2 tv1 tv2\n--   n j\n--   (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n--   (tvv : relV2 \u03c3 tv1 tv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n--   v1 \u2192\n--   (eqv1 : apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1) \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n \u2115- Fin.suc j)))\n-- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 n j svv tvv v1 eqv1 with svv _ \u2264-refl tv1 tv2 tvv v1\n-- ... | r = {!r !}\n-- -- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc zero) Fin.zero tt tt v1 eqv1 = {!CAN'T BE FILLED!}\n-- -- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc (suc n)) Fin.zero svv tvv v1 eqv1 = {!!}\n-- -- relV2-apply-go (closure st1 \u03c11) (closure st2 \u03c12) tv1 tv2 (suc n) (Fin.suc j) svv tvv v1 eqv1 = {!!}\n\n-- relV2-apply : \u2200 {\u03c3 \u03c4 sv1 sv2 tv1 tv2}\n--   n j\n--   (svv : relV2 (\u03c3 \u21d2 \u03c4) sv1 sv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n--   (tvv : relV2 \u03c3 tv1 tv2 (F.to\u2115 (n \u2115- Fin.suc j)))\n--   {v1} \u2192\n--   apply sv1 tv1 (suc (F.to\u2115 j)) \u2261 Done v1 \u2192\n--   \u03a3[ v2 \u2208 Val \u03c4 ] (apply sv2 tv2 (suc (F.to\u2115 j)) \u2261 Done v2 \u00d7 relV2 \u03c4 v1 v2 (F.to\u2115 (n \u2115- Fin.suc j)))\n-- relV2-apply n j svv tvv eqv1 = relV2-apply-go _ _ _ _ n j svv tvv _ eqv1\n\n-- fundamental2 : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c12 \u0393 \u03c11 \u03c12 n) \u2192 relT2 t t \u03c11 \u03c12 n\n-- fundamental2 t n \u03c11 \u03c12 \u03c1\u03c1 v1 Fin.zero ()\n-- fundamental2 (const (lit nv)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) (Fin.suc j) refl = intV nv , refl , relV-nat-refl (intV nv) (F.to\u2115 (n \u2115- Fin.suc j))\n-- fundamental2 (var x) n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl = fundamentalV2 x n \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) (Fin.suc j) refl\n-- fundamental2 (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) (Fin.suc j) refl = closure t \u03c12 , refl , res-valid (F.to\u2115 (n \u2115- Fin.suc j)) (ltn2 n j)\n--   -- foo (F.to\u2115 (n \u2115- Fin.suc j))\n--   where\n--     res-valid : \u2200 nj \u2192 nj \u2264 n \u2192 relV2 _ (closure t \u03c11) (closure t \u03c12) nj\n--     res-valid nj nj\u2264n k k\u2264nj v1 v2 vv v3 Fin.zero ()\n--     res-valid nj nj\u2264n k k\u2264nj v1 v2 vv v3 (Fin.suc j\u2081) eq = fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c12-mono _ _ (\u2264-trans k\u2264nj nj\u2264n) _ _ _ \u03c1\u03c1) v3 (Fin.suc j\u2081) eq\n\n--     -- foo : relV2 _ (closure t \u03c11) (closure t \u03c12) (F.to\u2115 (n \u2115- Fin.suc j))\n--     -- foo with (F.to\u2115 (n \u2115- Fin.suc j))\n--     -- foo | zero = tt\n--     -- foo | (suc nj) = \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192 {!!}\n\n--     -- \u03bb k k\u2264nj v1 v2 vv v3 j\u2081 eq \u2192\n--     -- fundamental2 t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264nj) _ _ _ \u03c1\u03c1)\n\n--     -- foo | zero = ?\n--     -- foo | (suc nj) k k\u2264nj v1 v2 vv = {!!}\n-- fundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc Fin.zero) ()\n-- fundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 with eval s \u03c11 (suc (F.to\u2115 j)) | inspect (eval s \u03c11) (suc (F.to\u2115 j))\n-- fundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] with eval t \u03c11 (suc (F.to\u2115 j)) | inspect (eval t \u03c11) (suc (F.to\u2115 j))\n-- fundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | Done tv1 | [ t\u03c11\u2193tv1 ] with eval s \u03c12 (suc (F.to\u2115 j)) | fundamental2 s n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) sv1 (Fin.suc j) s\u03c11\u2193sv1\n-- fundamental2 (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) appst-\u03c11\u2193v1 | Done sv1 | [ s\u03c11\u2193sv1 ] | (Done tv1) | [ t\u03c11\u2193tv1 ] | .(Done sv2) | (sv2 , refl , svv) with\n--   eval t \u03c12 (suc (F.to\u2115 j)) | fundamental2 t n \u03c11 \u03c12 (rel\u03c12-mono n (suc n) (\u2264-step \u2264-refl) _ _ _ \u03c1\u03c1) tv1 (Fin.suc j) t\u03c11\u2193tv1\n-- ... | .(Done tv2) | (tv2 , refl , tvv) = relV2-apply n j svv tvv appst-\u03c11\u2193v1\n-- fundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | Done sv1 | [ s\u03c11\u2193sv1 ] | TimeOut | [ t\u03c11\u2193tv1 ]\n-- fundamental2 (app s t) (suc _) \u03c11 \u03c12 \u03c1\u03c1 v1 (Fin.suc (Fin.suc j)) () | TimeOut | [ s\u03c11\u2193sv1 ]\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"16b5c9f27197deba9c913fcb95ad7ed35a650ee0","subject":"Import Everything in README.agda, like for Agda std-lib","message":"Import Everything in README.agda, like for Agda std-lib\n\nThis way, typechecking README is a sufficient test.\n\nOld-commit-hash: 80911907efcde157dff2d2b80039bcbc2f0fd6fa\n","repos":"inc-lc\/ilc-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\n\nimport Syntax.Constant.Popl14\nimport Syntax.Context.Plotkin\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.DeltaContext\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\nimport Syntax.Language.Popl14\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Atlas\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\n\nimport Everything\n","old_contents":"module README where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * multiple calculi\n\n\nimport Syntax.Constant.Popl14\nimport Syntax.Context.Plotkin\nimport Syntax.Context.Popl14\nimport Syntax.Context\nimport Syntax.DeltaContext\nimport Syntax.Derive.Canon-Popl14\nimport Syntax.Derive.Optimized-Popl14\nimport Syntax.FreeVars.Popl14\n{- Language definition of Calc. Atlas -}\nimport Syntax.Language.Atlas\nimport Syntax.Language.Calculus\nimport Syntax.Language.Popl14\n{- Terms of a calculus described in Plotkin style\n  - types are parametric in base types\n  - terms are parametric in constants\n  This style of language description is employed in:\n  G. D. Plotkin. \"LCF considered as a programming language.\"\n  Theoretical Computer Science 5(3) pp. 223--255, 1997.\n  http:\/\/dx.doi.org\/10.1016\/0304-3975(77)90044-5 -}\nimport Syntax.Term.Plotkin\nimport Syntax.Term.Popl14\nimport Syntax.Type.Atlas\nimport Syntax.Type.Plotkin\nimport Syntax.Type.Popl14\nimport Syntax.Vars\n\nimport Denotation.Change.Popl14\n{- Correctness theorem for canonical derivation of Calc. Popl14 -}\nimport Denotation.Derive.Canon-Popl14\n{- Correctness theorem for optimized derivation of Calc. Popl14 -}\nimport Denotation.Derive.Optimized-Popl14\nimport Denotation.Environment.Popl14\nimport Denotation.Environment\nimport Denotation.Evaluation.Popl14\nimport Denotation.FreeVars.Popl14\n{- The idea of implementing a denotational specification for Calc. Popl14 -}\nimport Denotation.Implementation.Popl14\nimport Denotation.Notation\n{- Denotation-as-specification for canonical derivation of Calc. Popl14 -}\nimport Denotation.Specification.Canon-Popl14\nimport Denotation.Specification.Optimized-Popl14\nimport Denotation.Value.Popl14\nimport experimental.DecidableEq\nimport experimental.FoldableBag\nimport experimental.FoldableBagParametric\nimport experimental.NormalizationByEvaluation\nimport experimental.OrdBag\nimport experimental.Sorting\nimport Postulate.Bag-Popl14\nimport Postulate.Extensionality\nimport Property.Uniqueness\nimport Structure.Bag.Popl14\nimport Structure.Tuples\n\nimport Theorem.CongApp\nimport Theorem.EqualityUnique\nimport Theorem.Groups-Popl14\nimport Theorem.IrrelevanceUnique-Popl14\nimport Theorem.ProductUnique\nimport Theorem.ValidityUnique-Popl14\nimport UNDEFINED\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"3a4947a7dbeece349539686f366edb8460b68402","subject":"Cosmetic changes.","message":"Cosmetic changes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/ConversionRules.agda","new_file":"src\/fot\/LTC-PCF\/Data\/Nat\/Inequalities\/ConversionRules.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion rules for inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for lt it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (lt-s\u2081,\n  -- lt-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q, e.g.\n  --\n  -- proof\u2081\u2192proof\u2082 : \u2200 m n \u2192 lt-s\u2082 m n \u2192 lt-s\u2083 m n.\n\n  -- The terms lt-00, lt-0S, lt-S0, and lt-SS show the use of the\n  -- states lt-s\u2081, lt-s\u2082, ..., and the proofs associated with the\n  -- execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt.\n\n  -- Initially, the conversion rule fix-eq is applied.\n  lt-s\u2081 : D \u2192 D \u2192 D\n  lt-s\u2081 m n = lth (fix lth) \u00b7 m \u00b7 n\n\n  -- First argument application.\n  lt-s\u2082 : D \u2192 D\n  lt-s\u2082 m = lam (\u03bb n \u2192\n               if (iszero\u2081 n)\n                  then false\n                  else (if (iszero\u2081 m)\n                           then true\n                           else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)))\n\n\n  -- Second argument application.\n  lt-s\u2083 : D \u2192 D \u2192 D\n  lt-s\u2083 m n = if (iszero\u2081 n)\n                 then false\n                 else (if (iszero\u2081 m)\n                          then true\n                          else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  lt-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2084 m n b = if b\n                   then false\n                   else (if (iszero\u2081 m)\n                            then true\n                            else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 true.\n  -- It should be\n  -- lt-s\u2085 : D \u2192 D \u2192 D\n  -- lt-s\u2085 m n = false\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  lt-s\u2085 : D \u2192 D \u2192 D\n  lt-s\u2085 m n = if (iszero\u2081 m)\n                 then true\n                 else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b.\n  lt-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2086 m n b = if b\n                   then true\n                   else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n  -- Reduction iszero\u2081 m \u2261 true.\n  -- It should be\n  -- lt-s\u2087 : D \u2192 D \u2192 D\n  -- lt-s\u2087 m n = true\n  -- but we do not give a name to this step.\n\n   -- Reduction iszero\u2081 m \u2261 false.\n  lt-s\u2087 : D \u2192 D \u2192 D\n  lt-s\u2087 m n = fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ m) \u2261 m.\n  lt-s\u2088 : D \u2192 D \u2192 D\n  lt-s\u2088 m n = fix lth \u00b7 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  lt-s\u2089 : D \u2192 D \u2192 D\n  lt-s\u2089 m n = fix lth \u00b7 m \u00b7 n\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n\n    To prove the execution steps, e.g.\n\n    proof\u2083\u2192proof\u2084 : \u2200 m n \u2192 lt-s\u2083 m n \u2192 lt-s\u2084 m n)\n\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n\n    We prove (1) using\n\n    subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\n    where\n      \u2022 P is given by \u03bb t \u2192 C [m] \u2261 C [t],\n      \u2022 x \u2261 y is given m \u2261 n, and\n      \u2022 P x is given by C [m] \u2261 C [m] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 m n \u2192 fix lth \u00b7 m \u00b7 n  \u2261 lt-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u00b7 m \u00b7 n  \u2261 lth (fix lth) \u00b7 m \u00b7 n )\n                       (sym (fix-eq lth ))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 m n \u2192 lt-s\u2081 m n \u2261 lt-s\u2082 m \u00b7 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u00b7 n \u2261 lt-s\u2082 m \u00b7 n)\n                       (sym (beta lt-s\u2082 m))\n                       refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 m n \u2192 lt-s\u2082 m \u00b7 n \u2261 lt-s\u2083 m n\n  proof\u2082\u208b\u2083 m n = beta (lt-s\u2083 m) n\n\n\n  -- Reduction iszero n \u2261 b using that proof.\n  proof\u2083\u208b\u2084 : \u2200 m n b \u2192 iszero\u2081 n \u2261 b \u2192 lt-s\u2083 m n \u2261 lt-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b h = subst (\u03bb x \u2192 lt-s\u2084 m n x \u2261 lt-s\u2084 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 n \u2261 true using the conversion rule if-true.\n  proof\u2084\u208a : \u2200 m n \u2192 lt-s\u2084 m n true \u2261 false\n  proof\u2084\u208a m n = if-true false\n\n  -- Reduction of iszero\u2081 n \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2084\u208b\u2085 : \u2200 m n \u2192 lt-s\u2084 m n false \u2261 lt-s\u2085 m n\n  proof\u2084\u208b\u2085 m n = if-false (lt-s\u2085 m n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b using that proof.\n  proof\u2085\u208b\u2086 : \u2200 m n b \u2192 iszero\u2081 m \u2261 b \u2192 lt-s\u2085 m n \u2261 lt-s\u2086 m n b\n  proof\u2085\u208b\u2086 m n b h = subst (\u03bb x \u2192 lt-s\u2086 m n x \u2261 lt-s\u2086 m n b )\n                           (sym h)\n                           refl\n\n  -- Reduction of iszero\u2081 m \u2261 true using the conversion rule if-true.\n  proof\u2086\u208a : \u2200 m n \u2192 lt-s\u2086 m n true \u2261 true\n  proof\u2086\u208a m n = if-true true\n\n  -- Reduction of iszero\u2081 m \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2086\u208b\u2087 : \u2200 m n \u2192 lt-s\u2086 m n false \u2261 lt-s\u2087 m n\n  proof\u2086\u208b\u2087 m n = if-false (lt-s\u2087 m n)\n\n  -- Reduction pred (succ m) \u2261 m using the conversion rule pred-S.\n  proof\u2087\u208b\u2088 : \u2200 m n \u2192 lt-s\u2087 (succ\u2081 m) n  \u2261 lt-s\u2088 m n\n  proof\u2087\u208b\u2088 m n = subst (\u03bb x \u2192 lt-s\u2088 x n \u2261 lt-s\u2088 m n)\n                       (sym (pred-S m ))\n                       refl\n\n  -- Reduction pred (succ n) \u2261 n using the conversion rule pred-S.\n  proof\u2088\u208b\u2089 : \u2200 m n \u2192 lt-s\u2088 m (succ\u2081 n)  \u2261 lt-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = subst (\u03bb x \u2192 lt-s\u2089 m x \u2261 lt-s\u2089 m n)\n                       (sym (pred-S n ))\n                       refl\n\n------------------------------------------------------------------------------\n\nprivate\n  X\u226e0 : \u2200 n \u2192 n \u226e zero\n  X\u226e0 n = fix lth \u00b7 n \u00b7 zero  \u2261\u27e8 proof\u2080\u208b\u2081 n zero \u27e9\n          lt-s\u2081 n zero        \u2261\u27e8 proof\u2081\u208b\u2082 n zero \u27e9\n          lt-s\u2082 n \u00b7 zero      \u2261\u27e8 proof\u2082\u208b\u2083 n zero \u27e9\n          lt-s\u2083 n zero        \u2261\u27e8 proof\u2083\u208b\u2084 n zero true iszero-0 \u27e9\n          lt-s\u2084 n zero true   \u2261\u27e8 proof\u2084\u208a n zero \u27e9\n          false              \u220e\n\nlt-00 : zero \u226e zero\nlt-00 = X\u226e0 zero\n\nlt-0S : \u2200 n \u2192 zero < succ\u2081 n\nlt-0S n =\n  fix lth \u00b7 zero \u00b7 (succ\u2081 n)    \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ\u2081 n) \u27e9\n  lt-s\u2081 zero (succ\u2081 n)          \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ\u2081 n) \u27e9\n  lt-s\u2082 zero \u00b7 (succ\u2081 n)        \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ\u2081 n) \u27e9\n  lt-s\u2083 zero (succ\u2081 n)          \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 zero (succ\u2081 n) false    \u2261\u27e8 proof\u2084\u208b\u2085 zero (succ\u2081 n) \u27e9\n  lt-s\u2085 zero (succ\u2081 n)          \u2261\u27e8 proof\u2085\u208b\u2086 zero (succ\u2081 n) true iszero-0 \u27e9\n  lt-s\u2086 zero (succ\u2081 n) true     \u2261\u27e8 proof\u2086\u208a  zero (succ\u2081 n) \u27e9\n  true                         \u220e\n\nlt-S0 : \u2200 n \u2192 succ\u2081 n \u226e zero\nlt-S0 n = X\u226e0 (succ\u2081 n)\n\nlt-SS : \u2200 m n \u2192 lt (succ\u2081 m) (succ\u2081 n) \u2261 lt m n\nlt-SS m n =\n  fix lth \u00b7 (succ\u2081 m) \u00b7 (succ\u2081 n)  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2081 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2082 (succ\u2081 m) \u00b7 (succ\u2081 n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2083 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 m) (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2085 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 m) (succ\u2081 n) false (iszero-S m) \u27e9\n  lt-s\u2086 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2086\u208b\u2087 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2087 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2087\u208b\u2088 m (succ\u2081 n) \u27e9\n  lt-s\u2088 m (succ\u2081 n)                \u2261\u27e8 proof\u2088\u208b\u2089 m n \u27e9\n  lt-s\u2089 m n                        \u220e\n","old_contents":"------------------------------------------------------------------------------\n-- Conversion rules for inequalities\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule LTC-PCF.Data.Nat.Inequalities.ConversionRules where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import LTC-PCF.Base\nopen import LTC-PCF.Data.Nat.Inequalities\n\n------------------------------------------------------------------------------\n\nprivate\n\n  -- Before to prove some properties for lt it is convenient\n  -- to descompose the behavior of the function step by step.\n\n  -- Initially, we define the possible states (lt-s\u2081,\n  -- lt-s\u2082, ...). Then we write down the proof for\n  -- the execution step from the state p to the state q, e.g.\n  --\n  -- proof\u2081\u2192proof\u2082 : \u2200 m n \u2192 lt-s\u2082 m n \u2192 lt-s\u2083 m n.\n\n  -- The terms lt-00, lt-0S, lt-S0, and lt-SS show the use of the\n  -- states lt-s\u2081, lt-s\u2082, ..., and the proofs associated with the\n  -- execution steps.\n\n  ----------------------------------------------------------------------\n  -- The steps of lt.\n\n  -- Initially, the conversion rule fix-eq is applied.\n  lt-s\u2081 : D \u2192 D \u2192 D\n  lt-s\u2081 m n = lth (fix lth) \u00b7 m \u00b7 n\n\n  -- First argument application.\n  lt-s\u2082 : D \u2192 D\n  lt-s\u2082 m = lam (\u03bb n \u2192\n               if (iszero\u2081 n)\n                  then false\n                  else (if (iszero\u2081 m)\n                           then true\n                           else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)))\n\n\n  -- Second argument application.\n  lt-s\u2083 : D \u2192 D \u2192 D\n  lt-s\u2083 m n = if (iszero\u2081 n)\n                 then false\n                 else (if (iszero\u2081 m)\n                          then true\n                          else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 b.\n  lt-s\u2084 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2084 m n b = if b\n                   then false\n                   else (if (iszero\u2081 m)\n                            then true\n                            else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n))\n\n  -- Reduction iszero\u2081 n \u2261 true.\n  -- It should be\n  -- lt-s\u2085 : D \u2192 D \u2192 D\n  -- lt-s\u2085 m n = false\n  -- but we do not give a name to this step.\n\n  -- Reduction iszero\u2081 n \u2261 false.\n  lt-s\u2085 : D \u2192 D \u2192 D\n  lt-s\u2085 m n = if (iszero\u2081 m)\n                 then true\n                 else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b.\n  lt-s\u2086 : D \u2192 D \u2192 D \u2192 D\n  lt-s\u2086 m n b = if b\n                   then true\n                   else (fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n)\n\n  -- Reduction iszero\u2081 m \u2261 true.\n  -- It should be\n  -- lt-s\u2087 : D \u2192 D \u2192 D\n  -- lt-s\u2087 m n = true\n  -- but we do not give a name to this step.\n\n   -- Reduction iszero\u2081 m \u2261 false.\n  lt-s\u2087 : D \u2192 D \u2192 D\n  lt-s\u2087 m n = fix lth \u00b7 pred\u2081 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ m) \u2261 m.\n  lt-s\u2088 : D \u2192 D \u2192 D\n  lt-s\u2088 m n = fix lth \u00b7 m \u00b7 pred\u2081 n\n\n  -- Reduction pred\u2081 (succ n) \u2261 n.\n  lt-s\u2089 : D \u2192 D \u2192 D\n  lt-s\u2089 m n = fix lth \u00b7 m \u00b7 n\n\n  ----------------------------------------------------------------------\n  -- The execution steps\n\n  {-\n\n    To prove the execution steps, e.g.\n\n    proof\u2083\u2192proof\u2084 : \u2200 m n \u2192 lt-s\u2083 m n \u2192 lt-s\u2084 m n)\n\n    we usually need to prove that\n\n                         C [m] \u2261 C [n]    (1)\n\n    given that\n                             m \u2261 n,       (2)\n\n    where (2) is a conversion rule usually.\n\n    We prove (1) using\n\n    subst : \u2200 {x y} (A : D \u2192 Set) \u2192 x \u2261 y \u2192 A x \u2192 A y\n\n    where\n      \u2022 P is given by \u03bb t \u2192 C [m] \u2261 C [t],\n      \u2022 x \u2261 y is given m \u2261 n, and\n      \u2022 P x is given by C [m] \u2261 C [m] (i.e. refl).\n  -}\n\n  -- Application of the conversion rule fix-eq.\n  proof\u2080\u208b\u2081 : \u2200 m n \u2192 fix lth \u00b7 m \u00b7 n  \u2261 lt-s\u2081 m n\n  proof\u2080\u208b\u2081 m n = subst (\u03bb x \u2192 x \u00b7 m \u00b7 n  \u2261\n                              lth (fix lth) \u00b7 m \u00b7 n )\n                       (sym (fix-eq lth ))\n                       refl\n\n  -- Application of the first argument.\n  proof\u2081\u208b\u2082 : \u2200 m n \u2192 lt-s\u2081 m n \u2261 lt-s\u2082 m \u00b7 n\n  proof\u2081\u208b\u2082 m n = subst (\u03bb x \u2192 x \u00b7 n \u2261 lt-s\u2082 m \u00b7 n)\n                       (sym (beta lt-s\u2082 m))\n                       refl\n\n  -- Application of the second argument.\n  proof\u2082\u208b\u2083 : \u2200 m n \u2192 lt-s\u2082 m \u00b7 n \u2261 lt-s\u2083 m n\n  proof\u2082\u208b\u2083 m n = beta (lt-s\u2083 m) n\n\n\n  -- Reduction iszero n \u2261 b using that proof.\n  proof\u2083\u208b\u2084 : \u2200 m n b \u2192 iszero\u2081 n \u2261 b \u2192 lt-s\u2083 m n \u2261 lt-s\u2084 m n b\n  proof\u2083\u208b\u2084 m n b prf = subst (\u03bb x \u2192 lt-s\u2084 m n x \u2261 lt-s\u2084 m n b )\n                             (sym prf )\n                             refl\n\n  -- Reduction of iszero\u2081 n \u2261 true using the conversion rule if-true.\n  proof\u2084\u208a : \u2200 m n \u2192 lt-s\u2084 m n true \u2261 false\n  proof\u2084\u208a m n = if-true false\n\n  -- Reduction of iszero\u2081 n \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2084\u208b\u2085 : \u2200 m n \u2192 lt-s\u2084 m n false \u2261 lt-s\u2085 m n\n  proof\u2084\u208b\u2085 m n = if-false (lt-s\u2085 m n)\n\n\n  -- Reduction iszero\u2081 m \u2261 b using that proof.\n  proof\u2085\u208b\u2086 : \u2200 m n b \u2192 iszero\u2081 m \u2261 b \u2192 lt-s\u2085 m n \u2261 lt-s\u2086 m n b\n  proof\u2085\u208b\u2086 m n b prf = subst (\u03bb x \u2192 lt-s\u2086 m n x \u2261 lt-s\u2086 m n b )\n                             (sym prf )\n                             refl\n\n  -- Reduction of iszero\u2081 m \u2261 true using the conversion rule if-true.\n  proof\u2086\u208a : \u2200 m n \u2192 lt-s\u2086 m n true \u2261 true\n  proof\u2086\u208a m n = if-true true\n\n  -- Reduction of iszero\u2081 m \u2261 false ... using the conversion rule\n  -- if-false.\n  proof\u2086\u208b\u2087 : \u2200 m n \u2192 lt-s\u2086 m n false \u2261 lt-s\u2087 m n\n  proof\u2086\u208b\u2087 m n = if-false (lt-s\u2087 m n)\n\n  -- Reduction pred (succ m) \u2261 m using the conversion rule pred-S.\n  proof\u2087\u208b\u2088 : \u2200 m n \u2192 lt-s\u2087 (succ\u2081 m) n  \u2261 lt-s\u2088 m n\n  proof\u2087\u208b\u2088 m n = subst (\u03bb x \u2192 lt-s\u2088 x n \u2261 lt-s\u2088 m n)\n                       (sym (pred-S m ))\n                       refl\n\n  -- Reduction pred (succ n) \u2261 n using the conversion rule pred-S.\n  proof\u2088\u208b\u2089 : \u2200 m n \u2192 lt-s\u2088 m (succ\u2081 n)  \u2261 lt-s\u2089 m n\n  proof\u2088\u208b\u2089 m n = subst (\u03bb x \u2192 lt-s\u2089 m x \u2261 lt-s\u2089 m n)\n                       (sym (pred-S n ))\n                       refl\n\n------------------------------------------------------------------------------\n\nprivate\n  X\u226e0 : \u2200 n \u2192 n \u226e zero\n  X\u226e0 n = fix lth \u00b7 n \u00b7 zero  \u2261\u27e8 proof\u2080\u208b\u2081 n zero \u27e9\n          lt-s\u2081 n zero        \u2261\u27e8 proof\u2081\u208b\u2082 n zero \u27e9\n          lt-s\u2082 n \u00b7 zero      \u2261\u27e8 proof\u2082\u208b\u2083 n zero \u27e9\n          lt-s\u2083 n zero        \u2261\u27e8 proof\u2083\u208b\u2084 n zero true iszero-0 \u27e9\n          lt-s\u2084 n zero true   \u2261\u27e8 proof\u2084\u208a n zero \u27e9\n          false              \u220e\n\nlt-00 : zero \u226e zero\nlt-00 = X\u226e0 zero\n\nlt-0S : \u2200 n \u2192 zero < succ\u2081 n\nlt-0S n =\n  fix lth \u00b7 zero \u00b7 (succ\u2081 n)    \u2261\u27e8 proof\u2080\u208b\u2081 zero (succ\u2081 n) \u27e9\n  lt-s\u2081 zero (succ\u2081 n)          \u2261\u27e8 proof\u2081\u208b\u2082 zero (succ\u2081 n) \u27e9\n  lt-s\u2082 zero \u00b7 (succ\u2081 n)        \u2261\u27e8 proof\u2082\u208b\u2083 zero (succ\u2081 n) \u27e9\n  lt-s\u2083 zero (succ\u2081 n)          \u2261\u27e8 proof\u2083\u208b\u2084 zero (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 zero (succ\u2081 n) false    \u2261\u27e8 proof\u2084\u208b\u2085 zero (succ\u2081 n) \u27e9\n  lt-s\u2085 zero (succ\u2081 n)          \u2261\u27e8 proof\u2085\u208b\u2086 zero (succ\u2081 n) true iszero-0 \u27e9\n  lt-s\u2086 zero (succ\u2081 n) true     \u2261\u27e8 proof\u2086\u208a  zero (succ\u2081 n) \u27e9\n  true                         \u220e\n\nlt-S0 : \u2200 n \u2192 succ\u2081 n \u226e zero\nlt-S0 n = X\u226e0 (succ\u2081 n)\n\nlt-SS : \u2200 m n \u2192 lt (succ\u2081 m) (succ\u2081 n) \u2261 lt m n\nlt-SS m n =\n  fix lth \u00b7 (succ\u2081 m) \u00b7 (succ\u2081 n)  \u2261\u27e8 proof\u2080\u208b\u2081 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2081 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2081\u208b\u2082 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2082 (succ\u2081 m) \u00b7 (succ\u2081 n)      \u2261\u27e8 proof\u2082\u208b\u2083 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2083 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2083\u208b\u2084 (succ\u2081 m) (succ\u2081 n) false (iszero-S n) \u27e9\n  lt-s\u2084 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2084\u208b\u2085 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2085 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2085\u208b\u2086 (succ\u2081 m) (succ\u2081 n) false (iszero-S m) \u27e9\n  lt-s\u2086 (succ\u2081 m) (succ\u2081 n) false  \u2261\u27e8 proof\u2086\u208b\u2087 (succ\u2081 m) (succ\u2081 n) \u27e9\n  lt-s\u2087 (succ\u2081 m) (succ\u2081 n)        \u2261\u27e8 proof\u2087\u208b\u2088 m (succ\u2081 n) \u27e9\n  lt-s\u2088 m (succ\u2081 n)                \u2261\u27e8 proof\u2088\u208b\u2089 m n \u27e9\n  lt-s\u2089 m n                        \u220e\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"f19d87d27799c2287a25be758f07afafeba640df","subject":"Agda ILC with nats: no more holes :)","message":"Agda ILC with nats: no more holes :)\n\nOld-commit-hash: 0cef17e2af890f459518031c44e31124303cbb49\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/A.agda","new_file":"nats\/A.agda","new_contents":"\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n\n","old_contents":"\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 (app t t\u2081) = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : as-\u0394 \u03c4\u2082 is dv\u2081 v\u2082 dv\u2082 ext-equiv-to v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      as-\u0394 \u03c4\u2082 is\n      \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n      ext-equiv-to\n      \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"456887b7b811ebbfcf89f5846339b3778163d905","subject":"Move nil-valid-change to use it in \u229e.","message":"Move nil-valid-change to use it in \u229e.\n\nWhat I learned from this project: To implement apply, you need nil.\n\nOld-commit-hash: 3af743a00cc7e7cad69e57ec30364639c1b5e928\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Validity.agda","new_file":"Parametric\/Change\/Validity.agda","new_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = ValidChange \u03c3 \u2192 Change \u03c4\n\n  before : ValidChange \u2286 \u27e6_\u27e7\n  before (cons v _ _) = v\n\n  after : ValidChange \u2286 \u27e6_\u27e7\n  after {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  nil-valid-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 ValidChange \u03c4\n  nil-valid-change \u03c4 v = cons v (v \u229f\u208d \u03c4 \u208e v) (R[v,u-v] {\u03c4} {v} {v})\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f (nil-valid-change \u03c3 v)\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u2192 g (after v) \u229f\u208d \u03c4 \u208e f (before v)\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : ValidChange \u03c3)\n    \u2192 valid {\u03c4} (f (before v)) (\u0394f v)\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after v) \u2261 f (before v) \u229e\u208d \u03c4 \u208e \u0394f v\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u2192\n    let\n      w\u2032 = after {\u03c3} w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v (before {\u03c3} w)}) \u27e9\n        v (before {\u03c3} w) \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w\n      \u220e) where open \u2261-Reasoning\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 before {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 after {\u03c4})\n","old_contents":"import Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Validity\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\n-- Changes for Calculus Popl14\n--\n-- Contents\n-- - Mutually recursive concepts: \u0394Val, validity.\n--     Under module Syntax, the corresponding concepts of\n--     \u0394Type and \u0394Context reside in separate files.\n--     Now they have to be together due to mutual recursiveness.\n-- - `diff` and `apply` on semantic values of changes:\n--     they have to be here as well because they are mutually\n--     recursive with validity.\n-- - The lemma diff-is-valid: it has to be here because it is\n--     mutually recursive with `apply`\n-- - The lemma apply-diff: it is mutually recursive with `apply`\n--     and `diff`\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Postulate.Extensionality\nopen import Data.Product hiding (map)\n\nimport Structure.Tuples as Tuples\nopen Tuples\n\nimport Base.Data.DependentList as DependentList\nopen DependentList\n\nopen import Relation.Unary using (_\u2286_)\n\nrecord Structure : Set\u2081 where\n  ----------------\n  -- Parameters --\n  ----------------\n\n  field\n    Change-base : Base \u2192 Set\n    valid-base : \u2200 {\u03b9} \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 Set\n    apply-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base\n    diff-change-base : \u2200 \u03b9 \u2192 \u27e6 \u03b9 \u27e7Base \u2192 \u27e6 \u03b9 \u27e7Base \u2192 Change-base \u03b9\n    R[v,u-v]-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 valid-base {\u03b9} v (diff-change-base \u03b9 u v)\n    v+[u-v]=u-base : \u2200 {\u03b9} {u v : \u27e6 \u03b9 \u27e7Base} \u2192 apply-change-base \u03b9 v (diff-change-base \u03b9 u v) \u2261 u\n\n  ---------------\n  -- Interface --\n  ---------------\n\n  Change : Type \u2192 Set\n  valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n\n  apply-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  diff-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n\n  infixl 6 apply-change diff-change -- as with + - in GHC.Num\n  syntax apply-change \u03c4 v dv = v \u229e\u208d \u03c4 \u208e dv\n  syntax diff-change \u03c4 u v = u \u229f\u208d \u03c4 \u208e v\n\n  -- Lemma diff-is-valid\n  R[v,u-v] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229f\u208d \u03c4 \u208e v)\n\n  -- Lemma apply-diff\n  v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192\n    v \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c4 \u208e v) \u2261 u\n\n  --------------------\n  -- Implementation --\n  --------------------\n\n  -- (Change \u03c4) is the set of changes of type \u03c4. This set is\n  -- strictly smaller than \u27e6 \u0394Type \u03c4\u27e7 if \u03c4 is a function type. In\n  -- particular, (Change (\u03c3 \u21d2 \u03c4)) is a function that accepts only\n  -- valid changes, while \u27e6 \u0394Type (\u03c3 \u21d2 \u03c4) \u27e7 accepts also invalid\n  -- changes.\n  --\n  -- Change \u03c4 is the target of the denotational specification \u27e6_\u27e7\u0394.\n  -- Detailed motivation:\n  --\n  -- https:\/\/github.com\/ps-mr\/ilc\/blob\/184a6291ac6eef80871c32d2483e3e62578baf06\/POPL14\/paper\/sec-formal.tex\n  ValidChange : Type \u2192 Set\n  ValidChange \u03c4 = Triple\n    \u27e6 \u03c4 \u27e7\n    (\u03bb _ \u2192 Change \u03c4)\n    (\u03bb v dv \u2192 valid {\u03c4} v dv)\n\n  -- Change : Type \u2192 Set\n  Change (base \u03b9) = Change-base \u03b9\n  Change (\u03c3 \u21d2 \u03c4) = ValidChange \u03c3 \u2192 Change \u03c4\n\n  before : ValidChange \u2286 \u27e6_\u27e7\n  before (cons v _ _) = v\n\n  after : ValidChange \u2286 \u27e6_\u27e7\n  after {\u03c4} (cons v dv R[v,dv]) = v \u229e\u208d \u03c4 \u208e dv\n\n  -- _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  n \u229e\u208d base \u03b9 \u208e \u0394n = apply-change-base \u03b9 n \u0394n\n  f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f = \u03bb v \u2192 f v \u229e\u208d \u03c4 \u208e \u0394f (cons v (v \u229f\u208d \u03c3 \u208e v) (R[v,u-v] {\u03c3}))\n\n  -- _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  m \u229f\u208d base \u03b9 \u208e n = diff-change-base \u03b9 m n\n  g \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e f = \u03bb v \u2192 g (after v) \u229f\u208d \u03c4 \u208e f (before v)\n\n  -- valid : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 Set\n  valid {base \u03b9} n \u0394n = valid-base {\u03b9} n \u0394n\n  valid {\u03c3 \u21d2 \u03c4} f \u0394f =\n    \u2200 (v : ValidChange \u03c3)\n    \u2192 valid {\u03c4} (f (before v)) (\u0394f v)\n    -- \u00d7 (f \u229e \u0394f) (v \u229e \u0394v) \u2261 f v \u229e \u0394f v \u0394v R[v,\u0394v]\n    \u00d7 (f \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e \u0394f) (after v) \u2261 f (before v) \u229e\u208d \u03c4 \u208e \u0394f v\n\n  -- v+[u-v]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u229e (u \u229f v) \u2261 u\n  v+[u-v]=u {base \u03b9} {u} {v} = v+[u-v]=u-base {\u03b9} {u} {v}\n  v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v} =\n      ext {-\u27e6 \u03c3 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4 \u27e7-} (\u03bb w \u2192\n        begin\n          (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\n        \u2261\u27e8 refl \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u (w \u229e\u208d \u03c3 \u208e (w \u229f\u208d \u03c3 \u208e w)) \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 cong (\u03bb hole \u2192 v w \u229e\u208d \u03c4 \u208e (u hole \u229f\u208d \u03c4 \u208e v w)) (v+[u-v]=u {\u03c3}) \u27e9\n          v w \u229e\u208d \u03c4 \u208e (u w \u229f\u208d \u03c4 \u208e v w)\n        \u2261\u27e8 v+[u-v]=u {\u03c4} \u27e9\n          u w\n        \u220e) where\n        open \u2261-Reasoning\n\n  R[v,u-v] {base \u03b9} {u} {v} = R[v,u-v]-base {\u03b9} {u} {v}\n  R[v,u-v] {\u03c3 \u21d2 \u03c4} {u} {v} = \u03bb w \u2192\n    let\n      w\u2032 = after {\u03c3} w\n    in\n      R[v,u-v] {\u03c4}\n      ,\n      (begin\n        (v \u229e\u208d \u03c3 \u21d2 \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v)) w\u2032\n      \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v+[u-v]=u {\u03c3 \u21d2 \u03c4} {u} {v}) \u27e9\n        u w\u2032\n      \u2261\u27e8 sym (v+[u-v]=u {\u03c4} {u w\u2032} {v (before {\u03c3} w)}) \u27e9\n        v (before {\u03c3} w) \u229e\u208d \u03c4 \u208e (u \u229f\u208d \u03c3 \u21d2 \u03c4 \u208e v) w\n      \u220e) where open \u2261-Reasoning\n\n  -- helpers\n  nil-valid-change : \u2200 \u03c4 \u2192 \u27e6 \u03c4 \u27e7 \u2192 ValidChange \u03c4\n  nil-valid-change \u03c4 v = cons v (v \u229f\u208d \u03c4 \u208e v) (R[v,u-v] {\u03c4} {v} {v})\n\n  -- syntactic sugar for implicit indices\n  infixl 6 _\u229e_ _\u229f_ -- as with + - in GHC.Num\n\n  _\u229e_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n  _\u229e_ {\u03c4} v dv = v \u229e\u208d \u03c4 \u208e dv\n\n  _\u229f_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 Change \u03c4\n  _\u229f_ {\u03c4} u v = u \u229f\u208d \u03c4 \u208e v\n\n  ------------------\n  -- Environments --\n  ------------------\n\n  open DependentList public using (\u2205; _\u2022_)\n  open Tuples public using (cons)\n\n  \u0394Env : Context \u2192 Set\n  \u0394Env = DependentList ValidChange\n\n  ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  ignore = map (\u03bb {\u03c4} \u2192 before {\u03c4})\n\n  update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n  update = map (\u03bb {\u03c4} \u2192 after {\u03c4})\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"c8e7e77f4929f4aaa77f55308dceda15c3d6faa9","subject":"Added 0\u00abx\u2192\u22a5, \u00ab-trans, and Sx\u00abSy\u2192x\u00aby.","message":"Added 0\u00abx\u2192\u22a5, \u00ab-trans, and Sx\u00abSy\u2192x\u00aby.\n\nIgnore-this: 37b919ef264426f65691542aec9b75e8\n\ndarcs-hash:20110215150922-3bd4e-f2de9ed726fa7c4e14ca2075eab5ff7f1d1967c8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/RelationATP.agda","new_file":"Draft\/McCarthy91\/RelationATP.agda","new_contents":"","old_contents":"","returncode":0,"stderr":"unknown","license":"mit","lang":"Agda"}
{"commit":"b711ec9e90294050d949174e0d13586c0c41ec26","subject":"Internalize Nat in example","message":"Internalize Nat in example\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  T : Set\n  T = Tag E\n\n  C : (A : El\u1d40 P) \u2192 T \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg T (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb A\n  \u2192 let D = Data.D R A\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R A t =\n  curryEl\u1d30 (Data.C R A t) (\u03bc (Data.D R A))\n    (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R =\n  icurryEl\u1d40 (Data.P R) (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n    (injUncurried R)\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115R : Data\n\u2115R = record\n  { P = End\n  ; I = \u03bb _ \u2192 End\n  ; E = \u2115E\n  ; B = \u03bb _ \u2192\n      End tt\n    , Rec tt (End tt)\n    , tt\n  }\n\n\u2115 : Set\n\u2115 = Form \u2115R\n\nzero : \u2115\nzero = inj \u2115R zeroT\n\nsuc : \u2115 \u2192 \u2115\nsuc = inj \u2115R sucT\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\n","old_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  T : Set\n  T = Tag E\n\n  C : (A : El\u1d40 P) \u2192 T \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg T (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb a \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nICurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nICurriedEl\u1d40 End X = X tt\nICurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 ICurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\nicurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 ICurriedEl\u1d40 T X\nicurryEl\u1d40 End X f = f tt\nicurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 icurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\niuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 ICurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\niuncurryEl\u1d40 End X x tt = x\niuncurryEl\u1d40 (Arg A B) X f (a , b) = iuncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb A\n  \u2192 let D = Data.D R A\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R A t =\n  curryEl\u1d30 (Data.C R A t) (\u03bc (Data.D R A))\n    (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 ICurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 let D = Data.D R p\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R =\n  icurryEl\u1d40 (Data.P R) (\u03bb p \u2192 let D = Data.D R p in CurriedEl\u1d30 D (\u03bc D))\n    (injUncurried R)\n\n----------------------------------------------------------------------\n\nFormUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 UncurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nFormUncurried R p i = \u03bc (Data.D R p) i\n\nForm : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb p\n  \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i\n  \u2192 Set\nForm R =\n  curryEl\u1d40 (Data.P R) (\u03bb p \u2192 CurriedEl\u1d40 (Data.I R p) \u03bb i \u2192 Set) \u03bb p \u2192\n  curryEl\u1d40 (Data.I R p) (\u03bb i \u2192 Set) \u03bb i \u2192\n  FormUncurried R p i\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec = Form VecR\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5a44e35722246a4542675d2ad19f2dced6fb8e9d","subject":"FromAdequate-sum: sum-ext","message":"FromAdequate-sum: sum-ext\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Explorable.agda","new_file":"lib\/Explore\/Explorable.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base\nopen import Data.Indexed\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin) renaming (zero to fzero)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product.NP renaming (map to \u00d7-map) hiding (first)\nopen import Data.Sum.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Nullary.NP\nopen import Relation.Binary\nopen import Relation.Binary.Sum using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise using (_\u00d7-cong_)\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen import HoTT\nopen Equivalences\nopen \u2261 using (_\u2261_)\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule _ {m a} {A : \u2605 a} where\n  open EM {a} m\n  gfilter-explore : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore m A \u2192 Explore m B\n  gfilter-explore f e\u1d2c = e\u1d2c >>= \u03bb x \u2192 maybe (\u03bb \u03b7 \u2192 point-explore \u03b7) empty-explore (f x)\n\n  filter-explore : (A \u2192 \ud835\udfda) \u2192 Explore m A \u2192 Explore m A\n  filter-explore p = gfilter-explore \u03bb x \u2192 [0: nothing 1: just x ] (p x)\n\n  -- monoidal exploration: explore A with a monoid M\n  explore-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-monoid e\u1d2c M = e\u1d2c \u03b5 _\u00b7_ where open Mon M renaming (_\u2219_ to _\u00b7_)\n\n  explore-endo : Explore m A \u2192 Explore m A\n  explore-endo e\u1d2c \u03b5 op f = e\u1d2c id _\u2218\u2032_ (op \u2218 f) \u03b5\n\n  explore-endo-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-endo-monoid = explore-monoid \u2218 explore-endo\n\n  explore-backward : Explore m A \u2192 Explore m A\n  explore-backward e\u1d2c \u03b5 _\u2219_ f = e\u1d2c \u03b5 (flip _\u2219_) f\n\n  -- explore-backward \u2218 explore-backward = id\n  -- (m : a comm monoid) \u2192 explore-backward m = explore m\n\nprivate\n  module FindForward {a} {A : \u2605 a} (explore : Explore a A) where\n    find? : Find? A\n    find? = explore nothing (M?._\u2223_ _)\n\n    first : Maybe A\n    first = find? just\n\n    findKey : FindKey A\n    findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule ExplorePlug {\u2113 a} {A : \u2605 a} where\n  record ExploreIndKit p (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (a \u2294 \u209b \u2113 \u2294 p) where\n    constructor mk\n    field\n      P\u03b5 : P empty-explore\n      P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081)\n      Pf : \u2200 x \u2192 P (point-explore x)\n\n  _$kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p} {e : Explore \u2113 A}\n           \u2192 ExploreInd p e \u2192 ExploreIndKit p P \u2192 P e\n  _$kit_ {P = P} ind (mk P\u03b5 P\u2219 Pf) = ind P P\u03b5 P\u2219 Pf\n\n  _,-kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p}{Q : Explore \u2113 A \u2192 \u2605 p}\n            \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\n  Pk ,-kit Qk = mk (P\u03b5 Pk , P\u03b5 Qk)\n                   (\u03bb x y \u2192 P\u2219 Pk (fst x) (fst y) , P\u2219 Qk (snd x) (snd y))\n                   (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n               where open ExploreIndKit\n\n  ExploreInd-Extra : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 _\n  ExploreInd-Extra p exp =\n    \u2200 (Q     : Explore \u2113 A \u2192 \u2605 p)\n      (Q-kit : ExploreIndKit p Q)\n      (P     : Explore \u2113 A \u2192 \u2605 p)\n      (P\u03b5    : P empty-explore)\n      (P\u2219    : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 Q e\u2080 \u2192 Q e\u2081 \u2192 P e\u2080 \u2192 P e\u2081\n               \u2192 P (merge-explore e\u2080 e\u2081))\n      (Pf    : \u2200 x \u2192 P (point-explore x))\n    \u2192 P exp\n\n  to-extra : \u2200 {p} {e : Explore \u2113 A} \u2192 ExploreInd p e \u2192 ExploreInd-Extra p e\n  to-extra e-ind Q Q-kit P P\u03b5 P\u2219 Pf =\n   snd (e-ind (Q \u00d7\u00b0 P)\n           (Q\u03b5 , P\u03b5)\n           (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n           (\u03bb x \u2192 Qf x , Pf x))\n   where open ExploreIndKit Q-kit renaming (P\u03b5 to Q\u03b5; P\u2219 to Q\u2219; Pf to Qf)\n\n  ExplorePlug : \u2200 {m} (M : Monoid \u2113 m) (e : Explore _ A) \u2192 \u2605 _\n  ExplorePlug M e = \u2200 f x \u2192 e\u2218 \u03b5 _\u2219_ f \u2219 x \u2248 e\u2218 x _\u2219_ f\n     where open Mon M\n           e\u2218 = explore-endo e\n\n  plugKit : \u2200 {m} (M : Monoid \u2113 m) \u2192 ExploreIndKit _ (ExplorePlug M)\n  plugKit M = mk (\u03bb _ \u2192 fst identity)\n                 (\u03bb Ps Ps' f x \u2192\n                    trans (\u2219-cong (! Ps _ _) refl)\n                          (trans (assoc _ _ _)\n                                 (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n                 (\u03bb x f _ \u2192 \u2219-cong (snd identity (f x)) refl)\n       where open Mon M\n\nmodule FromExplore\n    {a} {A : \u2605 a}\n    (explore : \u2200 {\u2113} \u2192 Explore \u2113 A) where\n\n  module _ {\u2113} where\n    with-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-monoid = explore-monoid explore\n\n    with\u2218 : Explore \u2113 A\n    with\u2218 = explore-endo explore\n\n    with-endo-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-endo-monoid = explore-endo-monoid explore\n\n    backward : Explore \u2113 A\n    backward = explore-backward explore\n\n    gfilter : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore \u2113 B\n    gfilter f = gfilter-explore f explore\n\n    filter : (A \u2192 \ud835\udfda) \u2192 Explore \u2113 A\n    filter p = filter-explore p explore\n\n  sum : Sum A\n  sum = explore 0 _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore 1 _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore 1\u2082 _\u2227_\n  and   = big-\u2227\n  all   = big-\u2227\n\n  big-\u2228 = explore 0\u2082 _\u2228_\n  or    = big-\u2228\n  any   = big-\u2228\n\n  big-xor = explore 0\u2082 _xor_\n\n  bin-tree : BinTree A\n  bin-tree = explore empty fork leaf\n\n  list : List A\n  list = explore List.[] _++_ List.[_]\n\n  module FindBackward = FindForward backward\n\n  findLast? : Find? A\n  findLast? = FindBackward.find?\n\n  last : Maybe A\n  last = FindBackward.first\n\n  findLastKey : FindKey A\n  findLastKey = FindBackward.findKey\n\n  open FindForward explore public\n\nmodule FromExploreInd\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (explore-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p explore)\n    where\n\n  open FromExplore explore public\n\n  module _ {\u2113 p} where\n    explore-mon-ext : ExploreMonExt {\u2113} p explore\n    explore-mon-ext m {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ _ f \u2248 s _ _ g) refl \u2219-cong f\u2248\u00b0g\n      where open Mon m\n\n    explore-mono : ExploreMono {\u2113} p explore\n    explore-mono _\u2286_ z\u2286 _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n      explore-ind (\u03bb e \u2192 e _ _ f \u2286 e _ _ g) z\u2286 _\u2219-mono_ f\u2286\u00b0g\n\n    open ExplorePlug {\u2113} {a} {A}\n\n    explore\u2218-plug : (M : Monoid \u2113 \u2113) \u2192 ExplorePlug M explore\n    explore\u2218-plug M = explore-ind $kit plugKit M\n\n    module _ (M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = explore-ind (\u03bb e \u2192 \u2200 z \u2192 e \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo e z _\u2219_ f)\n                                                (fst identity) (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _))) (\u03bb _ _ \u2192 refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n    explore\u2218-ind : \u2200 (M : Monoid \u2113 \u2113) \u2192 BigOpMonInd \u2113 M (with-endo-monoid M)\n    explore\u2218-ind M P P\u03b5 P\u2219 Pf P\u2248 =\n      snd (explore-ind (\u03bb e \u2192 ExplorePlug M e \u00d7 P (\u03bb f \u2192 e id _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n                 (const (fst identity) , P\u03b5)\n                 (\u03bb {e} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {e} {s'} (fst Ps) (fst Ps')\n                                    , P\u2248 (\u03bb f \u2192 fst Ps f _) (P\u2219 (snd Ps) (snd Ps')))\n                 (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                      , P\u2248 (\u03bb f \u2192 ! snd identity _) (Pf x)))\n      where open Mon M\n\n    explore-swap : \u2200 {b} \u2192 ExploreSwap {\u2113} p explore {b}\n    explore-swap mon {e\u1d2e} e\u1d2e-\u03b5 pf f =\n      explore-ind (\u03bb e \u2192 e _ _ (e\u1d2e \u2218 f) \u2248 e\u1d2e (e _ _ \u2218 flip f))\n                  (! e\u1d2e-\u03b5)\n                  (\u03bb p q \u2192 trans (\u2219-cong p q) (! pf _ _))\n                  (\u03bb _ \u2192 refl)\n      where open Mon mon\n\n    explore-\u03b5 : Explore\u03b5 {\u2113} p explore\n    explore-\u03b5 M = explore-ind (\u03bb e \u2192 e \u03b5 _ (const \u03b5) \u2248 \u03b5)\n                              refl\n                              (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (fst identity \u03b5))\n                              (\u03bb _ \u2192 refl)\n      where open Mon M\n\n    explore-hom : ExploreHom {\u2113} p explore\n    explore-hom cm f g = explore-ind (\u03bb e \u2192 e _ _ (f \u2219\u00b0 g) \u2248 e _ _ f \u2219 e _ _ g)\n                                     (! fst identity \u03b5)\n                                     (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _))\n                                     (\u03bb _ \u2192 refl)\n      where open CMon cm\n\n    explore-lin\u02e1 : ExploreLin\u02e1 {\u2113} p explore\n    explore-lin\u02e1 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e \u03b5 _\u2219_ f) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    explore-lin\u02b3 : ExploreLin\u02b3 {\u2113} p explore\n    explore-lin\u02b3 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e \u03b5 _\u2219_ f \u25ce k) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    module ProductMonoid\n        {M : \u2605\u2080} (\u03b5\u2098 : M) (_\u2295\u2098_ : Op\u2082 M)\n        {N : \u2605\u2080} (\u03b5\u2099 : N) (_\u2295\u2099_ : Op\u2082 N)\n        where\n        \u03b5 = (\u03b5\u2098 , \u03b5\u2099)\n        _\u2295_ : Op\u2082 (M \u00d7 N)\n        (x\u2098 , x\u2099) \u2295 (y\u2098 , y\u2099) = (x\u2098 \u2295\u2098 y\u2098 , x\u2099 \u2295\u2099 y\u2099)\n\n        explore-product-monoid :\n          \u2200 f\u2098 f\u2099 \u2192 explore \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (explore \u03b5\u2098 _\u2295\u2098_ f\u2098 , explore \u03b5\u2099 _\u2295\u2099_ f\u2099)\n        explore-product-monoid f\u2098 f\u2099 =\n          explore-ind (\u03bb e \u2192 e \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (e \u03b5\u2098 _\u2295\u2098_ f\u2098 , e \u03b5\u2099 _\u2295\u2099_ f\u2099)) \u2261.refl (\u2261.ap\u2082 _\u2295_) (\u03bb _ \u2192 \u2261.refl)\n  {-\n  empty-explore:\n    \u03b5 \u2261 (\u03b5\u2098 , \u03b5\u2099) \u2713\n  point-explore (x , y):\n    < f\u2098 , f\u2099 > (x , y) \u2261 (f\u2098 x , f\u2099 y) \u2713\n  merge-explore e\u2080 e\u2081:\n    e\u2080 \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2295 e\u2081 \u03b5 _\u2295_ < f\u2098 , f\u2099 >\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099) \u2295 (e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 \u2295 e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099 \u2295 e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n  -}\n\n  module _ {\u2113} where\n    reify : Reify {\u2113} explore\n    reify = explore-ind (\u03bb e\u1d2c \u2192 \u03a0\u1d49 e\u1d2c _) _ _,_\n\n    unfocus : Unfocus {\u2113} explore\n    unfocus = explore-ind Unfocus (\u03bb{ (lift ()) }) (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n    module _ {\u2113\u1d63 a\u1d63} {A\u1d63 : A \u2192 A \u2192 \u2605 a\u1d63}\n             (A\u1d63-refl : Reflexive A\u1d63) where\n      \u27e6explore\u27e7 : \u27e6Explore\u27e7 \u2113\u1d63 A\u1d63 (explore {\u2113}) (explore {\u2113})\n      \u27e6explore\u27e7 M\u1d63 z\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb e \u2192 M\u1d63 (e _ _ _) (e _ _ _)) z\u1d63 (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n    explore-ext : ExploreExt {\u2113} explore\n    explore-ext \u03b5 op = explore-ind (\u03bb e \u2192 e _ _ _ \u2261 e _ _ _) \u2261.refl (\u2261.ap\u2082 op)\n\n  module LiftHom\n       {m p}\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 p)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (zero : S)\n       (_+_  : Op\u2082 S)\n       (one  : T)\n       (_*_  : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom-0-1 : f zero \u2248 one)\n       (hom-+-* : \u2200 {x y} \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore zero _+_ g) \u2248 explore one _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb e \u2192 f (e zero _+_ g) \u2248 e one _*_ (f \u2218 g))\n                               hom-0-1\n                               (\u03bb p q \u2192 \u2248-trans hom-+-* (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  module _ {\u2113} {P : A \u2192 \u2605_ \u2113} where\n    open LiftHom {S = \u2605_ \u2113} {\u2605_ \u2113} (\u03bb A B \u2192 B \u2192 A) id _\u2218\u2032_\n                 (Lift \ud835\udfd8) _\u228e_ (Lift \ud835\udfd9) _\u00d7_\n                 (\u03bb f g \u2192 \u00d7-map f g) Dec P (const (no (\u03bb{ (lift ()) })))\n                 (uncurry Dec-\u228e)\n                 public renaming (lift-hom to lift-Dec)\n\n  module FromFocus {p} (focus : Focus {p} explore) where\n    Dec-\u03a3 : \u2200 {P} \u2192 \u03a0 A (Dec \u2218 P) \u2192 Dec (\u03a3 A P)\n    Dec-\u03a3 = map-Dec unfocus focus \u2218 lift-Dec \u2218 reify\n\n  lift-hom-\u2261 :\n      \u2200 {m} {S T : \u2605 m}\n        (zero : S)\n        (_+_  : Op\u2082 S)\n        (one  : T)\n        (_*_  : Op\u2082 T)\n        (f    : S \u2192 T)\n        (g    : A \u2192 S)\n        (hom-0-1 : f zero \u2261 one)\n        (hom-+-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore zero _+_ g) \u2261 explore one _*_ (f \u2218 g)\n  lift-hom-\u2261 z _+_ o _*_ = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans z _+_ o _*_ (\u2261.ap\u2082 _*_)\n\n  sum-ind : SumInd sum\n  sum-ind P P0 P+ Pf = explore-ind (\u03bb e \u2192 P (e 0 _+_)) P0 P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext 0 _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ z\u2264n _+-mono_\n\n  sum-swap' : SumSwap sum\n  sum-swap' {sum\u1d2e = s\u1d2e} s\u1d2e-0 hom f =\n    sum-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2261 s\u1d2e (s \u2218 flip f))\n            (! s\u1d2e-0)\n            (\u03bb p q \u2192 (ap\u2082 _+_ p q) \u2219 (! hom _ _)) (\u03bb _ \u2192 refl)\n    where open \u2261\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.ap\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sum-const : SumConst sum\n  sum-const x = sum-ext (\u03bb _ \u2192 ! snd \u2115\u00b0.*-identity x) \u2219 sum-lin (const 1) x \u2219 \u2115\u00b0.*-comm x Card\n    where open \u2261\n\n  exploreStableUnder\u2192sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  exploreStableUnder\u2192sumStableUnder SU-p = SU-p 0 _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sumStableUnder\u2192countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  sumStableUnder\u2192countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  diff-list = with-endo-monoid (List.monoid A) List.[_]\n\n  {-\n  list\u2261diff-list : list \u2261 diff-list\n  list\u2261diff-list = {!explore-endo-monoid-spec (List.monoid A) List.[_]!}\n  -}\n\n  lift-op\u2082 : \u2200 {a}{A : \u2605_ a}(op : Op\u2082 A){\u2113} \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A\n  lift-op\u2082 op (lift x) (lift y) = lift (op x y)\n\n  lift-sum : \u2200 \u2113 \u2192 Sum A\n  lift-sum \u2113 f = lower {\u2080} {\u2113} (explore (lift 0) (lift-op\u2082 _+_) (lift \u2218 f))\n\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin : \u2200 {{_ : UA}}(f : A \u2192 \u2115) \u2192 Fin (lift-sum _ f) \u2261 \u03a3\u1d49 explore (Fin \u2218 f)\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin f = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans (lift 0) (lift-op\u2082 _+_) (Lift \ud835\udfd8) _\u228e_ \u228e= (Fin \u2218 lower) (lift \u2218 f) (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id) (! Fin-\u228e-+)\n    where open \u2261\n\nmodule FromTwoExploreInd\n    {a} {A : \u2605 a}\n    {e\u1d2c : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (e\u1d2c-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2c)\n    {b} {B : \u2605 b}\n    {e\u1d2e : \u2200 {\u2113} \u2192 Explore \u2113 B}\n    (e\u1d2e-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2e)\n    where\n\n    module A = FromExploreInd e\u1d2c-ind\n    module B = FromExploreInd e\u1d2e-ind\n\n    module _ {c \u2113}(cm : CommutativeMonoid c \u2113) where\n        open CMon cm\n\n        op\u1d2c = e\u1d2c \u03b5 _\u2219_\n        op\u1d2e = e\u1d2e \u03b5 _\u2219_\n\n        -- TODO use lift-hom\n        explore-swap' : \u2200 f \u2192 op\u1d2c (op\u1d2e \u2218 f) \u2248 op\u1d2e (op\u1d2c \u2218 flip f)\n        explore-swap' = A.explore-swap m (B.explore-\u03b5 m) (B.explore-hom cm)\n\n    sum-swap : \u2200 f \u2192 A.sum (B.sum \u2218 f) \u2261 B.sum (A.sum \u2218 flip f)\n    sum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n\nmodule FromTwoAdequate-sum\n  {{_ : UA}}{{_ : FunExt}}\n  {A}{B}\n  {sum\u1d2c : Sum A}{sum\u1d2e : Sum B}\n  (open Adequacy _\u2261_)\n  (sum\u1d2c-adq : Adequate-sum sum\u1d2c)\n  (sum\u1d2e-adq : Adequate-sum sum\u1d2e) where\n\n  open \u2261\n  sumStableUnder : (p : A \u2243 B)(f : B \u2192 \u2115)\n                 \u2192 sum\u1d2c (f \u2218 \u00b7\u2192 p) \u2261 sum\u1d2e f\n  sumStableUnder p f = Fin-injective (sum\u1d2c-adq (f \u2218 \u00b7\u2192 p)\n                                      \u2219 \u03a3-fst\u2243 p _\n                                      \u2219 ! sum\u1d2e-adq f)\n\n  sumStableUnder\u2032 : (p : A \u2243 B)(f : A \u2192 \u2115)\n                 \u2192 sum\u1d2c f \u2261 sum\u1d2e (f \u2218 <\u2013 p)\n  sumStableUnder\u2032 p f = Fin-injective (sum\u1d2c-adq f\n                                      \u2219 \u03a3-fst\u2243\u2032 p _\n                                      \u2219 ! sum\u1d2e-adq (f \u2218 <\u2013 p))\n\nmodule FromAdequate-sum\n  {A}\n  {sum : Sum A}\n  (open Adequacy _\u2261_)\n  (sum-adq : Adequate-sum sum)\n  {{_ : UA}}{{_ : FunExt}}\n  where\n\n  open FromTwoAdequate-sum sum-adq sum-adq public\n  open \u2261\n\n  sum-ext : SumExt sum\n  sum-ext = ap sum \u2218 \u03bb=\n\n  private\n    count : Count A\n    count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  private\n    module M {p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) where\n      private\n\n        P = \u03bb x \u2192 p x \u2261 1\u2082\n        Q = \u03bb x \u2192 q x \u2261 1\u2082\n        \u00acP = \u03bb x \u2192 p x \u2261 0\u2082\n        \u00acQ = \u03bb x \u2192 q x \u2261 0\u2082\n\n        \u03c0 : \u03a3 A P \u2261 \u03a3 A Q\n        \u03c0 = ! \u03a3=\u2032 _ (count-\u2261 p)\n            \u2219 ! (sum-adq (\ud835\udfda\u25b9\u2115 \u2218 p))\n            \u2219 ap Fin same-count\n            \u2219 sum-adq (\ud835\udfda\u25b9\u2115 \u2218 q)\n            \u2219 \u03a3=\u2032 _ (count-\u2261 q)\n\n        lem1 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 qx \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 (not px) * \ud835\udfda\u25b9\u2115 qx\n        lem1 1\u2082 1\u2082 = \u2261.refl\n        lem1 1\u2082 0\u2082 = \u2261.refl\n        lem1 0\u2082 1\u2082 = \u2261.refl\n        lem1 0\u2082 0\u2082 = \u2261.refl\n\n        lem2 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 px \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 px * \ud835\udfda\u25b9\u2115 (not qx)\n        lem2 1\u2082 1\u2082 = \u2261.refl\n        lem2 1\u2082 0\u2082 = \u2261.refl\n        lem2 0\u2082 1\u2082 = \u2261.refl\n        lem2 0\u2082 0\u2082 = \u2261.refl\n\n        lemma1 : \u2200 px qx \u2192 (qx \u2261 1\u2082) \u2261 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 0\u2082 \u00d7 qx \u2261 1\u2082))\n        lemma1 px qx = ! Fin-\u2261-\u22611\u2082 qx\n                     \u2219 ap Fin (lem1 px qx)\n                     \u2219 ! Fin-\u228e-+\n                     \u2219 \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22610\u2082 px) (Fin-\u2261-\u22611\u2082 qx))\n\n        lemma2 : \u2200 px qx \u2192 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 1\u2082 \u00d7 qx \u2261 0\u2082)) \u2261 (px \u2261 1\u2082)\n        lemma2 px qx = ! \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22611\u2082 px) (Fin-\u2261-\u22610\u2082 qx)) \u2219 Fin-\u228e-+ \u2219 ap Fin (! lem2 px qx) \u2219 Fin-\u2261-\u22611\u2082 px\n\n        \u03c0' : (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192  P x \u00d7 \u00acQ x))\n           \u2261 (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192 \u00acP x \u00d7  Q x))\n        \u03c0' = \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n           \u2219 ! \u03a3\u228e-split\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma2 (p x) (q x))\n           \u2219 \u03c0\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma1 (p x) (q x))\n           \u2219 \u03a3\u228e-split\n           \u2219 ! \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n\n        \u03c0'' : \u03a3 A (P \u00d7\u00b0 \u00acQ) \u2261 \u03a3 A (\u00acP \u00d7\u00b0 Q)\n        \u03c0'' = Fin\u228e-injective (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u03c0'\n\n      open EquivalentSubsets \u03c0'' public\n\n  same-count\u2192iso : \u2200{p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) \u2192 p \u2261 q \u2218 M.\u03c0 {p} {q} same-count\n  same-count\u2192iso {p} {q} sc = M.prop {p} {q} sc\n\nmodule From\u27e6Explore\u27e7\n    {-a-} {A : \u2605\u2080 {- a-}}\n    {explore   : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (\u27e6explore\u27e7 : \u2200 {\u2113\u2080 \u2113\u2081} \u2113\u1d63 \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 _\u2261_ explore explore)\n    {{_ : UA}}\n    where\n  open FromExplore explore\n\n  module AlsoInFromExploreInd\n          {\u2113}(M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = \u27e6explore\u27e7 \u2080 {C} {C \u2192 C}\n                                              (\u03bb r s \u2192 \u2200 z \u2192 r \u2219 z \u2248 s z)\n                                              (fst identity)\n                                              (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _)))\n                                              (\u03bb x\u1d63 _ \u2192 \u2219-cong (reflexive (\u2261.ap f x\u1d63)) refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n  open \u2261\n  module _ (f : A \u2192 \u2115) where\n    sum\u21d2\u03a3\u1d49 : Fin (explore 0 _+_ f) \u2261 explore (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 f)\n    sum\u21d2\u03a3\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb n X \u2192 Fin n \u2261 X)\n                (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id)\n                (\u03bb p q \u2192 ! Fin-\u228e-+ \u2219 \u228e= p q)\n                (ap (Fin \u2218 f))\n\n    product\u21d2\u03a0\u1d49 : Fin (explore 1 _*_ f) \u2261 explore (Lift \ud835\udfd9) _\u00d7_ (Fin \u2218 f)\n    product\u21d2\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                    (\u03bb n X \u2192 Fin n \u2261 X)\n                    (Fin1\u2261\ud835\udfd9 \u2219 ! Lift\u2261id)\n                    (\u03bb p q \u2192 ! Fin-\u00d7-* \u2219 \u00d7= p q)\n                    (ap (Fin \u2218 f))\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    \u2713all-\u03a0\u1d49 : \u2713 (all f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    \u2713all-\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb b X \u2192 \u2713 b \u2261 X)\n                (! Lift\u2261id)\n                (\u03bb p q \u2192 \u2713-\u2227-\u00d7 _ _ \u2219 \u00d7= p q)\n                (ap (\u2713 \u2218 f))\n\n    \u2713any\u2192\u03a3\u1d49 : \u2713 (any f) \u2192 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    \u2713any\u2192\u03a3\u1d49 p = \u27e6explore\u27e7 {\u2080} {\u209b \u2080} \u2081\n                         (\u03bb b (X : \u2605\u2080) \u2192 Lift (\u2713 b) \u2192 X)\n                         (\u03bb x \u2192 lift (lower x))\n                         (\u03bb { {0\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inr (y\u1d63 z\u1d63)\n                            ; {1\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inl (x\u1d63 _) })\n                         (\u03bb x\u1d63 x \u2192 tr (\u2713 \u2218 f) x\u1d63 (lower x)) (lift p)\n\n  module FromAdequate-\u03a3\u1d49\n           (adequate-\u03a3\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-sum : Adequate-sum _\u2261_ sum\n    adequate-sum f = sum\u21d2\u03a3\u1d49 f \u2219 adequate-\u03a3\u1d49 (Fin \u2218 f)\n\n    open FromAdequate-sum adequate-sum public\n\n    adequate-any : Adequate-any -\u2192- any\n    adequate-any f e = coe (adequate-\u03a3\u1d49 (\u2713 \u2218 f)) (\u2713any\u2192\u03a3\u1d49 f e)\n\n  module FromAdequate-\u03a0\u1d49\n           (adequate-\u03a0\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-product : Adequate-product _\u2261_ product\n    adequate-product f = product\u21d2\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 (Fin \u2218 f)\n\n    adequate-all : Adequate-all _\u2261_ all\n    adequate-all f = \u2713all-\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 _\n\n    check! : (f : A \u2192 \ud835\udfda) {pf : \u2713 (all f)} \u2192 (\u2200 x \u2192 \u2713 (f x))\n    check! f {pf} = coe (adequate-all f) pf\n\n{-\nmodule ExplorableRecord where\n    record Explorable A : \u2605\u2081 where\n      constructor mk\n      field\n        explore     : Explore\u2080 A\n        explore-ind : ExploreInd\u2080 explore\n\n      open FromExploreInd explore-ind\n      field\n        adequate-sum     : Adequate-sum sum\n    --  adequate-product : AdequateProduct product\n\n      open FromExploreInd explore-ind public\n\n    open Explorable public\n\n    ExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\n    ExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\n    record Funable A : \u2605\u2082 where\n      constructor _,_\n      field\n        explorable : Explorable A\n        negative   : ExploreForFun A\n\n    module DistFun {A} (\u03bcA : Explorable A)\n                       (\u03bcA\u2192 : ExploreForFun A)\n                       {B} (\u03bcB : Explorable B){X}\n                       (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                       (0\u2032  : X)\n                       (_+_ : X \u2192 X \u2192 X)\n                       (_*_ : X \u2192 X \u2192 X) where\n\n      \u03a3\u1d2e = explore \u03bcB 0\u2032 _+_\n      \u03a0' = explore \u03bcA 0\u2032 _*_\n      \u03a3' = explore (\u03bcA\u2192 \u03bcB) 0\u2032 _+_\n\n      DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\n    DistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\n    DistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                       \u2200 _*_ \u2192 Zero _\u2248_ \u03b5 _*_ \u2192 _DistributesOver_ _\u2248_ _*_ _\u2219_ \u2192 _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                       \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ \u03b5 _\u2219_ _*_\n\n    DistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\n    DistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03bc-iso : \u2200 {A B} \u2192 (A \u2243 B) \u2192 Explorable A \u2192 Explorable B\n        \u03bc-iso {A}{B} A\u2243B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n          where\n            open \u2261\n            A\u2192B = \u2013> A\u2243B\n            ade = \u03bb f \u2192 adequate-sum \u03bcA (f \u2218 A\u2192B) \u2219 \u03a3-fst\u2243 A\u2243B _\n\n        -- I guess this could be more general\n        \u03bc-iso-preserve : \u2200 {A B} (A\u2243B : A \u2243 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2243B \u03bcA) (f \u2218 <\u2013 A\u2243B)\n        \u03bc-iso-preserve A\u2243B f \u03bcA = sum-ext \u03bcA (\u03bb x \u2192 ap f (! (<\u2013-inv-l A\u2243B x)))\n          where open \u2261\n\n          {-\n        \u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n        \u03bcLift = \u03bc-iso {!(! Lift\u2194id)!}\n          where open \u2261\n          -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"{-# OPTIONS --without-K #-}\n-- Constructions on top of exploration functions\n\nopen import Level.NP\nopen import Type hiding (\u2605)\nopen import Type.Identities\nopen import Function.NP\nopen import Function.Extensionality\nopen import Algebra.FunctionProperties.NP\nopen import Data.Two.Base\nopen import Data.Indexed\nopen import Data.Nat.NP hiding (_\u2294_)\nopen import Data.Nat.Properties\nopen import Data.Fin using (Fin) renaming (zero to fzero)\nopen import Data.Maybe.NP\nopen import Algebra\nopen import Data.Product.NP renaming (map to \u00d7-map) hiding (first)\nopen import Data.Sum.NP\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Tree.Binary\nimport Data.List as List\nopen List using (List; _++_)\nopen import Relation.Nullary.NP\nopen import Relation.Binary\nopen import Relation.Binary.Sum using (_\u228e-cong_)\nopen import Relation.Binary.Product.Pointwise using (_\u00d7-cong_)\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen import HoTT\nopen Equivalences\nopen \u2261 using (_\u2261_)\n\nopen import Explore.Core\nopen import Explore.Properties\nimport Explore.Monad as EM\n\nmodule Explore.Explorable where\n\nmodule _ {m a} {A : \u2605 a} where\n  open EM {a} m\n  gfilter-explore : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore m A \u2192 Explore m B\n  gfilter-explore f e\u1d2c = e\u1d2c >>= \u03bb x \u2192 maybe (\u03bb \u03b7 \u2192 point-explore \u03b7) empty-explore (f x)\n\n  filter-explore : (A \u2192 \ud835\udfda) \u2192 Explore m A \u2192 Explore m A\n  filter-explore p = gfilter-explore \u03bb x \u2192 [0: nothing 1: just x ] (p x)\n\n  -- monoidal exploration: explore A with a monoid M\n  explore-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-monoid e\u1d2c M = e\u1d2c \u03b5 _\u00b7_ where open Mon M renaming (_\u2219_ to _\u00b7_)\n\n  explore-endo : Explore m A \u2192 Explore m A\n  explore-endo e\u1d2c \u03b5 op f = e\u1d2c id _\u2218\u2032_ (op \u2218 f) \u03b5\n\n  explore-endo-monoid : \u2200 {\u2113} \u2192 Explore m A \u2192 ExploreMon m \u2113 A\n  explore-endo-monoid = explore-monoid \u2218 explore-endo\n\n  explore-backward : Explore m A \u2192 Explore m A\n  explore-backward e\u1d2c \u03b5 _\u2219_ f = e\u1d2c \u03b5 (flip _\u2219_) f\n\n  -- explore-backward \u2218 explore-backward = id\n  -- (m : a comm monoid) \u2192 explore-backward m = explore m\n\nprivate\n  module FindForward {a} {A : \u2605 a} (explore : Explore a A) where\n    find? : Find? A\n    find? = explore nothing (M?._\u2223_ _)\n\n    first : Maybe A\n    first = find? just\n\n    findKey : FindKey A\n    findKey pred = find? (\u03bb x \u2192 [0: nothing 1: just x ] (pred x))\n\nmodule ExplorePlug {\u2113 a} {A : \u2605 a} where\n  record ExploreIndKit p (P : Explore \u2113 A \u2192 \u2605 p) : \u2605 (a \u2294 \u209b \u2113 \u2294 p) where\n    constructor mk\n    field\n      P\u03b5 : P empty-explore\n      P\u2219 : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 P e\u2080 \u2192 P e\u2081 \u2192 P (merge-explore e\u2080 e\u2081)\n      Pf : \u2200 x \u2192 P (point-explore x)\n\n  _$kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p} {e : Explore \u2113 A}\n           \u2192 ExploreInd p e \u2192 ExploreIndKit p P \u2192 P e\n  _$kit_ {P = P} ind (mk P\u03b5 P\u2219 Pf) = ind P P\u03b5 P\u2219 Pf\n\n  _,-kit_ : \u2200 {p} {P : Explore \u2113 A \u2192 \u2605 p}{Q : Explore \u2113 A \u2192 \u2605 p}\n            \u2192 ExploreIndKit p P \u2192 ExploreIndKit p Q \u2192 ExploreIndKit p (P \u00d7\u00b0 Q)\n  Pk ,-kit Qk = mk (P\u03b5 Pk , P\u03b5 Qk)\n                   (\u03bb x y \u2192 P\u2219 Pk (fst x) (fst y) , P\u2219 Qk (snd x) (snd y))\n                   (\u03bb x \u2192 Pf Pk x , Pf Qk x)\n               where open ExploreIndKit\n\n  ExploreInd-Extra : \u2200 p \u2192 Explore \u2113 A \u2192 \u2605 _\n  ExploreInd-Extra p exp =\n    \u2200 (Q     : Explore \u2113 A \u2192 \u2605 p)\n      (Q-kit : ExploreIndKit p Q)\n      (P     : Explore \u2113 A \u2192 \u2605 p)\n      (P\u03b5    : P empty-explore)\n      (P\u2219    : \u2200 {e\u2080 e\u2081 : Explore \u2113 A} \u2192 Q e\u2080 \u2192 Q e\u2081 \u2192 P e\u2080 \u2192 P e\u2081\n               \u2192 P (merge-explore e\u2080 e\u2081))\n      (Pf    : \u2200 x \u2192 P (point-explore x))\n    \u2192 P exp\n\n  to-extra : \u2200 {p} {e : Explore \u2113 A} \u2192 ExploreInd p e \u2192 ExploreInd-Extra p e\n  to-extra e-ind Q Q-kit P P\u03b5 P\u2219 Pf =\n   snd (e-ind (Q \u00d7\u00b0 P)\n           (Q\u03b5 , P\u03b5)\n           (\u03bb { (a , b) (c , d) \u2192 Q\u2219 a c , P\u2219 a c b d })\n           (\u03bb x \u2192 Qf x , Pf x))\n   where open ExploreIndKit Q-kit renaming (P\u03b5 to Q\u03b5; P\u2219 to Q\u2219; Pf to Qf)\n\n  ExplorePlug : \u2200 {m} (M : Monoid \u2113 m) (e : Explore _ A) \u2192 \u2605 _\n  ExplorePlug M e = \u2200 f x \u2192 e\u2218 \u03b5 _\u2219_ f \u2219 x \u2248 e\u2218 x _\u2219_ f\n     where open Mon M\n           e\u2218 = explore-endo e\n\n  plugKit : \u2200 {m} (M : Monoid \u2113 m) \u2192 ExploreIndKit _ (ExplorePlug M)\n  plugKit M = mk (\u03bb _ \u2192 fst identity)\n                 (\u03bb Ps Ps' f x \u2192\n                    trans (\u2219-cong (! Ps _ _) refl)\n                          (trans (assoc _ _ _)\n                                 (trans (\u2219-cong refl (Ps' _ x)) (Ps _ _))))\n                 (\u03bb x f _ \u2192 \u2219-cong (snd identity (f x)) refl)\n       where open Mon M\n\nmodule FromExplore\n    {a} {A : \u2605 a}\n    (explore : \u2200 {\u2113} \u2192 Explore \u2113 A) where\n\n  module _ {\u2113} where\n    with-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-monoid = explore-monoid explore\n\n    with\u2218 : Explore \u2113 A\n    with\u2218 = explore-endo explore\n\n    with-endo-monoid : \u2200 {m} \u2192 ExploreMon \u2113 m A\n    with-endo-monoid = explore-endo-monoid explore\n\n    backward : Explore \u2113 A\n    backward = explore-backward explore\n\n    gfilter : \u2200 {B} \u2192 (A \u2192? B) \u2192 Explore \u2113 B\n    gfilter f = gfilter-explore f explore\n\n    filter : (A \u2192 \ud835\udfda) \u2192 Explore \u2113 A\n    filter p = filter-explore p explore\n\n  sum : Sum A\n  sum = explore 0 _+_\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  product : (A \u2192 \u2115) \u2192 \u2115\n  product = explore 1 _*_\n\n  big-\u2227 and big-\u2228 or big-xor : (A \u2192 \ud835\udfda) \u2192 \ud835\udfda\n\n  big-\u2227 = explore 1\u2082 _\u2227_\n  and   = big-\u2227\n  all   = big-\u2227\n\n  big-\u2228 = explore 0\u2082 _\u2228_\n  or    = big-\u2228\n  any   = big-\u2228\n\n  big-xor = explore 0\u2082 _xor_\n\n  bin-tree : BinTree A\n  bin-tree = explore empty fork leaf\n\n  list : List A\n  list = explore List.[] _++_ List.[_]\n\n  module FindBackward = FindForward backward\n\n  findLast? : Find? A\n  findLast? = FindBackward.find?\n\n  last : Maybe A\n  last = FindBackward.first\n\n  findLastKey : FindKey A\n  findLastKey = FindBackward.findKey\n\n  open FindForward explore public\n\nmodule FromExploreInd\n    {a} {A : \u2605 a}\n    {explore : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (explore-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p explore)\n    where\n\n  open FromExplore explore public\n\n  module _ {\u2113 p} where\n    explore-mon-ext : ExploreMonExt {\u2113} p explore\n    explore-mon-ext m {f} {g} f\u2248\u00b0g = explore-ind (\u03bb s \u2192 s _ _ f \u2248 s _ _ g) refl \u2219-cong f\u2248\u00b0g\n      where open Mon m\n\n    explore-mono : ExploreMono {\u2113} p explore\n    explore-mono _\u2286_ z\u2286 _\u2219-mono_ {f} {g} f\u2286\u00b0g =\n      explore-ind (\u03bb e \u2192 e _ _ f \u2286 e _ _ g) z\u2286 _\u2219-mono_ f\u2286\u00b0g\n\n    open ExplorePlug {\u2113} {a} {A}\n\n    explore\u2218-plug : (M : Monoid \u2113 \u2113) \u2192 ExplorePlug M explore\n    explore\u2218-plug M = explore-ind $kit plugKit M\n\n    module _ (M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = explore-ind (\u03bb e \u2192 \u2200 z \u2192 e \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo e z _\u2219_ f)\n                                                (fst identity) (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _))) (\u03bb _ _ \u2192 refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n    explore\u2218-ind : \u2200 (M : Monoid \u2113 \u2113) \u2192 BigOpMonInd \u2113 M (with-endo-monoid M)\n    explore\u2218-ind M P P\u03b5 P\u2219 Pf P\u2248 =\n      snd (explore-ind (\u03bb e \u2192 ExplorePlug M e \u00d7 P (\u03bb f \u2192 e id _\u2218\u2032_ (_\u2219_ \u2218 f) \u03b5))\n                 (const (fst identity) , P\u03b5)\n                 (\u03bb {e} {s'} Ps Ps' \u2192 ExploreIndKit.P\u2219 (plugKit M) {e} {s'} (fst Ps) (fst Ps')\n                                    , P\u2248 (\u03bb f \u2192 fst Ps f _) (P\u2219 (snd Ps) (snd Ps')))\n                 (\u03bb x \u2192 ExploreIndKit.Pf (plugKit M) x\n                      , P\u2248 (\u03bb f \u2192 ! snd identity _) (Pf x)))\n      where open Mon M\n\n    explore-swap : \u2200 {b} \u2192 ExploreSwap {\u2113} p explore {b}\n    explore-swap mon {e\u1d2e} e\u1d2e-\u03b5 pf f =\n      explore-ind (\u03bb e \u2192 e _ _ (e\u1d2e \u2218 f) \u2248 e\u1d2e (e _ _ \u2218 flip f))\n                  (! e\u1d2e-\u03b5)\n                  (\u03bb p q \u2192 trans (\u2219-cong p q) (! pf _ _))\n                  (\u03bb _ \u2192 refl)\n      where open Mon mon\n\n    explore-\u03b5 : Explore\u03b5 {\u2113} p explore\n    explore-\u03b5 M = explore-ind (\u03bb e \u2192 e \u03b5 _ (const \u03b5) \u2248 \u03b5)\n                              refl\n                              (\u03bb x\u2248\u03b5 y\u2248\u03b5 \u2192 trans (\u2219-cong x\u2248\u03b5 y\u2248\u03b5) (fst identity \u03b5))\n                              (\u03bb _ \u2192 refl)\n      where open Mon M\n\n    explore-hom : ExploreHom {\u2113} p explore\n    explore-hom cm f g = explore-ind (\u03bb e \u2192 e _ _ (f \u2219\u00b0 g) \u2248 e _ _ f \u2219 e _ _ g)\n                                     (! fst identity \u03b5)\n                                     (\u03bb p\u2080 p\u2081 \u2192 trans (\u2219-cong p\u2080 p\u2081) (\u2219-interchange _ _ _ _))\n                                     (\u03bb _ \u2192 refl)\n      where open CMon cm\n\n    explore-lin\u02e1 : ExploreLin\u02e1 {\u2113} p explore\n    explore-lin\u02e1 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 k \u25ce f x) \u2248 k \u25ce e \u03b5 _\u2219_ f) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    explore-lin\u02b3 : ExploreLin\u02b3 {\u2113} p explore\n    explore-lin\u02b3 m _\u25ce_ f k ide dist = explore-ind (\u03bb e \u2192 e \u03b5 _\u2219_ (\u03bb x \u2192 f x \u25ce k) \u2248 e \u03b5 _\u2219_ f \u25ce k) (! ide) (\u03bb x x\u2081 \u2192 trans (\u2219-cong x x\u2081) (! dist k _ _)) (\u03bb x \u2192 refl)\n      where open Mon m\n\n    module ProductMonoid\n        {M : \u2605\u2080} (\u03b5\u2098 : M) (_\u2295\u2098_ : Op\u2082 M)\n        {N : \u2605\u2080} (\u03b5\u2099 : N) (_\u2295\u2099_ : Op\u2082 N)\n        where\n        \u03b5 = (\u03b5\u2098 , \u03b5\u2099)\n        _\u2295_ : Op\u2082 (M \u00d7 N)\n        (x\u2098 , x\u2099) \u2295 (y\u2098 , y\u2099) = (x\u2098 \u2295\u2098 y\u2098 , x\u2099 \u2295\u2099 y\u2099)\n\n        explore-product-monoid :\n          \u2200 f\u2098 f\u2099 \u2192 explore \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (explore \u03b5\u2098 _\u2295\u2098_ f\u2098 , explore \u03b5\u2099 _\u2295\u2099_ f\u2099)\n        explore-product-monoid f\u2098 f\u2099 =\n          explore-ind (\u03bb e \u2192 e \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2261 (e \u03b5\u2098 _\u2295\u2098_ f\u2098 , e \u03b5\u2099 _\u2295\u2099_ f\u2099)) \u2261.refl (\u2261.ap\u2082 _\u2295_) (\u03bb _ \u2192 \u2261.refl)\n  {-\n  empty-explore:\n    \u03b5 \u2261 (\u03b5\u2098 , \u03b5\u2099) \u2713\n  point-explore (x , y):\n    < f\u2098 , f\u2099 > (x , y) \u2261 (f\u2098 x , f\u2099 y) \u2713\n  merge-explore e\u2080 e\u2081:\n    e\u2080 \u03b5 _\u2295_ < f\u2098 , f\u2099 > \u2295 e\u2081 \u03b5 _\u2295_ < f\u2098 , f\u2099 >\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099) \u2295 (e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n    \u2261\n    (e\u2080 \u03b5\u2098 _\u2295\u2098_ f\u2098 \u2295 e\u2081 \u03b5\u2098 _\u2295\u2098_ f\u2098 , e\u2080 \u03b5\u2099 _\u2295\u2099_ f\u2099 \u2295 e\u2081 \u03b5\u2099 _\u2295\u2099_ f\u2099)\n  -}\n\n  module _ {\u2113} where\n    reify : Reify {\u2113} explore\n    reify = explore-ind (\u03bb e\u1d2c \u2192 \u03a0\u1d49 e\u1d2c _) _ _,_\n\n    unfocus : Unfocus {\u2113} explore\n    unfocus = explore-ind Unfocus (\u03bb{ (lift ()) }) (\u03bb P Q \u2192 [ P , Q ]) (\u03bb \u03b7 \u2192 _,_ \u03b7)\n\n    module _ {\u2113\u1d63 a\u1d63} {A\u1d63 : A \u2192 A \u2192 \u2605 a\u1d63}\n             (A\u1d63-refl : Reflexive A\u1d63) where\n      \u27e6explore\u27e7 : \u27e6Explore\u27e7 \u2113\u1d63 A\u1d63 (explore {\u2113}) (explore {\u2113})\n      \u27e6explore\u27e7 M\u1d63 z\u1d63 \u2219\u1d63 f\u1d63 = explore-ind (\u03bb e \u2192 M\u1d63 (e _ _ _) (e _ _ _)) z\u1d63 (\u03bb \u03b7 \u2192 \u2219\u1d63 \u03b7) (\u03bb \u03b7 \u2192 f\u1d63 A\u1d63-refl)\n\n    explore-ext : ExploreExt {\u2113} explore\n    explore-ext \u03b5 op = explore-ind (\u03bb e \u2192 e _ _ _ \u2261 e _ _ _) \u2261.refl (\u2261.ap\u2082 op)\n\n  module LiftHom\n       {m p}\n       {S T : \u2605 m}\n       (_\u2248_ : T \u2192 T \u2192 \u2605 p)\n       (\u2248-refl : Reflexive _\u2248_)\n       (\u2248-trans : Transitive _\u2248_)\n       (zero : S)\n       (_+_  : Op\u2082 S)\n       (one  : T)\n       (_*_  : Op\u2082 T)\n       (\u2248-cong-* : _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_)\n       (f   : S \u2192 T)\n       (g   : A \u2192 S)\n       (hom-0-1 : f zero \u2248 one)\n       (hom-+-* : \u2200 {x y} \u2192 (f (x + y)) \u2248 (f x * f y))\n       where\n\n        lift-hom : f (explore zero _+_ g) \u2248 explore one _*_ (f \u2218 g)\n        lift-hom = explore-ind (\u03bb e \u2192 f (e zero _+_ g) \u2248 e one _*_ (f \u2218 g))\n                               hom-0-1\n                               (\u03bb p q \u2192 \u2248-trans hom-+-* (\u2248-cong-* p q))\n                               (\u03bb _ \u2192 \u2248-refl)\n\n  module _ {\u2113} {P : A \u2192 \u2605_ \u2113} where\n    open LiftHom {S = \u2605_ \u2113} {\u2605_ \u2113} (\u03bb A B \u2192 B \u2192 A) id _\u2218\u2032_\n                 (Lift \ud835\udfd8) _\u228e_ (Lift \ud835\udfd9) _\u00d7_\n                 (\u03bb f g \u2192 \u00d7-map f g) Dec P (const (no (\u03bb{ (lift ()) })))\n                 (uncurry Dec-\u228e)\n                 public renaming (lift-hom to lift-Dec)\n\n  module FromFocus {p} (focus : Focus {p} explore) where\n    Dec-\u03a3 : \u2200 {P} \u2192 \u03a0 A (Dec \u2218 P) \u2192 Dec (\u03a3 A P)\n    Dec-\u03a3 = map-Dec unfocus focus \u2218 lift-Dec \u2218 reify\n\n  lift-hom-\u2261 :\n      \u2200 {m} {S T : \u2605 m}\n        (zero : S)\n        (_+_  : Op\u2082 S)\n        (one  : T)\n        (_*_  : Op\u2082 T)\n        (f    : S \u2192 T)\n        (g    : A \u2192 S)\n        (hom-0-1 : f zero \u2261 one)\n        (hom-+-* : \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (explore zero _+_ g) \u2261 explore one _*_ (f \u2218 g)\n  lift-hom-\u2261 z _+_ o _*_ = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans z _+_ o _*_ (\u2261.ap\u2082 _*_)\n\n  sum-ind : SumInd sum\n  sum-ind P P0 P+ Pf = explore-ind (\u03bb e \u2192 P (e 0 _+_)) P0 P+ Pf\n\n  sum-ext : SumExt sum\n  sum-ext = explore-ext 0 _+_\n\n  sum-zero : SumZero sum\n  sum-zero = explore-\u03b5 \u2115+.monoid\n\n  sum-hom : SumHom sum\n  sum-hom = explore-hom \u2115\u00b0.+-commutativeMonoid\n\n  sum-mono : SumMono sum\n  sum-mono = explore-mono _\u2264_ z\u2264n _+-mono_\n\n  sum-swap' : SumSwap sum\n  sum-swap' {sum\u1d2e = s\u1d2e} s\u1d2e-0 hom f =\n    sum-ind (\u03bb s \u2192 s (s\u1d2e \u2218 f) \u2261 s\u1d2e (s \u2218 flip f))\n            (! s\u1d2e-0)\n            (\u03bb p q \u2192 (ap\u2082 _+_ p q) \u2219 (! hom _ _)) (\u03bb _ \u2192 refl)\n    where open \u2261\n\n  sum-lin : SumLin sum\n  sum-lin f zero    = sum-zero\n  sum-lin f (suc k) = \u2261.trans (sum-hom f (\u03bb x \u2192 k * f x)) (\u2261.ap\u2082 _+_ (\u2261.refl {x = sum f}) (sum-lin f k))\n\n  sum-const : SumConst sum\n  sum-const x = sum-ext (\u03bb _ \u2192 ! snd \u2115\u00b0.*-identity x) \u2219 sum-lin (const 1) x \u2219 \u2115\u00b0.*-comm x Card\n    where open \u2261\n\n  exploreStableUnder\u2192sumStableUnder : \u2200 {p} \u2192 StableUnder explore p \u2192 SumStableUnder sum p\n  exploreStableUnder\u2192sumStableUnder SU-p = SU-p 0 _+_\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (\u2261.cong \ud835\udfda\u25b9\u2115 \u2218 f\u2257g)\n\n  sumStableUnder\u2192countStableUnder : \u2200 {p} \u2192 SumStableUnder sum p \u2192 CountStableUnder count p\n  sumStableUnder\u2192countStableUnder sumSU-p f = sumSU-p (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  diff-list = with-endo-monoid (List.monoid A) List.[_]\n\n  {-\n  list\u2261diff-list : list \u2261 diff-list\n  list\u2261diff-list = {!explore-endo-monoid-spec (List.monoid A) List.[_]!}\n  -}\n\n  lift-op\u2082 : \u2200 {a}{A : \u2605_ a}(op : Op\u2082 A){\u2113} \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A \u2192 Lift {\u2113 = \u2113} A\n  lift-op\u2082 op (lift x) (lift y) = lift (op x y)\n\n  lift-sum : \u2200 \u2113 \u2192 Sum A\n  lift-sum \u2113 f = lower {\u2080} {\u2113} (explore (lift 0) (lift-op\u2082 _+_) (lift \u2218 f))\n\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin : \u2200 {{_ : UA}}(f : A \u2192 \u2115) \u2192 Fin (lift-sum _ f) \u2261 \u03a3\u1d49 explore (Fin \u2218 f)\n  Fin-lower-sum\u2261\u03a3\u1d49-Fin f = LiftHom.lift-hom _\u2261_ \u2261.refl \u2261.trans (lift 0) (lift-op\u2082 _+_) (Lift \ud835\udfd8) _\u228e_ \u228e= (Fin \u2218 lower) (lift \u2218 f) (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id) (! Fin-\u228e-+)\n    where open \u2261\n\nmodule FromTwoExploreInd\n    {a} {A : \u2605 a}\n    {e\u1d2c : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (e\u1d2c-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2c)\n    {b} {B : \u2605 b}\n    {e\u1d2e : \u2200 {\u2113} \u2192 Explore \u2113 B}\n    (e\u1d2e-ind : \u2200 {p \u2113} \u2192 ExploreInd {\u2113} p e\u1d2e)\n    where\n\n    module A = FromExploreInd e\u1d2c-ind\n    module B = FromExploreInd e\u1d2e-ind\n\n    module _ {c \u2113}(cm : CommutativeMonoid c \u2113) where\n        open CMon cm\n\n        op\u1d2c = e\u1d2c \u03b5 _\u2219_\n        op\u1d2e = e\u1d2e \u03b5 _\u2219_\n\n        -- TODO use lift-hom\n        explore-swap' : \u2200 f \u2192 op\u1d2c (op\u1d2e \u2218 f) \u2248 op\u1d2e (op\u1d2c \u2218 flip f)\n        explore-swap' = A.explore-swap m (B.explore-\u03b5 m) (B.explore-hom cm)\n\n    sum-swap : \u2200 f \u2192 A.sum (B.sum \u2218 f) \u2261 B.sum (A.sum \u2218 flip f)\n    sum-swap = explore-swap' \u2115\u00b0.+-commutativeMonoid\n\nmodule FromTwoAdequate-sum\n  {{_ : UA}}{{_ : FunExt}}\n  {A}{B}\n  {sum\u1d2c : Sum A}{sum\u1d2e : Sum B}\n  (open Adequacy _\u2261_)\n  (sum\u1d2c-adq : Adequate-sum sum\u1d2c)\n  (sum\u1d2e-adq : Adequate-sum sum\u1d2e) where\n\n  open \u2261\n  sumStableUnder : (p : A \u2243 B)(f : B \u2192 \u2115)\n                 \u2192 sum\u1d2c (f \u2218 \u00b7\u2192 p) \u2261 sum\u1d2e f\n  sumStableUnder p f = Fin-injective (sum\u1d2c-adq (f \u2218 \u00b7\u2192 p)\n                                      \u2219 \u03a3-fst\u2243 p _\n                                      \u2219 ! sum\u1d2e-adq f)\n\n  sumStableUnder\u2032 : (p : A \u2243 B)(f : A \u2192 \u2115)\n                 \u2192 sum\u1d2c f \u2261 sum\u1d2e (f \u2218 <\u2013 p)\n  sumStableUnder\u2032 p f = Fin-injective (sum\u1d2c-adq f\n                                      \u2219 \u03a3-fst\u2243\u2032 p _\n                                      \u2219 ! sum\u1d2e-adq (f \u2218 <\u2013 p))\n\nmodule FromAdequate-sum\n  {A}\n  {sum : Sum A}\n  (open Adequacy _\u2261_)\n  (sum-adq : Adequate-sum sum)\n  {{_ : UA}}{{_ : FunExt}}\n  where\n\n  open FromTwoAdequate-sum sum-adq sum-adq public\n  open \u2261\n\n  private\n    count : Count A\n    count f = sum (\ud835\udfda\u25b9\u2115 \u2218 f)\n\n  private\n    module M {p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) where\n      private\n\n        P = \u03bb x \u2192 p x \u2261 1\u2082\n        Q = \u03bb x \u2192 q x \u2261 1\u2082\n        \u00acP = \u03bb x \u2192 p x \u2261 0\u2082\n        \u00acQ = \u03bb x \u2192 q x \u2261 0\u2082\n\n        \u03c0 : \u03a3 A P \u2261 \u03a3 A Q\n        \u03c0 = ! \u03a3=\u2032 _ (count-\u2261 p)\n            \u2219 ! (sum-adq (\ud835\udfda\u25b9\u2115 \u2218 p))\n            \u2219 ap Fin same-count\n            \u2219 sum-adq (\ud835\udfda\u25b9\u2115 \u2218 q)\n            \u2219 \u03a3=\u2032 _ (count-\u2261 q)\n\n        lem1 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 qx \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 (not px) * \ud835\udfda\u25b9\u2115 qx\n        lem1 1\u2082 1\u2082 = \u2261.refl\n        lem1 1\u2082 0\u2082 = \u2261.refl\n        lem1 0\u2082 1\u2082 = \u2261.refl\n        lem1 0\u2082 0\u2082 = \u2261.refl\n\n        lem2 : \u2200 px qx \u2192 \ud835\udfda\u25b9\u2115 px \u2261 (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) + \ud835\udfda\u25b9\u2115 px * \ud835\udfda\u25b9\u2115 (not qx)\n        lem2 1\u2082 1\u2082 = \u2261.refl\n        lem2 1\u2082 0\u2082 = \u2261.refl\n        lem2 0\u2082 1\u2082 = \u2261.refl\n        lem2 0\u2082 0\u2082 = \u2261.refl\n\n        lemma1 : \u2200 px qx \u2192 (qx \u2261 1\u2082) \u2261 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 0\u2082 \u00d7 qx \u2261 1\u2082))\n        lemma1 px qx = ! Fin-\u2261-\u22611\u2082 qx\n                     \u2219 ap Fin (lem1 px qx)\n                     \u2219 ! Fin-\u228e-+\n                     \u2219 \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22610\u2082 px) (Fin-\u2261-\u22611\u2082 qx))\n\n        lemma2 : \u2200 px qx \u2192 (Fin (\ud835\udfda\u25b9\u2115 (px \u2227 qx)) \u228e (px \u2261 1\u2082 \u00d7 qx \u2261 0\u2082)) \u2261 (px \u2261 1\u2082)\n        lemma2 px qx = ! \u228e= refl (! Fin-\u00d7-* \u2219 \u00d7= (Fin-\u2261-\u22611\u2082 px) (Fin-\u2261-\u22610\u2082 qx)) \u2219 Fin-\u228e-+ \u2219 ap Fin (! lem2 px qx) \u2219 Fin-\u2261-\u22611\u2082 px\n\n        \u03c0' : (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192  P x \u00d7 \u00acQ x))\n           \u2261 (Fin (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u228e \u03a3 A (\u03bb x \u2192 \u00acP x \u00d7  Q x))\n        \u03c0' = \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n           \u2219 ! \u03a3\u228e-split\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma2 (p x) (q x))\n           \u2219 \u03c0\n           \u2219 \u03a3=\u2032 _ (\u03bb x \u2192 lemma1 (p x) (q x))\n           \u2219 \u03a3\u228e-split\n           \u2219 ! \u228e= (sum-adq (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) refl\n\n        \u03c0'' : \u03a3 A (P \u00d7\u00b0 \u00acQ) \u2261 \u03a3 A (\u00acP \u00d7\u00b0 Q)\n        \u03c0'' = Fin\u228e-injective (sum (\u03bb x \u2192 \ud835\udfda\u25b9\u2115 (p x \u2227 q x))) \u03c0'\n\n      open EquivalentSubsets \u03c0'' public\n\n  same-count\u2192iso : \u2200{p q : A \u2192 \ud835\udfda}(same-count : count p \u2261 count q) \u2192 p \u2261 q \u2218 M.\u03c0 {p} {q} same-count\n  same-count\u2192iso {p} {q} sc = M.prop {p} {q} sc\n\nmodule From\u27e6Explore\u27e7\n    {-a-} {A : \u2605\u2080 {- a-}}\n    {explore   : \u2200 {\u2113} \u2192 Explore \u2113 A}\n    (\u27e6explore\u27e7 : \u2200 {\u2113\u2080 \u2113\u2081} \u2113\u1d63 \u2192 \u27e6Explore\u27e7 {\u2113\u2080} {\u2113\u2081} \u2113\u1d63 _\u2261_ explore explore)\n    {{_ : UA}}\n    where\n  open FromExplore explore\n\n  module AlsoInFromExploreInd\n          {\u2113}(M : Monoid \u2113 \u2113)\n             (open Mon M)\n             (f : A \u2192 C)\n             where\n        explore-endo-monoid-spec\u2032 : \u2200 z \u2192 explore \u03b5 _\u2219_ f \u2219 z \u2248 explore-endo explore z _\u2219_ f\n        explore-endo-monoid-spec\u2032 = \u27e6explore\u27e7 \u2080 {C} {C \u2192 C}\n                                              (\u03bb r s \u2192 \u2200 z \u2192 r \u2219 z \u2248 s z)\n                                              (fst identity)\n                                              (\u03bb P\u2080 P\u2081 z \u2192 trans (assoc _ _ _) (trans (\u2219-cong refl (P\u2081 z)) (P\u2080 _)))\n                                              (\u03bb x\u1d63 _ \u2192 \u2219-cong (reflexive (\u2261.ap f x\u1d63)) refl)\n\n        explore-endo-monoid-spec : with-monoid M f \u2248 with-endo-monoid M f\n        explore-endo-monoid-spec = trans (! snd identity _) (explore-endo-monoid-spec\u2032 \u03b5)\n\n  open \u2261\n  module _ (f : A \u2192 \u2115) where\n    sum\u21d2\u03a3\u1d49 : Fin (explore 0 _+_ f) \u2261 explore (Lift \ud835\udfd8) _\u228e_ (Fin \u2218 f)\n    sum\u21d2\u03a3\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb n X \u2192 Fin n \u2261 X)\n                (Fin0\u2261\ud835\udfd8 \u2219 ! Lift\u2261id)\n                (\u03bb p q \u2192 ! Fin-\u228e-+ \u2219 \u228e= p q)\n                (ap (Fin \u2218 f))\n\n    product\u21d2\u03a0\u1d49 : Fin (explore 1 _*_ f) \u2261 explore (Lift \ud835\udfd9) _\u00d7_ (Fin \u2218 f)\n    product\u21d2\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                    (\u03bb n X \u2192 Fin n \u2261 X)\n                    (Fin1\u2261\ud835\udfd9 \u2219 ! Lift\u2261id)\n                    (\u03bb p q \u2192 ! Fin-\u00d7-* \u2219 \u00d7= p q)\n                    (ap (Fin \u2218 f))\n\n  module _ (f : A \u2192 \ud835\udfda) where\n    \u2713all-\u03a0\u1d49 : \u2713 (all f) \u2261 \u03a0\u1d49 explore (\u2713 \u2218 f)\n    \u2713all-\u03a0\u1d49 = \u27e6explore\u27e7 {\u2080} {\u2081} \u2081\n                (\u03bb b X \u2192 \u2713 b \u2261 X)\n                (! Lift\u2261id)\n                (\u03bb p q \u2192 \u2713-\u2227-\u00d7 _ _ \u2219 \u00d7= p q)\n                (ap (\u2713 \u2218 f))\n\n    \u2713any\u2192\u03a3\u1d49 : \u2713 (any f) \u2192 \u03a3\u1d49 explore (\u2713 \u2218 f)\n    \u2713any\u2192\u03a3\u1d49 p = \u27e6explore\u27e7 {\u2080} {\u209b \u2080} \u2081\n                         (\u03bb b (X : \u2605\u2080) \u2192 Lift (\u2713 b) \u2192 X)\n                         (\u03bb x \u2192 lift (lower x))\n                         (\u03bb { {0\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inr (y\u1d63 z\u1d63)\n                            ; {1\u2082} {x\u2081} x\u1d63 {y\u2080} {y\u2081} y\u1d63 z\u1d63 \u2192 inl (x\u1d63 _) })\n                         (\u03bb x\u1d63 x \u2192 tr (\u2713 \u2218 f) x\u1d63 (lower x)) (lift p)\n\n  module FromAdequate-\u03a3\u1d49\n           (adequate-\u03a3\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a3 {\u2113} (\u03a3\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-sum : Adequate-sum _\u2261_ sum\n    adequate-sum f = sum\u21d2\u03a3\u1d49 f \u2219 adequate-\u03a3\u1d49 (Fin \u2218 f)\n\n    open FromAdequate-sum adequate-sum public\n\n    adequate-any : Adequate-any -\u2192- any\n    adequate-any f e = coe (adequate-\u03a3\u1d49 (\u2713 \u2218 f)) (\u2713any\u2192\u03a3\u1d49 f e)\n\n  module FromAdequate-\u03a0\u1d49\n           (adequate-\u03a0\u1d49 : \u2200 {\u2113} \u2192 Adequate-\u03a0 {\u2113} (\u03a0\u1d49 explore))\n          where\n    open Adequacy\n\n    adequate-product : Adequate-product _\u2261_ product\n    adequate-product f = product\u21d2\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 (Fin \u2218 f)\n\n    adequate-all : Adequate-all _\u2261_ all\n    adequate-all f = \u2713all-\u03a0\u1d49 f \u2219 adequate-\u03a0\u1d49 _\n\n    check! : (f : A \u2192 \ud835\udfda) {pf : \u2713 (all f)} \u2192 (\u2200 x \u2192 \u2713 (f x))\n    check! f {pf} = coe (adequate-all f) pf\n\n{-\nmodule ExplorableRecord where\n    record Explorable A : \u2605\u2081 where\n      constructor mk\n      field\n        explore     : Explore\u2080 A\n        explore-ind : ExploreInd\u2080 explore\n\n      open FromExploreInd explore-ind\n      field\n        adequate-sum     : Adequate-sum sum\n    --  adequate-product : AdequateProduct product\n\n      open FromExploreInd explore-ind public\n\n    open Explorable public\n\n    ExploreForFun : \u2605\u2080 \u2192 \u2605\u2081\n    ExploreForFun A = \u2200 {X} \u2192 Explorable X \u2192 Explorable (A \u2192 X)\n\n    record Funable A : \u2605\u2082 where\n      constructor _,_\n      field\n        explorable : Explorable A\n        negative   : ExploreForFun A\n\n    module DistFun {A} (\u03bcA : Explorable A)\n                       (\u03bcA\u2192 : ExploreForFun A)\n                       {B} (\u03bcB : Explorable B){X}\n                       (_\u2248_ : X \u2192 X \u2192 \u2605 \u2080)\n                       (0\u2032  : X)\n                       (_+_ : X \u2192 X \u2192 X)\n                       (_*_ : X \u2192 X \u2192 X) where\n\n      \u03a3\u1d2e = explore \u03bcB 0\u2032 _+_\n      \u03a0' = explore \u03bcA 0\u2032 _*_\n      \u03a3' = explore (\u03bcA\u2192 \u03bcB) 0\u2032 _+_\n\n      DistFun = \u2200 f \u2192 \u03a0' (\u03a3\u1d2e \u2218 f) \u2248 \u03a3' (\u03a0' \u2218 _\u02e2_ f)\n\n    DistFun : \u2200 {A} \u2192 Explorable A \u2192 ExploreForFun A \u2192 \u2605\u2081\n    DistFun \u03bcA \u03bcA\u2192 = \u2200 {B} (\u03bcB : Explorable B) c \u2192 let open CMon {\u2080}{\u2080} c in\n                       \u2200 _*_ \u2192 Zero _\u2248_ \u03b5 _*_ \u2192 _DistributesOver_ _\u2248_ _*_ _\u2219_ \u2192 _*_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n                       \u2192 DistFun.DistFun \u03bcA \u03bcA\u2192 \u03bcB _\u2248_ \u03b5 _\u2219_ _*_\n\n    DistFunable : \u2200 {A} \u2192 Funable A \u2192 \u2605\u2081\n    DistFunable (\u03bcA , \u03bcA\u2192) = DistFun \u03bcA \u03bcA\u2192\n\n    module _ {{_ : UA}}{{_ : FunExt}} where\n        \u03bc-iso : \u2200 {A B} \u2192 (A \u2243 B) \u2192 Explorable A \u2192 Explorable B\n        \u03bc-iso {A}{B} A\u2243B \u03bcA = mk (EM.map _ A\u2192B (explore \u03bcA)) (EM.map-ind _ A\u2192B (explore-ind \u03bcA)) ade\n          where\n            open \u2261\n            A\u2192B = \u2013> A\u2243B\n            ade = \u03bb f \u2192 adequate-sum \u03bcA (f \u2218 A\u2192B) \u2219 \u03a3-fst\u2243 A\u2243B _\n\n        -- I guess this could be more general\n        \u03bc-iso-preserve : \u2200 {A B} (A\u2243B : A \u2243 B) f (\u03bcA : Explorable A) \u2192 sum \u03bcA f \u2261 sum (\u03bc-iso A\u2243B \u03bcA) (f \u2218 <\u2013 A\u2243B)\n        \u03bc-iso-preserve A\u2243B f \u03bcA = sum-ext \u03bcA (\u03bb x \u2192 ap f (! (<\u2013-inv-l A\u2243B x)))\n          where open \u2261\n\n          {-\n        \u03bcLift : \u2200 {A} \u2192 Explorable A \u2192 Explorable (Lift A)\n        \u03bcLift = \u03bc-iso {!(! Lift\u2194id)!}\n          where open \u2261\n          -}\n-- -}\n-- -}\n-- -}\n-- -}\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"11d3b05674088fdf56c115b77a31d9aef9ae6034","subject":"README.agda","message":"README.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"README.agda","new_file":"README.agda","new_contents":"module README where\n\nopen import composition-sem-sec-reduction\n\nopen import one-time-semantic-security\n\nopen import flat-funs\nopen import composable\nopen import vcomp\nopen import forkable\n\nopen import program-distance\nopen import diff\n\nopen import flipbased\nopen import flipbased-implem\n\nopen import bit-guessing-game\n\nopen import circuit\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'README.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5d95a3734811ffcaf8c9508aca53f1f46a9739ee","subject":"Add FunUniverse\/Fin\/Op.agda","message":"Add FunUniverse\/Fin\/Op.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Fin\/Op.agda","new_file":"FunUniverse\/Fin\/Op.agda","new_contents":"open import Type\nopen import Function\nopen import Data.Nat.NP using (\u2115; module \u2115\u00b0)\nopen import Data.Fin.NP as Fin using (Fin; inject+; raise; bound; free)\nopen import Data.Vec.NP using (lookup)\nopen import FunUniverse.Data\nopen import FunUniverse.Core\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Category\nopen import FunUniverse.Category.Op\nopen import FunUniverse.Fin\n\nmodule FunUniverse.Fin.Op where\n\n_\u1da0\u2192_ : (i o : \u2115) \u2192 \u2605\ni \u1da0\u2192 o = Fin o \u2192 Fin i\n\nfinOpFunU : FunUniverse \u2115\nfinOpFunU = Bits-U , _\u1da0\u2192_\n\nfinOpCat : Category _\u1da0\u2192_\nfinOpCat = op finCat\n\nopen FunUniverse finOpFunU\nopen Cong-*1 _`\u2192_\n\nfinOpLinRewiring : LinRewiring finOpFunU\nfinOpLinRewiring = mk finOpCat first (\u03bb {x} \u2192 swap {x}) (\u03bb {x} \u2192 assoc {x}) id id <_\u00d7_> (\u03bb {x} \u2192 second {x}) id id id id\n  where\n    -- NOTE that doing:\n    --   swap {A} {B} rewrite \u2115\u00b0.+-comm A B = id\n    -- would be totally wrong.\n    swap : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n    swap {A} {B} x with Fin.cmp B A x\n    swap {A} {B} ._ | Fin.bound y = raise A y\n    swap {A} {B} ._ | Fin.free  y = inject+ B y\n\n    -- Since the order of elements is preserved this is OK to rewrite\n    assoc : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n    assoc {A} {B} {C} rewrite \u2115\u00b0.+-assoc A B C = id\n\n    <_\u00d7_>  : \u2200 {A B C D} \u2192 (A `\u2192 C) \u2192 (B `\u2192 D) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n    <_\u00d7_> {A} {B} {C} {D} f g x with Fin.cmp C D x\n    <_\u00d7_> {A} {B} {C} {D} f g ._ | Fin.bound y = inject+ B (f y)\n    <_\u00d7_> {A} {B} {C} {D} f g ._ | Fin.free  y = raise A   (g y)\n\n    id\u1da0 : \u2200 {A} \u2192 A `\u2192 A\n    id\u1da0 = id\n\n    first : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u00d7 C) `\u2192 (B `\u00d7 C)\n    first f = < f \u00d7 id >\n\n    second : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n    second {A} f = < id\u1da0 {A} \u00d7 f >\n\nfinOpRewiring : Rewiring finOpFunU\nfinOpRewiring = mk finOpLinRewiring (\u03bb ()) dup (\u03bb()) (\u03bb f g \u2192 dup \u204f < f \u00d7 g >) (inject+ _) (\u03bb {x} \u2192 raise x) cong-*1 (cong-*1 \u2218 flip lookup)\n  where\n    open LinRewiring finOpLinRewiring using (<_\u00d7_>; _\u204f_)\n    dup : \u2200 {A} \u2192 A `\u2192 (A `\u00d7 A)\n    dup {A} x with Fin.cmp A A x\n    dup    ._ | Fin.bound y = y\n    dup    ._ | Fin.free  y = y\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FunUniverse\/Fin\/Op.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b79ca184ba283d286eb4742713e9deadc4487e1","subject":"Forking-lemma: definitions","message":"Forking-lemma: definitions\n","repos":"crypto-agda\/crypto-agda","old_file":"forking-lemma.agda","new_file":"forking-lemma.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Function.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Vec\nopen import Data.Maybe\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two.Base hiding (_==_)\nopen import Data.Fin.NP as Fin hiding (_+_; _-_; _\u2264_)\nopen import Data.Product.NP\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nmodule forking-lemma where\n\n_\u22651 : \u2200 {n} \u2192 Fin n \u2192 \ud835\udfda\nzero  \u22651 = 0\u2082\nsuc _ \u22651 = 1\u2082\n\ninfix 8 _\u203c_\n_\u203c_ : \u2200 {n a}{A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_\u203c_ = flip lookup\n\nreplace : \u2200 {A : Type} {q} (I : Fin q)\n            (hs hs' : Vec A q) \u2192 Vec A q\nreplace zero    hs             hs' = hs'\nreplace (suc I) (h \u2237 hs) (_ \u2237 hs') = h \u2237 replace I hs hs'\n\ntest-replace : replace (suc zero) (40 \u2237 41 \u2237 42 \u2237 []) (60 \u2237 61 \u2237 62 \u2237 []) \u2261 40 \u2237 61 \u2237 62 \u2237 []\ntest-replace = refl\n\n\u2261-prefix : \u2200 {A : Type} {q} (I : Fin (suc q))\n             (v\u2080 v\u2081 : Vec A q) \u2192 Type\n\u2261-prefix zero    _         _         = \ud835\udfd9\n\u2261-prefix (suc I) (x\u2080 \u2237 v\u2080) (x\u2081 \u2237 v\u2081) = (x\u2080 \u2261 x\u2081) \u00d7 \u2261-prefix I v\u2080 v\u2081\n\n\u2261-prefix (suc ()) [] []\n\npostulate\n  -- Pub    : Type\n  Res    : Type -- Side output\n  RndAdv : Type -- Coin\n  {-RndIG  : Type-}\n\n  -- IG : RndIG \u2192 Pub\n\n  q     : \u2115\n\n  instance\n    q\u22651   : q \u2265 1\n\n  #h     : \u2115\n  #h\u22652   : #h \u2265 2\n\nH : Type\nH = Fin #h\n\nAdv = {-(x : Pub)-}(hs : Vec H q)(\u03c1 : RndAdv) \u2192 Fin (suc q) \u00d7 Res\n\npostulate\n  A : Adv\n-- A x hs \u03c1\n-- A x hs = \u03c1 \u2190$ ...H ; A x hs \u03c1\n\n-- IO : Type \u2192 Type\n-- print : String \u2192 IO ()\n-- #r : \u2115\n-- #r\u22651 : #r \u2265 1\n-- Rnd = Fin #r\n-- random : IO Rnd\n-- run : (A : Input \u2192 Rnd \u2192 Res) \u2192 Input \u2192 IO Res\n-- run A i = do r \u2190 random\n--              return (A i r)\n\n-- Not used yet\nwell-def : Adv \u2192 Type\nwell-def A =\n  \u2200 {-x-} hs \u03c1 \u2192\n    case A {-x-} hs \u03c1 of \u03bb { (I , \u03c3) \u2192\n    \u2200 hs' \u2192 \u2261-prefix I hs hs' \u2192  A {-x-} hs' \u03c1 \u2261 (I , \u03c3) }\n\nrecord \u03a9 : Type where\n  field\n    -- rIG : RndIG\n    hs hs' : Vec H q\n    \u03c1      : RndAdv\n\nEvent = \u03a9 \u2192 \ud835\udfda\n\nmodule M (r : \u03a9) where\n  open \u03a9 r\n\n  F' : (I-1 : Fin q)(\u03c3 : Res) \u2192 Maybe (Res \u00d7 Res)\n  F' I-1 \u03c3 =\n    case I=I' \u2227 h=h' of \u03bb\n    { 1\u2082 \u2192 just (\u03c3 , \u03c3')\n    ; 0\u2082 \u2192 nothing\n    }\n    module F' where\n      res' = A (replace I-1 hs hs') \u03c1\n      I'   = fst res'\n      \u03c3'   = snd res'\n      h    = hs  \u203c I-1\n      h'   = hs' \u203c I-1\n      I    = suc I-1\n      I=I' = I == I'\n      h=h' = h == h'\n\n  F : Maybe (Res \u00d7 Res)\n  F =\n    case I of \u03bb\n    { zero      \u2192 nothing\n    ; (suc I-1) \u2192 F' I-1 \u03c3\n    }\n    module F where\n      res  = A hs \u03c1\n      I    = fst res\n      \u03c3    = snd res\n      I\u22651  = not (I == zero)\n\n-- Acceptance event for A\nacc : Event\nacc r = J \u22651\n  where open \u03a9 r\n     -- x = IG rIG\n        J = fst (A {-x-} hs \u03c1)\n\nfrk : Event\nfrk r = case M.F r of \u03bb { (just _) \u2192 1\u2082 ; nothing \u2192 0\u2082 }\n\ninstance\n    #h\u22651 : #h \u2265 1\n    #h\u22651 = \u2115\u2264.trans (s\u2264s z\u2264n) #h\u22652\n\ninfix 0 _\u2265'_\ninfixr 2 _\u2265\u27e8_\u27e9_ _\u2261\u27e8_\u27e9_\ninfix 2 _\u220e\n\npostulate\n  [0,1] : Type\n  Pr : Event \u2192 [0,1]\n  _-_ _\u00b7_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  _\u2265'_ : [0,1] \u2192 [0,1] \u2192 Type\n  1\/_ : (d : \u2115){{_ : d \u2265 1}} \u2192 [0,1]\n\n  _\u2265\u27e8_\u27e9_ : \u2200 x {y} \u2192 x \u2265' y \u2192 \u2200 {z} \u2192 y \u2265' z \u2192 x \u2265' z\n  _\u2261\u27e8_\u27e9_ : \u2200 x {y} \u2192 x \u2261  y \u2192 \u2200 {z} \u2192 y \u2265' z \u2192 x \u2265' z\n  _\u220e : \u2200 x \u2192 x \u2265' x\n\n_\/_ : [0,1] \u2192 (d : \u2115){{_ : d \u2265 1}} \u2192 [0,1]\nx \/ y = x \u00b7 1\/ y\n\n{-\nlemma3 : Pr frk \u2265' Pr acc \u00b7 ((Pr acc \/ q) - (1\/ #h))\nlemma3 = Pr frk\n       \u2261\u27e8 {!!} \u27e9\n         Pr {!\u03bb r \u2192 I=I' r \u2227 I\u22651 r!}\n       \u2265\u27e8 {!!} \u27e9\n         Pr acc \u00b7 ((Pr acc \/ q) - (1\/ #h))\n       \u220e\n  where\n    open M.F\n    -- open M.F' I-1 \u03c3\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'forking-lemma.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"994928a8def21882935ee15d443e40e6595bc3c4","subject":"Darkwing Duck!","message":"Darkwing Duck!\n\nStarting to model the One-True-Data-Declaration(TM).\n\nHere datatype parameters are encoded as first-class telescopes,\nsimilar to descriptions. Indices are a function from the\ninterpretation of a parameters telescope to an indices telescope.\n\nFirst-class parameters allow for them to be treaty specially\nwhen defining generic constructors, interpreting the parametric\nconstructor arguments as implicit arguments.\n\nSubsequent commits should extend this development with my generic\neliminator construction, where parameters also become implicit arguments\nfor the generic eliminator.\n\nAdditionally, this development makes it obvious that we also need\na generic type former, to complement our generic constructors and eliminators.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_file":"formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.DarkwingDuck where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl\u1d30 (End j) X i = j \u2261 i\nEl\u1d30 (Rec j D) X i = X j \u00d7 El\u1d30 D X i\nEl\u1d30 (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X i)\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El\u1d30 D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nrecord Data : Set where\n  field\n    P : Tel\n    I : El\u1d40 P \u2192 Tel\n    E : Enum\n    B : (A : El\u1d40 P) \u2192 Branches E (\u03bb _ \u2192 Desc (El\u1d40 (I A)))\n\n  T : Set\n  T = Tag E\n\n  C : (A : El\u1d40 P) \u2192 T \u2192 Desc (El\u1d40 (I A))\n  C A = case (\u03bb _ \u2192 Desc (El\u1d40 (I A))) (B A)\n\n  D : (A : El\u1d40 P) \u2192 Desc (El\u1d40 (I A))\n  D A = Arg T (C A)\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nUncurriedEl\u1d40 T X = (xs : El\u1d40 T) \u2192 X xs\n\nCurriedEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set) \u2192 Set\nCurriedEl\u1d40 End X = X tt\nCurriedEl\u1d40 (Arg A B) X = {a : A} \u2192 CurriedEl\u1d40 (B a) (\u03bb b \u2192 X (a , b))\n\ncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 UncurriedEl\u1d40 T X \u2192 CurriedEl\u1d40 T X\ncurryEl\u1d40 End X f = f tt\ncurryEl\u1d40 (Arg A B) X f = \u03bb {a} \u2192 curryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) (\u03bb b \u2192 f (a , b))\n\nuncurryEl\u1d40 : (T : Tel) (X : El\u1d40 T \u2192 Set)\n  \u2192 CurriedEl\u1d40 T X \u2192 UncurriedEl\u1d40 T X\nuncurryEl\u1d40 End X x tt = x\nuncurryEl\u1d40 (Arg A B) X f (a , b) = uncurryEl\u1d40 (B a) (\u03bb b \u2192 X (a , b)) f b\n\n----------------------------------------------------------------------\n\nUncurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl\u1d30 D X = \u2200{i} \u2192 El\u1d30 D X i \u2192 X i\n\nCurriedEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl\u1d30 (End i) X = X i\nCurriedEl\u1d30 (Rec i D) X = (x : X i) \u2192 CurriedEl\u1d30 D X\nCurriedEl\u1d30 (Arg A B) X = (a : A) \u2192 CurriedEl\u1d30 (B a) X\n\ncurryEl\u1d30 : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl\u1d30 D X \u2192 CurriedEl\u1d30 D X\ncurryEl\u1d30 (End i) X cn = cn refl\ncurryEl\u1d30 (Rec i D) X cn = \u03bb x \u2192 curryEl\u1d30 D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl\u1d30 (Arg A B) X cn = \u03bb a \u2192 curryEl\u1d30 (B a) X (\u03bb xs \u2192 cn (a , xs))\n\n----------------------------------------------------------------------\n\ninjUncurried : (R : Data)\n  \u2192 UncurriedEl\u1d40 (Data.P R) \u03bb A\n  \u2192 let D = Data.D R A\n  in CurriedEl\u1d30 D (\u03bc D)\ninjUncurried R A t =\n  curryEl\u1d30 (Data.C R A t) (\u03bc (Data.D R A))\n    (\u03bb xs \u2192 init (t , xs))\n\ninj : (R : Data)\n  \u2192 CurriedEl\u1d40 (Data.P R) \u03bb A\n  \u2192 let D = Data.D R A\n  in CurriedEl\u1d30 D (\u03bc D)\ninj R = curryEl\u1d40 _ _ (injUncurried R)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\nVecR : Data\nVecR = record\n  { P = Arg Set (\u03bb _ \u2192 End)\n  ; I = \u03bb _ \u2192 Arg \u2115 (\u03bb _ \u2192 End)\n  ; E = VecE\n  ; B = \u03bb A \u2192\n      End (zero , tt)\n    , Arg \u2115 (\u03bb n \u2192 Arg (proj\u2081 A) \u03bb _ \u2192 Rec (n , tt) (End (suc n , tt)))\n    , tt\n  }\n\nVec : (A : Set) \u2192 \u2115 \u2192 Set\nVec A n = \u03bc (Data.D VecR (A , tt)) (n , tt)\n\nnil : {A : Set} \u2192 Vec A zero\nnil = inj VecR nilT\n\ncons : {A : Set} (n : \u2115) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons = inj VecR consT\n\n----------------------------------------------------------------------\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formalization\/agda\/Spire\/Examples\/DarkwingDuck.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cb30fe225c68b76d51597d47cea4171ca31667fd","subject":"sha1.agda","message":"sha1.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"sha1.agda","new_file":"sha1.agda","new_contents":"-- https:\/\/upload.wikimedia.org\/wikipedia\/commons\/e\/e2\/SHA-1.svg\n-- http:\/\/www.faqs.org\/rfcs\/rfc3174.html\nopen import Type\nopen import Data.Nat\n--open import Data.Product\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule sha1\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\nopen FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\nopen FunOps funOps hiding (_\u2218_)\n\nWord : T\nWord = `Bits 32\n\nmap\u00b2\u02b7 : (`\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u00d7 Word `\u2192 Word\nmap\u00b2\u02b7 = zipWith\n\nmap\u02b7 : (`\ud835\udfda `\u2192 `\ud835\udfda) \u2192 Word `\u2192 Word\nmap\u02b7 = map\n\nlift : \u2200 {\u0393 A B} \u2192 (A `\u2192 B) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B\nlift f g = g \u204f f\n\nlift\u2082 : \u2200 {\u0393 A B C} \u2192 (A `\u00d7 B `\u2192 C) \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\nlift\u2082 op\u2082 f\u2080 f\u2081 = < f\u2080 , f\u2081 > \u204f op\u2082\n\n`not : \u2200 {\u0393} (f : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n`not = lift (map\u02b7 not)\n\ninfixr 3 _`\u2295_\n_`\u2295_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2295_ = lift\u2082 (map\u00b2\u02b7 <xor>)\n\ninfixr 3 _`\u2227_\n_`\u2227_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2227_ = lift\u2082 (map\u00b2\u02b7 <and>)\n\ninfixr 2 _`\u2228_\n_`\u2228_ : \u2200 {\u0393} (f\u2080 f\u2081 : \u0393 `\u2192 Word) \u2192 \u0393 `\u2192 Word\n_`\u2228_ = lift\u2082 (map\u00b2\u02b7 <or>)\n\nopen import Solver.Linear\nopen import Data.Vec as Vec using ([]; _\u2237_)\n\nmodule STest n M = Syntax T _`\u00d7_ `\ud835\udfd9 _`\u2192_ id _\u204f_ fst <id,tt> snd <tt,id> <_\u00d7_> first second assoc\u2032 assoc swap n M\n\niter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\niter {zero}  F = <[]>\niter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n               \u204f (helper \u204f < F \u00d7 id > \u204f < swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n               \u204f <\u2237>\n\n  where\n    helper : \u2200 {A B D VA VB} \u2192 (D `\u00d7 (A `\u00d7 VA) `\u00d7 (B `\u00d7 VB)) `\u2192 (D `\u00d7 A `\u00d7 B) `\u00d7 (VA `\u00d7 VB)\n    helper {A} {B} {D} {VA} {VB} = solve (! 2 , (! 0 , ! 3) , (! 1 , ! 4))\n                                 ((! 2 , ! 0 , ! 1) , (! 3 , ! 4)) _ where\n      open STest 5 (\u03bb i \u2192 Vec.lookup i (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])) renaming (rewire to solve; #_ to !_)\n\n<\u229e> adder : Word `\u00d7 Word `\u2192 Word\nadder = <tt\u204f <0b> , id > \u204f iter full-adder\n<\u229e> = adder\n\ninfixl 4 _`\u229e_\n_`\u229e_ : \u2200 {A} (f g : A `\u2192 Word) \u2192 A `\u2192 Word\n_`\u229e_ = lift\u2082 <\u229e>\n  \n<_,_,_> : \u2200 {\u0393 A B C} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C)\n< f\u2080 , f\u2081 , f\u2082 > = < f\u2080 , < f\u2081 , f\u2082 > >\n\n<_,_,_,_,_> : \u2200 {\u0393 A B C D E} \u2192 \u0393 `\u2192 A \u2192 \u0393 `\u2192 B \u2192 \u0393 `\u2192 C\n                              \u2192 \u0393 `\u2192 D \u2192 \u0393 `\u2192 E\n                              \u2192 \u0393 `\u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D `\u00d7 E)\n< f\u2080 , f\u2081 , f\u2082 , f\u2083 , f\u2084 > = < f\u2080 , < f\u2081 , < f\u2082 , f\u2083 , f\u2084 > > >\n\n<<<\u2085 : Word `\u2192 Word\n<<<\u2085 = rot-left 5\n\n<<<\u2083\u2080 : Word `\u2192 Word\n<<<\u2083\u2080 = rot-left 30\n\nWord\u00b2 = Word `\u00d7 Word\nWord\u00b3 = Word `\u00d7 Word\u00b2\nWord\u2074 = Word `\u00d7 Word\u00b3\nWord\u2075 = Word `\u00d7 Word\u2074\n\niterate\u207f : \u2200 {A} n \u2192 (Fin n \u2192 A `\u2192 A) \u2192 A `\u2192 A\niterate\u207f zero    f = id\niterate\u207f (suc n) f = f zero \u204f iterate\u207f n (f \u2218 suc)\n\n_\u00b2\u2070 : \u2200 {A} \u2192 (Fin 20 \u2192 A `\u2192 A) \u2192 (A `\u2192 A)\n_\u00b2\u2070 = iterate\u207f 20\n\nA-E : T\nA-E = Word\u2075\n\nmodule _ (#\u02b7 : \u2115 \u2192 `\ud835\udfd9 `\u2192 Word) where\n\n  module Iterations\n    (A B C D E : A-E `\u2192 Word)\n    where\n\n    module _ (F : A-E `\u2192 Word)\n             (K : `\ud835\udfd9  `\u2192 Word)\n             (W : `\ud835\udfd9  `\u2192 Word) where\n        Iteration = < A' , A , (B \u204f <<<\u2083\u2080) , C , D >\n         where A' = F `\u229e E `\u229e (A \u204f <<<\u2085) `\u229e (tt \u204f W) `\u229e (tt \u204f K)\n\n    module _ \n        (W : Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) where\n\n        Iteration\u2078\u2070 =\n              (Iteration F\u2080 K\u2080 \u2218 W\u2080)\u00b2\u2070 \u204f\n              (Iteration F\u2082 K\u2082 \u2218 W\u2082)\u00b2\u2070 \u204f\n              (Iteration F\u2084 K\u2084 \u2218 W\u2084)\u00b2\u2070 \u204f\n              (Iteration F\u2086 K\u2086 \u2218 W\u2086)\u00b2\u2070\n          where\n            F\u2080 = B `\u2227 C `\u2228 `not B `\u2227 D\n            F\u2082 = B `\u2295 C `\u2295 D\n            F\u2084 = B `\u2227 C `\u2228 B `\u2227 D `\u2228 C `\u2227 D\n            F\u2086 = F\u2082\n\n            K\u2080 = #\u02b7 0x5A827999\n            K\u2082 = #\u02b7 0x6ED9EBA1\n            K\u2084 = #\u02b7 0x8F1BBCDC\n            K\u2086 = #\u02b7 0xCA62C1D6\n\n            W\u2080 W\u2082 W\u2084 W\u2086 : Fin 20 \u2192 `\ud835\udfd9 `\u2192 Word\n            W\u2080 = W \u2218 inject+ 60\n            W\u2082 = W \u2218 inject+ 40 \u2218 raise 20\n            W\u2084 = W \u2218 inject+ 20 \u2218 raise 40\n            W\u2086 = W              \u2218 raise 60\n\n  SHA1 : (Fin 80 \u2192 `\ud835\udfd9 `\u2192 Word) \u2192 A-E `\u2192 A-E\n  SHA1 = Iterations.Iteration\u2078\u2070 H0 H1 H2 H3 H4\n   where\n    H0 = tt \u204f #\u02b7 0x67452301\n    H1 = tt \u204f #\u02b7 0xEFCDAB89\n    H2 = tt \u204f #\u02b7 0x98BADCFE\n    H3 = tt \u204f #\u02b7 0x10325476\n    H4 = tt \u204f #\u02b7 0xC3D2E1F0\n\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'sha1.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"814b7fb656ae80b984ca2b495f770c0127ffd384","subject":"Added note on internal syntax.","message":"Added note on internal syntax.\n","repos":"asr\/apia,asr\/apia","old_file":"notes\/InternalSyntax.agda","new_file":"notes\/InternalSyntax.agda","new_contents":"------------------------------------------------------------------------------\n-- Agda internal syntax\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with Agda development version on 25 February 2013.\n\nmodule InternalSyntax where\n\n------------------------------------------------------------------------------\n-- Sets\n\npostulate A B C : Set\n\n-- El {getSort = Type (Max [ClosedLevel 1]), unEl = Sort (Type (Max []))}\n\npostulate A\u2081 : Set\u2081\n\n-- El {getSort = Type (Max [ClosedLevel 2]), unEl = Sort (Type (Max [ClosedLevel 1]))}\n\n------------------------------------------------------------------------------\n-- The term is Def\n\npostulate def : A\n\n-- El {getSort = Type (Max []), unEl = Def InternalSyntax.A []}\n\n------------------------------------------------------------------------------\n-- The term is Pi\n\npostulate fun\u2081 : A \u2192 B\n\n-- El {getSort = Type (Max []),\n--    unEl = Pi []r(El {getSort = Type (Max []), unEl = Def InternalSyntax.A []})\n--              (NoAbs \"_\" El {getSort = Type (Max []), unEl = Def InternalSyntax.B []})}\n\npostulate fun\u2082 : A \u2192 B \u2192 C\n\n-- El {getSort = Type (Max []),\n--    unEl = Pi []r(El {getSort = Type (Max []), unEl = Def InternalSyntax.A []})\n--           (NoAbs \"_\" El {getSort = Type (Max []),\n--                      unEl = Pi []r(El {getSort = Type (Max []), unEl = Def InternalSyntax.B []})\n--                            (NoAbs \"_\" El {getSort = Type (Max []), unEl = Def InternalSyntax.C []})})}\n\npostulate fun\u2083 : (a : A) \u2192 B\n\n-- El (Type (Max []))\n--    (Pi r(El (Type (Max []))\n--             (Def InternalSyntax.A []))\n--        (Abs \"a\" El (Type (Max []))\n--                 (Def InternalSyntax.B [])))\n\n-- El {getSort = Type (Max []),\n--    unEl = Pi []r(El {getSort = Type (Max []), unEl = Def InternalSyntax.A []})\n--              (NoAbs \"a\" El {getSort = Type (Max []), unEl = Def InternalSyntax.B []})}\n\npostulate P : A \u2192 Set\n\n-- El {getSort = Type (Max [ClosedLevel 1]),\n--    unEl = Pi []r(El {getSort = Type (Max []), unEl = Def InternalSyntax.A []})\n--              (NoAbs \"_\" El {getSort = Type (Max [ClosedLevel 1]), unEl = Sort (Type (Max []))})}\n\npostulate fun\u2085 : (a : A) \u2192 P a\n\n-- El {getSort = Type (Max []),\n--    unEl = Pi []r(El {getSort = Type (Max []), unEl = Def InternalSyntax.A []})\n--             (Abs \"a\" El {getSort = Type (Max []), unEl = Def InternalSyntax.P [[]r(Var 0 [])]})}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/InternalSyntax.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"5f639b12f0526c0e5dcb4f5691bfb46f43219360","subject":"Preparation for a draft.","message":"Preparation for a draft.\n\nIgnore-this: 947ddfe84d510e12ffaaad540d34ace7\n\ndarcs-hash:20120608053614-3bd4e-6bb3eb8d7cbb274965a86b3c321a524e68e1ee66.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/Induction\/NonAcc\/LexicographicI.agda","new_file":"Draft\/FOTC\/Data\/Nat\/Induction\/NonAcc\/LexicographicI.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the lexicographic order on natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From the thesis: The induction principle $\\Conid{Lexi-wfind}$ is\n-- proved by well-founded induction on the usual order $\\Conid{LT}$ on\n-- (partial) natural numbers which, in turn, can be proved by pattern\n-- matching on the proof that the numbers are total.\n\nmodule Draft.FOTC.Data.Nat.Induction.NonAcc.LexicographicI where\n\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\nopen import FOTC.Data.Nat.Induction.NonAcc.WellFoundedInductionI\nopen module WFI = FOTC.Data.Nat.Induction.NonAcc.WellFoundedInductionI.WFInd\n\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n\nLexi-wfind :\n  (A : D \u2192 D \u2192 Set) \u2192\n  (\u2200 {m\u2081 n\u2081} \u2192 N m\u2081 \u2192 N n\u2081 \u2192\n     (\u2200 {m\u2082 n\u2082} \u2192 N m\u2082 \u2192 N n\u2082 \u2192 Lexi m\u2082 n\u2082 m\u2081 n\u2081 \u2192 A m\u2082 n\u2082) \u2192 A m\u2081 n\u2081) \u2192\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192 A m n\nLexi-wfind A accH {m} Nm Nn = LT-wfind {!!} {!!} {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Nat\/Induction\/NonAcc\/LexicographicI.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"deffa1ae4a95dc535cbdbe8c6232a8a4787883e4","subject":"Added partial proof that parametricity implies causality (\"Causality for free!\")","message":"Added partial proof that parametricity implies causality (\"Causality for free!\")\n","repos":"agda\/agda-frp-js,agda\/agda-frp-js","old_file":"src\/agda\/FRP\/JS\/Model.agda","new_file":"src\/agda\/FRP\/JS\/Model.agda","new_contents":"open import FRP.JS.Level using ( Level ; _\u2294_ ) renaming ( zero to o ; suc to \u2191 )\nopen import FRP.JS.Time using ( Time ; _\u225f_ ; _\u2264_ ; _<_ )\nopen import FRP.JS.True using ( True )\nopen import FRP.JS.Nat using ( \u2115 ; zero ; suc )\n\nmodule FRP.JS.Model where\n\n-- Preliminaries\n\ninfixr 4 _+_\n\nrecord \u22a4 {\u03b1} : Set \u03b1 where\n  constructor tt\n\nopen \u22a4 public\n\nrecord \u03a3 {\u03b1 \u03b2} (A : Set \u03b1) (B : A \u2192 Set \u03b2) : Set (\u03b1 \u2294 \u03b2) where\n  constructor _,_\n  field\n    proj\u2081 : A\n    proj\u2082 : B proj\u2081\n\nopen \u03a3 public\n\n_\u00d7_ : \u2200 {\u03b1 \u03b2} \u2192 Set \u03b1 \u2192 Set \u03b2 \u2192 Set (\u03b1 \u2294 \u03b2)\nA \u00d7 B = \u03a3 A (\u03bb a \u2192 B)\n\ndata _\u2261_ {\u03b1} {A : Set \u03b1} (a : A) : A \u2192 Set \u03b1 where\n  refl : a \u2261 a\n\n-- Relations on Set\n\n_\u220b_\u2194_ : \u2200 \u03b1 \u2192 Set \u03b1 \u2192 Set \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u2194 B = A \u2192 B \u2192 Set \u03b1\n\n-- RSets and relations on RSet\n\nRSet : \u2200 \u03b1 \u2192 Set (\u2191 \u03b1)\nRSet \u03b1 = Time \u2192 Set \u03b1\n\nRSet\u2080 = RSet o\nRSet\u2081 = RSet (\u2191 o)\n\n_\u220b_\u21d4_ : \u2200 \u03b1 \u2192 RSet \u03b1 \u2192 RSet \u03b1 \u2192 Set (\u2191 \u03b1)\n\u03b1 \u220b A \u21d4 B = \u2200 t \u2192 (\u03b1 \u220b A t \u2194 B t)\n\n--- Level sequences\n\ndata Levels : Set where\n  \u03b5 : Levels\n  _,_ : \u2200 (\u03b1s : Levels) (\u03b1 : Level) \u2192 Levels\n\nmax : Levels \u2192 Level\nmax \u03b5 = o\nmax (\u03b1s , \u03b1) = max \u03b1s \u2294 \u03b1\n\n-- Sequences of Sets\n\nSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nSets \u03b5        = \u22a4\nSets (\u03b1s , \u03b1) = Sets \u03b1s \u00d7 Set \u03b1\n\n\u27e8_\u220b_\u27e9 : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Set (max \u03b1s)\n\u27e8 \u03b5 \u220b tt \u27e9 = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (As , A) \u27e9 = \u27e8 \u03b1s \u220b As \u27e9 \u00d7 A\n\n_\u220b_\u2194*_ : \u2200 \u03b1s \u2192 Sets \u03b1s \u2192 Sets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u2194* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u2194* (Bs , B) = (\u03b1s \u220b As \u2194* Bs) \u00d7 (\u03b1 \u220b A \u2194 B)\n\n\u27e8_\u220b_\u27e9\u00b2 : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u2194* Bs) \u2192 (max \u03b1s \u220b \u27e8 \u03b1s \u220b As \u27e9 \u2194 \u27e8 \u03b1s \u220b Bs \u27e9)\n\u27e8 \u03b5        \u220b tt       \u27e9\u00b2 tt       tt       = \u22a4\n\u27e8 (\u03b1s , \u03b1) \u220b (\u211cs , \u211c) \u27e9\u00b2 (as , a) (bs , b) = (\u27e8 \u03b1s \u220b \u211cs \u27e9\u00b2 as bs) \u00d7 (\u211c a b)\n\n-- Sequences of RSets\n\nRSets : \u2200 \u03b1s \u2192 Set (\u2191 (max \u03b1s))\nRSets \u03b5        = \u22a4\nRSets (\u03b1s , \u03b1) = RSets \u03b1s \u00d7 RSet \u03b1\n\n_\u220b_\u21d4*_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 RSets \u03b1s \u2192 Set (\u2191 (max \u03b1s))\n\u03b5        \u220b tt       \u21d4* tt       = \u22a4\n(\u03b1s , \u03b1) \u220b (As , A) \u21d4* (Bs , B) = (\u03b1s \u220b As \u21d4* Bs) \u00d7 (\u03b1 \u220b A \u21d4 B)\n\n-- Concatenation of sequences\n\n_+_ : Levels \u2192 Levels \u2192 Levels\n\u03b1s + \u03b5        = \u03b1s\n\u03b1s + (\u03b2s , \u03b2) = (\u03b1s + \u03b2s) , \u03b2\n\n_\u220b_++_\u220b_ : \u2200 \u03b1s \u2192 RSets \u03b1s \u2192 \u2200 \u03b2s \u2192 RSets \u03b2s \u2192 RSets (\u03b1s + \u03b2s)\n\u03b1s \u220b As ++ \u03b5        \u220b tt       = As\n\u03b1s \u220b As ++ (\u03b2s , \u03b2) \u220b (Bs , B) = ((\u03b1s \u220b As ++ \u03b2s \u220b Bs) , B)\n\n_\u220b_++\u00b2_\u220b_ : \u2200 \u03b1s {As Bs} \u2192 (\u03b1s \u220b As \u21d4* Bs) \u2192 \u2200 \u03b2s {Cs Ds} \u2192 (\u03b2s \u220b Cs \u21d4* Ds) \u2192 \n  ((\u03b1s + \u03b2s) \u220b (\u03b1s \u220b As ++ \u03b2s \u220b Cs) \u21d4* (\u03b1s \u220b Bs ++ \u03b2s \u220b Ds))\n\u03b1s \u220b \u211cs ++\u00b2 \u03b5        \u220b tt       = \u211cs\n\u03b1s \u220b \u211cs ++\u00b2 (\u03b2s , \u03b2) \u220b (\u2111s , \u2111) = ((\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) , \u2111)\n\n-- Type variables\n\ndata TVar : Levels \u2192 Set\u2081 where\n  zero : \u2200 {\u03b1s \u03b1} \u2192 TVar (\u03b1s , \u03b1)\n  suc : \u2200 {\u03b1s \u03b1} \u2192 (\u03c4 : TVar \u03b1s) \u2192 TVar (\u03b1s , \u03b1)\n\n\u03c4level : \u2200 {\u03b1s} \u2192 TVar \u03b1s \u2192 Level\n\u03c4level (zero {\u03b1 = \u03b1}) = \u03b1\n\u03c4level (suc \u03c4) = \u03c4level \u03c4\n\n\u03c4\u27e6_\u27e7 : \u2200 {\u03b1s} (\u03c4 : TVar \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (\u03c4level \u03c4)\n\u03c4\u27e6 zero \u27e7  (As , A) = A\n\u03c4\u27e6 suc \u03c4 \u27e7 (As , A) = \u03c4\u27e6 \u03c4 \u27e7 As\n\n\u03c4\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u03c4 : TVar \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (\u03c4level \u03c4 \u220b \u03c4\u27e6 \u03c4 \u27e7 As \u21d4 \u03c4\u27e6 \u03c4 \u27e7 Bs)\n\u03c4\u27e6 zero \u27e7\u00b2  (\u211cs , \u211c) t a b = \u211c t a b\n\u03c4\u27e6 suc \u03c4 \u27e7\u00b2 (\u211cs , \u211c) t a b = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t a b\n\n-- Types\n\ndata Typ (\u03b1s : Levels) : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 Typ \u03b1s\n  _\u2227_ _\u21d2_ : (T : Typ \u03b1s) \u2192 (U : Typ \u03b1s) \u2192 Typ \u03b1s\n  var : (\u03c4 : TVar \u03b1s) \u2192 Typ \u03b1s\n  univ : \u2200 \u03b1 \u2192 (T : Typ (\u03b1s , \u03b1)) \u2192 Typ \u03b1s\n\ntlevel : \u2200 {\u03b1s} \u2192 Typ \u03b1s \u2192 Level\ntlevel \u27e8 A \u27e9 = o\ntlevel (T \u2227 U) = tlevel T \u2294 tlevel U\ntlevel (T \u21d2 U) = tlevel T \u2294 tlevel U\ntlevel (var \u03c4) = \u03c4level \u03c4\ntlevel (univ \u03b1 T) = \u2191 \u03b1 \u2294 tlevel T\n\nT\u27e6_\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u2192 RSets \u03b1s \u2192 RSet (tlevel T)\nT\u27e6 \u27e8 A \u27e9 \u27e7    As t = A\nT\u27e6 T \u2227 U \u27e7    As t = T\u27e6 T \u27e7 As t \u00d7 T\u27e6 U \u27e7 As t\nT\u27e6 T \u21d2 U \u27e7    As t = T\u27e6 T \u27e7 As t \u2192 T\u27e6 U \u27e7 As t\nT\u27e6 var \u03c4 \u27e7    As t = \u03c4\u27e6 \u03c4 \u27e7 As t\nT\u27e6 univ \u03b1 T \u27e7 As t = \u2200 (A : RSet \u03b1) \u2192 T\u27e6 T \u27e7 (As , A) t\n\nT\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (T : Typ \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (tlevel T \u220b T\u27e6 T \u27e7 As \u21d4 T\u27e6 T \u27e7 Bs)\nT\u27e6 \u27e8 A \u27e9 \u27e7\u00b2    \u211cs t a       b       = a \u2261 b\nT\u27e6 T \u2227 U \u27e7\u00b2    \u211cs t (a , b) (c , d) = T\u27e6 T \u27e7\u00b2 \u211cs t a c \u00d7 T\u27e6 U \u27e7\u00b2 \u211cs t b d\nT\u27e6 T \u21d2 U \u27e7\u00b2    \u211cs t f       g       = \u2200 {a b} \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t a b \u2192 T\u27e6 U \u27e7\u00b2 \u211cs t (f a) (g b)\nT\u27e6 var \u03c4 \u27e7\u00b2    \u211cs t v       w       = \u03c4\u27e6 \u03c4 \u27e7\u00b2 \u211cs t v w\nT\u27e6 univ \u03b1 T \u27e7\u00b2 \u211cs t f       g       = \u2200 {A B} (\u211c : \u03b1 \u220b A \u21d4 B) \u2192 T\u27e6 T \u27e7\u00b2 (\u211cs , \u211c) t (f A) (g B)\n\n-- Contexts\n\ndata Ctxt (\u03b1s : Levels) : Set\u2081 where\n  \u03b5 : Ctxt \u03b1s\n  _,_at_ : (\u0393 : Ctxt \u03b1s) (T : Typ \u03b1s) (t : Time) \u2192 Ctxt \u03b1s\n\nclevels : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Levels\nclevels \u03b5            = \u03b5\nclevels (\u0393 , T at t) = (clevels \u0393 , tlevel T)\n\n\u0393\u27e6_\u27e7 : \u2200 {\u03b1s} (\u0393 : Ctxt \u03b1s) \u2192 RSets \u03b1s \u2192 Sets (clevels \u0393)\n\u0393\u27e6 \u03b5 \u27e7          As = tt\n\u0393\u27e6 \u0393 , T at t \u27e7 As = (\u0393\u27e6 \u0393 \u27e7 As , T\u27e6 T \u27e7 As t)\n\n\u0393\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} (\u0393 : Ctxt \u03b1s) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 (clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u2194* \u0393\u27e6 \u0393 \u27e7 Bs)\n\u0393\u27e6 \u03b5 \u27e7\u00b2          \u211cs = tt\n\u0393\u27e6 \u0393 , T at t \u27e7\u00b2 \u211cs = (\u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs , T\u27e6 T \u27e7\u00b2 \u211cs t)\n\n-- Weakening of type variables\n\n\u03c4weaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 TVar (\u03b1s + \u03b2s) \u2192 TVar ((\u03b1s , \u03b1) + \u03b2s)\n\u03c4weaken \u03b1 \u03b5        \u03c4       = suc \u03c4\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) zero    = zero\n\u03c4weaken \u03b1 (\u03b2s , \u03b2) (suc \u03c4) = suc (\u03c4weaken \u03b1 \u03b2s \u03c4)\n\n\u27e6\u03c4weaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) As A Bs t \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192 \u03c4\u27e6 \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b5        \u03c4       As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) zero    As A Bs       t a = a\n\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) As A (Bs , B) t a = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n\n\u27e6\u03c4weaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (\u03c4 : TVar (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4weaken \u03b1 \u03b2s \u03c4 \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Cs t a) (\u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 Bs B Ds t b)\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b5        \u03c4       \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) zero    \u211cs \u211c \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) (suc \u03c4) \u211cs \u211c (\u2111s , \u2111) t a\u211cb = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n\n-- Weakening of types\n\ntweaken : \u2200 {\u03b1s} \u03b1 \u03b2s \u2192 Typ (\u03b1s + \u03b2s) \u2192 Typ ((\u03b1s , \u03b1) + \u03b2s)\ntweaken \u03b1 \u03b2s \u27e8 A \u27e9      = \u27e8 A \u27e9\ntweaken \u03b1 \u03b2s (T \u2227 U)    = tweaken \u03b1 \u03b2s T \u2227 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (T \u21d2 U)    = tweaken \u03b1 \u03b2s T \u21d2 tweaken \u03b1 \u03b2s U\ntweaken \u03b1 \u03b2s (var \u03c4)    = var (\u03c4weaken \u03b1 \u03b2s \u03c4)\ntweaken \u03b1 \u03b2s (univ \u03b2 T) = univ \u03b2 (tweaken \u03b1 (\u03b2s , \u03b2) T)\n\nmutual\n\n  \u27e6tweaken\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192 T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (var \u03c4)    As A Bs t a       = \u27e6\u03c4weaken\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\n  \u27e6tweaken\u207b\u00b9\u27e7 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) As A Bs t \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7 ((\u03b1s , \u03b1) \u220b (As , A) ++ \u03b2s \u220b Bs) t \u2192 T\u27e6 T \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u27e8 B \u27e9      As A Bs t a       = a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u2227 U)    As A Bs t (a , b) = (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Bs t a , \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t b)\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (T \u21d2 U)    As A Bs t f       = \u03bb a \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s U As A Bs t (f (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Bs t a))\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (var \u03c4)    As A Bs t a       = \u27e6\u03c4weaken\u207b\u00b9\u27e7 \u03b1 \u03b2s \u03c4 As A Bs t a\n  \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s (univ \u03b2 T) As A Bs t f       = \u03bb B \u2192 \u27e6tweaken\u207b\u00b9\u27e7 \u03b1 (\u03b2s , \u03b2) T As A (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192 \n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (var \u03c4)    \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 \u03b2s (T : Typ (\u03b1s + \u03b2s)) {As Bs A B Cs Ds} \u211cs \u211c \u2111s t {a b} \u2192\n    T\u27e6 tweaken \u03b1 \u03b2s T \u27e7\u00b2 ((\u03b1s , \u03b1) \u220b (\u211cs , \u211c) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 T \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T As A Cs t a) (\u27e6tweaken\u207b\u00b9\u27e7 \u03b1 \u03b2s T Bs B Ds t b)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u27e8 B \u27e9      \u211cs \u211c \u2111s t a\u211cb         = a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u2227 U)    \u211cs \u211c \u2111s t (a\u211cb , c\u211cd) = (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb , \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t c\u211cd)\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (T \u21d2 U)    \u211cs \u211c \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s U \u211cs \u211c \u2111s t (f\u211cg (\u27e6tweaken\u27e7\u00b2 \u03b1 \u03b2s T \u211cs \u211c \u2111s t a\u211cb))\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (var \u03c4)    \u211cs \u211c \u2111s t a\u211cb         = \u27e6\u03c4weaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s \u03c4 \u211cs \u211c \u2111s t a\u211cb\n  \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 \u03b2s (univ \u03b2 T) \u211cs \u211c \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b1 (\u03b2s , \u03b2) T \u211cs \u211c (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Weakening of contexts\n\ncweaken : \u2200 {\u03b1s} \u03b1 \u2192 Ctxt \u03b1s \u2192 Ctxt (\u03b1s , \u03b1)\ncweaken \u03b1 \u03b5            = \u03b5\ncweaken \u03b1 (\u0393 , T at t) = (cweaken \u03b1 \u0393 , tweaken \u03b1 \u03b5 T at t)\n  \n\u27e6cweaken\u27e7 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) As A \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7 (As , A) \u27e9\n\u27e6cweaken\u27e7 \u03b1 \u03b5            As A tt       = tt\n\u27e6cweaken\u27e7 \u03b1 (\u0393 , T at t) As A (as , a) = (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as , \u27e6tweaken\u27e7 \u03b1 \u03b5 T As A tt t a)\n\n\u27e6cweaken\u27e7\u00b2 : \u2200 {\u03b1s} \u03b1 (\u0393 : Ctxt \u03b1s) {As Bs A B} \u211cs \u211c {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192\n  \u27e8 clevels (cweaken \u03b1 \u0393) \u220b \u0393\u27e6 cweaken \u03b1 \u0393 \u27e7\u00b2 (\u211cs , \u211c) \u27e9\u00b2 (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as) (\u27e6cweaken\u27e7 \u03b1 \u0393 Bs B bs)\n\u27e6cweaken\u27e7\u00b2 \u03b1 \u03b5            \u211cs \u211c tt            = tt\n\u27e6cweaken\u27e7\u00b2 \u03b1 (\u0393 , T at t) \u211cs \u211c (as\u211cbs , a\u211cb) = (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs , \u27e6tweaken\u27e7\u00b2 \u03b1 \u03b5 T \u211cs \u211c tt t a\u211cb)\n\n-- Substitution into type variables\n\n\u03c4subst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 TVar ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\n\u03c4subst T \u03b5        zero    = T\n\u03c4subst T \u03b5        (suc \u03c4) = var \u03c4\n\u03c4subst T (\u03b2s , \u03b2) zero    = var zero\n\u03c4subst T (\u03b2s , \u03b2) (suc \u03c4) = tweaken \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4)\n\n\u27e6\u03c4subst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6tweaken\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t \n    (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n  \u03c4\u27e6 \u03c4 \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        zero    As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T \u03b5        (suc \u03c4) As Bs       t a = a\n\u27e6\u03c4subst\u207b\u00b9\u27e7      T (\u03b2s , \u03b2) zero    As Bs       t a = a \n\u27e6\u03c4subst\u207b\u00b9\u27e7 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) As (Bs , B) t a = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t \n    (\u27e6tweaken\u207b\u00b9\u27e7 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b As ++ \u03b2s \u220b Bs) B tt t a)\n\n\u27e6\u03c4subst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6tweaken\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t \n    (\u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb)\n\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (\u03c4 : TVar ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n  T\u27e6 \u03c4subst T \u03b2s \u03c4 \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n  \u03c4\u27e6 \u03c4 \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a) (\u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 Cs Ds t b)\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        zero    \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T \u03b5        (suc \u03c4) \u211cs \u2111s       t a\u211cb = a\u211cb\n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2      T (\u03b2s , \u03b2) zero    \u211cs \u2111s       t a\u211cb = a\u211cb \n\u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 {\u03b1s} T (\u03b2s , \u03b2) (suc \u03c4) \u211cs (\u2111s , \u2111) t a\u211cb = \n  \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t \n    (\u27e6tweaken\u207b\u00b9\u27e7\u00b2 \u03b2 \u03b5 (\u03c4subst T \u03b2s \u03c4) (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) \u2111 tt t a\u211cb)\n\n-- Substitution into types\n\ntsubst : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s \u2192 Typ ((\u03b1s , tlevel T) + \u03b2s) \u2192 Typ (\u03b1s + \u03b2s)\ntsubst T \u03b2s \u27e8 A \u27e9 = \u27e8 A \u27e9\ntsubst T \u03b2s (U \u2227 V) = tsubst T \u03b2s U \u2227 tsubst T \u03b2s V\ntsubst T \u03b2s (U \u21d2 V) = tsubst T \u03b2s U \u21d2 tsubst T \u03b2s V\ntsubst T \u03b2s (var \u03c4) = \u03c4subst T \u03b2s \u03c4\ntsubst T \u03b2s (univ \u03b2 U) = univ \u03b2 (tsubst T (\u03b2s , \u03b2) U)\n\nmutual\n\n  \u27e6tsubst\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t \u2192 \n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u27e7 T \u03b2s (var \u03c4)    As Bs t a       = \u27e6\u03c4subst\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\n  \u27e6tsubst\u207b\u00b9\u27e7 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) As Bs t \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7 (\u03b1s \u220b As ++ \u03b2s \u220b Bs) t \u2192\n    T\u27e6 U \u27e7 ((\u03b1s , tlevel T) \u220b (As , T\u27e6 T \u27e7 As) ++ \u03b2s \u220b Bs) t\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s \u27e8 A \u27e9      As Bs t a       = a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u2227 V)    As Bs t (a , b) = (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a , \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t b)\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (U \u21d2 V)    As Bs t f       = \u03bb a \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s V As Bs t (f (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a))\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (var \u03c4)    As Bs t a       = \u27e6\u03c4subst\u207b\u00b9\u27e7 T \u03b2s \u03c4 As Bs t a\n  \u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s (univ \u03b2 U) As Bs t f       = \u03bb B \u2192 \u27e6tsubst\u207b\u00b9\u27e7 T (\u03b2s , \u03b2) U As (Bs , B) t (f B)\n\nmutual\n\n  \u27e6tsubst\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (var \u03c4)    \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 : \u2200 {\u03b1s} (T : Typ \u03b1s) \u03b2s (U : Typ ((\u03b1s , tlevel T) + \u03b2s)) {As Bs Cs Ds} \u211cs \u2111s t {a b} \u2192\n    T\u27e6 tsubst T \u03b2s U \u27e7\u00b2 (\u03b1s \u220b \u211cs ++\u00b2 \u03b2s \u220b \u2111s) t a b \u2192\n    T\u27e6 U \u27e7\u00b2 ((\u03b1s , tlevel T) \u220b (\u211cs , T\u27e6 T \u27e7\u00b2 \u211cs) ++\u00b2 \u03b2s \u220b \u2111s) t (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U As Bs t a) (\u27e6tsubst\u207b\u00b9\u27e7 T \u03b2s U Cs Ds t b)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u27e8 A \u27e9      \u211cs \u2111s t a\u211cb         = a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u2227 V)    \u211cs \u2111s t (a\u211cb , c\u211cd) = (\u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb , \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t c\u211cd)\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (U \u21d2 V)    \u211cs \u2111s t f\u211cg         = \u03bb a\u211cb \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s V \u211cs \u2111s t (f\u211cg (\u27e6tsubst\u27e7\u00b2 T \u03b2s U \u211cs \u2111s t a\u211cb))\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (var \u03c4)    \u211cs \u2111s t a\u211cb         = \u27e6\u03c4subst\u207b\u00b9\u27e7\u00b2 T \u03b2s \u03c4 \u211cs \u2111s t a\u211cb\n  \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T \u03b2s (univ \u03b2 U) \u211cs \u2111s t f\u211cg         = \u03bb \u2111 \u2192 \u27e6tsubst\u207b\u00b9\u27e7\u00b2 T (\u03b2s , \u03b2) U \u211cs (\u2111s , \u2111) t (f\u211cg \u2111)\n\n-- Variables\n\ndata Var {\u03b1s} (T : Typ \u03b1s) (t : Time) : (\u0393 : Ctxt \u03b1s) \u2192 Set\u2081 where\n  zero : \u2200 {\u0393 : Ctxt \u03b1s} \u2192 Var T t (\u0393 , T at t)\n  suc : \u2200 {\u0393 : Ctxt \u03b1s} {U : Typ \u03b1s} {u} \u2192 Var T t \u0393 \u2192 Var T t (\u0393 , U at u)\n\nv\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Var T t \u0393 \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\nv\u27e6 zero \u27e7  As (as , a) = a\nv\u27e6 suc v \u27e7 As (as , a) = v\u27e6 v \u27e7 As as\n\nv\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (\u03c4 : Var T t \u0393) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (v\u27e6 \u03c4 \u27e7 As as) (v\u27e6 \u03c4 \u27e7 Bs bs)\nv\u27e6 zero \u27e7\u00b2  \u211cs (as\u211cbs , a\u211cb) = a\u211cb\nv\u27e6 suc v \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb) = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\n\n-- Expressions\n\ndata Exp : \u2200 {\u03b1s} \u2192 Ctxt \u03b1s \u2192 Typ \u03b1s \u2192 RSet\u2081 where\n  const : \u2200 {\u03b1s \u0393 A t} \u2192 (a : A) \u2192 Exp {\u03b1s} \u0393 \u27e8 A \u27e9 t\n  pair : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 T t) \u2192 (f : Exp \u0393 U t) \u2192 Exp {\u03b1s} \u0393 (T \u2227 U) t\n  fst : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 T t\n  snd : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp \u0393 (T \u2227 U) t) \u2192 Exp {\u03b1s} \u0393 U t\n  abs : \u2200 {\u03b1s \u0393 T U t} \u2192 (e : Exp (\u0393 , T at t) U t) \u2192 Exp {\u03b1s} \u0393 (T \u21d2 U) t\n  app : \u2200 {\u03b1s \u0393 T U t} \u2192 (f : Exp \u0393 (T \u21d2 U) t) \u2192 (e : Exp \u0393 T t) \u2192 Exp {\u03b1s} \u0393 U t\n  var : \u2200 {\u03b1s \u0393 T t} \u2192 (v : Var T t \u0393) \u2192 Exp {\u03b1s} \u0393 T t\n  tabs : \u2200 {\u03b1s \u0393} \u03b1 {T t} \u2192 (e : Exp (cweaken \u03b1 \u0393) T t) \u2192 Exp {\u03b1s} \u0393 (univ \u03b1 T) t\n  tapp : \u2200 {\u03b1s \u0393 t} (T : Typ \u03b1s) {U} \u2192 (e : Exp \u0393 (univ (tlevel T) U) t) \u2192 Exp {\u03b1s} \u0393 (tsubst T \u03b5 U) t\n\ne\u27e6_\u27e7 : \u2200 {\u03b1s} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} \u2192 Exp \u0393 T t \u2192 \u2200 As \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7 As \u27e9 \u2192 T\u27e6 T \u27e7 As t\ne\u27e6 const a \u27e7                  As as = a\ne\u27e6 pair e f \u27e7                 As as = (e\u27e6 e \u27e7 As as , e\u27e6 f \u27e7 As as)\ne\u27e6 fst e \u27e7                    As as = proj\u2081 (e\u27e6 e \u27e7 As as)\ne\u27e6 snd e \u27e7                    As as = proj\u2082 (e\u27e6 e \u27e7 As as)\ne\u27e6 abs e \u27e7                    As as = \u03bb a \u2192 e\u27e6 e \u27e7 As (as , a)\ne\u27e6 app f e \u27e7                  As as = e\u27e6 f \u27e7 As as (e\u27e6 e \u27e7 As as)\ne\u27e6 var v \u27e7                    As as = v\u27e6 v \u27e7 As as\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7         As as = \u03bb A \u2192 e\u27e6 e \u27e7 (As , A) (\u27e6cweaken\u27e7 \u03b1 \u0393 As A as)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7 As as = \u27e6tsubst\u27e7 T \u03b5 U As tt t (e\u27e6 e \u27e7 As as (T\u27e6 T \u27e7 As))\n\ne\u27e6_\u27e7\u00b2 : \u2200 {\u03b1s As Bs} {\u0393 : Ctxt \u03b1s} {T : Typ \u03b1s} {t} (e : Exp \u0393 T t) (\u211cs : \u03b1s \u220b As \u21d4* Bs) \u2192 \n  \u2200 {as bs} \u2192 \u27e8 clevels \u0393 \u220b \u0393\u27e6 \u0393 \u27e7\u00b2 \u211cs \u27e9\u00b2 as bs \u2192 T\u27e6 T \u27e7\u00b2 \u211cs t (e\u27e6 e \u27e7 As as) (e\u27e6 e \u27e7 Bs bs)\ne\u27e6 const a \u27e7\u00b2                  \u211cs as\u211cbs = refl\ne\u27e6 pair e f \u27e7\u00b2                 \u211cs as\u211cbs = (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs , e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 fst e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2081 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 snd e \u27e7\u00b2                    \u211cs as\u211cbs = proj\u2082 (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 abs e \u27e7\u00b2                    \u211cs as\u211cbs = \u03bb a\u211cb \u2192 e\u27e6 e \u27e7\u00b2 \u211cs (as\u211cbs , a\u211cb)\ne\u27e6 app f e \u27e7\u00b2                  \u211cs as\u211cbs = e\u27e6 f \u27e7\u00b2 \u211cs as\u211cbs (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs)\ne\u27e6 var v \u27e7\u00b2                    \u211cs as\u211cbs = v\u27e6 v \u27e7\u00b2 \u211cs as\u211cbs\ne\u27e6 tabs {\u0393 = \u0393} \u03b1 e \u27e7\u00b2         \u211cs as\u211cbs = \u03bb \u211c \u2192 e\u27e6 e \u27e7\u00b2 (\u211cs , \u211c) (\u27e6cweaken\u27e7\u00b2 \u03b1 \u0393 \u211cs \u211c as\u211cbs)\ne\u27e6 tapp {t = t} T {U = U} e \u27e7\u00b2 \u211cs as\u211cbs = \u27e6tsubst\u27e7\u00b2 T \u03b5 U \u211cs tt t (e\u27e6 e \u27e7\u00b2 \u211cs as\u211cbs (T\u27e6 T \u27e7\u00b2 \u211cs))\n\n-- Surface level types\n\ndata STyp : Set\u2081 where\n  \u27e8_\u27e9 : (A : Set) \u2192 STyp\n  _\u2227_ _\u21d2_ : (T : STyp) \u2192 (U : STyp) \u2192 STyp\n\n-- Translation of surface level types into types\n\n\u27ea_\u27eb : STyp \u2192 Typ (\u03b5 , o)\n\u27ea \u27e8 A \u27e9 \u27eb = \u27e8 A \u27e9\n\u27ea T \u2227 U \u27eb = \u27ea T \u27eb \u2227 \u27ea U \u27eb\n\u27ea T \u21d2 U \u27eb = \u27ea T \u27eb \u21d2 \u27ea U \u27eb\n\nT\u27ea_\u27eb : STyp \u2192 RSet\u2080\nT\u27ea \u27e8 A \u27e9 \u27eb t = A\nT\u27ea T \u2227 U \u27eb t = T\u27ea T \u27eb t \u00d7 T\u27ea U \u27eb t\nT\u27ea T \u21d2 U \u27eb t = T\u27ea T \u27eb t \u2192 T\u27ea U \u27eb t\n\nWorld : RSet\u2080\nWorld t = \u22a4\n\nmutual\n\n  trans : \u2200 T {s} \u2192 T\u27ea T \u27eb s \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) s\n  trans \u27e8 A \u27e9   a       = a\n  trans (T \u2227 U) (a , b) = (trans T a , trans U b)\n  trans (T \u21d2 U) f       = \u03bb a \u2192 trans U (f (trans\u207b\u00b9 T a))\n\n  trans\u207b\u00b9 : \u2200 T {s} \u2192 T\u27e6 \u27ea T \u27eb \u27e7 (tt , World) s \u2192 T\u27ea T \u27eb s\n  trans\u207b\u00b9 \u27e8 A \u27e9   a       = a\n  trans\u207b\u00b9 (T \u2227 U) (a , b) = (trans\u207b\u00b9 T a , trans\u207b\u00b9 U b)\n  trans\u207b\u00b9 (T \u21d2 U) f       = \u03bb a \u2192 trans\u207b\u00b9 U (f (trans T a))\n\n-- Causality\n\n_at_\u220b_\u2248[_\u2235_]_ : \u2200 T s \u2192 T\u27ea T \u27eb s \u2192 \u2200 u \u2192 True (s \u2264 u) \u2192 T\u27ea T \u27eb s \u2192 Set\n\u27e8 A \u27e9   at s \u220b a       \u2248[ u \u2235 s\u2264u ] b       = a \u2261 b\n(T \u2227 U) at s \u220b (a , b) \u2248[ u \u2235 s\u2264u ] (c , d) = (T at s \u220b a \u2248[ u \u2235 s\u2264u ] c) \u00d7 (U at s \u220b b \u2248[ u \u2235 s\u2264u ] d)\n(T \u21d2 U) at s \u220b f       \u2248[ u \u2235 s\u2264u ] g       = \u2200 {a b} \u2192 (T at s \u220b a \u2248[ u \u2235 s\u2264u ] b) \u2192 (U at s \u220b f a \u2248[ u \u2235 s\u2264u ] g b)\n\nCausal : \u2200 T U s \u2192 T\u27ea T \u21d2 U \u27eb s \u2192 Set\nCausal T U s f = \u2200 u s\u2264u {a b} \u2192 (T at s \u220b a \u2248[ u \u2235 s\u2264u ] b) \u2192 (U at s \u220b f a \u2248[ u \u2235 s\u2264u ] f b)\n\n-- Parametricity implies causality\n\n\u211c[_] : Time \u2192 (o \u220b World \u21d4 World)\n\u211c[ u ] t tt tt = True (t \u2264 u)\n\nmutual\n\n  \u211c-impl-\u2248 : \u2200 T s u s\u2264u {a b} \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) s a b \u2192\n    (T at s \u220b trans\u207b\u00b9 T a \u2248[ u \u2235 s\u2264u ] trans\u207b\u00b9 T b)\n  \u211c-impl-\u2248 \u27e8 A \u27e9   s u s\u2264u a\u211cb         = a\u211cb\n  \u211c-impl-\u2248 (T \u2227 U) s u s\u2264u (a\u211cc , b\u211cd) = (\u211c-impl-\u2248 T s u s\u2264u a\u211cc , \u211c-impl-\u2248 U s u s\u2264u b\u211cd)\n  \u211c-impl-\u2248 (T \u21d2 U) s u s\u2264u f\u211cg         = \u03bb a\u2248b \u2192 \u211c-impl-\u2248 U s u s\u2264u (f\u211cg (\u2248-impl-\u211c T s u s\u2264u a\u2248b))\n\n  \u2248-impl-\u211c : \u2200 T s u s\u2264u {a b} \u2192\n    (T at s \u220b a \u2248[ u \u2235 s\u2264u ] b) \u2192\n    T\u27e6 \u27ea T \u27eb \u27e7\u00b2 (tt , \u211c[ u ]) s (trans T a) (trans T b)\n  \u2248-impl-\u211c \u27e8 A \u27e9   s u s\u2264u a\u2248b         = a\u2248b\n  \u2248-impl-\u211c (T \u2227 U) s u s\u2264u (a\u2248c , b\u2248d) = (\u2248-impl-\u211c T s u s\u2264u a\u2248c , \u2248-impl-\u211c U s u s\u2264u b\u2248d)\n  \u2248-impl-\u211c (T \u21d2 U) s u s\u2264u f\u2248g         = \u03bb a\u211cb \u2192 \u2248-impl-\u211c U s u s\u2264u (f\u2248g (\u211c-impl-\u2248 T s u s\u2264u a\u211cb))\n\n-- Every expression is causal\n\ne\u27ea_at_\u220b_\u27eb : \u2200 T t \u2192 Exp \u03b5 \u27ea T \u27eb t \u2192 T\u27ea T \u27eb t\ne\u27ea T at t \u220b e \u27eb = trans\u207b\u00b9 T (e\u27e6 e \u27e7 (tt , World) tt)\n\ncausality : \u2200 T U s f \u2192 Causal T U s e\u27ea (T \u21d2 U) at s \u220b f \u27eb \ncausality T U s f u s\u2264u = \u211c-impl-\u2248 (T \u21d2 U) s u s\u2264u (e\u27e6 f \u27e7\u00b2 (tt , \u211c[ u ]) tt)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/agda\/FRP\/JS\/Model.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"d82193ca90c884e90ee1747ff92407937693ad43","subject":"Start a new approach: encode Smyth in Agda.","message":"Start a new approach: encode Smyth in Agda.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda2\/Text\/Greek\/Smyth.agda","new_file":"agda2\/Text\/Greek\/Smyth.agda","new_contents":"module Text.Greek.Smyth where\n\nopen import Data.Char\nopen import Data.Maybe\n\n-- Smyth \u00a71\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\ndata LetterCase : Set where\n  capital small : LetterCase\n\nform : Letter \u2192 LetterCase \u2192 Char\nform \u03b1\u2032 capital = '\u0391'\nform \u03b1\u2032 small = '\u03b1'\nform \u03b2\u2032 capital = '\u0392'\nform \u03b2\u2032 small = '\u03b2'\nform \u03b3\u2032 capital = '\u0393'\nform \u03b3\u2032 small = '\u03b3'\nform \u03b4\u2032 capital = '\u0394'\nform \u03b4\u2032 small = '\u03b4'\nform \u03b5\u2032 capital = '\u0395'\nform \u03b5\u2032 small = '\u03b5'\nform \u03b6\u2032 capital = '\u0396'\nform \u03b6\u2032 small = '\u03b6'\nform \u03b7\u2032 capital = '\u0397'\nform \u03b7\u2032 small = '\u03b7'\nform \u03b8\u2032 capital = '\u0398'\nform \u03b8\u2032 small = '\u03b8'\nform \u03b9\u2032 capital = '\u0399'\nform \u03b9\u2032 small = '\u03b9'\nform \u03ba\u2032 capital = '\u039a'\nform \u03ba\u2032 small = '\u03ba'\nform \u03bb\u2032 capital = '\u039b'\nform \u03bb\u2032 small = '\u03bb'\nform \u03bc\u2032 capital = '\u039c'\nform \u03bc\u2032 small = '\u03bc'\nform \u03bd\u2032 capital = '\u039d'\nform \u03bd\u2032 small = '\u03bd'\nform \u03be\u2032 capital = '\u039e'\nform \u03be\u2032 small = '\u03be'\nform \u03bf\u2032 capital = '\u039f'\nform \u03bf\u2032 small = '\u03bf'\nform \u03c0\u2032 capital = '\u03a0'\nform \u03c0\u2032 small = '\u03c0'\nform \u03c1\u2032 capital = '\u03a1'\nform \u03c1\u2032 small = '\u03c1'\nform \u03c3\u2032 capital = '\u03a3'\nform \u03c3\u2032 small = '\u03c3'\nform \u03c4\u2032 capital = '\u03a4'\nform \u03c4\u2032 small = '\u03c4'\nform \u03c5\u2032 capital = '\u03a5'\nform \u03c5\u2032 small = '\u03c5'\nform \u03c6\u2032 capital = '\u03a6'\nform \u03c6\u2032 small = '\u03c6'\nform \u03c7\u2032 capital = '\u03a7'\nform \u03c7\u2032 small = '\u03c7'\nform \u03c8\u2032 capital = '\u03a8'\nform \u03c8\u2032 small = '\u03c8'\nform \u03c9\u2032 capital = '\u03a9'\nform \u03c9\u2032 small = '\u03c9'\n\n-- Smyth \u00a71 a\ndata PositionInWord : Set where\n  not-end : PositionInWord\n  end-of-word : PositionInWord\n\nform-in-word : Letter \u2192 LetterCase \u2192 PositionInWord \u2192 Char\nform-in-word \u03c3\u2032 small end-of-word = '\u03c2'\nform-in-word \u2113 c _ = form \u2113 c\n\n-- Smyth \u00a73\ndata OlderLetter : Set where\n  \u03dc\u2032 : OlderLetter\n  \u03d8\u2032 : OlderLetter\n  \u03e1\u2032 : OlderLetter\n\n-- Smyth \u00a74\ndata Vowel : Letter \u2192 Set where\n  \u03b1-vowel : Vowel \u03b1\u2032\n  \u03b5-vowel : Vowel \u03b5\u2032\n  \u03b7-vowel : Vowel \u03b7\u2032\n  \u03b9-vowel : Vowel \u03b9\u2032\n  \u03bf-vowel : Vowel \u03bf\u2032\n  \u03c5-vowel : Vowel \u03c5\u2032\n  \u03c9-vowel : Vowel \u03c9\u2032\n\ndata AlwaysShort : Letter \u2192 Set where\n  \u03b5-always-short : Vowel \u03b5\u2032 \u2192 AlwaysShort \u03b5\u2032\n  \u03bf-always-short : Vowel \u03bf\u2032 \u2192 AlwaysShort \u03bf\u2032\n\ndata AlwaysLong : Letter \u2192 Set where\n  \u03b7-always-long : Vowel \u03b7\u2032 \u2192 AlwaysLong \u03b7\u2032\n  \u03c9-always-long : Vowel \u03c9\u2032 \u2192 AlwaysLong \u03c9\u2032\n\n-- Smyth \u00a74 All vowels with the circumflex (149) are long\n\n-- Smyth \u00a7149\ndata Accent : Set where\n  acute circumflex grave : Accent\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda2\/Text\/Greek\/Smyth.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9da7bfa317d7cb29c6b18faa7586a165a187ea77","subject":"IDesc model: type-in-type, so it goes... :-(","message":"IDesc model: type-in-type, so it goes... :-(","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/IDesc_type_type.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"13c4b4c064a0a4ebee4104eb40461152750f44d5","subject":"Implement NbE for lambda calculus with base types (see #50).","message":"Implement NbE for lambda calculus with base types (see #50).\n\nOld-commit-hash: 26dd0dee4e73552fe51a52999922812354a03188\n","repos":"inc-lc\/ilc-agda","old_file":"NormalizationByEvaluation.agda","new_file":"NormalizationByEvaluation.agda","new_contents":"module NormalizationByEvaluation where\n\n-- NORMALIZATION BY EVALUATION\n--\n-- for the simply-typed \u03bb calculus with base types\n\nopen import Level hiding (Lift; lift)\nopen import Data.Bool\n\nopen import Relation.Binary hiding (_\u21d2_)\n\n-- The rest of this file is parametric in the set of base types.\n\nmodule Parametric (Base : Set) where\n\n  -- SYNTAX OF TYPES\n  --\n  -- We support function types and base types.\n\n  data Type : Set where\n    _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n    base : (b : Base) \u2192 Type\n\n  open import Syntactic.Contexts Type public\n\n  -- DOMAIN CONSTRUCTION\n  --\n  -- The meaning of types is defined using Kripke structures.\n\n  record IsKripke {w \u2113\u2081 \u2113\u2082} {World : Set w}\n                  (_\u2248_ : Rel World \u2113\u2081)\n                  (_\u2264_ : Rel World \u2113\u2082)\n                  (_\u22a9\u27e6_\u27e7Base : World \u2192 Base \u2192 Set (\u2113\u2082 \u2294 w))\n                  : Set (w \u2294 \u2113\u2081 \u2294 \u2113\u2082) where\n    Lift : \u2200 {A : Set} (_\u22a9\u27e6_\u27e7 : World \u2192 A \u2192 Set (\u2113\u2082 \u2294 w)) \u2192 Set (\u2113\u2082 \u2294 w)\n    Lift _\u22a9\u27e6_\u27e7 = \u2200 {a w\u2081 w\u2082} \u2192 w\u2081 \u2264 w\u2082 \u2192 w\u2081 \u22a9\u27e6 a \u27e7 \u2192 w\u2082 \u22a9\u27e6 a \u27e7\n\n    field\n      isPreorder : IsPreorder _\u2248_ _\u2264_\n      liftBase : Lift _\u22a9\u27e6_\u27e7Base\n\n    open IsPreorder isPreorder public\n\n    _\u22a9\u27e6_\u27e7Type : World \u2192 Type \u2192 Set (\u2113\u2082 \u2294 w)\n    w\u2081 \u22a9\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u2200 {w\u2082} \u2192 w\u2081 \u2264 w\u2082 \u2192 w\u2082 \u22a9\u27e6 \u03c4\u2081 \u27e7Type \u2192 w\u2082 \u22a9\u27e6 \u03c4\u2082 \u27e7Type\n    w\u2081 \u22a9\u27e6 base b \u27e7Type = w\u2081 \u22a9\u27e6 b \u27e7Base\n\n    liftType : Lift _\u22a9\u27e6_\u27e7Type\n    liftType {\u03c4\u2081 \u21d2 \u03c4\u2082} w\u2081\u2264w\u2082 w\u2081\u22a9\u03c4\u2081\u21d2\u03c4\u2082 = \u03bb w\u2082\u2264w\u2083 w\u2083\u22a9\u03c4\u2081 \u2192 w\u2081\u22a9\u03c4\u2081\u21d2\u03c4\u2082 (trans w\u2081\u2264w\u2082 w\u2082\u2264w\u2083) w\u2083\u22a9\u03c4\u2081\n    liftType {base b} w\u2081\u2264w\u2082 w\u2081\u22a9b = liftBase w\u2081\u2264w\u2082 w\u2081\u22a9b\n\n    module _ (w : World) where\n      open import Denotational.Environments Type (\u03bb \u03c4 \u2192 w \u22a9\u27e6 \u03c4 \u27e7Type) public\n        renaming (\u27e6_\u27e7Context to _\u22a9\u27e6_\u27e7Context; \u27e6_\u27e7Var to _\u22a9\u27e6_\u27e7Var)\n\n    liftContext : Lift {Context} _\u22a9\u27e6_\u27e7Context\n    liftContext {\u2205} w\u2081\u2264w\u2082 w\u2081\u22a9\u0393 = \u2205\n    liftContext {\u03c4 \u2022 \u0393} w\u2081\u2264w\u2082 (w\u2081\u22a9\u03c4 \u2022 w\u2081\u22a9\u0393) = liftType {\u03c4} w\u2081\u2264w\u2082 w\u2081\u22a9\u03c4 \u2022 liftContext w\u2081\u2264w\u2082 w\u2081\u22a9\u0393\n\n  record Kripke w \u2113\u2081 \u2113\u2082 : Set (suc (w \u2294 \u2113\u2081 \u2294 \u2113\u2082)) where\n    field\n      World : Set w\n      _\u2248_ : Rel World \u2113\u2081\n      _\u2264_ : Rel World \u2113\u2082\n      _\u22a9\u27e6_\u27e7Base : World \u2192 Base \u2192 Set (\u2113\u2082 \u2294 w)\n      isKripke : IsKripke _\u2248_ _\u2264_ _\u22a9\u27e6_\u27e7Base\n\n    open IsKripke isKripke public\n\n  -- TERMS\n  --\n  -- The syntax of terms is parametric in the set of constants.\n\n  module Terms (Constant : Type \u2192 Set) where\n    data Term : Context \u2192 Type \u2192 Set where\n      abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n      app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n      var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n      con : \u2200 {\u0393 \u03c4} \u2192 (c : Constant \u03c4) \u2192 Term \u0393 \u03c4\n\n    weaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\n    weaken \u0393\u2081\u227c\u0393\u2082 (abs t) = abs (weaken (keep _ \u2022 \u0393\u2081\u227c\u0393\u2082) t)\n    weaken \u0393\u2081\u227c\u0393\u2082 (app t\u2081 t\u2082) = app (weaken \u0393\u2081\u227c\u0393\u2082 t\u2081) (weaken \u0393\u2081\u227c\u0393\u2082 t\u2082)\n    weaken \u0393\u2081\u227c\u0393\u2082 (var x) = var (lift \u0393\u2081\u227c\u0393\u2082 x)\n    weaken \u0393\u2081\u227c\u0393\u2082 (con c) = con c\n\n  -- SYMBOLIC EXECUTION and REIFICATION\n  --\n  -- The Kripke structure of symbolic computation, using typing\n  -- contexts as worlds. Base types are interpreted as\n  -- terms. This is parametric in the set of available constants.\n\n  module Symbolic (Constant : Type \u2192 Set) where\n    open import Relation.Binary.PropositionalEquality\n    open Terms Constant\n\n    BaseTerm : Context \u2192 Base \u2192 Set\n    BaseTerm \u0393 b = Term \u0393 (base b)\n\n    isKripke : IsKripke _\u2261_ _\u227c_ BaseTerm\n    isKripke = record {\n      isPreorder = \u227c-isPreorder;\n      liftBase = weaken }\n\n    symbolic : Kripke zero zero zero\n    symbolic = record {\n      World = Context;\n      _\u2248_ = _\u2261_;\n      _\u2264_ = _\u227c_;\n      _\u22a9\u27e6_\u27e7Base = BaseTerm;\n      isKripke = isKripke }\n\n    open Kripke symbolic\n\n    \u2193 : \u2200 {\u0393} \u03c4 \u2192 \u0393 \u22a9\u27e6 \u03c4 \u27e7Type \u2192 Term \u0393 \u03c4\n    \u2191 : \u2200 {\u0393} \u03c4 \u2192 Term \u0393 \u03c4 \u2192 \u0393 \u22a9\u27e6 \u03c4 \u27e7Type\n\n    \u2193 (\u03c4\u2081 \u21d2 \u03c4\u2082) v = abs (\u2193 \u03c4\u2082 (v (drop \u03c4\u2081 \u2022 \u227c-refl) (\u2191 \u03c4\u2081 (var this))))\n    \u2193 (base b) v = v\n\n    \u2191 (\u03c4\u2081 \u21d2 \u03c4\u2082) t = \u03bb \u0393\u2081\u227c\u0393\u2082 v \u2192 \u2191 \u03c4\u2082 (app (weaken \u0393\u2081\u227c\u0393\u2082 t) (\u2193 \u03c4\u2081 v))\n    \u2191 (base b) t = t\n\n    \u0393\u22a9\u0393 : \u2200 {\u0393} \u2192 \u0393 \u22a9\u27e6 \u0393 \u27e7Context\n    \u0393\u22a9\u0393 {\u2205} = \u2205\n    \u0393\u22a9\u0393 {\u03c4 \u2022 \u0393} = \u2191 \u03c4 (var this) \u2022 liftContext (drop \u03c4 \u2022 \u227c-refl) \u0393\u22a9\u0393\n\n  -- EVALUATION\n  --\n  -- Evaluation is defined for every Kripke structure and every\n  -- set of constants. We have to provide the interpretation of\n  -- the constants in the choosen Kripke structure, though.\n  module _ {w \u2113\u2081 \u2113\u2082} (K : Kripke w \u2113\u2081 \u2113\u2082) where\n    open Kripke K\n\n    module Evaluation (Constant : Type \u2192 Set)\n                      (_\u22a9\u27e6_\u27e7Constant : \u2200 w {\u03c4} \u2192 Constant \u03c4 \u2192 w \u22a9\u27e6 \u03c4 \u27e7Type) where\n      open Terms Constant\n\n      _\u22a9\u27e6_\u27e7Term_ : \u2200 w {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 w \u22a9\u27e6 \u0393 \u27e7Context \u2192 w \u22a9\u27e6 \u03c4 \u27e7Type\n      w\u2081 \u22a9\u27e6 abs t \u27e7Term w\u2081\u22a9\u0393 = \u03bb {w\u2082} w\u2081\u2264w\u2082 w\u2082\u22a9\u03c4\u2081 \u2192 w\u2082 \u22a9\u27e6 t \u27e7Term (w\u2082\u22a9\u03c4\u2081 \u2022 liftContext w\u2081\u2264w\u2082 w\u2081\u22a9\u0393)\n      w\u2081 \u22a9\u27e6 app t\u2081 t\u2082 \u27e7Term w\u2081\u22a9\u0393 = (w\u2081 \u22a9\u27e6 t\u2081 \u27e7Term w\u2081\u22a9\u0393) refl (w\u2081 \u22a9\u27e6 t\u2082 \u27e7Term w\u2081\u22a9\u0393)\n      w\u2081 \u22a9\u27e6 var x \u27e7Term w\u2081\u22a9\u0393 = (w\u2081 \u22a9\u27e6 x \u27e7Var) w\u2081\u22a9\u0393\n      w\u2081 \u22a9\u27e6 con c \u27e7Term w\u2081\u22a9\u0393 = w\u2081 \u22a9\u27e6 c \u27e7Constant\n\n  -- NORMALIZATION\n  --\n  -- Given a set of constants and their interpretation in the\n  -- Kripke structure for symbolic computation, we can normalize\n  -- terms.\n\n  module _ (Constant : Type \u2192 Set) where\n    open Terms Constant\n    open Symbolic Constant\n    open Kripke symbolic\n\n    module Normalization (_\u22a9\u27e6_\u27e7Constant : \u2200 \u0393 {\u03c4} \u2192 Constant \u03c4 \u2192 \u0393 \u22a9\u27e6 \u03c4 \u27e7Type) where\n      open Evaluation symbolic Constant _\u22a9\u27e6_\u27e7Constant\n\n      norm : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n      norm {\u0393} {\u03c4} t = \u2193 \u03c4 (\u0393 \u22a9\u27e6 t \u27e7Term \u0393\u22a9\u0393)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'NormalizationByEvaluation.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"7733c00a89f89d8d2ab941d498d88e4b5f59d2ef","subject":"add a \"simple\" circuit adder capable of 11+31=42","message":"add a \"simple\" circuit adder capable of 11+31=42\n","repos":"crypto-agda\/crypto-agda","old_file":"adder.agda","new_file":"adder.agda","new_contents":"open import Type\nopen import Data.Nat.NP\nopen import Data.Bool using (if_then_else_)\nimport Data.Vec as V\nopen V using (Vec; []; _\u2237_)\nopen import Function.NP hiding (id)\nopen import FunUniverse.Core hiding (_,_)\nopen import Data.Fin using (Fin; zero; suc; #_; inject+; raise) renaming (to\u2115 to Fin\u25b9\u2115)\n\nmodule adder where\n\nmodule FunAdder\n  {t}\n  {T : \u2605_ t}\n  {funU : FunUniverse T}\n  (funOps : FunOps funU)\n  where\n\n    open FunUniverse funU renaming (`\u22a4 to `\ud835\udfd9; `Bit to `\ud835\udfda)\n    open FunOps funOps renaming (_\u2218_ to _`\u2218_)\n\n    open import Solver.Linear\n\n    module LinSolver = Syntax\u1da0 linRewiring\n\n    --iter : \u2200 {n A B S} \u2192 (S `\u00d7 A `\u2192 S `\u00d7 B) \u2192 S `\u00d7 `Vec A n `\u2192 `Vec B n\n\n    iter : \u2200 {n A B C D} \u2192 (D `\u00d7 A `\u00d7 B `\u2192 D `\u00d7 C) \u2192 D `\u00d7 `Vec A n `\u00d7 `Vec B n `\u2192 `Vec C n\n    iter {zero}  F = <[]>\n    iter {suc n} F = < id \u00d7 < uncons \u00d7 uncons > >\n                  \u204f (helper \u204f < F \u204f swap \u00d7 id > \u204f assoc \u204f < id \u00d7 iter F >)\n                  \u204f <\u2237>\n\n      where\n        open LinSolver\n        helper = \u03bb {A} {B} {D} {VA} {VB} \u2192\n          rewire\u1da0 (A \u2237 B \u2237 D \u2237 VA \u2237 VB \u2237 [])\n                  (\u03bb a b d va vb \u2192 (d , (a , va) , (b , vb)) \u21a6 (d , a , b) , (va , vb))\n\n    -- TODO reverses all over the places... switch to lsb first?\n\n    adder : \u2200 {n} \u2192 `Bits n `\u00d7 `Bits n `\u2192 `Bits n\n    adder = <tt\u204f <0b> , < reverse \u00d7 reverse > > \u204f iter full-adder \u204f reverse\n\n    open import Data.Digit\n\n    bits : \u2200 \u2113 \u2192 \u2115 \u2192 `\ud835\udfd9 `\u2192 `Bits \u2113\n    bits \u2113 n\u2080 = constBits (V.reverse (L\u25b9V (L.map F\u25b9\ud835\udfda (proj\u2081 (toDigits 2 n\u2080)))))\n      where open import Data.List as L\n            open import Data.Product\n            open import Data.Two\n            L\u25b9V : \u2200 {n} \u2192 List \ud835\udfda \u2192 Vec \ud835\udfda n\n            L\u25b9V {zero} xs = []\n            L\u25b9V {suc n} [] = V.replicate 0'\n            L\u25b9V {suc n} (x \u2237 xs) = x \u2237 L\u25b9V xs\n            F\u25b9\ud835\udfda : Fin 2 \u2192 \ud835\udfda\n            F\u25b9\ud835\udfda zero    = 0'\n            F\u25b9\ud835\udfda (suc _) = 1'\n\nopen import IO\nimport IO.Primitive\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Coinduction\nopen import FunUniverse.Agda\nopen FunAdder agdaFunOps\nopen FunOps agdaFunOps\nputBit : \ud835\udfda \u2192 IO \ud835\udfd9\nputBit 1' = putStr \"1\"\nputBit 0' = putStr \"0\"\nputBits : \u2200 {n} \u2192 Vec \ud835\udfda n \u2192 IO \ud835\udfd9\nputBits [] = return _\nputBits (x \u2237 bs) = \u266f putBit x >> \u266f putBits bs\narg1   = bits 8 0x0b _\narg2   = bits 8 0x1f _\nresult = adder (arg1 , arg2)\nmainIO : IO \ud835\udfd9\nmainIO = \u266f putBits arg1 >>\n      \u266f (\u266f putStr \" + \" >>\n      \u266f (\u266f putBits arg2 >>\n      \u266f (\u266f putStr \" = \" >>\n         \u266f putBits result)))\nmain : IO.Primitive.IO \ud835\udfd9\nmain = IO.run mainIO\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'adder.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"07c10d40da46eb3d08007d62d71d03805caeeffd","subject":"Added test on polymorphic lists.","message":"Added test on polymorphic lists.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_file":"notes\/FOT\/FOTC\/Polymorphism\/List.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing polymorphic lists\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --universal-quantified-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Polymorphism.List where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\n\n------------------------------------------------------------------------------\n\n-- NB. The data type list is in *Set\u2081*.\ndata List (A : D \u2192 Set) : D \u2192 Set\u2081 where\n  lnil  :                              List A []\n  lcons : \u2200 {x xs} \u2192 A x \u2192 List A xs \u2192 List A (x \u2237 xs)\n{-# ATP axiom lnil lcons #-}\n\npostulate foo : \u2200 x \u2192 x \u2261 x\n{-# ATP prove foo #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Polymorphism\/List.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"a2b7895a12419279c164feddc936a8d79561f28c","subject":"Equalizer unit, counit naturality","message":"Equalizer unit, counit naturality\n","repos":"zeitraffer\/agda-cats","old_file":"src\/FreeNaive\/FiniteComplete.agda","new_file":"src\/FreeNaive\/FiniteComplete.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"012dda4330563386cda786daf4b5846bee5178d1","subject":"Added draft: Non-terminating version of the GCD.","message":"Added draft: Non-terminating version of the GCD.\n\nIgnore-this: bd7f27c02daff7cbc875dcd3a08191f4\n\ndarcs-hash:20110324170647-3bd4e-44433eb9e26bf92661aac951099b570218ac7f9b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/GCD\/GCD.agda","new_file":"Draft\/FOTC\/Program\/GCD\/GCD.agda","new_contents":"------------------------------------------------------------------------------\nmodule GCD where\n\nopen import Data.Nat\nopen import Relation.Nullary\n\ngcd : \u2115 \u2192 \u2115 \u2192 \u2115\ngcd 0       0       = 0\ngcd (suc m) 0       = suc m\ngcd 0       (suc n) = suc n\ngcd (suc m) (suc n) with (suc m \u2264? suc n)\ngcd (suc m) (suc n) | yes p = gcd (suc m) (suc n \u2238 suc m)\ngcd (suc m) (suc n) | no \u00acp = gcd (suc m \u2238 suc n) (suc n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Program\/GCD\/GCD.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"a53621c4fb2c570384071d883f4d8039a38fd29e","subject":"Added Everything.","message":"Added Everything.\n\nIgnore-this: ea7f485a93e2d8c0701a88a1f3cf6b0b\n\ndarcs-hash:20110220154508-3bd4e-207ea1b168d00935008b1197f4208c2b14c0d47d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/Everything.agda","new_file":"Draft\/Mirror\/Everything.agda","new_contents":"------------------------------------------------------------------------------\n-- All the mirror function modules\n------------------------------------------------------------------------------\n\nmodule Draft.Mirror.Everything where\n\nopen import Draft.Mirror.ListTree.Closures\nopen import Draft.Mirror.ListTree.PropertiesATP\nopen import Draft.Mirror.ListTree.PropertiesI\nopen import Draft.Mirror.Mirror.ConsistencyTest\nopen import Draft.Mirror.Mirror\nopen import Draft.Mirror.PropertiesATP\nopen import Draft.Mirror.PropertiesI\nopen import Draft.Mirror.Tree.Closures\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Mirror\/Everything.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"4861392af27ac6baa6e64eb733db79b06963a619","subject":"Control\/Protocol\/End.agda","message":"Control\/Protocol\/End.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/End.agda","new_file":"Control\/Protocol\/End.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Level.NP\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; refl)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nmodule Control.Protocol.End where\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\n\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\nEnd-equiv : End \u2243 \ud835\udfd9\nEnd-equiv = equiv {\u2080} _ _ (\u03bb _ \u2192 refl) (\u03bb _ \u2192 refl)\n\nEnd-uniq : {{_ : UA}} \u2192 End \u2261 \ud835\udfd9\nEnd-uniq = ua End-equiv\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/End.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a78efbf449144f3446d24ee96ed279265763cdf9","subject":"Model: label for indexed descriptions","message":"Model: label for indexed descriptions\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/ILabel.agda","new_file":"models\/ILabel.agda","new_contents":"{-# OPTIONS --type-in-type #-}\n\n-- Yeah, that's type-in-type. I'm a sissy, I want to use the stdlib, I\n-- don't want to reimplement \u2115, \u22a4, etc. in Set1.\n\nmodule ILabel where\n\nopen import Data.String\n\nopen import Data.Empty\nopen import Data.Unit\n\nopen import Data.Product\n\nopen import Data.Maybe\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\n\n--****************************************************************\n-- A universe of labels\n--****************************************************************\n\ninfixr 5 _\u2237_\n\ndata AllowedBy : Set -> Set where\n  \u03b5    : forall {T} -> \n         AllowedBy T\n  _\u2237_  : {S : Set}{T : S -> Set} -> \n         (s : S)(ts : AllowedBy (T s)) -> AllowedBy ((x : S) -> T x)\n\nUId = String\n\n-- The universe (that's a tiny one):\n\ndata Label : Set where\n  label : (u : UId)(T : Set)(ts : AllowedBy T) -> Label\n\n\n--****************************************************************\n-- IDesc & co.\n--****************************************************************\n\ndata IDesc (I : Set1) : Set1 where\n  var    : I -> IDesc I\n  const  : Set -> IDesc I\n  prod   : IDesc I -> IDesc I -> IDesc I\n  sigma  : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi     : (S : Set) -> (S -> IDesc I) -> IDesc I\n\ndesc : {I : Set1} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P \u00d7 desc D' P\ndesc (sigma S T) P = \u03a3 S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n-- We (may) stuck a label on the fix-point:\n\ndata IMu {I : Set1}(L : Maybe (I -> Label))(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu L R j) -> IMu L R i\n\n\n\n--****************************************************************\n-- Example: Nat\n--****************************************************************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc \u22a4\nnatCases Ze = const \u22a4\nnatCases Suc = var _\n\nNatD : \u22a4 \u2192 IDesc \u22a4\nNatD _ = sigma NatConst natCases\n\nNat : Set\nNat = IMu (just (\\ _ -> label \"Nat\" Set \u03b5)) NatD _\n\n--****************************************************************\n-- Example: Vector\n--****************************************************************\n\ndata VecConst : \u2115 -> Set where\n  Vnil : VecConst 0\n  Vcons : {n : \u2115} -> VecConst (1 + n)\n\nvecDChoice : Set -> (n : \u2115) -> VecConst n -> IDesc \u2115\nvecDChoice X 0 Vnil = const \u22a4\nvecDChoice X (suc n) Vcons = prod (const X) (var n)\n\nvecD : Set -> \u2115 -> IDesc \u2115\nvecD X n = sigma (VecConst n) (vecDChoice X n)\n\nvec : Set -> \u2115 -> Set\nvec X n = IMu (just (\\n -> label \"Vec\" (\u2115 -> Set) (n \u2237 \u03b5))) (vecD X) n\n\n--****************\n-- Example: Fin\n--****************\n\ndata FinConst : \u2115 -> Set where\n  Fz : {n : \u2115} -> FinConst (1 + n)\n  Fs : {n : \u2115} -> FinConst (1 + n)\n\nfinDChoice : (n : \u2115) -> FinConst n -> IDesc \u2115\nfinDChoice 0 ()\nfinDChoice (suc n) Fz = const \u22a4\nfinDChoice (suc n) Fs = var n\n\nfinD : \u2115 -> IDesc \u2115\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : \u2115 -> Set\nfin n = IMu (just (\\ n -> label \"Fin\" (\u2115 -> Set) (n \u2237 \u03b5))) finD n\n\n\n--****************\n-- Example: Identity type\n--****************\n\n-- data Id (A : Set) (a : A) : A -> Set := (refl : Id A a a)\n\ndata IdConst : Set where\n  Refl : IdConst \n\nidD : (A : Set)(a : A)(b : A) -> IDesc A\nidD A a b = sigma IdConst (\\ _ -> const (a \u2261 b))\n\nid : (A : Set)(a : A)(b : A) -> Set\nid A a b = IMu (just (\\ b -> label \"Id\" ((A : Set)(a : A)(b : A) -> Set) (A \u2237 a \u2237 b \u2237 \u03b5))) (idD A a) b\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/ILabel.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"a630fe1b88e0aeeab6aa693c339d19ad2ca21597","subject":"Added a simple solver for equations involving max and plus.","message":"Added a simple solver for equations involving max and plus.\n","repos":"crypto-agda\/crypto-agda","old_file":"Add-Max-Solver.agda","new_file":"Add-Max-Solver.agda","new_contents":"module Add-Max-Solver where\n\nimport Algebra\n\nopen import Data.Bool using (Bool ; true ; false)\nopen import Data.Nat.NP\nimport Data.Nat.Properties as Props\nopen import Data.Fin using (Fin ; Ordering ; less ; equal ; greater)\nopen import Data.List \nopen import Data.Vec hiding ([_] ; _++_)\n\n\nopen import Relation.Binary.PropositionalEquality hiding ([_])\n\nopen import Relation.Nullary using (Dec ; yes ; no)\n\nmodule \u2115P = Algebra.CommutativeSemiring Props.commutativeSemiring \nmodule \u2115\u2294 = Algebra.CommutativeSemiringWithoutOne Props.\u2294-\u2293-0-commutativeSemiringWithoutOne\n\n\n\n+-distr-\u2294 : \u2200 a m n \u2192 a + (m \u2294 n) \u2261 (a + m) \u2294 (a + n)\n+-distr-\u2294 zero m n = refl\n+-distr-\u2294 (suc a) m n = cong suc (+-distr-\u2294 a m n)\n\n+-distr-\u2294' : \u2200 m n a \u2192 (m \u2294 n) + a \u2261 (m + a) \u2294 (n + a)\n+-distr-\u2294' m n a = trans (\u2115P.+-comm (m \u2294 n) a) (trans (+-distr-\u2294 a m n) \n               (cong\u2082 _\u2294_ (\u2115P.+-comm a m) (\u2115P.+-comm a n)))\n\nmodule NatSyntax (nrVars : \u2115)(G : Vec \u2115 nrVars) where\n\n  Env = Vec \u2115 nrVars\n  Var = Fin nrVars\n\n  data Syn : Set where\n    con : \u2115 \u2192 Syn \n    var : Var \u2192 Syn\n    _:+_ : Syn \u2192 Syn \u2192 Syn \n    _:u_ : Syn \u2192 Syn \u2192 Syn\n\n  infixl 4 _:+_\n\n  eval : Syn \u2192 \u2115\n  eval (con x) = x\n  eval (var x) = lookup x G\n  eval (syn :+ syn\u2081) = eval syn + eval syn\u2081\n  eval (syn :u syn\u2081) = eval syn \u2294 eval syn\u2081\n\n\n  data N1 : Syn \u2192 Set where \n    con_::[] : \u2200 x \u2192 N1 (con x)\n    var_::_ : \u2200 {S} (i : Var) \u2192 N1 S \u2192 N1 (var i :+ S) \n\n  data N2 : Syn \u2192 Set where\n    lift : \u2200 {S} \u2192 N1 S \u2192 N2 S\n    max  : \u2200 {S S'} \u2192 N1 S \u2192 N2 S' \u2192 N2 (S :u S')\n\n  record N1Proof S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : N1 S'\n      proof : eval S \u2261 eval S' \n\n  record N2Proof S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : N2 S'\n      proof : eval S \u2261 eval S'\n\n  liftP : \u2200 {S} \u2192 N1Proof S \u2192 N2Proof S\n  liftP (term \u22a2 proof) = (lift term) \u22a2 proof\n\n  incr : \u2200 {S'} S \u2192 N1 S' \u2192 N1Proof (S :+ S')\n  incr S con x ::[] = con eval S + x ::[] \u22a2 refl\n  incr {var .i :+ S'} S (var i :: s1) with incr S s1\n  ... | s1' \u22a2 p = (var i :: s1') \u22a2 trans (sym (\u2115P.+-assoc (eval S) (lookup i G) (eval S'))) \n                                   (trans(cong\u2082 _+_ (\u2115P.+-comm (eval S) (lookup i G)) (refl {x = eval S'}))\n                                   (trans (\u2115P.+-assoc (lookup i G) (eval S) (eval S')) \n                                   (cong (_+_ (lookup i G)) p)))\n\n  append : \u2200 {S S'} \u2192 N1 S \u2192 N1 S' \u2192 N1Proof (S :+ S')\n  append con x ::[] s2 = incr _ s2\n  append (var i :: s1) s2 with append s1 s2\n  ... |\u00a0 s3 \u22a2 proof = var i :: s3 \u22a2 trans (\u2115P.+-assoc (lookup i G) _ _) (cong (_+_ (lookup i G)) proof)\n\n  insert : \u2200 {S} i \u2192 N1 S \u2192 N1Proof (var i :+ S)\n  insert i con x ::[] = (var i :: con x ::[]) \u22a2 refl\n  insert {var .j :+ S} i (var j :: s1) with Data.Fin.to\u2115 i \u2264? Data.Fin.to\u2115 j\n  ... |\u00a0yes _ = var i :: var j :: s1 \u22a2 refl\n  ... |\u00a0no  _ with insert i s1\n  ... | t \u22a2 proof = var j :: t \u22a2 trans (sym (\u2115P.+-assoc (lookup i G) (lookup j G) (eval S))) \n                                       (trans\n                                          (cong\u2082 _+_ (\u2115P.+-comm (lookup i G) (lookup j G))\n                                           (refl {x = eval S}))\n                                       (trans (\u2115P.+-assoc (lookup j G) _ _) (cong (_+_ (lookup j G)) proof)))\n\n  merge : \u2200 {S S'} \u2192 N1 S \u2192 N1 S' \u2192 N1Proof (S :+ S')\n  merge con x ::[] s2 = incr _ s2\n  merge (var i :: s1) s2 with merge s1 s2\n  ... |\u00a0t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 trans (trans (\u2115P.+-assoc (lookup i G) _ _) (cong (_+_ (lookup i G)) p1)) p2\n\n  mapN1-N2 : \u2200 {S S'} \u2192 N1 S \u2192 N2 S' \u2192 N2Proof (S :+ S')\n  mapN1-N2 s1 (lift x) = liftP (merge s1 x)\n  mapN1-N2 {S} s1 (max x s2) with merge s1 x | mapN1-N2 s1 s2\n  ... |\u00a0t1 \u22a2 p1 |\u00a0t2 \u22a2 p2 = (max t1 t2) \u22a2 trans (+-distr-\u2294 (eval S) _ _) (cong\u2082 _\u2294_ p1 p2)\n\n  u-merge : \u2200 {S S'} \u2192 N2 S \u2192 N2 S' \u2192 N2Proof (S :u S')\n  u-merge (lift x)   s2 = max x s2 \u22a2 refl\n  u-merge {S :u S'} (max x s1) s2 with u-merge s1 s2\n  ... |\u00a0t \u22a2 p = (max x t) \u22a2 trans (\u2115\u2294.+-assoc (eval S) _ _) (cong (_\u2294_ (eval S)) p)\n\n  +-merge : \u2200 {S S'} \u2192 N2 S \u2192 N2 S' \u2192 N2Proof (S :+ S')\n  +-merge (lift x)   s2 = mapN1-N2 x s2\n  +-merge {S :u S'} (max x s1) s2 with mapN1-N2 x s2 |\u00a0 +-merge s1 s2\n  ... |\u00a0tx \u22a2 px | ts \u22a2 ps with u-merge tx ts\n  ... | t \u22a2 p = t \u22a2 trans (trans (+-distr-\u2294' (eval S) _ _) (cong\u2082 _\u2294_ px ps)) p\n\n  _\u2264\u2115_ : \u2115 \u2192 \u2115 \u2192 Bool\n  m \u2264\u2115 n with m \u2264? n\n  ... |\u00a0yes _ = true\n  ... | no  _ = false\n\n  _\u22641_ : \u2200 {S S'} \u2192 N1 S \u2192 N1 S' \u2192 Bool\n  con x ::[] \u22641 con x\u2081 ::[] = x \u2264\u2115 x\u2081\n  con x ::[] \u22641 (var i :: ys) = true\n  (var i :: xs) \u22641 con x ::[] = false\n  (var i :: xs) \u22641 (var j :: ys) with Data.Fin.compare i j\n  (var .(Data.Fin.inject least) :: xs) \u22641 (var j :: ys) | less .j least = true\n  (var .j :: xs) \u22641 (var j :: ys) | equal .j = xs \u22641 ys\n  (var i :: xs) \u22641 (var .(Data.Fin.inject least) :: ys) | greater .i least = false\n\n  N2-insert : \u2200 {S S'} \u2192 N1 S \u2192 N2 S' \u2192 N2Proof (S :u S')\n  N2-insert {S} x (lift x\u2081) with x \u22641 x\u2081\n  ... | true = max x (lift x\u2081) \u22a2 refl\n  ... | false = max x\u2081 (lift x) \u22a2 \u2115\u2294.+-comm (eval S) _\n  N2-insert {S} {S' :u S''} x (max x\u2081 s) with x \u22641 x\u2081\n  ... |\u00a0true  = (max x (max x\u2081 s)) \u22a2 refl\n  ... |\u00a0false with N2-insert x s\n  ... |\u00a0t \u22a2 p = (max x\u2081 t) \u22a2 trans (trans (sym (\u2115\u2294.+-assoc (eval S) (eval S') (eval S'')))\n                                      (trans (cong\u2082 _\u2294_ (\u2115\u2294.+-comm (eval S) _) (refl {x = eval S''}))\n                                       (\u2115\u2294.+-assoc (eval S') _ _))) (cong (_\u2294_ (eval S')) p) \n\n  sort : \u2200 {S} \u2192 N2 S \u2192 N2Proof S\n  sort (lift x) = lift x \u22a2 refl\n  sort {S :u S'} (max x s1) with sort s1\n  ... |\u00a0s2 \u22a2 p2 with N2-insert x s2\n  ... | s3 \u22a2 p3 = s3 \u22a2 (trans (cong (_\u2294_ (eval S)) p2) p3)  \n\n  norm2 : (x : Syn) \u2192 N2Proof x\n  norm2 (con x) = (lift con x ::[]) \u22a2 refl\n  norm2 (var x) = lift (var x :: con 0 ::[]) \u22a2 \u2115P.+-comm 0 (lookup x G)\n  norm2 (x :+ x\u2081) with norm2 x | norm2 x\u2081\n  ... |\u00a0s1 \u22a2 p1 | s2 \u22a2 p2 with +-merge s1 s2\n  ... | s3 \u22a2 p3 = s3 \u22a2 (trans (cong\u2082 _+_ p1 p2) p3)\n  norm2 (x :u x\u2081) with norm2 x | norm2 x\u2081\n  ... |\u00a0s1 \u22a2 p1 |\u00a0s2 \u22a2 p2 with u-merge s1 s2\n  ... | s3 \u22a2 p3 = s3 \u22a2 trans (cong\u2082 _\u2294_ p1 p2) p3\n\n  norm : (x : Syn) \u2192 N2Proof x\n  norm x with norm2 x\n  ... |\u00a0t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 trans p1 p2\n\n  proof : (x y : Syn) \u2192 eval (N2Proof.S' (norm x)) \u2261 eval (N2Proof.S' (norm y))\n            \u2192 eval x \u2261 eval y\n  proof x y eq = trans (N2Proof.proof (norm x)) (trans eq (sym (N2Proof.proof (norm y))))\n\ntest : \u2200 (x y : \u2115) \u2192 5 + x + (9 + y + 10) \u2261 12 + y + x + 12\ntest x y = proof (con 5 :+ x' :+ (con 9 :+ y' :+ con 10)) (con 12 :+ y' :+ x' :+ con 12) refl where\n  open NatSyntax 2 (x \u2237 y \u2237 [])\n  x' = var Data.Fin.zero\n  y' = var (Data.Fin.suc Data.Fin.zero)\n\ntest' : \u2200 (x y : \u2115) \u2192 ((x + 3) \u2294 (2 + y)) + 7 \u2261 (2 + y + 7) \u2294 (5 + x + 5)\ntest' x y = proof LHS RHS refl where\n  open NatSyntax 2 (x \u2237 y \u2237 [])\n  #_ = con\n  x' = var Data.Fin.zero\n  y' = var (Data.Fin.suc Data.Fin.zero)\n  LHS = ((x' :+ # 3) :u (# 2 :+ y')) :+ # 7 \n  RHS = (# 2 :+ y' :+ # 7) :u (# 5 :+ x' :+ # 5) \n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Add-Max-Solver.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"de43590c6fcfb0c099b9076450f3c5281a68150b","subject":"First go at a clean model of IDesc. With too much universe polymorphism for now.","message":"First go at a clean model of IDesc. With too much universe polymorphism for now.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule IDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      zero : Level\n      suc  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO zero  #-}\n{-# BUILTIN LEVELSUC  suc   #-}\n\nmax : Level -> Level -> Level\nmax  zero    m      = m\nmax (suc n)  zero   = suc n\nmax (suc n) (suc m) = suc (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma {i j : Level}(A : Set i) (B : A -> Set j) : Set (max i j) where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : {i : Level}(A B : Set i) -> Set i\nA * B = Sigma A \\_ -> B\n\nfst : {i : Level}{A : Set i}{B : A -> Set i} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {i : Level}{A : Set i}{B : A -> Set i} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero {i : Level} : Set i where\nrecord One  {i : Level} : Set i where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ {i : Level}(A B : Set i) : Set i where\n  l : A -> A + B\n  r : B -> A + B\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc {i : Level}(I : Set i) : Set (suc i) where\n  var : I -> IDesc I\n  const : Set i -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set i) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set i) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : {i : Level}(I : Set i) -> IDesc I -> (I -> Set i) -> Set i\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {i : Level}(I : Set i)(R : I -> IDesc I) : IDesc I -> Set (suc i) where\n  rec : (i : I) -> IMu I R (R i) -> IMu I R (var i)\n  lambda : (S : Set i)(D : S -> IDesc I) -> ((s : S) -> IMu I R (D s)) -> IMu I R (pi S D)\n  pair : (S : Set i)(D : S -> IDesc I) -> (Sigma S (\\s -> IMu I R (D s))) -> IMu I R (sigma S D)","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/IDesc.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"070620c8ef646eb1491c1cd4de175a22c11b01f9","subject":"lib\/Relation\/Nullary\/NP.agda","message":"lib\/Relation\/Nullary\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Nullary\/NP.agda","new_file":"lib\/Relation\/Nullary\/NP.agda","new_contents":"module Relation.Nullary.NP where\n\nopen import Type\nopen import Data.Sum\nopen import Relation.Nullary public\n\nmodule _ {a b} {A : \u2605_ a} {B : \u2605_ b} where\n    Dec-\u228e : Dec A \u2192 Dec B \u2192 Dec (A \u228e B)\n    Dec-\u228e (yes p) _       = yes (inj\u2081 p)\n    Dec-\u228e (no \u00acp) (yes p) = yes (inj\u2082 p)\n    Dec-\u228e (no \u00acp) (no \u00acq) = no [ \u00acp , \u00acq ]\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Relation\/Nullary\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8096cbcde6b6e9660ddee4e2eab7aa5eddd4cf3b","subject":"Added draft: Numerical proofs.","message":"Added draft: Numerical proofs.\n\nIgnore-this: 78616fa321c24ccfa77f80bbe1a5deac\n\ndarcs-hash:20110208154828-3bd4e-b37f7651ed156539f7454aa6c2966a328340e785.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/Arithmetic.agda","new_file":"Draft\/McCarthy91\/Arithmetic.agda","new_contents":"module Draft.McCarthy91.Arithmetic where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.Numbers\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\n\npostulate\n  1+1>1 : GT (one + one) one\n{-# ATP prove 1+1>1 #-}\n\npostulate\n  10+90\u2261100 : ten + ninety \u2261 one-hundred\n{-# ATP prove 10+90\u2261100 #-}\n\npostulate\n  90+10\u2261100 : ninety + ten \u2261 one-hundred\n{-# ATP prove 90+10\u2261100 #-}\n\npostulate\n  100-10\u226190 : one-hundred \u2238 ten \u2261 ninety\n{-# ATP prove 100-10\u226190 #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/McCarthy91\/Arithmetic.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"47b209921f45703cf94d3d932d9f486d9d6f46e9","subject":"Natural numbers using our own equality module","message":"Natural numbers using our own equality module\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat1.agda","new_file":"agda\/Nat1.agda","new_contents":"module Nat1 where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero   + b = b\nsucc a + b = succ (a + b)\n\nopen import Equality\none   = succ zero\ntwo   = succ one\nthree = succ two\n\n0-is-id : \u2200 (n : \u2115) \u2192 (n + zero) \u2261 n\n0-is-id zero     =\n  begin\n    (zero + zero) \u2248 zero by definition\n  \u220e\n0-is-id (succ y) =\n  begin\n    (succ y + zero) \u2248 succ (y + zero) by definition\n                    \u2248 succ y          by cong succ (0-is-id y)\n  \u220e\n\n+-assoc : \u2200 (a b c : \u2115) \u2192 (a + b) + c \u2261 a + (b + c)\n+-assoc zero     b c     = definition\n+-assoc (succ a) b c     =\n  begin ((succ a + b) + c)\n               \u2248 succ (a + b)  + c   by definition\n               \u2248 succ ((a + b) + c)  by definition\n               \u2248 succ (a  + (b + c)) by cong succ (+-assoc a b c)\n               \u2248 succ a   + (b + c)  by definition\n  \u220e\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/Nat1.agda' did not match any file(s) known to git\n","license":"unlicense","lang":"Agda"}
{"commit":"72c4baef0b4762d0e68731f5561c74025613aeca","subject":"Add Nat.Primality.NP","message":"Add Nat.Primality.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Primality\/NP.agda","new_file":"lib\/Data\/Nat\/Primality\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.Nat.Primality.NP where\n\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Nat.Divisibility.NP\nopen import Data.Nat.Primality public\n  hiding (prime?) -- was slow\n\nopen import Data.Fin.Dec\nopen import Relation.Unary\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Nullary.Negation\nopen import Relation.Nullary\n\n-- same code but imports the fast _\u2223?_\nprime? : Decidable Prime\nprime? 0             = no \u03bb()\nprime? 1             = no \u03bb()\nprime? (suc (suc n)) = all? \u03bb _ \u2192 \u00ac? (_ \u2223? _)\n\nprime-\u22652 : \u2200 {n} \u2192 Prime n \u2192 n \u2265 2\nprime-\u22652 {0}    ()\nprime-\u22652 {1}    ()\nprime-\u22652 {2+ _} _ = s\u2264s (s\u2264s z\u2264n)\n\nprime-\u22620 : \u2200 {n} \u2192 Prime n \u2192 n \u2262 0\nprime-\u22620 {0}    () _\nprime-\u22620 {1+ _} _  ()\n\nprime-\u22621 : \u2200 {n} \u2192 Prime n \u2192 n \u2262 1\nprime-\u22621 {0}    () _\nprime-\u22621 {1}    () _\nprime-\u22621 {2+ _} _  ()\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Nat\/Primality\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5802d87cb34a4e92af208f25be49692945544675","subject":"Model: label for indexed descriptions","message":"Model: label for indexed descriptions","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/ILabel.agda","new_file":"models\/ILabel.agda","new_contents":"{-# OPTIONS --type-in-type #-}\n\n-- Yeah, that's type-in-type. I'm a sissy, I want to use the stdlib, I\n-- don't want to reimplement \u2115, \u22a4, etc. in Set1.\n\nmodule ILabel where\n\nopen import Data.String\n\nopen import Data.Empty\nopen import Data.Unit\n\nopen import Data.Product\n\nopen import Data.Maybe\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\n\n--****************************************************************\n-- A universe of labels\n--****************************************************************\n\ninfixr 5 _\u2237_\n\ndata AllowedBy : Set -> Set where\n  \u03b5    : forall {T} -> \n         AllowedBy T\n  _\u2237_  : {S : Set}{T : S -> Set} -> \n         (s : S)(ts : AllowedBy (T s)) -> AllowedBy ((x : S) -> T x)\n\nUId = String\n\n-- The universe (that's a tiny one):\n\ndata Label : Set where\n  label : (u : UId)(T : Set)(ts : AllowedBy T) -> Label\n\n\n--****************************************************************\n-- IDesc & co.\n--****************************************************************\n\ndata IDesc (I : Set1) : Set1 where\n  var    : I -> IDesc I\n  const  : Set -> IDesc I\n  prod   : IDesc I -> IDesc I -> IDesc I\n  sigma  : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi     : (S : Set) -> (S -> IDesc I) -> IDesc I\n\ndesc : {I : Set1} -> IDesc I -> (I -> Set) -> Set\ndesc (var i) P = P i\ndesc (const X) P = X\ndesc (prod D D') P = desc D P \u00d7 desc D' P\ndesc (sigma S T) P = \u03a3 S (\\s -> desc (T s) P)\ndesc (pi S T) P = (s : S) -> desc (T s) P\n\n-- We (may) stuck a label on the fix-point:\n\ndata IMu {I : Set1}(L : Maybe (I -> Label))(R : I -> IDesc I)(i : I) : Set where\n  con : desc (R i) (\\j -> IMu L R j) -> IMu L R i\n\n\n\n--****************************************************************\n-- Example: Nat\n--****************************************************************\n\ndata NatConst : Set where\n  Ze : NatConst\n  Su : NatConst\n\nnatCases : NatConst -> IDesc \u22a4\nnatCases Ze = const \u22a4\nnatCases Suc = var _\n\nNatD : \u22a4 \u2192 IDesc \u22a4\nNatD _ = sigma NatConst natCases\n\nNat : Set\nNat = IMu (just (\\ _ -> label \"Nat\" Set \u03b5)) NatD _\n\n--****************************************************************\n-- Example: Vector\n--****************************************************************\n\ndata VecConst : \u2115 -> Set where\n  Vnil : VecConst 0\n  Vcons : {n : \u2115} -> VecConst (1 + n)\n\nvecDChoice : Set -> (n : \u2115) -> VecConst n -> IDesc \u2115\nvecDChoice X 0 Vnil = const \u22a4\nvecDChoice X (suc n) Vcons = prod (const X) (var n)\n\nvecD : Set -> \u2115 -> IDesc \u2115\nvecD X n = sigma (VecConst n) (vecDChoice X n)\n\nvec : Set -> \u2115 -> Set\nvec X n = IMu (just (\\n -> label \"Vec\" (\u2115 -> Set) (n \u2237 \u03b5))) (vecD X) n\n\n--****************\n-- Example: Fin\n--****************\n\ndata FinConst : \u2115 -> Set where\n  Fz : {n : \u2115} -> FinConst (1 + n)\n  Fs : {n : \u2115} -> FinConst (1 + n)\n\nfinDChoice : (n : \u2115) -> FinConst n -> IDesc \u2115\nfinDChoice 0 ()\nfinDChoice (suc n) Fz = const \u22a4\nfinDChoice (suc n) Fs = var n\n\nfinD : \u2115 -> IDesc \u2115\nfinD n = sigma (FinConst n) (finDChoice n) \n\nfin : \u2115 -> Set\nfin n = IMu (just (\\ n -> label \"Fin\" (\u2115 -> Set) (n \u2237 \u03b5))) finD n\n\n\n--****************\n-- Example: Identity type\n--****************\n\n-- data Id (A : Set) (a : A) : A -> Set := (refl : Id A a a)\n\ndata IdConst : Set where\n  Refl : IdConst \n\nidD : (A : Set)(a : A)(b : A) -> IDesc A\nidD A a b = sigma IdConst (\\ _ -> const (a \u2261 b))\n\nid : (A : Set)(a : A)(b : A) -> Set\nid A a b = IMu (just (\\ b -> label \"Id\" ((A : Set)(a : A)(b : A) -> Set) (A \u2237 a \u2237 b \u2237 \u03b5))) (idD A a) b\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/ILabel.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c1ff882e5b2b8919dbe2d907c96072bff5801f96","subject":"Nand.Properties: proving adequacy of the nand based operations automatically using exploration functions","message":"Nand.Properties: proving adequacy of the nand based operations automatically using exploration functions\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Nand\/Properties.agda","new_file":"FunUniverse\/Nand\/Properties.agda","new_contents":"open import Level.NP\nopen import Type\nopen import Data.Two\nopen import Data.Product\nopen import Function.NP\nopen import Relation.Binary\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\nimport Explore.Explorable\nopen import Explore.Universe\n\nopen import FunUniverse.Nand\nopen import FunUniverse.Agda\n\nmodule FunUniverse.Nand.Properties where\n\nmodule Test {B : \u2605} (_\u225f_ : Decidable {A = B} _\u2261_)\n            (A : Ty)\n            {f g : El A \u2192 B} where\n  module _ {\u2113} where\n    A\u1d49 : Explore \u2113 (El A)\n    A\u1d49 = exploreU A\n    A\u2071 : ExploreInd \u2113 A\u1d49\n    A\u2071 = exploreU-ind A\n  A\u02e1 : Lookup {\u2080} A\u1d49\n  A\u02e1 = lookupU A\n\n  Check! = A\u1d49 _\u00d7_ \u03bb x \u2192 \u2713 \u230a f x \u225f g x \u230b\n\n  check! : {p\u2713 : Check!} \u2192 f \u2257 g\n  check! {p\u2713} x = toWitness (A\u02e1 p\u2713 x)\n\n  {- Unused\n  open Explore.Explorable.Explorable\u2080 A\u2071\n  test-\u2227 = big-\u2227 \u03bb x \u2192 \u230a f x \u225f g x \u230b\n  -}\n\nmodule Test22 where\n  nand nand' : \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\n\n  nand  (x , y) = not (x \u2227 y)\n  nand' (x , y) = not x \u2228 not y\n\n  module N = FromNand funRewiring nand\n  module T = Test Data.Two._\u225f_\n\n  module UnOp where\n    open T \ud835\udfda\u2032\n\n    pf-not : N.not \u2257 not\n    pf-not = check! \n\n  module BinOp where\n    open T (\ud835\udfda\u2032 \u00d7\u2032 \ud835\udfda\u2032)\n\n    pf-nand : nand \u2257 nand'\n    pf-nand = check!\n\n    pf-and : N.and \u2257 uncurry _\u2227_\n    pf-and = check!\n\n    pf-or : N.or \u2257 uncurry _\u2228_\n    pf-or = check!\n\n    pf-nor : N.nor \u2257 (not \u2218 uncurry _\u2228_)\n    pf-nor = check!\n\n    pf-xor : N.xor \u2257 uncurry _xor_\n    pf-xor = check!\n\n    pf-xnor : N.xnor \u2257 uncurry _==_\n    pf-xnor = check!\n\n  module TriOp where\n    open T (\ud835\udfda\u2032 \u00d7\u2032 (\ud835\udfda\u2032 \u00d7\u2032 \ud835\udfda\u2032))\n\n    fork : \ud835\udfda \u00d7 \ud835\udfda \u00d7 \ud835\udfda \u2192 \ud835\udfda\n    fork (c , e\u1d62) = proj e\u1d62 c\n\n    pf-fork : N.fork \u2257 fork\n    pf-fork = check!\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FunUniverse\/Nand\/Properties.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"206b77832dd7f0a09dc687d45d945f256e9baa9a","subject":"Added a simple solver for the linear functions (those who don't duplicate)","message":"Added a simple solver for the linear functions (those who don't duplicate)\n\nThis make it possible to get functions like (A x B) x C -> (B x A) x C\nbe derived.\n","repos":"crypto-agda\/crypto-agda","old_file":"linear-solver.agda","new_file":"linear-solver.agda","new_contents":"module linear-solver where\n\nopen import Data.Nat as \u2115 using (\u2115)\nopen import Data.Fin as F using (Fin)\nimport Data.Fin.Props as FP\n\nmodule Syntax {a} (A : Set a)(_x_ : A \u2192 A \u2192 A)(T : A)\n  (R' : A \u2192 A \u2192 Set)\n  (id' : \u2200 {A} \u2192 R' A A)\n  (_\u223b'_ : \u2200 {A B C} \u2192 R' A B \u2192 R' B C \u2192 R' A C)\n  (<id,tt>' : \u2200 {A} \u2192 R' (A x T) A)\n  (<id,tt>\u207b\u00b9' : \u2200 {A} \u2192 R' A (A x T))\n  (<tt,id>' : \u2200 {A} \u2192 R' (T x A) A)\n  (<tt,id>\u207b\u00b9' : \u2200 {A} \u2192 R' A (T x A))\n  (\u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 R' A C \u2192 R' B D \u2192 R' (A x B) (C x D))\n  (first'    : \u2200 {A B C} \u2192 R' A B \u2192 R' (A x C) (B x C))\n  (second'   : \u2200 {A B C} \u2192 R' B C \u2192 R' (A x B) (A x C))\n  (assoc'   : \u2200 {A B C} \u2192 R' (A x (B x C)) ((A x B) x C))\n  (assoc\u207b\u00b9' : \u2200 {A B C} \u2192 R' ((A x B) x C) (A x (B x C)))\n  (swap'    : \u2200 {A B}   \u2192 R' (A x B) (B x A))\n  nrVars (!_ : Fin nrVars \u2192 A) where\n\n  Var = Fin nrVars\n\n  open import Relation.Nullary using (yes ; no)\n  open import Relation.Nullary.Decidable\n\n  data Syn : Set where\n   var : Var \u2192 Syn\n   tt  : Syn\n   _,_ : Syn \u2192 Syn \u2192 Syn\n\n  #_ : \u2200 m {m<n : True (\u2115.suc m \u2115.\u2264? nrVars)} \u2192 Syn\n  #_ m {p} = var (F.#_ m {nrVars} {p})\n\n  eval : Syn \u2192 A\n  eval (var x) = ! x\n  eval tt = T\n  eval (s , s\u2081) = eval s x eval s\u2081\n\n  data R : Syn \u2192 Syn \u2192 Set where\n    _\u223b''_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n    <id,tt> : \u2200 {A} \u2192 R (A , tt) A\n    <tt,id> : \u2200 {A} \u2192 R (tt , A) A\n    <tt,id>\u207b\u00b9 : \u2200 {A} \u2192 R A (tt , A)\n    <id,tt>\u207b\u00b9 : \u2200 {A} \u2192 R A (A , tt)\n    \u27e8_\u00d7''_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n    assoc : \u2200 {A B C} \u2192 R (A , (B , C)) ((A , B) , C)\n    assoc\u207b\u00b9 : \u2200 {A B C} \u2192 R ((A , B) , C) (A , (B , C))\n    id : \u2200 {A} \u2192 R A A\n    swap : \u2200 {A B} \u2192 R (A , B) (B , A)\n\n  \u27e8_\u00d7_\u27e9 : \u2200 {A B C D} \u2192 R A C \u2192 R B D \u2192 R (A , B) (C , D)\n  \u27e8 id \u00d7 id \u27e9 = id\n  \u27e8 r\u2081 \u00d7 r\u2082 \u27e9 = \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9\n\n  _\u223b_ : \u2200 {A B C} \u2192 R A B \u2192 R B C \u2192 R A C\n  id \u223b r\u2082 = r\u2082\n  r\u2081 \u223b id = r\u2081\n  <tt,id>\u207b\u00b9 \u223b <tt,id> = id\n  <id,tt>\u207b\u00b9 \u223b <id,tt> = id\n  <tt,id> \u223b <tt,id>\u207b\u00b9 = id\n  <id,tt> \u223b <id,tt>\u207b\u00b9 = id\n  swap \u223b <id,tt> = <tt,id>\n  swap \u223b <tt,id> = <id,tt>\n  <id,tt>\u207b\u00b9 \u223b swap = <tt,id>\u207b\u00b9\n  <tt,id>\u207b\u00b9 \u223b swap = <id,tt>\u207b\u00b9\n  assoc \u223b assoc\u207b\u00b9 = id\n  assoc \u223b (assoc\u207b\u00b9 \u223b'' r) = r\n  assoc\u207b\u00b9 \u223b assoc = id\n  assoc\u207b\u00b9 \u223b (assoc \u223b'' r) = r\n  swap \u223b swap = id\n  swap \u223b (swap \u223b'' r) = r\n  (r\u2081 \u223b'' r\u2082) \u223b r\u2083 = r\u2081 \u223b (r\u2082 \u223b r\u2083)\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b \u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 = \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  \u27e8 r\u2081 \u00d7'' r\u2082 \u27e9 \u223b (\u27e8 r\u2083 \u00d7'' r\u2084 \u27e9 \u223b'' r\u2085) with \u27e8 r\u2081 \u223b r\u2083 \u00d7 r\u2082 \u223b r\u2084 \u27e9\n  ... |\u00a0id = r\u2085\n  ... |\u00a0r\u2086 = r\u2086 \u223b'' r\u2085\n  r\u2081 \u223b r\u2082 = r\u2081 \u223b'' r\u2082 \n\n  sym : \u2200 {S S'} \u2192 R S S' \u2192 R S' S\n  sym (r \u223b'' r\u2081) = sym r\u2081 \u223b sym r\n  sym <id,tt> = <id,tt>\u207b\u00b9\n  sym <tt,id> = <tt,id>\u207b\u00b9\n  sym <id,tt>\u207b\u00b9 = <id,tt>\n  sym <tt,id>\u207b\u00b9 = <tt,id>\n  sym \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 sym r \u00d7 sym r\u2081 \u27e9\n  sym assoc = assoc\u207b\u00b9\n  sym assoc\u207b\u00b9 = assoc\n  sym id = id\n  sym swap = swap\n\n  proof\u2081 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S) (eval S')\n  proof\u2081 (r \u223b'' r\u2081) = proof\u2081 r \u223b' proof\u2081 r\u2081\n  proof\u2081 <id,tt> = <id,tt>'\n  proof\u2081 <tt,id> = <tt,id>'\n  proof\u2081 <id,tt>\u207b\u00b9 = <id,tt>\u207b\u00b9'\n  proof\u2081 <tt,id>\u207b\u00b9 = <tt,id>\u207b\u00b9'\n  proof\u2081 \u27e8 id \u00d7'' r \u27e9 = second' (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' id \u27e9 = first'  (proof\u2081 r)\n  proof\u2081 \u27e8 r \u00d7'' r\u2081 \u27e9 = \u27e8 proof\u2081 r \u00d7' proof\u2081 r\u2081 \u27e9\n  proof\u2081 assoc = assoc'\n  proof\u2081 assoc\u207b\u00b9 = assoc\u207b\u00b9'\n  proof\u2081 id = id'\n  proof\u2081 swap = swap'\n\n  proof\u2082 : \u2200 {S S'} \u2192 R S S' \u2192 R' (eval S') (eval S)\n  proof\u2082 r = proof\u2081 (sym r)\n\n  data NF : Syn \u2192 Set where\n    tt : NF tt\n    var : (x : Var) \u2192 NF (var x)\n    var_::_ : \u2200 {S}(i : Var) \u2192 NF S \u2192 NF (var i , S)\n\n  record NFP S : Set where\n    constructor _\u22a2_\n    field\n      {S'} : Syn\n      term : NF S'\n      proof : R S' S\n  \n \n  merge : \u2200 {S S'} \u2192 NF S \u2192 NF S' \u2192 NFP (S , S')\n  merge tt n2 = n2 \u22a2 <tt,id>\u207b\u00b9\n  merge (var i) n2 = (var i :: n2) \u22a2 id\n  merge (var i :: n1) n2 with merge n1 n2\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b assoc)\n\n  norm : (x : Syn) \u2192 NFP x\n  norm (var x) = (var x) \u22a2 id\n  norm tt = tt \u22a2 id\n  norm (x , x\u2081) with norm x | norm x\u2081\n  ... | t1 \u22a2 p1 | t2 \u22a2 p2 with merge t1 t2\n  ... | t3 \u22a2 p3 = t3 \u22a2 (p3 \u223b \u27e8 p1 \u00d7 p2 \u27e9)\n\n  insert : \u2200 {S} \u2192 (x : Var) \u2192 NF S \u2192 NFP (var x , S)\n  insert y tt = (var y) \u22a2 <id,tt>\u207b\u00b9\n  insert y (var i) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: var i) \u22a2 id\n  ... | no  _ = (var i :: var y) \u22a2 swap\n  insert y (var i :: n1) with (F.to\u2115 y) \u2115.\u2264? (F.to\u2115 i)\n  ... | yes _ = (var y :: (var i :: n1)) \u22a2 id\n  ... | no  _ with insert y n1\n  ... | t \u22a2 p = (var i :: t) \u22a2 (\u27e8 id \u00d7 p \u27e9 \u223b (assoc \u223b (\u27e8 swap \u00d7 id \u27e9 \u223b assoc\u207b\u00b9)))\n\n  sort : \u2200 {x : Syn} \u2192 NF x \u2192 NFP x\n  sort tt = tt \u22a2 id\n  sort (var i) = var i \u22a2 id\n  sort (var i :: n1) with sort n1\n  ... | t1 \u22a2 p1 with insert i t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b \u27e8 id \u00d7 p1 \u27e9)\n\n  normal : (x : Syn) \u2192 NFP x\n  normal x with norm x\n  ... | t1 \u22a2 p1 with sort t1\n  ... | t2 \u22a2 p2 = t2 \u22a2 (p2 \u223b p1)\n\n  open import Relation.Binary.PropositionalEquality using (_\u2261_ ; refl)\n  open import Relation.Nullary\n\n  import Data.Unit\n  import Data.Empty\n\n  id\u2261 : \u2200 {S S'} \u2192 S \u2261 S' \u2192 R S S'\n  id\u2261 refl = id\n\n  _\u2262_ : \u2200 {A : Set} \u2192 A \u2192 A \u2192 Set\n  x \u2262 y = x \u2261 y \u2192 Data.Empty.\u22a5\n\n  \u2262-cong : \u2200 {A B}{x y : A}(f : A \u2192 B) \u2192 f x \u2262 f y \u2192 x \u2262 y\n  \u2262-cong f fr refl = fr refl\n\n  var-inj : \u2200 {i j : Fin nrVars} \u2192 i \u2262 j \u2192 Syn.var i \u2262 var j\n  var-inj p refl = p refl\n\n  ,-inj\u2081 : \u2200 {x y a b} \u2192 x \u2262 y \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2081 p refl = p refl\n\n  ,-inj\u2082 : \u2200 {x y a b} \u2192 a \u2262 b \u2192 (x Syn., a) \u2262 (y , b)\n  ,-inj\u2082 p refl = p refl\n\n  _\u225f_ : (x y : Syn) \u2192 Dec (x \u2261 y)\n  var x \u225f var x\u2081 with x FP.\u225f x\u2081\n  var .x\u2081 \u225f var x\u2081 | yes refl = yes refl\n  ... | no  p    = no (var-inj p)\n  var x \u225f tt = no (\u03bb ())\n  var x \u225f (y , y\u2081) = no (\u03bb ())\n  tt \u225f var x = no (\u03bb ())\n  tt \u225f tt = yes refl\n  tt \u225f (y , y\u2081) = no (\u03bb ())\n  (x , x\u2081) \u225f var x\u2082 = no (\u03bb ())\n  (x , x\u2081) \u225f tt = no (\u03bb ())\n  (x , x\u2081) \u225f (y , y\u2081) with x \u225f y | x\u2081 \u225f y\u2081\n  (x , x\u2081) \u225f (.x , .x\u2081) | yes refl | yes refl = yes refl\n  (x , x\u2081) \u225f (y , y\u2081) | yes p | no \u00acp = no (,-inj\u2082 \u00acp)\n  (x , x\u2081) \u225f (y , y\u2081) | no \u00acp | q = no (,-inj\u2081 \u00acp)\n\n  CHECK : Syn \u2192 Syn \u2192 Set\n  CHECK s1 s2 with s1 \u225f s2\n  ... | yes p = Data.Unit.\u22a4\n  ... | no  p = Data.Empty.\u22a5\n\n  rewire : (S\u2081 S\u2082 : Syn) \u2192 CHECK (NFP.S' (normal S\u2081)) (NFP.S' (normal S\u2082)) \u2192 R' (eval S\u2081) (eval S\u2082)\n  rewire s\u2081 s\u2082 eq with NFP.S' (normal s\u2081) \u225f NFP.S' (normal s\u2082)\n  ... | yes p = proof\u2081\n                  ((sym (NFP.proof (normal s\u2081)) \u223b id\u2261 p) \u223b NFP.proof (normal s\u2082))\n  rewire _ _ () | no _\n    -- proof\u2082 (NFP.proof (normal s\u2081)) \u223b' (eq \u223b' proof\u2081 (NFP.proof (normal s\u2082)))\n\n\nmodule example where\n\n  open import Data.Vec\n\n  open import Data.Product\n  open import Data.Unit\n\n  open import Function\n\n  -- need to etaexpand this because otherwise we get an error\n  module STest n M = Syntax Set _\u00d7_ \u22a4 (\u03bb x x\u2081 \u2192 x \u2192 x\u2081) (\u03bb x \u2192 x) \n         (\u03bb x x\u2081 x\u2082 \u2192 x\u2081 (x x\u2082)) (\u03bb x \u2192 proj\u2081 x) (\u03bb x \u2192 x , tt) \n         (\u03bb x \u2192 proj\u2082 x) (\u03bb x \u2192 tt , x) (\u03bb x x\u2081 x\u2082 \u2192 (x (proj\u2081 x\u2082)) , (x\u2081 (proj\u2082 x\u2082))) \n         (\u03bb x x\u2081 \u2192 (x (proj\u2081 x\u2081)) , (proj\u2082 x\u2081)) (\u03bb x x\u2081 \u2192 (proj\u2081 x\u2081) , (x (proj\u2082 x\u2081))) \n         (\u03bb x \u2192 ((proj\u2081 x) , (proj\u2081 (proj\u2082 x))) , (proj\u2082 (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2081 (proj\u2081 x)) , ((proj\u2082 (proj\u2081 x)) , (proj\u2082 x))) \n         (\u03bb x \u2192 (proj\u2082 x) , (proj\u2081 x)) n M\n\n  test : (A B C : Set) \u2192 (A \u00d7 B) \u00d7 C \u2192 (B \u00d7 A) \u00d7 C\n  test A B C = rewire LHS RHS _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n\n    LHS = (# 0 , # 1) , # 2\n    RHS = (# 1 , # 0) , # 2\n\n\nmodule example\u2082 where\n\n  open import Data.Vec\n\n  data Ty : Set where\n    _\u00d7_ : Ty \u2192 Ty \u2192 Ty\n    \u22a4 : Ty\n\n  infix 4 _\u27f6_ \n\n  data _\u27f6_ : Ty \u2192 Ty \u2192 Set where\n    id' : \u2200 {A} \u2192 A \u27f6 A\n    _\u223b'_ : \u2200 {A B C} \u2192 A \u27f6 B \u2192 B \u27f6 C \u2192 A \u27f6 C\n    <id,tt>' : \u2200 {A} \u2192 (A \u00d7 \u22a4) \u27f6 A\n    <id,tt>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (A \u00d7 \u22a4)\n    <tt,id>' : \u2200 {A} \u2192 (\u22a4 \u00d7 A) \u27f6 A\n    <tt,id>\u207b\u00b9' : \u2200 {A} \u2192 A \u27f6 (\u22a4 \u00d7 A)\n    \u27e8_\u00d7'_\u27e9   : \u2200 {A B C D} \u2192 A \u27f6 C \u2192 B \u27f6 D \u2192 (A \u00d7 B) \u27f6 (C \u00d7 D)\n    first    : \u2200 {A B C} \u2192 A \u27f6 B \u2192 A \u00d7 C \u27f6 B \u00d7 C\n    second   : \u2200 {A B C} \u2192 B \u27f6 C \u2192 A \u00d7 B \u27f6 A \u00d7 C \n    assoc'   : \u2200 {A B C} \u2192 (A \u00d7 (B \u00d7 C)) \u27f6 ((A \u00d7 B) \u00d7 C)\n    assoc\u207b\u00b9' : \u2200 {A B C} \u2192 ((A \u00d7 B) \u00d7 C) \u27f6 (A \u00d7 (B \u00d7 C))\n    swap'    : \u2200 {A B}   \u2192 (A \u00d7 B) \u27f6 (B \u00d7 A)\n  \n\n  module STest n M = Syntax Ty _\u00d7_ \u22a4 _\u27f6_ id' _\u223b'_ <id,tt>' <id,tt>\u207b\u00b9' <tt,id>' <tt,id>\u207b\u00b9' \u27e8_\u00d7'_\u27e9 first second assoc' assoc\u207b\u00b9' swap' n M\n\n  test2 : (A B C : Ty) \u2192 (A \u00d7 B) \u00d7 C \u27f6  (B \u00d7 A) \u00d7 C\n  test2 A B C = rewire ((# 0 , # 1) , # 2) ((# 1 , # 0) , # 2) _ where\n    open STest 3 (\u03bb i \u2192 lookup i (A \u2237 B \u2237 C \u2237 []))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'linear-solver.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ae579f2e62727f46e3ac66d3595279791223a9cb","subject":"Added draft about well-founded induction.","message":"Added draft about well-founded induction.\n\nIgnore-this: f722421546ab538ab5e9b43c374886ff\n\ndarcs-hash:20110301042354-3bd4e-2d2931394d9a7b3ef2a548048ac1509561498b32.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/WellFoundedInductionI.agda","new_file":"Draft\/WellFoundedInductionI.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers\n----------------------------------------------------------------------------\n\nmodule Draft.WellFoundedInductionI where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n-- Well-founded induction on the natural numbers.\n\nwfInd-LT : (P : D \u2192 Set) \u2192\n           (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 LT m n \u2192 P m) \u2192 P n) \u2192\n           \u2200 {n} \u2192 N n \u2192 P n\nwfInd-LT P accH zN      = accH zN (\u03bb Nm m<0 \u2192 \u22a5-elim (x<0\u2192\u22a5 Nm m<0))\nwfInd-LT P accH (sN Nn) = accH (sN Nn)\n                               (\u03bb Nm m<Sn \u2192 helper Nm Nn (wfInd-LT P accH Nn)\n                                                   (x<Sy\u2192x\u2264y Nm Nn m<Sn))\n  where\n    helper : \u2200 {n m} \u2192 N n \u2192 N m \u2192 P m \u2192 LE n m \u2192 P n\n    helper {n} {m} Nn Nm Pm n\u2264m =\n      [ (\u03bb n<m \u2192 {!!} )\n      , (\u03bb n\u2261m \u2192 helper\u2081 n\u2261m Pm)\n      ] (x\u2264y\u2192x<y\u2228x\u2261y Nn Nm n\u2264m)\n      where\n        helper\u2081 : \u2200 {a b} \u2192 a \u2261 b \u2192 P b \u2192 P a\n        helper\u2081 refl Pb = Pb\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/WellFoundedInductionI.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"70108e838a25f768a1cf2e664e2906b13c3c277d","subject":"Added a coproduct Sum universe","message":"Added a coproduct Sum universe\n","repos":"crypto-agda\/crypto-agda","old_file":"sum.agda","new_file":"sum.agda","new_contents":"open import Function\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Sum\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Relation.Binary.PropositionalEquality\n\nmodule sum where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\n_\u2264\u00b0_ : \u2200 {A : Set}(f g : A \u2192 \u2115) \u2192 Set\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMon : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMon sum\u1d2c = \u2200 f g \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nrecord SumProp A : \u2605 where\n  constructor mk\n  field\n    sum : Sum A\n    sum-ext : SumExt sum\n    sum-lin : SumLin sum\n    sum-hom : SumHom sum\n    sum-mon : SumMon sum\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\nsum\u22a4 : Sum \u22a4\nsum\u22a4 f = f _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk sum\u22a4 sum\u22a4-ext sum\u22a4-lin sum\u22a4-hom sum\u22a4-mon\n  where\n    sum\u22a4-ext : SumExt sum\u22a4\n    sum\u22a4-ext f\u2257g = f\u2257g _\n\n    sum\u22a4-lin : SumLin sum\u22a4\n    sum\u22a4-lin f k = refl\n\n    sum\u22a4-hom : SumHom sum\u22a4\n    sum\u22a4-hom f g = refl\n\n    sum\u22a4-mon : SumMon sum\u22a4\n    sum\u22a4-mon f g f\u2264\u00b0g = f\u2264\u00b0g tt\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk sumBit sumBit-ext sumBit-lin sumBit-hom sumBit-mon\n  where\n    sumBit-ext : SumExt sumBit\n    sumBit-ext f\u2257g rewrite f\u2257g 0b | f\u2257g 1b = refl\n\n    sumBit-lin : SumLin sumBit\n    sumBit-lin f k\n      rewrite \u2115\u00b0.*-comm k (f 0b)\n            | \u2115\u00b0.*-comm k (f 1b)\n            | \u2115\u00b0.*-comm k (f 0b + f 1b)\n            | \u2115\u00b0.distrib\u02b3 k (f 0b) (f 1b)\n            = refl\n\n    sumBit-hom : SumHom sumBit\n    sumBit-hom f g = +-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n\n    sumBit-mon : SumMon sumBit\n    sumBit-mon f g f\u2264\u00b0g = f\u2264\u00b0g 0b +-mono f\u2264\u00b0g 1b\n\ninfixr 4 _+Sum_\n\n_+Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c +Sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\n_+\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u228e B)\n(\u03bcA +\u03bc \u03bcB) = mk +-\u03bc +\u03bc-ext +\u03bc-lin +\u03bc-hom +\u03bc-mon\n  where\n    +-\u03bc = sum \u03bcA +Sum sum \u03bcB\n    +\u03bc-ext : SumExt +-\u03bc \n    +\u03bc-ext f\u2257g rewrite sum-ext \u03bcA (f\u2257g \u2218 inj\u2081) | sum-ext \u03bcB (f\u2257g \u2218 inj\u2082) = refl\n\n    +\u03bc-lin : SumLin +-\u03bc\n    +\u03bc-lin f k rewrite sum-lin \u03bcA (f \u2218 inj\u2081) k | sum-lin \u03bcB (f \u2218 inj\u2082) k \n          = sym (proj\u2081 \u2115\u00b0.distrib k (sum \u03bcA (f \u2218 inj\u2081)) (sum \u03bcB (f \u2218 inj\u2082)))\n\n    +\u03bc-hom : SumHom +-\u03bc\n    +\u03bc-hom f g rewrite sum-hom \u03bcA (f \u2218 inj\u2081) (g \u2218 inj\u2081) | sum-hom \u03bcB (f \u2218 inj\u2082) (g \u2218 inj\u2082)\n          = +-interchange (sum \u03bcA (f \u2218 inj\u2081)) (sum \u03bcA (g \u2218 inj\u2081))\n              (sum \u03bcB (f \u2218 inj\u2082)) (sum \u03bcB (g \u2218 inj\u2082))\n          \n    +\u03bc-mon : SumMon +-\u03bc\n    +\u03bc-mon f g f\u2264\u00b0g = sum-mon \u03bcA (f \u2218 inj\u2081) (g \u2218 inj\u2081) (f\u2264\u00b0g \u2218 inj\u2081)\n               +-mono sum-mon \u03bcB (f \u2218 inj\u2082) (g \u2218 inj\u2082) (f\u2264\u00b0g \u2218 inj\u2082)\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n_\u00d7Sum-ext_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} \u2192\n              SumExt sum\u1d2c \u2192 SumExt sum\u1d2e \u2192 SumExt (sum\u1d2c \u00d7Sum sum\u1d2e)\n(sumA-ext \u00d7Sum-ext sumB-ext) f\u2257g = sumA-ext (\u03bb x \u2192 sumB-ext (\u03bb y \u2192 f\u2257g (x , y)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB)\n   = mk (sum \u03bcA \u00d7Sum sum \u03bcB) (sum-ext \u03bcA \u00d7Sum-ext sum-ext \u03bcB) lin hom mon\n   where\n     lin : SumLin (sum \u03bcA \u00d7Sum sum \u03bcB)\n     lin f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-lin \u03bcB (\u03bb y \u2192 f (x , y)) k) = sum-lin \u03bcA (sum \u03bcB \u2218 curry f) k\n\n     hom : SumHom (sum \u03bcA \u00d7Sum sum \u03bcB)\n     hom f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-hom \u03bcB (\u03bb y \u2192 f (x , y)) (\u03bb y \u2192 g (x , y))) \n         = sum-hom \u03bcA (sum \u03bcB \u2218 curry f) (sum \u03bcB \u2218 curry g)\n\n     mon : SumMon (sum \u03bcA \u00d7Sum sum \u03bcB)\n     mon f g f\u2264\u00b0g = sum-mon \u03bcA (sum \u03bcB \u2218 curry f) (sum \u03bcB \u2218 curry g) (\u03bb x \u2192 sum-mon \u03bcB (curry f x) (curry g x) \n                      (\u03bb x\u2081 \u2192 f\u2264\u00b0g (x , x\u2081)))\n\nswapS : \u2200 {A B} \u2192 Sum (A \u00d7 B) \u2192 Sum (B \u00d7 A)\nswapS sumA\u00d7B f = sumA\u00d7B (f \u2218 swap)\n\n{-\n\u00d7Sum-swap : \u2200 {A B} (sum\u1d2c : Sum A) (sum\u1d2e : Sum B) \u2192\n              (sum\u1d2c \u00d7Sum sum\u1d2e) \u2248Sum swapS (sum\u1d2e \u00d7Sum sum\u1d2c)\n\u00d7Sum-swap sum\u1d2c sum\u1d2e f = {!!}\n\n-- wrong so far: \u00d7Sum-swap (const 42) (const 1)\n-}\n\n-- sum (\u03bb x \u2192 f x + g x) \u2261 sum f + sum g\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k \nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum)\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n))\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin n)\n\u03bcFin n = mk (sumFin n) sumFin-ext sumFin-lin sumFin-hom sumFin-mon\n  module SumFin where\n    sumFin-ext : SumExt (sumFin n)\n    sumFin-ext f\u2257g rewrite map-ext f\u2257g (allFin n) = refl\n\n    sumFin-lin : SumLin (sumFin n)\n    sumFin-lin f x = sum-distrib\u02e1 f x (allFin n)\n    \n    sumFin-hom : SumHom (sumFin n)\n    sumFin-hom f g = sum-linear f g (allFin n)\n\n    sumFin-mon : SumMon (sumFin n)\n    sumFin-mon f g f\u2264\u00b0g = sum-mono f g f\u2264\u00b0g (allFin n)\n","old_contents":"open import Function\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Unit hiding (_\u2264_)\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Bool.NP as Bool\nopen import Relation.Binary.PropositionalEquality\n\nmodule sum where\n\nprivate\n    \u2605\u2081 : Set\u2082\n    \u2605\u2081 = Set\u2081\n\n    \u2605 : \u2605\u2081\n    \u2605 = Set\n\n_\u2264\u00b0_ : \u2200 {A : Set}(f g : A \u2192 \u2115) \u2192 Set\nf \u2264\u00b0 g = \u2200 x \u2192 f x \u2264 g x\n\nSum : \u2605 \u2192 \u2605\nSum A = (A \u2192 \u2115) \u2192 \u2115\n\nCount : \u2605 \u2192 \u2605\nCount A = (A \u2192 Bit) \u2192 \u2115\n\nSumExt : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumExt sum\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 sum\u1d2c f \u2261 sum\u1d2c g\n\nCountExt : \u2200 {A} \u2192 Count A \u2192 \u2605\nCountExt count\u1d2c = \u2200 {f g} \u2192 f \u2257 g \u2192 count\u1d2c f \u2261 count\u1d2c g\n\nSumLin : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumLin sum\u1d2c = \u2200 f k \u2192 sum\u1d2c (\u03bb x \u2192 k * f x) \u2261 k * sum\u1d2c f\n\nSumHom : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumHom sum\u1d2c = \u2200 f g \u2192 sum\u1d2c (\u03bb x \u2192 f x + g x) \u2261 sum\u1d2c f + sum\u1d2c g\n\nSumMon : \u2200 {A} \u2192 Sum A \u2192 \u2605\nSumMon sum\u1d2c = \u2200 f g \u2192 f \u2264\u00b0 g \u2192 sum\u1d2c f \u2264 sum\u1d2c g\n\nrecord SumProp A : \u2605 where\n  constructor mk\n  field\n    sum : Sum A\n    sum-ext : SumExt sum\n    sum-lin : SumLin sum\n    sum-hom : SumHom sum\n    sum-mon : SumMon sum\n\n  Card : \u2115\n  Card = sum (const 1)\n\n  count : Count A\n  count f = sum (Bool.to\u2115 \u2218 f)\n\n  count-ext : CountExt count\n  count-ext f\u2257g = sum-ext (cong Bool.to\u2115 \u2218 f\u2257g)\n\nopen SumProp public\n\n_\u2248Sum_ : \u2200 {A} \u2192 (sum\u2080 sum\u2081 : Sum A) \u2192 \u2605\nsum\u2080 \u2248Sum sum\u2081 = \u2200 f \u2192 sum\u2080 f \u2261 sum\u2081 f\n\nsum\u22a4 : Sum \u22a4\nsum\u22a4 f = f _\n\n\u03bc\u22a4 : SumProp \u22a4\n\u03bc\u22a4 = mk sum\u22a4 sum\u22a4-ext sum\u22a4-lin sum\u22a4-hom sum\u22a4-mon\n  where\n    sum\u22a4-ext : SumExt sum\u22a4\n    sum\u22a4-ext f\u2257g = f\u2257g _\n\n    sum\u22a4-lin : SumLin sum\u22a4\n    sum\u22a4-lin f k = refl\n\n    sum\u22a4-hom : SumHom sum\u22a4\n    sum\u22a4-hom f g = refl\n\n    sum\u22a4-mon : SumMon sum\u22a4\n    sum\u22a4-mon f g f\u2264\u00b0g = f\u2264\u00b0g tt\n\nsumBit : Sum Bit\nsumBit f = f 0b + f 1b\n\n\u03bcBit : SumProp Bit\n\u03bcBit = mk sumBit sumBit-ext sumBit-lin sumBit-hom sumBit-mon\n  where\n    sumBit-ext : SumExt sumBit\n    sumBit-ext f\u2257g rewrite f\u2257g 0b | f\u2257g 1b = refl\n\n    sumBit-lin : SumLin sumBit\n    sumBit-lin f k\n      rewrite \u2115\u00b0.*-comm k (f 0b)\n            | \u2115\u00b0.*-comm k (f 1b)\n            | \u2115\u00b0.*-comm k (f 0b + f 1b)\n            | \u2115\u00b0.distrib\u02b3 k (f 0b) (f 1b)\n            = refl\n\n    sumBit-hom : SumHom sumBit\n    sumBit-hom f g = +-interchange (f 0b) (g 0b) (f 1b) (g 1b)\n\n    sumBit-mon : SumMon sumBit\n    sumBit-mon f g f\u2264\u00b0g = f\u2264\u00b0g 0b +-mono f\u2264\u00b0g 1b\n\ninfixr 4 _\u00d7Sum_\n\n-- liftM2 _,_ in the continuation monad\n_\u00d7Sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u00d7 B)\n(sum\u1d2c \u00d7Sum sum\u1d2e) f = sum\u1d2c (\u03bb x\u2080 \u2192\n                       sum\u1d2e (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n_\u00d7Sum-ext_ : \u2200 {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} \u2192\n              SumExt sum\u1d2c \u2192 SumExt sum\u1d2e \u2192 SumExt (sum\u1d2c \u00d7Sum sum\u1d2e)\n(sumA-ext \u00d7Sum-ext sumB-ext) f\u2257g = sumA-ext (\u03bb x \u2192 sumB-ext (\u03bb y \u2192 f\u2257g (x , y)))\n\ninfixr 4 _\u00d7\u03bc_\n\n_\u00d7\u03bc_ : \u2200 {A B} \u2192 SumProp A\n               \u2192 SumProp B\n               \u2192 SumProp (A \u00d7 B)\n(\u03bcA \u00d7\u03bc \u03bcB)\n   = mk (sum \u03bcA \u00d7Sum sum \u03bcB) (sum-ext \u03bcA \u00d7Sum-ext sum-ext \u03bcB) lin hom mon\n   where\n     lin : SumLin (sum \u03bcA \u00d7Sum sum \u03bcB)\n     lin f k rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-lin \u03bcB (\u03bb y \u2192 f (x , y)) k) = sum-lin \u03bcA (sum \u03bcB \u2218 curry f) k\n\n     hom : SumHom (sum \u03bcA \u00d7Sum sum \u03bcB)\n     hom f g rewrite sum-ext \u03bcA (\u03bb x \u2192 sum-hom \u03bcB (\u03bb y \u2192 f (x , y)) (\u03bb y \u2192 g (x , y))) \n         = sum-hom \u03bcA (sum \u03bcB \u2218 curry f) (sum \u03bcB \u2218 curry g)\n\n     mon : SumMon (sum \u03bcA \u00d7Sum sum \u03bcB)\n     mon f g f\u2264\u00b0g = sum-mon \u03bcA (sum \u03bcB \u2218 curry f) (sum \u03bcB \u2218 curry g) (\u03bb x \u2192 sum-mon \u03bcB (curry f x) (curry g x) \n                      (\u03bb x\u2081 \u2192 f\u2264\u00b0g (x , x\u2081)))\n\nswapS : \u2200 {A B} \u2192 Sum (A \u00d7 B) \u2192 Sum (B \u00d7 A)\nswapS sumA\u00d7B f = sumA\u00d7B (f \u2218 swap)\n\n{-\n\u00d7Sum-swap : \u2200 {A B} (sum\u1d2c : Sum A) (sum\u1d2e : Sum B) \u2192\n              (sum\u1d2c \u00d7Sum sum\u1d2e) \u2248Sum swapS (sum\u1d2e \u00d7Sum sum\u1d2c)\n\u00d7Sum-swap sum\u1d2c sum\u1d2e f = {!!}\n\n-- wrong so far: \u00d7Sum-swap (const 42) (const 1)\n-}\n\n-- sum (\u03bb x \u2192 f x + g x) \u2261 sum f + sum g\n\nsum-const : \u2200 {A} (\u03bcA : SumProp A) \u2192 \u2200 k \u2192 sum \u03bcA (const k) \u2261 Card \u03bcA * k \nsum-const \u03bcA k\n  rewrite \u2115\u00b0.*-comm (Card \u03bcA) k\n        | sym (sum-lin \u03bcA (const 1) k)\n        | proj\u2082 \u2115\u00b0.*-identity k = refl\n\ninfixr 4 _\u00d7Sum-proj\u2081_ _\u00d7Sum-proj\u2081'_ _\u00d7Sum-proj\u2082_ _\u00d7Sum-proj\u2082'_\n\n_\u00d7Sum-proj\u2081_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 Card \u03bcB * sum \u03bcA f\n(\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n  rewrite sum-ext \u03bcA (sum-const \u03bcB \u2218 f)\n        = sum-lin \u03bcA f (Card \u03bcB)\n\n_\u00d7Sum-proj\u2082_ : \u2200 {A B}\n                 (\u03bcA : SumProp A)\n                 (\u03bcB : SumProp B)\n                 f \u2192\n                 sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 Card \u03bcA * sum \u03bcB f\n(\u03bcA \u00d7Sum-proj\u2082 \u03bcB) f = sum-const \u03bcA (sum \u03bcB f)\n\n_\u00d7Sum-proj\u2081'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcA f \u2261 sum \u03bcA g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2081) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2081)\n(\u03bcA \u00d7Sum-proj\u2081' \u03bcB) {f} {g} sumf\u2261sumg\n  rewrite (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) f\n        | (\u03bcA \u00d7Sum-proj\u2081 \u03bcB) g\n        | sumf\u2261sumg = refl\n\n_\u00d7Sum-proj\u2082'_ : \u2200 {A B}\n                  (\u03bcA : SumProp A) (\u03bcB : SumProp B)\n                  {f} {g} \u2192\n                  sum \u03bcB f \u2261 sum \u03bcB g \u2192\n                  sum (\u03bcA \u00d7\u03bc \u03bcB) (f \u2218 proj\u2082) \u2261 sum (\u03bcA \u00d7\u03bc \u03bcB) (g \u2218 proj\u2082)\n(\u03bcA \u00d7Sum-proj\u2082' \u03bcB) sumf\u2261sumg = sum-ext \u03bcA (const sumf\u2261sumg)\n\nopen import Data.Fin hiding (_+_)\nopen import Data.Vec.NP as Vec renaming (map to vmap; sum to vsum)\n\nsumFin : \u2200 n \u2192 Sum (Fin n)\nsumFin n f = vsum (vmap f (allFin n))\n\n\u03bcFin : \u2200 n \u2192 SumProp (Fin n)\n\u03bcFin n = mk (sumFin n) sumFin-ext sumFin-lin sumFin-hom sumFin-mon\n  module SumFin where\n    sumFin-ext : SumExt (sumFin n)\n    sumFin-ext f\u2257g rewrite map-ext f\u2257g (allFin n) = refl\n\n    sumFin-lin : SumLin (sumFin n)\n    sumFin-lin f x = sum-distrib\u02e1 f x (allFin n)\n    \n    sumFin-hom : SumHom (sumFin n)\n    sumFin-hom f g = sum-linear f g (allFin n)\n\n    sumFin-mon : SumMon (sumFin n)\n    sumFin-mon f g f\u2264\u00b0g = sum-mono f g f\u2264\u00b0g (allFin n)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ee61ddea3e8eac499a032455cd51b646fa624d97","subject":"Added issue.","message":"Added issue.\n\nIgnore-this: 76bbff01b674d2413dffeddea8b4e57e\n\ndarcs-hash:20110923235106-3bd4e-3df2784e27525650905ef511fa96bcc49badd2f5.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/Agda365.agda","new_file":"Issues\/Agda365.agda","new_contents":"------------------------------------------------------------------------------\n-- Issue in the translation inspired by the Agda issue 365\n------------------------------------------------------------------------------\n\nmodule Issues.Agda365 where\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n\nfoo : D \u2192 D\n-- foo d = d  -- Works!\nfoo = \u03bb d \u2192 d  -- Doesn't work!\n{-# ATP definition foo #-}\n\npostulate bar : \u2200 d \u2192 d \u2261 foo d\n{-# ATP prove bar #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Issues\/Agda365.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"86091d67801f801efb69b14b9700beeaaf2300b2","subject":"Added missing file.","message":"Added missing file.\n\nIgnore-this: 845e884251e48165cbb2f45f5b3dab57\n\ndarcs-hash:20100908153610-3bd4e-e20fc7f615975afb00093672d6118d768d2a3b37.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/List\/Core.agda","new_file":"LTC\/Data\/List\/Core.agda","new_contents":"------------------------------------------------------------------------------\n-- Lists (some core definitions)\n------------------------------------------------------------------------------\n\n-- The definitions in this file are reexported by LTC.Data.List\n\nmodule LTC.Data.List.Core where\n\nopen import LTC.Minimal.Core using ( D )\n\ninfixr 5 _\u2237_\n\n------------------------------------------------------------------------------\n-- List terms\npostulate\n  []   : D\n  _\u2237_  : D \u2192 D \u2192 D\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'LTC\/Data\/List\/Core.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"bea4cea999556aa2df4e0a46b9e7eb9c06f0bfaa","subject":"FMSB: agda model of lambda madnesses","message":"FMSB: agda model of lambda madnesses\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/FMSB.agda","new_file":"models\/FMSB.agda","new_contents":"module FMSB where\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Lam (X : Nat -> Set)(n : Nat) : Set where\n  var : X n -> Lam X n\n  app : Lam X n -> Lam X n -> Lam X n\n  lam : Lam X (su n) -> Lam X n\n\n_->>_ : {I : Set}(X Y : I -> Set) -> Set\nX ->> Y = {i : _} -> X i -> Y i\n\nfmsub : {X Y : Nat -> Set} -> (X ->> Lam Y) -> Lam X ->> Lam Y\nfmsub sb (var x)   = sb x\nfmsub sb (app f s) = app (fmsub sb f) (fmsub sb s)\nfmsub sb (lam b)   = lam (fmsub sb b)\n\ndata Fin : Nat -> Set where\n  ze : {n : Nat} -> Fin (su n)\n  su : {n : Nat} -> Fin n -> Fin (su n)\n\nD : (Nat -> Set) -> Nat -> Set\nD F n = F (su n)\n\nweak : {F : Nat -> Set} -> (Fin ->> F) -> (F ->> D F) ->\n       {m n : Nat}-> (Fin m -> F n) -> D Fin m -> D F n\nweak v w f ze      = v ze\nweak v w f (su x)  = w (f x)\n\n_+_ : Nat -> Nat -> Nat\nze    + y = y\nsu x  + y = su (x + y)\n\n_!+_ : (Nat -> Set) -> Nat -> (Nat -> Set)\nF !+ n = \\ m -> F (m + n)\n\nweaks : {F : Nat -> Set} -> (Fin ->> F) -> (F ->> D F) ->\n        {m n : Nat}-> (Fin m -> F n) -> (Fin !+ m) ->> (F !+ n)\nweaks v w f {ze}    = f\nweaks v w f {su i}  = weak v w (weaks v w f {i})\n\njing : {m n : Nat} -> Lam Fin (m + n) -> Lam (Fin !+ n) m\njing (var x) = var x\njing (app f s) = app (jing f) (jing s)\njing (lam t) = lam (jing t)\n\ngnij : {m n : Nat} -> Lam (Fin !+ n) m -> Lam Fin (m + n)\ngnij (var x) = var x\ngnij (app f s) = app (gnij f) (gnij s)\ngnij (lam t) = lam (gnij t)\n\nren : {m n : Nat} -> (Fin m -> Fin n) -> Lam Fin m -> Lam Fin n\nren f t = gnij (fmsub (\\ {i} x -> var (weaks (\\ z -> z) su f {i} x)) {ze} (jing t))\n\nsub : {m n : Nat} -> (Fin m -> Lam Fin n) -> Lam Fin m -> Lam Fin n\nsub {m}{n} f t = gnij {ze} (fmsub (\\ {i} x -> jing {i} (weaks {Lam Fin} var (ren su) f {i} x)) (jing {ze} t))","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/FMSB.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"a22358b5e658d944b72f4ecb719b036e857e9f95","subject":"Add Data.One.Param.Binary","message":"Add Data.One.Param.Binary\n","repos":"np\/agda-parametricity","old_file":"lib\/Data\/One\/Param\/Binary.agda","new_file":"lib\/Data\/One\/Param\/Binary.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type.Param.Binary\nopen import Data.Unit renaming (\u22a4 to \ud835\udfd9; tt to 0\u2081)\n\nmodule Data.One.Param.Binary where\n\nrecord \u27e6\ud835\udfd9\u27e7 (x\u2081 x\u2082 : \ud835\udfd9) : Set\u2080 where\n  constructor \u27e60\u2081\u27e7\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/One\/Param\/Binary.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5562c5799eb809b2061aa409631dd5998bcf944a","subject":"Added an issue related to lazyness.","message":"Added an issue related to lazyness.\n\nIgnore-this: 78e6522f8fbc0febe67f3a9e58e8962c\n\ndarcs-hash:20110818183110-3bd4e-6c6348300e0e28fc130a4c545ea7dcb38068e3cd.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Issues\/Lazyness.agda","new_file":"Issues\/Lazyness.agda","new_contents":"module Issues.Lazyness where\n\n{-\n$ agda2atp Issues\/Lazyness.agda\n\nyields\n\nAn internal error has occurred. Please report this as a bug.\nLocation of the error: src\/FOL\/Translation\/Internal\/Terms.hs:486\n\nbut\n\n$ agda2atp -v 20 Issues\/Lazyness.agda\n\nyields\n\nAn internal error has occurred. Please report this as a bug.\nLocation of the error: src\/AgdaLib\/Syntax\/DeBruijn.hs:152\n\nThe internal error should be the same.\n-}\n\npostulate\n  D          : Set\n  true false : D\n  not        : D \u2192 D\n\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\ndata Bool : D \u2192 Set where\n  tB : Bool true\n  fB : Bool false\n\nfoo : \u2200 {b0 b1 b2} \u2192\n      Bool b1 \u2192\n      Bool b2 \u2192\n      b0 \u2261 not b1 \u2192\n      b2 \u2261 b2\nfoo      tB Bb2 b0-eq = refl\nfoo {b0} fB Bb2 b0-eq = refl\n  where\n  as : D\n  as = b0\n\n  bs : D\n  bs = not as\n  {-# ATP definition bs #-}\n\n  postulate bar : bs \u2261 bs\n  {-# ATP prove bar #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Issues\/Lazyness.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"b98e575a2ab3dede3805927eb714fea18c6a7764","subject":"Model some simple descriptions in Agda.","message":"Model some simple descriptions in Agda.\n","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/Ornament.agda","new_file":"models\/Ornament.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\nmodule Ornament where\n\nopen import Data.Nat using (\u2115)\nopen import Data.Fin \nopen import Data.Bool\nopen import Data.Product\nopen import Data.Unit\n\ninfix 4 _\u2261_\n\ndata _\u2261_ {a} {A : Set a} (x : A) : A \u2192 Set a where\n  refl : x \u2261 x\n\n--\n\n-- Modified from Conor's original HOAS presentation\n\ndata Desc (I : Set) : Set\u2081 where\n  -- read field in A; continue, given its value\n  -- I don't think this is really what we want -- want parametricity\n  arg : (a : Set) -> (a -> Desc I) -> Desc I\n\n  -- read an indexed subnode; continue regardless\n  rec : I -> Desc I -> Desc I\n\n  -- stop reading and return the node\u2019s index\n  ret : I -> Desc I\n\n  -- choice between n different constructors\n  choice : (n : \u2115) -> (Fin n -> Desc I) -> Desc I\n\n-- The semantics (\"extension\") of an indexed container\n\u27e6_\u27e7 : \u2200{I} -> Desc I -> (I -> Set) -> I -> Set\n\u27e6 arg A D \u27e7 R i = \u03a3 A \u03bb a \u2192 \u27e6 D a \u27e7 R i\n\u27e6 rec h D \u27e7 R i = R h \u00d7 \u27e6 D \u27e7 R i\n\u27e6 ret o \u27e7 R i = o \u2261 i\n\u27e6 choice n D \u27e7 R i =  \u03a3 (Fin n) \u03bb #n -> \u27e6 D #n \u27e7 R i\n\n\n-- This is really the least fixpoint operator\ndata Data {I}(D : Desc I)(i : I) : Set where\n  \u27e8_\u27e9 : \u27e6 D \u27e7 (Data D) i -> Data D i\n\n-- TODO think about codata \/ greatest fixed point\n-- data CoData {I}(D : Desc I)(i : I) : Set where\n--   \u27e9_\u27e8 : \n\n--\n\nNatChoiceDesc : Desc \u22a4\nNatChoiceDesc = choice 2 \u03bb\n  { zero -> ret tt\n  ; (suc n) -> rec tt (ret tt)\n  }\n\nNatChoice : Set\nNatChoice = Data NatChoiceDesc tt\n\nzeChoice : NatChoice\nzeChoice = \u27e8 zero , refl \u27e9\n\nsuChoice : NatChoice -> NatChoice\nsuChoice n = \u27e8 suc zero , n , refl \u27e9\n\n--\n\nNatDesc : Desc \u22a4\nNatDesc = arg Bool \u03bb\n  { true -> ret tt\n  ; false -> rec tt (ret tt)\n  }\n\nNat : Set\nNat = Data NatDesc tt\n\nze : Nat\nze = \u27e8 true , refl \u27e9\n\nsu : Nat -> Nat\nsu n = \u27e8 false , n , refl \u27e9\n\n--\n\nVecDesc : Set -> Desc Nat\nVecDesc X = arg Bool \u03bb\n  { false -> ret ze\n  ; true -> arg X \u03bb _ -> arg Nat \u03bb n -> rec n (ret (su n))\n  }\n\nVec : Set -> Nat -> Set\nVec X n = Data (VecDesc X) n\n\nnil : \u2200{X} -> Vec X ze\nnil = \u27e8 false , refl \u27e9\n\ncons : \u2200{X} -> (n : Nat) -> X -> Vec X n -> Vec X (su n)\ncons n x xs = \u27e8 true , x , n , xs , refl \u27e9\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Ornament.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"77cd5ac1628c7180c9d1e84bd3b372a7f15e6f48","subject":"Evens","message":"Evens\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"livecode\/Evens.agda","new_file":"livecode\/Evens.agda","new_contents":"open import Base\n\nopen import Nat\n\nopen import Logic\n\nmodule Evens  where\n\ndata Even : \u2115 \u2192 Type where \n  0even : Even 0\n  _+2even : {n : \u2115} \u2192 Even n \u2192 Even (succ (succ n))\n\n4even = (0even +2even) +2even \n\n1odd : Even 1 \u2192 False\n1odd ()\n\nmodule n|n+1even where\n  P : \u2115 \u2192 Type\n  P n = (Even n) \u2295 (Even (succ n))\n\n  step : {n : \u2115} \u2192 P n \u2192 P (succ n)\n  step (\u03b9\u2081 p) = \u03b9\u2082(p +2even)\n  step (\u03b9\u2082 p) = \u03b9\u2081 p\n\n  thm : (n : \u2115) \u2192 P n\n  thm 0 = \u03b9\u2081 0even\n  thm (succ n) = step (thm n) \n\nvalue : {n : \u2115} \u2192 Even n \u2192 \u2115\nvalue 0even = 0\nvalue (p +2even) = succ (succ (value p))\n\nhalf : (n : \u2115) \u2192 Even n \u2192 \u2115\nhalf .0 0even = 0\nhalf (succ (succ n)) (p +2even) = succ (half n p)\n\ndouble : (n : \u2115) \u2192 \u03a3 \u2115 Even\ndouble 0 = [ 0 , 0even ]\ndouble (succ n) = [ (succ (succ (proj\u2081 (double n)))) , (proj\u2082 (double n)) +2even ]\n\nmodule Collatz where\n  frompf : (n : \u2115) \u2192 n|n+1even.P n \u2192 \u2115\n  frompf n (\u03b9\u2081 p) = half n p\n  frompf n (\u03b9\u2082 p) = (3 * n) + 1\n\n  fn = \u03bb n \u2192 frompf n (n|n+1even.thm n) \n","old_contents":"","returncode":1,"stderr":"error: pathspec 'livecode\/Evens.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"2ca99e026dbcd963f37302ade1efc287a937bdf2","subject":"Added missing file.","message":"Added missing file.\n\nIgnore-this: 5677ad139f7c982e3b099d3b8d5c5236\n\ndarcs-hash:20100713210639-3bd4e-665913e7dccafb00169212791811bbcd4be0c90d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Data\/Nat\/List\/Properties.agda","new_file":"LTC\/Data\/Nat\/List\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with lists of natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Data.Nat.List.Properties where\n\nopen import LTC.Minimal\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.List.Type\nopen import LTC.Data.List\n\n------------------------------------------------------------------------------\n\n++-ListN : {ds es : D} \u2192 ListN ds \u2192 ListN es \u2192 ListN (ds ++ es)\n++-ListN {es = es} nilLN esL = prf\n  where\n    postulate prf : ListN ([] ++ es)\n    {-# ATP prove prf #-}\n\n++-ListN {es = es} (consLN {d} {ds} Nd LNds) LNes = prf (++-ListN LNds LNes)\n  where\n    postulate prf : ListN (ds ++ es) \u2192\n                    ListN ((d \u2237 ds) ++ es)\n    {-# ATP prove prf #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'LTC\/Data\/Nat\/List\/Properties.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"6d8a9d6c66d19e8055c938becd40237a317a6895","subject":"Isomorphism.agda","message":"Isomorphism.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Isomorphism.agda","new_file":"lib\/Algebra\/Group\/Isomorphism.agda","new_contents":"{- How this proof can be used for crypto, in particular ElGamal to DDH\n\n  the Group A is \u2124q with modular addition as operation\n  the Group B is the cyclic group with order q\n\n  \u03c6 is g^, the proof only need that it is a group homomorphism\n  and that it has a right inverse\n\n  we require that the explore (for type A) function (should work with only summation)\n  is Stable under addition of A (notice that we have flip in there that is so that\n  we don't need commutativity\n\n  finally we require that the explore function respects extensionality\n\n  This proof adds \u03c6\u207b\u00b9 m, because adding a constant is stable under the\n  big op F, this addition can then be pulled homomorphically through\n  f, to become a, multiplication by m.\n-}\nopen import Type using (Type_)\nopen import Level.NP\nopen import Function.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nmodule Algebra.Group.Isomorphism where\n\nmodule M\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  {c}{C  : Type c}\n  (\u03c6     : A \u2192 B)\n  (\u03c6\u207b\u00b9   : B \u2192 A)\n  (\u03c6-sur : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x)\n  (_+_   : Op\u2082 A)\n  (_*_   : Op\u2082 B)\n  (\u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y)\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F\u03c6+   : \u2200 {k} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k))\n  {m     : B}\n  where\n\n  *-stable : F \u03c6 \u2261 F (_*_ m \u2218 \u03c6)\n  *-stable =\n    F \u03c6                  \u2261\u27e8 F\u03c6+ \u27e9\n    F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 m))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 m + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                       \u03c6 (\u03c6\u207b\u00b9 m) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-sur idp \u27e9\n                                       m * \u03c6 x         \u220e) \u27e9\n    F (_*_ m \u2218 \u03c6)        \u220e\n\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\n\nrecord GroupIsomorphism \n         {a}{A  : Type a}\n         {b}{B  : Type b}\n         (G+ : Group A)\n         (G* : Group B)\n          : Set(a \u2294 b) where\n  open Additive-Group G+\n  open Multiplicative-Group G*\n\n  field\n    \u03c6 : A \u2192 B -- TODO\n    -- hom : GroupHomomorphism G+ G* \u03c6\n    \u03c6-+-* : \u2200 {x y} \u2192 \u03c6 (x + y) \u2261 \u03c6 x * \u03c6 y\n    \u03c6\u207b\u00b9   : B \u2192 A\n    \u03c6-sur : \u2200 {x} \u2192 \u03c6 (\u03c6\u207b\u00b9 x) \u2261 x\n    -- \u03c6-inj\n\nmodule N\n  {a}{A  : Type a}\n  {b}{B  : Type b}\n  {c}{C  : Type c}\n  (G+ : Group A)\n  (G* : Group B)\n  (\u03c6-iso : GroupIsomorphism G+ G*)\n  (open Additive-Group G+)\n  (open Multiplicative-Group G*)\n  (open GroupIsomorphism \u03c6-iso)\n  (F     : (A \u2192 B) \u2192 C)\n  (F=    : \u2200 {f g : A \u2192 B} \u2192 f \u2257 g \u2192 F f \u2261 F g)\n  (F+    : \u2200 {k \u03c6} \u2192 F \u03c6 \u2261 F (\u03c6 \u2218 _+_ k))\n  {m     : B}\n  where\n\n  *-stable : F \u03c6 \u2261 F (_*_ m \u2218 \u03c6)\n  *-stable =\n    F \u03c6                  \u2261\u27e8 F+ \u27e9\n    F (\u03c6 \u2218 _+_ (\u03c6\u207b\u00b9 m))  \u2261\u27e8 F= (\u03bb x \u2192 \u03c6 (\u03c6\u207b\u00b9 m + x)   \u2261\u27e8 \u03c6-+-* \u27e9\n                                       \u03c6 (\u03c6\u207b\u00b9 m) * \u03c6 x \u2261\u27e8 ap\u2082 _*_ \u03c6-sur idp \u27e9\n                                       m * \u03c6 x         \u220e) \u27e9\n    F (_*_ m \u2218 \u03c6)        \u220e\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/Group\/Isomorphism.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"caebf5bbbef0d6b532ca1f77a9b29221fd58ee9e","subject":"Added draft: Bug related with the general hints.","message":"Added draft: Bug related with the general hints.\n\nIgnore-this: 267b8d101ed62652325039228cd3388e\n\ndarcs-hash:20110207142449-3bd4e-100c2748092b672c228f89a114d3c8c79e94a6aa.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bugs\/N0.agda","new_file":"Draft\/Bugs\/N0.agda","new_contents":"module Draft.Bugs.N0 where\n\nopen import LTC.Base\nopen import LTC.Data.Nat\n\n-- TODO: Bug. The agda2atp tool does not translate the general hints.\n\n{-# ATP hint zN #-}\n{-# ATP hint sN #-}\n\npostulate\n N0 : N zero\n{-# ATP prove N0 #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Bugs\/N0.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"416d9451ca5387c8748106a46b3a786327895043","subject":"Ok Naive\/FC","message":"Ok Naive\/FC\n","repos":"zeitraffer\/agda-cats","old_file":"src\/FreeNaive\/FiniteComplete.agda","new_file":"src\/FreeNaive\/FiniteComplete.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"f68b16673b1ae5c21053f29f895f86918f0e379e","subject":"Added note on the identity type from refl and J.","message":"Added note on the identity type from refl and J.\n\nIgnore-this: bcfc109f1e68f635d7368060daf1fb3e\n\ndarcs-hash:20111001132925-3bd4e-c3c10b7438f8b4fd9502b8ac2b8f0276332d6c76.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/IdentityType.agda","new_file":"Draft\/IdentityType.agda","new_contents":"------------------------------------------------------------------------------\n-- The identity type\n------------------------------------------------------------------------------\n\n-- We can prove the properties of equality used in the formalization\n-- of FOTC, from refl and J.\n\nmodule IdentityType where\n\ninfix 7 _\u2261_\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 {x} \u2192 x \u2261 x\n  J    : (C : \u2200 x y \u2192 x \u2261 y \u2192 Set) \u2192\n         (\u2200 x \u2192 (C x x refl)) \u2192\n         \u2200 x y \u2192 (c : x \u2261 y) \u2192 C x y c\n\n-- From Thorsten's slides: A short history of equality.\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym {x} {y} = J (\u03bb x' y' _ \u2192 y' \u2261 x') (\u03bb x' \u2192 refl) x y\n\n-- From Thorsten's slides: A short history of equality.\ntrans : \u2200 {x y z} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans {x} {y} {z} = J (\u03bb x' y' _ \u2192 y' \u2261 z \u2192 x' \u2261 z) (\u03bb x' pr \u2192 pr) x y\n\nsubst : \u2200 {x} {y} \u2192 (P : D \u2192 Set) \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst {x} {y} P x\u2261y = J (\u03bb x' y' _ \u2192 P x' \u2192 P y') (\u03bb x' pr \u2192 pr) x y x\u2261y\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/IdentityType.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1d68f90eb1b88672a325396f5c379d97ff43b925","subject":"Control\/Protocol\/Alternate.agda","message":"Control\/Protocol\/Alternate.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Alternate.agda","new_file":"Control\/Protocol\/Alternate.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_) renaming (proj\u2081 to fst)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!; subst)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\nopen import Type\n\nopen import Control.Protocol.InOut\nopen import Control.Protocol.End\nimport Control.Protocol.Core as C\nopen C using (end;com)\n\nmodule Control.Protocol.Alternate where\n\ndata Proto : InOut \u2192 \u2605\u2081 where\n  end : \u2200 {io} \u2192 Proto io\n  com : (io : InOut){M : \u2605\u2080}(P : M \u2192 Proto (dual\u1d35\u1d3c io)) \u2192 Proto io\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 io {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n             {P\u2080 P\u2081}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 Proto.com io P\u2080 \u2261 com io P\u2081\n    com= io refl P= = ap (com _) (\u03bb= P=)\n\n    module _ io\n             {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 P\u2081}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io P\u2080 \u2261 com io P\u2081\n        com\u2243 = com= io (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    send= = com= Out\n    send\u2243 = com\u2243 Out\n    recv= = com= In\n    recv\u2243 = com\u2243 In\n \n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= io refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto _}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\nP\u27e6_\u27e7 : \u2200 {io} \u2192 Proto io \u2192 C.Proto\nP\u27e6 end      \u27e7 = end\nP\u27e6 com io P \u27e7 = com io \u03bb m \u2192 P\u27e6 P m \u27e7\n\n\u27e6_\u27e7 : \u2200 {io} \u2192 Proto io \u2192 \u2605\n\u27e6 P \u27e7 = C.\u27e6 P\u27e6 P \u27e7 \u27e7\n\ndual : \u2200 {io} \u2192 Proto io \u2192 Proto (dual\u1d35\u1d3c io)\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\nmodule _ {io}(P : Proto io) where\n    \u27e6_\u22a5\u27e7 : \u2605\n    \u27e6_\u22a5\u27e7 = \u27e6 dual P \u27e7\n\n    Log : \u2605\n    Log = C.\u27e6 C.source-of P\u27e6 P \u27e7 \u27e7\n\nmodule _ {{_ : FunExt}} where\n    P\u27e6_\u22a5\u27e7 : \u2200 {io}(P : Proto io) \u2192 C.dual P\u27e6 P \u27e7 \u2261 P\u27e6 dual P \u27e7\n    P\u27e6 end      \u22a5\u27e7 = refl\n    P\u27e6 com io P \u22a5\u27e7 = C.com= refl refl \u03bb m \u2192 P\u27e6 P m \u22a5\u27e7\n\n    telecom : \u2200 {io} (P : Proto io) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\n    telecom P t u = C.telecom P\u27e6 P \u27e7 t (subst C.\u27e6_\u27e7 (! P\u27e6 P \u22a5\u27e7) u)\n\nsend\u2032 : \u2605 \u2192 Proto In \u2192 Proto Out\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto Out \u2192 Proto In\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule _ {{_ : FunExt}} where\n    dual-involutive : \u2200 {io} (P : Proto io) \u2192 subst Proto (dual\u1d35\u1d3c-involutive io) (dual (dual P)) \u2261 P\n    dual-involutive {In}  end      = refl\n    dual-involutive {Out} end      = refl\n    dual-involutive       (send P) = com=\u2032 _ \u03bb m \u2192 dual-involutive (P m)\n    dual-involutive       (recv P) = com=\u2032 _ \u03bb m \u2192 dual-involutive (P m)\n\n    {-\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv dual-involutive\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n    -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Alternate.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5da781f2aa8648144fe7d2b0419f76a66d87f954","subject":"one-time-semantic-security.agda","message":"one-time-semantic-security.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"one-time-semantic-security.agda","new_file":"one-time-semantic-security.agda","new_contents":"module one-time-semantic-security where\n\nopen import Function\nopen import Data.Nat\nopen import flat-funs\nopen import Data.Vec using (splitAt)\nopen import program-distance\nopen import Data.Bits\nopen import flipbased-implem\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nCoins = \u2115\n\n-- module AbsSemSec (|M| |C| : \u2115) {FunPower : Set} {t} {T : Set t}\n--                  (\u266dFuns : PoweredFlatFuns FunPower T) where\nmodule AbsSemSec (|M| |C| : \u2115) {t} {T : Set t}\n                 (\u266dFuns : FlatFuns T) where\n\n  open FlatFuns \u266dFuns\n\n  M = `Bits |M|\n  C = `Bits |C|\n\n  record AbsSemSecAdv (|R| : Coins) : Set where\n    constructor mk\n\n    field\n      {|S|} : \u2115\n\n    S = `Bits |S|\n    R = `Bits |R|\n\n    field\n      step\u2080 : R `\u2192 (M `\u00d7 M) `\u00d7 S\n      step\u2081 : C `\u00d7 S `\u2192 `Bit\n      -- step\u2080 : \u27e8 p\u2080 \u27e9 R \u219d (M `\u00d7 M) `\u00d7 S\n      -- step\u2081 : \u27e8 p\u2081 \u27e9 C `\u00d7 S \u219d `Bit\n\n  SemSecReduction : \u2200 (f : Coins \u2192 Coins) \u2192 Set\n  SemSecReduction f = \u2200 {c} \u2192 AbsSemSecAdv c \u2192 AbsSemSecAdv (f c)\n\n  -- runAdv is not the good name\n  -- runAdv : \u2200 {p} (A : AbsSemSecAdv p) \u2192 \u27e8 ? \u27e9 (C `\u00d7 AbsSemSecAdv.R A) \u219d (M `\u00d7 M) `\u00d7 Bit\n  -- runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n{-\nmodule FFF |M| |C| {\u266dFuns : FlatFuns \u22a4 Set}\n               (\u266dops : FlatFunsOps _ \u266dFuns)\n               (run : \u2200 {i o} \u2192 FlatFuns.\u27e8_\u27e9_\u219d_ \u266dFuns _ i o \u2192 i \u2192 o) where\n  open AbsSemSec |M| |C| \u266dFuns\n  module FunSemSecAdv {|R|} (A : AbsSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n    open FlatFunsOps \u266dops\n    step\u2080O : \u27e8 _ \u27e9 R \u219d ((\u27e8 _ \u27e9 Bit \u219d M) \u00d7 S)\n    step\u2080O = {!run step\u2080!} >>> {!proj!} *** idO\n-}\n\n-- Here we use Agda functions for FlatFuns.\nmodule FunSemSec (prgDist : PrgDist) (|M| |C| : \u2115) where\n  open PrgDist prgDist\n  open AbsSemSec |M| |C| fun\u266dFuns\n  open FlatFunsOps fun\u266dOps\n\n  M\u00b2 = Bit \u2192 M\n\n  Enc : \u2200 cc \u2192 Set\n  Enc cc = M \u2192 \u21ba cc C\n\n  Tr : (cc\u2080 cc\u2081 : Coins) \u2192 Set\n  Tr cc\u2080 cc\u2081 = Enc cc\u2080 \u2192 Enc cc\u2081\n\n  FunSemSecAdv : Coins \u2192 Set\n  -- FunSemSecAdv |R| = AbsSemSecAdv (mk _ _ |R|)\n  FunSemSecAdv = AbsSemSecAdv\n\n  module FunSemSecAdv {|R|} (A : FunSemSecAdv |R|) where\n    open AbsSemSecAdv A public\n\n    step\u2080F : R \u2192 (M\u00b2 \u00d7 S)\n    step\u2080F = step\u2080 >>> proj *** idO\n\n    step\u2080\u21ba : \u21ba |R| (M\u00b2 \u00d7 S)\n    step\u2080\u21ba = mk step\u2080F\n\n    step\u2081F : S \u2192 C \u2192 Bit\n    step\u2081F s c = step\u2081 (c , s)\n\n  -- Returing 0 means Chal wins, Adv looses\n  --          1 means Adv  wins, Chal looses\n  runSemSec : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  runSemSec E A b\n    = A-step\u2080 >>= \u03bb { (m , s) \u2192 map\u21ba (A-step\u2081 s) (E (m b)) }\n    where open FunSemSecAdv A renaming (step\u2080\u21ba to A-step\u2080; step\u2081F to A-step\u2081)\n\n  _\u21c4_ : \u2200 {cc ca} (E : Enc cc) (A : FunSemSecAdv ca) b \u2192 \u21ba (ca + cc) Bit\n  _\u21c4_ = runSemSec\n\n  runAdv : \u2200 {|R|} \u2192 FunSemSecAdv |R| \u2192 C \u2192 Bits |R| \u2192 (M \u00d7 M) \u00d7 Bit\n  runAdv (mk A-step\u2080 A-step\u2081) C = A-step\u2080 >>> id *** (const C &&& id >>> A-step\u2081)\n\n  _\u2257A_ : \u2200 {p} (A\u2081 A\u2082 : FunSemSecAdv p) \u2192 Set\n  A\u2080 \u2257A A\u2081 = \u2200 C R \u2192 runAdv A\u2080 C R \u2261 runAdv A\u2081 C R\n\n  change-adv : \u2200 {cc ca} {E : Enc cc} {A\u2081 A\u2082 : FunSemSecAdv ca} \u2192 A\u2081 \u2257A A\u2082 \u2192 (E \u21c4 A\u2081) \u2257\u2141 (E \u21c4 A\u2082)\n  change-adv {ca = ca} {A\u2081 = _} {_} pf b R with splitAt ca R\n  change-adv {E = E} {A\u2081} {A\u2082} pf b ._ | pre \u03a3., post , refl = trans (cong proj\u2082 (helper\u2080 A\u2081)) helper\u2082\n     where open FunSemSecAdv\n           helper\u2080 = \u03bb A \u2192 pf (run\u21ba (E (proj (proj\u2081 (step\u2080 A pre)) b)) post) pre\n           helper\u2082 = cong (\u03bb m \u2192 step\u2081 A\u2082 (run\u21ba (E (proj (proj\u2081 m) b)) post , proj\u2082 (step\u2080 A\u2082 pre)))\n                          (helper\u2080 A\u2082)\n\n  module Unused (Power : Set) (coins : Power \u2192 Coins) where\n  -- NP: not sure about these two. such a function has acess to the two encryption\n\n    SemBroken : \u2200 {cc} (E : Enc cc) \u2192 Power \u2192 Set\n    SemBroken E power = \u2203 (\u03bb (A : FunSemSecAdv (coins power)) \u2192 breaks (E \u21c4 A))\n\n    -- functions.\n    -- SemSecReduction p\u2080 p\u2081 E\u2080 E\u2081:\n    --   security of E\u2080 reduces to security of E\u2081\n    --   breaking E\u2081 reduces to breaking E\u2080\n    SemSecReduction' : \u2200 (p\u2080 p\u2081 : Power) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n    SemSecReduction' p\u2080 p\u2081 E\u2080 E\u2081 = SemBroken E\u2081 p\u2081 \u2192 SemBroken E\u2080 p\u2080\n\n  SafeSemSecReduction : \u2200 (f : Coins \u2192 Coins) {cc\u2080 cc\u2081} (E\u2080 : Enc cc\u2080) (E\u2081 : Enc cc\u2081) \u2192 Set\n  SafeSemSecReduction f E\u2080 E\u2081 =\n     \u2203 \u03bb (red : SemSecReduction f) \u2192\n       \u2200 {c} A \u2192 (E\u2080 \u21c4 A) \u21d3 (E\u2081 \u21c4 red {c} A)\n\n  SemSecTr : \u2200 {cc\u2080 cc\u2081} (f : Coins \u2192 Coins) (tr : Tr cc\u2080 cc\u2081) \u2192 Set\n  SemSecTr {cc\u2080} f tr = \u2200 {E : Enc cc\u2080} \u2192 SafeSemSecReduction f (tr E) E\n\n  -- SemSecReductionToOracle : \u2200 (p\u2080 p\u2081 : Power) {cc} (E : Enc cc) \u2192 Set\n  -- SemSecReductionToOracle = SemBroken E p\u2080 \u2192 Oracle p\u2081\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'one-time-semantic-security.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"54d8dc027f6ebe86986e35cd77afa0637fa7c890","subject":"Data\/Tree\/Binary\/Perfect\/Any.agda","message":"Data\/Tree\/Binary\/Perfect\/Any.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Data\/Tree\/Binary\/Perfect\/Any.agda","new_file":"Data\/Tree\/Binary\/Perfect\/Any.agda","new_contents":"module Data.Tree.Binary.Perfect.Any where\n\nopen import Level\nopen import Relation.Binary.PropositionalEquality\nopen import prefect-bintree\n\ndata Any {a p} {A : Set a} (P : A \u2192 Set p) : \u2200 {n} \u2192 Tree A n \u2192 Set (a \u2294 p) where\n  leaf  : \u2200 {x} (Px : P x) \u2192 Any P (leaf x)\n  left  : \u2200 {n} {t\u2080 t\u2081 : Tree A n} \u2192 Any P t\u2080 \u2192 Any P (fork t\u2080 t\u2081)\n  right : \u2200 {n} {t\u2080 t\u2081 : Tree A n} \u2192 Any P t\u2081 \u2192 Any P (fork t\u2080 t\u2081)\n\npattern here = Any.leaf\n\ndata All {a p} {A : Set a} (P : A \u2192 Set p) : \u2200 {n} \u2192 Tree A n \u2192 Set (a \u2294 p) where\n  leaf  : \u2200 {x} (Px : P x) \u2192 All P (leaf x)\n  fork  : \u2200 {n} {t\u2080 t\u2081 : Tree A n} \u2192 All P t\u2080 \u2192 All P t\u2081 \u2192 All P (fork t\u2080 t\u2081)\n\n_\u2208_ : \u2200 {a} {A : Set a} (x : A) {n} \u2192 Tree A n \u2192 Set a\nx \u2208 t = Any (_\u2261_ x) t\n\nlookup : \u2200 x t \u2192 Dec (x \u2208 t)\n\nP \u21d2 Q \u2192 All P t \u2192 All Q (map f t)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Data\/Tree\/Binary\/Perfect\/Any.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"543017049d45e83c1fe56c307cfcb64b0c651d76","subject":"added Darais","message":"added Darais\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"src\/extra\/Darais.agda","new_file":"src\/extra\/Darais.agda","new_contents":"-- Author: David Darais\n--\n-- This is a dependent de Bruijn encoding of STLC with proofs for\n-- progress and preservation. This file has zero dependencies and is\n-- 100% self-contained.\n--\n-- Because there is only a notion of well-typed terms (non-well-typed\n-- terms do not exist), preservation is merely by observation that\n-- substitution (i.e., cut) can be defined.\n--\n-- A small-step evaluation semantics is defined after the definition of cut.\n--\n-- Progress is proved w.r.t. the evaluation semantics.\n--\n-- Some ideas for extensions or homeworks are given at the end.\n--\n-- A few helper definitions are required.\n-- * Basic Agda constructions (like booleans, products, dependent sums,\n--   and lists) are defined first in a Prelude module which is\n--   immediately opened.\n-- * Binders (x : \u03c4 \u2208 \u0393) are proofs that the element \u03c4 is contained in\n--   the list of types \u0393. Helper functions for weakening and\n--   introducing variables into contexts which are reusable are\n--   defined in the Prelude. Helpers for weakening terms are defined\n--   below. Some of the non-general helpers (like cut[\u2208]) could be\n--   defined in a generic way to be reusable, but I decided against\n--   this to keep things simple.\n\nmodule Darais where\n\nopen import Agda.Primitive using (_\u2294_)\n\nmodule Prelude where\n\n  infixr 3 \u2203\ud835\udc60\ud835\udc61\n  infixl 5 _\u2228_\n  infix 10 _\u2208_\n  infixl 15 _\u29fa_\n  infixl 15 _\u229f_\n  infixl 15 _\u223e_\n  infixr 20 _\u2237_\n  \n  data \ud835\udd39 : Set where\n    T : \ud835\udd39\n    F : \ud835\udd39\n\n  data _\u2228_ {\u2113\u2081 \u2113\u2082} (A : Set \u2113\u2081) (B : Set \u2113\u2082) : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    Inl : A \u2192 A \u2228 B\n    Inr : B \u2192 A \u2228 B\n\n  syntax \u2203\ud835\udc60\ud835\udc61 A (\u03bb x \u2192 B) = \u2203 x \u2982 A \ud835\udc60\ud835\udc61 B\n  data \u2203\ud835\udc60\ud835\udc61 {\u2113\u2081 \u2113\u2082} (A : Set \u2113\u2081) (B : A \u2192 Set \u2113\u2082) : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n    \u27e8\u2203_,_\u27e9 : \u2200 (x : A) \u2192 B x \u2192 \u2203 x \u2982 A \ud835\udc60\ud835\udc61 B x\n\n  data \u27ec_\u27ed {\u2113} (A : Set \u2113) : Set \u2113 where\n    [] : \u27ec A \u27ed\n    _\u2237_ : A \u2192 \u27ec A \u27ed \u2192 \u27ec A \u27ed\n\n  _\u29fa_ : \u2200 {\u2113} {A : Set \u2113} \u2192 \u27ec A \u27ed \u2192 \u27ec A \u27ed \u2192 \u27ec A \u27ed\n  [] \u29fa ys = ys\n  x \u2237 xs \u29fa ys = x \u2237 (xs \u29fa ys)\n\n  _\u223e_ : \u2200 {\u2113} {A : Set \u2113} \u2192 \u27ec A \u27ed \u2192 \u27ec A \u27ed \u2192 \u27ec A \u27ed\n  [] \u223e ys = ys\n  (x \u2237 xs) \u223e ys = xs \u223e x \u2237 ys\n\n  data _\u2208_ {\u2113} {A : Set \u2113} (x : A) : \u2200 (xs : \u27ec A \u27ed) \u2192 Set \u2113 where\n    Z : \u2200 {xs} \u2192 x \u2208 x \u2237 xs\n    S : \u2200 {x\u2032 xs} \u2192 x \u2208 xs \u2192 x \u2208 x\u2032 \u2237 xs\n\n  _\u229f_ : \u2200 {\u2113} {A : Set \u2113} (xs : \u27ec A \u27ed) {x} \u2192 x \u2208 xs \u2192 \u27ec A \u27ed\n  x \u2237 xs \u229f Z = xs\n  x \u2237 xs \u229f S \u03b5 = x \u2237 (xs \u229f \u03b5)\n\n  wk[\u2208] : \u2200 {\u2113} {A : Set \u2113} {xs : \u27ec A \u27ed} {x : A} xs\u2032 \u2192 x \u2208 xs \u2192 x \u2208 xs\u2032 \u223e xs\n  wk[\u2208] [] x = x\n  wk[\u2208] (x\u2032 \u2237 xs) x = wk[\u2208] xs (S x)\n\n  i[\u2208][_] : \u2200 {\u2113} {A : Set \u2113} {xs : \u27ec A \u27ed} {x x\u2032 : A} (\u03b5\u2032 : x\u2032 \u2208 xs) \u2192 x \u2208 xs \u229f \u03b5\u2032 \u2192 x \u2208 xs\n  i[\u2208][ Z ] x = S x \n  i[\u2208][ S \u03b5\u2032 ] Z = Z\n  i[\u2208][ S \u03b5\u2032 ] (S x) = S (i[\u2208][ \u03b5\u2032 ] x)\n\nopen Prelude\n\n-- ============================ --\n-- Simply Typed Lambda Calculus --\n-- ============================ --\n\n-- A dependent de Bruijn encoding\n-- Or, the dynamic semantics of natural deduction as seen through Curry-Howard\n\n-- types --\n\ndata type : Set where\n  \u27e8\ud835\udd39\u27e9 : type\n  _\u27e8\u2192\u27e9_ : type \u2192 type \u2192 type\n\n-- terms --\n\ninfix 10 _\u22a2_\ndata _\u22a2_ : \u2200 (\u0393 : \u27ec type \u27ed) (\u03c4 : type) \u2192 Set where\n  \u27e8\ud835\udd39\u27e9 : \u2200 {\u0393}\n    (b : \ud835\udd39)\n    \u2192 \u0393 \u22a2 \u27e8\ud835\udd39\u27e9\n  \u27e8if\u27e9_\u2774_\u2775\u2774_\u2775 : \u2200 {\u0393 \u03c4} \n    (e\u2081 : \u0393 \u22a2 \u27e8\ud835\udd39\u27e9)\n    (e\u2082 : \u0393 \u22a2 \u03c4)\n    (e\u2083 : \u0393 \u22a2 \u03c4)\n    \u2192 \u0393 \u22a2 \u03c4\n  Var : \u2200 {\u0393 \u03c4}\n    (x : \u03c4 \u2208 \u0393)\n    \u2192 \u0393 \u22a2 \u03c4\n  \u27e8\u03bb\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082}\n    (e : \u03c4\u2081 \u2237 \u0393 \u22a2 \u03c4\u2082)\n    \u2192 \u0393 \u22a2 \u03c4\u2081 \u27e8\u2192\u27e9 \u03c4\u2082\n  _\u27e8\u22c5\u27e9_ : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082}\n    (e\u2081 : \u0393 \u22a2 \u03c4\u2081 \u27e8\u2192\u27e9 \u03c4\u2082)\n    (e\u2082 : \u0393 \u22a2 \u03c4\u2081)\n    \u2192 \u0393 \u22a2 \u03c4\u2082\n\n-- introducing a new variable to the context --\n\ni[\u22a2][_] : \u2200 {\u0393 \u03c4 \u03c4\u2032} (x\u2032 : \u03c4\u2032 \u2208 \u0393) \u2192 \u0393 \u229f x\u2032 \u22a2 \u03c4 \u2192 \u0393 \u22a2 \u03c4\ni[\u22a2][ x\u2032 ] (\u27e8\ud835\udd39\u27e9 b) = \u27e8\ud835\udd39\u27e9 b\ni[\u22a2][ x\u2032 ] \u27e8if\u27e9 e\u2081 \u2774 e\u2082 \u2775\u2774 e\u2083 \u2775 = \u27e8if\u27e9 i[\u22a2][ x\u2032 ] e\u2081 \u2774 i[\u22a2][ x\u2032 ] e\u2082 \u2775\u2774 i[\u22a2][ x\u2032 ] e\u2083 \u2775\ni[\u22a2][ x\u2032 ] (Var x) = Var (i[\u2208][ x\u2032 ] x)\ni[\u22a2][ x\u2032 ] (\u27e8\u03bb\u27e9 e) = \u27e8\u03bb\u27e9 (i[\u22a2][ S x\u2032 ] e)\ni[\u22a2][ x\u2032 ] (e\u2081 \u27e8\u22c5\u27e9 e\u2082) = i[\u22a2][ x\u2032 ] e\u2081 \u27e8\u22c5\u27e9 i[\u22a2][ x\u2032 ] e\u2082\n\ni[\u22a2] : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 \u0393 \u22a2 \u03c4 \u2192 \u03c4\u2032 \u2237 \u0393 \u22a2 \u03c4\ni[\u22a2] = i[\u22a2][ Z ]\n\n-- substitution for variables --\n\ncut[\u2208]<_>[_] : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (x : \u03c4\u2081 \u2208 \u0393) \u0393\u2032 \u2192 \u0393\u2032 \u223e (\u0393 \u229f x) \u22a2 \u03c4\u2081 \u2192 \u03c4\u2082 \u2208 \u0393 \u2192 \u0393\u2032 \u223e (\u0393 \u229f x) \u22a2 \u03c4\u2082\ncut[\u2208]< Z >[ \u0393\u2032 ] e Z = e\ncut[\u2208]< Z >[ \u0393\u2032 ] e (S y) = Var (wk[\u2208] \u0393\u2032 y)\ncut[\u2208]< S x\u2032 >[ \u0393\u2032 ] e Z = Var (wk[\u2208] \u0393\u2032 Z)\ncut[\u2208]< S x\u2032 >[ \u0393\u2032 ] e (S x) = cut[\u2208]< x\u2032 >[ _ \u2237 \u0393\u2032 ] e x\n\ncut[\u2208]<_> : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (x : \u03c4\u2081 \u2208 \u0393) \u2192 \u0393 \u229f x \u22a2 \u03c4\u2081 \u2192 \u03c4\u2082 \u2208 \u0393 \u2192 \u0393 \u229f x \u22a2 \u03c4\u2082\ncut[\u2208]< x\u2032 > = cut[\u2208]< x\u2032 >[ [] ]\n\n-- substitution for terms --\n\ncut[\u22a2][_] : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (x : \u03c4\u2081 \u2208 \u0393) \u2192 \u0393 \u229f x \u22a2 \u03c4\u2081 \u2192 \u0393 \u22a2 \u03c4\u2082 \u2192 \u0393 \u229f x \u22a2 \u03c4\u2082\ncut[\u22a2][ x\u2032 ] e\u2032 (\u27e8\ud835\udd39\u27e9 b) = \u27e8\ud835\udd39\u27e9 b\ncut[\u22a2][ x\u2032 ] e\u2032 \u27e8if\u27e9 e\u2081 \u2774 e\u2082 \u2775\u2774 e\u2083 \u2775 = \u27e8if\u27e9 cut[\u22a2][ x\u2032 ] e\u2032 e\u2081 \u2774 cut[\u22a2][ x\u2032 ] e\u2032 e\u2082 \u2775\u2774 cut[\u22a2][ x\u2032 ] e\u2032 e\u2083 \u2775\ncut[\u22a2][ x\u2032 ] e\u2032 (Var x) = cut[\u2208]< x\u2032 > e\u2032 x\ncut[\u22a2][ x\u2032 ] e\u2032 (\u27e8\u03bb\u27e9 e) = \u27e8\u03bb\u27e9 (cut[\u22a2][ S x\u2032 ] (i[\u22a2] e\u2032) e)\ncut[\u22a2][ x\u2032 ] e\u2032 (e\u2081 \u27e8\u22c5\u27e9 e\u2082) = cut[\u22a2][ x\u2032 ] e\u2032 e\u2081 \u27e8\u22c5\u27e9 cut[\u22a2][ x\u2032 ] e\u2032 e\u2082\n\ncut[\u22a2] : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 \u0393 \u22a2 \u03c4\u2081 \u2192 \u03c4\u2081 \u2237 \u0393 \u22a2 \u03c4\u2082 \u2192 \u0393 \u22a2 \u03c4\u2082\ncut[\u22a2] = cut[\u22a2][ Z ]\n\n-- values --\n\ndata value {\u0393} : \u2200 {\u03c4} \u2192 \u0393 \u22a2 \u03c4 \u2192 Set where\n  \u27e8\ud835\udd39\u27e9 : \u2200 b \u2192 value (\u27e8\ud835\udd39\u27e9 b)\n  \u27e8\u03bb\u27e9 : \u2200 {\u03c4 \u03c4\u2032} (e : \u03c4\u2032 \u2237 \u0393 \u22a2 \u03c4) \u2192 value (\u27e8\u03bb\u27e9 e)\n\n-- CBV evaluation for terms --\n-- (borrowing some notation and style from Wadler: https:\/\/wenkokke.github.io\/sf\/Stlc)\n\ninfix 10 _\u219d_\ndata _\u219d_ {\u0393 \u03c4} : \u0393 \u22a2 \u03c4 \u2192 \u0393 \u22a2 \u03c4 \u2192 Set where\n  \u03be\u22c5\u2081 : \u2200 {\u03c4\u2032} {e\u2081 e\u2081\u2032 : \u0393 \u22a2 \u03c4\u2032 \u27e8\u2192\u27e9 \u03c4} {e\u2082 : \u0393 \u22a2 \u03c4\u2032}\n    \u2192 e\u2081 \u219d e\u2081\u2032\n    \u2192 e\u2081 \u27e8\u22c5\u27e9 e\u2082 \u219d e\u2081\u2032 \u27e8\u22c5\u27e9 e\u2082\n  \u03be\u22c5\u2082 : \u2200 {\u03c4\u2032} {e\u2081 : \u0393 \u22a2 \u03c4\u2032 \u27e8\u2192\u27e9 \u03c4} {e\u2082 e\u2082\u2032 : \u0393 \u22a2 \u03c4\u2032}\n    \u2192 value e\u2081\n    \u2192 e\u2082 \u219d e\u2082\u2032\n    \u2192 e\u2081 \u27e8\u22c5\u27e9 e\u2082 \u219d e\u2081 \u27e8\u22c5\u27e9 e\u2082\u2032\n  \u03b2\u03bb : \u2200 {\u03c4\u2032} {e\u2081 : \u03c4\u2032 \u2237 \u0393 \u22a2 \u03c4} {e\u2082 : \u0393 \u22a2 \u03c4\u2032}\n    \u2192 value e\u2082\n    \u2192 \u27e8\u03bb\u27e9 e\u2081 \u27e8\u22c5\u27e9 e\u2082 \u219d cut[\u22a2] e\u2082 e\u2081\n  \u03beif : \u2200 {e\u2081 e\u2081\u2032 : \u0393 \u22a2 \u27e8\ud835\udd39\u27e9} {e\u2082 e\u2083 : \u0393 \u22a2 \u03c4}\n    \u2192 e\u2081 \u219d e\u2081\u2032\n    \u2192 \u27e8if\u27e9 e\u2081 \u2774 e\u2082 \u2775\u2774 e\u2083 \u2775 \u219d \u27e8if\u27e9 e\u2081\u2032 \u2774 e\u2082 \u2775\u2774 e\u2083 \u2775\n  \u03beif-T : \u2200 {e\u2082 e\u2083 : \u0393 \u22a2 \u03c4}\n    \u2192 \u27e8if\u27e9 \u27e8\ud835\udd39\u27e9 T \u2774 e\u2082 \u2775\u2774 e\u2083 \u2775 \u219d e\u2082\n  \u03beif-F : \u2200 {e\u2082 e\u2083 : \u0393 \u22a2 \u03c4}\n    \u2192 \u27e8if\u27e9 \u27e8\ud835\udd39\u27e9 F \u2774 e\u2082 \u2775\u2774 e\u2083 \u2775 \u219d e\u2083\n\n-- progress --\n\nprogress : \u2200 {\u03c4} (e : [] \u22a2 \u03c4) \u2192 value e \u2228 (\u2203 e\u2032 \u2982 [] \u22a2 \u03c4 \ud835\udc60\ud835\udc61 e \u219d e\u2032)\nprogress (\u27e8\ud835\udd39\u27e9 b) = Inl (\u27e8\ud835\udd39\u27e9 b)\nprogress \u27e8if\u27e9 e \u2774 e\u2081 \u2775\u2774 e\u2082 \u2775 with progress e\n\u2026 | Inl (\u27e8\ud835\udd39\u27e9 T) = Inr \u27e8\u2203 e\u2081 , \u03beif-T \u27e9\n\u2026 | Inl (\u27e8\ud835\udd39\u27e9 F) = Inr \u27e8\u2203 e\u2082 , \u03beif-F \u27e9\n\u2026 | Inr \u27e8\u2203 e\u2032 , \u03b5 \u27e9 = Inr \u27e8\u2203 \u27e8if\u27e9 e\u2032 \u2774 e\u2081 \u2775\u2774 e\u2082 \u2775 , \u03beif \u03b5 \u27e9\nprogress (Var ())\nprogress (\u27e8\u03bb\u27e9 e) = Inl (\u27e8\u03bb\u27e9 e)\nprogress (e\u2081 \u27e8\u22c5\u27e9 e\u2082) with progress e\u2081 \n\u2026 | Inr \u27e8\u2203 e\u2081\u2032 , \u03b5 \u27e9 = Inr \u27e8\u2203 e\u2081\u2032 \u27e8\u22c5\u27e9 e\u2082 , \u03be\u22c5\u2081 \u03b5 \u27e9\n\u2026 | Inl (\u27e8\u03bb\u27e9 e) with progress e\u2082\n\u2026 | Inl x = Inr \u27e8\u2203 cut[\u22a2] e\u2082 e , \u03b2\u03bb x \u27e9\n\u2026 | Inr \u27e8\u2203 e\u2082\u2032 , \u03b5 \u27e9 = Inr \u27e8\u2203 e\u2081 \u27e8\u22c5\u27e9 e\u2082\u2032 , \u03be\u22c5\u2082 (\u27e8\u03bb\u27e9 e) \u03b5 \u27e9 \n\n-- Some ideas for possible extensions or homework assignments\n-- 1. A. Write a conversion from the dependent de Bruijn encoding (e : \u0393 \u22a2 \u03c4)\n--       to the untyped lambda calculus (e : term).\n--    B. Prove that the image of this conversion is well typed.\n--    C. Write a conversion from well-typed untyped lambda calculus\n--       terms ([e : term] and [\u03b5 : (\u0393 \u22a2 e \u2982 \u03c4)] into the dependent de\n--       Bruijn encoding (e : \u0393 \u22a2 \u03c4)\n-- 2. A. Write a predicate analogous to 'value' for strongly reduced\n--       terms (reduction under lambda)\n--    B. Prove \"strong\" progress: A term is either fully beta-reduced\n--       (including under lambda) or it can take a step\n-- 3. Relate this semantics to a big-step semantics.\n-- 4. Prove strong normalization.\n-- 5. Relate this semantics to a definitional interpreter into Agda's Set.\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/extra\/Darais.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"af28fb5fc8600f8fe31e581fd3c6a9b2c0906e99","subject":"Added note on ordinals.","message":"Added note on ordinals.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Ordinals.agda","new_file":"notes\/Ordinals.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nFrom: Hancock (2008). (Ordinal-theoretic) proof theory.\n\nmodule Ordinals where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata \u03a9 : Set where\n  zero : \u03a9\n  succ : \u03a9 \u2192 \u03a9\n  limN : (\u2115 \u2192 \u03a9) \u2192 \u03a9\n\ninj : \u2115 \u2192 \u03a9\ninj zero     = zero\ninj (succ n) = succ (inj n)\n\n\u03c9 : \u03a9\n\u03c9 = limN (\u03bb n \u2192 inj n)\n\n\u03c9+1 : \u03a9\n\u03c9+1 = succ \u03c9\n\ndata \u03a9\u2082 : Set where\n  zero : \u03a9\u2082\n  succ : \u03a9\u2082 \u2192 \u03a9\u2082\n  limN : (\u2115 \u2192 \u03a9\u2082) \u2192 \u03a9\u2082\n  lim\u03a9 : (\u03a9 \u2192 \u03a9\u2082) \u2192 \u03a9\u2082\n\ninj\u2081 : \u03a9 \u2192 \u03a9\u2082\ninj\u2081 zero     = zero\ninj\u2081 (succ o) = succ (inj\u2081 o)\ninj\u2081 (limN f) = limN (\u03bb n \u2192 inj\u2081 (f n))\n\n\u03c9\u2081 : \u03a9\u2082\n\u03c9\u2081 = lim\u03a9 (\u03bb o \u2192 inj\u2081 o)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/Ordinals.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1b0c01042c20c8220d71a8a84c759c4187dd6aff","subject":"proof sketch of otp-lem","message":"proof sketch of otp-lem\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-lem.agda","new_file":"otp-lem.agda","new_contents":"module otp-lem where\n\nopen import Level\nopen import Algebra.FunctionProperties\nopen import Data.Product\n\nopen import Function using (_\u2218_ ; flip)\nopen import Function.Inverse as Inv using (_\u2194_; module Inverse)\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Search.Type\nopen import Search.Sum\n\nrecord Group (G : Set) : Set where\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    -_  : G \u2192 G\n\n  -- laws\n  field\n    assoc : Associative _\u2261_ _\u2219_\n    identity : Identity _\u2261_ \u03b5 _\u2219_\n    inverse  : Inverse _\u2261_ \u03b5 -_ _\u2219_\n\n\nGroupHomomorphism : \u2200 {A B : Set} \u2192 Group A \u2192 Group B \u2192(A \u2192 B) \u2192 Set\nGroupHomomorphism GA GB f = \u2200 x y \u2192 f (x + y) \u2261 f x * f y\n  where\n    open Group GA renaming (_\u2219_ to _+_)\n    open Group GB renaming (_\u2219_ to _*_)\n\nmodule _ {A B}(GA : Group A)(GB : Group B)(f : A \u2192 B)(searchA : Search zero A)(f-homo : GroupHomomorphism GA GB f)([f] : B \u2192 A)(f-sur : \u2200 b \u2192 f ([f] b) \u2261 b)\n              (sui : \u2200 k \u2192 StableUnder searchA (flip (Group._\u2219_ GA) k))(search-ext : SearchExt searchA) where\n  open Group GA using (-_) renaming (_\u2219_ to _+_ ; \u03b5 to 0g)\n  open Group GB using ()   renaming (_\u2219_ to _*_ ; \u03b5 to 1g ; -_ to 1\/_)\n\n  help : \u2200 x y \u2192 x \u2261 (x * y) * 1\/ y\n  help x y = x\n           \u2261\u27e8 sym (proj\u2082 identity x) \u27e9\n             x * 1g\n           \u2261\u27e8 cong (_*_ x) (sym (proj\u2082 inverse y)) \u27e9\n             x * (y * 1\/ y)\n           \u2261\u27e8 sym (assoc x y (1\/ y)) \u27e9\n             (x * y) * (1\/ y)\n           \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  unique-1g : \u2200 x y \u2192 x * y \u2261 y \u2192 x \u2261 1g\n  unique-1g x y eq = x\n                   \u2261\u27e8 help x y \u27e9\n                     (x * y) * 1\/ y\n                   \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                     y * 1\/ y\n                   \u2261\u27e8 proj\u2082 inverse y \u27e9\n                     1g\n                   \u220e\n    where open \u2261-Reasoning\n          open Group GB\n\n  unique-\/ : \u2200 x y \u2192 x * y \u2261 1g \u2192 x \u2261 1\/ y\n  unique-\/ x y eq = x\n                  \u2261\u27e8 help x y \u27e9\n                    (x * y) * 1\/ y\n                  \u2261\u27e8 cong (flip _*_ (1\/ y)) eq \u27e9\n                    1g * 1\/ y\n                  \u2261\u27e8 proj\u2081 identity (1\/ y) \u27e9\n                    1\/ y\n                  \u220e\n    where open \u2261-Reasoning\n          open Group GB\n    \n  f-pres-\u03b5 : f 0g \u2261 1g\n  f-pres-\u03b5 = unique-1g (f 0g) (f 0g) part\n    where open \u2261-Reasoning\n          open Group GA\n          part = f 0g * f 0g\n               \u2261\u27e8 sym (f-homo 0g 0g) \u27e9\n                 f (0g + 0g)\n               \u2261\u27e8 cong f (proj\u2081 identity 0g) \u27e9\n                 f 0g\n               \u220e\n\n  f-pres-inv : \u2200 x \u2192 f (- x) \u2261 1\/ f x\n  f-pres-inv x = unique-\/ (f (- x)) (f x) part\n    where open \u2261-Reasoning\n          open Group GA hiding (-_)\n          part = f (- x) * f x\n               \u2261\u27e8 sym (f-homo (- x) x) \u27e9\n                 f (- x + x)\n               \u2261\u27e8 cong f (proj\u2081 inverse x) \u27e9\n                 f 0g\n               \u2261\u27e8 f-pres-\u03b5 \u27e9\n                 1g\n               \u220e\n  \n  -- this proof isn't actually any hard..\n  thm : \u2200 {X}(op : X \u2192 X \u2192 X)(O : B \u2192 X) m\u2080 m\u2081 \u2192 searchA op (\u03bb x \u2192 O (f x * m\u2080)) \u2261 searchA op (\u03bb x \u2192 O (f x * m\u2081))\n  thm op O m\u2080 m\u2081 = searchA op (\u03bb x \u2192 O (f x * m\u2080))\n                 \u2261\u27e8 sui (- [f] m\u2080) op (\u03bb x \u2192 O (f x * m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + - [f] m\u2080)  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong (\u03bb p \u2192 O (p * m\u2080)) (f-homo x (- [f] m\u2080))) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * f(- [f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O ((f x * p) * m\u2080))) (f-pres-inv ([f] m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O ((f x * 1\/ f([f] m\u2080))  * m\u2080))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (Group.assoc GB (f x) (1\/ f ([f] m\u2080)) m\u2080)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ f([f] m\u2080)  * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * (1\/ p * m\u2080)))) (f-sur m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * (1\/ m\u2080 * m\u2080)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (proj\u2081 (Group.inverse GB) m\u2080) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * 1g))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (proj\u2082 (Group.identity GB) (f x))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x ))\n                 \u2261\u27e8 sui ([f] m\u2081) op (\u03bb x \u2192 O (f x)) \u27e9\n                   searchA op (\u03bb x \u2192 O (f (x + [f] m\u2081)))\n                 \u2261\u27e8 search-ext op (\u03bb x \u2192 cong O (f-homo x ([f] m\u2081))) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * f ([f] m\u2081)))\n                 \u2261\u27e8 cong (\u03bb p \u2192 searchA op (\u03bb x \u2192 O (f x * p))) (f-sur m\u2081) \u27e9\n                   searchA op (\u03bb x \u2192 O (f x * m\u2081))\n                 \u220e\n    where\n      open \u2261-Reasoning\n      \n","old_contents":"","returncode":1,"stderr":"error: pathspec 'otp-lem.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"096a62a0da559691d35111bb2b739d783415b815","subject":"Initial commit: Embed validity proofs in \u0394-terms","message":"Initial commit: Embed validity proofs in \u0394-terms\n\nReason: Deal with nonfull abstraction by removing elements from\nthe domain of \u0394-types.\n\nCheckpoint: Untangle syntax and semantics of \u0394-terms via de-facto\nexistential type that hides the semantics function in the\ndefinition of \u0394-terms.\n\nOld-commit-hash: 598647f579bd017f335f1ce5f36414a6442dc4e9\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/TaggedDeltaTypes.agda","new_file":"experimental\/TaggedDeltaTypes.agda","new_contents":"{-\nThe goal of this file is to make the domain of \u0394 types smaller\nso as to be nearer to full abstraction and hopefully close\nenough for the purpose of having explicit nil changes.\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\nprivate data \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms,\n-- (because it embeds validity proofs)\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\nprivate\n  \u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\nabstract\n data \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192(ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                                 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {_} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {_} {u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car    : A\n    cadr   : B car\n    caddr  : C car cadr\n    cadddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nabstract\n \u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n \u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n \u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n \u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n \u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n   \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n \u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n   let\n     (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n     (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n   in\n     (old-s + old-t , new-s + new-t)\n \u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n   let\n     v  = \u27e6 b \u27e7 (ignore \u03c1)\n     h  = \u27e6 f \u27e7 (ignore \u03c1)\n     dv = \u27e6 db \u27e7\u0394 \u03c1\n     dh = \u27e6 df \u27e7\u0394 \u03c1\n   in\n     mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n \u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag b\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\nvalidity {nats} {\u0393} {nat n} {\u03c1} = {!!}\n\nvalidity {bags} {\u0393} {bag b} = {!!}\nvalidity {\u03c4} {\u0393} {var x} = {!!}\nvalidity {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} = {!!}\nvalidity {\u03c4} {\u0393} {app t t\u2081} = {!!}\nvalidity {nats} {\u0393} {add t t\u2081} = {!!}\nvalidity {bags} {\u0393} {map t t\u2081} = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'experimental\/TaggedDeltaTypes.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"59c5dcff1353ec62903946d0b10d49966fea58a7","subject":"no hello3","message":"no hello3\n","repos":"nfjinjing\/chu2","old_file":"src\/Hello3.agda","new_file":"src\/Hello3.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"05c45a6f311fdafa06d6fa0b08032bf897d701d7","subject":"models: \"algebraic\" model for IIR","message":"models: \"algebraic\" model for IIR\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/Alg-IIR.agda","new_file":"models\/Alg-IIR.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"error: RPC failed; HTTP 403 curl 22 The requested URL returned error: 403\nfatal: the remote end hung up unexpectedly\n","license":"mit","lang":"Agda"}
{"commit":"13ce41b77ff02dd50d175e6e44700ef12c66c573","subject":"Added draft: Comparing equational reasoning.","message":"Added draft: Comparing equational reasoning.\n\nIgnore-this: a34e9564240c152ee9826930fb0b857d\n\ndarcs-hash:20110119161619-3bd4e-8db9a3cb20dd00eca07f42f8e7e0115f97cd378c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/ComparingEqReasoning.agda","new_file":"Draft\/ComparingEqReasoning.agda","new_contents":"------------------------------------------------------------------------------\n-- Comparing Ulf and Nils styles for equational reasoning\n------------------------------------------------------------------------------\n\nmodule ComparingEqReasoning where\n\ninfix  7 _\u2261_\ninfixl 9  _+_\n\ndata _\u2261_ {A : Set}(x : A) : A \u2192 Set where\n  refl : x \u2261 x\n\ntrans : {A : Set}{x y z : A} \u2192 x \u2261 y \u2192 y \u2261 z \u2192 x \u2261 z\ntrans refl refl = refl\n\npostulate\n  \u2115               : Set\n  zero            : \u2115\n  _+_             : \u2115 \u2192 \u2115 \u2192 \u2115\n  +-comm          : (m n : \u2115) \u2192 m + n \u2261 n + m\n  +-rightIdentity : (n : \u2115) \u2192 n + zero \u2261 n\n\ninfix 7 _\u2243_\n\nprivate\n  data _\u2243_ (x y : \u2115) : Set where\n    prf : x \u2261 y \u2192 x \u2243 y\n\nmodule Ulf where\n  -- Adapted from the Ulf's thesis (p. 112).\n\n  infix  5 chain>_\n  infixl 5 _===_by_\n  infix  4 _qed\n\n  chain>_ : (x : \u2115) \u2192 x \u2243 x\n  chain> x = prf refl\n\n  _===_by_ : {x y : \u2115} \u2192 x \u2243 y \u2192 (z : \u2115) \u2192 y \u2261 z \u2192 x \u2243 z\n  prf p === z by q = prf (trans {_} {_} {_} p q)\n\n  _qed : {x y : \u2115} \u2192 x \u2243 y \u2192 x \u2261 y\n  prf p qed = p\n\n  -- Example\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    chain> zero + n\n       === n + zero by +-comm zero n\n       === n        by +-rightIdentity n\n    qed\n\nmodule Nils where\n  -- Adapted from the standard library (Relation.Binary.PreorderReasoning).\n\n  infix  4 begin_\n  infixr 5 _\u2261\u27e8_\u27e9_\n  infix  5 _\u220e\n\n  begin_ : {x y : \u2115} \u2192 x \u2243 y \u2192 x \u2261 y\n  begin prf x\u2261y = x\u2261y\n\n  _\u2261\u27e8_\u27e9_ : (x : \u2115){y z : \u2115} \u2192 x \u2261 y \u2192 y \u2243 z \u2192 x \u2243 z\n  _ \u2261\u27e8 x\u2261y \u27e9 prf y\u2261z = prf (trans x\u2261y y\u2261z)\n\n  _\u220e : (x : \u2115) \u2192 x \u2243 x\n  _\u220e _ = prf refl\n\n  -- Example.\n  +-leftIdentity : (n : \u2115) \u2192 zero + n \u2261 n\n  +-leftIdentity n =\n    begin\n      zero + n \u2261\u27e8 +-comm zero n \u27e9\n      n + zero \u2261\u27e8 +-rightIdentity n \u27e9\n      n\n    \u220e\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/ComparingEqReasoning.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"6a44501700af2b960ad356e01da82e5f0c0ef384","subject":"Added draft: Theory T from the paper.","message":"Added draft: Theory T from the paper.\n\nIgnore-this: 38207a8310ffe6760d66da34a4cf8be7\n\ndarcs-hash:20110125160655-3bd4e-f62a7cdd9887f5298d9e07fbbae290e3c39a73ac.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/T.agda","new_file":"Draft\/T.agda","new_contents":"------------------------------------------------------------------------------\n-- Theory T from the paper\n------------------------------------------------------------------------------\n\nmodule T where\n\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n\npostulate\n  T        : Set\n  zero K S : T\n  succ     : T \u2192 T\n  _\u00b7_      : T \u2192 T \u2192 T\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : T) : T \u2192 Set where\n  refl : x \u2261 x\n\n-- Conversion rules\npostulate\n  cK : \u2200 x y \u2192 K \u00b7 x \u00b7 y         \u2261 x\n  cS : \u2200 x y z \u2192 S \u00b7 x \u00b7  y \u00b7  z \u2261 x \u00b7  z \u00b7  (y \u00b7  z)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/T.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"976303ac7ed25341c88e835653aa475f51214ce2","subject":"Control\/Protocol\/Sequence.agda","message":"Control\/Protocol\/Sequence.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Sequence.agda","new_file":"Control\/Protocol\/Sequence.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_; first)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Sequence where\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\nmodule _ {{_ : FunExt}} where\n    >>=-right-unit : \u2200 P \u2192 (P >> end) \u2261 P\n    >>=-right-unit end       = refl\n    >>=-right-unit (com q P) = com= refl refl \u03bb m \u2192 >>=-right-unit (P m)\n\n    >>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n                \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261 ((P >>= Q) >>= R)\n    >>=-assoc end       Q R = refl\n    >>=-assoc (com q P) Q R = com= refl refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\n    dual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261 dual P >> dual Q\n    dual->> end        Q = refl\n    dual->> (com io P) Q = com= refl refl \u03bb m \u2192 dual->> (P m) Q\n\n    {- My coe-ap-fu is lacking...\n    dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261 dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n    dual->>= end Q = refl\n    dual->>= (com io P) Q = com= refl refl \u03bb m \u2192 dual->>= (P m) (Q \u2218 _,_ m) \u2219 ap (_>>=_ (dual (P m))) (\u03bb= \u03bb ms \u2192 ap (\u03bb x \u2192 dual (Q x)) (pair= {!!} {!!}))\n    -}\n\n    dual-replicate\u1d3e : \u2200 n P \u2192 dual (replicate\u1d3e n P) \u2261 replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    P = refl\n    dual-replicate\u1d3e (suc n) P = dual->> P (replicate\u1d3e n P) \u2219 ap (_>>_ (dual P)) (dual-replicate\u1d3e n P)\n\n{- An incremental telecom function which makes processes communicate\n   during a matching initial protocol. -}\n>>=-telecom : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n            \u2192 \u27e6 P >>= Q  \u27e7\n            \u2192 \u27e6 P >>= R \u22a5\u27e7\n            \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-telecom end      p0       p1       = _ , p0 , p1\n>>=-telecom (send P) (m , p0) p1       = first (_,_ m) (>>=-telecom (P m) p0 (p1 m))\n>>=-telecom (recv P) p0       (m , p1) = first (_,_ m) (>>=-telecom (P m) (p0 m) p1)\n\n>>-telecom : (P : Proto){Q R : Proto}\n           \u2192 \u27e6 P >> Q  \u27e7\n           \u2192 \u27e6 P >> R \u22a5\u27e7\n           \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-telecom P p q = >>=-telecom P p q\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Sequence.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e4ec4ba8abfa4f76052e2ec072a292d38f6f718a","subject":"+Algebra\/Field","message":"+Algebra\/Field\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field.agda","new_file":"lib\/Algebra\/Field.agda","new_contents":"module Algebra.Field where\n\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq as FP\n\nrecord IsField {\u2113} (A : Set \u2113) : Set \u2113 where\n  open FP {\u2113} {A}\n\n  infixl 6 _+_ _\u2212_\n  infixl 7 _*_ _\/_\n\n  field\n    _+_ _*_     : A \u2192 A \u2192 A\n    +-assoc     : Associative _+_\n    *-assoc     : Associative _*_\n    +-comm      : Commutative _+_\n    *-comm      : Commutative _*_\n    0\u1da0 1\u1da0        : A\n    0\u22621         : 0\u1da0 \u2262 1\u1da0\n    0+-identity : LeftIdentity 0\u1da0 _+_\n    1*-identity : LeftIdentity 1\u1da0 _*_\n    0-_         : A \u2192 A\n    0--inverse  : LeftInverse 0\u1da0 0-_ _+_\n    _\u207b\u00b9         : A \u2192 A\n    \u207b\u00b9-inverse  : LeftInverse 1\u1da0 _\u207b\u00b9 _*_\n    *-+-distr   : _*_ DistributesOver\u02e1 _+_\n\n  _\u2212_ _\/_ : A \u2192 A \u2192 A\n  a \u2212 b = a + 0- b\n  a \/ b = a * b \u207b\u00b9\n\n  0--right-inverse : RightInverse 0\u1da0 0-_ _+_\n  0--right-inverse = +-comm \u2219 0--inverse\n\n  +0-identity : RightIdentity 0\u1da0 _+_\n  +0-identity = +-comm \u2219 0+-identity\n\n  *1-identity : RightIdentity 1\u1da0 _*_\n  *1-identity = *-comm \u2219 1*-identity\n\n  += : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x + y \u2261 x' + y'\n  += {x} {y' = y'} p q = ap (_+_ x) q \u2219 ap (\u03bb z \u2192 z + y') p\n\n  *= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x * y \u2261 x' * y'\n  *= {x} {y' = y'} p q = ap (_*_ x) q \u2219 ap (\u03bb z \u2192 z * y') p\n\n  0-c+c+x : \u2200 {c x} \u2192 0- c + c + x \u2261 x\n  0-c+c+x = += 0--inverse refl \u2219 0+-identity\n\n  +-left-cancel : LeftCancel _+_\n  +-left-cancel p = ! 0-c+c+x \u2219 +-assoc\n                  \u2219 ap (\u03bb z \u2192 0- _ + z) p\n                  \u2219 ! +-assoc \u2219 0-c+c+x\n\n  +-right-cancel : RightCancel _+_\n  +-right-cancel p = +-left-cancel (+-comm \u2219 p \u2219 +-comm)\n\n  0--involutive : Involutive 0-_\n  0--involutive = +-left-cancel (0--right-inverse \u2219 ! 0--inverse)\n\nopen import Reflection.NP\nopen import Data.List\nopen import Data.Maybe.NP\n\nmodule TermField {a} {A : Set a} (F : IsField A) where\n  open IsField F\n  pattern _`+_ t u = con (quote _+_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern _`*_ t u = con (quote _*_) (arg\u1d5b\u02b3 t \u2237 arg\u1d5b\u02b3 u \u2237 [])\n  pattern `0-_ t = con\u1d5b\u02b3 (quote 0-_) t\n  pattern _`\u207b\u00b9 t = con\u1d5b\u02b3 (quote _\u207b\u00b9) t\n  pattern `0\u1da0 = con (quote 0\u1da0) []\n  pattern `1\u1da0 = con (quote 1\u1da0) []\n\n  module Decode-Field {b}{B : Set b}(G : IsField B)(default : Decode-Term B) where\n    module G = IsField G\n    decode-Field : Decode-Term B\n    decode-Field `0\u1da0 = just G.0\u1da0\n    decode-Field `1\u1da0 = just G.1\u1da0\n    decode-Field (t `\u207b\u00b9) = map? G._\u207b\u00b9 (decode-Field t)\n    decode-Field (`0- t) = map? G.0-_ (decode-Field t)\n    decode-Field (t `+ u) = \u27ea G._+_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field (t `* u) = \u27ea G._*_ \u00b7 decode-Field t \u00b7 decode-Field u \u27eb?\n    decode-Field t = default t\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/Field.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9d0b9c2735b9edccfdb18cb94fe21955cd1cc2c2","subject":"For those interested: a \"polynomial types\" DSL in Agda.","message":"For those interested: a \"polynomial types\" DSL in Agda.\n","repos":"DSLsofMath\/DSLsofMath","old_file":"L\/01\/TypeDSL.agda","new_file":"L\/01\/TypeDSL.agda","new_contents":"-- A DSL example in the language Agda: \"polynomial types\"\nmodule TypeDSL where\nopen import Data.Empty\nopen import Data.Unit\nopen import Data.Sum\nopen import Data.Product\nopen import Data.Nat\n\ndata E : Set1 where\n  Add   : E -> E -> E\n  Mul   : E -> E -> E\n  Zero  : E\n  One   : E\n\neval : E -> Set\neval (Add x y)  = (eval x) \u228e (eval y)\neval (Mul x y)  = (eval x) \u00d7 (eval y)\neval Zero       = \u22a5\neval One        = \u22a4\n\ntwo   = Add One One\nthree = Add One two\n\ntest1 : eval One\ntest1 = tt\n\nfalse : eval two\nfalse = inj\u2081 tt\ntrue : eval two\ntrue = inj\u2082 tt\n\ntest3 : eval three\ntest3 = inj\u2081 tt\n\ncard : E -> \u2115\ncard (Add x y)  = card x + card y\ncard (Mul x y)  = card x * card y\ncard Zero       = 0\ncard One        = 1\n\nopen import Data.Vec as V\n\nvariable\n    m n : \u2115\n    A B : Set\n\nenumAdd : Vec A m -> Vec B n -> Vec (A \u228e B) (m + n)\nenumAdd xs ys = V.map inj\u2081 xs ++ V.map inj\u2082 ys\n\n-- cartesianProduct\nenumMul : Vec A m \u2192 Vec B n \u2192 Vec (A \u00d7 B) (m * n)\nenumMul xs ys = concat (V.map  (\\a -> V.map  ((a ,_)) ys)  xs)\n\nenumerate : (t : E) -> Vec (eval t) (card t)\nenumerate (Add x y)  =  enumAdd  (enumerate x) (enumerate y)\nenumerate (Mul x y)  =  enumMul  (enumerate x) (enumerate y) \nenumerate Zero       =  []\nenumerate One        =  [ tt ]\n\ntest : Vec (eval three) 3\ntest = enumerate three\n-- inj\u2081 tt \u2237 inj\u2082 (inj\u2081 tt) \u2237 inj\u2082 (inj\u2082 tt) \u2237 []\n\n-- Exercise: add a constructor for function types to the syntax E \n","old_contents":"","returncode":1,"stderr":"error: pathspec 'L\/01\/TypeDSL.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6012f70eda087d9a7f5b8dd63326874c02a1cd0d","subject":"Control\/Protocol\/Multiplicative.agda","message":"Control\/Protocol\/Multiplicative.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Multiplicative.agda","new_file":"Control\/Protocol\/Multiplicative.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u2203;\u03a3;_\u00d7_;_,_;first;second)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Sequence\n\nmodule Control.Protocol.Multiplicative where\n\n_\u214b_ : Proto \u2192 Proto \u2192 Proto\nend    \u214b Q      = Q\nrecv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\nP      \u214b end    = P\nP      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\nsend P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                       ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n_\u2297_ : Proto \u2192 Proto \u2192 Proto\nend    \u2297 Q      = Q\nsend P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\nP      \u2297 end    = P\nP      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\nrecv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                       ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  -- absorption\n  \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n  \u22a4-\u214b P = \u03bb()\n\n_\u22b8_ : (P Q : Proto) \u2192 Proto\nP \u22b8 Q = dual P \u214b Q\n\n_o-o_ : (P Q : Proto) \u2192 Proto\nP o-o Q = (P \u22b8 Q) \u2297 (Q \u22b8 P)\n\nmodule _ {{_ : FunExt}} where\n  \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n  \u2297-endR end      = refl\n  \u2297-endR (recv _) = refl\n  \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n  \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n  \u214b-endR end      = refl\n  \u214b-endR (send _) = refl\n  \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n  \u2297-sendR end      Q = refl\n  \u2297-sendR (recv P) Q = refl\n  \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n  \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n  \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n  \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n  \u2297-comm (recv P) end      = refl\n  \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n  \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                              ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n  \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n  \u2297-assoc end      Q        R        = refl\n  \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n  \u2297-assoc (recv P) end      R        = refl\n  \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n  \u2297-assoc (recv P) (recv Q) end      = refl\n  \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n  \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                              ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                              ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n  \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n  \u214b-recvR end      Q = refl\n  \u214b-recvR (send P) Q = refl\n  \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n  \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n  \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n  \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n  \u214b-comm (send P) end      = refl\n  \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n  \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                              ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n  \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n  \u214b-assoc end      Q        R        = refl\n  \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n  \u214b-assoc (send P) end      R        = refl\n  \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n  \u214b-assoc (send P) (send Q) end      = refl\n  \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n  \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                            \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                              ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                              ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\nmodule _ {P Q R}{{_ : FunExt}} where\n  dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n  dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n  dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n  dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n  module _ {{_ : UA}} where\n      dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n      dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                \u2219 \u2295\u2261\u228e\n                \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                \u2219 ! \u2295\u2261\u228e\n\n      dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n      dist-\u214b-& = \u214b-comm P (Q & R)\n                \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                \u2219 &\u2261\u00d7\n                \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                \u2219 ! &\u2261\u00d7\n\n  -- sends can be pulled out of tensor\n  source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n  source->>=-\u2297 end       Q R = refl\n  source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n  -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n  -- recvs can be pulled out of par\n  sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n  sink->>=-\u214b end       Q R = refl\n  sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n  -- source-\u214b : source-of P \u214b source-of Q \u2261 \n\n  -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\nLog-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\nLog-\u214b-\u00d7 {end}   {Q}      q           = end , q\nLog-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\nLog-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\nLog-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\nLog-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\nLog-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\nmodule _ {{_ : FunExt}} where\n  \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n  \u2297\u214b-dual end Q = refl\n  \u2297\u214b-dual (recv P) Q = com=\u2032 _ \u03bb m \u2192 \u2297\u214b-dual (P m) _\n  \u2297\u214b-dual (send P) end = refl\n  \u2297\u214b-dual (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)\n  \u2297\u214b-dual (send P) (send Q) = com=\u2032 _\n    [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (send Q))\n    ,inr: (\u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)) ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\n  \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n  \u214b-sendR end      m p = m , p\n  \u214b-sendR (send P) m p = inr m , p\n  \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n  \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n  \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n  \u214b-id : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n  \u214b-id end      = end\n  \u214b-id (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (\u214b-id (P x))\n  \u214b-id (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (\u214b-id (P x))\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n\u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n\u214b-apply end      Q        s           p       = s\n\u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n\u214b-apply (send P) end      s           p       = _\n\u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n\u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n\u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n\u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n\u2297-pair {end}    {Q}      p q       = q\n\u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n\u2297-pair {recv P} {end}    p end     = p\n\u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n\u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                      ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n\u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n\u2297-fst end      Q        pq       = _\n\u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n\u2297-fst (recv P) end      pq       = pq\n\u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n\u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n\u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n\u2297-snd end      Q        pq       = pq\n\u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n\u2297-snd (recv P) end      pq       = end\n\u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n\u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\nmodule _ {{_ : FunExt}} where\n    \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n    \u2297-pair-fst end      Q        p q       = refl\n    \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n    \u2297-pair-fst (recv P) end      p q       = refl\n    \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n    \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n    \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n    \u2297-pair-snd end      Q        p q       = refl\n    \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n    \u2297-pair-snd (recv P) end      p q       = refl\n    \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n    \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\nmodule _ P Q where\n  \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n  \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n  module _ {{_ : FunExt}} where\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n{- WRONG\n\u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n\u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n\u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n-}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL' end      Q        R        q            r       = \u2297-pair {Q} {R} q r\n    switchL' (send P) end      R        p            r       = par (send P) R p r\n    switchL' (send P) (send Q) R        (inl m , pq) r       = inl m , switchL' (P m) (send Q) R pq r\n    switchL' (send P) (send Q) R        (inr m , pq) r       = inr m , switchL' (send P) (Q m) R pq r\n    switchL' (send P) (recv Q) end      pq           r       = pq\n    switchL' (send P) (recv Q) (send R) pq           (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n    switchL' (send P) (recv Q) (recv R) pq           r       = [inl: (\u03bb m \u2192 switchL' (send P) (Q m) (recv R) (pq m) r)\n                                                                ,inr: (\u03bb m \u2192 switchL' (send P) (recv Q) (R m) pq (r m)) ]\n    switchL' (recv P) Q        R        pq           r       = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n    switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n    switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n    \u22b8-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    \u22b8-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (! (dual-involutive P)) p)\n\n    o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n    o-o-apply P Q Po-oQ p = \u22b8-apply {P} {Q} (\u2297-fst (P \u22b8 Q) (Q \u22b8 P) Po-oQ) p\n\n    o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n    o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n    -- multiplicative mix (left-biased)\n    mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n    mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Multiplicative.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"517043dd3ff3b1549d01e5baf4aa341c7d5ce4d1","subject":"Add exercises for Brutal Metaintroduction to Agda","message":"Add exercises for Brutal Metaintroduction to Agda\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/BrutalMetaIntro.agda","new_file":"agda\/BrutalMetaIntro.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"1000130700500fc9f0c74f3feded244488fba784","subject":"Added thesis' example.","message":"Added thesis' example.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/report\/CombiningProofs\/ConatPropertiesATP.agda","new_file":"notes\/thesis\/report\/CombiningProofs\/ConatPropertiesATP.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"fatal: unable to access 'https:\/\/github.com\/asr\/fotc.git\/': The requested URL returned error: 403\n","license":"mit","lang":"Agda"}
{"commit":"c838b5d3d047273d8b46736db7c5e6353d903cbd","subject":"Add Crypto.Sig.Lamport.OneBit.Abstract","message":"Add Crypto.Sig.Lamport.OneBit.Abstract\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/Lamport\/OneBit\/Abstract.agda","new_file":"Crypto\/Sig\/Lamport\/OneBit\/Abstract.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Type.Eq using (Eq?; _==_; \u2261\u21d2==; ==\u21d2\u2261)\nopen import Data.Product.NP using (_\u00d7_; _,_; fst; snd; map)\nopen import Data.Bit using (Bit; 0b; 1b; proj\u2032; \u2713\u2192\u2261; \u2261\u2192\u2713)\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\n\nmodule Crypto.Sig.Lamport.OneBit.Abstract\n          (Secret : Type)\n          (Digest : Type)\n          (hash   : Secret \u2192 Digest)\n          (eq?    : Eq? Digest)\n          where\n\nSignKey    = Secret \u00d7 Secret\nSignature  = Secret\nVerifKey   = Digest \u00d7 Digest\n\nmodule signkey (sk : SignKey) where\n  skL = fst sk\n  skH = snd sk\n  vkL = hash skL\n  vkH = hash skH\n\n-- Derive the public key by hashing each secret\nverif-key : SignKey \u2192 VerifKey\nverif-key sk = vkL , vkH\n  where open signkey sk\n\nmodule verifkey (vk : VerifKey) where\n  vkL = fst vk\n  vkH = snd vk\n\n-- Key generation\nkey-gen : SignKey \u2192 VerifKey \u00d7 SignKey\nkey-gen sk = vk , sk\n  module key-gen where\n    vk = verif-key sk\n\n-- Sign a single bit message\nsign : SignKey \u2192 Bit \u2192 Signature\nsign = proj\u2032\n\n-- Verify the signature of a single bit message\nverify : VerifKey \u2192 Bit \u2192 Signature \u2192 Bit\nverify vk b sig = proj\u2032 vk b == hash sig\n\nverify-correct-sig : \u2200 sk b \u2192 verify (verif-key sk) b (sign sk b) \u2261 1b\nverify-correct-sig sk 0b = \u2713\u2192\u2261 (\u2261\u21d2== (hash (fst sk) \u220e))\nverify-correct-sig sk 1b = \u2713\u2192\u2261 (\u2261\u21d2== (hash (snd sk) \u220e))\n\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\n\nmodule Assuming-injectivity (hash-inj : Injective hash) where\n\n  -- If one considers the hash function injective, then, so is verif-key\n  verif-key-inj : \u2200 {sk1 sk2} \u2192 verif-key sk1 \u2261 verif-key sk2 \u2192 sk1 \u2261 sk2\n  verif-key-inj {sk1} {sk2} e\n    = ap\u2082 _,_ (hash-inj (ap fst e)) (hash-inj (ap snd e))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n\n  -- Therefor under this assumption, different secret keys means different public keys\n  verif-key-corrolary : \u2200 {sk1 sk2} \u2192 sk1 \u2262 sk2 \u2192 verif-key sk1 \u2262 verif-key sk2\n  verif-key-corrolary {sk1} {sk2} sk\u2262 vk= = sk\u2262 (verif-key-inj vk=)\n\n  -- If one considers the hash function injective then there is\n  -- only one signing key which can sign correctly all (0 and 1)\n  -- the messages\n  signkey-uniqness :\n        \u2200 sk1 sk2 \u2192\n          (\u2200 b \u2192 verify (verif-key sk1) b (sign sk2 b) \u2261 1b) \u2192\n          sk1 \u2261 sk2\n  signkey-uniqness sk1 sk2 e\n    = ap\u2082 _,_ (hash-inj (==\u21d2\u2261 (\u2261\u2192\u2713 (e 0b))))\n              (hash-inj (==\u21d2\u2261 (\u2261\u2192\u2713 (e 1b))))\n\nmodule Assuming-invertibility\n         (unhash : Digest \u2192 Secret)\n         (unhash-hash : \u2200 x \u2192 unhash (hash x) \u2261 x)\n         (hash-unhash : \u2200 x \u2192 hash (unhash x) \u2261 x)\n         where\n\n  -- EXERCISES\n  {-\n  recover-signkey : VerifKey \u2192 SignKey\n  recover-signkey = {!!}\n\n  recover-signkey-correct : \u2200 sk \u2192 recover-signkey (verif-key sk) \u2261 sk\n  recover-signkey-correct = {!!}\n\n  forgesig : VerifKey \u2192 Bit \u2192 Signature\n  forgesig = {!!}\n\n  forgesig-correct : \u2200 vk b \u2192 verify vk b (forgesig vk b) \u2261 1b\n  forgesig-correct = {!!}\n  -}\n\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Crypto\/Sig\/Lamport\/OneBit\/Abstract.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8da59660a259b5b26f47e6d23bd1d66836c726c4","subject":"Readd UNDEFINED","message":"Readd UNDEFINED\n\nThis was deleted in 5bde6429aaee189aecc700b69ff698946d226090: undefined\nis not appropriate for a complete development, but it is for a\nwork-in-progress (like the current one).\n","repos":"inc-lc\/ilc-agda","old_file":"UNDEFINED.agda","new_file":"UNDEFINED.agda","new_contents":"-- Allow holes in modules to import, by introducing a single general postulate.\n\nmodule UNDEFINED where\n\nopen import Data.Unit.Core using (Hidden)\nopen import Data.Unit.Core public using (reveal)\n\npostulate\n  -- If this postulate would produce a T, it could be used as instance argument, triggering many ambiguities.\n  -- This postulate is named in capitals because it introduces an inconsistency.\n  -- Hence, this must be used through `reveal UNDEFINED`.\n  UNDEFINED : \u2200 {\u2113} \u2192 {T : Set \u2113} \u2192 Hidden T\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'UNDEFINED.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"7a8739a45fc9ba53a4a65f923a1a2033ee67369f","subject":"Desc@ICFP: draft of a cleaner Hutton's aftershave","message":"Desc@ICFP: draft of a cleaner Hutton's aftershave\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/RevisedHutton.agda","new_file":"models\/RevisedHutton.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\n-- Open term: holes are either values or variables in a context\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = \n    subst (cong (IMu closeTerm) (snd variable)) \n          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/RevisedHutton.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"d245f73509a8c7b6ddee9bdcd869c6144bf6d4c8","subject":"Add FunUniverse\/Nand\/Function.agda","message":"Add FunUniverse\/Nand\/Function.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Nand\/Function.agda","new_file":"FunUniverse\/Nand\/Function.agda","new_contents":"open import Data.Product\n\nmodule FunUniverse.Nand.Function where\n\nmodule FromNand {A} (nand : A \u00d7 A \u2192 A) where\n\n    open import FunUniverse.Agda\n    open import FunUniverse.Nand (Abstract\ud835\udfda.funRewiring A)\n\n    open FromNand nand public\n\nopen import Data.Two\n\nmodule Nand\ud835\udfda = FromNand (uncurry nand)\n\nopen import Data.Tree.Binary\n\nmodule NandTree {A} = FromNand {BinTree A} (uncurry fork)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FunUniverse\/Nand\/Function.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b8767797645bef8303f453fd827e0ad1b95816b0","subject":"Added note on Agda parameters.","message":"Added note on Agda parameters.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Parameters.agda","new_file":"notes\/Parameters.agda","new_contents":"module Parameters where\n\ndata List (A : Set) : Set where\n  []  : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\n-- Error: The type of the constructor does not fit in the sort of the\n-- datatype, since Set\u2081 is not less or equal than Set when checking\n-- the constructor [] in the declaration of List\u2081\n--\n-- data List\u2081 : Set \u2192 Set where\n--   []  : {A : Set} \u2192 List\u2081 A\n--   _\u2237_ : {A : Set} \u2192 A \u2192 List\u2081 A \u2192 List\u2081 A\n\ndata List\u2081 : Set \u2192 Set\u2081 where\n  []  : {A : Set} \u2192 List\u2081 A\n  _\u2237_ : {A : Set} \u2192 A \u2192 List\u2081 A \u2192 List\u2081 A\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/Parameters.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"cf87a127259d824c11c3fb4c235421022c1fc846","subject":"Add SemiRing","message":"Add SemiRing\n","repos":"DSLsofMath\/DSLsofMath","old_file":"FLABloM\/SemiRingRecord.agda","new_file":"FLABloM\/SemiRingRecord.agda","new_contents":"module SemiRingRecord where\n\nimport Algebra.FunctionProperties\n  using (LeftIdentity; RightIdentity)\nimport Function using (_on_)\nimport Relation.Binary.EqReasoning as EqReasoning\nimport Relation.Binary.On using (isEquivalence)\nimport Algebra.Structures using (module IsCommutativeMonoid; IsCommutativeMonoid)\nopen import Relation.Binary\n  using (module IsEquivalence; IsEquivalence; _Preserves\u2082_\u27f6_\u27f6_ ; Setoid)\nopen import Data.Product renaming (_,_ to _,,_) -- just to avoid clash with other commas\n\nopen import Preliminaries\n\nopen import SemiNearRingRecord\n\nrecord SemiRing : Set\u2081 where\n  field\n    snr : SemiNearRing\n\n  open SemiNearRing snr\n\n  field\n    1s : s\n\n  open Algebra.FunctionProperties _\u2243s_ using (LeftIdentity; RightIdentity)\n\n  field\n    \u2219-identityl : LeftIdentity 1s _\u2219s_\n    \u2219-identityr : RightIdentity 1s _\u2219s_\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FLABloM\/SemiRingRecord.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6ddddf5e75a6269e99b20220d34b62e6f15eaa9f","subject":"Testing an alternative definition of subtraction.","message":"Testing an alternative definition of subtraction.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/SubtractionSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/SubtractionSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing an alternative definition of subtraction\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.SubtractionSL where\n\nopen import Data.Nat hiding ( _\u2238_ )\nopen import Relation.Binary.PropositionalEquality\n\n-- First definition (from the standard library).\n_\u2238\u2081_ : \u2115 \u2192 \u2115 \u2192 \u2115\nm     \u2238\u2081 zero  = m\nzero  \u2238\u2081 suc n = zero\nsuc m \u2238\u2081 suc n = m \u2238\u2081 n\n\n-- Second definition.\n_\u2238\u2082_ : \u2115 \u2192 \u2115 \u2192 \u2115\nm     \u2238\u2082 zero  = m\nzero  \u2238\u2082 n     = zero\nsuc m \u2238\u2082 suc n = m \u2238\u2082 n\n\n-- Both definitions are equivalents.\nthm : \u2200 m n \u2192 m \u2238\u2081 n \u2261 m \u2238\u2082 n\nthm m       zero    = refl\nthm zero    (suc n) = refl\nthm (suc m) (suc n) = thm m n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/SubtractionSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"36c679f1ab638dc793267d0f871ac65499d60f2e","subject":"Testing the use of numeric literals.","message":"Testing the use of numeric literals.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/NumericLiterals.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/NumericLiterals.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the use of numeric literals\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.NumericLiterals where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n------------------------------------------------------------------------------\n\ndata \u2115 : Set where\n  z : \u2115\n  s : \u2115 \u2192 \u2115\n\n{-# BUILTIN NATURAL \u2115 #-}\n{-# BUILTIN ZERO    z #-}\n{-# BUILTIN SUC     s #-}\n\ntoD : \u2115 \u2192 D\ntoD z     = zero\ntoD (s n) = succ\u2081 (toD n)\n\nten : D\nten = toD 10\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/NumericLiterals.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"6294baecf1175f6fee228b750d819181a4e0cd9e","subject":"added red-black tree","message":"added red-black tree\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/RedBlack\/RedBlack.agda","new_file":"agda\/RedBlack\/RedBlack.agda","new_contents":"module RedBlack where\n\nopen import Data.Nat\n\n-- | The colour of a node.\ndata Color : Set where\n  red   : Color\n  black : Color\n\n-- | The red black tree\ndata Tree  (A : Set) : \u2115       -- black height\n                     \u2192 Color   -- color of the root\n                     \u2192 Set\n     where\n\n  -- Empty node is always black\n  empty :  Tree A 0 black\n\n  -- Both the children of the red node are black.\n  red   :  {d : \u2115}\n        \u2192 Tree A d black  -- left child\n        \u2192 A               -- element on the node\n        \u2192 Tree A d black  -- right child\n        \u2192 Tree A d red    -- The black height does not change\n\n  -- The children can be of either color\n  black : {d : \u2115} {cL cR : Color}\n        \u2192 Tree A d       cL    -- left child\n        \u2192 A                    -- element\n        \u2192 Tree A d       cR    -- right child\n        \u2192 Tree A (d + 1) black -- black height goes up by one.\n\n-- Converts a tree with red root to a black tree. The type encodes the\n-- fact that the height of the tree goes up by 1.\nblacken : {A : Set} {d : \u2115}\n        \u2192 Tree A d       red\n        \u2192 Tree A (d + 1) black\nblacken (red lt x rt) = black lt x rt\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/RedBlack\/RedBlack.agda' did not match any file(s) known to git\n","license":"unlicense","lang":"Agda"}
{"commit":"3e95b3dcae62210c76f5088ffae7da49775ad763","subject":"Rolled back a test.","message":"Rolled back a test.\n\nIgnore-this: 6f7b957c439302714d52ae0596db654c\n\ndarcs-hash:20110318123636-3bd4e-deb0e738289da6f28d61ad88ca1531eb846a9c49.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/Conjectures\/LogicalConstants.agda","new_file":"Test\/Succeed\/Conjectures\/LogicalConstants.agda","new_contents":"module Test.Succeed.Conjectures.LogicalConstants where\n\ninfix  5 \u00ac_\ninfixr 4 _\u2227_\ninfixr 3 _\u2228_\ninfixr 2 _\u21d2_\ninfixr 1 _\u2194_\n\n------------------------------------------------------------------------------\n-- Propositional logic\n\n-- The logical connectives\n\n-- The logical connectives are hard-coded in our implementation,\n-- i.e. the following symbols must be used.\npostulate\n  \u22a5 \u22a4     : Set -- N.B. the name of the tautology symbol is \"\\top\", not T.\n  \u00ac_      : Set \u2192 Set -- N.B. the right hole.\n  _\u2227_ _\u2228_ : Set \u2192 Set \u2192 Set\n  _\u21d2_ _\u2194_ : Set \u2192 Set \u2192 Set\n\n-- We postulate some propositional variables (which are translated as\n-- 0-ary predicates).\npostulate\n  P Q R : Set\n\n-- The introduction and elimination rules for the propositional\n-- connectives are theorems.\npostulate\n  \u2192I  : (P \u2192 Q) \u2192 P \u21d2 Q\n  \u2192E  : (P \u21d2 Q) \u2192 P \u2192 Q\n  \u2227I  : P \u2192 Q \u2192 P \u2227 Q\n  \u2227E\u2081 : P \u2227 Q \u2192 P\n  \u2227E\u2082 : P \u2227 Q \u2192 Q\n  \u2228I\u2081 : P \u2192 P \u2228 Q\n  \u2228I\u2082 : Q \u2192 P \u2228 Q\n  \u2228E  : (P \u21d2 R) \u2192 (Q \u21d2 R) \u2192 P \u2228 Q \u2192 R\n  \u22a5E  : \u22a5 \u2192 P\n  \u00acE : (\u00ac P \u2192 \u22a5 ) \u2192 P\n{-# ATP prove \u2192I #-}\n{-# ATP prove \u2192E #-}\n{-# ATP prove \u2227I #-}\n{-# ATP prove \u2227E\u2081 #-}\n{-# ATP prove \u2227E\u2082 #-}\n{-# ATP prove \u2228I\u2081 #-}\n{-# ATP prove \u2228I\u2082 #-}\n{-# ATP prove \u2228E #-}\n{-# ATP prove \u22a5E #-}\n{-# ATP prove \u00acE #-}\n\n------------------------------------------------------------------------------\n-- Predicate logic\n\n-- The universe of discourse.\npostulate\n  D : Set\n\n-- The quantifiers are hard-coded in our implementation, i.e. the\n-- following symbols must be used.\npostulate\n  \u2203 : (P : D \u2192 Set) \u2192 Set\n  \u22c0 : (P : D \u2192 Set) \u2192 Set\n\n-- We postulate some predicate symbols.\npostulate\n  P\u2070    : Set\n  P\u00b9 Q\u00b9 : D \u2192 Set\n  P\u00b2    : D \u2192 D \u2192 Set\n\n-- The introduction and elimination rules for the quantifiers are theorems.\npostulate\n  \u2200I : ((x : D) \u2192 P\u00b9 x) \u2192 \u22c0 P\u00b9\n  \u2200E : \u22c0 P\u00b9 \u2192 (t : D) \u2192 P\u00b9 t\n  -- This elimination rule cannot prove in Coq because in Coq we can\n  -- have empty domains. We do not have this problem because the ATPs\n  -- assume a non-empty domain.\n  \u2203I : ((t : D) \u2192 P\u00b9 t) \u2192 \u2203 P\u00b9\n  -- TODO: \u2203E : (x : D) \u2192 \u2203 P\u00b9 \u2192 (P\u00b9 x \u2192 Q\u00b9 x) \u2192 Q\u00b9 x\n{-# ATP prove \u2200I #-}\n{-# ATP prove \u2200E #-}\n{-# ATP prove \u2203I #-}\n-- {-# ATP prove \u2203E #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/Conjectures\/LogicalConstants.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"5bbe3cdf70711a6d013914b2406923d04d1045a2","subject":"Added a draft about bisimilarity and streams via postulates","message":"Added a draft about bisimilarity and streams via postulates\n\nIgnore-this: bbdcec20cf94f65036bec7d31faf95a4\n\ndarcs-hash:20101021034205-3bd4e-3ea18ecd84a8600bbedc524ad0de6b3577ad5a0e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bisimilarity\/Stream\/Postulates.agda","new_file":"Draft\/Bisimilarity\/Stream\/Postulates.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the bisimilar relation with streams via postulates\n------------------------------------------------------------------------------\n\nmodule Draft.Bisimilarity.Stream.Postulates where\n\nopen import LTC.Minimal\nopen import LTC.MinimalER using ( subst )\n\nimport Lib.Relation.Binary.EqReasoning\nopen module Postulates-ER =\n  Lib.Relation.Binary.EqReasoning.StdLib _\u2261_ refl trans\n\nopen import LTC.Minimal.Properties using ( \u2237-injective ; \u2261-stream )\n\nopen import LTC.Data.Stream.Bisimulation hiding ( -\u2248-\u2261 )\n\n------------------------------------------------------------------------\n-- The LTC stream type.\n\npostulate\n  Stream   : D \u2192 Set\n  consS    : (x : D){xs : D} \u2192 Stream xs \u2192 Stream (x \u2237 xs)\n  invConsS : (x : D){xs : D} \u2192 Stream (x \u2237 xs) \u2192 Stream xs\n\n------------------------------------------------------------------------\n-- The bisimilar and the equality relation.\n-\u2248-\u2261 : {xs ys : D} \u2192 Stream xs \u2192 Stream ys \u2192 xs \u2248 ys \u2192 xs \u2261 ys\n-\u2248-\u2261 {xs} {ys} Sxs Sys xs\u2248ys =\n  begin\n    xs       \u2261\u27e8 xs\u2261x'\u2237xs' \u27e9\n    x' \u2237 xs' \u2261\u27e8 \u2261-stream x'\u2261y' (-\u2248-\u2261 Sxs' Sys' xs'\u2248ys') \u27e9\n    y' \u2237 ys' \u2261\u27e8 sym ys\u2261y'\u2237ys' \u27e9\n    ys\n  \u220e\n\n  where\n    aux : BIS _\u2248_ xs ys\n    aux = \u2248-GFP-eq\u2081 xs\u2248ys\n\n    x' : D\n    x' = \u2203D-proj\u2081 aux\n\n    y' : D\n    y' = \u2203D-proj\u2081 (\u2203D-proj\u2082 aux)\n\n    xs' : D\n    xs' = \u2203D-proj\u2081 (\u2203D-proj\u2082 (\u2203D-proj\u2082 aux))\n\n    ys' : D\n    ys' = \u2203D-proj\u2081 (\u2203D-proj\u2082 (\u2203D-proj\u2082 (\u2203D-proj\u2082 aux)))\n\n    x'\u2261y' : x' \u2261 y'\n    x'\u2261y' = \u2227-proj\u2081 (\u2203D-proj\u2082\n                    (\u2203D-proj\u2082\n                    (\u2203D-proj\u2082\n                    (\u2203D-proj\u2082 aux))))\n\n    xs'\u2248ys' : xs' \u2248 ys'\n    xs'\u2248ys' = \u2227-proj\u2081 (\u2227-proj\u2082 (\u2203D-proj\u2082\n                               (\u2203D-proj\u2082\n                               (\u2203D-proj\u2082\n                               (\u2203D-proj\u2082 aux)))))\n\n    xs\u2261x'\u2237xs' : xs \u2261 x' \u2237 xs'\n    xs\u2261x'\u2237xs' =\n      \u2227-proj\u2081 (\u2227-proj\u2082 (\u2227-proj\u2082 (\u2203D-proj\u2082\n                                (\u2203D-proj\u2082\n                                (\u2203D-proj\u2082\n                                (\u2203D-proj\u2082 aux))))))\n\n    ys\u2261y'\u2237ys' : ys \u2261 y' \u2237 ys'\n    ys\u2261y'\u2237ys' =\n      \u2227-proj\u2082 (\u2227-proj\u2082 (\u2227-proj\u2082 (\u2203D-proj\u2082\n                                (\u2203D-proj\u2082\n                                (\u2203D-proj\u2082\n                                (\u2203D-proj\u2082 aux))))))\n\n    Sxs' : Stream xs'\n    Sxs' = invConsS x' (subst (\u03bb t \u2192 Stream t) xs\u2261x'\u2237xs' Sxs)\n\n    Sys' : Stream ys'\n    Sys' = invConsS y' (subst (\u03bb t \u2192 Stream t) ys\u2261y'\u2237ys' Sys)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Bisimilarity\/Stream\/Postulates.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"2d1a447ac5baa546af0537db934c8f97d44c4197","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Hello2.agda","new_file":"src\/Hello2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d1edc229757c913944c238d1996e75049530cb15","subject":"natural numbers added\u2388","message":"natural numbers added\u2388\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Nat.agda","new_file":"agda\/Nat.agda","new_contents":"module Nat where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/Nat.agda' did not match any file(s) known to git\n","license":"unlicense","lang":"Agda"}
{"commit":"da40f434cb7b69079eedad7c55d088078b22eb6d","subject":"Added thesis' example.","message":"Added thesis' example.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/CombinedProofs\/CommAddGlobalHints.agda","new_file":"notes\/thesis\/CombinedProofs\/CommAddGlobalHints.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule CombinedProofs.CommAddGlobalHints where\n\nopen import PA.Axiomatic.Standard.Base\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = PA\u2083\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity = PA-ind A A0 is\n  where\n  A : M \u2192 Set\n  A i = i + zero \u2261 i\n  {-# ATP definition A #-}\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is #-}\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] m n = PA-ind A A0 is m\n  where\n  A : M \u2192 Set\n  A i = i + succ n \u2261 succ (i + n)\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 #-}\n\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is #-}\n\n-- Global hints\n{-# ATP hint x+Sy\u2261S[x+y] +-rightIdentity #-}\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm m n = PA-ind A A0 is m\n  where\n  A : M \u2192 Set\n  A i = i + n \u2261 n + i\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 #-}\n\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/CombinedProofs\/CommAddGlobalHints.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c32177cde9f8c8174a3e0af222a934941dba92ca","subject":"isos-examples.agda","message":"isos-examples.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"isos-examples.agda","new_file":"isos-examples.agda","new_contents":"module isos-examples where\n\nopen import Function\nimport Function.Inverse as FI\nopen FI using (_\u2194_)\nopen import Type hiding (\u2605)\nopen import Data.Product\nopen import Data.Bool\nopen import Data.Bits\nopen import Relation.Binary\n\n_\u2248\u2082_ : \u2200 {a} {A : \u2605 a} (f g : A \u2192 Bit) \u2192 \u2605 _\n_\u2248\u2082_ {A = A} f g = \u03a3 A (T \u2218 f) \u2194 \u03a3 A (T \u2218 g)\n\nmodule _ {a b c} {A : \u2605 a} {B : \u2605 b} {C : \u2605 c} where\n  _\u2248_ : (f : A \u2192 C) (g : B \u2192 C) \u2192 \u2605 _\n  f \u2248 g = \u2200 (O : C \u2192 \u2605 c) \u2192\n            (B \u00d7 \u03a3 A (O \u2218 f)) \u2194 (A \u00d7 \u03a3 B (O \u2218 g))\n\nmodule _ {a c} {A : \u2605 a} {C : \u2605 c} where\n  \u2248-refl : Reflexive {A = A \u2192 C} _\u2248_\n  \u2248-refl _ = FI.id\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'isos-examples.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0a1f721adc2df175c4277260c85bcaa8f0ab6c61","subject":"pisearch.agda","message":"pisearch.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"pisearch.agda","new_file":"pisearch.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"error: RPC failed; HTTP 403 curl 22 The requested URL returned error: 403\nfatal: the remote end hung up unexpectedly\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0804736c6e457437d5e4dad8eeae21a51a16abe4","subject":"Unfinished VectorSpace module","message":"Unfinished VectorSpace module\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/VectorSpace.agda","new_file":"lib\/Algebra\/VectorSpace.agda","new_contents":"{-# OPTIONS --without-K #-}\n\nopen import Level.NP\nopen import Function.Extensionality\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nopen import Algebra.Field\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Algebra.VectorSpace {{_ : FunExt}} where\n\nrecord VectorSpace {\u2113f}{F : Set \u2113f}(\ud835\udd3d : Field F)\n                   {\u2113v}(V : Set \u2113v) : Set (\u2113f \u2294 \u2113v) where\n\n  module \ud835\udd3d = Field {\u2113f} {F} \ud835\udd3d\n\n  field\n\n    V-grp : Group V\n\n    _\u00b7_   : F \u2192 V \u2192 V\n\n{-\n    _\u00b7_ : (F \u00d7 V) \u2192 V\n    _\u00b7_ : F \u2192 (V \u2192 V)\n    _\u00b7_ : F \u2192 Endo V\n-}\n\n    \u00b7-+ : MonoidHomomorphism \ud835\udd3d.+-mon (Pointwise\u2032.mon V (Group.mon V-grp)) _\u00b7_\n    \u00b7-* : MonoidHomomorphism \ud835\udd3d.*-mon (\u2218-mon V) _\u00b7_\n\n  open \ud835\udd3d\n  open Additive-Group V-grp using () renaming (_+_ to _\u2295_)\n\n  module Alternative\n    (\u00b7-+ : \u2200 v \u2192 GroupHomomorphism \ud835\udd3d.+-grp V-grp (\u03bb a \u2192 a \u00b7 v))\n    where\n    \u00b7-+' : \u2200 {v a b} \u2192 (a + b) \u00b7 v \u2261 a \u00b7 v \u2295 b \u00b7 v\n    \u00b7-+' = GroupHomomorphism.hom (\u00b7-+ _)\n\n  \u00b7-+' : \u2200 {v a b} \u2192 (a + b) \u00b7 v \u2261 a \u00b7 v \u2295 b \u00b7 v\n  \u00b7-+' {v} = ap (\u03bb f \u2192 f v) (MonoidHomomorphism.+-hom-* \u00b7-+)\n\n  \u00b7-*' : \u2200 {v a b} \u2192 (a * b) \u00b7 v \u2261 a \u00b7 (b \u00b7 v)\n  \u00b7-*' {v} = ap (\u03bb f \u2192 f v) (MonoidHomomorphism.+-hom-* \u00b7-*)\n  {-\n  _\u00b7_(a * b) \u2261 _\u00b7_(a) \u2218 _\u00b7_(b)\n  \u2200 v \u2192 _\u00b7_(a * b) v \u2261 (_\u00b7_(a) \u2218 _\u00b7_(b)) v\n  \u2200 v \u2192 (a * b) \u00b7 v \u2261 a \u00b7 (b \u00b7 v)\n  -}\n\n  \u00b7-1 : \u2200 {v} \u2192 1# \u00b7 v \u2261 v\n  \u00b7-1 {v} = ap (\u03bb f \u2192 f v) (MonoidHomomorphism.0-hom-1 \u00b7-*)\n  {-\n  _\u00b7_ 1# \u2261 id\n  \u2200 v \u2192 _\u00b7_ 1# v \u2261 id v\n  \u2200 v \u2192 1# \u00b7 v \u2261 v\n  -}\n\n  \u00b7-\u2295 : \u2200 {v w a} \u2192 a \u00b7 (v \u2295 w) \u2261 a \u00b7 v \u2295 a \u00b7 w\n  \u00b7-\u2295 {v} {w} {a} = {!!}\n  {-\n  _\u00b7_(a + b) \u2261 _\u00b7_(a) \u2218 _\u00b7_(b)\n  \u2200 v \u2192 _\u00b7_(a + b) v \u2261 (_\u00b7_(a) \u2218 _\u00b7_(b)) v\n  \u2200 v \u2192 (a + b) \u00b7 v \u2261 a \u00b7 (b \u00b7 v)\n  -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/VectorSpace.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f548d3a50a403325ccffbc9e2e00fdd45c5259ac","subject":"languages\/test.agda","message":"languages\/test.agda\n","repos":"sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis","old_file":"languages\/test.agda","new_file":"languages\/test.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/sharkspeed\/dororis.git\/'\n","license":"bsd-2-clause","lang":"Agda"}
{"commit":"6db81aa5c11142f9be2ecf5889e7147ee3302e3c","subject":"Added Agsy test on distributive laws.","message":"Added Agsy test on distributive laws.\n\nIgnore-this: 764a5b06812ad55cc3c070f1357ab83a\n\ndarcs-hash:20101213173849-3bd4e-0478a0141552920f27964e67c8cb72fd2013df15.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Agsy\/DistributiveLaws.agda","new_file":"Draft\/Agsy\/DistributiveLaws.agda","new_contents":"------------------------------------------------------------------------------\n-- Asgy test on distributive laws on a binary operation\n------------------------------------------------------------------------------\n\nmodule DistributiveLaws where\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _\u2219_\n\n------------------------------------------------------------------------------\n-- Distributive laws axioms\n\npostulate\n  D   : Set        -- The universe\n  _\u2219_ : D \u2192 D \u2192 D  -- The binary operation.\n\n  leftDistributive  : \u2200 x y z \u2192 x \u2219 (y \u2219 z) \u2261 (x \u2219 y) \u2219 (x \u2219 z)\n  rightDistributive : \u2200 x y z \u2192 (x \u2219 y) \u2219 z \u2261 (x \u2219 z) \u2219 (y \u2219 z)\n\n-- Properties\n\nStanovsky1 : \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n                        (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n                        x \u2219 z \u2219 (y \u2219 u)\nStanovsky1 u x y z = {!-t 20 -m!}  -- Agsy fails\n\nStanovsky : \u2200 u x y z \u2192 (x \u2219 y \u2219 (z \u2219 u)) \u2219\n                        (( x \u2219 y \u2219 ( z \u2219 u)) \u2219 (x \u2219 z \u2219 (y \u2219 u))) \u2261\n                        x \u2219 z \u2219 (y \u2219 u)\nStanovsky u x y z =\n-- The numbering of the justifications correspond to the numbers of\n-- the proof using only equational reasoning (see\n-- DistributiveLaws.Interactive.Properties).\n  begin\n    xy\u2219zu \u2219 (xy\u2219zu \u2219 xz\u2219yu)                                         \u2261\u27e8 j\u2081 \u27e9\n\n    xy\u2219zu \u2219 (xz \u2219 xu\u2219yu \u2219 (y\u2219zu \u2219 xz\u2219yu))                           \u2261\u27e8 j\u2085 \u27e9\n\n    xy\u2219zu \u2219 (xz \u2219 xyu \u2219 (yxz \u2219 yu))                                 \u2261\u27e8 j\u2089 \u27e9\n\n    xz \u2219 xyu \u2219 (yz \u2219 xyu) \u2219 (xz \u2219 xyu \u2219 (y\u2219xu \u2219 z\u2219yu))              \u2261\u27e8 j\u2081\u2084 \u27e9\n\n    xz \u2219 xyu \u2219 (y\u2219xu \u2219 (y\u2219yu \u2219 z\u2219yu) \u2219 (z \u2219 xu\u2219yu \u2219 (y\u2219xu \u2219 z\u2219yu))) \u2261\u27e8 j\u2082\u2080 \u27e9\n\n    xz \u2219 xyu \u2219 (y \u2219 xu\u2219zu \u2219 (z \u2219 xu\u2219yu \u2219 (y\u2219xu \u2219 z\u2219yu)))            \u2261\u27e8 j\u2082\u2083 \u27e9\n\n    (xz \u2219 xyu) \u2219 (y \u2219 xu\u2219zu \u2219 (z\u2219xu \u2219 y\u2219xu \u2219 z\u2219yu))                 \u2261\u27e8 j\u2082\u2085 \u27e9\n\n    xz \u2219 xyu \u2219 (y\u2219zy \u2219 xzu)                                         \u2261\u27e8 j\u2083\u2080 \u27e9\n\n    xz\u2219yu\n  \u220e\n  where\n    -- Two variables abbreviations\n\n    xz = x \u2219 z\n    yu = y \u2219 u\n    yz = y \u2219 z\n\n    -- Three variables abbreviations\n\n    xyu = x \u2219 y \u2219 u\n    xyz = x \u2219 y \u2219 z\n    xzu = x \u2219 z \u2219 u\n    yxz = y \u2219 x \u2219 z\n\n    y\u2219xu = y \u2219 (x \u2219 u)\n    y\u2219yu = y \u2219 (y \u2219 u)\n    y\u2219zu = y \u2219 (z \u2219 u)\n    y\u2219zy = y \u2219 (z \u2219 y)\n\n    z\u2219xu = z \u2219 (x \u2219 u)\n    z\u2219yu = z \u2219 (y \u2219 u)\n\n    -- Four variables abbreviations\n\n    xu\u2219yu = x \u2219 u \u2219 (y \u2219 u)\n    xu\u2219zu = x \u2219 u \u2219 (z \u2219 u)\n\n    xy\u2219zu = x \u2219 y \u2219 (z \u2219 u)\n\n    xz\u2219yu = x \u2219 z \u2219 (y \u2219 u)\n\n    -- Steps justifications\n\n    j\u2081 : xy\u2219zu \u2219 (xy\u2219zu \u2219 xz\u2219yu) \u2261\n         xy\u2219zu \u2219 (xz \u2219 xu\u2219yu \u2219 (y\u2219zu \u2219 xz\u2219yu))\n    j\u2081 = {!-t 20 -m !}  -- Agsy fails\n\n    j\u2085 : xy\u2219zu \u2219 (xz \u2219 xu\u2219yu \u2219 (y\u2219zu \u2219 xz\u2219yu)) \u2261\n         xy\u2219zu \u2219 (xz \u2219 xyu \u2219 (yxz \u2219 yu))\n    j\u2085 = {!-t 20 -m !}  -- Agsy fails\n\n    j\u2089 : xy\u2219zu \u2219 (xz \u2219 xyu \u2219 (yxz \u2219 yu)) \u2261\n         xz \u2219 xyu \u2219 (yz \u2219 xyu) \u2219 (xz \u2219 xyu \u2219 (y\u2219xu \u2219 z\u2219yu))\n    j\u2089 = {!-t 20 -m!}  -- Agsy fails\n\n    j\u2081\u2084 : xz \u2219 xyu \u2219 (yz \u2219 xyu) \u2219 (xz \u2219 xyu \u2219 (y\u2219xu \u2219 z\u2219yu)) \u2261\n          xz \u2219 xyu \u2219 (y\u2219xu \u2219 (y\u2219yu \u2219 z\u2219yu) \u2219 (z \u2219 xu\u2219yu \u2219 (y\u2219xu \u2219 z\u2219yu)))\n    j\u2081\u2084 = {!-t 20 -m!}  -- Agsy fails\n\n    j\u2082\u2080 : xz \u2219 xyu \u2219 (y\u2219xu \u2219 (y\u2219yu \u2219 z\u2219yu) \u2219 (z \u2219 xu\u2219yu \u2219 (y\u2219xu \u2219 z\u2219yu))) \u2261\n          xz \u2219 xyu \u2219 (y \u2219 xu\u2219zu \u2219 (z \u2219 xu\u2219yu \u2219 (y\u2219xu \u2219 z\u2219yu)))\n    j\u2082\u2080 = {!-t 20 -m!}  -- Agsy fails\n\n    j\u2082\u2083 : xz \u2219 xyu \u2219 (y \u2219 xu\u2219zu \u2219 (z \u2219 xu\u2219yu \u2219 (y\u2219xu \u2219 z\u2219yu))) \u2261\n          (xz \u2219 xyu) \u2219 (y \u2219 xu\u2219zu \u2219 (z\u2219xu \u2219 y\u2219xu \u2219 z\u2219yu))\n    j\u2082\u2083 = {!-t 20 -m!}  -- Agsy fails\n\n    j\u2082\u2085 : (xz \u2219 xyu) \u2219 (y \u2219 xu\u2219zu \u2219 (z\u2219xu \u2219 y\u2219xu \u2219 z\u2219yu)) \u2261\n          xz \u2219 xyu \u2219 (y\u2219zy \u2219 xzu)\n\n    j\u2082\u2085 = {!-t 20 -m!}  -- Agsy fails\n\n    j\u2083\u2080 : xz \u2219 xyu \u2219 (y\u2219zy \u2219 xzu) \u2261\n          xz\u2219yu\n    j\u2083\u2080 = {!-t 20 -m!}  -- Agsy fails\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Agsy\/DistributiveLaws.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"347bb71d0d851b87e8bad058f999182824c96ad9","subject":"Added incomple draft.","message":"Added incomple draft.\n\nIgnore-this: 914ebd400ea224ce249e5dd8bc1e7e63\n\ndarcs-hash:20120319121849-3bd4e-e2a2f7cab1269de8886051af48d637d604382a2c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Collatz\/CollatzSL.agda","new_file":"Draft\/FOTC\/Program\/Collatz\/CollatzSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Reasoning about a function without a termination proof\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of the standard library on\n-- 19 March 2012.\n\nmodule Draft.FOTC.Program.Collatz.CollatzSL where\n\nopen import Data.Bool\nopen import Data.Empty\nopen import Data.Nat renaming ( suc to succ )\nopen import Data.Product\nopen import Data.Sum\nopen import Relation.Binary.PropositionalEquality hiding ( [_] )\nopen import Relation.Nullary\n\n------------------------------------------------------------------------------\n\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nn ^ zero   = 1\nm ^ succ n = m * (m ^ n)\n\nisEven : \u2115 \u2192 Bool\nisEven 0 = true\nisEven 1 = false\nisEven (succ (succ n)) = isEven n\n\n{-# NO_TERMINATION_CHECK #-}\ncollatz : \u2115 \u2192 \u2115\ncollatz zero       = 1\ncollatz (succ zero) = 1\ncollatz n with isEven n\n... | true = collatz \u230a n \/2\u230b\n... | false = collatz (3 * n + 1)\n\nxy\u22610\u2192x\u22610\u2228y\u22610 : (m n : \u2115) \u2192 m * n \u2261 zero \u2192 m \u2261 zero \u228e n \u2261 zero\nxy\u22610\u2192x\u22610\u2228y\u22610 zero n h = inj\u2081 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (succ m) zero h = inj\u2082 refl\nxy\u22610\u2192x\u22610\u2228y\u22610 (succ m) (succ n) ()\n\n+\u22382 : \u2200 {n} \u2192 \u00ac (n \u2261 zero) \u2192 \u00ac (n \u2261 1) \u2192 n \u2261 succ (succ (n \u2238 2))\n+\u22382 {zero} 0\u22620 _ = \u22a5-elim (0\u22620 refl)\n+\u22382 {succ zero} _ 1\u22621 = \u22a5-elim (1\u22621 refl)\n+\u22382 {succ (succ n)} _ _ = refl\n\n2^x\u22620 : (n : \u2115) \u2192 \u00ac (2 ^ n \u2261 zero)\n2^x\u22620 zero ()\n2^x\u22620 (succ n) h = [ (\u03bb ())\n                   , (\u03bb 2^n\u22610 \u2192 \u22a5-elim (2^x\u22620 n 2^n\u22610))\n                   ]\u2032 (xy\u22610\u2192x\u22610\u2228y\u22610 2 (2 ^ n) h)\n\n2^[x+1]\/2\u22612^x : \u2200 n \u2192 \u230a 2 ^ succ n \/2\u230b \u2261 2 ^ n\n2^[x+1]\/2\u22612^x zero = refl\n2^[x+1]\/2\u22612^x (succ zero) = refl\n2^[x+1]\/2\u22612^x (succ (succ n)) = {!!}\n\ncollatz-2^x : \u2200 n \u2192 \u03a3 \u2115 (\u03bb k \u2192 n \u2261 2 ^ k) \u2192 collatz n \u2261 1\ncollatz-2^x zero h = refl\ncollatz-2^x (succ n) (zero , proj\u2082)   = subst (\u03bb t \u2192 collatz t \u2261 1) (sym proj\u2082) refl\ncollatz-2^x (succ n) (succ k , proj\u2082) = subst (\u03bb t \u2192 collatz t \u2261 1) (sym proj\u2082) {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Program\/Collatz\/CollatzSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"13ed0dbfcc658b371147819a1242853e7137e197","subject":"Playing around with powersets.","message":"Playing around with powersets.\n","repos":"heades\/AUGL","old_file":"power-set-algebra\/powerset.agda","new_file":"power-set-algebra\/powerset.agda","new_contents":"module powerset where\n\nopen import level\nopen import list\nopen import bool\nopen import product\nopen import empty\nopen import unit\nopen import sum\nopen import eq\nopen import functions renaming (id to id-set) public\n\n-- Extensionality will be used when proving equivalences of morphisms.\npostulate ext-set : \u2200{l1 l2 : level} \u2192 extensionality {l1} {l2}\n-- These are isomorphisms, but Agda has no way to prove these as\n-- equivalences.  They are consistent to adopt as equivalences by\n-- univalence:\npostulate \u228e-unit-r : \u2200{X : Set} \u2192 (X \u228e \u22a5) \u2261 X\npostulate \u228e-unit-l : \u2200{X : Set} \u2192 (\u22a5 \u228e X) \u2261 X\npostulate \u2227-unit : \u2200{\u2113}{A : Set \u2113} \u2192 A \u2261 ((\u22a4 {\u2113}) \u2227 A)\npostulate \u2227-sym : \u2200{\u2113}{A B : Set \u2113} \u2192 (A \u2227 B) \u2261 (B \u2227 A)\npostulate \u2227-unit-r : \u2200{\u2113}{A : Set \u2113} \u2192 A \u2261 (A \u2227 (\u22a4 {\u2113}))\npostulate \u2227-assoc : \u2200{\u2113}{A B C : Set \u2113} \u2192  (A \u2227 (B \u2227 C)) \u2261 ((A \u2227 B) \u2227 C)\n\n\u2119 : Set \u2192 Set\u2081\n\u2119 X = X \u2192 Set\n\n\u2119\u2090 : {X Y : Set} \u2192 (X \u2192 Y) \u2192 \u2119 Y \u2192 \u2119 X\n\u2119\u2090 f s x = s (f x)\n\n_\u222a_ : {X : Set} \u2192 \u2119 X \u2192 \u2119 X \u2192 \u2119 X\n(s\u2081 \u222a s\u2082) x = (s\u2081 x) \u228e (s\u2082 x)\n\n_\u2229_ : {X : Set} \u2192 \u2119 X \u2192 \u2119 X \u2192 \u2119 X\n(s\u2081 \u2229 s\u2082) x = (s\u2081 x) \u00d7 (s\u2082 x)\n\n\u00ac_ : {X : Set} \u2192 \u2119 X \u2192 \u2119 X\n(\u00ac s\u2081) x = (s\u2081 x) \u2192 \u22a5\n\n\u2205 : {X : Set} \u2192 \u2119 X\n\u2205 _ = \u22a5\n\ncarrier : {X : Set} \u2192 \u2119 X\ncarrier _ = \u22a4\n\n\u222a-unit\u2081 : \u2200{X : Set}{s : \u2119 X} \u2192 s \u222a \u2205 \u2261 s\n\u222a-unit\u2081 {X}{s} = ext-set aux\n where\n  aux : {a : X} \u2192 (s a \u228e \u22a5) \u2261 s a\n  aux {x} = \u228e-unit-r {s x}\n\n\u222a-unit\u2082 : \u2200{X : Set}{s : \u2119 X} \u2192 \u2205 \u222a s \u2261 s\n\u222a-unit\u2082 {X}{s} = ext-set aux\n where\n   aux : {a : X} \u2192 (\u22a5 \u228e s a) \u2261 s a\n   aux {x} = \u228e-unit-l\n\n\u2229-unit\u2081 : \u2200{X : Set}{s : \u2119 X} \u2192 s \u2229 carrier \u2261 s\n\u2229-unit\u2081 {X}{s} = ext-set aux\n where\n   aux : {a : X} \u2192 \u03a3 (s a) (\u03bb x \u2192 \u22a4) \u2261 s a\n   aux {x} = sym (\u2227-unit-r {_}{s x})\n\n\u2229-unit\u2082 : \u2200{X : Set}{s : \u2119 X} \u2192 carrier \u2229 s \u2261 s\n\u2229-unit\u2082 {X}{s} = ext-set aux\n where\n   aux : {a : X} \u2192 \u03a3 \u22a4 (\u03bb x \u2192 s a) \u2261 s a\n   aux {x} = sym (\u2227-unit {_}{s x})\n\ni\u2081 : {X Y : Set} \u2192 \u2119 X \u2192 \u2119 (X \u00d7 Y)\ni\u2081 = \u2119\u2090 fst\n\ni\u2082 : {X Y : Set} \u2192 \u2119 Y \u2192 \u2119 (X \u00d7 Y)\ni\u2082 = \u2119\u2090 snd\n\ncp-ar : {X Y Z : Set} \u2192 (Z \u2192 X) \u2192 (Z \u2192 Y) \u2192 \u2119 (X \u00d7 Y) \u2192 \u2119 Z\ncp-ar f g = \u2119\u2090 (trans-\u00d7 f g)\n\ncp-diag\u2081 : {X Y Z : Set}{f : Z \u2192 X}{g : Z \u2192 Y} \u2192 cp-ar f g \u2218 i\u2081 \u2261 \u2119\u2090 f\ncp-diag\u2081 {X}{Y}{Z}{f}{g} = refl\n\ncp-diag\u2082 : {X Y Z : Set}{f : Z \u2192 X}{g : Z \u2192 Y} \u2192 cp-ar f g \u2218 i\u2082 \u2261 \u2119\u2090 g\ncp-diag\u2082 {X}{Y}{Z}{f}{g} = refl\n\nco-curry : {A B C : Set} \u2192 ((A \u00d7 B) \u2192 C) \u2192 \u2119 (B \u2192 C) \u2192 \u2119 A\nco-curry {A}{B}{C} f = \u2119\u2090 {A}{B \u2192 C} (curry f)\n\nco-uncurry : {A B C : Set} \u2192 (A \u2192 B \u2192 C) \u2192 \u2119 C \u2192 \u2119 (A \u00d7 B)\nco-uncurry {A}{B}{C} f = \u2119\u2090 {A \u00d7 B} {C} (uncurry f)\n\nlift\u2119 : {A B : Set} \u2192 (A \u2192 \u2119 B) \u2192 \u2119 A \u2192 \u2119 B\nlift\u2119 {A} f s b = \u2200(a : A) \u2192 (s a) \u00d7 (f a b)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'power-set-algebra\/powerset.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"0d4aad96662c5c3c96874965507848f6954cb489","subject":"Comments","message":"Comments\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/WhyDependentTypesMatter.agda","new_file":"agda\/WhyDependentTypesMatter.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"4f7754ad07654f17376bfff57d68a8936f5dfa5f","subject":"Added test.","message":"Added test.\n\nIgnore-this: 8285ded7fae8d2e4f0a41678130fd2b\n\ndarcs-hash:20120531154627-3bd4e-4b2d7d451f2079aca2ed50457b493d389f18f81d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/FOL\/TermToFOLTerm\/ConTerm.agda","new_file":"Test\/Succeed\/FOL\/TermToFOLTerm\/ConTerm.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the function FOL.Translation.Terms.termToFOLTerm : Con term\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test.Succeed.FOL.TermToFOLTerm.ConTerm where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _+_\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n\ndata M : Set where\n  zero :     M\n  succ : M \u2192 M\n\ndata _\u2261_ (m : M) : M \u2192 Set where\n  refl : m \u2261 m\n\nsym : \u2200 {x y} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\n_+_ : M \u2192 M \u2192 M\nzero   + n = n\nsucc m + n = succ (m + n)\n\n+-Sx : \u2200 m n \u2192 succ m + n \u2261 succ (m + n)\n+-Sx m n = refl\n{-# ATP hint +-Sx #-}\n\npostulate\n  +-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n  x+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm zero     n = sym (+-rightIdentity n)\n+-comm (succ m) n = prf (+-comm m n)\n  where\n  postulate prf : m + n \u2261 n + m \u2192  -- IH.\n                  succ m + n \u2261 n + succ m\n  {-# ATP prove prf x+Sy\u2261S[x+y] #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/FOL\/TermToFOLTerm\/ConTerm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ddc36dae24c43d057f2b8f67f617631d371130bf","subject":"Add Control.Process.Type","message":"Add Control.Process.Type\n","repos":"audreyt\/agda-libjs,crypto-agda\/agda-libjs","old_file":"lib\/Control\/Process\/Type.agda","new_file":"lib\/Control\/Process\/Type.agda","new_contents":"{-\n  On the one hand this module has nothing to do with JS, on the other\n  hand our JS binding relies on this particular type definition.\n-}\nmodule Control.Process.Type where\n\nopen import Data.String.Base using (String)\n\ndata Proc (C {-channels-} : Set\u2080) (M {-messages-} : Set\u2080) : Set\u2080 where\n  end   : Proc C M\n  send  : (d : C) (m : M) (p : Proc C M) \u2192 Proc C M\n  recv  : (d : C) (p : M \u2192 Proc C M) \u2192 Proc C M\n  spawn : (p : C \u2192 Proc C M)\n          (q : C \u2192 Proc C M) \u2192 Proc C M\n  error : (err : String) \u2192 Proc C M\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Control\/Process\/Type.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fac230e3f80892464ba98eddf684b4174887002b","subject":"clean import","message":"clean import\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2\/Handler\/SnapServer.agda","new_file":"src\/Chu2\/Handler\/SnapServer.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"65ee37be8479081234a4f7463c886ba0c0460f7c","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2\/FFI.agda","new_file":"src\/Chu2\/FFI.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f735f8c61d7c09bceb9116b8ab3df3f76cdaa934","subject":"lib\/Category\/Monad\/NP.agda","message":"lib\/Category\/Monad\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Monad\/NP.agda","new_file":"lib\/Category\/Monad\/NP.agda","new_contents":"","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Category\/Monad\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"30ebaf47303b0b75227e010e104e1e802e3156e0","subject":"thrms about ~ and \\~","message":"thrms about ~ and \\~\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"core.agda","new_file":"core.agda","new_contents":"open import Nat\nopen import Prelude\n\nmodule core where\n  -- types\n  data \u03c4\u0307 : Set where\n    num   : \u03c4\u0307\n    \u2987\u2988    : \u03c4\u0307\n    _==>_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2295_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n    _\u2297_   : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 \u03c4\u0307\n\n  -- expressions, prefixed with a \u00b7 to distinguish name clashes with agda\n  -- built-ins\n  data e\u0307 : Set where\n    _\u00b7:_   : e\u0307 \u2192 \u03c4\u0307 \u2192 e\u0307\n    X      : Nat \u2192 e\u0307\n    \u00b7\u03bb     : Nat \u2192 e\u0307 \u2192 e\u0307\n    N      : Nat \u2192 e\u0307\n    _\u00b7+_   : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    \u2987\u2988[_]  : Nat \u2192 e\u0307\n    \u2987_\u2988[_] : e\u0307 \u2192 Nat \u2192 e\u0307\n    _\u2218_    : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    inl    : e\u0307 \u2192 e\u0307\n    inr    : e\u0307 \u2192 e\u0307\n    case   : e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 Nat \u2192 e\u0307 \u2192 e\u0307\n    [_,_]  : e\u0307 \u2192 e\u0307 \u2192 e\u0307\n    fst    : e\u0307 \u2192 e\u0307\n    snd    : e\u0307 \u2192 e\u0307\n\n  data subst : Set where -- todo\n\n  data \u00eb : Set where\n    X        : Nat \u2192 \u00eb\n    \u00b7\u03bb       : Nat \u2192 \u00eb \u2192 \u00eb\n    N        : Nat \u2192 \u00eb\n    _\u00b7+_     : \u00eb \u2192 \u00eb \u2192 \u00eb\n    \u2987\u2988[_,_]  : Nat \u2192 subst \u2192 \u00eb\n    \u2987_\u2988[_,_] : \u00eb \u2192 Nat \u2192 subst \u2192 \u00eb\n    _\u2218_      : \u00eb \u2192 \u00eb \u2192 \u00eb\n    inl      : \u00eb \u2192 \u00eb\n    inr      : \u00eb \u2192 \u00eb\n    case     : \u00eb \u2192 Nat \u2192 \u00eb \u2192 Nat \u2192 \u00eb \u2192 \u00eb\n    [_,_]    : \u00eb \u2192 \u00eb \u2192 \u00eb\n    fst      : \u00eb \u2192 \u00eb\n    snd      : \u00eb \u2192 \u00eb\n    <_>_     : \u00eb \u2192 \u03c4\u0307 \u2192 \u00eb\n\n  -- type consistency\n  data _~_ : (t1 : \u03c4\u0307) \u2192 (t2 : \u03c4\u0307) \u2192 Set where\n    TCRefl : {t : \u03c4\u0307} \u2192 t ~ t\n    TCHole1 : {t : \u03c4\u0307} \u2192 t ~ \u2987\u2988\n    TCHole2 : {t : \u03c4\u0307} \u2192 \u2987\u2988 ~ t\n    TCArr : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 ==> t2) ~ (t1' ==> t2')\n    TCSum : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2295 t2) ~ (t1' \u2295 t2')\n    TCPro : {t1 t2 t1' t2' : \u03c4\u0307} \u2192\n               t1 ~ t1' \u2192\n               t2 ~ t2' \u2192\n               (t1 \u2297 t2) ~ (t1' \u2297 t2')\n\n\n  -- type inconsistency\n  data _~\u0338_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    ICNumArr1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 ==> t2)\n    ICNumArr2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 num\n    ICArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n    ICArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 ==> t2) ~\u0338 (t3 ==> t4)\n\n    ICNumSum1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2295 t2)\n    ICNumSum2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 num\n    ICSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2295 t2) ~\u0338 (t3 \u2295 t4)\n    ICSumArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 ==> t4)\n    ICSumArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2295 t4)\n\n    ICNumPro1 : {t1 t2 : \u03c4\u0307} \u2192 num ~\u0338 (t1 \u2297 t2)\n    ICNumPro2 : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 num\n    ICPro1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t1 ~\u0338 t3 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICPro2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192\n               t2 ~\u0338 t4 \u2192\n               (t1 \u2297 t2) ~\u0338 (t3 \u2297 t4)\n    ICProArr1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 ==> t4)\n    ICProArr2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 ==> t2) ~\u0338 (t3 \u2297 t4)\n\n    ICProSum1 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2297 t2) ~\u0338 (t3 \u2295 t4)\n    ICProSum2 : {t1 t2 t3 t4 : \u03c4\u0307} \u2192 (t1 \u2295 t2) ~\u0338 (t3 \u2297 t4)\n\n\n  --- matching for arrows, sums, and products\n  data _\u25b8arr_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MAHole : \u2987\u2988 \u25b8arr (\u2987\u2988 ==> \u2987\u2988)\n    MAArr  : {t1 t2 : \u03c4\u0307} \u2192 (t1 ==> t2) \u25b8arr (t1 ==> t2)\n\n  data _\u25b8sum_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MSHole  : \u2987\u2988 \u25b8sum (\u2987\u2988 \u2295 \u2987\u2988)\n    MSPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2295 t2) \u25b8sum (t1 \u2295 t2)\n\n  data _\u25b8pro_ : \u03c4\u0307 \u2192 \u03c4\u0307 \u2192 Set where\n    MPHole  : \u2987\u2988 \u25b8pro (\u2987\u2988 \u2297 \u2987\u2988)\n    MPPlus  : {t1 t2 : \u03c4\u0307} \u2192 (t1 \u2297 t2) \u25b8pro (t1 \u2297 t2)\n\n  -- matching produces unique answers for arrows, sums, and products\n  \u25b8arr-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8arr t2 \u2192\n                 t \u25b8arr t3 \u2192\n                 t2 == t3\n  \u25b8arr-unicity MAHole MAHole = refl\n  \u25b8arr-unicity MAArr MAArr = refl\n\n  \u25b8sum-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8sum t2 \u2192\n                 t \u25b8sum t3 \u2192\n                 t2 == t3\n  \u25b8sum-unicity MSHole MSHole = refl\n  \u25b8sum-unicity MSPlus MSPlus = refl\n\n  \u25b8pro-unicity : \u2200{ t t2 t3 } \u2192\n                 t \u25b8pro t2 \u2192\n                 t \u25b8pro t3 \u2192\n                 t2 == t3\n  \u25b8pro-unicity MPHole MPHole = refl\n  \u25b8pro-unicity MPPlus MPPlus = refl\n\n\n  -- if an arrow matches, then it's consistent with the least restrictive\n  -- function type\n  matchconsist : \u2200{t t'} \u2192\n                 t \u25b8arr t' \u2192\n                 t ~ (\u2987\u2988 ==> \u2987\u2988)\n  matchconsist MAHole = TCHole2\n  matchconsist MAArr = TCArr TCHole1 TCHole1\n\n  matchnotnum : \u2200{t1 t2} \u2192 num \u25b8arr (t1 ==> t2) \u2192 \u22a5\n  matchnotnum ()\n\n  ---- contexts and some operations on them\n\n  -- variables are named with naturals in e\u0307. therefore we represent\n  -- contexts as functions from names for variables (nats) to possible\n  -- bindings.\n  \u00b7ctx : Set\n  \u00b7ctx = Nat \u2192 Maybe \u03c4\u0307\n\n  -- convenient shorthand for the (unique up to fun. ext.) empty context\n  \u2205 : \u00b7ctx\n  \u2205 _ = None\n\n  -- add a new binding to the context, clobbering anything that might have\n  -- been there before.\n  _,,_ : \u00b7ctx \u2192 (Nat \u00d7 \u03c4\u0307) \u2192 \u00b7ctx\n  (\u0393 ,, (x , t)) y with natEQ x y\n  (\u0393 ,, (x , t)) .x | Inl refl = Some t\n  (\u0393 ,, (x , t)) y  | Inr neq  = \u0393 y\n\n  -- membership, or presence, in a context\n  _\u2208_ : (p : Nat \u00d7 \u03c4\u0307) \u2192 (\u0393 : \u00b7ctx) \u2192 Set\n  (x , t) \u2208 \u0393 = (\u0393 x) == Some t\n\n  -- apartness for contexts, so that we can follow barendregt's convention\n  _#_ : (n : Nat) \u2192 (\u0393 : \u00b7ctx) \u2192 Set\n  x # \u0393 = (\u0393 x) == None\n\n  -- without: remove a variable from a context\n  _\/_ : \u00b7ctx \u2192 Nat \u2192 \u00b7ctx\n  (\u0393 \/ x) y with natEQ x y\n  (\u0393 \/ x) .x | Inl refl = None\n  (\u0393 \/ x) y  | Inr neq  = \u0393 y\n\n  -- bidirectional type checking judgements for e\u0307\n  mutual\n    -- synthesis\n    data _\u22a2_=>_ : (\u0393 : \u00b7ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      SAsc    : {\u0393 : \u00b7ctx} {e : e\u0307} {t : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e <= t \u2192\n                 \u0393 \u22a2 (e \u00b7: t) => t\n      SVar    : {\u0393 : \u00b7ctx} {t : \u03c4\u0307} {n : Nat} \u2192\n                 (n , t) \u2208 \u0393 \u2192\n                 \u0393 \u22a2 X n => t\n      SAp     : {\u0393 : \u00b7ctx} {e1 e2 : e\u0307} {t t' t2 : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e1 => t \u2192\n                 t \u25b8arr (t2 ==> t') \u2192\n                 \u0393 \u22a2 e2 <= t2 \u2192\n                 \u0393 \u22a2 (e1 \u2218 e2) => t'\n      SNum    :  {\u0393 : \u00b7ctx} {n : Nat} \u2192\n                 \u0393 \u22a2 N n => num\n      SPlus   : {\u0393 : \u00b7ctx} {e1 e2 : e\u0307}  \u2192\n                 \u0393 \u22a2 e1 <= num \u2192\n                 \u0393 \u22a2 e2 <= num \u2192\n                 \u0393 \u22a2 (e1 \u00b7+ e2) => num\n      SEHole  : {\u0393 : \u00b7ctx} {u : Nat} \u2192 \u0393 \u22a2 \u2987\u2988[ u ] => \u2987\u2988 -- todo: uniqueness of n?\n      SNEHole : {\u0393 : \u00b7ctx} {e : e\u0307} {t : \u03c4\u0307} {u : Nat} \u2192 -- todo: uniqueness of n?\n                 \u0393 \u22a2 e => t \u2192\n                 \u0393 \u22a2 \u2987 e \u2988[ u ] => \u2987\u2988\n\n    --todo: add rules for products\n\n    -- analysis\n    data _\u22a2_<=_ : (\u0393 : \u00b7ctx) \u2192 (e : e\u0307) \u2192 (t : \u03c4\u0307) \u2192 Set where\n      ASubsume : {\u0393 : \u00b7ctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                 \u0393 \u22a2 e => t' \u2192\n                 t ~ t' \u2192\n                 \u0393 \u22a2 e <= t\n      ALam : {\u0393 : \u00b7ctx} {e : e\u0307} {t t1 t2 : \u03c4\u0307} {x : Nat} \u2192\n                 x # \u0393 \u2192\n                 t \u25b8arr (t1 ==> t2) \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 (\u00b7\u03bb x e) <= t\n      AInl : {\u0393 : \u00b7ctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t1 \u2192\n                 \u0393 \u22a2 inl e <= t+\n      AInr : {\u0393 : \u00b7ctx} {e : e\u0307} {t+ t1 t2 : \u03c4\u0307} \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e <= t2 \u2192\n                 \u0393 \u22a2 inr e <= t+\n      ACase : {\u0393 : \u00b7ctx} {e e1 e2 : e\u0307} {t t+ t1 t2 : \u03c4\u0307} {x y : Nat} \u2192\n                 x # \u0393 \u2192\n                 y # \u0393 \u2192\n                 t+ \u25b8sum (t1 \u2295 t2) \u2192\n                 \u0393 \u22a2 e => t+ \u2192\n                 (\u0393 ,, (x , t1)) \u22a2 e1 <= t \u2192\n                 (\u0393 ,, (y , t2)) \u22a2 e2 <= t \u2192\n                 \u0393 \u22a2 case e x e1 y e2 <= t\n\n  ----- some theorems about the rules and judgement presented so far.\n\n  -- a variable is apart from any context from which it is removed\n  aar : (\u0393 : \u00b7ctx) (x : Nat) \u2192 x # (\u0393 \/ x)\n  aar \u0393 x with natEQ x x\n  aar \u0393 x | Inl refl = refl\n  aar \u0393 x | Inr x\u2260x  = abort (x\u2260x refl)\n\n  -- contexts give at most one binding for each variable\n  ctxunicity : {\u0393 : \u00b7ctx} {n : Nat} {t t' : \u03c4\u0307} \u2192\n               (n , t) \u2208 \u0393 \u2192\n               (n , t') \u2208 \u0393 \u2192\n               t == t'\n  ctxunicity {n = n} p q with natEQ n n\n  ctxunicity p q | Inl refl = someinj (! p \u00b7 q)\n  ctxunicity _ _ | Inr x\u2260x = abort (x\u2260x refl)\n\n  -- type consistency is symmetric\n  ~sym : {t1 t2 : \u03c4\u0307} \u2192 t1 ~ t2 \u2192 t2 ~ t1\n  ~sym TCRefl = TCRefl\n  ~sym TCHole1 = TCHole2\n  ~sym TCHole2 = TCHole1\n  ~sym (TCArr p1 p2) = TCArr (~sym p1) (~sym p2)\n  ~sym (TCSum p1 p2) = TCSum (~sym p1) (~sym p2)\n  ~sym (TCPro p1 p2) = TCPro (~sym p1) (~sym p2)\n\n  -- type consistency isn't transitive\n  not-trans : ((t1 t2 t3 : \u03c4\u0307) \u2192 t1 ~ t2 \u2192 t2 ~ t3 \u2192 t1 ~ t3) \u2192 \u22a5\n  not-trans t with t (num ==> num) \u2987\u2988 num TCHole1 TCHole2\n  ... | ()\n\n  --  every pair of types is either consistent or not consistent\n  ~dec : (t1 t2 : \u03c4\u0307) \u2192 ((t1 ~ t2) + (t1 ~\u0338 t2))\n    -- this takes care of all hole cases, so we don't consider them below\n  ~dec _ \u2987\u2988 = Inl TCHole1\n  ~dec \u2987\u2988 _ = Inl TCHole2\n    -- num cases\n  ~dec num num = Inl TCRefl\n  ~dec num (t2 ==> t3) = Inr ICNumArr1\n    -- arrow cases\n  ~dec (t1 ==> t2) num = Inr ICNumArr2\n  ~dec (t1 ==> t2) (t3 ==> t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCArr x y)\n  ... | Inl _ | Inr y = Inr (ICArr2 y)\n  ... | Inr x | _     = Inr (ICArr1 x)\n    -- plus cases\n  ~dec (t1 \u2295 t2) num = Inr ICNumSum2\n  ~dec (t1 \u2295 t2) (t3 ==> t4) = Inr ICSumArr1\n  ~dec (t1 \u2295 t2) (t3 \u2295 t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCSum x y)\n  ... | _     | Inr x = Inr (ICSum2 x)\n  ... | Inr x | Inl _ = Inr (ICSum1 x)\n  ~dec num (y \u2295 y\u2081) = Inr ICNumSum1\n  ~dec (x ==> x\u2081) (y \u2295 y\u2081) = Inr ICSumArr2\n  ~dec num (t2 \u2297 t3) = Inr ICNumPro1\n  ~dec (t1 ==> t2) (t3 \u2297 t4) = Inr ICProArr2\n  ~dec (t1 \u2295 t2) (t3 \u2297 t4) = Inr ICProSum2\n  ~dec (t1 \u2297 t2) num = Inr ICNumPro2\n  ~dec (t1 \u2297 t2) (t3 ==> t4) = Inr ICProArr1\n  ~dec (t1 \u2297 t2) (t3 \u2295 t4) = Inr ICProSum1\n  ~dec (t1 \u2297 t2) (t3 \u2297 t4) with ~dec t1 t3 | ~dec t2 t4\n  ... | Inl x | Inl y = Inl (TCPro x y)\n  ... | _     | Inr x = Inr (ICPro2 x)\n  ... | Inr x | Inl _ = Inr (ICPro1 x)\n\n  -- no pair of types is both consistent and not consistent\n  ~apart : {t1 t2 : \u03c4\u0307} \u2192 (t1 ~\u0338 t2) \u2192 (t1 ~ t2) \u2192 \u22a5\n  ~apart ICNumArr1 ()\n  ~apart ICNumArr2 ()\n  ~apart (ICArr1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICArr1 x) (TCArr y y\u2081) = ~apart x y\n  ~apart (ICArr2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICArr2 x) (TCArr y y\u2081) = ~apart x y\u2081\n  ~apart ICNumSum1 ()\n  ~apart ICNumSum2 ()\n  ~apart (ICSum1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICSum1 x) (TCSum y y\u2081) = ~apart x y\n  ~apart (ICSum2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICSum2 x) (TCSum y y\u2081) = ~apart x y\u2081\n  ~apart ICSumArr1 ()\n  ~apart ICSumArr2 ()\n  ~apart ICNumPro1 ()\n  ~apart ICNumPro2 ()\n  ~apart (ICPro1 x) TCRefl = ~apart x TCRefl\n  ~apart (ICPro1 x) (TCPro y y\u2081) = ~apart x y\n  ~apart (ICPro2 x) TCRefl = ~apart x TCRefl\n  ~apart (ICPro2 x) (TCPro y y\u2081) = ~apart x y\u2081\n  ~apart ICProArr1 ()\n  ~apart ICProArr2 ()\n  ~apart ICProSum1 ()\n  ~apart ICProSum2 ()\n\n  -- synthesis only produces equal types. note that there is no need for an\n  -- analagous theorem for analytic positions because we think of\n  -- the type as an input\n  synthunicity : {\u0393 : \u00b7ctx} {e : e\u0307} {t t' : \u03c4\u0307} \u2192\n                  (\u0393 \u22a2 e => t)\n                \u2192 (\u0393 \u22a2 e => t')\n                \u2192 t == t'\n  synthunicity (SAsc _) (SAsc _) = refl\n  synthunicity {\u0393 = G} (SVar in1) (SVar in2) = ctxunicity {\u0393 = G} in1 in2\n  synthunicity (SAp D1 MAHole b) (SAp D2 MAHole y) = refl\n  synthunicity (SAp D1 MAHole b) (SAp D2 MAArr y) with synthunicity D1 D2\n  ... | ()\n  synthunicity (SAp D1 MAArr b) (SAp D2 MAHole y) with synthunicity D1 D2\n  ... | ()\n  synthunicity (SAp D1 MAArr b) (SAp D2 MAArr y) with synthunicity D1 D2\n  ... | refl = refl\n  synthunicity SNum SNum = refl\n  synthunicity (SPlus _ _ ) (SPlus _ _ ) = refl\n  synthunicity SEHole SEHole = refl\n  synthunicity (SNEHole _) (SNEHole _) = refl\n\n\n  ----- the zippered form of the forms above and the rules for actions on them\n\n  -- those types without holes anywhere\n  tcomplete : \u03c4\u0307 \u2192 Set\n  tcomplete num         = \u22a4\n  tcomplete \u2987\u2988        = \u22a5\n  tcomplete (t1 ==> t2) = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2295 t2)   = tcomplete t1 \u00d7 tcomplete t2\n  tcomplete (t1 \u2297 t2)   = tcomplete t1 \u00d7 tcomplete t2\n\n  -- similarly to the complete types, the complete expressions\n  ecomplete : e\u0307 \u2192 Set\n  ecomplete (e1 \u00b7: t)  = ecomplete e1 \u00d7 tcomplete t\n  ecomplete (X _)      = \u22a4\n  ecomplete (\u00b7\u03bb _ e1)  = ecomplete e1\n  ecomplete (N x)      = \u22a4\n  ecomplete (e1 \u00b7+ e2) = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete \u2987\u2988[ u ]       = \u22a5\n  ecomplete \u2987 e1 \u2988[ u ]   = \u22a5\n  ecomplete (e1 \u2218 e2)  = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (inl e)    = ecomplete e\n  ecomplete (inr e)    = ecomplete e\n  ecomplete (case e x e1 y e2)  = ecomplete e \u00d7 ecomplete e1 \u00d7 ecomplete e2\n  ecomplete [ e1 , e2 ] = ecomplete e1 \u00d7 ecomplete e2\n  ecomplete (fst e) = ecomplete e\n  ecomplete (snd e) = ecomplete e\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'core.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"070d85b03a26d7ad1acae813690fd2fc43895ce8","subject":"Type.Eq: classify types with an equality test","message":"Type.Eq: classify types with an equality test\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Type\/Eq.agda","new_file":"lib\/Type\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Type.Eq where\n\nopen import Data.Two\n              renaming (_==_ to _==\ud835\udfda_)\nopen import Relation.Binary.PropositionalEquality\n              renaming (cong to ap; cong\u2082 to ap\u2082)\nopen import Data.Product\n              renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nrecord Eq? {a} (A : Set a) : Set a where\n  field\n    _==_ : A \u2192 A \u2192 \ud835\udfda\n    \u2261\u21d2== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n    ==\u21d2\u2261 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y\n\nopen Eq? {{...}} public\n\n\ud835\udfda-Eq? : Eq? \ud835\udfda\n\ud835\udfda-Eq? = record { _==_ = _==\ud835\udfda_\n               ; \u2261\u21d2== = \u03bb { {0\u2082} .{0\u2082} refl \u2192 _\n                          ; {1\u2082} .{1\u2082} refl \u2192 _\n                          }\n               ; ==\u21d2\u2261 = \u03bb { {0\u2082} {0\u2082} _ \u2192 refl\n                          ; {1\u2082} {1\u2082} _ \u2192 refl\n                          ; {0\u2082} {1\u2082} ()\n                          ; {1\u2082} {0\u2082} ()\n                          }\n               }\n\nmodule _ {a b}{A : Set a}{B : Set b}\n         {{A-eq? : Eq? A}}\n         {{B-eq? : Eq? B}} where\n  instance\n    \u00d7-Eq? : Eq? (A \u00d7 B)\n    \u00d7-Eq? = record\n      { _==_ = \u03bb x y \u2192 fst x == fst y \u2227 snd x == snd y\n      ; \u2261\u21d2== = \u03bb e \u2192 \u2713\u2227 (\u2261\u21d2== (ap fst e)) (\u2261\u21d2== (ap snd e))\n      ; ==\u21d2\u2261 = \u03bb e \u2192 let p = \u2713\u2227\u00d7 e in\n                      ap\u2082 _,_ (==\u21d2\u2261 (fst p))\n                              (==\u21d2\u2261 (snd p))\n      }\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Type\/Eq.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fe897e6f70171b8764c0c0e70d4b24674c191f69","subject":"universe.agda","message":"universe.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"universe.agda","new_file":"universe.agda","new_contents":"module universe where\n\nimport Level as L\nopen import Data.Nat.NP using (\u2115; _+_; _*_; _^_)\nopen import Data.Bits using (Bit)\nopen import Data.Unit using (\u22a4)\nopen import Data.Product using (_\u00d7_; _,_) renaming (zip to \u00d7-zip)\nopen import Data.Vec using (Vec)\n\nrecord Universe {t} (T : Set t) : Set t where\n  constructor mk\n  field\n    `\u22a4    : T\n    `Bit  : T\n    _`\u00d7_  : T \u2192 T \u2192 T\n    _`^_  : T \u2192 \u2115 \u2192 T\n\n  `Vec : T \u2192 \u2115 \u2192 T\n  `Vec A n = A `^ n\n\n  `Bits : \u2115 \u2192 T\n  `Bits n = `Bit `^ n\n\n  infixr 2 _`\u00d7_\n  infixl 2 _`^_\n\nSet-U : Universe Set\nSet-U = mk \u22a4 Bit _\u00d7_ Vec\n\nwidth-U : Universe \u2115\nwidth-U = mk 0 1 _+_ _*_\n\ncard-U : Universe \u2115\ncard-U = mk 1 2 _*_ _^_\n\n\u22a4-U : Universe \u22a4\n\u22a4-U = mk _ _ _ _\n\n\u00d7-U : \u2200 {s t} {S : Set s} {T : Set t} \u2192 Universe S \u2192 Universe T \u2192 Universe (S \u00d7 T)\n\u00d7-U S-U T-U = mk (S.`\u22a4 , T.`\u22a4) (S.`Bit , T.`Bit) (\u00d7-zip S._`\u00d7_ T._`\u00d7_)\n                 (\u03bb { (A\u2080 , A\u2081) n \u2192 S.`Vec A\u2080 n , T.`Vec A\u2081 n })\n  where module S = Universe S-U\n        module T = Universe T-U\n\nSym-U : \u2200 t \u2192 Set (L.suc t)\nSym-U t = \u2200 {T : Set t} \u2192 Universe T\n\ndata U : Set where\n  \u22a4\u2032 Bit\u2032 : U\n  _\u00d7\u2032_    : U \u2192 U \u2192 U\n  _^\u2032_    : U \u2192 \u2115 \u2192 U\n\nsyn-U : Universe U\nsyn-U = mk \u22a4\u2032 Bit\u2032 _\u00d7\u2032_ _^\u2032_\n\nfold-U : \u2200 {t} {T : Set t} \u2192 Universe T \u2192 U \u2192 T\nfold-U {_} {T} uni = go\n  where open Universe uni\n        go : U \u2192 T\n        go \u22a4\u2032         = `\u22a4\n        go Bit\u2032       = `Bit\n        go (t\u2080 \u00d7\u2032 t\u2081) = go t\u2080 `\u00d7 go t\u2081\n        go (t ^\u2032 n)   = go t `^ n\n\nUniOp : \u2200 {s t} (S : Set s) (T : Set t) \u2192 Set _\nUniOp S T = Universe S \u2192 Universe T\n\nUniOp\u2082 : \u2200 {r s t} (R : Set r) (S : Set s) (T : Set t) \u2192 Set _\nUniOp\u2082 R S T = Universe R \u2192 Universe S \u2192 Universe T\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'universe.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"27bb114888acef2e399d9013b1180d94e33cabb3","subject":"Added note on possible type classes.","message":"Added note on possible type classes.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/TypeClasses\/Test.agda","new_file":"notes\/FOT\/FOTC\/TypeClasses\/Test.agda","new_contents":"------------------------------------------------------------------------------\n-- Note on type classes\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.TypeClasses.Test where\n\nopen import FOTC.Base\nopen import FOTC.Data.List.Type\nopen import FOTC.Data.Nat.Type\n\n------------------------------------------------------------------------------\n\n-- 14 December 2012. Type classes for FOTC means type classes for the\n-- inductive predicates because them represent the Haskell types. In this\n-- case, we are not working in first-order logic.\n\ndata _\u2248_ {A : Set\u2081}(x : A) : A \u2192 Set\u2081 where\n  refl : x \u2248 x\n\nfoo : \u2200 {m n} \u2192 m \u2261 n \u2192 N m \u2248 N n\nfoo refl = refl\n\nbar : \u2200 {xs ys} \u2192 xs \u2261 ys \u2192 List xs \u2248 List ys\nbar refl = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/TypeClasses\/Test.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"cb9fbef09ea7bf13700b8138fa0a12b4bf47d1f7","subject":"md5.agda","message":"md5.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"md5.agda","new_file":"md5.agda","new_contents":"module md5 where\n\nopen import Type\nopen import Data.Fin using (Fin)\nopen import Data.Bits using (Bits)\nopen import Data.Nat\n\ninfix 0 _\u2325_\ninfixr 1 _\u204f_\ninfix 0 _`\u2192_\n_`\u00d7_ = _+_\n`Bit = 1\n`Bits : \u2115 \u2192 \u2115\n`Bits n = n\npostulate\n  -- \u2115 : Set\n  _\u229e_ : \u2200 {q} \u2192 Fin q \u2192 Fin q \u2192 Fin q\n  _\u2325_ : \u2115 \u2192 \u2115 \u2192 \u2605\n\n_`\u2192_ = _\u2325_\npostulate\n  rewire : \u2200 {i o} \u2192 (Fin o \u2192 Fin i) \u2192 i \u2325 o\n    -- cost: 0 time, o space\n  _\u2218_    : \u2200 {m n o} \u2192 (n \u2325 o) \u2192 (m \u2325 n) \u2192 (m \u2325 o)\n  _\u204f_    : \u2200 {m n o} \u2192 (m \u2325 n) \u2192 (n \u2325 o) \u2192 (m \u2325 o)\n  dup : \u2200 {n} \u2192 n \u2325 (n + n)\n    -- cost: sum time and space\n  <_\u00d7_>  : \u2200 {m n o p} \u2192 (m \u2325 o) \u2192 (n \u2325 p) \u2192 (m + n) \u2325 (o + p)\n    -- cost: max time and sum space\n  cond   : \u2200 {n} \u2192 (1 + (n + n)) \u2325 n\n    -- cost: 1 time, 1 space\n  bits   : \u2200 {n _\u22a4} \u2192 Bits n \u2192 _\u22a4 \u2325 n\n    -- cost: 0 time, n space\n  --xor    : \u2200 {n} \u2192 Bits n \u2192 n \u2325 n\n    -- cost: 1 time, n space\n  id        : \u2200 {A} \u2192 A `\u2192 A\n  fst       : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 A\n  snd       : \u2200 {A B} \u2192 A `\u00d7 B `\u2192 B\n  swap      : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 (B `\u00d7 A)\n  assoc     : \u2200 {A B C} \u2192 ((A `\u00d7 B) `\u00d7 C) `\u2192 (A `\u00d7 (B `\u00d7 C))\n  {-\n  tt        : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `\u22a4\n  <[]>      : \u2200 {_\u22a4 A} \u2192 _\u22a4 `\u2192 `Vec A 0\n  <\u2237>       : \u2200 {n A} \u2192 (A `\u00d7 `Vec A n) `\u2192 `Vec A (1 + n)\n  uncons    : \u2200 {n A} \u2192 `Vec A (1 + n) `\u2192 (A `\u00d7 `Vec A n)\n  <tt,id>   : \u2200 {A} \u2192 A `\u2192 `\u22a4 `\u00d7 A\n  snd<tt,>  : \u2200 {A} \u2192 `\u22a4 `\u00d7 A `\u2192 A\n  tt\u2192[]     : \u2200 {A} \u2192 `\u22a4 `\u2192 `Vec A 0\n  []\u2192tt     : \u2200 {A} \u2192 `Vec A 0 `\u2192 `\u22a4\n  rewireTbl : \u2200 {i o} \u2192 RewireTbl i o   \u2192 `Bits i `\u2192 `Bits o\n  -}\n  first     : \u2200 {A B C} \u2192 (A `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (C `\u00d7 B)\n  second    : \u2200 {A B C} \u2192 (B `\u2192 C) \u2192 (A `\u00d7 B) `\u2192 (A `\u00d7 C)\n  <_,_>     : \u2200 {A B C} \u2192 (A `\u2192 B) \u2192 (A `\u2192 C) \u2192 A `\u2192 B `\u00d7 C\n  bijFork   : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 `Bit `\u00d7 B\n  fork      : \u2200 {A B} (f g : A `\u2192 B) \u2192 `Bit `\u00d7 A `\u2192 B\n  <0b> <1b> : \u2200 {_\u22a4} \u2192 _\u22a4 `\u2192 `Bit\n  xor : \u2200 {n} \u2192 (n + n) \u2325 n\n  and : \u2200 {n} \u2192 (n + n) \u2325 n\n  or  : \u2200 {n} \u2192 (n + n) \u2325 n\n  add : \u2200 {A} \u2192 (A + A) `\u2192 A\n  neg : \u2200 {n} \u2192 n \u2325 n\n\nrecord Proj A B : Set where\n  field\n    get : A `\u2192 B\n    on  : (A `\u00d7 B) `\u2192 B \u2192 A `\u2192 A\nopen Proj\n\nmodule Ops\n  (F G : \u2115 \u2192 \u2605)\n  (lift\u2081 : \u2200 {A B} \u2192 A `\u2192 B \u2192 F A \u2192 G B)\n  (lift\u2082 : \u2200 {A B C} \u2192 (A `\u00d7 B) `\u2192 C \u2192 F A \u2192 F B \u2192 G C) where\n\n    infixl 6 _`+_ _\u2295_\n    _`+_ : \u2200 {A} \u2192 F A \u2192 F A \u2192 G A\n    _`+_ = lift\u2082 add\n\n    _\u2295_ : \u2200 {A} \u2192 F A \u2192 F A \u2192 G A\n    _\u2295_ = lift\u2082 xor\n\n    _\u2227_ : \u2200 {A} \u2192 F A \u2192 F A \u2192 G A\n    _\u2227_ = lift\u2082 and\n\n    _\u2228_ : \u2200 {A} \u2192 F A \u2192 F A \u2192 G A\n    _\u2228_ = lift\u2082 or\n\n    \u00ac_ : \u2200 {A} \u2192 F A \u2192 G A\n    \u00ac_ = lift\u2081 neg\n\nrol : \u2200 {n} \u2192 Fin n \u2192 n `\u2192 n\nrol x = rewire (_\u229e_ x)\n\n_+=_ : \u2200 {A B} \u2192 Proj A B \u2192 A `\u2192 B \u2192 A `\u2192 A\np += c = on p (first c \u204f add)\n\nmodule SOps\n  (F : \u2115 \u2192 \u2605)\n  (lift\u2081 : \u2200 {A B} \u2192 A `\u2192 B \u2192 F A \u2192 F B)\n  (lift\u2082 : \u2200 {A B C} \u2192 (A `\u00d7 B) `\u2192 C \u2192 F A \u2192 F B \u2192 F C)\n  where\n\n    open Ops F F lift\u2081 lift\u2082 public\n\n    Op3 : \u2115 \u2192 \u2605\n    Op3 A = F A \u2192 F A \u2192 F A \u2192 F A\n\n    R : \u2200 {A B : \u2115} {I : \u2605} \u2192 (I \u2192 A `\u2192 B) \u2192 Op3 B \u2192 (a b c d : Proj A B) (i : I) (k : Bits B) (s : Fin B) \u2192 _\n    R {A} {B} w f a b c d i k s =\n      a += ({!f (get b) (get c) (get d) {- `+ w i `+ bits k -}!}) \u204f on a (snd {A} \u204f rol s) \u204f a += get b\n\n    module f1,2,3,4 (A : \u2115) where\n\n        f1 : Op3 A\n        f2 : Op3 A\n        f3 : Op3 A\n        f4 : Op3 A\n\n        f1 x y z = z \u2295 (x \u2227 (y \u2295 z))\n        f2 x y z = f1 z x y -- y \u2295 (z \u2227 (x \u2295 y))\n        f3 x y z = x \u2295 y \u2295 z\n        f4 x y z = y \u2295 (x \u2228 \u00ac z)\n\n    {-\nlift\u2081 : \u2200 {A B C} \u2192 B `\u2192 C \u2192 Proj A B \u2192 A `\u2192 C\nlift\u2081 op p = get p \u204f op\n\nlift\u2082 : \u2200 {A B C D} \u2192 (B `\u00d7 C) `\u2192 D \u2192 Proj A B \u2192 Proj A C \u2192 A `\u2192 D\nlift\u2082 op p\u2080 p\u2081 = dup \u204f < get p\u2080 \u00d7 get p\u2081 > \u204f op\n\nmodule Ops' {A} = Ops (Proj A) (_`\u2192_ A) lift\u2081 lift\u2082\nopen Ops'\n-}\n\nlift\u2081 : \u2200 {A B C} \u2192 B `\u2192 C \u2192 A `\u2192 B \u2192 A `\u2192 C\nlift\u2081 op p = p \u204f op\n\nlift\u2082 : \u2200 {A B C D} \u2192 (B `\u00d7 C) `\u2192 D \u2192 A `\u2192 B \u2192 A `\u2192 C \u2192 A `\u2192 D\nlift\u2082 op p\u2080 p\u2081 = dup \u204f < p\u2080 \u00d7 p\u2081 > \u204f op\n\nmodule SOps' {A} = SOps (_`\u2192_ A) lift\u2081 lift\u2082\n-- open SOps'\n\n{-\nx : \u2200 {A B} \u2192 (A `\u00d7 B) `\u2192 A\nx = fst\ny : \u2200 {A B C} \u2192 (A `\u00d7 B `\u00d7 C) `\u2192 B\ny = snd \u204f fst\nz : \u2200 {A B C D} \u2192 (A `\u00d7 B `\u00d7 C `\u00d7 D) `\u2192 C\nz = snd \u204f snd \u204f fst\n-}\n\n{-\npartofmd5 =\n  R f1 a b c d 0 0xd76aa478 7  \u204f\n  R f1 d a b c 1 0xe8c7b756 12 \u204f\n  R f1 c d a b 2 0x242070db 17 \u204f\n  R f1 b c d a 3 0xc1bdceee 22\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n  -- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'md5.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a4abeacd622ab3122b0f0fa9dbfb4ac73bf0a371","subject":"Added draft: Conversion functions i, j, and p.","message":"Added draft: Conversion functions i, j, and p.\n\nIgnore-this: bd94fe9db721fc35f983fdb1f4eab4c9\n\ndarcs-hash:20110205045233-3bd4e-ce1165f0c88f3408a8ac80324d2f766c2f79f04c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/IJP.agda","new_file":"Draft\/IJP.agda","new_contents":"------------------------------------------------------------------------------\n-- Conversion functions between inductive natural numbers and LTC\n-- natural numbers\n------------------------------------------------------------------------------\n\nmodule IJP where\n\nopen import LTC.Base\n\nopen import LTC.Data.Nat\n\n------------------------------------------------------------------------------\n\ndata IN : Set where\n  zeroIN : IN\n  succIN : IN \u2192 IN\n\ni : IN \u2192 D\ni zeroIN     = zero\ni (succIN n) = succ (i n)\n\nj : (n : IN) \u2192 N (i n)\nj zeroIN     = zN\nj (succIN n) = sN (j n)\n\np : {n : D} \u2192 N n \u2192 IN\np zN      = zeroIN\np (sN Nn) = succIN (p Nn)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/IJP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c0c4dc65187a85dc78a9b0d8c4a8554d90599cf4","subject":"lib\/Function\/Extensionality\/Implicit.agda","message":"lib\/Function\/Extensionality\/Implicit.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Extensionality\/Implicit.agda","new_file":"lib\/Function\/Extensionality\/Implicit.agda","new_contents":"{-# OPTIONS --without-K #-}\n\nopen import Function.NP\nimport Function.Extensionality as FE\nopen FE using (FunExt)\nopen import Relation.Binary.PropositionalEquality.NP\n\n-- All of this should be derived from FE together with the iso\n-- between explicit \u03a0 and implicit \u03a0\nmodule Function.Extensionality.Implicit {a b}{A : Set a}{B : A \u2192 Set b} where\n\nprivate\n  T    = {x : A} \u2192 B x\n  _\u2261'_ = _\u2261_ {A = T}\n\ninfix 4 _\u223c_\n\n_\u223c_   : (f\u2080 f\u2081 : {x : A} \u2192 B x) \u2192 Set _\nf\u2080 \u223c f\u2081 = (x : A) \u2192 f\u2080 {x} \u2261 f\u2081 {x}\n\nmodule _ {f\u2080 f\u2081 : T} where\n  happly : f\u2080 \u2261' f\u2081 \u2192 f\u2080 \u223c f\u2081\n  happly p x = ap (\u03bb f\u2080 \u2192 f\u2080 {x}) p\n\n  module _ {{fe : FunExt}} where\n    postulate\n      \u03bb= : (f= : f\u2080 \u223c f\u2081) \u2192 f\u2080 \u2261' f\u2081\n\n      happly-\u03bb= : (f= : f\u2080 \u223c f\u2081) \u2192 happly (\u03bb= f=) \u2261 f=\n\n      \u03bb=-happly : (\u03b1 : f\u2080 \u2261' f\u2081) \u2192 \u03bb= (happly \u03b1) \u2261 \u03b1\n\n      -- This should be derivable if I had a proper proof of \u03bb=\n      tr-\u03bb= : \u2200 {p x}(P : B x \u2192 Set p)(f= : f\u2080 \u223c f\u2081)\n              \u2192 tr (\u03bb f\u2080 \u2192 P (f\u2080 {x})) (\u03bb= f=) \u2261 tr P (f= x)\n\n!-\u03b1-\u03bb= : \u2200 {f\u2080 f\u2081 : T}{{fe : FunExt}}\n           (\u03b1 : f\u2080 \u2261' f\u2081) \u2192 ! \u03b1 \u2261 \u03bb= (!_ \u2218 happly \u03b1)\n!-\u03b1-\u03bb= refl = ! \u03bb=-happly refl\n\nmodule _ {f\u2080 f\u2081 : T}{{fe : FunExt}} where\n  !-\u03bb= : (f= : f\u2080 \u223c f\u2081) \u2192 ! (\u03bb= f=) \u2261 \u03bb= (!_ \u2218 f=)\n  !-\u03bb= f= = !-\u03b1-\u03bb= (\u03bb= f=) \u2219 ap \u03bb= (FE.\u03bb= (\u03bb x \u2192 ap !_ (FE.happly (happly-\u03bb= f=) x)))\n\n  \u03bb=\u2071 : (f= : \u2200 {x} \u2192 f\u2080 {x} \u2261 f\u2081 {x}) \u2192 f\u2080 \u2261' f\u2081\n  \u03bb=\u2071 f= = \u03bb= \u03bb x \u2192 f= {x}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Function\/Extensionality\/Implicit.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"57fc66a45b304872a157b174f33e534a218635f4","subject":"Preliminary work on datalog termination","message":"Preliminary work on datalog termination\n","repos":"chubbymaggie\/holmes,maurer\/holmes,maurer\/holmes,tempbottle\/holmes,BinaryAnalysisPlatform\/holmes","old_file":"formal\/Datalog.agda","new_file":"formal\/Datalog.agda","new_contents":"module Datalog where\nopen import Data.Nat as \u2115\nopen import Data.List\nopen import Data.Integer as \u2124\nopen import Relation.Binary.PropositionalEquality\nopen import Data.List.Any\n\nmodule Term where\n  data tt : Set where\n    \u03c4-int : tt\n\n  open Membership (setoid tt)\n\n  ctx = List tt\n\n  data Lit : tt \u2192 Set where\n    int : \u2124 \u2192 Lit \u03c4-int\n\n  data Expr {\u03c4 : tt} {{ \u0393 : ctx }} {{ \u0394 : ctx }} : Set where\n    lit   : (Lit \u03c4) \u2192 Expr {\u03c4}\n    e-var : (\u03c4 \u2208 \u0393) \u2192 Expr {\u03c4}\n    a-var : (\u03c4 \u2208 \u0394) \u2192 Expr {\u03c4}\n\nmodule Program where\n  \n  at = List Term.tt\n\n  open Membership (setoid at)\n\n  ctx = List at\n\n  data PartialAtom {\u0393 : Term.ctx} {\u0394 : Term.ctx} {\u03a1 : ctx} (\u03c1 : at) : Set where\n    predicate : (\u03c1 \u2208 \u03a1) \u2192 PartialAtom \u03c1\n    apply : \u2200 {\u03c4 : Term.tt} \u2192 PartialAtom {\u0393} {\u0394} {\u03a1} (\u03c4 \u2237 \u03c1) \u2192 Term.Expr {\u03c4} \u2983 \u0393 \u2984 \u2983 \u0394 \u2984 \u2192 PartialAtom {\u0393} {\u0394} {\u03a1} \u03c1\n\n  data Atom {\u0393 : Term.ctx} {\u0394 : Term.ctx} \u2983 \u03a1 : ctx \u2984 : Set where\n    close : PartialAtom {\u0393} {\u0394} {\u03a1} [] \u2192 Atom\n\n  data Rule {\u03a1 : ctx} : Set where\n    -- Here, we should also be checking that the rhs uses \u0393 to make it a relevant ctx\n    -- when used in the universal position\n    -- I don't think I need that property for proofs yet, so it can wait\n    rule : {\u0393 : Term.ctx} \u2192 Atom {\u0393} {[]} \u2192 List (Atom {[]} {\u0393}) \u2192 Rule\n\n  Program : {\u03a1 : ctx} \u2192 Set\n  Program {\u03a1} = List (Rule {\u03a1})\n  \n  data Derivable {\u03a1} : Program {\u03a1} \u2192 Atom {[]} {[]} \u2192 Set where\n\n{-\n  Proof sketch:\n  0.) match as an assumed-correct-primitive\n  1.) Proof of upper bound : prog, \u2200 r \u2208 prog. \u2200 db. \u2200 s \u2208 match db r.body. subst s r.head \u2208 upper\n  2.) semantics : p : Program \u2192 db \u2192 \u2200 a \u2208 db. Derivable \u03a1 prog a \u2192 upper \u2192 (\u2200 r \u2208 p. \u2200 vdb. \u2200 s \u2208 match vdb r.body. let h = subst s r.head in (h \u2208 upper) || (h \u2208 db))\n      a.) Go through prog, trying to generate the proof of termination\n      b.) If on any rule you fail, take the failing fact:\n          i.) Move the fact from upper to db\n          ii.) Add derivability proof for the fact you just derived (look up the matched facts derivation and build up)\n          iii.) Fix up the completeness term by demonstrating that if all that happened moved from upper to db, the property is just as true\n  3.) Proof of correctness is:\n      db, \u2200 a \u2208 db. Derivable {\u03a1} prog a, \u2200 r \u2208 prog. \u2200 s \u2208 match db r.body. subst s r.head \u2208 db\n      e.g. database, proof that database is sound, proof that database is complete\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formal\/Datalog.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"6d4991bbf727723f28c5ee052f549dda6a4303d7","subject":"lib\/Lens\/Getter.agda","message":"lib\/Lens\/Getter.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Lens\/Getter.agda","new_file":"lib\/Lens\/Getter.agda","new_contents":"-- Inspired from the haskell 'lens' package\nmodule Lens.Getter where\n\nopen import Type\nopen import Lens.Type\nopen import Function\n\ndata Accessor (R : \u2605) (A : \u2605) : \u2605 where\n  mkAccessor : R \u2192 Accessor R A\n\nprivate\n  coerceAccessor : \u2200 {R A B} \u2192 Accessor R A \u2192 Accessor R B\n  coerceAccessor (mkAccessor x) = mkAccessor x\n  runAccessor : \u2200 {R A} \u2192 Accessor R A \u2192 R\n  runAccessor (mkAccessor x) = x\n\nGetting : (R S T A B : \u2605) \u2192 \u2605\nGetting R = LensLike (Accessor R)\n  -- Getting R S T A B \u2261 (A \u2192 Accessor R B) \u2192 S \u2192 Accessor R T\n\n-- unlike Control.Lens.Getter we have no Gettable type-class\nGetter : (S A : \u2605) \u2192 \u2605\u2081\nGetter S A = \u2200 {R} \u2192 Getting R S S A A\n\ninfixl 8 _^\u00b7_ _^&_\ninfixl 1 _&_\ninfixr 0 _^$_\n \n_^\u00b7_ : \u2200 {S T A B} \u2192 S \u2192 Getting A S T A B \u2192 A\ns ^\u00b7 \u2113 = runAccessor (\u2113 mkAccessor s)\n \n_^$_ : \u2200 {S T A B} \u2192 Getting A S T A B \u2192 S \u2192 A\n_^$_ = flip _^\u00b7_\n\nto : \u2200 {S A} \u2192 (S \u2192 A) \u2192 Getter S A\nto f g = coerceAccessor \u2218 g \u2218 f\n\n_&_ : \u2200 {A B : \u2605} \u2192 A \u2192 (A \u2192 B) \u2192 B\na & f = f a\n\n_^&_ : \u2200 {A B : \u2605} \u2192 A \u2192 (A \u2192 B) \u2192 B\na ^& f = f a\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Lens\/Getter.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"df9ba28b1c0a9a58c46415bcc5c8d15fb5f021cf","subject":"qualified import and clean up","message":"qualified import and clean up\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4c4d72cc858167ee81993a990e6d74fa521c636a","subject":"A simpler type for terms, mainly for debugging, includes a printer","message":"A simpler type for terms, mainly for debugging, includes a printer\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Simple.agda","new_file":"lib\/Reflection\/Simple.agda","new_contents":"open import Reflection.NP hiding (app)\nopen import Function\nopen import Data.Nat\nopen import Data.Nat.Show renaming (show to showNat)\nopen import Data.List hiding (_++_)\nopen import Data.String\nopen import Data.Float\nopen import Data.Char\n\nmodule Reflection.Simple where\n\ndata Var A : Set where\n  var : (x : A) \u2192 Var A\n  con : (c : Name) \u2192 Var A\n  def : (f : Name) \u2192 Var A\n\ndata Tm A : Set where\n  app     : (v : Var A) (args : List (Tm A)) \u2192 Tm A\n  lam     : (s : A) (t : Tm A) \u2192 Tm A\n  pi      : (s : A) (t\u2081 : Tm A) (t\u2082 : Tm A) \u2192 Tm A\n  sort    : \u2115 \u2192 Tm A\n  lit     : (l : Literal) \u2192 Tm A\n  unknown : Tm A\n\nTms : Set \u2192 Set\nTms A = List (Tm A)\n\nmodule Level where\n    Env : Set\n    Env = \u2115\n    v\u00e5r : Env \u2192 Var \u2115 \u2192 Var \u2115\n    v\u00e5r depth (var ix) = var (depth \u2238 suc ix)\n    v\u00e5r depth (con c)  = con c\n    v\u00e5r depth (def d)  = def d\n    term  : Env \u2192 Tm \u2115 \u2192 Tm \u2115\n    terms : Env \u2192 Tms \u2115 \u2192 Tms \u2115\n    term \u0393 (app x args) = app (v\u00e5r \u0393 x) (terms \u0393 args)\n    term \u0393 (lam _ t) = lam \u0393 (term (suc \u0393) t)\n    term \u0393 (pi _ t t\u2081) = pi \u0393 (term \u0393 t) (term (suc \u0393) t\u2081)\n    term \u0393 (sort l) = sort l\n    term \u0393 (lit l) = lit l\n    term \u0393 unknown = unknown\n    terms \u0393 [] = []\n    terms \u0393 (x \u2237 ts) = term \u0393 x \u2237 terms \u0393 ts\n\nmodule Simplify where\n    term : Term \u2192 Tm \u2115\n    t\u00ffpe : Type \u2192 Tm \u2115\n    \u00e5rgs : Args \u2192 List (Tm \u2115)\n    term (var x args) = app (var x) (\u00e5rgs args)\n    term (con c args) = app (con c) (\u00e5rgs args)\n    term (def f args) = app (def f) (\u00e5rgs args)\n    term (lam v t) = lam 42 (term t)\n    term (pat-lam cs args) = unknown\n    term (pi (arg _ t\u2081) t\u2082) = pi 42 (t\u00ffpe t\u2081) (t\u00ffpe t\u2082)\n    term (sort (lit l)) = sort l\n    term (sort _) = unknown\n    term (lit l) = lit l\n    term unknown = unknown\n    t\u00ffpe (el s t) = term t\n    \u00e5rgs [] = []\n    \u00e5rgs (arg i x \u2237 as) = term x \u2237 \u00e5rgs as\n\nsimple : Term \u2192 Tm \u2115\nsimple = Level.term 0 \u2218 Simplify.term\n\nopen import Text.Printer\nmodule Printer where\n    open PrEnv\n\n    top\u1d38 = mk 2\n    app\u1d38 = mk 1\n    atm\u1d38 = mk 0\n\n    prTm : PrEnv \u2192 Pr (Tm \u2115)\n    prV   : Pr \u2115\n    prVar : Pr (Var \u2115)\n    prTms : Pr (Tms \u2115)\n\n    prV x = ` \"x\" \u2218 `(showNat x)\n\n    prVar (var x) = prV x\n    prVar (con c) = `(showName c)\n    prVar (def f) = `(showName f)\n\n    prTm \u2113 (lam b t)  = paren top\u1d38 \u2113 (` \"\u03bb\" \u2218 prV b \u2218 ` \". \" \u2218 prTm top\u1d38 t)\n    prTm \u2113 (app v []) = prVar v\n    prTm \u2113 (app v as) = paren app\u1d38 \u2113 (prVar v \u2218 prTms as)\n    prTm \u2113 (pi s t u) = paren top\u1d38 \u2113 (` \"(\" \u2218 prV s \u2218 ` \" : \" \u2218 prTm top\u1d38 t \u2218 ` \") \u2192 \" \u2218 prTm top\u1d38 u)\n    prTm \u2113 (sort l)   = ` \"Set\" \u2218 ` showNat l\n    prTm \u2113 (lit l)    = `(showLiteral l)\n    prTm \u2113 unknown    = ` \"unknown\"\n\n    prTms [] = id\n    prTms (a \u2237 as) = ` \" \" \u2218 prTm atm\u1d38 a \u2218 prTms as\n\nshowTerm : Term \u2192 String\nshowTerm t = prTm top\u1d38 (simple t) \"\" where open Printer\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Reflection\/Simple.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"629f3a2f0578a2f8dd5745d5cffe560212a86e5a","subject":"Dependently typed versions of curry\u207f and $\u207f and one property","message":"Dependently typed versions of curry\u207f and $\u207f and one property\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/N-ary\/NP.agda","new_file":"lib\/Data\/Vec\/N-ary\/NP.agda","new_contents":"module Data.Vec.N-ary.NP where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Vec.N-ary public renaming (curry\u207f to curry\u207f\u2032; _$\u207f_ to _$\u207f\u2032_)\nopen import Relation.Binary.PropositionalEquality\n\ncurry\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192\n         ((xs : Vec A n) \u2192 B $\u207f\u2032 xs) \u2192 \u2200\u207f n B\ncurry\u207f {zero}  f = f []\ncurry\u207f {suc n} f = \u03bb x \u2192 curry\u207f (f \u2218 _\u2237_ x)\n\n_$\u207f_ : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} \u2192 \u2200\u207f n B \u2192 ((xs : Vec A n) \u2192 B $\u207f\u2032 xs)\nf $\u207f []       = f\nf $\u207f (x \u2237 xs) = f x $\u207f xs\n\ncurry-$\u207f : \u2200 {n a b} {A : Set a} {B : N-ary n A (Set b)} (f : (xs : Vec A n) \u2192 B $\u207f\u2032 xs) (xs : Vec A n)\n           \u2192 curry\u207f f $\u207f xs \u2261 f xs\ncurry-$\u207f f []       = refl\ncurry-$\u207f f (x \u2237 xs) = curry-$\u207f (f \u2218 _\u2237_ x) xs\n\ncurry-$\u207f\u2032 : \u2200 {n a b} {A : Set a} {B : Set b} (f : Vec A n \u2192 B) (xs : Vec A n)\n           \u2192 curry\u207f\u2032 f $\u207f\u2032 xs \u2261 f xs\ncurry-$\u207f\u2032 f []       = refl\ncurry-$\u207f\u2032 f (x \u2237 xs) = curry-$\u207f\u2032 (f \u2218 _\u2237_ x) xs\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Vec\/N-ary\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"736bf543885bea300642ba50075a1cacbe8ef968","subject":"Working in the length of streams.","message":"Working in the length of streams.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Stream\/Length.agda","new_file":"notes\/FOT\/FOTC\/Data\/Stream\/Length.agda","new_contents":"module FOT.FOTC.Data.Stream.Length where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen FOTC.Base.BList\nopen import FOTC.Base.PropertiesI\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.List\nopen import FOTC.Data.List.PropertiesI\nopen import FOTC.Data.Stream\n\nmodule Version\u2081 where\n  {-# NO_TERMINATION_CHECK #-}\n  Stream-length : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2261 \u221e\n  Stream-length {xs} Sxs with (Stream-unf Sxs)\n  ... | x' , xs' , Sxs' , h =\n    length xs\n      \u2261\u27e8 cong length h \u27e9\n    length (x' \u2237 xs')\n      \u2261\u27e8 lg-xs\u2261\u221e\u2192lg-x\u2237xs\u2261\u221e x' xs' (Stream-length Sxs') \u27e9\n    \u221e \u220e\n\n-- module Version\u2082 where\n--   Stream-length : \u2200 {xs} \u2192 Stream xs \u2192 length xs \u2248N \u221e\n--   Stream-length xs = \u2248N-coind {!!} {!!} {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Stream\/Length.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1cae5f18582f8cf58b28083fd77dfb7071bf9928","subject":"ZK.GroupHom.JSChal: basic setup, many postulates left","message":"ZK.GroupHom.JSChal: basic setup, many postulates left\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom\/JSChal.agda","new_file":"ZK\/GroupHom\/JSChal.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type)\nopen import Type.Eq\nopen import Relation.Binary\n\nopen import ZK.GroupHom.NumChal\nopen import FFI.JS.BigI\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nopen import Data.Bool.Base renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality\n\nmodule ZK.GroupHom.JSChal\n  (G+ G* : Type)\n  (\ud835\udd3e+ : Group G+)\n  (\ud835\udd3e* : Group G*)\n  {{eq?-G* : Eq? G*}}\n  (\u03c6 : G+ \u2192 G*)\n  (\u03c6-hom : GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* \u03c6)\n  (Y : G*)\n\n  (q : BigI)\n  where\n\ninfixl 6 _+\u207f_ _\u2238\u207f_\ninfixl 7 _*\u207f_\n\nmodule \ud835\udd3e+ = Group \ud835\udd3e+\nopen module \ud835\udd3e* = Multiplicative-Group \ud835\udd3e*\n\npostulate\n  BigIPrime : BigI \u2192 Type\n\n0\u207f   = 0I\n1\u207f   = 1I\n_+\u207f_ = add\n_\u2238\u207f_ = subtract\n_*\u207f_ = multiply\n\n_<BigI_ : (x y : BigI) \u2192 Type\nx <BigI y = \u2713 (x <I y)\n\n_div-q _mod-q inv-mod-q : BigI \u2192 BigI\nx div-q     = divide x q\nx mod-q     = mod x q\ninv-mod-q x = modInv x q\n\npostulate\n  _\u2297\u207f_ : G+ \u2192 BigI \u2192 G+\n--_\u2297\u207f_ = {!!}\n  _^\u207f_ : G* \u2192 BigI \u2192 G*\n--_^\u207f_ = {!!}\n\n-- TODO\npostulate\n  -- should be turned into a dynamic test!\n  q-prime : BigIPrime q\n\n  -- should be turned into a dynamic test!\n  G*-order : Y ^\u207f q \u2261 \ud835\udd3e*.1#\n\n-- We can hope to reduce these two more basic facts\npostulate\n  1^\u207f : \u2200 x \u2192 \ud835\udd3e*.1# ^\u207f x \u2261 \ud835\udd3e*.1#\n  \u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297\u207f c) \u2261 \u03c6 x ^\u207f c\n\npostulate\n  compare : Trichotomous _\u2261_ _<BigI_\n  <-\u2238\u22620 : \u2200 x y \u2192 y <BigI x \u2192 x \u2238\u207f y \u2262 0\u207f\n  div-mod-q-prop\u207f : \u2200 d \u2192 d \u2261 d mod-q +\u207f (d div-q) *\u207f q\n  inv-mod-q-prop : \u2200 x \u2192 x \u2262 0\u207f \u2192 (inv-mod-q x *\u207f x)mod-q \u2261 1\u207f\n\n  ^\u207f1\u207f+\u207f : \u2200 {x} \u2192 Y ^\u207f(1\u207f +\u207f x) \u2261 Y * Y ^\u207f x\n  ^\u207f-*  : \u2200 {x y} \u2192 Y ^\u207f(y *\u207f x) \u2261 (Y ^\u207f x)^\u207f y\n  ^\u207f-\u2238\u207f : \u2200 {c\u2080 c\u2081}(c> : \u2713 (c\u2081 <I c\u2080)) \u2192 Y ^\u207f(c\u2080 \u2238\u207f c\u2081) \u2261 (Y ^\u207f c\u2080) \/ (Y ^\u207f c\u2081)\n\nJSPackage : Package\nJSPackage = record\n              { Num = BigI\n              ; Prime = BigIPrime\n              ; _<_ = _<BigI_\n              ; 0\u207f = 0\u207f\n              ; 1\u207f = 1\u207f\n              ; _+\u207f_ = _+\u207f_\n              ; _\u2238\u207f_ = _\u2238\u207f_\n              ; _*\u207f_ = _*\u207f_\n              ; compare = compare\n              ; <-\u2238\u22620 = <-\u2238\u22620\n              ; G+ = G+\n              ; G* = G*\n              ; \ud835\udd3e+ = \ud835\udd3e+\n              ; \ud835\udd3e* = \ud835\udd3e*\n              ; _\u2297\u207f_ = _\u2297\u207f_\n              ; _^\u207f_ = _^\u207f_\n              ; 1^\u207f = 1^\u207f\n              ; \u03c6 = \u03c6\n              ; \u03c6-hom = \u03c6-hom\n              ; \u03c6-hom-iterated = \u03c6-hom-iterated\n              ; q = q\n              ; q-prime = q-prime\n              ; _div-q = _div-q\n              ; _mod-q = _mod-q\n              ; div-mod-q-prop\u207f = div-mod-q-prop\u207f\n              ; inv-mod-q = inv-mod-q\n              ; inv-mod-q-prop = inv-mod-q-prop\n              ; Y = Y\n              ; G*-order = G*-order\n              ; ^\u207f1\u207f+\u207f = ^\u207f1\u207f+\u207f\n              ; ^\u207f-* = ^\u207f-*\n              ; ^\u207f-\u2238\u207f = ^\u207f-\u2238\u207f\n              }\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/GroupHom\/JSChal.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e0c6a8e5b17986de3b0562105b17b44fb570453b","subject":"Add FiniteField\/FinImplem.agda (unfinished)","message":"Add FiniteField\/FinImplem.agda (unfinished)\n","repos":"crypto-agda\/crypto-agda","old_file":"FiniteField\/FinImplem.agda","new_file":"FiniteField\/FinImplem.agda","new_contents":"-- Implements \u2124q with Data.Fin\nopen import Type\nopen import Data.Nat\nopen import Data.Fin.NP as Fin\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\nopen import Explore.Explorable\n\nopen Fin.Modulo renaming (sucmod to [suc]; sucmod-inj to [suc]-inj)\n\nmodule FiniteField.FinImplem (q-1 : \u2115) ([0]' [1]' : Fin (suc q-1)) where\n  -- open Sum\n  q : \u2115\n  q = suc q-1\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  \u03bc\u2124q : Explorable \u2124q\n  \u03bc\u2124q = {!\u03bcFinSuc q-1!}\n\n  sum\u2124q : Sum \u2124q\n  sum\u2124q = sum \u03bc\u2124q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  [suc]-stable : SumStableUnder (sum \u03bc\u2124q) [suc]\n  [suc]-stable = {!\u03bcFinSUI [suc] [suc]-inj!}\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n\n  \u2115\u229e-inj : \u2200 n {x y} \u2192 n \u2115\u229e x \u2261 n \u2115\u229e y \u2192 x \u2261 y\n  \u2115\u229e-inj zero    eq = eq\n  \u2115\u229e-inj (suc n) eq = [suc]-inj (\u2115\u229e-inj n eq)\n\n  \u2115\u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u2115\u229e_ m)\n  \u2115\u229e-stable m = {!\u03bcFinSUI (_\u2115\u229e_ m) (\u2115\u229e-inj m)!}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin\u25b9\u2115 m \u2115\u229e n\n\n  \u229e-inj : \u2200 m {x y} \u2192 m \u229e x \u2261 m \u229e y \u2192 x \u2261 y\n  \u229e-inj m = \u2115\u229e-inj (Fin\u25b9\u2115 m)\n\n  \u229e-stable : \u2200 m \u2192 SumStableUnder (sum \u03bc\u2124q) (_\u229e_ m)\n  \u229e-stable m = {!\u03bcFinSUI (_\u229e_ m) (\u229e-inj m)!}\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin\u25b9\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin\u25b9\u2115 n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FiniteField\/FinImplem.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4e2b70ef3927ce5eb0105e927845f24225ef4a73","subject":"FunUniverse\/Loop.agda","message":"FunUniverse\/Loop.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Loop.agda","new_file":"FunUniverse\/Loop.agda","new_contents":"open import Type\nopen import FunUniverse.Core\n\nmodule FunUniverse.Loop where\n\nmodule LoopType {t} {T : \u2605_ t} (funU : FunUniverse T) where\n  open FunUniverse funU\n  Exit? = `\ud835\udfda\n\n  record Loop A I B O : \u2605_ t where\n    constructor loop[_\u204f_\u204f_]\n    field\n      {S} : T\n\n    Init = A `\u2192 S\n    Body = I `\u00d7 S `\u2192 Exit? `\u00d7 O `\u00d7 S\n    Exit = S `\u2192 B\n\n    field\n      init : Init\n      body : Body\n      exit : Exit\n\nmodule Foldl {t} {T : \u2605_ t} {funU : FunUniverse T}\n             (funOps : FunOps funU)\n             {I O}\n             (z : let open FunUniverse funU in\n                  `\ud835\udfd9 `\u2192 O)\n             (f : let open FunUniverse funU in\n                  I `\u00d7 O `\u2192 O) where\n    open LoopType funU\n    open FunUniverse funU\n    open FunOps funOps hiding (foldl)\n    foldl : Loop `\ud835\udfd9 I `\ud835\udfd9 O\n    foldl = loop[ z \u204f (f \u204f <tt\u204f <0\u2082> , dup >) \u204f tt ]\n\nopen import FunUniverse.Agda\nopen import Data.Product\nopen import Data.One\nopen import Data.Two\nopen import Data.List\nopen LoopType agdaFunU\nopen FunOps agdaFunOps hiding (init)\n\nmodule RunList {A I B O} (loop : Loop A I B O) where\n\n  open Loop loop\n\n  runList\u2080 : List I \u00d7 S \u2192 List O \u00d7 S\n  runList\u2081 : List I \u00d7 Exit? \u00d7 O \u00d7 S \u2192 List O \u00d7 S\n\n  runList\u2081 (is , 0\u2082 , o , s) = first (_\u2237_ o) (runList\u2080 (is , s))\n  runList\u2081 (is , 1\u2082 , o , s) = (o \u2237 [] , s)\n\n  runList\u2080 ([]     , s) = ([] , s)\n  runList\u2080 (i \u2237 is , s) = runList\u2081 (is , body (i , s))\n\n  runList : List I \u00d7 A \u2192 List O \u00d7 B\n  runList = second init \u204f runList\u2080 \u204f second exit\n\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FunUniverse\/Loop.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5fd1a21ec05df0b1dcab6cfdffe90459276b5360","subject":"Bug reproducer for Agda 2.3.2","message":"Bug reproducer for Agda 2.3.2\n\nOld-commit-hash: c47ccb8a19ecffd481c881027a572370b1c961c3\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/Bug.agda","new_file":"nats\/Bug.agda","new_contents":"-- Agda 2.3.2 bug reproducer.\n-- On type check, expect to see:\n--\n-- An internal error has occured. Please report this\n-- as a bug. Location of error:\n-- src\/full\/agda\/TypeChecking\/Records.hs:216\n\nmodule Bug where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- Question: Would it not have been better if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`?\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df v dv ?\n  --Putting `df=f\u2295df\u229df v dv ?` on RHS triggers Agda bug.\n  --TODO: Report it as Agda tells us to.\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'nats\/Bug.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"130e86f9642d0efdae47c3ecf166ab9e8777489e","subject":" [ note ] Testing timeout for arithmetical properties.","message":" [ note ] Testing timeout for arithmetical properties.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing normalisation on arithmetic properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.Inequalities.ArithmeticPropertiesSL where\n\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\n\n101\u2261100+11\u223810 : 101 \u2261 100 + 11 \u2238 10\n101\u2261100+11\u223810 = refl\n\n91\u2261100+11\u223810\u223810 : 91 \u2261 100 + 11 \u2238 10 \u2238 10\n91\u2261100+11\u223810\u223810 = refl\n\n-- Agsy failed using a timeout of 5 minutes and it succeed with a\n-- timeout of 10 minutes.\n100+11>100 : 100 + 11 > 100\n100+11>100 = s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      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z\u2264n))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/Inequalities\/ArithmeticPropertiesSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e0278308bb8d5b8c3118e867e23d2840dbdaaec8","subject":"Add substitution (and subst. lemma)","message":"Add substitution (and subst. lemma)\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/Subst.agda","new_file":"Thesis\/Subst.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"fatal: unable to access 'https:\/\/github.com\/inc-lc\/ilc-agda.git\/': The requested URL returned error: 403\n","license":"mit","lang":"Agda"}
{"commit":"97e6f40aa1024c96e893b3b85de369f8ef3915c8","subject":"Added draft on inductive predicates for representating FOTC functions.","message":"Added draft on inductive predicates for representating FOTC functions.\n\nIgnore-this: a8a69744dc25bf0d0428a6f96bb7bbb\n\ndarcs-hash:20120313144819-3bd4e-1c025eb4919a5c7f1ec28208bd7854b1e6934ed9.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Fun.agda","new_file":"Draft\/FOTC\/Data\/Fun.agda","new_contents":"------------------------------------------------------------------------------\n-- An inductive predicate for representing functions\n------------------------------------------------------------------------------\n\n-- Tested with FOT on 13 March 2012.\n\nmodule Draft.FOTC.Data.Fun where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n-- 2012-03-13. I don't see how we can distinguish between data\n-- elements and functions in FOTC. The following inductive predicate\n-- is true for any element d : D.\ndata Fun : D \u2192 Set where\n  fun : (f : D) \u2192 Fun f\n\n-- But using a lambda-abstraction we could make a distinguish:\npostulate lam : (D \u2192 D) \u2192 D  -- LTC-PCF abstraction.\n\ndata Fun\u2081 : D \u2192 Set where\n  fun\u2081 : (f : D \u2192 D) \u2192 Fun\u2081 (lam f)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Fun.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9a4aca17a376421707c40b49ca333c8f8369ff27","subject":"Added a bug.","message":"Added a bug.\n\nIgnore-this: c70dea8476b482629247a87e2ec3066e\n\ndarcs-hash:20120314150816-3bd4e-6472f54bd5cb96331c848d448b2b0ff54b2ab9b6.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Bugs\/BadErase.agda","new_file":"Bugs\/BadErase.agda","new_contents":"------------------------------------------------------------------------------\n-- The translation is badly erasing the universal quantification\n------------------------------------------------------------------------------\n\n-- Found on 14 March 2012.\n\nmodule Bugs.BadErase where\n\npostulate\n  D : Set\n  A : Set\n\npostulate bad : ((x : D) \u2192 A) \u2192 A\n{-# ATP prove bad #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Bugs\/BadErase.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"5b07e91ed000cf1fea26e02e28775031626b8539","subject":"lib\/Algebra\/Group.agda","message":"lib\/Algebra\/Group.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group.agda","new_file":"lib\/Algebra\/Group.agda","new_contents":"open import Function\nopen import Data.Product.NP\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\nopen import Algebra.FunctionProperties.Eq\nopen \u2261-Reasoning\n\nmodule Algebra.Group where\n\nrecord Group-Ops {\u2113} (G : Set \u2113) : Set \u2113 where\n  infixl 7 _\u2219_ _\/_\n\n  field\n    _\u2219_ : G \u2192 G \u2192 G\n    \u03b5   : G\n    _\u207b\u00b9 : G \u2192 G\n\n  _\/_ : G \u2192 G \u2192 G\n  x \/ y = x \u2219 y \u207b\u00b9\n\nrecord Group-Struct {\u2113} {G : Set \u2113} (grp-ops : Group-Ops G) : Set \u2113 where\n  open Group-Ops grp-ops\n\n  -- laws\n  field\n    assoc    : Associative _\u2219_\n    identity : Identity \u03b5 _\u2219_\n    inverse  : Inverse \u03b5 _\u207b\u00b9 _\u2219_\n\n  \u2219= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \u2219 y \u2261 x' \u2219 y'\n  \u2219= {x} {y' = y'} p q = ap (_\u2219_ x) q \u2666 ap (\u03bb z \u2192 z \u2219 y') p\n\n  \/= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 x \/ y \u2261 x' \/ y'\n  \/= {x} {y' = y'} p q = ap (_\/_ x) q \u2666 ap (\u03bb z \u2192 z \/ y') p\n\n  \u2219-\/ : \u2200 {x y} \u2192 x \u2261 (x \u2219 y) \/ y\n  \u2219-\/ {x} {y}\n    = x            \u2261\u27e8 ! snd identity \u27e9\n      x \u2219 \u03b5        \u2261\u27e8 ap (_\u2219_ x) (! snd inverse) \u27e9\n      x \u2219 (y \/ y)  \u2261\u27e8 ! assoc \u27e9\n      (x \u2219 y) \/ y  \u220e\n\n  \/-\u2219 : \u2200 {x y} \u2192 x \u2261 (x \/ y) \u2219 y\n  \/-\u2219 {x} {y}\n    = x               \u2261\u27e8 ! snd identity \u27e9\n      x \u2219 \u03b5           \u2261\u27e8 ap (_\u2219_ x) (! fst inverse) \u27e9\n      x \u2219 (y \u207b\u00b9 \u2219 y)  \u2261\u27e8 ! assoc \u27e9\n      (x \/ y) \u2219 y     \u220e\n\n  unique-\u03b5 : \u2200 {x y} \u2192 x \u2219 y \u2261 y \u2192 x \u2261 \u03b5\n  unique-\u03b5 {x} {y} eq\n    = x            \u2261\u27e8 \u2219-\/ \u27e9\n      (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n      y \/ y        \u2261\u27e8 snd inverse \u27e9\n      \u03b5            \u220e\n\n  unique-\u207b\u00b9 : \u2200 {x y} \u2192 x \u2219 y \u2261 \u03b5 \u2192 x \u2261 y \u207b\u00b9\n  unique-\u207b\u00b9 {x} {y} eq\n    = x            \u2261\u27e8 \u2219-\/ \u27e9\n      (x \u2219 y) \/ y  \u2261\u27e8 \/= eq idp \u27e9\n      \u03b5 \/ y        \u2261\u27e8 fst identity \u27e9\n      y \u207b\u00b9         \u220e\n\n  cancels-\u2219 : \u2200 {x y z} \u2192 x \u2219 y \u2261 x \u2219 z \u2192 y \u2261 z\n  cancels-\u2219 {x} {y} {z} e\n    = y              \u2261\u27e8 ! fst identity \u27e9\n      \u03b5 \u2219 y          \u2261\u27e8 \u2219= (! fst inverse) idp \u27e9\n      x \u207b\u00b9 \u2219 x \u2219 y   \u2261\u27e8 assoc \u27e9\n      x \u207b\u00b9 \u2219 (x \u2219 y) \u2261\u27e8 \u2219= idp e \u27e9\n      x \u207b\u00b9 \u2219 (x \u2219 z) \u2261\u27e8 ! assoc \u27e9\n      x \u207b\u00b9 \u2219 x \u2219 z   \u2261\u27e8 \u2219= (fst inverse) idp \u27e9\n      \u03b5 \u2219 z          \u2261\u27e8 fst identity \u27e9\n      z \u220e\n\n  \u207b\u00b9-hom\u2032 : \u2200 {x y} \u2192 (x \u2219 y)\u207b\u00b9 \u2261 y \u207b\u00b9 \u2219 x \u207b\u00b9\n  \u207b\u00b9-hom\u2032 {x} {y} = cancels-\u2219 {x \u2219 y}\n     ((x \u2219 y) \u2219 (x \u2219 y)\u207b\u00b9     \u2261\u27e8 snd inverse \u27e9\n      \u03b5                       \u2261\u27e8 ! snd inverse \u27e9\n      x \u2219 x \u207b\u00b9                \u2261\u27e8 ap (_\u2219_ x) (! fst identity) \u27e9\n      x \u2219 (\u03b5 \u2219 x \u207b\u00b9)          \u2261\u27e8 \u2219= idp (\u2219= (! snd inverse) idp) \u27e9\n      x \u2219 ((y \u2219 y \u207b\u00b9) \u2219 x \u207b\u00b9) \u2261\u27e8 ap (_\u2219_ x) assoc \u27e9\n      x \u2219 (y \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9)) \u2261\u27e8 ! assoc \u27e9\n      (x \u2219 y) \u2219 (y \u207b\u00b9 \u2219 x \u207b\u00b9) \u220e)\n\nrecord Group (G : Set) : Set where\n  field\n    grp-ops    : Group-Ops G\n    grp-struct : Group-Struct grp-ops\n  open Group-Ops    grp-ops    public\n  open Group-Struct grp-struct public\n\nrecord Abelian-Group-Struct {\u2113} {G : Set \u2113} (grp-ops : Group-Ops G) : Set \u2113 where\n  open Group-Ops grp-ops\n  field\n    grp-struct : Group-Struct grp-ops\n    comm : Commutative _\u2219_\n  open Group-Struct grp-struct\n\n  assoc-comm : \u2200 {x y z} \u2192 x \u2219 (y \u2219 z) \u2261 y \u2219 (x \u2219 z)\n  assoc-comm = ! assoc \u2666 \u2219= comm idp \u2666 assoc\n\n  interchange : Interchange _\u2219_ _\u2219_\n  interchange = InterchangeFromAssocComm.\u00b7-interchange _\u2219_ assoc comm\n\n  \u207b\u00b9-hom : \u2200 {x y} \u2192 (x \u2219 y)\u207b\u00b9 \u2261 x \u207b\u00b9 \u2219 y \u207b\u00b9\n  \u207b\u00b9-hom = \u207b\u00b9-hom\u2032 \u2666 comm\n\n  split-\/-\u2219 : \u2200 {x y z t} \u2192 (x \u2219 y) \/ (z \u2219 t) \u2261 (x \/ z) \u2219 (y \/ t)\n  split-\/-\u2219 {x} {y} {z} {t}\n    = (x \u2219 y)    \/ (z \u2219 t)       \u2261\u27e8by-definition\u27e9\n      (x \u2219 y)    \u2219 (z \u2219 t)\u207b\u00b9     \u2261\u27e8 \u2219= idp \u207b\u00b9-hom \u27e9\n      (x \u2219 y)    \u2219 (z \u207b\u00b9 \u2219 t \u207b\u00b9) \u2261\u27e8 assoc \u27e9\n      x \u2219 (y    \u2219 (z \u207b\u00b9 \u2219 t \u207b\u00b9)) \u2261\u27e8 \u2219= idp assoc-comm \u27e9\n      x \u2219 (z \u207b\u00b9 \u2219 (y \u2219 t \u207b\u00b9))    \u2261\u27e8 ! assoc \u27e9\n      (x \u2219 z \u207b\u00b9) \u2219 (y \u2219 t \u207b\u00b9)    \u2261\u27e8by-definition\u27e9\n      (x \/ z) \u2219 (y \/ t) \u220e\n\n  cancels-\/ : \u2200 {x y z} \u2192 (x \u2219 y) \/ (x \u2219 z) \u2261 y \/ z\n  cancels-\/ {x} {y} {z}\n    = (x \u2219 y) \/ (x \u2219 z) \u2261\u27e8 split-\/-\u2219 \u27e9\n      (x \/ x) \u2219 (y \/ z) \u2261\u27e8 ap (\u03bb u \u2192  u \u2219 (y \/ z)) (snd inverse) \u27e9\n      \u03b5 \u2219 (y \/ z)       \u2261\u27e8 fst identity \u27e9\n      y \/ z \u220e\n\nrecord Abelian-Group (G : Set) : Set where\n  field\n    grp-ops    : Group-Ops G\n    grp-comm   : Abelian-Group-Struct grp-ops\n  open Group-Ops    grp-ops    public\n  open Abelian-Group-Struct grp-comm public\n  open Group-Struct grp-struct public\n  grp : Group G\n  grp = record { grp-struct = grp-struct }\n\n-- A renaming of Group with additive notation\nmodule Additive-Group {G} (grp : Group G) = Group grp\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d4d; _\u207b\u00b9 to 0-_; _\/_ to _\u2212_\n             ; assoc to +-assoc; identity to +-identity\n             ; inverse to 0--inverse\n             ; \u2219-\/ to +-\u2212; \/-\u2219 to \u2212-+; unique-\u03b5 to unique-0\u1d4d; unique-\u207b\u00b9 to unique-0-\n             ; cancels-\u2219 to cancels-+\n             ; \u207b\u00b9-hom\u2032 to 0--hom\u2032\n             ; \u2219= to +=; \/= to \u2212=)\n\n-- A renaming of Group with multiplicative notation\nmodule Multiplicative-Group {G} (grp : Group G) = Group grp\n    using    ( _\u207b\u00b9; unique-\u207b\u00b9; _\/_; \/=; \u207b\u00b9-hom\u2032 )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d4d\n             ; assoc to *-assoc; identity to *-identity\n             ; inverse to \u207b\u00b9-inverse\n             ; \u2219-\/ to *-\/; \/-\u2219 to \/-*; unique-\u03b5 to unique-1\u1d4d\n             ; cancels-\u2219 to cancels-*\n             ; \u2219= to *= )\n\nmodule Additive-Abelian-Group {G} (grp-comm : Abelian-Group G)\n  = Abelian-Group grp-comm\n    renaming ( _\u2219_ to _+_; \u03b5 to 0\u1d4d; _\u207b\u00b9 to 0-_; _\/_ to _\u2212_\n             ; assoc to +-assoc; identity to +-identity\n             ; inverse to 0--inverse\n             ; \u2219-\/ to +-\u2212; \/-\u2219 to \u2212-+; unique-\u03b5 to unique-0\u1d4d; unique-\u207b\u00b9 to unique-0-\n             ; \u2219= to +=; \/= to \u2212=\n             ; assoc-comm to +-assoc-comm\n             ; interchange to +-interchange\n             ; \u207b\u00b9-hom to 0--hom\n             ; split-\/-\u2219 to split-\u2212-+\n             ; cancels-\/ to cancels-\u2212)\n\nmodule Multiplicative-Abelian-Group {G} (grp : Abelian-Group G) = Abelian-Group grp\n    using    ( _\u207b\u00b9; unique-\u207b\u00b9; _\/_; \/=; \u207b\u00b9-hom\u2032\n             ; \u207b\u00b9-hom\n             ; cancels-\/ )\n    renaming ( _\u2219_ to _*_; \u03b5 to 1\u1d4d\n             ; assoc to *-assoc; identity to *-identity\n             ; inverse to \u207b\u00b9-inverse\n             ; \u2219-\/ to *-\/; \/-\u2219 to \/-*; unique-\u03b5 to unique-1\u1d4d\n             ; cancels-\u2219 to cancels-*\n             ; \u2219= to *=\n             ; assoc-comm to *-assoc-comm\n             ; interchange to *-interchange\n             ; split-\/-\u2219 to split-\/-* )\n\nmodule _ {A B : Set}(grpA0+ : Group A)(grpB1* : Group B) where\n  open Additive-Group grpA0+\n  open Multiplicative-Group grpB1*\n\n  GroupHomomorphism : (A \u2192 B) \u2192 Set\n  GroupHomomorphism f = \u2200 {x y} \u2192 f (x + y) \u2261 f x * f y\n\n  -- TODO\n  -- If you are looking for a proof of:\n  --   f (\u03a3(x\u1d62\u2208A) g(x\u2081)) \u2261 \u03a0(x\u1d62\u2208A) (f(g(x\u1d62)))\n  -- Have a look to:\n  --   https:\/\/github.com\/crypto-agda\/explore\/blob\/master\/lib\/Explore\/GroupHomomorphism.agda\n\n  module GroupHomomorphismProp {f}(f-homo : GroupHomomorphism f) where\n    f-pres-unit : f 0\u1d4d \u2261 1\u1d4d\n    f-pres-unit = unique-1\u1d4d part\n      where part = f 0\u1d4d * f 0\u1d4d  \u2261\u27e8 ! f-homo \u27e9\n                   f (0\u1d4d + 0\u1d4d)  \u2261\u27e8 ap f (fst +-identity) \u27e9\n                   f 0\u1d4d         \u220e\n\n    f-pres-0\u1d4d-1\u1d4d = f-pres-unit\n\n    f-pres-inv : \u2200 {x} \u2192 f (0- x) \u2261 (f x)\u207b\u00b9\n    f-pres-inv {x} = unique-\u207b\u00b9 part\n      where part = f (0- x) * f x  \u2261\u27e8 ! f-homo \u27e9\n                   f (0- x + x)    \u2261\u27e8 ap f (fst 0--inverse) \u27e9\n                   f 0\u1d4d            \u2261\u27e8 f-pres-unit \u27e9\n                   1\u1d4d              \u220e\n\n    f-0--\u207b\u00b9 = f-pres-inv\n\n    f-\u2212-\/ : \u2200 {x y} \u2192 f (x \u2212 y) \u2261 f x \/ f y\n    f-\u2212-\/ {x} {y} = f (x \u2212 y)       \u2261\u27e8 f-homo \u27e9\n                    f x * f (0- y)  \u2261\u27e8 ap (_*_ (f x)) f-pres-inv \u27e9\n                    f x \/ f y       \u220e\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/Group.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"192d8ddf3d92d2d1a90e4bc2f650b666b1c18aa3","subject":"Experiment: Example 3 with `inc` as a primitive Speed-up is conceivable: The derivative of `inc` operates on the change alone and not on the original bag (see line 374):","message":"Experiment: Example 3 with `inc` as a primitive\nSpeed-up is conceivable: The derivative of `inc` operates on the\nchange alone and not on the original bag (see line 374):\n\n    derive (inc b) = inc (derive b)\n\nwhere `b` is a term of type `bags`.\n\nOld-commit-hash: d8eb6ddab9bc78bbd97ab1d3596eeb59630b4fa8\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/IncBag.agda","new_file":"experimental\/IncBag.agda","new_contents":"{-\nThe goal of this file is to make the 3rd example\ndescribed in \/examples.md, \"Map.mapValues\", fast,\nby having `inc` as a primitive, before tackling\nthe somewhat harder task of having some form of\n`isNil` available during the derivation process.\n\n    inc :: Bag Integer -> Bag Integer\n    inc = map (+1)\n\n    old = fromList [1, 2 .. n - 1, n]\n    res = inc old = [2, 3 .. n, n + 1]\n-}\n\nmodule IncBag where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\n\n-- This module has holes and can't be imported.\n-- We postulate necessary properties now to be able\n-- to work on derivation and remove them later.\n--\n-- Perhaps proving that bags form a group with\n-- emptyBag, ++, \\\\ is a necessity?\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  map-over-\\\\ : \u2200 {b d f} \u2192\n    mapBag f (b \\\\ d) \u2261 mapBag f b \\\\ mapBag f d\n\n\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\nimport Data.Product as Product\n\npostulate extensionality : Extensionality Level.zero Level.zero\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n  inc : \u2200 {\u0393} \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\n  -- Change to bags = a summand\n  -- b\u2081 -> b\u2082 ::= b\u2082 \\\\ b\u2081\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (inc b) = inc (weaken subctx b)\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 inc b \u27e7Term \u03c1 = mapBag suc (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (inc b) \u03c1 = cong (mapBag suc) (weaken-sound b \u03c1)\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n--\n-- Changes on bags are bags. They allow negative multiplicities\n-- to begin with.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type bags = bags\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 \u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\nfst snd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd = abs (abs (var this))\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\u27e6derive\u27e7 {bags} b = emptyBag\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n_\u27e6\u229d\u27e7_ {bags} b\u2081 b\u2082 = b\u2081 \\\\ b\u2082\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n_\u27e6\u2295\u27e7_ {bags} b db = b ++ db\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {bags} b = b\\\\b=\u2205\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {bags} b\u2081 b\u2082 = b++[d\\\\b]=d {b\u2081} {b\u2082}\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {bags} b = b++\u2205=b\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {bags} b db = \u22a4 -- all bags are vald for all other bags\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {bags} b\u2081 b\u2082 = tt\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- derive(bag) = \u2205\nderive (bag b) = bag emptyBag\n\n-- derive(inc b) = inc (derive b)\nderive (inc b) = inc (derive b)\n\n\n-- Extensional equivalence for changes\ndata Extensionally-equivalent-as-changes :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          Extensionally-equivalent-as-changes \u03c4 df dg\n\nsyntax Extensionally-equivalent-as-changes \u03c4 df dg = df \u2248 dg :\u0394 \u03c4\n\n-- Question: How to declare fixity for infix syntax?\n-- infix 4 _\u2248_:\u0394_ -- same as \u2261\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- It would not be necessary if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`.\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    df \u2248 dg :\u0394 \u03c4 \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 df \u2248 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) :\u0394 \u03c4\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-n-dn =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\n-- Case bags: rely on set-theoretic interpretation of bags.\ndf=f\u2295df\u229df {bags} b db valid-b-db = ext-\u0394 (\u03bb d _ _ \u2192\n  begin -- Reasoning done in 1 step. Here for clarity only.\n    d \u27e6\u2295\u27e7 db\n  \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_ {x = d} refl (sym ([b++d]\\\\b=d {b} {db})) \u27e9\n    d \u27e6\u2295\u27e7 (b \u27e6\u2295\u27e7 db \u27e6\u229d\u27e7 b)\n  \u220e) where open \u2261-Reasoning\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  (\u27e6 deriveVar x \u27e7 \u03c1)\n  \u2248\n  (\u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\nmodule Foo where\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n\n    (\u27e6 derive t \u27e7 \u03c1)\n  \u2248 (\u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) :\u0394 \u03c4\n\ncorrectness-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = correctness-of-derive \u03c1 {consistency} t\n\n-- Mutually recursive lemmas\n\n-- diff-apply holds with propositional equality on bags\n\ndb=b\u2295db\u229db : \u2200 {\u0393 : Context} \u2192\n  (b : Term \u0393 bags) \u2192\n  {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u27e6 derive b \u27e7 \u03c1 \u2261 \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\ndb=b\u2295db\u229db b {\u03c1 = \u03c1} {consistency = consistency} =\n  begin\n    \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 sym (\u2205++b=b {{\u27e6 derive b \u27e7 \u03c1}}) \u27e9\n    emptyBag ++ \u27e6 derive b \u27e7 \u03c1\n  \u2261\u27e8 extract-\u0394equiv\n       (correctness-of-derive \u03c1 {consistency} b)\n       emptyBag tt tt \u27e9\n    emptyBag ++ (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 \u2205++b=b {{\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)}} \u27e9\n    \u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1)\n  \u220e where open \u2261-Reasoning\n\n-- Derivatives are valid\n\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 weaken \u0393\u227c\u0394\u0393 t \u27e7 \u03c1) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-derive\u2032 \u03c1 {consistency} t\n  rewrite weaken-sound {subctx = \u0393\u227c\u0394\u0393} t \u03c1\n  = validity-of-derive \u03c1 {consistency} t\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] (v \u27e6\u2295\u27e7 dv)) consistency}\n             t)\n           ((R[f,g\u229df] (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                       update \u03c1 {consistency})))) \u27e9\n         \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 f\u2295[g\u229df]=g (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                   (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                          update \u03c1 {consistency})) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv \u27e6\u2295\u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv))) \u2022\n                  update \u03c1 {consistency})\n      \u2261\u27e8 cong \u27e6 t \u27e7 (cong\u2082 _\u2022_ (f\u2295\u0394f=f (v \u27e6\u2295\u27e7 dv)) refl) \u27e9\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n      \u2261\u27e8 sym (f\u2295[g\u229df]=g (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                        (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency}))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})\n           \u27e6\u229d\u27e7\n           \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n      \u2261\u27e8 sym (extract-\u0394equiv\n           (correctness-of-derive\n             ((dv \u2022 v \u2022 \u03c1))\n             {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n           (validity-of-derive\n             (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n             t)\n           (R[f,g\u229df] (\u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n                     (\u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 update \u03c1 {consistency})))) \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} t\u2081 t\u2082)\n  = R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7]\n  where\n    open \u2261-Reasoning\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 : \u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1) \u2261 v\u2081 v\u2082\n    \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 rewrite (sym v\u2082=old-v\u2082) = refl\n\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 : \u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1 \u2261 dv\u2081 v\u2082 dv\u2082\n    \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = refl\n\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] : valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (dv\u2081 v\u2082 dv\u2082)\n    R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082] rewrite \u27e6t\u2081t\u2082\u27e7=v\u2081v\u2082 = R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    -- What I want to write:\n    {-\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n          valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n        R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] rewrite \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 = R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n    -}\n\n    -- What I have to write:\n\n    R : {\u03c4 : Type} \u2192 {v : \u27e6 \u03c4 \u27e7} \u2192 {dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n        dv\u2081 \u2261 dv\u2082 \u2192 valid-\u0394 v dv\u2081 \u2192 valid-\u0394 v dv\u2082\n\n    R {nats} dv\u2081=dv\u2082 refl = cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl\n\n    R {bags} dv\u2081=dv\u2082 _ = tt\n\n    --R {\u03c4\u2081 \u21d2 \u03c4\u2082} dv\u2081=dv\u2082 valid-dv\u2081 rewrite dv\u2081=dv\u2082 = {!valid-dv\u2081!}\n    R {\u03c4\u2081 \u21d2 \u03c4\u2082} {v} {dv\u2081} {dv\u2082} dv\u2081=dv\u2082 valid-dv\u2081 =\n      \u03bb s ds R[s,ds] \u2192\n        R {\u03c4\u2082} {v s} {dv\u2081 s ds} {dv\u2082 s ds}\n           (cong\u2082 (\u03bb f x \u2192 f x)\n                  (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl)\n           (proj\u2081 (valid-dv\u2081 s ds R[s,ds]))\n        ,\n        (begin\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2082 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v (s \u27e6\u2295\u27e7 ds)} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) (sym dv\u2081=dv\u2082) refl) refl) \u27e9\n          v (s \u27e6\u2295\u27e7 ds) \u27e6\u2295\u27e7 dv\u2081 (s \u27e6\u2295\u27e7 ds) (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 sym (proj\u2082 (valid-dv\u2081\n               (s \u27e6\u2295\u27e7 ds)\n               (\u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n               (R[f,\u0394f] (s \u27e6\u2295\u27e7 ds)))) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds \u27e6\u2295\u27e7 \u27e6derive\u27e7 (s \u27e6\u2295\u27e7 ds))\n        \u2261\u27e8 cong (v \u27e6\u2295\u27e7 dv\u2081) (f\u2295\u0394f=f (s \u27e6\u2295\u27e7 ds)) \u27e9\n          (v \u27e6\u2295\u27e7 dv\u2081) (s \u27e6\u2295\u27e7 ds)\n        \u2261\u27e8 proj\u2082 (valid-dv\u2081 s ds R[s,ds]) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2081 s ds\n        \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n                 {x = v s} refl\n                 (cong\u2082 (\u03bb f x \u2192 f x)\n                        (cong\u2082 (\u03bb f x \u2192 f x) dv\u2081=dv\u2082 refl) refl) \u27e9\n          v s \u27e6\u2295\u27e7 dv\u2082 s ds\n        \u220e) where open \u2261-Reasoning\n\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] :\n      valid-\u0394 (\u27e6 app t\u2081 t\u2082 \u27e7 (ignore \u03c1)) (\u27e6 derive (app t\u2081 t\u2082) \u27e7 \u03c1)\n    R[\u27e6t\u2081t\u2082\u27e7,\u27e6\u0394[t\u2081t\u2082]\u27e7] = R \u27e6\u0394[t\u2081t\u2082]\u27e7=dv\u2081v\u2082dv\u2082 R[\u27e6t\u2081t\u2082\u27e7,dv\u2081v\u2082dv\u2082]\n\n-- Validity of deriving bag-typed terms is trivial.\n\nvalidity-of-derive \u03c1 (bag b) = tt\n\nvalidity-of-derive \u03c1 (inc b) = tt\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 extract-\u0394equiv ext-\u0394[t\u2081t\u2082] f R[f,\u0394t] R[f,t\u2032\u229dt] \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e)\n  where\n    open \u2261-Reasoning\n\n    v\u2081 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081 = \u27e6 t\u2081 \u27e7 (ignore \u03c1)\n    v\u2082 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082 = \u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1\n\n    dv\u2081 : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    dv\u2081 = \u27e6 derive t\u2081 \u27e7 \u03c1\n    dv\u2082 : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv\u2082 = \u27e6 derive t\u2082 \u27e7 \u03c1\n\n    v\u2081\u2032 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    v\u2081\u2032 = \u27e6 t\u2081 \u27e7 (update \u03c1 {consistency})\n    v\u2082\u2032 : \u27e6 \u03c4\u2081 \u27e7\n    v\u2082\u2032 = \u27e6 t\u2082 \u27e7 (update \u03c1 {consistency})\n\n    v\u2082=old-v\u2082 : v\u2082 \u2261 \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n    v\u2082=old-v\u2082 = weaken-sound {subctx = \u0393\u227c\u0394\u0393} t\u2082 \u03c1\n\n    valid-dv\u2081 : valid-\u0394 v\u2081 dv\u2081\n    valid-dv\u2081 = validity-of-derive \u03c1 {consistency} t\u2081\n  \n    valid-dv\u2082 : valid-\u0394 v\u2082 dv\u2082\n    valid-dv\u2082 rewrite v\u2082=old-v\u2082 =\n      validity-of-derive \u03c1 {consistency} t\u2082\n\n    v\u2081\u2295dv\u2081=v\u2081\u2032 : v\u2081 \u27e6\u2295\u27e7 dv\u2081 \u2261 v\u2081\u2032\n    v\u2081\u2295dv\u2081=v\u2081\u2032 =\n      begin\n        v\u2081 \u27e6\u2295\u27e7 dv\u2081\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2081)\n           v\u2081 valid-dv\u2081 (R[f,g\u229df] v\u2081 v\u2081\u2032) \u27e9\n        v\u2081 \u27e6\u2295\u27e7 (v\u2081\u2032 \u27e6\u229d\u27e7 v\u2081)\n      \u2261\u27e8 f\u2295[g\u229df]=g v\u2081 v\u2081\u2032 \u27e9\n        v\u2081\u2032\n      \u220e\n\n    -- TODO: remove code duplication.\n    v\u2082\u2295dv\u2082=v\u2082\u2032 : v\u2082 \u27e6\u2295\u27e7 dv\u2082 \u2261 v\u2082\u2032\n    v\u2082\u2295dv\u2082=v\u2082\u2032 rewrite v\u2082=old-v\u2082 =\n      begin\n        old-v\u2082 \u27e6\u2295\u27e7 dv\u2082\n      \u2261\u27e8 extract-\u0394equiv\n           (correctness-of-derive \u03c1 {consistency} t\u2082)\n           old-v\u2082\n           (validity-of-derive \u03c1 {consistency} t\u2082)\n           (R[f,g\u229df] old-v\u2082 v\u2082\u2032) \u27e9\n        old-v\u2082 \u27e6\u2295\u27e7 (v\u2082\u2032 \u27e6\u229d\u27e7 old-v\u2082)\n      \u2261\u27e8 f\u2295[g\u229df]=g old-v\u2082 v\u2082\u2032 \u27e9\n        v\u2082\u2032\n      \u220e\n      where old-v\u2082 = \u27e6 t\u2082 \u27e7 (ignore \u03c1)\n\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] : v\u2081\u2032 v\u2082\u2032 \u2261 (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082)\n    v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] = sym (cong\u2082 (\u03bb f x \u2192 f x) v\u2081\u2295dv\u2081=v\u2081\u2032 v\u2082\u2295dv\u2082=v\u2082\u2032)\n\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 :\n      (v\u2081 \u27e6\u2295\u27e7 dv\u2081) (v\u2082 \u27e6\u2295\u27e7 dv\u2082) \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082\n    [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082 = proj\u2082 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082 \u2261 v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082\n    v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      cong\u2082 _\u27e6\u229d\u27e7_\n        (trans v\u2081\u2032v\u2082\u2032=[v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082] [v\u2081\u2295dv\u2081][v\u2082\u2295dv\u2082]=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082)\n        refl\n\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] : valid-\u0394 (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082)\n    R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082] = proj\u2081 (valid-dv\u2081 v\u2082 dv\u2082 valid-dv\u2082)\n\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 :\n      (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081 v\u2082 \u27e6\u2295\u27e7 dv\u2081 v\u2082 dv\u2082 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      df=f\u2295df\u229df (v\u2081 v\u2082) (dv\u2081 v\u2082 dv\u2082) R[v\u2081v\u2082,dv\u2081v\u2082dv\u2082]\n\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 : (dv\u2081 v\u2082 dv\u2082) \u2248 (v\u2081\u2032 v\u2082\u2032 \u27e6\u229d\u27e7 v\u2081 v\u2082) :\u0394 \u03c4\u2082\n    dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082 rewrite v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082 =\n      dv\u2081v\u2082dv\u2082=v\u2081v\u2082\u2295dv\u2081v\u2082dv\u2082\u229dv\u2081v\u2082\n\n    ext-\u0394[t\u2081t\u2082] :\n      (\u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1))\n      \u2248\n      (\u27e6 t\u2081 \u27e7 (update \u03c1 {consistency}) (\u27e6 t\u2082 \u27e7 (update \u03c1 {consistency}))\n      \u27e6\u229d\u27e7 \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n      :\u0394 \u03c4\u2082\n    ext-\u0394[t\u2081t\u2082] rewrite sym v\u2082=old-v\u2082 = dv\u2081v\u2082dv\u2082=v\u2081\u2032v\u2082\u2032\u229dv\u2081v\u2082\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive \u03c1 (bag b) = ext-\u0394 (\u03bb d _ _ \u2192\n begin\n   d ++ emptyBag\n  \u2261\u27e8 cong\u2082 _++_ {x = d} refl (sym (b\\\\b=\u2205)) \u27e9\n   d ++ (b \\\\ b)\n \u220e) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 {consistency} (inc b) = ext-\u0394 (\u03bb d _ _ \u2192\n  begin\n    d ++ increment (\u27e6 derive b \u27e7 \u03c1)\n  \u2261\u27e8 cong (\u03bb hole \u2192 d ++ increment hole)\n          (db=b\u2295db\u229db b {\u03c1} {consistency}) \u27e9\n    d ++ increment (\u27e6 b \u27e7 (update \u03c1 {consistency}) \\\\ \u27e6 b \u27e7 (ignore \u03c1))\n  \u2261\u27e8 cong (\u03bb hole \u2192 d ++ hole)\n          (map-over-\\\\ {\u27e6 b \u27e7 (update \u03c1)} {\u27e6 b \u27e7 (ignore \u03c1)}) \u27e9\n    d ++ increment (\u27e6 b \u27e7 (update \u03c1)) \\\\ increment (\u27e6 b \u27e7 (ignore \u03c1))\n  \u220e) where\n    open \u2261-Reasoning\n    increment : Bag \u2192 Bag\n    increment = mapBag suc\n\ncorrectness-on-closed-terms : \u2200 {\u03c4\u2081 \u03c4\u2082} \u2192\n  \u2200 (f : Term \u2205 (\u03c4\u2081 \u21d2 \u03c4\u2082)) \u2192\n  \u2200 (s : Term \u2205 \u03c4\u2081) (ds : Term \u2205 (\u0394-Type \u03c4\u2081))\n    {R[v,dv] : valid-\u0394 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)} \u2192\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205 \u27e6\u2295\u27e7 \u27e6 ds \u27e7 \u2205)\n    \u2261\n    \u27e6 f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) \u27e6\u2295\u27e7 \u27e6 derive f \u27e7 \u2205 (\u27e6 s \u27e7 \u2205) (\u27e6 ds \u27e7 \u2205)\n\ncorrectness-on-closed-terms {\u03c4\u2081} {\u03c4\u2082} f s ds {R[v,dv]} =\n  begin\n    h (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n           (sym (f\u2295[g\u229df]=g h h))\n           refl \u27e9\n    (h \u27e6\u2295\u27e7 (h \u27e6\u229d\u27e7 h)) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 cong\u2082 (\u03bb f x \u2192 f x)\n      (sym (extract-\u0394equiv\n        (correctness-of-derive \u2205 f)\n        h\n        (validity-of-derive \u2205 {d\u03c1=\u2205} f)\n        (R[f,g\u229df] h h)))\n      refl \u27e9\n    (h \u27e6\u2295\u27e7 \u0394h) (v \u27e6\u2295\u27e7 dv)\n  \u2261\u27e8 proj\u2082 (validity-of-derive \u2205 {d\u03c1=\u2205} f v dv R[v,dv]) \u27e9\n    h v \u27e6\u2295\u27e7 \u0394h v dv\n  \u220e\n  where\n    open \u2261-Reasoning\n    h : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7\n    h = \u27e6 f \u27e7 \u2205\n    \u0394h : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\n    \u0394h = \u27e6 derive f \u27e7 \u2205\n    v : \u27e6 \u03c4\u2081 \u27e7\n    v = \u27e6 s \u27e7 \u2205\n    dv : \u27e6 \u0394-Type \u03c4\u2081 \u27e7\n    dv = \u27e6 ds \u27e7 \u2205\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'experimental\/IncBag.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"70233f15272b6a6ecf9d74b2254d91df4576eac8","subject":"clean up","message":"clean up\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"80516389760b8a704422d85e8bf6d5b682422d9d","subject":"Implement delta-observational equivalence (#244, #299)","message":"Implement delta-observational equivalence (#244, #299)\n\nThis proof shows that delta-observational equivalence is an equivalence\nrelation.\n\nKnown limitations of this commit:\n* it does not show that any function respects this relation;\n* it does not rephrase existing results;\n* this might better belong in Base.Change.Equivalence.\n\nOld-commit-hash: 46789e08b7a8bbd95738f9bf14b3ff8644151772\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Change\/Equivalence.agda","new_file":"Parametric\/Change\/Equivalence.agda","new_contents":"------------------------------------------------------------------------\n-- INCREMENTAL \u03bb-CALCULUS\n--\n-- Delta-observational equivalence\n------------------------------------------------------------------------\nimport Parametric.Syntax.Type as Type\nimport Parametric.Denotation.Value as Value\n\nmodule Parametric.Change.Equivalence\n    {Base : Type.Structure}\n    (\u27e6_\u27e7Base : Value.Structure Base)\n  where\n\nopen Type.Structure Base\nopen Value.Structure Base \u27e6_\u27e7Base\n\nopen import Base.Denotation.Notation public\n\nopen import Relation.Binary.PropositionalEquality\nopen import Base.Change.Algebra\nopen import Level\nopen import Data.Unit\n\n-- Extension Point: None (currently). Do we need to allow plugins to customize\n-- this concept?\nStructure : Set\nStructure = Unit\n\nmodule Structure (unused : Structure) where\n\n  module _ {\u2113} {A} {{ca : ChangeAlgebra \u2113 A}} {x : A} where\n    -- Delta-observational equivalence: these asserts that two changes\n    -- give the same result when applied to a base value.\n    _\u2259_ : \u2200 dx dy \u2192 Set\n    dx \u2259 dy = x \u229e dx \u2261 x \u229e dy\n\n    -- _\u2259_ is indeed an equivalence relation:\n    \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n    \u2259-refl = refl\n\n    \u2259-symm : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n    \u2259-symm \u2259 = sym \u2259\n\n    \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n    \u2259-trans \u2259\u2081 \u2259\u2082 = trans \u2259\u2081 \u2259\u2082\n\n    -- TODO: we want to show that all functions of interest respect\n    -- delta-observational equivalence, so that two d.o.e. changes can be\n    -- substituted for each other freely. That should be true for functions\n    -- using changes parametrically, for derivatives and function changes, and\n    -- for functions using only the interface to changes (including the fact\n    -- that function changes are functions). Stating the general result, though,\n    -- seems hard, we should rather have lemmas proving that certain classes of\n    -- functions respect this equivalence.\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Parametric\/Change\/Equivalence.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"924c73a4c7e26dac22cd0024c0142358cb5b5b41","subject":"Added Fair-Stream.","message":"Added Fair-Stream.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/Fair\/Properties.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/Fair\/Properties.agda","new_contents":"------------------------------------------------------------------------------\n-- Fair properties\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.Fair.Properties where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Fair.PropertiesI\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nFair-Stream : \u2200 {os} \u2192 Fair os \u2192 Stream os\nFair-Stream {os} Fos = Stream-coind A h refl\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 xs\n\n  h : A os \u2192 \u2203[ o' ] \u2203[ os' ] os \u2261 o' \u2237 os' \u2227 A os'\n  h _ with head-tail-Fair Fos\n  h x\u2081 | inj\u2081 prf = T , tail\u2081 os , prf , refl\n  h x\u2081 | inj\u2082 prf = F , tail\u2081 os , prf , refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Program\/ABP\/Fair\/Properties.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"70e498402482ffb501a7d1f81db553871aa93734","subject":"Model capnproto types","message":"Model capnproto types\n","repos":"maurer\/holmes,BinaryAnalysisPlatform\/holmes,maurer\/holmes,tempbottle\/holmes,chubbymaggie\/holmes","old_file":"formal\/capnproto.agda","new_file":"formal\/capnproto.agda","new_contents":"open import Data.List\nopen import Data.List.Any\nopen import Data.Nat as \u2115\ndata capn-kind : Set where\n cKStruct : capn-kind\n ckInter  : capn-kind\nopen import Relation.Binary.PropositionalEquality\nopen Membership (setoid capn-kind)\ncapn-ctx = List capn-kind\ndata capn-\u03c4 : {\u0393 : capn-ctx} \u2192 Set\ndata capn-field : {\u0393 : capn-ctx} \u2192 Set\ndata capn-method : {\u0393 : capn-ctx} \u2192 Set\n\n--While the actual capnp schema supports declaration of structs in any order \/ arbitrary recursion,\n--I am limiting this to reference to structs-declared-so-far. This makes the model simpler, without\n--removing any of the power (I think), though it would be less convenient to write code for.\n\ndata capn-\u03c4 where\n  cVoid      : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393}\n  cBool      : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393}\n  cInt8      : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cInt16     : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cInt32     : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cInt64     : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cUInt8     : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cUInt16    : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cUInt32    : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cUInt64    : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cFloat32   : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cFloat64   : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cData      : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \n  cText      : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393}\n  cList      : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \u2192 capn-\u03c4 {\u0393}\n  cStruct    : \u2200 {\u0393} \u2192 List (capn-field {cKStruct \u2237 \u0393}) \u2192 capn-\u03c4 {cKStruct \u2237 \u0393}\n  cEnum      : \u2200 {\u0393} \u2192 \u2115 \u2192 capn-\u03c4 {\u0393}\n  cVar       : \u2200 {\u0393 k} \u2192 (k \u2208 \u0393) \u2192 capn-\u03c4 {\u0393}\n  cInterface : \u2200 {\u0393} \u2192 List (capn-method {ckInter \u2237 \u0393}) \u2192 capn-\u03c4 {ckInter \u2237 \u0393}\n\n--I'm ignoring field names, because fields can be indexed by list position, which is enough for proofs\n--I'm also ignoring groups, since they are just an addressing convenience\ndata capn-field where\n  cField   : \u2200 {\u0393} \u2192 capn-\u03c4 {\u0393} \u2192 capn-field {\u0393}\n  cUnion   : \u2200 {\u0393} \u2192 List (capn-field {\u0393})\u2192 capn-field {\u0393}\n  cAny     : \u2200 {\u0393} \u2192 capn-field {\u0393}\n\ndata capn-method where\n  cMeth : \u2200 {\u0393} \u2192 List (capn-\u03c4 {\u0393}) \u2192 List (capn-\u03c4 {\u0393}) \u2192 capn-method {\u0393}\n","old_contents":"","returncode":128,"stderr":"fatal: invalid reference: FETCH_HEAD^\n","license":"mit","lang":"Agda"}
{"commit":"bdd1d432f9a137173391fa38a3c97003b87398e0","subject":"Nils' example about databases for Agsy.","message":"Nils' example about databases for Agsy.\n\nIgnore-this: de83605d4ea73cfa5979791e4351319a\n\ndarcs-hash:20110527160613-3bd4e-38150bcac1d06f53a4112e22fd3626561d5f3d56.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Agsy\/DataBase.agda","new_file":"Draft\/Agsy\/DataBase.agda","new_contents":"-- Nils' idea about databases in the Agda mailing list.\n-- http:\/\/thread.gmane.org\/gmane.comp.lang.agda\/2911\/focus=2917\n\n-- Tested with the development version of the standard library on\n-- 27 May 2011.\n\nmodule DataBase where\n\nopen import Data.Nat\nopen import Data.Product\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\npostulate\n  right-identity : \u2200 n \u2192 n + 0 \u2261 n\n  move-suc       : \u2200 m n \u2192 suc (m + n) \u2261 m + suc n\n\ndb = right-identity ,\u2032 move-suc\n\ncomm : \u2200 m n \u2192 m + n \u2261 n + m  -- Via auto {!-c db!}\ncomm zero n     = sym (proj\u2081 db n)\ncomm (suc n) n' =\n  begin\n    suc (n + n') \u2261\u27e8 cong suc (comm n n') \u27e9\n    suc (n' + n) \u2261\u27e8 proj\u2082 db n' n \u27e9 (n' + suc n)\n  \u220e\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Agsy\/DataBase.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"edd963e5e9c5f62dce44dbdf311eb583945136af","subject":"exn-arrow.agda","message":"exn-arrow.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"exn-arrow.agda","new_file":"exn-arrow.agda","new_contents":"A `\u2192? B = A `\u2192 Bit `\u00d7 B\n\n{-\n    id?\n   +---+\n   |  0|- Bit\nA \u2261|\u2261\u2261\u2261|\u2261 A\n   +---+\n-}\nid? : A `\u2192? A\nid? = {!< zero \u00d7 id >!}\n\n{-\n           f ;? g\n   +---------------------+\n   |       +--------(OR)-|- Bit\n   |   f   |     g    |  |\n   | +---+ |   +---+  |  |\n   | |  ?|-+   |  ?|--+  |\nA \u2261|\u2261|\u2261\u2261\u2261|\u2261\u2261B\u2261\u2261|\u2261\u2261\u2261|\u2261\u2261\u2261\u2261\u2261|\u2261 C\n   | +---+     +---+     |\n   +---------------------+\n-}\n_\u204f?_ : A `\u2192? B \u2192 B `\u2192? C \u2192 A `\u2192? C\n\n{-\n    fail!\n    +---+\n    | 1-|- Bit\n A \u2261|  \u2261|\u2261 B\n    +---+\n\n-}\nfail! : A `\u2192? B\n\n{-\n     lift f\n   +--------+\n   |   f  0-|- Bit\n   | +--+   |\nA \u2261|\u2261|  |\u2261\u2261\u2261|\u2261 B\n   | +--+   |\n   +--------+\n-}\nlift : A `\u2192 B \u2192 A `\u2192? B\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'exn-arrow.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1bd71d88d6f657b2dc52bad48d51ed05ba788f70","subject":"[ test-suite ] Added missing file.","message":"[ test-suite ] Added missing file.\n","repos":"asr\/apia,asr\/apia","old_file":"test\/fail\/errors\/NoATPs\/Test.agda","new_file":"test\/fail\/errors\/NoATPs\/Test.agda","new_contents":"module Test where\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'test\/fail\/errors\/NoATPs\/Test.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"19f49e03e9f7dac4474daea57550925c09f186df","subject":"Agda: Brutal Meta-introduction","message":"Agda: Brutal Meta-introduction\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/BrutalMetaIntro.agda","new_file":"agda\/BrutalMetaIntro.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"b3ca1943a3a5292f73fff95dbd40c8d5b75984f0","subject":"Adeed NestSL.","message":"Adeed NestSL.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/Nest\/NestSL.agda","new_file":"notes\/FOT\/FOTC\/Program\/Nest\/NestSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Example of a nested recursive function using the standard library\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. vol 2152 LNCS. 2001.\n\nmodule FOT.FOTC.Program.Nest.NestSL where\n\nopen import Data.Nat renaming (suc to succ)\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n-- The non-terminating function.\n{-# NO_TERMINATION_CHECK #-}\nnest : \u2115 \u2192 \u2115\nnest 0        = 0\nnest (succ n) = nest (nest n)\n\n-- The non-terminating property.\n{-# NO_TERMINATION_CHECK #-}\nnest-x\u22610 : \u2200 n \u2192 nest n \u2261 zero\nnest-x\u22610 zero     = refl\nnest-x\u22610 (succ n) = nest-x\u22610 (nest n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Program\/Nest\/NestSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"8b2a1d7e9f973ee3a38c1cdff82a8c9fcf07efc4","subject":"A(nother) model of Containers","message":"A(nother) model of Containers\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/Containers.agda","new_file":"models\/Containers.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check\n  #-}\n\nmodule Containers where\n\n  record Sigma (A : Set) (B : A -> Set) : Set where\n    field fst : A\n          snd : B fst \n\n  open Sigma\n\n  _*_ : (A B : Set) -> Set\n  A * B = Sigma A \\_ -> B\n\n  data Zero : Set where\n \n  record One : Set where\n\n  data PROP : Set where\n    Absu : PROP\n    Triv : PROP\n    _\/\\_ : PROP -> PROP -> PROP\n    All : (S : Set) -> (S -> PROP) -> PROP\n\n  Prf : PROP -> Set\n  Prf Absu = Zero\n  Prf Triv = One\n  Prf (P \/\\ Q) = Prf P * Prf Q\n  Prf (All S P) = (x : S) -> Prf (P x)\n\n  _,_ : forall {A B} -> (a : A) -> B a -> Sigma A B\n  a , b = record { fst = a ; snd = b }\n\n  split : \u2200 {A B} {C : Sigma A B -> Set} -> \n    (f : (a : A) -> (b : B a) -> C (a , b)) -> (ab : Sigma A B) -> C ab \n  split f ab = f (fst ab) (snd ab)\n\n  data _+_ (A B : Set) : Set where\n    l : A -> A + B\n    r : B -> A + B\n\n  cases : \u2200 {A B} {C : A + B -> Set} -> \n            ((a : A) -> C (l a)) -> ((b : B) -> C (r b)) -> \n            (ab : A + B) -> C ab\n  cases f g (l a) = f a\n  cases f g (r b) = g b \n\n  rearrange : \u2200 {A B X} (C : A \u2192 Set) (D : B \u2192 Set) (f : Sigma A C \u2192 X) \u2192 \n              Sigma (A + B) (cases C D) \u2192 X + Sigma B D\n  rearrange {A} {B} {X} C D f = split {A + B} (cases (\\a c \u2192 l (f (a , c))) (\\b d \u2192 r (b , d)))\n\n  data CON (I : Set) : Set where\n    ?? : I -> CON I\n    PrfC : PROP -> CON I\n    _*C_ : CON I -> CON I -> CON I\n    PiC : (S : Set) -> (S -> CON I) -> CON I\n    SiC : (S : Set) -> (S -> CON I) -> CON I\n    MuC : (O : Set) -> (O -> CON (I + O)) -> O -> CON I\n\n  mutual\n \n    data Mu {I : Set} (O : Set) (D : O -> CON (I + O)) \n            (X : I -> Set) : O -> Set where\n      inm : {o : O} -> [| D o |] (cases X (Mu O D X)) ->  Mu O D X o\n\n    [|_|]_ : {I : Set} -> (CON I) -> (I -> Set) -> Set\n    [| ?? i |] X = X i\n    [| PrfC p |] X = Prf p\n    [| C *C D |] X = [| C |] X * [| D |] X\n    [| PiC S C |] X = (s : S) -> [| C s |] X\n    [| SiC S C |] X = Sigma S \\s -> [| C s |] X\n    [| MuC O C o |] X = Mu O C X o\n\n\n  outm : {I O : Set} {D : O -> CON (I + O)} {X : I -> Set} {o : O} ->\n    Mu O D X o -> [| D o |]  (cases X (Mu O D X))\n  outm (inm x) = x \n\n\n  Everywhere : {I : Set} -> (D : CON I) -> (J : I -> Set) -> (X : Set) ->  \n               (Sigma I J -> X) ->  [| D |] J -> CON X\n  Everywhere (?? i) J X c t = ?? (c (i , t))\n  Everywhere (PrfC p) J X c t = PrfC Triv\n  Everywhere (C *C D) J X c t = \n    Everywhere C J X c (fst t) *C Everywhere D J X c (snd t)\n  Everywhere (PiC S C) J X c t = PiC S (\\s -> Everywhere (C s) J X c (t s))\n  Everywhere (SiC S C) J X c t = Everywhere (C (fst t)) J X c (snd t)\n  Everywhere (MuC O C o) J X c t = MuC (Sigma O (Mu O C J)) (\\ot' -> \n        Everywhere (C (fst ot')) \n                   (cases J (Mu O C J)) \n                   (X + (Sigma O (Mu O C J)))\n                   (split (cases(\\i j -> l (c (i , j))) \n                                (\\o'' t'' -> r (o'' , t'')))) \n                   (outm (snd ot'))) \n     (o , t)\n\n  everywhere : {I : Set} -> (D : CON I) -> (J : I -> Set) -> (X : Set) ->\n               (c : Sigma I J -> X) -> (Y : X -> Set) -> \n               (f : (ij : Sigma I J) -> Y (c ij)) -> (t : [| D |] J) -> \n               [| Everywhere D J X c t |] Y\n  everywhere (?? i) J X c Y f t = f (i , t)\n  everywhere (PrfC p) J X c Y f t = _\n  everywhere (C *C D) J X c Y f t = \n    everywhere C J X c Y f (fst t) , everywhere D J X c Y f (snd t)\n  everywhere (PiC S C) J X c Y f t = \\s -> everywhere (C s) J X c Y f (t s)\n  everywhere (SiC S C) J X c Y f t = everywhere (C (fst t)) J X c Y f (snd t)\n  everywhere (MuC O C o) J X c Y f (inm t) = \n    inm (everywhere (C o) (cases J (Mu O C J)) (X + Sigma O (Mu O C J)) (\n        (split (cases (\\i j -> l (c (i , j))) (\\o t -> r (o , t))))) \n        (cases Y (split \\o'' t'' -> [| Everywhere (MuC O C o'') J X c t'' |] Y))\n        (split (cases (\\i j -> f (i , j)) \n                      (\\o'' t'' -> everywhere (MuC O C o'') J X c Y f t''))) \n        t)\n \n  induction : {I O : Set} (D : O -> CON (I + O)) (J : I -> Set) (X : Set) \n              (c : Sigma I J -> X) (Y : X -> Set) (f : (ij : Sigma I J) -> Y (c ij)) \n              (P : Sigma O (Mu O D J) -> Set) ->\n              ({o : O} (v : [| D o |] cases J (Mu O D J)) -> \n               [| Everywhere (D o) _ (X + Sigma O (Mu O D J)) \n                             (split (cases (\\i j -> l (c (i , j))) \n                             (\\o v -> r (o , v)))) v |] cases Y P -> \n               P (o , inm v)) ->\n              (o : O) (v : Mu O D J o) -> P (o , v)\n  induction {I} {O} D J X c Y f P p o (inm v) = \n    p v (everywhere {I + O} (D o) \n                    (\u03bb x -> cases {I} {O} {\\_ -> Set} J \n                    (Mu O D J) x) ((X + Sigma O (Mu O D J))) \n                    (rearrange J (Mu O D J) c) (cases Y P) \n                    (split {I + O} (cases (\\i j -> f (i , j)) \n                    (\\o v -> induction D J X c Y f P p o v))) v) ","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Containers.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e408183cd0007cba52eef982b2abbaa2c7463d08","subject":"Added thesis' example.","message":"Added thesis' example.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/CombiningProofs\/ForallExistSchema.agda","new_file":"notes\/thesis\/CombiningProofs\/ForallExistSchema.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --schematic-propositional-functions #-}\n{-# OPTIONS --without-K #-}\n\nmodule CombiningProofs.ForallExistSchema where\n\nopen import Common.FOL.FOL-Eq\n\nA : D \u2192 Set\nA x = x \u2261 x \u2228 x \u2261 x\n{-# ATP definition A #-}\n\npostulate \u2200\u2192\u2203\u2081 : (\u2200 {x} \u2192 A x) \u2192 \u2203 A\n{-# ATP prove \u2200\u2192\u2203\u2081 #-}\n\npostulate \u2200\u2192\u2203\u2082 : {A : D \u2192 Set} \u2192 (\u2200 {x} \u2192 A x) \u2192 \u2203 A\n{-# ATP prove \u2200\u2192\u2203\u2082 #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/CombiningProofs\/ForallExistSchema.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"d3979feae629def02857aa80ca51b901d65d0530","subject":"DivMod.NP: simpler, more properties","message":"DivMod.NP: simpler, more properties\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/DivMod\/NP.agda","new_file":"lib\/Data\/Nat\/DivMod\/NP.agda","new_contents":"module Data.Nat.DivMod.NP where\n\nopen import Data.Fin as Fin using (Fin; to\u2115; #_)\nimport Data.Fin.Properties as FinP\nopen import Data.Product\nopen import Data.Nat.NP as Nat\nopen import Data.Nat.Properties as NatP\nopen import Data.Nat.Properties.Simple\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.PropositionalEquality as P using (_\u2261_)\nopen import Relation.Binary.PropositionalEquality.TrustMe.NP\n  using (erase)\nopen import Relation.Binary\n\n{-\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits.FromAssocComm _+_ +-assoc +-comm\n  renaming ( assoc-comm to +-assoc-comm )\n-}\n\nopen NatP.SemiringSolver\nopen P.\u2261-Reasoning\n\ninfixl 7 _div_ -- _mod\u2115_ -- _mod_ -- _divMod_\n\n-- A specification of integer division.\n\nrecord DivMod (dividend divisor : \u2115) : Set where\n  constructor result\n  field\n    quotient  : \u2115\n    remainder : Fin divisor\n    property  : dividend \u2261 to\u2115 remainder + quotient * divisor\n\n-- Integer division.\n\nprivate\n\n  div-helper : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  div-helper ack s zero    n       = ack\n  div-helper ack s (suc d) zero    = div-helper (suc ack) s d s\n  div-helper ack s (suc d) (suc n) = div-helper ack       s d n\n\n  {-# BUILTIN NATDIVSUCAUX div-helper #-}\n\ndiv-helper' : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\ndiv-helper' s zero    n       = 0\ndiv-helper' s (suc d) zero    = suc (div-helper' s d s)\ndiv-helper' s (suc d) (suc n) =      div-helper' s d n\n\n{-\ndiv-helper-prop' : \u2200 ack s d n \u2192 div-helper ack s d n \u2261 div-helper' s d n + ack\ndiv-helper-prop' ack s zero n = P.refl\ndiv-helper-prop' ack s (suc d) zero = {!div-helper-prop' (suc ack) s d s!}\ndiv-helper-prop' ack s (suc d) (suc n) = div-helper-prop' ack s d n\n-}\n\ndiv-helper-prop : \u2200 ack s d n \u2192 div-helper ack s d n \u2261 div-helper' s d n +\u1d43 ack\ndiv-helper-prop ack s zero n = P.refl\ndiv-helper-prop ack s (suc d) zero = div-helper-prop (suc ack) s d s\ndiv-helper-prop ack s (suc d) (suc n) = div-helper-prop ack s d n\n\n_div_ : (dividend divisor : \u2115) \u2192 \u2115\n(d div 0)     = 0\n(d div suc s) = div-helper 0 s d s\n\n-- The remainder after integer division.\n\nprivate\n\n  mod-helper : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n  mod-helper ack s zero    n       = ack\n  mod-helper ack s (suc d) zero    = mod-helper zero      s d s\n  mod-helper ack s (suc d) (suc n) = mod-helper (suc ack) s d n\n\n  {-# BUILTIN NATMODSUCAUX mod-helper #-}\n\n  -- The remainder is not too large.\n\n  mod-lemma : (ack d n : \u2115) \u2192\n              let s = ack + n in\n              mod-helper ack s d n \u2264 s\n\n  mod-lemma ack zero n = m\u2264m+n ack n\n\n  mod-lemma ack (suc d) zero = mod-lemma zero d (ack + 0)\n\n  mod-lemma ack (suc d) (suc n) =\n    P.subst (\u03bb x \u2192 mod-helper (suc ack) x d n \u2264 x)\n            (P.sym (+-suc ack n))\n            (mod-lemma (suc ack) d n)\n\n  mod-helper-+\u1d43 : \u2200 ack {d n} \u2192 d \u2264 n \u2192 mod-helper ack (ack +\u1d43 n) d n \u2261 d +\u1d43 ack\n  mod-helper-+\u1d43 ack z\u2264n = P.refl\n  mod-helper-+\u1d43 ack (s\u2264s d\u2264n) = mod-helper-+\u1d43 (suc ack) d\u2264n\n\n  mod-helper0-+\u1d43 : \u2200 {d n} \u2192 d \u2264 n \u2192 mod-helper 0 n d n \u2261 d\n  mod-helper0-+\u1d43 d\u2264n = P.trans (mod-helper-+\u1d43 0 d\u2264n) (+\u1d430-identity _)\n\n  mod-helper-+\u1d43-\u2238-+\u1d43 : \u2200 ack {s d} \u2192 s \u2264 d \u2192 mod-helper 0 (ack +\u1d43 s) (d \u2238 s) (ack +\u1d43 s) \u2261 mod-helper ack (ack +\u1d43 s) (suc d) s\n  mod-helper-+\u1d43-\u2238-+\u1d43 ack z\u2264n = P.refl\n  mod-helper-+\u1d43-\u2238-+\u1d43 ack (s\u2264s p) = mod-helper-+\u1d43-\u2238-+\u1d43 (suc ack) p\n\n  _+\u1d43-mono_ : _+\u1d43_ Preserves\u2082 _\u2264_ \u27f6 _\u2264_ \u27f6 _\u2264_\n  _+\u1d43-mono_ {x} {y} {u} {v} p q rewrite +\u1d43-+ x u | +\u1d43-+ y v = p +-mono q\n\n{-\n  mod-helper0-+\u1d43-id-+\u1d43 : \u2200 ack d s \u2192 d \u2264 ack \u2192 mod-helper 0 (ack +\u1d43 s) d (ack +\u1d43 s) \u2261 d\n  mod-helper0-+\u1d43-id-+\u1d43 ack d zero    p rewrite +\u1d430-identity ack = mod-helper0-+\u1d43 {d} {ack} p\n  mod-helper0-+\u1d43-id-+\u1d43 ack d (suc s) p = mod-helper0-+\u1d43-id-+\u1d43 (suc ack) d s (\u2264-step p)\n\n  mod-helper0-+\u1d43-id-+\u1d43' : \u2200 k ack s \u2192 mod-helper 0 (k + ack +\u1d43 s) (k +\u1d43 s) (k + ack +\u1d43 s) \u2261 k +\u1d43 s\n  mod-helper0-+\u1d43-id-+\u1d43' k ack s = mod-helper0-+\u1d43-id-+\u1d43 (k + ack) (k +\u1d43 s) s {!!}\n-}\n\n  mod-helper0-+\u1d43-+\u1d43-+\u1d43 : \u2200 a k s \u2192 mod-helper 0 (a + k +\u1d43 s) (a +\u1d43 s) (a + k +\u1d43 s) \u2261 a +\u1d43 s\n  mod-helper0-+\u1d43-+\u1d43-+\u1d43 a k zero    rewrite +\u1d430-identity a | +\u1d430-identity (a + k)\n                                        = mod-helper0-+\u1d43 {a} {a + k} (\u2264-steps\u2032 {a} _)\n  mod-helper0-+\u1d43-+\u1d43-+\u1d43 a k (suc s) = mod-helper0-+\u1d43-+\u1d43-+\u1d43 (suc a) k s\n\n  -- seems like \"mod-helper0-+\u1d43 : \u2200 {d n} \u2192 d \u2264 n \u2192 mod-helper 0 n d n \u2261 d\"\n  -- but with n \u2261 k + d\n  mod-helper0-+\u1d43-id-+\u1d43 : \u2200 k s \u2192 mod-helper 0 (k +\u1d43 s) s (k +\u1d43 s) \u2261 s\n  mod-helper0-+\u1d43-id-+\u1d43 k s = mod-helper0-+\u1d43-+\u1d43-+\u1d43 0 k s\n\n{-\n  mod-helper0-+\u1d43-id-+\u1d43 : \u2200 k ack s \u2192 mod-helper 0 (k +\u1d43 ack) (k +\u1d43 s) (k +\u1d43 ack) \u2261 k +\u1d43 s\n  mod-helper0-+\u1d43-id-+\u1d43 k ack zero rewrite +\u1d430-identity k = {!mod-helper0-+\u1d43 {k} {k + ack} (\u2264-steps\u2032 {k} _)!}\n  mod-helper0-+\u1d43-id-+\u1d43 k ack (suc s) = {!mod-helper0-+\u1d43-id-+\u1d43 (suc k) ack s!}\n\n  mod-helper-s\u2238 : \u2200 a a' {d s} \u2192 d \u2264 s \u2192 mod-helper 0 (a +\u1d43 s) (a' +\u1d43 s \u2238 d) (a +\u1d43 s) \u2261 a' +\u1d43 s \u2238 (mod-helper a (a' +\u1d43 s) d s)\n  mod-helper-s\u2238 a a' (z\u2264n {s}) = {!P.trans (mod-helper0-+\u1d43-id-+\u1d43 a' a s)\n                               {!(P.trans (P.sym (m+n\u2238n\u2261m s a))\n                               (P.cong (\u03bb z \u2192 z \u2238 a) (P.trans (+-comm s a) (P.sym (+\u1d43-+ a s)))))!}!}\n  mod-helper-s\u2238 a a' (s\u2264s p) = {!mod-helper-s\u2238 (suc a) (suc a') p!}\n  -}\n\n  mod-helper-s\u2238 : \u2200 ack {d s} \u2192 d \u2264 s \u2192 mod-helper 0 (ack +\u1d43 s) (s \u2238 d) (ack +\u1d43 s) \u2261 ack +\u1d43 s \u2238 (mod-helper ack (ack +\u1d43 s) d s)\n  mod-helper-s\u2238 ack (z\u2264n {s}) = P.trans (mod-helper0-+\u1d43-id-+\u1d43 ack s)\n                               (P.trans (P.sym (m+n\u2238n\u2261m s ack))\n                               (P.cong (\u03bb z \u2192 z \u2238 ack) (P.trans (+-comm s ack) (P.sym (+\u1d43-+ ack s)))))\n  mod-helper-s\u2238 ack (s\u2264s p) = mod-helper-s\u2238 (suc ack) p\n\n  mod-helper0-s\u2238 : \u2200 {d s} \u2192 d \u2264 s \u2192 mod-helper 0 s (s \u2238 d) s \u2261 s \u2238 (mod-helper 0 s d s)\n  mod-helper0-s\u2238 = mod-helper-s\u2238 0\n\n-- Notice: d mod\u2115 0 \u2261 d\n-- One can argue that if one cannot divide by 0 then what remains is the whole thing.\n-- Of course the remainder is not smaller than the divisor in this case.\n_mod\u2115_ : (dividend divisor : \u2115) \u2192 \u2115\n(d mod\u2115 0)     = d\n(d mod\u2115 suc s) = mod-helper 0 s d s\n\nmod\u2115<divisor : (dividend divisor : \u2115) {\u22620 : False (divisor \u225f 0)} \u2192 dividend mod\u2115 divisor < divisor\nmod\u2115<divisor d 0 {}\nmod\u2115<divisor d (suc s) = s\u2264s (mod-lemma 0 d s)\n\n_mod_ : (dividend divisor : \u2115) {\u22620 : False (divisor \u225f 0)} \u2192 Fin divisor\n(d mod s) {\u22620} = Fin.from\u2115\u2264 {d mod\u2115 s} (mod\u2115<divisor d s {\u22620})\n\nap-from\u2115\u2264 : \u2200 {x y n}{p : x < n}{q : y < n} \u2192 x \u2261 y \u2192 Fin.from\u2115\u2264 {x} {n} p \u2261 Fin.from\u2115\u2264 {y} {n} q\nap-from\u2115\u2264 {x} {y} {n} {p} {q} e = FinP.to\u2115-injective (begin\n     to\u2115 (Fin.from\u2115\u2264 p)  \u2261\u27e8 FinP.to\u2115-from\u2115\u2264 p \u27e9\n     x                   \u2261\u27e8 e \u27e9\n     y                   \u2261\u27e8 P.sym (FinP.to\u2115-from\u2115\u2264 q) \u27e9\n     to\u2115 (Fin.from\u2115\u2264 q)  \u220e)\n\n\u2238s-mod : \u2200 {d s}{\u22620 : False (s \u225f 0)} \u2192 s \u2264 d \u2192 ((d \u2238 s)mod s){\u22620} \u2261 (d mod s){\u22620}\n\u2238s-mod z\u2264n = P.refl\n\u2238s-mod (s\u2264s s\u2264d) = ap-from\u2115\u2264 (mod-helper-+\u1d43-\u2238-+\u1d43 0 s\u2264d)\n\nmod\u2115-<s : \u2200 {d s}(p : d < s) \u2192 d mod\u2115 s \u2261 d\nmod\u2115-<s (s\u2264s p) = mod-helper0-+\u1d43 p\n\nmod\u2115-\u2238s : \u2200 {d s} \u2192 s \u2264 d \u2192 (d \u2238 s)mod\u2115 s \u2261 d mod\u2115 s\nmod\u2115-\u2238s z\u2264n = P.refl\nmod\u2115-\u2238s (s\u2264s s\u2264d) = mod-helper-+\u1d43-\u2238-+\u1d43 0 s\u2264d\n\nbar : \u2200 x s \u2192 s \u2238 x \u2264 s\nbar zero s = \u2115\u2264.refl\nbar (suc x) zero = \u2115\u2264.refl\nbar (suc x) (suc s) = \u2264-step (bar x s)\n\ngar : \u2200 {x s} \u2192 0 < x \u2192 0 < s \u2192 s \u2238 x < s\ngar (s\u2264s (z\u2264n {x})) (s\u2264s (z\u2264n {s})) = s\u2264s (bar x s)\n\nmod\u2115-s\u2238 : \u2200 {x s} \u2192 0 < x \u2192 x < s \u2192 (s \u2238 x)mod\u2115 s \u2261 s \u2238 (x mod\u2115 s)\nmod\u2115-s\u2238 {x} {s} 0<x x<s = begin\n  (s \u2238 x)mod\u2115 s  \u2261\u27e8 mod\u2115-<s (gar 0<x (\u2115\u2264.trans (s\u2264s z\u2264n) x<s)) \u27e9\n  s \u2238 x          \u2261\u27e8 P.cong (_\u2238_ s) (P.sym (mod\u2115-<s x<s)) \u27e9\n  s \u2238 (x mod\u2115 s) \u220e\n\n{-\nmod-helper0-+\u1d43-suc-+\u1d43 : \u2200 k a s \u2192 mod-helper 0 (a +\u1d43 k +\u1d43 s) (a + suc s) (a +\u1d43 k +\u1d43 s) \u2261 suc s\nmod-helper0-+\u1d43-suc-+\u1d43 k zero s = {!!}\nmod-helper0-+\u1d43-suc-+\u1d43 k (suc a) s = {!mod-helper0-+\u1d43-suc-+\u1d43 (suc k) a s!}\n\nGoal : \u2200 a x s \u2192 x \u2264 s \u2192 mod-helper 0 (a +\u1d43 s) (suc s \u2238 x) (a +\u1d43 s) \u2261 a +\u1d43 suc s \u2238 mod-helper a (a +\u1d43 s) x s\nGoal a .0 s z\u2264n = {!mod-helper0-+\u1d43-+\u1d43-+\u1d43 0!}\nGoal a ._ ._ (s\u2264s p) = Goal (suc a) _ _ p\n\n-- mod\u2115-s\u2238 : \u2200 {x s} \u2192 x \u2264 s \u2192 (s \u2238 x)mod\u2115 s \u2261 s \u2238 (x mod\u2115 s)\nmod\u2115-s\u2238 : \u2200 {x s} \u2192 x < s \u2192 (s \u2238 x)mod\u2115 s \u2261 s \u2238 (x mod\u2115 s)\nmod\u2115-s\u2238 {s = zero} ()\nmod\u2115-s\u2238 {x} {s = suc s} (s\u2264s p) = Goal 0 x s p\n-}\n\n{-\n-- with \u2264\u21d2\u2203 p\n--... | k , e {-rewrite P.sym e-} =\n  begin\n  mod-helper 0 s (suc s \u2238 x) s \u2261\u27e8 {!mod-helper0-+\u1d43-id-+\u1d43 (suc x) (suc s \u2238 x)!} \u27e9 -- mod-helper0-+\u1d43 {suc s \u2238 x} {s} {!\u2261+-\u2265 (suc s \u2238 x) ? s!} \u27e9\n  suc s \u2238 x \u2261\u27e8 P.cong (\u03bb z \u2192 suc s \u2238 z) (P.sym (mod-helper0-+\u1d43 {x} {s} p)) \u27e9\n  suc s \u2238 mod-helper 0 s x s \u220e\n  {-\nbegin\n  mod-helper 0 s (s \u2238 x) s \u2261\u27e8 {!!} \u27e9\n  mod-helper 0 s k s \u2261\u27e8 mod-helper0-+\u1d43 {k} {s} (\u2261+-\u2265 s x k (P.sym (P.cong pred e))) \u27e9\n  k \u2261\u27e8 {!!} \u27e9\n  suc x + k \u2238 x \u2261\u27e8 P.cong (\u03bb z \u2192 suc x + k \u2238 z) (P.sym (mod-helper0-+\u1d43 {x} {s} {!!})) \u27e9\n  suc s \u2238 mod-helper 0 s x s \u220e\n\n{-\nsuc s \u2261 k + x\nmod-helper 0 s (k + x \u2238 x) s \u2261 suc s \u2238 mod-helper 0 s x s\n \n--{!P.trans ? ()!}\n-}\n-}\n-}\n\n-- Integer division with remainder.\n\nprivate\n\n  -- The quotient and remainder are related to the dividend and\n  -- divisor in the right way.\n\n  division-lemma :\n    (mod-ack div-ack d n : \u2115) \u2192\n    let s = mod-ack + n in\n    mod-ack + div-ack * suc s + d\n      \u2261\n    mod-helper mod-ack s d n + div-helper div-ack s d n * suc s\n\n  division-lemma mod-ack div-ack zero n = begin\n\n    mod-ack + div-ack * suc s + zero  \u2261\u27e8 +-right-identity _ \u27e9\n\n    mod-ack + div-ack * suc s         \u220e\n\n    where s = mod-ack + n\n\n  division-lemma mod-ack div-ack (suc d) zero = begin\n\n    mod-ack + div-ack * suc s + suc d                \u2261\u27e8 solve 3\n                                                              (\u03bb mod-ack div-ack d \u2192\n                                                                   let s = mod-ack :+ con 0 in\n                                                                   mod-ack :+ div-ack :* (con 1 :+ s) :+ (con 1 :+ d)\n                                                                     :=\n                                                                   (con 1 :+ div-ack) :* (con 1 :+ s) :+ d)\n                                                              P.refl mod-ack div-ack d \u27e9\n    suc div-ack * suc s + d                          \u2261\u27e8 division-lemma zero (suc div-ack) d s \u27e9\n\n    mod-helper zero          s      d  s    +\n    div-helper (suc div-ack) s      d  s    * suc s  \u2261\u27e8\u27e9\n\n    mod-helper      mod-ack  s (suc d) zero +\n    div-helper      div-ack  s (suc d) zero * suc s  \u220e\n\n    where s = mod-ack + 0\n\n  division-lemma mod-ack div-ack (suc d) (suc n) = begin\n\n    mod-ack + div-ack * suc s + suc d                   \u2261\u27e8 solve 4\n                                                                 (\u03bb mod-ack div-ack n d \u2192\n                                                                      mod-ack :+ div-ack :* (con 1 :+ (mod-ack :+ (con 1 :+ n))) :+ (con 1 :+ d)\n                                                                        :=\n                                                                      con 1 :+ mod-ack :+ div-ack :* (con 2 :+ mod-ack :+ n) :+ d)\n                                                                 P.refl mod-ack div-ack n d \u27e9\n    suc mod-ack + div-ack * suc s\u2032 + d                  \u2261\u27e8 division-lemma (suc mod-ack) div-ack d n \u27e9\n\n    mod-helper (suc mod-ack) s\u2032 d n +\n    div-helper      div-ack  s\u2032 d n * suc s\u2032            \u2261\u27e8 P.cong (\u03bb s \u2192 mod-helper (suc mod-ack) s d n +\n                                                                         div-helper div-ack s d n * suc s)\n                                                                  (P.sym (+-suc mod-ack n)) \u27e9\n\n    mod-helper (suc mod-ack) s d n +\n    div-helper      div-ack  s d n * suc s              \u2261\u27e8\u27e9\n\n    mod-helper      mod-ack  s (suc d) (suc n) +\n    div-helper      div-ack  s (suc d) (suc n) * suc s  \u220e\n\n    where\n    s  = mod-ack + suc n\n    s\u2032 = suc mod-ack + n\n\ndivModProp\u2115 : \u2200 d s \u2192 d \u2261 d mod\u2115 s + (d div s) * s\ndivModProp\u2115 _ zero    = P.sym (+-comm _ 0)\ndivModProp\u2115 d (suc s) = erase (division-lemma 0 0 d s)\n\ndivModProp : \u2200 d s {\u22620 : False (s \u225f 0)} \u2192 d \u2261 to\u2115 ((d mod s){\u22620}) + (d div s) * s\ndivModProp d (s) {\u22620} = erase (begin\n    d                                       \u2261\u27e8 divModProp\u2115 d (s) \u27e9\n    d mod\u2115 s + (d div s) * s                \u2261\u27e8 P.cong\u2082 _+_ (P.sym (FinP.to\u2115-from\u2115\u2264 lemma)) P.refl \u27e9\n    to\u2115 (Fin.from\u2115\u2264 lemma) + (d div s) * s  \u220e)\n  where lemma = mod\u2115<divisor d s {\u22620}\n\n_divMod_ : (dividend divisor : \u2115) {\u22620 : False (divisor \u225f 0)} \u2192\n           DivMod dividend divisor\n(d divMod s) {\u22620} = result (d div s) (d mod s) (divModProp d s {\u22620})\n\ndiv<\u22610 : \u2200 {d s} \u2192 d < s \u2192 d div s \u2261 0\ndiv<\u22610 {d} {zero}  d<s = P.refl\ndiv<\u22610 {d} {suc s} d<s = helper (\u2264-pred d<s)\n  where\n    helper : \u2200 {d s n} \u2192 d \u2264 n \u2192 div-helper 0 s d n \u2261 0\n    helper z\u2264n       = P.refl\n    helper (s\u2264s d\u2264n) = helper d\u2264n\n\nd\/d\u22611 : \u2200 d {\u22620 : False (d \u225f 0)} \u2192 d div d \u2261 1\nd\/d\u22611 zero {}\nd\/d\u22611 (suc d) = helper d\n  where\n    helper : \u2200 {s} d \u2192 div-helper 0 s (suc d) d \u2261 1\n    helper zero    = P.refl\n    helper (suc x) = helper x\n\n1-mod\u2115 : \u2200 {s} \u2192 1 < s \u2192 1 mod\u2115 s \u2261 1\n1-mod\u2115 = mod\u2115-<s\n\nmod\u2115-prop : \u2200 x {s} \u2192 x mod\u2115 s \u2261 x \u2238 (x div s) * s\nmod\u2115-prop x {s} = +-\u2238 x (x mod\u2115 s) (x div s * s) (divModProp\u2115 x s)\n\n{-\nmod\u2115-ns\u2238 : \u2200 {n x s} \u2192 0 < x \u2192 x < n * s \u2192 (n * s \u2238 x)mod\u2115 s \u2261 s \u2238 (x mod\u2115 s)\n{-\nmod\u2115-ns\u2238 {n} {x} {s} 0<x x<s =\n    begin\n    (n * s \u2238 x)mod\u2115 s      \u2261\u27e8 {!!} \u27e9\n    s \u2238 (x \u2238 q1 * s)  \u2261\u27e8 P.cong (_\u2238_ s) (P.sym (mod-prop x {s})) \u27e9\n    s \u2238 (x mod\u2115 s)         \u220e\n    where q1 = x div s\n-}\nmod\u2115-ns\u2238 {n} {x} {zero}  0<x x<s rewrite \u2115\u00b0.*-comm n 0 = P.refl\nmod\u2115-ns\u2238 {n} {x} {suc s} 0<x x<s = helper n x s 0<x x<s\n  where helper : \u2200 n x s \u2192\n          0 < x \u2192\n          x < n * suc s \u2192\n          mod-helper 0 s (n * suc s \u2238 x) s \u2261 suc s \u2238 mod-helper 0 s x s\n        helper n zero    s p q = {!!}\n        helper n (suc x) s p q = {!helper n x s!}\n\n{-\nsuc s * n = n + s * n\n\n(n + sn) \u2238 x\n\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Nat\/DivMod\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"96fadd7950bdddd78ac3e076ecb968c0cce04ee1","subject":"Added initial support for talking about probability.","message":"Added initial support for talking about probability.\n","repos":"crypto-agda\/crypto-agda","old_file":"Prob.agda","new_file":"Prob.agda","new_contents":"module Prob where\n\nopen import Data.Bool\nopen import Data.Nat\nopen import Data.Vec\n\nopen import Function\n\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Nullary\n\nrecord ]0,1]-ops (]0,1] : Set) : Set where\n  field     \n    1E : ]0,1]\n    _+E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    _\u00b7E_ : ]0,1] \u2192 ]0,1] \u2192 ]0,1]\n    1\/E_ : ]0,1] \u2192 ]0,1]\n\nmodule [0,1] {]0,1]}(]0,1]R : ]0,1]-ops ]0,1]) where\n\n  open ]0,1]-ops ]0,1]R public\n\n  infixl 6 _+I_\n  infix 4 _\u2264I_\n\n  data [0,1] : Set where\n   0I : [0,1]\n   _I : ]0,1] \u2192 [0,1]\n\n  1I : [0,1]\n  1I = 1E I\n\n  Pos : [0,1] \u2192 Set\n  Pos x = \u00ac (x \u2261 0I)\n\n  _<_> : (x : [0,1]) \u2192 Pos x \u2192 ]0,1] \n  0I < pos > with pos refl\n  ... | ()\n  (x I) < pos > = x\n\n  \n\n  _+I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I +I x = x\n  x I +I 0I = x I\n  x I +I x\u2081 I = (x +E x\u2081) I\n\n  _\u00b7I_ : [0,1] \u2192 [0,1] \u2192 [0,1]\n  0I \u00b7I y = 0I\n  (x I) \u00b7I 0I = 0I\n  (x I) \u00b7I (x\u2081 I) = (x \u00b7E x\u2081) I\n\n  1\/I_ : ]0,1] \u2192 [0,1]\n  1\/I x = (1\/E x) I\n\n  _\/I_ : [0,1] \u2192 ]0,1] \u2192 [0,1]\n  x \/I y = x \u00b7I (1\/I y)\n\n  +I-identity : (x : [0,1]) \u2192 x \u2261 0I +I x\n  +I-identity x = refl\n\n  postulate\n    +I-sym : (x y : [0,1]) \u2192 x +I y \u2261 y +I x\n    +I-assoc : (x y z : [0,1]) \u2192 x +I y +I z \u2261 x +I (y +I z)\n\n    _\u2264I_ : [0,1] \u2192 [0,1] \u2192 Set\n    \u2264I-refl : {x : [0,1]} \u2192 x \u2264I x\n\n    \u2264I-trans : {x y z : [0,1]} \u2192 x \u2264I y \u2192 y \u2264I z \u2192 x \u2264I z\n    \u2264I-mono  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 y \u2264I z +I x\n    \u2264I-pres  : (x : [0,1]){y z : [0,1]} \u2192 y \u2264I z \u2192 x +I y \u2264I x +I z\n\nmodule Univ ( ]0,1] : Set)(]0,1]R : ]0,1]-ops ]0,1])\n            (U : Set)\n            (size-1 : \u2115) \n            (allU : Vec U (suc size-1))\n            (x\u2208allU : (x : U) \u2192 x \u2208 allU)  where \n\n  open [0,1] ]0,1]R\n\n  sumP : {n : \u2115} \u2192 (U \u2192 [0,1]) \u2192 Vec U n \u2192 [0,1]\n  sumP P []       = 0I\n  sumP P (x \u2237 xs) = (P x) +I (sumP P xs)\n\n  module Prob (P : U \u2192 [0,1])\n              (sumP\u22611 : sumP P allU \u2261 1I) where\n    \n    Event : Set\n    Event = U \u2192 Bool \n  \n    pr_\u220b_ : Event \u2192 U \u2192 [0,1]\n    pr A \u220b x = if A x then P x else 0I\n\n    _\u222a_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u222a A\u2082 = \u03bb x \u2192 A\u2081 x \u2228 A\u2082 x\n\n    _\u2229_ : Event \u2192 Event \u2192 Event\n    A\u2081 \u2229 A\u2082 = \u03bb x \u2192 A\u2081 x \u2227 A\u2082 x\n\n    \u2102[_] : Event \u2192 Event\n    \u2102[ A ] = not \u2218 A\n\n    Pr[_] : Event \u2192 [0,1]\n    Pr[ A ] = sumP (pr_\u220b_ A) allU\n\n    Pr[_\u2223_]<_> : (A B : Event)(pf : Pos Pr[ B ]) \u2192 [0,1]   \n    Pr[ A \u2223 B ]< pr > = Pr[ A \u2229 B ] \/I Pr[ B ] < pr >  \n\n    _ind_ : (A B : Event) \u2192 Set\n    A ind B = Pr[ A ] \u00b7I Pr[ B ] \u2261 Pr[ A \u2229 B ]\n\n    union-bound : (A\u2081 A\u2082 : Event) \u2192 Pr[ (A\u2081 \u222a A\u2082) ] \u2264I Pr[ A\u2081 ] +I Pr[ A\u2082 ]\n    union-bound A\u2081 A\u2082 = go allU where\n      sA\u2081 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2081 = \u03bb xs \u2192 sumP (pr_\u220b_ A\u2081) xs\n      sA\u2082 : {n : \u2115} \u2192 Vec U n \u2192 [0,1]\n      sA\u2082 = sumP (pr_\u220b_ A\u2082)\n      go : {n : \u2115}(xs : Vec U n) \u2192 sumP (pr_\u220b_ (A\u2081 \u222a A\u2082)) xs \u2264I sumP (pr_\u220b_ A\u2081) xs +I sumP (pr_\u220b_ A\u2082) xs\n      go [] = \u2264I-refl\n      go (x \u2237 xs) with A\u2081 x | A\u2082 x \n      ... | true  | true  rewrite +I-assoc (P x) (sA\u2081 xs) (P x +I sA\u2082 xs)\n                                | +I-sym (P x) (sA\u2082 xs)\n                                | sym (+I-assoc (sA\u2081 xs) (sA\u2082 xs) (P x))\n                                = \u2264I-pres (P x) (\u2264I-mono (P x) (go xs)) \n      ... | true  | false rewrite +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | true  rewrite sym (+I-assoc (sA\u2081 xs) (P x) (sA\u2082 xs))\n                                | +I-sym (sA\u2081 xs) (P x)\n                                | +I-assoc (P x) (sA\u2081 xs) (sA\u2082 xs)\n                                = \u2264I-pres (P x) (go xs)\n      ... | false | false = go xs \n\n    module RandomVar (V : Set) (_==V_ : V \u2192 V \u2192 Bool) where\n       RV : Set\n       RV = U \u2192 V\n\n       _^-1_ : RV \u2192 V \u2192 Event\n       RV ^-1 v = \u03bb x \u2192 RV x ==V v \n\n       _\u2261r_ : RV \u2192 V \u2192 Event \n       RV \u2261r v = RV ^-1 v\n \n       -- sugar, socker et sucre\n       Pr[_\u2261_] : RV \u2192 V \u2192 [0,1]\n       Pr[ X \u2261 v ] = Pr[ X \u2261r v ]\n \n       -- \u2200 (a b) . Pr[X = a and Y = b] = Pr[X = a] \u00b7 Pr[Y = b]\n       _indRV_ : RV \u2192 RV \u2192 Set\n       X indRV Y = (a b : V) \u2192 (X ^-1 a) ind (Y ^-1 b) ","old_contents":"","returncode":1,"stderr":"error: pathspec 'Prob.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f58fb12a605924ada6a2cc1645dc725167a301ea","subject":"Add missing file.","message":"Add missing file.\n\nForgot to add this file in commit ba1d8a8be0778b4d917c4d98c7db8f78f8f3cbbc.\n\nOld-commit-hash: 02fcec1035aa2e0190d1ba3fc335cecbade97c9b\n","repos":"inc-lc\/ilc-agda","old_file":"meaning.agda","new_file":"meaning.agda","new_contents":"module meaning where\n\nopen import Level\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    Semantics : Set \u2113\n    \u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'meaning.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"594410fa7b159df8d53a46639d96244fb28f3280","subject":"BigDistr.agda","message":"BigDistr.agda\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/BigDistr.agda","new_file":"lib\/Explore\/BigDistr.agda","new_contents":"-- Unifinished\n\n\nopen import Type\nopen import Function.NP\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Fin using (Fin)\nopen import Explore.Type\nimport Function.Related as FR\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\nopen import Function.Related.TypeIsomorphisms.NP\n\nmodule Explore.BigDistr\n                {A B : \u2605\u2080}\n                {explore\u1d2c : Explore\u2080 A} \n                {explore\u1d2e : Explore\u2080 B} \n                {explore\u1d2c\u1d2e : Explore\u2080 (A \u2192 B)}\n                (F : A \u2192 B \u2192 \u2115)\n                where\n  product\u1d2c = explore\u1d2c _*_\n  sum\u1d2e = explore\u1d2e _+_\n  sum\u1d2c\u1d2e = explore\u1d2c\u1d2e _+_\n\n  module _\n    (adequate-product\u1d2c : AdequateProduct product\u1d2c)\n    (adequate-sum\u1d2e : AdequateSum sum\u1d2e)\n    (adequate-sum\u1d2c\u1d2e : AdequateSum sum\u1d2c\u1d2e)\n    where\n\n    open FR.EquationalReasoning\n    big-distr :\n      product\u1d2c (sum\u1d2e \u2218 F) \u2261\n      sum\u1d2c\u1d2e (\u03bb f \u2192 product\u1d2c (F \u02e2 f))\n    big-distr = Fin-injective (\n        Fin (product\u1d2c (sum\u1d2e \u2218 F))\n      \u2194\u27e8 adequate-product\u1d2c (sum\u1d2e \u2218 F) \u27e9\n        \u03a0 A (Fin \u2218 sum\u1d2e \u2218 F)\n      \u2194\u27e8 {!!} \u27e9\n        \u03a0 A (\u03bb x \u2192 \u03a3 B (Fin \u2218 F x))\n      \u2194\u27e8 dep-choice-iso _ \u27e9\n        \u03a3 (\u03a0 A (const B)) (\u03bb f \u2192 \u03a0 A (\u03bb x \u2192 Fin (F x (f x))))\n      \u2194\u27e8 second-iso (\u03bb f \u2192 sym (adequate-product\u1d2c (F \u02e2 f))) \u27e9\n        \u03a3 (\u03a0 A (const B)) (\u03bb f \u2192 Fin (product\u1d2c (F \u02e2 f)))\n      \u2194\u27e8 sym (adequate-sum\u1d2c\u1d2e (\u03bb f \u2192 product\u1d2c (F \u02e2 f))) \u27e9\n        Fin (sum\u1d2c\u1d2e (\u03bb f \u2192 product\u1d2c (F \u02e2 f)))\n      \u220e)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Explore\/BigDistr.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"777309b3675f9e338a2109af173e707c6e8e6555","subject":"lib\/Explore\/Examples.agda","message":"lib\/Explore\/Examples.agda\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Examples.agda","new_file":"lib\/Explore\/Examples.agda","new_contents":"module Explore.Examples where\n\nopen import Type\nopen import Level.NP\nopen import Data.Maybe.NP\nopen import Data.List\nopen import Data.Two\n\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Monad \u2080 renaming (map to map-explore)\n\nmodule E where\n  open FromExplore\u2080 public\n  open FromExplore public\n\nmodule M {Msg Digest : \u2605}\n         (_==_ : Digest \u2192 Digest \u2192 \ud835\udfda)\n         (H : Msg \u2192 Digest)\n         (exploreMsg : Explore \u2080 Msg)\n         (d : Digest) where\n\n  module V1 where\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = exploreMsg [] _++_ (\u03bb m \u2192 [0: [] 1: [ m ] ] (H m == d))\n\n  module ExploreMsg = FromExplore\u2080 exploreMsg\n\n  module V2 where\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = ExploreMsg.findKey (\u03bb m \u2192 H m == d)\n\n  module V3 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 \u03b5 _\u2295_ f = exploreMsg \u03b5 _\u2295_ (\u03bb m \u2192 [0: \u03b5 1: f m ] (H m == d))\n\n  module V4 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 = exploreMsg >>= \u03bb m \u2192 [0: empty-explore 1: point-explore m ] (H m == d)\n\n  module V5 where\n\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 = filter-explore (\u03bb m \u2192 H m == d) exploreMsg\n\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = E.list explore-H\u207b\u00b9\n\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = E.first explore-H\u207b\u00b9\n\n  module V6 where\n    explore-H\u207b\u00b9 : Explore \u2080 Msg\n    explore-H\u207b\u00b9 = explore-endo (filter-explore (\u03bb m \u2192 H m == d) exploreMsg)\n\n    list-H\u207b\u00b9 : List Msg\n    list-H\u207b\u00b9 = E.list explore-H\u207b\u00b9\n\n    first-H\u207b\u00b9 : Maybe Msg\n    first-H\u207b\u00b9 = E.first explore-H\u207b\u00b9\n\n    last-H\u207b\u00b9 : Maybe Msg\n    last-H\u207b\u00b9 = E.last explore-H\u207b\u00b9\n\nopen import Data.Two\nopen import Data.Product\nopen import Explore.Two\nopen import Explore.Product\nopen Explore.Product.Operators\nopen import Relation.Binary.PropositionalEquality\nMsg = \ud835\udfda \u00d7 \ud835\udfda\nDigest = \ud835\udfda\n-- _==_ : Digest \u2192 Digest \u2192 \ud835\udfda\nH : Msg \u2192 Digest\nH (x , y) = x xor y\nMsg\u1d49 : Explore \u2080 Msg\nMsg\u1d49 = \ud835\udfda\u1d49 \u00d7\u1d49 \ud835\udfda\u1d49\nmodule N5 = M.V5 _==_ H Msg\u1d49\nmodule N6 = M.V6 _==_ H Msg\u1d49\ntest5 = N5.list-H\u207b\u00b9\ntest6-list : N6.list-H\u207b\u00b9 0\u2082 \u2261 0\u2082 , 0\u2082 \u2237 1\u2082 , 1\u2082 \u2237 []\ntest6-list = refl\ntest6-rev-list : E.list (E.backward (N6.explore-H\u207b\u00b9 0\u2082)) \u2261 1\u2082 , 1\u2082 \u2237 0\u2082 , 0\u2082 \u2237 []\ntest6-rev-list = refl\ntest6-first : N6.first-H\u207b\u00b9 0\u2082 \u2261 just (0\u2082 , 0\u2082)\ntest6-first = refl\ntest6-last : N6.last-H\u207b\u00b9 0\u2082 \u2261 just (1\u2082 , 1\u2082)\ntest6-last = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Explore\/Examples.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1dcf0ea422d896f718d19d3d18e2f66c222f645f","subject":"IDesc model: type-in-type, so it goes... :-(","message":"IDesc model: type-in-type, so it goes... :-(\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IDesc_type_type.agda","new_file":"models\/IDesc_type_type.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule IDesc_type_type where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\ndesc : (I : Set) -> IDesc I -> (I -> Set) -> Set\ndesc I (var i) P = P i\ndesc I (const X) P = X\ndesc I (prod D D') P = desc I D P * desc I D' P\ndesc I (sigma S T) P = Sigma S (\\s -> desc I (T s) P)\ndesc I (pi S T) P = (s : S) -> desc I (T s) P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu (I : Set)(R : I -> IDesc I)(i : I) : Set where\n  con : desc I (R i) (\\j -> IMu I R j) -> IMu I R i\n\n--********************************************\n-- Predicate: Box\n--********************************************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (var i)     P x        = var (i , x)\nbox I (const X)   P x        = const X\nbox I (prod D D') P (d , d') = prod (box I D P d) (box I D' P d')\nbox I (sigma S T) P (a , b)  = box I (T a) P b\nbox I (pi S T)    P f        = pi S (\\s -> box I (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    induction : (i : I)(x : IMu I R i) -> P (i , x)\n    induction i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : desc I D (IMu I R)) -> \n           desc (Sigma I (IMu I R)) (box I D (IMu I R) xs) P\n    hyps (var i) x = induction i x\n    hyps (const X) x = x -- ??\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\ninduction : (I : Set)\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu I R) -> Set)\n            (m : (i : I)\n                 (xs : desc I (R i) (IMu I R))\n                 (hs : desc (Sigma I (IMu I R)) (box I (R i) (IMu I R) xs) P) ->\n                 P ( i , con xs)) ->\n            (i : I)(x : IMu I R i) -> P ( i , x )\ninduction = Elim.induction\n\n\n--********************************************\n-- DescD\n--********************************************\n\ndata DescDConst : Set where\n  lvar   : DescDConst\n  lconst : DescDConst\n  lprod  : DescDConst\n  lpi    : DescDConst\n  lsigma : DescDConst\n\ndescDChoice : Set -> DescDConst -> IDesc Unit\ndescDChoice I lvar = const I\ndescDChoice _ lconst = const Set\ndescDChoice _ lprod = prod (var Void) (var Void)\ndescDChoice _ lpi = sigma Set (\\S -> pi S (\\s -> var Void))\ndescDChoice _ lsigma = sigma Set (\\S -> pi S (\\s -> var Void))\n\ndescD : (I : Set) -> IDesc Unit\ndescD I = sigma DescDConst (descDChoice I)\n\nIDescl : (I : Set) -> Set\nIDescl I = IMu Unit (\\_ -> descD I) Void\n\nvarl : (I : Set)(i : I) -> IDescl I\nvarl I i = con (lvar ,  i)\n\nconstl : (I : Set)(X : Set) -> IDescl I\nconstl I X = con (lconst , X)\n\nprodl : (I : Set)(D D' : IDescl I) -> IDescl I\nprodl I D D' = con (lprod , (D , D'))\n\npil : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\npil I S T = con (lpi , ( S , T))\n\nsigmal : (I : Set)(S : Set)(T : S -> IDescl I) -> IDescl I\nsigmal I S T = con (lsigma , ( S , T))","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/IDesc_type_type.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"5daf05cb2ea0c6820deb958c5acb7c505754e48e","subject":"Added draft for the existential quantifier syntax.","message":"Added draft for the existential quantifier syntax.\n\nIgnore-this: 524214e2db728613cf79637177679098\n\ndarcs-hash:20111129214314-3bd4e-189edf1e0b969900d1c8b5ee6123a5c1a663d485.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/ExistentialSyntax.agda","new_file":"notes\/ExistentialSyntax.agda","new_contents":"module ExistentialSyntax where\n\ninfixr 4 _,_\n\npostulate\n  D : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  refl : \u2200 {d} \u2192 d \u2261 d\n  d : D\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (d : D) \u2192 P d \u2192 \u2203 P\n\nsyntax \u2203 (\u03bb x \u2192 e)  = \u2203[ x ] e\n\nt1 : \u2203 \u03bb x \u2192 x \u2261 x\nt1 = d , refl\n\nt2 : \u2203[ x ] (x \u2261 x)\nt2 = d , refl\n\nt3 : \u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 x \u2261 y\nt3 = d , d , refl\n\nt4 : \u2203[ x ] \u2203[ y ] x \u2261 y\nt4 = d , d ,  refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/ExistentialSyntax.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"85cbb9d9644ad82149497406dd594b410e246115","subject":"bintree-proba.agda","message":"bintree-proba.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"bintree-proba.agda","new_file":"bintree-proba.agda","new_contents":"module bintree-proba\n  ([0,1] : Set)\nwhere\n\nopen import Data.Nat\nopen import Data.Bits\n\ndata Tree : \u2115 \u2192 Set where\n  leaf : \u2200 {n} \u2192 Bits n \u2192 Tree n\n  fork : \u2200 {n} (proba : [0,1]) (left right : Tree n) \u2192 Tree (1 + n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'bintree-proba.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"48dc53b065d6c0975e009b46a46fe29be82ad046","subject":"Added some cpo (alea) stuff","message":"Added some cpo (alea) stuff\n","repos":"crypto-agda\/crypto-agda","old_file":"alea\/cpo.agda","new_file":"alea\/cpo.agda","new_contents":"module alea.cpo where\n\nimport Data.Nat.NP as Nat\n\nimport Relation.Binary.PropositionalEquality as \u2261\n\nrecord Order (A : Set) : Set\u2081 where\n  constructor mk\n  infix 4 _\u2261_ _\u2264_\n  field\n    _\u2264_ : A \u2192 A \u2192 Set\n    _\u2261_ : A \u2192 A \u2192 Set\n    reflexive : \u2200 {x} \u2192 x \u2264 x\n    antisym : \u2200 {x y} \u2192 x \u2264 y \u2192 y \u2264 x \u2192 x \u2261 y\n    transitive : \u2200 {x y z} \u2192 x \u2264 y \u2192 y \u2264 z \u2192 x \u2264 z\n\n  \u2261-reflexive : \u2200 {x} \u2192 x \u2261 x\n  \u2261-reflexive = antisym reflexive reflexive\n    \n\u2115 : Order Nat.\u2115\n\u2115 = mk Nat._\u2264_ \u2261._\u2261_ Nat.\u2115\u2264.refl Nat.\u2115\u2264.antisym Nat.\u2115\u2264.trans\n\nfunOrder : \u2200 {A B : Set} \u2192 Order B \u2192 Order (A \u2192 B)\nfunOrder or = mk (\u03bb f g  \u2192 \u2200 x \u2192 f x \u2264 g x) (\u03bb f g \u2192 \u2200 x \u2192 f x \u2261 g x)\n              (\u03bb x\u2081 \u2192 reflexive)\n              (\u03bb f g x \u2192 antisym (f x) (g x)) \n              (\u03bb f g x  \u2192 transitive (f x) (g x))\n  where open Order or\n\n_o\u2192_ : \u2200 A {B} \u2192 Order B \u2192 Order (A \u2192 B)\nA o\u2192 ob = funOrder {A} ob\n  \nrecord _\u2192m_ {A B}(oa : Order A)(ob : Order B) : Set where\n  constructor mk\n  field\n    _$_ : A \u2192 B\n    mon : \u2200 {x y} \u2192 Order._\u2264_ oa x y \u2192 Order._\u2264_ ob (_$_ x) (_$_ y)\nopen _\u2192m_ using (_$_ ; mon)\n\n_o\u2192m_ : \u2200 {A B}(oa : Order A)(ob : Order B) \u2192 Order (oa \u2192m ob)\n_o\u2192m_ {A} oa ob = mk (\u03bb f g \u2192 _$_ f \u2264 _$_ g)\n                     (\u03bb f g \u2192 _$_ f \u2261 _$_ g)\n                     reflexive antisym transitive\n  where open Order (A o\u2192 ob)\n\n_\u2218m_ : \u2200 {A B C}{oa : Order A}{ob : Order B}{oc : Order C}\n     \u2192 (ob \u2192m oc) \u2192 (oa \u2192m ob) \u2192 (oa \u2192m oc)\nf \u2218m g = mk (\u03bb x \u2192 f $ (g $ x)) (\u03bb x \u2192 mon f (mon g x))\n\n_$m_ : \u2200 {A B C}{oa : Order A}{oc : Order C} \u2192 oa \u2192m (funOrder {B} oc)\n     \u2192 B \u2192 oa \u2192m oc\n_$m_ {oa = oa}{oc} f b = mk (\u03bb a \u2192 (f $ a) b) (\u03bb x\u2264y \u2192 mon f x\u2264y b )\n\nUB : \u2200 {A} \u2192 Order A \u2192 A \u2192 Set\nUB or a = \u2200 x \u2192 x \u2264 a\n  where open Order or\n\nrecord cpo {D} (o : Order D) : Set where\n  constructor mk\n  open Order o\n\n  field\n    d0 : D\n    lub : (f : \u2115 \u2192m o) \u2192 D\n\n  -- proofs\n  field\n    Dbot : \u2200 x \u2192 d0 \u2264 x\n    lubUB : \u2200 (f : \u2115 \u2192m o) n \u2192 (f $ n) \u2264 lub f\n    lubLUB : \u2200 (f : \u2115 \u2192m o) x \u2192 (\u2200 n \u2192 (f $ n) \u2264 x) \u2192 lub f \u2264 x\n\nfuncpo : \u2200 {A D}{od : Order D} \u2192 cpo od \u2192 cpo (A o\u2192 od)\nfuncpo cpod = mk (\u03bb x \u2192 d0) (\u03bb f x \u2192 lub (f $m x)) (\u03bb x x\u2081 \u2192 Dbot (x x\u2081)) (\u03bb f n x \u2192 lubUB (f $m x) n)\n  (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (f $m x\u2082) (x x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n  where\n    open cpo cpod\n    open Nat\n    open \u2261 using (_\u2261_)\n\n_c\u2192m_ : \u2200 {A D}(oa : Order A){od : Order D} \u2192 cpo od \u2192 cpo (oa o\u2192m od)\n_c\u2192m_ {A} oa {od} cpod = mk (mk (\u03bb x \u2192 d0) (\u03bb x\u2081 \u2192 Order.reflexive od))\n                        (\u03bb f \u2192 mk (cpo.lub (funcpo {A} cpod)\n                                     (mk (\u03bb n a \u2192 (f $ n) $ a) (mon f)))\n                                  (\u03bb x\u2264y \u2192 lubLUB _ _ (\u03bb n \u2192 transitive (mon (f $ n) x\u2264y) (lubUB _ n)))) \n                        (\u03bb x x\u2081 \u2192 Dbot (x $ x\u2081))\n                        (\u03bb f n x \u2192 lubUB (mk (\u03bb m \u2192 (f $ m) $ x) (\u03bb x\u2082 \u2192 mon f x\u2082 x)) n)\n                        (\u03bb f x x\u2081 x\u2082 \u2192 lubLUB (mk (\u03bb n \u2192 _$_ (f $ n)) (mon f) $m x\u2082) (x $ x\u2082) (\u03bb n \u2192 x\u2081 n x\u2082))\n  where\n    open cpo cpod\n    open Order od renaming (_\u2264_ to _\u2264\u1d48_)\n    open Order oa using () renaming (_\u2264_ to _\u2264\u1d43_)\n\nrecord continous {A B}{oa : Order A}{ob : Order B}\n                 (c1 : cpo oa)(c2 : cpo ob)(f : oa \u2192m ob) : Set where\n  constructor mk\n  open cpo c1 renaming (lub to luba ; lubUB to lubUBa ; lubLUB to lubLUBa)\n  open cpo c2\n  open Order ob\n  field\n    contIntro : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2264 lub (f \u2218m h)\n\n  contIntro' : \u2200 (h : \u2115 \u2192m oa) \u2192 lub (f \u2218m h) \u2264 f $ luba h\n  contIntro' h = lubLUB (f \u2218m h) (f $ luba h) (\u03bb n \u2192 mon f (lubUBa _ n))\n\n  contEq : \u2200 (h : \u2115 \u2192m oa) \u2192 (f $ (luba h)) \u2261 lub (f \u2218m h)\n  contEq h = antisym (contIntro h) (contIntro' h)\n  \nopen continous\n\npostulate\n  Ur  : Set -- the set [0,1]\n  oUr : Order Ur\n  U   : cpo oUr\n\nrecord distr A : Set where\n  constructor mk\n  field\n    \u03bc : (A o\u2192 oUr) \u2192m oUr\n    muContinous : continous (funcpo U) U \u03bc\n\nopen distr\n\ndistrOrder : \u2200 A \u2192 Order (distr A)\ndistrOrder A = mk (\u03bb f g \u2192 \u03bc f \u2264 \u03bc g) (\u03bb f g \u2192 \u03bc f \u2261 \u03bc g)\n  (\u03bb {x} \u2192 reflexive {\u03bc x}) (\u03bb {x} {y} \u2192 antisym {\u03bc x} {\u03bc y}) (\u03bb {x} {y} {z} \u2192 transitive {\u03bc x} {\u03bc y} {\u03bc z}) \n  where\n    open Order ((A o\u2192 oUr) o\u2192m oUr)\n\ndistrcpo : \u2200 A \u2192 cpo (distrOrder A)\ndistrcpo A = mk (mk d0 (mk (\u03bb h \u2192 lubUB (mk (\u03bb z \u2192 mk (_) (_)) (_)) {!!} {!!})))\n                (\u03bb f \u2192 mk (lub (mk (\u03bb x \u2192 \u03bc (f $ x)) (mon f))) (mk (\u03bb h \u2192 {!!})))\n                {!!}\n                {!!}\n                {!!}\n  where\n     open cpo ((A o\u2192 oUr) c\u2192m U)\n    \nmodule FIXPOINTS {d} {D : Order d} {c : cpo D} where\n  open cpo c\n  open Order D\n\n  iter : (f : D \u2192m D) \u2192 Nat.\u2115 \u2192 d\n  iter f Nat.zero = d0\n  iter f (Nat.suc n) = f $ iter f n\n\n  iter-mon : (f : D \u2192m D) \u2192 \u2115 \u2192m D\n  iter-mon f = mk (iter f) mono\n    where\n      mono : \u2200 {x y} \u2192 x Nat.\u2264 y \u2192 iter f x \u2264 iter f y\n      mono Nat.z\u2264n = Dbot _\n      mono (Nat.s\u2264s prf) = mon f (mono prf)\n\n  fix : (f : D \u2192m D) \u2192 d\n  fix f = lub (iter-mon f)\n\n  fix-le : (F : D \u2192m D) \u2192 fix F \u2264 F $ (fix F)\n  fix-le F = lubLUB _ _ prf\n    where\n      prf : (n : Nat.\u2115) \u2192 iter-mon F $ n \u2264 F $ fix F\n      prf Nat.zero = Dbot (F $ fix F)\n      prf (Nat.suc n) = mon F (lubUB (iter-mon F) n)\n  \nmodule _ where\n\n  open Order oUr\n  open cpo U\n\n  Munit : \u2200 {A} \u2192 A \u2192 distr A\n  Munit x = mk (mk (\u03bb f \u2192 f x) (\u03bb x\u2264y \u2192 x\u2264y x)) (mk (\u03bb h \u2192 reflexive))\n\n  Mlet : \u2200 {A B} \u2192 distr A \u2192 (A \u2192 distr B) \u2192 distr B\n  Mlet m k = mk (mk (\u03bb f \u2192 \u03bc m $ (\u03bb x \u2192 \u03bc (k x) $ f)) (\u03bb x\u2264y \u2192 mon (\u03bc m) (\u03bb x \u2192 mon (\u03bc (k x)) x\u2264y)))\n     (mk (\u03bb h \u2192 transitive (mon (\u03bc m) (\u03bb x \u2192 contIntro (muContinous (k x)) h)) (contIntro (muContinous m) _)))\n\n  Munit-simpl : \u2200 {A}(q : A \u2192 Ur) x \u2192 \u03bc (Munit x) $ q \u2261 q x\n  Munit-simpl q x = \u2261-reflexive\n\n  Mlet-simpl : \u2200 {A B}(m : distr A)(M : A \u2192 distr B)(f : B \u2192 Ur)\n             \u2192 \u03bc (Mlet m M) $ f \u2261 \u03bc m $ (\u03bb x \u2192 \u03bc (M x) $ f) \n  Mlet-simpl m M f = \u2261-reflexive\n\n  Mlet-le-compat : \u2200 {A B} (m1 m2 : distr A)(M1 M2 : A \u2192 distr B)\n    \u2192 Order._\u2264_ (distrOrder A) m1 m2\n    \u2192 Order._\u2264_ (A o\u2192 distrOrder B) M1 M2\n    \u2192 Order._\u2264_ (distrOrder B) (Mlet m1 M1) (Mlet m2 M2)\n  Mlet-le-compat m1 m2 M1 M2 m1\u2264m2 M1\u2264M2 f = transitive (mon (\u03bc m1) (\u03bb x \u2192 M1\u2264M2 x f)) (m1\u2264m2 _)\n\n  -- Monad laws\n\n  Mlet-unit : \u2200 {A B} (x : A)(m : A \u2192 distr B)\n            \u2192 Order._\u2261_ (distrOrder B) (Mlet (Munit x) m) (m x)\n  Mlet-unit x m f = \u2261-reflexive\n\n  Mlet-ext : \u2200 {A}(m : distr A) \u2192 Order._\u2261_ (distrOrder A) (Mlet m Munit) m\n  Mlet-ext m f = \u2261-reflexive\n\n  Mlet-assoc : \u2200 {A B C}(m1 : distr A)(m2 : A \u2192 distr B)(m3 : B \u2192 distr C)\n             \u2192 Order._\u2261_ (distrOrder C)\n                (Mlet (Mlet m1 m2) m3)\n                (Mlet m1 (\u03bb x \u2192 Mlet (m2 x) m3))\n  Mlet-assoc m1 m2 m3 f = \u2261-reflexive\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'alea\/cpo.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b4f467da96d5de9050f58b41ac10fd9a73ac84df","subject":"Version of propositional desc examples with inferred arguments.","message":"Version of propositional desc examples with inferred arguments.\n\nIt is probably not acceptable to include the complexity of implicit\narguments and unification in the core type theory.\nHowever, breaking up the grammar to allow for more bidirectional\nsynthesis is probably okay and can reduce the size of core terms.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/InferredPropositionalDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/InferredPropositionalDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.InferredPropositionalDesc where\n\n----------------------------------------------------------------------\n\nelimEq : {A : Set} (x : A) (P : (y : A) \u2192 x \u2261 y \u2192 Set)\n  \u2192 P x refl\n  \u2192 (y : A) (p : x \u2261 y) \u2192 P y p\nelimEq .x P prefl x refl = prefl\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nCases : {E : Enum} (P : Tag E \u2192 Set) \u2192 Set\nCases {[]} P = \u22a4\nCases {l \u2237 E} P = P here \u00d7 Cases \u03bb t \u2192 P (there t)\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Cases P) (t : Tag E) \u2192 P t\ncase {l \u2237 E} P (c , cs) here = c\ncase {l \u2237 E} P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `End : (i : I) \u2192 Desc I\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n  `RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 ISet I\nEl (`End j) X i = j \u2261 i\nEl (`Rec j D) X i = X j \u00d7 El D X i\nEl (`Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\nEl (`RecFun A B D) X i = ((a : A) \u2192 X (B a)) \u00d7 El D X i\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  con : El D (\u03bc D) i \u2192 \u03bc D i\n\nAll : {I : Set} (D : Desc I) {X : ISet I} (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nAll (`End j) P i q = \u22a4\nAll (`Rec j D) P i (x , xs) = P j x \u00d7 All D P i xs\nAll (`Arg A B) P i (a , b) = All (B a) P i b\nAll (`RecFun A B D) P i (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 All D P i xs\n\ncaseD : (E : Enum) (I : Set) (cs : Cases (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  {I : Set}\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (pcon : (i : I) (xs : El D (\u03bc D) i) \u2192 All D P i xs \u2192 P i (con xs))\n  {i : I}\n  (x : \u03bc D i)\n  \u2192 P i x\n\nhyps :\n  {I : Set}\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El D\u2081 (\u03bc D\u2081) i) \u2192 All D\u2081 P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  {i : I}\n  (xs : El D\u2082 (\u03bc D\u2081) i)\n  \u2192 All D\u2082 P i xs\n\nind D P pcon (con xs) = pcon _ xs (hyps D P pcon D xs)\n\nhyps D P pcon (`End j) q = tt\nhyps D P pcon (`Rec j A) (x , xs) = ind D P pcon x , hyps D P pcon A xs\nhyps D P pcon (`Arg A B) (a , b) = hyps D P pcon (B a) b\nhyps D P pcon (`RecFun A B E) (f , xs) = (\u03bb a \u2192 ind D P pcon (f a)) , hyps D P pcon E xs\n\n----------------------------------------------------------------------\n\nmodule Desugared where\n\n  \u2115T : Enum\n  \u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n  \n  VecT : Enum\n  VecT = \"nil\" \u2237 \"cons\" \u2237 []\n  \n  \u2115C : Tag \u2115T \u2192 Desc \u22a4\n  \u2115C = caseD \u2115T \u22a4\n    ( `End tt\n    , `Rec tt (`End tt)\n    , tt\n    )\n  \n  \u2115D : Desc \u22a4\n  \u2115D = `Arg (Tag \u2115T) \u2115C\n  \n  \u2115 : \u22a4 \u2192 Set\n  \u2115 = \u03bc \u2115D\n  \n  zero : \u2115 tt\n  zero = con (here , refl)\n  \n  suc : \u2115 tt \u2192 \u2115 tt\n  suc n = con (there here , n , refl)\n  \n  VecC : (A : Set) \u2192 Tag VecT \u2192 Desc (\u2115 tt)\n  VecC A = caseD VecT (\u2115 tt)\n    ( `End zero\n    , `Arg (\u2115 tt) (\u03bb n \u2192 `Arg A \u03bb _ \u2192 `Rec n (`End (suc n)))\n    , tt\n    )\n  \n  VecD : (A : Set) \u2192 Desc (\u2115 tt)\n  VecD A = `Arg (Tag VecT) (VecC A)\n  \n  Vec : (A : Set) (n : \u2115 tt) \u2192 Set\n  Vec A n = \u03bc (VecD A) n\n  \n  nil : (A : Set) \u2192 Vec A zero\n  nil A = con (here , refl)\n  \n  cons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\n  cons A n x xs = con (there here , n , x , xs , refl)\n \n----------------------------------------------------------------------\n\n  module Induction where\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = ind \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case\n        (\u03bb t \u2192 (c : El (\u2115C t) \u2115 u)\n               (ih : All \u2115D (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 n)\n        , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = ind \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb u t,c \u2192 case\n        (\u03bb t \u2192 (c : El (\u2115C t) \u2115 u)\n               (ih : All \u2115D (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c))\n               \u2192 \u2115 u \u2192 \u2115 u\n        )\n        ( (\u03bb q ih n \u2192 zero)\n        , (\u03bb m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n    append A m = ind (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb m t,c \u2192 case\n        (\u03bb t \u2192 (c : El (VecC A t) (Vec A) m)\n               (ih : All (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m (t , c))\n               (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n        )\n        ( (\u03bb q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n        , (\u03bb m',x,xs,q ih,tt n ys \u2192\n            let m' = proj\u2081 m',x,xs,q\n                x = proj\u2081 (proj\u2082 m',x,xs,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n    \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m n = ind (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n      (\u03bb n t,c \u2192 case\n        (\u03bb t \u2192 (c : El (VecC (Vec A m) t) (Vec (Vec A m)) n)\n               (ih : All (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n (t , c))\n               \u2192 Vec A (mult n m)\n        )\n        ( (\u03bb q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n        , (\u03bb n',xs,xss,q ih,tt \u2192\n            let n' = proj\u2081 n',xs,xss,q\n                xs = proj\u2081 (proj\u2082 n',xs,xss,q)\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',xs,xss,q))\n                ih = proj\u2081 ih,tt\n            in\n            subst (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n  module Eliminator where\n\n    elim\u2115 : (P : (\u2115 tt) \u2192 Set)\n      (pzero : P zero)\n      (psuc : (m : \u2115 tt) \u2192 P m \u2192 P (suc m))\n      (n : \u2115 tt)\n      \u2192 P n\n    elim\u2115 P pzero psuc = ind \u2115D (\u03bb u n \u2192 P n)\n      (\u03bb u t,c \u2192 case\n        (\u03bb t \u2192 (c : El (\u2115C t) \u2115 u)\n               (ih : All \u2115D (\u03bb u n \u2192 P n) u (t , c))\n               \u2192 P (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq tt (\u03bb u q \u2192 P (con (here , q)))\n              pzero\n              u q\n          )\n        , (\u03bb n,q ih,tt \u2192\n            elimEq tt (\u03bb u q \u2192 P (con (there here , proj\u2081 n,q , q)))\n              (psuc (proj\u2081 n,q) (proj\u2081 ih,tt))\n              u (proj\u2082 n,q)\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n    elimVec : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n      (pnil : P zero (nil A))\n      (pcons : (n : \u2115 tt) (a : A) (xs : Vec A n) \u2192 P n xs \u2192 P (suc n) (cons A n a xs))\n      (n : \u2115 tt)\n      (xs : Vec A n)\n      \u2192 P n xs\n    elimVec A P pnil pcons n = ind (VecD A) (\u03bb n xs \u2192 P n xs)\n      (\u03bb n t,c \u2192 case\n        (\u03bb t \u2192 (c : El (VecC A t) (Vec A) n)\n               (ih : All (VecD A) (\u03bb n xs \u2192 P n xs) n (t , c))\n               \u2192 P n (con (t , c))\n        )\n        ( (\u03bb q ih \u2192\n            elimEq zero (\u03bb n q \u2192 P n (con (here , q)))\n              pnil\n              n q\n          )\n        , (\u03bb n',x,xs,q ih,tt \u2192\n            let n' = proj\u2081 n',x,xs,q\n                x = proj\u2081 (proj\u2082 n',x,xs,q)\n                xs = proj\u2081 (proj\u2082 (proj\u2082 n',x,xs,q))\n                q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n                ih = proj\u2081 ih,tt\n            in\n            elimEq (suc n') (\u03bb n q \u2192 P n (con (there here , n' , x , xs , q)))\n              (pcons n' x xs ih )\n              n q\n          )\n        , tt\n        )\n        (proj\u2081 t,c)\n        (proj\u2082 t,c)\n      )\n\n----------------------------------------------------------------------\n\n    add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    add = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 n)\n      (\u03bb m ih n \u2192 suc (ih n))\n  \n    mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n    mult = elim\u2115 (\u03bb _ \u2192 \u2115 tt \u2192 \u2115 tt)\n      (\u03bb n \u2192 zero)\n      (\u03bb m ih n \u2192 add n (ih n))\n  \n    append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n    append A = elimVec A (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n      (\u03bb n ys \u2192 ys)\n      (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n  \n    concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n    concat A m = elimVec (Vec A m) (\u03bb n xss \u2192 Vec A (mult n m))\n      (nil A)\n      (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n    \n----------------------------------------------------------------------\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formalization\/agda\/Spire\/Examples\/InferredPropositionalDesc.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"71d92c377670506164bb8938afb56549153ead77","subject":"redid comm, even exists proof","message":"redid comm, even exists proof\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"src\/extra\/Even.agda","new_file":"src\/extra\/Even.agda","new_contents":"import Relation.Binary.PropositionalEquality as Eq\nopen Eq using (_\u2261_; refl; sym; trans; cong; cong-app)\nopen Eq.\u2261-Reasoning\nopen import Data.Nat using (\u2115; zero; suc; _+_; _*_)\nopen import Data.Product using (\u2203; _,_)\n\n+-identity : \u2200 (m : \u2115) \u2192 m + zero \u2261 m\n+-identity zero =\n  begin\n    zero + zero\n  \u2261\u27e8\u27e9\n    zero\n  \u220e\n+-identity (suc m) =\n  begin\n    suc m + zero\n  \u2261\u27e8\u27e9\n    suc (m + zero)\n  \u2261\u27e8 cong suc (+-identity m) \u27e9\n    suc m\n  \u220e\n\n+-suc : \u2200 (m n : \u2115) \u2192 m + suc n \u2261 suc (m + n)\n+-suc zero n =\n  begin\n    zero + suc n\n  \u2261\u27e8\u27e9\n    suc n\n  \u2261\u27e8\u27e9\n    suc (zero + n)\n  \u220e\n+-suc (suc m) n =\n  begin\n    suc m + suc n\n  \u2261\u27e8\u27e9\n    suc (m + suc n)\n  \u2261\u27e8 cong suc (+-suc m n) \u27e9\n    suc (suc (m + n))\n  \u2261\u27e8\u27e9\n    suc (suc m + n)\n  \u220e\n\n+-comm : \u2200 (m n : \u2115) \u2192 m + n \u2261 n + m\n+-comm m zero =\n  begin\n    m + zero\n  \u2261\u27e8 +-identity m \u27e9\n    m\n  \u2261\u27e8\u27e9\n    zero + m\n  \u220e\n+-comm m (suc n) =\n  begin\n    m + suc n\n  \u2261\u27e8 +-suc m n \u27e9\n    suc (m + n)\n  \u2261\u27e8 cong suc (+-comm m n) \u27e9\n    suc (n + m)\n  \u2261\u27e8\u27e9\n    suc n + m\n  \u220e\n\ndata even : \u2115 \u2192 Set where\n  ev0 : even zero\n  ev+2 : \u2200 {n : \u2115} \u2192 even n \u2192 even (suc (suc n))\n\nlemma : \u2200 (m : \u2115) \u2192 2 * suc m \u2261 suc (suc (2 * m))\nlemma m =\n  begin\n    2 * suc m\n  \u2261\u27e8\u27e9\n    suc m + (suc m + zero)\n  \u2261\u27e8\u27e9\n    suc (m + (suc (m + zero)))\n  \u2261\u27e8 cong suc (+-suc m (m + zero)) \u27e9\n    suc (suc (m + (m + zero)))\n  \u2261\u27e8\u27e9\n    suc (suc (2 * m))\n  \u220e\n  \nev-ex : \u2200 {n : \u2115} \u2192 even n \u2192 \u2203(\u03bb (m : \u2115) \u2192 2 * m \u2261 n)\nev-ex ev0 =  (0 , refl)\nev-ex (ev+2 ev) with ev-ex ev\n... | (m , refl) = (suc m , lemma m)\n\nex-ev : \u2200 {n : \u2115} \u2192 \u2203(\u03bb (m : \u2115) \u2192 2 * m \u2261 n) \u2192 even n\nex-ev (zero , refl) = ev0\nex-ev (suc m , refl) rewrite lemma m = ev+2 (ex-ev (m , refl))\n\n\n-- I can't see how to avoid using rewrite in the second proof,\n-- or how to use rewrite in the first proof!\n\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/extra\/Even.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"8860928197baec5438ec687a44d5f26d0eea9d6f","subject":"Added missing file.","message":"Added missing file.\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/fot\/FOTC\/Program\/McCarthy91\/WF-Relation\/Induction\/Acc\/WF-ATP.agda","new_file":"src\/fot\/FOTC\/Program\/McCarthy91\/WF-Relation\/Induction\/Acc\/WF-ATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Well-founded induction on the relation _\u226a_\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- N.B This module does not contain combined proofs, but it imports\n-- modules which contain combined proofs.\n\nmodule FOTC.Program.McCarthy91.WF-Relation.Induction.Acc.WF-ATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\nimport FOTC.Data.Nat.Induction.Acc.WF-ATP\nopen FOTC.Data.Nat.Induction.Acc.WF-ATP.WF-<\n\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesATP\nopen import FOTC.Data.Nat.UnaryNumbers\nopen import FOTC.Induction.WF\nopen import FOTC.Program.McCarthy91.WF-Relation\nopen import FOTC.Program.McCarthy91.WF-Relation.PropertiesATP\n\n-- Parametrized modules\nopen module InvImg =\n  FOTC.Induction.WF.InverseImage {N} {N} {_<_} \u226a-fn-N\n\n------------------------------------------------------------------------------\n-- The relation _\u226a_ is well-founded (using the inverse image combinator).\nwf-\u226a : WellFounded _\u226a_\nwf-\u226a Nn = wellFounded wf-< Nn\n\n-- Well-founded induction on the relation _\u226a_.\nwfInd-\u226a : (A : D \u2192 Set) \u2192\n          (\u2200 {n} \u2192 N n \u2192 (\u2200 {m} \u2192 N m \u2192 m \u226a n \u2192 A m) \u2192 A n) \u2192\n          \u2200 {n} \u2192 N n \u2192 A n\nwfInd-\u226a A = WellFoundedInduction wf-\u226a\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/fot\/FOTC\/Program\/McCarthy91\/WF-Relation\/Induction\/Acc\/WF-ATP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"105df8ae981ceacb5b579cb038e4a8cc1098b446","subject":"refactor","message":"refactor\n","repos":"zeitraffer\/agda-cats","old_file":"src\/CategoryTheory\/Initial\/Trade-off\/FiniteComplete-Module.agda","new_file":"src\/CategoryTheory\/Initial\/Trade-off\/FiniteComplete-Module.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"b29e74f8334531f4ab67ec79f9fadf96ea6c4d1d","subject":"Vec: drop\u2032, take\u2032, \u229b","message":"Vec: drop\u2032, take\u2032, \u229b\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public hiding (_\u229b_; zipWith; zip; map)\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product hiding (map; zip)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nmodule waiting-for-a-fix-in-the-stdlib where\n\n    infixl 4 _\u229b_\n\n    _\u229b_ : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec (A \u2192 B) n \u2192 Vec A n \u2192 Vec B n\n    _\u229b_ {n = zero}  fs xs = []\n    _\u229b_ {n = suc n} fs xs = head fs (head xs) \u2237 (tail fs \u229b tail xs)\n\n    map : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          (A \u2192 B) \u2192 Vec A n \u2192 Vec B n\n    map f xs = replicate f \u229b xs\n\n    zipWith : \u2200 {a b c n} {A : Set a} {B : Set b} {C : Set c} \u2192\n              (A \u2192 B \u2192 C) \u2192 Vec A n \u2192 Vec B n \u2192 Vec C n\n    zipWith _\u2295_ xs ys = replicate _\u2295_ \u229b xs \u229b ys\n\n    zip : \u2200 {a b n} {A : Set a} {B : Set b} \u2192\n          Vec A n \u2192 Vec B n \u2192 Vec (A \u00d7 B) n\n    zip = zipWith _,_\n\nopen waiting-for-a-fix-in-the-stdlib public\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\ndrop\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A n\ndrop\u2032 zero    = id\ndrop\u2032 (suc m) = drop\u2032 m \u2218 tail\n\ndrop\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 drop\u2032 {A = A} m {n} \u2257 drop m {n}\ndrop\u2032-spec zero xs = refl\ndrop\u2032-spec (suc m) (x \u2237 xs) rewrite drop\u2032-spec m xs | drop-\u2237 m x xs = refl\n\ntake\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m\ntake\u2032 zero    _  = []\ntake\u2032 (suc m) xs = head xs \u2237 take\u2032 m (tail xs)\n\ntake\u2032-spec : \u2200 {a} {A : Set a} m {n} \u2192 take\u2032 {A = A} m {n} \u2257 take m {n}\ntake\u2032-spec zero xs = refl\ntake\u2032-spec (suc m) (x \u2237 xs) rewrite take\u2032-spec m xs | take-\u2237 m x xs = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","old_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero; _+_)\nopen import Data.Fin renaming (_+_ to _+\u1da0_)\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Data.Product hiding (map)\nopen import Function\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nvuncurry : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 Vec A n \u2192 B) \u2192 Vec A (1 + n) \u2192 B\nvuncurry f (x \u2237 xs) = f x xs\n\ncount\u1da0 : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount\u1da0 pred = foldr (Fin \u2218 suc) (\u03bb x \u2192 if pred x then suc else inject\u2081) zero\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 \u2115\ncount pred = to\u2115 \u2218 count\u1da0 pred\n\ncount-\u2218 : \u2200 {n a b} {A : Set a} {B : Set b} (f : A \u2192 B) (pred : B \u2192 Bool) \u2192\n            count {n} (pred \u2218 f) \u2257 count pred \u2218 map f\ncount-\u2218 f pred [] = refl\ncount-\u2218 f pred (x \u2237 xs) with pred (f x)\n... | true rewrite count-\u2218 f pred xs = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (map f xs))\n                  | inject\u2081-lemma (count\u1da0 (pred \u2218 f) xs)\n                  | count-\u2218 f pred xs = refl\n\ncount-++ : \u2200 {m n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A m) (ys : Vec A n)\n            \u2192 count pred (xs ++ ys) \u2261 count pred xs + count pred ys\ncount-++ pred [] ys = refl\ncount-++ pred (x \u2237 xs) ys with pred x\n... | true  rewrite count-++ pred xs ys = refl\n... | false rewrite inject\u2081-lemma (count\u1da0 pred (xs ++ ys))\n                  | inject\u2081-lemma (count\u1da0 pred xs) | count-++ pred xs ys = refl\n\next-count\u1da0 : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count\u1da0 f xs \u2261 count\u1da0 g xs\next-count\u1da0 f\u2257g [] = refl\next-count\u1da0 f\u2257g (x \u2237 xs) rewrite ext-count\u1da0 f\u2257g xs | f\u2257g x = refl\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (count pred xs)\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count\u1da0 pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n\nuncons : \u2200 {n a} {A : Set a} \u2192 Vec A (1 + n) \u2192 (A \u00d7 Vec A n)\nuncons (x \u2237 xs) = x , xs\n\nsplitAt\u2032 : \u2200 {a} {A : Set a} m {n} \u2192 Vec A (m + n) \u2192 Vec A m \u00d7 Vec A n\nsplitAt\u2032 m xs = case splitAt m xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n++-decomp : \u2200 {m n a} {A : Set a} {xs zs : Vec A m} {ys ts : Vec A n}\n             \u2192 (xs ++ ys) \u2261 (zs ++ ts) \u2192 (xs \u2261 zs \u00d7 ys \u2261 ts)\n++-decomp {zero} {xs = []} {[]} p = refl , p\n++-decomp {suc m} {xs = x \u2237 xs} {z \u2237 zs} eq with ++-decomp {m} {xs = xs} {zs} (cong tail eq)\n... | (q\u2081 , q\u2082) = (cong\u2082 _\u2237_ (cong head eq) q\u2081) , q\u2082\n\n{-\nopen import Data.Vec.Equality\n\nmodule Here {a} {A : Set a} where\n  open Equality (\u2261.setoid A)\n  \u2248-splitAt : \u2200 {m n} {xs zs : Vec A m} {ys ts : Vec A n}\n               \u2192 (xs ++ ys) \u2248 (zs ++ ts) \u2192 (xs \u2248 zs \u00d7 ys \u2248 ts)\n  \u2248-splitAt {zero} {xs = []} {[]} p = []-cong , p\n  \u2248-splitAt {suc m} {xs = x \u2237 xs} {z \u2237 zs} (x\u00b9\u2248x\u00b2 \u2237-cong p) with \u2248-splitAt {m} {xs = xs} {zs} p\n  ... | (q\u2081 , q\u2082) = x\u00b9\u2248x\u00b2 \u2237-cong q\u2081 , q\u2082\n-}\n\n++-inj\u2081 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 xs \u2261 ys\n++-inj\u2081 xs ys eq = proj\u2081 (++-decomp eq)\n\n++-inj\u2082 : \u2200 {m n} {a} {A : Set a} (xs ys : Vec A m) {zs ts : Vec A n} \u2192 xs ++ zs \u2261 ys ++ ts \u2192 zs \u2261 ts\n++-inj\u2082 xs ys eq = proj\u2082 (++-decomp {xs = xs} {ys} eq)\n\ntake-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\ntake-\u2237 n x xs with splitAt n xs\ntake-\u2237 _ _ ._ | _ , _ , refl = refl\n\ndrop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\ndrop-\u2237 n x xs with splitAt n xs\ndrop-\u2237 _ _ ._ | _ , _ , refl = refl\n\ntake-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 take m (xs ++ ys) \u2261 xs\ntake-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ntake-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2081 xs\u2081 xs eq)\n\ndrop-++ : \u2200 m {n} {a} {A : Set a} (xs : Vec A m) (ys : Vec A n) \u2192 drop m (xs ++ ys) \u2261 ys\ndrop-++ m xs ys with xs ++ ys | inspect (_++_ xs) ys\n... | zs | eq with splitAt m zs\ndrop-++ m xs\u2081 ys\u2081 | .(xs ++ ys) | Reveal_is_.[_] eq | xs , ys , refl = sym (++-inj\u2082 xs\u2081 xs eq)\n\ntake-drop-lem : \u2200 m {n} {a} {A : Set a} (xs : Vec A (m + n)) \u2192 take m xs ++ drop m xs \u2261 xs\ntake-drop-lem m xs with splitAt m xs\ntake-drop-lem m .(ys ++ zs) | ys , zs , refl = refl\n\ntake-them-all : \u2200 n {a} {A : Set a} (xs : Vec A (n + 0)) \u2192 take n xs ++ [] \u2261 xs\ntake-them-all n xs with splitAt n xs\ntake-them-all n .(ys ++ []) | ys , [] , refl = refl\n\nrewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\nrewire f v = tabulate (flip lookup v \u2218 f)\n\nRewireTbl : (i o : \u2115) \u2192 Set\nRewireTbl i o = Vec (Fin i) o\n\nrewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\nrewireTbl tbl v = map (flip lookup v) tbl\n\non\u1d62 : \u2200 {a} {A : Set a} (f : A \u2192 A) {n} (i : Fin n) \u2192 Vec A n \u2192 Vec A n\non\u1d62 f zero    (x \u2237 xs) = f x \u2237 xs\non\u1d62 f (suc i) (x \u2237 xs) = x \u2237 on\u1d62 f i xs\n\n-- Exchange elements at positions 0 and 1 of a given vector\n-- (this only apply if the vector is long enough).\n0\u21941 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n0\u21941 (x\u2080 \u2237 x\u2081 \u2237 xs) = x\u2081 \u2237 x\u2080 \u2237 xs\n0\u21941 xs = xs\n\nmap-tail : \u2200 {m n a} {A : Set a} \u2192 (Vec A m \u2192 Vec A n) \u2192 Vec A (suc m) \u2192 Vec A (suc n)\nmap-tail f (x \u2237 xs) = x \u2237 f xs\n\nmap-tail-id : \u2200 {n a} {A : Set a} \u2192 map-tail id \u2257 id {A = Vec A (suc n)}\nmap-tail-id (x \u2237 xs) = \u2261.refl\n\n-- \u27e8 i \u2194+1\u27e9: Exchange elements at position i and i + 1.\n\u27e8_\u2194+1\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194+1\u27e9 = 0\u21941\n\u27e8 suc i \u2194+1\u27e9 = map-tail \u27e8 i \u2194+1\u27e9\n\n-- rot-up-to i (x\u2080 \u2237 x\u2081 \u2237 x\u2082 \u2237 \u2026 \u2237 x\u1d62 \u2237 xs)\n--           \u2261 (x\u2081 \u2237 x\u2082 \u2237 x\u2083 \u2237 \u2026 \u2237 x\u2080 \u2237 xs)\nrot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot-up-to zero    = id\nrot-up-to (suc i) = map-tail (rot-up-to i) \u2218 0\u21941\n\n-- Inverse of rot-up-to\nrot\u207b\u00b9-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\nrot\u207b\u00b9-up-to zero    = id\nrot\u207b\u00b9-up-to (suc i) = 0\u21941 \u2218 map-tail (rot\u207b\u00b9-up-to i)\n\nrot\u207b\u00b9-up-to\u2218rot-up-to : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 rot\u207b\u00b9-up-to i \u2218 rot-up-to i \u2257 id {a} {Vec A n}\nrot\u207b\u00b9-up-to\u2218rot-up-to zero            _ = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i) {A = A} (x\u2080 \u2237 []) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i {A = A} [] = \u2261.refl\nrot\u207b\u00b9-up-to\u2218rot-up-to (suc i)         (x\u2080 \u2237 x\u2081 \u2237 xs) rewrite rot\u207b\u00b9-up-to\u2218rot-up-to i (x\u2080 \u2237 xs) = \u2261.refl\n\n-- \u27e80\u2194 i \u27e9: Exchange elements at position 0 and i.\n\u27e80\u2194_\u27e9 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194 zero  \u27e9 = id\n\u27e80\u2194 suc i \u27e9 = rot\u207b\u00b9-up-to (inject\u2081 i) \u2218 rot-up-to (suc i)\n\n\u27e80\u2194zero\u27e9 : \u2200 {n a} {A : Set a} \u2192 \u27e80\u2194 zero \u27e9 \u2257 id {A = Vec A (suc n)}\n\u27e80\u2194zero\u27e9 _ = \u2261.refl\n\n-- \u27e8 i \u2194 j \u27e9: Exchange elements at position i and j.\n\u27e8_\u2194_\u27e9 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 i \u2194 j \u27e9 = \u27e80\u2194 i \u27e9 \u2218 \u27e80\u2194 j \u27e9 \u2218 \u27e80\u2194 i \u27e9\n\n\u27e80\u2194\u27e9-spec : \u2200 {n a} {A : Set a} (i : Fin (suc n)) \u2192 \u27e80\u2194 i \u27e9 \u2257 \u27e8 zero \u2194 i \u27e9 {A = A}\n\u27e80\u2194\u27e9-spec _ _ = \u2261.refl\n\n\u27e8_\u2194_\u27e9\u2032 : \u2200 {n} (i j : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e8 zero  \u2194 j     \u27e9\u2032 = \u27e80\u2194 j \u27e9\n\u27e8 i     \u2194 zero  \u27e9\u2032 = \u27e80\u2194 i \u27e9\n\u27e8 suc i \u2194 suc j \u27e9\u2032 = map-tail \u27e8 i \u2194 j \u27e9\u2032\n\n\u27e8\u2194\u27e9\u2032-comm : \u2200 {n a} {A : Set a} (i j : Fin n) \u2192 \u27e8 i \u2194 j \u27e9\u2032 \u2257 \u27e8 j \u2194 i \u27e9\u2032 {A = A}\n\u27e8\u2194\u27e9\u2032-comm zero    zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm zero    (suc _) _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc _) zero    _ = \u2261.refl\n\u27e8\u2194\u27e9\u2032-comm (suc i) (suc j) (x \u2237 xs) rewrite \u27e8\u2194\u27e9\u2032-comm i j xs = \u2261.refl\n\n\u27e8\u2194+1\u27e9-spec : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 \u27e8 inject\u2081 i \u2194+1\u27e9 \u2257 \u27e8 inject\u2081 i \u2194 suc i \u27e9\u2032 {A = A}\n\u27e8\u2194+1\u27e9-spec zero    xs       rewrite map-tail-id (0\u21941 xs) = \u2261.refl\n\u27e8\u2194+1\u27e9-spec (suc i) (x \u2237 xs) rewrite \u27e8\u2194+1\u27e9-spec i xs = \u2261.refl\n\n\u27e80\u2194_\u27e9\u2032 : \u2200 {n} (i : Fin n) {a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u27e80\u2194_\u27e9\u2032 {zero}  i xs = xs\n\u27e80\u2194_\u27e9\u2032 {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n","returncode":0,"stderr":"","license":"bsd-3-clause","lang":"Agda"}
{"commit":"76d991953cbabb55b45e63dbd70ca8232b1f1e53","subject":"Added note on proof terms.","message":"Added note on proof terms.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/ProofTerm.agda","new_file":"notes\/ProofTerm.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Discussion about the inductive approach for representing\n-- first-order theories.\n\n-- Peter: If you take away the proof objects (as you do when you go to\n-- predicate logic) the K axiom doesn't give you any extra power.\n\nmodule ProofTerm where\n\ninfix 7  _\u2261_\ninfixr 7 _,_\n\npostulate D : Set\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\nfoo : (\u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 x \u2261 y) \u2192 (\u2203 \u03bb x \u2192 \u2203 \u03bb y \u2192 y \u2261 x)\nfoo (x , .x , refl) = x , x , refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/ProofTerm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"66fb5875d3959855b604ac040917f9cff4462844","subject":"Minimal example demonstrating sized types bug in Agda","message":"Minimal example demonstrating sized types bug in Agda\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/SizedMerge.agda","new_file":"agda\/SizedMerge.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"d2cfd1914bee83c8735e3d9b6a2e436c459c5003","subject":"First use of CTT ('Function Extensionality')","message":"First use of CTT ('Function Extensionality')\n","repos":"xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow,xekoukou\/sparrow","old_file":"agda\/CTT.agda","new_file":"agda\/CTT.agda","new_contents":"module CTT where\n\nopen import PathPrelude\nopen import Data.Empty\nopen import Relation.Nullary\n\n\n\n\u00acfun-eq : \u2200{\u2113} \u2192 {A : Set \u2113} \u2192 (f g : \u00ac A) \u2192 f \u2261 g\n\u00acfun-eq f g = funExt (\u00acfun-eq' f g) where\n  \u00acfun-eq' : \u2200{\u2113} \u2192 {A : Set \u2113} \u2192 (f g : \u00ac A) \u2192 (x : A) \u2192 f x \u2261 g x\n  \u00acfun-eq' f g x = \u22a5-elim (f x)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/CTT.agda' did not match any file(s) known to git\n","license":"mpl-2.0","lang":"Agda"}
{"commit":"33e6a45b5923007052c40b6e879a0ba7baf1b36f","subject":"Start labelled descriptions examples, and use eliminators exclusively.","message":"Start labelled descriptions examples, and use eliminators exclusively.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/LabelledDesc.agda","new_file":"formalization\/agda\/Spire\/Examples\/LabelledDesc.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nmodule Spire.Examples.LabelledDesc where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\ndata Cases2 (A : Set) : Enum \u2192 Set where -- Vec, Enum is \u2115, Tag is Fin\n  [] : Cases2 A []\n  _\u2237_ : \u2200{l E} \u2192 A \u2192 Cases2 A E \u2192 Cases2 A (l \u2237 E)\n\n-- lookup\ncase2 : (E : Enum) (A : Set) \u2192 Cases2 A E \u2192 Tag E \u2192 A\ncase2 (l \u2237 E) A (c \u2237 cs) here = c\ncase2 (l \u2237 E) A (c \u2237 cs) (there t) = case2 E A cs t\n\nCases : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nCases [] P = \u22a4\nCases (l \u2237 E) P = P here \u00d7 Cases E \u03bb t \u2192 P (there t)\n\ncase : (E : Enum) (P : Tag E \u2192 Set) (cs : Cases E P) (t : Tag E) \u2192 P t\ncase (l \u2237 E) P (c , cs) here = c\ncase (l \u2237 E) P (c , cs) (there t) = case E (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  `Rec : (i : I) (D : Desc I) \u2192 Desc I\n  `End : (i : I) \u2192 Desc I\n  `\u03a3 `\u03a0 : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : (I : Set) (D : Desc I) (X : ISet I) \u2192 ISet I\nEl I (`End j)   X i = i \u2261 j\nEl I (`Rec j D) X i = X j \u00d7 El I D X i\nEl I (`\u03a3 A B)   X i = \u03a3 A (\u03bb a \u2192 El I (B a) X i)\nEl I (`\u03a0 A B)   X i = (a : A) \u2192 El I (B a) X i\n\ndata \u03bc (I : Set) (D : Desc I) (i : I) : Set where\n  con : El I D (\u03bc I D) i \u2192 \u03bc I D i\n\nAll : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El I D X i) \u2192 Set\nAll I (`Rec j D) X P i (x , xs) = P j x \u00d7 All I D X P i xs\nAll I (`End j) X P i q = \u22a4\nAll I (`\u03a3 A B) X P i (a , b) = All I (B a) X P i b\nAll I (`\u03a0 A B) X P i f = (a : A) \u2192 All I (B a) X P i (f a)\n\ncaseD : (E : Enum) (I : Set) (cs : Cases E (\u03bb _ \u2192 Desc I)) (t : Tag E) \u2192 Desc I\ncaseD E I cs t = case E (\u03bb _ \u2192 Desc I) cs t\n\n----------------------------------------------------------------------\n\nind :\n  (I : Set)\n  (D : Desc I)\n  (P : (i : I) \u2192 \u03bc I D i \u2192 Set)\n  (pcon : (i : I) (xs : El I D (\u03bc I D) i) \u2192 All I D (\u03bc I D) P i xs \u2192 P i (con xs))\n  (i : I)\n  (x : \u03bc I D i)\n  \u2192 P i x\n\nhyps :\n  (I : Set)\n  (D\u2081 : Desc I)\n  (P : (i : I) \u2192 \u03bc I D\u2081 i \u2192 Set)\n  (pcon : (i : I) (xs : El I D\u2081 (\u03bc I D\u2081) i) \u2192 All I D\u2081 (\u03bc I D\u2081) P i xs \u2192 P i (con xs))\n  (D\u2082 : Desc I)\n  (i : I)\n  (xs : El I D\u2082 (\u03bc I D\u2081) i)\n  \u2192 All I D\u2082 (\u03bc I D\u2081) P i xs\n\nind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)\n\nhyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs\nhyps I D P pcon (`End j) i q = tt\nhyps I D P pcon (`\u03a3 A B) i (a , b) = hyps I D P pcon (B a) i b\nhyps I D P pcon (`\u03a0 A B) i f = \u03bb a \u2192 hyps I D P pcon (B a) i (f a)\n\n----------------------------------------------------------------------\n\n\u2115T : Enum\n\u2115T = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecT : Enum\nVecT = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115C : Tag \u2115T \u2192 Desc \u22a4\n\u2115C = caseD \u2115T \u22a4\n  ( `End tt\n  , `Rec tt (`End tt)\n  , tt\n  )\n\n\u2115D : Desc \u22a4\n\u2115D = `\u03a3 (Tag \u2115T) \u2115C\n\n\u2115D2 : Desc \u22a4\n\u2115D2 = `\u03a3 (Tag \u2115T) (case2 \u2115T (Desc \u22a4)\n  ( `End tt\n  \u2237 `Rec tt (`End tt)\n  \u2237 []\n  ))\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u22a4 \u2115D\n\n-- zero : \u2115 tt\npattern zero = con (here , refl)\n\n-- suc : \u2115 tt \u2192 \u2115 tt\npattern suc n = con (there here , n , refl)\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = `\u03a3 (Tag VecT) (caseD VecT (\u2115 tt)\n  ( `End zero\n  , `\u03a3 (\u2115 tt) (\u03bb n \u2192 `\u03a3 A \u03bb _ \u2192 `Rec n (`End (suc n)))\n  , tt\n  ))\n\nVec : (A : Set) (n : \u2115 tt) \u2192 Set\nVec A n = \u03bc (\u2115 tt) (VecD A) n\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = con (here , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = con (there here , n , x , xs , refl)\n\n----------------------------------------------------------------------\n\naddC : (u : \u22a4) (xs : El \u22a4 \u2115D \u2115 u)\n  (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u xs)\n  \u2192 \u2115 u \u2192 \u2115 u\naddC u xs = case \u2115T\n  (\u03bb t \u2192 (c : El \u22a4 (\u2115C t) \u2115 u) (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u (t , c)) \u2192 \u2115 u \u2192 \u2115 u )\n  ( (\u03bb q ih n \u2192 n) \n  , (\u03bb m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n  , tt\n  )\n  (proj\u2081 xs)\n  (proj\u2082 xs)\n\n-- addC : (u : \u22a4) (xs : El \u22a4 \u2115D \u2115 u)\n--   (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u xs)\n--   \u2192 \u2115 u \u2192 \u2115 u\n-- addC tt (here , q) ih n = n\n-- addC tt (there here , m , q) (ih , tt) n = suc (ih n)\n-- addC tt (there (there ()) , q) ih n\n\nadd : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nadd = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt) addC tt\n\n-- multC : (u : \u22a4) (xs : El \u22a4 \u2115D \u2115 u)\n--   (ih : All \u22a4 \u2115D \u2115 (\u03bb u n \u2192 \u2115 u \u2192 \u2115 u) u xs)\n--   \u2192 \u2115 u \u2192 \u2115 u\n-- multC tt (`zero , q) tt n = zero\n-- multC tt (`suc , m , q) (ih , tt) n = add n (ih n)\n\n-- mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n-- mult = ind \u22a4 \u2115D (\u03bb _ _ \u2192 \u2115 tt \u2192 \u2115 tt) multC tt\n\n-- appendC : (u : \u22a4) (A : Set) (m : \u2115 u) (xs : El (\u2115 u) (VecD A) (Vec A) m)\n--   (ih : All (\u2115 u) (VecD A) (Vec A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) m xs)\n--   (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n-- appendC tt A .(con (`zero , refl)) (`nil , refl) ih n ys = ys\n-- appendC tt A .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys = cons A (add m n) x (ih n ys)\n\n-- append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n-- append A = ind (\u2115 tt) (VecD A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)) (appendC tt A)\n\n-- concatC : (u : \u22a4) (A : Set) (m n : \u2115 u) (xss : El (\u2115 u) (VecD (Vec A m)) (Vec (Vec A m)) n)\n--   (ih : All (\u2115 u) (VecD (Vec A m)) (Vec (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) n xss)\n--   \u2192 Vec A (mult n m)\n-- concatC tt A m .(con (`zero , refl)) (`nil , refl) tt = nil A\n-- concatC tt A m .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) = append A m xs (mult n m) ih\n\n-- concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n-- concat A m = ind (\u2115 tt) (VecD (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m)) (concatC tt A m)\n\n-- -------------------------------------------------------------------------------\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formalization\/agda\/Spire\/Examples\/LabelledDesc.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"83cf253f51de35e348c7518c997d50fcf7d024e7","subject":"lib\/Function\/Bijection\/SyntaxKit.agda","message":"lib\/Function\/Bijection\/SyntaxKit.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Bijection\/SyntaxKit.agda","new_file":"lib\/Function\/Bijection\/SyntaxKit.agda","new_contents":"module Function.Bijection.SyntaxKit where\n\nimport Level as L\nopen import Function.NP\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Unit\nopen import Relation.Binary.PropositionalEquality\n\nrecord BijKit b {a} (A : Set a) : Set (L.suc b L.\u2294 a) where\n  constructor mk\n  field\n    Bij   : Set b\n    eval  : Bij \u2192 Endo A\n    _\u207b\u00b9   : Endo Bij\n    idBij : Bij\n    _\u204fBij_ : Bij \u2192 Bij \u2192 Bij\n\n  eval\u207b\u00b9 : Bij \u2192 Endo A\n  eval\u207b\u00b9 f = eval (f \u207b\u00b9)\n\n  _\u2257Bij_ : Bij \u2192 Bij \u2192 Set _\n  _\u2257Bij_ = \u03bb f g \u2192 \u2200 (x : A) \u2192 eval f x \u2261 eval g x\n\n  field\n    id-spec : eval idBij \u2257 id\n    _\u204f-spec_ : \u2200 f g \u2192 eval (f \u204fBij g) \u2257 eval g \u2218 eval f\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 eval (f \u207b\u00b9) \u2218 eval f \u2257 id\n    _\u207b\u00b9-involutive : \u2200 f \u2192 (f \u207b\u00b9) \u207b\u00b9 \u2257Bij f\n\n  _\u207b\u00b9-inverseBij : \u2200 f \u2192 (f \u204fBij f \u207b\u00b9) \u2257Bij idBij\n  (f \u207b\u00b9-inverseBij) x rewrite (f \u204f-spec (f \u207b\u00b9)) x\n                            | id-spec x\n                            | (f \u207b\u00b9-inverse)\n                            x = refl\n\n  _\u207b\u00b9-inverse\u2032 : \u2200 f \u2192 (eval f \u2218 eval (f \u207b\u00b9)) \u2257 id\n  (f \u207b\u00b9-inverse\u2032) x with ((f \u207b\u00b9) \u207b\u00b9-inverse) x\n  ... | p rewrite (f \u207b\u00b9-involutive) (eval (f \u207b\u00b9) x) = p\n\n  _\u207b\u00b9-inverseBij\u2032 : \u2200 f \u2192 (f \u207b\u00b9 \u204fBij f) \u2257Bij idBij\n  (f \u207b\u00b9-inverseBij\u2032) x rewrite (f \u207b\u00b9 \u204f-spec f) x\n                             | id-spec x\n                             | (f \u207b\u00b9-inverse\u2032)\n                            x = refl\n\nmodule BoolBijection where\n  data BoolBij : Set where\n    `id `not : BoolBij\n\n  bool-bijKit : BijKit L.zero Bool\n  bool-bijKit = mk BoolBij eval id `id _\u204fBij_ (\u03bb _ \u2192 refl) _\u204f-spec_ _\u207b\u00b9-inverse (\u03bb _ _ \u2192 refl) where\n    eval  : BoolBij \u2192 Endo Bool\n    eval `id = id\n    eval `not = not\n\n    _\u204fBij_ : BoolBij \u2192 BoolBij \u2192 BoolBij\n    `id  \u204fBij g    = g\n    f    \u204fBij `id  = f\n    `not \u204fBij `not = `id\n\n    _\u204f-spec_ : \u2200 f g \u2192 eval (f \u204fBij g) \u2257 eval g \u2218 eval f\n    `id  \u204f-spec g    = \u03bb _ \u2192 refl\n    `not \u204f-spec `id  = \u03bb _ \u2192 refl\n    `not \u204f-spec `not = sym \u2218 not-involutive\n\n    _\u207b\u00b9-inverse : \u2200 f \u2192 eval f \u2218 eval f \u2257 id\n    (`id  \u207b\u00b9-inverse) = \u03bb _ \u2192 refl\n    (`not \u207b\u00b9-inverse) = not-involutive\n\n  module BoolBijKit = BijKit bool-bijKit\n\nmodule 1-Bijection {a} (A : Set a) where\n    1-bij : BijKit L.zero A\n    1-bij = mk \u22a4 (\u03bb _ \u2192 id) _ _ _ (\u03bb _ \u2192 refl) (\u03bb _ _ _ \u2192 refl) (\u03bb _ _ \u2192 refl) (\u03bb _ _ \u2192 refl)\n\n    module 1-BijKit = BijKit 1-bij\n\nmodule UnitBijection where\n    open 1-Bijection\n    \u22a4-bijKit : BijKit L.zero \u22a4\n    \u22a4-bijKit = 1-bij \u22a4\n\n    module UnitBijKit = 1-BijKit \u22a4\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Function\/Bijection\/SyntaxKit.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"144880127b8c4657d30ef48685b4a38719e51fd1","subject":"cont.agda","message":"cont.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"cont.agda","new_file":"cont.agda","new_contents":"{-# OPTIONS --copatterns #-}\n\nopen import Type\nopen import Function\nopen import Data.Product hiding (map)\n\nmodule cont (T : \u2605) where \n\nContA : \u2605 \u2192 \u2605\nContA A = (A \u2192 T) \u2192 T \n\nrecord Cont (A : \u2605) : \u2605 where\n  field\n    runCont : (A \u2192 T) \u2192 T \n\nopen Cont\n\nreturn : \u2200 {A} \u2192 A \u2192 Cont A\nrunCont (return x) k = k x\n\nmap : \u2200 {A B} \u2192 (A \u2192 B) \u2192 (Cont A \u2192 Cont B)\nrunCont (map f mx) k = runCont mx \u03bb x \u2192 k (f x)\n\n_>>=_ : \u2200 {A B} \u2192 Cont A \u2192 (A \u2192 Cont B) \u2192 Cont B\nrunCont (mx >>= f) k = runCont mx \u03bb x \u2192 runCont (f x) k\n\njoin : \u2200 {A} \u2192 Cont (Cont A) \u2192 Cont A\njoin mmx = mmx >>= id\n\n_\u229b_ : \u2200 {A B} \u2192 Cont (A \u2192 B) \u2192 Cont A \u2192 Cont B\nmf \u229b mx = mf >>= \u03bb f \u2192 map f mx\n\n\u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192 (A \u2192 B \u2192 C) \u2192 Cont A \u2192 Cont B \u2192 Cont C\n\u27ea f \u00b7 mx \u00b7 my \u27eb = map f mx \u229b my\n\n_\u27e8,\u27e9_ : \u2200 {A B} \u2192 Cont A \u2192 Cont B \u2192 Cont (A \u00d7 B)\nmx \u27e8,\u27e9 my = \u27ea _,_ \u00b7 mx \u00b7 my \u27eb\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'cont.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4a314ad9b30e30302c114c4966d12715b2c53d4b","subject":"new cool way","message":"new cool way\n","repos":"zeitraffer\/agda-cats","old_file":"test\/0-Arrow.agda","new_file":"test\/0-Arrow.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"1489b38badd5c789c64f534bb2ee8245ae4fe719","subject":"[ test-suite ] Added common definitions.","message":"[ test-suite ] Added common definitions.\n","repos":"asr\/apia,asr\/apia","old_file":"test\/succeed\/fol-theorems\/Common\/FOL.agda","new_file":"test\/succeed\/fol-theorems\/Common\/FOL.agda","new_contents":"------------------------------------------------------------------------------\n-- Common FOL definitions\n------------------------------------------------------------------------------\n\n{-# OPTIONS --exact-split              #-}\n{-# OPTIONS --no-sized-types           #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K                #-}\n\nmodule Common.FOL where\n\ninfix  6 \u2203\ninfix  4 _\u2261_\ninfixr 4 _,_\n\npostulate D : Set\n\n-- The existential quantifier type on D.\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'test\/succeed\/fol-theorems\/Common\/FOL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e64ab79bbda6bc1ad26f63e370e9074c5eb6b04e","subject":"some trials","message":"some trials\n","repos":"siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces,siddhartha-gadgil\/LogicTypesSpaces","old_file":"code\/GoaTest.agda","new_file":"code\/GoaTest.agda","new_contents":"open import Base\n\nmodule GoaTest where\n\ndata Bool : Type where\n  true : Bool\n  false : Bool\n\nnot : Bool \u2192 Bool\nnot true = false\nnot false = true\n\nnotnot : Bool \u2192 Bool\nnotnot x = not (not x)\n\nand : Bool \u2192 Bool \u2192 Bool\nand true y = y\nand false y = false\n\ndata \u2115 : Type where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\neven : \u2115 \u2192 Bool\neven zero = true\neven (succ x) = not (even x)\n\nadd : \u2115 \u2192 \u2115 \u2192 \u2115\nadd zero y = y\nadd (succ x) y = succ (add x y)\n\n{-# BUILTIN NATURAL \u2115 #-}\n\ndata True : Type where\n  qed : True\n\n\ndata False : Type where\n\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'code\/GoaTest.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ec92dc6c37ff8e76b46127390d1176bc32c6a028","subject":"Another bug generator for Agda 2.3.2","message":"Another bug generator for Agda 2.3.2\n\nOld-commit-hash: 42373aea0f9da90ca9060e42aa9d40627ffacf75\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/Bug2.agda","new_file":"experimental\/Bug2.agda","new_contents":"{-\nBug generator for Agda 2.3.2\n\nTypechecking this file yields the following message:\n\nAn internal error has occurred. Please report this as a bug.\nLocation of the error: src\/full\/Agda\/TypeChecking\/Conversion.hs:428\n-}\n\nmodule TaggedDeltaTypes where\n\nopen import Data.NatBag renaming\n  (map to mapBag ; empty to emptyBag ; update to updateBag)\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Nat\nopen import Data.Unit using (\u22a4 ; tt)\nimport Data.Integer as \u2124\nimport Data.Product as Product\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\nimport Level\n\n-- Postulates: Extensionality and bag properties (#55)\npostulate extensionality : Extensionality Level.zero Level.zero\n--\n-- open import Data.NatBag.Properties\npostulate b\\\\b=\u2205 : \u2200 {{b : Bag}} \u2192 b \\\\ b \u2261 emptyBag\npostulate b++\u2205=b : \u2200 {{b : Bag}} \u2192 b ++ emptyBag \u2261 b\npostulate \u2205++b=b : \u2200 {{b : Bag}} \u2192 emptyBag ++ b \u2261 b\npostulate b++[d\\\\b]=d : \u2200 {b d} \u2192 b ++ (d \\\\ b) \u2261 d\npostulate [b++d]\\\\b=d : \u2200 {b d} \u2192 (b ++ d) \\\\ b \u2261 d\npostulate\n  [a++b]\\\\[c++d]=[a\\\\c]++[b\\\\d] : \u2200 {a b c d} \u2192\n    (a ++ b) \\\\ (c ++ d) \u2261 (a \\\\ c) ++ (b \\\\ d)\npostulate\n  [a\\\\b]\\\\[c\\\\d]=[a\\\\c]\\\\[b\\\\d] : \u2200 {a b c d} \u2192\n    (a \\\\ b) \\\\ (c \\\\ d) \u2261 (a \\\\ c) \\\\ (b \\\\ d)\n\n------------------------\n-- Syntax of programs --\n------------------------\n\ndata Type : Set where\n  nats : Type\n  bags : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u03c4 \u0393} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u03c4 \u03c4\u2032 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  bag : \u2200 {\u0393} \u2192 (b : Bag) \u2192 Term \u0393 bags\n\n  var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (s : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  add : \u2200 {\u0393} \u2192 (s : Term \u0393 nats) \u2192 (t : Term \u0393 nats) \u2192 Term \u0393 nats\n  map : \u2200 {\u0393} \u2192 (f : Term \u0393 (nats \u21d2 nats)) \u2192 (b : Term \u0393 bags) \u2192 Term \u0393 bags\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u0393\u227c\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0393\n\u0393\u227c\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0393 {\u03c4 \u2022 \u0393} = keep \u03c4 \u2022 \u0393\u227c\u0393 {\u0393}\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\nweaken subctx (bag b) = bag b\nweaken subctx (add t\u2081 t\u2082) = add (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (map f b) = map (weaken subctx f) (weaken subctx b)\n\n---------------------------\n-- Semantics of programs --\n---------------------------\n\nrecord Meaning (Syntax : Set) {\u2113 : Level.Level} : Set (Level.suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 nats \u27e7Type = \u2115\n\u27e6 bags \u27e7Type = Bag\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nidentity-weakening : \u2200 {\u0393} {\u03c1 : \u27e6 \u0393 \u27e7} \u2192 \u27e6 \u0393\u227c\u0393 {\u0393} \u27e7 \u03c1 \u2261 \u03c1\nidentity-weakening {\u2205} {\u2205} = refl\nidentity-weakening {\u03c4 \u2022 \u0393} {v \u2022 \u03c1} =\n  cong\u2082 _\u2022_ {x = v} refl (identity-weakening {\u0393} {\u03c1})\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\u27e6 bag b \u27e7Term \u03c1 = b\n\u27e6 add m n \u27e7Term \u03c1 = \u27e6 m \u27e7Term \u03c1 + \u27e6 n \u27e7Term \u03c1\n\u27e6 map f b \u27e7Term \u03c1 = mapBag (\u27e6 f \u27e7Term \u03c1) (\u27e6 b \u27e7Term \u03c1)\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n_\u27e8$\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n_\u27e8$\u27e9_ = cong\u2082 (\u03bb x y \u2192 x y)\n\n-- infix 0 $ in Haskell\ninfixl 0 _\u27e8$\u27e9_\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = weaken-sound t\u2081 \u03c1 \u27e8$\u27e9 weaken-sound t\u2082 \u03c1\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\nweaken-sound (bag b) \u03c1 = refl\nweaken-sound (add m n) \u03c1 = cong\u2082 _+_ (weaken-sound m \u03c1) (weaken-sound n \u03c1)\nweaken-sound (map f b) \u03c1 =\n  cong\u2082 mapBag (weaken-sound f \u03c1) (weaken-sound b \u03c1)\n\n----------------------------------------------------------\n-- Syntax and semantics of changes (they are entangled) --\n----------------------------------------------------------\n\ndata \u0394-Type : Set where\n  \u0394 : (\u03c4 : Type) \u2192 \u0394-Type\n\ndata \u0394-Context : Set where\n  \u0394 : (\u0393 : Context) \u2192 \u0394-Context\n\nEnv : Context \u2192 Set\nEnv \u0393 = \u27e6 \u0393 \u27e7Context\n\nprivate data \u0394-Term : \u0394-Context \u2192 \u0394-Type \u2192 Set\n\n-- Syntax of \u0394-Types\n-- ...   is mutually recursive with semantics of \u0394-Terms,\n-- (because it embeds validity proofs)\n-- which is mutually recursive with validity and \u0394-environments,\n-- which is mutually recursive with \u2295,\n-- which is mutually recursive with \u229d and R[v,v\u229dv],\n-- which is mutually recursive with v\u2295[u\u229dv]=v, et cetera.\n\u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u0394-Env : Context \u2192 Set\n\u27e6_\u27e7\u0394 : \u2200 {\u03c4 : Type} {\u0393 : Context}\n    \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n_\u2295_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 \u27e6 \u03c4 \u27e7\n_\u229d_ : \u2200 {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nvalid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4 \u2192 Set\nR[v,u\u229dv] : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 valid {\u03c4} v (u \u229d v)\nv\u2295[u\u229dv]=u : \u2200 {\u03c4 : Type} {u v : \u27e6 \u03c4 \u27e7} \u2192 v \u2295 (u \u229d v) \u2261 u\nignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\nupdate : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 \u27e6 \u0393 \u27e7\n\ninfixl 6 _\u2295_ _\u229d_ -- as with + - in GHC.Num\n\nabstract\n data \u0394-Term where\n  -- changes to numbers are replacement pairs\n  \u0394nat : \u2200 {\u0393} \u2192 (old : \u2115) \u2192 (new : \u2115) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- changes to bags are bags\n  \u0394bag : \u2200 {\u0393} \u2192 (db : Bag) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  -- changes to variables are variables\n  \u0394var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n  -- changes to abstractions are binders of x and dx\n  \u0394abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : \u0394-Term (\u0394 (\u03c4\u2081 \u2022 \u0393)) (\u0394 \u03c4\u2082)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082))\n  -- changes to applications are applications of a value and a change\n  \u0394app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192(ds : \u0394-Term (\u0394 \u0393) (\u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)))\n                    \u2192 ( t :   Term    \u0393      \u03c4\u2081)\n                    \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394  \u03c4\u2081))\n                    \u2192 (R[t,dt] : {\u03c1 : \u0394-Env \u0393} \u2192 -- 'Tis but a proof.\n                                 valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 \u03c4\u2082)\n  -- changes to addition are changes to their components\n  \u0394add : \u2200 {\u0393} \u2192 (ds : \u0394-Term (\u0394 \u0393) (\u0394 nats))\n               \u2192 (dt : \u0394-Term (\u0394 \u0393) (\u0394 nats)) \u2192\n         \u0394-Term (\u0394 \u0393) (\u0394 nats)\n  -- There are two kinds of changes to maps:\n  -- 0. recomputation,\n  -- 1. mapping over changes,\n  -- the latter used only with some form of isNil available.\n  \u0394map\u2080 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (df : \u0394-Term (\u0394 \u0393) (\u0394 (nats \u21d2 nats)))\n                \u2192 ( b :   Term    \u0393     bags)\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n  \u0394map\u2081 : \u2200 {\u0393} \u2192 ( f :   Term    \u0393     (nats \u21d2 nats))\n                \u2192 (db : \u0394-Term (\u0394 \u0393) (\u0394 bags))\n          \u2192 \u0394-Term (\u0394 \u0393) (\u0394 bags)\n\n-- \u27e6_\u27e7\u0394\u03c4 : \u0394-Type \u2192 Set\n\u27e6 \u0394 nats \u27e7\u0394\u03c4 = \u2115 \u00d7 \u2115\n\u27e6 \u0394 bags \u27e7\u0394\u03c4 = Bag\n\u27e6 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7\u0394\u03c4 =\n  (v : \u27e6 \u03c4\u2081 \u27e7) \u2192 (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) \u2192 valid v dv \u2192 \u27e6 \u0394 \u03c4\u2082 \u27e7\u0394\u03c4\n\nmeaning-\u0394\u03c4 : Meaning \u0394-Type\nmeaning-\u0394\u03c4 = meaning \u27e6_\u27e7\u0394\u03c4\n\n_\u2295_ {nats}   n dn = proj\u2082 dn\n_\u2295_ {bags}   b db = b ++ db\n_\u2295_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u2295 df v (v \u229d v) R[v,u\u229dv]\n\n_\u229d_ {nats}   m n = (n , m)\n_\u229d_ {bags}   b d = b \\\\ d\n_\u229d_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb v dv R[v,dv] \u2192 f (v \u2295 dv) \u229d g v\n\n-- valid : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid {nats} n dn = n \u2261 proj\u2081 dn\nvalid {bags} b db = \u22a4\nvalid {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (v : \u27e6 \u03c4\u2081 \u27e7) (dv : \u27e6 \u0394 \u03c4\u2081 \u27e7\u0394\u03c4) (R[v,dv] : valid v dv)\n  \u2192 valid (f v) (df v dv R[v,dv])\n  \u00d7 (f \u2295 df) (v \u2295 dv) \u2261 f v \u2295 df v dv R[v,dv]\n\neq1 : \u2200 {\u03c4\u2081 \u03c4\u2082 : Type} {f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {df} {x : \u27e6 \u03c4\u2081 \u27e7} {dx} \u2192\n      valid f df \u2192 (R[x,dx] : valid x dx) \u2192\n      (f \u2295 df) (x \u2295 dx) \u2261 f x \u2295 df x dx R[x,dx]\n\neq1 R[f,df] R[x,dx] = proj\u2082 (R[f,df] _ _ R[x,dx])\n\nv\u2295[u\u229dv]=u {nats}   {u} {v} = refl\nv\u2295[u\u229dv]=u {bags}   {u} {v} = b++[d\\\\b]=d {v} {u}\nv\u2295[u\u229dv]=u {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = extensionality (\u03bb w \u2192\n  begin\n    (v \u2295 (u \u229d v)) w\n  \u2261\u27e8 refl \u27e9 -- for clarity\n    v w \u2295 (u (w \u2295 (w \u229d w)) \u229d v w)\n  \u2261\u27e8 cong (\u03bb hole \u2192 v w \u2295 (u hole \u229d v w)) v\u2295[u\u229dv]=u \u27e9\n    v w \u2295 (u w \u229d v w)\n  \u2261\u27e8 v\u2295[u\u229dv]=u \u27e9\n    u w\n  \u220e) where open \u2261-Reasoning\n\nR[v,u\u229dv] {nats} {u} {v} = refl\nR[v,u\u229dv] {bags} {u} {v} = tt\nR[v,u\u229dv] {\u03c4\u2081 \u21d2 \u03c4\u2082} {u} {v} = \u03bb w dw R[w,dw] \u2192\n  let\n    w\u2032 = w \u2295 dw\n  in\n    R[v,u\u229dv] -- NOT a self recursion: implicit arguments are different.\n    ,\n    (begin\n      (v \u2295 (u \u229d v)) w\u2032\n    \u2261\u27e8 cong (\u03bb hole \u2192 hole w\u2032) (v\u2295[u\u229dv]=u {_} {u} {v}) \u27e9\n      u w\u2032\n    \u2261\u27e8 sym (v\u2295[u\u229dv]=u {_} {u w\u2032} {v w}) \u27e9\n      v w \u2295 (u \u229d v) w dw R[w,dw]\n    \u220e) where open \u2261-Reasoning\n\nrecord Quadruple\n  (A : Set) (B : A \u2192 Set) (C : (a : A) \u2192 B a \u2192 Set)\n  (D : (a : A) \u2192 (b : B a) \u2192 (c : C a b) \u2192 Set): Set where\n  constructor cons\n  field\n    car    : A\n    cadr   : B car\n    caddr  : C car cadr\n    cadddr : D car cadr caddr\n\nopen Quadruple public\n\n-- The type of environments ensures their consistency.\n-- \u0394-Env : Context \u2192 Set\n\u0394-Env \u2205 = EmptySet\n\u0394-Env (\u03c4 \u2022 \u0393) = Quadruple\n  \u27e6 \u03c4 \u27e7\n  (\u03bb v \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4)\n  (\u03bb v dv \u2192 valid v dv)\n  (\u03bb v dv R[v,dv] \u2192 \u0394-Env \u0393)\n\n-- ignore : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nignore {\u2205} \u03c1 = \u2205\nignore {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = v \u2022 ignore \u03c1\n\n-- update : \u2200 {\u0393 : Context} \u2192 (\u03c1 : \u0394-Env \u0393) \u2192 Env \u0393\nupdate {\u2205} \u03c1 = \u2205\nupdate {\u03c4 \u2022 \u0393} (cons v dv R[v,dv] \u03c1) = (v \u2295 dv) \u2022 update \u03c1\n\n\u27e6_\u27e7\u0394Var : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\n\u27e6 this   \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = dv\n\u27e6 that x \u27e7\u0394Var (cons v dv R[v,dv] \u03c1) = \u27e6 x \u27e7\u0394Var \u03c1\n\n-- \u27e6_\u27e7\u0394Var is not put into the Meaning type class\n-- because its argument is identical to that of \u27e6_\u27e7Var.\n\n-- \u27e6_\u27e7\u0394 : \u2200 {\u03c4 \u0393} \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4) \u2192 \u0394-Env \u0393 \u2192 \u27e6 \u0394 \u03c4 \u27e7\u0394\u03c4\nabstract\n \u27e6 \u0394nat old new \u27e7\u0394 \u03c1 = (old , new)\n \u27e6 \u0394bag db \u27e7\u0394 \u03c1 = db\n \u27e6 \u0394var x \u27e7\u0394 \u03c1 = \u27e6 x \u27e7\u0394Var \u03c1\n \u27e6 \u0394abs t \u27e7\u0394 \u03c1 = \u03bb v dv R[v,dv] \u2192 \u27e6 t \u27e7\u0394 (cons v dv R[v,dv] \u03c1)\n \u27e6 \u0394app ds t dt R[dt,t] \u27e7\u0394 \u03c1 =\n   \u27e6 ds \u27e7\u0394 \u03c1 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 dt \u27e7\u0394 \u03c1) R[dt,t]\n \u27e6 \u0394add ds dt \u27e7\u0394 \u03c1 =\n   let\n     (old-s , new-s) = \u27e6 ds \u27e7\u0394 \u03c1\n     (old-t , new-t) = \u27e6 dt \u27e7\u0394 \u03c1\n   in\n     (old-s + old-t , new-s + new-t)\n \u27e6 \u0394map\u2080 f df b db \u27e7\u0394 \u03c1 =\n   let\n     v  = \u27e6 b \u27e7 (ignore \u03c1)\n     h  = \u27e6 f \u27e7 (ignore \u03c1)\n     dv = \u27e6 db \u27e7\u0394 \u03c1\n     dh = \u27e6 df \u27e7\u0394 \u03c1\n   in\n     mapBag (h \u2295 dh) (v \u2295 dv) \\\\ mapBag h v\n \u27e6 \u0394map\u2081 f db \u27e7\u0394 \u03c1 = mapBag (\u27e6 f \u27e7 (ignore \u03c1)) (\u27e6 db \u27e7\u0394 \u03c1)\n\nmeaning-\u0394Term : \u2200 {\u03c4 \u0393} \u2192 Meaning (\u0394-Term (\u0394 \u0393) (\u0394 \u03c4))\nmeaning-\u0394Term = meaning \u27e6_\u27e7\u0394\n\n--------------------------------------------------------\n-- Program transformation and correctness (entangled) --\n--------------------------------------------------------\n\nderive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\n\nvalidity : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394-Env \u0393} \u2192\n  valid (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\n-- derive : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 \u0394-Term (\u0394 \u0393) (\u0394 \u03c4)\nderive (nat n) = \u0394nat n n\nderive (bag b) = \u0394bag b\nderive (var x) = \u0394var x\nderive (abs t) = \u0394abs (derive t)\nderive (app s t) = \u0394app (derive s) t (derive t) validity\nderive (add s t) = \u0394add (derive s) (derive t)\nderive (map f b) = \u0394map\u2080 f (derive f) b (derive b)\n\nvalidity-var : \u2200 {\u03c4 \u0393} \u2192 (x : Var \u0393 \u03c4) \u2192\n  \u2200 {\u03c1 : \u0394-Env \u0393} \u2192 valid (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 x \u27e7\u0394Var \u03c1)\n\nvalidity-var this {cons v dv R[v,dv] \u03c1} = R[v,dv]\nvalidity-var (that x) {cons v dv R[v,dv] \u03c1} = validity-var x\n\n--validity {nats} {\u0393} {nat n} {\u2205} = ?\nvalidity {nats} {\u2205}    {nat n} {\u2205} =\n  begin\n    n\n  \u2261\u27e8 ? \u27e9\n    proj\u2081 (n , n)\n  \u2261\u27e8 {!!} \u27e9\n    proj\u2081 (\u27e6 \u0394nat n n \u27e7 \u2205)\n  \u220e where open \u2261-Reasoning\nvalidity {nats} {\u03c4 \u2022 \u0393} {nat n} {\u03c1} = {!!}\nvalidity {bags} {\u0393} {bag b} = {!!}\nvalidity {\u03c4} {\u0393} {var x} = {!!}\nvalidity {\u03c4\u2081 \u21d2 \u03c4\u2082} {\u0393} {abs t} = {!!}\nvalidity {\u03c4} {\u0393} {app t t\u2081} = {!!}\nvalidity {nats} {\u0393} {add t t\u2081} = {!!}\nvalidity {bags} {\u0393} {map t t\u2081} = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'experimental\/Bug2.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"14b8f096b525d8a615d01965c700b05596b5d603","subject":"Add lemmas to prove function changes valid","message":"Add lemmas to prove function changes valid\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Equivalence.agda","new_file":"New\/Equivalence.agda","new_contents":"module New.Equivalence where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nopen import Data.Product\nopen import Postulate.Extensionality\n\nopen import New.Changes\n\nmodule _ {a} {A : Set a} {{CA : ChAlg A}} {x : A} where\n  -- Delta-observational equivalence: these asserts that two changes\n  -- give the same result when applied to a base value.\n\n  -- To avoid unification problems, use a one-field record (a Haskell \"newtype\")\n  -- instead of a \"type synonym\".\n  record _\u2259_ (dx dy : Ch A) : Set a where\n    -- doe = Delta-Observational Equivalence.\n    constructor doe\n    field\n      proof : x \u2295 dx \u2261 x \u2295 dy\n\n  open _\u2259_ public\n\n  -- Same priority as \u2261\n  infix 4 _\u2259_\n\n  open import Relation.Binary\n\n  -- _\u2259_ is indeed an equivalence relation:\n  \u2259-refl : \u2200 {dx} \u2192 dx \u2259 dx\n  \u2259-refl = doe refl\n\n  \u2259-sym : \u2200 {dx dy} \u2192 dx \u2259 dy \u2192 dy \u2259 dx\n  \u2259-sym \u2259 = doe $ sym $ proof \u2259\n\n  \u2259-trans : \u2200 {dx dy dz} \u2192 dx \u2259 dy \u2192 dy \u2259 dz \u2192 dx \u2259 dz\n  \u2259-trans \u2259\u2081 \u2259\u2082 = doe $ trans (proof \u2259\u2081) (proof \u2259\u2082)\n\n  -- That's standard congruence applied to \u2259\n  \u2259-cong  : \u2200 {b} {B : Set b}\n       (f : A \u2192 B) {dx dy} \u2192 dx \u2259 dy \u2192 f (x \u2295 dx) \u2261 f (x \u2295 dy)\n  \u2259-cong f da\u2259db = cong f $ proof da\u2259db\n\n  \u2259-isEquivalence : IsEquivalence (_\u2259_)\n  \u2259-isEquivalence = record\n    { refl  = \u2259-refl\n    ; sym   = \u2259-sym\n    ; trans = \u2259-trans\n    }\n\n  \u2259-setoid : Setoid a a\n  \u2259-setoid = record\n    { Carrier       = Ch A\n    ; _\u2248_           = _\u2259_\n    ; isEquivalence = \u2259-isEquivalence\n    }\n\n\u2259-syntax : \u2200 {a} {A : Set a} {{CA : ChAlg A}} (x : A) (dx\u2081 dx\u2082 : Ch A) \u2192 Set a\n\u2259-syntax x dx\u2081 dx\u2082 = _\u2259_ {x = x} dx\u2081 dx\u2082\nsyntax \u2259-syntax x dx\u2081 dx\u2082 = dx\u2081 \u2259[ x ] dx\u2082\n\n\nmodule BinaryValid\n  {A : Set} {{CA : ChAlg A}}\n  {B : Set} {{CB : ChAlg B}}\n  {C : Set} {{CC : ChAlg C}}\n  (f : A \u2192 B \u2192 C) (df : A \u2192 Ch A \u2192 B \u2192 Ch B \u2192 Ch C)\n  where\n  open import Data.Product\n\n  binary-valid-preserve-hp =\n    \u2200 a da (ada : valid a da)\n      b db (bdb : valid b db)\n    \u2192 valid (f a b) (df a da b db)\n\n  binary-valid-eq-hp =\n    \u2200 a da (ada : valid a da)\n      b db (bdb : valid b db)\n    \u2192 (f \u2295 df) (a \u2295 da) (b \u2295 db) \u2261 f a b \u2295 df a da b db\n\n  binary-valid :\n    binary-valid-preserve-hp \u2192\n    binary-valid-eq-hp \u2192\n    valid f df\n  binary-valid ext-valid proof a da ada =\n      (\u03bb b db bdb \u2192 ext-valid a da ada b db bdb , lem2 b db bdb)\n    , ext lem1\n    where\n      lem1 : \u2200 b \u2192 f (a \u2295 da) b \u2295 df (a \u2295 da) (nil (a \u2295 da)) b (nil b) \u2261\n        f a b \u2295 df a da b (nil b)\n      lem1 b\n        rewrite sym (update-nil b)\n        | proof a da ada b (nil b) (nil-valid b)\n        | update-nil b = refl\n      lem2 : \u2200 b (db : Ch B) (bdb : valid b db) \u2192\n        f a (b \u2295 db) \u2295 df a da (b \u2295 db) (nil (b \u2295 db)) \u2261\n        f a b \u2295 df a da b db\n      lem2 b db bdb\n        rewrite sym (proof a da ada (b \u2295 db) (nil (b \u2295 db)) (nil-valid (b \u2295 db)))\n        | update-nil (b \u2295 db) = proof a da ada b db bdb\n\nmodule TernaryValid\n  {A : Set} {{CA : ChAlg A}}\n  {B : Set} {{CB : ChAlg B}}\n  {C : Set} {{CC : ChAlg C}}\n  {D : Set} {{CD : ChAlg D}}\n  (f : A \u2192 B \u2192 C \u2192 D) (df : A \u2192 Ch A \u2192 B \u2192 Ch B \u2192 C \u2192 Ch C \u2192 Ch D)\n  where\n\n\n  ternary-valid-preserve-hp =\n    \u2200 a da (ada : valid a da)\n      b db (bdb : valid b db)\n      c dc (cdc : valid c dc)\n    \u2192 valid (f a b c) (df a da b db c dc)\n\n  -- These are explicit definitions only to speed up typechecking.\n  CA\u2192B\u2192C\u2192D : ChAlg (A \u2192 B \u2192 C \u2192 D)\n  CA\u2192B\u2192C\u2192D = funCA\n  f\u2295df = (_\u2295_ {{CA\u2192B\u2192C\u2192D}} f df)\n\n  -- Already this definition takes a while to typecheck.\n  ternary-valid-eq-hp =\n    \u2200 a (da : Ch A {{CA}}) (ada : valid {{CA}} a da)\n      b (db : Ch B {{CB}}) (bdb : valid {{CB}} b db)\n      c (dc : Ch C {{CC}}) (cdc : valid {{CC}} c dc)\n    \u2192 f\u2295df (a \u2295 da) (b \u2295 db) (c \u2295 dc) \u2261 f a b c \u2295 df a da b db c dc\n\n  ternary-valid :\n    ternary-valid-preserve-hp \u2192\n    ternary-valid-eq-hp \u2192\n    valid f df\n  ternary-valid ext-valid proof a da ada =\n    binary-valid\n      (\u03bb b db bdb c dc cdc \u2192 ext-valid a da ada b db bdb c dc cdc)\n      lem2\n    , ext (\u03bb b \u2192 ext (lem1 b))\n    where\n      open BinaryValid (f a) (df a da)\n      lem1 : \u2200 b c \u2192 f\u2295df (a \u2295 da) b c \u2261 (f a \u2295 df a da) b c\n      lem1 b c\n        rewrite sym (update-nil b)\n        | sym (update-nil c)\n        |\n          proof\n            a da ada\n            b (nil b) (nil-valid b)\n            c (nil c) (nil-valid c)\n        | update-nil b\n        | update-nil c = refl\n        -- rewrite\n        --   sym\n        --   (proof\n        --     (a \u2295 da) (nil (a \u2295 da)) (nil-valid (a \u2295 da))\n        --     b (nil b) (nil-valid b)\n        --     c (nil c) (nil-valid c))\n        -- | update-nil (a \u2295 da)\n        -- | update-nil b\n        -- | update-nil c = {! !}\n      lem2 : \u2200 b db (bdb : valid b db)\n               c dc (cdc : valid c dc) \u2192\n                 (f a \u2295 df a da) (b \u2295 db) (c \u2295 dc)\n               \u2261 f a b c \u2295 df a da b db c dc\n      lem2 b db bdb c dc cdc\n        rewrite sym\n          (proof\n            a da ada\n            (b \u2295 db) (nil (b \u2295 db)) (nil-valid (b \u2295 db))\n            (c \u2295 dc) (nil (c \u2295 dc)) (nil-valid (c \u2295 dc))\n          )\n        | update-nil (b \u2295 db)\n        | update-nil (c \u2295 dc) = proof a da ada b db bdb c dc cdc\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'New\/Equivalence.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"3ab3c3e53bbcd80487956cd8903717341cf7d73b","subject":"Rand.agda","message":"Rand.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Rand.agda","new_file":"Rand.agda","new_contents":"{- This file illustrate that if we were given a (non-indexed)\n   monad to write randomized programs then we could embed\n   randomized programs modeled as deterministic programs taking\n   their randomness as an extra argument. -}\n\nopen import Type\nopen import Function\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Inverse using (_\u2194_; module Inverse; sym)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Product\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Sum\nopen import Data.Nat using (\u2115; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat.Positive\nopen import Data.Unit\nopen import Data.Maybe.NP\nopen import Relation.Binary.PropositionalEquality as \u2261 using (\u2192-to-\u27f6)\n\nmodule Rand\n  (M       : \u2605 \u2192 \u2605)\n  (]0,1[   : \u2605)\n  (_\/num+_ : \u2115\u207a \u2192 \u2115\u207a \u2192 ]0,1[)\n  (unit    : \u2200 {A} \u2192 A \u2192 M A)\n  (_>>=_   : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B)\n  (map     : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B)\n  (choose  : \u2200 {A} \u2192 ]0,1[ \u2192 M A \u2192 M A \u2192 M A)\n where\n\nCardinality = \u2115\u207a\n\nRand : \u2605 \u2192 \u2605\nRand A = Cardinality \u00d7 M A\n\nembed : \u2200 {R A B : \u2605} \u2192 Rand R \u2192 (R \u2192 A \u2192 B) \u2192 A \u2192 M B\nembed (_ , rand\u1d3f) f x = rand\u1d3f >>= \u03bb r \u2192 unit (f r x)\n\n1\/2 : ]0,1[\n1\/2 = [ 1 ] \/num+ [ 1 ]\n\nrand\u22a4 : Rand \u22a4\nrand\u22a4 = [ 1 ] , unit _\n\nrandBit : Rand Bit\nrandBit = [ 2 ] , choose 1\/2 (unit false) (unit true)\n\n_\u00d7Rand_ : \u2200 {A B} \u2192 Rand A \u2192 Rand B \u2192 Rand (A \u00d7 B)\n(c\u1d2c , m\u1d2c) \u00d7Rand (c\u1d2e , m\u1d2e) = c\u1d2c\u02e3\u1d2e , m\u1d2c\u02e3\u1d2e\n  where\n  c\u1d2c\u02e3\u1d2e = c\u1d2c * c\u1d2e\n  m\u1d2c\u02e3\u1d2e = m\u1d2c >>= \u03bb x \u2192 m\u1d2e >>= \u03bb y \u2192 unit (x , y)\n\n_\u228eRand_ : \u2200 {A B} \u2192 Rand A \u2192 Rand B \u2192 Rand (A \u228e B)\n(c\u1d2c , m\u1d2c) \u228eRand (c\u1d2e , m\u1d2e) = c\u1d2c\u207a\u1d2e , m\u1d2c\u207a\u1d2e\n  where\n  c\u1d2c\u207a\u1d2e    = c\u1d2c + c\u1d2e\n  c\u1d2c\/c\u1d2c\u207a\u1d2e = c\u1d2c \/num+ c\u1d2e\n  m\u1d2c\u207a\u1d2e    = choose c\u1d2c\/c\u1d2c\u207a\u1d2e (map inj\u2081 m\u1d2c) (map inj\u2082 m\u1d2e)\n\nmapRand : \u2200 {A B} \u2192 A \u2194 B \u2192 Rand A \u2192 Rand B\nmapRand f (c\u1d2c , m\u1d2c) = c\u1d2c , map f\u2032 m\u1d2c\n  where f\u2032 = _\u27e8$\u27e9_ (Inverse.to f)\n\nrand^ : \u2200 {A} \u2192 Rand A \u2192 (n : \u2115) \u2192 Rand (A ^ n)\nrand^ rand\u1d2c zero    = rand\u22a4\nrand^ rand\u1d2c (suc n) = rand\u1d2c \u00d7Rand (rand^ rand\u1d2c n)\n\nrandVec : \u2200 {A} \u2192 Rand A \u2192 (n : \u2115) \u2192 Rand (Vec A n)\nrandVec rand\u1d2c n = mapRand (^\u2194Vec n) (rand^ rand\u1d2c n)\n\nrandBits : \u2200 n \u2192 Rand (Bits n)\nrandBits = randVec randBit\n\nrandMaybe : \u2200 {A} \u2192 Rand A \u2192 Rand (Maybe A)\nrandMaybe rand\u1d2c = mapRand (sym Maybe\u2194\u22a4\u228e) (rand\u22a4 \u228eRand rand\u1d2c)\n\nrandMaybe^ : \u2200 n \u2192 Rand (Maybe^ n \u22a4)\nrandMaybe^ zero    = rand\u22a4\nrandMaybe^ (suc n) = randMaybe (randMaybe^ n)\n\nrandFin1+ : \u2200 n \u2192 Rand (Fin (suc n))\nrandFin1+ n = mapRand (Maybe^\u22a4\u2194Fin1+ n) (randMaybe^ n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Rand.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"07bef33fd71a2ed3625cdbe978dcef5202c184ac","subject":"BinTree.agda","message":"BinTree.agda\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/BinTree.agda","new_file":"lib\/Explore\/BinTree.agda","new_contents":"module Explore.BinTree where\n\nopen import Data.Tree.Binary\nopen import Explore.Type\n\nfromBinTree : \u2200 {m A} \u2192 BinTree A \u2192 Explore m A\nfromBinTree (leaf x)   _   f = f x\nfromBinTree (fork \u2113 r) _\u2219_ f = fromBinTree \u2113 _\u2219_ f \u2219 fromBinTree r _\u2219_ f\n\nfromBinTree-ind : \u2200 {m p A} (t : BinTree A) \u2192 ExploreInd p (fromBinTree {m} t)\nfromBinTree-ind (leaf x)   P P\u2219 Pf = Pf x\nfromBinTree-ind (fork \u2113 r) P P\u2219 Pf = P\u2219 (fromBinTree-ind \u2113 P P\u2219 Pf)\n                                        (fromBinTree-ind r P P\u2219 Pf)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Explore\/BinTree.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7cd325e6fbf37a9335ba68457aab13668a3e88cb","subject":"Added draft.","message":"Added draft.\n\nIgnore-this: 407d8f291bd3f81e1fd716785e70b062\n\ndarcs-hash:20120317171605-3bd4e-9126a34c681bcdca472afce25dc8175b3105ac18.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/AddPartialRightIdentity.agda","new_file":"Draft\/FOTC\/Data\/Nat\/AddPartialRightIdentity.agda","new_contents":"------------------------------------------------------------------------------\n-- Reasoning partially about functions\n------------------------------------------------------------------------------\n\n-- We cannot reasoning partially about partial functions intended to\n-- operate in total values.\n\n-- Tested with FOT and agda2atp on 17 March 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Data.Nat.AddPartialRightIdentity where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n-- How proceed?\n+-partialRightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-partialRightIdentity n = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Nat\/AddPartialRightIdentity.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ebd8a5cc70b2d7a47b1d17c86210d5a848572c26","subject":"Added a consistency test.","message":"Added a consistency test.\n\nIgnore-this: 366798ddd1e890a4e0023f7957bf578c\n\ndarcs-hash:20110205132837-3bd4e-1ea1501f1785c59b2b75ff4e6d70907c9c170f06.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/McCarthy91\/ConsistencyTest.agda","new_file":"Draft\/McCarthy91\/McCarthy91\/ConsistencyTest.agda","new_contents":"------------------------------------------------------------------------------\n-- Test the consistency of Draft.McCarthy91.McCarthy91\n------------------------------------------------------------------------------\n\n-- In the module Draft.McCarthy91.McCarthy91 we declare Agda\n-- postulates as FOL axioms. We test if it is possible to prove an\n-- unprovable theorem from these axioms.\n\nmodule Draft.McCarthy91.McCarthy91.ConsistencyTest where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\n\n------------------------------------------------------------------------------\n\npostulate\n  impossible : \u2200 d e \u2192 d \u2261 e\n{-# ATP prove impossible #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/McCarthy91\/McCarthy91\/ConsistencyTest.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9ee7cfa733c921faca5b88c3d715233fc6dd1e8e","subject":"models\/DescFix.agda: generic recursion principle for Desc (not structurally recursive yet)","message":"models\/DescFix.agda: generic recursion principle for Desc (not structurally recursive yet)","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/DescFix.agda","new_file":"models\/DescFix.agda","new_contents":"{-# OPTIONS --type-in-type #-}\n\nmodule DescFix where\n\nopen import DescTT\n\n\naux : (C : Desc)(D : Desc)(P : Mu C -> Set)(x : [| D |] (Mu C)) -> Set\naux C id          P (con y) = P (con y) * aux C C P y \naux C (const K)   P k       = Unit\naux C (prod D D') P (s , t) = aux C D P s * aux C D' P t\naux C (sigma S T) P (s , t) = aux C (T s) P t\naux C (pi S T)    P f       = (s : S) -> aux C (T s) P (f s)\n\n\ngen : (C : Desc)(D : Desc)(P : Mu C -> Set)\n  (rec : (y : [| C |] Mu C) -> aux C C P y -> P (con y))\n  (x : [| D |] Mu C) -> aux C D P x\ngen C id          P rec (con x) = rec x (gen C C P rec x) , gen C C P rec x\ngen C (const K)   P rec k       = Void\ngen C (prod D D') P rec (s , t) = gen C D P rec s , gen C D' P rec t\ngen C (sigma S T) P rec (s , t) = gen C (T s) P rec t\ngen C (pi S T)    P rec f       = \\ s -> gen C (T s) P rec (f s)\n\n\nfix : (D : Desc)(P : Mu D -> Set)\n  (rec : (y : [| D |] Mu D) -> aux D D P y -> P (con y))\n  (x : Mu D) -> P x\nfix D P rec (con x) = rec x (gen D D P rec x)\n\n\n\nplus : Nat -> Nat -> Nat\nplus (con (Ze , Void)) n = n\nplus (con (Suc , m)) n = suc (plus m n)\n\nfib : Nat -> Nat\nfib = fix NatD (\\ _ -> Nat) help\n  where\n    help : (m : [| NatD |] Nat) -> aux NatD NatD (\\ _ -> Nat) m -> Nat\n    help (Ze , x) a = suc ze\n    help (Suc , con (Ze , _)) a = suc ze\n    help (Suc , con (Suc , con n)) (fib-n , (fib-sn , a)) = plus fib-n fib-sn","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/DescFix.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"d466e7494f141e32f104b27b092200f883382418","subject":"Added Kovacs strong normalisation proof to extra","message":"Added Kovacs strong normalisation proof to extra\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"extra\/KovacsSTLCnorm.agda","new_file":"extra\/KovacsSTLCnorm.agda","new_contents":"{-# OPTIONS --without-K #-}\n\nopen import Relation.Binary.PropositionalEquality\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\nopen import Function\n\n-- some HoTT-inspired combinators\n\n_&_ = cong\n_\u207b\u00b9 = sym\n_\u25fe_ = trans\n\ncoe : {A B : Set} \u2192 A \u2261 B \u2192 A \u2192 B\ncoe refl a = a\n\n_\u2297_ : \u2200 {A B : Set}{f g : A \u2192 B}{a a'} \u2192 f \u2261 g \u2192 a \u2261 a' \u2192 f a \u2261 g a'\nrefl \u2297 refl = refl\n\ninfix 6 _\u207b\u00b9\ninfixr 4 _\u25fe_\ninfixl 9 _&_\ninfixl 8 _\u2297_\n\n-- Syntax\n--------------------------------------------------------------------------------\n\ninfixr 4 _\u21d2_\ninfixr 4 _,_\n\ndata Ty : Set where\n  \u03b9   : Ty\n  _\u21d2_ : Ty \u2192 Ty \u2192 Ty\n\ndata Con : Set where\n  \u2219   : Con\n  _,_ : Con \u2192 Ty \u2192 Con\n\ndata _\u2208_ (A : Ty) : Con \u2192 Set where\n  vz : \u2200 {\u0393} \u2192 A \u2208 (\u0393 , A)\n  vs : \u2200 {B \u0393} \u2192 A \u2208 \u0393 \u2192 A \u2208 (\u0393 , B)\n\ndata Tm \u0393 : Ty \u2192 Set where\n  var : \u2200 {A} \u2192 A \u2208 \u0393 \u2192 Tm \u0393 A\n  lam : \u2200 {A B} \u2192 Tm (\u0393 , A) B \u2192 Tm \u0393 (A \u21d2 B)\n  app : \u2200 {A B} \u2192 Tm \u0393 (A \u21d2 B) \u2192 Tm \u0393 A \u2192 Tm \u0393 B\n\n-- Embedding\n--------------------------------------------------------------------------------\n\n-- Order-preserving embedding\ndata OPE : Con \u2192 Con \u2192 Set where\n  \u2219    : OPE \u2219 \u2219\n  drop : \u2200 {A \u0393 \u0394} \u2192 OPE \u0393 \u0394 \u2192 OPE (\u0393 , A) \u0394\n  keep : \u2200 {A \u0393 \u0394} \u2192 OPE \u0393 \u0394 \u2192 OPE (\u0393 , A) (\u0394 , A)\n\n-- OPE is a category\nid\u2091 : \u2200 {\u0393} \u2192 OPE \u0393 \u0393\nid\u2091 {\u2219}     = \u2219\nid\u2091 {\u0393 , A} = keep (id\u2091 {\u0393})\n\nwk : \u2200 {A \u0393} \u2192 OPE (\u0393 , A) \u0393\nwk = drop id\u2091\n\n_\u2218\u2091_ : \u2200 {\u0393 \u0394 \u03a3} \u2192 OPE \u0394 \u03a3 \u2192 OPE \u0393 \u0394 \u2192 OPE \u0393 \u03a3\n\u03c3      \u2218\u2091 \u2219      = \u03c3\n\u03c3      \u2218\u2091 drop \u03b4 = drop (\u03c3 \u2218\u2091 \u03b4)\ndrop \u03c3 \u2218\u2091 keep \u03b4 = drop (\u03c3 \u2218\u2091 \u03b4)\nkeep \u03c3 \u2218\u2091 keep \u03b4 = keep (\u03c3 \u2218\u2091 \u03b4)\n\nidl\u2091 : \u2200 {\u0393 \u0394}(\u03c3 : OPE \u0393 \u0394) \u2192 id\u2091 \u2218\u2091 \u03c3 \u2261 \u03c3\nidl\u2091 \u2219        = refl\nidl\u2091 (drop \u03c3) = drop & idl\u2091 \u03c3\nidl\u2091 (keep \u03c3) = keep & idl\u2091 \u03c3\n\nidr\u2091 : \u2200 {\u0393 \u0394}(\u03c3 : OPE \u0393 \u0394) \u2192 \u03c3 \u2218\u2091 id\u2091 \u2261 \u03c3\nidr\u2091 \u2219        = refl\nidr\u2091 (drop \u03c3) = drop & idr\u2091 \u03c3\nidr\u2091 (keep \u03c3) = keep & idr\u2091 \u03c3\n\nass\u2091 :\n  \u2200 {\u0393 \u0394 \u03a3 \u039e}(\u03c3 : OPE \u03a3 \u039e)(\u03b4 : OPE \u0394 \u03a3)(\u03bd : OPE \u0393 \u0394)\n  \u2192 (\u03c3 \u2218\u2091 \u03b4) \u2218\u2091 \u03bd \u2261 \u03c3 \u2218\u2091 (\u03b4 \u2218\u2091 \u03bd)\nass\u2091 \u03c3        \u03b4        \u2219        = refl\nass\u2091 \u03c3        \u03b4        (drop \u03bd) = drop & ass\u2091 \u03c3 \u03b4 \u03bd\nass\u2091 \u03c3        (drop \u03b4) (keep \u03bd) = drop & ass\u2091 \u03c3 \u03b4 \u03bd\nass\u2091 (drop \u03c3) (keep \u03b4) (keep \u03bd) = drop & ass\u2091 \u03c3 \u03b4 \u03bd\nass\u2091 (keep \u03c3) (keep \u03b4) (keep \u03bd) = keep & ass\u2091 \u03c3 \u03b4 \u03bd\n\n\u2208\u2091 : \u2200 {A \u0393 \u0394} \u2192 OPE \u0393 \u0394 \u2192 A \u2208 \u0394 \u2192 A \u2208 \u0393\n\u2208\u2091 \u2219        v      = v\n\u2208\u2091 (drop \u03c3) v      = vs (\u2208\u2091 \u03c3 v)\n\u2208\u2091 (keep \u03c3) vz     = vz\n\u2208\u2091 (keep \u03c3) (vs v) = vs (\u2208\u2091 \u03c3 v)\n\n\u2208-id\u2091 : \u2200 {A \u0393}(v : A \u2208 \u0393) \u2192 \u2208\u2091 id\u2091 v \u2261 v\n\u2208-id\u2091 vz     = refl\n\u2208-id\u2091 (vs v) = vs & \u2208-id\u2091 v\n\n\u2208-\u2218\u2091 : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : OPE \u0394 \u03a3)(\u03b4 : OPE \u0393 \u0394)(v : A \u2208 \u03a3) \u2192 \u2208\u2091 (\u03c3 \u2218\u2091 \u03b4) v \u2261 \u2208\u2091 \u03b4 (\u2208\u2091 \u03c3 v)\n\u2208-\u2218\u2091 \u2219        \u2219        v      = refl\n\u2208-\u2218\u2091 \u03c3        (drop \u03b4) v      = vs & \u2208-\u2218\u2091 \u03c3 \u03b4 v\n\u2208-\u2218\u2091 (drop \u03c3) (keep \u03b4) v      = vs & \u2208-\u2218\u2091 \u03c3 \u03b4 v\n\u2208-\u2218\u2091 (keep \u03c3) (keep \u03b4) vz     = refl\n\u2208-\u2218\u2091 (keep \u03c3) (keep \u03b4) (vs v) = vs & \u2208-\u2218\u2091 \u03c3 \u03b4 v\n\nTm\u2091 : \u2200 {A \u0393 \u0394} \u2192 OPE \u0393 \u0394 \u2192 Tm \u0394 A \u2192 Tm \u0393 A\nTm\u2091 \u03c3 (var v)   = var (\u2208\u2091 \u03c3 v)\nTm\u2091 \u03c3 (lam t)   = lam (Tm\u2091 (keep \u03c3) t)\nTm\u2091 \u03c3 (app f a) = app (Tm\u2091 \u03c3 f) (Tm\u2091 \u03c3 a)\n\nTm-id\u2091 : \u2200 {A \u0393}(t : Tm \u0393 A) \u2192 Tm\u2091 id\u2091 t \u2261 t\nTm-id\u2091 (var v)   = var & \u2208-id\u2091 v\nTm-id\u2091 (lam t)   = lam & Tm-id\u2091 t\nTm-id\u2091 (app f a) = app & Tm-id\u2091 f \u2297 Tm-id\u2091 a\n\nTm-\u2218\u2091 : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : OPE \u0394 \u03a3)(\u03b4 : OPE \u0393 \u0394)(t : Tm \u03a3 A) \u2192 Tm\u2091 (\u03c3 \u2218\u2091 \u03b4) t \u2261 Tm\u2091 \u03b4 (Tm\u2091 \u03c3 t)\nTm-\u2218\u2091 \u03c3 \u03b4 (var v)   = var & \u2208-\u2218\u2091 \u03c3 \u03b4 v\nTm-\u2218\u2091 \u03c3 \u03b4 (lam t)   = lam & Tm-\u2218\u2091 (keep \u03c3) (keep \u03b4) t\nTm-\u2218\u2091 \u03c3 \u03b4 (app f a) = app & Tm-\u2218\u2091 \u03c3 \u03b4 f \u2297 Tm-\u2218\u2091 \u03c3 \u03b4 a\n\n-- Theory of substitution & embedding\n--------------------------------------------------------------------------------\n\ninfixr 6 _\u2091\u2218\u209b_ _\u209b\u2218\u2091_ _\u2218\u209b_\n\ndata Sub (\u0393 : Con) : Con \u2192 Set where\n  \u2219   : Sub \u0393 \u2219\n  _,_ : \u2200 {A : Ty}{\u0394 : Con} \u2192 Sub \u0393 \u0394 \u2192 Tm \u0393 A \u2192 Sub \u0393 (\u0394 , A)\n\n_\u209b\u2218\u2091_ : \u2200 {\u0393 \u0394 \u03a3} \u2192 Sub \u0394 \u03a3 \u2192 OPE \u0393 \u0394 \u2192 Sub \u0393 \u03a3\n\u2219       \u209b\u2218\u2091 \u03b4 = \u2219\n(\u03c3 , t) \u209b\u2218\u2091 \u03b4 = \u03c3 \u209b\u2218\u2091 \u03b4 , Tm\u2091 \u03b4 t\n\n_\u2091\u2218\u209b_ : \u2200 {\u0393 \u0394 \u03a3} \u2192 OPE \u0394 \u03a3 \u2192 Sub \u0393 \u0394 \u2192 Sub \u0393 \u03a3\n\u2219      \u2091\u2218\u209b \u03b4       = \u03b4\ndrop \u03c3 \u2091\u2218\u209b (\u03b4 , t) = \u03c3 \u2091\u2218\u209b \u03b4\nkeep \u03c3 \u2091\u2218\u209b (\u03b4 , t) = \u03c3 \u2091\u2218\u209b \u03b4 , t\n\ndrop\u209b : \u2200 {A \u0393 \u0394} \u2192 Sub \u0393 \u0394 \u2192 Sub (\u0393 , A) \u0394\ndrop\u209b \u03c3 = \u03c3 \u209b\u2218\u2091 wk\n\nkeep\u209b : \u2200 {A \u0393 \u0394} \u2192 Sub \u0393 \u0394 \u2192 Sub (\u0393 , A) (\u0394 , A)\nkeep\u209b \u03c3 = drop\u209b \u03c3 , var vz\n\n\u231c_\u231d\u1d52\u1d56\u1d49 : \u2200 {\u0393 \u0394} \u2192 OPE \u0393 \u0394 \u2192 Sub \u0393 \u0394\n\u231c \u2219      \u231d\u1d52\u1d56\u1d49 = \u2219\n\u231c drop \u03c3 \u231d\u1d52\u1d56\u1d49 = drop\u209b \u231c \u03c3 \u231d\u1d52\u1d56\u1d49\n\u231c keep \u03c3 \u231d\u1d52\u1d56\u1d49 = keep\u209b \u231c \u03c3 \u231d\u1d52\u1d56\u1d49\n\n\u2208\u209b : \u2200 {A \u0393 \u0394} \u2192 Sub \u0393 \u0394 \u2192 A \u2208 \u0394 \u2192 Tm \u0393 A\n\u2208\u209b (\u03c3 , t) vz     = t\n\u2208\u209b (\u03c3  , t)(vs v) = \u2208\u209b \u03c3 v\n\nTm\u209b : \u2200 {A \u0393 \u0394} \u2192 Sub \u0393 \u0394 \u2192 Tm \u0394 A \u2192 Tm \u0393 A\nTm\u209b \u03c3 (var v)   = \u2208\u209b \u03c3 v\nTm\u209b \u03c3 (lam t)   = lam (Tm\u209b (keep\u209b \u03c3) t)\nTm\u209b \u03c3 (app f a) = app (Tm\u209b \u03c3 f) (Tm\u209b \u03c3 a)\n\nid\u209b : \u2200 {\u0393} \u2192 Sub \u0393 \u0393\nid\u209b {\u2219}     = \u2219\nid\u209b {\u0393 , A} = (id\u209b {\u0393} \u209b\u2218\u2091 drop id\u2091) , var vz\n\n_\u2218\u209b_ : \u2200 {\u0393 \u0394 \u03a3} \u2192 Sub \u0394 \u03a3 \u2192 Sub \u0393 \u0394 \u2192 Sub \u0393 \u03a3\n\u2219       \u2218\u209b \u03b4 = \u2219\n(\u03c3 , t) \u2218\u209b \u03b4 = \u03c3 \u2218\u209b \u03b4 , Tm\u209b \u03b4 t\n\nass\u209b\u2091\u2091 :\n  \u2200 {\u0393 \u0394 \u03a3 \u039e}(\u03c3 : Sub \u03a3 \u039e)(\u03b4 : OPE \u0394 \u03a3)(\u03bd : OPE \u0393 \u0394)\n  \u2192 (\u03c3 \u209b\u2218\u2091 \u03b4) \u209b\u2218\u2091 \u03bd \u2261 \u03c3 \u209b\u2218\u2091 (\u03b4 \u2218\u2091 \u03bd)\nass\u209b\u2091\u2091 \u2219       \u03b4 \u03bd = refl\nass\u209b\u2091\u2091 (\u03c3 , t) \u03b4 \u03bd = _,_ & ass\u209b\u2091\u2091 \u03c3 \u03b4 \u03bd \u2297 (Tm-\u2218\u2091 \u03b4 \u03bd t \u207b\u00b9)\n\nass\u2091\u209b\u2091 :\n  \u2200 {\u0393 \u0394 \u03a3 \u039e}(\u03c3 : OPE \u03a3 \u039e)(\u03b4 : Sub \u0394 \u03a3)(\u03bd : OPE \u0393 \u0394)\n  \u2192 (\u03c3 \u2091\u2218\u209b \u03b4) \u209b\u2218\u2091 \u03bd \u2261 \u03c3 \u2091\u2218\u209b (\u03b4 \u209b\u2218\u2091 \u03bd)\nass\u2091\u209b\u2091 \u2219        \u03b4       \u03bd = refl\nass\u2091\u209b\u2091 (drop \u03c3) (\u03b4 , t) \u03bd = ass\u2091\u209b\u2091 \u03c3 \u03b4 \u03bd\nass\u2091\u209b\u2091 (keep \u03c3) (\u03b4 , t) \u03bd = (_, Tm\u2091 \u03bd t) & ass\u2091\u209b\u2091 \u03c3 \u03b4 \u03bd\n\nidl\u2091\u209b : \u2200 {\u0393 \u0394}(\u03c3 : Sub \u0393 \u0394) \u2192 id\u2091 \u2091\u2218\u209b \u03c3 \u2261 \u03c3\nidl\u2091\u209b \u2219       = refl\nidl\u2091\u209b (\u03c3 , t) = (_, t) & idl\u2091\u209b \u03c3\n\nidl\u209b\u2091 : \u2200 {\u0393 \u0394}(\u03c3 : OPE \u0393 \u0394) \u2192 id\u209b \u209b\u2218\u2091 \u03c3 \u2261 \u231c \u03c3 \u231d\u1d52\u1d56\u1d49\nidl\u209b\u2091 \u2219        = refl\nidl\u209b\u2091 (drop \u03c3) =\n      ((id\u209b \u209b\u2218\u2091_) \u2218 drop) & idr\u2091 \u03c3 \u207b\u00b9\n    \u25fe ass\u209b\u2091\u2091 id\u209b \u03c3 wk \u207b\u00b9\n    \u25fe drop\u209b & idl\u209b\u2091 \u03c3\nidl\u209b\u2091 (keep \u03c3) =\n  (_, var vz) &\n      (ass\u209b\u2091\u2091 id\u209b wk (keep \u03c3)\n    \u25fe ((id\u209b \u209b\u2218\u2091_) \u2218 drop) & (idl\u2091 \u03c3 \u25fe idr\u2091 \u03c3 \u207b\u00b9)\n    \u25fe ass\u209b\u2091\u2091 id\u209b \u03c3 wk \u207b\u00b9\n    \u25fe (_\u209b\u2218\u2091 wk) & idl\u209b\u2091 \u03c3 )\n\nidr\u2091\u209b : \u2200 {\u0393 \u0394}(\u03c3 : OPE \u0393 \u0394) \u2192 \u03c3 \u2091\u2218\u209b id\u209b \u2261 \u231c \u03c3 \u231d\u1d52\u1d56\u1d49\nidr\u2091\u209b \u2219        = refl\nidr\u2091\u209b (drop \u03c3) = ass\u2091\u209b\u2091 \u03c3 id\u209b wk \u207b\u00b9 \u25fe drop\u209b & idr\u2091\u209b \u03c3\nidr\u2091\u209b (keep \u03c3) = (_, var vz) & (ass\u2091\u209b\u2091 \u03c3 id\u209b wk \u207b\u00b9 \u25fe (_\u209b\u2218\u2091 wk) & idr\u2091\u209b \u03c3)\n\n\u2208-\u2091\u2218\u209b : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : OPE \u0394 \u03a3)(\u03b4 : Sub \u0393 \u0394)(v : A \u2208 \u03a3) \u2192 \u2208\u209b (\u03c3 \u2091\u2218\u209b \u03b4) v \u2261 \u2208\u209b \u03b4 (\u2208\u2091 \u03c3 v)\n\u2208-\u2091\u2218\u209b \u2219        \u03b4       v      = refl\n\u2208-\u2091\u2218\u209b (drop \u03c3) (\u03b4 , t) v      = \u2208-\u2091\u2218\u209b \u03c3 \u03b4 v\n\u2208-\u2091\u2218\u209b (keep \u03c3) (\u03b4 , t) vz     = refl\n\u2208-\u2091\u2218\u209b (keep \u03c3) (\u03b4 , t) (vs v) = \u2208-\u2091\u2218\u209b \u03c3 \u03b4 v\n\nTm-\u2091\u2218\u209b : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : OPE \u0394 \u03a3)(\u03b4 : Sub \u0393 \u0394)(t : Tm \u03a3 A) \u2192 Tm\u209b (\u03c3 \u2091\u2218\u209b \u03b4) t \u2261 Tm\u209b \u03b4 (Tm\u2091 \u03c3 t)\nTm-\u2091\u2218\u209b \u03c3 \u03b4 (var v) = \u2208-\u2091\u2218\u209b \u03c3 \u03b4 v\nTm-\u2091\u2218\u209b \u03c3 \u03b4 (lam t) =\n  lam & ((\u03bb x \u2192 Tm\u209b (x , var vz) t) & ass\u2091\u209b\u2091 \u03c3 \u03b4 wk \u25fe Tm-\u2091\u2218\u209b (keep \u03c3) (keep\u209b \u03b4) t)\nTm-\u2091\u2218\u209b \u03c3 \u03b4 (app f a) = app & Tm-\u2091\u2218\u209b \u03c3 \u03b4 f \u2297 Tm-\u2091\u2218\u209b \u03c3 \u03b4 a\n\n\u2208-\u209b\u2218\u2091 : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : Sub \u0394 \u03a3)(\u03b4 : OPE \u0393 \u0394)(v : A \u2208 \u03a3) \u2192 \u2208\u209b (\u03c3 \u209b\u2218\u2091 \u03b4) v \u2261 Tm\u2091 \u03b4 (\u2208\u209b \u03c3 v)\n\u2208-\u209b\u2218\u2091 (\u03c3 , _) \u03b4 vz     = refl\n\u2208-\u209b\u2218\u2091 (\u03c3 , _) \u03b4 (vs v) = \u2208-\u209b\u2218\u2091 \u03c3 \u03b4 v\n\nTm-\u209b\u2218\u2091 : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : Sub \u0394 \u03a3)(\u03b4 : OPE \u0393 \u0394)(t : Tm \u03a3 A) \u2192 Tm\u209b (\u03c3 \u209b\u2218\u2091 \u03b4) t \u2261 Tm\u2091 \u03b4 (Tm\u209b \u03c3 t)\nTm-\u209b\u2218\u2091 \u03c3 \u03b4 (var v)   = \u2208-\u209b\u2218\u2091 \u03c3 \u03b4 v\nTm-\u209b\u2218\u2091 \u03c3 \u03b4 (lam t)   =\n  lam &\n      ((\u03bb x \u2192 Tm\u209b (x , var vz) t) &\n          (ass\u209b\u2091\u2091 \u03c3 \u03b4 wk\n        \u25fe (\u03c3 \u209b\u2218\u2091_) & (drop & (idr\u2091 \u03b4 \u25fe idl\u2091 \u03b4 \u207b\u00b9))\n        \u25fe ass\u209b\u2091\u2091 \u03c3 wk (keep \u03b4) \u207b\u00b9)\n    \u25fe Tm-\u209b\u2218\u2091 (keep\u209b \u03c3) (keep \u03b4) t)\nTm-\u209b\u2218\u2091 \u03c3 \u03b4 (app f a) = app & Tm-\u209b\u2218\u2091 \u03c3 \u03b4 f \u2297 Tm-\u209b\u2218\u2091 \u03c3 \u03b4 a\n\nass\u209b\u2091\u209b :\n  \u2200 {\u0393 \u0394 \u03a3 \u039e}(\u03c3 : Sub \u03a3 \u039e)(\u03b4 : OPE \u0394 \u03a3)(\u03bd : Sub \u0393 \u0394)\n  \u2192 (\u03c3 \u209b\u2218\u2091 \u03b4) \u2218\u209b \u03bd \u2261 \u03c3 \u2218\u209b (\u03b4 \u2091\u2218\u209b \u03bd)\nass\u209b\u2091\u209b \u2219       \u03b4 \u03bd = refl\nass\u209b\u2091\u209b (\u03c3 , t) \u03b4 \u03bd = _,_ & ass\u209b\u2091\u209b \u03c3 \u03b4 \u03bd \u2297 (Tm-\u2091\u2218\u209b \u03b4 \u03bd t \u207b\u00b9)\n\nass\u209b\u209b\u2091 :\n  \u2200 {\u0393 \u0394 \u03a3 \u039e}(\u03c3 : Sub \u03a3 \u039e)(\u03b4 : Sub \u0394 \u03a3)(\u03bd : OPE \u0393 \u0394)\n  \u2192 (\u03c3 \u2218\u209b \u03b4) \u209b\u2218\u2091 \u03bd \u2261 \u03c3 \u2218\u209b (\u03b4 \u209b\u2218\u2091 \u03bd)\nass\u209b\u209b\u2091 \u2219       \u03b4 \u03bd = refl\nass\u209b\u209b\u2091 (\u03c3 , t) \u03b4 \u03bd = _,_ & ass\u209b\u209b\u2091 \u03c3 \u03b4 \u03bd \u2297 (Tm-\u209b\u2218\u2091 \u03b4 \u03bd t \u207b\u00b9)\n\n\u2208-id\u209b : \u2200 {A \u0393}(v : A \u2208 \u0393) \u2192 \u2208\u209b id\u209b v \u2261 var v\n\u2208-id\u209b vz     = refl\n\u2208-id\u209b (vs v) = \u2208-\u209b\u2218\u2091 id\u209b wk v \u25fe Tm\u2091 wk & \u2208-id\u209b v \u25fe (var \u2218 vs) & \u2208-id\u2091 v\n\n\u2208-\u2218\u209b : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : Sub \u0394 \u03a3)(\u03b4 : Sub \u0393 \u0394)(v : A \u2208 \u03a3) \u2192 \u2208\u209b (\u03c3 \u2218\u209b \u03b4) v \u2261 Tm\u209b \u03b4 (\u2208\u209b \u03c3 v)\n\u2208-\u2218\u209b (\u03c3 , _) \u03b4 vz     = refl\n\u2208-\u2218\u209b (\u03c3 , _) \u03b4 (vs v) = \u2208-\u2218\u209b \u03c3 \u03b4 v\n\nTm-id\u209b : \u2200 {A \u0393}(t : Tm \u0393 A) \u2192 Tm\u209b id\u209b t \u2261 t\nTm-id\u209b (var v)   = \u2208-id\u209b v\nTm-id\u209b (lam t)   = lam & Tm-id\u209b t\nTm-id\u209b (app f a) = app & Tm-id\u209b f \u2297 Tm-id\u209b a\n\nTm-\u2218\u209b : \u2200 {A \u0393 \u0394 \u03a3}(\u03c3 : Sub \u0394 \u03a3)(\u03b4 : Sub \u0393 \u0394)(t : Tm \u03a3 A) \u2192 Tm\u209b (\u03c3 \u2218\u209b \u03b4) t \u2261 Tm\u209b \u03b4 (Tm\u209b \u03c3 t)\nTm-\u2218\u209b \u03c3 \u03b4 (var v)   = \u2208-\u2218\u209b \u03c3 \u03b4 v\nTm-\u2218\u209b \u03c3 \u03b4 (lam t)   =\n  lam &\n      ((\u03bb x \u2192 Tm\u209b (x , var vz) t) &\n          (ass\u209b\u209b\u2091 \u03c3 \u03b4 wk\n        \u25fe (\u03c3 \u2218\u209b_) & (idl\u2091\u209b  (drop\u209b \u03b4) \u207b\u00b9) \u25fe ass\u209b\u2091\u209b \u03c3 wk (keep\u209b \u03b4) \u207b\u00b9)\n    \u25fe Tm-\u2218\u209b (keep\u209b \u03c3) (keep\u209b \u03b4) t)\nTm-\u2218\u209b \u03c3 \u03b4 (app f a) = app & Tm-\u2218\u209b \u03c3 \u03b4 f \u2297 Tm-\u2218\u209b \u03c3 \u03b4 a\n\nidr\u209b : \u2200 {\u0393 \u0394}(\u03c3 : Sub \u0393 \u0394) \u2192 \u03c3 \u2218\u209b id\u209b \u2261 \u03c3\nidr\u209b \u2219       = refl\nidr\u209b (\u03c3 , t) = _,_ & idr\u209b \u03c3 \u2297 Tm-id\u209b t\n\nidl\u209b : \u2200 {\u0393 \u0394}(\u03c3 : Sub \u0393 \u0394) \u2192 id\u209b \u2218\u209b \u03c3 \u2261 \u03c3\nidl\u209b \u2219       = refl\nidl\u209b (\u03c3 , t) = (_, t) & (ass\u209b\u2091\u209b id\u209b wk (\u03c3 , t) \u25fe (id\u209b \u2218\u209b_) & idl\u2091\u209b \u03c3 \u25fe idl\u209b \u03c3)\n\n-- Reduction\n--------------------------------------------------------------------------------\n\ndata _~>_ {\u0393} : \u2200 {A} \u2192 Tm \u0393 A \u2192 Tm \u0393 A \u2192 Set where\n  \u03b2    : \u2200 {A B}(t : Tm (\u0393 , A) B) t'  \u2192 app (lam t) t' ~> Tm\u209b (id\u209b , t') t\n  lam  : \u2200 {A B}{t t' : Tm (\u0393 , A) B}     \u2192 t ~> t' \u2192  lam t   ~> lam t'\n  app\u2081 : \u2200 {A B}{f}{f' : Tm \u0393 (A \u21d2 B)}{a} \u2192 f ~> f' \u2192  app f a ~> app f' a\n  app\u2082 : \u2200 {A B}{f : Tm \u0393 (A \u21d2 B)} {a a'} \u2192 a ~> a' \u2192  app f a ~> app f  a'\ninfix 3 _~>_\n\n~>\u209b : \u2200 {\u0393 \u0394 A}{t t' : Tm \u0393 A}(\u03c3 : Sub \u0394 \u0393) \u2192 t ~> t' \u2192 Tm\u209b \u03c3 t ~> Tm\u209b \u03c3 t'\n~>\u209b \u03c3 (\u03b2 t t') =\n  coe ((app (lam (Tm\u209b (keep\u209b \u03c3) t)) (Tm\u209b \u03c3 t') ~>_) &\n      (Tm-\u2218\u209b (keep\u209b \u03c3) (id\u209b , Tm\u209b \u03c3 t') t \u207b\u00b9\n    \u25fe (\u03bb x \u2192 Tm\u209b (x , Tm\u209b \u03c3 t') t) &\n        (ass\u209b\u2091\u209b \u03c3 wk (id\u209b , Tm\u209b \u03c3 t')\n      \u25fe ((\u03c3 \u2218\u209b_) & idl\u2091\u209b id\u209b \u25fe idr\u209b \u03c3) \u25fe idl\u209b \u03c3 \u207b\u00b9)\n    \u25fe Tm-\u2218\u209b (id\u209b , t') \u03c3 t))\n  (\u03b2 (Tm\u209b (keep\u209b \u03c3) t) (Tm\u209b \u03c3 t'))\n~>\u209b \u03c3 (lam  step) = lam  (~>\u209b (keep\u209b \u03c3) step)\n~>\u209b \u03c3 (app\u2081 step) = app\u2081 (~>\u209b \u03c3 step)\n~>\u209b \u03c3 (app\u2082 step) = app\u2082 (~>\u209b \u03c3 step)\n\n~>\u2091 : \u2200 {\u0393 \u0394 A}{t t' : Tm \u0393 A}(\u03c3 : OPE \u0394 \u0393) \u2192 t ~> t' \u2192 Tm\u2091 \u03c3 t ~> Tm\u2091 \u03c3 t'\n~>\u2091 \u03c3 (\u03b2 t t')    =\n  coe ((app (lam (Tm\u2091 (keep \u03c3) t)) (Tm\u2091 \u03c3 t') ~>_)\n      & (Tm-\u2091\u2218\u209b (keep \u03c3) (id\u209b , Tm\u2091 \u03c3 t') t \u207b\u00b9\n      \u25fe (\u03bb x \u2192 Tm\u209b (x , Tm\u2091 \u03c3 t') t) & (idr\u2091\u209b \u03c3 \u25fe idl\u209b\u2091 \u03c3 \u207b\u00b9)\n      \u25fe Tm-\u209b\u2218\u2091 (id\u209b , t') \u03c3 t))\n  (\u03b2 (Tm\u2091 (keep \u03c3) t) (Tm\u2091 \u03c3 t'))\n~>\u2091 \u03c3 (lam step)  = lam (~>\u2091 (keep \u03c3) step)\n~>\u2091 \u03c3 (app\u2081 step) = app\u2081 (~>\u2091 \u03c3 step)\n~>\u2091 \u03c3 (app\u2082 step) = app\u2082 (~>\u2091 \u03c3 step)\n\nTm\u2091~> :\n  \u2200 {\u0393 \u0394 A}{t : Tm \u0393 A}{\u03c3 : OPE \u0394 \u0393}{t'}\n  \u2192 Tm\u2091 \u03c3 t ~> t' \u2192 \u2203 \u03bb t'' \u2192 (t ~> t'') \u00d7 (Tm\u2091 \u03c3 t'' \u2261 t')\nTm\u2091~> {t = var x} ()\nTm\u2091~> {t = lam t} (lam step) with Tm\u2091~> step\n... | t'' , (p , refl) = lam t'' , lam p , refl\nTm\u2091~> {t = app (var v) a} (app\u2081 ())\nTm\u2091~> {t = app (var v) a} (app\u2082 step) with Tm\u2091~> step\n... | t'' , (p , refl) = app (var v) t'' , app\u2082 p , refl\nTm\u2091~> {t = app (lam f) a} {\u03c3} (\u03b2 _ _) =\n  Tm\u209b (id\u209b , a) f , \u03b2 _ _ ,\n      Tm-\u209b\u2218\u2091 (id\u209b , a) \u03c3 f \u207b\u00b9\n    \u25fe (\u03bb x \u2192  Tm\u209b (x , Tm\u2091 \u03c3 a) f) & (idl\u209b\u2091 \u03c3 \u25fe idr\u2091\u209b \u03c3 \u207b\u00b9)\n    \u25fe Tm-\u2091\u2218\u209b (keep \u03c3) (id\u209b , Tm\u2091 \u03c3 a) f\nTm\u2091~> {t = app (lam f) a}     (app\u2081 (lam step)) with Tm\u2091~> step\n... | t'' , (p , refl) = app (lam t'') a , app\u2081 (lam p) , refl\nTm\u2091~> {t = app (lam f) a}     (app\u2082 step) with Tm\u2091~> step\n... | t'' , (p , refl) = app (lam f) t'' , app\u2082 p , refl\nTm\u2091~> {t = app (app f a) a'}  (app\u2081 step) with Tm\u2091~> step\n... | t'' , (p , refl) = app t'' a' , app\u2081 p , refl\nTm\u2091~> {t = app (app f a) a''} (app\u2082 step) with Tm\u2091~> step\n... | t'' , (p , refl) = app (app f a) t'' , app\u2082 p , refl\n\n-- Strong normalization\/neutrality definition\n--------------------------------------------------------------------------------\n\ndata SN {\u0393 A} (t : Tm \u0393 A) : Set where\n  sn : (\u2200 {t'} \u2192 t ~> t' \u2192 SN t') \u2192 SN t\n\nSN\u2091\u2192 : \u2200 {\u0393 \u0394 A}{t : Tm \u0393 A}(\u03c3 : OPE \u0394 \u0393) \u2192 SN t \u2192 SN (Tm\u2091 \u03c3 t)\nSN\u2091\u2192 \u03c3 (sn s) = sn \u03bb {t'} step \u2192\n  let (t'' , (p , q)) = Tm\u2091~> step in coe (SN & q) (SN\u2091\u2192 \u03c3 (s p))\n\nSN\u2091\u2190 : \u2200 {\u0393 \u0394 A}{t : Tm \u0393 A}(\u03c3 : OPE \u0394 \u0393) \u2192 SN (Tm\u2091 \u03c3 t) \u2192 SN t\nSN\u2091\u2190 \u03c3 (sn s) = sn \u03bb step \u2192 SN\u2091\u2190 \u03c3 (s (~>\u2091 \u03c3 step))\n\nSN-app\u2081 : \u2200 {\u0393 A B}{f : Tm \u0393 (A \u21d2 B)}{a} \u2192 SN (app f a) \u2192 SN f\nSN-app\u2081 (sn s) = sn \u03bb f~>f' \u2192 SN-app\u2081 (s (app\u2081 f~>f'))\n\nneu : \u2200 {\u0393 A} \u2192 Tm \u0393 A \u2192 Set\nneu (lam _)   = \u22a5\nneu _         = \u22a4\n\nneu\u2091 : \u2200 {\u0393 \u0394 A}(\u03c3 : OPE \u0394 \u0393)(t : Tm \u0393 A) \u2192 neu t \u2192 neu (Tm\u2091 \u03c3 t)\nneu\u2091 \u03c3 (lam t)   nt = nt\nneu\u2091 \u03c3 (var v)   nt = tt\nneu\u2091 \u03c3 (app f a) nt = tt\n\n-- The actual proof, by Kripke logical predicate\n--------------------------------------------------------------------------------\n\nTm\u1d3e : \u2200 {\u0393 A} \u2192 Tm \u0393 A \u2192 Set\nTm\u1d3e {\u0393}{\u03b9}     t = SN t\nTm\u1d3e {\u0393}{A \u21d2 B} t = \u2200 {\u0394}(\u03c3 : OPE \u0394 \u0393){a} \u2192 Tm\u1d3e a \u2192 Tm\u1d3e (app (Tm\u2091 \u03c3 t) a)\n\ndata Sub\u1d3e {\u0393} : \u2200 {\u0394} \u2192 Sub \u0393 \u0394 \u2192 Set where\n  \u2219   : Sub\u1d3e \u2219\n  _,_ : \u2200 {A \u0394}{\u03c3 : Sub \u0393 \u0394}{t : Tm \u0393 A}(\u03c3\u1d3e : Sub\u1d3e \u03c3)(t\u1d3e : Tm\u1d3e t) \u2192 Sub\u1d3e (\u03c3 , t)\n\nTm\u1d3e\u2091 : \u2200 {\u0393 \u0394 A}{t : Tm \u0393 A}(\u03c3 : OPE \u0394 \u0393) \u2192 Tm\u1d3e t \u2192 Tm\u1d3e (Tm\u2091 \u03c3 t)\nTm\u1d3e\u2091 {A = \u03b9}        \u03c3 t\u1d3e = SN\u2091\u2192 \u03c3 t\u1d3e\nTm\u1d3e\u2091 {A = A \u21d2 B}{t} \u03c3 t\u1d3e \u03b4 a\u1d3e rewrite Tm-\u2218\u2091 \u03c3 \u03b4 t \u207b\u00b9 = t\u1d3e (\u03c3 \u2218\u2091 \u03b4) a\u1d3e\n\nSub\u1d3e\u2091 : \u2200 {\u0393 \u0394 \u03a3}{\u03c3 : Sub \u0394 \u03a3}(\u03b4 : OPE \u0393 \u0394) \u2192 Sub\u1d3e \u03c3 \u2192 Sub\u1d3e (\u03c3 \u209b\u2218\u2091 \u03b4)\nSub\u1d3e\u2091 \u03c3 \u2219        = \u2219\nSub\u1d3e\u2091 \u03c3 (\u03b4 , t\u1d3e) = Sub\u1d3e\u2091 \u03c3 \u03b4 , Tm\u1d3e\u2091 \u03c3 t\u1d3e\n\n~>\u1d3e : \u2200 {\u0393 A}{t t' : Tm \u0393 A} \u2192 t ~> t' \u2192 Tm\u1d3e t \u2192 Tm\u1d3e t'\n~>\u1d3e {A = \u03b9}     t~>t' (sn t\u02e2\u207f) = t\u02e2\u207f t~>t'\n~>\u1d3e {A = A \u21d2 B} t~>t' t\u1d3e       = \u03bb \u03c3 a\u1d3e \u2192 ~>\u1d3e (app\u2081 (~>\u2091 \u03c3 t~>t')) (t\u1d3e \u03c3 a\u1d3e)\n\nmutual\n  -- quote\n  q\u1d3e : \u2200 {\u0393 A}{t : Tm \u0393 A} \u2192 Tm\u1d3e t \u2192 SN t\n  q\u1d3e {A = \u03b9}     t\u1d3e = t\u1d3e\n  q\u1d3e {A = A \u21d2 B} t\u1d3e = SN\u2091\u2190 wk $ SN-app\u2081 (q\u1d3e $ t\u1d3e wk (u\u1d3e (var vz) (\u03bb ())))\n\n  -- unquote\n  u\u1d3e : \u2200 {\u0393 A}(t : Tm \u0393 A){nt : neu t} \u2192 (\u2200 {t'} \u2192 t ~> t' \u2192 Tm\u1d3e t') \u2192 Tm\u1d3e t\n  u\u1d3e {\u0393} {A = \u03b9} t      f     = sn f\n  u\u1d3e {\u0393} {A \u21d2 B} t {nt} f {\u0394} \u03c3 {a} a\u1d3e =\n    u\u1d3e (app (Tm\u2091 \u03c3 t) a) (go (Tm\u2091 \u03c3 t) (neu\u2091 \u03c3 t nt) f' a a\u1d3e (q\u1d3e a\u1d3e))\n    where\n      f' : \u2200 {t'} \u2192 Tm\u2091 \u03c3 t ~> t' \u2192 Tm\u1d3e t'\n      f' step \u03b4 a\u1d3e with Tm\u2091~> step\n      ... | t'' , step' , refl rewrite Tm-\u2218\u2091 \u03c3 \u03b4 t'' \u207b\u00b9 = f step' (\u03c3 \u2218\u2091 \u03b4) a\u1d3e\n\n      go :\n        \u2200 {\u0393 A B}(t : Tm \u0393 (A \u21d2 B)) \u2192 neu t \u2192 (\u2200 {t'} \u2192 t ~> t' \u2192 Tm\u1d3e t')\n        \u2192 \u2200 a \u2192 Tm\u1d3e a \u2192 SN a \u2192 \u2200 {t'} \u2192 app t a ~> t' \u2192 Tm\u1d3e t'\n      go _ () _ _ _ _ (\u03b2 _ _)\n      go t nt f a a\u1d3e sna (app\u2081 {f' = f'} step) =\n        coe ((\u03bb x \u2192 Tm\u1d3e (app x a)) & Tm-id\u2091 f') (f step id\u2091 a\u1d3e)\n      go t nt f a a\u1d3e (sn a\u02e2\u207f) (app\u2082 {a' = a'} step) =\n        u\u1d3e (app t a') (go t nt f a' (~>\u1d3e step a\u1d3e) (a\u02e2\u207f step))\n\nfundThm-\u2208 : \u2200 {\u0393 A}(v : A \u2208 \u0393) \u2192 \u2200 {\u0394}{\u03c3 : Sub \u0394 \u0393} \u2192 Sub\u1d3e \u03c3 \u2192 Tm\u1d3e (\u2208\u209b \u03c3 v)\nfundThm-\u2208 vz     (\u03c3\u1d3e , t\u1d3e) = t\u1d3e\nfundThm-\u2208 (vs v) (\u03c3\u1d3e , t\u1d3e) = fundThm-\u2208 v \u03c3\u1d3e\n\nfundThm-lam :\n  \u2200 {\u0393 A B}\n    (t : Tm (\u0393 , A) B)\n  \u2192 SN t\n  \u2192 (\u2200 {a} \u2192 Tm\u1d3e a \u2192 Tm\u1d3e (Tm\u209b (id\u209b , a) t))\n  \u2192 \u2200 a \u2192 SN a \u2192 Tm\u1d3e a \u2192 Tm\u1d3e (app (lam t) a)\nfundThm-lam {\u0393} t (sn t\u02e2\u207f) hyp a (sn a\u02e2\u207f) a\u1d3e = u\u1d3e (app (lam t) a)\n  \u03bb {(\u03b2 _ _) \u2192 hyp a\u1d3e;\n     (app\u2081 (lam {t' = t'} t~>t')) \u2192\n       fundThm-lam t' (t\u02e2\u207f t~>t') (\u03bb a\u1d3e \u2192 ~>\u1d3e (~>\u209b _ t~>t') (hyp a\u1d3e)) a (sn a\u02e2\u207f) a\u1d3e;\n     (app\u2082 a~>a') \u2192\n       fundThm-lam t (sn t\u02e2\u207f) hyp _ (a\u02e2\u207f a~>a') (~>\u1d3e a~>a' a\u1d3e)}\n\nfundThm : \u2200 {\u0393 A}(t : Tm \u0393 A) \u2192 \u2200 {\u0394}{\u03c3 : Sub \u0394 \u0393} \u2192 Sub\u1d3e \u03c3 \u2192 Tm\u1d3e (Tm\u209b \u03c3 t)\nfundThm (var v) \u03c3\u1d3e = fundThm-\u2208 v \u03c3\u1d3e\nfundThm (lam {A} t) {\u03c3 = \u03c3} \u03c3\u1d3e \u03b4 {a} a\u1d3e\n  rewrite Tm-\u209b\u2218\u2091 (keep\u209b \u03c3) (keep \u03b4) t \u207b\u00b9 | ass\u209b\u2091\u2091 \u03c3 (wk {A}) (keep \u03b4) | idl\u2091 \u03b4\n  = fundThm-lam\n      (Tm\u209b (\u03c3 \u209b\u2218\u2091 drop \u03b4 , var vz) t)\n      (q\u1d3e (fundThm t (Sub\u1d3e\u2091 (drop \u03b4) \u03c3\u1d3e , u\u1d3e (var vz) (\u03bb ()))))\n      (\u03bb a\u1d3e \u2192 coe (Tm\u1d3e & sub-sub-lem) (fundThm t (Sub\u1d3e\u2091 \u03b4 \u03c3\u1d3e , a\u1d3e)))\n      a (q\u1d3e a\u1d3e) a\u1d3e\n  where\n    sub-sub-lem : \u2200 {a} \u2192 Tm\u209b (\u03c3 \u209b\u2218\u2091 \u03b4 , a) t \u2261 Tm\u209b (id\u209b , a) (Tm\u209b (\u03c3 \u209b\u2218\u2091 drop \u03b4 , var vz) t)\n    sub-sub-lem {a} =\n        (\u03bb x \u2192 Tm\u209b (x , a) t) &\n          (idr\u209b (\u03c3 \u209b\u2218\u2091 \u03b4) \u207b\u00b9 \u25fe ass\u209b\u2091\u209b \u03c3 \u03b4 id\u209b \u25fe ass\u209b\u2091\u209b \u03c3 (drop \u03b4) (id\u209b , a) \u207b\u00b9)\n      \u25fe Tm-\u2218\u209b (\u03c3 \u209b\u2218\u2091 drop \u03b4 , var vz) (id\u209b , a) t\nfundThm (app f a) {\u03c3 = \u03c3} \u03c3\u1d3e\n  rewrite Tm-id\u2091 (Tm\u209b \u03c3 f) \u207b\u00b9\n  = fundThm f \u03c3\u1d3e id\u2091 (fundThm a \u03c3\u1d3e)\n\nid\u209b\u1d3e : \u2200 {\u0393} \u2192 Sub\u1d3e (id\u209b {\u0393})\nid\u209b\u1d3e {\u2219}     = \u2219\nid\u209b\u1d3e {\u0393 , A} = Sub\u1d3e\u2091 wk id\u209b\u1d3e , u\u1d3e (var vz) (\u03bb ())\n\nstrongNorm : \u2200 {\u0393 A}(t : Tm \u0393 A) \u2192 SN t\nstrongNorm t = q\u1d3e (coe (Tm\u1d3e & Tm-id\u209b t) (fundThm t id\u209b\u1d3e))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'extra\/KovacsSTLCnorm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"f47287f62994ad3c2e51b3d5d67c81705388e684","subject":"Alternative way to perform levitation.","message":"Alternative way to perform levitation.\n\nDagand typically levitates Desc and elimDesc,\nkeeping El, Hyps, and prove primitive.\n\nInstead, we can levitate El, Hyps, and prove,\nkeeping Desc and elimDesc primitive.\n\nThis alternative version has the advantage that it can be\nperformed in an intrinsically typed theory, whereas Dagand\nlevitation is limited to extrinsic typing.\n\nThe disadvantage is that Desc values must be eliminated with\nelimDesc instead of ind.\n\nNote that Agda cannot tell that Fix is strictly positive.\nEl expands to elimDesc, and Agda cannot see that the \"Rec\"\nbranch of elimDesc is strictly positive. In fact, the helper\nfunction El' is defined such that its arguments do not contain\noccurrences of Fix, but its return type does. This is a positive\noccurrence in the constructor,\nsimiliar to \"foo : (Bar \u2192 Baz \u2192 Foo) \u2192 Foo\".\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev2.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev2.agda","new_contents":"{-# OPTIONS --type-in-type --no-positivity-check #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev2 where\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nelimDesc : {I : Set} (P : Desc I \u2192 Set)\n  (pend : (i : I) \u2192 P (End i))\n  (prec : (i : I) (D : Desc I) (pd : P D) \u2192 P (Rec i D))\n  (parg : (A : Set) (B : A \u2192 Desc I) (pb : (a : A) \u2192 P (B a)) \u2192 P (Arg A B))\n  (D : Desc I) \u2192 P D\nelimDesc P pend prec parg (End i) = pend i\nelimDesc P pend prec parg (Rec i D) = prec i D (elimDesc P pend prec parg D)\nelimDesc P pend prec parg (Arg A B) = parg A B (\u03bb a \u2192 elimDesc P pend prec parg (B a))\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl' : {I : Set} (X : ISet I) (i : I) (D : Desc I) \u2192 Set\nEl' X i = elimDesc _\n  (\u03bb j \u2192 j \u2261 i)\n  (\u03bb j D ih \u2192 X j \u00d7 ih)\n  (\u03bb A B ih \u2192 \u03a3 A ih)\n\nEl : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nEl D X i = El' X i D\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps = elimDesc _\n  (\u03bb j X P i q \u2192 \u22a4)\n  (\u03bb j D ih X P i x,xs \u2192 P j (proj\u2081 x,xs) \u00d7 ih X P i (proj\u2082 x,xs))\n  (\u03bb A B ih X P i a,b \u2192 ih (proj\u2081 a,b) X P i (proj\u2082 a,b))\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (D : Desc I) (i : I) : Set where\n  init : El D (\u03bc D) i \u2192 \u03bc D i\n\n----------------------------------------------------------------------\n\nprove : {I : Set} (D : Desc I) (X : I \u2192 Set)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (i : I) (x : X i) \u2192 P i x)\n  (i : I) (xs : El D X i) \u2192 Hyps D X P i xs\nprove = elimDesc _\n  (\u03bb j X P \u03b1 i q \u2192 tt)\n  (\u03bb j D ih X P \u03b1 i x,xs \u2192 \u03b1 j (proj\u2081 x,xs) , ih X P \u03b1 i (proj\u2082 x,xs))\n  (\u03bb A B ih X P \u03b1 i a,xs \u2192 ih (proj\u2081 a,xs) X P \u03b1 i (proj\u2082 a,xs))\n\n{-# NO_TERMINATION_CHECK #-}\nind : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El D (\u03bc D) i) (ihs : Hyps D (\u03bc D) P i xs) \u2192 P i (init xs))\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nind D P \u03b1 i (init xs) = \u03b1 i xs $\n  prove D (\u03bc D) P (ind D P \u03b1) i xs\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\nData : Set\nData =\n  \u03a3 Set \u03bb I \u2192\n  \u03a3 Enum \u03bb E \u2192\n  Branches E (\u03bb _ \u2192 Desc I)\n\nDataI : Data \u2192 Set\nDataI (I , E , B) = I\n\nDataE : Data \u2192 Enum\nDataE (I , E , B) = E\n\nDataT : Data \u2192 Set\nDataT R = Tag (DataE R)\n\nDataB : (R : Data) \u2192 Branches (DataE R) (\u03bb _ \u2192 Desc (DataI R))\nDataB (I , E , B) = B\n\nDataD : (R : Data) \u2192 Desc (DataI R)\nDataD R = Arg (DataT R) (case (\u03bb _ \u2192 Desc (DataI R)) (DataB R))\n\ncaseR : (R : Data) \u2192 DataT R \u2192 Desc (DataI R)\ncaseR R = case (\u03bb _ \u2192 Desc (DataI R)) (DataB R)\n\n----------------------------------------------------------------------\n\ninj : (R : Data)\n  \u2192 let D = DataD R\n  in CurriedEl D (\u03bc D)\ninj R = let D = DataD R\n  in curryEl D (\u03bc D) init\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (D : Desc I)\n  (P : (i : I) \u2192 \u03bc D i \u2192 Set)\n  (f : CurriedHyps D (\u03bc D) P init)\n  (i : I)\n  (x : \u03bc D i)\n  \u2192 P i x\nindCurried D P f i x = ind D P (uncurryHyps D (\u03bc D) P init f) i x\n\nSumCurriedHyps : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 DataT R \u2192 Set\nSumCurriedHyps R P t =\n  CurriedHyps (caseR R t) (\u03bc (DataD R)) P (\u03bb xs \u2192 init (t , xs))\n\nelimUncurried : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 Branches (DataE R) (SumCurriedHyps R P)\n  \u2192 \u2200 i (x : \u03bc D i) \u2192 P i x\nelimUncurried R P cs i x =\n  indCurried (DataD R) P\n    (case (SumCurriedHyps R P) cs)\n    i x\n\nelim : (R : Data)\n  \u2192 let D = DataD R in\n  (P : \u2200 i \u2192 \u03bc D i \u2192 Set)\n  \u2192 CurriedBranches (DataE R)\n      (SumCurriedHyps R P)\n      (\u2200 i (x : \u03bc D i) \u2192 P i x)\nelim R P = curryBranches (elimUncurried R P)\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115R : Data\n\u2115R = \u22a4 , \u2115E\n  , End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = DataD \u2115R\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115D\n\nzero : \u2115 tt\nzero = init (zeroT , refl)\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init (sucT , n , refl)\n\nVecR : (A : Set) \u2192 Data\nVecR A = (\u2115 tt) , VecE\n  , End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = DataD (VecR A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc (VecD A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init (nilT , refl)\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init (consT , n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj (VecR A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj (VecR A) consT\n\n----------------------------------------------------------------------\n\nadd : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nadd = elim \u2115R _\n  (\u03bb n \u2192 n)\n  (\u03bb m ih n \u2192 suc (ih n))\n  tt\n\nmult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\nmult = elim \u2115R _\n  (\u03bb n \u2192 zero)\n  (\u03bb m ih n \u2192 add n (ih n))\n  tt\n\nappend : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\nappend A = elim (VecR A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n  (\u03bb n ys \u2192 ys)\n  (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\nconcat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\nconcat A m = elim (VecR (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n  (nil A)\n  (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formalization\/agda\/Spire\/Examples\/PropLev2.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d6569c4a9a96591877a709d814d4447336de1484","subject":"lib\/Data\/ShapePolymorphism.agda","message":"lib\/Data\/ShapePolymorphism.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/ShapePolymorphism.agda","new_file":"lib\/Data\/ShapePolymorphism.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Function.NP\nopen import Level.NP\n\nmodule Data.ShapePolymorphism where\n\ndata \u2610 {a}(A : \u2605_ a) : \u2605_ a where\n  [_] : ..(x : A) \u2192 \u2610 A\n\nun\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} \u2192 (..(x : A) \u2192 B [ x ]) \u2192 \u03a0 (\u2610 A) B\nun\u2610 f [ x ] = f x\n\n\u03a0\u2610 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : \u2610 A \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u2610 A B = (x : \u2610 A) \u2192 B x\n\n_>>\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2610 A \u2192 \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 B [ x ]) \u2192 B x\n[ x ] >>\u2610 f = f x\n\n-- This is not a proper map since the function takes a ..A.\nmap\u2610 : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} \u2192 (..(x : A) \u2192 B) \u2192 \u2610 A \u2192 \u2610 B\nmap\u2610 f [ x ] = [ f x ]\n\n-- This does not work since a \u2610 has to be relevant when eliminated.\n-- join\u2610 : \u2200 {a}{A : \u2605_ a} \u2192 \u2610 (\u2610 A) \u2192 \u2610 A\n\n{- This is not a proper bind either.\n_>>=\u2610_ : \u2200 {a b}{A : \u2605_ a}{B : \u2605_ b} (x : \u2610 A) \u2192 (..(x : A) \u2192 \u2610 B) \u2192 \u2610 B\n_>>=\u2610_ = _>>\u2610_\n-}\n\ndata _\u2261\u2610_ {a} {A : \u2605_ a} (x : A) : ..(y : A) \u2192 \u2605_ a where\n  refl : x \u2261\u2610 x\n\nrecord S\u27e8_\u27e9 {a} {A : \u2605_ a} ..(x : A) : \u2605_ a where\n  constructor S[_\u2225_]\n  field\n    unS : A\n    isS : unS \u2261\u2610 x\nopen S\u27e8_\u27e9 public\n\npattern S[_] x = S[ x \u2225 refl ]\n{-\nS[_] : \u2200 {a}{A : \u2605_ a} (x : A) \u2192 S\u27e8 x \u27e9\nS[ x ] = S[ x \u2225 refl ]\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/ShapePolymorphism.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e905765b366d5d345d3b5a29d4924ee9010d15b8","subject":"+Opaque","message":"+Opaque\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Opaque.agda","new_file":"lib\/Opaque.agda","new_contents":"open import Data.String using (String)\nopen import Relation.Binary.PropositionalEquality\n\nmodule Opaque {a} {A : Set a} where\n\n{-\nabstract\n  opaque : String \u2192 A \u2192 A\n  opaque _ = \u03bb x \u2192 x\n\n  opaque-rule : \u2200 {s} x \u2192 opaque s x \u2261 x\n  opaque-rule x = refl\n-}\n\npostulate\n  opaque : String \u2192 A \u2192 A\n  opaque-rule : \u2200 {s} x \u2192 opaque s x \u2261 x\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Opaque.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c0cca81d816bde552b2bf2c28cda87d57722aa98","subject":"Helios.agda","message":"Helios.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Helios.agda","new_file":"Helios.agda","new_contents":"open import Type\nopen import Data.List\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\n{-\nopen import Data.Zero\nopen import Data.One\nopen import Data.Nat\n-}\nopen import Data.Two\nopen import Control.Strategy\n\nmodule Helios\n  (VoterId : \u2605)\n  (PublicKey : \u2605)\n  (SecretKey : \u2605)\n  (Vote   : \u2605)\n  (Ballot : \u2605)\n  (Decryption-share : \u2605)\n  (PublicInfo : \u2605)\n  where\n\nBallots = List (VoterId \u00d7 Ballot)\nDecryption-shares = List Decryption-share\n\nrecord BB : \u2605 where\n  constructor <_,,_,,_>\n  field\n    Y-pk              : PublicKey\n    ballots           : Ballots\n    decryption-shares : Decryption-shares\nopen BB public\n\nmodule Nested\n  (R-setup : \u2605)\n  (setup : R-setup \u2192 PublicKey \u00d7 {-List-} SecretKey)\n  (R-vote : \u2605)\n\n  (vote : PublicKey \u2192 R-vote \u2192 VoterId \u2192 Vote \u2192 Ballot)\n  (validate : BB \u2192 VoterId \u00d7 Ballot \u2192 \ud835\udfda)\n\n  (Tally : \u2605) -- could be a valid tally or a special symbol \u22a5\n  (dec-shares : PublicKey \u2192 Ballots \u2192 {-List-} SecretKey \u2192 Decryption-shares)\n  (result : BB \u2192 Tally)\n\n  (R-adversary : \u2605)\n  where\n\n  tally : BB \u2192 {-List-} SecretKey \u2192 BB\n  tally < Y ,, bs ,, ds > sks = < Y ,, bs ,, dec-shares Y bs sks >\n\n  data Q : \u2605 where\n    ask-vote   : (v_ : Vote \u00b2) \u2192 Q\n    ask-ballot : (b  : Ballot) \u2192 Q\n\n  VotingPhase = List (BB \u2192 VoterId \u00d7 Q)\n\n  add-ballot : VoterId \u2192 Ballot \u2192 BB \u2192 BB\n  add-ballot vid b bb = record bb { ballots = (vid , b) \u2237 ballots bb }\n\n  module Run (\u03b2 : \ud835\udfda)\n             (pk : PublicKey)\n             (r-votes : VoterId \u2192 R-vote \u00b2)\n           where\n\n    run-query : VoterId \u2192 Q \u2192 BB \u00b2 \u2192 BB \u00b2\n    run-query vid (ask-vote v_)  bb_ i = case validate (bb_ i) (vid , b)\n                                          0: bb_ i\n                                          1: add-ballot vid b (bb_ i)\n      where\n        b : Ballot\n        b = vote pk (r-votes vid i) vid (v i)\n\n    run-query vid (ask-ballot b) bb_ i =\n        case validate (bb_ \u03b2) (vid , b)\n          0: bb_ i\n          1: case validate (bb_ (not \u03b2)) (vid , b)\n               0: case i\n                    0: add-ballot vid b (bb_ 0\u2082)\n                    1: bb_ 1\u2082\n               1: add-ballot vid b (bb_ i)\n\n    run-voting-phase : VotingPhase \u2192 BB \u00b2 \u2192 BB \u00b2\n    run-voting-phase []       bb_ = bb_\n    run-voting-phase (q \u2237 qs) bb_ = run-voting-phase qs (uncurry run-query (q (bb_ \u03b2)) bb_)\n\n  Adversary = R-adversary \u2192\n              PublicKey \u2192\n              VotingPhase \u00d7\n              (BB \u2192\n              \ud835\udfda)\n\n  module Game (\u03b2 : \ud835\udfda)\n              (r-setup : R-setup)\n              (r-votes : VoterId \u2192 R-vote \u00b2)\n              (r-adversary : R-adversary)\n              (A : Adversary)\n              where\n    pksk = setup r-setup\n    pk = fst pksk\n    sk = snd pksk\n\n    BB-setup : BB\n    BB-setup = < pk ,, [] ,, [] >\n\n    BB\u00b2-setup : BB \u00b2\n    BB\u00b2-setup _ = BB-setup\n\n    A-setup = A r-adversary pk\n\n    A-voting-phase : VotingPhase\n    A-voting-phase = fst A-setup\n\n    open Run \u03b2 pk r-votes\n\n    exp : \ud835\udfda\n    exp = snd A-setup (run-voting-phase A-voting-phase BB\u00b2-setup \u03b2)\n\n    game = exp == \u03b2\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Helios.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e9c1c0a01bb344e3c264d116f3888c2e274e2366","subject":"use lambda literal","message":"use lambda literal\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"02b7dc1fcde2d5ea5944dc426ae4920689962b5d","subject":"add example of non-regular traversable functor","message":"add example of non-regular traversable functor\n","repos":"yfcai\/CREG","old_file":"experiments\/NonregularTraversable.agda","new_file":"experiments\/NonregularTraversable.agda","new_contents":"-- A traversable functor that is not regular.\n\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Product\n\nrecord Applicative : Set\u2081 where\n  constructor\n    appl\n  field\n    Map   : Set \u2192 Set\n    pure  : \u2200 {A} \u2192 A \u2192 Map A\n    call  : \u2200 {A B} \u2192 Map (A \u2192 B) \u2192 Map A \u2192 Map B\n\n\n\n\nTraversable : (Set \u2192 Set) \u2192 Set\u2081\nTraversable F = (G : Applicative) \u2192\n                \u2200 {A B} \u2192 (A \u2192 Map G B) \u2192 F A \u2192 Map G (F B)\n  where open Applicative\n\n\ninfixr 10 _^_\n\n-- exponents for naturals\n_^_ : \u2115 \u2192 \u2115 \u2192 \u2115\nm ^ zero = 1\nm ^ suc n = m * (m ^ n)\n\n-- a regular container\nRegCon : Set \u2192 Set\nRegCon A = \u03a3 \u2115 (Vec A)\n\n-- a regular traversal\nregTrav : Traversable RegCon\n\nregTrav G f (0 , []) = pure G ((0 , []))\n  where open Applicative\n\nregTrav G f (suc n , x \u2237 xs) =\n  call G (call G\n    (pure G (\u03bb y nys \u2192 (suc (proj\u2081 nys) , y \u2237 proj\u2082 nys)))\n    (f x)) (regTrav G f (n , xs))\n  where open Applicative\n\n-- a nonregular language: 0, 10, 1100, 111000, ....\nnonreg : \u2115 \u2192 \u2115\nnonreg n = let 2n = 2 ^ n in 2n * pred 2n\n\n-- a nonregular finitary container\nNonregCon : Set \u2192 Set\nNonregCon A = \u03a3 \u2115 (\u03bb n \u2192 Vec A (nonreg n))\n\nfmap : \u2200 {A B} (G : Applicative) \u2192 (A \u2192 B) \u2192\n       let open Applicative in Map G A \u2192 Map G B\nfmap G f gx = call G (pure G f) gx\n  where open Applicative\n\n-- a nonregular traversable functor\nnonregTrav : Traversable NonregCon\nnonregTrav G f (n , xs) =\n  let\n    gnnys = regTrav G f ((nonreg n , xs))\n  in\n    fmap G (\u03bb{(nn , ys) \u2192 (n , {!ys!})}) gnnys\n    -- should be well-typed but is not\n\n\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'experiments\/NonregularTraversable.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9488a5a17110cc138c52b801cc5f812ab7bc640c","subject":"Added a README file","message":"Added a README file\n","repos":"crypto-agda\/explore","old_file":"lib\/EXPLORE-README.agda","new_file":"lib\/EXPLORE-README.agda","new_contents":"module EXPLORE-README where\n\nopen import Explore.Type\nopen import Explore.Explorable\nopen import Explore.Sum\n\n-- Examples\nopen import Explore.Sum\nopen import Explore.Product\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/EXPLORE-README.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3dc7752903a7b2b2720fac842be8cd0a05c79342","subject":"markov.agda","message":"markov.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"markov.agda","new_file":"markov.agda","new_contents":"Trans R S = S \u2192 M R S\n          = S \u2192 R \u2192 S\n          = W R S \u2192 S\n          = R \u00d7 S \u2192 S\n\nTrans composition\n\nTrans choice\n\nTransFunction R S = R \u2192 S\nTransFunction = M\n\nif<_then_else_ : [0,1] \u2192 S \u2192 S \u2192 ([0,1] \u2192 S)\n(if< p then x else y) q = if q <? p then x else y\n\nwithPr[_]then_else_ : [0,1] \u2192 Trans R S \u2192 Trans R S \u2192 Trans ([0,1] \u00d7 R) S\nwithPr[ p ]then x else y = rand >>= if< p then x else y\n\ndata Machine R S ...\n\nrun : Machine R S \u2192 Trans R S\nrun ...\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'markov.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"00d73fc830a215c3cd9c60fc79da7cefd6f48bf0","subject":"cyclic10.agda","message":"cyclic10.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"cyclic10.agda","new_file":"cyclic10.agda","new_contents":"module cyclic10 where\n\n-- {{{\nopen import Function.NP\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Explicits\nopen import Relation.Binary.PropositionalEquality.NP\n-- }}}\n\n-- Addition is modulo 11\n-- Multiplication is modulo 11\n\ndata M10 : Set where\n  1# 2# 3# 4# 5# 6# 7# 8# 9# A# : M10\n\ndata C11 : Set where\n  0# 1# 2# 3# 4# 5# 6# 7# 8# 9# A# : C11\n\npattern 10# = A#\n\n1+_ 2+_ 3+_ 4+_ 5+_ 6+_ 7+_ 8+_ 9+_ A+_ : Op\u2081 C11\n1+ 0# = 1#\n1+ 1# = 2#\n1+ 2# = 3#\n1+ 3# = 4#\n1+ 4# = 5#\n1+ 5# = 6#\n1+ 6# = 7#\n1+ 7# = 8#\n1+ 8# = 9#\n1+ 9# = A#\n1+ A# = 0#\n\n2+ x = 1+ 1+ x\n3+ x = 1+ 2+ x\n4+ x = 1+ 3+ x\n5+ x = 1+ 4+ x\n6+ x = 1+ 5+ x\n7+ x = 1+ 6+ x\n8+ x = 1+ 7+ x\n9+ x = 1+ 8+ x\nA+ x = 1+ 9+ x\n\ninfixl 4 _+_\n_+_ : Op\u2082 C11\n0# + y =    y\n1# + y = 1+ y\n2# + y = 2+ y\n3# + y = 3+ y\n4# + y = 4+ y\n5# + y = 5+ y\n6# + y = 6+ y\n7# + y = 7+ y\n8# + y = 8+ y\n9# + y = 9+ y\nA# + y = A+ y\n\n2*_ 3*_ 4*_ 5*_ 6*_ 7*_ 8*_ 9*_ A*_ : Op\u2081 C11\n\n2* 0# = 0#\n2* 1# = 2#\n2* 2# = 4#\n2* 3# = 6#\n2* 4# = 8#\n2* 5# = A#\n2* 6# = 1#\n2* 7# = 3#\n2* 8# = 5#\n2* 9# = 7#\n2* A# = 9#\n\n3* 0# = 0#\n3* 1# = 3#\n3* 2# = 6#\n3* 3# = 9#\n3* 4# = 1#\n3* 5# = 4#\n3* 6# = 7#\n3* 7# = A#\n3* 8# = 2#\n3* 9# = 5#\n3* A# = 8#\n\n5* 0# = 0#\n5* 1# = 5#\n5* 2# = A#\n5* 3# = 4#\n5* 4# = 9#\n5* 5# = 3#\n5* 6# = 8#\n5* 7# = 2#\n5* 8# = 7#\n5* 9# = 1#\n5* A# = 6#\n\n7* 0# = 0#\n7* 1# = 7#\n7* 2# = 3#\n7* 3# = A#\n7* 4# = 6#\n7* 5# = 2#\n7* 6# = 9#\n7* 7# = 5#\n7* 8# = 1#\n7* 9# = 8#\n7* A# = 4#\n\n4* x = 2* 2* x\n6* x = 2* 3* x\n8* x = 2* 4* x\n9* x = 3* 3* x\nA* x = 2* 5* x\n\n2*-spec : \u2200 x \u2192 2* x \u2261 x + x\n2*-spec 0# = refl\n2*-spec 1# = refl\n2*-spec 2# = refl\n2*-spec 3# = refl\n2*-spec 4# = refl\n2*-spec 5# = refl\n2*-spec 6# = refl\n2*-spec 7# = refl\n2*-spec 8# = refl\n2*-spec 9# = refl\n2*-spec A# = refl\n\n3*-spec : \u2200 x \u2192 3* x \u2261 x + x + x\n-- {{{\n3*-spec 0# = refl\n3*-spec 1# = refl\n3*-spec 2# = refl\n3*-spec 3# = refl\n3*-spec 4# = refl\n3*-spec 5# = refl\n3*-spec 6# = refl\n3*-spec 7# = refl\n3*-spec 8# = refl\n3*-spec 9# = refl\n3*-spec A# = refl\n-- }}}\n\n_*_ : Op\u2082 C11\n0# * y = 0#\n1# * y = y\n2# * y = 2* y\n3# * y = 3* y\n4# * y = 4* y\n5# * y = 5* y\n6# * y = 6* y\n7# * y = 7* y\n8# * y = 8* y\n9# * y = 9* y\nA# * y = A* y\n\n*0-zero : \u2200 x \u2192 x * 0# \u2261 0#\n*0-zero 0# = refl\n*0-zero 1# = refl\n*0-zero 2# = refl\n*0-zero 3# = refl\n*0-zero 4# = refl\n*0-zero 5# = refl\n*0-zero 6# = refl\n*0-zero 7# = refl\n*0-zero 8# = refl\n*0-zero 9# = refl\n*0-zero A# = refl\n\n1*-id : \u2200 x \u2192 x * 1# \u2261 x\n1*-id 0# = refl\n1*-id 1# = refl\n1*-id 2# = refl\n1*-id 3# = refl\n1*-id 4# = refl\n1*-id 5# = refl\n1*-id 6# = refl\n1*-id 7# = refl\n1*-id 8# = refl\n1*-id 9# = refl\n1*-id A# = refl\n\n1*-comm : \u2200 x \u2192 1# * x \u2261 x * 1#\n1*-comm x = ! 1*-id x\n\n2*-comm : \u2200 x \u2192 2* x \u2261 x * 2#\n2*-comm 0# = refl\n2*-comm 1# = refl\n2*-comm 2# = refl\n2*-comm 3# = refl\n2*-comm 4# = refl\n2*-comm 5# = refl\n2*-comm 6# = refl\n2*-comm 7# = refl\n2*-comm 8# = refl\n2*-comm 9# = refl\n2*-comm A# = refl\n\n3*-comm : \u2200 x \u2192 3* x \u2261 x * 3#\n3*-comm 0# = refl\n3*-comm 1# = refl\n3*-comm 2# = refl\n3*-comm 3# = refl\n3*-comm 4# = refl\n3*-comm 5# = refl\n3*-comm 6# = refl\n3*-comm 7# = refl\n3*-comm 8# = refl\n3*-comm 9# = refl\n3*-comm A# = refl\n\n4*-comm : \u2200 x \u2192 4* x \u2261 x * 4#\n4*-comm 0# = refl\n4*-comm 1# = refl\n4*-comm 2# = refl\n4*-comm 3# = refl\n4*-comm 4# = refl\n4*-comm 5# = refl\n4*-comm 6# = refl\n4*-comm 7# = refl\n4*-comm 8# = refl\n4*-comm 9# = refl\n4*-comm A# = refl\n\n5*-comm : \u2200 x \u2192 5* x \u2261 x * 5#\n5*-comm 0# = refl\n5*-comm 1# = refl\n5*-comm 2# = refl\n5*-comm 3# = refl\n5*-comm 4# = refl\n5*-comm 5# = refl\n5*-comm 6# = refl\n5*-comm 7# = refl\n5*-comm 8# = refl\n5*-comm 9# = refl\n5*-comm A# = refl\n\n6*-comm : \u2200 x \u2192 6* x \u2261 x * 6#\n6*-comm 0# = refl\n6*-comm 1# = refl\n6*-comm 2# = refl\n6*-comm 3# = refl\n6*-comm 4# = refl\n6*-comm 5# = refl\n6*-comm 6# = refl\n6*-comm 7# = refl\n6*-comm 8# = refl\n6*-comm 9# = refl\n6*-comm A# = refl\n\n7*-comm : \u2200 x \u2192 7* x \u2261 x * 7#\n7*-comm 0# = refl\n7*-comm 1# = refl\n7*-comm 2# = refl\n7*-comm 3# = refl\n7*-comm 4# = refl\n7*-comm 5# = refl\n7*-comm 6# = refl\n7*-comm 7# = refl\n7*-comm 8# = refl\n7*-comm 9# = refl\n7*-comm A# = refl\n\n8*-comm : \u2200 x \u2192 8* x \u2261 x * 8#\n8*-comm 0# = refl\n8*-comm 1# = refl\n8*-comm 2# = refl\n8*-comm 3# = refl\n8*-comm 4# = refl\n8*-comm 5# = refl\n8*-comm 6# = refl\n8*-comm 7# = refl\n8*-comm 8# = refl\n8*-comm 9# = refl\n8*-comm A# = refl\n\n9*-comm : \u2200 x \u2192 9* x \u2261 x * 9#\n9*-comm 0# = refl\n9*-comm 1# = refl\n9*-comm 2# = refl\n9*-comm 3# = refl\n9*-comm 4# = refl\n9*-comm 5# = refl\n9*-comm 6# = refl\n9*-comm 7# = refl\n9*-comm 8# = refl\n9*-comm 9# = refl\n9*-comm A# = refl\n\nA*-comm : \u2200 x \u2192 A* x \u2261 x * A#\nA*-comm 0# = refl\nA*-comm 1# = refl\nA*-comm 2# = refl\nA*-comm 3# = refl\nA*-comm 4# = refl\nA*-comm 5# = refl\nA*-comm 6# = refl\nA*-comm 7# = refl\nA*-comm 8# = refl\nA*-comm 9# = refl\nA*-comm A# = refl\n\n2*2*3*-id : \u2200 x \u2192 2* 2* 3* x \u2261 x\n2*2*3*-id 0# = refl\n2*2*3*-id 1# = refl\n2*2*3*-id 2# = refl\n2*2*3*-id 3# = refl\n2*2*3*-id 4# = refl\n2*2*3*-id 5# = refl\n2*2*3*-id 6# = refl\n2*2*3*-id 7# = refl\n2*2*3*-id 8# = refl\n2*2*3*-id 9# = refl\n2*2*3*-id A# = refl\n\n2*7*-3* : \u2200 x \u2192 2* 7* x \u2261 3* x\n2*7*-3* 0# = refl\n2*7*-3* 1# = refl\n2*7*-3* 2# = refl\n2*7*-3* 3# = refl\n2*7*-3* 4# = refl\n2*7*-3* 5# = refl\n2*7*-3* 6# = refl\n2*7*-3* 7# = refl\n2*7*-3* 8# = refl\n2*7*-3* 9# = refl\n2*7*-3* A# = refl\n\n5*-2*8* : \u2200 y \u2192 5* y \u2261 2* (2* (2* (2* y)))\n5*-2*8* 0# = refl\n5*-2*8* 1# = refl\n5*-2*8* 2# = refl\n5*-2*8* 3# = refl\n5*-2*8* 4# = refl\n5*-2*8* 5# = refl\n5*-2*8* 6# = refl\n5*-2*8* 7# = refl\n5*-2*8* 8# = refl\n5*-2*8* 9# = refl\n5*-2*8* A# = refl\n\n7*-2*9* : \u2200 y \u2192 7* y \u2261 2* (3* (3* y))\n7*-2*9* 0# = refl\n7*-2*9* 1# = refl\n7*-2*9* 2# = refl\n7*-2*9* 3# = refl\n7*-2*9* 4# = refl\n7*-2*9* 5# = refl\n7*-2*9* 6# = refl\n7*-2*9* 7# = refl\n7*-2*9* 8# = refl\n7*-2*9* 9# = refl\n7*-2*9* A# = refl\n\n9*-20* : \u2200 y \u2192 3* (3* y) \u2261 2* (2* (5* y))\n9*-20* 0# = refl\n9*-20* 1# = refl\n9*-20* 2# = refl\n9*-20* 3# = refl\n9*-20* 4# = refl\n9*-20* 5# = refl\n9*-20* 6# = refl\n9*-20* 7# = refl\n9*-20* 8# = refl\n9*-20* 9# = refl\n9*-20* A# = refl\n\n*-comm : Commutative _*_\n*-comm 0# = !_ \u2218 *0-zero\n*-comm 1# = 1*-comm\n*-comm 2# = 2*-comm\n*-comm 3# = 3*-comm\n*-comm 4# = 4*-comm\n*-comm 5# = 5*-comm\n*-comm 6# = 6*-comm\n*-comm 7# = 7*-comm\n*-comm 8# = 8*-comm\n*-comm 9# = 9*-comm\n*-comm A# = A*-comm\n\n{-\n2*-assoc : \u2200 x y \u2192 (2* x) * y \u2261 2*(x * y)\n2*-assoc 0# _ = refl\n2*-assoc 1# _ = refl\n2*-assoc 2# _ = refl\n2*-assoc 3# _ = refl\n2*-assoc 4# _ = refl\n2*-assoc 5# _ = refl\n2*-assoc 6# = !_ \u2218 2*2*3*-id\n2*-assoc 7# = !_ \u2218 2*7*-3*\n2*-assoc 8# = 5*-2*8*\n2*-assoc 9# = 7*-2*9*\n2*-assoc A# = 9*-20*\n\n3*-assoc : \u2200 x y \u2192 (3* x) * y \u2261 3*(x * y)\n3*-assoc 0# y = refl\n3*-assoc 1# y = refl\n3*-assoc 2# y = {!!}\n3*-assoc 3# y = refl\n3*-assoc 4# y = {!!}\n3*-assoc 5# y = {!!}\n3*-assoc 6# y = {!!}\n3*-assoc 7# y = {!!}\n3*-assoc 8# y = {!!}\n3*-assoc 9# y = {!!}\n3*-assoc A# y = {!!}\n\n4*-assoc : \u2200 x y \u2192 (4* x) * y \u2261 4*(x * y)\n4*-assoc x _ = 2*-assoc (2* x) _ \u2219 ap 2*_ (2*-assoc x _)\n\n5*-assoc : \u2200 x y \u2192 (5* x) * y \u2261 5*(x * y)\n5*-assoc 0# y = refl\n5*-assoc 1# y = refl\n5*-assoc 2# y = {!!}\n5*-assoc 3# y = {!!}\n5*-assoc 4# y = {!!}\n5*-assoc 5# y = {!!}\n5*-assoc 6# y = {!!}\n5*-assoc 7# y = {!!}\n5*-assoc 8# y = {!!}\n5*-assoc 9# y = {!!}\n5*-assoc A# y = {!!}\n-}\n\n*-assoc : Associative _*_\n*-assoc 0# y z = refl\n*-assoc 1# y z = refl\n*-assoc 2# 0# z = refl\n*-assoc 2# 1# z = refl\n*-assoc 2# 2# z = refl\n*-assoc 2# 3# z = refl\n*-assoc 2# 4# z = refl\n*-assoc 2# 5# z = refl\n*-assoc 2# 6# 0# = refl\n*-assoc 2# 6# 1# = refl\n*-assoc 2# 6# 2# = refl\n*-assoc 2# 6# 3# = refl\n*-assoc 2# 6# 4# = refl\n*-assoc 2# 6# 5# = refl\n*-assoc 2# 6# 6# = refl\n*-assoc 2# 6# 7# = refl\n*-assoc 2# 6# 8# = refl\n*-assoc 2# 6# 9# = refl\n*-assoc 2# 6# A# = refl\n*-assoc 2# 7# 0# = refl\n*-assoc 2# 7# 1# = refl\n*-assoc 2# 7# 2# = refl\n*-assoc 2# 7# 3# = refl\n*-assoc 2# 7# 4# = refl\n*-assoc 2# 7# 5# = refl\n*-assoc 2# 7# 6# = refl\n*-assoc 2# 7# 7# = refl\n*-assoc 2# 7# 8# = refl\n*-assoc 2# 7# 9# = refl\n*-assoc 2# 7# A# = refl\n*-assoc 2# 8# 0# = refl\n*-assoc 2# 8# 1# = refl\n*-assoc 2# 8# 2# = refl\n*-assoc 2# 8# 3# = refl\n*-assoc 2# 8# 4# = refl\n*-assoc 2# 8# 5# = refl\n*-assoc 2# 8# 6# = refl\n*-assoc 2# 8# 7# = refl\n*-assoc 2# 8# 8# = refl\n*-assoc 2# 8# 9# = refl\n*-assoc 2# 8# A# = refl\n*-assoc 2# 9# 0# = refl\n*-assoc 2# 9# 1# = refl\n*-assoc 2# 9# 2# = refl\n*-assoc 2# 9# 3# = refl\n*-assoc 2# 9# 4# = refl\n*-assoc 2# 9# 5# = refl\n*-assoc 2# 9# 6# = refl\n*-assoc 2# 9# 7# = refl\n*-assoc 2# 9# 8# = refl\n*-assoc 2# 9# 9# = refl\n*-assoc 2# 9# A# = refl\n*-assoc 2# A# 0# = refl\n*-assoc 2# A# 1# = refl\n*-assoc 2# A# 2# = refl\n*-assoc 2# A# 3# = refl\n*-assoc 2# A# 4# = refl\n*-assoc 2# A# 5# = refl\n*-assoc 2# A# 6# = refl\n*-assoc 2# A# 7# = refl\n*-assoc 2# A# 8# = refl\n*-assoc 2# A# 9# = refl\n*-assoc 2# A# A# = refl\n*-assoc 3# 0# z = refl\n*-assoc 3# 1# z = refl\n*-assoc 3# 2# 0# = refl\n*-assoc 3# 2# 1# = refl\n*-assoc 3# 2# 2# = refl\n*-assoc 3# 2# 3# = refl\n*-assoc 3# 2# 4# = refl\n*-assoc 3# 2# 5# = refl\n*-assoc 3# 2# 6# = refl\n*-assoc 3# 2# 7# = refl\n*-assoc 3# 2# 8# = refl\n*-assoc 3# 2# 9# = refl\n*-assoc 3# 2# A# = refl\n*-assoc 3# 3# z = refl\n*-assoc 3# 4# 0# = refl\n*-assoc 3# 4# 1# = refl\n*-assoc 3# 4# 2# = refl\n*-assoc 3# 4# 3# = refl\n*-assoc 3# 4# 4# = refl\n*-assoc 3# 4# 5# = refl\n*-assoc 3# 4# 6# = refl\n*-assoc 3# 4# 7# = refl\n*-assoc 3# 4# 8# = refl\n*-assoc 3# 4# 9# = refl\n*-assoc 3# 4# A# = refl\n*-assoc 3# 5# 0# = refl\n*-assoc 3# 5# 1# = refl\n*-assoc 3# 5# 2# = refl\n*-assoc 3# 5# 3# = refl\n*-assoc 3# 5# 4# = refl\n*-assoc 3# 5# 5# = refl\n*-assoc 3# 5# 6# = refl\n*-assoc 3# 5# 7# = refl\n*-assoc 3# 5# 8# = refl\n*-assoc 3# 5# 9# = refl\n*-assoc 3# 5# A# = refl\n*-assoc 3# 6# 0# = refl\n*-assoc 3# 6# 1# = refl\n*-assoc 3# 6# 2# = refl\n*-assoc 3# 6# 3# = refl\n*-assoc 3# 6# 4# = refl\n*-assoc 3# 6# 5# = refl\n*-assoc 3# 6# 6# = refl\n*-assoc 3# 6# 7# = refl\n*-assoc 3# 6# 8# = refl\n*-assoc 3# 6# 9# = refl\n*-assoc 3# 6# A# = refl\n*-assoc 3# 7# 0# = refl\n*-assoc 3# 7# 1# = refl\n*-assoc 3# 7# 2# = refl\n*-assoc 3# 7# 3# = refl\n*-assoc 3# 7# 4# = refl\n*-assoc 3# 7# 5# = refl\n*-assoc 3# 7# 6# = refl\n*-assoc 3# 7# 7# = refl\n*-assoc 3# 7# 8# = refl\n*-assoc 3# 7# 9# = refl\n*-assoc 3# 7# A# = refl\n*-assoc 3# 8# 0# = refl\n*-assoc 3# 8# 1# = refl\n*-assoc 3# 8# 2# = refl\n*-assoc 3# 8# 3# = refl\n*-assoc 3# 8# 4# = refl\n*-assoc 3# 8# 5# = refl\n*-assoc 3# 8# 6# = refl\n*-assoc 3# 8# 7# = refl\n*-assoc 3# 8# 8# = refl\n*-assoc 3# 8# 9# = refl\n*-assoc 3# 8# A# = refl\n*-assoc 3# 9# 0# = refl\n*-assoc 3# 9# 1# = refl\n*-assoc 3# 9# 2# = refl\n*-assoc 3# 9# 3# = refl\n*-assoc 3# 9# 4# = refl\n*-assoc 3# 9# 5# = refl\n*-assoc 3# 9# 6# = refl\n*-assoc 3# 9# 7# = refl\n*-assoc 3# 9# 8# = refl\n*-assoc 3# 9# 9# = refl\n*-assoc 3# 9# A# = refl\n*-assoc 3# A# 0# = refl\n*-assoc 3# A# 1# = refl\n*-assoc 3# A# 2# = refl\n*-assoc 3# A# 3# = refl\n*-assoc 3# A# 4# = refl\n*-assoc 3# A# 5# = refl\n*-assoc 3# A# 6# = refl\n*-assoc 3# A# 7# = refl\n*-assoc 3# A# 8# = refl\n*-assoc 3# A# 9# = refl\n*-assoc 3# A# A# = refl\n*-assoc 4# 0# z = refl\n*-assoc 4# 1# z = refl\n*-assoc 4# 2# z = refl\n*-assoc 4# 3# 0# = refl\n*-assoc 4# 3# 1# = refl\n*-assoc 4# 3# 2# = refl\n*-assoc 4# 3# 3# = refl\n*-assoc 4# 3# 4# = refl\n*-assoc 4# 3# 5# = refl\n*-assoc 4# 3# 6# = refl\n*-assoc 4# 3# 7# = refl\n*-assoc 4# 3# 8# = refl\n*-assoc 4# 3# 9# = refl\n*-assoc 4# 3# A# = refl\n*-assoc 4# 4# 0# = refl\n*-assoc 4# 4# 1# = refl\n*-assoc 4# 4# 2# = refl\n*-assoc 4# 4# 3# = refl\n*-assoc 4# 4# 4# = refl\n*-assoc 4# 4# 5# = refl\n*-assoc 4# 4# 6# = refl\n*-assoc 4# 4# 7# = refl\n*-assoc 4# 4# 8# = refl\n*-assoc 4# 4# 9# = refl\n*-assoc 4# 4# A# = refl\n*-assoc 4# 5# 0# = refl\n*-assoc 4# 5# 1# = refl\n*-assoc 4# 5# 2# = refl\n*-assoc 4# 5# 3# = refl\n*-assoc 4# 5# 4# = refl\n*-assoc 4# 5# 5# = refl\n*-assoc 4# 5# 6# = refl\n*-assoc 4# 5# 7# = refl\n*-assoc 4# 5# 8# = refl\n*-assoc 4# 5# 9# = refl\n*-assoc 4# 5# A# = refl\n*-assoc 4# 6# 0# = refl\n*-assoc 4# 6# 1# = refl\n*-assoc 4# 6# 2# = refl\n*-assoc 4# 6# 3# = refl\n*-assoc 4# 6# 4# = refl\n*-assoc 4# 6# 5# = refl\n*-assoc 4# 6# 6# = refl\n*-assoc 4# 6# 7# = refl\n*-assoc 4# 6# 8# = refl\n*-assoc 4# 6# 9# = refl\n*-assoc 4# 6# A# = refl\n*-assoc 4# 7# 0# = refl\n*-assoc 4# 7# 1# = refl\n*-assoc 4# 7# 2# = refl\n*-assoc 4# 7# 3# = refl\n*-assoc 4# 7# 4# = refl\n*-assoc 4# 7# 5# = refl\n*-assoc 4# 7# 6# = refl\n*-assoc 4# 7# 7# = refl\n*-assoc 4# 7# 8# = refl\n*-assoc 4# 7# 9# = refl\n*-assoc 4# 7# A# = refl\n*-assoc 4# 8# 0# = refl\n*-assoc 4# 8# 1# = refl\n*-assoc 4# 8# 2# = refl\n*-assoc 4# 8# 3# = refl\n*-assoc 4# 8# 4# = refl\n*-assoc 4# 8# 5# = refl\n*-assoc 4# 8# 6# = refl\n*-assoc 4# 8# 7# = refl\n*-assoc 4# 8# 8# = refl\n*-assoc 4# 8# 9# = refl\n*-assoc 4# 8# A# = refl\n*-assoc 4# 9# 0# = refl\n*-assoc 4# 9# 1# = refl\n*-assoc 4# 9# 2# = refl\n*-assoc 4# 9# 3# = refl\n*-assoc 4# 9# 4# = refl\n*-assoc 4# 9# 5# = refl\n*-assoc 4# 9# 6# = refl\n*-assoc 4# 9# 7# = refl\n*-assoc 4# 9# 8# = refl\n*-assoc 4# 9# 9# = refl\n*-assoc 4# 9# A# = refl\n*-assoc 4# A# 0# = refl\n*-assoc 4# A# 1# = refl\n*-assoc 4# A# 2# = refl\n*-assoc 4# A# 3# = refl\n*-assoc 4# A# 4# = refl\n*-assoc 4# A# 5# = refl\n*-assoc 4# A# 6# = refl\n*-assoc 4# A# 7# = refl\n*-assoc 4# A# 8# = refl\n*-assoc 4# A# 9# = refl\n*-assoc 4# A# A# = refl\n*-assoc 5# 0# z = refl\n*-assoc 5# 1# z = refl\n*-assoc 5# 2# 0# = refl\n*-assoc 5# 2# 1# = refl\n*-assoc 5# 2# 2# = refl\n*-assoc 5# 2# 3# = refl\n*-assoc 5# 2# 4# = refl\n*-assoc 5# 2# 5# = refl\n*-assoc 5# 2# 6# = refl\n*-assoc 5# 2# 7# = refl\n*-assoc 5# 2# 8# = refl\n*-assoc 5# 2# 9# = refl\n*-assoc 5# 2# A# = refl\n*-assoc 5# 3# 0# = refl\n*-assoc 5# 3# 1# = refl\n*-assoc 5# 3# 2# = refl\n*-assoc 5# 3# 3# = refl\n*-assoc 5# 3# 4# = refl\n*-assoc 5# 3# 5# = refl\n*-assoc 5# 3# 6# = refl\n*-assoc 5# 3# 7# = refl\n*-assoc 5# 3# 8# = refl\n*-assoc 5# 3# 9# = refl\n*-assoc 5# 3# A# = refl\n*-assoc 5# 4# 0# = refl\n*-assoc 5# 4# 1# = refl\n*-assoc 5# 4# 2# = refl\n*-assoc 5# 4# 3# = refl\n*-assoc 5# 4# 4# = refl\n*-assoc 5# 4# 5# = refl\n*-assoc 5# 4# 6# = refl\n*-assoc 5# 4# 7# = refl\n*-assoc 5# 4# 8# = refl\n*-assoc 5# 4# 9# = refl\n*-assoc 5# 4# A# = refl\n*-assoc 5# 5# 0# = refl\n*-assoc 5# 5# 1# = refl\n*-assoc 5# 5# 2# = refl\n*-assoc 5# 5# 3# = refl\n*-assoc 5# 5# 4# = refl\n*-assoc 5# 5# 5# = refl\n*-assoc 5# 5# 6# = refl\n*-assoc 5# 5# 7# = refl\n*-assoc 5# 5# 8# = refl\n*-assoc 5# 5# 9# = refl\n*-assoc 5# 5# A# = refl\n*-assoc 5# 6# 0# = refl\n*-assoc 5# 6# 1# = refl\n*-assoc 5# 6# 2# = refl\n*-assoc 5# 6# 3# = refl\n*-assoc 5# 6# 4# = refl\n*-assoc 5# 6# 5# = refl\n*-assoc 5# 6# 6# = refl\n*-assoc 5# 6# 7# = refl\n*-assoc 5# 6# 8# = refl\n*-assoc 5# 6# 9# = refl\n*-assoc 5# 6# A# = refl\n*-assoc 5# 7# 0# = refl\n*-assoc 5# 7# 1# = refl\n*-assoc 5# 7# 2# = refl\n*-assoc 5# 7# 3# = refl\n*-assoc 5# 7# 4# = refl\n*-assoc 5# 7# 5# = refl\n*-assoc 5# 7# 6# = refl\n*-assoc 5# 7# 7# = refl\n*-assoc 5# 7# 8# = refl\n*-assoc 5# 7# 9# = refl\n*-assoc 5# 7# A# = refl\n*-assoc 5# 8# 0# = refl\n*-assoc 5# 8# 1# = refl\n*-assoc 5# 8# 2# = refl\n*-assoc 5# 8# 3# = refl\n*-assoc 5# 8# 4# = refl\n*-assoc 5# 8# 5# = refl\n*-assoc 5# 8# 6# = refl\n*-assoc 5# 8# 7# = refl\n*-assoc 5# 8# 8# = refl\n*-assoc 5# 8# 9# = refl\n*-assoc 5# 8# A# = refl\n*-assoc 5# 9# 0# = refl\n*-assoc 5# 9# 1# = refl\n*-assoc 5# 9# 2# = refl\n*-assoc 5# 9# 3# = refl\n*-assoc 5# 9# 4# = refl\n*-assoc 5# 9# 5# = refl\n*-assoc 5# 9# 6# = refl\n*-assoc 5# 9# 7# = refl\n*-assoc 5# 9# 8# = refl\n*-assoc 5# 9# 9# = refl\n*-assoc 5# 9# A# = refl\n*-assoc 5# A# 0# = refl\n*-assoc 5# A# 1# = refl\n*-assoc 5# A# 2# = refl\n*-assoc 5# A# 3# = refl\n*-assoc 5# A# 4# = refl\n*-assoc 5# A# 5# = refl\n*-assoc 5# A# 6# = refl\n*-assoc 5# A# 7# = refl\n*-assoc 5# A# 8# = refl\n*-assoc 5# A# 9# = refl\n*-assoc 5# A# A# = refl\n*-assoc 6# 0# z = refl\n*-assoc 6# 1# z = refl\n*-assoc 6# 2# 0# = refl\n*-assoc 6# 2# 1# = refl\n*-assoc 6# 2# 2# = refl\n*-assoc 6# 2# 3# = refl\n*-assoc 6# 2# 4# = refl\n*-assoc 6# 2# 5# = refl\n*-assoc 6# 2# 6# = refl\n*-assoc 6# 2# 7# = refl\n*-assoc 6# 2# 8# = refl\n*-assoc 6# 2# 9# = refl\n*-assoc 6# 2# A# = refl\n*-assoc 6# 3# 0# = refl\n*-assoc 6# 3# 1# = refl\n*-assoc 6# 3# 2# = refl\n*-assoc 6# 3# 3# = refl\n*-assoc 6# 3# 4# = refl\n*-assoc 6# 3# 5# = refl\n*-assoc 6# 3# 6# = refl\n*-assoc 6# 3# 7# = refl\n*-assoc 6# 3# 8# = refl\n*-assoc 6# 3# 9# = refl\n*-assoc 6# 3# A# = refl\n*-assoc 6# 4# 0# = refl\n*-assoc 6# 4# 1# = refl\n*-assoc 6# 4# 2# = refl\n*-assoc 6# 4# 3# = refl\n*-assoc 6# 4# 4# = refl\n*-assoc 6# 4# 5# = refl\n*-assoc 6# 4# 6# = refl\n*-assoc 6# 4# 7# = refl\n*-assoc 6# 4# 8# = refl\n*-assoc 6# 4# 9# = refl\n*-assoc 6# 4# A# = refl\n*-assoc 6# 5# 0# = refl\n*-assoc 6# 5# 1# = refl\n*-assoc 6# 5# 2# = refl\n*-assoc 6# 5# 3# = refl\n*-assoc 6# 5# 4# = refl\n*-assoc 6# 5# 5# = refl\n*-assoc 6# 5# 6# = refl\n*-assoc 6# 5# 7# = refl\n*-assoc 6# 5# 8# = refl\n*-assoc 6# 5# 9# = refl\n*-assoc 6# 5# A# = refl\n*-assoc 6# 6# 0# = refl\n*-assoc 6# 6# 1# = refl\n*-assoc 6# 6# 2# = refl\n*-assoc 6# 6# 3# = refl\n*-assoc 6# 6# 4# = refl\n*-assoc 6# 6# 5# = refl\n*-assoc 6# 6# 6# = refl\n*-assoc 6# 6# 7# = refl\n*-assoc 6# 6# 8# = refl\n*-assoc 6# 6# 9# = refl\n*-assoc 6# 6# A# = refl\n*-assoc 6# 7# 0# = refl\n*-assoc 6# 7# 1# = refl\n*-assoc 6# 7# 2# = refl\n*-assoc 6# 7# 3# = refl\n*-assoc 6# 7# 4# = refl\n*-assoc 6# 7# 5# = refl\n*-assoc 6# 7# 6# = refl\n*-assoc 6# 7# 7# = refl\n*-assoc 6# 7# 8# = refl\n*-assoc 6# 7# 9# = refl\n*-assoc 6# 7# A# = refl\n*-assoc 6# 8# 0# = refl\n*-assoc 6# 8# 1# = refl\n*-assoc 6# 8# 2# = refl\n*-assoc 6# 8# 3# = refl\n*-assoc 6# 8# 4# = refl\n*-assoc 6# 8# 5# = refl\n*-assoc 6# 8# 6# = refl\n*-assoc 6# 8# 7# = refl\n*-assoc 6# 8# 8# = refl\n*-assoc 6# 8# 9# = refl\n*-assoc 6# 8# A# = refl\n*-assoc 6# 9# 0# = refl\n*-assoc 6# 9# 1# = refl\n*-assoc 6# 9# 2# = refl\n*-assoc 6# 9# 3# = refl\n*-assoc 6# 9# 4# = refl\n*-assoc 6# 9# 5# = refl\n*-assoc 6# 9# 6# = refl\n*-assoc 6# 9# 7# = refl\n*-assoc 6# 9# 8# = refl\n*-assoc 6# 9# 9# = refl\n*-assoc 6# 9# A# = refl\n*-assoc 6# A# 0# = refl\n*-assoc 6# A# 1# = refl\n*-assoc 6# A# 2# = refl\n*-assoc 6# A# 3# = refl\n*-assoc 6# A# 4# = refl\n*-assoc 6# A# 5# = refl\n*-assoc 6# A# 6# = refl\n*-assoc 6# A# 7# = refl\n*-assoc 6# A# 8# = refl\n*-assoc 6# A# 9# = refl\n*-assoc 6# A# A# = refl\n*-assoc 7# 0# z = refl\n*-assoc 7# 1# z = refl\n*-assoc 7# 2# 0# = refl\n*-assoc 7# 2# 1# = refl\n*-assoc 7# 2# 2# = refl\n*-assoc 7# 2# 3# = refl\n*-assoc 7# 2# 4# = refl\n*-assoc 7# 2# 5# = refl\n*-assoc 7# 2# 6# = refl\n*-assoc 7# 2# 7# = refl\n*-assoc 7# 2# 8# = refl\n*-assoc 7# 2# 9# = refl\n*-assoc 7# 2# A# = refl\n*-assoc 7# 3# 0# = refl\n*-assoc 7# 3# 1# = refl\n*-assoc 7# 3# 2# = refl\n*-assoc 7# 3# 3# = refl\n*-assoc 7# 3# 4# = refl\n*-assoc 7# 3# 5# = refl\n*-assoc 7# 3# 6# = refl\n*-assoc 7# 3# 7# = refl\n*-assoc 7# 3# 8# = refl\n*-assoc 7# 3# 9# = refl\n*-assoc 7# 3# A# = refl\n*-assoc 7# 4# 0# = refl\n*-assoc 7# 4# 1# = refl\n*-assoc 7# 4# 2# = refl\n*-assoc 7# 4# 3# = refl\n*-assoc 7# 4# 4# = refl\n*-assoc 7# 4# 5# = refl\n*-assoc 7# 4# 6# = refl\n*-assoc 7# 4# 7# = refl\n*-assoc 7# 4# 8# = refl\n*-assoc 7# 4# 9# = refl\n*-assoc 7# 4# A# = refl\n*-assoc 7# 5# 0# = refl\n*-assoc 7# 5# 1# = refl\n*-assoc 7# 5# 2# = refl\n*-assoc 7# 5# 3# = refl\n*-assoc 7# 5# 4# = refl\n*-assoc 7# 5# 5# = refl\n*-assoc 7# 5# 6# = refl\n*-assoc 7# 5# 7# = refl\n*-assoc 7# 5# 8# = refl\n*-assoc 7# 5# 9# = refl\n*-assoc 7# 5# A# = refl\n*-assoc 7# 6# 0# = refl\n*-assoc 7# 6# 1# = refl\n*-assoc 7# 6# 2# = refl\n*-assoc 7# 6# 3# = refl\n*-assoc 7# 6# 4# = refl\n*-assoc 7# 6# 5# = refl\n*-assoc 7# 6# 6# = refl\n*-assoc 7# 6# 7# = refl\n*-assoc 7# 6# 8# = refl\n*-assoc 7# 6# 9# = refl\n*-assoc 7# 6# A# = refl\n*-assoc 7# 7# 0# = refl\n*-assoc 7# 7# 1# = refl\n*-assoc 7# 7# 2# = refl\n*-assoc 7# 7# 3# = refl\n*-assoc 7# 7# 4# = refl\n*-assoc 7# 7# 5# = refl\n*-assoc 7# 7# 6# = refl\n*-assoc 7# 7# 7# = refl\n*-assoc 7# 7# 8# = refl\n*-assoc 7# 7# 9# = refl\n*-assoc 7# 7# A# = refl\n*-assoc 7# 8# 0# = refl\n*-assoc 7# 8# 1# = refl\n*-assoc 7# 8# 2# = refl\n*-assoc 7# 8# 3# = refl\n*-assoc 7# 8# 4# = refl\n*-assoc 7# 8# 5# = refl\n*-assoc 7# 8# 6# = refl\n*-assoc 7# 8# 7# = refl\n*-assoc 7# 8# 8# = refl\n*-assoc 7# 8# 9# = refl\n*-assoc 7# 8# A# = refl\n*-assoc 7# 9# 0# = refl\n*-assoc 7# 9# 1# = refl\n*-assoc 7# 9# 2# = refl\n*-assoc 7# 9# 3# = refl\n*-assoc 7# 9# 4# = refl\n*-assoc 7# 9# 5# = refl\n*-assoc 7# 9# 6# = refl\n*-assoc 7# 9# 7# = refl\n*-assoc 7# 9# 8# = refl\n*-assoc 7# 9# 9# = refl\n*-assoc 7# 9# A# = refl\n*-assoc 7# A# 0# = refl\n*-assoc 7# A# 1# = refl\n*-assoc 7# A# 2# = refl\n*-assoc 7# A# 3# = refl\n*-assoc 7# A# 4# = refl\n*-assoc 7# A# 5# = refl\n*-assoc 7# A# 6# = refl\n*-assoc 7# A# 7# = refl\n*-assoc 7# A# 8# = refl\n*-assoc 7# A# 9# = refl\n*-assoc 7# A# A# = refl\n*-assoc 8# 0# z = refl\n*-assoc 8# 1# z = refl\n*-assoc 8# 2# 0# = refl\n*-assoc 8# 2# 1# = refl\n*-assoc 8# 2# 2# = refl\n*-assoc 8# 2# 3# = refl\n*-assoc 8# 2# 4# = refl\n*-assoc 8# 2# 5# = refl\n*-assoc 8# 2# 6# = refl\n*-assoc 8# 2# 7# = refl\n*-assoc 8# 2# 8# = refl\n*-assoc 8# 2# 9# = refl\n*-assoc 8# 2# A# = refl\n*-assoc 8# 3# 0# = refl\n*-assoc 8# 3# 1# = refl\n*-assoc 8# 3# 2# = refl\n*-assoc 8# 3# 3# = refl\n*-assoc 8# 3# 4# = refl\n*-assoc 8# 3# 5# = refl\n*-assoc 8# 3# 6# = refl\n*-assoc 8# 3# 7# = refl\n*-assoc 8# 3# 8# = refl\n*-assoc 8# 3# 9# = refl\n*-assoc 8# 3# A# = refl\n*-assoc 8# 4# 0# = refl\n*-assoc 8# 4# 1# = refl\n*-assoc 8# 4# 2# = refl\n*-assoc 8# 4# 3# = refl\n*-assoc 8# 4# 4# = refl\n*-assoc 8# 4# 5# = refl\n*-assoc 8# 4# 6# = refl\n*-assoc 8# 4# 7# = refl\n*-assoc 8# 4# 8# = refl\n*-assoc 8# 4# 9# = refl\n*-assoc 8# 4# A# = refl\n*-assoc 8# 5# 0# = refl\n*-assoc 8# 5# 1# = refl\n*-assoc 8# 5# 2# = refl\n*-assoc 8# 5# 3# = refl\n*-assoc 8# 5# 4# = refl\n*-assoc 8# 5# 5# = refl\n*-assoc 8# 5# 6# = refl\n*-assoc 8# 5# 7# = refl\n*-assoc 8# 5# 8# = refl\n*-assoc 8# 5# 9# = refl\n*-assoc 8# 5# A# = refl\n*-assoc 8# 6# 0# = refl\n*-assoc 8# 6# 1# = refl\n*-assoc 8# 6# 2# = refl\n*-assoc 8# 6# 3# = refl\n*-assoc 8# 6# 4# = refl\n*-assoc 8# 6# 5# = refl\n*-assoc 8# 6# 6# = refl\n*-assoc 8# 6# 7# = refl\n*-assoc 8# 6# 8# = refl\n*-assoc 8# 6# 9# = refl\n*-assoc 8# 6# A# = refl\n*-assoc 8# 7# 0# = refl\n*-assoc 8# 7# 1# = refl\n*-assoc 8# 7# 2# = refl\n*-assoc 8# 7# 3# = refl\n*-assoc 8# 7# 4# = refl\n*-assoc 8# 7# 5# = refl\n*-assoc 8# 7# 6# = refl\n*-assoc 8# 7# 7# = refl\n*-assoc 8# 7# 8# = refl\n*-assoc 8# 7# 9# = refl\n*-assoc 8# 7# A# = refl\n*-assoc 8# 8# 0# = refl\n*-assoc 8# 8# 1# = refl\n*-assoc 8# 8# 2# = refl\n*-assoc 8# 8# 3# = refl\n*-assoc 8# 8# 4# = refl\n*-assoc 8# 8# 5# = refl\n*-assoc 8# 8# 6# = refl\n*-assoc 8# 8# 7# = refl\n*-assoc 8# 8# 8# = refl\n*-assoc 8# 8# 9# = refl\n*-assoc 8# 8# A# = refl\n*-assoc 8# 9# 0# = refl\n*-assoc 8# 9# 1# = refl\n*-assoc 8# 9# 2# = refl\n*-assoc 8# 9# 3# = refl\n*-assoc 8# 9# 4# = refl\n*-assoc 8# 9# 5# = refl\n*-assoc 8# 9# 6# = refl\n*-assoc 8# 9# 7# = refl\n*-assoc 8# 9# 8# = refl\n*-assoc 8# 9# 9# = refl\n*-assoc 8# 9# A# = refl\n*-assoc 8# A# 0# = refl\n*-assoc 8# A# 1# = refl\n*-assoc 8# A# 2# = refl\n*-assoc 8# A# 3# = refl\n*-assoc 8# A# 4# = refl\n*-assoc 8# A# 5# = refl\n*-assoc 8# A# 6# = refl\n*-assoc 8# A# 7# = refl\n*-assoc 8# A# 8# = refl\n*-assoc 8# A# 9# = refl\n*-assoc 8# A# A# = refl\n*-assoc 9# 0# z = refl\n*-assoc 9# 1# z = refl\n*-assoc 9# 2# 0# = refl\n*-assoc 9# 2# 1# = refl\n*-assoc 9# 2# 2# = refl\n*-assoc 9# 2# 3# = refl\n*-assoc 9# 2# 4# = refl\n*-assoc 9# 2# 5# = refl\n*-assoc 9# 2# 6# = refl\n*-assoc 9# 2# 7# = refl\n*-assoc 9# 2# 8# = refl\n*-assoc 9# 2# 9# = refl\n*-assoc 9# 2# A# = refl\n*-assoc 9# 3# 0# = refl\n*-assoc 9# 3# 1# = refl\n*-assoc 9# 3# 2# = refl\n*-assoc 9# 3# 3# = refl\n*-assoc 9# 3# 4# = refl\n*-assoc 9# 3# 5# = refl\n*-assoc 9# 3# 6# = refl\n*-assoc 9# 3# 7# = refl\n*-assoc 9# 3# 8# = refl\n*-assoc 9# 3# 9# = refl\n*-assoc 9# 3# A# = refl\n*-assoc 9# 4# 0# = refl\n*-assoc 9# 4# 1# = refl\n*-assoc 9# 4# 2# = refl\n*-assoc 9# 4# 3# = refl\n*-assoc 9# 4# 4# = refl\n*-assoc 9# 4# 5# = refl\n*-assoc 9# 4# 6# = refl\n*-assoc 9# 4# 7# = refl\n*-assoc 9# 4# 8# = refl\n*-assoc 9# 4# 9# = refl\n*-assoc 9# 4# A# = refl\n*-assoc 9# 5# 0# = refl\n*-assoc 9# 5# 1# = refl\n*-assoc 9# 5# 2# = refl\n*-assoc 9# 5# 3# = refl\n*-assoc 9# 5# 4# = refl\n*-assoc 9# 5# 5# = refl\n*-assoc 9# 5# 6# = refl\n*-assoc 9# 5# 7# = refl\n*-assoc 9# 5# 8# = refl\n*-assoc 9# 5# 9# = refl\n*-assoc 9# 5# A# = refl\n*-assoc 9# 6# 0# = refl\n*-assoc 9# 6# 1# = refl\n*-assoc 9# 6# 2# = refl\n*-assoc 9# 6# 3# = refl\n*-assoc 9# 6# 4# = refl\n*-assoc 9# 6# 5# = refl\n*-assoc 9# 6# 6# = refl\n*-assoc 9# 6# 7# = refl\n*-assoc 9# 6# 8# = refl\n*-assoc 9# 6# 9# = refl\n*-assoc 9# 6# A# = refl\n*-assoc 9# 7# 0# = refl\n*-assoc 9# 7# 1# = refl\n*-assoc 9# 7# 2# = refl\n*-assoc 9# 7# 3# = refl\n*-assoc 9# 7# 4# = refl\n*-assoc 9# 7# 5# = refl\n*-assoc 9# 7# 6# = refl\n*-assoc 9# 7# 7# = refl\n*-assoc 9# 7# 8# = refl\n*-assoc 9# 7# 9# = refl\n*-assoc 9# 7# A# = refl\n*-assoc 9# 8# 0# = refl\n*-assoc 9# 8# 1# = refl\n*-assoc 9# 8# 2# = refl\n*-assoc 9# 8# 3# = refl\n*-assoc 9# 8# 4# = refl\n*-assoc 9# 8# 5# = refl\n*-assoc 9# 8# 6# = refl\n*-assoc 9# 8# 7# = refl\n*-assoc 9# 8# 8# = refl\n*-assoc 9# 8# 9# = refl\n*-assoc 9# 8# A# = refl\n*-assoc 9# 9# 0# = refl\n*-assoc 9# 9# 1# = refl\n*-assoc 9# 9# 2# = refl\n*-assoc 9# 9# 3# = refl\n*-assoc 9# 9# 4# = refl\n*-assoc 9# 9# 5# = refl\n*-assoc 9# 9# 6# = refl\n*-assoc 9# 9# 7# = refl\n*-assoc 9# 9# 8# = refl\n*-assoc 9# 9# 9# = refl\n*-assoc 9# 9# A# = refl\n*-assoc 9# A# 0# = refl\n*-assoc 9# A# 1# = refl\n*-assoc 9# A# 2# = refl\n*-assoc 9# A# 3# = refl\n*-assoc 9# A# 4# = refl\n*-assoc 9# A# 5# = refl\n*-assoc 9# A# 6# = refl\n*-assoc 9# A# 7# = refl\n*-assoc 9# A# 8# = refl\n*-assoc 9# A# 9# = refl\n*-assoc 9# A# A# = refl\n*-assoc A# 0# z = refl\n*-assoc A# 1# z = refl\n*-assoc A# 2# 0# = refl\n*-assoc A# 2# 1# = refl\n*-assoc A# 2# 2# = refl\n*-assoc A# 2# 3# = refl\n*-assoc A# 2# 4# = refl\n*-assoc A# 2# 5# = refl\n*-assoc A# 2# 6# = refl\n*-assoc A# 2# 7# = refl\n*-assoc A# 2# 8# = refl\n*-assoc A# 2# 9# = refl\n*-assoc A# 2# A# = refl\n*-assoc A# 3# 0# = refl\n*-assoc A# 3# 1# = refl\n*-assoc A# 3# 2# = refl\n*-assoc A# 3# 3# = refl\n*-assoc A# 3# 4# = refl\n*-assoc A# 3# 5# = refl\n*-assoc A# 3# 6# = refl\n*-assoc A# 3# 7# = refl\n*-assoc A# 3# 8# = refl\n*-assoc A# 3# 9# = refl\n*-assoc A# 3# A# = refl\n*-assoc A# 4# 0# = refl\n*-assoc A# 4# 1# = refl\n*-assoc A# 4# 2# = refl\n*-assoc A# 4# 3# = refl\n*-assoc A# 4# 4# = refl\n*-assoc A# 4# 5# = refl\n*-assoc A# 4# 6# = refl\n*-assoc A# 4# 7# = refl\n*-assoc A# 4# 8# = refl\n*-assoc A# 4# 9# = refl\n*-assoc A# 4# A# = refl\n*-assoc A# 5# 0# = refl\n*-assoc A# 5# 1# = refl\n*-assoc A# 5# 2# = refl\n*-assoc A# 5# 3# = refl\n*-assoc A# 5# 4# = refl\n*-assoc A# 5# 5# = refl\n*-assoc A# 5# 6# = refl\n*-assoc A# 5# 7# = refl\n*-assoc A# 5# 8# = refl\n*-assoc A# 5# 9# = refl\n*-assoc A# 5# A# = refl\n*-assoc A# 6# 0# = refl\n*-assoc A# 6# 1# = refl\n*-assoc A# 6# 2# = refl\n*-assoc A# 6# 3# = refl\n*-assoc A# 6# 4# = refl\n*-assoc A# 6# 5# = refl\n*-assoc A# 6# 6# = refl\n*-assoc A# 6# 7# = refl\n*-assoc A# 6# 8# = refl\n*-assoc A# 6# 9# = refl\n*-assoc A# 6# A# = refl\n*-assoc A# 7# 0# = refl\n*-assoc A# 7# 1# = refl\n*-assoc A# 7# 2# = refl\n*-assoc A# 7# 3# = refl\n*-assoc A# 7# 4# = refl\n*-assoc A# 7# 5# = refl\n*-assoc A# 7# 6# = refl\n*-assoc A# 7# 7# = refl\n*-assoc A# 7# 8# = refl\n*-assoc A# 7# 9# = refl\n*-assoc A# 7# A# = refl\n*-assoc A# 8# 0# = refl\n*-assoc A# 8# 1# = refl\n*-assoc A# 8# 2# = refl\n*-assoc A# 8# 3# = refl\n*-assoc A# 8# 4# = refl\n*-assoc A# 8# 5# = refl\n*-assoc A# 8# 6# = refl\n*-assoc A# 8# 7# = refl\n*-assoc A# 8# 8# = refl\n*-assoc A# 8# 9# = refl\n*-assoc A# 8# A# = refl\n*-assoc A# 9# 0# = refl\n*-assoc A# 9# 1# = refl\n*-assoc A# 9# 2# = refl\n*-assoc A# 9# 3# = refl\n*-assoc A# 9# 4# = refl\n*-assoc A# 9# 5# = refl\n*-assoc A# 9# 6# = refl\n*-assoc A# 9# 7# = refl\n*-assoc A# 9# 8# = refl\n*-assoc A# 9# 9# = refl\n*-assoc A# 9# A# = refl\n*-assoc A# A# 0# = refl\n*-assoc A# A# 1# = refl\n*-assoc A# A# 2# = refl\n*-assoc A# A# 3# = refl\n*-assoc A# A# 4# = refl\n*-assoc A# A# 5# = refl\n*-assoc A# A# 6# = refl\n*-assoc A# A# 7# = refl\n*-assoc A# A# 8# = refl\n*-assoc A# A# 9# = refl\n*-assoc A# A# A# = refl\n\n{-\n*-assoc : Associative _*_\n*-assoc 0# y z = refl\n*-assoc 1# y z = refl\n*-assoc 2# y z = 2*-assoc y z\n*-assoc 3# y z = 3*-assoc y z\n*-assoc 4# y z = 4*-assoc y z\n*-assoc 5# y z = 5*-assoc y z\n*-assoc 6# y z = 2*-assoc (3* y) _ \u2219 ap 2*_ (3*-assoc y _)\n*-assoc 7# y z = {!!}\n*-assoc 8# y z = 2*-assoc (4* y) _ \u2219 ap 2*_ (4*-assoc y _)\n*-assoc 9# y z = 3*-assoc (3* y) _ \u2219 ap 3*_ (3*-assoc y _)\n*-assoc A# y z = 2*-assoc (5* y) _ \u2219 ap 2*_ (5*-assoc y _)\n-}\n\nopen import Data.Nat.Base using (\u2115; _\u2238_)\n\nC\u25b9\u2115 : C11 \u2192 \u2115\nC\u25b9\u2115 0# = 0\nC\u25b9\u2115 1# = 1\nC\u25b9\u2115 2# = 2\nC\u25b9\u2115 3# = 3\nC\u25b9\u2115 4# = 4\nC\u25b9\u2115 5# = 5\nC\u25b9\u2115 6# = 6\nC\u25b9\u2115 7# = 7\nC\u25b9\u2115 8# = 8\nC\u25b9\u2115 9# = 9\nC\u25b9\u2115 A# = 10\n\nopen import Data.Fin.NP\n  renaming (Fin to \ud835\udd3d) using (#0; #1; #2; #3; #4; #5; #6; #7; #8; #9; #A; #B+)\n\nC\u25b9\ud835\udd3d : C11 \u2192 \ud835\udd3d 11\nC\u25b9\ud835\udd3d 0# = #0\nC\u25b9\ud835\udd3d 1# = #1\nC\u25b9\ud835\udd3d 2# = #2\nC\u25b9\ud835\udd3d 3# = #3\nC\u25b9\ud835\udd3d 4# = #4\nC\u25b9\ud835\udd3d 5# = #5\nC\u25b9\ud835\udd3d 6# = #6\nC\u25b9\ud835\udd3d 7# = #7\nC\u25b9\ud835\udd3d 8# = #8\nC\u25b9\ud835\udd3d 9# = #9\nC\u25b9\ud835\udd3d A# = #A\n\n\ud835\udd3d\u25b9C : \ud835\udd3d 11 \u2192 C11\n\ud835\udd3d\u25b9C #0 = 0#\n\ud835\udd3d\u25b9C #1 = 1#\n\ud835\udd3d\u25b9C #2 = 2#\n\ud835\udd3d\u25b9C #3 = 3#\n\ud835\udd3d\u25b9C #4 = 4#\n\ud835\udd3d\u25b9C #5 = 5#\n\ud835\udd3d\u25b9C #6 = 6#\n\ud835\udd3d\u25b9C #7 = 7#\n\ud835\udd3d\u25b9C #8 = 8#\n\ud835\udd3d\u25b9C #9 = 9#\n\ud835\udd3d\u25b9C #A = A#\n\ud835\udd3d\u25b9C (#B+ ())\n\nopen import Relation.Nullary.Decidable using (from-yes)\nopen import Data.Nat.Divisibility\nopen import Data.Nat.Primality\nopen import Data.Nat.GCD\n\npostulate\n  11-is-prime : Prime 11\n  -- SLOW 11-is-prime = from-yes (prime? 11)\n\nopen import Data.Nat.DivMod.NP\nopen import Data.Nat.ModInv 11 11-is-prime\n\n\u2115\u25b9\ud835\udd3d : \u2115 \u2192 \ud835\udd3d 11\n\u2115\u25b9\ud835\udd3d x = x mod 11\n\n\u2115\u25b9C : \u2115 \u2192 C11\n\u2115\u25b9C = \ud835\udd3d\u25b9C \u2218 \u2115\u25b9\ud835\udd3d\n\n1#\/_ : Op\u2081 C11\n1#\/ x = \u2115\u25b9C (1\/ (C\u25b9\u2115 x))\n\nopen import Data.Zero\nopen import Data.One\n\n_\u22620 : C11 \u2192 Set\n0# \u22620 = \ud835\udfd8\n1# \u22620 = \ud835\udfd9\n2# \u22620 = \ud835\udfd9\n3# \u22620 = \ud835\udfd9\n4# \u22620 = \ud835\udfd9\n5# \u22620 = \ud835\udfd9\n6# \u22620 = \ud835\udfd9\n7# \u22620 = \ud835\udfd9\n8# \u22620 = \ud835\udfd9\n9# \u22620 = \ud835\udfd9\nA# \u22620 = \ud835\udfd9\n\n1\/-inv : \u2200 x {x\u22620 : x \u22620} \u2192 (1#\/ x) * x \u2261 1#\n1\/-inv 0# {}\n1\/-inv 1# = refl\n1\/-inv 2# = refl\n1\/-inv 3# = refl\n1\/-inv 4# = refl\n1\/-inv 5# = refl\n1\/-inv 6# = refl\n1\/-inv 7# = refl\n1\/-inv 8# = refl\n1\/-inv 9# = refl\n1\/-inv A# = refl\n\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'cyclic10.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"81528449b32c7ba698b86e98484eb236dc6386fd","subject":"drafts\/dsa.agda","message":"drafts\/dsa.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"drafts\/dsa.agda","new_file":"drafts\/dsa.agda","new_contents":"-- Draft: we should move to something provably secure, like Schnorr signature.\n\nopen import Function\nopen import Data.Bool using (Bool; true; false; if_then_else_; _\u2228_)\nopen import Data.Nat using (\u2115)\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Maybe using (Maybe; nothing; just)\nopen import Data.Product using (_\u00d7_; _,_)\n\nmodule dsa\n\n-- (N : \u2115) -- = 256\n-- (L : \u2115) -- = 2048\n\n-- (q : \u2115) -- Bits N, Prime q\n-- (p : \u2115) -- Bits L, Prime p, st. \u2203 k, q * k = p-1\n\n(\u2124q : Set)\n\n(G : Set) -- = \u27e8\u2124p\u27e9\u2605 should for a group of order q\n(g : G)   -- generator of the group G\n(_^_  : G \u2192 \u2124q \u2192 G)\n(_\u2219pq_  : G \u2192 G \u2192 \u2124q)\n\n(g^_  : \u2124q \u2192 \u2124q)\n(_\u229e_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n(_\u2219_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n(_\/_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n\n(\u2124q\u2605 : Set)\n(refine? : \u2124q \u2192 Maybe \u2124q\u2605)\n(weaken : \u2124q\u2605 \u2192 \u2124q)\n(_==_ : \u2124q \u2192 \u2124q \u2192 Bool)\n(_\u207b\u00b9 : \u2124q\u2605 \u2192 \u2124q)\n\n(M : Set)\n(\u210b : M \u2192 \u2124q)\n\nwhere\n\nSig : Set\nSig = \u2124q\u2605 \u00d7 \u2124q\u2605\n\nSign : (m : M) (x : \u2124q) (r : \u2124q) \u2192 Maybe Sig\nSign m x r = case (refine? R , refine? S) of \u03bb\n                { (just R\u2032 , just S\u2032) \u2192 just (R\u2032 , S\u2032)\n                ; _                   \u2192 nothing\n                }\n  where R = g^ r\n        S = (\u210b m \u229e (x \u2219 R)) \/ r\n\n-- Reject the signature if 0 < r < q or 0 < s < q is not satisfied.\nVer : (m : M) (g\u02e3 : G) (sig : Sig) \u2192 Bool\nVer m g\u02e3 (R\u2032 , S\u2032) = R == v\n  where R = weaken R\u2032\n        S = weaken S\u2032\n        w = S\u2032 \u207b\u00b9\n        u\u2081 = \u210b m \u2219 w -- mod q\n        u\u2082 = R \u2219 w    -- mod q\n        v = (g ^ u\u2081) \u2219pq (g\u02e3 ^ u\u2082) {-mod p [mod-q] -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'drafts\/dsa.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ccc91a1d485cba9d42a78ea4ff80597f0f7e7849","subject":"Add bytecode.agda to experiment with circuits","message":"Add bytecode.agda to experiment with circuits\n","repos":"crypto-agda\/crypto-agda","old_file":"bytecode.agda","new_file":"bytecode.agda","new_contents":"-- This module is an example of the use of the circuit building library.\n-- The point is to start with a tiny bytecode evaluator for bit operations.\nmodule bytecode where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Vec\nopen import Data.Product.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Algebra.FunctionProperties\n\ndata I : \u2115 \u2192 Set where\n  `[]  : I 1\n  `op\u2080 : \u2200 {i} (b : Bit) (is : I (1 + i)) \u2192 I i\n  `op\u2081 : \u2200 {i} (op\u2081 : Op\u2081 Bit) (is : I (1 + i)) \u2192 I (1 + i)\n  `op\u2082 : \u2200 {i} (op\u2082 : Op\u2082 Bit) (is : I (1 + i)) \u2192 I (2 + i)\n\neval : \u2200 {i} \u2192 I i \u2192 Bits i \u2192 Bit\neval `[] (x \u2237 []) = x\neval (`op\u2080 b is) st = eval is (b \u2237 st)\neval (`op\u2081 op\u2081 is) (b \u2237 st) = eval is (op\u2081 b \u2237 st)\neval (`op\u2082 op\u2082 is) (b\u2080 \u2237 b\u2081 \u2237 st) = eval is (op\u2082 b\u2080 b\u2081 \u2237 st)\n\neval\u2080 : I 0 \u2192 Bit\neval\u2080 i\u2080 = eval i\u2080 []\n\nopen import circuit\n\nmodule Ck {C} (cb : CircuitBuilder C) where\n  open CircuitBuilder cb\n\n  data IC : \u2115 \u2192 Set where\n    `[] : IC 1\n    `op : \u2200 {i ki ko} (op : C ki ko) (is : IC (ko + i)) \u2192 IC (ki + i)\n\n  module Eval (runC : RunCircuit C) where\n    module CF = CircuitBuilder bitsFunCircuitBuilder\n    ckIC : \u2200 {i} \u2192 IC i \u2192 C i 1\n    ckIC `[]         = idC\n    ckIC (`op op is) = op *** idC >>> ckIC is\n\n    evalIC : \u2200 {i} \u2192 IC i \u2192 Bits i \u2192 Bit\n    evalIC is bs = head (runC (ckIC is) bs)\n\n  ck : \u2200 {i} \u2192 I i \u2192 C i 1\n  ck `[]           = idC\n  ck (`op\u2080 b   is) = bit b *** idC >>> ck is\n  ck (`op\u2081 op\u2081 is) = unOp op\u2081 *** idC >>> ck is\n  ck (`op\u2082 op\u2082 is) = binOp op\u2082 *** idC >>> ck is\n\n  I2IC : \u2200 {i} \u2192 I i \u2192 IC i\n  I2IC `[] = `[]\n  I2IC (`op\u2080 b is) = `op (bit b) (I2IC is)\n  I2IC (`op\u2081 op\u2081 is) = `op (unOp op\u2081) (I2IC is)\n  I2IC (`op\u2082 op\u2082 is) = `op (binOp op\u2082) (I2IC is)\n\n  ck\u2080 : I 0 \u2192 C 0 1\n  ck\u2080 = ck\n\nmodule CheckCk where\n  open Ck bitsFunCircuitBuilder\n  open CircuitBuilder bitsFunCircuitBuilder\n  open import Relation.Binary.PropositionalEquality\n\n  ck-eval : \u2200 {i} (is : I i) bs \u2192 ck is bs \u2261 eval is bs \u2237 []\n  ck-eval `[] (x \u2237 []) = refl\n  ck-eval (`op\u2080 b is) bs = ck-eval is (b \u2237 bs)\n  ck-eval (`op\u2081 op\u2081 is) (b \u2237 bs) = ck-eval is (op\u2081 b \u2237 bs)\n  ck-eval (`op\u2082 op\u2082 is) (b\u2080 \u2237 b\u2081 \u2237 bs) = ck-eval is (op\u2082 b\u2080 b\u2081 \u2237 bs)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'bytecode.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8d59de63949b568bec30d0510248d1d09645ac36","subject":"Added rewrite of Inference by Prabhakar","message":"Added rewrite of Inference by Prabhakar\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"extra\/842Inference.agda","new_file":"extra\/842Inference.agda","new_contents":"module 842Inference where\nimport Relation.Binary.PropositionalEquality as Eq\nopen Eq using (_\u2261_; refl; sym; trans; cong; cong\u2082; _\u2262_)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Nat using (\u2115; zero; suc; _+_)\nopen import Data.String using (String; _\u225f_)\nopen import Data.Product using (_\u00d7_; \u2203; \u2203-syntax) renaming (_,_ to \u27e8_,_\u27e9)\nopen import Relation.Nullary using (\u00ac_; Dec; yes; no)\nimport plfa.part2.DeBruijn as DB\n\n-- Syntax.\n\ninfix   4  _\u220b_\u2982_\ninfix   4  _\u22a2_\u2191_\ninfix   4  _\u22a2_\u2193_\ninfixl  5  _,_\u2982_\n\ninfixr  7  _\u21d2_\n\ninfix   5  \u019b_\u21d2_\ninfix   5  \u03bc_\u21d2_\ninfix   6  _\u2191\ninfix   6  _\u2193_\ninfixl  7  _\u00b7_\ninfix   8  `suc_\ninfix   9  `_\n\nId : Set\nId = String\n\ndata Type : Set where\n  `\u2115    : Type\n  _\u21d2_   : Type \u2192 Type \u2192 Type\n\ndata Context : Set where\n  \u2205     : Context\n  _,_\u2982_ : Context \u2192 Id \u2192 Type \u2192 Context\n\ndata Term\u207a : Set\ndata Term\u207b : Set\n\ndata Term\u207a where\n  `_                        : Id \u2192 Term\u207a\n  _\u00b7_                       : Term\u207a \u2192 Term\u207b \u2192 Term\u207a\n  _\u2193_                       : Term\u207b \u2192 Type \u2192 Term\u207a\n\ndata Term\u207b where\n  \u019b_\u21d2_                     : Id \u2192 Term\u207b \u2192 Term\u207b\n  `zero                    : Term\u207b\n  `suc_                    : Term\u207b \u2192 Term\u207b\n  `case_[zero\u21d2_|suc_\u21d2_]    : Term\u207a \u2192 Term\u207b \u2192 Id \u2192 Term\u207b \u2192 Term\u207b\n  \u03bc_\u21d2_                     : Id \u2192 Term\u207b \u2192 Term\u207b\n  _\u2191                       : Term\u207a \u2192 Term\u207b\n\n-- Examples of terms.\n\ntwo : Term\u207b\ntwo = `suc (`suc `zero)\n\nplus : Term\u207a\nplus = (\u03bc \"p\" \u21d2 \u019b \"m\" \u21d2 \u019b \"n\" \u21d2\n          `case (` \"m\") [zero\u21d2 ` \"n\" \u2191\n                        |suc \"m\" \u21d2 `suc (` \"p\" \u00b7 (` \"m\" \u2191) \u00b7 (` \"n\" \u2191) \u2191) ])\n            \u2193 (`\u2115 \u21d2 `\u2115 \u21d2 `\u2115)\n\n2+2 : Term\u207a\n2+2 = plus \u00b7 two \u00b7 two\nCh : Type\nCh = (`\u2115 \u21d2 `\u2115) \u21d2 `\u2115 \u21d2 `\u2115\n\ntwo\u1d9c : Term\u207b\ntwo\u1d9c = (\u019b \"s\" \u21d2 \u019b \"z\" \u21d2 ` \"s\" \u00b7 (` \"s\" \u00b7 (` \"z\" \u2191) \u2191) \u2191)\n\nplus\u1d9c : Term\u207a\nplus\u1d9c = (\u019b \"m\" \u21d2 \u019b \"n\" \u21d2 \u019b \"s\" \u21d2 \u019b \"z\" \u21d2\n           ` \"m\" \u00b7 (` \"s\" \u2191) \u00b7 (` \"n\" \u00b7 (` \"s\" \u2191) \u00b7 (` \"z\" \u2191) \u2191) \u2191)\n             \u2193 (Ch \u21d2 Ch \u21d2 Ch)\n\nsuc\u1d9c : Term\u207b\nsuc\u1d9c = \u019b \"x\" \u21d2 `suc (` \"x\" \u2191)\n\n2+2\u1d9c : Term\u207a\n2+2\u1d9c = plus\u1d9c \u00b7 two\u1d9c \u00b7 two\u1d9c \u00b7 suc\u1d9c \u00b7 `zero\n\n-- Lookup judgments.\n\ndata _\u220b_\u2982_ : Context \u2192 Id \u2192 Type \u2192 Set where\n\n  Z : \u2200 {\u0393 x A}\n      --------------------\n    \u2192 \u0393 , x \u2982 A \u220b x \u2982 A\n\n  S : \u2200 {\u0393 x y A B}\n    \u2192 x \u2262 y\n    \u2192 \u0393 \u220b x \u2982 A\n      -----------------\n    \u2192 \u0393 , y \u2982 B \u220b x \u2982 A\n\n-- Synthesis and inheritance.\n\ndata _\u22a2_\u2191_ : Context \u2192 Term\u207a \u2192 Type \u2192 Set\ndata _\u22a2_\u2193_ : Context \u2192 Term\u207b \u2192 Type \u2192 Set\n\ndata _\u22a2_\u2191_ where\n\n  \u22a2` : \u2200 {\u0393 A x}\n    \u2192 \u0393 \u220b x \u2982 A\n      -----------\n    \u2192 \u0393 \u22a2 ` x \u2191 A\n\n  _\u00b7_ : \u2200 {\u0393 L M A B}\n    \u2192 \u0393 \u22a2 L \u2191 A \u21d2 B\n    \u2192 \u0393 \u22a2 M \u2193 A\n      -------------\n    \u2192 \u0393 \u22a2 L \u00b7 M \u2191 B\n\n  \u22a2\u2193 : \u2200 {\u0393 M A}\n    \u2192 \u0393 \u22a2 M \u2193 A\n      ---------------\n    \u2192 \u0393 \u22a2 (M \u2193 A) \u2191 A\n\ndata _\u22a2_\u2193_ where\n\n  \u22a2\u019b : \u2200 {\u0393 x N A B}\n    \u2192 \u0393 , x \u2982 A \u22a2 N \u2193 B\n      -------------------\n    \u2192 \u0393 \u22a2 \u019b x \u21d2 N \u2193 A \u21d2 B\n\n  \u22a2zero : \u2200 {\u0393}\n      --------------\n    \u2192 \u0393 \u22a2 `zero \u2193 `\u2115\n\n  \u22a2suc : \u2200 {\u0393 M}\n    \u2192 \u0393 \u22a2 M \u2193 `\u2115\n      ---------------\n    \u2192 \u0393 \u22a2 `suc M \u2193 `\u2115\n\n  \u22a2case : \u2200 {\u0393 L M x N A}\n    \u2192 \u0393 \u22a2 L \u2191 `\u2115\n    \u2192 \u0393 \u22a2 M \u2193 A\n    \u2192 \u0393 , x \u2982 `\u2115 \u22a2 N \u2193 A\n      -------------------------------------\n    \u2192 \u0393 \u22a2 `case L [zero\u21d2 M |suc x \u21d2 N ] \u2193 A\n\n  \u22a2\u03bc : \u2200 {\u0393 x N A}\n    \u2192 \u0393 , x \u2982 A \u22a2 N \u2193 A\n      -----------------\n    \u2192 \u0393 \u22a2 \u03bc x \u21d2 N \u2193 A\n\n  \u22a2\u2191 : \u2200 {\u0393 M A B}\n    \u2192 \u0393 \u22a2 M \u2191 A\n    \u2192 A \u2261 B\n      -------------\n    \u2192 \u0393 \u22a2 (M \u2191) \u2193 B\n\n\n\n-- PLFA exercise: write the term for multiplication (from Lambda)\n\n-- PLFA exercise: extend the rules to support products (from More)\n\n-- PLFA exercise (stretch): extend the rules to support the other\n-- constructs from More\n\n-- Equality of types.\n\n_\u225fTp_ : (A B : Type) \u2192 Dec (A \u2261 B)\n`\u2115      \u225fTp `\u2115              =  yes refl\n`\u2115      \u225fTp (A \u21d2 B)         =  no \u03bb()\n(A \u21d2 B) \u225fTp `\u2115              =  no \u03bb()\n(A \u21d2 B) \u225fTp (A\u2032 \u21d2 B\u2032)\n  with A \u225fTp A\u2032 | B \u225fTp B\u2032\n...  | no A\u2262    | _         =  no \u03bb{refl \u2192 A\u2262 refl}\n...  | yes _    | no B\u2262     =  no \u03bb{refl \u2192 B\u2262 refl}\n...  | yes refl | yes refl  =  yes refl\n\n-- Helpers: domain and range of equal function types are equal,\n-- `\u2115 is not a function type.\n\ndom\u2261 : \u2200 {A A\u2032 B B\u2032} \u2192 A \u21d2 B \u2261 A\u2032 \u21d2 B\u2032 \u2192 A \u2261 A\u2032\ndom\u2261 refl = refl\n\nrng\u2261 : \u2200 {A A\u2032 B B\u2032} \u2192 A \u21d2 B \u2261 A\u2032 \u21d2 B\u2032 \u2192 B \u2261 B\u2032\nrng\u2261 refl = refl\n\n\u2115\u2262\u21d2 : \u2200 {A B} \u2192 `\u2115 \u2262 A \u21d2 B\n\u2115\u2262\u21d2 ()\n\n-- Lookup judgments are unique.\n\nuniq-\u220b : \u2200 {\u0393 x A B} \u2192 \u0393 \u220b x \u2982 A \u2192 \u0393 \u220b x \u2982 B \u2192 A \u2261 B\nuniq-\u220b Z Z                 =  refl\nuniq-\u220b Z (S x\u2262y _)         =  \u22a5-elim (x\u2262y refl)\nuniq-\u220b (S x\u2262y _) Z         =  \u22a5-elim (x\u2262y refl)\nuniq-\u220b (S _ \u220bx) (S _ \u220bx\u2032)  =  uniq-\u220b \u220bx \u220bx\u2032\n\n-- A synthesized type is unique.\n\nuniq-\u2191 : \u2200 {\u0393 M A B} \u2192 \u0393 \u22a2 M \u2191 A \u2192 \u0393 \u22a2 M \u2191 B \u2192 A \u2261 B\nuniq-\u2191 (\u22a2` \u220bx) (\u22a2` \u220bx\u2032)       =  uniq-\u220b \u220bx \u220bx\u2032\nuniq-\u2191 (\u22a2L \u00b7 \u22a2M) (\u22a2L\u2032 \u00b7 \u22a2M\u2032)  =  rng\u2261 (uniq-\u2191 \u22a2L \u22a2L\u2032)\nuniq-\u2191 (\u22a2\u2193 \u22a2M) (\u22a2\u2193 \u22a2M\u2032)       =  refl \n\n-- Failed lookups still fail if a different binding is added.\n\next\u220b : \u2200 {\u0393 B x y}\n  \u2192 x \u2262 y\n  \u2192 \u00ac \u2203[ A ]( \u0393 \u220b x \u2982 A )\n    -----------------------------\n  \u2192 \u00ac \u2203[ A ]( \u0393 , y \u2982 B \u220b x \u2982 A )\next\u220b x\u2262y _  \u27e8 A , Z \u27e9       =  x\u2262y refl\next\u220b _   \u00ac\u2203 \u27e8 A , S _ \u22a2x \u27e9  =  \u00ac\u2203 \u27e8 A , \u22a2x \u27e9\n\n-- Decision procedure for lookup judgments.\n\nlookup : \u2200 (\u0393 : Context) (x : Id)\n    -----------------------\n  \u2192 Dec (\u2203[ A ](\u0393 \u220b x \u2982 A))\nlookup \u2205 x                        =  no  (\u03bb ())\nlookup (\u0393 , y \u2982 B) x with x \u225f y\n... | yes refl                    =  yes \u27e8 B , Z \u27e9\n... | no x\u2262y with lookup \u0393 x\n...             | no  \u00ac\u2203          =  no  (ext\u220b x\u2262y \u00ac\u2203)\n...             | yes \u27e8 A , \u22a2x \u27e9  =  yes \u27e8 A , S x\u2262y \u22a2x \u27e9\n\n-- Helpers for promoting a failure to type.\n\n\u00acarg : \u2200 {\u0393 A B L M}\n  \u2192 \u0393 \u22a2 L \u2191 A \u21d2 B\n  \u2192 \u00ac \u0393 \u22a2 M \u2193 A\n    -------------------------\n  \u2192 \u00ac \u2203[ B\u2032 ](\u0393 \u22a2 L \u00b7 M \u2191 B\u2032)\n\u00acarg \u22a2L \u00ac\u22a2M \u27e8 B\u2032 , \u22a2L\u2032 \u00b7 \u22a2M\u2032 \u27e9 rewrite dom\u2261 (uniq-\u2191 \u22a2L \u22a2L\u2032) = \u00ac\u22a2M \u22a2M\u2032\n\n\u00acswitch : \u2200 {\u0393 M A B}\n  \u2192 \u0393 \u22a2 M \u2191 A\n  \u2192 A \u2262 B\n    ---------------\n  \u2192 \u00ac \u0393 \u22a2 (M \u2191) \u2193 B\n\u00acswitch \u22a2M A\u2262B (\u22a2\u2191 \u22a2M\u2032 A\u2032\u2261B) rewrite uniq-\u2191 \u22a2M \u22a2M\u2032 = A\u2262B A\u2032\u2261B\n\n-- Mutually-recursive synthesize and inherit functions.\n\nsynthesize : \u2200 (\u0393 : Context) (M : Term\u207a)\n    -----------------------\n  \u2192 Dec (\u2203[ A ](\u0393 \u22a2 M \u2191 A))\n\ninherit : \u2200 (\u0393 : Context) (M : Term\u207b) (A : Type)\n    ---------------\n  \u2192 Dec (\u0393 \u22a2 M \u2193 A)\n\nsynthesize \u0393 (` x) with lookup \u0393 x\n... | no  \u00ac\u2203              =  no  (\u03bb{ \u27e8 A , \u22a2` \u220bx \u27e9 \u2192 \u00ac\u2203 \u27e8 A , \u220bx \u27e9 })\n... | yes \u27e8 A , \u220bx \u27e9      =  yes \u27e8 A , \u22a2` \u220bx \u27e9\nsynthesize \u0393 (L \u00b7 M) with synthesize \u0393 L\n... | no  \u00ac\u2203              =  no  (\u03bb{ \u27e8 _ , \u22a2L  \u00b7 _  \u27e9  \u2192  \u00ac\u2203 \u27e8 _ , \u22a2L \u27e9 })\n... | yes \u27e8 `\u2115 ,    \u22a2L \u27e9  =  no  (\u03bb{ \u27e8 _ , \u22a2L\u2032 \u00b7 _  \u27e9  \u2192  \u2115\u2262\u21d2 (uniq-\u2191 \u22a2L \u22a2L\u2032) })\n... | yes \u27e8 A \u21d2 B , \u22a2L \u27e9 with inherit \u0393 M A\n...    | no  \u00ac\u22a2M          =  no  (\u00acarg \u22a2L \u00ac\u22a2M)\n...    | yes \u22a2M           =  yes \u27e8 B , \u22a2L \u00b7 \u22a2M \u27e9  \nsynthesize \u0393 (M \u2193 A) with inherit \u0393 M A\n... | no  \u00ac\u22a2M             =  no  (\u03bb{ \u27e8 _ , \u22a2\u2193 \u22a2M \u27e9  \u2192  \u00ac\u22a2M \u22a2M })\n... | yes \u22a2M              =  yes \u27e8 A , \u22a2\u2193 \u22a2M \u27e9\n\ninherit \u0393 (\u019b x \u21d2 N) `\u2115      =  no  (\u03bb())\ninherit \u0393 (\u019b x \u21d2 N) (A \u21d2 B) with inherit (\u0393 , x \u2982 A) N B\n... | no \u00ac\u22a2N                =  no  (\u03bb{ (\u22a2\u019b \u22a2N)  \u2192  \u00ac\u22a2N \u22a2N })\n... | yes \u22a2N                =  yes (\u22a2\u019b \u22a2N)\ninherit \u0393 `zero `\u2115          =  yes \u22a2zero\ninherit \u0393 `zero (A \u21d2 B)     =  no  (\u03bb())\ninherit \u0393 (`suc M) `\u2115 with inherit \u0393 M `\u2115\n... | no \u00ac\u22a2M                =  no  (\u03bb{ (\u22a2suc \u22a2M)  \u2192  \u00ac\u22a2M \u22a2M })\n... | yes \u22a2M                =  yes (\u22a2suc \u22a2M)\ninherit \u0393 (`suc M) (A \u21d2 B)  =  no  (\u03bb())\ninherit \u0393 (`case L [zero\u21d2 M |suc x \u21d2 N ]) A with synthesize \u0393 L\n... | no \u00ac\u2203                 =  no  (\u03bb{ (\u22a2case \u22a2L  _ _) \u2192 \u00ac\u2203 \u27e8 `\u2115 , \u22a2L \u27e9})\n... | yes \u27e8 _ \u21d2 _ , \u22a2L \u27e9    =  no  (\u03bb{ (\u22a2case \u22a2L\u2032 _ _) \u2192 \u2115\u2262\u21d2 (uniq-\u2191 \u22a2L\u2032 \u22a2L) })   \n... | yes \u27e8 `\u2115 ,    \u22a2L \u27e9 with inherit \u0393 M A\n...    | no \u00ac\u22a2M             =  no  (\u03bb{ (\u22a2case _ \u22a2M _) \u2192 \u00ac\u22a2M \u22a2M })\n...    | yes \u22a2M with inherit (\u0393 , x \u2982 `\u2115) N A\n...       | no \u00ac\u22a2N          =  no  (\u03bb{ (\u22a2case _ _ \u22a2N) \u2192 \u00ac\u22a2N \u22a2N })\n...       | yes \u22a2N          =  yes (\u22a2case \u22a2L \u22a2M \u22a2N)\ninherit \u0393 (\u03bc x \u21d2 N) A with inherit (\u0393 , x \u2982 A) N A\n... | no \u00ac\u22a2N                =  no  (\u03bb{ (\u22a2\u03bc \u22a2N) \u2192 \u00ac\u22a2N \u22a2N })\n... | yes \u22a2N                =  yes (\u22a2\u03bc \u22a2N)\ninherit \u0393 (M \u2191) B with synthesize \u0393 M\n... | no  \u00ac\u2203                =  no  (\u03bb{ (\u22a2\u2191 \u22a2M _) \u2192 \u00ac\u2203 \u27e8 _ , \u22a2M \u27e9 })\n... | yes \u27e8 A , \u22a2M \u27e9 with A \u225fTp B\n...   | no  A\u2262B             =  no  (\u00acswitch \u22a2M A\u2262B)\n...   | yes A\u2261B             =  yes (\u22a2\u2191 \u22a2M A\u2261B)\n\n-- Copied from Lambda.\n\n_\u2260_ : \u2200 (x y : Id) \u2192 x \u2262 y\nx \u2260 y  with x \u225f y\n...       | no  x\u2262y  =  x\u2262y\n...       | yes _    =  \u22a5-elim impossible\n  where postulate impossible : \u22a5\n\n-- Computed by evaluating 'synthesize \u2205 2+2' and editing.\n\n\u22a22+2 : \u2205 \u22a2 2+2 \u2191 `\u2115\n\u22a22+2 =\n  (\u22a2\u2193\n   (\u22a2\u03bc\n    (\u22a2\u019b\n     (\u22a2\u019b\n      (\u22a2case (\u22a2` (S (\u03bb()) Z)) (\u22a2\u2191 (\u22a2` Z) refl)\n       (\u22a2suc\n        (\u22a2\u2191\n         (\u22a2`\n          (S (\u03bb())\n           (S (\u03bb())\n            (S (\u03bb()) Z)))\n          \u00b7 \u22a2\u2191 (\u22a2` Z) refl\n          \u00b7 \u22a2\u2191 (\u22a2` (S (\u03bb()) Z)) refl)\n         refl))))))\n   \u00b7 \u22a2suc (\u22a2suc \u22a2zero)\n   \u00b7 \u22a2suc (\u22a2suc \u22a2zero))\n\n-- Check that synthesis is correct (more below).\n\n_ : synthesize \u2205 2+2 \u2261 yes \u27e8 `\u2115 , \u22a22+2 \u27e9\n_ = refl\n\n-- Example: 2+2 for Church numerals.\n\n\u22a22+2\u1d9c : \u2205 \u22a2 2+2\u1d9c \u2191 `\u2115\n\u22a22+2\u1d9c =\n  \u22a2\u2193\n  (\u22a2\u019b\n   (\u22a2\u019b\n    (\u22a2\u019b\n     (\u22a2\u019b\n      (\u22a2\u2191\n       (\u22a2`\n        (S (\u03bb())\n         (S (\u03bb())\n          (S (\u03bb()) Z)))\n        \u00b7 \u22a2\u2191 (\u22a2` (S (\u03bb()) Z)) refl\n        \u00b7\n        \u22a2\u2191\n        (\u22a2`\n         (S (\u03bb())\n          (S (\u03bb()) Z))\n         \u00b7 \u22a2\u2191 (\u22a2` (S (\u03bb()) Z)) refl\n         \u00b7 \u22a2\u2191 (\u22a2` Z) refl)\n        refl)\n       refl)))))\n  \u00b7\n  \u22a2\u019b\n  (\u22a2\u019b\n   (\u22a2\u2191\n    (\u22a2` (S (\u03bb()) Z) \u00b7\n     \u22a2\u2191 (\u22a2` (S (\u03bb()) Z) \u00b7 \u22a2\u2191 (\u22a2` Z) refl)\n     refl)\n    refl))\n  \u00b7\n  \u22a2\u019b\n  (\u22a2\u019b\n   (\u22a2\u2191\n    (\u22a2` (S (\u03bb()) Z) \u00b7\n     \u22a2\u2191 (\u22a2` (S (\u03bb()) Z) \u00b7 \u22a2\u2191 (\u22a2` Z) refl)\n     refl)\n    refl))\n  \u00b7 \u22a2\u019b (\u22a2suc (\u22a2\u2191 (\u22a2` Z) refl))\n  \u00b7 \u22a2zero\n\n_ : synthesize \u2205 2+2\u1d9c \u2261 yes \u27e8 `\u2115 , \u22a22+2\u1d9c \u27e9\n_ = refl\n\n-- Testing error cases.\n\n_ : synthesize \u2205 ((\u019b \"x\" \u21d2 ` \"y\" \u2191) \u2193 (`\u2115 \u21d2 `\u2115)) \u2261 no _\n_ = refl\n\n-- Bad argument type.\n\n_ : synthesize \u2205 (plus \u00b7 suc\u1d9c) \u2261 no _\n_ = refl\n\n-- Bad function types.\n\n_ : synthesize \u2205 (plus \u00b7 suc\u1d9c \u00b7 two) \u2261 no _\n_ = refl\n\n_ : synthesize \u2205 ((two \u2193 `\u2115) \u00b7 two) \u2261 no _\n_ = refl\n\n_ : synthesize \u2205 (two\u1d9c \u2193 `\u2115) \u2261 no _\n_ = refl\n\n-- A natural can't have a function type.\n\n_ : synthesize \u2205 (`zero \u2193 `\u2115 \u21d2 `\u2115) \u2261 no _\n_ = refl\n\n_ : synthesize \u2205 (two \u2193 `\u2115 \u21d2 `\u2115) \u2261 no _\n_ = refl\n\n-- Can't hide a bad type.\n\n_ : synthesize \u2205 (`suc two\u1d9c \u2193 `\u2115) \u2261 no _\n_ = refl\n\n-- Can't case on a function type.\n\n_ : synthesize \u2205\n      ((`case (two\u1d9c \u2193 Ch) [zero\u21d2 `zero |suc \"x\" \u21d2 ` \"x\" \u2191 ] \u2193 `\u2115) ) \u2261 no _\n_ = refl\n\n-- Can't hide a bad type inside case.\n\n_ : synthesize \u2205\n      ((`case (two\u1d9c \u2193 `\u2115) [zero\u21d2 `zero |suc \"x\" \u21d2 ` \"x\" \u2191 ] \u2193 `\u2115) ) \u2261 no _\n_ = refl\n\n-- Mismatch of inherited and synthesized types.\n\n_ : synthesize \u2205 (((\u019b \"x\" \u21d2 ` \"x\" \u2191) \u2193 `\u2115 \u21d2 (`\u2115 \u21d2 `\u2115))) \u2261 no _\n_ = refl\n\n\n-- Erasure: Taking the evidence provided by synthesis on a decorated term\n-- and producing the corresponding inherently-typed term.\n\n-- Erasing a type.\n\n\u2225_\u2225Tp : Type \u2192 DB.Type\n\u2225 `\u2115 \u2225Tp             =  DB.`\u2115\n\u2225 A \u21d2 B \u2225Tp          =  \u2225 A \u2225Tp DB.\u21d2 \u2225 B \u2225Tp\n\n-- Erasing a context.\n\n\u2225_\u2225Cx : Context \u2192 DB.Context\n\u2225 \u2205 \u2225Cx              =  DB.\u2205\n\u2225 \u0393 , x \u2982 A \u2225Cx      =  \u2225 \u0393 \u2225Cx DB., \u2225 A \u2225Tp\n\n-- Erasing a lookup judgment.\n\n\u2225_\u2225\u220b : \u2200 {\u0393 x A} \u2192 \u0393 \u220b x \u2982 A \u2192 \u2225 \u0393 \u2225Cx DB.\u220b \u2225 A \u2225Tp\n\u2225 Z \u2225\u220b               =  DB.Z\n\u2225 S x\u2262 \u22a2x \u2225\u220b         =  DB.S \u2225 \u22a2x \u2225\u220b\n\n-- Mutually-recursive functions to erase typing judgments.\n\n\u2225_\u2225\u207a : \u2200 {\u0393 M A} \u2192 \u0393 \u22a2 M \u2191 A \u2192 \u2225 \u0393 \u2225Cx DB.\u22a2 \u2225 A \u2225Tp\n\u2225_\u2225\u207b : \u2200 {\u0393 M A} \u2192 \u0393 \u22a2 M \u2193 A \u2192 \u2225 \u0393 \u2225Cx DB.\u22a2 \u2225 A \u2225Tp\n\n\u2225 \u22a2` \u22a2x \u2225\u207a           =  DB.` \u2225 \u22a2x \u2225\u220b\n\u2225 \u22a2L \u00b7 \u22a2M \u2225\u207a         =  \u2225 \u22a2L \u2225\u207a DB.\u00b7 \u2225 \u22a2M \u2225\u207b\n\u2225 \u22a2\u2193 \u22a2M \u2225\u207a           =  \u2225 \u22a2M \u2225\u207b\n\n\u2225 \u22a2\u019b \u22a2N \u2225\u207b           =  DB.\u019b \u2225 \u22a2N \u2225\u207b\n\u2225 \u22a2zero \u2225\u207b           =  DB.`zero\n\u2225 \u22a2suc \u22a2M \u2225\u207b         =  DB.`suc \u2225 \u22a2M \u2225\u207b\n\u2225 \u22a2case \u22a2L \u22a2M \u22a2N \u2225\u207b  =  DB.case \u2225 \u22a2L \u2225\u207a \u2225 \u22a2M \u2225\u207b \u2225 \u22a2N \u2225\u207b\n\u2225 \u22a2\u03bc \u22a2M \u2225\u207b           =  DB.\u03bc \u2225 \u22a2M \u2225\u207b\n\u2225 \u22a2\u2191 \u22a2M refl \u2225\u207b      =  \u2225 \u22a2M \u2225\u207a\n\n-- Tests that erasure works.\n\n_ : \u2225 \u22a22+2 \u2225\u207a \u2261 DB.2+2\n_ = refl\n\n_ : \u2225 \u22a22+2\u1d9c \u2225\u207a \u2261 DB.2+2\u1d9c\n_ = refl\n\n-- PLFA exercise: demonstrate that synthesis on your decorated multiplication\n-- followed by erasure gives your inherently-typed multiplication.\n\n-- PLFA exercise: extend the above to include products.\n\n-- PLFA exercise (stretch): extend the above to include\n-- the rest of the features added in More.\n\n-- Additions by PR:\n\n-- From Lambda, with type annotation added\n\ndata Term : Set where\n  `_                      :  Id \u2192 Term\n  \u019b_\u21d2_                    :  Id \u2192 Term \u2192 Term\n  _\u00b7_                     :  Term \u2192 Term \u2192 Term\n  `zero                   :  Term\n  `suc_                   :  Term \u2192 Term\n  `case_[zero\u21d2_|suc_\u21d2_]    :  Term \u2192 Term \u2192 Id \u2192 Term \u2192 Term\n  \u03bc_\u21d2_                    :  Id \u2192 Term \u2192 Term\n  _\u2982_                     :  Term \u2192 Type \u2192 Term\n\n-- Mutually-recursive decorators.\n\ndecorate\u207b : Term \u2192 Term\u207b\ndecorate\u207a : Term \u2192 Term\u207a\n\ndecorate\u207b (` x) = ` x \u2191\ndecorate\u207b (\u019b x \u21d2 M) = \u019b x \u21d2 decorate\u207b M\ndecorate\u207b (M \u00b7 M\u2081) = (decorate\u207a M) \u00b7 (decorate\u207b M\u2081) \u2191\ndecorate\u207b `zero = `zero\ndecorate\u207b (`suc M) = `suc (decorate\u207b M)\ndecorate\u207b `case M [zero\u21d2 M\u2081 |suc x \u21d2 M\u2082 ] \n = `case (decorate\u207a M) [zero\u21d2 (decorate\u207b M\u2081) |suc x \u21d2 (decorate\u207b M\u2082) ]\ndecorate\u207b (\u03bc x \u21d2 M) = \u03bc x \u21d2 decorate\u207b M\ndecorate\u207b (M \u2982 x) = decorate\u207b M\n\ndecorate\u207a (` x) = ` x\ndecorate\u207a (\u019b x \u21d2 M) = \u22a5-elim impossible\n where postulate impossible : \u22a5\ndecorate\u207a (M \u00b7 M\u2081) = (decorate\u207a M) \u00b7 (decorate\u207b M\u2081)\ndecorate\u207a `zero = \u22a5-elim impossible\n where postulate impossible : \u22a5\ndecorate\u207a (`suc M) = \u22a5-elim impossible\n where postulate impossible : \u22a5\ndecorate\u207a `case M [zero\u21d2 M\u2081 |suc x \u21d2 M\u2082 ] = \u22a5-elim impossible\n where postulate impossible : \u22a5\ndecorate\u207a (\u03bc x \u21d2 M) = \u22a5-elim impossible\n where postulate impossible : \u22a5\ndecorate\u207a (M \u2982 x) = decorate\u207b M \u2193 x\n\nltwo : Term\nltwo = `suc `suc `zero\n\nlplus : Term\nlplus = (\u03bc \"p\" \u21d2 \u019b \"m\" \u21d2 \u019b \"n\" \u21d2\n          `case ` \"m\"\n            [zero\u21d2 ` \"n\"\n            |suc \"m\" \u21d2 `suc (` \"p\" \u00b7 ` \"m\" \u00b7 ` \"n\") ])\n         \u2982 (`\u2115 \u21d2 `\u2115 \u21d2 `\u2115)\n\nl2+2 : Term\nl2+2 = lplus \u00b7 ltwo \u00b7 ltwo\n\nltwo\u1d9c : Term\nltwo\u1d9c =  \u019b \"s\" \u21d2 \u019b \"z\" \u21d2 ` \"s\" \u00b7 (` \"s\" \u00b7 ` \"z\")\n\nlplus\u1d9c : Term\nlplus\u1d9c =  (\u019b \"m\" \u21d2 \u019b \"n\" \u21d2 \u019b \"s\" \u21d2 \u019b \"z\" \u21d2\n         ` \"m\" \u00b7 ` \"s\" \u00b7 (` \"n\" \u00b7 ` \"s\" \u00b7 ` \"z\"))\n         \u2982 (Ch \u21d2 Ch \u21d2 Ch)\n\nlsuc\u1d9c : Term\nlsuc\u1d9c = \u019b \"x\" \u21d2 `suc (` \"x\")\n\nl2+2\u1d9c : Term\nl2+2\u1d9c = lplus\u1d9c \u00b7 ltwo\u1d9c \u00b7 ltwo\u1d9c \u00b7 lsuc\u1d9c \u00b7 `zero\n\n_ : decorate\u207b ltwo \u2261 two\n_ = refl\n\n_ : decorate\u207a lplus \u2261 plus\n_ = refl\n\n_ : decorate\u207a l2+2 \u2261 2+2\n_ = refl\n\n_ : decorate\u207b ltwo\u1d9c \u2261 two\u1d9c\n_ = refl\n\n_ : decorate\u207a lplus\u1d9c \u2261 plus\u1d9c\n_ = refl\n\n_ : decorate\u207b lsuc\u1d9c \u2261 suc\u1d9c\n_ = refl\n\n_ : decorate\u207a l2+2\u1d9c \u2261 2+2\u1d9c\n_ = refl\n\n\n{- \n  Unicode used in this chapter:\n\n  \u2193  U+2193:  DOWNWARDS ARROW (\\d)\n  \u2191  U+2191:  UPWARDS ARROW (\\u)\n  \u2225  U+2225:  PARALLEL TO (\\||)\n\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'extra\/842Inference.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"4318b59549a56c704d45525e65a6595e5ad42ba0","subject":"Unfinished Group.Endomorphism module","message":"Unfinished Group.Endomorphism module\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Group\/Endomorphism.agda","new_file":"lib\/Algebra\/Group\/Endomorphism.agda","new_contents":"{-# OPTIONS --without-K --copatterns #-}\nopen import Function.NP hiding (_\u2218_) renaming (_\u2218\u2032_ to _\u2218_)\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Homomorphism\nopen import Algebra.Group\nopen import Algebra.Group.Abelian\nopen import Algebra.Group.Homomorphism\nopen import Relation.Binary.PropositionalEquality.NP renaming (_\u2219_ to _\u2666_)\n\nimport Algebra as A\nimport Algebra.Structures as AS\n\n{- without commutativity\nmodule Algebra.Group.Endomorphism {{_ : FunExt}}{\u2113}{G : Set \u2113}(\ud835\udd3e : Group G)\n  where\n\nopen Group \ud835\udd3e public\n\nGroupEndomorphism : \u2200 {\u2113}{G : Set \u2113} \u2192 Group G \u2192 Endo G \u2192 Set \u2113\nGroupEndomorphism \ud835\udd3e = GroupHomomorphism \ud835\udd3e \ud835\udd3e\n\nEndoG = \u03a3 (Endo G) (GroupEndomorphism \ud835\udd3e)\n\n_\u2299_ : Endo G \u2192 Endo G \u2192 Endo G\n(f \u2299 g) x = f x \u2219 g x\n\n_\u207d\u207b\u00b9\u207e : Endo G \u2192 Endo G\n(f \u207d\u207b\u00b9\u207e) x = (f x)\u207b\u00b9\n\n\u27e8\u03b5\u27e9 : Endo G\n\u27e8\u03b5\u27e9 = \u03bb _ \u2192 \u03b5\n\n\u2299-mon-ops : Monoid-Ops (Endo G)\n\u2299-mon-ops = _\u2299_ , \u27e8\u03b5\u27e9\n\n\u2299-mon-struct : Monoid-Struct \u2299-mon-ops\n\u2299-mon-struct = (\u03bb=\u2071 assoc , \u03bb=\u2071 !assoc) , \u03bb=\u2071 \u03b5\u2219-identity , \u03bb=\u2071 \u2219\u03b5-identity\n\n\u2299-grp-ops : Group-Ops (Endo G)\n\u2299-grp-ops = \u2299-mon-ops , _\u207d\u207b\u00b9\u207e\n\n\u2299-grp-struct : Group-Struct \u2299-grp-ops\n\u2299-grp-struct = \u2299-mon-struct , \u03bb=\u2071 (fst inverse) , \u03bb=\u2071 (snd inverse)\n\ndistr : _\u2218_ DistributesOver\u02e1 _\u2299_\ndistr {f} {g} {h} = \u03bb= \u03bb x \u2192 {!!}\n\nring : A.Ring _ _\nring = record\n         { Carrier = Endo G\n         ; _\u2248_ = _\u2261_\n         ; _+_ = _\u2299_\n         ; _*_ = _\u2218_\n         ; -_ = _\u207d\u207b\u00b9\u207e\n         ; 0# = \u27e8\u03b5\u27e9\n         ; 1# = id\n         ; isRing =\n           record\n           { +-isAbelianGroup =\n             record\n             { isGroup =\n               record\n               { isMonoid =\n                 record\n                 { isSemigroup =\n                   record { isEquivalence = isEquivalence\n                          ; assoc = \u03bb _ _ _ \u2192 \u03bb=\u2071 assoc\n                          ; \u2219-cong = ap\u2082 _\u2299_ }\n                 ; identity = (\u03bb _ \u2192 \u03bb=\u2071 \u03b5\u2219-identity)\n                            , (\u03bb _ \u2192 \u03bb=\u2071 \u2219\u03b5-identity) }\n               ; inverse = (\u03bb _ \u2192 \u03bb=\u2071 (fst inverse))\n                         , (\u03bb _ \u2192 \u03bb=\u2071 (snd inverse))\n               ; \u207b\u00b9-cong = ap _\u207d\u207b\u00b9\u207e }\n             ; comm = {!!} }\n           ; *-isMonoid =\n             record\n             { isSemigroup =\n               record { isEquivalence = isEquivalence\n                      ; assoc = \u03bb _ _ _ \u2192 idp\n                      ; \u2219-cong = ap\u2082 _\u2218_ }\n             ; identity = (\u03bb _ \u2192 idp) , (\u03bb _ \u2192 idp) }\n           ; distrib = {!!} }\n         }\n-}\n\nmodule Algebra.Group.Endomorphism {{_ : FunExt}}{\u2113}{G : Set \u2113} -- (\ud835\udd3e : Group G)\n  where\n\npostulate\n  \ud835\udd3e : Abelian-Group G\n\nmodule \ud835\udd3e = Abelian-Group \ud835\udd3e\n\nopen \ud835\udd3e\n\n\ud835\udd3e' : Group G\n\ud835\udd3e' = grp\n\nopen GroupHomomorphism\n\nGroupEndomorphism : \u2200 {\u2113}{G : Set \u2113} \u2192 Group G \u2192 Endo G \u2192 Set \u2113\nGroupEndomorphism \ud835\udd3e = GroupHomomorphism \ud835\udd3e \ud835\udd3e\n\nEndoG = \u03a3 (Endo G) (GroupEndomorphism grp)\n\nendoG= : {f g : EndoG} \u2192 (\u2200 {x} \u2192 fst f x \u2261 fst g x) \u2192 f \u2261 g\nendoG= {f} {g} fg = ?\n\n_\u2299_ : Endo G \u2192 Endo G \u2192 Endo G\n(f \u2299 g) x = f x \u2219 g x\n\n_\u2299'_ : EndoG \u2192 EndoG \u2192 EndoG\nfst (f \u2299' g) = fst f \u2299 fst g\nhom (snd (f \u2299' g)) = \u2219= (hom (snd f)) (hom (snd g)) \u2666 interchange\n\n_\u207d\u207b\u00b9\u207e : Endo G \u2192 Endo G\n(f \u207d\u207b\u00b9\u207e) x = (f x)\u207b\u00b9\n\n_\u207d\u207b\u00b9\u207e' : EndoG \u2192 EndoG\nfst (f \u207d\u207b\u00b9\u207e') x = (fst f x)\u207b\u00b9\nhom (snd (f \u207d\u207b\u00b9\u207e')) = ap _\u207b\u00b9 (hom (snd f)) \u2666 \u207b\u00b9-hom\n\n\u27e8\u03b5\u27e9 : Endo G\n\u27e8\u03b5\u27e9 = \u03bb _ \u2192 \u03b5\n\n\u27e8\u03b5\u27e9' : EndoG\nfst \u27e8\u03b5\u27e9'       = \u27e8\u03b5\u27e9\nhom (snd \u27e8\u03b5\u27e9') = ! \u03b5\u2219-identity\n\n\u2299-mon-ops : Monoid-Ops (Endo G)\n\u2299-mon-ops = _\u2299_ , \u27e8\u03b5\u27e9\n\n\u2299-mon-ops' : Monoid-Ops EndoG\n\u2299-mon-ops' = _\u2299'_ , \u27e8\u03b5\u27e9'\n\n\u2299-mon-struct : Monoid-Struct \u2299-mon-ops\n\u2299-mon-struct = (\u03bb=\u2071 assoc , \u03bb=\u2071 !assoc) , \u03bb=\u2071 \u03b5\u2219-identity , \u03bb=\u2071 \u2219\u03b5-identity\n\n\u2299-mon-struct' : Monoid-Struct \u2299-mon-ops'\n\u2299-mon-struct' = (endoG= assoc , endoG= !assoc) , endoG= \u03b5\u2219-identity , endoG= \u2219\u03b5-identity\n\n\u2299-grp-ops : Group-Ops (Endo G)\n\u2299-grp-ops = \u2299-mon-ops , _\u207d\u207b\u00b9\u207e\n\n\u2299-grp-ops' : Group-Ops EndoG\n\u2299-grp-ops' = \u2299-mon-ops' , _\u207d\u207b\u00b9\u207e'\n\n\u2299-grp-struct : Group-Struct \u2299-grp-ops\n\u2299-grp-struct = \u2299-mon-struct , \u03bb=\u2071 (fst inverse) , \u03bb=\u2071 (snd inverse)\n\n\u2299-grp-struct' : Group-Struct \u2299-grp-ops'\n\u2299-grp-struct' = \u2299-mon-struct' , endoG= (fst inverse) , endoG= (snd inverse)\n\nid' : EndoG\nfst id' = id\nhom (snd id') = idp\n\n_\u2218'_ : EndoG \u2192 EndoG \u2192 EndoG\nfst (f \u2218' g) = fst f \u2218 fst g\nhom (snd (f \u2218' g)) = ap (fst f) (hom (snd g)) \u2666 hom (snd f)\n\n-- NOTE: only the the first function needs to be an homomorphism\ndistr : _\u2218'_ DistributesOver\u02e1 _\u2299'_\ndistr {_ , mk f-hom} = endoG= f-hom\n\n{-\nring : A.Ring _ _\nring = record\n         { Carrier = EndoG\n         ; _\u2248_ = _\u2261_\n         ; _+_ = _\u2299'_\n         ; _*_ = _\u2218'_\n         ; -_ = _\u207d\u207b\u00b9\u207e'\n         ; 0# = \u27e8\u03b5\u27e9'\n         ; 1# = {!id!}\n         ; isRing =\n           record\n           { +-isAbelianGroup =\n             record\n             { isGroup =\n               record\n               { isMonoid =\n                 record\n                 { isSemigroup =\n                   record { isEquivalence = isEquivalence\n                          ; assoc = {!\u03bb _ _ _ \u2192 \u03bb=\u2071 assoc!}\n                          ; \u2219-cong = ap\u2082 _\u2299'_ }\n                 ; identity = (\u03bb _ \u2192 \u03bb=\u2071 \u03b5\u2219-identity)\n                            , (\u03bb _ \u2192 \u03bb=\u2071 \u2219\u03b5-identity) }\n               ; inverse = (\u03bb _ \u2192 \u03bb=\u2071 (fst inverse))\n                         , (\u03bb _ \u2192 \u03bb=\u2071 (snd inverse))\n               ; \u207b\u00b9-cong = ap _\u207d\u207b\u00b9\u207e }\n             ; comm = \u03bb _ _ \u2192 \u03bb=\u2071 comm }\n           ; *-isMonoid =\n             record\n             { isSemigroup =\n               record { isEquivalence = isEquivalence\n                      ; assoc = \u03bb _ _ _ \u2192 idp\n                      ; \u2219-cong = ap\u2082 _\u2218_ }\n             ; identity = (\u03bb _ \u2192 idp) , (\u03bb _ \u2192 idp) }\n           ; distrib = (\u03bb _ _ _ \u2192 \u03bb=\u2071 {!!}) , {!\u03bb _ _ _ \u2192 \u03bb=\u2071 distr!} }\n         }\n{-\nring : A.Ring _ _\nring = record\n         { Carrier = Endo G\n         ; _\u2248_ = _\u2261_\n         ; _+_ = _\u2299_\n         ; _*_ = _\u2218_\n         ; -_ = _\u207d\u207b\u00b9\u207e\n         ; 0# = \u27e8\u03b5\u27e9\n         ; 1# = id\n         ; isRing =\n           record\n           { +-isAbelianGroup =\n             record\n             { isGroup =\n               record\n               { isMonoid =\n                 record\n                 { isSemigroup =\n                   record { isEquivalence = isEquivalence\n                          ; assoc = \u03bb _ _ _ \u2192 \u03bb=\u2071 assoc\n                          ; \u2219-cong = ap\u2082 _\u2299_ }\n                 ; identity = (\u03bb _ \u2192 \u03bb=\u2071 \u03b5\u2219-identity)\n                            , (\u03bb _ \u2192 \u03bb=\u2071 \u2219\u03b5-identity) }\n               ; inverse = (\u03bb _ \u2192 \u03bb=\u2071 (fst inverse))\n                         , (\u03bb _ \u2192 \u03bb=\u2071 (snd inverse))\n               ; \u207b\u00b9-cong = ap _\u207d\u207b\u00b9\u207e }\n             ; comm = \u03bb _ _ \u2192 \u03bb=\u2071 comm }\n           ; *-isMonoid =\n             record\n             { isSemigroup =\n               record { isEquivalence = isEquivalence\n                      ; assoc = \u03bb _ _ _ \u2192 idp\n                      ; \u2219-cong = ap\u2082 _\u2218_ }\n             ; identity = (\u03bb _ \u2192 idp) , (\u03bb _ \u2192 idp) }\n           ; distrib = (\u03bb _ _ _ \u2192 \u03bb=\u2071 {!!}) , {!\u03bb _ _ _ \u2192 \u03bb=\u2071 distr!} }\n         }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/Group\/Endomorphism.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"96af5a6390c506714c8c59dde82be1a95e179cab","subject":"Control\/Protocol\/CLL.agda","message":"Control\/Protocol\/CLL.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/CLL.agda","new_file":"Control\/Protocol\/CLL.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type\nopen import Control.Protocol\nopen import Control.Protocol.Additive\nopen import Control.Protocol.Multiplicative\n\nmodule Control.Protocol.CLL where\n{-\nA `\u2297 B 'times', context chooses how A and B are used\nA `\u214b B 'par', \"we\" chooses how A and B are used\nA `\u2295 B 'plus', select from A or B\nA `& B 'with', offer choice of A or B\n`! A   'of course!', server accept\n`? A   'why not?', client request\n`1     unit for `\u2297\n`\u22a5     unit for `\u214b\n`0     unit for `\u2295\n`\u22a4     unit for `&\n-}\ndata CLL : \u2605 where\n  `1 `\u22a4 `0 `\u22a5 : CLL\n  _`\u2297_ _`\u214b_ _`\u2295_ _`&_ : (A B : CLL) \u2192 CLL\n  -- `!_ `?_ : (A : CLL) \u2192 CLL\n\n_\u22a5 : CLL \u2192 CLL\n`1 \u22a5 = `\u22a5\n`\u22a5 \u22a5 = `1\n`0 \u22a5 = `\u22a4\n`\u22a4 \u22a5 = `0\n(A `\u2297 B)\u22a5 = A \u22a5 `\u214b B \u22a5\n(A `\u214b B)\u22a5 = A \u22a5 `\u2297 B \u22a5\n(A `\u2295 B)\u22a5 = A \u22a5 `& B \u22a5\n(A `& B)\u22a5 = A \u22a5 `\u2295 B \u22a5\n{-\n(`! A)\u22a5 = `?(A \u22a5)\n(`? A)\u22a5 = `!(A \u22a5)\n-}\n\nCLL-proto : CLL \u2192 Proto\nCLL-proto `1       = end\nCLL-proto `\u22a5       = end\nCLL-proto `0       = P0\nCLL-proto `\u22a4       = P\u22a4\nCLL-proto (A `\u2297 B) = CLL-proto A \u2297 CLL-proto B\nCLL-proto (A `\u214b B) = CLL-proto A \u214b CLL-proto B\nCLL-proto (A `\u2295 B) = CLL-proto A \u2295 CLL-proto B\nCLL-proto (A `& B) = CLL-proto A & CLL-proto B\n\n{- The point of this could be to devise a particular equivalence\n   relation for processes. It could properly deal with \u214b. -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/CLL.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"095f53c09c64b3abe9688067512e48f0db0bf52b","subject":"Add ZK.SigmaProtocol.KnowStatement","message":"Add ZK.SigmaProtocol.KnowStatement\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_file":"ZK\/SigmaProtocol\/KnownStatement.agda","new_contents":"open import Type                 using (Type)\nopen import Data.Bool.Base       using (Bool; true)\nopen import Relation.Binary.Core using (_\u2261_; _\u2262_)\n\n-- This module assumes the exact statement of the Zero-Knowledge proof to be\n-- known by all parties.\nmodule ZK.SigmaProtocol.KnownStatement\n          (Commitment : Type) -- Prover commitments\n          (Challenge  : Type) -- Verifier challenges, picked at random\n          (Response   : Type) -- Prover responses\/proofs to challenges\n          (Randomness : Type) -- Prover's randomness\n          (Witness    : Type) -- Prover's witness\n          (ValidWitness : Witness \u2192 Type) -- Valid witness for the statement\n       where\n\n-- The prover is made of two steps.\n-- * First, send the commitment.\n-- * Second, receive a challenge and send back the response.\n--\n-- Therefor one represents a prover as a pair (a record) made\n-- of a commitment (get-A) and a function (get-f) from challenges\n-- to responses.\n--\n-- Note that one might wish the function get-f to be partial.\n--\n-- This covers any kind of prover, either honest or dishonest\n-- Notice as well that such should depend on some randomness.\n-- So such a prover as type: SomeRandomness \u2192 Prover\nrecord Prover-Interaction : Type where\n  constructor _,_\n  field\n    get-A : Commitment\n    get-f : Challenge \u2192 Response\n\nProver : Type\nProver = (r : Randomness)(w : Witness) \u2192 Prover-Interaction\n\n-- A transcript of the interaction between the prover and the\n-- verifier.\n--\n-- Note that in the case of an interactive zero-knowledge\n-- proof the transcript alone does not prove anything. It can usually\n-- be simulated if one knows the challenge before the commitment.\n--\n-- The only way to be convinced by an interactive prover is to be\/trust the verifier,\n-- namely to receive the commitment first and send a randomly picked challenge\n-- thereafter. So the proof is not transferable.\n--\n-- The other way is by making the zero-knowledge proof non-interactive, which forces the\n-- challenge to be a cryptographic hash of the commitment and the statement.\n-- See the StrongFiatShamir module and the corresponding paper for more details.\nrecord Transcript : Type where\n  constructor mk\n  field\n    get-A : Commitment\n    get-c : Challenge\n    get-f : Response\n\n-- The actual behavior of an interactive verifier is as follows:\n-- * Given a statement\n-- * Receive a commitment from the prover\n-- * Send a randomly chosen challenge to the prover\n-- * Receive a response to that challenge\n-- Since the first two points are common to all \u03a3-protocols we describe\n-- the verifier behavior as a computable test on the transcript.\nVerifier : Type\nVerifier = (t : Transcript) \u2192 Bool\n\n-- To run the interaction, one only needs the prover and a randomly\n-- chosen challenge. The returned transcript is then checked by the\n-- verifier afterwards.\nrun : Prover-Interaction \u2192 Challenge \u2192 Transcript\nrun (A , f) c = mk A c (f c)\n\n-- A \u03a3-protocol is made of the code for honest prover\n-- and honest verifier.\nrecord \u03a3-Protocol : Type where\n  constructor _,_\n  field\n    prover   : Prover\n    verifier : Verifier\n\n  Verified : (t : Transcript) \u2192 Type\n  Verified t = verifier t \u2261 true\n\n-- Correctness (also called completeness in some papers): a \u03a3-protocol is said\n-- to be correct if for any challenge, the (honest) verifier accepts what\n-- the (honest) prover returns.\nCorrect : \u03a3-Protocol \u2192 Type\nCorrect (prover , verifier) = \u2200 r {w} c \u2192 ValidWitness w \u2192 verifier (run (prover r w) c) \u2261 true\n\n-- A simulator takes a challenge and a response and returns a commitment.\n--\n-- As defined next, a correct simulator picks the commitment such that\n-- the transcript is accepted by the verifier.\n--\n-- Notice that generally, to make a valid looking transcript one should\n-- randomly pick the challenge and the response.\nSimulator : Type\nSimulator = (c : Challenge) (s : Response) \u2192 Commitment\n\n-- A correct simulator always convinces the honest verifier.\nCorrect-simulator : Verifier \u2192 Simulator \u2192 Type\nCorrect-simulator verifier simulator\n  = \u2200 c s \u2192 let A = simulator c s in\n            verifier (mk A c s) \u2261 true\n\n{-\n  A \u03a3-protocol, more specifically a verifier which is equipped with\n  a correct simulator is said to be Special Honest Verifier Zero Knowledge.\n  This property is one of the condition to apply the Strong Fiat Shamir\n  transformation.\n\n  The Special part of Special-Honest-Verifier-Zero-Knowledge is covered by the\n  simulator being correct.\n\n  The Honest part is not covered yet, the definition is informally adapted from\n  the paper \"How not to prove yourself\":\n\n  Furthermore, if the challenge c and response s where chosen uniformly at random from their respective\n  domains then the triple (A, c, s) is distributed identically to that of an execution\n  between the (honest) prover and the (honest) verifier (run prover c).\n\n  where A = simulator c s\n-}\nrecord Special-Honest-Verifier-Zero-Knowledge (\u03a3-proto : \u03a3-Protocol) : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n    simulator         : Simulator\n    correct-simulator : Correct-simulator verifier simulator\n\n-- A pair of \"Transcript\"s such that the commitment is shared\n-- and the challenges are different.\nrecord Transcript\u00b2 (verifier : Verifier) : Type where\n  constructor mk\n  field\n    -- The commitment is shared\n    get-A         : Commitment\n\n    -- The challenges...\n    get-c\u2080 get-c\u2081 : Challenge\n\n    -- ...are different\n    c\u2080\u2262c\u2081         : get-c\u2080 \u2262 get-c\u2081\n\n    -- The responses\/proofs are arbitrary\n    get-f\u2080 get-f\u2081 : Response\n\n  -- The two transcripts\n  t\u2080 : Transcript\n  t\u2080 = mk get-A get-c\u2080 get-f\u2080\n  t\u2081 : Transcript\n  t\u2081 = mk get-A get-c\u2081 get-f\u2081\n\n  field\n    -- The transcripts verify\n    verify\u2080 : verifier t\u2080 \u2261 true\n    verify\u2081 : verifier t\u2081 \u2261 true\n\n-- Remark: What if the underlying witnesses are different? Nothing is enforced here.\n-- At least in the case of the Schnorr protocol it does not matter and yield a unique\n-- witness.\nExtractor : Verifier \u2192 Type\nExtractor verifier = Transcript\u00b2 verifier \u2192 Witness\n\nExtract-Valid-Witness : (verifier : Verifier) \u2192 Extractor verifier \u2192 Type\nExtract-Valid-Witness verifier extractor = \u2200 t\u00b2 \u2192 ValidWitness (extractor t\u00b2)\n\n-- A \u03a3-protocol, more specifically a verifier which is equipped with\n-- a correct extractor is said to have the Special Soundness property.\n-- This property is one of the condition to apply the Strong Fiat Shamir\n-- transformation.\nrecord Special-Soundness \u03a3-proto : Type where\n  open \u03a3-Protocol \u03a3-proto\n  field\n  -- TODO Challenge should exp large wrt the security param\n    extractor             : Extractor verifier\n    extract-valid-witness : Extract-Valid-Witness verifier extractor\n\nrecord Special-\u03a3-Protocol : Type where\n  field\n    \u03a3-protocol : \u03a3-Protocol\n    correct : Correct \u03a3-protocol\n    shvzk   : Special-Honest-Verifier-Zero-Knowledge \u03a3-protocol\n    ssound  : Special-Soundness \u03a3-protocol\n  open \u03a3-Protocol \u03a3-protocol public\n  open Special-Honest-Verifier-Zero-Knowledge shvzk public\n  open Special-Soundness ssound public\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/SigmaProtocol\/KnownStatement.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"70a6427689c02b499a86f2d2abab783abf36c132","subject":"start Terminal object structure as adjoint pair","message":"start Terminal object structure as adjoint pair\n","repos":"zeitraffer\/agda-cats","old_file":"src\/CategoryTheory\/Initial\/Trade-off\/FiniteComplete-Module.agda","new_file":"src\/CategoryTheory\/Initial\/Trade-off\/FiniteComplete-Module.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"a807c0e262cd6f414912f280c7f7dcc14670105c","subject":"lib\/Relation\/Binary\/Logical\/LEM.agda","message":"lib\/Relation\/Binary\/Logical\/LEM.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Logical\/LEM.agda","new_file":"lib\/Relation\/Binary\/Logical\/LEM.agda","new_contents":"-- Why I don't want LEM\n\nopen import Type\nopen import Function\nopen import Data.Two\nopen import Relation.Binary.Logical\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Relation.Binary.Logical.LEM\n  (LEM : (P : \u2605\u2081) \u2192 Dec P) where\n\nbroken-id : \u2200 (A : \u2605) \u2192 A \u2192 A\nbroken-id A with LEM (A \u2261 \ud835\udfda)\n... | yes p = coe! p \u2218 not \u2218 coe p\n... | no \u00acp = id\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Relation\/Binary\/Logical\/LEM.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f1c5cc65687a41bae580be0ccf21df84c0344c3e","subject":"ZK\/Types.agda","message":"ZK\/Types.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Types.agda","new_file":"ZK\/Types.agda","new_contents":"open import Type\nopen import Data.Bool.Minimal\n\nmodule ZK.Types where\n\n-- Minimal interface needed for cyclic group + base field\nrecord Cyclic-group (G \u2124q : \u2605) : \u2605 where\n  field\n    g    : G\n    _^_  : G  \u2192 \u2124q \u2192 G\n    _\u00b7_  : G  \u2192 G  \u2192 G\n    _\/_  : G  \u2192 G  \u2192 G\n    _+_  : \u2124q \u2192 \u2124q \u2192 \u2124q\n    _*_  : \u2124q \u2192 \u2124q \u2192 \u2124q\n    _==_ : (x y : G) \u2192 Bool\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/Types.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f335a7dd1955713ec918f050df88b2bb18b81bbb","subject":"Search\/Searchable\/Fin.agda","message":"Search\/Searchable\/Fin.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Fin.agda","new_file":"Search\/Searchable\/Fin.agda","new_contents":"module Search.Searchable.Fin where\n\nopen import Function\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Nat\n\nopen import Search.Type\nopen import Search.Searchable\n\nmodule _ {A : Set}(\u03bcA : Searchable A) where\n\n  sA = search \u03bcA\n\n  extend : \u2200 {n} \u2192 A \u2192 (Fin n \u2192 A) \u2192 Fin (suc n) \u2192 A\n  extend x g zero    = x\n  extend x g (suc i) = g i\n\n  \u00acFin0 : Fin 0 \u2192 A\n  \u00acFin0 ()\n\n  -- There is one function Fin 0 \u2192 A (called abs) so this should be fine\n  -- if not there is a version below that forces the domain to be non-empty\n  sFun : \u2200 n \u2192 Search _ (Fin n \u2192 A)\n  sFun zero    op f = f \u00acFin0\n  sFun (suc n) op f = sA op (\u03bb x \u2192 sFun n op (f \u2218 extend x))\n\n  ind : \u2200 n \u2192 SearchInd _ (sFun n)\n  ind zero    P P\u2219 Pf = Pf _\n  ind (suc n) P P\u2219 Pf =\n    search-ind \u03bcA (\u03bb sa \u2192 P (\u03bb op f \u2192 sa op (\u03bb x \u2192 sFun n op (f \u2218 extend x))))\n      P\u2219\n      (\u03bb x \u2192 ind n (\u03bb sf \u2192 P (\u03bb op f \u2192 sf op (f \u2218 extend x)))\n        P\u2219 (Pf \u2218 extend x))\n\n  \u03bcFun : \u2200 {n} \u2192 Searchable (Fin n \u2192 A)\n  \u03bcFun = mk _ (ind _) {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Search\/Searchable\/Fin.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8aa62105470a9596dc331a89be03179e6a5b5f01","subject":"The embed procedure, pending correctness proof (toward #62)","message":"The embed procedure, pending correctness proof (toward #62)\n\nOld-commit-hash: 3c509c159e02c62a037765e323a4b6f6445c5df9\n","repos":"inc-lc\/ilc-agda","old_file":"experimental\/Embed62.agda","new_file":"experimental\/Embed62.agda","new_contents":"module Embed62 where\n\nopen import TaggedDeltaTypes\nopen import Data.Product using (_\u00d7_ ; _,_ ; proj\u2081 ; proj\u2082)\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats -- Church pairs\n\u0394-Type bags = bags\n\u0394-Type (\u03c3 \u21d2 \u03c4) = \u03c3 \u21d2 \u0394-Type \u03c3 \u21d2 \u0394-Type \u03c4\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\n\u27e6fst\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6fst\u27e7 a b = a\n\u27e6snd\u27e7 : \u2200 {A : Set} \u2192 A \u2192 A \u2192 A\n\u27e6snd\u27e7 a b = b\n\ntranslate : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u0394Val \u03c4\ntranslate {nats} dn = dn \u27e6fst\u27e7 , dn \u27e6snd\u27e7\ntranslate {bags} db = db\ntranslate {\u03c3 \u21d2 \u03c4} df = \u03bb v dv R[v,dv] \u2192 translate (df v {!dv!})\n-- TODO figure out what to do to dv to put it in the hole\n\nconsistent : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 Set\nconsistent {\u2205} \u2205 = EmptySet\nconsistent {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = couple (valid v (translate dv)) (consistent \u03c1)\n\nrecall : \u2200 {\u0393} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (\u03b3 : consistent \u03c1)  \u2192 \u0394Env \u0393\nrecall {\u2205} \u2205 \u2205 = \u2205\nrecall {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) (cons R[v,dv] \u03b3 _ _) = {!!}\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nweak : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nweak = weaken \u0393\u227c\u0394\u0393\n\nderiveVar : \u2200 {\u03c4 \u0393} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nabsurd : \u2200 {A B : Set} \u2192 A \u2192 A \u2192 B \u2192 B\nabsurd _ _ b = b\n\nfst : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nfst = abs (abs (var (that this)))\nsnd : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\nsnd = abs (abs (var this))\n\ndifff : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 (\u0394-Type \u03c4)\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4) \u2192 Term \u0393 \u03c4 \u2192 Term \u0393 \u03c4\n\napply {nats} dt t = app dt snd\napply {bags} dt t = union t dt\napply {\u03c3 \u21d2 \u03c4} dt t = abs (apply\n  (app (app (weaken (drop _ \u2022 \u0393\u227c\u0393) dt)\n    (var this)) (difff (var this) (var this)))\n  (app (weaken (drop _ \u2022 \u0393\u227c\u0393) t) (var this)))\n\ndifff {nats} s t = abs (app (app (var this)\n  (weaken (drop _ \u2022 \u0393\u227c\u0393) s)) (weaken (drop _ \u2022 \u0393\u227c\u0393) t))\ndifff {bags} s t = diff s t\ndifff {\u03c3 \u21d2 \u03c4} s t = abs (abs (difff\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) s)\n    (apply (var this) (var (that this))))\n  (app (weaken (drop _ \u2022 drop _ \u2022 \u0393\u227c\u0393) t) (var (that this)))))\n\nembed : \u2200 {\u03c4 \u0393} \u2192 \u0394Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nembed (\u0394nat old new) = abs (app (app (var this) (nat old)) (nat new))\nembed (\u0394bag db) = bag db\nembed (\u0394var x) = var (deriveVar x)\nembed (\u0394abs dt) = abs (abs (embed dt))\nembed (\u0394app ds t dt) = app (app (embed ds) (weak t)) (embed dt)\nembed (\u0394add ds dt) = abs (app (app (var this)\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) fst)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) fst)))\n  (add (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed ds)) snd)\n       (app (weaken (drop _ \u2022 \u0393\u227c\u0393) (embed dt)) snd)))\nembed (\u0394map\u2080 s ds t dt) = diff\n  (map (apply (embed ds) (weak s)) (apply (embed dt) (weak t)))\n  (weak (map s t))\nembed (\u0394map\u2081 s dt) = map (weak s) (embed dt)\nembed (\u0394union ds dt) = union (embed ds) (embed dt)\nembed (\u0394diff ds dt) = diff (embed ds) (embed dt)\n\n-- embed-correct : \u2200 {\u03c4 \u0393} {t : Term \u0393 \u03c4} {\u03c1 : \u0394Env}\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'experimental\/Embed62.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"042ba6652c20d6022b5bb2125ea27935b9d4dd5e","subject":"Search\/Searchable\/Fun.agda","message":"Search\/Searchable\/Fun.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Fun.agda","new_file":"Search\/Searchable\/Fun.agda","new_contents":"module Search.Searchable.Fun where\n\nopen import Type hiding (\u2605)\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\nopen import Data.Bits\nopen import Function.NP\nopen import Search.Type\nopen import Search.Searchable\nopen import Search.Searchable.Product\nopen import Search.Searchable.Sum\nimport Function.Inverse.NP as FI\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\n\nprivate -- unused\n    S\u03a0\u03a3\u207b : \u2200 {m A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n           \u2192 Search m ((x : A) (y : B x) \u2192 C (x , y))\n           \u2192 Search m (\u03a0 (\u03a3 A B) C)\n    S\u03a0\u03a3\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry)\n\n    S\u03a0\u03a3\u207b-ind : \u2200 {m p A} {B : A \u2192 \u2605 _} {C : \u03a3 A B \u2192 \u2605 _}\n               \u2192 {s : Search m ((x : A) (y : B x) \u2192 C (x , y))}\n               \u2192 SearchInd p s\n               \u2192 SearchInd p (S\u03a0\u03a3\u207b s)\n    S\u03a0\u03a3\u207b-ind ind P P\u2219 Pf = ind (P \u2218 S\u03a0\u03a3\u207b) P\u2219 (Pf \u2218 uncurry)\n\n    S\u00d7\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 B \u2192 C) \u2192 Search m (A \u00d7 B \u2192 C)\n    S\u00d7\u207b = S\u03a0\u03a3\u207b\n\n    S\u00d7\u207b-ind : \u2200 {m p A B C}\n              \u2192 {s : Search m (A \u2192 B \u2192 C)}\n              \u2192 SearchInd p s\n              \u2192 SearchInd p (S\u00d7\u207b s)\n    S\u00d7\u207b-ind = S\u03a0\u03a3\u207b-ind\n\n    S\u03a0\u228e\u207b : \u2200 {m A B} {C : A \u228e B \u2192 \u2605 _}\n           \u2192 Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n           \u2192 Search m (\u03a0 (A \u228e B) C)\n    S\u03a0\u228e\u207b s _\u2219_ f = s _\u2219_ (f \u2218 uncurry [_,_])\n\n    S\u03a0\u228e\u207b-ind : \u2200 {m p A B} {C : A \u228e B \u2192 \u2605 _}\n                 {s : Search m (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))}\n                 (i : SearchInd p s)\n               \u2192 SearchInd p (S\u03a0\u228e\u207b {C = C} s) -- A sB)\n    S\u03a0\u228e\u207b-ind i P P\u2219 Pf = i (P \u2218 S\u03a0\u228e\u207b) P\u2219 (Pf \u2218 uncurry [_,_])\n\n    {- For each A\u2192C function\n       and each B\u2192C function\n       an A\u228eB\u2192C function is yield\n     -}\n    S\u228e\u207b : \u2200 {m A B C} \u2192 Search m (A \u2192 C) \u2192 Search m (B \u2192 C)\n                      \u2192 Search m (A \u228e B \u2192 C)\n    S\u228e\u207b sA sB =  S\u03a0\u228e\u207b (sA \u00d7-search sB)\n\n\u03bc\u03a0\u03a3\u207b : \u2200 {A B}{C : \u03a3 A B \u2192 \u2605\u2080} \u2192 Searchable ((x : A)(y : B x) \u2192 C (x , y)) \u2192 Searchable (\u03a0 (\u03a3 A B) C)\n\u03bc\u03a0\u03a3\u207b = \u03bc-iso (FI.sym curried)\n\n\u03a3-Fun : \u2200 {A B} \u2192 Funable A \u2192 Funable B \u2192 Funable (A \u00d7 B)\n\u03a3-Fun (\u03bcA , \u03bcA\u2192) FB  = \u03bc\u03a3 \u03bcA (searchable FB) , (\u03bb x \u2192 \u03bc\u03a0\u03a3\u207b (\u03bcA\u2192 (negative FB x)))\n  where open Funable\n\n{-\n\u03bc\u03a0\u228e\u207b : \u2200 {A B : \u2605\u2080}{C : A \u228e B \u2192 \u2605 _} \u2192 Searchable (\u03a0 A (C \u2218 inj\u2081) \u00d7 \u03a0 B (C \u2218 inj\u2082))\n     \u2192 Searchable (\u03a0 (A \u228e B) C)\n\u03bc\u03a0\u228e\u207b = \u03bc-iso ?\n\n_\u228e-Fun_ : \u2200 {A B : \u2605\u2080} \u2192 Funable A \u2192 Funable B \u2192 Funable (A \u228e B)\n_\u228e-Fun_ (\u03bcA , \u03bcA\u2192) (\u03bcB , \u03bcB\u2192) = (\u03bcA \u228e-\u03bc \u03bcB) , (\u03bb X \u2192 \u03bc\u03a0\u228e\u207b (\u03bcA\u2192 X \u00d7-\u03bc \u03bcB\u2192 X))\n-}\n\nS\u22a4 : \u2200 {m A} \u2192 Search m A \u2192 Search m (\u22a4 \u2192 A)\nS\u22a4 sA _\u2219_ f = sA _\u2219_ (f \u2218 const)\n\nS\u03a0Bit : \u2200 {m A} \u2192 Search m (A 0b) \u2192 Search m (A 1b)\n                \u2192 Search m (\u03a0 Bit A)\nS\u03a0Bit sA\u2080 sA\u2081 _\u2219_ f = sA\u2080 _\u2219_ \u03bb x \u2192 sA\u2081 _\u2219_ \u03bb y \u2192 f \u03bb {true \u2192 y; false \u2192 x}\n\n\u00d7-Dist : \u2200 {A B} FA FB \u2192 DistFunable {A} FA \u2192 DistFunable {B} FB \u2192 DistFunable (\u03a3-Fun FA FB)\n\u00d7-Dist FA FB FA-dist FB-dist \u03bcX c _\u2299_ distrib _\u2299-cong_ f\n  = \u03a0\u1d2c (\u03bb x \u2192 \u03a0\u1d2e (\u03bb y \u2192 \u03a3' (f (x , y))))\n  \u2248\u27e8 \u27e6search\u27e7 (searchable FA){_\u2261_} \u2261.refl _\u2248_ (\u03bb x y \u2192 x \u2299-cong y)\n       (\u03bb { {x} {.x} \u2261.refl \u2192 FB-dist \u03bcX c _\u2299_ distrib _\u2299-cong_ (curry f x)}) \u27e9\n    \u03a0\u1d2c (\u03bb x \u2192 \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb y \u2192 f (x , y) (fb y))))\n  \u2248\u27e8 FA-dist (negative FB \u03bcX) c _\u2299_ distrib _\u2299-cong_\n       (\u03bb x fb \u2192 search (searchable FB) _\u2299_ (\u03bb y \u2192 f (x , y) (fb y))) \u27e9\n    \u03a3\u1d2c\u1d2e (\u03bb fab \u2192 \u03a0\u1d2c (\u03bb x \u2192 \u03a0\u1d2e (\u03bb y \u2192 f (x , y) (fab x y))))\n  \u220e\n  where\n    open CMon c\n    open Funable\n\n    \u03a3' = search \u03bcX _\u2219_\n\n    \u03a0\u1d2c = search (searchable FA) _\u2299_\n    \u03a0\u1d2e = search (searchable FB) _\u2299_\n\n    \u03a3\u1d2c\u1d2e = search (negative FA (negative FB \u03bcX)) _\u2219_\n    \u03a3\u1d2e  = search (negative FB \u03bcX) _\u2219_\n\n    {-\n\u228e-Dist : \u2200 {A B : \u2605\u2080} FA FB \u2192 DistFunable {A} FA \u2192 DistFunable {B} FB \u2192 DistFunable (FA \u228e-Fun FB)\n\u228e-Dist FA FB FA-dist FB-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ f\n = \u03a0\u1d2c (\u03a3' \u2218 f \u2218 inj\u2081) \u25ce \u03a0\u1d2e (\u03a3' \u2218 f \u2218 inj\u2082)\n \u2248\u27e8 {!FA-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ (f \u2218 inj\u2081) \u25ce-cong FB-dist \u03bcX c _\u25ce_ distrib _\u25ce-cong_ (f \u2218 inj\u2082)!} \u27e9\n {-\n   \u03a3\u1d2c (\u03bb fa \u2192 \u03a0\u1d2c (\u03bb i \u2192 f (inj\u2081 i) (fa i))) \u25ce \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb i \u2192 f (inj\u2082 i) (fb i)))\n \u2248\u27e8 sym (search-lin\u02b3 (negative FA \u03bcX) monoid _\u25ce_ _ _ (proj\u2082 distrib)) \u27e9\n   \u03a3\u1d2c (\u03bb fa \u2192 \u03a0\u1d2c (\u03bb i \u2192 f (inj\u2081 i) (fa i)) \u25ce \u03a3\u1d2e (\u03bb fb \u2192 \u03a0\u1d2e (\u03bb i \u2192 f (inj\u2082 i) (fb i))))\n \u2248\u27e8 search-sg-ext (negative FA \u03bcX) semigroup (\u03bb fa \u2192 sym (search-lin\u02e1 (negative FB \u03bcX) monoid _\u25ce_ _ _ (proj\u2081 distrib))) \u27e9\n -}\n   (\u03a3\u1d2c \u03bb fa \u2192 \u03a3\u1d2e \u03bb fb \u2192 \u03a0\u1d2c ((f \u2218 inj\u2081) \u02e2 fa) \u25ce \u03a0\u1d2e ((f \u2218 inj\u2082) \u02e2 fb))\n \u220e\n where\n    open CMon c\n    open Funable\n\n    \u03a3' = search \u03bcX _\u2219_\n\n    \u03a0\u1d2c = search (searchable FA) _\u25ce_\n    \u03a0\u1d2e = search (searchable FB) _\u25ce_\n\n    \u03a3\u1d2c = search (negative FA \u03bcX) _\u2219_\n    \u03a3\u1d2e = search (negative FB \u03bcX) _\u2219_\n-}\n\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Search\/Searchable\/Fun.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3d92504d386bf78fa68d19fab2f618ac1def28e6","subject":"Added Agda examples.","message":"Added Agda examples.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/agda-introduction\/Introduction.agda","new_file":"notes\/thesis\/agda-introduction\/Introduction.agda","new_contents":"module Introduction where\n\ndata \u2115 : Set where\n  zero : \u2115\n  succ : \u2115 \u2192 \u2115\n\ndata List (A : Set) : Set where\n  [] : List A\n  _\u2237_ : A \u2192 List A \u2192 List A\n\ndata Fin : \u2115 \u2192 Set where\n  zero : {n : \u2115} \u2192 Fin (succ n)\n  succ : {n : \u2115} \u2192 Fin n \u2192 Fin (succ n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/agda-introduction\/Introduction.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"6419e6a499edf075accabe01063360bc7c867ef8","subject":"Fix internalized as a computational description.","message":"Fix internalized as a computational description.\n\nThe idea is that the core could have computatial descriptions,\nwhich do not mention things like which arguments are implicit,\nand Desc can be internalized as a description like the current\nprimitive (non-computational) description that also mentions\nwhich arguments are implicit. Then you can define the fixpoint\nof that description as a computational description, parameterized\nby (I : Set) (D : DescI) and indexed by (i : I) (E : Desc I).\n\nIt's cool that you can do this but I think it's one step too far\nin the direction of making the core type theory simple in exchange\nfor more complicated derivable programs.\n\nIf anything, defining `Desc, making an interpretation function for\nit, along with a toDesc function might be okay. But, again I think\ndefining `Fix is probably not worthwhile.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/CompLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/CompLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.CompLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\n----------------------------------------------------------------------\n\ndata Tel : Set where\n  End : Tel\n  Arg : (A : Set) (B : A \u2192 Tel) \u2192 Tel\n\nEl\u1d40  : Tel \u2192 Set\nEl\u1d40 End = \u22a4\nEl\u1d40 (Arg A B) = \u03a3 A (\u03bb a \u2192 El\u1d40 (B a))\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set where\n  End : Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  RecFun : (A : Set) (B : A \u2192 I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\n----------------------------------------------------------------------\n\nISet : Set \u2192 Set\nISet I = I \u2192 Set\n\nEl\u1d30 : {I : Set} (D : Desc I) \u2192 ISet I \u2192 Set\nEl\u1d30 End X = \u22a4\nEl\u1d30 (Rec j D) X = X j \u00d7 El\u1d30 D X\nEl\u1d30 (RecFun A B D) X = ((a : A) \u2192 X (B a)) \u00d7 El\u1d30 D X\nEl\u1d30 (Arg A B) X = \u03a3 A (\u03bb a \u2192 El\u1d30 (B a) X)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (xs : El\u1d30 D X) \u2192 Set\nHyps End X P tt = \u22a4\nHyps (Rec i D) X P (x , xs) = P i x \u00d7 Hyps D X P xs\nHyps (RecFun A B D) X P (f , xs) = ((a : A) \u2192 P (B a) (f a)) \u00d7 Hyps D X P xs\nHyps (Arg A B) X P (a , xs) = Hyps (B a) X P xs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (R : I \u2192 Desc I) (i : I) : Set where\n  init : El\u1d30 (R i) (\u03bc R) \u2192 \u03bc R i\n\n----------------------------------------------------------------------\n\nind : {I : Set} (R : I \u2192 Desc I)\n  (M : (i : I) \u2192 \u03bc R i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 (R i) (\u03bc R)) (ihs : Hyps (R i) (\u03bc R) M xs) \u2192 M i (init xs))\n  (i : I)\n  (x : \u03bc R i)\n  \u2192 M i x\n\nprove : {I : Set} (D : Desc I) (R : I \u2192 Desc I)\n  (M : (i : I) \u2192 \u03bc R i \u2192 Set)\n  (\u03b1 : \u2200 i (xs : El\u1d30 (R i) (\u03bc R)) (ihs : Hyps (R i) (\u03bc R) M xs) \u2192 M i (init xs))\n  (xs : El\u1d30 D (\u03bc R)) \u2192 Hyps D (\u03bc R) M xs\n\nind R M \u03b1 i (init xs) = \u03b1 i xs (prove (R i) R M \u03b1 xs)\n\nprove End R M \u03b1 tt = tt\nprove (Rec j D) R M \u03b1 (x , xs) = ind R M \u03b1 j x , prove D R M \u03b1 xs\nprove (RecFun A B D) R M \u03b1 (f , xs) = (\u03bb a \u2192 ind R M \u03b1 (B a) (f a)) , prove D R M \u03b1 xs\nprove (Arg A B) R M \u03b1 (a , xs) = prove (B a) R M \u03b1 xs\n\n----------------------------------------------------------------------\n\nDescE : Enum\nDescE = \"End\" \u2237 \"Rec\" \u2237 \"Arg\" \u2237 []\n\nDescT : Set\nDescT = Tag DescE\n\n-- EndT : DescT\npattern EndT = here\n\n-- RecT : DescT\npattern RecT = there here\n\n-- ArgT : DescT\npattern ArgT = there (there here)\n\nDescR : Set \u2192 \u22a4 \u2192 Desc \u22a4\nDescR I tt = Arg (Tag DescE)\n  (case (\u03bb _ \u2192 Desc \u22a4) (\n      (Arg I \u03bb i \u2192 End)\n    , (Arg I \u03bb i \u2192 Rec tt End)\n    , (Arg Set \u03bb A \u2192 RecFun A (\u03bb a \u2192 tt) End)\n    , tt\n  ))\n\n`Desc : (I : Set) \u2192 Set\n`Desc I = \u03bc (DescR I) tt\n\n-- `End : {I : Set} (i : I) \u2192 `Desc I\npattern `End i = init (EndT , i , tt)\n\n-- `Rec : {I : Set} (i : I) (D : `Desc I) \u2192 `Desc I\npattern `Rec i D = init (RecT , i , D , tt)\n\n-- `Arg : {I : Set} (A : Set) (B : A \u2192 `Desc I) \u2192 `Desc I\npattern `Arg A B = init (ArgT , A , B , tt)\n\n----------------------------------------------------------------------\n\nFixI : (I : Set) \u2192 Set\nFixI I = I \u00d7 `Desc I\n\nFixR' : (I : Set) (D : `Desc I) \u2192 I \u2192 `Desc I \u2192 Desc (FixI I)\nFixR' I D i (`End j) = Arg (j \u2261 i) \u03bb q \u2192 End\nFixR' I D i (`Rec j E) = Rec (j , D) (FixR' I D i E)\nFixR' I D i (`Arg A B) = Arg A \u03bb a \u2192 FixR' I D i (B a)\nFixR' I D i (init (there (there (there ())) , xs))\n\nFixR : (I : Set) (D : `Desc I) \u2192 FixI I \u2192 Desc (FixI I)\nFixR I D E,i = FixR' I D (proj\u2081 E,i) (proj\u2082 E,i)\n\n`Fix : (I : Set) (D : `Desc I) (i : I) \u2192 Set\n`Fix I D i = \u03bc (FixR I D) (i , D)\n\n----------------------------------------------------------------------\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formalization\/agda\/Spire\/Examples\/CompLev.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3be44d190ddb71b95417f9028965e046e7938081","subject":"Add an initial start for an Agda model.","message":"Add an initial start for an Agda model.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Language\/Greek.agda","new_file":"agda\/Language\/Greek.agda","new_contents":"module Language.Greek where\n\ndata Maybe A : Set where\n  none : Maybe A\n  just : A \u2192 Maybe A\n\ndata Unicode : Set where\n  \u0391 \u0392 \u0393 \u0394 \u0395 \u0396 \u0397 \u0398 \u0399 \u039a \u039b \u039c \u039d \u039e \u039f \u03a0 \u03a1 \u03a3 \u03a4 \u03a5 \u03a6 \u03a7 \u03a8 \u03a9 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c2 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : Unicode\n  grave acute diaeresis smooth rough circumflex iotaSubscript : Unicode\n\npostulate\n  Char : Set\n\n{-# BUILTIN CHAR Char #-}\n{-# COMPILED_TYPE Char Char #-}\n\ncharToUnicode : Char -> Maybe Unicode\ncharToUnicode '\u0391' = just \u0391\ncharToUnicode '\u0392' = just \u0392\ncharToUnicode '\u0393' = just \u0393\ncharToUnicode '\u0394' = just \u0394\ncharToUnicode '\u0395' = just \u0395\ncharToUnicode '\u0396' = just \u0396\ncharToUnicode '\u0397' = just \u0397\ncharToUnicode '\u0398' = just \u0398\ncharToUnicode '\u0399' = just \u0399\ncharToUnicode '\u039a' = just \u039a\ncharToUnicode '\u039b' = just \u039b\ncharToUnicode '\u039c' = just \u039c\ncharToUnicode '\u039d' = just \u039d\ncharToUnicode '\u039e' = just \u039e\ncharToUnicode '\u039f' = just \u039f\ncharToUnicode '\u03a0' = just \u03a0\ncharToUnicode '\u03a1' = just \u03a1\ncharToUnicode '\u03a3' = just \u03a3\ncharToUnicode '\u03a4' = just \u03a4\ncharToUnicode '\u03a5' = just \u03a5\ncharToUnicode '\u03a6' = just \u03a6\ncharToUnicode '\u03a7' = just \u03a7\ncharToUnicode '\u03a8' = just \u03a8\ncharToUnicode '\u03a9' = just \u03a9\ncharToUnicode '\u03b1' = just \u03b1\ncharToUnicode '\u03b2' = just \u03b2\ncharToUnicode '\u03b3' = just \u03b3\ncharToUnicode '\u03b4' = just \u03b4\ncharToUnicode '\u03b5' = just \u03b5\ncharToUnicode '\u03b6' = just \u03b6\ncharToUnicode '\u03b7' = just \u03b7\ncharToUnicode '\u03b8' = just \u03b8\ncharToUnicode '\u03b9' = just \u03b9\ncharToUnicode '\u03ba' = just \u03ba\ncharToUnicode '\u03bb' = just \u03bb\u2032\ncharToUnicode '\u03bc' = just \u03bc\ncharToUnicode '\u03bd' = just \u03bd\ncharToUnicode '\u03be' = just \u03be\ncharToUnicode '\u03bf' = just \u03bf\ncharToUnicode '\u03c0' = just \u03c0\ncharToUnicode '\u03c1' = just \u03c1\ncharToUnicode '\u03c2' = just \u03c2\ncharToUnicode '\u03c3' = just \u03c3\ncharToUnicode '\u03c4' = just \u03c4\ncharToUnicode '\u03c5' = just \u03c5\ncharToUnicode '\u03c6' = just \u03c6\ncharToUnicode '\u03c7' = just \u03c7\ncharToUnicode '\u03c8' = just \u03c8\ncharToUnicode '\u03c9' = just \u03c9\ncharToUnicode '\\x0300' = just grave          -- COMBINING GRAVE ACCENT\ncharToUnicode '\\x0301' = just acute          -- COMBINING ACUTE ACCENT\ncharToUnicode '\\x0308' = just diaeresis      -- COMBINING DIAERESIS\ncharToUnicode '\\x0313' = just smooth         -- COMBINING COMMA ABOVE\ncharToUnicode '\\x0314' = just rough          -- COMBINING REVERSED COMMA ABOVE\ncharToUnicode '\\x0342' = just circumflex     -- COMBINING GREEK PERISPOMENI\ncharToUnicode '\\x0345' = just iotaSubscript  -- COMBINING GREEK YPOGEGRAMMENI\ncharToUnicode _ = none\n\ndata UnicodeCategory : Set where\n  letter mark : UnicodeCategory\n\nunicodeToCategory : Unicode \u2192 UnicodeCategory\nunicodeToCategory \u0391 = letter\nunicodeToCategory \u0392 = letter\nunicodeToCategory \u0393 = letter\nunicodeToCategory \u0394 = letter\nunicodeToCategory \u0395 = letter\nunicodeToCategory \u0396 = letter\nunicodeToCategory \u0397 = letter\nunicodeToCategory \u0398 = letter\nunicodeToCategory \u0399 = letter\nunicodeToCategory \u039a = letter\nunicodeToCategory \u039b = letter\nunicodeToCategory \u039c = letter\nunicodeToCategory \u039d = letter\nunicodeToCategory \u039e = letter\nunicodeToCategory \u039f = letter\nunicodeToCategory \u03a0 = letter\nunicodeToCategory \u03a1 = letter\nunicodeToCategory \u03a3 = letter\nunicodeToCategory \u03a4 = letter\nunicodeToCategory \u03a5 = letter\nunicodeToCategory \u03a6 = letter\nunicodeToCategory \u03a7 = letter\nunicodeToCategory \u03a8 = letter\nunicodeToCategory \u03a9 = letter\nunicodeToCategory \u03b1 = letter\nunicodeToCategory \u03b2 = letter\nunicodeToCategory \u03b3 = letter\nunicodeToCategory \u03b4 = letter\nunicodeToCategory \u03b5 = letter\nunicodeToCategory \u03b6 = letter\nunicodeToCategory \u03b7 = letter\nunicodeToCategory \u03b8 = letter\nunicodeToCategory \u03b9 = letter\nunicodeToCategory \u03ba = letter\nunicodeToCategory \u03bb\u2032 = letter\nunicodeToCategory \u03bc = letter\nunicodeToCategory \u03bd = letter\nunicodeToCategory \u03be = letter\nunicodeToCategory \u03bf = letter\nunicodeToCategory \u03c0 = letter\nunicodeToCategory \u03c1 = letter\nunicodeToCategory \u03c3 = letter\nunicodeToCategory \u03c2 = letter\nunicodeToCategory \u03c4 = letter\nunicodeToCategory \u03c5 = letter\nunicodeToCategory \u03c6 = letter\nunicodeToCategory \u03c7 = letter\nunicodeToCategory \u03c8 = letter\nunicodeToCategory \u03c9 = letter\nunicodeToCategory grave = mark\nunicodeToCategory acute = mark\nunicodeToCategory diaeresis = mark\nunicodeToCategory smooth = mark\nunicodeToCategory rough = mark\nunicodeToCategory circumflex = mark\nunicodeToCategory iotaSubscript = mark\n\ndata List A : Set where\n  [] : List A\n  _\u2237_ : A \u2192 List A -> List A\n\ninfixr 6 _\u2237_\n\nfoldr : {A B : Set} \u2192 (A \u2192 B \u2192 B) \u2192 B \u2192 List A \u2192 B\nfoldr f y [] = y\nfoldr f y (x \u2237 xs) = f x (foldr f y xs)\n\ndata _And_ : (A : Set) \u2192 (B : Set) \u2192 Set\u2081 where\n  _and_ : {A B : Set} \u2192 A \u2192 B \u2192 A And B\n\ndata ConcreteLetter : Set where\n  \u0391 \u0392 \u0393 \u0394 \u0395 \u0396 \u0397 \u0398 \u0399 \u039a \u039b \u039c \u039d \u039e \u039f \u03a0 \u03a1 \u03a3 \u03a4 \u03a5 \u03a6 \u03a7 \u03a8 \u03a9 \u03b1 \u03b2 \u03b3 \u03b4 \u03b5 \u03b6 \u03b7 \u03b8 \u03b9 \u03ba \u03bb\u2032 \u03bc \u03bd \u03be \u03bf \u03c0 \u03c1 \u03c3 \u03c2 \u03c4 \u03c5 \u03c6 \u03c7 \u03c8 \u03c9 : ConcreteLetter\n\ndata ConcreteMark : Set where\n  grave acute diaeresis smooth rough circumflex iotaSubscript : ConcreteMark\n \n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/Language\/Greek.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"eb40272391aa09b5a5a42ac30183b073f3201ba7","subject":"Added relation LTM.","message":"Added relation LTM.\n\nIgnore-this: 6ba30eed4255786ce93e42c2812d9d4a\n\ndarcs-hash:20110210035613-3bd4e-daec2d205ad19e2446b212c728c4f25507a20cd4.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/InequalitiesATP.agda","new_file":"Draft\/McCarthy91\/InequalitiesATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Inequalities used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.InequalitiesATP where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.Arithmetic\n\nopen import LTC.Data.Nat\nopen import LTC.Data.Nat.Inequalities\nopen import LTC.Data.Nat.Inequalities.PropertiesATP\nopen import LTC.Data.Nat.Numbers\n\n------------------------------------------------------------------------------\n\nLMC : D \u2192 D \u2192 Set\nLMC m n = LT (hundred-one \u2238 m) (hundred-one \u2238 n)\n\nLMC-prop : \u2200 {n} \u2192 N n \u2192 LE n one-hundred \u2192 LMC (n + eleven) n\nLMC-prop zN 0\u2264100 = prf\n  where\n    postulate\n      prf : (hundred-one \u2238 (zero + eleven)) < (hundred-one \u2238 zero) \u2261 true\n    {-# ATP prove prf #-}\nLMC-prop (sN {n} Nn) Sn\u2264100 = prf (LMC-prop Nn n\u2264100)\n  where\n    n\u2264100 : LE n one-hundred\n    n\u2264100 = \u2264-trans Nn (sN Nn) N100 (x<y\u2192x\u2264y Nn (sN Nn) (x<Sx Nn)) Sn\u2264100\n\n    postulate\n      prf : (hundred-one \u2238 (n + eleven)) < (hundred-one \u2238 n) \u2261 true \u2192  -- IH.\n            (hundred-one \u2238 (succ n + eleven)) < (hundred-one \u2238 succ n) \u2261 true\n    {-# ATP prove prf #-} -- Fail with all the ATPs.\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/McCarthy91\/InequalitiesATP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"d93ecb74191bf6c6337155c917014e1aa07fa99e","subject":" [ notes ] Testing a type synonymous for Set.","message":" [ notes ] Testing a type synonymous for Set.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/SetType.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/SetType.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing a type synonymous for Set\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.SetType where\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\n\n-- Type synonymous for Set.\nType = Set\n\ndata N' : D \u2192 Type where\n  nzero' : N' zero\n  nsucc' : \u2200 {n} \u2192 N' n \u2192 N' (succ\u2081 n)\n{-# ATP axiom nzero' nsucc' #-}\n\npred-N' : \u2200 {n} \u2192 N' n \u2192 N' (pred\u2081 n)\npred-N' nzero' = prf\n  where postulate prf : N' (pred\u2081 zero)\n        {-# ATP prove prf #-}\n\npred-N' (nsucc' {n} N'n) = prf\n  where postulate prf : N' (pred\u2081 (succ\u2081 n))\n        {-# ATP prove prf #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/SetType.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"a4557a2034fef4033363c5f31e51df97b0a1fa09","subject":"use field as string alias","message":"use field as string alias\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b0125f1c18874052e72444389391f063c1be6a2","subject":"Start adding Greek script model in Agda.","message":"Start adding Greek script model in Agda.\n","repos":"scott-fleischman\/greek-grammar,scott-fleischman\/greek-grammar","old_file":"agda\/Text\/Greek\/Script.agda","new_file":"agda\/Text\/Greek\/Script.agda","new_contents":"module Text.Greek.Script where\n\nopen import Data.Maybe\nopen import Data.Vec\nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Binary.PropositionalEquality using (_\u2262_)\n\ndata Case : Set where\n  lower upper : Case\n\ndata Letter : Set where\n  \u03b1\u2032 \u03b2\u2032 \u03b3\u2032 \u03b4\u2032 \u03b5\u2032 \u03b6\u2032 \u03b7\u2032 \u03b8\u2032 \u03b9\u2032 \u03ba\u2032 \u03bb\u2032 \u03bc\u2032 \u03bd\u2032 \u03be\u2032 \u03bf\u2032 \u03c0\u2032 \u03c1\u2032 \u03c3\u2032 \u03c4\u2032 \u03c5\u2032 \u03c6\u2032 \u03c7\u2032 \u03c8\u2032 \u03c9\u2032 : Letter\n\nvowels : Vec _ _\nvowels = \u03b1\u2032 \u2237 \u03b5\u2032 \u2237 \u03b7\u2032 \u2237 \u03b9\u2032 \u2237 \u03bf\u2032 \u2237 \u03c5\u2032 \u2237 \u03c9\u2032 \u2237 []\n\nalways-short-letters : Vec _ _\nalways-short-letters = \u03b5\u2032 \u2237 \u03bf\u2032 \u2237 []\n\ndiaeresis-letters : Vec _ _\ndiaeresis-letters = \u03b9\u2032 \u2237 \u03c5\u2032 \u2237 []\n\niota-subscript-letters : Vec _ _\niota-subscript-letters = \u03b1\u2032 \u2237 \u03b7\u2032 \u2237 \u03c9\u2032 \u2237 []\n\ndata _vowel : Letter \u2192 Set where\n  is-vowel : \u2200 {v} \u2192 v \u2208 vowels \u2192 v vowel\n\ndata _always-short : Letter \u2192 Set where\n  is-always-short : \u2200 {v} \u2192 v vowel \u2192 v \u2208 always-short-letters \u2192 v always-short\n\ndata _diaeresis : Letter \u2192 Set where\n  add-diaeresis : \u2200 {v} \u2192 v vowel \u2192 v \u2208 diaeresis-letters \u2192 v diaeresis\n\ndata _\u27e6_\u27e7-final : Letter \u2192 Case \u2192 Set where\n  make-final : \u03c3\u2032 \u27e6 lower \u27e7-final\n\ndata _\u27e6_\u27e7-smooth : Letter \u2192 Case \u2192 Set where\n  add-smooth-lower-vowel : \u2200 {v} \u2192 v vowel \u2192 v \u27e6 lower \u27e7-smooth\n  add-smooth-\u03c1 : \u03c1\u2032 \u27e6 lower \u27e7-smooth\n  add-smooth-upper-vowel-not-\u03a5 : \u2200 {v} \u2192 v vowel \u2192 v \u2262 \u03c5\u2032 \u2192 v \u27e6 upper \u27e7-smooth\n\ndata _rough : Letter \u2192 Set where\n  add-rough-vowel : \u2200 {v} \u2192 v vowel \u2192 v rough\n  add-rough-\u03c1 : \u03c1\u2032 rough\n\ndata _iota-subscript : Letter \u2192 Set where\n  add-iota-subscript : \u2200 {v} \u2192 v vowel \u2192 v \u2208 iota-subscript-letters \u2192 v iota-subscript\n\ndata _\u27e6_\u27e7-breathing : Letter \u2192 Case \u2192 Set where\n  add-smooth : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-smooth \u2192 \u2113 \u27e6 c \u27e7-breathing\n  add-rough : \u2200 {\u2113 c} \u2192 \u2113 rough \u2192 \u2113 \u27e6 c \u27e7-breathing\n\ndata _long-vowel : Letter \u2192 Set where\n  make-long-vowel : \u2200 {v} \u2192 v vowel \u2192 \u00ac v always-short \u2192 v long-vowel\n\ndata _accent : Letter \u2192 Set where\n  add-acute : \u2200 {\u2113} \u2192 \u2113 vowel \u2192 \u2113 accent\n  add-grave : \u2200 {\u2113} \u2192 \u2113 vowel \u2192 \u2113 accent\n  add-circumflex : \u2200 {\u2113} \u2192 \u2113 long-vowel \u2192 \u2113 accent\n\ndata Token : Letter \u2192 Case \u2192 Set where\n  unmarked : \u2200 {\u2113 c} \u2192 Token \u2113 c\n  with-accent : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 Token \u2113 c\n  with-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Token \u2113 c\n  with-accent-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-diaeresis : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 diaeresis \u2192 Token \u2113 c\n  with-accent-iota : \u2200 {\u2113 c} \u2192 \u2113 accent \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-breathing-iota : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  with-iota : \u2200 {\u2113 c} \u2192 \u2113 iota-subscript \u2192 Token \u2113 c\n  final : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-final \u2192 Token \u2113 c\n\n-- Constructions\n\n-- \u0391 \u03b1\n\u03b1-vowel : \u03b1\u2032 vowel\n\u03b1-vowel = is-vowel here\n\n\u00ac\u03b1-always-short : \u00ac \u03b1\u2032 always-short\n\u00ac\u03b1-always-short (is-always-short \u03b1-vowel (there (there ())))\n\n\u03b1-long-vowel : \u03b1\u2032 long-vowel\n\u03b1-long-vowel = make-long-vowel \u03b1-vowel \u00ac\u03b1-always-short\n\n\u03b1-acute : \u03b1\u2032 accent\n\u03b1-acute = add-acute \u03b1-vowel\n\n\u03b1-grave : \u03b1\u2032 accent\n\u03b1-grave = add-grave \u03b1-vowel\n\n\u03b1-circumflex : \u03b1\u2032 accent\n\u03b1-circumflex = add-circumflex \u03b1-long-vowel\n\n\u03b1-smooth : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-smooth = add-smooth (add-smooth-lower-vowel \u03b1-vowel)\n\n\u0391-smooth : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b1-vowel (\u03bb ()))\n\n\u03b1-rough : \u03b1\u2032 \u27e6 lower \u27e7-breathing\n\u03b1-rough = add-rough (add-rough-vowel \u03b1-vowel)\n\n\u0391-rough : \u03b1\u2032 \u27e6 upper \u27e7-breathing\n\u0391-rough = add-rough (add-rough-vowel \u03b1-vowel)\n\n\u03b1-iota-subscript : \u03b1\u2032 iota-subscript\n\u03b1-iota-subscript = add-iota-subscript \u03b1-vowel here\n\n-- \u0395 \u03b5\n\u03b5-vowel : \u03b5\u2032 vowel\n\u03b5-vowel = is-vowel (there here)\n\n\u03b5-always-short : \u03b5\u2032 always-short\n\u03b5-always-short = is-always-short \u03b5-vowel here\n\n\u03b5-acute : \u03b5\u2032 accent\n\u03b5-acute = add-acute \u03b5-vowel\n\n\u03b5-grave : \u03b5\u2032 accent\n\u03b5-grave = add-grave \u03b5-vowel\n\n\u03b5-smooth : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-smooth = add-smooth (add-smooth-lower-vowel \u03b5-vowel)\n\n\u0395-smooth : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b5-vowel (\u03bb ()))\n\n\u03b5-rough : \u03b5\u2032 \u27e6 lower \u27e7-breathing\n\u03b5-rough = add-rough (add-rough-vowel \u03b5-vowel)\n\n\u0395-rough : \u03b5\u2032 \u27e6 upper \u27e7-breathing\n\u0395-rough = add-rough (add-rough-vowel \u03b5-vowel)\n\n-- \u0397 \u03b7\n\u03b7-vowel : \u03b7\u2032 vowel\n\u03b7-vowel = is-vowel (there (there here))\n\n\u00ac\u03b7-always-short : \u00ac \u03b7\u2032 always-short\n\u00ac\u03b7-always-short (is-always-short \u03b7-vowel (there (there ())))\n\n\u03b7-long-vowel : \u03b7\u2032 long-vowel\n\u03b7-long-vowel = make-long-vowel \u03b7-vowel \u00ac\u03b7-always-short\n\n\u03b7-acute : \u03b7\u2032 accent\n\u03b7-acute = add-acute \u03b7-vowel\n\n\u03b7-grave : \u03b7\u2032 accent\n\u03b7-grave = add-grave \u03b7-vowel\n\n\u03b7-circumflex : \u03b7\u2032 accent\n\u03b7-circumflex = add-circumflex \u03b7-long-vowel\n\n\u03b7-smooth : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-smooth = add-smooth (add-smooth-lower-vowel \u03b7-vowel)\n\n\u0397-smooth : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b7-vowel (\u03bb ()))\n\n\u03b7-rough : \u03b7\u2032 \u27e6 lower \u27e7-breathing\n\u03b7-rough = add-rough (add-rough-vowel \u03b7-vowel)\n\n\u0397-rough : \u03b7\u2032 \u27e6 upper \u27e7-breathing\n\u0397-rough = add-rough (add-rough-vowel \u03b7-vowel)\n\n\u03b7-iota-subscript : \u03b7\u2032 iota-subscript\n\u03b7-iota-subscript = add-iota-subscript \u03b7-vowel (there here)\n\n-- \u0399 \u03b9\n\ninstance\n  \u03b9-vowel : \u03b9\u2032 vowel\n  \u03b9-vowel = is-vowel (there (there (there here)))\n\n  \u00ac\u03b9-always-short : \u00ac \u03b9\u2032 always-short\n  \u00ac\u03b9-always-short (is-always-short \u03b9-vowel (there (there ())))\n\n  \u03b9-long-vowel : \u03b9\u2032 long-vowel\n  \u03b9-long-vowel = make-long-vowel \u03b9-vowel \u00ac\u03b9-always-short\n\n  \u03b9-not-\u03c5 : \u03b9\u2032 \u2262 \u03c5\u2032\n  \u03b9-not-\u03c5 ()\n\n\u03b9-acute : \u03b9\u2032 accent\n\u03b9-acute = add-acute \u03b9-vowel\n\n\u03b9-grave : \u03b9\u2032 accent\n\u03b9-grave = add-grave \u03b9-vowel\n\n\u03b9-circumflex : \u03b9\u2032 accent\n\u03b9-circumflex = add-circumflex \u03b9-long-vowel\n\n\u03b9-smooth : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-smooth = add-smooth (add-smooth-lower-vowel \u03b9-vowel)\n\n\u0399-smooth : \u03b9\u2032 \u27e6 upper \u27e7-breathing\n\u0399-smooth = add-smooth (add-smooth-upper-vowel-not-\u03a5 \u03b9-vowel (\u03bb ()))\n\n\u03b9-rough : \u03b9\u2032 \u27e6 lower \u27e7-breathing\n\u03b9-rough = add-rough (add-rough-vowel \u03b9-vowel)\n\n-- \u039f \u03bf\nf : \u2200 {\u2113} \u2192 {{ p : \u2113 vowel }} \u2192 \u2113 accent\nf {{ p }} = add-acute p\n\n\u03b9-acutes : Token \u03b9\u2032 lower\n\u03b9-acutes = with-accent f\n\n-- Unicode\n-- U+0390 - U+03CE\n\u0390 : Token \u03b9\u2032 lower -- U+0390 GREEK SMALL LETTER IOTA WITH DIALYTIKA AND TONOS\n\u0390 = unmarked\n\u0391 : Token \u03b1\u2032 upper -- U+0391 GREEK CAPITAL LETTER ALPHA\n\u0391 = unmarked\n\u0392 : Token \u03b2\u2032 upper -- U+0392 GREEK CAPITAL LETTER BETA\n\u0392 = unmarked\n\u0393 : Token \u03b3\u2032 upper -- U+0393 GREEK CAPITAL LETTER GAMMA\n\u0393 = unmarked\n\u0394 : Token \u03b4\u2032 upper -- U+0394 GREEK CAPITAL LETTER DELTA\n\u0394 = unmarked\n\u0395 : Token \u03b5\u2032 upper -- U+0395 GREEK CAPITAL LETTER EPSILON\n\u0395 = unmarked\n\u0396 : Token \u03b6\u2032 upper -- U+0396 GREEK CAPITAL LETTER ZETA\n\u0396 = unmarked\n\u0397 : Token \u03b7\u2032 upper -- U+0397 GREEK CAPITAL LETTER ETA\n\u0397 = unmarked\n\u0398 : Token \u03b8\u2032 upper -- U+0398 GREEK CAPITAL LETTER THETA\n\u0398 = unmarked\n\u0399 : Token \u03b9\u2032 upper -- U+0399 GREEK CAPITAL LETTER IOTA\n\u0399 = unmarked\n\u039a : Token \u03ba\u2032 upper -- U+039A GREEK CAPITAL LETTER KAPPA\n\u039a = unmarked\n\u039b : Token \u03bb\u2032 upper -- U+039B GREEK CAPITAL LETTER LAMDA\n\u039b = unmarked\n\u039c : Token \u03bc\u2032 upper -- U+039C GREEK CAPITAL LETTER MU\n\u039c = unmarked\n\u039d : Token \u03bd\u2032 upper -- U+039D GREEK CAPITAL LETTER NU\n\u039d = unmarked\n\u039e : Token \u03be\u2032 upper -- U+039E GREEK CAPITAL LETTER XI\n\u039e = unmarked\n\u039f : Token \u03bf\u2032 upper -- U+039F GREEK CAPITAL LETTER OMICRON\n\u039f = unmarked\n\u03a0 : Token \u03c0\u2032 upper -- U+03A0 GREEK CAPITAL LETTER PI\n\u03a0 = unmarked\n\u03a1 : Token \u03c1\u2032 upper -- U+03A1 GREEK CAPITAL LETTER RHO\n\u03a1 = unmarked\n                  -- U+03A2 <reserved>\n\u03a3 : Token \u03c3\u2032 upper -- U+03A3 GREEK CAPITAL LETTER SIGMA\n\u03a3 = unmarked\n\u03a4 : Token \u03c4\u2032 upper -- U+03A4 GREEK CAPITAL LETTER TAU\n\u03a4 = unmarked\n\u03a5 : Token \u03c5\u2032 upper -- U+03A5 GREEK CAPITAL LETTER UPSILON\n\u03a5 = unmarked\n\u03a6 : Token \u03c6\u2032 upper -- U+03A6 GREEK CAPITAL LETTER PHI\n\u03a6 = unmarked\n\u03a7 : Token \u03c7\u2032 upper -- U+03A7 GREEK CAPITAL LETTER CHI\n\u03a7 = unmarked\n\u03a8 : Token \u03c8\u2032 upper -- U+03A8 GREEK CAPITAL LETTER PSI\n\u03a8 = unmarked\n\u03a9 : Token \u03c9\u2032 upper -- U+03A9 GREEK CAPITAL LETTER OMEGA\n\u03a9 = unmarked\n\u03aa : Token \u03b9\u2032 upper -- U+03AA GREEK CAPITAL LETTER IOTA WITH DIALYTIKA\n\u03aa = unmarked\n\u03ab : Token \u03c5\u2032 upper -- U+03AB GREEK CAPITAL LETTER UPSILON WITH DIALYTIKA\n\u03ab = unmarked\n\u03ac : Token \u03b1\u2032 lower -- U+03AC GREEK SMALL LETTER ALPHA WITH TONOS\n\u03ac = unmarked\n\u03ad : Token \u03b5\u2032 lower -- U+03AD GREEK SMALL LETTER EPSILON WITH TONOS\n\u03ad = unmarked\n\u03ae : Token \u03b7\u2032 lower -- U+03AE GREEK SMALL LETTER ETA WITH TONOS\n\u03ae = unmarked\n\u03af : Token \u03b9\u2032 lower -- U+03AF GREEK SMALL LETTER IOTA WITH TONOS\n\u03af = unmarked\n\u03b0 : Token \u03c5\u2032 lower -- U+03B0 GREEK SMALL LETTER UPSILON WITH DIALYTIKA AND TONOS\n\u03b0 = unmarked\n\u03b1 : Token \u03b1\u2032 lower -- U+03B1 GREEK SMALL LETTER ALPHA\n\u03b1 = unmarked\n\u03b2 : Token \u03b2\u2032 lower -- U+03B2 GREEK SMALL LETTER BETA\n\u03b2 = unmarked\n\u03b3 : Token \u03b3\u2032 lower -- U+03B3 GREEK SMALL LETTER GAMMA\n\u03b3 = unmarked\n\u03b4 : Token \u03b4\u2032 lower -- U+03B4 GREEK SMALL LETTER DELTA\n\u03b4 = unmarked\n\u03b5 : Token \u03b5\u2032 lower -- U+03B5 GREEK SMALL LETTER EPSILON\n\u03b5 = unmarked\n\u03b6 : Token \u03b6\u2032 lower -- U+03B6 GREEK SMALL LETTER ZETA\n\u03b6 = unmarked\n\u03b7 : Token \u03b7\u2032 lower -- U+03B7 GREEK SMALL LETTER ETA\n\u03b7 = unmarked\n\u03b8 : Token \u03b8\u2032 lower -- U+03B8 GREEK SMALL LETTER THETA\n\u03b8 = unmarked\n\u03b9 : Token \u03b9\u2032 lower -- U+03B9 GREEK SMALL LETTER IOTA\n\u03b9 = unmarked\n\u03ba : Token \u03ba\u2032 lower -- U+03BA GREEK SMALL LETTER KAPPA\n\u03ba = unmarked\n\u2219\u03bb : Token \u03bb\u2032 lower -- U+03BB GREEK SMALL LETTER LAMDA\n\u2219\u03bb = unmarked\n\u03bc : Token \u03bc\u2032 lower -- U+03BC GREEK SMALL LETTER MU\n\u03bc = unmarked\n\u03bd : Token \u03bd\u2032 lower -- U+03BD GREEK SMALL LETTER NU\n\u03bd = unmarked\n\u03be : Token \u03be\u2032 lower -- U+03BE GREEK SMALL LETTER XI\n\u03be = unmarked\n\u03bf : Token \u03bf\u2032 lower -- U+03BF GREEK SMALL LETTER OMICRON\n\u03bf = unmarked\n\u03c0 : Token \u03c0\u2032 lower -- U+03C0 GREEK SMALL LETTER PI\n\u03c0 = unmarked\n\u03c1 : Token \u03c1\u2032 lower -- U+03C1 GREEK SMALL LETTER RHO\n\u03c1 = unmarked\n\u03c2 : Token \u03c3\u2032 lower -- U+03C2 GREEK SMALL LETTER FINAL SIGMA\n\u03c2 = unmarked\n\u03c3 : Token \u03c3\u2032 lower -- U+03C3 GREEK SMALL LETTER SIGMA\n\u03c3 = unmarked\n\u03c4 : Token \u03c4\u2032 lower -- U+03C4 GREEK SMALL LETTER TAU\n\u03c4 = unmarked\n\u03c5 : Token \u03c5\u2032 lower -- U+03C5 GREEK SMALL LETTER UPSILON\n\u03c5 = unmarked\n\u03c6 : Token \u03c6\u2032 lower -- U+03C6 GREEK SMALL LETTER PHI\n\u03c6 = unmarked\n\u03c7 : Token \u03c7\u2032 lower -- U+03C7 GREEK SMALL LETTER CHI\n\u03c7 = unmarked\n\u03c8 : Token \u03c8\u2032 lower -- U+03C8 GREEK SMALL LETTER PSI\n\u03c8 = unmarked\n\u03c9 : Token \u03c9\u2032 lower -- U+03C9 GREEK SMALL LETTER OMEGA\n\u03c9 = unmarked\n\u03ca : Token \u03b9\u2032 lower -- U+03CA GREEK SMALL LETTER IOTA WITH DIALYTIKA\n\u03ca = unmarked\n\u03cb : Token \u03c5\u2032 lower -- U+03CB GREEK SMALL LETTER UPSILON WITH DIALYTIKA\n\u03cb = unmarked\n\u03cc : Token \u03bf\u2032 lower -- U+03CC GREEK SMALL LETTER OMICRON WITH TONOS\n\u03cc = unmarked\n\u03cd : Token \u03c5\u2032 lower -- U+03CD GREEK SMALL LETTER UPSILON WITH TONOS\n\u03cd = unmarked\n\u03ce : Token \u03c9\u2032 lower -- U+03CE GREEK SMALL LETTER OMEGA WITH TONOS\n\u03ce = unmarked\n\n\n\u1f00 : Token \u03b1\u2032 lower\n\u1f00 = with-breathing \u03b1-smooth\n\u1f01 : Token \u03b1\u2032 lower\n\u1f01 = with-breathing \u03b1-rough\n\u1f02 : Token \u03b1\u2032 lower\n\u1f02 = with-accent-breathing \u03b1-grave \u03b1-smooth\n\u1f03 : Token \u03b1\u2032 lower\n\u1f03 = with-accent-breathing \u03b1-grave \u03b1-rough\n\u1f04 : Token \u03b1\u2032 lower\n\u1f04 = with-accent-breathing \u03b1-acute \u03b1-smooth\n\u1f05 : Token \u03b1\u2032 lower\n\u1f05 = with-accent-breathing \u03b1-acute \u03b1-rough\n\u1f06 : Token \u03b1\u2032 lower\n\u1f06 = with-accent-breathing \u03b1-circumflex \u03b1-smooth\n\u1f07 : Token \u03b1\u2032 lower\n\u1f07 = with-accent-breathing \u03b1-circumflex \u03b1-rough\n\u1f08 : Token \u03b1\u2032 upper\n\u1f08 = with-breathing \u0391-smooth\n\u1f09 : Token \u03b1\u2032 upper\n\u1f09 = with-breathing \u0391-rough\n\u1f0a : Token \u03b1\u2032 upper\n\u1f0a = with-accent-breathing \u03b1-grave \u0391-smooth\n\u1f0b : Token \u03b1\u2032 upper\n\u1f0b = with-accent-breathing \u03b1-grave \u0391-rough\n\u1f0c : Token \u03b1\u2032 upper\n\u1f0c = with-accent-breathing \u03b1-acute \u0391-smooth\n\u1f0d : Token \u03b1\u2032 upper\n\u1f0d = with-accent-breathing \u03b1-acute \u0391-rough\n\u1f0e : Token \u03b1\u2032 upper\n\u1f0e = with-accent-breathing \u03b1-circumflex \u0391-smooth\n\u1f0f : Token \u03b1\u2032 upper\n\u1f0f = with-accent-breathing \u03b1-circumflex \u0391-rough\n\n-- U+1F7x\n\u1f70 : Token \u03b1\u2032 lower\n\u1f70 = with-accent \u03b1-grave\n\u1f71 : Token \u03b1\u2032 lower\n\u1f71 = with-accent \u03b1-acute\n\n-- U+1F8x\n\u1f80 : Token \u03b1\u2032 lower\n\u1f80 = with-breathing-iota \u03b1-smooth \u03b1-iota-subscript\n\u1f81 : Token \u03b1\u2032 lower\n\u1f81 = with-breathing-iota \u03b1-rough \u03b1-iota-subscript\n\u1f82 : Token \u03b1\u2032 lower\n\u1f82 = with-accent-breathing-iota \u03b1-grave \u03b1-smooth \u03b1-iota-subscript\n\u1f83 : Token \u03b1\u2032 lower\n\u1f83 = with-accent-breathing-iota \u03b1-grave \u03b1-rough \u03b1-iota-subscript\n\u1f84 : Token \u03b1\u2032 lower\n\u1f84 = with-accent-breathing-iota \u03b1-acute \u03b1-smooth \u03b1-iota-subscript\n\u1f85 : Token \u03b1\u2032 lower\n\u1f85 = with-accent-breathing-iota \u03b1-acute \u03b1-rough \u03b1-iota-subscript\n\u1f86 : Token \u03b1\u2032 lower\n\u1f86 = with-accent-breathing-iota \u03b1-circumflex \u03b1-smooth \u03b1-iota-subscript\n\u1f87 : Token \u03b1\u2032 lower\n\u1f87 = with-accent-breathing-iota \u03b1-circumflex \u03b1-rough \u03b1-iota-subscript\n\u1f88 : Token \u03b1\u2032 upper\n\u1f88 = with-breathing-iota \u0391-smooth \u03b1-iota-subscript\n\u1f89 : Token \u03b1\u2032 upper\n\u1f89 = with-breathing-iota \u0391-rough \u03b1-iota-subscript\n\u1f8a : Token \u03b1\u2032 upper\n\u1f8a = with-accent-breathing-iota \u03b1-grave \u0391-smooth \u03b1-iota-subscript\n\u1f8b : Token \u03b1\u2032 upper\n\u1f8b = with-accent-breathing-iota \u03b1-grave \u0391-rough \u03b1-iota-subscript\n\u1f8c : Token \u03b1\u2032 upper\n\u1f8c = with-accent-breathing-iota \u03b1-acute \u0391-smooth \u03b1-iota-subscript\n\u1f8d : Token \u03b1\u2032 upper\n\u1f8d = with-accent-breathing-iota \u03b1-acute \u0391-rough \u03b1-iota-subscript\n\u1f8e : Token \u03b1\u2032 upper\n\u1f8e = with-accent-breathing-iota \u03b1-circumflex \u0391-smooth \u03b1-iota-subscript\n\u1f8f : Token \u03b1\u2032 upper\n\u1f8f = with-accent-breathing-iota \u03b1-circumflex \u0391-rough \u03b1-iota-subscript\n\n-- U+1FBx\n-- \u1fb0\n-- \u1fb1\n\u1fb2 : Token \u03b1\u2032 lower\n\u1fb2 = with-accent-iota \u03b1-grave \u03b1-iota-subscript\n\u1fb3 : Token \u03b1\u2032 lower\n\u1fb3 = with-iota \u03b1-iota-subscript\n\u1fb4 : Token \u03b1\u2032 lower\n\u1fb4 = with-accent-iota \u03b1-acute \u03b1-iota-subscript\n\u1fb6 : Token \u03b1\u2032 lower\n\u1fb6 = with-accent \u03b1-circumflex\n\u1fb7 : Token \u03b1\u2032 lower\n\u1fb7 = with-accent-iota \u03b1-circumflex \u03b1-iota-subscript\n-- \u1fb8\n-- \u1fb9\n\u1fba : Token \u03b1\u2032 upper\n\u1fba = with-accent \u03b1-grave\n\u1fbb : Token \u03b1\u2032 upper\n\u1fbb = with-accent \u03b1-acute\n\u1fbc : Token \u03b1\u2032 upper\n\u1fbc = with-iota \u03b1-iota-subscript\n\n-- Mapping\n\ndata Accent : Set where\n  acute-mark grave-mark circumflex-mark : Accent\n\nletter-to-accent : \u2200 {v} \u2192 v accent \u2192 Accent\nletter-to-accent (add-acute x) = acute-mark\nletter-to-accent (add-grave x) = grave-mark\nletter-to-accent (add-circumflex x) = circumflex-mark\n\nget-accent : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Accent\nget-accent (with-accent a) = just (letter-to-accent a)\nget-accent (with-accent-breathing a _) = just (letter-to-accent a)\nget-accent (with-accent-breathing-iota a _ _) = just (letter-to-accent a)\nget-accent (with-accent-diaeresis a _) = just (letter-to-accent a)\nget-accent (with-accent-iota a _) = just (letter-to-accent a)\nget-accent _ = nothing\n\ndata Breathing : Set where\n  smooth-mark rough-mark : Breathing\n\nletter-to-breathing : \u2200 {\u2113 c} \u2192 \u2113 \u27e6 c \u27e7-breathing \u2192 Breathing\nletter-to-breathing (add-smooth x) = smooth-mark\nletter-to-breathing (add-rough x) = rough-mark\n\nget-breathing : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Breathing\nget-breathing (with-breathing x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing _ x) = just (letter-to-breathing x)\nget-breathing (with-accent-breathing-iota _ x _) = just (letter-to-breathing x)\nget-breathing (with-breathing-iota x _) = just (letter-to-breathing x)\nget-breathing _ = nothing\n\ndata IotaSubscript : Set where\n  iota-subscript-mark : IotaSubscript\n\nget-iota-subscript : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe IotaSubscript\nget-iota-subscript (with-accent-breathing-iota _ _ _) = just iota-subscript-mark\nget-iota-subscript (with-accent-iota _ _) = just iota-subscript-mark\nget-iota-subscript (with-breathing-iota _ _) = just iota-subscript-mark\nget-iota-subscript _ = nothing\n\ndata Diaeresis : Set where\n  diaeresis-mark : Diaeresis\n\nget-diaeresis : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe Diaeresis\nget-diaeresis (with-diaeresis _) = just diaeresis-mark\nget-diaeresis (with-accent-diaeresis _ _) = just diaeresis-mark\nget-diaeresis _ = nothing\n\ndata FinalForm : Set where\n  final-form : FinalForm\n\nget-final-form : \u2200 {\u2113 c} \u2192 Token \u2113 c \u2192 Maybe FinalForm\nget-final-form (final _) = just final-form\nget-final-form _ = nothing\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/Text\/Greek\/Script.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"7a5845c6998b66f1c5b4f6fdc608b18e2b7f3c3a","subject":"fun-state.agda","message":"fun-state.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"fun-state.agda","new_file":"fun-state.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"fatal: unable to access 'https:\/\/github.com\/crypto-agda\/crypto-agda.git\/': The requested URL returned error: 403\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"96569369ee92be395860f342f96f002ca6617345","subject":"Added definition of equation using a recursive equation.","message":"Added definition of equation using a recursive equation.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/FOTC\/SubtractionRecursiveEquation.agda","new_file":"notes\/thesis\/FOTC\/SubtractionRecursiveEquation.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of subtraction using a recursive equation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOTC.SubtractionRecursiveEquation where\n\nopen import FOTC.Base\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u2238_\n\n------------------------------------------------------------------------------\n\npostulate\n  _\u2238_  : D \u2192 D \u2192 D\n  \u2238-eq : \u2200 m n \u2192 m \u2238 n \u2261\n         if (iszero\u2081 n)\n            then m\n            else if (iszero\u2081 m)\n                    then zero\n                    else pred\u2081 m \u2238 pred\u2081 n\n{-# ATP axiom \u2238-eq #-}\n\npostulate\n  \u2238-x0 : \u2200 n \u2192 n \u2238 zero \u2261 n\n{-# ATP prove \u2238-x0 #-}\n\npostulate\n  \u2238-0S : \u2200 n \u2192 zero \u2238 succ\u2081 n \u2261 zero\n{-# ATP prove \u2238-0S #-}\n\npostulate\n  \u2238-SS : \u2200 m n \u2192 succ\u2081 m \u2238 succ\u2081 n \u2261 m \u2238 n\n{-# ATP prove \u2238-SS #-}\n\n-- asr@pc:~\/fotc\/notes\/thesis$ agda2atp -i ~\/fotc\/src\/fot\/  -i. FOTC\/SubtractionRecursiveEquation.agda\n-- Proving the conjecture in \/tmp\/FOTC\/SubtractionRecursiveEquation\/32-8760-0S.tptp ...\n-- E 1.6 Tiger Hill proved the conjecture in \/tmp\/FOTC\/SubtractionRecursiveEquation\/32-8760-0S.tptp\n-- Proving the conjecture in \/tmp\/FOTC\/SubtractionRecursiveEquation\/36-8760-SS.tptp ...\n-- E 1.6 Tiger Hill proved the conjecture in \/tmp\/FOTC\/SubtractionRecursiveEquation\/36-8760-SS.tptp\n-- Proving the conjecture in \/tmp\/FOTC\/SubtractionRecursiveEquation\/28-8760-x0.tptp ...\n-- E 1.6 Tiger Hill proved the conjecture in \/tmp\/FOTC\/SubtractionRecursiveEquation\/28-8760-x0.tptp\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/FOTC\/SubtractionRecursiveEquation.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"289e0679c14f26ff2519ae38a84cdced062cb380","subject":"prg.agda","message":"prg.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"prg.agda","new_file":"prg.agda","new_contents":"module prg where\n\nopen import fun-universe\n\nopen import Function\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Bool.NP hiding (_==_)\nopen import Data.Nat.Properties\nopen import Data.Bool.Properties\nopen import Data.Bits\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product\nopen import Data.Empty\nopen import Data.Vec\nopen import Relation.Binary.PropositionalEquality.NP\n\nPRGFun : (k n : \u2115) \u2192 Set\nPRGFun k n = Bits k \u2192 Bits n\n\n==-refl : \u2200 {n} (xs : Bits n) \u2192 (xs == xs) \u2261 1b\n==-refl [] = refl\n==-refl (true \u2237 xs) = ==-refl xs\n==-refl (false \u2237 xs) = ==-refl xs\n\n_|\u2228|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2228|_ f g x = f x \u2228 g x\n\n_|\u2227|_ : \u2200 {n} \u2192 (f g : Bits n \u2192 Bit) \u2192 Bits n \u2192 Bit\n_|\u2227|_ f g x = f x \u2227 g x\n\nmodule PRG\u2141\u2081 {k n} (PRG : PRGFun k n) where\n  -- TODO: give the adv some rand\n  Adv : Set\n  Adv = Bits n \u2192 Bit\n\n  chal : Bit \u2192 Bits (n + k) \u2192 Bits n\n  chal b R = if b then take n R else PRG (drop n R)\n\n  prg\u2141 : Adv \u2192 Bit \u2192 Bits (n + k) \u2192 Bit\n  prg\u2141 adv b R = adv (chal b R)\n\n  -- ``Looks pseudo-random if the message is the output of the PRG some key''\n  brute\u2032 : Adv\n  brute\u2032 m = search {k} _\u2228_ ((_==_ m) \u2218 PRG)\n\n  -- ``Looks random if the message is not a possible output of the PRG''\n  brute : Adv\n  brute m = search {k} _\u2227_ (not \u2218 (_==_ m) \u2218 PRG)\n\nprivate\n  brute-PRG-K\u22610b-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n) K\n                           \u2192 PRG\u2141\u2081.brute PRG (PRG K) \u2261 0b\n  brute-PRG-K\u22610b-aux {zero}  {n} PRG [] = cong not (==-refl (PRG []))\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (true \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 1\u2237_) K\n      = Bool\u00b0.+-comm (search _\u2227_ (not \u2218 _==_ (PRG (1\u2237 K)) \u2218 PRG \u2218 0\u2237_)) 0b\n  brute-PRG-K\u22610b-aux {suc k} {n} PRG (false \u2237 K)\n    rewrite brute-PRG-K\u22610b-aux (PRG \u2218 0\u2237_) K = refl\n\n\n  always : \u2200 n \u2192 Bits n \u2192 Bit\n  always _ _ = 1b\n  never  : \u2200 n \u2192 Bits n \u2192 Bit\n  never _ _ = 0b\n\n  search-\u00b7-\u03b5\u2261\u03b5 : \u2200 {a} {A : Set a} \u03b5 (_\u00b7_ : A \u2192 A \u2192 A)\n                   (\u03b5\u00b7\u03b5 : \u03b5 \u00b7 \u03b5 \u2261 \u03b5) n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n  search-\u00b7-\u03b5\u2261\u03b5 \u03b5 _\u00b7_ \u03b5\u00b7\u03b5 = go\n    where\n      go : \u2200 n \u2192 search {n} _\u00b7_ (const \u03b5) \u2261 \u03b5\n      go zero = refl\n      go (suc n) rewrite go n = \u03b5\u00b7\u03b5\n\n  #never\u22610 : \u2200 n \u2192 #\u27e8 never n \u27e9 \u2261 0\n  #never\u22610 = search-\u00b7-\u03b5\u2261\u03b5 _ _ refl\n\n  #always\u22612^_ : \u2200 n \u2192 #\u27e8 always n \u27e9 \u2261 2^ n\n  #always\u22612^_ zero = refl\n  #always\u22612^_ (suc n) = cong\u2082 _+_ pf pf where pf = #always\u22612^ n\n\n  ==-comm : \u2200 {n} (xs ys : Bits n) \u2192 xs == ys \u2261 ys == xs\n  ==-comm [] [] = refl\n  ==-comm (x \u2237 xs) (x\u2081 \u2237 ys) rewrite Xor\u00b0.+-comm x x\u2081 | ==-comm xs ys = refl\n\n  count\u1d47 : Bit \u2192 \u2115\n  count\u1d47 b = if b then 1 else 0\n\n  #\u27e8==_\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 _==_ xs \u27e9 \u2261 1\n  #\u27e8== [] \u27e9 = refl\n  #\u27e8==_\u27e9 {suc n} (true \u2237 xs)  rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n  #\u27e8==_\u27e9 {suc n} (false \u2237 xs) rewrite #never\u22610 n | #\u27e8== xs \u27e9 = refl\n\n  \u2257-cong-search : \u2200 {n a} {A : Set a} op {f g : Bits n \u2192 A} \u2192 f \u2257 g \u2192 search op f \u2261 search op g\n  \u2257-cong-search {zero}  op f\u2257g = f\u2257g []\n  \u2257-cong-search {suc n} op f\u2257g = cong\u2082 op (\u2257-cong-search op (f\u2257g \u2218 0\u2237_))\n                                          (\u2257-cong-search op (f\u2257g \u2218 1\u2237_))\n\n  \u2257-cong-# : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\n  \u2257-cong-# f g f\u2257g = \u2257-cong-search _+_ (cong count\u1d47 \u2218 f\u2257g)\n\n  #-+ : \u2200 {n a b} (f : Bits (suc n) \u2192 Bit) \u2192 #\u27e8 f \u2218 0\u2237_ \u27e9 \u2261 a \u2192 #\u27e8 f \u2218 1\u2237_ \u27e9 \u2261 b \u2192 #\u27e8 f \u27e9 \u2261 a + b\n  #-+ f f0 f1 rewrite f0 | f1 = refl\n\n  take-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 take (suc n) (x \u2237 xs) \u2261 x \u2237 take n xs\n  take-\u2237 n x xs with splitAt n xs\n  take-\u2237 _ _ ._ | _ , _ , refl = refl\n  \n  drop-\u2237 : \u2200 {m a} {A : Set a} n x (xs : Vec A (n + m)) \u2192 drop (suc n) (x \u2237 xs) \u2261 drop n xs\n  drop-\u2237 n x xs with splitAt n xs\n  drop-\u2237 _ _ ._ | _ , _ , refl = refl\n  \n  #-take-drop : \u2200 m n (f : Bits m \u2192 Bit) (g : Bits n \u2192 Bit)\n                  \u2192 #\u27e8 (f \u2218 take m) |\u2227| (g \u2218 drop m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n  #-take-drop zero n f g with f []\n  ... | true rewrite \u2115\u00b0.+-comm #\u27e8 g \u27e9 0 = refl\n  ... | false = #never\u22610 n\n  #-take-drop (suc m) n f g = trans (#-+ {a = #\u27e8 f \u2218 0\u2237_ \u27e9 * #\u27e8 g \u27e9} ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)))\n                                    (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 0\u2237_)\n                                                  ((f \u2218 0\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                  (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 0b x) (drop-\u2237 m 0b x)))\n                                         (#-take-drop m n (f \u2218 0\u2237_) g))\n                                    (trans (\u2257-cong-# ((f \u2218 take (suc m)) |\u2227| (g \u2218 drop (suc m)) \u2218 1\u2237_)\n                                                  ((f \u2218 1\u2237_ \u2218 take m) |\u2227| (g \u2218 drop m))\n                                                  (\u03bb x \u2192 cong\u2082 (\u03bb x y \u2192 f x \u2227 g y) (take-\u2237 m 1b x) (drop-\u2237 m 1b x)))\n                                         (#-take-drop m n (f \u2218 1\u2237_) g)))\n                             (sym (proj\u2082 \u2115\u00b0.distrib #\u27e8 g \u27e9 #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9))\n\n  #-drop-take : \u2200 m n (f : Bits n \u2192 Bit) (g : Bits m \u2192 Bit)\n                  \u2192 #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9 \u2261 #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n  #-drop-take m n f g =\n             #\u27e8 (f \u2218 drop m) |\u2227| (g \u2218 take m) \u27e9\n           \u2261\u27e8 \u2257-cong-# ((f \u2218 drop m) |\u2227| (g \u2218 take m)) ((g \u2218 take m) |\u2227| (f \u2218 drop m)) (\u03bb x \u2192 Bool\u00b0.+-comm (f (drop m x)) _) \u27e9\n             #\u27e8 (g \u2218 take m) |\u2227| (f \u2218 drop m) \u27e9\n           \u2261\u27e8 #-take-drop m n g f \u27e9\n             #\u27e8 g \u27e9 * #\u27e8 f \u27e9\n           \u2261\u27e8 \u2115\u00b0.*-comm #\u27e8 g \u27e9 _ \u27e9\n             #\u27e8 f \u27e9 * #\u27e8 g \u27e9\n           \u220e\n    where open \u2261-Reasoning\n\n  #-take : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 take m {n} \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n  #-take m n f = #\u27e8 f \u2218 take m {n} \u27e9\n               \u2261\u27e8 #-drop-take m n (always n) f \u27e9\n                 #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n               \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                 2^ n * #\u27e8 f \u27e9\n               \u220e\n    where open \u2261-Reasoning\n\n  #-drop : \u2200 m n (f : Bits m \u2192 Bit) \u2192 #\u27e8 f \u2218 drop n \u27e9 \u2261 2^ n * #\u27e8 f \u27e9\n  #-drop m n f = #\u27e8 f \u2218 drop n \u27e9\n               \u2261\u27e8 #-take-drop n m (always n) f \u27e9\n                 #\u27e8 always n \u27e9 * #\u27e8 f \u27e9\n               \u2261\u27e8 cong (flip _*_ #\u27e8 f \u27e9) (#always\u22612^ n) \u27e9\n                 2^ n * #\u27e8 f \u27e9\n               \u220e\n    where open \u2261-Reasoning\n\n  #\u27e8_==\u27e9 : \u2200 {n} (xs : Bits n) \u2192 #\u27e8 flip _==_ xs \u27e9 \u2261 1\n  #\u27e8 xs ==\u27e9 = trans (\u2257-cong-# (flip _==_ xs) (_==_ xs) (flip ==-comm xs)) #\u27e8== xs \u27e9\n\n  #\u21d2 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 (\u2200 x \u2192 T (f x) \u2192 T (g x)) \u2192 #\u27e8 f \u27e9 \u2264 #\u27e8 g \u27e9\n  #\u21d2 {zero} f g f\u21d2g with f [] | g [] | f\u21d2g []\n  ... | true  | true  | _ = s\u2264s z\u2264n\n  ... | true  | false | p = \u22a5-elim (p _)\n  ... | false | _     | _ = z\u2264n\n  #\u21d2 {suc n} f g f\u21d2g = #\u21d2 (f \u2218 0\u2237_) (g \u2218 0\u2237_) (f\u21d2g \u2218 0\u2237_)\n                  +-mono #\u21d2 (f \u2218 1\u2237_) (g \u2218 1\u2237_) (f\u21d2g \u2218 1\u2237_)\n\n  #-\u2227-\u2228\u1d47 : \u2200 x y \u2192 count\u1d47 (x \u2227 y) + count\u1d47 (x \u2228 y) \u2261 count\u1d47 x + count\u1d47 y\n  #-\u2227-\u2228\u1d47 true y rewrite \u2115\u00b0.+-comm (count\u1d47 y) 1 = refl\n  #-\u2227-\u2228\u1d47 false y = refl\n\n  #-\u2227-\u2228 : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2227| g \u27e9 + #\u27e8 f |\u2228| g \u27e9 \u2261 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n  #-\u2227-\u2228 {zero} f g = #-\u2227-\u2228\u1d47 (f []) (g [])\n  #-\u2227-\u2228 {suc n} f g =\n    trans\n      (trans\n         (helper #\u27e8 (f \u2218 0\u2237_) |\u2227| (g \u2218 0\u2237_) \u27e9\n                 #\u27e8 (f \u2218 1\u2237_) |\u2227| (g \u2218 1\u2237_) \u27e9\n                 #\u27e8 (f \u2218 0\u2237_) |\u2228| (g \u2218 0\u2237_) \u27e9\n                 #\u27e8 (f \u2218 1\u2237_) |\u2228| (g \u2218 1\u2237_) \u27e9)\n         (cong\u2082 _+_ (#-\u2227-\u2228 (f \u2218 0\u2237_) (g \u2218 0\u2237_))\n                    (#-\u2227-\u2228 (f \u2218 1\u2237_) (g \u2218 1\u2237_))))\n      (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9)\n      where open SemiringSolver\n            helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n            helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n  #\u2228' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 #\u27e8 f |\u2228| g \u27e9 \u2264 #\u27e8 f \u27e9 + #\u27e8 g \u27e9\n  #\u2228' {zero} f g with f []\n  ... | true  = s\u2264s z\u2264n\n  ... | false = \u2115\u2264.refl\n  #\u2228' {suc _} f g = \u2115\u2264.trans (#\u2228' (f \u2218 0\u2237_) (g \u2218 0\u2237_) +-mono\n                               #\u2228' (f \u2218 1\u2237_) (g \u2218 1\u2237_))\n                      (\u2115\u2264.reflexive\n                        (helper #\u27e8 f \u2218 0\u2237_ \u27e9 #\u27e8 g \u2218 0\u2237_ \u27e9 #\u27e8 f \u2218 1\u2237_ \u27e9 #\u27e8 g \u2218 1\u2237_ \u27e9))\n      where open SemiringSolver\n            helper : \u2200 x y z t \u2192 x + y + (z + t) \u2261 x + z + (y + t)\n            helper = solve 4 (\u03bb x y z t \u2192 x :+ y :+ (z :+ t) := x :+ z :+ (y :+ t)) refl\n\n  #\u2228 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 (\u03bb x \u2192 f x \u2228 g x) \u27e9 \u2264 (m + n)\n  #\u2228 {m} {n} {o} {f} {g} pf pg = \u2115\u2264.trans (#\u2228' f g) (pf +-mono pg)\n\n  \u2227\u21d2\u2228 : \u2200 x y \u2192 T (x \u2227 y) \u2192 T (x \u200c\u2228 y)\n  \u2227\u21d2\u2228 true y = _\n  \u2227\u21d2\u2228 false y = \u03bb ()\n\n  #\u2227 : \u2200 {m n o} {f g : Bits o \u2192 Bit} \u2192 #\u27e8 f \u27e9 \u2264 m \u2192 #\u27e8 g \u27e9 \u2264 n \u2192 #\u27e8 f |\u2227| g \u27e9 \u2264 (m + n)\n  #\u2227 {f = f} {g} pf pg = \u2115\u2264.trans (#\u21d2 (f |\u2227| g) (f |\u2228| g) (\u03bb x \u2192 \u2227\u21d2\u2228 (f x) (g x))) (#\u2228 {f = f} pf pg)\n\n  lem' : \u2200 {n} (f g : Bits n \u2192 Bit) \u2192 f |\u2228| g \u2257 not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g))\n  lem' f g x with f x\n  ... | true = refl\n  ... | false = sym (not-involutive _)\n\n  lem : \u2200 {n} op (f g : Bits n \u2192 Bit) \u2192 search op (f |\u2228| g) \u2261 search op (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))\n  lem op f g = \u2257-cong-search op {-(f |\u2228| g) (not \u2218 ((not \u2218 f) |\u2227| (not \u2218 g)))-} (lem' f g) \n\n  lem'' :\n    \u2200 {n a b}\n      {A : Set a} {B : Set b}\n      (_+_ : A \u2192 A \u2192 A)\n      (_*_ : B \u2192 B \u2192 B)\n      (f : A \u2192 B)\n      (p : Bits n \u2192 A)\n      (hom : \u2200 x y \u2192 f (x + y) \u2261 f x * f y)\n      \u2192 f (search _+_ p) \u2261 search _*_ (f \u2218 p)\n  lem'' {zero} _+_ _*_ f p hom = refl\n  lem'' {suc n} _+_ _*_ f p hom = trans (hom _ _) (cong\u2082 _*_ (lem'' {n} _+_ _*_ f (p \u2218 0\u2237_) hom) (lem'' _+_ _*_ f (p \u2218 1\u2237_) hom))\n\n  [0\u2194_] : \u2200 {n} \u2192 Fin n \u2192 Bits n \u2192 Bits n\n  [0\u2194_] {zero}  i xs = xs\n  [0\u2194_] {suc n} i xs = lookup i xs \u2237 tail (xs [ i ]\u2254 head xs)\n\n  [0\u21941] : Bits 2 \u2192 Bits 2\n  [0\u21941] = [0\u2194 suc zero ]\n\n  [0\u21941]-spec : [0\u21941] \u2257 (\u03bb { (x \u2237 y \u2237 []) \u2192 y \u2237 x \u2237 [] })\n  [0\u21941]-spec (x \u2237 y \u2237 []) = refl\n\n  brute\u2032-bound-aux : \u2200 {k n} (PRG : Bits k \u2192 Bits n)\n                       \u2192 let open PRG\u2141\u2081 PRG in\n                          #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound-aux {zero}  PRG = \u2115\u2264.reflexive #\u27e8 PRG [] ==\u27e9\n  brute\u2032-bound-aux {suc k} PRG = #\u2228 {f = \u03bb x \u2192 search _\u2228_ (_==_ x \u2218 PRG \u2218 0\u2237_)}\n                                    (brute\u2032-bound-aux (PRG \u2218 0\u2237_))\n                                    (brute\u2032-bound-aux (PRG \u2218 1\u2237_))\n\nmodule PRG\u2141 {k n} (PRG : PRGFun k n) where\n  open PRG\u2141\u2081 PRG public\n\n  brute-PRG-K\u22610b : \u2200 K \u2192 brute (PRG K) \u2261 0b\n  brute-PRG-K\u22610b = brute-PRG-K\u22610b-aux PRG\n\n  prg\u2141-brute-0 : prg\u2141 brute 0b \u2257 (const 0b)\n  prg\u2141-brute-0 R = brute-PRG-K\u22610b (drop n R)\n\n  brute\u2032-bound : #\u27e8 brute\u2032 \u27e9 \u2264 2^ k\n  brute\u2032-bound = brute\u2032-bound-aux PRG\n\n  de-morgan : \u2200 x y \u2192 not (x \u2228 y) \u2261 not x \u2227 not y\n  de-morgan true  _ = refl\n  de-morgan false _ = refl\n\n  not\u2218brute\u2032\u2257brute : not \u2218 brute\u2032 \u2257 brute\n  not\u2218brute\u2032\u2257brute x = lem'' _\u2228_ _\u2227_ not (_==_ x \u2218 PRG) de-morgan\n\n  brute\u2257not\u2218brute\u2032 : brute\u2032 \u2257 not \u2218 brute\n  brute\u2257not\u2218brute\u2032 x = trans (sym (not-involutive _)) (cong not (not\u2218brute\u2032\u2257brute x))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'prg.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"99b3cc18b6bf76004ef16a5815e47a1686f0cb1a","subject":"Added draft version of the classical existential elimination.","message":"Added draft version of the classical existential elimination.\n\nIgnore-this: 7c94b8879a8bf77b6c0f1ef968364764\n\ndarcs-hash:20120216050548-3bd4e-5000cb63aeb450f74e667af510c1917f0b16eb50.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Common\/Data\/Product.agda","new_file":"Draft\/Common\/Data\/Product.agda","new_contents":"-----------------------------------------------------------------------------\n-- Classical existential elimination\n-----------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.Common.Data.Product where\n\n-- We add 3 to the fixities of the standard library.\ninfixr 7 _,_\n\n-----------------------------------------------------------------------------\n-- Existential elimination\n--     \u2203x.P(x)   P(x) \u2192 Q\n--  ------------------------\n--             Q\n\npostulate\n  D : Set\n\n-- The existential quantifier type on D.\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 ((x : D) \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Common\/Data\/Product.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"16e409065b78c4560e4bb6b6cb70b8aef18c1c50","subject":"FMSB: agda model of lambda madnesses","message":"FMSB: agda model of lambda madnesses","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/FMSB.agda","new_file":"models\/FMSB.agda","new_contents":"module FMSB where\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Lam (X : Nat -> Set)(n : Nat) : Set where\n  var : X n -> Lam X n\n  app : Lam X n -> Lam X n -> Lam X n\n  lam : Lam X (su n) -> Lam X n\n\n_->>_ : {I : Set}(X Y : I -> Set) -> Set\nX ->> Y = {i : _} -> X i -> Y i\n\nfmsub : {X Y : Nat -> Set} -> (X ->> Lam Y) -> Lam X ->> Lam Y\nfmsub sb (var x)   = sb x\nfmsub sb (app f s) = app (fmsub sb f) (fmsub sb s)\nfmsub sb (lam b)   = lam (fmsub sb b)\n\ndata Fin : Nat -> Set where\n  ze : {n : Nat} -> Fin (su n)\n  su : {n : Nat} -> Fin n -> Fin (su n)\n\nD : (Nat -> Set) -> Nat -> Set\nD F n = F (su n)\n\nweak : {F : Nat -> Set} -> (Fin ->> F) -> (F ->> D F) ->\n       {m n : Nat}-> (Fin m -> F n) -> D Fin m -> D F n\nweak v w f ze      = v ze\nweak v w f (su x)  = w (f x)\n\n_+_ : Nat -> Nat -> Nat\nze    + y = y\nsu x  + y = su (x + y)\n\n_!+_ : (Nat -> Set) -> Nat -> (Nat -> Set)\nF !+ n = \\ m -> F (m + n)\n\nweaks : {F : Nat -> Set} -> (Fin ->> F) -> (F ->> D F) ->\n        {m n : Nat}-> (Fin m -> F n) -> (Fin !+ m) ->> (F !+ n)\nweaks v w f {ze}    = f\nweaks v w f {su i}  = weak v w (weaks v w f {i})\n\njing : {m n : Nat} -> Lam Fin (m + n) -> Lam (Fin !+ n) m\njing (var x) = var x\njing (app f s) = app (jing f) (jing s)\njing (lam t) = lam (jing t)\n\ngnij : {m n : Nat} -> Lam (Fin !+ n) m -> Lam Fin (m + n)\ngnij (var x) = var x\ngnij (app f s) = app (gnij f) (gnij s)\ngnij (lam t) = lam (gnij t)\n\nren : {m n : Nat} -> (Fin m -> Fin n) -> Lam Fin m -> Lam Fin n\nren f t = gnij (fmsub (\\ {i} x -> var (weaks (\\ z -> z) su f {i} x)) {ze} (jing t))\n\nsub : {m n : Nat} -> (Fin m -> Lam Fin n) -> Lam Fin m -> Lam Fin n\nsub {m}{n} f t = gnij {ze} (fmsub (\\ {i} x -> jing {i} (weaks {Lam Fin} var (ren su) f {i} x)) (jing {ze} t))","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/FMSB.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e0c2ee98d7f0e60a360eee0c42d0766506ad24c8","subject":"otp-sem-sec.agda","message":"otp-sem-sec.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"otp-sem-sec.agda","new_file":"otp-sem-sec.agda","new_contents":"module otp-sem-sec where\n\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec\nopen import Data.Product\nopen import circuit\nopen import Relation.Binary.PropositionalEquality\n\n-- dup from Game\nEXP : \u2200 {a b} {A : Set a} {B : Set b} \u2192 Bit \u2192 (A \u2192 B) \u2192 (A \u00d7 A \u2192 B)\nEXP 0b f (x\u2080 , x\u2081) = f x\u2080\nEXP 1b f (x\u2080 , x\u2081) = f x\u2081\n\nCoins = \u2115\nPorts = \u2115\n\nmodule F\n  (Size Time : Set)\n  (FCp : Coins \u2192 Size \u2192 Time \u2192 Ports \u2192 Ports \u2192 Set)\n  (Cp : Ports \u2192 Ports \u2192 Set)\n  (builder : CircuitBuilder Cp)\n  (toC : \u2200 {c s t i o} \u2192 FCp c s t i o \u2192 Cp (c + i) o)\n  (runC : \u2200 {i o} \u2192 Cp i o \u2192 Bits i \u2192 Bits o)\n\n  (|M| |C| : Ports)\n\n  (Proba : Set)\n  -- (Pr[_\u22611] : \u2200 {c} (EXP : \u21ba c Bit) \u2192 Proba)\n  (Pr[_\u22611] : \u2200 {c} (EXP : Bits c \u2192 Bit) \u2192 Proba)\n  (negligible-advantage-proba : Proba \u2192 Proba \u2192 Set)\n\n  where\n\n  record Power : Set where\n    constructor mk\n    field\n      coins : Coins\n      size  : Size\n      time  : Time\n  open Power public\n\n  M = Bits |M|\n  C = Bits |C|\n\n  \u21ba : Coins \u2192 Set \u2192 Set\n  \u21ba c A = Bits c \u2192 A\n\n  record SemSecAdv power : Set where\n    constructor mk\n\n    open Power power renaming (coins to c; size to s; time to t)\n    field\n      fcp : FCp c s t |C| (1 + (|M| + |M|))\n\n      {c\u2080 c\u2081}   : Coins\n      c\u2261c\u2080+c\u2081  : c \u2261 c\u2080 + c\u2081\n\n    R\u2080 = Bits c\u2080\n    R\u2081 = Bits c\u2081\n\n    open CircuitBuilder builder\n\n    cp : Cp ((c\u2080 + c\u2081) + |C|) (1 + (|M| + |M|))\n    cp rewrite sym c\u2261c\u2080+c\u2081 = toC fcp\n\n    cp\u2081 : Cp c\u2080 (|M| + |M|)\n    cp\u2081 = padR c\u2081 >>> padR |C| >>> cp >>> tailC\n\n    cp\u2082 : Cp ((c\u2080 + c\u2081) + |C|) 1\n    cp\u2082 = cp >>> headC\n\n    beh : \u21ba c\u2080 ((M \u00d7 M) \u00d7 (C \u2192 \u21ba c\u2081 Bit))\n    beh R\u2080 = (case splitAt |M| (runC cp\u2081 R\u2080) of \u03bb { (x , y , _) \u2192 (x , y) })\n           , (\u03bb C R\u2081 \u2192 head (runC cp\u2082 ((R\u2080 ++ R\u2081) ++ C)))\n\n  runSemSec : \u2200 (E : M \u2192 C) {p} (A : SemSecAdv p) (b : Bit) \u2192 \u21ba (coins p) Bit\n  runSemSec E {mk c s t} A b R\n      = case SemSecAdv.beh A R\u2080 of \u03bb\n          { (m0m1 , kont) \u2192 {- b == -} kont (EXP b E m0m1) R\u2081 }\n    where R\u2080 = take (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n          R\u2081 = drop (SemSecAdv.c\u2080 A) (subst Bits (SemSecAdv.c\u2261c\u2080+c\u2081 A) R)\n\n  negligible-advantage : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  negligible-advantage EXP = negligible-advantage-proba Pr[ EXP 0b \u22611] Pr[ EXP 1b \u22611]\n\n  SemSec : \u2200 (E : M \u2192 C) \u2192 Power \u2192 Set\n  SemSec E power = \u2200 (A : SemSecAdv power) \u2192 negligible-advantage (runSemSec E A)\n\n  |K| = |M|\n  K = Bits |K|\n  -- OTP : K \u2192 M \u2192 C\n  -- OTP = _\u2295_\n\n  -- OTP-SemSec : SemSec (OTP \n\n  Enc = M \u2192 C\n\n  Tr = Enc \u2192 Enc\n\n  -- In general the power might change\n  SemSecTr : Tr \u2192 Set\n  SemSecTr tr = \u2200 {E p} \u2192 SemSec E p \u2192 SemSec (tr E) p\n\n  neg-pres-sem-sec : SemSecTr (_\u2218_ vnot)\n  neg-pres-sem-sec {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = vnot \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n\n{-\n  \u2295-pres-sem-sec : \u2200 mask \u2192 SemSecTr (_\u2218_ (_\u2295_ mask))\n  \u2295-pres-sem-sec mask {E} {p} E-sec A-breaks-E' = pf\n     where E' : Enc\n           E' = (_\u2295_ mask) \u2218 E\n           pf : negligible-advantage (runSemSec E' A-breaks-E')\n           pf = {!!}\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'otp-sem-sec.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b2f3f134e623643fbaeb91725627462195c32406","subject":"experiment with simulator and protocol","message":"experiment with simulator and protocol\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/CoreBind.agda","new_file":"Control\/Protocol\/CoreBind.agda","new_contents":"-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality\n\nmodule Control.Protocol.CoreBind where\nopen import Control.Strategy renaming (Strategy to Client) public\n\ndata Inf : \u2605 where\n  mu nu : Inf\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (A : \u2605)(B : A \u2192 Proto) \u2192 Proto\n  later : Inf \u2192 \u221e Proto \u2192 Proto\n\n{-\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' A \u03bb _ \u2192 B\n\ndualInf : Inf \u2192 Inf\ndualInf mu = nu\ndualInf nu = mu\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' A B) = \u03a3' A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' A B) = \u03a0' A (\u03bb x \u2192 dual (B x))\ndual (later i P) = later (dualInf i) (\u266f (dual (\u266d P)))\n\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n{-\ndata Process (A : \u2605): Proto \u2192 \u2605\u2081 where\n  send : \u2200 {M P} (m : M) \u2192 Process A (P m) \u2192 Process A (\u03a3' M P)\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 Process A (P m)) \u2192 Process A (\u03a0' M P)\n  end  : A \u2192 Process A end\n  -}\nelInf : Inf \u2192 \u2605\u2081 \u2192 \u2605\u2081\nelInf mu x = x\nelInf nu x = \u221e x\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  send : \u2200 {M P} (m : M) \u2192 this (P m) \u2192 ProcessF this (\u03a3' M P)\n  recv : \u2200 {M P} \u2192 ((m : M) \u2192 this (P m)) \u2192 ProcessF this (\u03a0' M P)\n  latr : \u2200 {P} i \u2192 elInf i (this (\u266d P)) \u2192 ProcessF this (later i P)\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\ndata Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  left  : \u2200 {P Q} \u2192 ProcessF (flip Sim Q) P \u2192 Sim P Q\n  right : \u2200 {P Q} \u2192 ProcessF (Sim P) Q \u2192 Sim P Q\n  end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' A B) = right (recv (\u03bb m \u2192 left (send m (sim-id (B m)))))\nsim-id (\u03a3' A B) = left (recv (\u03bb m \u2192 right (send m (sim-id (B m)))))\nsim-id (later mu P) = right (latr mu (left (latr nu (\u266f sim-id (\u266d P)))))\nsim-id (later nu P) = left (latr mu (right (latr nu (\u266f (sim-id (\u266d P))))))\n\nsim-comp : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\nsim-comp (left (send m x)) QR = left (send m (sim-comp x QR))\nsim-comp (left (recv x)) QR = left (recv (\u03bb m \u2192 sim-comp (x m) QR))\nsim-comp (left (latr mu x)) QR = left (latr mu (sim-comp x QR))\nsim-comp (left (latr nu x)) QR = left (latr nu (\u266f sim-comp (\u266d x) QR))\nsim-comp (right (send m x)) (left (recv x\u2081)) = sim-comp x (x\u2081 m)\nsim-comp (right (recv x)) (left (send m x\u2081)) = sim-comp (x m) x\u2081\nsim-comp (right (latr mu x)) (left (latr .nu x\u2081)) = sim-comp x (\u266d x\u2081)\nsim-comp (right (latr nu x)) (left (latr .mu x\u2081)) = sim-comp (\u266d x) x\u2081\nsim-comp PQ (right (send m x)) = right (send m (sim-comp PQ x))\nsim-comp PQ (right (recv x)) = right (recv (\u03bb m \u2192 sim-comp PQ (x m)))\nsim-comp PQ (right (latr mu x)) = right (latr mu (sim-comp PQ x))\nsim-comp PQ (right (latr nu x)) = right (latr nu (\u266f sim-comp PQ (\u266d x)))\nsim-comp end QR = QR\n{-\nsim-comp end QR = QR\nsim-comp (right (send m x)) (left (recv x\u2081)) = sim-comp x (x\u2081 m)\nsim-comp (right (recv x)) (left (send m x\u2081)) = sim-comp (x m) x\u2081\nsim-comp (left (send m x)) QR = left (send m (sim-comp x QR))\nsim-comp (left (recv x)) QR = left (recv (\u03bb m \u2192 sim-comp (x m) QR))\nsim-comp PQ (right (send m\u2081 x\u2081)) = right (send m\u2081 (sim-comp PQ x\u2081))\nsim-comp PQ (right (recv x\u2081)) = right (recv (\u03bb m\u2081 \u2192 sim-comp PQ (x\u2081 m\u2081)))\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = sim-comp\n\nsim-sym : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-sym (left (send m x)) = right (send m (sim-sym x))\nsim-sym (left (recv x)) = right (recv (\u03bb m \u2192 sim-sym (x m)))\nsim-sym (right (send m x)) = left (send m (sim-sym x))\nsim-sym (right (recv x)) = left (recv (\u03bb m \u2192 sim-sym (x m)))\nsim-sym end = end\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (send m x)) = do (send m (sim-unit x))\nsim-unit (right (recv x)) = do (recv (\u03bb m \u2192 sim-unit (x m)))\nsim-unit end = end 0\u2081\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n{-\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = {!!}\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/CoreBind.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3350e86784a1ac1565247a5cef8441f026222afe","subject":"Initial commit: agda ILC with nats","message":"Initial commit: agda ILC with nats\n\nOld-commit-hash: 25b473eeff3f5286dafba706c0d7c64b886d29b3\n","repos":"inc-lc\/ilc-agda","old_file":"nats\/A.agda","new_file":"nats\/A.agda","new_contents":"-- TODO: Include proof of\n-- \"not all correctly-typed values are valid changes\"\n\nmodule A where\n\nopen import Data.Product\nopen import Data.Nat using (\u2115)\nopen import Data.Unit using (\u22a4)\n\nopen import Relation.Binary.PropositionalEquality\n\nopen import Level using\n  (zero ; Level ; suc)\nopen import Relation.Binary using\n  (Reflexive ; Transitive ; Preorder ; IsPreorder)\n\npostulate extensionality : Extensionality zero zero\n\ndata Type : Set where\n  nats : Type\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n\u27e8\u2205\u27e9 : \u2205 \u2261 \u2205\n\u27e8\u2205\u27e9 = refl\n\n_\u27e8\u2022\u27e9_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393\u2081 \u0393\u2082} \u2192 \u03c4\u2081 \u2261 \u03c4\u2082 \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u03c4\u2081 \u2022 \u0393\u2081 \u2261 \u03c4\u2082 \u2022 \u0393\u2082\n_\u27e8\u2022\u27e9_ = cong\u2082 _\u2022_\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\ndata Term : Context -> Type -> Set where\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n  nat : \u2200 {\u0393} \u2192 (n : \u2115) \u2192 Term \u0393 nats\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081)\n                   \u2192 Term \u0393 \u03c4\u2082\n\n  -- Change to nats = replacement Church pairs\n  -- 3 -> 5 ::= \u03bbf. f 3 5\n\ninfix 8 _\u227c_\n\ndata _\u227c_ : (\u0393\u2081 \u0393\u2082 : Context) \u2192 Set where\n  \u2205\u227c\u2205 : \u2205 \u227c \u2205\n  keep_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u03c4 \u2022 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n  drop_\u2022_ : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 (\u03c4 : Type) \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u0393\u2081 \u227c \u03c4 \u2022 \u0393\u2082\n\n\u227c-reflexivity : Reflexive _\u227c_\n\u227c-reflexivity {\u2205} = \u2205\u227c\u2205\n\u227c-reflexivity {\u03c4 \u2022 x} = keep \u03c4 \u2022 \u227c-reflexivity\n\n\u227c-reflexive : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u2261 \u0393\u2082 \u2192 \u0393\u2081 \u227c \u0393\u2082\n\u227c-reflexive refl = \u227c-reflexivity\n\n\u227c-transitive : Transitive _\u227c_\n\u227c-transitive \u2205\u227c\u2205 rel1 = rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  keep \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (keep \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (keep \u03c4 \u2022 rel0) rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (keep .\u03c4 \u2022 rel1) =\n  drop \u03c4 \u2022 \u227c-transitive rel0 rel1\n\u227c-transitive (drop \u03c4 \u2022 rel0) (drop \u03c4\u2081 \u2022 rel1) =\n  drop \u03c4\u2081 \u2022 \u227c-transitive (drop \u03c4 \u2022 rel0) rel1\n\n\u227c-isPreorder : IsPreorder _\u2261_ _\u227c_\n\u227c-isPreorder = record\n  { isEquivalence = isEquivalence\n  ; reflexive = \u227c-reflexive\n  ; trans = \u227c-transitive\n  }\n\n\u227c-preorder : Preorder _ _ _\n\u227c-preorder = record\n  { Carrier = Context\n  ; _\u2248_ = _\u2261_\n  ; isPreorder = \u227c-isPreorder\n  }\n\nmodule \u227c-reasoning where\n  open import Relation.Binary.PreorderReasoning \u227c-preorder public\n    renaming\n      ( _\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_\n      ; _\u223c\u27e8_\u27e9_ to _\u227c\u27e8_\u27e9_\n      ; _\u2248\u27e8\u27e9_ to _\u2261\u27e8\u27e9_\n      )\n\nweakenVar : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 Var \u0393\u2081 \u03c4 \u2192 Var \u0393\u2082 \u03c4\nweakenVar \u2205\u227c\u2205 x = x\nweakenVar (keep \u03c4\u2081 \u2022 subctx) this = this\nweakenVar (keep \u03c4\u2081 \u2022 subctx) (that y) = that (weakenVar subctx y)\nweakenVar (drop \u03c4\u2081 \u2022 subctx) x = that (weakenVar subctx x)\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 (subctx : \u0393\u2081 \u227c \u0393\u2082) \u2192 Term \u0393\u2081 \u03c4 \u2192 Term \u0393\u2082 \u03c4\nweaken subctx (abs {\u03c4} t) = abs (weaken (keep \u03c4 \u2022 subctx) t)\nweaken subctx (app t\u2081 t\u2082) = app (weaken subctx t\u2081) (weaken subctx t\u2082)\nweaken subctx (var x) = var (weakenVar subctx x)\nweaken subctx (nat x) = nat x\n\nrecord Meaning (Syntax : Set) {\u2113 : Level} : Set (suc \u2113) where\n  constructor\n    meaning\n  field\n    {Semantics} : Set \u2113\n    \u27e8_\u27e9\u27e6_\u27e7 : Syntax \u2192 Semantics\n\nopen Meaning {{...}} public\n  renaming (\u27e8_\u27e9\u27e6_\u27e7 to \u27e6_\u27e7)\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\u27e6 nats \u27e7Type = \u2115\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning \u27e6_\u27e7Type\n\ndata EmptySet : Set where\n  \u2205 : EmptySet\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\n\u27e6_\u27e7Context : Context \u2192 Set\n\u27e6 \u2205 \u27e7Context = EmptySet\n\u27e6 \u03c4 \u2022 \u0393 \u27e7Context = Bind \u27e6 \u03c4 \u27e7 \u27e6 \u0393 \u27e7Context\n\nmeaningOfContext : Meaning Context\nmeaningOfContext = meaning \u27e6_\u27e7Context\n\n\u27e6_\u27e7Var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 this \u27e7Var (v \u2022 \u03c1) = v\n\u27e6 that x \u27e7Var (v \u2022 \u03c1) = \u27e6 x \u27e7Var \u03c1\n\nmeaningOfVar : \u2200 {\u0393 \u03c4} \u2192 Meaning (Var \u0393 \u03c4)\nmeaningOfVar = meaning \u27e6_\u27e7Var\n\n\u27e6_\u27e7\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 \u0393\u2081 \u227c \u0393\u2082 \u2192 \u27e6 \u0393\u2082 \u27e7 \u2192 \u27e6 \u0393\u2081 \u27e7\n\u27e6 \u2205\u227c\u2205 \u27e7\u227c \u2205 = \u2205\n\u27e6 keep \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = v \u2022 \u27e6 subctx \u27e7\u227c \u03c1\n\u27e6 drop \u03c4 \u2022 subctx \u27e7\u227c (v \u2022 \u03c1) = \u27e6 subctx \u27e7\u227c \u03c1\n\nmeaningOf\u227c : \u2200 {\u0393\u2081 \u0393\u2082} \u2192 Meaning (\u0393\u2081 \u227c \u0393\u2082)\nmeaningOf\u227c = meaning \u27e6_\u27e7\u227c\n\nweakenVar-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} (subctx : \u0393\u2081 \u227c \u0393\u2082) (x : Var \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weakenVar subctx x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweakenVar-sound \u2205\u227c\u2205 () \u03c1\nweakenVar-sound (keep \u03c4 \u2022 subctx) this (v \u2022 \u03c1) = refl\nweakenVar-sound (keep \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx x \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) this (v \u2022 \u03c1) =\n  weakenVar-sound subctx this \u03c1\nweakenVar-sound (drop \u03c4 \u2022 subctx) (that x) (v \u2022 \u03c1) =\n  weakenVar-sound subctx (that x) \u03c1\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 nat n \u27e7Term \u03c1 = n\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm = meaning \u27e6_\u27e7Term\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = cong\u2082 (\u03bb x y \u2192 x y)\n\nweaken-sound : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} {subctx : \u0393\u2081 \u227c \u0393\u2082} (t : Term \u0393\u2081 \u03c4) \u2192\n  \u2200 (\u03c1 : \u27e6 \u0393\u2082 \u27e7) \u2192 \u27e6 weaken subctx t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (\u27e6 subctx \u27e7 \u03c1)\nweaken-sound (abs t) \u03c1 = extensionality (\u03bb v \u2192 weaken-sound t (v \u2022 \u03c1))\nweaken-sound (app t\u2081 t\u2082) \u03c1 = \u2261-app (weaken-sound t\u2081 \u03c1) (weaken-sound t\u2082 \u03c1)\nweaken-sound {subctx = subctx} (var x) \u03c1 = weakenVar-sound subctx x \u03c1\nweaken-sound (nat n) \u03c1 = refl\n\n-- Changes to \u2115 are replacement Church pairs. The only arguments\n-- of conern are `fst` and `snd`, so the Church pairs don't have\n-- to be polymorphic.\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type nats = (nats \u21d2 nats \u21d2 nats) \u21d2 nats\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\n-- It is clear that \u27e6\u229d\u27e7 exists on the semantic level:\n-- there exists an Agda value to describe the change between any\n-- two Agda values denoted by terms. If we have (not dependently-\n-- typed) arrays, no term denotes the change between two arrays\n-- of different lengths. Thus no full abstraction.\n\n\u27e6derive\u27e7 : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n_\u27e6\u2295\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n_\u27e6\u229d\u27e7_ : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ninfixl 6 _\u27e6\u2295\u27e7_\ninfixl 6 _\u27e6\u229d\u27e7_\n\n\u27e6fst\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\u27e6snd\u27e7 : \u2115 \u2192 \u2115 \u2192 \u2115\n\n\u27e6derive\u27e7 {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 f (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f v\n\u27e6derive\u27e7 {nats} n = \u03bb f \u2192 f n n\n\n_\u27e6\u229d\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 f\u2081 (v \u27e6\u2295\u27e7 dv) \u27e6\u229d\u27e7 f\u2082 v\n_\u27e6\u229d\u27e7_ {nats} m n = \u03bb f \u2192 f n m\n-- m \u27e6\u229d\u27e7 n ::= replace n by m\n\n_\u27e6\u2295\u27e7_ {\u03c4\u2081 \u21d2 \u03c4\u2082} f df = \u03bb v \u2192 f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n_\u27e6\u2295\u27e7_ {nats} n dn = dn \u27e6snd\u27e7\n\n\u27e6fst\u27e7 m n = m\n\u27e6snd\u27e7 m n = n\n\n-- Strong validity!\ndata \u27e6Valid\u0394\u27e7 : {\u03c4 : Type} \u2192 (v : \u27e6 \u03c4 \u27e7) \u2192 (dv : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  -- Following Pierce's case names `T-Var`, `T-Abs`, `T-App`\n  -- with Agda's capitalization convention\n  -- generalized to the semantic (value) domain\n  v-nat : (n : \u2115) \u2192 (dn : (\u2115 \u2192 \u2115 \u2192 \u2115) \u2192 \u2115) \u2192\n          n \u2261 dn \u27e6fst\u27e7 \u2192\n          \u27e6Valid\u0394\u27e7 n dn\n\n  v-fun : {\u03c4\u2081 \u03c4\u2082 : Type} \u2192\n          (f : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) \u2192 (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) \u2192\n          -- A strong antecedent: f and df map invalid changes to\n          -- valid changes! So long as invalid changes exist,\n          -- this is NOT satisfied by\n          --\n          --      f = \u27e6 \u03bb x. x \u27e7     = identity\n          --     df = \u27e6 \u03bb x dx. dx \u27e7 = \u27e6snd\u27e7,\n          --\n          -- negating any hope of \u27e6Valid\u0394\u27e7 \u27e6 t \u27e7 \u27e6 derive t \u27e7.\n          (\u2200 {s ds} \u2192 {-Valid\u0394 s ds \u2192-} \u27e6Valid\u0394\u27e7 (f s) (df s ds)) \u2192\n          (\u2200 {s ds} \u2192 (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 (f s \u27e6\u2295\u27e7 df s ds)) \u2192\n          \u27e6Valid\u0394\u27e7 f df\n\nabsurdity-of-0=1 : 0 \u2261 1 \u2192 (\u2200 {A : Set} \u2192 A)\nabsurdity-of-0=1 ()\n\nit-is-absurd-that : \u27e6Valid\u0394\u27e7 {nats} 0 (\u03bb f \u2192 f 1 1) \u2192 (\u2200 {A : Set} \u2192 A)\nit-is-absurd-that (v-nat .0 .(\u03bb f \u2192 f 1 1) 0=1) = absurdity-of-0=1 0=1\n\n-- If (\u03bb x dx \u2192 dx) is a \u27e6Valid\u0394\u27e7 to (\u03bb x \u2192 x),\n-- then impossible is nothing.\nno-way : \u27e6Valid\u0394\u27e7 {nats \u21d2 nats} (\u03bb x \u2192 x) (\u03bb x dx \u2192 dx) \u2192\n         (\u2200 {A : Set} \u2192 A)\nno-way (v-fun .(\u03bb x \u2192 x) .(\u03bb x dx \u2192 dx) validity correctness) =\n  it-is-absurd-that (validity {0} {\u03bb f \u2192 f 1 1})\n\n-- Cool lemmas\n\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u229d\u27e7 f \u2261 \u27e6derive\u27e7 f\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {nats} f = refl\nf\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = refl\n\nf\u2295[g\u229df]=g : \u2200 {\u03c4 : Type} (f g : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) \u2261 g\n\nf\u2295\u0394f=f : \u2200 {\u03c4 : Type} (f : \u27e6 \u03c4 \u27e7) \u2192 f \u27e6\u2295\u27e7 (\u27e6derive\u27e7 f) \u2261 f\n\nf\u2295[g\u229df]=g {nats} m n = refl\nf\u2295[g\u229df]=g {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n            {x = f x} {y = f x} refl\n            (cong\u2082 _\u27e6\u229d\u27e7_ (cong g (f\u2295\u0394f=f x)) refl) \u27e9\n      f x \u27e6\u2295\u27e7 (g x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (g x) \u27e9\n      g x\n    \u220e\n  ) where open \u2261-Reasoning\n\nf\u2295\u0394f=f {nats} n = refl\nf\u2295\u0394f=f {\u03c4\u2081 \u21d2 \u03c4\u2082} f = extensionality (\u03bb x \u2192\n    begin\n      (f \u27e6\u2295\u27e7 \u27e6derive\u27e7 f) x\n    \u2261\u27e8 refl \u27e9\n      f x \u27e6\u2295\u27e7 (f (x \u27e6\u2295\u27e7 \u27e6derive\u27e7 x) \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 cong (\u03bb hole \u2192 f x \u27e6\u2295\u27e7 (f hole \u27e6\u229d\u27e7 f x)) (f\u2295\u0394f=f x) \u27e9\n      f x \u27e6\u2295\u27e7 (f x \u27e6\u229d\u27e7 f x)\n    \u2261\u27e8 f\u2295[g\u229df]=g (f x) (f x) \u27e9\n      f x\n    \u220e\n  ) where open \u2261-Reasoning\n\nvalid-\u0394 : {\u03c4 : Type} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 Set\nvalid-\u0394 {nats} n dn = n \u2261 dn \u27e6fst\u27e7\nvalid-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} f df =\n  \u2200 (s : \u27e6 \u03c4\u2081 \u27e7) (ds : \u27e6 \u0394-Type \u03c4\u2081 \u27e7) (R[s,ds] : valid-\u0394 s ds) \u2192\n    valid-\u0394 (f s) (df s ds) \u00d7              -- (valid-\u0394:1)\n    (f \u27e6\u2295\u27e7 df) (s \u27e6\u2295\u27e7 ds) \u2261 f s \u27e6\u2295\u27e7 df s ds -- (valid-\u0394:2)\n\nR[f,g\u229df] : \u2200 {\u03c4} (f g : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (g \u27e6\u229d\u27e7 f)\nR[f,g\u229df] {nats} m n = refl\nR[f,g\u229df] {\u03c4\u2081 \u21d2 \u03c4\u2082} f g = \u03bb x dx R[x,dx] \u2192\n  R[f,g\u229df] {\u03c4\u2082} (f x) (g (x \u27e6\u2295\u27e7 dx)) -- (valid-\u0394:1)\n  , -- tuple constructor\n  (begin\n    (f \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f)) (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 refl \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7\n      (g ((x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 \u27e6derive\u27e7 (x \u27e6\u2295\u27e7 dx)) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 cong (\u03bb hole \u2192 f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g hole \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx)))\n          (f\u2295\u0394f=f (x \u27e6\u2295\u27e7 dx)) \u27e9\n    f (x \u27e6\u2295\u27e7 dx) \u27e6\u2295\u27e7 (g (x \u27e6\u2295\u27e7 dx) \u27e6\u229d\u27e7 f (x \u27e6\u2295\u27e7 dx))\n  \u2261\u27e8 f\u2295[g\u229df]=g (f (x \u27e6\u2295\u27e7 dx)) (g (x \u27e6\u2295\u27e7 dx)) \u27e9\n    g (x \u27e6\u2295\u27e7 dx)\n  \u2261\u27e8 sym (f\u2295[g\u229df]=g (f x) (g (x \u27e6\u2295\u27e7 dx))) \u27e9\n      f x \u27e6\u2295\u27e7 (g \u27e6\u229d\u27e7 f) x dx\n  \u220e)\n  where open \u2261-Reasoning\n\nR[f,\u0394f] : \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 {\u03c4} f (\u27e6derive\u27e7 f)\nR[f,\u0394f] f rewrite sym (f\u27e6\u229d\u27e7f=\u27e6deriv\u27e7f f) = R[f,g\u229df] f f\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393 -- push \u03c4, then push \u0394\u03c4.\n\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394-Context \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\u227c\u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394-Type \u03c4 \u2022 (keep \u03c4 \u2022 \u0393\u227c\u0394\u0393)\n\n-- Data type to hold proofs that environments are valid\ndata Valid-\u0394env : {\u0393 : Context} (\u03c1 : \u27e6 \u0393 \u27e7) (\u0394\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is its own valid \u0394-environment.\n  \u03c1=\u2205 : Valid-\u0394env {\u2205} \u2205 \u2205\n  -- Induction case: the change introduced therein should be valid,\n  -- and the smaller \u0394-environment should be valid as well.\n  \u03c1=v\u2022\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n            {\u0393\u2080 : Context} {\u03c1\u2080 : \u27e6 \u0393\u2080 \u27e7} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n            valid-\u0394 v dv \u2192 Valid-\u0394env \u03c1\u2080 d\u03c1\u2080 \u2192\n            Valid-\u0394env {\u03c4 \u2022 \u0393\u2080} (v \u2022 \u03c1\u2080) (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- Data type to hold proofs that a \u0394-env is consistent with self\ndata Consistent-\u0394env : {\u0393 : Context} (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 Set\n where\n  -- Base case: the empty environment is consistent with itself\n  d\u03c1=\u2205 : Consistent-\u0394env {\u2205} \u2205\n  -- Induction case: the change introduced at top level\n  -- should be valid.\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 : \u2200 {\u03c4 : Type} {v : \u27e6 \u03c4 \u27e7} {dv : \u27e6 \u0394-Type \u03c4 \u27e7}\n                 {\u0393\u2080 : Context} {d\u03c1\u2080 : \u27e6 \u0394-Context \u0393\u2080 \u27e7} \u2192\n                 valid-\u0394 v dv \u2192 Consistent-\u0394env d\u03c1\u2080 \u2192\n                 Consistent-\u0394env {\u03c4 \u2022 \u0393\u2080} (dv \u2022 v \u2022 d\u03c1\u2080)\n\n-- If a \u0394-environment is valid for some other environment,\n-- then it is also consistent with itself.\n\nvalid-\u0394env-is-consistent :\n   \u2200 {\u0393 : Context} {\u03c1 : \u27e6 \u0393 \u27e7} {d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} \u2192\n    Valid-\u0394env \u03c1 d\u03c1 \u2192 Consistent-\u0394env d\u03c1\n\nvalid-\u0394env-is-consistent \u03c1=\u2205 = d\u03c1=\u2205\nvalid-\u0394env-is-consistent (\u03c1=v\u2022\u03c1\u2080 valid[v,dv] valid[\u03c1\u2080,d\u03c1\u2080]) =\n  d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] (valid-\u0394env-is-consistent valid[\u03c1\u2080,d\u03c1\u2080])\n\n-- finally, update and ignore\n\nupdate : \u2200 {\u0393} \u2192 (d\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 {_ : Consistent-\u0394env d\u03c1} \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 {d\u03c1=\u2205} = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 d\u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 valid[v,dv] consistent[d\u03c1\u2080]} =\n  (v \u27e6\u2295\u27e7 dv) \u2022 update d\u03c1 {consistent[d\u03c1\u2080]}\n\n-- Ignorance is bliss (don't have to pass consistency proofs around :D)\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore = \u27e6 \u0393\u227c\u0394\u0393 \u27e7 -- Using a proof to describe computation\n\n-- Naming scheme follows weakenVar\/weaken\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- derive(n) = \u03bbf. f n n\nderive (nat n) = abs (app (app (var this) (nat n)) (nat n))\n\n-- derive(x) = dx\nderive (var x) = var (deriveVar x)\n\n-- derive(\u03bbx. t) = \u03bbx. \u03bbdx. derive(t)\nderive (abs t) = abs (abs (derive t))\n\n-- derive(f s) = derive(f) s derive(s)\nderive (app f s) = app (app (derive f) (weaken \u0393\u227c\u0394\u0393 s)) (derive s)\n\n-- Extensional equivalence for changes\ndata as-\u0394_is_ext-equiv-to_ :\n  \u2200 (\u03c4 : Type) \u2192 (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 (dg : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192 Set where\n  ext-\u0394 : \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n          (\u2200 (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192\n             (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)) \u2192\n          as-\u0394 \u03c4 is df ext-equiv-to dg\n\ninfix 3 as-\u0394_is_ext-equiv-to_\n\n-- Extractor for extensional-equivalence-as-changes:\n-- Given a value of the data type holding the proof,\n-- returns the proof in applicable form.\n--\n-- Question: Would it not have been better if such\n-- proof-holding types were defined as a function in\n-- the first place, say in the manner of `valid-\u0394`?\n--\nextract-\u0394equiv :\n  \u2200 {\u03c4 : Type} {df dg : \u27e6 \u0394-Type \u03c4 \u27e7} \u2192\n    as-\u0394 \u03c4 is df ext-equiv-to dg \u2192\n    (f : \u27e6 \u03c4 \u27e7) \u2192 valid-\u0394 f df \u2192 valid-\u0394 f dg \u2192 (f \u27e6\u2295\u27e7 df) \u2261 (f \u27e6\u2295\u27e7 dg)\n\nextract-\u0394equiv (ext-\u0394 proof-method) = proof-method\n\n-- Distribution lemmas of validity over \u27e6\u229d\u27e7 and \u27e6\u2295\u27e7\n\napplication-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v) \u2261 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\napplication-over-\u2295-and-\u229d f g df v =\n  begin\n    f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 cong\u2082 _\u27e6\u229d\u27e7_\n       (cong\u2082 _\u27e6\u2295\u27e7_\n         (cong f ((sym (f\u2295\u0394f=f v))))\n         (cong\u2082 df\n           (sym (f\u2295\u0394f=f v))\n           (cong \u27e6derive\u27e7 (sym (f\u2295\u0394f=f v)))))\n       refl \u27e9\n    f (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v)\n    \u27e6\u2295\u27e7 df (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v) (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 \u27e6derive\u27e7 v))\n    \u27e6\u229d\u27e7 f v\n  \u2261\u27e8 refl \u27e9\n   (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n  \u220e where open \u2261-Reasoning\n\nvalidity-over-\u2295-and-\u229d :\n  \u2200 {\u03c4\u2081 \u03c4\u2082 : Type}\n    (f g : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7) (df : \u27e6 \u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) \u27e7) (v : \u27e6 \u03c4\u2081 \u27e7) \u2192\n    valid-\u0394 g (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) \u2192\n    valid-\u0394 (g v) (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\nvalidity-over-\u2295-and-\u229d f g df v R[g,f\u2295df]\n  rewrite application-over-\u2295-and-\u229d f g df v =\n    proj\u2081 (R[g,f\u2295df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))\n\n-- diff-apply\ndf=f\u2295df\u229df :\n  \u2200 {\u03c4} (f : \u27e6 \u03c4 \u27e7) (df : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n    valid-\u0394 f df \u2192 as-\u0394 \u03c4 is df ext-equiv-to f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f\n\n-- Case nat: this REFL is more obvious to Agda than to a human.\ndf=f\u2295df\u229df {nats} n dn valid-f-df =\n  ext-\u0394 (\u03bb m valid-m-dn valid-rhs \u2192 refl)\n\ndf=f\u2295df\u229df {\u03c4\u2081 \u21d2 \u03c4\u2082} f df R[f,df] = ext-\u0394 (\n  \u03bb g R[g,df] R[g,f\u2295df\u229df] \u2192\n    extensionality (\u03bb v \u2192\n      begin\n        g v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v)\n      \u2261\u27e8 extract-\u0394equiv\n           (df=f\u2295df\u229df\n             (f v) (df v (\u27e6derive\u27e7 v))\n             (proj\u2081 (R[f,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v))))\n           (g v)\n           (proj\u2081 (R[g,df] v (\u27e6derive\u27e7 v) (R[f,\u0394f] v)))\n           (validity-over-\u2295-and-\u229d f g df v R[g,f\u2295df\u229df]) \u27e9\n        g v \u27e6\u2295\u27e7 (f v \u27e6\u2295\u27e7 df v (\u27e6derive\u27e7 v) \u27e6\u229d\u27e7 f v)\n      \u2261\u27e8 cong\u2082 _\u27e6\u2295\u27e7_\n           {x = g v} {y = g v} refl\n           (application-over-\u2295-and-\u229d f g df v) \u27e9\n        g v \u27e6\u2295\u27e7 (f \u27e6\u2295\u27e7 df \u27e6\u229d\u27e7 f) v (\u27e6derive\u27e7 v)\n      \u220e\n    )\n  )\n  where open \u2261-Reasoning\n\n\n\ncorrectness-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 deriveVar x \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 x \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 x \u27e7 (ignore \u03c1)\n\ncorrectness-of-deriveVar {\u03c4 \u2022 \u0393\u2080} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  this = df=f\u2295df\u229df {\u03c4} v dv R[v,dv]\n  --Putting `df=f\u2295df\u229df v dv ?` on RHS triggers Agda bug.\n  --TODO: Report it as Agda tells us to.\n\ncorrectness-of-deriveVar {\u03c4\u2080 \u2022 \u0393\u2080} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 {.\u03c4\u2080} {.v} {.dv} {.\u0393\u2080} {.\u03c1} R[v,dv] _}\n  (that x) = correctness-of-deriveVar \u03c1 x\n\n\n\ncorrectness-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  as-\u0394 \u03c4 is\n  \u27e6 derive t \u27e7 \u03c1\n  ext-equiv-to\n  \u27e6 t \u27e7 (update \u03c1 {consistency}) \u27e6\u229d\u27e7 \u27e6 t \u27e7 (ignore \u03c1)\n\n-- Mutually recursive lemma: derivatives are valid\nvalidity-of-derive : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (t : Term \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 t \u27e7 (ignore \u03c1)) (\u27e6 derive t \u27e7 \u03c1)\n\nvalidity-of-deriveVar : \u2200 {\u0393 \u03c4} \u2192\n  \u2200 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) {consistency : Consistent-\u0394env \u03c1} \u2192\n  \u2200 (x : Var \u0393 \u03c4) \u2192\n  valid-\u0394 (\u27e6 x \u27e7 (ignore \u03c1)) (\u27e6 deriveVar x \u27e7 \u03c1)\n\nvalidity-of-deriveVar {\u03c4 \u2022 \u0393} {.\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  this = R[v,dv]\n\nvalidity-of-deriveVar {\u03c4\u2080 \u2022 \u0393} {\u03c4}\n  (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency}\n  (that x) = validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 {consistency} (var x) =\n  validity-of-deriveVar \u03c1 {consistency} x\n\nvalidity-of-derive \u03c1 (nat n) = refl\n\nvalidity-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs t) = \u03bb v dv R[v,dv] \u2192\n    validity-of-derive (dv \u2022 v \u2022 \u03c1) {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 R[v,dv] consistency} t\n    , -- tuple constructor\n      (begin\n        \u27e6 t \u27e7 ((v \u27e6\u2295\u27e7 dv) \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) \u27e6\u2295\u27e7\n          \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n      \u2261\u27e8 {!!} \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)\n        \u27e6\u2295\u27e7 {!!}\n        \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 (v \u27e6\u2295\u27e7 dv) \u2022 (v \u27e6\u2295\u27e7 dv) \u2022 \u03c1)\n\n      \u2261\u27e8 {!!} \u27e9\n        \u27e6 t \u27e7 (v \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c \u03c1) \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (dv \u2022 v \u2022 \u03c1)\n      \u220e)\n  where open \u2261-Reasoning\n\nvalidity-of-derive \u03c1 (app t t\u2081) = {!!}\n\ncorrectness-of-derive \u03c1 (var x) = correctness-of-deriveVar \u03c1 x\n\ncorrectness-of-derive \u03c1 (nat n) = ext-\u0394 (\u03bb _ _ _ \u2192 refl)\n\ncorrectness-of-derive {\u0393} {\u03c4\u2081 \u21d2 \u03c4\u2082}\n  \u03c1 {consistency} (abs {.\u03c4\u2081} {.\u03c4\u2082} t) =\n  ext-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082}\n    (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n      extensionality {\u27e6 \u03c4\u2081 \u27e7} {\u03bb _ \u2192 \u27e6 \u03c4\u2082 \u27e7} (\u03bb x \u2192\n        begin\n          f x \u27e6\u2295\u27e7 \u27e6 derive t \u27e7 (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n        \u2261\u27e8 extract-\u0394equiv\n            (correctness-of-derive {\u03c4\u2081 \u2022 \u0393} {\u03c4\u2082}\n               (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n               {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency}\n               t)\n            (f x)\n            (proj\u2081 (R[f,\u0394t] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x)))\n            (proj\u2081 (R[f,t\u2032\u229dt] x (\u27e6derive\u27e7 x) (R[f,\u0394f] x))) \u27e9\n          f x\n          \u27e6\u2295\u27e7\n          (\u27e6 t \u27e7 (update (\u27e6derive\u27e7 x \u2022 x \u2022 \u03c1)\n                {d\u03c1=dv\u2022v\u2022d\u03c1\u2080 (R[f,\u0394f] x) consistency})\n          \u27e6\u229d\u27e7 \u27e6 t \u27e7 (x \u2022 \u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1))\n        \u220e\n  )) where open \u2261-Reasoning\n\ncorrectness-of-derive \u03c1 (app {\u03c4\u2081} {\u03c4\u2082} {\u0393} t\u2081 t\u2082) =\n  ext-\u0394 {\u03c4\u2082}\n  (\u03bb f R[f,\u0394t] R[f,t\u2032\u229dt] \u2192\n    begin\n      f \u27e6\u2295\u27e7 \u27e6 derive t\u2081 \u27e7 \u03c1 (\u27e6 weaken \u0393\u227c\u0394\u0393 t\u2082 \u27e7 \u03c1) (\u27e6 derive t\u2082 \u27e7 \u03c1)\n    \u2261\u27e8 {!!} \u27e9\n      f \u27e6\u2295\u27e7\n        (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1))\n         \u27e6\u229d\u27e7\n         \u27e6 t\u2081 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1) (\u27e6 t\u2082 \u27e7 (\u27e6 \u0393\u227c\u0394\u0393 \u27e7 \u03c1)))\n    \u220e\n  ) where open \u2261-Reasoning\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'nats\/A.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"128eaa91a085c49e694bd83c87a5135240c2e386","subject":"Control\/Protocol\/Additive.agda","message":"Control\/Protocol\/Additive.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Additive.agda","new_file":"Control\/Protocol\/Additive.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (_\u00d7_; _,_)\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.LR\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap)\nopen import Function.Extensionality\nopen import HoTT\nopen Equivalences\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Additive where\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto){{_ : FunExt}}{{_ : UA}} where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n    P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n    P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n    P\u22a4-uniq = \u03a0\ud835\udfd8-uniq _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} {P Q} where\n    dual-\u2295 : dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} P Q where\n    &-comm : P & Q \u2261 Q & P\n    &-comm = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm = send\u2243 LR!-equiv [L: refl R: refl ]\n\n    -- additive mix (left-biased)\n    amixL : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixL pq = (`L , pq `L)\n\n    amixR : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    amixR pq = (`R , pq `R)\n\nmodule _ {P Q R S}(f : \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7)(g : \u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) where\n    \u2295-map : \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map (`L , pr) = `L , f pr\n    \u2295-map (`R , pr) = `R , g pr\n\n    &-map : \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map p `L = f (p `L)\n    &-map p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 = [inl: _,_ `L ,inr: _,_ `R ]\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = equiv \u2295\u2192\u228e \u228e\u2192\u2295 [inl: (\u03bb _ \u2192 refl) ,inr: (\u03bb _ \u2192 refl) ] \u228e\u2192\u2295\u2192\u228e\n      where\n        \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n        \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n        \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = &\u2192\u00d7 , record { linv = \u00d7\u2192& ; is-linv = \u00d7\u2192&\u2192\u00d7 ; rinv = \u00d7\u2192& ; is-rinv = &\u2192\u00d7\u2192& }\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ P {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\nmodule _ P Q R {{_ : FunExt}}{{_ : UA}} where\n    &-assoc : \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Additive.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"01a43c75410a327981096d502d5243aa1850d7c4","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5b8bd1e8ab8c9ba80c7526cbad450c56792b54e0","subject":"Testing a Collatz equation.","message":"Testing a Collatz equation.\n\nIgnore-this: 948c916df99a2e916db18694313cee79\n\ndarcs-hash:20110519193855-3bd4e-77ed59433535c117a35a8a9aed5a2339cb0fb479.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/Collatz\/EquationsATP.agda","new_file":"Draft\/FOTC\/Program\/Collatz\/EquationsATP.agda","new_contents":"------------------------------------------------------------------------------\n-- Equations for the Collatz function\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Program.Collatz.EquationsATP where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.UnaryNumbers\n\nopen import FOTC.Program.Collatz.Collatz\nopen import FOTC.Program.Collatz.Data.Nat\n\n------------------------------------------------------------------------------\n\npostulate\n  collatz-even    : \u2200 {n} \u2192 GT n one \u2192 Even n \u2192\n                    collatz n \u2261 collatz (n \/ two)\n-- The ATPs cannot prove this equation.\n{-# ATP prove collatz-even #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Program\/Collatz\/EquationsATP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e977efcdfa72ca72208ea15ed42086001c6d63e0","subject":"ElGamal: Homomorphic encryption and re-encryption","message":"ElGamal: Homomorphic encryption and re-encryption\n","repos":"crypto-agda\/crypto-agda","old_file":"Cipher\/ElGamal\/Homomorphic.agda","new_file":"Cipher\/ElGamal\/Homomorphic.agda","new_contents":"open import Function\nopen import Type using (\u2605)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Cipher.ElGamal.Homomorphic\n  (\u2124q  : \u2605)\n  (G   : \u2605)\n  (g   : G)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (_\u2666_ : G \u2192 G \u2192 G)\n  (_\/_ : G \u2192 G \u2192 G)\n  where\n\nopen import Cipher.ElGamal.Generic G \u2124q G g _^_ _\u2666_ _\/_\n\nCombine : CipherText \u2192 CipherText \u2192 CipherText\nCombine (a , b) (c , d) = (a \u2666 c , b \u2666 d)\n\nReenc : PubKey \u2192 CipherText \u2192 (r\u2091 : \u2124q) \u2192 CipherText\nReenc pk (g\u02b8 , \u03b6) r\u2091 = (g ^ r\u2091) \u2666 g\u02b8 , (pk ^ r\u2091) \u2666 \u03b6\n\nmodule HomomorphicCorrectness\n    (\/-\u2666    : \u2200 {x y} \u2192 (x \u2666 y) \/ x \u2261 y)\n    (comm-^ : \u2200 {\u03b1 x y} \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    (interchange-\u2666 : \u2200 {a b c d} \u2192 (a \u2666 b) \u2666 (c \u2666 d) \u2261 ((a \u2666 c) \u2666 (b \u2666 d)))\n    (^-\u229e : \u2200 {\u03b1 x y} \u2192 (\u03b1 ^ x) \u2666 (\u03b1 ^ y) \u2261 \u03b1 ^ (x \u229e y))\n    where\n\n  open FunctionalCorrectness \/-\u2666 comm-^\n\n  homomorphic-correctness : \u2200 sk m\u2080 m\u2081 r\u2080 r\u2081 \u2192\n    let pk = PubKeyGen sk in\n    Dec sk (Combine (Enc pk m\u2080 r\u2080) (Enc pk m\u2081 r\u2081)) \u2261 m\u2080 \u2666 m\u2081\n  homomorphic-correctness r\u2096 m\u2080 m\u2081 r\u2080 r\u2081 =\n    (ap (flip _\/_ D) p \u2219 ap (_\/_ N \u2218 flip _^_ r\u2096) ^-\u229e) \u2219 functional-correctness r\u2096 (r\u2080 \u229e r\u2081) (m\u2080 \u2666 m\u2081)\n    where D = ((g ^ r\u2080) \u2666 (g ^ r\u2081))^ r\u2096\n          N = ((g ^ r\u2096) ^ (r\u2080 \u229e r\u2081)) \u2666 (m\u2080 \u2666 m\u2081)\n          p = interchange-\u2666 \u2219 ap (flip _\u2666_ (m\u2080 \u2666 m\u2081)) ^-\u229e\n\nmodule ReencCorrectness\n    (\/-\u2666    : \u2200 {x y} \u2192 (x \u2666 y) \/ x \u2261 y)\n    (comm-^ : \u2200 {\u03b1 x y} \u2192 (\u03b1 ^ x)^ y \u2261 (\u03b1 ^ y)^ x)\n    (interchange-\u2666 : \u2200 {a b c d} \u2192 (a \u2666 b) \u2666 (c \u2666 d) \u2261 ((a \u2666 c) \u2666 (b \u2666 d)))\n    (^-\u229e : \u2200 {\u03b1 x y} \u2192 (\u03b1 ^ x) \u2666 (\u03b1 ^ y) \u2261 \u03b1 ^ (x \u229e y))\n    (g\u2070 : G)\n    (g\u2070-idl : \u2200 x \u2192 g\u2070 \u2666 x \u2261 x)\n    (g\u2070-idr : \u2200 x \u2192 x \u2666 g\u2070 \u2261 x)\n    where\n  open HomomorphicCorrectness \/-\u2666 comm-^ interchange-\u2666 ^-\u229e\n\n  Reenc-Combine-g\u2070 : \u2200 pk c r \u2192 Reenc pk c r \u2261 Combine (Enc pk g\u2070 r) c\n  Reenc-Combine-g\u2070 pk c r = cong\u2082 _,_ refl (ap (flip _\u2666_ (proj\u2082 c)) (! (g\u2070-idr _)))\n\n  Reenc-correct : \u2200 sk m r\u2080 r\u2081 \u2192\n                  let pk = PubKeyGen sk in\n                  Dec sk (Reenc pk (Enc pk m r\u2081) r\u2080) \u2261 m\n  Reenc-correct r\u2096 m r\u2080 r\u2081 = ap (\u03bb z \u2192 (z \u2666 (((g ^ r\u2096) ^ r\u2081) \u2666 m)) \/ D)\n                                (! (g\u2070-idr _))\n                           \u2219 homomorphic-correctness r\u2096 g\u2070 m r\u2080 r\u2081\n                           \u2219 g\u2070-idl _\n    where D = ((g ^ r\u2080) \u2666 (g ^ r\u2081))^ r\u2096\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Cipher\/ElGamal\/Homomorphic.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"8eb3bf726fb7d12e74f6c4c5fd3b4b9b641cfefd","subject":"Add Nearring+1","message":"Add Nearring+1\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Nearring.agda","new_file":"lib\/Algebra\/Nearring.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Relation.Binary.PropositionalEquality.NP\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\nopen import Function.Extensionality\nopen import Data.Product.NP\nopen import Data.Nat.NP using (\u2115; zero; fold) renaming (suc to 1+)\nopen import Data.Integer using (\u2124; +_; -[1+_])\nopen import Algebra.Raw\nopen import Algebra.Monoid\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Group\nopen import Algebra.Group.Abelian\nopen import HoTT\n\nmodule Algebra.Nearring where\n\nrecord Nearring+1-Struct {\u2113} {A : Set \u2113} (rng-ops : Ring-Ops A) : Set \u2113 where\n  open Ring-Ops rng-ops\n\n  open \u2261-Reasoning\n\n  field\n    +-grp-struct : Group-Struct +-grp-ops\n    *-mon-struct : Monoid-Struct *-mon-ops\n    *-+-distr\u02b3   : _*_ DistributesOver\u02b3 _+_\n\n  open Additive-Group-Struct +-grp-struct public\n  -- 0\u2212-involutive   : Involutive 0\u2212_\n  -- cancels-+-left  : LeftCancel _+_\n  -- cancels-+-right : RightCancel _+_\n  -- ...\n\n  open Multiplicative-Monoid-Struct *-mon-struct public\n\n  open From-Ring-Ops rng-ops\n  open From-+Group-1*Identity-DistributesOver\u02b3\n             +-assoc\n             0+-identity\n             +0-identity\n             (snd 0\u2212-inverse)\n             1*-identity\n             *-+-distr\u02b3\n             public\n\n  +-grp : Group A\n  +-grp = +-grp-ops , +-grp-struct\n\n  module +-Grp = Group +-grp\n\n  *-mon : Monoid A\n  *-mon = *-mon-ops , *-mon-struct\n\n  module *-Mon = Monoid *-mon\n\nrecord Nearring+1 {\u2113} (A : Set \u2113) : Set \u2113 where\n  field\n    rng-ops           : Ring-Ops A\n    nearring+1-struct : Nearring+1-Struct rng-ops\n  open Ring-Ops          rng-ops           public\n  open Nearring+1-Struct nearring+1-struct public\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/Nearring.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9f362692e1ed95439a993df21e55081f1976df0c","subject":"Monad.agda","message":"Monad.agda\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Monad.agda","new_file":"lib\/Explore\/Monad.agda","new_contents":"open import Function\nopen import Relation.Binary.PropositionalEquality\n\nopen import Explore.Type\n\nmodule Explore.Monad \u2113 where\n\nM = Explore \u2113\n\nmodule _ {A : Set} where\n    return : A \u2192 M A\n    return x _\u2219_ f = f x\n\n    return-ind : \u2200 x \u2192 ExploreInd\u2080 (return x)\n    return-ind x P _P\u2219_ Pf = Pf x\n\nmodule _ {A B : Set} where\n\n    infixl 1 _>>=_ _>>=-ind_\n    _>>=_ : M A \u2192 (A \u2192 M B) \u2192 M B\n    (exp\u1d2c >>= exp\u1d2e) _\u2219_ f = exp\u1d2c _\u2219_ \u03bb x \u2192 exp\u1d2e x _\u2219_ f\n\n    _>>=-ind_ : \u2200 {exp\u1d2c : M A} \u2192 ExploreInd\u2080 exp\u1d2c \u2192\n                 {exp\u1d2e : A \u2192 M B} \u2192 (\u2200 x \u2192 ExploreInd\u2080 (exp\u1d2e x))\n               \u2192 ExploreInd\u2080 (exp\u1d2c >>= exp\u1d2e)\n    _>>=-ind_ {exp\u1d2c} P\u1d2c {exp\u1d2e} P\u1d2e P _P\u2219_ Pf\n      = P\u1d2c (\u03bb e \u2192 P (e >>= exp\u1d2e)) _P\u2219_ \u03bb x \u2192 P\u1d2e x P _P\u2219_ Pf\n\n    map : (A \u2192 B) \u2192 M A \u2192 M B\n    map f e = e >>= return \u2218 f\n\n    map-ind : (f : A \u2192 B) {exp\u1d2c : M A} \u2192 ExploreInd\u2080 exp\u1d2c \u2192 ExploreInd\u2080 (map f exp\u1d2c)\n    map-ind f P\u1d2c = P\u1d2c >>=-ind return-ind \u2218 f\n\n    private\n        {- This law is for free: map f e = e >>= return \u2218 f -}\n        map' : (A \u2192 B) \u2192 M A \u2192 M B\n        map' f exp _\u2219_ g = exp _\u2219_ (g \u2218 f)\n\n        map'\u2261map : map' \u2261 map\n        map'\u2261map = refl\n\n-- universe issues...\n-- join : M (M A) \u2192 M A\n\nmodule _ {A B C : Set} (m : M A) (f : A \u2192 M B) (g : B \u2192 M C) where\n    infix 0 _\u2261e_\n    _\u2261e_ = \u03bb {A} \u2192 _\u2261_ {A = M A}\n\n    left-identity : f \u2261 (\u03bb x \u2192 return x >>= f)\n    left-identity = refl\n\n    right-identity : m \u2261e m >>= return\n    right-identity = refl\n\n    associativity : m >>= (\u03bb x \u2192 f x >>= g) \u2261e (m >>= f) >>= g\n    associativity = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Explore\/Monad.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9a262e122b631721ae0643303f49b4b448162809","subject":"Proved a non-admissible property from the HALO paper.","message":"Proved a non-admissible property from the HALO paper.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/UnguardedCorecursion\/HALO.agda","new_file":"notes\/FOT\/FOTC\/UnguardedCorecursion\/HALO.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing a non-admissible property from the HALO paper\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- References:\n--\n-- \u2022 Vytiniotis, D., Peyton Jones, S., Ros\u00e9n, D. and Claessen,\n--   K. (2013). HALO: Haskell to Logic thorugh Denotational\n--   Semantics. In: Proceedings of the 40th annual ACM SIGPLAN-SIGACT\n--   symposium on Principles of programming languages (POPL\u201913),\n--   pp. 431\u2013442.\n\nmodule FOT.FOTC.UnguardedCorecursion.HALO where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Stream.Equality.PropertiesI\nopen import FOTC.Relation.Binary.Bisimilarity\nopen import FOTC.Relation.Binary.Bisimilarity.PropertiesI\n\n------------------------------------------------------------------------------\n-- The HALO paper states the property\n--\n-- \u2200 x . f x \u2260 ones\n--\n-- cannot be proved because it is a non-admissible predicate.\n\npostulate\n  ones    : D\n  ones-eq : ones \u2261 succ\u2081 zero \u2237 ones\n\npostulate\n  f     : D \u2192 D\n  f-eq\u2081 : \u2200 n \u2192 f (succ\u2081 n) \u2261 succ\u2081 zero \u2237 f n\n  f-eq\u2082 : f zero \u2261 zero \u2237 []\n\n-- A proof using the bisimilarity relation relation.\nthm : \u2200 {n} \u2192 N n \u2192 \u00ac (f n \u2248 ones)\nthm nzero h with \u2248-unf h\n... | x' , xs' , ys' , prf\u2081 , prf\u2082 , prf\u2083 = 0\u2262S (trans (sym x'\u22610) x'\u22611)\n  where\n  x'\u22610 : x' \u2261 zero\n  x'\u22610 = \u2227-proj\u2081 (\u2237-injective (trans (sym prf\u2081) f-eq\u2082))\n\n  x'\u22611 : x' \u2261 succ\u2081 zero\n  x'\u22611 = \u2227-proj\u2081 (\u2237-injective (trans (sym prf\u2082) ones-eq))\n\nthm (nsucc {n} Nn) h with \u2248-unf h\n... | x' , xs' , ys' , prf\u2081 , prf\u2082 , prf\u2083 = thm Nn (\u2237-injective\u2248 helper)\n  where\n  xs'\u2261fn : xs' \u2261 f n\n  xs'\u2261fn = \u2227-proj\u2082 (\u2237-injective (trans (sym prf\u2081) (f-eq\u2081 n)))\n\n  ys'\u2261ones : ys' \u2261 ones\n  ys'\u2261ones = \u2227-proj\u2082 (\u2237-injective (trans (sym prf\u2082) ones-eq))\n\n  helper : x' \u2237 f n \u2248 x' \u2237 ones\n  helper = \u2248-pre-fixed ( x'\n                       , xs'\n                       , ys'\n                       , \u2237-rightCong (sym xs'\u2261fn)\n                       , \u2237-rightCong (sym ys'\u2261ones)\n                       , prf\u2083\n                       )\n\n-- A proof using the propositional equality.\nthm' : \u2200 {n} \u2192 N n \u2192 f n \u2262 ones\nthm' nzero h = 0\u2262S (\u2227-proj\u2081 (\u2237-injective helper))\n  where\n  helper : zero \u2237 [] \u2261 succ\u2081 zero \u2237 ones\n  helper = trans\u2082 (sym f-eq\u2082) h ones-eq\n\nthm' (nsucc {n} Nn) h = thm' Nn (\u2227-proj\u2082 (\u2237-injective helper))\n  where\n  helper : succ\u2081 zero \u2237 f n \u2261 succ\u2081 zero \u2237 ones\n  helper = trans\u2082 (sym (f-eq\u2081 n)) h ones-eq\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/UnguardedCorecursion\/HALO.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ec36da07d429e649c05472185540f9f60aeda0d8","subject":"Try a different method","message":"Try a different method\n","repos":"louisswarren\/hieretikz","old_file":"arrow.agda","new_file":"arrow.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"error: RPC failed; HTTP 403 curl 22 The requested URL returned error: 403\nfatal: the remote end hung up unexpectedly\n","license":"mit","lang":"Agda"}
{"commit":"b5129ee5c0b5ffae12387ee403ca85c9cabd2fb1","subject":"Add FFI.JS.BigI","message":"Add FFI.JS.BigI\n","repos":"crypto-agda\/agda-libjs,audreyt\/agda-libjs","old_file":"lib\/FFI\/JS\/BigI.agda","new_file":"lib\/FFI\/JS\/BigI.agda","new_contents":"open import FFI.JS hiding (toString)\n\nmodule FFI.JS.BigI where\n\nabstract\n  BigI : Set\n  BigI = JSValue\n\n  BigI\u25b9JSValue : BigI \u2192 JSValue\n  BigI\u25b9JSValue x = x\n\n  unsafe-JSValue\u25b9BigI : BigI \u2192 JSValue\n  unsafe-JSValue\u25b9BigI x = x\n\npostulate\n  bigI     : (x base : String) \u2192 BigI\n  add      : (x y : BigI) \u2192 BigI\n  multiply : (x y : BigI) \u2192 BigI\n  mod      : (x y : BigI) \u2192 BigI\n  modPow   : (this e m : BigI) \u2192 BigI\n  modInv   : (this m   : BigI) \u2192 BigI\n  equals   : (x y : BigI) \u2192 Bool\n  toString : (x : BigI) \u2192 String\n  fromHex  : String \u2192 BigI\n  toHex    : BigI \u2192 String\n\n{-# COMPILED_JS bigI       function(x) { return function (y) { return require(\"bigi\")(x,y); }; } #-}\n{-# COMPILED_JS add        function(x) { return function (y) { return x.add(y); }; } #-}\n{-# COMPILED_JS multiply   function(x) { return function (y) { return x.multiply(y); }; } #-}\n{-# COMPILED_JS mod        function(x) { return function (y) { return x.mod(y); }; } #-}\n{-# COMPILED_JS modPow     function(x) { return function (y) { return function (z) { return x.modPow(y,z); }; }; } #-}\n{-# COMPILED_JS modInv     function(x) { return function (y) { return x.modInverse(y); }; } #-}\n{-# COMPILED_JS equals     function(x) { return function (y) { return x.equals(y); }; } #-}\n{-# COMPILED_JS toString   function(x) { return x.toString(); } #-}\n{-# COMPILED_JS fromHex    require(\"bigi\").fromHex #-}\n{-# COMPILED_JS toHex      function(x) { return x.toHex(); } #-}\n\n0I 1I 2I : BigI\n0I = bigI \"0\" \"10\"\n1I = bigI \"1\" \"10\"\n2I = bigI \"2\" \"10\"\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/FFI\/JS\/BigI.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7964095ebfae9648dc4c14b392249a83cf67e13e","subject":"Algebra.NP: \"flip\"ping structures","message":"Algebra.NP: \"flip\"ping structures\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/NP.agda","new_file":"lib\/Algebra\/NP.agda","new_contents":"module Algebra.NP where\n\nopen import Algebra public\nopen import Algebra.Structures\nopen import Algebra.FunctionProperties\nopen import Function\nopen import Data.Product\nopen import Relation.Binary\n\nmodule Flip where\n    module Structures {c} {C : Set c} {\u2113} (_\u2248_ : C \u2192 C \u2192 Set \u2113) (_\u2219_ : Op\u2082 C) where \n        flipIsSemigroup : IsSemigroup _\u2248_ _\u2219_ \u2192 IsSemigroup _\u2248_ (flip _\u2219_)\n        flipIsSemigroup isSg = record { isEquivalence = isEquivalence\n                                      ; assoc = \u03bb x y z \u2192 sym (assoc z y x)\n                                      ; \u2219-cong = \u03bb p q \u2192 sym (\u2219-cong (sym q) (sym p)) }\n            where open IsSemigroup isSg\n\n        flipIsMonoid : \u2200 {\u03b5} \u2192 IsMonoid _\u2248_ _\u2219_ \u03b5 \u2192 IsMonoid _\u2248_ (flip _\u2219_) \u03b5\n        flipIsMonoid isMon = record { isSemigroup = flipIsSemigroup isSemigroup\n                                    ; identity    = swap identity }\n            where open IsMonoid isMon\n\n        flipIsCommutativeMonoid : \u2200 {\u03b5} \u2192 IsCommutativeMonoid _\u2248_ _\u2219_ \u03b5\n                                        \u2192 IsCommutativeMonoid _\u2248_ (flip _\u2219_) \u03b5\n        flipIsCommutativeMonoid isCMon\n          = record { isSemigroup = flipIsSemigroup isSemigroup\n                   ; identity\u02e1 = proj\u2082 identity; comm = flip comm }\n            where open IsCommutativeMonoid isCMon\n\n        flipIsGroup : \u2200 {\u03b5} {_\u207b\u00b9} \u2192 IsGroup _\u2248_ _\u2219_ \u03b5 _\u207b\u00b9 \u2192 IsGroup _\u2248_ (flip _\u2219_) \u03b5 _\u207b\u00b9\n        flipIsGroup isGr = record { isMonoid = flipIsMonoid isMonoid\n                                  ; inverse = swap inverse\n                                  ; \u207b\u00b9-cong = \u207b\u00b9-cong }\n            where open IsGroup isGr\n\n        flipIsAbelianGroup : \u2200 {\u03b5} {_\u207b\u00b9} \u2192 IsAbelianGroup _\u2248_ _\u2219_ \u03b5 _\u207b\u00b9\n                                         \u2192 IsAbelianGroup _\u2248_ (flip _\u2219_) \u03b5 _\u207b\u00b9\n        flipIsAbelianGroup isAGr = record { isGroup = flipIsGroup isGroup\n                                          ; comm    = flip comm }\n            where open IsAbelianGroup isAGr\n\n    flipSemigroup : \u2200 {c \u2113} \u2192 Semigroup c \u2113 \u2192 Semigroup c \u2113\n    flipSemigroup sg = record { isSemigroup = flipIsSemigroup isSemigroup }\n      where open Semigroup sg\n            open Structures _\u2248_ _\u2219_\n\n    flipMonoid : \u2200 {c \u2113} \u2192 Monoid c \u2113 \u2192 Monoid c \u2113\n    flipMonoid m = record { isMonoid = flipIsMonoid isMonoid }\n      where open Monoid m\n            open Structures _\u2248_ _\u2219_\n\n    flipCommutativeMonoid : \u2200 {c \u2113} \u2192 CommutativeMonoid c \u2113 \u2192 CommutativeMonoid c \u2113\n    flipCommutativeMonoid cm\n      = record { isCommutativeMonoid = flipIsCommutativeMonoid isCommutativeMonoid }\n      where open CommutativeMonoid cm\n            open Structures _\u2248_ _\u2219_\n\n    flipGroup : \u2200 {c \u2113} \u2192 Group c \u2113 \u2192 Group c \u2113\n    flipGroup gr = record { isGroup = flipIsGroup isGroup }\n      where open Group gr\n            open Structures _\u2248_ _\u2219_\n\n    flipAbelianGroup : \u2200 {c \u2113} \u2192 AbelianGroup c \u2113 \u2192 AbelianGroup c \u2113\n    flipAbelianGroup agr = record { isAbelianGroup = flipIsAbelianGroup isAbelianGroup }\n      where open AbelianGroup agr\n            open Structures _\u2248_ _\u2219_\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ca2771eab9bcbf70b14cccf2d6fa815119f0e3c2","subject":"Added an issue related with the eta-expansion of definitions.","message":"Added an issue related with the eta-expansion of definitions.\n\nIgnore-this: 9c9a69a8db6a0e8fb1efc562fa205c73\n\ndarcs-hash:20110726163740-3bd4e-d1f2d2388fef48710a9388f0aa1d157f3b113bf7.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Issues\/Eta.agda","new_file":"Test\/Issues\/Eta.agda","new_contents":"-- Issue: At the moment, we don't eta-expand the equations in the ATP\n-- definitions.\n\nmodule Test.Issues.Eta where\n\npostulate\n  D       : Set\n  binaryP : D \u2192 D \u2192 Set\n\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (witness : D) \u2192 P witness \u2192 \u2203 P\n\n-- Because Agda eta-reduces the equations, the internal representation\n-- of P corresponds to the function\n--\n-- P xs = \u2203 (binary xs)\n--\n-- Due to it, we cannot proof the conjecture bar.\n\nP : D \u2192 Set\nP xs = \u2203 \u03bb ys \u2192 binaryP xs ys\n{-# ATP definition P #-}\n\npostulate\n  foo : \u2200 {xs} \u2192 (\u2203 \u03bb ys \u2192 binaryP xs ys) \u2192 (\u2203 \u03bb ys \u2192 binaryP xs ys)\n{-# ATP prove foo #-}\n\npostulate\n  barP : \u2200 {xs} \u2192 P xs \u2192 (\u2203 \u03bb ys \u2192 binaryP xs ys)\n{-# ATP prove bar #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Issues\/Eta.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"be20bde7489d10b36a94fc724ed9599bedf32bd0","subject":"Control\/Protocol\/Parametricity.agda","message":"Control\/Protocol\/Parametricity.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Parametricity.agda","new_file":"Control\/Protocol\/Parametricity.agda","new_contents":"module Control.Protocol.Parametricity where\n\nopen import Type\nopen import Relation.Binary.Logical\n\ndata Proto : \u2605\u2081 where\n  end  : Proto\n  send : {M : \u2605}(P : M \u2192 Proto) \u2192 Proto\n  recv : {M : \u2605}(P : M \u2192 Proto) \u2192 Proto\n\ndata \u27e6Proto\u27e7 : \u27e6\u2605\u2081\u27e7 Proto Proto where\n  \u27e6end\u27e7  : \u27e6Proto\u27e7 end end\n  \u27e6send\u27e7 : (\u2200\u27e8 M\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 (M\u1d63 \u27e6\u2192\u27e7 \u27e6Proto\u27e7) \u27e6\u2192\u27e7 \u27e6Proto\u27e7) send send\n  \u27e6recv\u27e7 : (\u2200\u27e8 M\u1d63 \u2236 \u27e6\u2605\u2080\u27e7 \u27e9\u27e6\u2192\u27e7 (M\u1d63 \u27e6\u2192\u27e7 \u27e6Proto\u27e7) \u27e6\u2192\u27e7 \u27e6Proto\u27e7) recv recv\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Parametricity.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f373eb938985ebe5872d7cfa4ea09cc8560b3716","subject":"Text.Parser.Heap: sweet but slow :(","message":"Text.Parser.Heap: sweet but slow :(\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Text\/Parser\/Heap.agda","new_file":"lib\/Text\/Parser\/Heap.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Inspired from https:\/\/www.fpcomplete.com\/user\/edwardk\/heap-of-successes\n--               https:\/\/gist.github.com\/ekmett\/578eaf3e5a37f7315e6c\n--\n-- Too slow so far: see an example in github.com\/crypto-agda\/crypto-agda Solver.Linear.Parser\nopen import Type using (Type; Type_)\nopen import Data.Maybe.NP using (_\u2192?_; map?)\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; _\u2238_; _+_) renaming (_==_ to _==\u2115_)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Char.NP renaming (_==_ to _==\u1d9c_)\nopen import Data.List.NP as L using (List; []; _\u2237_; null; filter)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.BoundedVec using (BoundedVec) renaming ([] to []\u1d47\u1d5b; _\u2237_ to _\u2237\u1d47\u1d5b_)\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Bool using (Bool; if_then_else_; false)\nopen import Data.String using (String) renaming (toList to S\u25b9L; fromList to L\u25b9S)\nopen import Function.NP\nopen import Level.NP\n\nmodule Text.Parser.Heap where\n\n-- Loosely typed assoc list with sorted keys\nHeap : Type \u2192 Type\nHeap A = List (\u2115 \u00d7 A)\n\nmodule Heap {A : Type} where\n  gather : Heap A \u2192 Heap (List A)\n  gather [] = []\n  gather ((i\u2080 , a\u2080) \u2237 as\u2080) = go i\u2080 (a\u2080 \u2237 []) as\u2080 where\n    go : \u2115 \u2192 _ \u2192 List _ \u2192 _\n    go i acc [] = (i , acc) \u2237 []\n    go i acc ((j , a) \u2237 as)\n      = if i ==\u2115 j then go i (a \u2237 acc) as\n                   else (i , acc) \u2237 go j (a \u2237 []) as\n\n  -- fair interleaving, because, well, why not?\n  merge : Heap A \u2192 Heap A \u2192 Heap A\n  merge [] as = as\n  merge as [] = as\n  merge (a \u2237 as) (b \u2237 bs) =\n    if fst a <= fst b then a \u2237 merge (b \u2237 bs) as\n                      else b \u2237 merge bs (a \u2237 as)\n\n  sing : \u2115 \u2192 A \u2192 Heap A\n  sing n a = (n , a) \u2237 []\n\n  map : \u2200 {B} \u2192 (\u2115 \u2192 \u2115) \u2192 (A \u2192 B) \u2192 Heap A \u2192 Heap B\n  map f g = L.map (\u00d7-map f g)\n\n  reset : Heap A \u2192 Heap A\n  reset = L.map (first (const 0))\n \nParser : (I O : Type) \u2192 Type\nParser I O = I \u2192 Heap O\n\nParserL : (I O : Type) \u2192 Type\nParserL I O = Parser (List I) O\n\nParserLC = ParserL Char\n\nmodule _ {I O : Type} where\n  pure : O \u2192 Parser I O\n  pure o i = Heap.sing 0 o\n\n  empty : Parser I O\n  empty _ = []\n\n  infixr 3 _\u27e8|\u27e9_\n  _\u27e8|\u27e9_ : (f g : Parser I O) \u2192 Parser I O\n  (f \u27e8|\u27e9 g) i = Heap.merge (f i) (g i)\n\n  choices : List (Parser I O) \u2192 Parser I O\n  choices = L.foldr _\u27e8|\u27e9_ empty\n\n-- Parser is a Profunctor\ndimap : \u2200{I I' O O'}(f : I' \u2192 I)(g : O \u2192 O') \u2192 Parser I O \u2192 Parser I' O'\ndimap f g p s = L.map (second g) (p (f s))\n\n-- The remaning input is dropped, you may want to first\n-- combine your parser `p' with `eof': `p <* eof'\nrunParser : \u2200 {A} \u2192 ParserLC A \u2192 List Char \u2192? A\nrunParser p = map? snd \u2218 L.listToMaybe \u2218 p\n\nrunParser\u02e2 : \u2200 {A} \u2192 ParserLC A \u2192 String \u2192? A\nrunParser\u02e2 p = runParser p \u2218 S\u25b9L\n\nlookSat : \u2200 {I} \u2192 (I \u2192 Bool) \u2192 Parser I \ud835\udfd9\nlookSat p i = if p i then (0 , _) \u2237 [] else []\n\nnotFollowedBy : \u2200 {I O} \u2192 Parser I O \u2192 Parser I \ud835\udfd9\nnotFollowedBy m = lookSat (null \u2218 m)\n\nlookAhead : \u2200 {I O} \u2192 Parser I O \u2192 Parser I O\nlookAhead p = Heap.reset \u2218 p\n\nmodule _ {I : Type} where\n  private\n    M = ParserL I\n\n  sat : (I \u2192 Bool) \u2192 M I\n  sat pred (x \u2237 xs) = if pred x then (1 , x) \u2237 [] else []\n  sat _    []       = []\n\n  eof : M \ud835\udfd9\n  eof = lookSat null\n\n  infixr 0 _>>=_\n  _>>=_ : \u2200 {A B} \u2192 ParserL I A \u2192 (A \u2192 ParserL I B) \u2192 ParserL I B\n  (ma >>= amb) s\u2080 = go 0 s\u2080 (ma s\u2080) [] where\n    go : _ \u2192 _ \u2192 List _ \u2192 _ \u2192 _\n    go i s ((j , a) \u2237 fss) acc =\n      let s' = L.drop (j \u2238 i) s in\n      go j s' fss (Heap.merge acc (L.map (first (_+_ j)) (amb a s')))\n    go _ _ [] acc = acc\n\n  infixl 0 _=<<_\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 ParserL I B) \u2192 ParserL I A \u2192 ParserL I B\n  _=<<_ f pa = pa >>= f\n\n  infixl 4 _\u229b_ _*>_ _<*_\n  _\u229b_ : \u2200 {A B} \u2192 ParserL I (A \u2192 B) \u2192 ParserL I A \u2192 ParserL I B\n  (mf \u229b ma) s\u2080 = go 0 s\u2080 (Heap.gather (mf s\u2080)) [] where\n    go : _ \u2192 _ \u2192 List _ \u2192 _ \u2192 _\n    go i s ((j , fs) \u2237 fss) acc with L.drop (j \u2238 i) s\n    ... | s' =\n      go j s' fss (Heap.merge acc (L.concatMap (\u03bb { (k , a) \u2192 L.map (\u03bb f -> (j + k , f a)) fs }) (ma s')))\n    go _ _ [] acc = acc\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 ParserL I A \u2192 ParserL I B\n  map f pa s\u2080 = Heap.map id f (pa s\u2080)\n\n  {-\n  map-spec : \u2200 {A B}{f : A \u2192 B}{pa : ParserL I A} \u2192\n             map f pa \u2261 pure f \u229b pa\n  map-spec rewrite ... concatMap-map = refl\n  -}\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  \u27ea f \u00b7 x \u27eb = map f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} \u2192\n              (A \u2192 B \u2192 C) \u2192 M A \u2192 M B \u2192 M C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map f x \u229b y\n\n  \u27ea_\u00b7_\u00b7_\u00b7_\u27eb : \u2200 {A B C D} \u2192 (A \u2192 B \u2192 C \u2192 D)\n              \u2192 M A \u2192 M B \u2192 M C \u2192 M D\n  \u27ea f \u00b7 x \u00b7 y \u00b7 z \u27eb = map f x \u229b y \u229b z\n\n  _*>_ : \u2200 {A B} \u2192 M A \u2192 M B \u2192 M B\n  p\u2081 *> p\u2082 = \u27ea const id \u00b7 p\u2081 \u00b7 p\u2082 \u27eb\n\n  _<*_ : \u2200 {A B} \u2192 M A \u2192 M B \u2192 M A\n  p\u2081 <* p\u2082 = \u27ea const \u00b7 p\u2081 \u00b7 p\u2082 \u27eb\n\n  mapM : \u2200 {a} {A : Type_ a} {B} \u2192 (A \u2192 M B) \u2192 List A \u2192 M (List B)\n  mapM f []       = pure []\n  mapM f (x \u2237 xs) = \u27ea _\u2237_ \u00b7 f x \u00b7 mapM f xs \u27eb\n\n  mapMvoid : \u2200 {a} {A : Type_ a} {B} \u2192 (A \u2192 M B) \u2192 List A \u2192 M \ud835\udfd9\n  mapMvoid f []       = pure _\n  mapMvoid f (x \u2237 xs) = f x *> mapMvoid f xs\n\n  module _ {A : Type} where\n    join : ParserL I (ParserL I A) \u2192 ParserL I A\n    join = _=<<_ id\n\n    vec : \u2200 n \u2192 M A \u2192 M (Vec A n)\n    vec zero    p = pure []\n    vec (suc n) p = \u27ea _\u2237_ \u00b7 p \u00b7 vec n p \u27eb\n\n    -- These are producing bounded vectors mainly for termination\n    -- reasons.\n    manyBV : \u2200 n \u2192 M A \u2192 M (BoundedVec A n)\n    someBV : \u2200 n \u2192 M A \u2192 M (BoundedVec A (suc n))\n\n    manyBV 0       _ = empty\n    manyBV (suc n) p = someBV n p \u27e8|\u27e9 pure []\u1d47\u1d5b\n\n    someBV n p = \u27ea _\u2237\u1d47\u1d5b_ \u00b7 p \u00b7 manyBV n p \u27eb\n\n    many : \u2200 n \u2192 M A \u2192 M (List A)\n    some : \u2200 n \u2192 M A \u2192 M (List A)\n\n    many 0       _ = empty\n    many (suc n) p = some n p \u27e8|\u27e9 pure []\n\n    some n p = \u27ea _\u2237_ \u00b7 p \u00b7 many n p \u27eb\n\nmanySat : (Char \u2192 Bool) \u2192 ParserLC (List Char)\nmanySat p []       = (0 , []) \u2237 []\nmanySat p (x \u2237 xs) = if p x then Heap.map suc (_\u2237_ x) (manySat p xs) else (0 , []) \u2237 []\n\nmanySat\u02e2 : (Char \u2192 Bool) \u2192 ParserLC String\nmanySat\u02e2 = map L\u25b9S \u2218 manySat\n\nsomeSat : (Char \u2192 Bool) \u2192 ParserLC (List Char)\nsomeSat p = pure _\u2237_ \u229b sat p \u229b manySat p\n\nsomeSat\u02e2 : (Char \u2192 Bool) \u2192 ParserLC String\nsomeSat\u02e2 = map L\u25b9S \u2218 someSat\n\nchar : Char \u2192 ParserLC \ud835\udfd9\nchar c = sat (_==\u1d9c_ c) *> pure _\n\nstring : String \u2192 ParserLC \ud835\udfd9\nstring = mapMvoid char \u2218 S\u25b9L\n\noneOf : List Char \u2192 ParserLC Char\noneOf cs = sat (\u03bb c \u2192 c `elem` cs)\n\noneOf\u02e2 : String \u2192 ParserLC Char\noneOf\u02e2 = oneOf \u2218 S\u25b9L\n\nnoneOf : List Char \u2192 ParserLC Char\nnoneOf cs = sat (\u03bb c \u2192 c `notElem` cs)\n\nnoneOf\u02e2 : String \u2192 ParserLC Char\nnoneOf\u02e2 = noneOf \u2218 S\u25b9L\n\nsomeOneOf : List Char \u2192 ParserLC (List Char)\nsomeOneOf cs = someSat (\u03bb c \u2192 c `elem` cs)\n\nsomeOneOf\u02e2 : String \u2192 ParserLC String\nsomeOneOf\u02e2 = map L\u25b9S \u2218 someOneOf \u2218 S\u25b9L\n\nsomeNoneOf : List Char \u2192 ParserLC (List Char)\nsomeNoneOf cs = someSat (\u03bb c \u2192 c `notElem` cs)\n\nsomeNoneOf\u02e2 : String \u2192 ParserLC String\nsomeNoneOf\u02e2 = map L\u25b9S \u2218 someNoneOf \u2218 S\u25b9L\n\nmanyOneOf : List Char \u2192 ParserLC (List Char)\nmanyOneOf cs = manySat (\u03bb c \u2192 c `elem` cs)\n\nmanyOneOf\u02e2 : String \u2192 ParserLC String\nmanyOneOf\u02e2 = map L\u25b9S \u2218 manyOneOf \u2218 S\u25b9L\n\nmanyNoneOf : List Char \u2192 ParserLC (List Char)\nmanyNoneOf cs = manySat (\u03bb c \u2192 c `notElem` cs)\n\nmanyNoneOf\u02e2 : String \u2192 ParserLC String\nmanyNoneOf\u02e2 = map L\u25b9S \u2218 manyNoneOf \u2218 S\u25b9L\n\nlookSatHead : (Char \u2192 Bool) \u2192 ParserLC \ud835\udfd9\nlookSatHead p\u1d9c = lookSat p\n    where p : List Char \u2192 Bool\n          p (x \u2237 xs) = p\u1d9c x\n          p []       = false\n\nlookHeadNoneOf : List Char \u2192 ParserLC \ud835\udfd9\nlookHeadNoneOf cs = lookSatHead (\u03bb c \u2192 c `notElem` cs)\n\nlookHeadNoneOf\u02e2 : String \u2192 ParserLC \ud835\udfd9\nlookHeadNoneOf\u02e2 = lookHeadNoneOf \u2218 S\u25b9L\n\nlookHeadSomeOf : List Char \u2192 ParserLC \ud835\udfd9\nlookHeadSomeOf cs = lookSatHead (\u03bb c \u2192 c `elem` cs)\n\nlookHeadSomeOf\u02e2 : String \u2192 ParserLC \ud835\udfd9\nlookHeadSomeOf\u02e2 = lookHeadSomeOf \u2218 S\u25b9L\n\nmodule _ {A} where\n    between : \u2200 {_V} (begin end : ParserLC _V) \u2192 Endo (ParserLC A)\n    between begin end p = begin *> p <* end\n\n    between\u1d9c : (begin end : Char) \u2192 Endo (ParserLC A)\n    between\u1d9c begin end = between (char begin) (char end)\n\n    between\u02e2 : (begin end : String) \u2192 Endo (ParserLC A)\n    between\u02e2 begin end = between (string begin) (string end)\n\n    parens   = between\u1d9c '(' ')'\n    braces   = between\u1d9c '{' '}'\n    brackets = between\u1d9c '[' ']'\n    angles   = between\u1d9c '<' '>'\n    oxford   = between\u1d9c '\u27e6' '\u27e7'\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Text\/Parser\/Heap.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6cad5d969ef02deb0f2a829e45be27ff210452c6","subject":"Comments in WhyDependentTypesMatter","message":"Comments in WhyDependentTypesMatter\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/WhyDependentTypesMatter.agda","new_file":"agda\/WhyDependentTypesMatter.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"5ab348198754f43b1a227c2f20a0a9130abfa3d7","subject":"Added MinBC.agda example.","message":"Added MinBC.agda example.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/Min\/MinBC.agda","new_file":"notes\/FOT\/FOTC\/Program\/Min\/MinBC.agda","new_contents":"------------------------------------------------------------------------------\n-- Example of a partial function using the Bove-Capretta method\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From: Ana Bove and Venanzio Capretta. Nested general recursion and\n-- partiality in type theory. vol 2152 LNCS. 2001.\n\nmodule FOT.FOTC.Program.Min.MinBC where\n\nopen import Data.Nat\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\ndata minAcc : (\u2115 \u2192 \u2115) \u2192 Set where\n  minacc0 : (f : \u2115 \u2192 \u2115) \u2192 f 0 \u2261 0 \u2192 minAcc f\n  minacc1 : (f : \u2115 \u2192 \u2115) \u2192 f 0 \u2262 0 \u2192 minAcc (\u03bb n \u2192 f (suc n)) \u2192 minAcc f\n\nmin : (f : \u2115 \u2192 \u2115) \u2192 minAcc f \u2192 \u2115\nmin f (minacc0 .f q)   = 0\nmin f (minacc1 .f q h) = suc (min (\u03bb n \u2192 f (suc n)) h)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Program\/Min\/MinBC.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"27acbf899d88340ad1d03042fa744d1a3b783e0c","subject":"Implement change structure for sum types","message":"Implement change structure for sum types\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Sums.agda","new_file":"Base\/Change\/Sums.agda","new_contents":"module Base.Change.Sums where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\nopen import Base.Change.Equivalence\nopen import Postulate.Extensionality\nopen import Data.Sum\n\nmodule SumChanges \u2113 (X Y : Set \u2113) {{CX : ChangeAlgebra \u2113 X}} {{CY : ChangeAlgebra \u2113 Y}} where\n  open \u2261-Reasoning\n\n  data SumChange : X \u228e Y \u2192 Set \u2113 where\n    ch\u2081 : \u2200 {x} \u2192 (dx : \u0394 x) \u2192 SumChange (inj\u2081 x)\n    rp\u2081\u2082 : \u2200 {x} \u2192 (y : Y) \u2192 SumChange (inj\u2081 x)\n    ch\u2082 : \u2200 {y} \u2192 (dy : \u0394 y) \u2192 SumChange (inj\u2082 y)\n    rp\u2082\u2081 : \u2200 {y} \u2192 (x : X) \u2192 SumChange (inj\u2082 y)\n\n  _\u2295_ : (v : X \u228e Y) \u2192 SumChange v \u2192 X \u228e Y\n  inj\u2081 x \u2295 ch\u2081 dx = inj\u2081 (x \u229e dx)\n  inj\u2082 y \u2295 ch\u2082 dy = inj\u2082 (y \u229e dy)\n  inj\u2081 x \u2295 rp\u2081\u2082 y = inj\u2082 y\n  inj\u2082 y \u2295 rp\u2082\u2081 x = inj\u2081 x\n\n  _\u229d_ : \u2200 (v\u2082 v\u2081 : X \u228e Y) \u2192 SumChange v\u2081\n  inj\u2081 x\u2082 \u229d inj\u2081 x\u2081 = ch\u2081 (x\u2082 \u229f x\u2081)\n  inj\u2082 y\u2082 \u229d inj\u2082 y\u2081 = ch\u2082 (y\u2082 \u229f y\u2081)\n  inj\u2082 y\u2082 \u229d inj\u2081 x\u2081 = rp\u2081\u2082 y\u2082\n  inj\u2081 x\u2082 \u229d inj\u2082 y\u2081 = rp\u2082\u2081 x\u2082\n\n  s-nil : (v : X \u228e Y) \u2192 SumChange v\n  s-nil (inj\u2081 x) = ch\u2081 (nil x)\n  s-nil (inj\u2082 y) = ch\u2082 (nil y)\n\n  s-update-diff : \u2200 (u v : X \u228e Y) \u2192 v \u2295 (u \u229d v) \u2261 u\n  s-update-diff (inj\u2081 x\u2082) (inj\u2081 x\u2081) = cong inj\u2081 (update-diff x\u2082 x\u2081)\n  s-update-diff (inj\u2082 y\u2082) (inj\u2082 y\u2081) = cong inj\u2082 (update-diff y\u2082 y\u2081)\n  s-update-diff (inj\u2081 x\u2082) (inj\u2082 y\u2081) = refl\n  s-update-diff (inj\u2082 y\u2082) (inj\u2081 x\u2081) = refl\n\n  s-update-nil : \u2200 v \u2192 v \u2295 (s-nil v) \u2261 v\n  s-update-nil (inj\u2081 x) = cong inj\u2081 (update-nil x)\n  s-update-nil (inj\u2082 y) = cong inj\u2082 (update-nil y)\n\n  changeAlgebra : ChangeAlgebra \u2113 (X \u228e Y)\n  changeAlgebra = record\n    { Change = SumChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; nil = s-nil\n    ; isChangeAlgebra = record\n      { update-diff = s-update-diff\n      ; update-nil = s-update-nil\n      }\n    }\n\n  inj\u2081\u2032 : (x : X) \u2192 (dx : \u0394 x) \u2192 \u0394 (inj\u2081 x)\n  inj\u2081\u2032 x dx = ch\u2081 dx\n\n  inj\u2081\u2032Derivative : Derivative inj\u2081 inj\u2081\u2032\n  inj\u2081\u2032Derivative x dx = refl\n\n  inj\u2082\u2032 : (y : Y) \u2192 (dy : \u0394 y) \u2192 \u0394 (inj\u2082 y)\n  inj\u2082\u2032 y dy = ch\u2082 dy\n\n  inj\u2082\u2032Derivative : Derivative inj\u2082 inj\u2082\u2032\n  inj\u2082\u2032Derivative y dy = refl\n\n  -- Elimination form for sums. This is a less dependently-typed version of\n  -- [_,_].\n  match : \u2200 {Z : Set \u2113} \u2192 (X \u2192 Z) \u2192 (Y \u2192 Z) \u2192 X \u228e Y \u2192 Z\n  match f g (inj\u2081 x) = f x\n  match f g (inj\u2082 y) = g y\n\n  module _ {Z : Set \u2113} {{CZ : ChangeAlgebra \u2113 Z}} where\n    X\u2192Z = FunctionChanges.changeAlgebra X Z\n    --module \u0394X\u2192Z = FunctionChanges X Z {{CX}} {{CZ}}\n    Y\u2192Z = FunctionChanges.changeAlgebra Y Z\n    --module \u0394Y\u2192Z = FunctionChanges Y Z {{CY}} {{CZ}}\n    X\u228eY\u2192Z = FunctionChanges.changeAlgebra (X \u228e Y) Z\n    Y\u2192Z\u2192X\u228eY\u2192Z = FunctionChanges.changeAlgebra (Y \u2192 Z) (X \u228e Y \u2192 Z)\n\n    open FunctionChanges using (apply; correct)\n\n    match\u2032\u2080-realizer : (f : X \u2192 Z) \u2192 \u0394 f \u2192 (g : Y \u2192 Z) \u2192 \u0394 g \u2192 (s : X \u228e Y) \u2192 \u0394 s \u2192 \u0394 (match f g s)\n    match\u2032\u2080-realizer f df g dg (inj\u2081 x) (ch\u2081 dx) = apply df x dx\n    match\u2032\u2080-realizer f df g dg (inj\u2081 x) (rp\u2081\u2082 y) = ((g \u229e dg) y) \u229f (f x)\n    match\u2032\u2080-realizer f df g dg (inj\u2082 y) (rp\u2082\u2081 x) = ((f \u229e df) x) \u229f (g y)\n    match\u2032\u2080-realizer f df g dg (inj\u2082 y) (ch\u2082 dy) = apply dg y dy\n\n    match\u2032\u2080-realizer-correct : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192 (s : X \u228e Y) \u2192 (ds : \u0394 s) \u2192 match f g (s \u2295 ds) \u229e match\u2032\u2080-realizer f df g dg (s \u2295 ds) (nil (s \u2295 ds)) \u2261 match f g s \u229e match\u2032\u2080-realizer f df g dg s ds\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2081 x) (ch\u2081 dx) = correct df x dx\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2082 y) (ch\u2082 dy) = correct dg y dy\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2081 x) (rp\u2081\u2082 y) rewrite update-diff ((g \u229e dg) y) (f x) = refl\n    match\u2032\u2080-realizer-correct f df g dg (inj\u2082 y) (rp\u2082\u2081 x) rewrite update-diff ((f \u229e df) x) (g y) = refl\n\n    match\u2032\u2080 : (f : X \u2192 Z) \u2192 \u0394 f \u2192 (g : Y \u2192 Z) \u2192 \u0394 g \u2192 \u0394 (match f g)\n    match\u2032\u2080 f df g dg = record { apply = match\u2032\u2080-realizer f df g dg ; correct = match\u2032\u2080-realizer-correct f df g dg }\n\n    match\u2032-realizer-correct-body : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192 (s : X \u228e Y) \u2192\n      (match f (g \u229e dg) \u229e match\u2032\u2080 f df (g \u229e dg) (nil (g \u229e dg))) s \u2261 (match f g \u229e match\u2032\u2080 f df g dg) s\n\n    match\u2032-realizer-correct-body f df g dg (inj\u2081 x) = refl\n    -- refl doesn't work here. That seems a *huge* bad smell. However, that's simply because we're only updating g, not f\n    match\u2032-realizer-correct-body f df g dg (inj\u2082 y) rewrite update-nil y = update-diff (g y \u229e apply dg y (nil y)) (g y \u229e apply dg y (nil y))\n\n    match\u2032-realizer-correct : (f : X \u2192 Z) \u2192 (df : \u0394 f) \u2192 (g : Y \u2192 Z) \u2192 (dg : \u0394 g) \u2192\n      (match f (g \u229e dg)) \u229e match\u2032\u2080 f df (g \u229e dg) (nil (g \u229e dg)) \u2261 match f g \u229e match\u2032\u2080 f df g dg\n    match\u2032-realizer-correct f df g dg = ext (match\u2032-realizer-correct-body f df g dg)\n    match\u2032 : (f : X \u2192 Z) \u2192 \u0394 f \u2192 \u0394 (match f)\n    match\u2032 f df = record { apply = \u03bb g dg \u2192 match\u2032\u2080 f df g dg ; correct = match\u2032-realizer-correct f df }\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Base\/Change\/Sums.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ea9aaed69820b55adeed093c4b8a2ca5a1cf73de","subject":"circuit.agda","message":"circuit.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"circuit.agda","new_file":"circuit.agda","new_contents":"module circuit where\n\nopen import Function\nopen import Data.Nat.NP hiding (_\u225f_; compare)\nopen import Data.Bits\nopen import Data.Bool hiding (_\u225f_)\nopen import Data.Product hiding (swap; map)\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc; inject+; raise; #_)\nopen import Data.List using (List; []; _\u2237_)\nopen import Data.Vec using (Vec; []; _\u2237_; foldr; _[_]\u2254_; lookup; _++_; splitAt; tabulate; allFin) renaming (map to vmap)\nopen import Relation.Nullary.Decidable hiding (map)\nopen import Relation.Binary.PropositionalEquality\n\nCircuitType : Set\u2081\nCircuitType = (i o : \u2115) \u2192 Set\n\nmodule Rewire where\n  RewireFun : CircuitType\n  RewireFun i o = Fin o \u2192 Fin i\n\n  RewireTbl : CircuitType\n  RewireTbl i o = Vec (Fin i) o\n\n  rewire : \u2200 {a i o} {A : Set a} \u2192 (Fin o \u2192 Fin i) \u2192 Vec A i \u2192 Vec A o\n  rewire f v = tabulate (flip lookup v \u2218 f)\n\n  rewireTbl : \u2200 {a i o} {A : Set a} \u2192 RewireTbl i o \u2192 Vec A i \u2192 Vec A o\n  rewireTbl tbl v = vmap (flip lookup v) tbl\n\n  runRewireFun : \u2200 {i o} \u2192 RewireFun i o \u2192 Bits i \u2192 Bits o\n  runRewireFun = rewire\n\n  runRewireTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 Bits i \u2192 Bits o\n  runRewireTbl = rewireTbl\n\nopen Rewire using (RewireTbl; RewireFun)\n\nrecord RewiringBuilder (C : CircuitType) : Set where\n  constructor mk\n\n  infixr 1 _>>>_\n  infixr 3 _***_\n\n  field\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n\n    _>>>_  : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n\n    _***_  : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n\n  rewireWithTbl : \u2200 {i o} \u2192 RewireTbl i o \u2192 C i o\n  rewireWithTbl = rewire \u2218 flip lookup\n\n  idC : \u2200 {i} \u2192 C i i\n  idC = rewire id\n\n  sink : \u2200 i \u2192 C i 0\n  sink _ = rewire (\u03bb())\n\n  dup : \u2200 o \u2192 C 1 o\n  dup _ = rewire (const zero)\n\n  dup\u2082 : C 1 2\n  dup\u2082 = dup 2\n\n  vcat : \u2200 {i o n} \u2192 Vec (C i o) n \u2192 C (n * i) (n * o)\n  vcat []       = idC\n  vcat (x \u2237 xs) = x *** vcat xs\n\n  coerce : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 i\u2080 \u2261 i\u2081 \u2192 o\u2080 \u2261 o\u2081 \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081\n  coerce refl refl = id\n\n  dup\u207f : \u2200 {n} k \u2192 C n (k * n)\n  dup\u207f {n} k = coerce (proj\u2082 \u2115\u00b0.*-identity n) (\u2115\u00b0.*-comm n k) (vcat {n = n} (replicate (dup _)))\n\n  ext-before : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) (k + o)\n  ext-before {k} c = idC {k} *** c\n\n  ext-after : \u2200 {k i o} \u2192 C i o \u2192 C (i + k) (o + k)\n  ext-after c = c *** idC\n\n  dropC : \u2200 {k i o} \u2192 C i o \u2192 C (k + i) o\n  dropC {k} c = sink k *** c\n\n  dropC' : \u2200 {k i} \u2192 C (k + i) i\n  dropC' {k} = dropC {k} idC\n\n  swap : \u2200 {i} (x y : Fin i) \u2192 C i i\n  -- swap x y = arr (\u03bb xs \u2192 (xs [ x ]\u2254 (lookup y xs)) [ y ]\u2254 (lookup x xs))\n  swap x y = rewire (Fin.swap x y)\n\n  rev : \u2200 {i} \u2192 C i i\n  rev = rewire Fin.reverse\n\n  swap\u2082 : C 2 2\n  swap\u2082 = rev\n  -- swap\u2082 = swap (# 0) (# 1)\n\n  Perm : \u2115 \u2192 Set\n  Perm n = List (Fin n \u00d7 Fin n)\n\n  perm : \u2200 {i} \u2192 Perm i \u2192 C i i\n  perm [] = idC\n  perm ((x , y) \u2237 \u03c0) = swap x y >>> perm \u03c0\n\nrecord CircuitBuilder (C : CircuitType) : Set where\n  constructor mk\n  field\n    isRewiringBuilder : RewiringBuilder C\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n\n  open RewiringBuilder isRewiringBuilder public\n\n  bit : Bit \u2192 C 0 1\n  bit b = arr (const (b \u2237 []))\n\n  0\u02b7 : C 0 1\n  0\u02b7 = bit 0b\n\n  1\u02b7 : C 0 1\n  1\u02b7 = bit 1b\n\n  unOp : (Bit \u2192 Bit) \u2192 C 1 1\n  unOp op = arr (\u03bb { (x \u2237 []) \u2192 (op x) \u2237 [] })\n\n  notC : C 1 1\n  notC = unOp not\n\n  binOp : (Bit \u2192 Bit \u2192 Bit) \u2192 C 2 1\n  binOp op = arr (\u03bb { (x \u2237 y \u2237 []) \u2192 (op x y) \u2237 [] })\n\n  xorC : C 2 1\n  xorC = binOp _xor_\n\n  eqC : C 2 1\n  eqC = binOp _==\u1d47_\n\n  orC : C 2 1\n  orC = binOp _\u2228_\n\n  andC : C 2 1\n  andC = binOp _\u2227_\n\n  norC : C 2 1\n  norC = binOp (\u03bb x y \u2192 not (x \u2228 y))\n\n  nandC : C 2 1\n  nandC = binOp (\u03bb x y \u2192 not (x \u2227 y))\n\n_\u2192\u1d47_ : CircuitType\ni \u2192\u1d47 o = Bits i \u2192 Bits o\n\n_\u2192\u1da0_ : CircuitType\ni \u2192\u1da0 o = Fin o \u2192 Fin i\n\nfinFunRewiringBuilder : RewiringBuilder _\u2192\u1da0_\nfinFunRewiringBuilder = mk id _\u2218\u2032_ _***_\n  where\n    C = _\u2192\u1da0_\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {o\u2080 = o\u2080} f g x with Fin.cmp o\u2080 _ x\n    _***_      f _ ._ | Fin.bound x = inject+ _ (f x)\n    _***_ {i\u2080} _ g ._ | Fin.free  x = raise i\u2080 (g x)\n\ntblRewiringBuilder : RewiringBuilder RewireTbl\ntblRewiringBuilder = mk tabulate _>>>_ _***_\n  where\n    open Rewire\n    C = RewireTbl\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    c\u2080 >>> c\u2081 = rewireTbl c\u2081 c\u2080\n    -- c\u2080 >>> c\u2081 = tabulate (flip lookup c\u2080 \u2218 flip lookup c\u2081)\n    -- c\u2080 >>> c\u2081 = vmap (flip lookup c\u2080) c\u2081\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    _***_ {i\u2080} c\u2080 c\u2081 = vmap (inject+ _) c\u2080 ++ vmap (raise i\u2080) c\u2081\n\nbitsFunRewiringBuilder : RewiringBuilder _\u2192\u1d47_\nbitsFunRewiringBuilder = mk Rewire.rewire _>>>_ _***_\n  where\n    C = _\u2192\u1d47_\n\n    _>>>_ : \u2200 {i m o} \u2192 C i m \u2192 C m o \u2192 C i o\n    f >>> g = g \u2218 f\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) xs with splitAt _ xs\n    ... | ys , zs , _ = f ys ++ g zs\n\nbitsFunCircuitBuilder : CircuitBuilder _\u2192\u1d47_\nbitsFunCircuitBuilder = mk bitsFunRewiringBuilder id\n\nopen import bintree\nopen import flipbased-tree\n\nTreeBits : CircuitType\nTreeBits i o = Tree (Bits o) i\n\nmodule moretree where\n  _>>>_ : \u2200 {m n a} {A : Set a} \u2192 Tree (Bits m) n \u2192 Tree A m \u2192 Tree A n\n  f >>> g = map (flip bintree.lookup g) f\n\ntreeBitsRewiringBuilder : RewiringBuilder TreeBits\ntreeBitsRewiringBuilder = mk rewire moretree._>>>_ _***_\n  where\n    C = TreeBits\n\n    rewire : \u2200 {i o} \u2192 RewireFun i o \u2192 C i o\n    rewire f = fromFun (Rewire.rewire f)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 o\u2080 \u2192 C i\u2081 o\u2081 \u2192 C (i\u2080 + i\u2081) (o\u2080 + o\u2081)\n    (f *** g) = f >>= \u03bb xs \u2192 map (_++_ xs) g\n\ntreeBitsCircuitBuilder : CircuitBuilder TreeBits\ntreeBitsCircuitBuilder = mk treeBitsRewiringBuilder arr\n  where\n    C = TreeBits\n\n    arr : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 C i o\n    arr = fromFun\n\nRewiringTree : CircuitType\nRewiringTree i o = Tree (Fin i) o\n\nmodule RewiringWith2^Outputs where\n    C_\u27e82^_\u27e9 = RewiringTree\n\n    rewire : \u2200 {i o} \u2192 RewireFun i (2 ^ o) \u2192 C i \u27e82^ o \u27e9\n    rewire f = fromFun (f \u2218 toFin)\n\n    lookupFin : \u2200 {i o} \u2192 C i \u27e82^ o \u27e9 \u2192 Fin (2 ^ o) \u2192 Fin i\n    lookupFin c x = bintree.lookup (fromFin x) c\n\n    _>>>_ : \u2200 {i m o} \u2192 C i \u27e82^ m \u27e9 \u2192 C (2 ^ m) \u27e82^ o \u27e9 \u2192 C i \u27e82^ o \u27e9\n    f >>> g = rewire (lookupFin f \u2218 lookupFin g)\n\n    _***_ : \u2200 {i\u2080 i\u2081 o\u2080 o\u2081} \u2192 C i\u2080 \u27e82^ o\u2080 \u27e9 \u2192 C i\u2081 \u27e82^ o\u2081 \u27e9 \u2192 C (i\u2080 + i\u2081) \u27e82^ (o\u2080 + o\u2081) \u27e9\n    f *** g = f >>= \u03bb x \u2192 map (Fin._+\u2032_ x) g\n\nmodule Test where\n  open import Data.Bits.Bits4\n\n  tbl : RewireTbl 4 4\n  tbl = # 1 \u2237 # 0 \u2237 # 2 \u2237 # 2 \u2237 []\n\n  fun : RewireFun 4 4\n  fun zero = # 1\n  fun (suc zero) = # 0\n  fun (suc (suc zero)) = # 2\n  fun (suc (suc (suc x))) = # 2\n\n-- swap x y \u2248 swap y x\n-- reverse \u2218 reverse \u2248 id\n\n  abs : \u2200 {C} \u2192 RewiringBuilder C \u2192 C 4 4\n  abs builder = swap\u2082 *** dup\u2082 *** sink 1\n    where open RewiringBuilder builder\n\n  tinytree : Tree (Fin 4) 2\n  tinytree = fork (fork (leaf (# 1)) (leaf (# 0))) (fork (leaf (# 2)) (leaf (# 2)))\n\n  bigtree : Tree (Bits 4) 4\n  bigtree = fork (fork (fork (same 0000b) (same 0011b)) (fork (same 1000b) (same 1011b)))\n                 (fork (fork (same 0100b) (same 0111b)) (fork (same 1100b) (same 1111b)))\n    where same : \u2200 {n} {A : Set} \u2192 A \u2192 Tree A (suc n)\n          same x = fork (leaf x) (leaf x)\n\n  test\u2081 : tbl \u2261 tabulate fun\n  test\u2081 = refl\n\n  test\u2082 : tbl \u2261 abs tblRewiringBuilder\n  test\u2082 = refl\n\n  test\u2083 : tabulate fun \u2261 tabulate (abs finFunRewiringBuilder)\n  test\u2083 = refl\n\n  test\u2084 : bigtree \u2261 abs treeBitsRewiringBuilder\n  test\u2084 = refl\n\n  open RewiringWith2^Outputs\n  test\u2085 : tabulate (lookupFin tinytree) \u2261 tbl\n  test\u2085 = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'circuit.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c7fa12e2b9c6033d6b382b51db4593141bbd538d","subject":"lib\/Data\/Fin\/NP.agda","message":"lib\/Data\/Fin\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Fin\/NP.agda","new_file":"lib\/Data\/Fin\/NP.agda","new_contents":"module Data.Fin.NP where\n\nopen import Function\nopen import Data.Fin public\nopen import Data.Nat.NP using (\u2115; zero; suc; _<=_; module \u2115\u00b0) renaming (_+_ to _+\u2115_)\nopen import Data.Bool\nopen import Data.Sum as Sum\n\n_+\u2032_ : \u2200 {m n} (x : Fin m) (y : Fin n) \u2192 Fin (m +\u2115 n)\n_+\u2032_ {suc m} {n} zero y rewrite \u2115\u00b0.+-comm (suc m) n = inject+ _ y\nsuc x +\u2032 y = suc (x +\u2032 y)\n\n_==_ : \u2200 {n} (x y : Fin n) \u2192 Bool\nx == y = helper (compare x y) where\n  helper : \u2200 {n} {i j : Fin n} \u2192 Ordering i j \u2192 Bool\n  helper (equal _) = true\n  helper _         = false\n\nswap : \u2200 {i} (x y : Fin i) \u2192 Fin i \u2192 Fin i\nswap x y z = if x == z then y else if y == z then x else z\n\ndata FinSum m n : Fin (m +\u2115 n) \u2192 Set where\n  bound : (x : Fin m) \u2192 FinSum m n (inject+ n x)\n  free  : (x : Fin n) \u2192 FinSum m n (raise m x)\n\nopen import Relation.Binary.PropositionalEquality\n\ncmp : \u2200 m n (x : Fin (m +\u2115 n)) \u2192 FinSum m n x\ncmp zero n x = free x\ncmp (suc m) n zero = bound zero\ncmp (suc m) n (suc x) with cmp m n x\ncmp (suc m) n (suc .(inject+ n x)) | bound x = bound (suc x)\ncmp (suc m) n (suc .(raise m x))   | free x  = free x\n\n-- reverse x = n \u2238 (1 + x)\nreverse : \u2200 {n} \u2192 Fin n \u2192 Fin n\nreverse zero    = from\u2115 _\nreverse (suc x) = inject\u2081 (reverse x)\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Fin.Props renaming (reverse to reverse-old)\n\nreverse-from\u2115 : \u2200 n \u2192 reverse (from\u2115 n) \u2261 zero\nreverse-from\u2115 zero = refl\nreverse-from\u2115 (suc n) rewrite reverse-from\u2115 n = refl\n\nreverse-inject\u2081 : \u2200 {n} (x : Fin n) \u2192 reverse (inject\u2081 x) \u2261 suc (reverse x)\nreverse-inject\u2081 zero = refl\nreverse-inject\u2081 (suc x) rewrite reverse-inject\u2081 x = refl\n\nreverse-involutive : \u2200 {n} (x : Fin n) \u2192 reverse (reverse x) \u2261 x\nreverse-involutive zero = reverse-from\u2115 _\nreverse-involutive (suc x) rewrite reverse-inject\u2081 (reverse x) | reverse-involutive x = refl\n\nreverse-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-lem zero = to-from _\nreverse-lem (suc x) rewrite inject\u2081-lemma (reverse x) = reverse-lem x\n\nto\u2115-\u2115-lem : \u2200 {n} (x : Fin (suc n)) \u2192 to\u2115 (n \u2115- x) \u2261 n \u2238 to\u2115 x\nto\u2115-\u2115-lem zero = to-from _\nto\u2115-\u2115-lem {zero} (suc ())\nto\u2115-\u2115-lem {suc n} (suc x) = to\u2115-\u2115-lem x\n\nreverse-old-lem : \u2200 {n} (x : Fin n) \u2192 to\u2115 (reverse-old x) \u2261 n \u2238 suc (to\u2115 x)\nreverse-old-lem {zero} ()\nreverse-old-lem {suc n} x rewrite inject\u2264-lemma (n \u2115- x) (n\u2238m\u2264n (to\u2115 x) (suc n)) = to\u2115-\u2115-lem x\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Fin\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"98776944ea4983b6faeeced4116bc7798c5e3ebc","subject":"Add Crypto.Sig.Lamport","message":"Add Crypto.Sig.Lamport\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/Lamport.agda","new_file":"Crypto\/Sig\/Lamport.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (_\u2218_)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nimport Crypto.Sig.LamportOneBit\n\nmodule Crypto.Sig.Lamport\n          (#digest  : \u2115)\n          (#seed    : \u2115)\n          (#secret  : \u2115)\n          (#message : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule OTS1 = Crypto.Sig.LamportOneBit #secret #digest hash-secret\n\nH = hash-secret\n\n#seckey1    = 2* #secret\n#pubkey1    = 2* #digest\n#signature1 = #secret\n#seckey     = #message * #seckey1\n#pubkey     = #message * #pubkey1\n#signature  = #message * #signature1\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #seckey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #pubkey\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = map* #message OTS1.verif-key\n\nmodule verifkey (vk : VerifKey) where\n  vk1s = group #message #pubkey1 vk\n\nmodule signkey (sk : SignKey) where\n  sk1s = group #message #seckey1 sk\n  vk   = verif-key sk\n  open verifkey vk public\n\nmodule signature (sig : Signature) where\n  sig1s = group #message #signature1 sig\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sig1s = map OTS1.sign sk1s \u229b m\n    sig   = concat sig1s\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = and (map OTS1.verify vk1s \u229b m \u229b sig1s)\n  module verify where\n    open verifkey  vk  public\n    open signature sig public\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig = lemma\n  where\n    module lemma {#m} sk b m where\n      skL  = take #seckey1 sk\n      skH  = drop #seckey1 sk\n      vkL  = OTS1.verif-key skL\n      vkH  = map* #m OTS1.verif-key skH\n      sigL = OTS1.sign skL b\n      sigH = concat (map OTS1.sign (group #m #seckey1 skH) \u229b m)\n\n    lemma : \u2200 {#m} sk m \u2192 and (map OTS1.verify (group #m #pubkey1 (map* #m OTS1.verif-key sk)) \u229b m \u229b group #m #signature1\n                               (concat (map OTS1.sign (group #m #seckey1 sk) \u229b m))) \u2261 1b\n    lemma sk [] = refl\n    lemma sk (b \u2237 m)\n      rewrite (let open lemma sk b m in take-++ #pubkey1 vkL vkH)\n            | (let open lemma sk b m in drop-++ #pubkey1 vkL vkH)\n            | (let open lemma sk b m in take-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in drop-++ #signature1 sigL sigH)\n            | (let open lemma sk b m in OTS1.verify-correct-sig skL b)\n            | (let open lemma sk b m in lemma skH m)\n            = refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Crypto\/Sig\/Lamport.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e88f09e5f57023f575e645b8f8091a16491dceab","subject":"Added test.","message":"Added test.\n\nIgnore-this: 4a27bd1d22f7045b1df8c402c24da53d\n\ndarcs-hash:20120531151144-3bd4e-4a67b54d15a300e09d0e40094f99c0323e4b60da.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/FOL\/VarNamesLamTerm.agda","new_file":"Test\/Succeed\/FOL\/VarNamesLamTerm.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing an instance of the class VarNames: Lam term\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Test.Succeed.FOL.VarNamesLamTerm where\n\n-- We add 3 to the fixities of the standard library.\ninfixr 8 _\u2237_\ninfixr 7 _,_\ninfix  7 _\u2261_ _\u2248_\ninfixr 5 _\u2227_\n\n------------------------------------------------------------------------------\n\npostulate\n  D      : Set\n  _\u2237_    : D \u2192 D \u2192 D\n  Stream : D \u2192 Set\n  _\u2248_    : D \u2192 D \u2192 Set\n\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\ndata \u2203 (A : D \u2192 Set) : Set where\n  _,_ : (t : D) \u2192 A t \u2192 \u2203 A\n\nsyntax \u2203 (\u03bb x \u2192 e) = \u2203[ x ] e\n\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\npostulate\n  Stream-gfp\u2082 : (P : D \u2192 Set) \u2192\n                -- P is post-fixed point of StreamF.\n                (\u2200 {xs} \u2192 P xs \u2192 \u2203[ x' ] \u2203[ xs' ] P xs' \u2227 xs \u2261 x' \u2237 xs') \u2192\n                -- Stream is greater than P.\n                \u2200 {xs} \u2192 P xs \u2192 Stream xs\n\npostulate\n  \u2248-gfp\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192\n           \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs' \u2248 ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 x' \u2237 ys'\n{-# ATP axiom \u2248-gfp\u2081 #-}\n\n\u2248\u2192Stream : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 Stream ys\n\u2248\u2192Stream {xs} {ys} xs\u2248ys = Stream-gfp\u2082 P helper (xs , xs\u2248ys)\n  where\n  P : D \u2192 Set\n  P zs = \u2203[ ws ] ws \u2248 zs\n  {-# ATP definition P #-}\n\n  postulate\n    helper : \u2200 {zs} \u2192 P zs \u2192 \u2203[ z' ] \u2203[ zs' ] P zs' \u2227 zs \u2261 z' \u2237 zs'\n  {-# ATP prove helper #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/FOL\/VarNamesLamTerm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9c0a33ea3755b9b14ca033d85a6898394e005e34","subject":"+lib\/Relation\/Binary\/Logical\/Iso.agda","message":"+lib\/Relation\/Binary\/Logical\/Iso.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/Logical\/Iso.agda","new_file":"lib\/Relation\/Binary\/Logical\/Iso.agda","new_contents":"module Relation.Binary.Logical.Iso where\n\nopen import Type using (\u2605_)\nopen import Level using () renaming (zero to \u2080)\nopen import Data.Product.NP using (_,_)\nopen import Relation.Binary using (Setoid; module Setoid)\nopen import Function.Inverse.NP using (_$\u2081_;_$\u2082_;id;_\u2208_) renaming (Inverse to _\u2245_; module Inverse to \u2245; inverses to isomorphism)\nopen import Function.Equality as FE using (_\u27f6_; _\u21e8_; _\u27e8$\u27e9_; cong)\n\nmodule _ {a\u2081 a\u2082 a\u1d63} {A\u2081 : Setoid a\u2081 a\u1d63} {A\u2082 : Setoid a\u2082 a\u1d63} (A\u1d63 : A\u2081 \u2245 A\u2082)\n         {b\u2081 b\u2082 b\u1d63} {B\u2081 : Setoid b\u2081 b\u1d63} {B\u2082 : Setoid b\u2082 b\u1d63} (B\u1d63 : B\u2081 \u2245 B\u2082)\n         where\n\n    private\n        module A\u2081\u02e2 = Setoid A\u2081\n        module A\u2082\u02e2 = Setoid A\u2082\n        module B\u2081\u02e2 = Setoid B\u2081\n        module B\u2082\u02e2 = Setoid B\u2082\n        S = A\u2081 \u27f6 B\u2081\n        T = A\u2082 \u27f6 B\u2082\n        to : S \u2192 T\n        to f = \u2245.to B\u1d63 FE.\u2218 f FE.\u2218 \u2245.from A\u1d63\n        from : T \u2192 S\n        from f = \u2245.from B\u1d63 FE.\u2218 f FE.\u2218 \u2245.to A\u1d63\n        to\u2218from : \u2200 (f : T) x \u2192 to (from f) \u27e8$\u27e9 x B\u2082\u02e2.\u2248 f \u27e8$\u27e9 x\n        to\u2218from f x = B\u2082\u02e2.trans (\u2245.right-inverse-of B\u1d63 (f \u27e8$\u27e9 (A\u1d63 $\u2081 (A\u1d63 $\u2082 x)))) (cong f (\u2245.right-inverse-of A\u1d63 x))\n        from\u2218to : \u2200 (f : S) x \u2192 from (to f) \u27e8$\u27e9 x B\u2081\u02e2.\u2248 f \u27e8$\u27e9 x\n        from\u2218to f x = B\u2081\u02e2.trans (\u2245.left-inverse-of B\u1d63 (f \u27e8$\u27e9 (A\u1d63 $\u2082 (A\u1d63 $\u2081 x)))) (cong f (\u2245.left-inverse-of A\u1d63 x))\n\n{-\n    infixr 1 _\u27ea\u2192\u27eb_\n\n    _\u27ea\u2192\u27eb_ : (A\u2081 \u21e8 B\u2081) \u2245 (A\u2082 \u21e8 B\u2082)\n    _\u27ea\u2192\u27eb_ = record { to = record { _\u27e8$\u27e9_ = to\n                                 ; cong = {!!} }\n                   ; from = record { _\u27e8$\u27e9_ = from\n                                   ; cong = {!!} }\n                   ; inverse-of = record { left-inverse-of = \u03bb f s \u2192 {!from\u2218to f ? !}\n                                         ; right-inverse-of = {!!} } }\n-}\n\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nopen import Data.Two using (\ud835\udfda; 0'; 1'; not)\nopen import Data.Bool.Properties\n\n\ud835\udfda\u02e2 : Setoid \u2080 \u2080\n\ud835\udfda\u02e2 = \u2261.setoid \ud835\udfda\n\u27ea\ud835\udfda\u27eb : \ud835\udfda\u02e2 \u2245 \ud835\udfda\u02e2\n\u27ea\ud835\udfda\u27eb = id\n\u27ea0'\u27eb : (0' , 0') \u2208 \u27ea\ud835\udfda\u27eb\n\u27ea0'\u27eb = refl\n\u27ea1'\u27eb : (1' , 1') \u2208 \u27ea\ud835\udfda\u27eb\n\u27ea1'\u27eb = refl\n--\u27eanot\u27eb' : (\u0394 (\u2192-to-\u27f6 not)) \u2208 (\u27ea\ud835\udfda\u27eb \u27ea\u2192\u27eb \u27ea\ud835\udfda\u27eb)\n--\u27eanot\u27eb' refl = refl\n\n-- 'not' is an isomorphism on '\ud835\udfda' and so can be used as an \u201cequality\u201d on '\ud835\udfda'\n\u27eanot\u27eb : \ud835\udfda\u02e2 \u2245 \ud835\udfda\u02e2\n\u27eanot\u27eb = isomorphism not not not-involutive not-involutive\n\n\u27ea0'1'\u27eb : (0' , 1') \u2208 \u27eanot\u27eb\n\u27ea0'1'\u27eb = refl\n\u27ea1'0'\u27eb : (0' , 1') \u2208 \u27eanot\u27eb\n\u27ea1'0'\u27eb = refl\n\n--\u27eanot\u27eb'' : (\u0394 (\u2192-to-\u27f6 not)) \u2208 (\u27eanot\u27eb \u27ea\u2192\u27eb \u27eanot\u27eb)\n--\u27eanot\u27eb'' refl = not-involutive _\n\n-- since \ud835\udfda\u02b3 is not reflexive it cannot be an equivalence relation and\n-- thus cannot be used to build a setoid.\ndata \ud835\udfda\u02b3 : \ud835\udfda \u2192 \ud835\udfda \u2192 \u2605 \u2080 where\n  0'1' : \ud835\udfda\u02b3 0' 1'\n  1'0' : \ud835\udfda\u02b3 1' 0'\n\nopen import Data.Nat.NP using (\u2115; zero; suc; \u2115\u02e2)\n\u27ea\u2115\u27eb : \u2115\u02e2 \u2245 \u2115\u02e2\n\u27ea\u2115\u27eb = id\n\u27eazero\u27eb : (zero , zero) \u2208 \u27ea\u2115\u27eb\n\u27eazero\u27eb = refl\n--\u27easuc\u27eb : (\u2192-to-\u27f6 suc , \u2192-to-\u27f6 suc) \u2208 (\u27ea\u2115\u27eb \u27ea\u2192\u27eb \u27ea\u2115\u27eb)\n--\u27easuc\u27eb refl = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Relation\/Binary\/Logical\/Iso.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fb674e513254ccae5e4d3fc11299283a0101360d","subject":"A first step towards a field solver","message":"A first step towards a field solver\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/Field\/Solver.agda","new_file":"lib\/Algebra\/Field\/Solver.agda","new_contents":"import Algebra.Field as Field renaming (Field-Ops to RawField ; module Field-Ops to RawField)\nimport Algebra.RingSolver as RS\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Level\nopen import Data.Nat as Nat using (\u2115 ; suc ; zero)\nopen import Data.Fin\nopen import Data.One\nopen import Data.Product\n\nopen import Function\n\nmodule Algebra.Field.Solver\n  {f}{F : Set f}\n  (\ud835\udd3d : Field.RawField F) where\n\nmodule \ud835\udd3d = Field.RawField \ud835\udd3d\n\n\ndata R-Op : Set where\n  [+] [*] [-] : R-Op\n\ndata F-Op : Set where\n  [+] [*] [-] [\/] : F-Op\n\nF-Op-eval : F-Op \u2192 F \u2192 F \u2192 F\nF-Op-eval [+] x y = x \ud835\udd3d.+ y\nF-Op-eval [*] x y = x \ud835\udd3d.* y\nF-Op-eval [-] x y = x \ud835\udd3d.\u2212 y\nF-Op-eval [\/] x y = x \ud835\udd3d.\/ y\n\nR-include : R-Op \u2192 F-Op\nR-include [+] = [+]\nR-include [*] = [*]\nR-include [-] = [-]\n\nR-Op-eval : R-Op \u2192 F \u2192 F \u2192 F\nR-Op-eval op = F-Op-eval (R-include op)\n\n\ndata Tm (Op : Set)(m : \u2115) : Set f where\n  op : (o : Op) (t\u2081 t\u2082 : Tm Op m) \u2192 Tm Op m\n  con : F \u2192 Tm Op m\n  var : Fin m \u2192 Tm Op m\n\ninfixl 6 _:+_ _:-_\ninfixl 7 _:*_ _:\/_\npattern _:+_ x y = op [+] x y\npattern _:*_ x y = op [*] x y\npattern _:-_ x y = op [-] x y\npattern _:\/_ x y = op [\/] x y\n\nmodule _ {m}{Op : Set}(evalOp : Op \u2192 F \u2192 F \u2192 F)(\u03c1 : Fin m \u2192 F) where\n  eval : Tm Op m \u2192 F\n  eval (op o tm tm\u2081) = evalOp o (eval tm) (eval tm\u2081)\n  eval (con x) = x\n  eval (var x) = \u03c1 x\n\ninfix 2 _\/_:[_,_]\nrecord Div {m} \u03c1 (tm : Tm F-Op m) : Set f where\n  constructor _\/_:[_,_]\n  field\n    N : Tm R-Op m\n    D : Tm R-Op m\n    .nonZero-D : eval R-Op-eval \u03c1 D \u2262 \ud835\udd3d.0\u1da0\n    .is-Correct : eval F-Op-eval \u03c1 tm \u2261 eval R-Op-eval \u03c1 N \ud835\udd3d.\/ eval R-Op-eval \u03c1 D\n\n\nmodule _ (FS : Field.Field-Struct \ud835\udd3d) where\n  open Field.Field-Struct {{...}}\n\n  {-\n  toRingOp : \u2200 {m} \u2192 F-Op \u2192 Div {m} \u2192 Div {m} \u2192 Div {m}\n  toRingOp [+] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* D' :+ N' :* D  \/ D :* D'\n                                                :[ (\u03bb \u03c1 \u2192 noZeroDivisor (P \u03c1) (P' \u03c1) ) ]\n  toRingOp [*] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* N' \/ D :* D' :[ (\u03bb \u03c1 \u2192 noZeroDivisor (P \u03c1) (P' \u03c1)) ]\n  toRingOp [-] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* D' :- N' :* D \/ D :* D'\n                                                :[ (\u03bb \u03c1 \u2192 noZeroDivisor (P \u03c1) (P' \u03c1)) ]\n  toRingOp [\/] (N \/ D :[ P ]) (N' \/ D' :[ P' ]) = N :* D' \/ N' :* D :[ {!!} ]\n  -}\n\n\n  module _ {m} (\u03c1 : Fin m \u2192 F) where\n    Assumptions : Tm F-Op m \u2192 Set f\n    Assumptions (op [+] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081\n    Assumptions (op [*] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081\n    Assumptions (op [-] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081\n    Assumptions (op [\/] tm tm\u2081) = Assumptions tm \u00d7 Assumptions tm\u2081 \u00d7 eval F-Op-eval \u03c1 tm\u2081 \u2262 0\u1da0\n    Assumptions (con x) = Lift \ud835\udfd9\n    Assumptions (var x) = Lift \ud835\udfd9\n\n    toRing : \u2200 (tm : Tm F-Op m) \u2192 Assumptions tm \u2192 Div {m} \u03c1 tm\n    toRing (op [+] tm tm\u2081) (atm , atm\u2081) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = N :* D\u2081 :+ N\u2081 :* D \/ D :* D\u2081\n                                               :[ noZeroDivisor nD nD\u2081 , += isC isC\u2081 \u2219 +-quotient nD nD\u2081 ]\n    toRing (op [*] tm tm\u2081) (atm , atm\u2081) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = N :* N\u2081 \/ D :* D\u2081\n                                               :[ noZeroDivisor nD nD\u2081 , *= isC isC\u2081 \u2219 *-quotient nD nD\u2081 ]\n    toRing (op [-] tm tm\u2081) (atm , atm\u2081) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = N :* D\u2081 :- N\u2081 :* D \/ D :* D\u2081\n                                               :[ noZeroDivisor nD nD\u2081 , \u2212= isC isC\u2081 \u2219 \u2212-quotient nD nD\u2081 ]\n    toRing (op [\/] tm tm\u2081) (atm , atm\u2081 , tm\u2081\/=0) with toRing tm atm | toRing tm\u2081 atm\u2081\n    ... | N \/ D :[ nD , isC ] | N\u2081 \/ D\u2081 :[ nD\u2081 , isC\u2081 ] = (N :* D\u2081) \/ (D :* N\u2081)\n                                               :[ noZeroDivisor nD nN\u2081 , \/= isC isC\u2081 \u2219 \/-quotient nN\u2081 nD nD\u2081 ]\n      where .nN\u2081 : eval R-Op-eval \u03c1 N\u2081 \u2262 0\u1da0\n            nN\u2081 p = tm\u2081\/=0 (isC\u2081 \u2219 *= p refl \u2219 0*-zero)\n    toRing (con x) _ = con x \/ con \ud835\udd3d.1\u1da0 :[ 0\u22621 \u2218 !_ , ! (*= refl 1\u207b\u00b9\u22611 \u2219 *1-identity) ]\n    toRing (var x) _ = var x \/ con \ud835\udd3d.1\u1da0 :[ 0\u22621 \u2218 !_ , ! (*= refl 1\u207b\u00b9\u22611 \u2219 *1-identity) ]\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/Field\/Solver.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"799dea6615e2c558c80d8a2b706f680108c6a947","subject":"added agda model of list-style datatypes","message":"added agda model of list-style datatypes\n","repos":"larrytheliquid\/pigit","old_file":"models\/LLev.agda","new_file":"models\/LLev.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule LLev where\n\n--****************\n-- Universe polymorphism\n--****************\n\ndata Level : Set where\n      ze : Level\n      su  : Level -> Level\n\n{-# BUILTIN LEVEL     Level #-}\n{-# BUILTIN LEVELZERO ze  #-}\n{-# BUILTIN LEVELSUC  su   #-}\n\nmax : Level -> Level -> Level\nmax  ze     m     = m\nmax (su n)  ze    = su n\nmax (su n) (su m) = su (max n m)\n\n{-# BUILTIN LEVELMAX max #-}\n\nrecord Up {l : Level} (A : Set l) : Set (su l) where\n  constructor up\n  field down : A\nopen Up\n\nrecord Sg {l : Level}(S : Set l)(T : S -> Set l) : Set l where\n  constructor _,_\n  field\n    fst : S\n    snd : T fst\nopen Sg\n_*_ : {l : Level} -> Set l -> Set l -> Set l\nS * T = Sg S \\ _ -> T\ninfixr 4 _*_ _,_\n\ndata Zero : Set where\n\nrecord One {l : Level} : Set l where\n  constructor <>\n\ndata Desc {l : Level}(I : Set l) : Set (su l) where\n  var : (i : I) -> Desc I\n  con : (A : Set l) -> Desc I\n  sg pi : (S : Set l)(T : S -> Desc I) -> Desc I\n  _**_ : (S T : Desc I) -> Desc I\ninfixr 4 _**_\n\n[!_!] : {l : Level}{I : Set l} -> Desc I -> (I -> Set l) -> Set l\n[! var i !] X = X i\n[! con A !] X = A\n[! sg S T !] X = Sg S \\ s -> [! T s !] X\n[! pi S T !] X = (s : S) -> [! T s !] X\n[! S ** T !] X = [! S !] X * [! T !] X\n\n{-\ndata Mu {l : Level}(I : Set l)(F : I -> Desc I)(i : I) : Set l where\n  <_> : [! F i !] (Mu I F) -> Mu I F i\n-}\n\nAll : {l : Level}{I : Set l}\n      (D : Desc I)(X : I -> Set l) -> [! D !] X -> Desc (Sg I X)\nAll (var i) X x = var (i , x)\nAll (con A) X d = con One\nAll (sg S T) X (s , t) = All (T s) X t\nAll (pi S T) X f = pi S \\ s -> All (T s) X (f s)\nAll (S ** T) X (s , t) = All S X s ** All T X t\n\nall : {l : Level}{I : Set l}\n      (D : Desc I)(X : I -> Set l)(P : Sg I X -> Set l) ->\n      ((ix : Sg I X) -> P ix) ->\n      (d : [! D !] X) -> [! All D X d !] P\nall (var i) X P p x = p (i , x)\nall (con A) X P p d = _\nall (sg S T) X P p (s , t) = all (T s) X P p t\nall (pi S T) X P p f = \\ s -> all (T s) X P p (f s)\nall (S ** T) X P p (s , t) = all S X P p s , all T X P p t\n\n{-\ninduction :  {l : Level}{I : Set l}{F : I -> Desc I}\n             (P : Sg I (Mu I F) -> Set l) ->\n             ((i : I)(d : [! F i !] (Mu I F)) ->\n               [! All (F i) (Mu I F) d !] P -> P (i , < d >)) ->\n             (ix : Sg I (Mu I F)) -> P ix\ninduction {F = F} P p (i , < d >) = p i d (all (F i) (Mu _ F) P (induction P p) d)\n-}\n\n{-\nmutual\n  induction :  {l : Level}{I : Set l}{F : I -> Desc I}\n               (P : Sg I (Mu I F) -> Set l) ->\n               ((i : I)(d : [! F i !] (Mu I F)) ->\n                 [! All (F i) (Mu I F) d !] P -> P (i , < d >)) ->\n               {i : I}(x : Mu I F i) -> P (i , x)\n  induction {F = F} P p {i} < d > = p i d (allInduction F P p (F i) d)\n  allInduction : {l : Level}{I : Set l}(F : I -> Desc I)\n                 (P : Sg I (Mu I F) -> Set l) ->\n                 ((i : I)(d : [! F i !] (Mu I F)) ->\n                    [! All (F i) (Mu I F) d !] P -> P (i , < d >)) ->\n                 (D : Desc I) ->\n                 (d : [! D !] (Mu I F)) -> [! All D (Mu I F) d !] P\n  allInduction F P p (var i) d = induction P p d\n  allInduction F P p (con A) d = _\n  allInduction F P p (sg S T) (s , t) = allInduction F P p (T s) t\n  allInduction F P p (pi S T) f = \\ s -> allInduction F P p (T s) (f s)\n  allInduction F P p (S ** T) (s , t) = allInduction F P p S s , allInduction F P p T t\n-}\n\ndata List {l : Level}(X : Set l) : Set l where\n  [] : List X\n  _::_ : X -> List X -> List X\ninfixr 3 _::_\n\nmap : {k l : Level}{X : Set k}{Y : Set l} -> (X -> Y) -> List X -> List Y\nmap f [] = []\nmap f (x :: xs) = f x :: map f xs\n\ndata UId {l : Level} : Set l where\n  ze : UId\n  su : UId {l} -> UId\n\ndata # {l : Level} : List {l} UId -> Set l where\n  ze : forall {x xs} -> # (x :: xs)\n  su : forall {x xs} -> # xs -> # (x :: xs)\n\nConstrs : {l : Level} -> Set l -> Set (su l)\nConstrs I = List (Up UId * Desc I)\n\nConstr : {l : Level}{I : Set l} -> Constrs I -> Set l\nConstr uDs = # (map (\\ uD -> down (fst uD)) uDs)\n\nConD : {l : Level}{I : Set l}(uDs : Constrs I) -> Constr uDs -> Desc I\nConD [] ()\nConD (uD :: _)  ze     = snd uD\nConD (_ :: uDs) (su c) = ConD uDs c\n\ndata Data {l : Level}{I : Set l}(F : I -> Constrs I)(i : I) : Set l where\n  _\/_ : (u : Constr (F i))(d : [! ConD (F i) u !] (Data F)) -> Data F i\n\n{-\nDataD : {l : Level}{I : Set l} -> Constrs I -> Desc I\nDataD uDs = sg (Constr uDs) (ConD uDs)\n\nData : {l : Level}{I : Set l}(F : I -> Constrs I)(i : I) -> Set l\nData F = Mu _ \\ i -> DataD (F i)\n-}\n\nMethodD : {l : Level}{I : Set l}(X : I -> Set l)\n          (uDs : Constrs I)\n          (P : Sg I X -> Set l)\n          (i : I) -> ((u : Constr uDs)(d : [! ConD uDs u !] X) -> X i) -> Set l\nMethodD X [] P i c = One\nMethodD X ((u , D) :: uDs) P i c\n  = ((d : [! D !] X) -> [! All D X d !] P -> P (i , c ze d))\n  * MethodD X uDs P i (\\ u d -> c (su u) d)\n\nmutual\n  induction : {l : Level}{I : Set l}{F : I -> Constrs I}{i : I}(x : Data F i) \n              (P : Sg I (Data F) -> Set l)\n              (ps : (i : I) -> MethodD (Data F) (F i) P i _\/_)\n              -> P (i , x)\n  induction {F = F}{i = i}(u \/ d) P ps = indMethod P ps (F i) _\/_ (ps i) u d\n  indMethod : {l : Level}{I : Set l}{F : I -> Constrs I}{i : I}\n              (P : Sg I (Data F) -> Set l)\n              (ps : (i : I) -> MethodD (Data F) (F i) P i _\/_)\n              (uDs : Constrs I)\n              (c : (u : Constr uDs) -> [! ConD uDs u !] (Data F) -> Data F i)\n              (ms : MethodD (Data F) uDs P i c)\n              (u : Constr uDs)\n              (d : [! ConD uDs u !] (Data F))\n              -> P (i , c u d)\n  indMethod P ps [] c ms () d\n  indMethod P ps ((u , D) :: _) c (p , ms) ze d = p d (indHyps P ps D d)\n  indMethod P ps ((_ , _) :: uDs) c (m , ms) (su u) d\n    = indMethod P ps uDs _ ms u d\n  indHyps : {l : Level}{I : Set l}{F : I -> Constrs I}\n            (P : Sg I (Data F) -> Set l)\n            (ps : (i : I) -> MethodD (Data F) (F i) P i _\/_)\n            (D : Desc I)\n            (d : [! D !] (Data F))\n            -> [! All D (Data F) d !] P\n  indHyps P ps (var i) x = induction x P ps\n  indHyps P ps (con A) a = _\n  indHyps P ps (sg S T) (s , t) = indHyps P ps (T s) t\n  indHyps P ps (pi S T) f = \\ s -> indHyps P ps (T s) (f s)\n  indHyps P ps (S ** T) (s , t) = indHyps P ps S s , indHyps P ps T t\n\nzi : {l : Level} -> UId {l}\nzi = ze\n\nsi : {l : Level} -> UId {l}\nsi = su ze\n\ndata UQ {l : Level}(x : UId {l}) : UId {l} -> Set l where\n  yes : UQ x x\n  no  : {y : UId {l}} -> UQ x y\n\nuq : {l : Level}(x y : UId {l}) -> UQ x y\nuq ze ze = yes\nuq ze (su y) = no\nuq (su x) ze = no\nuq (su x) (su y) with uq x y\nuq (su x) (su .x) | yes = yes\nuq (su x) (su y)  | no  = no\n\nUIn : {l : Level}(x : UId {l}) -> List (UId {l}) -> Set\nUIn x [] = Zero\nUIn x (y :: ys) with uq x y\nUIn x (.x :: _) | yes = One\nUIn x (_ :: ys) | no = UIn x ys\n\nuin : {l : Level}(x : UId {l})(ys : List (UId {l})) -> UIn x ys -> # ys\nuin x [] ()\nuin x (y :: ys) p with uq x y\nuin x (.x :: _) _ | yes = ze\nuin x (_ :: ys) p | no  = su (uin x ys p)\n\n_!_ : {l : Level}{I : Set l}{F : I -> Constrs I}{i : I} ->\n      let us = map (\\ uD -> down (fst uD)) (F i) in\n      (u : UId {l}){p : UIn u us} ->\n      [! ConD (F i) (uin u us p) !] (Data F) ->\n      Data F i\n_!_ {l}{I}{F}{i} u {p} d = uin u (map (\\ uD -> down (fst uD)) (F i)) p \/ d\n\ninfixr 3 _!_\n\n\nNAT : {l : Level} -> One {l} -> Constrs {l} One\nNAT _ = up zi , con One \n     :: up si , var _ ** con One\n     :: []\n\nZE : {l : Level} -> Data {l} NAT <>\nZE = zi ! <>\n\nSU : {l : Level} -> Data {l} NAT <> -> Data {l} NAT <>\nSU n = si ! n , <>\n\nvari : {l : Level} -> UId {l}\nvari = ze\nconi : {l : Level} -> UId {l}\nconi = su ze\nsgi : {l : Level} -> UId {l}\nsgi = su (su ze)\npii : {l : Level} -> UId {l}\npii = su (su (su ze))\nasti : {l : Level} -> UId {l}\nasti = su (su (su (su ze)))\n\nDESC : {l : Level}(I : Set l) -> One {su l} -> Constrs {su l} One\nDESC I _ = up vari , con (Up I) ** con One\n        :: up coni , con (Set _) ** con One\n        :: up sgi , (sg (Set _) \\ S -> (pi (Up S) \\ _ -> var _) ** con One)\n        :: up pii , (sg (Set _) \\ S -> (pi (Up S) \\ _ -> var _) ** con One)\n        :: up asti , var _ ** var _ ** con One\n        :: []\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/LLev.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"bb09f298de2551b1970837adf473a34d86a73c50","subject":"clean up","message":"clean up\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1d65636ce909fb65b816a417e4121440292c192b","subject":"Added a example of combined proof instantiating the induction principle.","message":"Added a example of combined proof instantiating the induction principle.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/PA\/Axiomatic\/Standard\/AddComm.agda","new_file":"notes\/FOT\/PA\/Axiomatic\/Standard\/AddComm.agda","new_contents":"------------------------------------------------------------------------------\n-- Commutativity of addition\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.PA.Axiomatic.Standard.AddComm where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import PA.Axiomatic.Standard.Base\n\n------------------------------------------------------------------------------\n-- Auxiliary functions\n\nsuccCong : \u2200 {m n} \u2192 m \u2261 n \u2192 succ m \u2261 succ n\nsuccCong = cong succ\n\n+-leftIdentity : \u2200 n \u2192 zero + n \u2261 n\n+-leftIdentity = PA\u2083\n\n+-rightIdentity : \u2200 n \u2192 n + zero \u2261 n\n+-rightIdentity = PA-ind A A0 is\n  where\n  A : M \u2192 Set\n  A i = i + zero \u2261 i\n\n  A0 : A zero\n  A0 = +-leftIdentity zero\n\n  is : \u2200 i \u2192 A i \u2192 A (succ i)\n  is i ih = succ i + zero   \u2261\u27e8 PA\u2084 i zero \u27e9\n            succ (i + zero) \u2261\u27e8 succCong ih \u27e9\n            succ i          \u220e\n\nx+Sy\u2261S[x+y] : \u2200 m n \u2192 m + succ n \u2261 succ (m + n)\nx+Sy\u2261S[x+y] m n = PA-ind A A0 is m\n  where\n  A : M \u2192 Set\n  A i = i + succ n \u2261 succ (i + n)\n\n  A0 : A zero\n  A0 = zero + succ n   \u2261\u27e8 +-leftIdentity (succ n) \u27e9\n       succ n          \u2261\u27e8 succCong (sym (+-leftIdentity n)) \u27e9\n       succ (zero + n) \u220e\n\n  is : \u2200 i \u2192 A i \u2192 A (succ i)\n  is i ih = succ i + succ n     \u2261\u27e8 PA\u2084 i (succ n) \u27e9\n            succ (i + succ n)   \u2261\u27e8 succCong ih \u27e9\n            succ (succ (i + n)) \u2261\u27e8 succCong (sym (PA\u2084 i n)) \u27e9\n            succ (succ i + n)   \u220e\n\n------------------------------------------------------------------------------\n-- Interactive proof\n\n+-comm\u2081 : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm\u2081 m n = PA-ind A A0 is m\n  where\n  A : M \u2192 Set\n  A i = i + n \u2261 n + i\n\n  A0 : A zero\n  A0 = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n       n        \u2261\u27e8 sym (+-rightIdentity n) \u27e9\n       n + zero \u220e\n\n  is : \u2200 i \u2192 A i \u2192 A (succ i)\n  is i ih = succ i + n   \u2261\u27e8 PA\u2084 i n \u27e9\n            succ (i + n) \u2261\u27e8 succCong ih \u27e9\n            succ (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n i) \u27e9\n            n + succ i   \u220e\n\n------------------------------------------------------------------------------\n-- Combined proof\n\n+-comm\u2082 : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm\u2082 m n = PA-ind A A0 is m\n  where\n  A : M \u2192 Set\n  A i = i + n \u2261 n + i\n  {-# ATP definition A #-}\n\n  postulate A0 : A zero\n  {-# ATP prove A0 +-rightIdentity #-}\n\n  postulate is : \u2200 i \u2192 A i \u2192 A (succ i)\n  {-# ATP prove is x+Sy\u2261S[x+y] #-}\n\n------------------------------------------------------------------------------\n-- Interactive proof instantiating the induction principle\n\n+-comm-ind : \u2200 n \u2192\n             (zero + n \u2261 n + zero) \u2192\n             (\u2200 m \u2192 m + n \u2261 n + m \u2192 succ m + n \u2261 n + succ m) \u2192\n             \u2200 m \u2192 m + n \u2261 n + m\n+-comm-ind n = PA-ind (\u03bb i \u2192 i + n \u2261 n + i)\n\n+-comm\u2083 : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm\u2083 m n = +-comm-ind n base is m\n  where\n  base : zero + n \u2261 n + zero\n  base = zero + n \u2261\u27e8 +-leftIdentity n \u27e9\n         n        \u2261\u27e8 sym (+-rightIdentity n) \u27e9\n         n + zero \u220e\n\n  is : \u2200 i \u2192 i + n \u2261 n + i \u2192 succ i + n \u2261 n + succ i\n  is i ih = succ i + n   \u2261\u27e8 PA\u2084 i n \u27e9\n            succ (i + n) \u2261\u27e8 succCong ih \u27e9\n            succ (n + i) \u2261\u27e8 sym (x+Sy\u2261S[x+y] n i) \u27e9\n            n + succ i   \u220e\n\n------------------------------------------------------------------------------\n-- Combined proof instantiating the induction principle\n\n-- TODO. 25 October 2012. Why is it not necessary the hypothesis\n-- +-rightIdentity ?\npostulate +-comm\u2084 : \u2200 m n \u2192 m + n \u2261 n + m\n{-# ATP prove +-comm\u2084 +-comm-ind x+Sy\u2261S[x+y] #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/PA\/Axiomatic\/Standard\/AddComm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"fce524078fe45e85654e0404a6e1f33f5050e4a1","subject":"normal tuple syntax","message":"normal tuple syntax\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2\/FFI.agda","new_file":"src\/Chu2\/FFI.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"907115ffb2fdc0ca292efc2520ea3691fe8c61f6","subject":"how internal flipping the coin affects the count","message":"how internal flipping the coin affects the count\n","repos":"crypto-agda\/crypto-agda","old_file":"dist-props.agda","new_file":"dist-props.agda","new_contents":"module dist-props where\n\nopen import Data.Bool\nopen import Data.Nat.NP\nopen import Data.Product\n\nopen import Function\n\nopen import Search.Type\nopen import Search.Sum\n\nopen import Search.Searchable.Sum\nopen import Search.Searchable.Product\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule GameFlipping (R : Set)(sum : Sum R)(sum-ind : SumInd sum)(X Y : R \u2192 Bool) where\n\n  R' = Bool \u00d7 R\n  sum' : Sum R'\n  sum' = searchBit _+_ \u00d7-sum sum\n\n  open FromSum sum' renaming (count to count')\n  open FromSumInd sum-ind\n\n  _==\u1d2e_ : Bool \u2192 Bool \u2192 Bool\n  true  ==\u1d2e y = y\n  false ==\u1d2e y = not y\n\n  G : R' \u2192 Bool\n  G (b , r) = b ==\u1d2e (if b then X r else Y r)\n\n  1\/2 : R' \u2192 Bool\n  1\/2 = proj\u2081\n\n  lemma : \u2200 X \u2192 sum (const 1) \u2261 count (not \u2218 X) + count X\n  lemma X = sum-ind P (\u03bb {a}{b} \u2192 part1 {a}{b}) part2\n    where\n      # = FromSum.count\n      \n      P = \u03bb s \u2192 s (const 1) \u2261 # s (not \u2218 X) + # s X\n      \n      part1 : \u2200 {s\u2080 s\u2081} \u2192 P s\u2080 \u2192 P s\u2081 \u2192 P (\u03bb f \u2192 s\u2080 f + s\u2081 f)\n      part1 {s\u2080} {s\u2081} Ps\u2080 Ps\u2081 rewrite Ps\u2080 | Ps\u2081 = +-interchange (# s\u2080 (not \u2218 X)) (# s\u2080 X) (# s\u2081 (not \u2218 X)) (# s\u2081 X)  \n\n      part2 : \u2200 x \u2192 P (\u03bb f \u2192 f x)\n      part2 x with X x\n      part2 x | true  = refl\n      part2 x | false = refl\n      \n\n  thm : dist (count' G) (count' 1\/2) \u2261 dist (count X) (count Y)\n  thm = dist (count' G) (count' 1\/2)\n      \u2261\u27e8 refl \u27e9\n        dist (count (not \u2218 Y) + count X) (sum (const 0) + sum (const 1))\n      \u2261\u27e8 cong (\u03bb p \u2192 dist (count (not \u2218 Y) + count X) (p + sum (const 1))) sum-zero \u27e9\n        dist (count (not \u2218 Y) + count X) (sum (const 1))\n      \u2261\u27e8 cong (dist (count (not \u2218 Y) + count X)) (lemma Y) \u27e9\n        dist (count (not \u2218 Y) + count X) (count (not \u2218 Y) + count Y)\n      \u2261\u27e8 dist-x+ (count (not \u2218 Y)) (count X) (count Y) \u27e9\n        dist (count X) (count Y)\n      \u220e\n    where\n      open \u2261-Reasoning\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'dist-props.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"128af4af6ee42b630e845312b9e05e5b42880cae","subject":"Add Control.Strategy.Utils","message":"Add Control.Strategy.Utils\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Strategy\/Utils.agda","new_file":"Control\/Strategy\/Utils.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.Product\nopen import Control.Strategy renaming (map to mapS)\n\nmodule Control.Strategy.Utils where\n\nrecord Proto : \u2605\u2081 where\n  constructor P[_,_]\n  field\n    Q : \u2605\n    R : Q \u2192 \u2605\n\nClient : Proto \u2192 \u2605 \u2192 \u2605\nClient P[ Q , R ] A = Strategy Q R A\n\n{-\ndata Bisim {P P' A A'} (RA : A \u2192 A' \u2192 \u2605) : Client P A \u2192 Client P' A' \u2192 \u2605 where\n  ask-nop : \u2200 {q? cont clt} r\n            \u2192 Bisim RA (cont r) clt\n            \u2192 Bisim RA (ask q? cont) clt\n  ask-ask : \u2200 q\u2080 q\u2081 cont\u2080 cont\u2081 r\u2080 r\u2081\n            \u2192 ({!!} \u2192 Bisim RA (cont\u2080 r\u2080) (cont\u2081 r\u2081))\n            \u2192 Bisim RA (ask q\u2080 cont\u2080) (ask q\u2081 cont\u2081)\n-}\nmodule Unused where\n  mapS' : \u2200 {A B Q Q' R} (f : A \u2192 B) (g : Q \u2192 Q') \u2192 Strategy Q (R \u2218 g) A \u2192 Strategy Q' R B\n  mapS' f g (ask q cont) = ask (g q) (\u03bb r \u2192 mapS' f g (cont r))\n  mapS' f g (done x)     = done (f x)\n\n[_,_]=<<_ : \u2200 {A B Q Q' R} (f : A \u2192 Strategy Q' R B) (g : Q \u2192 Q') \u2192 Strategy Q (R \u2218 g) A \u2192 Strategy Q' R B\n[ f , g ]=<< ask q? cont = ask (g q?) (\u03bb r \u2192 [ f , g ]=<< cont r)\n[ f , g ]=<< done x      = f x\n\nmodule Rec {A B Q : \u2605} {R : Q \u2192 \u2605} {M : \u2605 \u2192 \u2605}\n           (runAsk : \u2200 {A} \u2192 (q : Q) \u2192 (R q \u2192 M A) \u2192 M A)\n           (runDone : A \u2192 M B) where\n  runMM : Strategy Q R A \u2192 M B\n  runMM (ask q? cont) = runAsk q? (\u03bb r \u2192 runMM (cont r))\n  runMM (done x)      = runDone x\n\nmodule MM {A B Q : \u2605} {R : Q \u2192 \u2605} {M : \u2605 \u2192 \u2605}\n         (_>>=M_ : \u2200 {A B : \u2605} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B)\n         (runAsk : (q : Q) \u2192 M (R q))\n         (runDone : A \u2192 M B) where\n  runMM : Strategy Q R A \u2192 M B\n  runMM (ask q? cont) = runAsk q? >>=M (\u03bb r \u2192 runMM (cont r))\n  runMM (done x)      = runDone x\n\nmodule _ {A B Q Q' : \u2605} {R : Q \u2192 \u2605} {R'}\n         (f : (q : Q) \u2192 Strategy Q' R' (R q))\n         (g : A \u2192 Strategy Q' R' B) where\n  [_,_]=<<'_ : Strategy Q R A \u2192 Strategy Q' R' B\n  [_,_]=<<'_ (ask q? cont) = f q? >>= \u03bb r \u2192 [_,_]=<<'_ (cont r)\n  [_,_]=<<'_ (done x)      = g x\n\nrecord ServerResp (P : Proto) (q : Proto.Q P) (A : \u2605\u2080) : \u2605\u2080 where\n  coinductive\n  open Proto P\n  field\n      srv-resp : R q\n      srv-cont : \u2200 q \u2192 ServerResp P q A\nopen ServerResp\n\nServer : Proto \u2192 \u2605\u2080 \u2192 \u2605\u2080\nServer P A = \u2200 q \u2192 ServerResp P q A\n\nOracleServer : Proto \u2192 \u2605\u2080\nOracleServer P[ Q , R ] = (q : Q) \u2192 R q\n\nmodule _ {P : Proto} {A : \u2605} where\n  open Proto P\n  com : Server P A \u2192 Client P A \u2192 A\n  com srv (ask q \u03bac) = com (srv-cont r) (\u03bac (srv-resp r)) where r = srv q\n  com srv (done x)   = x\n\nmodule _ {P P' : Proto} {A A' : \u2605} where\n  module P = Proto P\n  module P' = Proto P'\n  record MITM : \u2605 where\n    coinductive\n    field\n      hack-query : (q' : P'.Q) \u2192 Client P (P'.R q' \u00d7 MITM)\n      hack-result : A' \u2192 Client P A\n  open MITM\n\n  module WithOutBind where\n      hacked-com-client : Server P A \u2192 MITM \u2192 Client P' A' \u2192 A\n      hacked-com-mitm : \u2200 {q'} \u2192 Server P A \u2192 Client P (P'.R q' \u00d7 MITM) \u2192 (P'.R q' \u2192 Client P' A') \u2192 A\n      hacked-com-srv-resp : \u2200 {q q'} \u2192 ServerResp P q A \u2192 (P.R q \u2192 Client P (P'.R q' \u00d7 MITM)) \u2192 (P'.R q' \u2192 Client P' A') \u2192 A\n\n      hacked-com-srv-resp r mitm clt = hacked-com-mitm (srv-cont r) (mitm (srv-resp r)) clt\n\n      hacked-com-mitm srv (ask q? mitm) clt = hacked-com-srv-resp (srv q?) mitm clt\n      hacked-com-mitm srv (done (r' , mitm)) clt = hacked-com-client srv mitm (clt r')\n\n      hacked-com-client srv mitm (ask q' \u03bac) = hacked-com-mitm srv (hack-query mitm q') \u03bac\n      hacked-com-client srv mitm (done x) = com srv (hack-result mitm x)\n\n  mitm-to-client-trans : MITM \u2192 Client P' A' \u2192 Client P A\n  mitm-to-client-trans mitm (ask q? cont) = hack-query mitm q? >>= \u03bb { (r' , mitm') \u2192 mitm-to-client-trans mitm' (cont r') }\n  mitm-to-client-trans mitm (done x)      = hack-result mitm x\n\n  hacked-com : Server P A \u2192 MITM \u2192 Client P' A' \u2192 A\n  hacked-com srv mitm clt = com srv (mitm-to-client-trans mitm clt)\n\nmodule _ (P : Proto) (A : \u2605) where\n  open Proto P\n  open MITM\n  honest : MITM {P} {P} {A} {A}\n  hack-query  honest q = ask q \u03bb r \u2192 done (r , honest)\n  hack-result honest a = done a\n\nmodule _ (P : Proto) (A : \u2605) (Oracle : OracleServer P) where\n  oracle-server : Server P A\n  srv-resp (oracle-server q) = Oracle q\n  srv-cont (oracle-server q) = oracle-server\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Strategy\/Utils.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"daa9df855db5381725da8718d075b29836d96cdd","subject":"Desc@ICFP: draft of a cleaner Hutton's aftershave","message":"Desc@ICFP: draft of a cleaner Hutton's aftershave","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/RevisedHutton.agda","new_file":"models\/RevisedHutton.agda","new_contents":" {-# OPTIONS --type-in-type #-}\n\nmodule RevisedHutton where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set) (B : A -> Set) : Set where\n  _,_ : (x : A) (y : B x) -> Sigma A B\n\n_*_ : (A : Set)(B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\ndata Zero : Set where\ndata Unit  : Set where\n  Void : Unit\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A : Set)(B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Equality\n--****************\n\ndata _==_ {A : Set}(x : A) : A -> Set where\n  refl : x == x\n\nsubst : forall {x y} -> x == y -> x -> y\nsubst refl x = x\n\ncong : {A B : Set}(f : A -> B){x y : A} -> x == y -> f x == f y\ncong f refl = refl\n\ncong2 : {A B C : Set}(f : A -> B -> C){x y : A}{z t : B} -> \n        x == y -> z == t -> f x z == f y t\ncong2 f refl refl = refl\n\npostulate \n  reflFun : {A B : Set}(f : A -> B)(g : A -> B)-> ((a : A) -> f a == g a) -> f == g \n\n--****************\n-- Meta-language\n--****************\n\n-- Note that we could define Nat, Bool, and the related operations in\n-- IDesc. But it is awful to code with them, in Agda.\n\ndata Nat : Set where\n  ze : Nat\n  su : Nat -> Nat\n\ndata Bool : Set where\n  true : Bool\n  false : Bool\n\nplus : Nat -> Nat -> Nat\nplus ze n' = n'\nplus (su n) n' = su (plus n n')\n\nle : Nat -> Nat -> Bool\nle ze _ = true\nle (su _) ze = false\nle (su n) (su n') = le n n'\n\ndata Vec (A : Set) : Nat -> Set where\n   vnil : Vec A ze\n   vcons : {n : Nat} -> A -> Vec A n -> Vec A (su n)\n\ndata Fin : Nat -> Set where\n  fze : {n : Nat} -> Fin (su n)\n  fsu : {n : Nat} -> Fin n -> Fin (su n)\n\n\n\n--********************************************\n-- Desc code\n--********************************************\n\ndata IDesc (I : Set) : Set where\n  var : I -> IDesc I\n  const : Set -> IDesc I\n  prod : IDesc I -> IDesc I -> IDesc I\n  sigma : (S : Set) -> (S -> IDesc I) -> IDesc I\n  pi : (S : Set) -> (S -> IDesc I) -> IDesc I\n\n\n--********************************************\n-- Desc interpretation\n--********************************************\n\n[|_|] : {I : Set} -> IDesc I -> (I -> Set) -> Set\n[| var i |] P = P i\n[| const X |] P = X\n[| prod D D' |] P = [| D |] P * [| D' |] P\n[| sigma S T |] P = Sigma S (\\s -> [| T s |] P)\n[| pi S T |] P = (s : S) -> [| T s |] P\n\n--********************************************\n-- Fixpoint construction\n--********************************************\n\ndata IMu {I : Set}(R : I -> IDesc I)(i : I) : Set where\n  con : [| R i |] (\\j -> IMu R j) -> IMu R i\n\n--********************************************\n-- Predicate: All\n--********************************************\n\nAll : {I : Set}(D : IDesc I)(P : I -> Set) -> [| D |] P -> IDesc (Sigma I P)\nAll (var i)     P x        = var (i , x)\nAll (const X)   P x        = const Unit\nAll (prod D D') P (d , d') = prod (All D P d) (All D' P d')\nAll (sigma S T) P (a , b)  = All (T a) P b\nAll (pi S T)    P f        = pi S (\\s -> All (T s) P (f s))\n\n--********************************************\n-- Elimination principle: induction\n--********************************************\n\nmodule Elim {I : Set}\n            (R : I -> IDesc I)\n            (P : Sigma I (IMu R) -> Set)\n            (m : (i : I)\n                 (xs : [| R i |] (IMu R))\n                 (hs : [| All (R i) (IMu R) xs |] P) ->\n                 P ( i , con xs ))\n       where\n\n  mutual\n    indI : (i : I)(x : IMu R i) -> P (i , x)\n    indI i (con xs) = m i xs (hyps (R i) xs)\n\n    hyps : (D : IDesc I) -> \n           (xs : [| D |] (IMu R)) -> \n           [| All D (IMu R) xs |] P\n    hyps (var i) x = indI i x\n    hyps (const X) x = Void\n    hyps (prod D D') (d , d') =  hyps D d , hyps D' d'\n    hyps (pi S R) f = \\ s -> hyps (R s) (f s)\n    hyps (sigma S R) ( a , b ) = hyps (R a) b\n\n\nindI : {I : Set}\n       (R : I -> IDesc I)\n       (P : Sigma I (IMu R) -> Set)\n       (m : (i : I)\n            (xs : [| R i |] (IMu R))\n            (hs : [| All (R i) (IMu R) xs |] P) ->\n            P ( i , con xs)) ->\n       (i : I)(x : IMu R i) -> P ( i , x )\nindI = Elim.indI\n\n\n--********************************************\n-- Enumerations (hard-coded)\n--********************************************\n\n-- Unlike in Desc.agda, we don't carry the levitation of finite sets\n-- here. We hard-code them and manipulate with standard Agda\n-- machinery. Both presentation are isomorph but, in Agda, the coded\n-- one quickly gets unusable.\n\ndata EnumU : Set where\n  nilE : EnumU\n  consE : EnumU -> EnumU\n\ndata EnumT : (e : EnumU) -> Set where\n  EZe : {e : EnumU} -> EnumT (consE e)\n  ESu : {e : EnumU} -> EnumT e -> EnumT (consE e)\n\nspi : (e : EnumU)(P : EnumT e -> Set) -> Set\nspi nilE P = Unit\nspi (consE e) P = P EZe * spi e (\\e -> P (ESu e))\n\nswitch : (e : EnumU)(P : EnumT e -> Set)(b : spi e P)(x : EnumT e) -> P x\nswitch nilE P b ()\nswitch (consE e) P b EZe = fst b\nswitch (consE e) P b (ESu n) = switch e (\\e -> P (ESu e)) (snd b) n\n\n_++_ : EnumU -> EnumU -> EnumU\nnilE ++ e' = e'\n(consE e) ++ e' = consE (e ++ e')\n\n-- A special switch, for tagged descriptions. Switching on a\n-- concatenation of finite sets:\nsswitch : (e : EnumU)(e' : EnumU)(P : Set)\n       (b : spi e (\\_ -> P))(b' : spi e' (\\_ -> P))(x : EnumT (e ++ e')) -> P\nsswitch nilE nilE P b b' ()\nsswitch nilE (consE e') P b b' EZe = fst b'\nsswitch nilE (consE e') P b b' (ESu n) = sswitch nilE e' P b (snd b') n\nsswitch (consE e) e' P b b' EZe = fst b\nsswitch (consE e) e' P b b' (ESu n) = sswitch e e' P (snd b) b' n\n\n--********************************************\n-- Tagged indexed description\n--********************************************\n\nFixMenu : Set -> Set\nFixMenu I = Sigma EnumU (\\e -> (i : I) -> spi e (\\_ -> IDesc I))\n\nSensitiveMenu : Set -> Set\nSensitiveMenu I = Sigma (I -> EnumU) (\\F -> (i : I) -> spi (F i) (\\_ -> IDesc I))\n\nTagIDesc : Set -> Set\nTagIDesc I = FixMenu I * SensitiveMenu I\n\ntoIDesc : (I : Set) -> TagIDesc I -> (I -> IDesc I)\ntoIDesc I ((E , ED) , (F , FD)) i = sigma (EnumT (E ++ F i)) \n                                          (\\x -> sswitch E (F i) (IDesc I) (ED i) (FD i) x)\n\n--********************************************\n-- Catamorphism\n--********************************************\n\ncata : (I : Set)\n       (R : I -> IDesc I)\n       (T : I -> Set) ->\n       ((i : I) -> [| R i |] T -> T i) ->\n       (i : I) -> IMu R i -> T i\ncata I R T phi i x = indI R (\\it -> T (fst it)) (\\i xs ms -> phi i (replace (R i) T xs ms)) i x\n  where replace : (D : IDesc I)(T : I -> Set)\n                  (xs : [| D |] (IMu R))\n                  (ms : [| All D (IMu R) xs |] (\\it -> T (fst it))) -> \n                  [| D |] T\n        replace (var i) T x y = y\n        replace (const Z) T z z' = z\n        replace (prod D D') T (x , x') (y , y') = replace D T x y , replace D' T x' y'\n        replace (sigma A B) T (a , b) t = a , replace (B a) T b t\n        replace (pi A B) T f t = \\s -> replace (B s) T (f s) (t s)\n\n--********************************************\n-- Hutton's razor\n--********************************************\n\n--********************************\n-- Types code\n--********************************\n\ndata Type : Set where\n  nat : Type\n  bool : Type\n  pair : Type -> Type -> Type\n\n\n--********************************************\n-- Free monad construction\n--********************************************\n\n_**_ : {I : Set} (R : TagIDesc I)(X : I -> Set) -> TagIDesc I\n((E , ED) , FFD) ** X = ((( consE E ,  \\ i -> ( const (X i) , ED i ))) , FFD) \n\n\n--********************************************\n-- Substitution\n--********************************************\n\napply : {I : Set}\n        (R : TagIDesc I)(X Y : I -> Set) ->\n        ((i : I) -> X i -> IMu (toIDesc I (R ** Y)) i) ->\n        (i : I) -> \n        [| toIDesc I (R ** X) i |] (IMu (toIDesc I (R ** Y))) ->\n        IMu (toIDesc I (R ** Y)) i\napply (( E , ED) , (F , FD)) X Y sig i (EZe , x) = sig i x\napply (( E , ED) , (F , FD)) X Y sig i (ESu n , t) = con (ESu n , t)\n\nsubstI : {I : Set} (X Y : I -> Set)(R : TagIDesc I)\n         (sigma : (i : I) -> X i -> IMu (toIDesc I (R ** Y)) i)\n         (i : I)(D : IMu (toIDesc I (R ** X)) i) ->\n         IMu (toIDesc I (R ** Y)) i\nsubstI {I} X Y R sig i term = cata I (toIDesc I (R ** X)) (IMu (toIDesc I (R ** Y))) (apply R X Y sig) i term \n\n\n--********************************************\n-- Hutton's razor is free monad\n--********************************************\n\n-- Fix menu:\nexprFreeFixMenu : FixMenu Type\nexprFreeFixMenu = ( consE nilE , \n                    \\ty -> (prod (var bool) (prod (var ty) (var ty)), -- if b then t1 else t2\n                           Void))\n\n-- Index-dependent menu:\nchoiceFreeMenu : Type -> EnumU\nchoiceFreeMenu nat = consE nilE\nchoiceFreeMenu bool = consE nilE\nchoiceFreeMenu (pair x y) = nilE\n\nchoiceFreeDessert : (ty : Type) -> spi (choiceFreeMenu ty) (\\ _ -> IDesc Type)\nchoiceFreeDessert nat = (prod (var nat) (var nat) , Void)     -- plus x y\nchoiceFreeDessert bool = (prod (var nat) (var nat) , Void )   -- le x y\nchoiceFreeDessert (pair x y) = Void\n\nexprFreeSensitiveMenu : SensitiveMenu Type\nexprFreeSensitiveMenu = ( choiceFreeMenu ,  choiceFreeDessert )\n\n-- Tagged description of expressions\nexprFree : TagIDesc Type\nexprFree = exprFreeFixMenu , exprFreeSensitiveMenu\n\n-- Free monadic expressions\nexprFreeC : (Type -> Set) -> TagIDesc Type\nexprFreeC X = exprFree ** X\n\n--********************************\n-- Closed terms\n--********************************\n\nVal : Type -> Set\nVal nat = Nat\nVal bool = Bool\nVal (pair x y) = (Val x) * (Val y)\n\ncloseTerm : Type -> IDesc Type\ncloseTerm = toIDesc Type (exprFree ** Val)\n\n--********************************\n-- Closed term evaluation\n--********************************\n\neval : {ty : Type} -> IMu closeTerm ty -> Val ty\neval {ty} term = cata Type closeTerm Val evalOneStep ty term\n        where evalOneStep : (ty : Type) -> [| closeTerm ty |] Val -> Val ty\n              evalOneStep _ (EZe , t) = t\n              evalOneStep _ ((ESu EZe) , (true , ( x , _))) = x\n              evalOneStep _ ((ESu EZe) , (false , ( _ , y ))) = y\n              evalOneStep nat ((ESu (ESu EZe)) , (x , y)) = plus x y\n              evalOneStep nat ((ESu (ESu (ESu ()))) , t) \n              evalOneStep bool ((ESu (ESu EZe)) , (x , y) ) =   le x y\n              evalOneStep bool ((ESu (ESu (ESu ()))) , _) \n              evalOneStep (pair x y) (ESu (ESu ()) , _)\n\n\n--********************************\n-- [Open terms]\n--********************************\n\n-- Context and context lookup \nContext : Nat -> Set\nContext n = Vec (Sigma Type (\\ty -> IMu closeTerm ty)) n\n\ntypeAt : {n : Nat}(c : Context n) -> Fin n -> Type\ntypeAt {ze} c ()\ntypeAt {su _} (vcons x _) fze = fst x\ntypeAt {su _} (vcons _ xs) (fsu y) = typeAt xs y\n\nlookup : {n : Nat}(c : Context n)(i : Fin n) -> IMu closeTerm (typeAt c i)\nlookup {ze} c ()\nlookup {su _} (vcons x _) fze = snd x\nlookup {su _} (vcons _ xs) (fsu y) = lookup xs y\n\n-- A variable is an index into the context *and* a proof that the\n-- context contains the expected stuff\nVar : {n : Nat} -> Context n -> Type -> Set\nVar {n} c ty = Sigma (Fin n) (\\i -> typeAt c i == ty)\n\n-- Open term: holes are either values or variables in a context\nopenTerm : {n : Nat} -> Context n -> Type -> IDesc Type\nopenTerm c = toIDesc Type (exprFree ** (\\ty -> Val ty + Var c ty))\n\n--********************************\n-- Evaluation of open terms\n--********************************\n\n-- |discharge| is the local substitution expected by |substI|. It is\n-- just sugar around context lookup, taking care of transmitting the\n-- proof\ndischarge : {n : Nat}\n            {context : Context n}\n            (ty : Type) ->\n            (Val ty + Var context ty) ->\n            IMu closeTerm ty\ndischarge ty (l value) = con (EZe , value)\ndischarge {n} {c} ty (r variable) = \n    subst (cong (IMu closeTerm) (snd variable)) \n          (lookup c (fst variable))\n\n-- |substExpr| is the specialized |substI| to expressions. We get it\n-- generically from the free monad construction.\nsubstExpr : {n : Nat}\n            {ty : Type}\n            (context : Context n)\n            (sigma : (ty : Type) ->\n                     (Val ty + Var context ty) ->\n                     IMu closeTerm ty) ->\n            IMu (openTerm context) ty ->\n            IMu closeTerm ty\nsubstExpr {n} {ty} c sig term = \n    substI (\\ty -> Val ty + Var c ty) Val exprFree sig ty term\n\n-- By first doing substitution to close the term, we can use\n-- evaluation of closed terms, obtaining evaluation of open terms\n-- under a valid context.\nevalOpen : {n : Nat}{ty : Type}\n           (context : Context n) ->\n           IMu (openTerm context) ty ->\n           Val ty\nevalOpen context tm = eval (substExpr context discharge tm)\n\n--********************************\n-- Tests\n--********************************\n\n-- V 0 :-> true, V 1 :-> 2, V 2 :-> ( false , 1 )\ntestContext : Context _\ntestContext =  vcons (bool , con (EZe , true )) \n              (vcons (nat , con (EZe , su (su ze)) ) \n              (vcons (pair bool nat , con (EZe , ( false , su ze )))\n               vnil))\n\n-- V 1\ntest1 : IMu (openTerm testContext) nat\ntest1 = con (EZe , r ( fsu fze , refl ) )\n\ntestSubst1 : IMu closeTerm nat\ntestSubst1 = substExpr testContext \n                       discharge \n                       test1\n-- = 2\ntestEval1 : Val nat\ntestEval1 = evalOpen testContext test1\n\n-- add 1 (V 1)\ntest2 : IMu (openTerm testContext) nat\ntest2 = con (ESu (ESu EZe) , (con (EZe , l (su ze)) , con ( EZe , r (fsu fze , refl) )) )\n\ntestSubst2 : IMu closeTerm nat\ntestSubst2 = substExpr testContext \n                        discharge \n                        test2\n\n-- = 3\ntestEval2 : Val nat\ntestEval2 = evalOpen testContext test2\n\n-- if (V 0) then (V 1) else 0\ntest3 : IMu (openTerm testContext) nat\ntest3 = con (ESu EZe , (con (EZe , r (fze , refl)) ,\n                       (con (EZe , r (fsu fze , refl)) ,\n                        con (EZe , l ze))))\n\ntestSubst3 : IMu closeTerm nat\ntestSubst3 = substExpr testContext \n                       discharge \n                       test3\n\n-- = 2\ntestEval3 : Val nat\ntestEval3 = evalOpen testContext test3\n\n-- V 2\ntest4 : IMu (openTerm testContext) (pair bool nat)\ntest4 = con (EZe , r ( fsu (fsu fze) , refl ) )\n\ntestSubst4 : IMu closeTerm (pair bool nat)\ntestSubst4 = substExpr testContext \n                       discharge \n                       test4\n-- = (false , 1)\ntestEval4 : Val (pair bool nat)\ntestEval4 = evalOpen testContext test4\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/RevisedHutton.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"50bceb19499a5e0a942f0bc77967922777ceabe4","subject":"Added draft on +-assoc using Mendelson's axioms.","message":"Added draft on +-assoc using Mendelson's axioms.\n\nIgnore-this: 9e025a50e62c86b02912e79316a6ae25\n\ndarcs-hash:20120222142143-3bd4e-43509a674b220dd8d4cf127220b2081b5f4a4abe.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/PA\/Axiomatic\/Mendelson\/AssocAdditionATP.agda","new_file":"Draft\/PA\/Axiomatic\/Mendelson\/AssocAdditionATP.agda","new_contents":"-- Tested with FOT on 22 February 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.PA.Axiomatic.Mendelson.AssocAdditionATP where\n\nopen import PA.Axiomatic.Mendelson.Base\n\npostulate +-leftCong : \u2200 {m n o} \u2192 m \u2250 n \u2192 m + o \u2250 n + o\n\n+-asocc : \u2200 m n o \u2192 m + n + o \u2250 m + (n + o)\n+-asocc m n o = S\u2089 P P0 is m\n  where\n  P : M \u2192 Set\n  P i = i + n + o \u2250 i + (n + o)\n  {-# ATP definition P #-}\n\n  postulate P0 : P zero\n  -- {-# ATP prove P0 +-leftCong #-}\n\n  -- (2012-02-22) E 1.4. CPU time limit exceeded, terminating (180 sec).\n  -- (2012-02-22) Metis 2.3 (release 20110926). SZS status Unknown (180 sec).\n  -- (2012-02-22) Vampire 0.6 (revision 903). Time limit reached! (180 sec).\n  postulate is : \u2200 i \u2192 P i \u2192 P (succ i)\n  {-# ATP prove is +-leftCong #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/PA\/Axiomatic\/Mendelson\/AssocAdditionATP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"31c1b48f6bacc9ea1bc19276577514994806c12c","subject":"added models","message":"added models","repos":"kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram","old_file":"models\/Data.agda","new_file":"models\/Data.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check\n #-}\n\nmodule Data where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Two : Set where\n  tt : Two\n  ff : Two\n\ncond : {S : Two -> Set} -> S tt -> S ff -> (b : Two) -> S b\ncond t f tt = t\ncond t f ff = f\n\n_+_ : Set -> Set -> Set\nS + T = Sig Two (cond S T)\n\n[_,_] : {S T : Set}{P : S + T -> Set} ->\n        ((s : S) -> P (tt , s)) -> ((t : T) -> P (ff , t)) ->\n        (x : S + T) -> P x\n[ f , g ] = sig (cond f g)\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (I : Set) : Set where\n  ?? : I -> CON I\n  PrfC : PROP -> CON I\n  _*C_ : CON I -> CON I -> CON I\n  PiC : (S : Set) -> (S -> CON I) -> CON I\n  SiC : (S : Set) -> (S -> CON I) -> CON I\n  MuC : (O : Set) -> (O -> CON (I + O)) -> O -> CON I\n  NuC : (O : Set) -> (O -> CON (I + O)) -> O -> CON I\n\nmutual\n  [|_|] : {I : Set} -> CON I -> (I -> Set) -> Set\n  [| ?? i |]       X = X i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {I O : Set}(C : O -> CON (I + O))(X : I -> Set)(o : O) : Set where\n    con : [| C o |] [ X , Mu C X ] -> Mu C X o\n  codata Nu {I O : Set}(C : O -> CON (I + O))(X : I -> Set)(o : O) : Set where\n    con : [| C o |] [ X , Nu C X ] -> Nu C X o\n\nnoc :  {I O : Set}{C : O -> CON (I + O)}{X : I -> Set}{o : O} ->\n       Nu C X o -> [| C o |] [ X , Nu C X ]\nnoc (con xs) = xs\n\nmutual\n  map : {I : Set}{X Y : I -> Set} -> (C : CON I) -> ((i : I) -> X i -> Y i) ->\n        [| C |] X -> [| C |] Y\n  map (?? i)       f = f i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = fold C (Mu C _) (\\ o xs -> con (map (C o) [ f , konst id ] xs)) o\n  map (NuC O C o)  f = unfold C (Nu C _) (\\ o ss -> map (C o) [ f , konst id ] (noc ss)) o\n\n  fold :  {I O : Set}(C : O -> CON (I + O)){X : I -> Set}(Y : O -> Set) ->\n          ((o : O) -> [| C o |] [ X , Y ] -> Y o) ->\n          (o : O) -> Mu C X o -> Y o\n  fold C Y f o (con xs) = f o (map (C o) [ konst id , fold C Y f ] xs)\n\n  unfold :  {I O : Set}(C : O -> CON (I + O)){X : I -> Set}(Y : O -> Set) ->\n            ((o : O) -> Y o -> [| C o |] [ X , Y ]) ->\n            (o : O) -> Y o -> Nu C X o\n  unfold C Y f o y = con (map (C o) [ konst id , unfold C Y f ] (f o y))\n\ninduction : {I O : Set}(C : O -> CON (I + O)){X : I -> Set}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] [ X , (\\ o -> Sig (Mu C X o) (P o)) ]) ->\n             P o (con (map (C o) [ konst id , (\\ o -> fst) ] xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction C P p o (con xys) =\n  p o (map (C o) [ konst id , (\\ o y -> (y , induction C P p o y)) ] xys)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Data.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"f4f6e7a3cad61d0a276de2905272032cf125b5eb","subject":"add Data.Product.Param.Binary","message":"add Data.Product.Param.Binary\n","repos":"np\/agda-parametricity","old_file":"lib\/Data\/Product\/Param\/Binary.agda","new_file":"lib\/Data\/Product\/Param\/Binary.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Level\nopen import Data.Product\n  renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nopen import Function.Param.Binary\n\nmodule Data.Product.Param.Binary where\n\nrecord \u27e6\u03a3\u27e7 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n           {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n           {B\u2081 : A\u2081 \u2192 Set b\u2081}\n           {B\u2082 : A\u2082 \u2192 Set b\u2082}\n           (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n           (B\u1d63 : {x\u2081 : A\u2081} {x\u2082 : A\u2082} (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n                \u2192 B\u2081 x\u2081 \u2192 B\u2082 x\u2082 \u2192 Set b\u1d63)\n           (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) : Set (a\u1d63 \u2294 b\u1d63) where\n  constructor _\u27e6,\u27e7_\n  field\n    \u27e6fst\u27e7 : A\u1d63 (fst p\u2081) (fst p\u2082)\n    \u27e6snd\u27e7 : B\u1d63 \u27e6fst\u27e7 (snd p\u2081) (snd p\u2082)\nopen \u27e6\u03a3\u27e7 public\ninfixr 4 _\u27e6,\u27e7_\n\nsyntax \u27e6\u03a3\u27e7 A\u1d63 (\u03bb x\u1d63 \u2192 e) = [ x\u1d63 \u2236 A\u1d63 ]\u27e6\u00d7\u27e7[ e ]\n\n\u27e6\u2203\u27e7 : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n        {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n        {A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63}\n        {B\u2081 : A\u2081 \u2192 Set b\u2081}\n        {B\u2082 : A\u2082 \u2192 Set b\u2082}\n        (B\u1d63 : \u27e6Pred\u27e7 b\u1d63 A\u1d63 B\u2081 B\u2082)\n        (p\u2081 : \u03a3 A\u2081 B\u2081) (p\u2082 : \u03a3 A\u2082 B\u2082) \u2192 Set _\n\u27e6\u2203\u27e7 = \u27e6\u03a3\u27e7 _\n\nsyntax \u27e6\u2203\u27e7 (\u03bb x\u1d63 \u2192 e) = \u27e6\u2203\u27e7[ x\u1d63 ] e\n\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 b\u2082 b\u2081 a\u1d63 b\u1d63}\n          {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n          {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082}\n          (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 Set (a\u1d63 \u2294 b\u1d63)\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u27e6\u03a3\u27e7 A\u1d63 (\u03bb _ \u2192 B\u1d63)\n\n{-\n_\u27e6\u00d7\u27e7_ : \u2200 {a\u2081 a\u2082 a\u1d63} {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082} (A\u1d63 : A\u2081 \u2192 A\u2082 \u2192 Set a\u1d63)\n          {b\u2081 b\u2082 b\u1d63} {B\u2081 : Set b\u2081} {B\u2082 : Set b\u2082} (B\u1d63 : B\u2081 \u2192 B\u2082 \u2192 Set b\u1d63)\n          (p\u2081 : A\u2081 \u00d7 B\u2081) (p\u2082 : A\u2082 \u00d7 B\u2082) \u2192 Set _\n_\u27e6\u00d7\u27e7_ A\u1d63 B\u1d63 = \u03bb p\u2081 p\u2082 \u2192 A\u1d63 (fst p\u2081) (fst p\u2082) \u00d7\n                        B\u1d63 (snd p\u2081) (snd p\u2082)\n-}\n\n{- One can give these two types to \u27e6_,_\u27e7:\n\n\u27e6_,_\u27e7' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n       {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082}\n       {A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {x\u2081 x\u2082 y\u2081 y\u2082}\n       (x\u1d63 : A\u1d63 x\u2081 x\u2082)\n       (y\u1d63 : B\u1d63 x\u1d63 y\u2081 y\u2082)\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 (x\u2081 , y\u2081) (x\u2082 , y\u2082)\n\u27e6_,_\u27e7' = \u27e6_,_\u27e7\n\n\u27e6_,_\u27e7'' : \u2200 {a\u2081 a\u2082 b\u2081 b\u2082 a\u1d63 b\u1d63}\n       {A\u2081 : Set a\u2081} {A\u2082 : Set a\u2082}\n       {B\u2081 : A\u2081 \u2192 Set b\u2081} {B\u2082 : A\u2082 \u2192 Set b\u2082}\n       {A\u1d63 : \u27e6Set\u27e7 a\u1d63 A\u2081 A\u2082}\n       {B\u1d63 : \u27e6Pred\u27e7 A\u1d63 b\u1d63 B\u2081 B\u2082}\n       {p\u2081 p\u2082}\n       (\u27e6fst\u27e7 : A\u1d63 (fst p\u2081) (fst p\u2082))\n       (\u27e6snd\u27e7 : B\u1d63 \u27e6fst\u27e7 (snd p\u2081) (snd p\u2082))\n     \u2192 \u27e6\u03a3\u27e7 A\u1d63 B\u1d63 p\u2081 p\u2082\n\u27e6_,_\u27e7'' = \u27e6_,_\u27e7\n-}\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Product\/Param\/Binary.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9d43feb67f096246bbfefae15f85bbef0cc0cc6a","subject":"Control\/Protocol\/Lift.agda","message":"Control\/Protocol\/Lift.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Lift.agda","new_file":"Control\/Protocol\/Lift.agda","new_contents":"open import Level.NP\nopen import Data.Product.NP\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Lift a {\u2113} where\n    lift\u1d3e : Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\n    lift\u1d3e end        = end\n    lift\u1d3e (com io P) = com io \u03bb m \u2192 lift\u1d3e (P (lower {\u2113 = a} m))\n\n    lift-proc : (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e P \u27e7\n    lift-proc end      end     = end\n    lift-proc (send P) (m , p) = lift m , lift-proc (P m) p\n    lift-proc (recv P) p       = \u03bb { (lift m) \u2192 lift-proc (P m) (p m) }\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Lift.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"eb12f9ca6e7009aad1748696c53d9bc1da0a774f","subject":"lower for function name","message":"lower for function name\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5a23ed5a45d5f64acc54379ebc46facac5b9f707","subject":"Reflection.Scoped.Translation: from unscoped to scoped terms","message":"Reflection.Scoped.Translation: from unscoped to scoped terms\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Scoped\/Translation.agda","new_file":"lib\/Reflection\/Scoped\/Translation.agda","new_contents":"module Reflection.Scoped.Translation where\n{-\nopen import Level\nopen import Data.List\nopen import Reflection\nopen import Relation.Binary.PropositionalEquality\npostulate\n  f : (a : Level) \u2192 Set a\ntest : type (quote f) \u2261 el unknown (pi (arg (arg-info visible relevant) (el (lit 0) (def (quote Level) []))) (el unknown (sort (set (var 0 [])))))\ntest = refl\n-}\n\nopen import Level\nopen import Data.Zero\nopen import Data.One\nopen import Data.Sum\nopen import Data.Product\nopen import Data.Nat\nopen import Data.List\nopen import Reflection.NP as R\nopen import Reflection.Scoped as S\n\nmodule RS where\n    V\u00e5r : Set \u2192 Set\n    V\u00e5r A = \u2115 \u2192 S.Args A \u2192 S.Term A\n\n    record Env (A : Set) : Set where\n      field\n        v\u00e5r  : V\u00e5r A\n    open Env public\n\n    \u03b5 : \u2200 {A} \u2192 Env A\n    \u03b5 = record { v\u00e5r = \u03bb _ _ \u2192 unknown }\n\n    v\u00e5r\u2191 : \u2200 {A} \u2192 V\u00e5r A \u2192 Scope \ud835\udfd9 V\u00e5r A\n    v\u00e5r\u2191 f zero    as = var (inj\u2082 0\u2081) as\n    v\u00e5r\u2191 f (suc n) as = S.app (Map.term inj\u2081 (f n [])) as\n\n    \u2191 : \u2200 {A} \u2192 Env A \u2192 Scope \ud835\udfd9 Env A\n    \u2191 \u0393 = record { v\u00e5r = v\u00e5r\u2191 (v\u00e5r \u0393) }\n\n    term : \u2200 {A} \u2192 Env A \u2192 R.Term \u2192 S.Term A\n    t\u00ffpe : \u2200 {A} \u2192 Env A \u2192 R.Type \u2192 S.Type A\n    args : \u2200 {A} \u2192 Env A \u2192 R.Args \u2192 S.Args A\n    s\u00f8rt : \u2200 {A} \u2192 Env A \u2192 R.Sort \u2192 S.Sort A\n\n    term \u0393 (var x as) = v\u00e5r \u0393 x (args \u0393 as)\n    term \u0393 (con c as) = con c (args \u0393 as)\n    term \u0393 (def d as) = def d (args \u0393 as)\n    term \u0393 (lam v t) = lam v (term (\u2191 \u0393) t)\n    term \u0393 (pat-lam cs as) = unknown -- TODO\n    term \u0393 (pi (arg i t\u2081) t\u2082) = pi (arg i (t\u00ffpe \u0393 t\u2081)) (t\u00ffpe (\u2191 \u0393) t\u2082)\n    term \u0393 (sort x) = sort (s\u00f8rt \u0393 x)\n    term \u0393 (lit x) = lit x\n    term \u0393 unknown = unknown\n\n    t\u00ffpe \u0393 (el s t) = el (s\u00f8rt \u0393 s) (term \u0393 t)\n\n    s\u00f8rt \u0393 (set t) = set (term \u0393 t)\n    s\u00f8rt \u0393 (lit n) = lit n\n    s\u00f8rt \u0393 _ = unknown\n\n    args \u0393 []             = []\n    args \u0393 (arg i t \u2237 as) = arg i (term \u0393 t) \u2237 args \u0393 as\n\n\nmodule SR where\n    V\u00e5r : Set \u2192 Set\n    V\u00e5r A = A \u2192 R.Args \u2192 R.Term\n\n    record Env (A : Set) : Set where\n      field\n        v\u00e5r  : V\u00e5r A\n    open Env public\n\n    \u03b5 : Env \ud835\udfd8\n    \u03b5 = record { v\u00e5r = \u03bb () }\n\n    v\u00e5r\u2191 : \u2200 {A} \u2192 V\u00e5r A \u2192 Scope \ud835\udfd9 V\u00e5r A\n    v\u00e5r\u2191 f (inj\u2081 x) as = R.app (liftTerm (f x [])) as\n    v\u00e5r\u2191 f (inj\u2082 y) as = var 0 as\n\n    \u2191 : \u2200 {A} \u2192 Env A \u2192 Scope \ud835\udfd9 Env A\n    \u2191 \u0393 = record { v\u00e5r = v\u00e5r\u2191 (v\u00e5r \u0393) }\n\n    term : \u2200 {A} \u2192 Env A \u2192 S.Term A \u2192 R.Term\n    t\u00ffpe : \u2200 {A} \u2192 Env A \u2192 S.Type A \u2192 R.Type\n    args : \u2200 {A} \u2192 Env A \u2192 S.Args A \u2192 R.Args\n    s\u00f8rt : \u2200 {A} \u2192 Env A \u2192 S.Sort A \u2192 R.Sort\n\n    term \u0393 (var x as) = v\u00e5r \u0393 x (args \u0393 as)\n    term \u0393 (con c as) = con c (args \u0393 as)\n    term \u0393 (def d as) = def d (args \u0393 as)\n    term \u0393 (lam v t) = lam v (term (\u2191 \u0393) t)\n    term \u0393 (pi (arg i t\u2081) t\u2082) = pi (arg i (t\u00ffpe \u0393 t\u2081)) (t\u00ffpe (\u2191 \u0393) t\u2082)\n    term \u0393 (sort x) = sort (s\u00f8rt \u0393 x)\n    term \u0393 (lit x) = lit x\n    term \u0393 unknown = unknown\n\n    t\u00ffpe \u0393 (el s t) = el (s\u00f8rt \u0393 s) (term \u0393 t)\n\n    s\u00f8rt \u0393 (set t) = set (term \u0393 t)\n    s\u00f8rt \u0393 (lit n) = lit n\n    s\u00f8rt \u0393 unknown = unknown\n\n    args \u0393 []             = []\n    args \u0393 (arg i t \u2237 as) = arg i (term \u0393 t) \u2237 args \u0393 as\n\nmodule Test where\n  open import Relation.Binary.PropositionalEquality\n  open import Function\n\n  COMP : Set _\n  COMP = {A : Set} {B : A \u2192 Set} {C : {x : A} \u2192 B x \u2192 Set} \u2192\n         (\u2200 {x} (y : B x) \u2192 C y) \u2192 (g : (x : A) \u2192 B x) \u2192\n         ((x : A) \u2192 C (g x))\n\n         {-\n  _\u2218_ : COMP\n  _\u2218_ = \u03bb f g x \u2192 f (g x)\n  -}\n\n  comp : {A : Set} {B : A \u2192 Set} {C : {x : A} \u2192 B x \u2192 Set} \u2192\n         (\u2200 {x} (y : B x) \u2192 C y) \u2192 (g : (x : A) \u2192 B x) \u2192\n         ((x : A) \u2192 C (g x))\n  comp = \u03bb f g x \u2192 f (g x)\n\n         {-\n  _\u2218\u2032_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} \u2192\n             (B \u2192 C) \u2192 (A \u2192 B) \u2192 (A \u2192 C)\n  _\u2218\u2032_ = \u03bb f g x \u2192 f (g x)\n  -}\n  {-\n  \u2218\u2032-def = get-term-def (definition (quote _\u2218\u2032_))\n  test : S.Term \ud835\udfd8\n  test = RS.term RS.\u03b5 \u2218\u2032-def\n  test' : SR.term SR.\u03b5 test \u2261 \u2218\u2032-def\n  test' = refl\n  -}\n\n  check-term : R.Term \u2192 Set\n  check-term t = SR.term SR.\u03b5 (RS.term RS.\u03b5 t) \u2261 t\n\n  strip-univ-poly : R.Term \u2192 R.Term\n  strip-univ-poly (pi (arg i (el s (def (quote Level) []))) (el _ t)) = strip-univ-poly t\n  strip-univ-poly t = t\n\n  check : Name \u2192 Set\n  check n = check-term (Get-term.from-name n)\n          \u00d7 check-term (R.unEl (type n))\n\n  test-\u2218\u2032 : check (quote _\u2218_)\n  test-\u2218\u2032 = refl , refl\n\n  test-\u2218 : check (quote _\u2218\u2032_)\n  test-\u2218 = refl , refl\n\n  test-comp : check (quote comp)\n  test-comp = refl , refl\n\n  test-COMP : check (quote COMP)\n  test-COMP = refl , refl\n\n  test-flip : check (quote flip)\n  test-flip = refl , refl\n\n  test-[] : check (quote _-[_]-_)\n  test-[] = refl , refl\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Reflection\/Scoped\/Translation.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"50aba82fa071077bdeb246e6a948beb832c0832e","subject":"Added draft about the bisimilarity relation.","message":"Added draft about the bisimilarity relation.\n\nIgnore-this: 6fad54c52f5b705de073c3fcb7b1dff6\n\ndarcs-hash:20101027171125-3bd4e-cbfedf7a085dcba6213e863b086a47f85a7aca44.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Bisimilarity\/BisimilarityPostFixedPoint.agda","new_file":"Draft\/Bisimilarity\/BisimilarityPostFixedPoint.agda","new_contents":"module BisimilarityPostFixedPoint where\n\n-- open import LTC.Minimal\n-- open import LTC.MinimalER\n\ninfixr 5 _\u2237_\ninfix 4 _\u2248_\ninfixr 4 _,_\ninfix  3 _\u2261_\ninfixr 2 _\u2227_\n\n------------------------------------------------------------------------------\n-- LTC stuff\n\n-- The universal domain.\npostulate D : Set\n\n  -- LTC lists.\npostulate\n  _\u2237_  : D \u2192 D \u2192 D\n\n-- The existential quantifier type on D.\ndata \u2203D (P : D \u2192 Set) : Set where\n  _,_ : (d : D) (Pd : P d) \u2192 \u2203D P\n\n-- The identity type on D.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\n-- Identity properties\n\nsym : {x y : D} \u2192 x \u2261 y \u2192 y \u2261 x\nsym refl = refl\n\nsubst : (P : D \u2192 Set){x y : D} \u2192 x \u2261 y \u2192 P x \u2192 P y\nsubst P refl px = px\n\n-- The conjunction data type.\ndata _\u2227_ (A B : Set) : Set where\n  _,_ : A \u2192 B \u2192 A \u2227 B\n\n------------------------------------------------------------------------------\nBISI : (D \u2192 D \u2192 Set) \u2192 D \u2192 D \u2192 Set\nBISI R xs ys =\n  \u2203D (\u03bb x' \u2192\n  \u2203D (\u03bb y' \u2192\n  \u2203D (\u03bb xs' \u2192\n  \u2203D (\u03bb ys' \u2192\n     x' \u2261 y' \u2227 R xs' ys' \u2227 xs \u2261 x' \u2237 xs' \u2227 ys \u2261 y' \u2237 ys'))))\n\n-- From Peter email:\n\n-- For the case of bisimilarity \u2248 we have\n\n-- (i) a post-fixed point of BISI, is the following first order axiom\n\n-- forall xs,ys,x,y. x = y & xs \u2248 ys -> x :: xs \u2248 y :: ys\n\npostulate\n  -- The bisimilarity relation.\n  _\u2248_ : D \u2192 D \u2192 Set\n\n  -- The bisimilarity relation is a post-fixed point of BISI.\n  -\u2248-gfp\u2081 : {x y xs ys : D} \u2192 x \u2261 y \u2227 xs \u2248 ys \u2192 x \u2237 xs \u2248 y \u2237 ys\n\nfoo : (xs ys : D) \u2192 xs \u2248 ys \u2192 BISI _\u2248_ xs ys\nfoo xs ys xs\u2248ys = {!!}\n\nbar : (xs ys : D) \u2192 BISI _\u2248_ xs ys \u2192 xs \u2248 ys\nbar xs ys (x' , y' , xs' , ys' , x'\u2261y' , xs'\u2248ys' , xs\u2261x'\u2237xs' , ys\u2261y'\u2237ys)\n  = subst (\u03bb t\u2081 \u2192 t\u2081 \u2248 ys)\n          (sym xs\u2261x'\u2237xs')\n          (subst (\u03bb t\u2082 \u2192 x' \u2237 xs' \u2248 t\u2082)\n                 (sym ys\u2261y'\u2237ys)\n                 (-\u2248-gfp\u2081 (x'\u2261y' , xs'\u2248ys')))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Bisimilarity\/BisimilarityPostFixedPoint.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"848bb1054ebda65c1c858c47c247ae4a19e26925","subject":"Added stream example (draft).","message":"Added stream example (draft).\n\nIgnore-this: 9047faf72f7e39b3894cb3750ff52ca\n\ndarcs-hash:20110803160607-3bd4e-01e8331e2b03a9cbfd89520090f414b362d2c596.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Stream\/Examples.agda","new_file":"Draft\/FOTC\/Data\/Stream\/Examples.agda","new_contents":"------------------------------------------------------------------------------\n-- Stream examples\n------------------------------------------------------------------------------\n\nmodule Draft.FOTC.Data.Stream.Examples where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.Stream.Type\n\n------------------------------------------------------------------------------\n\n-- We cannot use the expected definition of zeros because Agda hangs.\nzeros : D\nzeros = zero \u2237 {!!} -- zeros\n\nP : D \u2192 Set\nP xs = xs \u2261 zeros\n\nzerosS : Stream zeros\nzerosS = Stream-gfp\u2082 P (\u03bb {xs} Pxs \u2192 zero , zeros , refl , trans Pxs refl) refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Stream\/Examples.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"40b13998dcea8078971392142e22ffcfaff7968e","subject":"Agda model for Desc.","message":"Agda model for Desc.\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n-- Bringing in an hand-made Prop universe\n\ndata PROP : Set where\n  All : (S : Set) -> (S -> PROP) -> PROP\n  And : PROP -> PROP -> PROP\n  Trivial : PROP\n  Absurd : PROP\n\nPrf : PROP -> Set\nPrf Absurd = \u22a5\nPrf Trivial = \u22a4\nPrf (All S P) = (x : S) -> Prf (P x)\nPrf (And A B) = \u03a3 (Prf A)  (\\ _ -> Prf B)\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  Prf ( All H (\\y -> p (proj\u2081 v y)))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) ->\n           (Prf (All D (\\y -> bp y))) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n-- On the Ind case, the Pig code is saying something along those lines:\n-- mapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (p?! (proj\u2082 v))\n\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> PROP) ->\n         (Prf (All (descOp d (Mu d)) (\\ x -> (All (boxOp d (Mu d) bp x) (\\ _ -> bp (Con x)))))) ->\n         (v : Mu d) -> Prf (bp v)\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Desc.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"b0287a9a5b9dac181e24674987da1469303dfc13","subject":"Examples of tactics needing to match against function calls.","message":"Examples of tactics needing to match against function calls.\n","repos":"spire\/spire","old_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_file":"proposal\/examples\/TacticsMatchingFunctionCalls.agda","new_contents":"{- \nThis file demonstrates a technique that simulates the ability to\npattern match against function calls rather than just variables\nor constructors.\n\nThis technique is necessary to write certain kinds of tactics\nas generic functions.\n-}\n\nmodule TacticsMatchingFunctionCalls where\nopen import Data.Bool hiding ( _\u225f_ )\nopen import Data.Nat\nopen import Data.Fin hiding ( _+_ )\nopen import Data.Product hiding ( map )\nopen import Data.List\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\nopen import Function\n\n----------------------------------------------------------------------\n\n{-\nZero is the right identity of addition on the natural numbers.\n-}\n\nplusrident : (n : \u2115) \u2192 n \u2261 n + 0\nplusrident zero = refl\nplusrident (suc n) = cong suc (plusrident n)\n\n----------------------------------------------------------------------\n\n{-\nArith is a universe of codes representing the natural numbers\nalong with a limited set of functions over them. We would like to write\ntactics that pattern match against types indexed by applications of\nthese functions.\n-}\n\ndata Arith : Set where\n  `zero : Arith\n  `suc : (a : Arith) \u2192 Arith\n  _`+_ _`*_ : (a\u2081 a\u2082 : Arith) \u2192 Arith\n\neval : Arith \u2192 \u2115\neval `zero = zero\neval (`suc a) = suc (eval a)\neval (a\u2081 `+ a\u2082) = eval a\u2081 + eval a\u2082\neval (a\u2081 `* a\u2082) = eval a\u2081 * eval a\u2082\n\n----------------------------------------------------------------------\n\n{-\n\u03a6 (standing for *function*-indexed types)\nrepresents an indexed type, but we *can* pattern\nmatch on function calls in its index position.\n\nA - The type of codes, including constructors and functions.\n  i.e. Arith\nA\u2032 - The model underlying the codes.\n  i.e. \u2115\nel - The evaluation function from code values to model values.\n  i.e. eval\nB - The genuine indexed datatype that \u03a6 represents.\n  i.e. Fin\na - The index value as a code, which may be a function call and hence supports\n    pattern matching.\n  i.e. (a `+ `zero)\n-}\n\ndata \u03a6 (A A\u2032 : Set) (el : A \u2192 A\u2032) (B : A\u2032 \u2192 Set) (a : A) : Set where\n  \u03c6 : (b : B (el a)) \u2192 \u03a6 A A\u2032 el B a\n\n----------------------------------------------------------------------\n\n{-\nAlternatively, we can specialize \u03a6 to our Arith universe. This may be\na good use of our ability to cheaply define new datatypes via Desc.\n-}\n\ndata \u03a6Arith (B : \u2115 \u2192 Set) (a : Arith) : Set where\n  \u03c6 : (b : B (eval a)) \u2192 \u03a6Arith B a\n\n----------------------------------------------------------------------\n\n{-\nA small universe of dependent types that is sufficient for the\nexamples given below.\n-}\n\ndata Type : Set\n\u27e6_\u27e7 : Type \u2192 Set\n\ndata Type where\n  `Bool `\u2115 `Arith : Type\n  `Fin : (n : \u2115) \u2192 Type\n  `\u03a6 : (A A\u2032 : Type)\n    (el : \u27e6 A \u27e7 \u2192 \u27e6 A\u2032 \u27e7)\n    (B : \u27e6 A\u2032 \u27e7 \u2192 Type)\n    (a : \u27e6 A \u27e7)\n    \u2192 Type\n  `\u03a6Arith : (B : \u2115 \u2192 Type) (a : Arith) \u2192 Type\n  `\u03a0 `\u03a3 : (A : Type) (B : \u27e6 A \u27e7 \u2192 Type) \u2192 Type\n  `Id : (A : Type) (x y : \u27e6 A \u27e7) \u2192 Type\n\n\u27e6 `Bool \u27e7 = Bool\n\u27e6 `\u2115 \u27e7 = \u2115\n\u27e6 `Arith \u27e7 = Arith\n\u27e6 `Fin n \u27e7 = Fin n\n\u27e6 `\u03a6 A A\u2032 el B a \u27e7 = \u03a6 \u27e6 A \u27e7 \u27e6 A\u2032 \u27e7 el (\u03bb a \u2192 \u27e6 B a \u27e7) a\n\u27e6 `\u03a6Arith B a \u27e7 = \u03a6Arith (\u03bb x \u2192 \u27e6 B x \u27e7) a\n\u27e6 `\u03a0 A B \u27e7 = (a : \u27e6 A \u27e7) \u2192 \u27e6 B a \u27e7\n\u27e6 `\u03a3 A B \u27e7 = \u03a3 \u27e6 A \u27e7 (\u03bb a \u2192 \u27e6 B a \u27e7)\n\u27e6 `Id A x y \u27e7 = x \u2261 y\n\n_`\u2192_ : (A B : Type) \u2192 Type\nA `\u2192 B = `\u03a0 A (const B)\n\n_`\u00d7_ : (A B : Type) \u2192 Type\nA `\u00d7 B = `\u03a3 A (const B)\n\n----------------------------------------------------------------------\n\n{-\nTactics are represented as generic functions taking a Dynamic value (of\nany type), and returning a Dynamic value (at a possibly different type).\nThe convention is to behave like the identity function if the tactic\ndoesn't match the current value or fails.\n\nIn practice, the context will be a List of Dynamic values, and a tactic\nwill map a Context to a Context. For simplicity, the tactics I give below\nonly map a Dynamic to a Dynamic.\n-}\n\nDynamic : Set\nDynamic = \u03a3 Type \u27e6_\u27e7\n\nTactic : Set\nTactic = Dynamic \u2192 Dynamic\n\n----------------------------------------------------------------------\n\n{-\nHere is an example of a tactic that only needs to pattern match on a\nvariable or constructor, but not a function. We don't encounter\nany problems when writing this kind of a tactic.\n\nThe tactic below changes a `Fin n` value into a `Fin (n + 0)` value.\n-}\n\nadd-plus0-Fin : Tactic\nadd-plus0-Fin (`Fin n , i) = `Fin (n + 0) , subst Fin (plusrident n) i\nadd-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nIn contrast, it is not straightforward to write tactics that need\nto pattern match on function calls.\nFor example, below is ideally what we want to write to change\na value of `Fin (n + 0)` to a value of `Fin n`.\n-}\n\n-- rm-plus0-Fin : Tactic\n-- rm-plus0-Fin (`Fin (n + 0) , i) = `Fin n , subst Fin (sym (plusrident n)) i\n-- rm-plus0-Fin x = x\n\n----------------------------------------------------------------------\n\n{-\nInstead of matching directly on `Fin, we can match on a `\u03a6 that\nrepresents `Fin. This allows us to match on the the `+ function call\nin the index position.\n\nBecause we cannot match against the function `el` (we would like to match\nit against `eval`), we add to our return type that proves the\nextentional equality between `el` and `eval`.\n\nAlso note that the tactic below can be used for any type indexed by\n\u2115, not just Fin.\n-}\n\nrm-plus0 : Tactic\nrm-plus0 (`\u03a6 `Arith `\u2115 el B (a `+ `zero) , \u03c6 b) =\n  `\u03a0 `Arith (\u03bb x \u2192 `Id `\u2115 (el x) (eval x))\n  `\u2192\n  `\u03a6 `Arith `\u2115 el B a\n  ,\n  \u03bb f \u2192 \u03c6\n  (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n         (sym (f a))\n         (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                (sym (plusrident (eval a)))\n                (subst (\u03bb x \u2192 \u27e6 B x \u27e7)\n                       (f (a `+ `zero))\n                       b)))\nrm-plus0 x = x\n\n----------------------------------------------------------------------\n\n{-\nNow it is possible to apply the generic tactic to a specific\nvalue and have the desired behavior occur. Notice that the generated\nextentional equality premise is trivially provable because\neval is always equal to eval.\n-}\n\neg-rm-plus0 : (a : Arith)\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6 `Arith `\u2115 eval `Fin a \u27e7\neg-rm-plus0 a (\u03c6 i) = proj\u2082\n  (rm-plus0 (`\u03a6 `Arith `\u2115 eval `Fin (a `+ `zero) , \u03c6 i))\n  (\u03bb _ \u2192 refl)\n\n----------------------------------------------------------------------\n\n{-\nHere is the same tactic but using the specialized \u03a6Arith type.\n-}\n\nrm-plus0-Arith : Tactic\nrm-plus0-Arith (`\u03a6Arith B (a `+ `zero) , \u03c6 b) =\n  `\u03a6Arith B a\n  ,\n  \u03c6 (subst (\u03bb x \u2192 \u27e6 B x \u27e7) (sym (plusrident (eval a))) b)\nrm-plus0-Arith x = x\n\n----------------------------------------------------------------------\n\n{-\nAnd the same example usage, this time without needing to prove a premise.\n-}\n\neg-rm-plus0-Arith : (a : Arith)\n  \u2192 \u27e6 `\u03a6Arith `Fin (a `+ `zero) \u27e7\n  \u2192 \u27e6 `\u03a6Arith `Fin a \u27e7\neg-rm-plus0-Arith a (\u03c6 i) = proj\u2082\n  (rm-plus0-Arith (`\u03a6Arith `Fin (a `+ `zero) , \u03c6 i))\n\n----------------------------------------------------------------------\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'proposal\/examples\/TacticsMatchingFunctionCalls.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"44c8e054ae8f0d4fe681b3438a30c51e9cff8c00","subject":"Basic logic in agda","message":"Basic logic in agda\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Logic.agda","new_file":"agda\/Logic.agda","new_contents":"module Logic where\n\n-- The true proposition.\ndata \u22a4 : Set where\n  true : \u22a4 -- The proof of truth.\n\n-- The false proposition.\ndata \u22a5 : Set where\n  -- There is nothing here so one can never prove false.\n\n-- The AND of two statments.\ndata _\u2227_ (A B : Set)  : Set where\n  -- The only way to construct a proof of A \u2227 B is by pairing a a\n  -- proof of A with a proof of and B.\n  \u27e8_,_\u27e9 : (a : A)  -- Proof of A\n        \u2192 (b : B)  -- Proof of B\n        \u2192 A \u2227 B    -- Proof of A \u2227 B\n\n-- The OR of two statements.\ndata _\u2228_ (A B : Set) : Set where\n  -- There are two ways of constructing a proof of A \u2228 B.\n  inl : (a : A) \u2192  A \u2228 B   -- From a proof of A by left introduction\n  inr : (b : B) \u2192  A \u2228 B   -- From a proof of B by right introduction\n\n-- The not of statement A\n\u00ac_ : (A : Set) \u2192 Set\n\u00ac A = A \u2192 \u22a5  -- Given a proof of A one should be able to get a proof\n             -- of \u22a5.\n\n-- The statement A \u2194 B are equivalent.\n_\u2194_ : (A B : Set) \u2192 Set\nA \u2194 B = (A \u2192 B) -- If\n           \u2227    -- and\n        (B \u2192 A) -- only if\n\n\ninfixr 1 _\u2227_\ninfixr 1 _\u2228_\ninfixr 0 _\u2194_\ninfix  2 \u00ac_\n\n-- Function composition\n_\u2218_   : {A B C : Set} \u2192 (B \u2192 C) \u2192 (A \u2192 B) \u2192 A \u2192 C\n(f \u2218 g) x = f (g x)\n\n-- Double negation\ndoubleNegation : \u2200 {A : Set} \u2192 A \u2192 \u00ac (\u00ac A)\ndoubleNegation a negNegA = negNegA a\n\n{-\n\ndoubleNegation' : \u2200 {A : Set} \u2192 \u00ac ( \u00ac (\u00ac A)) \u2192 \u00ac A\n\n-}\n\n\ndeMorgan1 : \u2200 (A B : Set) \u2192 \u00ac (A \u2228 B) \u2192 \u00ac A \u2227 \u00ac B\ndeMorgan1 A B notAorB = \u27e8 notAorB \u2218 inl , notAorB \u2218 inr \u27e9\n\ndeMorgan2 : \u2200 (A B : Set) \u2192 \u00ac A \u2227 \u00ac B \u2192 \u00ac (A \u2228 B)\ndeMorgan2 A B \u27e8 notA , notB \u27e9 (inl a) = notA a\ndeMorgan2 A B \u27e8 notA , notB \u27e9 (inr b) = notB b\n\n\ndeMorgan : \u2200 (A B : Set) \u2192 \u00ac (A \u2228 B) \u2194 \u00ac A \u2227 \u00ac B\ndeMorgan A B = \u27e8 deMorgan1 A B , deMorgan2 A B \u27e9\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/Logic.agda' did not match any file(s) known to git\n","license":"unlicense","lang":"Agda"}
{"commit":"ad7f16ba82e55407834239db7f8d3d5cdddd3f1e","subject":"add middleware type","message":"add middleware type\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ae0e7b37c5f8e2c8dcaad20f13df0d1a9c06bbfa","subject":"Added note on the universal quantifier.","message":"Added note on the universal quantifier.\n\nIgnore-this: 228858b33ec27bac0a3f1ae83ea96982\n\ndarcs-hash:20120508160306-3bd4e-ac9c9e9ef9bbaea37ab26205c32451f4ec7301b8.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Universal.agda","new_file":"notes\/thesis\/logical-framework\/Universal.agda","new_contents":"-- Tested with FOT on 08 May 2012.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Universal where\n\npostulate D : Set\n\n-- The universal quantifier type on D.\ndata Forall (A : D \u2192 Set) : Set where\n  Forall-intro : ((t : D) \u2192 A t) \u2192 Forall A\n\nForall-elim : {A : D \u2192 Set} \u2192 Forall A \u2192 (t : D) \u2192 A t\nForall-elim (Forall-intro h) t = h t\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/logical-framework\/Universal.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"92c5af0d64d61f2cd71d643d44a4ba58c064798e","subject":"SOme work on Norell's and Chappman's Agda tutorial","message":"SOme work on Norell's and Chappman's Agda tutorial\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/AgdaBasics.agda","new_file":"agda\/AgdaBasics.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"9727b3f963598e797355204ba5b9dfe32408b723","subject":"Added wfInd-RMC.","message":"Added wfInd-RMC.\n\nIgnore-this: 3c178ec1427644c6853cfd6a2ca9d952\n\ndarcs-hash:20110210053633-3bd4e-82684e8337c912506aab5666fa21a0a385c3a76b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/McCarthy91\/WellFoundedInduction.agda","new_file":"Draft\/McCarthy91\/WellFoundedInduction.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on the relation RMC\n----------------------------------------------------------------------------\n\nmodule Draft.McCarthy91.WellFoundedInduction where\n\nopen import LTC.Base\n\nopen import Draft.McCarthy91.McCarthy91\n\nopen import LTC.Data.Nat\n\n----------------------------------------------------------------------------\n\n-- TODO: To remove the postulate.\npostulate\n  wfInd-RMC :\n   (P : D \u2192 Set) \u2192\n   (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 RMC n m \u2192 P n) \u2192 P m) \u2192\n   \u2200 {n} \u2192 N n \u2192 P n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/McCarthy91\/WellFoundedInduction.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"2e42e10181734edf03d6310a619c412c08a7e1fa","subject":"prefect-bintree.agda","message":"prefect-bintree.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree.agda","new_file":"prefect-bintree.agda","new_contents":"module prefect-bintree where\n\nopen import Function\nopen import Data.Nat\nopen import Data.Bool\nopen import Data.Bits\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a where\n  leaf : (x : A) \u2192 Tree A zero\n  fork : \u2200 {n} (left right : Tree A n) \u2192 Tree A (suc n)\n\nfromFun : \u2200 {n a} {A : Set a} \u2192 (Bits n \u2192 A) \u2192 Tree A n\nfromFun {zero} f = leaf (f [])\nfromFun {suc n} f = fork (fromFun (f \u2218 0\u2237_)) (fromFun (f \u2218 1\u2237_))\n\ntoFun : \u2200 {n a} {A : Set a} \u2192 Tree A n \u2192 Bits n \u2192 A\ntoFun (leaf x) _ = x\ntoFun (fork left right) (b \u2237 bs) = toFun (if b then right else left) bs\n\nlookup : \u2200 {n a} {A : Set a} \u2192 Bits n \u2192 Tree A n \u2192 A\nlookup = flip toFun\n\nlookup' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A (m + n) \u2192 Tree A n\nlookup' [] t = t\nlookup' (b \u2237 bs) (fork t t\u2081) = lookup' bs (if b then t\u2081 else t)\n\n{-\nupdate' : \u2200 {m n a} {A : Set a} \u2192 Bits m \u2192 Tree A n \u2192 Tree A (m + n) \u2192 Tree A (m + n)\nupdate' = ?\n-}\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 Tree A n \u2192 Tree B n\nmap f (leaf x) = leaf (f x)\nmap f (fork t\u2080 t\u2081) = fork (map f t\u2080) (map f t\u2081)\n\nopen import Relation.Binary\nopen import Data.Star\n\ndata Swp {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  swp\u2081 : \u2200 {n} {left right : Tree A n} \u2192\n         Swp (fork left right) (fork right left)\n  swp\u2082 : \u2200 {n} {t\u2080\u2080 t\u2080\u2081 t\u2081\u2080 t\u2081\u2081 : Tree A n} \u2192\n         Swp (fork (fork t\u2080\u2080 t\u2080\u2081) (fork t\u2081\u2080 t\u2081\u2081)) (fork (fork t\u2081\u2081 t\u2080\u2081) (fork t\u2081\u2080 t\u2080\u2080))\n\nSwp\u2605 : \u2200 {a} {A : Set a} {n} (left right : Tree A n) \u2192 Set a\nSwp\u2605 = Star Swp\n\nSwp-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Swp {A = A} {n})\nSwp-sym swp\u2081 = swp\u2081\nSwp-sym swp\u2082 = swp\u2082\n\ndata Rot {a} {A : Set a} : \u2200 {n} (left right : Tree A n) \u2192 Set a where\n  leaf : \u2200 x \u2192 Rot (leaf x) (leaf x)\n  fork : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 left\u2081 \u2192\n         Rot right\u2080 right\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n  krof : \u2200 {n} {left\u2080 left\u2081 right\u2080 right\u2081 : Tree A n} \u2192\n         Rot left\u2080 right\u2081 \u2192\n         Rot right\u2080 left\u2081 \u2192\n         Rot (fork left\u2080 right\u2080) (fork left\u2081 right\u2081)\n\nRot-refl : \u2200 {n a} {A : Set a} \u2192 Reflexive (Rot {A = A} {n})\nRot-refl {x = leaf x} = leaf x\nRot-refl {x = fork left right} = fork Rot-refl Rot-refl\n\nRot-sym : \u2200 {n a} {A : Set a} \u2192 Symmetric (Rot {A = A} {n})\nRot-sym (leaf x) = leaf x\nRot-sym (fork p\u2080 p\u2081) = fork (Rot-sym p\u2080) (Rot-sym p\u2081)\nRot-sym (krof p\u2080 p\u2081) = krof (Rot-sym p\u2081) (Rot-sym p\u2080)\n\nRot-trans : \u2200 {n a} {A : Set a} \u2192 Transitive (Rot {A = A} {n})\nRot-trans (leaf x) q = q\nRot-trans (fork p\u2080 p\u2081) (fork q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (fork p\u2080 p\u2081) (krof q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2080) (Rot-trans p\u2081 q\u2081)\nRot-trans (krof p\u2080 p\u2081) (fork q\u2080 q\u2081) = krof (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\nRot-trans (krof p\u2080 p\u2081) (krof q\u2080 q\u2081) = fork (Rot-trans p\u2080 q\u2081) (Rot-trans p\u2081 q\u2080)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'prefect-bintree.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"97ca1b2a5a53c08f5ee2852e6782b7a89405e039","subject":"Sketch fundamental theorem with big-step semantics","message":"Sketch fundamental theorem with big-step semantics\n\nOK, honestly that was much easier than with functional big-step semantics.\nThough probably a clever strategy would simplify those functional proofs.\n\nBeware I haven't proven any lemmas about the semantics (other than a decider\nbased on the evaluator).\n\nAlso missing: lots of inequalities about step-indexes differences. Proving them\nin Agda is a pain. But they all look true.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStepSILR2.agda","new_file":"Thesis\/BigStepSILR2.agda","new_contents":"{-# OPTIONS --allow-unsolved-metas #-}\nmodule Thesis.BigStepSILR2 where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat -- using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\nopen import Data.Nat.Properties\nopen import Data.Nat.Properties.Simple\nopen DecTotalOrder Data.Nat.decTotalOrder using () renaming (refl to \u2264-refl; trans to \u2264-trans)\nopen import Thesis.FunBigStepSILR2\n\n-- Standard relational big-step semantics.\ndata _\u22a2_\u2193[_]_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u2115 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193[ 0 ] closure t \u03c1\n  app : \u2200 n1 n2 n3 {\u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193[ n1 ] closure t\u2032 \u03c1\u2032 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193[ n2 ] v\u2082 \u2192\n    (v\u2082 \u2022 \u03c1\u2032) \u22a2 t\u2032 \u2193[ n3 ] v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193[ suc n1 + n2 + n3 ] v\u2032\n  var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n    \u03c1 \u22a2 var x \u2193[ 0 ] (\u27e6 x \u27e7Var \u03c1)\n  lit : \u2200 n \u2192\n    \u03c1 \u22a2 const (lit n) \u2193[ 0 ] intV n\n\n-- Silly lemmas for eval-dec-sound\nmodule _ where\n  Done-inj : \u2200 {\u03c4} m n {v1 v2 : Val \u03c4} \u2192 Done v1 m \u2261 Done v2 n \u2192 m \u2261 n\n  Done-inj _ _ refl = refl\n\n  lem1 : \u2200 n d \u2192 d \u2262 suc (n + d)\n  lem1 n d eq rewrite +-comm n d = m\u22621+m+n d eq\n\n  subd : \u2200 n d \u2192 n + d \u2261 d \u2192 n \u2261 0\n  subd zero d eq = refl\n  subd (suc n) d eq = \u22a5-elim (lem1 n d (sym eq))\n\n  comp\u2238 : \u2200 a b \u2192 b \u2264 a \u2192 suc a \u2238 b \u2261 suc (a \u2238 b)\n  comp\u2238 a zero le = refl\n  comp\u2238 zero (suc b) ()\n  comp\u2238 (suc a) (suc b) (s\u2264s le) = comp\u2238 a b le\n\n  rearr\u2238 : \u2200 a b c \u2192 c \u2264 b \u2192 a + (b \u2238 c) \u2261 (a + b) \u2238 c\n  rearr\u2238 a b zero c\u2264b = refl\n  rearr\u2238 a zero (suc c) ()\n  rearr\u2238 a (suc b) (suc c) (s\u2264s c\u2264b) rewrite +-suc a b = rearr\u2238 a b c c\u2264b\n\n  cancel\u2238 : \u2200 a b \u2192 b \u2264 a \u2192 a \u2238 b + b \u2261 a\n  cancel\u2238 a zero b\u2264a = +-right-identity a\n  cancel\u2238 zero (suc b) ()\n  cancel\u2238 (suc a) (suc b) (s\u2264s b\u2264a) rewrite +-suc (a \u2238 b) b = cong suc (cancel\u2238 a b b\u2264a)\n\n  lemR : \u2200 {\u03c4} n m {v1 v2 : Val \u03c4} \u2192 Done v1 m \u2261 Done v2 n \u2192 m \u2261 n\n  lemR n .n refl = refl\n\n  lem2 : \u2200 n n3 r \u2192 n3 \u2264 n \u2192 n + n3 \u2261 suc r \u2192 \u2203 \u03bb s \u2192 (n \u2261 suc s)\n  lem2 zero .0 r z\u2264n ()\n  lem2 (suc n) n3 .(n + n3) le refl = n , refl\n\n{-# TERMINATING #-}\neval-dec-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 \u2200 \u03c1 v m n \u2192 eval t \u03c1 m \u2261 Done v n \u2192 \u03c1 \u22a2 t \u2193[ m \u2238 n ] v\neval-dec-sound (const (lit x)) \u03c1 (intV v) m n eq with lemR n m eq\neval-dec-sound (const (lit x)) \u03c1 (intV .x) m .m refl | refl rewrite n\u2238n\u22610 m = lit x\neval-dec-sound (var x) \u03c1 v m n eq with lemR n m eq\neval-dec-sound (var x) \u03c1 .(\u27e6 x \u27e7Var \u03c1) m .m refl | refl rewrite n\u2238n\u22610 m  = var x\neval-dec-sound (abs t) \u03c1 v m n eq with lemR n m eq\neval-dec-sound (abs t) \u03c1 .(closure t \u03c1) m .m refl | refl rewrite n\u2238n\u22610 m  = abs\neval-dec-sound (app s t) \u03c1 v zero n ()\neval-dec-sound (app s t) \u03c1 v (suc r) n eq with eval s \u03c1 r | inspect (eval s \u03c1) r\neval-dec-sound (app s t) \u03c1 v (suc r) n eq | (Done sv n1) | [ seq ] with eval t \u03c1 n1 | inspect (eval t \u03c1) n1\neval-dec-sound (app s t) \u03c1 v (suc r) n eq | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] with eval st (tv \u2022 s\u03c1) n2 | inspect (eval st (tv \u2022 s\u03c1)) n2\neval-dec-sound (app s t) \u03c1 .v (suc r) .n3 refl | (Done sv@(closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | (Done v n3) | [ veq ] = body\n  where\n    n1\u2264r : n1 \u2264 r\n    n1\u2264r = eval-dec s \u03c1 sv r n1 seq\n    n2\u2264n1 : n2 \u2264 n1\n    n2\u2264n1 = eval-dec t \u03c1 tv n1 n2 teq\n    n3\u2264n2 : n3 \u2264 n2\n    n3\u2264n2 = eval-dec st (tv \u2022 s\u03c1) v n2 n3 veq\n    eval1 = eval-dec-sound s \u03c1 sv r n1 seq\n    eval2 = eval-dec-sound t \u03c1 tv n1 n2 teq\n    eval3 = eval-dec-sound st (tv \u2022 s\u03c1) v n2 n3 veq\n    l1 : suc r \u2238 n3 \u2261 suc (r \u2238 n3)\n    l1 = comp\u2238 r n3 (\u2264-trans n3\u2264n2 (\u2264-trans n2\u2264n1 n1\u2264r))\n    l2 : suc r \u2238 n3 \u2261 suc (((r \u2238 n1) + (n1 \u2238 n2)) + (n2 \u2238 n3))\n    l2\n      rewrite rearr\u2238 (r \u2238 n1) n1 n2 n2\u2264n1 | cancel\u2238 r n1 n1\u2264r\n      | rearr\u2238 (r \u2238 n2) n2 n3 n3\u2264n2 | cancel\u2238 r n2 (\u2264-trans n2\u2264n1 n1\u2264r) = l1\n    foo : \u03c1 \u22a2 app s t \u2193[ suc (r \u2238 n1 + (n1 \u2238 n2) + (n2 \u2238 n3)) ] v\n    foo = app (r \u2238 n1) (n1 \u2238 n2) (n2 \u2238 n3) {t\u2081 = s} {t\u2082 = t} {t\u2032 = st} eval1 eval2 eval3\n    body : \u03c1 \u22a2 app s t \u2193[ suc r \u2238 n3 ] v\n    body rewrite l2 = foo\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | Error | [ veq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done (closure st s\u03c1) n1) | [ seq ] | (Done tv n2) | [ teq ] | TimeOut | [ veq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done sv n1) | [ seq ] | Error | [ teq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | (Done sv n1) | [ seq ] | TimeOut | [ teq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | Error | [ seq ]\neval-dec-sound (app s t) \u03c1 v (suc r) n () | TimeOut | [ seq ]\n-- \u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n--   ((Den.\u27e6 x \u27e7Var) (\u27e6 \u03c1 \u27e7Context)) \u2261 (\u27e6 (\u27e6 x \u27e7Var) \u03c1 \u27e7Val)\n-- \u21a6-sound (px \u2022 \u03c1) this = refl\n-- \u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n-- \u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n--   \u03c1 \u22a2 t \u2193 v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n-- \u2193-sound abs = refl\n-- \u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n-- \u2193-sound (var x) = \u21a6-sound _ x\n-- \u2193-sound (lit n) = refl\n\nimport Data.Integer as I\nopen I using (\u2124)\nmutual\n  rrelT : \u2200 {\u03c4 \u0393} (t1 : Term \u0393 \u03c4) (t2 : Term \u0393 \u03c4) (\u03c11 : \u27e6 \u0393 \u27e7Context) (\u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\n  -- To compare this definition, note that the original k is suc n here.\n  rrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n    (v1 : Val \u03c4) \u2192\n    -- Originally we have 0 \u2264 j < k, so j < suc n, so k - j = suc n - j.\n    -- It follows that 0 < k - j \u2264 k, hence suc n - j \u2264 suc n, or n - j \u2264 n.\n    -- Here, instead of binding j we bind n-j = n - j, require n - j \u2264 n, and\n    -- use suc n-j instead of k - j.\n    \u2200 j (j<n : j < n) \u2192\n    -- The next assumption is important. This still says that evaluation consumes j steps.\n    -- Since j \u2264 n, it is OK to start evaluation with n steps.\n    -- Starting with (suc n) and getting suc n-j is equivalent, per eval-mono\n    -- and eval-strengthen. But in practice this version is easier to use.\n    (\u03c11\u22a2t1\u2193[j]v1 : \u03c11 \u22a2 t1 \u2193[ j ] v1) \u2192\n    \u03a3[ v2 \u2208 Val \u03c4 ] \u03a3[ n2 \u2208 \u2115 ] \u03c12 \u22a2 t2 \u2193[ n2 ] v2 \u00d7 rrelV \u03c4 v1 v2 (n \u2238 j)\n\n  rrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\n  rrelV nat (intV v1) (intV v2) n = \u03a3[ dv \u2208 \u2124 ] dv I.+ (I.+ v1) \u2261 (I.+ v2)\n  rrelV (\u03c3 \u21d2 \u03c4) (closure {\u03931} t1 \u03c11) (closure {\u03932} t2 \u03c12) n =\n    \u03a3 (\u03931 \u2261 \u03932) \u03bb { refl \u2192\n      \u2200 (k : \u2115) (k<n : k < n) v1 v2 \u2192\n      (vv : rrelV \u03c3 v1 v2 k) \u2192\n      rrelT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\n    }\n\nrrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = rrelV \u03c4 v1 v2 n \u00d7 rrel\u03c1 \u0393 \u03c11 \u03c12 n\n\n\u27e6_\u27e7RelVar : \u2200 {\u0393 \u03c4 n} (x : Var \u0393 \u03c4) {\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context} \u2192\n  rrel\u03c1 \u0393 \u03c11 \u03c12 n \u2192\n  rrelV \u03c4 (\u27e6 x \u27e7Var \u03c11) (\u27e6 x \u27e7Var \u03c12) n\n\u27e6 this \u27e7RelVar {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = vv\n\u27e6 that x \u27e7RelVar {v1 \u2022 \u03c11} {v2 \u2022 \u03c12} (vv , \u03c1\u03c1) = \u27e6 x \u27e7RelVar \u03c1\u03c1\n\nrrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 rrelV \u03c4 v1 v2 n \u2192 rrelV \u03c4 v1 v2 m\nrrelV-mono m n m\u2264n nat (intV v1) (intV v2) vv = vv\nrrelV-mono m n m\u2264n (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (refl , ff) = refl , \u03bb k k\u2264m \u2192 ff k (\u2264-trans k\u2264m m\u2264n)\n\nrrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rrel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rrel\u03c1 \u0393 \u03c11 \u03c12 m\nrrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = rrelV-mono m n m\u2264n _ v1 v2 vv , rrel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\nrfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rrel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 rrelT (var x) (var x) \u03c11 \u03c12 n\nrfundamentalV x n \u03c11 \u03c12 \u03c1\u03c1 .(\u27e6 x \u27e7Var \u03c11) .0 j<n (var .x) = \u27e6 x \u27e7Var \u03c12 , 0 , (var x) , \u27e6 x \u27e7RelVar \u03c1\u03c1\n\nrfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rrel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 rrelT t t \u03c11 \u03c12 n\nrfundamental (var x) n \u03c11 \u03c12 \u03c1\u03c1 = rfundamentalV x n \u03c11 \u03c12 \u03c1\u03c1\nrfundamental (const (lit x)) n \u03c11 \u03c12 \u03c1\u03c1 .(intV x) .0 j<n (lit .x) = intV x , 0 , lit x , I.+ 0 , refl\nrfundamental (abs t) n \u03c11 \u03c12 \u03c1\u03c1 .(closure t \u03c11) .0 j<n abs = closure t \u03c12 , 0 , abs , refl , (\u03bb k k<n v1 v2 vv v3 j j<k \u03c11\u22a2t1\u2193[j]v1 \u2192 rfundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rrel\u03c1-mono k n (lt1 k<n) _ _ _ \u03c1\u03c1) v3 j j<k \u03c11\u22a2t1\u2193[j]v1 )\nrfundamental (app s t) n \u03c11 \u03c12 \u03c1\u03c1 v1 .(suc (n1 + n2 + n3)) j<n (app n1 n2 n3 \u03c11\u22a2t1\u2193[j]v1 \u03c11\u22a2t1\u2193[j]v2 \u03c11\u22a2t1\u2193[j]v3) = body\n  where\n    n1<n : n1 < n\n    n1<n = {!!}\n    n2<n : n2 < n\n    n2<n = {!!}\n    n-n1-n2<n-n1 : n \u2238 n1 \u2238 n2 < n \u2238 n1\n    n-n1-n2<n-n1 = {!!}\n    n-n1-n2\u2264n-n2 : n \u2238 n1 \u2238 n2 \u2264 n \u2238 n2\n    n-n1-n2\u2264n-n2 = {!!}\n    n3<n-n1-n2 : n3 < n \u2238 n1 \u2238 n2\n    n3<n-n1-n2 = {!!}\n    n-[1+n1+n2+n3]\u2264n-n1-n2-n3 : n \u2238 suc (n1 + n2 + n3) \u2264 n \u2238 n1 \u2238 n2 \u2238 n3\n    n-[1+n1+n2+n3]\u2264n-n1-n2-n3 = {!!}\n    body : \u03a3[ v2 \u2208 Val _ ] \u03a3[ tn \u2208 \u2115 ] \u03c12 \u22a2 app s t \u2193[ tn ] v2 \u00d7 rrelV _ v1 v2 (n \u2238 suc (n1 + n2 + n3))\n    body with rfundamental s n \u03c11 \u03c12 \u03c1\u03c1 _ n1 n1<n \u03c11\u22a2t1\u2193[j]v1 | rfundamental t n \u03c11 \u03c12 \u03c1\u03c1 _ n2 n2<n \u03c11\u22a2t1\u2193[j]v2\n    ... | sv2@(closure st2 s\u03c12) , sn2 , \u03c12\u22a2s\u2193 , refl , sv1v2 | tv2 , tn2 , \u03c12\u22a2t\u2193 , tv1v2 with sv1v2 (n \u2238 n1 \u2238 n2) n-n1-n2<n-n1 _ tv2 (rrelV-mono (n \u2238 n1 \u2238 n2) (n \u2238 n2) n-n1-n2\u2264n-n2 _ _ tv2 tv1v2 ) v1 n3 n3<n-n1-n2 \u03c11\u22a2t1\u2193[j]v3\n    ... | v2 , stn , \u03c12\u22a2st2\u2193 , vv = v2 , suc (sn2 + tn2 + stn) , app _ _ _ \u03c12\u22a2s\u2193 \u03c12\u22a2t\u2193  \u03c12\u22a2st2\u2193 , rrelV-mono (n \u2238 suc (n1 + n2 + n3)) (n \u2238 n1 \u2238 n2 \u2238 n3) n-[1+n1+n2+n3]\u2264n-n1-n2-n3 _ v1 v2 vv\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Thesis\/BigStepSILR2.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"67310e93bb7ce8516de3fbabd32f0479cc2c64b7","subject":"Terminal-0-Groupoid","message":"Terminal-0-Groupoid\n","repos":"zeitraffer\/agda-cats","old_file":"src\/CategoryTheory\/Operations\/Terminal-0-Groupoid-Module.agda","new_file":"src\/CategoryTheory\/Operations\/Terminal-0-Groupoid-Module.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"fatal: unable to access 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/': The requested URL returned error: 403\n","license":"unlicense","lang":"Agda"}
{"commit":"8fd23013cafc4153d1b4a63b41e95bc6ecbbac7e","subject":"Add Crypto.Sig.LamportOneBit","message":"Add Crypto.Sig.LamportOneBit\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/LamportOneBit.agda","new_file":"Crypto\/Sig\/LamportOneBit.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Product.NP hiding (map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Crypto.Sig.LamportOneBit\n          (#secret : \u2115)\n          (#digest : \u2115)\n          (hash    : Bits #secret \u2192 Bits #digest)\n          where\n\n#signkey   = 2* #secret\n#verifkey  = 2* #digest\n#signature = #secret\n\nDigest     = Bits #digest\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nmodule signkey (sk : SignKey) where\n  skL = take #secret sk\n  skH = drop #secret sk\n  vkL = hash skL\n  vkH = hash skH\n\n-- Derive the public key by hashing each secret\nverif-key : SignKey \u2192 VerifKey\nverif-key = map2* #secret #digest hash\n\nmodule verifkey (vk : VerifKey) where\n  vkL = take #digest vk\n  vkH = drop #digest vk\n\n-- Key generation\nkey-gen : SignKey \u2192 VerifKey \u00d7 SignKey\nkey-gen sk = vk , sk\n  module key-gen where\n    vk = verif-key sk\n\n-- Sign a single bit message\nsign : SignKey \u2192 Bit \u2192 Signature\nsign sk b = take2* _ b sk\n\n-- Verify the signature of a single bit message\nverify : VerifKey \u2192 Bit \u2192 Signature \u2192 Bit\nverify vk b sig = take2* _ b vk == hash sig\n\nverify-correct-sig : \u2200 sk b \u2192 verify (verif-key sk) b (sign sk b) \u2261 1b\nverify-correct-sig sk 0b = ==-reflexive (take-++ #digest (hash (take #secret sk)) _)\nverify-correct-sig sk 1b = ==-reflexive (drop-++ #digest (hash (take #secret sk)) _)\n\nimport Algebra.FunctionProperties.Eq\nopen Algebra.FunctionProperties.Eq.Implicits\n\nmodule Assuming-injectivity (hash-inj : Injective hash) where\n\n  -- If one considers the hash function injective, then, so is verif-key\n  verif-key-inj : \u2200 {sk1 sk2} \u2192 verif-key sk1 \u2261 verif-key sk2 \u2192 sk1 \u2261 sk2\n  verif-key-inj {sk1} {sk2} e\n    = take-drop= #secret sk1 sk2\n        (hash-inj (++-inj\u2081 e))\n        (hash-inj (++-inj\u2082 sk1.vkL sk2.vkL e))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n\n  -- Therefor under this assumption, different secret keys means different public keys\n  verif-key-corrolary : \u2200 {sk1 sk2} \u2192 sk1 \u2262 sk2 \u2192 verif-key sk1 \u2262 verif-key sk2\n  verif-key-corrolary {sk1} {sk2} sk\u2262 vk= = sk\u2262 (verif-key-inj vk=)\n\n  -- If one considers the hash function injective then there is\n  -- only one signing key which can sign correctly all (0 and 1)\n  -- the messages\n  signkey-uniqness :\n        \u2200 sk1 sk2 \u2192\n          (\u2200 b \u2192 verify (verif-key sk1) b (sign sk2 b) \u2261 1b) \u2192\n          sk1 \u2261 sk2\n  signkey-uniqness sk1 sk2 e\n    = take-drop= #secret sk1 sk2 (lemmaL (e 0b)) (lemmaH (e 1b))\n    where\n      module sk1 = signkey sk1\n      module sk2 = signkey sk2\n      lemmaL : verify (verif-key sk1) 0b (sign sk2 0b) \u2261 1b \u2192\n              sk1.skL \u2261 sk2.skL\n      lemmaL e0\n        rewrite take-++ #digest sk1.vkL sk1.vkH\n              | hash-inj (==\u21d2\u2261 e0)\n              = refl\n\n      lemmaH : verify (verif-key sk1) 1b (sign sk2 1b) \u2261 1b \u2192\n               sk1.skH \u2261 sk2.skH\n      lemmaH e1\n        rewrite drop-++ #digest sk1.vkL sk1.vkH\n              | hash-inj (==\u21d2\u2261 e1)\n              = refl\n\nmodule Assuming-invertibility\n         (unhash : Digest \u2192 Secret)\n         (unhash-hash : \u2200 x \u2192 unhash (hash x) \u2261 x)\n         (hash-unhash : \u2200 x \u2192 hash (unhash x) \u2261 x)\n         where\n\n  -- EXERCISES\n  {-\n  recover-signkey : VerifKey \u2192 SignKey\n  recover-signkey = {!!}\n\n  recover-signkey-correct : \u2200 sk \u2192 recover-signkey (verif-key sk) \u2261 sk\n  recover-signkey-correct = {!!}\n\n  forgesig : VerifKey \u2192 Bit \u2192 Signature\n  forgesig = {!!}\n\n  forgesig-correct : \u2200 vk b \u2192 verify vk b (forgesig vk b) \u2261 1b\n  forgesig-correct = {!!}\n  -}\n\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Crypto\/Sig\/LamportOneBit.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fefd4590ac23e7cdf91cbc2277dcce1be57d736e","subject":"Add Control.Strategy","message":"Add Control.Strategy\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Strategy.agda","new_file":"Control\/Strategy.agda","new_contents":"module Control.Strategy where\n\nopen import Function\nopen import Type using (\u2605)\nopen import Category.Monad\nopen import Relation.Binary.PropositionalEquality.NP\n\ndata Strategy (Q R A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\ninfix 2 _\u2248_\ndata _\u2248_ {Q R A} : (s\u2080 s\u2081 : Strategy Q R A) \u2192 \u2605 where\n  ask-ask : \u2200 {k\u2080 k\u2081} \u2192 (q? : Q) (cont : \u2200 r \u2192 k\u2080 r \u2248 k\u2081 r) \u2192 ask q? k\u2080 \u2248 ask q? k\u2081\n  done-done : \u2200 x \u2192 done x \u2248 done x\n\n\u2248-refl : \u2200 {Q R A} {s : Strategy Q R A} \u2192 s \u2248 s\n\u2248-refl {s = ask q? cont} = ask-ask q? (\u03bb r \u2192 \u2248-refl)\n\u2248-refl {s = done x}      = done-done x\n\nmodule _ {Q R : \u2605} where\n  private\n    M : \u2605 \u2192 \u2605\n    M = Strategy Q R\n\n  _=<<_ : \u2200 {A B} \u2192 (A \u2192 M B) \u2192 M A \u2192 M B\n  f =<< ask q? cont = ask q? (\u03bb r \u2192 f =<< cont r)\n  f =<< done x      = f x\n\n  return : \u2200 {A} \u2192 A \u2192 M A\n  return = done\n\n  map : \u2200 {A B} \u2192 (A \u2192 B) \u2192 M A \u2192 M B\n  map f s = (done \u2218 f) =<< s\n\n  join : \u2200 {A} \u2192 M (M A) \u2192 M A\n  join s = id =<< s\n\n  _>>=_ : \u2200 {A B} \u2192 M A \u2192 (A \u2192 M B) \u2192 M B\n  _>>=_ = flip _=<<_\n\n  rawMonad : RawMonad M\n  rawMonad = record { return = return; _>>=_ = _>>=_ }\n\n  return-=<< : \u2200 {A} (m : M A) \u2192 return =<< m \u2248 m\n  return-=<< (ask q? cont) = ask-ask q? (\u03bb r \u2192 return-=<< (cont r))\n  return-=<< (done x)      = done-done x\n\n  return->>= : \u2200 {A B} (x : A) (f : A \u2192 M B) \u2192 return x >>= f \u2261 f x\n  return->>= _ _ = refl\n\n  >>=-assoc : \u2200 {A B C} (mx : M A) (my : A \u2192 M B) (f : B \u2192 M C) \u2192 (mx >>= \u03bb x \u2192 (my x >>= f)) \u2248 (mx >>= my) >>= f\n  >>=-assoc (ask q? cont) my f = ask-ask q? (\u03bb r \u2192 >>=-assoc (cont r) my f)\n  >>=-assoc (done x)      my f = \u2248-refl\n\n  map-id : \u2200 {A} (s : M A) \u2192 map id s \u2248 s\n  map-id = return-=<<\n\n  map-\u2218 : \u2200 {A B C} (f : A \u2192 B) (g : B \u2192 C) s \u2192 map (g \u2218 f) s \u2248 (map g \u2218 map f) s\n  map-\u2218 f g s = >>=-assoc s (done \u2218 f) (done \u2218 g)\n\n  module EffectfulRun\n           (E : \u2605 \u2192 \u2605)\n           (_>>=E_  : \u2200 {A B} \u2192 E A \u2192 (A \u2192 E B) \u2192 E B)\n           (returnE : \u2200 {A} \u2192 A \u2192 E A)\n           (Oracle  : Q \u2192 E R) where\n    runE : \u2200 {A} \u2192 M A \u2192 E A\n    runE (ask q? cont) = Oracle q? >>=E (\u03bb r \u2192 runE (cont r))\n    runE (done x)      = returnE x\n\n  module _ (Oracle : Q \u2192 R) where\n    run : \u2200 {A} \u2192 M A \u2192 A\n    run (ask q? cont) = run (cont (Oracle q?))\n    run (done x)      = x\n\n    module _ {A B : \u2605} (f : A \u2192 M B) where\n        run->>= : (s : M A) \u2192 run (s >>= f) \u2261 run (f (run s))\n        run->>= (ask q? cont) = run->>= (cont (Oracle q?))\n        run->>= (done x)      = refl\n\n    module _ {A B : \u2605} (f : A \u2192 B) where\n        run-map : (s : M A) \u2192 run (map f s) \u2261 f (run s)\n        run-map = run->>= (return \u2218 f) \n\n    module _ {A B : \u2605} where\n        run-\u2248 : {s\u2080 s\u2081 : M A} \u2192 s\u2080 \u2248 s\u2081 \u2192 run s\u2080 \u2261 run s\u2081\n        run-\u2248 (ask-ask q? cont) = run-\u2248 (cont (Oracle q?))\n        run-\u2248 (done-done x)     = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Strategy.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"64208706bec60f97145580dc8aca67cafa324c37","subject":"Control\/Protocol\/Relation.agda","message":"Control\/Protocol\/Relation.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Relation.agda","new_file":"Control\/Protocol\/Relation.agda","new_contents":"open import Type\nopen import Function.NP\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u03a3;_,_)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Protocol\n\nmodule Control.Protocol.Relation where\n\n\u211b\u27e6_\u27e7 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u211b\u27e6 end    \u27e7 p q = End\n\u211b\u27e6 recv P \u27e7 p q = \u2200 m \u2192 \u211b\u27e6 P m \u27e7 (p m) (q m)\n\u211b\u27e6 send P \u27e7 p q = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u211b\u27e6 P (fst q) \u27e7 (subst (\u27e6_\u27e7 \u2218 P) e (snd p)) (snd q)\n\n\u211b\u27e6_\u27e7-refl : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Reflexive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-refl     = end\n\u211b\u27e6 recv P \u27e7-refl     = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-refl\n\u211b\u27e6 send P \u27e7-refl {x} = refl , \u211b\u27e6 P (fst x) \u27e7-refl\n\n\u211b\u27e6_\u27e7-sym : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Symmetric \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-sym p          = end\n\u211b\u27e6 recv P \u27e7-sym p          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-sym (p m)\n\u211b\u27e6 send P \u27e7-sym (refl , q) = refl , \u211b\u27e6 P _ \u27e7-sym q    -- TODO HoTT\n\n\u211b\u27e6_\u27e7-trans : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Transitive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-trans p          q          = end\n\u211b\u27e6 recv P \u27e7-trans p          q          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-trans (p m) (q m)\n\u211b\u27e6 send P \u27e7-trans (refl , p) (refl , q) = refl , \u211b\u27e6 P _ \u27e7-trans p q    -- TODO HoTT\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Relation.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9606ac119b0c77c52a621066f104838025a497bc","subject":"Added draft about the conversion from\/to naturals numbers to\/from co-naturals numbers.","message":"Added draft about the conversion from\/to naturals numbers to\/from co-naturals numbers.\n\nIgnore-this: 7217f8ea2eb263ea7e03f8dcfbfbb4cc\n\ndarcs-hash:20120321171751-3bd4e-cf577bf55571a38494dd3c7a5e31fed9bd446c94.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Conat\/PropertiesSL.agda","new_file":"Draft\/FOTC\/Data\/Conat\/PropertiesSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the conversion from\/to natural numbers to\/from co-natural numbers\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with the development version of the standard library on\n-- 21 March 2012.\n\nmodule Draft.FOTC.Data.Conat.PropertiesSL where\n\nopen import Coinduction\nopen import Data.Conat renaming ( suc to csuc )\nopen import Data.Nat\n\n------------------------------------------------------------------------------\n\n\u2115\u2192Co\u2115 : \u2115 \u2192 Co\u2115\n\u2115\u2192Co\u2115 zero    = zero\n\u2115\u2192Co\u2115 (suc n) = csuc (\u266f (\u2115\u2192Co\u2115 n))\n\nCo\u2115\u2192\u2115 : Co\u2115 \u2192 \u2115\nCo\u2115\u2192\u2115 zero     = zero\nCo\u2115\u2192\u2115 (csuc n) = suc (Co\u2115\u2192\u2115 (\u266d n))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Conat\/PropertiesSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"2e1c3e4316fc4449d1a70e490cd5ad3f713438aa","subject":"Add partensor experiment","message":"Add partensor experiment\n","repos":"crypto-agda\/protocols","old_file":"partensor\/Terms.agda","new_file":"partensor\/Terms.agda","new_contents":"open import Function using (_\u2218_ ; id ; const ; flip)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Nat.NP\nopen import Data.List\nopen import Data.Product using (_,_ ; \u03a3 ; \u2203 ; _\u00d7_ ; proj\u2081 ; proj\u2082)\nopen import Data.Sum using (_\u228e_ ; inj\u2081 ; inj\u2082)\n  renaming (map to smap ; [_,_] to either ; [_,_]\u2032 to either\u2032)\n\nopen import Level\nopen import Size\n\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_ ; refl ; !_ ; ap ; tr ; _\u2219_ ; _\u2262_)\n\nmodule partensor.Terms where\n\ndata Com : Set where IN OUT : Com\n\ndata Session : Set\u2081 where\n  end : Session\n  com : Com \u2192 {M : Set} (P : M \u2192 Session) \u2192 Session\n\ndata Ended : Session \u2192 Set\u2081 where\n  end : Ended end\n\npattern send P = com OUT P\npattern recv P = com IN P\n\nmapSession : (Com \u2192 Com) \u2192 Session \u2192 Session\nmapSession f end = end\nmapSession f (com c P) = com (f c) (\u03bb m \u2192 mapSession f (P m))\n\ndualC : Com \u2192 Com\ndualC IN = OUT\ndualC OUT = IN\n\ndual : Session \u2192 Session\ndual = mapSession dualC\n\ninfix 5 _\u214b_ _\u2297_\ndata Proto : Set\u2081 where\n  act : Session \u2192 Proto\n  _\u214b_ _\u2297_ : Proto \u2192 Proto \u2192 Proto\n\ndual' : Proto \u2192 Proto\ndual' (act x) = act (dual x)\ndual' (P \u214b P\u2081) = dual' P \u2297 dual' P\u2081\ndual' (P \u2297 P\u2081) = dual' P \u214b dual' P\u2081\n\ndata Dual : (P Q : Proto) \u2192 Set\u2081 where\n  act  : \u2200 {P} \u2192 Dual (act P) (act (dual P))\n  act' : \u2200 {P} \u2192 Dual (act (dual P)) (act P)\n  \u2297\u214b : \u2200 {A A' B B'}\n     \u2192 Dual A A' \u2192 Dual A' A\n     \u2192 Dual B B' \u2192 Dual B' B\n     \u2192 Dual (A \u2297 B) (A' \u214b B')\n  \u214b\u2297 : \u2200 {A A' B B'}\n     \u2192 Dual A A' \u2192 Dual A' A\n     \u2192 Dual B B' \u2192 Dual B' B\n     \u2192 Dual (A \u214b B) (A' \u2297 B')\n\nsymDual : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nsymDual act = act'\nsymDual act' = act\nsymDual (\u2297\u214b x x\u2081 x\u2082 x\u2083) = \u214b\u2297 x\u2081 x x\u2083 x\u2082\nsymDual (\u214b\u2297 x x\u2081 x\u2082 x\u2083) = \u2297\u214b x\u2081 x x\u2083 x\u2082\n\nmkDual : \u2200 P \u2192 Dual P (dual' P)\nmkDual (act x) = act\nmkDual (P \u214b P\u2081) = \u214b\u2297 (mkDual P) (symDual (mkDual P)) (mkDual P\u2081) (symDual (mkDual P\u2081))\nmkDual (P \u2297 P\u2081) = \u2297\u214b (mkDual P) (symDual (mkDual P)) (mkDual P\u2081) (symDual (mkDual P\u2081))\n\ninfix 4 _\u2208'_ -- _\u2208_\ndata _\u2208'_ : Proto \u2192 Proto \u2192 Set\u2081 where\n  here   : \u2200 {S} \u2192 S \u2208' S\n  left  : \u2200 {P Q S} \u2192 S \u2208' P \u2192 S \u2208' (P \u214b Q)\n  right : \u2200 {P Q S} \u2192 S \u2208' Q \u2192 S \u2208' (P \u214b Q)\n\ninfix 4 _\u2286_\ndata _\u2286_ : (\u03a8 : Proto) \u2192 Proto \u2192 Set\u2081 where\n  \u2286-in : \u2200 {A B \u0393} \u2192 (A \u2297 B) \u2208' \u0393 \u2192 (A \u2297 B) \u2286 \u0393\n\n\u2286-left : \u2200 {P Q \u03a8} \u2192 \u03a8 \u2286 P \u2192 \u03a8 \u2286 P \u214b Q\n\u2286-left (\u2286-in x) = \u2286-in (left x)\n\ninfix 6 _\/_\n\n_[_\u2254_]' : {x : Proto}(\u0394 : Proto) \u2192 x \u2208' \u0394 \u2192 Proto \u2192 Proto\nx [ here \u2254 S' ]' = S'\n(\u0394 \u214b \u0394\u2081) [ left l \u2254 S' ]' = \u0394 [ l \u2254 S' ]' \u214b \u0394\u2081\n(\u0394 \u214b \u0394\u2081) [ right l \u2254 S' ]' = \u0394 \u214b \u0394\u2081 [ l \u2254 S' ]'\n\n_\/_ : {x : Proto} (\u0394 : Proto) \u2192 x \u2286 \u0394 \u2192 Proto\n\u0394 \/ (\u2286-in l) = \u0394 [ l \u2254 act end ]'\n\ndata PEnded : Proto \u2192 Set\u2081 where\n  \u03b5 : PEnded (act end)\n  P\u214b : \u2200 {P Q} \u2192 PEnded P \u2192 PEnded Q \u2192 PEnded (P \u214b Q)\n\ndata PEnded' : Proto \u2192 Set\u2081 where\n  \u03b5 : PEnded' (act end)\n  P\u214b : \u2200 {P Q} \u2192 PEnded' P \u2192 PEnded' Q \u2192 PEnded' (P \u214b Q)\n  P\u2297 : \u2200 {P Q} \u2192 PEnded' P \u2192 PEnded' Q \u2192 PEnded' (P \u2297 Q)\n\n_\u2208_ : Session \u2192 Proto \u2192 Set\u2081\nS \u2208 \u0393 = act S \u2208' \u0393\n\n\n_[_\u2254_] : {x : Session}(\u0394 : Proto) \u2192 x \u2208 \u0394 \u2192 Session \u2192 Proto\n\u0394 [ l \u2254 S ] = \u0394 [ l \u2254 act S ]'\n\ninfix 4 _\u2248_ _\u2248'_\ninfixr 4 _\u00b7_\n\nmodule _ {x M P}(let S = com x {M} P) where\n  \u2209-PEnd : \u2200 {P} \u2192 PEnded P \u2192 act S \u2208' P \u2192 \ud835\udfd8\n  \u2209-PEnd () here\n  \u2209-PEnd (P\u214b p p\u2081) (left l) = \u2209-PEnd p l\n  \u2209-PEnd (P\u214b p p\u2081) (right l) = \u2209-PEnd p\u2081 l\n\ndata _\u2248'_ : Proto \u2192 Proto \u2192 Set\u2081 where\n  _\u00b7_ : \u2200 {P Q R} \u2192 P \u2248' Q \u2192 Q \u2248' R \u2192 P \u2248' R\n  \u214b-cong\u2097 : \u2200 {P P' Q} \u2192 P \u2248' P' \u2192 P \u214b Q \u2248' P' \u214b Q\n  \u214b\u03b5 : \u2200 {P} \u2192 P \u214b act end \u2248' P\n  \u214b\u03b5' : \u2200 {P} \u2192 P \u2248' P \u214b act end\n  \u214b-comm : \u2200 {P Q} \u2192 P \u214b Q \u2248' Q \u214b P\n  \u214b-assoc : \u2200 {P Q R} \u2192 (P \u214b Q) \u214b R \u2248' P \u214b (Q \u214b R)\n\ndata _\u2248_ : Proto \u2192 Proto \u2192 Set\u2081 where\n  _\u00b7_ : \u2200 {P Q R} \u2192 P \u2248 Q \u2192 Q \u2248 R \u2192 P \u2248 R\n\n  \u214b-cong\u2097 : \u2200 {P P' Q} \u2192 P \u2248 P' \u2192 P \u214b Q \u2248 P' \u214b Q\n  \u214b\u03b5 : \u2200 {P} \u2192 P \u214b act end \u2248 P\n  \u214b\u03b5' : \u2200 {P} \u2192 P \u2248 P \u214b act end\n  \u214b-comm : \u2200 {P Q} \u2192 P \u214b Q \u2248 Q \u214b P\n  \u214b-assoc : \u2200 {P Q R} \u2192 (P \u214b Q) \u214b R \u2248 P \u214b (Q \u214b R)\n\n  \u2297-cong\u2097 : \u2200 {P P' Q} \u2192 P \u2248 P' \u2192 P \u2297 Q \u2248 P' \u2297 Q\n  \u2297\u03b5 : \u2200 {P} \u2192 P \u2297 act end \u2248 P\n  \u2297-comm : \u2200 {P Q} \u2192 P \u2297 Q \u2248 Q \u2297 P\n  \u2297-assoc : \u2200 {P Q R} \u2192 (P \u2297 Q) \u2297 R \u2248 P \u2297 (Q \u2297 R)\n\n\u2192\u2248' : \u2200 {P Q} \u2192 P \u2248' Q \u2192 P \u2248 Q\n\u2192\u2248' (x \u00b7 x\u2081) = \u2192\u2248' x \u00b7 \u2192\u2248' x\u2081\n\u2192\u2248' (\u214b-cong\u2097 x) = \u214b-cong\u2097 (\u2192\u2248' x)\n\u2192\u2248' \u214b\u03b5 = \u214b\u03b5\n\u2192\u2248' \u214b\u03b5' = \u214b\u03b5'\n\u2192\u2248' \u214b-comm = \u214b-comm\n\u2192\u2248' \u214b-assoc = \u214b-assoc\n\nid'\u2248 : \u2200 {P} \u2192 P \u2248' P\nid'\u2248 = \u214b\u03b5' \u00b7 \u214b\u03b5\n\n!'_ : \u2200 {P Q} \u2192 P \u2248' Q \u2192 Q \u2248' P\n!' (e \u00b7 e\u2081) = !' e\u2081 \u00b7 !' e\n!' \u214b-cong\u2097 e = \u214b-cong\u2097 (!' e)\n!' \u214b\u03b5 = \u214b\u03b5'\n!' \u214b\u03b5' = \u214b\u03b5\n!' \u214b-comm = \u214b-comm\n!' \u214b-assoc = \u214b-comm \u00b7 \u214b-cong\u2097 \u214b-comm \u00b7 \u214b-assoc \u00b7 \u214b-comm \u00b7 \u214b-cong\u2097 \u214b-comm\n\n\u214b'-cong\u1d63 : \u2200 {P P' Q} \u2192 P \u2248' P' \u2192 Q \u214b P \u2248' Q \u214b P'\n\u214b'-cong\u1d63 e = \u214b-comm \u00b7 \u214b-cong\u2097 e \u00b7 \u214b-comm\n\n\u214b'-cong : \u2200 {P P' Q Q'} \u2192 Q \u2248' Q' \u2192 P \u2248' P' \u2192 Q \u214b P \u2248' Q' \u214b P'\n\u214b'-cong e e' = \u214b-cong\u2097 e \u00b7 \u214b'-cong\u1d63 e'\n\n\u2248'-PEnd : \u2200 {P Q} \u2192 P \u2248' Q \u2192 PEnded P \u2192 PEnded Q\n\u2248'-PEnd (e \u00b7 e\u2081) p = \u2248'-PEnd e\u2081 (\u2248'-PEnd e p)\n\u2248'-PEnd (\u214b-cong\u2097 e) (P\u214b x x\u2081) = P\u214b (\u2248'-PEnd e x) x\u2081\n\u2248'-PEnd \u214b\u03b5 (P\u214b x x\u2081) = x\n\u2248'-PEnd \u214b\u03b5' p = P\u214b p \u03b5\n\u2248'-PEnd \u214b-comm (P\u214b x x\u2081) = P\u214b x\u2081 x\n\u2248'-PEnd \u214b-assoc (P\u214b (P\u214b p p\u2081) p\u2082) = P\u214b p (P\u214b p\u2081 p\u2082)\n\nPEnd-\u2248' : \u2200 {P Q} \u2192 PEnded P \u2192 PEnded Q \u2192 P \u2248' Q\nPEnd-\u2248' \u03b5 \u03b5 = id'\u2248\nPEnd-\u2248' \u03b5 (P\u214b q q\u2081) = \u214b\u03b5' \u00b7 \u214b'-cong (PEnd-\u2248' \u03b5 q) (PEnd-\u2248' \u03b5 q\u2081)\nPEnd-\u2248' (P\u214b p p\u2081) \u03b5 = \u214b'-cong (PEnd-\u2248' p \u03b5) (PEnd-\u2248' p\u2081 \u03b5) \u00b7 \u214b\u03b5\nPEnd-\u2248' (P\u214b p p\u2081) (P\u214b q q\u2081) = \u214b'-cong (PEnd-\u2248' p q) (PEnd-\u2248' p\u2081 q\u2081)\n\n\n\u2288-PEnd : \u2200 {P Q} \u2192 PEnded Q \u2192 P \u2286 Q \u2192 \ud835\udfd8\n\u2288-PEnd \u03b5 (\u2286-in ())\n\u2288-PEnd (P\u214b p p\u2081) (\u2286-in (left x)) = \u2288-PEnd p (\u2286-in x)\n\u2288-PEnd (P\u214b p p\u2081) (\u2286-in (right x)) = \u2288-PEnd p\u2081 (\u2286-in x)\n\ndata \u27ea_\u27eb (\u0394 : Proto) : Set\u2081 where\n  end : PEnded \u0394 \u2192 \u27ea \u0394 \u27eb\n\n  input : \u2200 {M P} (l : recv P \u2208 \u0394)\n    \u2192 ((m : M) \u2192 \u27ea \u0394 [ l \u2254 P m ] \u27eb)\n    \u2192 \u27ea \u0394 \u27eb\n\n  output : \u2200 {M P} (l : send P \u2208 \u0394)\n    \u2192 (m : M) \u2192 \u27ea \u0394 [ l \u2254 P m ] \u27eb\n    \u2192 \u27ea \u0394 \u27eb\n\n  pair : \u2200 {\u0393 \u0393' A B}\n    \u2192 (l : A \u2297 B \u2286 \u0394) \u2192 (\u0394 \/ l) \u2248' (\u0393 \u214b \u0393')\n    \u2192 \u27ea \u0393 \u214b A \u27eb \u2192 \u27ea \u0393' \u214b B \u27eb\n    \u2192 \u27ea \u0394 \u27eb\n\n\ndata NotPar : Proto \u2192 Set\u2081 where\n  act1 : \u2200 {x M P} \u2192 NotPar (act (com x {M} P))\n  ten : \u2200 {A B} \u2192 NotPar (A \u2297 B)\n\ndata NotPar' : Proto \u2192 Set\u2081 where\n  act0 : \u2200 {S} \u2192 NotPar' (act S)\n  ten : \u2200 {A B} \u2192 NotPar' (A \u2297 B)\n\n\u2254-same : \u2200 {P Q}(l : P \u2208' Q) \u2192 Q \u2248' Q [ l \u2254 P ]'\n\u2254-same here = id'\u2248\n\u2254-same (left x) = \u214b-cong\u2097 (\u2254-same x)\n\u2254-same (right x) = \u214b'-cong\u1d63 (\u2254-same x)\n\n\u2208'-conv : \u2200 {P Q \u0393} \u2192 NotPar \u0393 \u2192 P \u2248' Q \u2192 \u0393 \u2208' P \u2192 \u0393 \u2208' Q\n\u2208'-conv np (e \u00b7 e\u2081) l = \u2208'-conv np e\u2081 (\u2208'-conv np e l)\n\u2208'-conv () (\u214b-cong\u2097 e) here\n\u2208'-conv np (\u214b-cong\u2097 e) (left l) = left (\u2208'-conv np e l)\n\u2208'-conv np (\u214b-cong\u2097 e) (right l) = right l\n\u2208'-conv () \u214b\u03b5 here\n\u2208'-conv np \u214b\u03b5 (left l) = l\n\u2208'-conv () \u214b\u03b5 (right here)\n\u2208'-conv np \u214b\u03b5' l = left l\n\u2208'-conv () \u214b-comm here\n\u2208'-conv np \u214b-comm (left l) = right l\n\u2208'-conv np \u214b-comm (right l) = left l\n\u2208'-conv () \u214b-assoc here\n\u2208'-conv () \u214b-assoc (left here)\n\u2208'-conv np \u214b-assoc (left (left l)) = left l\n\u2208'-conv np \u214b-assoc (left (right l)) = right (left l)\n\u2208'-conv np \u214b-assoc (right l) = right (right l)\n\n\u2254'-conv : \u2200 {P Q \u0393 S'}(np : NotPar \u0393)(e : P \u2248' Q)(l : \u0393 \u2208' P)\n  \u2192 P [ l \u2254 S' ]' \u2248' Q [ \u2208'-conv np e l \u2254 S' ]'\n\u2254'-conv np (e \u00b7 e\u2081) l = \u2254'-conv np e l \u00b7 \u2254'-conv np e\u2081 _\n\u2254'-conv () (\u214b-cong\u2097 e) here\n\u2254'-conv np (\u214b-cong\u2097 e) (left l) = \u214b-cong\u2097 (\u2254'-conv np e l)\n\u2254'-conv np (\u214b-cong\u2097 e) (right l) = \u214b-cong\u2097 e\n\u2254'-conv () \u214b\u03b5 here\n\u2254'-conv np \u214b\u03b5 (left l) = \u214b\u03b5\n\u2254'-conv () \u214b\u03b5 (right here)\n\u2254'-conv np \u214b\u03b5' l = \u214b\u03b5'\n\u2254'-conv () \u214b-comm here\n\u2254'-conv np \u214b-comm (left l) = \u214b-comm\n\u2254'-conv np \u214b-comm (right l) = \u214b-comm\n\u2254'-conv () \u214b-assoc here\n\u2254'-conv () \u214b-assoc (left here)\n\u2254'-conv np \u214b-assoc (left (left l)) = \u214b-assoc\n\u2254'-conv np \u214b-assoc (left (right l)) = \u214b-assoc\n\u2254'-conv np \u214b-assoc (right l) = \u214b-assoc\n\n\u2286-conv : \u2200 {P Q \u0393} \u2192 P \u2248' Q \u2192 \u0393 \u2286 P \u2192 \u0393 \u2286 Q\n\u2286-conv e (\u2286-in x) = \u2286-in (\u2208'-conv ten e x)\n\n\/-conv : \u2200 {P Q \u0393}(e : P \u2248' Q)(l : \u0393 \u2286 P) \u2192 P \/ l \u2248' Q \/ \u2286-conv e l\n\/-conv e (\u2286-in x) = \u2254'-conv ten e x\n\nmodule _ {x M P}(let S = com x {M} P) where\n\n  \u2208-conv : \u2200 {P Q} \u2192 P \u2248' Q \u2192 S \u2208 P \u2192 S \u2208 Q\n  \u2208-conv e l = \u2208'-conv act1 e l\n\n  \u2254-conv : \u2200 {P Q S'}(e : P \u2248' Q)(l : S \u2208 P) \u2192 P [ l \u2254 S' ] \u2248' Q [ \u2208-conv e l \u2254 S' ]\n  \u2254-conv e l = \u2254'-conv act1 e l\n\n\nconv : \u2200 {P Q} \u2192 P \u2248' Q \u2192 \u27ea P \u27eb \u2192 \u27ea Q \u27eb\nconv e (end x) = end (\u2248'-PEnd e x)\nconv e (input l x) = input (\u2208-conv e l) (\u03bb m \u2192 conv (\u2254-conv e l) (x m))\nconv e (output l m p) = output (\u2208-conv e l) m (conv (\u2254-conv e l) p)\nconv e (pair l x p p\u2081) = pair (\u2286-conv e l ) (!' \/-conv e l \u00b7 x) p p\u2081\n\nfwd-S : \u2200 S \u2192 \u27ea act S \u214b act (dual S) \u27eb\nfwd-S end = end (P\u214b \u03b5 \u03b5)\nfwd-S (send P) = input (right here) \u03bb m \u2192\n                 output (left here) m (fwd-S (P m))\nfwd-S (recv P) = input (left here) \u03bb m \u2192\n                 output (right here) m (fwd-S (P m))\n\nfwd : \u2200 \u0393 \u2192 \u27ea \u0393 \u214b dual' \u0393 \u27eb\nfwd (act x) = fwd-S x\nfwd (\u0393 \u214b \u0393\u2081) = pair (\u2286-in (right here)) \u214b\u03b5 (fwd \u0393) (fwd \u0393\u2081)\nfwd (\u0393 \u2297 \u0393\u2081) = pair (\u2286-in (left here)) (\u214b-comm \u00b7 \u214b\u03b5) (conv \u214b-comm (fwd \u0393)) (conv \u214b-comm (fwd \u0393\u2081))\n\nsame-var : \u2200 {\u0394 \u0393 \u0393'}(np : NotPar' \u0393)(np' : NotPar' \u0393')(l : \u0393 \u2208' \u0394)(l' : \u0393' \u2208' \u0394) \u2192\n  (\u2203 \u03bb (nl : \u2200 {S'} \u2192 \u0393 \u2208' \u0394 [ l' \u2254 S' ]')\n  \u2192 \u2203 \u03bb (nl' : \u2200 {S} \u2192 \u0393' \u2208' \u0394 [ l \u2254 S ]')\n  \u2192 \u2200 {S S'} \u2192 ((\u0394 [ l' \u2254 S' ]') [ nl \u2254 S ]') \u2248' (\u0394 [ l \u2254 S ]') [ nl' \u2254 S' ]')\n  \u228e \u2203 \u03bb (p : \u0393 \u2261 \u0393') \u2192 tr _ p l \u2261 l'\nsame-var np np' here here = inj\u2082 (refl , refl)\nsame-var () np' here (left l')\nsame-var () np' here (right l')\nsame-var np () (left l) here\nsame-var np np' (left l) (left l') with same-var np np' l l'\nsame-var np np' (left l) (left l') | inj\u2081 (nl , nl' , s)\n  = inj\u2081 (left nl , left nl' , \u214b-cong\u2097 s)\nsame-var np np' (left l) (left .l) | inj\u2082 (refl , refl) = inj\u2082 (refl , refl)\nsame-var np np' (left l) (right l') = inj\u2081 (left l , right l' , id'\u2248)\nsame-var np () (right l) here\nsame-var np np' (right l) (left l') = inj\u2081 (right l , left l' , id'\u2248)\nsame-var np np' (right l) (right l') with same-var np np' l l'\nsame-var np np' (right l) (right l') | inj\u2081 (nl , nl' , s)\n  = inj\u2081 (right nl , right nl' , \u214b'-cong\u1d63 s)\nsame-var np np' (right l) (right .l) | inj\u2082 (refl , refl) = inj\u2082 (refl , refl)\n\nsame-\u2297var : \u2200 {\u0394 \u0393 \u0393'}(l : \u0393 \u2286 \u0394)(l' : \u0393' \u2286 \u0394) \u2192\n  (\u2203 \u03bb (nl : \u0393 \u2286 \u0394 \/ l') \u2192 \u2203 \u03bb (nl' : \u0393' \u2286 \u0394 \/ l) \u2192 (\u0394 \/ l') \/ nl \u2248' (\u0394 \/ l) \/ nl') \u228e \u2203 \u03bb (p : \u0393 \u2261 \u0393') \u2192 tr _ p l \u2261 l'\nsame-\u2297var (\u2286-in l) (\u2286-in l') with same-var ten ten l l'\nsame-\u2297var (\u2286-in l) (\u2286-in l') | inj\u2081 (nl , nl' , s) = inj\u2081 (\u2286-in nl , \u2286-in nl' , s)\nsame-\u2297var (\u2286-in l) (\u2286-in .l) | inj\u2082 (refl , refl) = inj\u2082 (refl , refl)\n\n\n\u2208'-\u2254' : \u2200 {\u0394 \u0393 \u0393' S}(l : \u0393 \u2208' \u0394) \u2192 \u0393' \u2208' \u0394 \u2192 NotPar' \u0393 \u2192 NotPar' \u0393'\n  \u2192 \u0393 \u2262 \u0393' \u2192 \u0393' \u2208' \u0394 [ l \u2254 S ]'\n\u2208'-\u2254' here here np np' e = \ud835\udfd8-elim (e refl)\n\u2208'-\u2254' (left l) here np () e\n\u2208'-\u2254' (right l) here np () e\n\u2208'-\u2254' here (left l') () np' e\n\u2208'-\u2254' (left l) (left l') np np' e = left (\u2208'-\u2254' l l' np np' e)\n\u2208'-\u2254' (right l) (left l') np np' e = left l'\n\u2208'-\u2254' here (right l') () np' e\n\u2208'-\u2254' (left l) (right l') np np' e = right l'\n\u2208'-\u2254' (right l) (right l') np np' e = right (\u2208'-\u2254' l l' np np' e)\n\n\u2254'-swap : \u2200 {\u0394 \u0393 \u0393' M M'}(l : \u0393 \u2208' \u0394)(l' : \u0393' \u2208' \u0394)\n    (np : NotPar' \u0393)(np' : NotPar' \u0393')(e : \u0393 \u2262 \u0393')(e' : \u0393' \u2262 \u0393)\n  \u2192 (\u0394 [ l' \u2254 M' ]') [ \u2208'-\u2254' l' l np' np e' \u2254 M ]'\n  \u2248' (\u0394 [ l \u2254 M ]') [ \u2208'-\u2254' l l' np np' e \u2254 M' ]'\n\u2254'-swap here here np np' e e' = \ud835\udfd8-elim (e refl)\n\u2254'-swap here (left l') () np' e e'\n\u2254'-swap here (right l') () np' e e'\n\u2254'-swap (left l) here np () e e'\n\u2254'-swap (left l) (left l') np np' e e' = \u214b-cong\u2097 (\u2254'-swap l l' np np' e e')\n\u2254'-swap (left l) (right l') np np' e e' = id'\u2248\n\u2254'-swap (right l) here np () e e'\n\u2254'-swap (right l) (left l') np np' e e' = id'\u2248\n\u2254'-swap (right l) (right l') np np' e e' = \u214b'-cong\u1d63 (\u2254'-swap l l' np np' e e')\n\nmodule _ {S} where\n  \u2208-\/ : \u2200 {\u0394 \u0393}(l : \u0393 \u2286 \u0394) \u2192 S \u2208 \u0394 \u2192 S \u2208 (\u0394 \/ l)\n  \u2208-\/ (\u2286-in l) l' = \u2208'-\u2254' l l' ten act0 (\u03bb ())\n\n  \u2286-\u2254 : \u2200 {\u0393 \u0394 M} \u2192 \u0393 \u2286 \u0394 \u2192 (l : S \u2208 \u0394) \u2192 \u0393 \u2286 \u0394 [ l \u2254 M ]\n  \u2286-\u2254 (\u2286-in l) l' = \u2286-in (\u2208'-\u2254' l' l act0 ten (\u03bb ()))\n\n  \u2254\/ : \u2200 {\u0393 \u0394 M}(l : \u0393 \u2286 \u0394)(v : S \u2208 \u0394) \u2192 \u0394 [ v \u2254 M ] \/ \u2286-\u2254 l v \u2248' (\u0394 \/ l) [ \u2208-\/ l v \u2254 M ]\n  \u2254\/ (\u2286-in l) l' = \u2254'-swap l l' ten act0 (\u03bb ()) (\u03bb ())\n\nin-sub : \u2200 {\u0393 \u0393' \u0394}(l : \u0393 \u2208' \u0394) \u2192 \u0393' \u2208' \u0394 [ l \u2254 \u0393' ]'\nin-sub here = here\nin-sub (left x) = left (in-sub x)\nin-sub (right x) = right (in-sub x)\n\nsub-twice : \u2200 {\u0393 \u0393' \u0393'' \u0394}(l : \u0393 \u2208' \u0394) \u2192\n  (\u0394 [ l \u2254 \u0393' ]') [ in-sub l \u2254 \u0393'' ]'\n  \u2248' \u0394 [ l \u2254 \u0393'' ]'\nsub-twice here = id'\u2248\nsub-twice (left x) = \u214b-cong\u2097 (sub-twice x)\nsub-twice (right x) = \u214b'-cong\u1d63 (sub-twice x)\n\nact\u2260\u2297 : \u2200 {S A B} \u2192 act S \u2262 A \u2297 B\nact\u2260\u2297 ()\n\n\u2297\u2260act : \u2200 {S A B} \u2192 A \u2297 B \u2262 act S\n\u2297\u2260act ()\n\nmix : \u2200 {P Q} \u2192 \u27ea P \u27eb \u2192 \u27ea Q \u27eb \u2192 \u27ea P \u214b Q \u27eb\nmix (end x) q = conv (\u214b\u03b5' \u00b7 \u214b-comm \u00b7 \u214b-cong\u2097 (PEnd-\u2248' \u03b5 x)) q\nmix (input l x) q = input (left l) \u03bb m \u2192 mix (x m) q\nmix (output l m p) q = output (left l) m (mix p q)\nmix (pair (\u2286-in l) x p p\u2081) q = pair (\u2286-in (left l)) (\u214b-cong\u2097 x \u00b7 \u214b-assoc)\n  p (conv (\u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc) (mix p\u2081 q))\n\nend' : \u2200 {\u0394} \u2192 PEnded' \u0394 \u2192 \u27ea \u0394 \u27eb\nend' \u03b5 = end \u03b5\nend' (P\u214b p p\u2081) = mix (end' p) (end' p\u2081)\nend' (P\u2297 p p\u2081) = pair (\u2286-in here) \u214b\u03b5' (conv (\u214b\u03b5' \u00b7 \u214b-comm) (end' p)) (conv (\u214b\u03b5' \u00b7 \u214b-comm) (end' p\u2081))\n\ncut\u2081 : \u2200 {\u0394 \u0394' S}(l : S \u2208 \u0394)(l' : dual S \u2208 \u0394') \u2192 \u27ea \u0394 \u27eb \u2192 \u27ea \u0394' \u27eb \u2192 \u27ea \u0394 [ l \u2254 end ] \u214b \u0394' [ l' \u2254 end ] \u27eb\ncut\u2081 {S = end} cl cl' p q = conv (\u214b'-cong (\u2254-same cl) (\u2254-same cl')) (mix p q)\ncut\u2081 {S = com x P} cl cl' (pair (\u2286-in tl) s p p\u2081) q\n with \u2208'-\u2254' {S = act end} cl tl act0 ten act\u2260\u2297\n    | \u2208'-\u2254' {S = act end} tl cl ten act0 \u2297\u2260act\n    | \u2254'-swap {M = act end} {M' = act end} cl tl act0 ten act\u2260\u2297 \u2297\u2260act\n... | ntl | ncl | sub with \u2208-conv s ncl | \u2254-conv {S' = end} s ncl\n... | left gncl | sub' = pair (\u2286-in (left ntl))\n                            (\u214b-cong\u2097 (!' sub \u00b7 sub')\n                            \u00b7 \u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc)\n                            (conv (\u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc)\n                              (cut\u2081 (left gncl) cl' p q))\n                            p\u2081\n... | right gncl | sub' = pair (\u2286-in (left ntl))\n       (\u214b-cong\u2097 (!' sub \u00b7 sub') \u00b7 \u214b-assoc)\n       p\n       (conv (\u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc)\n         (cut\u2081 (left gncl) cl' p\u2081 q))\ncut\u2081 {S = com x P} cl cl' p (pair (\u2286-in tl) s q q\u2081)\n with \u2208'-\u2254' {S = act end} cl' tl act0 ten act\u2260\u2297\n    | \u2208'-\u2254' {S = act end} tl cl' ten act0 \u2297\u2260act\n    | \u2254'-swap {M = act end} {M' = act end} cl' tl act0 ten act\u2260\u2297 \u2297\u2260act\n... | ntl | ncl' | sub with \u2208-conv s ncl' | \u2254-conv {S' = end} s ncl'\n... | left gncl' | sub' = pair (\u2286-in (right ntl))\n  (\u214b'-cong\u1d63 (!' sub \u00b7 sub') \u00b7 !' \u214b-assoc)\n  (conv (!' \u214b-assoc) (cut\u2081 cl (left gncl') p q))\n  q\u2081\n... | right gncl' | sub' = pair (\u2286-in (right ntl))\n  (\u214b'-cong\u1d63 (!' sub \u00b7 sub') \u00b7 !' \u214b-assoc \u00b7 \u214b-cong\u2097 \u214b-comm \u00b7 \u214b-assoc)\n  q\n  (conv (!' \u214b-assoc) (cut\u2081 cl (left gncl') p q\u2081))\ncut\u2081 {S = com x P} cl cl' (end p) q = \ud835\udfd8-elim (\u2209-PEnd p cl)\ncut\u2081 {S = com x P} cl cl' p (end q) = \ud835\udfd8-elim (\u2209-PEnd q cl')\n\ncut\u2081 cl cl' (input l x) (input l' x\u2081) with same-var act0 act0 cl l | same-var act0 act0 cl' l'\ncut\u2081 cl cl' (input l x\u2081) (input l' x\u2082) | inj\u2081 (ncl , nl , s) | Q = input (left nl) \u03bb m\n  \u2192 conv (\u214b-cong\u2097 s) (cut\u2081 ncl cl' (x\u2081 m) (input l' x\u2082))\ncut\u2081 cl cl' (input l x\u2081) (input l' x\u2082) | inj\u2082 y | inj\u2081 (ncl' , nl' , s) = input (right nl') \u03bb m\n  \u2192 conv (\u214b'-cong\u1d63 s) (cut\u2081 cl ncl' (input l x\u2081) (x\u2082 m))\ncut\u2081 cl cl' (input l x) (input l' x\u2081) | inj\u2082 (refl , proj\u2082) | inj\u2082 (() , proj\u2084)\n\ncut\u2081 cl cl' (input l p) (output l' m q) with same-var act0 act0 cl l | same-var act0 act0 cl' l'\ncut\u2081 cl cl' (input l p) (output l' m q) | inj\u2081 (ncl , nl , s) | Q = input (left nl) \u03bb m'\n \u2192 conv (\u214b-cong\u2097 s) (cut\u2081 ncl cl' (p m') (output l' m q))\ncut\u2081 cl cl' (input l p) (output l' m q) | inj\u2082 y | inj\u2081 (ncl' , nl' , s) = output (right nl') m\n  (conv (\u214b'-cong\u1d63 s) (cut\u2081 cl ncl' (input l p) q))\ncut\u2081 cl cl' (input .cl p) (output .cl' m q) | inj\u2082 (refl , refl) | inj\u2082 (refl , refl)\n  = conv (\u214b'-cong (sub-twice cl) (sub-twice cl')) (cut\u2081 (in-sub cl) (in-sub cl') (p m) q)\n\ncut\u2081 cl cl' (output l m p) (input l' q) with same-var act0 act0 cl l | same-var act0 act0 cl' l'\ncut\u2081 cl cl' (output l m p) (input l' q) | inj\u2081 (ncl , nl , s) | Q = output (left nl) m\n  (conv (\u214b-cong\u2097 s) (cut\u2081 ncl cl' p (input l' q)))\ncut\u2081 cl cl' (output l m p) (input l' q) | inj\u2082 y | inj\u2081 (ncl' , nl' , s) = input (right nl') \u03bb m' \u2192\n  conv (\u214b'-cong\u1d63 s) (cut\u2081 cl ncl' (output l m p) (q m'))\ncut\u2081 cl cl' (output .cl m p) (input .cl' q) | inj\u2082 (refl , refl) | inj\u2082 (refl , refl)\n  = conv (\u214b'-cong (sub-twice cl) (sub-twice cl')) (cut\u2081 (in-sub cl) (in-sub cl') p (q m))\n\ncut\u2081 cl cl' (output l m p) (output l' m\u2081 q) with same-var act0 act0 cl l | same-var act0 act0 cl' l'\ncut\u2081 cl cl' (output l m p) (output l' m\u2081 q) | inj\u2081 (ncl , nl , s) | Q = output (left nl) m (conv (\u214b-cong\u2097 s)\n  (cut\u2081 ncl cl' p (output l' m\u2081 q)))\ncut\u2081 cl cl' (output l m p) (output l' m\u2081 q) | inj\u2082 y | inj\u2081 (ncl' , nl' , s) = output (right nl') m\u2081\n  (conv (\u214b'-cong\u1d63 s) (cut\u2081 cl ncl' (output l m p) q))\ncut\u2081 cl cl' (output l m p) (output l' m\u2081 q) | inj\u2082 (refl , proj\u2082) | inj\u2082 (() , proj\u2084)\n\nthe-cut\u2081 : \u2200 {\u0394 \u0394' S} \u2192 \u27ea \u0394 \u214b act S \u27eb \u2192 \u27ea act (dual S) \u214b \u0394' \u27eb \u2192 \u27ea \u0394 \u214b \u0394' \u27eb\nthe-cut\u2081 p q = conv (\u214b-cong\u2097 \u214b\u03b5 \u00b7 \u214b'-cong\u1d63 (\u214b-comm \u00b7 \u214b\u03b5)) (cut\u2081 (right here) (left here) p q)\n\n\nmutual\n  cut\u2297 : \u2200 {\u0394 \u0394' A A' B B'} \u2192 Dual A A' \u2192 Dual B B' \u2192 (l : (A \u2297 B) \u2286 \u0394) \u2192\n    \u27ea \u0394 \u27eb \u2192 \u27ea (A' \u214b B') \u214b \u0394' \u27eb \u2192 \u27ea \u0394 \/ l \u214b \u0394' \u27eb\n  cut\u2297 da db cl (end x) q = \ud835\udfd8-elim (\u2288-PEnd x cl)\n  cut\u2297 da db cl (input l x) q = input (left (\u2208-\/ cl l)) (\u03bb m \u2192 conv (\u214b-cong\u2097 (\u2254\/ cl l)) (cut\u2297 da db (\u2286-\u2254 cl l) (x m) q))\n  cut\u2297 da db cl (output l m p) q = output (left (\u2208-\/ cl l)) m (conv (\u214b-cong\u2097 (\u2254\/ cl l)) (cut\u2297 da db (\u2286-\u2254 cl l) p q))\n  cut\u2297 da db cl (pair l x p p\u2081) q with same-\u2297var cl l\n  cut\u2297 da db cl (pair l x\u2081 p p\u2081) q | inj\u2081 (cl' , l' , s) with \u2286-conv x\u2081 cl' | \/-conv x\u2081 cl'\n  cut\u2297 da db cl (pair (\u2286-in l) x\u2081 p p\u2081) q | inj\u2081 (cl' , \u2286-in l' , s) | \u2286-in (left P\u2081) | sub\n     = pair (\u2286-in (left l'))\n     (\u214b-cong\u2097  (!' s \u00b7 sub) \u00b7 \u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc)\n     (conv (\u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc) (cut\u2297 da db (\u2286-in (left P\u2081)) p q)) p\u2081\n  cut\u2297 da db cl (pair (\u2286-in l) x\u2081 p p\u2081) q | inj\u2081 (cl' , \u2286-in l' , s) | \u2286-in (right P\u2081) | sub\n     = pair (\u2286-in (left l')) (\u214b-cong\u2097 (!' s \u00b7 sub) \u00b7 \u214b-assoc) p\n     (conv (\u214b-assoc \u00b7 \u214b'-cong\u1d63 \u214b-comm \u00b7 !' \u214b-assoc) (cut\u2297 da db (\u2286-in (left P\u2081)) p\u2081 q ))\n  cut\u2297 da db cl (pair .cl x p p\u2081) q | inj\u2082 (refl , refl) = conv\n    (!' \u214b-assoc \u00b7 \u214b-cong\u2097 (\u214b-comm \u00b7 !' x))\n    (let X = cut da p (conv \u214b-assoc q)\n      in cut db p\u2081 (conv (!' \u214b-assoc \u00b7 \u214b-cong\u2097 \u214b-comm \u00b7 \u214b-assoc) X))\n\n  cut : \u2200 {\u0394 \u0394' \u03a8 \u03a8'} \u2192 Dual \u03a8 \u03a8' \u2192 \u27ea \u0394 \u214b \u03a8 \u27eb \u2192 \u27ea \u03a8' \u214b \u0394' \u27eb \u2192 \u27ea \u0394 \u214b \u0394' \u27eb\n  cut act p q = the-cut\u2081 p q\n  cut act' p q = conv \u214b-comm (the-cut\u2081 (conv \u214b-comm q) (conv \u214b-comm p))\n  cut (\u2297\u214b d d\u2081 d\u2082 d\u2083) p q = conv (\u214b-cong\u2097 \u214b\u03b5) (cut\u2297 d d\u2082 (\u2286-in (right here)) p q)\n  cut (\u214b\u2297 d d\u2081 d\u2082 d\u2083) p q = conv (\u214b-cong\u2097 (\u214b-comm \u00b7 \u214b\u03b5) \u00b7 \u214b-comm)\n                            (cut\u2297 d\u2081 d\u2083 (\u2286-in (left here)) q (conv \u214b-comm p))\n\nthe-cut : \u2200 {\u0394 \u0394'} \u03a8 \u2192 \u27ea \u0394 \u214b \u03a8 \u27eb \u2192 \u27ea dual' \u03a8 \u214b \u0394' \u27eb \u2192 \u27ea \u0394 \u214b \u0394' \u27eb\nthe-cut \u03a8 = cut (mkDual \u03a8)\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'partensor\/Terms.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"35043fdf0f8261102f21acdf3484d89c7deb2be6","subject":"TrustMe.NP","message":"TrustMe.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/PropositionalEquality\/TrustMe\/NP.agda","new_file":"lib\/Relation\/Binary\/PropositionalEquality\/TrustMe\/NP.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Relation.Binary.PropositionalEquality.TrustMe.NP where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary.PropositionalEquality.TrustMe public\n\nerase : \u2200 {a} {A : Set a} {x y : A} \u2192 x \u2261 y \u2192 x \u2261 y\nerase _ = trustMe\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Relation\/Binary\/PropositionalEquality\/TrustMe\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cd9d3b220f9c8d39ec3b5b0d5f463bc5eeea3168","subject":"Added note on bisimulations.","message":"Added note on bisimulations.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Relation\/Binary\/Bisimilarity\/NonBisimulation.agda","new_file":"notes\/FOT\/FOTC\/Relation\/Binary\/Bisimilarity\/NonBisimulation.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Relation.Binary.Bisimilarity.NonBisimulation where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\nh : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2261 xs \u2192\n      \u2203[ x' ] \u2203[ xs' ] \u2203[ ys' ] xs \u2261 x' \u2237 xs' \u2227 xs \u2261 x' \u2237 ys' \u2227 xs' \u2261 xs'\nh Sxs _ with Stream-unf Sxs\n... | x' , xs' , xs\u2261x'\u2237xs' , _ = x' , xs' , xs' , xs\u2261x'\u2237xs' , xs\u2261x'\u2237xs' , refl\n\n\u2248-refl : \u2200 {xs} \u2192 Stream xs \u2192 xs \u2248 xs\n\u2248-refl {xs} Sxs = \u2248-coind (\u03bb ys _ \u2192 ys \u2261 ys) (h Sxs) refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Relation\/Binary\/Bisimilarity\/NonBisimulation.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"0ba9448cbcb6075a7a254c2984eea58c1995f8b1","subject":"second attempt at induction via map","message":"second attempt at induction via map","repos":"kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/Data1.agda","new_file":"models\/Data1.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check --no-positivity-check\n #-}\n\nmodule Data1 where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Cx : Set where\n  E : Cx\n  _\/_ : Cx -> Set -> Cx\n\ndata _:>_ : Cx -> Set -> Set where\n  ze : {G : Cx}{S : Set} -> (G \/ S) :> S\n  su : {G : Cx}{S T : Set} -> G :> T -> (G \/ S) :> T\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (G : Cx) : Set where\n  ?? : {S : Set} -> G :> S -> S -> CON G\n  PrfC : PROP -> CON G\n  _*C_ : CON G -> CON G -> CON G\n  PiC : (S : Set) -> (S -> CON G) -> CON G\n  SiC : (S : Set) -> (S -> CON G) -> CON G\n  MuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n  NuC : (O : Set) -> (O -> CON (G \/ O)) -> O -> CON G\n\nBox : (G : Cx) -> ((S : Set) -> (G :> S) -> Set) -> Set\nBox E F = One\nBox (G \/ S) F = Box G (\\ S n -> F S (su n)) * F S ze\n\npr : {G : Cx}{S : Set}{F : (S : Set) -> (G :> S) -> Set} -> Box G F -> (n : G :> S) -> F S n\npr fsf ze = snd fsf\npr fsf (su n) = pr (fst fsf) n\n\nPay : Cx -> Set\nPay G = Box G (\\ I _ -> I -> Set)\n\nArr : (G : Cx) -> Pay G -> Pay G -> Set\nArr G X Y = Box G (\\ S n -> (s : S) -> pr X n s -> pr Y n s)\n\n\n\nmutual\n  [|_|] : {G : Cx} -> CON G -> Pay G -> Set\n  [| ?? n i |]     X = pr X n i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Mu C X ) -> Mu C X o\n  codata Nu {G : Cx}{O : Set}(C : O -> CON (G \/ O))(X : Pay G)(o : O) : Set where\n    con : [| C o |] ( X , Nu C X ) -> Nu C X o\n\nnoc :  {G : Cx}{O : Set}{C : O -> CON (G \/ O)}{X : Pay G}{o : O} ->\n       Nu C X o -> [| C o |] ( X , Nu C X )\nnoc (con xs) = xs\n\nmutual\n  map : {G : Cx}{X Y : Pay G} -> (C : CON G) ->\n        Arr G X Y ->\n        [| C |] X -> [| C |] Y\n  map (?? n i)     f = pr f n i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = foldMap C (Mu C _) f (\\ o -> con) o\n  map (NuC O C o)  f = unfoldMap C (Nu C _) f (\\ o -> noc) o\n\n  foldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n          Arr G X Y ->\n          ((o : O) -> [| C o |] ( Y , Z ) -> Z o) ->\n          (o : O) -> Mu C X o -> Z o\n  foldMap C Z f alg o (con xs) = alg o (map (C o) (f , foldMap C Z f alg) xs)\n\n  unfoldMap :  {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X Y : Pay G}(Z : O -> Set) ->\n            Arr G X Y ->\n            ((o : O) -> Z o -> [| C o |] ( X , Z )) ->\n            (o : O) -> Z o -> Nu C Y o\n  unfoldMap C Z f coalg o y = con (map (C o) (f , unfoldMap C Z f coalg) (coalg o y))\n\nida : (G : Cx) -> (X : Pay G) -> Arr G X X\nida E = id\nida (G \/ S) = sig \\ Xs X -> ida G Xs , \\ s x -> x\n\ncompa : (G : Cx) -> (X Y Z : Pay G) -> Arr G Y Z -> Arr G X Y -> Arr G X Z\ncompa E = \\ _ _ _ _ _ -> _\ncompa (G \/ S) = sig \\ Xs X -> sig \\ Ys Y -> sig \\ Zs Z -> sig \\ fs f -> sig \\ gs g ->\n  compa G Xs Ys Zs fs gs , \\ s x -> f s (g s x)\n\ninduction : {G : Cx}{O : Set}(C : O -> CON (G \/ O)){X : Pay G}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] (X , (\\ o -> Sig (Mu C X o) (P o)))) ->\n             P o (con (map (C o) (ida G X , konst fst) xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction {G} C {X} P p o (con xys) =\n  p o (map (C o) (ida G X , (\\ o y -> (y , induction C P p o y))) xys)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Data1.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"2f7c08fd74bea61b080979aab90459d79f2c6288","subject":"Control\/Protocol\/IP.agda","message":"Control\/Protocol\/IP.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/IP.agda","new_file":"Control\/Protocol\/IP.agda","new_contents":"open import Type\nopen import Function\nopen import Control.Protocol.Choreography\nopen import Data.One\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Maybe\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nmodule Control.Protocol.IP (Query : \u2605) (Resp : Query \u2192 \u2605) where\n\n-- One round from the point of view of the verifier\nVerifierRound : Proto\nVerifierRound = \u03a3\u1d3e Query    \u03bb q \u2192\n                \u03a0\u1d3e (Resp q) \u03bb r \u2192\n                end\n\nProverRound : Proto\nProverRound = dual VerifierRound\n\nmodule Rounds (#rounds : \u2115) where\n  Verifier\u1d3e : Proto\n  Verifier\u1d3e = replicate\u1d3e #rounds VerifierRound >> (\u03a3\u1d3e Accept? \u03bb _ \u2192 end)\n\n  Prover\u1d3e   : Proto\n  Prover\u1d3e   = dual Verifier\u1d3e\n\n  Transcript\u1d3e : Proto\n  Transcript\u1d3e = source-of Verifier\u1d3e\n\n  Prover   : \u2605\n  Prover   = \u27e6 Prover\u1d3e \u27e7\n\n  Verifier : \u2605\n  Verifier = \u27e6 Verifier\u1d3e \u27e7\n\n  Transcript : \u2605\n  Transcript = \u27e6 Transcript\u1d3e \u27e7\n\n  _\u21c6_ : Prover \u2192 Verifier \u2192 Accept?\n  _\u21c6_ = \u03bb p v \u2192 case >>-com (replicate\u1d3e #rounds VerifierRound) v p of \u03bb { (_ , (a , _) , _) \u2192 a }\n\nrecord IP (#rounds : \u2115) {A : \u2605} (\u2112 : A \u2192 \u2605) : \u2605 where\n  open Rounds #rounds public\n\n  field\n    verifier : Verifier\n\n  Convincing : Prover \u2192 \u2605\n  Convincing = \u03bb p \u2192 Is-accept (p \u21c6 verifier)\n\n  Complete : \u2605\n  Complete = \u2200 x \u2192 \u2112 x \u2192 \u03a3 Prover Convincing\n\n  Sound : \u2605\n  Sound = \u2200 x \u2192 \u00ac(\u2112 x) \u2192 (p : Prover) \u2192 \u00ac(Convincing p)\n\nNP-Verifier = \u03a3 Query \u03bb q \u2192 Resp q \u2192 Accept? \u00d7 \ud835\udfd9\n\nNP-Verifier\u2261Verifier1 : NP-Verifier \u2261 Rounds.Verifier 1\nNP-Verifier\u2261Verifier1 = refl\n\nNP : {A : \u2605} (\u2112 : A \u2192 \u2605) \u2192 \u2605\nNP = IP 1\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/IP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2a3e2eab275766112367365c9989c813962995c1","subject":"languages\/agda\/test.agda","message":"languages\/agda\/test.agda\n","repos":"sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis","old_file":"languages\/agda\/test.agda","new_file":"languages\/agda\/test.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/sharkspeed\/dororis.git\/'\n","license":"bsd-2-clause","lang":"Agda"}
{"commit":"04d4693c8280b9c25e5b8c04f0489228136bcd98","subject":"Added draft: The relation LTL is well-founded (using the image inverse combinator from the standard library).","message":"Added draft: The relation LTL is well-founded (using the image inverse combinator from the standard library).\n\nIgnore-this: 9e870170aec2a84a8400f4bc3d41951d\n\ndarcs-hash:20110305070736-3bd4e-5e3478597e514100f8302001b24ae92a76da00cc.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/List\/LT-LengthSL.agda","new_file":"Draft\/List\/LT-LengthSL.agda","new_contents":"------------------------------------------------------------------------------\n-- The relation LTL is well-founded using the image inverse combinator\n-- from the standard library\n------------------------------------------------------------------------------\n\nmodule LT-LengthSL {A : Set} where\n\nopen import Data.List\nopen import Data.Nat\n\nopen import Induction.Nat\nopen import Induction.WellFounded\n\nopen module II =\n  Induction.WellFounded.Inverse-image length\n\nLTL : List A \u2192 List A \u2192 Set\nLTL xs ys = (length xs) <\u2032 (length ys)\n\nwfLTL : Well-founded LTL\nwfLTL = well-founded <-well-founded\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/List\/LT-LengthSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"50a1f5ec44d0086ddb857f1f7cb14b579b4fb00f","subject":"Added a test.","message":"Added a test.\n\nIgnore-this: 5c764a0c6b26f4cca506d737df64e61d\n\ndarcs-hash:20110406204621-3bd4e-2d12be8ec0f0528007464d7e2f9d424628e42e79.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/Conjectures\/RepeatedConjecture.agda","new_file":"Test\/Succeed\/Conjectures\/RepeatedConjecture.agda","new_contents":"module Test.Succeed.Conjectures.RepeatedConjecture where\n\npostulate\n  D    : Set\n  _\u2261_  : D \u2192 D \u2192 Set\n  refl : \u2200 d \u2192 d \u2261 d\n\n-- The ATPs only try to prove the last ATP pragma.\npostulate\n  sym : \u2200 d e \u2192 d \u2261 e \u2192 e \u2261 d\n{-# ATP prove sym #-}\n{-# ATP prove sym refl #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/Conjectures\/RepeatedConjecture.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"284650d8ccddc04021187acb40ec38ce02b6fb93","subject":"Added mutual recursive functions example.","message":"Added mutual recursive functions example.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/MutualRecursiveFunctions.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/MutualRecursiveFunctions.agda","new_contents":"------------------------------------------------------------------------------\n-- Mutual recursive functions\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with agda2atp on 29 August 2012.\n\nmodule FOT.FOTC.Data.Nat.MutualRecursiveFunctions where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  even : D \u2192 D\n  odd  : D \u2192 D\n\n  even-0 :       even zero      \u2261 true\n  even-S : \u2200 d \u2192 even (succ\u2081 d) \u2261 odd d\n\n  odd-0 :       odd zero      \u2261 false\n  odd-S : \u2200 d \u2192 odd (succ\u2081 d) \u2261 even d\n{-# ATP axiom even-0 even-S odd-0 odd-S #-}\n\npostulate even-2 : even (succ\u2081 (succ\u2081 zero)) \u2261 true\n{-# ATP prove even-2 #-}\n\npostulate even-3 : even (succ\u2081 (succ\u2081 (succ\u2081 zero))) \u2261 false\n{-# ATP prove even-3 #-}\n\npostulate odd-2 : odd (succ\u2081 (succ\u2081 zero)) \u2261 false\n{-# ATP prove odd-2 #-}\n\npostulate odd-3 : odd (succ\u2081 (succ\u2081 (succ\u2081 zero))) \u2261 true\n{-# ATP prove odd-3 #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/MutualRecursiveFunctions.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e77b2c8bd596a2b15d946864c5f39cc2378cc4cd","subject":"First go at a clean model of IDesc. With too much universe polymorphism for now.","message":"First go at a clean model of IDesc. With too much universe polymorphism for now.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/IDesc.agda","new_file":"models\/IDesc.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"fatal: unable to access 'https:\/\/github.com\/brixen\/Epigram.git\/': The requested URL returned error: 403\n","license":"mit","lang":"Agda"}
{"commit":"06bfe669048b9c8b84213881923adf5a070481c5","subject":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","message":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_file":"lib\/Function\/Related\/TypeIsomorphisms\/NP.agda","new_contents":"module Function.Related.TypeIsomorphisms.NP where\n\nimport Level as L\nopen import Algebra\nopen L using (Lift; lower; lift)\nopen import Type\nimport Function as F\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat using (\u2115; zero; suc)\nopen import Data.Maybe.NP\nopen import Data.Product\n--open import Data.Product.N-ary\nopen import Data.Sum renaming (map to map\u228e)\nopen import Data.Unit\nopen import Data.Empty\nopen import Function.Equality using (_\u27e8$\u27e9_)\nopen import Function.Related\nopen import Function.Related.TypeIsomorphisms public\nopen import Function.Inverse using (_\u2194_; _\u2218_; sym; id; module Inverse)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (\u2192-to-\u27f6)\n\nMaybe\u2194\u22a4\u228e : \u2200 {a} {A : Set a} \u2192 Maybe A \u2194 (\u22a4 \u228e A)\nMaybe\u2194\u22a4\u228e\n  = record { to   = \u2192-to-\u27f6 (maybe inj\u2082 (inj\u2081 _))\n           ; from = \u2192-to-\u27f6 [ F.const nothing , just ]\n           ; inverse-of\n           = record { left-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl\n                    ; right-inverse-of = [ (\u03bb _ \u2192 \u2261.refl) , (\u03bb _ \u2192 \u2261.refl) ] } }\n\nMaybe-cong : \u2200 {a b} {A : Set a} {B : Set b} \u2192 A \u2194 B \u2192 Maybe A \u2194 Maybe B \nMaybe-cong A\u2194B = sym Maybe\u2194\u22a4\u228e \u2218 id \u228e-cong A\u2194B \u2218 Maybe\u2194\u22a4\u228e\n\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe : \u2200 {A} n \u2192 Maybe (Maybe^ n A) \u2194 Maybe^ n (Maybe A)\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe zero    = id\nMaybe\u2218Maybe^\u2194Maybe^\u2218Maybe (suc n) = Maybe-cong (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n)\n\nMaybe^-cong : \u2200 {A B} n \u2192 A \u2194 B \u2192 Maybe^ n A \u2194 Maybe^ n B\nMaybe^-cong zero    = F.id\nMaybe^-cong (suc n) = Maybe-cong F.\u2218 Maybe^-cong n\n\nVec0\u2194\u22a4 : \u2200 {a} {A : Set a} \u2192 Vec A 0 \u2194 \u22a4\nVec0\u2194\u22a4 = record { to         = \u2192-to-\u27f6 _\n                ; from       = \u2192-to-\u27f6 (F.const [])\n                ; inverse-of = record { left-inverse-of  = \u03bb { [] \u2192 \u2261.refl }\n                                      ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nVec\u2218suc\u2194A\u00d7Vec : \u2200 {a} {A : Set a} {n} \u2192 Vec A (suc n) \u2194 (A \u00d7 Vec A n)\nVec\u2218suc\u2194A\u00d7Vec\n  = record { to         = \u2192-to-\u27f6 (\u03bb { (x \u2237 xs) \u2192 x , xs })\n           ; from       = \u2192-to-\u27f6 (uncurry _\u2237_)\n           ; inverse-of = record { left-inverse-of  = \u03bb { (x \u2237 xs) \u2192 \u2261.refl }\n                                 ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\ninfix 8 _^_\n\n_^_ : \u2605 \u2192 \u2115 \u2192 \u2605\nA ^ 0     = \u22a4\nA ^ suc n = A \u00d7 A ^ n\n\n^\u2194Vec : \u2200 {A} n \u2192 (A ^ n) \u2194 Vec A n\n^\u2194Vec zero    = sym Vec0\u2194\u22a4\n^\u2194Vec (suc n) = sym Vec\u2218suc\u2194A\u00d7Vec \u2218 (id \u00d7-cong (^\u2194Vec n))\n\nFin0\u2194\u22a5 : Fin 0 \u2194 \u22a5\nFin0\u2194\u22a5 = record { to   = \u2192-to-\u27f6 \u03bb()\n                ; from = \u2192-to-\u27f6 \u03bb()\n                ; inverse-of = record { left-inverse-of  = \u03bb()\n                                      ; right-inverse-of = \u03bb() } }\n\nFin\u2218suc\u2194Maybe\u2218Fin : \u2200 {n} \u2192 Fin (suc n) \u2194 Maybe (Fin n)\nFin\u2218suc\u2194Maybe\u2218Fin {n}\n  = record { to   = \u2192-to-\u27f6 to\n           ; from = \u2192-to-\u27f6 (maybe suc zero)\n           ; inverse-of\n           = record { left-inverse-of = \u03bb { zero \u2192 \u2261.refl ; (suc n) \u2192 \u2261.refl }\n                    ; right-inverse-of = maybe (\u03bb _ \u2192 \u2261.refl) \u2261.refl } }\n  where to : Fin (suc n) \u2192 Maybe (Fin n)\n        to zero = nothing\n        to (suc n) = just n\n\nLift\u2194id : \u2200 {a} {A : Set a} \u2192 Lift {a} {a} A \u2194 A\nLift\u2194id = \u03bb {a} {A} \u2192 record { to = \u2192-to-\u27f6 lower\n                             ; from = \u2192-to-\u27f6 lift\n                             ; inverse-of = record { left-inverse-of = \u03bb { (lift x) \u2192 \u2261.refl }\n                                                   ; right-inverse-of = \u03bb _ \u2192 \u2261.refl } }\n\nMaybe\u22a5\u2194\u22a4 : Maybe \u22a5 \u2194 \u22a4\nMaybe\u22a5\u2194\u22a4 = (proj\u2082 CMon.identity \u22a4 \u2218 id \u228e-cong (sym (Lift\u2194id {A = \u22a5}))) \u2218 Maybe\u2194\u22a4\u228e\n  where module CMon = CommutativeMonoid (\u228e-CommutativeMonoid bijection L.zero)\n\nMaybe^\u22a5\u2194Fin : \u2200 n \u2192 Maybe^ n \u22a5 \u2194 Fin n\nMaybe^\u22a5\u2194Fin zero    = sym Fin0\u2194\u22a5\nMaybe^\u22a5\u2194Fin (suc n) = sym Fin\u2218suc\u2194Maybe\u2218Fin \u2218 Maybe-cong (Maybe^\u22a5\u2194Fin n)\n\nMaybe^\u22a4\u2194Fin1+ : \u2200 n \u2192 Maybe^ n \u22a4 \u2194 Fin (suc n)\nMaybe^\u22a4\u2194Fin1+ n = Maybe^\u22a5\u2194Fin (suc n) \u2218 sym (Maybe\u2218Maybe^\u2194Maybe^\u2218Maybe n) \u2218 Maybe^-cong n (sym Maybe\u22a5\u2194\u22a4)\n\nFin\u2218suc\u2194\u22a4\u228eFin : \u2200 {n} \u2192 Fin (suc n) \u2194 (\u22a4 \u228e Fin n)\nFin\u2218suc\u2194\u22a4\u228eFin = Maybe\u2194\u22a4\u228e \u2218 Fin\u2218suc\u2194Maybe\u2218Fin\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Function\/Related\/TypeIsomorphisms\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"049ca80e6601ce5fb84f3c4c2667dd8813ac121c","subject":"Added draft on conat.","message":"Added draft on conat.\n\nIgnore-this: dbc067167cfc6e616f4179393950c00d\n\ndarcs-hash:20120320161638-3bd4e-c7da32d17a45b600fe008dbb9ae8b223d936e716.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Conat\/PropertiesI.agda","new_file":"Draft\/FOTC\/Data\/Conat\/PropertiesI.agda","new_contents":"{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with FOT 20 March 2012.\n\nmodule Draft.FOTC.Data.Conat.PropertiesI where\n\nopen import FOTC.Base\nopen import FOTC.Data.Conat\nopen import FOTC.Data.Conat.Equality\nopen import FOTC.Data.Nat\n\n\u2248N-refl : \u2200 {n} \u2192 Conat n \u2192 n \u2248N n\n\u2248N-refl {n} h = \u2248N-gfp\u2082 P helper\u2081 helper\u2082\n  where\n  P : D \u2192 D \u2192 Set\n  P a b = Conat a \u2227 Conat b \u2227 a \u2261 b\n\n  helper\u2081 : \u2200 {a b} \u2192 P a b \u2192 a \u2261 zero \u2227 b \u2261 zero\n            \u2228 (\u2203 (\u03bb a' \u2192 \u2203 (\u03bb b' \u2192 P a' b' \u2227 a \u2261 succ\u2081 a' \u2227 b \u2261 succ\u2081 b')))\n  helper\u2081 {a} {.a} (Ca , Cb , refl) with Conat-gfp\u2081 Ca\n  ... | inj\u2081 prf = inj\u2081 (prf , prf)\n  ... | inj\u2082 (a' , Ca' , prf) = inj\u2082 (a' , a' , (Ca' , Ca' , refl) , (prf , prf))\n\n  helper\u2082 : Conat n \u2227 Conat n \u2227 n \u2261 n\n  helper\u2082 = h , h , refl\n\n\u2261\u2192\u2248N : \u2200 {m n} \u2192 Conat m \u2192 Conat n \u2192 m \u2261 n \u2192 m \u2248N n\n\u2261\u2192\u2248N h\u2081 h\u2082 refl = \u2248N-refl h\u2081\n\n{-# NO_TERMINATION_CHECK #-}\nConat\u2192N : \u2200 {n} \u2192 Conat n \u2192 N n\nConat\u2192N Cn with Conat-gfp\u2081 Cn\n... | inj\u2081 prf = subst N (sym prf) zN\n... | inj\u2082 (n' , Cn' , prf) = subst N (sym prf) (sN {!Conat\u2192N Cn'!})\n-- Agda error: Failed to give (Conat\u2192N Cn')\n\nN\u2192Conat : \u2200 {n} \u2192 N n \u2192 Conat n\nN\u2192Conat zN          = Conat-gfp\u2083 (inj\u2081 refl)\nN\u2192Conat (sN {n} Nn) = Conat-gfp\u2083 (inj\u2082 (n , ({!N\u2192Conat Nn!} , refl)))\n-- Agda error: Failed to give N\u2192Conat Nn\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Conat\/PropertiesI.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"4e3d82e4e02d3ecbf70ef14e694a240315f5bf6f","subject":"Added a note on the normalization of conditional expression.","message":"Added a note on the normalization of conditional expression.\n\nIgnore-this: 628fc598478e377a98efa1cafb381885\n\ndarcs-hash:20120504220208-3bd4e-15e074241f642af80f0aa4a3cc9e9172ac5a3a5a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Norm.agda","new_file":"notes\/Norm.agda","new_contents":"-- Tested with the development version of Agda on 04 May 2012.\n\n-- From: Peter Dybjer. Comparing integrated and external logics of\n-- functional programs. Science of Computer Programming, 14:59\u201379,\n-- 1990\n\nmodule Norm where\n\npostulate Char : Set\n\ndata Atom : Set where\n  True False : Atom\n  Var        : Char \u2192 Atom\n\ndata Exp : Set where\n  at : Atom \u2192 Exp\n  if : Exp \u2192 Exp \u2192 Exp \u2192 Exp\n\n-- Agda doesn't recognize the termination of the function.\nnorm : Exp \u2192 Exp\nnorm (at a)              = at a\nnorm (if (at a) y z)     = if (at a) (norm y) (norm z)\nnorm (if (if u v w) y z) = norm (if u (if v y z) (if w y z))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/Norm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"962dded37d6c55fa91fe570fbfe0821e7c37241a","subject":"Create ttfp1.agda","message":"Create ttfp1.agda\n\nnotes while learning ttfp","repos":"wangbj\/excises,wangbj\/excises,wangbj\/excises","old_file":"ttfp1.agda","new_file":"ttfp1.agda","new_contents":"-- some excises for learning ttfp (type theory for functional programming).\n-- most of the code is *borrowed* from learn you an agda shamelessly\n\n-- Function composition\n_\u2218_ : {A : Set} { B : A -> Set} { C : (x : A) -> B x -> Set}\n      (f : {x : A}(y : B x) -> C x y) (g : (x : A) -> B x)\n      (x : A) -> C x (g x)\n(f \u2218 g) x = f (g x)\n\n-- Conjuntion, \u220f-type in haskell\n\ndata _\u2227_ (\u03b1 : Set) (\u03b2 : Set) : Set where\n  \u2227-intro : \u03b1 \u2192 \u03b2 \u2192 (\u03b1 \u2227 \u03b2)\n\n\u2227-elim\u2081 : {\u03b1 \u03b2 : Set} \u2192 (\u03b1 \u2227 \u03b2) \u2192 \u03b1\n\u2227-elim\u2081 (\u2227-intro p q) = p\n\n\u2227-elim\u2082 : {\u03b1 \u03b2 : Set} \u2192 (\u03b1 \u2227 \u03b2) \u2192 \u03b2\n\u2227-elim\u2082 (\u2227-intro p q) = q\n\n_\u21d4_ : (\u03b1 : Set) \u2192 (\u03b2 : Set) \u2192 Set\na \u21d4 b = (a \u2192 b) \u2227 (b \u2192 a)\n\n\u2227-comm\u2032 : {\u03b1 \u03b2 : Set} \u2192 (\u03b1 \u2227 \u03b2) \u2192 (\u03b2 \u2227 \u03b1)\n\u2227-comm\u2032 (\u2227-intro a b) = \u2227-intro b a\n\n\u2227-comm : {\u03b1 \u03b2 : Set} \u2192 (\u03b1 \u2227 \u03b2) \u21d4 (\u03b2 \u2227 \u03b1)\n\u2227-comm = \u2227-intro \u2227-comm\u2032 \u2227-comm\u2032\n\n-- type signature cannot be eliminated, was expecting type could be infered but it wasn't the case.\ne\u2081e\u2081 : { P Q R : Set} \u2192 ( (P \u2227 Q) \u2227 R ) \u2192 P\ne\u2081e\u2081 = \u2227-elim\u2081 \u2218 \u2227-elim\u2081\ne\u2082e\u2081 : { P Q R : Set} \u2192 ( (P \u2227 Q) \u2227 R) \u2192 Q\ne\u2082e\u2081 = \u2227-elim\u2082 \u2218 \u2227-elim\u2081\n\n\u2227-assoc\u2081 : { P Q R : Set } \u2192 ((P \u2227 Q) \u2227 R) \u2192 (P \u2227 (Q \u2227 R))\n\u2227-assoc\u2081 = \u03bb x \u2192 \u2227-intro (\u2227-elim\u2081 (\u2227-elim\u2081 x)) (\u2227-intro (\u2227-elim\u2082 (\u2227-elim\u2081 x)) (\u2227-elim\u2082 x))\n\n\u2227-assoc\u2082 : {P Q R : Set} \u2192 (P \u2227 (Q \u2227 R)) \u2192 ((P \u2227 Q) \u2227 R)\n\u2227-assoc\u2082 = \u03bb x \u2192 \u2227-intro (\u2227-intro (\u2227-elim\u2081 x) (\u2227-elim\u2081 (\u2227-elim\u2082 x))) (\u2227-elim\u2082 (\u2227-elim\u2082 x))\n\n\u2227-assoc : {P Q R : Set} \u2192 ((P \u2227 Q) \u2227 R) \u21d4 (P \u2227 (Q \u2227 R))\n\u2227-assoc = \u2227-intro \u2227-assoc\u2081 \u2227-assoc\u2082\n\n-- Disjunctions, \u2211-type in haskell\n\ndata _\u2228_ (P Q : Set) : Set where\n  \u2228-intro\u2081 : P \u2192 P \u2228 Q\n  \u2228-intro\u2082 : Q \u2192 P \u2228 Q\n\n\u2228-elim : {A B C : Set} \u2192 (A \u2228 B) \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 C\n\u2228-elim (\u2228-intro\u2081 a) ac bc = ac a\n\u2228-elim (\u2228-intro\u2082 b) ac bc = bc b\n\n\u2228-comm\u2032 : {P Q : Set} \u2192 (P \u2228 Q) \u2192 (Q \u2228 P)\n\u2228-comm\u2032 (\u2228-intro\u2081 p) = \u2228-intro\u2082 p\n\u2228-comm\u2032 (\u2228-intro\u2082 q) = \u2228-intro\u2081 q\n\n\u2228-comm : {P Q : Set} \u2192 (P \u2228 Q) \u21d4 (Q \u2228 P)\n\u2228-comm = \u2227-intro \u2228-comm\u2032 \u2228-comm\u2032\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ttfp1.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"51d460665c07c190bfbaaa8aba89baf0df8ace6c","subject":"Progress on Agda formalization (see #21).","message":"Progress on Agda formalization (see #21).\n\nOld-commit-hash: 01bccf0894187bf2dce188a42cd81a5a40ffc777\n","repos":"inc-lc\/ilc-agda","old_file":"total.agda","new_file":"total.agda","new_contents":"module total where\n\n-- INCREMENTAL \u03bb-CALCULUS\n--   with total derivatives\n--\n-- Features:\n--   * changes and derivatives are unified (following Cai)\n--   * \u0394 e describes how e changes when its free variables or its arguments change\n--   * denotational semantics including semantics of changes\n--\n-- Work in Progress:\n--   * lemmas about behavior of changes\n--   * lemmas about behavior of \u0394\n--   * correctness proof for symbolic derivation\n\nimport Relation.Binary as B\n\nopen import Relation.Binary using\n  (IsEquivalence; Setoid; Reflexive; Symmetric; Transitive)\nimport Relation.Binary.EqReasoning as EqR\n\nopen import meaning\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Type : Type -> Set\n\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7Type = \u27e6 \u03c4\u2081 \u27e7Type \u2192 \u27e6 \u03c4\u2082 \u27e7Type\n\nmeaningOfType : Meaning Type\nmeaningOfType = meaning Set \u27e6_\u27e7Type\n\n-- Value Equivalence\n\ndata _\u2261_ : \u2200 {\u03c4} \u2192 (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192 Set where\n  ext : \u2200 {\u03c4\u2081 \u03c4\u2082} {f\u2081 f\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} \u2192\n    (\u2200 v \u2192 f\u2081 v \u2261 f\u2082 v) \u2192\n    f\u2081 \u2261 f\u2082\n\n\u2261-refl : \u2200 {\u03c4} {v : \u27e6 \u03c4 \u27e7} \u2192\n  v \u2261 v\n\u2261-refl {\u03c4\u2081 \u21d2 \u03c4\u2082} = ext (\u03bb v \u2192 \u2261-refl)\n\n\u2261-sym : \u2200 {\u03c4} {v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2081\n\u2261-sym {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2261) = ext (\u03bb v \u2192 \u2261-sym (\u2261 v))\n\n\u2261-trans : \u2200 {\u03c4} {v\u2081 v\u2082 v\u2083 : \u27e6 \u03c4 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2082 \u2261 v\u2083 \u2192 v\u2081 \u2261 v\u2083\n\u2261-trans {\u03c4\u2081 \u21d2 \u03c4\u2082} {f} (ext \u2261\u2081) (ext \u2261\u2082) =\n  ext (\u03bb v \u2192 \u2261-trans (\u2261\u2081 v) (\u2261\u2082 v))\n\n\u2261-cong : \u2200 {\u03c4\u2082 \u03c4\u2081 v\u2081 v\u2082} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 f v\u2081 \u2261 f v\u2082\n\u2261-cong {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261 = ext (\u03bb v \u2192 \u2261-cong (\u03bb x \u2192 f x v) \u2261)\n\n\u2261-cong\u2082 : \u2200 {\u03c4\u2083 \u03c4\u2081 \u03c4\u2082 v\u2081 v\u2082 v\u2083 v\u2084} (f : \u27e6 \u03c4\u2081 \u27e7 \u2192 \u27e6 \u03c4\u2082 \u27e7 \u2192 \u27e6 \u03c4\u2083 \u27e7) \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 f v\u2081 v\u2083 \u2261 f v\u2082 v\u2084\n\u2261-cong\u2082 {\u03c4\u2081 \u21d2 \u03c4\u2082} f \u2261\u2081 \u2261\u2082 = ext (\u03bb v \u2192 \u2261-cong\u2082 (\u03bb x y \u2192 f x y v) \u2261\u2081 \u2261\u2082)\n\n\u2261-app : \u2200 {\u03c4\u2081 \u03c4\u2082} {v\u2081 v\u2082 : \u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7} {v\u2083 v\u2084 : \u27e6 \u03c4\u2081 \u27e7} \u2192\n  v\u2081 \u2261 v\u2082 \u2192 v\u2083 \u2261 v\u2084 \u2192 v\u2081 v\u2083 \u2261 v\u2082 v\u2084\n\u2261-app = \u2261-cong\u2082 (\u03bb x y \u2192 x y)\n\n\u2261-isEquivalence : \u2200 {\u03c4} \u2192 IsEquivalence (_\u2261_ {\u03c4})\n\u2261-isEquivalence = record\n  { refl  = \u2261-refl\n  ; sym   = \u2261-sym\n  ; trans = \u2261-trans\n  }\n\n\u2261-setoid : Type \u2192 Setoid _ _\n\u2261-setoid \u03c4 = record\n  { Carrier       = \u27e6 \u03c4 \u27e7\n  ; _\u2248_           = _\u2261_\n  ; isEquivalence = \u2261-isEquivalence\n  }\n\nmodule \u2261-Reasoning where\n  module _ {\u03c4 : Type} where\n    open EqR (\u2261-setoid \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\nderive : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\napply : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\ndiff : \u2200 {\u03c4} \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\n\ndiff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = \u03bb v dv \u2192 diff (f\u2081 (apply dv v)) (f\u2082 v)\n\nderive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = \u03bb v dv \u2192 diff (f (apply dv v)) (f v)\n\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = \u03bb v \u2192 apply (df v (derive v)) (f v)\n\ncompose : \u2200 {\u03c4} \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7 \u2192 \u27e6 \u0394-Type \u03c4 \u27e7\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 = \u03bb v dv \u2192 compose (df\u2081 v dv) (df\u2082 v dv)\n\n-- PROPERTIES of changes\n\ndiff-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff v v \u2261 derive v\ndiff-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} v = \u2261-refl\n\ndiff-apply : \u2200 {\u03c4} (dv : \u27e6 \u0394-Type \u03c4 \u27e7) (v : \u27e6 \u03c4 \u27e7) \u2192\n  diff (apply dv v) v \u2261 dv\ndiff-apply {\u03c4\u2081 \u21d2 \u03c4\u2082} df f = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  {!!}))\n\napply-diff : \u2200 {\u03c4} (v\u2081 v\u2082 : \u27e6 \u03c4 \u27e7) \u2192\n  apply (diff v\u2082 v\u2081) v\u2081 \u2261 v\u2082\napply-diff {\u03c4\u2081 \u21d2 \u03c4\u2082} f\u2081 f\u2082 = ext (\u03bb v \u2192 apply-diff (f\u2081 v) (f\u2082 v))\n\napply-derive : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) \u2192\n  apply (derive v) v \u2261 v\napply-derive {\u03c4\u2081 \u21d2 \u03c4\u2082} f = ext (\u03bb v \u2192\n  begin\n    apply (derive f) f v\n  \u2261\u27e8 \u2261-refl \u27e9\n    apply (diff (f (apply (derive v) v)) (f v)) (f v)\n  \u2261\u27e8 \u2261-cong (\u03bb x \u2192 apply (diff (f x) (f v)) (f v)) (apply-derive v) \u27e9\n    apply (diff (f v) (f v)) (f v)\n  \u2261\u27e8 apply-diff (f v) (f v)\u27e9\n    f v\n  \u220e) where open \u2261-Reasoning\n\napply-compose : \u2200 {\u03c4} (v : \u27e6 \u03c4 \u27e7) (dv\u2081 dv\u2082 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  apply (compose dv\u2081 dv\u2082) v \u2261 apply dv\u2081 (apply dv\u2082 v)\napply-compose {\u03c4\u2081 \u21d2 \u03c4\u2082} f df\u2081 df\u2082 = ext (\u03bb v \u2192\n  apply-compose (f v) (df\u2082 v (derive v)) (df\u2082 v (derive v)))\n\ncompose-assoc : \u2200 {\u03c4} (dv\u2081 dv\u2082 dv\u2083 : \u27e6 \u0394-Type \u03c4 \u27e7) \u2192\n  compose dv\u2081 (compose dv\u2082 dv\u2083) \u2261 compose (compose dv\u2081 dv\u2082) dv\u2083\ncompose-assoc {\u03c4\u2081 \u21d2 \u03c4\u2082} df\u2081 df\u2082 df\u2083 = ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  compose-assoc (df\u2081 v dv) (df\u2082 v dv) (df\u2083 v dv)))\n\n-- TYPING CONTEXTS, VARIABLES and WEAKENING\n\nopen import binding Type \u27e6_\u27e7Type\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u0394-Type \u03c4 \u2022 \u03c4 \u2022 \u0394-Context \u0393\n\nupdate : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nupdate {\u2205} \u2205 = \u2205\nupdate {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = apply dv v \u2022 update \u03c1\n\nignore : \u2200 {\u0393} \u2192 \u27e6 \u0394-Context \u0393 \u27e7 \u2192 \u27e6 \u0393 \u27e7\nignore {\u2205} \u2205 = \u2205\nignore {\u03c4 \u2022 \u0393} (dv \u2022 v \u2022 \u03c1) = v \u2022 ignore \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n  -- `\u0394 t` describes how t changes if its free variables or arguments change\n  \u0394 : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\n\n-- Denotational Semantics\n\n\u27e6_\u27e7Term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7\n\u27e6 abs t \u27e7Term \u03c1 = \u03bb v \u2192 \u27e6 t \u27e7Term (v \u2022 \u03c1)\n\u27e6 app t\u2081 t\u2082 \u27e7Term \u03c1 = (\u27e6 t\u2081 \u27e7Term \u03c1) (\u27e6 t\u2082 \u27e7Term \u03c1)\n\u27e6 var x \u27e7Term \u03c1 = \u27e6 x \u27e7 \u03c1\n\u27e6 \u0394 t \u27e7Term \u03c1 = diff (\u27e6 t \u27e7Term (update \u03c1)) (\u27e6 t \u27e7Term (ignore \u03c1))\n\nmeaningOfTerm : \u2200 {\u0393 \u03c4} \u2192 Meaning (Term \u0393 \u03c4)\nmeaningOfTerm {\u0393} {\u03c4} = meaning (\u27e6 \u0393 \u27e7 \u2192 \u27e6 \u03c4 \u27e7) \u27e6_\u27e7Term\n\n-- Term Equivalence\n\nmodule _ {\u0393} {\u03c4} where\n  data _\u2248_ (t\u2081 t\u2082 : Term \u0393 \u03c4) : Set where\n    ext :\n      (\u2200 \u03c1 \u2192 \u27e6 t\u2081 \u27e7 \u03c1 \u2261 \u27e6 t\u2082 \u27e7 \u03c1) \u2192\n      t\u2081 \u2248 t\u2082\n\n  \u2248-refl : Reflexive _\u2248_\n  \u2248-refl = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n  \u2248-sym : Symmetric _\u2248_\n  \u2248-sym (ext \u2248) = ext (\u03bb \u03c1 \u2192 \u2261-sym (\u2248 \u03c1))\n\n  \u2248-trans : Transitive _\u2248_\n  \u2248-trans (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192 \u2261-trans (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n  \u2248-isEquivalence : IsEquivalence _\u2248_\n  \u2248-isEquivalence = record\n    { refl  = \u2248-refl\n    ; sym   = \u2248-sym\n    ; trans = \u2248-trans\n    }\n\n\u2248-setoid : Context \u2192 Type \u2192 Setoid _ _\n\u2248-setoid \u0393 \u03c4 = record\n  { Carrier       = Term \u0393 \u03c4\n  ; _\u2248_           = _\u2248_\n  ; isEquivalence = \u2248-isEquivalence\n  }\n\n\u2248-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2083 t\u2084 : Term \u0393 \u03c4\u2081} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 t\u2083 \u2248 t\u2084 \u2192 app t\u2081 t\u2083 \u2248 app t\u2082 t\u2084\n\u2248-app (ext \u2248\u2081) (ext \u2248\u2082) = ext (\u03bb \u03c1 \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2248\u2081 \u03c1) (\u2248\u2082 \u03c1))\n\n\u2248-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} {t\u2081 t\u2082 : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 abs t\u2081 \u2248 abs t\u2082\n\u2248-abs (ext \u2248) = ext (\u03bb \u03c1 \u2192\n  ext (\u03bb v \u2192 \u2248 (v \u2022 \u03c1)))\n\n\u2248-\u0394 : \u2200 {\u03c4 \u0393} {t\u2081 t\u2082 : Term \u0393 \u03c4} \u2192\n  t\u2081 \u2248 t\u2082 \u2192 \u0394 t\u2081 \u2248 \u0394 t\u2082\n\u2248-\u0394 {\u03c4\u2081 \u21d2 \u03c4\u2082} (ext \u2248) = ext (\u03bb \u03c1 \u2192 ext (\u03bb v \u2192 ext (\u03bb dv \u2192\n  \u2261-cong\u2082 (\u03bb x y \u2192 diff (x v) (y v)) (\u2248 (update \u03c1)) (\u2248 (ignore \u03c1)))))\n\nmodule \u2248-Reasoning where\n  module _ {\u0393 : Context} {\u03c4 : Type} where\n    open EqR (\u2248-setoid \u0393 \u03c4) public\n      hiding (_\u2261\u27e8_\u27e9_)\n\n-- LIFTING terms into \u0394-Contexts\n\nlift-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) \u03c4\nlift-var this = that this\nlift-var (that x) = that (that (lift-var x))\n\nlift-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) \u03c4\nlift-term (abs t) = abs {!!}\nlift-term (app t\u2081 t\u2082) = app (lift-term t\u2081) (lift-term t\u2082)\nlift-term (var x) = var (lift-var x)\nlift-term (\u0394 t) = \u0394 (lift-term t)\n\n-- PROPERTIES of lift-term\n\nlift-var-ignore : \u2200 {\u0393 \u03c4} (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) (x : Var \u0393 \u03c4) \u2192\n  \u27e6 lift-var x \u27e7 \u03c1 \u2261 \u27e6 x \u27e7 (ignore \u03c1)\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) this = \u2261-refl\nlift-var-ignore (v \u2022 dv \u2022 \u03c1) (that x) = lift-var-ignore \u03c1 x\n\nlift-term-ignore : \u2200 {\u0393 \u03c4} {\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7} (t : Term \u0393 \u03c4) \u2192\n  \u27e6 lift-term t \u27e7 \u03c1 \u2261 \u27e6 t \u27e7 (ignore \u03c1)\nlift-term-ignore (abs t) = {!!}\nlift-term-ignore (app t\u2081 t\u2082) = \u2261-app (lift-term-ignore t\u2081) (lift-term-ignore t\u2082)\nlift-term-ignore (var x) = lift-var-ignore _ x\nlift-term-ignore (\u0394 t) = {!!}\n\n-- PROPERTIES of \u0394\n\n\u0394-abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192\n  \u0394 (abs t) \u2248 abs (abs (\u0394 t))\n\u0394-abs t = ext (\u03bb \u03c1 \u2192 \u2261-refl)\n\n\u0394-app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192\n  \u0394 (app t\u2081 t\u2082) \u2248 app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n\u0394-app t\u2081 t\u2082 = \u2248-sym (ext (\u03bb \u03c1 \u2192\n  begin\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 lift-term t\u2082 \u27e7 \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 lift-term t\u2082 \u27e7 \u03c1))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff\n          (\u27e6 t\u2081 \u27e7 (update \u03c1)\n           (apply (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) x))\n          (\u27e6 t\u2081 \u27e7 (ignore \u03c1) x))\n       (lift-term-ignore {\u03c1 = \u03c1} t\u2082) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1)\n       (apply\n         (diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n         (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u2261\u27e8 \u2261-cong\n       (\u03bb x \u2192\n          diff (\u27e6 t\u2081 \u27e7 (update \u03c1) x) (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))))\n       (apply-diff (\u27e6 t\u2082 \u27e7 (update \u03c1)) (\u27e6 t\u2082 \u27e7 (ignore \u03c1))) \u27e9\n    diff\n      (\u27e6 t\u2081 \u27e7 (update \u03c1) (\u27e6 t\u2082 \u27e7 (update \u03c1)))\n      (\u27e6 t\u2081 \u27e7 (ignore \u03c1) (\u27e6 t\u2082 \u27e7 (ignore \u03c1)))\n  \u220e)) where open \u2261-Reasoning\n\n-- SYMBOLIC DERIVATION\n\nderive-var : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-var this = this\nderive-var (that x) = that (that (derive-var x))\n\nderive-term : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394-Context \u0393) (\u0394-Type \u03c4)\nderive-term (abs t) = abs (abs (derive-term t))\nderive-term (app t\u2081 t\u2082) = app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\nderive-term (var x) = var (derive-var x)\nderive-term (\u0394 t) = \u0394 (derive-term t)\n\n-- CORRECTNESS of derivation\n\nderive-var-correct : \u2200 {\u0393 \u03c4} \u2192 (\u03c1 : \u27e6 \u0394-Context \u0393 \u27e7) \u2192 (x : Var \u0393 \u03c4) \u2192\n  diff (\u27e6 x \u27e7 (update \u03c1)) (\u27e6 x \u27e7 (ignore \u03c1)) \u2261\n  \u27e6 derive-var x \u27e7 \u03c1\nderive-var-correct (dv \u2022 v \u2022 \u03c1) this = diff-apply dv v\nderive-var-correct (dv \u2022 v \u2022 \u03c1) (that x) = derive-var-correct \u03c1 x\n\nderive-term-correct : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n   \u0394 t \u2248 derive-term t\nderive-term-correct {\u0393} (abs t) =\n  begin\n    \u0394 (abs t)\n  \u2248\u27e8 \u0394-abs t \u27e9\n    abs (abs (\u0394 t))\n  \u2248\u27e8 \u2248-abs (\u2248-abs (derive-term-correct t)) \u27e9\n    abs (abs (derive-term t))\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (abs t)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (app t\u2081 t\u2082) =\n  begin\n    \u0394 (app t\u2081 t\u2082)\n  \u2248\u27e8 \u0394-app t\u2081 t\u2082 \u27e9\n    app (app (\u0394 t\u2081) (lift-term t\u2082)) (\u0394 t\u2082)\n  \u2248\u27e8 \u2248-app (\u2248-app (derive-term-correct t\u2081) \u2248-refl) (derive-term-correct t\u2082) \u27e9\n    app (app (derive-term t\u2081) (lift-term t\u2082)) (derive-term t\u2082)\n  \u2248\u27e8 \u2248-refl \u27e9\n    derive-term (app t\u2081 t\u2082)\n  \u220e where open \u2248-Reasoning\nderive-term-correct (var x) = ext (\u03bb \u03c1 \u2192 derive-var-correct \u03c1 x)\nderive-term-correct (\u0394 t) = \u2248-\u0394 (derive-term-correct t)\n\n-- NATURAL SEMANTICS\n\n-- (without support for \u0394 for now)\n\n-- Syntax\n\ndata Env : Context \u2192 Set\ndata Val : Type \u2192 Set\n\ndata Val where\n  \u27e8abs_,_\u27e9 : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) (\u03c1 : Env \u0393) \u2192 Val (\u03c4\u2081 \u21d2 \u03c4\u2082)\n\ndata Env where\n  \u2205 : Env \u2205\n  _\u2022_ : \u2200 {\u0393 \u03c4} \u2192 Val \u03c4 \u2192 Env \u0393 \u2192 Env (\u03c4 \u2022 \u0393)\n\n-- Lookup\n\ninfixr 8 _\u22a2_\u2193_ _\u22a2_\u21a6_\n\ndata _\u22a2_\u21a6_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Var \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {v : Val \u03c4} \u2192\n    v \u2022 \u03c1 \u22a2 this \u21a6 v\n  that : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 x} {\u03c1 : Env \u0393} {v\u2081 : Val \u03c4\u2081} {v\u2082 : Val \u03c4\u2082} \u2192\n    \u03c1 \u22a2 x \u21a6 v\u2082 \u2192\n    v\u2081 \u2022 \u03c1 \u22a2 that x \u21a6 v\u2082\n\n-- Reduction\n\ndata _\u22a2_\u2193_ : \u2200 {\u0393 \u03c4} \u2192 Env \u0393 \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082 \u03c1} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 \u27e8abs t , \u03c1 \u27e9\n  app : \u2200 {\u0393 \u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 \u27e8abs t\u2032 , \u03c1\u2032 \u27e9 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    v\u2082 \u2022 \u03c1\u2032 \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u0393 \u03c4 x} {\u03c1 : Env \u0393} {v : Val \u03c4}\u2192\n    \u03c1 \u22a2 x \u21a6 v \u2192\n    \u03c1 \u22a2 var x \u2193 v\n\n-- SOUNDNESS of natural semantics\n\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 Env \u0393 \u2192 \u27e6 \u0393 \u27e7\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 \u27e8abs t , \u03c1 \u27e9 \u27e7Val = \u03bb v \u2192 \u27e6 t \u27e7 (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\nmeaningOfEnv : \u2200 {\u0393} \u2192 Meaning (Env \u0393)\nmeaningOfEnv {\u0393} = meaning \u27e6 \u0393 \u27e7 \u27e6_\u27e7Env\n\nmeaningOfVal : \u2200 {\u03c4} \u2192 Meaning (Val \u03c4)\nmeaningOfVal {\u03c4} = meaning \u27e6 \u03c4 \u27e7 \u27e6_\u27e7Val\n\n\u21a6-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {x : Var \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 x \u21a6 v \u2192\n  \u27e6 x \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u21a6-sound this = \u2261-refl\n\u21a6-sound (that \u21a6) = \u21a6-sound \u21a6\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7 \u27e6 \u03c1 \u27e7 \u2261 \u27e6 v \u27e7\n\u2193-sound abs = \u2261-refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) =\n  \u2261-trans\n    (\u2261-cong\u2082 (\u03bb x y \u2192 x y) (\u2193-sound \u2193\u2081) (\u2193-sound \u2193\u2082))\n    (\u2193-sound \u2193\u2032)\n\u2193-sound (var \u21a6) = \u21a6-sound \u21a6\n\n-- WEAKENING\n\n-- Weaken a term to a super context\n\n{-\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\nweaken {\u0393\u2081} {\u0393\u2082} (\u0394 t) = ?\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'total.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9a6fe0c66576eb6467697ed44bdf2cd2c3e16c2c","subject":"combination.agda","message":"combination.agda\n","repos":"crypto-agda\/explore","old_file":"examples\/combination.agda","new_file":"examples\/combination.agda","new_contents":"module combination where\n\nopen import Type\nopen import Data.Nat.NP\nopen import Data.Nat.DivMod\nopen import Data.Fin using (Fin; Fin\u2032; zero; suc) renaming (to\u2115 to Fin\u25b9\u2115)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Sum\nopen import Data.Product\nopen import Data.Vec\nopen import Function.NP\nopen import Function.Related.TypeIsomorphisms.NP\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; sym; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from; id to idI; _\u2218_ to _\u2218I_)\nopen import Relation.Binary.Product.Pointwise\nopen import Relation.Binary.Sum\nimport Relation.Binary.PropositionalEquality as \u2261\n\nopen import factorial\n\n_\/_ : \u2115 \u2192 \u2115 \u2192 \u2115\nx \/ y = _div_ x y {{!!}}\n\nmodule v0 where\n\n  C : \u2115 \u2192 \u2115 \u2192 \u2115\n  C n       zero    = 1\n  C zero    (suc k) = 0\n  C (suc n) (suc k) = C n k + C n (suc k)\n\n  D : \u2115 \u2192 \u2115 \u2192 \u2605\n  D n       zero    = \ud835\udfd9\n  D zero    (suc k) = \ud835\udfd8\n  D (suc n) (suc k) = D n k \u228e D n (suc k)\n\n  FinC\u2245D : \u2200 n k \u2192 Fin (C n k) \u2194 D n k\n  FinC\u2245D n       zero    = Fin1\u2194\u22a4\n  FinC\u2245D zero    (suc k) = Fin0\u2194\u22a5\n  FinC\u2245D (suc n) (suc k) =  FinC\u2245D n k \u228e-cong FinC\u2245D n (suc k)\n                         \u2218I sym (Fin-\u228e-+ (C n k) (C n (suc k)))\n\nmodule v! where\n\n  C : \u2115 \u2192 \u2115 \u2192 \u2115\n  C n k = n ! \/ (k ! * (n \u2238 k)!)\n\nmodule v1 where\n\n  C : \u2115 \u2192 \u2115 \u2192 \u2115\n  C n 0 = 1\n  C n (suc k) = (C n k * (n \u2238 k)) \/ suc k\n\nmodule v2 where\n\n  C : \u2115 \u2192 \u2115 \u2192 \u2115\n  C n       zero    = 1\n  C zero    (suc k) = 0\n  C (suc n) (suc k) = (C n k * suc n) \/ suc k\n\n\n{- \u03a3_{0\u2264k<n} (C n k) \u2261 2\u207f -}\n\n\u03a3\ud835\udfd9F\u2194F : \u2200 (F : \ud835\udfd9 \u2192 \u2605) \u2192 \u03a3 \ud835\udfd9 F \u2194 F _\n\u03a3\ud835\udfd9F\u2194F F = inverses proj\u2082 ,_ (\u03bb _ \u2192 \u2261.refl) (\u03bb _ \u2192 \u2261.refl)\n\nBits = Vec \ud835\udfda\nBits1\u2194\ud835\udfda : Bits 1 \u2194 \ud835\udfda\nBits1\u2194\ud835\udfda = {!!}\n\nopen v0\nfoo : \u2200 n \u2192 \u03a3 (Fin (suc n)) (D (suc n) \u2218 Fin\u25b9\u2115) \u2194 Bits (suc n)\nfoo zero = ((sym Bits1\u2194\ud835\udfda \u2218I {!!}) \u2218I \u03a3\ud835\udfd9F\u2194F _) \u2218I first-iso Fin1\u2194\u22a4\nfoo (suc n) = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'examples\/combination.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"38dc0e74c8050099800069c05d5282fdff28dced","subject":"Implement STLC in AGDA (fix #13).","message":"Implement STLC in AGDA (fix #13).\n\nOld-commit-hash: 8c8d0d1e3a7f67f1a6640d7526542321b3a5a424\n","repos":"inc-lc\/ilc-agda","old_file":"incremental.agda","new_file":"incremental.agda","new_contents":"module incremental where\n\n-- SIMPLE TYPES\n\n-- Syntax\n\ndata Type : Set where\n  _\u21d2_ : (\u03c4\u2081 \u03c4\u2082 : Type) \u2192 Type\n\ninfixr 5 _\u21d2_\n\n-- Semantics\n\nDom\u27e6_\u27e7 : Type -> Set\nDom\u27e6 \u03c4\u2081 \u21d2 \u03c4\u2082 \u27e7 = Dom\u27e6 \u03c4\u2081 \u27e7 \u2192 Dom\u27e6 \u03c4\u2082 \u27e7\n\n-- TYPING CONTEXTS\n\n-- Syntax\n\ndata Context : Set where\n  \u2205 : Context\n  _\u2022_ : (\u03c4 : Type) (\u0393 : Context) \u2192 Context\n\ninfixr 9 _\u2022_\n\n-- Semantics\n\ndata Empty : Set where\n  \u2205 : Empty\n\ndata Bind A B : Set where\n  _\u2022_ : (v : A) (\u03c1 : B) \u2192 Bind A B\n\nEnv\u27e6_\u27e7 : Context \u2192 Set\nEnv\u27e6 \u2205 \u27e7 = Empty\nEnv\u27e6 \u03c4 \u2022 \u0393 \u27e7 = Bind Dom\u27e6 \u03c4 \u27e7 Env\u27e6 \u0393 \u27e7\n\n-- VARIABLES\n\n-- Syntax\n\ndata Var : Context \u2192 Type \u2192 Set where\n  this : \u2200 {\u0393 \u03c4} \u2192 Var (\u03c4 \u2022 \u0393) \u03c4\n  that : \u2200 {\u0393 \u03c4 \u03c4\u2032} \u2192 (x : Var \u0393 \u03c4) \u2192 Var (\u03c4\u2032 \u2022 \u0393) \u03c4\n\n-- Semantics\n\nlookup\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\nlookup\u27e6 this \u27e7 (v \u2022 \u03c1) = v\nlookup\u27e6 that x \u27e7 (v \u2022 \u03c1) = lookup\u27e6 x \u27e7 \u03c1\n\n-- TERMS\n\n-- Syntax\n\ndata Term : Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082) \u2192 Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u0393 \u03c4\u2081 \u03c4\u2082} \u2192 (t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u03c4\u2081) \u2192 Term \u0393 \u03c4\u2082\n  var : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u03c4\n\n-- Semantics\n\neval\u27e6_\u27e7 : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Env\u27e6 \u0393 \u27e7 \u2192 Dom\u27e6 \u03c4 \u27e7\neval\u27e6 abs t \u27e7 \u03c1 = \u03bb v \u2192 eval\u27e6 t \u27e7 (v \u2022 \u03c1)\neval\u27e6 app t\u2081 t\u2082 \u27e7 \u03c1 = (eval\u27e6 t\u2081 \u27e7 \u03c1) (eval\u27e6 t\u2082 \u27e7 \u03c1)\neval\u27e6 var x \u27e7 \u03c1 = lookup\u27e6 x \u27e7 \u03c1\n\n-- WEAKENING\n\n-- Extend a context to a super context\n\ninfixr 10 _\u22ce_\n\n_\u22ce_ : (\u0393\u2081 \u0393\u2081 : Context) \u2192 Context\n\u2205 \u22ce \u0393\u2082 = \u0393\u2082\n(\u03c4 \u2022 \u0393\u2081) \u22ce \u0393\u2082 = \u03c4 \u2022 \u0393\u2081 \u22ce \u0393\u2082\n\n-- Lift a variable to a super context\n\nlift : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Var (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Var (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nlift {\u2205} {\u2205} x = x\nlift {\u2205} {\u03c4 \u2022 \u0393\u2082} x = that (lift {\u2205} {\u0393\u2082} x)\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} this = this\nlift {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} (that x) = that (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- Weaken a term to a super context\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0393\u2083 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2083) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082 \u22ce \u0393\u2083) \u03c4\nweaken {\u0393\u2081} {\u0393\u2082} (abs  {\u03c4\u2081 = \u03c4} t) = abs (weaken {\u03c4 \u2022 \u0393\u2081} {\u0393\u2082} t)\nweaken {\u0393\u2081} {\u0393\u2082} (app t\u2081 t\u2082) = app (weaken {\u0393\u2081} {\u0393\u2082} t\u2081) (weaken {\u0393\u2081} {\u0393\u2082} t\u2082)\nweaken {\u0393\u2081} {\u0393\u2082} (var x) = var (lift {\u0393\u2081} {\u0393\u2082} x)\n\n-- CHANGE TYPES\n\n\u0394-Type : Type \u2192 Type\n\u0394-Type (\u03c4\u2081 \u21d2 \u03c4\u2082) = \u03c4\u2081 \u21d2 \u0394-Type \u03c4\u2082\n\napply : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u03c4 \u21d2 \u03c4)\napply {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app apply (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbf.  \u03bbx.           apply (     df                       x        )  (     f                 x        )\n\ncompose : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 (\u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4 \u21d2 \u0394-Type \u03c4)\ncompose {\u03c4\u2081 \u21d2 \u03c4\u2082} =\n    abs (abs (abs (app (app compose (app (var (that (that this))) (var this))) (app (var (that this)) (var this)))))\n -- \u03bbdf. \u03bbdg. \u03bbx.           compose (     df                       x         ) (     dg                x      )\n\n-- Hey, apply is \u03b1-equivalent to compose, what's going on?\n-- Oh, and `\u0394-Type` is the identity function?\n\n-- CHANGE CONTEXTS\n\n\u0394-Context : Context \u2192 Context\n\u0394-Context \u2205 = \u2205\n\u0394-Context (\u03c4 \u2022 \u0393) = \u03c4 \u2022 \u0394-Type \u03c4 \u2022 \u0393\n\n-- CHANGING TERMS WHEN THE ENVIRONMENT CHANGES\n\n\u0394-term : \u2200 {\u0393\u2081 \u0393\u2082 \u03c4} \u2192 Term (\u0393\u2081 \u22ce \u0393\u2082) \u03c4 \u2192 Term (\u0393\u2081 \u22ce \u0394-Context \u0393\u2082) (\u0394-Type \u03c4)\n\u0394-term {\u0393} (abs {\u03c4\u2081 = \u03c4} t) = abs (\u0394-term {\u03c4 \u2022 \u0393} t)\n\u0394-term {\u0393} (app t\u2081 t\u2082) = {!!}\n\u0394-term {\u0393} (var x) = {!!}\n","old_contents":"","returncode":128,"stderr":"fatal: invalid reference: FETCH_HEAD^\n","license":"mit","lang":"Agda"}
{"commit":"50fd1a11e84433aae78978c163bfb8ba2ab32800","subject":"+Universe.NP","message":"+Universe.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Universe\/NP.agda","new_file":"lib\/Universe\/NP.agda","new_contents":"module Universe.NP where\n\nopen import Type\nopen import Type.Identities\nopen import Function.NP\nopen import Level.NP\nopen import Data.Product.NP\nopen import Data.Nat using (\u2115)\nopen import Data.Zero\nopen import Data.One\nopen import Data.Two\nopen import Data.Fin\nopen import Data.Sum.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen import HoTT\nopen Equivalences\n\nopen import Universe public\n\nmodule _ {u e} {U : \u2605_ u} (El : U \u2192 \u2605_ e) where\n    universe : Universe u e\n    universe = record { El = El }\n\nU\u2605_ : (\u2113 : Level) \u2192 Universe (\u209b \u2113) \u2113\nU\u2605 \u2113 = universe id\n\nU\u2605\u2080 = U\u2605_ \u2080\nU\u2605\u2081 = U\u2605_ \u2081\nU\u2605\u2082 = U\u2605_ \u2082\n\nUFin U\u2713 : Universe \u2080 \u2080\n\nUFin = record { El = Fin }\nU\u2713 = record { El = \u2713 }\n\nU\ud835\udfd8 : \u2200 {e} \u2192 Universe \u2080 e\nU\ud835\udfd8 = record { U = \ud835\udfd8; El = \u03bb() }\n\n-- Or Uconst\nU\ud835\udfd9 : \u2200 {\u2113} \u2192 \u2605_ \u2113 \u2192 Universe \u2080 \u2113\nU\ud835\udfd9 A = record { U = \ud835\udfd9; El = \u03bb _ \u2192 A }\n\nopen Universe.Universe renaming (U to Code)\n\nrecord Map {u u'}(U : Universe u u')\n           {v v'}(V : Universe v v') : \u2605_ (u \u2294 v \u2294 u' \u2294 v') where\n    constructor mk\n    field\n      map-Code : Code U \u2192 Code V\n      map-El   : \u2200 {u} \u2192 El U u \u2192 El V (map-Code u)\n\n    {- This one has contravariant mapping of El's -}\nrecord CoMap {u u'}(U : Universe u u')\n             {v v'}(V : Universe v v') : \u2605_ (u \u2294 v \u2294 u' \u2294 v') where\n    constructor mk\n    field\n      map-Code : Code U \u2192 Code V\n      comap-El : \u2200 {u} \u2192 El V (map-Code u) \u2192 El U u\n\nmodule _ \n  {e u v}(U : Universe u e)\n         (V : Universe v e)\n  where\n    record _\u228f_ : \u2605_ (u \u2294 v \u2294 \u209b e) where\n      constructor mk\n      field\n        \u228f-Code : Code U \u2192 Code V\n        \u228f-El   : \u2200 A \u2192 El U A \u2243 El V (\u228f-Code A)\n\nmodule _\n  {e u v}{U : Universe u e}\n         {V : Universe v e}\n  where\n  \u228f-Map : U \u228f V \u2192 Map U V\n  \u228f-Map (mk f e) = mk f (\u00b7\u2192 (e _))\n\n  \u228f-CoMap : U \u228f V \u2192 CoMap U V\n  \u228f-CoMap (mk f e) = mk f (\u00b7\u2190 (e _))\n\n\u2713\u228fFin : U\u2713 \u228f UFin\n\u2713\u228fFin = mk \ud835\udfda\u25b9\u2115 [0: \u2243-! Fin0\u2243\ud835\udfd8\n                 1: \u2243-! Fin1\u2243\ud835\udfd9 ]\n\nmodule _ {u e}(U : Universe u e) where\n    private\n      U\u2605 = U\u2605_ e\n\n    \u228f-refl : U \u228f U\n    \u228f-refl = mk _ \u03bb _ \u2192 \u2243-refl\n\n    \ud835\udfd8\u228f_ : U\ud835\udfd8 \u228f U\n    \ud835\udfd8\u228f_ = mk (\u03bb()) (\u03bb())\n\n    _\u228f\u2605 : U \u228f U\u2605\n    _\u228f\u2605 = mk (El U) \u03bb _ \u2192 \u2243-refl\n\n    -- derived...\n\n    idMap : Map U U\n    idMap = \u228f-Map \u228f-refl\n\n    idCoMap : CoMap U U\n    idCoMap = \u228f-CoMap \u228f-refl\n\n    Map\ud835\udfd8 : Map U\ud835\udfd8 U\n    Map\ud835\udfd8 = \u228f-Map \ud835\udfd8\u228f_\n\n    CoMap\ud835\udfd8 : CoMap U\ud835\udfd8 U\n    CoMap\ud835\udfd8 = \u228f-CoMap \ud835\udfd8\u228f_\n\n    Map\u2605 : Map U U\u2605\n    Map\u2605 = \u228f-Map _\u228f\u2605\n\n    CoMap\u2605 : CoMap U U\u2605\n    CoMap\u2605 = \u228f-CoMap _\u228f\u2605\n\nmodule _ {a b}{F : \u2605_ a \u2192 \u2605_ b} where\n    mapF : (\u2200 {A} \u2192 A \u2192 F A) \u2192 Map (U\u2605 a) (U\u2605 b)\n    mapF = mk F\n\n    coMapF : (\u2200 {A} \u2192 F A \u2192 A) \u2192 CoMap (U\u2605 a) (U\u2605 b)\n    coMapF = mk F\n\nmodule _ {a}{F : \u2605_ a \u2192 \u2605_ a} where\n    \u228fF : (\u2200 A \u2192 A \u2243 F A) \u2192 U\u2605 a \u228f U\u2605 a\n    \u228fF = mk F\n\nmodule _ {e u v w}{U : Universe u e}\n                  {V : Universe v e}\n                  {W : Universe w e} where\n  \u228f-trans : U \u228f V \u2192 V \u228f W \u2192 U \u228f W\n  \u228f-trans (mk _ UV) (mk _ VW) = mk _ \u03bb _ \u2192 \u2243-trans (UV _) (VW _)\n\n  -- could be more lax on univ levels\n  Map-\u2218 : Map U V \u2192 Map V W \u2192 Map U W\n  Map-\u2218 (mk _ UV) (mk _ VW) = mk _ (VW \u2218 UV)\n\n  -- could be more lax on univ levels\n  CoMap-\u2218 : CoMap U V \u2192 CoMap V W \u2192 CoMap U W\n  CoMap-\u2218 (mk _ UV) (mk _ VW) = mk _ (UV \u2218 VW)\n\nmodule _ {e u v}(U : Universe u e)(V : Universe v e) where\n  _\u228e\u1d41_ : Universe (u \u2294 v) e\n  _\u228e\u1d41_ = record { El = [inl: El U ,inr: El V ] }\n\n  _\u00d7\u1d41_ : Universe (u \u2294 v) e\n  _\u00d7\u1d41_ = record { El = \u03bb { (A , B) \u2192 El U A \u00d7 El V B } }\n\nmodule _ {e u v}(U : Universe u e)(V : Universe v e) where\n  inl\u1d41 : U \u228f (U \u228e\u1d41 V)\n  inl\u1d41 = mk inl \u03bb _ \u2192 \u2243-refl\n\n  inr\u1d41 : V \u228f (U \u228e\u1d41 V)\n  inr\u1d41 = mk inr \u03bb _ \u2192 \u2243-refl\n\n  fst\u1d41 : Map (U \u00d7\u1d41 V) U\n  fst\u1d41 = mk fst fst\n\n  snd\u1d41 : Map (U \u00d7\u1d41 V) V\n  snd\u1d41 = mk snd snd\n\nmodule _\n  {u e}(U : Universe u e)\n  {p}(P : Code U \u2192 \u2605_ p)\n  where\n    \u03a3\u1d41 : Universe (u \u2294 p) e\n    \u03a3\u1d41 = record { U = \u03a3 (Code U) P; El = El U \u2218 fst }\n\n    \u03a3\u1d41-fst : CoMap \u03a3\u1d41 U\n    \u03a3\u1d41-fst = mk fst id\n\n{-\nmodule _\n  {u e}(U : Universe u e)\n  {b}(B : \u2605_ b)\n  where\n    \u03a0\u1d41' : Universe u (e \u2294 b)\n    \u03a0\u1d41' = record { U = Code U ; El = \u03bb A \u2192 B \u2192 El U A }\n\n    todo' : B \u2192 CoMap U \u03a0\u1d41'\n    todo' b = mk id (\u03bb f \u2192 f b)\n\nmodule _\n  {u e}(U : Universe u e)\n  (F : \u2605_ e \u2192 \u2605_ e)\n  where\n  bla : (\u2200 {A} \u2192 El U A \u2192 F (El U A)) \u2192 Map U (record { El = F \u2218 El U })\n  bla = mk id\n\nmodule _\n  {u e}(U : Universe u e)\n  {p}(P : \u2200 {A} \u2192 El U A \u2192 \u2605_ p)\n  where\n    \u03a0\u1d41 : Universe u (e \u2294 p)\n    \u03a0\u1d41 = record { U = Code U ; El = \u03bb A \u2192 \u03a0 (El U A) P }\n\n    todo : Map \u03a0\u1d41 U\n    todo = mk id {!!}\n-}\n\nmodule _\n  {u e}(U : Universe u e)\n  {p}(P : \u2200 {A} \u2192 El U A \u2192 \u2605_ p)\n  where\n    \u03a3El : Code U \u2192 \u2605_ _\n    \u03a3El A = \u03a3 (El U A) P\n\n    \u03a3El\u1d41 : Universe u (e \u2294 p)\n    \u03a3El\u1d41 = record { El = \u03a3El }\n\n    \u03a3El-fst : CoMap U \u03a3El\u1d41\n    \u03a3El-fst = mk id fst\n\n-- TODO: Move elsewhere\nFinite : \u2605\u2080 \u2192 \u2605\u2081\nFinite A = \u03a3 \u2115 \u03bb n \u2192 A \u2261 Fin n\n\nUFinite : Universe _ _\nUFinite = \u03a3\u1d41 U\u2605\u2080 Finite\n\nfinite : CoMap UFinite U\u2605\u2080\nfinite = \u03a3\u1d41-fst U\u2605\u2080 Finite\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Universe\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2b799044d7a59fe246760d6c96122342c7c0eef0","subject":"lib\/Text\/Printer.agda","message":"lib\/Text\/Printer.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Text\/Printer.agda","new_file":"lib\/Text\/Printer.agda","new_contents":"module Text.Printer where\n\nopen import Data.String\nopen import Data.Nat\nopen import Data.Nat.Show\nopen import Data.Bool\nopen import Function\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n\nShowS : Set\nShowS = String \u2192 String\n\nPr : Set \u2192 Set\nPr A = A \u2192 ShowS\n\n`_ : String \u2192 ShowS\n(` s) tail = Data.String._++_ s tail\n\nparenBase : ShowS \u2192 ShowS\nparenBase doc = ` \"(\" \u2218 doc \u2218 ` \")\"\n\nrecord PrEnv : Set where\n  constructor mk\n  field\n    level : \u2115\n\n  withLevel : \u2115 \u2192 PrEnv\n  withLevel x = record { level = x }\n\nopen PrEnv\n\nparen : (PrEnv \u2192 PrEnv) \u2192 PrEnv \u2192 ShowS \u2192 ShowS\nparen f \u0394 = if \u230a level (f \u0394) \u2264? level \u0394 \u230b then id else parenBase\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Text\/Printer.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0f98de584e09e036bfb2f8536d0eacbf9fc1f7e1","subject":"rewind-on-success.agda","message":"rewind-on-success.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"rewind-on-success.agda","new_file":"rewind-on-success.agda","new_contents":"open import Type\nopen import Function\nopen import Data.Vec hiding (sum)\nimport Data.Fin as Fin\nopen Fin using (Fin; zero; suc)\nopen import Data.Nat.NP hiding (_\u2265_)\nopen import Data.Two hiding (_\u00b2; _==_)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Relation.Binary.PropositionalEquality.NP renaming (subst to tr)\nopen import HoTT\n\nmodule rewind-on-success where\n\nmodule _ {a} {A : \u2605_ a} where\n    infixr 4 _\u2666_\n    _\u2666_ : \u2200 {n} \u2192 Vec A n \u2192 (\u2115 \u2192 A) \u2192 (\u2115 \u2192 A)\n    ([]     \u2666 f) i = f i\n    (x \u2237 xs \u2666 f) zero    = x\n    (x \u2237 xs \u2666 f) (suc i) = (xs \u2666 f) i\n\n    \u2666-spec1 : \u2200 {n} (xs : Vec A n) (f : \u2115 \u2192 A) i \u2192 (xs \u2666 f) (Fin.to\u2115 i) \u2261 lookup i xs\n    \u2666-spec1 [] f ()\n    \u2666-spec1 (x \u2237 xs) f zero    = refl\n    \u2666-spec1 (x \u2237 xs) f (suc i) = \u2666-spec1 xs f i\n\n    \u2666-spec2 : \u2200 {n} (xs : Vec A n) (f : \u2115 \u2192 A) i \u2192 (xs \u2666 f) (n + i) \u2261 f i\n    \u2666-spec2 []       f i = refl\n    \u2666-spec2 (x \u2237 xs) f i = \u2666-spec2 xs f i\n\n    takeS : \u2200 n (f : \u2115 \u2192 A) \u2192 Vec A n\n    takeS zero    f = []\n    takeS (suc n) f = f 0 \u2237 takeS n (f \u2218 suc)\n\n    lookup-takeS : \u2200 {n} (f : \u2115 \u2192 A) i \u2192 f (Fin.to\u2115 i) \u2261 lookup i (takeS n f)\n    lookup-takeS f zero    = refl\n    lookup-takeS f (suc i) = lookup-takeS (f \u2218 suc) i\n\nmodule _\n  (Symbol Output : \u2605)\n  (success? : Output \u2192 \ud835\udfda)\n\n  (adversaryS : (r : \u2115 \u2192 Symbol) \u2192 \u2115)\n  (adversaryO : (r : \u2115 \u2192 Symbol) \u2192 Output)\n  where\n\n  Transcript = Vec Symbol\n\n  transcript : (r : \u2115 \u2192 Symbol) \u2192 Transcript (adversaryS r)\n  transcript r = takeS (adversaryS r) r\n\n  adversary : (r : \u2115 \u2192 Symbol) \u2192 \u2115 \u00d7 Output\n  adversary r = adversaryS r , adversaryO r\n\n  Adversary-spec = \u2200 (r s : \u2115 \u2192 Symbol) \u2192 adversary r \u2261 adversary (transcript r \u2666 s)\n\n  _\u2227\u00b0_ : \u2200 {\u03a9 : \u2605} (f g : \u03a9 \u2192 \ud835\udfda) \u2192 \u03a9 \u2192 \ud835\udfda\n  (f \u2227\u00b0 g) x = f x \u2227 g x\n\n  infix 2 _\u2194_\n  _\u2194_ : (A B : \u2605) \u2192 \u2605\n  A \u2194 B = (A \u2192 B) \u00d7 (B \u2192 A)\n\n  record PartitionV1 (\u03a9 : \u2605) : \u2605 where\n    field\n      B : \u2115 \u2192 \u03a9 \u2192 \ud835\udfda\n      B-\u2203 : \u2200 w \u2192 \u2203 \u03bb i \u2192 \u2713 (B i w)\n      B-! : \u2200 i j w \u2192 \u2713 (B i w) \u2192 \u2713 (B j w) \u2192 i \u2261 j\n\n  record Partition (\u03a9 : \u2605) : \u2605\u2081 where\n    field\n      F      : \u2115 \u2192 \u2605\n      F-part : \u03a9 \u2261 \u03a3 \u2115 F\n\n    v1 : PartitionV1 \u03a9\n    v1 = record {B = B; B-\u2203 = B-\u2203; B-! = B-!} where\n        g : \u03a9 \u2192 \u2115\n        g = fst \u2218 coe F-part\n\n        B : \u2115 \u2192 \u03a9 \u2192 \ud835\udfda\n        B i w = i == g w\n\n        B-\u2203 : \u2200 w \u2192 \u2203 \u03bb i \u2192 \u2713 (B i w)\n        B-\u2203 w = g w , ==.refl {g w}\n\n        B-! : \u2200 i j w \u2192 \u2713 (B i w) \u2192 \u2713 (B j w) \u2192 i \u2261 j\n        B-! i j w Biw Bjw = ==.sound i _ Biw \u2219 ! ==.sound j _ Bjw\n\n    open PartitionV1 v1 public\n\n  module _ {A : \u2605}{B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    pair='' : \u2200 {x y : \u03a3 A (\u03a3 B \u2218 C)}\n                (p : fst x \u2261 fst y)\n                (q : fst (snd x) \u2261 fst (snd y))\n              \u2192 tr (\u03bb x\u2081 \u2192 C x\u2081 (fst (snd y))) p (tr (C (fst x)) q (snd (snd x))) \u2261 snd (snd y) \u2192 x \u2261 y\n    pair='' refl refl = ap (\u03bb x \u2192 _ , _ , x)\n\n  \u2713-prop : (b : \ud835\udfda)(x y : \u2713 b) \u2192 x \u2261 y\n  \u2713-prop 1\u2082 pf pf' = refl\n  \u2713-prop 0\u2082 () pf'\n\n  module V1\u2192 {\u03a9} (v1 : PartitionV1 \u03a9) {{_ : UA}} where\n    open PartitionV1 v1\n    open Equivalences\n    part : Partition \u03a9\n    part = record { F = F ; F-part = F-part }\n      where\n        F : \u2115 \u2192 \u2605\n        F i = \u03a3 \u03a9 \u03bb w \u2192 \u2713 (B i w)\n\n        \u03a9\u2192 : \u03a9 \u2192 \u03a3 \u2115 F\n        \u03a9\u2192 w = fst (B-\u2203 w) , w , snd (B-\u2203 w)\n\n        \u2192\u03a9 : \u03a3 \u2115 F \u2192 \u03a9\n        \u2192\u03a9 (i , w , pf) = w\n\n        \u03a9\u2192\u03a9 : \u2200 w \u2192 \u2192\u03a9 (\u03a9\u2192 w) \u2261 w\n        \u03a9\u2192\u03a9 w = refl\n\n        \u2192\u03a9\u2192 : \u2200 x \u2192 \u03a9\u2192 (\u2192\u03a9 x) \u2261 x\n        \u2192\u03a9\u2192 (i , w , pf) = pair='' (B-! (fst (B-\u2203 w)) i w (snd (B-\u2203 w)) pf) refl (\u2713-prop (B i w) _ pf)\n\n        F-part : \u03a9 \u2261 \u03a3 \u2115 F\n        F-part = ua (equiv \u03a9\u2192 \u2192\u03a9 \u2192\u03a9\u2192 \u03a9\u2192\u03a9)\n\n  record ProbTheory : \u2605\u2081 where\n    field\n      \u211d : \u2605\n      0\u1d63 1\u1d63 : \u211d\n      _\u00b7_ _\/_ : \u211d \u2192 \u211d \u2192 \u211d\n    _\u00b2 : \u211d \u2192 \u211d\n    _\u00b2 = \u03bb x \u2192 x \u00b7 x\n    field\n      _\u2265_ : \u211d \u2192 \u211d \u2192 \u2605\n      Pr[_] : {\u03a9 : \u2605}(f : \u03a9 \u2192 \ud835\udfda) \u2192 \u211d\n      Pr\u22650 : \u2200 {\u03a9} (f : \u03a9 \u2192 \ud835\udfda) \u2192 Pr[ f ] \u2265 0\u1d63\n      Pr\u03a9\u22611 : \u2200 {\u03a9} \u2192 Pr[ (\u03bb (_ : \u03a9) \u2192 1\u2082) ] \u2261 1\u1d63\n      Pr[_\u2225_] : {\u03a9 : \u2605}(f g : \u03a9 \u2192 \ud835\udfda) \u2192 \u211d\n      Pr[_\u2225_]-spec : \u2200 {\u03a9} (f g : \u03a9 \u2192 \ud835\udfda) \u2192 Pr[ f \u2225 g ] \u2261 (Pr[ f \u2227\u00b0 g ] \/ Pr[ g ])\n      sum : {A : \u2605} \u2192 (A \u2192 \u211d) \u2192 \u211d\n      ttt : \u2200 {\u03a9}(f : \u03a9 \u2192 \ud835\udfda)\n                         (p : Partition \u03a9)\n                       \u2192 Pr[ sum \u03bb (n : \u2115) \u2192 Partition.B p n ] \u2261 sum \u03bb (n : \u2115) \u2192 Pr[ Partition.B p n ]\n      law-total-prob : \u2200 {\u03a9}(f : \u03a9 \u2192 \ud835\udfda)\n                         (p : Partition \u03a9)\n                       \u2192 Pr[ f ] \u2261 sum \u03bb (n : \u2115) \u2192 Pr[ Partition.B p n ] \u00b7 Pr[ f \u2225 Partition.B p n ]\n      E[_] : {\u03a9 : \u2605}(f : \u03a9 \u2192 \u211d) \u2192 \u211d\n    _\u00b2' : {\u03a9 : \u2605}(f : \u03a9 \u2192 \u211d) \u2192 \u03a9 \u2192 \u211d\n    _\u00b2' = \u03bb f x \u2192 (f x)\u00b2\n    field\n      E\u00b2 : \u2200 {\u03a9} (X : \u03a9 \u2192 \u211d) \u2192 E[ X \u00b2' ] \u2265 E[ X ] \u00b2\n\n  module _\n    (adversary-spec : Adversary-spec)\n\n    (process : \u2200 {n}(t : Transcript n){n\u2032}(t\u2032 : Transcript n\u2032) \u2192 {!!})\n    where\n\n    M : (r s : \u2115 \u2192 Symbol) \u2192 {!!}\n    M r s = case success? o\n              0: {!!}\n              1: process t t\u2032\n      where\n        n = adversaryS r\n        o = adversaryO r\n        t = transcript r\n        #h = {!!}\n        h  = takeS #h r\n        r\u2032 = h \u2666 s\n        n\u2032 = adversaryS r\u2032\n        o\u2032 = adversaryO r\u2032\n        t\u2032 = transcript r\u2032\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'rewind-on-success.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2824f8185e103bd7e2500937eff9fb9af1077e07","subject":"Add a broken attempt at simplifying Nat.GCD: Nat.GCD.NP","message":"Add a broken attempt at simplifying Nat.GCD: Nat.GCD.NP\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/GCD\/NP.agda","new_file":"lib\/Data\/Nat\/GCD\/NP.agda","new_contents":"module Data.Nat.GCD.NP where\n\nopen import Data.Nat\nopen import Data.Nat.Properties\nopen import Data.Nat.Divisibility as Div\nopen import Relation.Binary\nprivate module P = Poset Div.poset\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality as PropEq using (_\u2261_)\nopen import Induction\nopen import Induction.Nat\nopen import Induction.Lexicographic\nopen import Function\nopen import Data.Nat.GCD.Lemmas\n\n------------------------------------------------------------------------\n-- Greatest common divisor\n\nmodule GCD where\n\n  -- Specification of the greatest common divisor (gcd) of two natural\n  -- numbers.\n\n  record GCD (m n gcd : \u2115) : Set where\n    constructor is\n    field\n      -- The gcd is a common divisor.\n      commonDivisor : gcd \u2223 m \u00d7 gcd \u2223 n\n\n      -- All common divisors divide the gcd, i.e. the gcd is the\n      -- greatest common divisor according to the partial order _\u2223_.\n      greatest : \u2200 {d} \u2192 d \u2223 m \u00d7 d \u2223 n \u2192 d \u2223 gcd\n\n  open GCD public\n\n  -- The gcd is unique.\n\n  unique : \u2200 {d\u2081 d\u2082 m n} \u2192 GCD m n d\u2081 \u2192 GCD m n d\u2082 \u2192 d\u2081 \u2261 d\u2082\n  unique d\u2081 d\u2082 = P.antisym (GCD.greatest d\u2082 (GCD.commonDivisor d\u2081))\n                           (GCD.greatest d\u2081 (GCD.commonDivisor d\u2082))\n\n  -- The gcd relation is \"symmetric\".\n\n  sym : \u2200 {d m n} \u2192 GCD m n d \u2192 GCD n m d\n  sym g = is (swap $ GCD.commonDivisor g) (GCD.greatest g \u2218 swap)\n\n  -- The gcd relation is \"reflexive\".\n\n  refl : \u2200 {n} \u2192 GCD n n n\n  refl = is (P.refl , P.refl) proj\u2081\n\n  -- The GCD of 0 and n is n.\n\n  base : \u2200 {n} \u2192 GCD 0 n n\n  base {n} = is (n \u22230 , P.refl) proj\u2082\n\n  -- If d is the gcd of n and k, then it is also the gcd of n and\n  -- n\u00a0+\u00a0k.\n\n  step : \u2200 {n k d} \u2192 GCD n k d \u2192 GCD n (n + k) d\n  step g with GCD.commonDivisor g\n  step {n} {k} {d} g | (d\u2081 , d\u2082) = is (d\u2081 , \u2223-+ d\u2081 d\u2082) greatest\u2032\n    where\n    greatest\u2032 : \u2200 {d\u2032} \u2192 d\u2032 \u2223 n \u00d7 d\u2032 \u2223 n + k \u2192 d\u2032 \u2223 d\n    greatest\u2032 (d\u2081 , d\u2082) = GCD.greatest g (d\u2081 , \u2223-\u2238 d\u2082 d\u2081)\n\nopen GCD public using (GCD)\n\n------------------------------------------------------------------------\n-- Calculating the gcd\n\n-- The calculation also proves B\u00e9zout's lemma.\n\nmodule B\u00e9zout where\n\n  module Identity where\n\n    -- If m and n have greatest common divisor d, then one of the\n    -- following two equations is satisfied, for some numbers x and y.\n    -- The proof is \"lemma\" below (B\u00e9zout's lemma).\n    --\n    -- (If this identity was stated using integers instead of natural\n    -- numbers, then it would not be necessary to have two equations.)\n\n    data Identity (d m n : \u2115) : Set where\n    --  +- : (x y : \u2115) (eq : d + y * n \u2261 x * m) \u2192 Identity d m n\n      -+ : (x y : \u2115) (eq : d + x * m \u2261 y * n) \u2192 Identity d m n\n\n    -- Various properties about Identity.\n\n{-\n    sym : \u2200 {d} \u2192 Symmetric (Identity d)\n    sym (+- x y eq) = -+ y x eq\n    sym (-+ x y eq) = +- y x eq\n    -}\n\n    refl : \u2200 {d} \u2192 Identity d d d\n    refl = -+ 0 1 PropEq.refl\n\n    base : \u2200 {d} \u2192 Identity d 0 d\n    base = -+ 0 1 PropEq.refl\n\n    private\n      infixl 7 _\u2295_\n\n      _\u2295_ : \u2115 \u2192 \u2115 \u2192 \u2115\n      m \u2295 n = 1 + m + n\n\n    step : \u2200 {d n k} \u2192 Identity d n k \u2192 Identity d n (n + k)\n    {-\n    step {d}     (+-  x  y       eq) with compare x y\n    step {d}     (+- .x .x       eq) | equal x     = +- (2 * x)     x       (lem\u2082 d x   eq)\n    step {d}     (+- .x .(x \u2295 i) eq) | less x i    = +- (2 * x \u2295 i) (x \u2295 i) (lem\u2083 d x   eq)\n    step {d} {n} (+- .(y \u2295 i) .y eq) | greater y i = +- (2 * y \u2295 i) y       (lem\u2084 d y n eq)\n    -}\n    step {d}     (-+  x  y       eq) with compare x y\n    step {d}     (-+ .x .x       eq) | equal x     = -+ (2 * x)     x       (lem\u2085 d x   eq)\n    step {d}     (-+ .x .(x \u2295 i) eq) | less x i    = -+ (2 * x \u2295 i) (x \u2295 i) (lem\u2086 d x   eq)\n    step {d} {n} (-+ .(y \u2295 i) .y eq) | greater y i = -+ (2 * y \u2295 i) y       (lem\u2087 d y n eq)\n\n  open Identity public using (Identity) -- ; +-; -+)\n\n  module Lemma where\n\n    -- This type packs up the gcd, the proof that it is a gcd, and the\n    -- proof that it satisfies B\u00e9zout's identity.\n\n    data Lemma (m n : \u2115) : Set where\n      result : (d : \u2115) (g : GCD m n d) (b : Identity d m n) \u2192 Lemma m n\n\n    -- Various properties about Lemma.\n\n    {-\n    sym : Symmetric Lemma\n    sym (result d g b) = result d (GCD.sym g) (Identity.sym b)\n    -}\n\n    base : \u2200 d \u2192 Lemma 0 d\n    base d = result d GCD.base Identity.base\n\n    refl : \u2200 d \u2192 Lemma d d\n    refl d = result d GCD.refl Identity.refl\n\n    step\u02e1 : \u2200 {n k} \u2192 Lemma n (suc k) \u2192 Lemma n (suc (n + k))\n    step\u02e1 {n} {k} (result d g b) =\n      PropEq.subst (Lemma n) (lem\u2080 n k) $\n        result d (GCD.step g) (Identity.step b)\n\n    {-\n    step\u02b3 : \u2200 {n k} \u2192 Lemma (suc k) n \u2192 Lemma (suc (n + k)) n\n    step\u02b3 = sym \u2218 step\u02e1 \u2218 sym\n    -}\n\n  open Lemma public using (Lemma; result)\n\n  -- B\u00e9zout's lemma proved using some variant of the extended\n  -- Euclidean algorithm.\n\n  lemma : (m n : \u2115) \u2192 Lemma m n\n  lemma zero n = Lemma.base n\n  lemma m zero = {!Lemma.sym (Lemma.base m)!}\n  lemma (suc m) (suc n) = build [ <-rec-builder \u2297 <-rec-builder ] P gcd (m , n)\n    where\n    P : \u2115 \u00d7 \u2115 \u2192 Set\n    P (m , n) = Lemma (suc m) (suc n)\n\n    gcd : \u2200 p \u2192 (<-Rec \u2297 <-Rec) P p \u2192 P p\n    gcd (m , n           ) rec with compare m n\n    gcd (m , .m          ) rec | equal .m     = Lemma.refl (suc m)\n    gcd (m , .(suc m + k)) rec | less .m k    =\n                      -- \"gcd m k\"\n                      -- k < n\n                      -- m < n\n                      -- n = suc m + k\n                      -- dist m n = suc k\n\n                      -- m + k < m + suc m + k\n                      --    k < suc m + k\n                      --    0 < suc m\n      Lemma.step\u02e1 $ proj\u2081 rec k (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264-steps {k} {k} m (\u2264\u2032\u21d2\u2264 \u2264\u2032-refl))))\n    gcd (.(suc n + k) , n) rec | greater .n k =\n                      -- \"gcd k n\"\n                      -- m > n\n                      -- m \u2261 suc n + k\n                      -- dist m n = suc k\n\n                      -- k + n < suc n + k + n\n                      --    k < suc n + k\n                      --    0 < suc n\n                      -- gcd n k\n      {!Lemma.step\u02b3 $ proj\u2082 rec k (\u2264\u21d2\u2264\u2032 (s\u2264s (\u2264-steps {k} {k} n (\u2264\u2032\u21d2\u2264 \u2264\u2032-refl)))) n!}\n\n  -- B\u00e9zout's identity can be recovered from the GCD.\n\n  identity : \u2200 {m n d} \u2192 GCD m n d \u2192 Identity d m n\n  identity {m} {n} g with lemma m n\n  identity g | result d g\u2032 b with GCD.unique g g\u2032\n  identity g | result d g\u2032 b | PropEq.refl = b\n\n-- Calculates the gcd of the arguments.\n\ngcd : (m n : \u2115) \u2192 \u2203 \u03bb d \u2192 GCD m n d\ngcd m n with B\u00e9zout.lemma m n\ngcd m n | B\u00e9zout.result d g _ = (d , g)\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Nat\/GCD\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4995bc2a3bd988bcf90ed2efe5b614e17b5554a0","subject":"Add alternative defintion for IND CPA","message":"Add alternative defintion for IND CPA\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/IND-CPA-alt.agda","new_file":"Game\/IND-CPA-alt.agda","new_contents":"\n{-# OPTIONS --without-K #-}\nopen import Type\nopen import Data.Product\nopen import Data.Bit\n\nmodule Game.IND-CPA-alt\n  (PubKey     : \u2605)\n  (SecKey     : \u2605)\n  (Message    : \u2605)\n  (CipherText : \u2605)\n\n  -- randomness supply for: encryption, key-generation, adversary, extensions\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText)\n\nwhere\n\nM\u00b2 = Bit \u2192 Message\n\n-- IND-CPA adversary in two parts\nAdv : \u2605\nAdv = R\u2090 \u2192 PubKey \u2192 (M\u00b2 \u00d7 (CipherText \u2192 Bit))\n\n-- IND-CPA randomness supply\nR : \u2605\nR = (R\u2090 \u00d7 R\u2096 \u00d7 R\u2091)\n\n-- IND-CPA games:\n--   * input: adversary and randomness supply\n--   * output b: adversary claims we are in game \u2141 b\nGame : \u2605\nGame = Adv \u2192 R \u2192 Bit\n\n-- The game step by step:\n-- (pk) key-generation, only the public-key is needed\n-- (mb) send randomness, public-key and bit\n--      receive which message to encrypt\n-- (c)  encrypt the message\n-- (b\u2032) send randomness, public-key and ciphertext\n--      receive the guess from the adversary\n\u2141 : Bit \u2192 Game\n\u2141 b m (r\u2090 , r\u2096 , r\u2091) = b\u2032\n  where\n  pk = proj\u2081 (KeyGen r\u2096)\n  ad = m r\u2090 pk\n  mb = proj\u2081 ad b\n  c  = Enc pk mb r\u2091\n  b\u2032 = proj\u2082 ad c\n\n\u2141\u2080 \u2141\u2081 : Game\n\u2141\u2080 = \u2141 0b\n\u2141\u2081 = \u2141 1b\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Game\/IND-CPA-alt.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b39ba7eb464b3b10930618ae98141e24089339ef","subject":"Added draft about the accesibility relation.","message":"Added draft about the accesibility relation.\n\nIgnore-this: 4464636b2ef8f930696708c29bdaf37a\n\ndarcs-hash:20110221161001-3bd4e-55f9c66f8e4147045286838c732c521ac519dd40.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Accesibility\/WellFoundedInduction.agda","new_file":"Draft\/Accesibility\/WellFoundedInduction.agda","new_contents":"----------------------------------------------------------------------------\n-- Well-founded induction on LT using the accesibility relation\n----------------------------------------------------------------------------\n\n\nmodule Draft.Accesibility.WellFoundedInduction where\n\nopen import FOTC.Base\n\nopen import Common.Function\n\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\n\n----------------------------------------------------------------------------\n-- Accesibility stuff for the FOTC natural numbers\n\n-- Adapted from\n-- http:\/\/www.iis.sinica.edu.tw\/~scm\/2008\/well-founded-recursion-and-accessibility\/\n\ndata AccN (_<_ : D \u2192 D \u2192 Set) : D \u2192 Set where\n accN : \u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 y < x \u2192 AccN _<_ y) \u2192 AccN _<_ x\n\nacc-elimN : (_<_ : D \u2192 D \u2192 Set) {P : D \u2192 Set} \u2192\n            (\u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 y < x \u2192 AccN _<_ y) \u2192\n                 (\u2200 {y} \u2192 N y \u2192 y < x \u2192 P y) \u2192 P x) \u2192\n            \u2200 {x} \u2192 N x \u2192 AccN _<_ x \u2192 P x\nacc-elimN _<_ \u03a6 Nx (accN _ h) = \u03a6 Nx h (\u03bb Ny y<x \u2192 acc-elimN _<_ \u03a6 Ny (h Ny y<x))\n\nacc-foldN : (R : D \u2192 D \u2192 Set) {P : D \u2192 Set} \u2192\n            (\u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 R y x \u2192 P y) \u2192 P x) \u2192\n            \u2200 {x} \u2192 N x \u2192 AccN R x \u2192 P x\nacc-foldN R f Nx (accN _ h) = f Nx (\u03bb Ny yRx \u2192 acc-foldN R f Ny (h Ny yRx))\n\nwell-foundN : (D \u2192 D \u2192 Set) \u2192 Set\nwell-foundN _<_ = \u2200 {x} \u2192 N x \u2192 AccN _<_ x\n\nrec-wfN : {_<_ : D \u2192 D \u2192 Set} {P : D \u2192 Set} \u2192\n          well-foundN _<_ \u2192\n          (\u2200 {x} \u2192 N x \u2192 (\u2200 {y} \u2192 N y \u2192 y < x \u2192 P y) \u2192 P x) \u2192\n          \u2200 {x} \u2192 N x \u2192 P x\nrec-wfN {_<_} wf f Nx = acc-foldN _<_ f Nx (wf Nx)\n\n-- The LT relation is well-founded.\nLT-wf : well-foundN LT\nLT-wf Nn = accN Nn (helper Nn)\n\n  where\n    helper : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LT n m \u2192 AccN LT n\n    helper zN     Nn n<0  = \u22a5-elim $ x<0\u2192\u22a5 Nn n<0\n    helper (sN _) zN 0<Sm = accN zN (\u03bb Nn' n'<0 \u2192 \u22a5-elim $ x<0\u2192\u22a5 Nn' n'<0)\n\n    helper (sN {m} Nm) (sN {n} Nn) Sn<Sm = accN (sN Nn)\n      (\u03bb {n'} Nn' n'<Sn \u2192\n        let n<m : LT n m\n            n<m = Sx<Sy\u2192x<y Sn<Sm\n\n            n'<m : LT n' m\n            n'<m = [ (\u03bb n'<n \u2192 <-trans Nn' Nn Nm n'<n n<m)\n                   , (\u03bb n'\u2261n \u2192 x\u2261y\u2192y<z\u2192x<z n'\u2261n n<m)\n                   ] (x<Sy\u2192x<y\u2228x\u2261y Nn' Nn n'<Sn)\n\n        in  helper Nm Nn' n'<m\n      )\n\nwfInd-LT : (P : D \u2192 Set) \u2192\n           (\u2200 {m} \u2192 N m \u2192 (\u2200 {n} \u2192 N n \u2192 LT n m \u2192 P n) \u2192 P m) \u2192\n           \u2200 {n} \u2192 N n \u2192 P n\nwfInd-LT P = rec-wfN LT-wf\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Accesibility\/WellFoundedInduction.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"0741babb2a80d8c4597d19d43235f0bfeb4b1882","subject":"Added Empty version for LF.","message":"Added Empty version for LF.\n\nIgnore-this: 2cc68c08e10e526379c3ec6c4c331c06\n\ndarcs-hash:20111215163330-3bd4e-c65169a77e39acb5940ec3e70168e6c39f2cfc0a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Empty.agda","new_file":"notes\/thesis\/logical-framework\/Empty.agda","new_contents":"-- Tested with Agda 2.3.1 on 15 December 2011.\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Empty where\n\npostulate\n  \u22a5      : Set\n  \u22a5-elim : {A : Set} \u2192 \u22a5 \u2192 A\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/logical-framework\/Empty.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ed1d572450bb1d5cf21f506fad032b5356e0e6a2","subject":"alternative to BCD subtyping","message":"alternative to BCD subtyping\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"extra\/extra\/Value.agda","new_file":"extra\/extra\/Value.agda","new_contents":"open import Data.Nat using (\u2115; suc ; zero; _+_; _\u2264\u2032_; _<\u2032_; _<_; _\u2264_;\n    z\u2264n; s\u2264s; \u2264\u2032-refl; \u2264\u2032-step; _\u225f_) renaming (_\u2294_ to max)\nopen import Data.Nat.Properties\n  using (n\u22640\u21d2n\u22610; \u2264-refl; \u2264-trans; m\u2264m\u2294n; n\u2264m\u2294n; \u2264-step; \u2294-mono-\u2264;\n         +-mono-\u2264; +-mono-\u2264-<; +-mono-<-\u2264; +-comm; +-assoc; n\u22641+n; \n         \u2264-pred; m\u2264m+n; n\u2264m+n; \u2264-reflexive; \u2264\u2032\u21d2\u2264; \u2264\u21d2\u2264\u2032; +-suc)\nopen Data.Nat.Properties.\u2264-Reasoning using (begin_; _\u2264\u27e8_\u27e9_; _\u220e)\nopen import Data.Bool  using (Bool) renaming (_\u225f_ to _=?_)\nopen import Data.Product using (_\u00d7_; \u03a3; \u03a3-syntax; \u2203; \u2203-syntax; proj\u2081; proj\u2082)\n  renaming (_,_ to \u27e8_,_\u27e9)\nopen import Data.Sum using (_\u228e_; inj\u2081; inj\u2082)\nopen import Data.Empty using (\u22a5-elim) renaming (\u22a5 to Bot)\nopen import Data.Unit using (\u22a4; tt)\nopen import Data.Maybe\nopen import Data.List using (List ; _\u2237_ ; []; _++_) \nopen import Relation.Nullary using (\u00ac_)\nopen import Relation.Nullary using (Dec; yes; no)\nopen import Relation.Nullary.Negation using (contradiction)\nopen import Relation.Binary.PropositionalEquality using (_\u2261_; refl; sym; cong)\nopen Relation.Binary.PropositionalEquality.\u2261-Reasoning renaming (begin_ to start_; _\u220e to _\u25a1)\n\n\nmodule extra.Value where\n\ndata Base : Set where\n  Nat : Base\n  \ud835\udd39 : Base\n\ndata Prim : Set where\n  base : Base \u2192 Prim\n  _\u21d2_ : Base \u2192 Prim \u2192 Prim\n\nbase-rep : Base \u2192 Set \nbase-rep Nat = \u2115\nbase-rep \ud835\udd39 = Bool\n\nrep : Prim \u2192 Set\nrep (base b) = base-rep b\nrep (b \u21d2 p) = base-rep b \u2192 rep p\n\nbase-eq? : (B : Base) \u2192 (B' : Base) \u2192 Dec (B \u2261 B')\nbase-eq? Nat Nat = yes refl\nbase-eq? Nat \ud835\udd39 = no (\u03bb ())\nbase-eq? \ud835\udd39 Nat = no (\u03bb ())\nbase-eq? \ud835\udd39 \ud835\udd39 = yes refl\n\nbase-rep-eq? : \u2200{B} \u2192 (k : base-rep B) (k\u2032 : base-rep B) \u2192 Dec (k \u2261 k\u2032)\nbase-rep-eq? {Nat} k k\u2032 = k \u225f k\u2032\nbase-rep-eq? {\ud835\udd39} k k\u2032 = k =? k\u2032\n\ninfixr 7 _\u21a6_\ninfixl 6 _\u2294_\n\ndata Value : Set \n\ndata Value where\n  \u22a5 : Value\n  const : {b : Base} \u2192 base-rep b \u2192 Value\n  _\u21a6_ : Value \u2192 Value \u2192 Value\n  _\u2294_ : (u : Value) \u2192 (v : Value) \u2192 Value\n\ninfix 5 _\u2208_\n\n_\u2208_ : Value \u2192 Value \u2192 Set\nu \u2208 \u22a5 = u \u2261 \u22a5\nu \u2208 const {B} k = u \u2261 const {B} k\nu \u2208 v \u21a6 w = u \u2261 v \u21a6 w\nu \u2208 (v \u2294 w) = u \u2208 v \u228e u \u2208 w\n\ninfix 5 _\u2286_\n\n_\u2286_ : Value \u2192 Value \u2192 Set\nv \u2286 w = \u2200{u} \u2192 u \u2208 v \u2192 u \u2208 w\n\n\nAllFun : (u : Value) \u2192 Set\nAllFun \u22a5 = Bot\nAllFun (const x) = Bot\nAllFun (v \u21a6 w) = \u22a4\nAllFun (u \u2294 v) = AllFun u \u00d7 AllFun v \n\ndom : (u : Value) \u2192 Value\ndom \u22a5 = \u22a5\ndom (const k) = \u22a5\ndom (v \u21a6 w) = v\ndom (u \u2294 v) = dom u \u2294 dom v\n\ncod : (u : Value) \u2192 Value\ncod \u22a5  = \u22a5\ncod (const k) = \u22a5\ncod (v \u21a6 w) = w\ncod (u \u2294 v) = cod u \u2294 cod v\n\ninfix 4 _\u2291_\n\ndata _\u2291_ : Value \u2192 Value \u2192 Set where\n\n  \u2291-\u22a5 : \u2200 {v} \u2192 \u22a5 \u2291 v\n\n  \u2291-const : \u2200 {B}{k} \u2192 const {B} k \u2291 const {B} k\n\n  \u2291-conj-L : \u2200 {u v w}\n        \u2192 v \u2291 u\n        \u2192 w \u2291 u\n          -----------\n        \u2192 v \u2294 w \u2291 u\n\n  \u2291-conj-R1 : \u2200 {u v w}\n       \u2192 u \u2291 v\n         ------------------\n       \u2192 u \u2291 v \u2294 w\n\n  \u2291-conj-R2 : \u2200 {u v w}\n       \u2192 u \u2291 w\n         -----------\n       \u2192 u \u2291 v \u2294 w\n\n  \u2291-fun : \u2200 {u u\u2032 v w}\n       \u2192 u\u2032 \u2286 u\n       \u2192 AllFun u\u2032\n       \u2192 dom u\u2032 \u2291 v\n       \u2192 w \u2291 cod u\u2032\n         -------------------\n       \u2192 v \u21a6 w \u2291 u\n\n\n\u2291-refl : \u2200{v} \u2192 v \u2291 v\n\u2291-refl {\u22a5} = \u2291-\u22a5\n\u2291-refl {const k} = \u2291-const\n\u2291-refl {v \u21a6 w} = \u2291-fun{v \u21a6 w}{v \u21a6 w} (\u03bb {u} z \u2192 z) tt (\u2291-refl{v}) \u2291-refl\n\u2291-refl {v\u2081 \u2294 v\u2082} = \u2291-conj-L (\u2291-conj-R1 \u2291-refl) (\u2291-conj-R2 \u2291-refl)\n\nfactor : (u : Value) \u2192 (u\u2032 : Value) \u2192 (v : Value) \u2192 (w : Value) \u2192 Set\nfactor u u\u2032 v w = AllFun u\u2032 \u00d7 u\u2032 \u2286 u \u00d7 dom u\u2032 \u2291 v \u00d7 w \u2291 cod u\u2032\n\n\n\u2291-fun-inv : \u2200{u\u2081 u\u2082 v w}\n      \u2192 u\u2081 \u2291 u\u2082\n      \u2192 v \u21a6 w \u2208 u\u2081\n      \u2192 \u03a3[ u\u2083 \u2208 Value ] factor u\u2082 u\u2083 v w\n\u2291-fun-inv {.\u22a5} {u\u2082} {v} {w} \u2291-\u22a5 ()\n\u2291-fun-inv {.(const _)} {.(const _)} {v} {w} \u2291-const ()\n\u2291-fun-inv {u11 \u2294 u12} {u\u2082} {v} {w} (\u2291-conj-L u\u2081\u2291u\u2082 u\u2081\u2291u\u2083) (inj\u2081 x) =\n    \u2291-fun-inv u\u2081\u2291u\u2082 x\n\u2291-fun-inv {u11 \u2294 u12} {u\u2082} {v} {w} (\u2291-conj-L u\u2081\u2291u\u2082 u\u2081\u2291u\u2083) (inj\u2082 y) =\n    \u2291-fun-inv u\u2081\u2291u\u2083 y\n\u2291-fun-inv {u\u2081} {u21 \u2294 u22} {v} {w} (\u2291-conj-R1 u\u2081\u2291u\u2082) v\u21a6w\u2208u\u2081\n    with \u2291-fun-inv {u\u2081} {u21} {v} {w} u\u2081\u2291u\u2082 v\u21a6w\u2208u\u2081\n... | \u27e8 u\u2083 , \u27e8 afu\u2083 , \u27e8 u3\u2286u\u2081 , \u27e8 du\u2083\u2291v , w\u2291codu\u2083 \u27e9 \u27e9 \u27e9 \u27e9 =\n    \u27e8 u\u2083 , \u27e8 afu\u2083 , \u27e8 (\u03bb {x} x\u2081 \u2192 inj\u2081 (u3\u2286u\u2081 x\u2081)) , \u27e8 du\u2083\u2291v , w\u2291codu\u2083 \u27e9 \u27e9 \u27e9 \u27e9  \n\u2291-fun-inv {u\u2081} {u21 \u2294 u22} {v} {w} (\u2291-conj-R2 u\u2081\u2291u\u2082) v\u21a6w\u2208u\u2081\n    with \u2291-fun-inv {u\u2081} {u22} {v} {w} u\u2081\u2291u\u2082 v\u21a6w\u2208u\u2081\n... | \u27e8 u\u2083 , \u27e8 afu\u2083 , \u27e8 u3\u2286u\u2081 , \u27e8 du\u2083\u2291v , w\u2291codu\u2083 \u27e9 \u27e9 \u27e9 \u27e9 =\n    \u27e8 u\u2083 , \u27e8 afu\u2083 , \u27e8 (\u03bb {x} x\u2081 \u2192 inj\u2082 (u3\u2286u\u2081 x\u2081)) , \u27e8 du\u2083\u2291v , w\u2291codu\u2083 \u27e9 \u27e9 \u27e9 \u27e9  \n\u2291-fun-inv {u11 \u21a6 u21} {u\u2082} {v} {w} (\u2291-fun{u\u2032 = u\u2032} u\u2032\u2286u\u2082 afu\u2032 du\u2032\u2291u11 u21\u2291cu\u2032)\n    refl =\n      \u27e8 u\u2032 , \u27e8 afu\u2032 , \u27e8 u\u2032\u2286u\u2082 , \u27e8 du\u2032\u2291u11 , u21\u2291cu\u2032 \u27e9 \u27e9 \u27e9 \u27e9\n\n\nsub-inv-trans : \u2200{u\u2032 u\u2082 u : Value}\n    \u2192 AllFun u\u2032  \u2192  u\u2032 \u2286 u\n    \u2192 (\u2200{v\u2032 w\u2032} \u2192 v\u2032 \u21a6 w\u2032 \u2208 u\u2032 \u2192 \u03a3[ u\u2083 \u2208 Value ] factor u\u2082 u\u2083 v\u2032 w\u2032)\n      ---------------------------------------------------------------\n    \u2192 \u03a3[ u\u2083 \u2208 Value ] factor u\u2082 u\u2083 (dom u\u2032) (cod u\u2032)\nsub-inv-trans {\u22a5} {u\u2082} {u} () u\u2032\u2286u IH\nsub-inv-trans {const k} {u\u2082} {u} () u\u2032\u2286u IH\nsub-inv-trans {u\u2081\u2032 \u21a6 u\u2082\u2032} {u\u2082} {u} fu\u2032 u\u2032\u2286u IH = IH refl\nsub-inv-trans {u\u2081\u2032 \u2294 u\u2082\u2032} {u\u2082} {u} \u27e8 afu\u2081\u2032 , afu\u2082\u2032 \u27e9 u\u2032\u2286u IH\n    with sub-inv-trans {u\u2081\u2032} {u\u2082} {u} afu\u2081\u2032\n               (\u03bb {u\u2081} z \u2192 u\u2032\u2286u (inj\u2081 z)) (\u03bb {v\u2032} {w\u2032} z \u2192 IH (inj\u2081 z))\n    | sub-inv-trans {u\u2082\u2032} {u\u2082} {u} afu\u2082\u2032\n               (\u03bb {u\u2081} z \u2192 u\u2032\u2286u (inj\u2082 z)) (\u03bb {v\u2032} {w\u2032} z \u2192 IH (inj\u2082 z))\n... | \u27e8 u\u2083 , \u27e8 afu\u2083 , \u27e8 u\u2083\u2286 , \u27e8 du\u2083\u2291 , \u2291cu\u2083 \u27e9 \u27e9 \u27e9 \u27e9\n    | \u27e8 u\u2084 , \u27e8 afu\u2084 , \u27e8 u\u2084\u2286 , \u27e8 du\u2084\u2291 , \u2291cu\u2084 \u27e9 \u27e9 \u27e9 \u27e9 =\n\n      \u27e8 (u\u2083 \u2294 u\u2084) , \u27e8 \u27e8 afu\u2083 , afu\u2084 \u27e9 , \u27e8 G , \u27e8 H , I \u27e9 \u27e9 \u27e9 \u27e9\n    where\n    G : \u2200 {u\u2081} \u2192 u\u2081 \u2208 u\u2083 \u228e u\u2081 \u2208 u\u2084 \u2192 u\u2081 \u2208 u\u2082\n    G {u\u2081} (inj\u2081 x) = u\u2083\u2286 x\n    G {u\u2081} (inj\u2082 y) = u\u2084\u2286 y\n\n    H : dom u\u2083 \u2294 dom u\u2084 \u2291 dom u\u2081\u2032 \u2294 dom u\u2082\u2032\n    H = \u2291-conj-L (\u2291-conj-R1 du\u2083\u2291) (\u2291-conj-R2 du\u2084\u2291)\n\n    I : cod u\u2081\u2032 \u2294 cod u\u2082\u2032 \u2291 cod u\u2083 \u2294 cod u\u2084\n    I = \u2291-conj-L (\u2291-conj-R1 \u2291cu\u2083) (\u2291-conj-R2 \u2291cu\u2084)\n\n\n\u2294\u2291R : \u2200{B C A}\n    \u2192 B \u2294 C \u2291 A\n    \u2192 B \u2291 A\n\u2294\u2291R (\u2291-conj-L B\u2294C\u2291A B\u2294C\u2291A\u2081) = B\u2294C\u2291A\n\u2294\u2291R (\u2291-conj-R1 B\u2294C\u2291A) = \u2291-conj-R1 (\u2294\u2291R B\u2294C\u2291A)\n\u2294\u2291R (\u2291-conj-R2 B\u2294C\u2291A) = \u2291-conj-R2 (\u2294\u2291R B\u2294C\u2291A)\n\n\u2294\u2291L : \u2200{B C A}\n    \u2192 B \u2294 C \u2291 A\n    \u2192 C \u2291 A\n\u2294\u2291L (\u2291-conj-L B\u2294C\u2291A B\u2294C\u2291A\u2081) = B\u2294C\u2291A\u2081\n\u2294\u2291L (\u2291-conj-R1 B\u2294C\u2291A) = \u2291-conj-R1 (\u2294\u2291L B\u2294C\u2291A)\n\u2294\u2291L (\u2291-conj-R2 B\u2294C\u2291A) = \u2291-conj-R2 (\u2294\u2291L B\u2294C\u2291A)\n\n\nu\u2208v\u2291w\u2192u\u2291w : \u2200{B A C} \u2192 C \u2208 B \u2192 B \u2291 A \u2192 C \u2291 A\nu\u2208v\u2291w\u2192u\u2291w {\u22a5} C\u2208B B\u2291A  rewrite C\u2208B = B\u2291A\nu\u2208v\u2291w\u2192u\u2291w {const k} C\u2208B B\u2291A rewrite C\u2208B = B\u2291A\nu\u2208v\u2291w\u2192u\u2291w {B\u2081 \u21a6 B\u2082} C\u2208B B\u2291A rewrite C\u2208B = B\u2291A\nu\u2208v\u2291w\u2192u\u2291w {B\u2081 \u2294 B\u2082}{A}{C} (inj\u2081 C\u2208B\u2081) B\u2291A = u\u2208v\u2291w\u2192u\u2291w {B\u2081}{A}{C} C\u2208B\u2081 (\u2294\u2291R B\u2291A)\nu\u2208v\u2291w\u2192u\u2291w {B\u2081 \u2294 B\u2082}{A}{C} (inj\u2082 C\u2208B\u2082) B\u2291A = u\u2208v\u2291w\u2192u\u2291w {B\u2082}{A}{C} C\u2208B\u2082 (\u2294\u2291L B\u2291A)\n\nu\u2286v\u2291w\u2192u\u2291w : \u2200{u v w} \u2192 u \u2286 v \u2192 v \u2291 w \u2192 u \u2291 w\nu\u2286v\u2291w\u2192u\u2291w {\u22a5} {v} {w} u\u2286v v\u2291w = \u2291-\u22a5\nu\u2286v\u2291w\u2192u\u2291w {const k} {v} {w} u\u2286v v\u2291w\n    with u\u2286v refl\n... | k\u2208v = u\u2208v\u2291w\u2192u\u2291w k\u2208v v\u2291w\nu\u2286v\u2291w\u2192u\u2291w {u\u2081 \u21a6 u\u2082} {v} {w} u\u2286v v\u2291w\n    with u\u2286v refl\n... | u\u2081\u21a6u\u2082\u2208v = u\u2208v\u2291w\u2192u\u2291w u\u2081\u21a6u\u2082\u2208v v\u2291w\nu\u2286v\u2291w\u2192u\u2291w {u\u2081 \u2294 u\u2082} {v} {w} u\u2286v v\u2291w =\n    \u2291-conj-L (u\u2286v\u2291w\u2192u\u2291w u\u2081\u2286v v\u2291w) (u\u2286v\u2291w\u2192u\u2291w u\u2082\u2286v v\u2291w)\n    where\n    u\u2081\u2286v : u\u2081 \u2286 v\n    u\u2081\u2286v {u\u2032} u\u2032\u2208u\u2081 = u\u2286v (inj\u2081 u\u2032\u2208u\u2081)\n    u\u2082\u2286v : u\u2082 \u2286 v\n    u\u2082\u2286v {u\u2032} u\u2032\u2208u\u2082 = u\u2286v (inj\u2082 u\u2032\u2208u\u2082)\n\ndepth : (v : Value) \u2192 \u2115\ndepth \u22a5 = zero\ndepth (const k) = zero\ndepth (v \u21a6 w) = suc (max (depth v) (depth w))\ndepth (v\u2081 \u2294 v\u2082) = max (depth v\u2081) (depth v\u2082)\n\nsize : (v : Value) \u2192 \u2115\nsize \u22a5 = zero\nsize (const k) = zero\nsize (v \u21a6 w) = suc (size v + size w)\nsize (v\u2081 \u2294 v\u2082) = suc (size v\u2081 + size v\u2082)\n\n\u2208\u2192depth\u2264 : \u2200{v u : Value} \u2192 u \u2208 v \u2192 depth u \u2264 depth v\n\u2208\u2192depth\u2264 {\u22a5} {u} u\u2208v rewrite u\u2208v = _\u2264_.z\u2264n\n\u2208\u2192depth\u2264 {const x} {u} u\u2208v rewrite u\u2208v = _\u2264_.z\u2264n\n\u2208\u2192depth\u2264 {v \u21a6 w} {u} u\u2208v rewrite u\u2208v = \u2264-refl\n\u2208\u2192depth\u2264 {v\u2081 \u2294 v\u2082} {u} (inj\u2081 x) =\n    \u2264-trans (\u2208\u2192depth\u2264 {v\u2081} {u} x) (m\u2264m\u2294n (depth v\u2081) (depth v\u2082))\n\u2208\u2192depth\u2264 {v\u2081 \u2294 v\u2082} {u} (inj\u2082 y) =\n    \u2264-trans (\u2208\u2192depth\u2264 {v\u2082} {u} y) (n\u2264m\u2294n (depth v\u2081) (depth v\u2082))\n\nmax-lub : \u2200{x y z : \u2115} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 max x y \u2264 z\nmax-lub {.0} {y} {z} _\u2264_.z\u2264n y\u2264z = y\u2264z\nmax-lub {suc x} {.0} {suc z} (_\u2264_.s\u2264s x\u2264z) _\u2264_.z\u2264n = _\u2264_.s\u2264s x\u2264z\nmax-lub {suc x} {suc y} {suc z} (_\u2264_.s\u2264s x\u2264z) (_\u2264_.s\u2264s y\u2264z) =\n  let max-xy\u2264z = max-lub {x}{y}{z} x\u2264z y\u2264z in\n  _\u2264_.s\u2264s max-xy\u2264z\n\n\u2294\u2286-inv : \u2200{u v w : Value}\n       \u2192 (u \u2294 v) \u2286 w\n         ---------------\n       \u2192 u \u2286 w  \u00d7  v \u2286 w\n\u2294\u2286-inv uvw = \u27e8 (\u03bb x \u2192 uvw (inj\u2081 x)) , (\u03bb x \u2192 uvw (inj\u2082 x)) \u27e9\n\n\u2286\u2192depth\u2264 : \u2200{u v : Value} \u2192 u \u2286 v \u2192 depth u \u2264 depth v\n\u2286\u2192depth\u2264 {\u22a5} {v} u\u2286v = _\u2264_.z\u2264n\n\u2286\u2192depth\u2264 {const x} {v} u\u2286v = _\u2264_.z\u2264n\n\u2286\u2192depth\u2264 {u\u2081 \u21a6 u\u2082} {v} u\u2286v = \u2208\u2192depth\u2264 (u\u2286v refl)\n\u2286\u2192depth\u2264 {u\u2081 \u2294 u\u2082} {v} u\u2286v\n    with \u2294\u2286-inv u\u2286v\n... | \u27e8 u\u2081\u2286v , u\u2082\u2286v \u27e9 =\n    let u\u2081\u2264v = \u2286\u2192depth\u2264 u\u2081\u2286v in\n    let u\u2082\u2264v = \u2286\u2192depth\u2264 u\u2082\u2286v in\n    max-lub u\u2081\u2264v u\u2082\u2264v\n\ndom-depth-\u2264 : \u2200{u : Value} \u2192 depth (dom u) \u2264 depth u\ndom-depth-\u2264 {\u22a5} = _\u2264_.z\u2264n\ndom-depth-\u2264 {const k} = _\u2264_.z\u2264n\ndom-depth-\u2264 {v \u21a6 w} = \u2264-step (m\u2264m\u2294n (depth v) (depth w))\ndom-depth-\u2264 {u \u2294 v} =\n  let ih1 = dom-depth-\u2264 {u} in\n  let ih2 = dom-depth-\u2264 {v} in\n  \u2294-mono-\u2264 ih1 ih2\n\ncod-depth-\u2264 : \u2200{u : Value} \u2192 depth (cod u) \u2264 depth u\ncod-depth-\u2264 {\u22a5} = _\u2264_.z\u2264n\ncod-depth-\u2264 {const k} = _\u2264_.z\u2264n\ncod-depth-\u2264 {v \u21a6 w} = \u2264-step (n\u2264m\u2294n (depth v) (depth w))\ncod-depth-\u2264 {u \u2294 v} =\n  let ih1 = cod-depth-\u2264 {u} in\n  let ih2 = cod-depth-\u2264 {v} in\n  \u2294-mono-\u2264 ih1 ih2\n\n\u2264\u2032-trans : \u2200{x y z} \u2192 x \u2264\u2032 y \u2192 y \u2264\u2032 z \u2192 x \u2264\u2032 z\n\u2264\u2032-trans x\u2264\u2032y y\u2264\u2032z = \u2264\u21d2\u2264\u2032 (\u2264-trans (\u2264\u2032\u21d2\u2264 x\u2264\u2032y) (\u2264\u2032\u21d2\u2264 y\u2264\u2032z))\n\ndata _<<_ : \u2115 \u00d7 \u2115 \u2192 \u2115 \u00d7 \u2115 \u2192 Set where\n  fst : \u2200{x x' y y'} \u2192 x <\u2032 x' \u2192 \u27e8 x , y \u27e9 << \u27e8 x' , y' \u27e9\n  snd : \u2200{x x' y y'} \u2192 x \u2264\u2032 x' \u2192 y <\u2032 y' \u2192 \u27e8 x , y \u27e9 << \u27e8 x' , y' \u27e9\n\n<<-nat-wf : (P : \u2115 \u2192 \u2115 \u2192 Set) \u2192\n         (\u2200 x y \u2192 (\u2200 {x' y'} \u2192 \u27e8 x' , y' \u27e9 << \u27e8 x , y \u27e9 \u2192 P x' y') \u2192 P x y) \u2192\n         \u2200 x y \u2192 P x y\n<<-nat-wf P ih x y = ih x y (help x y)\n  where help : (x y : \u2115) \u2192 \u2200{ x' y'} \u2192 \u27e8 x' , y' \u27e9 << \u27e8 x , y \u27e9 \u2192 P x' y'\n        help .(suc x') y {x'}{y'} (fst \u2264\u2032-refl) =\n            ih x' y' (help x' y') \n        help .(suc x) y {x'}{y'} (fst (\u2264\u2032-step {x} q)) =\n            help x y {x'}{y'} (fst q)\n        help x .(suc y) {x'}{y} (snd x'\u2264x \u2264\u2032-refl) =\n            let h : \u2200 {x\u2081} {x\u2082} \u2192 (\u27e8 x\u2081 , x\u2082 \u27e9 << \u27e8 x , y \u27e9) \u2192 P x\u2081 x\u2082\n                h = help x y in\n            ih x' y G\n            where\n            G : \u2200 {x'' y'} \u2192 \u27e8 x'' , y' \u27e9 << \u27e8 x' , y \u27e9 \u2192 P x'' y'\n            G {x''} {y'} (fst x''<x') =\n               help x y (fst {y = y'}{y' = y} (\u2264\u2032-trans x''<x' x'\u2264x))\n            G {x''} {y'} (snd x''\u2264x' y'<y) =\n               help x y {x''}{y'} (snd (\u2264\u2032-trans x''\u2264x' x'\u2264x) y'<y)\n        help x .(suc y) {x'}{y'} (snd x\u2032\u2264x (\u2264\u2032-step {y} q)) =\n            help x y {x'}{y'} (snd x\u2032\u2264x q)\n\n\n\u2291-trans-P : \u2115 \u2192 \u2115 \u2192 Set\n\u2291-trans-P d s = \u2200{u v w} \u2192 d \u2261 depth u + depth w \u2192 s \u2261 size u + size v\n                         \u2192 u \u2291 v \u2192 v \u2291 w \u2192 u \u2291 w\n\n\u2291-trans-rec : \u2200 d s \u2192 \u2291-trans-P d s\n\u2291-trans-rec = <<-nat-wf \u2291-trans-P helper\n  where\n  helper : \u2200 x y \n         \u2192 (\u2200 {x' y'} \u2192 \u27e8 x' , y' \u27e9 << \u27e8 x , y \u27e9 \u2192 \u2291-trans-P x' y')\n         \u2192 \u2291-trans-P x y\n  helper d s IH {.\u22a5} {v} {w} d\u2261 s\u2261 \u2291-\u22a5 v\u2291w = \u2291-\u22a5\n  helper d s IH {.(const _)} {.(const _)} {w} d\u2261 s\u2261 \u2291-const v\u2291w = v\u2291w\n  helper d s IH {u\u2081 \u2294 u\u2082} {v} {w} d\u2261 s\u2261 (\u2291-conj-L u\u2081\u2291v u\u2082\u2291v) v\u2291w\n      rewrite d\u2261 | s\u2261 =\n      let u\u2081\u2291w = IH M1 {u\u2081}{v}{w} refl refl u\u2081\u2291v v\u2291w in\n      let u\u2082\u2291w = IH M2 {u\u2082}{v}{w} refl refl u\u2082\u2291v v\u2291w in\n      \u2291-conj-L u\u2081\u2291w u\u2082\u2291w\n      where\n      M1a = begin\n               depth u\u2081 + depth w\n             \u2264\u27e8 +-mono-\u2264 (m\u2264m\u2294n (depth u\u2081) (depth u\u2082)) \u2264-refl \u27e9\n               max (depth u\u2081) (depth u\u2082) + depth w\n            \u220e\n      M1b = begin\n               suc (size u\u2081 + size v)\n            \u2264\u27e8 s\u2264s (+-mono-\u2264 \u2264-refl (n\u2264m+n (size u\u2082) (size v))) \u27e9\n               suc (size u\u2081 + (size u\u2082 + size v))\n            \u2264\u27e8 s\u2264s (\u2264-reflexive (sym (+-assoc (size u\u2081) (size u\u2082) (size v)))) \u27e9\n               suc (size u\u2081 + size u\u2082 + size v)\n            \u220e \n      M1 : \u27e8 depth u\u2081 + depth w , size u\u2081 + size v \u27e9 <<\n           \u27e8 max (depth u\u2081) (depth u\u2082) + depth w ,\n             suc (size u\u2081 + size u\u2082 + size v) \u27e9\n      M1 = snd (\u2264\u21d2\u2264\u2032 M1a) (\u2264\u21d2\u2264\u2032 M1b)\n      M2a = begin\n               depth u\u2082 + depth w\n             \u2264\u27e8 +-mono-\u2264 (n\u2264m\u2294n (depth u\u2081) (depth u\u2082)) \u2264-refl \u27e9\n               max (depth u\u2081) (depth u\u2082) + depth w\n            \u220e\n      M2b = begin\n               suc (size u\u2082 + size v)\n            \u2264\u27e8 s\u2264s (+-mono-\u2264 (n\u2264m+n (size u\u2081) (size u\u2082)) \u2264-refl) \u27e9\n               suc ((size u\u2081 + size u\u2082) + size v)\n            \u220e \n      M2 : \u27e8 depth u\u2082 + depth w , size u\u2082 + size v \u27e9 <<\n           \u27e8 max (depth u\u2081) (depth u\u2082) + depth w ,\n             suc (size u\u2081 + size u\u2082 + size v) \u27e9\n      M2 = snd (\u2264\u21d2\u2264\u2032 M2a) (\u2264\u21d2\u2264\u2032 M2b)\n  helper d s IH {u} {v\u2081 \u2294 v\u2082} {w} d\u2261 s\u2261 (\u2291-conj-R1 u\u2291v\u2081) v\u2081\u2294v\u2082\u2291w \n      rewrite d\u2261 | s\u2261 =\n      let v\u2081\u2291w = \u2294\u2291R v\u2081\u2294v\u2082\u2291w in\n      IH M {u}{v\u2081}{w} refl refl u\u2291v\u2081 v\u2081\u2291w\n      where\n      Ma = begin\n              suc (size u + size v\u2081)\n           \u2264\u27e8 \u2264-reflexive (sym (+-suc (size u) (size v\u2081))) \u27e9\n              size u + suc (size v\u2081)\n           \u2264\u27e8 +-mono-\u2264 \u2264-refl (s\u2264s (m\u2264m+n (size v\u2081) (size v\u2082))) \u27e9\n              size u + suc (size v\u2081 + size v\u2082)\n           \u220e \n      M : \u27e8 depth u + depth w , size u + size v\u2081 \u27e9 <<\n          \u27e8 depth u + depth w , size u + suc (size v\u2081 + size v\u2082) \u27e9\n      M = snd (\u2264\u21d2\u2264\u2032 \u2264-refl) (\u2264\u21d2\u2264\u2032 Ma)\n  helper d s IH {u} {v\u2081 \u2294 v\u2082} {w} d\u2261 s\u2261 (\u2291-conj-R2 u\u2291v\u2082) v\u2081\u2294v\u2082\u2291w\n      rewrite d\u2261 | s\u2261 =\n      let v\u2082\u2291w = \u2294\u2291L v\u2081\u2294v\u2082\u2291w in\n      IH M {u}{v\u2082}{w} refl refl u\u2291v\u2082 v\u2082\u2291w\n      where\n      Ma = begin\n              suc (size u + size v\u2082)\n           \u2264\u27e8 \u2264-reflexive (sym (+-suc (size u) (size v\u2082))) \u27e9\n              size u + suc (size v\u2082)\n           \u2264\u27e8 +-mono-\u2264 \u2264-refl (s\u2264s (n\u2264m+n (size v\u2081) (size v\u2082))) \u27e9\n              size u + suc (size v\u2081 + size v\u2082)\n           \u220e \n      M : \u27e8 depth u + depth w , size u + size v\u2082 \u27e9 <<\n          \u27e8 depth u + depth w , size u + suc (size v\u2081 + size v\u2082) \u27e9\n      M = snd (\u2264\u21d2\u2264\u2032 \u2264-refl) (\u2264\u21d2\u2264\u2032 Ma)\n  helper d s IH {u\u2081 \u21a6 u\u2082}{v}{w}d\u2261 s\u2261 (\u2291-fun{u\u2032 = v\u2032}v\u2032\u2286v afv\u2032 dv\u2032\u2291u\u2081 u\u2082\u2291cv\u2032) v\u2291w\n      rewrite d\u2261 | s\u2261\n      with sub-inv-trans afv\u2032 v\u2032\u2286v\n                (\u03bb {v\u2081}{v\u2082} v\u2081\u21a6v\u2082\u2208v\u2032 \u2192 \u2291-fun-inv {v\u2032} {w} (u\u2286v\u2291w\u2192u\u2291w v\u2032\u2286v v\u2291w)\n                                         v\u2081\u21a6v\u2082\u2208v\u2032)\n  ... | \u27e8 w\u2032 , \u27e8 afw\u2032 , \u27e8 w\u2032\u2286w , \u27e8 dw\u2032\u2291dv\u2032 , cv\u2032\u2291cw\u2032 \u27e9 \u27e9 \u27e9 \u27e9 =\n        let dw\u2032\u2291u\u2081 = IH M1 {dom w\u2032}{dom v\u2032}{u\u2081} refl refl dw\u2032\u2291dv\u2032 dv\u2032\u2291u\u2081 in\n        let u\u2082\u2291cw\u2032 = IH M2 {u\u2082}{cod v\u2032}{cod w\u2032} refl refl u\u2082\u2291cv\u2032 cv\u2032\u2291cw\u2032 in\n        \u2291-fun{u\u2032 = w\u2032} w\u2032\u2286w afw\u2032 dw\u2032\u2291u\u2081 u\u2082\u2291cw\u2032\n      where\n      dw\u2032\u2264w : depth (dom w\u2032) \u2264 depth w\n      dw\u2032\u2264w = \u2264-trans (dom-depth-\u2264{w\u2032}) (\u2286\u2192depth\u2264 w\u2032\u2286w)\n      cw\u2032\u2264w : depth (cod w\u2032) \u2264 depth w\n      cw\u2032\u2264w = \u2264-trans (cod-depth-\u2264{w\u2032}) (\u2286\u2192depth\u2264 w\u2032\u2286w)\n\n      M1a = begin\n               suc (depth (dom w\u2032) + depth u\u2081)\n            \u2264\u27e8 s\u2264s (\u2264-reflexive (+-comm (depth (dom w\u2032)) (depth u\u2081))) \u27e9\n               suc (depth u\u2081 + depth (dom w\u2032))\n            \u2264\u27e8 s\u2264s (+-mono-\u2264 (m\u2264m\u2294n (depth u\u2081) (depth u\u2082)) dw\u2032\u2264w) \u27e9\n               suc (max (depth u\u2081) (depth u\u2082) + depth w)\n            \u220e \n      M1 : \u27e8 depth (dom w\u2032) + depth u\u2081 , size (dom w\u2032) + size (dom v\u2032) \u27e9\n        << \u27e8 suc (max (depth u\u2081) (depth u\u2082) + depth w) ,\n             suc (size u\u2081 + size u\u2082 + size v) \u27e9\n      M1 = fst (\u2264\u21d2\u2264\u2032 M1a)\n      M2a = begin\n               suc (depth u\u2082 + depth (cod w\u2032))\n            \u2264\u27e8 s\u2264s (+-mono-\u2264 (n\u2264m\u2294n (depth u\u2081) (depth u\u2082)) cw\u2032\u2264w) \u27e9\n               suc (max (depth u\u2081) (depth u\u2082) + depth w)\n            \u220e \n      M2 : \u27e8 depth u\u2082 + depth (cod w\u2032) ,\n             size u\u2082 + size (cod v\u2032) \u27e9\n        << \u27e8 suc (max (depth u\u2081) (depth u\u2082) + depth w) ,\n             suc (size u\u2081 + size u\u2082 + size v) \u27e9\n      M2 = fst (\u2264\u21d2\u2264\u2032 M2a)\n\n\n\u2291-trans : \u2200{u v w} \u2192 u \u2291 v \u2192 v \u2291 w \u2192 u \u2291 w\n\u2291-trans {u} {v} {w} u\u2291v v\u2291w =\n    \u2291-trans-rec (depth u + depth w) (size u + size v) refl refl u\u2291v v\u2291w\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'extra\/extra\/Value.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1208ecce5a401aac3b163c8b206a2f0c047c8027","subject":"init commit of ITyping-Norm","message":"init commit of ITyping-Norm\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/ITyping-Norm.agda","new_file":"Agda\/ITyping-Norm.agda","new_contents":"module ITyping-Norm where\n\nopen import Data.Empty\nopen import Data.List\nopen import Data.Nat\nopen import Data.Product\nopen import Data.Sum\nopen import Data.List.Properties\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.Core\nopen import Relation.Binary.PropositionalEquality using (sym)\n\nopen import Core\nopen import Core-Lemmas\nopen import ChurchRosser\nopen import Typing\nopen import Typed-Core\nopen import ITyping-Core\nopen import ITyping-Subst\n\n\ndata NF : \u2200 {A} -> \u039b A -> Set where\n  bvN : \u2200 {t i} -> NF (bv {t} i)\n  fvN : \u2200 {t x} -> NF (fv {t} x)\n  lamN : \u2200 L {A B} {m : \u039b B} ->\n    ( cf : \u2200 {x} -> (x\u2209L : x \u2209 L) -> NF (\u039b[ 0 >> fv {A} x ] m) ) -> NF (lam A m)\n  appBvN : \u2200 {t s i m} -> NF m -> NF (app {t} {s} (bv i) m)\n  appFvN : \u2200 {t s i m} -> NF m -> NF (app {t} {s} (fv i) m)\n  YN : \u2200 {t} -> NF (Y t)\n\nHasNF : \u2200 {A} (m : \u039b A) -> Set\nHasNF m = \u00ac (\u2203 (\u03bb m' -> m ->\u039b\u03b2 m'))\n\nbvHasNF : \u2200 {t i} -> HasNF (bv {t} i)\nbvHasNF (m' , ())\n\nfvHasNF : \u2200 {t i} -> HasNF (fv {t} i)\nfvHasNF (m' , ())\n\nNF->HasNF : \u2200 {A} {m : \u039b A} -> NF m -> HasNF m\nNF->HasNF bvN (m' , ())\nNF->HasNF fvN (m' , ())\nNF->HasNF (lamN L cf) (_ , (abs L' {m = m} {m'} cf')) = ih (\u039b[ 0 >> fv x ] m' , cf' x\u2209L')\n  where\n  x = \u2203fresh (L ++ L')\n\n  x\u2209 : x \u2209 L ++ L'\n  x\u2209 = \u2203fresh-spec (L ++ L')\n\n  x\u2209L : x \u2209 L\n  x\u2209L = \u2209-cons-l _ L' x\u2209\n\n  x\u2209L' : x \u2209 L'\n  x\u2209L' = \u2209-cons-r L L' x\u2209\n\n  ih : HasNF (\u039b[ 0 >> fv x ] m)\n  ih = NF->HasNF (cf x\u2209L)\nNF->HasNF (appBvN {i = i} _) (_ , redL {m' = m'} _ bvi->\u039b\u03b2m') = ih (m' , bvi->\u039b\u03b2m')\n  where\n  ih : HasNF (bv i)\n  ih = bvHasNF\nNF->HasNF (appBvN NFn) (_ , redR {n = n} {n'} _ n->\u039b\u03b2n') = ih (n' , n->\u039b\u03b2n')\n  where\n  ih : HasNF n\n  ih = NF->HasNF NFn\nNF->HasNF (appFvN {i = i} _) (_ , redL {m' = m'} _ bvi->\u039b\u03b2m') = ih (m' , bvi->\u039b\u03b2m')\n  where\n  ih : HasNF (fv i)\n  ih = fvHasNF\nNF->HasNF (appFvN NFn) (_ , redR {n = n} {n'} _ n->\u039b\u03b2n') = ih (n' , n->\u039b\u03b2n')\n  where\n  ih : HasNF n\n  ih = NF->HasNF NFn\nNF->HasNF YN (m' , ())\n\n--Type->\u2203IType : \u2200 A -> \u2203(\u03bb \u03c4 ->  \u03c4 \u2237' A)\n--Type->\u2203IType\u2097 : \u2200 A -> \u2203(\u03bb \u03c4 ->  \u03c4 \u2237'\u2097 A)\n\n--Type->\u2203IType \u03c3 = o , base\n--Type->\u2203IType (A \u27f6 A\u2081) = (\u03c9 ~> \u03c9) , arr nil nil\n--Type->\u2203IType\u2097 \u03c3 = \u03c9 , nil\n--Type->\u2203IType\u2097 (A \u27f6 A\u2081) = \u03c9 , nil\n\n\n_\u2208\u0393_ : \u2200 {A} \u2192 Atom \u2192 List (Atom \u00d7 A) \u2192 Set _\nx \u2208\u0393 xs = x \u2208 (dom xs)\n\n\n_\u2208\u0393?_ : \u2200 {A} \u2192 Decidable {A = Atom} (_\u2208\u0393_ {A})\nx \u2208\u0393? [] = no \u03bb ()\nx \u2208\u0393? ((x' , y) \u2237 xs) with x \u225f x'\nx \u2208\u0393? ((.x , y) \u2237 xs) | yes refl = yes (here refl)\nx \u2208\u0393? ((x' , y) \u2237 xs) | no _ with x \u2208\u0393? xs\nx \u2208\u0393? ((x' , y) \u2237 xs) | no _ | yes x\u2208xs = yes (there x\u2208xs)\nx \u2208\u0393? ((x' , y) \u2237 xs) | no x\u2260x' | no x\u2209xs =\n  no (\u03bb x\u2208x'\u2237xs \u2192 x\u2209xs (\u2208-\u2237-elim x (dom xs) (\u03bb x'\u2261x \u2192 x\u2260x' (sym x'\u2261x)) x\u2208x'\u2237xs))\n\n\n\u2286-list-trans : {A : Set}{xs ys zs : List A} -> xs \u2286 ys -> ys \u2286 zs -> xs \u2286 zs\n\u2286-list-trans {A} {xs} {ys} {zs} xs\u2286ys ys\u2286zs = \u03bb {x} z \u2192 ys\u2286zs (xs\u2286ys z)\n\n\u2208\u0393-\u2203 : \u2200 {x \u0393} -> x \u2208\u0393 \u0393 -> Wf-ICtxt \u0393 -> \u2203(\u03bb \u0393' -> \u2203(\u03bb \u03c4 -> \u2203(\u03bb A -> \u0393 \u2286 ((x , \u03c4 , A) \u2237 \u0393') \u00d7 ((x , \u03c4 , A) \u2237 \u0393') \u2286 \u0393 \u00d7 Wf-ICtxt ((x , \u03c4 , A) \u2237 \u0393')))) \n\u2208\u0393-\u2203 {\u0393 = []} () _\n\u2208\u0393-\u2203 {\u0393 = (_ , \u03c4 , A) \u2237 \u0393} (here refl) wf-\u0393 = \u0393 , \u03c4 , A , (\u03bb {x} z \u2192 z) , (\u03bb {x} z \u2192 z) , wf-\u0393\n\u2208\u0393-\u2203 {x} {(y , \u03c4 , A) \u2237 \u0393} (there x\u2208\u0393) (cons y\u2209 \u03c4\u2237A wf-\u0393) with \u2208\u0393-\u2203 x\u2208\u0393 wf-\u0393\n\u2208\u0393-\u2203 {x} {(y , \u03c4 , A) \u2237 \u0393} (there x\u2208\u0393) (cons y\u2209 \u03c4\u2237A wf-\u0393) |\n  \u0393' , \u03c4' , A' , \u0393\u2286x,\u03c4,A\u2237\u0393' , x,\u03c4,A\u2237\u0393'\u2286\u0393 , wf-x,\u03c4,A\u2237\u0393'  =\n  ((y , \u03c4 , A) \u2237 \u0393') ,\n  \u03c4' ,\n  A' ,\n  \u2286-list-trans (cons-\u2286 \u0393\u2286x,\u03c4,A\u2237\u0393') (help _ _ \u0393') ,\n  \u2286-list-trans (help _ _ \u0393') (cons-\u2286 x,\u03c4,A\u2237\u0393'\u2286\u0393) ,\n  wfI-\u0393-exchange (cons (\u03bb y\u2208x\u2237\u0393' \u2192 y\u2209 (help\u2082 y\u2208x\u2237\u0393' x,\u03c4,A\u2237\u0393'\u2286\u0393)) \u03c4\u2237A wf-x,\u03c4,A\u2237\u0393')\n  where\n  help : {A : Set} (x y : A) (xs : List A) -> x \u2237 y \u2237 xs \u2286 y \u2237 x \u2237 xs\n  help x y\u2081 xs (here refl) = there (here refl)\n  help x\u2081 y xs (there (here refl)) = here refl\n  help x\u2081 y xs (there (there x'\u2208x\u2237y\u2237xs)) = there (there x'\u2208x\u2237y\u2237xs)\n\n  help\u2081 : \u2200 {x} {\u0393 : ICtxt} -> x \u2208 dom \u0393 -> \u2203(\u03bb \u03c4 -> (x , \u03c4) \u2208 \u0393)\n  help\u2081 {\u0393 = []} ()\n  help\u2081 {\u0393 = (_ , \u03c4) \u2237 \u0393} (here refl) = \u03c4 , here refl\n  help\u2081 {\u0393 = y \u2237 \u0393} (there x\u2208dom\u0393) with help\u2081 x\u2208dom\u0393\n  help\u2081 {x} {y \u2237 \u0393} (there x\u2208dom\u0393) | \u03c4 , x\u2208\u0393 = \u03c4 , there x\u2208\u0393\n\n  help\u2082 : \u2200 {x} {\u0393 \u0393' : ICtxt} -> x \u2208 dom \u0393 -> \u0393 \u2286 \u0393' -> x \u2208 dom \u0393'\n  help\u2082 x\u2208dom\u0393 \u0393\u2286\u0393' with help\u2081 x\u2208dom\u0393\n  help\u2082 x\u2208dom\u0393 \u0393\u2286\u0393' | \u03c4 , x\u2208\u0393 = \u2208-imp-\u2208dom (\u0393\u2286\u0393' x\u2208\u0393)\n\nNF->\u22a9 : \u2200 {A} {m : \u039b A} -> NF m -> \u039bTerm m -> \u2203 (\u03bb \u0393 -> \u2203 (\u03bb \u03c4 -> \u0393 \u22a9 m \u2236 \u03c4))\nNF->\u22a9 bvN ()\nNF->\u22a9 {\u03c3} (fvN {x = x}) var =\n  [ (x , [ o ] , \u03c3) ] ,\n  o ,\n  var\n    (cons (\u03bb ()) (cons base nil) nil)\n    (here refl)\n    (cons (o , here refl , base) (nil (cons base nil)))\nNF->\u22a9 {A \u27f6 B} (fvN {x = x}) var =\n  [ (x , [ \u03c9 ~> \u03c9 ] , A \u27f6 B) ] ,\n  \u03c9 ~> \u03c9 ,\n  var\n    (cons (\u03bb ()) (cons (arr nil nil) nil) nil)\n    (here refl)\n     (cons ((\u03c9 ~> \u03c9) , here refl , arr (nil nil) (nil nil)) (nil (cons (arr nil nil) nil)))\nNF->\u22a9 (lamN L {A} {B} {m} cf) (lam L' cf') = body x\u2209FVm \u0393\u22a9m^x\u2236\u03c4\n  -- \u0393 ,\n  -- (\u03c9 ~> [ \u03c4 ]) ,\n  -- abs\n  --   (x \u2237 L ++ L')\n  --   (\u03bb x\u2209L'' \u2192\n  --     cons\n  --     {!\u0393\u22a9m^x\u2236\u03c4!}\n  --       (nil (cons {!x\u2209L''!} nil {!!})))\n  where\n  x = \u2203fresh (\u039bFV m ++ L ++ L')\n\n  x\u2209 : x \u2209 \u039bFV m ++ L ++ L'\n  x\u2209 = \u2203fresh-spec (\u039bFV m ++ L ++ L')\n\n  x\u2209FVm : x \u2209 \u039bFV m\n  x\u2209FVm = \u2209-cons-l _ _ x\u2209\n\n  x\u2209L : x \u2209 L\n  x\u2209L = \u2209-cons-l _ _ (\u2209-cons-r (\u039bFV m) _ x\u2209)\n\n  x\u2209L' : x \u2209 L'\n  x\u2209L' = \u2209-cons-r L L' (\u2209-cons-r (\u039bFV m) _ x\u2209)\n\n  ih : \u2203 (\u03bb \u0393 -> \u2203 (\u03bb \u03c4 -> \u0393 \u22a9 (\u039b[ 0 >> fv x ] m) \u2236 \u03c4))\n  ih = NF->\u22a9 (cf x\u2209L) (cf' x\u2209L')\n\n  \u0393 = proj\u2081 ih\n  \u03c4 = proj\u2081 (proj\u2082 ih)\n  \u0393\u22a9m^x\u2236\u03c4 = proj\u2082 (proj\u2082 ih)\n\n  body : \u2200 {\u0393 A B \u03c4 x} {m : \u039b B} -> x \u2209 \u039bFV m -> \u0393 \u22a9 (\u039b[ 0 >> fv {A} x ] m) \u2236 \u03c4 -> \u2203 (\u03bb \u0393 -> \u2203 (\u03bb \u03c4 -> \u0393 \u22a9 (lam A m) \u2236 \u03c4))\n  body {\u0393} {A} {B} {\u03c4} {x = x} x\u2209FVm \u0393\u22a9m^x\u2236\u03c4 with x \u2208\u0393? \u0393\n  body {\u0393} {\u03c4 = \u03c4} {x} {m} x\u2209FVm \u0393\u22a9m^x\u2236\u03c4 | yes x\u2208\u0393 with \u2208\u0393-\u2203 x\u2208\u0393 (\u22a9-wf-\u0393 \u0393\u22a9m^x\u2236\u03c4)\n  body {\u0393} {\u03c4 = \u03c4} {x} {m} x\u2209FVm \u0393\u22a9m^x\u2236\u03c4 | yes x\u2208\u0393 | \u0393' , \u03c4' , A' , \u0393\u2286x\u2237\u0393' , _ , wf-x\u2237\u0393' = {!!}\n    where\n    \u0393\u2286\u0393x\u2237\u0393' = \u2286\u0393-\u2286 wf-x\u2237\u0393' \u0393\u2286x\u2237\u0393'\n\n    x\u2237\u0393'\u22a9m^x\u2236\u03c4 : ((x , \u03c4' , A') \u2237 \u0393') \u22a9 \u039b[ 0 >> fv x ] m \u2236 \u03c4\n    x\u2237\u0393'\u22a9m^x\u2236\u03c4 = sub\u0393 \u0393\u22a9m^x\u2236\u03c4 \u0393\u2286\u0393x\u2237\u0393'\n\n  body {\u0393} {\u03c4 = \u03c4} {x} {m} x\u2209FVm \u0393\u22a9m^x\u2236\u03c4 | no x\u2209\u0393 =\n    \u0393 ,\n    (\u03c9 ~> [ \u03c4 ]) ,\n    (abs (x \u2237 (dom \u0393 ++ \u039bFV m)) (\u03bb {y} y\u2209L \u2192\n      cons\n        (aux {x = x}\n          m\n          (proj\u2082 (\u2203~T m))\n          (\u2209-\u2237-elim _ (\u2209-cons-l _ _ y\u2209L))\n          x\u2209FVm\n          (\u2209-cons-r (dom \u0393) (\u039bFV m) (\u2209-\u2237-elim _ y\u2209L))\n          (\u03bb x\u2261y \u2192 y\u2209L (here (sym x\u2261y)))\n          (sub\u0393 \u0393\u22a9m^x\u2236\u03c4 (\u2286\u0393-\u2286 (cons x\u2209\u0393 nil (\u22a9-wf-\u0393 \u0393\u22a9m^x\u2236\u03c4)) (\u03bb {x\u2081} \u2192 there))))\n          (nil (cons (\u2209-\u2237-elim _ (\u2209-cons-l _ _ y\u2209L)) nil (\u22a9-wf-\u0393 \u0393\u22a9m^x\u2236\u03c4)))))\n\nNF->\u22a9 (appBvN NFm) (app () trm-m)\nNF->\u22a9 (appFvN {A} {B} {x} {m} NFm) (app trm trm-m) = {!!}\n  -- (x , [ [ \u03c4 ] ~> \u03c9 ] , A \u27f6 B) \u2237 \u0393 ,\n  -- [ \u03c4 ] ~> \u03c9 ,\n  -- app\n  --   (var {!!} {!!} {!!})\n  --   (cons {!!} (nil {!!}))\n  --   {!!}\n  -- where\n  -- ih : \u2203 (\u03bb \u0393 -> \u2203 (\u03bb \u03c4 -> \u0393 \u22a9 m \u2236 \u03c4))\n  -- ih = NF->\u22a9 NFm trm-m\n\n  -- \u0393 = proj\u2081 ih\n  -- \u03c4 = proj\u2081 (proj\u2082 ih)\n  -- \u0393\u22a9m\u2236\u03c4 = proj\u2082 (proj\u2082 ih)\nNF->\u22a9 (YN {t = A}) Y =\n  [] ,\n  [ \u03c9 ~> \u03c9 ] ~> \u03c9 ,\n  Y {\u03c4 = \u03c9}\n    nil\n    (cons ((\u03c9 ~> \u03c9) , here refl , arr (nil nil) (nil nil)) (nil (cons (arr nil nil) nil)))\n    (nil nil)\n\n--HasNF->\u22a9 : \u2200 {A \u0393 \u03c4} {m n : \u039b A} -> NF n -> m ->\u039b\u03b2 n -> \u0393 \u22a9 n \u2236 \u03c4 -> \u0393 \u22a9 m \u2236 \u03c4\n--HasNF->\u22a9 = {!!}\n\n\nP : \u2200 A -> IType -> \u039b A -> Set\nP\u2097 : \u2200 {A} -> List IType -> \u039b A -> Set\n\nP \u03c3 o m = [] \u22a9 m \u2236 o \u00d7 NF m\nP \u03c3 (_ ~> _) _ = \u22a5\nP (_ \u27f6 _) o _ = \u22a5\nP (A \u27f6 B) (\u03c4 ~> \u03c4') m = [] \u22a9 m \u2236 (\u03c4 ~> \u03c4') \u00d7 (\u2200 {n : \u039b A} -> P\u2097 \u03c4 n -> P\u2097 \u03c4' (app m n))\n\nP\u2097 [] m = [] \u22a9\u2097 m \u2236 \u03c9\nP\u2097 {A} (\u03c4 \u2237 \u03c4\u1d62) m = P A \u03c4 m \u00d7 P\u2097 \u03c4\u1d62 m\n\n--data P where\n--  base : \u2200 {A} {m : \u039b A} -> [] \u22a9 m \u2236 o -> NF m -> P o m\n--  arr : \u2200 {A B \u03c4 \u03c4'} {m : \u039b (A \u27f6 B)} ->\n--    [] \u22a9 m \u2236 (\u03c4 ~> \u03c4') ->\n--    (\u2200 {n : \u039b A} -> P\u2097 \u03c4 n -> P\u2097 \u03c4' (app m n)) ->\n--    P (\u03c4 ~> \u03c4') m\n\n--data P\u2097 where\n--  nil : \u2200 {A} {m : \u039b A} -> [] \u22a9\u2097 m \u2236 \u03c9 -> P\u2097 \u03c9 m\n--  cons : \u2200 {A \u03c4 \u03c4\u1d62} {m : \u039b A} -> P \u03c4 m -> P\u2097 \u03c4\u1d62 m -> P\u2097 (\u03c4 \u2237 \u03c4\u1d62) m\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Agda\/ITyping-Norm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"cd3a589fae8d787c6f1f81513b0c63a0bfb33ecf","subject":"adding a stub theorem for what complete progress probably looks like.","message":"adding a stub theorem for what complete progress probably looks like.\n\n@cyrus- i'm not quite sure what complete preservation looks like. the\nstatement of preservation is actually pretty standard, so i don't know what\nthe adjustment would be. maybe that you just don't add any casts or\nsomething? there are no casts in hexps so a complete term won't have any;\nif we believe that casts are a thing that results from holes being\nindeterminate, we shouldn't add any. that's probably a theorem we should be\nproving but i'm not sure it's preservation.\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-progress.agda","new_file":"complete-progress.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-progress where\n\n  data ok : (d : dhexp) (\u0394 : hctx) \u2192 Set where\n    V : \u2200{d \u0394} \u2192 d val \u2192 ok d \u0394\n    S : \u2200{d \u0394} \u2192 \u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d') \u2192 ok d \u0394\n\n  -- nb: don't need (\u0394 , \u2205 \u22a2 d :: \u03c4) because of typed-expansion\n  complete-progress : {\u0394 : hctx} {d : dhexp} {\u03c4 : htyp} {e : hexp}\u2192\n                       ecomplete e \u2192\n                       \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n                       ok d \u0394\n  complete-progress = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'complete-progress.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"439fffb9a4e0b8f9255c3be111d87d658a58168f","subject":"Crypto\/Schemes.agda","message":"Crypto\/Schemes.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Schemes.agda","new_file":"Crypto\/Schemes.agda","new_contents":"module Crypto.Schemes where\n\nopen import Type\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\nrecord PublicKeyEncryption : \u2605\u2081 where\n  constructor mk\n  field\n    PubKey SecKey Message CipherText R\u2096 R\u2091 : \u2605\n    KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey\n    Enc : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\n    Dec : SecKey \u2192 CipherText \u2192 Message\n\n  FunctionallyCorrect : \u2605\n  FunctionallyCorrect =\n    \u2200 r\u2096 r\u2091 m \u2192 let (pk , sk) = KeyGen r\u2096 in\n                 Dec sk (Enc pk m r\u2091) \u2261 m\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Crypto\/Schemes.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f17b1a5782934e2520b4bfaaee28459ee0c11d46","subject":"clean spec","message":"clean spec\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"695e1cf0aefb93b8a7009273ba75ffdfef2389b5","subject":"Newest Desc model.","message":"Newest Desc model.\n\nIn the style of https:\/\/github.com\/larrytheliquid\/generic-elim.\nThis version has a specialized Mu that supports levitation.\n","repos":"spire\/spire","old_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_file":"formalization\/agda\/Spire\/Examples\/PropLev.agda","new_contents":"{-# OPTIONS --type-in-type #-}\nopen import Data.Unit\nopen import Data.Product hiding ( curry ; uncurry )\nopen import Data.List hiding ( concat )\nopen import Data.String\nopen import Relation.Binary.PropositionalEquality\nopen import Function\nmodule Spire.Examples.PropLev where\n\n----------------------------------------------------------------------\n\nLabel : Set\nLabel = String\n\nEnum : Set\nEnum = List Label\n\ndata Tag : Enum \u2192 Set where\n  here : \u2200{l E} \u2192 Tag (l \u2237 E)\n  there : \u2200{l E} \u2192 Tag E \u2192 Tag (l \u2237 E)\n\nBranches : (E : Enum) (P : Tag E \u2192 Set) \u2192 Set\nBranches [] P = \u22a4\nBranches (l \u2237 E) P = P here \u00d7 Branches E (\u03bb t \u2192 P (there t))\n\ncase : {E : Enum} (P : Tag E \u2192 Set) (cs : Branches E P) (t : Tag E) \u2192 P t\ncase P (c , cs) here = c\ncase P (c , cs) (there t) = case (\u03bb t \u2192 P (there t)) cs t\n\nUncurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nUncurriedBranches E P X = Branches E P \u2192 X\n\nCurriedBranches : (E : Enum) (P : Tag E \u2192 Set) (X : Set)\n  \u2192 Set\nCurriedBranches [] P X = X\nCurriedBranches (l \u2237 E) P X = P here \u2192 CurriedBranches E (\u03bb t \u2192 P (there t)) X\n\ncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 UncurriedBranches E P X \u2192 CurriedBranches E P X\ncurryBranches {[]} f = f tt\ncurryBranches {l \u2237 E} f = \u03bb c \u2192 curryBranches (\u03bb cs \u2192 f (c , cs))\n\nuncurryBranches : {E : Enum} {P : Tag E \u2192 Set} {X : Set}\n  \u2192 CurriedBranches E P X \u2192 UncurriedBranches E P X\nuncurryBranches {[]} x tt = x\nuncurryBranches {l \u2237 E} f (c , cs) = uncurryBranches (f c) cs\n\n----------------------------------------------------------------------\n\ndata Desc (I : Set) : Set\u2081 where\n  End : (i : I) \u2192 Desc I\n  Rec : (i : I) (D : Desc I) \u2192 Desc I\n  Arg : (A : Set) (B : A \u2192 Desc I) \u2192 Desc I\n\nISet : Set \u2192 Set\u2081\nISet I = I \u2192 Set\n\nEl : {I : Set} (D : Desc I) \u2192 ISet I \u2192 ISet I\nEl (End j) X i = j \u2261 i\nEl (Rec j D) X i = X j \u00d7 El D X i\nEl (Arg A B) X i = \u03a3 A (\u03bb a \u2192 El (B a) X i)\n\nHyps : {I : Set} (D : Desc I) (X : ISet I) (P : (i : I) \u2192 X i \u2192 Set) (i : I) (xs : El D X i) \u2192 Set\nHyps (End j) X P i q = \u22a4\nHyps (Rec j D) X P i (x , xs) = P j x \u00d7 Hyps D X P i xs\nHyps (Arg A B) X P i (a , b) = Hyps (B a) X P i b\n\n----------------------------------------------------------------------\n\nBranchesD : (I : Set) (E : Enum) \u2192 Set\nBranchesD I E = Branches E (\u03bb _ \u2192 Desc I)\n\ncaseD : {I : Set} {E : Enum} (cs : BranchesD I E) (t : Tag E) \u2192 Desc I\ncaseD = case (\u03bb _ \u2192 Desc _)\n\n----------------------------------------------------------------------\n\nUncurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nUncurriedEl D X = \u2200{i} \u2192 El D X i \u2192 X i\n\nCurriedEl : {I : Set} (D : Desc I) (X : ISet I) \u2192 Set\nCurriedEl (End i) X = X i\nCurriedEl (Rec i D) X = (x : X i) \u2192 CurriedEl D X\nCurriedEl (Arg A B) X = (a : A) \u2192 CurriedEl (B a) X\n\nCurriedEl' : {I : Set} (D : Desc I) (X : ISet I) (i : I) \u2192 Set\nCurriedEl' (End j) X i = j \u2261 i \u2192 X i\nCurriedEl' (Rec j D) X i = (x : X j) \u2192 CurriedEl' D X i\nCurriedEl' (Arg A B) X i = (a : A) \u2192 CurriedEl' (B a) X i\n\ncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 UncurriedEl D X \u2192 CurriedEl D X\ncurryEl (End i) X cn = cn refl\ncurryEl (Rec i D) X cn = \u03bb x \u2192 curryEl D X (\u03bb xs \u2192 cn (x , xs))\ncurryEl (Arg A B) X cn = \u03bb a \u2192 curryEl (B a) X (\u03bb xs \u2192 cn (a , xs))\n\nuncurryEl : {I : Set} (D : Desc I) (X : ISet I)\n  \u2192 CurriedEl D X \u2192 UncurriedEl D X\nuncurryEl (End i) X cn refl = cn\nuncurryEl (Rec i D) X cn (x , xs) = uncurryEl D X (cn x) xs\nuncurryEl (Arg A B) X cn (a , xs) = uncurryEl (B a) X (cn a) xs\n\n----------------------------------------------------------------------\n\nUncurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nUncurriedHyps D X P cn =\n  \u2200 i (xs : El D X i) (ihs : Hyps D X P i xs) \u2192 P i (cn xs)\n\nCurriedHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 Set\nCurriedHyps (End i) X P cn =\n  P i (cn refl)\nCurriedHyps (Rec i D) X P cn =\n  (x : X i) \u2192 P i x \u2192 CurriedHyps D X P (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps (Arg A B) X P cn =\n  (a : A) \u2192 CurriedHyps (B a) X P (\u03bb xs \u2192 cn (a , xs))\n\nCurriedHyps' : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (i : I)\n  (cn : El D X i \u2192 X i)\n  \u2192 Set\nCurriedHyps' (End j) X P i cn = (q : j \u2261 i) \u2192 P i (cn q)\nCurriedHyps' (Rec j D) X P i cn =\n  (x : X j) \u2192 P j x \u2192 CurriedHyps' D X P i (\u03bb xs \u2192 cn (x , xs))\nCurriedHyps' (Arg A B) X P i cn =\n  (a : A) \u2192 CurriedHyps' (B a) X P i (\u03bb xs \u2192 cn (a , xs))\n\ncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 UncurriedHyps D X P cn\n  \u2192 CurriedHyps D X P cn\ncurryHyps (End i) X P cn pf =\n  pf i refl tt\ncurryHyps (Rec i D) X P cn pf =\n  \u03bb x ih \u2192 curryHyps D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb i xs ihs \u2192 pf i (x , xs) (ih , ihs))\ncurryHyps (Arg A B) X P cn pf =\n  \u03bb a \u2192 curryHyps (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb i xs ihs \u2192 pf i (a , xs) ihs)\n\nuncurryHyps : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  \u2192 CurriedHyps D X P cn\n  \u2192 UncurriedHyps D X P cn\nuncurryHyps (End .i) X P cn pf i refl tt =\n  pf\nuncurryHyps (Rec j D) X P cn pf i (x , xs) (ih , ihs) =\n  uncurryHyps D X P (\u03bb ys \u2192 cn (x , ys)) (pf x ih) i xs ihs\nuncurryHyps (Arg A B) X P cn pf i (a , xs) ihs =\n  uncurryHyps (B a) X P (\u03bb ys \u2192 cn (a , ys)) (pf a) i xs ihs\n\n----------------------------------------------------------------------\n\ndata \u03bc {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) : ISet I where\n  init : (t : Tag E) \u2192 UncurriedEl (C t) (\u03bc E C)\n\ninj : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I) (t : Tag E) \u2192 CurriedEl (C t) (\u03bc E C)\ninj E C t = curryEl (C t) (\u03bc E C) (init t)\n\n----------------------------------------------------------------------\n\nind :\n  {I : Set}\n  (E : Enum)\n  (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\n\nhyps :\n  {I : Set}\n  (E : Enum)\n  (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (D : Desc I)\n  (i : I)\n  (xs : El D (\u03bc E C) i)\n  \u2192 Hyps D (\u03bc E C) P i xs\n\nind E C P \u03b1 i (init t xs) = \u03b1 t i xs (hyps E C P \u03b1 (Arg (Tag E) C) i (t , xs))\n\nhyps E C P \u03b1 (End j) i q = tt\nhyps E C P \u03b1 (Rec j A) i (x , xs) = ind E C P \u03b1 j x , hyps E C P \u03b1 A i xs\nhyps E C P \u03b1 (Arg A B) i (a , b) = hyps E C P \u03b1 (B a) i b\n\n----------------------------------------------------------------------\n\nindCurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (f : (t : Tag E) \u2192 CurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I)\n  (x : \u03bc E C i)\n  \u2192 P i x\nindCurried E C P f i x = ind E C P (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (f t)) i x\n\nSummer : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSummer E C X cn P t = CurriedHyps (C t) X P (cn t)\n\nSumCurriedHyps : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Tag E \u2192 Set\nSumCurriedHyps E C P t = Summer E C (\u03bc E C) init P t\n\nelimUncurried : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 Branches E (SumCurriedHyps E C P)\n  \u2192 (i : I) (x : \u03bc E C i) \u2192 P i x\nelimUncurried E C P cs i x =\n  indCurried E C P\n    (case (SumCurriedHyps E C P) cs)\n    i x\n\nelim : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  \u2192 CurriedBranches E\n      (SumCurriedHyps E C P)\n      ((i : I) (x : \u03bc E C i) \u2192 P i x)\nelim E C P = curryBranches (elimUncurried E C P)\n\n----------------------------------------------------------------------\n\nSoundness : Set\u2081\nSoundness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (cs : Branches E (SumCurriedHyps E C P))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb \u03b1\n  \u2192 elimUncurried E C P cs i x \u2261 ind E C P \u03b1 i x\n\nsound : Soundness\nsound E C P cs i xs =\n  let D = Arg (Tag E) C in\n  (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) cs t)) , refl\n\nCompleteness : Set\u2081\nCompleteness = {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 \u2203 \u03bb cs\n  \u2192 ind E C P \u03b1 i x \u2261 elimUncurried E C P cs i x\n\nuncurryHypsIdent : {I : Set} (D : Desc I) (X : ISet I)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (cn : UncurriedEl D X)\n  (\u03b1 : UncurriedHyps D X P cn)\n  (i : I) (xs : El D X i) (ihs : Hyps D X P i xs)\n  \u2192 \u03b1 i xs ihs \u2261 uncurryHyps D X P cn (curryHyps D X P cn \u03b1) i xs ihs\nuncurryHypsIdent (End .i) X P cn \u03b1 i refl tt = refl\nuncurryHypsIdent (Rec j D) X P cn \u03b1 i (x , xs) (p , ps) =\n  uncurryHypsIdent D X P (\u03bb xs \u2192 cn (x , xs)) (\u03bb k ys rs \u2192 \u03b1 k (x , ys) (p , rs)) i xs ps\nuncurryHypsIdent (Arg A B) X P cn \u03b1 i (a , xs) ps =\n  uncurryHypsIdent (B a) X P (\u03bb xs \u2192 cn (a , xs)) (\u03bb j ys \u2192 \u03b1 j (a , ys)) i xs ps\n\npostulate\n  ext4 : {A : Set} {B : A \u2192 Set} {C : (a : A) \u2192 B a \u2192 Set}\n    {D : (a : A) (b : B a) \u2192 C a b \u2192 Set}\n    {Z : (a : A) (b : B a) (c : C a b) \u2192 D a b c \u2192 Set}\n    (f g : (a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 Z a b c d)\n    \u2192 ((a : A) (b : B a) (c : C a b) (d : D a b c) \u2192 f a b c d \u2261 g a b c d)\n    \u2192 f \u2261 g\n\ntoBranches : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  \u2192 Branches E (Summer E C X cn P)\ntoBranches [] C X cn P \u03b1 = tt\ntoBranches (l \u2237 E) C X cn P \u03b1 =\n    curryHyps (C here) X P (\u03bb xs \u2192 cn here xs) (\u03bb i xs \u2192 \u03b1 here i xs)\n  , toBranches E (\u03bb t \u2192 C (there t)) X\n     (\u03bb t \u2192 cn (there t))\n     P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih)\n\nToBranches : {I : Set} {E : Enum} (C : Tag E \u2192 Desc I)\n  (X  : ISet I) (cn : (t : Tag E) \u2192 UncurriedEl (C t) X)\n  (P : (i : I) \u2192 X i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) X P (cn t))\n  (t : Tag E)\n  \u2192 let \u03b2 = toBranches E C X cn P \u03b1 in\n  case (Summer E C X cn P) \u03b2 t \u2261 curryHyps (C t) X P (cn t) (\u03b1 t)\nToBranches C X cn P \u03b1 here = refl\nToBranches C X cn P \u03b1 (there t)\n  with ToBranches (\u03bb t \u2192 C (there t)) X\n    (\u03bb t xs \u2192 cn (there t) xs)\n    P (\u03bb t i xs ih \u2192 \u03b1 (there t) i xs ih) t\n... | ih rewrite ih = refl\n\ncomplete\u03b1 : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (t : Tag E) (i : I) (xs : El (C t) (\u03bc E C) i) (ihs : Hyps (C t) (\u03bc E C) P i xs)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  \u03b1 t i xs ihs \u2261 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t) i xs ihs\ncomplete\u03b1 E C P \u03b1 t i xs ihs\n  with ToBranches C (\u03bc E C) init P \u03b1 t\n... | q rewrite q = uncurryHypsIdent (C t) (\u03bc E C) P (init t) (\u03b1 t) i xs ihs\n\ncomplete' : {I : Set} (E : Enum) (C : Tag E \u2192 Desc I)\n  (P : (i : I) \u2192 \u03bc E C i \u2192 Set)\n  (\u03b1 : (t : Tag E) \u2192 UncurriedHyps (C t) (\u03bc E C) P (init t))\n  (i : I) (x : \u03bc E C i)\n  \u2192 let \u03b2 = toBranches E C (\u03bc E C) init P \u03b1 in\n  ind E C P \u03b1 i x \u2261 elimUncurried E C P \u03b2 i x\ncomplete' E C P \u03b1 i (init t xs) = cong\n  (\u03bb f \u2192 ind E C P f i (init t xs))\n  (ext4 \u03b1\n    (\u03bb t \u2192 uncurryHyps (C t) (\u03bc E C) P (init t) (case (SumCurriedHyps E C P) \u03b2 t))\n    (complete\u03b1 E C P \u03b1)\n  )\n  where \u03b2 = toBranches E C (\u03bc E C) init P \u03b1\n\ncomplete : Completeness\ncomplete E C P \u03b1 i x =\n  let D = Arg (Tag E) C in\n    toBranches E C (\u03bc E C) init P \u03b1\n  , complete' E C P \u03b1 i x\n\n----------------------------------------------------------------------\n\n\u2115E : Enum\n\u2115E = \"zero\" \u2237 \"suc\" \u2237 []\n\nVecE : Enum\nVecE = \"nil\" \u2237 \"cons\" \u2237 []\n\n\u2115T : Set\n\u2115T = Tag \u2115E\n\nVecT : Set\nVecT = Tag VecE\n\nzeroT : \u2115T\nzeroT = here\n\nsucT : \u2115T\nsucT = there here\n\nnilT : VecT\nnilT = here\n\nconsT : VecT\nconsT = there here\n\n\u2115C : \u2115T \u2192 Desc \u22a4\n\u2115C = caseD $\n    End tt\n  , Rec tt (End tt)\n  , tt\n\n\u2115D : Desc \u22a4\n\u2115D = Arg \u2115T \u2115C\n\n\u2115 : \u22a4 \u2192 Set\n\u2115 = \u03bc \u2115E \u2115C\n\nzero : \u2115 tt\nzero = init zeroT refl\n\nsuc : \u2115 tt \u2192 \u2115 tt\nsuc n = init sucT (n , refl)\n\nVecC : (A : Set) \u2192 VecT \u2192 Desc (\u2115 tt)\nVecC A = caseD $\n    End zero\n  , Arg (\u2115 tt) (\u03bb n \u2192 Arg A \u03bb _ \u2192 Rec n (End (suc n)))\n  , tt\n\nnilD : (A : Set) \u2192 Desc (\u2115 tt)\nnilD A = End zero\n\nconsD : (A : Set) \u2192 Desc (\u2115 tt)\nconsD A = Arg (\u2115 tt) (\u03bb n \u2192 Arg A (\u03bb _ \u2192 Rec n (End (suc n))))\n\nVecD : (A : Set) \u2192 Desc (\u2115 tt)\nVecD A = Arg VecT (VecC A)\n\nVec : (A : Set) \u2192 \u2115 tt \u2192 Set\nVec A = \u03bc VecE (VecC A)\n\nNilEl : (A : Set) (n : \u2115 tt) \u2192 Set\nNilEl A n = El (nilD A) (Vec A) n\n\nConsEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nConsEl A n = El (consD A) (Vec A) n\n\nVecEl : (A : Set) \u2192 \u2115 tt \u2192 Set\nVecEl A n = El (VecD A) (Vec A) n\n\nNilHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : NilEl A n) \u2192 Set\nNilHyps A P n xs = Hyps (nilD A) (Vec A) P n xs\n\nConsHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : ConsEl A n) \u2192 Set\nConsHyps A P n xs = Hyps (consD A) (Vec A) P n xs\n\nVecHyps : (A : Set) (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set) (n : \u2115 tt) (xs : VecEl A n) \u2192 Set\nVecHyps A P n xs = Hyps (VecD A) (Vec A) P n xs\n\nConsUncurriedHyps : (A : Set)\n  (P : (n : \u2115 tt) \u2192 Vec A n \u2192 Set)\n  (cn : UncurriedEl (consD A) (Vec A)) \u2192 Set\nConsUncurriedHyps A P cn = UncurriedHyps (consD A) (Vec A) P cn\n\nnil : (A : Set) \u2192 Vec A zero\nnil A = init nilT refl\n\ncons : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons A n x xs = init consT (n , x , xs , refl)\n\nnil2 : (A : Set) \u2192 Vec A zero\nnil2 A = inj VecE (VecC A) nilT\n\ncons2 : (A : Set) (n : \u2115 tt) (x : A) (xs : Vec A n) \u2192 Vec A (suc n)\ncons2 A = inj VecE (VecC A) consT\n\n----------------------------------------------------------------------\n\nmodule Induction where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 n)\n      , (\u03bb u m,q ih,tt n \u2192 suc (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = ind \u2115E \u2115C _\n    (case (\u03bb t \u2192 UncurriedHyps (\u2115C t) \u2115 _ (init t))\n      ( (\u03bb u q ih n \u2192 zero)\n      , (\u03bb u m,q ih,tt n \u2192 add n (proj\u2081 ih,tt n))\n      , tt\n      )\n    )\n    tt\n  \n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n) \n  append A = ind VecE (VecC A) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC A t) (Vec A) _ (init t))\n      ( (\u03bb m q ih n ys \u2192 subst (\u03bb m \u2192 Vec A (add m n)) q ys)\n      , (\u03bb m m',x,xs,q ih,tt n ys \u2192\n          let m' = proj\u2081 m',x,xs,q\n              x = proj\u2081 (proj\u2082 m',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 m',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in\n          subst (\u03bb m \u2192 Vec A (add m n)) q (cons A (add m' n) x (ih n ys))\n        )\n      , tt\n      )\n    )\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = ind VecE (VecC (Vec A m)) _\n    (case (\u03bb t \u2192 UncurriedHyps (VecC (Vec A m) t) (Vec (Vec A m)) _ (init t))\n      ( (\u03bb n q ih \u2192 subst (\u03bb n \u2192 Vec A (mult n m)) q (nil A))\n      , (\u03bb n n',x,xs,q ih,tt \u2192\n          let n' = proj\u2081 n',x,xs,q\n              xs = proj\u2081 (proj\u2082 n',x,xs,q)\n              q = proj\u2082 (proj\u2082 (proj\u2082 n',x,xs,q))\n              ih = proj\u2081 ih,tt\n          in subst  (\u03bb n \u2192 Vec A (mult n m)) q (append A m xs (mult n' m) ih)\n        )\n      , tt\n      )\n    )\n\n----------------------------------------------------------------------\n\nmodule GenericElim where\n\n  add : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  add = elim \u2115E \u2115C _\n    (\u03bb n \u2192 n)\n    (\u03bb m ih n \u2192 suc (ih n))\n    tt\n\n  mult : \u2115 tt \u2192 \u2115 tt \u2192 \u2115 tt\n  mult = elim \u2115E \u2115C _\n    (\u03bb n \u2192 zero)\n    (\u03bb m ih n \u2192 add n (ih n))\n    tt\n\n  append : (A : Set) (m : \u2115 tt) (xs : Vec A m) (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n)\n  append A = elim VecE (VecC A) (\u03bb m xs \u2192 (n : \u2115 tt) (ys : Vec A n) \u2192 Vec A (add m n))\n    (\u03bb n ys \u2192 ys)\n    (\u03bb m x xs ih n ys \u2192 cons A (add m n) x (ih n ys))\n\n  concat : (A : Set) (m n : \u2115 tt) (xss : Vec (Vec A m) n) \u2192 Vec A (mult n m)\n  concat A m = elim VecE (VecC (Vec A m)) (\u03bb n xss \u2192 Vec A (mult n m))\n    (nil A)\n    (\u03bb n xs xss ih \u2192 append A m xs (mult n m) ih)\n\n----------------------------------------------------------------------\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'formalization\/agda\/Spire\/Examples\/PropLev.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2402e981ca1a3c0e11ab9d9838c22734784eef9a","subject":"Added unproven theorem.","message":"Added unproven theorem.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/MayorPremiseATP.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/MayorPremiseATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP mayor premise\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.ABP.MayorPremiseATP where\n\nopen import FOTC.Base\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Stream\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\npostulate\n  mayorPremise : \u2200 {b fs\u2080 fs\u2081 is} \u2192 Bit b \u2192 Fair fs\u2080 \u2192 Fair fs\u2081 \u2192 Stream is \u2192\n                 B is (transfer b fs\u2080 fs\u2081 is)\n-- {-# ATP prove mayorPremise #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Program\/ABP\/MayorPremiseATP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"0789c50adedeb31dfbd595fe9101a71c407a10d8","subject":"Added existential quantifier using postulates.","message":"Added existential quantifier using postulates.\n\nIgnore-this: 44b355c6d698e97868bb8a4040303fbf\n\ndarcs-hash:20111122164841-3bd4e-384e766b50a58a95260af860c8ad63006d6c06a0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/logical-framework\/Existential.agda","new_file":"notes\/thesis\/logical-framework\/Existential.agda","new_contents":"module Existential where\n\npostulate\n  D       : Set\n  \u2203       : (P : D \u2192 Set) \u2192 Set\n  _,_     : {P : D \u2192 Set}(d : D) \u2192 P d \u2192 \u2203 P\n  \u2203-proj\u2081 : {P : D \u2192 Set} \u2192 \u2203 P \u2192 D\n  \u2203-proj\u2082 : {P : D \u2192 Set}(p : \u2203 P) \u2192 P (\u2203-proj\u2081 p)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/logical-framework\/Existential.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"fd4e621a3d2fd47ac0c50f3806e1f3571d4cae77","subject":"lib\/Data\/Nat\/Positive.agda","message":"lib\/Data\/Nat\/Positive.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Positive.agda","new_file":"lib\/Data\/Nat\/Positive.agda","new_contents":"module Data.Nat.Positive where\n\nopen import Type\nopen import Data.Nat renaming (_*_ to _*\u2115_; _+_ to _+\u2115_)\n\ndata \u2115\u207a : \u2605 where\n  suc : \u2115 \u2192 \u2115\u207a\n\none : \u2115\u207a\none = suc zero\n\n_+_ : \u2115\u207a \u2192 \u2115\u207a \u2192 \u2115\u207a\nsuc x + suc y = suc (suc (x +\u2115 y))\n\n_*_ : \u2115\u207a \u2192 \u2115\u207a \u2192 \u2115\u207a\nsuc x * suc y = suc (x +\u2115 y +\u2115 x *\u2115 y)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Nat\/Positive.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4ec83ae70e8223c117d9974d77d6febc1cf8903f","subject":"Added recursive equation for addition.","message":"Added recursive equation for addition.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of addition using a recursive equation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- Tested with agda2atp on 17 August 2012.\n\nmodule FOT.FOTC.Data.Nat.AddRecursiveEquation where\n\nopen import FOTC.Base\n\n------------------------------------------------------------------------------\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-eq : \u2200 m n \u2192 m + n \u2261 if (iszero\u2081 m) then n else succ\u2081 (pred\u2081 m + n)\n{-# ATP axiom +-eq #-}\n\npostulate\n  +-0x : \u2200 n \u2192   zero    + n \u2261 n\n  +-Sx : \u2200 m n \u2192 succ\u2081 m + n \u2261 succ\u2081 (m + n)\n{-# ATP prove +-0x #-}\n{-# ATP prove +-Sx #-}\n\n-- $ agda2atp -i. -i ~\/fotc\/fot\/ FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation.agda\n-- Proving the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/23-43-Sx.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/23-43-Sx.tptp\n-- Proving the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/22-43-0x.tptp ...\n-- Vampire 0.6 (revision 903) proved the conjecture in \/tmp\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation\/22-43-0x.tptp\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/AddRecursiveEquation.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"eab26807f32081883e2bbffe0a08da7ad2babe45","subject":"flipbased.agda","message":"flipbased.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased.agda","new_file":"flipbased.agda","new_contents":"module flipbased where\n\nopen import Algebra\nimport Level as L\nopen L using () renaming (_\u2294_ to _L\u2294_)\nopen import Function using (id; _\u2218_; const; _$_)\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Product using (proj\u2081; proj\u2082; _,_; swap; _\u00d7_)\nopen import Data.Bits hiding (replicateM)\nopen import Data.Vec using (Vec; []; _\u2237_; take; drop)\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality as \u2261\n\nrecord M {a} n (A : Set a) : Set a where\n  constructor mk\n  field\n    run : Bits n \u2192 A\n\nflip\u2032 : M 1 Bit\nflip\u2032 = mk (\u03bb { (b \u2237 []) \u2192 b })\n\nreturn\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\nreturn\u2032 x = mk (\u03bb _ \u2192 x)\n\npure\u2032 : \u2200 {a} {A : Set a} \u2192 A \u2192 M 0 A\npure\u2032 = return\u2032\n\ncomap : \u2200 {m n a} {A : Set a} \u2192 (Bits n \u2192 Bits m) \u2192 M m A \u2192 M n A\ncomap f (mk g) = mk (g \u2218 f)\n\nweaken : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (m + n) A\nweaken {m} = comap (drop m)\n\nweaken\u2032 : \u2200 {m n a} {A : Set a} \u2192 M n A \u2192 M (n + m) A\nweaken\u2032 = comap (take _)\n\nprivate\n  take\u2264 : \u2200 {a} {A : Set a} {m n} \u2192 n \u2264 m \u2192 Vec A m \u2192 Vec A n\n  take\u2264 z\u2264n _ = []\n  take\u2264 (s\u2264s p) (x \u2237 xs) = x \u2237 take\u2264 p xs\n\nweaken\u2264 : \u2200 {m n a} {A : Set a} \u2192 n \u2264 m \u2192 M n A \u2192 M m A\nweaken\u2264 p = comap (take\u2264 p)\n\nflip : \u2200 {n} \u2192 M (1 + n) Bit\nflip = weaken\u2032 flip\u2032\n\nreturn : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\nreturn = weaken\u2032 \u2218 return\u2032\n\npure : \u2200 {n a} {A : Set a} \u2192 A \u2192 M n A\npure = return\n\n_>>=_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2081 + n\u2082) B\n_>>=_ {n\u2081} x f = mk (\u03bb bs \u2192 M.run (f (M.run x (take _ bs))) (drop n\u2081 bs))\n\n_>>=\u2032_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 (A \u2192 M n\u2082 B) \u2192 M (n\u2082 + n\u2081) B\n_>>=\u2032_ {n\u2081} {n\u2082} rewrite \u2115\u00b0.+-comm n\u2082 n\u2081 = _>>=_\n\n_>>_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 A \u2192 M n\u2082 B \u2192 M (n\u2081 + n\u2082) B\n_>>_ {n\u2081} x y = x >>= const y\n\nmap : \u2200 {n a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 M n A \u2192 M n B\nmap f x = mk (f \u2218 M.run x)\n-- map f x \u2257 x >>=\u2032 (return {0} \u2218 f)\n\njoin : \u2200 {n\u2081 n\u2082 a} {A : Set a} \u2192\n       M n\u2081 (M n\u2082 A) \u2192 M (n\u2081 + n\u2082) A\njoin x = x >>= id\n\ninfixl 4 _\u229b_\n_\u229b_ : \u2200 {n\u2081 n\u2082 a b} {A : Set a} {B : Set b} \u2192\n       M n\u2081 (A \u2192 B) \u2192 M n\u2082 A \u2192 M (n\u2081 + n\u2082) B\n_\u229b_ {n\u2081} mf mx = mf >>= \u03bb f \u2192 map (_$_ f) mx \n\nreplicateM : \u2200 {a} {A : Set a} {n m} \u2192 M m A \u2192 M (n * m) (Vec A n)\nreplicateM {n = zero}  _ = pure []\nreplicateM {n = suc n} {m} x rewrite \u2115\u00b0.+-comm (n * m) m = pure {0} _\u2237_ \u229b x \u229b replicateM x\n\nflips : \u2200 {n} \u2192 M n (Bits n)\n-- specialized version for now\nflips {zero} = pure []\nflips {suc n} = map _\u2237_ flip\u2032 \u229b flips {n}\n\n_\u27e8_\u27e9_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {m n} \u2192\n        M m A \u2192 (A \u2192 B \u2192 C) \u2192 M n B \u2192 M (m + n) C\nx \u27e8 f \u27e9 y = map f x \u229b y\n\nprivate --move to Data.Nat.NP\n  module \u2294\u00b0 = CommutativeSemiringWithoutOne \u2294-\u2293-0-commutativeSemiringWithoutOne\n\nrecord ProgEquiv a \u2113 : Set (L.suc \u2113 L\u2294 L.suc a) where\n  infix 2 _\u2248_\n  field\n    _\u2248_ : \u2200 {n} {A : Set a} \u2192 Rel (M n A) \u2113\n\n  _\u224b_ : \u2200 {n\u2081 n\u2082} {A : Set a} \u2192 M n\u2081 A \u2192 M n\u2082 A \u2192 Set \u2113\n  _\u224b_ {n\u2081} {n\u2082} p\u2081 p\u2082 = _\u2248_ {n = n\u2081 \u2294 n\u2082} (weaken\u2264 (m\u2264m\u2294n _ _) p\u2081) (weaken\u2264 (m\u2264n\u2294m _ n\u2081) p\u2082)\n    where m\u2264n\u2294m : \u2200 m n \u2192 m \u2264 n \u2294 m\n          m\u2264n\u2294m m n rewrite \u2294\u00b0.+-comm n m = m\u2264m\u2294n m n\n\nrecord PRG (prgEq : ProgEquiv L.zero L.zero) k n : Set where\n  open ProgEquiv prgEq\n  field\n    prg : Bits k \u2192 Bits n\n    sec : map prg flips \u224b flips\n\ninit : \u2200 {k a} {A : Set a} \u2192 (Bits k \u2192 A) \u2192 M k A\ninit f = map f flips\n\nmodule Examples (progEq : ProgEquiv L.zero L.zero) where\n  open ProgEquiv progEq\n  ex\u2081 = \u2200 {x} \u2192 flip\u2032 \u27e8 _xor_ \u27e9 pure\u2032 x \u2248 pure x\n\n  ex\u2082 = p \u2248 map swap p where p = flip\u2032 \u27e8 _,_ \u27e9 flip\u2032 \n\n  ex\u2083 = \u2200 {n} {xs : Bits n} \u2192 map (_\u2295_ xs) flips \u2248 flips\n\n  ex\u2084 = \u2200 {k n} (prg : PRG progEq k n) xs \u2192 map (_\u2295_ xs \u2218 PRG.prg prg) flips \u224b flips\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'flipbased.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f9510280666609fdd9b8b35645bab6c0e17092c4","subject":"Add FFI.JS","message":"Add FFI.JS\n","repos":"audreyt\/agda-libjs,crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS.agda","new_file":"lib\/FFI\/JS.agda","new_contents":"module FFI.JS where\n\nopen import Data.Char.Base   public using (Char)\nopen import Data.String.Base public using (String)\nopen import Data.Bool.Base   public using (Bool; true; false)\nopen import Data.List.Base   using (List)\nopen import Data.Product     using (_\u00d7_) renaming (proj\u2081 to fst; proj\u2082 to snd)\n\nopen import Control.Process.Type\n\n{-# COMPILED_JS Bool function (x,v) { return ((x)? v[\"true\"]() : v[\"false\"]()); } #-}\n{-# COMPILED_JS true  true #-}\n{-# COMPILED_JS false false #-}\n\ndata JSType : Set where\n  array object number string bool null : JSType\n\ninfixr 5 _++_\n\npostulate\n    Number      : Set\n    readNumber  : String  \u2192 Number\n    zero        : Number\n    one         : Number\n    _+_         : Number  \u2192 Number  \u2192 Number\n\n    _++_        : String \u2192 String \u2192 String\n    reverse     : String \u2192 String\n    sort        : String \u2192 String\n    take-half   : String \u2192 String\n    drop-half   : String \u2192 String\n    String\u25b9List : String \u2192 List Char\n    List\u25b9String : List Char \u2192 String\n    Number\u25b9String : Number \u2192 String\n\n    JSValue        : Set\n    _+JS_          : JSValue \u2192 JSValue \u2192 JSValue\n    _\u2264JS_          : JSValue \u2192 JSValue \u2192 Bool\n    _===_          : JSValue \u2192 JSValue \u2192 Bool\n    JSON-stringify : JSValue \u2192 String\n    JSON-parse     : String \u2192 JSValue\n    toString       : JSValue \u2192 String\n    fromString     : String \u2192 JSValue\n    fromChar       : Char   \u2192 JSValue\n    fromNumber     : Number \u2192 JSValue\n    objectFromList : {A : Set} \u2192 List A \u2192 (A \u2192 String) \u2192 (A \u2192 JSValue) \u2192 JSValue\n    fromJSArray    : {A : Set} \u2192 JSValue \u2192 (Number \u2192 JSValue \u2192 A) \u2192 List A\n    fromJSArrayString : JSValue \u2192 List String\n    castNumber     : JSValue \u2192 Number\n    castString     : JSValue \u2192 String\n    nullJS         : JSValue\n    trueJS         : JSValue\n    falseJS        : JSValue\n    readJSType     : String \u2192 JSType\n    showJSType     : JSType \u2192 String\n    typeof         : JSValue \u2192 JSType\n    _\u00b7[_]          : JSValue \u2192 JSValue \u2192 JSValue\n\n    onString : {A : Set} (f : String \u2192 A) \u2192 JSValue \u2192 A\n\n    trace : {A B : Set} \u2192 String \u2192 A \u2192 (A \u2192 B) \u2192 B\n\n{-# COMPILED_JS zero      0 #-}\n{-# COMPILED_JS one       1 #-}\n{-# COMPILED_JS readNumber Number #-}\n{-# COMPILED_JS _+_       function(x) { return function(y) { return x + y; }; }  #-}\n{-# COMPILED_JS _++_      function(x) { return function(y) { return x + y; }; }  #-}\n{-# COMPILED_JS _+JS_     function(x) { return function(y) { return x + y; }; }  #-}\n{-# COMPILED_JS reverse   function(x) { return x.split(\"\").reverse().join(\"\"); } #-}\n{-# COMPILED_JS sort      function(x) { return x.split(\"\").sort().join(\"\"); }    #-}\n{-# COMPILED_JS take-half function(x) { return x.substring(0,x.length\/2); }      #-}\n{-# COMPILED_JS drop-half function(x) { return x.substring(x.length\/2); }        #-}\n{-# COMPILED_JS List\u25b9String function(x) { return (require(\"libagda\").fromList(x, function(y) { return y; })).join(\"\"); } #-}\n{-# COMPILED_JS String\u25b9List function(x) { return require(\"libagda\").fromJSArrayString(x.split(\"\")); } #-}\n{-# COMPILED_JS fromJSArray       function (ty) { return require(\"libagda\").fromJSArray; } #-}\n{-# COMPILED_JS fromJSArrayString require(\"libagda\").fromJSArrayString #-}\n{-# COMPILED_JS _\u2264JS_     function(x) { return function(y) { return x <=  y; }; } #-}\n{-# COMPILED_JS _===_     function(x) { return function(y) { return x === y; }; } #-}\n{-# COMPILED_JS JSON-stringify JSON.stringify #-}\n{-# COMPILED_JS JSON-parse JSON.parse #-}\n{-# COMPILED_JS toString   function(x) { return x.toString(); } #-}\n{-# COMPILED_JS fromString function(x) { return x; } #-}\n{-# COMPILED_JS fromChar   function(x) { return x; } #-}\n{-# COMPILED_JS fromNumber function(x) { return x; } #-}\n{-# COMPILED_JS Number\u25b9String String #-}\n{-# COMPILED_JS castNumber Number #-}\n{-# COMPILED_JS castString String #-}\n{-# COMPILED_JS nullJS     null #-}\n{-# COMPILED_JS trueJS     true #-}\n{-# COMPILED_JS falseJS    false #-}\n{-# COMPILED_JS typeof     function(x) { return typeof(x); } #-}\n{-# COMPILED_JS _\u00b7[_]      require(\"libagda\").readProp #-}\n{-# COMPILED_JS onString   function(t) { return require(\"libagda\").onString; } #-}\n{-# COMPILED_JS trace      require(\"libagda\").trace #-}\n\ndata Value : Set\u2080 where\n  array  : List Value \u2192 Value\n  object : List (String \u00d7 Value) \u2192 Value\n  string : String \u2192 Value\n  number : Number  \u2192 Value\n  bool   : Bool \u2192 Value\n  null   : Value\n\npostulate\n    fromValue : Value \u2192 JSValue\n\n{-# COMPILED_JS fromValue require(\"libagda\").fromValue #-}\n\ndata ValueView : Set\u2080 where\n  array  : List JSValue \u2192 ValueView\n  object : List (String \u00d7 JSValue) \u2192 ValueView\n  string : String \u2192 ValueView\n  number : Number  \u2192 ValueView\n  bool   : Bool \u2192 ValueView\n  null   : ValueView\n\n{-\npostulate\n    fromJSValue : JSValue \u2192 ValueView\n-}\n\nfromBool : Bool \u2192 JSValue\nfromBool true  = trueJS\nfromBool false = falseJS\n\nObject = List (String \u00d7 JSValue)\n\nfromObject : Object \u2192 JSValue\nfromObject o = objectFromList o fst snd\n\n_\u2264Char_ : Char \u2192 Char \u2192 Bool\nx \u2264Char y = fromChar x \u2264JS fromChar y\n\n_\u2264String_ : String \u2192 String \u2192 Bool\nx \u2264String y = fromString x \u2264JS fromString y\n\n_\u2264Number_ : Number \u2192 Number \u2192 Bool\nx \u2264Number y = fromNumber x \u2264JS fromNumber y\n\n_\u00b7\u00ab_\u00bb : JSValue \u2192 String \u2192 JSValue\nv \u00b7\u00ab s \u00bb = v \u00b7[ fromString s ]\n\nabstract\n  URI = String\n  showURI : URI \u2192 String\n  showURI x = x\n  readURI : String \u2192 URI\n  readURI x = x\n\nJSProc = Proc URI JSValue\n\ndata JSCmd : Set where\n  server : (ip port  : String)\n           (proc     : URI \u2192 JSProc)\n           (callback : URI \u2192 JSCmd) \u2192 JSCmd\n  client : JSProc \u2192 JSCmd \u2192 JSCmd\n\n  end          : JSCmd\n  assert       : Bool \u2192 JSCmd \u2192 JSCmd\n  console_log  : String \u2192 JSCmd \u2192 JSCmd\n  process_argv : (List String \u2192 JSCmd) \u2192 JSCmd\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/FFI\/JS.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9d0d80e9463f941d47b743378a7bc31d63ba826b","subject":"Added missing file.","message":"Added missing file.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/agda-interface\/OptionPragmaCommandLine.agda","new_file":"notes\/agda-interface\/OptionPragmaCommandLine.agda","new_contents":"-- Tested with the development version of Agda on 17 October 2012.\n\n-- Testing how the pragmas are saved in the agda interface files (using\n-- the program dump-agdai) when they are used from the command-line:\n\n-- $ agda --no-termination-check OptionPragmaCommandLine.agda\n\n-- 17 October 2012. Because the PragmaOption --no-termination-check\n-- was used from the command-line it is *not* saved in the interface\n-- file.\n\n-- iPragmaOptions = []\n\nmodule OptionPragmaCommandLine where\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/agda-interface\/OptionPragmaCommandLine.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"b454a95ce096e2d2f5874dce86c0aedea0ae1571","subject":"Reflection.Scoped.Param: binary parametricity on scoped terms","message":"Reflection.Scoped.Param: binary parametricity on scoped terms\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Scoped\/Param.agda","new_file":"lib\/Reflection\/Scoped\/Param.agda","new_contents":"open import Relation.Binary.Logical\nopen import Data.Nat.Logical\nopen import Function\nopen import Opaque\nopen import Data.Zero\nopen import Data.One\nopen import Data.Nat\nopen import Data.List\nopen import Data.Sum.NP\nimport Reflection.NP as R\nopen R using (Name; Arg; arg; Arg-info; arg-info; Literal; Visibility; visible; hidden; Relevance; relevant; irrelevant; Pattern; arg\u1d5b\u02b3; arg\u02b0\u02b3; absTerm; Abs; module Abs; _,_)\nopen import Reflection.Scoped as S\nopen import Reflection.Scoped.Translation as T\n\nmodule Reflection.Scoped.Param where\n\ndata VarKind : Set where\n  K\u2080 K\u2081 K\u1d63 : VarKind\n\nrecord Env (A B : Set) : Set where\n  field\n    pVar : VarKind \u2192 A \u2192 B\n    pCon : Name \u2192 Name\n    pDef : Name \u2192 Name\nopen Env\n\n\u03b5 : \u2200 {B} \u2192 Env \ud835\udfd8 B\n\u03b5 = record { pVar = \u03bb _ ()\n           ; pCon = const (opaque \"\u03b5.pCon\" (quote \u03b5))\n           ; pDef = const (opaque \"\u03b5.pDef\" (quote \u03b5)) }\n\ndefDefEnv : Name \u2192 Name\ndefDefEnv (quote \u2115) = quote \u27e6\u2115\u27e7\ndefDefEnv n         = opaque \"defDefEnv\" n\n\ndefConEnv : Name \u2192 Name\ndefConEnv (quote \u2115.zero) = quote \u27e6\u2115\u27e7.zero\ndefConEnv (quote \u2115.suc)  = quote \u27e6\u2115\u27e7.suc\ndefConEnv n              = opaque \"defConEnv\" n\n\ndefEnv : \u2200 {B} \u2192 Env \ud835\udfd8 B\ndefEnv = record { pVar = \u03bb _ ()\n                ; pCon = defConEnv\n                ; pDef = defDefEnv }\n\n_\u228e\ud835\udfd9^_ : Set \u2192 \u2115 \u2192 Set\nA \u228e\ud835\udfd9^ 0       = A\nA \u228e\ud835\udfd9^ (suc n) = (A \u228e\ud835\udfd9^ n) \u228e \ud835\udfd9\n\ninl^ : \u2200 n {A} \u2192 A \u2192 A \u228e\ud835\udfd9^ n\ninl^ 0 x = x\ninl^ (suc n) x = inl (inl^ n x)\n\n# : \u2200 n {A} \u2192 A \u228e\ud835\udfd9^ (suc n)\n# 0       = inr 0\u2081\n# (suc n) = inl (# n)\n\non-pVar : \u2200 {A B C D} \u2192 (VarKind \u2192 (A \u2192 B) \u2192 (C \u2192 D)) \u2192 Env A B \u2192 Env C D\non-pVar f \u0393 = record\n  { pVar = \u03bb k \u2192 f k (pVar \u0393 k)\n  ; pCon = pCon \u0393\n  ; pDef = pDef \u0393 }\n\n_+1 : \u2200 {A B} \u2192 Env A B \u2192 Env A (B \u228e \ud835\udfd9)\n_+1 = on-pVar \u03bb k f \u2192 inl \u2218 f\n\n_+2 : \u2200 {A B} \u2192 Env A B \u2192 Env A (B \u228e\ud835\udfd9^ 2)\n\u0393 +2 = \u0393 +1 +1\n\n_+3 : \u2200 {A B} \u2192 Env A B \u2192 Env A (B \u228e\ud835\udfd9^ 3)\n\u0393 +3 = \u0393 +2 +1\n\n_\u21911 : \u2200 {A B} \u2192 Env A B \u2192 Env (A \u228e \ud835\udfd9) (B \u228e \ud835\udfd9)\n_\u21911 = on-pVar \u03bb k \u2192 Map.\u2191\n\n_\\3 : \u2200 {A B} \u2192 Env A B \u2192 Env (A \u228e \ud835\udfd9) (B \u228e\ud835\udfd9^ 3)\n_\\3 = on-pVar \u03bb { K\u2080 f \u2192 inl^ 2 \u2218 Map.\u2191 f\n                ; K\u2081 f \u2192 inl \u2218 Map.\u2191 (inl \u2218 f)\n                ; K\u1d63 f \u2192 Map.\u2191 (inl^ 2 \u2218 f) }\n\npW : \u2200 {A B}(\u0393 : Env A B)\n       (f : (\u0394 : Env A (B \u228e\ud835\udfd9^ 2))(a\u2080 a\u2081 : Term (B \u228e\ud835\udfd9^ 2)) \u2192 Term (B \u228e\ud835\udfd9^ 2))\n     \u2192 Term B\npW \u0393 f = lam visible (lam visible (f (\u0393 +2) (var (# 1) []) (var (# 0) [])))\n\npTerm  : \u2200 {A B}(\u0393 : Env A B)(t : Term A)  \u2192 Term B\npArgs  : \u2200 {A B}(\u0393 : Env A B)(a : Args A)  \u2192 Args B\npLit   : \u2200 {A B}(\u0393 : Env A B)(l : Literal) \u2192 Term B\npType\u2208 : \u2200 {A B}(\u0393 : Env A B)(t : Type A)(a\u2080 a\u2081 : Term B) \u2192 Type B\npTerm\u2208 : \u2200 {A B}(\u0393 : Env A B)(t : Term A)(a\u2080 a\u2081 : Term B) \u2192 Term B\npSort\u2208 : \u2200 {A B}(\u0393 : Env A B)(s : Sort A)(a\u2080 a\u2081 : Term B) \u2192 Term B\npPi\u2208   : \u2200 {A B}(\u0393 : Env A B)(t : Arg (Type A))(u : Scope \ud835\udfd9 Type A)(a\u2080 a\u2081 : Term B) \u2192 Term B\n\npTerm \u0393 (lam _ t) = lam hidden (lam hidden (lam visible (pTerm (\u0393 \\3) t)))\npTerm \u0393 (var v args) = var (pVar \u0393 K\u1d63 v) (pArgs \u0393 args)\npTerm \u0393 (con c args) = con (pCon \u0393 c) (pArgs \u0393 args)\npTerm \u0393 (def d args) = def (pDef \u0393 d) (pArgs \u0393 args)\npTerm \u0393 (lit l) = pLit \u0393 l\n-- pTerm \u0393 (pat-lam cs args) = unknown -- TODO\npTerm \u0393 unknown = unknown\n\npTerm \u0393 (sort s) = pW \u0393 \u03bb \u0394 \u2192 pSort\u2208 \u0394 s\npTerm \u0393 (pi s t) = pW \u0393 \u03bb \u0394 \u2192 pPi\u2208 \u0394 s t\n\n-- pTerm \u0393 t as' = opaque \"impossible: a type cannot be applied to arguments\" unknown\n\npPi\u2208 \u0393 (arg (arg-info _ r) s) t a\u2080 a\u2081 =\n  pi (arg (arg-info hidden r) (Map.t\u00ffpe (pVar \u0393 K\u2080) s)) $\n  `\u03a0 (arg (arg-info hidden r) (Map.t\u00ffpe (pVar (\u0393 +1) K\u2081) s)) $\n  `\u03a0 (arg (arg-info visible relevant)\n          (pType\u2208 (\u0393 +2) s (var (# 1) []) (var (# 0) [])))\n  (pType\u2208 (\u0393 \\3) t\n    (app (Map.term (inl^ 3) a\u2080) (arg\u1d5b\u02b3 (var (# 2) []) \u2237 []))\n    (app (Map.term (inl^ 3) a\u2081) (arg\u1d5b\u02b3 (var (# 1) []) \u2237 [])))\n\npTerm\u2208 \u0393 (sort s) = pSort\u2208 \u0393 s\npTerm\u2208 \u0393 (pi s t) = pPi\u2208 \u0393 s t\npTerm\u2208 \u0393 t a\u2080 a\u2081 = app (pTerm \u0393 t) (arg\u1d5b\u02b3 a\u2080 \u2237 arg\u1d5b\u02b3 a\u2081 \u2237 [])\n\npLit  \u0393 l  = unknown -- TODO\npArgs \u0393 [] = []\npArgs \u0393 (arg (arg-info _ r) t \u2237 as)\n  = arg (arg-info hidden  r) (opaque \"t0\" (Map.term (pVar \u0393 K\u2080)) t)\n  \u2237 arg (arg-info hidden  r) (opaque \"t1\" (Map.term (pVar \u0393 K\u2081)) t)\n  \u2237 arg (arg-info visible relevant) (pTerm \u0393 t)\n  \u2237 pArgs \u0393 as\npSort\u2208 \u0393 (lit n) a\u2080 a\u2081 = pi\u1d5b\u02b3 (el (lit n) a\u2080)\n                            (`\u03a0\u1d5b\u02b3 (el (lit n) (Map.term inl a\u2081)) ``\u2605\u2080)\npSort\u2208 \u0393 (set t) _ _ = unknown -- TODO\npSort\u2208 \u0393 unknown _ _ = unknown\n\npType\u2208 \u0393 (el (lit n) t) a\u2080 a\u2081 = el (lit n) (pTerm\u2208 \u0393 t a\u2080 a\u2081)\npType\u2208 \u0393 (el s t)       _ _  = el unknown unknown\n\n{-\npClause : \u2200 {A B} \u2192 Env A B \u2192 Clause A \u2192 Clause B\npClause \u0393 c = ?\n-}\n\nopen import Relation.Binary.PropositionalEquality\nmodule Test' where\n\n    param : R.Term \u2192 R.Term\n    param t = T.SR.term T.SR.\u03b5 (pTerm defEnv (T.RS.term T.RS.\u03b5 t))\n    --  test-lift : lift \n    p\u2115\u2192\u2115 = param (quoteTerm (\u2115 \u2192 \u2115))\n    \u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n    test-p\u2115\u2192\u2115 : unquote p\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\n    test-p\u2115\u2192\u2115 = refl\n    p\u2115\u2192\u2115\u2192\u2115 = param (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n    \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n    test-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261 quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\n    test-p\u2115\u2192\u2115\u2192\u2115 = refl\n    ZERO : Set\u2081\n    ZERO = (A : Set\u2080) \u2192 A\n    \u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n    \u27e6ZERO\u27e7 f\u2080 f\u2081 =\n        {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n        \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\n    pZERO = param (quoteTerm ZERO)\n    q\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\n    test-pZERO : pZERO \u2261 q\u27e6ZERO\u27e7\n    test-pZERO = refl\n    ID : Set\u2081\n    ID = (A : Set\u2080) \u2192 A \u2192 A\n    \u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n    \u27e6ID\u27e7 f\u2080 f\u2081 =\n        {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n        {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n        \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\n    pID = param (quoteTerm ID)\n    q\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\n    test-ID-abs : absTerm q\u27e6ID\u27e7 \u2261 absTerm pID\n    test-ID-abs = refl\n    test-param : q\u27e6ID\u27e7 \u2261 pID\n    test-param = refl\n\n    test-opaque : opaque \"a\" \"\" \u2261 opaque \"a\" \"\"\n    test-opaque = refl\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Reflection\/Scoped\/Param.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"aade6891f439c96fb5ba77472cc14d8b79acac47","subject":"composition-sem-sec-reduction.agda","message":"composition-sem-sec-reduction.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"composition-sem-sec-reduction.agda","new_file":"composition-sem-sec-reduction.agda","new_contents":"module otp-sem-sec where\n\nimport Level as L\nopen import Function\nopen import Data.Nat.NP\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Bool.Properties\nopen import Data.Vec hiding (_>>=_)\nopen import Data.Product.NP hiding (_\u27e6\u00d7\u27e7_)\nopen import Relation.Nullary\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\nopen import flipbased-implem\nopen \u2261-Reasoning\nopen import Data.Unit using (\u22a4)\nopen import composable\nopen import vcomp\nopen import forkable\nopen import flat-funs\nopen import program-distance\nopen import one-time-semantic-security\nimport bit-guessing-game\n\nmodule BitsExtra where\n  splitAt\u2032 : \u2200 k {n} \u2192 Bits (k + n) \u2192 Bits k \u00d7 Bits n\n  splitAt\u2032 k xs = case splitAt k xs of \u03bb { (ys , zs , _) \u2192 (ys , zs) }\n\n  vnot\u2218vnot : \u2200 {n} \u2192 vnot {n} \u2218 vnot \u2257 id\n  vnot\u2218vnot [] = refl\n  vnot\u2218vnot (x \u2237 xs) rewrite not-involutive x = cong (_\u2237_ x) (vnot\u2218vnot xs)\n\nopen BitsExtra\n\n\n\n\n\n\n\nmodule CompRed {t} {T : Set t}\n               {\u266dFuns : FlatFuns T}\n               (\u266dops : FlatFunsOps \u266dFuns)\n               (ep ep' : EncParams) where\n    open FlatFunsOps \u266dops\n    open AbsSemSec \u266dFuns\n\n    open EncParams ep using (|M|; |C|)\n    open EncParams ep' using () renaming (|M| to |M|'; |C| to |C|')\n\n    M = `Bits |M|\n    C = `Bits |C|\n    M' = `Bits |M|'\n    C' = `Bits |C|'\n\n    comp-red : (pre-E : M' `\u2192 M)\n               (post-E : C `\u2192 C') \u2192 SemSecReduction ep' ep _\n    comp-red pre-E post-E (mk A\u2080 A\u2081) = mk (A\u2080 >>> (pre-E *** pre-E) *** idO) (post-E *** idO >>> A\u2081)\n\nmodule Comp (ep : EncParams) (|M|' |C|' : \u2115) where\n  open EncParams ep\n\n  ep' : EncParams\n  ep' = EncParams.mk |M|' |C|' |R|e\n\n  open EncParams ep' using () renaming (Enc to Enc'; M to M'; C to C')\n\n  Tr = Enc \u2192 Enc'\n\n  open FlatFunsOps fun\u266dOps\n\n  comp : (pre-E : M' \u2192 M) (post-E : C \u2192 C') \u2192 Tr\n  comp pre-E post-E E = pre-E >>> E >>> {-weaken\u2264 |R|e\u2264|R|e' \u2218-} map\u21ba post-E\n\nmodule CompSec (prgDist : PrgDist) (ep : EncParams) (|M|' |C|' : \u2115) where\n  open PrgDist prgDist\n  open FlatFunsOps fun\u266dOps\n  open FunSemSec prgDist\n  open AbsSemSec fun\u266dFuns\n\n  open EncParams ep\n\n  ep' : EncParams\n  ep' = EncParams.mk |M|' |C|' |R|e\n\n  open EncParams ep' using () renaming (Enc to Enc'; M to M'; C to C')\n\n  module CompSec' (pre-E : M' \u2192 M) (post-E : C \u2192 C') where\n    open SemSecReductions ep ep'\n    open CompRed fun\u266dOps ep ep' hiding (M; C)\n    open Comp ep |M|' |C|'\n\n    comp-pres-sem-sec : SemSecTr id (comp pre-E post-E)\n    comp-pres-sem-sec = comp-red pre-E post-E , \u03bb E A \u2192 id\n\n  open SemSecReductions ep ep'\n  open CompRed fun\u266dOps ep ep' hiding (M; C)\n  open CompSec' public\n  module Comp' {x y z} = Comp x y z\n  open Comp' hiding (Tr)\n  comp-pres-sem-sec\u207b\u00b9 : \u2200 pre-E pre-E\u207b\u00b9\n                          (pre-E-right-inv : pre-E \u2218 pre-E\u207b\u00b9 \u2257 id)\n                          post-E post-E\u207b\u00b9\n                          (post-E-left-inv : post-E\u207b\u00b9 \u2218 post-E \u2257 id)\n                        \u2192 SemSecTr\u207b\u00b9 id (comp pre-E post-E)\n  comp-pres-sem-sec\u207b\u00b9 pre-E pre-E\u207b\u00b9 pre-E-inv post-E post-E\u207b\u00b9 post-E-inv =\n    SemSecTr\u2192SemSecTr\u207b\u00b9\n      (comp pre-E\u207b\u00b9 post-E\u207b\u00b9)\n      (comp pre-E post-E)\n      (\u03bb E m R \u2192 trans (post-E-inv _) (cong (\u03bb x \u2192 run\u21ba (E x) R) (pre-E-inv _)))\n      (comp-pres-sem-sec pre-E\u207b\u00b9 post-E\u207b\u00b9)\n\nmodule PostNegSec (prgDist : PrgDist) ep where\n  open EncParams ep\n  open Comp ep |M| |C| hiding (Tr)\n  open CompSec prgDist ep |M| |C|\n  open FunSemSec prgDist\n  open SemSecReductions ep ep\n\n  post-neg : Tr\n  post-neg = comp id vnot\n\n  post-neg-pres-sem-sec : SemSecTr id post-neg\n  post-neg-pres-sem-sec = comp-pres-sem-sec id vnot\n\n  post-neg-pres-sem-sec\u207b\u00b9 : SemSecTr\u207b\u00b9 id post-neg\n  post-neg-pres-sem-sec\u207b\u00b9 = comp-pres-sem-sec\u207b\u00b9 id id (\u03bb _ \u2192 refl) vnot vnot vnot\u2218vnot\n\nmodule Concrete k where\n  open program-distance.Implem k\n  module Guess' = bit-guessing-game prgDist\n  module FunSemSec' = FunSemSec prgDist\n  module CompSec' = CompSec prgDist\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'composition-sem-sec-reduction.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"36b66ccaa9e3355e0573a1c9bf80efa92670a25e","subject":"reverse each line by agda.","message":"reverse each line by agda.\n","repos":"notogawa\/agda-haskell-example","old_file":"src\/Main.agda","new_file":"src\/Main.agda","new_contents":"-- agda -c -isrc -i\/usr\/share\/agda-stdlib\/ src\/Main.agda\nmodule Main where\n\nopen import IO\nopen import Function\nopen import Coinduction\nopen import Data.String using (String; toList; fromList)\nopen import Example\n\n-- main = interact $ unline . reverse . lines\nmain = run (\u266f getContents >>= \u266f_ \u2218 eachline ( fromList \u2218 rev \u2218 toList) ) where\n\n  open import Data.Maybe\n  open import Data.Product\n  open import Data.List using ([]; _\u2237_; [_])\n  open import Data.Colist using (Colist; []; _\u2237_)\n  open import Data.String using (Costring; _++_)\n  open import Data.Unit using (\u22a4; tt)\n  open import Category.Monad.Partiality\n\n  takeLine : Costring \u2192 Maybe (String \u00d7 \u221e Costring) \u22a5\n  takeLine [] = now nothing -- EOF\n  takeLine xs = go \"\" xs where\n    go : String \u2192 Costring \u2192 Maybe (String \u00d7 \u221e Costring) \u22a5\n    go acc [] = now (just (acc , \u266f []))\n    go acc (x \u2237 xs) with fromList [ x ]\n    go acc (_ \u2237 xs) | \"\\n\" = now (just (acc , xs))\n    go acc (_ \u2237 xs) | last = later (\u266f go (acc ++ last) (\u266d xs))\n\n  eachline : (String \u2192 String) \u2192 Costring \u2192 IO \u22a4\n  eachline f = go \u2218 takeLine where\n    go : Maybe (String \u00d7 \u221e Costring) \u22a5 \u2192 IO \u22a4\n    go (now nothing) = return tt\n    go (now (just (line , xs))) = \u266f putStrLn (f line) >> \u266f go (takeLine (\u266d xs))\n    go (later x) = \u266f return tt >> \u266f go (\u266d x)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/Main.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"dabc14b2369be28bb7008e0933c8d6e5d005c198","subject":"Search\/Derived.agda","message":"Search\/Derived.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Derived.agda","new_file":"Search\/Derived.agda","new_contents":"module Search.Derived where\n\nopen import Function\nopen import Data.Nat.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_)\nopen import Data.Nat.Properties\nopen import Search.Type\nopen import Search.Searchable\n\nsum-lin\u21d2sum-zero : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumLin sum \u2192 SumZero sum\nsum-lin\u21d2sum-zero sum-lin = sum-lin (\u03bb _ \u2192 0) 0\n\nsum-mono\u21d2sum-ext : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumMono sum \u2192 SumExt sum\nsum-mono\u21d2sum-ext sum-mono f\u2257g = \u2115\u2264.antisym (sum-mono (\u2115\u2264.reflexive \u2218 f\u2257g)) (sum-mono (\u2115\u2264.reflexive \u2218 \u2261.sym \u2218 f\u2257g))\n\nsum-ext+sum-hom\u21d2sum-mono : \u2200 {A} \u2192 {sum : Sum A} \u2192 SumExt sum \u2192 SumHom sum \u2192 SumMono sum\nsum-ext+sum-hom\u21d2sum-mono {sum = sum} sum-ext sum-hom {f} {g} f\u2264\u00b0g =\n    sum f                         \u2264\u27e8 m\u2264m+n _ _ \u27e9\n    sum f + sum (\u03bb x \u2192 g x \u2238 f x) \u2261\u27e8 \u2261.sym (sum-hom _ _) \u27e9\n    sum (\u03bb x \u2192 f x + (g x \u2238 f x)) \u2261\u27e8 sum-ext (m+n\u2238m\u2261n \u2218 f\u2264\u00b0g) \u27e9\n    sum g \u220e where open \u2264-Reasoning\n\nsearch-swap' : \u2200 {A B} cm (\u03bcA : Searchable A) (\u03bcB : Searchable B) f \u2192\n               let open CMon cm\n                   s\u1d2c = search \u03bcA _\u2219_\n                   s\u1d2e = search \u03bcB _\u2219_ in\n               s\u1d2c (s\u1d2e \u2218 f) \u2248 s\u1d2e (s\u1d2c \u2218 flip f)\nsearch-swap' cm \u03bcA \u03bcB f = search-swap \u03bcA sg f (search-hom \u03bcB cm)\n  where open CMon cm\n\nsum-swap : \u2200 {A B} (\u03bcA : Searchable A) (\u03bcB : Searchable B) f \u2192\n           sum \u03bcA (sum \u03bcB \u2218 f) \u2261 sum \u03bcB (sum \u03bcA \u2218 flip f)\nsum-swap = search-swap' \u2115\u00b0.+-commutativeMonoid\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Search\/Derived.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1f1d3fb9411c82844b272871e93552dc3baf3940","subject":"agda-libjs.agda","message":"agda-libjs.agda\n","repos":"audreyt\/agda-libjs,crypto-agda\/agda-libjs","old_file":"agda-libjs.agda","new_file":"agda-libjs.agda","new_contents":"module agda-libjs where\nimport Control.Process.Type\nimport FFI.JS\nimport FFI.JS.BigI\nimport FFI.JS.Console\nimport FFI.JS.FS\nimport FFI.JS.Proc\nimport FFI.JS.Process\nimport FFI.JS.Properties\nimport FFI.JS.SHA1\nimport FFI.JS.Type\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda-libjs.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0e722b3cdf7825b9bb89c8e4b70209a45c3701ab","subject":"hash.agda","message":"hash.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"hash.agda","new_file":"hash.agda","new_contents":"module hash where\n\nopen import Data.Bits\nopen import Data.Fin as Fin hiding (_<_; from\u2115) renaming (_+_ to _+f_)\nopen import Data.Fin.Props\nopen import Data.Vec hiding (last)\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Bool\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Function\n\nrecord PaddingScheme n (M : Set) : Set where\n  constructor mk\n  field\n    padlen : M \u2192 \u2115\n    pad : (m : M) \u2192 Vec (Bits n) (padlen m)\n\nmodule Merkle-Damg\u00e5rd\u2081\n  {T : Set}             -- Tag space\n  {n}                   -- Block length\n  (h : T \u2192 Bits n \u2192 T) -- Compression function\n  where\n\n  H\u2081 : \u2200 {\u2113} \u2192 T \u2192 Vec (Bits n) \u2113 \u2192 T\n  H\u2081 tag []       = tag\n  H\u2081 tag (x \u2237 xs) = H\u2081 (h tag x) xs\n\nmodule Merkle-Damg\u00e5rd\u2082\n  {T : Set}             -- Tag space\n  {n}                   -- Block length\n  (h : T \u2192 Bits n \u2192 T) -- Compression function\n\n  (IV : T)              -- Initialization vector\n\n  {M : Set}\n  (paddingSch : PaddingScheme n M)\n  where\n\n  open Merkle-Damg\u00e5rd\u2081 h\n  open PaddingScheme paddingSch\n\n  H\u2082 : M \u2192 T\n  H\u2082 m = H\u2081 IV (pad m)\n\n{-\nlengthPadding\u2081 : \u2200 {\u2113 n} \u2192 PaddingScheme n (Bits \u2113)\nlengthPadding\u2081 = mk padlen pad where\n  padlen : \u2200 {\u2113} \u2192 Bits \u2113 \u2192 \u2115\n  padlen xs = {!!}\n\n  pad : \u2200 {\u2113 n} \u2192 (m : Bits \u2113) \u2192 Vec (Bits n) (padlen m)\n  pad [] = {!!}\n  pad (b \u2237 bs) = {!!}\n-}\n\n-- Fin (2 ^ n) \u2192 Bits n\n\nlengthPadding\u2080 : \u2200 {n s} \u2192 \u2115 \u2192 Bits (1 + n + s)\nlengthPadding\u2080 {n} {s} x = 1b \u2237 0\u2026 {n} ++ from\u2115 {s} x\n\ninfixl 6 _\u2115-\u2115'_\n\n-- adapted from Data.Fin, looks better this way:\n_\u2115-\u2115'_ : (n : \u2115) \u2192 Fin n \u2192 \u2115\nzero  \u2115-\u2115' ()\nsuc n \u2115-\u2115' zero   = suc n\nsuc n \u2115-\u2115' suc i  = n \u2115-\u2115' i\n\nn-m+m\u2261n : \u2200 n (m : Fin n) \u2192 (n \u2115-\u2115' m) + Fin.to\u2115 m \u2261 n\nn-m+m\u2261n zero ()\nn-m+m\u2261n (suc n) zero rewrite \u2115\u00b0.+-comm n 0 = refl\nn-m+m\u2261n (suc n) (suc m)\n   = (suc n \u2115-\u2115' suc m) + Fin.to\u2115 (suc m) \u2261\u27e8 \u2115\u00b0.+-comm (n \u2115-\u2115' m) (Fin.to\u2115 (suc m)) \u27e9\n     suc (Fin.to\u2115 m + (n \u2115-\u2115' m)) \u2261\u27e8 cong suc (\u2115\u00b0.+-comm (Fin.to\u2115 m) (n \u2115-\u2115' m)) \u27e9\n     suc ((n \u2115-\u2115' m) + Fin.to\u2115 m) \u2261\u27e8 cong suc (n-m+m\u2261n n m) \u27e9\n     suc n \u220e where open \u2261-Reasoning\n\nmodule LengthPadding\u2081\n  {n}\n  {s : Fin n}              -- Length of the size part of the padding block\n  (sizemsg : Bits (Fin.to\u2115 s))      -- Usually the size of the message\n  {\u2113 : Fin n}\n  where\n  lengthPadding\u2081 : PaddingScheme n (Bits (Fin.to\u2115 \u2113))\n  lengthPadding\u2081 = mk padlen pad where\n    padlen : \u2200 {\u2113} \u2192 Bits (Fin.to\u2115 \u2113) \u2192 \u2115\n    padlen xs = if suc n <= (Fin.to\u2115 \u2113 + Fin.to\u2115 s) then 2 else 1\n\n    pad : (m : Bits (Fin.to\u2115 \u2113)) \u2192 Vec (Bits n) (padlen m)\n    pad bs with suc n <= (Fin.to\u2115 \u2113 + Fin.to\u2115 s)\n    ... | true = helper (bounded _) bs \u2237 block\u2082 \u2237 []\n         where\n           helper : \u2200 {n' n} \u2192 n' < n \u2192 Bits n' \u2192 Bits n\n           helper (s\u2264s p) [] = 1b \u2237 0\u2026\n           helper (s\u2264s p) (b \u2237 bs) = b \u2237 helper p bs\n           block\u2082 : Bits n\n           block\u2082 = subst Bits (n-m+m\u2261n n s) (0\u2026 {n \u2115-\u2115' s} ++ sizemsg)\n    ... | false = helper' \u2237 [] where\n           helper : \u2200 {#0s #m} \u2192 Bits #m \u2192 Bits (#m + 1 + #0s + Fin.to\u2115 s)\n           helper {p} [] = 1b \u2237 0\u2026 {p} ++ sizemsg\n           helper (b \u2237 bs) = b \u2237 helper bs\n\n{-\n           pf = Fin.to\u2115 \u2113 + 1 + (n \u2238 (Fin.to\u2115 \u2113 + 1 + Fin.to\u2115 s)) + Fin.to\u2115 s \u2261\u27e8 ? \u27e9\n                (n \u2238 (Fin.to\u2115 \u2113 + 1 + Fin.to\u2115 s)) + Fin.to\u2115 \u2113 + 1 + Fin.to\u2115 s \u2261\u27e8 ? \u27e9\n                (n \u2238 Fin.to\u2115 (\u2113 +f suc s)) + Fin.to\u2115 \u2113 + 1 + Fin.to\u2115 s \u2261\u27e8 ? \u27e9\n                (n \u2238 Fin.to\u2115 (\u2113 +f suc s)) + Fin.to\u2115 (\u2113 +f suc s) \u2261\u27e8 ? \u27e9\n                (n \u2115-\u2115' (\u2113 +f suc s)) + Fin.to\u2115 (\u2113 +f suc s) \u2261\u27e8 ? \u27e9\n                n \u220e where open \u2261-Reasoning\n-}\n\n           postulate helper' : Bits n\n           -- helper' = subst Bits pf (helper {n \u2238 (Fin.to\u2115 \u2113 + 1 + Fin.to\u2115 s)} bs)\n\nrecord WithLast \u2113 n : Set where\n  constructor mk\n  field\n    main : Vec (Bits n) \u2113\n    {m}  : Fin n\n    last : Bits (Fin.to\u2115 m)\n\nlengthPadding\u2082 : \u2200 {\u2113 n} \u2192 (\u2200 m \u2192 PaddingScheme n (Bits (Fin.to\u2115 m))) \u2192 PaddingScheme n (WithLast \u2113 n)\nlengthPadding\u2082 {\u2113} {n} ps = mk padlen pad where\n  padlen : WithLast \u2113 n \u2192 \u2115\n  padlen (mk _ last) = \u2113 + PaddingScheme.padlen (ps _) last\n\n  pad : (m : WithLast \u2113 n) \u2192 Vec (Bits n) (padlen m)\n  pad (mk xs last) = xs ++ PaddingScheme.pad (ps _) last\n\n-- lengthPadding\u2083 : \u2200 {\u2113 n} \u2192 PaddingScheme n (WithLast \u2113 n)\n-- lengthPadding\u2083 = lengthPadding\u2082 (\u03bb _ \u2192 LengthPadding\u2081.lengthPadding\u2081)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'hash.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"da088ef717601aa3942604b5ee2ecde82f1699ad","subject":"flipbased-tree.agda","message":"flipbased-tree.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"flipbased-tree.agda","new_file":"flipbased-tree.agda","new_contents":"module flipbased-tree where\n\nopen import Function\nopen import Data.Bits\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\nopen import Data.Vec\n\nimport flipbased\n\ndata Tree {a} (A : Set a) : \u2115 \u2192 Set a\n\n\u21ba : \u2200 {a} n (A : Set a) \u2192 Set a\n\u21ba = flip Tree\n\n-- \u201c\u21ba n A\u201d reads like: \u201ctoss n coins and then return a value of type A\u201d\ndata Tree {a} (A : Set a) where\n  return\u21ba : \u2200 {c} \u2192 A \u2192 \u21ba c A\n  fork    : \u2200 {c} \u2192 (left right : \u21ba c A) \u2192 \u21ba (suc c) A\n\nrunDet : \u2200 {a} {A : Set a} \u2192 \u21ba 0 A \u2192 A\nrunDet (return\u21ba x) = x\n\ntoss : \u21ba 1 Bit\ntoss = fork (return\u21ba 0b) (return\u21ba 1b)\n\nmap\u21ba : \u2200 {c a b} {A : Set a} {B : Set b} \u2192 (A \u2192 B) \u2192 \u21ba c A \u2192 \u21ba c B\nmap\u21ba f (return\u21ba x) = return\u21ba (f x)\nmap\u21ba f (fork left right) = fork (map\u21ba f left) (map\u21ba f right)\n\nweaken\u2264 : \u2200 {m c a} {A : Set a} \u2192 m \u2264 c \u2192 \u21ba m A \u2192 \u21ba c A\nweaken\u2264 p (return\u21ba x) = return\u21ba x\nweaken\u2264 (s\u2264s p) (fork left right) = fork (weaken\u2264 p left) (weaken\u2264 p right)\n\nweaken+ : \u2200 {m c a} {A : Set a} \u2192 \u21ba m A \u2192 \u21ba (c + m) A\nweaken+ {m} {c} = weaken\u2264 (\u2115\u2264.trans (m\u2264m+n m c) (\u2115\u2264.reflexive (\u2115\u00b0.+-comm m c)))\n\njoin\u21ba : \u2200 {c\u2081 c\u2082 a} {A : Set a} \u2192 \u21ba c\u2081 (\u21ba c\u2082 A) \u2192 \u21ba (c\u2081 + c\u2082) A\njoin\u21ba {c} (return\u21ba x)      = weaken+ {_} {c} x\njoin\u21ba     (fork left right) = fork (join\u21ba left) (join\u21ba right)\n\nopen flipbased \u21ba toss weaken\u2264 return\u21ba map\u21ba join\u21ba public\n\nopen import Data.Bool\nopen import Data.Bits\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\n\ninfix 4 _\/2+_\/2\ninfix 6 _\/2\npostulate\n  [0,1] : Set\n  0\/1 : [0,1]\n  1\/1 : [0,1]\n  _\/2 : [0,1] \u2192 [0,1]\n  _\/2+_\/2 : [0,1] \u2192 [0,1] \u2192 [0,1]\n-- sym _\/2+_\/2\n-- 1 \/2+ 1 \/2 = 1\/1\n-- p \/2+ p \/2 = p\n-- p \/2+ (1- p) \/2 = 1\/2\n\n1\/2^_ : \u2115 \u2192 [0,1]\n1\/2^ zero = 1\/1\n1\/2^ suc n = (1\/2^ n)\/2\n\n1\/2 : [0,1]\n1\/2 = 1\/2^ 1\n1\/4 : [0,1]\n1\/4 = 1\/2^ 2\n\npostulate\n  Pr[_\u2261_] : \u2200 {c a} {A : Set a} \u2192 \u21ba c A \u2192 A \u2192 [0,1]\n\n_\u2248_ : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 Set a\np\u2081 \u2248 p\u2082 = \u2200 x \u2192 Pr[ p\u2081 \u2261 x ] \u2261 Pr[ p\u2082 \u2261 x ]\n\npostulate\n  fork-sym : \u2200 {c a} {A : Set a} (p\u2081 p\u2082 : \u21ba c A) \u2192 fork p\u2081 p\u2082 \u2248 fork p\u2082 p\u2081\n\n  Pr-return-\u2261 : \u2200 {c a} {A : Set a} (x : A) \u2192 Pr[ return\u21ba {c = c} x \u2261 x ] \u2261 1\/1\n\n  Pr-return-\u2262 : \u2200 {c a} {A : Set a} {x y : A} \u2192 x \u2262 y \u2192 Pr[ return\u21ba {c = c} x \u2261 y ] \u2261 0\/1\n\n  Pr-fork : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p q}\n            \u2192 Pr[ left \u2261 x ] \u2261 p\n            \u2192 Pr[ right \u2261 x ] \u2261 q\n            \u2192 Pr[ fork left right \u2261 x ] \u2261 p \/2+ q \/2\n\n  0\/2+p\/2\u2261p\/2 : \u2200 p \u2192 (0\/1 \/2+ p \/2) \u2261 p \/2\n\nPr-fork-0 : \u2200 {c a} {A : Set a} {left right : \u21ba c A} {x : A} {p}\n            \u2192 Pr[ left \u2261 x ] \u2261 0\/1\n            \u2192 Pr[ right \u2261 x ] \u2261 p\n            \u2192 Pr[ fork left right \u2261 x ] \u2261 p \/2\nPr-fork-0 {p = p} eq\u2081 eq\u2082 rewrite sym (0\/2+p\/2\u2261p\/2 p) = Pr-fork eq\u2081 eq\u2082\n\nex\u2081 : \u2200 x \u2192 Pr[ toss \u2261 x ] \u2261 1\/2\nex\u2081 1b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) (Pr-return-\u2261 1b)\nex\u2081 0b rewrite fork-sym {0} (return\u21ba 0b) (return\u21ba 1b) 0b = Pr-fork-0 (Pr-return-\u2262 (\u03bb ())) (Pr-return-\u2261 0b)\n\npostulate\n  ex\u2082 : \u2200 x y \u2192 Pr[ toss \u27e8,\u27e9 toss \u2261 (x , y) ] \u2261 1\/2^ 2\n\n  ex\u2083 : \u2200 {n} (x : Bits n) \u2192 Pr[ random \u2261 x ] \u2261 1\/2^ n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'flipbased-tree.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"591ca9633e15ca6ee064164b512842a8d2bde09f","subject":"models\/DescFix.agda: generic recursion principle for Desc (not structurally recursive yet)","message":"models\/DescFix.agda: generic recursion principle for Desc (not structurally recursive yet)\n","repos":"mietek\/epigram2,mietek\/epigram2,larrytheliquid\/pigit","old_file":"models\/DescFix.agda","new_file":"models\/DescFix.agda","new_contents":"{-# OPTIONS --type-in-type #-}\n\nmodule DescFix where\n\nopen import DescTT\n\n\naux : (C : Desc)(D : Desc)(P : Mu C -> Set)(x : [| D |] (Mu C)) -> Set\naux C id          P (con y) = P (con y) * aux C C P y \naux C (const K)   P k       = Unit\naux C (prod D D') P (s , t) = aux C D P s * aux C D' P t\naux C (sigma S T) P (s , t) = aux C (T s) P t\naux C (pi S T)    P f       = (s : S) -> aux C (T s) P (f s)\n\n\ngen : (C : Desc)(D : Desc)(P : Mu C -> Set)\n  (rec : (y : [| C |] Mu C) -> aux C C P y -> P (con y))\n  (x : [| D |] Mu C) -> aux C D P x\ngen C id          P rec (con x) = rec x (gen C C P rec x) , gen C C P rec x\ngen C (const K)   P rec k       = Void\ngen C (prod D D') P rec (s , t) = gen C D P rec s , gen C D' P rec t\ngen C (sigma S T) P rec (s , t) = gen C (T s) P rec t\ngen C (pi S T)    P rec f       = \\ s -> gen C (T s) P rec (f s)\n\n\nfix : (D : Desc)(P : Mu D -> Set)\n  (rec : (y : [| D |] Mu D) -> aux D D P y -> P (con y))\n  (x : Mu D) -> P x\nfix D P rec (con x) = rec x (gen D D P rec x)\n\n\n\nplus : Nat -> Nat -> Nat\nplus (con (Ze , Void)) n = n\nplus (con (Suc , m)) n = suc (plus m n)\n\nfib : Nat -> Nat\nfib = fix NatD (\\ _ -> Nat) help\n  where\n    help : (m : [| NatD |] Nat) -> aux NatD NatD (\\ _ -> Nat) m -> Nat\n    help (Ze , x) a = suc ze\n    help (Suc , con (Ze , _)) a = suc ze\n    help (Suc , con (Suc , con n)) (fib-n , (fib-sn , a)) = plus fib-n fib-sn","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/DescFix.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"7e742f6b7c04dad7eb96d9d38fc39ca8af9812d1","subject":"Added a note about fixed-points.","message":"Added a note about fixed-points.\n\nIgnore-this: 4291537fe55f220577beb75b04f5afd3\n\ndarcs-hash:20101004162158-3bd4e-157b42baf139d0cf6b3a57ed82f5e0659a2437a0.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"others\/FixedPoints.agda","new_file":"others\/FixedPoints.agda","new_contents":"{-# OPTIONS --universe-polymorphism #-}\n\nmodule FixedPoints where\n\nopen import Level renaming ( zero to zeroL )\n\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n  _\u2261_  : Set \u2192 Set \u2192 Set\n  _\u2263_  : (D \u2192 Set) \u2192 (D \u2192 Set) \u2192 Set  -- Equality on predicates.\n\n-- data _\u2261_ {l : Level}{A : Set l}(x : A) : A \u2192 Set l where\n--   refl : x \u2261 x\n\n-- The pure (i.e. non-inductive) LTC natural numbers.\npostulate\n  N\u2081  : D \u2192 Set\n  zN\u2081 : N\u2081 zero\n  sN\u2081 : (n : D) \u2192 N\u2081 n \u2192 N\u2081 (succ n)\n\n-- The LTC natural numbers.\ndata N\u2082 : D \u2192 Set where\n  zN\u2082 : N\u2082 zero\n  sN\u2082 : (n : D) \u2192 N\u2082 n \u2192 N\u2082 (succ n)\n\n-- Because N\u2082 is an inductive data type, we can defined it as the least\n-- fixed-point (Fix) of an appropriate functional.\npostulate\n  Fix  : ((D \u2192 Set) \u2192 D \u2192 Set) \u2192 D \u2192 Set\n  cFix : (f : (D \u2192 Set) \u2192 D \u2192 Set) \u2192 Fix f \u2263 f (Fix f)\n\n-- A version of FN if D was an inductive type\n-- FN : (D \u2192 Set) \u2192 D \u2192 Set\n-- FN N zero     = N zero\n-- FN N (succ n) = N n \u2192 N (succ N)\n\npostulate\n  FN     : (D \u2192 Set) \u2192 D \u2192 Set\n  zeroFN : (N : D \u2192 Set) \u2192        FN N zero     \u2261 N zero\n  succFN : (N : D \u2192 Set)(n : D) \u2192 FN N (succ n) \u2261 N n \u2192 N (succ n)\n\n-- The LTC natural numbers using Fix.\nN\u2083 : D \u2192 Set\nN\u2083 = Fix FN\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'others\/FixedPoints.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"3ddb61f6fbe230196abc5b673252ad621c26f43b","subject":"Added an old draft.","message":"Added an old draft.\n\nIgnore-this: cb330225429d806e0cc92de5faa8e151\n\ndarcs-hash:20110928210550-3bd4e-8d3a9422d2c1cf542cd829e127a667fe09a8463d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/old\/Aczel-FOTC.agda","new_file":"Draft\/old\/Aczel-FOTC.agda","new_contents":"------------------------------------------------------------------------------\n-- Formalization of Azcel's First Order Theory of Combinators (FOTC) [*]\n\n-- [*] Peter Aczel. The strength of Martin-L\u00f6f's intuitionistic type\n-- theory with one universe. In S. Miettinen and J. V\u00e4\u00e4nanen, editors,\n-- Proc. of the Symposium on Mathematical Logic (Oulu, 1974), Report\n-- No. 2, Department of Philosopy, University of Helsinki, Helsinki,\n-- pages 1\u201332, 1977.\n\n------------------------------------------------------------------------------\n\nmodule Aczel-FOTC where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  7 _\u2261_\n\n------------------------------------------------------------------------------\n\n-- The universe.\npostulate D : Set\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : D) : D \u2192 Set where\n  refl : x \u2261 x\n\npostulate\n  K S : D\n  _\u00b7_ : D \u2192 D \u2192 D\n\n-- Conversion rules\npostulate\n  Kax : \u2200 x y \u2192       K \u00b7 x \u00b7 y \u2261 x\n  Sax : \u2200 x y z \u2192 S \u00b7 x \u00b7 y \u00b7 z \u2261 x \u00b7 z \u00b7 (y \u00b7 z)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/old\/Aczel-FOTC.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"704493d17106cb857c980ad61451b7c1c3062fa7","subject":"Control\/Beh.agda","message":"Control\/Beh.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Beh.agda","new_file":"Control\/Beh.agda","new_contents":"open import Type\nopen import Data.One\n\nmodule Control.Beh where\n\ndata Beh (PId E S R I O A : \u2605) : \u2605 where\n  return : A \u2192 Beh PId E S R I O A\n  input  : (PId \u2192 I \u2192 Beh PId E S R I O A) \u2192 Beh PId E S R I O A\n  output : PId \u2192 O \u2192 Beh PId E S R I O A \u2192 Beh PId E S R I O A\n  get    : (S \u2192 Beh PId E S R I O A) \u2192 Beh PId E S R I O A\n  set    : S \u2192 Beh PId E S R I O A \u2192 Beh PId E S R I O A\n  ask    : (E \u2192 Beh PId E S R I O A) \u2192 Beh PId E S R I O A\n  rand   : (R \u2192 Beh PId E S R I O A) \u2192 Beh PId E S R I O A\n\nmodule _ {PId E S R I O : \u2605} where\n    end : Beh PId E S R I O \ud835\udfd9\n    end = return _\n\nmodule Sim[E=\ud835\udfd9,R=1] {PId S I O A : \u2605} where\n  E = \ud835\udfd9\n  R = \ud835\udfd9\n  Proc = Beh PId E S R I O A\n  data _\u2248_ : (P\u2080 P\u2081 : Proc) \u2192 \u2605 where\n    input-input :\n      \u2200 {P\u2080 P\u2081 : PId \u2192 I \u2192 Beh _ _ _ _ _ _ _}\n      \u2192 (\u2200 i m \u2192 P\u2080 i m \u2248 P\u2081 i m)\n      \u2192 input P\u2080 \u2248 input P\u2081\n    output-output :\n      \u2200 {i m P\u2080 P\u2081}\n      \u2192 P\u2080 \u2248 P\u2081\n      \u2192 output i m P\u2080 \u2248 output i m P\u2081\n    return-return :\n      \u2200 x\n      \u2192 return x \u2248 return x\n    nop-ask :\n      \u2200 {P\u2080 P\u2081}\n      \u2192 P\u2080 \u2248 P\u2081 _\n      \u2192 P\u2080 \u2248 ask P\u2081\n    nop-rand :\n      \u2200 {P\u2080 P\u2081}\n      \u2192 P\u2080 \u2248 P\u2081 _\n      \u2192 P\u2080 \u2248 rand P\u2081\n    -- rand rand, ask ask...\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Beh.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"feee99177c858567e2de5aa5a81539071bb07ba9","subject":"Control\/Protocol\/InOut.agda","message":"Control\/Protocol\/InOut.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/InOut.agda","new_file":"Control\/Protocol\/InOut.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (\u03a3; _\u00d7_; _,_) renaming (proj\u2081 to fst)\nopen import Data.One using (\ud835\udfd9)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\nopen import Type\n\nmodule Control.Protocol.InOut where\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual\u1d35\u1d3c-equiv : Is-equiv dual\u1d35\u1d3c\ndual\u1d35\u1d3c-equiv = self-inv-is-equiv dual\u1d35\u1d3c-involutive\n\ndual\u1d35\u1d3c-inj : \u2200 {x y} \u2192 dual\u1d35\u1d3c x \u2261 dual\u1d35\u1d3c y \u2192 x \u2261 y\ndual\u1d35\u1d3c-inj = Is-equiv.injective dual\u1d35\u1d3c-equiv\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113)(P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/InOut.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"9236c772d09686e999993e80d77ce8bd225d16f5","subject":"Add FFI.JS.SHA1","message":"Add FFI.JS.SHA1\n","repos":"audreyt\/agda-libjs,crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS\/SHA1.agda","new_file":"lib\/FFI\/JS\/SHA1.agda","new_contents":"module FFI.JS.SHA1 where\n\nopen import Data.String.Base using (String)\n\npostulate SHA1 : String \u2192 String\n{-# COMPILED_JS SHA1 function (x) { return require(\"sha1\")(x); } #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/FFI\/JS\/SHA1.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"bbf1caa7206ff781a22acaaf0464b85246c3102a","subject":"ReceiptFreeness.agda","message":"ReceiptFreeness.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/ReceiptFreeness.agda","new_file":"Game\/ReceiptFreeness.agda","new_contents":"{-# OPTIONS --without-K #-}\n\nopen import Function\nopen import Type\nopen import Data.Maybe\nopen import Data.Fin hiding (_+_)\nopen import Data.Product renaming (zip to zip-\u00d7)\n--open import Data.One\nopen import Data.Two\nopen import Data.List as L\n\nopen import Data.Nat.NP hiding (_==_)\n\n{-\nopen import Explore.Core\nopen import Explore.Explorable\nopen import Explore.Product\nopen Operators\nopen import Relation.Binary.PropositionalEquality\n-}\n--open import Control.Strategy renaming (run to runStrategy)\n\n\nmodule Game.ReceiptFreeness\n  (PubKey    : \u2605)\n  (SecKey    : \u2605)\n  -- Message = \ud835\udfda\n  (CipherText : \u2605)\n  (CheckedCipherText : \u2605)\n\n  (SerialNumber : \u2605)\n\n  -- randomness supply for, encryption, key-generation, adversary, adversary state\n  (R\u2091 R\u2096 R\u2090 : \u2605)\n  (KeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey)\n  (Enc    : let Message = \ud835\udfda in\n            PubKey \u2192 Message \u2192 R\u2091 \u2192 CheckedCipherText)\n  (forget : CheckedCipherText \u2192 CipherText)\n  (check  : CipherText \u2192 Maybe CheckedCipherText)\n  (Dec    : let Message = \ud835\udfda in\n            SecKey \u2192 CheckedCipherText \u2192 Message)\n\nwhere\n\ndata Strategy (Q : \u2605) (R : Q \u2192 \u2605) (A : \u2605) : \u2605 where\n  ask  : (q? : Q) (cont : R q? \u2192 Strategy Q R A) \u2192 Strategy Q R A\n  done : A \u2192 Strategy Q R A\n\nmodule _ {Q : \u2605} {R : Q \u2192 \u2605} (Oracle : (q : Q) \u2192 R q) where\n  private\n    M = Strategy Q R\n  run : \u2200 {A} \u2192 M A \u2192 A\n  run (ask q? cont) = run (cont (Oracle q?))\n  run (done x)      = x\n\nCandidate : \u2605\nCandidate = \ud835\udfda -- as in the paper: \"for simplicity\"\n\nalice bob : Candidate\nalice = 0\u2082\nbob   = 1\u2082\n\n-- candidate order\n-- also known as LHS or Message\n-- represented by who is the first candidate\nCO : \u2605\nCO = \ud835\udfda\n\nalice-then-bob bob-then-alice : CO\nalice-then-bob = 0\u2082\nbob-then-alice = 1\u2082\n\ndata CO-spec : CO \u2192 Candidate \u2192 Candidate \u2192 \u2605 where\n  01\u2082 : CO-spec 0\u2082 0\u2082 1\u2082\n  10\u2082 : CO-spec 1\u2082 1\u2082 0\u2082\n\nalice-then-bob-spec : CO-spec alice-then-bob alice bob\nalice-then-bob-spec = 01\u2082\n\nbob-then-alice-spec : CO-spec bob-then-alice bob alice\nbob-then-alice-spec = 10\u2082\n\nMarkedRHS : \u2605\nMarkedRHS = \ud835\udfda\n\ndata MarkedRHS-spec : MarkedRHS \u2192 \u2605 where\n  marked-on-first-cell  : MarkedRHS-spec 0\u2082\n  marked-on-second-cell : MarkedRHS-spec 1\u2082\n\nReceipt : \u2605\nReceipt = SerialNumber \u00d7 CipherText\n\nCheckedReceipt : \u2605\nCheckedReceipt = SerialNumber \u00d7 CheckedCipherText\n  \ndata MarkedRHS? : \u2605 where\n  not-marked : MarkedRHS?\n  marked     : MarkedRHS \u2192 MarkedRHS?\n  \nRHS : \u2605\nRHS = Receipt \u00d7 MarkedRHS?\n\nCheckedRHS : \u2605\nCheckedRHS = CheckedReceipt \u00d7 MarkedRHS?\n\nreceipt : RHS \u2192 Receipt\nreceipt = proj\u2081\n\nmarkedRHS? : RHS \u2192 MarkedRHS?\nmarkedRHS? = proj\u2082\n\n-- Marked when there is a 1\nmarked? : MarkedRHS? \u2192 \ud835\udfda\nmarked? not-marked = 0\u2082\nmarked? (marked x) = 1\u2082\n\nfirst-mark : MarkedRHS? \u2192 \ud835\udfda\nfirst-mark not-marked = 0\u2082\nfirst-mark (marked x) = x == 0\u2082\n\nsecond-mark : MarkedRHS? \u2192 \ud835\udfda\nsecond-mark not-marked = 0\u2082\nsecond-mark (marked x) = x == 1\u2082\n\nBallot : \u2605\nBallot = CO \u00d7 RHS\n\nCheckedBallot : \u2605\nCheckedBallot = CO \u00d7 RHS\n\nRgb : \u2605\nRgb = CO \u00d7 R\u2091\n\ngenBallot : Rgb \u2192 SerialNumber \u2192 PubKey \u2192 Ballot\ngenBallot (rCO , r\u2091) sn pk = rCO , ((sn , forget (Enc pk rCO r\u2091)) , not-marked)\n\nlhs : Ballot \u2192 CO\nlhs = proj\u2081\n\nrhs : Ballot \u2192 RHS\nrhs = proj\u2082\n\nBB : \u2605\nBB = List CheckedRHS\n\nTally : \u2605\nTally = \u2115 \u00d7 \u2115\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\n\n-- In the paper RBB is the returning the Tally and we\n-- return the BB, here RTally is returning the Tally\ndata Q : \u2605 where\n  REB RBB RTally : Q\n  RCO            : CheckedRHS \u2192 Q\n  Vote           : RHS \u2192 Q\n\nResp : Q \u2192 \u2605\nResp REB = Ballot\nResp (RCO x) = CO\nResp (Vote x) = Accept?\nResp RBB = BB\nResp RTally = Tally\n\nPhase : \u2605 \u2192 \u2605\nPhase = Strategy Q Resp\n\n-- A \u00d7 B   sends A, then behave as B\n-- A \u2192 B   receives A, then behave as B\n\nRFChallenge : \u2605 \u2192 \u2605\nRFChallenge Next = (\ud835\udfda \u2192 Receipt) \u2192 Next\n\nAdversary : \u2605\nAdversary = R\u2090 \u2192 PubKey \u2192 Phase -- Phase[I]\n                           (RFChallenge\n                             (Phase -- Phase[II]\n                               \ud835\udfda)) -- Adversary guess of whether the vote is for alice\n\n-- TODO adversary validity\n\ntallyMarkedRHS : CO \u2192 MarkedRHS \u2192 Tally\ntallyMarkedRHS co m = \ud835\udfda\u25b9\u2115 for-alice , \ud835\udfda\u25b9\u2115 (not for-alice)\n  where for-alice = co == m\n\n1:alice-0:bob 0:alice-1:bob : Tally\n1:alice-0:bob = 1 , 0\n0:alice-1:bob = 0 , 1\n\ndata TallyMarkedRHS-spec : CO \u2192 MarkedRHS \u2192 Tally \u2192 \u2605 where\n  -- CO is alice-then-bob\n  c1 : TallyMarkedRHS-spec alice-then-bob{-0-} 0\u2082 1:alice-0:bob\n  c2 : TallyMarkedRHS-spec alice-then-bob{-0-} 1\u2082 0:alice-1:bob\n\n  -- CO is bob-then-alice\n  c3 : TallyMarkedRHS-spec bob-then-alice{-1-} 0\u2082 0:alice-1:bob\n  c4 : TallyMarkedRHS-spec bob-then-alice{-1-} 1\u2082 1:alice-0:bob\n\ntallyMarkedRHS-ok : \u2200 co m \u2192 TallyMarkedRHS-spec co m (tallyMarkedRHS co m)\ntallyMarkedRHS-ok 1\u2082 1\u2082 = c4\ntallyMarkedRHS-ok 1\u2082 0\u2082 = c3\ntallyMarkedRHS-ok 0\u2082 1\u2082 = c2\ntallyMarkedRHS-ok 0\u2082 0\u2082 = c1\n\ntallyCheckedRHS : SecKey \u2192 CheckedRHS \u2192 Tally\ntallyCheckedRHS sk (r , not-marked) = 0 , 0\ntallyCheckedRHS sk ((sn , co) , marked x) = tallyMarkedRHS (Dec sk co) x\n\ntally : SecKey \u2192 BB \u2192 Tally\ntally sk bb = L.foldr (zip-\u00d7 _+_ _+_) (0 , 0) (L.map (tallyCheckedRHS sk) bb)\n\nR : \u2605\nR = R\u2096 \u00d7 R\u2090 \u00d7 (Fin {!!} \u2192 Rgb) \u00d7 {!!}\n\n-- Return\nGame : \u2605\nGame = Adversary \u2192 R \u2192 \ud835\udfda\n\nmodule _ (sk : SecKey) (pk : PubKey) (rgb : Rgb) (sn : SerialNumber) (bb : BB) where\n    Oracle : (q : Q) \u2192 Resp q\n    Oracle REB = genBallot rgb sn pk\n    Oracle RBB = bb\n    Oracle RTally = tally sk bb\n    Oracle (RCO ((sn , receipt) , m?)) = Dec sk receipt\n\n    -- do we check if the sn is already here?\n    Oracle (Vote ((sn , receipt) , m?)) = case check receipt of \u03bb { (just x) \u2192 accept ; nothing \u2192 reject }\n\n    OracleBB : Q \u2192 BB\n    OracleBB REB = bb\n    OracleBB RBB = bb\n    OracleBB RTally = bb\n    OracleBB (RCO _) = bb\n\n    -- do we check if the sn is already here?\n    OracleBB (Vote ((sn , receipt) , m?)) = case check receipt of \u03bb { (just x) \u2192 ((sn , x) , m?) \u2237 bb ; nothing \u2192 bb }\n\n\u2141 : Game\n\u2141 A (r\u2096 , r\u2090 , rgbs , _) =\n  -- setup\n  case KeyGen r\u2096 of \u03bb\n  { (pk , sk) \u2192\n    let bb = [] in\n    let Asetup = A r\u2090 pk in\n    let Aphase[I] = run (Oracle sk pk (rgbs {!!}) {!!} {!bb!}) Asetup in\n    {!!}\n  }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Game\/ReceiptFreeness.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2d4f5dd54df2d6c0a4264aae4bd00b09d8e2dcca","subject":"lib\/Category\/Applicative\/NP.agda","message":"lib\/Category\/Applicative\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Applicative\/NP.agda","new_file":"lib\/Category\/Applicative\/NP.agda","new_contents":"module Category.Applicative.NP where\n\nopen import Data.Vec using (Vec; []; _\u2237_)\nopen import Data.Nat\nopen import Category.Applicative public hiding (module RawApplicative)\n\nmodule RawApplicative {f} {F : Set f \u2192 Set f}\n                      (app : RawApplicative F) where\n    open Category.Applicative.RawApplicative app public\n\n    replicateM : \u2200 {n A} \u2192 F A \u2192 F (Vec A n)\n    replicateM {zero}  _ = pure []\n    replicateM {suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Category\/Applicative\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4e9c378c074c6cc340bac301c5b650486b83f58b","subject":"[ #22 ] Added example.","message":"[ #22 ] Added example.\n","repos":"asr\/apia,asr\/apia","old_file":"issues\/Issue22.agda","new_file":"issues\/Issue22.agda","new_contents":"module Issue22 where\n\npostulate\n  D   : Set\n  _\u2261_ : D \u2192 D \u2192 Set\n  d e : D\n\nfoo : \u2200 x \u2192 x \u2261 x\nfoo x = bar\n  where\n  postulate\n    d\u2261e : d \u2261 e\n\n  postulate\n    bar : x \u2261 x\n  {-# ATP prove bar d\u2261e #-}\n\n-- $ apia Bug.agda\n-- An internal error has occurred. Please report this as a bug.\n-- Location of the error: src\/Apia\/Utils\/AgdaAPI\/Interface.hs:307\n\n-- The error occurs because the dead code analysis removes `d\u2261e` from\n-- the the interfase file.\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'issues\/Issue22.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"f80f223c43f387a983258b53e011533ad5a8c87b","subject":"+GenChal","message":"+GenChal\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/GenChal.agda","new_file":"Game\/GenChal.agda","new_contents":"{-# OPTIONS --without-K --copatterns #-}\nopen import Type\nopen import Function\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\n\nopen import Control.Protocol.Core\n\nmodule Game.GenChal\n  -- could be Public Key\n  (Public    : \u2605)\n\n  -- could be:\n  --   * CPA:  Query _ = \ud835\udfd8\n  --   * CCA2: Query _ = Ciphertext\n  --   * CCA1: Query = [0: Ciphertext 1: \ud835\udfd8 ]\n  (Query : \ud835\udfda \u2192 \u2605)\n\n  -- could be:\n  --   * CPA:  Response _ ()\n  --   * CCA2: Response _ _ = Message\n  --   * CCA1: Response 0\u2082 _ = Message\n  --           Response 1\u2082 ()\n  (Response : {i : \ud835\udfda} \u2192 Query i \u2192 \u2605)\n\n  -- could be:\n  --   * CPA\/CCA2\/CCA1: ChallengeInput = Message \u00d7 Message\n  (ChallengeInput : \u2605)\n\n  -- could be:\n  --   * CPA\/CCA2\/CCA1: Challenge = Ciphertext\n  --   * CCA2\u2020: Challenge = Ciphertext \u00d7 Ciphertext\n  (Challenge : \u2605)\n\n  (ContProto : Proto)\n  where\n\nQuery\u2080 = Query 0\u2082\nQuery\u2081 = Query 1\u2082\n\nRound : (phase# : \ud835\udfda) \u2192 Proto \u2192 Proto\nRound i = Server' (Query i) Response\n\nRound\u2080 = Round 0\u2082\nRound\u2081 = Round 1\u2082\n\nChal : Proto \u2192 Proto\nChal X = ChallengeInput \u2192' (Challenge \u00d7' X)\n\n-- challenger implement this\nMain : Proto\nMain = Public \u00d7' Round\u2080 (Chal (Round\u2081 ContProto))\n\nmodule Implementation\n  -- could be Private Key\n  (Private   : \u2605)\n\n  -- could be \ud835\udfda for IND games\n  (Secret    : \u2605)\n\n  {X}\n  (cont      : El X ContProto)\n\n  {- {OracleRand : {i : \ud835\udfda} (q : Query i) \u2192 \u2605} -}\n  (oracle    : Private \u2192 {i : \ud835\udfda} (q : Query i) \u2192 {-OracleRand q \u2192-} Response q)\n  {ChalRand : \u2605}\n  (challenge : Public \u2192 Secret \u2192 ChallengeInput \u2192 ChalRand \u2192 Challenge)\n  where\n\n    module _ (b : Secret)(pk : Public)(sk : Private)(chal-rand : ChalRand) where\n\n      module _ {i : \ud835\udfda}{A}(ContDone : A) where\n        service : Server (Query i) Response A\n        srv-ask  service q = oracle sk q {-{!!}-} , service\n        srv-done service   = ContDone\n\n      phase\u2081 : Server Query\u2081 Response (El X ContProto)\n      phase\u2081 = service cont\n\n      exchange : El X (Chal (Round\u2081 ContProto))\n      exchange m = challenge pk b m chal-rand , phase\u2081\n\n      phase\u2080 : Server Query\u2080 Response (El X (Chal (Round\u2081 ContProto)))\n      phase\u2080 = service exchange\n\n      main : El X Main\n      main = pk , phase\u2080\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Game\/GenChal.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e823862a7e973b3c12fe1fadad51d366f331dbc1","subject":"Deriving N-ary parametricity using the reflection API","message":"Deriving N-ary parametricity using the reflection API\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Param.agda","new_file":"lib\/Reflection\/Param.agda","new_contents":"open import Function\nopen import Opaque\nopen import Type\nopen import Data.Zero\nopen import Data.One hiding (_\u225f_)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Two.Logical\nopen import Data.Nat hiding (_\u225f_)\nopen import Data.Fin using (Fin; zero; suc; to\u2115)\nopen import Data.Vec using (Vec; []; _\u2237_; replicate; tabulate; allFin; reverse; _\u229b_; toList) renaming (map to vmap)\nopen import Data.Nat.Logical hiding (_\u225f_) renaming (zero to \u27e6zero\u27e7; suc to \u27e6suc\u27e7)\nopen import Data.List hiding (replicate; reverse)\nopen import Data.Sum.NP\nopen import Reflection.NP\nopen import Relation.Unary.Logical\nopen import Relation.Binary.Logical\nopen import Relation.Nullary.NP\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Param where\n\nmodule Revelator (tyH : Type) where\n    tm : Term \u2192 \u2115 \u2192 Args \u2192 Term\n    tm (pi (arg (arg-info v _) t\u2081) (el _ t\u2082)) y as = lam visible (tm t\u2082 (suc y) (mapVarArgs suc as ++ arg\u02b3 v (var 0 []) \u2237 []))\n    tm (var x args) = var\n    tm (def f args) = var\n    tm (sort s)     = var\n    tm unknown _ _ = unknown\n    tm (con c args) _ _ = opaque \"revealator\/tm\/con: impossible\" unknown\n    tm (lam v ty) _ _ = opaque \"revealator\/tm\/lam: impossible\" unknown\n    tm (lit l) _ _ = opaque \"revealator\/tm\/lit: impossible\" unknown\n    tm (pat-lam cs args) _ _ = opaque \"revealator\/tm\/pat-lam: TODO\" unknown\n    t\u00ffpe : Type\n    t\u00ffpe = tyH `\u2192\u1d5b\u02b3 Reveal-args.t\u00ffpe tyH\n    term : Term\n    term = lam visible (tm (unEl tyH) 0 [])\n    fun : FunctionDef\n    fun = fun-def t\u00ffpe (clause (arg\u1d5b\u02b3 var \u2237 []) (tm (unEl tyH) 0 []) \u2237 [])\n\ndata VarKind (n : \u2115) : Set where\n  K\u1d62 : Fin n \u2192 VarKind n\n  K\u1d63 : VarKind n\n\nrecord Env (n : \u2115)(A B : Set) : Set where\n  field\n    pVar : VarKind n \u2192 A \u2192 B\n    pCon : Name \u2192 Name\n    pDef : Name \u2192 Name\nopen Env\n\nEnv' = \u03bb n \u2192 Env n \u2115 \u2115\n\n\u03b5 : \u2200 n \u2192 Env' n\n\u03b5 n = record { pVar = \u03bb _ _ \u2192 opaque \"\u03b5.pVar\" 0\n             ; pCon = const (opaque \"\u03b5.pCon\" (quote \u03b5))\n             ; pDef = const (opaque \"\u03b5.pDef\" (quote \u03b5)) }\n\nextDefEnv : ((Name \u2192 Name) \u2192 (Name \u2192 Name)) \u2192 \u2200 {n A B} \u2192 Env n A B \u2192 Env n A B\nextDefEnv ext \u0393 = record \u0393 { pDef = ext (pDef \u0393) }\n\next1DefEnv : (old new : Name) \u2192 \u2200 {n A B} \u2192 Env n A B \u2192 Env n A B\next1DefEnv old new = extDefEnv (\u03bb f x \u2192 [yes: (\u03bb _ \u2192 new) no: (\u03bb _ \u2192 f x) ]\u2032 (x \u225f-Name old))\n\n\u2191pVar : \u2115 \u2192 (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n\u2191pVar zero = id\n\u2191pVar (suc n) = \u2191pVar n \u2218 mapVar\u2191\n\non-pVar : \u2200 {n A B C D} \u2192 (VarKind n \u2192 (A \u2192 B) \u2192 (C \u2192 D)) \u2192 Env n A B \u2192 Env n C D\non-pVar f \u0393 = record\n  { pVar = \u03bb k \u2192 f k (pVar \u0393 k)\n  ; pCon = pCon \u0393\n  ; pDef = pDef \u0393 }\n\n_+\u2191 : \u2200 {n} \u2192 Env' n \u2192 Env' n\n_+\u2191 {n} = on-pVar goK\n  where\n    goK : VarKind n \u2192 (\u2115 \u2192 \u2115) \u2192 (\u2115 \u2192 \u2115)\n    goK (K\u1d62 x) f = _+_ (n \u2238 to\u2115 x) \u2218 \u2191pVar 1 (_+_ (to\u2115 x) \u2218 f)\n    goK K\u1d63     f = \u2191pVar 1 (_+_ n \u2218 f)\n\n_+'_ : \u2200 {w} \u2192 Env' w \u2192 \u2115 \u2192 Env' w\n\u0393 +' n = record { pVar = \u03bb k \u2192 _+_ n \u2218 pVar \u0393 k ; pCon = pCon \u0393 ; pDef = pDef \u0393 }\n\n_+1 : \u2200 {w} \u2192 Env' w \u2192 Env' w\n\u0393 +1 = \u0393 +' 1\n\npattern `\u27e6zero\u27e7  = con (quote \u27e6\u2115\u27e7.zero) []\npattern `\u27e6suc\u27e7 t = con (quote \u27e6\u2115\u27e7.suc) (arg\u1d5b\u02b3 t \u2237 [])\npNat : \u2115 \u2192 Term\npNat zero    = `\u27e6zero\u27e7\npNat (suc n) = `\u27e6suc\u27e7 (pNat n)\n\nlam^ : \u2115 \u2192 Visibility \u2192 Term \u2192 Term\nlam^ zero    v x = x\nlam^ (suc n) v x = lam v (lam^ n v x)\n\nlam\u2208 : \u2200 {n} \u2192 Env' n \u2192 (f : Env' n \u2192 Vec Term n \u2192 Term) \u2192 Term\nlam\u2208 {n} \u0393 f = lam^ n visible (f (\u0393 +' n) (reverse (tabulate (\u03bb x \u2192 var (to\u2115 x) []))))\n\napp-tabulate : \u2200 {n a} {A : Set a} \u2192 (Fin n \u2192 A) \u2192 List A \u2192 List A\napp-tabulate {zero}  f xs = xs\napp-tabulate {suc n} f xs = f zero \u2237 app-tabulate (f \u2218 suc) xs\n\np^args : \u2200 {n A} \u2192 Relevance \u2192 (Fin n \u2192 A) \u2192 A \u2192 List (Arg A) \u2192 List (Arg A)\np^args r f x args = app-tabulate (\u03bb k \u2192 arg\u02b0 r (f k)) (arg\u1d5b\u02b3 x \u2237 args)\n\np^lam : \u2115 \u2192 Term \u2192 Term\np^lam n t = lam^ n hidden (lam visible t)\n\n{-\npi^ : \u2115 \u2192 Arg Type \u2192 Type \u2192 Type\npi^ zero    i t = t\npi^ (suc n) i t = `\u03a0 i (pi^ n i t)\n-}\n\nallVarsFrom : \u2200 n \u2192 \u2115 \u2192 Vec Term n\nallVarsFrom zero    k = []\nallVarsFrom (suc n) k = var (k + n) [] \u2237 allVarsFrom n k\n\np3pi\u2208 : \u2200 {n} (\u0393 : Env' n) (r : Relevance)\n          (s : Type)\n          (t : Env' n \u2192 Vec Term n \u2192 Type)\n          (u : Env' n \u2192 Vec Term n \u2192 Type) \u2192 Term\np3pi\u2208 {n} \u0393 r s t u = unEl (go \u0393 (allFin n))\n  where\n  go : \u2200 {m} \u2192 Env' n \u2192 Vec (Fin n) m \u2192 Type\n  go \u0394 [] = \n      `\u03a0\u1d5b\u02b3 (t \u0394 (allVarsFrom n 0)) (u (\u0393 +\u2191) (allVarsFrom n 1))\n  go \u0394 (x \u2237 xs) =\n      `\u03a0 (arg\u02b0 r (mapVarType (pVar \u0394 (K\u1d62 x)) s))\n         (go (\u0394 +1) xs)\n\npSort\u2208 : \u2200 {n} \u2192 Sort \u2192 Vec Term n \u2192 Type\npSort\u2208 s = go 0\n  where\n    go : \u2115 \u2192 \u2200 {n} \u2192 Vec Term n \u2192 Type\n    go k []       = el (suc-sort s) (sort s)\n    go k (t \u2237 ts) = `\u03a0\u1d5b\u02b3 (mapVarType (_+_ k) (el s t)) (go (suc k) ts)\n\npEnvPats : List (Arg Pattern) \u2192 \u2200 {n} \u2192 Env' n \u2192 Env' n\npEnvPat : Pattern \u2192 \u2200 {n} \u2192 Env' n \u2192 Env' n\n\npEnvPat (con _ pats) = pEnvPats pats\npEnvPat dot = id\npEnvPat var = _+\u2191\npEnvPat (lit _) = id\npEnvPat (proj _) = opaque \"pEnvPats\/proj\" id\npEnvPat absurd = id\n\npEnvPats [] = id\npEnvPats (arg i p \u2237 ps) = pEnvPat p \u2218 pEnvPats ps\n\npPats : \u2200 {n} \u2192 Env' n \u2192 List (Arg Pattern) \u2192 List (Arg Pattern)\npPat  : \u2200 {n} \u2192 Env' n \u2192 Arg Pattern \u2192 List (Arg Pattern)\n\npPats \u0393 [] = []\npPats \u0393 (pat \u2237 ps) = pPat \u0393 pat ++ pPats \u0393 ps\n\nnodot = var\n\npPat {n} \u0393 (arg (arg-info _ r) (con c pats))\n  = p^args {n} r (const nodot) (con (pCon \u0393 c) (pPats \u0393 pats)) []\npPat {n} \u0393 (arg (arg-info _ r) dot)\n  = p^args {n} r (const dot) dot []\npPat {n} \u0393 (arg (arg-info _ r) var)\n  = p^args {n} r (const var) var []\npPat {n} \u0393 (arg i (lit (nat l))) = opaque \"pPat\/lit\/nat\" []\npPat {n} \u0393 (arg (arg-info _ r) (lit l))\n  = p^args {n} r go\n                 (con (quote refl) []) []\n      where\n        go : \u2200 {n} \u2192 Fin n \u2192 Pattern\n        go zero = lit l\n        go (suc _) = nodot\npPat {n} \u0393 (arg (arg-info _ r) absurd) = p^args {n} r (const var) absurd []\npPat {n} \u0393 (arg i (proj p)) = opaque \"pPat\/proj\" []\n\npLit : Literal \u2192 Term\npLit (nat n) = pNat n\npLit _       = con (quote refl) []\n\npTerm  : \u2200 {n} \u2192 Env' n \u2192 Term \u2192 Term\npArgs  : \u2200 {n} \u2192 Env' n \u2192 Args \u2192 Args\npType\u2208 : \u2200 {n} \u2192 Env' n \u2192 Type             \u2192 Vec Term n \u2192 Type\npTerm\u2208 : \u2200 {n} \u2192 Env' n \u2192 Term             \u2192 Vec Term n \u2192 Term\npPi\u2208   : \u2200 {n} \u2192 Env' n \u2192 Arg Type \u2192 Type \u2192 Vec Term n \u2192 Term\n\npType\u2208 \u0393 (el s t) as = el s (pTerm\u2208 \u0393 t as)\n\npTerm {n} \u0393 (lam _ t) = p^lam n (pTerm (\u0393 +\u2191) t)\npTerm \u0393 (var v args)  = var (pVar \u0393 K\u1d63 v) (pArgs \u0393 args)\npTerm \u0393 (con c args)  = con (pCon \u0393 c) (pArgs \u0393 args)\npTerm \u0393 (def d args)  = def (pDef \u0393 d) (pArgs \u0393 args)\npTerm \u0393 (lit l)       = pLit l\npTerm \u0393 (pat-lam _ _) = opaque \"pTerm\/pat-lam\" unknown\npTerm \u0393 unknown       = unknown\n\npTerm \u0393 (sort s) = lam\u2208 \u0393 \u03bb _ as \u2192 unEl (pSort\u2208 s as)\npTerm \u0393 (pi s t) = lam\u2208 \u0393 \u03bb \u0394 \u2192 pPi\u2208 \u0394 s t\n\npPi\u2208 {n} \u0393 (arg (arg-info _ r) s) t as =\n  p3pi\u2208 \u0393 r s (\u03bb \u0394 \u2192 pType\u2208 \u0394 s) (\u03bb \u0394 vs \u2192 pType\u2208 \u0394 t\n    (vmap (\u03bb a v \u2192 app (mapVar (_+_ (suc n)) a) (arg\u1d5b\u02b3 v \u2237 [])) as \u229b vs))\n\npTerm\u2208 \u0393 (sort s) as = unEl (pSort\u2208 s as)\npTerm\u2208 \u0393 (pi s t) = pPi\u2208 \u0393 s t\npTerm\u2208 \u0393 t as = app (pTerm \u0393 t) (toList (vmap arg\u1d5b\u02b3 as))\n\npArgs \u0393 [] = []\npArgs \u0393 (arg (arg-info _ r) t \u2237 as)\n  = p^args r (\u03bb k \u2192 mapVar (pVar (\u0393 +\u2191) (K\u1d62 k)) t) (pTerm \u0393 t) (pArgs \u0393 as)\n\npClause  : \u2200 {n} \u2192 Env' n \u2192 Clause \u2192 Clause\npClauses : \u2200 {n} \u2192 Env' n \u2192 Clauses \u2192 Clauses\n\npClause \u0393 (clause pats body) = clause (pPats \u0393 pats) (pTerm (pEnvPats pats \u0393) body)\npClause \u0393 (absurd-clause pats) = absurd-clause (pPats \u0393 pats)\n\npClauses \u0393 [] = []\npClauses \u0393 (c \u2237 cs) = pClause \u0393 c \u2237 pClauses \u0393 cs\n\npFunDef : \u2200 {n} \u2192 Env' n \u2192 Name \u2192 FunctionDef \u2192 FunctionDef\npFunDef \u0393 d (fun-def t cs)\n  = fun-def (pType\u2208 \u0393 t (replicate (def d []))) (pClauses \u0393 cs)\n\npostulate\n  IMPOSSIBLE : FunctionDef\n\npFunName : \u2200 {n} \u2192 Env' n \u2192 Name \u2192 FunctionDef\npFunName \u0393 x with definition x\n... | function d = pFunDef \u0393 x d\n... | _ = IMPOSSIBLE -- opaque \"pFunName\" (fun-def {!!} {!!})\n\npFunNameRec : \u2200 {n} \u2192 Env' n \u2192 (x x\u209a : Name) \u2192 FunctionDef\npFunNameRec \u0393 x x\u209a = pFunName (ext1DefEnv x x\u209a \u0393) x\n\n{-\npDefinition : Env' 2 \u2192 Name \u2192 Definition \u2192 Definition\npDefinition \u0393 n (function x) = function {!pFunDef!}\npDefinition \u0393 n (data-type x) = {!!}\npDefinition \u0393 n (record\u2032 x) = {!!}\npDefinition \u0393 n constructor\u2032 = {!!}\npDefinition \u0393 n axiom = {!!}\npDefinition \u0393 n primitive\u2032 = {!!}\n\npName : Env' 2 \u2192 Name \u2192 Definition\npName \u0393 n = pDefinition \u0393 n (definition n)\n-}\n\nopen import Data.String using (String)\nopen import Data.Float  using (Float)\n\ninfixr 1 _[\u2080\u2192\u2080]_\n_[\u2080\u2192\u2080]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2080} (B\u209a : B \u2192 Set\u2080)\n            (f : A \u2192 B) \u2192 Set\u2080\n_[\u2080\u2192\u2080]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2080\u2192\u2081]_\n_[\u2080\u2192\u2081]_ : \u2200 {A : Set\u2080} (A\u209a : A \u2192 Set\u2080)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2080\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2081]_\n_[\u2081\u2192\u2081]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2081} (B\u209a : B \u2192 Set\u2081)\n            (f : A \u2192 B) \u2192 Set\u2081\n_[\u2081\u2192\u2081]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\ninfixr 1 _[\u2081\u2192\u2082]_\n_[\u2081\u2192\u2082]_ : \u2200 {A : Set\u2081} (A\u209a : A \u2192 Set\u2081)\n            {B : Set\u2082} (B\u209a : B \u2192 Set\u2082)\n            (f : A \u2192 B) \u2192 Set\u2082\n_[\u2081\u2192\u2082]_ = \u03bb {A} A\u209a {B} B\u209a f \u2192 \u2200 {a} (a\u209a : A\u209a a) \u2192 B\u209a (f a)\n\n[[Set\u2080]] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080]\n[[Set\u2080]] = \u03bb A \u2192 A [\u2080\u2192\u2081] [Set\u2080]\n\n{-\nEqEnv = {!!}\nEqRes = {!!}\n\neqTerm : EqEnv \u2192 Term \u2192 Term \u2192 EqRes\neqTerm \u0393 (var x args) (var x\u2081 args\u2081) = {!!}\neqTerm \u0393 (def f\u2080 args\u2080) (def f\u2081 args\u2081) = {!!}\neqTerm \u0393 (con c\u2080 args\u2080) (con c\u2081 args\u2081) = {!!}\neqTerm \u0393 (lam v t) (lam v' t') = {!!}\neqTerm \u0393 (pi t\u2081 t\u2082) (pi u\u2081 u\u2082) = {!!}\neqTerm \u0393 (sort s\u2080) (sort s\u2081) = {!!}\neqTerm \u0393 (lit l\u2080) (lit l\u2081) = {!!}\neqTerm \u0393 unknown unknown = {!!}\neqTerm \u0393 (def f args) u = {!!}\n--eqTerm \u0393 (pat-lam cs args) u = {!!}\neqTerm _ _ = ?\n-}\n\nimport Reflection.Simple as Si\nopen Si using (var;con;def;lam;pi;sort;unknown;simple;showTerm)\n\np[Set\u2080] = pFunName (\u03b5 1) (quote [Set\u2080])\nq[[Set\u2080]] = definition (quote [[Set\u2080]]) -- quoteTerm [[Set\u2080]]\ntest-type-p[Set\u2080] : ([Set\u2080] [\u2081\u2192\u2082] [Set\u2081]) [Set\u2080] \u2261 unquote (unEl (Get-type.from-fun-def p[Set\u2080]))\ntest-type-p[Set\u2080] = refl\ntest-term-p[Set\u2080] : quoteTerm [[Set\u2080]] \u2261 Get-term.from-fun-def p[Set\u2080]\ntest-term-p[Set\u2080] = refl\nunquoteDecl u-p[Set\u2080] = p[Set\u2080]\n\n{-\np-[\u2192]-type = unEl (Get-type.from-fun-def p-[\u2192])\np-[\u2192]' : \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n                {A\u209a : A \u2192 Set\u2080} (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 Set\u2080)\n                {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n                {B\u209a : B \u2192 Set\u2080} (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 Set\u2080)\n                {f : A \u2192 B}    (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n              \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f \u2192\n              {C : Set\u2080} \u2192 (C \u2192 Set\u2080) \u2192 {t : {!!}} \u2192 (A \u2192 B) \u2192 Set\u2080 -- _[\u2080\u2192\u2080]_ {A} {-A\u2080\u209a} {A\u209a-} {-A\u2081\u209a-} {!!} {B} {!!} {-B\u2080\u209a} B\u2081\u209a-} f -- f\u209a\np-[\u2192]' = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n-}\n\n[String] : [\u2605\u2080] String\n[String] _ = \ud835\udfd9\n\n[Float] : [\u2605\u2080] Float\n[Float] _ = \ud835\udfd9\n\n\u27e6String\u27e7 : \u27e6\u2605\u2080\u27e7 String String\n\u27e6String\u27e7 = _\u2261_\n\n\u27e6Float\u27e7 : \u27e6\u2605\u2080\u27e7 Float Float\n\u27e6Float\u27e7 = _\u2261_\n\ndata [\ud835\udfda] : [Set\u2080] \ud835\udfda where\n  [0\u2082] : [\ud835\udfda] 0\u2082\n  [1\u2082] : [\ud835\udfda] 1\u2082\n\ndata [\u2115] : [Set\u2080] \u2115 where\n  [zero] : [\u2115] zero\n  [suc]  : ([\u2115] [\u2192] [\u2115]) suc\n\n[pred] : ([\u2115] [\u2192] [\u2115]) pred\n[pred] [zero]     = [zero]\n[pred] ([suc] x\u209a) = x\u209a\n\nrevealed-[\u2192] = Reveal-args.n\u00e5me (quote _[\u2080\u2192\u2080]_)\nunquoteDecl revealed-[\u2192]' = un-function revealed-[\u2192]\n\nunquoteDecl revelator-[\u2192] = Revelator.fun (type (quote _[\u2080\u2192\u2080]_))\n\n{-\np-[\u2192] = pFunName (\u03b5 1) (quote _[\u2080\u2192\u2080]_)\np-[\u2192]-type = unEl (Get-type.from-fun-def p-[\u2192])\n\np-[\u2192]' : (x0 : Set) \u2192 (x1 : (x1 : x0) \u2192 Set) \u2192 (x2 : (x2 : x0) \u2192 Set) \u2192 (x3 : (x3 : x0) \u2192 (x4 : x1 x3) \u2192 (x5 : x2 x3) \u2192 Set) \u2192 (x4 : Set) \u2192 (x5 : (x5 : x4) \u2192 Set) \u2192 (x6 : (x6 : x4) \u2192 Set) \u2192 (x7 : (x7 : x4) \u2192 (x8 : x5 x7) \u2192 (x9 : x6 x7) \u2192 Set) \u2192 (x8 : (x8 : x0) \u2192 x4) \u2192 (x9 : (x9 : x0) \u2192 (x10 : x1 x9) \u2192 x5 (x8 x9)) \u2192 (x10 : _[\u2080\u2192\u2080]_ {x0} x2 {x4} x6 x8) \u2192 Set\n{-\np-[\u2192]' : \u2200 {A : Set\u2080}       (A\u2080\u209a : A \u2192 Set\u2080)\n                {A\u209a : A \u2192 Set\u2080} (A\u2081\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 Set\u2080)\n                {B : Set\u2080}       (B\u2080\u209a : B \u2192 Set\u2080)\n                {B\u209a : B \u2192 Set\u2080} (B\u2081\u209a : {x : B} \u2192 B\u2080\u209a x \u2192 Set\u2080)\n                {f : A \u2192 B}    (f\u209a : {x : A} \u2192 A\u2080\u209a x \u2192 B\u2080\u209a (f x))\n              \u2192 (A\u209a [\u2080\u2192\u2080] B\u209a) f \u2192\n              {C : Set\u2080} \u2192 (C \u2192 Set\u2080) \u2192 {t : {!!}} \u2192 (A \u2192 B) \u2192 Set\u2080 -- _[\u2080\u2192\u2080]_ {A} {-A\u2080\u209a} {A\u209a-} {-A\u2081\u209a-} {!!} {B} {!!} {-B\u2080\u209a} B\u2081\u209a-} f -- f\u209a\n-}\np-[\u2192]' = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n\ntest : {!showTerm p-[\u2192]-type!}\ntest = {!unquote (Get-term.from-fun-def p-[\u2192])!}\n{-\n-- unquoteDecl _[[\u2192]]_ = p-[\u2192]\n\n{-\ndata [[\u2115]] : [[Set\u2080]] [\u2115] [\u2115] where\n  [[zero]] : [[\u2115]] [zero] [zero]\n  [[suc]]  : ([[\u2115]] [[\u2192]] [[\u2115]]) [suc] [suc]\n-}\n\ndefDefEnv1 : Name \u2192 Name\ndefDefEnv1 (quote \ud835\udfda) = quote [\ud835\udfda]\ndefDefEnv1 (quote \u2115) = quote [\u2115]\ndefDefEnv1 (quote String) = quote [String]\ndefDefEnv1 (quote Float) = quote [Float]\ndefDefEnv1 (quote \u2605\u2080) = quote [\u2605\u2080]\ndefDefEnv1 (quote \u2605\u2081) = quote [\u2605\u2081]\ndefDefEnv1 n         = opaque \"defDefEnv1\" n\n\ndefConEnv1 : Name \u2192 Name\ndefConEnv1 (quote 0\u2082) = quote [0\u2082]\ndefConEnv1 (quote 1\u2082) = quote [1\u2082]\ndefConEnv1 (quote \u2115.zero) = quote [zero]\ndefConEnv1 (quote \u2115.suc)  = quote [suc]\ndefConEnv1 n              = opaque \"defConEnv1\" n\n\ndefDefEnv2 : Name \u2192 Name\ndefDefEnv2 (quote \ud835\udfda) = quote \u27e6\ud835\udfda\u27e7\ndefDefEnv2 (quote \u2115) = quote \u27e6\u2115\u27e7\ndefDefEnv2 (quote String) = quote \u27e6String\u27e7\ndefDefEnv2 (quote Float) = quote \u27e6Float\u27e7\ndefDefEnv2 (quote \u2605\u2080) = quote \u27e6\u2605\u2080\u27e7\ndefDefEnv2 (quote \u2605\u2081) = quote \u27e6\u2605\u2081\u27e7\ndefDefEnv2 n         = opaque \"defDefEnv\" n\n\ndefConEnv2 : Name \u2192 Name\ndefConEnv2 (quote 0\u2082) = quote \u27e60\u2082\u27e7\ndefConEnv2 (quote 1\u2082) = quote \u27e61\u2082\u27e7\ndefConEnv2 (quote \u2115.zero) = quote \u27e6\u2115\u27e7.zero\ndefConEnv2 (quote \u2115.suc)  = quote \u27e6\u2115\u27e7.suc\ndefConEnv2 n              = opaque \"defConEnv2\" n\n\ndefEnv1 : Env' 1\ndefEnv1 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv1.pVar\" 0\n                 ; pCon = defConEnv1\n                 ; pDef = defDefEnv1 }\n\ndefEnv2 : Env' 2\ndefEnv2 = record { pVar = \u03bb _ _ \u2192 opaque \"defEnv2.pVar\" 0\n                 ; pCon = defConEnv2\n                 ; pDef = defDefEnv2 }\n\n{-\n_\u27e6+\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7) _+_ _+_\nzero  \u27e6+\u27e7 n = n\nsuc m \u27e6+\u27e7 n = suc (m \u27e6+\u27e7 n)\n-}\n\n_\/2 : \u2115 \u2192 \u2115\nzero        \/2 = zero\nsuc zero    \/2 = zero\nsuc (suc n) \/2 = suc (n \/2)\n\n_\u27e6\/2\u27e7 : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _\/2 _\/2\n\u27e6zero\u27e7         \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 \u27e6zero\u27e7    \u27e6\/2\u27e7 = \u27e6zero\u27e7\n\u27e6suc\u27e7 (\u27e6suc\u27e7 n) \u27e6\/2\u27e7 = \u27e6suc\u27e7 (n \u27e6\/2\u27e7)\n\n_+\u2115_ : \u2115 \u2192 \u2115 \u2192 \u2115\nzero  +\u2115 n = n\nsuc m +\u2115 n = suc (m +\u2115 n)\n\n_\u27e6+\u2115\u27e7_ : (\u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\u2115\u27e7) _+\u2115_ _+\u2115_\n\u27e6zero\u27e7  \u27e6+\u2115\u27e7 n = n\n\u27e6suc\u27e7 m \u27e6+\u2115\u27e7 n = \u27e6suc\u27e7 (m \u27e6+\u2115\u27e7 n)\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7 : (\u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2080\u27e7 \u27e6\u2081\u2192\u2082\u27e7 \u27e6Set\u2081\u27e7) \u27e6Set\u2080\u27e7 \u27e6Set\u2080\u27e7\n\u27e6\u27e6Set\u2080\u27e7\u27e7 = \u03bb A\u2080 A\u2081 \u2192 A\u2080 \u27e6\u2080\u2192\u2081\u27e7 A\u2081 \u27e6\u2080\u2192\u2081\u27e7 \u27e6Set\u2080\u27e7\n\n\u27e6\u27e6Set\u2080\u27e7\u27e7' : {x\u2081 x\u2082 : Set} (x\u1d63 : x\u2081 \u2192 x\u2082 \u2192 Set) {x\u2083 : Set} {x\u2084 : Set}\n            (x\u1d63\u2081 : x\u2083 \u2192 x\u2084 \u2192 Set) \u2192\n            (x\u2081 \u2192 x\u2083 \u2192 Set) \u2192 (x\u2082 \u2192 x\u2084 \u2192 Set) \u2192 Set\u2081\n\u27e6\u27e6Set\u2080\u27e7\u27e7' = \u03bb A\u2080 A\u2081 f\u2081 f\u2082 \u2192\n  \u2200 {x\u2081} {x\u2082} (x\u1d63 : A\u2080 x\u2081 x\u2082)\n    {x\u2083} {x\u2084} (x\u1d63\u2081 : A\u2081 x\u2083 x\u2084) \u2192\n    f\u2081 x\u2081 x\u2083 \u2192 f\u2082 x\u2082 x\u2084 \u2192 Set\n\n-- Since quoteTerm normalises\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 : quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7 \u2261 quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7'\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7 = refl\n\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type : unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7))) \u2261 unquote (unEl (type (quote \u27e6\u27e6Set\u2080\u27e7\u27e7')))\ntest-\u27e6\u27e6Set\u2080\u27e7\u27e7-type = refl\n\npSet\u2080 = pTerm defEnv2 `\u2605\u2080\nppSet\u2080 = pTerm defEnv2 pSet\u2080\np\u27e6Set\u2080\u27e7 = pFunName defEnv2 (quote \u27e6Set\u2080\u27e7)\ntest-pSet\u2080 : pSet\u2080 \u2261 quoteTerm \u27e6Set\u2080\u27e7\ntest-pSet\u2080 = refl\ntest-ppSet\u2080 : ppSet\u2080 \u2261 quoteTerm \u27e6\u27e6Set\u2080\u27e7\u27e7\ntest-ppSet\u2080 = refl\ntest-ppSet\u2080'' : ppSet\u2080 \u2261 Get-term.from-fun-def p\u27e6Set\u2080\u27e7\ntest-ppSet\u2080'' = refl\n\n-- unquoteDecl \u27e6\u27e6Set\u2080\u27e7\u27e7'' = p\u27e6Set\u2080\u27e7\n\ndata \u27e6\u27e6\ud835\udfda\u27e7\u27e7 : (\u27e6\u27e6Set\u2080\u27e7\u27e7 \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7) \u27e6\ud835\udfda\u27e7 \u27e6\ud835\udfda\u27e7 where\n  \u27e6\u27e60\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7 \u27e60\u2082\u27e7\n  \u27e6\u27e61\u2082\u27e7\u27e7 : \u27e6\u27e6\ud835\udfda\u27e7\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7 \u27e61\u2082\u27e7\n\n_\u2261clauses_ = \u03bb x y \u2192 Get-clauses.from-def x \u2261 Get-clauses.from-def y\n_\u2261type_ = \u03bb x y \u2192 Get-type.from-def x \u2261 Get-type.from-def y\n  \nmodule Test where\n  p1\u2115\u2192\u2115 = pTerm defEnv1 (quoteTerm (\u2115 \u2192 \u2115))\n  [\u2115\u2192\u2115] = [\u2115] [\u2192] [\u2115]\n  test-p1\u2115\u2192\u2115 : unquote p1\u2115\u2192\u2115 \u2261 [\u2115\u2192\u2115]\n  test-p1\u2115\u2192\u2115 = refl\n\n  p2\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115))\n  \u27e6\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n  test-p2\u2115\u2192\u2115 : unquote p2\u2115\u2192\u2115 \u2261 \u27e6\u2115\u2192\u2115\u27e7\n  test-p2\u2115\u2192\u2115 = refl\n\n  p\u2115\u2192\u2115\u2192\u2115 = pTerm defEnv2 (quoteTerm (\u2115 \u2192 \u2115 \u2192 \u2115))\n  \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7 = \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7 \u27e6\u2192\u27e7 \u27e6\u2115\u27e7\n  test-p\u2115\u2192\u2115\u2192\u2115 : p\u2115\u2192\u2115\u2192\u2115 \u2261 quoteTerm \u27e6\u2115\u2192\u2115\u2192\u2115\u27e7\n  test-p\u2115\u2192\u2115\u2192\u2115 = refl\n  ZERO : Set\u2081\n  ZERO = (A : Set\u2080) \u2192 A\n  \u27e6ZERO\u27e7 : ZERO \u2192 ZERO \u2192 Set\u2081\n  \u27e6ZERO\u27e7 f\u2080 f\u2081 =\n    {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n    \u2192 A\u1d63 (f\u2080 A\u2080) (f\u2081 A\u2081)\n  pZERO = pTerm (\u03b5 2) (quoteTerm ZERO)\n  q\u27e6ZERO\u27e7 = quoteTerm \u27e6ZERO\u27e7\n  test-pZERO : pZERO \u2261 q\u27e6ZERO\u27e7\n  test-pZERO = refl\n  ID : Set\u2081\n  ID = (A : Set\u2080) \u2192 A \u2192 A\n  \u27e6ID\u27e7 : ID \u2192 ID \u2192 Set\u2081\n  \u27e6ID\u27e7 f\u2080 f\u2081 =\n    {A\u2080 A\u2081 : Set\u2080} (A\u1d63 : A\u2080 \u2192 A\u2081 \u2192 Set\u2080)\n    {x\u2080 : A\u2080} {x\u2081 : A\u2081} (x : A\u1d63 x\u2080 x\u2081)\n    \u2192 A\u1d63 (f\u2080 A\u2080 x\u2080) (f\u2081 A\u2081 x\u2081)\n  pID = pTerm (\u03b5 2) (quoteTerm ID)\n  q\u27e6ID\u27e7 = quoteTerm \u27e6ID\u27e7\n  test-param : absTerm q\u27e6ID\u27e7 \u2261 absTerm pID\n  test-param = refl\n  test-ID : q\u27e6ID\u27e7 \u2261 pID\n  test-ID = refl\n\n  p-not = pFunName defEnv2 (quote not)\n  unquoteDecl P-not = p-not\n  test-not : \u2200 {x\u2080 x\u2081 : \ud835\udfda} (x\u1d63 : \u27e6\ud835\udfda\u27e7 x\u2080 x\u2081) \u2192 \u27e6not\u27e7 x\u1d63 \u2261 P-not x\u1d63\n  test-not \u27e60\u2082\u27e7 = refl\n  test-not \u27e61\u2082\u27e7 = refl\n\n  p1-pred = pFunName defEnv1 (quote pred)\n  q-[pred] = quoteTerm [pred]\n  unquoteDecl [pred]' = p1-pred\n  test-p1-pred : \u2200 {n} (n\u209a : [\u2115] n) \u2192 [pred]' n\u209a \u2261 [pred] n\u209a\n  test-p1-pred [zero] = refl\n  test-p1-pred ([suc] n\u209a) = refl\n\n  p2-pred = pFunName defEnv2 (quote pred)\n  q-\u27e6pred\u27e7 = definition (quote \u27e6pred\u27e7)\n  unquoteDecl \u27e6pred\u27e7' = p2-pred\n  test-p2-pred : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 \u27e6pred\u27e7' n\u1d63 \u2261 \u27e6pred\u27e7 n\u1d63\n  test-p2-pred \u27e6zero\u27e7 = refl\n  test-p2-pred (\u27e6suc\u27e7 n\u1d63) = refl\n\n  p\/2 = pFunNameRec defEnv2 (quote _\/2)\n  q\u27e6\/2\u27e7 = definition (quote _\u27e6\/2\u27e7)\n  unquoteDecl _\u27e6\/2\u27e7' = p\/2 _\u27e6\/2\u27e7'\n  {-\n  test-\/2 : function (p\/2 (quote _\u27e6\/2\u27e7)) \u2261 q\u27e6\/2\u27e7\n  test-\/2 = refl\n  -}\n  test-\/2 : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) \u2192 n\u1d63 \u27e6\/2\u27e7' \u2261 n\u1d63 \u27e6\/2\u27e7\n  test-\/2 \u27e6zero\u27e7 = refl\n  test-\/2 (\u27e6suc\u27e7 \u27e6zero\u27e7) = refl\n  test-\/2 (\u27e6suc\u27e7 (\u27e6suc\u27e7 n\u1d63)) = ap \u27e6suc\u27e7 (test-\/2 n\u1d63)\n\n  p+ = pFunNameRec defEnv2 (quote _+\u2115_)\n  q\u27e6+\u27e7 = definition (quote _\u27e6+\u2115\u27e7_)\n  unquoteDecl _\u27e6+\u27e7'_ = p+ _\u27e6+\u27e7'_\n  -- test-+ : function (p+ (quote _\u27e6+\u2115\u27e7_)) \u2261 q\u27e6+\u27e7\n  -- test-+ = refl\n  test-+ : \u2200 {n\u2080 n\u2081} (n\u1d63 : \u27e6\u2115\u27e7 n\u2080 n\u2081) {n'\u2080 n'\u2081} (n'\u1d63 : \u27e6\u2115\u27e7 n'\u2080 n'\u2081) \u2192 n\u1d63 \u27e6+\u27e7' n'\u1d63 \u2261 n\u1d63 \u27e6+\u2115\u27e7 n'\u1d63\n  test-+ \u27e6zero\u27e7    n'\u1d63 = refl\n  test-+ (\u27e6suc\u27e7 n\u1d63) n'\u1d63 = ap \u27e6suc\u27e7 (test-+ n\u1d63 n'\u1d63)\n\n  {-\n  is-good : String \u2192 \ud835\udfda\n  is-good \"good\" = 1\u2082\n  is-good _      = 0\u2082\n\n  \u27e6is-good\u27e7 : (\u27e6String\u27e7 \u27e6\u2080\u2192\u2080\u27e7 \u27e6\ud835\udfda\u27e7) is-good is-good\n  \u27e6is-good\u27e7 {\"good\"} refl = \u27e61\u2082\u27e7\n  \u27e6is-good\u27e7 {_}      refl = {!!}\n\n  p-is-good = pFunName defEnv2 (quote is-good)\n  unquoteDecl \u27e6is-good\u27e7' = p-is-good\n  test-is-good = {!!}\n  -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Reflection\/Param.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4de99690f3407a9e009fd1da49094aa69a73487f","subject":"Formalize change algebras.","message":"Formalize change algebras.\n\nOld-commit-hash: 19a6f7f8a08464d8f2e98dcc618ac9a06d50da78\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Algebra.agda","new_file":"Base\/Change\/Algebra.agda","new_contents":"module Base.Change.Algebra where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\n-- Change Algebras\n\nrecord IsChangeAlgebra\n    {c} {d}\n    {Carrier : Set c}\n    (Change : Carrier \u2192 Set d)\n    (update : (v : Carrier) (dv : Change v) \u2192 Carrier)\n    (diff : (u v : Carrier) \u2192 Change v) : Set (c \u2294 d) where\n  field\n    update-diff : \u2200 u v \u2192 update v (diff u v) \u2261 u\n\nrecord ChangeAlgebra {c} \u2113\n    (Carrier : Set c) : Set (c \u2294 suc \u2113) where\n  field\n    Change : Carrier \u2192 Set \u2113\n    update : (v : Carrier) (dv : Change v) \u2192 Carrier\n    diff : (u v : Carrier) \u2192 Change v\n\n    isChangeAlgebra : IsChangeAlgebra Change update diff\n\n\n  infixl 6 update diff\n\n  open IsChangeAlgebra isChangeAlgebra public\n\nopen ChangeAlgebra {{...}} public\n  using\n    ( update-diff\n    )\n  renaming\n    ( Change to \u0394\n    ; update to _\u229e_\n    ; diff to _\u229f_\n    )\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Base\/Change\/Algebra.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"87dafe419f3b26274f7c9dbbea6565a12605faf4","subject":"adding theorems that each branch of the progress are disjoint","message":"adding theorems that each branch of the progress are disjoint\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"progress-checks.agda","new_file":"progress-checks.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\nopen import lemmas-consistency\nopen import canonical-forms\nopen import type-assignment-unicity\n\n\n-- taken together, the theorems in this file argue that for any expression\n-- d, at most one summand of the labeled sum that results from progress may\n-- be true at any time, i.e. that values, indeterminates, errors, and\n-- expressions that step are pairwise disjoint. (note that as a consequence\n-- of currying and comutativity of products, this means that there are six\n-- theorems to prove)\nmodule progress-checks where\n  -- values and indeterminates are disjoint\n  vi : \u2200{d} \u2192 d val \u2192 d indet \u2192 \u22a5\n  vi VConst ()\n  vi VLam ()\n\n  -- values and errors are disjoint\n  ve : \u2200{d \u0394} \u2192 d val \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ve VConst ()\n  ve VLam ()\n\n  -- values and expressions that step are disjoint\n  vs : \u2200{d \u0394} \u2192 d val \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  vs VConst (d , Step (FHFinal x) () (FHFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () FHEHole)\n  vs VConst (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  vs VConst (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2082))\n  vs VConst (_ , Step (FHFinal x) () (FHCastFinal x\u2082))\n  vs VLam (d , Step (FHFinal x\u2081) () (FHFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () FHEHole)\n  vs VLam (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2083))\n  vs VLam (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2083))\n\n  -- indeterminates and errors are disjoint\n  ie : \u2200{d \u0394} \u2192 d indet \u2192 \u0394 \u22a2 d err \u2192 \u22a5\n  ie IEHole ()\n  ie (INEHole (FVal x)) (ENEHole e) = ve x e\n  ie (INEHole (FIndet x)) (ENEHole e) = ie x e\n  ie (IAp i x) (EAp1 e) = ie i e\n  ie (IAp i (FVal x)) (EAp2 e) = ve x e\n  ie (IAp i (FIndet x)) (EAp2 e) = ie x e\n\n  -- indeterminates and expressions that step are disjoint\n  is : \u2200{d \u0394} \u2192 d indet \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  is IEHole (d0' , Step (FHFinal x) () (FHFinal x\u2081))\n  is IEHole (d0' , Step FHEHole () (FHFinal x))\n  is IEHole (_ , Step (FHFinal x) () FHEHole)\n  is IEHole (_ , Step FHEHole () FHEHole)\n  is IEHole (_ , Step (FHFinal x) () FHNEHoleEvaled)\n  is IEHole (_ , Step FHEHole () FHNEHoleEvaled)\n  is IEHole (_ , Step (FHFinal x) () (FHNEHoleFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHNEHoleFinal x))\n  is IEHole (_ , Step (FHFinal x) () (FHCastFinal x\u2081))\n  is IEHole (_ , Step FHEHole () (FHCastFinal x))\n  is (INEHole x) (d , Step (FHFinal x\u2081) () (FHFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHEHole)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHNEHoleFinal x\u2082))\n  is (INEHole x) (_ , Step (FHFinal x\u2081) () (FHCastFinal x\u2082))\n  is (INEHole x) (d , Step FHNEHoleEvaled () (FHFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHEHole)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () FHNEHoleEvaled)\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHNEHoleFinal x\u2081))\n  is (INEHole x) (_ , Step FHNEHoleEvaled () (FHCastFinal x\u2081))\n  is (IAp i x) (d' , Step p q r) = {!p r!}\n\n  -- errors and expressions that step are disjoint\n  es : \u2200{d \u0394} \u2192 \u0394 \u22a2 d err \u2192 (\u03a3[ d' \u2208 dhexp ] (\u0394 \u22a2 d \u21a6 d')) \u2192 \u22a5\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FVal ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (d' , Step (FHFinal (FIndet ())) x\u2083 x\u2084)\n  es (ECastError x x\u2081) (_ , Step (FHCast x\u2082) x\u2083 (FHCast x\u2084)) = {!x\u2083!}\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FVal x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (ECastError x x\u2081) (d' , Step (FHCastFinal (FIndet x\u2082)) (ITCast x\u2083 x\u2084 x\u2085) x\u2086)\n    with type-assignment-unicity x x\u2084\n  ... | refl = ~apart x\u2081 x\u2085\n  es (EAp1 e) (d , Step (FHFinal (FVal x)) x\u2081 q) = vs x (d , Step (FHFinal (FVal x)) x\u2081 q)\n  es (EAp1 e) (d , Step (FHFinal (FIndet x)) x\u2081 q) = is x (d , Step (FHFinal (FIndet x)) x\u2081 q)\n  es (EAp1 e) (_ , Step (FHAp1 x x\u2081) x\u2082 (FHAp1 x\u2083 x\u2084)) = {!!}\n  es (EAp1 e) (_ , Step (FHAp2 x) x\u2081 (FHAp2 x\u2082)) = {!!}\n  es (EAp2 e) s = {!!}\n  es (ENEHole e) s = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FVal x)) (ITCast x\u2081 x\u2082 x\u2083) x\u2084) = {!!}\n  es (ECastProp e) (d' , Step (FHFinal (FIndet x)) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCast x) x\u2081 x\u2082) = {!!}\n  es (ECastProp e) (d' , Step (FHCastFinal x) x\u2081 x\u2082) = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'progress-checks.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"372353c6307b5867f7ca42b382b2f8f5f826fbf0","subject":"Added Agsy test about the totality of addition.","message":"Added Agsy test about the totality of addition.\n\nIgnore-this: a8258a4fd6d8c1e7fac155dd2f1e5068\n\ndarcs-hash:20110324140030-3bd4e-59b07b9164d8e8fb7afe865220ccc4721d839818.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Agsy\/AddTotality.agda","new_file":"Draft\/Agsy\/AddTotality.agda","new_contents":"module AddTotality where\n\nopen import Relation.Binary.PropositionalEquality\n\ninfixl 9  _+_\n\n------------------------------------------------------------------------------\n\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\npostulate\n  _+_  : D \u2192 D \u2192 D\n  +-0x : \u2200 d \u2192   zero   + d \u2261 d\n  +-Sx : \u2200 d e \u2192 succ d + e \u2261 succ (d + e)\n\n+-N : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N {m} {n} Nm Nn = {!-c -m -t 20!}  -- Agsy fails\n\n+-N\u2081 : \u2200 {m n} \u2192 N m \u2192 N n \u2192 N (m + n)\n+-N\u2081 zN Nn = {!-m!}  -- Agda bug?\n                     -- An internal error has occurred. Please report this as a bug.\n                     -- Location of the error: src\/full\/Agda\/Auto\/Convert.hs:444\n\n+-N\u2081 (sN Nm) Nn = {!-m!} -- Agda bug?\n                         -- An internal error has occurred. Please report this as a bug.\n                         -- Location of the error: src\/full\/Agda\/Auto\/Convert.hs:444\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Agsy\/AddTotality.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"462531057f6d44ae107133e5ddcc767c1af0069f","subject":"adding prelude from popl work","message":"adding prelude from popl work\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"Prelude.agda","new_file":"Prelude.agda","new_contents":"module Prelude where\n  open import Agda.Primitive using (Level; lzero; lsuc) renaming (_\u2294_ to lmax)\n\n  -- empty type\n  data \u22a5 : Set where\n\n  -- from false, derive whatever\n  abort : \u2200 {C : Set} \u2192 \u22a5 \u2192 C\n  abort ()\n\n  -- unit\n  data \u22a4 : Set where\n    <> : \u22a4\n\n  -- sums\n  data _+_ (A B : Set) : Set where\n    Inl : A \u2192 A + B\n    Inr : B \u2192 A + B\n\n  -- pairs\n  infixr 1 _,_\n  record \u03a3 {l1 l2 : Level} (A : Set l1) (B : A \u2192 Set l2) : Set (lmax l1 l2) where\n    constructor _,_\n    field\n      \u03c01 : A\n      \u03c02 : B \u03c01\n  open \u03a3 public\n\n  -- Sigma types, or dependent pairs, with nice notation.\n  syntax \u03a3 A (\\ x -> B) = \u03a3[ x \u2208 A ] B\n\n  _\u00d7_ : {l1 : Level} {l2 : Level} \u2192 (Set l1) \u2192 (Set l2) \u2192 Set (lmax l1 l2)\n  A \u00d7 B = \u03a3 A \u03bb _ \u2192 B\n\n  infixr 1 _\u00d7_\n\n  -- equality\n  data _==_ {l : Level} {A : Set l} (M : A) : A \u2192 Set l where\n    refl : M == M\n\n  infixr 9 _==_\n\n  {-# BUILTIN EQUALITY _==_ #-}\n  {-# BUILTIN REFL refl #-}\n\n  -- transitivity of equality\n  _\u00b7_ : {l : Level} {\u03b1 : Set l} {x y z : \u03b1} \u2192 x == y \u2192 y == z \u2192 x == z\n  refl \u00b7 refl = refl\n\n  -- symmetry of equality\n  ! : {l : Level} {\u03b1 : Set l} {x y : \u03b1} \u2192 x == y \u2192 y == x\n  ! refl = refl\n\n  -- ap, in the sense of HoTT, that all functions respect equality in their\n  -- arguments. named in a slightly non-standard way to avoid naming\n  -- clashes with hazelnut constructors.\n  ap1 : {l1 l2 : Level} {\u03b1 : Set l1} {\u03b2 : Set l2} {x y : \u03b1} (F : \u03b1 \u2192 \u03b2)\n          \u2192 x == y \u2192 F x == F y\n  ap1 F refl = refl\n\n  -- transport, in the sense of HoTT, that fibrations respect equality\n  tr : {l1 l2 : Level} {\u03b1 : Set l1} {x y : \u03b1}\n                    (B : \u03b1 \u2192 Set l2)\n                    \u2192 x == y\n                    \u2192 B x\n                    \u2192 B y\n  tr B refl x\u2081 = x\u2081\n\n  -- options\n  data Maybe (A : Set) : Set where\n    Some : A \u2192 Maybe A\n    None : Maybe A\n\n  -- the some constructor is injective. perhaps unsurprisingly.\n  someinj : {A : Set} {x y : A} \u2192 Some x == Some y \u2192 x == y\n  someinj refl = refl\n\n  --  some isn't none.\n  somenotnone : {A : Set} {x : A} \u2192 Some x == None \u2192 \u22a5\n  somenotnone ()\n\n  -- function extensionality, used to reason about contexts as finite\n  -- functions.\n  postulate\n     funext : {A : Set} {B : A \u2192 Set} {f g : (x : A) \u2192 (B x)} \u2192\n              ((x : A) \u2192 f x == g x) \u2192 f == g\n\n  -- non-equality is commutative\n  flip : {A : Set} {x y : A} \u2192 (x == y \u2192 \u22a5) \u2192 (y == x \u2192 \u22a5)\n  flip neq eq = neq (! eq)\n\n  -- two types are said to be equivalent, or isomorphic, if there is a pair\n  -- of functions between them where both round-trips are stable up to ==\n  _\u2243_ : Set \u2192 Set \u2192 Set\n  _\u2243_ A B = \u03a3[ f \u2208 (A \u2192 B) ] \u03a3[ g \u2208 (B \u2192 A) ]\n             (((a : A) \u2192 g (f a) == a) \u00d7 (((b : B) \u2192 f (g b) == b)))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Prelude.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"273956411f14f6d15979001a82eeaff100018e85","subject":"lib\/Data\/Vec\/NP.agda","message":"lib\/Data\/Vec\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Vec\/NP.agda","new_file":"lib\/Data\/Vec\/NP.agda","new_contents":"module Data.Vec.NP where\n\nopen import Data.Vec public\nopen import Data.Nat using (\u2115; suc; zero)\nopen import Data.Fin\nopen import Data.Fin.Props\nopen import Data.Bool\nopen import Function\n\ncount : \u2200 {n a} {A : Set a} \u2192 (A \u2192 Bool) \u2192 Vec A n \u2192 Fin (suc n)\ncount pred [] = zero\ncount pred (x \u2237 xs) = (if pred x then suc else inject\u2081) (count pred xs)\n\nfilter : \u2200 {n a} {A : Set a} (pred : A \u2192 Bool) (xs : Vec A n) \u2192 Vec A (to\u2115 (count pred xs))\nfilter pred [] = []\nfilter pred (x \u2237 xs) with pred x\n... | true  = x \u2237 filter pred xs\n... | false rewrite inject\u2081-lemma (count pred xs) = filter pred xs\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n\n\u03b7\u2032 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7\u2032 {zero}  = \u03bb _ \u2192 []\n\u03b7\u2032 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Vec\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"4c15e4c65beeb080a1ba712b3601b86c01bbdad0","subject":"Add Cipher.ElGamal.Group","message":"Add Cipher.ElGamal.Group\n","repos":"crypto-agda\/crypto-agda","old_file":"Cipher\/ElGamal\/Group.agda","new_file":"Cipher\/ElGamal\/Group.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"error: RPC failed; HTTP 403 curl 22 The requested URL returned error: 403\nfatal: the remote end hung up unexpectedly\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"748a4d24636ae695c8c1bd2cb0b69087549bb4a2","subject":"+prefect-bintree-sorting.agda","message":"+prefect-bintree-sorting.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"prefect-bintree-sorting.agda","new_file":"prefect-bintree-sorting.agda","new_contents":"module prefect-bintree-sorting where\nimport Level as L\nopen import Function.NP\nimport Data.Nat.NP as Nat\nopen Nat using (\u2115; zero; suc; 2^_; _+_; _\u2238_; module \u2115\u00b0; module \u2115\u2264; +-\u2264-inj; \u00acn+\u2264y<n; sucx\u2270x; \u00acn\u2264x<n; module \u2115cmp) renaming (module <= to \u2115<=; _<=_ to _\u2115<=_)\nopen import Data.Bool.NP hiding (to\u2115)\nopen import Data.Sum\nopen import Data.Bits\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082; \u2203; uncurry) renaming (swap to swap-\u00d7)\nopen import Data.Vec.NP using (Vec; _++_; module Alternative-Reverse; shallow-\u03b7)\nopen import Relation.Nullary\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; _\u2257_; module \u2261-Reasoning; [_])\nopen import Algebra.FunctionProperties\nimport Relation.Binary.ToNat as ToNat\nopen import prefect-bintree\nopen import Data.Nat.Properties\nopen Nat using (z\u2264n; s\u2264s; \u2264-pred; 2*_; 2*\u2032_; 2*-spec; 2*\u2032-spec; suc-injective) renaming (_\u2264_ to _\u2115\u2264_; _<_ to _\u2115<_)\n\nmodule MM where\n open Nat using (_\u2264_; _<_)\n split-\u2264 : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2261 y \u228e x < y\n split-\u2264 {zero} {zero} p = inj\u2081 \u2261.refl\n split-\u2264 {zero} {suc y} p = inj\u2082 (s\u2264s z\u2264n)\n split-\u2264 {suc x} {zero} ()\n split-\u2264 {suc x} {suc y} (s\u2264s p) with split-\u2264 {x} {y} p\n ... | inj\u2081 q rewrite q = inj\u2081 \u2261.refl\n ... | inj\u2082 q = inj\u2082 (s\u2264s q)\n\n <\u2192\u2264 : \u2200 {x y} \u2192 x < y \u2192 x \u2264 y\n <\u2192\u2264 (s\u2264s p) = \u2264-steps 1 p\n\n Monotone : \u2115 \u2192 Endo \u2115 \u2192 Set\n Monotone ub f = \u2200 {x y} \u2192 x \u2264 y \u2192 y < ub \u2192 f x \u2264 f y\n\n IsInj : \u2115 \u2192 Endo \u2115 \u2192 Set\n IsInj ub f = \u2200 {x y} \u2192 x < ub \u2192 y < ub \u2192 f x \u2261 f y \u2192 x \u2261 y\n\n Bounded : \u2115 \u2192 Endo \u2115 \u2192 Set\n Bounded ub f = \u2200 x \u2192 x < ub \u2192 f x < ub\n\n module M (f : \u2115 \u2192 \u2115) {ub}\n          (f-mono : Monotone ub f)\n          (f-inj : IsInj ub f)\n          (f-bounded : Bounded ub f) where\n\n  f-mono-< : \u2200 {x y} \u2192 x < y \u2192 y < ub \u2192 f x < f y\n  f-mono-< {x} {y} p y<ub with split-\u2264 (f-mono {x} {y} (<\u2192\u2264 p) y<ub)\n  ... | inj\u2081 q = \u22a5-elim (sucx\u2270x y (\u2261.subst (\u03bb z \u2192 suc z \u2115\u2264 y) (f-inj {x} {y} (<-trans p y<ub) y<ub q) p))\n  ... | inj\u2082 q = q\n\n  -- f-mono-<\u2032 : \u2200 {x} \u2192 suc x < ub \u2192 f x < f (suc x)\n\n  le : \u2200 n \u2192 suc n < ub \u2192 f (suc n) \u2264 n \u2192 f n < n\n  le n 1+n<ub p = \u2115\u2264.trans (f-mono-< {n} {suc n} \u2115\u2264.refl 1+n<ub) p\n\n  bo : \u2200 b \u2192 b < ub \u2192 \u00ac(f b < b)\n  bo zero _ ()\n  bo (suc b) b<ub (s\u2264s p) = bo b (\u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) b<ub) (le b b<ub p)\n\n  fp : \u2200 b \u2192 b < ub \u2192 Bounded (suc b) f \u2192 f b \u2261 b\n  fp b b<ub bub with split-\u2264 (bub b \u2115\u2264.refl)\n  ... | inj\u2081 p = suc-injective p\n  ... | inj\u2082 p = \u22a5-elim (bo b b<ub (\u2264-pred p))\n\n  ob : \u2200 b \u2192 b \u2264 ub \u2192 Bounded b f \u2192 \u2200 x \u2192 x < b \u2192 f x \u2261 x\n  ob zero b\u2264ub bub _ ()\n  ob (suc b) b\u2264ub bub x pf with split-\u2264 pf\n  ... | inj\u2081 p rewrite suc-injective p = fp b b\u2264ub bub\n  ... | inj\u2082 (s\u2264s p) = ob b (<\u2192\u2264 b\u2264ub) ((\u03bb y y<b \u2192 \u2115\u2264.trans (f-mono-< y<b b\u2264ub) (\u2115\u2264.reflexive (fp b b\u2264ub bub)))) x p\n\n  is-id : \u2200 x \u2192 x < ub \u2192 f x \u2261 x\n  is-id = ob ub \u2115\u2264.refl f-bounded\n\nopen BitsSorting\nopen OperationSyntax\nopen PermTreeProof\nopen EvalTree\nopen Alternative-Reverse\n\nmkBijT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij\nmkBijT = Perm.perm SPP.sort-proof\n\nmkBijT\u207b\u00b9 : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Bij\nmkBijT\u207b\u00b9 f = mkBijT f \u207b\u00b9\n\nmkBij : \u2200 {n} \u2192 Endo (Bits n) \u2192 Bij\nmkBij f = mkBijT (fromFun f) \u207b\u00b9\n\nmkBij-p : \u2200 {n} (t : Tree (Bits n) n) \u2192 t \u2261 evalTree (mkBijT t) (sort t)\nmkBij-p t = Perm.proof SPP.sort-proof t\n\nIsInj : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set\nIsInj f = \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n\nidT : \u2200 {n} \u2192 Tree (Bits n) n\nidT = fromFun id\n\ntoFun-idT : \u2200 {n} \u2192 toFun idT \u2257 id {A = Bits n}\ntoFun-idT x = toFun\u2218fromFun id x\n\nInjT : \u2200 {n} \u2192 Tree (Bits n) n \u2192 Set\nInjT = IsInj \u2218 toFun\n\n_\u207b\u00b9-inverseTree : \u2200 {n} f \u2192 evalTree (f `\u204f f \u207b\u00b9) \u2257 id {A = Tree (Bits n) n}\n(f \u207b\u00b9-inverseTree) t = evalTree (f `\u204f f \u207b\u00b9) t\n                     \u2261\u27e8 \u2261.sym (fromFun\u2218toFun (evalTree (f `\u204f f \u207b\u00b9) t)) \u27e9\n                       fromFun (toFun (evalTree (f `\u204f f \u207b\u00b9) t))\n                     \u2261\u27e8 fromFun-\u2257 (evalTree-eval\u2032 (f `\u204f f \u207b\u00b9) t) \u27e9\n                       fromFun (toFun t \u2218 eval (f \u207b\u00b9 \u207b\u00b9 `\u204f f \u207b\u00b9))\n                     \u2261\u27e8 fromFun-\u2257 (\u03bb x \u2192 \u2261.cong (toFun t) (VecBijKit._\u207b\u00b9-inverse\u2032 _ (f \u207b\u00b9) x)) \u27e9\n                       fromFun (toFun t)\n                     \u2261\u27e8 fromFun\u2218toFun t \u27e9\n                       t\n                     \u220e where open \u2261-Reasoning\n\nmkBij-p\u207b\u00b9 : \u2200 {n} (t : Tree (Bits n) n) \u2192 evalTree (mkBijT\u207b\u00b9 t) t \u2261 sort t\nmkBij-p\u207b\u00b9 t = \u2261.trans (\u2261.cong (evalTree (mkBijT\u207b\u00b9 t)) (mkBij-p t)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t))\n\nfoo : \u2200 {n} (t : Tree (Bits n) n) \u2192 toFun t \u2257 toFun (sort t) \u2218 eval (mkBijT\u207b\u00b9 t)\nfoo t xs = toFun t xs\n         \u2261\u27e8 \u2261.cong (\u03bb t \u2192 toFun t xs) (Perm.proof SPP.sort-proof t) \u27e9\n           toFun (evalTree (mkBijT t) (sort t)) xs\n         \u2261\u27e8 evalTree-eval (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t)) xs \u27e9\n           toFun (evalTree (mkBijT\u207b\u00b9 t) (evalTree (mkBijT t) (sort t))) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.refl \u27e9\n           toFun (evalTree (mkBijT t `\u204f mkBijT\u207b\u00b9 t) (sort t)) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 \u2261.cong (\u03bb u \u2192 toFun u (eval (mkBijT\u207b\u00b9 t) xs)) (((mkBijT t) \u207b\u00b9-inverseTree) (sort t)) \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u220e where open \u2261-Reasoning\n\n\nmodule SortedMonotonicFunctionsBits {n} where\n    open SortedMonotonicFunctions (_\u2264_ {n}) isPreorder public\nopen SortedMonotonicFunctionsBits hiding (Sorted)\n\n-- _\u2218-inj_ : \u2200 {a b c} {A : Set a} {B : Set b} {C : Set c} {f : B \u2192 C} {g : A \u2192 B} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\n_\u2218-inj_ : \u2200 {a b c} {f : Bits b \u2192 Bits c} {g : Bits a \u2192 Bits b} \u2192 IsInj f \u2192 IsInj g \u2192 IsInj (f \u2218 g)\nf-inj \u2218-inj g-inj = g-inj \u2218 f-inj\n\n_\u2237-inj : \u2200 {n} b \u2192 IsInj (_\u2237_ {n = n} b)\n(_ \u2237-inj) \u2261.refl = \u2261.refl\n\nlemvec : \u2200 {n} (xs ys : Bits (suc n)) \u2192 head xs \u2261 head ys \u2192 tail xs \u2261 tail ys \u2192 xs \u2261 ys\nlemvec (x \u2237 xs) (.x \u2237 .xs) \u2261.refl \u2261.refl = \u2261.refl\n\n\u2264\u2192\u2264\u1d2e : \u2200 {n} {x y : Bits n} \u2192 x \u2264 y \u2192 x \u2264\u1d2e y\n\u2264\u2192\u2264\u1d2e {x = []}         {[]}         p = []\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {true \u2237 ys}  p = there 1b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} (\u2115<=.complete {to\u2115 xs} {to\u2115 ys} (+-\u2264-inj (2^ n) (\u2115<=.sound _ _ p))))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {false \u2237 ys} p = there 0b (\u2264\u2192\u2264\u1d2e {x = xs} {ys} p)\n\u2264\u2192\u2264\u1d2e {suc n} {true \u2237 xs}  {false \u2237 ys} p = \u22a5-elim (2\u207f+\u2270to\u2115 ys (\u2115<=.sound (2^ n + to\u2115 xs) (to\u2115 ys) p))\n\u2264\u2192\u2264\u1d2e {x = false \u2237 xs} {true \u2237 ys}  p = 0-1 xs ys\n\n\u2264\u1d2e\u2192\u2264 : \u2200 {n} {x y : Bits n} \u2192 x \u2264\u1d2e y \u2192 x \u2264 y\n\u2264\u1d2e\u2192\u2264 [] = _\n\u2264\u1d2e\u2192\u2264 {suc n} (there true p) = \u2115<=.complete (\u2115\u2264.refl {2^ n} +-mono (\u2115<=.sound _ _ (\u2264\u1d2e\u2192\u2264 p)))\n\u2264\u1d2e\u2192\u2264 (there false p) = \u2264\u1d2e\u2192\u2264 p\n\u2264\u1d2e\u2192\u2264 (0-1 p q) = \u2115<=.complete (to\u2115\u22642\u207f+ p {to\u2115 q})\n\nMonotone' : \u2200 {i o} \u2192 (Bits i \u2192 Bits o) \u2192 Set _\nMonotone' {i} {o} f = \u2200 {p q : Bits i} \u2192 p \u2264\u1d2e q \u2192 f p \u2264\u1d2e f q\n\ntoEndo\u2115 : \u2200 {n} \u2192 Endo (Bits n) \u2192 Endo \u2115\ntoEndo\u2115 f = to\u2115 \u2218 f \u2218 from\u2115\n\n\u2264-steps\u2032 : \u2200 {x} y \u2192 x Nat.\u2264 x + y\n\u2264-steps\u2032 {x} y rewrite \u2115\u00b0.+-comm x y = \u2264-steps y \u2115\u2264.refl\n\n<=-steps\u2032 : \u2200 {x} y \u2192 T (x \u2115<= (x + y))\n<=-steps\u2032 {x} y = \u2115<=.complete (\u2264-steps\u2032 {x} y)\n\nopen Nat using (_\u2270_)\n2\u207f\u2270to\u2115 : \u2200 {n} (xs : Bits n) \u2192 2^ n \u2270 to\u2115 xs\n2\u207f\u2270to\u2115 xs p = \u00acn\u2264x<n _ p (to\u2115-bound xs)\n\nTnot2\u207f<=to\u2115 : \u2200 {n} (xs : Bits n) \u2192 T (not (2^ n \u2115<= (to\u2115 xs)))\nTnot2\u207f<=to\u2115 {n} xs with (2^ n) \u2115<= (to\u2115 xs) | \u2261.inspect (_\u2115<=_ (2^ n)) (to\u2115 xs)\n... | true  | [ p ] = 2\u207f\u2270to\u2115 xs (\u2115<=.sound (2^ n) (to\u2115 xs) (\u2261\u2192T p))\n... | false |   _   = _\n\nfrom\u2115\u2218to\u2115 : \u2200 {n} (x : Bits n) \u2192 from\u2115 (to\u2115 x) \u2261 x\nfrom\u2115\u2218to\u2115 [] = \u2261.refl\nfrom\u2115\u2218to\u2115 {suc n} (true \u2237 xs)\n  rewrite T\u2192\u2261 (<=-steps\u2032 {2^ n} (to\u2115 xs))\n        | \u2115\u00b0.+-comm (2^ n) (to\u2115 xs)\n        | m+n\u2238n\u2261m (to\u2115 xs) (2^ n)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\nfrom\u2115\u2218to\u2115 (false \u2237 xs)\n  rewrite Tnot\u2192\u2261 (Tnot2\u207f<=to\u2115 xs)\n        | from\u2115\u2218to\u2115 xs\n        = \u2261.refl\n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 : \u2200 x y \u2192 suc x \u2238 y \u2115\u2264 suc (x \u2238 y)\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x zero = \u2115\u2264.refl\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 zero (suc y) rewrite 0\u2238n\u22610 y = z\u2264n\nsucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 (suc x) (suc y) = sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y\n\n{-\nx\u22642y\u2032\u2192x\u2238y\u2264y : \u2200 x y \u2192 x \u2115\u2264 2*\u2032 y \u2192 x \u2238 y \u2115\u2264 y\nx\u22642y\u2032\u2192x\u2238y\u2264y x zero p = p\nx\u22642y\u2032\u2192x\u2238y\u2264y zero (suc y) p = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc zero) (suc y) (s\u2264s p) rewrite 0\u2238n\u22610 y = z\u2264n\nx\u22642y\u2032\u2192x\u2238y\u2264y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p)) = \u2115\u2264.trans (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y) (s\u2264s (x\u22642y\u2032\u2192x\u2238y\u2264y x y p))\n-}\n\nx<2y\u2032\u2192x\u2238y<y : \u2200 x y \u2192 x \u2115< 2*\u2032 y \u2192 x \u2238 y \u2115< y\nx<2y\u2032\u2192x\u2238y<y x zero p = p\nx<2y\u2032\u2192x\u2238y<y zero (suc y) p = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc zero) (suc y) (s\u2264s (s\u2264s p)) rewrite 0\u2238n\u22610 y = s\u2264s z\u2264n\nx<2y\u2032\u2192x\u2238y<y (suc (suc x)) (suc y) (s\u2264s (s\u2264s p))\n  = \u2115\u2264.trans (s\u2264s (sucx\u2238y\u2264suc\u27e8x\u2238y\u27e9 x y)) (s\u2264s (x<2y\u2032\u2192x\u2238y<y x y p))\n\nx<2y\u2192x\u2238y<y : \u2200 x y \u2192 x \u2115< 2* y \u2192 x \u2238 y \u2115< y\nx<2y\u2192x\u2238y<y x y p rewrite \u2261.sym (2*\u2032-spec y) = x<2y\u2032\u2192x\u2238y<y x y p\n\n\u2270\u2192< : \u2200 x y \u2192 x \u2270 y \u2192 y \u2115< x\n\u2270\u2192< x y p with \u2115cmp.compare (suc y) x\n\u2270\u2192< x y p | tri< a \u00acb \u00acc = \u2115\u2264.trans (s\u2264s (\u2264-step \u2115\u2264.refl)) a\n\u2270\u2192< x y p | tri\u2248 \u00aca b \u00acc = \u2115\u2264.reflexive b\n\u2270\u2192< x y p | tri> \u00aca \u00acb c = \u22a5-elim (p (\u2264-pred c))\n\nnot<=\u2192< : \u2200 x y \u2192 T (not (x \u2115<= y)) \u2192 T (suc y \u2115<= x)\nnot<=\u2192< x y p = \u2115<=.complete (\u2270\u2192< x y (T'not'\u00ac p \u2218 \u2115<=.complete))\n\nto\u2115\u2218from\u2115 : \u2200 {n} x \u2192 x \u2115< 2^ n \u2192 to\u2115 {n} (from\u2115 x) \u2261 x\nto\u2115\u2218from\u2115 {zero} .0 (s\u2264s z\u2264n) = \u2261.refl\nto\u2115\u2218from\u2115 {suc n} x x<2\u207f with 2^ n \u2115<= x | \u2261.inspect (_\u2115<=_ (2^ n)) x\n... | true  | [ p ] rewrite to\u2115\u2218from\u2115 {n} (x \u2238 2^ n) (x<2y\u2192x\u2238y<y x (2^ n) x<2\u207f) = m+n\u2238m\u2261n {2^ n} {x} (\u2115<=.sound (2^ n) x (\u2261\u2192T p))\n... | false | [ p ] = to\u2115\u2218from\u2115 {n} x (\u2115<=.sound (suc x) (2^ n) (not<=\u2192< (2^ n) x (\u2261\u2192Tnot p)))\n\nfrom\u2115-inj : \u2200 {n} {x y : \u2115} \u2192 x \u2115< 2^ n \u2192 y \u2115< 2^ n \u2192 from\u2115 {n} x \u2261 from\u2115 y \u2192 x \u2261 y \nfrom\u2115-inj {n} {x} {y} x< y< fx\u2261fy\n  = x\n  \u2261\u27e8 \u2261.sym (to\u2115\u2218from\u2115 {n} x x<) \u27e9\n    to\u2115 (from\u2115 {n} x)\n  \u2261\u27e8 \u2261.cong to\u2115 fx\u2261fy \u27e9\n    to\u2115 (from\u2115 {n} y)\n  \u2261\u27e8 to\u2115\u2218from\u2115 {n} y y< \u27e9\n    y\n  \u220e where open \u2261-Reasoning\n\nmodule ToEndo\u2115 {n} (f : Endo (Bits n)) where\n    f\u2115 = toEndo\u2115 f\n\n    from\u2115n = from\u2115 {n}\n\n    mono : Monotone f \u2192 MM.Monotone (2^ n) f\u2115\n    mono f-mono {x} {y} x\u2264y y<2\u207f\n       = \u2115<=.sound _ _\n          (f-mono\n            (\u2264\u2192\u2264\u1d2e (\u2115<=.complete {to\u2115 (from\u2115n x)} {to\u2115 (from\u2115n y)}\n                     (\u2115\u2264.trans (\u2115\u2264.reflexive (to\u2115\u2218from\u2115 {n} x (\u2115\u2264.trans (s\u2264s x\u2264y) y<2\u207f)))\n                        (\u2115\u2264.trans x\u2264y (\u2115\u2264.reflexive (\u2261.sym (to\u2115\u2218from\u2115 {n} y y<2\u207f))))))))\n    \n    inj : IsInj f \u2192 MM.IsInj (2^ n) f\u2115\n    inj f-inj {x} {y} x< y< = from\u2115-inj x< y< \u2218 f-inj \u2218 to\u2115-inj _ _\n\n    bounded : MM.Bounded (2^ n) f\u2115\n    bounded x x\u22642\u207f = to\u2115-bound (f (from\u2115 x))\n\nlem : \u2200 {n} (f : Endo (Bits n)) \u2192 Monotone f \u2192 IsInj f \u2192 f \u2257 id\nlem {n} f f-mono f-inj x = to\u2115-inj _ _ (\u2261.trans (\u2261.cong (to\u2115 \u2218 f) (\u2261.sym (from\u2115\u2218to\u2115 x))) (MM.M.is-id (toEndo\u2115 f) {2^ n} (mono f-mono) (inj f-inj) bounded (to\u2115 x) (to\u2115-bound x)))\n  where open ToEndo\u2115 f\n\nIsInj-toFunFromFun : \u2200 {n} (f : Endo (Bits n)) \u2192 IsInj f \u2192 IsInj (toFun (fromFun f))\nIsInj-toFunFromFun f f-inj {x} {y} eq\n  rewrite toFun\u2218fromFun f x\n        | toFun\u2218fromFun f y\n        = f-inj eq\n\nsort-spec\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 Sorted (evalTree (mkBijT\u207b\u00b9 t) t)\nsort-spec\u2032 t rewrite mkBij-p\u207b\u00b9 t = sort-spec t\n\nsort-id\u2032\u2032 : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted t \u2192 toFun t \u2257 id\nsort-id\u2032\u2032 t Pt st x = lem (toFun t) (DataSorted\u2192Sorted st) Pt x\n\ntoFun-eval-inj : \u2200 {n} f \u2192 IsInj (eval f {n})\ntoFun-eval-inj f {x} {y} eq = x\n                            \u2261\u27e8 \u2261.sym ((f \u207b\u00b9-inverse) x) \u27e9\n                              eval (f \u207b\u00b9) (eval f x)\n                            \u2261\u27e8 \u2261.cong (eval (f \u207b\u00b9)) eq \u27e9\n                              eval (f \u207b\u00b9) (eval f y)\n                            \u2261\u27e8 (f \u207b\u00b9-inverse) y \u27e9\n                              y\n                            \u220e where open \u2261-Reasoning\n\ntoFun-evalTree-inj : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 InjT (evalTree f t)\ntoFun-evalTree-inj f t Pt {x} {y} eq\n  rewrite evalTree-eval\u2032 f t x\n        | evalTree-eval\u2032 f t y = toFun-eval-inj (f \u207b\u00b9) (Pt eq)\n\nsort-id\u2032 : \u2200 {n} f (t : Tree (Bits n) n) \u2192 InjT t \u2192 Sorted (evalTree f t) \u2192 toFun (evalTree f t) \u2257 id\nsort-id\u2032 f t Pt = sort-id\u2032\u2032 (evalTree f t) (\u03bb {x} {y} eq \u2192 toFun-evalTree-inj f t Pt eq)\n\nsort-id : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun (sort t) \u2257 id\nsort-id t Pt rewrite \u2261.sym (mkBij-p\u207b\u00b9 t) = sort-id\u2032 (mkBijT\u207b\u00b9 t) t Pt (sort-spec\u2032 t)\n\nbar : \u2200 {n} (t : Tree (Bits n) n) \u2192 InjT t \u2192 toFun t \u2257 eval (mkBijT\u207b\u00b9 t)\nbar t Pt xs = toFun t xs\n         \u2261\u27e8 foo t xs \u27e9\n           toFun (sort t) (eval (mkBijT\u207b\u00b9 t) xs)\n         \u2261\u27e8 sort-id t Pt (eval (mkBijT\u207b\u00b9 t) xs) \u27e9\n           eval (mkBijT\u207b\u00b9 t) xs\n         \u220e where open \u2261-Reasoning\n\nthm : \u2200 {n} (f : Endo (Bits n)) (f-inj : IsInj f) \u2192\n        eval (mkBij f) \u2257 f\nthm f f-inj xs = eval (mkBij f) xs\n               \u2261\u27e8 \u2261.sym (bar (fromFun f) (IsInj-toFunFromFun f f-inj) xs) \u27e9\n                 toFun (fromFun f) xs\n               \u2261\u27e8 toFun\u2218fromFun f xs \u27e9\n                 f xs\n               \u220e where open \u2261-Reasoning\n\nthm# : \u2200 {n} (f : Bits n \u2192 Bit) (g : Endo (Bits n)) (g-inj : IsInj g) \u2192\n       #\u27e8 f \u2218 g \u27e9 \u2261 #\u27e8 f \u27e9\nthm# f g g-inj = #\u27e8 f \u2218 g \u27e9\n               \u2261\u27e8 #-\u2257 (f \u2218 g) (f \u2218 eval (mkBij g)) (\u03bb x \u2192 \u2261.cong f (\u2261.sym (thm g g-inj x))) \u27e9\n                 #\u27e8 f \u2218 eval (mkBij g) \u27e9\n               \u2261\u27e8 #-bij f (mkBij g) \u27e9\n                 #\u27e8 f \u27e9\n               \u220e where open \u2261-Reasoning\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'prefect-bintree-sorting.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"aacc0525f1e3341811285a4adced27b2202ce722","subject":"lib\/Data\/RGB.agda","message":"lib\/Data\/RGB.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/RGB.agda","new_file":"lib\/Data\/RGB.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Data.RGB where\n\nopen import Type\n\ndata RGB : \u2605 where\n  `R `G `B : RGB\n\n[R:_G:_B:_] : \u2200 {\u2113}{C : RGB \u2192 \u2605_ \u2113}(r : C `R)(g : C `G)(b : C `B)(rgb : RGB) \u2192 C rgb\n[R: r G: g B: b ] `R = r\n[R: r G: g B: b ] `G = g\n[R: r G: g B: b ] `B = b\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/RGB.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d5f6beafc85ff125a91c2f035d2d29dd6409f54e","subject":"Add ZK.GroupHom.ElGamal.JS, so far only known-enc-rnd","message":"Add ZK.GroupHom.ElGamal.JS, so far only known-enc-rnd\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom\/ElGamal\/JS.agda","new_file":"ZK\/GroupHom\/ElGamal\/JS.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import FFI.JS.BigI\nopen import Data.Bool.Base using (Bool)\nopen import SynGrp\nopen import ZK.GroupHom.Types\nimport ZK.GroupHom.ElGamal\nimport ZK.GroupHom.JS\n\nmodule ZK.GroupHom.ElGamal.JS (p q : BigI)(g : \u2124[ p ]\u2605) where\n\nmodule EG = ZK.GroupHom.ElGamal `\u2124[ q ]+ `\u2124[ p ]\u2605 _`^_ g\n\nmodule known-enc-rnd (pk : EG.PubKey)\n                     (M  : EG.Message)\n                     (ct : EG.CipherText) where\n  module ker = EG.Known-enc-rnd pk M ct\n  module zkh = `ZK-hom ker.zk-hom\n  module zkp = ZK.GroupHom.JS q zkh.`\u03c6 zkh.y\n  prover-commitment : (r : zkp.Randomness)(w : zkp.Witness) \u2192 zkp.Commitment\n  prover-commitment r w = zkp.Prover-Interaction.get-A (zkp.prover r w)\n  prover-response : (r : zkp.Randomness)(w : zkp.Witness)(c : zkp.Challenge) \u2192 zkp.Response\n  prover-response r w c = zkp.Prover-Interaction.get-f (zkp.prover r w) c\n  verify-transcript : (A : zkp.Commitment)(c : zkp.Challenge)(r : zkp.Response) \u2192 Bool\n  verify-transcript A c r = zkp.verifier (zkp.Transcript.mk A c r)\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/GroupHom\/ElGamal\/JS.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fdd69ff719f4c665bd9878af82c8271f2fbc3086","subject":"refactoring","message":"refactoring\n","repos":"zeitraffer\/agda-cats","old_file":"test\/0-Arrow.agda","new_file":"test\/0-Arrow.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"341f65611f28943e0432541f8aa65c0f74efae69","subject":"Added note about induction principles for N.","message":"Added note about induction principles for N.\n\nIgnore-this: 3f4594794619ba6d27827d2c894512cb\n\ndarcs-hash:20111102230806-3bd4e-52b0658622e9fe2c81ce207a5aef1937ef655764.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/Nat\/InductionPrinciples.agda","new_file":"Draft\/FOTC\/Data\/Nat\/InductionPrinciples.agda","new_contents":"module InductionPrinciples where\n\npostulate\n  D    : Set\n  zero : D\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  zN :               N zero\n  sN : \u2200 {n} \u2192 N n \u2192 N (succ n)\n\n-- The induction principle without the hypothesis N n in the inductive step.\nind\u2081 : Set\u2081\nind\u2081 = (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\n\n-- The induction principle with the hypothesis N n in the inductive step.\nind\u2082 : Set\u2081\nind\u2082 = (P : D \u2192 Set) \u2192\n       P zero \u2192\n       (\u2200 {n} \u2192 N n \u2192 P n \u2192 P (succ n)) \u2192\n       \u2200 {n} \u2192 N n \u2192 P n\n\n-- We couldn't prove ind\u2082 from ind\u2081\n1\u21922 : ind\u2081 \u2192 ind\u2082\n1\u21922 h P P0 is {n} Nn = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/Nat\/InductionPrinciples.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"232c592b1f304c377f3c87e54aeaabbfd786f4ac","subject":"improve FiniteComplete.agda","message":"improve FiniteComplete.agda\n","repos":"zeitraffer\/idris-cats","old_file":"src\/CategoryTheory\/Free\/FiniteComplete.agda","new_file":"src\/CategoryTheory\/Free\/FiniteComplete.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/idris-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"82dea663a7a394bb2bb4ea4a21550e72ed526a47","subject":"guessing-game.agda","message":"guessing-game.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"guessing-game.agda","new_file":"guessing-game.agda","new_contents":"module guessing-game where\n\nopen import Function.NP\nopen import Data.Nat.NP\nopen import Data.Product\nopen import Data.Fin hiding (Ordering; compare; _+_)\nopen import Data.Bool -- hiding (_\u225f_)\n\nrecord Beh (ChalMove AdvMove : Set) : Set where\n  constructor mk\n  field run : AdvMove \u2192 ChalMove \u00d7 Beh ChalMove AdvMove\n\nmodule OrderingTags where\n  data OrderingTag : Set where\n    LT EQ GT : OrderingTag\n\n  tag : \u2200 {m n} \u2192 Ordering m n \u2192 OrderingTag\n  tag (less _ _)    = LT\n  tag (equal _)     = EQ\n  tag (greater _ _) = GT\n\n  cmp : (m n : \u2115) \u2192 OrderingTag\n  cmp m n = tag (compare m n)\n\nopen OrderingTags\n\ndata Trace : Set where\n  say : (lastguess : \u2115) \u2192 Trace\n  ask : (guess : \u2115) (ordering : OrderingTag) \u2192 Trace \u2192 Trace\n\ndata Win answer : Trace \u2192 Set where\n  win  : Win answer (say answer)\n  step : \u2200 {guess trace} \u2192 Win answer trace \u2192 Win answer (ask guess (cmp guess answer) trace)\n\ndata ChalMove : Set where\n  chalStop : ChalMove\n  chalStep : OrderingTag \u2192 ChalMove\n\ndata AdvMove : Set where\n  advStop : (guess : \u2115) \u2192 AdvMove\n  advStep : (guess : \u2115) \u2192 AdvMove\n\nadvGuess : AdvMove \u2192 \u2115\nadvGuess (advStop guess) = guess\nadvGuess (advStep guess) = guess\n\nChal : Set\nChal = Beh ChalMove AdvMove\n\nAdv : Set\nAdv = Beh AdvMove ChalMove\n\nchalBeh : (answer : \u2115) \u2192 Chal\nchalBeh answer = chal where\n  chal : Chal\n  go : AdvMove \u2192 ChalMove \u00d7 Chal\n  go (advStop lastguess) = chalStop , chal\n  go (advStep guess) = chalStep (cmp guess answer) , chal\n  chal = mk go\n\nmiddle : (lo hi : \u2115) \u2192 \u2115\nmiddle lo hi = lo + \u230a hi \u2238 lo \/2\u230b\n\nadvStop' : \u2115 \u2192 AdvMove \u00d7 Adv\nadvStop' g = advStop g , mk (const (advStop' g))\n\nadv : \u2200 (lo g hi : \u2115) \u2192 ChalMove \u2192 AdvMove \u00d7 Adv\n\nadvStep' : \u2200 (lo g hi : \u2115) \u2192 AdvMove \u00d7 Adv\nadvStep' lo g hi = advStep g , mk (adv lo g hi)\n\nadv lo g hi chalStop = advStop' g\nadv lo g hi (chalStep LT) = advStep' g (middle g hi) hi\nadv lo g hi (chalStep EQ) = advStop' g\nadv lo g hi (chalStep GT) = advStep' lo (middle lo g) g\n\nplay : \u2115 \u2192 Chal \u2192 AdvMove \u00d7 Adv \u2192 Trace\nplay (suc n) (mk chal) (advStep g , mk adv) with chal (advStep g)\n... | chalStop , chalNext = say g\n... | chalStep o , chalNext = ask g o (play n chalNext (adv (chalStep o)))\nplay _ _ (advMove , _) = say (advGuess advMove)\n\nplay100 : Fin 100 \u2192 Trace\nplay100 a = play 100 (chalBeh (to\u2115 a)) (advStep' 0 50 100)\n\nex : Trace\nex = play100 (# 5)\n\nex-wining : Win 5 ex\nex-wining = step (step (step (step (step (step (step win))))))\n\n--  winner's-win : \u2200 answer \u2192 Win (to\u2115 answer) (play100 answer)\n-- winner's-win answer = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'guessing-game.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"58847feaa12a3520049784bd18467e45524bee90","subject":"Cipher\/ElGamal\/Generic.agda","message":"Cipher\/ElGamal\/Generic.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Cipher\/ElGamal\/Generic.agda","new_file":"Cipher\/ElGamal\/Generic.agda","new_contents":"open import Type\nopen import Data.Product\n\nmodule Cipher.ElGamal.Generic\n  (Message : \u2605)\n  (\u2124q      : \u2605)\n  (G       : \u2605)\n  (g       : G)\n  (_^_     : G \u2192 \u2124q \u2192 G)\n\n  -- Required for encryption\n  (_\u2219_     : G \u2192 Message \u2192 Message)\n\n  -- Required for decryption\n  (_\/_     : Message \u2192 G \u2192 Message)\nwhere\n\nPubKey     = G\nSecKey     = \u2124q\nCipherText = G \u00d7 Message\nR\u2096         = \u2124q\nR\u2091         = \u2124q\n\nKeyGen : R\u2096 \u2192 PubKey \u00d7 SecKey\nKeyGen x = (g ^ x , x)\n\nEnc : PubKey \u2192 Message \u2192 R\u2091 \u2192 CipherText\nEnc g\u02e3 m y = g\u02b8 , \u03b6 where\n  g\u02b8 = g  ^ y\n  \u03b4  = g\u02e3 ^ y\n  \u03b6  = \u03b4  \u2219 m\n\nDec : SecKey \u2192 CipherText \u2192 Message\nDec x (g\u02b8 , \u03b6) = \u03b6 \/ (g\u02b8 ^ x)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Cipher\/ElGamal\/Generic.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7364caf51fc5f3384f53485937de0b7eb5f1463a","subject":"lib\/Function\/Extensionality.agda","message":"lib\/Function\/Extensionality.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Extensionality.agda","new_file":"lib\/Function\/Extensionality.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule Function.Extensionality where\n\nopen import Relation.Binary.PropositionalEquality\n\npostulate\n    FunExt : Set\n    \u03bb= : \u2200 {a b}{A : Set a}{B : A \u2192 Set b}{f\u2080 f\u2081 : (x : A) \u2192 B x}(f= : \u2200 x \u2192 f\u2080 x \u2261 f\u2081 x){{fe : FunExt}} \u2192 f\u2080 \u2261 f\u2081\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Function\/Extensionality.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ab89b0e94bd9054523612d899cb869aac9573932","subject":"hash-param.agda","message":"hash-param.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"hash-param.agda","new_file":"hash-param.agda","new_contents":"module hash-param where\n\nopen import Type using (\u2605_)\nopen import Function\nopen import Reflection.Param\nopen import Data.List.NP as L renaming (dup to dupL)\nopen import Data.List.Properties.NP\nopen import Data.Zero\nopen import Data.Two\nopen import Data.One\nopen import Data.Product.NP\nopen import Data.Sum.NP\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Fin hiding (_+_; pred)\nimport Data.Vec.NP as V\nopen V using (Vec; []; _\u2237_)\nopen import Algebra.FunctionProperties.Eq\nopen import Relation.Binary.PropositionalEquality.NP\nopen import Relation.Binary\nopen import Relation.Nullary\nopen import Coinduction\n\nmodule _ {A : Set} where\n    tabulate : \u2200 {n} \u2192 (Fin n \u2192 A) \u2192 List A\n    tabulate = V.toList \u2218 V.tabulate\n\nInjective\u2082 : {A B C : Set}(f : A \u2192 B \u2192 C) \u2192 Set\nInjective\u2082 f = \u2200 {x} \u2192 Injective (f x)\n\nmodule PRNG where\n    module PRNG {B D : Set} (H : B \u2192 D) {n} (mkB : Fin n \u2192 B) where\n      prng : List D\n      prng = tabulate (H \u2218 mkB)\n\n    data DPRNG (B : Set) : Set where\n      dH : B \u2192 DPRNG B\n\n    dprng : \u2200 {B : Set} {n} (mkB : Fin n \u2192 B) \u2192 List (DPRNG B)\n    dprng = PRNG.prng dH\n\n    fprng : \u2200 {n} \u2192 List (Fin n)\n    fprng = PRNG.prng id id\n\n    fprng-allFin : \u2200 {n} \u2192 fprng {n} \u2261 V.toList (V.allFin n)\n    fprng-allFin = refl\n\n    module PRNG-Key {Key B D : Set} (H : Key \u2192 B \u2192 D) {n} (mkB : Fin n \u2192 B) where\n      prng : Key \u2192 List D\n      prng k = PRNG.prng (H k) mkB\n\nmodule HMAC where\n  module HMAC-mono\n    {B : Set}\n    (opad\u2295_ ipad\u2295_ : B \u2192 B)\n    (H : List B \u2192 B)\n    where\n    hmac : B \u2192 List B \u2192 B\n    hmac key msg = H (opad\u2295 key \u2237 H (ipad\u2295 key \u2237 msg) \u2237 [])\n\n  module HMAC\n    {Key B D : Set}\n    (opad\u2295_ ipad\u2295_ : Key \u2192 B)\n    (H : List B \u2192 D)\n    (E : D \u2192 List B) -- E as in \"embed\"\n    where\n    hmac : Key \u2192 List B \u2192 D\n    hmac key msg = H (opad\u2295 key \u2237 E (H (ipad\u2295 key \u2237 msg)))\n\n  module HMAC-id\n    {B : Set}\n    where\n    hmac-id : B \u2192 List B \u2192 List B\n    hmac-id = HMAC.hmac id id id id\n    hmac-id-code : hmac-id \u2261 \u03bb key msg \u2192 key \u2237 key \u2237 msg\n    hmac-id-code = refl\n\nmodule Signing\n  {VerifKey SignKey : Set}\n  (verifkey : SignKey \u2192 VerifKey)\n  {D Sig Rsign : Set}\n  (sign     : Rsign \u2192 SignKey \u2192 D \u2192 Sig)\n  (Verify   : VerifKey \u2192 D \u2192 Sig \u2192 Set)\n  (correct  : \u2200 r sk d \u2192 Verify (verifkey sk) d (sign r sk d))\n\n  \n  -- beware\n  (extractor : VerifKey \u2192 (D \u2192 Sig) \u2192 D \u2192 SignKey)\n  (extractor-prop\n            : \u2200 sk (f : D \u2192 Sig) d\n               (let vk = verifkey sk)\n             \u2192 Verify vk d (f d) \u2192 extractor vk f d \u2261 sk)\n\n  where\n\n  module SigH\n    (Msg : Set) (H : Msg \u2192 D) (E : D \u2192 Msg) where\n    signH     : Rsign \u2192 SignKey \u2192 Msg \u2192 Sig\n    signH r sk msg = sign r sk (H msg)\n    VerifyH   : VerifKey \u2192 Msg \u2192 Sig \u2192 Set\n    VerifyH vk msg sig = Verify vk (H msg) sig \n    correctH  : \u2200 r sk msg \u2192 VerifyH (verifkey sk) msg (signH r sk msg)\n    correctH r sk msg = correct r sk (H msg)\n\n    extractorH : VerifKey \u2192 (Msg \u2192 Sig) \u2192 Msg \u2192 SignKey\n    extractorH vk f msg = extractor vk (\u03bb d \u2192 f (E d)) (H msg)\n\n    module TrivialH (EH : \u2200 m \u2192 E (H m) \u2261 m) where\n        extractorH-prop\n                : \u2200 sk (f : Msg \u2192 Sig) d\n                   (let vk = verifkey sk)\n                 \u2192 VerifyH vk d (f d) \u2192 extractorH vk f d \u2261 sk\n        extractorH-prop sk f d vok = extractor-prop sk (f \u2218 E) (H d) (tr (\u03bb z \u2192 Verify (verifkey sk) (H d) (f z)) (! EH d) vok)\n\n  module Sig-id where\n    open SigH D id id\n    open TrivialH (\u03bb _ \u2192 refl)\n\nmodule Signing-data\n  (VerifKey SignKey Rsign : Set)\n  (verifkey : SignKey \u2192 VerifKey)\n  where\n\n  module Sig-abs (D : Set) where\n\n    record Sig : Set where\n      constructor sign\n      field\n        rsign   : Rsign\n        signkey : SignKey\n        d       : D\n\n    Verify : VerifKey \u2192 D \u2192 Sig \u2192 Set\n    Verify vk d (sign r sk d') = vk \u2261 verifkey sk \u00d7 d \u2261 d'\n\n    correct  : \u2200 r sk d \u2192 Verify (verifkey sk) d (sign r sk d)\n    correct r sk d = refl , refl\n\n    extractor : VerifKey \u2192 (D \u2192 Sig) \u2192 D \u2192 SignKey\n    extractor vk f d = Sig.signkey (f d)\n\n    module _\n      (verifkey-inj : Injective verifkey)\n      where\n      extractor-prop\n                : \u2200 sk (f : D \u2192 Sig) d\n                   (let vk = verifkey sk)\n                 \u2192 Verify vk d (f d) \u2192 extractor vk f d \u2261 sk\n      extractor-prop sk f d (proj\u2081 , proj\u2082) = ! verifkey-inj proj\u2081\n\n      open Signing verifkey sign Verify correct extractor extractor-prop public\n\n  module SigH-data\n    {Msg : Set}\n    where\n\n    data D : Set where\n      h : Msg \u2192 D\n\n    e : D \u2192 Msg\n    e (h msg) = msg\n\n    open Sig-abs D\n\n    module _\n        (verifkey-inj : Injective verifkey)\n        where\n        open SigH verifkey-inj Msg h e\n        open TrivialH (\u03bb m \u2192 refl)\n\nmodule _ {A B C D : Set} (f : A \u2192 B) (g : C \u2192 D)\n         (conflict-pres : Conflict f \u2192 Conflict g)\n         (_\u225f_ : Decidable {A = A} _\u2261_)\n         (inj-g : Injective g)\n  where\n  inj-pres : Injective f\n  inj-pres {x} {y} e with x \u225f y\n  ... | yes x\u2261y = x\u2261y\n  ... | no  x\u2262y = \ud835\udfd8-elim (Conflict-\u00acInjective (conflict-pres (x , y , x\u2262y , e)) inj-g)\n\nmodule _ {A B C : Set} (g : B \u2192 C) (f : A \u2192 B)\n         (_\u225f_ : Decidable {A = B} _\u2261_) where\n  private\n    h : A \u2192 C\n    h = g \u2218 f\n\n  conflict-composition : Conflict h \u2192 Conflict g \u228e Conflict f\n  conflict-composition (x , y , x\u2262y , e) with f x \u225f f y\n  ... | yes fx\u2261fy = inr (x , y , x\u2262y , fx\u2261fy)\n  ... | no  fx\u2262fy = inl (f x , f y , fx\u2262fy , e)\n\n  post-inj-pres-conflict : Injective f \u2192 Conflict h \u2192 Conflict g\n  post-inj-pres-conflict f-inj (x , y , x\u2262y , e) = f x , f y , x\u2262y \u2218 f-inj , e\n\n  -- Therefore all of these constructions preserves conflicts on g:\n  --   h(x) = g(k , x)\n  --   h(x) = g(k \u2237 x)\n  --   h(x) = g(k ++ x)\n  --   h(x) = g(k ++ x ++ k')\n  --   h(x) = g(k \u2295 x)\n  --   h(x) = g(x ++ x)\n  --   h(x) = g(x , x)\n\n  pre-inj-pres-conflict : Injective g \u2192 Conflict h \u2192 Conflict f\n  pre-inj-pres-conflict g-inj (x , y , x\u2262y , e) = x , y , x\u2262y , g-inj e\n\n  -- Therefore all of these constructions preserves conflicts on g:\n  --   h(x) = k , g(x)\n  --   h(x) = k \u2237 g(x)\n  --   h(x) = k ++ g(x)\n  --   h(x) = k ++ g(x) ++ k'\n  --   h(x) = k \u2295 g(x)\n\n-- A construct such as H(H(x)) is a particular case of the conflict-composition\nmodule Self-composition\n          {A : Set}\n          (h : A \u2192 A)\n          (_\u225f_ : Decidable {A = A} _\u2261_) where\n  self-comp-pres-conflict : Conflict (h \u2218 h) \u2192 Conflict h\n  self-comp-pres-conflict c = untag (conflict-composition h h _\u225f_ c)\n\nmodule _ {A B : Set} (h : A \u00d7 A \u2192 B) where\n  dupP : A \u2192 A \u00d7 A\n  dupP x = x , x\n\n  dupP-inj : Injective dupP\n  dupP-inj = ap fst\n\n  dupL-inj = dup-inj\n  dupV-inj = dup-inj\n\n-- WRONG\n{-\nmodule _ {A B C : Set} (f : A \u2192 B \u00d7 C) where\n  h = fst \u2218 f\n\n  foo : Conflict h \u2192 Conflict f\n  foo (x , y , i , e) = x , y , i , {!e!}\n  -}\n\nrecord I (B D : Set) : Set where\n  field\n    h   : List D \u2192 D\n    i   : B \u2192 D\n    _\u2295_ : D \u2192 D \u2192 D\n\ndata S (B : Set) : Set where\n  h\u02e2   : List (S B) \u2192 S B\n  i\u02e2   : B \u2192 S B\n  _\u2295\u02e2_ : S B \u2192 S B \u2192 S B\n\nmodule TinyEnc {B D : Set} (imp : I B D) where\n  open I imp\n  enc : (key msg : D) \u2192 D\n  enc key msg = h (key \u2237 []) \u2295 msg\n  dec : (key ct : D) \u2192 D\n  dec = enc\n\n  module _ (\u2295-ok : \u2200 {x y} \u2192 x \u2295 (x \u2295 y) \u2261 y) where\n    dec-enc : \u2200 {key msg} \u2192 dec key (enc key msg) \u2261 msg\n    dec-enc = \u2295-ok\n\n  Msg = D\n  CT  = D\n  Key = D\n\n  module _ (R\u2090 : Set) where \n    Adv = R\u2090 \u2192 Msg \u00b2 \u00d7 (CT \u2192 \ud835\udfda)\n\n    run : \ud835\udfda \u2192 Adv \u2192 (R\u2090 \u00d7 Key) \u2192 \ud835\udfda\n    run \u03b2 adv (ra , key) = let msgs , k = adv ra in k (enc key (msgs \u03b2))\n\n    run' : Adv \u2192 (R\u2090 \u00d7 Key \u00d7 \ud835\udfda) \u2192 \ud835\udfda\n    run' adv (r , key , \u03b2) = run \u03b2 adv (r , key) == \u03b2\n\n    -- Pr[ run 0 adv ] \u2248 Pr[ run 1 adv ]\n\n-- Sigma protocol\nmodule _ (Commit ChalQuery ChalResp Res : Set) where\n  Verifier = Commit \u2192\n            (ChalQuery \u00d7\n             ChalResp \u2192\n            (Res \u00d7 \ud835\udfd9))\n\n  {-\n\ndata Adv (Msg CT : Set) : Set where\n  query : (msg : Msg) (k : (ct : CT) \u2192 Adv Msg CT) \u2192 Adv Msg CT\n  chal  : (msgs : Msg \u00b2) (k : (ct : CT) \u2192 \ud835\udfda) \u2192 Adv Msg CT\nopen import Data.Tree.Binary\n-}\n\ndata T (A : Set) : Set where\n  empty : T A\n  fork  : T A \u2192 T A \u2192 T A\n  leaf  : A \u2192 T A\n\nH : Set\u2081\nH = {B : Set} (c : B \u2192 B \u2192 B)(\u03b5 : B) \u2192 List B \u2192 B\n\nH\u2713 : H \u2192 Set\u2081\nH\u2713 h = \u2200 {B : Set} (c : B \u2192 B \u2192 B) \u03b5\n     \u2192 Injective\u2082 c\n     \u2192 Conflict (h c \u03b5)\n     \u2192 Conflict (uncurry c)\n\nmodule _ {A : Set} {x y : A} {xs ys : List A} where\n  \u2262\u2237 : x List.\u2237 xs \u2262 y \u2237 ys \u2192 x \u2262 y \u228e (x \u2261 y \u00d7 xs \u2262 ys)\n  \u2262\u2237 = {!!}\n\n  {-\nc-inj\u2082 : \u2200 {t u v} \u2192 c t u \u2261 c t v \u2192 u \u2261 v\nc-inj\u2082 refl = refl\n-}\n\nmodule Fold where\n  h : H\n  h = foldr\n\n  module _ {B : Set} (c : B \u2192 B \u2192 B) \u03b5\n           (c-inj : \u2200 {x y z} \u2192 c x y \u2261 c x z \u2192 y \u2261 z) where\n    h\u2713' : \u2200 xs ys(is : xs \u2262 ys)(es : h c \u03b5 xs \u2261 h c \u03b5 ys)\n          \u2192 Conflict (uncurry c)\n\n    h\u2713' [] [] is es = \ud835\udfd8-elim (is refl)\n    h\u2713' [] (x \u2237 ys) is es = {!!}\n    h\u2713' (x \u2237 xs) [] is es = {!!}\n    h\u2713' (x \u2237 xs) (y \u2237 ys) is es with \u2262\u2237 is\n    ... | inl i = (x , h c \u03b5 xs) , (y , h c \u03b5 ys) , (\u03bb p \u2192 i (ap fst p)) , es\n    ... | inr (e , is') = h\u2713' xs ys is' (c-inj (tr (\u03bb z \u2192 c z (h c \u03b5 xs) \u2261 c y (h c \u03b5 ys)) e es))\n\n  h\u2713 : H\u2713 h\n  h\u2713 c \u03b5 c-inj (xs , ys , is , es) = h\u2713' c \u03b5 c-inj xs ys is es\n\n  {-\n_\u2295_ : H \u2192 H \u2192 H\n(h\u2080 \u2295 h\u2081) c xs = c (h\u2080 c xs) (h\u2081 c xs)\n\nH\u209a : H \u2192 Set\u2081\nH\u209a h = (B\u209a : H \u2192 Set)\n       (c\u209a : \u2200 {h\u2080 h\u2081 : H} \u2192 B\u209a h\u2080 \u2192 B\u209a h\u2081 \u2192 B\u209a (h\u2080 \u2295 h\u2081))\n     \u2192 B\u209a h\n\nmodule _ (h : H) (h\u209a : H\u209a h) {B : Set} (c : B \u2192 B \u2192 B)(\u03b5 : B) where\n\n  P\u2295 : \u2200 {h\u2080 h\u2081 : H} \u2192 P h\u2080 \u2192 P h\u2081 \u2192 P (h\u2080 \u2295 h\u2081)\n  P\u2295 h\u2080\u209a h\u2081\u209a ([] , [] , is , es) = \ud835\udfd8-elim (is refl)\n  P\u2295 h\u2080\u209a h\u2081\u209a ([] , x \u2237 ys , is , es) = {!!}\n  P\u2295 h\u2080\u209a h\u2081\u209a (x \u2237 xs , [] , is , es) = {!!}\n  P\u2295 h\u2080\u209a h\u2081\u209a (x \u2237 xs , y \u2237 ys , is , es) with \u2262\u2237 is\n  ... | inl i = x , y , i , {!es!}\n  ... | inr i = {!!}\n\n  ph : P h\n  ph = h\u209a P (\u03bb {x}{y} \u2192 P\u2295 {x} {y})\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'hash-param.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2ce36c584b70907f3e5c451c1871375187e4b4d5","subject":"[ #27 ] Added example.","message":"[ #27 ] Added example.\n\n[ci skip]\n","repos":"asr\/apia,asr\/apia","old_file":"issues\/Issue27.agda","new_file":"issues\/Issue27.agda","new_contents":"\nmodule Issue27 where\n\ninfix  4 _\u2261_\ninfixl 4 _>_ _<_\ninfixr 1 _\u2228_\n\npostulate \u211d : Set\n\n------------------------------------------------------------------------------\n-- Logic stuff\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : \u211d) : \u211d \u2192 Set where\n  refl : x \u2261 x\n\ndata _\u2228_ (A B : Set) : Set where\n  inj\u2081 : A \u2192 A \u2228 B\n  inj\u2082 : B \u2192 A \u2228 B\n\ncase : \u2200 {A B} \u2192 {C : Set} \u2192 (A \u2192 C) \u2192 (B \u2192 C) \u2192 A \u2228 B \u2192 C\ncase f g (inj\u2081 a) = f a\ncase f g (inj\u2082 b) = g b\n\n------------------------------------------------------------------------------\n-- Real numbers stuff\n\npostulate _>_ : \u211d \u2192 \u211d \u2192 Set\n\n_<_ : \u211d \u2192 \u211d \u2192 Set\ny < x = x > y\n{-# ATP definition _<_ #-}\n\npostulate\n  trichotomy : (x y : \u211d) \u2192 (x > y) \u2228 (x \u2261 y) \u2228 (x < y)\n{-# ATP axiom trichotomy #-}\n\n------------------------------------------------------------------------------\n\nrmin : \u211d \u2192 \u211d \u2192 \u211d\nrmin x y = case (\u03bb _ \u2192 y) (\u03bb h \u2192 case (\u03bb _ \u2192 x) (\u03bb _ \u2192 x) h) (trichotomy x y)\n{-# ATP definition rmin #-}\n\npostulate foo : \u2200 x y \u2192 rmin x y \u2261 rmin x y\n{-# ATP prove foo #-}\n\n{-\nError:\n$ apia --check Issue27.agda\napia: tptp4X found an error\/warning in the file \/tmp\/Issue27\/44-foo.fof\nPlease report this as a bug\n\nWARNING: Line 34 Char 159 Token \",\" : Multiple arity symbol n_62__46_8405600604384089447, arity 0 and now 2\nWARNING: Line 34 Char 259 Token \")\" : Multiple arity symbol n_60__48_8405600604384089447, arity 0 and now 2\nWARNING: Line 34 Char 387 Token \",\" : Multiple arity symbol n_60__48_8405600604384089447, arity 0 and now 2\nWARNING: Line 34 Char 460 Token \")\" : Multiple arity symbol case_32_8405600604384089447, arity 5 and now 6\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'issues\/Issue27.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"7ea1b4233060166b71a1d4b0bab34a1838fccde1","subject":"dice.agda","message":"dice.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"dice.agda","new_file":"dice.agda","new_contents":"open import Type\nopen import Search.Type\nopen import Search.Searchable\n\nmodule dice where\n\ndata Dice : \u2605\u2080 where\n  \u2680 \u2681 \u2682 \u2683 \u2684 \u2685 : Dice\n\nsearchDice : \u2200 m \u2192 Search m Dice\nsearchDice m _\u2219_ f = f \u2680 \u2219 (f \u2681 \u2219 (f \u2682 \u2219 (f \u2683 \u2219 (f \u2684 \u2219 f \u2685))))\n\nsearchDice-ind : \u2200 {m} p \u2192 SearchInd p (searchDice m)\nsearchDice-ind p P _\u2219_ f = f \u2680 \u2219 (f \u2681 \u2219 (f \u2682 \u2219 (f \u2683 \u2219 (f \u2684 \u2219 f \u2685))))\n\n\u03bcDice : Searchable Dice\n\u03bcDice = _ , searchDice-ind _\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'dice.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"43cea19e8ab1502790b9b459a58160c801cf7439","subject":"lib\/Relation\/Binary\/ToNat.agda","message":"lib\/Relation\/Binary\/ToNat.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Relation\/Binary\/ToNat.agda","new_file":"lib\/Relation\/Binary\/ToNat.agda","new_contents":"import Data.Nat.NP as \u2115\nopen \u2115 using (\u2115; zero; suc; module \u2115\u2264)\nopen import Data.Bool\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.Sum renaming (map to \u228e-map)\nopen import Function\nopen import Relation.Binary\nopen import Algebra.FunctionProperties\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_)\n\nmodule Relation.Binary.ToNat {a} {A : Set a} (f : A \u2192 \u2115) (f-inj : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y) where\n\n_<=_ : A \u2192 A \u2192 Bool\n_<=_ = \u2115._<=_ on f \n\n_\u2264_ : A \u2192 A \u2192 Set\nx \u2264 y = T (x <= y)\n\n_\u2293_ : A \u2192 A \u2192 A\nx \u2293 y = if x <= y then x else y\n\n_\u2294_ : A \u2192 A \u2192 A\nx \u2294 y = if x <= y then y else x\n\nisPreorder : IsPreorder _\u2261_ _\u2264_\nisPreorder = record { isEquivalence = \u2261.isEquivalence\n                    ; reflexive = reflexive\n                    ; trans = \u03bb {x} {y} {z} \u2192 trans {x} {y} {z} }\n  where open \u2115.<=\n        reflexive : \u2200 {i j} \u2192 i \u2261 j \u2192 i \u2264 j\n        reflexive {i} \u2261.refl = complete (\u2115\u2264.refl {f i})\n        trans : Transitive _\u2264_\n        trans {x} {y} {z} p q = complete (\u2115\u2264.trans (sound (f x) (f y) p) (sound (f y) (f z) q))\n\nisPartialOrder : IsPartialOrder _\u2261_ _\u2264_\nisPartialOrder = record { isPreorder = isPreorder; antisym = antisym }\n  where open \u2115.<= using (sound)\n        antisym : Antisymmetric _\u2261_ _\u2264_\n        antisym {x} {y} p q = f-inj (\u2115\u2264.antisym (sound (f x) (f y) p) (sound (f y) (f x) q))\n\nisTotalOrder : IsTotalOrder _\u2261_ _\u2264_\nisTotalOrder = record { isPartialOrder = isPartialOrder; total = \u03bb x y \u2192 \u228e-map complete complete (\u2115\u2264.total (f x) (f y)) }\n  where open \u2115.<= using (complete)\n\nopen IsTotalOrder isTotalOrder\n\n\u2294-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2294 y \u2261 y\n\u2294-spec {x} {y} p with x <= y\n\u2294-spec _    | true = \u2261.refl\n\u2294-spec ()   | false\n\n\u2293-spec : \u2200 {x y} \u2192 x \u2264 y \u2192 x \u2293 y \u2261 x\n\u2293-spec {x} {y} p with x <= y\n\u2293-spec _    | true = \u2261.refl\n\u2293-spec ()   | false\n\n\u2293-comm : Commutative _\u2261_ _\u2293_\n\u2293-comm x y with x <= y | y <= x | antisym {x} {y} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2294-comm : Commutative _\u2261_ _\u2294_\n\u2294-comm x y with x <= y | y <= x | antisym {y} {x} | total x y\n... | true | true   | p | _ = p _ _\n... | true | false  | _ | _ = \u2261.refl\n... | false | true  | _ | _ = \u2261.refl\n... | false | false | _ | p = \u22a5-elim ([ id , id ] p)\n\n\u2293-\u2264 : \u2200 x y \u2192 (x \u2293 y) \u2264 y\n\u2293-\u2264 x y with total x y\n\u2293-\u2264 x y | inj\u2081 p rewrite \u2293-spec p = p\n\u2293-\u2264 x y | inj\u2082 p rewrite \u2293-comm x y | \u2293-spec p = refl\n\n\u2264-<_,_> : \u2200 {x y z} \u2192 x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y \u2293 z)\n\u2264-<_,_> {x} {y} {z} x\u2264y y\u2264z with total y z\n... | inj\u2081 p rewrite \u2293-spec p = x\u2264y\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = y\u2264z\n\n\u2264-[_,_] : \u2200 {x y z} \u2192 x \u2264 z \u2192 y \u2264 z \u2192 (x \u2294 y) \u2264 z\n\u2264-[_,_] {x} {y} {z} x\u2264z y\u2264z with total x y\n... | inj\u2081 p rewrite \u2294-spec p = y\u2264z\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = x\u2264z\n\n\u2264-\u2293\u2080 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 y\n\u2264-\u2293\u2080 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = id\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = flip trans p\n\n\u2264-\u2293\u2081 : \u2200 {x y z} \u2192 x \u2264 (y \u2293 z) \u2192 x \u2264 z\n\u2264-\u2293\u2081 {x} {y} {z} with total y z\n... | inj\u2081 p rewrite \u2293-spec p = flip trans p\n... | inj\u2082 p rewrite \u2293-comm y z | \u2293-spec p = id\n\n\u2264-\u2294\u2080 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 x \u2264 z\n\u2264-\u2294\u2080 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = trans p\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = id\n\n\u2264-\u2294\u2081 : \u2200 {x y z} \u2192 (x \u2294 y) \u2264 z \u2192 y \u2264 z\n\u2264-\u2294\u2081 {x} {y} {z} with total x y\n... | inj\u2081 p rewrite \u2294-spec p = id\n... | inj\u2082 p rewrite \u2294-comm x y | \u2294-spec p = trans p\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Relation\/Binary\/ToNat.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"7d9724a56d9a9c4dd173a913198fb966ddfe4ee5","subject":"Added possible test case.","message":"Added possible test case.\n\nIgnore-this: 35250bb5e1176e1d4af513edf0ad0b04\n\ndarcs-hash:20110924002242-3bd4e-165ef85e8c9d6cdf2cf5c0df56d74c5f544dee31.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Issues\/Agda365.agda","new_file":"Draft\/Issues\/Agda365.agda","new_contents":"-- From LTC-PCF.Data.Nat.Inequalities.PropertiesATP\n\nmodule Draft.Issues.Agda365 where\n\nopen import LTC-PCF.Base\n\nopen import LTC-PCF.Data.Nat\nopen import LTC-PCF.Data.Nat.Inequalities\nopen import LTC-PCF.Data.Nat.Inequalities.EquationsATP\nopen import LTC-PCF.Data.Nat.PropertiesATP\n\n------------------------------------------------------------------------------\n\n-- After the Agda patch\n\n-- Thu Sep 15 14:15:05 COT 2011  ulfn@chalmers.se\n--  * fixed issue 365: different evaluation behaviour for point-free and pointed style\n\n-- we get the error\n\n-- Translating Draft\/Issues\/Agda365.agda ...\n-- An internal error has occurred. Please report this as a bug.\n-- Location of the error: src\/FOL\/Translation\/Internal\/Terms.hs:516\n--\n-- because we don't translate the Agda internal lambda terms. We fixed\n-- the issue using an abstract block to avoid the reduction (see\n-- LTC-PCF.Data.Nat.Inequalities). I am keeping this example as an\n-- possible test case for the translation of the Agda internal lambda\n-- terms.\n\nx\u2264y\u2192y\u2238x+x\u2261y : \u2200 {m n} \u2192 N m \u2192 N n \u2192 LE m n \u2192 (n \u2238 m) + m \u2261 n\nx\u2264y\u2192y\u2238x+x\u2261y (sN {m} Nm) (sN {n} Nn) Sm\u2264Sn = prf (x\u2264y\u2192y\u2238x+x\u2261y Nm Nn m\u2264n)\n  where\n  postulate m\u2264n : LE m n\n  {-# ATP prove m\u2264n <-SS #-}\n\n  postulate prf : (n \u2238 m) + m \u2261 n \u2192  -- IH.\n                  (succ n \u2238 succ m) + succ m \u2261 succ n\n  -- Metis 2.3 (release 20101019): SZS status Unknown (using timeout 180 sec).\n  {-# ATP prove prf +-comm \u2238-N +-Sx \u2238-SS <-SS #-}\n\n-- This is an incomplete case which does not matter.\nx\u2264y\u2192y\u2238x+x\u2261y {m} {n} Nm Nn m\u2264n = whatever\n  where\n  postulate whatever : n \u2238 m + m \u2261 n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Issues\/Agda365.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"197f470d70bb9ca0d5726b93a2dc20d1481a5110","subject":"Control\/Protocol\/Examples.agda","message":"Control\/Protocol\/Examples.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/Examples.agda","new_file":"Control\/Protocol\/Examples.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP using (_,_)\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.Two hiding (_\u225f_)\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; coe)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nopen import Control.Protocol\nopen import Control.Protocol.Lift\nopen import Control.Protocol.Multiplicative\n\nmodule Control.Protocol.Examples where\n\nmodule MonomorphicSky (P : Proto\u2080) where\n  Cloud : Proto\u2080\n  Cloud = recv \u03bb (t   : \u27e6 P  \u27e7) \u2192\n          recv \u03bb (u   : \u27e6 P \u22a5\u27e7) \u2192\n          send \u03bb (log : Log P)  \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud t u = telecom P t u , _\n\nmodule PolySky where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P : Proto\u2080) \u2192\n          lift\u1d3e \u2081 (MonomorphicSky.Cloud P)\n  cloud : \u27e6 Cloud \u27e7\n  cloud P = lift-proc \u2081 (MonomorphicSky.Cloud P) (MonomorphicSky.cloud P)\n\nmodule PolySky' where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P   : Proto\u2080) \u2192\n          recv \u03bb (t   : Lift \u27e6 P  \u27e7)  \u2192\n          recv \u03bb (u   : Lift \u27e6 P \u22a5\u27e7)  \u2192\n          send \u03bb (log : Lift (Log P)) \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud P (lift t) (lift u) = lift (telecom P t u) , _\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = send\u2610 \u03bb (s : Secret) \u2192\n             recv  \u03bb (g : Guess)  \u2192\n             send  \u03bb (_ : S\u27e8 s \u27e9) \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g [ m ] = g , _\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = send \u03bb (g\u02b3 : G)  \u2192 -- commitment\n                 recv \u03bb (c  : \u2124q) \u2192 -- challenge\n                 send \u03bb (s  : \u2124q) \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- He is honest but its type does not say it\n        -- The definition for \"prover\" below is derived from\n        -- a well typed version.\n        prover' : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover' x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        verifier : (c : \u2124q) \u2192 \u27e6 Verifier \u27e7\n        verifier c _ = c , _\n\n        module _ ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) where\n            Honest-Prover : Proto\n            Honest-Prover\n              = send\u2610 \u03bb (r  : \u2124q)                \u2192 -- ideal commitment\n                send  \u03bb (g\u02b3 : S\u27e8 g ^ r \u27e9)        \u2192 -- real  commitment\n                recv  \u03bb (c  : \u2124q)                \u2192 -- challenge\n                send  \u03bb (s  : S\u27e8 r + (c * x) \u27e9)  \u2192 -- response\n                recv  \u03bb (_  : Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u2192\n                end\n\n            Honest-Verifier : Proto\n            Honest-Verifier = dual Honest-Prover\n\n            honest-verifier' : (c : \u2124q) \u2192 \u27e6 Honest-Verifier \u27e7\n            honest-verifier' c [ r ] g\u02b3 = c , \u03bb s \u2192 _ \u225f _ , _\n\n            sim-honest-prover : \u27e6 Honest-Prover \u22b8 Prover \u27e7\n            sim-honest-prover [ r ] S[ g\u02b3 \u2225 g\u02b3= ] = inr g\u02b3 , \u03bb c \u2192\n                                                    inl c  , \u03bb { S[ s \u2225 s= ] \u2192\n                                                    inr s  ,\n                                                    _ \u225f _  ,\n                                                    end }\n\n            -- The next version is derived from sim-honest-prover\n            honest-prover\u2192prover' : \u27e6 Honest-Prover \u27e7 \u2192 \u27e6 Prover \u27e7\n            honest-prover\u2192prover' ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n            module _ {{_ : FunExt}}{{_ : UA}} where\n                sim-honest-verifier : \u27e6 Verifier \u22b8 Honest-Verifier \u27e7\n                sim-honest-verifier = coe (\u214b-comm Honest-Verifier _) sim-honest-prover\n\n                honest-prover\u2192prover : \u27e6 Honest-Prover \u27e7 \u2192 \u27e6 Prover \u27e7\n                honest-prover\u2192prover = \u22b8-apply {Honest-Prover} sim-honest-prover\n\n                verifier\u2192honest-verifier : \u27e6 Verifier \u27e7 \u2192 \u27e6 Honest-Verifier \u27e7\n                verifier\u2192honest-verifier = \u22b8-apply {Verifier} {Honest-Verifier} sim-honest-verifier\n\n                honest-verifier : (c : \u2124q) \u2192 \u27e6 Honest-Verifier \u27e7\n                honest-verifier c = verifier\u2192honest-verifier (verifier c)\n\n        module _ (x r : \u2124q) where\n            honest-prover : \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n            honest-prover = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n            -- agsy can do it\n\n            module _ {{_ : FunExt}}{{_ : UA}} where\n                prover : \u27e6 Prover \u27e7\n                prover = honest-prover\u2192prover x S[ g ^ x ] honest-prover\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Examples.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a3fdcdabaeac3da8c2d8207a14d3002194e23eab","subject":"refactor","message":"refactor\n","repos":"zeitraffer\/agda-cats","old_file":"src\/CategoryTheory\/Initial\/Trade-off\/FiniteComplete-Module.agda","new_file":"src\/CategoryTheory\/Initial\/Trade-off\/FiniteComplete-Module.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"2a0b652632c8447e5cf3ed5e8485fdde4a21a7d7","subject":"Testing \u2248\u2192\u2261.","message":"Testing \u2248\u2192\u2261.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda","new_file":"notes\/FOT\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties for the bisimilarity relation\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n-- {-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Relation.Binary.Bisimilarity.PropertiesI where\n\nopen import Common.FOL.Relation.Binary.EqReasoning\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.List.PropertiesI\nopen import FOTC.Relation.Binary.Bisimilarity\n\n------------------------------------------------------------------------------\n\n{-# NO_TERMINATION_CHECK #-}\n\u2248\u2192\u2261\u2081 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 xs \u2261 ys\n\u2248\u2192\u2261\u2081 {xs} {ys} xs\u2248ys with (\u2248-unf xs\u2248ys)\n... | x' , xs' , ys' , h\u2081 , h\u2082 , h\u2083 =\n  xs       \u2261\u27e8 h\u2082 \u27e9\n  x' \u2237 xs' \u2261\u27e8 \u2237-rightCong (\u2248\u2192\u2261\u2081 h\u2081) \u27e9\n  x' \u2237 ys' \u2261\u27e8 sym h\u2083 \u27e9\n  ys       \u220e\n\n-- Requires K.\n{-# NO_TERMINATION_CHECK #-}\n\u2248\u2192\u2261\u2082 : \u2200 {xs ys} \u2192 xs \u2248 ys \u2192 xs \u2261 ys\n\u2248\u2192\u2261\u2082 xs\u2248ys with (\u2248-unf xs\u2248ys)\n... | x' , xs' , ys' , h , refl , refl = \u2237-rightCong (\u2248\u2192\u2261\u2082 h)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Relation\/Binary\/Bisimilarity\/PropertiesI.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"760afaab3588d47c2605736d48230f8ae07d6db4","subject":"Renaming of FiniteField.JS","message":"Renaming of FiniteField.JS\n","repos":"crypto-agda\/agda-libjs","old_file":"lib\/FFI\/JS\/BigI\/FiniteField.agda","new_file":"lib\/FFI\/JS\/BigI\/FiniteField.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import FFI.JS using (\ud835\udfd9; Number; Bool; true; false; String; warn-check; check; trace-call; _++_)\nopen import FFI.JS.BigI\nopen import Data.List.Base hiding (sum; _++_)\n{-\nopen import Algebra.Raw\nopen import Algebra.Field\n-}\n\nmodule FFI.JS.BigI.FiniteField (q : BigI) where\n\nabstract\n  \ud835\udd3d : Set\n  \ud835\udd3d = BigI\n\n  private\n    mod-q : BigI \u2192 \ud835\udd3d\n    mod-q x = mod x q\n\n    check' : {A : Set}(pred : Bool)(errmsg : \ud835\udfd9 \u2192 String)(input : A) \u2192 A\n    -- check' = warn-check\n    check' = check\n\n  -- There is two ways to go from BigI to \u2124q: check and mod-q\n  -- Use check for untrusted input data and mod-q for internal\n  -- computation.\n  fromBigI : BigI \u2192 \ud835\udd3d\n  fromBigI = -- trace-call \"BigI\u25b9\u2124q \"\n    \u03bb x \u2192\n      (check' (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: q:\" ++ toString q ++ \" is less than x:\" ++ toString x)\n         (check' (x \u2265I 0I)\n            (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") x))\n\n  check-non-zero : \ud835\udd3d \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \ud835\udd3d \u2192 BigI\n  repr x = x\n\n  0# 1# : \ud835\udd3d\n  0# = 0I\n  1# = 1I\n\n  1\/_ : Op\u2081 \ud835\udd3d\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : Op\u2082 \ud835\udd3d\n  x ^ y = modPow (repr x) (repr y) q\n\n_+_ _\u2212_ _*_ _\/_ : Op\u2082 \ud835\udd3d\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = mod-q (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\n0\u2212_ : Op\u2081 \ud835\udd3d\n0\u2212 x = mod-q (negate (repr x))\n\n_==_ : (x y : \ud835\udd3d) \u2192 Bool\nx == y = equals (repr x) (repr y)\n\nsum prod : List \ud835\udd3d \u2192 \ud835\udd3d\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\n{-\n+-mon-ops : Monoid-Ops \ud835\udd3d\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \ud835\udd3d\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \ud835\udd3d\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \ud835\udd3d\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \ud835\udd3d\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \ud835\udd3d\nfld = fld-ops , fld-struct\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/FFI\/JS\/BigI\/FiniteField.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"26e18d972c15d9a583d5509a3ff2a2bd712b7773","subject":"infix priority","message":"infix priority\n","repos":"zeitraffer\/agda-cats","old_file":"src\/InitSmart\/MonoidModule.agda","new_file":"src\/InitSmart\/MonoidModule.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/zeitraffer\/agda-cats.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"88cceee36f26f4d62aaed48ad44239d906af023b","subject":"an equality module for Sets","message":"an equality module for Sets\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/Equality.agda","new_file":"agda\/Equality.agda","new_contents":"module Equality where\n\ndata _\u2261_  {A : Set} : (a b : A) \u2192 Set where\n  definition : {x : A} \u2192 x \u2261 x\n\nsym : {A : Set} {a b : A} \u2192 a \u2261 b \u2192 b \u2261 a\nsym definition = definition\n\ntrans : {A : Set} {a b c : A} \u2192 a \u2261 b \u2192 b \u2261 c \u2192 a \u2261 c\ntrans pAB definition = pAB\n-- trans definition pBC = pBC\n\ncong : {A B : Set} {a\u2080 a\u2081 : A} \u2192 (f : A \u2192 B) \u2192 a\u2080 \u2261 a\u2081 \u2192 f a\u2080 \u2261 f a\u2081\ncong f definition = definition\n\nbegin : {A : Set} (a : A) \u2192  a \u2261 a\nbegin a = definition\n\n\n_\u2248_by_ : {A : Set} {a b : A} \u2192 a \u2261 b \u2192 (c : A) \u2192 b \u2261 c \u2192 a \u2261 c\naEb \u2248 c by bEc = trans aEb bEc\n\n\n_\u220e : {A : Set} (a : A) \u2192 A\nx \u220e = x\n\ninfixl 1 _\u2248_by_\ninfixl 1 _\u2261_\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/Equality.agda' did not match any file(s) known to git\n","license":"unlicense","lang":"Agda"}
{"commit":"5ba183bf800bf31f0ee6424745440f638079b5f4","subject":"Control\/Protocol\/MultiParty.agda","message":"Control\/Protocol\/MultiParty.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/MultiParty.agda","new_file":"Control\/Protocol\/MultiParty.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import Type\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd) using (\u03a3;_,_)\nopen import Data.Zero\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two hiding (_\u225f_; nand)\nopen Data.Two.Indexed\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; !_; _\u2219_; refl; subst; ap; coe; coe!)\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\nopen import Control.Protocol\n\nmodule Control.Protocol.MultiParty where\n\nmodule _ (I : \u2605) where\n    data MProto : \u2605\u2081 where\n      _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 MProto) \u2192 MProto\n      _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 MProto) \u2192 MProto\n      end       : MProto\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I)(M : \u2605)(\u2102 : ..(m : M) \u2192 MProto I) \u2192 MProto I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    Group = I \u2192 \ud835\udfda\n\n    _\/_ : (\u2102 : MProto I) (\u03c6 : Group) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/ \u03c6 = case \u03c6 A\n                               0: case \u03c6 B\n                                    0: recv (\u03bb { [ m ] \u2192 \u2102 m \/ \u03c6 })\n                                    1: recv (\u03bb     m   \u2192 \u2102 m \/ \u03c6)\n                               1: send (\u03bb m \u2192 \u2102 m \/ \u03c6)\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/ \u03c6 = com (case \u03c6 A 0: In 1: Out) \u03bb m \u2192 \u2102 m \/ \u03c6\n    end               \/ \u03c6 = end\n\n    empty-group : Group\n    empty-group _ = 0\u2082\n\n    full-group : Group\n    full-group _ = 1\u2082\n\n    -- MultiParty Obs(erver)\n    MObs : MProto I \u2192 Proto\n    MObs \u2102 = \u2102 \/ empty-group\n\n    -- MultiParty Log\n    MLog : MProto I \u2192 Proto\n    MLog \u2102 = \u2102 \/ full-group\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (MLog \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = send' \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = send' \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (MObs \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = recv' \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = recv' \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data Nand : (x y : \ud835\udfda) \u2192 \u2605 where\n      n01 : Nand 0\u2082 1\u2082\n      n10 : Nand 1\u2082 0\u2082\n      n00 : Nand 0\u2082 0\u2082\n\n    or : \u2200 {x y} \u2192 Nand x y \u2192 \ud835\udfda\n    or n01 = 1\u2082\n    or n10 = 1\u2082\n    or n00 = 0\u2082\n\n    Nand\u00b0 : (\u03c6 \u03c8 : Group) \u2192 \u2605\n    Nand\u00b0 \u03c6 \u03c8 = \u2200 i \u2192 Nand (\u03c6 i) (\u03c8 i)\n\n    or\u00b0 : \u2200 {\u03c6 \u03c8} \u2192 Nand\u00b0 \u03c6 \u03c8 \u2192 Group\n    or\u00b0 r\u00b0 = or \u2218 r\u00b0\n\n    data R : (x y z : \ud835\udfda) \u2192 \u2605 where\n      r01 : R 0\u2082 1\u2082 1\u2082\n      r10 : R 1\u2082 0\u2082 1\u2082\n      r00 : R 0\u2082 0\u2082 0\u2082\n\n    R\u00b0 : (\u03c6 \u03c8 \u03be : Group) \u2192 \u2605\n    R\u00b0 \u03c6 \u03c8 \u03be = \u2200 i \u2192 R (\u03c6 i) (\u03c8 i) (\u03be i)\n\n    Nand-R : \u2200 {x y z} \u2192 R x y z \u2192 \u03a3 (Nand x y) (_\u2261_ z \u2218 or) \n    Nand-R r01 = n01 , refl\n    Nand-R r10 = n10 , refl\n    Nand-R r00 = n00 , refl\n\n    Nand-R\u00b0 : \u2200 {x y z}{{_ : FunExt}} \u2192 R\u00b0 x y z \u2192 \u03a3 (Nand\u00b0 x y) (_\u2261_ z \u2218 or\u00b0)\n    Nand-R\u00b0 p = fst \u2218 Nand-R \u2218 p , \u03bb= (\u03bb i \u2192 snd (Nand-R (p i)))\n\n    module _ {p q : I \u2192 \ud835\udfda}(pq : Nand\u00b0 p q) where\n        group-merge : (\u2102 : MProto I) \u2192 \u27e6 \u2102 \/ p \u27e7 \u2192 \u27e6 \u2102 \/ q \u27e7 \u2192 \u27e6 \u2102 \/ or\u00b0 pq \u27e7\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) \u2102p \u2102q with p A | q A | pq A | p B | q B | pq B\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | n01 |  1\u2082 | .0\u2082 | n10 = m , group-merge (\u2102 m) (\u2102p m) \u2102q\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | n01 |  0\u2082 |  _  | _   = m , group-merge (\u2102 m) (\u2102p [ m ]) \u2102q\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | n10 | .0\u2082 |  1\u2082 | n01 = m , group-merge (\u2102 m) \u2102p (\u2102q m)\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | n10 |  _  |  0\u2082 | _   = m , group-merge (\u2102 m) \u2102p (\u2102q [ m ])\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) \u2102p \u2102q       | .0\u2082 |  0\u2082 | n00 | .0\u2082 |  1\u2082 | n01 = \u03bb m \u2192 group-merge (\u2102 m) (\u2102p [ m ]) (\u2102q m)\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) \u2102p \u2102q       | .0\u2082 |  0\u2082 | n00 |  1\u2082 | .0\u2082 | n10 = \u03bb m \u2192 group-merge (\u2102 m) (\u2102p m) (\u2102q [ m ])\n        group-merge (A -[ M ]\u2192 B \u204f \u2102) \u2102p \u2102q       | .0\u2082 |  0\u2082 | n00 |  0\u2082 |  0\u2082 | n00 = \u03bb { [ m ] \u2192 group-merge (\u2102 m) (\u2102p [ m ]) (\u2102q [ m ]) }\n        group-merge (A -[ M ]\u2192\u2605\u204f   \u2102) \u2102p \u2102q with p A | q A | pq A\n        group-merge (A -[ M ]\u2192\u2605\u204f   \u2102) \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | n01 = m , group-merge (\u2102 m) (\u2102p m) \u2102q\n        group-merge (A -[ M ]\u2192\u2605\u204f   \u2102) (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | n10 = m , group-merge (\u2102 m) \u2102p (\u2102q m)\n        group-merge (A -[ M ]\u2192\u2605\u204f   \u2102) \u2102p \u2102q       | .0\u2082 |  0\u2082 | n00 = \u03bb m \u2192 group-merge (\u2102 m) (\u2102p m) (\u2102q m)\n        group-merge end \u2102p \u2102q = _\n\n    module _ {p q r : I \u2192 \ud835\udfda}(pqr : R\u00b0 p q r){{_ : FunExt}} where\n        group-merge' : (\u2102 : MProto I) \u2192 \u27e6 \u2102 \/ p \u27e7 \u2192 \u27e6 \u2102 \/ q \u27e7 \u2192 \u27e6 \u2102 \/ r \u27e7\n        group-merge' \u2102 p q with Nand-R\u00b0 pqr\n        ... | z , e = subst (\u27e6_\u27e7 \u2218 _\/_ \u2102) (! e) (group-merge z \u2102 p q)\n\n        {-\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : MProto I)(Rqr : Nand\u00b0 q r)(Rpq : Nand\u00b0 p q)(Rpqr : Nand\u00b0 p _)(Rpqr' : Nand\u00b0 _ r) \u2192\n                             (\u2102p : \u27e6 \u2102 \/ p \u27e7) (\u2102q : \u27e6 \u2102 \/ q \u27e7) (\u2102r : \u27e6 \u2102 \/ r \u27e7)\n                             \u2192 group-merge Rpqr \u2102 \u2102p (group-merge Rqr \u2102 \u2102q \u2102r)\n                             \u2261 subst (\u03bb x \u2192 \u27e6 \u2102 \/ nand\u00b0 x \u27e7) {!!}\n                               (group-merge Rpqr' \u2102 (group-merge Rpq \u2102 \u2102p \u2102q) \u2102r)\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (\u03c6 : I \u2192 \ud835\udfda) i \u2192 R (\u03c6 i) (not (\u03c6 i)) 1\u2082\n    R-p-\u00acp-1 \u03c6 i with \u03c6 i\n    R-p-\u00acp-1 \u03c6 i | 1\u2082 = r10\n    R-p-\u00acp-1 \u03c6 i | 0\u2082 = r01\n\n    module _ {{_ : FunExt}} where\n        choreo-bi : {\u03c6 : I \u2192 \ud835\udfda}(\u2102 : MProto I) \u2192 \u27e6 \u2102 \/ \u03c6 \u27e7 \u2192 \u27e6 \u2102 \/ (not \u2218 \u03c6) \u27e7 \u2192 \u27e6 MLog \u2102 \u27e7\n        choreo-bi {\u03c6} \u2102 \u2102p \u2102\u00acp = group-merge' (R-p-\u00acp-1 \u03c6) \u2102 \u2102p \u2102\u00acp\n\nmodule _ {{_ : FunExt}} where\n    choreo2 : (\u2102 : MProto \ud835\udfda) \u2192 \u27e6 \u2102 \/ id \u27e7 \u2192 \u27e6 \u2102 \/ not \u27e7 \u2192 \u27e6 MLog \u2102 \u27e7\n    choreo2 = choreo-bi\n\nmodule ThreeParty where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  R-1-2-\u00ac0 : R\u00b0 1\u2083? 2\u2083? (not \u2218 0\u2083?)\n  R-1-2-\u00ac0 0\u2083 = r00\n  R-1-2-\u00ac0 1\u2083 = r10\n  R-1-2-\u00ac0 2\u2083 = r01\n\n  {- While this does not scale -}\n  3com : (\u2102 : MProto \ud835\udfdb) \u2192 \u27e6 \u2102 \/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/ 2\u2083? \u27e7 \u2192 \u27e6 MLog \u2102 \u27e7\n  3com (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , 3com (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  3com (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , 3com (\u2102 m) p0 (p1 m) (p2 [ m ])\n  3com (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , 3com (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  3com (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , 3com (\u2102 m) (p0 m) p1 (p2 [ m ])\n  3com (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , 3com (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  3com (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , 3com (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  3com (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , 3com (\u2102 m) (p0 m) (p1 [ m ]) p2\n  3com (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , 3com (\u2102 m) (p0 [ m ]) (p1 m) p2\n  3com (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , 3com (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  3com (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , 3com (\u2102 m) p0 (p1 m) (p2 m)\n  3com (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , 3com (\u2102 m) (p0 m) p1 (p2 m)\n  3com (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , 3com (\u2102 m) (p0 m) (p1 m) p2\n  3com end p0 p1 p2 = _\n\n  {- This does scale -}\n  module _ {{_ : FunExt}} where\n    3com' : (\u2102 : MProto \ud835\udfdb) \u2192 \u27e6 \u2102 \/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/ 2\u2083? \u27e7 \u2192 \u27e6 MLog \u2102 \u27e7\n    3com' \u2102 p0 p1 p2 = group-merge' (R-p-\u00acp-1 0\u2083?) \u2102 p0 (group-merge' R-1-2-\u00ac0 \u2102 p1 p2)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/MultiParty.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"1ffd391240cdf755b930cf4216a8231b81aee07d","subject":"Add FunUniverse\/Fin\/Op\/Abstract.agda","message":"Add FunUniverse\/Fin\/Op\/Abstract.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Fin\/Op\/Abstract.agda","new_file":"FunUniverse\/Fin\/Op\/Abstract.agda","new_contents":"open import Type\nopen import Function\nopen import Data.Product using (proj\u2082)\nopen import Data.Two renaming (mux to mux\u2082)\nopen import Data.Nat.NP using (\u2115; zero; suc; _+_; _*_; module \u2115\u00b0)\nopen import Data.Fin.NP as Fin using (Fin; inject+; raise; bound; free; zero; suc)\nopen import Data.Vec.NP using (Vec; []; _\u2237_; lookup; tabulate)\nopen import Data.Vec.Properties using (lookup\u2218tabulate)\nopen import Data.Bits using (Bits; _\u2192\u1d47_; RewireTbl{-; 0\u207f; 1\u207f-})\nopen import Relation.Binary.PropositionalEquality\n\nopen import FunUniverse.Data\nopen import FunUniverse.Core\nopen import FunUniverse.Fin.Op\nopen import FunUniverse.Rewiring.Linear\nopen import FunUniverse.Category\nopen import Language.Simple.Interface\n\nmodule FunUniverse.Fin.Op.Abstract where\n\nprivate\n  module BF = Rewiring finOpRewiring\n\nmodule _ {E : \u2605 \u2192 \u2605} {Op} (lang : Lang Op \ud835\udfda E) where\n    open Lang lang\n\n    {-\n    bits : \u2200 {o} \u2192 Bits o \u2192 0 \u2192\u1d49 o\n    bits xs = \ud835\udfda\u25b9E \u2218 flip Vec.lookup xs\n    -}\n\n    efinFunU : FunUniverse \u2115\n    efinFunU = Bits-U , _\u2192\u1d49_\n\n    module EFinFunUniverse = FunUniverse efinFunU\n\n    efinCat : Category _\u2192\u1d49_\n    efinCat = input , _\u2218\u1d49_\n\n    private\n      open EFinFunUniverse\n      module _ {A B C D} (f : A `\u2192 C) (g : B `\u2192 D) where\n        <_\u00d7_>  : (A `\u00d7 B) `\u2192 (C `\u00d7 D)\n        <_\u00d7_> x with Fin.cmp C D x\n        <_\u00d7_> ._ | Fin.bound y = inject+ B <$> f y\n        <_\u00d7_> ._ | Fin.free  y = raise   A <$> g y\n\n    efinLin : LinRewiring efinFunU\n    efinLin = mk efinCat (flip <_\u00d7_> input)\n                         (\u03bb {A} \u2192 input \u2218 BF.swap {A})\n                         (\u03bb {A} \u2192 input \u2218 BF.assoc {A})\n                         input input <_\u00d7_> (\u03bb {A} \u2192 <_\u00d7_> {A} input)\n                         input input input input\n\n    efinRewiring : Rewiring efinFunU\n    efinRewiring = mk efinLin (\u03bb()) (input \u2218 BF.dup) (\u03bb()) (\u03bb f g \u2192 < f \u00d7 g > \u2218\u1d49 (input \u2218 BF.dup))\n                      (input \u2218 inject+ _) (\u03bb {A} \u2192 input \u2218 raise A)\n                      (\u03bb f \u2192 input \u2218 BF.rewire f) (\u03bb f \u2192 input \u2218 BF.rewireTbl f)\n\n    {-\n    efinFork : HasFork efinFunU\n    efinFork = cond , \u03bb f g \u2192 second {1} < f , g > \u204f cond\n      where\n        open Rewiring efinRewiring\n        cond : \u2200 {A} \u2192 `Bit `\u00d7 A `\u00d7 A `\u2192 A\n        cond {A} x = {!mux (return zero) (input (raise 1 (inject+ _ x))) (input (raise 1 (raise A x)))!}\n\n    \ud835\udfda\u25b9E = {!!}\n    efinOps : FunOps efinFunU\n    efinOps = mk efinRewiring efinFork (const (\ud835\udfda\u25b9E 0\u2082)) (const (\ud835\udfda\u25b9E 1\u2082))\n    open Cong-*1 _\u2192\u1d49_\n\n    reify : \u2200 {i o} \u2192 i \u2192\u1d47 o \u2192 i \u2192\u1d49 o\n    reify = cong-*1\u2032 \u2218 FunOps.fromBitsFun efinOps\n    -}\n\n      {-\n    eval-reify : eval (reify f)\n    eval-reify = ?\n    -}\n\n    -- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FunUniverse\/Fin\/Op\/Abstract.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3bccc4d031195db6666b0688d6497ecfeebc81ec","subject":"Reflection.Scoped: a subset of the reflection API, scoped using nested data types","message":"Reflection.Scoped: a subset of the reflection API, scoped using nested data types\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Reflection\/Scoped.agda","new_file":"lib\/Reflection\/Scoped.agda","new_contents":"open import Reflection.NP using (Name; Arg; arg; Arg-info; arg-info; Literal; Visibility; visible; hidden; Relevance; relevant; irrelevant; Pattern; arg\u1d5b\u02b3; arg\u02b0\u02b3)\nopen import Level.NP\nopen import Data.List\nopen import Data.Nat renaming (_\u2294_ to _\u2294\u2115_)\nopen import Data.Sum.NP hiding (map)\nopen import Data.One\nopen import Data.Maybe.NP\nopen import Opaque\n\nmodule Reflection.Scoped where\n \nScope : Set \u2192 (Set \u2192 Set) \u2192 Set \u2192 Set\nScope B F A = F (A \u228e B)\n\nmutual\n  data Term (A : Set) : Set where\n    -- Variable applied to arguments.\n    var     : (x : A) (args : List (Arg (Term A))) \u2192 Term A\n    -- Constructor applied to arguments.\n    con     : (c : Name) (args : List (Arg (Term A))) \u2192 Term A\n    -- Identifier applied to arguments.\n    def     : (f : Name) (args : List (Arg (Term A))) \u2192 Term A\n    -- Different kinds of \u03bb-abstraction.\n    lam     : (v : Visibility) (t : Scope \ud835\udfd9 Term A) \u2192 Term A\n    -- Pattern matching \u03bb-abstraction\n    -- pat-lam : (cs : List (Clause A)) (args : List (Arg (Term A))) \u2192 Term A\n    -- Pi-type.\n    pi      : (t\u2081 : Arg (Type A)) (t\u2082 : Scope \ud835\udfd9 Type A) \u2192 Term A\n    -- A sort.\n    sort    : Sort A \u2192 Term A\n    -- A literal.\n    lit     : Literal \u2192 Term A\n    -- Anything else.\n    unknown : Term A\n\n  data Type (A : Set) : Set where\n    el : (s : Sort A) (t : Term A) \u2192 Type A\n\n  data Sort (A : Set) : Set where\n    -- A Set of a given (possibly neutral) level.\n    set     : (t : Term A) \u2192 Sort A\n    -- A Set of a given concrete level.\n    lit     : (n : \u2115) \u2192 Sort A\n    -- Anything else.\n    unknown : Sort A\n\n    {-\n  data Clause (A : Set) : Set where\n    clause : (ps : List (Arg Pattern)) \u2192 Scope (BoundPats ps) Term A \u2192 Clause A\n    absurd-clause : List (Arg Pattern) \u2192 Clause A\n    -}\n\nArgs : Set \u2192 Set\nArgs A = List (Arg (Term A))\n\nunEl : \u2200 {A} \u2192 Type A \u2192 Term A\nunEl (el _ tm) = tm\n\ngetSort : \u2200 {A} \u2192 Type A \u2192 Sort A\ngetSort (el s _) = s\n\nunArg : \u2200 {A} \u2192 Arg A \u2192 A\nunArg (arg _ a) = a\n\n-- sort\u2080 : Sort\npattern sort\u2080 = lit 0\n\n-- sort\u2081 : Sort\npattern sort\u2081 = lit 1\n\n-- `\u2605_ : \u2115 \u2192 Term\npattern  `\u2605_ i = sort (lit i)\n\n-- `\u2605\u2080 : Term\npattern `\u2605\u2080 = `\u2605_ 0\n\n-- el\u2080 : Term \u2192 Type\npattern el\u2080 t = el sort\u2080 t\n\n-- Builds a type variable (of type \u2605\u2080)\n{-\n``var\u2080 : \u2115 \u2192 Args \u2192 Type\n``var\u2080 n args = el\u2080 (var n args)\n-}\n\n-- ``set : \u2115 \u2192 \u2115 \u2192 Type\npattern ``set i j = el (lit (suc j)) (`\u2605_ i)\n\n``\u2605_ : \u2200 {A} \u2192 \u2115 \u2192 Type A\n``\u2605_ i = el (lit (suc i)) (`\u2605_ i)\n\n-- ``\u2605\u2080 : Type\npattern ``\u2605\u2080 = ``set 0 0\n\ndecode-Sort : \u2200 {A} \u2192 Sort A \u2192 Maybe \u2115\n--decode-Sort `set\u2080 = just zero\n--decode-Sort (`set\u209b_ s) = map? suc (decode-Sort (set s))\n--decode-Sort (`set_ s) = decode-Sort s\ndecode-Sort (set _) = nothing\ndecode-Sort (lit n) = just n\ndecode-Sort unknown = nothing\n\npattern _`\u2294_ s\u2081 s\u2082 = set (def (quote _\u2294_) (arg\u1d5b\u02b3 (sort s\u2081) \u2237 arg\u1d5b\u02b3 (sort s\u2082) \u2237 []))\n\n{-\n_`\u2294`_ : \u2200 {A} \u2192 Sort A \u2192 Sort A \u2192 Sort A\ns\u2081 `\u2294` s\u2082 with decode-Sort s\u2081 | decode-Sort s\u2082\n...          | just n\u2081        | just n\u2082        = lit (n\u2081 \u2294\u2115 n\u2082)\n...          | _              | _              = s\u2081 `\u2294 s\u2082\n-}\n\n_`\u2294`_ : \u2200 {A} \u2192 Sort A \u2192 Scope \ud835\udfd9 Sort A \u2192 Sort A\ns\u2081 `\u2294` s\u2082 with decode-Sort s\u2081 | decode-Sort s\u2082\n...          | just n\u2081        | just n\u2082        = lit (n\u2081 \u2294\u2115 n\u2082)\n...          | _              | _              = s\u2081 -- `\u2294 s\u2082\n\npattern pi\u1d5b\u02b3 t u = pi (arg\u1d5b\u02b3 t) u\npattern pi\u02b0\u02b3 t u = pi (arg\u02b0\u02b3 t) u\n\n`\u03a0 : \u2200 {A} \u2192 Arg (Type A) \u2192 Scope \ud835\udfd9 Type A \u2192 Type A\n`\u03a0 t u = el (getSort (unArg t) `\u2294` getSort u) (pi t u)\n\n`\u03a0\u1d5b\u02b3 : \u2200 {A} \u2192 Type A \u2192 Scope \ud835\udfd9 Type A \u2192 Type A\n`\u03a0\u1d5b\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u1d5b\u02b3 t u)\n\n`\u03a0\u02b0\u02b3 : \u2200 {A} \u2192 Type A \u2192 Scope \ud835\udfd9 Type A \u2192 Type A\n`\u03a0\u02b0\u02b3 t u = el (getSort t `\u2294` getSort u) (pi\u02b0\u02b3 t u)\n\napp : \u2200 {A} \u2192 Term A \u2192 Args A \u2192 Term A\napp (var x args) args\u2081 = var x (args ++ args\u2081)\napp (con c args) args\u2081 = con c (args ++ args\u2081)\napp (def f args) args\u2081 = def f (args ++ args\u2081)\napp (lam v t) args = opaque \"app\/lam\" unknown\n-- app (pat-lam cs args) args\u2081 = opaque \"app\/pat-lam\" unknown\napp (pi t\u2081 t\u2082) args = opaque \"app\/pi\" unknown\napp (sort x) args = opaque \"app\/sort\" unknown\napp (lit x) args = opaque \"app\/lit\" unknown\napp unknown args = unknown\n\nmodule Map where\n    TermMap : (Set \u2192 Set) \u2192 Set\u2081\n    TermMap F = \u2200 {A B} \u2192 (A \u2192 B) \u2192 F A \u2192 F B\n\n    \u2191 : \u2200 {A B C : Set} \u2192 (A \u2192 B) \u2192 (A \u228e C) \u2192 (B \u228e C)\n    \u2191 \u0393 (inl x) = inl (\u0393 x)\n    \u2191 \u0393 (inr y) = inr y\n\n    term : TermMap Term\n    t\u00ffpe : TermMap Type\n    args : TermMap Args\n    s\u00f8rt : TermMap Sort\n\n    term \u0393 (var x as) = var (\u0393 x) (args \u0393 as)\n    term \u0393 (con c as) = con c (args \u0393 as)\n    term \u0393 (def d as) = def d (args \u0393 as)\n    term \u0393 (lam v t) = lam v (term (\u2191 \u0393) t)\n    term \u0393 (pi (arg i t\u2081) t\u2082) = pi (arg i (t\u00ffpe \u0393 t\u2081)) (t\u00ffpe (\u2191 \u0393) t\u2082)\n    term \u0393 (sort x) = sort (s\u00f8rt \u0393 x)\n    term \u0393 (lit x) = lit x\n    term \u0393 unknown = unknown\n\n    t\u00ffpe \u0393 (el s t) = el (s\u00f8rt \u0393 s) (term \u0393 t)\n\n    s\u00f8rt \u0393 (set t) = set (term \u0393 t)\n    s\u00f8rt \u0393 (lit n) = lit n\n    s\u00f8rt \u0393 unknown = unknown\n\n    args \u0393 []             = []\n    args \u0393 (arg i t \u2237 as) = arg i (term \u0393 t) \u2237 args \u0393 as\n\n-- Non-hereditary!\nmodule Subst where\n    VarSubst : Set \u2192 Set \u2192 Set\n    VarSubst A B = A \u2192 Args B \u2192 Term B\n\n    TermSubst : (Set \u2192 Set) \u2192 Set\u2081\n    TermSubst F = \u2200 {A B} \u2192 VarSubst A B \u2192 F A \u2192 F B\n\n    \u2191 : \u2200 {A B C} \u2192 VarSubst A B \u2192 VarSubst (A \u228e C) (B \u228e C)\n    \u2191 \u0393 (inl x) as = app (Map.term inl (\u0393 x [])) as\n    \u2191 \u0393 (inr y) as = var (inr y) as\n\n    term : TermSubst Term\n    t\u00ffpe : TermSubst Type\n    args : TermSubst Args\n    s\u00f8rt : TermSubst Sort\n\n    term \u0393 (var x as) = \u0393 x (args \u0393 as)\n    term \u0393 (con c as) = con c (args \u0393 as)\n    term \u0393 (def d as) = def d (args \u0393 as)\n    term \u0393 (lam v t) = lam v (term (\u2191 \u0393) t)\n    term \u0393 (pi (arg i t\u2081) t\u2082) = pi (arg i (t\u00ffpe \u0393 t\u2081)) (t\u00ffpe (\u2191 \u0393) t\u2082)\n    term \u0393 (sort x) = sort (s\u00f8rt \u0393 x)\n    term \u0393 (lit x) = lit x\n    term \u0393 unknown = unknown\n\n    t\u00ffpe \u0393 (el s t) = el (s\u00f8rt \u0393 s) (term \u0393 t)\n\n    s\u00f8rt \u0393 (set t) = set (term \u0393 t)\n    s\u00f8rt \u0393 (lit n) = lit n\n    s\u00f8rt \u0393 unknown = unknown\n\n    args \u0393 []             = []\n    args \u0393 (arg i t \u2237 as) = arg i (term \u0393 t) \u2237 args \u0393 as\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Reflection\/Scoped.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"aff8b247e9926293f4892dd23b9f7b239177ed50","subject":"Added note on Mendelson's axioms.","message":"Added note on Mendelson's axioms.\n\nIgnore-this: c104ea1898a2c73827f738bf188d4439\n\ndarcs-hash:20120220191820-3bd4e-fa646ab5d6d1eeab01c2ff4943b5e60843f3c58a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/MendelsonAxioms.agda","new_file":"notes\/MendelsonAxioms.agda","new_contents":"-- Tested with the development version of Agda on 20 February 2012.\n\nmodule MendelsonAxioms where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 10 _*_\ninfixl 9  _+_\ninfix 7   _\u2261_\n\npostulate\n  M       : Set\n  zero    : M\n  succ    : M \u2192 M\n  _+_ _*_ : M \u2192 M \u2192 M\n\n-- The identity type on the universe of discourse.\ndata _\u2261_ (x : M) : M \u2192 Set where\n  refl : x \u2261 x\n\n-- Impossible.\n-- S\u2084 : \u2200 {m n} \u2192 succ m \u2261 succ n \u2192 m \u2261 n\n-- S\u2084 h = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/MendelsonAxioms.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"58e4adeb6904900cda214b7bae6a9096329277d6","subject":"WIP sketch System L","message":"WIP sketch System L\n","repos":"inc-lc\/ilc-agda","old_file":"Parametric\/Syntax\/LType.agda","new_file":"Parametric\/Syntax\/LType.agda","new_contents":"import Parametric.Syntax.Type as Type\n\n{--\n-- Encoding types of polarized System L, as described in \"A dissection of L\".\n--}\n\nmodule Parametric.Syntax.LType where\n\n-- The treatment of base type seems questionable. We might want instead to\n-- distinguish base value types and base computation types. Or not, I'm not\n-- sure.\n\nmodule Structure (Base : Type.Structure) where\n  open Type.Structure Base\n\n  mutual\n    -- Polarized L.\n    --  Values (positive types)\n    data VType : Set where\n      vB : (\u03b9 : Base) \u2192 VType\n\n      _\u2297_ : (\u03c4\u2081 : VType) \u2192 (\u03c4\u2082 : VType) \u2192 VType\n      vUnit : VType\n\n      _\u2295_ : (\u03c4\u2081 : VType) \u2192 (\u03c4\u2082 : VType) \u2192 VType\n      vZero : VType\n\n      ! : VType \u2192 VType\n      \u2193 : (c : CType) \u2192 VType\n\n    data CType : Set where\n      cB : (\u03b9 : Base) \u2192 CType\n\n      _\u214b_ : CType \u2192 CType \u2192 CType\n      c\u22a5 : CType\n\n      _&_ : CType \u2192 CType \u2192 CType\n      c\u22a4 : CType\n\n      \u00bf : CType \u2192 CType\n      \u2191 : VType \u2192 CType\n\n  _\u2020v : VType \u2192 CType\n  _\u2020c : CType \u2192 VType\n\n  vB \u03b9 \u2020v = cB \u03b9\n  (v\u2081 \u2297 v\u2082) \u2020v = (v\u2081 \u2020v) \u214b (v\u2082 \u2020v)\n  vUnit \u2020v = c\u22a5\n  (v\u2081 \u2295 v\u2082) \u2020v = (v\u2081 \u2020v) & (v\u2082 \u2020v)\n  vZero \u2020v = c\u22a4\n  ! v \u2020v = \u00bf (v \u2020v)\n  \u2193 c \u2020v = \u2191 (c \u2020c)\n\n  cB \u03b9 \u2020c = vB \u03b9\n  (c\u2081 \u214b c\u2082) \u2020c = c\u2081 \u2020c \u2297 c\u2082 \u2020c\n  c\u22a5 \u2020c = vUnit\n  (c\u2081 & c\u2082) \u2020c = (c\u2081 \u2020c) \u2295 (c\u2082 \u2020c)\n  c\u22a4 \u2020c = vZero\n  \u00bf c \u2020c = ! (c \u2020c)\n  \u2191 v \u2020c = \u2193 (v \u2020v)\n\n  open import Relation.Binary.PropositionalEquality\n\n  -- This proof really calls for tactics, doesn't it?\n  invV : \u2200 {v} \u2192 v \u2261 v \u2020v \u2020c\n  invC : \u2200 {c} \u2192 c \u2261 c \u2020c \u2020v\n\n  invV {vB \u03b9} = refl\n  invV {v\u2081 \u2297 v\u2082} = cong\u2082 _\u2297_ invV invV\n  invV {vUnit} = refl\n  invV {v\u2081 \u2295 v\u2082} = cong\u2082 _\u2295_ invV invV\n  invV {vZero} = refl\n  invV { ! v} = cong ! invV\n  invV {\u2193 c} = cong \u2193 invC\n\n  invC {cB \u03b9} = refl\n  invC {c\u2081 \u214b c\u2082} = cong\u2082 _\u214b_ invC invC\n  invC {c\u22a5} = refl\n  invC {c\u2081 & c\u2082} = cong\u2082 _&_ invC invC\n  invC {c\u22a4} = refl\n  invC {\u00bf c} = cong \u00bf invC\n  invC {\u2191 x} = cong \u2191 invV\n\n  open import Base.Syntax.Context VType public\n    using ()\n    renaming\n      ( \u2205 to \u2205\u2205\n      ; _\u2022_ to _\u2022\u2022_\n      ; _\u22ce_ to _\u22ce\u22ce_\n      ; mapContext to mapVCtx\n      ; Var to VVar\n      ; Context to VContext\n      ; this to vThis; that to vThat)\n\n  fromVar : \u2200 {\u0393 \u03c4} \u2192 (f : Type \u2192 VType) \u2192 Var \u0393 \u03c4 \u2192 VVar (mapVCtx f \u0393) (f \u03c4)\n  fromVar {x \u2022 \u0393} f this = vThis\n  fromVar {x \u2022 \u0393} f (that v) = vThat (fromVar f v)\n\n  data VTerm : (\u039e : VContext) \u2192 (\u0393 : VContext) \u2192 VType \u2192 Set\n  data CTerm : (\u039e : VContext) \u2192 (\u0393 : VContext) \u2192 CType \u2192 Set\n\n  -- Each constructor of the following types encodes a rule from the paper (Fig.\n  -- 4); the original names are given in comments at the end of each row.\n\n  data Cmd : (\u039e : VContext) \u2192 (\u0393 : VContext) \u2192 Set where\n    -- I swapped the operands compared to the cut rule in the paper (Fig. 4),\n    -- but this should be purely cosmetic.\n    \u27e8_\u2223_\u27e9 : \u2200 {\u039e \u0393 \u0394 \u03c4} \u2192 (v : CTerm \u039e \u0394 (\u03c4 \u2020v)) \u2192 (u : VTerm \u039e \u0393 \u03c4) \u2192 Cmd \u039e (\u0393 \u22ce\u22ce \u0394) -- cut\n\n  data VTerm where\n    lvar : \u2200 {\u039e A} \u2192 VTerm \u039e (A \u2022\u2022 \u2205\u2205) A        -- id\n    vvar : \u2200 {\u039e A} \u2192 VVar \u039e A \u2192 VTerm \u039e \u2205\u2205 A   -- id'\n    \u03bc\u21d1_ : \u2200 {\u039e \u0393 N} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 \u0393) \u2192 VTerm \u039e \u0393 (\u2193 N) -- \u2193\n\n    -- Multiplicative fragment\n    _,_ : \u2200 {\u039e \u0393 \u0394 A B} \u2192 VTerm \u039e \u0393 A \u2192 VTerm \u039e \u0394 B \u2192 VTerm \u039e (\u0393 \u22ce\u22ce \u0394) (A \u2297 B) -- \u2297\n    \u27e8\u27e9 : \u2200 {\u039e} \u2192 VTerm \u039e \u2205\u2205 vUnit --1\n\n    -- Additive fragment\n    _\u2081as_ : \u2200 {\u039e \u0393 A} \u2192 VTerm \u039e \u0393 A \u2192 \u2200 B \u2192 VTerm \u039e \u0393 (A \u2295 B) -- \u2295L\n    _\u2082as_ : \u2200 {\u039e \u0393 B} \u2192 VTerm \u039e \u0393 B \u2192 \u2200 A \u2192 VTerm \u039e \u0393 (A \u2295 B) -- \u2295R\n\n    -- No rule for vZero\n\n    -- Exponential fragment (in the sense of linear logic)\n\n    \u2983\u2984_ : \u2200 {\u039e A} \u2192 VTerm \u039e \u2205\u2205 A \u2192 VTerm \u039e \u2205\u2205 (! A) -- !\n    -- : \u2200 {\u039e \u0393 A} \u2192\n\n  -- Compared to the paper, we avoid using \u2020 in indexes of the result, and\n  -- exploit the fact that \u2020 is involutive. See for instance \u03bc\n  data CTerm where\n    \u03bc : \u2200 {\u039e \u0393 N} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 \u0393) \u2192 CTerm \u039e \u0393 N\n    \u21d1 : \u2200 {\u039e \u0393 A} \u2192 VTerm \u039e \u0393 A \u2192 CTerm \u039e \u0393 (\u2191 A) -- \u2191\n\n    -- Multiplicative fragment\n    \u03bc, : \u2200 {\u039e \u0393 N M} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 M \u2020c \u2022\u2022 \u0393) \u2192 CTerm \u039e \u0393 (N \u214b M) -- \u214b\n    \u03bc\u27e8\u27e9 : \u2200 {\u039e \u0393} \u2192 Cmd \u039e \u0393 \u2192 CTerm \u039e \u0393 c\u22a5 -- \u22a5\n\n    -- Additive fragment\n    \u03bc\u2081_\u03bc\u2082_ : \u2200 {\u039e \u0393 N M} \u2192 Cmd \u039e (N \u2020c \u2022\u2022 \u0393) \u2192 Cmd \u039e (M \u2020c \u2022\u2022 \u0393) \u2192 CTerm \u039e \u0393 (N & M)\n\n    -- Exponential fragment (in the sense of linear logic)\n    \u03bc\u2983\u2984 : \u2200 {\u039e \u0393 N} \u2192 Cmd (N \u2020c \u2022\u2022 \u039e) \u0393 \u2192 CTerm \u039e \u0393 (\u00bf N) -- ?\n\n    -- : \u2200 {\u039e \u0393 \u03c4} \u2192\n\n  cast-invV : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 N \u2192 VTerm \u039e \u0393 ((N \u2020v) \u2020c)\n  cast-invV {\u039e} {\u0393} t = subst (VTerm \u039e \u0393) invV t\n\n  cast-invV2 : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 ((N \u2020v) \u2020c) \u2192 VTerm \u039e \u0393 N\n  cast-invV2 {\u039e} {\u0393} t = subst (VTerm \u039e \u0393) (sym invV) t\n\n  cast-invC : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 ((N \u2020c) \u2020v)\n  cast-invC {\u039e} {\u0393} t = subst (CTerm \u039e \u0393) invC t\n\n  -- Check that type rules match by doing eta-expansion.\n  -- (Is this checking harmony?)\n  \u03b7-\u03bc : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 N\n  \u03b7-\u03bc t = \u03bc \u27e8 cast-invC t \u2223 lvar \u27e9\n\n  maybe-\u03b7-\u03bc\u21d1 : \u2200 {\u039e \u0393 A} \u2192 VTerm \u039e \u0393 A \u2192 VTerm \u039e \u0393 (\u2193 (\u2191 A))\n  maybe-\u03b7-\u03bc\u21d1 t = \u03bc\u21d1 \u27e8 cast-invC (\u21d1 t) \u2223 lvar \u27e9\n\n  -- A \"thunk\" combinator (from paper)\n  \u21d3 : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 N \u2192 VTerm \u039e \u0393 (\u2193 N)\n  \u21d3 t = \u03bc\u21d1 \u27e8 cast-invC t \u2223 lvar \u27e9\n\n  -- from paper\n  force : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 (\u2193 N) \u2192 CTerm \u039e \u0393 N\n  force t = {! \u03bc \u27e8 \u21d1 lvar \u2223 t \u27e9 !}\n\n  -- XXX Here we \"just\" need to reorder variables in contexts. OMG, I screwed up\n  -- (which is bad), or they use implicit exchange (which would be worse).\n  \u03b7-\u03bc\u21d1 : \u2200 {\u039e \u0393 N} \u2192 VTerm \u039e \u0393 (\u2193 N) \u2192 VTerm \u039e \u0393 (\u2193 N)\n  \u03b7-\u03bc\u21d1 t = \u03bc\u21d1 {! \u27e8 \u21d1 lvar \u2223 t \u27e9 !}\n\n  n-\u03bc, : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 (N \u214b M) \u2192 CTerm \u039e \u0393 (N \u214b M)\n  n-\u03bc, t = \u03bc, \u27e8 cast-invC t \u2223 lvar , lvar \u27e9 -- The two lvars are in different contexts, so they have different types!\n\n  \u03b7-\u03bc\u27e8\u27e9 : \u2200 {\u039e \u0393} \u2192 CTerm \u039e \u0393 c\u22a5 \u2192 CTerm \u039e \u0393 c\u22a5\n  \u03b7-\u03bc\u27e8\u27e9 t = \u03bc\u27e8\u27e9 \u27e8 t \u2223 \u27e8\u27e9 \u27e9\n\n  \u03b7-\u03bc\u2983\u2984 : \u2200 {\u039e \u0393 N} \u2192 CTerm \u039e \u0393 (\u00bf N) \u2192 CTerm \u039e \u0393 (\u00bf N)\n  \u03b7-\u03bc\u2983\u2984 t = \u03bc\u2983\u2984 \u27e8  cast-invC {!weaken t!} \u2223 \u2983\u2984 (vvar vThis) \u27e9\n\n  \u03b2-expand-\u03bc\u27e8\u27e9 : \u2200 {\u039e \u0393} \u2192 Cmd \u039e \u0393 \u2192 Cmd \u039e \u0393\n  \u03b2-expand-\u03bc\u27e8\u27e9 c = \u27e8 \u03bc\u27e8\u27e9 c \u2223 \u27e8\u27e9 \u27e9\n\n  -- \u03b7-rules for sums look different.\n  maybe-\u03b7-\u03bc\u2081 : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 M \u2192 CTerm \u039e \u0393 (N & M)\n  maybe-\u03b7-\u03bc\u2081 t\u2081 t\u2082 = \u03bc\u2081 \u27e8 cast-invC t\u2081 \u2223 lvar \u27e9 \u03bc\u2082  \u27e8 cast-invC t\u2082 \u2223 lvar \u27e9\n\n  \u03b7-\u03bc\u2081 : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 M \u2192 CTerm \u039e \u0393 N\n  \u03b7-\u03bc\u2081 {M = M} t\u2081 t\u2082 =\n    \u03bc \u27e8\n      \u03bc\u2081 \u27e8 cast-invC (cast-invC t\u2081) \u2223 lvar \u27e9  \u03bc\u2082 \u27e8 cast-invC (cast-invC t\u2082) \u2223 lvar \u27e9\n    \u2223\n      lvar \u2081as (M \u2020c)\n    \u27e9\n\n  \u03b7-\u03bc\u2082 : \u2200 {\u039e \u0393 N M} \u2192 CTerm \u039e \u0393 N \u2192 CTerm \u039e \u0393 M \u2192 CTerm \u039e \u0393 M\n  \u03b7-\u03bc\u2082 {N = N} t\u2081 t\u2082 =\n    \u03bc \u27e8\n      \u03bc\u2081 \u27e8 cast-invC (cast-invC t\u2081) \u2223 lvar \u27e9  \u03bc\u2082 \u27e8 cast-invC (cast-invC t\u2082) \u2223 lvar \u27e9\n    \u2223\n      lvar \u2082as (N \u2020c)\n    \u27e9\n\n  -- Some syntactic sugar.\n  _\u22b8_ : VType \u2192 CType \u2192 CType\n  \u03c3 \u22b8 \u03c4 = (\u03c3 \u2020v) \u214b \u03c4\n\n  _\u21db_ : VType \u2192 CType \u2192 CType\n  \u03c3 \u21db \u03c4 = ! \u03c3 \u22b8 \u03c4\n\n\n  -- Not checked again yet. This code comes from CBPV and was refactored for\n  -- renamings done to get polarized L, so it might or might not work.\n\n  cbnToCType : Type \u2192 CType\n  cbnToCType (base \u03b9) = \u2191 (vB \u03b9)\n  cbnToCType (\u03c3 \u21d2 \u03c4) = \u2193 (cbnToCType \u03c3) \u21db cbnToCType \u03c4\n\n  cbvToVType : Type \u2192 VType\n  cbvToVType (base \u03b9) = vB \u03b9\n  cbvToVType (\u03c3 \u21d2 \u03c4) = \u2193 (cbvToVType \u03c3 \u21db \u2191 (cbvToVType \u03c4))\n\n  cbnToVType : Type \u2192 VType\n  cbnToVType \u03c4 = \u2193 (cbnToCType \u03c4)\n\n  cbvToCType : Type \u2192 CType\n  cbvToCType \u03c4 = \u2191 (cbvToVType \u03c4)\n\n  fromCBNCtx : Context \u2192 VContext\n  fromCBNCtx \u0393 = mapVCtx cbnToVType \u0393\n\n  fromCBVCtx : Context \u2192 VContext\n  fromCBVCtx \u0393 = mapVCtx cbvToVType \u0393\n\n  open import Data.List\n  open Data.List using (List) public\n  fromCBVToCompList : Context \u2192 List CType\n  fromCBVToCompList \u0393 = mapVCtx cbvToCType \u0393\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Parametric\/Syntax\/LType.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"d209b424fd44d3a701135c4b778c85eb87e58fd4","subject":"Added note on unguarded co-recursive functions.","message":"Added note on unguarded co-recursive functions.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/Alter.agda","new_file":"notes\/FOT\/FOTC\/Program\/Alter.agda","new_contents":"------------------------------------------------------------------------------\n-- Alter: An unguarded co-recursive function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Program.Alter where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Data.Stream\n\n------------------------------------------------------------------------------\n\n{-# NO_TERMINATION_CHECK #-}\nalter : D\nalter = true \u2237 false \u2237 alter\n\nalter-Stream : Stream (alter)\nalter-Stream = Stream-coind A h\u2081 h\u2082\n  where\n  A : D \u2192 Set\n  A xs = xs \u2261 xs\n\n  h\u2081 : A alter \u2192 \u2203[ x' ] \u2203[ xs' ] alter \u2261 x' \u2237 xs' \u2227 A xs'\n  h\u2081 _ = true , (false \u2237 alter) , refl , refl\n\n  h\u2082 : A alter\n  h\u2082 = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Program\/Alter.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"07a5dd764672d18324f8dfe1411c16757921c46c","subject":"Added note about the existential.","message":"Added note about the existential.\n\nIgnore-this: 3d9fb02a35a02c6d345a892aead4a148\n\ndarcs-hash:20120226033342-3bd4e-4b43cf7250b47b730b02a9952914e3605575ce84.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/Existential.agda","new_file":"notes\/Existential.agda","new_contents":"-- Tested with the development version of Agda on 25 February 2012.\n\nmodule Existential where\n\ninfixr 7 _,_\n\npostulate\n  D   : Set\n  P Q : D \u2192 Set\n  P\u2192Q : \u2200 {x} \u2192 P x \u2192 Q x\n\ndata \u2203 (P : D \u2192 Set) : Set where\n  _,_ : (x : D) \u2192 P x \u2192 \u2203 P\n\n\u2203-elim : {P : D \u2192 Set}{Q : Set} \u2192 \u2203 P \u2192 (\u2200 x \u2192 P x \u2192 Q) \u2192 Q\n\u2203-elim (x , p) h = h x p\n\nthm\u2081 : \u2203 P \u2192 \u2203 Q\nthm\u2081 h = \u2203-elim h (\u03bb x Px \u2192 x , P\u2192Q Px)\n\nthm\u2082 : \u2203 P \u2192 \u2203 Q\nthm\u2082 h = \u2203-elim h helper\n  where\n  helper : \u2200 x \u2192 P x \u2192 \u2203 Q\n  helper x Px = x , P\u2192Q Px\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/Existential.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"869c4391d7d2344704a1a2f793fe41473265628e","subject":"moved the typed terms into a separate file","message":"moved the typed terms into a separate file\n","repos":"goodlyrottenapple\/lamYcalc","old_file":"Agda\/Typed-Core.agda","new_file":"Agda\/Typed-Core.agda","new_contents":"module Typed-Core where\nopen import Data.List\nopen import Data.Nat\nopen import Data.List.Any as LAny\nopen LAny.Membership-\u2261\nopen import Relation.Nullary\nopen import Relation.Binary.Core\n\nopen import Core\nopen import Core-Lemmas\nopen import Typing\n\ndata \u039b : Type -> Set where\n  bv : \u2200 {A} -> (i : \u2115) -> \u039b A\n  fv : \u2200 {A} -> (x : Atom) -> \u039b A\n  lam : \u2200 {B} -> (A : Type) -> (e : \u039b B) -> \u039b (A \u27f6 B)\n  app : \u2200 {A B} -> (e\u2081 : \u039b (A \u27f6 B)) -> (e\u2082 : \u039b A) -> \u039b B\n  Y : (t : Type) -> \u039b ((t \u27f6 t) \u27f6 t)\n\n\ndata _~_ : \u2200{t} -> \u039b t -> PTerm -> Set where\n  bv : \u2200 {t i} -> (bv {t} i) ~ (bv i)\n  fv : \u2200 {t x} -> (fv {t} x) ~ (fv x)\n  lam : \u2200 {t s m m'} -> m ~ m' -> (lam {s} t m) ~ (lam m')\n  app : \u2200 {t s m n m' n'} -> m ~ m' -> n ~ n' -> (app {t} {s} m n) ~ (app m' n')\n  Y : \u2200 {t} -> (Y t) ~ (Y t)\n\n\n\u039b[_<<_] : \u2200 {t} -> \u2115 -> Atom -> \u039b t -> \u039b t\n\u039b[ k << x ] (bv i) = bv i\n\u039b[ k << x ] (fv y) with x \u225f y\n... | yes _ = bv k\n... | no _ = fv y\n\u039b[ k << x ] (lam t m) = lam t (\u039b[ (suc k) << x ] m)\n\u039b[ k << x ] (app t1 t2) = app (\u039b[ k << x ] t1) (\u039b[ k << x ] t2)\n\u039b[ k << x ] (Y t) = Y t\n\n\n\n\u22a2->\u039b : \u2200 {\u0393 m t} -> \u0393 \u22a2 m \u2236 t -> \u039b t\n\u22a2->\u039b {m = bv i} ()\n\u22a2->\u039b {m = fv x} {t} \u0393\u22a2m\u2236t = fv {t} x\n\u22a2->\u039b {m = lam m} (abs {_} {\u03c4\u2081} {\u03c4\u2082} L cf) =\n  lam \u03c4\u2081 ( \u039b[ 0 << \u2203fresh (L ++ FV m) ] (\u22a2->\u039b (cf (\u2209-cons-l _ _ (\u2203fresh-spec (L ++ FV m)) ))) )\n\u22a2->\u039b {m = app s t} (app \u0393\u22a2s \u0393\u22a2t) = app (\u22a2->\u039b \u0393\u22a2s) (\u22a2->\u039b \u0393\u22a2t)\n\u22a2->\u039b {m = Y \u03c4} (Y _) = Y \u03c4\n\n-- \u22a2->\u039b\u2261 : \u2200 {\u0393 m n t} -> m \u2261 n -> {\u0393\u22a2m : \u0393 \u22a2 m \u2236 t} -> {\u0393\u22a2n : \u0393 \u22a2 n \u2236 t} -> (\u22a2->\u039b \u0393\u22a2m) \u2261 (\u22a2->\u039b \u0393\u22a2m)\n-- \u22a2->\u039b\u2261 refl = \u03bb {\u0393\u22a2m} {\u0393\u22a2n} \u2192 refl\n\n\n\u039b*^-*^~ : \u2200 {\u03c4 x k} t t' -> _~_ {\u03c4} t t' -> \u039b[ k << x ] t ~ ([ k << x ] t')\n\u039b*^-*^~ _ _ bv = bv\n\u039b*^-*^~ {x = x} (fv y) _ fv with x \u225f y\n\u039b*^-*^~ (fv x) .(fv x) fv | yes _ = bv\n\u039b*^-*^~ (fv y) .(fv y) fv | no _ = fv\n\u039b*^-*^~ _ _ (lam {m = m} {m'} t~t') = lam (\u039b*^-*^~ m m' t~t')\n\u039b*^-*^~ _ _ (app {m = m} {n} {m'} {n'} t~t' t~t'') = app (\u039b*^-*^~ m m' t~t') (\u039b*^-*^~ n n' t~t'')\n\u039b*^-*^~ _ _ Y = Y\n\n\n\u22a2->\u039b~ : \u2200 {\u0393 t \u03c4} -> (\u0393\u22a2t : \u0393 \u22a2 t \u2236 \u03c4) -> (\u22a2->\u039b \u0393\u22a2t) ~ t\n\u22a2->\u039b~ {t = bv i} ()\n\u22a2->\u039b~ {t = fv x} (var _ _) = fv\n\u22a2->\u039b~ {t = lam t} (abs {\u03c4\u2081 = \u03c4\u2081} {\u03c4\u2082} L cf) = lam ih\n  where\n  x = \u2203fresh (L ++ FV t)\n  x\u2209 = \u2203fresh-spec (L ++ FV t)\n  x\u2237\u0393\u22a2t^'x = cf (\u2209-cons-l _ _ x\u2209)\n\n  sub : \u2200 {\u03c4 x m} -> x \u2209 FV t -> _~_ {\u03c4} m t \u2261 m ~ (* x ^ (t ^' x))\n  sub {_} {x} x\u2209 rewrite fv-^-*^-refl x t {0} x\u2209 = refl\n\n  ih : \u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) ~ t\n  ih rewrite sub {_} {x} {\u039b[ 0 << x ] (\u22a2->\u039b x\u2237\u0393\u22a2t^'x)} (\u2209-cons-r L _ x\u2209) =\n    \u039b*^-*^~ (\u22a2->\u039b x\u2237\u0393\u22a2t^'x) (t ^' x) (\u22a2->\u039b~ (cf (\u2209-cons-l _ _ x\u2209)))\n\u22a2->\u039b~ {t = app t t\u2081} (app \u0393\u22a2t \u0393\u22a2t\u2081) = app (\u22a2->\u039b~ \u0393\u22a2t) (\u22a2->\u039b~ \u0393\u22a2t\u2081)\n\u22a2->\u039b~ {t = Y t\u2081} (Y x) = Y\n\n\n\n\u039b[_>>_] : \u2200 {\u03c4 \u03c4'} -> \u2115 -> \u039b \u03c4' -> \u039b \u03c4 -> \u039b \u03c4\n\u039b[_>>_] {\u03c4} {\u03c4'} k u (bv i) with k \u225f i | \u03c4 \u225fT \u03c4'\n\u039b[ k >> u ] (bv i) | yes _ | yes refl = u\n... | yes _ | no _ = bv i\n... | no _ | _ = bv i\n\u039b[ k >> u ] (fv x) = fv x\n\u039b[ k >> u ] (lam A t) = lam A (\u039b[ (suc k) >> u ] t)\n\u039b[ k >> u ] (app t1 t2) = app (\u039b[ k >> u ] t1) (\u039b[ k >> u ] t2)\n\u039b[ k >> u ] (Y t) = Y t\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Agda\/Typed-Core.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"8184c69577c57aa76eb47180306e9f4a5f03c130","subject":"Added example for discussion on induction.","message":"Added example for discussion on induction.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Nat\/PredTotality.agda","new_file":"notes\/FOT\/FOTC\/Data\/Nat\/PredTotality.agda","new_contents":"------------------------------------------------------------------------------\n-- Totality of predecessor function\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Nat.PredTotality where\n\nopen import FOTC.Base\n\ndata N : D \u2192 Set where\n  nzero : N zero\n  nsucc : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n\n-- Induction principle generated by Coq when we define the data type N\n-- in Prop.\nN-ind\u2081 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 N n \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2081 A A0 h nzero      = A0\nN-ind\u2081 A A0 h (nsucc Nn) = h Nn (N-ind\u2081 A A0 h Nn)\n\n-- The induction principle removing the hypothesis N n from the\n-- inductive step.\nN-ind\u2082 : (A : D \u2192 Set) \u2192\n         A zero \u2192\n         (\u2200 {n} \u2192 A n \u2192 A (succ\u2081 n)) \u2192\n         \u2200 {n} \u2192 N n \u2192 A n\nN-ind\u2082 A A0 h nzero      = A0\nN-ind\u2082 A A0 h (nsucc Nn) = h (N-ind\u2082 A A0 h Nn)\n\n-- Proof by pattern matching.\npred-N : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N nzero          = subst N (sym pred-0) nzero\npred-N (nsucc {n} Nn) = subst N (sym (pred-S n)) Nn\n\n-- Proof using N-ind\u2081.\npred-N\u2081 : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\npred-N\u2081 = N-ind\u2081 A A0 is\n  where\n  A : D \u2192 Set\n  A i = N (pred\u2081 i)\n\n  A0 : A zero\n  A0 = subst N (sym pred-0) nzero\n\n  is : \u2200 {i} \u2192 N i \u2192 A i \u2192 A (succ\u2081 i)\n  is {i} Ni ih = subst N (sym (pred-S i)) Ni\n\n-- Proof using N-ind\u2082.\n-- pred-N\u2082 : \u2200 {n} \u2192 N n \u2192 N (pred\u2081 n)\n-- pred-N\u2082 = N-ind\u2082 A A0 is\n--   where\n--   A : D \u2192 Set\n--   A i = N (pred\u2081 i)\n\n--   A0 : A zero\n--   A0 = subst N (sym pred-0) nzero\n\n--   is : \u2200 {i} \u2192 A i \u2192 A (succ\u2081 i)\n--   is {i} ih = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Nat\/PredTotality.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"b05a3c591f643f5f9920c3b5131cef041156a7ce","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a51e92b8f9b2d99088fdab1c0b584492a5a5d12c","subject":"Initial version of Data.Bits","message":"Initial version of Data.Bits\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Bits.agda","new_file":"lib\/Data\/Bits.agda","new_contents":"module Data.Bits where\n\n-- cleanup\nopen import Category.Applicative\nopen import Category.Monad\nopen import Data.Nat.NP hiding (_\u2264_; _==_)\nopen import Data.Bool\nopen import Data.Fin using (Fin; zero; suc; to\u2115; #_; inject\u2081)\nopen import Data.Vec hiding (_\u229b_) renaming (map to vmap)\nopen import Data.Unit using (\u22a4)\nopen import Data.Empty using (\u22a5)\nopen import Data.Product using (_\u00d7_; _,_; uncurry)\nopen import Function.NP hiding (_\u2192\u27e8_\u27e9_)\nopen import Relation.Binary.PropositionalEquality\nopen import Algebra.FunctionProperties\nimport Data.List as L\n\nBit : Set\nBit = Bool\n\npattern 0b = false\npattern 1b = true\n{-\ndata Bit : Set where\n  0b 1b : Bit\n-}\n\nBits : \u2115 \u2192 Set\nBits = Vec Bit\n\n0\u2026 : \u2200 {n} \u2192 Bits n\n0\u2026 = replicate 0b\n\n-- Warning: 0\u2026 {0} \u2261 1\u2026 {0}\n1\u2026 : \u2200 {n} \u2192 Bits n\n1\u2026 = replicate 1b\n\n_!_ : \u2200 {a n} {A : Set a} \u2192 Vec A n \u2192 Fin n \u2192 A\n_!_ = flip lookup\n\n_==\u1d47_ : (b\u2080 b\u2081 : Bit) \u2192 Bool\nb\u2080 ==\u1d47 b\u2081 = not (b\u2080 xor b\u2081)\n\n_==_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bool\n[] == [] = true\n(b\u2080 \u2237 bs\u2080) == (b\u2081 \u2237 bs\u2081) = (b\u2080 ==\u1d47 b\u2081) \u2227 bs\u2080 == bs\u2081\n\ninfixr 5 _\u2295_\n_\u2295_ : \u2200 {n} (bs\u2080 bs\u2081 : Bits n) \u2192 Bits n\n_\u2295_ = zipWith _xor_\n\nvnot : \u2200 {n} \u2192 Endo (Bits n)\nvnot = _\u2295_ 1\u2026\n\npostulate\n  \u2295-assoc : \u2200 {n} \u2192 Associative _\u2261_ (_\u2295_ {n})\n  \u2295-comm  : \u2200 {n} \u2192 Commutative _\u2261_ (_\u2295_ {n})\n  \u2295-left-identity : \u2200 {n} \u2192 LeftIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n  \u2295-right-identity : \u2200 {n} \u2192 RightIdentity _\u2261_ (replicate 0b) (_\u2295_ {n})\n  \u2295-\u2261 : \u2200 {n} (x : Bits n) \u2192 x \u2295 x \u2261 replicate 0b\n\n  \u2295-\u2262 : \u2200 {n} (x : Bits n) \u2192 x \u2295 vnot x \u2261 replicate 1b\n  -- x \u2295 vnot x \u2261 x \u2295 (1\u2026 \u2295 x) \u2261 (x \u2295 x) \u2295 1\u2026 \u2261 0\u2026 \u2295 1\u2026 \u2261 1\u2026\n\nmsb : \u2200 {n} k \u2192 Bits (k + n) \u2192 Bits k\nmsb zero    bs       = []\nmsb (suc k) (b \u2237 bs) = b \u2237 msb k bs\n\nlsb : \u2200 {n} k \u2192 Bits (n + k) \u2192 Bits k\nlsb {n} k rewrite \u2115\u00b0.+-comm n k = reverse \u2218 msb k \u2218 reverse\n\nmsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nmsb\u2082 = msb 2\n\nlsb\u2082 : \u2200 {n} \u2192 Bits (2 + n) \u2192 Bits 2\nlsb\u2082 = reverse \u2218 msb 2 \u2218 reverse\n\n#1 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#1 [] = zero\n#1 (0b \u2237 bs) = inject\u2081 (#1 bs)\n#1 (1b \u2237 bs) = suc (#1 bs)\n\n#0 : \u2200 {n} \u2192 Bits n \u2192 Fin (suc n)\n#0 = #1 \u2218 vmap not\n\n-- _\u1d2e : (s : String) {pf : IsBitString s} \u2192 Bits (length s)\n-- _\u1d2e =\n\n11\u1d47 : Bits 2\n11\u1d47 = 1b \u2237 1b \u2237 []\n\nprivate\n module M {a} {A : Set a} {M} (appl : RawApplicative M) where\n  open RawApplicative appl\n\n  replicateM : \u2200 {n} \u2192 M A \u2192 M (Vec A n)\n  replicateM {n = zero}  _ = pure []\n  replicateM {n = suc n} x = pure _\u2237_ \u229b x \u229b replicateM x\n\nopen M public\n\nbits\u2074 : Vec (Bits 4) 16\nbits\u2074 = fromList $ replicateM rawIApplicative (toList (0b \u2237 1b \u2237 []))\n  where open RawMonad L.monad\n\n0000b = bits\u2074 ! # 0\n0001b = bits\u2074 ! # 1\n0010b = bits\u2074 ! # 2\n0011b = bits\u2074 ! # 3\n0100b = bits\u2074 ! # 4\n0101b = bits\u2074 ! # 5\n0110b = bits\u2074 ! # 6\n0111b = bits\u2074 ! # 7\n1000b = bits\u2074 ! # 8\n1001b = bits\u2074 ! # 9\n1010b = bits\u2074 ! # 10\n1011b = bits\u2074 ! # 11\n1100b = bits\u2074 ! # 12\n1101b = bits\u2074 ! # 13\n1110b = bits\u2074 ! # 14\n1111b = bits\u2074 ! # 15\n\n\u03b7 : \u2200 {n a} {A : Set a} \u2192 Vec A n \u2192 Vec A n\n\u03b7 = tabulate \u2218 flip lookup\n{- alternative\n\u03b7 {zero} = \u03bb _ \u2192 []\n\u03b7 {suc n} = \u03bb xs \u2192 head xs \u2237 \u03b7 (tail xs)\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Bits.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"f075fc555e07101cc4de51d4e5671861b77d7fe6","subject":"program-distance.agda","message":"program-distance.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"program-distance.agda","new_file":"program-distance.agda","new_contents":"module program-distance where\n\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Bool\nopen import Data.Vec.NP using (Vec; count)\nopen import Data.Nat.NP\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality\nopen import diff\nimport Data.Fin as Fin\n\nrecord PrgDist : Set\u2081 where\n  constructor mk\n  field\n    _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n    ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n\n  breaks : \u2200 {c} (EXP : Bit \u2192 \u21ba c Bit) \u2192 Set\n  breaks \u2141 = \u2141 0b ]-[ \u2141 1b\n\n  _\u2257\u2141_ : \u2200 {c} (\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit) \u2192 Set\n  \u2141\u2080 \u2257\u2141 \u2141\u2081 = \u2200 b \u2192 \u2141\u2080 b \u2257\u21ba \u2141\u2081 b\n\n  \u2257\u2141-trans : \u2200 {c} \u2192 Transitive (_\u2257\u2141_ {c})\n  \u2257\u2141-trans p q b R = trans (p b R) (q b R)\n\n  -- An wining adversary for game \u2141\u2080 reduces to a wining adversary for game \u2141\u2081\n  _\u21d3_ : \u2200 {c\u2080 c\u2081} (\u2141\u2080 : Bit \u2192 \u21ba c\u2080 Bit) (\u2141\u2081 : Bit \u2192 \u21ba c\u2081 Bit) \u2192 Set\n  \u2141\u2080 \u21d3 \u2141\u2081 = breaks \u2141\u2080 \u2192 breaks \u2141\u2081\n\n  extensional-reduction : \u2200 {c} {\u2141\u2080 \u2141\u2081 : Bit \u2192 \u21ba c Bit}\n                          \u2192 \u2141\u2080 \u2257\u2141 \u2141\u2081 \u2192 \u2141\u2080 \u21d3 \u2141\u2081\n  extensional-reduction same-games = ]-[-cong (same-games 0b) (same-games 1b)\n\next-count : \u2200 {n a} {A : Set a} {f g : A \u2192 Bool} \u2192 f \u2257 g \u2192 (xs : Vec A n) \u2192 count f xs \u2261 count g xs\next-count f\u2257g [] = refl\next-count f\u2257g (x \u2237 xs) rewrite ext-count f\u2257g xs | f\u2257g x = refl\n\next-# : \u2200 {c} {f g : Bits c \u2192 Bit} \u2192 f \u2257 g \u2192 #\u27e8 f \u27e9 \u2261 #\u27e8 g \u27e9\next-# f\u2257g = ext-count f\u2257g (allBits _)\n\nmodule Implem k where\n\n  --  diff (#1 f) (#1 g) \u2265 2^(-k) * 2^ c\n  --  diff (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --  dist (#1 f) (#1 g) \u2265 \u03b5 * 2^ c\n  --    where \u03b5 = 2^ -k\n  -- {!dist (#1 f \/ 2^ c) (#1 g \/ 2^ c) > \u03b5 !}\n  _]-[_ : \u2200 {c} (f g : \u21ba c Bit) \u2192 Set\n  _]-[_ {c} f g = diff (Fin.to\u2115 #\u27e8 run\u21ba f \u27e9) (Fin.to\u2115 #\u27e8 run\u21ba g \u27e9) \u2265 2^(c \u2238 k)\n\n  ]-[-cong : \u2200 {c} {f f' g g' : \u21ba c Bit} \u2192 f \u2257\u21ba g \u2192 f' \u2257\u21ba g' \u2192 f ]-[ f' \u2192 g ]-[ g'\n  ]-[-cong f\u2257g f'\u2257g' f]-[f' rewrite ext-# f\u2257g | ext-# f'\u2257g' = f]-[f'\n\n  prgDist : PrgDist\n  prgDist = mk _]-[_ ]-[-cong\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'program-distance.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"977c1594fffefac6ac1dd463272b5a9ed801e6bd","subject":"lib\/Data\/Nat\/Ack.agda","message":"lib\/Data\/Nat\/Ack.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/Ack.agda","new_file":"lib\/Data\/Nat\/Ack.agda","new_contents":"module Data.Nat.Ack where\n\nopen import Data.Nat.NP\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261\n\nopen ack-Props\nopen InflModule\n\npostulate\n  \u21913+ : \u2200 a n b \u2192 a \u2191\u27e8 3 + n \u27e9 b \u2261 fold (_^_ a) (fold 1) n b\n  -- mon\u21913+'' : \u2200 a b \u2192 Mon (\u03bb n \u2192 fold (_^_ a) (fold 1) n b)\n\n  mon\u21913+' : \u2200 b \u2192 Mon (\u03bb n \u2192 fold (_^_ 2) (fold 1) n (3 + b))\n-- mon\u21913+' b = {!!}\n\nmon\u21913+ : \u2200 b \u2192 Mon (\u03bb n \u2192 2 \u2191\u27e8 3 + n \u27e9 (3 + b))\nmon\u21913+ b {m} {n} rewrite \u21913+ 2 m (3 + b) | \u21913+ 2 n (3 + b) = mon\u21913+' b\n\nopen \u2191-Props\nlem>=3 : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 3 + m \u27e9 (3 + n)\nlem>=3 m n = 3 \u2264\u27e8 s\u2264s (s\u2264s (s\u2264s z\u2264n)) \u27e9\n             2 \u2191\u27e8 3 \u27e9 3 \u2261\u27e8 \u2191\u2083-^ 2 3 \u27e9\n             2 ^ 3 \u2264\u27e8 lem2^3 n \u27e9\n             2 ^ (3 + n) \u2261\u27e8 \u2261.sym (\u2191\u2083-^ 2 (3 + n)) \u27e9\n             2 \u2191\u27e8 3 \u27e9 (3 + n) \u2264\u27e8 mon\u21913+ n z\u2264n \u27e9\n             2 \u2191\u27e8 3 + m \u27e9 (3 + n) \u220e\n  where open \u2264-Reasoning\n\nlem>=3'' : \u2200 m n \u2192 3 \u2264 2 \u2191\u27e8 suc m \u27e9 (3 + n)\nlem>=3'' zero n = s\u2264s (s\u2264s (s\u2264s z\u2264n))\nlem>=3'' (suc zero) n rewrite \u2191\u2082-* 2 (3 + n) = s\u2264s (s\u2264s (s\u2264s z\u2264n))\nlem>=3'' (suc (suc m)) n = lem>=3 m n\nack-\u2191 : \u2200 m n \u2192 3 + ack m n \u2261 2 \u2191\u27e8 m \u27e9 (3 + n)\nack-\u2191 zero n = \u2261.refl\nack-\u2191 (suc m) zero = 3 + ack (suc m) 0   \u2261\u27e8 ack-\u2191 m 1 \u27e9\n                     2 \u2191\u27e8 m \u27e9 4          \u2261\u27e8 \u2261.cong (_\u2191\u27e8_\u27e9_ 2 m) (lem4212 m) \u27e9\n                     2 \u2191\u27e8 suc m \u27e9 3 \u220e\n  where open \u2261-Reasoning\nack-\u2191 (suc m) (suc n) = 3 + ack (suc m) (suc n)\n                      \u2261\u27e8 \u2261.refl \u27e9\n                        3 + ack m (ack (suc m) n)\n                      \u2261\u27e8 \u2261.refl \u27e9\n                        3 + ack m ((3 + ack (suc m) n) \u2238 3)\n                      \u2261\u27e8 \u2261.cong (\u03bb x \u2192 3 + ack m (x \u2238 3)) (ack-\u2191 (suc m) n) \u27e9\n                         3 + ack m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3)\n                      \u2261\u27e8 ack-\u2191 m (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3) \u27e9\n                         2 \u2191\u27e8 m \u27e9 (3 + (2 \u2191\u27e8 suc m \u27e9 (3 + n) \u2238 3))\n                      \u2261\u27e8 \u2261.cong (\u03bb x \u2192 2 \u2191\u27e8 m \u27e9 x) (lem\u2238 (lem>=3'' m n)) \u27e9\n                         2 \u2191\u27e8 m \u27e9 (2 \u2191\u27e8 suc m \u27e9 (3 + n))\n                      \u2261\u27e8 \u2261.refl \u27e9\n                        2 \u2191\u27e8 suc m \u27e9 (4 + n) \u220e \n  where open \u2261-Reasoning\n\npostulate\n  1+a^-infl< : \u2200 {a}  \u2192 Infl< (_^_ (1 + a))\n\n--  2+a*1+b-infl< : \u2200 a \u2192 Infl< (\u03bb x \u2192 (2 + a) * (1 + x))\n-- \u2200 a b \u2192 b < (2 + a) * b\n-- fold-a*-fold1 : \u2200 {n a} \u2192 Infl< (_\u21912+\u27e8_\u27e9_ (2 + a) n)\nfold-a^-fold1 : \u2200 {n a} \u2192 Infl< (fold (_^_ (1 + a)) (fold 1) n)\nfold-a^-fold1 {n} = fold-infl< 1+a^-infl< fold1+-inflT< {n}\n\n\u21913+-mon : \u2200 a n \u2192 Mon (fold (_^_ (1 + a)) (fold 1) n)\n\u21913+-mon a n = fold-mon' 1+a^-mon 1+a^-infl< (\u03bb \u03b7\u2081 \u03b7\u2082 \u2192 fold-mon \u03b7\u2081 \u03b7\u2082) fold1+-inflT< {n}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Nat\/Ack.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"068763230ff391f8e404a12cdfa1a28a29fa2f4f","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Hello2.agda","new_file":"src\/Hello2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"80ba4d5c4589aa86731022139c8a2dfc7ae7e14a","subject":"Added draft for the function mirror-Tree.","message":"Added draft for the function mirror-Tree.\n\nIgnore-this: 850e3a11b04c024a7a64ededdb1e6dbc\n\ndarcs-hash:20110303140728-3bd4e-4ee6830368cdb96e613044a1d827af3f22f5f16e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Mirror\/Tree\/ClosuresI.agda","new_file":"Draft\/Mirror\/Tree\/ClosuresI.agda","new_contents":"------------------------------------------------------------------------------\n-- Properties related with the closures of the tree type\n------------------------------------------------------------------------------\n\n-- {-# OPTIONS --no-termination-check #-}\n{-# OPTIONS --injective-type-constructors #-}\n\nmodule Draft.Mirror.Tree.ClosuresI where\n\nopen import FOTC.Base\n\nopen import FOTC.Data.List\n\nopen import FOTC.Program.Mirror.Mirror\nopen import FOTC.Program.Mirror.ListTree.Closures\nopen import FOTC.Program.Mirror.ListTree.PropertiesI\n\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n\nnode-ListTree : \u2200 {d ts} \u2192 Tree (node d ts) \u2192 ListTree ts\nnode-ListTree {d} (treeT .d LTts) = LTts\n\npostulate\n  reverse-\u2237' : \u2200 x ys \u2192 reverse (x \u2237 ys) \u2261 reverse ys ++ (x \u2237 [])\n\nmirror-Tree : \u2200 {t} \u2192 Tree t \u2192 Tree (mirror \u00b7 t)\nmirror-Tree (treeT d nilLT) =\n  subst Tree (sym (mirror-eq d [])) (treeT d helper\u2082)\n    where\n      helper\u2081 : rev (map mirror []) [] \u2261 []\n      helper\u2081 =\n        begin\n          rev (map mirror []) []\n          \u2261\u27e8 subst (\u03bb x \u2192 rev (map mirror []) [] \u2261 rev x [])\n                   (map-[] mirror)\n                   refl\n          \u27e9\n          rev [] []\n            \u2261\u27e8 rev-[] [] \u27e9\n          []\n        \u220e\n\n      helper\u2082 : ListTree (rev (map mirror []) [])\n      helper\u2082 = subst ListTree (sym helper\u2081) nilLT\n\nmirror-Tree (treeT d (consLT {t} {ts} Tt LTts)) =\n  subst Tree (sym (mirror-eq d (t \u2237 ts))) (treeT d helper)\n\n  where\n     h\u2081 : Tree (node d (reverse (map mirror ts)))\n     h\u2081 = subst Tree (mirror-eq d ts) (mirror-Tree (treeT d LTts))\n\n     h\u2082 : ListTree (reverse (map mirror ts))\n     h\u2082 = node-ListTree h\u2081\n\n     h\u2083 : ListTree ((reverse (map mirror ts)) ++ (mirror \u00b7 t \u2237 []))\n     h\u2083 = ++-ListTree h\u2082 (consLT (mirror-Tree Tt) nilLT)\n\n     h\u2084 : ListTree (reverse (mirror \u00b7 t \u2237 map mirror ts))\n     h\u2084 = subst ListTree ( sym (reverse-\u2237' (mirror \u00b7 t) (map mirror ts))) h\u2083\n\n     helper : ListTree (reverse (map mirror (t \u2237 ts)))\n     helper = subst ListTree\n              (subst (\u03bb x \u2192 reverse x \u2261 reverse (map mirror (t \u2237 ts)))\n                     (map-\u2237 mirror t ts)\n                     refl)\n              h\u2084\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Mirror\/Tree\/ClosuresI.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1f8ca472a406e19639d68975dc6221ef77e3268d","subject":"Update Agda tutorial code","message":"Update Agda tutorial code\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/AgdaBasics.agda","new_file":"agda\/AgdaBasics.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"3ad59ab088501878d8a2cf57d3e993dc9edc0d31","subject":"added models","message":"added models\n","repos":"mietek\/epigram2,larrytheliquid\/pigit,mietek\/epigram2","old_file":"models\/Data.agda","new_file":"models\/Data.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check\n #-}\n\nmodule Data where\n\n_-_ : {A : Set}{B : A -> Set}{C : (a : A) -> B a -> Set} ->\n      ({a : A}(b : B a) -> C a b) -> (f : (a : A) -> B a) ->\n      (a : A) -> C a (f a)\nf - g = \\ x -> f (g x)\n\nid : {X : Set} -> X -> X\nid x = x\n\nkonst : {S : Set}{T : S -> Set} -> ({x : S} -> T x) -> (x : S) -> T x\nkonst c x = c\n\ndata Zero : Set where\nrecord One : Set where\n\nrecord Sig (S : Set)(T : S -> Set) : Set where\n  field\n    fst : S\n    snd : T fst\n\nopen module Sig' {S : Set}{T : S -> Set} = Sig {S}{T}\n\n_,_ : forall {S T}(s : S) -> T s -> Sig S T\ns , t = record {fst = s; snd = t}\n\ninfixr 40 _,_\n\n_*_ : Set -> Set -> Set\nS * T = Sig S \\ _ -> T\n\nsig : forall {S T}{P : Sig S T -> Set} ->\n      ((s : S)(t : T s) -> P (s , t)) -> (st : Sig S T) -> P st\nsig f st = f (fst st) (snd st)\n\ndata Two : Set where\n  tt : Two\n  ff : Two\n\ncond : {S : Two -> Set} -> S tt -> S ff -> (b : Two) -> S b\ncond t f tt = t\ncond t f ff = f\n\n_+_ : Set -> Set -> Set\nS + T = Sig Two (cond S T)\n\n[_,_] : {S T : Set}{P : S + T -> Set} ->\n        ((s : S) -> P (tt , s)) -> ((t : T) -> P (ff , t)) ->\n        (x : S + T) -> P x\n[ f , g ] = sig (cond f g)\n\ndata PROP : Set where\n  Absu : PROP\n  Triv : PROP\n  _\/\\_ : PROP -> PROP -> PROP\n  All : (S : Set) -> (S -> PROP) -> PROP\n\nPrf : PROP -> Set\nPrf Absu = Zero\nPrf Triv = One\nPrf (P \/\\ Q) = Prf P * Prf Q\nPrf (All S P) = (x : S) -> Prf (P x)\n\ndata CON (I : Set) : Set where\n  ?? : I -> CON I\n  PrfC : PROP -> CON I\n  _*C_ : CON I -> CON I -> CON I\n  PiC : (S : Set) -> (S -> CON I) -> CON I\n  SiC : (S : Set) -> (S -> CON I) -> CON I\n  MuC : (O : Set) -> (O -> CON (I + O)) -> O -> CON I\n  NuC : (O : Set) -> (O -> CON (I + O)) -> O -> CON I\n\nmutual\n  [|_|] : {I : Set} -> CON I -> (I -> Set) -> Set\n  [| ?? i |]       X = X i\n  [| PrfC P |]     X = Prf P\n  [| C *C D |]     X = [| C |] X * [| D |] X\n  [| PiC S C |]    X = (s : S) -> [| C s |] X\n  [| SiC S C |]    X = Sig S \\ s -> [| C s |] X\n  [| MuC O C o |]  X = Mu C X o\n  [| NuC O C o |]  X = Nu C X o\n\n  data Mu {I O : Set}(C : O -> CON (I + O))(X : I -> Set)(o : O) : Set where\n    con : [| C o |] [ X , Mu C X ] -> Mu C X o\n  codata Nu {I O : Set}(C : O -> CON (I + O))(X : I -> Set)(o : O) : Set where\n    con : [| C o |] [ X , Nu C X ] -> Nu C X o\n\nnoc :  {I O : Set}{C : O -> CON (I + O)}{X : I -> Set}{o : O} ->\n       Nu C X o -> [| C o |] [ X , Nu C X ]\nnoc (con xs) = xs\n\nmutual\n  map : {I : Set}{X Y : I -> Set} -> (C : CON I) -> ((i : I) -> X i -> Y i) ->\n        [| C |] X -> [| C |] Y\n  map (?? i)       f = f i\n  map (PrfC P)     f = id\n  map (C *C D)     f = sig \\ c d -> map C f c , map D f d\n  map (PiC S C)    f = \\ c s -> map (C s) f (c s)\n  map (SiC S C)    f = sig \\ s c -> s , map (C s) f c\n  map (MuC O C o)  f = fold C (Mu C _) (\\ o xs -> con (map (C o) [ f , konst id ] xs)) o\n  map (NuC O C o)  f = unfold C (Nu C _) (\\ o ss -> map (C o) [ f , konst id ] (noc ss)) o\n\n  fold :  {I O : Set}(C : O -> CON (I + O)){X : I -> Set}(Y : O -> Set) ->\n          ((o : O) -> [| C o |] [ X , Y ] -> Y o) ->\n          (o : O) -> Mu C X o -> Y o\n  fold C Y f o (con xs) = f o (map (C o) [ konst id , fold C Y f ] xs)\n\n  unfold :  {I O : Set}(C : O -> CON (I + O)){X : I -> Set}(Y : O -> Set) ->\n            ((o : O) -> Y o -> [| C o |] [ X , Y ]) ->\n            (o : O) -> Y o -> Nu C X o\n  unfold C Y f o y = con (map (C o) [ konst id , unfold C Y f ] (f o y))\n\ninduction : {I O : Set}(C : O -> CON (I + O)){X : I -> Set}\n            (P : (o : O) -> Mu C X o -> Set) ->\n            ((o : O)(xyps : [| C o |] [ X , (\\ o -> Sig (Mu C X o) (P o)) ]) ->\n             P o (con (map (C o) [ konst id , (\\ o -> fst) ] xyps))) ->\n            (o : O)(y : Mu C X o) -> P o y\ninduction C P p o (con xys) =\n  p o (map (C o) [ konst id , (\\ o y -> (y , induction C P p o y)) ] xys)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Data.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"607f06d6ab4ee2bea51b2ad939f153be21100ec8","subject":"Lamport.Merkle: verifkey is a single hash, sig has half secrets and half hashes","message":"Lamport.Merkle: verifkey is a single hash, sig has half secrets and half hashes\n","repos":"crypto-agda\/crypto-agda","old_file":"Crypto\/Sig\/Lamport\/Merkle.agda","new_file":"Crypto\/Sig\/Lamport\/Merkle.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function using (id; _\u2218_; _$_; flip)\nopen import Data.Nat.NP hiding (_==_)\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Product.NP renaming (map to \u00d7-map)\nopen import Data.Bit hiding (_==_)\nopen import Data.Bits\nopen import Data.Bits.Properties\nopen import Data.Vec.NP\nopen import Relation.Binary.PropositionalEquality.NP\nopen \u2261-Reasoning\nimport Crypto.Sig.Lamport\n\nmodule Crypto.Sig.Lamport.Merkle\n          (#digest   : \u2115)\n          (#seed     : \u2115)\n          (#secret   : \u2115)\n          (#message  : \u2115)\n          (hash-secret    : Bits #secret \u2192 Bits #digest)\n          (hash-node      : Bits (2* #digest) \u2192 Bits #digest)\n          (hash-verif-key : Bits (#message * 2* #digest) \u2192 Bits #digest)\n          (seed-expansion : Bits #seed   \u2192 Bits (#message * 2* #secret))\n          where\n\nmodule B = Crypto.Sig.Lamport #digest #seed #secret #message hash-secret seed-expansion\nmodule B1 = B.OTS1\n\nopen B using (H\u00b2)\n\nH = hash-secret\n\n#signkey   = B.#signkey\n#verifkey  = #digest\n#signature = B.#signature + #message * #digest\n\nDigest     = Bits #digest\nSeed       = Bits #seed\nSecret     = Bits #secret\nSignKey    = Bits #signkey\nMessage    = Bits #message\nSignature  = Bits #signature\nVerifKey   = Bits #verifkey\n\nmodule signature (sig : Signature) where\n  sigB        = take B.#signature sig\n  module sigB = B.signature sigB\n  sigB1s      = sigB.sig1s\n  sigL1s      = map H sigB1s\n  sigH        = drop B.#signature sig\n  sigH1s      = group #message #digest sigH\n\nverif-key : SignKey \u2192 VerifKey\nverif-key = hash-verif-key\n          \u2218 B.verif-key\n\nmodule signkey (sk : SignKey) where\n  open B.signkey sk public renaming (vk to vkF) -- F\"ull\"\n  vk   = verif-key sk\n\nkey-gen : Seed \u2192 VerifKey \u00d7 SignKey\nkey-gen s = vk , sk\n  module key-gen where\n    sk = seed-expansion s\n    vk = verif-key sk\n\nco-sign : Bits (2* #secret) \u2192 Bit \u2192 Digest\nco-sign s b = H (B1.sign s (not b))\n\nsign : SignKey \u2192 Message \u2192 Signature\nsign sk m = sig\n  module sign where\n    open signkey sk public\n    sigL   = B.sign sk m\n    sigH1s = map co-sign sk1s \u229b m\n    sigB1s = map B1.sign sk1s \u229b m\n    sigL1s = map H sigB1s\n    sigH   = concat sigH1s\n    sig    = sigL ++ sigH\n    module sigL = B.sign sk m\n    module sig  = signature sig\n    sigH1s-spec : sig.sigH1s \u2261 sigH1s\n    sigH1s-spec = sig.sigH1s                  \u2261\u27e8 ap (group #message #digest) (drop-++ B.#signature sigL sigH) \u27e9\n                  group #message #digest sigH \u2261\u27e8 group-concat sigH1s \u27e9\n                  sigH1s                      \u220e\n    sigB1s-spec : sig.sigB1s \u2261 sigB1s\n    sigB1s-spec = sig.sigB1s                  \u2261\u27e8 ap (group #message #secret) (take-++ B.#signature sigL sigH) \u27e9\n                  group #message #secret sigL \u2261\u27e8 group-concat sigL.sig1s \u27e9\n                  sigB1s                      \u220e\n    sigL1s-spec : sig.sigL1s \u2261 sigL1s\n    sigL1s-spec = ap (map H) sigB1s-spec\n\ntwist-, : Bit \u2192 Digest \u2192 Digest \u2192 Digest \u00d7 Digest\ntwist-, b sL sH = proj\u2032 (sL , sH) b , proj\u2032 (sH , sL) b\n\ntwist-++ : \u2200 {n}(b : Bit)(sL sH : Bits n) \u2192 Bits (2* n)\ntwist-++ b sL sH = proj\u2032 (sL , sH) b ++ proj\u2032 (sH , sL) b\n\ntwist-++-map2* : \u2200 {m n}(f : Bits m \u2192 Bits n) b sL sH\n               \u2192 map2* m n f (twist-++ b sL sH) \u2261 twist-++ b (f sL) (f sH)\ntwist-++-map2* f 0b sL sH rewrite take-++ _ sL sH | drop-++ _ sL sH = refl\ntwist-++-map2* f 1b sL sH rewrite take-++ _ sH sL | drop-++ _ sH sL = refl\n\ntwist-++-not : \u2200 {n}(f : Bit \u2192 Bits n) b\n               \u2192 twist-++ b (f b) (f (not b)) \u2261 f 0b ++ f 1b\ntwist-++-not f 0b = refl\ntwist-++-not f 1b = refl\n\nverify : VerifKey \u2192 Message \u2192 Signature \u2192 Bit\nverify vk m sig = root-hash == vk\n  module verify where\n    open signature sig public\n    sig1s     = map twist-++ m \u229b sigL1s \u229b sigH1s\n    root-hash = hash-verif-key (concat sig1s)\n\nlemma1 : \u2200 sk b \u2192 twist-++ b (H (B1.sign sk b)) (H (B1.sign sk (not b))) \u2261 H\u00b2 sk\nlemma1 sk b =\n  twist-++ b (H (B1.sign sk b)) (H (B1.sign sk (not b)))\n      \u2261\u27e8 ! twist-++-map2* H b _ _ \u27e9\n  H\u00b2 (twist-++ b (B1.sign sk b) (B1.sign sk (not b)))\n      \u2261\u27e8 ap H\u00b2 (twist-++-not (B1.sign sk) b) \u27e9\n  H\u00b2 (B1.sign sk 0b ++ B1.sign sk 1b)\n      \u2261\u27e8 ap H\u00b2 (B1.sign-both-reveals-signkey sk) \u27e9\n  H\u00b2 sk\n      \u220e\n\nverify-correct-sig : \u2200 sk m \u2192 verify (verif-key sk) m (sign sk m) \u2261 1b\nverify-correct-sig sk m = ==-reflexive {xs = root-hash} lemma\n  where\n     vk = verif-key sk\n     sig = sign sk m\n     module sig = sign sk m\n     module sk = signkey sk\n     open verify vk m sig\n\n     lemma-pointwise : \u2200 (i : Fin #message) \u2192\n           (map twist-++ m \u229b sig.sigL1s \u229b sig.sigH1s \u203c i) \u2261 \n           (map H\u00b2 sig.sk1s \u203c i)\n     lemma-pointwise i =\n       (map twist-++ m \u229b sig.sigL1s \u229b sig.sigH1s \u203c i)\n           -- Pushes the lookup down the operations map\/\u229b\n           \u2261\u27e8 \u203c-\u229b= i {map twist-++ m \u229b sig.sigL1s}\n                (\u203c-\u229b= i {map twist-++ m}\n                   (\u203c-map= i twist-++ refl)\n                   (\u203c-map= i H (\u203c-\u229b= i (\u203c-map i _ _) refl)))\n                (\u203c-\u229b= i (\u203c-map i co-sign sig.sk1s) refl) \u27e9\n       twist-++ (m \u203c i) (H (B1.sign (sig.sk1s \u203c i) (m \u203c i))) (H (B1.sign (sig.sk1s \u203c i) (not (m \u203c i))))\n           \u2261\u27e8 lemma1 (sig.sk1s \u203c i) (m \u203c i) \u27e9\n       H\u00b2 (sig.sk1s \u203c i)\n           -- Wrap-up the lookup through the map\n           \u2261\u27e8 ! \u203c-map i H\u00b2 _ \u27e9\n       (map H\u00b2 sig.sk1s \u203c i)\n           \u220e\n\n     lemma : root-hash \u2261 vk\n     lemma = ap (hash-verif-key \u2218 concat)\n                (map twist-++ m \u229b sigL1s \u229b sigH1s\n                   -- Since the signature to be verified is known to come from\n                   -- the sign function, one can symbolically reduce the\n                   -- processing of the bit vector layout done on both parts of\n                   -- the signature (only take\/drop-++, group-concat).\n                   \u2261\u27e8 ap\u2082 _\u229b_ (ap\u2082 _\u229b_ refl sig.sigL1s-spec) sig.sigH1s-spec \u27e9\n                 map twist-++ m \u229b sig.sigL1s \u229b sig.sigH1s\n                   -- One proves this equality pointwise for all index in the\n                   -- message m.\n                   \u2261\u27e8 vec= lemma-pointwise \u27e9\n                 map B1.verif-key sk.sk1s \u220e)\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Crypto\/Sig\/Lamport\/Merkle.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a25c6bdce3cf37b5be3cecff23fd12d70e869018","subject":"Fixed Makefile.","message":"Fixed Makefile.\n\nIgnore-this: 31338435bfed927703838c661d7e459b\n\ndarcs-hash:20100409024937-3bd4e-9da79543efc45e9a8672175836d2da523fc3716a.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/OnlyAxioms\/RemoveQuantificationOverProofs.agda","new_file":"Test\/Succeed\/OnlyAxioms\/RemoveQuantificationOverProofs.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing the removal of the quantification over proofs of formulas\n------------------------------------------------------------------------------\n\nmodule Test.Succeed.OnlyAxioms.RemoveQuantificationOverProofs where\n\npostulate\n  D    : Set\n  succ : D \u2192 D\n\ndata N : D \u2192 Set where\n  -- zN : N zero\n  sN\u2081 : {n : D} \u2192  N n \u2192 N (succ n)\n  sN\u2082 : {n : D} \u2192 (Nn : N n) \u2192 N (succ n)\n\n-- The data constructors sN\u2081 and sN\u2082 must have the same translation,\n-- i.e. we must remove the quantification of the variable Nn on N n.\n\n{-# ATP hint sN\u2082 #-}\n\n-- The Agda internal type of sN\u2081\n{-\nEl (Type (Lit (LitLevel  0)))\n   (Pi {El (Type (Lit (LitLevel  0)))\n           (Def Test.RemoveQuantificationOverProofs.D [])}\n       (Abs \"n\" El (Type (Lit (LitLevel  0)))\n                   (Fun (El (Type (Lit (LitLevel  0)))\n                            (Def Test.RemoveQuantificationOverProofs.N [(Var 0 [])]))\n                        (El (Type (Lit (LitLevel  0)))\n                            (Def Test.RemoveQuantificationOverProofs.N\n                                 [(Def Test.RemoveQuantificationOverProofs.succ [(Var 0 [])])]\n                            )\n                        )\n                   )\n       )\n   )\n-}\n\n-- The Agda internal type of sN\u2082\n{-\nEl (Type (Lit (LitLevel  0)))\n   (Pi {El (Type (Lit (LitLevel  0)))\n           (Def Test.RemoveQuantificationOverProofs.D [])}\n       (Abs \"n\" El (Type (Lit (LitLevel  0)))\n                   (Pi (El (Type (Lit (LitLevel  0)))\n                           (Def Test.RemoveQuantificationOverProofs.N [(Var 0 [])]))\n                       (Abs \"Nn\" El (Type (Lit (LitLevel  0)))\n                                    (Def Test.RemoveQuantificationOverProofs.N\n                                         [(Def Test.RemoveQuantificationOverProofs.succ [(Var 1 [])])]\n                                    )\n                       )\n                   )\n       )\n   )\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/OnlyAxioms\/RemoveQuantificationOverProofs.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c65654d6762b588466c7c40ed08e6e6aab5275f7","subject":"Added Agsy test.","message":"Added Agsy test.\n\nIgnore-this: 8c40e2930df08b1bf59edbc3c2e7d27e\n\ndarcs-hash:20101203050959-3bd4e-f60a8e16080c611b7bdb958ee245e85297629e0c.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Agsy\/AgsyTest.agda","new_file":"Draft\/Agsy\/AgsyTest.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing Agsy using the Agda standard library\n------------------------------------------------------------------------------\n\nmodule AgsyTest where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\nopen \u2261-Reasoning\n\n------------------------------------------------------------------------------\n\n0+m\u2261m+0 : \u2200 m \u2192 0 + m \u2261 m + 0  -- via Agsy {-c}\n0+m\u2261m+0 zero    = refl\n0+m\u2261m+0 (suc n) = cong suc (0+m\u2261m+0 n)\n\n+-comm : \u2200 m n \u2192 m + n \u2261 n + m\n+-comm zero    zero    = refl -- via Agsy\n+-comm zero    (suc n) = sym (cong suc (+-comm n zero))  -- via Agsy\n+-comm (suc m) zero    = sym (cong suc (+-comm zero m))  -- via Agsy\n+-comm (suc m) (suc n) = {!-t 20!} -- Agsy: No solution found at time out (20s)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Agsy\/AgsyTest.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c5452f3f8cd9d3ac09a7b79754c7f06a512f0e12","subject":"ModInv: the final property is not finished","message":"ModInv: the final property is not finished\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Data\/Nat\/ModInv.agda","new_file":"lib\/Data\/Nat\/ModInv.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function\nopen import Data.Zero\nopen import Data.Nat.NP\nopen import Data.Nat.Properties\n-- open import Data.Nat.DivMod\nopen import Data.Nat.DivMod.NP\nopen import Data.Nat.Primality.NP\nopen import Data.Nat.Coprimality\nopen import Data.Nat.Divisibility\nopen import Data.Nat.GCD\nopen import Data.Product.NP\nopen import Data.Fin.NP using (Fin; Fin\u25b9\u2115; zero; suc; Fin\u25b9\u2115<)\nopen import Relation.Nullary.Decidable hiding (\u230a_\u230b)\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Data.Nat.ModInv (q : \u2115) (q-prime : Prime q) where\n\nprime\u22620 : \u2200 {x} \u2192 Prime x \u2192 False (x \u225f 0)\nprime\u22620 {zero}  ()\nprime\u22620 {suc _} _ = _\n\nq\u22620 : False (q \u225f 0)\nq\u22620 = prime\u22620 q-prime\n\n1<q : 1 < q\n1<q = prime-\u22652 q-prime\n\n0<q : 0 < q\n0<q = \u2115\u2264.trans (s\u2264s z\u2264n) 1<q\n\ngcd\u22611 : \u2200 n d \u2192 0 < n \u2192 n < q \u2192 GCD n q d \u2192 d \u2261 1\ngcd\u22611 n d 0<n n<q g\n  = GCD.unique g (GCD.sym (coprime-gcd (prime\u21d2coprime q q-prime n 0<n n<q)))\n\n\u230a_\u230b : \u2115 \u2192 \u2115\n\u230a x \u230b = x mod\u2115 q\n\n_\/q : \u2115 \u2192 \u2115\nx \/q = x div q\n\n\u2223\u2262\u2192< : \u2200 d q (q\u22620 : False (q \u225f 0)) \u2192 d \u2223 q \u2192 d \u2262 q \u2192 d < q\n\u2223\u2262\u2192< d zero () d\u2223q d\u2262q\n\u2223\u2262\u2192< d (suc q-1) q\u22620 d\u2223q d\u2262q = \u2264\u2262\u2192< (\u2223\u21d2\u2264 d\u2223q) d\u2262q\n\nopen \u2261-Reasoning\n\ndivMod-prop : \u2200 x \u2192 x \u2261 \u230a x \u230b + x \/q * q\ndivMod-prop x = divModProp\u2115 x q\n\nmod<q : \u2200 x \u2192 \u230a x \u230b < q\nmod<q x = mod\u2115<divisor x q {q\u22620}\n\n\u2265d*q : \u2200 x \u2192 x \u2265 (x \/q) * q\n\u2265d*q x = \u2261+-\u2265 x \u230a x \u230b (x \/q * q) (divMod-prop x)\n\nq\/q\u22611 : q \/q \u2261 1\nq\/q\u22611 = d\/d\u22611 q {q\u22620}\n\nmod-prop : \u2200 x \u2192 \u230a x \u230b \u2261 x \u2238 (x \/q) * q\nmod-prop x = mod\u2115-prop x {q}\n\nq-mod-q : \u230a q \u230b \u2261 0\nq-mod-q = mod-prop q \u2219\n  (q \u2238 q \/q * q \u2261\u27e8 ap (\u03bb z \u2192 q \u2238 z * q) q\/q\u22611 \u27e9\n   q \u2238 1 * q    \u2261\u27e8 ap (_\u2238_ q) (\u2115\u00b0.+-comm q 0) \u27e9\n   q \u2238 q        \u2261\u27e8 n\u2238n\u22610 q \u27e9\n   0 \u220e)\n\n0-mod : \u230a 0 \u230b \u2261 0\n0-mod = mod-prop 0 \u2219 0\u2238n\u22610 (0 \/q * q)\n\n{-\nq\u2238-mod : \u2200 {x} \u2192 q \u2265 x \u2192 \u230a q \u2238 x \u230b \u2261 q \u2238 \u230a x \u230b\nq\u2238-mod p = mod\u2115-s\u2238 {!!} {!p!}\n-}\n\n-- q+-mod : \u2200 {x} \u2192 \u230a q + x \u230b \u2261 \u230a x \u230b\n-- q+-mod = {!!}\n\n\u2238q-mod : \u2200 {x} \u2192 x \u2265 q \u2192 \u230a x \u2238 q \u230b \u2261 \u230a x \u230b\n\u2238q-mod = mod\u2115-\u2238s\n\n\u2238-*q-mod : \u2200 x y \u2192 x \u2265 y * q \u2192 \u230a x \u2238 y * q \u230b \u2261 \u230a x \u230b\n\u2238-*q-mod x zero H = idp\n\u2238-*q-mod x (suc y) H =\n  \u230a x \u2238 (q + y * q) \u230b \u2261\u27e8 ap \u230a_\u230b (! \u2238-assoc-+ x q _) \u27e9\n  \u230a x \u2238 q \u2238 y * q \u230b \u2261\u27e8 \u2238-*q-mod _ y (+\u2264\u2192\u2264\u2238 q H) \u27e9\n  \u230a x \u2238 q \u230b \u2261\u27e8 \u2238q-mod (+\u2264 q H) \u27e9\n  \u230a x \u230b \u220e\n\nmod-mod : \u2200 x \u2192 \u230a \u230a x \u230b \u230b \u2261 \u230a x \u230b\nmod-mod x = \u230a \u230a x \u230b \u230b \u2261\u27e8 ap \u230a_\u230b (mod-prop x) \u27e9\n            \u230a x \u2238 x \/q * q \u230b \u2261\u27e8 \u2238-*q-mod x (x \/q) (\u2265d*q x) \u27e9\n            \u230a x \u230b \u220e\n\n<-\/q-0 : \u2200 {x} \u2192 x < q \u2192 x \/q \u2261 0\n<-\/q-0 = div<\u22610\n\n<-mod : \u2200 x \u2192 x < q \u2192 \u230a x \u230b \u2261 x\n<-mod x x<q = mod-prop x \u2219 ap (\u03bb z \u2192 x \u2238 z * q) (<-\/q-0 x<q)\n\nk+-mod : \u2200 k {x} \u2192 \u230a k + \u230a x \u230b \u230b \u2261 \u230a k + x \u230b\nk+-mod k {x} =\n  \u230a k + \u230a x \u230b \u230b \u2261\u27e8 ap (\u03bb z \u2192 \u230a k + z \u230b) (mod-prop _) \u27e9\n  \u230a k + (x \u2238 x \/q * q) \u230b \u2261\u27e8 ap \u230a_\u230b (! +-\u2238-assoc k (\u2265d*q x)) \u27e9\n  \u230a k + x \u2238 x \/q * q \u230b \u2261\u27e8 \u2238-*q-mod (k + x) (x \/q)\n                           (let open \u2264-Reasoning in\n                            x \/q * q \u2264\u27e8 \u2265d*q x \u27e9\n                            x \u2264\u27e8 \u2264-steps\u2032 k \u27e9\n                            \u2115\u2264.reflexive (\u2115\u00b0.+-comm x k)) \u27e9\n  \u230a k + x \u230b \u220e\n\n+k-mod : \u2200 {x} k \u2192 \u230a \u230a x \u230b + k \u230b \u2261 \u230a x + k \u230b\n+k-mod {x} k =\n  \u230a \u230a x \u230b + k \u230b  \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.+-comm \u230a x \u230b k) \u27e9\n  \u230a k + \u230a x \u230b \u230b  \u2261\u27e8 k+-mod k \u27e9\n  \u230a k + x \u230b      \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.+-comm k x) \u27e9\n  \u230a x + k \u230b      \u220e\n\n+-mod : \u2200 x y \u2192 \u230a \u230a x \u230b + \u230a y \u230b \u230b \u2261 \u230a x + y \u230b\n+-mod x y = k+-mod \u230a x \u230b \u2219 +k-mod y\n\nsuc-mod : \u2200 x \u2192 \u230a suc \u230a x \u230b \u230b \u2261 \u230a suc x \u230b\nsuc-mod x = k+-mod 1\n\n1-mod : \u230a 1 \u230b \u2261 1\n1-mod = 1-mod\u2115 {q} 1<q \n\nk*-mod : \u2200 k {x} \u2192 \u230a k * \u230a x \u230b \u230b \u2261 \u230a k * x \u230b\nk*-mod zero = idp\nk*-mod (suc k) {x} =\n  \u230a \u230a x \u230b +   k * \u230a x \u230b   \u230b \u2261\u27e8 +k-mod (k * \u230a x \u230b) \u27e9\n  \u230a   x   +   k * \u230a x \u230b   \u230b \u2261\u27e8 ! k+-mod x \u27e9\n  \u230a   x   + \u230a k * \u230a x \u230b \u230b \u230b \u2261\u27e8 ap (\u03bb z \u2192 \u230a x + z \u230b) (k*-mod k {x}) \u27e9\n  \u230a   x   + \u230a k *   x   \u230b \u230b \u2261\u27e8 k+-mod x \u27e9\n  \u230a   x   +   k *   x     \u230b \u220e\n\n*k-mod : \u2200 {x} k \u2192 \u230a \u230a x \u230b * k \u230b \u2261 \u230a x * k \u230b\n*k-mod {x} k = \n  \u230a \u230a x \u230b * k \u230b  \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.*-comm \u230a x \u230b k) \u27e9\n  \u230a k * \u230a x \u230b \u230b  \u2261\u27e8 k*-mod k \u27e9\n  \u230a k * x \u230b      \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.*-comm k x) \u27e9\n  \u230a x * k \u230b      \u220e\n\n*-mod : \u2200 x y \u2192 \u230a \u230a x \u230b * \u230a y \u230b \u230b \u2261 \u230a x * y \u230b\n*-mod x y = k*-mod \u230a x \u230b \u2219 *k-mod y\n\nnq-mod-q : \u2200 n \u2192 \u230a n * q \u230b \u2261 0\nnq-mod-q n = \n  \u230a n * q \u230b     \u2261\u27e8 ! k*-mod n {q} \u27e9\n  \u230a n * \u230a q \u230b \u230b \u2261\u27e8 ap (\u230a_\u230b \u2218 _*_ n) q-mod-q \u27e9\n  \u230a n * 0     \u230b \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.*-comm n 0) \u27e9\n  \u230a 0 \u230b         \u2261\u27e8 0-mod \u27e9\n  0             \u220e\n\n+q-mod : \u2200 x \u2192 \u230a x + q \u230b \u2261 \u230a x \u230b\n+q-mod x = \n  \u230a x + q \u230b     \u2261\u27e8 ! k+-mod x {q} \u27e9\n  \u230a x + \u230a q \u230b \u230b \u2261\u27e8 ap (\u03bb z \u2192 \u230a x + z \u230b) q-mod-q \u27e9\n  \u230a x + 0     \u230b \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.+-comm x 0) \u27e9\n  \u230a x \u230b         \u220e\n\n+nq-mod : \u2200 x n \u2192 \u230a x + n * q \u230b \u2261 \u230a x \u230b\n+nq-mod x n =\n  \u230a x + n * q \u230b     \u2261\u27e8 ! k+-mod x {n * q} \u27e9\n  \u230a x + \u230a n * q \u230b \u230b \u2261\u27e8 ap (\u03bb z \u2192 \u230a x + z \u230b) (nq-mod-q n) \u27e9\n  \u230a x + 0         \u230b \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.+-comm x 0) \u27e9\n  \u230a x \u230b             \u220e\n\n{-\n\u2238k-mod : \u2200 k {x} \u2192 \u230a \u230a x \u230b \u2238 k \u230b \u2261 \u230a x \u2238 k \u230b\n\u2238k-mod k {x} = {!!}\n\nk\u2238-mod : \u2200 k {x} \u2192 \u230a k \u2238 \u230a x \u230b \u230b \u2261 \u230a k \u2238 x \u230b\nk\u2238-mod k {x} =\n  \u230a k \u2238 \u230a x \u230b \u230b \u2261\u27e8 {!!} \u27e9\n  \u230a k \u2238 (x \u2238 (x \/q) * q) \u230b \u2261\u27e8 {!!} \u27e9\n  \u230a k \u2238 (x \u2238 (x \/q) * q) \u230b \u2261\u27e8 {!!} \u27e9\n  \u230a k \u2238 x     \u230b \u220e\n-}\n\n+-*q-mod : \u2200 d y \u2192 \u230a d + y * q \u230b \u2261 \u230a d \u230b\n+-*q-mod d y =\n  \u230a d + y * q \u230b\n    \u2261\u27e8 ! k+-mod d \u27e9\n  \u230a d + \u230a y * q \u230b \u230b\n    \u2261\u27e8 ap (\u03bb x \u2192 \u230a d + x \u230b) (! k*-mod y) \u27e9\n  \u230a d + \u230a y * \u230a q \u230b \u230b \u230b\n    \u2261\u27e8 ap (\u03bb x \u2192 \u230a d + \u230a y * x \u230b \u230b) q-mod-q \u27e9\n  \u230a d + \u230a y * 0 \u230b \u230b\n    \u2261\u27e8 ap (\u03bb x \u2192 \u230a d + \u230a x \u230b \u230b) (\u2115\u00b0.*-comm y 0) \u27e9\n  \u230a d + \u230a 0 \u230b \u230b\n    \u2261\u27e8 ap (\u03bb x \u2192 \u230a d + x \u230b) 0-mod \u27e9\n  \u230a d + 0 \u230b\n    \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.+-comm _ 0) \u27e9\n  \u230a d \u230b \u220e\n\n1\/ : \u2115 \u2192 \u2115\n1\/ n with B\u00e9zout.lemma q n\n1\/ n | B\u00e9zout.result d g (B\u00e9zout.+- x y eq) = STUCK q n x y where postulate STUCK : (q n x y : \u2115) \u2192 \u2115 -- q \u2238 y\n1\/ n | B\u00e9zout.result d g (B\u00e9zout.-+ x y eq) = y\n\ngcd\u2262q : \u2200 n q d \u2192 n \u2262 0 \u2192 n < q \u2192 GCD n q d \u2192 d \u2262 q\ngcd\u2262q zero      q  _ n\u22620 n<q g _    = \ud835\udfd8-elim (n\u22620 refl)\ngcd\u2262q (suc n-1) q .q n\u22620 n<q g refl = no-<-> n<q q<n\n  where q<n = \u2223\u2262\u2192< q (suc n-1) _ (fst (GCD.commonDivisor g)) (<\u2192\u2262 n<q \u2218 !_)\n\nblah' : \u2200 x \u2192 suc x \u2238 x \u2261 1\nblah' zero = refl\nblah' (suc x) = blah' x\n\nblah : \u2200 x \u2192 0 < x \u2192 x \u2238 (x \u2238 1) \u2261 1\nblah zero ()\nblah (suc x) p = blah' x\n\n1<q+ : \u2200 x \u2192 1 < q + x\n1<q+ x = \u2115\u2264.trans (prime-\u22652 q-prime) (\u2264-steps\u2032 {q} x)\n\n{-\n*q-\u2238-mod : \u2200 x y \u2192 0 < y \u2192 y < x * q \u2192 \u230a x * q \u2238 y \u230b \u2261 q \u2238 \u230a y \u230b\n*q-\u2238-mod x y = mod\u2115-ns\u2238 {x} {y}\n-}\n\n{-\nTODO remove n < q assumption\n\n\n1\/-prop : \u2200 n \u2192 n \u2262 0 \u2192 n < q \u2192 \u230a 1\/ n * n \u230b \u2261 1\n1\/-prop n n\u22620 n<q with B\u00e9zout.lemma q n\n1\/-prop n n\u22620 n<q | B\u00e9zout.result d g (B\u00e9zout.-+ x y eq) =\n  \u230a y * n     \u230b \u2261\u27e8 ap \u230a_\u230b (! eq) \u27e9\n  \u230a d + x * q \u230b \u2261\u27e8 +-*q-mod d x \u27e9\n  \u230a d         \u230b \u2261\u27e8 ap \u230a_\u230b (gcd\u22611 n d (\u22620\u21d20< n n\u22620) n<q (GCD.sym g)) \u27e9\n  \u230a 1         \u230b \u2261\u27e8 1-mod \u27e9\n  1 \u220e\n1\/-prop n n\u22620 n<q | B\u00e9zout.result d g (B\u00e9zout.+- zero y eq) = {!!}\n1\/-prop n n\u22620 n<q | B\u00e9zout.result d g (B\u00e9zout.+- (suc x-1) y eq) =\n  \u230a (q \u2238 y)   * n       \u230b \u2261\u27e8 ap \u230a_\u230b (*-distrib-\u2238\u02b3 n q y) \u27e9\n  \u230a q * n \u2238 y * n       \u230b \u2261\u27e8 ap \u230a_\u230b (ap (_\u2238_ (q * n)) (+-\u2238' d (y * n) (x * q) eq)) \u27e9\n  \u230a q * n \u2238 (x * q \u2238 d) \u230b \u2261\u27e8 ap (\u03bb z \u2192 \u230a q * n \u2238 (x * q \u2238 z) \u230b) d\u22611 \u27e9\n  \u230a q * n \u2238 (x * q \u2238 1) \u230b \u2261\u27e8 ap (\u03bb z \u2192 \u230a z \u2238 (x * q \u2238 1) \u230b) (\u2115\u00b0.*-comm q n) \u27e9\n  \u230a n * q \u2238 (x * q \u2238 1) \u230b \u2261\u27e8 *q-\u2238-mod n _ {!!} xq\u22381<nq \u27e9\n  q \u2238 \u230a x * q \u2238 1 \u230b \u2261\u27e8 ap (_\u2238_ q) (*q-\u2238-mod x _ (s\u2264s z\u2264n) (1<q+ (x-1 * q))) \u27e9\n  q \u2238 (q \u2238 \u230a 1 \u230b) \u2261\u27e8 ap (\u03bb z \u2192 q \u2238 (q \u2238 z)) 1-mod \u27e9\n  q \u2238 (q \u2238 1) \u2261\u27e8 blah q 0<q \u27e9\n  {-\n  \u230a n * q \u2238 (x * q \u2238 1) \u230b \u2261\u27e8 *q-\u2238-mod n _ {!+\u2264!} \u27e9\n  \u230a x * q \u2238 1           \u230b \u2261\u27e8 *q-\u2238-mod x _ (+\u2264 1 (\u2115\u2264.reflexive e)) \u27e9\n  \u230a 1 \u230b                   \u2261\u27e8 1-mod \u27e9\n  -}\n  1                       \u220e\n  where d\u22611 = gcd\u22611 n d (\u22620\u21d20< n n\u22620) n<q (GCD.sym g)\n        e   = ap\u2082 _+_ (! d\u22611) idp \u2219 eq\n        x = suc x-1\n        xq<nq+1 : x * q < n * q + 1\n        xq<nq+1 = {!!}\n\n\n        xq\u22381<nq : x * q \u2238 1 < n * q\n        xq\u22381<nq = {!!}\n\n1\/-prop' : \u2200 n \u2192 n \u2262 0 \u2192 n < q \u2192 \u230a n * 1\/ n \u230b \u2261 1\n1\/-prop' n n\u22620 n<q = \u230a n * 1\/ n \u230b \u2261\u27e8 ap \u230a_\u230b (\u2115\u00b0.*-comm n (1\/ n)) \u27e9\n                     \u230a 1\/ n * n \u230b \u2261\u27e8 1\/-prop n n\u22620 n<q \u27e9\n                     1            \u220e\n-}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Data\/Nat\/ModInv.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"40d09d0cc473ebc134d6f07fa3aa34ee7ec84d61","subject":"A(nother) model of Containers","message":"A(nother) model of Containers","repos":"kwangkim\/pigment,kwangkim\/pigment,brixen\/Epigram,kwangkim\/pigment","old_file":"models\/Containers.agda","new_file":"models\/Containers.agda","new_contents":"{-# OPTIONS --type-in-type --no-termination-check\n  #-}\n\nmodule Containers where\n\n  record Sigma (A : Set) (B : A -> Set) : Set where\n    field fst : A\n          snd : B fst \n\n  open Sigma\n\n  _*_ : (A B : Set) -> Set\n  A * B = Sigma A \\_ -> B\n\n  data Zero : Set where\n \n  record One : Set where\n\n  data PROP : Set where\n    Absu : PROP\n    Triv : PROP\n    _\/\\_ : PROP -> PROP -> PROP\n    All : (S : Set) -> (S -> PROP) -> PROP\n\n  Prf : PROP -> Set\n  Prf Absu = Zero\n  Prf Triv = One\n  Prf (P \/\\ Q) = Prf P * Prf Q\n  Prf (All S P) = (x : S) -> Prf (P x)\n\n  _,_ : forall {A B} -> (a : A) -> B a -> Sigma A B\n  a , b = record { fst = a ; snd = b }\n\n  split : \u2200 {A B} {C : Sigma A B -> Set} -> \n    (f : (a : A) -> (b : B a) -> C (a , b)) -> (ab : Sigma A B) -> C ab \n  split f ab = f (fst ab) (snd ab)\n\n  data _+_ (A B : Set) : Set where\n    l : A -> A + B\n    r : B -> A + B\n\n  cases : \u2200 {A B} {C : A + B -> Set} -> \n            ((a : A) -> C (l a)) -> ((b : B) -> C (r b)) -> \n            (ab : A + B) -> C ab\n  cases f g (l a) = f a\n  cases f g (r b) = g b \n\n  rearrange : \u2200 {A B X} (C : A \u2192 Set) (D : B \u2192 Set) (f : Sigma A C \u2192 X) \u2192 \n              Sigma (A + B) (cases C D) \u2192 X + Sigma B D\n  rearrange {A} {B} {X} C D f = split {A + B} (cases (\\a c \u2192 l (f (a , c))) (\\b d \u2192 r (b , d)))\n\n  data CON (I : Set) : Set where\n    ?? : I -> CON I\n    PrfC : PROP -> CON I\n    _*C_ : CON I -> CON I -> CON I\n    PiC : (S : Set) -> (S -> CON I) -> CON I\n    SiC : (S : Set) -> (S -> CON I) -> CON I\n    MuC : (O : Set) -> (O -> CON (I + O)) -> O -> CON I\n\n  mutual\n \n    data Mu {I : Set} (O : Set) (D : O -> CON (I + O)) \n            (X : I -> Set) : O -> Set where\n      inm : {o : O} -> [| D o |] (cases X (Mu O D X)) ->  Mu O D X o\n\n    [|_|]_ : {I : Set} -> (CON I) -> (I -> Set) -> Set\n    [| ?? i |] X = X i\n    [| PrfC p |] X = Prf p\n    [| C *C D |] X = [| C |] X * [| D |] X\n    [| PiC S C |] X = (s : S) -> [| C s |] X\n    [| SiC S C |] X = Sigma S \\s -> [| C s |] X\n    [| MuC O C o |] X = Mu O C X o\n\n\n  outm : {I O : Set} {D : O -> CON (I + O)} {X : I -> Set} {o : O} ->\n    Mu O D X o -> [| D o |]  (cases X (Mu O D X))\n  outm (inm x) = x \n\n\n  Everywhere : {I : Set} -> (D : CON I) -> (J : I -> Set) -> (X : Set) ->  \n               (Sigma I J -> X) ->  [| D |] J -> CON X\n  Everywhere (?? i) J X c t = ?? (c (i , t))\n  Everywhere (PrfC p) J X c t = PrfC Triv\n  Everywhere (C *C D) J X c t = \n    Everywhere C J X c (fst t) *C Everywhere D J X c (snd t)\n  Everywhere (PiC S C) J X c t = PiC S (\\s -> Everywhere (C s) J X c (t s))\n  Everywhere (SiC S C) J X c t = Everywhere (C (fst t)) J X c (snd t)\n  Everywhere (MuC O C o) J X c t = MuC (Sigma O (Mu O C J)) (\\ot' -> \n        Everywhere (C (fst ot')) \n                   (cases J (Mu O C J)) \n                   (X + (Sigma O (Mu O C J)))\n                   (split (cases(\\i j -> l (c (i , j))) \n                                (\\o'' t'' -> r (o'' , t'')))) \n                   (outm (snd ot'))) \n     (o , t)\n\n  everywhere : {I : Set} -> (D : CON I) -> (J : I -> Set) -> (X : Set) ->\n               (c : Sigma I J -> X) -> (Y : X -> Set) -> \n               (f : (ij : Sigma I J) -> Y (c ij)) -> (t : [| D |] J) -> \n               [| Everywhere D J X c t |] Y\n  everywhere (?? i) J X c Y f t = f (i , t)\n  everywhere (PrfC p) J X c Y f t = _\n  everywhere (C *C D) J X c Y f t = \n    everywhere C J X c Y f (fst t) , everywhere D J X c Y f (snd t)\n  everywhere (PiC S C) J X c Y f t = \\s -> everywhere (C s) J X c Y f (t s)\n  everywhere (SiC S C) J X c Y f t = everywhere (C (fst t)) J X c Y f (snd t)\n  everywhere (MuC O C o) J X c Y f (inm t) = \n    inm (everywhere (C o) (cases J (Mu O C J)) (X + Sigma O (Mu O C J)) (\n        (split (cases (\\i j -> l (c (i , j))) (\\o t -> r (o , t))))) \n        (cases Y (split \\o'' t'' -> [| Everywhere (MuC O C o'') J X c t'' |] Y))\n        (split (cases (\\i j -> f (i , j)) \n                      (\\o'' t'' -> everywhere (MuC O C o'') J X c Y f t''))) \n        t)\n \n  induction : {I O : Set} (D : O -> CON (I + O)) (J : I -> Set) (X : Set) \n              (c : Sigma I J -> X) (Y : X -> Set) (f : (ij : Sigma I J) -> Y (c ij)) \n              (P : Sigma O (Mu O D J) -> Set) ->\n              ({o : O} (v : [| D o |] cases J (Mu O D J)) -> \n               [| Everywhere (D o) _ (X + Sigma O (Mu O D J)) \n                             (split (cases (\\i j -> l (c (i , j))) \n                             (\\o v -> r (o , v)))) v |] cases Y P -> \n               P (o , inm v)) ->\n              (o : O) (v : Mu O D J o) -> P (o , v)\n  induction {I} {O} D J X c Y f P p o (inm v) = \n    p v (everywhere {I + O} (D o) \n                    (\u03bb x -> cases {I} {O} {\\_ -> Set} J \n                    (Mu O D J) x) ((X + Sigma O (Mu O D J))) \n                    (rearrange J (Mu O D J) c) (cases Y P) \n                    (split {I + O} (cases (\\i j -> f (i , j)) \n                    (\\o v -> induction D J X c Y f P p o v))) v) ","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Containers.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c870f1b9765ad2e1d47422ec2bea71fd83994175","subject":"bit-guessing-game.agda","message":"bit-guessing-game.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"bit-guessing-game.agda","new_file":"bit-guessing-game.agda","new_contents":"open import program-distance\nopen import flipbased-implem\nopen import Data.Bits\nopen import Data.Product\nopen import Relation.Nullary\n\nmodule bit-guessing-game (prgDist : PrgDist) where\n  open PrgDist prgDist\n\n  GuessAdv : Coins \u2192 Set\n  GuessAdv c = \u21ba c Bit\n\n  runGuess\u2141 : \u2200 {ca} (A : GuessAdv ca) (b : Bit) \u2192 \u21ba ca Bit\n  runGuess\u2141 A _ = A\n\n  -- An oracle: an adversary who can break the guessing game.\n  Oracle : Coins \u2192 Set\n  Oracle ca = \u2203 (\u03bb (A : GuessAdv ca) \u2192 breaks (runGuess\u2141 A))\n\n  -- Any adversary cannot do better than a random guess.\n  GuessSec : Coins \u2192 Set\n  GuessSec ca = \u2200 (A : GuessAdv ca) \u2192 \u00ac(breaks (runGuess\u2141 A))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'bit-guessing-game.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6c4f1afe15f2e46bc0099cdf77a7d95a2eee39e0","subject":"start proving things for String Fiat Shamir","message":"start proving things for String Fiat Shamir\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/Strong-Fiat-Shamir.agda","new_file":"ZK\/Strong-Fiat-Shamir.agda","new_contents":"open import Type\nopen import Data.Maybe\nopen import Data.Two\nopen import Data.Product.NP\nopen import Relation.Nullary.Decidable\nopen import Relation.Nullary\nopen import Relation.Binary.PropositionalEquality\n\nopen import Control.Strategy\n\nmodule ZK.Strong-Fiat-Shamir\n  {W S : \u2605}{L : W \u2192 S \u2192 \u2605}{Prf : S \u2192 \u2605}\n  {RP RV RS : \u2605}{SimState : \u2605}{Q : \u2605}{Resp :    \u2605}\n  (L? : \u2200 w Y \u2192 Dec (L w Y))(L-to-Prf : \u2200 {w Y} \u2192 L w Y \u2192 Prf Y)\n  where\n\nRandom-Oracle : \u2605\nRandom-Oracle = Q \u2192 Resp\n\nState : (S A : \u2605) \u2192 \u2605\nState S A = S \u2192 A \u00d7 S\n\nrecord Proof-System : \u2605 where\n  field\n    Prove : RP \u2192  (w : W)(Y : S) \u2192 Prf Y\n    Verify : RV \u2192 (Y : S)(\u03c0 : Prf Y) \u2192 \ud835\udfda\n\nrecord Simulator (PF : Proof-System) : \u2605 where\n  open Proof-System PF\n  field\n    init : RS \u2192 SimState\n    H : Q \u2192 State SimState Resp\n    Simulate : (Y : S) \u2192 State SimState (Prf Y)\n    verify-sim-spec : (rv : RV)(Y : S)(s : SimState) \u2192 let \u03c0 = proj\u2081 (Simulate Y s)\n                     in Verify rv Y \u03c0 \u2261 1\u2082\n\nmodule Is-Zero-Knowledge\n  (PF : Proof-System)\n  (sim : Simulator PF)\n  where\n\n  open Simulator sim\n\n  data Adversary-Query : \u2605 where\n    query-H : (q : Q) \u2192 Adversary-Query\n    query-create-proof : (w : W)(Y : S) \u2192 Adversary-Query\n\n  Challenger-Resp : Adversary-Query \u2192 \u2605\n  Challenger-Resp (query-H s) = Resp\n  Challenger-Resp (query-create-proof w Y) = Maybe (Prf Y)\n\n  Adversary : \u2605\n  Adversary = Strategy Adversary-Query Challenger-Resp \ud835\udfda\n\n  simulated-Challenger : \u2200 q \u2192 State SimState (Challenger-Resp q)\n  simulated-Challenger (query-H q) s = H q s\n  simulated-Challenger (query-create-proof w Y) s with L? w Y\n  simulated-Challenger (query-create-proof w Y) s | yes p = first just (Simulate Y s)\n  simulated-Challenger (query-create-proof w Y) s | no \u00acp = nothing , s\n\n  open StatefulRun\n  simulated : SimState \u2192 Adversary \u2192 \ud835\udfda\n  simulated s adv = evalS simulated-Challenger adv s\n\n  module _ (ro : Random-Oracle) where\n\n    real-Challenger : \u2200 q \u2192 Challenger-Resp q\n    real-Challenger (query-H q) = ro q\n    real-Challenger (query-create-proof w Y) with L? w Y\n    ... | yes p = just (L-to-Prf p)\n    ... | no  _ = nothing\n\n    real : Adversary \u2192 \ud835\udfda\n    real adv = run real-Challenger adv\n\n    Experiment : \ud835\udfda \u2192 RS \u2192 Adversary \u2192 \ud835\udfda\n    Experiment 0\u2082 rs = real\n    Experiment 1\u2082 rs = simulated (init rs)\n\n{-\n-- there exists a simulator, such that for all adversaries they are clueless if\n-- they are in the real or simulated Experiment\nZero-Knowledge : Proof-System \u2192 \u2605\nZero-Knowledge PF = \u03a3 (Simulator PF) (\u03bb sim \u2192 {!!})\n-}\n\nmodule Sigma-Protocol\n  (Commitment Challenge : \u2605)\n  (\u03a3-Prf : S \u2192 \u2605)\n  {R\u03a3P : \u2605}\n  (PF : Proof-System)\n  where\n  open Proof-System PF\n\n  record \u03a3-Prover : \u2605 where\n    field\n      A : R\u03a3P \u2192 (Y : S) \u2192 Commitment\n      f : R\u03a3P \u2192 (Y : S) \u2192 (c : Challenge) \u2192 \u03a3-Prf Y\n\n  \u03a3-Verifier : \u2605\n  \u03a3-Verifier = (Y : S)(A : Commitment)(c : Challenge)\n             \u2192 \u03a3-Prf Y \u2192 \ud835\udfda\n\n  Correct : \u03a3-Verifier \u2192 \u03a3-Prover \u2192 \u2605\n  Correct v p = \u2200 {Y w} \u2192 L w Y \u2192 (r : R\u03a3P)(c : _) \u2192\n    let open \u03a3-Prover p\n    in v Y (A r Y) c (f r Y c) \u2261 1\u2082\n\n  module Special-Honest-Verifier-Zero-Knowledge\n    where\n\n    record shvzk : \u2605 where\n\n\nmodule Simulation-Sound-Extractability\n  (PF : Proof-System)\n  (sim : Simulator PF)\n  where\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/Strong-Fiat-Shamir.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6fa34070fc177fe60f86961ed9727d7cadbab004","subject":"schnorr.agda","message":"schnorr.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"schnorr.agda","new_file":"schnorr.agda","new_contents":"open import Data.Bool using (Bool; true; false) -- ; if_then_else_; _\u2228_)\n{-\nopen import Function\nopen import Data.Nat using (\u2115)\nopen import Data.Fin using (Fin; zero; suc)\n-}\n-- open import Data.Maybe using (Maybe; nothing; just)\nopen import Data.Product using (_\u00d7_; _,_; proj\u2081; proj\u2082)\n\nmodule schnorr\n-- (q : \u2115) -- Prime q\n\n(\u2124q : Set)\n\n(G : Set) -- = \u27e8\u2124p\u27e9\u2605 should for a group of order q\n(g : G)   -- generator of the group G\n(_^_  : G \u2192 \u2124q \u2192 G)\n(_\u2219_  : G \u2192 G \u2192 G)\n\n(_\u229f_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n(_\u22a0_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n\n(\u2124q\u2605 : Set) -- \u2124q excluding 0\n(weaken : \u2124q\u2605 \u2192 \u2124q)\n\n(_==_ : \u2124q \u2192 \u2124q \u2192 Bool)\n-- (_==\u1d33_ : G \u2192 G \u2192 Bool)\n\n(M : Set)\n(\u210b\u27e8_\u2225_\u27e9 : G \u2192 M \u2192 \u2124q)  -- I flipped the order of arguments to fit the\n                        -- \"random prefix\" assumption\nwhere\n\nSig : Set\nSig = \u2124q \u00d7 \u2124q\n\n-- Do not call directly this function, use Sign\nSign\u2605 : (m : M) (x : \u2124q) (k : \u2124q) \u2192 Sig\nSign\u2605 m x k = (s , e)\n\n  where r = g ^ k\n        e = \u210b\u27e8 r \u2225 m \u27e9\n        s = k \u229f (x \u22a0 e)\n\nSign : (m : M) (x : \u2124q\u2605) (k : \u2124q\u2605) \u2192 Sig\nSign m x\u2032 k\u2032 = Sign\u2605 m (weaken x\u2032) (weaken k\u2032)\n\n-- Reject the signature if 0 < r < q or 0 < s < q is not satisfied.\nVer : (m : M) (g\u02e3 : G) (sig : Sig) \u2192 Bool\nVer m g\u02e3 (s , e) = e\u1d65 == e\n\n  module Ver where r\u1d65 = (g ^ s) \u2219 (g\u02e3 ^ e)\n                   e\u1d65 = \u210b\u27e8 r\u1d65 \u2225 m \u27e9\n\nopen import Relation.Binary.PropositionalEquality\n\n_^\u2032_ : G \u2192 \u2124q\u2605 \u2192 G\ng ^\u2032 x = g ^ weaken x\n\nmodule Proof\n  (^-\u229f : \u2200 \u03b1 x y \u2192 (\u03b1 ^ (x \u229f y)) \u2219 (\u03b1 ^ y) \u2261 \u03b1 ^ x)\n  (^-\u22a0 : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 \u03b1 ^(x \u22a0 y))\n  (==-refl : \u2200 {x} \u2192 x == x \u2261 true)\n  (m : M) (x k : \u2124q) where\n\n  g\u02e3  = g ^ x\n  sig = Sign\u2605 m x k\n  s   = proj\u2081 sig\n  e   = proj\u2082 sig\n\n  r   = g ^ k\n\n  pf1 : e \u2261 \u210b\u27e8 r \u2225 m \u27e9\n  pf1 = refl\n  pf2 : s \u2261 k \u229f (x \u22a0 e)\n  pf2 = refl\n\n  open Ver m g\u02e3 s e\n\n  r\u1d65\u2261r : r\u1d65 \u2261 r\n  r\u1d65\u2261r rewrite ^-\u22a0 g x e\n             | ^-\u229f g k (x \u22a0 e) = refl\n             \n  e\u1d65\u2261e : e\u1d65 \u2261 e\n  e\u1d65\u2261e rewrite r\u1d65\u2261r = refl\n\n  pf : Ver m g\u02e3 (Sign\u2605 m x k) \u2261 true\n  pf rewrite e\u1d65\u2261e = ==-refl\n\nmodule HashFunctionRequirements\n    (Rd : Set)\n    (Message : Set)\n    (Hash : Set)\n    (_==\u1d34_ : Hash \u2192 Hash \u2192 Bool)\n    (\u210b\u27e8_\u2225_\u27e9 : Rd \u2192 Message \u2192 Hash)\n    where\n\n    RPPAdv : Set \u2192 Set\n    RPPAdv R\u2090 = (R\u2090 \u2192 Hash)\n              \u00d7 (R\u2090 \u2192 Rd \u2192 Message)\n\n    RPP-advantage : \u2200 {R\u2090} \u2192 RPPAdv R\u2090 \u2192 (R\u2090 \u00d7 Rd) \u2192 Bool\n    RPP-advantage (A\u2080 , A\u2081) (r\u2090 , rd) = \u210b\u27e8 rd \u2225 m \u27e9 ==\u1d34 h\n      where h = A\u2080 r\u2090\n            m = A\u2081 r\u2090 rd\n\n    RPSPAdv : Set \u2192 Set\n    RPSPAdv R\u2090 = (R\u2090 \u2192 Message)\n               \u00d7 (R\u2090 \u2192 Rd \u2192 Message)\n\n    RPSP-advantage : \u2200 {R\u2090} \u2192 RPSPAdv R\u2090 \u2192 (R\u2090 \u00d7 Rd) \u2192 Bool\n    RPSP-advantage (A\u2080 , A\u2081) (r\u2090 , rd) = \u210b\u27e8 rd \u2225 m\u2080 \u27e9 ==\u1d34 \u210b\u27e8 rd \u2225 m\u2081 \u27e9\n      where m\u2080 = A\u2080 r\u2090\n            m\u2081 = A\u2081 r\u2090 rd\n\nRd = G\nMessage = M\nHash = \u2124q\nopen HashFunctionRequirements Rd Message Hash _==_ \u210b\u27e8_\u2225_\u27e9\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'schnorr.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e74cb4372cf0a3adc5e6e63810d37a21160dfac8","subject":"IMDesc: Agda model.","message":"IMDesc: Agda model.\n","repos":"larrytheliquid\/pigit,mietek\/epigram2,mietek\/epigram2","old_file":"models\/IMDesc.agda","new_file":"models\/IMDesc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule IMDesc where\n\n--********************************************\n-- Prelude\n--********************************************\n\n-- Some preliminary stuffs, to avoid relying on the stdlib\n\n--****************\n-- Sigma and friends\n--****************\n\ndata Sigma (A : Set)(B : A -> Set) : Set where\n  _,_ : (x : A)(y : B x) -> Sigma A B\n\n_*_ : (A B : Set) -> Set\nA * B = Sigma A \\_ -> B\n\nfst : {A : Set}{B : A -> Set} -> Sigma A B -> A\nfst (a , _) = a\n\nsnd : {A : Set}{B : A -> Set} (p : Sigma A B) -> B (fst p)\nsnd (a , b) = b\n\n\ndata Zero : Set where\nrecord One : Set where\n\n--****************\n-- Sum and friends\n--****************\n\ndata _+_ (A B : Set) : Set where\n  l : A -> A + B\n  r : B -> A + B\n\n--****************\n-- Prop\n--****************\n\ndata EProp : Set where\n  Trivial : EProp\n  \nPrf : EProp -> Set\nPrf _ = {! !}\n\n--****************\n-- Enum\n--****************\n\ndata EnumU : Set where\n  \ndata EnumT (E : EnumU) : Set where\n\nBranches : (E : EnumU) -> (P : EnumT E -> Set) -> Set\nBranches  _ _ = {! !}\n\nswitch : (E : EnumU) -> (e : EnumT E) -> (P : EnumT E -> Set) -> Branches E P -> P e\nswitch = {! !}\n\n\n\n\n--********************************************\n-- IMDesc\n--********************************************\n\n\n--****************\n-- Code\n--****************\n\ndata IDesc (I : Set) : Set where\n  iat      : (i : I) -> IDesc I\n  ipi      : (S : Set)(T : S -> IDesc I) -> IDesc I\n  isigma   : (S : Set)(T : S -> IDesc I) -> IDesc I\n  iprf     : (Q : EProp) -> IDesc I\n  iprod    : (D : IDesc I)(D : IDesc I) -> IDesc I\n  ifsigma  : (E : EnumU)(f : Branches E (\\ _ -> IDesc I)) -> IDesc I\n\n--****************\n-- Interpretation\n--****************\n\ndesc : (I : Set)(D : IDesc I)(P : I -> Set) -> Set\ndesc I (iat i)        P = P i\ndesc I (ipi S T)      P = (s : S) -> desc I (T s) P\ndesc I (isigma S T)   P = Sigma S (\\ s -> desc I (T s) P)\ndesc I (iprf q)       P = Prf q\ndesc I (iprod D D')   P = Sigma (desc I D P) (\\ _ -> desc I D' P)\ndesc I (ifsigma E B)  P = Sigma (EnumT E) (\\ e -> desc I (switch E e (\\ _ -> IDesc I) B) P)\n\n--****************\n-- Curried and uncurried interpretations\n--****************\n\ncurryD : (I : Set)(D : IDesc I)(P : I -> Set)(R : desc I D P -> Set) -> Set\ncurryD I (iat i) P R = (x : P i) -> R x\ncurryD I (ipi S T) P R = (f : (s : S) -> desc I (T s) P) -> R f\ncurryD I (isigma S T) P R = (s : S) -> curryD I (T s) P (\\ d -> R (s , d))\ncurryD I (iprf Q) P R = (x : Prf Q) -> R x\ncurryD I (iprod D D') P R = curryD I D P (\\ d ->\n                             curryD I D' P (\\ d' -> R (d , d')))\ncurryD I (ifsigma E B) P R = Branches E (\\e -> curryD I (switch E e (\\_ -> IDesc I) B) P (\\d -> R (e , d)))\n\n\n\nuncurryD : (I : Set)(D : IDesc I)(P : I -> Set)(R : desc I D P -> Set) ->\n           curryD I D P R -> (xs : desc I D P) -> R xs\nuncurryD I (iat i) P R f xs = f xs\nuncurryD I (ipi S T) P R F f = F f\nuncurryD I (isigma S T) P R f (s , d) =  uncurryD I (T s) P (\\d -> R (s , d)) (f s) d\nuncurryD I (iprf Q) P R f q = f q \nuncurryD I (iprod D D') P R f (d , d') =  uncurryD I D' P ((\\ d' \u2192 R (d , d'))) \n                                           (uncurryD I D P ((\\ x \u2192 curryD I D' P (\\ d0 \u2192 R (x , d0)))) f d) d'\nuncurryD I (ifsigma E B) P R br (e , d) =  uncurryD I (switch E e (\\ _ -> IDesc I) B) P \n                                                      (\\ d -> R ( e , d )) \n                                                      (switch E e ( \\ e -> R ( e , d)) br) d\n\n\n--****************\n-- Everywhere\n--****************\n\nbox : (I : Set)(D : IDesc I)(P : I -> Set) -> desc I D P -> IDesc (Sigma I P)\nbox I (iat i)        P x         = iat (i , x)\nbox I (ipi S T)      P f         = ipi S (\\ s -> box I (T s) P (f s))\nbox I (isigma S T)   P (s , d)   = box I (T s) P d\nbox I (iprf Q)       P q         = iprf Trivial\nbox I (iprod D D')   P (d , d')  = iprod (box I D P d) (box I D' P d')\nbox I (ifsigma E B)  P (e , d)   = box I (switch E e (\\ _ -> IDesc I) B) P d\n\nmapbox : (I : Set)(D : IDesc I)(Z : I -> Set)(P : Sigma I Z -> Set) ->\n         ((x : Sigma I Z) -> P x) ->\n         ( d : desc I D Z) -> desc (Sigma I Z) (box I D Z d) P\nmapbox = {!!}\n\n\n--****************\n-- Fix-point\n--****************\n\n\ndata IMu (I : Set)(D : I -> IDesc I)(i : I) : Set where\n  Con : desc I (D i) (\\ j -> IMu I D j) -> IMu I D i\n\n\n--****************\n-- Induction, curried\n--****************\n\n\ninductionC : (I : Set)(D : I -> IDesc I)(i : I)(v : IMu I D i)\n             (P : Sigma I (IMu I D) -> Set)\n             (m : (i : I) -> \n                  curryD I (D i) (IMu I D) (\\xs ->\n                    curryD (Sigma I (IMu I D))\n                           (box I (D i) (IMu I D) xs) \n                           P \n                           (\\ _ -> P (i , Con xs)))) ->\n             P ( i , v )\ninductionC I D i (Con x) P m = \n  let t = uncurryD  I\n                    (D i)\n                    (IMu I D) \n                    (\\xs ->\n                    curryD (Sigma I (IMu I D))\n                           (box I (D i) (IMu I D) xs) \n                           P \n                           (\\ _ -> P (i , Con xs)))\n                    (m i)\n                    x \n  in\n  uncurryD (Sigma I (IMu I D))\n           ( box I (D i) (IMu I D) x)\n           P\n           (\\ _ -> P ( i , Con x))\n           t\n           (mapbox I ( D i) ( IMu I D) P ind x)\n           where ind : (x' : Sigma I (IMu I D)) -> P x'\n                 ind ( i , x) = inductionC I D i x P m\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/IMDesc.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"dbb81124c3e7d2945fee75620a3d37648e4c220f","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"2fcbf02a0fb50e23d31a8f1329f1b6b1ac03e4f7","subject":"clean","message":"clean\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"64dc1651c301b836b6841ecdf3f752abe76ae081","subject":"added Mutual","message":"added Mutual\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"src\/extra\/Mutual.agda","new_file":"src\/extra\/Mutual.agda","new_contents":"open import Data.Nat using (\u2115; zero; suc)\nopen import Data.Bool using (Bool; true; false)\n\ndata even : \u2115 \u2192 Set\ndata odd : \u2115 \u2192 Set\n\ndata even where\n  zero : even zero\n  suc : \u2200 {n : \u2115} \u2192 odd n \u2192 even (suc n)\ndata odd where\n  suc : \u2200 {n : \u2115} \u2192 even n \u2192 odd (suc n)\n\nmutual\n  data even\u2032 : \u2115 \u2192 Set where\n    zero : even\u2032 zero\n    suc : \u2200 {n : \u2115} \u2192 odd\u2032 n \u2192 even\u2032 (suc n)\n  data odd\u2032 : \u2115 \u2192 Set where\n    suc : \u2200 {n : \u2115} \u2192 even\u2032 n \u2192 odd\u2032 (suc n)\n\n{-\n\/Users\/wadler\/sf\/src\/extra\/Mutual.agda:3,6-10\nMissing definition for even\n-}\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/extra\/Mutual.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"f3d4938f8d32cbd9e5f30fde308278d822290f9f","subject":"learn you an agda -- peano numbers","message":"learn you an agda -- peano numbers\n","repos":"shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps,shouya\/thinking-dumps","old_file":"learyouanagda\/peano.agda","new_file":"learyouanagda\/peano.agda","new_contents":"module peano where\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_+_ : \u2115 \u2192 \u2115 \u2192 \u2115\n\nzero + zero  = zero\nzero + n     = n\n(suc n) + n\u2032 = suc (n + n\u2032)\n\n\ndata _even : \u2115 \u2192 Set where\n  ZERO : zero even\n  STEP : \u2200 x \u2192 x even \u2192 (suc (suc x)) even\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'learyouanagda\/peano.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"13d20ba11cedef09da23882fdeb0ffdaaf4f304d","subject":"Agda by example: Simply Typed Lambda Calculus","message":"Agda by example: Simply Typed Lambda Calculus\n","repos":"jstolarek\/sandbox,jstolarek\/sandbox,jstolarek\/sandbox","old_file":"agda\/LambdaCalculus.agda","new_file":"agda\/LambdaCalculus.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/jstolarek\/sandbox.git\/'\n","license":"unlicense","lang":"Agda"}
{"commit":"4954d8fb51a301fc4278bd5149f5ad5d6354804e","subject":"Algebra.DoubleAndAdd: Multplication in any monoid using \"double and add\"","message":"Algebra.DoubleAndAdd: Multplication in any monoid using \"double and add\"\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/DoubleAndAdd.agda","new_file":"lib\/Algebra\/DoubleAndAdd.agda","new_contents":"open import Relation.Binary.PropositionalEquality.NP\nopen import Data.Two.Base\nopen import Data.List\nopen import Function\nopen import Algebra.FunctionProperties.Eq\nopen Implicits\nopen import Algebra.Monoid.Commutative\nopen import Algebra.Ring\n\nmodule Algebra.DoubleAndAdd\n  {\ud835\udd3d : Set} (\ud835\udd3d-ring : Ring \ud835\udd3d)\n  {\u2119 : Set} (\u2119-monoid : Commutative-Monoid \u2119)\n  where\n\nopen module \ud835\udd3d = Ring \ud835\udd3d-ring\n\nopen module \u2295 = Commutative-Monoid \u2119-monoid\n  renaming\n    ( _\u2219_ to _\u2295_\n    ; \u2219= to \u2295=\n    ; \u03b5\u2219-identity to \u03b5\u2295-identity\n    ; \u2219\u03b5-identity to \u2295\u03b5-identity\n    ; _\u00b2 to 2\u00b7_\n    )\n\n2\u00b7-\u2295-distr : \u2200 {P Q} \u2192 2\u00b7 (P \u2295 Q) \u2261 2\u00b7 P \u2295 2\u00b7 Q\n2\u00b7-\u2295-distr = \u2295.interchange\n\n2\u00b7-\u2295 : \u2200 {P Q R} \u2192 2\u00b7 P \u2295 (Q \u2295 R) \u2261 (P \u2295 Q) \u2295 (P \u2295 R)\n2\u00b7-\u2295 = \u2295.interchange\n\n-- NOT used yet\nmultiply-bin : List \ud835\udfda \u2192 \u2119 \u2192 \u2119\nmultiply-bin scalar P = go scalar\n  where\n    go : List \ud835\udfda \u2192 \u2119\n    go []       = P\n    go (b \u2237 bs) = [0: x\u2080 1: x\u2081 ] b\n      where x\u2080 = 2\u00b7 go bs\n            x\u2081 = P \u2295 x\u2080\n\n{-\nmultiply : \ud835\udd3d \u2192 \u2119 \u2192 \u2119\nmultiply scalar P =\n  -- if scalar == 0 or scalar >= N: raise Exception(\"Invalid Scalar\/Private Key\")\n    multiply-bin (bin scalar) P\n\n_\u00b7_ = multiply\ninfixr 8 _\u00b7_\n-}\n\ninfixl 7 1+2*_\n1+2*_ = \u03bb x \u2192 1+ 2* x\n\ndata Parity-View : \ud835\udd3d \u2192 Set where\n  zero\u27e8_\u27e9    : \u2200 {n} \u2192 n \u2261 0# \u2192 Parity-View n\n  even_by\u27e8_\u27e9 : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 2* m    \u2192 Parity-View n\n  odd_by\u27e8_\u27e9  : \u2200 {m n} \u2192 Parity-View m \u2192 n \u2261 1+ 2* m \u2192 Parity-View n\n\ncast_by\u27e8_\u27e9 : \u2200 {x y} \u2192 Parity-View x \u2192 y \u2261 x \u2192 Parity-View y\ncast zero\u27e8 x\u2091 \u27e9       by\u27e8 y\u2091 \u27e9 = zero\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast even x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = even x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\ncast odd  x\u209a by\u27e8 x\u2091 \u27e9 by\u27e8 y\u2091 \u27e9 = odd  x\u209a by\u27e8 y\u2091 \u2219 x\u2091 \u27e9\n\ninfixr 8 _\u00b7\u209a_\n_\u00b7\u209a_ : \u2200 {n} (p : Parity-View n) \u2192 \u2119 \u2192 \u2119\nzero\u27e8 e \u27e9      \u00b7\u209a P = \u03b5\neven p by\u27e8 e \u27e9 \u00b7\u209a P = 2\u00b7 (p \u00b7\u209a P)\nodd  p by\u27e8 e \u27e9 \u00b7\u209a P = P \u2295 (2\u00b7 (p \u00b7\u209a P))\n\n_+2*_ : \ud835\udfda \u2192 \ud835\udd3d \u2192 \ud835\udd3d\n0\u2082 +2* m =   2* m\n1\u2082 +2* m = 1+2* m\n\n{-\npostulate\n    bin-2* : \u2200 {n} \u2192 bin (2* n) \u2261 0\u2082 \u2237 bin n\n    bin-1+2* : \u2200 {n} \u2192 bin (1+2* n) \u2261 1\u2082 \u2237 bin n\n\nbin-+2* : (b : \ud835\udfda)(n : \ud835\udd3d) \u2192 bin (b +2* n) \u2261 b \u2237 bin n\nbin-+2* 1\u2082 n = bin-1+2*\nbin-+2* 0\u2082 n = bin-2*\n-}\n\n-- (msb) most significant bit first\nbin\u209a : \u2200 {n} \u2192 Parity-View n \u2192 List \ud835\udfda\nbin\u209a zero\u27e8 e \u27e9      = []\nbin\u209a even p by\u27e8 e \u27e9 = 0\u2082 \u2237 bin\u209a p\nbin\u209a odd  p by\u27e8 e \u27e9 = 1\u2082 \u2237 bin\u209a p\n\nhalf : \u2200 {n} \u2192 Parity-View n \u2192 \ud835\udd3d\nhalf zero\u27e8 _ \u27e9            = 0#\nhalf (even_by\u27e8_\u27e9 {m} _ _) = m\nhalf (odd_by\u27e8_\u27e9  {m} _ _) = m\n\n{-\nbin-parity : \u2200 {n} (p : ParityView n) \u2192 bin n \u2261 parity p \u2237 bin (half p)\nbin-parity (even n) = bin-2*\nbin-parity (odd n)  = bin-1+2*\n-}\n\ninfixl 6 1+\u209a_ _+\u209a_\n1+\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (1+ x)\n1+\u209a zero\u27e8 e \u27e9      = odd zero\u27e8 refl \u27e9 by\u27e8 ap 1+_ (e \u2219 ! 0+-identity) \u27e9\n1+\u209a even p by\u27e8 e \u27e9 = odd p      by\u27e8 ap 1+_ e \u27e9\n1+\u209a odd  p by\u27e8 e \u27e9 = even 1+\u209a p by\u27e8 ap 1+_ e \u2219 ! +-assoc \u2219 +-interchange \u27e9\n\n_+\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x + y)\nzero\u27e8 x\u2091 \u27e9       +\u209a y\u209a        = cast y\u209a by\u27e8 ap (\u03bb z \u2192 z + _) x\u2091 \u2219 0+-identity \u27e9\nx\u209a              +\u209a zero\u27e8 y\u2091 \u27e9 = cast x\u209a by\u27e8 ap (_+_ _) y\u2091 \u2219 +-comm \u2219 0+-identity \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even     x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-interchange \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-comm \u2219 +-assoc \u2219 ap 1+_ (+-comm \u2219 +-interchange) \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = odd      x\u209a +\u209a y\u209a by\u27e8 += x\u2091 y\u2091 \u2219 +-assoc \u2219 ap 1+_ +-interchange \u27e9\nodd  x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9 = even 1+\u209a (x\u209a +\u209a y\u209a) by\u27e8 += x\u2091 y\u2091 \u2219 +-on-sides refl +-interchange \u27e9\n\ninfixl 7 2*\u209a_\n2*\u209a_ : \u2200 {x} \u2192 Parity-View x \u2192 Parity-View (2* x)\n2*\u209a x\u209a = x\u209a +\u209a x\u209a\n\nopen \u2261-Reasoning\n\nmodule _ {P Q} where\n    \u00b7\u209a-\u2295-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a (P \u2295 Q) \u2261 x\u209a \u00b7\u209a P \u2295 x\u209a \u00b7\u209a Q\n    \u00b7\u209a-\u2295-distr zero\u27e8 x\u2091 \u27e9       = ! \u03b5\u2295-identity\n    \u00b7\u209a-\u2295-distr even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u2295-distr x\u209a) \u2219 2\u00b7-\u2295-distr\n    \u00b7\u209a-\u2295-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap (\u03bb z \u2192 P \u2295 Q \u2295 2\u00b7 z) (\u00b7\u209a-\u2295-distr x\u209a)\n                               \u2219 ap (\u03bb z \u2192 P \u2295 Q \u2295 z) (! 2\u00b7-\u2295)\n                               \u2219 \u2295.interchange\n\nmodule _ {P} where\n    cast-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u2091 : y \u2261 x) \u2192 cast x\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P\n    cast-\u00b7\u209a-distr zero\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n    cast-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2081 \u27e9 y\u2091 = refl\n\n    1+\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (1+\u209a x\u209a) \u00b7\u209a P \u2261 P \u2295 x\u209a \u00b7\u209a P\n    1+\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9       = ap (_\u2295_ P) \u03b5\u2295-identity\n    1+\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 = refl\n    1+\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (1+\u209a-\u00b7\u209a-distr x\u209a) \u2219 \u2295.interchange \u2219 \u2295.assoc\n\n    +\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y)\n                 \u2192 (x\u209a +\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P\n    +\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = cast-\u00b7\u209a-distr y\u209a _ \u2219 ! \u03b5\u2295-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u2295\u03b5-identity\n\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr\n    +\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 \u2295.assoc-comm\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap (_\u2295_ P) (ap 2\u00b7_ (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u2295-distr) \u2219 ! \u2295.assoc\n    +\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9\n       = (odd x\u209a by\u27e8 x\u2091 \u27e9 +\u209a odd y\u209a by\u27e8 y\u2091 \u27e9) \u00b7\u209a P\n       \u2261\u27e8by-definition\u27e9\n         2\u00b7((1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P)\n       \u2261\u27e8 ap 2\u00b7_ helper \u27e9\n         2\u00b7(P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P))\n       \u2261\u27e8 ap (_\u2295_ (2\u00b7 P)) 2\u00b7-\u2295-distr \u27e9\n         2\u00b7 P \u2295 (2\u00b7(x\u209a \u00b7\u209a P) \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8 2\u00b7-\u2295 \u27e9\n         P \u2295 2\u00b7(x\u209a \u00b7\u209a P) \u2295 (P \u2295 2\u00b7(y\u209a \u00b7\u209a P))\n       \u2261\u27e8by-definition\u27e9\n         odd x\u209a by\u27e8 x\u2091 \u27e9 \u00b7\u209a P \u2295 odd y\u209a by\u27e8 y\u2091 \u27e9 \u00b7\u209a P\n       \u220e\n        where helper = (1+\u209a (x\u209a +\u209a y\u209a)) \u00b7\u209a P\n                     \u2261\u27e8 1+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) \u27e9\n                       P \u2295 ((x\u209a +\u209a y\u209a) \u00b7\u209a P)\n                     \u2261\u27e8 ap (_\u2295_ P) (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u27e9\n                       P \u2295 (x\u209a \u00b7\u209a P \u2295 y\u209a \u00b7\u209a P)\n                     \u220e\n\n*-1+-distr : \u2200 {x y} \u2192 x * (1+ y) \u2261 x + x * y\n*-1+-distr = *-+-distr\u02e1 \u2219 += *1-identity refl\n\n1+-*-distr : \u2200 {x y} \u2192 (1+ x) * y \u2261 y + x * y\n1+-*-distr = *-+-distr\u02b3 \u2219 += 1*-identity refl\n\ninfixl 7 _*\u209a_\n_*\u209a_ : \u2200 {x y} \u2192 Parity-View x \u2192 Parity-View y \u2192 Parity-View (x * y)\nzero\u27e8 x\u2091 \u27e9       *\u209a y\u209a              = zero\u27e8 *= x\u2091 refl \u2219 0*-zero \u27e9\nx\u209a              *\u209a zero\u27e8 y\u2091 \u27e9       = zero\u27e8 *= refl y\u2091 \u2219 *0-zero \u27e9\neven x\u209a by\u27e8 x\u2091 \u27e9 *\u209a y\u209a              = even (x\u209a *\u209a y\u209a) by\u27e8 *= x\u2091 refl \u2219 *-+-distr\u02b3 \u27e9\nx\u209a              *\u209a even y\u209a by\u27e8 y\u2091 \u27e9 = even (x\u209a *\u209a y\u209a) by\u27e8 *= refl y\u2091 \u2219 *-+-distr\u02e1 \u27e9\nodd x\u209a by\u27e8 x\u2091 \u27e9  *\u209a odd y\u209a by\u27e8 y\u2091 \u27e9  = odd  (x\u209a +\u209a y\u209a +\u209a 2*\u209a (x\u209a *\u209a y\u209a)) by\u27e8 *= x\u2091 y\u2091 \u2219 helper \u27e9\n   where\n     x = _\n     y = _\n     helper = (1+2* x)*(1+2* y)\n            \u2261\u27e8 1+-*-distr \u27e9\n              1+2* y + 2* x * 1+2* y\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2* y + z)\n                 (2* x * 1+2* y\n                 \u2261\u27e8 *-1+-distr \u27e9\n                 (2* x + 2* x * 2* y)\n                 \u2261\u27e8 += refl *-+-distr\u02e1 \u2219 +-interchange \u27e9\n                 (2* (x + 2* x * y))\n                 \u220e) \u27e9\n              1+2* y + 2* (x + 2* x * y)\n            \u2261\u27e8 +-assoc \u2219 ap 1+_ +-interchange \u27e9\n              1+2*(y + (x + 2* x * y))\n            \u2261\u27e8 ap 1+2*_ (! +-assoc \u2219 += +-comm refl) \u27e9\n              1+2*(x + y + 2* x * y)\n            \u2261\u27e8 ap (\u03bb z \u2192 1+2*(x + y + z)) *-+-distr\u02b3 \u27e9\n              1+2*(x + y + 2* (x * y))\n            \u220e\n\nmodule _ {P} where\n    2\u00b7-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 2\u00b7(x\u209a \u00b7\u209a P) \u2261 x\u209a \u00b7\u209a 2\u00b7 P\n    2\u00b7-\u00b7\u209a-distr x\u209a = ! \u00b7\u209a-\u2295-distr x\u209a\n\n    2*\u209a-\u00b7\u209a-distr : \u2200 {x} (x\u209a : Parity-View x) \u2192 (2*\u209a x\u209a) \u00b7\u209a P \u2261 2\u00b7(x\u209a \u00b7\u209a P)\n    2*\u209a-\u00b7\u209a-distr x\u209a = +\u209a-\u00b7\u209a-distr x\u209a x\u209a\n\n\u00b7\u209a-\u03b5 : \u2200 {x} (x\u209a : Parity-View x) \u2192 x\u209a \u00b7\u209a \u03b5 \u2261 \u03b5\n\u00b7\u209a-\u03b5 zero\u27e8 x\u2091 \u27e9       = refl\n\u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9 = ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\u00b7\u209a-\u03b5 odd x\u209a by\u27e8 x\u2091 \u27e9  = \u03b5\u2295-identity \u2219 ap 2\u00b7_ (\u00b7\u209a-\u03b5 x\u209a) \u2219 \u03b5\u2295-identity\n\n*\u209a-\u00b7\u209a-distr : \u2200 {x y} (x\u209a : Parity-View x) (y\u209a : Parity-View y) {P} \u2192 (x\u209a *\u209a y\u209a) \u00b7\u209a P \u2261 x\u209a \u00b7\u209a y\u209a \u00b7\u209a P\n*\u209a-\u00b7\u209a-distr zero\u27e8 x\u2091 \u27e9 y\u209a = refl\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 even x\u209a by\u27e8 x\u2091 \u27e9\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 zero\u27e8 y\u2091 \u27e9 = ! \u00b7\u209a-\u03b5 odd  x\u209a by\u27e8 x\u2091 \u27e9\n\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a even y\u209a by\u27e8 y\u2091 \u27e9)\n*\u209a-\u00b7\u209a-distr even x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 = ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a odd  y\u209a by\u27e8 y\u2091 \u27e9)\n\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 even y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr odd x\u209a by\u27e8 x\u2091 \u27e9 y\u209a) \u2219 2\u00b7-\u2295-distr \u2219 ap (\u03bb z \u2192 2\u00b7 (y\u209a \u00b7\u209a P) \u2295 2\u00b7 z) (2\u00b7-\u00b7\u209a-distr x\u209a)\n*\u209a-\u00b7\u209a-distr odd  x\u209a by\u27e8 x\u2091 \u27e9 odd  y\u209a by\u27e8 y\u2091 \u27e9 {P} =\n   ap (_\u2295_ P)\n     (ap 2\u00b7_\n        (+\u209a-\u00b7\u209a-distr (x\u209a +\u209a y\u209a) (2*\u209a (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= (+\u209a-\u00b7\u209a-distr x\u209a y\u209a) (2*\u209a-\u00b7\u209a-distr (x\u209a *\u209a y\u209a))\n        \u2219 \u2295= \u2295.comm (ap 2\u00b7_ (*\u209a-\u00b7\u209a-distr x\u209a y\u209a) \u2219 2\u00b7-\u00b7\u209a-distr x\u209a) \u2219 \u2295.assoc\n        \u2219 \u2295= refl (! \u00b7\u209a-\u2295-distr x\u209a) ) \u2219 2\u00b7-\u2295-distr)\n     \u2219 \u2295.!assoc\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/DoubleAndAdd.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"3887acc776498fc936c85a42b1ba8ed98dfe1593","subject":"Add Everything file","message":"Add Everything file\n","repos":"bch29\/agda-holes","old_file":"Everything.agda","new_file":"Everything.agda","new_contents":"module Everything where\n\nimport Holes.Prelude\nimport Holes.Term\nimport Holes.Cong.Limited\nimport Holes.Cong.General\nimport Holes.Cong.Propositional\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Everything.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1b63adc325b518b2cd1977fab328425563a76b43","subject":"scraps and a new file from some refactoring i forgot to check in. first cut at a slightly better readme for #15, although that will need to be updated before final submission. (i'll probably create more files and remove some, and some provisos will get dropped)","message":"scraps and a new file from some refactoring i forgot to check in. first cut\nat a slightly better readme for #15, although that will need to be updated\nbefore final submission. (i'll probably create more files and remove some,\nand some provisos will get dropped)\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"lemmas-subst-ta.agda","new_file":"lemmas-subst-ta.agda","new_contents":"module lemmas-subst-ta where\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lemmas-subst-ta.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"43e75dd87813edda5a9f848c7d0ae91f40843a4a","subject":"implementing rummy rules in types","message":"implementing rummy rules in types\n","repos":"piyush-kurur\/sample-code","old_file":"agda\/rummy.agda","new_file":"agda\/rummy.agda","new_contents":"module rummy where\n\nopen import Data.Nat\n\n-- The suit type\ndata Suit : Set where\n  \u2663 : Suit\n  \u2662 : Suit\n  \u2665 : Suit\n  \u2660 : Suit\n\n-- A card consists of a value and a suite.\ndata Card : \u2115 \u2192 Suit \u2192 Set where\n  _of_ : (n : \u2115) (s : Suit) \u2192 Card n s\n\n-- Common names for cards\n\nA : \u2115; A = 1\nK : \u2115; K = 13\nQ : \u2115; Q = 12\nJ : \u2115; J = 11\n\n\n--\n-- The clubs suit\n--\n\nA\u2663  : Card A  \u2663; A\u2663  = A  of \u2663\n2\u2663  : Card 2  \u2663; 2\u2663  = 2  of \u2663\n3\u2663  : Card 3  \u2663; 3\u2663  = 3  of \u2663\n4\u2663  : Card 4  \u2663; 4\u2663  = 4  of \u2663\n5\u2663  : Card 5  \u2663; 5\u2663  = 5  of \u2663\n6\u2663  : Card 6  \u2663; 6\u2663  = 6  of \u2663\n7\u2663  : Card 7  \u2663; 7\u2663  = 7  of \u2663\n8\u2663  : Card 8  \u2663; 8\u2663  = 8  of \u2663\n9\u2663  : Card 9  \u2663; 9\u2663  = 9  of \u2663\n10\u2663 : Card 10 \u2663; 10\u2663 = 10 of \u2663\nJ\u2663  : Card J  \u2663; J\u2663  = J  of \u2663\nQ\u2663  : Card Q  \u2663; Q\u2663  = Q  of \u2663\nK\u2663  : Card K  \u2663; K\u2663  = K  of \u2663\n\n--\n-- The diamond suit\n--\n\nA\u2662  : Card A  \u2662; A\u2662  = A  of \u2662\n2\u2662  : Card 2  \u2662; 2\u2662  = 2  of \u2662\n3\u2662  : Card 3  \u2662; 3\u2662  = 3  of \u2662\n4\u2662  : Card 4  \u2662; 4\u2662  = 4  of \u2662\n5\u2662  : Card 5  \u2662; 5\u2662  = 5  of \u2662\n6\u2662  : Card 6  \u2662; 6\u2662  = 6  of \u2662\n7\u2662  : Card 7  \u2662; 7\u2662  = 7  of \u2662\n8\u2662  : Card 8  \u2662; 8\u2662  = 8  of \u2662\n9\u2662  : Card 9  \u2662; 9\u2662  = 9  of \u2662\n10\u2662 : Card 10 \u2662; 10\u2662 = 10 of \u2662\nJ\u2662  : Card J  \u2662; J\u2662  = J  of \u2662\nQ\u2662  : Card Q  \u2662; Q\u2662  = Q  of \u2662\nK\u2662  : Card K  \u2662; K\u2662  = K  of \u2662\n\n--\n-- The heart suit\n--\n\nA\u2665  : Card A  \u2665; A\u2665  = A  of \u2665\n2\u2665  : Card 2  \u2665; 2\u2665  = 2  of \u2665\n3\u2665  : Card 3  \u2665; 3\u2665  = 3  of \u2665\n4\u2665  : Card 4  \u2665; 4\u2665  = 4  of \u2665\n5\u2665  : Card 5  \u2665; 5\u2665  = 5  of \u2665\n6\u2665  : Card 6  \u2665; 6\u2665  = 6  of \u2665\n7\u2665  : Card 7  \u2665; 7\u2665  = 7  of \u2665\n8\u2665  : Card 8  \u2665; 8\u2665  = 8  of \u2665\n9\u2665  : Card 9  \u2665; 9\u2665  = 9  of \u2665\n10\u2665 : Card 10 \u2665; 10\u2665 = 10 of \u2665\nJ\u2665  : Card J  \u2665; J\u2665  = J  of \u2665\nQ\u2665  : Card Q  \u2665; Q\u2665  = Q  of \u2665\nK\u2665  : Card K  \u2665; K\u2665  = K  of \u2665\n\n--\n-- The spade suit\n--\n\nA\u2660  : Card A  \u2660; A\u2660  = A  of \u2660\n2\u2660  : Card 2  \u2660; 2\u2660  = 2  of \u2660\n3\u2660  : Card 3  \u2660; 3\u2660  = 3  of \u2660\n4\u2660  : Card 4  \u2660; 4\u2660  = 4  of \u2660\n5\u2660  : Card 5  \u2660; 5\u2660  = 5  of \u2660\n6\u2660  : Card 6  \u2660; 6\u2660  = 6  of \u2660\n7\u2660  : Card 7  \u2660; 7\u2660  = 7  of \u2660\n8\u2660  : Card 8  \u2660; 8\u2660  = 8  of \u2660\n9\u2660  : Card 9  \u2660; 9\u2660  = 9  of \u2660\n10\u2660 : Card 10 \u2660; 10\u2660 = 10 of \u2660\nJ\u2660  : Card J  \u2660; J\u2660  = J  of \u2660\nQ\u2660  : Card Q  \u2660; Q\u2660  = Q  of \u2660\nK\u2660  : Card K  \u2660; K\u2660  = K  of \u2660\n\n\n-- A run should be at least of length 1.\ndata Run : Suit \u2192 (start : \u2115) \u2192 (length : \u2115) \u2192 Set where\n  [] : {suit : Suit} {start : \u2115} \u2192 Run  suit  start 0\n\n  _,_ : {suit : Suit} {n \u2113 : \u2115}\n       \u2192 Card n suit\n       \u2192 Run    suit (1 + n)     \u2113\n       \u2192 Run    suit    n     (1 + \u2113)\n\n\n-- A group should be at least of size on.\ndata Group : (value \u2113 : \u2115) \u2192 Set where\n  [] : {value : \u2115} \u2192 Group value 0\n\n  _,_ : {suit : Suit} {value \u2113 : \u2115}\n       \u2192 Card   value  suit\n       \u2192 Group  value    \u2113\n       \u2192 Group  value    (1 + \u2113)\n\n-- Some pretty functions for creating runs. You can create a run as\n-- follows:\n-- myrun  = \u27e8 A\u2663 , 2\u2663 , 3\u2663 \u27e9\n--\n\n\u27e8_ : {suit : Suit} {n \u2113 : \u2115} \u2192 Run suit n \u2113 \u2192 Run suit n \u2113\n\u27e8 r = r\n\n_\u27e9 : {suit : Suit} {n : \u2115} \u2192 Card n suit \u2192 Run suit n 1\nc \u27e9 = c , []\n\n\n-- Some pretty functions for creating groups. You can create a group\n-- group as follows:\n--\n-- mygroup = \u27e6 A\u2663 , A\u2665 , A\u2662 \u27e7\n\n\u27e6_ : {value \u2113 : \u2115} \u2192 Group value \u2113 \u2192 Group value \u2113\n\u27e6 g = g\n\n\n_\u27e7 : {suit : Suit} {n : \u2115} \u2192 Card n suit \u2192 Group n 1\nc \u27e7 = c , []\n\ninfixr 2 _\u27e9 _\u27e7 _,_\ninfixl 1 \u27e8_ \u27e6_\n\n\nmygroup = \u27e6 A\u2663 , A\u2663 , A\u2660 \u27e7\nmyrun   = \u27e8 A\u2663 , 2\u2663 , 3\u2663 \u27e9\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'agda\/rummy.agda' did not match any file(s) known to git\n","license":"unlicense","lang":"Agda"}
{"commit":"cc9dd1a76cf147a706102dd279d1279ce09dc59e","subject":"Added test for the Agda internal term Lam.","message":"Added test for the Agda internal term Lam.\n\nIgnore-this: 5070391dc904e0e9f9db84f3816717a2\n\ndarcs-hash:20120422152654-3bd4e-d4297b9ada9ca7421fbf78593aef650ede8867b3.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/LamTerm.agda","new_file":"Test\/Succeed\/LamTerm.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing Agda internal terms: Lam\n------------------------------------------------------------------------------\n\n-- The following conjecture uses the internal Agda term Lam.\n\nmodule Test.Succeed.LamTerm where\n\npostulate\n  D : Set\n  \u2203 : (D \u2192 Set) \u2192 Set  -- The existential quantifier type on D.\n  A : D \u2192 Set\n\npostulate \u2203-intro : (t : D) \u2192 A t \u2192 \u2203 A\n{-# ATP prove \u2203-intro #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/LamTerm.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"8b8fb52739a3f81035debb0221d43ac75734dc83","subject":"2nd version of the example","message":"2nd version of the example\n","repos":"toonn\/popartt","old_file":"koopa.agda","new_file":"koopa.agda","new_contents":"{-\n\n    Verified Koopa Troopa Movement\n             Toon Nolten\n\n-}\n\nmodule koopa where\nopen import Data.Nat\n\ndata Color : Set where\n  Green : Color\n  Red : Color\n\ndata KoopaTroopa Color : Set where\n  _KT : Color \u2192 KoopaTroopa Color\n\nrecord Position : Set where\n  constructor pos\n  field\n    x : \u2115\n    y : \u2115\n\ndata _follows_ : Position \u2192 Position \u2192 Set where\n  still : \u2200 {p} \u2192 p follows p\n  next  : \u2200 {x y} \u2192 pos (suc x) y follows pos x y\n  back  : \u2200 {x y} \u2192 pos x y follows pos (suc x) y\n  -- jump  : \u2200 {x y} \u2192 pos x (suc y) follows pos x y\n  fall  : \u2200 {x y} \u2192 pos x y follows pos x (suc y)\n\n\ndata Path (Koopa : KoopaTroopa Color) : Position \u2192 Position \u2192 Set where\n  []  : \u2200 {p} \u2192 Path Koopa p p\n  _\u21a0[_]_ : {q r : Position} \u2192 (p : Position) \u2192 q follows p\n           \u2192 (qs : Path Koopa q r) \u2192 Path Koopa p r\n\nex_path : Path (Red KT) (pos 0 0) (pos 0 0)\nex_path = pos 0 0 \u21a0[ next ] (pos 1 0 \u21a0[ back ] [])\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'koopa.agda' did not match any file(s) known to git\n","license":"bsd-2-clause","lang":"Agda"}
{"commit":"ec6ff43e5c7094d8bb93a9c044a5e0cc17f58967","subject":"lib\/Level\/NP.agda","message":"lib\/Level\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Level\/NP.agda","new_file":"lib\/Level\/NP.agda","new_contents":"module Level.NP where\n\nimport Level\n\nopen Level public renaming (zero to \u2080; suc to \u209b)\n\n\u2081 \u2082 \u2083 \u2084 : Level\n\u2081 = \u209b \u2080\n\u2082 = \u209b \u2081\n\u2083 = \u209b \u2082\n\u2084 = \u209b \u2083\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Level\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0e85571f18bc18d81013468d4d121214eb61e394","subject":"languages\/test.agda","message":"languages\/test.agda\n","repos":"sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis,sharkspeed\/dororis","old_file":"languages\/test.agda","new_file":"languages\/test.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/sharkspeed\/dororis.git\/'\n","license":"bsd-2-clause","lang":"Agda"}
{"commit":"22eab6635a40915cfce9e29a45fe31de8280bcbf","subject":"add list of theorems","message":"add list of theorems\n\nOld-commit-hash: ae20c5bded5fecd55d9723e7870ba03979cd0880\n","repos":"inc-lc\/ilc-agda","old_file":"PLDI14-List-of-Theorems.agda","new_file":"PLDI14-List-of-Theorems.agda","new_contents":"module PLDI14-List-of-Theorems where\n\nopen import Function\n\n-- List of theorems in PLDI submission\n--\n-- For hints about installation and execution, please refer\n-- to README.agda.\n--\n-- Agda modules corresponding to definitions, lemmas and theorems\n-- are listed here with the most important name. For example,\n-- after this file type checks (C-C C-L), placing the cursor\n-- on the purple \"Base.Change.Algebra\" and pressing M-. will\n-- bring you to the file where change structures are defined.\n-- The name for change structures in that file is\n-- \"ChangeAlgebra\", given in the using-clause.\n\n-- Definition 2.1 (Change structures)\n-- (d) ChangeAlgebra.diff\n-- (e) IsChangeAlgebra.update-diff\nopen import Base.Change.Algebra using (ChangeAlgebra)\n---- Carrier in record ChangeAlgebra                        --(a)\nopen Base.Change.Algebra.ChangeAlgebra   using (Change)     --(b)\nopen Base.Change.Algebra.ChangeAlgebra   using (update)     --(c)\nopen Base.Change.Algebra.ChangeAlgebra   using (diff)       --(d)\nopen Base.Change.Algebra.IsChangeAlgebra using (update-diff)--(e)\n\n-- Definition 2.2 (Nil change)\n-- IsChangeAlgebra.nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Lemma 2.3 (Behavior of nil)\n-- IsChangeAlgebra.update-nil\nopen Base.Change.Algebra using (IsChangeAlgebra)\n\n-- Definition 2.4 (Derivatives)\nopen Base.Change.Algebra using (Derivative)\n\n-- Definition 2.5 (Carrier set of function changes)\nopen Base.Change.Algebra.FunctionChanges\n\n-- Definition 2.6 (Operations on function changes)\n-- ChangeAlgebra.update FunctionChanges.changeAlgebra\n-- ChangeAlgebra.diff   FunctionChanges.changeAlgebra\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Theorem 2.7 (Function changes form a change structure)\n-- (In Agda, the proof of Theorem 2.7 has to be included in the\n-- definition of function changes, here\n-- FunctionChanges.changeAlgebra.)\nopen Base.Change.Algebra.FunctionChanges using (changeAlgebra)\n\n-- Lemma 2.8 (Incrementalization)\nopen Base.Change.Algebra.FunctionChanges using (incrementalization)\n\n-- Theorem 2.9 (Nil changes are derivatives)\nopen Base.Change.Algebra.FunctionChanges using (nil-is-derivative)\n\n-- For each plugin requirement, we include its definition and\n-- a concrete instantiation called \"Popl14\" with integers and\n-- bags of integers as base types.\n\n-- Plugin Requirement 3.1 (Domains of base types)\nopen import Parametric.Denotation.Value using (Structure)\nopen import     Popl14.Denotation.Value using (\u27e6_\u27e7Base)\n\n-- Definition 3.2 (Domains)\nopen Parametric.Denotation.Value.Structure using (\u27e6_\u27e7Type)\n\n-- Plugin Requirement 3.3 (Evaluation of constants)\nopen import Parametric.Denotation.Evaluation using (Structure)\nopen import     Popl14.Denotation.Evaluation using (\u27e6_\u27e7Const)\n\n-- Definition 3.4 (Environments)\nopen import Base.Denotation.Environment using (\u27e6_\u27e7Context)\n\n-- Definition 3.5 (Evaluation)\nopen Parametric.Denotation.Evaluation.Structure using (\u27e6_\u27e7Term)\n\n-- Plugin Requirement 3.6 (Changes on base types)\nopen import Parametric.Change.Validity using (Structure)\nopen import     Popl14.Change.Validity using (change-algebra-base-family)\n\n-- Definition 3.7 (Changes)\nopen Parametric.Change.Validity.Structure using (change-algebra)\n\n-- Definition 3.8 (Change environments)\nopen Parametric.Change.Validity.Structure using (environment-changes)\n\n-- Plugin Requirement 3.9 (Change semantics for constants)\nopen import Parametric.Change.Specification using (Structure)\nopen import     Popl14.Change.Specification using (specification-structure)\n\n-- Definition 3.10 (Change semantics)\nopen Parametric.Change.Specification.Structure using (\u27e6_\u27e7\u0394)\n\n-- Lemma 3.11 (Change semantics is the derivative of semantics)\nopen Parametric.Change.Specification.Structure using (correctness)\n\n-- Definition 3.12 (Erasure)\nimport Parametric.Change.Implementation\nopen Parametric.Change.Implementation.Structure using (_\u2248_)\nopen import Popl14.Change.Implementation using (implements-base)\n\n-- Lemma 3.13 (The erased version of a change is almost the same)\nopen Parametric.Change.Implementation.Structure using (carry-over)\n\n-- Lemma 3.14 (\u27e6 t \u27e7\u0394 erases to Derive(t))\nimport Parametric.Change.Correctness\nopen Parametric.Change.Correctness.Structure using (main-theorem)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'PLDI14-List-of-Theorems.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c6a0fb6c91188f7a139429b4b537753a505fb04e","subject":"adding a stab at what complete preseration looks like","message":"adding a stab at what complete preseration looks like\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"complete-preservation.agda","new_file":"complete-preservation.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\nopen import contexts\n\nmodule complete-preservation where\n  complete-preservation : {\u0394 : hctx} {e : hexp} {d d' : dhexp} {\u03c4 : htyp} \u2192\n             ecomplete e \u2192\n             \u2205 \u22a2 e \u21d2 \u03c4 ~> d \u22a3 \u0394 \u2192\n             \u0394 \u22a2 d \u21a6 d' \u2192\n             \u0394 , \u2205 \u22a2 d' :: \u03c4\n  complete-preservation = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'complete-preservation.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"04797f1632e450e8de8bf8b7656f35b05b83a61e","subject":"backed up Preorder","message":"backed up Preorder\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"src\/extra\/Preorder.agda","new_file":"src\/extra\/Preorder.agda","new_contents":"open import Stlc hiding (\u27f9*-Preorder; _\u27f9*\u27ea_\u27eb_; example\u2080; example\u2081)\nopen import Relation.Binary.PropositionalEquality as P using (_\u2261_; _\u2262_; refl)\nopen import Relation.Binary using (Preorder)\nimport Relation.Binary.PreorderReasoning as PreorderReasoning\n\n\u27f9*-Preorder : Preorder _ _ _\n\u27f9*-Preorder = record\n  { Carrier    = Term\n  ; _\u2248_        = _\u2261_\n  ; _\u223c_        = _\u27f9*_\n  ; isPreorder = record\n    { isEquivalence = P.isEquivalence\n    ; reflexive     = \u03bb {refl \u2192 \u27e8\u27e9}\n    ; trans         = _>>_\n    }\n  }\n\nopen PreorderReasoning \u27f9*-Preorder\n     using (_IsRelatedTo_; begin_; _\u220e) renaming (_\u2248\u27e8_\u27e9_ to _\u2261\u27e8_\u27e9_; _\u223c\u27e8_\u27e9_ to _\u27f9*\u27e8_\u27e9_)\n\ninfixr 2 _\u27f9*\u27ea_\u27eb_\n\n_\u27f9*\u27ea_\u27eb_ : \u2200 x {y z} \u2192 x \u27f9 y \u2192 y IsRelatedTo z \u2192 x IsRelatedTo z\nx \u27f9*\u27ea x\u27f9y \u27eb yz  =  x \u27f9*\u27e8 \u27e8 x\u27f9y \u27e9 \u27e9 yz\n\nexample\u2080 : not \u00b7 true \u27f9* false\nexample\u2080 =\n  begin\n    not \u00b7 true\n  \u27f9*\u27ea \u03b2\u21d2 value-true \u27eb\n    if true then false else true\n  \u27f9*\u27ea \u03b2\ud835\udd39\u2081 \u27eb\n    false\n  \u220e\n\nexample\u2081 : I\u00b2 \u00b7 I \u00b7 (not \u00b7 false) \u27f9* true\nexample\u2081 =\n  begin\n    I\u00b2 \u00b7 I \u00b7 (not \u00b7 false)\n  \u27f9*\u27ea \u03b3\u21d2\u2081 (\u03b2\u21d2 value-\u03bb) \u27eb\n    (\u03bb[ x \u2236 \ud835\udd39 ] I \u00b7 var x) \u00b7 (not \u00b7 false)                  \n  \u27f9*\u27ea \u03b3\u21d2\u2082 value-\u03bb (\u03b2\u21d2 value-false) \u27eb\n    (\u03bb[ x \u2236 \ud835\udd39 ] I \u00b7 var x) \u00b7 (if false then false else true)\n  \u27f9*\u27ea \u03b3\u21d2\u2082 value-\u03bb \u03b2\ud835\udd39\u2082 \u27eb\n    (\u03bb[ x \u2236 \ud835\udd39 ] I \u00b7 var x) \u00b7 true\n  \u27f9*\u27ea \u03b2\u21d2 value-true \u27eb\n    I \u00b7 true\n  \u27f9*\u27ea \u03b2\u21d2 value-true \u27eb\n    true\n  \u220e  \n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/extra\/Preorder.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"dd18fcc5ba9cbb6d8cbf62d171887cf3a22893f5","subject":"Define a change structure for products","message":"Define a change structure for products\n\nDerivatives of curried functions seem challenging.\nAccessing nested records too, but seems easy to fix.\n\nOld-commit-hash: d89e6171d447aae6292cc025dc6db95ce2370b88\n","repos":"inc-lc\/ilc-agda","old_file":"Base\/Change\/Products.agda","new_file":"Base\/Change\/Products.agda","new_contents":"module Base.Change.Products where\n\nopen import Relation.Binary.PropositionalEquality\nopen import Level\n\nopen import Base.Change.Algebra\n\nopen import Data.Product\n\n-- Also try defining sectioned change structures on the positives halves of\n-- groups? Or on arbitrary subsets?\n\n-- Restriction: we pair sets on the same level (because right now everything\n-- else would risk getting in the way).\nmodule ProductChanges \u2113 (A B : Set \u2113) {{CA : ChangeAlgebra \u2113 A}} {{CB : ChangeAlgebra \u2113 B}} where\n  open \u2261-Reasoning\n\n  -- The simplest possible definition of changes for products.\n\n  -- The following is probably bullshit:\n  -- Does not handle products of functions - more accurately, writing the\n  -- derivative of fst and snd for products of functions is hard: fst' p dp must return the change of fst p\n  PChange : A \u00d7 B \u2192 Set \u2113\n  PChange (a , b) = \u0394 a \u00d7 \u0394 b\n\n  _\u2295_ : (v : A \u00d7 B) \u2192 PChange v \u2192 A \u00d7 B\n  _\u2295_ (a , b) (da , db) = a \u229e da , b \u229e db\n  _\u229d_ : A \u00d7 B \u2192 (v : A \u00d7 B) \u2192 PChange v\n  _\u229d_ (aNew , bNew) (a , b) = aNew \u229f a , bNew \u229f b\n\n  p-update-diff : (u v : A \u00d7 B) \u2192 v \u2295 (u \u229d v) \u2261 u\n  p-update-diff (ua , ub) (va , vb) =\n    let u = (ua , ub)\n        v = (va , vb)\n    in\n      begin\n        v \u2295 (u \u229d v)\n      \u2261\u27e8\u27e9\n        (va \u229e (ua \u229f va) , vb \u229e (ub \u229f vb))\n        --v \u2295 ((ua \u229f va , ub \u229f vb))\n      \u2261\u27e8 cong\u2082 _,_ (update-diff ua va) (update-diff ub vb)\u27e9\n        (ua , ub)\n      \u2261\u27e8\u27e9\n        u\n      \u220e\n\n  changeAlgebra : ChangeAlgebra \u2113 (A \u00d7 B)\n  changeAlgebra = record\n    { Change = PChange\n    ; update = _\u2295_\n    ; diff = _\u229d_\n    ; isChangeAlgebra = record\n      { update-diff = p-update-diff\n      }\n    }\n\n  proj\u2081\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2081 v)\n  proj\u2081\u2032 (a , b) (da , db) = da\n\n  proj\u2081\u2032Derivative : Derivative proj\u2081 proj\u2081\u2032\n  -- Implementation note: we do not need to pattern match on v and dv because\n  -- they are records, hence Agda knows that pattern matching on records cannot\n  -- fail. Technically, the required feature is the eta-rule on records.\n  proj\u2081\u2032Derivative v dv = refl\n\n  -- An extended explanation.\n  proj\u2081\u2032Derivative\u2081 : Derivative proj\u2081 proj\u2081\u2032\n  proj\u2081\u2032Derivative\u2081 (a , b) (da , db) =\n    let v  = (a  , b)\n        dv = (da , db)\n    in\n      begin\n        proj\u2081 v \u229e proj\u2081\u2032 v dv\n      \u2261\u27e8\u27e9\n        a \u229e da\n      \u2261\u27e8\u27e9\n        proj\u2081 (v \u229e dv)\n      \u220e\n\n  -- Same for the second extractor.\n  proj\u2082\u2032 : (v : A \u00d7 B) \u2192 \u0394 v \u2192 \u0394 (proj\u2082 v)\n  proj\u2082\u2032 (a , b) (da , db) = db\n\n  proj\u2082\u2032Derivative : Derivative proj\u2082 proj\u2082\u2032\n  proj\u2082\u2032Derivative v dv = refl\n\n  -- We should do the same for uncurry instead.\n\n  -- What one could wrongly expect to be the derivative of the constructor:\n  _,_\u2032 : (a : A) \u2192 (da : \u0394 a) \u2192 (b : B) \u2192 (db : \u0394 b) \u2192 \u0394 (a , b)\n  _,_\u2032 a da b db = da , db\n\n  -- That has the correct behavior, in a sense, and it would be in the\n  -- subset-based formalization in the paper.\n  --\n  -- But the above is not even a change, because it does not contain a proof of its own validity, and because after application it does not contain a proof\n\n  -- As a consequence, proving that's a derivative seems too insanely hard. We\n  -- might want to provide a proof schema for curried functions at once,\n  -- starting from the right correctness equation.\n\n  B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} B (A \u00d7 B)\n  A\u2192B\u2192A\u00d7B = FunctionChanges.changeAlgebra {c = \u2113} {d = \u2113} A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n  module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n  module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n\n  _,_\u2032-real : \u0394 _,_\n  _,_\u2032-real = nil _,_\n  module FC\u0394A\u2192B\u2192A\u00d7B = \u0394A\u2192B\u2192A\u00d7B.FunctionChange\n  _,_\u2032-real-Derivative : Derivative {{CA}} {{B\u2192A\u00d7B}} _,_ (FC\u0394A\u2192B\u2192A\u00d7B.apply _,_\u2032-real)\n  _,_\u2032-real-Derivative =\n    FunctionChanges.nil-is-derivative A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}} _,_\n\n  {-\n\n  _,_\u2032\u2032 :  (a : A) \u2192 \u0394 a \u2192\n      \u0394 {{B\u2192A\u00d7B}} (\u03bb b \u2192 (a , b))\n  _,_\u2032\u2032 a da = record\n    { apply = _,_\u2032 a da\n    ; correct = \u03bb b db \u2192\n      begin\n        (a , b \u229e db) \u229e (_,_\u2032 a da) (b \u229e db) (nil (b \u229e db))\n      \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db))\n      \u2261\u27e8 cong (\u03bb \u25a1 \u2192  a \u229e da , \u25a1) (update-nil (b \u229e db)) \u27e9\n        a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9\n        (a , b) \u229e (_,_\u2032 a da) b db\n      \u220e\n    }\n\n  _,_\u2032\u2032\u2032 : \u0394 {{A\u2192B\u2192A\u00d7B}} _,_\n  _,_\u2032\u2032\u2032 = record\n    { apply = _,_\u2032\u2032\n    ; correct = \u03bb a da \u2192\n      begin\n        update\n        (_,_ (a \u229e da))\n        (_,_\u2032\u2032 (a \u229e da) (nil (a \u229e da)))\n      \u2261\u27e8 {!!} \u27e9\n        update (_,_ a) (_,_\u2032\u2032 a da)\n        --ChangeAlgebra.update B\u2192A\u00d7B (_,_ a) (a , da \u2032\u2032)\n      \u220e\n    }\n    where\n      open ChangeAlgebra B\u2192A\u00d7B hiding (nil)\n{-\n  {!\n      begin\n        (_,_ (a \u229e da)) \u229e _,_\u2032\u2032 (a \u229e da) (nil (a \u229e da))\n      {- \u2261\u27e8\u27e9\n        a \u229e da , b \u229e db \u229e (nil (b \u229e db)) -}\n      \u2261\u27e8 ? \u27e9\n        {-a \u229e da , b \u229e db\n      \u2261\u27e8\u27e9-}\n       (_,_ a) \u229e (_,_\u2032\u2032 a da)\n      \u220e!}\n    }\n-}\n  open import Postulate.Extensionality\n\n  _,_\u2032Derivative :\n    Derivative {{CA}} {{B\u2192A\u00d7B}} _,_  _,_\u2032\u2032\n  _,_\u2032Derivative a da =\n    begin\n      _\u229e_ {{B\u2192A\u00d7B}} (_,_ a) (_,_\u2032\u2032 a da)\n    \u2261\u27e8\u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.FunctionChange.apply (_,_\u2032\u2032 a da) b (nil b))\n    --ext (\u03bb b \u2192 cong (\u03bb \u25a1 \u2192  (a , b) \u229e \u25a1) (update-nil {{?}} b))\n    \u2261\u27e8 {!!} \u27e9\n      (\u03bb b \u2192 (a , b) \u229e \u0394BA\u00d7B.FunctionChange.apply (_,_\u2032\u2032 a da) b (nil b))\n    \u2261\u27e8 sym {!\u0394A\u2192B\u2192A\u00d7B.incrementalization _,_ _,_\u2032\u2032\u2032 a da!} \u27e9\n  --FunctionChanges.incrementalization A (B \u2192 A \u00d7 B) {{CA}} {{{!B\u2192A\u00d7B!}}} _,_ {!!} {!!} {!!}\n       _,_ (a \u229e da)\n    \u220e\n    where\n      --open FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n      module \u0394BA\u00d7B = FunctionChanges B (A \u00d7 B) {{CB}} {{changeAlgebra}}\n      module \u0394A\u2192B\u2192A\u00d7B = FunctionChanges A (B \u2192 A \u00d7 B) {{CA}} {{B\u2192A\u00d7B}}\n\n  -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Base\/Change\/Products.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e4816ddb3196a6d8facbc3b5fdc1712f607cb8a8","subject":"lib\/Category\/Profunctor.agda","message":"lib\/Category\/Profunctor.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Category\/Profunctor.agda","new_file":"lib\/Category\/Profunctor.agda","new_contents":"import Level as L\nopen import Type hiding (\u2605)\nopen import Function.NP\nopen import Category.Functor\n\nmodule Category.Profunctor {\u2113} where\n\nrecord Profunctor (_\u219d_ : \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113) : \u2605 (L.suc \u2113) where\n  constructor _,_\n  field\n    lmap : \u2200 {A B C} \u2192 (A \u2192 B) \u2192 B \u219d C \u2192 A \u219d C\n    rmap : \u2200 {A B C} \u2192 (B \u2192 C) \u2192 A \u219d B \u2192 A \u219d C\n\n\u2192Profunctor : Profunctor -\u2192-\n\u2192Profunctor = flip _\u2218\u2032_ , _\u2218\u2032_\n\nUpStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nUpStar F D C = D \u2192 F C\n\nupStarRawFunctor : \u2200 {F A} \u2192 RawFunctor F \u2192 RawFunctor (UpStar F A)\nupStarRawFunctor fun = record { _<$>_ = \u03bb k f x \u2192 k <$> f x }\n  where open RawFunctor fun\n\nupStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (UpStar F)\nupStarProfunctor fun = flip _\u2218\u2032_\n                     , RawFunctor._<$>_ (upStarRawFunctor fun)\n\nDownStar : (\u2605 \u2113 \u2192 \u2605 \u2113) \u2192 \u2605 \u2113 \u2192 \u2605 \u2113 \u2192 \u2605 \u2113\nDownStar F D C = F D \u2192 C\n\ndownStarProfunctor : \u2200 {F} \u2192 RawFunctor F \u2192 Profunctor (DownStar F)\ndownStarProfunctor fun = (\u03bb f g x \u2192 g (f <$> x)) , _\u2218\u2032_\n  where open RawFunctor fun\n\ndownStarRawFunctor : \u2200 {F A} \u2192 RawFunctor (DownStar F A)\ndownStarRawFunctor = record { _<$>_ = _\u2218\u2032_ }\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Category\/Profunctor.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"e345f8399aaeb1770b1600df6dd746ffada8b9e6","subject":"Search\/Searchable\/Sum.agda","message":"Search\/Searchable\/Sum.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Sum.agda","new_file":"Search\/Searchable\/Sum.agda","new_contents":"open import Function.NP\nopen import Data.Nat using (_+_)\nimport Level as L\nimport Function.Inverse.NP as FI\nimport Function.Related as FR\nopen FI using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP\nopen import Data.Product.NP\nopen import Data.Sum\nopen import Data.Fin using (Fin)\nopen import Search.Type\nopen import Search.Searchable\n\nopen import Relation.Binary.Sum\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_ ; module \u2261-Reasoning; cong)\n\nmodule Search.Searchable.Sum where\n\ninfixr 4 _\u228e-search_\n\n_\u228e-search_ : \u2200 {m A B} \u2192 Search m A \u2192 Search m B \u2192 Search m (A \u228e B)\n(search\u1d2c \u228e-search search\u1d2e) _\u2219_ f = search\u1d2c _\u2219_ (f \u2218 inj\u2081) \u2219 search\u1d2e _\u2219_ (f \u2218 inj\u2082)\n\n_\u228e-search-ind_ : \u2200 {m p A B} {s\u1d2c : Search m A} {s\u1d2e : Search m B}\n                 \u2192 SearchInd p s\u1d2c \u2192 SearchInd p s\u1d2e \u2192 SearchInd p (s\u1d2c \u228e-search s\u1d2e)\n(Ps\u1d2c \u228e-search-ind Ps\u1d2e) P P\u2219 Pf\n  = P\u2219 (Ps\u1d2c (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2081))) P\u2219 (Pf \u2218 inj\u2081))\n       (Ps\u1d2e (\u03bb s \u2192 P (\u03bb _ f \u2192 s _ (f \u2218 inj\u2082))) P\u2219 (Pf \u2218 inj\u2082))\n\ninfixr 4 _\u228e-sum_\n\n_\u228e-sum_ : \u2200 {A B} \u2192 Sum A \u2192 Sum B \u2192 Sum (A \u228e B)\n(sum\u1d2c \u228e-sum sum\u1d2e) f = sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)\n\nmodule _ {A B} {sum\u1d2c : Sum A} {sum\u1d2e : Sum B} where\n\n    sum\u1d2c\u1d2e = sum\u1d2c \u228e-sum sum\u1d2e\n\n    _\u228e-adequate-sum_ : AdequateSum sum\u1d2c \u2192 AdequateSum sum\u1d2e \u2192 AdequateSum sum\u1d2c\u1d2e\n    (asum\u1d2c \u228e-adequate-sum asum\u1d2e) f = (Fin (sum\u1d2c (f \u2218 inj\u2081) + sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 FI.sym (Fin-\u228e-+ _ _) \u27e9\n                                     (Fin (sum\u1d2c (f \u2218 inj\u2081)) \u228e Fin (sum\u1d2e (f \u2218 inj\u2082)))\n                                   \u2194\u27e8 asum\u1d2c (f \u2218 inj\u2081) \u228e-cong asum\u1d2e (f \u2218 inj\u2082) \u27e9\n                                     (\u03a3 A (Fin \u2218 f \u2218 inj\u2081) \u228e \u03a3 B (Fin \u2218 f \u2218 inj\u2082))\n                                   \u2194\u27e8 FI.sym \u03a3\u228e-distrib \u27e9\n                                     \u03a3 (A \u228e B) (Fin \u2218 f)\n                                   \u220e\n      where open FR.EquationalReasoning\n\n_\u228e-\u03bc_ : \u2200 {A B} \u2192 Searchable A \u2192 Searchable B \u2192 Searchable (A \u228e B)\n\u03bcA \u228e-\u03bc \u03bcB = mk _ (search-ind \u03bcA \u228e-search-ind search-ind \u03bcB)\n                 (adequate-sum \u03bcA \u228e-adequate-sum adequate-sum \u03bcB)\n\nmodule _ {A B} {s\u1d2c : Search\u2081 A} {s\u1d2e : Search\u2081 B} where\n  s\u1d2c\u207a\u1d2e = s\u1d2c \u228e-search s\u1d2e\n  _\u228e-focus_ : Focus s\u1d2c \u2192 Focus s\u1d2e \u2192 Focus s\u1d2c\u207a\u1d2e\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2081 x , y) = inj\u2081 (f\u1d2c (x , y))\n  (f\u1d2c \u228e-focus f\u1d2e) (inj\u2082 x , y) = inj\u2082 (f\u1d2e (x , y))\n\n  _\u228e-unfocus_ : Unfocus s\u1d2c \u2192 Unfocus s\u1d2e \u2192 Unfocus s\u1d2c\u207a\u1d2e\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2081 x) = first inj\u2081 (f\u1d2c x)\n  _\u228e-unfocus_ f\u1d2c f\u1d2e (inj\u2082 y) = first inj\u2082 (f\u1d2e y)\n\n  {-\n  _\u228e-focused_ : Focused s\u1d2c \u2192 Focused s\u1d2e \u2192 Focused {L.zero} s\u1d2c\u207a\u1d2e\n  _\u228e-focused_ f\u1d2c f\u1d2e {B} = inverses (to f\u1d2c \u228e-focus to f\u1d2e) (from f\u1d2c \u228e-unfocus from f\u1d2e) (\u21d2) (\u21d0)\n      where\n        \u21d2 : (x : \u03a3 (A \u228e {!!}) {!!}) \u2192 _\n        \u21d2 (x , y) = {!!}\n        \u21d0 : (x : s\u1d2c _\u228e_ (B \u2218 inj\u2081) \u228e s\u1d2e _\u228e_ (B \u2218 inj\u2082)) \u2192 _\n        \u21d0 (inj\u2081 x) = cong inj\u2081 {!!}\n        \u21d0 (inj\u2082 x) = cong inj\u2082 {!!}\n  -}\n\n  _\u228e-lookup_ : Lookup s\u1d2c \u2192 Lookup s\u1d2e \u2192 Lookup (s\u1d2c \u228e-search s\u1d2e)\n  (lookup\u1d2c \u228e-lookup lookup\u1d2e) (x , y) = [ lookup\u1d2c x , lookup\u1d2e y ]\n\n  _\u228e-reify_ : Reify s\u1d2c \u2192 Reify s\u1d2e \u2192 Reify (s\u1d2c \u228e-search s\u1d2e)\n  (reify\u1d2c \u228e-reify reify\u1d2e) f = (reify\u1d2c (f \u2218 inj\u2081)) , (reify\u1d2e (f \u2218 inj\u2082))\n\n -- -}\n -- -}\n -- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Search\/Searchable\/Sum.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"36f75949b7ea002abb1388845ca75076f08c9ae0","subject":"elgamal.agda","message":"elgamal.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"elgamal.agda","new_file":"elgamal.agda","new_contents":"{-# OPTIONS --copatterns #-}\nopen import Function\nopen import Data.Product\nopen import Data.Bool.NP as Bool\nopen import Data.Unit\nopen import Data.Maybe.NP\nimport Data.Fin.NP as Fin\nopen Fin using (Fin; zero; suc) renaming (#_ to ##_)\nopen import Data.Nat.NP hiding (_^_)\nopen import Data.Bits\nimport Data.Vec.NP as Vec\nopen Vec using (Vec; []; _\u2237_; _\u2237\u02b3_; allFin; lookup) renaming (map to vmap)\n--import Data.Vec.Properties as Vec\nopen import Algebra.FunctionProperties\nopen import Relation.Binary.PropositionalEquality as \u2261\n\nmodule elgamal where\n\n\u2605\u2081 : Set\u2082\n\u2605\u2081 = Set\u2081\n\n\u2605 : Set\u2081\n\u2605 = Set\n\n[0\u2192_,1\u2192_] : \u2200 {a} {A : Set a} \u2192 A \u2192 A \u2192 Bool \u2192 A\n[0\u2192 e\u2080 ,1\u2192 e\u2081 ] b = if b then e\u2081 else e\u2080\n\ncase_0\u2192_1\u2192_ : \u2200 {a} {A : Set a} \u2192 Bool \u2192 A \u2192 A \u2192 A\ncase b 0\u2192 e\u2080 1\u2192 e\u2081 = if b then e\u2081 else e\u2080\n\nsumBit : (Bit \u2192 \u2115) \u2192 \u2115\nsumBit f = f 0b + f 1b\n\ndata `\u2605 : \u2605 where\n  `\u22a4   : `\u2605\n  `X   : `\u2605\n  _`\u00d7_ : `\u2605 \u2192 `\u2605 \u2192 `\u2605\ninfixr 2 _`\u00d7_\n\nmodule Univ (X : \u2605) where\n    El : `\u2605 \u2192 \u2605\n    El `\u22a4         = \u22a4\n    El `X         = X\n    El (u\u2080 `\u00d7 u\u2081) = El u\u2080 \u00d7 El u\u2081\n\n    record \u21ba (R : `\u2605) (A : \u2605) : \u2605 where\n      constructor mk\n      field\n        run\u21ba : El R \u2192 A\n    open \u21ba public\n\n    EXP : (R : `\u2605) \u2192 \u2605\n    EXP R = \u21ba R Bit\n\n    Det : \u2605 \u2192 \u2605\n    Det = \u21ba `\u22a4\n\nmodule \u2124q-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : (\u2124q \u2192 \u2115) \u2192 \u2115)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  where\n\n  open Univ \u2124q\n  open `\u2605 public renaming (`X to `\u2124q)\n\n  sum : (u : `\u2605) \u2192 (El u \u2192 \u2115) \u2192 \u2115\n  sum `\u22a4         f = f _\n  sum `X         f = sum\u2124q f\n  sum (u\u2080 `\u00d7 u\u2081) f = sum u\u2080 (\u03bb x\u2080 \u2192\n                       sum u\u2081 (\u03bb x\u2081 \u2192\n                         f (x\u2080 , x\u2081)))\n\n  #_ : \u2200 {u} \u2192 \u21ba u Bit \u2192 \u2115\n  #_ {u} f = sum u (Bool.to\u2115 \u2218 run\u21ba f)\n\n  \u2047 : \u2200 R \u2192 \u21ba R (El R)\n  run\u21ba (\u2047 _) = id\n\n  pure\u21ba : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  run\u21ba (pure\u21ba x) r = x -- turning r to _ produce an error\n\n  \u27ea_\u27eb : \u2200 {R A} \u2192 A \u2192 \u21ba R A\n  \u27ea_\u27eb = pure\u21ba\n\n  {-\n  \u27ea_\u27eb\u1d30 : \u2200 {a} {A : Set a} \u2192 A \u2192 Det A\n  \u27ea_\u27eb\u1d30 = pure\u1d30\n  -}\n\n  map\u21ba : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  run\u21ba (map\u21ba f x) r = f (run\u21ba x r)\n\n  infixl 4 _\u229b_\n  _\u229b_ : \u2200 {R S A B} \u2192 \u21ba R (A \u2192 B) \u2192 \u21ba S A \u2192 \u21ba (R `\u00d7 S) B\n  run\u21ba (af \u229b ax) rs = run\u21ba af (proj\u2081 rs) (run\u21ba ax (proj\u2082 rs))\n\n  \u27ea_\u00b7_\u27eb : \u2200 {A B R} \u2192 (A \u2192 B) \u2192 \u21ba R A \u2192 \u21ba R B\n  \u27ea f \u00b7 x \u27eb = map\u21ba f x\n\n  \u27ea_\u00b7_\u00b7_\u27eb : \u2200 {A B C} {R S} \u2192\n              (A \u2192 B \u2192 C) \u2192 \u21ba R A \u2192 \u21ba S B \u2192 \u21ba (R `\u00d7 S) C\n  \u27ea f \u00b7 x \u00b7 y \u27eb = map\u21ba f x \u229b y\n\n  _\u27e8\u229e\u27e9_ : \u2200 {R S} \u2192 \u21ba R \u2124q \u2192 \u21ba S \u2124q \u2192 \u21ba (R `\u00d7 S) \u2124q\n  x \u27e8\u229e\u27e9 y = \u27ea _\u229e_ \u00b7 x \u00b7 y \u27eb\n\n  \u27e8_\u229e\u27e9_ : \u2200 {R} \u2192 \u2124q \u2192 \u21ba R \u2124q \u2192 \u21ba R \u2124q\n  \u27e8 x \u229e\u27e9 y = \u27ea _\u229e_ x \u00b7 y \u27eb\n\n  infix 4 _\u2248\u21ba_ _\u2248\u1d2c_\n  _\u2248\u21ba_ : \u2200 {R : `\u2605} (f g : EXP R) \u2192 \u2605\n  _\u2248\u21ba_ = _\u2261_ on #_\n\n  _\u2248\u1d2c_ : \u2200 {A R} (f g : \u21ba R A) \u2192 Set _\n  _\u2248\u1d2c_ {A} f g = \u2200 (Adv : A \u2192 Bit) \u2192 \u27ea Adv \u00b7 f \u27eb \u2248\u21ba \u27ea Adv \u00b7 g \u27eb\n\n  lem : \u2200 x \u2192 \u27e8 x \u229e\u27e9 (\u2047 `\u2124q) \u2248\u1d2c \u2047 _ \n  lem x Adv = sum\u2124q-lem (Bool.to\u2115 \u2218 Adv) x\n\n  -- \u2200 (A : \u2124q \u2192 Bit) \u2192 # (A \u2047) \n\nopen Fin.Modulo renaming (sucmod to [suc])\n\n{-\nmodule \u2124q-implem (q-2 : \u2115) where\n  q : \u2115\n  q = 2 + q-2\n\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = zero\n\n  [1] : \u2124q\n  [1] = suc zero\n-}\nmodule \u2124q-implem (q : \u2115) ([0]' [1]' : Fin q) where\n  \u2124q : \u2605\n  \u2124q = Fin q\n\n  [0] : \u2124q\n  [0] = [0]'\n\n  [1] : \u2124q\n  [1] = [1]'\n\n  _\u2115\u229e_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = m \u2115\u229e ([suc] n)\n  {-\n  zero  \u2115\u229e n = n\n  suc m \u2115\u229e n = [suc] (m \u2115\u229e n)\n  -}\n\n  _\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u229e n = Fin.to\u2115 m \u2115\u229e n\n\n  _\u2115\u22a0_ : \u2115 \u2192 \u2124q \u2192 \u2124q\n  zero  \u2115\u22a0 n = [0]\n  suc m \u2115\u22a0 n = n \u229e (m \u2115\u22a0 n)\n\n  _\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m \u22a0 n = Fin.to\u2115 m \u2115\u22a0 n\n\n  _[^]\u2115_ : \u2124q \u2192 \u2115 \u2192 \u2124q\n  m [^]\u2115 zero  = [1]\n  m [^]\u2115 suc n = m \u22a0 (m [^]\u2115 n)\n\n  _[^]_ : \u2124q \u2192 \u2124q \u2192 \u2124q\n  m [^] n = m [^]\u2115 (Fin.to\u2115 n)\n\n  all\u2124q : Vec \u2124q q\n  all\u2124q = allFin q\n\n  sum\u2124q : (\u2124q \u2192 \u2115) \u2192 \u2115\n  sum\u2124q f = Vec.sum (vmap f all\u2124q)\n\n  sum\u2124q-[suc]-lem : \u2200 f \u2192 sum\u2124q (f \u2218 [suc]) \u2261 sum\u2124q f\n  sum\u2124q-[suc]-lem f rewrite \u2261.sym (Vec.sum-map-rot\u2081 f all\u2124q)\n                          | Vec.map-\u2218 f [suc] all\u2124q\n                          | rot\u2081-map-sucmod q\n                          = refl\n\n  --  comm-[suc]-\u2115\u229e : \u2200 m n \u2192 [suc] (m \u2115\u229e n) \u2261 m \u2115\u229e ([suc] n)\n\n  sum\u2124q-\u2115\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u2115\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u2115\u229e-lem zero f = refl\n  sum\u2124q-\u2115\u229e-lem (suc m) f rewrite sum\u2124q-[suc]-lem (f \u2218 _\u2115\u229e_ m)\n                               | sum\u2124q-\u2115\u229e-lem m f = refl\n\n  sum\u2124q-\u229e-lem : \u2200 m f \u2192 sum\u2124q (f \u2218 _\u229e_ m) \u2261 sum\u2124q f\n  sum\u2124q-\u229e-lem = sum\u2124q-\u2115\u229e-lem \u2218 Fin.to\u2115\n\nmodule G-implem (p q : \u2115) (g' : Fin p) (0[p] 1[p] : Fin p) (0[q] 1[q] : Fin q) where\n  open \u2124q-implem q 0[q] 1[q] public\n  open \u2124q-implem p 0[p] 1[p] public using () renaming (\u2124q to G; _\u22a0_ to _\u2219_; _[^]\u2115_ to _^[p]_)\n\n  g : G\n  g = g'\n\n  _^_ : G \u2192 \u2124q \u2192 G\n  x ^ n = x ^[p] Fin.to\u2115 n\n\n  g^_ : \u2124q \u2192 G\n  g^ n = g ^ n\n\n  postulate\n    g^-inj : \u2200 m n \u2192 g^ m \u2261 g^ n \u2192 m \u2261 n\n\nmodule G-count\n  (\u2124q : \u2605)\n  (_\u229e_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (sum\u2124q : (\u2124q \u2192 \u2115) \u2192 \u2115)\n  (sum\u2124q-lem : \u2200 f x \u2192 sum\u2124q (f \u2218 _\u229e_ x) \u2261 sum\u2124q f)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  where\n\n  open Univ \u2124q\n  open \u2124q-count \u2124q _\u229e_ sum\u2124q sum\u2124q-lem\n\n  \u2047G : \u21ba `\u2124q G\n  run\u21ba \u2047G x = g ^ x\n\nmodule DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  where\n    DDHAdv : \u2605 \u2192 \u2605\n    DDHAdv R = G \u2192 G \u2192 G \u2192 R \u2192 Bool\n\n    \u2141\u2080 : \u2200 {R} {_I : \u2605} \u2192 DDHAdv R \u2192 (\u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bool\n    \u2141\u2080 D (x , y , _ , r) = D (g^ x) (g^ y) (g^ (x \u22a0 y)) r\n\n    \u2141\u2081 : \u2200 {R} \u2192 DDHAdv R \u2192 (\u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    \u2141\u2081 D (x , y , z , r) = D (g^ x) (g^ y) (g^ z) r\n\n    \u2141 : \u2200 {R} \u2192 DDHAdv R \u2192 Bool \u2192 (\u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    \u2141 D b = (if b then \u2141\u2080 else \u2141\u2081) D\n\n    \u2141\u2032 : \u2200 {R} \u2192 DDHAdv R \u2192 (Bool \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    \u2141\u2032 D (b , x , y , z , r) = \u2141 D b (x , y , z , r)\n\n    module With\u21ba where\n        open Univ \u2124q\n        open `\u2605 public renaming (`X to `\u2124q)\n        DDHAdv\u21ba : `\u2605 \u2192 \u2605\n        DDHAdv\u21ba R = G \u2192 G \u2192 G \u2192 \u21ba R Bool\n        \u2141\u2080\u21ba : \u2200 {R _I} \u2192 DDHAdv\u21ba R \u2192 \u21ba (`\u2124q `\u00d7 `\u2124q `\u00d7 _I `\u00d7 R) Bool\n        run\u21ba (\u2141\u2080\u21ba D) = \u2141\u2080 (\u03bb a b c \u2192 run\u21ba (D a b c))\n\nmodule El-Gamal-Generic\n  (\u2124q : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G : \u2605)\n  (g : G)\n  (_^_ : G \u2192 \u2124q \u2192 G)\n  (Message : \u2605)\n  (_\u2295_ : G \u2192 Message \u2192 Message)\n  (_\u2295\u207b\u00b9_ : G \u2192 Message \u2192 Message)\n  where\n\n    -- \u03b1 is the pk\n    -- \u03b1 = g ^ x\n    -- x is the sk\n\n    PubKey = G\n    SecKey = \u2124q\n    KeyPair = PubKey \u00d7 SecKey\n    CipherText = G \u00d7 Message\n\n    M = Message\n    C = CipherText\n\n    KeyGen : \u2124q \u2192 KeyPair\n    KeyGen x = (g ^ x , x)\n\n    -- KeyGen\u21ba : \u21ba \u2124q KeyPair\n    -- KeyGen\u21ba = mk KeyGen\n\n    Enc : PubKey \u2192 Message \u2192 \u2124q \u2192 CipherText\n    Enc \u03b1 m y = \u03b2 , \u03b6 where\n      \u03b2 = g ^ y\n      \u03b4 = \u03b1 ^ y\n      \u03b6 = \u03b4 \u2295 m\n\n    -- Enc\u21ba : PubKey \u2192 Message \u2192 \u21ba \u2124q CipherText\n    -- Enc\u21ba \u03b1 m = mk (Enc \u03b1 m)\n\n    Dec : SecKey \u2192 CipherText \u2192 Message\n    Dec x (\u03b2 , \u03b6) = (\u03b2 ^ x) \u2295\u207b\u00b9 \u03b6\n\n    EncAdv : \u2605 \u2192 \u2605\n    EncAdv R = PubKey \u2192 R \u2192 (Bool \u2192 M) \u00d7 (C \u2192 Bool)\n\n    Game0 : \u2200 {R _I : Set} \u2192 EncAdv R \u2192 (Bool \u00d7 \u2124q \u00d7 \u2124q \u00d7 _I \u00d7 R) \u2192 Bool\n    Game0 A (b , x , y , z , r) =\n      let (pk , sk) = KeyGen x\n          (m , D) = A pk r in\n      D (Enc pk (m b) y)\n\n    Game : (i : Bool) \u2192 \u2200 {R} \u2192 EncAdv R \u2192 (Bool \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q \u00d7 R) \u2192 Bool\n    Game i A (b , x , y , z , r) =\n      let (\u03b1 , sk) = KeyGen x\n          (m , D) = A \u03b1 r\n          \u03b2 = g ^ y\n          \u03b4 = \u03b1 ^ case i 0\u2192 y 1\u2192 z\n          \u03b6 = \u03b4 \u2295 m b\n      in {-b ==\u1d47-} D (\u03b2 , \u03b6)\n\n    Game-0b\u2261Game0 : \u2200 {R} \u2192 Game 0b \u2261 Game0 {R}\n    Game-0b\u2261Game0 = refl\n\n    open DDH \u2124q _\u22a0_ G (_^_ g) public\n\n    -- Game0 \u2248 Game 0b\n    -- Game1 = Game 1b\n\n    -- Game0 \u2264 \u03b5\n    -- Game1 \u2261 0\n\n    -- \u2047 \u2295 x \u2248 \u2047\n\n    -- g ^ \u2047 \u2219 x \u2248 g ^ \u2047\n\nmodule El-Gamal-Base\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    (_\u207b\u00b9 : G \u2192 G)\n    (_\u2219_ : G \u2192 G \u2192 G)\n    where\n\n    _\/_ : G \u2192 G \u2192 G\n    x \/ y = x \u2219 (y \u207b\u00b9)\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ G _\u2219_ (flip _\/_) public\n\n    TrA : \u2200 {R} \u2192 Bool \u2192 EncAdv R \u2192 DDHAdv R\n    TrA b A g\u02e3 g\u02b8 g? r = d (g\u02b8 , g? \u2219 m b)\n       where m,d = A g\u02e3 r\n             m = proj\u2081 m,d\n             d = proj\u2082 m,d\n\n    module Proof\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      (dist-^-\u22a0 : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      -- (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n      where\n\n        pf1 : \u2200 {R} A r \u2192 Game 0b {R} A r \u2261 \u2141\u2032 (TrA 1b A) r\n        pf1 A (true , x , y , z , r) rewrite dist-^-\u22a0 g x y = refl\n        pf1 A (false , x , y , z , r) = {!!}\n\n        pf2 : \u2200 {R} A r \u2192 Game 1b {R} A r \u2261 \u2141\u2032 (TrA 0b A) r\n        pf2 A (true , x , y , z , r)  = {!!}\n        pf2 A (false , x , y , z , r) = {!!}\n\nmodule El-Gamal-Hashed\n    (\u2124q : \u2605)\n    (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (G : \u2605)\n    (g : G)\n    (_^_ : G \u2192 \u2124q \u2192 G)\n    -- (HKey : \u2605)\n    (|M| : \u2115)\n    (\u210b : {-HKey \u2192-} G \u2192 Bits |M|) where\n\n    Message = Bits |M|\n\n    \u210b\u27e8_\u27e9\u2295_ : G \u2192 Message \u2192 Message\n    \u210b\u27e8 \u03b4 \u27e9\u2295 m = \u210b \u03b4 \u2295 m\n\n    open El-Gamal-Generic \u2124q _\u22a0_ G g _^_ Message \u210b\u27e8_\u27e9\u2295_\n\n    module Proof\n      -- (\u22a0-comm : Commutative _\u2261_ _\u22a0_)\n      -- (^-lem : \u2200 \u03b1 x y \u2192 \u03b1 ^ (x \u22a0 y) \u2261 (\u03b1 ^ x) ^ y)\n      (^-comm : \u2200 \u03b1 x y \u2192 (\u03b1 ^ x) ^ y \u2261 (\u03b1 ^ y) ^ x)\n      -- (x \u2219 x \u207b\u00b9 \u2261 0)\n      (\u211a : \u2605)\n      (dist : \u211a \u2192 \u211a \u2192 \u211a)\n      (1\/2 : \u211a)\n      (_\u2264_ : \u211a \u2192 \u211a \u2192 \u2605)\n      (\u03b5-DDH : \u211a)\n      where\n\n        Pr[S_] : Bool \u2192 \u211a\n        Pr[S b ] = {!!}\n\n        -- SS\n        -- SS = dist Pr[\n\n        Pr[S\u2080] = Pr[S 0b ]\n        Pr[S\u2081] = Pr[S 1b ]\n\n        pf1 : Pr[S\u2081] \u2261 1\/2\n        pf1 = {!!}\n\n        pf2 : dist Pr[S\u2080] 1\/2 \u2261 dist Pr[S\u2080] Pr[S\u2081]\n        pf2 = {!!}\n\n        pf3 : dist Pr[S\u2080] Pr[S\u2081] \u2264 \u03b5-DDH\n        pf3 = {!!}\n\n        pf4 : dist Pr[S\u2080] 1\/2 \u2264 \u03b5-DDH\n        pf4 = {!!}\n\nmodule G11 = G-implem 11 10 (## 2) (## 0) (## 1) (## 0) (## 1)\nopen G11\nmodule E11 = El-Gamal-Base _ _\u22a0_ G g _^_ {!!} _\u2219_\nopen E11\nopen Proof {!!}\n\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n        -- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'elgamal.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"50b0ec2faee73481dd79f4195a7d115f58e31e9e","subject":"Added code related to an Agda's unification question.","message":"Added code related to an Agda's unification question.\n\nIgnore-this: 2627f8de779caf0c7e0a2ec10669323c\n\ndarcs-hash:20110805141522-3bd4e-4f91309c869eb37735f00141e803aa0c0a26983f.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Data\/List\/PostulatesVersusDataTypes.agda","new_file":"Draft\/FOTC\/Data\/List\/PostulatesVersusDataTypes.agda","new_contents":"module Draft.FOTC.Data.List.PostulatesVersusDataTypes where\n\ndata U : Set where\n  _::_ : U \u2192 U \u2192 U\n\ndata ListU : U \u2192 Set where\n  cons : \u2200 x xs \u2192 ListU xs \u2192 ListU (x :: xs)\n\ntailU : \u2200 {x xs} \u2192 ListU (x :: xs) \u2192 ListU xs\ntailU {x} {xs} (cons .x .xs xsL) = xsL\n\npostulate\n  D   : Set\n  _\u2237_ : D \u2192 D \u2192 D\n\ndata List : D \u2192 Set where\n  cons : \u2200 x xs \u2192 List xs \u2192 List (x \u2237 xs)\n\ntail : \u2200 {x xs} \u2192 List (x \u2237 xs) \u2192 List xs\ntail l = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Data\/List\/PostulatesVersusDataTypes.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"df7ee245210e34a38e6948dc3146322bc179f971","subject":"Added Asgy example on inequalities.","message":"Added Asgy example on inequalities.\n\nIgnore-this: 1a5138c14a5f7f363798e9cd3583bc16\n\ndarcs-hash:20110326142057-3bd4e-b341b9e2ff2ba1b0ac31beaffa922f67cc0f331e.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"src\/Agsy\/McCarthy91\/Arithmetic.agda","new_file":"src\/Agsy\/McCarthy91\/Arithmetic.agda","new_contents":"------------------------------------------------------------------------------\n-- Arithmetic properties used by the McCarthy 91 function\n------------------------------------------------------------------------------\n\nmodule Agsy.McCarthy91.Arithmetic where\n\nopen import Data.Nat\n\nopen import Relation.Binary.PropositionalEquality\n\n------------------------------------------------------------------------------\n\n91\u2261[100+11\u223810]\u223810 : (100 + 11 \u2238 10) \u2238 10 \u2261 91\n91\u2261[100+11\u223810]\u223810 = refl\n\n20>19 : 20 > 19  -- via Agsy\n20>19 = s\u2264s\n          (s\u2264s\n           (s\u2264s\n            (s\u2264s\n             (s\u2264s\n              (s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s (s\u2264s (s\u2264s (s\u2264s (s\u2264s (s\u2264s (s\u2264s (s\u2264s (s\u2264s z\u2264n)))))))))))))))))))\n\n50>49 : 50 > 49  -- via Agsy {-t 30}\n50>49 = s\u2264s\n          (s\u2264s\n           (s\u2264s\n            (s\u2264s\n             (s\u2264s\n              (s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      (s\u2264s\n                                                       (s\u2264s\n                                                        (s\u2264s\n                                                         (s\u2264s\n                                                          (s\u2264s\n                                                           z\u2264n)))))))))))))))))))))))))))))))))))))))))))))))))\n\n75>74 : 75 > 74  -- via Agsy {-t 180}\n75>74 = s\u2264s\n          (s\u2264s\n           (s\u2264s\n            (s\u2264s\n             (s\u2264s\n              (s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      (s\u2264s\n                                                       (s\u2264s\n                                                        (s\u2264s\n                                                         (s\u2264s\n                                                          (s\u2264s\n                                                           (s\u2264s\n                                                            (s\u2264s\n                                                             (s\u2264s\n                                                              (s\u2264s\n                                                               (s\u2264s\n                                                                (s\u2264s\n                                                                 (s\u2264s\n                                                                  (s\u2264s\n                                                                   (s\u2264s\n                                                                    (s\u2264s\n                                                                     (s\u2264s\n                                                                      (s\u2264s\n                                                                       (s\u2264s\n                                                                        (s\u2264s\n                                                                         (s\u2264s\n                                                                          (s\u2264s\n                                                                           (s\u2264s\n                                                                            (s\u2264s\n                                                                             (s\u2264s\n                                                                              (s\u2264s\n                                                                               (s\u2264s\n                                                                                (s\u2264s\n                                                                                 (s\u2264s\n                                                                                  (s\u2264s\n                                                                                   (s\u2264s\n                                                                                    z\u2264n))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n\n101>100 : 101 > 100  -- via Agsy {-t 600}\n101>100 = s\u2264s\n          (s\u2264s\n           (s\u2264s\n            (s\u2264s\n             (s\u2264s\n              (s\u2264s\n               (s\u2264s\n                (s\u2264s\n                 (s\u2264s\n                  (s\u2264s\n                   (s\u2264s\n                    (s\u2264s\n                     (s\u2264s\n                      (s\u2264s\n                       (s\u2264s\n                        (s\u2264s\n                         (s\u2264s\n                          (s\u2264s\n                           (s\u2264s\n                            (s\u2264s\n                             (s\u2264s\n                              (s\u2264s\n                               (s\u2264s\n                                (s\u2264s\n                                 (s\u2264s\n                                  (s\u2264s\n                                   (s\u2264s\n                                    (s\u2264s\n                                     (s\u2264s\n                                      (s\u2264s\n                                       (s\u2264s\n                                        (s\u2264s\n                                         (s\u2264s\n                                          (s\u2264s\n                                           (s\u2264s\n                                            (s\u2264s\n                                             (s\u2264s\n                                              (s\u2264s\n                                               (s\u2264s\n                                                (s\u2264s\n                                                 (s\u2264s\n                                                  (s\u2264s\n                                                   (s\u2264s\n                                                    (s\u2264s\n                                                     (s\u2264s\n                                                      (s\u2264s\n                                                       (s\u2264s\n                                                        (s\u2264s\n                                                         (s\u2264s\n                                                          (s\u2264s\n                                                           (s\u2264s\n                                                            (s\u2264s\n                                                             (s\u2264s\n                                                              (s\u2264s\n                                                               (s\u2264s\n                                                                (s\u2264s\n                                                                 (s\u2264s\n                                                                  (s\u2264s\n                                                                   (s\u2264s\n                                                                    (s\u2264s\n                                                                     (s\u2264s\n                                                                      (s\u2264s\n                                                                       (s\u2264s\n                                                                        (s\u2264s\n                                                                         (s\u2264s\n                                                                          (s\u2264s\n                                                                           (s\u2264s\n                                                                            (s\u2264s\n                                                                             (s\u2264s\n                                                                              (s\u2264s\n                                                                               (s\u2264s\n                                                                                (s\u2264s\n                                                                                 (s\u2264s\n                                                                                  (s\u2264s\n                                                                                   (s\u2264s\n                                                                                    (s\u2264s\n                                                                                     (s\u2264s\n                                                                                      (s\u2264s\n                                                                                       (s\u2264s\n                                                                                        (s\u2264s\n                                                                                         (s\u2264s\n                                                                                          (s\u2264s\n                                                                                           (s\u2264s\n                                                                                            (s\u2264s\n                                                                                             (s\u2264s\n                                                                                              (s\u2264s\n                                                                                               (s\u2264s\n                                                                                                (s\u2264s\n                                                                                                 (s\u2264s\n                                                                                                  (s\u2264s\n                                                                                                   (s\u2264s\n                                                                                                    (s\u2264s\n                                                                                                     (s\u2264s\n                                                                                                      (s\u2264s\n                                                                                                       (s\u2264s\n                                                                                                        (s\u2264s\n                                                                                                         (s\u2264s\n                                                                                                          (s\u2264s\n                                                                                                           (s\u2264s\n                                                                                                            (s\u2264s\n                                                                                                             (s\u2264s\n                                                                                                              z\u2264n))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/Agsy\/McCarthy91\/Arithmetic.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"c69d737c5cde9d40803f5bc7f86993283dd35f54","subject":"lib\/Function\/Inverse\/NP.agda","message":"lib\/Function\/Inverse\/NP.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Function\/Inverse\/NP.agda","new_file":"lib\/Function\/Inverse\/NP.agda","new_contents":"module Function.Inverse.NP where\n\nopen import Function.Inverse public\nopen import Function.Equality\nopen import Relation.Binary\nopen import Data.Product\n\n_$\u2081_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Inverse From To \u2192 Setoid.Carrier From \u2192 Setoid.Carrier To\niso $\u2081 x = Inverse.to iso \u27e8$\u27e9 x\n\n_$\u2082_ : \u2200 {f\u2081 f\u2082 t\u2081 t\u2082}\n           {From : Setoid f\u2081 f\u2082} {To : Setoid t\u2081 t\u2082}\n         \u2192 Inverse From To \u2192 Setoid.Carrier To \u2192 Setoid.Carrier From\niso $\u2082 x = Inverse.from iso \u27e8$\u27e9 x\n\n_\u2208_ : \u2200  {a\u2081 a\u2082}\n         {A\u2081 A\u2082  : Setoid a\u2081 a\u2082}\n         (x\u1d62     : Setoid.Carrier A\u2081 \u00d7 Setoid.Carrier A\u2082)\n         (bij    : Inverse A\u2081 A\u2082)\n      \u2192  Set a\u2082\n_\u2208_ {A\u2082 = A\u2082} (x\u2081 , x\u2082) iso = iso $\u2081 x\u2081 \u2248 x\u2082\n  where open Setoid A\u2082\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Function\/Inverse\/NP.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"70a75a08ff05e0de26f0116ef2d992557aaa9417","subject":"Control\/Protocol\/GameSemantics.agda","message":"Control\/Protocol\/GameSemantics.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol\/GameSemantics.agda","new_file":"Control\/Protocol\/GameSemantics.agda","new_contents":"open import Algebra\nopen import Type\nopen import Data.One\nopen import Data.Two\nopen import Data.Nat\nopen import Data.Sum.NP hiding ([_,_]\u2032)\nopen import Data.Product.NP\nopen import Data.List\nopen import Data.List.Properties\nopen import HoTT\nopen import Function.NP\nopen import Function.Extensionality\nopen import Relation.Binary.PropositionalEquality.NP\n\nopen import Control.Protocol renaming (In to O; Out to J; dual\u1d35\u1d3c to O\u2194J)\n\nmodule Control.Protocol.GameSemantics where\n\n_\u2237=_ : \u2200 {a}{A : \u2605_ a}{x y : A}{xs ys} \u2192 (x \u2261 y) \u2192 (xs \u2261 ys) \u2192 (x \u2237 xs) \u2261 (y \u2237 ys)\n_\u2237=_ e e' = ap\u2082 _\u2237_ e e'\n\ndata Move s : \u2605 where\n  move : (ms : List (Move (O\u2194J s))) \u2192 Move s\n\nArena = List \u2218 Move\n\nmove= : \u2200 {s}{xs ys : Arena (O\u2194J s)} \u2192 xs \u2261 ys \u2192 move xs \u2261 move ys\nmove= = ap move\n\ndual\u1d39 : \u2200 {s} \u2192 Move s \u2192 Move (O\u2194J s)\ndual\u1d2c : \u2200 {s} \u2192 List (Move s) \u2192 List (Move (O\u2194J s))\ndual\u1d39 (move ms) = move (dual\u1d2c ms)\ndual\u1d2c []        = []\ndual\u1d2c (m \u2237 ms)  = dual\u1d39 m \u2237 dual\u1d2c ms\n\n-- 1-move : \u2200 {s} \u2192 Move (O\u2194J s) \u2192 Move s\npattern 1-move m = move (m \u2237 [])\n\n-- 2-moves : \u2200 {s}(a b : Move (O\u2194J s)) \u2192 Move s\npattern 2-moves a b = move (a \u2237 b \u2237 [])\n\n\u22a4\u1d2c : \u2200 {s} \u2192 Arena s\n\u22a4\u1d2c = []\n\n\u22a5\u1d2c : \u2200 {s} \u2192 Arena s\n\u22a5\u1d2c = move [] \u2237 []\n\n2\u1d2c : \u2200 {s} \u2192 Arena s\n2\u1d2c = move [] \u2237 move [] \u2237 []\n\n2\u1d2c' : \u2200 {s} \u2192 Arena s\n2\u1d2c' = move 2\u1d2c \u2237 []\n\n_\u00d7\u1d2c_ : \u2200 {s} \u2192 Arena s \u2192 Arena s \u2192 Arena s\na \u00d7\u1d2c b = a ++ b\n\n_\u214b\u1d2c_ : \u2200 {s} \u2192 Arena (O\u2194J s) \u2192 Arena s \u2192 Arena s\na \u214b\u1d2c []             = []\na \u214b\u1d2c (move ms \u2237 bs) = move (a \u00d7\u1d2c ms) \u2237 (a \u214b\u1d2c bs)\n\ninfixr 4 _\u2192\u1d2c_\n_\u2192\u1d2c_ : \u2200 {s} \u2192 Arena s \u2192 Arena s \u2192 Arena s\na \u2192\u1d2c b = dual\u1d2c a \u214b\u1d2c b\n\nmonoid\u1d2c : \u2200 {s} \u2192 Monoid _ _\nmonoid\u1d2c {s} = monoid (Move s)\nmodule Monoid\u1d2c {s} = Monoid (monoid\u1d2c {s})\n\nArena\u1d30\u1d30 : \u2200 {s} \u2192 Arena s \u2261 Arena (O\u2194J (O\u2194J s))\nArena\u1d30\u1d30 = ap Arena (! dual\u1d35\u1d3c-involutive _)\n\nMove\u1d30\u1d30 : \u2200 {s} \u2192 Move s \u2261 Move (O\u2194J (O\u2194J s))\nMove\u1d30\u1d30 = ap Move (! dual\u1d35\u1d3c-involutive _)\n\ndual\u1d2c-involutive : \u2200 {s}(a : Arena s) \u2192 dual\u1d2c (dual\u1d2c a) \u2261 coe Arena\u1d30\u1d30 a\ndual\u1d39-involutive : \u2200 {s}(m : Move s)  \u2192 dual\u1d39 (dual\u1d39 m) \u2261 coe Move\u1d30\u1d30  m\n\ndual\u1d2c-involutive {O} [] = refl\ndual\u1d2c-involutive {J} [] = refl\ndual\u1d2c-involutive {O} (m \u2237 a) = dual\u1d39-involutive m \u2237= dual\u1d2c-involutive a\ndual\u1d2c-involutive {J} (m \u2237 a) = dual\u1d39-involutive m \u2237= dual\u1d2c-involutive a\n\ndual\u1d39-involutive {O} (move ms) = move= (dual\u1d2c-involutive ms)\ndual\u1d39-involutive {J} (move ms) = move= (dual\u1d2c-involutive ms)\n\n\u00ac\u1d2c_  : \u2200 {s} \u2192 Arena s \u2192 Arena s\n\u00ac\u00ac\u1d2c_ : \u2200 {s} \u2192 Arena s \u2192 Arena s\n\n\u00ac\u1d2c  a = a \u2192\u1d2c \u22a5\u1d2c\n\u00ac\u00ac\u1d2c a = \u00ac\u1d2c (\u00ac\u1d2c a)\n\n\u00ac-spec : \u2200 {s} (a : Arena s) \u2192 a \u2192\u1d2c \u22a5\u1d2c \u2261 move (dual\u1d2c a) \u2237 []\n\u00ac-spec a = move= (snd Monoid\u1d2c.identity (dual\u1d2c a)) \u2237= refl\n\n\u00ac\u00ac-spec : \u2200 {s} (a : Arena s) \u2192 (a \u2192\u1d2c \u22a5\u1d2c) \u2192\u1d2c \u22a5\u1d2c \u2261 move (move (coe Arena\u1d30\u1d30 a) \u2237 []) \u2237 []\n\u00ac\u00ac-spec a = \u00ac-spec (a \u2192\u1d2c \u22a5\u1d2c) \u2219 move= (move= (ap dual\u1d2c (snd Monoid\u1d2c.identity (dual\u1d2c a)) \u2219 dual\u1d2c-involutive a) \u2237= refl) \u2237= refl\n\nmodule Examples where\n  a1 : Arena O\n  a1 = move \u22a5\u1d2c \u2237 2-moves (move \u22a5\u1d2c) (move []) \u2237 []\n\n  a2 : Arena O\n  a2 = move 2\u1d2c \u2237 1-move (move \u22a5\u1d2c) \u2237 []\n\n  a1\u2192a2 : Arena O\n  a1\u2192a2 = move (move \u22a5\u1d2c \u2237\n                2-moves (move \u22a5\u1d2c) (move []) \u2237\n                move [] \u2237\n                move [] \u2237 [])\n        \u2237 move (move \u22a5\u1d2c \u2237\n                2-moves (move \u22a5\u1d2c) (move []) \u2237\n                move \u22a5\u1d2c \u2237 []) \u2237 []\n\n  a1\u2192a2-ok : a1\u2192a2 \u2261 (a1 \u2192\u1d2c a2)\n  a1\u2192a2-ok = refl\n\n  2\u1d2c\u2261\u22a5\u2192\u22a5\u2192\u22a5 : 2\u1d2c' {O} \u2261 \u22a5\u1d2c \u2192\u1d2c \u22a5\u1d2c \u2192\u1d2c \u22a5\u1d2c\n  2\u1d2c\u2261\u22a5\u2192\u22a5\u2192\u22a5 = refl\n\nTy' : \u2605\n{-\ndata Ty' : \u2605 where\n  `\ud835\udfda : Ty'\n  _`\u2192_ : Ty' \u2192 Ty' \u2192 Ty'\n  _`\u00d7_ : Ty' \u2192 Ty' \u2192 Ty'\n-}\n\ndata Ty : \u2605 where\n  `\ud835\udfda : Ty\n  {- _`\u2192_ -}\n  _`\u00d7_ : Ty \u2192 Ty \u2192 Ty\n  _`\u2192_ : Ty \u2192 Ty \u2192 Ty\n\nTy' = Ty\n\n\u27e6_\u27e7Ty' : Ty' \u2192 \u2605\n{-\n\u27e6 `\ud835\udfda     \u27e7Ty' = \ud835\udfda\n\u27e6 t `\u2192 u \u27e7Ty' = \u27e6 t \u27e7Ty' \u2192 \u27e6 u \u27e7Ty'\n\u27e6 t `\u00d7 u \u27e7Ty' = \u27e6 t \u27e7Ty' \u00d7 \u27e6 u \u27e7Ty'\n-}\n\n\u27e6_\u27e7Ty : Ty \u2192 \u2605\n\u27e6 `\ud835\udfda     \u27e7Ty = \ud835\udfda\n\u27e6 t `\u2192 u \u27e7Ty = \u27e6 t \u27e7Ty \u2192 \u27e6 u \u27e7Ty\n\u27e6 t `\u00d7 u \u27e7Ty = \u27e6 t \u27e7Ty \u00d7 \u27e6 u \u27e7Ty\n\n\u27e6_\u27e7Ty' = \u27e6_\u27e7Ty\n\nreplicate\u1d3e-send : \u2200 n {A : \u2605}(f : \u2115 \u2192 A) \u2192 \u27e6 replicate\u1d3e n (send\u2032 A end) \u27e7\nreplicate\u1d3e-send zero    f = end\nreplicate\u1d3e-send (suc n) f = f n , replicate\u1d3e-send n f\n\nmodule Steps n where\n    P\ud835\udfda = replicate\u1d3e n (send\u2032 \ud835\udfda end)\n\n    const-P\ud835\udfda : \ud835\udfda \u2192 \u27e6 P\ud835\udfda \u27e7\n    const-P\ud835\udfda b = replicate\u1d3e-send n (const b)\n\n    ty'\u1d3e : Ty' \u2192 Proto\n    {-\n    ty'\u1d3e `\ud835\udfda       = P\ud835\udfda\n    ty'\u1d3e (t `\u2192 u) = ty'\u1d3e t \u22b8 ty'\u1d3e u\n    ty'\u1d3e (t `\u00d7 u) = ty'\u1d3e t \u2297 ty'\u1d3e u\n    -}\n\n    ty\u1d3e : Ty \u2192 Proto\n    ty\u1d3e `\ud835\udfda       = P\ud835\udfda\n    ty\u1d3e (t `\u2192 u) = ty'\u1d3e t \u22b8 ty\u1d3e u\n    ty\u1d3e (t `\u00d7 u) = ty\u1d3e t \u2297 ty\u1d3e u\n\n    ty'\u1d3e t = ty\u1d3e t\n\nopen Steps 1\n\nmodule SingleRound where\n    `_ : \u2605 \u2192 Proto\n    ` A = recv\u2032 \ud835\udfd9 (send\u2032 A end)\n\n    p-swap : \u2200 {A B} \u2192 \u27e6 (` A \u2297 ` B) \u22b8 (` B \u2297 ` A) \u27e7\n    p-swap (inl 0\u2081) = inl (inl 0\u2081) , \u03bb a \u2192 inl 0\u2081 , \u03bb b \u2192 b , \u03bb _ \u2192 a , end\n    p-swap (inr 0\u2081) = inl (inr 0\u2081) , \u03bb b \u2192 inl 0\u2081 , \u03bb a \u2192 a , \u03bb _ \u2192 b , end\n                -- OR inl (inl 0\u2081) , \u03bb a \u2192 inl 0\u2081 , \u03bb b \u2192 a , \u03bb _ \u2192 b , end\n\n    p-dup : \u2200 {A} \u2192 \u27e6 (` A) \u22b8 (` A \u2297 ` A) \u27e7\n    p-dup (inl 0\u2081) = inl 0\u2081 , \u03bb a \u2192 a , \u03bb _ \u2192 a , end\n    p-dup (inr 0\u2081) = inl 0\u2081 , \u03bb a \u2192 a , \u03bb _ \u2192 a , end\n\n    p-ap : \u2200 {A B} \u2192 \u27e6 ((` A \u22b8 ` B) \u2297 ` A) \u22b8 ` B \u27e7\n    p-ap 0\u2081 = inl (inl 0\u2081) , [inl: (\u03bb _   \u2192 inl (inr 0\u2081) , \u03bb a \u2192 inl a , \u03bb b \u2192 b , end)\n                             ,inr: (\u03bb b _ \u2192 inl (inr 0\u2081) , \u03bb a \u2192 inl a , b , end) ]\n                             -- ,inr: (\u03bb b _ \u2192 inr b , inr 0\u2081 , \u03bb a \u2192 a , end) ]\n\n                             {-\n    is-not : \u27e6 dual(` \ud835\udfda \u22b8 ` \ud835\udfda) \u214b ` \ud835\udfda \u27e7\n    is-not 0\u2081 = inl 0\u2081 , [inl: const (inl 0\u2082 , [0: 0\u2082 , end 1: 1\u2082 , end ])\n                         ,inr: [0: const (inr {!!} , {!!})\n                                1: const {!!} ] ]\n                                -}\n\nmodule SingleSend where\n    `_ : \u2605 \u2192 Proto\n    ` A = send\u2032 A end\n\n    p-swap : \u2200 {A B} \u2192 \u27e6 (` A \u2297 ` B) \u22b8 (` B \u2297 ` A) \u27e7\n    p-swap a b = b , a , end\n\n    -- Why what we have is not about linearity!\n    p-dup : \u2200 {A} \u2192 \u27e6 (` A) \u22b8 (` A \u2297 ` A) \u27e7\n    p-dup a = a , a , end\n\n    p-ap : \u2200 {A B} \u2192 \u27e6 ((` A \u22b8 ` B) \u2297 ` A) \u22b8 ` B \u27e7\n    p-ap a = inl a , \u03bb b \u2192 b , end\n\n    p-curry : \u2200 {A B C} \u2192 \u27e6 ((` A \u2297 ` B) \u22b8 ` C) \u22b8 (` A \u22b8 ` B \u22b8 ` C) \u27e7\n    p-curry a b = inl a , inl b , \u03bb c \u2192 c , end\n\n    p-arr : \u2200 {A B} \u2192 (A \u2192 B) \u2192 \u27e6 ` A \u22b8 ` B \u27e7\n    p-arr f a = f a , end\n\n    p-arr' : \u2200 {A B C} \u2192 (A \u2192 B \u2192 C) \u2192 \u27e6 (` A \u2297 ` B) \u22b8 ` C \u27e7\n    p-arr' f a b = f a b , end\n\n    p-arr'' : \u2200 {A B C} \u2192 (A \u2192 B \u00d7 C) \u2192 \u27e6 ` A \u22b8 (` B \u2297 ` C) \u27e7\n    p-arr'' f a = fst (f a) , snd (f a) , end\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n{-\n        opp : \u2200 t \u2192 \u27e6 ty\u1d3e t \u22a5\u27e7\n        opp `\ud835\udfda       = _\n        opp (t `\u2192 u) = {!!}\n        opp (t `\u00d7 u) = {!!}\n-}\n\n        \n\n        {-\n        embed : \u2200 t \u2192 \u27e6 t \u27e7Ty \u2192 \u27e6 ty\u1d3e t \u27e7\n        reify : \u2200 t \u2192 \u27e6 ty\u1d3e t \u27e7 \u2192 \u27e6 t \u27e7Ty\n\n        {-\n        \u27e6 dual (P \u2297 Q) \u214b R \u27e7\n        \u27e6 dual P \u214b dual Q \u214b R \u27e7\n        \u27e6 dual P \u214b (Q \u22b8 R) \u27e7\n        -}\n        embed\u2192 : \u2200 t u \u2192 (\u27e6 t \u27e7Ty' \u2192 \u27e6 ty\u1d3e u \u27e7) \u2192 \u27e6 ty\u1d3e (t `\u2192 u) \u27e7\n        embed\u2192 `\ud835\udfda        u f = f\n        embed\u2192 (t `\u2192 t\u2081) u f = {!\u03bb g \u2192 f (reify (t `\u2192 t\u2081) g)!}\n       --  [[A]\u214bB]\u214bC = ([[A]]\u2297[B])\u214bC = (A\u2297[B])\u214bC\n        embed\u2192 (t `\u00d7 t\u2081) u f = coe (ap \u27e6_\u27e7 Goal) (embed\u2192 t (t\u2081 `\u2192 u) (embed\u2192 t\u2081 u \u2218 curry f))\n          where open \u2261-Reasoning\n                Goal = dual (ty'\u1d3e t) \u214b (ty'\u1d3e t\u2081 \u22b8 ty\u1d3e u)\n                     \u2261\u27e8 \u214b-assoc (dual (ty'\u1d3e t)) _ _ \u27e9\n                       (dual (ty'\u1d3e t) \u214b dual (ty'\u1d3e t\u2081)) \u214b ty\u1d3e u\n                     \u2261\u27e8 ! ap (flip _\u214b_ (ty\u1d3e u)) (dual-\u2297 (ty'\u1d3e t) (ty'\u1d3e t\u2081)) \u27e9\n                       dual (ty'\u1d3e t \u2297 ty'\u1d3e t\u2081) \u214b ty\u1d3e u\n                     \u220e\n\n        embed `\ud835\udfda       b       = b , end\n        embed (t `\u2192 u) f       = embed\u2192 t u (embed u \u2218 f)\n        embed (t `\u00d7 u) (x , y) = \u2297-pair {ty\u1d3e t} {ty\u1d3e u} (embed t x) (embed u y)\n\n        reify `\ud835\udfda       (b , end) = b\n        reify (t `\u2192 u) p = \u03bb x \u2192 reify u (\u22b8-apply {ty\u1d3e t} {ty\u1d3e u} p (embed t x))\n        reify (t `\u00d7 u) p = reify t (\u2297-fst (ty\u1d3e t) (ty\u1d3e u) p) , reify u (\u2297-snd (ty\u1d3e t) (ty\u1d3e u) p)\n        -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/GameSemantics.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c3b86560014129a64426a3a322b3fea9e1055db7","subject":"bintree.agda","message":"bintree.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"bintree.agda","new_file":"bintree.agda","new_contents":"module bintree\n  ([0,1] : Set)\nwhere\n\nopen import Data.Nat\nopen import Data.Bits\n\ndata T : \u2115 \u2192 Set where\n  leaf : \u2200 {n} \u2192 Bits n \u2192 T n\n  fork : \u2200 {n} (proba : [0,1]) (left right : T n) \u2192 T (1 + n)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'bintree.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5fccf4fd6ff6f91ce19babef44a0e0fc7c9f08e1","subject":"Game\/DDH.agda","message":"Game\/DDH.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/DDH.agda","new_file":"Game\/DDH.agda","new_contents":"open import Type\nopen import Data.Bits\nopen import Data.Product\n\n-- Decisional Diffe-Hellman security game\nmodule Game.DDH\n  (\u2124q  : \u2605)\n  (_\u22a0_ : \u2124q \u2192 \u2124q \u2192 \u2124q)\n  (G   : \u2605)\n  (g^_ : \u2124q \u2192 G)\n  (R\u2090  : \u2605) where\n\n-- Decisional Diffe-Hellman smoothing adversary\nAdv : \u2605\nAdv = R\u2090 \u2192 G \u2192 G \u2192 G \u2192 Bit\n\n-- The randomness supply needed for the decisional\n-- Diffe-Hellman games\nR : \u2605\nR = R\u2090 \u00d7 \u2124q \u00d7 \u2124q \u00d7 \u2124q\n\n-- Decisional Diffe-Hellman game:\n--   * input: adversary and randomness supply\n--   * output b: adversary claims we are in game \u2141 b\nGame : \u2605\nGame = Adv \u2192 R \u2192 Bit\n\n-- The adversary is given correlated inputs\n\u2141\u2080 : Game\n\u2141\u2080 d (r , x , y , _) = d r (g^ x) (g^ y) (g^ (x \u22a0 y))\n\n-- The adversary is given independent inputs\n\u2141\u2081 : Game\n\u2141\u2081 d (r , x , y , z) = d r (g^ x) (g^ y) (g^ z)\n\n-- Package \u2141\u2080 and \u2141\u2081\n\u2141 : Bit \u2192 Game\n\u2141 = proj (\u2141\u2080 , \u2141\u2081)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Game\/DDH.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"977cad8a00658df07d789e537cb036aa297a6509","subject":"3to2.agda","message":"3to2.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"3to2.agda","new_file":"3to2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"error: RPC failed; HTTP 403 curl 22 The requested URL returned error: 403\nfatal: the remote end hung up unexpectedly\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"16e3c79c254b8edd7f64b46df9c7c746fa4194c2","subject":"Add Language.Simple.Free","message":"Add Language.Simple.Free\n","repos":"crypto-agda\/crypto-agda","old_file":"Language\/Simple\/Free.agda","new_file":"Language\/Simple\/Free.agda","new_contents":"{-# OPTIONS --without-K --copatterns #-}\nopen import Level.NP\nopen import Function hiding (_$_)\nopen import Type\nopen import Algebra\nopen import Algebra.FunctionProperties\nopen import Algebra.Structures\nopen import Data.Nat using (\u2115; zero; suc; _+_; _\u2264?_)\nopen import Data.Fin using (Fin; zero; suc)\nopen import Data.Vec using (Vec; []; _\u2237_; _++_)\nopen import Data.One\nopen import Data.Two\nopen import Data.Product.NP\nopen import Relation.Nullary.Decidable\nopen import Relation.Binary.NP\nopen import Relation.Binary.PropositionalEquality.NP as \u2261 using (_\u2261_; !_; _\u2257_; module \u2261-Reasoning)\nopen import Category.Monad.NP\nimport Language.Simple.Abstract\n\nmodule Language.Simple.Free where\n\nmodule _ {Op : \u2115 \u2192 \u2605} where\n\n  open Language.Simple.Abstract Op renaming (E to T)\n\n  data Ctx (X : \u2605) : \u2605 where\n     hole : Ctx X\n     op : \u2200 {a b} \u2192 Op (a + (1 + b)) \u2192 Vec (T X) a \u2192 Ctx X \u2192 Vec (T X) b \u2192 Ctx  X\n\n  module _ {X : \u2605} where\n     \u03b7 : \u2200 {X} \u2192 X \u2192 T X\n     \u03b7 = var\n\n     _\u00b7_ : Ctx X \u2192 T X \u2192 T X\n     hole \u00b7 e = e\n     op o ts E us \u00b7 e = op o (ts ++ E \u00b7 e \u2237 us)\n\n  open WithEvalOp\n  open Monad monad public renaming (liftM to map-T)\n  module Unused\n                 {X R : \u2605}\n                 (\u22a2_\u2261_ : T X \u2192 T X \u2192 \u2605)\n                 (eval-Op  : \u2200 {n} \u2192 Op n \u2192 Vec R n \u2192 R)\n                 (eval-X : X \u2192 R)\n                 (t u : T X)\n                 (\u22a2t\u2261u : \u22a2 t \u2261 u)\n                 where\n     ev = eval eval-Op eval-X\n     \u2248 = ev t \u2261 ev u\n  module \u211b {X : \u2605} (\u22a2_\u2261_ : T X \u2192 T X \u2192 \u2605) where\n\n     {-\n     data _\u224b*_ : \u2200 {a} (ts us : Vec (T X) a) \u2192 \u2605\n     data _\u224b_ : (t u : T X) \u2192 \u2605\n\n     data _\u224b*_ where\n       []    : [] \u224b* []\n       _\u2237_ : \u2200 {t u a} {ts us : Vec _ a}\n               \u2192 t \u224b u \u2192 ts \u224b* us \u2192 (t \u2237 ts) \u224b* (u \u2237 us)\n     data _\u224b_ where\n       \u03b7 : \u2200 {t u} \u2192 \u22a2 t \u2261 u \u2192 t \u224b u\n       !\u03b7 : \u2200 {t u} \u2192 \u22a2 t \u2261 u \u2192 u \u224b t\n       var : \u2200 x \u2192 var x \u224b var x\n       op : \u2200 {a} (o : Op a) {ts us} \u2192 ts \u224b* us \u2192 op o ts \u224b op o us\n     -}\n\n     module Unused2 where\n\n        data Succ (X : \u2605) : \u2605 where\n           old : X \u2192 Succ X\n           new : Succ X\n\n        infix 9 _$_\n\n        _$_ : T (Succ X) \u2192 T X \u2192 T X\n        E $ t = E >>= (\u03bb { new \u2192 t ; (old x) \u2192 var x })\n\n     infix 0 _\u2248_\n     data _\u2248_   : (t u : T X) \u2192 \u2605 where\n        refl    : Reflexive _\u2248_\n        sym     : Symmetric _\u2248_\n        trans   : Transitive _\u2248_\n        \u03b7\u2261      : \u2200 {t u} \u2192 \u22a2 t \u2261 u \u2192 t \u2248 u\n        congCtx : \u2200 E {t u} \u2192 t \u2248 u \u2192 E \u00b7 t \u2248 E \u00b7 u\n\n     \u2248-isEquivalence : IsEquivalence _\u2248_\n     \u2248-isEquivalence = record { refl = refl ; sym = sym ; trans = trans }\n\nmodule FreeSemigroup where\n\n    U = Semigroup.Carrier\n    record _\u2192\u209b_ (X Y : Semigroup \u2080 \u2080) : \u2605 where\n      open Semigroup X renaming (_\u2219_ to _\u2219x_)\n      open Semigroup Y renaming (_\u2219_ to _\u2219y_)\n      field\n        to : U X \u2192 U Y\n        to-\u2219-cong : \u2200 t u \u2192 to (t \u2219x u) \u2261 to t \u2219y to u\n    open _\u2192\u209b_ public\n\n    id\u209b : \u2200 {S} \u2192 S \u2192\u209b S\n    to id\u209b = id\n    to-\u2219-cong id\u209b _ _ = \u2261.refl\n\n    module _ {X Y Z} where\n        _\u2218\u209b_ : (Y \u2192\u209b Z) \u2192 (X \u2192\u209b Y) \u2192 X \u2192\u209b Z\n        to (f \u2218\u209b g) = to f \u2218 to g\n        to-\u2219-cong (f \u2218\u209b g) t u\n           = to (f \u2218\u209b g) (t \u2219x u)\n           \u2261\u27e8 \u2261.cong (to f) (to-\u2219-cong g _ _) \u27e9\n              to f (to g t \u2219y to g u)\n           \u2261\u27e8 to-\u2219-cong f _ _ \u27e9\n              to (f \u2218\u209b g) t \u2219z to (f \u2218\u209b g) u\n           \u220e\n          where\n            open \u2261-Reasoning\n            open Semigroup X renaming (_\u2219_ to _\u2219x_)\n            open Semigroup Y renaming (_\u2219_ to _\u2219y_)\n            open Semigroup Z renaming (_\u2219_ to _\u2219z_)\n\n    data Op : \u2115 \u2192 \u2605 where\n       `\u03bc : Op 2\n    open Language.Simple.Abstract Op renaming (E to T)\n    open Monad monad\n\n    module _ {X : \u2605} where\n        _\u2219_ : T X \u2192 T X \u2192 T X\n        t \u2219 u = op `\u03bc (t \u2237 u \u2237 [])\n\n    module _ (X : \u2605) where\n        infix 0 \u22a2_\u2261_\n        data \u22a2_\u2261_ : T X \u2192 T X \u2192 \u2605 where\n           \u22a2-\u2219-assoc : \u2200 x y z \u2192 \u22a2 ((x \u2219 y) \u2219 z) \u2261 (x \u2219 (y \u2219 z))\n\n        open \u211b \u22a2_\u2261_\n\n        \u2219-assoc : Associative _\u2248_ _\u2219_\n        \u2219-assoc x y z = \u03b7\u2261 (\u22a2-\u2219-assoc x y z)\n\n        open Equivalence-Reasoning \u2248-isEquivalence\n\n        \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n        \u2219-cong {x} {y} {u} {v} p q =\n           x \u2219 u\n             \u2248\u27e8 congCtx (op `\u03bc (x \u2237 []) hole []) q \u27e9\n           x \u2219 v\n             \u2248\u27e8 congCtx (op `\u03bc [] hole (v \u2237 [])) p \u27e9\n           y \u2219 v \u220e\n\n        T-IsSemigroup : IsSemigroup _\u2248_ _\u2219_\n        T-IsSemigroup = record { isEquivalence = \u2248-isEquivalence\n                                              ; assoc = \u2219-assoc\n                                              ; \u2219-cong = \u2219-cong }\n\n        T-Semigroup : Semigroup \u2080 \u2080\n        T-Semigroup = record { Carrier = T X\n                                           ; _\u2248_ = _\u2248_\n                                           ; _\u2219_ = _\u2219_\n                                           ; isSemigroup = T-IsSemigroup }\n\n    map-T-\u2219-hom : \u2200 {X Y : \u2605} (f : X \u2192 Y) t u\n                           \u2192 map-T f (t \u2219 u) \u2261 map-T f t \u2219 map-T f u\n    map-T-\u2219-hom _ _ _ = \u2261.refl\n\n    T\u209b = T-Semigroup\n\n    T\u209b-hom : \u2200 {X Y} (f : X \u2192 Y) \u2192 T\u209b X \u2192\u209b T\u209b Y\n    to (T\u209b-hom f) = map-T f\n    to-\u2219-cong (T\u209b-hom f) = map-T-\u2219-hom f\n\n    module WithSemigroup (S : Semigroup \u2080 \u2080) where\n       module S = Semigroup S\n       open S renaming (Carrier to S\u2080; _\u2219_ to _\u2219\u209b_)\n       eval-Op : \u2200 {n} \u2192 Op n \u2192 Vec S\u2080 n \u2192 S\u2080\n       eval-Op `\u03bc (x \u2237 y \u2237 []) = x \u2219\u209b y\n       eval-T : T S\u2080 \u2192 S\u2080\n       eval-T = WithEvalOp.eval eval-Op id\n\n       eval-T-var : \u2200 x \u2192 eval-T (var x) \u2261 x\n       eval-T-var x = \u2261.refl\n\n       eval-T-\u2219 : \u2200 t u \u2192 eval-T (t \u2219 u) \u2261 eval-T t \u2219\u209b eval-T u\n       eval-T-\u2219 t u = \u2261.refl\n\n       eval-T\u209b : T\u209b S\u2080 \u2192\u209b S\n       to eval-T\u209b = eval-T\n       to-\u2219-cong eval-T\u209b t u = \u2261.refl\n\n    {-\n    T\u209b : Sets \u27f6 Semigroups\n    U : Semigroups \u27f6 Sets\n    T\u209b \u22a3 U\n    Semigroups(T\u209b X, Y) \u2245 Sets(X, U Y)\n    -}\n    module _ {X : \u2605} {Y : Semigroup \u2080 \u2080} where\n        open WithSemigroup Y\n        open Semigroup Y renaming (Carrier to Y\u2080; _\u2219_ to _\u2219Y_)\n\n        \u03c6 : (T\u209b X \u2192\u209b Y) \u2192 X \u2192 U Y\n        \u03c6 f = to f \u2218 var\n        -- var is the unit (\u03b7)\n\n        \u03c8 : (f : X \u2192 U Y) \u2192 T\u209b X \u2192\u209b Y\n        \u03c8 f = eval-T\u209b \u2218\u209b T\u209b-hom f\n        -- eval-T\u209b is the co-unit (\u03b5)\n\n        \u03c6-\u03c8 : \u2200 f \u2192 \u03c6 (\u03c8 f) \u2261 f\n        \u03c6-\u03c8 f = \u2261.refl\n\n        \u03c8-\u03c6 : \u2200 f t \u2192 to (\u03c8 (\u03c6 f)) t \u2261 to f t\n        \u03c8-\u03c6 f (var x) = \u2261.refl\n        \u03c8-\u03c6 f (op `\u03bc (t \u2237 u \u2237 []))\n           = to (\u03c8 (\u03c6 f)) (t \u2219 u)\n           \u2261\u27e8 \u2261.refl \u27e9\n               eval-T (map-T (\u03c6 f) (t \u2219 u))\n           \u2261\u27e8 \u2261.refl \u27e9\n               eval-T (map-T (\u03c6 f) t \u2219 map-T (\u03c6 f) u)\n           \u2261\u27e8 \u2261.refl \u27e9\n               eval-T (map-T (\u03c6 f) t) \u2219Y eval-T (map-T (\u03c6 f) u)\n           \u2261\u27e8 \u2261.refl \u27e9\n               to (\u03c8 (\u03c6 f)) t \u2219Y to (\u03c8 (\u03c6 f)) u\n           \u2261\u27e8 \u2261.cong\u2082 _\u2219Y_ (\u03c8-\u03c6 f t) (\u03c8-\u03c6 f u) \u27e9\n               to f t \u2219Y to f u\n           \u2261\u27e8 ! (to-\u2219-cong f t u) \u27e9\n               to f (t \u2219 u)\n           \u220e\n           where open \u2261-Reasoning\n\nmodule FreeMonoid where\n    data Op : \u2115 \u2192 \u2605 where\n       `\u03b5 : Op 0\n       `\u03bc : Op 2\n    open Language.Simple.Abstract Op renaming (E to T)\n    \u03b5 : \u2200 {X} \u2192 T X\n    \u03b5 = op `\u03b5 []\n    _\u2219_ : \u2200 {X} \u2192 T X \u2192 T X \u2192 T X\n    t \u2219 u = op `\u03bc (t \u2237 u \u2237 [])\n\n    infix 0 _\u22a2_\u2261_\n    data _\u22a2_\u2261_ (X : \u2605) : T X \u2192 T X \u2192 \u2605 where\n       \u22a2-\u03b5\u2219x : \u2200 x \u2192 X \u22a2 \u03b5 \u2219 x \u2261 x\n       \u22a2-x\u2219\u03b5 : \u2200 x \u2192 X \u22a2 x \u2219 \u03b5 \u2261 x\n       \u22a2-\u2219-assoc : \u2200 x y z \u2192 X \u22a2 ((x \u2219 y) \u2219 z) \u2261 (x \u2219 (y \u2219 z))\n       \n    module _ (X : \u2605) where\n        open \u211b (_\u22a2_\u2261_ X)\n\n        {- TODO, share the proofs with FreeSemigroup above -}\n        \u2219-assoc : Associative _\u2248_ _\u2219_\n        \u2219-assoc x y z = \u03b7\u2261 (\u22a2-\u2219-assoc x y z)\n\n        open Equivalence-Reasoning \u2248-isEquivalence\n\n        \u2219-cong : _\u2219_ Preserves\u2082 _\u2248_ \u27f6 _\u2248_ \u27f6 _\u2248_\n        \u2219-cong {x} {y} {u} {v} p q =\n           x \u2219 u\n             \u2248\u27e8 congCtx (op `\u03bc (x \u2237 []) hole []) q \u27e9\n           x \u2219 v\n             \u2248\u27e8 congCtx (op `\u03bc [] hole (v \u2237 [])) p \u27e9\n           y \u2219 v \u220e\n\n        monoid : Monoid \u2080 \u2080\n        monoid = record { Carrier = T X ; _\u2248_ = _\u2248_ ; _\u2219_ = _\u2219_ ; \u03b5 = \u03b5\n                        ; isMonoid = record { isSemigroup = record { isEquivalence = \u2248-isEquivalence\n                                                                   ; assoc = \u2219-assoc\n                                                                   ; \u2219-cong = \u2219-cong }\n                                            ; identity = (\u03bb x \u2192 \u03b7\u2261 (\u22a2-\u03b5\u2219x x))\n                                                       , (\u03bb x \u2192 \u03b7\u2261 (\u22a2-x\u2219\u03b5 x)) } }\n\n\n   {-\n\n   Free monoid as a functor F : Sets \u2192 Mon\n\n   map-[]    : map f [] \u2261 []\n   map-++ : map f xs ++ map f ys = map f (xs ++ ys)\n\n   F X = (List X , [] , _++_)\n   F (f : Sets(X , Y))\n      : Mon(F X , F Y)\n      = (map f , map-[] , map-++)\n\n   -}\n\nmodule FreeGroup where\n    data Op : \u2115 \u2192 \u2605 where\n       `\u03b5 : Op 0\n       `i : Op 1\n       `\u03bc : Op 2\n    open Language.Simple.Abstract Op renaming (E to T)\n    \u03b5 : \u2200 {X} \u2192 T X\n    \u03b5 = op `\u03b5 []\n    infix 9 \u2212_\n    \u2212_ : \u2200 {X} \u2192 T X \u2192 T X\n    \u2212 t = op `i (t \u2237 [])\n    infixr 6 _\u2219_\n    _\u2219_ : \u2200 {X} \u2192 T X \u2192 T X \u2192 T X\n    t \u2219 u = op `\u03bc (t \u2237 u \u2237 [])\n\n    infix 0 _\u22a2_\u2261_\n    data _\u22a2_\u2261_ (X : \u2605) : T X \u2192 T X \u2192 \u2605 where\n       \u03b5\u2219x : \u2200 x \u2192 X \u22a2 \u03b5 \u2219 x \u2261 x\n       x\u2219\u03b5 : \u2200 x \u2192 X \u22a2 x \u2219 \u03b5 \u2261 x\n       \u2219-assoc : \u2200 x y z \u2192 X \u22a2 (x \u2219 (y \u2219 z)) \u2261 ((x \u2219 y) \u2219 z)\n       \u2212\u2219-inv : \u2200 x \u2192 X \u22a2 \u2212 x \u2219 x \u2261 \u03b5\n       \u2219\u2212-inv : \u2200 x \u2192 X \u22a2 x \u2219 \u2212 x \u2261 \u03b5\n\n-- \u2212 (\u2212 t) \u2261 t\n-- \u2212 t \u2219 \u2212 (\u2212 t) \u2261 \u03b5 \u2261 \u2212 t \u2219 t\n\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Language\/Simple\/Free.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"00ad06b5975119fc9a9b950f3c994a197714b826","subject":"specify utf-8 encoding in default","message":"specify utf-8 encoding in default\n","repos":"nfjinjing\/chu2","old_file":"src\/Chu2.agda","new_file":"src\/Chu2.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6693f1c42b3516894542b02056238ef53daca736","subject":"Control\/Protocol.agda","message":"Control\/Protocol.agda\n","repos":"crypto-agda\/protocols","old_file":"Control\/Protocol.agda","new_file":"Control\/Protocol.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level.NP\nopen import Data.Product.NP renaming (map to \u00d7-map; proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Zero\nopen import Data.Sum renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_]) hiding ([_,_]\u2032)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Two hiding (_\u225f_)\nopen import Data.Nat hiding (_\u2294_)\nopen import Data.LR\nopen Data.Two.Indexed\n\nopen import Relation.Binary\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; !_; _\u2219_; refl; subst; J; ap; coe; coe!; J-orig; _\u2262_)\n\nopen import Function.Extensionality\nopen import HoTT\nopen import Data.ShapePolymorphism\nopen Equivalences\n\nmodule Control.Protocol where\n\ndata InOut : \u2605 where\n  In Out : InOut\n\ndual\u1d35\u1d3c : InOut \u2192 InOut\ndual\u1d35\u1d3c In  = Out\ndual\u1d35\u1d3c Out = In\n\ndual\u1d35\u1d3c-involutive : \u2200 io \u2192 dual\u1d35\u1d3c (dual\u1d35\u1d3c io) \u2261 io\ndual\u1d35\u1d3c-involutive In = refl\ndual\u1d35\u1d3c-involutive Out = refl\n\ndual\u1d35\u1d3c-equiv : Is-equiv dual\u1d35\u1d3c\ndual\u1d35\u1d3c-equiv = self-inv-is-equiv dual\u1d35\u1d3c-involutive\n\ndual\u1d35\u1d3c-inj : \u2200 {x y} \u2192 dual\u1d35\u1d3c x \u2261 dual\u1d35\u1d3c y \u2192 x \u2261 y\ndual\u1d35\u1d3c-inj = Is-equiv.injective dual\u1d35\u1d3c-equiv\n\n{-\nmodule UniversalProtocols \u2113 {U : \u2605_(\u209b \u2113)}(U\u27e6_\u27e7 : U \u2192 \u2605_ \u2113) where\n-}\nmodule _ \u2113 where\n  U = \u2605_ \u2113\n  U\u27e6_\u27e7 = id\n  data Proto_ : \u2605_(\u209b \u2113) where\n    end : Proto_\n    com : (io : InOut){M : U}(P : U\u27e6 M \u27e7 \u2192 Proto_) \u2192 Proto_\n{-\nmodule U\u2605 \u2113 = UniversalProtocols \u2113 {\u2605_ \u2113} id\nopen U\u2605\n-}\n\nProto : \u2605\u2081\nProto = Proto_ \u2080\nProto\u2080 = Proto\nProto\u2081 = Proto_ \u2081\n\npattern recv P = com In  P\npattern send P = com Out P\n\nmodule _ {{_ : FunExt}} where\n    com= : \u2200 io {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081)\n                {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080))\n           \u2192 com io P\u2080 \u2261 com io P\u2081\n    com= io refl P= = ap (com io) (\u03bb= P=)\n\n    module _ io {M\u2080 M\u2081}(M\u2243 : M\u2080 \u2243 M\u2081)\n             {P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}\n             (P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (\u2013> M\u2243 m\u2080))\n             {{_ : UA}} where\n        com\u2243 : com io P\u2080 \u2261 com io P\u2081\n        com\u2243 = com= io (ua M\u2243) \u03bb m \u2192 P= m \u2219 ap P\u2081 (! coe-\u03b2 M\u2243 m)\n\n    module _ io {M N}(P : M \u00d7 N \u2192 Proto)\n             where\n        com\u2082 : Proto\n        com\u2082 = com io \u03bb m \u2192 com io \u03bb n \u2192 P (m , n)\n\n    {- Proving this would be awesome...\n    module _ io\n             {M\u2080 M\u2081 N\u2080 N\u2081 : \u2605}\n             (M\u00d7N\u2243 : (M\u2080 \u00d7 N\u2080) \u2243 (M\u2081 \u00d7 N\u2081))\n             {P\u2080 : M\u2080 \u00d7 N\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u00d7 N\u2081 \u2192 Proto}\n             (P= : \u2200 m,n\u2080 \u2192 P\u2080 m,n\u2080 \u2261 P\u2081 (\u2013> M\u00d7N\u2243 m,n\u2080))\n             {{_ : UA}} where\n        com\u2082\u2243 : com\u2082 io P\u2080 \u2261 com\u2082 io P\u2081\n        com\u2082\u2243 = {!com=!}\n    -}\n\n    -- send= : \u2200 {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081){P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080)) \u2192 send P\u2080 \u2261 send P\u2081\n    send= = com= Out\n    send\u2243 = com\u2243 Out\n\n    -- recv= : \u2200 {M\u2080 M\u2081}(M= : M\u2080 \u2261 M\u2081){P\u2080 : M\u2080 \u2192 Proto}{P\u2081 : M\u2081 \u2192 Proto}(P= : \u2200 m\u2080 \u2192 P\u2080 m\u2080 \u2261 P\u2081 (coe M= m\u2080)) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv= = com= In\n    recv\u2243 = com\u2243 In\n\n    com=\u2032 : \u2200 io {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 com io P\u2080 \u2261 com io P\u2081\n    com=\u2032 io = com= io refl\n\n    send=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 send P\u2080 \u2261 send P\u2081\n    send=\u2032 = send= refl\n\n    recv=\u2032 : \u2200 {M}{P\u2080 P\u2081 : M \u2192 Proto}(P= : \u2200 m \u2192 P\u2080 m \u2261 P\u2081 m) \u2192 recv P\u2080 \u2261 recv P\u2081\n    recv=\u2032 = recv= refl\n\npattern recvE M P = com In  {M} P\npattern sendE M P = com Out {M} P\n\nmodule ProtoRel (_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605) where\n    infix 0 _\u2248\u1d3e_\n    data _\u2248\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081 where\n      end : end \u2248\u1d3e end\n      com : \u2200 {io\u2080 io\u2081} (io : io\u2080 \u2248\u1d35\u1d3c io\u2081){M P Q} \u2192 (\u2200 (m : M) \u2192 P m \u2248\u1d3e Q m) \u2192 com io\u2080 P \u2248\u1d3e com io\u2081 Q\n\nmodule ProtoRelImplicit {_\u2248\u1d35\u1d3c_ : InOut \u2192 InOut \u2192 \u2605} = ProtoRel _\u2248\u1d35\u1d3c_\nopen ProtoRelImplicit hiding (_\u2248\u1d3e_)\nopen ProtoRel _\u2261_ public renaming (_\u2248\u1d3e_ to _\u2261\u1d3e_) using ()\n\ndata View-\u2261\u1d3e : (P Q : Proto) \u2192 P \u2261\u1d3e Q \u2192 \u2605\u2081 where\n  end : View-\u2261\u1d3e end end end\n  \u2261-\u03a3 : \u2200 {M P Q} (p\u2261q : (m : M) \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (send P) (send Q) (com refl p\u2261q)\n  \u2261-\u03a0 : \u2200 {M P Q} (p\u2261q : (m : M) \u2192 P m \u2261\u1d3e Q m) \u2192 View-\u2261\u1d3e (recv P) (recv Q) (com refl p\u2261q)\n\nview-\u2261\u1d3e : \u2200 {P Q} (p\u2261q : P \u2261\u1d3e Q) \u2192 View-\u2261\u1d3e P Q p\u2261q\nview-\u2261\u1d3e end = end\nview-\u2261\u1d3e (com {In}  refl _) = \u2261-\u03a0 _\nview-\u2261\u1d3e (com {Out} refl _) = \u2261-\u03a3 _\n\nrecv\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nrecv\u2610 P = recv (\u03bb { [ m ] \u2192 P m })\n\nsend\u2610 : {M : \u2605}(P : ..(_ : M) \u2192 Proto) \u2192 Proto\nsend\u2610 P = send (\u03bb { [ m ] \u2192 P m })\n\nsource-of : Proto \u2192 Proto\nsource-of end       = end\nsource-of (com _ P) = send \u03bb m \u2192 source-of (P m)\n\n{-\ndual : Proto \u2192 Proto\ndual end      = end\ndual (send P) = recv \u03bb m \u2192 dual (P m)\ndual (recv P) = send \u03bb m \u2192 dual (P m)\n-}\n\ndual : Proto \u2192 Proto\ndual end        = end\ndual (com io P) = com (dual\u1d35\u1d3c io) \u03bb m \u2192 dual (P m)\n\ndata IsSource : Proto \u2192 \u2605\u2081 where\n  end : IsSource end\n  com : \u2200 {M P} (PT : (m : M) \u2192 IsSource (P m)) \u2192 IsSource (send P)\n\ndata IsSink : Proto \u2192 \u2605\u2081 where\n  end : IsSink end\n  com : \u2200 {M P} (PT : (m : M) \u2192 IsSink (P m)) \u2192 IsSink (recv P)\n\ndata Proto\u2610 : Proto \u2192 \u2605\u2081 where\n  end : Proto\u2610 end\n  com : \u2200 q {M P} (P\u2610 : \u2200 (m : \u2610 M) \u2192 Proto\u2610 (P m)) \u2192 Proto\u2610 (com q P)\n\nrecord End_ \u2113 : \u2605_ \u2113 where\n  constructor end\n\nEnd : \u2200 {\u2113} \u2192 \u2605_ \u2113\nEnd = End_ _\n\nmodule _ {{_ : UA}} where\n    End-uniq : End \u2261 \ud835\udfd9\n    End-uniq = ua (equiv _ _ (\u03bb _ \u2192 refl) (\u03bb _ \u2192 refl))\n\n\u27e6_\u27e7\u1d35\u1d3c : InOut \u2192 \u2200{\u2113}(M : \u2605_ \u2113)(P : M \u2192 \u2605_ \u2113) \u2192 \u2605_ \u2113\n\u27e6_\u27e7\u1d35\u1d3c In  = \u03a0\n\u27e6_\u27e7\u1d35\u1d3c Out = \u03a3\n\n\u27e6_\u27e7 : \u2200 {\u2113} \u2192 Proto_ \u2113 \u2192 \u2605_ \u2113\n\u27e6 end     \u27e7 = End\n\u27e6 com q P \u27e7 = \u27e6 q \u27e7\u1d35\u1d3c _ \u03bb m \u2192 \u27e6 P m \u27e7\n\n\u27e6_\u22a5\u27e7 : Proto \u2192 \u2605\n\u27e6 P \u22a5\u27e7 = \u27e6 dual P \u27e7\n\n\u211b\u27e6_\u27e7 : \u2200{\u2113}(P : Proto_ \u2113) (p q : \u27e6 P \u27e7) \u2192 \u2605_ \u2113\n\u211b\u27e6 end    \u27e7 p q = End\n\u211b\u27e6 recv P \u27e7 p q = \u2200 m \u2192 \u211b\u27e6 P m \u27e7 (p m) (q m)\n\u211b\u27e6 send P \u27e7 p q = \u03a3 (fst p \u2261 fst q) \u03bb e \u2192 \u211b\u27e6 P (fst q) \u27e7 (subst (\u27e6_\u27e7 \u2218 P) e (snd p)) (snd q)\n\n\u211b\u27e6_\u27e7-refl : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Reflexive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-refl     = end\n\u211b\u27e6 recv P \u27e7-refl     = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-refl\n\u211b\u27e6 send P \u27e7-refl {x} = refl , \u211b\u27e6 P (fst x) \u27e7-refl\n\n\u211b\u27e6_\u27e7-sym : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Symmetric \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-sym p          = end\n\u211b\u27e6 recv P \u27e7-sym p          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-sym (p m)\n\u211b\u27e6 send P \u27e7-sym (refl , q) = refl , \u211b\u27e6 P _ \u27e7-sym q    -- TODO HoTT\n\n\u211b\u27e6_\u27e7-trans : \u2200 {\u2113}(P : Proto_ \u2113) \u2192 Transitive \u211b\u27e6 P \u27e7\n\u211b\u27e6 end    \u27e7-trans p          q          = end\n\u211b\u27e6 recv P \u27e7-trans p          q          = \u03bb m \u2192 \u211b\u27e6 P m \u27e7-trans (p m) (q m)\n\u211b\u27e6 send P \u27e7-trans (refl , p) (refl , q) = refl , \u211b\u27e6 P _ \u27e7-trans p q    -- TODO HoTT\n\nsend\u2032 : \u2605 \u2192 Proto \u2192 Proto\nsend\u2032 M P = send \u03bb (_ : M) \u2192 P\n\nrecv\u2032 : \u2605 \u2192 Proto \u2192 Proto\nrecv\u2032 M P = recv \u03bb (_ : M) \u2192 P\n\nmodule send\/recv-\ud835\udfd8 (P : \ud835\udfd8 \u2192 Proto){{_ : FunExt}}{{_ : UA}} where\n    P\u22a4 : Proto\n    P\u22a4 = recvE \ud835\udfd8 P\n\n    P0 : Proto\n    P0 = sendE \ud835\udfd8 P\n\n    P0-empty : \u27e6 P0 \u27e7 \u2261 \ud835\udfd8\n    P0-empty = ua (equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) }))\n\n    P\u22a4-uniq : \u27e6 P\u22a4 \u27e7 \u2261 \ud835\udfd9\n    P\u22a4-uniq = \u03a0\ud835\udfd8-uniq _\n\nopen send\/recv-\ud835\udfd8 (\u03bb _ \u2192 end) public\n\n\u2261\u1d3e-refl : \u2200 P \u2192 P \u2261\u1d3e P\n\u2261\u1d3e-refl end       = end\n\u2261\u1d3e-refl (com q P) = com refl \u03bb m \u2192 \u2261\u1d3e-refl (P m)\n\n\u2261\u1d3e-reflexive : \u2200 {P Q} \u2192 P \u2261 Q \u2192 P \u2261\u1d3e Q\n\u2261\u1d3e-reflexive refl = \u2261\u1d3e-refl _\n\n\u2261\u1d3e-sym : Symmetric _\u2261\u1d3e_\n\u2261\u1d3e-sym end = end\n\u2261\u1d3e-sym (com refl r) = com refl \u03bb m \u2192 \u2261\u1d3e-sym (r m)\n\n\u2261\u1d3e-trans : Transitive _\u2261\u1d3e_\n\u2261\u1d3e-trans end qr = qr\n\u2261\u1d3e-trans (com refl x) (com refl x\u2081) = com refl (\u03bb m \u2192 \u2261\u1d3e-trans (x m) (x\u2081 m))\n\n!\u1d3e = \u2261\u1d3e-sym\n_\u2219\u1d3e_ = \u2261\u1d3e-trans\n\ndual-involutive : \u2200 P \u2192 dual (dual P) \u2261\u1d3e P\ndual-involutive end       = end\ndual-involutive (com q P) = com (dual\u1d35\u1d3c-involutive q) \u03bb m \u2192 dual-involutive (P m)\n\nmodule _ {{_ : FunExt}} where\n    \u2261\u1d3e-sound : \u2200 {P Q} \u2192 P \u2261\u1d3e Q \u2192 P \u2261 Q\n    \u2261\u1d3e-sound end            = refl\n    \u2261\u1d3e-sound (com refl P\u2261Q) = ap (com _) (\u03bb= \u03bb m \u2192 \u2261\u1d3e-sound (P\u2261Q m))\n\n    \u2261\u1d3e-cong : \u2200 {P Q} (f : Proto \u2192 Proto) \u2192 P \u2261\u1d3e Q \u2192 f P \u2261\u1d3e f Q\n    \u2261\u1d3e-cong f P\u2261Q = \u2261\u1d3e-reflexive (ap f (\u2261\u1d3e-sound P\u2261Q))\n\n    dual-equiv : Is-equiv dual\n    dual-equiv = self-inv-is-equiv (\u2261\u1d3e-sound \u2218 dual-involutive)\n\n    dual-inj : \u2200 {P Q} \u2192 dual P \u2261 dual Q \u2192 P \u2261 Q\n    dual-inj = Is-equiv.injective dual-equiv\n\nsource-of-idempotent : \u2200 P \u2192 source-of (source-of P) \u2261\u1d3e source-of P\nsource-of-idempotent end       = end\nsource-of-idempotent (com _ P) = com refl \u03bb m \u2192 source-of-idempotent (P m)\n\nsource-of-dual-oblivious : \u2200 P \u2192 source-of (dual P) \u2261\u1d3e source-of P\nsource-of-dual-oblivious end       = end\nsource-of-dual-oblivious (com _ P) = com refl \u03bb m \u2192 source-of-dual-oblivious (P m)\n\nsink-of : Proto \u2192 Proto\nsink-of = dual \u2218 source-of\n\nSink : Proto \u2192 \u2605\nSink P = \u27e6 sink-of P \u27e7\n\nsink : \u2200 P \u2192 Sink P\nsink end         = _\nsink (com _ P) x = sink (P x)\n\nmodule _ {{_ : FunExt}} where\n    sink-contr : \u2200 P s \u2192 sink P \u2261 s\n    sink-contr end       s = refl\n    sink-contr (com _ P) s = \u03bb= \u03bb m \u2192 sink-contr (P m) (s m)\n\n    Sink-is-contr : \u2200 P \u2192 Is-contr (Sink P)\n    Sink-is-contr P = sink P , sink-contr P\n\n    \ud835\udfd9\u2243Sink : \u2200 P \u2192 \ud835\udfd9 \u2243 Sink P\n    \ud835\udfd9\u2243Sink P = Is-contr-to-Is-equiv.\ud835\udfd9\u2243 (Sink-is-contr P)\n\nLog : Proto \u2192 \u2605\nLog P = \u27e6 source-of P \u27e7\n\n_>>=_ : (P : Proto) \u2192 (Log P \u2192 Proto) \u2192 Proto\nend     >>= Q = Q _\ncom q P >>= Q = com q \u03bb m \u2192 P m >>= \u03bb ms \u2192 Q (m , ms)\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>= \u03bb _ \u2192 Q\n\nreplicate\u1d3e : \u2115 \u2192 Proto \u2192 Proto\nreplicate\u1d3e 0       P = end\nreplicate\u1d3e (suc n) P = P >> replicate\u1d3e n P\n\n++Log : \u2200 (P : Proto){Q : Log P \u2192 Proto} (xs : Log P) \u2192 Log (Q xs) \u2192 Log (P >>= Q)\n++Log end       _        ys = ys\n++Log (com q P) (x , xs) ys = x , ++Log (P x) xs ys\n\n>>=-right-unit : \u2200 P \u2192 (P >> end) \u2261\u1d3e P\n>>=-right-unit end       = end\n>>=-right-unit (com q P) = com refl \u03bb m \u2192 >>=-right-unit (P m)\n\n>>=-assoc : \u2200 (P : Proto)(Q : Log P \u2192 Proto)(R : Log (P >>= Q) \u2192 Proto)\n            \u2192 P >>= (\u03bb x \u2192 Q x >>= (\u03bb y \u2192 R (++Log P x y))) \u2261\u1d3e ((P >>= Q) >>= R)\n>>=-assoc end       Q R = \u2261\u1d3e-refl (Q _ >>= R)\n>>=-assoc (com q P) Q R = com refl \u03bb m \u2192 >>=-assoc (P m) (\u03bb ms \u2192 Q (m , ms)) (\u03bb ms \u2192 R (m , ms))\n\ndata Accept? : \u2605 where\n  accept reject : Accept?\ndata Is-accept : Accept? \u2192 \u2605 where\n  accept : Is-accept accept\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    extend-send : Proto \u2192 Proto\n    extend-send end      = end\n    extend-send (send P) = send [inl: (\u03bb m \u2192 extend-send (P m)) ,inr: A\u1d3e ]\n    extend-send (recv P) = recv \u03bb m \u2192 extend-send (P m)\n\n    extend-recv : Proto \u2192 Proto\n    extend-recv end      = end\n    extend-recv (recv P) = recv [inl: (\u03bb m \u2192 extend-recv (P m)) ,inr: A\u1d3e ]\n    extend-recv (send P) = send \u03bb m \u2192 extend-recv (P m)\n\nmodule _ {A : \u2605} (A\u1d3e : A \u2192 Proto) where\n    dual-extend-send : \u2200 P \u2192 dual (extend-send A\u1d3e P) \u2261\u1d3e extend-recv (dual \u2218 A\u1d3e) (dual P)\n    dual-extend-send end      = end\n    dual-extend-send (recv P) = com refl (\u03bb m \u2192 dual-extend-send (P m))\n    dual-extend-send (send P) = com refl [inl: (\u03bb m \u2192 dual-extend-send (P m))\n                                         ,inr: (\u03bb x \u2192 \u2261\u1d3e-refl (dual (A\u1d3e x))) ]\n\ndata Abort : \u2605 where abort : Abort\n\nAbort\u1d3e : Abort \u2192 Proto\nAbort\u1d3e _ = end\n\nadd-abort : Proto \u2192 Proto\nadd-abort = extend-send Abort\u1d3e\n\ntelecom : \u2200 P \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u22a5\u27e7 \u2192 Log P\ntelecom end      _       _       = _\ntelecom (recv P) p       (m , q) = m , telecom (P m) (p m) q\ntelecom (send P) (m , p) q       = m , telecom (P m) p (q m)\n\nlift\u1d3e : \u2200 a {\u2113} \u2192 Proto_ \u2113 \u2192 Proto_ (a \u2294 \u2113)\nlift\u1d3e a end        = end\nlift\u1d3e a (com io P) = com io \u03bb m \u2192 lift\u1d3e a (P (lower {\u2113 = a} m))\n\nlift-proc : \u2200 a {\u2113} (P : Proto_ \u2113) \u2192 \u27e6 P \u27e7 \u2192 \u27e6 lift\u1d3e a P \u27e7\nlift-proc a {\u2113} end      end     = end\nlift-proc a {\u2113} (send P) (m , p) = lift m , lift-proc a (P m) p\nlift-proc a {\u2113} (recv P) p       = \u03bb { (lift m) \u2192 lift-proc _ (P m) (p m) }\n\nmodule MonomorphicSky (P : Proto\u2080) where\n  Cloud : Proto\u2080\n  Cloud = recv \u03bb (t   : \u27e6 P  \u27e7) \u2192\n          recv \u03bb (u   : \u27e6 P \u22a5\u27e7) \u2192\n          send \u03bb (log : Log P)  \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud t u = telecom P t u , _\n\nmodule PolySky where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P : Proto\u2080) \u2192\n          lift\u1d3e \u2081 (MonomorphicSky.Cloud P)\n  cloud : \u27e6 Cloud \u27e7\n  cloud P = lift-proc \u2081 (MonomorphicSky.Cloud P) (MonomorphicSky.cloud P)\n\nmodule PolySky' where\n  Cloud : Proto_ \u2081\n  Cloud = recv \u03bb (P   : Proto\u2080) \u2192\n          recv \u03bb (t   : Lift \u27e6 P  \u27e7)  \u2192\n          recv \u03bb (u   : Lift \u27e6 P \u22a5\u27e7)  \u2192\n          send \u03bb (log : Lift (Log P)) \u2192\n          end\n  cloud : \u27e6 Cloud \u27e7\n  cloud P (lift t) (lift u) = lift (telecom P t u) , _\n\ndata Choreo (I : \u2605) : \u2605\u2081 where\n  _-[_]\u2192_\u204f_ : (A : I) (M : \u2605) (B : I) (\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n  _-[_]\u2192\u2605\u204f_ : (A : I) (M : \u2605)         (\u2102 :   (m : M) \u2192 Choreo I) \u2192 Choreo I\n  end       : Choreo I\n\nmodule _ {I : \u2605} where \n    _-[_]\u2192\u00f8\u204f_ : \u2200 (A : I)(M : \u2605)(\u2102 : ..(m : M) \u2192 Choreo I) \u2192 Choreo I\n    A -[ M ]\u2192\u00f8\u204f \u2102 = A -[ \u2610 M ]\u2192\u2605\u204f \u03bb { [ m ] \u2192 \u2102 m }\n\n    _\/\/_ : (\u2102 : Choreo I) (p : I \u2192 \ud835\udfda) \u2192 Proto\n    (A -[ M ]\u2192 B \u204f \u2102) \/\/ p = case p A\n                               0: case p B\n                                    0: recv (\u03bb { [ m ] \u2192 \u2102 m \/\/ p })\n                                    1: recv (\u03bb     m   \u2192 \u2102 m \/\/ p)\n                               1: send (\u03bb m \u2192 \u2102 m \/\/ p)\n    (A -[ M ]\u2192\u2605\u204f   \u2102) \/\/ p = com (case p A 0: In 1: Out) \u03bb m \u2192 \u2102 m \/\/ p\n    end               \/\/ p = end\n\n    \u2102Observer : Choreo I \u2192 Proto\n    \u2102Observer \u2102 = \u2102 \/\/ \u03bb _ \u2192 0\u2082\n\n    \u2102Log : Choreo I \u2192 Proto\n    \u2102Log \u2102 = \u2102 \/\/ \u03bb _ \u2192 1\u2082\n\n    \u2102Log-IsSource : \u2200 \u2102 \u2192 IsSource (\u2102Log \u2102)\n    \u2102Log-IsSource (A -[ M ]\u2192 B \u204f \u2102) = com \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource (A -[ M ]\u2192\u2605\u204f   \u2102) = com \u03bb m \u2192 \u2102Log-IsSource (\u2102 m)\n    \u2102Log-IsSource end               = end\n\n    \u2102Observer-IsSink : \u2200 \u2102 \u2192 IsSink (\u2102Observer \u2102)\n    \u2102Observer-IsSink (A -[ M ]\u2192 B \u204f \u2102) = com \u03bb { [ m ] \u2192 \u2102Observer-IsSink (\u2102 m) }\n    \u2102Observer-IsSink (A -[ M ]\u2192\u2605\u204f   \u2102) = com \u03bb m \u2192 \u2102Observer-IsSink (\u2102 m)\n    \u2102Observer-IsSink end = end\n\n    data R : (p q r : \ud835\udfda) \u2192 \u2605 where\n      R011 : R 0\u2082 1\u2082 1\u2082\n      R101 : R 1\u2082 0\u2082 1\u2082\n      R000 : R 0\u2082 0\u2082 0\u2082\n    R\u00b0 : \u2200 {I : \u2605} (p q r : I \u2192 \ud835\udfda) \u2192 \u2605\n    R\u00b0 p q r = \u2200 i \u2192 R (p i) (q i) (r i)\n\n    module _ {p q r : I \u2192 \ud835\udfda} where\n        choreo-merge : (\u2102 : Choreo I)(pqr : R\u00b0 p q r) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ q \u27e7 \u2192 \u27e6 \u2102 \/\/ r \u27e7\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A | p B | q B | r B | pqr B\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 |  0\u2082 |  _  |  _  | _    = m , choreo-merge (\u2102 m) pqr (\u2102p [ m ]) \u2102q\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 |  _  |  0\u2082 |  _  | _    = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 | .0\u2082 |  1\u2082 | .1\u2082 | R011 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q m)\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  1\u2082 | .0\u2082 | .1\u2082 | R101 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q [ m ])\n        choreo-merge (A -[ M ]\u2192 B \u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 |  0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb { [ m ] \u2192 choreo-merge (\u2102 m) pqr (\u2102p [ m ]) (\u2102q [ m ]) }\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q with p A | q A | r A | pqr A\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p (m , \u2102q) | .0\u2082 |  1\u2082 | .1\u2082 | R011 = m , choreo-merge (\u2102 m) pqr (\u2102p m) \u2102q\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr (m , \u2102p) \u2102q |  1\u2082 | .0\u2082 | .1\u2082 | R101 = m , choreo-merge (\u2102 m) pqr \u2102p (\u2102q m)\n        choreo-merge (A -[ M ]\u2192\u2605\u204f \u2102) pqr \u2102p \u2102q       | .0\u2082 |  0\u2082 | .0\u2082 | R000 = \u03bb m \u2192 choreo-merge (\u2102 m) pqr (\u2102p m) (\u2102q m)\n        choreo-merge end pqr \u2102p \u2102q = _\n\n        {-\n    module _ {p q r pq qr pqr : I \u2192 \ud835\udfda} where\n        choreo-merge-assoc : (\u2102 : Choreo I)(Rpqr : R\u00b0 p qr pqr)(Rqr : R\u00b0 q r qr)(Rpqr' : R\u00b0 pq r pqr)(Rpq : R\u00b0 p q pq) \u2192\n                             (\u2102p : \u27e6 \u2102 \/\/ p \u27e7) (\u2102q : \u27e6 \u2102 \/\/ q \u27e7) (\u2102r : \u27e6 \u2102 \/\/ r \u27e7)\n                             \u2192 choreo-merge \u2102 Rpqr \u2102p (choreo-merge \u2102 Rqr \u2102q \u2102r) \u2261 choreo-merge \u2102 Rpqr' (choreo-merge \u2102 Rpq \u2102p \u2102q) \u2102r\n        choreo-merge-assoc = {!!}\n        -}\n\n    R-p-\u00acp-1 : \u2200 (p : I \u2192 \ud835\udfda) i \u2192 R (p i) (not (p i)) 1\u2082\n    R-p-\u00acp-1 p i with p i\n    R-p-\u00acp-1 p i | 1\u2082 = R101\n    R-p-\u00acp-1 p i | 0\u2082 = R011\n\n    choreo-bi : {p : I \u2192 \ud835\udfda}(\u2102 : Choreo I) \u2192 \u27e6 \u2102 \/\/ p \u27e7 \u2192 \u27e6 \u2102 \/\/ (not \u2218 p) \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n    choreo-bi {p} \u2102 \u2102p \u2102\u00acp = choreo-merge \u2102 (R-p-\u00acp-1 p) \u2102p \u2102\u00acp\n\nchoreo2 : (\u2102 : Choreo \ud835\udfda) \u2192 \u27e6 \u2102 \/\/ id \u27e7 \u2192 \u27e6 \u2102 \/\/ not \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\nchoreo2 = choreo-bi\n\nmodule Choreo3 where\n  data \ud835\udfdb : \u2605 where\n    0\u2083 1\u2083 2\u2083 : \ud835\udfdb\n  0\u2083? 1\u2083? 2\u2083? : \ud835\udfdb \u2192 \ud835\udfda\n  0\u2083? 0\u2083 = 1\u2082\n  0\u2083? _  = 0\u2082\n  1\u2083? 1\u2083 = 1\u2082\n  1\u2083? _  = 0\u2082\n  2\u2083? 2\u2083 = 1\u2082\n  2\u2083? _  = 0\u2082\n\n  choreo3 : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3 (0\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 [ m ])\n  choreo3 (0\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 [ m ]) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 [ m ])\n  choreo3 (1\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 [ m ]) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192 0\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 [ m ]) p2\n  choreo3 (2\u2083 -[ M ]\u2192 1\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 m) p2\n  choreo3 (2\u2083 -[ M ]\u2192 2\u2083 \u204f \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 [ m ]) (p1 [ m ]) p2\n  choreo3 (0\u2083 -[ M ]\u2192\u2605\u204f    \u2102) (m , p0) p1 p2 = m , choreo3 (\u2102 m) p0 (p1 m) (p2 m)\n  choreo3 (1\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 (m , p1) p2 = m , choreo3 (\u2102 m) (p0 m) p1 (p2 m)\n  choreo3 (2\u2083 -[ M ]\u2192\u2605\u204f    \u2102) p0 p1 (m , p2) = m , choreo3 (\u2102 m) (p0 m) (p1 m) p2\n  choreo3 end p0 p1 p2 = _\n\n  choreo3' : (\u2102 : Choreo \ud835\udfdb) \u2192 \u27e6 \u2102 \/\/ 0\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 1\u2083? \u27e7 \u2192 \u27e6 \u2102 \/\/ 2\u2083? \u27e7 \u2192 \u27e6 \u2102Log \u2102 \u27e7\n  choreo3' \u2102 p0 p1 p2 = choreo-merge \u2102 (R-p-\u00acp-1 0\u2083?) p0 (choreo-merge \u2102 R-1-2-\u00ac0 p1 p2)\n     where R-1-2-\u00ac0 : \u2200 i \u2192 R (1\u2083? i) (2\u2083? i) (not (0\u2083? i))\n           R-1-2-\u00ac0 0\u2083 = R000\n           R-1-2-\u00ac0 1\u2083 = R101\n           R-1-2-\u00ac0 2\u2083 = R011\n\n>>=-com : (P : Proto){Q : Log P \u2192 Proto}{R : Log P \u2192 Proto}\n          \u2192 \u27e6 P >>= Q  \u27e7\n          \u2192 \u27e6 P >>= R \u22a5\u27e7\n          \u2192 \u03a3 (Log P) (\u03bb t \u2192 \u27e6 Q t \u27e7 \u00d7 \u27e6 R t \u22a5\u27e7)\n>>=-com end      p0       p1       = _ , p0 , p1\n>>=-com (send P) (m , p0) p1       = first (_,_ m) (>>=-com (P m) p0 (p1 m))\n>>=-com (recv P) p0       (m , p1) = first (_,_ m) (>>=-com (P m) (p0 m) p1)\n\n>>-com : (P : Proto){Q R : Proto}\n       \u2192 \u27e6 P >> Q  \u27e7\n       \u2192 \u27e6 P >> R \u22a5\u27e7\n       \u2192 Log P \u00d7 \u27e6 Q \u27e7 \u00d7 \u27e6 R \u22a5\u27e7\n>>-com P p q = >>=-com P p q\n\nmodule _ {{_ : FunExt}} where\n    ap->>= : \u2200 P {Q\u2080 Q\u2081} \u2192 (\u2200 {log} \u2192 \u27e6 Q\u2080 log \u27e7 \u2261 \u27e6 Q\u2081 log \u27e7) \u2192 \u27e6 P >>= Q\u2080 \u27e7 \u2261 \u27e6 P >>= Q\u2081 \u27e7\n    ap->>= end      Q= = Q=\n    ap->>= (send P) Q= = \u03a3=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n    ap->>= (recv P) Q= = \u03a0=\u2032 _ \u03bb m \u2192 ap->>= (P m) Q=\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    P2 = send\u2032 \ud835\udfda end\n\n    0\u2082\u22621\u2082 : 0\u2082 \u2262 1\u2082\n    0\u2082\u22621\u2082 ()\n\n    \ud835\udfd8\u2262 : \u2200 {A} (x : A) \u2192 \ud835\udfd8 \u2262 A\n    \ud835\udfd8\u2262 x e = coe! e x\n\n    \ud835\udfd8\u2262\ud835\udfd9 : \ud835\udfd8 \u2262 \ud835\udfd9\n    \ud835\udfd8\u2262\ud835\udfd9 = \ud835\udfd8\u2262 _\n\n    \ud835\udfd8\u2262\ud835\udfda : \ud835\udfd8 \u2262 \ud835\udfda\n    \ud835\udfd8\u2262\ud835\udfda = \ud835\udfd8\u2262 0\u2082\n\n\nmodule ClientServerV1 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) (P : Proto) where\n    Client : \u2115 \u2192 Proto\n    Client zero    = P\n    Client (suc n) = send \u03bb (q : Query) \u2192 recv \u03bb (r : Resp q) \u2192 Client n\n\n    Server : \u2115 \u2192 Proto\n    Server zero    = P\n    Server (suc n) = recv \u03bb (q : Query) \u2192 send \u03bb (r : Resp q) \u2192 Server n\n\nmodule ClientServerV2 (Query : \u2605\u2080) (Resp : Query \u2192 \u2605\u2080) where\n    ClientRound ServerRound : Proto\n    ClientRound = send \u03bb (q : Query) \u2192 recv \u03bb (r : Resp q) \u2192 end\n    ServerRound = dual ClientRound\n\n    Client Server : \u2115 \u2192 Proto\n    Client n = replicate\u1d3e n ClientRound\n    Server = dual \u2218 Client\n\n    DynamicServer StaticServer : Proto\n    DynamicServer = recv \u03bb n \u2192\n                    Server n\n    StaticServer  = send \u03bb n \u2192\n                    Server n\n\n    module PureServer (serve : \u03a0 Query Resp) where\n      server : \u2200 n \u2192 \u27e6 Server n \u27e7\n      server zero      = _\n      server (suc n) q = serve q , server n\n\nmodule _ {{_ : FunExt}} where\n  dual-Log : \u2200 P \u2192 Log (dual P) \u2261 Log P\n  dual-Log P = ap \u27e6_\u27e7 (\u2261\u1d3e-sound (source-of-dual-oblivious P))\n\ndual->> : \u2200 P Q \u2192 dual (P >> Q) \u2261\u1d3e dual P >> dual Q\ndual->> end      Q = \u2261\u1d3e-refl _\ndual->> (recv P) Q = com refl \u03bb m \u2192 dual->> (P m) Q\ndual->> (send P) Q = com refl \u03bb m \u2192 dual->> (P m) Q\n\n  {- ohoh!\n  dual->>= : \u2200 P (Q : Log P \u2192 Proto) \u2192 dual (P >>= Q) \u2261\u1d3e dual P >>= (dual \u2218 Q \u2218 coe (dual-Log P))\n  dual->>= end Q = \u2261\u1d3e-refl _\n  dual->>= (recv M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!dual->>= (P m) (Q \u2218 _,_ m)!})\n  dual->>= (send M P) Q = ProtoRel.com refl M (\u03bb m \u2192 {!!})\n  -}\n\nmodule _ {{_ : FunExt}} (P : Proto) where\n    dual-replicate\u1d3e : \u2200 n \u2192 dual (replicate\u1d3e n P) \u2261\u1d3e replicate\u1d3e n (dual P)\n    dual-replicate\u1d3e zero    = end\n    dual-replicate\u1d3e (suc n) = dual->> P (replicate\u1d3e n P) \u2219\u1d3e \u2261\u1d3e-cong (_>>_ (dual P)) (dual-replicate\u1d3e n)\n\n_\u2295_ : (l r : Proto) \u2192 Proto\nl \u2295 r = send [L: l R: r ]\n\n_&_ : (l r : Proto) \u2192 Proto\nl & r = recv [L: l R: r ]\n\nmodule _ {{_ : FunExt}} where\n    dual-\u2295 : \u2200 {P Q} \u2192 dual (P \u2295 Q) \u2261 dual P & dual Q\n    dual-\u2295 = recv=\u2032 [L: refl R: refl ]\n\n    dual-& : \u2200 {P Q} \u2192 dual (P & Q) \u2261 dual P \u2295 dual Q\n    dual-& = send=\u2032 [L: refl R: refl ]\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    &-comm : \u2200 P Q \u2192 P & Q \u2261 Q & P\n    &-comm P Q = recv\u2243 LR!-equiv [L: refl R: refl ]\n\n    \u2295-comm : \u2200 P Q \u2192 P \u2295 Q \u2261 Q \u2295 P\n    \u2295-comm P Q = send\u2243 LR!-equiv [L: refl R: refl ]\n\nmodule _ {P Q R S} where\n    \u2295-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P \u2295 R \u27e7 \u2192 \u27e6 Q \u2295 S \u27e7\n    \u2295-map f g (`L , pr) = `L , f pr\n    \u2295-map f g (`R , pr) = `R , g pr\n\n    &-map : (\u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7) \u2192 (\u27e6 R \u27e7 \u2192 \u27e6 S \u27e7) \u2192 \u27e6 P & R \u27e7 \u2192 \u27e6 Q & S \u27e7\n    &-map f g p `L = f (p `L)\n    &-map f g p `R = g (p `R)\n\nmodule _ {P Q} where\n    \u2295\u2192\u228e : \u27e6 P \u2295 Q \u27e7 \u2192 \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7\n    \u2295\u2192\u228e (`L , p) = inl p\n    \u2295\u2192\u228e (`R , q) = inr q\n\n    \u228e\u2192\u2295 : \u27e6 P \u27e7 \u228e \u27e6 Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n    \u228e\u2192\u2295 (inl p) = `L , p\n    \u228e\u2192\u2295 (inr q) = `R , q\n\n    \u228e\u2192\u2295\u2192\u228e : \u2200 x \u2192 \u228e\u2192\u2295 (\u2295\u2192\u228e x) \u2261 x\n    \u228e\u2192\u2295\u2192\u228e (`L , _) = refl\n    \u228e\u2192\u2295\u2192\u228e (`R , _) = refl\n\n    \u2295\u2192\u228e\u2192\u2295 : \u2200 x \u2192 \u2295\u2192\u228e (\u228e\u2192\u2295 x) \u2261 x\n    \u2295\u2192\u228e\u2192\u2295 (inl _) = refl\n    \u2295\u2192\u228e\u2192\u2295 (inr _) = refl\n\n    \u2295\u2243\u228e : \u27e6 P \u2295 Q \u27e7 \u2243 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2243\u228e = \u2295\u2192\u228e , record { linv = \u228e\u2192\u2295 ; is-linv = \u228e\u2192\u2295\u2192\u228e ; rinv = \u228e\u2192\u2295 ; is-rinv = \u2295\u2192\u228e\u2192\u2295 }\n\n    \u2295\u2261\u228e : {{_ : UA}} \u2192 \u27e6 P \u2295 Q \u27e7 \u2261 (\u27e6 P \u27e7 \u228e \u27e6 Q \u27e7)\n    \u2295\u2261\u228e = ua \u2295\u2243\u228e\n\n    &\u2192\u00d7 : \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    &\u2192\u00d7 p = p `L , p `R\n\n    \u00d7\u2192& : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P & Q \u27e7\n    \u00d7\u2192& (p , q) `L = p\n    \u00d7\u2192& (p , q) `R = q\n\n    &\u2192\u00d7\u2192& : \u2200 x \u2192 &\u2192\u00d7 (\u00d7\u2192& x) \u2261 x\n    &\u2192\u00d7\u2192& (p , q) = refl\n\n    module _ {{_ : FunExt}} where\n        \u00d7\u2192&\u2192\u00d7 : \u2200 x \u2192 \u00d7\u2192& (&\u2192\u00d7 x) \u2261 x\n        \u00d7\u2192&\u2192\u00d7 p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n        &\u2243\u00d7 : \u27e6 P & Q \u27e7 \u2243 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2243\u00d7 = &\u2192\u00d7 , record { linv = \u00d7\u2192& ; is-linv = \u00d7\u2192&\u2192\u00d7 ; rinv = \u00d7\u2192& ; is-rinv = &\u2192\u00d7\u2192& }\n\n        &\u2261\u00d7 : {{_ : UA}} \u2192 \u27e6 P & Q \u27e7 \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n        &\u2261\u00d7 = ua &\u2243\u00d7\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n    P\u22a4-& : \u2200 P \u2192 \u27e6 P\u22a4 & P \u27e7 \u2261 \u27e6 P \u27e7\n    P\u22a4-& P = &\u2261\u00d7 \u2219 ap (flip _\u00d7_ \u27e6 P \u27e7) P\u22a4-uniq \u2219 \u03a3\ud835\udfd9-snd\n\n    P0-\u2295 : \u2200 P \u2192 \u27e6 P0 \u2295 P \u27e7 \u2261 \u27e6 P \u27e7\n    P0-\u2295 P = \u2295\u2261\u228e \u2219 ap (flip _\u228e_ \u27e6 P \u27e7) \u03a3\ud835\udfd8-fst \u2219 \u228e-comm \u2219 ! \u228e\ud835\udfd8-inl\n\n    &-assoc : \u2200 P Q R \u2192 \u27e6 P & (Q & R) \u27e7 \u2261 \u27e6 (P & Q) & R \u27e7\n    &-assoc P Q R = &\u2261\u00d7 \u2219 (ap (_\u00d7_ \u27e6 P \u27e7) &\u2261\u00d7 \u2219 \u00d7-assoc \u2219 ap (flip _\u00d7_ \u27e6 R \u27e7) (! &\u2261\u00d7)) \u2219 ! &\u2261\u00d7\n\n    \u2295-assoc : \u2200 P Q R \u2192 \u27e6 P \u2295 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2295 Q) \u2295 R \u27e7\n    \u2295-assoc P Q R = \u2295\u2261\u228e \u2219 (ap (_\u228e_ \u27e6 P \u27e7) \u2295\u2261\u228e \u2219 \u228e-assoc \u2219 ap (flip _\u228e_ \u27e6 R \u27e7) (! \u2295\u2261\u228e)) \u2219 ! \u2295\u2261\u228e\n\nmodule _ where\n\n  _\u214b_ : Proto \u2192 Proto \u2192 Proto\n  end    \u214b Q      = Q\n  recv P \u214b Q      = recv \u03bb m \u2192 P m \u214b Q\n  P      \u214b end    = P\n  P      \u214b recv Q = recv \u03bb m \u2192 P \u214b Q m\n  send P \u214b send Q = send [inl: (\u03bb m \u2192 P m \u214b send Q)\n                         ,inr: (\u03bb n \u2192 send P \u214b Q n) ]\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    -- absorption\n    \u22a4-\u214b : \u2200 P \u2192 \u27e6 P\u22a4 \u214b P \u27e7\n    \u22a4-\u214b P = \u03bb()\n\n  _\u2297_ : Proto \u2192 Proto \u2192 Proto\n  end    \u2297 Q      = Q\n  send P \u2297 Q      = send \u03bb m \u2192 P m \u2297 Q\n  P      \u2297 end    = P\n  P      \u2297 send Q = send \u03bb m \u2192 P \u2297 Q m\n  recv P \u2297 recv Q = recv [inl: (\u03bb m \u2192 P m \u2297 recv Q)\n                         ,inr: (\u03bb n \u2192 recv P \u2297 Q n) ]\n\n  _-o_ : (P Q : Proto) \u2192 Proto\n  P -o Q = dual P \u214b Q\n\n  _o-o_ : (P Q : Proto) \u2192 Proto\n  P o-o Q = (P -o Q) \u2297 (Q -o P)\n\n  module _ {{_ : FunExt}} where\n    \u2297-endR : \u2200 P \u2192 P \u2297 end \u2261 P\n    \u2297-endR end      = refl\n    \u2297-endR (recv _) = refl\n    \u2297-endR (send P) = send=\u2032 \u03bb m \u2192 \u2297-endR (P m)\n\n    \u214b-endR : \u2200 P \u2192 P \u214b end \u2261 P\n    \u214b-endR end      = refl\n    \u214b-endR (send _) = refl\n    \u214b-endR (recv P) = recv=\u2032 \u03bb m \u2192 \u214b-endR (P m)\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    \u2297-sendR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u2297 send Q \u27e7 \u2261 (\u03a3 M \u03bb m \u2192 \u27e6 P \u2297 Q m \u27e7)\n    \u2297-sendR end      Q = refl\n    \u2297-sendR (recv P) Q = refl\n    \u2297-sendR (send P) Q = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-sendR (P m) Q) \u2219 \u03a3\u03a3-comm\n\n    \u2297-comm : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2261 \u27e6 Q \u2297 P \u27e7\n    \u2297-comm end      Q        = ! ap \u27e6_\u27e7 (\u2297-endR Q)\n    \u2297-comm (send P) Q        = (\u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (P m) Q) \u2219 ! \u2297-sendR Q P\n    \u2297-comm (recv P) end      = refl\n    \u2297-comm (recv P) (send Q) = \u03a3=\u2032 _ \u03bb m \u2192 \u2297-comm (recv P) (Q m)\n    \u2297-comm (recv P) (recv Q) = \u03a0\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u2297-comm (P m) (recv Q))\n                                               ,inr: (\u03bb m \u2192 \u2297-comm (recv P) (Q m)) ]\n\n    \u2297-assoc : \u2200 P Q R \u2192 P \u2297 (Q \u2297 R) \u2261 (P \u2297 Q) \u2297 R\n    \u2297-assoc end      Q        R        = refl\n    \u2297-assoc (send P) Q        R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (P m) Q R\n    \u2297-assoc (recv P) end      R        = refl\n    \u2297-assoc (recv P) (send Q) R        = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (Q m) R\n    \u2297-assoc (recv P) (recv Q) end      = refl\n    \u2297-assoc (recv P) (recv Q) (send R) = send=\u2032 \u03bb m \u2192 \u2297-assoc (recv P) (recv Q) (R m)\n    \u2297-assoc (recv P) (recv Q) (recv R) = recv\u2243 \u228e-assoc-equiv\n                                             \u03bb { (inl m)       \u2192 \u2297-assoc (P m) (recv Q) (recv R)\n                                               ; (inr (inl m)) \u2192 \u2297-assoc (recv P) (Q m) (recv R)\n                                               ; (inr (inr m)) \u2192 \u2297-assoc (recv P) (recv Q) (R m) }\n\n\n    \u214b-recvR : \u2200 P{M}(Q : M \u2192 Proto) \u2192 \u27e6 P \u214b recv Q \u27e7 \u2261 (\u03a0 M \u03bb m \u2192 \u27e6 P \u214b Q m \u27e7)\n    \u214b-recvR end      Q = refl\n    \u214b-recvR (send P) Q = refl\n    \u214b-recvR (recv P) Q = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-recvR (P m) Q) \u2219 \u03a0\u03a0-comm\n\n    \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2261 \u27e6 Q \u214b P \u27e7\n    \u214b-comm end      Q        = ! ap \u27e6_\u27e7 (\u214b-endR Q)\n    \u214b-comm (recv P) Q        = (\u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (P m) Q) \u2219 ! \u214b-recvR Q P\n    \u214b-comm (send P) end      = refl\n    \u214b-comm (send P) (recv Q) = \u03a0=\u2032 _ \u03bb m \u2192 \u214b-comm (send P) (Q m)\n    \u214b-comm (send P) (send Q) = \u03a3\u2243 \u228e-comm-equiv [inl: (\u03bb m \u2192 \u214b-comm (P m) (send Q))\n                                               ,inr: (\u03bb m \u2192 \u214b-comm (send P) (Q m)) ]\n\n    \u214b-assoc : \u2200 P Q R \u2192 P \u214b (Q \u214b R) \u2261 (P \u214b Q) \u214b R\n    \u214b-assoc end      Q        R        = refl\n    \u214b-assoc (recv P) Q        R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (P m) Q R\n    \u214b-assoc (send P) end      R        = refl\n    \u214b-assoc (send P) (recv Q) R        = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (Q m) R\n    \u214b-assoc (send P) (send Q) end      = refl\n    \u214b-assoc (send P) (send Q) (recv R) = recv=\u2032 \u03bb m \u2192 \u214b-assoc (send P) (send Q) (R m)\n    \u214b-assoc (send P) (send Q) (send R) = send\u2243 \u228e-assoc-equiv\n                                             \u03bb { (inl m)       \u2192 \u214b-assoc (P m) (send Q) (send R)\n                                               ; (inr (inl m)) \u2192 \u214b-assoc (send P) (Q m) (send R)\n                                               ; (inr (inr m)) \u2192 \u214b-assoc (send P) (send Q) (R m) }\n\n  module _ {P Q R}{{_ : FunExt}} where\n    dist-\u2297-\u2295\u2032 : (Q \u2295 R) \u2297 P \u2261 (Q \u2297 P) \u2295 (R \u2297 P)\n    dist-\u2297-\u2295\u2032 = send=\u2032 [L: refl R: refl ]\n\n    dist-\u214b-&\u2032 : (Q & R) \u214b P \u2261 (Q \u214b P) & (R \u214b P)\n    dist-\u214b-&\u2032 = recv=\u2032 [L: refl R: refl ]\n\n    module _ {{_ : UA}} where\n        dist-\u2297-\u2295 : \u27e6 P \u2297 (Q \u2295 R) \u27e7 \u2261 \u27e6 (P \u2297 Q) \u2295 (P \u2297 R) \u27e7\n        dist-\u2297-\u2295 = \u2297-comm P (Q \u2295 R)\n                 \u2219 ap \u27e6_\u27e7 dist-\u2297-\u2295\u2032\n                 \u2219 \u2295\u2261\u228e\n                 \u2219 \u228e= (\u2297-comm Q P) (\u2297-comm R P)\n                 \u2219 ! \u2295\u2261\u228e\n\n        dist-\u214b-& : \u27e6 P \u214b (Q & R) \u27e7 \u2261 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n        dist-\u214b-& = \u214b-comm P (Q & R)\n                 \u2219 ap \u27e6_\u27e7 dist-\u214b-&\u2032\n                 \u2219 &\u2261\u00d7\n                 \u2219 \u00d7= (\u214b-comm Q P) (\u214b-comm R P)\n                 \u2219 ! &\u2261\u00d7\n\n  -- P \u27e6\u2297\u27e7 Q \u2243 \u27e6 P \u2297 Q \u27e7\n  -- but potentially more convenient\n  _\u27e6\u2297\u27e7_ : Proto \u2192 Proto \u2192 \u2605\n  end    \u27e6\u2297\u27e7 Q      = \u27e6 Q \u27e7\n  send P \u27e6\u2297\u27e7 Q      = \u2203 \u03bb m \u2192 P m \u27e6\u2297\u27e7 Q\n  P      \u27e6\u2297\u27e7 end    = \u27e6 P \u27e7\n  P      \u27e6\u2297\u27e7 send Q = \u2203 \u03bb m \u2192 P \u27e6\u2297\u27e7 Q m\n  recv P \u27e6\u2297\u27e7 recv Q = (\u03a0 _ \u03bb m \u2192 P m    \u27e6\u2297\u27e7 recv Q)\n                    \u00d7 (\u03a0 _ \u03bb n \u2192 recv P \u27e6\u2297\u27e7 Q n)\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    \u27e6\u2297\u27e7-correct : \u2200 P Q \u2192 P \u27e6\u2297\u27e7 Q \u2261 \u27e6 P \u2297 Q \u27e7\n    \u27e6\u2297\u27e7-correct end      Q        = refl\n    \u27e6\u2297\u27e7-correct (send P) Q        = \u03a3=\u2032 _ \u03bb m \u2192 \u27e6\u2297\u27e7-correct (P m) Q\n    \u27e6\u2297\u27e7-correct (recv P) end      = refl\n    \u27e6\u2297\u27e7-correct (recv P) (send Q) = \u03a3=\u2032 _ \u03bb n \u2192 \u27e6\u2297\u27e7-correct (recv P) (Q n)\n    \u27e6\u2297\u27e7-correct (recv P) (recv Q) = ! dist-\u00d7-\u03a0\n                                  \u2219 \u03a0=\u2032 (_ \u228e _) \u03bb { (inl m)  \u2192 \u27e6\u2297\u27e7-correct (P m) (recv Q)\n                                                  ; (inr n) \u2192 \u27e6\u2297\u27e7-correct (recv P) (Q n) }\n\n  -- an alternative, potentially more convenient\n  _\u27e6\u214b\u27e7_ : Proto \u2192 Proto \u2192 \u2605\n  end    \u27e6\u214b\u27e7 Q      = \u27e6 Q \u27e7\n  recv P \u27e6\u214b\u27e7 Q      = \u2200 m \u2192 P m \u27e6\u214b\u27e7 Q\n  P      \u27e6\u214b\u27e7 end    = \u27e6 P \u27e7\n  P      \u27e6\u214b\u27e7 recv Q = \u2200 n \u2192 P \u27e6\u214b\u27e7 Q n\n  send P \u27e6\u214b\u27e7 send Q = (\u2203 \u03bb m \u2192 P m    \u27e6\u214b\u27e7 send Q)\n                    \u228e (\u2203 \u03bb n \u2192 send P \u27e6\u214b\u27e7 Q n)\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n    \u27e6\u214b\u27e7-correct : \u2200 P Q \u2192 P \u27e6\u214b\u27e7 Q \u2261 \u27e6 P \u214b Q \u27e7\n    \u27e6\u214b\u27e7-correct end      Q        = refl\n    \u27e6\u214b\u27e7-correct (recv P) Q        = \u03a0=\u2032 _ \u03bb m \u2192 \u27e6\u214b\u27e7-correct (P m) Q\n    \u27e6\u214b\u27e7-correct (send P) end      = refl\n    \u27e6\u214b\u27e7-correct (send P) (recv Q) = \u03a0=\u2032 _ \u03bb n \u2192 \u27e6\u214b\u27e7-correct (send P) (Q n)\n    \u27e6\u214b\u27e7-correct (send P) (send Q) = ! dist-\u228e-\u03a3\n                                  \u2219 \u03a3=\u2032 (_ \u228e _) \u03bb { (inl m) \u2192 \u27e6\u214b\u27e7-correct (P m) (send Q)\n                                                  ; (inr n) \u2192 \u27e6\u214b\u27e7-correct (send P) (Q n) }\n\n                                                  {-\n    -- sends can be pulled out of tensor\n    source->>=-\u2297 : \u2200 P Q R \u2192 (source-of P >>= Q) \u2297 R \u2261 source-of P >>= \u03bb log \u2192 (Q log \u2297 R)\n    source->>=-\u2297 end       Q R = refl\n    source->>=-\u2297 (com _ P) Q R = send=\u2032 \u03bb m \u2192 source->>=-\u2297 (P m) (Q \u2218 _,_ m) R\n\n    -- consequence[Q = const end]: \u2200 P R \u2192 source-of P \u2297 R \u2261 source-of P >> R\n\n    -- recvs can be pulled out of par\n    sink->>=-\u214b : \u2200 P Q R \u2192 (sink-of P >>= Q) \u214b R \u2261 sink-of P >>= \u03bb log \u2192 (Q log \u214b R)\n    sink->>=-\u214b end       Q R = refl\n    sink->>=-\u214b (com _ P) Q R = recv=\u2032 \u03bb m \u2192 sink->>=-\u214b (P m) (Q \u2218 _,_ m) R\n\n    -- consequence[Q = const end]: \u2200 P R \u2192 sink-of P \u214b R \u2261 sink-of P >> R\n\n  Log-\u214b-\u00d7 : \u2200 {P Q} \u2192 Log (P \u214b Q) \u2192 Log P \u00d7 Log Q\n  Log-\u214b-\u00d7 {end}   {Q}      q           = end , q\n  Log-\u214b-\u00d7 {recv P}{Q}      (m , p)     = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {Q} p\n  Log-\u214b-\u00d7 {send P}{end}    (m , p)     = (m , p) , end\n  Log-\u214b-\u00d7 {send P}{recv Q} (m , p)     = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n  Log-\u214b-\u00d7 {send P}{send Q} (inl m , p) = first  (_,_ m) $ Log-\u214b-\u00d7 {P m} {send Q} p\n  Log-\u214b-\u00d7 {send P}{send Q} (inr m , p) = second (_,_ m) $ Log-\u214b-\u00d7 {send P} {Q m} p\n\n  module _ {{_ : FunExt}} where\n    \u2297\u214b-dual : \u2200 P Q \u2192 dual (P \u214b Q) \u2261 dual P \u2297 dual Q\n    \u2297\u214b-dual end Q = refl\n    \u2297\u214b-dual (recv P) Q = com=\u2032 _ \u03bb m \u2192 \u2297\u214b-dual (P m) _\n    \u2297\u214b-dual (send P) end = refl\n    \u2297\u214b-dual (send P) (recv Q) = com=\u2032 _ \u03bb n \u2192 \u2297\u214b-dual (send P) (Q n)\n    \u2297\u214b-dual (send P) (send Q) = com=\u2032 _\n      [inl: (\u03bb m \u2192 \u2297\u214b-dual (P m) (send Q))\n      ,inr: (\u03bb n \u2192 \u2297\u214b-dual (send P) (Q n))\n      ]\n\n  data View-\u214b-proto : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    end-X     : \u2200 Q \u2192 View-\u214b-proto end Q\n    recv-X    : \u2200 {M}(P : M \u2192 Proto)Q \u2192 View-\u214b-proto (recv P) Q\n    send-send : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u214b-proto (send P) (send Q)\n    send-recv : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u214b-proto (send P) (recv Q)\n    send-end  : \u2200 {M}(P : M \u2192 Proto) \u2192 View-\u214b-proto (send P) end\n\n  view-\u214b-proto : \u2200 P Q \u2192 View-\u214b-proto P Q\n  view-\u214b-proto end      Q        = end-X Q\n  view-\u214b-proto (recv P) Q        = recv-X P Q\n  view-\u214b-proto (send P) end      = send-end P\n  view-\u214b-proto (send P) (recv Q) = send-recv P Q\n  view-\u214b-proto (send P) (send Q) = send-send P Q\n\n  data View-\u2297-proto : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    end-X     : \u2200 Q \u2192 View-\u2297-proto end Q\n    send-X    : \u2200 {M}(P : M \u2192 Proto)Q \u2192 View-\u2297-proto (send P) Q\n    recv-recv : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u2297-proto (recv P) (recv Q)\n    recv-send : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) \u2192 View-\u2297-proto (recv P) (send Q)\n    recv-end  : \u2200 {M}(P : M \u2192 Proto) \u2192 View-\u2297-proto (recv P) end\n\n  view-\u2297-proto : \u2200 P Q \u2192 View-\u2297-proto P Q\n  view-\u2297-proto end      Q        = end-X Q\n  view-\u2297-proto (send P) Q        = send-X P Q\n  view-\u2297-proto (recv P) end      = recv-end P\n  view-\u2297-proto (recv P) (recv Q) = recv-recv P Q\n  view-\u2297-proto (recv P) (send Q) = recv-send P Q\n\n  -- the terminology used for the constructor follows the behavior of the combined process\n  data View-\u214b : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u2605\u2081 where\n    sendL' : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto)(m  : M )(p : \u27e6 P m \u214b send Q \u27e7) \u2192 View-\u214b (send P) (send Q) (inl m  , p)\n    sendR' : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto)(n : N)(p : \u27e6 send P \u214b Q n \u27e7) \u2192 View-\u214b (send P) (send Q) (inr n , p)\n    recvL' : \u2200 {M} (P : M \u2192 Proto) Q (p : ((m : M) \u2192 \u27e6 P m \u214b Q \u27e7)) \u2192 View-\u214b (recv P) Q p\n    recvR' : \u2200 {M N} (P : M \u2192 Proto) (Q : N \u2192 Proto)(p : (n : N) \u2192 \u27e6 send P \u214b Q n \u27e7) \u2192 View-\u214b (send P) (recv Q) p\n    endL   : \u2200 Q (p : \u27e6 Q \u27e7) \u2192 View-\u214b end Q p\n    send'  : \u2200 {M}(P : M \u2192 Proto)(m : M)(p : \u27e6 P m \u27e7) \u2192 View-\u214b (send P) end (m , p)\n\n  view-\u214b : \u2200 P Q (p : \u27e6 P \u214b Q \u27e7) \u2192 View-\u214b P Q p\n  view-\u214b end Q p = endL Q p\n  view-\u214b (recv P) Q p = recvL' P Q p\n  view-\u214b (send P) end (m , p) = send' P m p\n  view-\u214b (send P) (recv Q) p = recvR' P Q p\n  view-\u214b (send P) (send Q) (inl x , p) = sendL' P Q x p\n  view-\u214b (send P) (send Q) (inr y , p) = sendR' P Q y p\n\n  {-\n  -- use coe (... \u214b-assoc P Q R)\n  \u214b-assoc : \u2200 P Q R \u2192 \u27e6 P \u214b (Q \u214b R) \u27e7 \u2192 \u27e6 (P \u214b Q) \u214b R \u27e7\n  \u214b-assoc end      Q        R         s                 = s\n  \u214b-assoc (recv P) Q        R         s m               = \u214b-assoc (P m) _ _ (s m)\n  \u214b-assoc (send P) end      R         s                 = s\n  \u214b-assoc (send P) (recv Q) R         s m               = \u214b-assoc (send P) (Q m) _ (s m)\n  \u214b-assoc (send P) (send Q) end       s                 = s\n  \u214b-assoc (send P) (send Q) (recv R)  s m               = \u214b-assoc (send P) (send Q) (R m) (s m)\n  \u214b-assoc (send P) (send Q) (send R) (inl m , s)       = inl (inl m) , \u214b-assoc (P m) (send Q) (send R) s\n  \u214b-assoc (send P) (send Q) (send R) (inr (inl m) , s) = inl (inr m) , \u214b-assoc (send P) (Q m) (send R) s\n  \u214b-assoc (send P) (send Q) (send R) (inr (inr m) , s) = inr m       , \u214b-assoc (send P) (send Q) (R m) s\n\n  -- use coe (\u214b-endR P) instead\n  \u214b-rend : \u2200 P \u2192 \u27e6 P \u214b end \u27e7  \u2192 \u27e6 P \u27e7\n  \u214b-rend end      p = p\n  \u214b-rend (send _) p = p\n  \u214b-rend (recv P) p = \u03bb m \u2192 \u214b-rend (P m) (p m)\n\n  -- use coe! (\u214b-endR P) instead\n  \u214b-rend! : \u2200 P  \u2192 \u27e6 P \u27e7 \u2192 \u27e6 P \u214b end \u27e7\n  \u214b-rend! end      p = p\n  \u214b-rend! (send _) p = p\n  \u214b-rend! (recv P) p = \u03bb m \u2192 \u214b-rend! (P m) (p m)\n\n  -- use coe! (\u214b-recvR P Q) instead\n  \u214b-isendR : \u2200 {N} P Q \u2192 \u27e6 P \u214b recv Q \u27e7 \u2192 (n : N) \u2192 \u27e6 P \u214b Q n \u27e7\n  \u214b-isendR end Q s n = s n\n  \u214b-isendR (recv P) Q s n = \u03bb m \u2192 \u214b-isendR (P m) Q (s m) n\n  \u214b-isendR (send P) Q s n = s n\n\n\n  -- see \u214b-recvR\n  \u214b-recvR : \u2200 {M} P Q \u2192 ((m : M) \u2192 \u27e6 P \u214b Q m \u27e7) \u2192 \u27e6 P \u214b recv Q \u27e7\n  \u214b-recvR end      Q s = s\n  \u214b-recvR (recv P) Q s = \u03bb x \u2192 \u214b-recvR (P x) Q (\u03bb m \u2192 s m x)\n  \u214b-recvR (send P) Q s = s\n  -}\n\n  module _ {{_ : FunExt}}{{_ : UA}} where\n\n    \u214b-sendR : \u2200 {M}P{Q : M \u2192 Proto}(m : M) \u2192 \u27e6 P \u214b Q m \u27e7 \u2192 \u27e6 P \u214b send Q \u27e7\n    \u214b-sendR end      m p = m , p\n    \u214b-sendR (send P) m p = inr m , p\n    \u214b-sendR (recv P) m p = \u03bb x \u2192 \u214b-sendR (P x) m (p x)\n\n    \u214b-sendL : \u2200 {M}{P : M \u2192 Proto} Q (m : M) \u2192 \u27e6 P m \u214b Q \u27e7 \u2192 \u27e6 send P \u214b Q \u27e7\n    \u214b-sendL {M} {P} Q m pq = coe (\u214b-comm Q (send P)) (\u214b-sendR Q m (coe (\u214b-comm (P m) Q) pq))\n\n    \u214b-id : \u2200 P \u2192 \u27e6 dual P \u214b P \u27e7\n    \u214b-id end      = end\n    \u214b-id (recv P) = \u03bb x \u2192 \u214b-sendL (P x) x (\u214b-id (P x))\n    \u214b-id (send P) = \u03bb x \u2192 \u214b-sendR (dual (P x)) x (\u214b-id (P x))\n\n  data View-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u2605\u2081 where\n    sendLL : \u2200 {M N}(P : M \u2192 Proto)(Q : N \u2192 Proto) R (m : M)(p : \u27e6 P m \u214b send Q \u27e7)(q : \u27e6 dual (send Q) \u214b R \u27e7)\n             \u2192 View-\u2218 (send P) (send Q) R (inl m , p) q\n    recvLL : \u2200 {M} (P : M \u2192 Proto) Q R\n               (p : ((m : M) \u2192 \u27e6 P m \u214b Q \u27e7))(q : \u27e6 dual Q \u214b R \u27e7)\n             \u2192 View-\u2218 (recv P) Q R p q\n    recvR-sendR : \u2200 {MP MQ MR}ioP(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (mR : MR)(p : \u27e6 com ioP P \u214b recv Q \u27e7)(q : \u27e6 dual (recv Q) \u214b R mR \u27e7)\n                    \u2192 View-\u2218 (com ioP P) (recv Q) (send R) p (inr mR , q)\n\n    recvRR : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n               (p : \u27e6 send P \u214b recv Q \u27e7)(q : (m : MR) \u2192 \u27e6 dual (recv Q) \u214b R m \u27e7)\n             \u2192 View-\u2218 (send P) (recv Q) (recv R) p q\n    sendR-recvL : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)R(m : MQ)\n                    (p : \u27e6 send P \u214b Q m \u27e7)(q : (m : MQ) \u2192 \u27e6 dual (Q m) \u214b R \u27e7)\n                  \u2192 View-\u2218 (send P) (send Q) R (inr m , p) q\n    recvR-sendL : \u2200 {MP MQ MR}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)(R : MR \u2192 Proto)\n                    (p : (m : MQ) \u2192 \u27e6 send P \u214b Q m \u27e7)(m : MQ)(q : \u27e6 dual (Q m) \u214b send R \u27e7)\n                  \u2192 View-\u2218 (send P) (recv Q) (send R) p (inl m , q)\n    endL : \u2200 Q R\n           \u2192 (q : \u27e6 Q \u27e7)(qr : \u27e6 dual Q \u214b R \u27e7)\n           \u2192 View-\u2218 end Q R q qr\n    sendLM : \u2200 {MP}(P : MP \u2192 Proto)R\n               (m : MP)(p : \u27e6 P m \u27e7)(r : \u27e6 R \u27e7)\n             \u2192 View-\u2218 (send P) end R (m , p) r\n    recvL-sendR : \u2200 {MP MQ}(P : MP \u2192 Proto)(Q : MQ \u2192 Proto)\n                    (m : MQ)(p : \u2200 m \u2192 \u27e6 send P \u214b Q m \u27e7)(q : \u27e6 dual (Q m) \u27e7)\n                  \u2192 View-\u2218 (send P) (recv Q) end p (m , q)\n\n  view-\u2218 : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7)(qr : \u27e6 dual Q \u214b R \u27e7) \u2192 View-\u2218 P Q R pq qr\n  view-\u2218 P Q R pq qr = view-\u2218-view (view-\u214b P Q pq) (view-\u214b (dual Q) R qr)\n   where\n    view-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b Q \u27e7}{qr : \u27e6 dual Q \u214b R \u27e7} \u2192 View-\u214b P Q pq \u2192 View-\u214b (dual Q) R qr \u2192 View-\u2218 P Q R pq qr\n    view-\u2218-view (sendL' _ _ _ _) _                 = sendLL _ _ _ _ _ _\n    view-\u2218-view (recvL' _ _ _)   _                 = recvLL _ _ _ _ _\n    view-\u2218-view (sendR' _ _ _ _) _                 = sendR-recvL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendL' ._ _ _ _) = recvR-sendL _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (sendR' ._ _ _ _) = recvR-sendR _ _ _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (recvR' ._ _ _)   = recvRR _ _ _ _ _\n    view-\u2218-view (recvR' _ _ _)   (send' ._ _ _)    = recvL-sendR _ _ _ _ _\n    view-\u2218-view (endL _ _)       _                 = endL _ _ _ _\n    view-\u2218-view (send' _ _ _)    _                 = sendLM _ _ _ _ _\n\n  \u214b-apply : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual P \u27e7 \u2192 \u27e6 Q \u27e7\n  \u214b-apply end      Q        s           p       = s\n  \u214b-apply (recv P) Q        s           (m , p) = \u214b-apply (P m) Q (s m) p\n  \u214b-apply (send P) end      s           p       = _\n  \u214b-apply (send P) (recv Q) s           p n     = \u214b-apply (send P) (Q n) (s n) p\n  \u214b-apply (send P) (send Q) (inl m , s) p       = \u214b-apply (P m) (send Q) s (p m)\n  \u214b-apply (send P) (send Q) (inr m , s) p       = m , \u214b-apply (send P) (Q m) s p\n\n  {-\n  -- see dist-\u214b-&\n  dist-\u214b-fst : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  dist-\u214b-fst (recv P) Q R p = \u03bb m \u2192 dist-\u214b-fst (P m) Q R (p m)\n  dist-\u214b-fst (send P) Q R p = p `L\n  dist-\u214b-fst end      Q R p = p `L\n\n  -- see dist-\u214b-&\n  dist-\u214b-snd : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n  dist-\u214b-snd (recv P) Q R p = \u03bb m \u2192 dist-\u214b-snd (P m) Q R (p m)\n  dist-\u214b-snd (send P) Q R p = p `R\n  dist-\u214b-snd end      Q R p = p `R\n\n  -- see dist-\u214b-&\n  dist-\u214b-\u00d7 : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 P \u214b Q \u27e7 \u00d7 \u27e6 P \u214b R \u27e7\n  dist-\u214b-\u00d7 P Q R p = dist-\u214b-fst P Q R p , dist-\u214b-snd P Q R p\n\n  -- see dist-\u214b-&\n  dist-\u214b-& : \u2200 P Q R \u2192 \u27e6 P \u214b (Q & R) \u27e7 \u2192 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n  dist-\u214b-& P Q R p = \u00d7\u2192& (dist-\u214b-\u00d7 P Q R p)\n\n  -- see dist-\u214b-&\n  factor-,-\u214b : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 P \u214b R \u27e7 \u2192 \u27e6 P \u214b (Q & R) \u27e7\n  factor-,-\u214b end      Q R pq pr = \u00d7\u2192& (pq , pr)\n  factor-,-\u214b (recv P) Q R pq pr = \u03bb m \u2192 factor-,-\u214b (P m) Q R (pq m) (pr m)\n  factor-,-\u214b (send P) Q R pq pr = [L: pq R: pr ]\n\n  -- see dist-\u214b-&\n  factor-\u00d7-\u214b : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u00d7 \u27e6 P \u214b R \u27e7 \u2192 \u27e6 P \u214b (Q & R) \u27e7\n  factor-\u00d7-\u214b P Q R (p , q) = factor-,-\u214b P Q R p q\n\n  -- see dist-\u214b-&\n  factor-&-\u214b : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7 \u2192 \u27e6 P \u214b (Q & R) \u27e7\n  factor-&-\u214b P Q R p = factor-\u00d7-\u214b P Q R (&\u2192\u00d7 p)\n\n  -- see dist-\u214b-&\n  module _ {{_ : FunExt}} where\n    dist-\u214b-fst-factor-&-, : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7)(pr : \u27e6 P \u214b R \u27e7)\n                            \u2192 dist-\u214b-fst P Q R (factor-,-\u214b P Q R pq pr) \u2261 pq\n    dist-\u214b-fst-factor-&-, (recv P) Q R pq pr = \u03bb= \u03bb m \u2192 dist-\u214b-fst-factor-&-, (P m) Q R (pq m) (pr m)\n    dist-\u214b-fst-factor-&-, (send P) Q R pq pr = refl\n    dist-\u214b-fst-factor-&-, end      Q R pq pr = refl\n\n    dist-\u214b-snd-factor-&-, : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7)(pr : \u27e6 P \u214b R \u27e7)\n                            \u2192 dist-\u214b-snd P Q R (factor-,-\u214b P Q R pq pr) \u2261 pr\n    dist-\u214b-snd-factor-&-, (recv P) Q R pq pr = \u03bb= \u03bb m \u2192 dist-\u214b-snd-factor-&-, (P m) Q R (pq m) (pr m)\n    dist-\u214b-snd-factor-&-, (send P) Q R pq pr = refl\n    dist-\u214b-snd-factor-&-, end      Q R pq pr = refl\n\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 : \u2200 P Q R \u2192 (factor-\u00d7-\u214b P Q R) LeftInverseOf (dist-\u214b-\u00d7 P Q R)\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (recv P) Q R p = \u03bb= \u03bb m \u2192 factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (P m) Q R (p m)\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 (send P) Q R p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n    factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 end      Q R p = \u03bb= \u03bb { `L \u2192 refl ; `R \u2192 refl }\n\n    module _ P Q R where\n        factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 : (factor-\u00d7-\u214b P Q R) RightInverseOf (dist-\u214b-\u00d7 P Q R)\n        factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 (x , y) = pair\u00d7= (dist-\u214b-fst-factor-&-, P Q R x y) (dist-\u214b-snd-factor-&-, P Q R x y)\n\n        dist-\u214b-\u00d7-\u2243 : \u27e6 P \u214b (Q & R) \u27e7 \u2243 (\u27e6 P \u214b Q \u27e7 \u00d7 \u27e6 P \u214b R \u27e7)\n        dist-\u214b-\u00d7-\u2243 = dist-\u214b-\u00d7 P Q R\n                   , record { linv = factor-\u00d7-\u214b P Q R; is-linv = factor-\u00d7-\u214b-linv-dist-\u214b-\u00d7 P Q R\n                            ; rinv = factor-\u00d7-\u214b P Q R; is-rinv = factor-\u00d7-\u214b-rinv-dist-\u214b-\u00d7 }\n\n        dist-\u214b-&-\u2243 : \u27e6 P \u214b (Q & R) \u27e7 \u2243 \u27e6 (P \u214b Q) & (P \u214b R) \u27e7\n        dist-\u214b-&-\u2243 = dist-\u214b-\u00d7-\u2243 \u2243-\u2219 \u2243-! &\u2243\u00d7\n  -}\n\nmodule _ {{_ : FunExt}}{{_ : UA}} where\n  \u2297-pair : \u2200 {P Q} \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n  \u2297-pair {end}    {Q}      p q       = q\n  \u2297-pair {send P} {Q}      (m , p) q = m , \u2297-pair {P m} p q\n  \u2297-pair {recv P} {end}    p end     = p\n  \u2297-pair {recv P} {send Q} p (m , q) = m , \u2297-pair {recv P} {Q m} p q\n  \u2297-pair {recv P} {recv Q} p q       = [inl: (\u03bb m \u2192 \u2297-pair {P m}    {recv Q} (p m) q)\n                                       ,inr: (\u03bb n \u2192 \u2297-pair {recv P} {Q n}    p     (q n)) ]\n\n  \u2297-fst : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7\n  \u2297-fst end      Q        pq       = _\n  \u2297-fst (send P) Q        (m , pq) = m , \u2297-fst (P m) Q pq\n  \u2297-fst (recv P) end      pq       = pq\n  \u2297-fst (recv P) (send Q) (_ , pq) = \u2297-fst (recv P) (Q _) pq\n  \u2297-fst (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-fst (P m) (recv Q) (pq (inl m))\n\n  \u2297-snd : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 Q \u27e7\n  \u2297-snd end      Q        pq       = pq\n  \u2297-snd (send P) Q        (_ , pq) = \u2297-snd (P _) Q pq\n  \u2297-snd (recv P) end      pq       = end\n  \u2297-snd (recv P) (send Q) (m , pq) = m , \u2297-snd (recv P) (Q m) pq\n  \u2297-snd (recv P) (recv Q) pq       = \u03bb m \u2192 \u2297-snd (recv P) (Q m) (pq (inr m))\n\n  \u2297-pair-fst : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-fst P Q (\u2297-pair {P} {Q} p q) \u2261 p\n  \u2297-pair-fst end      Q        p q       = refl\n  \u2297-pair-fst (send P) Q        (m , p) q = pair= refl (\u2297-pair-fst (P m) Q p q)\n  \u2297-pair-fst (recv P) end      p q       = refl\n  \u2297-pair-fst (recv P) (send Q) p (m , q) = \u2297-pair-fst (recv P) (Q m) p q\n  \u2297-pair-fst (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-fst (P m) (recv Q) (p m) q\n\n  \u2297-pair-snd : \u2200 P Q (p : \u27e6 P \u27e7)(q : \u27e6 Q \u27e7) \u2192 \u2297-snd P Q (\u2297-pair {P} {Q} p q) \u2261 q\n  \u2297-pair-snd end      Q        p q       = refl\n  \u2297-pair-snd (send P) Q        (m , p) q = \u2297-pair-snd (P m) Q p q\n  \u2297-pair-snd (recv P) end      p q       = refl\n  \u2297-pair-snd (recv P) (send Q) p (m , q) = pair= refl (\u2297-pair-snd (recv P) (Q m) p q)\n  \u2297-pair-snd (recv P) (recv Q) p q       = \u03bb= \u03bb m \u2192 \u2297-pair-snd (recv P) (Q m) p (q m)\n\n  module _ P Q where\n    \u2297\u2192\u00d7 : \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n    \u2297\u2192\u00d7 pq = \u2297-fst P Q pq , \u2297-snd P Q pq\n\n    \u00d7\u2192\u2297 : \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7 \u2192 \u27e6 P \u2297 Q \u27e7\n    \u00d7\u2192\u2297 (p , q) = \u2297-pair {P} {Q} p q\n\n    \u00d7\u2192\u2297\u2192\u00d7 : \u00d7\u2192\u2297 RightInverseOf \u2297\u2192\u00d7\n    \u00d7\u2192\u2297\u2192\u00d7 = \u03bb { (x , y) \u2192 pair\u00d7= (\u2297-pair-fst P Q x y) (\u2297-pair-snd P Q x y) }\n\n    \u2297\u2192\u00d7-has-rinv : Rinv \u2297\u2192\u00d7\n    \u2297\u2192\u00d7-has-rinv = record { rinv = \u00d7\u2192\u2297 ; is-rinv = \u00d7\u2192\u2297\u2192\u00d7 }\n\n  {- WRONG\n  \u2297\u2192\u00d7\u2192\u2297 : (\u00d7\u2192\u2297 P Q) LeftInverseOf (\u2297\u2192\u00d7 P Q)\n  \u2297\u2243\u00d7   : \u27e6 P \u2297 Q \u27e7 \u2243 \u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7\n  \u27e6\u2297\u27e7\u2261\u00d7 : P \u27e6\u2297\u27e7 Q \u2261 (\u27e6 P \u27e7 \u00d7 \u27e6 Q \u27e7)\n  -}\n\n  -o-apply : \u2200 {P Q} \u2192 \u27e6 dual P \u214b Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  -o-apply {P} {Q} pq p = \u214b-apply (dual P) Q pq (subst \u27e6_\u27e7 (\u2261.sym (\u2261\u1d3e-sound (dual-involutive P))) p)\n\n  o-o-apply : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7\n  o-o-apply P Q Po-oQ p = -o-apply {P} {Q} (\u2297-fst (P -o Q) (Q -o P) Po-oQ) p\n\n  o-o-comm : \u2200 P Q \u2192 \u27e6 P o-o Q \u27e7 \u2261 \u27e6 Q o-o P \u27e7\n  o-o-comm P Q = \u2297-comm (dual P \u214b Q) (dual Q \u214b P)\n\n  -- left-biased \u201cstrategy\u201d\n  par : \u2200 P Q \u2192 \u27e6 P \u27e7 \u2192 \u27e6 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  par (recv P) Q p       q = \u03bb m \u2192 par (P m) Q (p m) q\n  par (send P) Q (m , p) q = \u214b-sendL Q m (par (P m) Q p q)\n  par end      Q end     q = q\n\n  \u214b-\u2218 : \u2200 P Q R \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 dual Q \u214b R \u27e7 \u2192 \u27e6 P \u214b R \u27e7\n  \u214b-\u2218 P Q R pq qr = \u214b-\u2218-view (view-\u2218 P Q R pq qr)\n   where\n    \u214b-\u2218-view : \u2200 {P Q R}{pq : \u27e6 P \u214b Q \u27e7}{qr : \u27e6 dual Q \u214b R \u27e7} \u2192 View-\u2218 P Q R pq qr \u2192 \u27e6 P \u214b R \u27e7\n    \u214b-\u2218-view (sendLL P Q R m p qr)          = \u214b-sendL R m (\u214b-\u2218 (P m) (send Q) R p qr)\n    \u214b-\u2218-view (recvLL P Q R p qr)            = \u03bb m \u2192 \u214b-\u2218 (P m) Q R (p m) qr\n    \u214b-\u2218-view (recvR-sendR ioP P Q R m pq q) = \u214b-sendR (com ioP P) m (\u214b-\u2218 (com ioP P) (recv Q) (R m) pq q)\n    \u214b-\u2218-view (recvRR P Q R pq q)            = \u03bb m \u2192 \u214b-\u2218 (send P) (recv Q) (R m) pq (q m)\n    \u214b-\u2218-view (sendR-recvL P Q R m p q)      = \u214b-\u2218 (send P) (Q m) R p (q m)\n    \u214b-\u2218-view (recvR-sendL P Q R p m q)      = \u214b-\u2218 (send P) (Q m) (send R) (p m) q\n    \u214b-\u2218-view (endL Q R pq qr)               = -o-apply {Q} {R} qr pq\n    \u214b-\u2218-view (sendLM P R m pq qr)           = \u214b-sendL R m (par (P m) R pq qr)\n    \u214b-\u2218-view (recvL-sendR P Q m pq qr)      = \u214b-\u2218 (send P) (Q m) end (pq m) (coe! (ap \u27e6_\u27e7 (\u214b-endR (dual (Q m)))) qr)\n\n    {-\n  mutual\n    \u214b-comm : \u2200 P Q \u2192 \u27e6 P \u214b Q \u27e7 \u2192 \u27e6 Q \u214b P \u27e7\n    \u214b-comm P Q p = \u214b-comm-view (view-\u214b P Q p)\n\n    \u214b-comm-view : \u2200 {P Q} {pq : \u27e6 P \u214b Q \u27e7} \u2192 View-\u214b P Q pq \u2192 \u27e6 Q \u214b P \u27e7\n    \u214b-comm-view (sendL' P Q m p) = \u214b-sendR (send Q) m (\u214b-comm (P m) (send Q) p)\n    \u214b-comm-view (sendR' P Q n p) = inl n , \u214b-comm (send P) (Q n) p\n    \u214b-comm-view (recvL' P Q pq)  = \u214b-recvR Q P \u03bb m \u2192 \u214b-comm (P m) Q (pq m)\n    \u214b-comm-view (recvR' P Q pq)  = \u03bb n \u2192 \u214b-comm (send P) (Q n) (pq n)\n    \u214b-comm-view (endL Q pq)      = \u214b-rend! Q pq\n    \u214b-comm-view (send P m pq)    = m , pq\n  -}\n\n  switchL' : \u2200 P Q R (pq : \u27e6 P \u214b Q \u27e7) (r : \u27e6 R \u27e7) \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n  switchL' end      Q        R        q  r = \u2297-pair {Q} {R} q r\n  switchL' (send P) end      R        p  r = par (send P) R p r\n  switchL' (send P) (send Q) R        (inl m , pq) r = inl m , switchL' (P m) (send Q) R pq r\n  switchL' (send P) (send Q) R        (inr m , pq) r = inr m , switchL' (send P) (Q m) R pq r\n  switchL' (send P) (recv Q) end      pq r = pq\n  switchL' (send P) (recv Q) (send R) pq (m , r) = inr m , switchL' (send P) (recv Q) (R m) pq r\n  switchL' (send P) (recv Q) (recv R) pq r (inl m) = switchL' (send P) (Q m) (recv R) (pq m) r\n  switchL' (send P) (recv Q) (recv R) pq r (inr m) = switchL' (send P) (recv Q) (R m) pq (r m)\n  switchL' (recv P) Q R pq r = \u03bb m \u2192 switchL' (P m) Q R (pq m) r\n\n  switchL : \u2200 P Q R \u2192 \u27e6 (P \u214b Q) \u2297 R \u27e7 \u2192 \u27e6 P \u214b (Q \u2297 R) \u27e7\n  switchL P Q R pqr = switchL' P Q R (\u2297-fst (P \u214b Q) R pqr) (\u2297-snd (P \u214b Q) R pqr)\n\n  -- multiplicative mix (left-biased)\n  mmix : \u2200 P Q \u2192 \u27e6 P \u2297 Q \u27e7 \u2192 \u27e6 P \u214b Q \u27e7\n  mmix P Q pq = par P Q (\u2297-fst P Q pq) (\u2297-snd P Q pq)\n\n  -- additive mix (left-biased)\n  amix : \u2200 P Q \u2192 \u27e6 P & Q \u27e7 \u2192 \u27e6 P \u2295 Q \u27e7\n  amix P Q pq = (`L , pq `L)\n\n{-\nA `\u2297 B 'times', context chooses how A and B are used\nA `\u214b B 'par', \"we\" chooses how A and B are used\nA `\u2295 B 'plus', select from A or B\nA `& B 'with', offer choice of A or B\n`! A   'of course!', server accept\n`? A   'why not?', client request\n`1     unit for `\u2297\n`\u22a5     unit for `\u214b\n`0     unit for `\u2295\n`\u22a4     unit for `&\n-}\ndata CLL : \u2605 where\n  `1 `\u22a4 `0 `\u22a5 : CLL\n  _`\u2297_ _`\u214b_ _`\u2295_ _`&_ : (A B : CLL) \u2192 CLL\n  -- `!_ `?_ : (A : CLL) \u2192 CLL\n\n_\u22a5 : CLL \u2192 CLL\n`1 \u22a5 = `\u22a5\n`\u22a5 \u22a5 = `1\n`0 \u22a5 = `\u22a4\n`\u22a4 \u22a5 = `0\n(A `\u2297 B)\u22a5 = A \u22a5 `\u214b B \u22a5\n(A `\u214b B)\u22a5 = A \u22a5 `\u2297 B \u22a5\n(A `\u2295 B)\u22a5 = A \u22a5 `& B \u22a5\n(A `& B)\u22a5 = A \u22a5 `\u2295 B \u22a5\n{-\n(`! A)\u22a5 = `?(A \u22a5)\n(`? A)\u22a5 = `!(A \u22a5)\n-}\n\nCLL-proto : CLL \u2192 Proto\nCLL-proto `1       = end  -- TODO\nCLL-proto `\u22a5       = end  -- TODO\nCLL-proto `0       = send\u2032 \ud835\udfd8 end -- Alt: send \u03bb()\nCLL-proto `\u22a4       = recv\u2032 \ud835\udfd8 end -- Alt: recv \u03bb()\nCLL-proto (A `\u2297 B) = CLL-proto A \u2297 CLL-proto B\nCLL-proto (A `\u214b B) = CLL-proto A \u214b CLL-proto B\nCLL-proto (A `\u2295 B) = CLL-proto A \u2295 CLL-proto B\nCLL-proto (A `& B) = CLL-proto A & CLL-proto B\n\n{- The point of this could be to devise a particular equivalence\n   relation for processes. It could properly deal with \u214b. -}\n\nmodule Commitment {Secret Guess : \u2605} {R : ..(_ : Secret) \u2192 Guess \u2192 \u2605} where\n    Commit : Proto\n    Commit = send\u2610 \u03bb (s : Secret) \u2192\n             recv  \u03bb (g : Guess)  \u2192\n             send  \u03bb (_ : S\u27e8 s \u27e9) \u2192\n             end\n\n    commit : (s : Secret)  \u2192 \u27e6 Commit \u27e7\n    commit s = [ s ] , \u03bb g \u2192 S[ s ] , _\n\n    decommit : (g : Guess) \u2192 \u27e6 dual Commit \u27e7\n    decommit g = \u03bb { [ m ] \u2192 g , _ }\n\nopen import Relation.Nullary\nopen import Relation.Nullary.Decidable\n{-\ntest-sim : Sim (\ud835\udfd8 \u00d7' end) end\ntest-sim = end\n-}\n\n-- Prove knowledge of a discrete log\n-- Adapted here to have precise types\nmodule Shnorr-protocol\n    (G \u2124q : \u2605)\n    (g    : G) \n    (_^_  : G  \u2192 \u2124q \u2192 G)\n    (_\u00b7_  : G  \u2192 G  \u2192 G)\n    (_+_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_*_  : \u2124q \u2192 \u2124q \u2192 \u2124q)\n    (_\u225f_  : (x y : G) \u2192 Dec (x \u2261 y))\n    where\n    module Real where\n        Prover : Proto\n        Prover = send \u03bb (g\u02b3 : G)  \u2192 -- commitment\n                 recv \u03bb (c  : \u2124q) \u2192 -- challenge\n                 send \u03bb (s  : \u2124q) \u2192 -- response\n                 end\n\n        Verifier : Proto\n        Verifier = dual Prover\n\n        -- he is honest but its type does not say it\n        prover : (x r : \u2124q) \u2192 \u27e6 Prover \u27e7\n        prover x r = (g ^ r) , \u03bb c \u2192 (r + (c * x)) , _\n\n        Honest-Prover : ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) \u2192 Proto\n        Honest-Prover x y\n          = send\u2610 \u03bb (r  : \u2124q)                \u2192 -- ideal commitment\n            send  \u03bb (g\u02b3 : S\u27e8 g ^ r \u27e9)        \u2192 -- real  commitment\n            recv  \u03bb (c  : \u2124q)                \u2192 -- challenge\n            send  \u03bb (s  : S\u27e8 r + (c * x) \u27e9)  \u2192 -- response\n            recv  \u03bb (_  : Dec ((g ^ unS s) \u2261 (unS g\u02b3 \u00b7 (unS y ^ c)))) \u2192\n            end\n\n        Honest-Verifier : ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) \u2192 Proto\n        Honest-Verifier x y = dual (Honest-Prover x y)\n\n        honest-prover : (x r : \u2124q) \u2192 \u27e6 Honest-Prover x S[ g ^ x ] \u27e7\n        honest-prover x r = [ r ] , S[ g ^ r ] , \u03bb c \u2192 S[ r + (c * x) ] , _\n        -- agsy can do it\n\n        honest-verifier : ..(x : \u2124q) (y : S\u27e8 g ^ x \u27e9) (c : \u2124q) \u2192 \u27e6 Honest-Verifier x y \u27e7\n        honest-verifier x y c = \u03bb { [ r ] \u2192 \u03bb g\u02b3 \u2192 c , \u03bb s \u2192 (g ^ unS s) \u225f (unS g\u02b3 \u00b7 (unS y ^ c)) , _ }\n\n        honest-prover\u2192prover : ..(x : \u2124q)(y : S\u27e8 g ^ x \u27e9) \u2192 \u27e6 Honest-Prover x y \u27e7 \u2192 \u27e6 Prover \u27e7\n        honest-prover\u2192prover x y ([ r ] , g\u02b3 , p) = unS g\u02b3 , \u03bb c \u2192 case p c of \u03bb { (s , _) \u2192 unS s , _ }\n\n        {-\n        sim-honest-prover : ..(x : \u2124q)(y : S\u27e8 g ^ x \u27e9) \u2192 Sim (dual (Honest-Prover x y)) Prover\n        sim-honest-prover x y = recvL (\u03bb { [ r ] \u2192\n                                recvL \u03bb g\u02b3 \u2192\n                                sendR (unS g\u02b3) (\n                                recvR \u03bb c \u2192\n                                sendL c (recvL \u03bb s \u2192 sendR (unS s) (sendL {!!} {!!}) )) })\n                                -}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":128,"stderr":"fatal: invalid reference: FETCH_HEAD^\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"ca4b07c7bf5767039f5c201a62d71b561e4f1fe4","subject":"A tiny universe for explorations","message":"A tiny universe for explorations\n","repos":"crypto-agda\/explore","old_file":"lib\/Explore\/Universe.agda","new_file":"lib\/Explore\/Universe.agda","new_contents":"open import Level.NP\nopen import Type\nopen import Data.One\nopen import Data.Two\nopen import Data.Product\nopen import Explore.Type\nopen import Explore.One\nopen import Explore.Two\nopen import Explore.Product\n\nmodule Explore.Universe where\n\nopen Operators\n\n{-\nopen import FunUniverse.Data\n-}\ndata Ty : \u2605 where\n  \ud835\udfd9\u2032 \ud835\udfda\u2032 : Ty\n  _\u00d7\u2032_  : Ty \u2192 Ty \u2192 Ty\n\nEl : Ty \u2192 \u2605\nEl \ud835\udfd9\u2032 = \ud835\udfd9\nEl \ud835\udfda\u2032 = \ud835\udfda\nEl (u\u2080 \u00d7\u2032 u\u2081) = El u\u2080 \u00d7 El u\u2081\n--El (u ^\u2032 x) = {!Vec (El u) x!}\n\nmodule _ {\u2113} where\n\n    exploreU : \u2200 u \u2192 Explore \u2113 (El u)\n    exploreU \ud835\udfd9\u2032 = \ud835\udfd9\u1d49\n    exploreU \ud835\udfda\u2032 = \ud835\udfda\u1d49\n    exploreU (u \u00d7\u2032 u\u2081) = exploreU u \u00d7\u1d49 exploreU u\u2081\n    --exploreU (u ^\u2032 x) = {!!}\n\n    exploreU-ind : \u2200 u \u2192 ExploreInd \u2113 (exploreU u)\n    exploreU-ind \ud835\udfd9\u2032 = \ud835\udfd9\u2071\n    exploreU-ind \ud835\udfda\u2032 = \ud835\udfda\u2071\n    exploreU-ind (u \u00d7\u2032 u\u2081) = exploreU-ind u \u00d7\u2071 exploreU-ind u\u2081\n    --exploreU-ind (u ^\u2032 x) = {!!}\n\nmodule _ {\u2113} where\n    lookupU : \u2200 u \u2192 Lookup {\u2113} (exploreU {\u209b \u2113} u)\n    lookupU \ud835\udfd9\u2032 = \ud835\udfd9\u02e1\n    lookupU \ud835\udfda\u2032 = \ud835\udfda\u02e1\n    lookupU (u \u00d7\u2032 u\u2081) = lookup\u00d7 {_} {_} {_} {exploreU u} {exploreU u\u2081} (lookupU u) (lookupU u\u2081)\n    --lookupU (u ^\u2032 x) = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Explore\/Universe.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"327df0a778f2cc811302b3a95fd3c6c05aa4cd87","subject":"thrm statement","message":"thrm statement\n","repos":"hazelgrove\/hazelnut-dynamics-agda","old_file":"expandability.agda","new_file":"expandability.agda","new_contents":"open import Nat\nopen import Prelude\nopen import List\nopen import core\n\nmodule expandability where\n  mutual\n    expandability-synth : (\u0393 : \u00b7ctx) (e : e\u0307) (t : \u03c4\u0307) \u2192\n                          \u0393 \u22a2 e => t \u2192\n                          \u03a3[ e' \u2208 \u00eb ] \u03a3[ \u0394 \u2208 \u00b7ctx ]\n                            (\u0393 \u22a2 e \u21d2 t ~> e' \u22a3 \u0394)\n    expandability-synth = {!!}\n\n    expandability-ana : (\u0393 : \u00b7ctx) (e : e\u0307) (t : \u03c4\u0307) \u2192\n                         \u0393 \u22a2 e => t \u2192\n                          \u03a3[ e' \u2208 \u00eb ] \u03a3[ \u0394 \u2208 \u00b7ctx ] \u03a3[ t' \u2208 \u03c4\u0307 ]\n                            (\u0393 \u22a2 e \u21d0 t ~> e' :: t' \u22a3 \u0394)\n    expandability-ana = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'expandability.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e963cfdef26c2e78bc9d64522b66cee4faca5918","subject":"Try using agda","message":"Try using agda\n","repos":"louisswarren\/hieretikz","old_file":"hierarchy.agda","new_file":"hierarchy.agda","new_contents":"module hierarchy where\n\ndata Bool : Set where\n  true  : Bool\n  false : Bool\n\n_and_ : Bool \u2192 Bool \u2192 Bool\ntrue and true = true\ntrue and false = false\nfalse and true = false\nfalse and false = false\n\n\ndata \u2115 : Set where\n  zero : \u2115\n  suc  : \u2115 \u2192 \u2115\n\n_==_ : \u2115 \u2192 \u2115 \u2192 Bool\nzero == zero   = true\nsuc n == suc m = true\n_ == _         = false\n\n\n\ndata List (A : Set) : Set where\n  []   : List A\n  _::_ : A \u2192 List A \u2192 List A\n\n_\u2208_ : \u2115 \u2192 List \u2115 \u2192 Bool\nx \u2208 []      = false\nx \u2208 (y :: ys) with x == y\n...              | true  = true\n...              | false = x \u2208 ys\n\n_\u2282_ : List \u2115 \u2192 List \u2115 \u2192 Bool\n[] \u2282 ys      = true\n(x :: xs) \u2282 ys with x \u2208 ys\n...               | true  = xs \u2282 ys\n...               | false = false\n\n\n_++_ : List \u2115 \u2192 List \u2115 \u2192 List \u2115\n[] ++ ys        = ys\n(x :: xs) ++ ys with (x \u2208 ys)\n...               | true  = xs ++ ys\n...               | false = x :: (xs ++ ys)\n\n\n\ndata Arrow : Set where\n  _\u21d2_ : List \u2115 \u2192 \u2115 \u2192 Arrow\n\nSingleClosure : Arrow \u2192 List \u2115 \u2192 List \u2115\nSingleClosure (ts \u21d2 h) ps with ts \u2282 ps\n...                     | true  = h :: ps\n...                     | false = ps\n\nClosure' : List Arrow \u2192 List \u2115 \u2192 List \u2115\nClosure' [] roots = roots\nClosure' (c :: cs) roots = (SingleClosure c roots) ++ (Closure' cs roots)\n\nClosure : List Arrow \u2192 List \u2115 \u2192 List \u2115\nClosure arrows roots with (Closure' arrows roots) \u2282 roots\n...                     | true = roots\n...                     | false = Closure arrows (Closure' arrows roots)\n\n\n_,_\u22a2_ : List Arrow \u2192 List \u2115 \u2192 \u2115 \u2192 Bool\n_ , [] \u22a2 _ = false\nthms , p :: premises \u22a2 conc = true\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'hierarchy.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"1f178bf019b7eae0c066c104bcf755562b63b7e4","subject":"Agda model for Desc.","message":"Agda model for Desc.","repos":"brixen\/Epigram,kwangkim\/pigment,kwangkim\/pigment,kwangkim\/pigment","old_file":"models\/Desc.agda","new_file":"models\/Desc.agda","new_contents":"{-# OPTIONS --type-in-type\n            --no-termination-check\n            --no-positivity-check #-}\n\nmodule Desc where\n\nimport Data.Product\nimport Data.Unit\nimport Data.Empty\n\nopen import Data.Product\nopen import Data.Unit\nopen import Data.Empty\n\n-- Bringing in an hand-made Prop universe\n\ndata PROP : Set where\n  All : (S : Set) -> (S -> PROP) -> PROP\n  And : PROP -> PROP -> PROP\n  Trivial : PROP\n  Absurd : PROP\n\nPrf : PROP -> Set\nPrf Absurd = \u22a5\nPrf Trivial = \u22a4\nPrf (All S P) = (x : S) -> Prf (P x)\nPrf (And A B) = \u03a3 (Prf A)  (\\ _ -> Prf B)\n\n\n-- Then, we build Desc in Set\n-- it's all Set in Set here\n\n-- Desc code:\ndata Desc : Set where\n  Done : Desc\n  Arg : (X : Set) -> (X -> Desc) -> Desc\n  Ind : (X : Set) -> Desc -> Desc\n\ndescOp : Desc -> Set -> Set\ndescOp Done _ = \u22a4\ndescOp (Arg A f) D = \u03a3 A (\\ a -> descOp (f a) D)\ndescOp (Ind H x) D =  \u03a3 (H -> D) (\\ _ -> descOp x D)\n\n-- Followed by Mu:\n\ndata Mu (X : Desc) : Set where\n  Con : descOp X (Mu X) -> Mu X\n\n-- All of this is fully-furnished with:\n--   * boxOp\n--   * mapBoxOp\n--   * elimOp\n\nboxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) -> (v : descOp d D) -> Set\nboxOp Done _ _ _ =  \u22a4\nboxOp (Arg A f) d p v =  boxOp (f (proj\u2081 v)) d p (proj\u2082 v)\nboxOp (Ind H x) d p v =  Prf ( All H (\\y -> p (proj\u2081 v y)))\n                      \u00d7 boxOp x d p (proj\u2082 v)\n\n\nmapBoxOp : (d : Desc) -> (D : Set) -> (bp : D -> PROP) ->\n           (Prf (All D (\\y -> bp y))) -> (v : descOp d D) ->\n           (boxOp d D bp v)\nmapBoxOp Done _ _ _ _ =  tt\nmapBoxOp (Arg a f) d bp p v =  mapBoxOp (f (proj\u2081 v)) d bp p (proj\u2082 v)\nmapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (proj\u2082 v)\n\n-- On the Ind case, the Pig code is saying something along those lines:\n-- mapBoxOp (Ind H x) d bp p v =  (\\ y -> p (proj\u2081 v y)) ,  mapBoxOp x d bp p (p?! (proj\u2082 v))\n\n\n\nelimOp : (d : Desc) -> (bp : Mu d -> PROP) ->\n         (Prf (All (descOp d (Mu d)) (\\ x -> (All (boxOp d (Mu d) bp x) (\\ _ -> bp (Con x)))))) ->\n         (v : Mu d) -> Prf (bp v)\nelimOp d bp p (Con v) =  p v (mapBoxOp d (Mu d) bp (\\x -> elimOp d bp p x) v) ","old_contents":"","returncode":1,"stderr":"error: pathspec 'models\/Desc.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"820af659da893d60e5786154acd640ba52f72021","subject":"Added test for various arguments.","message":"Added test for various arguments.\n\nIgnore-this: 65825db26b50cc5e4fee1e0addff3694\n\ndarcs-hash:20111015073300-3bd4e-ded95ccd196a59d7ddb9b17c8ff7eec76aeacd2d.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Test\/Succeed\/VariousArguments.agda","new_file":"Test\/Succeed\/VariousArguments.agda","new_contents":"------------------------------------------------------------------------------\n-- Testing various arguments in the ATP pragmas\n------------------------------------------------------------------------------\n\nmodule Test.Succeed.VariousArguments where\n\npostulate\n  D       : Set\n  _\u2261_     : D \u2192 D \u2192 Set\n  a b c d : D\n  a\u2261b     : a \u2261 b\n  b\u2261c     : b \u2261 c\n  c\u2261d     : c \u2261 d\n{-# ATP axiom a\u2261b b\u2261c c\u2261d #-}\n\npostulate a\u2261d : a \u2261 d\n{-# ATP prove a\u2261d #-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Test\/Succeed\/VariousArguments.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"ac55e32900a6f72a4f838b5ac00093745ae210ab","subject":"Forgot a file.","message":"Forgot a file.\n","repos":"heades\/AUGL","old_file":"functions.agda","new_file":"functions.agda","new_contents":"module functions where\n\nopen import level\nopen import eq\nopen import product\n\n{- Note that the Agda standard library has an interesting generalization\n   of the following basic composition operator, with more dependent typing. -}\n\n_\u2218_ : \u2200{\u2113 \u2113' \u2113''}{A : Set \u2113}{B : Set \u2113'}{C : Set \u2113''} \u2192\n      (B \u2192 C) \u2192 (A \u2192 B) \u2192 (A \u2192 C)\nf \u2218 g = \u03bb x \u2192 f (g x)\n\n\u2218-assoc : \u2200{\u2113 \u2113' \u2113'' \u2113'''}{A : Set \u2113}{B : Set \u2113'}{C : Set \u2113''}{D : Set \u2113'''}\n           (f : C \u2192 D)(g : B \u2192 C)(h : A \u2192 B) \u2192 (f \u2218 g) \u2218 h \u2261 f \u2218 (g \u2218 h)\n\u2218-assoc f g h = refl\n\nid : \u2200{\u2113}{A : Set \u2113} \u2192 A \u2192 A\nid = \u03bb x \u2192 x\n\n\u2218-id : \u2200{\u2113 \u2113'}{A : Set \u2113}{B : Set \u2113'}(f : A \u2192 B) \u2192 f \u2218 id \u2261 f\n\u2218-id f = refl\n\nid-\u2218 : \u2200{\u2113 \u2113'}{A : Set \u2113}{B : Set \u2113'}(f : A \u2192 B) \u2192 id \u2218 f \u2261 f\nid-\u2218 f = refl\n\nextensionality : {\u2113\u2081 \u2113\u2082 : Level} \u2192 Set (lsuc \u2113\u2081 \u2294 lsuc \u2113\u2082)\nextensionality {\u2113\u2081}{\u2113\u2082} = \u2200{A : Set \u2113\u2081}{B : Set \u2113\u2082}{f g : A \u2192 B} \u2192 (\u2200{a : A} \u2192 f a \u2261 g a) \u2192 f \u2261 g\n\nrecord Iso {\u2113\u2081 \u2113\u2082 : Level} (A : Set \u2113\u2081) (B : Set \u2113\u2082) : Set (\u2113\u2081 \u2294 \u2113\u2082) where\n constructor isIso\n field\n   l-inv : A \u2192 B\n   r-inv : B \u2192 A\n   l-cancel : r-inv \u2218 l-inv \u2261 id\n   r-cancel : l-inv \u2218 r-inv \u2261 id \n\ncurry : \u2200{\u2113\u2081 \u2113\u2082 \u2113\u2083}{A : Set \u2113\u2081}{B : Set \u2113\u2082}{C : Set \u2113\u2083} \u2192 ((A \u00d7 B) \u2192 C) \u2192 (A \u2192 B \u2192 C)\ncurry f x y = f (x , y)\n\nuncurry : \u2200{\u2113\u2081 \u2113\u2082 \u2113\u2083}{A : Set \u2113\u2081}{B : Set \u2113\u2082}{C : Set \u2113\u2083} \u2192 (A \u2192 B \u2192 C) \u2192 ((A \u00d7 B) \u2192 C)\nuncurry f (x , y) = f x y\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'functions.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"f645c94ac3f7707191096d55afdc176e1170db6f","subject":"Nand: all computations based on nands","message":"Nand: all computations based on nands\n","repos":"crypto-agda\/crypto-agda","old_file":"FunUniverse\/Nand.agda","new_file":"FunUniverse\/Nand.agda","new_contents":"-- http:\/\/en.wikipedia.org\/wiki\/NAND_logic\nopen import Type\nopen import FunUniverse.Core\nopen import Level.NP\n--open import FunUniverse.Rewiring\n\nmodule FunUniverse.Nand {t} {T : \u2605_ t} {funU : FunUniverse T} (rewiring : Rewiring funU) where\n\nopen FunUniverse funU\nopen Rewiring rewiring\n\nmodule FromNand (nand : `\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda) where\n  not : `\ud835\udfda `\u2192 `\ud835\udfda\n  not  = dup \u204f nand \n\n  and or nor xor xnor : `\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda\n\n  and  = nand \u204f not\n  or   = < not \u00d7 not > \u204f nand\n  nor  = or \u204f not\n  xor  = < dup \u00d7 dup > \u204f inner2 (nand \u204f dup) \u204f < nand \u00d7 nand > \u204f nand\n  xnor = xor \u204f not\n\n  -- xnor is _==_\n\n  module _ {A B C D E}\n    (f : A `\u00d7 B `\u2192 D)\n    (g : A `\u00d7 C `\u2192 E) where\n\n    dispatch : A `\u00d7 B `\u00d7 C `\u2192 D `\u00d7 E\n    dispatch = first dup \u204f inner2 swap \u204f < f \u00d7 g > \n\n  --fork c e0 e1 = (not c \u2227 e0) \u2228 (c \u2227 e1)\n  fork : `\ud835\udfda `\u00d7 `\ud835\udfda `\u00d7 `\ud835\udfda `\u2192 `\ud835\udfda\n  fork = dispatch (< not \u00d7 id > \u204f and) and \u204f or\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FunUniverse\/Nand.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"a2284fe9263eadf1deb8ce96940c9f0cc72949bb","subject":"Implement modal logic","message":"Implement modal logic\n\nThis doesn't belong here, but it reuses @Toxaris modular library.\n\nOld-commit-hash: 2f12ca949c8cb58e2b65b773607282bafbccb5c9\n","repos":"inc-lc\/ilc-agda","old_file":"ModalLogic.agda","new_file":"ModalLogic.agda","new_contents":"module ModalLogic where\n\n-- Implement Frank Pfenning's lambda calculus based on modal logic S4, as\n-- described in \"A Modal Analysis of Staged Computation\".\n\nopen import Denotational.Notation\n\ndata BaseType : Set where\n  Base : BaseType\n  Next : BaseType \u2192 BaseType\n\ndata Type : Set where\n  base : BaseType \u2192 Type\n  _\u21d2_ : Type \u2192 Type \u2192 Type\n  \u25a1_ : Type \u2192 Type\n\n-- Reuse contexts, variables and weakening from Tillmann's library.\nopen import Syntactic.Contexts Type\n\n-- No semantics for these types.\n\ndata Term : Context \u2192 Context \u2192 Type \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t : Term (\u03c4\u2081 \u2022 \u0393) \u0394 \u03c4\u2082) \u2192 Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)\n  app : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (t\u2081 : Term \u0393 \u0394 (\u03c4\u2081 \u21d2 \u03c4\u2082)) (t\u2082 : Term \u0393 \u0394 \u03c4\u2081) \u2192 Term \u0393 \u0394 \u03c4\u2082\n  ovar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  mvar : \u2200 {\u0393 \u0394 \u03c4} \u2192 (u : Var \u0394 \u03c4) \u2192 Term \u0393 \u0394 \u03c4\n  -- Note the builtin weakening\n  box : \u2200 {\u0393 \u0394 \u03c4} \u2192 Term \u2205 \u0394 \u03c4 \u2192 Term \u0393 \u0394 (\u25a1 \u03c4)\n  let-box_in-_ : \u2200 {\u03c4\u2081 \u03c4\u2082 \u0393 \u0394} \u2192 (e\u2081 : Term \u0393 \u0394 (\u25a1 \u03c4\u2081)) \u2192 (e\u2082 : Term \u0393 (\u03c4\u2081 \u2022 \u0394) \u03c4\u2082) \u2192 Term \u0393 \u0394 \u03c4\u2082\n\n-- Lemma 1 includes weakening.\n-- Most of the proof can be generated by C-c C-a, but syntax errors must then be fixed.\n\nweaken : \u2200 {\u0393\u2081 \u0393\u2082 \u0394\u2081 \u0394\u2082 \u03c4} \u2192 (\u0393\u2032 : \u0393\u2081 \u227c \u0393\u2082) \u2192 (\u0394\u2032 : \u0394\u2081 \u227c \u0394\u2082) \u2192 Term \u0393\u2081 \u0394\u2081 \u03c4 \u2192 Term \u0393\u2082 \u0394\u2082 \u03c4\nweaken \u0393\u2032 \u0394\u2032 (abs {\u03c4\u2081} t) = abs (weaken (keep \u03c4\u2081 \u2022 \u0393\u2032) \u0394\u2032 t)\nweaken \u0393\u2032 \u0394\u2032 (app t t\u2081) = app (weaken \u0393\u2032 \u0394\u2032 t) (weaken \u0393\u2032 \u0394\u2032 t\u2081)\nweaken \u0393\u2032 \u0394\u2032 (ovar x) = ovar (lift \u0393\u2032 x)\nweaken \u0393\u2032 \u0394\u2032 (mvar u) = mvar (lift \u0394\u2032 u)\nweaken \u0393\u2032 \u0394\u2032 (box t) = box (weaken \u2205 \u0394\u2032 t)\nweaken \u0393\u2032 \u0394\u2032 (let-box_in-_ {\u03c4\u2081} t t\u2081) =\n  let-box weaken \u0393\u2032 \u0394\u2032 t in-\n          weaken \u0393\u2032 (keep \u03c4\u2081 \u2022 \u0394\u2032) t\u2081\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ModalLogic.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"3333d586899fc6e8c61983d340e5fc4c35df66c5","subject":"Add file for choreography","message":"Add file for choreography\n","repos":"crypto-agda\/crypto-agda","old_file":"Control\/Protocol\/Choreography.agda","new_file":"Control\/Protocol\/Choreography.agda","new_contents":"\n-- {-# OPTIONS --without-K #-}\nopen import Coinduction\nopen import Function.NP\nopen import Type\nopen import Level\nopen import Data.Product\nopen import Data.One\n\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule Control.Protocol.Choreography where\nopen import Control.Strategy renaming (Strategy to Client) public\n\n\u03a0\u00b7 : \u2200 {a b}(A : \u2605_ a) \u2192 (B : ..(_ : A) \u2192 \u2605_ b) \u2192 \u2605_ (a \u2294 b)\n\u03a0\u00b7 A B = ..(x : A) \u2192 B x\n\ndata Mod : \u2605 where S D : Mod\n\n\u2192M : \u2200 {a b} \u2192 Mod \u2192 \u2605_ a \u2192 \u2605_ b \u2192 \u2605_ (a \u2294 b)\n\u2192M S A B = ..(_ : A) \u2192 B\n\u2192M D A B = A \u2192 B\n\n\u03a0M : \u2200 {a b}(m : Mod) \u2192 (A : \u2605_ a) \u2192 (B : \u2192M m A (\u2605_ b)) \u2192 \u2605_ (a \u2294 b)\n\u03a0M S A B = \u03a0\u00b7 A B\n\u03a0M D A B = \u03a0 A B\n\n-- appM : \u2200 {a b}(m : Mod){A : \u2605_ a}{B : \u2192M m A (\u2605_ b)}(P : \u03a0M m A B)(x : A) \u2192 B\n\ndata Proto : \u2605\u2081 where\n  end : Proto\n  \u03a0' \u03a3' : (f : Mod)(A : \u2605)(B : \u2192M f A Proto) \u2192 Proto\n\n{-\nTele : Proto \u2192 \u2605\nTele end = \ud835\udfd9\nTele (\u03a0' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (\u03a3' A B) = \u03a3 A \u03bb x \u2192 Tele (B x)\nTele (later i P) = ?\n\n_>>\u2261_ : (P : Proto) \u2192 (Tele P \u2192 Proto) \u2192 Proto\nend >>\u2261 Q = Q _\n\u03a0' A B >>\u2261 Q = \u03a0' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\n\u03a3' A B >>\u2261 Q = \u03a3' A (\u03bb x \u2192 B x >>\u2261 (\u03bb xs \u2192 Q (x , xs)))\nlater i P >>\u2261 Q = ?\n\n++Tele : \u2200 (P : Proto)(Q : Tele P \u2192 Proto) \u2192 (x : Tele P) \u2192 Tele (Q x) \u2192 Tele (P >>\u2261 Q)\n++Tele end Q x y = y\n++Tele (\u03a0' M C) Q (m , x) y = m , ++Tele (C m) (\u03bb x\u2081 \u2192 Q (m , x\u2081)) x y\n++Tele (\u03a3' M C) Q (m , x) y = m , ++Tele (C m) _ x y\n++Tele (later i P) Q x y = ?\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : \u2605_ b}{f g : A \u2192 B} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  right-unit : \u2200 (P : Proto) \u2192 (P >>\u2261 \u03bb x \u2192 end) \u2261 P\n  right-unit end = refl\n  right-unit (\u03a0' M C) = let p = funExt (\u03bb x \u2192 right-unit (C x)) in cong (\u03a0' M) p\n  right-unit (\u03a3' M C) = cong (\u03a3' M) (funExt (\u03bb x \u2192 right-unit (C x)))\n  right-unit (later i P) = ?\n\n  assoc : \u2200 (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (P >>\u2261 Q) \u2192 Proto)\n        \u2192 P >>\u2261 (\u03bb x \u2192 Q x >>\u2261 (\u03bb y \u2192 R (++Tele P Q x y))) \u2261 ((P >>\u2261 Q) >>\u2261 R)\n  assoc end Q R = refl\n  assoc (\u03a0' M C\u2081) Q R = cong (\u03a0' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (\u03a3' M C\u2081) Q R = cong (\u03a3' M) (funExt (\u03bb x \u2192 assoc (C\u2081 x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb xs \u2192 R (x , xs))))\n  assoc (later i P) Q R = ?\n\n_>>_ : Proto \u2192 Proto \u2192 Proto\nP >> Q = P >>\u2261 \u03bb _ \u2192 Q\n-}\n\n_\u00d7'_ : Set \u2192 Proto \u2192 Proto\nA \u00d7' B = \u03a3' D A \u03bb _ \u2192 B\n\n_\u2192'_ : Set \u2192 Proto \u2192 Proto\nA \u2192' B = \u03a0' D A \u03bb _ \u2192 B\n\ndual : Proto \u2192 Proto\ndual end = end\ndual (\u03a0' S A B) = \u03a3' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a0' D A B) = \u03a3' D A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' S A B) = \u03a0' S A (\u03bb x \u2192 dual (B x))\ndual (\u03a3' D A B) = \u03a0' D A (\u03bb x \u2192 dual (B x))\n\ndata Dual : Proto \u2192 Proto \u2192 \u2605\u2081 where\n  end : Dual end end\n  \u03a0\u03a3'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' S A B) (\u03a3' S A B')\n  \u03a0\u03a3'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a0' D A B) (\u03a3' D A B')\n  \u03a3\u03a0'S : \u2200 {A B B'} \u2192 (\u2200 ..x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' S A B) (\u03a0' S A B')\n  \u03a3\u03a0'D : \u2200 {A B B'} \u2192 (\u2200 x \u2192 Dual (B x) (B' x)) \u2192 Dual (\u03a3' D A B) (\u03a0' D A B')\n\nDual-sym : \u2200 {P Q} \u2192 Dual P Q \u2192 Dual Q P\nDual-sym end = end\nDual-sym (\u03a0\u03a3'S f) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a0\u03a3'D f) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'S f) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-sym (f x))\nDual-sym (\u03a3\u03a0'D f) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-sym (f x))\n\nDual-spec : \u2200 P \u2192 Dual P (dual P)\nDual-spec end = end\nDual-spec (\u03a0' S A B) = \u03a0\u03a3'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a0' D A B) = \u03a0\u03a3'D (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' S A B) = \u03a3\u03a0'S (\u03bb x \u2192 Dual-spec (B x))\nDual-spec (\u03a3' D A B) = \u03a3\u03a0'D (\u03bb x \u2192 Dual-spec (B x))\n{-\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  dual-Tele : \u2200 P \u2192 Tele P \u2261 Tele (dual P)\n  dual-Tele end = refl\n  dual-Tele (\u03a0' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (\u03a3' A B) = cong (\u03a3 A) (funExt (\u03bb x \u2192 dual-Tele (B x)))\n  dual-Tele (later i P) = ?\n-}{-\nmodule _ X where\n  El : Proto \u2192 \u2605\n  El end = X\n  El (\u03a0' A B) = \u03a0 A \u03bb x \u2192 El (B x)\n  El (\u03a3' A B) = \u03a3 A \u03bb x \u2192 El (B x)\nmodule _ where\n  El : (P : Proto) \u2192 (Tele P \u2192 \u2605) \u2192 \u2605\n  El end X = X _\n  El (\u03a0' A B) X = \u03a0 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (\u03a3' A B) X = \u03a3 A \u03bb x \u2192 El (B x) (\u03bb y \u2192 X (x , y))\n  El (later i P) = ?\n\nmodule ElBind (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n\n  bind-spec : (P : Proto)(Q : Tele P \u2192 Proto)(X : Tele (P >>\u2261 Q) \u2192 \u2605) \u2192 El (P >>\u2261 Q) X \u2261 El P (\u03bb x \u2192 El (Q x) (\u03bb y \u2192 X (++Tele P Q x y)))\n  bind-spec end Q X = refl\n  bind-spec (\u03a0' A B) Q X = cong (\u03a0 A) (funExt (\u03bb x \u2192 bind-spec (B x) (\u03bb xs \u2192 Q (x , xs)) (\u03bb y \u2192 X (x , y))))\n  bind-spec (\u03a3' A B) Q X = cong (\u03a3 A) (funExt (\u03bb x \u2192 bind-spec (B x) _ _))\n  bind-spec (later i p) Q X = ?\n\n\nmodule _ {A B} where\n  com : (P : Proto) \u2192 El P (const A) \u2192 El (dual P) (const B) \u2192 A \u00d7 B\n  com end a b = a , b\n  com (\u03a0' A B) f (x , y) = com (B x) (f x) y\n  com (\u03a3' A B) (x , y) f = com (B x) y (f x)\n  com (later i P) x y = ?\n\nmodule _ (funExt : \u2200 {a b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)where\n  com-cont : (P : Proto){A : Tele P \u2192 \u2605}{B : Tele (dual P) \u2192 \u2605} \u2192 El P A \u2192 El (dual P) B \u2192 \u03a3 (Tele P) A \u00d7 \u03a3 (Tele (dual P)) B\n  com-cont end p q = (_ , p)  , (_ , q)\n  com-cont (\u03a0' A B) p (m , q) with com-cont (B m) (p m) q\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (\u03a3' A B) (m , p) q with com-cont (B m) p (q m)\n  ... | (X , a) , (Y , b) = ((m , X) , a) , (m , Y) , b\n  com-cont (later i P) p q = ?\n-}\n\ndata ProcessF (this : Proto \u2192 \u2605\u2081): Proto \u2192 \u2605\u2081 where\n  recvD : \u2200 {M P} \u2192 (\u03a0M D M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' D M P)\n  recvS : \u2200 {M P} \u2192 (\u03a0M S M \u03bb m \u2192 this (P m)) \u2192 ProcessF this (\u03a0' S M P)\n  sendD : \u2200 {M P} \u2192 \u03a0M D M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' D M P))\n  sendS : \u2200 {M P} \u2192 \u03a0M S M (\u03bb m \u2192 this (P m) \u2192 ProcessF this (\u03a3' S M P))\n\ndata Process (A : \u2605) : Proto \u2192 \u2605\u2081 where\n  do  : \u2200 {P} \u2192 ProcessF (Process A) P \u2192 Process A P\n  end : A \u2192 Process A end\n\nmutual\n  SimL : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimL P Q = ProcessF (flip Sim Q) P\n\n  SimR : Proto \u2192 Proto \u2192 \u2605\u2081\n  SimR P Q = ProcessF (Sim P) Q\n\n  data Sim : Proto \u2192 Proto \u2192 \u2605\u2081 where\n    left  : \u2200 {P Q} \u2192 SimL P Q \u2192 Sim P Q\n    right : \u2200 {P Q} \u2192 SimR P Q \u2192 Sim P Q\n    end   : Sim end end\n\n_\u27f9_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9 Q = Sim (dual P) Q\n\n_\u27f9\u1d3e_ : Proto \u2192 Proto \u2192 \u2605\u2081\nP \u27f9\u1d3e Q = \u2200 {A} \u2192 Process A (dual P) \u2192 Process A Q\n\nsim-id : \u2200 P \u2192 Sim (dual P) P\nsim-id end = end\nsim-id (\u03a0' S A B) = right (recvS (\u03bb x \u2192 left (sendS x (sim-id (B x)))))\nsim-id (\u03a0' D A B) = right (recvD (\u03bb x \u2192 left (sendD x (sim-id (B x)))))\nsim-id (\u03a3' S A B) = left (recvS (\u03bb x \u2192 right (sendS x (sim-id (B x)))))\nsim-id (\u03a3' D A B) = left (recvD (\u03bb x \u2192 right (sendD x (sim-id (B x)))))\n\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-compRL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimR P Q \u2192 SimL Q' R \u2192 Sim P R\nsim-compL : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 SimL P Q \u2192 Sim Q' R \u2192 SimL P R\nsim-compR : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 SimR Q' R \u2192 SimR P R\n\nsim-comp Q-Q' (left x) QR = left (sim-compL Q-Q' x QR)\nsim-comp Q-Q' (right x) (left x\u2081) = sim-compRL Q-Q' x x\u2081\nsim-comp Q-Q' (right x) (right x\u2081) = right (sim-compR Q-Q' (right x) x\u2081)\nsim-comp end (right x) end = right x\nsim-comp end end QR = QR\n\nsim-compRL end () QR\nsim-compRL (\u03a0\u03a3'S x\u2081) (recvS x) (sendS x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a0\u03a3'D x\u2081) (recvD x) (sendD x\u2082 x\u2083) = sim-comp (x\u2081 x\u2082) (x x\u2082) x\u2083\nsim-compRL (\u03a3\u03a0'S x) (sendS x\u2081 x\u2082) (recvS x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\nsim-compRL (\u03a3\u03a0'D x) (sendD x\u2081 x\u2082) (recvD x\u2083) = sim-comp (x x\u2081) x\u2082 (x\u2083 x\u2081)\n\nsim-compL Q-Q' (recvD PQ) QR = recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (recvS PQ) QR = recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR)\nsim-compL Q-Q' (sendD m PQ) QR = sendD m (sim-comp Q-Q' PQ QR)\nsim-compL Q-Q' (sendS m PQ) QR = sendS m (sim-comp Q-Q' PQ QR)\n\nsim-compR Q-Q' PQ (recvD QR)   = recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m))\nsim-compR Q-Q' PQ (recvS QR)   = recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x))\nsim-compR Q-Q' PQ (sendD m QR) = sendD m (sim-comp Q-Q' PQ QR)\nsim-compR Q-Q' PQ (sendS m QR) = sendS m (sim-comp Q-Q' PQ QR)\n\n{-\nsim-comp : \u2200 {P Q Q' R} \u2192 Dual Q Q' \u2192 Sim P Q \u2192 Sim Q' R \u2192 Sim P R\nsim-comp Q-Q' (left (recvD PQ)) QR = left (recvD (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (recvS PQ)) QR = left (recvS (\u03bb m \u2192 sim-comp Q-Q' (PQ m) QR))\nsim-comp Q-Q' (left (sendD m PQ)) QR = left (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' (left (sendS m PQ)) QR = left (sendS m (sim-comp Q-Q' PQ QR))\nsim-comp end (right ()) (left x\u2081)\nsim-comp end end QR = QR\nsim-comp end PQ end = PQ\nsim-comp (\u03a0\u03a3'S Q-Q') (right (recvS PQ)) (left (sendS m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a0\u03a3'D Q-Q') (right (recvD PQ)) (left (sendD m QR)) = sim-comp (Q-Q' m) (PQ m) QR\nsim-comp (\u03a3\u03a0'S Q-Q') (right (sendS m PQ)) (left (recvS QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp (\u03a3\u03a0'D Q-Q') (right (sendD m PQ)) (left (recvD QR)) = sim-comp (Q-Q' m) PQ (QR m)\nsim-comp Q-Q' PQ (right (recvD QR)) = right (recvD (\u03bb m \u2192 sim-comp Q-Q' PQ (QR m)))\nsim-comp Q-Q' PQ (right (recvS QR)) = right (recvS (\u03bb x \u2192 sim-comp Q-Q' PQ (QR x)))\nsim-comp Q-Q' PQ (right (sendD m QR)) = right (sendD m (sim-comp Q-Q' PQ QR))\nsim-comp Q-Q' PQ (right (sendS m QR)) = right (sendS m (sim-comp Q-Q' PQ QR))\n-}\n\n_\u2666_ : \u2200 {P Q R} \u2192 Sim P Q \u2192 Sim (dual Q) R \u2192 Sim P R\n_\u2666_ = sim-comp (Dual-spec _)\n\n\u27f9-comp : \u2200 {P Q R} \u2192 P \u27f9 Q \u2192 Q \u27f9 R \u2192 P \u27f9 R\n\u27f9-comp = _\u2666_\n\nsim-sym : \u2200 {P Q} \u2192 Sim P Q \u2192 Sim Q P\nsim-symL : \u2200 {P Q} \u2192 SimL P Q \u2192 SimR Q P\nsim-symR : \u2200 {P Q} \u2192 SimR P Q \u2192 SimL Q P\n\nsim-sym (left x) = right (sim-symL x)\nsim-sym (right x) = left (sim-symR x)\nsim-sym end = end\n\nsim-symL (recvD PQ)   = recvD (\u03bb m \u2192 sim-sym (PQ m))\nsim-symL (recvS PQ)   = recvS (\u03bb m \u2192 sim-sym (PQ m))\nsim-symL (sendD m PQ) = sendD m (sim-sym PQ)\nsim-symL (sendS m PQ) = sendS m (sim-sym PQ)\n\nsim-symR (recvD PQ)   = recvD (\u03bb m \u2192 sim-sym (PQ m))\nsim-symR (recvS PQ)   = recvS (\u03bb m \u2192 sim-sym (PQ m))\nsim-symR (sendD m PQ) = sendD m (sim-sym PQ)\nsim-symR (sendS m PQ) = sendS m (sim-sym PQ)\n\nsim-unit : \u2200 {P} \u2192 Sim end P \u2192 Process \ud835\udfd9 P\nsim-unit (left ())\nsim-unit (right (recvD P)) = do (recvD (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (recvS P)) = do (recvS (\u03bb m \u2192 sim-unit (P m)))\nsim-unit (right (sendD m P)) = do (sendD m (sim-unit P))\nsim-unit (right (sendS m P)) = do (sendS m (sim-unit P))\nsim-unit end = end 0\u2081\n\nmodule _ {P Q : Proto} where\n  infix 2 _\u223c_\n  _\u223c_ : (PQ PQ' : Sim P Q) \u2192 \u2605\u2081\n  PQ \u223c PQ' = \u2200 {P'} \u2192 (P'-P : Dual P' P) \u2192 (\u00f8P : Sim end P')\n           \u2192 sim-comp P'-P \u00f8P PQ \u2261 sim-comp P'-P \u00f8P PQ'\n\nmodule _\n  (funExtD : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  (funExtS : \u2200 {a}{b}{A : \u2605_ a}{B : ..(_ : A) \u2192 \u2605_ b}{f g : ..(x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g)\n  where\n  sim-comp-assoc-end : \u2200 {P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (\u00f8P : Sim end P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' \u00f8P PQ) QR\n    \u2261 sim-comp P-P' \u00f8P (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc-end P-P' Q-Q' (left ()) PQ QR\n  sim-comp-assoc-end end Q-Q' (right ()) (left PQ) QR\n  sim-comp-assoc-end (\u03a0\u03a3'S x\u2081) Q-Q' (right (recvS x)) (left (sendS x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a0\u03a3'D x\u2081) Q-Q' (right (recvD x)) (left (sendD x\u2082 x\u2083)) QR\n    = sim-comp-assoc-end (x\u2081 x\u2082) Q-Q' (x x\u2082) x\u2083 QR\n  sim-comp-assoc-end (\u03a3\u03a0'S x) Q-Q' (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end (\u03a3\u03a0'D x) Q-Q' (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083)) QR\n    = sim-comp-assoc-end (x x\u2081) Q-Q' x\u2082 (x\u2083 x\u2081) QR\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) (left x\u2081)\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'S x\u2081) (right \u00f8P) (right (recvS x)) (left (sendS x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a0\u03a3'D x\u2081) (right \u00f8P) (right (recvD x)) (left (sendD x\u2082 x\u2083))\n    = sim-comp-assoc-end P-P' (x\u2081 x\u2082) (right \u00f8P) (x x\u2082) x\u2083\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'S x) (right \u00f8P) (right (sendS x\u2081 x\u2082)) (left (recvS x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' (\u03a3\u03a0'D x) (right \u00f8P) (right (sendD x\u2081 x\u2082)) (left (recvD x\u2083))\n    = sim-comp-assoc-end P-P' (x x\u2081) (right \u00f8P) x\u2082 (x\u2083 x\u2081)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvD x\u2081))\n    = cong (right \u2218 recvD) (funExtD (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (recvS x\u2081))\n    = cong (right \u2218 recvS) (funExtS (\u03bb m \u2192 sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (x\u2081 m)))\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendD x\u2081 x\u2082))\n    = cong (right \u2218 sendD x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) (right (sendS x\u2081 x\u2082))\n    = cong (right \u2218 sendS x\u2081) (sim-comp-assoc-end P-P' Q-Q' (right \u00f8P) (right x) x\u2082)\n  sim-comp-assoc-end P-P' end (right \u00f8P) (right ()) end\n  sim-comp-assoc-end end end (right \u00f8P) end QR = refl\n  sim-comp-assoc-end end Q-Q' end PQ QR = refl\n\n  \u2666-assoc-end : \u2200 {P Q R}(\u00f8P : Sim end P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (\u00f8P \u2666 PQ) \u2666 QR \u2261 \u00f8P \u2666 (PQ \u2666 QR)\n  \u2666-assoc-end = sim-comp-assoc-end (Dual-spec _) (Dual-spec _)\n\n  open \u2261-Reasoning\n  sim-comp-assoc : \u2200 {W P P' Q Q' R}(P-P' : Dual P P')(Q-Q' : Dual Q Q')\n    (WP : Sim W P)(PQ : Sim P' Q)(QR : Sim Q' R)\n    \u2192 sim-comp Q-Q' (sim-comp P-P' WP PQ) QR\n    \u223c sim-comp P-P' WP (sim-comp Q-Q' PQ QR)\n  sim-comp-assoc P-P' Q-Q' WP PQ QR {W'} W'-W \u00f8W'\n    = sim-comp W'-W \u00f8W' (sim-comp Q-Q' (sim-comp P-P' WP PQ) QR)\n    \u2261\u27e8 sym (sim-comp-assoc-end W'-W Q-Q' \u00f8W' (sim-comp P-P' WP PQ) QR) \u27e9\n      sim-comp Q-Q' (sim-comp W'-W \u00f8W' (sim-comp P-P' WP PQ)) QR\n    \u2261\u27e8 cong (\u03bb X \u2192 sim-comp Q-Q' X QR) (sym (sim-comp-assoc-end W'-W P-P' \u00f8W' WP PQ)) \u27e9\n      sim-comp Q-Q' (sim-comp P-P' (sim-comp W'-W \u00f8W' WP) PQ) QR\n    \u2261\u27e8 sim-comp-assoc-end P-P' Q-Q' (sim-comp W'-W \u00f8W' WP) PQ QR \u27e9\n      sim-comp P-P' (sim-comp W'-W \u00f8W' WP) (sim-comp Q-Q' PQ QR)\n    \u2261\u27e8 sim-comp-assoc-end W'-W P-P' \u00f8W' WP (sim-comp Q-Q' PQ QR) \u27e9\n      sim-comp W'-W \u00f8W' (sim-comp P-P' WP (sim-comp Q-Q' PQ QR))\n    \u220e\n\n  \u2666-assoc : \u2200 {W P Q R}(WP : Sim W P)(PQ : Sim (dual P) Q)(QR : Sim (dual Q) R)\n    \u2192 (WP \u2666 PQ) \u2666 QR \u223c WP \u2666 (PQ \u2666 QR)\n  \u2666-assoc = sim-comp-assoc (Dual-spec _) (Dual-spec _)\n\n\nsim-comp-! : \u2200 {P Q Q' R}(Q-Q' : Dual Q Q')(PQ : Sim P Q)(Q'R : Sim Q' R)\n  \u2192 sim-comp (Dual-sym Q-Q') (sim-sym Q'R) (sim-sym PQ) \u2261 sim-sym (sim-comp Q-Q' PQ Q'R)\nsim-comp-! Q-Q' (left (recvD x)) (left x\u2081) = {!!}\nsim-comp-! Q-Q' (left (recvS x)) (left x\u2081) = {!!}\nsim-comp-! Q-Q' (left (sendD x x\u2081)) (left x\u2082) = cong (right \u2218 sendD x) (sim-comp-! Q-Q' x\u2081 (left x\u2082))\nsim-comp-! Q-Q' (left (sendS x x\u2081)) (left x\u2082) = cong (right \u2218 sendS x) (sim-comp-! Q-Q' x\u2081 (left x\u2082))\nsim-comp-! Q-Q' (left x) (right x\u2081) = {!!}\nsim-comp-! Q-Q' (left x) end = {!!}\nsim-comp-! Q-Q' (right x) QR = {!!}\nsim-comp-! end end (left x) = refl\nsim-comp-! end end (right x) = {!!}\nsim-comp-! end end end = refl\n{-\n\nunit-sim : \u2200 {P} \u2192 Process \ud835\udfd9 P \u2192 Sim end P\nunit-sim (do (send m x)) = right (send m (unit-sim x))\nunit-sim (do (recv x)) = right (recv (\u03bb m \u2192 unit-sim (x m)))\nunit-sim (end x) = end\n\n{-\ntoEl : \u2200 {P A} \u2192 Process A P \u2192 El P (const A)\ntoEl (end x) = x\ntoEl (do (recv f)) = \u03bb x \u2192 toEl (f x)\ntoEl (do (send m x)) = m , toEl x\n-}\n\nidP : \u2200 {A} \u2192 Process A (\u03a0' A (const end))\nidP = do (recv end)\n\ndual\u00b2 : \u2200 {P A} \u2192 Process A P \u2192 Process A (dual (dual P))\ndual\u00b2 (end x) = end x\ndual\u00b2 (do (recv x)) = do (recv (\u03bb m \u2192 dual\u00b2 (x m)))\ndual\u00b2 (do (send m x)) = do (send m (dual\u00b2 x))\n\napply-sim : \u2200 {P Q} \u2192 Sim P Q \u2192 P \u27f9\u1d3e Q\napply-sim (left (send m x)) (do (recv x\u2081)) = apply-sim x (x\u2081 m)\napply-sim (left (recv x)) (do (send m x\u2081)) = apply-sim (x m) x\u2081\napply-sim (right (send m x)) P\u2082 = do (send m (apply-sim x P\u2082))\napply-sim (right (recv x)) P\u2082 = do (recv (\u03bb m \u2192 apply-sim (x m) P\u2082))\napply-sim end P = P\n\napply-sim' : \u2200 {P Q} \u2192 Sim P Q \u2192 Q \u27f9\u1d3e P -- \u2200 {A} \u2192 Process A Q \u2192 Process A (dual P)\napply-sim' PQ P = apply-sim (sim-sym PQ) P\n\napply : \u2200 {P Q A} \u2192 P \u27f9 Q \u2192 Process A P \u2192 Process A Q\napply PQ P = apply-sim PQ (dual\u00b2 P)\n\nmodule _ (funExt : \u2200 {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b}{f g : (x : A) \u2192 B x} \u2192 (\u2200 x \u2192 f x \u2261 g x) \u2192 f \u2261 g) where\n  apply-comp : \u2200 {P Q R A}(PQ : Sim P Q)(QR : Sim (dual Q) R)(p : Process A (dual P)) \u2192 apply-sim (sim-comp PQ QR) p \u2261 apply QR (apply-sim PQ p)\n  apply-comp (left (send m x)) QR (do (recv x\u2081)) = apply-comp x QR (x\u2081 m)\n  apply-comp (left (recv x)) QR (do (send m x\u2081)) = apply-comp (x m) QR x\u2081\n  apply-comp (right (send m x)) (left (recv x\u2081)) p = apply-comp x (x\u2081 m) p\n  apply-comp (right (send m x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m' \u2192 apply-comp (right (send m x)) (x\u2081 m') p))\n  apply-comp (right (send m x)) (right (send m\u2081 x\u2081)) p\n    rewrite apply-comp (right (send m x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (left (send m x\u2081)) p = apply-comp (x m) x\u2081 p\n  apply-comp (right (recv x)) (right (send m x\u2081)) p\n    rewrite apply-comp (right (recv x)) x\u2081 p = refl\n  apply-comp (right (recv x)) (right (recv x\u2081)) p = cong (\u03bb X \u2192 do (recv X))\n    (funExt (\u03bb m \u2192 apply-comp (right (recv x)) (x\u2081 m) p))\n  apply-comp end QR (do ())\n  apply-comp end QR (end x) = refl\n\n{-\n_>>=P_ : \u2200 {A B P}{Q : Tele P \u2192 Proto} \u2192 Process A P \u2192 ((p : Tele P) \u2192 A \u2192 Process B (Q p)) \u2192 Process B (P >>\u2261 Q)\nsend m p >>=P k = send m (p >>=P (\u03bb p\u2081 \u2192 k (m , p\u2081)))\nrecv x >>=P k = recv (\u03bb m \u2192 x m >>=P (\u03bb p \u2192 k (m , p)))\nend x >>=P k = k 0\u2081 x\n\n\n  {-\nmodule _ where\n  bind-com : (P : Proto)(Q : Tele P \u2192 Proto)(R : Tele (dual P) \u2192 Proto)\n    (X : Tele (P >>\u2261 Q) \u2192 \u2605)(Y : Tele (dual P >>\u2261 R) \u2192 \u2605)\n    \u2192 El (P >>\u2261 Q) X \u2192 El (dual P >>\u2261 R) Y \u2192 ? \u00d7 ?\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Control\/Protocol\/Choreography.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c38b63b28862e0514cc3ec01c09e19ff4217233a","subject":"ZK\/GroupHom.agda","message":"ZK\/GroupHom.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom.agda","new_file":"ZK\/GroupHom.agda","new_contents":"open import Type using (Type)\nopen import Function using (flip)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum.NP\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import ZK.Types using (Cyclic-group; module Cyclic-group\n                           ; Cyclic-group-properties; module Cyclic-group-properties)\nimport ZK.SigmaProtocol.KnownStatement\n\n\nmodule ZK.GroupHom where\npostulate\n  G+ G* : Type\n  -- grp-G+-comm : Abelian-Group G+\n  grp-G+ : Group G+\n  -- grp-G* : Group G*\n  grp-G*-comm : Abelian-Group G*\nmodule _\n--     {G+ G* : Type} (grp-G+ : Group G+) (grp-G* : Group G*)\n\n    {C : Type} -- \u2124\n    (\u03c6 : G+ \u2192 G*) -- For instance \u03c6(x) = g^x\n    (y : G*)\n    (_==_ : G* \u2192 G* \u2192 Bool)\n    (_^_ : G* \u2192 C \u2192 G*)\n    (_\u2297_ : G+ \u2192 C \u2192 G+)\n\n    (_\u2212\u1d9c_ : C \u2192 C \u2192 C)\n    {-\n    -- Coprime m n \u2194 GCD m n 1\n    (Coprime : (m n : C) \u2192 Type)\n    (_<_ : C \u2192 C \u2192 Type)\n    (\u2262-< : \u2200 {c\u2080 c\u2081} \u2192 c\u2080 \u2262 c\u2081 \u2192 (c\u2080 < c\u2081) \u228e (c\u2081 < c\u2080))\n\n    (\u2113 : C)\n    (u : G+)(\u03c6u : \u03c6 u \u2261 y ^ \u2113)\n    (<-coprime : \u2200 {c\u2080 c\u2081} \u2192 c\u2081 < c\u2080 \u2192 Coprime (c\u2080 \u2212\u1d9c c\u2081) \u2113)\n    (egcd : (m n : C) \u2192 C \u00d7 C)\n    -- (egcd-coprime : \u2200 m n \u2192 let a , b = egcd m n in {!!})\n    -}\n    (modinv : C \u2192 C)\n    where\n\n  -- grp-G+ = Abelian-Group.grp grp-G+-comm\n  -- open Additive-Abelian-Group grp-G+-comm\n  open Additive-Group grp-G+\n  grp-G* = Abelian-Group.grp grp-G*-comm\n  -- open Multiplicative-Group grp-G*\n  open Multiplicative-Abelian-Group grp-G*-comm\n\n  Commitment = G*\n  Challenge  = C\n  Response   = G+\n  Witness    = G+\n  Randomness = G+\n  _\u2208y : Witness \u2192 Type\n  x \u2208y = \u03c6 x \u2261 y\n\n  open ZK.SigmaProtocol.KnownStatement Commitment Challenge Response Randomness Witness _\u2208y public\n\n  prover : Prover\n  prover a x = (\u03c6 a) , response\n     where response : Challenge \u2192 Response\n           response c = (a + (x \u2297 c))\n\n  verifier : Verifier\n  verifier (mk A c s) = (\u03c6 s) == (A * (y ^ c))\n\n  Schnorr : \u03a3-Protocol\n  Schnorr = (prover , verifier)\n\n  -- The simulator shows why it is so important for the\n  -- challenge to be picked once the commitment is known.\n  -- To fake a transcript, the challenge and response can\n  -- be arbitrarily chosen. However to be indistinguishable\n  -- from a valid proof they need to be picked at random.\n  simulator : Simulator\n  simulator c s = A\n      where\n        -- Compute A, such that the verifier accepts!\n        A = (\u03c6 s) \/ (y ^ c)\n\n  -- The extractor shows the importance of never reusing a\n  -- commitment. If the prover answers to two different challenges\n  -- comming from the same commitment then the knowledge of the\n  -- prover (the witness) can be extracted.\n  extractor : Extractor verifier\n  extractor t\u00b2 = x'\n      module Witness-extractor where\n        open Transcript\u00b2 t\u00b2\n        rd = get-f\u2080 \u2212 get-f\u2081\n        cd = get-c\u2080 \u2212\u1d9c get-c\u2081\n        {-\n        ab = egcd cd \u2113\n        a = fst ab\n        b = snd ab\n        x' = u \u2297 a + rd \u2297 b\n        -}\n        x' = rd \u2297 modinv cd\n  \n  module Proofs\n    (\u03c6-hom : GroupHomomorphism grp-G+ grp-G* \u03c6)\n    (\u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y))\n    (==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y)\n    (\u03c6\u207b\u00b9 : G* \u2192 G+)\n    .(\u03c6\u2218\u03c6\u207b\u00b9 : \u2200 y \u2192 \u03c6 (\u03c6\u207b\u00b9 y) \u2261 y)\n    (\u03c6-inj : \u2200 {x y} \u2192 \u03c6 x \u2261 \u03c6 y \u2192 x \u2261 y)\n    (\u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297 c) \u2261 \u03c6 x ^ c)\n    (^--  : \u2200 {b x y} \u2192 b ^(x \u2212\u1d9c y) \u2261 (b ^ x) \/ (b ^ y))\n    (left-*-to-right-\/ : \u2200 {x y c} \u2192 x \u2297 c \u2261 y \u2192 x \u2261 y \u2297 modinv c)\n     where\n    open \u2261-Reasoning\n    open GroupHomomorphismProp grp-G+ grp-G* {\u03c6} \u03c6-hom\n\n    correct : Correct Schnorr\n    correct x a c w\n      = \u2713-== (\u03c6(a + (x \u2297 c))\n           \u2261\u27e8 \u03c6-hom \u27e9\n              A * (\u03c6(x \u2297 c))\n           \u2261\u27e8 ap (\u03bb z \u2192 A * z) \u03c6-hom-iterated \u27e9\n              A * ((\u03c6 x) ^ c)\n           \u2261\u27e8 ap (\u03bb z \u2192 A * (z ^ c)) w \u27e9\n              A * (y ^ c)\n           \u220e)\n      where\n        A = \u03c6 a\n\n    shvzk : Special-Honest-Verifier-Zero-Knowledge Schnorr\n    shvzk = record { simulator = simulator\n                   ; correct-simulator = \u03bb _ _ \u2192 \u2713-== \/-* }\n\n    module _ (t\u00b2 : Transcript\u00b2 verifier) where\n      private\n        x = \u03c6\u207b\u00b9 y\n\n      open Transcript\u00b2 t\u00b2 renaming (get-A to A; get-c\u2080 to c\u2080; get-c\u2081 to c\u2081\n                                   ;get-f\u2080 to f\u2080; get-f\u2081 to f\u2081)\n      open Witness-extractor t\u00b2\n\n      .\u03c6xcd\u2261\u03c6rd : _\n      \u03c6xcd\u2261\u03c6rd = \u03c6(x \u2297 cd)\n                 \u2261\u27e8 \u03c6-hom-iterated \u27e9\n                   (\u03c6 x) ^ (c\u2080 \u2212\u1d9c c\u2081)\n                 \u2261\u27e8 ap\u2082 _^_ (\u03c6\u2218\u03c6\u207b\u00b9 y) idp \u27e9\n                   y ^ (c\u2080 \u2212\u1d9c c\u2081)\n                 \u2261\u27e8 ^-- \u27e9\n                   (y ^ c\u2080) \/ (y ^ c\u2081)\n                 \u2261\u27e8 ! cancels-\/ \u27e9\n                   (A * (y ^ c\u2080)) \/ (A * (y ^ c\u2081))\n                 \u2261\u27e8 \/= (! ==-\u2713 verify\u2080) (! ==-\u2713 verify\u2081) \u27e9\n                   (\u03c6 f\u2080) \/ (\u03c6 f\u2081)\n                 \u2261\u27e8 ! f-\u2212-\/ \u27e9\n                   \u03c6 rd\n                 \u220e\n\n      .xcd\u2261rd : _\n      xcd\u2261rd = \u03c6-inj \u03c6xcd\u2261\u03c6rd\n\n      -- The extracted x is correct\n      .extractor-ok : \u03c6 x' \u2261 y \n      extractor-ok = ! ap \u03c6 (left-*-to-right-\/ xcd\u2261rd) \u2219 \u03c6\u2218\u03c6\u207b\u00b9 y\n\n    special-soundness : Special-Soundness Schnorr\n    special-soundness = record { extractor = extractor\n                               ; extract-valid-witness = extractor-ok }\n\n    special-\u03a3-protocol : Special-\u03a3-Protocol\n    special-\u03a3-protocol = record { \u03a3-protocol = Schnorr ; correct = correct ; shvzk = shvzk ; ssound = special-soundness }\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/GroupHom.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"446cedbaaf03389bf155e5ee9e658f7bc4eb4d3a","subject":"add hello world app","message":"add hello world app\n","repos":"nfjinjing\/chu2","old_file":"src\/Hello.agda","new_file":"src\/Hello.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"remote: Support for password authentication was removed on August 13, 2021.\nremote: Please see https:\/\/docs.github.com\/en\/get-started\/getting-started-with-git\/about-remote-repositories#cloning-with-https-urls for information on currently recommended modes of authentication.\nfatal: Authentication failed for 'https:\/\/github.com\/nfjinjing\/chu2.git\/'\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"33c1a921bdd890151107191d88840ca8c646315b","subject":"lib\/HoTT.agda","message":"lib\/HoTT.agda\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/HoTT.agda","new_file":"lib\/HoTT.agda","new_contents":"{-# OPTIONS --without-K #-}\nmodule HoTT where\n\nopen import Type\nopen import Level.NP\nopen import Function.NP\nopen import Function.Extensionality\nopen import Data.Zero using (\ud835\udfd8)\nopen import Data.One using (\ud835\udfd9)\nopen import Data.Product.NP renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum using (_\u228e_) renaming (inj\u2081 to inl; inj\u2082 to inr; [_,_] to [inl:_,inr:_])\nopen import Relation.Binary using (Reflexive; Symmetric; Transitive)\nimport Relation.Binary.PropositionalEquality.NP as \u2261\nopen \u2261 using (_\u2261_; ap; coe; coe!; !_; _\u2219_; J) renaming (subst to tr; refl to idp; cong\u2082 to ap\u2082)\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\nopen import Function.Related.TypeIsomorphisms.NP hiding (\u03a3-assoc)\n\n-- Contractible\nmodule _ {a}(A : \u2605_ a) where\n    Is-contr : \u2605_ a\n    Is-contr = \u03a3 A \u03bb x \u2192 \u2200 y \u2192 x \u2261 y\n\nmodule _ {a}{b}{A : \u2605_ a}{B : A \u2192 \u2605_ b} where\n    pair= : \u2200 {x y : \u03a3 A B} \u2192 (p : fst x \u2261 fst y) \u2192 tr B p (snd x) \u2261 snd y \u2192 x \u2261 y\n    pair= idp = ap (_,_ _)\n    snd= : \u2200 {x : A} {y y' : B x} \u2192 y \u2261 y' \u2192 _\u2261_ {A = \u03a3 A B} (x , y) (x , y')\n    snd= = pair= idp\nmodule _ {a}{b}{A : \u2605_ a}{B : \u2605_ b} where\n    pair\u00d7= : \u2200 {x x' : A}(p : x \u2261 x')\n               {y y' : B}(q : y \u2261 y')\n             \u2192 (x , y) \u2261 (x' , y')\n    pair\u00d7= idp q = snd= q\n\nmodule _ {a}(A : \u2605_ a){b}{B\u2080 B\u2081 : A \u2192 \u2605_ b}(B : (x : A) \u2192 B\u2080 x \u2261 B\u2081 x){{_ : FunExt}} where\n    \u03a3=\u2032 : \u03a3 A B\u2080 \u2261 \u03a3 A B\u2081\n    \u03a3=\u2032 = ap (\u03a3 A) (\u03bb= B)\n\n    \u03a0=\u2032 : \u03a0 A B\u2080 \u2261 \u03a0 A B\u2081\n    \u03a0=\u2032 = ap (\u03a0 A) (\u03bb= B)\n\nmodule _ {{_ : FunExt}} where\n    \u03a3= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}\n           (A : A\u2080 \u2261 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3= idp B = \u03a3=\u2032 _ B\n\n    \u03a0= : \u2200 {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b}\n           (A : A\u2080 \u2261 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (coe A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0= idp B = \u03a0=\u2032 _ B\n\nmodule _ {a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 B\u2081 : \u2605_ b}(A= : A\u2080 \u2261 A\u2081)(B= : B\u2080 \u2261 B\u2081) where\n    \u00d7= : (A\u2080 \u00d7 B\u2080) \u2261 (A\u2081 \u00d7 B\u2081)\n    \u00d7= = ap\u2082 _\u00d7_ A= B=\n\n    \u228e= : (A\u2080 \u228e B\u2080) \u2261 (A\u2081 \u228e B\u2081)\n    \u228e= = ap\u2082 _\u228e_ A= B=\n\nmodule Equivalences where\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    _LeftInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ a\n    linv LeftInverseOf f = \u2200 x \u2192 linv (f x) \u2261 x\n\n    _RightInverseOf_ : (B \u2192 A) \u2192 (A \u2192 B) \u2192 \u2605_ b\n    rinv RightInverseOf f = \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Linv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n\n    record Rinv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        rinv : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n    record Is-equiv (f : A \u2192 B) : \u2605_(a \u2294 b) where\n      field\n        linv : B \u2192 A\n        is-linv : \u2200 x \u2192 linv (f x) \u2261 x\n        rinv : B \u2192 A\n        is-rinv : \u2200 x \u2192 f (rinv x) \u2261 x\n\n      injective : \u2200 {x y} \u2192 f x \u2261 f y \u2192 x \u2261 y\n      injective {x} {y} p = !(is-linv x) \u2219 ap linv p \u2219 is-linv y\n\n      surjective : \u2200 y \u2192 \u2203 \u03bb x \u2192 f x \u2261 y\n      surjective y = rinv y , is-rinv y\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}{f : A \u2192 B}(f\u1d31 : Is-equiv f) where\n      open Is-equiv f\u1d31\n      inv : B \u2192 A\n      inv = linv \u2218 f \u2218 rinv\n\n      inv-is-equiv : Is-equiv inv\n      inv-is-equiv = record { linv = f\n                         ; is-linv = \u03bb x \u2192 ap f (is-linv (rinv x)) \u2219 is-rinv x\n                         ; rinv = f\n                         ; is-rinv = \u03bb x \u2192 ap linv (is-rinv (f x)) \u2219 is-linv x }\n\n  module _ {a}{A : \u2605_ a}{f : A \u2192 A}(f-inv : f LeftInverseOf f) where\n      self-inv-is-equiv : Is-equiv f\n      self-inv-is-equiv = record { linv = f ; is-linv = f-inv ; rinv = f ; is-rinv = f-inv }\n\n  module _ {a}{A : \u2605_ a} where\n    id\u1d31 : Is-equiv {A = A} id\n    id\u1d31 = self-inv-is-equiv \u03bb _ \u2192 idp\n\n  module _ {a b c}{A : \u2605_ a}{B : \u2605_ b}{C : \u2605_ c}{g : B \u2192 C}{f : A \u2192 B} where\n    _\u2218\u1d31_ : Is-equiv g \u2192 Is-equiv f \u2192 Is-equiv (g \u2218 f)\n    g\u1d31 \u2218\u1d31 f\u1d31 = record { linv = F.linv \u2218 G.linv ; is-linv = \u03bb x \u2192 ap F.linv (G.is-linv (f x)) \u2219 F.is-linv x\n                      ; rinv = F.rinv \u2218 G.rinv ; is-rinv = \u03bb x \u2192 ap g (F.is-rinv _) \u2219 G.is-rinv x }\n      where\n        module G = Is-equiv g\u1d31\n        module F = Is-equiv f\u1d31\n\n  module _ {a b} where\n    infix 4 _\u2243_\n    _\u2243_ : \u2605_ a \u2192 \u2605_ b \u2192 \u2605_(a \u2294 b)\n    A \u2243 B = \u03a3 (A \u2192 B) Is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b} where\n    \u2013> : (e : A \u2243 B) \u2192 (A \u2192 B)\n    \u2013> e = fst e\n\n    <\u2013 : (e : A \u2243 B) \u2192 (B \u2192 A)\n    <\u2013 e = Is-equiv.linv (snd e)\n\n    <\u2013-inv-l : (e : A \u2243 B) (a : A)\n              \u2192 (<\u2013 e (\u2013> e a) \u2261 a)\n    <\u2013-inv-l e a = Is-equiv.is-linv (snd e) a\n\n    {-\n    <\u2013-inv-r : (e : A \u2243 B) (b : B)\n                \u2192 (\u2013> e (<\u2013 e b) \u2261 b)\n    <\u2013-inv-r e b = Is-equiv.is-rinv (snd e) b\n    -}\n\n    -- Equivalences are \"injective\"\n    equiv-inj : (e : A \u2243 B) {x y : A}\n                \u2192 (\u2013> e x \u2261 \u2013> e y \u2192 x \u2261 y)\n    equiv-inj e {x} {y} p = ! (<\u2013-inv-l e x) \u2219 ap (<\u2013 e) p \u2219 <\u2013-inv-l e y\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           {f : A \u2192 B}(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    is-equiv : Is-equiv f\n    is-equiv = record { linv = g ; is-linv = g-f ; rinv = g ; is-rinv = f-g }\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}\n           (f : A \u2192 B)(g : B \u2192 A)\n           (f-g : (y : B) \u2192 f (g y) \u2261 y)\n           (g-f : (x : A) \u2192 g (f x) \u2261 x) where\n    equiv : A \u2243 B\n    equiv = f , is-equiv g f-g g-f\n\n  module _ {\u2113} where\n    \u2243-refl : Reflexive (_\u2243_ {\u2113})\n    \u2243-refl = _ , id\u1d31\n\n    \u2243-sym : Symmetric (_\u2243_ {\u2113})\n    \u2243-sym (_ , f\u1d31) = _ , inv-is-equiv f\u1d31\n\n    \u2243-trans : Transitive (_\u2243_ {\u2113})\n    \u2243-trans (_ , p) (_ , q) = _ , q \u2218\u1d31 p\n\n    \u2243-! = \u2243-sym\n    _\u2243-\u2219_ = \u2243-trans\n\n  module _ {a}{A : \u2605_ a} where\n    Paths : \u2605_ a\n    Paths = \u03a3 A \u03bb x \u2192 \u03a3 A \u03bb y \u2192 x \u2261 y\n\n    id-path : A \u2192 Paths\n    id-path x = x , x , idp\n\n    fst-rinv-id-path : \u2200 p \u2192 id-path (fst p) \u2261 p\n    fst-rinv-id-path (x , y , p) = snd= (pair= p (J (\u03bb {y} p \u2192 tr (_\u2261_ x) p idp \u2261 p) idp p))\n\n    id-path-is-equiv : Is-equiv id-path\n    id-path-is-equiv = record { linv = fst\n                              ; is-linv = \u03bb x \u2192 idp\n                              ; rinv = fst\n                              ; is-rinv = fst-rinv-id-path }\n\n    \u2243-Paths : A \u2243 Paths\n    \u2243-Paths = id-path , id-path-is-equiv\n\n  module _ {a b}{A : \u2605_ a}{B : \u2605_ b}(f : A \u2192 B) where\n    hfiber : (y : B) \u2192 \u2605_(a \u2294 b)\n    hfiber y = \u03a3 A \u03bb x \u2192 f x \u2261 y\n\n    Is-equiv-alt : \u2605_(a \u2294 b)\n    Is-equiv-alt = (y : B) \u2192 Is-contr (hfiber y)\n\n  module Is-contr-to-Is-equiv {a}{A : \u2605_ a}(A-contr : Is-contr A) where\n    const-is-equiv : Is-equiv (\u03bb (_ : \ud835\udfd9) \u2192 fst A-contr)\n    const-is-equiv = record { linv = _ ; is-linv = \u03bb _ \u2192 idp ; rinv = _ ; is-rinv = snd A-contr }\n    \ud835\udfd9\u2243 : \ud835\udfd9 \u2243 A\n    \ud835\udfd9\u2243 = _ , const-is-equiv\n  module Is-equiv-to-Is-contr {a}{A : \u2605_ a}(f : \ud835\udfd9 \u2192 A)(f-is-equiv : Is-equiv f) where\n    open Is-equiv f-is-equiv\n    A-contr : Is-contr A\n    A-contr = f _ , is-rinv\n\n  module _ {a}{A : \u2605_ a}{b}{B : \u2605_ b} where\n    iso-to-equiv : (A \u2194 B) \u2192 (A \u2243 B)\n    iso-to-equiv iso = to iso , record { linv = from iso ; is-linv = Inverse.left-inverse-of iso\n                                       ; rinv = from iso ; is-rinv = Inverse.right-inverse-of iso }\n\n    equiv-to-iso : (A \u2243 B) \u2192 (A \u2194 B)\n    equiv-to-iso (f , f-is-equiv) = inverses f (f\u1d31.linv \u2218 f \u2218 f\u1d31.rinv)\n                                             (\u03bb x \u2192 ap f\u1d31.linv (f\u1d31.is-rinv (f x)) \u2219 f\u1d31.is-linv x)\n                                             (\u03bb x \u2192 ap f (f\u1d31.is-linv (f\u1d31.rinv x)) \u2219 f\u1d31.is-rinv x)\n      where module f\u1d31 = Is-equiv f-is-equiv\n\n    {-\n    iso-to-equiv-to-iso : (iso : A \u2194 B) \u2192 equiv-to-iso (iso-to-equiv iso) \u2261 iso\n    iso-to-equiv-to-iso iso = {!!}\n      where module Iso = Inverse iso\n\n    iso-to-equiv-is-equiv : Is-equiv iso-to-equiv\n    iso-to-equiv-is-equiv = record { linv = equiv-to-iso ; is-linv = {!!} ; rinv = {!!} ; is-rinv = {!!} }\n    -}\nopen Equivalences\n\nmodule _ {\u2113}{A : \u2605_ \u2113} where\n    coe!-inv-r : \u2200 {B}(p : A \u2261 B) y \u2192 coe p (coe! p y) \u2261 y\n    coe!-inv-r idp y = idp\n\n    coe!-inv-l : \u2200 {B}(p : A \u2261 B) x \u2192 coe! p (coe p x) \u2261 x\n    coe!-inv-l idp x = idp\n\n    coe-equiv : \u2200 {B} \u2192 A \u2261 B \u2192 A \u2243 B\n    coe-equiv p = equiv (coe p) (coe! p) (coe!-inv-r p) (coe!-inv-l p)\n\npostulate\n  UA : \u2605\nmodule _ {\u2113}{A B : \u2605_ \u2113}{{_ : UA}} where\n  postulate\n    ua : (A \u2243 B) \u2192 (A \u2261 B)\n    coe-equiv-\u03b2 : (e : A \u2243 B) \u2192 coe-equiv (ua e) \u2261 e\n    ua-\u03b7 : (p : A \u2261 B) \u2192 ua (coe-equiv p) \u2261 p\n\n  ua-equiv : (A \u2243 B) \u2243 (A \u2261 B)\n  ua-equiv = equiv ua coe-equiv ua-\u03b7 coe-equiv-\u03b2\n\n  coe-\u03b2 : (e : A \u2243 B) (a : A) \u2192 coe (ua e) a \u2261 \u2013> e a\n  coe-\u03b2 e a = ap (\u03bb e \u2192 \u2013> e a) (coe-equiv-\u03b2 e)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{a}{A\u2080 A\u2081 : \u2605_ a}{b}{B\u2080 : A\u2080 \u2192 \u2605_ b}{B\u2081 : A\u2081 \u2192 \u2605_ b} where\n    \u03a3\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a3 A\u2080 B\u2080 \u2261 \u03a3 A\u2081 B\u2081\n    \u03a3\u2243 A B = \u03a3= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\n    \u03a0\u2243 : (A : A\u2080 \u2243 A\u2081)(B : (x : A\u2080) \u2192 B\u2080 x \u2261 B\u2081 (\u2013> A x))\n         \u2192 \u03a0 A\u2080 B\u2080 \u2261 \u03a0 A\u2081 B\u2081\n    \u03a0\u2243 A B = \u03a0= (ua A) \u03bb x \u2192 B x \u2219 ap B\u2081 (! coe-\u03b2 A x)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{A : \u2605}{B C : A \u2192 \u2605} where\n    \u03a3\u228e-split : (\u03a3 A (\u03bb x \u2192 B x \u228e C x)) \u2261 (\u03a3 A B \u228e \u03a3 A C)\n    \u03a3\u228e-split = ua (equiv (\u03bb { (x , inl y) \u2192 inl (x , y)\n                            ; (x , inr y) \u2192 inr (x , y) })\n                         (\u03bb { (inl (x , y)) \u2192 x , inl y\n                            ; (inr (x , y)) \u2192 x , inr y })\n                         (\u03bb { (inl (x , y)) \u2192 idp\n                            ; (inr (x , y)) \u2192 idp })\n                         (\u03bb { (x , inl y) \u2192 idp\n                            ; (x , inr y) \u2192 idp }))\n\nmodule _ {{_ : UA}}{{_ : FunExt}}{A B : \u2605}{C : A \u2192 \u2605}{D : B \u2192 \u2605} where\n    dist-\u228e-\u03a3 : (\u03a3 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a3 A C \u228e \u03a3 B D)\n    dist-\u228e-\u03a3 = ua (iso-to-equiv \u03a3\u228e-distrib)\n    dist-\u00d7-\u03a0 : (\u03a0 (A \u228e B) [inl: C ,inr: D ]) \u2261 (\u03a0 A C \u00d7 \u03a0 B D)\n    dist-\u00d7-\u03a0 = ua (iso-to-equiv (\u03a0\u00d7-distrib (\u03bb fg \u2192 \u03bb= fg)))\n\nmodule _ {A : \u2605}{B : A \u2192 \u2605}{C : (x : A) \u2192 B x \u2192 \u2605} where\n    \u03a3-assoc-equiv : (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2243 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc-equiv = equiv (\u03bb x \u2192 (fst x , fst (snd x)) , snd (snd x))\n                          (\u03bb x \u2192 fst (fst x) , snd (fst x) , snd x)\n                          (\u03bb y \u2192 idp)\n                          (\u03bb y \u2192 idp)\n\n    \u03a3-assoc : {{_ : UA}} \u2192 (\u03a3 A (\u03bb x \u2192 \u03a3 (B x) (C x))) \u2261 (\u03a3 (\u03a3 A B) (uncurry C))\n    \u03a3-assoc = ua \u03a3-assoc-equiv\n\nmodule _ {A B : \u2605} where\n    \u00d7-comm-equiv : (A \u00d7 B) \u2243 (B \u00d7 A)\n    \u00d7-comm-equiv = equiv swap swap (\u03bb y \u2192 idp) (\u03bb x \u2192 idp)\n\n    \u00d7-comm : {{_ : UA}} \u2192 (A \u00d7 B) \u2261 (B \u00d7 A)\n    \u00d7-comm = ua \u00d7-comm-equiv\n\n    \u228e-comm-equiv : (A \u228e B) \u2243 (B \u228e A)\n    \u228e-comm-equiv = equiv [inl: inr ,inr: inl ]\n                         [inl: inr ,inr: inl ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n                         [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ]\n\n    \u228e-comm : {{_ : UA}} \u2192 (A \u228e B) \u2261 (B \u228e A)\n    \u228e-comm = ua \u228e-comm-equiv\n\nmodule _ {A B : \u2605}{C : A \u2192 B \u2192 \u2605} where\n    \u03a0\u03a0-comm-equiv : ((x : A)(y : B) \u2192 C x y) \u2243 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm-equiv = equiv flip flip (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a0\u03a0-comm : {{_ : UA}} \u2192 ((x : A)(y : B) \u2192 C x y) \u2261 ((y : B)(x : A) \u2192 C x y)\n    \u03a0\u03a0-comm = ua \u03a0\u03a0-comm-equiv\n\n    \u03a3\u03a3-comm-equiv : (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2243 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm-equiv = equiv (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb { (x , y , z) \u2192 y , x , z })\n                          (\u03bb _ \u2192 idp)\n                          (\u03bb _ \u2192 idp)\n\n    \u03a3\u03a3-comm : {{_ : UA}} \u2192 (\u03a3 A \u03bb x \u2192 \u03a3 B \u03bb y \u2192 C x y) \u2261 (\u03a3 B \u03bb y \u2192 \u03a3 A \u03bb x \u2192 C x y)\n    \u03a3\u03a3-comm = ua \u03a3\u03a3-comm-equiv\n\nmodule _ {A B C : \u2605} where\n    \u00d7-assoc : {{_ : UA}} \u2192 (A \u00d7 (B \u00d7 C)) \u2261 ((A \u00d7 B) \u00d7 C)\n    \u00d7-assoc = \u03a3-assoc\n\n    \u228e-assoc-equiv : (A \u228e (B \u228e C)) \u2243 ((A \u228e B) \u228e C)\n    \u228e-assoc-equiv = equiv [inl: inl \u2218 inl ,inr: [inl: inl \u2218 inr ,inr: inr ] ]\n                          [inl: [inl: inl ,inr: inr \u2218 inl ] ,inr: inr \u2218 inr ]\n                          [inl: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ,inr: (\u03bb x \u2192 idp) ]\n                          [inl: (\u03bb x \u2192 idp) ,inr: [inl: (\u03bb x \u2192 idp) ,inr: (\u03bb x \u2192 idp) ] ]\n\n    \u228e-assoc : {{_ : UA}} \u2192 (A \u228e (B \u228e C)) \u2261 ((A \u228e B) \u228e C)\n    \u228e-assoc = ua \u228e-assoc-equiv\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \ud835\udfd8 \u2192 \u2605) where\n    \u03a0\ud835\udfd8-uniq : \u03a0 \ud835\udfd8 A \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq = ua (equiv _ (\u03bb _ ()) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 \u03bb= (\u03bb())))\n\nmodule _ {{_ : UA}}(A : \ud835\udfd9 \u2192 \u2605) where\n    \u03a0\ud835\udfd9-uniq : \u03a0 \ud835\udfd9 A \u2261 A _\n    \u03a0\ud835\udfd9-uniq = ua (equiv (\u03bb f \u2192 f _) (\u03bb x _ \u2192 x) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp))\n\nmodule _ {{_ : UA}}(A : \u2605) where\n    \u03a0\ud835\udfd9-uniq\u2032 : (\ud835\udfd9 \u2192 A) \u2261 A\n    \u03a0\ud835\udfd9-uniq\u2032 = \u03a0\ud835\udfd9-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : UA}}{{_ : FunExt}}(A : \u2605) where\n    \u03a0\ud835\udfd8-uniq\u2032 : (\ud835\udfd8 \u2192 A) \u2261 \ud835\udfd9\n    \u03a0\ud835\udfd8-uniq\u2032 = \u03a0\ud835\udfd8-uniq (\u03bb _ \u2192 A)\n\nmodule _ {{_ : FunExt}}(F G : \ud835\udfd8 \u2192 \u2605) where\n    -- also by \u03a0\ud835\udfd8-uniq twice\n    \u03a0\ud835\udfd8-uniq' : \u03a0 \ud835\udfd8 F \u2261 \u03a0 \ud835\udfd8 G\n    \u03a0\ud835\udfd8-uniq' = \u03a0=\u2032 \ud835\udfd8 (\u03bb())\n\nmodule _ {A : \ud835\udfd8 \u2192 \u2605} where\n    \u03a3\ud835\udfd8-fst-equiv : \u03a3 \ud835\udfd8 A \u2243 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst-equiv = equiv fst (\u03bb()) (\u03bb()) (\u03bb { (() , _) })\n\n    \u03a3\ud835\udfd8-fst : {{_ : UA}} \u2192 \u03a3 \ud835\udfd8 A \u2261 \ud835\udfd8\n    \u03a3\ud835\udfd8-fst = ua \u03a3\ud835\udfd8-fst-equiv\n\nmodule _ {A : \ud835\udfd9 \u2192 \u2605} where\n    \u03a3\ud835\udfd9-snd-equiv : \u03a3 \ud835\udfd9 A \u2243 A _\n    \u03a3\ud835\udfd9-snd-equiv = equiv snd (_,_ _) (\u03bb _ \u2192 idp) (\u03bb _ \u2192 idp)\n\n    \u03a3\ud835\udfd9-snd : {{_ : UA}} \u2192 \u03a3 \ud835\udfd9 A \u2261 A _\n    \u03a3\ud835\udfd9-snd = ua \u03a3\ud835\udfd9-snd-equiv\n\nmodule _ {A : \u2605} where\n    \u228e\ud835\udfd8-inl-equiv : A \u2243 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl-equiv = equiv inl [inl: id ,inr: (\u03bb()) ] [inl: (\u03bb _ \u2192 idp) ,inr: (\u03bb()) ] (\u03bb _ \u2192 idp)\n\n    \u228e\ud835\udfd8-inl : {{_ : UA}} \u2192 A \u2261 (A \u228e \ud835\udfd8)\n    \u228e\ud835\udfd8-inl = ua \u228e\ud835\udfd8-inl-equiv\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/HoTT.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d954c5c1e9b139492b16c9ae6c3642bfd2e86e7b","subject":"beginning of some simpler proof management","message":"beginning of some simpler proof management\n","repos":"crypto-agda\/crypto-agda","old_file":"tactic.agda","new_file":"tactic.agda","new_contents":"module tactic where\n\nopen import Type\nopen import Search.Type\nopen import Search.Searchable\nopen import Search.Searchable.Product\n\nopen import Data.Unit\nopen import Data.Product\nopen import Function\nopen import Relation.Binary.PropositionalEquality.NP\n\nmodule _ {R S} (sum-R : Sum R)(sum-R-ext : SumExt sum-R)(sum-S : Sum S) where\n  su-\u00d7 : \u2200 {f : R \u2192 R}{g : S \u2192 S} \u2192 SumStableUnder sum-R f \u2192 SumStableUnder sum-S g\n       \u2192 SumStableUnder (sum-R \u00d7-sum sum-S) (map f g)\n  su-\u00d7 {f}{g} su-f su-g F\n    = (sum-R \u00d7-sum sum-S) F\n    \u2261\u27e8 sum-R-ext (\u03bb x \u2192 su-g (\u03bb y \u2192 F ( x , y))) \u27e9\n      (sum-R \u00d7-sum sum-S) (F \u2218 map id g)\n    \u2261\u27e8 su-f (\u03bb x \u2192 sum-S (\u03bb y \u2192 F (map id g (x , y)))) \u27e9\n      (sum-R \u00d7-sum sum-S) (F \u2218 map f g)\n    \u220e where open \u2261-Reasoning\n\nmodule _ {R}(sum-R : Sum R) where\n  su-id : SumStableUnder sum-R id\n  su-id f = refl\n\n  su-\u2218 : \u2200 {f g} \u2192 SumStableUnder sum-R f \u2192 SumStableUnder sum-R g \u2192 SumStableUnder sum-R (f \u2218 g)\n  su-\u2218 {f} su-f su-g F = trans (su-f F) (su-g (F \u2218 f))\n\nmodule _ {R R' S'}(sum-R : Sum R)(sum-R' : Sum R')(sum-S' : Sum S')(sum-R'-ext : SumExt sum-R')\n           (h : R' \u2192 R')(su-h : SumStableUnder sum-R' h)\n           (to : R' \u2192 S' \u2192 R)(sum-same : \u2200 f \u2192 sum-R f \u2261 sum-R' (\u03bb r' \u2192 sum-S' (f \u2218 to r'))) where\n\n  principle : \u2200 {f g} \u2192 (\u2200 r' \u2192 sum-S' (f \u2218 to r') \u2261 sum-S' (g \u2218 to (h r'))) \u2192 sum-R f \u2261 sum-R g\n  principle {f}{g} prf\n    = sum-R f\n    \u2261\u27e8 sum-same f \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 f (to r' s')))\n    \u2261\u27e8 sum-R'-ext (\u03bb r' \u2192 prf r') \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to (h r') s')))\n    \u2261\u27e8 sym (su-h (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to r' s')))) \u27e9\n      sum-R' (\u03bb r' \u2192 sum-S' (\u03bb s' \u2192 g (to r' s')))\n    \u2261\u27e8 sym (sum-same g) \u27e9\n      sum-R g\n    \u220e\n    where open \u2261-Reasoning\n  \n","old_contents":"","returncode":1,"stderr":"error: pathspec 'tactic.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"cac936b513fea803107345cf5a1fa1bf072f3073","subject":"Added missing file.","message":"Added missing file.\n\nIgnore-this: f74085bc0060648712fb8cd9c6f11adc\n\ndarcs-hash:20100603041857-3bd4e-c1a3cff4fc9c542799499c761f9921a31a08829b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"LTC\/Relation\/InequalitiesPCF.agda","new_file":"LTC\/Relation\/InequalitiesPCF.agda","new_contents":"------------------------------------------------------------------------------\n-- Inequalities on partial natural numbers\n------------------------------------------------------------------------------\n\nmodule LTC.Relation.InequalitiesPCF where\n\nopen import LTC.Minimal\n\n------------------------------------------------------------------------------\n\n-- Version using lambda-abstraction.\n-- lth : D \u2192 D\n-- lth lt = lam (\u03bb m \u2192 lam (\u03bb n \u2192\n--               if (isZero n) then false\n--                  else (if (isZero m) then true\n--                           else (lt \u2219 (pred m) \u2219 (pred n)))))\n\n-- Version using lambda lifting via super-combinators.\n-- (Hughes. Super-combinators. 1982)\n\nlt-aux\u2081 : D \u2192 D \u2192 D \u2192 D\nlt-aux\u2081 d lt e = if (isZero e) then false\n                    else (if (isZero d) then true\n                          else (lt \u2219 (pred d) \u2219 (pred e)))\n{-# ATP definition lt-aux\u2081 #-}\n\nlt-aux\u2082 : D \u2192 D \u2192 D\nlt-aux\u2082 lt d = lam (lt-aux\u2081 d lt)\n{-# ATP definition lt-aux\u2082 #-}\n\nlth : D \u2192 D\nlth lt = lam (lt-aux\u2082 lt)\n{-# ATP definition lth #-}\n\nlt : D \u2192 D \u2192 D\nlt d e = fix lth \u2219 d \u2219 e\n{-# ATP definition lt #-}\n\ngt : D \u2192 D \u2192 D\ngt d e = lt e d\n{-# ATP definition gt #-}\n\n------------------------------------------------------------------------\n-- The data types\n\n-- infix 4 _\u2264_ _<_ _\u2265_ _>_\n\nGT : D \u2192 D \u2192 Set\nGT d e = lt e d \u2261 true\n{-# ATP definition GT #-}\n\nLT : D \u2192 D  \u2192 Set\nLT d e = lt d e \u2261 true\n{-# ATP definition LT #-}\n\nLE : D \u2192 D \u2192 Set\nLE d e = lt e d \u2261 false\n{-# ATP definition LE #-}\n\nGE : D \u2192 D \u2192 Set\nGE d e = lt d e \u2261 false\n{-# ATP definition GE #-}\n\n------------------------------------------------------------------------------\n-- The lexicographical order\nLT\u2082 : D \u2192 D \u2192 D \u2192 D \u2192 Set\nLT\u2082 x\u2081 y\u2081 x\u2082 y\u2082 = LT x\u2081 x\u2082 \u2228 x\u2081 \u2261 x\u2082 \u2227 LT y\u2081 y\u2082\n{-# ATP definition LT\u2082 #-}\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'LTC\/Relation\/InequalitiesPCF.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"2ab10e65cca9b9f249ee976598bdd341e2b73aff","subject":"Some work on thre ballot","message":"Some work on thre ballot\n","repos":"crypto-agda\/crypto-agda","old_file":"ThreeBallot.agda","new_file":"ThreeBallot.agda","new_contents":"module ThreeBallot\n   -- ()\n   where\n\nopen import Data.Fin using (Fin ; zero ; suc )\nopen import Data.Nat\nopen import Data.Bool.NP using (to\u2115) renaming (false to 0b ; true to 1b)\n\nopen import Data.List using (List ; [] ; _\u2237_ ; length)\nopen import Data.Bits hiding (to\u2115 ; 0b ; 1b)\n\nopen import Data.Vec.NP renaming (map to vmap)\nopen import Data.Product\n\nimport Function.Inverse.NP as Inv\nopen Inv using (_\u2194_; _\u2218_; sym; id; inverses; module Inverse) renaming (_$\u2081_ to to; _$\u2082_ to from)\n\nimport Relation.Binary.PropositionalEquality as \u2261\nopen \u2261 using (_\u2261_ ; refl)\n\ndata Cand : Set where\n  Nice'Guy Bad'Guy : Cand\n\nVote = Bits 2\nTag  = \u2115 \n\nrecord Ballot : Set where\n  constructor mk\n  field\n    vote : Vote\n    tag  : Tag\n\nopen Ballot\n    \nrecord 3'_ (A : Set) : Set where\n  constructor [_][_][_]\n  field\n    1t 2t 3t : A\n\n3Vote = 3' Vote\n3Ballot = 3' Ballot\n    \n[_][_] : Bit \u2192 Bit \u2192 Bits 2\n[ x ][ y ] = x \u2237 y \u2237 []\n\nsvote : Vote\nsvote = [ 0b ][ 1b ]\n\nexample : 3Vote\nexample = [ [ 1b ][ 0b ] ][ [ 0b ][ 1b ] ][ [ 0b ][ 1b ] ] \n\nrow-const : Bit \u2192 3' Bit \u2192 Set\nrow-const b [ 1t ][ 2t ][ 3t ] = 1 + to\u2115 b \u2261 to\u2115 1t + to\u2115 2t + to\u2115 3t\n\nvoteFor : Vote \u2192 3Vote \u2192 Set\nvoteFor (n \u2237 b \u2237 []) [ x \u2237 x\u2081 \u2237 [] ][ x\u2082 \u2237 x\u2083 \u2237 [] ][ x\u2084 \u2237 x\u2085 \u2237 [] ] = row-const n [ x ][ x\u2082 ][ x\u2084 ]\n                                                                     \u00d7 row-const b [ x\u2081 ][ x\u2083 ][ x\u2085 ]\n\nvote-example : voteFor svote example\nvote-example = refl , refl\n\ncast : (_ _ : Vote) \u2192 Vote\ncast i r = vnot r\n\n\nvote-any : (intent receipt : Vote) \u2192 voteFor intent [ intent ][ receipt ][ cast intent receipt ]\nvote-any (true \u2237 true \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (true \u2237 true \u2237 []) (true \u2237 false \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (false \u2237 true \u2237 []) = refl , refl\nvote-any (true \u2237 true \u2237 []) (false \u2237 false \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (true \u2237 false \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (false \u2237 true \u2237 []) = refl , refl \nvote-any (true \u2237 false \u2237 []) (false \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (true \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (false \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 true \u2237 []) (false \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (true \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (true \u2237 false \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (false \u2237 true \u2237 []) = refl , refl \nvote-any (false \u2237 false \u2237 []) (false \u2237 false \u2237 []) = refl , refl   \n\n{-\n\nNP :\n\n  (R?\u00d7V)^n \u2192 (3V)^n \u2192 (3B)^n \u2192 B^3n -\u27e8 shuffle \u27e9\u2192 b^3n \u2192 T\n\n-}\n\nPerm : \u2115 \u2192 Set\nPerm n = Fin n \u2194 Fin n\n\nshuffle : \u2200 {n}{A : Set} \u2192 Perm n \u2192 Vec A n \u2192 Vec A n\nshuffle p xs = tabulate (\u03bb i \u2192 lookup (to p i) xs)\n\nmodule _\n  (tally : \u2200 {n} \u2192 Vec Vote n \u2192 \u2115)\n  (tally-stable : \u2200 {n} vs \u03c0 \u2192 tally {n} vs \u2261 tally (shuffle \u03c0 vs)) \n  where\n\n  3-tally : \u2200 {n} \u2192 Vec Ballot (3 * n) \u2192 \u2115\n  3-tally {n} bs = tally (vmap vote bs) \u2238 n\n\n\n  lookup-map : \u2200 {n}{A B : Set}(f : A \u2192 B)(i : Fin n)(xs : Vec A n)\n             \u2192 f (lookup i xs) \u2261 lookup i (vmap f xs)\n  lookup-map f zero (x \u2237 xs) = refl \n  lookup-map f (suc i) (x \u2237 xs) = lookup-map f i xs\n  \n  -- map g (tabulate f) = tabulate (g \u2218 f)\n  -- f (lookup i xs) \u2261 lookup i (map f xs)\n  3-tally-stable : \u2200 {n} bs \u03c0 \u2192 3-tally {n} bs \u2261 3-tally {n} (shuffle \u03c0 bs)\n  3-tally-stable {n} bs p rewrite tally-stable (vmap vote bs) p\n                           | \u2261.sym (tabulate-ext (\u03bb i \u2192 lookup-map vote (to p i) bs))\n                           = \u2261.cong (\u03bb p\u2081 \u2192 tally p\u2081 \u2238 n) (tabulate-\u2218 _ _)\n                           \n{-\n  Ballot privacy \u263a\n-}\n\nmodule BallotPrivacy\n  (RA : Set)\n  where\n\n  data Msg : Set where\n    vote! : (intent receipt : Vote) \u2192 Msg\n    stop! : Msg\n  \n  Resp = List Ballot\n    \n  Adv\u2080 = RA \u2192 Vote\n  Adv\u2081 = RA \u2192 Resp \u2192 Msg\n  Adv\u2082 = RA \u2192 (l : Resp) \u2192 Vec Ballot (3 * length l) \u2192 Bit\n  \n  Adv = Adv\u2080 \u00d7 Adv\u2081 \u00d7 Adv\u2082\n  \n  \n  G : Adv \u2192 (\u2200 {n} \u2192 Perm n) \u2192 RA \u2192 Bit \u2192  Bit\n  G (A\u2080 , A\u2081 , A\u2082) \u03c0 Ra b = {!!} where\n    \u03b5 = A\u2080 Ra\n\n    go : (r : Resp) \u2192 Vec Ballot (3 * length r) \u2192 Bit\n    go r bb with A\u2081 Ra r | b\n    go r bb | vote! intent receipt | 1b = {!!}\n    go r bb | vote! intent receipt | 0b = {!!}\n    go r bb | stop!                | _ = A\u2082 Ra r (shuffle \u03c0 bb) \n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n3<4 : 3 < 4\n3<4 =  s\u2264s (s\u2264s (s\u2264s (s\u2264s z\u2264n)))\n\n\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ThreeBallot.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"6f0c4a50db5aaa4e31efff2cf6ca0ca966340027","subject":"+Algebra\/FunctionProperties\/Eq","message":"+Algebra\/FunctionProperties\/Eq\n","repos":"crypto-agda\/agda-nplib","old_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_file":"lib\/Algebra\/FunctionProperties\/Eq.agda","new_contents":"{-# OPTIONS --without-K #-}\n-- Like Algebra.FunctionProperties but specialized to _\u2261_ and using implict arguments\n\n-- These properties can (for instance) be used to define algebraic\n-- structures.\n\nopen import Level\nopen import Data.Product\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP\n\n-- The properties are specified using the following relation as\n-- \"equality\".\n\nmodule Algebra.FunctionProperties.Eq {a} {A : Set a} where\n\n------------------------------------------------------------------------\n-- Unary and binary operations\n\nopen import Algebra.FunctionProperties.Core public\n\n------------------------------------------------------------------------\n-- Properties of operations\n\nAssociative : Op\u2082 A \u2192 Set _\nAssociative _\u00b7_ = \u2200 {x y z} \u2192 ((x \u00b7 y) \u00b7 z) \u2261 (x \u00b7 (y \u00b7 z))\n\nCommutative : Op\u2082 A \u2192 Set _\nCommutative _\u00b7_ = \u2200 {x y} \u2192 (x \u00b7 y) \u2261 (y \u00b7 x)\n\nLeftIdentity : A \u2192 Op\u2082 A \u2192 Set _\nLeftIdentity e _\u00b7_ = \u2200 {x} \u2192 (e \u00b7 x) \u2261 x\n\nRightIdentity : A \u2192 Op\u2082 A \u2192 Set _\nRightIdentity e _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 e) \u2261 x\n\nIdentity : A \u2192 Op\u2082 A \u2192 Set _\nIdentity e \u00b7 = LeftIdentity e \u00b7 \u00d7 RightIdentity e \u00b7\n\nLeftZero : A \u2192 Op\u2082 A \u2192 Set _\nLeftZero z _\u00b7_ = \u2200 {x} \u2192 (z \u00b7 x) \u2261 z\n\nRightZero : A \u2192 Op\u2082 A \u2192 Set _\nRightZero z _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 z) \u2261 z\n\nZero : A \u2192 Op\u2082 A \u2192 Set _\nZero z \u00b7 = LeftZero z \u00b7 \u00d7 RightZero z \u00b7\n\nLeftInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nLeftInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u207b\u00b9 \u00b7 x) \u2261 e\n\nRightInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nRightInverse e _\u207b\u00b9 _\u00b7_ = \u2200 {x} \u2192 (x \u00b7 (x \u207b\u00b9)) \u2261 e\n\nInverse : A \u2192 Op\u2081 A \u2192 Op\u2082 A \u2192 Set _\nInverse e \u207b\u00b9 \u00b7 = LeftInverse e \u207b\u00b9 \u00b7 \u00d7 RightInverse e \u207b\u00b9 \u00b7\n\n_DistributesOver\u02e1_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02e1 _+_ =\n  \u2200 {x y z} \u2192 (x * (y + z)) \u2261 ((x * y) + (x * z))\n\n_DistributesOver\u02b3_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_*_ DistributesOver\u02b3 _+_ =\n  \u2200 {x y z} \u2192 ((y + z) * x) \u2261 ((y * x) + (z * x))\n\n_DistributesOver_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n* DistributesOver + = (* DistributesOver\u02e1 +) \u00d7 (* DistributesOver\u02b3 +)\n\n_IdempotentOn_ : Op\u2082 A \u2192 A \u2192 Set _\n_\u00b7_ IdempotentOn x = (x \u00b7 x) \u2261 x\n\nIdempotent : Op\u2082 A \u2192 Set _\nIdempotent \u00b7 = \u2200 {x} \u2192 \u00b7 IdempotentOn x\n\nIdempotentFun : Op\u2081 A \u2192 Set _\nIdempotentFun f = \u2200 {x} \u2192 f (f x) \u2261 f x\n\n_Absorbs_ : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\n_\u00b7_ Absorbs _\u2218_ = \u2200 {x y} \u2192 (x \u00b7 (x \u2218 y)) \u2261 x\n\nAbsorptive : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nAbsorptive \u00b7 \u2218 = (\u00b7 Absorbs \u2218) \u00d7 (\u2218 Absorbs \u00b7)\n\nInvolutive : Op\u2081 A \u2192 Set _\nInvolutive f = \u2200 {x} \u2192 f (f x) \u2261 x\n\nInterchange : Op\u2082 A \u2192 Op\u2082 A \u2192 Set _\nInterchange _\u00b7_ _\u2218_ = \u2200 {x y z t} \u2192 ((x \u00b7 y) \u2218 (z \u00b7 t)) \u2261 ((x \u2218 z) \u00b7 (y \u2218 t))\n\nLeftCancel : Op\u2082 A \u2192 Set _\nLeftCancel _\u00b7_ = \u2200 {c x y} \u2192 c \u00b7 x \u2261 c \u00b7 y \u2192 x \u2261 y\n\nRightCancel : Op\u2082 A \u2192 Set _\nRightCancel _\u00b7_ = \u2200 {c x y} \u2192 x \u00b7 c \u2261 y \u00b7 c \u2192 x \u2261 y\n\nmodule InterchangeFromAssocComm\n         (_\u00b7_     : Op\u2082 A)\n         (\u00b7-assoc : Associative _\u00b7_)\n         (\u00b7-comm  : Commutative _\u00b7_)\n         where\n\n    open \u2261-Reasoning\n\n    \u00b7= : \u2200 {x x' y y'} \u2192 x \u2261 x' \u2192 y \u2261 y' \u2192 (x \u00b7 y) \u2261 (x' \u00b7 y')\n    \u00b7= refl refl = refl\n\n    \u00b7-interchange : Interchange _\u00b7_ _\u00b7_\n    \u00b7-interchange {x} {y} {z} {t}\n                = (x \u00b7 y) \u00b7 (z \u00b7 t)\n                \u2261\u27e8 \u00b7-assoc \u27e9\n                  x \u00b7 (y \u00b7 (z \u00b7 t))\n                \u2261\u27e8 \u00b7= refl (! \u00b7-assoc) \u27e9\n                  x \u00b7 ((y \u00b7 z) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl (\u00b7= \u00b7-comm refl) \u27e9\n                  x \u00b7 ((z \u00b7 y) \u00b7 t)\n                \u2261\u27e8 \u00b7= refl \u00b7-assoc \u27e9\n                  x \u00b7 (z \u00b7 (y \u00b7 t))\n                \u2261\u27e8 ! \u00b7-assoc \u27e9\n                  (x \u00b7 z) \u00b7 (y \u00b7 t)\n                \u220e\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'lib\/Algebra\/FunctionProperties\/Eq.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"c4b95b94ffa1aa379ab3b7f9b88afe12b8b15bef","subject":"Added setoids draft.","message":"Added setoids draft.\n\nIgnore-this: 972580c96ceed8e6215d40030ce46100\n\ndarcs-hash:20110325220333-3bd4e-876f32f3d7f06dac74e8b78ec17c08dfc48b947b.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/Setoid.agda","new_file":"Draft\/Setoid.agda","new_contents":"------------------------------------------------------------------------------\n-- Using setoids to formalize the FOTC\n------------------------------------------------------------------------------\n\nmodule Setoid where\n\n-- We add 3 to the fixities of the standard library.\ninfixl 9 _\u00b7_  -- The symbol is '\\cdot'.\ninfix  7 _\u2250_\n\n------------------------------------------------------------------------------\n\ndata D : Set where\n  K S : D\n  _\u00b7_ : D \u2192 D \u2192 D\n\ndata _\u2250_ : D \u2192 D \u2192 Set where\n  refl  : \u2200 x \u2192                                       x \u2250 x\n  sym   : \u2200 {x y} \u2192 x \u2250 y \u2192                           y \u2250 x\n  trans : \u2200 {x y z} \u2192 x \u2250 y \u2192 y \u2250 z \u2192                 x \u2250 z\n  cong  : \u2200 {x\u2081 x\u2082 y\u2081 y\u2082} \u2192 x\u2081 \u2250 x\u2082 \u2192 y\u2081 \u2250 y\u2082 \u2192 x\u2081 \u00b7 y\u2081 \u2250 x\u2082 \u00b7 y\u2082\n  Kax   : \u2200 x y \u2192                            K \u00b7 x \u00b7 y  \u2250 x\n  Sax   : \u2200 x y z \u2192                      S \u00b7 x \u00b7 y \u00b7 z  \u2250 x \u00b7 z \u00b7 (y \u00b7 z)\n\n-- It seems we cannot define the identity elimination using the setoid\n-- equality\nsubst : \u2200 {x y} (P : D \u2192 Set) \u2192 x \u2250 y \u2192 P x \u2192 P y\nsubst P x\u2250y Px = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/Setoid.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"006fd9c825ad2011f95bb01a40c7f000d4fd4f30","subject":"Agda: add another bug example","message":"Agda: add another bug example\n\nOld-commit-hash: 82f681165ed59ccc36685cbf9c376026811e127f\n","repos":"inc-lc\/ilc-agda","old_file":"bugs\/Future.agda","new_file":"bugs\/Future.agda","new_contents":"module Future where\n\nopen import Denotational.Notation\n\n-- Uses too much memory.\n-- open import Data.Integer\n\nopen import Data.Product\nopen import Data.Sum\nopen import Data.Unit\n\npostulate \u27e6Int\u27e7 : Set\npostulate \u27e6Bag\u27e7 : Set \u2192 Set\n\n--mutual\n-- First, define a language of types, their denotations, and then the whole algebra of changes in the semantic domain\ndata Type : Set\n\u27e6_\u27e7Type : Type \u2192 Set\n\ndata Type where\n  Int : Type\n  --Bag : Type \u2192 Type\n  _\u21d2_ {- _\u22a0_ _\u229e_ -} : Type \u2192 Type \u2192 Type\n  \u0394 : (\u03c4 : Type) \u2192 (old : \u27e6 \u03c4 \u27e7Type) \u2192 Type\n\n-- meaningOfType : Meaning Type\n-- meaningOfType = meaning \u27e6_\u27e7Type\n\n-- Try out \u0394 old here\n\nChange : (\u03c4 : Type) \u2192 \u27e6 \u03c4 \u27e7Type \u2192 Set\nChange Int old\u2081 = {!!}\n-- Change (Bag \u03c4\u2081) old\u2081    = \u27e6 Bag \u03c4\u2081 \u27e7Type \u228e (\u03a3[ old \u2208 (\u27e6 \u03c4\u2081 \u27e7Type) ] (\u27e6Bag\u27e7 \u27e6 \u0394 \u03c4\u2081 old \u27e7Type))\nChange (\u03c4\u2081 \u21d2 \u03c4\u2082) oldF   = \u27e6 \u0394 \u03c4\u2082 (oldF ?) \u27e7Type\n-- oldF ? -- oldF ? -- (oldInput : \u27e6 \u03c4\u2081 \u27e7Type) \u2192 \u27e6 \u0394 \u03c4\u2081 oldInput \u27e7Type \u2192 \u27e6 \u0394 \u03c4\u2082 (oldF oldInput) \u27e7Type \n-- Change (\u03c4\u2081 \u22a0 \u03c4\u2082) old\u2081 = {!\u27e6 \u03c4\u2081 \u27e7Type \u00d7 \u27e6 \u03c4\u2082 \u27e7Type!}\n-- Change (\u03c4\u2081 \u229e \u03c4\u2082) old\u2081 = {!!}\nChange (\u0394 \u03c4\u2081 old\u2081) old\u2082 = {!!}\n\n\u27e6 Int \u27e7Type = \u27e6Int\u27e7\n-- \u27e6 Bag \u03c4 \u27e7Type = \u27e6Bag\u27e7 \u27e6 \u03c4 \u27e7Type\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 \u03c3 \u22a0 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u00d7 \u27e6 \u03c4 \u27e7Type\n-- \u27e6 \u03c3 \u229e \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u228e \u27e6 \u03c4 \u27e7Type\n\u27e6 \u0394 \u03c4 old \u27e7Type = {- \u22a4 \u228e \u27e6 \u03c4 \u27e7Type \u228e -} Change \u03c4 old\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'bugs\/Future.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"9bb44c3e31a11725d740b2f59d40ebbe7e0917a2","subject":"Add FiniteField\/JS","message":"Add FiniteField\/JS\n","repos":"crypto-agda\/crypto-agda","old_file":"FiniteField\/JS.agda","new_file":"FiniteField\/JS.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Function.NP\nopen import FFI.JS using (Number; Bool; true; false; String; warn-check; check; trace-call; _++_)\nopen import FFI.JS.BigI\nopen import Data.List.Base hiding (sum; _++_)\n{-\nopen import Algebra.Raw\nopen import Algebra.Field\n-}\n\nmodule FiniteField.JS (q : BigI) where\n\nabstract\n  \u2124q : Set\n  \u2124q = BigI\n\n  private\n    mod-q : BigI \u2192 \u2124q\n    mod-q x = mod x q\n\n  -- There is two ways to go from BigI to \u2124q: check and mod-q\n  -- Use check for untrusted input data and mod-q for internal\n  -- computation.\n  BigI\u25b9\u2124q : BigI \u2192 \u2124q\n  BigI\u25b9\u2124q = -- trace-call \"BigI\u25b9\u2124q \"\n    \u03bb x \u2192\n    mod-q\n      (warn-check (x <I q)\n         (\u03bb _ \u2192 \"Not below the modulus: \" ++ toString q ++ \" < \" ++ toString x)\n         (warn-check (x \u2265I 0I)\n            (\u03bb _ \u2192 \"Should be positive: \" ++ toString x ++ \" < 0\") x))\n\n  check-non-zero : \u2124q \u2192 BigI\n  check-non-zero = -- trace-call \"check-non-zero \"\n    \u03bb x \u2192 check (x >I 0I) (\u03bb _ \u2192 \"Should be non zero\") x\n\n  repr : \u2124q \u2192 BigI\n  repr x = x\n\n  0# 1# : \u2124q\n  0# = 0I\n  1# = 1I\n\n  1\/_ : Op\u2081 \u2124q\n  1\/ x = modInv (check-non-zero x) q\n\n  _^_ : Op\u2082 \u2124q\n  x ^ y = modPow (repr x) (repr y) q\n\n_+_ _\u2212_ _*_ _\/_ : Op\u2082 \u2124q\n\nx + y = mod-q (add      (repr x) (repr y))\nx \u2212 y = mod-q (subtract (repr x) (repr y))\nx * y = mod-q (multiply (repr x) (repr y))\nx \/ y = x * 1\/ y\n\n0\u2212_ : Op\u2081 \u2124q\n0\u2212 x = mod-q (negate (repr x))\n\n_==_ : (x y : \u2124q) \u2192 Bool\nx == y = equals (repr x) (repr y)\n\nsum prod : List \u2124q \u2192 \u2124q\nsum  = foldr _+_ 0#\nprod = foldr _*_ 1#\n\n{-\n+-mon-ops : Monoid-Ops \u2124q\n+-mon-ops = _+_ , 0#\n\n+-grp-ops : Group-Ops \u2124q\n+-grp-ops = +-mon-ops , 0\u2212_\n\n*-mon-ops : Monoid-Ops \u2124q\n*-mon-ops = _*_ , 1#\n\n*-grp-ops : Group-Ops \u2124q\n*-grp-ops = *-mon-ops , 1\/_\n\nfld-ops : Field-Ops \u2124q\nfld-ops = +-grp-ops , *-grp-ops\n\npostulate\n  fld-struct : Field-Struct fld-ops\n\nfld : Field \u2124q\nfld = fld-ops , fld-struct\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'FiniteField\/JS.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"d9ddffd55ed97b1b9a7c1fa3e4ca46126455e411","subject":"created Head to ask for help","message":"created Head to ask for help\n","repos":"wenkokke\/sf,wenkokke\/sf","old_file":"src\/extra\/Head.agda","new_file":"src\/extra\/Head.agda","new_contents":"import Relation.Binary.PropositionalEquality as Eq\nopen Eq using (_\u2261_; refl; sym; trans; cong; cong\u2082; _\u2262_)\nopen import Data.Empty using (\u22a5; \u22a5-elim)\nopen import Data.List using (List; []; _\u2237_; dropWhile)\nopen import Data.Char using (Char)\nopen import Data.Bool using (Bool; true; false)\nimport Data.Char as Char using (_\u225f_)\nopen import Relation.Nullary using (\u00ac_; Dec; yes; no)\nopen import Relation.Nullary.Decidable using (\u230a_\u230b)\n\n\ndata Head {A : Set} (P : A \u2192 Bool) : List A \u2192 Set where\n  head : \u2200 (c : A) (s : List A) \u2192 P c \u2261 true \u2192 Head P (c \u2237 s)\n\nprime : Char\nprime = '\u2032'\n\nisPrime : Char \u2192 Bool\nisPrime c = \u230a c Char.\u225f prime \u230b\n\nhead-lemma : \u2200 (s : List Char) \u2192 \u00ac Head isPrime (dropWhile isPrime s)\nhead-lemma [] = \u03bb()\nhead-lemma (c \u2237 s) with isPrime c \n...                   | true       =  head-lemma s\n...                   | false      =  \u00ach\n  where\n  \u00ach : \u00ac Head isPrime (c \u2237 s)\n  \u00ach (head c s eqn\u2032) = {!!}\n\n{-\nGoal: \u22a5\n\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\ns    : List Char\nc    : Char\neqn\u2032 : \u230a (c Char.\u225f '\u2032' | .Agda.Builtin.Char.primCharEquality c '\u2032')\n       \u230b\n       \u2261 true\ns    : List Char\nc    : Char\n-}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'src\/extra\/Head.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"54f69647ad33989519972503b591c574d9a0024e","subject":"Search\/Searchable\/Maybe.agda","message":"Search\/Searchable\/Maybe.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"Search\/Searchable\/Maybe.agda","new_file":"Search\/Searchable\/Maybe.agda","new_contents":"open import Function.Related.TypeIsomorphisms.NP\nopen import Function.Inverse.NP\nopen import Data.Maybe.NP\nopen import Data.Nat\n\nopen import Search.Type\nopen import Search.Searchable\nopen import Search.Searchable.Sum\n\nmodule Search.Searchable.Maybe where\n\n\u03bcMaybe : \u2200 {A} \u2192 Searchable A \u2192 Searchable (Maybe A)\n\u03bcMaybe \u03bcA = \u03bc-iso (sym Maybe\u2194\u22a4\u228e) (\u03bc\u22a4 \u228e-\u03bc \u03bcA)\n\n\u03bcMaybe^ : \u2200 {A} n \u2192 Searchable A \u2192 Searchable (Maybe^ n A)\n\u03bcMaybe^ zero    \u03bcA = \u03bcA\n\u03bcMaybe^ (suc n) \u03bcA = \u03bcMaybe (\u03bcMaybe^ n \u03bcA)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Search\/Searchable\/Maybe.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"b87e1f71f434b29021fa06987ba417281fdf2366","subject":"ZK.GroupHom.NumChal: using an abstract type of natural number","message":"ZK.GroupHom.NumChal: using an abstract type of natural number\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/GroupHom\/NumChal.agda","new_file":"ZK\/GroupHom\/NumChal.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import Type using (Type; Type\u2081)\nopen import Function using (flip)\nopen import Data.Product renaming (proj\u2081 to fst; proj\u2082 to snd)\nopen import Data.Sum.NP\nopen import Data.Zero\nopen import Data.Fin.NP using (Fin\u25b9\u2115)\nopen import Data.Bool.Base using (Bool) renaming (T to \u2713)\nopen import Relation.Binary\nopen import Relation.Binary.PropositionalEquality.NP using (_\u2261_; _\u2262_; idp; ap; ap\u2082; !_; _\u2219_; module \u2261-Reasoning)\nopen import Algebra.Group\nopen import Algebra.Group.Homomorphism\nimport Data.Nat.ModInv\nimport ZK.SigmaProtocol.KnownStatement\nimport ZK.GroupHom\n\nmodule ZK.GroupHom.NumChal where\n\nrecord Package : Type\u2081 where\n  infixl 6 _+\u207f_ _\u2238\u207f_\n  infixl 7 _*\u207f_\n  field\n    Num : Type\n    Prime : Num \u2192 Type\n    _<_ : (x y : Num) \u2192 Type\n    0\u207f  : Num\n    1\u207f  : Num\n    _+\u207f_ : (x y : Num) \u2192 Num\n    _\u2238\u207f_ : (x y : Num) \u2192 Num\n    _*\u207f_ : (x y : Num) \u2192 Num\n    compare : Trichotomous _\u2261_ _<_\n    <-\u2238\u22620 : \u2200 x y \u2192 y < x \u2192 x \u2238\u207f y \u2262 0\u207f\n\n    G+ G* : Type\n    \ud835\udd3e+ : Group G+\n    \ud835\udd3e* : Group G*\n\n  _>_ : (x y : Num) \u2192 Type\n  _>_ = flip _<_\n\n  open Additive-Group \ud835\udd3e+ public\n  open Multiplicative-Group \ud835\udd3e* public\n\n  field\n    _==_ : G* \u2192 G* \u2192 Bool\n    \u2713-== : \u2200 {x y} \u2192 x \u2261 y \u2192 \u2713 (x == y)\n    ==-\u2713 : \u2200 {x y} \u2192 \u2713 (x == y) \u2192 x \u2261 y\n\n    _\u2297\u207f_ : G+ \u2192 Num \u2192 G+\n    _^\u207f_ : G* \u2192 Num \u2192 G*\n    1^\u207f : \u2200 x \u2192 1# ^\u207f x \u2261 1#\n\n    \u03c6 : G+ \u2192 G*\n    \u03c6-hom : GroupHomomorphism \ud835\udd3e+ \ud835\udd3e* \u03c6\n    \u03c6-hom-iterated : \u2200 {x c} \u2192 \u03c6 (x \u2297\u207f c) \u2261 \u03c6 x ^\u207f c\n\n    q : Num\n    q-prime : Prime q\n\n    _div-q : Num \u2192 Num\n    _mod-q : Num \u2192 Num\n    div-mod-q-prop\u207f : \u2200 d \u2192 d \u2261 d mod-q +\u207f (d div-q) *\u207f q\n\n    inv-mod-q : Num \u2192 Num\n    inv-mod-q-prop : \u2200 x \u2192 x \u2262 0\u207f \u2192 (inv-mod-q x *\u207f x)mod-q \u2261 1\u207f\n\n    Y : G*\n\n    -- Can be turned into a dynamic test!\n    G*-order : Y ^\u207f q \u2261 1#\n\n    ^\u207f1\u207f+\u207f : \u2200 {x} \u2192 Y ^\u207f(1\u207f +\u207f x) \u2261 Y * Y ^\u207f x\n    ^\u207f-*  : \u2200 {x y} \u2192 Y ^\u207f(y *\u207f x) \u2261 (Y ^\u207f x)^\u207f y\n    ^\u207f-\u2238\u207f : \u2200 {c\u2080 c\u2081}(c> : c\u2080 > c\u2081) \u2192 Y ^\u207f(c\u2080 \u2238\u207f c\u2081) \u2261 (Y ^\u207f c\u2080) \/ (Y ^\u207f c\u2081)\n\nmodule FromPackage (p : Package) where\n  open Package p\n  open \u2261-Reasoning\n\n  private\n    module \u03c6 = GroupHomomorphism \u03c6-hom\n    module \ud835\udd3e* = Group \ud835\udd3e*\n\n  Y^-^-1\/-id : \u2200 {x y} \u2192 x > y \u2192 (Y ^\u207f(x \u2238\u207f y))^\u207f(inv-mod-q(x \u2238\u207f y)) \u2261 Y\n  Y^-^-1\/-id {x} {y} x>y\n    = (Y ^\u207f d)^\u207f(inv-mod-q d) \u2261\u27e8 ! ^\u207f-* \u27e9\n      Y ^\u207f(inv-mod-q d *\u207f d)  \u2261\u27e8 ap (_^\u207f_ Y) (div-mod-q-prop\u207f e) \u27e9\n      Y ^\u207f(r +\u207f m *\u207f q)        \u2261\u27e8 ap (\u03bb z \u2192 Y ^\u207f(z +\u207f m *\u207f q)) (inv-mod-q-prop d (<-\u2238\u22620 x y x>y)) \u27e9\n      Y ^\u207f(1\u207f +\u207f(m *\u207f q))      \u2261\u27e8 ^\u207f1\u207f+\u207f \u27e9\n      Y * Y ^\u207f(m *\u207f q)        \u2261\u27e8 *= idp ^\u207f-* \u27e9\n      Y * (Y ^\u207f q)^\u207f m        \u2261\u27e8 ap (\u03bb z \u2192 Y * z ^\u207f m) G*-order \u27e9\n      Y * 1# ^\u207f m            \u2261\u27e8 *= idp (1^\u207f m) \u27e9\n      Y * 1#                 \u2261\u27e8 *1-identity \u27e9\n      Y \u220e\n      where d = x \u2238\u207f y\n            e = inv-mod-q d *\u207f d\n            m = e div-q\n            r = e mod-q\n\n  swap? : {c\u2080 c\u2081 : Num} \u2192 c\u2080 \u2262 c\u2081 \u2192 (c\u2080 > c\u2081) \u228e (c\u2081 > c\u2080)\n  swap? {x} {y} i with compare x y\n  swap? i | tri< a \u00acb \u00acc = inr a\n  swap? i | tri> \u00aca \u00acb c = inl c\n  swap? i | tri\u2248 \u00aca b \u00acc = \ud835\udfd8-elim (i b)\n\n  open ZK.GroupHom \ud835\udd3e+ \ud835\udd3e* _==_ \u2713-== ==-\u2713 _>_ swap? _\u2297\u207f_ _^\u207f_ _\u2238\u207f_ inv-mod-q\n                   \u03c6 \u03c6-hom \u03c6-hom-iterated\n                   Y\n                   ^\u207f-\u2238\u207f\n                   Y^-^-1\/-id\n    public\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/GroupHom\/NumChal.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"5cdc1f71f8724f78cbf9a922007d2ab5b16ebb3f","subject":"Try proving fundamental theorem in simpler context","message":"Try proving fundamental theorem in simpler context\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/FunBigStepSILR.agda","new_file":"Thesis\/FunBigStepSILR.agda","new_contents":"-- Step-indexed logical relations based on functional big-step semantics.\n--\n-- Goal for now: just prove the fundamental theorem of logical relations,\n-- relating a term to itself in a different environments.\n--\n-- Because of closures, we need relations across different terms with different\n-- contexts and environments.\n--\n-- This development is strongly inspired by \"A verified framework for\n-- higher-order uncurrying optimizations\" (and a bit by \"Functional Big-Step\n-- Semantics\"), though I deviate somewhere.\nmodule Thesis.FunBigStepSILR where\n\nopen import Data.Empty\nopen import Data.Unit.Base hiding (_\u2264_)\nopen import Data.Product\nopen import Relation.Binary.PropositionalEquality\nopen import Relation.Binary hiding (_\u21d2_)\nopen import Data.Nat using (\u2115; zero; suc; decTotalOrder; _<_; _\u2264_)\n\ndata Type : Set where\n  _\u21d2_ : (\u03c3 \u03c4 : Type) \u2192 Type\n  nat : Type\n\n\u27e6_\u27e7Type : Type \u2192 Set\n\u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type = \u27e6 \u03c3 \u27e7Type \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6 nat \u27e7Type = \u2115\n\nopen import Base.Syntax.Context Type public\nopen import Base.Syntax.Vars Type public\n\ndata Const : (\u03c4 : Type) \u2192 Set where\n  lit : \u2115 \u2192 Const nat\n  -- succ : Const (int \u21d2 int)\n\ndata Term (\u0393 : Context) :\n  (\u03c4 : Type) \u2192 Set where\n  -- constants aka. primitives\n  const : \u2200 {\u03c4} \u2192\n    (c : Const \u03c4) \u2192\n    Term \u0393 \u03c4\n  var : \u2200 {\u03c4} \u2192\n    (x : Var \u0393 \u03c4) \u2192\n    Term \u0393 \u03c4\n  app : \u2200 {\u03c3 \u03c4}\n    (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n    (t : Term \u0393 \u03c3) \u2192\n    Term \u0393 \u03c4\n  -- we use de Bruijn indices, so we don't need binding occurrences.\n  abs : \u2200 {\u03c3 \u03c4}\n    (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n    Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2115) \u2192 Val nat\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool! (Commented out until I paste that semantics here.)\n-- eval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n--   eval t \u03c1 n \u2261 Done v \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\n-- apply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n--   \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n--   apply f a n \u2261 Done v \u2192\n--   \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\n-- apply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\n-- eval-sound \u03c1 v zero t ()\n-- eval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\n-- eval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\n-- eval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n--   let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\n-- eval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n-- -- eval-sound n (const c) eq = {!!}\n-- -- eval-sound n (var x) eq = {!!}\n-- -- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- -- eval-sound n (abs t) eq = {!!}\n\nrelV : \u2200 \u03c4 (v1 v2 : Val \u03c4) \u2192 \u2115 \u2192 Set\nrel\u03c1 : \u2200 \u0393 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 \u2115 \u2192 Set\nrel\u03c1 \u2205 \u2205 \u2205 n = \u22a4\nrel\u03c1 (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) n = relV \u03c4 v1 v2 n \u00d7 rel\u03c1 \u0393 \u03c11 \u03c12 n\n\n-- Indexing not according to source. But I can't quite write the correct\n-- indexing without changing the definitions a lot.\nrelT : \u2200 {\u03c4 \u03931 \u03932} (t1 : Term \u03931 \u03c4) (t2 : Term \u03932 \u03c4) (\u03c11 : \u27e6 \u03931 \u27e7Context) (\u03c12 : \u27e6 \u03932 \u27e7Context) \u2192 \u2115 \u2192 Set\nrelT {\u03c4} t1 t2 \u03c11 \u03c12 n =\n  (v1 : Val \u03c4) \u2192\n  eval t1 \u03c11 n \u2261 Done v1 \u2192\n  \u03a3[ v2 \u2208 Val \u03c4 ] (eval t2 \u03c12 n \u2261 Done v2 \u00d7 relV \u03c4 v1 v2 n)\n\nrelV \u03c4 v1 v2 zero = \u22a4\nrelV (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) =\n  \u2200 (k : \u2115) (k\u2264n : k \u2264 n) \u2192\n  \u2200 v1 v2 \u2192 relV \u03c3 v1 v2 k \u2192 relT t1 t2 (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) k\nrelV nat v1 v2 (suc n) = v1 \u2261 v2\n\nopen import Data.Nat.Properties\n\nrelV-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u03c4 v1 v2 \u2192 relV \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 m\nrelV-mono zero n m\u2264n \u03c4 v1 v2 vv = tt\nrelV-mono (suc m) zero () \u03c4 v1 v2 vv\nrelV-mono (suc m) (suc n) m\u2264n nat v1 v2 vv = vv\nrelV-mono (suc m) (suc n) (_\u2264_.s\u2264s m\u2264n) (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) vv k k\u2264m = vv k (DecTotalOrder.trans decTotalOrder k\u2264m m\u2264n)\n\nrel\u03c1-mono : \u2200 m n \u2192 m \u2264 n \u2192 \u2200 \u0393 \u03c11 \u03c12 \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 n \u2192 rel\u03c1 \u0393 \u03c11 \u03c12 m\nrel\u03c1-mono m n m\u2264n \u2205 \u2205 \u2205 tt = tt\nrel\u03c1-mono m n m\u2264n (\u03c4 \u2022 \u0393) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = relV-mono m n m\u2264n _ v1 v2 vv , rel\u03c1-mono m n m\u2264n \u0393 \u03c11 \u03c12 \u03c1\u03c1\n\n-- relV-mono \u03c4 v1 v2 vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k ?\n\n-- relV-mono : \u2200 \u03c4 v1 v2 n \u2192 relV \u03c4 v1 v2 (suc n) \u2192 relV \u03c4 v1 v2 n\n-- relV-mono \u03c4 v1 v2 zero vv = tt\n-- relV-mono nat v1 v2 (suc n) vv = vv\n-- relV-mono (\u03c3 \u21d2 \u03c4) (closure t1 \u03c11) (closure t2 \u03c12) (suc n) vv k k\u2264n = vv k (\u2264-step k\u2264n)\n\n-- fundamental lemma of logical relations.\nfundamentalV : \u2200 {\u0393 \u03c4} (x : Var \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT (var x) (var x) \u03c11 \u03c12 n\nfundamentalV x zero _ _ _ _ ()\nfundamentalV this (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) .v1 refl = v2 , refl , vv\nfundamentalV (that x) (suc n) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , \u03c1\u03c1) = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1\n\nfundamental : \u2200 {\u0393 \u03c4} (t : Term \u0393 \u03c4) \u2192 (n : \u2115) \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) (\u03c1\u03c1 : rel\u03c1 \u0393 \u03c11 \u03c12 n) \u2192 relT t t \u03c11 \u03c12 n\nfundamental t zero \u03c11 \u03c12 \u03c1\u03c1 v ()\n-- XXX trivial case for constants.\nfundamental (const (lit nv)) (suc n) \u03c11 \u03c12 \u03c1\u03c1 .(intV nv) refl = intV nv , refl , refl\nfundamental (var x) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 refl = fundamentalV x (suc n) \u03c11 \u03c12 \u03c1\u03c1 (\u27e6 x \u27e7Var \u03c11) refl\nfundamental (abs t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 (closure .t .\u03c11) refl =\n  closure t \u03c12 , refl , \u03bb k k\u2264n v1 v2 vv \u2192 fundamental t k (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (vv , rel\u03c1-mono k _ (\u2264-step k\u2264n) _ _ _ \u03c1\u03c1)\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 with eval s \u03c11 n | inspect (eval s \u03c11) n | eval t \u03c11 n\n-- TODO: match sv2 before matching on fundamental s.\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq ] | Done tv1 with fundamental s n \u03c11 \u03c12 (rel\u03c1-mono n _ (n\u22641+n n) _ _ _ \u03c1\u03c1) sv1 eq\nfundamental {\u03c4 = \u03c4} (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 t\u03c11\u2193v1 | Done sv1 | [ eq ] | Done tv1 | (sv2 , s\u03c12\u2193sv2 , svv) = v2 , {!!}\n  where\n    v2 : Val \u03c4\n    v2 = {!!}\n    -- t\u03c12\u2193v2 : apply sv2 tv2 n \u2261 Done v1\n    -- t\u03c12\u2193v2 = ?\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | Done sv | _ | TimeOut\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | Done tv1\nfundamental (app s t) (suc n) \u03c11 \u03c12 \u03c1\u03c1 v1 () | TimeOut | _ | TimeOut\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Thesis\/FunBigStepSILR.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"3134ec26ae49d7f2c2e60ab08a8b332405cd195a","subject":"NCE: draft of non-commiting encryption","message":"NCE: draft of non-commiting encryption\n","repos":"crypto-agda\/crypto-agda","old_file":"Game\/NCE.agda","new_file":"Game\/NCE.agda","new_contents":"{- Separating Random Oracle Proofs from\n   Complexity Theoretic Proofs:\n   The Non-committing Encryption Case\n\nNCE: Non-committing Encryption\n-}\n\nopen import Function\nopen import Type using (\u2605)\nopen import Data.Product\nopen import Data.One\nopen import Data.Two\nopen import Control.Beh\n\nmodule Game.NCE\n  (MId PId Msg Len : \u2605)\n  (length : Msg \u2192 Len)\n  (A : PId) -- adversary PId\n  (F : PId) -- functionality PId (Fnce)\n  where\n\nS : \ud835\udfda \u2192 \u2605\nS = [0: MId \u00d7 PId \u00d7 Msg\n     1: MId \u00d7 PId \u00d7 PId \u00d7 Len ]\n\nIdeal-Fnce' : Beh PId \ud835\udfd9 \ud835\udfd9 \ud835\udfd9 (S 0\u2082) (\u03a3 \ud835\udfda S) \ud835\udfd9\nIdeal-Fnce' = input (\u03bb { i (mid , j , msg) \u2192\n                 output j (0\u2082 , mid , i , msg)\n                  (output A (1\u2082 , mid , i , j , length msg)\n                   end) })\n\nIdeal-Fnce : Beh PId \ud835\udfd9 \ud835\udfd9 \ud835\udfd9 (\u03a3 \ud835\udfda S) (\u03a3 \ud835\udfda S) \ud835\udfd9\nIdeal-Fnce = input (\u03bb i m \u2192\n               case m of \u03bb\n               { (0\u2082 , mid , j , msg) \u2192\n                 output j (0\u2082 , mid , i , msg)\n                  (output A (1\u2082 , mid , i , j , length msg)\n                   end)\n               ; (1\u2082 , _) \u2192 end\n               })\n\nPlainText = \u03a3 \ud835\udfda S\nmodule Real\n  (SecKey PubKey R\u2091 CipherText : \u2605)\n  (pk  : PId \u2192 PubKey)\n  (Enc : PubKey \u2192 R\u2091 \u2192 PlainText \u2192 CipherText)\n  (Dec : SecKey \u2192 CipherText \u2192 PlainText)\n  where\n\n    Real-Fnce : Beh PId SecKey \ud835\udfd9 R\u2091 CipherText CipherText \ud835\udfd9\n    Real-Fnce =\n     input \u03bb i c \u2192\n      ask \u03bb sk \u2192\n       case Dec sk c of \u03bb\n       { (0\u2082 , (mid , j , m)) \u2192\n        rand \u03bb r\u2091 \u2192\n         output j (Enc (pk j) r\u2091 (0\u2082 , mid , i , m))\n          (rand \u03bb r\u2091 \u2192\n            output A\n              (Enc (pk A) r\u2091 (1\u2082 , mid , i , j , length m))\n            end)\n       ; (1\u2082 , _) \u2192 end\n       }\n\nmodule Sim where\n  open Sim[E=\ud835\udfd9,R=1] {PId} {\ud835\udfd9} {\u03a3 \ud835\udfda S} {\u03a3 \ud835\udfda S} {\ud835\udfd9}\n    public\n\n  end-end : end \u2248 end\n  end-end = return-return _\n\nmodule DummyCrypto where\n  open Real \ud835\udfd9 \ud835\udfd9 \ud835\udfd9 (\u03a3 \ud835\udfda S) (const _) (\u03bb _ _ m \u2192 m) (\u03bb _ m \u2192 m)\n  open Sim\n\n  p : Ideal-Fnce \u2248 Real-Fnce\n  p = input-input (\u03bb i m \u2192 nop-ask (helper i m)) where\n    helper : (i : PId) (m : \u03a3 \ud835\udfda S) \u2192 _ \u2248 _\n    helper i (0\u2082 , m) =\n      nop-rand\n        (output-output\n           (nop-rand\n              (output-output end-end)))\n    helper i (1\u2082 , m) =\n      end-end\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Game\/NCE.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"56812c473938f4ef72f3a928595e478d8fe9fb0a","subject":"Added information for the thesis.","message":"Added information for the thesis.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/thesis\/CombiningProofs\/Erasing.agda","new_file":"notes\/thesis\/CombiningProofs\/Erasing.agda","new_contents":"module Erasing where\n\npostulate\n  D     : Set\n  succ\u2081 : D \u2192 D\n\ndata N : D \u2192 Set where\n  nsucc\u2081 : \u2200 {n} \u2192 N n \u2192 N (succ\u2081 n)\n  nsucc\u2082 : \u2200 {n} \u2192 (Nn : N n) \u2192 N (succ\u2081 n)\n\n-- The types of nsucc\u2081 and nsucc\u2082 are the same.\n\n-- (18 April 2013)\n-- $ dump-agdai Erasing\n\n-- Qname: Erasing.N.nsucc\u2081\n-- Type: El {getSort = Type (Max []), unEl = Pi []r{El {getSort = Type (Max []), unEl = Def Erasing.D []}} (Abs \"n\" El {getSort = Type (Max []), unEl = Pi []r(El {getSort = Type (Max []), unEl = Def Erasing.N [[]r(Var 0 [])]}) (NoAbs \"_\" El {getSort = Type (Max []), unEl = Def Erasing.N [[]r(Def Erasing.succ\u2081 [[]r(Var 0 [])])]})})}\n\n-- Qname: Erasing.N.nsucc\u2082\n-- Type: El {getSort = Type (Max []), unEl = Pi []r{El {getSort = Type (Max []), unEl = Def Erasing.D []}} (Abs \"n\" El {getSort = Type (Max []), unEl = Pi []r(El {getSort = Type (Max []), unEl = Def Erasing.N [[]r(Var 0 [])]}) (NoAbs \"Nn\" El {getSort = Type (Max []), unEl = Def Erasing.N [[]r(Def Erasing.succ\u2081 [[]r(Var 0 [])])]})})}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/thesis\/CombiningProofs\/Erasing.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"0bcd1e5cf152745d8a7996f634f7788d390822f9","subject":"Drop superseded TODO","message":"Drop superseded TODO\n\nThe NewNew proof is MUCH simpler.\n","repos":"inc-lc\/ilc-agda","old_file":"New\/Correctness.agda","new_file":"New\/Correctness.agda","new_contents":"module New.Correctness where\n\nopen import Function hiding (const)\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\nopen import New.FunctionLemmas\nopen import New.Unused\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1 \u03c1d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t \u03c1 d\u03c1 \u03c1d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1 \u03c1d\u03c1))\n  (sym (weaken-sound t _))\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 v' \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\n-- module _ {t1 t2 t3 : Type}\n--   (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n--   (df : Cht (t1 \u21d2 t3))\n--   (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n--   (dg : Cht (t2 \u21d2 t3))\n--   where\n--   private\n--     \u0393 = sum t1 t2 \u2022\n--       (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n--       (t2 \u21d2 t3) \u2022\n--       (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n--       (t1 \u21d2 t3) \u2022\n--       sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n--       sum t1 t2 \u2022 \u2205\n--   module _ where\n--     private\n--       \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n--       \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n--     changeMatchSem-lem1 :\n--       \u2200 a1 b2 \u2192\n--         \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n--       \u2261\n--         g b2 \u2295 dg b2 (nil b2) \u229d f a1\n--     changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n--   module _ where\n--     private\n--       \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n--       \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n--     changeMatchSem-lem2 :\n--       \u2200 b1 a2 \u2192\n--         \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n--       \u2261\n--         f a2 \u2295 df a2 (nil a2) \u229d g b1\n--     changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\n-- correctDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\n-- correctDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\n-- correctDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n--   where\n--     lemma : \u2200 s f g \u2192\n--       \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n--       (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n--     lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n--     lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\n-- validDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\n-- validDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\n-- validDeriveConst (match {t1} {t2} {t3}) =\n--   ternary-valid dmatch-valid dmatch-eq\n--   where\n--     open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n--     dmatch-valid : ternary-valid-preserve-hp\n--     dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n--     dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n--     dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem1 f df g dg a1 b2\n--       = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n--     dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem2 f df g dg b1 a2\n--       = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n--     dmatch-eq : ternary-valid-eq-hp\n--     dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n--       rewrite update-nil (a \u2295 da)\n--       | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n--     dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n--       rewrite update-nil (b \u2295 db)\n--       | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n--     dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem1 f df g dg a1 b2\n--       | update-nil b2\n--       | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n--       | update-nil (g b2 \u2295 dg b2 (nil b2))\n--       = refl\n--     dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem2 f df g dg b1 a2\n--       | update-nil a2\n--       | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n--       | update-nil (f a2 \u2295 df a2 (nil a2))\n--       = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1 \u03c1d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term d\u03c1\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext $ \u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid a , refl , \u03c1d\u03c1\n  in\n    -- equal-future-expand-derivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term (correctDerive t)\n    --   \u03c11 d\u03c11 \u03c11d\u03c11\n    --   (a \u2022 (\u03c1 \u2295 d\u03c1))\n    --   (cong (_\u2022 \u03c1 \u2295 d\u03c1) (sym (update-nil a)))\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term d\u03c1\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term d\u03c1\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1 \u03c1d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , refl , \u03c1d\u03c1\n    \u03c12d\u03c12 = ada , refl , \u03c1d\u03c1\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     equal-future-derivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term (correctDerive t)\n       \u03c11 d\u03c11 \u03c11d\u03c11\n       \u03c12 d\u03c12 \u03c12d\u03c12\n       (cong (\u03bb a\u2032 \u2192 (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)))\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","old_contents":"module New.Correctness where\n\nopen import Function hiding (const)\nopen import New.Lang\nopen import New.Changes\nopen import New.Derive\nopen import New.LangChanges\nopen import New.LangOps\nopen import New.FunctionLemmas\nopen import New.Unused\n\n\u27e6\u0393\u227c\u0394\u0393\u27e7 : \u2200 {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n  \u03c1 \u2261 \u27e6 \u0393\u227c\u0394\u0393 \u27e7\u227c d\u03c1\n\u27e6\u0393\u227c\u0394\u0393\u27e7 \u2205 \u2205 tt = refl\n\u27e6\u0393\u227c\u0394\u0393\u27e7 (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = cong\u2082 _\u2022_ refl (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1 \u03c1d\u03c1)\n\nfit-sound : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n  \u27e6 t \u27e7Term \u03c1 \u2261 \u27e6 fit t \u27e7Term d\u03c1\nfit-sound t \u03c1 d\u03c1 \u03c1d\u03c1 = trans\n  (cong \u27e6 t \u27e7Term (\u27e6\u0393\u227c\u0394\u0393\u27e7 \u03c1 d\u03c1 \u03c1d\u03c1))\n  (sym (weaken-sound t _))\n\n-- XXX Should try to simply relate the semantics to the nil change, and prove\n-- that validity can be carried over, instead of proving separately validity and\n-- correctness; elsewhere this does make things simpler.\n\nvalidDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192\n  valid\u0393 \u03c1 d\u03c1 \u2192 valid (\u27e6 x \u27e7Var \u03c1) (\u27e6 x \u27e7\u0394Var \u03c1 d\u03c1)\n\nvalidDeriveVar this (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = vdv\nvalidDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\ncorrectDeriveVar : \u2200 {\u0393 \u03c4} \u2192 (x : Var \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 x \u27e7Var (\u27e6 x \u27e7\u0394Var)\ncorrectDeriveVar this (v \u2022 \u03c1) (dv \u2022 v' \u2022 d\u03c1) \u03c1d\u03c1 = refl\ncorrectDeriveVar (that x) (v \u2022 \u03c1) (dv \u2022 .v \u2022 d\u03c1) (vdv , refl , \u03c1d\u03c1) = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\n\nvalidDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : eCh \u0393) \u2192 valid\u0393 \u03c1 d\u03c1 \u2192\n    valid (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\ncorrectDerive : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192\n  IsDerivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term\n\n\n-- module _ {t1 t2 t3 : Type}\n--   (f : \u27e6 t1 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n--   (df : Cht (t1 \u21d2 t3))\n--   (g : \u27e6 t2 \u27e7Type \u2192 \u27e6 t3 \u27e7Type)\n--   (dg : Cht (t2 \u21d2 t3))\n--   where\n--   private\n--     \u0393 = sum t1 t2 \u2022\n--       (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022\n--       (t2 \u21d2 t3) \u2022\n--       (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022\n--       (t1 \u21d2 t3) \u2022\n--       sum (sum (\u0394t t1) (\u0394t t2)) (sum t1 t2) \u2022\n--       sum t1 t2 \u2022 \u2205\n--   module _ where\n--     private\n--       \u0393\u2032 = t2 \u2022 (t2 \u21d2 \u0394t t2 \u21d2 \u0394t t3) \u2022 (t2 \u21d2 t3) \u2022 \u0393\n--       \u0393\u2032\u2032 = t2 \u2022 \u0393\u2032\n--     changeMatchSem-lem1 :\n--       \u2200 a1 b2 \u2192\n--         \u27e6 match \u27e7\u0394Const (inj\u2081 a1) (inj\u2082 (inj\u2082 b2)) f df g dg\n--       \u2261\n--         g b2 \u2295 dg b2 (nil b2) \u229d f a1\n--     changeMatchSem-lem1 a1 b2 rewrite ominus\u03c4-equiv-ext t2 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n--   module _ where\n--     private\n--       \u0393\u2032 = t1 \u2022 (t1 \u21d2 \u0394t t1 \u21d2 \u0394t t3) \u2022 (t1 \u21d2 t3) \u2022 \u0393\n--       \u0393\u2032\u2032 = t1 \u2022 \u0393\u2032\n--     changeMatchSem-lem2 :\n--       \u2200 b1 a2 \u2192\n--         \u27e6 match \u27e7\u0394Const (inj\u2082 b1) (inj\u2082 (inj\u2081 a2)) f df g dg\n--       \u2261\n--         f a2 \u2295 df a2 (nil a2) \u229d g b1\n--     changeMatchSem-lem2 b1 a2 rewrite ominus\u03c4-equiv-ext t1 \u0393\u2032\u2032 | oplus\u03c4-equiv-ext t3 \u0393\u2032 | ominus\u03c4-equiv-ext t3 \u0393 = refl\n\n\nvalidDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 valid \u27e6 c \u27e7Const (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst : \u2200 {\u03c4} (c : Const \u03c4) \u2192 \u27e6 c \u27e7Const \u2261 \u27e6 c \u27e7Const \u2295 (\u27e6_\u27e7\u0394Const c)\ncorrectDeriveConst plus = ext (\u03bb m \u2192 ext (lemma m))\n  where\n    lemma : \u2200 m n \u2192 m + n \u2261 m + n + (m + - m + (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m + n) = refl\n\ncorrectDeriveConst minus = ext (\u03bb m \u2192 ext (\u03bb n \u2192 lemma m n))\n  where\n    lemma : \u2200 m n \u2192 m - n \u2261 m - n + (m + - m - (n + - n))\n    lemma m n rewrite right-inv-int m | right-inv-int n | right-id-int (m - n) = refl\ncorrectDeriveConst cons = ext (\u03bb v1 \u2192 ext (\u03bb v2 \u2192 sym (update-nil (v1 , v2))))\ncorrectDeriveConst fst = ext (\u03bb vp \u2192 sym (update-nil (proj\u2081 vp)))\ncorrectDeriveConst snd = ext (\u03bb vp \u2192 sym (update-nil (proj\u2082 vp)))\n-- correctDeriveConst linj = ext (\u03bb va \u2192 sym (cong inj\u2081 (update-nil va)))\n-- correctDeriveConst rinj = ext (\u03bb vb \u2192 sym (cong inj\u2082 (update-nil vb)))\n-- correctDeriveConst (match {t1} {t2} {t3}) = ext\u00b3 lemma\n--   where\n--     lemma : \u2200 s f g \u2192\n--       \u27e6 match {t1} {t2} {t3} \u27e7Const s f g \u2261\n--       (\u27e6 match \u27e7Const \u2295 \u27e6 match \u27e7\u0394Const) s f g\n--     lemma (inj\u2081 x) f g rewrite update-nil x | update-nil (f x) = refl\n--     lemma (inj\u2082 y) f g rewrite update-nil y | update-nil (g y) = refl\n\nvalidDeriveConst {\u03c4 = t1 \u21d2 t2 \u21d2 pair .t1 .t2} cons = binary-valid (\u03bb a da ada b db bdb \u2192 (ada , bdb)) dcons-eq\n  where\n    open BinaryValid \u27e6 cons {t1} {t2} \u27e7Const (\u27e6 cons \u27e7\u0394Const)\n    dcons-eq : binary-valid-eq-hp\n    dcons-eq a da ada b db bdb rewrite update-nil (a \u2295 da) | update-nil (b \u2295 db) = refl\nvalidDeriveConst fst (a , b) (da , db) (ada , bdb) = ada , update-nil (a \u2295 da)\nvalidDeriveConst snd (a , b) (da , db) (ada , bdb) = bdb , update-nil (b \u2295 db)\n\nvalidDeriveConst plus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dplus-eq\n  where\n    open BinaryValid \u27e6 plus \u27e7Const (\u27e6 plus \u27e7\u0394Const)\n    dplus-eq : binary-valid-eq-hp\n    dplus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da + (b + db)) = mn\u00b7pq=mp\u00b7nq {a} {da} {b} {db}\nvalidDeriveConst minus = binary-valid (\u03bb a da ada b db bdb \u2192 tt) dminus-eq\n  where\n    open BinaryValid \u27e6 minus \u27e7Const (\u27e6 minus \u27e7\u0394Const)\n    dminus-eq : binary-valid-eq-hp\n    dminus-eq a da ada b db bdb rewrite right-inv-int (a + da) | right-inv-int (b + db) | right-id-int (a + da - (b + db)) | sym (-m\u00b7-n=-mn {b} {db}) = mn\u00b7pq=mp\u00b7nq {a} {da} { - b} { - db}\n-- validDeriveConst linj a da ada = sv\u2081 a da ada , cong inj\u2081 (update-nil (a \u2295 da))\n-- validDeriveConst rinj b db bdb = sv\u2082 b db bdb , cong inj\u2082 (update-nil (b \u2295 db))\n-- validDeriveConst (match {t1} {t2} {t3}) =\n--   ternary-valid dmatch-valid dmatch-eq\n--   where\n--     open TernaryValid {{chAlgt (sum t1 t2)}} {{chAlgt (t1 \u21d2 t3)}} {{chAlgt (t2 \u21d2 t3)}} {{chAlgt t3}} \u27e6 match \u27e7Const (\u27e6 match \u27e7\u0394Const)\n\n--     dmatch-valid : ternary-valid-preserve-hp\n--     dmatch-valid .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg = proj\u2081 (fdf a da ada)\n--     dmatch-valid .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg = proj\u2081 (gdg b db bdb)\n--     dmatch-valid .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem1 f df g dg a1 b2\n--       = \u229d-valid (f a1) (g b2 \u2295 dg b2 (nil b2))\n--     dmatch-valid .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem2 f df g dg b1 a2\n--       = \u229d-valid (g b1) (f a2 \u2295 df a2 (nil a2))\n--     dmatch-eq : ternary-valid-eq-hp\n--     dmatch-eq .(inj\u2081 a) .(inj\u2081 (inj\u2081 da)) (sv\u2081 a da ada) f df fdf g dg gdg\n--       rewrite update-nil (a \u2295 da)\n--       | update-nil (f (a \u2295 da) \u2295 df (a \u2295 da) (nil (a \u2295 da))) = proj\u2082 (fdf a da ada)\n--     dmatch-eq .(inj\u2082 b) .(inj\u2081 (inj\u2082 db)) (sv\u2082 b db bdb) f df fdf g dg gdg\n--       rewrite update-nil (b \u2295 db)\n--       | update-nil (g (b \u2295 db) \u2295 dg (b \u2295 db) (nil (b \u2295 db))) = proj\u2082 (gdg b db bdb)\n--     dmatch-eq .(inj\u2081 a1) .(inj\u2082 (inj\u2082 b2)) (svrp\u2081 a1 b2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem1 f df g dg a1 b2\n--       | update-nil b2\n--       | update-diff (g b2 \u2295 dg b2 (nil b2)) (f a1)\n--       | update-nil (g b2 \u2295 dg b2 (nil b2))\n--       = refl\n--     dmatch-eq .(inj\u2082 b1) .(inj\u2082 (inj\u2081 a2)) (svrp\u2082 b1 a2) f df fdf g dg gdg\n--       rewrite changeMatchSem-lem2 f df g dg b1 a2\n--       | update-nil a2\n--       | update-diff (f a2 \u2295 df a2 (nil a2)) (g b1)\n--       | update-nil (f a2 \u2295 df a2 (nil a2))\n--       = refl\n\ncorrectDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = correctDeriveConst c\ncorrectDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = correctDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\ncorrectDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite sym (fit-sound t \u03c1 d\u03c1 \u03c1d\u03c1) =\n  let\n    open \u2261-Reasoning\n    a0 = \u27e6 t \u27e7Term \u03c1\n    da0 = \u27e6 derive t \u27e7Term d\u03c1\n    a0da0 = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n  in\n    begin\n      \u27e6 s \u27e7Term (\u03c1 \u2295 d\u03c1) (\u27e6 t \u27e7Term (\u03c1 \u2295 d\u03c1))\n    \u2261\u27e8 correctDerive s \u03c1 d\u03c1 \u03c1d\u03c1 \u27e8$\u27e9 correctDerive t \u03c1 d\u03c1 \u03c1d\u03c1 \u27e9\n      (\u27e6 s \u27e7Term \u03c1 \u2295 \u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1 \u2295 \u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u2261\u27e8 proj\u2082 (validDerive s \u03c1 d\u03c1 \u03c1d\u03c1 a0 da0 a0da0)  \u27e9\n      \u27e6 s \u27e7Term \u03c1 (\u27e6 t \u27e7Term \u03c1) \u2295 (\u27e6 s \u27e7\u0394Term \u03c1 d\u03c1) (\u27e6 t \u27e7Term \u03c1) (\u27e6 t \u27e7\u0394Term \u03c1 d\u03c1)\n    \u220e\n  where\n    open import Theorem.CongApp\ncorrectDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 = ext $ \u03bb a \u2192\n  let\n    open \u2261-Reasoning\n    \u03c11 = a \u2022 \u03c1\n    d\u03c11 = nil a \u2022 a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid a , refl , \u03c1d\u03c1\n  in\n    -- equal-future-expand-derivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term (correctDerive t)\n    --   \u03c11 d\u03c11 \u03c11d\u03c11\n    --   (a \u2022 (\u03c1 \u2295 d\u03c1))\n    --   (cong (_\u2022 \u03c1 \u2295 d\u03c1) (sym (update-nil a)))\n    begin\n      \u27e6 t \u27e7Term (a \u2022 \u03c1 \u2295 d\u03c1)\n    \u2261\u27e8 cong (\u03bb a\u2032 \u2192 \u27e6 t \u27e7Term (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (sym (update-nil a)) \u27e9\n      \u27e6 t \u27e7Term (\u03c11 \u2295 d\u03c11)\n    \u2261\u27e8 correctDerive t \u03c11 d\u03c11 \u03c11d\u03c11 \u27e9\n      \u27e6 t \u27e7Term \u03c11 \u2295 \u27e6 t \u27e7\u0394Term \u03c11 d\u03c11\n    \u220e\n\nvalidDerive (app s t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  let\n    f = \u27e6 s \u27e7Term \u03c1\n    df = \u27e6 derive s \u27e7Term d\u03c1\n    v = \u27e6 t \u27e7Term \u03c1\n    dv = \u27e6 derive t \u27e7Term d\u03c1\n    vdv = validDerive t \u03c1 d\u03c1 \u03c1d\u03c1\n    fdf = validDerive s \u03c1 d\u03c1 \u03c1d\u03c1\n    fvdfv = proj\u2081 (fdf v dv vdv)\n  in subst (\u03bb v\u2032 \u2192 valid (f v) (df v\u2032 dv)) (fit-sound t \u03c1 d\u03c1 \u03c1d\u03c1) fvdfv\nvalidDerive (abs t) \u03c1 d\u03c1 \u03c1d\u03c1 =\n  \u03bb a da ada \u2192\n  let\n    \u03c11 = a \u2295 da \u2022 \u03c1\n    d\u03c11 = nil (a \u2295 da) \u2022 (a \u2295 da) \u2022 d\u03c1\n    \u03c12 = a \u2022 \u03c1\n    d\u03c12 = da \u2022 a \u2022 d\u03c1\n    \u03c11d\u03c11 = nil-valid (a \u2295 da) , refl , \u03c1d\u03c1\n    \u03c12d\u03c12 = ada , refl , \u03c1d\u03c1\n    rdr = validDerive t \u03c12 d\u03c12 \u03c12d\u03c12\n    open \u2261-Reasoning\n   in\n     rdr ,\n     equal-future-derivative \u27e6 t \u27e7Term \u27e6 t \u27e7\u0394Term (correctDerive t)\n       \u03c11 d\u03c11 \u03c11d\u03c11\n       \u03c12 d\u03c12 \u03c12d\u03c12\n       (cong (\u03bb a\u2032 \u2192 (a\u2032 \u2022 \u03c1 \u2295 d\u03c1)) (update-nil (a \u2295 da)))\nvalidDerive (var x) \u03c1 d\u03c1 \u03c1d\u03c1 = validDeriveVar x \u03c1 d\u03c1 \u03c1d\u03c1\nvalidDerive (const c) \u03c1 d\u03c1 \u03c1d\u03c1 rewrite \u27e6 c \u27e7\u0394Const-rewrite \u03c1 d\u03c1 = validDeriveConst c\n","returncode":0,"stderr":"","license":"mit","lang":"Agda"}
{"commit":"ba8c3608b8cfce772326b2cb41a1e7e98bd087a6","subject":"ZK.JSChecker.Proof: shows that the 2 definitions for the verifier of PoK-CP-EG-rnd are equivalent","message":"ZK.JSChecker.Proof: shows that the 2 definitions for the verifier of PoK-CP-EG-rnd are equivalent\n","repos":"crypto-agda\/crypto-agda","old_file":"ZK\/JSChecker\/Proof.agda","new_file":"ZK\/JSChecker\/Proof.agda","new_contents":"{-# OPTIONS --without-K #-}\nopen import ZK.JSChecker using (verify-PoK-CP-EG-rnd; PoK-CP-EG-rnd; module PoK-CP-EG-rnd)\nopen import FFI.JS.BigI using (BigI)\nopen import Data.Product using (_,_)\nopen import Crypto.JS.BigI.FiniteField\nopen import Crypto.JS.BigI.CyclicGroup\nopen import Relation.Binary.PropositionalEquality.Base using (_\u2261_; refl)\nimport ZK.GroupHom.ElGamal.JS\n\nmodule ZK.JSChecker.Proof (p q : BigI) (pok : PoK-CP-EG-rnd \u2124[ q ] \u2124[ p ]\u2605) where\n\nopen module ^-h = ^-hom p {q} using (_^_)\nopen module pok = PoK-CP-EG-rnd pok\n\npf : ZK.GroupHom.ElGamal.JS.known-enc-rnd.verify-transcript p q g y (g ^ m) (\u03b1 , \u03b2) (A , B) (\u2124q\u25b9BigI q c) s\n   \u2261 verify-PoK-CP-EG-rnd p q pok\npf = refl\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ZK\/JSChecker\/Proof.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"fc353f2a4a5ff86e967d4d77b4db17f2c3211647","subject":"ECC\/ec-linkable-ring-sig.agda","message":"ECC\/ec-linkable-ring-sig.agda\n","repos":"crypto-agda\/crypto-agda","old_file":"ECC\/ec-linkable-ring-sig.agda","new_file":"ECC\/ec-linkable-ring-sig.agda","new_contents":"module ec-linkable-ring-sig where\n\nopen import Data.Nat  using (\u2115;   zero; suc)\nopen import Data.Fin  using (Fin; zero; suc; from\u2115)\nopen import Data.Vec  using (Vec; tabulate; lookup)\nopen import Data.Bool using (Bool; true; false; if_then_else_)\n\n_\u203c_ : \u2200 {a} {A : Set a} {n} \u2192 Vec A n \u2192 Fin n \u2192 A\nv \u203c i = lookup i v\n\ninfix 5 _+_ _-_\ninfix 6 _\u00b7_ _*_\npostulate\n    Point    : Set\n    Msg      : Set\n    FieldN   : Set\n    FieldP   : Set\n\n    _\u00b7_ : FieldN \u2192 Point \u2192 Point\n    _+_ : Point \u2192 Point \u2192 Point\n    _*_ : FieldN \u2192 FieldN \u2192 FieldN\n    _-_ : FieldN \u2192 FieldN \u2192 FieldN\n    _==_ : FieldN \u2192 FieldN \u2192 Bool\n    G   : Point\n    N-1 : \u2115\n    map-to-point : FieldP \u2192 Point\n\nN : \u2115\nN = suc N-1\n\nPrvKey   : Set\nPrvKey   = FieldN\nPubKey   : Set\nPubKey   = Point\n[PubKey] : Set\n[PubKey] = Vec PubKey N\n[FieldN] : Set\n[FieldN] = Vec FieldN N\nIndex    : Set\nIndex    = Fin N\n\nmax : Index\nmax = from\u2115 N-1\n\nrecord Sig : Set where\n  constructor sig\n  field\n    c* : FieldN\n    s* : [FieldN]\n    Y* : Point\n\nmodule Core-sign (H* : Point)\n                 (Y  : Index \u2192 PubKey)\n                 (Y* : Point)\n                 (H\u2081 : Point \u2192 Point \u2192 FieldN)\n                 (u  : FieldN)\n                 (s  : [FieldN])\n                 (x\u03c0 : FieldN)\n                 (\u03c0  : Index)\n                 where\n\n    c : Index \u2192 FieldN\n    c i = {!!}\n      where\n        s\u1d62 = s \u203c i\n        c-suc-\u03c0 = H\u2081 (u \u00b7 G) (u \u00b7 H*)\n        c-suc-i = H\u2081 (s\u1d62 \u00b7 G + c i \u00b7 Y i) (s\u1d62 \u00b7 H* + c i \u00b7 Y*)\n\n    s\u03c0  = u - x\u03c0 * c \u03c0\n\n    s* : Index \u2192 FieldN\n    s* i = {!if i \u225f \u03c0 then s\u03c0 else s \u203c i!}\n\n    core-sign : Sig\n    core-sign = record\n      { c* = c zero\n      ; s* = tabulate s*\n      ; Y* = Y* }\n\nmodule _\n  (H\u2081      : [PubKey] \u2192 Point \u2192 Msg \u2192 Point \u2192 Point \u2192 FieldN)\n  (H\u2082      : [PubKey] \u2192 FieldP)\n  (pubKeys : [PubKey])\n  (m       : Msg)\n  where\n\n    private\n        Y : Index \u2192 Point\n        Y i = pubKeys \u203c i\n\n        H* = map-to-point (H\u2082 pubKeys)\n\n    sign : Index \u2192 PrvKey \u2192 [FieldN] \u2192 Sig\n    sign \u03c0 x\u03c0 s{-random-} = sg\n      where\n        Y*  = x\u03c0 \u00b7 H*\n        u   = s \u203c \u03c0\n        H\u2081C = H\u2081 pubKeys Y* m\n        sg  = Core-sign.core-sign H* Y Y* H\u2081C u s x\u03c0 \u03c0\n\n    verify : Sig \u2192 Bool\n    verify (sig c* s* Y*) = res\n      where\n        H\u2081C = H\u2081 pubKeys Y* m\n        module M i where\n            Y\u1d62   = Y i\n            s\u1d62   = s* \u203c i\n            c\u1d62   = {!!}\n            Z\u1d62   = s\u1d62 \u00b7 G  + c\u1d62 \u00b7 Y\u1d62\n            T\u1d62   = s\u1d62 \u00b7 H* + c\u1d62 \u00b7 Y*\n            c\u1d62+1 = H\u2081C Z\u1d62 T\u1d62\n        res = c* == H\u2081C (M.Z\u1d62 max) (M.T\u1d62 max)\n\n{-\nsign-using-ecdsa pubKeys m \u03c0 x\u03c0 s =\n  h = ?\n  r = ?\n  Rx , s = ecdsa.sign h r x\u03c0\n-}\n\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n-- -}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'ECC\/ec-linkable-ring-sig.agda' did not match any file(s) known to git\n","license":"bsd-3-clause","lang":"Agda"}
{"commit":"0fc17f9922dbb50a9bd90a19b52b87c0b357826c","subject":"Added gcd-comm draft.","message":"Added gcd-comm draft.\n\nIgnore-this: ebe434527351634ba618f9a6abd88da4\n\ndarcs-hash:20120124153045-3bd4e-156751b5a8eec9d4b0e2bcf8796b69cc1991ed96.gz\n","repos":"asr\/fotc,asr\/fotc","old_file":"Draft\/FOTC\/Program\/GCD\/Total\/CommutativeI.agda","new_file":"Draft\/FOTC\/Program\/GCD\/Total\/CommutativeI.agda","new_contents":"------------------------------------------------------------------------------\n-- The gcd is commutative\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule Draft.FOTC.Program.GCD.Total.CommutativeI where\n\nopen import Common.Function\n\nopen import FOTC.Base\nopen import FOTC.Data.Nat\nopen import FOTC.Data.Nat.Induction.NonAcc.LexicographicI\nopen import FOTC.Data.Nat.Inequalities\nopen import FOTC.Data.Nat.Inequalities.EliminationProperties\nopen import FOTC.Data.Nat.Inequalities.PropertiesI\nopen import FOTC.Data.Nat.PropertiesI\nopen import FOTC.Program.GCD.Total.EquationsI\nopen import FOTC.Program.GCD.Total.GCD\nopen import FOTC.Relation.Binary.EqReasoning\n\n------------------------------------------------------------------------------\n-- Informal proof:\n-- 1. gcd 0 n = n        -- gcd def\n--            = gcd n 0  -- gcd def\n\n-- 2. gcd n 0 = n        -- gcd def\n--            = gcd 0 n  -- gcd def\n\n-- 3.1. Case: S m > S n\n-- gcd (S m) (S n) = gcd (S m - S n) (S n)  -- gcd def\n--                 = gcd (S n) (S m - S n)  -- IH\n--                 = gcd (S n) (S m)        -- gcd def\n\n-- 3.2. Case: S m \u226e S n\n-- gcd (S m) (S n) = gcd (S m) (S n - S m)  -- gcd def\n--                 = gcd (S n - S m) (S m)  -- IH\n--                 = gcd (S n) (S m)        -- gcd def\n\n------------------------------------------------------------------------------\n-- Commutativity property.\nComm : D \u2192 D \u2192 Set\nComm d\u2081 d\u2082 = gcd d\u2081 d\u2082 \u2261 gcd d\u2082 d\u2081\n{-# ATP definition Comm #-}\n\nx>y\u2192y\u226fx : \u2200 {m n} \u2192 N m \u2192 N n \u2192 GT m n \u2192 NGT n m\nx>y\u2192y\u226fx zN          Nn          0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\nx>y\u2192y\u226fx Nm          zN          _     = 0\u226fx Nm\nx>y\u2192y\u226fx (sN {m} Nm) (sN {n} Nn) Sm>Sn =\n  trans (<-SS m n) (x>y\u2192y\u226fx Nm Nn (trans (sym $ <-SS n m) Sm>Sn))\n\npostulate\n  x\u226fSy\u2192Sy>x : \u2200 {m n} \u2192 N m \u2192 N n \u2192 NGT m (succ\u2081 n) \u2192 GT (succ\u2081 n) m\n-- x\u226fSy\u2192Sy>x {n = n} zN      Nn _ = <-0S n\n-- x\u226fSy\u2192Sy>x {n = n} (sN {m} Nm) Nn h = {!!}\n\n------------------------------------------------------------------------------\n-- gcd 0 0 is commutative.\ngcd-00-comm : Comm zero zero\ngcd-00-comm = refl\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 n) 0 is commutative.\ngcd-S0-comm : \u2200 n \u2192 Comm (succ\u2081 n) zero\ngcd-S0-comm n = trans (gcd-S0 n) (sym (gcd-0S n))\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 m) (succ\u2081 n) when succ\u2081 m > succ\u2081 n is commutative.\ngcd-S>S-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n               Comm (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n) \u2192\n               GT (succ\u2081 m) (succ\u2081 n) \u2192\n               Comm (succ\u2081 m) (succ\u2081 n)\ngcd-S>S-comm {m} {n} Nm Nn ih Sm>Sn =\n  begin\n    gcd (succ\u2081 m) (succ\u2081 n)\n      \u2261\u27e8 gcd-S>S m n Sm>Sn \u27e9\n    gcd (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n)\n      \u2261\u27e8 ih \u27e9\n    gcd (succ\u2081 n) (succ\u2081 m \u2238 succ\u2081 n)\n      \u2261\u27e8 sym (gcd-S\u226fS n m (x>y\u2192y\u226fx (sN Nm) (sN Nn) Sm>Sn)) \u27e9\n    gcd (succ\u2081 n) (succ\u2081 m)\n  \u220e\n\n------------------------------------------------------------------------------\n-- gcd (succ\u2081 m) (succ\u2081 n) when succ\u2081 m \u226f succ\u2081 n is commutative.\ngcd-S\u226fS-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192\n               Comm (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m) \u2192\n               NGT (succ\u2081 m) (succ\u2081 n) \u2192\n               Comm (succ\u2081 m) (succ\u2081 n)\ngcd-S\u226fS-comm {m} {n} Nm Nn ih Sm\u226fSn =\n  begin\n    gcd (succ\u2081 m) (succ\u2081 n)\n      \u2261\u27e8 gcd-S\u226fS m n Sm\u226fSn \u27e9\n    gcd (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m)\n      \u2261\u27e8 ih \u27e9\n    gcd (succ\u2081 n \u2238 succ\u2081 m) (succ\u2081 m)\n      \u2261\u27e8 sym (gcd-S>S n m (x\u226fSy\u2192Sy>x (sN Nm) Nn Sm\u226fSn)) \u27e9\n    gcd (succ\u2081 n) (succ\u2081 m)\n  \u220e\n\n------------------------------------------------------------------------------\n-- gcd m n when m > n is commutative.\ngcd-x>y-comm :\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192\n  (\u2200 {o p} \u2192 N o \u2192 N p \u2192 LT\u2082 o p m n \u2192 Comm o p) \u2192\n  GT m n \u2192\n  Comm m n\ngcd-x>y-comm zN          Nn          _    0>n   = \u22a5-elim $ 0>x\u2192\u22a5 Nn 0>n\ngcd-x>y-comm (sN {n} _)  zN          _    _     = gcd-S0-comm n\ngcd-x>y-comm (sN {m} Nm) (sN {n} Nn) accH Sm>Sn = gcd-S>S-comm Nm Nn ih Sm>Sn\n  where\n  -- Inductive hypothesis.\n  ih : Comm (succ\u2081 m \u2238 succ\u2081 n) (succ\u2081 n)\n  ih = accH {succ\u2081 m \u2238 succ\u2081 n}\n            {succ\u2081 n}\n            (\u2238-N (sN Nm) (sN Nn))\n            (sN Nn)\n            ([Sx\u2238Sy,Sy]<[Sx,Sy] Nm Nn)\n\n------------------------------------------------------------------------------\n-- gcd m n when m \u226f n is commutative.\ngcd-x\u226fy-comm :\n  \u2200 {m n} \u2192 N m \u2192 N n \u2192\n  (\u2200 {o p} \u2192 N o \u2192 N p \u2192 LT\u2082 o p m n \u2192 Comm o p) \u2192\n  NGT m n \u2192\n  Comm m n\ngcd-x\u226fy-comm zN          zN          _    _     = gcd-00-comm\ngcd-x\u226fy-comm zN          (sN {n} _)  _    _     = sym (gcd-S0-comm n)\ngcd-x\u226fy-comm (sN {m} Nm) zN          _    Sm\u226f0  = \u22a5-elim $ S\u226f0\u2192\u22a5 Sm\u226f0\ngcd-x\u226fy-comm (sN {m} Nm) (sN {n} Nn) accH Sm\u226fSn = gcd-S\u226fS-comm Nm Nn ih Sm\u226fSn\n  where\n  -- Inductive hypothesis.\n  ih : Comm (succ\u2081 m) (succ\u2081 n \u2238 succ\u2081 m)\n  ih = accH {succ\u2081 m}\n            {succ\u2081 n \u2238 succ\u2081 m}\n            (sN Nm)\n            (\u2238-N (sN Nn) (sN Nm))\n            ([Sx,Sy\u2238Sx]<[Sx,Sy] Nm Nn)\n\n------------------------------------------------------------------------------\n-- gcd is commutative.\ngcd-comm : \u2200 {m n} \u2192 N m \u2192 N n \u2192 Comm m n\ngcd-comm = wfInd-LT\u2082 P istep\n  where\n  P : D \u2192 D \u2192 Set\n  P i j = Comm i j\n\n  istep : \u2200 {i j} \u2192 N i \u2192 N j \u2192 (\u2200 {k l} \u2192 N k \u2192 N l \u2192 LT\u2082 k l i j \u2192 P k l) \u2192\n          P i j\n  istep Ni Nj accH =\n    [ gcd-x>y-comm Ni Nj accH\n    , gcd-x\u226fy-comm Ni Nj accH\n    ] (x>y\u2228x\u226fy Ni Nj)\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Draft\/FOTC\/Program\/GCD\/Total\/CommutativeI.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"e85d1cb3f7830809e06f08177c95ba6a9f3db5dc","subject":"Added note on structurally smaller.","message":"Added note on structurally smaller.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/StructurallySmaller.agda","new_file":"notes\/StructurallySmaller.agda","new_contents":"","old_contents":"","returncode":128,"stderr":"error: RPC failed; HTTP 403 curl 22 The requested URL returned error: 403\nfatal: the remote end hung up unexpectedly\n","license":"mit","lang":"Agda"}
{"commit":"9fe4c6cd99ceb139133e6e3bd4aedafb79c66147","subject":"Start ILC proof with functional big-step semantics","message":"Start ILC proof with functional big-step semantics\n\nThere's old cruft to remove here.\n","repos":"inc-lc\/ilc-agda","old_file":"Thesis\/BigStep.agda","new_file":"Thesis\/BigStep.agda","new_contents":"module Thesis.BigStep where\n\nopen import Thesis.Syntax hiding (suc)\nopen import Thesis.Lang hiding (\u27e6_\u27e7Context; \u27e6_\u27e7Var; suc)\n\nopen import Relation.Binary.PropositionalEquality\n\n-- open import Base.Syntax.Context Type public\n-- open import Base.Syntax.Vars Type public\n\n-- data Const : (\u03c4 : Type) \u2192 Set where\n--   lit : \u2124 \u2192 Const int\n--   -- succ : Const (int \u21d2 int)\n\n-- data Term (\u0393 : Context) :\n--   (\u03c4 : Type) \u2192 Set where\n--   -- constants aka. primitives\n--   const : \u2200 {\u03c4} \u2192\n--     (c : Const \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   var : \u2200 {\u03c4} \u2192\n--     (x : Var \u0393 \u03c4) \u2192\n--     Term \u0393 \u03c4\n--   app : \u2200 {\u03c3 \u03c4}\n--     (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) \u2192\n--     (t : Term \u0393 \u03c3) \u2192\n--     Term \u0393 \u03c4\n--   -- we use de Bruijn indices, so we don't need binding occurrences.\n--   abs : \u2200 {\u03c3 \u03c4}\n--     (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192\n--     Term \u0393 (\u03c3 \u21d2 \u03c4)\n\ndata Val : Type \u2192 Set\nopen import Base.Denotation.Environment Type Val public\nopen import Base.Data.DependentList public\nopen import Data.Nat.Base using (\u2115; zero; suc)\n\ndata Val where\n  closure : \u2200 {\u0393 \u03c3 \u03c4} \u2192 (t : Term (\u03c3 \u2022 \u0393) \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  intV : \u2200 (n : \u2124) \u2192 Val int\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : Val \u03c3 \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u21d2 \u03c4 \u27e7Type) \u2192 Val (\u03c3 \u21d2 \u03c4)\n  pairV : \u2200 {\u03c3 \u03c4} \u2192 Val \u03c3 \u2192 Val \u03c4 \u2192 Val (pair \u03c3 \u03c4)\n  -- prim : \u2200 {\u03c3 \u03c4} \u2192 (f : \u27e6 \u03c3 \u27e7Type \u2192 Val \u03c4) \u2192 Val (\u03c3 \u21d2 \u03c4)\n\n-- TODO: add defunctionalized interpretations for primitives. Yes, annoying but mechanical.\n\n-- Doesn't work\n-- data Val2 : \u2115 \u2192 Type \u2192 Set where\n--   prim : \u2200 {\u03c3 \u03c4 n} \u2192 (f : Val2 n \u03c3 \u2192 Val2 n \u03c4) \u2192 Val2 (\u2115.suc n) (\u03c3 \u21d2 \u03c4)\n\n-- Standard relational big-step semantics.\ndata _\u22a2_\u2193_ {\u0393} (\u03c1 : \u27e6 \u0393 \u27e7Context) : \u2200 {\u03c4} \u2192 Term \u0393 \u03c4 \u2192 Val \u03c4 \u2192 Set where\n  abs : \u2200 {\u03c4\u2081 \u03c4\u2082} {t : Term (\u03c4\u2081 \u2022 \u0393) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 abs t \u2193 closure t \u03c1\n  app : \u2200 {\u0393\u2032 \u03c4\u2081 \u03c4\u2082 \u03c1\u2032 v\u2082 v\u2032} {t\u2081 : Term \u0393 (\u03c4\u2081 \u21d2 \u03c4\u2082)} {t\u2082 : Term \u0393 \u03c4\u2081} {t\u2032 : Term (\u03c4\u2081 \u2022 \u0393\u2032) \u03c4\u2082} \u2192\n    \u03c1 \u22a2 t\u2081 \u2193 closure t\u2032 \u03c1\u2032 \u2192\n    \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n    (v\u2082 \u2022 \u03c1\u2032) \u22a2 t\u2032 \u2193 v\u2032 \u2192\n    \u03c1 \u22a2 app t\u2081 t\u2082 \u2193 v\u2032\n  var : \u2200 {\u03c4} (x : Var \u0393 \u03c4) \u2192\n    \u03c1 \u22a2 var x \u2193 (\u27e6 x \u27e7Var \u03c1)\n  lit : \u2200 n \u2192\n    \u03c1 \u22a2 const (lit n) \u2193 intV n\n  plus : \u2200 {t1 t2 v1 v2} \u2192\n    \u03c1 \u22a2 t1 \u2193 (intV v1) \u2192\n    \u03c1 \u22a2 t2 \u2193 (intV v2) \u2192\n    \u03c1 \u22a2 app (app (const plus) t1) t2 \u2193 intV (v1 + v2)\n  minus : \u2200 {t1 t2 v1 v2} \u2192\n    \u03c1 \u22a2 t1 \u2193 (intV v1) \u2192\n    \u03c1 \u22a2 t2 \u2193 (intV v2) \u2192\n    \u03c1 \u22a2 app (app (const minus) t1) t2 \u2193 intV (v1 - v2)\n  -- cond-true : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2082 : Val \u03c4} \u2192\n  --   \u03c1 \u22a2 t\u2081 \u2193 true \u2192\n  --   \u03c1 \u22a2 t\u2082 \u2193 v\u2082 \u2192\n  --   \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2082\n  -- cond-false : \u2200 {\u0393 \u03c4} {\u03c1 : Env \u0393} {t\u2081 t\u2082 t\u2083} {v\u2083 : Val \u03c4} \u2192\n  --   \u03c1 \u22a2 t\u2081 \u2193 false \u2192\n  --   \u03c1 \u22a2 t\u2083 \u2193 v\u2083 \u2192\n  --   \u03c1 \u22a2 cond t\u2081 t\u2082 t\u2083 \u2193 v\u2083\n\nimport Base.Denotation.Environment\n-- Den stands for Denotational semantics.\nmodule Den = Base.Denotation.Environment Type \u27e6_\u27e7Type\n\n\u27e6_\u27e7Val : \u2200 {\u03c4} \u2192 Val \u03c4 \u2192 \u27e6 \u03c4 \u27e7Type\n\u27e6_\u27e7Env : \u2200 {\u0393} \u2192 \u27e6 \u0393 \u27e7Context \u2192 Den.\u27e6 \u0393 \u27e7Context\n\n\u27e6 \u2205 \u27e7Env = \u2205\n\u27e6 v \u2022 \u03c1 \u27e7Env = \u27e6 v \u27e7Val \u2022 \u27e6 \u03c1 \u27e7Env\n\n\u27e6 closure t \u03c1 \u27e7Val = \u03bb v \u2192 (\u27e6 t \u27e7Term) (v \u2022 \u27e6 \u03c1 \u27e7Env)\n\u27e6 intV n \u27e7Val = n\n-- \u27e6 prim f \u27e7Val = \u03bb v \u2192 \u27e6 f v \u27e7Val\n\u27e6 pairV a b \u27e7Val = (\u27e6 a \u27e7Val , \u27e6 b \u27e7Val)\n-- \u27e6 true \u27e7Val = true\n-- \u27e6 false \u27e7Val = false\n\n\u21a6-sound : \u2200 {\u0393 \u03c4} \u03c1 (x : Var \u0393 \u03c4) \u2192\n  Den.\u27e6 x \u27e7Var \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 \u27e6 x \u27e7Var \u03c1 \u27e7Val\n\u21a6-sound (px \u2022 \u03c1) this = refl\n\u21a6-sound (px \u2022 \u03c1) (that x) = \u21a6-sound \u03c1 x\n\n\u2193-sound : \u2200 {\u0393 \u03c4 \u03c1 v} {t : Term \u0393 \u03c4} \u2192\n  \u03c1 \u22a2 t \u2193 v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\u2193-sound abs = refl\n\u2193-sound (app \u2193\u2081 \u2193\u2082 \u2193\u2032) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 | \u2193-sound \u2193\u2032 = refl\n\u2193-sound (var x) = \u21a6-sound _ x\n\u2193-sound (lit n) = refl\n\u2193-sound (plus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\u2193-sound (minus \u2193\u2081 \u2193\u2082) rewrite \u2193-sound \u2193\u2081 | \u2193-sound \u2193\u2082 = refl\n\n--\n-- Functional big-step semantics\n--\n\n-- Termination is far from obvious to Agda once we use closures. So we use\n-- step-indexing with a fuel value, following \"Type Soundness Proofs with\n-- Definitional Interpreters\". Since we focus for now on STLC, unlike that\n-- paper, we can avoid error values by keeping types.\n--\n-- One could drop types and add error values, to reproduce what they do.\n\ndata ErrVal (\u03c4 : Type) : Set where\n  Done : (v : Val \u03c4) \u2192 ErrVal \u03c4\n  TimeOut : ErrVal \u03c4\n\nRes : Type \u2192 Set\nRes \u03c4 = \u2115 \u2192 ErrVal \u03c4\n\nevalConst : \u2200 {\u03c4} \u2192 Const \u03c4 \u2192 Res \u03c4\nevalConst (lit v) n = Done (intV v)\nevalConst c n = {!!}\n\n-- evalConst plus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a + b))))\n-- evalConst minus n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 intV (a - b))))\n-- evalConst cons n = Done (prim (\u03bb a \u2192 prim (\u03bb b \u2192 {!pairV !})))\n-- evalConst fst n = {!!}\n-- evalConst snd n = {!!}\n-- evalConst linj n = {!!}\n-- evalConst rinj n = {!!}\n-- evalConst match n = {!!}\n-- evalConst zero = intV (+ 0)\n-- evalConst succ = {!!}\n\neval : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 \u27e6 \u0393 \u27e7Context \u2192 Res \u03c4\n\napply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u03c3 \u21d2 \u03c4) \u2192 Val \u03c3 \u2192 Res \u03c4\napply (closure t \u03c1) a n = eval t (a \u2022 \u03c1) n\n-- apply (prim f) a n = Done (f \u27e6 a \u27e7Val)\n\neval t \u03c1 zero = TimeOut\neval (const c) \u03c1 (suc n) = evalConst c n\neval (var x) \u03c1 (suc n) = Done (\u27e6 x \u27e7Var \u03c1)\neval (abs t) \u03c1 (suc n) = Done (closure t \u03c1)\neval (app s t) \u03c1 (suc n) with eval s \u03c1 n | eval t \u03c1 n\n... | Done f | Done a = apply f a n\n... | _ | _ = TimeOut\n-- eval t0 \u03c1 (suc n) with t0\n-- ... | const c = evalConst c n\n-- ... | var x = Done (\u27e6 x \u27e7Var \u03c1)\n-- ... | abs t = Done (closure t \u03c1)\n-- ... | app s t with eval s \u03c1 n | eval t \u03c1 n\n-- ... | Done f | Done a = apply f a n\n-- ... | _ | _ = TimeOut\n\n-- Can we prove eval sound wrt. our reference denotational semantics? Yes! Very\n-- cool!\neval-sound : \u2200 {\u0393 \u03c4} \u03c1 v n (t : Term \u0393 \u03c4) \u2192\n  eval t \u03c1 n \u2261 Done v \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 v \u27e7Val\n\napply-sound : \u2200 {\u0393 \u03c3 \u03c4} \u03c1 v f a n (s : Term \u0393 (\u03c3 \u21d2 \u03c4)) t \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 f \u27e7Val \u2192\n  \u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env \u2261 \u27e6 a \u27e7Val \u2192\n  apply f a n \u2261 Done v \u2192\n  \u27e6 s \u27e7Term \u27e6 \u03c1 \u27e7Env (\u27e6 t \u27e7Term \u27e6 \u03c1 \u27e7Env) \u2261 \u27e6 v \u27e7Val\napply-sound _ v (closure ft \u03c1) a n s t feq aeq eq rewrite feq | aeq = eval-sound (a \u2022 \u03c1) v n ft eq\n\neval-sound \u03c1 v zero t ()\neval-sound \u03c1 v (\u2115.suc n) (const c) eq = {!!}\neval-sound \u03c1 v (\u2115.suc n) (var x) refl = \u21a6-sound \u03c1 x\neval-sound \u03c1 v (\u2115.suc n) (abs t) refl = refl\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq with eval s \u03c1 n | inspect (eval s \u03c1) n | eval t \u03c1 n | inspect (eval t \u03c1) n\neval-sound \u03c1 v (\u2115.suc n) (app s t) eq | Done f | [ feq ] | Done a | [ aeq ] =\n  let feq = eval-sound \u03c1 f n s feq; aeq = eval-sound \u03c1 a n t aeq in apply-sound \u03c1 v f a n s t feq aeq eq\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | Done f | _ | TimeOut | _\neval-sound \u03c1 v (\u2115.suc n) (app s t) () | TimeOut | _ | _ | _\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n-- eval-sound n (const c) eq = {!!}\n-- eval-sound n (var x) eq = {!!}\n-- eval-sound n (app s t) eq = {!!} -- with eval s \u03c1 n\n-- eval-sound n (abs t) eq = {!!}\n\n-- Next, we can try defining change structures and validity.\n\ndapply : \u2200 {\u03c3 \u03c4} \u2192 Val (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 Val \u03c3 \u2192 Val (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply df a1 da zero = TimeOut\ndapply df a1 da (suc zero) = TimeOut\ndapply df a1 da (suc (suc n)) with apply df a1 n\n... | Done dfa = apply dfa da (suc n)\n... | TimeOut = TimeOut\n\n-- apply (closure (abs (derive t)) d\u03c1) a1 (suc (suc n))\n-- eval (derive (abs t)) d\u03c1 (suc n) \u2261\n-- dapply-sound-0\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) n\n\ndapply' : \u2200 {\u03c3 \u03c4} \u2192 ErrVal (\u0394t (\u03c3 \u21d2 \u03c4)) \u2192 ErrVal \u03c3 \u2192 ErrVal (\u0394t \u03c3) \u2192 Res (\u0394t \u03c4)\ndapply' (Done df) (Done v) (Done dv) n = dapply df v dv n\ndapply' _ _ _ _ = TimeOut\n\n-- Which step-indexes should we use in dapply's specification and\n-- implementation? That's highly non-obvious. To come up with them, I had to\n-- step through the evaluation of eval (app (app ds t) dt), and verify that I\n-- never invoke eval or apply on the same inputs at different step-indexes.\n--\n-- However, I expect I wouldn't be able to get a working proof if I got the\n-- indexes wrong.\ndapply-sound : \u2200 {\u0393 \u03c3 \u03c4} n \u03c1 (ds : Term \u0393 (\u0394t (\u03c3 \u21d2 \u03c4))) t dt \u2192 eval (app (app ds t) dt) \u03c1 (suc (suc n)) \u2261 dapply' (eval ds \u03c1 n) (eval t \u03c1 n) (eval dt \u03c1 (suc n)) (suc (suc n))\ndapply-sound zero \u03c1 ds t dt = refl\ndapply-sound (suc n) \u03c1 ds t dt with eval ds \u03c1 (suc n) | eval t \u03c1 (suc n) | eval dt \u03c1 (suc (suc n))\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | Done dv with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut  = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | Done v | TimeOut with apply df v (suc n)\n... | Done dfv = refl\n... | TimeOut = refl\ndapply-sound (suc n) \u03c1 ds t dt | Done df | TimeOut | dv = refl\ndapply-sound (suc n) \u03c1 ds t dt | TimeOut | v | dv = refl\n\nopen import Data.Empty\nopen import Data.Unit\n\n\u0394\u0393 : Context \u2192 Context\n\u0394\u0393 \u2205 = \u2205\n\u0394\u0393 (\u03c4 \u2022 \u0393) = \u0394t \u03c4 \u2022 \u03c4 \u2022 \u0394\u0393 \u0393\n\nCh\u0393 : \u2200 (\u0393 : Context) \u2192 Set\nCh\u0393 \u0393 = \u27e6 \u0394\u0393 \u0393 \u27e7Context\n\u0393\u227c\u0394\u0393 : \u2200 {\u0393} \u2192 \u0393 \u227c \u0394\u0393 \u0393\n\u0393\u227c\u0394\u0393 {\u2205} = \u2205\n\u0393\u227c\u0394\u0393 {\u03c4 \u2022 \u0393} = drop \u0394t \u03c4 \u2022 keep \u03c4 \u2022 \u0393\u227c\u0394\u0393\n\nfit : \u2200 {\u03c4 \u0393} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) \u03c4\nfit = weaken \u0393\u227c\u0394\u0393\n\nderiveConst : \u2200 {\u03c4} \u2192\n  Const \u03c4 \u2192\n  Term \u2205 (\u0394t \u03c4)\nderiveConst = {!!}\n\nderiveVar : \u2200 {\u0393 \u03c4} \u2192 Var \u0393 \u03c4 \u2192 Var (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderiveVar this = this\nderiveVar (that x) = that (that (deriveVar x))\n\nderive : \u2200 {\u0393 \u03c4} \u2192 Term \u0393 \u03c4 \u2192 Term (\u0394\u0393 \u0393) (\u0394t \u03c4)\nderive (const c) = weaken (\u2205\u227c\u0393 {\u0394\u0393 _}) (deriveConst c)\nderive (var x) = var (deriveVar x)\nderive (app s t) = app (app (derive s) (fit t)) (derive t)\nderive (abs t) = abs (abs (derive t))\n\n\n-- open import Thesis.Derive\nCh\u03c4 : Type \u2192 Set\nCh\u03c4 \u03c4 = Val (\u0394t \u03c4)\n\ndeval : \u2200 {\u0393 \u03c4} \u2192 (t : Term \u0393 \u03c4) \u2192 (\u03c1 : \u27e6 \u0393 \u27e7Context) (d\u03c1 : Ch\u0393 \u0393) \u2192 Res (\u0394t \u03c4)\ndeval t \u03c1 d\u03c1 = eval (derive t) d\u03c1\n\n\ndapply-equiv : \u2200 {\u0393 \u03c3 \u03c4} n\n  (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n  (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n  dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc (suc n))) \u2261\n  eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc (suc n))\ndapply-equiv n t d\u03c1 \u03c11 da a1 = refl\n\n-- Failed variants, the step-indexes are wrong:\n-- dapply-equiv-2 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc (suc n)) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-2 (suc n) t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 zero t d\u03c1 \u03c11 da a1 = refl\n-- -- dapply-equiv-2 (suc zero) (const c) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (var x) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (app t t\u2081) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc zero) (abs t) d\u03c1 \u03c11 da a1 = {!!}\n-- -- dapply-equiv-2 (suc (suc n)) t d\u03c1 \u03c11 da a1 = refl\n\n-- Worse:\n-- dapply-equiv-0 : \u2200 {\u0393 \u03c3 \u03c4} n\n--   (t : Term (\u03c3 \u2022 \u0393) \u03c4)\n--   (d\u03c1 : Ch\u0393 \u0393) (\u03c11 : \u27e6 \u0393 \u27e7Context)\n--   (da : Val (\u0394t \u03c3)) (a1 : Val \u03c3) \u2192\n--   dapply (closure (abs (derive t)) d\u03c1) a1 da (suc n) \u2261\n--   eval (derive t) (da \u2022 a1 \u2022 d\u03c1) (suc n)\n-- dapply-equiv-0 zero t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc zero) t d\u03c1 \u03c11 da a1 = {!!}\n-- dapply-equiv-0 (suc (suc n)) t d\u03c1 \u03c11 da a1 = {!!}\n\n-- Target statement:\n\n_[_]\u03c4_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 Val (\u0394t \u03c4) \u2192 Val \u03c4 \u2192 Val \u03c4 \u2192 Set\n_[_]\u03c4'_from_to_ : (n : \u2115) \u2192 \u2200 \u03c4 \u2192 ErrVal (\u0394t \u03c4) \u2192 ErrVal \u03c4 \u2192 ErrVal \u03c4 \u2192 Set\nn [ \u03c4 ]\u03c4' Done dv from Done v1 to Done v2 = n [ \u03c4 ]\u03c4 dv from v1 to v2\nn [ \u03c4 ]\u03c4' TimeOut from Done v1 to Done v2 = \u22a5\nn [ \u03c4 ]\u03c4' dv from _ to _ = \u22a4\n\n_[_]\u03c4'_fromToTimeOut : \u2200 n \u03c4 edv \u2192 n [ \u03c4 ]\u03c4' edv from TimeOut to TimeOut \u2261 \u22a4\nn [ \u03c4 ]\u03c4' Done v fromToTimeOut = refl\nn [ \u03c4 ]\u03c4' TimeOut fromToTimeOut = refl\n-- n [ \u03c4 ]\u03c4' dv from Done v1 to TimeOut = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to Done v2 = \u22a4\n-- n [ \u03c4 ]\u03c4' dv from TimeOut to TimeOut = \u22a4\n\n-- In the static-caching paper, the specification we use is similar to this one:\n-- n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n--    \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n--        dapply df a1 da n \u2261 Done dv\n--      \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n-- I defined _[_]\u03c4'_from_to_ to be able to write an equivalent (hopefully) specification more quickly.\n\nzero [ \u03c4 ]\u03c4 dv from v1 to v2 = \u22a4\n-- suc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 =\n--   \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n--    n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n--    n [ \u03c4 ]\u03c4' dapply df a1 da n from apply f1 a1 n to apply f2 a2 n\n\nsuc n [ \u03c3 \u21d2 \u03c4 ]\u03c4 df from f1 to f2 = \u2200 (da : Val (\u0394t \u03c3)) \u2192 (a1 a2 : Val \u03c3) \u2192\n   n [ \u03c3 ]\u03c4 da from a1 to a2 \u2192\n   \u2200 v1 v2 \u2192 apply f1 a1 n \u2261 Done v1 \u2192 apply f2 a2 n \u2261 Done v2 \u2192\n   \u03a3[ dv \u2208 Val (\u0394t \u03c4) ]\n       dapply df a1 da (suc (suc n)) \u2261 Done dv\n     \u00d7 n [ \u03c4 ]\u03c4 dv from v1 to v2\n\nsuc n [ int ]\u03c4 intV dn from intV n1 to intV n2 = n1 + dn \u2261 n2\nsuc n [ pair \u03c3 \u03c4 ]\u03c4 pairV da db from pairV a1 b1 to pairV a2 b2 =\n  suc n [ \u03c3 ]\u03c4 da from a1 to a2 \u00d7 suc n [ \u03c4 ]\u03c4 db from b1 to b2\nsuc n [ sum \u03c3 \u03c4 ]\u03c4 () from v1 to v2\n\ndata _[_]\u0393_from_to_ : \u2200 \u2115 \u0393 \u2192 Ch\u0393 \u0393 \u2192 (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 Set where\n  v\u2205 : \u2200 {n} \u2192 n [ \u2205 ]\u0393 \u2205 from \u2205 to \u2205\n  _v\u2022_ : \u2200 {n \u03c4 \u0393 dv v1 v2 d\u03c1 \u03c11 \u03c12} \u2192\n    (dvv : n [ \u03c4 ]\u03c4 dv from v1 to v2) \u2192\n    (d\u03c1\u03c1 : n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12) \u2192\n    n [ \u03c4 \u2022 \u0393 ]\u0393 (dv \u2022 v1 \u2022 d\u03c1) from (v1 \u2022 \u03c11) to (v2 \u2022 \u03c12)\n\nfromtoDeriveVar : \u2200 {\u0393} \u03c4 n \u2192 (x : Var \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4 (\u27e6 deriveVar x \u27e7Var d\u03c1) from \u27e6 x \u27e7Var \u03c11 to \u27e6 x \u27e7Var \u03c12\nfromtoDeriveVar \u03c4 n this (dv \u2022 .(v1 \u2022 _)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = dvv\nfromtoDeriveVar \u03c4 n (that x) (dv \u2022 (.v1 \u2022 d\u03c1)) (v1 \u2022 \u03c11) (v2 \u2022 \u03c12) (dvv v\u2022 d\u03c1\u03c1) = fromtoDeriveVar \u03c4 n x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\n\nfromtoDerive : \u2200 {\u0393} \u03c4 n \u2192 (t : Term \u0393 \u03c4) \u2192\n  (d\u03c1 : Ch\u0393 \u0393) (\u03c11 \u03c12 : \u27e6 \u0393 \u27e7Context) \u2192 n [ \u0393 ]\u0393 d\u03c1 from \u03c11 to \u03c12 \u2192\n    n [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 n) from eval t \u03c11 n to eval t \u03c12 n\nfromtoDerive \u03c4 zero t d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 rewrite zero [ \u03c4 ]\u03c4' (deval t \u03c11 d\u03c1 (suc zero)) fromToTimeOut = tt\nfromtoDerive \u03c4 (suc n) (const c) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\nfromtoDerive \u03c4 (suc n) (var x) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = fromtoDeriveVar \u03c4 (suc n) x d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1\nfromtoDerive \u03c4 (suc n) (app s t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc n) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   rewrite dapply-equiv n t d\u03c1 \u03c11 da a1 =\n--   fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 d\u03c1\u03c1)\n\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc zero) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 () eqv2\nfromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc n)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa v1 v2 eqv1 eqv2 = {!fromtoDerive \u03c4 (suc n) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) ?!}\n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc zero)) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa = {!!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa\n--   with eval (abs (derive t)) (a1 \u2022 d\u03c1) n | inspect (eval (abs (derive t)) (a1 \u2022 d\u03c1)) n\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | Done v | [ eq ] = {!fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) _!}\n-- fromtoDerive (\u03c3 \u21d2 \u03c4) (suc (suc (suc n))) (abs t) d\u03c1 \u03c11 \u03c12 d\u03c1\u03c1 da a1 a2 daa | TimeOut | _ = {!!}\n-- --   fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) (daa v\u2022 ?)\n-- -- fromtoDerive \u03c4 (suc (suc n)) t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n--   -- fromtoDerive \u03c4 n t (da \u2022 a1 \u2022 d\u03c1) (a1 \u2022 \u03c11) (a2 \u2022 \u03c12) {! daa v\u2022 d\u03c1\u03c1!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'Thesis\/BigStep.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"a48ff25823603032db6b1a26e4605512aefd0561","subject":"Add incomplete note on co-inductive natural numbers.","message":"Add incomplete note on co-inductive natural numbers.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda","new_file":"notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda","new_contents":"------------------------------------------------------------------------------\n-- Definition of FOTC Conat using Agda's co-inductive combinators\n------------------------------------------------------------------------------\n\n{-# OPTIONS --allow-unsolved-metas #-}\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\nmodule FOT.FOTC.Data.Conat.ConatSL where\n\nopen import FOTC.Base\nopen import Coinduction\n\n------------------------------------------------------------------------------\n\ndata Conat : D \u2192 Set where\n  cozero : Conat zero\n  cosucc : \u2200 {n} \u2192 (\u221e (Conat n)) \u2192 Conat (succ\u2081 n)\n\nConat-unf : \u2200 {n} \u2192 Conat n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')\nConat-unf cozero = inj\u2081 refl\nConat-unf (cosucc {n} Cn) = inj\u2082 (n , \u266d Cn , refl)\n\nConat-pre-fixed : \u2200 {n} \u2192\n                  (n \u2261 zero \u2228 (\u2203[ n' ] Conat n' \u2227 n \u2261 succ\u2081 n')) \u2192\n                  Conat n\nConat-pre-fixed (inj\u2081 h) = subst Conat (sym h) cozero\nConat-pre-fixed (inj\u2082 (n , Cn , h)) = subst Conat (sym h) (cosucc (\u266f Cn))\n\nConat-coind : \u2200 (A : D \u2192 Set) {n} \u2192\n              (A n \u2192 n \u2261 zero \u2228 (\u2203[ n' ] A n' \u2227 n \u2261 succ\u2081 n')) \u2192\n              A n \u2192 Conat n\nConat-coind A h An = {!!}\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Data\/Conat\/ConatSL.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}
{"commit":"7901c93c035a96fb8df26abb488c38c80a35ef7b","subject":"[ draft ] ABP's lemma1 without helper function.","message":"[ draft ] ABP's lemma1 without helper function.\n","repos":"asr\/fotc,asr\/fotc","old_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1WithoutHelperATP.agda","new_file":"notes\/FOT\/FOTC\/Program\/ABP\/Lemma1WithoutHelperATP.agda","new_contents":"------------------------------------------------------------------------------\n-- ABP lemma 1\n------------------------------------------------------------------------------\n\n{-# OPTIONS --no-universe-polymorphism #-}\n{-# OPTIONS --without-K #-}\n\n-- From Dybjer and Sander's paper: The first lemma states that given a\n-- start state S of the ABP, we will arrive at a state S', where the\n-- message has been received by the receiver, but where the\n-- acknowledgement has not yet been received by the sender.\n\nmodule FOT.FOTC.Program.ABP.Lemma1WithoutHelperATP where\n\nopen import FOTC.Base\nopen import FOTC.Base.List\nopen import FOTC.Base.Loop\nopen import FOTC.Data.Bool\nopen import FOTC.Data.Bool.PropertiesATP using ( x\u2262not-x )\nopen import FOTC.Data.List\nopen import FOTC.Program.ABP.ABP\nopen import FOTC.Program.ABP.Fair.Type\nopen import FOTC.Program.ABP.Fair.PropertiesATP\nopen import FOTC.Program.ABP.Terms\n\n------------------------------------------------------------------------------\n\nas^ : \u2200 b i' is' ds \u2192 D\nas^ b i' is' ds = await b i' is' ds\n{-# ATP definition as^ #-}\n\nbs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nbs^ b i' is' ds os\u2081^ = corrupt os\u2081^ \u00b7 (as^ b i' is' ds)\n{-# ATP definition bs^ #-}\n\ncs^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\ncs^ b i' is' ds os\u2081^ = ack b \u00b7 (bs^ b i' is' ds os\u2081^)\n{-# ATP definition cs^ #-}\n\nds^ : D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D \u2192 D\nds^ b i' is' ds os\u2081^ os\u2082^ = corrupt os\u2082^ \u00b7 cs^ b i' is' ds os\u2081^\n{-# ATP definition ds^ #-}\n\nos\u2081^ : D \u2192 D \u2192 D\nos\u2081^ os\u2081' ft\u2081^ = ft\u2081^ ++ os\u2081'\n{-# ATP definition os\u2081^ #-}\n\nos\u2082^ : D \u2192 D\nos\u2082^ os\u2082 = tail\u2081 os\u2082\n{-# ATP definition os\u2082^ #-}\n\n-- Helper function for the ABP lemma 1\n-- module Helper where\n--   -- 30 November 2013. If we don't have the following definitions\n--   -- outside the where clause, the ATPs cannot prove the theorems.\n\n--   helper : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n--            Bit b \u2192\n--            Fair os\u2082 \u2192\n--            S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n--            \u2200 ft\u2081 os\u2081' \u2192 F*T ft\u2081 \u2192 Fair os\u2081' \u2192 os\u2081 \u2261 ft\u2081 ++ os\u2081' \u2192\n--            \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--              Fair os\u2081'\n--              \u2227 Fair os\u2082'\n--              \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n--              \u2227 js \u2261 i' \u2237 js'\n--   helper b i' is' os\u2081 os\u2082 as bs cs ds js\n--          Bb Fos\u2082 s .(T \u2237 []) os\u2081' f*tnil Fos\u2081' os\u2081-eq = prf\n--     where\n--     postulate\n--       prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n--               Fair os\u2081'\n--               \u2227 Fair os\u2082'\n--               \u2227 (as' \u2261 await b i' is' ds'\n--                  \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n--                  \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n--                  \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n--                  \u2227 js' \u2261 out (not b) \u00b7 bs')\n--               \u2227 js \u2261 i' \u2237 js'\n--     {-# ATP prove prf #-}\n\n--   -- TODO (25 January 2014): Why Agda cannot refine using the helper\n--   -- function? Agda bug?\n--   helper b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2082 s\n--          .(F \u2237 ft\u2081^) os\u2081' (f*tcons {ft\u2081^} FTft\u2081^) Fos\u2081' os\u2081-eq =\n--          helper b i' is'\n--                 (ft\u2081^ ++ os\u2081')\n--                 (tail\u2081 os\u2082)\n--                 (await b i' is' ds)\n--                 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n--                 (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n--                 (corrupt (tail\u2081 os\u2082) \u00b7\n--                    (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n--                 js\n--                 Bb (tail-Fair Fos\u2082) ihS ft\u2081^ os\u2081' FTft\u2081^ Fos\u2081' refl\n--     where\n--     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n--     {-# ATP prove os\u2081-eq-helper #-}\n\n--     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n--     {-# ATP prove as-eq #-}\n\n--     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n--     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n--     {-# ATP prove cs-eq bs-eq #-}\n\n--     postulate\n--       ds-eq-helper\u2081 :\n--         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n--         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n--     postulate\n--       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n--                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n--     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n--                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n--                  (head-tail-Fair Fos\u2082)\n\n--     postulate\n--       as^-eq-helper\u2081 :\n--         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n--         as^ b i' is' ds \u2261\n--           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n--     postulate\n--       as^-eq-helper\u2082 :\n--         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n--         as^ b i' is' ds \u2261\n--           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     {-# ATP prove as^-eq-helper\u2082 #-}\n\n--     as^-eq : as^ b i' is' ds \u2261\n--              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n--     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n--     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n--     {-# ATP prove js-eq bs-eq #-}\n\n--     ihS : S b\n--             (i' \u2237 is')\n--             (os\u2081^ os\u2081' ft\u2081^)\n--             (os\u2082^ os\u2082)\n--             (as^ b i' is' ds)\n--             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n--             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n--             js\n--     ihS = as^-eq , refl , refl , refl , js-eq\n\n------------------------------------------------------------------------------\n-- From Dybjer and Sander's paper: From the assumption that os\u2081\u00a0\u2208\u00a0Fair\n-- and hence by unfolding Fair, we conclude that there are ft\u2081\u00a0:\u00a0 F*T\n-- and os\u2081'\u00a0:\u00a0Fair, such that os\u2081\u00a0=\u00a0ft\u2081\u00a0++ os\u2081'.\n--\n-- We proceed by induction on ft\u2081\u00a0:\u00a0F*T using helper.\n\n-- open Helper\nlemma\u2081 : \u2200 b i' is' os\u2081 os\u2082 as bs cs ds js \u2192\n         Bit b \u2192\n         Fair os\u2081 \u2192\n         Fair os\u2082 \u2192\n         S b (i' \u2237 is') os\u2081 os\u2082 as bs cs ds js \u2192\n         \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n           Fair os\u2081'\n           \u2227 Fair os\u2082'\n           \u2227 S' b i' is' os\u2081' os\u2082' as' bs' cs' ds' js'\n           \u2227 js \u2261 i' \u2237 js'\nlemma\u2081 b i' is' os\u2081 os\u2082 as bs cs ds js Bb Fos\u2081 Fos\u2082 s with Fair-out Fos\u2081\n... | .(true \u2237 []) , os\u2081' , f*tnil , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' = prf\n  where\n  postulate\n    prf : \u2203[ os\u2081' ] \u2203[ os\u2082' ] \u2203[ as' ] \u2203[ bs' ] \u2203[ cs' ] \u2203[ ds' ] \u2203[ js' ]\n            Fair os\u2081'\n            \u2227 Fair os\u2082'\n            \u2227 (as' \u2261 await b i' is' ds'\n              \u2227 bs' \u2261 corrupt os\u2081' \u00b7 as'\n              \u2227 cs' \u2261 ack (not b) \u00b7 bs'\n              \u2227 ds' \u2261 corrupt os\u2082' \u00b7 (b \u2237 cs')\n              \u2227 js' \u2261 out (not b) \u00b7 bs')\n            \u2227 js \u2261 i' \u2237 js'\n  {-# ATP prove prf #-}\n\n... | .(F \u2237 ft\u2081^) , os\u2081' , f*tcons {ft\u2081^} FTft\u2081 , os\u2081\u2261ft\u2081++os\u2081' , Fos\u2081' =\n  lemma\u2081 b i' is'\n         (ft\u2081^ ++ os\u2081')\n         (tail\u2081 os\u2082)\n         (await b i' is' ds)\n         (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)\n         (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds))\n         (corrupt (tail\u2081 os\u2082) \u00b7\n           (ack b \u00b7 (corrupt (ft\u2081^ ++ os\u2081') \u00b7 await b i' is' ds)))\n         js Bb ft\u2081^++-os\u2081'-Fair ((tail-Fair Fos\u2082)) ihS\n     where\n     ft\u2081^++-os\u2081'-Fair : Fair (ft\u2081^ ++ os\u2081')\n     ft\u2081^++-os\u2081'-Fair = Fair-in (ft\u2081^ , os\u2081' , FTft\u2081 , refl , Fos\u2081')\n\n     postulate os\u2081-eq-helper : os\u2081 \u2261 F \u2237 os\u2081^ os\u2081' ft\u2081^\n     {-# ATP prove os\u2081-eq-helper #-}\n\n     postulate as-eq : as \u2261 < i' , b > \u2237 (as^ b i' is' ds)\n     {-# ATP prove as-eq #-}\n\n     postulate bs-eq : bs \u2261 error \u2237 (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n     {-# ATP prove bs-eq os\u2081-eq-helper as-eq #-}\n\n     postulate cs-eq : cs \u2261 not b \u2237 cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove cs-eq bs-eq #-}\n\n     postulate\n       ds-eq-helper\u2081 :\n         os\u2082 \u2261 T \u2237 tail\u2081 os\u2082 \u2192\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2081 cs-eq #-}\n\n     postulate\n       ds-eq-helper\u2082 : os\u2082 \u2261 F \u2237 tail\u2081 os\u2082 \u2192\n                       ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove ds-eq-helper\u2082 cs-eq #-}\n\n     ds-eq : ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n             \u2228 ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     ds-eq = case (\u03bb h \u2192 inj\u2081 (ds-eq-helper\u2081 h))\n                  (\u03bb h \u2192 inj\u2082 (ds-eq-helper\u2082 h))\n                  (head-tail-Fair Fos\u2082)\n\n     postulate\n       as^-eq-helper\u2081 :\n         ds \u2261 ok (not b) \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2081 x\u2262not-x #-}\n\n     postulate\n       as^-eq-helper\u2082 :\n         ds \u2261 error \u2237 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082) \u2192\n         as^ b i' is' ds \u2261\n           send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     {-# ATP prove as^-eq-helper\u2082 #-}\n\n     as^-eq : as^ b i' is' ds \u2261\n              send b \u00b7 (i' \u2237 is') \u00b7 ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082)\n     as^-eq = case as^-eq-helper\u2081 as^-eq-helper\u2082 ds-eq\n\n     postulate js-eq : js \u2261 out b \u00b7 bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^)\n     {-# ATP prove js-eq bs-eq #-}\n\n     ihS : S b\n             (i' \u2237 is')\n             (os\u2081^ os\u2081' ft\u2081^)\n             (os\u2082^ os\u2082)\n             (as^ b i' is' ds)\n             (bs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (cs^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^))\n             (ds^ b i' is' ds (os\u2081^ os\u2081' ft\u2081^) (os\u2082^ os\u2082))\n             js\n     ihS = as^-eq , refl , refl , refl , js-eq\n","old_contents":"","returncode":1,"stderr":"error: pathspec 'notes\/FOT\/FOTC\/Program\/ABP\/Lemma1WithoutHelperATP.agda' did not match any file(s) known to git\n","license":"mit","lang":"Agda"}